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abstract: 'We consider the facility location problem in two dimensions. In particular, we consider a setting where agents have Euclidean preferences, defined by their ideal points, for a facility to be located in $\mathbb{R}^2$. For the utilitarian objective and an odd number of agents, we show that the coordinate-wise median mechanism (CM) has a worst-case approximation ratio (WAR) of $\sqrt{2}\frac{\sqrt{n^2+1}}{n+1}$. Further, we show that CM has the lowest WAR for this objective in the class of strategyproof, anonymous, continuous mechanism . For the $p-norm$ social welfare objective, we find that the WAR for CM is bounded above by $2^{\frac{3}{2}-\frac{2}{p}}$ for $p\geq 2$. Since it follows from previous results in one-dimension that any deterministic strategyproof mechanism must have WAR at least $2^{1-\frac{1}{p}}$, our upper bound guarantees that the CM mechanism is very close to being the best deterministic strategyproof mechanism for $p\geq 2$.'
author:
- 'Sumit Goel Wade Hann-Caruthers'
bibliography:
- 'refs.bib'
nocite: '[@*]'
title: 'Coordinate-wise Median: Not Bad, Not Bad, Pretty Good[^1]'
---
Introduction
============
We consider the problem of locating a single facility in $\R^2$, given a finite set of agents who have Euclidean costs defined by their private ideal locations. A central authority wishes to choose a facility location that optimizes some measure of social welfare. Since the ideal points are private information, the mechanism choosing the facility location based on reported ideal points must be strategyproof. Hence, the problem is to check if the mechanism choosing a socially optimal location is strategyproof, and if not, to find a strategyproof mechanism that best approximates the optimal social cost.
This problem has been extensively studied in the literature known as *Approximate Mechanism Design without money*. It was first introduced by Procaccia and Tennenholtz [@procaccia2009approximate] who studied the setting of locating a single facility on a real line under the utilitarian (sum of individual costs) and egalitarian (maximum of individual costs) objectives. Since then, the problem has received much attention, with extensions to alternative objective functions ([@feigenbaum2017approximately], [@feldman2013strategyproof],[@fotakis2013strategyproof], [@cai2016facility]), multiple facilities ([@lu2010asymptotically], [@escoffier2011strategy],[@procaccia2013approximate]), obnoxious facilities [@cheng2013strategy], different networks ([@alon2010strategyproof], [@dokow2012mechanism], [@meir2019strategyproof2]), and other variations. In the class of deterministic strategyproof mechanisms for locating a facility, the median mechanism has been shown to be optimal under various objectives and domains. [@meir2018strategic] provides a good survey of results on approximation ratios achieved in several of these settings.
There is also a large literature in social choice theory on characterising the set of strategyproof mechanisms under different assumptions on preference domains ([@moulin1980strategy], [@schummer2002strategy]). In multiple dimensions, the characterizations typically include or are completely described by the coordinate-wise median mechanism ([@kim1984nonmanipulability], [@border1983straightforward], [@peters1992pareto], [@barbera1993generalized]). The strong axiomatic foundations of the coordinate-wise median mechanism, together with the nice properties of the median in one dimension, motivate our study of the coordinate-wise median mechanism against the optimal mechanism in two dimensions under a variety of social cost functions.
There has been some recent work in extending the facility location problem to multiple dimensions. [@kyropoulou2019mechanism] characterises regions in the Euclidean plane that would lead to the optimal facility location being strategyproof when agents may want the facility to be close or far from their location on a real segment. [@sui2013analysis] and [@sui2015approximately] show that the generalized median mechanisms are not group-strategyproof in Euclidean space and the incentive of group misreport is unbounded. The most closely related work is Meir ([@meir2019strategyproof]), who uses techniques different from ours to find the approximation ratio for the coordinate-wise median mechanism for the case of three agents.
Our contribution
----------------
Our main contribution is to show that for $n$ odd, the coordinate-wise median mechanism has a worst-case approximation ratio of $\sqrt{2}\frac{\sqrt{n^2+1}}{n+1}$ under the minisum objective. Using the worst-case profile and the characterisation from [@kim1984nonmanipulability], we establish that there is no deterministic, strategyproof, anonymous and continuous mechanism that does better. Using the one-dimensional analysis of [@feigenbaum2017approximately], we show that for the $p-norm$ objective, the worst-case approximation ratio for the coordinate-wise median mechanism is bounded above by $2^{\frac{3}{2}-\frac{2}{p}}$ for $p\geq 2$. Since it follows from [@feigenbaum2017approximately] that any deterministic strategyproof mechanism must have approximation ratio at least $2^{1-\frac{1}{p}}$, our bounds suggest that the coordinate-wise median mechanism is (at worst) very nearly optimal.
Preliminaries
=============
We study the problem in which a facility is to be located in $\R^2$. Let $N=\{1,2, \dots, n\}$ be the set of agents. Each agent $i$ has a *Euclidean cost*; that is, there is a point $x_i = (a_i, b_i)$, called agent $i$’s *ideal point*, such that her cost for locating the facility at $y$ is $C(y, x_i) = d(y, x_i)$. We denote by $\vec{x}$ the profile of ideal points: $\vec{x} = (x_1, \dots, x_n)$.
We assume there is a welfare objective summarized by a *social cost function* $sc$, in the sense that locating the facility at $y$ is (weakly) better than locating the facility at $z$ if and only if $sc(y, \vec{x}) \leq sc(z, \vec{x})$.
Mechanisms
----------
A *mechanism* is a function $f : (\R^2)^n \to \R^2$. A mechanism is said to be *strategyproof* if no agent can benefit by misreporting her ideal point, *regardless* of the reports of the other agents. Formally:
A mechanism $f$ is strategyproof if for all $i \in N$, $x_i, x_i' \in \R^2$, $x_{-i} \in (\R^2)^{n-1}$, $$C(f(x_i,x_{-i}), x_i) \leq C(f(x_i',x_{-i}), x_i).$$
A mechanism $f$ is anonymous if for any permutation $\pi: [n] \to [n]$, $$f(x_1, \dots, x_n)=f(x_{\pi(1)}, \dots, x_{\pi(n)})$$
A mechanism $f$ is *continuous* if it is a continuous function. A *social cost function* is a function $sc : \R^2 \times (\R^2)^n \to \R$. We denote by $OPT(sc, \vec{x})$ the set of minimizers for $sc$ given $\vec{x}$: $$OPT(sc, \vec{x}) = \text{argmin}_y sc(y, \vec{x}).$$ When $OPT(sc, \cdot)$ is singleton-valued, we will abuse notation and use $OPT(sc, \vec{x})$ to refer to the unique element contained therein. When $sc$ is clear from context, we will suppress the first argument and write $OPT(sc, \vec{x})$ simply as $OPT(\vec{x})$.
To measure how closely a strategyproof mechanism approximates the optimal social cost for a given preference profile, we use the notion of *approximation ratio*.
The approximation ratio of a mechanism $f$ at a profile $\vec{x} \in (\mathbb{R}^2)^n$ is $$AR(f,\vec{x})=\dfrac{sc(f(\vec{x}), \vec{x})}{sc(OPT(\vec{x}), \vec{x})} \cdot$$
To measure how closely a strategyproof mechanism approximates the optimal social cost *in the worst case*, we use the *worst-case approximation ratio*.
The worst case approximation ratio of a mechanism $f$ is $$WAR(f) = \sup_{\vec{x} \in (\R^2)^n} AR(f,\vec{x}).$$
### The coordinate-wise median mechanism
The coordinate-wise median mechanism is given by:[^2] $$c(\vec{x}) = (\text{median}(a_1,a_2, \dots, a_n),\text{median}(b_1,b_2, \dots, b_n)).$$
The coordinate-wise median has strong axiomatic foundations in the literature (see, e.g., [@border1983straightforward], [@peters1992pareto], [@kim1984nonmanipulability]). In particular, the strategyproofness of $CM$ follows from the following result of [@kim1984nonmanipulability]:
\[kim\] A mechanism $f$ is strategyproof, anonymous and continuous if and only if there exist points $p_1, p_2, \dots, p_{n+1} \in (\{-\infty, \infty\} \cup \R)^2$ such that $f(\vec{x})=c(x_1, \dots, x_n, p_1, \dots, p_{n+1})$.
Note that for $n$ odd, the coordinate-wise median mechanism is obtained by taking, e.g., $p_1 = \cdots = p_{\frac{n+1}{2}} = (-\infty, -\infty)$ and $p_{\frac{n+1}{2} + 1} = \cdots = p_{n+1} = (\infty, \infty)$.
### Geometric median {#sec:gm}
A point minimizing the sum of distances from a finite set of points in a Euclidean space is known as a *geometric median* for that set of points. The geometric median is characterised by the following result:[^3]
\[lem:gm\] Let $X$ be a Euclidean space. Given $\vec{x} \in X^n$, a point $y \in X$ is a geometric median for $\vec{x}$ if and only if there are vectors $u_1, \dots, u_n$ such that $$\sum_{i=1}^{n} u_i=0$$ where for $x_i \neq y$, $u_i=\frac{x_i-y}{{\left | x_i-y \right |}}$ and for $x_i=y$, ${\left | u_i \right |} \leq 1$.
This characterisation yields conditions under which changing a profile of points does not change the geometric median, as summarized in the following corollary:
Let $\vec{x}\in X^n$, and denote by $y$ the geometric median of $\vec{x}$. For any $i$, if $x_i \neq y$ and if $x_i' \in \{ y + t(x_i - y) \, | \, t \in \R_{\geq 0} \} $, then the geometric median for the profile $(x_i', x_{-i})$ is also $y$.
Informally, moving a point directly away from or directly towards (but not past) the geometric median leaves the geometric median unchanged. We will use this observation repeatedly in the sequel and note here that in fact it will be the only characteristic of the geometric median that we use for much of the paper.
For the special case of $n=3$, a more explicit characterisation is easily obtained from Lemma \[lem:gm\]:[^4] if any angle of the triangle formed by the three points is at least $120^o$, $g(\vec{x})$ lies on the vertex of that angle; otherwise, it is the unique point inside the triangle that subtends an angle of $120^o$ to all three pairs of vertices.
Notation
--------
We refer to the coordinates of points in $\R^2$ by $a$ and $b$. We refer to the sets $\R \times \{0\}$ and $\{0\} \times \R$ as the $a$-axis and the $b$-axis, respectively. We refer to the sets $\pm \R_{\geq 0} \times \{0\}$ and $\{0\} \times \pm \R_{\geq 0}$ as the $\pm a$-axes and $\pm b$-axes, respectively. We refer to the geometric median by $g(\vec{x})$[^5]. We use the notation $a_{g}(\vec{x})$ and $b_{g}(\vec{x})$ to denote the first and second coordinates of $g(\vec{x})$, respectively. Similarly, we denote by $c(\vec{x})$ the coordinate-wise median of $\vec{x}$. We denote by $a_c(\cdot)$ and $b_c(\cdot)$ the coordinates of $c(\cdot)$, so that $c(\vec{x}) = (a_c(\vec{x}), b_c(\vec{x}))$.
We use the notation $[\vec{y} \vec{z}]$ to denote the line segment joining $\vec{y}$ and $\vec{z}$: $\{t \vec{y} + (1 - t) \vec{z} \, : \, t \in [0, 1]\}$. Similarly, we denote by $(\vec{y}, \vec{z})$ the set $[\vec{y} \vec{z}] \setminus \{y,z\}$.
Coordinate-wise Median mechanism’s WAR for the minisum objective is not bad
===========================================================================
In this section, we consider the objective $sc(y,\vec{x})=(\sum_{i=1}^n {\left | y-x_i \right |})$. Assume that $n$ is odd and so the geometric median is unique. Consider the geometric median mechanism, which chooses the geometric median $g(\vec{x})$ at any profile $\vec{x}$. Since this mechanism is anonymous, continuous, and not a (generalized) coordinate-wise median mechanism, it follows from Lemma \[kim\] that it is not strategyproof.
Note that by definition, the geometric median mechanism has constant (and hence worst-case) approximation ratio of $1$. However, the question remains what the best possible strategyproof approximation to the geometric median mechanism is, in the sense of having the minimum possible $WAR$. Due to its strong axiomatic foundations ([@peters1992pareto], [@kim1984nonmanipulability], [@border1983straightforward], [@plott1967notion], [@shepsle1981structure] etc.), we consider the coordinate-wise median mechanism as a candidate and investigate the problem of finding its $WAR$.
We begin by stating our results. We then provide a full proof for the case that $n=3$, as we find the approach taken in its proof to be simple enough to be digestible yet sufficiently similar to the more nuanced approach required for arbitrary odd $n$ as to be illuminating. We then provide a sketch of the proof for all odd $n$, relegating the formal proof for this case to the appendix.
Results
-------
\[thm:main-theorem\]
For $n$ odd, the worst-case approximation ratio for the coordinate-wise median mechanism is given by: $$WAR(CM) = \sqrt{2}\dfrac{\sqrt{n^2+1}}{n+1}.$$
The argument for obtaining the exact value of $WAR(CM)$ is rather involved. Interestingly, establishing an (asymptotically tight[^6]) upper bound of $\sqrt{2}$ for any odd $n$ is fairly straightforward:
\[bound\] For any profile $\vec{x} \in (\R^2)^n$, $$\begin{aligned}
sc(c(\vec{x}), \vec{x}) \leq \sqrt{2} \cdot sc(OPT(\vec{x}),\vec{x}).\end{aligned}$$
The proof follows from observing that the median is optimal in one dimension and that for any right triangle, the sum of the lengths of the legs is at most $\sqrt{2}$ times the length of the hypotenuse.
\[neven\] When $n=2m$ is even, the version of the coordinate-wise median mechanism given by $c(\vec{x}) = (\text{median}(-\infty,\vec{a}),\text{median}( -\infty,\vec{b}))$ has worst-case approximation ratio *equal* to $\sqrt{2}$. This follows from the bound in the previous lemma and the worst-case profile $\vec{x}$ where $x_1=x_2 \dots x_m=(1,0)$ and $x_{m+1}=x_{m+2} \dots x_{2m}=(0,1)$.
In both the $n=3$ case and the general case, the key to the proof is to reduce the search space for the worst-case profile from $(\mathbb{R}^2)^n$ to a much smaller space of profiles that have a simple structure. In many cases, this involves “transforming" one profile into another profile that has a higher approximation ratio and a simpler structure. One important transformation that helps in significantly reducing the search space involves moving a point $x_i$ directly towards $g(\vec{x})$, getting as close as possible to $g(\vec{x})$ without changing $c(\vec{x})$. Because this transformation will be used repeatedly throughout this section, we provide here a proof that this transformation leads to a profile $(x_i', x_{-i})$ with a weakly higher approximation ratio.
\[togm\] Let $\vec{x}$ be a profile and $i \in N$, and let $\vec{x}'$ be any profile such that
1. $x_i' \in [x_i, g(\vec{x})]$,
2. for all $j \neq i$, $x_j' = x_j$, and
3. $c(\vec{x}') = c(\vec{x})$.
Then $AR(\vec{x}') \geq AR(\vec{x})$.
By definition, $g(\vec{x}')=g(\vec{x})$ and $c(\vec{x}')=c(\vec{x})$. The change in optimal social cost is given by ${\left | x_i-x_i' \right |}$ while the change in social cost with respect to coordinate-wise median is ${\left | c(\vec{x})-x_i' \right |}-{\left | c(\vec{x})-x_i \right |}$. By triangle inequality, ${\left | x_i-x_i' \right |} \geq {\left | c(\vec{x})-x_i' \right |}-{\left | c(\vec{x})-x_i \right |} $. Thus, the $sc(OPT(\cdot),\cdot)$ reduces by a greater amount than $sc(CM(\cdot), \cdot)$ as we move $x_i$ to $x_i'$. Since the ratio is always at least $1$, it follows that $AR(\vec{x}') \geq AR(\vec{x})$.
Proof for n=3
-------------
Define the set of *Centered perpendicular (CP)* profiles as follows: $$CP=\{\vec{x} \in (\mathbb{R}^{2})^3: c(\vec{x})=(0,0) \text{ and } \forall i, \text{ either } a_i=0 \text{ or } b_i=0\}.$$ In words, a profile is in $CP$ if the coordinate-wise median is at the origin and all points in $\vec{x}$ are on the axes.
Define the set of *Isosceles-centered perpendicular (I-CP)* profiles as follows: $$I-CP=\{\vec{x} \in CP: \exists t \text{ such that } \vec{x}=((t,0), (-t, 0), (0,1)) \text{ and } g(\vec{x})=(0,1)\}$$ In words, a profile is in $I-CP$ if there are two points on the $a$-axis equidistant from the origin and the third point is at $(0,1)$, which is also the geometric median.
We first show that we can reduce the search space for the worst-case profile from $(\mathbb{R}^2)^3$ to $CP$.
\[lem:cp\]
For any profile $\vec{x} \in (\mathbb{R}^2)^3$, there is a profile $\vec{\chi} \in CP$ such that $AR(\vec{\chi}) \geq AR (\vec{x})$.
Let $\vec{x} \in (\mathbb{R}^2)^3$ be a profile. Let $\vec{x}'$ be the profile where $x_i' = x_i - c(\vec{x})$. Then $\vec{x}'$ has the same approximation ratio as $\vec{x}$ and $c(\vec{x}') = (0, 0)$. Denote $A = \{i: a_i=0\}$ and $B = \{i: b_i=0\}$. Note that since $c(\vec{x}') = (0, 0)$, it follows from the definition of $c(\vec{x}')$ that $A \neq \emptyset$ and $B \neq \emptyset$. For each $i$, define $x_i''$ as follows. Let $\G = \{(a, b) \in \R^2 \, : \, a = 0 \text{ or } b = 0\}$. If $i \in A \cup B$, let $x_i'' = x_i'$; otherwise, let $x_i''$ be the point in $[x_i', g(\vec{x}')] \cap \G$[^7] that is closest to $x_i'$. Then $x_i'' \in \G$ for all $i$ and $c(\vec{x}'') = (0, 0)$, so $\vec{x}'' \in CP$. Further, it follows from Lemma \[togm\] that $AR(\vec{x}'') \geq AR(\vec{x}') = AR(\vec{x})$; hence, taking $\vec{\chi} = \vec{x''}$ completes the proof.
\[fig:move\]
(1,3) circle\[radius=2pt\]; (4,-1) circle\[radius=2pt\]; (-2,-3) circle\[radius=2pt\];
(1,-1) circle\[radius=2pt\]; (1.523, -0.08) circle\[radius=2pt\]; (0.4,-1) circle\[radius=2pt\];
(1,3) – (1,-4); (4,-1) – (-2,-1); (-2,-3) – (1.523, -0.08); (-2,-3) – (1,-1);
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Now we show that we can further reduce the search space from $CP$ to $I-CP$.
\[lem:icp\]
For any profile $\vec{x} \in CP$, there exists a profile $\vec{\chi} \in I-CP$ such that $AR(\vec{\chi}) \geq AR (\vec{x})$.
Let $\vec{x}$ be a profile in $CP$.
Without loss of generality, we may assume that all $x_i$ are weakly above the $a$-axis and there are at least two $x_i$ on the $a$-axis, since reflecting a profile in $CP$ across the $a$-axis, the $b$-axis, or the line $a = b$ gives a profile in $CP$ with the same approximation ratio. Hence, we can label the points such that $x_1 = (-a, 0)$, $x_2 = (b, 0)$, and $x_3 = (0, c)$, for some $a, b, c \geq 0$.
If $c = 0$, then $AR(\vec{x}) = 1$, and so every profile has approximation ratio weakly greater than $\vec{x}$. Hence, we may further assume that $c > 0$.
Since $x_1$ and $x_2$ are on the $a$-axis, it follows from the characterization of the geometric median for three points given in Section \[sec:gm\] that $-a \leq a_g(\vec{x}) \leq b$ and $0 < b_{g}(\vec{x}) \leq c$. Hence, moving $x_3$ to $g(\vec{x})$ then (if necessary) translating all points by the same vector so that the coordinate-wise median is at the origin yields a profile in $CP$ which has higher approximation ratio. Hence, we may further assume that $g(\vec{x}) = x_3$.
Let $\vec{x}'$ be the profile where $x_1' = (-(a + b)/2, 0)$, $x_2' = ((a + b)/2, 0)$, and $x_3' = (0, c)$. By definition, $sc(g(\vec{x}'), \vec{x}') \leq sc(g(\vec{x}), \vec{x}')$ and by an argument that exploits the convexity of the distance function, $sc(g(\vec{x}), \vec{x}') \leq sc(g(\vec{x}), \vec{x})$. Combining these inequalities gives $sc(g(\vec{x}'), \vec{x}') \leq sc(g(\vec{x}), \vec{x})$, and a simple calculation shows that $sc(c(\vec{x}'), \vec{x}')=sc(c(\vec{x}) ,\vec{x})$. Thus, $AR(\vec{x}') \geq AR(\vec{x})$.
Note that under $\vec{x}'$, $g(\vec{x}')=(0,k)$ for some $k \leq c$. Define $\vec{x}''$ to be the profile with $x_1''=x_1'$, $x_2''=x_2'$, and $x_3''=g(\vec{x}')$. Then, by Lemma \[togm\], $AR(\vec{x}'') \geq AR(\vec{x}')$.
Finally, define $\vec{x}'''$ such that $x_i''' = \frac{1}{c} x_i''$ for each $i$. Then since $AR(\cdot)$ is homogeneous of degree $0$, $AR(\vec{x}''') = AR(\vec{x}'')$, and so $AR(\vec{x}''') \geq AR(\vec{x})$. Further, $c(\vec{x}''') = (0,0)$, $x_1''' = (-t, 0)$, $x_2''' = (t, 0)$, and $x_3''' = (0, 1)$ for some $t \geq 0$; in fact, it follows from the characterisation of the geometric median that $t \geq \sqrt{3}$. Hence, $\vec{x}''' \in I-CP$, and so taking $\vec{\chi} = \vec{x}'''$ completes the proof.
Denote by $\vec{\eta}_t = ((t,0), (-t, 0), (0,1)).$ It follows from the arguments in the proof of Lemma \[lem:icp\] that $I-CP=\{\vec{\eta}_t \, : \, t \geq \sqrt{3}\}$. Let $\a(t) = \frac{2t + 1}{2 \sqrt{t^2 + 1}}$. A simple calculation shows that for $t \geq \sqrt{3}$, $AR(\vec{\eta}_t) = \a(t)$. In particular, it follows that $WAR(CM)$ is equal to by $\sup_{t \geq \sqrt{3}}{\a(t)}$. Since $\a(t)$ achieves its global maximum at $t^* = 2 > \sqrt{3}$, $WAR(CM) = AR(\vec{\eta}_2) = \a(2)$. Since $\a(2) = \sqrt{2}\dfrac{\sqrt{3^2+1}}{3+1}$, the result follows.
Outline for general (odd) $n$
-----------------------------
We now consider the case of $n=2m+1$ agents. We begin by defining classes of profiles analogous to those used in the proof for $n=3$.
We define the class of Centered Perpendicular (CP) profiles as all profiles $\vec{x} \in (\mathbb{R}^2)^n$ such that
- $c(\vec{x})=(0,0)$
- for all $i$, either $a_i=0$ or $b_i=0$ or $x_i=g(\vec{x})$
- if $x_i' \in (x_i, g(\vec{x}))$, then $c(x_i', x_{-i}) \neq (0,0)$
Since the last condition is slightly more subtle than the others and will be important in the sequel, we describe it now in words. This condition says that *any* (nonzero) movement of *any* $x_i$ towards the geometric median would result in a change in the coordinate-wise median.
We define the class of Isosceles-Centered Perpendicular (I-CP) profiles as all $\vec{x} \in CP$ for which there exists $t \geq 0$ such that
- $x_1= \dots =x_m=(t,0)$
- $x_{m+1}=(-t, 0)$
- $x_{m+2}= \dots= x_{2m+1}=(0,1)$
- $g(\vec{x})=(0,1)$.
The proof proceeds much as in the proof for $n=3$. We first show that for every profile, there is some profile in $CP$ with weakly higher approximation ratio. The approach used in the $n=3$ case extends naturally here: first, translate the profile $\vec{x} \in (\R^2)^n$ so that coordinate-wise median moves to the origin; then, starting from $i = 1$ and going to $i = n$, move $x_i$ directly towards the geometric median until either it reaches the geometric median or moving it further would move the coordinate-wise median. The resulting profile is in $CP$ and has an approximation ratio that is weakly greater than $\vec{x}$’s.
\[fig:npoints\]
(-3,0) – (3,0); (0,-3) – (0,3);
(1,0) circle\[radius=2pt\]; (2,0) circle\[radius=2pt\]; (-0.8,0) circle\[radius=2pt\]; (0,0.8) circle\[radius=2pt\]; (0,1.4) circle\[radius=2pt\]; (0,-0.5) circle\[radius=2pt\]; (0,-2) circle\[radius=2pt\]; (-2,0) circle\[radius=2pt\];
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Next, we show that for any profile in $CP$, there is some profile in $I-CP$ with weakly higher approximation ratio. The approach used in the $n=3$ case for this step *does not* extend in a straightforward manner to the general case—the main obstruction arises from the fact that for a profile $\vec{x}$ in $CP$, there may be $i \in N$ such that $x_i = g(\vec{x})$, which may not be on either axis. The next subsection is devoted to giving an overview of the procedure used to transform a profile in $CP$ to one in $I-CP$ with weakly higher approximation ratio.
Finally, the approach used to calculate the worst-case approximation ratio for profiles in $I-CP$ has much the same structure as in the $n=3$ case. We define $\vec{\eta}_t = (x_1^t, \dots, x_{2m+1}^t),$ where $$\begin{aligned}
x_i^t =
\begin{cases}
(t, 0), & i = 1, \dots, m \\
(-t, 0), & i = m+1 \\
(0, 1), & i = m+2, \dots, 2m+1
\end{cases}\end{aligned}$$ and we show that $I-CP = \left \{ \vec{\eta}_t \, : \, t \geq \sqrt{\frac{2m + 1}{2m - 1}} \right \}$. Defining $\a(t) = \frac{(m+1)t + m}{(m+1) \sqrt{t^2 + 1}}$, we show that for $t \geq \sqrt{\frac{2m + 1}{2m - 1}}$, $AR(\vec{\eta}_t) = \a(t)$, and that $\a(t)$ has a global maximum at $t^* = \frac{m+1}{m} > \sqrt{\frac{2m + 1}{2m - 1}}$, from which it follows that $$\begin{aligned}
WAR(CM) = \a \left( \frac{m+1}{m} \right ) = \sqrt{2}\dfrac{\sqrt{(2m+1)^2+1}}{(2m+1)+1} = \sqrt{2}\dfrac{\sqrt{n^2+1}}{n+1}.\end{aligned}$$
\[fig:worst\]
(-3,0) – (3,0); (0,-3) – (0,3);
(3,0) circle\[radius=4pt\];
(0,2) circle\[radius=4pt\]; (0,2) circle\[radius=2pt\];
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Reduction from $CP$ to $I-CP$
-----------------------------
Next, we discuss informally some transformations that allow us to deal with the profiles in $CP$. Without loss of generality (using reflections if necessary as in the $n=3$ case), we may restrict consideration to profiles $\vec{x} \in CP$ with $g(\vec{x})=(a_g,b_g)$ such that $a_g \geq 0$, $b_g \geq 0$, and $b_g \geq a_g$.
1. **Reducing axes**: In this step, we move all points on $-b$-axis to $-a$-axis while keeping them equidistant from $c(\vec{x})=(0,0)$. This works because the $sc(c(\cdot),\cdot)$ remains the same while $sc(g(\cdot),\cdot)$ reduces, as the points move closer to the old geometric median. Thus, we get a profile in which all points are either on one of the $+a$-, $+b$-, or $-a$-axes or at $g(\vec{x})$.
2. **Convexity**: Consider a profile obtained after applying step 1. Transform the profile so that all points on the $+a$-, $+b$-, and $-a$-axes are at their mean coordinates on the $+a$-, $+b$-, and $-a$-axes respectively. Again, $sc(c(\cdot),\cdot)$ remains the same while $sc(g(\cdot),\cdot)$ falls because of convexity of the distance function. Thus, we get a profile with weakly higher approximation ratio which has $k$ points at $(-b, 0)$, $m+1-k$ points at $(0,c)$, $m+1-k$ points at $(a,0)$ and $k-1$ points at $g(x)$. Note that we are able to pin down the exact cardinalities of these sets because of the third condition in the definition of $CP$, which requires that if any of the points were to move towards $g(\vec{x})$, then $c(\vec{x})$ would change.
3. **Double Rotation**: Consider a profile obtained after applying step 2. Transform the profile by moving the $k-1$ points at $g(\vec{x})$ to $(0, \alpha)$, where $\alpha=d(c(\vec{x}),g(\vec{x}))$, and moving $k-1$ of the $k$ points at $(-b,0)$ to $(\beta, 0)$, where $\beta$ is the unique positive number such that $d(g(\vec{x}), (\beta, 0))=d(g(\vec{x}), (-b,0))$. In this case, one can show that the increase in $sc(c(\cdot),\cdot)$ is at least $\sqrt{2}$ times the increase in $sc(g(\cdot),\cdot)$ and therefore, by Lemma \[bound\], it follows that the approximation ratio weakly increases. Applying convexity again, we get a profile such that there is one point at $(-b, 0)$, $m$ points at $(0,c)$ and $m$ points at $(a,0)$. Note that $g(\vec{x})$ may still not be on the axes.
4. **Geometric to axis**: Consider a profile obtained after applying step 3. Transform the profile so that the $m$ points at $(0,c)$ are at $g(\vec{x})$, then translate all points by the same amount so that the coordinate-wise median is back to the origin. Doing so weakly increases the approximation ratio and yields a profile where one point is at $(-b,0)$, $m$ points are at $(0,c)$, $m$ points are at $(a,0)$ and $g(\vec{x})=(0,c)$.
From here, we apply a transformation similar to step 2 to get a profile in $I-CP$. Note that we have suppressed some details (especially when the same transformation must be used repeatedly) in order to make the exposition as clear as possible—see the appendix for a rigorous proof.
Coordinate-wise Median mechanism’s WAR for the $p$-norm objective is not bad
============================================================================
In this section, we consider the objective $sc(y,\vec{x})=(\sum_{i=1}^n {\left | y-x_i \right |}^p)^\frac{1}{p}$ for $p \geq 2$. We refer to this objective as the $p$-norm objective. Feigenbaum et al. [@feigenbaum2017approximately] consider this objective for the one-dimensional problem and obtain the following result:
\[1d\] Suppose there are $n$ agents with ideal points $a=(a_1,a_2, \dots, a_n) \in \mathbb{R}^n$. Then the social cost incurred by the median mechanism under the $p$-norm is at most $2^{1-\frac{1}{p}}$ times the optimal social cost. Further, there are no deterministic strategyproof mechanisms with a lower worst-case approximation ratio.
In particular, if $a_c$ is the median of $(a_1, a_2,\dots, a_n)$ and $OPT(a)$ is the optimal location, then $\sum_{i=1}^n {\left | a_c-a_i \right |}^p \leq 2^{p-1} \sum_{i=1}^n {\left | OPT(a)-a_i \right |}^p$
We consider the problem of quantifying the WAR for the coordinate-wise median mechanism under the $p$-norm objective when agents have ideal points in $\mathbb{R}^2$.
\[2d\] For the $p$-norm objective with $p \geq 2$, $2^{1-\frac{1}{p}} \leq WAR(CM) \leq 2^{\frac{3}{2}-\frac{2}{p}}$
The lower bound follows directly from Lemma \[1d\]. The upper bounds are obtained by using the following inequalities, together with Lemma \[1d\]: $$\begin{aligned}
(\alpha^2+\beta^2)^{\frac{p}{2}} \geq (\alpha^p+\beta^p) \;\;\;\;\;\;\;\;\;\;\;\; \alpha^p+\beta^p \geq 2^{1-\frac{p}{2}}(\alpha^2+\beta^2)^{\frac{p}{2}}.\end{aligned}$$
Coordinate-wise Median mechanism’s WAR is pretty good
=====================================================
In this section, we aim to compare the performance of the coordinate-wise median mechanism against other strategyproof mechanisms. We begin by stating the conjecture that motivates this work on the coordinate-wise median:
\[conj:CM-very-best\] For $n$ odd and $p\geq 1$, $CM$ has the lowest $WAR$ for the $p$-norm among all deterministic strategyproof mechanisms.
For the case $p=1$, which corresponds to the minisum objective, we have the following result, which we view as progress towards the conjecture.
\[thm:CM-best-in-subclass\] For $n$ odd and the minisum objective, CM has the lowest WAR among all deterministic, strategyproof, anonymous, and continuous mechanisms.
We note that our proof of Theorem \[thm:CM-best-in-subclass\] makes heavy use of the worst case profile $\vec{w}$ derived in Theorem \[thm:main-theorem\]. Specifically, the proof uses the characterisation of Lemma \[kim\] and shows that for any generalized median mechanism $f$, we can find a profile $\vec{w}'$, where $w_i'=w_i+\theta$ for some fixed $\th \in \R^2$, such that $f(\vec{w}') \in P + \th$, where $P = \{ \frac{-(m+1)}{m}, 0,\frac{-(m+1)}{m} \} \times \{0, 1\}$. Then, we show that for any of these six locations, we can find a profile which is the same as $\vec{w}$ up to translation and reflection, and whose AR for the mechanism under consideration is at least as high as $WAR(CM)$. The formal proof is in appendix.\
For general $p$, we are able to show quantitatively that $CM$ can not be much worse than the optimal deterministic strategyproof mechanism. As we show in Theorem \[2d\], $WAR(CM)$ for the $p$-norm is bounded above by $2^{\frac{3}{2}-\frac{2}{p}}$ for $p\geq 2$. In addition, Lemma \[1d\] gives a lower bound on $WAR$ for *any* deterministic strategyproof mechanism, as summarized in the following corollary:
Any deterministic strategyproof mechanism for facility location in $\mathbb{R}^2$ has WAR at least $2^{1-\frac{1}{p}}$ for the $p$-norm objective.
Since the ratio of $WAR(CM)$ and this lower bound is at most $\sqrt{2}$ for $p \geq 2$, it follows that for such $p$ no deterministic strategyproof mechanism has a worst-case approximation ratio that is better than $CM$ by more than a factor of $\sqrt{2}$. This implies that the coordinate-wise median mechanism is already very close to being optimal. While Theorem \[2d\] gives bounds on $WAR(CM)$ for arbitrary $p$, more precise results follow directly for $p=2$ and $p=\infty$ from previous work in the one-dimensional setting. Even though these results extend in a straightforward way, we are not aware of anyone stating them explicitly, and so we mention them here for completeness. For $p=2$ (also referred to in the literature as the miniSOS objective), both the bounds in Theorem \[2d\] are equal to $\sqrt{2}$ and thus $WAR(CM)= \sqrt{2}$. Thus, it follows from the second part of Lemma \[1d\] that there is no deterministic strategyproof mechanism with a better $WAR$ for $p=2$. For $p = \infty$ (also referred to in the literature as the minimax objective), any deterministic strategyproof mechanism has $WAR \geq 2$ [@procaccia2009approximate]. Also, any pareto optimal mechanism has $WAR \leq 2$. Together, we get that $WAR(CM)=2$ for the minimax objective, and hence again there is no deterministric strategyproof mechanism that does better.
Conclusion
==========
We show that the social cost of the coordinate-wise median is always within $\sqrt{2}\dfrac{\sqrt{n^2+1}}{n+1}$ of the social cost obtained under the optimal mechanism. Using the worst case profile, we further show that there is no deterministic, strategyproof, anonymous, and continuous mechanism that does better. For the $p$-norm objectives, we find that the worst-case approximation ratio for the coordinate-wise median mechanism is bounded above by $2^{\frac{1}{2}-\frac{1}{p}} \cdot 2^{1-\frac{1}{p}}$ for $p \geq 2$, and we observe that, by [@feigenbaum2017approximately], the worst-case approximation ratio for *any* deterministic strategyproof mechanism is bounded below by $2^{1-\frac{1}{p}}$. Thus, our bounds show that in these cases, at worst the coordinate-wise median mechanism is within a factor of $\sqrt{2}$ of being optimal.\
Work on higher-dimensional facility location problems remains hitherto relatively limited, but we are optimistic that the results and methods used in this paper will encourage further research in this fundamental domain. We wish to stress, in particular, that the simple structure of the worst-case profile can provide a ready tool for researchers looking to advance progress on the conjectural optimality of the coordinate-wise median mechanism. We also hope that the techniques used in the proof of Theorem \[thm:main-theorem\] can be adapted to the problem for other $p$-norms or more general families of social welfare objectives.
Appendix
========
Let $x_i=(a_i,b_i)$, $ c(\vec{x})=(a_c(\vec{x}),b_c(\vec{x}))$ and $ g(\vec{x})=(a_g(\vec{x}),b_g(\vec{x}))$. For any right triangle, the sum of the lengths of the legs is at most $\sqrt{2}$ times the length of the hypotenuse. Thus, together with the optimality of the median mechanism in one dimension we get $$\begin{aligned}
\sqrt{2} \sum_{i=1}^n{{\left | x_i-g(\vec{x}) \right |}} &\geq \sum_{i=1}^n{|a_i - a_g|} + \sum_{i=1}^n{|b_i - b_g|}\\
& \geq \sum_{i=1}^n{|a_i - a_c|} +\sum_{i=1}^n{|b_i - b_c|}\\
& \geq \sum_{i=1}^n {\left | x_i-c(\vec{x}) \right |}\end{aligned}$$
Hence, the worst-case approximation ratio for the coordinate-wise median mechanism is at most $\sqrt{2}$.
Define Centered Perpendicular (CP) profiles as all profiles $\vec{x} \in (\mathbb{R}^2)^n$ such that
- $c(\vec{x})=(0,0)$
- for all $i$, either $a_i=0$ or $b_i=0$ or $x_i=g(\vec{x})$
- if $x_i' \in (x_i, g(\vec{x}))$, then $c(x_i', x_{-i}) \neq (0,0)$
\[cpn\] For any profile $\vec{x} \in (\mathbb{R}^2)^n$, there exists a profile $\vec{\chi} \in CP$ such that $AR(\vec{\chi}) \geq AR (\vec{x})$.
Let $\vec{x} \in (\mathbb{R}^2)^n$ be a profile. Let $\vec{x}'$ be the profile where $x_i' = x_i - c(\vec{x})$. Then $\vec{x}'$ has the same approximation ratio and $c(\vec{x}') = (0,0)$. Denote $A = \{i: a_i=0\}$ and $B = \{i: b_i=0\}$. Note that since $c(\vec{x}') = (0, 0)$, it follows from the definition of $c(\vec{x}')$ that $A \neq \emptyset$ and $B \neq \emptyset$. Let $\G = \{(a, b) \, : \, a = 0 \text{ or } b = 0\} \cup g(\vec{x}')$. Starting from $i=1$ and going till $n$, define $x_i''$ to be the point in $[x_i', g(\vec{x}')] \cap \G$ that is closest to $g(\vec{x}')$ under the constraint that $c(x_1'', x_2'', \dots, x_i'', x_{i+1}, x_n)=(0,0)$. Then $\vec{x}'' \in CP$. Further, by lemma \[togm\] $AR(\vec{x}'') \geq AR(\vec{x}') = AR(\vec{x})$; hence, taking $\vec{\chi} = \vec{\vec{x}''}$ completes the proof.
Define Isosceles-Centered Perpendicular (I-CP) profiles as all $\vec{x} \in CP$ for which there exists $t \geq 0$ such that
- $x_1= \dots =x_m=(t,0)$
- $x_{m+1}=(-t, 0)$
- $x_{m+2}= \dots= x_{2m+1}=(0,1)$
- $g(\vec{x})=(0,1)$.
Next, we prove some lemmas that will be useful in reducing the search space for the worst-case profile from $CP$ to $I-CP$.
First, we show that we can reduce the number of half-axes that the points lie on from (at most) four to (at most) three.
\[Reduce axes\] Suppose $\vec{x}$ and $\vec{x}'$ are profiles which differ only at $i$ where for some $a > 0$, $x_i = (0, -a)$ and $x_i' = (-a, 0)$, and for which $c(\vec{x}) = c(\vec{x}') = (0, 0)$ and $b_g(\vec{x}) \geq a_g(\vec{x}) \geq 0$. Then $AR(\vec{x}') \geq AR(\vec{x})$.
Again $c(\vec{x}')=c(\vec{x})$ and $sc(c(\vec{x}'), \vec{x}')=sc(c(\vec{x}), \vec{x})$. Thus, it is sufficient to show that $sc(g(\vec{x}'), \vec{x}') \leq sc(g(\vec{x}), \vec{x})$. For this, we just need to show that $d(x_i', g(\vec{x})) \leq d(x_i, g(\vec{x}))$. This follows from the following simple calculation: $$\begin{aligned}
d(x_i', g(\vec{x}))^2 &= (a_g(\vec{x}) + a)^2 + b_g(\vec{x})^2\\
&= a_g(\vec{x})^2 + 2 a_g(\vec{x})a + a^2 + b_g(\vec{x})^2\\
&\leq a_g(\vec{x})^2 + b_g(\vec{x})^2 + 2a b_g(\vec{x}) + a^2\\
&= a_g(\vec{x})^2 + (b_g(\vec{x}) + a)^2\\
&= d(x_i, g(\vec{x}))^2.\end{aligned}$$
Next, we show that we can combine points on each of the three half-axes while weakly increasing the approximation ratio.
\[convexity\] Let $\vec{x} \in CP$ and let $S \subseteq N$ be such that for all $i \in S$, $a_i > 0$ and $b_i = 0$. Let $x_S$ be the mean of the $x_i$ across $i \in S$. Let $\vec{x}'$ be the profile where
1. $x_j' = x_j$ for $j \notin S$ and
2. $x_j' = x_S$ for $j \in S$.
Then $AR(\vec{x}') \geq AR(\vec{x})$.
It is immediate that $c(\vec{x}') = c(\vec{x})$. Hence, it will be sufficient to show that AR for $\vec{x}'$ with $c(\vec{x})$ and $g(\vec{x})$ instead of $c(\vec{x}')$ and $g(\vec{x}')$ is at least as big as $AR(\vec{x})$. Indeed, $sc(c(\vec{x}),\vec{x}')=sc(c(\vec{x}),\vec{x}))$ and $sc(g(\vec{x}'),\vec{x}') <sc(g(\vec{x}),\vec{x}') <sc(g(\vec{x}),\vec{x})$ where the last inequality follows from convexity of the distance function.
The same argument applies for any of the other strict half axes.
Next, we show that we can move all the points that are on the geometric median to the axis in a way that weakly increases the approximation ratio.
\[Double Rotation\] Let $\vec{x}$ and $\vec{x}'$ be profiles that differ only at $i_1$ and $i_2$, such that for some $a \geq 0$
- $c(\vec{x}) = (0, 0)$,
- $b_g(\vec{x}) \geq a_g(\vec{x}) > 0$,
- $x_{i_1} = (-a, 0)$,
- $x_{i_1}' = (a + 2 a_g(\vec{x}), 0)$,
- $x_{i_2} = g(\vec{x})$, and
- $x_{i_2}' = (0 , d(g(\vec{x}), (0, 0)))$.
Then $c(\vec{x}') = (0, 0)$ and $AR(\vec{x}') \geq AR(\vec{x})$.
The first claim is immediate.
For the second claim, let $$\begin{aligned}
A &= \sum_{i \neq i_1}{d(x_i, c(\vec{x}))}\\
B &= \sum_{i \neq i_2}{d(x_i, g(\vec{x}))}.\end{aligned}$$ By a previous result, $$\begin{aligned}
A + d(x_{i_1}, c(\vec{x})) \leq \sqrt{2} B.\end{aligned}$$ Hence, it follows that $$\begin{aligned}
[A + d(x_{i_1}, c(\vec{x}))] d(x_{i_2}', g(\vec{x})) \leq \sqrt{2} B d(x_{i_2}', g(\vec{x})).\end{aligned}$$ But since $b_g(\vec{x}) \geq a_g(\vec{x})$, it follows that $d(x_{i_2}', g(\vec{x})) \leq \sqrt{2} a_g(\vec{x})$. Hence, $$\begin{aligned}
[A + d(x_{i_1}, c(\vec{x}))] d(x_{i_2}', g(\vec{x})) &\leq 2 B a_g(\vec{x})\\
&= B (d(x_{i_1}', c(\vec{x})) - d(x_{i_2}', c(\vec{x}))).\end{aligned}$$ From this it follows that $$\begin{aligned}
(A + d(x_{i_1}, c(\vec{x}))) (B + d(x_{i_2}', g(\vec{x}))) &= AB + B d(x_{i_1}, c(\vec{x})) + [A + d(x_{i_1}, c(\vec{x}))] d(x_{i_2}', g(\vec{x}))\\
&\leq AB + B d(x_{i_1}', c(\vec{x}))\\
&= (A + d(x_{i_1}', c(\vec{x}))) B\end{aligned}$$ and hence $$\begin{aligned}
AR(\vec{x}) &= \frac{A + d(x_{i_1}, c(\vec{x}))}{B}\\
&\leq \frac{A + d(x_{i_1}', c(\vec{x}))}{B + d(x_{i_2}', g(\vec{x}))}\\
&= \frac{A + d(x_{i_1}', c(\vec{x}'))}{B + d(x_{i_2}', g(\vec{x}))}\\
&\leq AR(\vec{x}').\end{aligned}$$
Once we have all the points on the three half-axes, we now show that we can move the geometric median to the axis as well.
\[Geometric to axis\] Suppose that $\vec{x}$ is a profile such that there are $a \geq 0$ and $b, c > 0$ and subsets $L, R, U \subseteq N$ with $L \cap R = L \cap U = R \cap U = \emptyset$, $L \cup R \cup U = N$, $|L| = 1$, $|U| = |R| = m$, and
- $x_i = (0, -a)$ for $i \in L$
- $x_i = (0, b)$ for $i \in U$
- $x_i = (c, 0)$ for $i \in R$
and so that $b_g(\vec{x}) \geq a_g(\vec{x}) > 0$.
Let $\vec{x}'$ be the profile which is the same as $\vec{x}$ for $i \notin U$ and which has $x_i' = g(\vec{x})$ for $i \in U$. Then $AR(\vec{x}') \geq AR(\vec{x})$.
Define $$\begin{aligned}
h(t) = \frac{(a + (1-t)a_g(\vec{x})) + m(c - (1-t)a_g(\vec{x})) + mb}{d((-a, 0), g(\vec{x})) + m d((c,0), g(\vec{x})) + mt d((0, b), g(\vec{x}))}.\end{aligned}$$ Then $AR(\vec{x}) = h(1)$ and $AR(\vec{x}') = h(0)$. Hence, it will be sufficient to show that $h(1) \leq h(0)$.
To see this, note that since the denominator of $h(t)$ is strictly positive for $t \geq 0$ and since both the numerator and the denominator are linear in $t$, $h(t)$ is monotonic on $[0, \infty)$. Now, note that since the approximation ratio is always at least $1$, $h(0) = AR(\vec{x}') \geq 1$. Further, $$\begin{aligned}
\lim_{t \to \infty}{h(t)} &= \frac{(m-1) a_g(\vec{x})}{m d((0, b), g(\vec{x}))}\\
&< \frac{a_g(\vec{x})}{d((0, b), g(\vec{x}))}\\
&< 1.\end{aligned}$$ Hence, there is some $t > 0$ such that $h(t) < 1 \leq h(0)$, and so since $h(t)$ is monotonic on $[0, \infty)$, it follows that $h(t)$ is decreasing on $[0, \infty)$. Thus, $AR(\vec{x}') = h(0) \geq h(1) = AR(\vec{x})$.
Finally, the following lemma shows that we can use convexity to make the triangle formed by the three groups of points isosceles.
\[Isosceles\] Let $\vec{x}$ be a profile such for which are $m$ points at $(a,0)$, $1$ point at $(-b,0)$ and $m$ points at $(0,c)$, and for which $g(\vec{x})=(0,c)$ and $c(\vec{x})=(0,0)$. Let $\vec{x}'$ be the profile where there are $m$ points at $\left(\dfrac{ma+b}{m+1},0\right)$, $1$ point at $\left(-\dfrac{ma+b}{m+1},0\right)$, and $m$ points at $(0,c)$. Then, $AR(\vec{x}') \geq AR(\vec{x})$.
Note that $c(\vec{x})=c(\vec{x}')=(0,0)$. Since $m*a+b=m*\frac{(ma+b)}{m+1}+\frac{ma+b}{m+1}$, we get that the numerator in $AR(\vec{x})$ and $AR(\vec{x}')$ remains the same. Thus, we only need to argue that the denominator goes down as we go from $AR(\vec{x})$ to $AR(\vec{x}')$.
Even though $g(\vec{x}')$ may not be equal to $g(\vec{x})$ we have that $sc(g(\vec{x}),\vec{x}') \leq sc(g(\vec{x}),\vec{x})$ by the convexity of the distance function which would imply $sc(g(\vec{x}'),\vec{x}') \leq sc(g(\vec{x}),\vec{x})$ by definition of $g(\vec{x})$. Thus, we have that $AR(\vec{x}') \geq AR(\vec{x})$.
Now, we use above lemmas to reduce the search space to I-CP.
\[icp2\] For every $\vec{x} \in CP$, there exists $\chi \in I-CP$ such that $AR(\chi) \geq AR(\vec{x})$.
Without loss of generality, consider any profile $\vec{x} \in CP$ such that $b_g(\vec{x}) \geq a_{g}(\vec{x}) \geq 0$. Applying Lemma \[Reduce axes\] to all points on the negative b axis gives a profile $\vec{x}'$ with a weakly higher approximation ratio. In $\vec{x}'$, we have all points on positive a, negative a, positive b and the geometric median. Using lemma \[convexity\], we can combine the points on positive a, negative a, positive b to some $(a,0), (0,b), (-c, 0)$ while weakly increasing AR. Let this profile be $\vec{x}''$. Now, we use lemma \[Double Rotation\] to move points on the geometric median to $+b$-axis. Using \[convexity\] again, we get a profile $\vec{x}'''$ with $m$ points on some $(a,0)$, 1 point on $(-c, 0)$ and $m$ points on $(0,b)$. Now we use lemma \[Geometric to axis\] to move the geometric median to the axis. Then, we use lemma \[Isosceles\] which gives a profile $\vec{x}'''$ such that $\vec{x}'''' \in I-CP$ and $AR(\vec{x}'''') \geq AR(\vec{x})$. Setting $\chi=\vec{x}''''$ completes the proof.
Using Lemma \[icp2\], we can now restrict attention to profiles in $I-CP$. Define $$\begin{aligned}
\vec{\eta}_t = (x_1^t, \dots, x_{2m+1}^t),\end{aligned}$$ where $$\begin{aligned}
x_i^t =
\begin{cases}
(t, 0) & i = 1, \dots, m \\
(-t, 0) & i = m+1 \\
(0, 1) & i = m+2, \dots, 2m+1
\end{cases}\end{aligned}$$ Then, $I-CP = \left \{ \vec{\eta}_t \, : \, t \geq \sqrt{\frac{2m + 1}{2m - 1}} \right \}$. Defining $\a(t) = \frac{(m+1)t + m}{(m+1) \sqrt{t^2 + 1}}$, we get that for $t \geq \sqrt{\frac{2m + 1}{2m - 1}}$, $AR(\vec{\eta}_t) = \a(t)$, and that $\a(t)$ is maximized at $t^* = \frac{m+1}{m} > \sqrt{\frac{2m + 1}{2m - 1}}$, from which it follows that $$\begin{aligned}
WAR(CM) = \a \left( \frac{m+1}{m} \right ) = \sqrt{2}\dfrac{\sqrt{(2m+1)^2+1}}{(2m+1)+1} = \sqrt{2}\dfrac{\sqrt{n^2+1}}{n+1}.\end{aligned}$$
Thus, we get that $WAR(CM)=\sqrt{2}\dfrac{\sqrt{n^2+1}}{n+1}$ as required.
The lower bound follows directly from lemma \[1d\].
Consider any profile $\vec{x}=(a_i, b_i) \in (\mathbb{R}^2)^n$. Let $g(\vec{x})=(a_g(\vec{x}), b_g(\vec{x}))$ and $c(\vec{x})=(a_c(\vec{x}),b_c(\vec{x}))$. Then, we have that
$$\begin{aligned}
sc(g(\vec{x}), \vec{x})^p&=\sum_{i=1}^n {\left | g(\vec{x})-x_i \right |}^p\\
&\geq \left(\sum_{i=1}^n {\left | a_g(\vec{x})-a_i \right |}^p+\sum_{i=1}^n {\left | b_g(\vec{x})-b_i \right |}^p\right)\\
&\geq \left(\sum_{i=1}^n {\left | OPT(a)-a_i \right |}^p+\sum_{i=1}^n {\left | OPT(b)-b_i \right |}^p\right)\\
&\geq \dfrac{1}{2^{p-1}}\left(\sum_{i=1}^n {\left | c_a-a_i \right |}^p+\sum_{i=1}^n {\left | c_b-b_i \right |}^p\right)\\
&\geq \dfrac{2^{1-\frac{p}{2}}}{2^{p-1}}\sum_{i=1}^n {\left | c(\vec{x})-x_i \right |}^p\\
&=2^{2-\frac{3p}{2}}sc(c(\vec{x}),\vec{x})^p\end{aligned}$$
Thus, we get $WAR(CM) \leq 2^{\frac{3}{2}-\frac{2}{p}}$ for $p \geq 2$ as required.
(Theorem \[thm:CM-best-in-subclass\]) Using Lemma \[kim\], we know that every deterministic, strategyproof, anonymous, and continuous mechanism $f$ is defined by points $p_1, p_2, \dots, p_{n+1}$ such that $f(\vec{x})=c(\vec{x},p)$. Consider any arbitrary such mechanism $f$. The worst case profile $\vec{w}$ defines six important points which are $P=\{(a,b)\in \mathbb{R}^2: a \in \{ \frac{-(m+1)}{m}, 0,\frac{-(m+1)}{m} \}, b \in \{0, 1\}\} $ as illustrated in figure \[fig:CMisgood\].
Observe that if $f(\vec{w}) \notin P$, there exists some $\theta \in \mathbb{R}^2$ such that $f(w+\theta) \in P +\theta$. Thus, without loss of generality, we restrict attention to the case where $f(\vec{w}) \in P$ and show that no matter which point $f$ chooses under $\vec{w}$ in $P$, we can find a profile $\vec{x}'$ such that $AR(f,\vec{x}') \geq WAR(CM)=AR(\vec{w})$.
If $f(\vec{w}) \in \{(\frac{-(m+1)}{m}, 0), (0,0)\}$, then we set $\vec{x}'=\vec{w}$ and we are done. If $f(\vec{w})=(\frac{m+1}{m}, 0)$, consider the $\vec{w'}$ obtained by reflecting $\vec{w}$ around the $b$-axis. It follows that $f(\vec{w'}) \in \{(\frac{(m+1)}{m}, 0), (0,0)\}$ where $(\frac{(m+1)}{m}, 0)$ only has 1 agent on it in $\vec{w'}$. Thus, setting $\vec{x}'=\vec{w'}$, we are done.
Now, if $f(\vec{w}) \in \{(\frac{-(m+1)}{m}, 1), (0,1), (\frac{(m+1)}{m}, 1)\}$, consider the $\vec{w'}$ obtained by reflecting $\vec{w}$ around $a$-axis. It follows by definition of $f$ that $f(\vec{w'}) \in \{(\frac{-(m+1)}{m}, 0), (0,0), (\frac{(m+1)}{m}, 0)\}$. This is same as the previous case and hence, we get that there is no deterministic, strategyproof, anonymous and continuous mechanism with a better WAR than the coordinate-wise median mechanism.
\[fig:CMisgood\]
(-3,0) – (3,0); (0,-1) – (0,3);
(0,-1) – (0,3); (-3,0) – (3,0); (-3,-1) – (-3,3); (3,-1) – (3,3); (-3,2) – (3,2);
(3,0) circle\[radius=4pt\];
(0,2) circle\[radius=4pt\];
(-3,0) circle\[radius=2pt\];
(-3,2) circle\[radius=2pt\]; (3,2) circle\[radius=2pt\]; (0,0) circle\[radius=2pt\];
; ;
;
;
;
;
;
;
[^1]: We are grateful for the feedback provided by seminar audiences at the Caltech Theory Lunch, SISL seminar, ISI Delhi, summer school of Econometric society at DSE.
[^2]: Note that this is only well-defined in the case that $n$ is odd. We will also discuss mechanisms which are adaptations of the coordinate-wise median mechanism to even $n$.
[^3]: See, e.g., [@wiki:gm]
[^4]: This characterisation was obtained by Torricelli (1646) and is referred to as the *Torricelli point*.
[^5]: The geometric median is unique whenever $n$ is odd or points are not collinear.
[^6]: Note that the $WAR$ is monotonically increasing and converges to $\sqrt{2}$ as $n\to \infty$.
[^7]: The set $[x_i', g(\vec{x}')] \cap \G$ is non-empty because $g(\vec{x}')$ cannot be in the same quadrant as $x_i'$. Any point in the same quadrant as $x_i'$ subtends an angle of less than $90^o$ with the other two points and hence it cannot be the Torricelli point.
|
{
"pile_set_name": "ArXiv"
}
|
---
---
[****]{} [ Oleg Lebedev\
]{}
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I study the weak basis CP-violating invariants in supersymmetric models, in particular those which cannot be expressed in terms of the Jarlskog–type invariants, and find basis–independent conditions for CP conservation. With an example of the $K - \bar K$ mixing, I clarify what are the combinations of supersymmetric parameters which are constrained by experiment.
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Introduction.
=============
The Standard Model possesses only one CP–odd quantity invariant under a quark basis transformation (apart from $\bar \theta_{\rm QCD}$), which is known as the Jarlskog invariant [@Jarlskog:1985ht]: $$\begin{aligned}
J&=&{\rm Im~\biggl(~ Det} \left[ Y^u Y^{u\dagger}, Y^d Y^{d\dagger} \right]~
\biggr) \nonumber\\
&\propto& (m_t^2-m_u^2)(m_t^2-m_c^2)(m_c^2-m_u^2)
(m_b^2-m_d^2)(m_b^2-m_s^2)(m_s^2-m_d^2)\nonumber\\
&\times& {\rm Im}(V_{11}V_{22}V_{12}^* V_{21}^*)\;,
\label{jar}\end{aligned}$$ where $Y_{ij}^a$ are the Yukawa matrices and $V_{ij}$ is the CKM matrix. In supersymmetric models, there are many additional sources of CP violation as well as new flavor structures [@Haber:1984rc]. In this paper, I will concentrate on the quark–squark sector and will ignore leptonic effects for simplicity. Then, the relevant superpotential and the soft SUSY breaking terms are written as follows: $$\begin{aligned}
\Delta W&=&-\hat{H}_2 Y^u_{ij} \hat{Q}_i \hat{U}_j
+\hat{H}_1 Y^d_{ij} \hat{Q}_i \hat{D}_j
-\mu\hat{H}_1\hat{H}_2 \;, \nonumber\\
\Delta V_{\rm s.b.} &=& M^{2 q_{_L}}_{ij} \tilde q_{Li} \tilde q_{Lj}^*+
M^{2 u_{_R}}_{ij} \tilde u_{Ri} \tilde u_{Rj}^*+
M^{2 d_{_R}}_{ij} \tilde d_{Ri} \tilde d_{Rj}^* + m_1^2 \vert H_1 \vert^2
+ m_2^2 \vert H_2 \vert^2
\nonumber \\ & + &
\left( -H_2 A^u_{ij} \tilde q_i \tilde{u}_j^* +
H_1 A^d_{ij} \tilde q_i \tilde{d}_j^*
- B\mu H_1 H_2 + {\rm h.c.} \right)
-{1\over 2} \sum_i M_i \lambda_i \lambda_i .
\label{l}\end{aligned}$$ Clearly, all quantities with flavor indices can contain flavor–dependent and, possibly, flavor–independent CP violating phases. The latter (e.g. overall phases of the A-terms, $\mu$, $B\mu$, $M_i$) have been studied scrupulously in the past, while the former (e.g. the off–diagonal phases of $M^{2 q_{_L}}$, etc.) have not received as much attention. One of the reasons besides cumbersomeness is that such phases are basis–dependent and thus should be treated with care. In the case of the Standard Model, the flavor–dependent CKM phase can be expressed in terms of the Jarlskog invariant. An important question to address is what is the generalization of the Jarlskog invariant for supersymmetric models and how it is related to the SUSY CP phases.
The class of supersymmetric models under consideration possesses the following symmetries $$U(3)_{\hat Q_L} \times U(3)_{\hat U_R} \times U(3)_{\hat D_R}
\label{symmetry}$$ acting on the quark superfields, which preserve the structure of the supergauge interactions. The CP transformation acts on the Yukawa and mass matrices as the complex conjugation: $$M \stackrel{CP}{\longrightarrow} M^*\;,$$ where $M=\{ Y^u,Y^d,M^{2 q_{_L}},M^{2 u_{_R}},M^{2 d_{_R}},A^u,A^d \}$. If this can be “undone” with the symmetry transformation (\[symmetry\]), the $physical$ flavor–dependent CP phases vanish. Supersymmetric models also possess the Peccei-Quinn and R symmetries $U(1)_{\rm PQ}$ and $U(1)_{\rm R}$, which allow us to eliminate two of the flavor–independent phases (see e.g. [@Abel:2001vy]). Then, in order for CP to be conserved, the invariant CP–phases $${\rm Arg}\left[ (B\mu)^* \mu M_i \right] \;\;,\;\; {\rm Arg}\left[A^*_\alpha M_i \right]$$ have to vanish too. Here Arg$\left[ A_\alpha \right], \alpha=u,d$, denotes the “overall” phases of the A-terms which can be defined in a basis–independent way as $${\rm Arg}\left[ A_\alpha \right] \equiv {1\over 3} {\rm Arg} \left( {\rm Det }\left[ A_\alpha Y^\dagger_\alpha \right] \right) \;,$$ provided this determinant is non–zero. The flavor–independent phases are not affected by the quark superfield basis transformation and thus are physically meaningful. The discussion of the flavor–dependent phases is much more involved. The main subject of this paper is to find the $physical$ CP–phases, i.e. those which are invariant under phase redefinitions of the quark superfields in analogy with the CKM phase, and the corresponding basis–invariant quantities similar to the Jarlskog invariant.
The paper is organized as follows. In section 2 I build up necessary techniques to handle the issues of CP violation in theories with many flavor structures. In section 3 I apply these methods to the Minimal Supersymmetric Standard Model and provide some examples of how observable quantities can be written in manifestly reparametrization invariant form.
Auxiliary Construction.
========================
The case of three matrices.
---------------------------
In the Standard Model, the CP–odd invariant is built on the hermitian quantities $Y^u Y^{u\dagger}$ and $Y^d Y^{d\dagger}$ which transform in the same way under a basis transformation, i.e. $Y^u Y^{u\dagger} \rightarrow U_L Y^u Y^{u\dagger} U_L^\dagger$ and $Y^d Y^{d\dagger} \rightarrow U_L Y^d Y^{d\dagger} U_L^\dagger$. Suppose, in addition to these, we have another quantity with the same transformation property, for instance, $M^{2 q_{_L}}$. Denoting $A\equiv Y^u Y^{u\dagger}$, $B \equiv Y^d Y^{d\dagger}$, and $C \equiv M^{2 q_{_L}}$, we have $$\begin{aligned}
&& A \rightarrow U_L ~A~ U_L^\dagger \;\;,\;\; B \rightarrow U_L ~B~ U_L^\dagger \;\;,\;\; C \rightarrow U_L ~C~ U_L^\dagger\;,\end{aligned}$$ where $U_L$ is a $U(3)$ quark superfield transformation $\hat Q_L \rightarrow (U_L)^T \hat Q_L$ (clearly, these quantities are invariant under the right–handed superfield transformations). What are the invariant CP–violating quantities and the physical CP–phases in this case?
Taking advantage of the unitary symmetry, let us go over to the basis where one of the matrices, say $A$, is diagonal. In this basis, $$\begin{aligned}
{A= \left( \matrix{a_1 & 0 & 0 \cr
0 & a_2 & 0 \cr
0 & 0 & a_3 } \right) \;,\;
B=\left( \matrix{b_{11} & b_{12} & b_{13} \cr
b_{12}^* & b_{22} & b_{23} \cr
b_{13}^* & b_{23}^* & b_{33} } \right) \;,\;
C=\left( \matrix{c_{11} & c_{12} & c_{13} \cr
c_{12}^* & c_{22} & c_{23} \cr
c_{13}^* & c_{23}^* & c_{33} } \right) \;.} \label{ABC}\end{aligned}$$ The residual symmetry is associated with the $U(3)$ generators commuting with the diagonal matrix $A$. These are two $SU(3)$ Cartan subalgebra generators and the generator proportional to the unit matrix. This means that $B$ and $C$ are defined up to the phase transformation $B,C \rightarrow U_1~ B,C~ U_1^\dagger$ with $$U_1= {\rm diag}\left( e^{i \delta_1}, e^{i \delta_2}, e^{i \delta_3} \right) \;.$$ Under this phase transformation the matrix elements transform as $$b_{ij} \longrightarrow b_{ij}~e^{i(\delta_i-\delta_j)}$$ and similarly for $c_{ij}$. Physically this freedom corresponds to the arbitrariness in the choice of the quark superfield phases. The quark and squark fields are to be transformed with the same phases in order not to pick up CP phases in the interaction vertices such as $\tilde q^* q \tilde g$. In our basis, $A={\rm diag}(m_u^2,m_c^2,m_t^2)/v_2^2$, $B=V{\rm diag}(m_d^2,m_s^2,m_b^2)V^\dagger/v_1^2$, where $V$ is the CKM matrix and $v_{1,2}$ are the Higgs VEVs. The supergauge vertices are diagonal and the flavor mixing is contained in the propagators (and the non–gauge vertices). The residual rephasing symmetry implies that all physical quantities must be invariant under a phase redefinition of the quark superfields.
If we have only two matrices $A$ and $B$, the only reparametrization–invariant CP phase we can construct is the CKM–type phase $$\phi_0 = {\rm Arg}(b_{12} b_{13}^* b_{23})\;.$$ In the case of three matrices, there are 3 additional [*non–CKM–type*]{} invariant phases $$\phi_i = \epsilon_{ijk}~ {\rm Arg}(b_{jk} c_{jk}^*)\;,$$ i.e. Arg$(b_{12} c_{12}^*)$, etc. In the non–degenerate case, the other physical CP–phases can be expressed in terms of these 4 phases. For instance, the CKM–type phase for the matrix C, $\phi_0' \equiv$Arg$(c_{12} c_{13}^* c_{23})$, is given by $\phi_0'=\phi_0-\phi_1-\phi_2-\phi_3$ (yet, this is not true in the degenerate case, e.g. when some $b_{ij}=0$). The number of physical phases can also be computed by a simple parameter counting. Three hermitian matrices have nine phases. A $U(3)$ transformation has six phases, of which one leaves all hermitian matrices invariant. Thus the number of non–removable phases is 9-5=4. An interesting feature here is that a new class of reparametrization–invariant CP–phases, not expressible in terms of those of the CKM type, arises.
Consequently, the necessary and sufficient conditions for CP conservation are $$\phi_0= \phi_0' = \phi_i =0 \;\; {\rm mod}\;\; \pi \;\;\; (i=1,2,3)\;.$$
Having identified a set of the physical CP phases, one may ask what are the weak basis CP–odd invariants associated with such phases. In the case of two matrices, there is only one independent CP–odd invariant which can be written as $$J_{AB}= {\rm Im} {\rm Tr [A,B]^3} \;.$$ It is proportional to the Jarlskog invariant of Eq.(\[jar\]) and $\sin\phi_0$. In the case of three matrices with the same transformation properties, one can construct a number of CP–odd invariants such as[^1] $$K_{ABC}(p,q,r)={\rm Im} {\rm Tr} [A^p,B^q]C^r$$ with integer $p,q,r$. This invariant can also be written as ${\rm Im} {\rm Tr} A^p [B^q,C^r]$ or an imaginary part of the trace of the completely antisymmetric product of $A^p,B^q$, and $C^r$. For $p=q=r=1$ it is proportional to a linear combination of $\sin \phi_i$ ($i=1,2,3$): $$K_{ABC}(1,1,1)=2(a_1-a_2) \vert b_{12} c_{12} \vert \sin \phi_3 +
2(a_2-a_3) \vert b_{23} c_{23} \vert \sin \phi_1 +
2(a_3-a_1) \vert b_{13} c_{13} \vert \sin \phi_2 \;.$$ It is worth emphasising that the $K$-invariants are entirely $new$ objects which $cannot$ be expressed in terms of the Jarlskog invariants and vice versa. The simplest way to see that is to imagine that $A$ and $B$ (and maybe $C$) have 2 degenerate eigenvalues. Then all Jarlskog invariants $J_{AB}$, $J_{BC}$, and $J_{CA}$ vanish. Yet, the $K$-invariants can be nonzero. And conversely, suppose that $A$ is proportional to the unit matrix. Then all $K$-invariants vanish, while $J_{BC}$ can be nonzero.
It is instructive to express these invariants in terms of the eigenvalues and the mutual CKM–type matrices. That is, write Eq.(\[ABC\]) as $$A= {\rm diag}(a_1,a_2,a_3)\;,\;
B= V ~{\rm diag}(b_1,b_2,b_3)~ V^\dagger \;,\;
C= U ~{\rm diag}(c_1,c_2,c_3)~ U^\dagger \;.$$ Then, $$K_{ABC}(p,q,r)= \sum_{ijkl} (a^p_i-a_j^p) b_k^q c_l^r V_{ik} V_{jk}^* U_{jl} U_{il}^* \;.$$ Note the appearance of the rephasing invariant quantities $V_{ik} V_{jk}^* U_{jl} U_{il}^*$ which generalize the CKM–type combination $V_{ik} V_{jk}^* V_{jl} V_{il}^*$. To be exact, there are three independent invariant quantities $$\begin{aligned}
&& \phi_1= {\rm Arg } \left( \sum_i b_i V_{2i} V_{3i}^* \right) \left(\sum_i c_i U_{2i} U_{3i}^* \right)^* \;, \nonumber\\
&& \phi_2= -{\rm Arg } \left(\sum_i b_i V_{1i} V_{3i}^* \right) \left(\sum_i c_i U_{1i} U_{3i}^* \right)^* \;, \nonumber\\
&& \phi_3= {\rm Arg } \left(\sum_i b_i V_{1i} V_{2i}^* \right) \left(\sum_i c_i U_{1i} U_{2i}^* \right)^* \;, \end{aligned}$$ while the other are functions of these and $\phi_0$. Indeed, let us consider an example of $K_{ABC}(1,2,1)$. The arising invariant quantity is, e.g. $${\rm Im }\left(\sum_i b_i^2 V_{1i} V_{2i}^* \right) \left(\sum_i c_i U_{1i} U_{2i}^* \right)^*\;.
\label{example}$$ Rewriting $$\sum_i b_i^2 V_{1i} V_{2i}^* = \sum_{ijk} \left( b_i V_{1i} V_{ki}^* \right) \left( b_j V_{kj} V_{2j}^* \right)\;,$$ and extracting terms with different $k$, Eq.(\[example\]) can be brought to the form $$\begin{aligned}
&& \sin\phi_3 \sum_i b_i \left[ \vert V_{1i}\vert^2 + \vert V_{2i}\vert^2 \right]
\left\vert \sum_i b_i V_{1i} V_{2i}^*\right\vert \left\vert \sum_i c_i U_{1i} U_{2i}^*\right\vert + \
\nonumber\\
&& \sin(\phi_3 - \phi_0) \left\vert \sum_i b_i V_{1i} V_{3i}^* \right\vert
\left\vert \sum_i b_i V_{3i} V_{2i}^* \right\vert \left\vert \sum_i c_i U_{1i} U_{2i}^*\right\vert \;. \end{aligned}$$ This can be seen even more easily in terms of the original matrix elements $b_{ij}\equiv \sum_k b_k V_{ik} V_{jk}^*$ and $c_{ij}\equiv \sum_k c_k U_{ik} U_{jk}^*$.
We therefore see that, as expected, not all of the $K$-invariants are independent. From the above exercise it is clear that, in the non-degenerate case, one can choose $$J_{AB}\;,\; K_{ABC}(1,1,1)\;,\; K_{ABC}(2,1,1) \;,\; K_{ABC}(1,2,1)
\label{basis}$$ as the basis of the CP–odd invariants. Given the values of these four invariants, the mixing angles, and the eigenvalues, one can solve for the physical CP–phases $\phi_i$ ($i=0..3$). A complication here, compared to the Standard Model, is that the $K$–invariants generally are non–trivial functions of these CP–phases. The necessary and sufficient conditions for CP conservation can be expressed as $${\rm Im} {\rm Tr [A,B]^3}={\rm Im} {\rm Tr [B,C]^3}={\rm Im} {\rm Tr [C,A]^3}=
{\rm Im} {\rm Tr [A^p,B^q]C^r} =0$$ for any $p,q,r$.
### Degenerate case.
So far we have been assuming that all of the physical phases are non–zero. It is important to find out under which circumstances some of them vanish. To do that, let us go back to Eq.(\[ABC\]). Now, suppose that two eigenvalues of matrix $A$ are degenerate, $a_1=a_2$. In this case, one of the physical phases will disappear. Indeed, the residual symmetry in this case is $U(2)\times U(1)$. Using this symmetry the upper left 2$\times$2 block of matrix B can be diagonalized: $$U_2 \; \left( \matrix{ b_{11} & b_{12} \cr
b_{12}^* & b_{22} } \right) \; U_2^\dagger \;\;\;\; \longrightarrow \;\;\;\;
\left( \matrix{ b_{11}' & 0 \cr
0 & b_{22}' } \right)\;,$$ where $U_2$ is a $U(2)$ matrix. As a result, the invariant phase $\phi_3$ vanishes. Of course, the CKM–type phase $\phi_0$ also vanishes, but it can be replaced with an analogous phase built from the matrix elements of $C$: $\phi_0' = {\rm Arg}(c_{12} c_{13}^* c_{23})$. Thus, the basis for the remaining physical phases can be chosen as $$\phi_0' \;,\; \phi_1 \;,\; \phi_2 \;.$$
Now suppose that two of the eigenvalues of matrix $B$ are also degenerate. By a unitary (permutational) transformation these can be made $b_1$ and $b_2$. This introduces an additional $U(2)$ symmetry, so the CKM matrix between $A$ and $B$ is now defined only up to a $U(2)\times U(1)$ $biunitary$ transformation, i.e. $$A= {\rm diag}(a,a,a_3)\;\;,\;\; B= U_2 V \tilde U_2^\dagger ~{\rm diag}(b,b,b_3)~ \tilde U_2 V^\dagger U_2^\dagger\;,$$ where $U_2$ and $\tilde U_2$ have a $U(2)$ block in upper left corner and a phase in the (33) position. Since any matrix can be diagonalized by a biunitary transformation, the upper left block of $V$ can be brought to a diagonal form, i.e. $V_{12}'=V_{21}'=0$. The unitarity of $V'$ then requires $V_{13}'=V_{31}'=0$ and $\vert V_{11}' \vert =1$ (or $V_{23}'=V_{32}'=0$ and $\vert V_{22}' \vert =1$ ), $$V'= \left( \matrix{ e^{i \theta } & 0& 0 \cr
0 & V_{22}' & V_{23}' \cr
0 & V_{32}' & V_{33}'}
\right)\;.$$ This results in $b_{12}=b_{13}=0$. We therefore see that the residual symmetry allows us to eliminate two of the physical phases. The remaining CP phases are $$\phi_0' \;,\;\phi_1 \;.$$ Finally, if two eigenvalues of $C$ are also degenerate, $\phi_0'=0$ and the only physical phase is $$\phi_1 \;.$$ Note that in this case Tr$[A,B]C \propto \sin \phi_1$.
Now let us briefly discuss the case of three degenerate eigenvalues. If $A \propto $ [**I**]{}, the residual symmetry is $U(3)$ which allows us to diagonalize another matrix, say $B$. Then the only invariant phase is $$\phi_0'$$ and $J_{BC}\propto \sin\phi_0'$. It is clear that if either $B$ or $C$ have degenerate eigenvalues, no CP violation occurs.
These results can also be obtained by a naive parameter counting. $U(2)$ has an extra phase parameter compared to $U(1)\times U(1) $, so enlarging the residual symmetry from $U(1)\times U(1) $ to $U(2)$ will reduce the number of physical phases by one. Similarly, $U(3)$ will allow us to eliminate three more phases compared to $U(1)\times U(1) \times U(1)$.
The number of the physical phases also reduces if some of the mixings are zero. The mixing angles are defined through the following parametrization of a unitary matrix $V$ (up to a phase transformation) [@Hagiwara:pw]: $$V= \left( \matrix{ \mathcal C_{12} \mathcal C_{13} & \mathcal S_{12} \mathcal C_{13} &
\mathcal S_{13} e^{-i \delta_{13}} \cr
-\mathcal S_{12} \mathcal C_{23}- \mathcal C_{12} \mathcal S_{23} \mathcal S_{13} e^{i \delta_{13}} & \mathcal C_{12} \mathcal C_{23} -\mathcal S_{12}\mathcal S_{23} \mathcal S_{13} e^{i \delta_{13}} &
\mathcal S_{23} \mathcal C_{13} \cr
\mathcal S_{12} \mathcal S_{23}- \mathcal C_{12} \mathcal C_{23} \mathcal S_{13} e^{i \delta_{13}} &
-\mathcal C_{12} \mathcal S_{23} -\mathcal S_{12}\mathcal C_{23} \mathcal S_{13} e^{i \delta_{13}} &
\mathcal C_{23} \mathcal C_{13} }
\right)\;,$$ where $\delta_{13}$ is a phase; $\mathcal S_{ij}=\sin \theta_{ij}$, $\mathcal C_{ij}=\cos \theta_{ij}$, and $\theta_{12}, \theta_{13}, \theta_{23}$ are the mixing angles. It is easy to see that one vanishing mixing angle annuls one phase, say the CKM–type phase $\phi_0$. Another vanishing mixing eliminates $\phi_0'$ which is equivalent to saying $\phi_1 +\phi_2 +\phi_3=0$. If both of these mixings are in the same matrix, say $V$, then this implies that two of the elements $\{ b_{12},b_{13},b_{23} \}$ vanish, so again two of the physical phases disappear. Further, the third zero mixing would eliminate another phase. The next step, however, is nontrivial. If one matrix contains three zero mixings and the other – one, then no CP violation is possible. Indeed, this means that two matrices, e.g. $A$ and $B$, are diagonalizable simultaneously so that all $K$–invariants vanish, whereas the $J$–invariants vanish due to a single zero mixing. On the other hand, if $V$ and $U$ have two zero mixings each, then CP violation is still possible. As mentioned above, in this case two of $\{ b_{12},b_{13},b_{23} \}$ and two of $\{ c_{12},c_{13},c_{23} \}$ vanish, so that one can have $$\begin{aligned}
{A= \left( \matrix{a_1 & 0 & 0 \cr
0 & a_2 & 0 \cr
0 & 0 & a_3 } \right) \;,\;
B=\left( \matrix{b_{11} & b_{12} & 0 \cr
b_{12}^* & b_{22} & 0 \cr
0 & 0 & b_{33} } \right) \;,\;
C=\left( \matrix{c_{11} & c_{12} & 0 \cr
c_{12}^* & c_{22} & 0 \cr
0 & 0 & c_{33} } \right) \;.} \end{aligned}$$ The surviving invariant phase is $\phi_3={\rm Arg}(b_{12}c_{12}^*)$ and Tr$[A,B]C \propto \sin \phi_3$. If the non–vanishing off–diagonal entries are misaligned, CP is conserved. No CP violation can occur if five of the mixing angles are zero.
Generalization to more than three matrices.
-------------------------------------------
Suppose we have $N$ hermitian matrices $H_1, H_2,..H_N$ with the same transformation properties, $H_i \rightarrow U_L H_i U_L^\dagger$. How can the results of the previous subsection be generalized for this case?
The generalization is quite straightforward. Using the unitary freedom, we bring $H_1$ to the diagonal form: $$\begin{aligned}
H_1= \left( \matrix{(H_1)_1 & 0 & 0 \cr
0 & (H_1)_2 & 0 \cr
0 & 0 & (H_1)_3 } \right) \;,\;
H_2=\left( \matrix{(H_2)_{11} & (H_2)_{12} & (H_2)_{13} \cr
(H_2)_{12}^* & (H_2)_{22} & (H_2)_{23} \cr
(H_2)_{13}^* & (H_2)_{23}^* & (H_2)_{33} } \right) \;,\; ...\end{aligned}$$ The reparametrization invariant phases can be constructed by taking cyclic products of the elements of the same matrix or by taking products of elements in the same positions in different matrices. In the non–degenerate case, the $3N-5$ independent phases can be chosen as $$\phi_0= {\rm Arg}\left[ (H_2)_{12} (H_2)_{13}^* (H_2)_{23} \right]\;\;,\;\;
\phi_i^a= \epsilon_{ijk} {\rm Arg} \left[ (H_2)_{jk} (H_a)_{jk}^* \right]\;,\; a=3...N\;.
\label{phases1}$$ Any other physical phase can be expressed in terms of these basis phases. The corresponding weak basis invariants are $$J_{H_1 H_2} \;\;,\;\; K_{H_1 H_2 H_a}(p,q,r) \;,\;a=3...N\;,
\label{basis1}$$ with $( p,q,r)=\{ (1,1,1); (2,1,1); (1,2,1) \}$.
In the degenerate case, the discussion of the previous subsection equally applies. Any additional $U(2)$ symmetry, i.e. the presence of two degenerate eigenvalues, eliminates one physical phase which can be taken to be the CKM–type phase for this matrix. An extra $U(3)$ eliminates three physical phases, for instance $\phi^a_{1,2,3}$. A vanishing mixing angle typically entails one vanishing phase, yet there are subtleties discussed above.
The necessary and sufficient conditions for CP conservation can be written as $${\rm Im Tr}[H_i,H_j]^3={\rm Im Tr}H^p_{[i}H^q_j H^r_{k]}=0$$ for any $i,j,k$ and $p,q,r$. Here the square brackets denote antisymmetrization with respect to the indices. These conditions amount to $${\rm Arg}\left[ (H_\alpha)_{12} (H_\alpha)_{13}^* (H_\alpha)_{23} \right]=
\epsilon_{ijk} {\rm Arg} \left[ (H_\alpha)_{jk} (H_\beta)_{jk}^* \right]=0 \; {\rm mod}\; \pi
\label{phases2}$$ for all $\alpha,\beta,$ and $i$.
One may wonder whether it is possible to construct CP–odd $K$– type invariants with more than three matrices under the trace. This is certainly possible, yet they will be functions of the basic reparametrization invariant phases (\[phases1\]) or, in a more general case, (\[phases2\]), and thus will not provide independent CP violating quantities.
The Minimal Supersymmetric Standard Model.
==========================================
The general technology of the previous section can be applied (with some reservations) to the case of the Minimal Supersymmetric Standard Model (MSSM). The MSSM has a number of flavor structures which transform under the $U(3)_{\hat Q_L} \times U(3)_{\hat U_R} \times U(3)_{\hat D_R}$ symmetry. In particular, the transformation properties are given by $$\begin{aligned}
Y^u &\longrightarrow& U_L Y^u U_{u_{_R}}^\dagger \;,\nonumber\\
Y^d &\longrightarrow& U_L Y^d U_{d_{_R}}^\dagger \;,\nonumber\\
A^u &\longrightarrow& U_L A^u U_{u_{_R}}^\dagger \;,\nonumber\\
A^d &\longrightarrow& U_L A^d U_{d_{_R}}^\dagger \;,\nonumber\\
M^{2 q_{_L}} &\longrightarrow& U_L M^{2 q_{_L}} U_{L}^\dagger \;,\nonumber\\
M^{2 u_{_R}} &\longrightarrow& U_{u_{_R}} M^{2 u_{_R}} U_{u_{_R}}^\dagger \;,\nonumber\\
M^{2 d_{_R}} &\longrightarrow& U_{d_{_R}} M^{2 d_{_R}} U_{d_{_R}}^\dagger \;.\end{aligned}$$ To construct the weak basis invariants, it is necessary to identify hermitian objects which transform under one of the unitary groups. These are given in Table 1. Not all of them are, however, independent. In particular, only three matrices out of $$A^u A^{u\dagger} \;,\; A^{u\dagger} A^{u} \;,\; A^u Y^{u\dagger}+{\rm h.c.} \;,\;
A^{u\dagger} Y^{u}+{\rm h.c.}$$ contain independent phases in the off-diagonal elements. This can be seen as follows. Let us go over to the basis where $Y^u$ is diagonal, $Y^u \rightarrow U_L Y^u U_{u_{_R}}^\dagger ={\rm diag}(m_u,m_c,m_t)/v_2$. Given $A^u A^{u\dagger}$ and $ A^{u\dagger} A^{u}$, we can find $A^u$ up to a phase transformation. Indeed, $A^u A^{u\dagger}$ and $ A^{u\dagger} A^{u}$ fix the diagonalization matrices of $A^u$: $$\begin{aligned}
A^u A^{u\dagger} &\longrightarrow& \tilde U_L A^u A^{u\dagger} \tilde U_{L}^\dagger \;\;\; =
{\rm diag} (a_1^{u2},a_2^{u2},a_3^{u2})\;,\nonumber\\
A^{u\dagger} A^u &\longrightarrow& \tilde U_{u_{_R}} A^{u\dagger} A^u \tilde U_{u_{_R}}^\dagger =
{\rm diag} (a_1^{u2},a_2^{u2},a_3^{u2})\;, \end{aligned}$$ so that $A^u$ is given by $$A^u =\tilde U_{u_{_R}}^\dagger {\rm diag} (a_1^{u},a_2^{u},a_3^{u}) \tilde U_L \;.$$ Note that both $\tilde U_{u_{_R}}$ and $\tilde U_L$ are only defined up to a diagonal phase transformation, $$\tilde U_{u_{_R}} \sim {\rm diag}(e^{i\delta_1}, e^{i\delta_2},e^{i\delta_3}) ~\tilde U_{u_{_R}}\;\;,\;\;
\tilde U_{L} \sim {\rm diag}(e^{i\phi_1}, e^{i\phi_2},e^{i\phi_3})~ \tilde U_{L} \;.$$ This introduces a phase ambiguity in the matrix elements of $A^u$. A phase transformation with $\delta_i = \phi_i $ leaves $A^u$ intact, whereas that with $\delta_i = -\phi_i $ changes it. This remaining ambiguity is eliminated by fixing $A^u Y^{u\dagger}+{\rm h.c.}$ such that $A^u$ and $A^{u\dagger} Y^{u}+{\rm h.c.}$ can be determined unambiguously. This can also be understood by parameter counting: $A^u$ has nine phases which, in the non–degenerate case, can be found from the nine phases of the three hermitian matrices $A^u A^{u\dagger}$, $ A^{u\dagger} A^{u}$, and $A^u Y^{u\dagger}+{\rm h.c.}$ Of course, similar considerations apply to $A^{d\dagger} Y^{d}+{\rm h.c.}$
This argument can be generalized to an arbitrary number of generations $N$. $N^2$ phases of $A^u$ can be found via $N(N-1)$ phases of $A^u A^{u\dagger}$ and $ A^{u\dagger} A^{u}$, and $N$ phases of $A^u Y^{u\dagger}+{\rm h.c.}$ Although $A^u Y^{u\dagger}+{\rm h.c.}$ has $N(N-1)/2$ phases, only $N$ of them are independent and correspond to the residual phase freedom with $\tilde U_L = \tilde U_{u_{_R}}^\dagger ={\rm diag }(e^{i\delta_1},..,e^{i\delta_N} )$. This, however, does not work if $N>N(N-1)/2$ in which case $\delta_i$ cannot be determined unambiguously from the off–diagonal phases of $A^u Y^{u\dagger}+{\rm h.c.}$ So, for $N=1$ and $N=2$, additional information besides the hermitian quantities is needed which can be, for instance, the anti–hermitian matrix $A^u Y^{u\dagger}-{\rm h.c.}$
Let us now identify the physical CP phases. First of all, the physical phases must be invariant under the $U(1)_{\rm PQ}$ and $U(1)_{\rm R}$ symmetries. By an $R$–rotation, the gluino mass can be made real, so henceforth the phases of the A–terms will be assumed to be relative to the gluino phase. Similarly, by a Peccei–Quinn transformation the $B\mu$ term can be made real. Thus we have three CP phases in the flavor independent objects $\mu$,$M_1$, and $M_2$. The phases of the flavor–dependent objects can be easily counted: out of the original 45 phases of the flavor objects 17 can be eliminated by the $U(3)^3$ symmetry (one of the $U(1)$ transformations leaves all flavor structures intact), leaving 28 physical phases. These can be expressed in terms of the reparametrization invariant phases of the hermitian matrices. The three columns of Table 1 form three separate sequences. Using the unitary freedom, the first matrix in each column can be made diagonal. Then the strategy of section 2 can be applied. Each column has $3N-5$ physical phases, where $N$ is the number of matrices in the column. However, as I argued above, the two matrices $A^{u\dagger} Y^{u}+{\rm h.c.}$ and $A^{d\dagger} Y^{d}+{\rm h.c.}$ are not independent. In particular, that means that, in the second column, the CKM–type phase for $A^{u\dagger} Y^{u}+{\rm h.c.}$ is not an independent phase, whereas the phase differences of the off–diagonal elements of $A^{u\dagger} Y^{u}+{\rm h.c.}$ and $ M^{2 u_{_R}} $, and $A^{u\dagger} Y^{u}+{\rm h.c.}$ and $A^{u\dagger} A^u $ are independent. Therefore, instead of 7 phases in the second column we should only count 6, and similarly for the third column. Thus, we end up with 16+6+6=28 physical independent phases, as expected.
$U(3)_{\hat Q_L}$ $U(3)_{\hat U_R} $ $ U(3)_{\hat D_R} $
------------------------------- ----------------------------------- -----------------------------------
$Y^u Y^{u\dagger} $ $ Y^{u\dagger} Y^u $ $ Y^{d\dagger} Y^d $
$Y^d Y^{d\dagger} $ $ M^{2 u_{_R}} $ $ M^{2 d_{_R}} $
$M^{2 q_{_L}}$ $A^{u\dagger} A^u $ $A^{d\dagger} A^d $
$A^u A^{u\dagger}$ $ A^{u\dagger} Y^{u}+{\rm h.c.} $ $ A^{d\dagger} Y^{d}+{\rm h.c.} $
$A^d A^{d\dagger}$ $ $ $ $
$A^u Y^{u\dagger}+{\rm h.c.}$ $ $ $ $
$A^d Y^{d\dagger}+{\rm h.c.}$ $ $ $ $
: Hermitian objects of the MSSM transforming under the unitary flavor symmetries. []{data-label="table1"}
In the basis where $Y^u Y^{u\dagger}$, $Y^{u\dagger} Y^u$, and $Y^{d\dagger} Y^d$ are diagonal, the 28 independent physical phases can be chosen as follows: $$\begin{aligned}
&&\phi_0={\rm Arg}\left[ (Y^d Y^{d\dagger})_{12} (Y^d Y^{d\dagger})_{13}^* (Y^d Y^{d\dagger})_{23} \right] \;,\nonumber\\
&& \phi_i^a= \epsilon_{ijk} {\rm Arg} \biggl[ (Y^d Y^{d\dagger})_{jk} (F^a)_{jk}^* \biggr] \;,\; \nonumber\\
&& \chi_i^a= \epsilon_{ijk} {\rm Arg} \biggl[ ( A^{u\dagger} Y^u +{\rm h.c.})_{jk} (G^a)_{jk}^* \biggr] \;,\; \nonumber\\
&& \xi_i^a= \epsilon_{ijk} {\rm Arg} \biggl[ ( A^{d\dagger} Y^d +{\rm h.c.})_{jk} (H^a)_{jk}^* \biggr] \;,\;
\label{allphases}\end{aligned}$$ where $$\begin{aligned}
&&F^a= \{ M^{2 q_{_L}},A^u A^{u\dagger},A^d A^{d\dagger},A^u Y^{u\dagger}+{\rm h.c.},A^d Y^{d\dagger}+{\rm h.c.} \} \;, \nonumber\\
&&G^a= \{ M^{2 u_{_R}}, A^{u\dagger} A^u \} \;, \nonumber\\
&&H^a= \{ M^{2 d_{_R}}, A^{d\dagger} A^d \} \;.\end{aligned}$$ Here I have assumed that there are no degenerate eigenvalues and the mixing angles are non–zero. The corresponding weak basis CP–odd invariants are given by Eq.(\[basis1\]).
In the degenerate case, i.e. when some mixing angles are zero and/or there are degenerate eigenvalues, the situation becomes more complicated. In particular, the intrinsically non–hermitian objects such as $A^\alpha$ may have CP–phases which cannot be “picked up” by the hermitian quantities in the degenerate case. This occurs, for instance, when only a 2$\times$2 block of $A^\alpha$ is non–zero such that we effectively deal with two generations. Another example is the case when $A^u$ and $Y^u$ can be diagonalized simultaneously. Then, in the basis where they are diagonal, the reparametrization invariant CP–phases are $$\rho_i^u= {\rm Arg} (A_{ii}^u Y_{ii}^{u *})
\label{rho}$$ and similarly for $A^d$. On the other hand, the hermitian matrices $A^u A^{u\dagger},A^{u\dagger} A^u $ and $A^u Y^{u\dagger}+{\rm h.c.}$ are diagonal (and real), so there are no CP phases associated with them. The CP violating invariants corresponding to the phases $\rho_i^u$ are based on the anti–hermitian matrices: $$L_{A^u Y^u}(p)={\rm Im Tr}\left[ (A^u Y^{u\dagger})^p- {\rm h.c.} \right]\;,$$ where $p$ is an integer. For $p=1$, this becomes $$L_{A^u Y^u}(1)=2 \sum_i \vert A_{ii}^u m_i^u/v_2 \vert \sin\rho_i^u \;.$$ Quantities of the type ${\rm Im Tr} (A^{\alpha\dagger} Y^{\alpha})^p$ do not provide independent CP–violating invariants. Further, if all $A_{ij}^u$ are non–zero in the basis where $Y^u$ is diagonal, $\rho_i^u$ are not independent and are functions of the phases (\[allphases\]).
The necessary and sufficient conditions for CP conservation are $$\begin{aligned}
&& {\rm Im Tr}[M_i,M_j]^3={\rm Im Tr}M^p_{[i}M^q_j M^r_{k]}=0 \;, \nonumber \\
&& {\rm Im Tr} (A^\alpha Y^{\alpha\dagger})^p =0\end{aligned}$$ for any $i,j,k$ and $p,q,r$; $\alpha=\{ u,d \}$. Here $M_i$ are hermitian matrices belonging to the same column of Table 1 and these conditions are to be satisfied for each column. In addition, one, of course, has to require that the gaugino and the $\mu$-term phases vanish. In this case, the full MSSM will conserve CP. If all of the physical phases (\[allphases\]) and (\[rho\]) are much smaller than one, CP is an approximate symmetry (yet, this is unrealistic as the CKM phase is of order one experimentally). In terms of basis–independent quantities, this implies that, when the mixing angles and the eigenvalues are fixed, the $J$-, $K$-, and $L$-invariants are close to zero (in the appropriate units).
If we are to require that the CKM phase be the only source of CP violation, all invariants apart from ${\rm Im Tr}[Y^u Y^{u\dagger},Y^d Y^{d\dagger}]^3$ must vanish. This occurs, for instance, when all SUSY flavor structures are proportional to the unit matrix and the A–terms are real. It is important to note that it is not sufficient to require that the SUSY flavor structures be real in some basis because the presence of CP phases in the Yukawas can make some of the invariants, apart from the Jarlskog one, non–zero. For example, when $Y^u Y^{u\dagger}$ is diagonal, $Y^d Y^{d\dagger}=V{\rm diag}(m_d^2,m_s^2,m_b^2)V^\dagger/v_1^2$, and $M^{2 q_{_L}}$ is real, the invariant phases $\phi_i^1$ and ${\rm Im Tr}[Y^u Y^{u\dagger},Y^d Y^{d\dagger}] M^{2 q_{_L}}$ are nonzero. This creates certain difficulties for realistic string models with low energy supersymmetry [@Abel:2001cv]. The reason is that such models generally predict non–universal A–terms. Then even if the SUSY breaking F–terms are real, the complex phases of the type (\[rho\]) are generated by the basis transformation which brings the Yukawa couplings to the diagonal form. Equivalently, this means that some of the $L$– or $K$–invariants are non–zero. As a result, large electric dipole moments of fermions are induced, in conflict with experiment.
Finally, it should be noted that after the electroweak symmetry breaking a number of corrections to the Lagrangian (\[l\]) will appear. In particular, $M^{2 q_{_L}}$ will split into $M^{2 u_{_L}}$ and $M^{2 d_{_L}}$ due to the isospin breaking corrections. Since no additional sources of CP violation arise in this process, the consequent CP phases will be functions of the original phases (\[allphases\]) and (\[rho\]).
Examples.
---------
Let us consider a few phenomenological examples.
[**i. Kaon mixing.**]{} A first example is a supersymmetric contribution to the $K -\bar K$ mixing. This is conveniently expressed in terms of the mass insertions [@Hall:1985dx]. Suppose that we work in the super–CKM basis, i.e. the basis where the quark mass matrices are diagonal, and that the only non–vanishing mass insertion is $(\delta_{LL}^d)_{12}\equiv ( M^{2 d_{_L}})_{12}/\tilde m^2$ where $\tilde m^2$ is the average squark mass. Then the gluino–mediated contribution to the $\varepsilon_K$ parameter is usually written as [@Gabbiani:1996hi] $ (\varepsilon_K)_{_{\rm SUSY}} \propto {\rm Im} (\delta_{LL}^d)_{12}^2\;.$ Clearly, this result is $not$ rephasing invariant. The super–CKM basis is defined only up to a phase transformation, which physics should be independent of. It can be shown that, in fact[^2], $$(\varepsilon_K)_{_{\rm SUSY}} \propto {\rm Im} \Bigl[ (\delta_{LL}^d)_{12} V_{21} V_{22}^*
\Bigr]^2\;.$$ As a result, the $K -\bar K$ mixing constrains the invariant quantity $(\delta_{LL}^d)_{12} V_{21} V_{22}^*$ and even a $real$ mass insertion can lead to CP violation. Ref.[@Gabbiani:1996hi] assumes a special form of the CKM matrix in which $V_{21} V_{22}^*$ is real. Note, however, that, unlike in the Standard Model, different CKM “conventions” are $not$ $physically$ $equivalent$. This is due to the presence of the additional invariant phases (\[phases1\]). In other words, diagonalization of the Yukawa matrices leads to the CKM matrix in some generic phase convention. To bring it to a special form, one rotates the quark and squark fields simultaneously thereby inducing CP–phases in $M^{2 d_{_L}}$ even if it was real initially. Equivalently, the $K$-invariants such as ${\rm Im Tr}[Y^d Y^{d\dagger},Y^u Y^{u\dagger}]M^{2 d_{_L}}$ may not vanish even if $M^{2 d_{_L}}$ is real. Thus, in all phenomenological analyses the definition of the super-CKM basis must be supplemented with the specification of the phase convention of the CKM matrix.
The reparametrization invariant phases can be identified as follows. By choosing an appropriate $U_L$, let us go to the basis where $$Y^d Y^{d\dagger}={\rm diag}(m_d^2,m_s^2,m_b^2)/v_1^2 \;\;,\;\;
Y^u Y^{u\dagger}=V^\dagger {\rm diag}(m_u^2,m_c^2,m_t^2)V /v_2^2 \;,$$ where $V$ is the CKM matrix. In this basis $M^{2 d_{_L}}$ is the same as in the super–CKM basis. The reparametrization invariant CP phases are the relative phases between the matrix elements of $Y^u Y^{u\dagger}$ and $M^{2 d_{_L}} $ in the same position. In particular, the relevant to the $K-\bar K$ mixing invariant phase is $$\delta = {\rm Arg}\left( \sum_i \vert m_i^{u}\vert^2 V_{i1} V_{i2}^* \right) \left( M^{2 d_{_L}} \right)_{12} \;.$$ Here I have used the absolute values of the masses to stress their invariance with respect to the phase transformations. If $(\delta_{LL}^d)_{12}$ is the only non–zero mass insertion, there is only one additional invariant phase – the CKM phase. Then, $\varepsilon_K$ is a function of these two phases. Specifically, the relevant CP phase is expressed as $${\rm Arg} \Bigl[ (\delta_{LL}^d)_{12} V_{21} V_{22}^* \Bigr]=
\delta - \phi (\delta_{_{\rm CKM}})\;,$$ where $\phi (\delta_{_{\rm CKM}})$ is an invariant phase defined by $$\phi (\delta_{_{\rm CKM}})={\rm Arg} \left[ 1+ {\vert m_u \vert^2 V_{11} V_{12}^* \over
\vert m_c \vert^2 V_{21} V_{22}^* } + {\vert m_t \vert^2 V_{31} V_{32}^* \over
\vert m_c \vert^2 V_{21} V_{22}^* }
\right]\;.$$
The physical phases can be written in terms of the basis–independent quantities. In particular, if $(\delta_{LL}^d)_{12}$ is the only non–zero mass insertion then $${\rm Im Tr}[Y^d Y^{d\dagger},Y^u Y^{u\dagger}]M^{2 d_{_L}}=
2 {m_s^2-m_d^2 \over v_1^2 v_2^2} \biggl\vert \sum_i \vert m_i^{u}\vert^2 V_{i1} V_{i2}^*
\left( M^{2 d_{_L}} \right)_{12} \biggr\vert ~\sin\delta \;.$$ In a more general case, the invariant phases and the magnitudes of the matrix elements of $M^{2 d_{_L}}$ can be found via 3 CP–violating and 6 CP–conserving weak basis invariants $$\begin{aligned}
&& {\rm Im Tr}[Y^d Y^{d\dagger},Y^u Y^{u\dagger}]M^{2 d_{_L}} \;,
{\rm Im Tr}[(Y^d Y^{d\dagger})^2,Y^u Y^{u\dagger}]M^{2 d_{_L}} \;,
{\rm Im Tr}[Y^d Y^{d\dagger},(Y^u Y^{u\dagger})^2]M^{2 d_{_L}} ,\nonumber\\
&& {\rm Tr~} Y^u Y^{u\dagger}M^{2 d_{_L}} \;,\; {\rm Tr~} (Y^u Y^{u\dagger})^2M^{2 d_{_L}} \;,\;
{\rm Tr~} Y^u Y^{u\dagger}(M^{2 d_{_L}})^2 \;,\; \nonumber\\
&& {\rm Tr~} Y^d Y^{d\dagger}M^{2 d_{_L}} \;,\; {\rm Tr~} (Y^d Y^{d\dagger})^2M^{2 d_{_L}} \;,\;
{\rm Tr~} Y^d Y^{d\dagger}(M^{2 d_{_L}})^2 \;. \end{aligned}$$
[**ii. EDMs.**]{} Another example is a SUSY contribution to the electric dipole moments of the quarks. For the down quark, the relevant gluino–mediated contribution is typically written as [@Gabbiani:1996hi] $$(d_{\rm d})_{_{\rm SUSY}} \propto {\rm Im} (\delta_{LR}^d)_{11} M_3^* \;,
\label{edm}$$ with $(\delta_{LR}^d)_{11} \sim v_1 A^d_{11}/ \tilde m^2$ (omitting the $\mu$-term contribution). Again, it is clear that this result is not rephasing invariant.
To rectify this problem, one may use the strategy advocated above. For the purpose of illustation, let us assume the following simple form of $A^d$ in the super–CKM basis: $$A^d= \left( \matrix{ A^d_{11}& A^d_{12} & 0 \cr
0 & 0 & 0 \cr
0 & 0 & 0 } \right) \;.$$ The relevant physical phase can be expressed via the hermitian quantities of Table 1, column 3: $$\xi= {\rm Arg}\left( A^{d\dagger}Y^d + {\rm h.c.}\right)_{12} \left( A^{d \dagger} A^d\right)_{12}^*=
{\rm Arg~} A^{d}_{11}m_d^* \;.$$ Since $m_d$ has the same transformation properties under the left and right rephasings as $A^d_{11}$, this expression is manifestly reparametrization invariant. The corresponding weak basis invariant is ${\rm Im Tr}[Y^{d\dagger} Y^d, ( A^{d\dagger}Y^d + {\rm h.c.})]A^{d \dagger} A^d \propto \sin\xi$. Thus, Eq.(\[edm\]) is to be modified as $$(\delta_{LR}^d)_{11} \longrightarrow \Bigl\vert (\delta_{LR}^d)_{11} \Bigr\vert e^{i \xi} \;.$$ Clearly, if $m_d=0$, the SUSY EDM contribution vanishes.
The invariant phase associated with $A^d_{12}$ can similarly be found from $A^{d}Y^{d\dagger} + {\rm h.c.}$ and $Y^{u } Y^{u\dagger}$. It is important to note that if $A^d_{12}$ vanishes, $A^d$ and $Y^d$ are diagonal simultaneously, and the phase of $A^d_{11}$ $cannot$ be extracted from hermitian quantities. In this case, the relevant phase is given by Eq.(\[rho\]) and the corresponding weak basis invariant is $L_{A^d Y^d}(1)$.
Summary.
========
In this work, I have studied the CP–odd weak basis invariants in supersymmetric models and the corresponding reparametrization invariant CP phases. I have shown that, in SUSY models, a new class of CP–odd invariants, not expressible in terms of the Jarlskog–type invariants, appears. I have also obtained basis independent conditions for CP conservation and clarified the issues of rephasing invariance of observable quantities. This work was supported by PPARC.
[99]{}
C. Jarlskog, Phys. Rev. Lett. [**55**]{}, 1039 (1985); Z. Phys. C [**29**]{}, 491 (1985). See also I. Dunietz, O. W. Greenberg and D. d. Wu, Phys. Rev. Lett. [**55**]{}, 2935 (1985); D. d. Wu, Phys. Rev. D [**33**]{}, 860 (1986); F. J. Botella and L. L. Chau, Phys. Lett. B [**168**]{}, 97 (1986). For more than three generations, see J. Bernabeu, G. C. Branco and M. Gronau, Phys. Lett. B [**169**]{}, 243 (1986); M. Gronau, A. Kfir and R. Loewy, Phys. Rev. Lett. [**56**]{}, 1538 (1986). For beyond the Standard Model, see e.g. G. C. Branco and V. A. Kostelecky, Phys. Rev. D [**39**]{}, 2075 (1989); A. I. Sanda, Phys. Rev. D [**32**]{}, 2992 (1985); M. Gluck, Phys. Rev. D [**33**]{}, 3470 (1986); A. Mendez and A. Pomarol, Phys. Lett. B [**272**]{}, 313 (1991); L. Lavoura and J. P. Silva, Phys. Rev. D [**50**]{}, 4619 (1994); F. J. Botella and J. P. Silva, Phys. Rev. D [**51**]{}, 3870 (1995); J. A. Aguilar-Saavedra, J. Phys. G [**24**]{}, L31 (1998).
For a review, see H. E. Haber and G. L. Kane, Phys. Rept. [**117**]{}, 75 (1985).
S. Dimopoulos and S. Thomas, Nucl. Phys. B [**465**]{}, 23 (1996). See also D. A. Demir, Phys. Rev. D [**62**]{}, 075003 (2000); S. Abel, S. Khalil and O. Lebedev, Nucl. Phys. B [**606**]{}, 151 (2001).
K. Hagiwara [*et al.*]{} \[Particle Data Group Collaboration\], Phys. Rev. D [**66**]{}, 010001 (2002).
S. Abel, S. Khalil and O. Lebedev, Phys. Rev. Lett. [**89**]{}, 121601 (2002).
L. J. Hall, V. A. Kostelecky and S. Raby, Nucl. Phys. B [**267**]{}, 415 (1986).
F. Gabbiani, E. Gabrielli, A. Masiero and L. Silvestrini, Nucl. Phys. B [**477**]{}, 321 (1996).
[^1]: The commutator under the trace can also be raised to an odd power. This will not provide independent invariants and I omit its discussion for brevity.
[^2]: This follows from the penguin dominance in the kaon decay amplitude.
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{
"pile_set_name": "ArXiv"
}
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---
author:
- |
Kai Zheng\
Peking University\
`[email protected]`
- |
Tianle Cai\
Peking University\
`[email protected]`
- |
Weiran Huang\
Huawei Noah’s Ark Lab\
`[email protected]`
- |
Zhenguo Li\
Huawei Noah’s Ark Lab\
`[email protected]`
- |
Liwei Wang\
Peking University\
`[email protected]`
bibliography:
- 'reference.bib'
title: 'Locally Differentially Private (Contextual) Bandits Learning'
---
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Let $\mathfrak{g}$ be a basic Lie superalgebra. A weight module $M$ over $\mathfrak{g}$ is called finite if all of its weight spaces are finite dimensional, and it is called bounded if there is a uniform bound on the dimension of a weight space. The minimum bound is called the degree of $M$. For $\mathfrak{g} = D(2,1,\alpha)$, we prove that every simple weight module $M$ is bounded and has degree less than or equal to $8$. This bound is attained by a cuspidal module $M$ if and only if $M$ belongs to a $(\mathfrak{g},\mathfrak{g}_{\bar{0}})$-coherent family $L(\lambda)_{\Gamma}^{\mu}$ for some typical module $L(\lambda)$. Cuspidal modules which correspond to atypical modules have degree less than or equal to $6$ and greater than or equal to $2$.'
author:
- 'Crystal Hoyt[^1]'
date: 'July 30, 2013'
title: 'Weight modules of $D(2,1,\alpha)$'
---
\^\#2\_\#3[@[-\^>]{}@<.5ex>\[\#1\]\^[\#2]{} @[\_<-]{}@<-.5ex>\[\#1\]\_[\#3]{}]{}
Introduction
============
Basic Lie superalgebras are a natural generalization of simple finite dimensional Lie algebras, and their finite dimensional modules have been studied extensively [@K77; @M12]. In this case, every simple $\mathfrak{g}$-module is a highest weight module with respect to each choice of simple roots of $\mathfrak{g}$, and moreover, there exist various character formulas which help one “count” the dimension of each weight space of the module. A natural generalization of this setting is to the study of (possibly) infinite dimensional modules that have finite dimensional weight spaces, namely, finite weight modules. However, since these modules are not necessarily highest weight with respect to any choice of the set of simple roots, the question arises how to characterize and classify all such simple modules.
In [@M00], Mathieu gave an answer to this problem for simple Lie algebras by relating to each simple finite weight module $M$ a corresponding simple highest weight module $L(\lambda)$ such that $M$ is the “twisted localization” of $L(\lambda)$, that is, $M\cong L(\lambda)_{\Gamma}^{\mu}$. Grantcharov extended this result to cover classical Lie superalgebras in [@Gr09]. Using this characterization one can gain information about a simple finite weight module $M$ from the corresponding simple highest weight module $L(\lambda)$, including the calculation of its degree (see (\[degree\])). Moreover, this is a major step towards the classification of simple finite weight modules.
In addition, one must determine which simple highest weight modules $L(\lambda)$ can appear in this correspondence. These are the modules which have uniformly bounded weight multiplicities, the so called “bounded modules”. Then one should determine the simplicity conditions for the modules $L(\lambda)_{\Gamma}^{\mu}$ and the isomorphisms between them. For simple Lie algebras this problem was completely solved by Mathieu in [@M00], but the general problem remains open for basic Lie superalgebras. For modules with a strongly typical central character, this description can be derived from results of Gorelik, Penkov and Serganova [@G01; @P94; @PS92], however the situation is not surprisingly more difficult when the central character is atypical.
One can further reduce the classification problem to that of classifying “cuspidal modules” using Theorem \[cusp\] (Fernando [@F90]; Dimitrov, Mathieu, Penkov [@DMP]). A cuspidal module is a simple finite weight $\mathfrak{g}$-module that is not parabolically induced from any proper parabolic subalgebra $\mathfrak{p}\subset\mathfrak{g}$. These modules can be characterized in terms of their support and in terms of the action of $\mathfrak{g}$.
In this paper, we focus on the family of Lie superalgebras $D(2,1,\alpha)$ which are defined by one complex parameter $\alpha\in\mathbb{C}\setminus\{0,-1\}$ and study their finite weight modules. For $\mathfrak{g}=D(2,1,\alpha)$, every simple weight module is a finite weight module, and moreover it is bounded! Indeed, here $\mathfrak{g}_{\bar{0}}=\mathfrak{sl}_2\times\mathfrak{sl}_2\times\mathfrak{sl}_2$, so a simple weight module $V$ for $\mathfrak{g}_{\bar{0}}$ is just the tensor product $V=V_1\otimes V_2\otimes V_3$ of simple $\mathfrak{sl}_2$ weight modules. Now each simple $\mathfrak{sl}_2$ weight module $V_i$ must have one dimensional weight spaces, since the Casimir element $h^2+2h+fe$ acts by a scalar. Hence, $V$ also has one dimensional weight spaces. Now any simple weight module of $\mathfrak{g}$ can be realized as the quotient of an induced module $\mbox{Ind}_{\mathfrak{g}_{\bar{0}}}^{\mathfrak{g}}V=U(\mathfrak{g})\otimes_{\mathfrak{g}_{\bar{0}}}V$, where $V$ is a simple $\mathfrak{g}_{\bar{0}}$ weight module. The claim then follows from the fact that $U(\mathfrak{g}_{\bar{1}})$ is finite dimensional (here $\mbox{dim }U(\mathfrak{g}_{\bar{1}})=256$).
For $\mathfrak{g}=D(2,1,\alpha)$, we prove that every Verma module is bounded and has degree less than or equal to $8$. It then follows from a result of Grantcharov that simple (finite) weight modules are also bounded of degree less than or equal to $8$. We show that the dimensions of the weight spaces of a cuspidal $\mathfrak{g}$-module are constant on $Q_{\bar{0}}$-cosets and we calculate this degree. We prove that a cuspidal $\mathfrak{g}$-module has degree $8$ if and only if it is “typical”. We prove that if $M$ is an “atypical” cuspidal $\mathfrak{g}$-module then $2\leq\mbox{deg }M\leq 6$. We determine the conditions on $\lambda$ and $\Gamma$ that are necessary for $L(\lambda)_{\Gamma}^{\mu}$ to be a cuspidal module. It remains to determine the restrictions on $\mu\in\mathbb{C}^{3}$ and isomorphisms between these modules.
Basic Lie superalgebras
=======================
A simple finite dimensional Lie superalgebra $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ is called [*basic*]{} if $\mathfrak{g}_{\bar{0}}$ is a reductive Lie algebra, and there exists an even non-degenerate (symmetric) invariant bilinear form on $\mathfrak{g}$. These are the Lie superalgebras: $\mathfrak{sl}(m|n)$ for $m\neq n$, $\mathfrak{psl}(n|n)$, $\mathfrak{osp}(m|2n)$, $F(4)$, $G(3)$ and $D(2,1,\alpha)$, $a\in\mathbb{C}\setminus \{0,-1\}$, and finite-dimensional simple Lie algebras. A basic Lie superalgebra can be represented by a Dynkin diagram, though not uniquely.
Let $\mathfrak{g}$ be a basic Lie superalgebra, and fix a Cartan subalgebra $\mathfrak{h}\subset\mathfrak{g}_{\bar{0}}\subset\mathfrak{g}$. We have a root space decomposition $\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\alpha\in\Delta}\mathfrak{g}_{\alpha}$. For each set of simple roots $\Pi\subset\Delta$, we have a corresponding set of positive roots $\Delta^{+}=\Delta^{+}_{\bar{0}}\cup\Delta^{+}_{\bar{1}}$ and a triangular decomposition $\mathfrak{g}=\mathfrak{n}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}^{+}$. This induces a triangular decomposition of $\mathfrak{g}_{\bar{0}}$, namely, $\mathfrak{g}_{\bar{0}}=\mathfrak{n}_{\bar{0}}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}_{\bar{0}}^{+}$. Let $\Pi_{\bar{0}}$ denote the corresponding set of simple roots for $\mathfrak{g}_{\bar{0}}$. The root lattice of $\mathfrak{g}$ (resp. $\mathfrak{g}_{\bar{0}}$) is defined to be $Q=\sum_{\alpha\in\Pi} \mathbb{Z} \alpha$ (resp. $Q_{\bar{0}}=\sum_{\alpha\in\Pi_{\bar{0}}} \mathbb{Z} \alpha$). Let $\rho_{0}=\frac{1}{2}\sum_{\alpha\in\Delta^{+}_{\bar{0}}}\alpha$, $\rho_{1}=\frac{1}{2}\sum_{\alpha\in\Delta^{+}_{\bar{1}}}\alpha$ and $\rho=\rho_{0}-\rho_{1}$. Denote by $U(\mathfrak{g})$ (resp. $U(\mathfrak{g}_{\bar{0}})$) the universal enveloping algebra of $\mathfrak{g}$ (resp. $\mathfrak{g}_{\bar{0}}$). See [@K77; @M12] for definitions and more details.
Finite weight modules
---------------------
A $\mathfrak{g}$-module $M$ is called a [*weight module*]{} if it decomposes into a direct sum of weight spaces $M=\oplus_{\mu\in\mathfrak{h}^{*}} M_{\mu}$, where $M_{\mu}= \{m\in M \mid h.m=\mu(h)m \text{ for all }h\in\mathfrak{h}\}$. A weight module $M$ is called [*finite*]{} if $\mbox{dim }M_{\mu} < \infty$ for all $\mu\in\mathfrak{h}^{*}$. Define the [*support of $M$*]{} to be the set $$\mbox{Supp }M=\{\mu\in\mathfrak{h}^{*}\mid \mbox{dim }M_{\mu}\neq 0\}.$$
The module $M$ is called [*bounded*]{} if the exists a constant $c$ such that $\mbox{dim }M_{\mu} < c$ for all $\mu\in\mathfrak{h}^{*}$. Recall that a $\mathfrak{g}$-module $M=M_{\bar{0}}\oplus M_{\bar{1}}$ is also $\mathbb{Z}/2\mathbb{Z}$-graded. The [*degree of M*]{} is defined to be $$\label{degree}
\mbox{deg }M=max_{\mu\in\mathfrak{h}^*} \mbox{dim}(M)_{\mu},$$ and we define the [*graded degree of M*]{} to be $(d_{0},d_{1})$, where $$d_i=max_{\mu\in\mathfrak{h}^*} \mbox{dim}(M_{\bar{i}})_{\mu} \text{ for }i\in\{0,1\}.$$
Clearly, $\mbox{max}\{d_0,d_1\} \leq \mbox{deg }M\leq d_0+d_1$. However, if $M$ is a weight module that can be generated by a single weight vector (i.e. simple or highest weight module), then each weight space of $M$ is either purely even or purely odd, and so in this case $\mbox{deg }M=\mbox{max}\{d_0,d_1\}$.
Let $M(\lambda)$ denote the Verma module of highest weight $\lambda\in\mathfrak{h}^{*}$ with respect to a set of simple roots $\Pi$, and let $L(\lambda)$ denote its unique simple quotient [@M12]. It is clear that $M(\lambda)$ and $L(\lambda)$ are finite weight modules, but they are not always bounded.
For each $\beta\in\Pi$ an odd isotropic root (i.e. $(\beta,\beta)=0$), we have an odd reflection of the set of simple roots $r_{\beta}:\Pi \rightarrow \Pi'$ satisfying $\Pi'=(\Pi\setminus\{\beta\})\cup\{-\beta\}$ [@LSS]. Moreover, for a simple highest weight module $L_{\Pi}(\lambda)$ there exists $\lambda'\in\mathfrak{h}^*$ such that $L_{\Pi'}(\lambda')=L_{\Pi}(\lambda)$. In particular, $\lambda'=\lambda-\beta$ if $(\lambda,\beta)\neq 0$, while $\lambda'=\lambda$ otherwise [@KW]. Using even and odd reflections one can move between all the different choices of simple roots for a basic Lie superalgebra $\mathfrak{g}$ [@S11]. Moreover, one can move between two different Dynkin diagrams of $\mathfrak{g}$ using only odd reflections.
A simple highest weight module $L(\lambda)$ is called [*typical*]{} if $(\lambda+\rho,\alpha)\neq 0$ for all $\alpha\in\Delta_{\bar{1}}$, and [*atypical*]{} otherwise. The notion of typicality is preserved by an odd reflection of the set of simple roots, that is, given an odd reflection $r_{\beta}:\Pi \rightarrow \Pi'$ and $L_{\Pi'}(\lambda')=L_{\Pi}(\lambda)$, then $L_{\Pi'}(\lambda')$ is typical iff $L_{\Pi}(\lambda)$ is typical [@KW; @S11].
It was shown by Penkov and Serganova that if $\mathfrak{g}$ is a basic Lie superalgebra, then the category of representations of $\mathfrak{g}$ with a fixed generic typical central character is equivalent to the category of representations of $\mathfrak{g}_{\bar{0}}$ with a certain corresponding central character [@P94; @PS92]. This equivalence of categories was extended to representations of $\mathfrak{g}$ with a fixed strongly typical central character by Gorelik in [@G01]. In the case that the root system of $\mathfrak{g}$ is reduced ($\alpha,k\alpha\in\Delta$ implies $k=\pm 1$) all typical central characters are strongly typical.
Cuspidal modules
----------------
A [*$\mathbb{Z}$-grading*]{} of $\mathfrak{g}$ is a decomposition $ \mathfrak{g}=\oplus_{j\in\mathbb{Z}} \mathfrak{g}(j)$ satisfying $[\mathfrak{g}(i),\mathfrak{g}(j)]\subset\mathfrak{g}(i+j)$ and $\mathfrak{h}\subset\mathfrak{g}(0)$. A subalgebra $\mathfrak{p}\subset\mathfrak{g}$ is called a [*parabolic subalgebra*]{} if there exists a $\mathbb{Z}$-grading of $\mathfrak{g}$ such that $\mathfrak{p}=\oplus_{j\geq 0} \mathfrak{g}(j)$. In this case, $\mathfrak{l}=\mathfrak{g}(0)$ is a [*Levi subalgebra*]{} and $\mathfrak{n}=\oplus_{j\geq 1} \mathfrak{g}(j)$ is the nilradical of $\mathfrak{p}$.
Let $\mathfrak{p}=\mathfrak{l}\oplus\mathfrak{n}$ be a parabolic subalgebra of $\mathfrak{g}$, and let $S$ be a simple $\mathfrak{p}$-module. Then $M_{\mathfrak{p}}(S):=\mbox{Ind}_{\mathfrak{p}}^{\mathfrak{g}}S$ has a unique simple quotient $L_{\mathfrak{p}}(S)$. The module $L_{\mathfrak{p}}(S)$ is said to be [*parabolically induced*]{}. A simple $\mathfrak{g}$-module is called [*cuspidal*]{} if it is not parabolically induced from any proper parabolic subalgebra $\mathfrak{p}\subset\mathfrak{g}$.
\[cusp\] Let $\mathfrak{g}$ be a basic Lie superalgebra. Any simple finite weight $\mathfrak{g}$-module is obtained by parabolic induction from a cuspidal module of a Levi subalgebra.
This theorem is an important step towards the classification of all simple finite weight modules. It reduces the general classification problem to that of classifying cuspidal modules.
If $\mathfrak{g}$ is a finite dimensional simple Lie algebra that admits a cuspidal module, then $\mathfrak{g}$ is of type A or C.
Only the following basic Lie superalgebras admit a cuspidal module: $\mathfrak{psl}(n|n)$, $\mathfrak{osp}(m|2n)$ with $m\leq 6$, $D(2,1,\alpha)$ with $\alpha\in\mathbb{C}\setminus\{0,-1\}$, $\mathfrak{sl}(n)$, $\mathfrak{sp}(2n)$.
The following theorem gives a characterization of cuspidal $\mathfrak{g}_{\bar{0}}$-modules.
Let $\mathfrak{g}_{\bar{0}}$ be a reductive Lie algebra, and let $M$ be a simple finite weight $\mathfrak{g}_{\bar{0}}$-module. Then $M$ is cuspidal iff $\mbox{Supp }M$ is exactly one $Q$ coset iff $\mbox{ad }x_{\alpha}$ is injective for all $\alpha\in\Delta$, $x_{\alpha}\in\mathfrak{g}_{\alpha}$.
Let $\mathfrak{g}_{\bar{0}}$ be a reductive Lie algebra. If $M$ is a cuspidal $\mathfrak{g}_{\bar{0}}$-module, then $M$ is bounded and $\mbox{dim }M_{\mu}=\mbox{deg }M$ for all $\mu\in\mbox{Supp }M$.
These conditions are too strict when $\mathfrak{g}$ is a basic Lie superalgebra, and so the following definitions were introduced in [@DMP]. A finite weight module $M$ is called [*torsion-free*]{} if the monoid generated by $$\mbox{inj }M=\{\alpha\in\Delta_{\bar{0}}\mid x_{\alpha}\in\mathfrak{g}_{\alpha} \text{ acts injectively on }M\}$$ is a subgroup of finite index in $Q$. A finite weight module $M$ is called [*dense*]{} if $\mbox{Supp }M$ is a finite union of $Q'$-cosets, for some subgroup $Q'$ of finite index in $Q$.
\[equivalence\] Let $\mathfrak{g}$ be a basic Lie superalgebra, and let $M$ be a simple finite weight $\mathfrak{g}$-module. Then $M$ is cuspidal iff $M$ is dense iff $M$ is torsion free.
The following lemmas are proven in [@M00].
Let $\mathfrak{g}_{\bar{0}}$ be a reductive Lie algebra. Any bounded $\mathfrak{g}_{\bar{0}}$-module has finite length.
Let $\mathfrak{g}$ be a basic Lie superalgebra. If $M$ is a simple finite weight $\mathfrak{g}$-module, then for each $\alpha\in\Delta_{\bar{0}}$ the action of $x\in\mathfrak{g}_{\alpha}$ on $M$ is either injective or locally nilpotent.
For each $\mathfrak{g}_{\bar{0}}$-module $N$, let $$\widetilde{N} := \mbox{Ind}_{\mathfrak{g}_{\bar{0}}}^{\mathfrak{g}}(N).$$
\[decompose\] (i) For any finite cuspidal $\mathfrak{g}_{\bar{0}}$-module $N$, the module $\widetilde{N}$ contains at least one and only finitely many non-isomorphic cuspidal submodules.\
(ii) For any finite cuspidal $\mathfrak{g}$-module $M$, there is at least one and only finitely many non-isomorphic cuspidal $\mathfrak{g}_{\bar{0}}$-modules such that $M\subset \widetilde{N}$.
Coherent families
-----------------
Let $C(\mathfrak{h})$ denote the centralizer of $\mathfrak{h}$ in $U(\mathfrak{g}_{\bar{0}})$. A [*$(\mathfrak{g},\mathfrak{g}_{\bar{0}})$-coherent family of degree $d$*]{} is a finite weight $\mathfrak{g}$-module $M$ such that $\mbox{dim }M_{\mu}=d$ for all $\lambda\in\mathfrak{h}^{*}$ and the function $\mu\mapsto \mbox{Tr }u|_{M_{\mu}}$ is polynomial in $\mu$, for all $u\in C(\mathfrak{h})$ [@Gr03; @Gr06; @M00].
\[sl2\] Let $\mathfrak{g}=\mathfrak{sl}_2$ and fix $a\in\mathbb{C}$. Define a module $V(a)=\oplus_{s\in\mathbb{C}}\mathbb{C}x^{s}$ with the following $\mathfrak{sl}_2$ action. $$\begin{aligned}
&e\mapsto x^{2}\ d/dx +ax &e.x^{s}=(a+s)x^{s+1}\\
&f\mapsto -d/dx +a(1/x) &f.x^{s}=(a-s)x^{s-1}\\
&h\mapsto 2x\ d/dx &h.x^{s}=(2s)x^{s}\end{aligned}$$
For each $a\in\mathbb{C}$, $V(a)$ is a $\mathfrak{sl}_2$-coherent family. For each $[\mu]\in\mathbb{C}/\mathbb{Z}$ with representative $\mu\in\mathfrak{h}^{*}$, $$V(a)^{[\mu]}:=\oplus_{n\in\mathbb{Z}}\mathbb{C}x^{\mu+n}$$ is a submodule, which is simple and cuspidal if and only if $\mu\pm a \not\in \mathbb{Z}$.
Let $\Gamma=\{\gamma_1,\ldots,\gamma_k\}\subset\Delta_{\bar{0}}^{+}$ be a set of commuting roots, and for each $\gamma_i\in\Gamma$ choose $f_{i}\in\mathfrak{g}_{-\gamma_i}$. Let $U_{\Gamma}$ be the localization of $U(\mathfrak{g})$ at the set $\{f_{i}^{n}\mid n\in\mathbb{N}, \gamma_i\in\Gamma\}$. If $\Gamma\subset \mbox{inj }L(\lambda)$, define the [*localization of $L(\lambda)$ at $\Gamma$*]{} to be the module $L(\lambda)_{\Gamma}:=U_{\Gamma}\otimes_{U(\mathfrak{g})} L(\lambda)$. Then $L(\lambda)$ is a submodule of $L(\lambda)_{\Gamma}$ and $\mbox{deg }L(\lambda)_{\Gamma}=\mbox{deg }L(\lambda)$.
Now for each $\mu\in\mathbb{C}^{k}$, we define a new module $L(\lambda)_{\Gamma}^{\mu}$ whose underlying vector space is $L(\lambda)_{\Gamma}$, but with a new action of $\mathfrak{g}$ defined as follows. For $u\in U_{\Gamma}$ and $x\in L(\lambda)_{\Gamma}^{\mu}$, $$u\cdot x:= \Phi^{\mu}_{\Gamma}(u)v,$$ where $$\Phi^{\mu}_{\Gamma}(u)=\sum_{0\leq i_1,\ldots,i_k} {\mu_1 \choose i_1} \dots {\mu_k \choose i_k} \mbox{ad}(f_1)^{i_1}\dots\mbox{ad}(f_k)^{i_k}(u)f_1^{-i_1}\dots f_k^{-i_k}.$$ Note that this sum is finite for each choice of $u$. The module $L(\lambda)_{\Gamma}^{\mu}$ is called the [*twisted localization of $L(\lambda)$ with respect to $\Gamma$ and $\mu$*]{}, and it is a $(\mathfrak{g},\mathfrak{g}_{\bar{0}})$-coherent family of degree $d=\mbox{deg }L(\lambda)$.
\[simple\] Let $\mathfrak{g}$ be a basic Lie superalgebra. Each simple finite weight $\mathfrak{g}$-module $M$ is a twisted localization of a simple highest weight module $L_{\mathfrak{b}}(\lambda)$ for some Borel subalgebra $\mathfrak{b}\subset\mathfrak{g}$ and $\lambda\in\mathfrak{h}^{*}$. In particular, $M\cong L_{\mathfrak{b}}(\lambda)_{\Gamma}^{\mu}$ for some $\mu\in\mathbb{C}$ and set of commuting even roots $\Gamma$.
If $M$ is a cuspidal or bounded module, then $L_{\mathfrak{b}}(\lambda)$ is necessarily bounded.
The Lie superalgebra $D(2,1,\alpha)$
====================================
For each $\alpha\in\mathbb{C}\setminus\{0,-1\}$, the Lie superalgebra $\mathfrak{g}=D(2,1,\alpha)$ can be realized as a contragredient Lie superalgebra $\mathfrak{g}(A)$ with Cartan matrix $$\label{A} A=\left(\begin{array}{ccc}0 & 1 & \alpha\\
1 & 0 & -\alpha-1\\
\alpha & -\alpha-1 & 0\end{array}\right),$$ set of simple roots $\Pi=\{\beta_1,\beta_2,\beta_3\}$ with parity $(1,1,1)$, generating set $$\{e_i\in\mathfrak{g}_{\beta_i}, f_i\in\mathfrak{g}_{-\beta_i}, h_i\in\mathfrak{h}\mid i=1,2,3\}$$ and defining relations [@K77]. Then $\mathfrak{g}_{\bar{0}}=\mathfrak{sl}_2 \times \mathfrak{sl}_2 \times \mathfrak{sl}_2$ is 9-dimensional with $$\Pi_{\bar{0}}=\{\beta_1+\beta_2,\beta_1+\beta_3,\beta_2+\beta_3\},$$ and $\mathfrak{g}_{\bar{1}}$ is the 8-dimensional $\mathfrak{g}_{\bar{0}}$-module given by tensoring three copies of the standard representation of $\mathfrak{sl}_2$. Our choice of $\Pi$ induces a triangular decomposition $\Delta=\Delta^{+}\cup\Delta^{-}$ where
$$\label{triangleD} \Delta_{\bar{0}}^{+}=\{\beta_1+\beta_2,\beta_1+\beta_3,\beta_2+\beta_3\} \text{ and } \Delta_{\bar{1}}^{+}=\{\beta_1,\beta_2,\beta_3,\beta_1+\beta_2+\beta_3\}.$$
\[lattice\] For $\mathfrak{g}=D(2,1,\alpha)$, the even root lattice $Q_{\bar{0}}$ is a sublattice of index 2 in $Q$.
$D(2,1,\alpha)$ has four different Dynkin diagrams.
$$\label{diagrams}
\begin{array}{c}\xymatrix{& {\bigotimes}\AW[ldd]^{1}_{1} \AW[rdd]^{\alpha}_{\alpha} & \\
& & \\ {\bigotimes}\AW[rr]^{-1-\alpha}_{-1-\alpha} & & {\bigotimes}} \end{array}
\begin{array}{c}
\xymatrix{ {\bigcirc}\AW[r]^{-1}_{1} & {\bigotimes}\AW[r]^{\alpha}_{-1} & {\bigcirc}}\\
\xymatrix{ {\bigcirc}\AW[r]^{-1}_{-1-\alpha} & {\bigotimes}\AW[r]^{\alpha}_{-1} & {\bigcirc}}\\
\xymatrix{ {\bigcirc}\AW[r]^{-1}_{1} & {\bigotimes}\AW[r]^{-1-\alpha}_{-1} & {\bigcirc}}\end{array}$$
The Cartan matrix $A$ in (\[A\]) is equivalent to the diagram on the left, and corresponds to $\Delta^+$ given in (\[triangleD\]). Most of our computations will be with respect to this choice of the set of simple roots, since here $\rho=0$ and since no set of simple roots of $\mathfrak{g}=D(2,1,\alpha)$ contains a set of simple roots for $\mathfrak{g}_{\bar{0}}$.
Finite weight modules for $D(2,1,\alpha)$ {#cuspD}
-----------------------------------------
In this section we study finite weight modules for the basic Lie superalgebra $D(2,1,\alpha)$.
It was shown in [@DMP] that every simple finite weight module is obtained by parabolic induction from a cuspidal module $L_{\mathfrak{p}}(S)$ with $\mathfrak{p}=\mathfrak{l}\oplus\mathfrak{n}$, such that either $\mathfrak{l}$ is a proper reductive subalgebra of $\mathfrak{g}_{\bar{0}}=\mathfrak{sl}_2\times\mathfrak{sl}_2\times\mathfrak{sl}_2$ or $\mathfrak{l}=\mathfrak{g}$. Cuspidal modules for $\mathfrak{sl}_2$ are classified in Example \[sl2\], and all cuspidal modules for $\mathfrak{sl}_2\times\mathfrak{sl}_2$ can be obtained by tensoring two cuspidal $\mathfrak{sl}_2$-modules together, namely, $$V(a_1)^{[\mu_{1}]}\otimes V(a_2)^{[\mu_2]}\text{ with }\mu_i\pm a_i,\not\in\mathbb{Z},\ i=1,2.$$ So it remains to describe the cuspidal modules for $\mathfrak{g}=D(2,1,\alpha)$. The following theorems will help us realize these modules.
\[thmVerma\] Let $\mathfrak{g}=D(2,1,\alpha)$. For each set of positive roots $\Delta^{+}$ and each $\lambda\in\mathfrak{h}^{*}$, the Verma module $M(\lambda)$ is bounded. $M(\lambda)$ has degree $8$ and graded degree $(8,8)$.
The dimensions of the weight spaces of $M(\lambda)$ are given by the coefficients of the character formula $\mbox{ch }M(\lambda)=\frac{e^{\lambda+\rho}}{e^{\rho}R}=e^{\lambda}\frac{R_{1}}{R_{0}}$, where $R_{0}=\Pi_{\alpha\in\Delta_{\bar{0}}^{+}}(1-e^{-\alpha})$ and $R_{1}=\Pi_{\alpha\in\Delta_{\bar{1}}^{+}}(1+e^{-\alpha})$. Since $e^{\rho}R$ is invariant under odd reflections, it is sufficient to prove the theorem with respect to the set of positive roots $\Delta^{+}$ from (\[triangleD\]). $$\begin{aligned}
\mbox{ch }M(\lambda)&=e^{\lambda} \frac{(1+e^{-\beta_1})(1+e^{-\beta_2})(1+e^{-\beta_3})(1+e^{-\beta_1-\beta_2-\beta_3})}
{(1-e^{-\beta_1-\beta_2})(1-e^{-\beta_1-\beta_3})(1-e^{-\beta_2-\beta_3})}\\
&=e^{\lambda}(1+e^{-\beta_1})(1+e^{-\beta_2})(1+e^{-\beta_3})(1+e^{-\beta_1-\beta_2-\beta_3})\cdot\left(\sum_{\substack{k_1,k_2,k_3\in\mathbb{N}:\\
k_1+k_2+k_3\text{ is even}}}e^{-k_1\beta_1-k_2\beta_2-k_3\beta_3}\right)
\end{aligned}$$ So for $m_1,m_2,m_3\in\mathbb{N}$ sufficiently large, the $(\lambda-m_1\beta_1-m_2\beta_2-m_3\beta_3)$ weight space of $M(\lambda)$ is purely even with dimension ${4\choose 0} + {4\choose 2} + {4\choose 4}=8$ when $m_1+m_2+m_3$ is even, (which corresponds to a choice of an even number of odd roots from $\Delta_{\bar{1}}^{-}$), and it is purely odd with dimension ${4\choose 1} +{4\choose 3}=8$ when $m_1+m_2+m_3$ is odd, (corresponding to a choice of an odd number of odd roots from $\Delta_{\bar{1}}^{-}$).
\[corHW\] A highest weight module is bounded and has degree less than or equal to $8$.
Since every simple weight module of $D(2,1,\alpha)$ is a finite weight module, we can combine Theorem \[simple\] with Corollary \[corHW\] to obtain the following.
\[small\] For $\mathfrak{g}=D(2,1,\alpha)$, any simple weight module $M$ is bounded and has degree less than or equal to $8$.
In Theorem \[small\], we do not assume that $M$ is a highest weight module.
Let $\Delta^{+}$ be as in (\[triangleD\]), and let $\alpha_1=\beta_2+\beta_3$, $\alpha_2=\beta_1+\beta_3$, $\alpha_3=\beta_1+\beta_2$, so that $\Pi_{\bar{0}}=\{\alpha_1,\alpha_2,\alpha_3\}$. Then for each $\alpha_i\in\Pi_{\bar{0}}$, $i=1,2,3$, we have an $\mathfrak{sl}_2$-triple $\{E_{i},F_{i},H_{i}\}$ with $E_{i}\in\mathfrak{g}_{\alpha_{i}}$, $F_{i}\in\mathfrak{g}_{-\alpha_{i}}$ and $H_i=[E_i,F_i]$ satisfying $\alpha_{i}(H_{i})=2$. For each $\lambda\in\mathfrak{h}^{*}$, $i=1,2,3$, let $\lambda_i=\lambda(h_i)$ and $c_i=\lambda(H_i)$. Then $$\label{cs}
c_{1}=\frac{\lambda_2+\lambda_3}{-\alpha-1},\ c_2=\frac{\lambda_1+\lambda_3}{\alpha},\ c_3=\lambda_1+\lambda_2.$$
\[conditions\] Let $\mathfrak{g}=D(2,1,\alpha)$ and let $\Delta^+$ be as in (\[triangleD\]). Then for each $\lambda\in\mathfrak{h}^{*}$, we have that $\mbox{inj }L(\lambda)=\Delta_{\bar{0}}^{-}$ if and only if $c_1,c_2,c_3\not\in\mathbb{Z}_{\geq 1}$ and at most one of $\lambda_1,\lambda_2,\lambda_3$ equals zero.
Let $\Pi$ denote the set of simple roots of $\Delta^{+}$. Then $F_i$ acts injectively on $L(\lambda)$ if and only if given a set of simple roots $\Pi'$ containing $\alpha_i$ which can be obtained by a sequence of odd reflections from $\Pi$, we have that $\lambda'(H_i)\not\in\mathbb{Z}_{\geq 0}$, where $\lambda'\in\mathfrak{h}^{*}$ satisfies $L_{\Pi'}(\lambda')=L_{\Pi}(\lambda)$.
Now if $\lambda_i,\lambda_j=0$, then $f_iv=0$ and $f_jv=0$ imply $F_kv=[f_i,f_j]v=0$. Hence, if $F_1,F_2,F_3$ act injectively on $L(\lambda)$, then at most one of $\lambda_1,\lambda_2,\lambda_3$ equals zero. By reflecting at $\beta_j$ we get $\Pi'=\{-\beta_j,\alpha_i,\alpha_k\}$ where $i\neq j\neq k$. If $\beta_j$ is a typical root ($\lambda_j\neq0$) then $F_i$ acts injectively iff $(\lambda-\beta_j)(H_i)=c_i-1\not\in\mathbb{Z}_{\geq 0}$, and $F_k$ acts injectively iff $(\lambda-\beta_j)(H_k)=c_k-1\not\in\mathbb{Z}_{\geq 0}$. Hence, if at least two of $\lambda_1,\lambda_2,\lambda_3$ are non-zero, it follows that $F_1,F_2,F_3$ act injectively iff $c_1,c_2,c_3\not\in\mathbb{Z}_{\geq 1}$.
Let $\Delta^+$ be as in (\[triangleD\]), then $L(\lambda)$ is typical if and only if $\lambda_1,\lambda_2,\lambda_3\neq 0$ and $\lambda_1+\lambda_2+\lambda_3\neq 0$.
For $\mathfrak{g}=D(2,1,\alpha)$, suppose that $L(\lambda)$ is a simple highest weight module satisfying $\mbox{inj }L(\lambda)=\Delta_{\bar{0}}^{-}$, then $L(\lambda)= M(\lambda)$ iff $L(\lambda)$ is typical.
Since each of these properties is a module property that is preserved under odd reflections, it suffices to prove the theorem with respect to $\Delta^{+}$ in (\[triangleD\]). By Theorem \[conditions\], we have $\mbox{inj }L(\lambda)=\Delta_{\bar{0}}^{-}$ implies $\lambda(H_i)\not\in\mathbb{Z}_{\geq 1}$ for each $\alpha_i\in\Delta_{\bar{0}}^{+}$. Hence, $$\frac{2(\lambda+\rho,\alpha_i)}{(\alpha_i,\alpha_i)}=\frac{2(\lambda,\alpha_i)}{(\alpha_i,\alpha_i)}=\lambda(H_i)\quad\not\in\mathbb{Z}_{\geq 1},$$ where the first equality is due to the fact that $\rho=0$ for our choice of $\Delta^{+}$, and the second equality is given by the identification of $\mathfrak{h}$ with $\mathfrak{h}^*$. The claim now follows from computing the Shapovalov determinant using the formula given in [@GK07 Section 1.2.8]. Since $(\beta,\beta)=0$ for $\beta\in\Delta_{\bar{1}}$, we conclude that $M(\lambda)$ is simple if and only if $(\lambda+\rho,\beta)\neq0$ for all $\beta\in\Delta_{\bar{1}}$.
Cuspidal modules for $D(2,1,\alpha)$
------------------------------------
The following theorem gives a characterization of cuspidal modules for $D(2,1,\alpha)$.
\[characterize\] Let $\mathfrak{g}=D(2,1,\alpha)$, and let $M$ be a simple weight $\mathfrak{g}$-module. The following are equivalent
1. $M$ is cuspidal;
2. $\mbox{Supp }M$ is exactly one $Q$ coset;
3. $x_{\alpha}$ acts injectively for all $\alpha\in\Delta_{\bar{0}}$, $x_{\alpha}\in\mathfrak{g}_{\alpha}$.
Let $\mathfrak{g}=D(2,1,\alpha)$. If $M$ is a cuspidal $\mathfrak{g}$-module, then $\mbox{dim }M_{\lambda}=\mbox{dim }M_{\mu}$ for all $\lambda - \mu \in Q_{\bar{0}}$.
For $\mathfrak{g}=D(2,1,\alpha)$ it follows from [@G01], that for typical central characters we have a $1-1$ correspondence between cuspidal $\mathfrak{g}$-modules and cuspidal $\mathfrak{g}_{\bar{0}}$-modules. Here we describe cuspidal $\mathfrak{g}_{\bar{0}}$-modules.
Let $\mathfrak{g}_{\bar{0}}=\mathfrak{sl}_2\times \mathfrak{sl}_2\times \mathfrak{sl}_2$. Then the cuspidal $\mathfrak{g}_{\bar{0}}$-modules are as follows. $$\label{V} V_a^{\mu}:=V(a_1)^{[\mu_{1}]}\otimes V(a_2)^{[\mu_2]}\otimes V(a_3)^{[\mu_3]}\quad a,\mu\in\mathbb{C}^{3},\ \mu_i\pm a_i,\not\in\mathbb{Z},\ i=1,2,3$$ Moreover, $\mbox{Supp }V_a^{\mu}=Q+\mu$ and $\mbox{deg }V_a^{\mu}=1$.
We have two ways to realize cuspidal modules. The first method is by decomposing the modules $\widetilde{N}$ appearing in Theorem \[decompose\], since each simple subquotient of $\widetilde{N}$ is cuspidal. It follows from the PBW theorem that $\mbox{deg }\widetilde{V_a^{\mu}}=2^7$, so we see from Theorem \[small\] that $\widetilde{V_a^{\mu}}$ is far from simple. Alternatively, one could determine simplicity conditions for the modules $L(\lambda)_{\Gamma}^{\mu}$ appearing in Theorem \[simple\].
Here we calculate the degree of a cuspidal $\mathfrak{g}$-module $L(\lambda)_{\Gamma}^{\mu}$ using the results from Section \[cuspD\] and Shapovalov determinants for the module $M(\lambda)$ [@G04; @GK07; @K78].
Let $\mathfrak{g}=D(2,1,\alpha)$, and suppose that $L(\lambda)_{\Gamma}^{\mu}$ is a (simple) cuspidal $\mathfrak{g}$-module for $\lambda\in\mathfrak{h}^{*}$, $\Delta=\Delta^+\cup\Delta^-$, $\Gamma\subset\Delta^{-}_{\bar{0}}$ and $\mu\in\mathbb{C}^{3}$. Then
1. $\Gamma=\mbox{inj }L(\lambda)=\Delta_{\bar{0}}^{-}$, and if $\Delta^+$ is as in (\[triangleD\]) then Theorem \[conditions\] applies;
2. $\mbox{deg }L(\lambda)_{\Gamma}^{\mu}=8$ iff $L(\lambda)$ is typical iff $L(\lambda)$ has graded degree $(8,8)$;
3. if $L(\lambda)$ is atypical, then $2\leq \mbox{deg }L(\lambda)_{\Gamma}^{\mu}\leq 6$;
4. if $(\lambda,\beta)=0$ for some simple odd root $\beta$, then $\mbox{deg }L(\lambda)_{\Gamma}^{\mu}\leq 4$.
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank Maria Gorelik, Ivan Penkov and Vera Serganova for helpful discussions. Supported in part at the Technion by an Aly Kaufman Fellowship.
I. Dimitrov, O. Mathieu, I. Penkov [*On the structure of weight modules*]{}, Trans. Amer. Math. Soc. 352 (2000), 2857–2869.
S. Fernando, [*Lie algebra modules with finite-dimensional weight spaces, I*]{}, Trans. Amer. Math. Soc. 322 (1990), 757–781.
M. Gorelik, [*Strongly typical representations of the basic classical Lie superalgebras*]{}, J. Amer. Math. Soc. 15 (2001), 167–184.
M. Gorelik, [*The Kac construction of the centre of $U(\mathfrak{g})$ for Lie superalgebras*]{}, JNMP 11 (2004), 325–349.
M. Gorelik, V. Kac, [*On Simplicity of Vacuum modules*]{}, Advances in Math. 211 (2007), 621-677.
D. Grantcharov, [*Coherent families of weight modules of Lie superalgebras and an explicit description of the simple admissible $\mathfrak{sl}(n+1|1)$-modules*]{}, J. of Algebra 265 (2003), 711–733.
D. Grantcharov, [*On the structure and character of weight modules*]{}, Forum Math. 18 (2006), 933–950.
D. Grantcharov, [*Explicit realizations of simple weight modules of classical Lie superalgebras*]{}, Contemp. Math. 499 (2009) 141–148.
V.G. Kac, [*Lie superalgebras*]{}, Advances in Math. 26 (1977) 8–96.
V.G. Kac, [*Representations of classical Lie superalgebras*]{}, Lect. Notes Math. 676, Springer- Verlag (1978), 597–626.
V. G. Kac and M. Wakimoto, [*Integrable highest weight modules over affine superalgebras and number theory*]{}, Lie Theory and Geometry, Progress in Math. 123, (1994), 415–456.
D. Leites, M. Savel’ev, V. Serganova, [*Embeddings of Lie superalgebra $\mathfrak{osp}(1|2)$ and nonlinear supersymmetric equations*]{}, Group Theoretical Methods in Physics, vol. 1 (1986), 377–394.
O. Mathieu, [*Classification of irreducible weight modules*]{}, Ann. Inst. Fourier 50 (2000), 537–592.
I. Musson, [*Lie superalgebras and enveloping algebras*]{}, Graduate Studies in Mathematics, vol. 131, 2012.
I. Penkov, [*Generic representations of classical Lie superalgebras and their localization*]{}, Monatshefte f. Math. 118 (1994) p.267–313.
I. Penkov, V. Serganova, [*Representation of classical Lie superalgebras of type I*]{}, Indag. Mathem., N.S. 3 (1992), p.419–466.
V. Serganova, [*Kac-Moody superalgebras and integrability*]{}, in Developments and Trends in Infinite-Dimensional Lie Theory, Progress in Mathematics, Vol. 288, Birkh¨auser, Boston, 2011, 169–218.
[^1]: Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel; [email protected].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper, we investigate topological properties of the ground state of the SU($N$) Heisenberg chain, which is argued to be relevant to the Mott-insulating phase of alkaline-earth cold fermions in a one-dimensional optical lattice. By calculating the entanglement spectrum, we show that the ground state is in one of the topological phases protected by SU($N$) symmetry. We then discuss an alternative characterization of it with non-local string order parameters. We also consider how the reduction of the protecting symmetry affects the topological phase paying particular attention to the entanglement spectrum.'
author:
- 'K. Tanimoto'
- 'K. Totsuka'
title: |
Symmetry-protected topological order in SU([*N*]{}) Heisenberg magnets\
– quantum entanglement and non-local order parameters
---
Introduction
============
Symmety in physics not only is the key to understanding phases of matter but also play a vital role in unifying seemingly different things and uncovering fundamental principles underlying them. In particular, unitary groups have been playing very important roles in quantum mechanics as the orthogonal groups in classical mechanics. For instance, SU(3) is the fundamental symmetry underlying the quantum chromodynamics (QCD) of strong interactions. In traditional condensed-matter physics, however, high symmetry like SU($N$) is usually realized, aside from few exceptions, only in rather idealized situations and has been mainly used as mathematical convenience that makes problems tractable. For instance, in the large-$N$ approximations, we replace the physical symmetry SU(2) with SU($N$) and use $1/N$ as the (small) control parameter of the approximation hoping that there is a smooth crossover down to $N=2$.
Recent suggestions[@Cazalilla-H-U-09; @Gorshkov-et-al-10] that SU($N$)-symmetric fermion systems could be simulated using the alkaline-earth atoms and their cousins (${}^{171}$Yb, ${}^{173}$Yb, ${}^{87}$Sr, etc.) loaded in optical lattices opened a new era of SU($N$) physics [@Kitagawa-et-al-PRA-08; @DeSalvo-Y-M-M-K-10] (see, e.g., Refs. for recent reviews). For instance, the SU($N$) generalization of quantum magnetism is of direct relevance to the Mott-insulating regime of these systems. The SU($N$) “spin” models provide us with examples of underconstrained systems that yield, on top of usual “magnetically ordered” states, various unconventional states, e.g., deconfined criticalities[@Kaul-S-12; @Harada-S-O-M-L-W-T-K-13], an algebraic spin liquid[@Corboz-L-L-P-M-12] and a chiral spin liquid[@Hermele-G-11].
On the other hand, topological states of matter[@Wen-book-04] have been subjects of extensive research for the past decade. Since the advent of topological insulators and superconductors[@Qi-Z-RMP-11], it has been widely realized that there exists a special class of “topological” phases that is stable [*only*]{} in the presence of certain symmetries[@Gu-W-09; @Pollmann-T-B-O-10; @Chen-G-W-11; @Chen-G-L-W-12; @Vishwanath-S-13]. This class of topological phases is called “symmetry-protected topological (SPT)”[@Gu-W-09] as it is topologically protected only when we impose symmetries on the system in question, and otherwise they reduce to trivial ones. The catalogue of possible topological phases depends crucially on the symmetry we impose and different lists of possible phases may be obtained for different protecting symmetries (see, e.g., Ref. for a catalogue of SPT phases). One defining property of SPT phases is the existence of gapless boundary excitations (edge states) that are intrinsically different from those in the gapped bulk. A modern mathematical way of observing the edge states would be to use the entanglement spectrum[@Li-H-08] that is obtained solely from the ground-state wave function. In the following, we heavily use the entanglement spectrum in characterizing topological phases.
Despite the recent effort[@Chen-G-W-11] in systematically enumerating possible SPT phases in one dimension, not much is known, except for a few examples, about how to observe those phases in realistic settings. Recently, it has been suggested[@Nonne-M-C-L-T-13; @Bois-C-L-M-T-15] that a class of SPT phases is realized in the Mott-insulating region of the alkaline-earth cold fermions, and this is one of the motivations of our study here. Specifically, deep inside the Mott phase at half-filling, the low-energy physics of a system of alkaline-earth fermions is described by an SU($N$) “spin” model (see Secs. \[sec:relation-to-AE\] and \[sec:strong-coupling\]) whose ground state is expected to be in one of the topological phases predicted in Ref. . Therefore the alkaline-earth fermions provide us with a unique arena for the realization of new SPT phases in a very controlled manner. Our goal is to clarify the nature of the ground state of the above SU($N$) spin Hamiltonian in several complementary ways and demonstrate the use of non-local string order parameters to detect the phase.
The outline of this paper is as follows. In Sec. \[sec:model\], we introduce the SU($N$) Heisenberg model and sketch how it is derived as the effective Hamiltonian for the Mott-insulating phase of the alkaline-earth cold fermions on a one-dimensional optical lattice. A variant of the Heisenberg model that gives useful insights about the topological properties of the original model is introduced as well. After briefly summarizing the minimal background of SPT phases expected for our SU($N$) spin systems, we try, in Sec. \[sec:SPT\], to characterize the topological properties of the ground state of the SU($N$) Heisenberg model using its entanglement spectrum. By carefully investigating the structure of the spectrum obtained for $N=4$, we present a strong evidence that the ground state of the SU(4) Heisenberg model is in one of the SU(4) topological phases. In Sec. \[sec:string-orer-parameter\], we present an alternative way of characterizing the SU($N$) SPT phases using [*non-local*]{} string order parameters.
Although the alkaline-earth fermions, that motivated our study, possess very precise SU($N$) symmetry, it would be interesting theoretically to consider the situations where the original SU($N$) symmetry gets lowered. We investigate this problem in Sec. \[sec:symmetry-reduction\] to find that, depending on $N$, the system remains topological even after the SU($N$) symmetry is relaxed. Summary of the main results is given in Sec. \[sec:summary\].
Model {#sec:model}
=====
In this paper, we consider the ground-state properties of the following Hamiltonian $$\mathcal{H}_{\text{Heis}} = \mathcal{J} \sum_{A=1}^{N^{2}-1}
\mathcal{S}_{i}^{A} \mathcal{S}_{i+1}^{A}
\label{eqn:SUN-Heisenberg}$$ where $\mathcal{S}_{i}^{A}$ ($A=1,\ldots, N^{2}-1$) denote the SU($N$) generators. In SU($N$), instead of fixing spin $S$, one has to specify the irreducible representation(s) to which the generators $\mathcal{S}_{i}^{A}$ belong. In the following, $\mathcal{S}_{i}^{A}$ ($A=1,\ldots, N^{2}-1$) denote, unless otherwise stated, the SU($N$) generators in the irreducible representation characterized by the following Young diagram with $N/2$ rows and two columns: $$\text{\scriptsize $N/2$} \left\{
\yng(2,2,2)
\right. \quad (N=\text{even}) \; .
\label{eqn:Young-diagram-GS}$$
It is well-known that the low-energy physics of the SU($N$) Heisenberg model depends crucially on the representation(s) we put on the individual lattice sites. For the fully-symmetrized representation ${\tiny \yng(2)\cdots \yng(1)}$ ($n_{\text{c}}$ boxes), the exact Bethe-ansatz solutions are available[@Sutherland-75; @Andrei-J-84; @Johannesson-86]; the ground state is known to be gapless and described by the level-$n_{\text{c}}$ SU($N$) Wess-Zumino-Witten conformal field theory with the central charge $c=n_{\text{c}}(N^{2}-1)/(N+n_{\text{c}})$[@Alcaraz-M-SUN-89]. For [*any*]{} translationally invariant choice of representations (i.e., the same representation is assigned on every site), we can show that the SU($N$) chain, which has a unique (finite-size) ground state[^1], is either gapless or has degenerate ground states (with broken symmetries) provided that the number of boxes $n_{\text{Y}}$ in the Young diagram is [*not*]{} divisible by $N$[@Affleck-L-86]. In other words, except for the cases of $n_{\text{Y}}=0$ (mod $N$) \[including the one shown in Eq. which is relevant to our spin chain\], this statement excludes the possibility of gapped topological ground states. Remarkably, this is perfectly consistent with the recent group-cohomology classification of the gapped SPT phases[@Duivenvoorden-Q-13] (see Sec. \[sec:SUN-SPT\] for the detail). There is also an attempt[@Greiter-R-07] at summarizing these observations into a “generalized” Haldane conjecture.
Some insights about the nature of the ground state of are gained from the large-$N$ analysis[@Marston-A-89; @Read-S-NP-89; @Read-S-90] as well. For $N/2$ rows but with a single column, the ground state is expected dimerized[@Marston-A-89], while, for two columns, we may have a gapped translationally invariant ground state[@Read-S-NP-89; @Read-S-90], which we will argue to be topological.
Relation to cold fermion systems {#sec:relation-to-AE}
--------------------------------
It has been argued in Refs. that the Hamiltonian emerges as the effective Hamiltonian in the Mott-insulating region of the alkaline-earth cold fermions loaded in a one-dimensional optical lattice at half-filling. To emphasize the relevance of our results to experimentally realizable systems, we sketch how the model $\mathcal{H}_{\text{Heis}}$ is derived from the cold-fermion systems in the Mott region.
It is known that the decoupling between the nuclear spin ($I$) and the total electron angular momentum makes it possible to organize the $(2I+1)$ nuclear-spin states of each atom into a multiplet of larger SU($2I+1$)-symmetry. Specifically, the interaction between two like alkaline-earth atoms does [*not*]{} depend on the nuclear-spin states of each and hence is SU($2I+1$)-symmetric[@Cazalilla-H-U-09; @Gorshkov-et-al-10]. Moreover, one can add one more degree of freedom ([*orbital*]{}) by taking into account the first meta-stable excited states (in $^{3}P_{0}$; denoted as “$e$”) as well as the atomic ground state in $^{1}S_{0}$ (“$g$”).[^2] That this SU($2I+1$)-symmetry holds for both orbitals with very high accuracy has been verified in recent scattering-length measurements[@Kitagawa-et-al-PRA-08; @Zhang-et-al-Sr-14; @Scazza-et-al-14].
When loaded into a one-dimensional optical lattice, the system of alkaline-earth cold fermions is described by the following Hubbard-like Hamiltonian[@Gorshkov-et-al-10] $$\begin{split}
\mathcal{H}_{\text{G}} =&
- \sum_{i}\sum_{m=g,e} t^{(m)} \sum_{\alpha=1}^{N}
\left(c_{m\alpha,\,i}^\dag c_{m\alpha,\,i+1} + \text{h.c.}\right) \\
& -\sum_{m=g,e}\mu^{(m)}_{\text{G}} \sum_i n_{m,i}
+\sum_{i}\sum_{m=g,e} \frac{U^{(m)}_{\text{G}}}{2} n_{m,\,i}(n_{m,\,i}-1) \\
& +V_{\text{G}} \sum_i n_{g,\,i} n_{e,\,i}
+ V_{\text{ex}}^{g\text{-}e} \sum_{i,\alpha \beta}
c_{g\alpha,\,i}^\dag c_{e\beta,\,i}^\dag
c_{g\beta ,\,i} c_{e\alpha,\,i} ,
\end{split}
\label{eqn:Gorshkov-Ham}$$ where $N=2I+1$ denotes the number of nuclear-spin states and the operator $c_{m\alpha,\,i}^{\dag}$ creates an atom in the internal state $(\alpha,m)$ ($\alpha=1,\ldots,N$, $m=g,e$) at the site $i$. The number operators are defined as $n_{m\alpha,\,i} = c_{m\alpha,\,i}^{\dag}c_{m\alpha,\,i}$ and $n_{m,\,i} = \sum_{\alpha = 1}^N n_{m\alpha,\,i}$. As the two orbitals are not symmetry-related, the hopping amplitudes $t^{(m)}$ ($m=g,e$), the chemical potential $\mu^{(m)}_{\text{G}}$, and the intra-orbital interaction $U^{(m)}_{\text{G}}$ in general are different for the two orbitals. The inter-orbital exchange (or, Hund coupling) $V_{\text{ex}}^{g\text{-}e}$ is crucial in determining the nature of the Mott-insulating phases[@Bois-C-L-M-T-15].
Clearly, the Hamiltonian is invariant under the SU($N$) transformation $$c_{m\alpha,i} \to \sum_{\beta=1}^{N} \mathcal{U}_{\alpha\beta} c_{m\beta,i}
\quad [\, \mathcal{U} \in \text{SU($N$)} \, ]$$ as well as the multiplication of a global U(1) phase: $$c_{m\alpha,i} \to {\mathrm{e}}^{i \theta} c_{m\alpha,i} \; .
\label{eqn:U1-gauge}$$ Borrowing a terminology from the electron systems, we call, in the rest of this paper, the degree of freedom associated with “charge”, although the fermions $c_{m\alpha,\,i}$ are charge-neutral in the cold-atom context. This and the related systems have been investigated extensively both for SU(2)[@Nonne-B-C-L-10; @Nonne-B-C-L-11; @Kobayashi-O-O-Y-M-12; @Kobayashi-O-O-Y-M-14] and for SU($N$)[@Nonne-M-C-L-T-13; @Bois-C-L-M-T-15; @Szirmai-13].
Strong-coupling limit {#sec:strong-coupling}
---------------------
Recently, it has been argued[@Nonne-M-C-L-T-13; @Bois-C-L-M-T-15] that for large positive $U^{(m)}_{\text{G}}$ and $V_{\text{ex}}^{g\text{-}e}$, there exists a topological Mott phase protected by SU(4)-symmetry.[^3] In order to consider the Mott-insulating phases, it is convenient to start from the strong-coupling limit $U_{\text{G}}^{(m)}, V_{\text{G}}, V_{\text{ex}}^{g\text{-}e} \ll t^{(m)}$. In this limit, charge fluctuations are strongly suppressed and the SU($N$) “spin” and orbital dominate the low-energy physics. One may introduce the psuedo-spin operator $T_i^a = \frac{1}{2}\sum_{\alpha,\beta,m}c_{m\alpha,i}^{\dagger}\sigma_{\alpha\beta}^ac_{m\beta, i}$ ($a=x,y,z$) for each orbital to rewrite the single-site (i.e., $t^{(m)}=0$) part of the Hamiltonian as $$\begin{split}
& \mathcal{H}_{\text{G}}(t^{(m)}=0) = \sum_{i} h_{\text{atomic}}(i) \\
& h_{\text{atomic}}(i) \equiv
-\frac{1}{2}\left(\mu _e + \mu _g\right) n_{i} +\frac{U}{2} n_i^2 \\
& \phantom{h_{\text{atomic}}(i) =}
+J \left\{ (T_i^x)^2 + (T_i^y)^2\right\} + J_z (T_i^z)^2 \\
& \phantom{h_{\text{atomic}}(i) =}
- \left(\mu _g -\mu _e\right) T^{z}_{i} + U_{\text{diff}} T^{z}_{i} n_{i}
\end{split}
\label{eqn:atmic-limit-Ham}$$ with the following coupling constants $$\begin{split}
& U=\frac{1}{4} (U_{\text{G}}^{(g)}+ U_{\text{G}}^{(e)}
+2 V_{\text{G}}), \;\;
U_{\text{diff}} = \frac{1}{2}(U_{\text{G}}^{(g)} - U_{\text{G}}^{(e)}) , \\
& J=V^{g\text{-}e}_{\text{ex}} , \;\;
J_{z} = \frac{1}{2}(U_{\text{G}}^{(e)}+ U_{\text{G}}^{(g)} -2 V_{\text{G}}), \\
& \mu_{m} = \frac{1}{2} (2 \mu_{\text{G}}^{(m)}+ U_{\text{G}}^{(m)}+V^{g\text{-}e}_{\text{ex}}) \quad
(m=g,e) \; .
\end{split}
\label{eqn:Hund-by-Gorshkov}$$
Let us consider the case of half-filling where each site is occupied by $N$ fermions on average. The Fermi statistics allows $(2N)!/(N!)^{2}$ states and, out of them, the optimal ones are chosen by the orbital-dependent terms \[the last four terms in $h_{\text{atomic}}(i)$\]; when $N$ is even and $V^{g\text{-}e}_{\text{ex}}$ is positive, the states that transform under SU($N$) as the irreducible representation are the ground states of $h_{\text{atomic}}(i)$[@Bois-C-L-M-T-15]. When $N=4$, they form the 20-dimensional representation of SU(4). For these states, the orbital pseudo-spin $\mathbf{T}_{i}$ is quenched and only the SU($N$) degree of freedom remains. When $N$ is odd, on the other hand, [*both*]{} SU($N$) spin and orbital are active and we obtain, in general, SU($N$)-orbital-coupled models. In the following, we consider only the case with even-$N$ where pure spin models are obtained.
Interactions among the remaining SU($N$) spins are derived by the second-order perturbation in $t^{(m)}$ as[@Bois-C-L-M-T-15] $$\frac{1}{2} \left\{
\frac{{t^{(g)}}^{2}}{U+U_{\text{diff}}+J+\frac{J_z}{2}}
+ \frac{{t^{(e)}}^{2}}{U-U_{\text{diff}}+J+\frac{J_z}{2}}
\right\} \mathcal{S}_i \cdot \mathcal{S}_{i+1} \; ,
\label{eqn:2nd-order-effective-Ham-Gorshkov}$$ where we have introduced a short-hand notation $\mathcal{S}_i \cdot \mathcal{S}_{i+1} \equiv \sum_{A=1}^{N^{2}-1} \mathcal{S}_{i}^{A}\mathcal{S}_{i+1}^{A}$ with $\mathcal{S}_{i}^{A}$ being the SU($N$) generators in the irreducible representation specified by the Young diagram . Therefore, one sees that the model $\mathcal{H}_{\text{Heis}}$ \[eq.\] describes the low-energy physics of the alkaline-earth cold fermions \[eq.\] in the Mott-insulating phase (for $J=V^{g\text{-}e}_{\text{ex}}>0$).
Solvable Hamiltonian {#sec:solvable-Ham}
--------------------
Unfortunately, the Heisenberg Hamiltonian cannot be solved exactly. However, one can design a solvable model Hamiltonian whose ground state may share important properties with that of the original Heisenberg model . Clearly, when $N=2$, the Affleck-Kennedy-Lieb-Tasaki (AKLT) model proposed in Refs. will do the job: $$\mathcal{H}^{N=2}_{\text{VBS}} = \sum_i \left\{
\mathbf{S}_i \cdot \mathbf{S}_{i+1}
+ \frac{1}{3} \left(\mathbf{S}_i \cdot \mathbf{S}_{i+1}\right)^2 \right\} \; ,
\label{eqn:AKLT}$$ where $\mathbf{S}_i$ denote the spin-1 operators. Its (rigorous) ground state, dubbed the valence-bond solid (VBS) state, is constructed[@Affleck-K-L-T-87; @Affleck-K-L-T-88] by first decomposing an $S=1$ on each site into a pair of $S=1/2$s, forming uniform tiling of dimer singlets (‘valence-bond solid’) among the neighboring sites, and then fusing the $S=1/2$ pairs back to the original spin-1s.
Suggested by the above construction of the VBS ground state, we can think of constructing the model ground state by first preparing two auxiliary ‘spins’ $$\text{\scriptsize $N/2$} \left\{
\yng(1,1,1)
\right. \quad (N=\text{even})
\label{eqn:Young-diagram-ancilla}$$ on each site and pairing such spins on the adjacent sites into SU($N$) singlets (see Fig. \[fig:SU(4)VBS\]). The VBS ground state is obtained by projecting the product of the two fictitious spins on each site onto the physical Hilbert space characterized by the Young diagram in (see Fig. \[fig:SU(4)VBS\]). In the following, we call this kind of states the SU($N$) VBS states.[^4] The parent Hamiltonians for these states read, e.g., for $N=4$[@Nonne-M-C-L-T-13; @Bois-C-L-M-T-15] and for $N=6$ as[^5] $$\begin{split}
& \mathcal{H}^{N=4}_{\text{VBS}} \\
&= \sum_i \left\{
\mathcal{S}_i \cdot \mathcal{S}_{i+1}
+ \frac{13}{108} \left(\mathcal{S}_i \cdot \mathcal{S}_{i+1}\right)^2
+ \frac{1}{216}\left(\mathcal{S}_i \cdot \mathcal{S}_{i+1}\right)^3 \right\}
\end{split}
\label{eqn:SU4-VBS}$$ and $$\begin{split}
\mathcal{H}^{N=6}_{\text{VBS}} =& \sum_{i} \biggl\{
\mathcal{S}_i \cdot \mathcal{S}_{i+1}
+ \frac{47}{508}(\mathcal{S}_i \cdot \mathcal{S}_{i+1} )^2 \\
& +\frac{17}{4572} (\mathcal{S}_i \cdot \mathcal{S}_{i+1} )^3
+\frac{1}{18288} (\mathcal{S}_i \cdot \mathcal{S}_{i+1} )^4
\biggr\} \; ,
\end{split}
\label{eqn:SU6-VBS}$$ respectively.[^6] (In writing down the above expressions, we have normalized the generators $\mathcal{S}_{i}$ in such a way that the lengths of the simple roots are all $\sqrt{2}$.) The dimensions of the physical SU($N$) ‘spin’ multiplet on each site are 20 and 175 for $N=4$ and $6$, respectively. In Refs. , the ground state wave function of $\mathcal{H}^{N=4}_{\text{VBS}}$ has been obtained in a matrix-product-state (MPS) form (see Appendix \[sec:MPS-matrices\]). Clearly, the higher-order terms are rapidly suppressed as we go to larger-$N$. This suggests that the larger $N$ is, the better the VBS state shown in Fig. \[fig:SU(4)VBS\] approximates the ground state of the original Heisenberg model . This is quite natural in view of the large-$N$ results[@Read-S-NP-89; @Read-S-90]. These models will serve as an ideal starting point for the study of the topological properties.
![(Color online) (a) Ground state of SU(4) VBS model \[eq.\]. Two 6-dimensional representations (‘fictitious spins’) are projected onto a physical 20-dimensional representations. (b) Similar construction applies to the cases with larger $N$ as well. \[fig:SU(4)VBS\]](SU4-VBS-20){width="0.8\columnwidth"}
Symmetry-Protected Topological Phases {#sec:SPT}
=====================================
In this section, we try to characterize the nature of the ground state of the SU($N$) spin chain . Specifically, in Sec. \[sec:ES\], we show that the ground state of the model shares essentially the same properties with that of the solvable VBS models and that it is in fact in one of the SPT phases. Being topological, this class of topological phases defies the traditional characterization with broken symmetries and the associated local order parameters. One way is to use the [*physical*]{} edge states to distinguish between topological phases from trivial ones. However, this approach is not quite satisfactory in the following respects. First, even topologically [*trivial*]{} states may have certain structures around the edges of the system, as, e.g., the spin-2 Heisenberg chain does[@Nishiyama-T-H-S-95; @Qin-N-S-95]. Second, in order to see the edge excitations, it is necessary to consider the excitation spectrum, while the topological properties are intrinsic to the ground state itself and should be seen only by examining the ground-state wave function.
Recently, the use of the entanglement spectrum in characterizing topological phases has been suggested in Ref. . This is based on the observation that the entanglement spectrum [*resembles*]{} the spectrum of the physical edge excitations. The idea has been successfully applied to various systems[@Pollmann-T-B-O-10; @Pollmann-B-T-O-12; @Fidkowski-K-11; @Turner-P-B-11; @Zheng-Z-X-L-11; @Lou-T-K-K-11] and enabled us to characterize topological phases and quantum phase transitions among them. In this section, we present a clear evidence from the entanglement spectrum that the ground state of the SU(4) Heisenberg model is indeed in the SPT phases protected by SU(4) \[PSU(4), precisely\] symmetry.
Haldane phase –an SPT primer
----------------------------
To understand the nature of the SPT phases in the case of SU($N$) symmetry, it is convenient to begin with the simplest case $N=2$. In 1983, Haldane conjectured[@Haldane-PLA-83; @Haldane-PRL-83] that the ground-state properties of the spin-$S$ Heisenberg chain are qualitatively different according to the parity of $2S$; when $2S=\text{even}$, the ground state is in a featureless non-magnetic phase ([*Haldane phase*]{}) with the gapped triplon excitations in the bulk, while, for odd $2S$, we have a gapless (i.e., algebraic) ground state with spinon excitations. This conjecture has been later confirmed both by the construction of a rigorous example[@Affleck-K-L-T-87; @Affleck-K-L-T-88; @Arovas-A-H-88] \[Eq. \] and by extensive numerical simulations[@White-H-93; @Schollwock-G-J-96; @Todo-K-01]. Soon after, it has been pointed out that the featureless gapped ground state of the integer-$S$ spin chains may have a [*hidden*]{} “topological” order characterized by non-local order parameters[@denNijs-R-89; @Girvin-A-89; @Kennedy-T-92-PRB; @Kennedy-T-92-CMP] [*at least*]{} when $S$ is an odd integer[@Oshikawa-92].
However, it was not until the concept of SPT phases was established that the true meaning of “topological order” in the Haldane phase was understood[@Gu-W-09]. Now it is realized that the gapped phases in integer-spin chains with some protecting symmetry (e.g., time-reversal, reflection) are further categorized into topological phases and the other trivial ones. To understand the difference, it is useful to consider how the ground state in question transforms under the symmetry operation. As the ground state is assumed symmetric, [*the bulk*]{} does not respond to the symmetry operation but the edges do. As the consequence, the symmetry operation gets [*fractionalized*]{} into two pieces; one acts on the left edge and the other on the right. For instance, the VBS ground state $|S=1\text{ VBS}\rangle_{\alpha,\beta}$ of the spin-1 AKLT model hosts two [*emergent*]{} $S=\frac{1}{2}$ spins (i.e., $\alpha,\beta=\uparrow,\downarrow$) on both edges and hence transforms under the SO(3) rotation as $$|S=1\text{ VBS}\rangle_{\alpha,\beta} \xrightarrow{\text{SO(3)}}
\sum_{\alpha^{\prime},\beta^{\prime}}U^{\dagger}_{\alpha,\alpha^{\prime}}U_{\beta,\beta^{\prime}}
|S=1\text{ VBS}\rangle_{\alpha^{\prime},\beta^{\prime}} \; ,$$ where $U$ is the $S=\frac{1}{2}$ rotation matrix of SU(2). Putting it another way, $U$ serves as the mathematical labeling of the physical edge states. It is important to note that $U$ in the above in general is a projective representation of SO(3) as both $U^{\dagger}$ and $U$ appear simultaneously in the equation.
Since this $U$ belongs to a non-trivial projective representation that is intrinsically different from any irreducible representations of the original SO(3), one sees that $|S=1\text{ VBS}\rangle_{\alpha,\beta}$ is in a non-trivial topological phase with emergent edge states. On the other hand, one can construct another exact ground state of a spin-1 chain which transforms as above but with $U$ belonging to the spin-1 representation. Since the spin-1 representation is trivial in the sense of projective representation of SO(3), one can kill the would-be edge states by continuously deforming the Hamiltonian[@Pollmann-B-T-O-12] and this ground state is in a trivial phase. This reasoning may be readily generalized; when $U$ transforms like a half-odd-integer spin, the phase is topological, while when $U$ transforms in an integer-spin representation \[i.e., linear representation of SO(3)\], the system is in a trivial phase. What is crucial in the topological properties is not the bulk spins at the individual sites but the [*edge*]{} spins.
For later convenience, we summarize the situation in terms of Young diagrams. The spin-$S$ representation of SU(2) is represented by the following Young diagram: $$\underbrace{\yng(2) \cdots \yng(1)
}_{\text{$2S$ boxes}} \; .$$ With this in mind, the above result may be summarized as follows; when $U$ belongs to the representations $$\yng(1) \, , \; \yng(3) \, , \ldots \; ,$$ the state represented by the corresponding MPS is topologically non-trivial, while the phase is trivial for $U$ transforming in $$\;\;
\yng(2)\, , \; \yng(4) \, , \ldots \; .$$ That is, the number of boxes (mod 2) in the Young diagram for the representation to which $U$ belongs labels the topological classes protected by SO(3) and leads to the $\mathbb{Z}_{2}$ classification of the SO(3) SPT phases[@Chen-G-W-11].
SU($N$) topological phases {#sec:SUN-SPT}
--------------------------
Using the MPS representation[@Garcia-V-W-C-07] of the gapped ground state in one dimension, the above “physical” idea can be generalized and made mathematically precise. In fact, when a given ground state that is represented by an MPS $$\sum_{\{m_{i}\}}A(m_1)A(m_2) \cdots A(m_L)|m_1\rangle{\otimes}\cdots {\otimes} |m_{L}\rangle$$ is invariant under some symmetry $G$, a $D$-dimensional unitary matrix $U_{g}$ ($g \in G$) exists such that[@Garcia-W-S-V-C-08] $$A(m_i) \xrightarrow{G} {\mathrm{e}}^{i\phi_{g}} U^{\dagger}_{g} A(m_i) U_{g} \; ,
\label{eqn:MPS-UAU}$$ where $A(m_i)$ denotes the $D{\times}D$ MPS matrices corresponding to the local physical state $|m_i\rangle$ and ${\mathrm{e}}^{i\phi_{G}}$ is a phase that depends on $G$. As has been mentioned above, the unitary matrix $U_{g}$ is in fact a projective representation of the symmetry $G$, that corresponds to the physical edge states[@Pollmann-T-B-O-10]. Therefore, the enumeration of topologically stable phases in the presence of symmetry $G$ boils down to counting the possible (non-trivial) projective representations of $G$.[@Chen-G-W-11]
This problem was solved for SU($N$) and other Lie groups in Ref. and the picture in the previous section basically generalizes to the case of SU($N$) with some mathematical complications. Now the role of SO(3) in the previous section is played by $\text{PSU($N$)} \simeq \text{SU($N$)}/\mathbb{Z}_{N}$ \[note $\text{SO(3)}\simeq \text{PSU(2)}$\]. Considering $\text{PSU($N$)}$ instead of SU($N$) amounts to restricting ourselves only to the irreducible representations of SU($N$) specified by Young diagrams with the number of boxes $n_{\text{Y}}$ divisible by $N$ \[i.e., $n_{\text{Y}}=N k$ ($k=0,1,\ldots$)\]. This subset of irreducible representations roughly corresponds to the integer-spin ones in the SU(2) case. As in the previous section, the topological class of a given ground state (typically written as an MPS) is determined by looking at to which projective representation the unitary $U_{g}$ of the state belongs. Since inequivalent projective representations of PSU($N$) are labeled by $n_{\text{Y}}$ (mod $N$)[@Duivenvoorden-Q-13], there are $N-1$ non-trivial topological classes (as well as one trivial one) specified by the $\mathbb{Z}_{N}$ label $n_{\text{top}}=n_{\text{Y}}$ (mod $N$). In the following, we use the name “class-$n_{\text{top}}$” for these topological classes (the class-0 corresponds to trivial phases). For instance, one can readily see that the “VBS states” (which are different from ours) investigated in Refs. fall into the class-1 and $N-1$ of the PSU($N$) SPT phases (see [Supplementary Material]()). Quite recently, the class-1,2 phases as well as other (conventional) phases of SU(3)-invariant spin chains were investigated from the SPT point of view[@Morimoto-U-M-F-14].
A remark is in order about the definition of the topological class. In contrast to the SU(2) case where all the irreducible representations are self-conjugate, we must distinguish between an irreducible representation and its conjugate in SU($N$). The relation suggests that if we have the edge state transforming under the projective representation $\mathcal{R}$ on the right edge, we necessarily have its conjugate $\bar{\mathcal{R}}$ on the other. This means that when we talk about the topological class we must first fix which edge state we use to label the topological phases. Throughout this paper, we define the topological class by the [*right edge state*]{} \[i.e., $U_{g}$ acting from the right in Eq. \]. Now it is easy to see that the SU($N$) VBS state introduced in Sec. \[sec:solvable-Ham\] belongs to class-$N/2$.
Entanglement spectrum {#sec:ES}
---------------------
Remarkably, the above-mentioned difference in the projective representation $U_g$ can be seen in the entanglement spectrum[@Pollmann-T-B-O-10]. In order to define the entanglement spectrum, we first divide the system into two subsystems A and B. Then, the entanglement spectrum $\{\xi_{\alpha}(\geq 0)\}$ is defined through the Schmidt decomposition of the ground state $|\psi\rangle$ of the entire system: $$|\psi\rangle= \sum_{\alpha = 1}^{\chi}
{\mathrm{e}}^{-\frac{\xi_{\alpha}}{2}} |\phi_{\alpha}^{\text{A}} \rangle \otimes |\phi_{\alpha}^{\text{B}} \rangle,$$ where $\{|\phi_{\alpha}^{\text{A}} \rangle\}$ and $\{|\phi_{\alpha}^{\text{B}} \rangle\}$ are orthonormal basis sets for the subsystems satisfying $\langle \phi_{\alpha}^{\text{A,B}}|\phi_{\beta}^{\text{A,B}}\rangle = \delta_{\alpha\beta}$ and the number $\chi$ of finite $\xi_{\alpha}(< \infty)$ defines the Schmidt number.
According to Ref. , the entanglement spectrum of a given system exhibits a structure quite similar to that of the (energy) spectrum of the physical edge state of the same system and might be useful in characterizing topological states of matter. In one dimension, the edge states are not dispersive and we expect a discrete set of degenerate levels to appear in the entanglement spectrum reflecting the physical gapless edge modes. In fact, in accordance with the degeneracy in the entanglement spectrum, the projective representation $U_{g}$ assumes a block-diagonal structure[@Sanz-W-G-C-09], where each block corresponds to an irreducible representation of SU($N$) compatible with the topological class. For instance, in a ground state in the class-2 topological phase of SU(4), each entanglement level should exhibit the degeneracy corresponding to an SU(4) irreducible representation with $n_{\text{Y}}=2$ (mod 4). In Table \[tab:Young\], the Young diagrams as well as their dimensions are listed for some typical irreducible representations compatible with the class-2 topological phase \[i.e., $n_{\text{Y}}=2$ (mod 4)\].
### VBS point
To investigate the topological phase protected by PSU(4) symmetry, we begin with the simplest case. The ground state of the SU(4) VBS Hamiltonian (\[eqn:SU4-VBS\]) can be given exactly in the form of an MPS[@Nonne-M-C-L-T-13; @Bois-C-L-M-T-15] and its entanglement spectrum is readily obtained by rendering the MPS into the canonical form (for the expressions of the matrices, see Appendix \[sec:MPS-matrices\]).
Reflecting the existence of the 6-dimensional (physical) edge states ${\tiny \yng(1,1)}$ ($n_{\text{top}}=n_{\text{Y}}=2$), the only entanglement level indeed is 6-fold degenerate indicating the class-2 phase[@Nonne-M-C-L-T-13]: $\xi_{\alpha}=\log 6$ ($\alpha=1,\ldots,6$; $\chi=6$). This is in perfect agreement with the above argument.
### Heisenberg point
In order to check if the ground state of the SU(4) Heisenberg chain is in the class-2 topological phase, we calculated the entanglement spectrum with the infinite time-evolving block decimation (iTEBD) algorithm[@Vidal-iTEBD-07; @Orus-V-08]. which enables us to directly access the entanglement spectrum.
The simulations were done using the MPS with the bond dimensions up to 150 and the spectrum obtained is shown in Fig. \[fig:ES\_a=00\]. The degrees of degeneracy seen in Fig. \[fig:ES\_a=00\] are $\{6, 64, 6, 50\}$ from the bottom to the top. Clearly, this pattern perfectly fits into the dimensions in Table \[tab:Young\]; the edge state transform under the four (self-conjugate) irreducible representations shown in Fig. \[fig:ES\_a=00\]. All these have $n_{\text{Y}}=2$ (mod $4$) and, from the discussion in Sec. \[sec:SUN-SPT\], this ground state is classified as the topological class 2.
Here a remark is in order. As the bosonic SU(4) Heisenberg model is obtained as the effective Hamiltonian in the Mott phase of the [*fermionic*]{} model , one may suspect that the same degeneracy structure could have been obtained for the original fermion model as well. However, this is not necessarily the case. In fact, in models where both bosonic and fermionic modes coexist, the entanglement spectrum contains the contribution from the fermionic sector as well as that from the bosonic one, and some of the levels may not obey the degeneracy rule that is obtained for the [*purely*]{} bosonic models[@Hasebe-T-13]. This is the reason why we simulated the effective bosonic model .
$n_{\text{Y}}$ Young diagram dimension
---------------- --------------- ----------- --
2
6
10
: \[tab:Young\] Typical Young diagrams with the number of boxes $n_{\text{Y}}\equiv 2$ (mod 4) and their dimensions in SU(4).
![(Color online) Entanglement spectrum of an infinite SU(4) Heisenberg chain calculated by iTEBD. The degeneracy $\{6, 64, 6, 50\}$ may be understood in terms of the SU(4) irreducible representations shown in the figure. (inset) Zoom-up of the lowest six-fold-degenerate entanglement level. \[fig:ES\_a=00\]](ES_a=00){width="0.9\columnwidth"}
### Continuity between Heisenberg and VBS points
In the previous sections, we have seen, by inspecting the entanglement spectra, that the original SU(4) Heisenberg model and the solvable SU(4) VBS model share the same topological properties in common. Next, we consider adiabatic connection between the Heisenberg point and the solvable VBS point to show that they belong to the same unique phase in the sense that they are connected to each other without quantum phase transitions[@Chen-G-W-10]. To connect the two Hamiltonians, we use the following one-parameter family of Hamiltonians $$\begin{split}
& \mathcal{H}(a) \\
& = \sum_i \left\{ \mathcal{S}_i {\cdot} \mathcal{S}_{i+1}
+a\left[ \frac{13}{108} \left(\mathcal{S}_i {\cdot} \mathcal{S}_{i+1}\right)^2
+ \frac{1}{216}\left(\mathcal{S}_i {\cdot} \mathcal{S}_{i+1}\right)^3 \right]
\right \} \; ,
\end{split}
\label{eqn:Ha}$$ where $a$ is an interpolating parameter changing from 0 \[Heisenberg point: Eq. \] to 1 \[VBS point: Eq.\]. We calculated the entanglement spectrum of the ground state of $\mathcal{H}(a)$ for $a=0.0$, $0.1$, $0.3$, $0.5$, $0.7$, $0.9$, and $1.0$ with iTEBD and the results are shown in Fig. \[fig:SU4-ES-all\]. It is evident that the structure of the entanglement spectrum (including the six-fold degeneracy in the lowest level) is preserved all the way from the Heisenberg point up to the VBS point showing that the two models indeed belong to the same class-2 topological phase.
![(Color online) Evolution of entanglement spectrum as we interpolate between SU(4) Heisenberg model \[; $a=0$\] and SU(4) VBS model \[; $a=1$\]. Numbers shown next to the levels are degrees of degeneracy. \[fig:SU4-ES-all\]](SU4-ES-all){width="1.0\columnwidth"}
Non-local string order parameters {#sec:string-orer-parameter}
=================================
In Sec. \[sec:ES\], we have seen that the structure of the entanglement spectrum helps us to identify the topological class of a given ground state provided that we have enough information on the protecting symmetry of the system in advance. However, in general, the degeneracy structure alone does [*not*]{} uniquely identify the topological class. For instance, the class-2 phase of PSU(4)-symmetric systems has doubly-degenerate entanglement levels (see Appendix \[sec:PSUN-to-ZnxZn\]), that are reminiscent of the Haldane phase protected by $\mathbb{Z}_{2}{\times}\mathbb{Z}_{2}$, although these two phases are essentially different as we will see in Sec. \[sec:symmetry-reduction\]. Furthermore, despite some recent proposals[@Guhne-H-B-E-L-M-S-02; @Abanin-D-12; @Daley-P-S-Z-12; @Pichler-B-D-L-Z-13], it is not very straightforward to directly measure entanglement in experiments. In fact, what is more fundamental in identifying SPT phases is the projective representation $U_{g}$. Therefore, “order parameters” that have more direct access to $U_{g}$ is desirable.
Several order parameters for SPT phases, including a gauge-invariant product of $U_{g}$s, were proposed recently[@Pollmann-T-12] (for discussion of the detection of SPTs using the response of the physical edge states to external perturbations, see Ref. ). However, these order parameters are written directly in terms of the projective representation $U_{g}$ and are not accessible in experiments in spite of their use in numerical simulations. Therefore, for the purpose of the detection of SPT phases in experiments, the characterization with order parameters, that are written in terms of [*measurable*]{} quantities, is still useful. In this section, we introduce a set of non-local string order parameters for our SU($N$) spin system to characterize the topological phases.
$\mathbb{Z}_{N}{\times}\mathbb{Z}_{N}$ and SPT phases {#sec:def-ZnxZn}
-----------------------------------------------------
In Ref. , a set of generalized string order parameters based on the symmetry $\mathbb{Z}_{N}{\times}\mathbb{Z}_{N}$ was introduced for generic $\mathbb{Z}_{N}{\times}\mathbb{Z}_{N}$-invariant systems and its connection to the $\mathbb{Z}_{N}{\times}\mathbb{Z}_{N}$ SPT phases was discussed. As PSU($N$) and $\mathbb{Z}_{N}{\times}\mathbb{Z}_{N}$ have the same cohomology group[@Duivenvoorden-Q-ZnxZn-13; @Chen-G-L-W-13] $H^{2}(\text{PSU($N$)},\text{U(1)})=H^{2}(\mathbb{Z}_{N}{\times}\mathbb{Z}_{N},\text{U(1)})=\mathbb{Z}_{N}$ in common, we may expect that we can characterize our topological phase by using these string order parameters. In order to adapt the string order parameters, that was introduced in Ref. in the context of $\mathbb{Z}_{N}{\times}\mathbb{Z}_{N}$-invariant systems, to our SU($N$) case, we have to first identify the two [*commuting*]{} $\mathbb{Z}_{N}$s in SU($N$). The construction of a pair of $\mathbb{Z}_{N}$s itself does not rely on a particular choice of the irreducible representation. In fact, we do not need the explicit expressions of the generators which depend on the choice of the basis and representation; the commutation relations among the generators suffice for our purpose. The most convenient way is to use the Cartan-Weyl basis $\{ H_{a},E_{\alpha}\}$ that satisfy[@Georgi-book-99] $$\begin{split}
& [H_{a},H_{b}]=0 \, , \;\;
[H_{a},E_{\alpha}] = (\alpha)_{a}E_{\alpha} \, , \\
& [E_{\alpha},E_{-\alpha}] = \sum_{a=1}^{3} (\alpha)_{a}H_{a} \, , \;
\text{Tr}\, (H_{a}H_{b})= \kappa \delta_{ab} ,
\\
&(a,b=1,\ldots,N-1)
\end{split}
\label{eqn:Cartan-Weyl-commutation-rel}$$ where $\alpha$ denotes the $N^{2}-N$ roots of SU($N$) normalized as $|\alpha|=\sqrt{2}$ which are generated by the simple roots $\alpha_{i}$ ($i=1,2,3$). The normalization $\kappa$ depends on the representation and set to 1 for the $N$-dimensional fundamental representation ${\tiny \yng(1)}$ \[e.g., $\kappa=16$ for the 20-dimensional representation ${\tiny \yng(2,2)}$ of SU(4) considered here\]. In the actual calculations, one may use, e.g., the generators and the weights given in Sec. 13.1 of Ref. with due modification of the normalization.
Now let us look for the operators $G_{Q}$ and $G_{P}$ that generate the two $\mathbb{Z}_{N}$s. Regardless of $N$, the first generator $G_{Q}$, which is diagonal and plays the role of $S^{z}$ in SU(2), is given simply by $$G_{Q} = \sum_{k=1}^{N-1} (\vec{\rho})_{k} H_{k} \; ,
\label{eqn:def-G_Q}$$ where $H_{k}$ are the $N-1$ Cartan generators and $\vec{\rho}$ is the Weyl vector of SU($N$). The generator $G_{Q}$ has the following simple commutation relations with the simple roots $\alpha$: $$[ G_{Q}, E_{\pm \alpha}]= \pm E_{\alpha} \; ,
\label{eqn:comm-rel-Gq-E}$$ which guarantee integer-spaced eigenvalues of $G_{Q}$ (for the fundamental representation $\boldsymbol{N}$, they are essentially $1,2,\cdots, N$). With this, the first $\mathbb{Z}_{N}$ is generated as $$Q = c_{N} \exp \left(i\frac{2\pi}{N} G_{Q}\right) \; ,$$ where the phase $c_{N}$ has been introduced so that $Q$ satisfy $Q^{N}=1$. The expression of the other generator $G_{P}$ depends on $N$ and, in the following, we will explicitly work it out for $N=4$.
The first $\mathbb{Z}_{4}$-generator $Q$ is defined in terms of the two commuting SU(4) generators (the Cartan generators) as $$\begin{split}
& Q \equiv {\mathrm{e}}^{i\frac{3\pi}{4}} \exp\left( i \frac{2\pi}{4}G_{Q} \right) , \;\; Q^{4}=1 \\
& G_{Q} \equiv 2H_{1} + H_{2} \; .
\end{split}
\label{eqn:Q-by-SU4}$$ The generator $G_{Q}$ satisfies Eq. . On the other hand, the second $\mathbb{Z}_{4}$ is generated by $$\begin{split}
& P \equiv {\mathrm{e}}^{i\frac{3\pi}{4}} \exp\left( i \frac{2\pi}{4}G_{P} \right) , \;\; P^{4}=1 \\
& G_{P} \equiv - \frac{1}{2} \sum_{\alpha} E_{\alpha} + \frac{i}{2} \left( \sum_{i=1}^{3}E_{\alpha_{i}}
- E_{\alpha_{1}+\alpha_{2}+\alpha_{3}} \right) \\
& \phantom{G_{P} \equiv}
- \frac{i}{2} \left( \sum_{i=1}^{3}E_{-\alpha_{i}}- E_{-\alpha_{1}-\alpha_{2}-\alpha_{3}} \right) \; .
\end{split}
\label{eqn:P-by-SU4}$$ The summation $\sum_{\alpha}$ runs over all the twelve non-zero roots $\alpha$ of SU(4). Here we do not give the explicit expressions of the generators which depend on a particular choice of the basis, since giving the commutation relations suffices to define $\mathbb{Z}_{4}{\times}\mathbb{Z}_{4}$ (see [Supplementary Material]() for the expressions in a particular basis set that are more convenient for the actual calculations). It is important to note that the two operators $Q$ and $P$ constructed here generate $\mathbb{Z}_{4}{\times}\mathbb{Z}_{4}$ (i.e., $[Q,P]=0$) [*only*]{} when the number of boxes in the Young diagram is an integer multiple of 4. In other words, what we have defined is the $\mathbb{Z}_{4}{\times}\mathbb{Z}_{4}$ subgroup of PSU(4). This is reminiscent of that the two $\pi$-rotations along the $x$ and $z$ axes generate $\mathbb{Z}_{2}{\times}\mathbb{Z}_{2}$ only for $\text{SO(3)}\simeq \text{SU(2)}/\mathbb{Z}_{2}$. In Appendix \[sec:ZnxZn\], we present the expressions of $G_{P}$ and $G_{Q}$ for other $N$s.
Having explicitly constructed a $\mathbb{Z}_{N}{\times}\mathbb{Z}_{N}$ subgroup of PSU($N$), we now consider how the existence of this subgroup leads to $(N-1)$ SPT phases. Consider the two $\mathbb{Z}_{N}$ generators $P$ and $Q$ satisfying $$(Q)^{N}=(P)^{N}=1 \; .
\label{eqn:U-to-N-1}$$ As has been shown above, we can explicitly construct $P$ and $Q$ using the generators of SU($N$). By carefully choosing the gauge, we can always make the corresponding projective representations $U_{P}$ and $U_{Q}$ satisfy $$\begin{split}
& U_{1}=\mathbf{1} \, , \; (U_{P})^{N}= (U_{Q})^{N}=\mathbf{1} \, ,
\\
& U_{P^{n}}=(U_{P})^{n} \, , \; U_{Q^{n}}=(U_{Q})^{n} \quad (n=1,\ldots, N-1) \; .
\end{split}
\label{eqn:ZnxZn-gauge-choice}$$ If one requires that $QP=PQ$ hold when both sides act on the MPS in question, one obtains from Eq. $$A(m) = (U_{Q}U_{P}U_{Q}^{\dagger}U_{P}^{\dagger})A(m)
(U_{P}U_{Q}U_{P}^{\dagger}U_{Q}^{\dagger}) \; .$$ When the MPS in question is pure and canonical, this implies $$U_{P}U_{Q} = {\mathrm{e}}^{- i\Phi_{QP}} U_{Q}U_{P} \; .
\label{eqn:U1-U2-exchange-Zn}$$ On the other hand, combining $(U_{P})^{N}=1$ and $U_{P}U_{Q}U_{P}^{\dagger}U_{Q}^{\dagger} = {\mathrm{e}}^{ - i\Phi_{QP}} \mathbf{1}$ obtained above, we obtain another relation: $$\begin{split}
U_{Q}U_{P}^{N-1} &= U_{Q}U_{P}^{\dagger}
= U_{P}^{N-1}(U_{P}U_{Q}U_{P}^{\dagger}U_{Q}^{\dagger})U_{Q}
\\
& = {\mathrm{e}}^{ - i\Phi_{QP}}U_{P}^{N-1}U_{Q} \; .
\end{split}$$ Using , the right-hand side may be rewritten as $$\begin{split}
{\mathrm{e}}^{ -i\Phi_{QP}}U_{P}^{N-1}U_{Q} &= ({\mathrm{e}}^{-i\Phi_{QP}})^{2}U_{P}^{N-2}U_{Q}U_{P} \\
&= ({\mathrm{e}}^{-i\Phi_{QP}})^{N}U_{Q}U_{P}^{N-1} \; .
\end{split}$$ Therefore, we arrive at the conclusion that ${\mathrm{e}}^{i\Phi_{QP}}$ is the $\mathbb{Z}_{N}$ phase[@Duivenvoorden-Q-ZnxZn-13]: $${\mathrm{e}}^{i\Phi_{QP}}= {\mathrm{e}}^{i \frac{2\pi}{N}n_{\text{top}}}
= \omega^{n_{\text{top}}} \;\; (n_{\text{top}}=0,1,\ldots, N-1) \; .
\label{eqn:exchange-phase}$$ To see that $\Phi_{QP}$ is in fact given by $(2\pi/N) n_{\text{top}}[=(2\pi/N) n_{\text{Y}}]$, we just note that $U_{Q}U_{P}={\mathrm{e}}^{i\frac{2\pi}{N}} U_{P}U_{Q}$ for the $N$-dimensional representation ${\tiny \yng(1)}$ and that other representations are constructed by tensoring ${\tiny \yng(1)}$ $n_{\text{Y}}$ times. Eq. implies that the exchange phase between $U_{P}$ and $U_{Q}$ carries the information on the topological class $n_{\text{top}}$.
Definition
----------
Next, we define another set of operators $\hat{X}_{P}$ and $\hat{X}_{Q}$ satisfying the following commutation relations with $\hat{P}$ and $\hat{Q}$ introduced in the previous section $$\begin{split}
& \hat{Q}^{\dagger}\hat{X}_{Q} \hat{Q} = \omega \hat{X}_{Q} \;\; , \quad
\hat{P}^{\dagger}\hat{X}_{Q} \hat{P} = \hat{X}_{Q} \\
& \hat{Q}^{\dagger}\hat{X}_{P} \hat{Q} = \hat{X}_{P} \;\; , \quad
\hat{P}^{\dagger}\hat{X}_{P} \hat{P} = \omega^{-1}\hat{X}_{P}
\quad
(\omega={\mathrm{e}}^{i\frac{2\pi}{N}})
\end{split}
\label{eqn:Z4xZ4-XpXq}$$ for [*any*]{} irreducible representations of SU($N$). Using the commutation relations , one sees that the operators $\hat{X}_{Q}$ and $\hat{X}_{P}$ for $N=4$ can be expressed by the SU(4) generators as $$\begin{split}
& \hat{X}_{Q} = \frac{1}{\sqrt{2}}( E_{-\alpha_1}+E_{-\alpha_2}+E_{-\alpha_3} + E_{-\alpha_4}) \\
& \hat{X}_{P} = H_{1} - i H_{3} \; ,
\end{split}
\label{eqn:XQ-XP-by-SU4}$$ where $\alpha_4 \equiv - \alpha_1 -\alpha_2 -\alpha_3$ and the normalization has been chosen such that $\text{Tr}\,({\hat{X}_{Q}}^{\dagger}\hat{X}_{Q})= \text{Tr}\,({\hat{X}_{P}}^{\dagger}\hat{X}_{P})$. From these operators, we define the following string operators:
$$\begin{aligned}
V_{P}(m,n;i) & \equiv \hat{Q}^{\dagger}(1)^{n} \cdots \hat{Q}^{\dagger}(i-1)^{n}
\left(\hat{X}_{P}(i)\right)^{m}
\label{eqn:string-op-1} \\
V_{Q}(m,n;i) &\equiv \left(\hat{X}_{Q}(i)\right)^{m} \hat{P}(i+1)^{n} \cdots \hat{P}(L)^{n} \; .
\label{eqn:string-op-2} \end{aligned}$$
Then, the string-order parameters (SOP) are (infinite-distance limits of) the two-point functions of these string operators:
$$\begin{aligned}
\begin{split}
& \mathcal{O}_{1}(m,n) \equiv
\lim_{|i-j|\nearrow \infty} \langle V_{P}(m,n;i)V_{P}^{\dagger}(m,n;j) \rangle \\
& = \lim_{|i-j|\nearrow \infty}
\Biggl\langle \left\{\hat{X}_{P}(i)\right\}^{m} \left\{
\prod_{i\leq k <j} \hat{Q}(k)^{n}
\right\} \left\{ \hat{X}_{P}^{\dagger}(j) \right\}^{m} \Biggr\rangle
\label{eqn:def-stringOP-1b}
\end{split}
\\
\begin{split}
& \mathcal{O}_{2}(m,n) \equiv \lim_{|i-j|\nearrow \infty} \langle V_{Q}(m,n;i)V_{Q}^{\dagger}(m,n;j) \rangle\\
& = \lim_{|i-j|\nearrow \infty}
\Biggl\langle
\left\{ \hat{X}_{Q}(i) \right\}^{m}
\left\{ \prod_{i < k \leq j} \hat{P}(k)^{n}
\right\} \left\{\hat{X}_{Q}^{\dagger}(j) \right\}^{m} \Biggr\rangle \\
& \qquad (0 \leq m,n < N)
\; .
\end{split}
\label{eqn:def-stringOP-2b}\end{aligned}$$
The subscripts 1 and 2 refer to the SOP corresponding to the two commuting $\mathbb{Z}_{N}$’s (associated with $Q$ and $P$, respectively).
It is important to note that when the model is realized in the cold-atom system , the SOP $\mathcal{O}_{1}(m,n)$ are expressed [*only*]{} in terms of the local fermion numbers $n_{\alpha,i}=c^{\dagger}_{g\alpha,i}c_{g\alpha,i}+c^{\dagger}_{e\alpha,i}c_{e\alpha,i}$. In fact, the expressions involves only the diagonal generators $\{ H_{a} \}$ \[see, e.g., Eqs. and \] which, when second-quantized, can be written only with the local fermion densities $n_{\alpha,i}$. This property is desirable in view of future detection of the non-local order with the site-resolved-imaging techniques.[@Endres-etal-stringOP-11; @Gross-B-review-15]
As is seen in , the second SOP $\mathcal{O}_{2}(m,n)$ contain the off-diagonal generators \[see Eqs. Eqs. and \] and are more complicated; in order to express them in terms of the fermions, we first second-quantize the (off-diagonal) generators, e.g., as $$\hat{E}_{\alpha,i} = c^{\dagger}_{g\beta,i}(\mathcal{E}_{\alpha})_{\beta\gamma}c_{g\gamma,i}
+ c^{\dagger}_{e\beta,i}(\mathcal{E}_{\alpha})_{\beta\gamma}c_{e\gamma,i} \; ,$$ where the $4{\times}4$ matrices $\mathcal{E}_{\alpha}$ are four-dimensional fundamental representations of the generators $E_{\alpha}$ (see [Supplementary Material]() for the expressions). Acting on the states in , the second-quantized generators $\hat{E}_{\alpha}$ reproduce the ones appearing in .
The merit of using the SOP is that they carry the information on the projective representation $U_{P}$ and $U_{Q}$ that determine the topological class[@Pollmann-T-12; @Hasebe-T-13] (see Sec. \[sec:def-ZnxZn\]). To show this, we first note that the SOP decouple into the product of the boundary contributions: $$\begin{split}
& \mathcal{O}_{1}(m,n) \xrightarrow{|i-j|\nearrow \infty}
\sum_{\alpha,\beta}\left\{
(T_{Q}^{X_P} \mathbf{V}^{(Q)}_{\text{R},1})
(\mathbf{V}^{(Q)}_{\text{L},1}T^{X_P})
\right\}_{\alpha,\alpha;\beta,\beta} \\
&= \sum_{\alpha,\beta}\left\{
\left(T_{Q}^{X_P} \left\{ \mathbf{1}{\otimes} (U_{Q}^{\dagger})^{n} \right\}\mathbf{1} \right)
(\mathbf{1}\left\{ \mathbf{1}{\otimes} (U_{Q})^{n} \right\}T^{X_P})
\right\}_{\alpha,\alpha;\beta,\beta} \\
&= \raisebox{-5.0ex}{\includegraphics[scale=0.5]{../figures/string-in-MPS-SUN}}
\equiv \mathcal{O}_{1,\text{L}}(m,n) \mathcal{O}_{1,\text{R}}(m,n) \; ,
\end{split}
\label{eqn:string-in-MPS-SUN}$$ where $U_{Q}$ is the projective representation of $Q$ and the transfer matrices are defined as $$\begin{split}
& [T^{X_P}]_{\bar{\alpha},\alpha;\bar{\beta},\beta} \equiv \sum_{a,b=1}^{d}
\left[A^{\ast}(a)\right]_{\bar{\alpha},\bar{\beta}}
\left[A(b)\right]_{\alpha,\beta}
\langle a | (\hat{X}^{\dagger}_{P})^{m}| b \rangle \\
& [T_{Q}^{X_P}]_{\bar{\alpha},\alpha;\bar{\beta},\beta} \equiv \sum_{a,b=1}^{d}
\left[A^{\ast}(a)\right]_{\bar{\alpha},\bar{\beta}}
\left[A(b)\right]_{\alpha,\beta}
\langle a |(\hat{X}_{P})^{m}\hat{Q}^{n} | b\rangle \; .
\end{split}$$ The $\mathbf{V}^{(Q)}_{\text{L},1}$ ($\mathbf{V}^{(Q)}_{\text{R},1}$) denotes the largest left (right) eigenvector of the following transfer matrix: $$[T_{Q}]_{\bar{\alpha},\alpha;\bar{\beta},\beta} \equiv \sum_{a,b=1}^{d}
\left[A^{\ast}(a)\right]_{\bar{\alpha},\bar{\beta}}
\left[A(b)\right]_{\alpha,\beta}
\langle a |\hat{Q}^{n}|b\rangle \; .$$ Using the properties of the canonical MPS[@Garcia-W-S-V-C-08], we can show that the right boundary term $\mathcal{O}_{1,\text{R}}(m,n)$ in Eq. satisfies the following identity[@Pollmann-T-12; @Hasebe-T-13; @Duivenvoorden-Q-ZnxZn-13] (see Fig. \[fig:string-boundary-SUN\]): $$\mathcal{O}_{1,\text{R}}(m,n) =
\omega^{-l(m+n \, n_{\text{top}})} \mathcal{O}_{1,\text{R}}(m,n) \quad
(l=1,\ldots, N-1) \; .$$ That is, if $\omega^{-l(m+n \, n_{\text{top}})} \neq 1$ for some $l$, $\mathcal{O}_{1,\text{R}}(m,n)=\mathcal{O}_{1}(m,n)=0$ [*solely*]{} by symmetry. A similar identity is obtained for $\mathcal{O}_{2}(m,n)$ as well. Then, these idendity imply that when [*both*]{} $ \mathcal{O}_{1}(m,n)$ and $ \mathcal{O}_{2}(m,n)$ are non-zero, the topological index $n_{\text{top}}$ necessarily satisfies $$\omega^{-(m+n \, n_{\text{top}})} = 1 \; .
\label{eqn:string-OP-constraint}$$ For $N=4$, we can use the set of $\mathcal{O}_{1,2}(m,n)$ with $$(m,n)=(1,3) \; (\text{class-1)}, \;\;
(2,1) \; (\text{class-2)}, \; \;
(1,1) \; (\text{class-3)}
\label{eqn:Z4xZ4-string-OP}$$ to distinguish between the three topological phases (as well as one trivial one). In the SU(4) class-2 phase we discuss here, we expect $$\begin{split}
& \mathcal{O}_{1,2}(2,1) \neq 0 \, , \\
& \mathcal{O}_{1,2}(1,3) = \mathcal{O}_{1,2}(1,1) = 0 \; .
\end{split}$$ In fact, for the solvable SU(4) VBS state discussed in Sec. \[sec:solvable-Ham\], we have $$\mathcal{O}_{1,2}(m,n)
=
\begin{cases}
0 & (m,n)=(1,3) \\
1 & (m,n)=(2,1) \\
0 & (m,n)=(1,1) \; ,
\end{cases}$$ which clearly indicate the class-2 topological phase.
In general, we need a set of $2(N-1)$ SOPs $\mathcal{O}_{1,2}(m,n)$ to identify the PSU($N$) topological phases. Note that the non-vanishing SOP is the [*sufficient*]{} condition for the corresponding topological class. In other words, even if the system is in the topological phase, the corresponding SOP might be zero for some other special reasons.
![(Color online) Boundary term $\mathcal{O}_{1,\text{R}}(m,n)$ carries the information on the exchange phase $\omega^{n_{\text{top}}}$ between $U_{P}$ and $U_{Q}$ \[see Eq. \]. Here a trivial identity $(\hat{X}_{P})^{m}=(\hat{P}^{\dagger})^{l}\left\{ (\hat{P})^{l}(\hat{X}_{P})^{m}(\hat{P}^{\dagger})^{l}\right\}
(\hat{P})^{l}$ ($l$: arbitrary) has been used. \[fig:string-boundary-SUN\]](string-boundary-SUN){width="0.9\columnwidth"}
Reflection
----------
In contrast to the SU(2) case where the operators $\hat{X}_{P}=S^{z}$ and $\hat{X}_{Q}=S^{x}$ are hermitian (see Appendix \[sec:ZnxZn\]), $\mathcal{O}_{1,2}(m,n)$ are not invariant under reflection symmetry $\mathcal{I}$ (with respect to a site or a bond) for SU($N$) with $N\geq 3$. In fact, reflection $\mathcal{I}$ takes them to $$\mathcal{O}_{1,2}(m,n) \xrightarrow{\mathcal{I}}
\widetilde{\mathcal{O}}_{1,2}(m,N-n)^{\ast} \; ,$$ where $\widetilde{\mathcal{O}}_{1,2}$ here are defined as $$\begin{split}
& \widetilde{\mathcal{O}}_{1}(m,n) \\
& \equiv
\lim_{|i-j|\nearrow \infty}
\Biggl\langle \left\{ \hat{X}_{P}(i) \right\}^{m} \left\{
\prod_{i < k \leq j } \hat{Q}(k)^{n}
\right\} \left\{\hat{X}^{\dagger}_{P}(j)\right\}^{m} \Biggr\rangle \\
& \widetilde{\mathcal{O}}_{2}(m,n) \\
& \equiv
\lim_{|i-j|\nearrow \infty}
\Biggl\langle \left\{ \hat{X}_{Q}(i) \right\}^{m} \left\{
\prod_{i \leq k < j} \hat{P}(k)^{n}
\right\} \left\{\hat{X}^{\dagger}_{Q}(j)\right\}^{m} \Biggr\rangle \; .
\end{split}$$ The new order parameters $\widetilde{\mathcal{O}}_{1,2}(m,n)$ look similar to the original SOP $\mathcal{O}_{1,2}(m,n)$ but are different in the relative position between the string and the end points \[see Eqs. and \]. Now one can repeat the preceding argument \[see Eq. and Fig. \[fig:string-boundary-SUN\]\] on the boundary terms to obtain exactly the same selection rule . Therefore, one sees that when both $\mathcal{O}_{1}(m,n)$ and $\mathcal{O}_{2}(m,n)$ are non-vanishing in a given ground state $|\psi\rangle$, its parity partner $\mathcal{I} |\psi\rangle$ has finite $\widetilde{\mathcal{O}}_{1}(m,N-n)$ and $\widetilde{\mathcal{O}}_{2}(m,N-n)$, and hence is in another topological phase characterized by $\mathcal{O}_{1,2}(m,N-n)$. For instance, the SU(4) class-1 topological phase characterized by $\mathcal{O}_{1,2}(1,3)$ is the parity partner of the class-3 phase characterized by $\mathcal{O}_{1,2}(1,4-3)=\mathcal{O}_{1,2}(1,1)$ (Fig. \[fig:KT-ZnxZn-string-OP\]; see [Supplementary Material]() for the explicit demonstration).
Numerical results
-----------------
To demonstrate the use of the SOP in detecting the SU($N$) topological phases, we plot the value of the SOP $\mathcal{O}_{1}(m,n)$ for the model obtained using iTEBD. Note that by the SU(4)-symmetry, we do not need to calculate $\mathcal{O}_{2}(m,n)$. That $\mathcal{O}_{1}(2,1)$ is non-vanishing for $\mathcal{H}(a)$ from $a=0$ to $a=1$ gives a strong evidence of the class-2 topological phase.
![(Color online) Plot of SOP for $\mathcal{H}(a)$. $\mathcal{O}_{1}(2,1)$ is non-zero between the Heisenberg point ($a=0$) and the VBS point ($a=1$) giving additional evidence for the topological nature. \[fig:string-order-Ha\]](Ostring-iTEBD){width="0.9\columnwidth"}
Non-local transformation
------------------------
Before concluding this section, we give a remark on the connection between the SOP and the non-local unitary transformation (generalized Kennedy-Tasaki transformation) eliminating the entanglement of the SPT phase that was first introduced in Refs. for the SO(3)-based spin chains (see also Refs. for recent discussions in the context of disentangler). A straightforward generalization of the above non-local unitary transformation to the PSU($N$) case may be given by[@Duivenvoorden-Q-ZnxZn-13] $$U_{\text{KT}} = \exp\left\{
i \frac{2\pi}{N} \sum_{k<j} G_{P}(k) G_{Q}(j)
\right\} \; .
\label{eqn:def-Kennedy-Tasaki-ZnxZn}$$ Then, it is easy to see that the string operators defined in Eqs. and transform (up to phase) as $$\begin{split}
& U_{\text{KT}}^{\dagger} V_{P}(m,n;i) U_{\text{KT}} = V_{P}(m,m+n;i) \\
& U_{\text{KT}}^{\dagger} V_{Q}(m,n;i) U_{\text{KT}} = V_{Q}(m,m+n;i) \; .
\end{split}
\label{eqn:UKT-vs-V}$$ This and Eq. imply that repeated applications of $U_{\text{KT}}$ take the system from one topological phase to another (see Fig. \[fig:KT-ZnxZn-string-OP\]). In particular, the class-1 and 3 phases can be reduced to conventional phase with (spontaneously-broken) local orders, while the class-2 is not.
![(Color online) Generalized Kennedy-Tasaki transformation $U_{\text{KT}}$ and three PSU(4) SPT phases. The class-2 state are mapped onto the state of the same topological class by $U_{\text{KT}}$. Note that both $\mathcal{O}_{1,2}(2,1)$ and $\mathcal{O}_{1,2}(2,3)$ characterize the same class-2 phase \[see Eq. \]. \[fig:KT-ZnxZn-string-OP\]](KT-ZnxZn-string-OP){width="0.9\columnwidth"}
Symmetry Reduction {#sec:symmetry-reduction}
==================
In SPT phases, the list of possible topological phases is closely tied to the symmetry we impose on the system, and a phase which is topological under a certain symmetry may not be so when we consider a lower symmetry. Although the protecting symmetry $\text{PSU($N$)}$ is automatically (i.e., without fine tuning) guaranteed almost perfectly in alkaline-earth cold fermions[@Gorshkov-et-al-10; @Scazza-et-al-14; @Zhang-et-al-Sr-14], it would be interesting, from the theoretical point of view, to consider the fate of the topological phases when $\text{PSU($N$)}$ gets reduced.
Systems only with reflection symmetry {#sec:PSUN-to-reflection}
-------------------------------------
We begin with the case where the PSU($N$) symmetry is broken down to reflection symmetry with respect to the middle of a bond ([*link-parity*]{} $\mathcal{I}$). As is emphasized in Refs. , symmetry operations (whether local or non-local) which keep a given state (which we assume is represented as an MPS) invariant are expressed in the form of Eq. : $$A(m_i) \xrightarrow{\mathcal{I}} A(m_i)^{\text{T}} =
{\mathrm{e}}^{i\phi_{\mathcal{I}}}
U^{\dagger}_{\mathcal{I}} A(m_i) U_{\mathcal{I}} \; ,
\label{eqn:MPS-UAU-inv}$$ where $U_{\mathcal{I}}$ satisfies $U_{\mathcal{I}}^{\text{T}} = \pm U_{\mathcal{I}}$. Depending on the sign appearing on the right-hand, there are two classes for systems with link-parity $\mathcal{I}$ (topological when $-1$ and trivial if $+1$)[@Pollmann-T-B-O-10].
Now let us determine the sign for the SU($N$) VBS state shown in Fig. \[fig:SU(4)VBS\]. To this end, we first note that the MPS matrices $A(m_i)$ is written as $$A(m_i) = \mathcal{R} P(m_i) \; ,$$ where $P(m_i)$ is the projection operators from the two fractional objects $|\alpha\rangle_{i}$ and $|\beta\rangle_{i} $ \[in our SU(4) case they are two [**6**]{} representations ${\tiny \yng(1,1)}$\] at site $i$ onto the physical states $|m_i\rangle$: $$\left[ P(m_i)\right]_{\alpha\beta} \equiv \langle m_i
|\alpha\rangle_{i}{\otimes}|\beta\rangle_{i} \; .$$ The metric matrix $\mathcal{R}$ creates the SU($N$)-singlet out of the two fractional objects $|\alpha\rangle_{i}$ and $|\beta\rangle_{i+1}$ on the adjacent site as (see Fig. \[fig:SU(4)VBS\]): $$|\text{singlet}\rangle = \mathcal{R}_{\alpha\beta}|\alpha\rangle_{i} |\beta\rangle_{i+1} \; .$$ Then, we can show that $U_{\mathcal{I}}$ is given by the matrix $\mathcal{R}$: $$\begin{split}
& \mathcal{R}^{\dagger} A(m_i) \mathcal{R} =
\mathcal{R}^{\dagger} \left(\mathcal{R}P(m_i)\right) \mathcal{R} \\
&= P(m_i) \mathcal{R}
= {\mathrm{e}}^{-i \phi_{I}}\left\{ \mathcal{R} P(m_i) \right\}^{\text{T}}
= {\mathrm{e}}^{-i \phi_{I}}A(m_i)^{\text{T}} \; ,
\end{split}$$ where $\phi_{I}$ is 0 when both $P(m_i)$ and $\mathcal{R}$ are symmetric/anti-symmetric, and $\pi$ otherwise \[in our case, $P(m_i)$ are symmetric by construction\]. Therefore, in order to know if $U_{\mathcal{I}}$ is antisymmetric or not, we have only to know how the SU($N$)-singlet is constructed out of $|\alpha\rangle_{i}$ and $|\beta\rangle_{i+1}$.
The SU($N$)-singlet is written as the following fully-antisymmetrized product of $N=2n$ states in the fundamental representation $\mathbf{N}$: $$\begin{split}
& |\text{singlet}\rangle \\
&= \sum_{\{i_k,j_k\}}
\epsilon_{i_1 i_2 \cdots i_n j_1 j_2 \cdots j_n}
|v_{i_1}\rangle \cdots |v_{i_n}\rangle
|v_{j_1}\rangle \cdots |v_{j_n}\rangle \\
& = \sum_{\text{partition}}
C_{\{i_k\};\{j_k\}}
\left\{
\sum_{\{i_k\}}
\epsilon_{i_1 i_2 \cdots i_n}|v_{i_1}\rangle |v_{i_2}\rangle \cdots |v_{i_n}\rangle
\right\} \\
& \times \left\{
\sum_{\{j_k\}}
\epsilon_{j_1 j_2 \cdots j_n}
|v_{j_1}\rangle |v_{j_2}\rangle \cdots |v_{j_n}\rangle
\right\} \; .
\end{split}$$ As the states inside the braces transform like $$\text{\scriptsize $n=N/2$} \left\{
\yng(1,1,1)
\right. \; ,$$ the symmetry of $\mathcal{R}$ is encoded in that of the coefficient $C_{\{i_k\};\{j_k\}}(=\pm 1)$. From the antisymmetry of $\epsilon_{i_1 i_2 \cdots i_n j_1 j_2 \cdots j_n}$, one imediatetely sees $$C_{\{i_k\};\{j_k\}} = (-1)^{n^{2}}C_{\{j_k\};\{i_k\}} = (-1)^{n}C_{\{j_k\};\{i_k\}} \; .$$ Therefore, under the symmtry-lowering perturbation, the class-2 SPT phase crosses over to the topological Haldane phase (a trivial phase) when $n=N/2=\text{odd}$ (even) (see Fig. \[fig:SUN-topo-phases\]).
\_[N]{}\_[N]{} \_[2]{}\_[2]{} {#sec:PSUN-to-Z2xZ2}
------------------------------
As has been seen in Sec. \[sec:def-ZnxZn\], we may regard the $N-1$ PSU($N$) topological phases as protected by the subgroup $\mathbb{Z}_{N}{\times}\mathbb{Z}_{N}$ (see also Appendix \[sec:PSUN-to-ZnxZn\]). In that case, the following commutation relation determines the topological classes[@Duivenvoorden-Q-ZnxZn-13]: $$U_P U_Q = {\mathrm{e}}^{i\frac{2\pi}{N}n_{\text{top}}} U_P U_Q \quad (n_{\text{top}}=0,1,\ldots, N-1) \; .
\label{eqn:U1-U2-exchange-Zn-2}$$ As the above $\mathbb{Z}_{N}{\times}\mathbb{Z}_{N}$ contains the $\mathbb{Z}_{2}{\times}\mathbb{Z}_{2}$ subgroup generated by $Q^{N/2}$ and $P^{N/2}$ when $N$ is even, we may consider the symmetry reduction $\mathbb{Z}_{N}{\times}\mathbb{Z}_{N} \mapsto \mathbb{Z}_{2}{\times}\mathbb{Z}_{2}$. From the relation $$\begin{split}
U_Q (U_P)^{N/2} &= \left( {\mathrm{e}}^{i\frac{2\pi}{N}n_{\text{top}}}\right)^{N/2} (U_P)^{N/2} U_Q \\
&= (-1)^{n_{\text{top}}} (U_P)^{N/2} U_Q \; ,
\end{split}$$ one can easily see that the projective representations of the two $\mathbb{Z}_{2}$ generators satisfy $$(U_Q)^{N/2} (U_P)^{N/2} = (-1)^{\frac{1}{2}N n_{\text{top}}} (U_P)^{N/2} (U_Q)^{N/2} \; .$$ It is known [@Pollmann-T-B-O-10; @Pollmann-B-T-O-12] that in the presence of $\mathbb{Z}_{2}{\times}\mathbb{Z}_{2}$-symmetry, the phase is topologically non-trivial when the projective representations of the two $\mathbb{Z}_{2}$s are anti-commuting, i.e., $(-1)^{\frac{1}{2}N n_{\text{top}}}=-1$. This is possible only when $$N=2(2k+1) \;\; (k \in \mathbb{Z}) \quad \text{and} \quad n_{\text{top}}=\text{odd} \; .$$ Since our SU($N$) ($N$: even) topological phase corresponds to $n_{\text{top}}=N/2$, it remains topological (i.e., Haldane phase) even after the symmetry gets reduced down to $\mathbb{Z}_{2}{\times}\mathbb{Z}_{2}$ when $N=2,6,10,\ldots$. When $N=0$ (mod 4), on the other hand, the topological phases considered here ($n_{\text{top}}=N/2$) smoothly cross over to trivial ones. In Fig. \[fig:SUN-topo-phases\], we summarize the crossover predicted here.
![(Color online) Fate of SPT phases protected by PSU($2n$) when the symmetry is reduced down to link-parity $\mathcal{I}$ (Sec. \[sec:PSUN-to-reflection\]) or $\mathbb{Z}_{2}{\times}\mathbb{Z}_{2}$ (Sec. \[sec:PSUN-to-Z2xZ2\]). All these phases are labelled according to the irreducible representation(s) under which edge states transform. The label “$n$” of the SPT classes stands for the number $n_{\text{Y}}$ (see the text) corresponding to the projective representations. When only the link-parity or $\mathbb{Z}_{2}{\times}\mathbb{Z}_{2}$ is imposed, only a part of them remains topological (warped arrows). \[fig:SUN-topo-phases\]](../figures/SUN-topo-phases){width="1.0\columnwidth"}
Conclusion and outlook {#sec:summary}
======================
The possibility of realizing SU($N$) symmetry using alkaline-earth cold atoms provides a new arena for the symmetry-protected topological phases. In this paper, we have studied the topological properties of the ground state of the SU($N$) Heisenberg chain with the “spins” at each site, especially for $N=4$. This model is interesting as it is expected to describe the Mott-insulating region of the two-orbital SU($N$) Hubbard model . From the analysis of the ground state of the solvable VBS Hamiltonian , we have suspected that the ground state of belongs to one of the three topological phases predicted for the SU(4)-invariant systems. To substantiate this, we have calculated the entanglement spectrum of an infinite-size system with iTEBD and found that the degeneracy structure is perfectly consistent with that expected for the topological class (called [*class-2*]{} in the text). In order to establish the adiabatic continuity between the Heisenberg model and the solvable VBS model, we have considered a simple one-parameter deformation $\mathcal{H}(a)$ of the Hamiltonian. The entanglement spectrum preserves its degeneracy structure all the way between the two models thus establishing the continuity.
Then, we have investigated how the entanglement spectrum changes when the protecting symmetry gets lowered. Specifically, we have considered the situations where the original SU($N$) symmetry (which is perfect in the alkaline-earth cold atoms) is reduced to (i) link-parity and (ii) $\mathbb{Z}_{2}{\times}\mathbb{Z}_{2}$. In both cases, the stability of our topological phase depends on the value of $N$; when $N=4n+2$ (i.e., $N=2,6,10,\ldots$), we expect a crossover from our SU($N$) topological phase to the Haldane phase.
Although the entanglement spectrum gives useful insights about the nature of topological phases, it may not fully characterize it. In fact, what is more fundamental is, at least from the group-cohomology point of view, the projective representations which is the mathematical representation of the physical edge states. The non-local string-order parameter (SOP) is appealing since it contains the information of the projective representation in a manner that may be accessible in experiments. We have numerically calculated the SOP $\mathcal{O}_{1}(2,1)$ with iTEBD and observed that it stays finite in our topological phase. This gives another support to our claim that the ground state of the SU(4) Heisenberg model is in the class-2 topological phase.
At least two interesting questions remain to be answered. One is about the quantum phase transition(s) out of the topological phase discussed here. In fact, an SU($N$) dimerized phase (called “spin-Peierls”) is observed numerically in Ref. next to (i.e., on the smaller-$U$ side of) the SPT phase. As the inclusion of higher-order terms in $t/U$ may be mimicked by adding terms higher order in $(\mathcal{S}_{i}{\cdot}\mathcal{S}_{i+1})$ to , we may include an extra term that favors dimerization to study the topological-dimerized quantum phase transition.
Another interesting problem would be the nature of the strong-coupling (Mott) phase of the model with [*odd*]{}-$N$. In this case, the orbital degree of freedom is not fully quenched and we obtain an effective Hamiltonian different from , where the SU($N$) “spin” are highly entangled with the orbital degree of freedom[@Bolens-C-L-T-15]. As the nature of the effective Hamiltonian, which is reminiscent of the Kugel-Khomskii-type model[@Kugel-K-82] for manganese, is not understood, it would be interesting to investigate it by the strategy used here.
Acknowledgements {#acknowledgements .unnumbered}
================
One of the authors (K.T.) has benefitted from stimulating discussions with A. Bolens, S. Capponi, P. Lecheminant, and K. Penc on related projects. He was also supported in part by JSPS KAKENHI Grant No. 24540402 and No. 15K05211 and by the PICS grant from CNRS France.
MPS matrices for SU(4) VBS state {#sec:MPS-matrices}
================================
In this appendix, we give the matrices necessary for the MPS representation of the SU(4) VBS state in Sec. \[sec:solvable-Ham\]. The MPS for the SU(4) VBS state is given by the following product of six-dimensional matrices $A(\mathbf{m}_i) = \Lambda \Gamma(\mathbf{m}_i) = \Gamma(\mathbf{m}_i) \Lambda$ (we follow the notations used in Ref. ): $$\begin{split}
& |\text{VBS}\rangle \\
& =\sum_{\{\mathbf{m}_{i}\}} A(\mathbf{m}_1)A(\mathbf{m}_2) \cdots A(\mathbf{m}_L)
|\mathbf{m}_1\rangle\otimes |\mathbf{m}_2\rangle \otimes \cdots
\otimes |\mathbf{m}_L\rangle \; ,
\end{split}$$ where the summation is taken over all the weights $\mathbf{m}_{i}=(m_{i}^1,m_{i}^2,m_{i}^3)$ of the 20-dimensional representation of SU(4) and $\Lambda$ is a diagonal matrix with non-negative diagonal elements. Throughout this paper, we assume infinite-size systems where the MPS is given by infinite-product of matrices $A(\mathbf{m}_{i})$. For several reasons, it is convenient to use the canonical form of the above MPS, where the transfer matrix satisfies certain conditions. One possible choice of the canonical MPS is[^7])
$$\Lambda =
\frac{1}{\sqrt{6}}
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1
\end{pmatrix}$$
$$\Gamma (2,0,0) = \sqrt{6}
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \; , \;\;
\Gamma (1,1,0) = \sqrt{3}
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
-1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0
\end{pmatrix} \; , \;\;
\Gamma (1,0,-1) = \sqrt{3}
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0
\end{pmatrix} \; , \;\;$$
$$\Gamma (0,2,0) = \sqrt{6}
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \; , \;\;
\Gamma (1,0,1) = \sqrt{3}
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0
\end{pmatrix} \; , \;\;
\Gamma (0,1,-1) = \sqrt{3}
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \; , \;\;$$
$$\Gamma (1,-1,0) = \sqrt{3}
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 \\
-1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0
\end{pmatrix} \; , \;\;
\Gamma (0,1,1) = \sqrt{3}
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \; , \;\;
\Gamma (0,0,-2) = \sqrt{6}
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \; , \;\;$$
$$\Gamma (0,0,0)_{\text{A}} = \sqrt{6}
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & -\frac{1}{2} & 0 & 0 & 0 & 0 \\
0 & 0 & \frac{1}{2} & 0 & 0 & 0 \\
0 & 0 & 0 & \frac{1}{2} & 0 & 0 \\
0 & 0 & 0 & 0 & -\frac{1}{2} & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \; , \;\;
\Gamma (0,0,0)_{\text{B}} = \sqrt{2}
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & -\frac{1}{2} & 0 & 0 & 0 & 0 \\
0 & 0 & -\frac{1}{2} & 0 & 0 & 0 \\
0 & 0 & 0 & -\frac{1}{2} & 0 & 0 \\
0 & 0 & 0 & 0 & -\frac{1}{2} & 0 \\
0 & 0 & 0 & 0 & 0 & 1
\end{pmatrix} \; , \;\;$$
$$\Gamma (0,0,2) = \sqrt{6}
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \; , \;\;
\Gamma (0,-1,-1) = \sqrt{3}
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \; , \;\;
\Gamma (-1,1,0) = \sqrt{3}
\begin{pmatrix}
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -1 \\
0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \; , \;\;$$
$$\Gamma (0,-1,1) = \sqrt{3}
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \; , \;\;
\Gamma (-1,0,-1) = \sqrt{3}
\begin{pmatrix}
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \; , \;\;
\Gamma (0,-2,0) = \sqrt{6}
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \; , \;\;$$
$$\Gamma (-1,0,1) = \sqrt{3}
\begin{pmatrix}
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \; , \;\;
\Gamma (-1,-1,0) = \sqrt{3}
\begin{pmatrix}
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & -1 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \; , \;\;
\Gamma (-2,0,0) = \sqrt{6}
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \; .$$
Note that the diagonal elements of $\Lambda$ are related to the entanglement spectrum $\{\xi_{\alpha}\}$ by $[\Lambda]_{\alpha\alpha}= {\mathrm{e}}^{-\xi_{\alpha}/2}$.
Construction of $\mathbb{Z}_{N}{\times}\mathbb{Z}_{N}$ {#sec:ZnxZn}
======================================================
In Sec. \[sec:def-ZnxZn\], we have explicitly constructed the $\mathbb{Z}_{4}{\times}\mathbb{Z}_{4}$ subgroup of PSU(4) using the generators of the latter. Below, we give the expressions of the $\mathbb{Z}_{N}{\times}\mathbb{Z}_{N}$ generators in terms of PSU($N$) for $N=2$ and $3$.
Regardless of $N$, the first generator $G_{Q}$ is given simply by $$G_{Q} = \sum_{k=1}^{N-1} (\vec{\rho})_{k} H_{k} \; ,
\label{eqn:def-G_Q}$$ where $H_{k}$ are the $N-1$ Cartan generators and $\vec{\rho}$ is the Weyl vector of SU($N$). The generator $G_{Q}$ has the following simple commutation relations with the simple roots $\alpha$: $$[ G_{Q}, E_{\alpha}]= E_{\alpha} \; , \quad
[ G_{Q}, E_{-\alpha}]= - E_{-\alpha} \; ,$$ which guarantee integer-spaced eigenvalues of $G_{Q}$ (for the fundamental representation $\boldsymbol{N}$, they are essentially $1,2,\cdots, N$). With this, the first $\mathbb{Z}_{N}$ is generated as $$Q = c_{N} \exp \left(i\frac{2\pi}{N} G_{Q}\right) \; ,$$ where the phase $c_{N}$ has been introduced so that $Q$ satisfy $Q^{N}=1$.
The expression of the other generator $G_{P}$ depends on $N$. For $\mathbb{Z}_{2}{\times}\mathbb{Z}_{2}$, we recover the well-known results[@Kennedy-T-92-PRB; @Kennedy-T-92-CMP]
$$\begin{aligned}
& G_{Q} = \rho H = S^{z} \quad (H=\sqrt{2} S^{z}, \;
\rho=1/\sqrt{2}) \\
& G_{P} = -\frac{1}{2} E_{\alpha} -\frac{1}{2} E_{-\alpha} = - S^{x} \; .\end{aligned}$$
The operators $\hat{X}_{P}$ and $\hat{X}_{Q}$ satisfying are obtained as $$\hat{X}_{P} = S^{z} \, , \;\;
\hat{X}_{Q} = S^{x} \; .$$
For $\mathbb{Z}_{3}{\times}\mathbb{Z}_{3}$, we have
$$\begin{aligned}
& G_{Q} = \rho_1 H_1 + \rho_2 H_2 = \sqrt{2} H_1 , \;
(\vec{\rho}=(\sqrt{2},0) ) \\
& G_{P} =
-\frac{i}{\sqrt{3}}\sum_{k=1}^{3}(E_{\alpha_k} - E_{- \alpha_k}) \; ,\end{aligned}$$
where $\alpha_{1,2}$ are the simple roots of SU(3) and $\alpha_{3}$ is defined by $\alpha_{3}\equiv -\alpha_1 - \alpha_2$. The operators $X_{P}$ and $X_{Q}$ satisfying Eq. are given by
$$\begin{aligned}
& X_{P} = H_1 - i H_2 \\
& X_{Q}
= \sqrt{\frac{2}{3}}\left\{
E_{-\alpha_1} + E_{-\alpha_2} + E_{-\alpha_3}
\right\} \quad (\alpha_3 \equiv -\alpha_1 - \alpha_2) \; .\end{aligned}$$
$\text{PSU(\texorpdfstring{$\boldsymbol{N}$}{N})}$ and $\mathbb{Z}_{N}{\times}\mathbb{Z}_{N}$ {#sec:PSUN-to-ZnxZn}
=============================================================================================
Since PSU($N$) and $\mathbb{Z}_{N}{\times}\mathbb{Z}_{N} [\subset \text{PSU($N$)}]$ share the same cohomology group $H^{2}(\text{PSU($N$)},\text{U(1)})=H^{2}(\mathbb{Z}_{N}{\times}\mathbb{Z}_{N},\text{U(1)})=\mathbb{Z}_{N}$, a phase which is topological under PSU($N$) may remain so even if we weakly break down to $\mathbb{Z}_{N}{\times}\mathbb{Z}_{N}$. As we have seen in Sec. \[sec:SPT\], when the system has the full PSU($N$)-symmetry, the entanglement spectrum exhibits the degeneracy pattern that is compatible with SU($N$)-symmetry. That is, the degeneracy of each entanglement level should find the corresponding entry in TABLE \[tab:Young\]. Now let us consider how the reduction of the symmetry down to a subgroup $\mathbb{Z}_{N}{\times}\mathbb{Z}_{N}$ changes the entanglement spectrum.
As the unitary matrices $U_{P,Q}$ assume block-diagonal forms reflecting the structure of the entanglement levels, the relation holds for each block corresponding to the degenerate entanglement levels $\lambda$ $$U_{P}(\lambda)U_{P}(\lambda) = {\mathrm{e}}^{i\Phi_{QP}} U_{Q}(\lambda)U_{P}(\lambda)
= {\mathrm{e}}^{i \frac{2\pi}{N}n_{\text{top}}} U_{Q}(\lambda)U_{P}(\lambda) \; .$$ This restricts the degree of degeneracy $D_{\lambda}$ of each entanglement level[@Bois-C-L-M-T-15]. Calculating the determinant of both sides of the above equation, one obtains $$\begin{split}
\text{det}\, (U_{P}(\lambda)U_{Q}(\lambda)) &= \text{det} \,U_{P}(\lambda)\text{det}\,U_{Q}(\lambda) \\
& = ({\mathrm{e}}^{i \frac{2\pi}{N}n_{\text{top}}})^{D_{\lambda}} \text{det}\,U_{P}(\lambda)\text{det}\,U_{Q}(\lambda) \; ,
\end{split}
\label{eqn:det-U1-U2-Zn}$$ which immediately implies $({\mathrm{e}}^{i \frac{2\pi}{N}n_{\text{top}}})^{D_{\lambda}}=1$. When $N$ and $n_{\text{top}}$ are mutually co-prime, $D_{\lambda}$ should be integer multiple of $N$. Otherwise, $D_{\lambda}$ of each level may be smaller. In particular, the entanglement spectrum of the class-1 PST phase exhibits the $N$-fold degenerate structure for any $N(\geq 2)$, which is consistent with the results of the explicit calculation[@Katsura-H-K-08] for the SU($N$) VBS chain based on another representation[@Affleck-K-L-T-87; @Affleck-K-L-T-88].
For $N=4$ ($\mathbb{Z}_{4}{\times}\mathbb{Z}_{4}$), there are three topological phases (i) class-1 ($n_{\text{top}}=1$), (ii) class-2 ($n_{\text{top}}=2$), and (iii) class-3 ($n_{\text{top}}=3$). In the class-1 and 3 phases, $D_{\lambda}=0$ (mod 4), while [*any even integers*]{} are allowed for $D_{\lambda}$ in the class-2 phase. Therefore, the degeneracy pattern observed in Sec. \[sec:ES\] in general may be modified when we relax the full PSU(4) symmetry down to $\mathbb{Z}_{4}{\times}\mathbb{Z}_{4}$, although the system still stays in the same phase. For instance, the lowest six-fold-degenerate level might be split into, e.g., three two-fold-degenerate levels.
For instance, we may add to the original Hamiltonian the following $\mathbb{Z}_{N}{\times}\mathbb{Z}_{N}$-invariant perturbation \[see Eq. \] $$\mathcal{V}_{N}
= g_{N} \sum_{i} \left\{
\left( \hat{X}_{P}(i) \right)^{N} + \left( {\hat{X}_{P}}^{\dagger}(i) \right)^{N}
\right\} \; ,$$ which is a generalization of the well-known single-ion anisotropy $D\sum_{i}(S_{i}^{z})^{2}$ in the usual spin chains \[note $\hat{X}_{P}(i)={\hat{X}_{P}}^{\dagger}(i)=S^{z}$ for $N=2$\].
[^1]: The proof of the existence of low-lying states works regardless of whether the ground state is unique or not. However, unless the (finite-size) ground state is unique, the proof does not imply anything about [*excited*]{} states.
[^2]: A remark is in order here about the use of the terminology ‘orbital’ here. In the case of electrons in crystals, orbital is closely tied to the spatial structure of the wave function and often allows pair-hopping processes that break continuous orbital symmetry down to a discrete one. The two orbitals $g$ and $e$, on the other hand, are internal degrees of freedom and, in the absence of the internal conversion between $g$ and $e$, the system retains at least orbital U(1) symmetry.
[^3]: To be precise, the protecting symmetry is not SU(4) but $\text{PSU(4)}\simeq \text{SU(4)}/\mathbb{Z}_{4}$.
[^4]: In fact, there is another way of generalizing the spin-1 SU(2) VBS state. Instead of using two copies of the self-conjugate representations , we may use the $n$-dimensional defining representation $\mathbf{n}$ and its conjugate $\bar{\mathbf{n}}$. This type of SU($N$) “VBS state” has been already discussed in the AKLT paper (Refs. ).
[^5]: In fact, the expression of the parent Hamiltonian is [*not*]{} unique. There are 3 (6) free positive parameters in the parent Hamiltonian of the SU(4) \[SU(6)\] VBS state. The ones shown in the text are obtained when we require that they be of lowest degree in $\mathcal{S}{\cdot}\mathcal{S}$ and that the coefficient of the linear term be 1.
[^6]: For $N \ge 8$, the parent Hamiltonians are not always written [*only*]{} in terms of $\left(\mathcal{S}_i \cdot \mathcal{S}_{i+1}\right)$, as $\left(\mathcal{S}_i \cdot \mathcal{S}_{i+1}\right)$ alone cannot always distinguish among all the irreducible representations.
[^7]: The derivation is sketched in Supplementary Material at <http://www.example.com/>.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We find a new polynomial identity in characteristic 2: $$\prod_{w\in\F_q^\x} (D_{q+1}(wX)-Y)
= X^{q^2-1} + \left(\sum_{i=1}^{n} Y^{2^{n}-2^i}\right) X^{q-1} + Y^{q-1},$$ where $q = 2^n$ and $D_k$ is a Dickson polynomial, defined by $D_k(u+u^{-1})=u^k + u^{-k}$. Using this identity, we prove that if $F$ is a field of characteristic 2 and $a$ is a nonzero element of $F$, then for $q=2^n>2$, the two polynomials $x^{q+1}+x+1/a$ and $C(x)+a$ have the same splitting field over $F$, where $C(x) = x (\sum_{i=0}^{n-1} x^{2^i-1})^{q+1}$ is a Müller–Cohen–Matthews polynomial of degree $(q^2-q)/2$. We find explicit formulas for how the roots of the two polynomials are related, and for the action of the Galois group. As a result, we can describe precisely how the factorizations of the two polynomials are related in the case where $F$ is finite. In addition, we obtain a new proof of the known result that $C(x)$ induces a permutation on $\F_{2^m}$ if $2m$ and $n$ are relatively prime.
author:
- 'Antonia W. Bluher'
title: '**A New Identity of Dickson Polynomials** '
---
Introduction {#sec:Intro}
============
We find a new polynomial identity in characteristic 2: $$\label{identity}
\prod_{w\in\F_q^\x} (D_{q+1}(wX)-Y)
= X^{q^2-1} + \left(\sum_{i=1}^{n} Y^{2^{n}-2^i}\right) X^{q-1} + Y^{q-1},$$ where $q = 2^n$ and $D_k$ is a Dickson polynomial, defined by $D_k(u+u^{-1})=u^k + u^{-k}$. Using this identity, we prove that if $F$ is a field of characteristic 2 and $a$ is a nonzero element of $F$, then for $q=2^n>2$, the two polynomials $x^{q+1}+x+1/a$ and $C(x)+a$ have the same splitting field over $F$, where $C(x) = x (\sum_{i=0}^{n-1} x^{2^i-1})^{q+1}$ is a Müller–Cohen–Matthews polynomial of degree $(q^2-q)/2$. We find explicit formulas relating the roots of the two polynomials, and we describe the Galois action. As a result, when $F$ is finite, related factorizations of the two polynomials can be explained. We also found a new proof that $C(x)$ induces a permutation on $\F_{2^m}$ if $(2m,n)=1$. (See [@CM] for the original proof. A polynomial that induces a permutation on infinitely many finite fields is said to be [*exceptional*]{}.)
A first draft of this article was written in the 2001–2004 timeframe, but was left and forgotten for over a decade. The project was resumed and completed in 2016, with the following improvements. A hypothesis that the field $F$ must be perfect was removed, the new proof of exceptionality of $C(x)$ was added, a simpler formula was found for the roots of $C(x)$, and a simpler description of dihedral subgroups of $\PGL_2(q)$ was obtained (see Proposition 6.1; an analogous proposition holds for $q$ odd as well.) Finally, references were updated to reflect advances in the understanding of exceptional polynomials that occurred in the intervening decade.
A few remarks are in order. First, the equality of the splitting fields of the two polynomials $x^{q+1}+x+1/a$ and $C(x)+a$ can be derived from work of Zieve [@Z] and Lenstra and Zieve [@LZ], at least in the case where $a$ is transcendental. Many calculations in this paper could perhaps be done more expediently with their methods, which utilize group theory. However, the author was unaware of these methods at the time that she carried out her work, and as a result she used different techniques and was motivated by a different set of questions. We hope that this new perspective will complement the existing literature.
The polynomial identity involving Dickson polynomials in characteristic 2 is new. It seems to apply only to characteristic 2. Bob Guralnick points out that the Dickson polynomials are ramified at the prime 2, thus it is not surprising to find formulas that are special to characteristic 2.
The results in our paper seem related to but different from results in Abhyankar, Cohen, and Zieve [@ACZ]. Both our paper and theirs give a factorization of $x^{q^2-1}-a(y)x^{q-1}-b(y)$ in terms of Dickson polynomials and use it to deduce information about the Galois groups of certain polynomials. However, the functions $a(y)$ and $b(y)$ differ, and so do the Galois groups that are involved. Their identity generalizes to all characteristics, whereas ours applies only to characteristic 2. A precise statement of the identity in [@ACZ] is given in the remark preceding Lemma \[distinctLemma\]. It would be interesting to understand more fully how the two polynomial identities are related.
Finally, we mention that in one case, our work fits nicely with results of Dummit [@Dummit] on solvable quintics. Namely, $x^{q+1}+x+1/a$ is a quintic when $q=4$. Dummit notes that a quintic (over any field) is solvable if and only if its Galois group is contained in a group that is conjugate to $F_{20}$, where $F_{20}\subset S_5$ is generated by the permutations $(1 2 3 4 5)$ and $(2354)$. It turns out that an invariant $\theta$ for $F_{20}$ is given by: $$\begin{aligned}
\theta &=& x_1^2 x_2 x_5 + x_1^2 x_3 x_4 + x_2^2 x_1 x_3 + x_2^2 x_4 x_5
+ x_3^2 x_1 x_5 \\
&& + x_3^2 x_2 x_4 + x_4^2 x_1 x_2 + x_4^2 x_3 x_5 + x_5^2 x_1 x_4 +
x_5^2 x_2 x_3,\end{aligned}$$ where $x_i$ are the roots of the quintic. Let $\g_1,\ldots,\g_6$ be coset representatives for $S_5/F_{20}$. Then $\prod_{i=1}^6 (x - \g_i(\theta))$ is a sextic. In the case of the polynomial $x^5+x+1/a$ in characteristic 2, the sextic is equal to $x^6+a^{-4}x+a^{-4}$. Setting $y=1/x$, we see that this is equivalent (has the same splitting field) to $y^6+y^5+a^4$. If we set $y=z^4$ and then take a fourth root, this reduces to our function $z^5(z+1) + a$. The substitution $z \mapsto z+1$ brings it to the form $C(z)+a$. The above expression for $\theta$ shows explicitly in this case how the roots of $C(x)+a$ are related to the roots of the quintic.
The relation between the roots causes a relation between the factorizations of the two polynomials. Write $f \sim [n_1,n_2,...,n_t]$ if $f$ factors into irreducibles of degrees $n_1, n_2, \ldots , n_t$. When $q=4$ and $F=\F_{2^k}$, we will prove the following in Section \[galoisSec\].
\[quinticSextic\] For $k$ even, $x^5+x+1/a$ has one of these factorization types: $[1,1,1,1,1]$, $[1,1,3]$, $[1,2,2]$, or $[5]$. We have $$\begin{aligned}
x^5 + x + 1/a \sim [1,1,1,1,1] &\iff& x(1+x)^5 + a \sim [1,1,1,1,1,1] \\
x^5 + x + 1/a \sim [1,1,3] &\iff& x(1+x)^5 + a \sim [3,3] \\
x^5 + x + 1/a \sim [1,2,2] &\iff& x(1+x)^5 + a \sim [1,1,2,2] \\
x^5 + x + 1/a \sim [5] &\iff& x(1+x)^5 + a \sim [1,5]. \end{aligned}$$ For $k$ odd, $x^5+x+1/a$ has factorization type $[1,1,1,2]$, $[1,4]$, or $[2,3]$, and we have $$\begin{aligned}
x^5 + x + 1/a \sim [1,1,1,2] &\iff& x(1+x)^5 + a \sim [2,2,2] \\
x^5 + x + 1/a \sim [1,4] &\iff& x(1+x)^5 + a \sim [1,1,4] \\
x^5 + x + 1/a \sim [2,3] &\iff& x(1+x)^5 + a \sim [6]. \end{aligned}$$
The author wishes to thank John Dillon and Mike Zieve for their encouragement and for stimulating discussions. As a newcomer to the study of Galois groups and exceptional polynomials, the author found their expertise to be invaluable.
[**Notation.**]{} If $R$ is a ring then $R^\times$ denotes the group of units in $R$, and $R[x]$ denotes the ring of polynomials in the indeterminate $x$ with coefficients in $R$. The separable algebraic closure of a field $F$ is denoted $\cj F$. For a nonzero polynomial $f\in F[x]$, $\deg(f)$ is its degree, and $f_{rev}(x) = x^{\deg(f)}f(1/x)$ is the reverse of $f$. The splitting field of $f$ over $F$ is written $\SF(f;F)$; this is the subfield of $\cj F$ that is generated by $F$ and by all the roots of $f$. The Galois group of $\SF(f;F)$ over $F$ is denoted by $\Gal(f;F)$; it is the group of automorphisms of the field $\SF(f;F)$ that fix the subfield $F$. If $\ell$ is a prime power, then $\F_\ell$ denotes the (unique) field with $\ell$ elements. For $k\ge 1$, $D_k(x)$ denotes the $k$-th Dickson polynomial, which is the unique monic polynomial of degree $k$ such that $D_k(x+1/x)=x^k + 1/x^k$.
Because the expression $x+1/x$ arises so frequently in this article, we introduce the special notation: $$\langle x \rangle = x + 1/x.$$ Then the defining property of the Dickson polynomial may be written as $$D_k(\langle x \rangle) = \langle x^k \rangle.$$ Note that $$\begin{aligned}
\ang {1/x} &=& \ang x \\ \nonumber
\label{angProperty1}
\langle x \rangle \langle y \rangle &=& \langle x y \rangle
+ \langle x/y \rangle \\
\label{angProperty2}
\ang x \ang{y/z} &=& \ang{xy}\ang z + \ang{xz} \ang y \qquad\text{in char.~2}
\\
\langle x^{p^i} \rangle &=& \langle x \rangle^{p^i}\qquad\text{ in char.~$p$.}\end{aligned}$$
We use the following notation that is specific to this article: $$q = 2^n, \qquad{\rm where\ }n > 1.$$ $$T(x) = \sum_{i=0}^{n-1} x^{2^i-1},\qquad
T_{rev}(x)=\sum_{i=0}^{n-1}x^{(q/2)-2^i}$$ $$C(x) = x\cdot T(x)^{q+1}\quad\text{(a M\"uller--Cohen--Matthews polynomial)}$$ $$F\ \text{is a field of char.~2 and $a$ is a fixed nonzero element of $F$}$$ $$\mu_{k} = \{\zeta\in\cj \F_2^\x : \zeta^{k}=1\}$$ $$K=\SF(C(x)+a;F)$$ $$L = \SF(x^{q+1}+ x + 1/a;F).$$ $$\F_{q,1} = \{\,d \in \F_q : \Tr_{\F_q/\F_2}(d) = 1 \},\qquad
\F_{q,0} = \{\,d \in \F_q : \Tr_{\F_q/\F_2}(d) = 0 \}.$$ If $f(x)$ is a polynomial and its irreducible factors have degrees $d_1,d_2,\ldots,d_k$ then we write $f\sim [d_1,d_2,\ldots,d_k]$.
In Section \[qplus1Sec\] only we allow $q=p^n$ and $F$ has char. $p$, where $p$ is any prime. We make frequent use of elements $\zeta\in\mu_{q+1}$. Since $q+1$ divides $q^2-1$, we know $\zeta\in \F_{q^2}$. Note that $\Norm_{\F_{q^2}/\F_q}(\zeta)=\zeta\zeta^q=1$, and $\Tr_{\F_{q^2}/\F_q}(\zeta)=\zeta+\zeta^q=\zeta+\zeta^{-1}=\ang\zeta$.
[**Remark.**]{} If one assumes that $F$ is perfect, then replacing all the coefficients of $f$ by their $2^i$-th powers does not affect the splitting field or Galois group. For example, the substitution $x \mapsto x/a$ transforms the polynomial $x^{q+1} + x + 1/a$ into $x^{q+1} + a^q x + a^q$, and this has the same automorphism group as the polynomial $x^{q+1} + a x + a$. In this article, we do not assume that $F$ is perfect, but we are still able to show in Proposition \[bpLemma\] that the polynomials $x^{q+1} + x + 1/a$ and $x^{q+1} + ax + a$ have the same splitting field. The latter polynomial turns out to be the most convenient for the purpose of proving that these polynomials have the same splitting field as $C(x)+a$.
The article is organized as follows. Section \[sec:Identity\] proves the Dickson polynomial identity. Section \[qplus1Sec\] concerns $x^{q+1}+ax+a$. It reviews results from [@qplus1] and proves a few additional results. For example, we show that $x^{q+1}+x+1/a$ has the same splitting field as $x^{q+1}+ax+a$, without having to assume that $F$ is perfect. Sections \[sec:RootsOfC\] and \[sec:Equality\] prove that $K = L$ by explicitly writing the roots of $C(x)+a$ as a rational function of the roots of $x^{q+1}+ax+a$, and explicitly writing the roots of $x^{q+1}+ax+a$ in terms of the roots of $C(x)+a$. Also the following polynomial identity is derived: $$\prod_{c\in\F_q^\x,\ j \in \F_{q,1}} (c y^2+y+j/c) = 1 + (y^q+y)^{q-1}.$$ Section \[galoisSec\] considers the Galois group and shows how the factorizations of $x^{q+1}+x+1/a$ and $C(x)+a$ are related. For example, we prove the related factorizations between $x^5 + x + 1/a$ and $x(x+1)^5 + a$ that were asserted in Proposition \[quinticSextic\]. Section \[dihedralSec\] investigates dihedral groups of order $2(q+1)$ and shows that such groups fix a root of $C(x)+a$ in the geometric case. This is used in Section \[sec:Exceptional\] to give a new proof that $C(x)$ is exceptional over $\F_2$ when $n$ is odd.
An identity of Dickson polynomials {#sec:Identity}
==================================
The $k$th Dickson polynomial is the monic polynomial with integer coefficients such that the formal identity holds, $D_k(u+1/u) = u^k + 1/u^k$. To see that such a polynomial exists, note that $u^k +v^k$ is a polynomial in $u+v$ and $uv$ by the Theorem of Symmetric Functions, say $u^k+v^k=F_k(u+v,uv)$. By setting $v=1/u$, we find that $D_k(x) = F_k(x,1)$. It is easy to see that $D_k(x)$ has degree $k$, and $D_k(-x)=(-1)^k D_k(x)$. Thus, if $k$ is even then $D_k(x)=E_k(x^2)$, and if $k$ is odd then $D_k(x) = x E_k(x^2)$, where $E_k$ has degree $\lfloor k/2 \rfloor$. A useful relation is $$\label{DkDl}
D_k(x) D_\ell(x) = D_{k+\ell}(x) + D_{k-\ell}(x),$$ as can be seen from the identity $\ang{u^k}\ang{u^\ell} =
\ang{u^{k+\ell}} + \ang{u^{k-\ell}}$.
Since $D_k$ has integral coefficients, it can be considered over any field $F$, in any characteristic. If the characteristic is $p$, then $$\label{Dkp}
D_{kp^r}(x) = D_k(x)^{p^r}.$$ The complete set of roots of $D_k(x) - c$ is easy to construct. Namely, we find $u$ so that $c=\ang{u^k}$; then $\ang{u}$ will be a root, and the other roots will be $\ang{\zeta u}$ for $\zeta\in\mu_k$. To find $u$, first solve the quadratic $v+1/v = c$, then solve $v=u^k$.
We will need some well-known formulas for $D_{q-1}$ and $D_{q+1}$ in characteristic 2, where $q=2^n$. For the reader’s convenience, we include their proof below.
\[Dicksonqm1\] If $q=p^n$, then in characteristic $p$, $$D_{q+1}(Y) = Y^{q+1} - D_{q-1}(Y).$$ If $q=2^n$, then in characteristic 2, $$D_{q-1}(Y) = \sum_{i=1}^n Y^{q-2^i+1}.$$
By (\[DkDl\]), $D_{q}(Y)D_1(Y)=D_{q+1}(Y)+D_{q-1}(Y)$. By (\[Dkp\]), $D_q(Y)D_1(Y)=Y^{q+1}$. Thus $D_{q+1}(Y)=Y^{q+1}-D_{q-1}(Y)$.
Now apply (\[DkDl\]) with $k=1$ and $\ell=q-1$. We find that $YD_{q-1}(Y) = Y^q + D_{q-2}(Y)$. If $p=2$, then this becomes $$Y^q = Y D_{q-1}(Y) + D_{(q/2)-1}(Y)^2.$$ Let $f_m = D_{2^m-1}(Y)$ and $g_m =\sum_{i=1}^m Y^{2^m-2^i+1}$. Then $f_1=g_1=Y$, $Yf_{m}=Y^{2^m} + f_{m-1}^2$ for $m\ge2$, and $Yg_{m}=Y^{2^m} + g_{m-1}^2$ for $m\ge2$. Thus $f_n=g_n$.
\[DicksonIdentity\] Let $q=2^n$. In the polynomial ring $\F_q[X,Y]$ we have the identity: $$\label{DicksonEq}
\prod_{w\in\F_q^\x} (D_{q+1}(wX)-Y)
= X^{q^2-1} + \left(\sum_{i=1}^{n} Y^{2^{n}-2^i}\right) X^{q-1} + Y^{q-1}.$$
Let $U$ be transcendental over $\F_2$ and $$Y=\ang{U^{q+1}}$$ (where we recall $\ang{u}$ is shorthand for $u+1/u$). Then $Y$ is also transcendental, and $\F_2(Y)\subset\F_2(U)$. Let $L(X)$ and $R(X)$ denote the left-hand and right-hand sides of (\[DicksonEq\]) respectively, considered as elements of $\cj\F_2(U)[X]$. Note that $L$ and $R$ are both monic polynomials in $X$ of degree $q^2-1$, and so $\deg_X(L-R) < q^2-1$. Thus, to prove $L-R$ is identically zero it suffices to find $q^2-1$ distinct roots in $\cj\F_2(U)$. We claim that these roots are $$\{\,\ang{\zeta U}/w : \zeta\in\mu_{q+1} {\rm\ and\ } w\in\F_q^\x\,\}.
\label{identityRoots}$$ (The proof that these are distinct will be shown in Lemma \[distinctLemma\] below.) In fact we will show that $L$ and $R$ each vanish at these values. Let $x$ denote one of these values: $$x=\ang{\zeta U}/w.$$ First, $L(x)=0$ because $$D_{q+1}(wx)-Y= D_{q+1}(\ang{\zeta U}) -Y
= \ang{(\zeta U)^{q+1}} - \ang{U^{q+1}} = 0.$$ Next we show $R(x)=0$. Set $V=\zeta U$; then $$x = \ang{V}/w \qquad{\rm and } \qquad Y = \ang{V^{q+1}}.
\label{wxY}$$ Note that $\ang{V}Y=\ang{V}\ang{V^{q+1}}$ is nonzero, since $V$ is transcendental. Thus, it will suffice to show that $\ang V Y\,R(x)=0$. Noting that $w^{q-1}=1$ and invoking Lemma \[Dicksonqm1\], we have $$\begin{aligned}
\ang V Y\,R(x) &=&
Y{\ang V}^{q^2} + \left(\sum_{i=1}^n Y^{2^n-2^i+1}\right) {\ang V}^q
+ \ang V Y^q \\
&=& \ang{V^{q+1}} \ang{V^{q^2}} + D_{q-1}(Y) \ang{V^q} + \ang V \ang{V^{q(q+1)}}.\end{aligned}$$ Now $D_{q-1}(Y)=D_{q-1}(\ang{V^{q+1}}) = \ang{(V^{q+1})^{q-1} } =
\ang{V^{q^2-1}}.$ Using this observation and (\[DkDl\]), we obtain $$\begin{aligned}
\ang V Y\,R(x)
&=& \ang{V^{q+1}} \ang{V^{q^2}} + \ang{V^{q^2-1}} \ang{V^q} + \ang V \ang{V^{q(q+1)}} \\
&=& \ang{V^{q^2+q+1}} + \ang{V^{q^2-q-1}} + \ang{V^{q^2+q-1}} + \ang{V^{q^2-q-1}} + \ang{V^{q^2+q+1}} + \ang{V^{q^2+q-1}} \\
&=& 0.\end{aligned}$$ Thus $R(x)=0$, as claimed.
[**Remark.**]{} Our identity (\[DicksonEq\]) is tantalizingly similar to an identity in Theorem (1.1) from the article by Abhyankar, Cohen, and Zieve [@ACZ]. Their identity is $$X^{q^2-1} - E_q(Y,1) X^{q-1} + E_{q-1}(Y,1)
=(X^{2q-2} - Y X^{q-1} + 1)\left(
\prod_{w\in\F_q^\x} (D_{q-1}(X,w) - Y) \right),$$ where $q=p^n$, $D_n(X,a)$ is defined by $D_n(U_1+U_2,U_1U_2)=U_1^n+U_2^n$, and $E_i(Y,a)$ is defined by $E_i(U_1+U_2,U_1U_2) = (U_1^{n+1}-U_2^{n+1})/(U_1-U_2)$. Using the relations (2.20) and (2.9) of [@ACZ], this identity can be rewritten when $p=2$ as: $$X^{q^2-1} + (D_{q+1}(Y)/Y) X^{q-1} + Y^{q-1}
=(X^{2q-2} - Y X^{q-1} + 1)\left(
\prod_{w\in\F_q^\x} (D_{q-1}(wX) - Y) \right),$$ and using Lemma 2.1, our identity can be rewritten as $$X^{q^2-1} + (D_{q-1}(Y)/Y) X^{q-1} + Y^{q-1}
= \prod_{w\in\F_q^\x} \left(D_{q+1}(wX) - Y \right).$$ In both this article and [@ACZ], the identity is used to compute a certain Galois group, which in this article turns out to be $\PSL_2(q)$ and in [@ACZ] turns out to be an orthogonal group. Bob Guralnick points out that the Dickson polynomials are ramified at the prime 2, thus it is not surprising to find formulas that are special to characteristic 2.
The following result was needed in the proof of Theorem \[DicksonIdentity\].
\[distinctLemma\] Let $M$ be a field of characteristic 2 that strictly contains $\F_{q^2}$, and let $u\in M \setminus \F_{q^2}$. Then the values $$\{\,\ang{\zeta u}/w : \zeta\in\mu_{q+1} {\rm\ and\ } w\in\F_q^\x\,\}$$ are distinct.
Let $\zeta,\lambda \in \mu_{q+1}$ and $w,w' \in \F_q$, and suppose that $\ang{\zeta u}/w = \ang{\lambda u}/w'$. Then $w/w'=\ang{\zeta u}/\ang{\lambda u}$, and so $$\frac{\zeta w}{\lambda w'} = \frac{\zeta \ang{\zeta u}}{\lambda \ang{\lambda u}} = \frac{\zeta^2 u^2 + 1}{\lambda^2 u^2 + 1}.$$ If $\zeta \ne \lambda$, then we can solve for $u^2$ in terms of $\zeta,\lambda,w,w'$, but this contradicts the hypothesis that $u \not \in \F_{q^2}$. Thus, $\zeta = \lambda$, and consequently $w=w'$ also. We have shown that $\ang{\zeta u}/w=\ang{\lambda u}/w'$ implies $\zeta=\zeta'$ and $w=w'$, so the roots are distinct, as claimed.
For future use, we record the following lemma.
\[DicksonIdLemma\] Let $y$ be a nonzero element of a field $M$ of characteristic 2, and let $$f(x) = \prod_{w\in\F_q^\x}(D_{q+1}(wx)-y).$$ Let $u\in \cj M$ satisfy $u^{q+1} + 1/u^{q+1} = y$. The complete set of roots of $f$ is $$\{\,w \ang{\zeta u} : w\in\F_q^\x, \zeta\in\mu_{q+1}\,\}$$ and these roots are distinct.
By (\[DicksonEq\]), $f = x^{q^2-1} + T_{rev}(y^2) x^{q-1} + y^{q-1}$. The roots of $f$ are distinct, because $f-xf'=y^{q-1}\ne 0$ shows that $\GCD(f,f')=1$. Also the roots are nonzero, since the constant term of $f$ is $y^{q-1}$. Now $D_{q+1}(wx)-y$ vanishes at $w^{-1}\ang{\zeta u}$, for all $\zeta\in\mu_{q+1}$. We claim the values $\ang{\zeta u}$ are distinct. If not, then $\ang{\zeta u} = \ang{\zeta' u}$ for distinct $\zeta,\zeta'\in \mu_{q+1}$. One finds that $u^2=(1/\zeta' + 1/\zeta)/(\zeta
+\zeta') = 1/(\zeta \zeta')$. Then $y^2=\ang{u^{2(q+1)}}=\ang1=0$, contrary to the hypothesis that $y$ is nonzero. This establishes that the $q+1$ roots of $D_{q+1}(wx)-y$ given by $\{\ang{\zeta u}/w:\zeta\in\mu_{q+1}\}$ are distinct. Now the roots of $D_{q+1}(wx)-y$ must be disjoint from the roots of $D_{q+1}(w'x)-y$ when $w\ne w'$, since we already observed that $f$ has no repeated roots. Thus, the $q^2-1$ roots given in the statement of the lemma are distinct, and since $\deg(f)=q^2-1$, we have found all the roots.
Now we show how the Dickson polynomial leads to a relation between the polynomials $C(x)+a$ and $x^{q+1} + a x + a$, where we recall that $C$ is defined by $$C(x) = x\cdot T(x)^{q+1},\qquad T(x)=\sum_{i=0}^{n-1} x^{2^i-1}.$$
$C(x)+a$ has distinct roots over $\cj F$, all nonzero.
A polynomial has distinct roots over the algebraic closure if and only if it is relatively prime to its derivative. Since $C=xT^{q+1}$ and $xT'=T+1$, we see that $C'=T^q(xT'+T)=T^q$, which divides $C$. Setting $G(x) = C(x)+a$, we have $G' = C' = C/(xT)$, and so $G - x T G' = (C + a) - C = a$. This proves that $\GCD(G,G')=1$, and so $C(x)+a$ has no repeated roots. Since $C(0)+a=a\ne0$, the roots are nonzero.
For the remainder of this section, let $e$ be an arbitrary root of $C(x)+a$: $$C(e) = a.$$
\[eRelation\] Let $u\in\cj F$ satisfy $$u^{q+1} + 1/u^{q+1} = 1/e.$$ Then the complete set of roots of $x^{q+1} + a^2 x + a^2$ is $$\left\{\,e T(e)^2 \ang{\zeta u }^{q-1}: \zeta\in\mu_{q+1}\,\right\}.$$
Substitute $Y=1/e$ into the identity (\[DicksonEq\]) and leave $X$ as an indeterminate. We find: $$\prod_{w\in\F_q^\times} (D_{q+1}(wX)-1/e)
= X^{q^2-1} + \alpha X^{q-1} + \beta,$$ where $$\alpha = \sum_{i=1}^n e^{2^i-q} = (e T(e))^2/e^q = e^{2-q} T(e)^2,\qquad \beta = e^{1-q}.$$ Let $$R = (\alpha/\beta) X^{q-1} = e T(e)^2 X^{q-1}.$$ Then, the right side of the identity can be written as $(\beta/\alpha)^{q+1}(R^{q+1} + \alpha^{q+1}/\beta^q R + \alpha^{q+1}/\beta^q)$. Now $\alpha^{q+1}/\beta^q = e^2 T(e)^{2(q+1)} = C(e)^2 = a^2$, and so we have the identity: $$\prod_{w\in\F_q^\times} (D_{q+1}(wX)-1/e) = (\beta/\alpha)^{q+1}
(R^{q+1}+a^2 R + a^2).$$ By Lemma \[DicksonIdLemma\], the roots of the left side are $w\ang{\zeta u}$ for $\zeta\in\mu_{q+1}$ and $w\in\F_q^\x$, and these are distinct. Denote the set of these roots by $S$; we have $|S|=q^2-1$, and also $s\in S$ implies $ws\in S$ for all $w\in\F_q^\x$. If $s_1,s_2\in S$ then $s_1^{q-1}=s_2^{q-1}$ if and only if $s_1/s_2\in\F_q^\x$. Thus, each power $s^{q-1}$ has exactly $q-1$ preimages in $S$, namely $\{ws : w \in\F_q^\x\}$. It follows that there are exactly $q+1$ distinct values $\{s^{q-1}:s\in S\}=\{\ang{\zeta u}^{q-1}:
\zeta\in\mu_{q+1}\}$. This shows that the values $\ang{\zeta u}^{q-1}$ are distinct. Since $R=e T(e)^2 X^{q-1}$, it follows that $e T(e)^2 \ang{\zeta u}^{q-1}$ are roots of $R^{q+1}+a^2 R + a^2$, and since they are distinct, all $q+1$ roots are accounted for.
Splitting field of $x^{q+1} - b x + b$. {#qplus1Sec}
========================================
For this section only, we will consider both even and odd characteristic. Let $p$ be a prime and $q=p^n$. The polynomial $f(x) = x^{q+1}-bx+b$ in characteristic $p$ (where $b\ne 0$) was studied in [@qplus1]. More generally, one could begin with $x^{q+1} + A x^q + B x + C$ (where $(AB- C)(B-A^q)\ne 0$), and the substitution $x=(AB-C)(B-A^q)^{-1}x_1-A$ brings us to the “standard” form $x_1^{q+1} - b x_1 + b$, where $b\ne 0$. The polynomial $x^{q+1}+a^2x+a^2$ that arises in Proposition \[eRelation\] is just the special case $p=2$, $b=a^2$. The article [@qplus1] gives explicit formulas for the splitting field and for the Galois action, which we recall in Theorems \[qplus1Thm\] and \[qplus1Thm2\] below. Theorem \[qplus1Thm\] is illustrated in Figure \[splitting\_diagram\].
\[qplus1Thm\] ([@qplus1]) Let $q=p^n$ and let $b$ be a nonzero of a field $F$ in char. $p$. Let $r$, $r_0$, $r_1$ be distinct roots of $x^{q+1}-bx+b$, and define $$\label{zyxi}
z=r_0/r,\qquad
y=(r_1-r)/(r_1-r_0),\qquad \xi=y^q-y.$$ Then $$y^{q-1}=z,\qquad \xi=y(z-1),\quad {\rm and}\quad
z(z-1)^{q-1}=1/(r-1)=\xi^{q-1}.$$ We have $y^{q^2}-y=\xi^q+\xi\ne 0$.
Although the proof can be found in [@qplus1], we include it here because it is so short. First, $rr_0(r-r_0)^q=r_0r^{q+1}-rr_0^{q+1}=r_0b(r-1)-rb(r_0-1)=b(r-r_0)$, so $$b=rr_0(r-r_0)^{q-1}.$$ Dividing through by $r^{q+1}$ and using $r^{q+1}=b(r-1)$, we find $1/(r-1)=z(z-1)^{q-1}$. Next, from $b=r_1r_0(r_1-r_0)^{q-1}=r_1r(r_1-r)^{q-1}$ we find $1=(r_1r_0/r_1r)((r_1-r_0)/(r_1-r))^{q-1}=zy^{1-q}$, so $z=y^{q-1}$. We have $\xi=y^q-y=y(z-1)$, and so $\xi^{q-1}=y^{q-1}(z-1)^{q-1}=
z(z-1)^{q-1}=1/(r-1)$. Note that $\xi\ne0$ and $\xi^{q-1}\ne-1$, since $\xi^{q-1}=1/(r-1)$. Thus $\xi^q+\xi\ne0$.
\[splitting\_diagram\] $$\xymatrix{
& F(y)=F(r,r_0,r_1) \ar@{-}[dl]^{q-1} \ar@{-}[dr]_q &\\
F(z)=F(r,r_0) \ar@{-}[dr]^{q}& &
F(\xi) \ar@{-}[dl]_{q-1} \\
&F(r) \ar@{-}[d]^{q+1} & \\
&F&}$$
\[qplus1Thm2\] Let $f(x)=x^{q+1}-bx+b$. The complete set of roots of $f$ is $\{r_w : w \in \P^1(q)\}$, where $r_\infty=r$ and $r_w=r(y-w)^{q-1}$ for $w\in\F_{q}$. The roots are distinct. The splitting field over $\F_p(b)$ is $\F_{q}(y)$. If $\sigma\in\Gal(f/\F_p(b))$ then there is a unique $\gamma\in \PGL_2(q)$ such that $\sigma(y)=\gamma^{-1}(y)$. We have $\sigma(r_w)=r_{\gamma(\sigma w)}$, where $\gamma$ has the usual action by linear fractional transformations on $\P^1(q)$. For any $\gamma \in \PGL_2(\F_q)$ we have $$\gamma^{-1} y = \frac {r_{\gamma(1)} - r_{\gamma(\infty)}}{ r_{\gamma(1)} - r_{\gamma(0)}}. \label{gammay}$$
All the above results were proved in [@qplus1] except for (\[gammay\]), which we will prove here. First, we show it in a few special cases. As above, let $y=(r_1-r)/(r_1-r_0)$.
- If $\gamma= \textmatrix 0 1 1 0$ then $$\frac{r_{\gamma(1)} - r_{\gamma(\infty)} } { r_{\gamma(1)} - r_{\gamma(0)}} =\frac{r_1-r_0}{r_1-r_{\infty}} = 1/y = \gamma^{-1} y.$$
- If $\gamma = \textmatrix w 0 0 1$ with $w \in \F_q^\x$ then $$\begin{aligned}
\frac{r_{\gamma(1)} - r_{\gamma(\infty)} } { r_{\gamma(1)} - r_{\gamma(0)}} &=& \frac{r_w-r_\infty}{r_{w}-r_{0}} \\
&=& \frac{ r(y-w)^{q-1} - r}{ r(y-w)^{q-1} - ry^{q-1} } \times \frac{y-w}{y-w} \\
&=& \frac{ (y-w)^q-(y-w)} { (y-w)^q - y^{q-1}(y-w)} \\
&=& \frac{y^q-y}{-w + w y^{q-1} } = y/w = \gamma^{-1} y.\end{aligned}$$
- If $\gamma = \textmatrix 1 w 0 1$ with $w \in \F_q$, then $$\begin{aligned}
\frac{r_{\gamma(1)} - r_{\gamma(\infty)} } { r_{\gamma(1)} - r_{\gamma(0)}} &=& \frac{r_{1+w}-r_\infty}{r_{1+w}-r_{w}} \\
&=& \frac{ r(y-w-1)^{q-1} - r } {r(y-w-1)^{q-1}- r(y-w)^{q-1} } \times \frac{y-w-1}{y-w-1} \\
&=& \frac{ (y-w-1)^q - (y-w-1) } { (y-w-1)^q - (y-w)^{q-1}(y-w-1) } \\
&=& \frac{y^q-y} { (y - w)^q - 1 - (y-w)^q + (y-w)^{q-1}} \times \frac{y-w}{y-w}\\
&=& \frac{(y^q-y)(y-w)} {-(y-w) + (y-w)^q} \\
&=& \frac{(y^q-y)(y-w)} {y^q-y} = y-w = \gamma^{-1} y.\end{aligned}$$
Since the above three matrices generate $\PGL_2(\F_q)$, to complete the proof we need only show that if (\[gammay\]) is true for $\gamma$ and $\delta$ then it is true for $\gamma \delta$. Define $s_w = r_{\gamma w}$. Since (\[gammay\]) is true for $\gamma$, we have $$\gamma^{-1} y = \frac{s_1-s_\infty}{s_1-s_0}.$$ Since (\[gammay\]) is true for $\delta$, we have $$\delta^{-1} (\gamma^{-1} y) = \frac{s_{\delta 1}-s_{\delta \infty}} {s_{\delta 1} - s_{\delta 0} }.$$ The left side is $(\gamma\delta)^{-1} y$. Since $s_w=r_{\gamma w}$ for all $w\in \P^1(\F_q)$, the right side is $$\frac{r_{\gamma \delta 1} - r_{\gamma \delta \infty}} {r_{\gamma \delta 1} - r_{\gamma \delta 0}}.$$ This shows that (\[gammay\]) is true for $\gamma\delta$ and completes the proof.
[**Remark. **]{} If $w\in\F_q$, then $w$ can explicitly be expressed in terms of the roots of $x^{q+1}-bx+b$ as follows. Let $\gamma = \textmatrix 1{ w} 01$. Then $$\begin{aligned}
w &=& y - (y-w) = y - \gamma^{-1} y \\
&=& \frac{r_1-r_\infty}{r_1-r_0} - \frac{r_{\gamma 1}-r_{\gamma\infty}}{r_{\gamma 1}-r_{\gamma 0}} \\
&=&\frac{r_1-r_\infty}{r_1-r_0} - \frac{r_{w+1}-r}{r_{w+1}-r_w}.\end{aligned}$$
\[bpLemma\] Let $p$ be any prime (even or odd) and $q=p^n$, let $F$ be a field of characteristic $p$ (not necessarily perfect), and $0\ne b \in F$. Then $x^{q+1}-bx+b$ and $x^{q+1} - x + 1/b$ have the same splitting field over $F$. Also, the polynomials $x^{q+1} - b^{p^i} x + b^{p^i}$ have the same splitting field over $F$ for all $i\ge 0$.
We begin by proving that $x^{q+1} -b x + b$ and $x^{q+1} - b^p x + b^p$ have the same splitting field over $F$. Denote these splitting fields by $L$ and $L_1$, respectively. Let $r,r_0,r_1$ be distinct roots of $x^{q+1}-bx+b$, and let $y=(r_1-r)/(r_1-r_0)$. Then $r^p$, $r_0^p$, $r_1^p$ are distinct roots of $x^{q+1} - b^p x + b^p$, and $y^p = (r_1^p-r^p)/(r_1^p-r_0^p)$. By Theorem \[qplus1Thm\], $$L = F \circ \F_q(y), \qquad L_1 = F \circ \F_q(y^p).$$ To prove equality of these fields, it will suffice to show that $$F_q(b,y) = \F_q(b,y^p),$$ or equivalently, that $y \in \F_q(b,y^p)$. First we express $b$ in terms of $y$, using formulas from Theorem \[qplus1Thm\]: $$\begin{aligned}
b &=& r^{q+1}/(r-1) \qquad \text{ (since $r^{q+1} - b r + b = 0$)} \\
&=& (1+\xi^{1-q})^{q+1} \xi^{q-1} \qquad \text{ (since $\xi^{q-1}=1/(r-1)$ )} \\
&=& (1+\xi^{1-q})^q (\xi^{q-1} + 1),\end{aligned}$$ where $\xi = y^q-y$. Noting that $\xi^q \in \F_q(b,y^p)$, we see that $$\xi^{q-1}+1 = b \left(1+\xi^{q(1-q)}\right)^{-1} \in\F_q(b,y^p).$$ Subtracting one from both sides, taking the reciprocal, and then multiplying by $\xi^q$ shows that $\xi \in \F_q(b,y^p)$. Finally, since $\xi = y^q-y$, we conclude that $y = y^q-\xi \in \F_q(b,y^p)$ as required.
We showed that $x^{q+1} - b^p x + b^p$ has the same splitting field over $F$ as $x^{q+1} - b x + b$. Repeating the argument with $b^p$ in place of $b$, we see that $x^{q+1}-b^{p^2} x + b^{p^2}$ has the same splitting field over $F$ as $x^{q+1} - b^p x + b^p$. By induction on $i$, all fields $x^{q+1}-b^{p^i} x + b^{p^i}$ have the same splitting field over $F$.
It remains to prove that $x^{q+1} - x + 1/b$ has the same splitting field as well. If $r$ is a root of $x^{q+1}-x+1/b$, then $br$ is a root of $x^{q+1}-b^q x + b^q$, because $$(br)^{q+1} - b^q (br) + b^q = b^{q+1} (r^{q+1} - r + 1/b) = 0.$$ This shows $\SF(x^{q+1}-x+1/b;F) = \SF(x^{q+1}-b^q x + b^q;F)$.
Expressing roots of $C(x)+a$ in terms of $\SF(x^{q+1}+ax+a;F)$ {#sec:RootsOfC}
==============================================================
Now we apply the theory from Section \[qplus1Sec\] to derive formulas expressing the roots of $C(x)+a$ in terms of the roots of $x^{q+1}+ax+a$, where $0\ne a \in F$. For the remainder of this article, we are working in characteristic 2; in particular $q=2^n>2$.
Let $e,u\in\cj F$ satisfy $$C(e)=a,\qquad 1/e = \ang{u^{q+1}}.$$ Proposition \[eRelation\] showed that the roots of $x^{q+1} + a^2 x + a^2$ are $$\text{$\{\,\lambda \ang{\zeta u}^{q-1} : \zeta \in \mu_{q+1}\}$, \quad where
$\lambda = e\, T(e)^2$.}$$
Let $r$, $r_0$, $r_1$ be any three distinct roots of $x^{q+1} + a x + a$. Then $r^2$, $r_0^2$, and $r_1^2$ are distinct roots of $x^{q+1} + a^2 x + a^2$. After rescaling $u$ by an element of $\mu_{q+1}$, we can arrange that $r^2 = \lambda \ang u^{q-1}$, while still keeping the condition $1/e=\ang{u^{q+1}}$. Next, there are $\zeta,\rho \in \mu_{q+1} \setminus \{1\}$ such that $$r^2 = \lambda \ang{u}^{q-1} \qquad
r_0^2 = \lambda \ang{\zeta^2 u}^{q-1} \qquad
r_1^2 = \lambda \ang{\rho^2 u }^{q-1}.$$ Let $$y = (r_1-r)/(r_1-r_0).$$ By Theorem \[qplus1Thm\], the splitting field of $x^{q+1}+ax+a$ is $L=F \circ \F_q[y]$.
\[sevenFormulas\] Let $y$, $e$, $\zeta$, $\rho$ be as above, and let $$c = \frac{\ang{\zeta/\rho}}{\ang\zeta\ang\rho}\qquad{\rm and}\qquad d = \frac 1 {\ang\zeta}. \label{cDef}$$ The following formulas hold. $$y^2 = \frac{\ang{\rho^2} \ang{\zeta^2 u} } {\ang{\zeta^2/\rho^2} \ang u } \label{formula1}$$ $$y = \frac { \ang{\rho} (\zeta u + 1/\zeta) } {\ang{\zeta/\rho} (u + 1) } \label{formula2}$$ $$u = \frac {\ang{\zeta/\rho} y + \ang{\rho}/\zeta } {\ang{\zeta/\rho} y + \ang{\rho}\zeta } \label{formula3}$$ $$\ang{u} = \frac {1}{(cy)^2 + c y + d^2} \label{formula4}$$ $$1/e = D_{q+1}\left(\frac{1}{(cy)^2+cy+d^2}\right) \label{formula5}$$ $$(y^q+y)^2 = \frac { \ang {u^{q+1}} } {c^2{\ang u}^{q+1} }
\label{formula6}$$ $$e = \left(cy^2+y+\frac {d^2}{c} \right)^{q+1} \cdot (y^q+y)^{-2}. \label{formula7}$$
$$\begin{aligned}
y^2 &=& \frac{r_1^2-r^2}{r_1^2-r_0^2} \\
&=& \frac{\ang{\rho^2 u}^{q-1} - \ang{u}^{q-1}}{\ang{\rho^2 u}^{q-1} - \ang{\zeta^2 u}^{q-1}} \times
\frac{ \ang{u} \ang{\rho^2 u} \ang{\zeta^2 u} }{ \ang{u} \ang{\rho^2 u} \ang{\zeta^2 u}} \\
&=& \frac{ \ang{\rho^2 u}^q \ang{u} + \ang{u}^q \ang{\rho^2 u} } {\ang{\rho^2 u}^q \ang{\zeta^2 u} + \ang{\zeta^2 u}^q \ang{\rho^2 u} }
\times \frac {\ang{\zeta^2 u}} {\ang{u}} \\
&=& \frac{ \ang{\rho^{-2} u^q} \ang{u} + \ang{u^q} \ang{\rho^2 u} } {\ang{\rho^{-2} u^q} \ang{\zeta^2 u} + \ang{\zeta^{-2} u^q} \ang{\rho^2 u} }
\times \frac {\ang{\zeta^2 u}} {\ang{u}}. \\\end{aligned}$$
In the first fraction, the numerator and denominator can be rewritten as follows: $$\begin{aligned}
\ang{\rho^{-2} u^q} \ang{u} + \ang{u^q} \ang{\rho^2 u} &=& (\rho^{-2} u^q + \rho^2 u^{-q}) (u+1/u) + (u^q + u^{-q})(\rho^2 u + \rho^{-2}u^{-1}) \\
&=& \ang{\rho^{2}} \ang{ u^{q+1}}; \\
\ang{\rho^{-2} u^q} \ang{\zeta^2 u} + \ang{\zeta^{-2} u^q} \ang{\rho^2 u} &=& \ang{\zeta^2/\rho^2} \ang{u^{q+1}}.\end{aligned}$$ After canceling $\ang{u^{q+1}}$, we obtain the formula (\[formula1\]). To obtain (\[formula2\]), multiply the right side of (\[formula1\]) by $u/u$ and then take the square root. Now (\[formula2\]) shows that $u$ and $y$ are related by a linear fractional transformation over $\F_{q^2}$. Solving for $u$ in terms of $y$ gives (\[formula3\]). To derive (\[formula4\]), let $A=\ang{\zeta/\rho} y + \ang{\rho}/\zeta$, $B=\ang{\zeta/\rho}y+\ang{\rho}\zeta$, so $u=A/B$. Then $\ang u = A/B+B/A=(A^2+B^2)/(AB)$. It is easy to compute that $A^2+B^2 = \ang{\rho^2}\ang{\zeta^2}$ and $AB=\ang{\zeta/\rho}^2y^2+\ang{\zeta/\rho}\ang\zeta \ang\rho y + \ang{\rho}^2$, and formula (\[formula4\]) follows. We have $1/e=\ang{u^{q+1}}=D_{q+1}(\ang{u})$, which proves (\[formula5\]). For (\[formula6\]), we have $$y = w \frac{\zeta u + 1/\zeta}{u+1},\qquad y^q = w \frac{\zeta^{-1}u^q+\zeta}{u^q+1},$$ where $w=\ang{\rho}/\ang{\zeta/\rho}\in\F_q$. Thus, $$\begin{aligned}
y^q+y&=&w \cdot \frac{ (\zeta u + 1/\zeta)(u^q+1) + (\zeta^{-1} u^q + \zeta)(u+1)} {(u^q+1)(u+1) } \\
&=& \frac{w \ang{\zeta}(u^{q+1}+1)}{(u+1)^{q+1}} =
\frac{(u^{q+1}+1)}{c(u+1)^{q+1}}, \end{aligned}$$ where $c$ is defined in (\[cDef\]). Now square both sides and multiply on the right by $u^{-(q+1)}/u^{-(q+1)}$ to obtain (\[formula6\]). Finally, (\[formula7\]) is obtained by substituting $1/e=\ang{u^{q+1}}$ and $1/\ang u = c^2 y^2 + c y + d^2$ into (\[formula6\]).
On account of Lemma \[sevenFormulas\], Figure \[splitting\_diagram\] can be extended to incorporate other subfields of the splitting field, as shown in Figure \[splitting\_diagram2\].
\[splitting\_diagram2\] $$\xymatrix{
&& F(y)=F(r,r_0,r_1) \ar@{-}[ddll]^{q-1} \ar@{-}[d]_2 \ar@{-}[ddr]^{2(q+1)}&\\
&& F(\ang{u}) \ar@{-}[d]_{q/2} \ar@{-}[dr]_{q+1} & \\
F(z)=F(r,r_0) \ar@{-}[dr]^{q} & &
F(\xi) \ar@{-}[dl]_{q-1} &
F(e) \ar@{-}[ddl]^{(q/2)(q-1)} \\
&F(r) \ar@{-}[dr]^{q+1} && \\
&&F=\F_q(a)&}$$
We will need the following lemma that distinguishes the elements $1/\ang\zeta$ for $\zeta \in \mu_{q+1}\setminus \{1\}$.
\[Fq1Lemma\] Let $\F_{q,1}$ denote the elements of $\F_q$ having absolute trace 1. Then $$\{1/\ang\zeta : \zeta \in \mu_{q+1}, \zeta\ne 1 \} = \F_{q,1}.$$
If $a \in \F_q^\x$, then $x^2+ax+1$ has a root $r\in\F_q^\x$ if and only if $1/a^2=(r/a)^2 + (r/a)$. Thus, $x^2+ax+1$ is reducible if and only if $\Tr_{\F_q/\F_2}(1/a) = 0$. Since $\zeta\not\in\F_q$, its minimal polynomial $x^2 + \ang\zeta x + 1$ is irreducible, and therefore $\Tr_{\F_q/\F_2}(1/\ang\zeta)=1$, [*i.e.*]{} $1/\ang{\zeta}\in \F_{q,1}$. There are exactly $q/2$ elements of $\F_{q,1}$ and exactly $q/2$ elements $1/\ang{\zeta}$, so the two sets coincide.
\[KsubsetL\] Let $y=(r_1-r)/(r_1-r_0)$, where $r,r_0,r_1$ are three distinct roots of $x^{q+1}+ax+a$. Then the distinct roots of $C(x)+a$ are $$\begin{aligned}
{\cal E} &=& \left\{\, D_{q+1}\left( \frac{1}{c^2y^2+cy+j} \right)^{-1} : c \in\F_q^\x,\ j \in \F_{q,1} \,\right\} \\
&=& \left\{\, \frac{ ( cy^2 + y + j/c)^{q+1} }{(y^q + y )^2} : c \in \F_q^\x,\ j \in \F_{q,1} \, \right\}.\end{aligned}$$ If $X$ is an indeterminate then $$\prod_{c \in \F_q^\x,\ j \in \F_{q,1}} \left(X - \frac{ (c y^2 + y + j/c)^{q+1} }{(y^q+y)^2}\right) = C(X) + a.\label{CXaIdentity}$$
By Lemma \[sevenFormulas\], if $C(e)=a$ then we can write $1/e=D_{q+1}(1/((cy)^2+cy+j))$, where $c=\ang{\zeta/\rho}/(\ang\zeta \ang\rho)$ and $j=1/\ang{\zeta^2}$, and $1,\zeta,\rho$ are distinct elements of $\mu_{q+1}$. By Lemma \[Fq1Lemma\], $j\in\F_{q,1}$. There are $(q-1)$ choices for $c$ and $q/2$ choices for $j$, giving a total of $(q/2)(q-1)$ pairs. This is exactly the degree of $C$. Since $C$ has distinct roots, each pair $(c,j)$ must occur. The last sentence follows from formula (\[formula7\]), combined with $$C(X)+a = \prod_{e \in {\cal E}} (X-e).$$
The following identity holds for all $y$: $$\prod_{c\in\F_q^\x,\ j \in \F_{q,1} } (c y^2 + y + j/c) = 1 + (y^q+y)^{q-1}.$$
We express $a$ in two ways. First, if we substitute $X=0$ into (\[CXaIdentity\]), we obtain $$\prod_{c \in \F_q^\x,\ j \in \F_{q,1}} \frac{ (c y^2 + y + j/c)^{q+1} }{(y^q+y)^2}= a.\label{aIdentity}$$ Second, by Theorem \[qplus1Thm\] $\xi = y^q-y$, $\xi^{q-1}=1/(r+1)$, and $r^{q+1}+a r + a = 0$. Hence, $$a=\frac{r^{q+1}}{r+1} = r^{q+1}\xi^{q-1} = (1+\xi^{1-q})^{q+1}\xi^{q-1},
\quad{\rm where}\quad \xi = y^q-y.$$ Comparing the two expressions, we find that $$\frac{\prod_{c\in\F_q^\x,\ j\in\F_{q,1}} (c y^2 + y + j/c)^{q+1}} { \xi^{q(q-1)} } = r^{q+1} \xi^{q-1}.$$ If $a$ (and hence also $y$) is transcendental, then this may be interpreted as an identity in the ring $\F_q(y)$. On multiplying through by $\xi^{q(q-1)}$ and then taking the unique $(q+1)$th root belonging to $\F_q(y)$, we obtain: $$\prod_{c\in\F_q^\x,\ j \in \F_{q,1}} (cy^2 + y + j/c) = \xi^{q-1} r = \xi^{q-1}(1+\xi^{1-q}) =1+\xi^{q-1} = 1+(y^q+y)^{q-1}.$$
Equality of splitting fields {#sec:Equality}
============================
In this section, we prove one of our main results, that $x^{q+1}+x-1/a$ and $C(x)+a$ have the same splitting field. This will be accomplished by explicitly writing the roots of each polynomial in terms of the roots of the other.
From here on, let $${\cal Y} = \left\{\frac{r_1-r}{r_1-r_0} : r,r_0,r_1\text{\ are distinct roots of $x^{q+1}+ax+x$} \right\}.$$ Note that if $y \in {\cal Y}$ and $\gamma \in \PGL_2(\F_q)$, then $\gamma^{-1}(y)\in {\cal Y}$ by (\[gammay\]). For $y\in {\cal Y}$, $c\in \F_q^\x$ and $j\in \F_{q,1}$, define $$e(y,c,j) =
\frac{ ( cy^2 + y + j/c)^{q+1} }{(y^q - y )^2}. \label{eycdDef}$$ By Theorem \[KsubsetL\], for a fixed $y$, the values $\{ e(y,c,j) : c \in \F_q^\x,\ j \in \F_{q,1}\}$ are the distinct roots of $C(x)+a$.
\[KeqL\] For $q=2^n \ge 4$, we have $\SF(C(x) + a; F) = \SF(x^{q+1} + x + 1/a;F)= \SF(x^{q+1}+ax+a;F)$.
The equality $\SF(x^{q+1}+x+1/a;F)=\SF(x^{q+1}+ax+a;F)$ was shown in Proposition \[bpLemma\], so it suffices to show that $K=L$, where $$K=\SF(C(x)+a;F),\qquad L = \SF(x^{q+1}+ax+a;F).$$ Theorem \[KsubsetL\] explicitly expresses each root of $C(x)+a$ in terms of the roots of $x^{q+1} + ax + a$. (See the remark following Theorem \[qplus1Thm\] to see how $c,j$ can be written in terms of roots of $x^{q+1}+ax+a$.) This implies that $K\subset L$. To show $L\subset K$, we will express an arbitrary root of $x^{q+1}+ax+a$ in terms of the roots of $C(x)+a$.
Let $r$ be an arbitrary root of $x^{q+1}+ax+a$. Select any other two roots $r_0$ and $r_1$ and define $y=(r_1-r)/(r_1-r_0)$, $\xi=y^q-y$. By Theorem \[qplus1Thm\], $r = 1+\xi^{1-q}$, so it suffices to express $\xi$ in terms of the roots of $C(x)+a$. By Theorem \[KsubsetL\], these roots are $\{\,e(y,c,j) : c \in \F_q,\ j \in \F_{q,1}\,\}$, where $e(y,c,j) = (cy^2 + y + j/c)^{q+1}/\xi^2$.
First assume that $q>4$. We have $$\begin{aligned}
e(y,c,j) &=& (cy^{2q}+y^q+j/c)(cy^2+y+j/c)/\xi^2 \\
&=& \frac{c^2 y^{2q+2} + c(y^{2q+1} + y^{q+2}) + y^{q+1} +j \xi^2 + (j/c) \xi + (j/c)^2}{\xi^2}.\end{aligned}$$ We claim that there are $c_1,c_2,c_3,c_4\in \F_q^\x$ such that $\sum_{i=1}^4 c_i = 1$ and $\sum_{i=1}^4 1/c_i = 0$. To see this, select $\alpha \in \F_q \setminus \F_4$; such $\alpha$ exists because $q>4$. Let $w_1 = 1$, $w_2 = \alpha$, $w_3 = 1/\alpha$, and note that $w_1 + w_2 + w_3 = 1 + \alpha + \alpha^{-1} = 1/w_1 + 1/w_2 + 1/w_3$. Furthermore, this value does not belong to $\F_2$, because the only solutions to $x + 1/x + 1 \in \{0,1\}$ belong to $\F_4$. Set $w_4=1/(1+\alpha+\alpha^{-1})$. Then $1/w_1 + 1/w_2 + 1/w_3 + 1/w_4 = 0$. Since $w_4 \not \in \F_2$, we know $w_4 \ne w_4^{-1}$, and so $w_1 + w_2 + w_3 + w_4 = 1 + \alpha + \alpha^{-1} + w_4 = 1/w_4 + w_4 \ne 0$. Setting $c_i = w_i/\sum w_i$, we find that $\sum c_i = 1$ and $\sum c_i^{-1} = 0$. This establishes the claim. Let $$t_1(c,j) = \sum_{i=1}^4 e(y,cc_i,j) = \frac { c^2y^{2q+2} + c(y^{2q+1} + y^{q+2}) } {\xi^2},$$ $$t_2(c,j) = e(y,c,j) - t_1(c,j) = \frac{ y^{q+1} + j \xi^2 + (j/c) \xi + (j/c)^2} {\xi^2}$$ and note that these both belong to $K$. Since $t_1(c,j)$ does not depend on $j$, we may denote it simply $t_1(c)$. For any $j_1,j_2 \in \F_{q,1}$ we have $$t_2(j_1,j_1) + t_2(j_2,j_2) = j_1 + j_2 \in K.$$ Note that $j_1 + j_2$ represents an arbitrary element of $\F_{q,0}$. In addition, we have for any $j\in\F_{q,1}$ and $b\in F\setminus \F_2$: $$t_2(j,j) + t_2(j/(b+1),j) = \frac b \xi + \frac {b^2} {\xi^2} \in K. \label{bxi}$$ Also, $$t_2(j/(b+1),j) + t_2(j/b,j) = \frac 1 \xi + \frac 1 {\xi^2} \in K.$$ Combining this with (\[bxi\]), we see that in fact $b/\xi + b^2/\xi^2 \in K$ for all $b \in \F_q$.
Next we show that $\xi \in K$. Select distinct values $d_1,d_2,d_3 \in \F_{q,1}$ such that $d_i + d_j \ne 1$ for each $i,j$. To see that these exist, note that if $n$ is odd, then $1\in \F_{q,1}$, and so the sum of two elements of $\F_{q,1}$ is never one and it suffices to select $d_1,d_2,d_3$ to be distinct. Since $|\F_{q,1}| = q/2 \ge 4$, this selection is possible. If $n$ is even, then $q/2 \ge 8$, so there are at least eight choices for $d_1\in \F_{q,1}$, six choices for $d_2\in\F_{q,1}\setminus \{d_1,d_1+1\}$, and four choices for $d_3 \in \F_{q,1} \setminus \{d_1,d_2,d_1+1,d_2+1\}$. This shows again that $d_1,d_2,d_3$ can be selected so that $d_i+d_j \ne 1$ for each pair $(i,j)$. Let $\tau_1 = d_1 + d_2$ and $\tau_2 = d_1 + d_3$, and note that $\{\tau_1,\tau_2,\tau_1+\tau_2\} \cap \F_2 = \emptyset$. Let $$c_1 = \frac 1 {\tau_1(\tau_1+\tau_2)},\qquad
c_2 = \frac 1 {\tau_2(\tau_1+\tau_2)}$$ and observe that $$\tau_1 c_1 + \tau_2 c_2 = 0,\qquad \tau_1^2 c_1 + \tau_2^2 c_2 = 1.$$ Since $\tau_i$ and $c_i/\xi + c_i^2/\xi^2$ are in $K$, so is $$\tau_1^2 (c_1/\xi + c_1^2/\xi^2) +\tau_2^2 (c_2/\xi + c_2^2/\xi^2) = 1/\xi.$$ This shows $\xi \in K$, so $r\in K$. Since $r$ is an arbitrary root of $x^{q+1} + ax + b$, this completes the proof when $q>4$.
If $q=4$, then let $\alpha$ be a cube root of unity in $\F_4$. By direct calculation, $$1/\xi = e(y,1,\alpha) + e(y,1,\alpha^2) + \frac{ \left(e(y,\alpha,\alpha) + e(y,\alpha,\alpha^2) \right)
\left(e(y,\alpha^2,\alpha) + e(y,\alpha^2,\alpha^2)\right) }
{e(y,1,\alpha) + e(y,1,\alpha^2)}.$$ Thus, $\xi \in K$ and consequently $r\in K$, as desired.
Galois action and related factorizations {#galoisSec}
========================================
Since the roots of the two polynomials $C(x)+a$ and $x^{q+1} + a x + a$ belong to the same field $L=F \circ \F_q(y)$, any element of the Galois group $\Gal(L/F)$ simultaneously permutes the roots of $C(x)+a$ and of $x^{q+1} + a x + a$. For this reason, the factorizations of these two polynomials are related. This section explores this.
We recall from Section \[qplus1Sec\] that an ordered triple $(r,r_0,r_1)$ of distinct roots of $x^{q+1}+ax+a$ determines $y=(r_1-r)/(r_1-r_0)$. If we selected a different triple of roots, then the cross-ratio $y'$ that they determine satisfies $y'=\gamma^{-1} y$ for a unique $\gamma \in \PGL_2(\F_q)$. It will be useful to see the effect of such transformations on the roots of $C(x)+a$.
Recall that the distinct roots of $C(x)+a$ are $\{\,e(y,c,j) : c \in \F_q^\x,\ j\in \F_{q,1} \}$, where $e(y,c,j) = (c y^2 + y + j/c)^{q+1}/(y^q+y)^2$.
\[galCLemma\] If $b \in \F_q^\x$ then $$\begin{aligned}
e(y+b,c,j) &=& e(y,c,j+bc+(bc)^2) \label{yplusb} \\
e(by,c,j) &=& e(y,bc,j) \label{yb} \\
e(1/y,c,j) &=& e(y,j/c,j) \label{recipy}\end{aligned}$$
For the first formula, $(y+b)^q-(y+b)=y^q-y$ and $c(y+b)^2+(y+b)+j/c
= cy^2 + y + (j+cb+c^2b^2)/c$. Note that $j+cb+c^2b^2 \in \F_{q,1}$, because $$\Tr(j+bc+b^2c^2) = \Tr(j) + 2\Tr(bc) = \Tr(j) = 1,$$ where Tr denotes the trace from $\F_q$ to $\F_2$. For the second formula, $$e(by,c,j)=\frac{(cb^2y^2 + by + j/c)^{q+1}}{((by)^q+by)^2} = \frac{ b^{q+1} (cby^2+y+j/(cb))^{q+1} }
{ b^2 (y^q+y)^2 } = e(y,bc,j).$$ Finally, $$\begin{aligned}
e(1/y,c,j) &=& \frac{ (c y^{-2} + y^{-1} + j/c)^{q+1} } {(y^{-q}+y^{-1})^2 } \times \frac{ y^{2(q+1)} }{ y^{2(q+1)} }
= \frac{(c+y+(j/c) y^2)^{q+1}} {(y+y^q)^2} = e(y,j/c,j).\end{aligned}$$
\[sigmaFacThm\] Let $L$ be the splitting field of $x^{q+1}+x+1/a$ (which is also the splitting field of $C(x)+a$ by Theorem \[KeqL\]), and let $\sigma \in \Gal(L/F\circ \F_q)$.
- If $\sigma$ fixes at least three roots of $x^{q+1} + x + 1/a$, then it fixes all roots of $x^{q+1}+x+1/a$ and all roots of $C(x)+a$.
- If $\sigma$ fixes exactly two roots of $x^{q+1}+x+1/a$, then the permutation induced by $\sigma$ on the roots has orbits of size $[1,1,\delta,\delta,\ldots,\delta]$ where $\delta$ divides $q-1$. The permutation induced by $\sigma$ on the roots of $C(x)+a$ has all its orbits of size $\delta$.
- If $\sigma$ fixes exactly one root of $x^{q+1} + x + 1/a$, then the remaining roots fall into $\sigma$-orbits of size 2. Also, $\sigma$ fixes exactly $q/2$ roots of $C(x)+a$, and the remaining roots fall into exactly $q/2(q/2-1)$ $\sigma$-orbits of size two.
- If $\sigma$ fixes no roots of $x^{q+1} + x + 1/a$, then all $\sigma$-orbits of $x^{q+1}+x+1/a$ have the same size $\delta$, where $\delta$ divides $q+1$. Also, $\sigma$ fixes exactly one root of $C(x)+a$, and all remaining roots belong to $\sigma$-orbits of size $\delta$.
Since $x^{q+1}+x+1/a$ and $x^{q+1} + a x + a$ have the same splitting field and their roots are in bijection by a Galois-invariant map, we can instead work with $x^{q+1}+ax+a$.
In case ([*i*]{}), let $r,r_0,r_1$ be three roots of $x^{q+1}+ax+a$ that are fixed by $\sigma$. Then $y=(r_1-r)/(r_1-r_0) $ is also fixed by $\sigma$. By Theorem \[qplus1Thm\], $L=F\circ \F_q(y)$. Since $\sigma\in\Gal(L/F\circ \F_q)$ and it fixes $y$, it follows that $\sigma$ is the identity, and so it fixes all roots of both polynomials.
In case ([*ii*]{}), let $r$ and $r_0$ be roots of $x^{q+1}+ax+a$ that are fixed by $\sigma$, and select a third root $r_1$ with which to form $y$. Let $\gamma$ be the element of $\PGL_2(q)$ such that $\sigma(r_w)=r_{\gamma(w)}$ and $\sigma(y)=\gamma^{-1}y$. (Such $\gamma$ exists by Theorem \[qplus1Thm2\].) Since $\gamma$ fixes $\infty$ and 0, it must be of the form $\textmatrix b001$ with $b\in\F_q^\x$. Let $\delta$ be the multiplicative order of $b$. Then the orbits of $\gamma$ acting on $\P^1(q) \setminus \{\infty,0\}$ are of the form $\{w,bw,b^2w,\ldots, b^{\delta-1}w\}$ showing that the non-singleton orbits all have the same order $\delta$. Consequently, the $\sigma$-orbits on roots of $x^{q+1}+x+1/a$ have sizes $[1,1,\delta,\delta,\ldots,\delta]$. The action on roots of $C(x)+a$ is $$\sigma\left( e(y,c,j)\right)=e(by,c,j)=e(y,bc,j).$$ Thus, each $\sigma$-orbit has size exactly $\delta$.
In case ([*iii*]{}), $\sigma$ fixes exactly one root of $x^{q+1}+ax+a$ which we may assume is $r$. The elements of $\PGL_2(q)$ that fix only $\infty$ are of the form $\textmatrix 1b01$ with $b\in\F_q^\x$. Then $$\sigma\left( e(y,c,j)\right)=e(y+b,c,j)=e(y,c,j+(bc)+(bc)^2).$$ Since $b$ and $c$ are nonzero, we can have $(c,j)=(c,j+(bc)+(bc)^2)$ if and only if $c=1/b$. So the $q/2$ roots $e(y,1/b,j)$ are fixed, and the other roots belong to $\sigma$-orbits of size 2.
In case ([*iv*]{}), define $y$ with respect to three roots $r$, $r_0$, $r_1$ of $x^{q+1}+ax+a$ such that $\sigma(r) = r_0$ and $\sigma(r_0) = r_1$. By Theorem \[qplus1Thm2\], there is a unique $\gamma \in \PGL_2(\F_q)$ such that $\sigma(y)=\gamma^{-1}(y)$ and $\sigma(r_w)=r_{\gamma(w)}$ for all $w\in \P^1(\F_q)$. Since $\gamma$ takes $\infty$ to 0 and 0 to 1, it has the form $\gamma = \textmatrix 0 1 k 1$. By hypothesis, $\sigma$ fixes no roots, and so $1/(kw+1)=w$ has no solutions in $\F_q$. This is equivalent to $(kw)^2 + (kw) + 1$ having no rational roots, which is equivalent to $k \in \F_{q,1}$. Let $\delta$ be the order of $\gamma$. We will show in Proposition \[prop:dihedral\] (or see [@Dickson]) that $\delta$ divides $q+1$ and that $\gamma$ has no fixed points in $\P^1(\F_q)$. (In the notation of (\[DjCDef\]), $\gamma$ belongs to the group ${\cal C}_{\sqrt k,\sqrt k}$, which is cyclic of order $q+1$.) Thus, the orbits of $\sigma$ on the roots of $x^{q+1}+ax+a$ all have the same size, $\delta$.
We have $\gamma^{-1} y = (1/y+1)/k$, and so $$\begin{aligned}
\sigma(e(y,c,j)) &=& e((1/y+1)/k,c,j) \\
&=& e(1/y+1,c/k,j) \qquad \text{by (\ref{yb})} \\
&=& e(1/y,c/k,j+c/k+c^2/k^2)\qquad \text{by (\ref{yplusb})} \\
&=& e(y,jk/c+1+c/k,j+c/k+c^2/k^2)\qquad \text{by (\ref{recipy}).} \end{aligned}$$ This can equal $e(y,c,j)$ only if $(c/k)+(c/k)^2=0$, [*i.e.*]{} $c=k$. In that case, we have $$\sigma(e(y,k,j))=e(y,j,j).$$ So, for the root to be fixed by $\sigma$ we also need $j=k$. Thus, we find there is exactly one fixed root, namely $e(y,j,j)$. The other roots must belong to orbits of size dividing $\delta$, where $\delta$ is the order of $\gamma$. We claim the orbits have size exactly $\delta$. Indeed, suppose that $\sigma$ had an orbit of size $i$, where $i$ strictly divides $\delta$, and consider $\sigma^i$. This fixes no roots of $x^{q+1}+ax+a$, so it fixes exactly one root of $C(x)+a$, which must be $e(y,j,j)$. But it also fixes the points on the $\sigma$-orbit of size $i$, a contradiction. So the roots of $C(x)+a$ fall into $\sigma$-orbits of size $\left[1,\delta,\delta,\ldots,\delta\right]$.
\[facTypes\] For a polynomial $g\in F[x]$, write $g \sim [n_1,n_2,...,n_t]$ if $g$ factors into irreducibles of degrees $n_1, n_2, \ldots , n_t$. Let $0\ne a\in F=\F_{q^m}$, where $q=2^n>2$. Let $f(x)=x^{q+1}+x$ and let $C(x)=xT(x)^{q+1}$.
- If $f(x)+1/a$ has at least three roots in $F$ then both $f(x) + 1/a$ and $C(x)+a$ have all their roots in $F$.
- If $f(x)+1/a$ has exactly two roots in $F$, then $f(x)+1/a\sim[1,1,\d,\d,\ldots,\d]$ and $C(x)+a\sim[\d,\d,\ldots,\d]$, where $\d|q-1$.
- If $f(x)+1/a$ has exactly one root in $F$ then $f(x)+1/a\sim[1,2,2,\ldots,2]$ and $C(x)+a\sim[1,1\ldots,1,2,2,\ldots,2]$, where $C(x)+a$ has $q/2$ linear factors and $(q^2-2q)/4$ irreducible quadratic factors.
- If $f(x)+1/a$ has no roots in $F$, then $f(x)+1/a\sim[\d,\d,\ldots,\d]$ and $C(x)+a\sim[1,\d,\d,\ldots,\d]$, where $\d|q+1$.
Apply Theorem \[sigmaFacThm\], taking $\sigma$ to be the Frobenius map: $\sigma(u) = u^{|F|}$. The sizes of the $\sigma$-orbits acting on the roots of $f(x)+1/a$ or $C(x)+a$ are the degrees of the irreducible factors over $F$.
If $F = \F_{q^m}$ and $0\ne a \in F$ then $C(x)+a$ has exactly 0, 1, $q/2$, or $(q/2)(q-1)$ roots in $F$. Let $c_i$ denote the number of $a \in F^\x$ for which $C(x) + a$ has exactly $i$ roots in $F$. If $m$ is even, then $$c_0 = \frac{(q-2)(q^m-1)}{2(q-1)},\qquad c_1 = \frac{q^{m+1}-q}{2(q+1)},\qquad c_{q/2} = q^{m-1},\qquad c_{(q/2)(q-1)} = \frac{q^{m-1}-q}{q^2-1}.$$ If $m$ is odd, then $$c_0 = \frac{(q-2)(q^m-1)}{2(q-1)},\qquad c_1 = \frac{q^{m+1}+q}{2(q+1)},\qquad c_{q/2} = q^{m-1}-1,\qquad c_{(q/2)(q-1)} = \frac{q^{m-1}-1}{q^2-1}.$$
For $i\in \{0,1,2,q+1\}$, let $N_i$ denote the number of $a \in \F^\x$ such that $x^{q+1}+ax+a$ has exactly $i$ roots. By Corollary \[facTypes\], we have $N_0=c_1$, $N_1 = c_{q/2}$, $N_2 = c_{0}$, and $N_{q+1} = c_{(q/2)(q-1)}$. The $N_i$’s are computed in [@qplus1 Theorem 5.6]. The result follows.
We conclude this section by proving Proposition \[quinticSextic\], which gives the related factorizations of $x^{q+1}+x+1/a$ and $C(x)+a$ when $q=4$ and $F=\F_{2^k}$. Note that we are not assuming that $\F_q \subset F$. The polynomials are $x^5+x+1/a$ and $x(x+1)^5+a$. Since $x^5+ax+a$ has the same splitting field and factorization type as $x^5+x+1/a$, we may study it instead. Let $L$ denote the splitting field and let $\sigma \in \Gal(L/F)$ denote the Frobenius element, $\sigma(b)=b^{|F|}$; then $\sigma$ generates $\Gal(L/F)$.
If $k$ is even, then $\F_q\subset F$. In that case, Proposition \[quinticSextic\] follows from Corollary \[facTypes\].
Now assume $k$ is odd, and we must show that one of the following cases holds. $$\text{$x^5+x+1/a \sim [1,1,1,2]$ and $x(x+1)^5 + a \sim [2,2,2]$}$$ $$\text{$x^5+x+1/a \sim [1,4]$ and $x(x+1)^5 + a\sim [1,1,4]$}$$ $$\text{$x^5+x+1/a \sim [2,3]$ and $x(x+1)^5 + a\sim [6]$}.$$ Note that $\sigma(c)=c^2$ for $c\in\F_4$.
If $\sigma$ fixes at least three roots of $x^{q+1}+ax+a$, then we can arrange for $y$ to be rational. Then $\sigma(r_w) = r_{\sigma(w)}$ for $w\in \P^1(\F_4)$. The conjugate pair $\alpha$ and $\alpha^2$ are exchanged, while all other elements are fixed, and so $x^5+ax+a \sim [1,1,1,2]$. The roots of $C(x)+a$ are $e(y,c,j)$ for $c \in \{1,\alpha,\alpha^2\}$ and $j \in \{\alpha,\alpha^2\}$. Since $y$ is fixed, we have $\sigma(e(y,c,j)) = e(y,c^2,j^2)$. There are three orbits of size 2, so $C(x)+a \sim [2,2,2]$.
Now suppose that $\sigma$ fixes exactly one or two roots, so there is at least one rational root $r$. Select $r_0$ to be any root that is not fixed by $\sigma$, and let $r_1 = \sigma(r_0)$. Set $y=(r_1-r)/(r_1-r_0)$ and $r_w = r(y-w)^{q-1}$. By Theorem \[qplus1Thm2\] there is a unique $\gamma \in \PGL_2(\F_q)$ such that $\sigma(r_w) = r_{\gamma(w^2)}$ and $\sigma(y)=\gamma^{-1}(y)$. Let $\sigma(r_1) = r_{1+b}$, so $b \in \F_4^\x$. Now $\gamma(\infty)=\infty$, $\gamma(0)=1$, and $\gamma(1)=1+b$. From this we see that $\gamma = \textmatrix b 1 0 1$. We have $\sigma(r_w) = r_{b w^2+1}$. If $b=1$ then $\sigma$ fixes $\infty$, $\alpha$, and $\alpha^2$, contradicting that $\sigma$ fixes exactly one or two roots. Thus, $b=\alpha$ or $b=\alpha^2$. Let us assume that $b=\alpha$, as the other case is similar. Then, $\sigma(r_0)=r_1$, $\sigma(r_1) = r_{\alpha+1}$, $\sigma(r_{\alpha+1})=r_\alpha$, and $\sigma(r_\alpha)=r_0$, thus $x^5+ax+a\sim[1,4]$. The action on roots of $C(x)+a$ is given by $$\begin{aligned}
\sigma\left(e(y,c,j)\right)&=& e(\gamma^{-1}y,c^2,j^2)=e((y+1)/b,c^2,j^2)=e(y+1,c^2/b,j^2)\\
&=& e(y,c^2/b,j^2+c^2/b+c/b^2).\end{aligned}$$ Setting $b=\alpha$, the $\sigma$-orbits are as follows: $$e(y,1,\alpha) \to e(y,\alpha^2,\alpha) \to e(y,1,\alpha^2) \to e(y,\alpha^2,\alpha^2) \to e(y,1,\alpha)$$ while $e(y,\alpha,\alpha)$ and $e(y,\alpha,\alpha^2)$ are fixed. Thus, $C(x)+a\sim [1,1,4]$, as claimed.
It remains to consider the case where $x^5+ax+a$ has no rational roots. Select three roots as follows. Let $r_\infty$ belong to an orbit of odd order, let $r_0 = \sigma(r_\infty)$, and let $r_1=\sigma(r_0)$. Either $\sigma(r_1)=r_\infty$ or $\sigma(r_1)=r_c$ with $c\in \{\alpha,\alpha^2\}$. Since $\sigma(r_w)=r_{\gamma \sigma(w)}$, we know $\gamma$ takes $(\infty,0,1)$ to $(0,1,\infty)$ or $(0,1,c)$. In the former case, $\gamma = \textmatrix 0 1 1 1$, and in the latter case, $\gamma = \textmatrix 0 1 c 1$, where $c\in \{\alpha,\alpha^2\}$. In the latter case, $r_{c+1}$ is fixed, because $\sigma(r_{c+1}) = r_{\gamma(\sigma(c+1))} = r_{\gamma(c)} = r_{1/(c^2+1)} = r_{c+1}.$ Since we were assuming no rational roots, this case can be eliminated from consideration. Thus, we have $\gamma=\textmatrix 0111$.
For $\gamma = \textmatrix 0 1 1 1$, we have $\sigma(r_w)=r_{\gamma(w^2)}$, and the $\sigma$-orbits on $\{r_w : w \in \P^1(q)\}$ are $(\infty\ 0\ 1)(\alpha\ \alpha+1)$. The action on roots of $C(x)+a$ is $$\begin{aligned}
\sigma(e(y,c,j)) &=& \sigma(\gamma^{-1}(y),c^2,j^2) = \sigma((1/y)+1,c^2,j^2) = \sigma(1/y,c^2,j^2 + c^2 + c)\\
&=& \sigma(y,(j^2/c^2)+1+1/c,j^2+c^2+c) =\sigma(y,cj^2+1+c^2,j^2+c^2+c),\end{aligned}$$ where we used $c^3=j^3=1$ since $|\F_q^\x|=3$. There is a single $\sigma$-orbit: $$e(y,1,\alpha)\to e(y,\alpha^2,\alpha^2) \to e(y,\alpha,\alpha^2) \to e(y,1,\alpha^2) \to e(y,\alpha,\alpha) \to e(y,\alpha^2,\alpha) \to e(y,1,\alpha).$$ So in this case, $x^5+x+1/a \sim [3,2]$ and $x(x+1)^5 + a \sim [6]$.
Dihedral group {#dihedralSec}
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Let $L$ be the splitting field of $x^{q+1}+ax+a$ and let $e\in L$ be a root of $C(x)+a$. As shown in Figure \[splitting\_diagram2\], when $F=\F_q(a)$ with $a$ transcendental we have $[L:F(e)]=2(q+1)$. Thus, $\Gal(L/F(e))$ is a subgroup of order $2(q+1)$ in $\PGL_2(\F_q)$, and by [@Dickson Chapter XII], the only such subgroup is a dihedral group. In this section, we give explicit formulas for this dihedral group. Later, we will use these formulas to give a new proof that $C(x)$ is exceptional when $n$ is odd.
First, we discuss $\PGL_2(\F_q)$ (where $q=2^n$) in more detail. Dickson [@Dickson Chapter XII] showed that all nontrivial elements of $\PGL_2(\F_q)$ have order 2, or have order dividing $q-1$, or have order dividing $q+1$. In fact, he enumerated these:
- $q^2-1$ elements have order 2
- $(q+1)(q/2)(q-2)$ elements have order dividing $q-1$
- $q^2(q-1)/2$ elements have order dividing $q+1$.
Including also the trivial element, these numbers add up correctly to the full cardinality of $\PGL_2(\F_q)$: $$q^2-1 + (q+1)(q/2)(q-2) + q(q-1)(q/2) + 1 = q(q-1)(q+1).$$ We would like to explicitly describe these elements in a simple manner. If an element $\gamma\in \PGL_2(\F_q)$ is normalized to have determinant 1, then its order divides $q+1$ if and only if it is conjugate to $\diag\{\rho,\rho^{-1}\}$ with $\rho \in \mu_{q+1}$. In that case, its trace is $\ang\rho$. By Lemma \[Fq1Lemma\], such values are characterized by the fact that $1/\ang{\rho} \in \F_{q,1}$. Thus, for any nontrivial $\gamma=\textmatrix ABCD$ we have $$\text{${\rm order}(\gamma)$ divides $q+1$ iff $A+D\ne 0$ and $\Tr_{\F_q/\F_2}\left((AD-BC)/(A^2+D^2)\right) = 1$.}$$ The order divides $q-1$ if and only if it is conjugate to $\diag\{w,1/w\}$ with $w\ne w^{-1}\in \F_q^\x$. In that case, $\ang{w}=w+1/w$ is nonzero and $\Tr_{\F_q/\F_2}(1/\ang w) = 0$. Thus, $$\text{${\rm order}(\gamma)$ divides $q-1$ iff $A+D\ne 0$ and $\Tr_{\F_q/\F_2}\left((AD-BC)/(A^2+D^2)\right) = 0$.}$$ Finally, if $\gamma$ has order 2 then its eigenvalues are equal, so its (matrix) trace is zero: $$\text{${\rm order}(\gamma)=2$ iff $A=D$ and $B$ or $C$ is nonzero.}$$
To count the matrices of order 2, we note that if we include the identity matrix then each can be written uniquely as either $\textmatrix 1bc1$ with $bc \ne 1$, or as $\textmatrix 0bc0$ with $bc=1$, so the total number of matrices equals the number of pairs $(b,c)$, which is $q^2$. Excluding the identity, there are exactly $q^2-1$ matrices of order 2.
In the remaining cases we have $A+D\ne 0$. Consider $(AD-BC)/(A+D)^2$. The absolute trace of this quantity determines whether the order of $\gamma$ divides $q-1$ or $q+1$. If $BC=0$, then the absolute trace is always zero because $$\frac{AD}{A^2+D^2} = \frac D{A+D} + \left(\frac D{A+D} \right)^2.$$ If $BC \ne 0$, then we choose to normalize so that $BC=1$, and we have $$\frac{AD+1}{A^2+D^2} =\frac D{A+D} + \left(\frac D{A+D} \right)^2 + \frac 1{(A+D)^2}.$$ Set $j=1/(A+D)\in\F_q^\x$. We have proved that if $$\gamma = \begin{pmatrix} A & 1/C \\ C & A + 1/j \end{pmatrix} \in \PGL_2(\F_q) \label{j_form}$$ then the order of $\gamma$ divides $q+1$ if and only if $j\in \F_{q,1}$, and otherwise the order of $\gamma $ divides $q-1$. Note that $j^2\det(\gamma) = j^2 A^2 + j A + j^2$. If $\Tr_{\F_q/\F_2}(j)=1$, this has no rational solutions for $A$, and so we obtain a matrix for every triple $(A,C,j) \in \F_q \x \F_q^\x \x \F_{q,1}$. There are exactly $q(q-1)(q/2)$ such triples, which agrees with Dickson’s count.
By a direct computation, we find that if $A_1+A_2+1/j\ne 0$ then $$\begin{pmatrix} A_1 & 1/C \\ C & A_1 + 1/j \end{pmatrix}
\begin{pmatrix} A_2 & 1/C \\ C & A_2 + 1/j \end{pmatrix}
=\begin{pmatrix} A_3 & 1/C \\ C & A_3 + 1/j \end{pmatrix},\quad \text{where $A_3 = \frac {1+A_1A_2}{A_1+A_2+1/j}$.}$$ Thus, if $C,j$ are held fixed then the elements of the form (\[j\_form\]), together with the identity, are closed under multiplication and so they form a group which we denote by ${\cal C}_{j,C}$. (Here we must exclude the matrices of determinant zero.) We may associate the identity element with “$A=\infty$”.
It is useful to observe that $$\begin{pmatrix} C&0\\ 0&1 \end{pmatrix} \begin{pmatrix} A & 1/C \\ C & A + 1/j \end{pmatrix}
\begin{pmatrix} 1/C&0\\ 0&1 \end{pmatrix} = \begin{pmatrix} A & 1 \\ 1 & A + 1/j \end{pmatrix} \label{conjC}$$ Often this makes it easy to reduce to the case $C=1$. We let ${\cal C}_j $ denote the cyclic group consisting of matrices $M_A=\textmatrix A 1 1 {A+1/j}$. Note that the value $j$ does not change under the above conjugation. This is an advantage of the normalization $BC=1$.
Also we note the formula: $$\begin{pmatrix} 1&\frac 1{jC}\\ 0&1 \end{pmatrix} \begin{pmatrix} A & \frac 1 C \\ C & A + \frac 1 j \end{pmatrix}
\begin{pmatrix} 1&\frac 1{jC} \\ 0&1 \end{pmatrix} = \begin{pmatrix} A & \frac 1 C \\ C & A + \frac 1 j \end{pmatrix}^{-1}.$$ Thus, the group generated by ${\cal C}_{j,C}$ and $\textmatrix {1\ }{1/(jC)} 0 1$ is dihedral of order $2(q-1)$ (if $j\in \F_{q,0}$) or $2(q+1)$ (if $j\in \F_{q,1}$).
Another point is worth making. Let ${\cal B}\subset \PGL_2(\F_q)$ denote the matrices that fix $\infty$, , matrices of the form $\textmatrix ab01$, where $a\in \F_q^\x$ and $b\in\F_q$. Suppose $j\in\F_{q,1}$, so ${\cal C}_j$ has order $q+1$. Since ${\cal C}_j \cap {\cal B} = \{\textmatrix 1001\}$, and $|{\cal C}_j| \cdot |{\cal B}| = (q+1)\cdot q(q-1) = \PGL_2(\F_q)$, we see that each element of $\PGL_2(\F_q)$ can be decomposed uniquely as $\delta \beta$ with $\delta \in {\cal C}_j$ and $\beta \in {\cal B}$. Alternatively, each element may be decomposed uniquely as $\beta \delta$.
Let $M_A \in {\cal C}_j$, where $j\in\F_{q,1}$, and suppose we wish to compute all conjugates, $\gamma^{-1} M_A \gamma$. Decompose $\gamma = \delta \beta$, where $\delta \in {\cal C}_j$ and $\beta\in {\cal B}$. Then $\gamma^{-1} M_A\gamma= \beta^{-1} M_A \beta$. We may decompose $\beta$ as $$\beta = \begin{pmatrix} 1&b \\ 0 & 1 \end{pmatrix} \begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix}.$$
Conjugating $M_A$ by $\textmatrix 1b01$ gives $$\begin{pmatrix} 1 & b \\ 1 & 0 \end{pmatrix} \begin{pmatrix} A & 1 \\ 1 & A + 1/j \end{pmatrix}
\begin{pmatrix} 1 & b \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} A+b & u \\ 1 & A+b+1/j \end{pmatrix},\quad \text{
where $u=1+b^2+b/j$.}$$ Since $\Tr(uj^2) = \Tr(j^2+ (bj)^2+(bj)) = \Tr(j^2)$, we see that if $j\in\F_{q,1}$ then $u\ne 0$. As explained above, it is useful to normalize to make the product of the off-diagonal entries equal to 1, which in this case amounts to dividing each entry by $\sqrt u$. Then $$\begin{pmatrix} 1 & b \\ 1 & 0 \end{pmatrix} \begin{pmatrix} A & 1 \\ 1 & A + 1/j \end{pmatrix}
\begin{pmatrix} 1 & b \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} (A+b)/\sqrt u & \sqrt u \\ 1/\sqrt u & (A+b+1/j)/\sqrt u \end{pmatrix}
= \begin{pmatrix} A' & C \\ 1/C & A' + 1/J \end{pmatrix}$$ where $$A' = (A+b)/(1+b+\sqrt{b/j}), \qquad C = 1+b+\sqrt{b/j},\qquad J = j(1+b+\sqrt{b/j}) = j + \sqrt{bj} + bj.$$ Note that $\Tr(J)=1$, since $\Tr(\sqrt{bj}+bj)=0$. Conjugating this result by $\textmatrix a001$ fixes $J$ but changes $C$ to $aC$. It was pointed out by Dickson that all the cyclic subgroups of order $q+1$ are conjugate. The above formulas make this explicit.
We summarize part of our discussion in the following proposition, which shows that the dihedral groups of order $2q+2$ are naturally parameterized by $\F_{q,1} \x \F_q^\x$.
\[prop:dihedral\] Let $q=2^n$, $C\in \F_q^\x$, and $j\in \F_{q,1}$, , $\Tr_{\F_q/\F_2}(j)=1$. For $A\in \P^1(\F_q)$, define $M_A \in \PGL_2(\F_q)$ by the formula $$M_A = \begin{pmatrix} A & 1/C \\ C & A+1/j \end{pmatrix} \qquad\text{if $A\in\F_q$, }\qquad
M_\infty = \textmatrix 1001.$$ Define $${\cal C}_{j,C} = \{ M_A : A \in \P^1(\F_q) \},\qquad {\cal D}_{j,C} = {\cal C}_{j,C} \cup \left\{\, M_A \begin{pmatrix} 1&\frac 1 {jC}\\0&1\end{pmatrix}
: A \in \P^1(\F_q) \,\right\}.\label{DjCDef}$$ Then ${\cal C}_{j,C}$ is a cyclic group of order $q+1$, and ${\cal D}_{j,C}$ is a dihedral group of order $2(q+1)$. Any nontrivial element of this group has no fixed points in $\P^1(\F_q)$. We have $$\text{$M_A M_B = M_K$, with $K=\frac{1+A B}{1/j+A +B}$,}
\label{kappaEqn}$$ $$\begin{pmatrix} 1&\frac 1 {jC}\\0&1 \end{pmatrix} M_A \begin{pmatrix} 1&\frac 1{jC}\\0&1 \end{pmatrix}= M_A^{-1} = M_{A+1/j}.\label{dihedralRelation}$$
$M_A$ is invertible because $j^2\det(M_A) = (jA)^2+(jA)+j^2 \in \F_{q,1}$. The fact that $M_A$ has no fixed points on $\P^1(\F_q)$ when $A\in \F_q$ is shown as follows. If $M_A(w)=w$ with $w\in \F_q$ then $(Aw+1/C)/(Cw+A+1/j)=w$, which is equivalent to $(Cjw)^2+(Cjw)+j^2=0$. But this would imply $\Tr_{\F_q/\F_2}(j)=0$, a contradiction. Also $M_A(\infty)=A/C \ne \infty$ and this shows $M_A$ has no fixed points in $\P^1(\F_q)$. Next, we show that $M_A M_B = M_K$, where $K = (1+A B)/(1/j+A+B)$. If $A = \infty$ then $K=B$ and $M_A = \textmatrix 1001$, so the claim is true; similarly if $B=\infty$. If $A $ and $B$ are both finite, then $$M_A M_B = \begin{pmatrix} 1+AB & (A+B+1/j)/C \\ (A+B+1/j)C & 1+AB + \frac{A+B+1/j}j \end{pmatrix}.$$ If $A + B + 1/j=0$ then this is the identity, and also $K=\infty$ so the claim is true. If $A+B+1/j\ne0$, then on dividing through by that constant we find $$M_A M_B = \begin{pmatrix} \frac{1+A B}{A+B+1/j} & 1/C \\ C & \frac{1+AB}{A+B+1/j} + 1/j \end{pmatrix}
= M_K.$$ To see that ${\cal C}_{j,C}$ is cyclic of order $q+1$, we present an isomorphism with $\mu_{q+1}$. By Lemma \[Fq1Lemma\], we can select $\zeta\in\mu_{q+1}$ such that $\ang{\zeta} = 1/j$. Let $f(M_A) = (A+1/\zeta)/(A+\zeta)$. The reader can verify that if $K = (1+A B)/(A + B + 1/j)$, then $$\frac{ A + 1/\zeta } {A + \zeta} \times \frac{ B + 1/\zeta} {B + \zeta} = \frac{K + 1/\zeta}{K + \zeta},$$ and so $f$ is a homomorphism. Also, it is one-to-one since $A \mapsto (A+1/\zeta)/(A+\zeta)$ is invertible. Finally, we note that if $\rho = (A+1/\zeta)/(A+\zeta)$, then since $\zeta^q=\zeta^{-1}$ we have $\rho^q = \rho^{-1}$. Thus, $f(M_A) \in \mu_{q+1}$. Since $|{\cal C}_{j,C}|=q+1=\mu_{q+1}$ and we exhibited an injective homomorphism from ${\cal C}_{j,C}$ to $\mu_{q+1}$, it must be an isomorphism and so ${\cal C}_{j,C}$ is cyclic. The relation (\[dihedralRelation\]) is straightforward to check, and this verifies the claim that ${\cal D}_{j,C}$ is a dihedral group of order $2(q+1)$.
For the remainder of this section, let $0\ne a \in F$ where $F$ has characteristic 2.
\[fixedE\] Let $r,r_0,r_1$ be distinct roots of $x^{q+1}+ax+a$ and $y=(r_1-r)/(r_1-r_0)$. Recall from Theorem \[KsubsetL\] that each root of $C(x)+a$ can be written uniquely as $e(y,c,j)$ for some $c\in\F_q^\x$ and $j\in \F_{q,1}$, where $$e(y,c,j) = (c y^2 + y + j/c)^{q+1}/(y^q+y)^2.$$ For $\gamma\in \PGL_2(\F_q)$, we have $$e(\gamma^{-1} y ,c,j) = e(y,c,j) \iff \gamma \in {\cal D}_{d,c/d},\quad \text{where $d=\sqrt j$.}$$ Here, ${\cal D}_{j,C}$ is the dihedral group of order $2(q+1)$ that is defined by (\[DjCDef\]).
At the beginning of Section \[sec:RootsOfC\], we showed that if $e$ is any root of $C(x)+a$, then there is $u\in \cj F$ and $\zeta,\rho\in \mu_{q+1}$ satisfying $$1/e = \ang{u^{q+1}},\qquad r^2 = eT(e)^2 \ang u^{q-1},\qquad r_0^2 = eT(e)^2 \ang{\zeta u}^{q-1},\qquad
r_1^2 = eT(e)^2 \ang{\rho u}^{q-1}.$$ (Note that in these formulas, we may replace $(u,\zeta,\rho)$ by $(1/u,1/\zeta,1/\rho)$ without affecting $e$, $r$, $r_0$, and $r_1$.) Further, we proved that $$e=e(y,c,d^2),\quad \text{where
$c= \frac{\ang{\zeta/\rho}}{\ang\zeta \ang \rho }$ and $d=\frac 1{\ang\zeta}$.}$$ By (\[formula3\]) of Lemma \[sevenFormulas\], $$u = T_{\zeta,\rho} y, \qquad \text{where $T_{\zeta,\rho} = \begin{pmatrix} \ang{\zeta/\rho} & \ang{\rho}/\zeta \\ \ang{\zeta/\rho} &
\ang{\rho}\zeta \end{pmatrix}$.} \label{uyReln}$$
Let $\gamma \in \PGL_2(\F_q)$. Recall from Theorem \[qplus1Thm2\] that $\gamma^{-1}(y) = (r_{\gamma(1)}-r_{\gamma(\infty)})/
(r_{\gamma(1)}-r_{\gamma(0)})$. By applying the above reasoning to $\gamma^{-1}y$, we see that there are $\widetilde u$, $\widetilde \zeta$ and $\widetilde \rho$ such that $1/e = \ang{\widetilde u^{q+1}}$ and $$e = e(\gamma^{-1} y,\widetilde c, \widetilde d^2),\qquad
\widetilde u = T_{\widetilde \zeta, \widetilde \rho} \gamma^{-1} y, \label{uyTildeReln}$$ where $\widetilde c$ and $\widetilde d$ have the same formula as $c$ and $d$, but with $\zeta$ and $\rho$ replaced by $\widetilde \zeta$ and $\widetilde \rho$. Recall also that we are free to replace $(\widetilde u, \widetilde \zeta, \widetilde \rho)$ by $(1/\widetilde u,1/\widetilde \zeta, 1/\widetilde \rho)$. We are trying to find the condition on $\gamma$ such that $(c,d) = (\widetilde c, \widetilde d)$. So, assume $c=\widetilde c$ and $d=\widetilde d$. It it easy to see that $d=\widetilde d$ if and only if $\widetilde \zeta \in \{\zeta,\zeta^{-1}\}$. By possibly replacing $\widetilde u$ with its reciprocal, we may arrange that $\widetilde \zeta = \zeta$. Then $\widetilde c = c$ implies $\widetilde \rho = \rho$.
Because $\ang{u^{q+1}} = \ang{\widetilde u^{q+1}} = 1/e$, there is $\nu \in \mu_{q+1}$ such that $$\text{$\widetilde u = \nu^2 u$ or $\widetilde u= \nu^2/u$.}$$ We write this as $\widetilde u = \nu^2 u^\varepsilon$, where $\varepsilon \in \{1,-1\}$. Then $$\widetilde u = N u,\qquad \text{where $N = \begin{pmatrix} \nu & 0 \\ 0 & \nu^{-1} \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0
\end{pmatrix}^\varepsilon$.}$$ By (\[uyReln\]) and (\[uyTildeReln\]), $$\gamma^{-1} y = T_{\zeta,\rho}^{-1} N u = T_{\zeta,\rho}^{-1} N T_{\zeta,\rho} y.\label{gammaDelta}$$ Let $\delta^{-1}=T_{\zeta,\rho}^{-1} N T_{\zeta,\rho}$. Since $y \not\in \F_{q^2}$ (by Theorem \[qplus1Thm\]), an equality $\gamma^{-1} y = \delta^{-1} y$ with $\gamma,\delta \in\PGL_2(\F_q)$ implies $\gamma= \delta$. We claim that $\delta^{-1} \in \PGL_2(\F_q)$. Indeed, a direct computation shows that $$\begin{aligned}
T_{\zeta,\rho}^{-1} \begin{pmatrix} \nu &0\\0&\nu^{-1} \end{pmatrix} T_{\zeta,\rho}
&=& \begin{pmatrix} \ang\rho \zeta &\ang\rho/\zeta \\ \ang{\zeta/\rho} & \ang{\zeta/\rho} \end{pmatrix}
\begin{pmatrix} \nu \ang{\zeta/\rho} & \nu \ang{\rho}/\zeta \\ \nu^{-1}\ang{\zeta/\rho} & \nu^{-1} \ang\rho \zeta \end{pmatrix} \\
&=& \begin{pmatrix} \ang{\rho}\ang{\zeta/\rho}\ang{\zeta\nu} & \ang{\rho}^2 \ang\nu \\
\ang{\zeta/\rho}^2\ang\nu & \ang{\zeta/\rho} \ang\rho \ang{\zeta/\nu} \end{pmatrix},\end{aligned}$$ $$\begin{aligned}
T_{\zeta,\rho}^{-1} \begin{pmatrix} 0 &1\\1&0 \end{pmatrix} T_{\zeta,\rho}
&=& \begin{pmatrix} \ang\rho \zeta &\ang\rho/\zeta \\ \ang{\zeta/\rho} & \ang{\zeta/\rho} \end{pmatrix}
\begin{pmatrix} \ang{\zeta/\rho} & \ang\rho \zeta \\ \ang{\zeta/\rho} & \ang\rho/\zeta \end{pmatrix}\\
&=& \begin{pmatrix} \ang\rho\ang{\zeta/\rho}\ang\zeta & \ang\rho^2\ang\zeta^2 \\ 0 & \ang\rho\ang{\zeta/\rho}\ang\zeta
\end{pmatrix} \\
&=& \begin{pmatrix} 1 & 1/c \\ 0 & 1 \end{pmatrix}, \qquad \text{where $c= \frac{\ang{\zeta/\rho}}{\ang\zeta \ang\rho}$.}\end{aligned}$$ As can be seen, all entries of the above two matrices are rational and so we do indeed have $\delta^{-1} \in \PGL_2(\F_q)$. Consequently, (\[gammaDelta\]) implies that $$\gamma^{-1} = T_{\zeta,\rho}^{-1} N T_{\zeta,\rho}.$$
Our next goal is to show that $\gamma^{-1}$ belongs to ${\cal D}_{c/d,d}$. We do this separately for the two matrices that comprise $\gamma^{-1}$: $M_1 = T_{\zeta,\rho}^{-1} \diag\{\nu,\nu^{-1}\} T_{\zeta,\rho}$ and $M_2 = T_{\zeta,\rho}^{-1} \textmatrix 0110 T_{\zeta,\rho}$. As in the discussion at the beginning of Section \[dihedralSec\], we normalize $M_1$ to make $BC=1$ and find: $$T_{\zeta,\rho}^{-1} \begin{pmatrix} \nu &0\\0&\nu^{-1} \end{pmatrix} T_{\zeta,\rho} = \begin{pmatrix} A & 1/C \\ C & D
\end{pmatrix},\quad \text{where $A = \frac{\ang{\zeta\nu}}{\ang{\nu}}$, $C = \frac{\ang{\zeta/\rho}}{\ang{\rho}}$,
$D = \frac{\ang{\zeta/\nu}}{\ang\nu}$.}$$ Using the formula $\ang x \ang y = \ang{xy} + \ang{x/y}$, we have $$A+D=\frac{\ang{\zeta \nu} + \ang{\zeta/\nu}}{\ang{\nu}} = \frac{\ang\zeta \ang \nu} {\ang\nu} = \ang\zeta = 1/d.$$ Also notice that $C = c \ang\zeta = c/d$. Thus, $M_1 \in {\cal D}_{d,c/d}$ as required. Since $M_2 = \textmatrix 1 {1/c} 0 1$, it is immediate from (\[DjCDef\]) that $M_2\in {\cal D}_{d,c/d}$ as well.
We have shown that $$\text{$e(\gamma^{-1}y,c,d^2)=e(y,c,d^2)$ implies $\gamma\in{\cal D}_{d,c/d}$. } \label{implies}$$ To prove the converse, we use a counting argument. Let ${\cal E}$ denote the roots of $C(x)+a$. For $e'\in{\cal E}$, let $H_{e'} = \{ \gamma \in \PGL_2(\F_q) : e(\gamma^{-1} y ,c,d) = e' \}$. Then $\PGL_2(\F_q)$ is the disjoint union of $H_{e'}$, for $e' \in {\cal E}$, and so the average size of $H_{e'}$ is $|\PGL_2(\F_q)|/|{\cal E}| = q(q+1)(q-1)/\deg(C) = 2(q+1)$. On the other hand, if $\gamma_1,\gamma_2 \in H_{e'}$ then setting $y'=\gamma_2^{-1} y$ we see that $$e'=e(\gamma_1^{-1} \gamma_2 y',c,d^2) = e(y',c,d^2),$$ and so $\gamma_1^{-1} \gamma_2 \in {\cal D}_{d,c/d}$ by (\[implies\]). Thus, $|H_{e'}| \le |{\cal D}_{d,c/d}| = 2(q+1)$. Since the average size of $H_{e'}$ is $2(q+1)$, it must be that $|H_{e'}|=2(q+1)$. In particular, $|H_e|=2(q+1)$. By (\[implies\]), we know $H_e \subset {\cal D}_{d,c/d}$, and by comparing cardinalities, equality must hold.
Exceptionality of $C(x)$ {#sec:Exceptional}
========================
A polynomial $P(x)\in\F_r[x]$ is said to be [*exceptional*]{} over a finite field $\F_r$ if it induces a permutation on $K$ for infinitely many extension fields $K=\F_{r^m}$. It was proved in [@CM] that $C(x)$ and some related polynomials are exceptional over $\F_2$ when $n$ is odd. Specifically, $C(x)$ induces a permutation on $\F_{2^m}$ if and only if $m$ and $n$ are relatively prime. The first polynomials in this family were found by P. M" uller with $q=8$, degree 28. Müller’s search was motivated by some deep work by Fried, Guralnick and Saxl suggesting that new examples of permutation polynomials might be found in characteristic $p=2$ or $p=3$ having degree $(q/2)(q-1)$, where $q=p^n$ and $n$ is odd.
In this section we give a new proof that $C(x)$ is exceptional. We emphasize that the next proposition is known, and only the proof is new.
If $q=2^n$, then $C(x)=xT(x)^{q+1}$ induces a permutation on $\F_{2^m}$ if and only if $(n,2m)=1$.
Note that $xT(x)= x + x^2 + \ldots x^{2^{n-1}}$, and its roots are precisely $\F_{2^n,0}$. Since $C(x)=xT(x)^{q+1}$, the set of roots of $C(x)$ is also $\F_{2^n,0}$. In particular, if $\F_{2^m} \cap \F_{2^n,0}\ne \{0\}$, then $C(x)$ is not a permutation polynomial on $\F_{2^m}$. Now $\F_{2^m}\cap \F_{2^n} = \F_{2^k}$ where $k=(m,n)$. Since $\F_{2^k,0}\subset \F_{2^n,0}$, we see that for $C(x)$ to be a permutation polynomial on $\F_{2^m}$, we must have $\F_{2^k,0} = \{0\}$, which forces $k=1$, [*i.e.*]{}, $(m,n)=1$. If $n$ is even then $1\in \F_{2^m}\cap \F_{2^n,0}$, so another necessary condition for $C(x)$ to be a permutation polynomial on $\F_{2^m}$ is that $n$ is odd. Together, these necessary conditions may be written as $(2m,n)=1$. Assuming this condition, then $\F_{2^m}\cap \F_q=\F_2$ and $\F_{2^m}\cap \F_{q,0} = \F_2\cap\F_{q,0}=\{0\}$, so $C(x)$ has no roots in $\F_{2^m}^\x$. Thus, $C(x)$ sends $\F_{2^m}^\x$ to $\F_{2^m}^\x$.
From here on, let $F = \F_{2^m}$, where $(2m,n)=1$. To prove that $C(x)$ is a permutation polynomial, it suffices to show that if $e \in F^\x$ and $a=C(e)$, then $e$ is the unique root of $C(x)+a$. Suppose that $e'$ also satisfies $a=C(e')$, and we will show that $e=e'$.
By Theorem \[KsubsetL\], we have $e = e(y_0,c_0,j_0)$ for some $c_0\in \F_q^\x$ and $j_0 \in \F_{q,1}$, where $y_0\in{\cal Y}$ is the cross-ratio of any three distinct roots of $x^{q+1}+ax+a$. Since $n$ is odd, we know $\Tr_{\F_q/\F_2}(1)=1$, and so we may write $j_0 = 1 + b_0^2 + b_0$ for some $b_0\in \F_q$. By Lemma \[galCLemma\], we have $e(y_0,c_0,j_0)=e(c_0y_0,1,j_0)=e(c_0y_0+b_0,1,1)= e(y,1,1)$, where $y=c_0y+b_0\in {\cal Y}$. Then, as noted at the beginning of Section \[sec:Equality\], $e'$ may be written as $e(y,c,j)$ where $c\in \F_q^\x$ and $j\in \F_{q,1}$. Writing $j=1+b+b^2$, we see that $e'=e(cy+b,1,1)$. Thus, $$e = e(y,1,1), \qquad e' = e(cy+b,1,1).$$
Since $e$ and $e'$ are rational, they are fixed by every element of $\Gal(L/F)$, where $L=F \circ \F_q(y)$ is the splitting field of $C(x)+a$. Since $L$ is finite, $\Gal(L/F)$ is generated by the Frobenius element: $$\sigma(u) = u^{|F|}.$$ Since $\F_q \cap F = \F_2$, we know that if $w \in \F_q$, then $\sigma(w)=w$ if and only if $w\in \F_2$. Let $\gamma$ be the element of $\PGL_2(\F_q)$ such that $\sigma(y)=\gamma^{-1}(y)$. (Such $\gamma$ exists and is unique by Theorem \[qplus1Thm2\].) We have $$\sigma(e) = e(\gamma^{-1} y,1,1), \qquad \sigma(e') = e(\sigma(c) \gamma^{-1}y + \sigma(b),1,1).$$ Since $\sigma$ fixes $e$ and $e'$, we have $$e=e(y,1,1) = e(\gamma^{-1}y,1,1), \qquad e'=e(cy+b,1,1) = e(\sigma(c)\gamma^{-1}y+\sigma(b),1,1).$$ By Proposition \[fixedE\], the first equality implies that $\gamma \in {\cal D}_{1,1}$; that is, $\gamma = 1$, $\gamma = \textmatrix 1101$, or $$\gamma = \begin{pmatrix} A & 1 \\ 1 & A + 1 \end{pmatrix}\begin{pmatrix} 1 & \varepsilon \\ 0 & 1 \end{pmatrix},$$ where $A \in \F_q$ and $\varepsilon \in \F_2$. Let $y' = cy+b$, so that $e'=e(y',1,1)$. Then $$\sigma(c) \gamma^{-1}y + \sigma(b) = \delta^{-1}y',\qquad {\rm where}\quad \delta^{-1} = \begin{pmatrix} \sigma(c) & \sigma(b) \\ 0 & 1 \end{pmatrix} \gamma^{-1}
\begin{pmatrix} c & b \\ 0 & 1 \end{pmatrix}^{-1}.$$ The fact that $e'=e(y',1,1) = e(\delta^{-1}y',1,1)$ implies that $\delta \in {\cal D}_{1,1}$ as well.
We may write $$\gamma^{-1} = M_A \begin{pmatrix} 1& \varepsilon_1\\ 0&1\end{pmatrix},\qquad \delta^{-1} = M_B \begin{pmatrix} 1& \varepsilon_2\\ 0& 1\end{pmatrix}$$ where $$M_A = \begin{pmatrix} A & 1 \\ 1 & A+1 \end{pmatrix}\qquad \text{if $A\in \F_q$, } M_\infty = \begin{pmatrix} 1&0 \\ 0 & 1 \end{pmatrix}.$$
First, if $A = \infty$ (so $M_A$ is the identity), then our relation is $$M_B = \begin{pmatrix} \sigma(c) & \sigma(b) \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1&\varepsilon_1 \\ 0 & 1 \end{pmatrix} {\textmatrix cb01}^{-1}\begin{pmatrix} 1&\varepsilon_2 \\ 0 & 1 \end{pmatrix}.$$ Since the bottom left entry is 0, we have $M_B = \textmatrix 1001$. The right side is $$\begin{pmatrix} \sigma(c) & \sigma(c)(\varepsilon_2+b) + c\left(\varepsilon_1 \sigma(c) + \sigma(b)\right) \\ 0 & c \end{pmatrix}.$$ For this to be the identity, we need $c=\sigma(c)$. Since the fixed field of $\sigma$ is $F$, we obtain $c \in F^\x \cap \F_q = \F_2^\x = \{1\}$. Setting $c=1$, we find the top-right entry is $\varepsilon_1 + \varepsilon_2 + b + \sigma(b)$, and so $b + \sigma(b) \in \F_2$. If $b+\sigma(b)=1$, then we’d have $\Tr_{\F_q/\F_2}(b+\sigma(b)) = \Tr(1)$, or $0=1$, a contradiction. So $b=\sigma(b)$, and $b\in \F_2$. Then $j = 1 + b + b^2 = 1$, and so $e'=e(y,c,j)=e(y,1,1)=e$, as we wanted to show.
Now suppose $A\in \F_q$. Then we obtain an equation $$M_B = \begin{pmatrix} \sigma(c) & \sigma(b) \\ 0 & 1 \end{pmatrix} M_A \begin{pmatrix} 1&\varepsilon_1 \\ 0 & 1 \end{pmatrix} {\textmatrix cb01}^{-1}\begin{pmatrix} 1&\varepsilon_2 \\ 0 & 1 \end{pmatrix}.$$ The right side is $$\begin{pmatrix} \sigma(c) A + \sigma(b)\ & \left(\sigma(c) A + \sigma(b)\right)(\varepsilon_1 c + b + \varepsilon_2) + \left(\sigma(c)+\sigma(b) A + \sigma(b)\right)c \\
1 & \varepsilon_1 c + b + \varepsilon_2 + Ac + c \end{pmatrix}.$$ To be of the form $M_B$, we require that the top-right entry is 1 and the trace is 1. This gives the two equalities: $$\left(\sigma(c) A + \sigma(b)\right)(\varepsilon_1 c + b + \varepsilon_2) + \left(\sigma(c)+\sigma(b) A + \sigma(b) \right)c = 1 \label{star}$$ $$(c + \sigma(c)) A + (\varepsilon_1+1) c + \sigma(b) + b + \varepsilon_2 = 1. \label{star2}$$
First, if $c+\sigma(c)=0$, then we will have $c\in F^\x \cap \F_q = \F_2^\x$. Substituting $c=1$, we find as before that $b\in\F_2$ and obtain $e=e'$ in the same way as before.
Next assume $c+\sigma(c)\ne 0$, and we will obtain a contradiction. Multiply through (\[star\]) by $c+\sigma(c)$, and then use $(c+\sigma(c)) A = (\varepsilon_1+1)c+\sigma(b)+b+ \varepsilon_2 + 1$ in order to eliminate $A$ from (\[star\]). After simplifying, we obtain: $$\sigma(b)^2 c + b^2 \sigma(c) + \sigma(b) c + b \sigma(c) + (b + \varepsilon_1 + \sigma(b) + \varepsilon_2) c \sigma(c) +
(c+\sigma(c)) (c \sigma(c)+1) = 0.$$ On dividing by $c \sigma(c)$, we find that $x + \sigma(x) = \varepsilon_1 + \varepsilon_2 \in \F_2$, where $$x = b^2/c + b/c + b + c + 1/c.$$ Since $\Tr_{\F_q/\F_2}(x+\sigma(x)+\varepsilon_1+\varepsilon_2)=\varepsilon_1+\varepsilon_2$, we must have $\varepsilon_1=\varepsilon_2$. Thus, $\sigma(x)=x$ and consequently $x \in \F_2$.
If $x=0$, we find that $b^2 + b(c+1) + c^2 + 1 = 0$. Since we are assuming $c\ne \sigma(c)$, we may divide through by $(c+1)^2$ and this gives $$\left( \frac b {c+1} \right)^2 + \frac b {c+1} + 1 = 0.$$ Then $\Tr_{\F_q/\F_2}(1)=0$, a contradiction since $n$ is odd.
If $x=1$, we find that $b^2 + bc+b + c^2 + 1 = c$. If $b=c$ then $c^2+c^2+c+c^2+ 1 = c$, so $c=1$, a contradiction since we are in the case $c \ne \sigma(c)$. So we may divide through by $(b+c)^2$, and we find $$0 = \frac {b^2 + c^2 + b + c + 1 + bc } {(b+c)^2} = 1 + \frac {c+1}{b+c} + \left(\frac {c+1}{b+c}\right)^2.$$ On taking the trace, we obtain $0=\Tr_{\F_q/\F_2}(1)$, a contradiction since $n$ is odd. The contradiction shows that $c+\sigma(c)$ must be 0, and we already showed that implies $e=e'$. We conclude that $C(e)+a=C(e')+a$ implies $e=e'$, so $C(x)$ is indeed a permutation polynomial when $(2n,m)=1$.
[9]{}
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We numerically construct asymptotically Anti-de Sitter charged black hole solutions of $(4+1)$-dimensional $SU(2)$ Einstein-Yang-Mills theory that, for sufficiently low temperature, develop vector hair. Via gauge-gravity duality, these solutions describe a strongly-coupled conformal field theory at finite temperature and density that undergoes a phase transition to a superfluid state with spontaneously broken rotational symmetry (a p-wave superfluid state). The bulk theory has a single free parameter, the ratio of the five-dimensional gravitational constant to the Yang-Mills coupling, which we denote as $\a$. Previous analyses have shown that in the so-called probe limit, where $\a$ goes to zero and hence the gauge fields are ignored in Einstein’s equation, the transition to the superfluid state is second order. We construct fully back-reacted solutions, where $\a$ is finite and the gauge fields are included in Einstein’s equation, and find that for values of $\a$ above a critical value $\a_c = 0.365 \pm 0.001$ in units of the AdS radius, the transition becomes first order.'
address: |
Max-Planck-Institut für Physik (Werner-Heisenberg-Institut)\
Föhringer Ring 6, 80805 München, Germany
author:
- Martin Ammon
- Johanna Erdmenger
- Viviane Grass
- Patrick Kerner
- 'Andy O’Bannon'
title: 'On Holographic p-wave Superfluids with Back-reaction'
---
Gauge/gravity duality, Black Holes, Phase transitions 11.25.Tq, 04.70.Bw, 05.70.Fh
Introduction
============
The Anti-de Sitter/Conformal Field Theory correspondence (AdS/CFT) [@Maldacena:1997re] provides a novel method for studying strongly-coupled systems at finite density. As such, it may have useful applications in condensed matter physics, especially for studying scale-invariant strongly-coupled systems, for example, low-temperature systems near quantum criticality (see for example refs.[@Herzog:2009xv; @Hartnoll:2009sz; @McGreevy:2009xe] and references therein). Such systems are not purely theoretical: the (thermo)dynamics of some high-$T_c$ superconductors may be controlled by a quantum critical point.
AdS/CFT is a holographic duality: it equates a weakly-coupled theory of gravity in $(d+1)$-dimensional AdS space with a strongly-coupled $d$-dimensional CFT “living” at the AdS boundary. CFT states with finite temperature are dual to black hole geometries, where the Hawking temperature of the black hole is identified with the temperature in the CFT [@Witten:1998zw]. The current of a global $U(1)$ symmetry in the CFT will be dual to a $U(1)$ gauge field in AdS space. AdS/CFT thus allows us to compute observables in a strongly-coupled CFT, in states with finite temperature and density, by studying asymptotically AdS charged black holes. By now AdS/CFT can model many basic phenomena in condensed matter physics, such as the quantum Hall effect [@Davis:2008nv], non-relativistic scale-invariance [@Son:2008ye; @Balasubramanian:2008dm], and Fermi surfaces [@Rey:2008zz; @Liu:2009dm; @Cubrovic:2009ye].
AdS/CFT can also describe phase transitions to superfluid states, that is, phase transitions in which a sufficiently large $U(1)$ charge density triggers spontaneous breaking of the $U(1)$ symmetry: an operator charged under the $U(1)$ acquires a nonzero expectation value [@Gubser:2008px; @Hartnoll:2008vx; @Hartnoll:2008kx]. We will refer to this as the operator “condensing.” The simplest bulk action that can describe such a transition is Einstein-Maxwell theory coupled to a charged scalar. In the bulk, a charged black hole develops scalar hair. In the CFT, a charged scalar operator condenses.
A simple bulk action has one great virtue, namely a kind of universality: the results may be true for many different dual CFT’s, independent of the details of their dynamics. For the Einstein-Maxwell-scalar case, a fruitful exercise is to study various functional forms for the scalar potential and to scan through values of couplings in that potential [@Franco:2009yz; @Franco:2009if; @Aprile:2009ai]. Generally speaking, scanning through values of these parameters corresponds to scanning through many different dual CFT’s. As shown in refs. [@Franco:2009yz; @Franco:2009if; @Aprile:2009ai], such changes can have a dramatic effect, for example, the phase transition can change from second to first order.
AdS/CFT can also describe superfluid states in which the condensing operator is a vector and hence rotational symmetry is broken, that is, p-wave superfluid states [@Gubser:2008wv; @Herzog:2009ci]. Here the CFT has a global $SU(2)$ symmetry and hence three conserved currents $J^{\mu}_a$, where $a=1,2,3$ label the generators of $SU(2)$. For a sufficiently large charge density for some $U(1)$ subgroup of $SU(2)$, say a sufficiently large $\langle J^t_3\rangle$, holographic calculations reveal that, of the known available states, those with lowest free energy have a nonzero $\langle J^x_1 \rangle$. Not only is the $U(1)$ broken, but spatial rotational symmetry is also broken to some subgroup.
On the AdS side, a simple bulk action that can describe such a transition is Einstein-Yang-Mills theory with gauge group $SU(2)$. CFT states with nonzero $\langle J^t_3\rangle$ are dual to black hole solutions with nonzero vector field $A_t^3(r)$ in the bulk, where $r$ is the radial coordinate of AdS space. States with nonzero $\langle J^x_1\rangle$ are dual to black hole solutions with a nontrivial $A_x^1(r)$. The superfluid phase transition is thus dual to charged AdS black holes developing vector hair. A string theory realization for this model is given in refs. [@Ammon:2008fc; @Basu:2008bh; @Ammon:2009fe; @Peeters:2009sr].
Unlike the Einstein-Maxwell-scalar case, $SU(2)$ Einstein-Yang-Mills theory has only a *single* free parameter, $\alpha\equiv \k / \hat{g}$, where $\k$ is the gravitational constant (we will work in $(4+1)$ dimensions, hence the subscript) and $\hat{g}$ is the Yang-Mills coupling. The Yang-Mills source terms on the right-hand-side of Einstein’s equation are proportional to $\alpha^2$. To date, most analyses of the holographic p-wave superfluid transition have employed the so-called probe limit, which consists of taking $\alpha \ra 0$ so that the gauge fields have no effect on the geometry, which becomes simply AdS-Schwarzschild. The probe limit was sufficient to show that a superfluid phase transition occurs and is second order.
Our goal is to study the effect of finite $\alpha$, that is, to study the back-reaction of the gauge fields on the metric. We will work with $(4+1)$-dimensional $SU(2)$ Einstein-Yang-Mills theory, with finite $\alpha$. We will numerically construct asymptotically AdS charged black hole solutions with vector hair (for similar studies see refs. [@Gubser:2008zu; @Basu:2009vv; @Manvelyan:2008sv]). Our principal result is that for a sufficiently large value of $\alpha$ the phase transition becomes first order. More specifically, we find a critical value $\alpha_c=0.365\pm0.001$ in units of the AdS radius, such that the transition is second order when $\alpha<\alpha_c$ and first order when $\alpha>\alpha_c$.
We can provide some intuition for what increasing $\alpha$ means, in CFT terms, as follows. Generically, in AdS/CFT $1/\k^2 \propto c$, where $c$ is the central charge of the CFT [@Henningson:1998gx; @Balasubramanian:1999re], which, roughly speaking, counts the total number of degrees of freedom in the CFT. Correlation functions involving the $SU(2)$ current will generically be proportional to $1/\hat{g}^2$ [@Herzog:2009xv; @Hartnoll:2009sz; @McGreevy:2009xe]. We may (again roughly) think of $1/\hat{g}^2$ as counting the number of degrees of freedom in the CFT that carry $SU(2)$ charge. Intuitively, then, in the CFT increasing $\alpha$ means increasing the ratio of charged degrees of freedom to total degrees of freedom.
The paper is organized as follows. In section \[setup\] we write the action of our model and discuss our ansatz for the bulk fields. In section \[thermo\] we describe how to extract thermodynamic information from our solutions. In section \[phasetrans\] we present numerical results demonstrating that increasing $\alpha$ changes the order of the phase transition. We conclude with some discussion and suggestions for future research in section \[discuss\].
Holographic Setup {#setup}
=================
We consider $SU(2)$ Einstein-Yang-Mills theory in $(4+1)$-dimensional asymptotically AdS space. The action is $$\label{eq:action}
S = \int\!{\mathrm{d}}^5x\,\sqrt{-g} \, \left [ \frac{1}{2\k^2} \left( R -\Lambda\right) - \frac{1}{4\hat g^2} \, F^a_{\mu\nu} F^{a\mu \nu} \right] + S_{bdy}\,,$$ where $\k$ is the five-dimensional gravitational constant, $\Lambda = - \frac{12}{L^2}$ is the cosmological constant, with $L$ being the AdS radius, and $\hat g$ is the Yang-Mills coupling constant. The $SU(2)$ field strength $F^a_{\mu\nu}$ is $$F^a_{\mu\nu}=\del_\mu A^a_\nu -\del_\nu A^a_\mu + \epsilon^{abc}A^b_\mu A^c_\nu \,,$$ where $\mu, \nu = \{t,r,x,y,z\}$, with $r$ being the AdS radial coordinate, and $\epsilon^{abc}$ is the totally antisymmetric tensor with $\epsilon^{123}=+1$. The $A^a_\mu$ are the components of the matrix-valued gauge field, $A=A^a_\mu\tau^a dx^\mu$, where the $\tau^a$ are the $SU(2)$ generators, which are related to the Pauli matrices by $\tau^a=\sigma^a/2i$. $S_{bdy}$ includes boundary terms that do not affect the equations of motion, namely the Gibbons-Hawking boundary term as well as counterterms required for the on-shell action to be finite. We will write $S_{bdy}$ explicitly in section \[thermo\].
The Einstein and Yang-Mills equations derived from the above action are $$\begin{aligned}
\label{eq:einsteinEOM}
R_{\mu \nu}+\frac{4}{L^2}g_{\mu \nu}&=\k^2\left(T_{\mu\nu}-\frac{1}{3}{T_{\rho}}^{\rho}g_{\mu\nu}\right)\,, \\
\label{eq:YangMillsEOM}
\nabla_\mu F^{a\mu\nu}&=-\epsilon^{abc}A^b_\mu F^{c\mu\nu} \,,\end{aligned}$$ where the Yang-Mills stress-energy tensor $T_{\mu\nu}$ is $$\label{eq:energymomentumtensor}
T_{\mu \nu}=\frac{1}{\hat{g}^2}{\rm tr}\left(F_{\rho\mu}{F^{\rho}}_{\nu}-\frac{1}{4}g_{\mu\nu} F_{\rho\sigma}F^{\rho\sigma}\right)\,.$$
Following ref. [@Gubser:2008wv], to construct charged black hole solutions with vector hair we choose a gauge field ansatz $$\label{eq:gaugefieldansatz}
A=\phi(r)\tau^3{\mathrm{d}}t+w(r)\tau^1{\mathrm{d}}x\,.$$ The motivation for this ansatz is as follows. In the field theory we will introduce a chemical potential for the $U(1)$ symmetry generated by $\tau^3$. We will denote this $U(1)$ as $U(1)_3$. The bulk operator dual to the $U(1)_3$ density is $A^3_t$, hence we include $A^3_t(r) \equiv \phi(r)$ in our ansatz. We want to allow for states with a nonzero $\langle J^x_1\rangle$, so in addition we introduce $A^1_x(r) \equiv w(r)$. With this ansatz for the gauge field, the Yang-Mills stress-energy tensor in eq. is diagonal. Solutions with nonzero $w(r)$ will preserve only an $SO(2)$ subgroup of the $SO(3)$ rotational symmetry, so our metric ansatz will respect only $SO(2)$. Furthermore, given that the Yang-Mills stress-energy tensor is diagonal, a diagonal metric is consistent. We will also pattern our metric ansatz after the ones used in ref. [@Manvelyan:2008sv] since these tame singular points in the equations of motion. Our metric ansatz is $$\label{eq:metricansatz}
{\mathrm{d}}s^2 = -N(r)\sigma(r)^2{\mathrm{d}}t^2 + \frac{1}{N(r)}{\mathrm{d}}r^2 +r^2 f(r)^{-4}{\mathrm{d}}x^2 + r^2f(r)^2\left({\mathrm{d}}y^2 + {\mathrm{d}}z^2\right)\,,$$ with $N(r)=-\frac{2m(r)}{r^2}+\frac{r^2}{L^2}$. For our black hole solutions we will denote the position of the horizon as $r_h$. The AdS boundary will be at $r\rightarrow\infty$.
Inserting our ansatz into the Einstein and Yang-Mills equations yields five equations of motion for $m(r),\,\sigma(r),\,f(r),\,\phi(r),\,w(r)$ and one constraint equation from the $rr$ component of the Einstein equations. The dynamical equations can be recast as (prime denotes $\frac{\partial}{\partial r}$)
$$\label{eom}
\begin{split}
m' &= \frac{\alpha^2 r f^4 w^2 \phi^2}{6 N \sigma^2} + \frac{\alpha^2 r^3 {\phi'}^2}{6 \sigma^2} + N\left(\frac{r^3{f'}^2}{f^2} + \frac{\alpha^2}{6} r f^4 {w'}^2\right) \,, \\ \sigma' &= \frac{\alpha^2 f^4 w^2 \phi^2}{3 r N^2 \sigma} + \sigma\left(\frac{2 r {f'}^2}{f^2} + \frac{\alpha^2 f^4 {w'}^2}{3 r}\right) \,, \\ f'' &= -\frac{\alpha^2 f^5 w^2 \phi^2}{3 r^2 N^2 \sigma^2} + \frac{\alpha^2 f^5 {w'}^2}{3 r^2} - f'\left(\frac{3}{r} - \frac{f'}{f} + \frac{N'}{N} +\frac{\sigma'}{\sigma}\right) \,, \\ \phi'' &= \frac{f^4 w^2 \phi}{r^2 N} - \phi'\left(\frac{3}{r} + \frac{\sigma'}{\sigma}\right) \,, \\ w'' &= -\frac{w \phi^2}{N^2 \sigma^2} - w'\left( \frac{1}{r} + \frac{4 f'}{f} + \frac{N'}{N} + \frac{\sigma'}{\sigma} \right). \,
\end{split}$$
The equations of motion are invariant under four scaling transformations (invariant quantities are not shown), $$\begin{aligned}
(I) & \sigma\rightarrow \lambda\sigma, \qquad \phi\rightarrow \lambda\phi,& \nonumber \\ (II)& f\rightarrow \lambda f, \qquad w\rightarrow \lambda^{-2} w,& \nonumber \\ (III) & r\rightarrow \lambda r\,, \quad m\rightarrow \lambda^4 m \,, \quad w\rightarrow \lambda w \,, \quad \phi\rightarrow \lambda\phi, \, & \nonumber \\ (IV) & r\rightarrow \lambda r\,,\quad m\rightarrow \lambda^2 m\,, \quad L\rightarrow \lambda L\,,\quad \phi\rightarrow \frac{\phi}{\lambda}\,, \quad \alpha\rightarrow \lambda \alpha, & \nonumber\end{aligned}$$ where in each case $\lambda$ is some real positive number. Using (I) and (II) we can set the boundary values of both $\sigma(r)$ and $f(r)$ to one, so that the metric will be asymptotically AdS. We are free to use (III) to set $r_h$ to be one, but we will retain $r_h$ as a bookkeeping device. We will use (IV) to set the AdS radius $L$ to one.
A known analytic solution of the equations of motion is an asymptotically AdS Reissner-Nordström black hole, which has $\phi(r)=\mu - q/r^2$, $w(r)=0$, $\sigma(r)=f(r)=1$, and $N(r)= \left(r^2 - \frac{2m_0}{r^2} + \frac{2\alpha^2 q^2}{3 r^4}\right)$, where $m_0=\frac{r_h^4}{2}+\frac{\alpha^2 q^2}{3r_h^2}$ and $q= \mu r^2_h$. Here $\mu$ is the value of $\phi(r)$ at the boundary, which in CFT terms is the $U(1)_3$ chemical potential.
To find solutions with nonzero $w(r)$ we resort to numerics. We will solve the equations of motion using a shooting method. We will vary the values of functions at the horizon until we find solutions with suitable values at the AdS boundary. We thus need the asymptotic form of solutions both near the horizon $r=r_h$ and near the boundary $r=\infty$.
Near the horizon, we define $\epsilon_h\equiv\frac{r}{r_h}-1\ll 1$ and then expand every function in powers of $\epsilon_h$ with some constant coefficients. Two of these we can fix as follows. We determine $r_h$ by the condition $N(r_h)=0$, which gives that $m(r_h)=r_h^4/2$. Additionally, we must impose $A^3_t(r_h)=\phi(r_h)=0$ for $A$ to be well-defined as a one-form (see for example ref. [@Kobayashi:2006sb]). The equations of motion then impose relations among all the coefficients. A straightforward exercise shows that only four coefficients are independent, $$\left\{\phi^h_1, \sigma^h_0, f^h_0, w^h_0\right\} \,,$$ where the subscript denotes the order of $\epsilon_h$ (so $\sigma^h_0$ is the value of $\sigma(r)$ at the horizon, etc.). All other near-horizon coefficients are determined in terms of these four.
Near the boundary $r=\infty$ we define $\epsilon_b\equiv \left(\frac{r_h}{r}\right)^2\ll1$ and then expand every function in powers of $\epsilon_b$ with some constant coefficients. The equations of motion again impose relations among the coefficients. The independent coefficients are $$\label{coeffb}
\left\{m^b_0, \mu, \phi^b_1, w^b_1, f^b_2\right\} \,,$$ where here the subscript denotes the power of $\epsilon_b$. All other near-boundary coefficients are determined in terms of these.
We used scaling symmetries to set $\sigma_0^b = f_0^b=1$. Our solutions will also have $w_0^b=0$ since we do not want to source the operator $J^x_1$ in the CFT ($U(1)_3$ will be *spontaneously* broken). In our shooting method we choose a value of $\mu$ and then vary the four independent near-horizon coefficients until we find a solution which produces the desired value of $\mu$ and has $\sigma_0^b = f_0^b=1$ and $w_0^b=0$.
In what follows we will often work with dimensionless coefficients by scaling out factors of $r_h$. We thus define the dimensionless functions $\mt(r)\equiv m(r)/r_h^4$, $\tilde\phi(r)\equiv \phi(r)/r_h$ and $\tilde w(r)\equiv w(r)/r_h$, while $f(r)$ and $\sigma(r)$ are already dimensionless.
Thermodynamics {#thermo}
==============
Next we will describe how to extract thermodynamic information from our solutions. Our solutions describe thermal equilibrium states in the dual CFT. We will work in the grand canonical ensemble, with fixed chemical potential $\mu$.
We can obtain the temperature and entropy from horizon data. The temperature $T$ is given by the Hawking temperature of the black hole, $$\label{eq:temperature}
T=\frac{\kappa}{2\pi}=\frac{\sigma^h_0}{12\pi}\left(12-\alpha^2 \frac{{(\phit^h_1)}^2}{{\sigma^h_0}^2}\right)\,r_h\,.$$ Here $\kappa=\left . \sqrt{\del_\mu \xi \del^\mu \xi} \right |_{r_h}$ is the surface gravity of the black hole, with $\xi$ being the norm of the timelike Killing vector, and in the second equality we write $T$ in terms of near-horizon coefficients. In what follows we will often convert from $r_h$ to $T$ simply by inverting the above equation. The entropy $S$ is given by the Bekenstein-Hawking entropy of the black hole, $$\label{eq:entropy}
S=\frac{2\pi}{\k^2}A_h=\frac{2\pi V}{\k^2}r_h^3= \frac{2\pi^4}{\k^2}VT^3
\frac{12^3{\sigma_0^h}^3}{\left(12{\sigma_0^h}^2-{(\phit_1^h)}^2\alpha^2\right)^3}\,,$$ where $A_h$ denotes the area of the horizon and $V = \int\!{\mathrm{d}}^3x$.
The central quantity in the grand canonical ensemble is the grand potential $\Omega$. In AdS/CFT we identify $\Omega$ with $T$ times the on-shell bulk action in Euclidean signature. We thus analytically continue to Euclidean signature and compactify the time direction with period $1/T$. We denote the Euclidean bulk action as $I$ and $I_{\text{on-shell}}$ as its on-shell value (and similarly for other on-shell quantities). Our solutions will always be static, hence $I_{\text{on-shell}}$ will always include an integration over the time direction, producing a factor of $1/T$. To simplify expressions, we will define $I \equiv \tilde{I}/T$. Starting now, we will refer to $\tilde{I}$ as the action. $\tilde{I}$ includes a bulk term, a Gibbons-Hawking boundary term, and counterterms, $$\label{eq:renomaction}
\tilde{I}=\tilde{I}_{\text{bulk}}+\tilde{I}_{\text{GH}}+\tilde{I}_{\text{CT}}\,.$$ $\tilde{I}_{\text{bulk}}^{\text{on-shell}}$ and $\tilde{I}_{GH}^{\text{on-shell}}$ exhibit divergences, which are canceled by the counterterms in $\tilde{I}_{\text{CT}}$. To regulate these divergences we introduce a hypersurface $r=r_{bdy}$ with some large but finite $r_{bdy}$. We will always ultimately remove the regulator by taking $r_{bdy}\rightarrow\infty$. Using the equations of motion, for our ansatz $\tilde{I}_{\text{bulk}}^{\text{on-shell}}$ is $$\tilde{I}_{\text{bulk}}^{\text{on-shell}}=\frac{V}{\k^2} \,\frac{1}{2f^2}r N \sigma (r^2 f^2)' \Big|_{r=r_{\text{bdy}}} \,.$$ For our ansatz, the Euclidean Gibbons-Hawking term is $$\label{eq:GHterm}
\tilde{I}_{\text{GH}}^{\text{on-shell}}=-\frac{1}{\k^2}\int\!{\mathrm{d}}^3x\sqrt{\gamma}\,\nabla_\mu n^\mu=-\frac{V}{\k^2}N\sigma r^3\left(\frac{N'}{2N}+\frac{\sigma'}{\sigma}+\frac3r\right)\Big|_{r=r_{\text{bdy}}}\,,$$ where $\gamma$ is the induced metric on the $r=r_{bdy}$ hypersurface and $n_\mu{\mathrm{d}}x^\mu=1/\sqrt{N(r)}\,{\mathrm{d}}r$ is the outward-pointing normal vector. The only divergence in $\tilde{I}_{\text{bulk}}^{\text{on-shell}} + \tilde{I}_{\text{GH}}^{\text{on-shell}}$ comes from the infinite volume of the asymptotically AdS space, hence, for our ansatz, the only nontrivial counterterm is $$\label{eq:counterterms}
\tilde{I}_{\text{CT}}^{\text{on-shell}}=\frac{3}{\k^2}\int\!{\mathrm{d}}^3x\sqrt{\gamma}=\frac{3 V}{\k^2}r^3\sqrt{N}\sigma\Big|_{r=r_{\text{bdy}}}\,.$$ Finally, $\Omega$ is related to the on-shell action, $\tilde{I}_{\text{on-shell}}$, as $$\Omega = \lim_{r_{bdy}\rightarrow\infty} \tilde{I}_{\text{on-shell}}.$$
The chemical potential $\mu$ is simply the boundary value of $A^3_t(r) = \phi(r)$. The charge density $\langle J^t_3\rangle$ of the dual field theory can be extracted from $\tilde{I}_{\text{on-shell}}$ by $$\begin{aligned}
\langle J^t_3\rangle= \frac{1}{V} \, \lim_{r_{bdy}\rightarrow \infty} \frac{\delta \tilde{I}_{\text{on-shell}}}{\delta A_t^3(r_{bdy})} =-\frac{2\pi^3\alpha^2}{\k^2}T^3
\frac{12^3{\sigma_0^h}^3}{\left(12{\sigma_0^h}^2-{(\phit_1^h)}^2\alpha^2\right)^3} \, \tilde{\phi}^b_1 \,.\end{aligned}$$ Similarly, the current density $\langle J^x_1\rangle$ is $$\begin{aligned}
\langle J^x_1\rangle= \frac{1}{V} \, \lim_{r_{bdy}\rightarrow \infty} \frac{\delta \tilde{I}_{\text{on-shell}}}{\delta A_x^1(r_{bdy})} =+\frac{2\pi^3\alpha^2}{\k^2}T^3
\frac{12^3{\sigma_0^h}^3}{\left(12{\sigma_0^h}^2-{(\phit_1^h)}^2\alpha^2\right)^3} \, \tilde{w}^b_1 \,.\end{aligned}$$
The expectation value of the stress-energy tensor of the CFT is [@Balasubramanian:1999re; @deHaro:2000xn] $$\label{eq:energymombdy}
\langle T_{ij}\rangle=\lim_{r_{bdy}\rightarrow \infty} \frac{2}{\sqrt{\gamma}}\frac{\delta \tilde{I}_{\text{on-shell}}}{\delta \gamma^{ij}}= \lim_{r_{bdy}\rightarrow \infty} \left [ \frac{r^2}{\k^2}
\left(-K_{ij}+{K^l}_l\gamma_{ij}-3\,\gamma_{ij}\right) \right ]_{r=r_{\text{bdy}}} \,,$$ where $i, j, l = \{t,x,y,z\}$ and $K_{ij}= \frac{1}{2} \sqrt{N(r)} \, \partial_r \gamma_{ij}$ is the extrinsic curvature. We find $$\label{eq:cftstressenergytensor}
\begin{split}
\langle T_{tt} \rangle&=3\frac{\pi^4}{\k^2}VT^4\frac{12^4{\sigma_0^h}^4}{\left(12{\sigma_0^h}^2-{(\phit_1^h)}^2\alpha^2\right)^4}\,\mt^b_0 \,,\\
\langle T_{xx} \rangle&= \frac{\pi^4}{\k^2}VT^4\frac{12^4{\sigma^h_0}^4}{\left(12{\sigma^h_0}^2-{(\phit_1^h)}^2\alpha^2\right)^4}\left(\mt^b_0-8f_2^b\right)\,,\\
\langle T_{yy} \rangle = \langle T_{zz} \rangle&= \frac{\pi^4}{\k^2}VT^4\frac{12^4{\sigma^h_0}^4}{\left(12{\sigma^h_0}^2-{(\phit_1^h)}^2\alpha^2\right)^4}\left(\mt^b_0+4f_2^b\right)\,.
\end{split}$$ Notice that $\langle T_{tx} \rangle = \langle T_{ty} \rangle = \langle T_{tz}
\rangle = 0$. Even in phases where the current $\langle J_1^x \rangle$ is nonzero, the fluid will have zero net momentum. Indeed, this result is guaranteed by our ansatz for the gauge field which implies a diagonal Yang-Mills stress-energy tensor and a diagonal metric (the spacetime is static).
For $\mt^b_0=\frac12+\frac{\alpha^2 \tilde\mu^2}{3}$, $\sigma_0^h=1$, ${\phit_1^h}=2\tilde\mu$, $f_2^b=0$, and $\tilde\phi^b_1=-\tilde\mu$ we recover the correct thermodynamic properties of the Reissner-Nordström black hole, which preserves the $SO(3)$ rotational symmetry. For example, we find that $\langle T_{xx} \rangle=\langle T_{yy} \rangle = \langle T_{zz} \rangle$ and $\Omega=-\langle T_{yy}\rangle$. For solutions with nonzero $\langle J_1^x\rangle$, the $SO(3)$ is broken to $SO(2)$. In these cases, we find that $\langle T_{xx} \rangle \neq \langle T_{yy} \rangle = \langle T_{zz} \rangle$. Just using the equations above, we also find $\Omega = - \langle T_{yy} \rangle$. In the superfluid phase, both the nonzero $\langle J^x_1 \rangle$ and the stress-energy tensor indicate breaking of $SO(3)$.
Tracelessness of the stress-energy tensor (in Lorentzian signature) implies $\langle T_{tt} \rangle= \langle T_{xx} \rangle + \langle T_{yy} \rangle + \langle T_{zz} \rangle$, which is indeed true for eq. (\[eq:cftstressenergytensor\]), so in the dual CFT we always have a conformal fluid. The only physical parameter in the CFT is thus the ratio $\mu/T$.
Phase Transitions {#phasetrans}
=================
In this section we present our numerical results. We scanned through values of $\alpha$ from $\alpha=0.032$ to $\alpha=0.548$. Typical solutions for the metric and gauge field functions appear in Figure \[fig:metricgaugealpha0.1\]. The solutions for other values of $\alpha$ are qualitatively similar. Notice that all boundary conditions are met: at the horizon $\phit(r)$ vanishes, and at the boundary$f_0^b=\sigma_0^b=1$ and $\tilde{w}_0^b=0$.
For every value of $\alpha$ that we use, we find Reissner-Nordström solutions for all temperatures, and for sufficiently low temperatures we always find additional solutions, with nonzero $w(r)$, that are thermodynamically preferred to the Reissner-Nordström solution. In other words, for every value of $\alpha$ that we use, we find a phase transition, at some temperature $T_c$, in which a charged black hole grows vector hair, which in the CFT is a p-wave superfluid phase transition. Our numerical results show that the phase transition is second order for $\alpha<\alpha_c$ and first order for $\alpha>\alpha_c$ where $\alpha_c\approx 0.365\pm0.001$.
For example, for $\alpha=0.316<\alpha_c$, we only find solutions with $\langle J^x_1 \rangle = 0$ until a temperature $T_c$ where a second set of solutions, with nonzero $\langle J^x_1\rangle$, appears. Figure \[fig:cond\] shows that $\langle J^x_1 \rangle$ rises continuously from zero as we decrease $T$ below $T_c$.
![The order parameter $\langle J^x_1\rangle$, multiplied by $\k^2/(2\pi^3\alpha^2T_c^3)$, versus the rescaled temperature $T/T_c$ for different $\alpha$: $\alpha=0.032<\alpha_c$ (green dotted), $\alpha=0.316<\alpha_c$ (blue solid) and $\alpha=0.447>\alpha_c$ (red dashed). The black dot-dashed curve is the function $a(1-T/T_c)^{1/2}$ with $a=160$. The green dotted curve is scaled up by a factor of $8$ while the red dashed curve is scaled down by a factor of $5$ such that $a$, which depends on $\alpha$, coincides for the green dotted and blue solid curves. If we decrease $T$ toward $T_c$, entering the figure from the right, we see that the blue solid and the green dotted curves rise continuously and monotonically from zero at $T=T_c$, signaling a second-order phase transition. The close agreement with the black dot-dashed curve suggests that these grow from zero as $\left(1-T/T_c\right)^{1/2}$. In the $\alpha=0.447$ case, the red dashed curve becomes multi-valued at $T=1.061\,T_c$. In this case, at $T=T_c$, the value of $\k^2\langle J^x_1 \rangle/(2\pi^3\alpha^2T_c^3)$ jumps from zero to the upper part of the red dashed curve, signaling a first-order transition.[]{data-label="fig:cond"}](./condensate.pdf){width="0.7\linewidth"}
Figure \[fig:thermoalpha0.1\] (a) shows the grand potential $\Omega$, divided by $\pi^4 V T_c^4/\k^2$, versus the rescaled temperature $T/T_c$ for $\alpha=0.316$. The blue solid curve in Figure \[fig:thermoalpha0.1\] (a) comes from solutions with $\langle J^x_1 \rangle = 0$ and the red dashed curve comes from solutions with $\langle J^x_1 \rangle \neq 0$.
We see clearly that at $T<T_c$ the states with $\langle J^x_1 \rangle\neq0$ have the lower $\k^2\Omega/\left(\pi^4 V T_c^4\right)$ and hence are thermodynamically preferred. We thus conclude that a phase transition occurs at $T=T_c$. The nonzero $\langle J^x_1\rangle$ indicates spontaneous breaking of $U(1)_3$ and of $SO(3)$ rotational symmetry down to $SO(2)$, and hence is an order parameter for the transition. Figure \[fig:thermoalpha0.1\] (b) shows the entropy $S$, divided by $2\pi^4 V T_c^3/\k^2$, versus the rescaled temperature $T/T_c$ for $\alpha=0.316$. The blue solid curve and the red dashed curve have the same meaning as in Figure \[fig:thermoalpha0.1\] (a). Here we see that $\k^2S/\left(2\pi^4 V T_c^3\right)$ is continuous but has a kink, *i.e.* a discontinuous first derivative, clearly indicating a second-order transition. For other values of $\alpha<\alpha_c$, the figures are qualitatively similar.
A good question concerning these second-order transitions is: what are the critical exponents? In the probe limit, $\alpha=0$, an analytic solution for the gauge fields exists for $T$ near $T_c$ [@Basu:2008bh], which was used in ref. [@Herzog:2009ci] to show that for $T \lesssim T_c$, $\langle J^x_1 \rangle
\propto \left( 1- T/T_c \right)^{1/2}$. In other words, in the probe limit the critical exponent for $\langle J^x_1\rangle$ takes the mean-field value $1/2$. Does increasing $\alpha$ change the critical exponent? Our numerical evidence suggests that the answer is no: for all $\alpha < \alpha_c$, we appear to find $\langle J^x_1 \rangle \propto \left( 1-T/T_c\right)^{1/2}$ (see Figure \[fig:cond\]).
As $\alpha$ increases past $\alpha_c=0.365\pm0.001$, we see a qualitative change in the thermodynamics. Consider for example $\alpha = 0.447$. Here again we only find solutions with $\langle J^x_1 \rangle=0$ down to some temperature where *two* new sets of solutions appear, both with nonzero $\langle J^x_1\rangle$. In other words, three states are available to the system: one with $\langle J^x_1\rangle=0$ and two with nonzero $\langle J^x_1 \rangle$. Figure \[fig:cond\] shows that as we cool the system, $\langle J^x_1 \rangle$ becomes multi-valued at $T=1.061 \, T_c$. To determine which state is thermodynamically preferred, we compute the grand potential $\Omega$. Figure \[fig:thermoalpha0.2\] (a) shows $\k^2\Omega/\left(\pi^4 V T_c^4\right)$ versus $T/T_c$. The blue solid curve and the red dashed curve have the same meanings as in Figure \[fig:thermoalpha0.1\].
We immediately see the characteristic “swallowtail” shape of a first-order phase transition. If we decrease $T$, entering the figure along the blue solid curve from the right, we reach the temperature $T=1.061\,T_c$ where the new solutions appear (as the red dashed curve). The blue solid curve still has the lowest $\k^2\Omega/\left(\pi^4 V T_c^4\right)$ until $T=T_c$ (by definition). If we continue reducing $T$ below $T_c$, then the red curve has the lowest $\k^2\Omega/\left(\pi^4 V T_c^4\right)$. The transition is clearly first order: $\k^2\Omega/\left(\pi^4 V T_c^4\right)$ has a kink at $T=T_c$. We can also see from the entropy that the transition is first order. Figure \[fig:thermoalpha0.2\] (b) shows $\k^2S/\left(2\pi^4 V T_c^3\right)$ versus $T/T_c$. The entropy, like the grand potential, is multi-valued, and jumps discontinuously from the blue solid curve to the lowest part of the red dashed curve at $T=T_c$, indicating a first-order transition.
Notice that a crucial difference between $\alpha<\alpha_c$ (second order) and $\alpha>\alpha_c$ (first order) is that for $\alpha>\alpha_c$ the critical temperature $T_c$ is not simply the temperature at which $\langle J^x_1\rangle$ becomes nonzero. We need more information to determine $T_c$ when $\alpha>\alpha_c$, for example we can study $\Omega$.
A good question is: how does increasing $\alpha$ change $T_c$? Table \[table:alphatc\] shows several values of $\alpha$ and the associated $T_c$ in units of fixed $\mu$. In the probe limit, $\alpha=0$, we have the analytic result from ref. [@Herzog:2009ci] that $T_c/\mu = 1/4\pi \approx 7.96 \times 10^{-2}$. For finite $\alpha$, the general trend is that $T_c$ decreases as we increase $\alpha$.
$\alpha$ $T_c/\mu \times 10^2$
---------- -----------------------
$0$ $7.96$
$0.032$ $7.92$
$0.316$ $4.57$
$0.364$ $3.62$
$0.447$ $2.18$
: Values of $T_c/\mu$, scaled up by a factor of $10^2$, for various values of $\alpha$. Recall that the critical value of $\alpha$ is $\alpha_c = 0.365\pm0.001$.[]{data-label="table:alphatc"}
Discussion and Outlook {#discuss}
======================
We studied asymptotically AdS charged black holes in $(4+1)$-dimensional $SU(2)$ Einstein-Yang-Mills theory with finite $\alpha = \k/\hat{g}$, that is, with back-reaction of the gauge fields. Our numerical solutions show that, for a given value of $\alpha$, as the temperature decreases the black holes grow vector hair. Via AdS/CFT, this process appears as a phase transition to a p-wave superfluid state in a strongly-coupled CFT. We have shown that the order of the phase transition depends on the value of $\alpha$: for values below $\alpha_c=0.365\pm0.001$, the transition is second order, while for larger values the transition is first order.
As we mentioned in the introduction, intuitively we may think of increasing $\alpha$ as increasing the ratio of charged degrees of freedom to total degrees of freedom in the CFT. To make that intuition precise, we can consider a specific system. One string theory realization of $SU(2)$ gauge fields in AdS space is type IIB supergravity in $(4+1)$-dimensional AdS space (times a five-sphere) plus two coincident D7-branes that provide the $SU(2)$ gauge fields [@Ammon:2008fc; @Basu:2008bh; @Ammon:2009fe; @Peeters:2009sr]. The dual field theory is $\N=4$ supersymmetric $SU(N_c)$ Yang-Mills theory, in the limits of large $N_c$ and large ’t Hooft coupling, coupled to a number $N_f=2$ of massless $\N=2$ supersymmetric hypermultiplets in the $N_c$ representation of $SU(N_c)$, *i.e.* flavor fields. The global $SU(N_f)=SU(2)$ is an isospin symmetry. Translating from gravity to field theory quantities, we have $1/\k^2\propto N_c^2$ and $1/\hat{g}^2\propto N_fN_c$, hence $\alpha \propto \sqrt{N_f/N_c}$, which supports our intuition. We must be cautious, however. In the field theory, the probe limit consists of neglecting quantum effects due to the flavor fields because these are suppressed by powers of $N_f/N_c$. If $N_f/N_c$ becomes finite, then, for example, in the field theory the coupling would run, the dual statement being that in type IIB supergravity the dilaton would run, which is an effect absent in our model. We should not draw too close an analogy between our simple model and this particular string theory system.
Returning to our simple system, with fully back-reacted solutions we can potentially answer many questions:
What happens as we increase $\alpha$ further? As $\alpha$ increases, $T_c$ appears to decrease. An obvious question is whether $T_c$ ever becomes zero. The second-order transition occurs because of an instability of the gauge field in the Reissner-Nordström background [@Gubser:2008wv]. At $T=0$, that instability only exists for values of $\alpha$ below some upper bound [@Basu:2009vv]. If the transition was always second-order, then we would conclude that the transition is only possible for $\alpha$ below the bound: if $\alpha$ is greater than the bound, then the instability never appears, even if we cool the system to $T=0$. If the transition becomes first-order, however, then we must rethink the bound: now Reissner-Nordström becomes metastable, and a phase transition occurs, at temperatures above those where the instability appears, so now $T_c$ may go to zero for some $\alpha$ above the bound (if at all).
Does a new scaling symmetry emerge at zero temperature? The zero-temperature analysis of ref. [@Basu:2009vv] suggests that the metric exhibits a new, emergent, scaling symmetry. Our preliminary numerical results suggest that indeed, in the approach to zero temperature, an emergent scaling symmetry appears, of the form suggested in ref. [@Basu:2009vv]. The $T\rightarrow0$ limit is numerically challenging, however, so a firm answer must wait.
What about the transport properties of the dual conformal fluid, for example the electrical conductivity, which at zero temperature should exhibit a “hard gap,” as explained in ref. [@Basu:2009vv]? What is the fluid’s response to nonzero superfluid velocities? In similar systems, sufficiently large superfluid velocities also changed the transition from second to first order [@Basu:2008st; @Herzog:2008he]. What is the speed of sound, which need not be the same in all directions since rotational symmetry is broken, or the speeds of second and fourth sounds [@Herzog:2009ci; @Yarom:2009uq; @Herzog:2009md]?
We plan to investigate these and related questions in the future.
#### ACKNOWLEDGEMENTS
We are grateful to A. Buchel and M. Rangamani for discussions. This work was supported in part by [*The Cluster of Excellence for Fundamental Physics - Origin and Structure of the Universe*]{}. M. A. would also like to thank the Studienstiftung des deutschen Volkes for financial support.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
One way of obtaining a version of quantum mechanics without observers, and thus of solving the paradoxes of quantum mechanics, is to modify the Schrödinger evolution by implementing spontaneous collapses of the wave function. An explicit model of this kind was proposed in 1986 by Ghirardi, Rimini, and Weber (GRW), involving a nonlinear, stochastic evolution of the wave function. We point out how, by focussing on the essential mathematical structure of the GRW model and a clear ontology, it can be generalized to (regularized) quantum field theories in a simple and natural way.
PACS numbers: 03.65.Ta; 03.70.+k. Key words: quantum field theory without observers; Ghirardi–Rimini–Weber model; identical particles; second quantization.
author:
- 'Roderich Tumulka[^1]'
date: 'December 14, 2005'
title: On Spontaneous Wave Function Collapse and Quantum Field Theory
---
Introduction
============
John S. Bell concluded from the quantum measurement problem that “either the wave function, as given by the Schrödinger equation, is not everything or it is not right” [@Belljumps]. Let us assume, for the purpose of this paper, the second option of the alternative: that the Schrödinger equation should be modified in such a way that superpositions of macroscopically different states, as exemplified by Schrödinger’s cat, either cannot arise or cannot persist for more than a fraction of a second. Theories of this kind have come to be known under the names of “dynamical reduction”, “spontaneous localization”, or “spontaneous wave function collapse,” and have been advocated and studied by various authors [@BB66; @Pe76; @Pe79; @Gi84; @grw; @Belljumps; @diosi; @pearle90; @Pen00; @adler; @leggett; @dowker1]; see [@BG03] for an overview. The merit of such theories, which I shall call “collapse theories” in the following, is that they provide, instead of statements about what observers would see if they were to make certain experiments, a possible story about what events objectively occur: they are, in other words, quantum theories without observers. In collapse theories, observations are merely special cases of the objective events, for which the theory, if it is to be empirically adequate, predicts the same distribution of outcomes as the standard quantum formalism. An example of a no-collapse quantum theory without observers is Bohmian mechanics [@DGZ04; @Bellbook].
An explicit collapse theory has been proposed by Ghirardi, Rimini, and Weber (GRW) in 1986 [@grw] for nonrelativistic quantum mechanics of $N$ distinguishable particles. While the testable predictions of the GRW model differ in principle from the quantum formulas, as yet no experiment could decide between the two, [[provided one chooses the collapse rates proportional to the masses,]{}]{} since the differences are too tiny for standard experiments and the experiments leading to noticeable differences are hard to carry out [@BG03]. [[In the GRW model, the wave function obeys the unitary Schrödinger evolution until, at an unforeseeable, random time, it changes discontinuously in an unforeseeable, random way—it collapses. In order to obtain similar models for quantum field theories (QFTs), one path of research, which has been followed under the name “continuous spontaneous localization” (CSL) [@Pe89; @GPR90], is based on the idea of incessant mild collapses, so that the quantum state vector follows a diffusion process in Hilbert space. For a CSL model for identical particles, see [@Pe89].]{}]{}
I will show how [[the GRW model can be extended to QFT in a more direct way; the resulting collapse QFT]{}]{} is as hard to distinguish experimentally from standard QFT as the GRW model from standard quantum mechanics, while solving the quantum measurement problem in the same way as the GRW model. My proposal retains (and indeed is based on) the discreteness of the GRW model; it gets along with indistinguishable particles, both fermions and bosons, and with particle creation and annihilation. However, I will not try here to make the theory Lorentz-invariant; I hope to be able in a future work to combine the construction I present here with the Lorentz-invariant version of the GRW model for $N$ particles that I have described recently [@Tum05]. [[When applied to a system of $N$ identical particles, my proposal yields a collapse process proposed already in 1995 by Dove and Squires [@dovethesis; @DS95], which, however, seems to have received little or no attention so far.]{}]{}
Mathematical Framework of the GRW Model
=======================================
I now describe the GRW model in an unusual way that emphasizes the abstract mathematical structure it is based on and uses an ontology proposed by Bell [@Belljumps; @Bellexact]. According to this *flash ontology*, matter consists of millions of so-called *flashes*, physical events that are mathematically represented by space-time points. The flashes can be thought of as replacing the continuous particle trajectories in space-time postulated by classical mechanics. The flashes are random, thus forming what probabilists would call a point process in space-time, with a distribution determined by the (initial) wave function. The reader may wonder why, in a collapse theory, there is any need at all to introduce space-time objects such as flashes, a question that I will take up in Section \[sec:collapses\]. For now I ask for patience and suggest to regard it as the sole role of the wave function to determine the distribution of the flashes.
We begin with an (arbitrary) Hilbert space ${\mathscr{H}}$ with scalar product ${\langle \phi | \psi \rangle}$, which in the GRW model is $L^2({\mathbb{R}}^{3N})$, and an (arbitrary) self-adjoint operator $H$ on ${\mathscr{H}}$, the Hamiltonian, which in the GRW model is the usual Schrödinger operator $-\tfrac{\hbar^2}{2} \Delta +V$. The third mathematical object that we need, besides ${\mathscr{H}}$ and $H$, is a $$\text{positive-operator-valued function }\Lambda({\boldsymbol{x}})$$ on physical space ${\mathbb{R}}^3$ acting on ${\mathscr{H}}$. (For mathematicians I should add the postulate that there is a dense domain in ${\mathscr{H}}$ on which all of the $\Lambda({\boldsymbol{x}})$ are defined.) $\Lambda$ forms the link between Hilbert space and physical 3-space and can be thought of as representing the (smeared-out) position observable, and in particular as representing a preferred basis. In the GRW model, $\Lambda({\boldsymbol{x}})$ is a multiplication operator $$\label{LambdaGRW}
\Lambda({\boldsymbol{x}}) \, \psi({\boldsymbol{r}}_1, \ldots, {\boldsymbol{r}}_N) = {\mathcal{N}}\,
{\mathrm{e}}^{-({\boldsymbol{x}}- {\boldsymbol{r}}_i)^2/a^2} \, \psi({\boldsymbol{r}}_1, \ldots, {\boldsymbol{r}}_N) \,,$$ multiplying by a Gaussian with width $a/\sqrt{2}$ (in Bell’s [@Belljumps] notation) [[and center ${\boldsymbol{x}}$, where ${\mathcal{N}}$ and $a$ are new constants of nature with suggested values of]{}]{} $${\mathcal{N}}\approx 10^5\, \mathrm{sec}^{-1}\mathrm{m}^{-3}, \quad
a \approx 10^{-7} \, \mathrm{m} \,.$$ [[We are considering for the moment only flashes correspoding to the $i$-th particle coordinate ${\boldsymbol{r}}_i$, $i \in \{1,\ldots,N\}$, and will later consider flashes due to all particles.]{}]{}
The role of $\Lambda$ is to define the rate of a flash to occur, i.e., the probability per unit time. That is why I will call $\Lambda({\boldsymbol{x}})$ the *flash rate operators*. The rate at time $t=0$ of a flash in the set $B \subseteq {\mathbb{R}}^3$ is $$\label{rateB}
{\langle \psi | \Lambda(B)|\psi \rangle} \,,$$ where $\Lambda(B)$ is short for $$\Lambda(B) = \int_B {\mathrm{d}}^3 {\boldsymbol{x}}\, \Lambda({\boldsymbol{x}}) \,.$$ This shows [[why]{}]{} $\Lambda(B)$ must have dimension $1/$time. In the GRW model, the total flash rate is independent of the quantum state $\psi$ since $\Lambda({\mathbb{R}}^3) = \pi^{3/2}{\mathcal{N}}a^3 I$ is a multiple of the identity operator $I$ on ${\mathscr{H}}$. The constant in front of $I$, or, equivalently, the total flash rate, is called $1/\tau$ in Bell’s [@Belljumps] notation, with $$\tau \approx 10^{15} \, \mathrm{sec} \,.$$ [[It has been pointed out by Pearle and Squires [@PS94] that a fixed $\tau$ would contradict the observed stability of nucleons; they suggest instead that the flash rate constant ${\mathcal{N}}a^3$ be proportional to the mass of a particle, rather than a universal constant.]{}]{} In the QFT model we will devise, [[however,]{}]{} $\Lambda({\mathbb{R}}^3)$ will not be a multiple of the identity, and this is the only aspect in which we essentially generalize the mathematical structure of the GRW model.
We now define the probability distribution of the first flash $({{\boldsymbol{X}}}_1,T_1)$ with random location ${{\boldsymbol{X}}}_1$ and random time $T_1$, a probability distribution on the space-time region with $t>0$. The distribution is quadratic in $\psi$. The wish that the rate be given by $\Lambda$ and that the evolution before the flash be given essentially by $H$ leads us to the following form for the distribution: $$\label{1flashdist}
{\mathbb{P}}\bigl( {{\boldsymbol{X}}}_1 \in {\mathrm{d}}^3 {\boldsymbol{x}}, T_1 \in {\mathrm{d}}t \bigr) =
{\langle \psi | W_t^* \Lambda({\boldsymbol{x}}) W_t|\psi \rangle} \, {\mathrm{d}}^3{\boldsymbol{x}}\, {\mathrm{d}}t \,,$$ where the star denotes the adjoint operator, and $$\label{Wdef}
W_t = {\mathrm{e}}^{-\frac{1}{2} \Lambda({\mathbb{R}}^3)t - \frac{{\mathrm{i}}}{\hbar} Ht}
\text{ for }t\geq 0\,, \quad W_t = 0 \text{ for } t<0 \,.$$ Without the $\Lambda({\mathbb{R}}^3)$ term, this would be the ordinary unitary evolution; we need the additional term to keep track of the probability that time $t$ is reached without a flash. Indeed, the definition implies that is a probability distribution:
\[calculation\] $$\begin{aligned}
&\int{\mathrm{d}}^3 {\boldsymbol{x}}\int\limits_0^\infty {\mathrm{d}}t \, {\langle \psi | W_t^* \Lambda({\boldsymbol{x}}) W_t|\psi \rangle} =
\int\limits_0^\infty {\mathrm{d}}t \, {\langle \psi | W_t^* \Lambda({\mathbb{R}}^3) W_t|\psi \rangle} = \\
&= -\int\limits_0^\infty {\mathrm{d}}t \, {\langle \psi | W_t^* \Bigl(- \tfrac{1}{2} \Lambda({\mathbb{R}}^3) +
\tfrac{{\mathrm{i}}}{\hbar} H- \tfrac{1}{2} \Lambda({\mathbb{R}}^3) -
\tfrac{{\mathrm{i}}}{\hbar} H \Bigr) W_t|\psi \rangle} =\\
&= -\int\limits_0^\infty {\mathrm{d}}t \, {\langle \psi | (\dot{W}_t^* W_t +
W_t^* \dot{W}_t ) |\psi \rangle} =\\
&= -\int\limits_0^\infty {\mathrm{d}}t \, \frac{{\mathrm{d}}}{{\mathrm{d}}t} {\langle \psi | W_t^* W_t |\psi \rangle} =
{\langle \psi | W_0^* W_0|\psi \rangle} = {\langle \psi | \psi \rangle} = 1 \,,\end{aligned}$$
provided $W_t \to 0$ as $t \to \infty$, which is the case if the spectrum of $\Lambda({\mathbb{R}}^3)$ is bounded away from zero (as in the GRW case).[^2] The same calculation for a time integral from 0 to $t$ shows that the probability of a flash before $t$ equals $1-\|W_t \psi\|^2$; in particular we can see why $W_t$ should not be unitary. In the GRW case, since $\Lambda({\mathbb{R}}^3)$ is a multiple of the identity and thus commutes with $H$, we find that the exponential splits into a product of two exponentials, $$W_t = {\mathrm{e}}^{-t/2\tau} {\mathrm{e}}^{-{\mathrm{i}}Ht/\hbar} \text{ for }t \geq 0\,.$$
The joint distribution of the first $n$ flashes is defined to be $$\begin{gathered}
\label{nflashdist}
{\mathbb{P}}\bigl( {{\boldsymbol{X}}}_1\in {\mathrm{d}}^3 {\boldsymbol{x}}_1, T_1 \in {\mathrm{d}}t_1, \ldots,
{{\boldsymbol{X}}}_n \in {\mathrm{d}}^3 {\boldsymbol{x}}_n, T_n \in {\mathrm{d}}t_n \bigr) =\\
\bigl\| {K}_n(0,{\boldsymbol{x}}_1, t_1, \ldots, {\boldsymbol{x}}_n, t_n) \, \psi \bigr\|^2
\, {\mathrm{d}}^3 {\boldsymbol{x}}_1 \, {\mathrm{d}}t_1 \cdots {\mathrm{d}}^3 {\boldsymbol{x}}_n \, {\mathrm{d}}t_n \,,\end{gathered}$$ where ${K}_n$ is an operator-valued function on (space-time)$^n$ defined by $$\begin{gathered}
\label{Kndef}
{K}_n(t_0,{\boldsymbol{x}}_1, t_1, \ldots, {\boldsymbol{x}}_n, t_n) =\\
\Lambda({\boldsymbol{x}}_n)^{1/2} \,W_{t_n-t_{n-1}} \Lambda({\boldsymbol{x}}_{n-1})^{1/2} \,W_{t_{n-1}-t_{n-2}}
\cdots \Lambda({\boldsymbol{x}}_1)^{1/2} \, W_{t_1-t_0} \,.\end{gathered}$$ The square-roots exist since the $\Lambda({\boldsymbol{x}})$ are positive operators. Observing that, by a reasoning analogous to , $$\int {\mathrm{d}}^3 {\boldsymbol{x}}_n \int\limits_{t_{n-1}}^\infty {\mathrm{d}}t_n \, {K}^*_n \,{K}_n =
{K}^*_{n-1} \, {K}_{n-1} \,,$$ we see two things: firstly that the right hand side of is a probability distribution on $(\text{space-time})^n$, and secondly that these distributions, for different values of $n$, are marginals of each other, thus forming a consistent family and arising from a joint distribution of infinitely many random variables ${{\boldsymbol{X}}}_1, T_1, {{\boldsymbol{X}}}_2, T_2, \ldots$.
The original GRW model contains one further complication: that collapses can act on different coordinates, as encoded in the particle index $i$ in . Reflecting the fact that the model should account for the quantum mechanics of $N$ *distinguishable* particles, we simply postulate that there are $N$ different *types of flashes*, or, equivalently, that each flash is labeled by an index $i \in \{1,\ldots,N\}$. Correspondingly, we need to be given $N$ positive-operator-valued functions $\Lambda_i({\boldsymbol{x}})$, while ${\mathscr{H}}$ and $H$ are the same for all types of flashes. Thus, with every flash $({{\boldsymbol{X}}}_k,T_k)$ is associated a random label $I_k \in \{1, \ldots, N\}$, and the joint distribution is defined to be $$\begin{gathered}
\label{Nnflashdist}
{\mathbb{P}}\bigl( {{\boldsymbol{X}}}_1\in {\mathrm{d}}^3 {\boldsymbol{x}}_1, T_1 \in {\mathrm{d}}t_1, I_1 = i_1, \ldots,
{{\boldsymbol{X}}}_n \in {\mathrm{d}}^3 {\boldsymbol{x}}_n, T_n \in {\mathrm{d}}t_n, I_n = i_n \bigr) =\\
\bigl\| {K}_n(0,{\boldsymbol{x}}_1, t_1,i_1, \ldots, {\boldsymbol{x}}_n, t_n,i_n) \, \psi \bigr\|^2
\, {\mathrm{d}}^3 {\boldsymbol{x}}_1 \, {\mathrm{d}}t_1 \cdots {\mathrm{d}}^3 {\boldsymbol{x}}_n \, {\mathrm{d}}t_n \,,\end{gathered}$$ where ${K}_n$ is now an operator-valued function on $\bigl[(\text{space-time}) \times \{1,\ldots,N\}\bigr]^n$ defined by $$\begin{gathered}
\label{Knidef}
{K}_n(t_0,{\boldsymbol{x}}_1, t_1,i_1, \ldots, {\boldsymbol{x}}_n, t_n, i_n) =\\
\Lambda_{i_n}({\boldsymbol{x}}_n)^{1/2} \,W_{t_n-t_{n-1}} \Lambda_{i_{n-1}}({\boldsymbol{x}}_{n-1})^{1/2}
\,W_{t_{n-1}-t_{n-2}}
\cdots \Lambda_{i_1}({\boldsymbol{x}}_1)^{1/2} \, W_{t_1-t_0} \,,\end{gathered}$$ with $$\label{WNdef}
W_t = \exp\Bigl(-\tfrac{1}{2} \sum_{i=1}^N \Lambda_i ({\mathbb{R}}^3)t
- \tfrac{{\mathrm{i}}}{\hbar} Ht\Bigr)
\text{ for }t\geq 0\,, \quad W_t = 0 \text{ for } t<0 \,.$$ In the same way as before, one checks that $$\sum_{i_n=1}^N \int {\mathrm{d}}^3 {\boldsymbol{x}}_n \int\limits_{t_{n-1}}^\infty {\mathrm{d}}t_n \, {K}^*_n \,{K}_n =
{K}^*_{n-1} \, {K}_{n-1}$$ and $$\sum_{i_1=1}^N \int {\mathrm{d}}^3 {\boldsymbol{x}}_1 \int\limits_{0}^\infty {\mathrm{d}}t_1 \, {K}^*_1 \,{K}_1 = I \,,$$ implying that is a consistent family of probability distributions. This completes our definition of a theory from ${\mathscr{H}}$, $H$, and $\Lambda_1({\boldsymbol{x}}), \ldots, \Lambda_N({\boldsymbol{x}})$, including the GRW model.
QFT
===
Since ${\mathscr{H}}$ can be taken to be the Hilbert space of a QFT and $H$ its (regularized) Hamiltonian, we get a collapse version of that QFT as soon as we have the flash rate operators $\Lambda({\boldsymbol{x}})$. The model is then defined by , , and . If the QFT contains several species of particles, we may wish to introduce one type of flash for every species, and use eq.s , , and instead.
So what would be a natural choice of $\Lambda({\boldsymbol{x}})$? Since the operator $\Lambda(B)$ determines the flash rate in $B$, it should represent the amount of matter in $B$, smeared out by a Gaussian with width $a/\sqrt{2}$. [[Two natural choices are the particle number density operator $N({\boldsymbol{y}})$ and the mass density operator $M({\boldsymbol{y}})$, that is,]{}]{} $$\label{LambdaN}
\Lambda({\boldsymbol{x}}) = \int {\mathrm{d}}^3 {\boldsymbol{y}}\, {\mathcal{N}}\, {\mathrm{e}}^{-({\boldsymbol{x}}-{\boldsymbol{y}})^2/a^2} \, N({\boldsymbol{y}})\,,$$ [[respectively]{}]{} $$\label{LambdaM}
\Lambda({\boldsymbol{x}}) = \int {\mathrm{d}}^3 {\boldsymbol{y}}\, {\mathcal{M}}\, {\mathrm{e}}^{-({\boldsymbol{x}}-{\boldsymbol{y}})^2/a^2} \, M({\boldsymbol{y}})\,,$$ [[with ${\mathcal{M}}$ a suitable constant. The operators have been considered already in [@GPR90] for CSL in connection with identical particles, albeit apparently in a role more analogous to $\Lambda({\boldsymbol{x}})^{1/2}$ than to $\Lambda({\boldsymbol{x}})$. The choice of instead of corresponds to the proposal of Pearle and Squires [@PS94] to choose the collapse rate proportional to the mass.]{}]{}
[[In general, the number operator $N({\boldsymbol{y}})$ can be expressed in terms of suitable annihilation operators $a({\boldsymbol{y}})$ and creation operators $a^*({\boldsymbol{y}})$ in the position representation, acting on a suitable Fock space ${\mathscr{H}}$, by]{}]{} $$N({\boldsymbol{y}}) = a^*({\boldsymbol{y}}) \, a({\boldsymbol{y}}),$$ [[where the product involves, when appropriate, summation over spin indices. In a nonrelativistic QFT, $a({\boldsymbol{y}})$ is simply the field operator at the location ${\boldsymbol{y}}$. (Of course, since in nonrelativistic quantum theories usually the Hamiltonian does not contain terms creating and annihilating particles, we would have to add such terms artificially to the Hamiltonian to obtain a model with non-conserved particle number.)]{}]{}
For several species [[of particles]{}]{} corresponding to several [[quantum]{}]{} fields, we [[thus]{}]{} obtain several rate density operators $\Lambda_i({\boldsymbol{x}})$.
Second Quantization
===================
I would like to describe another way of constructing flash rate operators $\Lambda({\boldsymbol{x}})$ for QFT. It will turn out equivalent to . It is based on the second quantization algorithm for forming a (bosonic or fermionic) Fock space ${\mathscr{H}}$ from a one-particle Hilbert space ${\mathscr{H}}_{(1)}$, and consists of an algorithm for forming flash rate operators $\Lambda({\boldsymbol{x}})$ acting on Fock space ${\mathscr{H}}$ from flash rate operators $\Lambda_{(1)}({\boldsymbol{x}})$ acting on ${\mathscr{H}}_{(1)}$. This algorithm in turn is based on two procedures, one concerning direct sums of Hilbert spaces and the other tensor products.
On the direct sum ${\mathscr{H}}_{1} \oplus {\mathscr{H}}_{2}$ of two Hilbert spaces, each equipped with a positive-operator-valued function $\tilde\Lambda_{i}({\boldsymbol{x}})$, $i=1,2$, the natural way of obtaining a positive-operator-valued function is $$\label{Lambdasum}
\Lambda({\boldsymbol{x}}) = \tilde\Lambda_{1}({\boldsymbol{x}}) \oplus \tilde\Lambda_{2}({\boldsymbol{x}}) \,.$$
On the tensor product space ${\mathscr{H}}_{1} \otimes {\mathscr{H}}_{2}$, it is natural to consider the two functions $$\label{2Lambdas}
\Lambda_1 ({\boldsymbol{x}}) = \tilde\Lambda_{1}({\boldsymbol{x}}) \otimes I_{2}\text{ and }
\Lambda_2({\boldsymbol{x}}) = I_{1} \otimes \tilde\Lambda_{2}({\boldsymbol{x}})\,,$$ defining a theory with two types of flashes. A relevant property of this choice is that if the two physical systems corresponding to ${\mathscr{H}}_{1}$ and ${\mathscr{H}}_{2}$ do not interact, $H = H_{1} \otimes I_{2} + I_{1} \otimes H_{2}$, and if the initial state vector factorizes, $\psi = \psi_{1} \otimes \psi_{2}$, then type-1 and type-2 flashes are independent of each other. Indeed, the type-1 flashes are also independent of $H_{2}$, $\Lambda_{2}$, and $\psi_{2}$, and vice versa. Each of the two systems behaves as if it was alone in the world, obeying its own version of the law , and that is a reasonable behavior.
Suppose we do not want two types of flashes, but one. Observe that, [[by , the rate (at time $t=0$) for a flash *of any type* to occur in a set $B\subseteq {\mathbb{R}}^3$ is $\langle \psi |\Lambda_1(B)|\psi \rangle + \langle \psi |\Lambda_2(B)|\psi \rangle$, the same as the rate of flashes of only one type with rate operators]{}]{} $$\label{Lambdaprod}
\Lambda({\boldsymbol{x}}) = \tilde\Lambda_{1}({\boldsymbol{x}}) \otimes I_{2} +
I_{1} \otimes \tilde\Lambda_{2}({\boldsymbol{x}}) \,,$$ which is thus a natural choice of a positive-operator-valued function on the tensor product.
On the $N$-th tensor power ${\mathscr{H}}_{(1)}^{\otimes N}$ [[of the one-particle Hilbert space ${\mathscr{H}}_{(1)}$]{}]{}, the [[flash rate operator corresponding to ,]{}]{} $$\Lambda({\boldsymbol{x}}) = \sum_{i=1}^N I^{\otimes (i-1)} \otimes \Lambda_{(1)}({\boldsymbol{x}})
\otimes I^{\otimes (N-i)}\,,$$ can be written using the permutation operators $U_\sigma$ on ${\mathscr{H}}_{(1)}^{\otimes N}$ for $\sigma$ in the permutation group $S_N$, $$\label{symmLambda}
\Lambda({\boldsymbol{x}}) = \frac{1}{(N-1)!} \sum_{\sigma \in S_N} U_\sigma^*
\bigl( \Lambda_{(1)}({\boldsymbol{x}})
\otimes I^{\otimes (N-1)} \bigr) U_\sigma \,,$$ and therefore assumes values in the symmetric operators, mapping in particular symmetric (bosonic) vectors to symmetric ones and anti-symmetric (fermionic) vectors to anti-symmetric ones, thus defining two positive-operator-valued functions $\Lambda_{(N)}^{\pm}({\boldsymbol{x}})$ acting on the bosonic ($+$) respectively fermionic ($-$) $N$-particle Hilbert space: [[$\Lambda_{(N)}^{\pm}({\boldsymbol{x}})$ is the restriction of the $\Lambda({\boldsymbol{x}})$ given by to the bosonic respectively fermionic subspace of ${\mathscr{H}}_{(1)}^{\otimes N}$. The flash theory for $N$ bosons or fermions with these flash rate operators is closely related, in a way that will become clear in the next section, to the collapse process proposed by Dove and Squires [@dovethesis; @DS95].]{}]{}
[[Adding the]{}]{} $\Lambda^\pm_{(N)}({\boldsymbol{x}})$ in the sense of from $N=0$ to $\infty$ yields two functions $\Lambda^{\pm}({\boldsymbol{x}})$ acting on the bosonic respectively fermionic Fock space. This completes our construction for the “second quantization" of $\Lambda_{(1)}({\boldsymbol{x}})$. If we take $\Lambda_{(1)}({\boldsymbol{x}})$ to be the multiplication operator with $N=1$ (and $i=1$) and $a({\boldsymbol{x}})$ the canonical annihilation operator on Fock space, then $\Lambda({\boldsymbol{x}})$ coincides with .
Collapses {#sec:collapses}
=========
After talking so much about flashes, I should point out what they have to do with collapses of the wave function. Suppose that $n$ flashes have occurred between time $0$ and time $t$, with the $k$-th flash at time $t_k$ and location ${\boldsymbol{x}}_k$. Then the distribution of the next $m$ flashes after time $t$, conditional on the history of flashes between $0$ and $t$, is, as a consequence of , given by $$\begin{gathered}
\label{condflashdist}
{\mathbb{P}}\Bigl( {{\boldsymbol{X}}}_{n+1} \in {\mathrm{d}}^3 {\boldsymbol{x}}_{n+1}, T_{n+1} \in {\mathrm{d}}t_{n+1}, \ldots,
{{\boldsymbol{X}}}_{n+m} \in {\mathrm{d}}^3 {\boldsymbol{x}}_{n+m}, T_{n+m} \in {\mathrm{d}}t_{n+m} \Big|\\
{{\boldsymbol{X}}}_1 = {\boldsymbol{x}}_1, T_1 = t_1, \ldots, {{\boldsymbol{X}}}_n = {\boldsymbol{x}}_n, T_n = t_n\leq t, T_{n+1}\geq t \Bigr) =\\
\bigl \| {K}_m(t, {\boldsymbol{x}}_{n+1}, t_{n+1}, \ldots, {\boldsymbol{x}}_{n+m},t_{n+m}) \psi_t \|^2 \,
{\mathrm{d}}^3 {\boldsymbol{x}}_{n+1} \, {\mathrm{d}}t_{n+1} \cdots {\mathrm{d}}^3 {\boldsymbol{x}}_{n+m} \, {\mathrm{d}}t_{n+m} \,,\end{gathered}$$ where $\psi_t$ is the *conditional wave function* $$\label{condpsidef}
\psi_t = \frac{ W_{t-t_n} \, {K}_n (0,{\boldsymbol{x}}_1,t_1, \ldots, {\boldsymbol{x}}_n,t_n) \, \psi }
{\| W_{t-t_n} \, {K}_n (0,{\boldsymbol{x}}_1,t_1, \ldots, {\boldsymbol{x}}_n,t_n) \, \psi \|} \,.$$ [[As a corollary, the flash rate at time $t$ in a set $B \subseteq {\mathbb{R}}^3$ is]{}]{} $$\label{rateBt}
\langle \psi_t | \Lambda (B)| \psi_t \rangle\,.$$ It is $\psi_t$ that collapses whenever a flash occurs, say at $({{\boldsymbol{X}}},T)$, according to $$\label{collapse}
\psi_{T+} = \frac{\Lambda({{\boldsymbol{X}}})^{1/2} \, \psi_{T-}}
{\|\Lambda({{\boldsymbol{X}}})^{1/2} \, \psi_{T-} \|} \,,$$ and evolves deterministically between the flashes according to the operators $W_t$ (up to normalization). [[More explicitly, for the example case of $N$ identical particles, the (anti)symmetric wave function $\psi({\boldsymbol{r}}_1, \ldots, {\boldsymbol{r}}_N)$ collapses to (a normalization factor times)]{}]{} $$\Lambda_{(N)}^{\pm}({{\boldsymbol{X}}})^{1/2} \, \psi({\boldsymbol{r}}_1, \ldots, {\boldsymbol{r}}_N) = \biggl(
{\mathcal{N}}\sum_{i=1}^N
{\mathrm{e}}^{-({{\boldsymbol{X}}}- {\boldsymbol{r}}_i)^2/a^2} \biggr)^{1/2} \, \psi({\boldsymbol{r}}_1, \ldots, {\boldsymbol{r}}_N)$$ [[for bosons $(+)$ respectively fermions $(-)$. It is this formula for the collapsed wave function that Dove and Squires proposed [@DS95].]{}]{}
[[Since the conditional wave function $\psi_t$ is random, one can form the density matrix]{}]{} $$\rho_t = \int\limits_{\mathscr{H}}{\mathbb{P}}(\psi_t \in {\mathrm{d}}\phi) \, |\phi\rangle \langle \phi|$$ [[of its distribution, in other words the density matrix of an ensemble of systems, each of which started with the same initial wave function $\psi$ but experienced flashes independently of the other systems. For the sake of completeness we note that it can be computed to be]{}]{} $$\rho_t = \sum_{n=0}^\infty \int {\mathrm{d}}^3 {\boldsymbol{x}}_1 \cdots {\mathrm{d}}^3{\boldsymbol{x}}_n \int\limits_0^t {\mathrm{d}}t_1
\cdots \int\limits_{t_{n-1}}^t {\mathrm{d}}t_n \, W_{t-t_n} \, K_n |\psi \rangle \langle \psi | K_n^*
\, W^*_{t-t_n}$$ [[with $K_n = K_n(0,{\boldsymbol{x}}_1,t_1, \ldots, {\boldsymbol{x}}_n,t_n)$, and obeys the master equation]{}]{} $$\frac{{\mathrm{d}}\rho_t}{{\mathrm{d}}t} = -\tfrac{{\mathrm{i}}}{\hbar} [H,\rho_t] - \tfrac{1}{2}
\{\Lambda({\mathbb{R}}^3), \rho_t \} + \int {\mathrm{d}}^3 {\boldsymbol{x}}\, \Lambda ({\boldsymbol{x}})^{1/2} \,
\rho_t \, \Lambda({\boldsymbol{x}})^{1/2}$$ [[with $[\,,]$ the commutator and $\{\,,\}$ the anti-commutator.]{}]{}
It is tempting to regard the collapsed wave function $\psi_t$ as the ontology, but I insist that the flashes form the ontology. This is a subtle point. After all, since for example the wave function of Schrödinger’s cat, $(|\text{alive}\rangle + |\text{dead}\rangle)/\sqrt{2}$, quickly collapses into essentially either $|\text{alive}\rangle$ or $|\text{dead}\rangle$, it may seem that the collapsed wave function represents reality. However, that this view is problematic becomes evident when we note that even after the collapse into $\psi_t \approx |\text{alive}\rangle$, the coefficient of $|\text{dead}\rangle$ in $\psi_t$ is tiny but not zero. How small would it have to be to make the cat alive? The more fundamental problem with this view is that while the wave function may *govern* the behavior of matter, it *is* not matter; instead, matter corresponds to variables *in space and time* [@AGTZ], called “local beables” by Bell [@Bellbook] and “primitive ontology” by Dürr, Goldstein, and Zanghì [@DGZ04].
In what I described in the previous sections, the flashes form the primitive ontology. But other choices are possible, and this fact underlines that the theory is not completely specified by the stochastic evolution law for $\psi_t$ alone. An example of a different primitive ontology, instead of flashes, is the *matter density ontology*, a continuous distribution of matter in space with density $$\label{mdef}
m({\boldsymbol{x}},t) = {\langle \psi_t | \Lambda({\boldsymbol{x}})|\psi_t \rangle}$$ in our notation. While the two theories (using the same wave function with either the flash ontology or the matter density ontology) cannot be distinguished empirically, they differ metaphysically and physically. For example, the equations I considered in [@Tum05] define a Lorentz-invariant theory with the flash ontology but not with the matter density ontology, and strong superselection rules [@ssr] can hold with the flash ontology but not with the matter density ontology. For further discussion of the concept of primitive ontology see [@AGTZ].
Predictions
===========
The empirically testable predictions of the collapse QFT model we described agree with the standard predictive rules of QFT to the same extent as the GRW model [[(say, with mass-dependent collapse rate)]{}]{} agrees with standard quantum mechanics. To see this, recall first from the paragraph containing eq. that when a system (defined, e.g., by a region in 3-space) can be regarded as decoupled and disentangled from its environment then its flash process is independent of the environment. Next note that, [[by ]{}]{}, the total flash rate is ${\langle \psi_t | \Lambda({\mathbb{R}}^3)|\psi_t \rangle}$, proportional to [[either]{}]{} the (value regarded in quantum theory as the) average number of particles (relative to $\psi_t$) [[or the average net mass]{}]{}. As a consequence, as with the GRW model, a system containing fewer than a thousand particles experiences no more flashes than once in 100,000 years. Up to the first flash, the deviation of $\psi_t$ from the Schrödinger evolution is small for $t \ll \tau/\Delta N$ if $\Delta N$ is a bound on the spread in particle number of ${\mathrm{e}}^{-{\mathrm{i}}Hs/\hbar} \,\psi$, $0\leq s \leq t$. A macroscopic piece of matter, in contrast, with over $10^{22}$ particles, experiences millions of flashes every second. A macroscopic superposition essentially breaks down with the first flash (with consequences for the distribution of the future flashes) to one of the contributions, and for the same reasons as in the GRW model the random choice of the surviving contribution occurs with almost exactly the quantum theoretical probabilities. And like in the GRW model, this entails that experiments that “measure” any quantum “observable” on a microscopic system have almost exactly the quantum theoretical distribution of outcomes. [[Dove and Squires [@DS95] provide a discussion of the consistency of their collapse process for identical particles with those experimental data that yield restrictions on the possibility of spontaneous collapse.]{}]{}
Some collapse theories imply the possibility of superluminal (i.e., faster than light) signalling; even if the theory is hard to distinguish empirically from standard quantum theories, those experiments sensitive enough to detect the deviation can allow signalling using EPR–Bell pairs. Such collapse theories are therefore unlikely to possess a Lorentz-invariant version. In contrast, the collapse QFT developed here and the GRW model exclude superluminal signalling; this follows essentially from their property that the distribution of the flashes is quadratic in $\psi$. Indeed, if two systems are entangled but decoupled, $H = H_{1} \otimes I_{2} + I_{1} \otimes H_{2}$, and the flash rate operators are additive according to or , then, as a consequence of , the marginal distribution of the flashes belonging to system 1 depends on $H$, $\Lambda$, and $\psi$ only through $H_{1}$, $\tilde\Lambda_{1}$, and the reduced density matrix $\mathrm{tr}_{2} |\psi \rangle \langle \psi|$ of system 1. [[To see this, consider first the case , in which there are two types of flashes for the two systems; in this situation the claim follows from the fact that the operators ${K}_n$ defined by decompose into ${K}_{1,n_{1}} \otimes {K}_{2,n_{2}}$. In the case of a single type of flashes it is not obvious which flashes are to be attributed to which system, unless the two systems have disjoint supports ($S_1,S_2 \subseteq {\mathbb{R}}^3$ with $S_1 \cap S_2 = \emptyset$) and we count the flashes in $S_1$ for system 1. But then we can in fact regard the flashes as labeled, the label being a function of the location, corresponding to $\Lambda_i({\boldsymbol{x}}) = 1_{S_i}({\boldsymbol{x}}) \, \Lambda({\boldsymbol{x}})$ with $1_{S_i}$ the characteristic function of $S_i$, which brings us back to the previous case.]{}]{}
Literature
==========
GianCarlo Ghirardi sometimes suggests in his writings that identical particles or QFT cannot be treated in the framework of the GRW model in a satisfactory way [@Ghi98 page 118], [@BG03 pages 312 and 382], but require a diffusion process in Hilbert space; I think that the model I have presented [[(respectively, as far as $N$ identical particles are concerned, the model of Dove and Squires [@DS95])]{}]{} is a counterexample. [[Part of the reason why the model of Dove and Squires has not received enough attention may be that they have not made clear enough, in my view, its naturalness and simplicity, and that they have presented it on equal footing with another, much less natural, proposal.]{}]{}
I know of [[five]{}]{} variants of the GRW model for identical particles that have been proposed, [[apart from the one discussed in this paper]{}]{}: One was introduced by Ghirardi, Nicrosini, Rimini, and Weber in 1988 [@GNRW88], which, however, appears theoretically unsatisfactory since it prescribes that the flashes of a system of $N$ identical particles occur in clusters of $N$ simultaneous ones, leaving no hope for a Lorentz-invariant version. It is also presumably empirically inadequate since it predicts that a superposition of two wave packets for the same single particle at a distance greater than 10 kilometers collapses within $10^{-7} \, \mathrm{sec}$. The second proposal was made by Kent in 1989 [@kent], in which, however, the distribution of the flashes is not quadratic in $\psi$, thus allowing superluminal signalling. [[The same is true of the third proposal, which is contained as well in the paper of Dove and Squires [@DS95].]{}]{} The [[fourth]{}]{} proposal was made by Bell in 1987, but never published, [[except in a brief description in Sec. IV C of [@GPR90]]{}]{}. I have learnt about it from Alberto Rimini on a recent conference in honor of GianCarlo Ghirardi’s 70th birthday at Trieste and Mali Losinj; there I have learnt as well that equations similar to and had already been considered in a different context, namely for models of coupling classical and quantum systems [@BJ95]. [[The fifth variant was mentioned by Ghirardi, Pearle, and Rimini in 1990 as a side remark in Sec. IV C of [@GPR90]: The process for the wave function is similar in spirit to the proposal of [@GNRW88], involving a simultaneous collapse for all particles, but differs in that it cannot be associated with flashes, as the collapse involves a continuous distribution function $n({\boldsymbol{x}})$ on, rather than a set of $N$ points in, 3-space.]{}]{}
*Acknowledgments.* [[I thank Philip Pearle for his detailed comments on a previous version of this article.]{}]{}
[\[20\]]{}
Adler, S. L., Brun, T. A.: “Generalized stochastic Schrödinger equations for state vector collapse”, J. Phys. A: Math. Gen. **34**, 4797–4809 (2001) and quant-ph/0103037
Allori, V., Goldstein, S., Tumulka, R., Zanghì, N.: “On the Common Structure of Bohmian Mechanics and the Ghirardi–Rimini–Weber Theory”, in preparation
Bassi, A., Ghirardi, G. C.: “Dynamical Reduction Models”, Phys. Rep. **379**, 257–427 (2003) and quant-ph/0302164
Bell, J. S.: “Are there quantum jumps?”, in *Schrödinger. Centenary of a polymath*, p. 41–52. Cambridge: Cambridge University Press (1987). Reprinted in [@Bellbook], p. 201–212.
Bell, J. S.: *Speakable and unspeakable in quantum mechanics*. Cambridge: Cambridge University Press (1987)
Bell, J. S.: “Towards An Exact Quantum Mechanics”, in *Themes in contemporary physics, II*, S. Deser and R. J. Finkelstein (eds.), p. 1–26. Teaneck, NJ: World Scientific (1989)
Blanchard, P., Jadczyk, A.: “Events and piecewise deterministic dynamics in event-enhanced quantum theory”, Phys. Lett. A **203**, 260–266 (1995)
Bohm, D., Bub, J.: “A proposed solution of the measurement problem in quantum mechanics by a hidden variable theory”, Rev. Modern Phys. **38**, 453–469 (1966)
Colin, S., Durt, T., Tumulka, R.: “On Superselection Rules in Bohm–Bell Theories”, quant-ph/0509177
Diósi, L.: “Quantum stochastic processes as models for state vector reduction”, J. Phys. A: Math. Gen. **21**, 2885–2898 (1988)
Dove, C.: “Explicit Wavefunction Collapse and Quantum Measurement”, Ph.D. thesis, Department of Mathematical Sciences, University of Durham (1996)
Dove, C., Squires, E. J.: “Symmetric versions of explicit wavefunction collapse models”, Found. Phys. **25**, 1267–1282 (1995)
Dowker, F., Henson, J.: “Spontaneous Collapse Models on a Lattice”, J. Statist. Phys. **115**, 1327–1339 (2004) and quant-ph/0209051
Dürr, D., Goldstein, S., Zanghì, N.: “Quantum Equilibrium and the Role of Operators as Observables in Quantum Theory”, J. Statist. Phys. **116**, 959–1055 (2004) and quant-ph/0308038
Ghirardi, G. C.: “Some Lessons from Relativistic Reduction Models”, in *Open Systems and Measurement in Relativistic Quantum Theory*, H.-P. Breuer, F. Petruccione (ed.s), p. 117–152. Berlin: Springer-Verlag (1999)
Ghirardi, G. C., Nicrosini, O., Rimini, A., Weber, T.: “Spontaneous localization of a system of identical particles”, Il Nuovo Cimento B **102(4)**, 383–396 (1988)
Ghirardi, G. C., Pearle, P., Rimini, A.: “Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles”, Phys. Rev. A (3) **42**, 78–89 (1990)
Ghirardi, G. C., Rimini, A., Weber, T.: “Unified dynamics for microscopic and macroscopic systems”, Phys. Rev. D **34**, 470–491 (1986)
Gisin, N.: “Quantum Measurements and Stochastic Processes”, Phys. Rev. Lett. **52**, 1657–1660 (1984)
Kent, A.: ““Quantum jumps” and indistinguishability”, Modern Phys. Lett. A **4(19)**, 1839–1845 (1989)
Leggett, A. J.: “Testing the limits of quantum mechanics: motivation, state of play, prospects", J. Phys. CM **14**, R415–R451 (2002)
Pearle, P.: “Reduction of the state vector by a nonlinear Schrödinger equation”, Phys. Rev. D (3) **13**, no. 4, 857–868 (1976)
Pearle, P.: “Toward explaining why events occur”, Int. J. Theor. Phys. **18**, 489–518 (1979)
Pearle, P.: “Combining stochastic dynamical state-vector reduction with spontaneous localization”, Phys. Rev. A **39**, 2277–2289 (1989)
Pearle, P.: “Toward a Relativistic Theory of Statevector Reduction”, in *Sixty-Two Years of Uncertainty: Historical, Philosophical, and Physical Inquiries into the Foundations of Quantum Physics*, A.I. Miller (ed.), volume 226 of *NATO ASI Series B*, p. 193–214. New York: Plenum Press (1990)
Pearle, P., Squires, E.: “Bound state excitation, nucleon decay experiments and models of wave function collapse”, Phys. Rev. Lett. **73**, 1–5 (1994)
Penrose, R.: “Wavefunction collapse as a real gravitational effect”, in *Mathematical Physics 2000* (ed. A. Fokas, T. W. B. Kibble, A. Grigoriou, B. Zegarlinski), pp. 266–282. London: Imperial College Press (2000)
Tumulka, R.: “A Relativistic Version of the Ghirardi–Rimini–Weber Model”, quant-ph/0406094
[^1]: Mathematisches Institut, Eberhard-Karls-Unversität, Auf der Morgenstelle 10, 72076 Tübingen, Germany. E-mail: [email protected]
[^2]: Otherwise, the theory is not ill-defined, but rather the flash rate is so low for some $\psi$ that there is positive probability that no flash occurs.
|
{
"pile_set_name": "ArXiv"
}
|
\
[**General asymptotic supnorm estimates for**]{}\
\
[**solutions of one-dimensional advection-diffusion**]{}\
\
[**equations in heterogeneous media, I**]{} [\
]{}\
[\
]{}[**José A. Barrionuevo, Lucas S. Oliveira and Paulo R. Zingano**]{}\
\
[Departamento de Matemática Pura e Aplicada]{}\
\
[Universidade Federal do Rio Grande do Sul]{}\
\
[Porto Alegre, RS 91509-900, Brazil]{}\
[\
]{}\
[**Abstract**]{}\
\
[We derive general bounds for the large time size of supnorm values $ {\displaystyle
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\!
} $\
of solutions to one-dimensional advection-diffusion equations\
\
$$\notag
u_t \;\!+\, (\;\! b(x,t) \;\!u \;\!)_{x} \;\!=\;
u_{xx},
\qquad x \in \mathbb{R}, \; t > 0$$\
with initial data $ {\displaystyle
u(\cdot,0) \in L^{p_{\mbox{}_{\!\;\!0}}}(\mathbb{R}) \cap L^{\infty}(\mathbb{R})
} $ for some $ 1 \leq p_{\mbox{}_{\!\;\!0}} \!< \infty $, and arbitrary\
bounded advection speeds $ b(x,t) $, introducing new techniques based on suitable\
energy arguments. Some open problems and related results are also given.\
]{}
[\
]{}\
[\
]{}[AMS Mathematics Subject Classification:]{} 35B40 (primary), 35B45, 35K15 (secondary)\
[\
]{}\
[Key words:]{} advection-diffusion equations, initial value problem, energy method, heterogeneous media, forced advection, supnorm estimates, large time behavior.\
\
[**General asymptotic supnorm estimates for**]{}\
\
[**solutions of one-dimensional advection-diffusion**]{}\
\
[**equations in heterogeneous media, I**]{} [\
]{}\
[\
]{}[**José A. Barrionuevo, Lucas S. Oliveira and Paulo R. Zingano**]{}\
\
[Departamento de Matemática Pura e Aplicada]{}\
\
[Universidade Federal do Rio Grande do Sul]{}\
\
[Porto Alegre, RS 91509-900, Brazil]{}\
[\
]{}\
[**Abstract**]{}\
\
[We derive general bounds for the large time size of supnorm values $ {\displaystyle
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\!
} $\
of solutions to one-dimensional advection-diffusion equations\
\
$$\notag
u_t \;\!+\, (\;\! b(x,t) \;\!u \;\!)_{x} \;\!=\;
u_{xx},
\qquad x \in \mathbb{R}, \; t > 0$$\
with initial data $ {\displaystyle
u(\cdot,0) \in L^{p_{\mbox{}_{\!\;\!0}}}(\mathbb{R}) \cap L^{\infty}(\mathbb{R})
} $ for some $ 1 \leq p_{\mbox{}_{\!\;\!0}} \!< \infty $, and arbitrary\
bounded advection speeds $ b(x,t) $, introducing new techniques based on suitable\
energy arguments. Some open problems and related results are also given.\
]{}
[\
]{}
[**§1. Introduction**]{}\
In this work, we obtain very general large time estimates for supnorm values of solutions $ u(\cdot,t) $ to parabolic initial value problems of the form\
\
$$\tag{1.1$a$}
u_t \;\!+\, (\;\!b(x,t) \;\!u \;\!)_{x}
\;\!=\;
u_{xx},
\qquad
x \in \mathbb{R}, \; t > 0,$$\
$$\tag{1.1$b$}
u(\cdot,0) \,=\, u_{\mbox{}_{\!\;\!0}} \in
L^{p_{\mbox{}_{\!\;\!0}}}(\mathbb{R}) \cap L^{\infty}(\mathbb{R}),
\qquad
1 \leq p_{\mbox{}_{\!\;\!0}} \!< \infty,$$\
for arbitrary continuously differentiable advection fields $ \;\!b \in L^{\infty}(\mathbb{R} \times [\;\!0, \infty\;\![\;\!) $. Here, by [*solution*]{} to (1.1) in some time interval $ [\;\!0, T_{\mbox{}_{\scriptstyle \!\ast}}[ $, $ \:\!0 < T_{\mbox{}_{\scriptstyle \!\ast}} \!\leq \infty $, we mean a function $ {\displaystyle
u \!\:\!:\;\! \mathbb{R} \times [\;\!0, T_{\mbox{}_{\scriptstyle \!\ast}} [
\;\rightarrow \mathbb{R}
} $ which is bounded in each strip $ S_{{\scriptstyle T}} \!\:\!=\, \mathbb{R} \times [\;\!0, T\:\!] $, $ 0 < T \!\,\!< T_{\mbox{}_{\scriptstyle \!\ast}} $, solves equation (1.1$a$) in the classical sense for $ \;\!0 < t < T_{\mbox{}_{\scriptstyle \!\ast}} $, and satisfies $ u(\cdot,t) \rightarrow u_{\mbox{}_{\!\;\!0}} \!\;\!$ in $ L^{1}_{\tt loc}(\mathbb{R}) $ as $ t \rightarrow 0 $. It follows from the a priori estimates given in Section 2 below that all solutions of problem (1.1$a$), (1.1$b$) are actually globally defined ($ T_{\mbox{}_{\scriptstyle \!\ast}} \!= \infty $), with $ {\displaystyle
u(\cdot,t) \in
C^{0}(\;\![\;\!0, \infty \:\! [, L^{p}(\mathbb{R}))
} $ for each $ \:\!p \geq p_{\mbox{}_{\!\;\!0}} \!\:\!$ finite. Given $ \;\!b \in L^{\infty}(\mathbb{R} \times [\;\!0, \infty\;\![\;\!) $, what then can be said about the size of supnorm values $ {\displaystyle
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
} $ for $ t \gg 1 $?
[\
]{}\
When $ \partial b/\partial x \geq 0 \;\!$ for all $ x \in \mathbb{R}, t \geq 0 $, it is well known that, for each $ \:\!p_{\mbox{}_{\!\;\!0}} \!\leq p \leq \;\!\!\infty $, $ {\displaystyle
\:\!
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{p}(\mathbb{R})}}
\!
} $ is monotonically decreasing in $t$, with\
\
$$\tag{1.2}
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\leq\;\!
K\!\;\!(p_{\mbox{}_{\!\;\!0}}) \,
\|\, u_{\mbox{}_{0}} \;\!
\|_{\mbox{}_{\scriptstyle L^{p_{\mbox{}_{\!\;\!0}}}(\mathbb{R})}} \;\!
t^{\mbox{}^{\scriptstyle \! - \,
\frac{\scriptstyle 1}
{\scriptstyle \;\!2 \;\! p_{\mbox{}_{\mbox{}_{\!0}}}\!\:\!} }}
\qquad
\forall \;\, t > 0
\qquad
\;\;(\;\! b_x \geq 0 \;\!)\!\!\!\!$$\
for some constant $\;\!0 <\!\;\! K\!\;\!(p_{\mbox{}_{\!\;\!0}}) <
\,\! 2^{\mbox{}^{\scriptstyle \!-\;\!1/p_{\mbox{}_{\!\;\!0}} }} \!$ that depends only on $ p_{\mbox{}_{\!\;\!0}} \;\!\!$, see e.g.$\;$[@AmickBonaSchonbek1989; @BrazSchutzZingano2013; @EscobedoZuazua1991; @Porzio2009; @Schonbek1986]. For general $ b(x,t)$, however, estimating $ {\displaystyle
\;\!
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\!
} $ is much harder. To see why, let us illustrate with the important case $ p_{\mbox{}_{\!\;\!0}} \!= 1 $, where one has\
\
$$\tag{1.3}
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R})}}
\!\;\!\leq\;
\|\, u_{\mbox{}_{0}} \;\!\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R})}}
\qquad
\forall \;\, t > 0,$$\
as recalled in Theorem 2.1 below. Writing equation (1.1$a$) as\
\
$$\tag{1.4}
u_t \;\!+\, b(x,t) \;\!u_x
\;\!=\;
u_{xx} \;\!-\, b_{x}(x,t) \;\!u,$$\
we observe on the righthand side of (1.4) that $ |\,u(x,t)\,| $ is pushed to grow at points $ (x,t) $ where $ b_{x}(x,t) < 0 $. If this condition persists long enough, large values of $ |\,u(x,t)\,| $ might be generated, particularly at sites where . Now, because of the constraint (1.3), any persistent growth in solution size will eventually create long thin structures as shown in Fig.$\,$1, which, in turn, tend to be effectively dissipated by viscosity. The final overall behavior that ultimately results from such competition is not immediately clear, either on physical or mathematical grounds.\
\
\
[ Solution profiles showing typical growth in regions with $\;\! b_x \!< 0 $, where\
$ b \:\!=\:\! 5\:\!\cos \:\!x $. After reaching maximum height, solution starts decaying very slowly\
due to its spreading and mass conservation. (Decay rate is not presently known.) ]{}
[\
]{}\
As shown by equation (1.4), it is not the [magnitude]{} of $ b(x,t) $ itself but instead its [*oscillation*]{} that is relevant in determining $ {\displaystyle
\;\!
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\!\;\!
} $. Accordingly, we introduce the quantity $ B(t) $ defined by\
\
$$\tag{1.5}
B(t) \,=\,
\frac{\mbox{\small $1$}}{\;\!\mbox{\small $2$}\;\!} \,
\Bigl(\,
\sup_{x \,\in\, \mathbb{R}} \;\!b(x,t)
\;\;\!-\:
\inf_{x \,\in\, \mathbb{R}} \!\;\!b(x,t)
\,\Bigr),
\qquad
t \geq 0,$$\
which plays a fundamental role in the analysis. Our main result is now easily stated. [\
]{}[**Main Theorem.**]{} [*For each $ \;\! p \geq p_{\mbox{}_{\!\;\!0}} $, we have*]{}[^1]\
\
$$\tag{1.6}
\limsup_{t\,\rightarrow\,\infty}\;\!
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\leq\,
\Bigl(\;\! \frac{\mbox{\small $\;\!3 \;\! \sqrt{\,\!3\;\!} \;$}}
{\mbox{\small $ 2 \:\! \pi $} } \: p \;\!
\Bigr)^{\mbox{}^{\scriptstyle \!\!
\frac{\scriptstyle 1}{\scriptstyle p} }}
\!\!\cdot\,
{\cal B}^{\mbox{}^{\scriptstyle
\frac{\scriptstyle 1}{\scriptstyle p} }}
\!\!\cdot\:
\limsup_{t\,\rightarrow\,\infty}\;\!
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{p}(\mathbb{R})}}
\!,$$\
[*where* ]{} $ {\displaystyle
{\cal B} =\;\!
\limsup_{t\,\rightarrow \;\!\infty}
\:\!
B(t)
} $.\
[\
]{}In particular, in the important case $ p_{\mbox{}_{\!0}} \!= 1 $ considered above, we obtain, using (1.3),\
\
$$\tag{1.7}
\limsup_{t\,\rightarrow\,\infty}\;\!
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\leq\,
\Bigl(\;\! \frac{\mbox{\small $\;\!3 \;\! \sqrt{\,\!3\;\!} \;$}}
{\mbox{\small $ 2 \:\! \pi $} } \;\!
\Bigr)
\cdot\,
{\cal B}
\cdot\:
\|\, u_{\mbox{}_{0}} \;\!\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R})}}
\!,$$\
so that $ u(\cdot,t) $ stays uniformly bounded for all time in this case.[^2] Estimates similar to (1.6) can be also shown to hold for the $n$-dimensional problem\
\
$$\tag{1.8}
u_t \,+\;
\mbox{\tt div}\,
(\;\!\mbox{\boldmath $b$}(x,t) \;\!u \;\!)
\:=\:
\Delta \:\!u,
\qquad
u(\cdot,0) \in
L^{p}(\mathbb{R}^{n}) \cap L^{\infty}(\mathbb{R}^{n}),$$\
but to simplify our discussion we consider here the case $ n = 1 $ only. Our derivation of (1.6), which improves some unpublished results by the third author, uses the 1-D inequality\
\
$$\tag{1.9}
\|\; \mbox{v}\;\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\leq\;
C_{\mbox{}_{\!\infty}} \,
\|\; \mbox{v}\;\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R})}}
^{\mbox{}^{\scriptstyle 1/3}}
\|\; \mbox{v}_{x}\,\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}
^{\mbox{}^{\scriptstyle 2/3}}
\!,
\qquad
\mbox{v} \in L^{1}(\mathbb{R}) \cap H^{1}(\mathbb{R}),$$\
where $ C_{\mbox{}_{\!\infty}} \!= (\;\!3/4\;\!)^{\mbox{}^{\scriptstyle 2/3}} \!$, and can be readily extended to other problems of interest like 1-D systems of viscous conservation laws ([@Melo2011], Ch.$\,$9) or the more general equation\
\
$$\tag{1.10}
u_t \;\!+\, (\;\!b(x,t,u) \;\!u \;\!)_{x}
\;\!=\;
(\;\!a(x,t,u) \;\!u_x \;\!)_{x},
\qquad
a(x,t,u) \geq \mu(t) > 0,$$\
with bounded values $ b(x,t,u) $, provided that we assume $ {\displaystyle
\!\;\!
\int^{\infty} \!\!\!\! \mu(t) \, dt \,=\, \infty
} $: using a\
similar argument, we get the estimate$^{1}$ ([@Oliveira2013], Ch.$\;$2)\
\
$$\tag{1.11}
\limsup_{t\,\rightarrow\,\infty}\;\!
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\leq\,
\Bigl(\;\! \frac{\mbox{\small $\;\!3 \;\! \sqrt{\,\!3\;\!} \;$}}
{\mbox{\small $ 2 \:\! \pi $} } \: p \;\!
\Bigr)^{\mbox{}^{\scriptstyle \!\!
\frac{\scriptstyle 1}{\scriptstyle p} }}
\!\!\cdot\,
{\cal B}_{\mu}^{\mbox{}^{\scriptstyle \,
\frac{\scriptstyle 1}{\scriptstyle p} }}
\!\!\cdot\:
\limsup_{t\,\rightarrow\,\infty}\;\!
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{p}(\mathbb{R})}}
\!,$$\
for each $ \;\! p \geq p_{\mbox{}_{\!\;\!0}} \!\;\!$, where\
\
$$\tag{1.12$a$}
{\cal B}_{\mu} \;\!=\;
\limsup_{t\,\rightarrow \;\!\infty}
\;
\frac{\;\!B(t)\;\!}{\mu(t)},$$\
$$\tag{1.12$b$}
B(t) \,=\,
\frac{\mbox{\small $1$}}{\;\!\mbox{\small $2$}\;\!} \,
\Bigl(\,
\sup_{x \,\in\, \mathbb{R}} \;\!b(x,t,u(x,t))
\;\;\!-\:
\inf_{x \,\in\, \mathbb{R}} \!\;\!b(x,t,u(x,t))
\,\Bigr).$$\
More involving applications, such as problems with superlinear advection, where solutions may blow up in finite time, will be described in a sequel to this work.\
\
[**§2. A priori estimates**]{}\
This section contains some preliminary results on the solutions of problem (1.1) needed later for our derivation of estimate (1.6), which is completed in Section 3. ($\,\!$Recall that a solution on some given time interval $ [\;\!0, \mbox{\small $T$}_{\!\ast} [ $, $ 0 < \mbox{\small $T$}_{\!\ast} \!\leq \infty $, is a function $ {\displaystyle
u(\cdot,t) \in
L^{\infty}_{\tt loc}
(\:\![\;\!0, \mbox{\small $T$}_{\!\ast}[, L^{\infty}(\mathbb{R})\:\!)
} $ which is smooth ($C^{2}$ in $x$, $C^{1}$ in $t$) in $ \mathbb{R} \;\!\times\, ]\;\!0, \mbox{\small $T$}_{\!\ast} \,\![ $ and solves equation (1.1$a$) there, verifying the initial condition in the sense of $L^{1}_{\tt loc}(\mathbb{R}) $, i.e., $ {\displaystyle
\|\, u(\cdot,t) - u_{\mbox{}_{\!\,\!0}} \;\!
\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{K})}}
\!\!\!\;\!\rightarrow 0
\;\!
} $ as $ \,\!t \rightarrow 0 \,\!$ for each compact $ \mathbb{K} \!\;\!\subset \mathbb{R} $. Local existence theory can be found in e.g.$\;$[@Serre1999], Ch.$\:$6.$\,\!$) We start with a simple Gronwall-type estimate for $\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}} \!$, $ p_{\mbox{}_{\!\;\!0}} \!\,\!\leq q < \infty $. The corresponding result for the supnorm ($q = \infty$) is more difficult to obtain and will be given at the end of Section 2, see Theorem 2.4. [\
]{}[**Theorem 2.1.**]{} *If $ {\displaystyle
\,
u(\cdot,t) \in
L^{\infty}_{\tt loc}(\:\![\;\!0, T_{\ast}[, L^{\infty}(\mathbb{R})\:\!)
\:\!
} $ solves problem $\:\!(1.1a)$, $(1.1b)$, then $ {\displaystyle
u(\cdot,t) \in
C^{0}(\:\![\;\!0, T_{\ast}[, L^{q}(\mathbb{R})\:\!)
\:\!
} $ for each $ \;\! p_{\mbox{}_{\!\;\!0}} \!\leq q < \infty $, and*\
\
$$\tag{2.1}
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\leq\;
\|\, u(\cdot,0) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\!\cdot\,
\exp \,\Bigl\{\,
\mbox{\small $ {\displaystyle \frac{\small 1}{\small 2} }$}
\, (q-1) \!
\int_{0}^{\mbox{\footnotesize $\:\!t$}} \!\!
B(\tau)^{2}
\; d\tau \,\Bigr\}$$\
*for all $ \,0 < t < T_{\ast} $.*
\
[ The proof is standard, so we will only sketch the basic steps. Taking $ S \in C^{1}(\mathbb{R}) $ such that $ S^{\prime}({\tt v}) \geq 0 $ for all ${\tt v}$, $ S(0) = 0 $, $ S({\tt v}) = \mbox{sgn} \;\!({\tt v}) $ for $ | \,{\tt v} \,| \geq 1 $, let (given $ \delta > 0 $) $ L_{\delta}({\tt u}) = \int_{0}^{\mbox{\footnotesize ${\tt u}$}}
\! S({\tt v}/\delta) \, d{\tt v} $, so that $ L_{\delta}({\tt u}) \rightarrow |\,{\tt u}\,| $ as $ \delta \rightarrow 0 $, uniformly in $ {\tt u} $. Let $ \Phi_{\delta}({\tt u}) = L_{\delta}({\tt u})^{\mbox{\footnotesize $q$}} \!\;\!$. Given $ \mbox{\footnotesize $R$} > 0 $, $ 0 < \epsilon \leq 1 $, let $ \zeta_{\mbox{}_{R}}(\cdot) $ be the cut-off function $ \;\!\zeta_{\mbox{}_{R}}(x) = 0 \;\!$ for $ \;\!|\,x\,| \geq R $, $ \,\zeta_{\mbox{}_{R}}(x) = $ $ {\displaystyle
\exp\;\!\{\;\!-\:\epsilon \,\sqrt{1 + x^{2}\,}\,\} \;\!-\:
\exp\;\!\{\;\!-\:\epsilon \,\sqrt{1 + \mbox{\footnotesize $R$}^{2}\,}\,\}
\;\!
} $ for $ \;\!|\,x\,| < \mbox{\footnotesize $R$} $. Multiplying equation (1.1$a$) by $ \Phi_{\delta}^{\prime}(u(x,t)) \cdot \zeta_{\mbox{}_{R}}(x) \;\! $ if $ q \neq 2 $, or $ \,\!u(x,t) \cdot \zeta_{\mbox{}_{R}}(x) \;\! $ if $ q = 2 $, and integrating the result on $ \mathbb{R} \!\;\!\times\!\;\! [\;\!0, t\;\!] $, we obtain, letting $ \delta \rightarrow 0 $ and then $ \mbox{\footnotesize $R$} \rightarrow \infty $, since $ {\displaystyle
\;\!
u \in
L^{\infty}(\mathbb{R} \!\;\!\times\!\;\! [\;\!0, t\;\!])
} $:\
\
$$\tag{2.2$a$}
{\tt U}_{\epsilon}(t) \,+\, V_{\epsilon}(t)
\;\leq\;
{\tt U}_{\epsilon}(0)
\,+
\int_{0}^{\mbox{\footnotesize $\:\!t$}} \!\!\:\!
G_{\epsilon}(\tau) \, {\tt U}_{\epsilon}(\tau) \: d\tau,
\quad
\;
{\tt U}_{\epsilon}(t) \,=
\int_{\mathbb{R}} \!\;\!
|\, u(x,t) \,|^{\mbox{}^{\mbox{\scriptsize $q$}}} \;\!
w_{\epsilon}(x) \: dx,
$$\
where $ {\displaystyle
\,
w_{\epsilon}(x) \;\!=\;\! \exp\;\!\{\;\!-\:\epsilon \,\sqrt{1 + x^{2}\,}\,\}
} $, $ {\displaystyle
\;\!
G_{\epsilon}(t) \;\!=\;\!
\frac{1}{2} \, q \;\!(q-1) \;\! B(t)^{2}
+\,\!
\epsilon \;\! 2 \;\! q \,\!\cdot \!\!\!
\sup_{0\,\leq\,\tau\,\leq\,t} \!\!\!
\|\,u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
} $\
$ {\displaystyle
+\,
\epsilon
} $, $\;\!$and\
\
$$\tag{2.2$b$}
{\tt V}_{\epsilon}(t) \,=\:
\left\{\,
\begin{array}{lll}
\mbox{$ {\displaystyle
\frac{1}{2} \: q \,(q-1) \!
\int_{0}^{\mbox{\footnotesize $\:\!t$}} \!\!\;\!
\int_{\mbox{\scriptsize $\:\!u \neq 0$}} \hspace{-0.500cm}
|\: u(x,\tau) \,|^{\mbox{}^{\scriptstyle \;\!q\;\!-\;\!2}} \;\!
|\:u_{x}(x,\tau)\,|^{\mbox{}^{\scriptstyle 2}} \;\!
w_{\epsilon}(x) \; dx \, d\tau
} $},
& \mbox{} & \mbox{if }\; q \neq 2, \\
\mbox{} \vspace{-0.250cm} \\
\mbox{$ {\displaystyle
\int_{0}^{\mbox{\footnotesize $\:\!t$}} \!\!\;\!
\int_{\mathbb{R}} \:\!
|\:u_{x}(x,\tau)\,|^{\mbox{}^{\scriptstyle 2}} \;\!
w_{\epsilon}(x) \; dx \, d\tau
} $},
& \mbox{} & \mbox{if }\; q = 2.
\end{array}
\right.$$\
By Gronwall’s lemma, (2.2) gives $ \,
{\tt U}_{\epsilon}(t) \leq\;\!
{\tt U}_{\epsilon}(0) \cdot
\exp\,\big\{ \!\;\!\int_{0}^{\mbox{\footnotesize $\:\!t$}}
\!\;\!G_{\epsilon}(\tau) \, d\tau \,\!\bigr\}
$, from which we obtain (2.1) by simply letting $ \epsilon \rightarrow 0 $. This shows, in particular, that $ {\displaystyle
\;\!
u(\cdot,t) \in
L^{\infty}_{\tt loc}(\:\![\;\!0, T_{\ast}[, L^{q}(\mathbb{R})\:\!)
} $ if $ p_{\mbox{}_{\!\;\!0}} \!\leq q < \infty $. Now, to get $ {\displaystyle
u(\cdot,t) \in
C^{0}(\:\![\;\!0, T_{\ast}[, L^{q}(\mathbb{R})\:\!)
} $, it is sufficient to show that, given $\;\!\varepsilon > 0 \;\!$ and $ \;\! 0 < \mbox{\footnotesize $T$} < \mbox{\footnotesize $T$}_{\ast} $ arbitrary, we can find $ {\displaystyle
\;\!\mbox{\footnotesize $R$} \;\!=\;\!
\mbox{\footnotesize $R$}(\varepsilon,\mbox{\footnotesize $T$})
\gg 1 \;\!
} $ large enough so that we have $ {\displaystyle
\;\!
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\;\!|\,x\,| \,>\, R\;\!)}}
\!\!\;\!< \varepsilon
\:\!
} $ for any $ \;\!0 \leq t \leq \mbox{\footnotesize $T$} $. Taking $ \psi \in C^{2}(\mathbb{R}) $ with $ 0 \leq \psi \leq 1 $ and $ \psi(x) = 0 $ for all $ x \leq 0 $, $ \psi(x) = 1 $ for all $ x \geq 1 $, let $ \Psi_{\scriptstyle \!\:\!R,\,M} \in C^{2}(\mathbb{R}) $ be the cut-off function given by $ \;\!\Psi_{\scriptstyle \!\:\!R,\,M} (x) = 0 \;\!$ if $ |\,x\,| \leq \mbox{\footnotesize $R$} - 1 $, $ \;\!\Psi_{\scriptstyle \!\:\!R,\,M} (x) =
\psi(\;\!|\,x\,| - \mbox{\footnotesize $R$} + 1 ) \;\!$ if $ \mbox{\footnotesize $R$} - 1 < |\,x\,| < \mbox{\footnotesize $R$} $, and $ \;\!\Psi_{\scriptstyle \!\:\!R,\,M} (x) = 1 \;\!$ if $ \mbox{\footnotesize $R$} \leq |\,x\,| \leq
\mbox{\footnotesize $R$} + \mbox{\footnotesize $M$} $, $ \;\!\Psi_{\scriptstyle \!\:\!R,\,M} (x) =
\psi(\;\!\mbox{\footnotesize $R$} + \mbox{\footnotesize $M$} + 1 - |\,x\,| \;\!) \;\!$ if $ \mbox{\footnotesize $R$} + \mbox{\footnotesize $M$} < |\,x\,|
< \mbox{\footnotesize $R$} + \mbox{\footnotesize $M$} + 1 $, $ \;\!\Psi_{\scriptstyle \!\:\!R,\,M} (x) = 0 \;\!$ if $ \;\!|\,x\,| \geq \mbox{\footnotesize $R$} + \mbox{\footnotesize $M$} + 1 $, $\;\!$where $ \mbox{\footnotesize $R$} > 1 $, $ \mbox{\footnotesize $M$} > 0 $ are given. Multiplying (1.1$a$) by $ \Phi_{\delta}^{\prime}(u(x,t)) \cdot
\Psi_{\scriptstyle \!\:\!R,\,M} (x) \;\! $ if $ q \neq 2 $, or $ \,\!u(x,t) \cdot \Psi_{\scriptstyle \!\:\!R,\,M} (x) \;\! $ if $ q = 2 $, and integrating the result on $ \mathbb{R} \!\;\!\times\!\;\! [\;\!0, t\;\!] $, $ 0 < t \leq \mbox{\footnotesize $T$} $, we obtain, as in (2.2), by letting $ \delta \rightarrow 0 $, $ \mbox{\footnotesize $M$} \rightarrow \infty $, that $ {\displaystyle
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\;\!|\,x\,| \,>\, R\;\!)}}
\!\!\;\!< \varepsilon/2 \,+\,
\|\, u(\cdot,0) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\;\!|\,x\,| \,>\, R-1\;\!)}}
\:\!
} $ for all $ 0 \leq t \leq \mbox{\footnotesize $T$} $, provided that we take $ R > 1 $ sufficiently large. This gives the continuity result, and the proof is complete. $ \Box$\
]{}\
An important by-product of the proof above is that we have (letting $ \epsilon \rightarrow 0 $ in (2.2), and using (2.1)), for each $ {\displaystyle
\;\!
0 < \mbox{\footnotesize $T$} < \mbox{\footnotesize $T$}_{\ast}
} $ and $ \;\!q \geq \max\;\!\{\;\! p_{\mbox{}_{\!\;\!0}}, 2 \;\!\} $,\
\
$$\tag{2.3}
\int_{0}^{\mbox{\scriptsize $\;\!T$}} \!\!\!
\int_{\mathbb{R}} \,
|\, u(x,\tau) \,|^{\mbox{}^{\scriptstyle q-2}} \,
|\, u_x(x,\tau) \,|^{\mbox{}^{\scriptstyle 2}}
\;\! dx \, d\tau
\,< \infty.$$\
Therefore, if we repeat the steps above leading to (2.2), we obtain (letting $ \delta \rightarrow 0 $, $ \mbox{\small $R$} \rightarrow \infty $, $ \epsilon \rightarrow 0 $, in this order, taking (2.1), (2.3) into account) the identity\
\
$ {\displaystyle
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
^{\mbox{}^{\scriptstyle \;\!q}}
+\;
q \, (q-1) \!
\int_{0}^{\mbox{\footnotesize $\:\!t$}} \!\!\;\!
\int_{\mathbb{R}}
|\, u(x,\tau) \,|^{\mbox{}^{\scriptstyle q - 2}} \,
|\, u_x(x,\tau) \,|^{\mbox{}^{\scriptstyle 2}}
\;\! dx \, d\tau
\;=
} $\
\
(2.4)\
\
$ {\displaystyle
\mbox{}\;\,=\;\,
\|\, u(\cdot,0) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
^{\mbox{}^{\scriptstyle \;\!q}}
+\;
q \, (q-1) \!
\int_{0}^{\mbox{\footnotesize $\:\!t$}} \!\!
\int_{\mathbb{R}}
\bigl(\;\! b(x,\tau) - \beta(\tau) \:\!\bigr) \,
|\, u(x,\tau) \,|^{\mbox{}^{\scriptstyle q - 2}}
\hspace{-0.300cm}
u(x,\tau) \,
u_x(x,\tau)
\; dx \, d\tau
} $\
\
for every $ \;\!0 < t < \mbox{\small $T$}_{\!\ast} $ and $ \;\!\max\;\!\{\;\! p_{\mbox{}_{\!\;\!0}}, 2 \;\!\} \leq q < \infty $, where\
\
$$\tag{2.5}
\beta(t)
\,=\,
\frac{\mbox{\small $1$}}{\;\!\mbox{\small $2$}\;\!} \,
\Bigl(\,
\sup_{x \,\in\, \mathbb{R}} \;\!b(x,t)
\;\;\!+\:
\inf_{x \,\in\, \mathbb{R}} \!\;\!b(x,t)
\,\Bigr),
\qquad
t \geq 0.$$\
The core of the difficulty in the analysis of (1.1) is apparent here: under the sole assumption that $ b $ is bounded, it is not much clear how one should go about the last term in (2.4) in order to get more than (2.1) above. Actually, it will be convenient to consider (2.4) in the (equivalent) differential form, i.e.,\
\
$ {\displaystyle
\frac{d}{d \:\!t} \:
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
^{\mbox{}^{\scriptstyle \;\!q}}
+\;
q \, (q-1) \!
\int_{\mathbb{R}}
|\, u(x,t) \,|^{\mbox{}^{\scriptstyle q - 2}} \,
|\, u_x(x,t) \,|^{\mbox{}^{\scriptstyle 2}}
\;\! dx
\;=
} $\
\
(2.6)\
\
$ {\displaystyle
\mbox{}\;\,=\;
q \, (q-1) \!
\int_{\mathbb{R}}
\bigl(\;\! b(x,t) - \beta(t) \:\!\bigr) \,
|\, u(x,t) \,|^{\mbox{}^{\scriptstyle q - 2}}
\hspace{-0.300cm}
u(x,t) \,
u_x(x,t)
\; dx
} $\
\
for all $ {\displaystyle
t \in \:\![\;\!0, \mbox{\small $T$}_{\!\ast} [ \,\setminus\;\! E_{q}
} $, where $ E_{q} \!\;\!\subset [\;\!0, \mbox{\small $T$}_{\!\ast}[ $ has zero measure. We then readily obtain, using (1.9) and the one-dimensional Nash inequality [@Nash1958]\
\
$$\tag{2.7}
\|\: {\tt v} \:\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}
\leq\,
C_{\mbox{}_{2}} \,
\|\: {\tt v} \:\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R})}}
^{\mbox{}^{\scriptstyle \;\!2/3}}
\;\!
\|\: {\tt v}_{x} \,\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}
^{\mbox{}^{\scriptstyle \:\!1/3}}
\!,
\qquad
C_{\mbox{}_{2}}
=\,
\Bigl(\;\!
\mbox{\small $ {\displaystyle
\frac{\small \;\!3 \;\!\sqrt{\:\!3\;}\,}{\small 4 \:\! \pi} }$}
\;\!\Bigr)^{\mbox{}^{\scriptstyle \!\!\!\,\! 1/3}}
\!\!\!,$$\
where the value given above for $C_{\mbox{}_{2}}$ is optimal [@CarlenLoss1993], the following result:\
\
[**Theorem 2.2.**]{} *$\!\:\!$Let $ \:\!q \geq 2 \:\!p_{\mbox{}_{\!\;\!0}} $. $\!$If $ {\displaystyle
\;\!
\hat{t} \in
\:\![\;\!0, \mbox{\small $T$}_{\!\ast} [ \,\setminus\;\! E_{q}
\!\;\!
} $ is such that $ {\displaystyle
\;\!
\mbox{\footnotesize $ {\displaystyle \frac{d}{d\:\!t} }$} \,
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
^{\mbox{}^{\scriptstyle \;\!q}}
{\mbox{}_{\bigr|}}_{\mbox{}_{\mbox{\footnotesize $t = \:\!\hat{t}$}}}
\hspace{-0.700cm}
\geq\: 0
} $,\
then*\
$$\tag{2.8$a$}
\|\, u(\cdot,\hat{t}\:\!) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\;\!\leq\,
\Bigl(\;\!
\mbox{\small $ {\displaystyle \frac{\;\!q\;\!}{2} }$} \,
C_{\mbox{}_{\!\;\!2}}^{\mbox{}^{\scriptstyle \;\!3}}
\:\!\Bigr)^{\mbox{}^{\scriptstyle \!\!\!\;\!1/q}}
\!
B(\:\!\hat{t}\:\!)^{\mbox{}^{\scriptstyle \!\:\!1/q}}
\,
\|\, u(\cdot,\hat{t}\:\!) \,
\|_{\mbox{}_{\scriptstyle L^{q/2}(\mathbb{R})}}$$\
[*and*]{}\
\
$$\tag{2.8$b$}
\|\, u(\cdot,\hat{t}\:\!) \,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\;\!\leq\,
\Bigl(\;\!
\mbox{\small $ {\displaystyle \frac{\;\!q\;\!}{2} }$} \,
C_{\mbox{}_{\!\;\!2}} \;\! C_{\mbox{}_{\!\infty}}
\Bigr)^{\mbox{}^{\scriptstyle \!\!2/q}}
B(\:\!\hat{t}\:\!)^{\mbox{}^{\scriptstyle \!\;\!2/q}}
\,
\|\, u(\cdot,\hat{t}\:\!) \,
\|_{\mbox{}_{\scriptstyle L^{q/2}(\mathbb{R})}}
\!\,\!.$$\
\
[ Consider (2.8$a$) first. From (1.5), (2.5) and (2.6), we have\
\
$$\notag
\int_{\mathbb{R}} \!\;\!
|\, u(x,\hat{t}\:\!) \,|^{\mbox{}^{\scriptstyle q - 2}}
\;\!
|\, u_x(x,\hat{t}\:\!) \,|^{\mbox{}^{\scriptstyle 2}}
\;\!dx
\;\leq\;
B(\:\!\hat{t}\:\!) \!
\int_{\mathbb{R}} \!\;\!
|\, u(x,\hat{t}\:\!) \,|^{\mbox{}^{\scriptstyle q - 1}}
\;\!
|\;\! u_x(x,\hat{t}\:\!) \,|
\;dx.$$\
This gives\
\
$$\notag
\int_{\mathbb{R}} \!\;\!
|\, u(x,\hat{t}\:\!) \,|^{\mbox{}^{\scriptstyle q - 2}}
\,
|\, u_x(x,\hat{t}\:\!) \,|^{\mbox{}^{\scriptstyle 2}}
\;\!dx
\;\leq\;
B(\:\!\hat{t}\:\!)^{\mbox{}^{\scriptstyle 2}}
\;\!
\|\, u(\cdot,\hat{t}\:\!) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
^{\mbox{}^{\scriptstyle \;\!q}}
\!,$$\
or, in terms of $ {\displaystyle
\;\!
\hat{v} \in L^{1}(\mathbb{R}) \cap L^{\infty}(\mathbb{R})
} $ defined by $ {\displaystyle
\;\!
\hat{v}(x) =
|\,u(x,\hat{t}\:\!)\,|^{\mbox{}^{\scriptstyle \;\!q/2}}
\!\;\!
} $ if $ q > 2 $, $ {\displaystyle
\;\!
\hat{v}(x) = u(x,\hat{t}\:\!)
\;\!
} $ if $ q = 2 $,\
\
$$\notag
\|\: \hat{v}_{x} \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}
\,\leq\;
\frac{\;\!q\;\!}{2} \,
B(\:\!\hat{t}\:\!) \;
\|\: \hat{v} \:
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}\!\:\!.$$\
Using (2.7), we then get $ {\displaystyle
\;\!
\|\: \hat{v} \:
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}
^{\mbox{}^{\scriptstyle \;\!2}}
\leq\,
\frac{\;\!q\;\!}{2} \:
C_{\mbox{}_{\!2}}^{\mbox{}^{\scriptstyle \;\!3}}
\;\!
B(\:\!\hat{t}\:\!) \:
\|\: \hat{v} \:
\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R})}}
^{\mbox{}^{\scriptstyle \;\!2}}
\!\:\!
} $, which is equivalent to (2.8$a$).\
Similarly, (2.8$b$) can be obtained, using (1.9). $ \Box$\
]{}\
Thus, we can use (2.8) when $ {\displaystyle
\;\!
\|\,u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\!
} $ is not decreasing. If it is decreasing, (2.6) becomes useless but at least we know in such case that $ {\displaystyle
\;\!
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
} $ is not increasing, which should be useful too. Different values of $q$ have different scenarios, which we will have to piece together in some way. The next result shows us just how. To this end, it is convenient to introduce the quantities $ \mathbb{B}(t_0\:\!; t) $, $ \mathbb{U}_{\!\;\!p}(t_0\:\!; t) $ defined by\
\
$$\tag{2.9}
\mathbb{B}(t_0\:\!; t)
\;=\;\;\!
\sup\:\Bigl\{\;\!
B(\tau)\!\;\!:
\; t_0 \!\leq \tau \leq t \;\Bigr\},$$\
$$\tag{2.10}
\mbox{} \;\;
\mathbb{U}_{p}(t_0\:\!; t)
\;=\;\;\!
\sup\:\Bigl\{\;\!
\|\, u(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{p}(\mathbb{R})}}
\!\!\;\!:
\; t_0 \!\leq \tau \leq t \;\Bigr\},$$\
given $ \;\!p \geq p_{\mbox{}_{\!\;\!0}} \!\;\!$, $ \;\! 0 \leq t_0 \!\;\!\leq t < \mbox{\small $T$}_{\!\!\;\!\ast} $ arbitrary.\
[\
]{}[**Theorem 2.3.**]{} *Let $\;\! q \geq 2 \:\!p_{\mbox{}_{\!\;\!0}} $. For each $ \,0 \leq t_0 \!\;\!< \mbox{\small $T$}_{\!\!\;\!\ast} $, we have*\
\
$$\tag{2.11}
\mathbb{U}_{\!\;\!q}(t_0\:\!; t)
\;\leq\;\;\!
\max\,\biggl\{\;\!
\|\, u(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}};
\,
\Bigl(\;\! \frac{\;\!q\;\!}{2} \,
C_{\mbox{}_{\!2}}^{\mbox{}^{\scriptstyle \;\!3}}
\,\!\Bigr)^{\mbox{}^{\scriptstyle \!\!\!
\frac{\scriptstyle 1}{\scriptstyle q} }}
\mathbb{B}(t_0\:\!; t)^{\mbox{}^{\scriptstyle \!
\frac{\scriptstyle 1}{\scriptstyle q} }}
\;\!
\mathbb{U}_{\mbox{}_{\scriptstyle \!
\frac{\scriptstyle q}{\scriptscriptstyle 2}}}\!(t_0\:\!; t)
\,\biggr\}$$\
*for all* $\;\! t_0 \!\;\! \leq t < \mbox{\small $T$}_{\!\!\;\!\ast} $.\
\
[ Set $ {\displaystyle
\;\!
\lambda_{q}(t)
\;\!=\;\!
\Bigl(\;\! \frac{\;\!q\;\!}{2} \,
C_{\mbox{}_{\!2}}^{\mbox{}^{\scriptstyle \;\!3}}
\,\!\Bigr)^{\mbox{}^{\scriptstyle \!\!\!
\frac{\scriptstyle 1}{\scriptstyle q} }}
\mathbb{B}(t_0\:\!; t)^{\mbox{}^{\scriptstyle \!
\frac{\scriptstyle 1}{\scriptstyle q} }}
\;\!
\mathbb{U}_{\mbox{}_{\scriptstyle \!
\frac{\scriptstyle q}{\scriptscriptstyle 2}}}\!(t_0\:\!; t)
} $. There are three cases to consider:\
\
[Case I:]{} $ {\displaystyle
\|\, u(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\!> \lambda_{q}(t)
\;\!
} $ for all $\;\! t_0 \!\;\!\leq \tau \leq t $. $\;\!$By (2.8$a$), Theorem 2.2, we must then have $ {\displaystyle
\;\!
d/d\tau \;\!
\|\, u(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
^{{\scriptstyle \;\!q}}
\!\!\;\!< 0
\;\!
} $ for all $ \;\!\tau \!\;\!\in \!\;\![\;\!t_0, t \;\!] \:\!\setminus\,\! E_{q} $, so that $ {\displaystyle
\;\!
\|\, u(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\!
} $ is monotonically decreasing in $ [\;\!t_0, t \;\!] $. In particular, $ {\displaystyle
\;\!
\mathbb{U}_{\!\;\!q}(t_0\:\!; t)
\!\;\!=\;\!
\|\, u(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\!
} $ in this case, and (2.11) holds.\
\
[Case II:]{} $ {\displaystyle
\|\, u(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\!> \lambda_{q}(t)
\;\!
} $ and $ {\displaystyle
\;\!
\|\, u(\cdot,t_1) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\!\leq \lambda_{q}(t)
\;\!
} $ for some $ t_1 \!\in\; ]\;\!t_0, t\;\!] $.\
\
In this case, let $ t_2 \!\in\; ]\;\!t_0, t\;\!] $ be such that we have $ {\displaystyle
\;\!
\|\, u(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\!>\!\;\! \lambda_{q}(t)
\;\!
} $ for all $ t_0 \!\leq \tau < t_2 $, while $ {\displaystyle
\;\!
\|\, u(\cdot,t_2) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\!= \lambda_{q}(t)
} $. We claim that $ {\displaystyle
\;\!
\|\, u(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\!\leq\!\;\! \lambda_{q}(t)
\;\!
} $ for every $ t_2 \!\leq \tau \leq t $: in fact, if this were not true, we could then find $ t_3, t_4 $ with $ t_2 \!\leq t_3 \!< t_4 \!\leq t $ such that $ {\displaystyle
\|\, u(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\!>\!\;\! \lambda_{q}(t)
\;\!
} $ for all $ t_3 \!< \tau \leq t_4 $, $ {\displaystyle
\;\!
\|\, u(\cdot,t_3) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\!= \lambda_{q}(t)
} $. By (2.8$a$), Theorem 2.2, this would require $ {\displaystyle
\,
d/d\tau \;\!
\|\, u(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
^{{\scriptstyle \;\!q}}
\!\! < 0
\;\!
} $ for all $ \;\!\tau \!\;\!\in \;]\;\!t_3, t_4 \,\!] \:\!\setminus\,\! E_{q} $, so that $ {\displaystyle
\;\!
\|\, u(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\!
} $ could not increase anywhere on $ \;\![\;\!t_3, t_4 \,\!] $. This contradicts $ {\displaystyle
\;\!
\|\, u(\cdot,t_3) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\!<\;\!
\|\, u(\cdot,t_4) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\!\;\!
} $, and so we have $ {\displaystyle
\;\!
\|\, u(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\!\leq\!\;\! \lambda_{q}(t)
\;\!
} $ for every $ t_2 \!\leq \tau \leq t $, as claimed. On the other hand, by (2.8$a$), $ {\displaystyle
\|\, u(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\!
} $ has to be monotonically decreasing on $ \;\![\;\!t_0, t_2 \,\!] $, just as in [Case]{} [I]{}. Therefore, we have $ {\displaystyle
\;\!
\mathbb{U}_{\!\;\!q}(t_0\:\!; t)
\!\;\!=\;\!
\|\, u(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\!
} $ in this case again, which shows (2.11).\
\
[Case III:]{} $ {\displaystyle
\|\, u(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\!\leq \lambda_{q}(t)
} $. This gives $ {\displaystyle
\;\!
\|\, u(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\!\leq\!\;\! \lambda_{q}(t)
\;\!
} $ for every $ t_0 \!\leq \tau \leq t $, by repeating the argument used on the interval $\;\![\;\!t_2, t \;\!] \;\!$ in [Case II]{} above. It follows that we must have $ {\displaystyle
\;\!
\mathbb{U}_{\!\;\!q}(t_0\:\!; t)
\leq
\lambda_{q}(t)
} $ in this case, and the proof of Theorem 2.3 is complete. $\Box$\
]{}\
An important application of Theorem 2.3 is the following result.\
[\
]{}[**Theorem 2.4.**]{} *Let $\, p_{\mbox{}_{\!\;\!0}} \!\;\!\leq p < \infty $, $\;\! 0 \leq t_0 \!\;\!< \mbox{\small $T$}_{\!\ast} $. Then*\
\
$$\tag{2.12}
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\;\!\leq\:
\bigl(\;\! 2 \:\! p \;\!
\bigr)^{\mbox{}^{\scriptstyle \!\!\:\!\frac{1}{\scriptstyle p}}}
\!\cdot\;
\max\;\!
\biggl\{\,
\|\, u(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\,\!;\;
\mathbb{B}(t_0\:\!; t)^{\mbox{}^{\scriptstyle \!\!\;\!\frac{1}{\scriptstyle p}}}
\!\:
\mathbb{U}_{\!\;\!p}(t_0\:\!; t)
\,\biggr\}$$\
*for any $ \;\!t_0 \leq t < \mbox{\small $T$}_{\!\ast} $, $\;\!$where $ {\displaystyle
\;\!
\mathbb{B}(t_0\:\!; t)
} $, $ {\displaystyle
\mathbb{U}_{\!\;\!p}(t_0\:\!; t)
\;\!
} $ are given in $\;\!(2.9)$, $(2.10)$ above.*\
\
[ Let $\;\!k \in \mathbb{Z} $, $ k \geq 2 $. Applying (2.11) successively with $ {\displaystyle
\;\!q \:\!=\:\!
2 \:\!p, \;\! 4\:\!p, ...\:\!, \;\! 2^{k}p
} $, we obtain\
\
$ {\displaystyle
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2^{k}\!p}(\mathbb{R})}}
\!\:\!\leq\;
\max \,\biggl\{\,
\|\, \mbox{\boldmath $u$}(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{2^{k}p}(\mathbb{R})}}
\!\:\!;\:
K\!\:\!(k,\ell)^{\mbox{}^{\scriptstyle \!\:\!\frac{1}{\scriptstyle p}}}
\!\!\!\cdot\;\!
\mathbb{B}(t_0\:\!; t)^{\mbox{}^{\scriptstyle \!\:\!\frac{1}{\scriptstyle p}
\bigl(\,\!2^{\mbox{}^{\!-\ell}} \!\!\!\!\;\!-\; 2^{\mbox{}^{\!-k}}\,\!\bigr)}}
\hspace{-1.100cm} \cdot \hspace{0.770cm}
\|\, u(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{2^{\ell}\!p}(\mathbb{R})}}
\!\,\!,
} $\
\
$ 1 \leq \ell \leq k - 1 \:\!; $\
\
$ {\displaystyle
K\!\:\!(k,0)^{\mbox{}^{\scriptstyle \!\:\!\frac{1}{\scriptstyle p}}}
\!\!\!\:\!\cdot\;\!
\mathbb{B}(t_0\:\!; t)^{\mbox{}^{\scriptstyle \!\:\!\frac{1}{\scriptstyle p}
\bigl(\:\!1\;\!-\: 2^{\mbox{}^{\!-k}}\,\!\bigr)}}
\hspace{-0.990cm} \cdot \hspace{0.700cm}
\mathbb{U}_{\!\;\!p}(t_0\:\!; t)
\,\biggr\}\,\!
} $, (2.13$a$)\
\
where\
\
$ {\displaystyle
K\!\:\!(k,\ell) \;\,=\,
\hspace{-0.250cm}
\prod_{\mbox{}\;\;j\,=\,\ell\;\!+\;\!1}^{k}
\!\!\!\!\;\!
\bigl(\;\! 2^{\mbox{}^{\scriptstyle \:\!j - 1}}
\!\:\!p \;C_{\mbox{}_{\!2}}^{\mbox{}^{\scriptstyle \;\!3}}
\;\!\bigr)^{\mbox{}^{\scriptstyle \!\,\! 2^{\mbox{}^{\!\;\!-\;\!j}} }}
\!\!\!\!\!,
\mbox{} \hspace{0.550cm}
0 \leq \ell \leq k - 1
} $. (2.13$b$)\
\
Now, for $\;\!1 \leq \ell \leq k - 1 $:\
\
$ {\displaystyle
\mathbb{B}(t_0\:\!; t)^{\mbox{}^{\scriptstyle \!\:\!\frac{1}{\scriptstyle p}
\bigl(\,\!2^{\mbox{}^{\!-\ell}} \!\!\!\!\:\!-\; 2^{\mbox{}^{\!-k}}\,\!\bigr)}}
\hspace{-1.040cm} \cdot \hspace{0.770cm}
\|\, u(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{2^{\ell}\!p}(\mathbb{R})}}
} $\
\
$ {\displaystyle
\leq\;
\mathbb{B}(t_0\:\!; t)^{\mbox{}^{\scriptstyle \!\:\!\frac{1}{\scriptstyle p}
\bigl(\,\!2^{\mbox{}^{\!-\ell}} \!\!\!\!\:\!-\; 2^{\mbox{}^{\!-k}}\,\!\bigr)}}
\hspace{-1.040cm} \cdot \hspace{0.770cm}
\|\, u(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{p}(\mathbb{R})}}
^{\mbox{}^{\scriptstyle \!
\frac{\scriptstyle \;\!
2^{\mbox{}^{\!-\ell}} \!\!\!-\; 2^{\mbox{}^{\!-k}} }
{\scriptstyle 1 \;\!-\: 2^{\mbox{}^{\!-k}} } }}
\hspace{-0.350cm} \cdot \;\,
\|\, u(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{2^{k}\!p}(\mathbb{R})}}
^{\mbox{}^{\scriptstyle \!
\frac{\scriptstyle \;\!1 \;\!-\: 2^{\mbox{}^{\!-\ell}} }
{\scriptstyle \;\!1 \;\!-\: 2^{\mbox{}^{\!-k}} } }}
} $\
\
$ {\displaystyle
\leq\;
\max\,\biggl\{\,
\|\, u(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{2^{k}\!p}(\mathbb{R})}}
\!\:\!;\;\;\!
\mathbb{B}(t_0\:\!; t)^{\mbox{}^{\scriptstyle \!\:\!\frac{1}{\scriptstyle p}
\bigl(\:\!1 \;\!-\, 2^{\mbox{}^{\!-k}}\,\!\bigr)}}
\hspace{-1.000cm} \cdot \hspace{0.670cm}
\|\, u(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{p}(\mathbb{R})}}
\;\!\biggr\}
} $\
\
by Young’s inequality (see e.g.$\;$[@Evans2002], p.$\;$622); in particular, we get, from (2.13),\
\
$ {\displaystyle
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2^{k}\!p}(\mathbb{R})}}
\!\;\!\leq\;
\bigl(\;\! 2 \:\! p \,
\bigr)^{\mbox{}^{\scriptstyle \!\!\:\!\frac{1}{\scriptstyle p}}}
\!\!\!\:\!\cdot\,
\max\,
\biggl\{\,
\|\, u(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{2^{k}\!p}(\mathbb{R})}}
\!\:\!;\;\,
\mathbb{B}(t_0\:\!; t)^{\mbox{}^{\scriptstyle \!\:\!\frac{1}{\scriptstyle p}
\bigl(\:\!1 \;\!-\, 2^{\mbox{}^{\!-k}}\,\!\bigr)}}
\hspace{-1.000cm} \cdot \hspace{0.670cm}
\mathbb{U}_{p}(t_0\:\!; t)
\,\biggr\}
} $,\
\
since $ {\displaystyle
\:\!
K\!\;\!(k,\ell)
\leq
2 \:\! p \,
} $ for all $ \;\! 0 \leq \ell \leq k - 1 $. Letting $ k \rightarrow \infty $, (2.12) is obtained. $\Box$\
]{}\
It follows from Theorems 2.1 and 2.4 that $ u(\cdot,t) $ is globally defined ($ \mbox{\small $T$}_{\!\ast} \!= \infty $). Now, from (2.12), we immediately obtain, letting $ \;\! t \rightarrow \infty $,\
\
$ {\displaystyle
\limsup_{t\,\rightarrow\,\infty} \;\!
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\leq\;\!
\bigl(\;\! 2 \;\! p \;\!
\bigr)^{\mbox{}^{\scriptstyle \!\!\:\!\frac{1}{\scriptstyle p}}}
\!\!\cdot\;
\max\,
\biggr\{\;\!
\|\, u(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\!\:\!; \;\;\!
\mathbb{B}(t_0)^{\mbox{}^{\scriptstyle \!\!\;\!\frac{1}{\scriptstyle p}}}
\!\,
\mathbb{U}_{\!\;\!p}(t_0)
\;\!\biggr\}
} $ (2.14)\
\
for any $ t_0 \geq 0 $, where $ \mathbb{B}(t_0) $, $ \mathbb{U}_{\!\;\!p}(t_0) $ are given by\
\
$$\tag{2.15}
\mathbb{B}(t_0)
\;=\;\;\!
\sup\:\Bigl\{\;\!
B(t)
\!\;\!:
\; t \geq t_0 \,\Bigr\},$$\
$$\tag{2.16}
\mbox{} \;\;
\mathbb{U}_{p}(t_0)
\;=\;\;\!
\sup\:\Bigl\{\,
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{p}(\mathbb{R})}}
\!\!\:\!:
\; t \geq t_0 \,\Bigr\}.$$\
Taking $ (\:\! t_0^{(n)} )_{n} \!\;\!$ such that $ \:\!t_0^{(n)} \!\rightarrow \infty \,$ and $ {\displaystyle
\,
\|\, u(\cdot,t_0^{(n)}) \,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\!\rightarrow\,
\liminf_{t\,\rightarrow\,\infty} \,
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\!\:\!
} $,\
and applying (2.14) with $ \:\!t_0^{\mbox{}} \!=\,\! t_0^{(n)} $ for each $n$, we then obtain, letting $ n \rightarrow \infty $,\
\
$ {\displaystyle
\limsup_{t\,\rightarrow\,\infty} \;\!
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\!\;\!\leq\;\!
\bigl(\;\! 2 \:\! p \;\!
\bigr)^{\mbox{}^{\scriptstyle \!\!\:\!\frac{1}{\scriptstyle p}}}
\!\!\!\;\!\cdot\;\!
\max\,
\biggr\{\!\;\!
\liminf_{t\,\rightarrow\,\infty} \:\!
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\!\;\!; \;\;\!
{\cal B}^{\mbox{}^{\scriptstyle \:\!\frac{1}{\scriptstyle p}}}
\!\!\!\,\!\cdot\;\!\!\;\!
{\cal U}_{p}
\;\!\biggr\},
} $ (2.17)\
\
where $ \;\!{\cal B} $, $ {\cal U}_{p} $ are given by\
\
$$\tag{2.18}
{\cal B}
\;\!=\:
\limsup_{t\,\rightarrow\,\infty}
\,
B(t),
\qquad
{\cal U}_{p}
\;\!=\:
\limsup_{t\,\rightarrow\,\infty}
\;\!
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{p}(\mathbb{R})}}
\!\;\!.$$ [\
]{}\
[**§3. Large time estimates**]{}\
In this section, we use the results obtained above to derive two basic large time estimates (given in Theorems 3.1 and 3.2 below) for solutions $ u(\cdot,t) $ of problem (1.1$a$), (1.1$b$), which represent important intermediate steps that will ultimately lead to the main result stated in Theorem 3.3.\
[\
]{}[**Theorem 3.1.**]{} *Let $ \;\!q \geq 2 \:\!p_{\mbox{}_{\!\;\!0}} \!\:\!$, and $ \;\!{\cal B} \!\;\!\geq 0 \:\!$ be as defined in $(2.18)$. Then*\
\
$$\tag{3.1}
\limsup_{t\,\rightarrow\,\infty} \;\!
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R})}}
\leq\,
\Bigl(\;\! \frac{\;\!q\;\!}{\mbox{\small $2$}}
\, C_{\mbox{}_{\!\;\!2}}^{\;\!3} \;\!
\Bigr)^{\scriptstyle \!\!
\frac{1}{\scriptstyle q} } \!\cdot\,
{\cal B}^{\mbox{}^{\scriptstyle \:\!
\frac{1}{\scriptstyle q} }} \!\cdot\;
\limsup_{t\,\rightarrow\,\infty} \;\!
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{q/2}(\mathbb{R})}}
\!\:\!,
$$\
*where $ {\displaystyle
\;\!C_{\mbox{}_{\!\;\!2}}
\!\:\!=\,\!
\bigl(\;\! 3 \;\!\sqrt{\:\!3\,} /\;\!(4 \:\!\pi) \:\!\bigr)^{\!1/3}
\!
} $ is the constant in the Nash inequality $\;\!(2.7)$.*\
\
[ We set $ \;\! p = q/2 \;\!$ and assume that $ {\cal U}_{p} $ is finite. As in the proof of Theorem 2.2, we take $ {\displaystyle
\;\! v \in L^{\infty}(\mathbb{R} \times [\;\!0, \infty\:\![)
\;\!
} $ given by $ {\displaystyle
v(x,t) = |\, u(x,t) \,|^{{\scriptstyle p}}
} $ if $ p > 1 $, $ {\displaystyle
v(x,t) = u(x,t)
\;\!
} $ if $ p = 1 $. It follows that\
\
$ {\displaystyle
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}^{2}
\!\;\!=\;
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2p}(\mathbb{R})}}^{2p}
\!
} $,\
\
$ {\displaystyle
\|\, v_{x}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}^{2}
\!\;\!=\;
p^{2} \!
\int_{\mathbb{R}} \!\;\!
|\, u(x,t) \,|^{\mbox{}^{\scriptstyle \;\!2\;\!
\mbox{\footnotesize $p$} \;\!-\;\! 2}}
\;\!
|\, u_x(x,t) \,|^{\mbox{}^{\scriptstyle \;\!2}}
\, dx
} $.\
\
Therefore, from (2.6), we have, for some null set $ {\displaystyle
E_{\mbox{}_{2\:\!\mbox{\scriptsize $p$}}}
\!\subset [\;\!0, \infty \;\![
} $,\
\
$ {\displaystyle
\frac{d}{d\:\!t} \,
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}^{2}
\:\!+\;
4 \,\Bigl( 1 - \frac{1}{2\:\!p} \Bigr)
\,
\|\, v_{x}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}^{2}
} $\
\
$ {\displaystyle
\leq\;
4 \,p\,
\Bigl( 1 - \frac{1}{2\:\!p} \Bigr)
\, B(t) \:
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}
\:\!
\|\, v_{x}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}
} $\
\
for all $ {\displaystyle
\;\! t \in [\;\!0, \infty\;\![ \,\setminus\;\!
E_{\mbox{}_{2\:\!\mbox{\scriptsize $p$}}}
} $, and so, by (2.7),\
\
$ {\displaystyle
\frac{d}{d\:\!t} \,
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}^{2}
\:\!+\;
4 \,\Bigl( 1 - \frac{1}{2\:\!p} \Bigr)
\,
\|\, v_{x}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}^{2}
} $\
\
$ {\displaystyle
\leq\;
4 \,p \:C_{\mbox{}_{\!\;\!2}} \:\!
\Bigl( 1 - \frac{1}{2\:\!p} \Bigr)
\;\! B(t) \:
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R})}}^{\;\!2/3}
\:\!
\|\, v_{x}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}^{\;\!4/3}
\!\:\!
} $.\
\
This gives, by Young’s inequality ([@Evans2002], p.$\;$622), for all $ {\displaystyle
\;\! t \in [\;\!0, \infty\;\![ \,\setminus\;\!
E_{\mbox{}_{2\:\!\mbox{\scriptsize $p$}}}
} $,\
\
$ {\displaystyle
\frac{d}{d\:\!t} \,
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}^{2}
\:\!+\;
\frac{\;\!4\;\!}{3} \;\!
\Bigl( 1 - \frac{1}{2\:\!p} \Bigr)
\,
\|\, v_{x}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}^{2}
\:\!\leq
} $\
\
(3.2)\
\
$ {\displaystyle
\leq\;
\frac{\;\!4\;\!}{3} \,
\Bigl( 1 - \frac{1}{2\:\!p} \Bigr)
\;\!
\bigl(\;\! p \:C_{\mbox{}_{\!\;\!2}} \:\!
\bigr)^{\!\;\!3}
\,
B(t)^{3} \,
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R})}}^{\;\!2}
\!\:\!
} $.\
\
Setting\
\
$ {\displaystyle
\lambda_{p} \;\!=\;
\limsup_{t\,\rightarrow\,\infty} \,
g(t),
\qquad
g(t) \,=\,
\bigl(\, p \:C_{\mbox{}_{\!\;\!2}}^{\;\!3}
\:\!\bigr)^{\mbox{}^{\scriptstyle \!\!\:\!1/2}}
\!\;\!B(t)^{\mbox{}^{\scriptstyle \!\!1/2}} \,
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R})}}
\!\:\!
} $,\
\
we claim that\
\
$$\tag{3.3}
\limsup_{t\,\rightarrow\,\infty} \;\!
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}
\leq\,
\lambda_{p}
\:\!.
\;\;\;$$\
In fact, let us argue by contradiction. If (3.3) is false, we can pick $ {\displaystyle
\;\!0 < \eta \ll 1
\;\!
} $ and a sequence $ (\;\!t_{j} \:\!)_{\mbox{}_{\scriptstyle \!\;\!j \,\geq\,0}} $, $ t_{j} \rightarrow \infty $, such that $ {\displaystyle
\;\!
\|\, v(\cdot,t_{j}) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}
\!>\!\;\!
\lambda_{p} \!\;\!+ \eta
\,
} $ (for all $ j \geq 0 $) and $ {\displaystyle
\;\!
g(t) \leq
\lambda_{p} \!\;\!+ \eta/2
\;\!
} $ for all $ \;\!t \geq t_0 $. From (2.8$a$), Theorem 2.2, it will then follow that\
\
$$\tag{3.4}
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}
>\:\!
\lambda_{p} +\;\! \eta,
\qquad
\forall \;\, t \geq t_0 \:\!.$$\
In fact, suppose that (3.4) were false, so that we had $ {\displaystyle
\;\!
\|\, v(\cdot, \tilde{t}) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}
\!\!\:\!\leq\!\;\!
\lambda_{p} \!+ \eta
\;\!
} $ for some $ \;\!\tilde{t} > t_0 $. Taking $ j \gg 1 $ with $ t_{j} \!> \tilde{t} $, we could then find $ \;\!\hat{t} \!\;\!\in [\,\tilde{t}, t_{j} \:\![ \:\! $ such that $ {\displaystyle
\;\!
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}
\!\!\;\!>\!\;\!
\lambda_{p} +\;\! \eta
\;\!
} $ for all $ \;\!t \!\:\!\in \:]\,\hat{t}, t_{j} \:\!] $, while $ {\displaystyle
\;\!
\|\, v(\cdot,\hat{t}\:\!) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}
\!=
\lambda_{p} +\;\! \eta
} $, and so there would exist $ {\displaystyle
t_{\ast} \!\in
[\,\hat{t}, t_{j} \:\!]
\:\!\setminus E_{\mbox{}_{2\;\!\mbox{\scriptsize $p$}}}
} $ with $ {\displaystyle
\;\!
d/d\:\!t \,
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}^{2}
\!
} $ positive at $ \;\!t = t_{\ast} $. By (2.8$a$), we would have $ {\displaystyle
\;\!
\|\, v(\cdot,t_{\ast}) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}
\!\leq
\lambda_{p}
} $, but this would contradict the fact that $ {\displaystyle
\;\!
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}
\!\geq
\lambda_{p} \!\,\!+ \eta
\,
} $ everywhere on $ \;\![\, \hat{t}, t_{j} \,\!] $. Thus, we conclude that (3.4) cannot be false, as claimed. $\!$We then obtain, from (2.7), (3.2), (3.4),\
$ {\displaystyle
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}^{\;\!6}
\;\!\leq\;
C_{\mbox{}_{\!2}}^{\;\!6} \,
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R})}}^{\;\!4}
\,
\|\, v_{x}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}^{\;\!2}
} $\
\
$ {\displaystyle
\leq\;
g(t)^{6}
\:+\;
\frac{2\:\!p}{\:\!2\:\!p - 1\:\!} \:
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R})}}^{\;\!4}
\!\;\!
\Bigl(\;\!- \; \frac{d}{d\:\!t} \,
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}^{\;\!2}
\:\! \Bigr)
} $\
\
for all $ {\displaystyle
\, t \!\;\!\in [\;\!t_0, \infty\;\![ \,\setminus\;\!
E_{\mbox{}_{2\:\!\mbox{\scriptsize $p$}}}
} $. Recalling that $ {\displaystyle
\;\!
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}
\!>
\lambda_{p} \!\;\!+ \eta
} $, $ {\displaystyle
\:
g(t) \;\!\leq
\lambda_{p} \!\;\!+ \eta/2
\;\!
} $, $ \; \forall \; t \geq t_0 $, this gives\
\
$$\notag
- \; \frac{d}{d\:\!t} \,
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}^{\;\!2}
\geq\:
K\!\:\!(\eta),
\qquad
\forall \;\,
t \in [\,t_0, \infty\;\![ \;\setminus\,
E_{\mbox{}_{2\:\!\mbox{\scriptsize $p$}}}$$\
for some constant $ K\!\:\!(\eta) > 0\;\! $ independent of $\;\!t$, which cannot be, since this implies\
\
$$\notag
\|\, v(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}^{\;\!2}
\geq\:
K\!\:\!(\eta) \cdot (\;\!t - t_0)
\qquad
\forall \;\,
t > t_0 \:\!.$$\
This contradiction shows (3.3), which is equivalent to (3.1), and the proof is complete. $\Box$\
]{}\
Applying (3.1) successively with $ {\displaystyle
q =\:\!
2\:\!p, 4\:\!p, ... \:\!, 2^{k}p
} $, we get\
\
$$\tag{3.5}
\limsup_{t\,\rightarrow\,\infty} \;\!
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{2^{k}\!\:\!p}(\mathbb{R})}}
\!\;\!\leq\,
\biggl[\;\;\!
\prod_{j\,=\,1}^{k} \;\!
\bigl(\;\! 2^{j-1} \:\!p\;\!
\,C_{\mbox{}_{\!2}}^{\;\!3} \:\!
\bigr)^{\mbox{}^{\scriptstyle \!\!\;\! 2^{\mbox{}^{\!-\;\!j}}}}
\biggr]^{\mbox{}^{\scriptstyle \!\frac{1}{\scriptstyle p}}}
\!\!\!\cdot\,
{\cal B}^{\mbox{}^{\scriptstyle
\;\!\frac{1}{\scriptstyle p}
\!\;\!\bigl( 1 \;\!-\: 2^{-k}\bigr)}}
\hspace{-0.950cm} \cdot \hspace{0.640cm}
{\cal U}_{p}
$$\
for $ k \geq 1 $ arbitrary, where $ {\displaystyle
\;\!
{\cal U}_{p} \!\;\!=\;\!
\limsup_{t\,\rightarrow\,\infty} \;\!
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{p}(\mathbb{R})}}
\!\;\!
} $. Letting $ k \rightarrow \infty$, this suggests\
\
$$\tag{3.6$a$}
\limsup_{t\,\rightarrow\,\infty} \;\!
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\leq\,
K\!\:\!(p)
\,\!\cdot\;\!
{\cal B}^{\mbox{}^{\scriptstyle
\frac{1}{\scriptstyle p}}}
\hspace{-0.150cm} \cdot \hspace{0.060cm}
\limsup_{t\,\rightarrow\,\infty} \;\!
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{p}(\mathbb{R})}}
\!\;\!,$$\
where\
\
$$\tag{3.6$b$}
K\!\:\!(p)
\;=\;
\biggl[\;\;\!
\prod_{j\,=\,1}^{\infty} \;\!
\bigl(\;\! 2^{j-1} \:\!p\;\!
\,C_{\mbox{}_{\!2}}^{\;\!3} \:\!
\bigr)^{\mbox{}^{\scriptstyle \!\!\;\! 2^{\mbox{}^{\!-\;\!j}}}}
\biggr]^{\mbox{}^{\scriptstyle \!\frac{1}{\scriptstyle p}}}
\!=\;
\Bigl(\;\! \frac{\mbox{\small $\;\!3 \;\! \sqrt{\,\!3\;\!} \;$}}
{\mbox{\small $ 2 \:\! \pi $} } \: p \;\!
\Bigr)^{\mbox{}^{\scriptstyle \!\!
\frac{\scriptstyle 1}{\scriptstyle p} }}
\!,$$\
cf.$\;$(1.6) above, as long as the limit processes $ k \rightarrow \infty$, $ t \rightarrow \infty $ can be interchanged. That this is indeed the case is a consequence of (2.17) and the following result.\
[\
]{}[**Theorem 3.2.**]{} *Let $ \;\!p \geq p_{\mbox{}_{\!\;\!0}} \!\:\!$. Then*\
\
$$\tag{3.7}
\liminf_{t\,\rightarrow\,\infty} \;\!
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\leq\,
\bigl(\, p \,
C_{\mbox{}_{\!\;\!2}} \:\! C_{\mbox{}_{\!\!\;\!\infty}}
\bigr)^{\mbox{}^{\scriptstyle \!\!\;\!
\frac{1}{\scriptstyle p} }} \!\!\!\;\!\cdot\,
{\cal B}^{\mbox{}^{\scriptstyle \:\!
\frac{1}{\scriptstyle p} }} \!\cdot\;
\limsup_{t\,\rightarrow\,\infty} \;\!
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{p}(\mathbb{R})}}
\!\;\!,
$$\
*where $ {\displaystyle
\;\!C_{\mbox{}_{\!\;\!2}}, \, C_{\mbox{}_{\!\infty}}
\!\:\!
} $ are the constants given in $\;\!(2.7)$, $(1.9)$.*\
\
[ Again, assuming $ {\displaystyle
\;\!
{\cal U}_{p} \!\;\!
} $ finite (otherwise, (3.7) is obvious, cf.$\;$footnote 1), we introduce, as in the previous proof, $ {\displaystyle
\;\!
v \in L^{\infty}(\mathbb{R} \times [\;\!0, \infty\:\![)
\;\!
} $ given by $ {\displaystyle
v(x,t) = |\, u(x,t) \,|^{{\scriptstyle p}}
} $ if $ p > 1 $, and $ {\displaystyle
v(x,t) = u(x,t)
\;\!
} $ if $ p = 1 $. Thus, (3.2) is valid, and setting $ \lambda_{p} \!\in \mathbb{R} $, $ \;\!g \in L^{\infty}([\;\!0, \infty\;\![\:\!) \;\!$ by\
\
$$\notag
\lambda_{p} \;\!=\;
\limsup_{t\,\rightarrow\,\infty} \,
g(t),
\qquad
g(t) \,=\,
p \:C_{\mbox{}_{\!\;\!2}} \,
B(t) \,
\|\, \mbox{\boldmath $v$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R})}}
\!\:\!,$$\
we have that (3.7) is obtained if we show that\
\
$$\tag{3.8}
\liminf_{t\,\rightarrow\,\infty} \;\!
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\;\!\leq\;
C_{\mbox{}_{\!\infty}} \!\!\cdot \lambda_{p}
\:\!.
\;\;\;$$\
We argue by contradiction and assume that (3.8) is false. Taking then $ {\displaystyle
\;\!0 < \eta \ll 1,
\; t_0 \gg 1
} $ so that $ {\displaystyle
\;\!
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\!\geq\;\!
C_{\mbox{}_{\!\infty}} \!\!\cdot
(\lambda_{p} \!\;\!+ \eta \:\!)
\,
} $ and $ {\displaystyle
\:
g(t) \;\!\leq
\lambda_{p} \!\;\!+ \eta/2
\;\!
} $ hold for all $ t \geq t_0 $, we get, by (1.9), (3.2),\
\
$ {\displaystyle
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}^{\;\!3}
\;\!\leq\;
C_{\mbox{}_{\!\infty}}^{\;\!3} \,
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R})}}
\,
\|\, v_{x}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}^{\;\!2}
} $\
\
$ {\displaystyle
\leq\;
C_{\mbox{}_{\!\!\;\!\infty}}^{\,3} \: g(t)^{3}
\:+\;
C_{\mbox{}_{\!\!\;\!\infty}}^{\,3} \;
\frac{2\:\!p}{\:\!2\:\!p - 1\:\!} \;
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R})}}
\:\!
\Bigl(\;\!- \; \frac{d}{d\:\!t} \,
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}^{\;\!2}
\:\! \Bigr)
} $\
\
for all $ {\displaystyle
\;\! t \in [\;\!t_0, \infty\;\![ \,\setminus\;\!
E_{\mbox{}_{2\:\!\mbox{\scriptsize $p$}}}
} $. Since $ {\displaystyle
\;\!
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\!\geq\;\!
C_{\mbox{}_{\!\infty}} \!\!\cdot
(\lambda_{p} \!\;\!+ \eta \:\!)
} $, $ {\displaystyle
\:
g(t) \;\!\leq
\lambda_{p} \!\;\!+ \eta/2
\;\!
} $, $\:\!$this gives\
\
$$\notag
- \; \frac{d}{d\:\!t} \,
\|\, v(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}^{\;\!2}
\;\!\geq\;
K\!\:\!(\eta),
\qquad
\forall \;\,
t \in [\,t_0, \infty\;\![ \:\setminus\,
E_{\mbox{}_{2\:\!\mbox{\scriptsize $p$}}}$$\
for some constant $ K\!\;\!(\eta) > 0\;\! $ independent of $\,t$. As before, this implies that $ {\displaystyle
\,
\|\, v(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}^{\;\!2}
\!\geq
} $ $ {\displaystyle
K\!\:\!(\eta) \cdot (\:\! t - t_0)
\;\!
} $ for all $ \;\! t \geq t_0 $, which is impossible because $ {\displaystyle
\|\, v(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R})}}
\!
} $ is finite. This contradiction establishes (3.8) above, completing the proof of Theorem 3.2. $\Box$\
]{}\
We are finally in good position to derive (1.6), (3.6). $\!$Combining (2.17) and (3.7) above, we obtain\
\
$$\tag{3.9}
\limsup_{t\,\rightarrow\,\infty} \;\!
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\leq\,
\bigl(\, 2 \;\! p^{2} \:\!
\bigr)^{\mbox{}^{\scriptstyle \!\!\;\!
\frac{1}{\scriptstyle p} }} \!\!\!\,\cdot\,
{\cal B}^{\mbox{}^{\scriptstyle \:\!
\frac{1}{\scriptstyle p} }} \!\cdot\;
{\cal U}_{p}
$$\
for each $ p \!\;\!\geq p_{\mbox{}_{\!\;\!0}} $, so that we have, in particular,\
\
$$\tag{3.10}
\limsup_{t\,\rightarrow\,\infty} \;\!
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\leq\,
\bigl(\;\! 2^{2\:\! k \,+\, 1} \;\!p^{2}
\bigr)^{\mbox{}^{\scriptstyle \!\!\;\!
\frac{1}{\scriptstyle 2^{k}\!\:\!p} }}
\hspace{-0.380cm} \cdot\;
{\cal B}^{\mbox{}^{\scriptstyle \:\!
\frac{1}{\scriptstyle 2^{k}\!\:\!p} }} \!\cdot\;
{\cal U}_{2^{k}\!\:\!p}
$$\
for each $ k \geq 0 $. By (3.5), we then get\
\
$$\tag{3.11}
\limsup_{t\,\rightarrow\,\infty} \;\!
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\leq\,
\biggl\{\;\!
\bigl(\;\! 2^{2\:\! k \,+\, 1} \;\!p^{2}
\bigr)^{\mbox{}^{\scriptstyle \!\!\;\!2^{\mbox{}^{\!-\;\!k}} }}
\hspace{-0.400cm} \:\!\cdot\;
\prod_{j\,=\,1}^{k} \;\!
\bigl(\;\! 2^{j-1} \:\!p\;\!
\,C_{\mbox{}_{\!2}}^{\;\!3} \:\!
\bigr)^{\mbox{}^{\scriptstyle \!\!\;\! 2^{\mbox{}^{\!-\;\!j}}}}
\biggr\}^{\mbox{}^{\scriptstyle \!\!\frac{1}{\scriptstyle p} }}
\hspace{-0.150cm} \cdot\:
{\cal B}^{\mbox{}^{\scriptstyle \:\!
\frac{1}{\scriptstyle p} }} \!\cdot\;
{\cal U}_{p}
$$\
for all $\;\! k $. Letting $ \;\!k \rightarrow \infty $, Theorem 3.3 is obtained, and our argument is complete.\
[\
]{}[**Theorem 3.3.**]{} *Let $ \;\!p \geq p_{\mbox{}_{\!\;\!0}} \!\;\!$. Assuming $ {\displaystyle
\;\!b \in L^{\infty}(\mathbb{R} \times [\;\!0, \infty\;\![\;\!)
} $, then $(1.6)$, $(3.6)$ hold.*\
\
It is worth noticing that the corresponding estimate for the $n$-dimensional problem (1.8), namely,\
\
$$\tag{3.12}
\limsup_{t\,\rightarrow\,\infty}\;\!
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R}^{n})}}
\leq\,
K\!\:\!(n,p) \;\!\cdot\;\!
{\cal B}^{\mbox{}^{\scriptstyle
\;\!\frac{\scriptstyle n}{\scriptstyle p} }}
\!\!\cdot\:
\limsup_{t\,\rightarrow\,\infty}\;\!
\|\, u(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{p}(\mathbb{R}^{n})}}
\!\:\!,$$\
where $ {\cal B} \geq 0 $ is similarly defined, can be also derived in arbitrary dimension $ n > 1 $.
\
[**§4. Concluding remarks**]{}\
We close our discussion of the problem (1.1$a$), (1.1$b$), given $ {\displaystyle
\;\!b \in L^{\infty}(\mathbb{R} \times [\;\!0, \infty\;\![\;\!)
} $, $ 1 \leq p_{\mbox{}_{\!\;\!0}} \!< \infty $, indicating a few questions which were not answered by our analysis:\
\
([*a*]{})
characterize all $ {\displaystyle
\;\!b \in L^{\infty}(\mathbb{R} \times [\;\!0, \infty\;\![\;\!)
} $ for which it is true that $ {\displaystyle
\,
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\!\!\rightarrow 0
} $ (as $ t \rightarrow \infty $) for every solution $ u(\cdot,t) $ of problem (1.1);
[\
]{}\
([*b*]{})
same question as ([*a*]{}) above, but requiring only that $ {\displaystyle
\;\!
\limsup
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\!\!\;\!< \infty
} $ (as $ t \rightarrow \infty $) for every solution $ u(\cdot,t) $ of problem (1.1), in case $ \;\!p_{\mbox{}_{\!\;\!0}} \!> 1 $;
[\
]{}\
([*c*]{})
given $ p_{\mbox{}_{\!\;\!0}} > 1 $, characterize all $ {\displaystyle
\;\!b \in L^{\infty}(\mathbb{R} \times [\;\!0, \infty\;\![\;\!)
} $ such that $ {\displaystyle
\,
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{p_{\mbox{}_{\!\;\!0}}}(\mathbb{R})}}
\!\!\rightarrow 0
} $ (as $ t \rightarrow \infty $) for every solution $ u(\cdot,t) $ of problem (1.1);
[\
]{}\
([*d*]{})
same question as ([*c*]{}) above, but requiring only that $ {\displaystyle
\;\!
\limsup
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{p_{\mbox{}_{\!\;\!0}}}(\mathbb{R})}}
\!\!\;\!< \infty
} $ (as $ t \rightarrow \infty $) for every solution $ u(\cdot,t) $ of problem (1.1);
[\
]{}\
([*e*]{})
for $ p_{\mbox{}_{\!\;\!0}} = 1 $, characterize all $ {\displaystyle
\;\!b \in L^{\infty}(\mathbb{R} \times [\;\!0, \infty\;\![\;\!)
} $ such that $ {\displaystyle
\,
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R})}}
\!\!\rightarrow |\,m\,|
} $ (as $ \,\!t \rightarrow \infty $) for every solution $ u(\cdot,t) $, where $ \,\!m = \!\;\!\int_\mathbb{R} \!u_0(x)\;\!dx\;\! $ is the solution mass;
[\
]{}\
([*f*]{})
for $ p_{\mbox{}_{\!\;\!0}} = 1 $, and $ {\displaystyle
\;\!b \in L^{\infty}(\mathbb{R} \times [\;\!0, \infty\;\![\;\!)
} $ not satisfying property ([*e*]{}) above, what are the values of $ {\displaystyle
\lim_{t\,\rightarrow\,\infty} \;\!
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R})}}
} $ in case of initial states that change sign?
[\
]{}\
These questions can be similarly posed for solutions $ u(\cdot,t) $ of autonomous problems\
\
$$\tag{4.1}
u_t \;\!+\, (\;\!b(x) \;\!u \;\!)_{x}
\;\!=\;
u_{xx},
\qquad
u(\cdot,0) \in
L^{p_{\mbox{}_{\!\;\!0}}}(\mathbb{R} \cap L^{\infty}(\mathbb{R})$$\
where $ b \in L^{\infty}(\mathbb{R}) $ does not depend on the time variable. For (4.1), question ([*e*]{}) has been answered in [@Rudnicki1993]. (See also [@BrzezniakSzafirski1991]). Another interesting question is the following:\
\
([*g*]{})
when (4.1) admits no stationary solutions other than the trivial solution $ u = 0 $, is it true that $ {\displaystyle
\lim_{t\,\rightarrow\,\infty}
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\! =\;\! 0
\,
} $ for every solution $ u(\cdot,t) $?
[\
]{}\
Moreover, for solutions $ u(\cdot,t) $ of (1.1) or (4.1) with $ {\displaystyle
\|\, u(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R})}}
\!\! \rightarrow 0
\;\!
} $ as $ \;\!t \rightarrow \infty $, there is the question of determining the proper decay rate.[^3] As suggested by Fig.$\,$1, solution decay may sometimes happen at remarkably slow rates.\
[\
]{}[\
]{}\
[**Acknowledgements.**]{} The authors would like to thank [CNPq]{} (Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil) for their financial support.\
\
[00]{}
<span style="font-variant:small-caps;">C. J. Amick, J. L. Bona and M. E. Schonbek</span>, Decay of solutions of some nonlinear wave equations, *J. Diff. Eqs.*, **81** (1989), 1$\,$–$\,$49.
<span style="font-variant:small-caps;">P. Braz e Silva, L. Scütz and P. R. Zingano</span>, On some energy inequalities and supnorm estimates for advection-diffusion equations in $\mathbb{R}^{n}\!$, Nonl. Anal., **93** (2013), 90$\,$–$\,$96.
<span style="font-variant:small-caps;">Z. Brzeźniak and B. Szafirski</span>, Asymptotic behaviour of $ L^{1}$ norm of solutions to parabolic equations, *Bull. Polish Acad. Sci. Math.*, [**39**]{} (1991), 1$\,$–$\,$10.
<span style="font-variant:small-caps;">E. A. Carlen and M. Loss</span>, Sharp constant in Nash’s inequality, *Internat. Math. Res. Notices*, 1993, 213$\,$–$\,$215.
<span style="font-variant:small-caps;">M. Escobedo and E. Zuazua</span>, Large time behavior for convection-diffusion equations in $\mathbb{R}^{N}\!\!\:\!$, *J. Funct. Anal.*, **100** (1991), 119$\,$–$\,$161.
<span style="font-variant:small-caps;">L. C. Evans</span>, Partial Differential Equations, American Mathematical Society, Providence, 2002.
<span style="font-variant:small-caps;">W. G. Melo</span>, A priori estimates for various systems of advection-diffusion equations (Portuguese), PhD Thesis, Universidade Federal de Pernambuco, Recife, Brazil, 2011.
<span style="font-variant:small-caps;">L. S. Oliveira</span>, Two results in Classical Analysis (Portuguese), PhD Thesis, Graduate School in Applied and Computational Mathematics, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil, 2013.
<span style="font-variant:small-caps;">J. Nash</span>, Continuity of solutions of parabolic and elliptic equations, *Amer. J. Math.*, **80** (1958), 931$\,$–$\,$954.
<span style="font-variant:small-caps;">M. M. Porzio</span>, On decay estimates, *J. Evol. Equations*, **9** (2009), 561$\,$–$\,$591.
<span style="font-variant:small-caps;">R. Rudnicki</span>, Asymptotic stability in $L^{1}$ of $\,$parabolic equations, *J. Diff. Equations*, **102** (1993), 391$\,$–$\,$401.
<span style="font-variant:small-caps;">M. E. Schonbek</span>, Uniform decay rates for parabolic conservation laws, *Nonlinear Anal.$\;$T.$\:$M.$\:$A*, **10** (1986), 943$\,$–$\,$956.
<span style="font-variant:small-caps;">D. Serre</span>, [Systems of Conservation Laws]{}, vol.$\;$1, Cambridge University Press, Cambridge, 1999.
[\
]{}\
[\
]{}
<span style="font-variant:small-caps;">José Afonso Barrionuevo</span>\
Departamento de Matemática Pura e Aplicada\
Universidade Federal do Rio Grande do Sul\
Porto Alegre, RS 91509-900, Brazil\
E-mail: [[email protected]]{}
[\
]{}[\
]{}\
<span style="font-variant:small-caps;">Lucas da Silva Oliveira</span>\
Departamento de Matemática Pura e Aplicada\
Universidade Federal do Rio Grande do Sul\
Porto Alegre, RS 91509-900, Brazil\
E-mail: [[email protected]]{}
[\
]{}[\
]{}\
<span style="font-variant:small-caps;">Paulo Ricardo Zingano</span>\
Departamento de Matemática Pura e Aplicada\
Universidade Federal do Rio Grande do Sul\
Porto Alegre, RS 91509-900, Brazil\
E-mail: [[email protected]]{}
[^1]: In (1.6), (1.11) and other similar expressions in the text, it is assumed that $ 0 \cdot \infty = \infty $.
[^2]: The constants $(3\sqrt{3}p/(2\pi))^{1/p}$ in (1.6), (1.7) are not optimal; minimal values are not known.
[^3]: $\:\!$In case we have $ b_x \geq 0 $ for all $x$, $t$, the answer is given in (1.2) above.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A lattice calculation shows that the Casimir scaling hypothesis is well verified in QCD, that is to say that the potential between two opposite color charges in a color singlet is proportional to the value of the quadratic Casimir operator. On the other hand, in a bag model calculation for the same system, a scaling of the string tension with the square root of the quadratic Casimir operator is obtained. It is shown that, within the same formalism but with the assumption that the width of the string is independent of the color charges, the string tension is proportional to value of the quadratic Casimir operator. Some considerations about the color behavior of the total interaction are given.'
author:
- Claude
title: About the Casimir scaling hypothesis
---
[^1]
The Casimir scaling hypothesis means that the potential between two opposite colour charges in a colour singlet is proportional to the value of the quadratic Casimir operator. A lattice calculation [@bali00] exclude any violations of this hypothesis that exceed 5% for charge separations of up to 1 fm. Nevertheless, other models do not predict such a colour behaviour. For instance, a scaling of the string tension with the square root of the quadratic Casimir operator is obtained in a bag model calculation [@john76]. We show here that the Casimir scaling can be obtained if the fundamental assumption in the bag model, the existence of a confining pressure $B$, is replaced by the hypothesis of the existence of a universal string section in the rest frame of the charges.
We use the same formalism as in Ref. [@john76]. Let us consider two opposite colour charges with zero mass, moving attached by a string, in a colour singlet. The colour electric flux $\vec E_a$ which leaves a colour charge has the strength $$\label{ea}
|\vec E_a| A = g \lambda_a,$$ where $A$ is the cross section of the string, and $\lambda_a$ are the colour matrices. If $x$ is the distance from the centre of mass (middle of the string), a point of the string moves with the speed $$\label{v}
v=\frac{2}{L} x,$$ where $L$ is the length of the string. The colour charges at the extremities move at the speed of light. The colour magnetic field, which is produced by the rotation of the colour electric field, is given by $$\label{ba}
\vec B_a = \vec v \times \vec E_a,$$ at a point of the string which moves with velocity $\vec v$. The quadratic Casimir operator $C$ is $$\label{c}
C = \frac{1}{4}\sum_a \lambda_a^2.$$
In Ref. [@john76], the section of the string is determined by the surface equation of the bag containing the coloured particles. This implies that its section $A$ is proportional to $\sqrt{C}$. In this work, we assume that the section of the string is a constant $A_0$, independent of $C$, in the rest frame of the string. Consequently, our model is not a bag model, and no confining pressure $B$ is introduced. When the string rotates, the section undergoes a Lorentz contraction $$\label{av}
A = A_0 \sqrt{1-v^2}.$$
To calculate the mass $M$ of the colour singlet system, let us first compute the strength fields $$\label{ea2}
\sum_a E_a^2 = \frac{4g^2C}{A^2} \quad \text{and} \quad
\sum_a B_a^2 = \frac{4g^2C}{A^2} v^2,$$ the speed $\vec v$ of a point of the string being always perpendicular to $\vec E_a$. All volume integrals are replaced by $$\label{int}
\int d^3 x \rightarrow 2\int_{0}^{L/2} A dx = L \int_{0}^{1} A dv.$$ The energy of the coloured flux lines is [@john76] $$\label{ef1}
E_f = \frac{1}{2} \int d^3 x \sum_a \left( E_a^2 + B_a^2 \right).$$ With the notations defined above, we obtain $$\label{ef2}
E_f =2 g^2 C \frac{L}{A_0} \int_{0}^{1} \frac{1+v^2}{\sqrt{1-v^2}} dv.$$ The angular momentum of the coloured flux lines is [@john76] $$\label{jf1}
\vec J_f = \int d^3 x \sum_a \vec r \times \left( \vec E_a \times \vec
B_a \right).$$ Thus, we obtain $$\label{jf2}
J_f = 2 g^2 C \frac{L^2}{A_0} \int_{0}^{1} \frac{v^2}{\sqrt{1-v^2}} dv.$$ Classically, a massless colour charge does not carry nor energy neither momentum [@laco89]. Consequently, the mass $M$ of the state is equal to $E_f$ and the total angular momentum $J$ is equal to $J_f$. We then obtain $$\label{m2}
M^2 = \frac{9 \pi}{2} \frac{g^2}{A_0} C \, J = 18 \pi^2
\frac{\alpha_S}{A_0} C \, J,$$ with $\alpha_S=g^2/4\pi$ the strong coupling constant. Let us note that, in Ref. [@john76], the mass is determined from the condition $\partial M/\partial L=0$. But this implies also that the contributions of the massless colour charges to energy and momentum are vanishing.
We obtain the linear Regge trajectories, but with a slope–that is to say a string tension–proportional to $C$, and not to $\sqrt{C}$. This result has already been obtained in Ref. [@hans86], but with a different technique. With the more phenomenological approach used here, we find that the energy density of the flux tube is given by $$\label{mol}
\frac{M}{L} = 6 \pi^2 \frac{\alpha_S\, C}{A_0},$$ which is quite different from the result of Ref. [@hans86].
In order to check the relevance of formula (\[m2\]), let us consider the case of a meson, for which $C=4/3$. The relativistic flux tube model [@laco89] predict that $$\label{rft}
M^2 = 2\pi a \, J,$$ where $a$ is the usual string tension. It is then possible to link the section $A_0$ of the string to its tension $a$ and the strong coupling constant $\alpha_S$ $$\label{a0}
A_0=12 \pi \frac{\alpha_s}{a}.$$ The radius $R_0$ of the string is given by $\sqrt{A_0/\pi}$, assuming a cylindrical form for the string. For reasonable values of the QCD parameters, $\alpha_S \in [0.1-0.4]$ and $a \in [0.17-0.20]$ GeV$^2$, we find $R_0$ in the range 0.5-1.0 fm [@hans86]. A lattice calculation predicts a gaussian string width with a mean radius around 0.35 fm [@bali95]. Given the simplicity of our model, the agreement is quite reasonable.
We can expect that our model is relevant only if $L> 2R_0$. This condition is satisfied if $$\label{condj}
J > 8\pi C \alpha_s.$$ Small values for $J$ are acceptable if the product $C \alpha_S$ is not too large.
The key ingredient of this work is the assumption that the width of the string is independent of the colour charges. Such a possibility is also studied in recent works [@luci01; @shos03]. It could be interesting to test this hypothesis with lattice calculations.
Besides the confinement, a one-gluon exchange process exists between the two particles. The colour dependence of this interaction is given by $$\label{oge}
\frac{1}{4}\sum_a \lambda_a(1) \lambda_a(2) =
\frac{1}{2} \left( 0-C-C \right) =-C.$$ So we find again a colour scaling given by $C$. A constant potential plays an important role in the hadron spectroscopy. In various approaches [@grom81; @simo01], this constant is proportional to the string tension. In this case the colour scaling is also given by $C$. Finally, we can expect that the total potential between two opposite colour charges in a colour singlet is proportional to the quadratic Casimir operator, and not to its square root.
[aa]{} G. S. Bali, Phys. Rev. D [**62**]{}, 114503 (2000) \[hep-lat/0006022\]. K. Johnson and C. B. Thorn, Phys. Rev. D [**13**]{}, 1934 (1976). D. LaCourse and M. G. Olsson, Phys. Rev. D **39**, 2751 (1989). T. H. Hansson, Phys. Lett. B [**166**]{}, 343 (1986). G. S. Bali, C. Schlichter, and K. Schilling, Phys. Rev. D [**51**]{}, 5165 (1995) \[hep-lat/9409005\]. B. Lucini and M. Teper, Phys. Rev. D **64**, 105019 (2001) \[hep-lat/0107007\]. A. I. Shoshi, F. D. Steffen, H. G. Dosch, and H. J. Pirner, Phys. Rev. D **68**, 074004 (2003) \[hep-ph/0211287\]. D. Gromes, Z. Phys. C [**11**]{}, 147 (1981); erratum [**14**]{}, 94 (1982). Yu. A. Simonov, Phys. Lett. B [**515**]{}, 137 (2001) \[hep-ph/0105141\].
[^1]: FNRS Research Associate
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper, we present a proof of Schauder estimate on Euclidean space and use it to generalize Donaldson’s Schauder estimate on space with conical singularities in the following two directions. The first is that we allow the total cone angle to be larger than 2$\pi$ and the second is that we discuss higher order estimates.'
address:
- 'Hao Yin, School of Mathematical Sciences, University of Science and Technology of China, Hefei, China'
- 'Yaoting Gui, School of Mathematical Sciences, University of Science and Technology of China, Hefei, China'
author:
- Yaoting Gui and Hao Yin
bibliography:
- 'foo.bib'
title: Schauder estimates on smooth and singular spaces
---
Introduction
============
In this paper, we discuss the classical Schauder estimate on Euclidean space ${\mathbb R}^n$ and on some singular space with conical type singularities. The discussion contained in this paper should apply with minor modification to a class of conical type singular spaces, however, for simplicity, we restrict ourselves to a special case, namely, ${\mathbb R}^{2}\times {\mathbb R}^{n-2}$ with a singular metric $$\label{eqn:metric1}
g_\beta= {\left\vertz\right\vert}^{2\beta-2} dz^2 + d\xi^2, \quad \beta>0.$$ where $z$ is in $\mathbb C$ (identified with ${\mathbb R}^2$) and $\xi$ is in ${\mathbb R}^{n-2}$. The geometry is nothing but the product of a 2-dimensional cone with cone angle $2\pi \beta$ and ${\mathbb R}^{n-2}$. Our attention is drawn to this space because of the recent study of conical Kähler geometry proposed by [@tian1996kahler] and [@donaldson2012]. In the rest of this paper, we denote this space together with the metric by $X_\beta$.
We shall restrict ourselves to the interior estimates only and hopefully, the boundary value problem will be discussed in the future. Hence by Schauder estimate in ${\mathbb R}^n$, we mean the inequality $${\left\Vertu\right\Vert}_{C^{2,\alpha}(B_{1/2})}\leq C(n,\alpha) ({\left\Vertf\right\Vert}_{C^\alpha(B_1)}+ {\left\Vertu\right\Vert}_{C^0(B_1)})$$ if $\triangle u =f$ on $B_1$. If $f$ is in $C^{k,\alpha}$ for $k\in \mathbb N$, we can bound ${\left\Vertu\right\Vert}_{C^{k+2,\alpha}}$ by successively taking derivatives and applying the above $C^{2,\alpha}$ estimate.
Besides the classical proof of potential theory, there are many different proofs by Campanato [@campanato], Peetre [@peetre], Trudinger [@trudinger], Simon [@simon], Safonov [@safonov1; @safonov2], Caffarelli [@caffarelli; @CC] and Wang [@wang2006]. We refer the readers to [@wang2006] for brief comments on these proofs. Many of the above proofs have important applications to the study of nonlinear (or even fully nonlinear) elliptic and parabolic equations. The proof given below is motivated by the study of regularity problem on spaces with conical singularity. The ideas used here are related to the above mentioned proofs, for example, we shall use a characterization of Hölder continuous function known to Campanato and we shall compare the solution to polynomials as Caffarelli did in [@caffarelli; @CC]. Moreover, the idea of pointwise Schauder estimate, due to Han [@han1998; @han2000], is particularly useful and effective for conical singularities. In the first part of this paper, we give a proof of the Schauder estimate on ${\mathbb R}^n$. The proof is by far not as simple as the above mentioned ones. We need it, first to illustrate the basic idea of this paper, and second to prove some theorem that will be needed for the proof of Schauder estimate on $X_\beta$.
We start with an equivalent formulation of the Hölder space and the Hölder norm on ${\mathbb R}^n$. Thanks to the Taylor expansion theorem, if $u$ is $C^{k,\alpha}$ in a neighborhood of $x$, then there exists a polynomial $P_x$ of degree $k$ such that $$u(x+h)= P_x(h)+ O_x({\left\verth\right\vert}^{k+\alpha}),\qquad \text{for} \quad {\left\verth\right\vert}<\delta_x.$$ It is natural to ask about the reverse: if a function $u$ has the above expansion around each point $x$ in an open set $\Omega$, is it true that $u\in C^{k,\alpha}(\Omega)$? As shown by the function $$u(x)= x^2 \sin \frac{1}{x}, \qquad x\in {\mathbb R},$$ which is not $C^1$, we can not expect a positive answer without putting more restrictions to the expansion. It turns out that we need to ask the expansion to be uniform in the following sense: for some positive constants $\Lambda$ and $\delta$ independent of $x\in \Omega$,
- the coefficients of the polynomial $P_x$ are bounded by $\Lambda$;
- the constant in the definition of $O_x({\left\verth\right\vert}^{k+\alpha})$ is bounded by $\Lambda$;
- $\delta_x>\delta.$
The function $u$ that satisfies the above assumption is then said to have [**uniformly bounded expansion**]{}, or UBE for simplicity. It will be shown in Section \[sec:space\] that the set of UBE functions is the same as $C^{k,\alpha}$ functions if one does not mind shrinking the domain a little, which is not a problem since we are only concerned with interior estimate in this paper. This allows us to translate the classical Schauder estimate on ${\mathbb R}^n$ into a theorem about UBE functions.
A feature of the UBE characterization is that it seems to be a pointwise property. The proof of the Schauder estimate then reduces to showing that if $\triangle u =f$ and $f$ has an expansion of order $k+\alpha$ at $0$ bounded by $\Lambda$ in the above sense, then $u$ has an expansion at $0$ up to order $k+\alpha+2$ bounded by a constant multiple of $\Lambda$ and its own $C^0$ norm. This is exactly what we do in Section \[sec:another\].
Similar to what Han did in [@han1998; @han2000], an important step (Lemma \[lem:key\]) is the following: let $f$ be $O({\left\vertx\right\vert}^{k+\alpha})$, then there exists $u$ that is $O({\left\vertx\right\vert}^{k+2+\alpha})$ satisfying $$\triangle u =f \qquad \text{on} \quad B$$ and $$\sup_{B\setminus {\left\{0\right\}}} \frac{{\left\vertu\right\vert}}{{\left\vertx\right\vert}^{k+2+\alpha}} \leq C \sup_{B\setminus {\left\{0\right\}}} \frac{{\left\vertf\right\vert}}{{\left\vertx\right\vert}^{k+\alpha}}.$$ This was proved by using the potential in [@han1998; @han2000] and it is our intention to avoid using the potential, because the analysis of Green’s function on $X_\beta$ could be complicated. Hence, we provide a proof of Lemma \[lem:key\] using only the fact that harmonic functions are polynomials. This argument generalizes well on $X_\beta$.
We then move on to the discussion of the singular space $X_\beta$. If $n$ is even and we identify ${\mathbb R}^2\times {\mathbb R}^{n-2}$ with $\mathbb C \times \mathbb C^{\frac{n-2}{2}}$, $X_\beta$ together with $g_\beta$ is also a (noncomplete) Kähler manifold. For $\beta\in (0,1)$ and $\alpha\in (0,\min{\left\{1,\frac{1}{\beta}-1\right\}})$, one can define $C^\alpha$ function, using the Riemannian distance as usual. Donaldson [@donaldson2012] observed that if one defines $C^{2,\alpha}_\beta$ space by requiring the function $u$, its gradient (in Riemannian geometric sense), its complex Hessian (in the above mentioned Kähler structure) to be $C^\alpha$, there is still a Schauder estimate. This estimate plays an important role in the study of conical Kähler geometry. Its original proof due to Donaldson is by potential theory and recently, there is another proof (without potential theory) of the same estimate by Guo and Song [@guo2016schauder]. Moreover, there is also a parabolic version of Donaldson’s estimate due to Chen and Wang [@chen2015bessel].
It is the main purpose of this paper to generalize the above Schauder estimate due to Donaldson in two directions (with a different proof). First, we allow any $\beta>0$ instead of $\beta\in (0,1)$. Second, we study regularity beyond second order derivatives. There are indeed situations in the study of conical Kähler geometry in which higher order regularity is necessary. See [@li2016uniqueness; @zheng2017geodesics] for example. The case $\beta>1$ is useful, because the induced metric of some algebraic variety from its ambient space [**can be**]{} conic with integer $\beta>1$. Applications along this line will be pursued in a separate paper.
This generalization is achieved by defining a new function space $\mathcal U^q$ on $X_\beta$(in Section \[sec:Holder\]). Here $q$ is some positive number, taking the place of $k+\alpha$ for the usual $C^{k,\alpha}$ space. Briefly speaking, the definition is a combination of two ideas: first we require the function to be $C^{k,\alpha}$ for $q=k+\alpha$ away from the singularities; second, we use the idea in the first part of this paper, namely, we use [**uniformly bounded expansion**]{} up to order $q$ to describe the regularity of $f$ at the singular points; finally, we need to take care of the transition between the two point of views. See (H1-H3) in Definition \[defn:big\] for details.
We need to be clear about the type of expansion that is used near a singular point of $X_\beta$, because the Taylor expansion is not available here. This is the topic of Section \[subsec:formal\]. On one hand, we need the expansion to be general so that it can be used to describe the regularity of the solution that we care; on the other hand, we want the expansion to be very special so that it contains as much information as possible. A choice of the expansion is a balance of the above two considerations. Our previous experience on the regularity issue of PDE’s on conical spaces [@yin2016analysis; @yin2016expansion] suggests that the good choice depends both on the parameter $\beta$ and on the type of equations that we are interested in. In order not to distract the attention of the readers, we give one particular choice in Section \[subsec:formal\] by defining the $\mathcal T$-polynomial. This choice is sufficient to present the idea of our proof and it is general enough to have Donaldson’s Schauder estimate as a special case.
For future applications, we list a family of properties (P1-P4). As long as the definition of $\mathcal T$-polynomial satisfies (P1-P4), the Schauder estimate holds.
Our choice is justified by the following theorems. They are the main results of this paper. The first is the Schauder estimate. Here $\mathcal D$ is a countable and discrete set of positive numbers (see Section \[sec:Holder\]), $\hat{B}_r$ is the metric ball in $X_\beta$ with the origin as its center and $\mathcal U^q$ is the new space of functions given by Definition \[defn:big\].
\[thm:main1\] Let $q>0$ and $q, q+2\notin \mathcal D$. Suppose $f\in \mathcal U^{q}(\hat{B}_2)$ and $u$ is a bounded weak solution to $$\triangle_\beta u =f.$$ Then $u$ is in ${\mathcal U}^{q+2}(\hat{B}_1)$ and $${\left\Vertu\right\Vert}_{{\mathcal U}^{q+2}(\hat{B}_1)}\leq C\left( {\left\Vertu\right\Vert}_{C^0(\hat{B}_2)} + {\left\Vertf\right\Vert}_{{\mathcal U}^{q}(\hat{B}_2)} \right).$$
The second is a comparison between the newly defined space $\mathcal U^q$ and the Donaldson’s $C^{2,\alpha}_\beta$ space, whose definition we recall in Section \[subsec:compare\].
\[thm:main2\] Suppose $0<\beta<1$ and $0<\alpha<\min {\left\{1,\frac{1}{\beta}-1\right\}}$. If we write $\mathcal X$ for $C^\alpha$ ($\mathcal U^\alpha$, $C^{2,\alpha}_\beta$ and $\mathcal U^{2+\alpha}$ respectively) and $\mathcal Y$ for $\mathcal U^\alpha$ ($C^\alpha$, $\mathcal U^{2+\alpha}$ and $C^{2,\alpha}_\beta$ respectively), then $u\in \mathcal X(\hat{B}_2)$ implies that $u\in \mathcal Y(\hat{B}_1)$ and $${\left\Vertu\right\Vert}_{\mathcal Y(\hat{B}_1)} \leq C {\left\Vertu\right\Vert}_{\mathcal X(\hat{B}_2)}.$$
With Theorem \[thm:main2\], Theorem \[thm:main1\] implies the Schauder estimate of Donaldson in [@donaldson2012].
The rest of the paper is organized as follows. In Section \[sec:space\], we give a characterization of the $C^{k,\alpha}$ space on ${\mathbb R}^n$ using uniformly bounded expansion. This was known to Campanato back to the 1960’s. We include a proof for completeness, which may be omitted for a first reading. In Section \[sec:another\], we prove the Schauder estimate on ${\mathbb R}^n$. These two sections form the first part of the paper. We then move on to the study on $X_\beta$. We first set up some notations and recall some easy facts about Poisson equations on $X_\beta$ in Section \[sec:pre\]. In Section \[sec:harmonic\], we study bounded harmonic functions on $X_\beta$, which is key to the proof of Theorem \[thm:main1\]. In Section \[sec:Holder\], we define the space $\mathcal U^q$ and prove Theorem \[thm:main2\]. In the final section, we prove Theorem \[thm:main1\].
[**Acknowledgement.**]{} The second author would like to thank Professor Xinan Ma for bringing the references [@han1998; @han2000] to his attention.
Hölder space on ${\mathbb R}^n$ {#sec:space}
===============================
In this section, we define a new space of functions that satisfy the uniform Taylor expansion condition and prove that it is equivalent to the usual Hölder space $C^{k,\alpha}$. As remarked in the introduction, this result is not new and the proofs are included for completeness.
We write $B_r$ for the ball of radius $r$ centered at the origin in ${\mathbb R}^n$.
\[defn:ube\] Suppose $r$ and $\delta$ are two positive real numbers. For a function $u$ defined on $B_{r+\delta}$, we say that it has [**uniformly bounded expansion**]{} (or UBE for simplicity) up to order $q$ on $B_r$ with scale $\delta$ if there exists some $\Lambda>0$ such that for any $x\in B_r$ and $h\in B_\delta$ $$u(x+h)= p_x(h)+ O_x(q),$$ where $p_x(h)$ is a polynomial of $h$ whose coefficients (depending on $x$) are uniformly bounded by $\Lambda$ and $O_x(q)$ is also a function of $h$ satisfying $$\label{eqn:normO}
{\left\vertO_x(q)(h)\right\vert}\leq \Lambda {\left\verth\right\vert}^q,\quad \forall {\left\verth\right\vert}<\delta, \quad \forall x\in B_r.$$
Related to the above definition, we define the following notations:
(a) The infimum of all $\Lambda$ satisfying is denoted by $[O_x(q)]_{O_q,B_\delta}$, which is nothing but $$\sup_{h\in B_\delta, h\ne 0} \frac{{\left\vertO_x(q)\right\vert}}{{\left\verth\right\vert}^q}.$$
(b) The set of all functions that have UBE up to order $q$ on $B_r$ with scale $\delta$ is denoted by $\mathcal U^{q,\delta}(B_r)$.
(c) For $u\in \mathcal U^{q,\delta}(B_r)$, the infimum of $\Lambda$ in the above definition is defined to be the norm of $u$, denoted by ${\left\Vertu\right\Vert}_{\mathcal U^{q,\delta}(B_r)}$.
It turns out that $\mathcal U^{q,\delta}$ is equivalent to the usual Hölder space $C^{k,\alpha}$ with $q=k+\alpha$ in the following sense.
\[prop:rn\]Suppose $q=k+\alpha$ for some $k\in \mathbb N\cup {\left\{0\right\}}$ and $\alpha\in (0,1)$.
\(i) If $u\in C^{k,\alpha}(B_{r+\delta})$, then $u\in \mathcal U^{q,\delta}(B_r)$ and $${\left\Vertu\right\Vert}_{\mathcal U^{q,\delta}(B_r)}\leq C(\delta,q,r,n) {\left\Vertu\right\Vert}_{C^{k,\alpha}(B_{r+\delta})}.$$
\(ii) If $u\in \mathcal U^{q,\delta}(B_{r+\eta})$ for some $\eta>0$, then $u\in C^{k,\alpha}(B_{r})$ and $${\left\Vertu\right\Vert}_{C^{k,\alpha}(B_{r})}\leq C(\eta,q,\delta,r,n) {\left\Vertu\right\Vert}_{\mathcal U^{q,\delta}(B_{r+\eta})}.$$
The rest of this section is devoted to the proof of this proposition. The first part follows trivially from the Taylor theorem with integral remainder.
The proof of (ii) is by induction and we assume without loss of generality that $r=1$. The starting point of the induction is the observation that the claim holds trivially true when $k=0$. For $k>0$, the expansion in Definition \[defn:ube\] implies that $u$ is differentiable for each $x\in B_{1+\eta}$. Therefore, the proof of Proposition \[prop:rn\] reduces to
[**Claim 1:**]{} If $u\in \mathcal U^{q,\delta}(B_{1+\eta})$, then for $i=1,\cdots,n$, some $\eta'\in (0,\eta)$ and some $\delta'>0$, $${\frac{\partial u}{\partial x_i}} \in \mathcal U^{q-1,\delta'}(B_{1+\eta'}) \quad \text{and} \quad {\left\Vert{\frac{\partial u}{\partial x_i}}\right\Vert}_{\mathcal U^{q-1,\delta'}(B_{1+\eta'})} \leq C(\delta,q,\eta,n) {\left\Vertu\right\Vert}_{\mathcal U^{q,\delta}(B_{1+\eta})}.$$
For the proof of the claim, we recall some notations. Let $\epsilon=(\epsilon_1,\cdots,\epsilon_n)$ be a multi-index. For $h\in {\mathbb R}^n$, we write $$h^\epsilon= h_1^{\epsilon_1}\cdots h_n^{\epsilon_n}.$$ For some $\delta'>0$ to be determined in a minute, we [**fix**]{} $h=(h_1,\cdots,h_n)$ satisfying ${\left\verth\right\vert}<\delta'$ and $h_i\ne 0$ for all $i$. Given this $h$ and a multi-index $\epsilon$ with ${\left\vert\epsilon\right\vert}<q+2$, we define the difference quotient operator $P_{\epsilon,h}$ which maps a function defined on $B_{1+\eta}$ to a function defined on $B_{1+\eta'}$ as follows. For $\epsilon=(0,\cdots,1,\cdots,0)$, where the only $1$ is at the $i$-th position, $$P_{\epsilon,h}[f](y):=\frac{f(y+h_ie_i)-f(y)}{h_i}$$ where $e_i$ is the natural basis of ${\mathbb R}^n$. For $\epsilon=\epsilon'+e_i$, we define $$P_{\epsilon,h}[f](y)=\frac{P_{\epsilon',h}[f](y+h_ie_i)-P_{\epsilon',h}[f](y)}{h_i}.$$ Since ${\left\vert\epsilon\right\vert}$ is bounded by $q+2$, by choosing $\delta'$ small (say, $\delta'= \frac{\eta-\eta'}{2(q+2)}$, so that ${\left\verth\right\vert}$ small) depending on $\eta$ and $\eta'$, $P_{\epsilon,h}[f]$ is a function defined on $B_{1+\eta'}$.
\[lem:P\] $P_{\epsilon,h}$ is well defined, i.e., it is independent of the order of induction in its definition. Moreover, we have $$P_{\epsilon,h}[f](y)= \frac{1}{h^\epsilon}\sum_{0\leq \gamma\leq \epsilon} (-1)^{{\left\vert\gamma\right\vert}+1} C^\gamma_\epsilon f(y+\gamma h).
\label{eqn:P}$$ Here
\(a) $\gamma$ is a multi-index and $0\leq \gamma\leq \epsilon$ means that for each $i=1,\cdots,n$, we have $0\leq \gamma_i\leq \epsilon_i$;
\(b) $\gamma h= (\gamma_1h_1,\cdots,\gamma_n h_n)$;
\(c) $C^\gamma_\epsilon:= C^{\gamma_1}_{\epsilon_1} \cdots C^{\gamma_n}_{\epsilon_n}$.
The proof is very elementary and omitted. We shall also need the following lemma about combinatorics.
\[lem:Pkill\] For any multi-index $\sigma$, set $$Q^\sigma_\epsilon= \sum_{0\leq \gamma\leq \epsilon} (-1)^{{\left\vert\gamma\right\vert}+1} C^\gamma_\epsilon \gamma^\sigma.$$ If $\sigma_i< \epsilon_i$ for some $i=1,\cdots,n$, then $Q^\sigma_\epsilon=0$. As a consequence, if we denote the multi-index $(\gamma_1-1,\cdots,\gamma_n-1)$ by $\gamma-1$, then for the same $\sigma$ and $\epsilon$ above $$Q^\sigma_\epsilon= \sum_{0\leq \gamma\leq \epsilon} (-1)^{{\left\vert\gamma\right\vert}+1} C^\gamma_\epsilon (\gamma-1)^\sigma.$$
We only prove the first claim of the lemma. An easy observation is that $$Q^\sigma_\epsilon=- \prod^n_{i=1} \sum_{0\leq \gamma_i\leq \epsilon_i} (-1)^{\gamma_i} C^{\gamma_i}_{\epsilon_i}\gamma_i^{\sigma_i}.$$ To show the product is zero, it suffices to show that the $i$-th factor is zero if $\sigma_i<\epsilon_i$. Consider the polynomial $$f(y_i)=(1-y_i)^{\epsilon_i}=\sum_{0\leq \gamma_i\leq \epsilon_i} C^{\gamma_i}_{\epsilon_i} (-y_i)^{\gamma_i}.$$ For each $j=0,\cdots,\sigma_i$, since $j<\epsilon_i$, we have $$(\partial_{y_i})^{j} f|_{y_i=1}=0,$$ which gives $$\sum_{j\leq \gamma_i\leq \epsilon_i} (-1)^{\gamma_i} C^{\gamma_i}_{\epsilon_i} \frac{\gamma_i !}{(\gamma_i-j)!} =0.$$ By setting $$F(\gamma;j)=\gamma \cdot (\gamma-1) \cdot \cdots \cdot(\gamma-j+1),$$ we have $$\label{eqn:qsb}
\sum_{0\leq \gamma_i\leq \epsilon_i} (-1)^{\gamma_i} C^{\gamma_i}_{\epsilon_i} F(\gamma_i;j) =0.$$ $F(\gamma;j)$ is a polynomial of $\gamma$ of degree $j$. Since $\sigma_i<\epsilon_i$, $\gamma_i^{\sigma_i}$ is then a linear combination of $F(\gamma_i;j)$, $j=0,\cdots,\epsilon_i$. With , this concludes the proof of Lemma \[lem:Pkill\].
The above lemma is related to the fact that difference quotient kills polynomials.
Now, we come back to the proof of Claim 1, which consists of two steps. In the first step, we restrict ourselves to a special type of $h$ satisfying $$\label{eqn:betterh}
{\left\verth_i\right\vert}\geq \frac{1}{2\sqrt{n}} {\left\verth\right\vert},\quad \text{for}\, i=1,\cdots,n.$$ We denote the set of such $h$ by $\Omega$. The reason will be clear in a minute.
\[defn:pube\] Suppose $r$ and $\delta$ are two positive real numbers. A function $u:B_{r+\delta}\to {\mathbb R}$ is said to have [**partially uniformly bounded expansion**]{} with respect to $\Omega$, up to order $q$ and with scale $\delta$, if the assumptions in Definition \[defn:ube\] hold with $h\in B_\delta$ replaced by $h\in \Omega\cap B_\delta$.
The space of these functions is denoted by $\mathcal U^{p,\delta}_\Omega(B_{r})$ and its norm by ${\left\Vert\cdot\right\Vert}_{\mathcal U^{p,\delta}_\Omega}$.
The goal of the first step is the following claim, which is a partial version of Claim 1.
[**Claim 2:**]{} If $u\in \mathcal U_\Omega^{q,\delta}(B_{1+\eta})$, then for $i=1,\cdots,n$, some $\eta'\in (0,\eta)$ and some $\delta'>0$, $${\frac{\partial u}{\partial x_i}} \in \mathcal U_\Omega^{q-1,\delta'}(B_{1+\eta'}) \quad \text{and} \quad {\left\Vert{\frac{\partial u}{\partial x_i}}\right\Vert}_{\mathcal U_\Omega^{q-1,\delta'}(B_{1+\eta'})} \leq C(\delta,q,\eta,n,\Omega) {\left\Vertu\right\Vert}_{\mathcal U_\Omega^{q,\delta}(B_{1+\eta})}.$$
In the second step, we shall derive Claim 1 from Claim 2. For the proof of Claim 2, recall that $$\label{eqn:Pbeta}
P_{\epsilon,h}[f](x)=\frac{1}{h^\epsilon} \sum_{0\leq \gamma\leq \epsilon} (-1)^{{\left\vert\gamma\right\vert}+1} C^\gamma_\epsilon f(x+\gamma h).$$ Using the partial UBE assumption, we may expand $f(x+\gamma h)$ into a polynomial centered at $x$, $$f(x+\gamma h)=f(x)+\sum_{{\left\vert\sigma\right\vert}<q} a_\sigma(x) (\gamma h)^\sigma + O({\left\verth\right\vert}^q).
\label{eqn:fx}$$ We may also use the expansion centered at $x+h$ to get $$f(x+\gamma h) =f(x+h)+\sum_{{\left\vert\sigma\right\vert}<q} a_\sigma(x+h) ( (\gamma-1) h)^\sigma + O({\left\verth\right\vert}^q).
\label{eqn:fxh}$$ If we plug both and into and notice that $(\gamma h)^\sigma= \gamma^\sigma h^\sigma$, Lemma \[lem:Pkill\] implies that $$\label{eqn:gooda}
\sum_{\epsilon\leq \sigma, {\left\vert\sigma\right\vert}<q} a_\sigma(x) Q^\sigma_\epsilon h^{\sigma-\epsilon} = \sum_{\epsilon\leq \sigma, {\left\vert\sigma\right\vert}<q} a_\sigma(x+h) Q^\sigma_\epsilon h^{\sigma-\epsilon} + O({\left\verth\right\vert}^{q-{\left\vert\epsilon\right\vert}}).$$ Here we also used the fact that $$\frac{O({\left\verth\right\vert}^q)}{h^\epsilon}$$ is an $O({\left\verth\right\vert}^{q-{\left\vert\epsilon\right\vert}})$, which is true because $h\in \Omega$. In fact, this is the only place we use the restriction $h\in \Omega$ in the proof of Claim 2.
We learn from that for any multi-index $\epsilon$ with $1\leq {\left\vert\epsilon\right\vert}<q$, $$\label{eqn:bettera}
a_\epsilon(x+h) = a_\epsilon(x) + \bar{P}_{x,\epsilon}(h) + O({\left\verth\right\vert}^{q-{\left\vert\epsilon\right\vert}}),$$ where $\bar{P}_{x,\epsilon}$ is a polynomial of $h$, whose coefficients (depending on $x$ and $\epsilon$) is uniformly bounded. When ${\left\vert\epsilon\right\vert}=[q]$, is the same as (with $\bar{P}_{x,\epsilon}=0$). When ${\left\vert\epsilon\right\vert}<[q]$, we prove by induction and assume that is known for all $a_\sigma$ if ${\left\vert\sigma\right\vert}>{\left\vert\epsilon\right\vert}$. We rewrite $$Q^\epsilon_\epsilon(a_\epsilon(x+h)-a_\epsilon(x))= \sum_{\epsilon\lneq \sigma, {\left\vert\sigma\right\vert}<q} \left( a_\sigma(x)-a_\sigma(x+h) \right) Q^\sigma_\epsilon h^{\sigma-\epsilon} + O({\left\verth\right\vert}^{q-{\left\vert\epsilon\right\vert}}).$$ By induction hypothesis, we insert for $\sigma\gneq \epsilon$ into the above equation to get for $\epsilon$.
If ${\left\vert\epsilon\right\vert}=1$, then $a_\epsilon(x)$ is nothing but the partial derivative of $u$ at $x$ and is the desired estimate in Claim 2.
Next, we show how to obtain Claim 1 from Claim 2, exploiting the rotational symmetry of the statement. For any orthogonal $n\times n$ matrix $A$, set $$\tilde{f}(y)=f(Ay), \quad \text{or equivalently,} \quad f(x)=\tilde{f}(A^{-1} x).$$ Assume $f$ is in $\mathcal U^{q,\delta}(B_{1+\eta})$ (as assumed in Claim 1). Then $\tilde{f}\in \mathcal U^{q,\delta}(B_{1+\eta})\subset \mathcal U^{q,\delta}_\Omega(B_{1+\eta})$. By Claim 2, which has been proved, for any $y\in B_{1+\eta'}$ and $h\in \Omega\cap B_{\delta'}$, we have, uniformly, $$\label{eqn:goodpartial}
{\frac{\partial \tilde{f}}{\partial y_i}}(y+h) = {\frac{\partial \tilde{f}}{\partial y_i}}(y) + \sum_{0<{\left\vert\epsilon\right\vert}<q} \tilde{a}_\epsilon(y) h^\epsilon + O({\left\verth\right\vert}^{q-1}).$$ By Claim 2, the chain rule and , as long as $A^{-1}h\in \Omega$, we have $$\begin{aligned}
{\frac{\partial f}{\partial x_j}}(x+h)&=& (A^{-1})^i_j {\frac{\partial \tilde{f}}{\partial y_i}} \left( A^{-1} x + A^{-1}h \right) \\
&=& (A^{-1})^i_j \left[ {\frac{\partial \tilde{f}}{\partial y_i}}(A^{-1} x) + \sum_{0<{\left\vert\epsilon\right\vert}<q} \tilde{a}_\epsilon(A^{-1}x) (A^{-1} h)^\epsilon + O({\left\verth\right\vert}^{q-1}) \right] \\
&=& {\frac{\partial f}{\partial x_j}}(x) + \sum_{0<{\left\vert\epsilon\right\vert}<q} \hat{a}_{\epsilon,A}(x) h^\epsilon + O({\left\verth\right\vert}^{q-1}).\end{aligned}$$
In summary, we have proved that ${\frac{\partial f}{\partial x_i}}$ is partially UBE with respect to $A\Omega$ up to order $q-1$. Now, we take orthogonal matrices $A_1,\cdots,A_l$ such that $${\mathbb R}^n = A_1\Omega \bigcup \cdots \bigcup A_l \Omega.$$ Then the partial UBE conditions for each $k$ combine to be UBE if we can justify that $$\tilde{a}_{\epsilon,A_k}(x)$$ is independent of $k=1,\cdots,l$. This is true because we can choose $A_k$ so that $A_{k_1}\Omega \cap A_{k_2}\Omega$ is either empty or has non-empty interior.
A proof of the Schauder estimates on ${\mathbb R}^n$ {#sec:another}
====================================================
We give another proof to the well-known interior Schauder estimate in this section.
Given Proposition \[prop:rn\], it suffices to prove
\[thm:rn\]Suppose that $f\in \mathcal U^{q,\delta}(B_1)$ for some $q=k+\alpha$ with $k\in \mathbb N$ and $\alpha\in (0,1)$. If $u$ is a bounded solution to $\triangle u =f$ on $B_1$, then $u$ lies in $\mathcal U^{q+2,\delta}(B_{1-\delta})$ and $$\label{eqn:circlestar}
{\left\Vertu\right\Vert}_{\mathcal U^{q+2,\delta}(B_{1-\delta})} \leq C(n,q,\delta) ({\left\Vertf\right\Vert}_{\mathcal U^{q,\delta}(B_1)} + {\left\Vertu\right\Vert}_{C^0(B_1)}).$$
For the proof, we need the following lemma (see Lemma 2.1 in [@han2000]),
\[lem:key\] If $f:B_{r}\to {\mathbb R}$ is $O(q)$ and $q$ is not an integer, then there exists some $u\in O(2+q)$ satisfying $$\triangle u =f\quad \text{on} \quad B_r.$$ Moreover, for some $C>0$ depending on $n,q,r$, $$[u]_{O_{q+2},B_r}\leq C [f]_{O_q,B_r}.$$
Before the proof of Lemma \[lem:key\], we show how Theorem \[thm:rn\] follows from it. For any $x\in B_{1-\delta}$ fixed, there exists a polynomial $p_{f,x}(h)$ (of order $k$) such that $$f(x+h) = p_{f,x}(h) + e_{f,x}(h) \quad \text{on} \quad B_\delta,$$ where $e_{f,x}$ is $O(q)$. By the definition, all the coefficients of $p_{f,x}$ are bounded by ${\left\Vertf\right\Vert}_{\mathcal U^{q,\delta}(B_1)}$. Hence there exists another polynomial $p_{u,x}(h)$ (of order $k+2$, not unique) whose coefficients are bounded by $C(n,q) {\left\Vertf\right\Vert}_{\mathcal U^{q,\delta}(B_1)}$ such that $$\triangle p_{u,x}= p_{f,x} \quad \text{on} \quad B_\delta.$$
By Lemma \[lem:key\], there is some $e_{u,x} \in O(q+2)$ such that $\triangle e_{u,x}= e_{f,x}$ on $B_\delta$. Therefore, $$\label{eqn:upe}
\triangle (u(x+h) - p_{u,x}(h) - e_{u,x}(h)) =0\quad \text{on} \quad B_\delta.$$ Moreover, also by Lemma \[lem:key\], $$[e_{u,x}]_{O_{q+2},B_\delta} \leq C [e_{f,x}]_{O_q,B_\delta}.$$ In particular, ${\left\Verte_{u,x}\right\Vert}_{C^0(B_\delta)}$ is bounded by a multiple of ${\left\Vertf\right\Vert}_{\mathcal U^{q,\delta}(B_1)}$.
By , $u(x+h)-p_{u,x}(h)-e_{u,x}(h)$ is a bounded harmonic function that is bounded on $B_\delta$ by the right hand side of . By well-known properties of harmonic functions, $$u(x+h)-p_{u,x}(h)-e_{u,x}(h)=\tilde{p}(h) + \tilde{e}(h),$$ where $\tilde{p}(h)$ is a polynomial of order $k+2$ and $\tilde{e}(h)$ is $O_{q+2,B_\delta}$ and again, the coefficients of $\tilde{p}$ and $[\tilde{e}]_{O_{q+2,B_\delta}}$ is bounded by the right hand side of .
By setting $\tilde{p}_{u,x}=p_{u,x}+\tilde{p}$ and $\tilde{e}_{u,x}=e_{u,x}+\tilde{e}$, we have $$u(x+h)=\tilde{p}_{u,x}(h)+ \tilde{e}_{u,x}(h).$$ Notice that the constants in the above argument are independent of $x\in B_{1-\delta}$, hence we have verified that $u\in \mathcal U^{q+2,\delta}(B_{1-\delta})$ with the desired bound.
The rest of this section is devoted to the proof of Lemma \[lem:key\]. Without loss of generality, we assume $\delta=1$.
We decompose $B_1$ into the union of a sequence of annulus $$A_l:= B_{2^{-l}} \setminus B_{2^{-l-1}} \quad \text{for} \quad l=0,1,2,\cdots.$$ Set $$f_l= f\cdot \chi_{A_l},$$ where $\chi_{A_l}$ is the characteristic function of $A_l$. For simplicity, in the rest of this proof, we write $\Lambda_f$ for $[f]_{O_q,B_\delta}$. By definition, $${\left\vertf_l(x)\right\vert}\leq \Lambda_f {\left\vertx\right\vert}^q \chi_{A_l} \leq \Lambda_f 2^{-lq} \quad \text{on} \quad {\mathbb R}^n.$$ Let $w_l$ be the unique solution (vanishing at the infinity) to the Poisson equation $\triangle w_l =f_l$ on ${\mathbb R}^n$. Obviously, $w_l$ is harmonic in the complement of $A_l$. Moreover, we have the uniform bound $$\label{eqn:wl}
\sup_{{\mathbb R}^n} {\left\vertw_l\right\vert} \leq C \Lambda_f 2^{-l (q+2)}.$$ Since $w_l$ is harmonic in $B_{2^{-l-1}}$, it is a converging power series there and let $P_l$ be the polynomial that is the part of this series with order strictly smaller than $q+2$, namely, if $$w_l(x)= \sum_\epsilon a_\epsilon x^\epsilon$$ then $$\label{eqn:defnPl}
P_l(x) = \sum_{{\left\vert\epsilon\right\vert}<q+2} a_\epsilon x^\epsilon.$$ Using the fact that $w_l$ is harmonic on $B_{2^{-l-1}}$ and bounded by $C\Lambda_f 2^{-l(q+2)}$, we estimate $$\label{eqn:abeta}
{\left\verta_\epsilon\right\vert}\leq C_\epsilon \Lambda_f 2^{-l(q+2-{\left\vert\epsilon\right\vert})}.$$ Here $C_\epsilon$ is a constant depending on $n$ and $\epsilon$.
Setting $$u_l(x) = w_l(x) - P_l(x),$$ we have the following estimates,
\[lem:ul\] There exists a constant $C_q$ depending on $n$ and $q$ such that
\(i) on $B_{2^{-l-1}}$, $${\left\vertu_l(x)\right\vert} \leq C_q\Lambda_f 2^{(1+[q]-q)l}{\left\vertx\right\vert}^{[q]+3};$$
\(ii) on $A_l$, $${\left\vertu_l(x)\right\vert} \leq C_q \Lambda_f 2^{-(q+2)l}$$ or equivalently $${\left\vertu_l(x)\right\vert} \leq C_q \Lambda_f {\left\vertx\right\vert}^{q+2};$$
\(iii) on $B_1\setminus B_{2^{-l}}$, $${\left\vertu_l(x)\right\vert} \leq C_q \Lambda_f \left[ 2^{-l(q+2)} + \sum_{{\left\vert\epsilon\right\vert}<q+2} 2^{-l(q+2-{\left\vert\epsilon\right\vert})} {\left\vertx\right\vert}^{{\left\vert\epsilon\right\vert}} \right].$$
Here and in the following proof, $C_q$ may vary from line to line, as long as it depends only on $n$ and $q$.
First, we estimate $P_l(x)$ in $B_{2^{-l}}$ as follows $$\begin{aligned}
{\left\vertP_l(x)\right\vert} &\leq& \sum_{ {\left\vert\epsilon\right\vert}<q+2} {\left\verta_\epsilon\right\vert} {\left\vertx\right\vert}^{{\left\vert\epsilon\right\vert}} \\
&\leq& C_q \Lambda_f \sum_{{\left\vert\epsilon\right\vert}<q+2} 2^{-l(q+2-{\left\vert\epsilon\right\vert})} 2^{-l{\left\vert\epsilon\right\vert}} \\
&\leq& C_q \Lambda_f 2^{-l(q+2)}.
\end{aligned}$$ Here we have used and ${\left\vertx\right\vert}\leq 2^{-l}$. Together with , this implies that $${\left\vertu_l(x)\right\vert} \leq C_q \Lambda_f 2^{-(q+2)l}\qquad \text{on}\quad B_{2^{-l}},
\label{eqn:ul2}$$ which in particular proves (ii) of Lemma \[lem:ul\].
By the definition of $P_l$, $u_l$ is a harmonic function on $B_{2^{-l}}$ which vanishes at the origin up to order $[q]+2$. Hence, (i) of Lemma \[lem:ul\] follows from and a scaled version of the following fact:
If $u$ is a harmonic function on $B_1$ bounded by $1$ and vanishes at the origin up to order $k$, then there is a universal constant depending only on the dimension such that $${\left\vertu(x)\right\vert}\leq C_n {\left\vertx\right\vert}^{k+1}.$$
The last part of Lemma \[lem:ul\] is a trivial combination of and .
With these preparations, we claim that $$\label{eqn:defu}
u(x) =\sum_{l=0}^\infty u_l(x)$$ converges on $B_1$ and gives the desired solution $u$ in Lemma \[lem:key\].
Since $w_l$ is harmonic in a neighborhood of $0\in {\mathbb R}^n$, $P_l$ defined in is a harmonic polynomial on the entire ${\mathbb R}^n$. As a consequence, $$\triangle u_l = f_l \quad \text{on} \quad {\mathbb R}^n.$$ Hence, to show Lemma \[lem:key\], it suffices to prove $$\label{eqn:convergeu}
\sum_{l=0}^\infty {\left\vertu_l\right\vert}(x) \leq C_q \Lambda_f {\left\vertx\right\vert}^{q+2} \qquad \text{on} \quad B_1,$$ which not only implies the convergence of , but also gives the expected bound of $u$ in Lemma \[lem:key\]. For each $x\in B_1\setminus {\left\{0\right\}}$, all but finitely many $u_l$’s are harmonic in a neighborhood of $x$, hence, the convergence is smooth and $u$ satisfies the Poisson equation $\triangle u=f$.
For each fixed $x\in B_1\setminus {\left\{0\right\}}$, let $l_0$ be given by the condition that $$x\in A_{l_0}.$$ In other words, ${\left\vertx\right\vert} < 2^{-l_0} \leq 2{\left\vertx\right\vert}$.
To estimate the left hand side of , we compute $$\begin{aligned}
\sum_{l=0}^{l_0-1} {\left\vertu_l\right\vert}(x) &\leq& C_q \Lambda_f \left( \sum_{l=0}^{l_0-1} 2^{(1+[q]-q)l} \right) {\left\vertx\right\vert}^{[q]+3} \\
&\leq& C_q \Lambda_f 2^{(1+[q]-q)l_0} {\left\vertx\right\vert}^{[q]+3} \\
&\leq& C_q \Lambda_f {\left\vertx\right\vert}^{q+2},\end{aligned}$$ where we used (i) in Lemma \[lem:ul\]. (ii) of Lemma \[lem:ul\] implies $${\left\vertu_{l_0}(x)\right\vert} \leq C_q \Lambda_f {\left\vertx\right\vert}^{q+2}.$$ Similarly, using (iii) of Lemma \[lem:ul\], we have $$\begin{aligned}
\sum_{l>l_0} {\left\vertu_l\right\vert}(x) &\leq& C_q \Lambda_f \sum_{l>l_0} \left[ 2^{-l(q+2)} + \sum_{{\left\vert\epsilon\right\vert}<q+2} 2^{-l(q+2-{\left\vert\epsilon\right\vert})} {\left\vertx\right\vert}^{{\left\vert\epsilon\right\vert}} \right]\\
&\leq& C_q \Lambda_f {\left\vertx\right\vert}^{q+2}.\end{aligned}$$ This finishes the proof of and hence the proof of Lemma \[lem:key\].
Preliminaries about $X_\beta$ {#sec:pre}
=============================
In this section, we first define some notations and then recall some basic properties about the Poisson equation on $X_\beta$ whose proofs are omitted.
Notations {#subsec:notation}
---------
Aside from the natural coordinates $(x,y,\xi)$ of $X_\beta= {\mathbb R}^2\times {\mathbb R}^{n-2}$, there is a global coordinate system $(\rho,\theta,\xi)$ on the smooth part of $X_\beta$ given by $$\rho=\frac{1}{\beta} r^{\beta}, \quad x=r\cos\theta,\quad y=r\sin \theta.$$ In terms of $(\rho,\theta,\xi)$, the metric $g_\beta$ in becomes $$g_\beta= d\rho^2 + \rho^2 \beta^2 d\theta^2 + d\xi^2.$$
Here is a list of notations that are useful.
(i) The singular set, denoted by $\mathcal S$, corresponds to ${\left\{\rho=0\right\}}$ and can be parametrized by $\xi$.
(ii) $d(x,y)$ is the Riemannian distance (given by $g_\beta$) between $x$ and $y$ in $X_\beta$.
(iii) $\mathcal S_\delta$ is the set of points whose distance to $\mathcal S$ is smaller than $\delta>0$.
(iv) $\Omega_\delta= X_\beta \setminus \mathcal S_\delta$.
(v) For a point $x\in X_\beta$, $\hat{B}_r(x)$ is the set of points whose distance to $x$ is smaller than $r$.
(vi) $x_0$ is the origin of $X_\beta$, i.e. the point with $\rho=0$ and $\xi=0$. $\hat{B}_1(x_0)$ is the unit ball, which for simplicity is often denoted by $\hat{B}_1$.
(vii) $d(x,x_0)$ is usually denoted by $d(x)$.
(viii) $\tilde{x}_0$ is the point in $X_\beta$ with $\rho=1$, $\theta=0$ and $\xi=0$. Then there is a constant $c_\beta$ depending only on $\beta$ such that $\hat{B}_{c_\beta}(\tilde{x}_0)$ is topologically a ball and that the restriction of $g_\beta$ to it is comparable with the flat metric on $B_{c_\beta}(0)\subset {\mathbb R}^n$. Throughout this paper, we fix this $c_\beta$ and denote $\hat{B}_{c_\beta}(\tilde{x}_0)$ by $\tilde{B}$, which also serves as a unit ball. We also write $\tilde{B}_{r}$ for $\hat{B}_{c_\beta r}(\tilde{x}_0)$
For each $x\in X_\beta\setminus \mathcal S$, there is a natural scaling map $\Psi_x$ which maps $\hat{B}_{c_\beta \rho(x)}(x)$ to $\tilde{B}$, where $\rho(x)$ is the $\rho$ coordinate of $x$, i.e. the distance to $\mathcal S$. If $x=(\rho_0,\theta_0,\xi_0)$, then $$\Psi_x(\rho,\theta,\xi)= (\frac{\rho}{\rho_0}, \theta-\theta_0, \frac{\xi-\xi_0}{\rho_0}).$$ Here $\theta-\theta_0$ is understood as the natural minus operation of the group $S^1$. The scaling $\Psi_x$ induces a pushforward of functions, which we denote by $S_x$. More precisely, if $u$ is a function defined on $\hat{B}_{c_\beta \rho(x)}(x)$, then $$S_x(u)(y)= u(\Psi_x^{-1}(y))\qquad \forall y\in \tilde{B}.$$
Basics on the Poisson equation {#subsec:basic}
------------------------------
We collect a few basic facts about PDEs on $X_\beta$.
We start with an observation that is known and utilised by many authors. Consider another copy of ${\mathbb R}^n$, whose coordinates are given by $(w,v,\xi)$, where $w,v\in {\mathbb R}$ and $\xi\in {\mathbb R}^{n-2}$. The Euclidean metric on ${\mathbb R}^n$ is given by $$dw^2+dv^2+d\xi^2.$$ One can check by direct computation that the mapping $$(\rho,\theta,\xi) \mapsto (\rho\cos\theta, \rho \sin\theta, \xi)\in {\mathbb R}^{n}$$ is bi-Lipschitzian from $X_\beta$ to ${\mathbb R}^n$. Hence, the Sobolev space $W^{1,2}(X_\beta)$ ($L^p(X_\beta)$) is the same set of functions as $W^{1,2}({\mathbb R}^n)$ ($L^p({\mathbb R}^n)$). Moreover, the Sobolev inequality on $X_\beta$ holds with a different constant.
One can prove the following by the usual variation method and Moser iteration. Please note that we state it in a scaling invariant form.
\[lem:poisson\] Let $f$ be an $L^\infty$ function supported in $\hat{B}_r$. There exists a solution $u\in W^{1,2}_{loc}(X_\beta)\cap L^\infty(X_\beta)$ to the Poisson equation $$\triangle_\beta u =f$$ with the bound $${\left\Vertu\right\Vert}_{L^\infty(X_\beta)}\leq C r^2 {\left\Vertf\right\Vert}_{L^\infty(\hat{B}_r)}.$$
Bounded harmonic functions on $X_\beta$ {#sec:harmonic}
=======================================
Suppose that $u$ is a bounded harmonic function on $\hat{B}_2\subset X_\beta$, i.e., $$\triangle_\beta u =0.$$ We discuss in this section the regularity of $u$ in $\hat{B}_{1}$. Before our discussion on $X_\beta$, we recall that if $u$ is a harmonic function on $B_1\subset {\mathbb R}^n$, then we can bound any derivatives of $u$ on $B_{1/2}$ by the $C^0$ norm of $u$ on $B_1$. Equivalently, there is the Taylor expansion of $u$ at $0$, $$\label{eqn:expansion1}
u(x)= \sum_{{\left\vert\sigma\right\vert}\leq k} a_\sigma x^\sigma + O({\left\vertx\right\vert}^{k+1}),$$ where $\sigma$ is a multi-index and the $a_\sigma$’s and the constant in the definition of $O({\left\vertx\right\vert}^{k+1})$ are bounded by the $C^0$ norm of $u$ on $B_1$. The first goal of this section is to prove a generalization of this result for harmonic functions on $X_\beta$. More precisely, we prove an analog of for harmonic function on $X_\beta$ and provide estimates for the coefficients in the expansion.
Although it is very likely that the expansion we prove below (see Proposition \[prop:stronghf\]) gives a converging series if we trace the bound for coefficients in the expansion, we do not pursue it here. However, we need the fact that the approximation polynomial of a harmonic function (as given in Proposition \[prop:stronghf\]) is still harmonic as in the case of ${\mathbb R}^n$. This is the second goal of this section and is contained in the second subsection.
The expansion {#sub:expansion}
-------------
The first thing we need to do is to generalize the concept of polynomial (or monomial) used in the Taylor expansion. For the time being, we concentrate on the expansion describing the regularity of harmonic functions. In terms of the polar coordinates $(\rho,\theta,\xi)$, the (monic) [**$X_\beta$-monomials**]{} are defined to be $$\rho^{2j+\frac{k}{\beta}} \cos k\theta \xi^\sigma, \rho^{2j+\frac{k}{\beta}}\sin k \theta \xi^\sigma,$$ for each multi-index $\sigma$ of ${\mathbb R}^{n-2}$ and any $j,k\in \mathbb N\cup {\left\{0\right\}}$. Then sum $2j+\frac{k}{\beta}+{\left\vert\sigma\right\vert}$ is called the [**degree of the $X_\beta$-monomial**]{} and an [**$X_\beta$-polynomial**]{} is a finite linear combination of monomials.
Throughout this paper, unless stated otherwise, the range of $j,k$ and $\sigma$ in a summation is understood as above.
With these definitions, we can now state the main result of this subsection.
\[prop:stronghf\] If $u$ is a bounded harmonic function in $\hat{B}_1$, then for any $q>0$ and $d(x)<1/2$, $$\label{eqn:hfexpansion}
u(x)= \sum_{2j+\frac{k}{\beta}+{\left\vert\sigma\right\vert}<q} \rho^{2j+\frac{k}{\beta}} \left( a_{j,k}^\sigma \cos k\theta + b_{j,k}^\sigma \sin k\theta \right) \xi^\sigma + O(d(x)^q),$$ where by definition ${\left\vertO(d(x)^q)\right\vert}\leq \Lambda d(x)^q$. Moreover, $${\left\verta_{j,k}^\sigma\right\vert}, {\left\vertb_{j,k}^\sigma\right\vert}, \Lambda \leq C(q) {\left\Vertu\right\Vert}_{C^0(\hat{B}_1)}.$$
The rest of this subsection is devoted to the proof of this result. First, we notice that the regularity of $u$ in $\xi$-direction is not a problem. This is summarized in the next lemma, whose proof is omitted.
\[lem:xigood\] Suppose that $u$ is a weak harmonic function on $\hat{B}_2$. Then for any multi-index $\sigma$, $\partial_\xi^\sigma u$ is also a weak harmonic function on $\hat{B}_2$ and $${\left\Vert\partial_\xi^\sigma u\right\Vert}_{C^0(\hat{B}_1)}\leq C({\left\vert\sigma\right\vert}) {\left\Vertu\right\Vert}_{C^0(\hat{B}_2)}.$$
When $\xi$ is fixed, we write $u(\xi)$ for $u$ as a function of $\rho,\theta$. The regularity of $u(\xi)$ is essentially a two dimensional problem that has been studied in [@yin2016analysis]. The proof in [@yin2016analysis] yields
\[lem:rhogood\] For ${\left\vert\xi\right\vert}<1/2$, $$\label{eqn:expansionu}
u(\xi)= \sum_{2j+\frac{k}{\beta}<q} \rho^{2j+\frac{k}{\beta}} \left( a_{j,k}(\xi) \cos k\theta + b_{j,k}(\xi) \sin k\theta \right) + O(\rho^q)\qquad \text{for} \quad \rho<1/2.$$
For any multi-index $\sigma$, $$\label{eqn:expansionusigma}
\partial^\sigma_\xi u(\xi)= \sum_{2j+\frac{k}{\beta}<q} \rho^{2j+\frac{k}{\beta}} \left( a_{j,k,\sigma}(\xi) \cos k\theta + b_{j,k,\sigma}(\xi) \sin k\theta \right) + O(\rho^q)\qquad \text{for} \quad \rho<1/2.$$ Moreover, $a_{j,k}(\xi)$, $b_{j,k}(\xi)$, $a_{j,k,\sigma}(\xi)$, $b_{j,k,\sigma}(\xi)$ and the constants in the definition of $O(q)$ are uniformly (independent of ${\left\vert\xi\right\vert}<\frac{1}{2}$) bounded by a multiple of ${\left\Vertu\right\Vert}_{C^0(\hat{B}_2)}$.
For completeness, we include a proof of Lemma \[lem:rhogood\] in the appendix.
Please note the difference between $a_{j,k,\sigma}$ in this lemma and $a_{j,k}^\sigma$ in Proposition \[prop:stronghf\]. Lemma \[lem:rhogood\] does not claim any relation between $a_{j,k}(\xi)$ and $a_{j,k,\sigma}(\xi)$. It will be clear in a minute that $$a_{j,k,\sigma}(\xi)= \partial_\xi^\sigma a_{j,k}.$$ The same applies to $b_{j,k}$ and $b_{j,k,\sigma}$.
Given Lemma \[lem:rhogood\] and Lemma \[lem:xigood\], we claim that:
[**Claim:**]{} the $a_{j,k}(\xi)$ and $b_{j,k}(\xi)$ in are smooth functions of $\xi$ on ${\left\{{\left\vert\xi\right\vert}<1/2\right\}}$. Moreover, for any multi-index $\sigma$, $$\sup_{{\left\vert\xi\right\vert}<1/2} {\left\vert\partial_\xi^\sigma a_{j,k}\right\vert} + {\left\vert\partial_\xi^\sigma b_{j,k}\right\vert} \leq C(j,k,\sigma) {\left\Vertu\right\Vert}_{C^0(\hat{B}_2)}.
\label{eqn:claim}$$
Proposition \[prop:stronghf\] follows from the claim, because we can expand $a_{j,k}(\xi)$ and $b_{j,k}(\xi)$ (in ) into the sum of a Taylor polynomial of $\xi$ and a remainder $O({\left\vert\xi\right\vert}^q)$. Then the $O(d^q)$ in Proposition \[prop:stronghf\] is the sum of $O(\rho^q)$ in , a sum of $X_\beta$-monomials with degree no less than $q$ and $$\rho^{2j+\frac{k}{\beta}} O({\left\vert\xi\right\vert}^q), \qquad 2j+\frac{k}{\beta}<q$$ in the expansion of $a_{j,k}(\xi)$ and $b_{j,k}(\xi)$.
0.5cm
For the proof of the claim, we start with $a_{0,0}(\xi)$. By , $$a_{0,0}(\xi)=\lim_{\rho\to 0} u.$$ By Lemma \[lem:xigood\] ${\left\vert\partial_\xi^\sigma u\right\vert}$ is uniformly bounded for any ${\left\vert\xi\right\vert}<\frac{1}{2}$, hence the convergence is in fact in $C^l$ for any $l$, then our claim for $a_{0,0}$ follows.
Now for a fixed $\sigma$, gives $$a_{0,0,\sigma}(\xi)=\lim_{\rho\to 0} \partial_\xi^\sigma u.$$ As before, since $\lim_{\rho\to 0} u$ is in $C^l$ topology for any $l>0$, we have $$\lim_{\rho\to 0} \partial_\xi^\sigma u = \partial_\xi^\sigma (\lim_{\rho\to 0} u),$$ which implies that $$\label{eqn:tricky}
a_{0,0,\sigma}(\xi)=\partial_\xi^\sigma a_{0,0}(\xi).$$
Since our claim for $a_{0,0}$ is proved, we may assume that it is zero at the very beginning of the proof by replacing $u$ with $u-a_{0,0}(\xi)$. Thanks to , a consequence of this assumption is that $$\label{eqn:a00}
a_{0,0,\sigma}\equiv 0,\qquad \forall \sigma.$$
Next, suppose that $2j_1+\frac{k_1}{\beta}$ is the next (smallest) nonzero power in the expansion, namely, $j_1=1$ and $k_1=0$ when $\beta<1/2$ or $j_1=0$ and $k_1=1$ when $\beta\geq 1/2$. [^1] and imply $$\frac{\partial_\xi^\sigma u}{\rho^{2j_1+\frac{k_1}{\beta}}}$$ is uniformly bounded (w.r.t. $\xi$) by a constant depending on $\sigma$. Hence, the convergence $$a_{j_1,k_1}(\xi) = \lim_{\rho\to 0} \frac{1}{\pi}\int_0^{2\pi} \frac{u}{\rho^{2j_1+\frac{k_1}{\beta}}} \cos k_1\theta d\theta$$ is uniform in any $C^l$ topology. This implies that our claim for $a_{j_1,k_1}$ holds, which enables us to assume $a_{j_1,k_1}(\xi)=0$ at the beginning. Notice that we also have (by the same reason) $$\partial_\xi^\sigma a_{j_1,k_1}(\xi) = a_{j_1,k_1,\sigma}(\xi),$$ so that we can repeat the argument to prove the claim for any $a_{j,k}$ and $b_{j,k}$.
The approximating $X_\beta$-polynomial is still harmonic {#sub:approx}
--------------------------------------------------------
We prove in this section
\[prop:truncation\] Suppose that $u$ satisfies the assumptions of Proposition \[prop:stronghf\] and therefore has an expansion given by . Then $$\label{eqn:truncation}
\triangle_\beta \left[ \sum_{2j+\frac{k}{\beta}+{\left\vert\sigma\right\vert}<q} \rho^{2j+\frac{k}{\beta}} \left( a_{j,k}^\sigma \cos k\theta + b_{j,k}^\sigma \sin k\theta \right) \xi^\sigma \right] =0.$$
To prove this proposition, it suffices to justify that the $\triangle_\beta$ of the remainder $O(d^q)$ (in ) is an $O(d^{q-2})$. In fact, letting $T$ be the $X_\beta$-polynomial in (or equivalently, in ), we have $$\label{eqn:dueto}
\triangle_\beta u = \triangle_\beta T + O(d^{q-2})=0,$$ [**if**]{} our claim for $O(d^q)$ holds. One can check that $\triangle_\beta T$ is an $X_\beta$-polynomial of degree smaller than $q-2$, then it vanishes due to .
For the claimed property of $O(d^q)$, recall that, by the proof of Proposition \[prop:stronghf\], it is the sum of
(1) $\rho^{2j+\frac{k}{\beta}}\cos k\theta \xi^\sigma$ for $2j+\frac{k}{\beta}+{\left\vert\sigma\right\vert}>q$; (there is a similar term with $\sin$ replacing $\cos$)
(2) $\rho^{2j+\frac{k}{\beta}}\cos k\theta O({\left\vert\xi\right\vert}^q)$;
(3) the $O(\rho^q)$ term in .
It is trivial that the desired property is true for functions in (1). For (2), notice that $O({\left\vert\xi\right\vert}^q)$ comes from the Taylor expansion of $a_{j,k}(\xi)$ and $b_{j,k}(\xi)$ that are smooth in $\xi$ and hence $\partial_\xi^2 O({\left\vert\xi\right\vert}^q)$ is $O({\left\vert\xi\right\vert}^{q-2})$. Given this, it is straightforward to check that $$\triangle_\beta \left( \rho^{2j+\frac{k}{\beta}}\cos k\theta O({\left\vert\xi\right\vert}^q) \right) = O(d^{q-2}), \qquad \text{where} \quad d^2 =\rho^2 + {\left\vert\xi\right\vert}^2.$$
It remains to study the remainder $O(\rho^q)$ in . In the proof of Proposition \[prop:stronghf\], we have shown that $a_{j,k}(\xi)$ and $b_{j,k}(\xi)$ in are smooth functions and for any multi-index $\sigma$, $$\partial_\xi^\sigma a_{j,k}(\xi)=a_{j,k,\sigma}(\xi)\quad \text{and} \quad
\partial_\xi^\sigma b_{j,k}(\xi)=b_{j,k,\sigma}(\xi).$$ By comparing and , we notice that any $\xi$-derivative of the $O(\rho^q)$ in is still a $O(\rho^q)$.
Since $u$ is a harmonic function, it is not difficult to prove by the interior Schauder estimate and Lemma \[lem:xigood\] that $$\label{eqn:weighted}
\sup_{\hat{B}_1\setminus \mathcal{S}} {\left\vert(\rho \partial_\rho)^{k_1} (\partial_\theta)^{k_2} u\right\vert} \leq C(k_1,k_2).$$ Using equation , the estimate holds for $O(\rho^q)$, which implies that $$\left( \partial_\rho^2 + \frac{1}{\rho}\partial_\rho + \frac{1}{\beta^2 \rho^2}\partial_\theta^2 \right) O(\rho^q) = O(\rho^{q-2}).$$ This concludes the proof of Proposition \[prop:truncation\] by noticing that $\rho\leq d$ and $O(\rho^{q-2})$ is $O(d^{q-2})$.
Generalized Hölder space on $X_\beta$ {#sec:Holder}
=====================================
In this section, we define the $X_\beta$ counterpart of $C^{k,\alpha}$ function on ${\mathbb R}^n$ and compare it with Donaldson’s $C^{2,\alpha}_\beta$ space when $0<\beta<1$ and $0<\alpha<\min {\left\{\frac{1}{\beta}-1,1\right\}}$.
Formal discussion {#subsec:formal}
-----------------
The basic idea of our definition as illustrated by Section \[sec:space\] is to use generalized ‘polynomial’ to describe the regularity near a singular point. In this subsection, we are concerned with the question of what is the correct ‘polynomial’ for $X_\beta$.
Recall that in Proposition \[prop:stronghf\], for the expansion of harmonic functions, we defined $X_\beta$-polynomials, which are finite linear combinations of $$\rho^{2j+\frac{k}{\beta}}\cos k\theta \xi^\sigma, \rho^{2j+\frac{k}{\beta}}\cos k\theta \xi^\sigma$$ where $j,k\in \mathbb N \cup {\left\{0\right\}}$ and $\sigma$ is a multi-index of ${\mathbb R}^{n-2}$.
$X_\beta$-polynomials are not enough for the study of more complicated PDE solutions, because the product of two $X_\beta$-polynomials is [**not**]{} $X_\beta$-polynomial. This motivates the following definition.
Suppose that $j,k,m\in \mathbb N\cup {\left\{0\right\}}$ satisfy $\frac{k-m}{2}\in \mathbb N\cup {\left\{0\right\}}$ and $\sigma$ is a multi-index of dimension $n-2$. The functions $$\rho^{2j+\frac{k}{\beta}}\cos m\theta \xi^\sigma,\rho^{2j+\frac{k}{\beta}}\sin m\theta \xi^\sigma$$ are called (monic) [**$\mathcal T$-monomials**]{} of degree $2j+\frac{k}{\beta}+{\left\vert\sigma\right\vert}$. A [**$\mathcal T$-polynomial**]{} is a finite linear combination of $\mathcal T$-monomials.
It is elementary to check that the product of $\mathcal T$-polynomials is $\mathcal T$-polynomial.
Our previous experience in PDE’s with conical singularity suggests that the regularity of solutions near a singular point (like the cone singularity in $X_\beta$) depends both on the singularity of space and on the type of PDE that we are working with.
While the above definition of $\mathcal T$-polynomial suffices (see [@yin2016analysis]) for the study of nonlinear equations like $$\triangle_\beta u =F(u).$$ It is not enough for the conical complex Monge-Ampere equation studied in [@yin2016expansion]. To minimize the difficulty of understanding this paper, we refrain from working in that generality. Instead, we will list below a family of properties that should be satisfied by $\mathcal T$-polynomials. It will be clear in the proof that follows, if these properties hold, the main result of this paper remains true for other definitions of $\mathcal T$-polynomials.
Our definition of $\mathcal T$-polynomial satisfies a family of properties, which we summarize in the form of a lemma.
\[lem:PPP\] (P1) Let $f$ be a monic $\mathcal T$-monomial of degree $q$. For any $q'<q$, $l\in \mathbb N\cup {\left\{0\right\}}$ and any point $x\in X_\beta$ satisfying $\rho(x)<1/2$ and $\xi(x)=0$, $${\left\VertS_x(f)\right\Vert}_{C^l(\tilde{B})}\leq C(q,l,q') \rho(x)^{q'}.$$
(P2) Let $f$ be a monic $\mathcal T$-monomial of degree $q$. There is a constant depending only on $q$ and $\delta>0$ such that for any $l \in \mathbb N\cup {\left\{0\right\}}$, $${\left\Vertf\right\Vert}_{C^l(\Omega_\delta \cap \hat{B}_1)}\leq C(q,l,\delta).$$
(P3) Let $f$ be a $\mathcal T$-polynomial of degree $q$. There is a constant $C$ depending only on $q$ such that $${\left\Vertf\right\Vert}_{C^{l}(\tilde{B}_{1/8}(z))} \leq C(l,q) {\left\Vertf\right\Vert}_{C^0(\tilde{B}_{1/8}(z))}$$ for any $l \in \mathbb N\cup {\left\{0\right\}}$ and all $z\in \tilde{B}_{1/2}$.
(P4) Let $f$ be any $\mathcal T$-polynomial of degree $q$. There exists a $\mathcal T$-polynomial $u$ of degree $q+2$ (not unique) such that $$\triangle_\beta u =f.$$ Moreover, the coefficients of $u$ are bounded by a multiple (depending on $q$) of the coefficients of $f$.
(P1) and (P2) can be checked by direct computation. Notice that (P1) is true even if $q'=q$ and we have stated it in this weaker form, because we have in our mind monomials involving $\log$ terms (for example $\rho \log \rho$), which may appear in applications.
For (P3), recall that the number of (monic) $\mathcal T$-monomials with degree no more than $q$ is finite and that they are linearly independent functions so that the $C^0$ norm of a $\mathcal T$-polynomial on any open set bounds every coefficient.
The proof of (P4) is an induction argument based on the fact that $\triangle_\beta \rho^{\gamma+2}\cos m\theta \xi^\sigma$ is a linear combination of $\rho^\gamma \cos m\theta \xi^\sigma$ and a $\mathcal T$-polynomial whose order in $\xi$ is smaller than ${\left\vert\sigma\right\vert}$ by $2$.
The definition.
---------------
In this section, we define the generalized Hölder space $\mathcal U^q$. We assume that $q$ is any positive number that is [**not in**]{} $$\mathcal D={\left\{j+\frac{k}{\beta}|\quad j,k\in \mathbb N\cup {\left\{0\right\}}\right\}},$$ which is the set of degrees of $\mathcal T$-monomials.
The overall idea is to require that $u$ is $C^{k,\alpha}$ in the usual sense in $\Omega_\delta \cap \hat{B}_1$ and for each $x\in \mathcal S\cap \hat{B}_1$, $u$ has some uniform expansion (using $\mathcal T$-polynomials) in a ball of size $\delta$.
\[defn:big\] Suppose that $q=k+\alpha$ for some $k\in \mathbb N\cup {\left\{0\right\}}$ and $\alpha\in (0,1)$. $u$ is said to be in $\mathcal U^q(\hat{B}_1)$ if and only if there is some $\Lambda>0$ such that
(H1) $u$ is $C^{k,\alpha}$ on $\hat{B}_1\cap \Omega_\delta$ and $${\left\Vertu\right\Vert}_{C^{k,\alpha}(\hat{B}_1\cap \Omega_\delta)}\leq \Lambda.$$
(H2) For each $x\in \mathcal S\cap \hat{B}_1$, there is a $\mathcal T$-polynomial $P_x$ such that (i) the degree of $P_x$ is smaller than $q$ and the coefficients of $P_x$ is bounded by $\Lambda$ and (ii) $$u(x+y)=P_x(y) + O(d(y)^q),\quad \forall y\in \hat{B}_\delta,$$ where $O(d(y)^q)$ above satisfies $${\left\vertO(d(y)^q)\right\vert} \leq \Lambda d(y)^q.$$
(H3) For each $x\in S_\delta\cap \hat{B}_1$, let $\tilde{x}$ be the projection of $x$ to $\mathcal S$. $$\label{eqn:h3}
{\left\VertS_x(u-P_{\tilde{x}}(\cdot -\tilde{x}))\right\Vert}_{C^{k,\alpha}(\tilde{B}_{1/2})}\leq \Lambda \rho(x)^q.$$
The infimum of $\Lambda$ such that (H1-H3) hold for some $u\in \mathcal U^q(\hat{B}_1)$ is defined to be the norm of $u$, denoted by ${\left\Vertu\right\Vert}_{\mathcal U^q(\hat{B}_1)}$.
(H1-H2) is in line with the overall idea mentioned before Definition \[defn:big\]. (H3) is necessary to describe the behavior of the functions in $\mathcal U^q(\hat{B}_1)$ at those points that are closer and closer to the singular set. Definition \[defn:big\] may look unusual, but it will be justified when we show that Donaldson’s $C^{2,\alpha}_\beta$ is a special case of $\mathcal U^q$ in Section \[subsec:compare\] and when we prove a Schauder estimate in Section \[sec:schauder\].
Several remarks are helpful in understanding the defnition.
\[rem:smaller\] We need to check that for $0<q_1<q_2$ ($q_1,q_2\notin \mathcal D$), $\mathcal U^{q_2}(\hat{B}_1)\subset \mathcal U^{q_1}(\hat{B}_1)$, which is not totally trivial from the definition above. To see this, let $u$ be in $\mathcal U^{q_2}(\hat{B}_1)$, it suffices to check (H2) and (H3) in the definition of $\mathcal U^{q_1}(\hat{B}_1)$. For (H2), let $x\in \mathcal S\cap \hat{B}_1$, then there is some $\mathcal T$-polynomial $P_x$ such that $$u(x+y)=P_x(y)+O(d(y)^{q_2}) \qquad \forall y\in \hat{B}_\delta.$$ Let $P'_x$ be the part of $P_x$ that involves only monomials of degree smaller than $q_1$. Let $Q_x=P_x-P'_x$. Obviously, $$u(x+y)=P'_x(y) + O(d(y)^{q_1}) \qquad \forall y\in \hat{B}_\delta.$$ For (H3), it suffices to check that for each $x\in \mathcal S_\delta\cap \hat{B}_1$ with $\tilde{x}$ being its projection to $\mathcal S$, $Q_{\tilde{x}}$ satisfies $$\label{eqn:checkQ}
{\left\VertS_x(Q_{\tilde{x}}(\cdot -\tilde{x}))\right\Vert}_{C^{k_1,\alpha_1}(\tilde{B}_{1/2})}\leq C\Lambda \rho(x)^{q_1},$$ if $q_1=k_1+\alpha_1$. In fact, $Q_{\tilde{x}}$ is a finite linear combination of $\mathcal T$-monomials of degree strictly larger than $q_1$ and (H2) implies that the coefficients of the combination are bounded by $\Lambda$. Hence, follows from (P1).
\[rem:delta\] Due to (P2) above, the definition of $\mathcal U^q(\hat{B}_1)$ is independent of the constant $\delta$. A different choice of $\delta$ yields an equivalent norm ${\left\Vert\cdot\right\Vert}_{\mathcal U^q}$.
\[rem:final\] Finally, we remark that in (H3) of Definition \[defn:big\] can be replaced by $$\label{eqn:h3p}
{\left\VertS_x(u-P_{\tilde{x}}(\cdot-\tilde{x}))(z)\right\Vert}_{C^{k,\alpha}(\tilde{B}_{3/8})}\leq \Lambda \rho(x)^q.$$
This may look plausible, however, we give a detailed proof, because we shall need it explicitly in the proof of Theorem \[thm:schauder\]. In the sequel, we shall use (H3’) for the assumption (H3) with replaced by . Assume that we have a function $u$ satisfying (H1), (H2) and (H3’).
Since (H3) only matters when $\rho(x)$ is small, we fix $x\in S_{\delta/2} \cap \hat{B}_1$. By (H2), $$\label{eqn:SPx}
\sup_{\tilde{B}} {\left\vertS_x(u-P_{\tilde{x}}(\cdot-\tilde{x}))\right\vert} = \sup_{\hat{B}_{c_\beta \rho(x)}(x)} {\left\vertu- P_{\tilde{x}}(\cdot-\tilde{x})\right\vert} \leq C\Lambda \rho(x)^q.$$ For any $y\in \hat{B}_{c_\beta \rho(x)/2}(x)=\tilde{B}_{1/2}(x)$, (H3’) implies that $${\left\VertS_y (u-P_{\tilde{y}}(\cdot-\tilde{y}))(z)\right\Vert}_{C^{k,\alpha}(\tilde{B}_{3/8})}\leq \Lambda \rho(y)^q \leq C\Lambda \rho(x)^q.$$ By the definition of $S_x$ and $S_y$, $$\label{eqn:Sxy}
\begin{split}
S_x(u-P_{\tilde{y}}(\cdot-\tilde{y}))(z) &= (u-P_{\tilde{y}}(\cdot-\tilde{y}))(\Psi_x^{-1}(z)) \\
&= (u-P_{\tilde{y}}(\cdot-\tilde{y}))(\Psi_y^{-1} \circ \Psi_y\circ \Psi_x^{-1}(z)) \\
&= S_y(u-P_{\tilde{y}}(\cdot-\tilde{y})) (\Psi_y\circ \Psi_x^{-1}(z)).
\end{split}$$ If $z=(\rho_z,\theta_z,\xi_z)$, we compute explicitly $$\label{eqn:psi}
\begin{split}
\Psi_y\circ \Psi_x^{-1} (z) &= \Psi_y (\rho_x \rho_z, \theta_x+\theta_z, \xi_x+\rho_x\xi_z) \\
&= ( \frac{\rho_x}{\rho_y}\rho_z, \theta_z +\theta_x-\theta_y, \frac{\rho_x \xi_z+\xi_x-\xi_y}{\rho_y}).
\end{split}$$ Obviously when $z=\Psi_x(y)\in \tilde{B}_{1/2}$, $\Psi_y \circ \Psi_x^{-1}(z)=\tilde{x}_0$. Since $y\in \hat{B}_{c_\beta \rho(x)/2}(x)$ (so that $1/2< \rho_x/\rho_y<2$), we know $\Psi_y\circ \Psi_x^{-1}$ is a Lipschitz map with Lipschitz constant smaller than $2$, which implies that $$\Psi_y\circ \Psi_x^{-1} (\tilde{B}_{1/8}(z))\subset \tilde{B}_{3/8}.$$ Noticing that the map $\Psi_y\circ \Psi_x^{-1}$ is uniformly bounded (for all $y\in \tilde{B}_{1/2}(x)$) in any $C^k$ norm on $\tilde{B}_{1/8}(z)$, implies that $$\label{eqn:SPy}
{\left\VertS_x(u-P_{\tilde{y}}(\cdot-\tilde{y}))\right\Vert}_{C^{k,\alpha}(\tilde{B}_{1/8}(z))} \leq C\Lambda \rho(x)^q.$$ We claim that $$\label{eqn:SP}
{\left\VertS_x(u-P_{\tilde{x}}(\cdot-\tilde{x}))\right\Vert}_{C^{k,\alpha}(\tilde{B}_{1/8}(z))} \leq C\Lambda \rho(x)^q.$$ (H3) follows from because $y$ is any point in $\tilde{B}_{1/2}(x)$, $z=\Psi_x(y)$ and $$\tilde{B}_{1/2} \subset \bigcup_{y\in \tilde{B}_{1/2(x)}} \tilde{B}_{1/8}(z).$$ By comparing and , the proof of the above claim reduces to $$\label{eqn:down1}
{\left\VertS_x(P_{\tilde{y}}(\cdot-\tilde{y})-P_{\tilde{x}}(\cdot-\tilde{x}))\right\Vert}_{C^{k,\alpha}(\tilde{B}_{1/8}(z))} \leq C\Lambda \rho(x)^q.$$ Since $S_x(P_{\tilde{y}}(\cdot-\tilde{y})-P_{\tilde{x}}(\cdot-\tilde{x}))$ is a $\mathcal T$-polynomial with degree smaller than $q$, (P3) implies that follows from $$\label{eqn:down2}
{\left\VertS_x(P_{\tilde{y}}(\cdot-\tilde{y})-P_{\tilde{x}}(\cdot-\tilde{x}))\right\Vert}_{C^{0}(\tilde{B}_{1/8}(z))} \leq C\Lambda \rho(x)^q.$$ Recall that $z=\Psi_x(y)$, hence $\Psi_x$ maps $\tilde{B}_{1/8}(z)$ to $\tilde{B}_{1/8}(y)\subset \tilde{B}(x)$. (H2) implies respectively $${\left\Vertu-P_{\tilde{y}(\cdot-\tilde{y})}\right\Vert}_{C^0(\tilde{B}_{1/8}(y))} \leq C\Lambda \rho(y)^q \leq C\Lambda \rho(x)^q$$ and $${\left\Vertu-P_{\tilde{x}(\cdot-\tilde{x})}\right\Vert}_{C^0(\tilde{B}(x))}\leq C\Lambda \rho(x)^q.$$ Then follows from a combination of the above two inequalities and the fact that $\tilde{B}_{1/8}(y)\subset \tilde{B}(x)$.
Comparison with Donaldson’s $C^{2,\alpha}_\beta$ space {#subsec:compare}
------------------------------------------------------
Donaldson’s definition requires both $0<\beta<1$ and $\alpha< \min{\left\{\frac{1}{\beta}-1,1\right\}}$, which we assume in this subsection only. It is the purpose of this subsection to show that the space $\mathcal U^{2+\alpha}$ is equivalent to $C^{2,\alpha}_\beta$ when the above restriction to $\beta$ and $\alpha$ applies.
In this case, the only monic $\mathcal T$-monomials whose order is smaller than $2+\alpha$ are $$\label{eqn:monic}
1, \rho^{\frac{1}{\beta}}\cos \theta, \rho^{\frac{1}{\beta}}\sin \theta,\rho^2, \xi, \xi^2.$$
First, let’s recall the definition of $C^{2,\alpha}_\beta$ space defined by Donaldson. It is defined to be the set of functions satisfying
1. $u$ is in $C^\alpha(\hat{B}_1)$;
2. $\partial_\rho u$, $\frac{1}{\rho} \partial_\theta u$ and $\partial_\xi u$ are in $C^\alpha(\hat{B}_1)$;
3. $\partial_\xi^2 u$, $\partial_\xi \partial_\rho u$, $\frac{1}{\rho}\partial_\xi \partial_\theta u$ and $\tilde{\triangle}_\beta u$ are in $C^{\alpha}(\hat{B}_1)$, where $\tilde{\triangle}_\beta= \partial_\rho^2 + \frac{1}{\rho}\partial_\rho + \frac{1}{\beta^2 \rho^2} \partial_\theta^2$ is the Laplacian on the cone surface of cone angle $2\pi \beta$.
Here $C^\alpha(\hat{B}_1)$ is the space of Hölder continuous functions with respect to the distance $d$ of $X^\beta$. Moreover, the $C^{2,\alpha}_\beta$ norm is the sum of all $C^\alpha$ norms mentioned above.
We start the comparison with the following lemma,
\[lem:c1\] (i) If $u$ is $C^\alpha(\hat{B}_2)$, then $u$ is in $\mathcal U^{\alpha}(\hat{B}_1)$ with $$\label{eqn:c1}
{\left\Vertu\right\Vert}_{\mathcal U^\alpha(\hat{B}_1)}\leq C {\left\Vertu\right\Vert}_{C^\alpha(\hat{B}_2)};$$ and (ii) if $u$ is in $\mathcal U^\alpha (\hat{B}_2)$, then $u$ is in $C^\alpha(\hat{B}_1)$ with $$\label{eqn:c2}
{\left\Vertu\right\Vert}_{C^\alpha(\hat{B}_1)}\leq C {\left\Vertu\right\Vert}_{\mathcal U^\alpha(\hat{B}_2)}.$$
\(i) We notice that (H1) is trivial and (H2) is just the definition of Hölder continuous if we choose $P_x$ to be the constant $u(x)$. For $x\in \mathcal S_\delta\cap \hat{B}_1$ and any $z\in \tilde{B}_{1/2}$, since $P_{\tilde{x}}$ is the constant $u(\tilde{x})$, we have $$\begin{aligned}
S_x(u-P_{\tilde{x}}(\cdot - \tilde{x}))(z)&=& (u-P_{\tilde{x}}(\cdot-\tilde{x}))(\Psi_x^{-1}(z))\\
&=& u(\Psi_x^{-1}(z)) - u(\tilde{x}).\end{aligned}$$ For any $z\in \tilde{B}_{1/2}$, the distance from $\Psi_x^{-1}(z)$ to $\tilde{x}$ is at most $2\rho(x)$ and hence $$\label{eqn:c11}
{\left\VertS_x(u-P_{\tilde{x}}(\cdot-\tilde{x}))\right\Vert}_{C^0(\tilde{B}_{1/2})}\leq C {\left\Vertu\right\Vert}_{C^\alpha(\hat{B}_2)}\rho(x)^\alpha.$$ For any $z_1,z_2$ in $\tilde{B}_{1/2}$, $$\label{eqn:c12}
\begin{split}
&{\left\vertS_x(u-P_{\tilde{x}}(\cdot-\tilde{x}))(z_1)-S_x(u-P_{\tilde{x}}(\cdot-\tilde{x}))(z_2)\right\vert}\\
=& {\left\vertu(\Psi_x^{-1}(z_1)) - u(\Psi_x^{-1}(z_2))\right\vert} \\
\leq& {\left\Vertu\right\Vert}_{C^\alpha(\hat{B}_2)} \left( \rho(x)d(z_1,z_2)\right)^\alpha.
\end{split}$$ Hence, (H3) and then follow from and .
\(ii) By (H1), it suffices to consider $x_1$ and $x_2$ in $\mathcal S_\delta\cap \hat{B}_1$. Assume that $\rho(x_1)\geq \rho(x_2)$.
[**If $d(x_1,x_2)\geq \frac{c_\beta \rho(x_1)}{2}$:**]{}
Let $\tilde{x}_1$ and $\tilde{x}_2$ be the projections of $x_1$ and $x_2$ to $\mathcal S$ respectively. The triangle inequality and (H2) imply that $$\begin{aligned}
{\left\vertu(x_1)-u(x_2)\right\vert} &\leq& {\left\vertu(x_1)-u(\tilde{x}_1)\right\vert} + {\left\vertu(x_2)-u(\tilde{x}_2)\right\vert} + {\left\vertu(\tilde{x}_1)- u(\tilde{x}_2)\right\vert} \\
&\leq& {\left\Vertu\right\Vert}_{\mathcal U^\alpha (\hat{B}_2)} \left( \rho(x_1)^\alpha + \rho(x_2)^\alpha + d(\tilde{x}_1,\tilde{x}_2)^\alpha \right) \\
&\leq & C {\left\Vertu\right\Vert}_{\mathcal U^\alpha (\hat{B}_2)} d(x_1,x_2)^\alpha.\end{aligned}$$
[**If $d(x_1,x_2)< \frac{c_\beta \rho(x_1)}{2}$:**]{} denote $\Psi_{x_1}(x_2)$ by $z$, which is a point in $\tilde{B}_{1/2}$.
$$\begin{aligned}
{\left\vertu(x_1)-u(x_2)\right\vert} &=& {\left\vertS_{x_1}(u)(\tilde{x}_0) - S_{x_1}(u)(z)\right\vert} \\
&\leq& {\left\vertS_{x_1}(u-P_{\tilde{x}_1}(\cdot -\tilde{x}_1))(\tilde{x}_0)-S_{x_1}(u-P_{\tilde{x}_1}(\cdot -\tilde{x}_1))(z) \right\vert} \\
&\leq & C {\left\Vertu\right\Vert}_{\mathcal U^\alpha(\hat{B}_2)} \rho(x_1)^\alpha d(\tilde{x}_0,z)^\alpha \\
&\leq& C {\left\Vertu\right\Vert}_{\mathcal U^\alpha(\hat{B}_2)} d(x_1,x_2)^\alpha.\end{aligned}$$
Here in the second line above, we used (H3).
To conclude this section, we prove
\[lem:c2\] If $u$ is in $\mathcal U^{2+\alpha}(\hat{B}_2)$, then $u$ is in $C^{2,\alpha}_\beta(\hat{B}_1)$ and $${\left\Vertu\right\Vert}_{C^{2,\alpha}_\beta(\hat{B}_1)} \leq C {\left\Vertu\right\Vert}_{\mathcal U^{2+\alpha}(\hat{B}_2)}.$$
It is natural to ask about the other direction of Lemma \[lem:c2\]. While it is possible to give a direct proof, we omit it because it follows from Lemma \[lem:c1\] and the Schauder estimate (Theorem \[thm:schauder\]), which is to be proved in the next section. So we conclude that $C^{2,\alpha}_\beta$ and $\mathcal U^{2+\alpha}$ are the same (in the sense above).
The proof consists of several steps.
[**Step 1:**]{} By Remark \[rem:smaller\] and Lemma \[lem:c1\], we have $u\in C^\alpha(\hat{B}_1)$.
[**Step 2:**]{} All derivatives listed in (D2) and (D3) are bounded. Since the proofs are the same, we prove the claim for $\partial_\rho u$ only. Thanks to (H1), it suffices to consider $x\in \mathcal S_\delta\cap \hat{B}_1$. Let $\tilde{x}$ be the projection of $x$ onto $\mathcal S$, then for the $\mathcal T$-polynomial $P_{\tilde{x}}$ (with order smaller than $2+\alpha$) in (H2), we have $$\begin{aligned}
\label{eqn:decompose}
\partial_\rho u(x) &=& \partial_\rho \left( u - P_{\tilde{x}}(\cdot-\tilde{x}) \right)(x) + \partial_\rho P_{\tilde{x}} (x-\tilde{x}).
\end{aligned}$$ The second term above is bounded by a multiple of ${\left\Vertu\right\Vert}_{\mathcal U^{2+\alpha}(\hat{B}_2)}$, because of (H2) and the fact that the $\partial_\rho$ of each monic $\mathcal T$-monomial (listed in ) is bounded. The first term is bounded by $$\label{eqn:firstterm}
\frac{1}{\rho} {\left\vert\nabla S_x(u-P_{\tilde{x}}(\cdot-\tilde{x}))\right\vert}(\tilde{x}_0),$$ which is in turn bounded by $C{\left\Vertu\right\Vert}_{\mathcal U^{2+\alpha}(\hat{B}_2)}$ due to (H3).
[**Step 3:**]{} All derivatives of $u$ listed in (D2) and (D3) are in $C^\alpha$.
As before, due to (H1), we may assume that $x_1,x_2\in \mathcal S_\delta\cap \hat{B}_1$ and $\rho(x_1)\geq \rho(x_2)$.
[**If $d(x_1,x_2)< c_\beta\rho(x_1)/2$:**]{} Let $\tilde{x}_1$ be the projection of $x_1$ onto $\mathcal S$ and $z=\Psi_{x_1}(x_2)\in \tilde{B}_{1/2}$. $$\begin{aligned}
&& {\left\vert\partial_\rho u(x_1)- \partial_\rho u(x_2)\right\vert} \\
&\leq & {\left\vert\partial_\rho (u-P_{\tilde{x}_1}(\cdot-\tilde{x}_1)) (x_1)- \partial_\rho (u-P_{\tilde{x}_1}(\cdot-\tilde{x}_1)) (x_2)\right\vert} + {\left\vert\partial_\rho P_{\tilde{x}_1}(x_1-\tilde{x}_1) - \partial_\rho P_{\tilde{x}_1}(x_2-\tilde{x}_1) \right\vert} \\
&\leq & \frac{1}{\rho(x_1)} {\left\vert\nabla S_{x_1}(u-P_{\tilde{x}_1}(\cdot-\tilde{x}_1))(\tilde{x}_0)-\nabla S_{x_1}(u-P_{\tilde{x}_1}(\cdot-\tilde{x}_1))(z) \right\vert} + C{\left\Vertu\right\Vert}_{\mathcal U^{2+\alpha}(\hat{B}_2)} d(x_1,x_2)^\alpha\\
&\leq & \frac{1}{\rho(x_1)}\cdot C{\left\Vertu\right\Vert}_{\mathcal U^{2+\alpha}(\hat{B}_2)} \rho(x_1)^{2+\alpha} d(\tilde{x}_0,z)^\alpha + C{\left\Vertu\right\Vert}_{\mathcal U^{2+\alpha}(\hat{B}_2)} d(x_1,x_2)^\alpha\\
&\leq & C {\left\Vertu\right\Vert}_{\mathcal U^{2+\alpha}(\hat{B}_2)}d(x_1,x_2)^\alpha.
\end{aligned}$$ For the estimate of the second term in the second line above, we check that for each monic monomial $f$ in , there holds $${\left\vert\partial_\rho f(x_1)- \partial_\rho f(x_2)\right\vert}\leq C d(x_1,x_2)^\alpha.$$ For the first term in the third line above, we used (H3).
Again, the same argument works for all other derivatives in (D2) and (D3) in the case $d(x_1,x_2)<c_\beta\frac{\rho(x_1)}{2}$.
[**If $d(x_1,x_2)\geq c_\beta\rho(x_1)/2$:**]{} We study $\partial_\rho u$, $\frac{1}{\rho} \partial_\theta u$ and $\tilde{\triangle}_\beta u$ first. In this case, (H3) implies that $$\label{eqn:ux1}
{\left\vert\partial_\rho (u(x_1)-P_{\tilde{x}_1}(x_1-\tilde{x}_1))\right\vert}\leq \frac{1}{\rho(x_1)}{\left\vert\nabla S_{x_1}(u-P_{\tilde{x}_1}(\cdot-\tilde{x}_1))\right\vert}(\tilde{x}_0) \leq C{\left\Vertu\right\Vert}_{\mathcal U^{2+\alpha}(\hat{B}_2)} \rho(x_1)^{1+\alpha}$$ and $$\label{eqn:ux2}
{\left\vert\partial_\rho (u(x_2)-P_{\tilde{x}_2}(x_2-\tilde{x}_2))\right\vert}\leq \frac{1}{\rho(x_2)}{\left\vert\nabla S_{x_2}(u-P_{\tilde{x}_2}(\cdot-\tilde{x}_2))\right\vert}(\tilde{x}_0) \leq C{\left\Vertu\right\Vert}_{\mathcal U^{2+\alpha}(\hat{B}_2)} \rho(x_2)^{1+\alpha}.$$ By checking the monic monomials in one by one, we find that for either $x=x_1$ or $x=x_2$, $$\label{eqn:ux12}
{\left\vert\partial_\rho P_{\tilde{x}}(x-\tilde{x})\right\vert}\leq C {\left\Vertu\right\Vert}_{\mathcal U^{2+\alpha}(\hat{B}_2)} \rho(x)^\alpha.$$ By , and and the fact that $\rho(x_2)\leq \rho(x_1)\leq \frac{2}{c_\beta}d(x_1,x_2)$, we get $$\begin{aligned}
{\left\vert\partial_\rho u(x_1) - \partial_\rho u(x_2)\right\vert} \leq C{\left\Vertu\right\Vert}_{\mathcal U^{2+\alpha}(\hat{B}_2)} d(x_1,x_2)^\alpha.\end{aligned}$$ The same argument works for $\frac{1}{\rho}\partial_\theta u$ and $\tilde{\triangle}_\beta u$.
Now, let’s turn to the proof for $\partial_\xi u$, $\partial_\rho\partial_\xi u$, $\frac{1}{\rho}\partial_\theta \partial_\xi u$ and $\partial_\xi^2 u$. We prove for $\partial_\xi^2 u$ only and the same proof works for the other three functions. Similar to , we have $$\partial_\xi^2 u (x) = \partial_\xi^2 (u-P_{\tilde{x}}(\cdot-\tilde{x}))(x) + \partial_\xi^2 P_{\tilde{x}}(x-\tilde{x}).
\label{eqn:decompose2}$$ By , we have $$\begin{aligned}
&& {\left\vert\partial_\xi^2 u(x_1) - \partial_\xi^2 u(x_2)\right\vert} \\
&\leq& {\left\vert\partial_\xi^2 (u-P_{\tilde{x}_1}(\cdot-\tilde{x}_1))(x_1)\right\vert} + {\left\vert\partial_\xi^2(u-P_{\tilde{x}_2}(\cdot-\tilde{x}_2))(x_2)\right\vert} + {\left\vert\partial_\xi^2 P_{\tilde{x}_1}(x_1-\tilde{x}_1)- \partial_\xi^2 P_{\tilde{x}_2}(x_2-\tilde{x}_2)\right\vert}.\end{aligned}$$ Using (H3) and the inequality $\rho(x_2)\leq \rho(x_1)\leq \frac{2}{c_\beta} d(x_1,x_2)$ as before, we can bound the first two terms by $C {\left\Vertu\right\Vert}_{\mathcal U^{2+\alpha}(\hat{B}_2)} d(x_1,x_2)^\alpha$. It remains to estimate the third term above.
For each $f$ in , the mixed second derivatives $\partial_\rho\partial_\xi$ and $\frac{1}{\rho}\partial_\theta \partial_\xi$ vanish. The proofs are done in these two cases.
Again by checking the monic monomials in , we notice that $$\partial_\xi^2 P_{\tilde{x}_i}(x_i-\tilde{x}_i)= \partial_\xi^2 P_{\tilde{x}_i}(0), \qquad i=1,2.$$ The proof will be done, if we can show $$\label{eqn:final}
{\left\vert\partial_\xi^2 P_{\tilde{x}_1}(0)- \partial_\xi^2 P_{\tilde{x}_2}(0)\right\vert}\leq C {\left\vert\tilde{x}_1-\tilde{x}_2\right\vert}^\alpha.$$ To see this, we define a function $\tilde{u}$ on $\mathcal S\cap \hat{B}_1$ by $$\tilde{u}(\tilde{x})= u(\tilde{x}) = P_{\tilde{x}}(0).$$ By (H2) (restricted to $\mathcal S$ direction), $\tilde{u}$ satisfies the assumptions in Definition \[defn:ube\] in Section \[sec:space\]. Proposition \[prop:rn\] then implies that $\tilde{u}$ is $C^{2,\alpha}$ (on ${\mathbb R}^{n-2}$) in the usual sense and that $$(\partial_\xi^2 P_{\tilde{x}})(0) = (\partial_\xi^2 \tilde{u})(\tilde{x}),$$ from which follows.
Schauder estimates for $X_\beta$ {#sec:schauder}
================================
We prove here an estimate that by our perspective should be called the Schauder estimate on $X_\beta$. Recall that $\mathcal D$ is the set of degrees of all $\mathcal T$-monomials and we have defined generalized Hölder spaces $\mathcal U^q$ only for $q\notin \mathcal D$.
\[thm:schauder\] Assume that $q>0$ and $q,q+2\notin \mathcal D$. Suppose $f\in \mathcal U^{q}(\hat{B}_2)$ and $u$ is a bounded weak solution to $$\triangle_\beta u =f.$$ Then $u$ is in ${\mathcal U}^{q+2}(\hat{B}_1)$ and $${\left\Vertu\right\Vert}_{{\mathcal U}^{q+2}(\hat{B}_1)}\leq C\left( {\left\Vertu\right\Vert}_{C^0(\hat{B}_2)} + {\left\Vertf\right\Vert}_{{\mathcal U}^{q}(\hat{B}_2)} \right).$$
This is the main result of this paper. The key step in its proof is an analog of Lemma \[lem:key\].
\[lem:hkey\] For $q>0$ and $q+2 \notin D$, if $f$ is $O(d^q)$ on $\hat{B}_r$, then there exists some $u$ that is $O(d^{q+2})$ and satisfies $$\triangle_\beta u =f$$ and $$[u]_{{O}_{q+2},\hat{B}_r}\leq C [f]_{{O}_q,\hat{B}_r}.$$
The rest of this section consists of two parts. In the first part, we prove Lemma \[lem:hkey\], following the proof of Lemma \[lem:key\] and utilizing many facts about harmonic functions on $X_\beta$ proved in Section \[sec:harmonic\]. In the second part, we complete the proof of Theorem \[thm:schauder\].
Proof of Lemma \[lem:hkey\]
---------------------------
Without loss of generality, we assume $r=1$.
Setting $$\hat{A}_l:=\hat{B}_{2^{-l}}\setminus \hat{B}_{2^{-l-1}}, \quad \text{for} \quad l=0,1,2,\cdots$$ and $$f_l= f\cdot \chi_{\hat{A}_l},$$ we have $${\left\vertf_l\right\vert}(x)\leq \Lambda_f d(x)^q \chi_{\hat{A}_l}\leq \Lambda_f 2^{-lq}\qquad \text{on} \quad X_\beta,$$ where as before $\Lambda_f=[f]_{\hat{O}_q,\hat{B}_1}$.
Let $w_l$ be the solution to the Poisson equation $\triangle_\beta w_l=f_l$ on $X_\beta$ in Lemma \[lem:poisson\], which satisfies that $$\sup_{X_\beta} {\left\vertw_l\right\vert} \leq C\Lambda_f 2^{-l(q+2)}.
\label{eqn:hwl}$$
Again, $w_l$ is harmonic in $\hat{B}_{2^{-l-1}}$. Let $P_l$ be the $X_\beta$-polynomial in Proposition \[prop:stronghf\] applied to $w_l$ in $\hat{B}_{2^{-l-1}}$ with $q+2$ replacing $q$. Our plan is to set $$u_l(x)=w_l(x)-P_l(x)$$ and to show that the series $$u(x)=\sum_{l=0}^\infty u_l(x)$$ converges and gives the solution needed in Lemma \[lem:hkey\]. Notice that by Proposition \[prop:truncation\], $P_l$ is harmonic on the entire $X_\beta$ and $u_l$ is harmonic outside $\hat{A}_l$.
For that purpose, we need an analog of Lemma \[lem:ul\].
\[lem:hul\] There exists a constant $C_q$ depending on $n,\beta$ and $q$ such that
\(i) on $\hat{B}_{2^{-l-1}}$, $${\left\vertu_l(x)\right\vert} \leq C_q\Lambda_f 2^{( (q+2)^*-(q+2))l}d(x)^{ (q+2)^*};$$ Here $(q+2)^*$ is the smallest number in $\mathcal D$ that is larger than $q+2$.
\(ii) on $\hat{A}_l$, $${\left\vertu_l(x)\right\vert} \leq C_q \Lambda_f 2^{-(q+2)l}$$ or equivalently $${\left\vertu_l(x)\right\vert} \leq C_q \Lambda_f d(x)^{q+2};$$
\(iii) on $\hat{B}_1\setminus \hat{B}_{2^{-l}}$, $${\left\vertu_l(x)\right\vert} \leq C_q \Lambda_f \left[ 2^{-l(q+2)} + \sum_{2j+\frac{k}{\beta}+ {\left\vert\sigma\right\vert}<q+2} 2^{-l(q+2-(2j+\frac{k}{\beta}+{\left\vert\sigma\right\vert}) )} d(x)^{2j+\frac{k}{\beta}+{\left\vert\sigma\right\vert}} \right].$$
The following are immediate corollaries of Proposition \[prop:stronghf\].
(a) Recall that $P_l(x)$ is the $X_\beta$-polynomial in Proposition \[prop:stronghf\] applied to $w_l$, namely, $$P_l(x) = \sum_{2j+\frac{k}{\beta}+{\left\vert\sigma\right\vert}<q+2} \rho^{2j+\frac{k}{\beta}} \left( a_{j,k,\sigma}^l \cos k\theta + b_{j,k,\sigma}^l \sin k\theta \right) \xi^\sigma,$$ then $${\left\verta_{j,k,\sigma}^l\right\vert} + {\left\vertb_{j,k,\sigma}^l\right\vert} \leq C\Lambda_f 2^{-l(q+2-(2j+\frac{k}{\beta}+{\left\vert\sigma\right\vert}))};$$
(b) $u_l(x)$ is ${O}(d^{(q+2)^*})$ in $\hat{B}_{2^{-l-1}}$ and $$[u_l(x)]_{{O}_{(q+2)^{*}},\hat{B}_{2^{-l-1}}}\leq C\Lambda_f 2^{-l(q+2-(q+2)^{*})}.$$
Given the $C^0$ bound in , (a) follows from a scaled version of Proposition \[prop:stronghf\] applied to the harmonic function $w_l$ on $\hat{B}_{2^{-l-1}}$. For (b), we notice that $u_l$ is exactly the $O(d^q)$ term in if $q$ in Proposition \[prop:stronghf\] is taken to be $(q+2)^*$ here. The factor $2^{-l(q+2)}$ comes from the $C^0$ bound of $w_l$ and $2^{(l(q+2)^*)}$ is due to scaling.
With (a) and (b) above, we notice that (1) is the same as (b); (2) is the same as ; (3) is an easy combination of (a) and .
With this lemma, a similar computation as in the ${\mathbb R}^n$ case verifies that the series $\sum_{l=1}^\infty u_l$ converges and therefore gives the solution we need in Lemma \[lem:hkey\].
Proof of Theorem \[thm:schauder\]
---------------------------------
By the usual Schauder estimate, to prove an estimate of $u$ in $\mathcal U^{q+2}(\hat{B}_1)$, we do not need to worry about (H1). For (H2), let $x\in \mathcal S\cap \hat{B}_1$, there is a $\mathcal T$-polynomial $P_f$ (order less than $q$) such that $$f(y)=P_f(y-x) + O(d(x,y)^q),\quad \forall y\in \hat{B}_\delta(x).$$ By our choice of $\mathcal T$ (see (P4) in Lemma \[lem:PPP\]), there is some $\mathcal T$-polynomial $\tilde{P}_u$ (order less than $q+2$) such that $$\label{eqn:puf}
\triangle_\beta \tilde{P}_u =P_f.$$ Notice that $\tilde{P}_u$ is not uniquely determined by $P_f$, since we may add any harmonic $\mathcal T$-polynomial to it.
If we denote the $O(d(x,y)^q)$ term by $e_f(y)$, Lemma \[lem:hkey\] implies the existence of $e_u(y)$ defined on $\hat{B}_\delta(x)$ such that $$\triangle_\beta e_u(y) = e_f(y)$$ and $$[e_u]_{O_{q+2},\hat{B}_\delta(x)} \leq C [e_f]_{O_q,\hat{B}_{\delta}(x)}.$$ Therefore, $u-\tilde{P}_u(\cdot-x)-e_u$ is a harmonic function $v$ bounded by $C({\left\Vertu\right\Vert}_{C^0(\hat{B}_2)}+ {\left\Vertf\right\Vert}_{\mathcal U^q(\hat{B}_2)})$. Proposition \[prop:stronghf\] implies the existence of some $X_\beta$-polynomial $h_u$ of order less than $q+2$ such that $$v(y)=h_u(y-x)+O(d(x,y)^{q+2}).$$ Then (H2) is verified by setting $P_{\tilde{x}}=\tilde{P}_u + h_u$.
For (H3), let $x\in \mathcal S_\delta$ and $\tilde{x}$ be its projection to $\mathcal S$. On one hand, (H2), which is proved above, implies the existence of $P_{\tilde{x}}$ such that $$\label{eqn:qq1}
{\left\VertS_x(u-P_{\tilde{x}}(\cdot-\tilde{x}))\right\Vert}_{C^0(\tilde{B})}\leq C \rho(x)^{q+2}.$$ On the other hand, by and the definition of $P_{\tilde{x}}$, $$\triangle_\beta (S_x(u-P_{\tilde{x}}(\cdot-\tilde{x}))) = \rho(x)^{2} S_x (\triangle_\beta (u-P_{\tilde{x}}(\cdot-\tilde{x}))) = \rho(x)^{2} S_x (f-P_f(\cdot-\tilde{x})).$$ By (H3) for $f$ and the above equation, $$\label{eqn:qq2}
{\left\Vert \triangle_\beta S_x(u-P_{\tilde{x}}(\cdot-\tilde{x}))\right\Vert}_{C^{k,\alpha}(\tilde{B}_{1/2})}\leq C \rho(x)^{q+2}.$$ The usual interior Schauder estimate on $\tilde{B}_{1/2}$, together with and , implies that $${\left\VertS_x(u-P_{\tilde{x}}(\cdot-\tilde{x}))\right\Vert}_{C^{k+2,\alpha}(\tilde{B}_{3/8})}\leq C \rho(x)^{q+2}.$$ Now the proof of Theorem \[thm:schauder\] is concluded by Remark \[rem:final\].
Proof of Lemma \[lem:rhogood\]
==============================
The proof of this lemma consists of a bootstrapping argument of a family of Poisson equations on cone surface, which was used in [@yin2016analysis]. More precisely, by Lemma \[lem:xigood\], $\partial_\sigma u$ is harmonic, which implies that $$\tag{$E_\sigma$} \tilde{\triangle}_\beta (\partial_\xi^\sigma u)=- \triangle_\xi (\partial_\xi^\sigma u).$$ Here $\tilde{\triangle}_\beta$ is the Laplacian of the two dimensional cone surface $X_\beta^2$, parametrized by $(\rho,\theta)$ and equipped with the cone metric $$g_\beta^2 = d\rho^2 + \beta^2 \rho^2 d\theta^2.$$ Also by Lemma \[lem:xigood\], the right hand side of ($E_\sigma$) is bounded by some constant depending on $\sigma$. It then follows that $\partial_\xi^\sigma u$ (with $\xi$ fixed) is Hölder continuous function with respect to the distance of $X_\beta^2$. To see this, recall that in terms of the $(u,v)$ coordinates, where $u=\rho\cos \theta$ and $v=\rho\sin \theta$, $\tilde{\triangle}_\beta$ is a uniformly elliptic operator with bounded coefficients and hence the Hölder continuity follows from De Giorgi’s iteration. (see [@yin2016analysis] for detail). This is the starting point of the bootstrapping.
A function $w$ is said to have $\mathcal T_h$-expansion up to order $q>0$ if $$w(\rho,\theta)= \sum_{2j+\frac{k}{\beta}<q} \rho^{2j+\frac{k}{\beta}}(A_{j,k} \cos k\theta + B_{j,k} \sin k\theta) + O(\rho^q)$$ for $\rho<1/2$. The expansion is said to be bounded by $\Lambda$ if the coefficients $A_{j,k}$, $B_{j,k}$ and the constant in the definition of $O(\rho^q)$ are bounded by $\Lambda$.
\[rem:diff\] Note that the expansion above is different from the one used in [@yin2016analysis], where we also included $$\rho^{2j+\frac{k}{\beta}}\cos m\theta \quad \text{and} \quad \rho^{2j+\frac{k}{\beta}}\sin m\theta, \qquad \text{for} \quad \frac{k-m}{2}\in \mathbb N\cup {\left\{0\right\}}.$$ This is because in [@yin2016analysis], we dealt with nonlinear equations, while here we are essentially working with linear equations. The product of harmonic functions is not necessarily harmonic and hence there is no need to require the formal series to be multiplicatively closed.
The Hölder continuity of $\partial_\xi^\sigma u$ means that $\partial_\xi^\sigma u(\xi)$ has an expansion up to some order $q\in (0,1)$ uniformly (independent of $\xi$) bounded by $\Lambda=\Lambda(q,\sigma)$. The proof of Lemma \[lem:rhogood\] is now reduced to the following claim. Notice that we prove and simultaneously.
[**Claim.**]{} Let $u$ be a bounded solution to $$\tilde{\triangle}_\beta u =f$$ on the unit ball centered at the unique singular point of $X^2_\beta$. If $f$ has a $\mathcal T_h$-expansion up to order $q$ bounded by $\Lambda$ for $q\ne 2j+\frac{k}{\beta}$ for any $k,j\in \mathbb N\cup {\left\{0\right\}}$, then $u$ has a $\mathcal T_h$-expansion up to order $q+2$ bounded by a multiple of $\Lambda$ and $C^0$ norm of $u$ on the ball.
This is nothing but Lemma 6.9 in [@yin2016analysis]. The difference pointed out in Remark \[rem:diff\] does not cause a problem because for the proof, we only require that for each $\mathcal T_h$-polynomial of order $q'$, there exists a $\mathcal T_h$-polynomial of order $q'+2$ that is mapped to the given one by $\tilde{\triangle}_\beta$.
As a final remark, we notice that Lemma 6.9 in [@yin2016analysis] relies on Lemma 6.10 there, which is the precursor of Lemma \[lem:key\] and Lemma \[lem:hkey\] in this paper. We find the proof of Lemma 6.10 in [@yin2016analysis], which depends on Fourier series, hard to generalize to higher dimensions. The new proof here of course can be used to prove Lemma 6.10.
[^1]: Think about $\beta=1/2$. Annoying discussion.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Segmentation is a key step in analyzing and processing medical images. Due to the low fault tolerance in medical imaging, manual segmentation remains the de facto standard in this domain. Besides, efforts to automate the segmentation process often rely on large amounts of manually labeled data. While existing software supporting manual segmentation is rich in features and delivers accurate results, the necessary time to set it up and get comfortable using it can pose a hurdle for the collection of large datasets.
This work introduces a client/server based online environment, referred to as Studierfenster, that can be used to perform manual segmentations directly in a web browser. The aim of providing this functionality in the form of a web application is to ease the collection of ground truth segmentation datasets. Providing a tool that is quickly accessible and usable on a broad range of devices, offers the potential to accelerate this process. The manual segmentation workflow of Studierfenster consists of dragging and dropping the input file into the browser window and slice-by-slice outlining the object under consideration. The final segmentation can then be exported as a file storing its contours and as a binary segmentation mask.
In order to evaluate the usability of Studierfenster, a user study was performed. The user study resulted in a mean of 6.3 out of 7.0 possible points given by users, when asked about their overall impression of the tool. The evaluation also provides insights into the results achievable with the tool in practice, by presenting two ground truth segmentations performed by physicians.
author:
- Paper09
- 'Daniel Wild[^1]'
- 'Maximilian Weber[^2]'
- 'Jan Egger[^3]'
bibliography:
- 'cescg\_09.bib'
title: 'A Client/Server Based Online Environment for Manual Segmentation of Medical Images'
---
\[2005/11/29 v0.1.4 CESCG proceedings sample file\]
Introduction
============
Image segmentation is an important step in the analysis of medical images. It helps to study the anatomical structure of body parts and is useful for treatment planning and monitoring of diseases over time. Segmentation can either be done manually by outlining the regions of an image by hand or with the help of semi-automatic or automatic algorithms. A lot of research focuses on semi-automatic and automatic algorithms, as manual segmentation can be quite tedious and time-consuming work. But even for the development of automated segmentation systems, some ground truth has to be found, telling the system what exactly constitutes a correct segmentation. As the avoidance of errors is of especially high importance in the medical domain, this ground truth is typically delivered by physicians, as they have the necessary expert knowledge to reliably decide what should be part of a segmentation and what not. To form the ground truth physicians can make use of manual or semi-automated segmentation techniques.
Software supporting such techniques usually requires a local installation on the computer of the user and some training time to get comfortable with the extensive features available. This necessary preparation time can be a hurdle for the collection of a big amount of data, which is typically needed to develop a robust automated segmentation algorithm. Web-based tools have the advantage that they require no installation time and can be adapted quickly to the needs of a particular data collection task, without having to distribute updates to each user individually.
This work describes the development of a client/server based online environment for manual segmentation, referred to as Studierfenster ([www.studierfenster.at](www.studierfenster.at) or <http://studierfenster.tugraz.at>) in the rest of the work, that can be used directly in a web browser. Thus there is no need to install any software, as the tool is readily available in the web browser of the user. Another advantage is the platform independence of Studierfenster, which keeps its potential user base as big as possible.
Following this first outline of the motivation behind the development of Studierfenster, briefly discusses related work. Sections \[architecture\] and \[workflow\] then go into more detail on the implementation of Studierfenster. The expert evaluation described in and the user study described in constitute the evaluation of the tool. The final gives a short conclusion and suggests areas of improvement and approaches for future work.
Related Work
============
´\[relatedwork\] There is a wide range of offline software tools available that offer sophisticated medical image analysis and processing capabilities, including tools for manual segmentation. Examples include 3D Slicer [@slicerGBM] and MeVisLab [@MeVisLabOpenIGT], [@MeVisLabVive]. The drawback of these tools is that they require a local installation and are thus not readily available on any device.
Examples of actively supported web-based tools useable for medical imaging include the OHIF Viewer (<https://viewer.ohif.org>), Paraview Glance (<https://kitware.github.io/paraview-glance>) and Slice:Drop [@slicedrop], the medical image viewer used as the basis for the development of Studierfenster. While some tools, like the OHIF Viewer, provide simple analysis tools like a ruler for length measurements or annotation tools, others, like Paraview Glance and Slice:Drop, provide only visualization capabilities. What is missing is a tool that provides more sophisticated analysis tools or even processing routines directly in the web-browser.
Software Architecture {#architecture}
=====================
This section gives a high-level overview of the architectural model behind the segmentation tool developed in this work.
Slice:Drop, which is the medical image viewer this tool is built upon, uses a purely client-oriented approach. This means that all the necessary computations for visualization are carried out on the client using JavaScript and no files have to be uploaded to the host server. The most recent version of Slice:Drop at the time of writing is limited in its functionality, however. While it does provide a volume rendering view and three 2D views in the axial, sagittal and coronal direction, it does not provide any image processing capabilities. For the presented segmentation extension of Slice:Drop, an additional server backend was developed. An exemplary use case of the server backend is the conversion of the segmentation contours to a filled Nearly Raw Raster Data (NRRD) volume mask. The architectural scheme behind this conversion can be seen in . In the following, this conversion process will be used to describe the architecture behind the segmentation tool in more detail.
![An architectural model of the mask generation process.[]{data-label="arch"}](figures/architecture.pdf)
Client
------
The process starts on the client side with the user opening up a web browser and navigating to the domain that hosts the HTML, CSS and JavaScript files of the segmentation tool. These files are then transferred to the user so that the segmentation tool can be used directly in the web browser.
Now the user is able to load a volume file in the viewer and carry out a segmentation of some area of interest. The resulting segmentation contours are all stored in JavaScript variables for now. If the user decides to save the segmentation contours directly as a file in the Visualization Toolkit (VTK) [@vtkbook] format and without conversion to an NRRD volume mask, no additional communication with the server is necessary. The process of carrying out a segmentation and saving only its contours as a VTK file thus makes no use of the server backend.
Server
------
The primary component of the server backend is the Python Flask server instance, as it is responsible for handling all the communication with the client. The first moment this Python Flask server comes into play is during the process of converting the segmentation contours to an NRRD volume mask. If the user decides to do so, the segmentation contours are internally written to a VTK file on the client. This file is then immediately uploaded to the server, however. To be able to construct an NRRD volume mask with the same size, spacing, and orientation as the original file, the corresponding information of the original file has to be uploaded additionally. In order to not having to upload the entire original file, this metadata is extracted from the file on the client and then uploaded to the server as a small separate JSON file.
Once the segmentation and the metadata file are both uploaded, the client asks the server to invoke the C`++` volume mask generation program, which is explained in more detail in . During the conversion process, the C`++` program continuously reports the progress to the Python Flask server, which then hands over the progress information to the client.
As soon as the conversion is finished, the C`++` program writes the NRRD volume to the file system of the server, from where it is handed over by the Python Flask server to the client for downloading.
Due to the modular design of the server backend, it is also possible to integrate other processing capabilities into it. An exemplary additional server module, which was integrated to prove said modularity, is a volume file converter. The converter takes volume files as input and produces a volume file in the NRRD file format. Additionally, the volume files are transformed into a world coordinate system with a Right Anterior Superior (RAS) basis. An explanation of this coordinate system is given in .
Segmentation Workflow {#workflow}
=====================
![This figure shows the workflow to obtain both a VTK file of the segmentation contours and a filled NRRD volume mask.[]{data-label="flowchart"}](figures/workflow.pdf)
The following section will explore the segmentation workflow of the tool developed in this work in more technical detail. The flowchart in illustrates the necessary steps to obtain both a mesh file containing the segmentation contours and a volume file containing a filled segmentation mask. The process can be grouped into three main parts: The manual outlining of the object to be segmented in the web interface, the conversion of these outlines to segmentation contours stored in a VTK file and the generation of a filled segmentation mask on the server backend.
Manual Segmentation in Web Interface
------------------------------------
![Contours of a glioblastoma segmentation visualized in Studierfenster.[]{data-label="interop"}](figures/testseg_sf.PNG)
The interface used for drawing the manual segmentation contours is developed as an extension to the web interface of Slice:Drop, the medical image viewer Studierfenster is based upon.
Slice:Drop’s core feature is loading and viewing volume files directly in the web browser. So for the first step in the segmentation workflow, which is loading the NRRD volume file of interest into the segmentation tool, the standard loading method of Slice:Drop can be used. Selecting a file to load can either be done by a simple drag&drop interaction or by using the file finder of the web browser. Once a file is selected Slice:Drop calls the NRRD file parser of the X Toolkit (XTK) [@slicedrop] to perform the loading of the file. Some changes had to be made to this file parser to allow Slice:Drop to extract the space and orientation information from the header of the NRRD file. This information is needed later on during the mask generation process in order to reconstruct a mask with the same origin, spacing, and size as the original volume file.
As soon as the parsing is complete, the volume is displayed in the predefined views of Slice:Drop. Slice:Drop makes use of the renderer classes of XTK to perform the necessary reslicing and volume ray casting for 2D and 3D visualization respectively. In the default setting the 3D view is enlarged and the three 2D views are miniaturized on the right side. As the segmentation extension only supports segmenting in the 2D axial view, the user now has to enlarge this view by clicking on it.
On the axial view, one can now navigate through the different slices of the volume file by either using the mouse wheel or the slider on top of the view. Once the first slice featuring an element of interest is in view, the segmentation mode can be started by selecting “Start Segmentation" in the segmentation menu on the left side. In order to display the contours, a second HTML canvas with a transparent background was superimposed on the canvas displaying the current slice. As the HTML canvas itself does not provide any drawing capabilities, the standard drawing methods of its `getContext(2D)` object are used to display the points of a segmentation and its connecting lines on the drawing canvas. Entering the segmentation mode brings the drawing canvas into the foreground.
The user can now start to draw a segmentation contour by holding the left mouse button pressed while moving along the border of the object of interest. A new point is added to the segmentation contour once the current distance of the mouse cursor to the previous point surpasses a given threshold. The segmentation contour can be finished by moving the mouse cursor close to the first point of the contour and releasing the left mouse button. To contrast the newly segmented region from the rest of the slice, the segmented region is filled with a light red color.
If the user is not completely satisfied with the accuracy of the contour, individual points can be deleted and reset to a new location. This can be done by clicking on the erroneous points and subsequently clicking on the correct location. Should the user wish to start over with a segmentation contour completely, the whole contour can be deleted by selecting “Delete Slice" and clicking on a point lying on the contour to be deleted.
As soon as every region of interest on one slice is segmented with satisfying accuracy, the user can navigate to the next slice using the mouse wheel and start over with the process of drawing and refining the segmentation contours. Once the segmentation on all individual slices containing the object of interest is finished, the user can download the combined contours directly as a VTK file or convert them to a filled volume mask in the NRRD format. The technical details behind those two options are explained in the following two sections.
Download of Segmentation Contours {#contours}
---------------------------------
The quickest way to export the contours of a segmentation is to download them as a VTK file. As explained in , this can be done without using the server backend. The contours of the segmentation are written to a VTK file directly on the client. Before starting to write contours to the VTK file, however, one has to take care to align the coordinates of their points with the coordinate system of the segmented volume file.
During the segmentation process, a point is internally stored in index space coordinates. Described informally, in index space the coordinates of a point directly refer to indices into the volume file. Indexing into the volume file with the coordinates of a point would thus yield the voxel at which the point is located. The volume file itself, however, is stored in a coordinate system called world space. This coordinate system describes the position and orientation of a patient relative to the medical scanner used to acquire the image volume file. It is defined by the origin, which is the position of the first voxel of the volume file in millimeters and its basis vectors. A commonly used basis in neuroimaging, which is also used by Slice:Drop as the reference frame to display volume files in, is the RAS basis. Its axes are related to the patient being scanned, with the R axis increasing from left to right, the A axis increasing from posterior to anterior and the S axis increasing from inferior to superior [@coordinates].
In order to align the segmentation contours with the volume file, the points of the contours now have to be positioned in the world space as well. This is done by multiplying each point with the `IJKToRAS` Matrix provided by XTK, which describes an affine transformation from the index coordinate system to the RAS coordinate system.
Once the transformation of a point is completed, its coordinates are written to the first part of the VTK file. While the first part of the file comprises the coordinates of all points in the segmentation, it gives no information on which contour they belong to yet. The connectivity information is defined by the second part of the VTK file. Each line in this part describes one closed contour, by listing the indices of the points in the first part of the file in the order they are connected with one another [@vtkguide].
shows exemplary segmentation contours which were exported as a VTK file and visualized in Studierfenster.
Generation of Segmentation Mask {#cplusplus}
-------------------------------
The second way to export the segmentation is as a filled volume mask in the NRRD format. Such a mask has the same dimension and orientation as the original volume file which was segmented and can thus be superimposed on it. In the volume mask, the voxels lying inside the segmentation contours are white and the voxels outside the contour are black. Thus the goal is to convert the segmentation contours to such a filled volume mask. This process is handled by a C`++` program on the server backend, that makes use of both the Insight Toolkit (ITK) and the VTK library. The details of how the client communicates with the server and transmits the necessary files for the conversion are given in . Assuming that the two necessary files for the conversion, namely the VTK file containing the segmentation contours and the JSON file containing the space metadata, are present on the server, the conversion process can be started.
The first step is loading the JSON file containing the space metadata and, using said metadata, constructing an all black volume with the same dimensions, origin, and spacing as the original volume file. For this, the `Image` class of ITK is used.
The next thing to take care of is the loading of the segmentation contours. As those were transmitted to the server as a VTK file containing polygons, the easiest way to load them is to use the `vtkPolyDataReader`. The polygons are given in index space coordinates, however, and thus do not align with the just created volume. So before continuing the points of the polygons have to be transformed into the same world coordinate system the volume uses. This is done by applying the `TransformContinuousIndexToPhysicalPoint` method of the ITK Image object the volume is stored as to each individual point of the polygons.
The next, at the first glance seemingly unnecessary step, is to triangulate the polygons using the `vtkTriangleFilter`. This is done because later on the `PointInPolygon` method of the `vtkPolygon` objects is used to check whether a voxel lies within a contour. During development this method yielded bad results with more complex contour shapes if they were used as one big polygon. Splitting the contours up in individual triangles results in the method working reliably, however.
Before now starting to iterate through the volume and checking whether every voxel on every slice lies within one of the polygons, the bounding box of the segmentation is calculated. Checking if a voxel lies within the bounding box is less time consuming than checking whether it lies within one of the polygons and so incorporating this prior check results in a considerable performance increase. Only if a voxel lies within the bounding box, an inclusion test with the polygons lying on the voxels slice is performed. This is done with the `PointInPolygon` method of the `vtkPolygon` class. Every voxel that lies within one of the polygons is colored white. Once all the voxels have been iterated through, the resulting volume is thus a binary mask, that marks the area inside the segmented region with white colored voxels. This volume mask can now be downloaded in the NRRD file format.
Expert Evaluation {#expert}
=================
In order to test the segmentation capabilities of Studierfenster, two manual segmentations have been performed by physicians. The resulting segmentation files are then checked for their validity and compatibility with other medical imaging platforms.
Datasets {#datasets-expert}
--------
The first segmentation was done on an expansive basalioma of the left midface, found in the initial Magnetic Resonance Imaging (MRI) scan of the patient suffering from it. The tumor showed intra orbital growth and was initially unresectable due to its large size. After administration of medicines, the tumor shrank in its size, which eventually allowed surgeons to completely remove it.
The second segmentation was performed on the MRI of a female, 75-year-old patient with a glioblastoma in the left hemisphere.
Results
-------
The resulting contours of the basalioma segmentation, which took 35 minutes to perform, can be seen in . The visualization is done using the `View3D` module of MeVisLab, which can be used to superimpose the contours on the original MRI dataset.
![Basalioma segmentation contours visualized in MeVisLab. The red point cloud represents the resulting segmentation contours of the basalioma superimposed on the original MRI dataset.[]{data-label="basalmevis"}](figures/basal_mevis.PNG)
shows the resulting segmentation contours of the second performed segmentation, which was the one of the glioblastoma, visualized in Studierfenster.
Discussion
----------
The discussion of the obtained segmentation results focuses on two main aspects: The accuracy of the resulting segmentation contours and the compatibility of the file format used to store them with different medical imaging platforms. Both aspects are important prerequisites for the practical application of the tool. To stress their importance one can take the use case of ground truth data collection as an example. Here, wrongly aligned or calculated coordinates of the segmentation contours would render the obtained datasets worthless and annihilate all the effort that has gone into collecting them. Therefore it is important that the segmentation contours in the exported VTK file are at the exact location the expert performing the segmentation intended them to be. The aspect of file compatibility also comes into play when thinking about the data collection use case. To make the most use of the resulting data, it should be possible to analyze and process the resulting files with already established software without much effort.
One empirical way to verify the two aforementioned aspects is to use a different medical imaging software like MeVisLab to superimpose the exported segmentation contours on the datasets they originate from. That way one can check whether the calculation of the coordinates of the points constituting the segmentation contours has been done correctly and whether the coordinate system of the contours matches the one of the original dataset. This verification was done using the two collected expert segmentations, namely the one of the basalioma seen in and the one of the glioblastoma seen in . From the screenshots taken in MeVisLab and Studierfenster, it is evident that both expert segmentations are correctly aligned with the datasets they were performed on. Under the assumption that MeVisLab handles the VTK format correctly, this also demonstrates that Studierfenster exports the contours as valid VTK files, that can be loaded in different medical imaging tools as well.
User Study {#studyofusers}
==========
The goal of this work is to develop a browser-based manual segmentation tool. From a user perspective, major usability improvements of a browser-based solution compared to a desktop solution include its faster accessibility due to the missing installation and update process. This advantage is, of course, neglectable if the rest of the tool is not perceived well in terms of usability. Thus in order to evaluate the usability of the presented segmentation tool, a user study was conducted.
Dataset {#datasets-userstudy}
-------
The ground truth reference for the user study is the expert segmentation of the glioblastoma described in more detail in . It serves as the reference to which the segmentations of the participants of the user study are compared to.
A separate dataset [@dataset] was used during the introduction of Studierfenster that the participants received. The dataset used for this purpose originates from a clinical evaluation of segmentation algorithms [@mandbone], [@ctdata]. It includes ten Computed Tomography (CT) images in the NRRD format of patients without teeth, which were randomly chosen from a bigger dataset.
![Result of the user study visualized as a bar chart. The bars represent the mean of the ratings of all users grouped per question.[]{data-label="barplot"}](figures/barplot.pdf)
Methodology {#methodology-userstudy}
-----------
The design of the user study was derived from the one performed in the context of a cranial implant planning tool for MeVisLab in the work by Egger et al. [@userstudy]. In our case, the test users were first given a short initial introduction to the segmentation tool during which they could explore the features of the platform freely. For this, the CT scan of *patient six* of the mandibular dataset described in was used. For participants with no medical background, the introduction included an explanation of the purpose of the tool and the importance of segmentation for medical imaging. All participants were then guided through the segmentation and metric calculation process.
The actual task for the user was to then segment the same glioblastoma as the one in the reference segmentation seen in . This allowed the user to later compare the resulting segmentation with the one of the neurosurgeon. The two comparison metrics, namely the Dice Score [@dice] and Hausdorff Distance [@hausdorff], were obtained with the help of the calculation tool that was developed as a second use case for the Studierfenster platform by a colleague in parallel to this work. Calculating these two metrics and saving them as a PDF file was the last step of the user’s task. After the user finished the task, a questionnaire was used to capture the impressions of the user regarding the usability of the segmentation tool. The questions were taken from the work by Egger et al. [@userstudy], where questions derived from the ISONORM 9241/10 were used. Answers were given on a Likert scale ranging from 1 to 7, where 1 is the worst rating and 7 the best. The questions presented to the user were as follows:
1. The software does not need a lot of training time.
2. The software is adjusted well to achieve a satisfying result.
3. The software provides all necessary functions to achieve the goal.
4. The software is not complicated to use.
5. How satisfied are you with the UI surface?
6. How satisfied are you with the presented result?
7. How satisfied have you been with the time consumed?
8. How is your Overall impression?
The test users chosen for the user study consisted of 10 users in total. This group was further split into 5 users that are familiar with the medical use of segmentation and the status quo of existing solutions and 5 users that had no prior experience with medical image segmentation.
Results {#results}
-------
visualizes the mean ratings given in the questionnaire and the corresponding standard error as a bar chart.
In the Dice scores of users with a medical background can be seen in comparison with the Dice scores of users with no medical background. also shows a comparison of the Hausdorff distances.
Discussion
----------
The first part of the user study, which was the short introduction to the tool, with a chance for participants to explore the tool freely, was well received. On average this initial training took about five minutes.
For the actual segmentation task no timing constraints were given to participants. As can be seen in the time participants took to finish the task varied between 3 and 15 minutes. It is notable that most of the variance stems from the medical group, which took an average of 8 minutes and 47 seconds, with participants well distributed between the 3 and 15 minutes mark. In contrary all participants but one of the non-medical group took between 3 and 4 minutes for the segmentation task. The one visible outlier took 14 minutes and 32 seconds. The most probable cause for the time difference between the two groups is that participants of the medical group edited their segmentation contours more frequently.
Once the participants were content with the quality of their segmentation, they were asked to calculate the Hausdorff Distance and the Dice Score between their segmentation and the reference segmentation of the physician. The resulting Hausdorff Distances and the Dice Scores can be seen in . Hausdorff Distances range between $2.81$ and $4.93$ and Dice Scores between $0.82$ and $0.87$. A comparison between the medical and the non-medical group again reveals differences in values, albeit them being less apparent than those found in the segmentation times. The difference between the two mean Hausdorff distances per group is $0.287$, with the medical group having the lower mean distance of $3.518$ compared to the mean distance of $3.805$ found in the non-medical group. Dice Scores also show slightly better results for the medical group. Here the mean of the medical group was $0.852$ and the mean of the non-medical group $0.836$, yielding a small difference of $0.016$. Surprisingly the non-medical group did thus not perform much worse than the medical group, even though they took less time to complete the manual segmentation of the glioblastoma.
In the last part of the user study, participants were handed the usability questionnaire described in . As can be seen in the bar chart in , the ratings were overall positive, ranging between a minimum mean rating of $5.6$ given for questions 1, 3 and 5 and a maximum mean rating of $6.4$ for question 7. Question 1 was about the necessary training time for the tool and question 5 about the satisfaction with the UI surface. Putting this results in the context of the segmentation interface suggests that there is still work to be done to make it more intuitive to use. One specific area to improve would be the presentation of the buttons on the left-hand side. Adding icons to them would improve their look and giving additional information on their function when hovering over them would reduce the necessary training time for the tool. The mean rating of $6.3$ given to question 4 still suggests that the tool is not too complicated to use. Question 3 asked participants whether the tool provided all necessary functions to achieve the manual segmentation and the calculation of the results. One suggestion regarding an additional feature that was repeatedly given over the course of the user study was the integration of a zoom function. This would ease the segmentation of small structures and in turn improve the segmentation results. The next highest result of the questionnaire was the mean rating of $5.9$ given to the second question asking whether the tool is well adjusted to achieve a satisfying result. Although this rating is already high, it could for example also be improved by including the aforementioned zoom function. However, as the high mean rating of $6.1$ given to question 6 indicates, the tool already allows users to produce satisfying results for sufficiently big structures.
The on average highest rated question shows that participants were content with the time consumed to complete the task and, with a mean rating of $6.3$, the overall impression was also reported as very good.
Conclusion and Future Outlook {#conclusion}
=============================
This work presented the implementation of a web-based tool supporting manual segmentation directly in the browser. The motivation behind focusing on web technologies was to ease the collection of ground truth segmentation datasets, by providing an easily accessible tool to create them. Considering the positive feedback from the user study and the evaluation of the expert segmentations acquired with the tool, it can be concluded that this goal was reached. Exported segmentation contours correctly align with their dataset of origin and are compatible with established medical imaging tools such as MeVisLab.
Nevertheless, there remain areas to improve upon in future work, like including a zoom function and other features enabling more advanced segmentation tasks. One could also think about ways to eliminate the need to upload files to the web server in order to analyze and process them. At the time of writing, this is necessary for generating the segmentation masks and for calculating the segmentation metrics. A promising solution to this could be JavaScript enabled versions of ITK and VTK, which are, at the time of writing, under heavy development by Kitware, the company behind the two libraries, but not yet finalized and completely documented. Including these libraries and the aforementioned features will be considered in the ongoing efforts to expand and improve the presented tool.
Acknowledgments
===============
This work received funding from the Austrian Science Fund (FWF) KLI 678-B31: “enFaced: Virtual and Augmented Reality Training and Navigation Module for 3D-Printed Facial Defect Reconstructions” and the TU Graz Lead Project (Mechanics, Modeling and Simulation of Aortic Dissection). Moreover, this work was supported by CAMed (COMET K-Project 871132) which is funded by the Austrian Federal Ministry of Transport, Innovation and Technology (BMVIT) and the Austrian Federal Ministry for Digital and Economic Affairs (BMDW) and the Styrian Business Promotion Agency (SFG).
[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a flexible framework for learning predictive models that approximately satisfy the equalized odds notion of fairness. This is achieved by introducing a general discrepancy functional that rigorously quantifies violations of this criterion. This differentiable functional is used as a penalty driving the model parameters towards equalized odds. To rigorously evaluate fitted models, we develop a formal hypothesis test to detect whether a prediction rule violates this property, the first such test in the literature. Both the model fitting and hypothesis testing leverage a resampled version of the sensitive attribute obeying equalized odds, by construction. We demonstrate the applicability and validity of the proposed framework both in regression and multi-class classification problems, reporting improved performance over state-of-the-art methods. Lastly, we show how to incorporate techniques for equitable uncertainty quantification—unbiased for each group under study—to communicate the results of the data analysis in exact terms.'
author:
- |
Yaniv Romano\
Department of Statistics\
Stanford University\
Stephen Bates\
Department of Statistics\
Stanford University\
Emmanuel J. Candès\
Departments of Mathematics\
and of Statistics\
Stanford University\
bibliography:
- 'MyBib.bib'
title: Achieving Equalized Odds by Resampling Sensitive Attributes
---
Introduction
============
Machine learning algorithms are now frequently used to inform high-stakes decisions—and even to make them outright. As such, society has become increasingly critical of the ethical implications of automated decision making, and researchers in algorithmic fairness are responding with new tools. [While fairness is context dependent and may mean different things to different people]{}, a suite of recent work has given rise to a useful vocabulary for discussing fairness in automated systems [@dwork2012fairness; @chouldechova2017fair; @kleinberg2017inherent; @barocas2017fairness; @chouldechova2018frontiers]. Fairness constraints can often be articulated as conditional independence relations, and in this work we will focus on the *equalized odds* criterion [@hardt2016equality], defined as $$\label{eq:eqodds_def}
\hat{Y} {\protect\mathpalette{\protect\independenT}{\perp}}A \mid Y$$ [where the relationship above applies to test points]{}; here, $Y$ is the response variable, $A$ is a sensitive attribute (e.g. gender), $X$ is a vector of features that may also contain $A$, and $\hat{Y} = \hat{f}(X)$ is the [prediction]{} obtained with a [[*fixed*]{}]{} prediction rule $\hat{f}(\cdot)$. While the idea that a [prediction rule obeying]{} the equalized odds property is desirable has gained traction, actually finding such [a rule]{} for a real-valued or multi-class response is a relatively open problem. Indeed, there are only a few recent works attempting this task [@zhang2018mitigating; @mary2019fairness]. Moreover, there are no existing methods to rigorously check whether [a learned model achieves this property.]{}
We address these two questions by introducing a novel training scheme to fit models that approximately satisfy the equalized odds criterion and a hypothesis test to detect when a prediction rule violates this same criterion. Both solutions build off of one key idea: we create a synthetic version $\tilde A$ of the [sensitive]{} attribute such that the triple $(\hat Y, \tilde A, Y)$ obeys with $\tilde A$ in lieu of $A$. To achieve equitable model fits, we regularize our models toward the distribution of the synthetic data. Similarly, to test whether equalized odds holds, we compare the observed data to a collection of artificial data sets. The synthetic data is straightforward to sample, making our framework both simple to implement and modular in that it [works together with any loss function, architecture, training algorithm, and so on. Based on real data experiments on both regression and multi-class classification tasks, we find improved performance compared to state-of-the-art methods.]{}
A synthetic example {#sec:synth}
-------------------
[0.24]{}
[0.24]{}
[0.24]{}
[0.24]{}
To set the stage for our methodology, we first present an experiment demonstrating the challenges of making equitable predictions as well as a preview of our method’s results. We simulate a regression data set with a binary [sensitive]{} attribute and two features: $$(X_1,X_2) \mid (A=0) \ \eqd \ (Z_1, 3Z_2) \quad \text{and} \quad (X_1,X_2) \mid (A=1) \ \eqd \ (3Z_1, Z_2),$$ where $Z_1,Z_2 \sim \mathcal{N}(0,1)$ is a pair of independent standard normal variables, and the symbol $\eqd$ denotes equality in distribution. [We create a population where 90% of the observations are from the group $A=1$ in order to investigate a setting with a large majority group.]{} After conditioning on $A$, the model for $Y \mid X$ is linear: $Y~=~X^{\top}\beta_{A}~+~\epsilon,$ with noise $\epsilon \sim \mathcal{N}(0,1)$ and coefficients $\beta_{0} = (0,3)$ and $\beta_{1} = (3,0)$. [We designed the model in this way so that the distribution of $Y$ given $X$ is the same for the two groups, up to a permutation of the coordinates. (In some settings, we might say that both groups are therefore equally deserving.) Consequently, the best model has equal performance in both groups. We therefore find it reasonable to search for a fitted model that achieves equalized odds in this setting.]{} To serve as an initial point of comparison, we first fit a classic linear regression model with coefficients $\hat{\beta}\in\RR^2$ on the training data, minimizing the mean squared error. Figures and show the performance of the fitted model for each group [on a separate test set]{}. The fitted model performs significantly better on the samples from the majority group $A=1$ than those from the minority group $A=0$. [This is not surprising since the model seeks to minimize the overall prediction error.]{} Here, the overall [root]{} mean squared error [(RMSE)]{} [evaluated on test points]{} is equal to [2.29]{}, with an average value of [4.96]{} for group $A=0$ and of [1.79]{} for group $A=1$. [It is visually clear that for any vertical slice of the graph at a fixed value of $Y$, the distribution of $\hat{Y}$ is different in the two classes, i.e. the equalized odds property in is violated.]{} This fact can be checked formally with our hypothesis test for described later in Section \[sec:crt\]. The resulting p-value [on the test set]{} is $0.001$ providing rigorous evidence that equalized odds is violated in this case.
Next, we apply our proposed fitting method (see Section \[sec:achieving\_eo\]) on this data set. Rather than a naive least squares fit, we instead fit a linear regression model that approximately satisfies the equalized odds criterion. The new predictions are displayed in Figures and . In contrast to the naive fit, the new predictive model achieves [a more balanced performance]{} across the two groups: the blue points are dispersed similarly in these two panels. This observation is consistent with the results of our hypothesis test; the p-value [on the test set]{} is equal to $0.452$, which provides no indication that the equalized odds property is violated. Turning to the statistical efficiency, the equitable model has improved performance for observations in the minority group $A=0$ with an [RMSE equal to 3.07]{}, at the price of reduced performance in the majority group $A=1$, where the [RMSE rises to 3.35]{}. The overall RMSE is 3.33, larger than that of the baseline model.
Related work {#sec:related_work}
------------
The notion of equalized odds as a criterion for algorithmic fairness was introduced in [@hardt2016equality]. In the special case of a binary target variables and a binary response variable, the aforementioned work offered a procedure to post-process any predictive model to construct a new model achieving equalized odds, possibly at the cost of reduced accuracy. Building off this notion of fairness, [@zafar2017fairnessconstraints] and [@donini2018empirical] show how to fit linear and kernel classifiers that are aligned with this criterion as well—these methods apply when the response and sensitive attribute are both binary. Similarly, building on the Hirschfeld-Gebelein-Renyi (HGR) Maximum Correlation Coefficient, [@mary2019fairness] introduces a penalization scheme to fit neural networks that approximately obey equalized odds, applying to continuous targets and [sensitive]{} attributes. Coming at the problem from a different angle, [@louppe2017learning; @zhang2018mitigating] fit models with an equalized odds penalty using an adversarial learning scheme. The main idea behind this method is to maximize the prediction accuracy while minimizing the adversary’s ability to predict the [sensitive]{} attribute. [Our method has the same objective as the latter two, but uses a new subsampling technique for regularization, which also leads to the first formal test of the equalized odds property in the literature.]{}
Fitting fair models {#sec:achieving_eo}
===================
Regularization with fair dummies {#subsection:regularization}
--------------------------------
This section presents a method for fitting a predictive function $\hat{f}(\cdot)$ on i.i.d. training data $\{(X_i, A_i, Y_i)\}$ indexed by $i \in \mathcal{I}_{\mathrm{train}}$ that approximately satisfies the equalized odds property . [In regression settings, [let]{} $\hat{Y}=\hat{f}(X) \in \RR $ [be the predicted value of]{} the continuous response $Y \in \RR$. In multi-class classification problems where the response variable $Y \in \{1,\dots,L\}$ is discrete, we take the output of the classifier to be $ \hat{Y} = \hat{f}(X) \in \RR^L$, [a vector whose entries are estimated probabilities that an observation with $X=x$ belongs to class $Y=y$.]{}]{} [We use this formulation of $\hat{Y}$ because it is the typical information available to the user when deploying a neural network for regression or classification, and our methods will use neural networks as the underlying predictive model. Nonetheless, the material in this subsection holds for any formulation of $\hat{Y}$, such as an estimated class label.]{}
Our procedure starts by constructing a *fair dummy* [sensitive]{} attribute $\tilde{A}_i$ for each training sample: $$\begin{aligned}
\tilde{A}_i \sim P_{A|Y}\left(A_i \mid Y_i\right), \quad i \in \mathcal{I}_{\mathrm{train}}, $$ where $P_{A|Y}$ denotes the conditional distribution of $A_i$ given $Y_i$. This sampling is straightforward; see below. Importantly, we generate $\tilde{A}_i$ without looking at $\hat{Y}_i$ so that we have the following property: $$\begin{aligned}
\label{eq:control}
\hat{Y}_i {\protect\mathpalette{\protect\independenT}{\perp}}\tilde{A}_i \mid Y_i, \quad i \in \mathcal{I}_{\mathrm{train}}.
$$ Notice that the above is exactly the equalized odds relation in , with a crucial difference that the original [sensitive]{} attribute $A_i$ is replaced by the artificial one $\tilde{A}_i$. We will leverage this fair, synthetic data for both model fitting and hypothesis testing in the remainder of this work.
[Motivated by]{} , we propose the following objective function for equalized odds model fitting: $$\label{eq:epi_optimization}
\hat{f}(x) = \underset{f \in \mathcal{F}}{\mathrm{argmin}} \ \frac{1-\lambda}{|\mathcal{I}_{\mathrm{train}}|}\sum_{i \in \mathcal{I}_{\mathrm{train}} } \ell(Y_i,f(X_i)) + \lambda \mathcal{D}\left( ( \hat{\Y}, \A, \Y ), ( \hat{\Y}, \tilde{\A}, \Y ) \right).$$ Here, $\ell(\cdot)$ is a loss function that measures the prediction error, such as the mean squared error for regression, or the cross-entropy for multi-class classification. The second term on the right hand side is a penalty promoting the equalized odds property, described in detail soon. The hyperparameter $\lambda$ trades off accuracy versus equalized odds. Above, the $i$th row of $\hat{\Y}\in\RR^{|\mathcal{I}_{\mathrm{train}}| \times k}$ is $f(X_i) \in \R^k$, [with $k=1$ in regression and $k=L$ in multi-class classification.]{} Similarly, we define $\mathbf{X}\in \RR^{|\mathcal{I}_{\mathrm{train}}| \times p}$ $\A\in\RR^{|\mathcal{I}_{\mathrm{train}}|}$, $\tilde{\A}\in\RR^{|\mathcal{I}_{\mathrm{train}}|}$, and $\Y\in\RR^{|\mathcal{I}_{\mathrm{train}}|}$, whose entries correspond to the features, [sensitive]{} attributes, fair dummies, and labels, respectively. As a result, both $(\hat{\Y}, \A, \Y)$ and $(\hat{\Y}, \tilde{\A}, \Y)$ are matrices of size $|\mathcal{I}_{\mathrm{train}}| \times (k + 2)$. The function $\mathcal{D}(\U,\V)$ is any measure of the discrepancy between two probability distributions $P_{U}$ and $P_{V}$ based on the samples $\U$ and $\V$, summarizing the differences between the two samples into a real-valued score. A large value suggests that $P_U$ and $P_V$ are distinct, whereas a small value suggests that they are similar. We give a concrete choice based on adversarial classification in Section \[sec:impl\_two\_sample\]. Since $(\hat{\Y}, \tilde{\A}, \Y )$ obeys the equalized odds property by construction, making the discrepancy with $(\hat{\Y}, \A, \Y )$ small forces the latter to approximately obey equalized odds.
[Take $(X, A, Y) \sim P_{XAY}$ and set $\hat{Y} = \hat{f}(X)$ for some fixed $\hat{f}(\cdot)$ (again, $X$ may include $A$). Let $\tilde A$ be sampled indpendently from $P_{A|Y}(A|Y)$.[^1] Then, $\hat Y {\protect\mathpalette{\protect\independenT}{\perp}}A \mid Y$ if and only if $(Y,A,\hat{Y}) \eqd (Y, \tilde A, Y)$. ]{} \[prop:fair\_dummies\_eo\]
The proof of this proposition as well as all other proofs are in Appendix \[app:proofs\]. We argue that this equivalence is particularly fruitful: indeed, if we find a prediction rule $\hat{f}(\cdot)$ such that $(\hat{\Y}, \A, \Y )$ has the same distribution as $(\hat{\Y}, \tilde{\A}, \Y )$ (treating the prediction rule as fixed), then $\hat{f}(\cdot)$ exactly satisfies equalized odds. Motivated by this, our penalty drives the model to a point where these two distributions are close based on the training set. When this happens, then, informally speaking, we expect that equalized odds approximately holds for future observations.
The regularization term in can be used with essentially any existing machine learning framework, allowing us to fit a predictive model that is aligned with the equalized odds criterion, no matter whether the response is discrete, continuous, or multivariate. It remains to formulate an effective discriminator $\mathcal{D}(\cdot)$ to capture the difference between the two distributions, which we turn to next.
The discrepancy measure {#sec:impl_two_sample}
-----------------------
A good discrepancy measure $\mathcal{D}(\cdot)$ should detect differences in distribution between the training data and the fair dummies in order to better promote equalized odds. Many examples have already been developed for the purpose of two-sample tests; examples include the Friedman-Rafsky test [@friedman1979multivariate], the popular maximum mean discrepancy (MMD) [@gretton2012kernel], the energy test [@szekely2013energy], and classifier two-sample tests [@friedman1983graph; @lopez2016revisiting]. The latter are tightly connected to the idea of generative adversarial networks [@goodfellow2014generative] which serves as the foundation of our procedure.
To motivate our proposal, suppose we are given two independent data sets $\{U_i\}$ and $\{V_i\}$: the first contains samples of the form $U_i = (\hat{Y}_i, A_i, Y_i)$, and the second includes $ V_i = (\hat{Y}_i, \tilde{A}_i, Y_i)$. Our goal is to design a function that can distinguish between the two sets, so we assign a positive (resp. negative) label to each $U_i$ (resp. $V_i$) and fit a binary classifier $\hat{d}(\cdot)$. Under the null hypothesis that $P_U = P_V$, the classification accuracy of $\hat{d}(\cdot)$ on hold-out points should be close to $1/2$, while larger values provide evidence against the null. To turn this idea into a training scheme, we repeat the following two steps: first, we fit a classifier $\hat{d}(\cdot)$ whose goal is to recognize any difference in distribution between $U$ and $V$, and second, we fit a prediction function $\hat{f}(\cdot)$ that attempts to “fool” the classifier $\hat{d}(\cdot)$ while also minimizing the prediction error. In our experiment, the function $\hat{d}(\cdot)$ is formulated as a deep neural network with a differentiable loss function, so as the two models—$\hat{f}(\cdot)$ and $\hat{d}(\cdot)$—can be simultaneously trained via stochastic gradient descent.
While adversarial training is powerful, it can be sensitive to the choice of parameters and requires delicate tuning [@louppe2017learning; @zhang2018mitigating]. To improve stability, we add an additional penalty that forces the relevant second moments of $U$ and $V$ to approximately match; [we penalize by $\|\cov(\hat{\mathbf{Y}}, \mathbf{A}) - \cov(\hat{\mathbf{Y}}, \tilde{\mathbf{A}})\|^2$ [where $\tilde{\mathbf{A}}$ is as in and]{} $\cov$ denotes the covariance, since under equalized odds this would be zero in the population [(because $(\hat Y, A) \eqd (\hat Y, \tilde{A})$ by Proposition \[prop:fair\_dummies\_eo\])]{}.]{} Combining all of the above elements, we can now give the full proposed procedure in Algorithm \[alg:fit\].
**Input**: Data $\{(X_i, A_i, Y_i)\}_{i\in\mathcal{I}_{\mathrm{train}}}$; predictive model $\hat{f}_{\theta_f}(\cdot)$ and discriminator $\hat{d}_{\theta_d}(\cdot)$.
Sample fair dummies $\tilde{A}_i \sim P_{A|Y}(A_i \mid Y_i),
i \in \mathcal{I}_{\mathrm{train}}$. [See Section \[subsection:sampling\] for details.]{} Update the discriminator parameters $\theta_d$ by repeating the following for $N_g$ gradient steps: $$\begin{aligned}
& \! \! \! \! \! \mathcal{J}_d(\theta_d) = \frac{1}{|\mathcal{I}_{\mathrm{train}}|}\sum_{i \in \mathcal{I}_{\mathrm{train}}} \bigg[ \log\left(\hat{d}_{\theta_d}\left(\hat{f}_{\theta_f}(X_i), A_i, Y_i\right)\right) + \log\left(1-\hat{d}_{\theta_d}\left(\hat{f}_{\theta_f}(X_i), \tilde{A}_i, Y_i\right)\right)\bigg] \\
& \! \! \! \! \! \theta_d \leftarrow \theta_d - \mu \nabla_{\theta_d}\mathcal{J}_d(\theta_d)\end{aligned}$$ Update the predictive model parameters $\theta_f$ by repeating the following for $N_g$ gradient steps: $$\begin{aligned}
& \! \! \! \! \! \mathcal{J}_f(\theta_f) = \frac{1-\lambda}{| \mathcal{I}_{\mathrm{train}}|}\sum_{i \in \mathcal{I}_{\mathrm{train}}} \ell\left(Y
_i,\hat{f}_{\theta_f}(X_i)\right) + \lambda\gamma \|\cov(\hat{\mathbf{Y}}, \mathbf{A}) - \cov(\hat{\mathbf{Y}}, \tilde{\mathbf{A}})\|^2 \\
& \! \! \! \! \! \quad \quad \quad + \frac{\lambda}{| \mathcal{I}_{\mathrm{train}}|}\sum_{i \in \mathcal{I}_{\mathrm{train}}} \bigg[ \log\left(\hat{d}_{\theta_d}\left(\hat{f}_{\theta_f}(X_i), \tilde{A}_i, Y_i\right) \right) +
\log\left(1-\hat{d}_{\theta_d}\left(\hat{f}_{\theta_f}(X_i), A_i, Y_i\right)\right)\bigg] \\
& \! \! \! \! \! \theta_f \leftarrow \theta_f - \mu \nabla_{\theta_f}\mathcal{J}_f(\theta_f)\end{aligned}$$
**Output**: Predictive model $\hat{f}_{\theta_f}(\cdot)$ approximately satisfying equalized odds.
Sampling fair dummies {#subsection:sampling}
---------------------
To apply the proposed framework we must sample fair dummies $\tilde{A}$ from the distribution $P_{A|Y}$. Since this distribution is typically unknown, we use the training examples $\{(A_i,Y_i)\}_{i \in \mathcal{I}_{\mathrm{train}}}$ to estimate the conditional density of $A \mid Y$. For example, when the [sensitive]{} attribute of interest is binary, we apply Bayes’ rule and obtain $$\begin{aligned}
\PP\{A=1|Y=y\} = \frac{\PP\{Y=y \mid A=1\}\PP\{A=1\}}{\PP\{Y=y \mid A=1\}\PP\{A=1\} + \PP\{Y=y \mid A=0\}\PP\{A=0\}}.
\label{eq:sampling_dummies}\end{aligned}$$ All the terms in the above equation are straightforward to estimate; in practice, we approximate terms of the form $\PP\{Y=y \mid A=a\}$ using a linear kernel density estimation.
Validating equalized odds {#sec:crt}
=========================
Once we have a fixed predictive model $\hat{f}(\cdot)$ in hand (for example, a model fit on a separate training set), it is important to carefully evaluate whether equalized odds is violated on test points $\{(X_i,A_i,Y_i)\}_{i\in \mathcal{I}_{\mathrm{test}}}$. To this end, we develop a hypothesis test for the relation . Our test leverages once again the fair dummies $\Atilde_i$, but we emphasize that it applies to any prediction rule, not just those trained with our proposed fitting method. [The idea is straightforward: we generate many instances of the test fair dummies $\bAtilde$ and compare the observed test data $(\bYhat, \bA , \bY)$ to those with the dummy attributes $(\bYhat, \bAtilde , \bY)$, since the latter triple obeys equalized odds.]{} One can compare these distributions with any test statistic to obtain a valid hypothesis test; this is a special case of the conditional randomization test of [@candes2018panning]. In Algorithm \[alg:crt\] below, [we present a version of this general test using [@tansey2018holdout] to form test statistic based on a deep neural network $\hat{r}(\cdot)$.]{} Invoking [@candes2018panning], the output of the test is a p-value for the hypothesis that equalized odds holds:
\[thm:crt\_valid\] [Suppose the test observations $(Y_i, X_i, A_i)$ for $i \in \mathcal{I}_{\mathrm{test}}$ are i.i.d.. [Set $\hat{Y}_i = \hat{f}(X_i)$ for a fixed function $\hat{f}(\cdot)$ and construct independently distributed fair dummies $\tilde{A}_i$ as in Proposition \[prop:fair\_dummies\_eo\]. If equalized odds holds for each $i$, i.e., $\hat Y_i {\protect\mathpalette{\protect\independenT}{\perp}}A_i \mid Y_i$, then the distribution of the output $p_v$ of Algorithm \[alg:crt\] stochastically dominates the uniform distribution; in other words, it is a valid p-value.]{}]{}
**Input**: Data $\{(\hat{Y}_i, A_i, Y_i)\}$, $i \in \mathcal{I}_{\mathrm{test}}$
Split $\mathcal{I}_{\mathrm{test}}$ into disjoint subsets $\mathcal{I}_1$ and $\mathcal{I}_2$. Fit a model $\hat{r}(A_i, Y_i)$ on $\{(\hat{Y}_i, A_i, Y_i): i\in \mathcal{I}_1\}$, aiming to predict $\hat{Y}_i$ given $(A_i, Y_i)$. Compute the test statistic on the validation set: $t^* = \frac{1}{|\mathcal{I}_2|}\sum_{i\in\mathcal{I}_2} T(\hat{Y}_i,Y_i,\hat{r}(A_i, Y_i))$. Sample a fresh copy of the fair dummies $\tilde{A}_i \sim P_{A|Y}(A_i \mid Y_i), \ i\in \mathcal{I}_2$. Compute the test statistic using the fair dummies: $t^{(k)} = \frac{1}{|\mathcal{I}_2|}\sum_{i\in\mathcal{I}_2} T(\hat{Y}_i,Y_i,\hat{r}(\tilde{A}_i, Y_i))$. Compute the quantile of the true statistic $t^*$ among the fair dummy statistics $t_1, \dots, t_K$: $$p_v = \frac{1 + \#\{k: t^* \le t^{(k)}\}}{ K+ 1}.$$
**Output**: A p-value $p_v$ for the hypothesis that holds, valid under the assumptions of Proposition \[thm:crt\_valid\].
We reiterate that this holds for any choice of the test statistic $T(\cdot)$, so we next discuss a good all-around choice. For problems with a continuous response $Y\in\RR$ [and prediction $\hat{Y}\in\RR$]{}, we define the test statistic as the squared error function, $
T(\hat{Y}_i,Y_i,\hat{r}(A_i,Y_i)) = (\hat{Y}_i - \hat{r}(A_i,Y_i))^2.
$ Here, $\hat{r}(\cdot)$ can be any model predicting $\hat{Y}_i \in \RR$ from $(A_i,Y_i)$[; we use a two-layer neural network in our experiments.]{} We describe a similar test statistic for multi-class classification in Appendix \[app:multiclass\_crt\_stat\].
Experiments {#sec:experiments}
===========
We now evaluate our proposed fitting method in real data experiments. We compare our approach to two recently published methods, adversarial debiasing [@zhang2018mitigating] and HGR [@mary2019fairness], demonstrating moderately improved performance. While our fitting algorithm also applies to binary classification, we only consider regression and multi-class classification tasks here because there are very few available techniques for such problems. In all experiments, we randomly split the data into a training set (60%), a hold-out set (20%) to fit the test statistic for the fair-dummies test, and a test set (20%) to evaluate their performance. For reproducibility, all software is available at <https://github.com/yromano/fair_dummies>.
Real data: regression {#sec:experiments-reg}
---------------------
[1]{} \[subfig:crimes\]
We begin with experiments on two data sets with real-valued responses: the 2016 Medical Expenditure Panel Survey (MEPS), where we seek to predict medical usage based on demographic variables, and the widely used UCI Communities and Crime data set, where we seek to predict violent crime levels from census and police data. See Appendix \[app:data\_regression\] for more details. [Decision makers may wish to predict medical usage or crime rates to better allocate medical funding, social programs, police resources and so on [e.g., @henderson2010predicting], but such information must be treated carefully. For both data sets we use race information as a binary [sensitive]{} attribute, and it is not used as a covariate for the predictive model. An equalized odds model in this context can add a layer of protection against possible misuse of the model predictions by downstream agents: any two people (neighborhoods) with the same underlying medical usage (crime rate) would be treated the same by the model, regardless of racial makeup. Further care is still required to ensure that such a model is deployed ethically, but equalized odds serves as a useful safeguard.]{}
We will consider two base predictors: a linear model and a neural network. As fairness-unaware baselines, we fit each of the above by minimizing the MSE, without any fairness promoting penalty. We also use each of the base regression models together with the [*adversarial debiasing*]{} method [@zhang2018mitigating], the [*HGR*]{} method [@mary2019fairness], and our proposed method; see Appendix \[app:learning\] for technical details. The methods that promote equalized odds, including our own, each have many hyperparameters, and we find it challenging to automate the task of finding a set of parameters that maximizes accuracy while approximately achieving equalized odds, as also observed in [@zhang2018mitigating]. Therefore, we choose to tune the set of parameters of each method only once and treat the chosen set as fixed in future experiments; see Appendix \[app:hyper\] for a full description of the tuning of each method. The performance of these methods is summarized in Figure \[fig:res\_reg\_mse\]. We observe that the p-values of the two fairness-unaware baseline algorithms are small, indicating that the underlying predictions may not satisfy the equalized odds requirement. In contrast, adversarial debiasing, HGR, and our approach are all better aligned with the equalized odds criterion as the p-values of the fair dummies test are dispersed on the $[0,1]$ range. Turning to the predictive accuracy, we find that that the fairness-aware methods perform similarly to each other, although our proposed methods perform a little better than the alternatives. Each of the fairness-aware models have slightly worse [RMSE]{} than the corresponding fairness-unaware baselines, as expected.
Real data: multi-class classification {#sec:experiments-class}
-------------------------------------
[Next, we consider a multi-class classification example using the UCI Nursery data set, where we aim to rank nursery school applications based on family information. The response has four classes and we use financial standing as a binary [sensitive]{} attribute. See Appendix \[app:data\_classification\] for more details.]{} Similar to our regression experiments, we use a linear multi-class logistic regression and neural network as fairness-unaware baseline algorithms. As before, we also fit predictive models using our proposed method and compare the results to those from adversarial debiasing and HGR. The latter only handles one-dimensional $\hat{Y}$, so we adapted it to the multi-class setting by evaluating the penalty separately on each element of the vector of class-probabilities $\hat{Y}\in\RR^L$ and summing all $L$ of the penalty scores. See Appendix \[app:learning\] for additional details.
[r]{}[0.5]{} \[subfig:nursery\_test\]
We report the results in Figure \[fig:res\_class\_acc\]. The p-values that correspond to the fairness-unaware baseline algorithms are close to zero, indicating that these methods violate the equalized odds requirement. In contrast, HGR, [adversarial debiasing, and our method]{} lead to a nice spread of the p-values over the $[0,1]$ range, with the exception of [adversarial debiasing with the linear model]{} which appears to violate equalized odds. Turning to the prediction error, when forcing the equalized odds criterion the statistical efficiency is significantly reduced compared to the fairness-unaware baselines, and since the linear [adversarial debiasing]{} method violates the equalized odds property, our method has the best performance among procedures that seem to satisfy equalized odds.
Evaluating performance with uncertainty sets {#subsec:eq_cov_sim}
============================================
Quantifying uncertainty in predictive modeling is essential, and, as a final case study, we revisit the previous data set with a new metric based on prediction sets. In particular, using the [*equalized coverage*]{} method [@romano2019malice], we create predictive sets $C(X, A)\subseteq\{1,2,\dots,L\}$ that are guaranteed to contain the unknown response $Y$ with probability $90\%$. To ensure the prediction sets are unbiased to the [sensitive]{} attribute, the coverage property is made to hold identically across values of $A=a$: $$\PP\{Y \in C(X,A) \mid A = a\} \geq 90\% \qquad \mathrm{for \ all } \ a \in \{0, 1\}.
\label{eq:eq_cov_intervals}$$ Such sets can be created using any base predictor, and we report on these sets for the methods previously discussed in Figure \[fig:res\_class\]; see Appendix \[app:ec\_details\]. We observe that all methods obtain exactly $90\%$ coverage per group, as guaranteed by the theory [@romano2019malice]. To compare the statistical efficiency, we look at the size of the prediction sets; smaller size corresponds to more precise predictions. Among the prediction rules that approximately satisfy equalized odds, a neural network trained with our proposed penalty performs the best (recall from Figure \[fig:res\_class\_acc\] that the linear method with adversarial debiasing violates equalized odds in this case).
[1]{}
Discussion
==========
In this work we presented a novel method for fitting models that approximately satisfy the equalized odds criterion, as well as a rigorous statistical test to detect violations of this property. The latter is the first of its kind, and we view it as an important step toward understanding the equalized odds property with complex models. Returning to the former, a handful of other approaches have been proposed, and we demonstrated similar or better performance to state-the-art methods in our numerical experiments. Beyond statistical efficiency, we wish to highlight the flexibility of our proposed approach. Our penalization scheme can be used with any discriminator or two sample test, any loss function, any architecture, any training algorithm, and so on, with minimal modification. Moreover, the inclusion of the second moment penalty makes our scheme stable, alleviating the sensitivity to the choice of hyperparameters. [From a mathematical perspective, the synthetic data allows us to translate the problem of promoting and testing a conditional independence relation to the potentially more tractable problem of promoting and testing equality in distribution of two samples. [We expect this reframing will be useful broadly within algorithmic fairness. ]{}]{} Lastly, we point out our procedure applies more generally to the task of fitting a predictive model while promoting a conditional independence relation [e.g., @louppe2017learning], and leveraging this same technique in domains other than algorithmic fairness is a promising direction for future work.
We conclude with a critical discussion of the role of the equalized odds criterion in algorithmic fairness. We view our proposal as a way to [*move beyond mean-squared error*]{}; with modern flexible methods, there are often many prediction rules that achieve indistinguishable predictive performance, but they may have different properties with respect to robustness, fairness, and so on. When there is a rich enough set of good prediction rules, we can choose one that approximately satisfies the equalized odds property. Nonetheless, we point out two potential problems with exclusively focusing on the equalized odds criterion. First, it is well-known that forcing a learned model to satisfy the equalized odds can lead to decreased predictive performance [@kamiran2012data; @feldman2015certifying; @chouldechova2017fair; @kleinberg2017inherent; @menon2018cost; @chen2018why]. Demanding that the equalized odds is exactly satisfied may force us to intentionally destroy information, as clearly seen in the algorithms for binary prediction rules in [@hardt2016equality; @zafar2017fairnessconstraints; @donini2018empirical; @zhang2018mitigating; @mary2019fairness], and as implicitly happens in some of our experiments. Second, for regression and multi-class classification problems, there is no known way to certify that a prediction rule exactly satisfies equalized odds or to precisely bound the violation from this ideal, so the resulting prediction rules do not come with any formal guarantee. Both of these issues are alleviated when we return uncertainty intervals that satisfy the equalized coverage property, as shown in Section \[subsec:eq\_cov\_sim\]. With this approach, we regularize models towards equalized odds to the extent desired, while returning uncertainty sets valid for each group separately to accurately convey any difference in performance across the groups. Importantly, this gives an interpretable, finite-sample fairness guarantee only relying on the assumption of i.i.d. data. For these reasons, we see the combination of an (approximately) equalized odds model with equalized coverage predictive sets as an attractive combination for predictive models in high-stakes deployments.
E. C. was partially supported by the Office of Naval Research grant N00014-20-12157, and by the National Science Foundation grants DMS 1712800 and OAC 1934578. He thanks Rina Barber and Chiara Sabatti for useful discussions related to this project. S. B. was supported by NSF under grant DMS 1712800 and a Ric Weiland Graduate Fellowship. Y. R. was supported by the Army Research Office (ARO) under grant W911NF-17-1-0304. Y. R. thanks the Zuckerman Institute, ISEF Foundation, the Viterbi Fellowship, Technion, and the Koret Foundation, for providing additional research support.
Proofs {#app:proofs}
======
The “if” direction is immediate. For the reverse direction, taking discrete random variables for simplicity, we have $$\begin{aligned}
\PP(\hat{Y} = \hat{y}, A = a, Y = y) &= \PP(\hat{Y} = \hat{y}, A = a \mid Y = y) \cdot \PP(Y = y) \\
&= \PP(\hat{Y} = \hat{y} \mid Y = y) \cdot \PP(A = a \mid Y = y) \cdot \PP(Y = y) \\
&= \PP(\hat{Y} = \hat{y} \mid Y = y) \cdot \PP(\tilde{A} = a \mid Y = y) \cdot \PP(Y = y) \\
&= \PP(\hat{Y} = \hat{y}, \tilde{A} = a \mid Y = y) \cdot \PP(Y = y) \\
&= \PP(\hat{Y} = \hat{y}, \tilde{A} = a, Y = y)
\end{aligned}$$
The proposed test is an instance of the Holdout Randomization Test [@tansey2018holdout], which is in turn a special case of the Conditional Randomization Test [@candes2018panning], so the result follows directly from Lemma 4.1 of [@candes2018panning].
Test statistics for multi-class classification {#app:multiclass_crt_stat}
==============================================
[In this section, we give the details of the fair dummies test (Algorithm \[alg:crt\]) for multi-class classification. Here, with response $Y \in \{1,\dots,L\}$ and class probability estimates $\hat{Y} \in \RR^L$, let $\hat{Y}^{Y} \in \RR$ be the variable located in the $Y^{\text{th}}$ entry of $\hat{Y}$.]{} Similar to the regression case, we fit a predictive model [$\hat{r}({A}_i,Y_i) \in \RR$]{}, aiming to predict [the estimated class probability $\hat{Y}_i^{Y_i}$]{} given the pair $(A_i,Y_i)$ by minimizing the cross entropy loss function. [(We use a one-hot encoding for $Y_i$.)]{} This function is then used to formulate our final test statistic: $$\begin{aligned}
\label{eq:crt_class}
T(\hat{Y}_i, Y_i, \hat{r}(A_i,Y_i)) = -\hat{Y}_i^{Y_i}\log(\hat{r}(A_i,Y_i)) - (1-\hat{Y}_i^{Y_i})\log(1-\hat{r}(A_i,Y_i)). \end{aligned}$$ Another reasonable statistic for this setting would be to use the whole vector of class probabilities together with the multi-class cross-entropy loss, but we found that the above is more powerful at detecting violations of equalized odds.
Data sets
=========
Regression {#app:data_regression}
----------
For regression problems, we compare the performance of our methods to adversarial debiasing [@zhang2018mitigating] and HGR [@mary2019fairness] on the following two data sets:
- The 2016 Medical Expenditure Panel Survey (MEPS).[^2] Here, the goal is to predict the utilization of medical services based on features such as the individual’s age, marital status, race, poverty status, and functional limitations. [After pre-processing the data as in [@romano2019conformalized],]{} there are $15656$ samples and $138$ features. We take race as the binary [sensitive]{} attribute—there are $9640$ white individuals and $6016$ non-white individuals. [Note that MEPS data is subject to usage rules. We downloaded the data set using conformalized quantile regression [@romano2019conformalized] software package, available online.[^3]]{}
- Communities and Crime data set.[^4] The goal is to estimate the number of violent crimes for U.S. cities given the median family income, per capita number of police officers, percent of officers assigned to drug units, and so on. [We clean the data according to [@mary2019fairness], resulting in $1994$ observations of $121$ variables.]{} Race information is again used as the as [sensitive]{} attribute, with $784$ observations from communities whose percentage of African American is above 10% and $1210$ observations from other communities.
Multi-class classification {#app:data_classification}
--------------------------
The Nursery data contains information on nursery school applicants.[^5] The task is to rank applications based on features such as the parents’ occupation, family structure, and financial standing. The original data set contains five classes, however, after cleaning and rearranging the data we remain with four classes: children who are (1) “not recommended”, (2) “very recommended”, (3) “prioritized”, and (4) “specifically prioritized” to join the nursery. In total, the data set contains $12958$ examples and $13$ features. We use the financial status as a [sensitive]{} attribute; applicants with “inconvenient” standing are assigned to group $A=0$ ($6478$ samples) and those with “convenient” status are assigned to group $A=1$ ($6480$ samples).
Further information about the learning algorithms {#app:learning}
=================================================
Hyper-parameter tuning {#app:hyper}
----------------------
To successfully deploy the learning algorithms presented in Section \[sec:experiments\], we must tune various hyperparameters, such as the equalized odds penalty weight, learning rate, batch size, and number of epochs. This task is particularly challenging because we have a multi-criteria objective: the goal is not only to maximize accuracy but also to pass the fair dummies test, i.e. approximately achieve equalized odds. In our experiments, we find the best set of parameters using $10$ fold cross validation, optimizing the accuracy-fairness objective. Since this process is computationally expensive and partly manual, in practice, we tune the hyperparameters only once using cross validation on the entire data set and then treat the chosen set as fixed for the rest of the experiments. The drawback of this approach is that it may suffer from over-fitting, since we test on the same data used to tune the hyperparameters. To mitigate this problem, in Section \[sec:experiments\], we compare the performance metrics of the different algorithms on data splits that are different than the ones used to tune the parameters; some optimism, however, remains. In any case, this same tuning scheme is used for all methods, ensuring that the comparisons are meaningful.
Implementation details {#app:learning_details}
----------------------
### Regression {#regression .unnumbered}
Our regression experiments build on two base learning algorithms, which are then combined with HGR, adversarial debiasing, and our framework to yield eight methods:
- [Baseline Linear]{}: we fit a linear model by minimizing the MSE loss function, using the stochastic gradient descent optimizer with a learning rate and number of epochs in $\{0.01, 0.1\}$ and $\{100, 200, 400, 600, 1000, 2000, 3000, 4000\}$, respectively. [We normalize the features to have zero mean and unit variance using the training data.]{}
- [Baseline NNet]{}: we fit a two layer neural network with a 64-dimensional hidden layer and ReLU nonlinearity function. The network is optimized by minimizing the MSE, following the same fitting strategy described in Baseline Linear.
- Debiasing Linear and Debiasing NNet: the predictors are formulated as described in the baseline algorithms. Here, we follow the implementation provided in <https://github.com/equialgo/fairness-in-ml> and design the adversary as a four-layer neural network with hidden layers of size $32$ and ReLU nonlinearities. Since the [ sensitive]{} attribute is binary, we apply the sigmoid function on the output of the last layer. We use the Adam optimizer [@kingma2014adam] for training, with a learning rate in $\{0.001, 0.01, 0.1\}$ and a minibatch size in $\{64, 128\}$. We also follow the pre-training strategy suggested in [@zhang2018mitigating] and fit separately the predictor and adversary for a number of epochs in $\{2, 4, 10, 20, 30, 40\}$. Then, the two pre-trained models are fitted interchangeably for additional $\{50, 100, 200, 300, 400\}$ epochs. The weight on the equalized odds penalty is selected from $\{0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99\}$.
- HGR Linear and HGR NNet: we again use architectures identical to those of the baseline models. As suggested in [@mary2019fairness], we use the Adam optimizer with a mini-batch size in $\{128, 256\}$, learning rate in $\{0.001, 0.01\}$, and the number of epochs in $\{10, 15, 20, 30, 40, 50, 80, 100\}$. The `HGR` function is implemented in <https://github.com/criteo-research/continuous-fairness> and we select the weight penalty from the $\{0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9\}$ range.
- Fair-Dummies Linear and Fair-Dummies NNet: we fit predictors that have the same structure as the baseline algorithms, with our proposed regularization. The discriminator is implemented as a two-layer neural network with a hidden layer of size $30$ and ReLU nonlinearities. We use the stochastic gradient descent optimizer, with a fixed learning rate of $0.01$. We use the same optimizer for the classifier, with the same learning rate, except for the addition of a momentum term with value $0.9$. The number of epochs is chosen from the $\{20, 30, 40 ,50, 80, 100\}$ range, and the number of gradient steps ($N_g$ in Algorithm \[alg:fit\]) is selected from the range of $\{40, 50, 60, 70, 80\}$. The weight on the equalized odds penalty is selected from $\{0.4, 0.5, 0.6, 0.7, 0.8, 0.9\}$ ($\lambda$ in Algorithm \[alg:fit\]), and the second moment term ($\gamma$ in Algorithm \[alg:fit\]) is chosen from $\{1, 10, 20\}$.
The predictive model $\hat{r}(\cdot)$, defining the test statistics in the fair dummies test (see Section \[sec:crt\]), is formulated as a two-layer neural network, with a hidden dimension of size 64, and dropout layer with rate $1/2$. We use stochastic gradient descent to fit the network, run for $200$ epochs with a minibatch of size $128$ and a fixed momentum term with weight $0.9$.
### Multi-class classification {#multi-class-classification .unnumbered}
Our experiments are again based on two underlying predictive models which are regularized using fairness-aware methodologies:
- Baseline Linear: we fit a linear model by minimizing the cross entropy loss function. We use the Adam optimizer, with a minibatch size of 32. We choose the learning rate, and number of epochs from the range of $\{0.001, 0.01, 0.1\}$, and $\{20, 40, 60, 80, 100\}$, respectively. [We normalize the features to have zero mean and unit variance using the training data.]{}
- Baseline NNet: we fit a two layer neural network with a 64-dimensional hidden layer, ReLU nonlinearity function, and dropout regularization with rate $1/2$. We use the same optimization strategy as above above.
- Debiasing Linear and Debiasing NNet: we form classifiers as in the baseline algorithms. Similarly to the regression setting, we rely on the implementation from <https://github.com/equialgo/fairness-in-ml>. We use the same adversary as described in the regression setting. Training is done via the Adam optimizer, with a fixed learning rate that is equal to $0.5$ and minibatches of size $32$. We again apply the pre-training strategy [@zhang2018mitigating] and fit separately the predictor and adversary for number of epochs from the range of $\{1, 2\}$. The adversarial training is then repeated for $\{20, 40, 60, 100, 200\}$ epochs. The weight on the equalized odds penalty is selected from $\{0.9, 0.99, 0.999, 0.9999, 0.99999, 0.999999\}$.
- HGR Linear and HGR NNet: we again take classifiers as in the baseline models. To fit them, we apply the Adam optimizer with a mini-batch size of $128$, learning rate in the range of $\{0.001, 0.01\}$, number of epochs selected from $\{10, 20, 30, 40, 50\}$. The HGR penalty weight is selected in the range of $\{0.9, 0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99\}$.
- [Fair-Dummies Linear]{} and Fair-Dummies NNet: we again take classifiers with the same structure as the baseline algorithms. The adversary is implemented as a four-layer neural network with a 32-dimensional hidden layer and ReLU nonlinearity. We use the Adam optimizer, with a fixed learning rate that is equal to $0.5$. The number of epochs is fixed and equal to $50$. The number of gradient steps $N_g$ is selected in the range of $\{1,2\}$, and the weight on the equalized odds penalty is selected from $\{0.9, 0.99, 0.999, 0.9999, 0.99999, 0.999999\}$ for $\lambda$ and from $\{0.01, 0.001, 0001, 0.00001\}$ for $\gamma$.
The fair dummies test statistics is again evaluated using a predictive model $\hat{r}(\cdot)$ that is implemented as a neural network. We use the same architecture and learning strategy as in the regression setup, with the addition of a sigmoid function as the last layer.
Further details on equalized coverage {#app:ec_details}
=====================================
We now turn to a few details of the equalized coverage prediction sets from Section \[subsec:eq\_cov\_sim\]. In our experiments, we use the software package provided by [@romano2019malice], which is available online at <https://github.com/yromano/cqr>. While equalized coverage [@romano2019malice] is presented for regression problems, it is straightforward to extend this method to multi-class classification tasks. To this end, we follow split conformal prediction [@vovk2005algorithmic] and randomly split the data into a proper training set (60%), a hold-out calibration set (20%), and a test set (20%). We use the same predictive models from Section \[sec:experiments-class\], which are fitted to the whole proper training data, providing estimates for class probabilities. The examples $\{(X_i,A_i,Y_i)\}$ that belong to the calibration set are then used to construct the prediction sets for the test points. Specifically, following the notations from Section \[sec:crt\], we deploy the popular inverse probability conformity score [@shafer2008tutorial], given by $1 - \hat{Y}_i^{Y_i}$. Here, $\hat{Y}_i=\hat{f}(X_i) \in \RR^L$ and the variable $\hat{Y_i}^{Y_i} \in \RR$ is the estimated probability that the calibration example $
X_i$ belongs to class $Y_i$.
[^1]: This means that we can write $\tilde{A} = h(Y, \epsilon)$ for some function $h(\cdot)$, where the random variable $\epsilon$ is independent of everything else.
[^2]: <https://meps.ahrq.gov/mepsweb/data_stats/download_data_files_detail.jsp?cboPufNumber=HC-192>
[^3]: <https://github.com/yromano/cqr>
[^4]: <http://archive.ics.uci.edu/ml/datasets/communities+and+crime>
[^5]: <https://archive.ics.uci.edu/ml/datasets/nursery>
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study the asymptotic behavior of solutions to stochastic evolution equations with monotone drift and multiplicative Poisson noise in the variational setting, thus covering a large class of (fully) nonlinear partial differential equations perturbed by jump noise. In particular, we provide sufficient conditions for the existence, ergodicity, and uniqueness of invariant measures. Furthermore, under mild additional assumptions, we prove that the Kolmogorov equation associated to the stochastic equation with additive noise is solvable in $L_1$ spaces with respect to an invariant measure.'
address:
- 'Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany'
- 'Dipartimento di Matematica, Università di Trento, via Sommarive 14, I-38123 Trento, Italy'
author:
- Carlo Marinelli
- Giacomo Ziglio
bibliography:
- 'ref.bib'
date: 20 September 2009
title: Ergodicity for nonlinear stochastic evolution equations with multiplicative Poisson noise
---
[^1]
Introduction
============
This paper is devoted to the study of asymptotic properties of the solution to an infinite dimensional stochastic differential equation of the type $$\label{eq:caro}
\left\{
\begin{aligned}
&du(t)+Au(t)dt=\int_Z G(u(t-),z)\,\bar{\mu}(dt,dz)\\
&u(0)=x
\end{aligned}
\right.$$ where $A$ is a nonlinear monotone operator defined on an evolution triple $V \subset H \subset V'$ (see e.g. the classical works [@Pard; @KR-spde]), and $\bar{\mu}$ is a compensated Poisson measure. Precise assumptions on the data of the problem will be given below. In particular, $A$ may be chosen as the $p$-Laplace operator, as well as the porous media diffusion operator $-\Delta\beta(\cdot)$, thus covering a wide class of nonlinear partial differential equations with discontinuous random perturbations.
While existence and uniqueness of solutions for (\[eq:caro\]) has been established in [@Gyo-semimg] (in fact allowing $\bar{\mu}$ to be a general compensated random measure), we are not aware of any result on the asymptotic behavior of the solutions to such equations. Furthermore, as we show in this paper, invariant measures provide a suitable class of reference measures with respect to which one can study infinite dimensional Kolmogorov equations of non-local type, thus extending results that, to the best of our knowledge, were available only for second-order (local) Kolmogorov equations (see e.g. [@DP-K]).
Let us briefly describe our main results in more detail: we first prove the existence of an invariant measure for the Markovian semigroup associated to (\[eq:caro\]), under the (standing) assumption that $V$ is compactly embedded in $H$. Moreover, suitable a priori estimates on any invariant measure imply the existence of an ergodic invariant measure, and an extra superlinearity assumption on $A$ yields exponential mixing, hence uniqueness. Finally, we prove that the (non-local) Kolmogorov operator $L$ associated to (\[eq:caro\]), with $G$ independent of $u$, is essentially $m$-dissipative in $L_1(H,\nu)$, with $\nu$ an infinitesimally invariant measure for $L$. The last result in particular is equivalent to the solvability in $L_1(H,\nu)$ of the (elliptic) integro-differential Kolmogorov equation associated to (\[eq:caro\]).
We should mention that the case where the right-hand side in (\[eq:caro\]) is replaced by an additive Gaussian noise has been considered in [@BDP-erg], where sufficient conditions for the existence and the uniqueness of invariant measures are given. Moreover, the authors study the Kolmogorov equation associated to (\[eq:caro\]) in $L_2(H,\nu)$, assuming that $A$ is differentiable and its differential satisfies a certain polynomial growth condition. Our $L_1$ approach does not require any such hypothesis. Moreover, combining the results in [@BDP-erg] with ours and appealing to the Lévy-Itô decomposition theorem, one could rather easily obtain corresponding results for evolution equations driven by general (locally) square-integrable Lévy noise.
In this regard, let us also recall that results on existence and uniqueness of invariant measures for semilinear evolution equations driven by Lévy noise can be found in the recent monograph [@PZ-libro], as well as in [@cm:rd]. However, the authors work in the mild setting, hence equations with fully nonlinear drift (i.e. without a leading linear operator generating a strongly continuous semigroup) cannot be covered.
The rest of the paper is organized as follows: results on existence, uniqueness, and ergodicity of invariant measures $\nu$ are contained in Section \[sec:inv.meas\]. In Section \[sec:Kol\], assuming that $G$ does not depend on $u$ and that $A$ satisfies a (mild) “regularizability” hypothesis, we prove that the Kolmogorov operator associated to the stochastic equation (\[eq:caro\]) is dissipative, hence closable, and its closure is $m$-dissipative in $L_1(H,\nu)$. Equivalently, this amounts to saying that the (elliptic) infinite-dimensional non-local Kolmogorov equation associated to (\[eq:caro\]) is uniquely solvable in $L_1(H,\nu)$. In Section \[sec:ex\] we show that our abstract results apply to several situations of interest. In particular, we concentrate on equations with non-linear drift in divergence form (thus including the $p$-Laplace operator) and on the generalized porous media equations with pure-jump noise.
Notation
--------
Given a Banach (or Hilbert) space $E$, its norm will be denoted by $|\cdot|_E$. We shall denote the space of all Borel measureable bounded functions from $X$ to ${\mathbb{R}}$ by $B_b(E)$. Given another Banach space $F$, the space of $k$-times continuously differentiable functions from $E$ to $F$ will be denoted by $C^k(E \to F)$, and $C^{k,1}(E \to F)$ stands for the subset of $C^k(E \to F)$ whose elements posses a Lipschitz continuous $k$-th derivative. We shall add a subscript $\cdot_b$ if the functions themselves and all their derivatives are bounded. If $\phi:E \to F$ is Lipschitz continuous, we shall write $\phi \in \dot{C}^{0,1}(E \to F)$, and we define $$|\phi|_{\dot{C}^{0,1}(E \to F)} :=
\sup_{x,y\in E, x\neq y} \frac{|\phi(x)-\phi(y)|_F}{|x-y|_E}.$$ If $F={\mathbb{R}}$, we shall simply write $C^k(E)$ etc. Sometimes we shall just write $C^k$ etc. if it is obvious what $E$ and $F$ are. By $\mathcal{M}_1(E)$ we shall indicate the space of probability measures on $E$, endowed with the $\sigma(\mathcal{M}_1(E),C^0_b(E))$ topology. Weak convergence (of functions and measures) will be denoted by by ${\rightharpoonup}$, without explicit reference to the underlying topology if no confusion arises.
If $X \leq NY$ for some positive constant $N$, we shall equivalently write $X \lesssim Y$. If $N$ depends on a set of parameters $p_1,\ldots,p_n$, we shall also write $N=N(p_1,\ldots,p_n)$ and $X
\lesssim_{p_1,\ldots,p_n} Y$.
Invariant measures and ergodicity {#sec:inv.meas}
=================================
Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},{\mathbb{P}})$ be a filtered probability space satisfying the usual conditions, and ${\mathbb{E}}$ denote expectation with respect to ${\mathbb{P}}$. All stochastic elements will be defined on this stochastic basis, unless otherwise specified. Let $(Z,\mathcal{Z},m)$ be a measure space with a $\sigma$-finite measure $m$ and $\mu$ a Poisson random measure on ${\mathbb{R}}_+ \times Z$ with compensator $\mathrm{Leb}\otimes m$, and set $\bar{\mu}:=\mu-\mathrm{Leb}\otimes m$ (Leb stands for Lebesgue measure on ${\mathbb{R}}$). Let $H$ be a real separable Hilbert space, and $G: H \times Z \to H$ a measurable function such that $$|G(x,\cdot)|_m^2 := \int_Z |G(x,z)|_H^2\,m(dz) < \infty
\qquad \forall x \in H.$$ Let $V$ and $V'$ be a reflexive Banach space and its dual, respectively, such that $V \hookrightarrow H \hookrightarrow V'$ with dense and continous embeddings. Thanks to Asplund’s renorming theorem [@asplund], we shall assume without loss of generality that both $V$ and $V'$ are strictly convex. Furthermore, we shall assume that $V
\hookrightarrow H$ is compact. Both the duality pairing between $V$ and $V'$ and the inner product in $H$ will be denoted by ${\langle \cdot,\cdot \rangle}$.
The operator $A:V \to V'$ is assumed to be demicontinuous (i.e. strongly-weakly closed) and to satisfy the monotonicity condition $$\label{eq:mon}
2{\langle Ax-Ay,x-y \rangle} - |G(x,\cdot)-G(y,\cdot)|_m^2 \geq 0
\qquad \forall x,\,y \in V,$$ as well as the following coercivity and growth conditions: $$\begin{aligned}
2{\langle Ax,x \rangle} - |G(x,\cdot)|_m^2 + \alpha_0|x|^2_H &\geq \alpha_1|x|^p_V
-C_0 &\forall x\in V, \label{eq:milch}\\
|Ax|_{V'} &\leq C_1|x|_V^{p-1}+C_2 &\forall x\in V, \label{eq:kofi}\end{aligned}$$ for some constants $\alpha_0\geq0$, $\alpha_1>0$, $C_0,C_1>0$, $C_2
\in {\mathbb{R}}$ and $p>2$. Instead of (\[eq:milch\]) one could assume that there exists a constant $\alpha_1>0$ such that $$2{\langle Ax,x \rangle}- |G(x,\cdot)|_m^2 \geq \alpha_1|x|^2_V \qquad \forall x \in V.$$ Note that, by (\[eq:milch\]) and (\[eq:kofi\]), one has $$\label{weissbier}
|G(x,\cdot)|^2_m\leq 2C_1|x|^p_V+\alpha_0|x|^2_H+2C_2|x|_V+C_0
\qquad \forall x \in V.$$ All assumptions stated so far will be in force throughout the paper and will be used without further mention.
Let us recall the following well-posedness result for (\[eq:caro\]) due to Gy[ő]{}ngy [@Gyo-semimg Thm. 2.10]. Here and in the following we shall denote the space of $H$-valued random variables with finite $p$-th moment by $\mathbb{L}_p(H)$, and the space of adapted processes $X:[0,T] \to H$ such that ${\mathbb{E}}\sup_{t \leq T}
|X(t)|_H^p < \infty$ by $\mathbb{H}_p(T)$.
Let $x \in \mathbb{L}_2(H)$ and $T \geq 0$. Then equation (\[eq:caro\]) admits a unique strong solution $u$ such that $u(t)
\in V$ $\mathbb{P}$-a.s. for a.a. $t \in [0,T]$, $t \mapsto u(t)$ is càdlàg in $H$, and satisfies $${\mathbb{E}}\sup_{t \leq T} |u(t)|_H^2 + {\mathbb{E}}\int_0^T|u(t)|^p_V\,dt < \infty.$$ Moreover, $u$ is a Markov process, and the solution map $x \mapsto u$ is Lipschitz continuous from $\mathbb{L}_2$ to $\mathbb{H}_2(T)$.
The solution to (\[eq:caro\]) generates a Markovian semigroup $P_t$ on $B_b(H)$ by the usual prescription $P_t\phi(x)={\mathbb{E}}\phi(u(t,x))$, $\phi \in B_b(H)$. The continuity of the solution map ensures that $P_t$ is Feller.
In the following subsection we establish the existence and uniqueness of an ergodic invariant measure for $P_t$ under an assumption stronger than (\[eq:mon\]). The proof is adapted from a classical method used for stochastic evolution equations with Wiener noise in the mild setting (see e.g. [@DZ96]). This simple result is included only for completeness, while the main results of this section are contained in [§]{}\[subs:gen.case\].
Strictly dissipative case {#subs:str.diss.}
-------------------------
Throughout this subsection we assume that there exists $\alpha\in(0,\infty)$ such that $$\label{aperol}
2{\langle Ax-Ay,x-y \rangle}-|G(x,\cdot)-G(y,\cdot)|_m^2
\geq \alpha |x-y|_H^2 \qquad \forall x,y\in V.$$ We shall need a few preparatory results. The following inequality can be obtained by a simple computation based on (\[weissbier\]), (\[aperol\]), and Young’s inequality (see e.g. [@PreRoeck [§]{}4.3] for a related case).
Let $\eta\in(0,\alpha)$. There exist $\delta_\eta\in(0,\infty)$ such that $$\label{sake}
2{\langle Ax,x \rangle}-|G(x,\cdot)|_m^2\geq\eta|x|_H^2-\delta_\eta
\qquad \forall x \in V.$$
Let us define the random measure $\mu_1$ on ${\mathbb{R}}\times Z$ as $$\mu_1(t,A) :=
\left\{
\begin{array}{ll}
\mu(t,A), & t \geq 0,\quad A \in \mathcal{Z},\\
\mu_0(-t,A), & t< 0,\quad A \in \mathcal{Z},
\end{array}
\right.$$ with $\mu_0$ an independent copy of $\mu$, on the naturally associated filtration $(\tilde{\mathcal{F}}_t)_{t\in{\mathbb{R}}}$. Let us also define the compensated measure measure $\tilde{\mu} := \mu_1 -
\mathrm{Leb}\otimes m$.
For $s\in{\mathbb{R}}$, consider the equation $$\label{orange}
\left\{
\begin{aligned}
&du(t)+Au(t)dt=\int_Z G(u(t-),z)\tilde{\mu}(dt,dz),\qquad t \geq s,\\
&u(s)=x.
\end{aligned}
\right.$$ It is clear that (\[orange\]) admits a unique solution $u(t,s,x)$ which generates a semigroup $P_{s,t}$ on $B_b(H)$, exactly as above.
\[augustiner\] Let $s\in(-\infty,0]$ and $x \in \mathbb{L}_2(H)$. There exists $\eta\in\mathbb{L}_2(H)$, independent of $x$, such that $$\lim_{s\rightarrow-\infty} {\mathbb{E}}|u(0,s,x) - \eta|_H^2 = 0.$$ Moreover, one has $${\mathbb{E}}|u(0,s,x)-\eta|_H^2 \lesssim e^{\alpha s}(1+{\mathbb{E}}|x|^2_H).$$
For $s_1,s_2\in(-\infty,0]$, $s_1\leq s_2$ and $x\in
\mathbb{L}_2(H)$, we have $$\begin{aligned}
u(0,s_1,x)-u(0,s_2,x)=&-\int_{s_2}^0[Au(r,s_1,x)-Au(r,s_2,x)]dr\\
&+\int_{s_2}^0\int_Z [G(u(r,s_1,x),z)-G(u(r,s_2,x),z)]\tilde{\mu}(dr,dz)\\
&+u(s_2,s_1,x)-x.
\end{aligned}$$ Appealing to Itô’s formula for the square of the norm (see [@KryGyo2]) and recalling (\[sake\]) we obtain $$\label{eq:trinka}
{\mathbb{E}}|u(0,s_1,x)-u(0,s_2,x)|^2_H \leq
\Big(\frac{\delta_\eta}{\eta}+2{\mathbb{E}}|x|_H^2\Big) e^{\alpha s_2}.$$ Letting $s_2$ tend to $-\infty$, it follows that there exists $\eta(x)\in \mathbb{L}_2(H)$ such that $$\lim_{s\rightarrow-\infty}{\mathbb{E}}|u(0,s,x)-\eta(x)|^2=0.$$ By the same arguments one can prove that $$\lim_{s\rightarrow-\infty}{\mathbb{E}}|u(0,s,x)-u(0,s,y)|^2=0$$ for all $x,y\in \mathbb{L}_2(H)$, hence that $\eta$ is independent of $x\in \mathbb{L}_2(H)$. Letting $s_1$ tend to $-\infty$ in (\[eq:trinka\]) one obtains the exponential convergence.
We can now prove the main result of this subsection.
\[paulaner\] Assume that (\[aperol\]) holds and that $x \in
\mathbb{L}_2(H)$. There exists a unique invariant measure $\nu$ for $P_t$. Moreover, one has $$\int|y|^2_H\,\nu(dy) < \infty,$$ and $$\left|P_t\varphi(y)-\int \varphi(w)\,\nu(dw)\right|\leq
e^{-\frac\alpha2t} |\varphi|_{\dot{C}^{0,1}}\int|y-w|_H\,\nu(dw)$$ for all $t \geq 0$, $y \in H$, and $\varphi \in C_b^{0,1}(H)$.
Let $\nu$ be the law of the random variable $\eta$ constructed in Lemma \[augustiner\]. In particular, $\int |y|^2\,\nu(dy)<\infty$ is equivalent to $\eta \in \mathbb{L}_2(H)$. Similarly, the previous lemma immediately yields $P^*_{s,0}\delta_y {\rightharpoonup}\nu$ for all $y \in H$ in $\mathcal{M}_1(H)$ as $s \to -\infty$. Moreover, for any $\varphi \in C_b^{0,1}(H \to {\mathbb{R}})$, we have $$\int (P_{0,t}\varphi)\,d\nu =
\lim_{s\rightarrow-\infty} \int(P_{0,t}\varphi)\,d(P^*_{s,0}\delta_y)
= \lim_{s\rightarrow-\infty}(P_{s-t,0}\varphi)(y)
= \int \varphi\,d\nu,$$ i.e. $\nu$ is invariant for $P_t$. Moreover, if $\nu$ is an invariant measure for $P_t$, we have $$\Big| P_t\varphi(y) - \int\varphi\,d\nu \Big|
= \Big| \int (P_t\varphi(y)-P_t\varphi(w))\,\nu(dw) \Big|
\leq e^{-\frac{\alpha}{2}t} |\varphi|_{{\dot{C}^{0,1}}} \int|y-w|_H\,\nu(dw)$$ for all $t \geq 0$.
General case {#subs:gen.case}
------------
We can still prove the existence of an ergodic invariant measure without the assumption that the couple $(A,G)$ is strictly dissipative, using an argument based on Krylov-Bogoliubov’s theorem.
\[thm:exim\] There exists an invariant measure $\nu$ for $P_t$. Moreover, $\nu$ is concentrated on $V$, i.e. $\nu(V)=1$.
We assume $p>2$, since the proof for the case $p=2$ is completely similar. Let $x \in \mathbb{L}_2(H)$. By Itô’s formula for the square of the norm in $H$ (see [@KryGyo2]) we have $$\begin{aligned}
|u(t,x)|^2_H &- |x|_H^2 = 2\int_0^t {\langle u(s-,x),du(s,x) \rangle} + [u](t)
\nonumber\\
&= -2\int_0^t {\langle Au(s,x),u(s,x) \rangle}\,ds
+2\int_0^t\!\int_Z{\langle u(s-,x),G(u(s-,x),z) \rangle}\,\bar{\mu}(ds,dz)
\nonumber\\
&\quad +\int_0^t\!\int_Z|G(u(s-,x),z)|_H^2\,\mu(ds,dz).\label{eq:ito}
\end{aligned}$$ Taking expectations on both side and recalling that the compensator of $\mu$ is $\mathrm{Leb}\otimes m$, we obtain $${\mathbb{E}}|u(t,x)|^2_H = -2{\mathbb{E}}\int_0^t {\langle Au(s,x),u(s,x) \rangle}\,ds + {\mathbb{E}}|x|_H^2
+ \int_0^t |G(u(s,x),\cdot)|^2_{m}\,ds,$$ hence, thanks to (\[eq:milch\]), $$\label{lemonade}
{\mathbb{E}}|u(t,x)|^2_H \leq \alpha_0{\mathbb{E}}\int_0^t|u(s,x)|_H^2\,ds
-\alpha_1{\mathbb{E}}\int_0^t|u(s,x)|_V^p ds+{\mathbb{E}}|x|_H^2+tC_0.$$ Since $V \hookrightarrow H$ is continuous, there exists a constant $c>0$ such that $|v|_H\leq c|v|_V$ for all $v \in V$, hence $${\mathbb{E}}|u(t,x)|^2_H \leq \alpha_0{\mathbb{E}}\int_0^t|u(s,x)|_H^2\,ds -
\frac{\alpha_1}{c^p} {\mathbb{E}}\int_0^t|u(s,x)|_H^p\,ds +{\mathbb{E}}|x|_H^2 + tC_0.$$ The elementary inequality $\varepsilon^2 |y|^2 \leq \varepsilon^p
|y|^p+1$ (with $\varepsilon>0$ and $p\geq2$) yields $$-{\mathbb{E}}|u(t,x)|_H^p \leq -\varepsilon^{2-p}{\mathbb{E}}|u(t,x)|_H^2 +
\varepsilon^{-p},$$ thus also $$\begin{aligned}
\label{apfelschorle}
{\mathbb{E}}|u(t,x)|^2_H &\leq -\left(\frac{\alpha_1\varepsilon^{2-p}}{c^p}
- \alpha_0\right) \int_0^t{\mathbb{E}}|u(s,x)|_H^2\,ds
+ t\left(\frac{\alpha_1\varepsilon^{-p}}{c^p}+C_0\right)+{\mathbb{E}}|x|_H^2\nonumber\\
&=-\gamma\int_0^t{\mathbb{E}}|u(s,x)|_H^2 ds+{\mathbb{E}}|x|^2_H+tC
\end{aligned}$$ where $$\gamma := \frac{\alpha_1\varepsilon^{2-p}}{c^p}-\alpha_0, \qquad
C := \frac{\alpha_1\varepsilon^{-p}}{c^p}+C_0.$$ Choosing $\varepsilon$ so that $\gamma>0$ and applying Gronwall’s inequality to (\[apfelschorle\]), it follows that $$\label{apfelsaft}
{\mathbb{E}}|u(t,x)|^2_H\leq{\mathbb{E}}|x|^2_He^{-\gamma t}+K\qquad\forall t\geq0$$ where $K$ is a constant independent of $t$. Moreover, by (\[lemonade\]) and (\[apfelsaft\]) we obtain $$\begin{aligned}
\label{cola}
{\mathbb{E}}\int_0^t|u(s,x)|_V^p\,ds &\leq \frac{1}{\alpha_1}
\left(\alpha_0{\mathbb{E}}\int_0^t|u(s,x)|_H^2 ds + {\mathbb{E}}|x|_H^2
+ tC_0\right)\nonumber\\
&\leq \frac{1}{\alpha_1} \left[\left(\frac{\alpha_0}{\gamma}+1\right)
{\mathbb{E}}|x|_H^2 + t(\alpha_0K+C_0)\right]
\end{aligned}$$ for all $t>0$.
We shall now use the estimates just obtained to prove the tightness of the sequence of measures $$\nu_n := \frac{1}{n}\int_0^n \lambda_t\,dt, \qquad n \in {\mathbb{N}},$$ where $\lambda_t$ stands for the law of the random variable $u(t,0)$, so that $$\int_H \varphi\,d\nu_n = \frac1n \int_0^n {\mathbb{E}}\varphi(u(t,0))\,dt$$ for all $\varphi \in B_b(H)$. By (\[cola\]) we obtain $${\mathbb{E}}\int_0^t|u(s,0)|_V^p\,ds \lesssim 1 + t \qquad \forall t>0,$$ which in turn implies $$\label{sahne}
\int_H |y|_V^p\,\nu_n(dy) = \frac1n \int_0^n {\mathbb{E}}|u(s,0)|_V^p\,ds \lesssim 1
\qquad\forall n\in{\mathbb{N}}.$$ By Markov’s inequality we thus obtain $$\sup_{n\in{\mathbb{N}}}\nu_n(|y|_V\geq R) \leq \sup_{n\in{\mathbb{N}}}\frac{1}{nR^p}
\int_0^n {\mathbb{E}}|u(s,0)|_V^p\,ds \lesssim \frac{1}{R^p},$$ which converges to zero as $R \to \infty$. Since the ball $B_R:=\{y \in H: |y|_V \leq R\}$ is bounded in $V$, and $V
\hookrightarrow H$ is compact, it follows that, for any given $\varepsilon$, there exists $\bar{R}\in{\mathbb{R}}_+$ such that $\nu_n(B_{\bar{R}}) > 1 - \varepsilon$ uniformly over $n$, with $B_{\bar{R}}$ a compact subset of $H$. In other words, the sequence $\nu_n$ is tight, and Prohorov’s theorem yields the existence of a subsequence $\nu_{n_k}$ such that $\nu_{n_k}{\rightharpoonup}\nu$ in $\mathcal{M}_1(H)$. Furthermore, recalling that $P_t$ is Feller on $H$, $\nu$ is an invariant measure for $P_t$ by Krylov-Bogoliubov’s theorem.
Let us now show that $\nu$ is concentrated on $V$. To this end, let $\Theta:H\rightarrow[0,\infty]$ be a lower semicontinuous function such that $$\Theta(y) =
\begin{cases}
|y|_V, &y \in V,\\
+\infty, &y \in H \setminus V
\end{cases}$$ and $\Theta(y)=\sup_{k\in{\mathbb{N}}} \big| {\langle \ell_k,y \rangle} \big|$, where $\{\ell_k\}_{k\in{\mathbb{N}}}$ is a countable dense subset of $B_1^{V'}
\cap H$ in the topology of $H$ (see e.g. [@PreRoeck p. 74]), and $B_1^{V^\prime}$ is the closed unit ball in $V^\prime$. Then $(\ref{sahne})$ implies $$\begin{aligned}
\int_H \Theta(y)^p\,\nu(dy) &= \lim_{L\to\infty}\lim_{M\to\infty}
\int_H \big(\sup_{k\leq L} |{\langle \ell_k,y \rangle}|^p\wedge M \big)\,\nu(dy)\\
&= \sup_{L,M\in{\mathbb{N}}} \lim_{h\to\infty}
\int_H \big(\sup_{k\leq L} |{\langle \ell_k,y \rangle}|^p \wedge M
\big)\,\nu_{n_h}(dy)\\
&\leq \liminf_{h\to\infty} \sup_{L,M\in{\mathbb{N}}}\int_H\big(
\sup_{k\leq L} |{\langle \ell_k,y \rangle}|^p \wedge M \big)\nu_{n_h}(dy)\\
&= \liminf_{h\to\infty}\int_H |y|^p_V\,\nu_{n_h}(dy) < \infty,
\end{aligned}$$ hence $\Theta<\infty$ $\nu$-a.e., thus also $\nu(V)=1$ since $\{
y \in H:\,\Theta(y)<\infty\}=V$.
\[thm:intega\] Let $\nu$ be an invariant measure for $P_t$. Then $\nu$ satisfies the estimate $$\int_H \big( |x|_H^2 + |x|_V^p \big)\,\nu(dx) < \infty.$$
Let $x \in H$ and consider the one dimensional process $U(t):=|u(t,x)|_H^2$, which can be written, in view of \[eq:ito\]), as $$U(t) = |x|_H^2 + \int_0^t F_1(s)\,ds + \int_0^t\!\int_Z
F_2(s,z)\,\bar{\mu}(ds,dz) + \int_0^t\!\int_Z F_3(s,z)\,\mu(ds,dz),$$ where $F_1$, $F_2$, $F_3$ are defined in the obvious way. Let $\chi
\in C^1_b({\mathbb{R}}_+,{\mathbb{R}})$ be a smooth cutoff function with $\chi(x)=1$ for all $x\in[0,1]$, $\chi(x)=0$ for all $x \geq 2$, and $\chi'(x)\leq 0$ for all $x\in{\mathbb{R}}_+$. Setting $\chi_N(x)=\chi(x/N)$ and $\varphi_N(x)=\int_0^y \chi_N(y)\,dy$ for all $x \in {\mathbb{R}}_+$, Itô’s formula yields, suppressing the $\cdot_H$ subscript for semplicity of notation, $$\begin{aligned}
\varphi_N(U(t)) &= \varphi_N(|x|^2)
+ \int_0^t \varphi'_N(U(s-))\,dU(s)\nonumber\\
&\quad + \sum_{s\leq t} \big[ \varphi_N(U(s-)+\Delta U(s))
- \varphi_N(U(s-)) - \varphi'_N(U(s-))\Delta U(s) \big]
\label{eq:rata}
\end{aligned}$$ By Taylor’s formula, there exists $\theta \in (0,1)$ such that the summand in the last term on the right-hand side can be written as $$\varphi''_N\big(U(s-) + \theta\Delta U(s)\big) \, |\Delta U(s)|^2,$$ which is negative ${\mathbb{P}}$-a.s. because $\varphi''_N(x)=\chi'_N(x)=N^{-1}\chi'(x/N) \leq 0$ for all $x \in
{\mathbb{R}}_+$. Moreover, the second term on the right-hand side of (\[eq:rata\]) can be written as $$\begin{gathered}
-2\int_0^t \chi_N(|u(s)|^2){\langle Au(s),u(s) \rangle}\,ds
+ 2\int_0^t\!\int_Z \chi_N(|u(s-)|^2){\langle u(s-),G(u(s-),z) \rangle}
\,\bar{\mu}(ds,dz)\\
+ \int_0^t\!\int_Z \chi_N(|u(s-)|^2)|G(u(s-),z)|^2\,\mu(ds,dz).
\end{gathered}$$ Therefore, taking expectation on both sides of (\[eq:rata\]), recalling that the compensator of $\mu$ is $\mathrm{Leb}\otimes m$, we are left with $$\begin{aligned}
{\mathbb{E}}\varphi_N(|u(t)|^2) &\leq {\mathbb{E}}\varphi_N(|x|^2)
-2{\mathbb{E}}\int_0^t \chi_N(|u(s)|^2){\langle Au(s),u(s) \rangle}\,ds\\
&\quad + {\mathbb{E}}\int_0^t\!\int_Z \chi_N(|u(s)|^2)|G(u(s),z)|^2\,m(dz)\,ds.
\end{aligned}$$ Recalling (\[eq:milch\]), Tonelli’s theorem yields $$\begin{aligned}
&{\mathbb{E}}\varphi_N(|u(t)|^2)
+ \alpha_1\int_0^t {\mathbb{E}}\chi_N(|u(s)|^2) |u(s)|_V^p\,ds\\
&\qquad \leq {\mathbb{E}}\varphi_N(|x|^2)
+ \alpha_0\int_0^t {\mathbb{E}}\chi_N(|u(s)|^2) |u(s)|^2\,ds + tC.
\end{aligned}$$ for all $t\geq0$. Integrating both sides with respect to $\nu$ on $H$, applying again Tonelli’s theorem, the definition of invariant measure, and setting $t=1$, we obtain $$\label{eq:ced}
\alpha_1 \int_H \chi_N(|x|^2) |x|_V^p\,\nu(dx) \leq
\alpha_0 \int_H \chi_N(|x|^2) |x|^2\,\nu(dx) + C.$$ By the inequality $\varepsilon^2|x|^2 \leq \varepsilon^p|x|^p+1$ and the continuity of $V \hookrightarrow H$, we have $$\int_H \chi_N(|x|^2) |x|^2\,\nu(dx) \leq
\varepsilon^{p-2}c^p \int_H \chi_N(|x|^2) |x|_V^p\,\nu(dx)
+ \varepsilon^{-2},$$ hence $$\int_H \chi_N(|x|^2) |x|^2\,\nu(dx) \leq \varepsilon^{-2}
+ \frac{\varepsilon^{p-2}c^p}{\alpha_1} \Big(
\alpha_0 \int_H \chi_N(|x|^2) |x|^2\,\nu(dx) + C \Big).$$ Choosing $\varepsilon$ sufficiently small we get $$\int_H \chi_N(|x|^2) |x|^2\,\nu(dx) \lesssim 1,$$ thus also, by the monotone convergence theorem, $\int_H
|x|^2\,\nu(dx)<\infty$. This immediately yields the result, in view of (\[eq:ced\]).
The estimates just established allow one to deduce the existence of an ergodic invariant measure.
There exists an ergodic invariant measure for the semigroup $P_t$.
The last estimate in the proof of the previous theorem and (\[eq:ced\]) allow to conclude that there exists a constant $N$, independent of $\nu$, such that $$\int_H |x|_V^p\,\nu(dx) < N$$ for any invariant measure $\nu$. Denoting by $\mathcal{N} \subset
\mathcal{M}_1(H)$ the set of invariant measures of $P_t$, Markov’s inequality yields $$\sup_{\nu\in\mathcal{N}} \nu(|x|_V>R) \leq
\frac1{R^p} \sup_{\nu\in\mathcal{N}} \int_H |x|_V^p\,\nu(dx)
< \frac{N}{R^p} \xrightarrow{R\to+\infty} 0.$$ Therefore, by the same argument used in the proof of Theorem \[thm:exim\], we conclude that $\mathcal{N}$ is tight, hence, thanks to Prohorov’s theorem, (relatively) compact in $\mathcal{M}_1(H)$. Since $\mathcal{N}$ is non-empty and convex, Krein-Milman’s theorem ensures that $\mathcal{N}$ has extreme points, which are ergodic invariant measures for $P_t$ by a well-known criterion (see e.g. [@AliBor thm. 19.25]).
Finally, we give a sufficient condition for uniqueness of an invariant measure under an extra superlinearity assumptions on the couple $(A,G)$.
Assume that there exist $\eta>0$ and $\delta>0$ such that $$2{\langle Av-Aw,v-w \rangle} - |G(v,\cdot)-G(w,\cdot)|^2_m
\geq \eta |v-w|_H^{2+\delta},
\qquad\forall v,w\in V.$$ Then $P_t$ has a unique strongly mixing invariant measure.
Let $x,y\in H$. Then Itô’s formula for the square of the norm in $H$ implies, after taking expectations, $$\begin{aligned}
{\mathbb{E}}|u(t,x)-u(t,y)|^2 &+ 2{\mathbb{E}}\int_0^t
{\langle Au(s,x)-Au(s,y),u(s,x)-u(s,y) \rangle}\,ds\\
&\leq |x-y|^2 + {\mathbb{E}}\int_0^t\int_Z|G(u(s,x),z)-G(u(s,y),z)|^2\,m(dz)\,ds\\
&\leq |x-y|^2 - \eta \int_0^t {\mathbb{E}}|u(s,x)-u(s,y)|^{2+\delta}\,ds\\
&\leq |x-y|^2 - \eta \int_0^t \big({\mathbb{E}}|u(s,x)-u(s,y)|^2\big)^{1+\delta/2}\,ds
\end{aligned}$$ for all $t>0$, where we have used Jensen’s inequality in the last step. Since the solution $\zeta:{\mathbb{R}}_+\to{\mathbb{R}}_+$ of the ordinary differential equation $$\zeta' = -\eta \zeta^{1+\delta/2}, \qquad \zeta(0)=|x-y|^2$$ is such that $\lim_{t\to\infty} \zeta(t)=0$ for all $x$, $y \in H$, we conclude by a standard comparison argument that ${\mathbb{E}}|u(t,x)-u(t,y)|^2 \to 0$ as $t\to\infty$.
Let $\nu$ be an invariant measure for $P_t$. Then for any $f \in
C^{0,1}_b(H)$ we have $$\begin{aligned}
\Big| P_tf(x) - \int_H f\,d\nu \Big|
&= \Big| \int_H P_tf(x)\,\nu(dy) - \int_H P_tf(y)\,\nu(dy) \Big|\\
&\leq \int_H |P_tf(x)-P_tf(y)|\,\nu(dy)\\
&\leq |f|_{{\dot{C}^{0,1}}} \int_H \big({\mathbb{E}}|u(t,x)-u(t,y)|^2\big)^{1/2}\,\nu(dy).
\end{aligned}$$ Since $({\mathbb{E}}|u(t,x)-u(t,y)|^2)^{1/2} \leq |x-y|$ and $\int_H
|x-y|\,\nu(dy)<\infty$, we can pass to the limit under the integral sign as $t \to \infty$ by the dominated convergence theorem, thus concluding that $|P_tf(x)-\int_Hf\,d\nu| \to 0$ as $t\to \infty$, and in particular that $\nu$ is the unique invariant measure. Moreover, since $C^1_b(H)$ is dense in $L_2(H,\nu)$, one has that for any $f \in L_2(H,\nu)$, $$\lim_{t \to \infty} P_tf(x) = \int_H f\,d\nu,\qquad x\in H,$$ i.e. $\nu$ is strongly mixing (in particular ergodic) as required.
Essential $m$-dissipativity of the Kolmogorov operator {#sec:Kol}
======================================================
Denoting by $u(\cdot,x)$ the solution to the stochastic equation (\[eq:caro\]), we have proved in the previous section that the semigroup $$P_tf(x) := {\mathbb{E}}f(u(t,x)), \qquad f \in B_b(H)$$ admits a (not necessarily unique) invariant measure $\nu$. As is well-known, $P_t$ can be extended to a strongly continuous Markovian semigroup of contractions on $L_p(H,\nu)$, $p \geq 1$. In the following we shall denote the extension of $P_t$ to $L_p(H,\nu)$ again by $P_t$.
Let us define the operator $(L,D(L))$ in $L_1(H,\nu)$ by $$\begin{aligned}
Lf(x) &= -{\langle Ax,Df(x) \rangle} + \mathcal{I}f(x), \qquad x \in V,\\
\mathcal{I}f(x) &= \int_Z \big[
f(x+G(z)) - f(x) - {\langle Df(x),G(z) \rangle} \big]\,m(dz),\\
D(L) &= \big\{ f \in C^2_b(H) \cap C^1_b(V): \; {\langle Ax,Df(x) \rangle} \in L_1(H,\nu)
\big\}.\end{aligned}$$ Note that the nonlocal term $\mathcal{I}f$ in the definition of $L$ is a well-defined element of $L_1(H,\nu)$ for $f \in C^{1,1}_b(H)$. In fact, the fundamental theorem of calculus yields $$\begin{aligned}
\big| f(x+G(z))&-f(x)-{\langle Df(x),G(z) \rangle} \big|\\
&\leq \Big| \int_0^1 {\langle Df(x+\theta G(z)),G(z) \rangle}\,d\theta
- {\langle Df(x),G(z) \rangle} \Big|
\leq |Df|_{{\dot{C}^{0,1}}} |G(z)|^2\end{aligned}$$ therefore, since $G \in L_2(Z,m)$, we have that $|\mathcal{I}f|
\lesssim 1$, thus also $\mathcal{I}f \in L_1(H,\nu)$.
By a computation based on Itô’s formula one can see that the infinitesimal generator of $P_t$ in $L_1(H,\nu)$ acts on smooth enough functions as the operator $L$ just defined. Since $P_t$ is a contraction for all $t \geq 0$, we have that $(L,D(L))$ is dissipative in $L_1(H,\nu)$. The question of $L_1$-uniqueness then arises naturally: is $P_t$ the only strongly continuous semigroup on $L_1(H,\nu)$ such that its infinitesimal generator extends $(L,D(L))$? Under a “regularizability” hypothesis on $A$, we shall give an affirmative answer to this question, proving that the closure of $L$ in $L_1(H,\nu)$ generates a strongly continuous semigroup. In fact, since $L$ is dissipative, this will imply that the semigroup coincides with $P_t$.
Throughout this section we shall assume that there exists a sequence of monotone operators $A^\varepsilon \in \dot{C}^{0,1}(H \to H) \cap
C^1_b(V \to V')$ such that $A^\varepsilon x \to Ax$ in $V'$ for all $x
\in V$ and $|A^\varepsilon x|_{V'} \leq N(|x|_V^{p-1}+1)$ with $N$ independent of $\varepsilon$.
We are going to prove that $L$ is dissipative in $L_1(H,\nu)$ just assuming that $\nu$ is an infinitesimally invariant for $L$ satisfying the integrability condition $$\label{eq:ippo}
x \mapsto |x|_V^p + |x|_H \in L_1(H,\nu).$$ More precisely, the assumption of $\nu$ being infinitesimally invariant amounts to assuming that $$\int_H Lf\,d\nu = 0 \qquad \forall f \in \mathcal{K},$$ where $\mathcal{K}:=C^{1,1}_b(H) \cap C^1_b(V')$. Note that and $f \in \mathcal{K}$ imply that $Lf \in
L_1(H,\nu)$, so that the above condition is meaningful. In fact, one has $\mathcal{I}f \in L_1(H,\nu)$ for all $f \in C^{1,1}_b(H)$, as seen above, and $$|{\langle Ax,Df(x) \rangle}| \leq |Ax|_{V'} \sup_{y\in V}|Df(y)| \lesssim |x|_V^p + 1
\in L_1(H,\nu).$$ Let us recall that any invariant measure is infinitesimally invariant, but the converse does not hold, in general. Moreover, any invariant measure for (\[eq:caro\]) satisfies the integrability condition (\[eq:ippo\]) thanks to Theorem \[thm:intega\].
The operator $(L,D(L))$ is dissipative, hence closable, in $L_1(H,\nu)$.
Let $f \in \mathcal{K}$ and $\gamma_\varepsilon \in C^2({\mathbb{R}})$ be a convex function with such that $\gamma'_\varepsilon$ is a smooth approximation of the signum graph $$\mathrm{sgn}(x) =
\begin{cases}
-1, & x<0,\\
[-1,1], & x=0,\\
1, & x>0.
\end{cases}$$ Then we have $\gamma_\varepsilon(f) \in \mathcal{K}$ and $$\label{eq:chino}
L\gamma_\varepsilon(f) = {\langle Ax,Df \rangle}\gamma'_\varepsilon(f)
+ \mathcal{I}\gamma_\varepsilon(f),$$ where, by a direct calculation, $$\begin{aligned}
&\mathcal{I}\gamma_\varepsilon(f) - \gamma'_\varepsilon(f) \mathcal{I}f\\
&\qquad = \int_Z \big[\gamma_\varepsilon(f(x+G(z))) - \gamma_\varepsilon(f(x))
- \gamma'_\varepsilon(f(x))\big(f(x+G(z))-f(x)\big)\big]\,m(dz)\\
&\qquad =: R_\varepsilon(f).
\end{aligned}$$ Since $\gamma_\varepsilon$ is convex and differentiable, we infer that $R_\varepsilon(f) \geq 0$. Therefore, taking the previous inequality into account and the infinitesimal invariance of $\nu$, one has, integrating (\[eq:chino\]) with respect to $\nu$, $$\int L\gamma_\varepsilon(f)\,d\nu = 0 = \int \gamma'_\varepsilon(f)\,Lf\,d\nu
+ \int R_\varepsilon(f)\,d\nu,$$ hence $\int \gamma'_\varepsilon(f)\,Lf\,d\nu \leq 0$, and passing to the limit as $\varepsilon \to 0$, $$\int Lf\,\xi\,d\nu \leq 0,$$ where $\xi \in L_\infty(H,\nu)$, $\xi \in \mathrm{sgn}(f)$ $\nu$-a.e. Since $L_1(H,\nu)'=L_\infty(H,\nu)$, recalling that the duality map $J:L_1(H,\nu) \to 2^{L_\infty(H,\nu)}$ is given by $$J: u \mapsto \big\{ v \in L_\infty(H,\nu): \;
v \in |u|_{L_1(H,\nu)} \mathrm{sgn}(u) \quad \nu\text{-a.e.} \big\},$$ we infer by the previous inequality that $L$ is dissipative in $L_1(H,\nu)$.
The following result gives a positive answer to the $L_1$-uniqueness question posed above.
Let $(\bar{L},D(\bar{L}))$ be the closure of the Kolmogorov operator $L$ in $L_1(H,\nu)$. Then $(\bar{L},D(\bar{L}))$ generates a strongly continuous Markovian semigroup of contractions $T_t$ in $L_1(H,\nu)$, for which $\nu$ is an invariant measure.
By the Lumer-Phillips theorem, $\bar{L}$ generates a strongly continuous semigroup of contractions if $R(\alpha I-\bar{L})$ is dense in $L_1(H,\nu)$ for some $\alpha>0$.
Consider the regularized equation $$\label{eq:reg}
du(t) + A^{\varepsilon\lambda} u\,dt = \int_Z G(z)\,d\bar{\mu}(dt,dz),
\qquad u(0)=x \in H,$$ with $$A^{\varepsilon\lambda} x := \int_H e^{\lambda C} A^\varepsilon(e^{\lambda C}x+y)
\gamma_{\frac{1}{2}C^{-1}(e^{2\lambda C}-1)}(dy), \qquad \lambda>0,$$ where $C: D(C) \subset V \to H$ is a self-adjoint, negative definite linear operator such that $C^{-1}$ is of trace class, and $\gamma_Q$ stands for a centered Gaussian measure on $H$ with covariance operator $Q$. Then, by the Cameron-Martin formula, one has $$A^{\varepsilon\lambda} \in C^\infty(H \to H),
\qquad
(A^{\varepsilon\lambda})' \in C_b^\infty(H \to \mathcal{L}(H\to H))$$ and $A^{\varepsilon\lambda}x \to A^\varepsilon x$ for all $x \in H$ as $\lambda \to 0$ (see e.g. [@DP-K [§]{}2.3-2.4] for details). Moreover, $A^{\varepsilon\lambda}$ inherits the monotonicity of $A^\varepsilon$, and $$(A^{\varepsilon\lambda})' x = \int_H e^{\lambda C}
(A^\varepsilon)'(e^{\lambda C}x+y) e^{\lambda C}
\gamma_{\frac{1}{2}C^{-1}(e^{2\lambda C}-1)}(dy),$$ so that $A^{\varepsilon\lambda} \in C^1_b(V \to V')$.
Since $A^{\varepsilon\lambda}$ is Lipschitz continuous on $H$, (\[eq:reg\]) admits a unique strong solution $u_{\varepsilon\lambda}$ (e.g. by [@Met thm. 34.7]). Set $$\label{eq:resolv}
f_{\varepsilon\lambda}(x) := {\mathbb{E}}\int_0^\infty e^{-\alpha t}
\varphi(u_{\varepsilon\lambda}(t,x))\,dt,\qquad x\in H,$$ where $\varphi \in \mathcal{K}$ and $\alpha>0$ are fixed. Since $A^{\varepsilon\lambda} \in C^1(H \to H)$, one has, thanks to [@Met thm. 36.9], that $x \mapsto u_{\varepsilon\lambda}(t,x)$ is Fréchet differentiable for all $t \geq 0$, and its Fréchet derivative acting on an arbitrary $y \in H$, denoted by $v^y_{\varepsilon\lambda}:=Du_{\varepsilon\lambda}[y]$, solves the initial value problem (in the ${\mathbb{P}}$-a.s. sense) $$\label{eq:deri}
\frac{d}{dt}v^y_{\varepsilon\lambda}
+ (A^{\varepsilon\lambda})'(u_{\varepsilon\lambda})
v^y_{\varepsilon\lambda}=0,
\qquad v^y_{\varepsilon\lambda}(0,x)=y.$$ A computation based on Itô’s lemma for the square of the norm and the monotonicity of $A^{\varepsilon\lambda}$ reveals that $x \mapsto
u_{\varepsilon\lambda}(\cdot,x) \in \dot{C}^{0,1}(H \to \mathbb{H}_2(T))$ for all $T\geq 0$, and $$\big| x \mapsto u_{\varepsilon\lambda}(t,x) \big|_{\dot{C}^{0,1}(H \to H)}
\leq 1
\qquad \forall t \geq 0.$$ This immediately implies that $|v^y_{\varepsilon\lambda}| \leq |y|$ for all $y \in H$, as the operator norm of the Fréchet derivative of a Lipschitz continuous function cannot exceed its Lipschitz constant. Moreover, since $(A^{\varepsilon\lambda})'(\xi) \in C^0_b(H \to H)$ for all $\xi \in H$, from (\[eq:deri\]) we infer that $x \mapsto
u_{\varepsilon\lambda}(t,x)$ is continuously differentiable ${\mathbb{P}}$-a.s. for all $t\geq 0$ (e.g. by [@dieudonne [§]{}X.8]). Applying the chain rule for Fréchet derivatives (see e.g. [@AmbPro Prop. 1.4]) in (\[eq:resolv\]), taking into account that $\varphi \in C^{1,1}_b(H)$ and $u_{\varepsilon\lambda}$ is Fréchet differentiable with $|Du_{\varepsilon\lambda}(t)|$ bounded uniformly over $t$, we get $$\label{eq:diffres}
Df_{\varepsilon\lambda}(x)[y] = {\mathbb{E}}\int_0^\infty e^{-\alpha t}
D\varphi(u_{\varepsilon\lambda}(t,x)) v^y_{\varepsilon\lambda}(t,x)\,dt$$ for all $y \in H$, which also immediately yields $$\label{eq:limuf}
\big| Df_{\varepsilon\lambda}(x)[y] \big| \lesssim |y|
\qquad \forall y \in H,$$ that is $f_{\varepsilon\lambda} \in C^1_b(H)$. In order to conclude that $f_{\varepsilon\lambda} \in C^{1,1}_b(H)$ we thus have to prove that $Df_{\varepsilon\lambda} \in \dot{C}^{0,1}(H \to H)$. Let us observe that we can write $$\begin{aligned}
& \big| Df_{\varepsilon\lambda}(x)[y] - Df_{\varepsilon\lambda}(x)[z] \big|\\
&\qquad\quad \leq {\mathbb{E}}\int_0^\infty e^{-\alpha t} \big|
D\varphi(u_{\varepsilon\lambda}(t,x)) v_{\varepsilon\lambda}^y(t,x)
- D\varphi(u_{\varepsilon\lambda}(t,z))v_{\varepsilon\lambda}^y(t,z)
\big|\,dt\\
&\qquad\quad \leq {\mathbb{E}}\int_0^\infty e^{-\alpha t} \big|
D\varphi(u_{\varepsilon\lambda}(t,x)) v_{\varepsilon\lambda}^y(t,x)
- D\varphi(u_{\varepsilon\lambda}(t,x))v_{\varepsilon\lambda}^y(t,z)
\big|\,dt\\
&\qquad\quad\quad + {\mathbb{E}}\int_0^\infty e^{-\alpha t} \big|
D\varphi(u_{\varepsilon\lambda}(t,x)) v_{\varepsilon\lambda}^y(t,z)
- D\varphi(u_{\varepsilon\lambda}(t,z))v_{\varepsilon\lambda}^y(t,z) \big|\,dt,
\end{aligned}$$ where, recalling that $x \mapsto u_{\varepsilon\lambda}(t,x)$ and $v_{\varepsilon\lambda}(t)$ are respectively Lipschitz and bounded uniformly over $\varepsilon$, $\lambda$ and $t$, and that $\varphi \in
C^{1,1}_b(H)$, $$\begin{aligned}
\big| D\varphi(u_{\varepsilon\lambda}(t,x)) v_{\varepsilon\lambda}^y(t,z) -
D\varphi(u_{\varepsilon\lambda}(t,z))v_{\varepsilon\lambda}^y(t,z) \big|
&\leq
|D\varphi|_{{\dot{C}^{0,1}}} |u_{\varepsilon\lambda}(t,x)-u_{\varepsilon\lambda}(t,z)| \,
|v_{\varepsilon\lambda}^y(t,z)|\\
&\lesssim |x-z| \, |y|.
\end{aligned}$$ Moreover, we also have $$\begin{aligned}
\big| D\varphi(u_{\varepsilon\lambda}(t,x)) v_{\varepsilon\lambda}^y(t,x) -
D\varphi(u_{\varepsilon\lambda}(t,x))v_{\varepsilon\lambda}^y(t,z) \big| \leq
|D\varphi|_{C^0(H\to H)} |v_{\varepsilon\lambda}^y(t,x) -
v_{\varepsilon\lambda}^y(t,z)|,
\end{aligned}$$ from which it follows that in order to show that $Df_{\varepsilon\lambda}$ is Lipschitz on $H$ it suffices to prove that $x \mapsto
v_{\varepsilon\lambda}(t,x)$ is Lipschitz on $H$. We have $$\frac{d}{dt} \big(v_{\varepsilon\lambda}^y(t,x)
- v_{\varepsilon\lambda}^y(t,z) \big)
+ (A^{\varepsilon\lambda})'(u_{\varepsilon\lambda}(t,x))
v_{\varepsilon\lambda}^y(t,x)
- (A^{\varepsilon\lambda})'(u_{\varepsilon\lambda}(t,z))
v_{\varepsilon\lambda}^y(t,z) = 0,$$ hence, taking scalar products with $v_{\varepsilon\lambda}^y(t,x) -
v_{\varepsilon\lambda}^y(t,z)$, $$\frac12 \frac{d}{dt} \big| v_{\varepsilon\lambda}^y(t,x) -
v_{\varepsilon\lambda}^y(t,z) \big|^2 + I = 0,$$ where $I \equiv I(\varepsilon,\lambda,t,x,z,y)$ satisfies $$\begin{aligned}
I &=
\big\langle(A^{\varepsilon\lambda})'(u_{\varepsilon\lambda}(t,x))
\big( v_{\varepsilon\lambda}^y(t,x) -
v_{\varepsilon\lambda}^y(t,z)\big),
v_{\varepsilon\lambda}^y(t,x) - v_{\varepsilon\lambda}^y(t,z) \big\rangle\\
&\quad + \big\langle
(A^{\varepsilon\lambda})'(u_{\varepsilon\lambda}(t,x))
v_{\varepsilon\lambda}^y(t,z) -
(A^{\varepsilon\lambda})'(u_{\varepsilon\lambda}(t,z))
v_{\varepsilon\lambda}^y(t,z),
v_{\varepsilon\lambda}^y(t,x) - v_{\varepsilon\lambda}^y(t,z) \big\rangle\\
&\geq \big\langle
(A^{\varepsilon\lambda})'(u_{\varepsilon\lambda}(t,x))
v_{\varepsilon\lambda}^y(t,z)
-
(A^{\varepsilon\lambda})'(u_{\varepsilon\lambda}(t,z))
v_{\varepsilon\lambda}^y(t,z),
v_{\varepsilon\lambda}^y(t,x) - v_{\varepsilon\lambda}^y(t,z)
\big\rangle,
\end{aligned}$$ once one takes into account that $(A^{\varepsilon\lambda})'(u_{\varepsilon\lambda}(t,x))$ is a positive linear operator, because $A^{\varepsilon\lambda}:H \to H$ is monotone and differentiable. Then we also get, recalling that $|v_{\varepsilon\lambda}^y(t,z)| \leq |y|$, $$\begin{aligned}
-I &\leq \frac12 \big| \big(
(A^{\varepsilon\lambda})'(u_{\varepsilon\lambda}(t,x)) -
(A^{\varepsilon\lambda})'(u_{\varepsilon\lambda}(t,z)) \big)
v_{\varepsilon\lambda}^y(t,z) \big|^2\\
&\quad + \frac12 \big| v_{\varepsilon\lambda}^y(t,x)
- v_{\varepsilon\lambda}^y(t,z) \big|^2\\
&\leq \frac12 |y|^2 \, [(A^{\varepsilon\lambda})']^2_1 \, \big|
u_{\varepsilon\lambda}(t,x) - u_{\varepsilon\lambda}(t,z) \big|^2
+ \frac12 \big| v_{\varepsilon\lambda}^y(t,x)
- v_{\varepsilon\lambda}^y(t,z) \big|^2\\
&\lesssim |y|^2 \, |x-z|^2 + \big| v_{\varepsilon\lambda}^y(t,x) -
v_{\varepsilon\lambda}^y(t,z) \big|^2.
\end{aligned}$$ In the last step we have used that $(A^{\varepsilon\lambda})' \in
C_b^{\infty}(H \to \mathcal{L}(H \to H)$ and that $x \mapsto
u_{\varepsilon\lambda}(t,x)$ is Lipschitz. Gronwall’s inequality then yields $$\big| v_{\varepsilon\lambda}(t,x) -
v_{\varepsilon\lambda}(t,z) \big| \lesssim |x-z|,$$ thus concluding the proof that $f_{\varepsilon\lambda} \in C^{1,1}_b(H)$.
Let us now prove that $f_{\varepsilon\lambda} \in C^1_b(V')$: in view of (\[eq:diffres\]), it is enough to prove that $|v_{\varepsilon\lambda}^y(x)|_{V'} \leq |y|_{V'}$. Here we regard $\varphi$ as a function from $V'$ to ${\mathbb{R}}$ and $x \mapsto u_{\varepsilon\lambda}(t,x)$ as a map from $V'$ to itself, so that $v_{\varepsilon\lambda}(t,x) \in
\mathcal{L}(V' \to V')$ and $v_{\varepsilon\lambda}^y(t,x) \in V'$. Let $J:V' \to
V'' \simeq V$ denote the duality map between $V'$ and $V$ (or equivalently, let $J=F^{-1}$, with $F$ the duality map between $V$ and $V'$). Multiplying both sides of (\[eq:deri\]) by $J(v_{\varepsilon\lambda}^y(t,x))$, in the sense of the duality pairing between $V'$ and $V$, we obtain, taking into account that $(A^{\varepsilon\lambda})'$ is positive, $|v_{\varepsilon\lambda}^y(x)|_{V'} \leq |y|_{V'}$. We have thus proved that $f_{\varepsilon\lambda} \in \mathcal{K}$. This in turn implies that $f_{\varepsilon\lambda}$ satisfies $$\begin{gathered}
\alpha f_{\varepsilon\lambda}(x) + {}_{V'}\big\langle A^{\varepsilon\lambda}x,
Df_{\varepsilon\lambda}(x)\big\rangle_V\\
- \int_Z \big[f_{\varepsilon\lambda}(x+G(z)) - f_{\varepsilon\lambda}(x)
- {\langle Df_{\varepsilon\lambda}(x),G(z) \rangle}\big]\,m(dz)
= \varphi(x), \qquad x \in H,
\end{gathered}$$ hence also $$\alpha f_{\varepsilon\lambda}(x) + {\langle Ax,Df_{\varepsilon\lambda}(x) \rangle}
- \mathcal{I}f_{\varepsilon\lambda}(x) = \varphi(x)
+ {\langle Ax-A^{\varepsilon\lambda}x,Df_{\varepsilon\lambda}(x) \rangle},$$ and $$\big| \alpha f_{\varepsilon\lambda} + {\langle Ax,Df_{\varepsilon\lambda} \rangle}
- \mathcal{I}f_{\varepsilon\lambda} \big|_{L_1(H,\nu)} \leq
|\varphi|_{L_1(H,\nu)}
+ \big| {\langle Ax-A^{\varepsilon\lambda}x,Df_{\varepsilon\lambda} \rangle}
\big|_{L_1(H,\nu)}.$$ Note that $|Df_{\varepsilon\lambda}(x)|_V \lesssim 1$ thanks to the above bound on $|v_{\varepsilon\lambda}(x)|_{V'}$, so that $$\begin{aligned}
&\int_H \big|{\langle Ax-A^{\varepsilon\lambda}x,Df_{\varepsilon\lambda}(x) \rangle}
\big|\,\nu(dx)\\
&\qquad\qquad \lesssim
\int_H |Ax-A^{\varepsilon}x|_{V'}\,\nu(dx)
+ \int_H |A^{\varepsilon}x-A^{\varepsilon\lambda}x|_{V'}\,\nu(dx)
\end{aligned}$$ which converges to $0$ as $\lambda \to 0$ and $\varepsilon \to 0$ by the dominated convergence theorem. In fact, thanks to the hypotheses on $A$ and $A^\varepsilon$, we have $|Ax-A^{\varepsilon}x|_{V'}
\lesssim |x|_V^p+1$ for all $x \in V$, and $\nu$ is concentrated on $V$ by (\[eq:ippo\]). Moreover, since $H \hookrightarrow V'$ is continuous and $|A^{\varepsilon\lambda}x| \leq |A^\varepsilon x|$ for all $x \in H$, we have $|A^{\varepsilon}x-A^{\varepsilon\lambda}x|_{V'}| \lesssim |x|_H + 1
\in L_1(H,\nu)$, because of (\[eq:ippo\]). We have thus shown that $$\lim_{\varepsilon\to 0}\,\lim_{\lambda\to 0}
\big( \alpha f_{\varepsilon\lambda} + {\langle Ax,Df_{\varepsilon\lambda} \rangle}
- \mathcal{I}f_{\varepsilon\lambda} \big) = \varphi$$ in $L_1(H,\nu)$, i.e. that $R(\alpha I - L)$ is dense in $L_1(H,\nu)$, because $\mathcal{K}$ is dense in $L_1(H,\nu)$. Since $L$ is also dissipative, we immediately infer that $\bar{L}$ is $m$-dissipative in $L_1(H,\nu)$.
Let us denote the strongly continuous semigroup of contractions on $L_1(H,\nu)$ with generator $\bar{L}$ by $T_t$. Let us now prove that $T_t$ is Markovian: for this it is enough to show that $$\int_H \bar{L}f \, 1_{\{f>1\}}\,d\nu \leq 0
\qquad \forall f \in D(\bar{L})$$ (see e.g. [@Stannat-L1 p. 109]). Let $\gamma_\varepsilon \in
C^2({\mathbb{R}})$ be a convex function such that $\gamma'_\varepsilon$ is a smooth approximation of $x \mapsto 1_{]1,+\infty[}(x)$. Then, proceeding as in the proof of the previous lemma, we obtain the claim for all $f \in \mathcal{K}$ first, and for all $f \in
D(\bar{L})$ by density.
In order to prove that $\nu$ is an invariant measure for $T_t$, let us observe that one has, by definition of infinitesimal invariance and by a density argument, $$\int_H \bar{L}f\,d\nu = 0 \qquad \forall f \in D(\bar{L}).$$ Since $T_tf \in D(\bar{L})$ for all $t \geq 0$ if $f \in
D(\bar{L})$, we have, by the infinitesimal invariance of $\nu$, $$\int_H T_tf\,d\nu = \int_H f\,d\nu + \int_0^t\int_H \bar{L}T_sf\,d\nu\,ds
= \int_H f\,d\nu$$ for all $f \in D(\bar{L})$, thus also for all $f \in L_1(H,\nu)$ by density.
The theorem implies that if $\nu$ is an invariant measure to the stochastic equation (\[eq:caro\]) satisfying the integrability condition, then for all $f \in B_b(H)$, one has that $T_tf$ is a $\nu$-version of $P_tf$ for all $t \geq 0$.
The dissipativity of $L$ in $L_2(H,\nu)$ is easier to prove: in fact, for $f \in \mathcal{K}$, we have $$L(f^2) = 2fLf + \Gamma(f,f),$$ where $$\Gamma(f,f) = \int_Z |f(x+G(z))-f(x)|^2\,m(dz) \geq 0$$ is the so-called carré du champ operator associated to $\mathcal{I}$, which is defined as $$\Gamma(f,g) := \mathcal{I}(fg) - f\mathcal{I}g - g\mathcal{I}f$$ and takes the form $$\Gamma(f,g) = \int_Z \big(f(x+G(z))-f(x)\big) \big(g(x+G(z))-g(x)\big)
\,m(dz).$$ In particular one has the integration by parts formula $$\int f\,Lf\,d\nu = -\int \Gamma(f,f)\,d\nu.$$ However, as one might expect, one needs stronger integrability assumptions on $\nu$ to prove the essential $m$-dissipativity of $L$, e.g. (roughly) of the type $x \mapsto |Ax|^2 \in
L_1(H,\nu)$. Such an assumption would in turn require the data of the problem to be much more regular.
Applications {#sec:ex}
============
SDEs with monotone drift
------------------------
If $V=H={\mathbb{R}}^d$, so that (\[eq:caro\]) reduces to an ordinary stochastic differential equation with monotone drift, our results on ergodicity can be recovered applying [@KryGyo1 Thm. 2], which provides existence and uniqueness of strong solutions (even in a more general situation than that treated here), and [@Sko-asympt Thm. I.25], which establishes boundedness in probability for the solution by a Lyapunov-type criterion. In our case one can choose as Lyapunov function simply $V(x)=|x|^2$.
Stochastic equations with drift in divergence form
--------------------------------------------------
Let $D \subset {\mathbb{R}}^d$ be a bounded domain with smooth boundary, and set $H:=L_2(D)$, $V=\mathring{W}_p^1(D)$, $V'=W_q^{-1}(D)$, with $p>2$, $p^{-1}+q^{-1}=1$. Note that $V \hookrightarrow H$ is compact by a Sobolev embedding theorem (see e.g. [@AmbPro Thm. 0.4]). Consider the operator $A: V \to V'$ defined by $$Au := -{\operatorname{div}}\big(a(\nabla u)\big),$$ which must be interpreted, as usual, as $${\langle Au,v \rangle} = \int_D {\langle a(\nabla u),\nabla v \rangle}_{{\mathbb{R}}^d}\,dx \qquad
\forall v \in V.$$ Here $a \in C^0({\mathbb{R}}^d \to {\mathbb{R}}^d)$ is a monotone function satisfying the polynomial growth condition $|a(x)| \lesssim |x|^{p-1}+1$ and the coercivity condition $xa(x) \gtrsim |x|^p-1$.
Let $a_\varepsilon \in \dot{C}^{0,1}({\mathbb{R}}^d \to {\mathbb{R}}^d)$, $\tilde{a}_\varepsilon(x)=\varepsilon^{-1}(x-(I+\varepsilon a)^{-1}x)$ be the Yosida approximation of $a$, set $a_\varepsilon =
\tilde{a}_\varepsilon \ast \zeta_\varepsilon$, where $\{\zeta_\varepsilon\}$ is a standard sequence of mollifier (in particular $a_\varepsilon \in C^\infty$, $a'_\varepsilon \in
C_b^\infty$), and define the operator $A^\varepsilon$ on smooth functions as $$A^\varepsilon u = -(I-\varepsilon \Delta)^{-1}
{\operatorname{div}}\big(a_\varepsilon(\nabla (I-\varepsilon \Delta)^{-1}u)\big),$$ where $\Delta$ stands for the Dirichlet Laplacian on $D$. We are going to show that $A^\varepsilon$ satisfies the assumptions of the previous section. For this we shall need some elliptic regularity results, which we recall here (see e.g. [@krylov-LectSob [§]{}8.5] for details).
\[lm:kry\] Let $f \in L_p(D)$, $p \geq 2$. Then there exists $\varepsilon_1$ such that, for all $\varepsilon < \varepsilon_1$, there exists a unique solution $u \in \mathring{W}_p^1$ to the equation $$u - \varepsilon \Delta u = f$$ on $D$ with Dirichlet boundary conditions. Moreover $u$ satisfies the estimate $$| u |_{L_p(D)} + \varepsilon^{1/2} | u |_{W_p^1(D)} \leq N
| f |_{L_p(D)},$$ where $N$ does not depend on $\varepsilon$.
Let us first show that $A^\varepsilon$ is well-defined both as an operator from $H$ to itself, as well as from $V$ to $V'$. Using the notation $$v^{(\varepsilon)} = (I-\varepsilon \Delta)^{-1}v,$$ we may write $$\label{eq:ape}
{\langle A^\varepsilon u,v \rangle} = \int_D
{\langle a_\varepsilon(\nabla {{u}^{(\varepsilon)}}),\nabla {{v}^{(\varepsilon)}} \rangle}_{{\mathbb{R}}^d}\,dx.$$ Note that if $v \in H$, then ${{v}^{(\varepsilon)}} \in \mathring{W}_2^1$ and $$|\nabla{{v}^{(\varepsilon)}}|_H \leq |{{v}^{(\varepsilon)}}|_{W_2^1(D)} \lesssim_\varepsilon |v|_H.$$ Moreover, since $a_\varepsilon$ is Lipschitz continuous, we have $$|a_\varepsilon(\nabla {{u}^{(\varepsilon)}})| \leq |a_\varepsilon(\nabla {{u}^{(\varepsilon)}})
- a_\varepsilon(0)| + |a_\varepsilon(0)|
\lesssim |\nabla {{u}^{(\varepsilon)}}| + |a_\varepsilon(0)|,$$ thus also $$\big| {\langle A^\varepsilon u,v \rangle} \big| \leq
|a_\varepsilon(\nabla {{u}^{(\varepsilon)}})|_H \, |\nabla {{v}^{(\varepsilon)}}|_H
\lesssim (|u|_H+a_\varepsilon(0))|v|_H,$$ which shows that $A^\varepsilon$ is well-defined from $H$ to itself. Similarly, if $u$, $v \in V=\mathring{W}_p^1(D)$, we have, by Hölder’s inequality, $$\big| {\langle A^\varepsilon u,v \rangle} \big| \leq
|a_\varepsilon(\nabla {{u}^{(\varepsilon)}})|_{L_q(D)} \, |\nabla {{v}^{(\varepsilon)}}|_{L_p(D)}
\lesssim (|u|_V+1)|v|_V,$$ where we have used again Lemma \[lm:kry\] and $\|\cdot\|_{L_q(D)}
\lesssim \|\cdot\|_{L_p(D)}$ for $p>q$ and $D$ bounded. We have thus shown that $A^\varepsilon$ is well-defined from $V$ to $V'$.
The monotonicity of $A^\varepsilon$, both as an operator from $H$ to itself and from $V$ to $V'$ is immediate by (\[eq:ape\]) and the monotonicity of $a_\varepsilon$.
Let us now show that $A^\varepsilon$ is Lipschitz continuous on $H$. In fact, taking into account Lemma \[lm:kry\], we have $$\begin{aligned}
\big| {\langle A^\varepsilon u - A^\varepsilon v,w \rangle} \big|
&= \big| \big\langle a_\varepsilon (\nabla{{u}^{(\varepsilon)}}) -
a_\varepsilon (\nabla{{v}^{(\varepsilon)}}), \nabla{{w}^{(\varepsilon)}} \big\rangle \big|\\
&\lesssim_\varepsilon | \nabla({{v}^{(\varepsilon)}} - {{w}^{(\varepsilon)}})| \, |\nabla{{w}^{(\varepsilon)}}|
\lesssim_\varepsilon |u - v|_H \, |w|_H.\end{aligned}$$ Since $a_\varepsilon \in C^1$, a direct computation yields that $A^\varepsilon$ is Gâteaux differentiable from $V$ to $V'$ with Gâteaux differential $$\label{eq:aped}
\big\langle (A^\varepsilon)'(u)[v],w \big\rangle =
\int_D \big\langle a'_\varepsilon(\nabla{{u}^{(\varepsilon)}}) \nabla{{v}^{(\varepsilon)}},\nabla{{w}^{(\varepsilon)}}
\big\rangle_{{\mathbb{R}}^d}\,dx$$ for all $u$, $v$, $w \in V$. Note that the integral is well defined because $|a'_\varepsilon(x)| \lesssim 1$ for all $x \in {\mathbb{R}}^d$, since $a_\varepsilon$ is Lipschitz continuous. By a well-known criterion, we can conclude that $A^\varepsilon \in C^1(V \to V')$ if we show that $(A^\varepsilon)'$ in (\[eq:aped\]) is continuous as a map $V \to \mathcal{L}(V \to V')$. Let $u_n \to u$ in $\mathring{W}_p^1(D)$ as $n \to \infty$: applying Hölder’s inequality and Lemma \[lm:kry\] repeatedly, we obtain $$\begin{aligned}
&\sup_{|v|_V \leq 1} \, \sup_{|w|_{V} \leq 1}
\big\langle (A^\varepsilon)'(u_n)[v]-(A^\varepsilon)'(u)[v],w
\big\rangle\\
&\qquad\qquad\leq |\nabla{{w}^{(\varepsilon)}}|_{L_p(D)} \,
\big| \big(a'_\varepsilon(\nabla{{u}^{(\varepsilon)}}_n)
- a'_\varepsilon(\nabla{{u}^{(\varepsilon)}})\big)\nabla{{v}^{(\varepsilon)}}
\big|_{L_{p/(p-1)}(D)}\\
&\qquad\qquad\lesssim |\nabla{{v}^{(\varepsilon)}}|_{L_p(D)}
\big| a'_\varepsilon(\nabla{{u}^{(\varepsilon)}}_n)
- a'_\varepsilon(\nabla{{u}^{(\varepsilon)}}) \big|_{L_{p/(p-2)}(D)}\\
&\qquad\qquad\lesssim \big| a'_\varepsilon(\nabla{{u}^{(\varepsilon)}}_n)
- a'_\varepsilon(\nabla{{u}^{(\varepsilon)}}) \big|_{L_{p/(p-2)}(D)}
\xrightarrow{n \to \infty} 0.\end{aligned}$$ In fact, since $a'_\varepsilon$ is Lipschitz, it follows that $|a'_\varepsilon(x)| \lesssim |x|^{p-2}+1$, and $\nabla{{u}^{(\varepsilon)}}_n \to
\nabla{{u}^{(\varepsilon)}}$ in $L_p$ implies convergence a.e. on a subsequence, from which we can conclude by the dominated convergence theorem (see e.g. [@AmbPro Thm. 1.2.6] for complete details in a similar situation). We have thus proved that $A^\varepsilon \in C^1(V
\to V')$.
We conclude proving that $$\lim_{\varepsilon \to 0} |A^\varepsilon u - Au|_{V'} = 0
\qquad \forall u \in V.$$ We have $$\begin{gathered}
\int_D \big| \big\langle a_\varepsilon(\nabla{{u}^{(\varepsilon)}}),\nabla{{w}^{(\varepsilon)}} \big\rangle
- \big\langle a(\nabla u),\nabla w \big\rangle \big|\,dx\\
\leq
\int_D \big| \big\langle a_\varepsilon(\nabla{{u}^{(\varepsilon)}}) - a(\nabla u),
\nabla w \big\rangle \big|\,dx
+ \int_D \big| \big\langle a_\varepsilon(\nabla{{u}^{(\varepsilon)}}),
\nabla{{w}^{(\varepsilon)}} - \nabla w \big\rangle \big|\,dx.\end{gathered}$$ Since $|a_\varepsilon(x)| \leq N(|x|^{p-1}+1)$ with $N$ independent of $\varepsilon$, the second term on the right-hand side can be majorized by $$\big| a_\varepsilon(\nabla{{u}^{(\varepsilon)}}) \big|_{L_q} \,
\big| \nabla{{w}^{(\varepsilon)}} - \nabla w \big|_{L_p}
\lesssim \big(\big| \nabla u \big|^{p-1}_{L_p} +1 \big)\,
\big| \nabla{{w}^{(\varepsilon)}} - \nabla w \big|_{L_p}
\xrightarrow{\varepsilon\to 0} 0,$$ where we have used once again Lemma \[lm:kry\]. Since $\nabla{{u}^{(\varepsilon)}} \to \nabla u$ in $L_p$ as $\varepsilon \to 0$, we can upgrade the convergence to a.e. convergence, passing to a subsequence, still denoted by $\varepsilon$. By Egorov’s theorem, there exists $D_\delta \subset D$, $|D \setminus D_\delta| \leq
\delta$, such that $\nabla{{u}^{(\varepsilon)}} \to \nabla u$ uniformly on $D_\delta$ as $\varepsilon \to 0$. Since $a_\varepsilon$ and its limit function $a$ are continuous on ${\mathbb{R}}^d$, we have $$\lim_{\varepsilon \to 0} \lim_{\eta \to 0}
a_\varepsilon(\nabla u^{(\eta)}) =
\lim_{\eta \to 0} \lim_{\varepsilon \to 0} a_\varepsilon(\nabla u^{(\eta)}) =
a(\nabla u)$$ pointwise on $D_\delta$, hence by a diagonal extraction argument, there exists a further subsequence of $\varepsilon$, still denoted by $\varepsilon$, such that, by the dominated convergence theorem, $$\big| a_\varepsilon(\nabla{{u}^{(\varepsilon)}}) - a(\nabla u) \big|_{L_q(D_\delta)}
\xrightarrow{\varepsilon\to 0} 0.$$ On the other hand, we have $$\begin{aligned}
\big| a_\varepsilon(\nabla{{u}^{(\varepsilon)}}) - a(\nabla u) \big|_{L_q(D \setminus D_\delta)}
&\lesssim \int_{D \setminus D_\delta} \big(|\nabla u|^p +1\big)\,dx\\
&\lesssim |D \setminus D_\delta| \, \big(|\nabla u|_{L_p}+1\big)
\leq \delta \big(|\nabla u|_{L_p}+1\big).\end{aligned}$$ Since $\delta$ is arbitrary, we conclude that the integral above converges to zero as $\varepsilon \to 0$, thus finishing the proof.
Stochastic porous media equations
---------------------------------
Let $D$, $\Delta$, $p$, $q$, and $\{\zeta_\varepsilon\}$ be defined as in the previous subsection. Set $V=L_p(D)$, $H=W_2^{-1}(D)$, $V'=\Delta(L_q(D))$, so that $V \hookrightarrow H$ compactly by a Sobolev embedding theorem (see e.g. [@Tri3 Prop. 4.6]). The norm in $\mathring{W}_2^{-1}(D)$ will be denoted by $|\cdot|_{-1}$. Consider the operator $$\begin{aligned}
A: V &\to V'\\
u &\mapsto -\Delta\beta(u),\end{aligned}$$ where $\beta \in C^0({\mathbb{R}})$ is increasing and satisfies $$x\beta(x) \gtrsim |x|^{p}-1, \qquad
|\beta(x)| \lesssim |x|^{p-1} + 1$$ for all $x \in {\mathbb{R}}$. Note that these conditions on $\beta$ imply that $A$ is well-defined (see e.g. [@PreRoeck [§]{}4.1] for details). Set $$\beta_\varepsilon(x) = \tilde{\beta}_\varepsilon \ast \zeta_\varepsilon,
\qquad
\tilde{\beta}_\varepsilon =
-\varepsilon^{-1} \vee \beta(x) \wedge \varepsilon^{-1},$$ so that $\beta_\varepsilon \in C^\infty_b$, and define the operator $$A^\varepsilon u := -\Delta(I-\varepsilon\Delta)^{-1}
\beta_\varepsilon\big((I-\varepsilon\Delta)^{-1}u\big)$$ on smooth functions. Then $A^\varepsilon$ is well-defined as an operator from $H$ to itself, since $${\langle A^\varepsilon u,w \rangle}_{-1} = \int_D \beta_\varepsilon({{u}^{(\varepsilon)}}) {{w}^{(\varepsilon)}}\,dx
\leq \big| {{w}^{(\varepsilon)}} \big|_{L_2(D)} \,
\big| \beta_\varepsilon({{u}^{(\varepsilon)}}) \big|_{L_2(D)}
\lesssim |w|_{-1} \big( |u|_{-1}+1 \big)$$ for all $u$, $w \in \mathring{W}_2^{-1}(D)$, because $\beta_\varepsilon$ is Lipschitz and $|{{u}^{(\varepsilon)}}|_{L_2(D)} \lesssim
|u|_{-1}$ (see e.g. [@barbu-pde Thm. 3.3.1]). A completely analogous computation also shows that $A^\varepsilon \in
\dot{C}^{0,1}(H \to H)$. Let us also show that $A^\varepsilon$ is well-defined as an operator from $V$ to $V'$: for $u$, $w \in L_p(D)$, Hölder’s inequality yields $$\begin{aligned}
{\langle A^\varepsilon u,w \rangle} = \int_D \beta_\varepsilon({{u}^{(\varepsilon)}}) {{w}^{(\varepsilon)}}\,dx
\leq \big| {{w}^{(\varepsilon)}} \big|_{L_p(D)}
\big| \beta_\varepsilon({{u}^{(\varepsilon)}}) \big|_{L_q(D)}
\lesssim |w|_{L_p(D)} \big( |u|_{L_p(D)} + 1 \big),\end{aligned}$$ where we have used Lemma \[lm:kry\] and the estimate $|\beta_\varepsilon(x)| \leq |\beta(x)| \lesssim |x|^{p-1} + 1$. The latter also immediately implies that $|A^\varepsilon x|_{V'} \leq
N(|x|_V^{p-1}+1)$, with $N$ independent of $\varepsilon$.
As in the previous subsection, it is not difficult to see that $A^\varepsilon$ is Gâteaux differentiable from $V$ to $V'$, with Gâteaux differential $$\big\langle (A^\varepsilon)'u[v],w \big\rangle =
\int_D \beta'_\varepsilon({{u}^{(\varepsilon)}}){{v}^{(\varepsilon)}}{{w}^{(\varepsilon)}},
\qquad u,\,v,\,w \in L_p(D).$$ The continuity of the Gâteaux differential (hence the Fréchet differentiability of $A^\varepsilon:V \to V'$) follows by an argument similar to the one used in the previous subsection, and we shall be more concise here: for $u_n \to u$ in $L_p(D)$, we have $$\begin{aligned}
\big\langle (A^\varepsilon)'(u_n)[v] - (A^\varepsilon)'(u)[v],w \big\rangle
&\leq \big| {{w}^{(\varepsilon)}} \big|_{L_p(D)} \big| [\beta'_\varepsilon({{u}^{(\varepsilon)}}_n)
- \beta'_\varepsilon({{u}^{(\varepsilon)}})] {{v}^{(\varepsilon)}} \big|_{L_{p/(p-1)}(D)}\\
&\lesssim |v|_{L_p(D)} \, |w|_{L_p(D)} \,
\big| \beta'_\varepsilon({{u}^{(\varepsilon)}}_n)-\beta'_\varepsilon({{u}^{(\varepsilon)}})
\big|_{L_{p/(p-2)}(D)}.\end{aligned}$$ We proceed now as above: since ${{u}^{(\varepsilon)}}_n \to {{u}^{(\varepsilon)}}$ a.e. along a subsequence, we can appeal to the dominated convergence theorem, in view of the obvious bound $|\beta'_\varepsilon(x)| \lesssim |x|^{p-2}+1$.
The proof that $A^\varepsilon u \to Au$ in $V'$ for all $u \in V$ as $\varepsilon \to 0$ is completely similar to the corresponding proof in the previous subsection, hence omitted.
[^1]: The work for this paper was carried out while the first-named author was visiting the Department of Statistics of Purdue University supported by a MOIF fellowship.
|
{
"pile_set_name": "ArXiv"
}
|
\[section\] \[section\] \[section\] \[theor\] \[section\]
**ON THE J-FLOW IN HIGHER DIMENSIONS**
**AND THE LOWER BOUNDEDNESS OF**
**THE MABUCHI ENERGY**
Ben Weinkove [^1]
Department of Mathematics, Harvard University
1 Oxford Street, Cambridge, MA 02138
[**Abstract.** ]{} The J-flow is a parabolic flow on Kähler manifolds. It was defined by Donaldson in the setting of moment maps and by Chen as the gradient flow of the J-functional appearing in his formula for the Mabuchi energy. It is shown here that under a certain condition on the initial data, the J-flow converges to a critical metric. This is a generalization to higher dimensions of the author’s previous work on Kähler surfaces. A corollary of this is the lower boundedness of the Mabuchi energy on Kähler classes satisfying a certain inequality when the first Chern class of the manifold is negative.
[**1. Introduction**]{}
The J-flow is a parabolic flow of potentials on Kähler manifolds with two Kähler classes. It was defined by Donaldson [@D1] in the setting of moment maps and by Chen [@C1] as the gradient flow for the J-functional appearing in his [@C1] formula for the Mabuchi energy [@Ma]. Chen [@C2] proved long-time existence of the flow for any smooth initial data, and proved a convergence result in the case of non-negative bisectional curvature.
The J-flow is defined as follows. Let $(M, \omega)$ be a compact Kähler manifold of complex dimension $n$ and let $\chi_0$ be another Kähler form on $M$. Let $\mathcal{H}$ be the space of Kähler potentials $$\mathcal{H} = \{ \phi \in C^{\infty}(M) \ | \ \chi_{\phi} = \chi_0 +
\frac{\sqrt{-1}}{2} \partial {\overline{\partial}}\phi >0 \}.$$ The J-flow is the flow in $\mathcal{H}$ given by $$\label{eqnJflow}
\left\{
\begin{array}{rcl}
{\displaystyle {\frac{\partial \phi_t}{\partial t}}} & = & \displaystyle{ c - \frac{\omega \wedge
\chi_{\phi_t}^{n-1}}{\chi_{\phi_t}^n}}
\\
\phi_0 & = & 0, \end{array} \right.$$ where $c$ is the constant given by $$c = \frac{ \int_M \omega \wedge \chi_0^{n-1}}{\int_M \chi_0^n}.$$ A critical point of the J-flow gives a Kähler metric $\chi$ satisfying $$\label{maineqn}
\omega \wedge \chi^{n-1} = c \chi^n.$$ Donaldson [@D1] showed that a necessary condition for a solution to (\[maineqn\]) in $[\chi_0]$ is that $[nc \chi_0 - \omega]$ be a positive class. He remarked that a natural conjecture would be that this condition be sufficient. Chen [@C1] proved this result if $n=2$, without using the J-flow. In [@W1], the author gave an alternative proof by showing that for $n=2$, the J-flow converges in $C^{\infty}$ to a critical metric under the condition $nc \chi_0 - \omega>0$.
We generalize this as follows.
[**Main Theorem.**]{} [*If the Kähler metrics $\omega$ and $\chi_0$ satisfy $$nc \chi_0 - (n-1)\omega >0,$$ then the J-flow (\[eqnJflow\]) converges in $C^{\infty}$ to a smooth critical metric.*]{}
This shows that (\[maineqn\]) has a solution in $[\chi_0]$ under the condition $$nc [\chi_0] - (n-1) [\omega]>0.$$
Recall that the Mabuchi energy is a functional on $\mathcal{H}$ defined by $$M_{\chi_0}(\phi) = - \int_0^1 \int_M {\frac{\partial \phi_t}{\partial t}} (R_t-\underline{R})
\frac{\chi_{\phi_t}^n}{n!} dt,$$ where $\{ \phi_t \}_{0 \le t \le 1}$ is a path in $\mathcal{H}$ between $0$ and $\phi$, $R_t$ is the scalar curvature of $\chi_{\phi_t}$ and $\underline{R}$ is the average of the scalar curvature. The critical points of this functional are metrics of constant scalar curvature.
If $c_1(M)<0$ then there exists a Kähler-Einstein metric in the class $-c_1(M)$ ([@Y1], [@Au]). It follows easily that the Mabuchi energy is bounded below in this class. Also, the result of Donaldson [@D3] implies that $M$ is asymptotically Chow stable with respect to the canonical bundle. More generally, for any class, it is expected that the lower boundedness of the Mabuchi energy should be equivalent to some notion of semistability in the sense of geometric invariant theory (see [@Y2], [@T2; @T3], [@PS] and [@D4]).
Chen [@C1] shows that if $c_1(M)$ is negative with $-\omega \in c_1(M)$ and if there is a solution of (\[maineqn\]) in $[\chi_0]$, then the Mabuchi energy is bounded below [*in the class $[\chi_0]$*]{}. This suggests that (\[maineqn\]) is related to how a change of polarization affects the condition of stability of a manifold. We immediately have the following corollary of the main theorem, generalizing the result of Chen for dimension $n=2$.
[**Corollary.**]{} [*Let $M$ be a compact Kähler manifold with $c_1(M)<0$. If the Kähler class $[\chi_0]$ satisfies the inequality $$\label{classinequality}
- n \frac{c_1(M) \cdot [\chi_0]^{n-1}}{[\chi_0]^n} [\chi_0] + (n-1) c_1(M) >0,$$ then the Mabuchi energy is bounded below on $[\chi_0]$.*]{}
Note that if $[\chi_0]=[-c_1(M)]$ then the inequality is more than adequately satisfied, and so the set of $[\chi_0]$ satisfying the condition is a reasonably large open set containing the canonical class.
In section 2, we prove a second order estimate of $\phi$ in terms of $\phi$ itself, and in section 3, we prove the zero order estimate and complete the proof of the main theorem. The techniques used are generalizations of those given in [@W1], and we will refer the reader to that paper for some of the calculations and arguments.
In the following, $C_0, C_1, C_2, \ldots$ will denote constants depending only on the initial data.
[**2. Second order estimate**]{}
We use the maximum principle to prove the following estimate on the second derivatives on $\phi$.
\[theoremC2\] Suppose that $$nc \chi_0 - (n-1) \omega >0.$$ Let $\phi=\phi_t$ be a solution of the J-flow (\[eqnJflow\]) on $[0,
\infty)$. Then there exist constants $A>0$ and $C>0$ depending only on the initial data such that for any time $t\ge 0$, $\chi = \chi_{\phi_t}$ satisfies $$\label{eqnC2}
\Lambda_{\omega} \chi \le C e^{A (\phi - \inf_{M \times [0,t]} \phi)}.$$
We will assume that $\omega$ has been scaled so that $c=1/n$. Choose $0 < \epsilon < 1/(n+1)$ to be sufficiently small so that $$\label{eqnepsilon}
\chi_0 \ge (n-1 + (n+1)\epsilon) \omega.$$ We will use the same notation as in [@W1]. In particular, the operator ${\tilde{\triangle}}$ acts on functions $f$ by $${\tilde{\triangle}}f = \frac{1}{n} {h^{k \overline{l}}} {\partial_{k}} {\partial_{\overline{l}}} f, \qquad \textrm{where } \qquad
{h^{k \overline{l}}} =
{\chi^{k \overline{j}}} {\chi^{i \overline{l}}} {g_{i \overline{j}}}.$$ We calculate the evolution of $(\log(\Lambda_{\omega} \chi) - A
\phi)$, where $A$ is a constant to be determined. From [@W1], we have $$\begin{aligned}
\lefteqn{({\tilde{\triangle}}- {\frac{\partial }{\partial t}}) (\log (\Lambda_{\omega} \chi) - A\phi)} \\
& \ge & \mbox{} \frac{1}{n} (-C_0
{h^{k \overline{l}}} {g_{k \overline{l}}} - \frac{1}{\Lambda_{\omega} \chi} {\chi^{k \overline{l}}} R_{k
\overline{l}} - 2A {\chi^{i \overline{j}}} {g_{i \overline{j}}} + A{h^{k \overline{l}}} {\chi_{0 \, k \overline{l}}} + A) \\
& = & \frac{1}{n}(-C_0
{h^{k \overline{l}}} {g_{k \overline{l}}} - \frac{1}{\Lambda_{\omega} \chi} {\chi^{k \overline{l}}} R_{k
\overline{l}} - 2A {\chi^{i \overline{j}}} {g_{i \overline{j}}} + (1-\epsilon) A{h^{k \overline{l}}} {\chi_{0 \, k \overline{l}}} \\
&& \mbox{} +
\epsilon A {h^{k \overline{l}}} {\chi_{0 \, k \overline{l}}} + A),\end{aligned}$$ where $C_0$ is a lower bound for the bisectional curvature of $\omega$, and $R_{k \overline{l}}$ is the Ricci curvature tensor of $\omega$. Recall from (2.4) in [@W1], that $\chi$ is uniformly bounded away from zero. Hence we can choose $A$ to be large enough so that $$\epsilon A {h^{k \overline{l}}} {\chi_{0 \, k \overline{l}}} \ge C_0 {h^{k \overline{l}}} {g_{k \overline{l}}} +
\frac{1}{\Lambda_{\omega} \chi} {\chi^{k \overline{l}}} R_{k
\overline{l}}.$$ Now fix a time $t>0$. There is a point $(x_0, t_0)$ in $M \times [0,t]$ at which the maximum of $(\log(\Lambda_{\omega} \chi) - A
\phi)$ is achieved. We may assume that $t_0>0$. Then at this point $(x_0,
t_0)$, we have $$1 + (1-\epsilon) {h^{k \overline{l}}} {\chi_{0 \, k \overline{l}}} - 2 {\chi^{i \overline{j}}} {g_{i \overline{j}}}\le 0.$$ From (\[eqnepsilon\]), we get $$1 + (n-1 + \epsilon) {h^{k \overline{l}}} {g_{k \overline{l}}} - 2{\chi^{i \overline{j}}} {g_{i \overline{j}}} \le 0.$$ We will compute in normal coordinates at $x_0$ for $\omega$ in which $\chi$ is diagonal and has eigenvalues $\lambda_1 \le \ldots \le \lambda_n$. The above inequality becomes $$\label{maininequality}
1 + (n-1 + \epsilon) \sum_{i=1}^n
\frac{1}{\lambda_i^2} - 2 \sum_{i=1}^{n} \frac{1}{\lambda_i} \le 0.$$ We claim that (\[maininequality\]) gives an upper bound for the $\lambda_i$. To see this, complete the square to obtain $$\sum_{i=1}^n \left(
\frac{1}{\sqrt{n-1+\epsilon}} - \frac{\sqrt{n-1+\epsilon}}{\lambda_i}
\right)^2 \le \frac{n}{n-1+\epsilon} -1.$$ Hence, for $i=1, \ldots, n$, $$\frac{1}{\sqrt{n-1+\epsilon}} -\frac{\sqrt{n-1+\epsilon}}{\lambda_i}\le
\frac{\sqrt{1-\epsilon}}{\sqrt{n-1+\epsilon}},$$ from which we obtain the upper bound $$\lambda_i \le \frac{n-1+\epsilon}{1-\sqrt{1-\epsilon}}.$$ Hence at the point $(x_0, t_0)$, we have a constant $C$ depending only on the initial data such that $$\Lambda_{\omega} \chi \le C.$$ Then, on $M \times [0,t]$, $$\log (\Lambda_{\omega} \chi) - A\phi \le \log C - A\inf_{M \times [0,t]}
\phi.$$ Exponentiating gives $$\Lambda_{\omega} \chi \le C e^{A(\phi - \inf_{M \times [0,t]} \phi)},$$ completing the proof of the theorem.
[**3. Proof of the Main Theorem**]{}
We know from [@C2] that the flow exists for all time. To prove the main theorem we need uniform estimates on $\phi_t$ and all of its derivatives. Given such estimates, the argument of section 5 of [@W1], which is valid for any dimension, shows that $\phi_t$ converges in $C^{\infty}$ to a smooth critical metric.
From Theorem \[theoremC2\], and standard parabolic methods, it suffices to have a uniform $C^0$ estimate on $\phi$. We prove this below, generalizing the method of [@W1], using the precise form of the estimate (\[eqnC2\]) and a Moser iteration type argument.
\[theoremC0\]
Suppose that $$nc \chi_0 - (n-1) \omega >0.$$ Let $\phi=\phi_t$ be a solution of the J-flow (\[eqnJflow\]) on $[0,
\infty)$. Then there exists a constant $\tilde{C}>0$ depending only on the initial data such that $$\| \phi_t \|_{C^0} \le \tilde{C}.$$
We begin with a lemma.
\[lemmasup\] $0 \le \sup_M \phi_t \le - C_1 \inf_M \phi_t +
C_2.$
We will use the functional $I_{\chi_0}$ defined on $\mathcal{H}$ by $$\label{eqnI}
I_{\chi_0}(\phi) = \int_0^1 \int_M {\frac{\partial \phi_t}{\partial t}} \frac{\chi_{\phi_t}^n}{n!}dt,$$ for $\{\phi_t\}$ a path between $0$ and $\phi$ (this is a well-known functional, see [@D2] for example). Taking the path $\phi_t = t\phi$, we obtain the formula: $$\begin{aligned}
\nonumber
I_{\chi_0}(\phi) & = & \frac{1}{n!} \int_0^1 \int_M \phi \, \chi_{t \phi}^n dt \\
\nonumber
& = & \frac{1}{n!} \int_0^1 \int_M \phi \, (t \chi_{\phi} + (1-t) \chi_0)^n dt \\
\label{eqnI2}
& = & \frac{1}{n!} \sum_{k=0}^n \left[ \left(\genfrac{}{}{0pt}{}{n}{k} \right)
\int_0^1 t^k(1-t)^{n-k}dt \right] \int_M \phi \, \chi_{\phi}^k \wedge
\chi_0^{n-k}.\end{aligned}$$ From (\[eqnI\]), we see that $I(\phi_t)=0$ along the flow. The first inequality then follows immediately, since the expression in the square brackets in (\[eqnI2\]) is a positive function of $n$ and $k$. The second inequality follows from (\[eqnI2\]), the fact that $\triangle_{\omega}
\phi_t > - \Lambda_{\omega} \chi_0$, and properties of the Green’s function of $\omega$.
From this lemma, it is sufficient to prove a lower bound for $\inf_M \phi_t$. If such a lower bound does not exist, then we can choose a sequence of times $t_i
\rightarrow \infty$ such that
1. $\inf_M \phi_{t_i} = \inf_{t \in [0,t_i]} \inf_M \phi_t$
2. $\inf_M \phi_{t_i} \rightarrow -\infty.$
We will find a contradiction. Set $B = A/(1-\delta)$ where $A$ is the constant from (\[eqnC2\]), and let $\delta$ be a small positive constant to be determined later. Let $$\psi_{i} = \phi_{t_i} - \sup_M \phi_{t_i},$$ and let $u= e^{-B\psi_{i}}$. We will show that $u$ is uniformly bounded from above, which will give the contradiction. First, we have the following lemma.
For any $p \ge 1$, $$\label{estimateu}
\int_M | \nabla u^{p/2} |^2 \frac{\omega^n}{n!} \le C_3 \, p \, \|u
\|_{C^0}^{1-\delta}
\int_M u^{p-(1-\delta)} \frac{\omega^n}{n!}.$$
The proof is given for $n=2$ in [@W1], and since the same argument works for any dimension, we will not reproduce it here. Crucially, the proof uses the estimate (\[eqnC2\]).
We will use the notation $$\| f \|_{c} = \left( \int_M |f|^c \frac{\omega^n}{n!} \right)^{1/c},$$ for $c>0$. It is not a norm for $0 < c <1$ but this fact is not important. The following lemma allows us to estimate the $C^0$ norm of $u$ using a Moser iteration type method (compare to [@Y1]).
If $u \ge 0$ satisfies the estimate (\[estimateu\]) for all $p \ge 1$, then for some constant $C'$ independent of $u$, $$\| u \|_{C^0} \le C' \| u \|_{\delta}.$$
For $\beta = n/(n-1)$, the Sobolev inequality for functions $f$ on $(M, \omega)$ is $$\| f \|_{2 \beta}^2 \le C_4 (\| \nabla f \|_2^2 + \|f \|_2^2).$$ Applying this to $u^{p/2}$ and making use of (\[estimateu\]) gives $$\|u \|_{p\beta} \le C_5^{1/p} p^{1/p} \|u \|_{C^0}^{\gamma/p} \| u
\|_{p-\gamma}^{(p - \gamma)/p},$$ for $\gamma = 1 - \delta$. By replacing $p$ with $p \beta + \gamma$ we obtain inductively $$\|u \|_{p_k \beta} \le C(k) \| u \|_{C^0}^{1-a(k)} \| u
\|_{p-\gamma}^{a(k)},$$ where $$\begin{aligned}
p_k & = & p \beta^k + \gamma (1 + \beta + \beta^2 + \cdots + \beta^{k-1}) \\
C(k) & = & C_5^{(1+ \beta + \cdots + \beta^k)/p_k} p_0^{\beta^k/p_k}
p_1^{\beta^{k-1}/p_k} \cdots p_k^{1/p_k} \\
a(k) & = & \frac{(p-\gamma)\beta^k}{p_k}.\end{aligned}$$ Set $p=1$. Note that for some fixed $l$, $\beta^k \le p_k \le
\beta^{k+l}$. It is easy to check that $C(k)
\le C_6$ for some constant $C_6$. As $k$ tends to infinity, $p_k\rightarrow
\infty$, $a(k) \rightarrow a \in (0,1)$, and the required estimate follows immediately.
We can now finish the proof of Theorem \[theoremC0\]. Since $u=e^{-B\psi_i}$ and $\psi_i$ satisfies $\sup_M \psi_i =0$ and $${\chi_{0 \, k \overline{l}}} + {\partial_{k}} {\partial_{\overline{l}}}\psi_i \ge 0,$$ we can apply Proposition 2.1 of [@T1] to get a bound on $\| u \|_{\delta}$ for $\delta$ small enough. This completes the proof of Theorem \[theoremC0\].
[**Acknowledgements.**]{} The author would like to thank his thesis advisor, D.H. Phong for his constant support, advice and encouragement. The author also thanks Jacob Sturm, Lijing Wang, Mao-Pei Tsui and Lei Ni for some helpful discussions. In addition, he is very grateful to S.-T. Yau for his support and advice. The results of this paper are contained in the author’s PhD thesis at Columbia University [@W2].
[99]{} Aubin, T. [*Equations du type Monge-Ampère sur les variétés Kähleriennes compacts*]{}, Bull. Sc. Math. (2) [**102**]{} (1978), no. 1, 63–95, MR0494932, Zbl 0374.53022 Chen, X. X. [*On the lower bound of the Mabuchi energy and its application*]{}, Internat. Math. Res. Notices [**12**]{} (2000), 607–623, MR1772078, Zbl 0980.58007 Chen, X. X. [*A new parabolic flow in Kähler manifolds*]{}, Comm. Anal. Geom. [**12**]{} (2004), no. 4, 837–852, MR2104078, Zbl pre02148045 Donaldson, S. K. [*Moment maps and diffeomorphisms.*]{} Asian J. Math. [**3**]{} (1999), no. 1, 1–15, MR1701920, Zbl 0999.53053 Donaldson, S. K. [*Symmetric spaces, Kähler geometry and Hamiltonian dynamics*]{}, in ‘Northern California Symplectic Geometry Seminar’ (Eliashberg et al eds.), Amer. Math. Soc. Transl. Ser. 2, 196 (1999), 13–33, MR1736211, Zbl 0972.53025 Donaldson, S. K. [*Scalar curvature and projective embeddings, I.*]{}, J. Differential Geom. [**59**]{} (2001), no. 3, 479–522, MR1916953, Zbl 1052.32017 Donaldson, S. K. [*Scalar curvature and stability of toric varieties*]{}, J. Diff. Geom. [**62**]{} (2002), no. 2, 289–349, MR1988506, Zbl pre02171919 Mabuchi, T. [*$K$-energy maps integrating Futaki invariants*]{}, Tohoku Math. J. (2) [**38**]{} (1986), no. 4, 575–593, MR0867064, Zbl 0619.53040 Phong, D. H. and Sturm, J. [*Stability, energy functionals, and Kähler-Einstein metrics*]{}, Comm. Anal. Geom. [**11**]{} (2003), no.3, 565–597, MR2015757 Tian, G. [*On Kähler-Einstein metrics on certain Kähler manifolds with $c_1(M)>0$*]{}, Invent. Math. [**89**]{} (1987), no. 2, 225–246, MR0894378, Zbl 0599.53046 Tian, G. [*The $K$-energy on hypersurfaces and stability*]{}, Comm. Anal. Geom. [**2**]{} (1994), no. 2, 239–265, MR1312688, Zbl 0846.32019 Tian, G. [*Kähler-Einstein metrics with positive scalar curvature*]{}, Invent. Math. [**137**]{} (1997), no. 1, 1–37, MR1471884, Zbl 0892.53027 Weinkove, B. [*Convergence of the J-flow on Kähler surfaces*]{}, Comm. Anal. Geom. [**12**]{} (2004), no. 4, 949–965, MR2104082, Zbl 1060.53072 Weinkove, B. [*The J-flow, the Mabuchi energy, the Yang-Mills flow and multiplier ideal sheaves*]{}, PhD thesis, Columbia University, 2004 Yau, S.-T. [*On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I*]{}, Comm. Pure Appl. Math. [**31**]{} (1978), no. 3, 339–411, MR0480350, Zbl 0369.53059 Yau, S.-T. [*Open problems in geometry*]{}, Proc. Symposia Pure Math. [**54**]{}, Part 1 (1993), 1–28, MR1216473, Zbl 0801.53001
[^1]: This work was carried out while the author was supported by a graduate fellowship at Columbia University.
|
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"pile_set_name": "ArXiv"
}
|
Introduction
============
The study of the angular dependence of vortex pinning in high temperature superconductors (HTSC) with tilted columnar defects has revealed a richer variety of phenomena and pinning regimes than originally expected. At high temperatures and magnetic fields, the uniaxial nature of pinning by CD dominates the vortex response[@civale91]. This is clearly seen, for instance, when isothermal magnetization loops ${\bf M(H)}$ are measured for different field orientations[@silhanek99a]. At fixed field modulus $H$, the irreversible magnetization $M_i=\frac 12 \Delta
M$ (where $\Delta M$ is the width of the hysteresis, proportional to the persistent current density $J$) exhibits a well defined maximum when ${\bf H} \parallel$ CD. For other orientations “staircase vortices” develop. In a previous study we have shown[@silhanek99a] that in YBa$_2$Cu$_3$O$_7$ (YBCO), and due to the simultaneous presence of CD, twin boundaries and crystallographic ab-planes, correlated pinning dominates over random pinning for all orientations, forming staircases of different configuration (i.e., with segments locked into different correlated structures) depending on the field direction.
An additional feature is the existence of a lock-in phase. When the angle between ${\bf H}$ and the CD is less than a lock-in angle $\varphi_{L}\left(H,T\right)$, it is energetically convenient for vortices to ignore the ${\bf H}$ orientation and to remain locked into the tracks[@nel-vin; @blatter94]. Since $\varphi_{L}$ scales as $1/H$, in practice this effect is only visible at low fields. An experimental manifestation of the lock-in regime is the existence[@silhanek99a] of a “plateau” in the irreversible magnetization, $M_i\left(\Theta\right) \approx const$, over a certain angular range. Here $\Theta$ is the angle between the normal to the platelet crystal, ${\bf n}$ (which coincides with the crystallographic c-axis) and ${\bf H}$, defined within the plane that contains the CD.
At low fields, an additional effect must be taken into account. Due to both the anisotropic superconducting response of the HTSC and the sample geometry, the direction of the internal field ${\bf B}$, that coincides with the direction of the vortices, differs from that of ${\bf H}$. As the uniaxial pinning of the CD maximizes when ${\bf B}$ (rather than ${\bf H}$) is aligned with the tracks, the maximum in $M_i\left(\Theta\right)$ occurs at an angle that progressively departs[@silhanek99a] from the orientation of the tracks, $\Theta_D$, as H decreases.
The low field misorientation between ${\bf B}$ and ${\bf H}$ poses a serious experimental concern. All studies of the pinning properties of tilted CD that are based solely on measurements at ${\bf H} \parallel$ CD, or on comparison of this orientation with a few others, give valid information at high fields, but are misleading at low fields; vortices are just not oriented in the right direction. To avoid this problem, a rather complete knowledge of the angular dependent response, $M_i \left(H, \Theta
\right)$, is required.
In this work we present a procedure that allows us to obtain directly $M_i\left(\Theta\right)$ by rotating the sample at fixed $H$ and $T$. This method has the advantage that a fine grid can be easily obtained in the angular ranges of interest, thus permitting the exploration of the various regimes with significantly improved angular resolution. We apply this experimental procedure to investigate the pinning produced by tilted CD in an ErBa$_2$Cu$_3$O$_7$ single crystal. We present a detailed analysis of the lock-in angle as a function of $H$ and $T$. The width of the lock-in regime is shown to follow a $1/H$ dependence over a wide temperature range, and from the temperature dependence of the slope of $\varphi_{L}$ vs $1/H$ we determine the entropic smearing function $f\left(T/T^*\right)$.
Experimental Details
====================
The sample used in this work is a rectangular ErBa$_2$Cu$_3$O$_{7-\delta}$ single crystal platelet of dimensions $0.44\times 0.33\times 0.01 mm^3$, grown by the self-flux method in a commercial yttria-stabilized-zirconia crucible [@avila00]. After growth it was annealed under oxygen atmosphere for 7 days at 450$^{\circ}$C. This sample was then irradiated with 309 MeV $Au^{26+}$ ions (whose penetration range in this material is $\sim 15\mu m$) at the TANDAR accelerator in Buenos Aires (Argentina), to introduce columnar defects at an angle $\Theta_D\approx 30^{\circ}$ off the [*c*]{} axis. The rotation axis, which is perpendicular to the plane formed by the c-axis and the track’s direction, is parallel to the largest crystal dimension. The irradiation dose was equivalent to a matching field of $B_{\Phi} = 1 T$. After irradiation the sample presented a superconducting transition temperature of $T_c = 90.0 K$ and transition width of $\Delta T < 1 K$.
Magnetization experiments were conducted on a commercial superconducting quantum interference device (SQUID) magnetometer (Quantum Design MPMS-5), equipped with two sets of detectors that allow to record both the longitudinal ($M_L$) and transverse ($M_T$) components of the magnetization vector ${\bf M}$, with respect to the longitudinally applied field ${\bf H}$. We have developed a sample rotation system (hardware and software) that solves the problems usually involved in the measurement of $M_T$, and thus allows us to study the response of the samples at arbitrary orientations. We have used that system in the past to measure magnetization loops $\bf {M}(H)$ at different field orientations in samples similar to the one investigated here.[@silhanek99a; @anomalous] In those experiments the sample is initially rotated to the desired $\Theta$, then zero-field-cooled (ZFC) from above $T_c$ to the desired measuring temperature $T$. Both $M_L(H)$ and $M_T(H)$ are then measured, and the separation between the upper and lower branches of both loops ($\Delta M_L(H)$ and $\Delta M_T(H)$) is used to calculate the amplitude $M_i=\frac 12 \sqrt{\Delta
M_L(H)^2 + \Delta M_T(H)^2}$ and direction $\Theta_M=arctan(\Delta M_T(H)/ \Delta M_L(H))$ of the irreversible magnetization vector ${\bf M_i}$.
The alternative rotating sample measurements presented here are performed by setting up a desired initial state $\left(
T,H,\Theta_H \right)$ and then recording $M_L(\Theta)$ and $M_T(\Theta)$ for fixed $T$ and $H$. The sample is rotated a given angle step (typically 1$^{\circ}$ to 3$^{\circ}$) and re-measured. Usually, the procedure is repeated until the crystal completes 2 or 3 full turns. This provides us with redundant information that contributes to improve the quality of the data. After careful subtraction of the signal of the plastic sample holder (which has only longitudinal component and is small, linear in $H$, almost temperature independent and, most importantly, angle independent) and of the reversible response, the irreversible components $M_{Li}(H)$ and $M_{Ti}(H)$ are used to determine $M_i= \sqrt{M_{Li}(H)^2 + M_{Ti}(H)^2}$ and $\Theta_M=arctan(M_{Ti}(H)/M_{Li}(H))$.
Rotating measurements
=====================
Meissner response
-----------------
We began this study with an analysis of the Meissner response. To that end we ZFC the crystal, then applied a field $H$ smaller than the lower critical field $H_{c1}\left(\Theta\right)$ for all $\Theta$, and subsequently performed the rotating measurements. Ideally, under those conditions there are no vortices in the crystal and the response depends neither on the material anisotropy nor on the pinning properties, it is totally determined by the sample geometry. As was previously shown[@candia99], $M_L\left(\Theta\right)$ and $M_T\left(\Theta\right)$ for a thin platelet should follow the dependencies
$$4\pi M_L\left(\Theta\right)=-H\left( \frac 1{2\nu }\cos ^2 \Theta+\frac 1{1-\nu }\sin^2 \Theta\right) \label{eq:meissL1}$$
$$4\pi M_T\left(\Theta\right)=-H\left( \frac 1{2\nu }-\frac 1{1-\nu }\right) \sin \Theta\cos \Theta \label{eq:meissT1}$$
where $\nu$ is the appropriate demagnetizing factor, which is essentially given by the thickness of the platelet ($t$) divided by its width ($W$). These equations can be easily rewritten as
$$M_L=-M_0-M_{2\Theta }\cos 2\Theta_H \label{eq:meissL2}$$
$$M_T=-M_{2\Theta }\sin 2\Theta_H \label{eq:meissT2}$$
where
$$4\pi M_0=\frac H2\left( \frac 1{2\nu }+\frac 1{1-\nu }\right) \label{eq:meissM0}$$
$$4\pi M_{2\Theta}=\frac H2\left( \frac 1{2\nu }-\frac 1{1-\nu }\right) \label{eq:meissM2}$$
Equations \[eq:meissL2\] and \[eq:meissT2\] indicate that the magnetization vector [**M**]{} can be visualized as the sum of a fixed contribution ${\bf M_0}$, anti-parallel to ${\bf H}$, and a rotating contribution ${\bf M_{2\Theta}}$ with a periodicity of $180^{\circ}$. This suggests that a convenient way to plot these data is on an $M_L$,$M_T$ plane. In this presentation the Meissner response is expected to lie on a circumference of radius $M_{2\Theta}$ centered at $\left(M_L,M_T\right)=\left(-M_0,0\right)$. One complete circumference is drawn by a rotation of $180^{\circ}$. An example of this procedure (for $T=60K$ and $H=50Oe$) is shown in Figure 1. The crystal was rotated by two complete turns, thus there are four sets of data points covering $180^{\circ}$ each, which are clearly separated in two groups. This is due to a small remnant magnetization ${\bf M_R}$, which originates from the small residual field that is usually present during the ZFC.[@candia99]
The vector ${\bf M_R}$ has fixed modulus and its direction remains fixed with respect to the sample during rotation,[@candia99] thus $M_{RL}=M_R\cos \left( \Theta+\Theta_R \right)$ and $M_{RT}=M_R\sin
\left( \Theta+\Theta_R \right)$, where $M_R$ and $\Theta_R$ are constants. Since ${\bf M_R}\left(\Theta\right)$ has a one fold periodicity, it breaks the Meissner two fold periodicity and splits the experimental data into two sets. Indeed, by fitting the data in Fig. 1 using a combination of the Meissner and remnant contributions we can easily determine and remove the remnant part and all points collapse on a single circumference (solid symbols). Fig. 1 is an extreme example of remnant influence, chosen to show that even in that case the Meissner response can be obtained. By carefully canceling the residual magnetic field in the ZFC procedure we can obtain a much smaller ${\bf M_R}$ such that both circumferences of raw data in Fig. 1 almost collapse on a single one.
Equations \[eq:meissL2\] and \[eq:meissT2\] were used to fit several measurements for different temperatures and fields, and used to calculate the sample volume $V\approx 1.45\times 10^{-6}
cm^3$ and demagnetization factor $\nu\approx 0.033$. Both results are in very good agreement with the values directly determined from crystal dimensions (V $\sim 1.46 \times 10^{-3} cm^3; \nu \approx 0.03$).
critical state
--------------
We now focus on the high field range, where the crystal is in the mixed state. Fig. 2(a) shows $M_L(\Theta)$ and $M_T(\Theta)$ for a rotation at $70K$ and $8kOe$, where the angle independent background due to the holder has already been removed from $M_L$. As the reversible magnetization of the superconductor \[$\sim
\left( \Phi_0 /32\pi^2\lambda^2 \right) \ln \left( H_{c2}/H
\right) \sim 5G$\] is negligible compared to $\bf{M_i}$, the response is dominated by vortex pinning. Curves in Fig. 2(a) exhibit a rich structure, due to the combination of crystalline anisotropy, directional vortex pinning and geometrical effects. In order to extract useful information from them, we must first establish the relation between $\bf{M_i}$ and the screening current $\bf J$ flowing through the crystal.
For simplicity, we will analyze the case of a thin infinite strip of aspect ratio $\nu=t/W \ll 1$, that can rotate around its axis, which is perpendicular to $\bf H$. Let’s assume that the strip was originally ZFC at an angle $\Theta$ and $H$ was subsequently applied (the initial condition in Fig. 2(a)). If $H$ is high enough we can consider[@clem-sanch] that a current density of uniform modulus $J_c(\Theta)$ flows over the whole volume.[@clem-sanch] This $\bf J$ is parallel to the strip axis and it reverses sign at the plane that contains the axis and $\bf H$.
It has been shown[@prozorov96; @zhukov97; @hasanain99] that, in this fully penetrated critical state and as long as $\nu\tan\Theta \ll 1$, the angle between ${\bf M_i}$ and the sample normal $\bf n$ is $\alpha \sim \arctan \left(\frac
{2}{3}\nu^2\tan\Theta\right) \ll \Theta$. That is, ${\bf M_i}$ remains almost locked to $\bf n$ due to a purely geometrical effect. For the particular crystal of the present study, $\alpha$ should be smaller than $1^{\circ}$ for $\Theta \le 80^{\circ}$. Another result[@zhukov97] is that, although in principle the geometrical factor relating $M_i$ with $J_c(\Theta)$ depends on $\Theta$, within that same angular range the variations are given by the factor $\left(1-\frac {2}{3}\nu^2\tan^2\Theta\right)$ and thus are negligible.
We now discuss what happens when the strip is rotated away from this initial state by a small angle $\delta \Theta$. The result will depend on the direction of rotation. If $\bf n$ approaches $\bf H$ (this corresponds to the angular ranges $90^{\circ}$ to $180^{\circ}$ and $270^{\circ}$ to $360^{\circ}$ in Fig. 2(a)), the normal component $H_{\perp}$ will increase, thus inducing screening currents at the edges of the crystal in the same direction as those already flowing. Vortices will then displace to satisfy the condition $J \le
J_c\left(\Theta+\delta\Theta\right)$ everywhere. If $J_c\left(\Theta+\delta\Theta\right)\le J_c(\Theta)$ the new distribution will be analogous to the initial one, with $J=J_c(\Theta+\delta\Theta)$ everywhere and the boundary of current reversal rotated by an angle $\delta \Theta$ in order to remain parallel to $\bf H$. On the contrary, if $J_c(\Theta+\delta\Theta) > J_c(\Theta)$, the new field profile will propagate all the way to the center of the sample only if $\delta H_{\perp} = H \sin(\Theta)\delta\Theta$, is larger than the maximum possible additional screening $\sim t\left[
J_c(\Theta+\delta\Theta)-J_c(\Theta) \right]$. The condition for the “full penetration of the rotational perturbation” is thus
$$H \sin(\Theta) \ge t \frac {dJ_c}{d\Theta} \label{eq:fullp}$$
If the inequality (\[eq:fullp\]) is satisfied, the vortex system will evolve under rotations maintaining a fully penetrated critical state with uniform $J$. In other words, the state at any $\Theta$ will be the same that would have formed by increasing $H$ after ZFC at that orientation. Then, as long as $\nu\tan\Theta
\ll 1$, the condition that ${\bf M_i}$ is almost parallel to $\bf
n$ is preserved. We have experimentally confirmed this fact: $\alpha \le 1^{\circ}$ for all measurements conducted in this work, except in a very narrow angular range around the ab-planes, where a flip in ${\bf M_i}$ occurs.[@zhukov97] Thus, from now on we will plot all the results as a function of $\Theta$. Furthermore, we can obtain $J_c(\Theta)$ by simply multiplying $M_i$ by the angle independent factor that corresponds to the relation valid for $\bf H \parallel \bf n$. If eq. (\[eq:fullp\]) is not satisfied, $J$ will be subcritical in part of the sample and this relation is no longer valid.
If the crystal is rotated in such a way that $\bf n$ moves away from $\bf H$ (so $H_{\perp}$ decreases), the new screening currents induced at the edges of the crystal will oppose to those already flowing. As the rotation progresses the boundary between the old and new $\bf J$ directions will move inwards, until eventually the new critical state propagates to the whole sample. From that point the situation will again be analogous to that already discussed, except that ${\bf M_i}$ will be paramagnetic instead of diamagnetic.
A rotation at fixed $H$ is to some extent analogous to a hysteresis loop[@prozorov96]. Rotating $\bf n$ towards $\bf H$ increases $H_{\perp}$, which is roughly equivalent to increasing $H$ at $\Theta=0^{\circ}$, moving along the lower (diamagnetic) branch of the loop. Decreasing $H_{\perp}$ (either by rotating $\bf n$ away from $\bf H$ or by crossing the $\bf H \parallel \bf
c$ condition), is equivalent to reversing the field sweep, thus producing a switch to the other branch of the loop. This is a useful analogy for the analysis of the rotations, although it should not be pushed too far.
A basic difference is that a rotation also produces a variation in the parallel field component, $\delta
H_{\parallel}=H\cos(\Theta)\delta\Theta$. This generates screening currents flowing in opposite directions on the upper and lower surfaces of the strip, which produce a tilting force on the vortices.[@clem82; @clem86; @perez90] If the perturbation propagates all the way to the central plane, the result is a rotation of the vortex direction following $\bf H$, the situation that we have implicitly assumed above. However, if pinning were strong enough it could preclude the propagation of the tilt beyond a certain depth, thus generating a critical state along the crystal thickness, with a central segment of the vortices remaining in the original direction.[@clem82; @clem86; @perez90; @goeckner94; @hasanain96; @hasan97; @obaidat97] If this effect were significant, as the rotation proceeded the orientation of the vortices would lag behind the field direction. In an extreme case, vortices deep inside the sample would rotate rigidly with it, a situation that has indeed been observed[@vlasko98; @obaidat98; @hasan99]. As we will show below, in the present case we have clear experimental evidence that the misorientation between the vortex direction and $\bf H$ due to this lag effect is negligible, so all this complication can be ignored.
We now analyze the curves shown in Fig. 2. The measurement starts at $\Theta \sim 30^{\circ}$ (point A) with $\bf n$ rotating away from $\bf H$. Thus, $\bf J$ initially undergoes a flip until the reversed fully penetrated critical state is formed (point B). From here the evolution of the system turns independent of the initial conditions and becomes two fold periodic. From point C ($\Theta = 90^{\circ}$) to point E ($\Theta = 180^{\circ}$) the system evolves in a fully penetrated critical state (in the hysteresis loop analogy, this is equivalent to increasing the field from zero to $H$). Clearly visible within this angular range is the peak in both $M_L$ and $M_T$ at $\Theta \sim
150^{\circ}$ (point D), that corresponds to the direction of the CD. At point E, $M_T$ is null as expected by symmetry, while $M_L$ begins a quick flip due to the reversal of the screening currents as $H_{\perp}$ reaches a maximum at $\bf H
\parallel \bf n$ and then starts to decrease. The end of this flip at point G indicates that the critical state is completely reversed. From G to C’ ($\Theta=270^{\circ}$) the evolution is analogous to a field decreasing portion of a loop.
Note that between E and G there is one unique angle (point F) where both $M_L$ and $M_T$ are null. This condition is equivalent to the unique $H$ value in the switch from the lower to the upper branch of a $M(H)$ loop where ${\bf M_i=0}$. The fact that the condition $M_L=0$ occurs at the same angle where $M_T=0$ confirms that the background signal has been correctly subtracted, and we have systematically made use of this checking procedure.
In Fig. 2(a) the direction of rotation is such that the conditions $\bf H \perp c$; $\bf H \parallel$ CD and $\bf H \parallel$ c proceed in that order. We define this as a clockwise (CW) rotation. In contrast, in a counter-clockwise (CCW) rotation the alignment occurs when $\bf
n$ is moving away from $\bf H$. The consequences of this difference are described below.
In Figure 2(b) the same CW data of Fig. 2(a) is shown in an $M_L$ vs $M_T$ polar graph (full symbols), together with the CCW rotation under the same conditions (open symbols). In both cases the initial behavior until the critical state is fully developed (portion A to B in the CW and P to Q in the CCW) and the subsequent $180^{\circ}$-periodic evolution in the critical state (covering approximately two periods of $180^{\circ}$) are clearly distinguished. Another feature that is apparent in this representation is that the magnetization vector passes through the origin (${\bf M_i}=0$) and reaches the opposite quadrant each time that (i) a rotation starts moving $\bf n$ away from $\bf H$; or (ii) the $\bf n \parallel \bf H$ condition is crossed.
Although the CW and CCW curves in Fig. 2(b) are similar (rotated in $180^{\circ}$ with respect to each other) they also exhibit some differences. The most obvious one is that the peak at the CD direction (dotted line) is prominently seen in the CW rotation (point D), while in the CCW rotation it is partially suppressed by the flip of ${\bf M_i}$. The flip starts at $\bf H \parallel
\bf n$, and ends at the angle $\Theta_F$ where the fully reversed critical state is achieved. Making use of the loop analogy, this requires a field decrease of $\sim 2H^*$, where $H^*(H,T)$ is the well known full penetration field, then
$$2H^* = H\left( 1-\cos\Theta_F \right)
\label{reverse}$$
This analysis indicates that there is a blind range in the rotation measurements, extending up to an angle $\Theta_F$ from $\bf n$, where the critical state is not fully developed and thus $J$ cannot be extracted. Depending on the direction of rotation, this blind range occurs either in the same quadrant of the CD (case CCW) or in the opposite (case CW). As $\Theta_F$ decreases with $H$, in CCW rotations the peak due to the CD is totally hidden at low fields but can be fully measured at high enough $H$.
The values of $\Theta_F$ are easily obtained from Figure 3(a), where $M_i$ is plotted as a function of $\Theta$ for the same two sets of data (CW and CCW) of Fig. 2(b). We observe here that the agreement between the CW and CCW data is excellent, thus they can complement each other to eliminate the blind region at low angles. Estimating $\Theta_F \sim 25^{\circ}$ for the CW rotation and $\Theta_F \sim 30^{\circ}$ for the CCW case, and using eq. (\[reverse\]) we obtain $H^* \sim 370 Oe$ and $\sim 540 Oe$ respectively. We can check the consistency of these estimates in two ways. First, we know that in a thin sample $H^* \sim J t$. Combining with the critical state relation $J \sim 60 M_i / W$ (valid for a square platelet) we have $H^* \sim 60 M_i t/W \sim
1.8 M_i$. From the figure we have $M_i(\Theta_F) \sim 200 G$ for the CW and $\sim 340 G$ for the CCW, so we get $H^* \sim 360 Oe$ and $\sim 610 Oe$ respectively, in very good agreement with the above estimates. On the other hand, we can compare the values of $H^*$ obtained from eq. (\[reverse\]) with those directly measured in hysteresis loops at the appropriate angles. We have done so for several temperatures and fields, and we have systematically obtained very good consistency.
Fig. 3(a) confirms that the condition (\[eq:fullp\]) is satisfied in this measurement. In fact, the largest slope $dM_i/d\Theta \sim 1 kG/rad$, that occurs at $\Theta \sim
33^{\circ}$, implies that $t dJ_c/d\Theta \sim 1.8 kG/rad$, which is indeed smaller than $H\sin(\Theta) \sim 4.4 kG$. This condition is also fulfilled in all the cases discussed in the next section.
In order to compare the data measured by sample rotations with those resulting from traditional loop measurements, in Fig. 3(a) we also included $M_i$ values at several $\Theta$ obtained in the latter way at the same $T$ and $H$ (large open diamonds). The agreement is very good over the full range of angles, except that the loop values tend to be somewhat smaller. This is a feature observed for all measured fields, and can be explained by the fact that a rotation step is a process that takes only a couple of seconds, while a field increase and stabilization typically requires more than 1 minute in our magnetometer, during which the $\bf M_i$ is already relaxing. Indeed, by performing short relaxation measurements we have verified that the rotation data approaches the loop data after 1-2 minutes. This results in another advantage of the rotations over the loops: measurements are made closer to the true initial critical state.
Finally, the coincidence of the CW and CCW rotations and the loops, particularly in the region of the peak due to the CD, rules out the possibility that vortices lag significantly behind the direction of $\bf H$ in our rotating sample experiments. In summary, the information obtained from our rotation measurements is essentially the same as that provided by hysteresis loops, with several advantages including the possibility to acquire significantly more data points for each field. This feature permits a more detailed analysis of the peak associated with the uniaxial pinning of the CD, as will be shown in the next section.
Determination of the lock-in angle
==================================
A complete set of rotations at $70K$ for different applied fields is shown in Figure 3(b). At high fields (above $\sim 6kOe$) a well-defined peak at the CD direction is observed. At lower fields ($1kOe \le H \le 5kOe$) the peak progressively broadens and transforms into a plateau (a certain angular range where $M_i(\Theta) \sim const.$), while it shifts towards the c-axis. We had previously reported[@silhanek99a] all these features in YBa$_2$Cu$_3$O$_7$ crystals.
The plateau represents the angular range of applied field over which it is energetically convenient for the vortices to remain locked into the columnar defects, thus its angular width is twice the lock-in angle $\varphi_L$. Below this angle, the vortices are subject to an invariant (and maximum) pinning force. According to theoretical models[@nel-vin; @blatter94]
$$\varphi_L \simeq \frac{ 4\pi\sqrt{2\varepsilon_l\varepsilon_r(T)} }{\Phi_0 H}
\label{eq:lockin}$$
where $\varepsilon_l$ is the vortex line tension and $\varepsilon_r(T)$ is the effective pinning energy.
Equation (\[eq:lockin\]) predicts that $\varphi_L$ should be inversely proportional to $H$. The improved resolution of the rotation measurements, that permits a much better determination of the width of the plateau, allows us to test this dependence. To that end, we have measured several other sets of data similar to Figure 3(b), for a wide range of temperatures ($35K$ to $85K$). A few examples of the observed plateaus are shown in Figure 4.
We then extracted the plateau width for every measurement which displayed such a feature. This procedure was done very carefully, including an over-zealous estimate of the errors involved. The results for all measurable $\varphi_L$ are plotted as a function of $H^{-1}$ in Figure 5. This figure clearly demonstrates the $H^{-1}$ dependence of $\varphi_L$, as evidenced by the solid lines which are the best linear fits to the data points for each temperature.
According to eq. (\[eq:lockin\]), the data in fig. 5 should extrapolate to the origin, what is clearly not the case. For all the temperatures where reliable extrapolations can be made ($35K$ to $80K$) the linear fits systematically give a [*positive*]{} value of $\varphi_L \sim 1.5^{\circ}$ to $3^{\circ}$ at $H^{-1}=0$, which is above the experimental error. There are at least two reasons for this discrepancy. In the first place, we experimentally determine $\varphi_L$ from the intersection of straight lines extrapolated from the plateau and the slopes at both sides of it (see fig. 4). Due to the rounded ends of the plateau, this definition tends to [*overestimate*]{} $\varphi_L$. Second, the natural splay of the tracks will tend to wash away the expected cusp-like behavior at high fields, thus also contributing to the overestimate of $\varphi_L$. It is clear, on the other hand, that the influence of the splay is not too dramatic, as we indeed observe a rather sharp peak at high fields as seen in fig. 3(b). TRIM calculations show that[@suppression], in our irradiation conditions, the median radian angle of splay slowly increases from zero at the entry surface of the crystal to $\sim 3^{\circ}$ at a depth of $8 \mu m$, and then grows faster to $\sim 6^{\circ}$ at the exit surface.
We now want to analyze whether eq. (\[eq:lockin\]) provides a satisfactory description of the temperature dependence of the lock-in effect. As this expression does not account for the nonzero extrapolation of $\varphi_L$ discussed in the previous paragraph, it would be incorrect to force a fit through the origin to determine the prefactor of $H^{-1}$. Instead, it is appropriate to identify such prefactor with the slopes $\alpha(T)=d\varphi_L/d(H^{-1})$ of the linear fits. Indeed, the splay of the CD is a geometrical feature independent of H and hence it should only add a constant width to the plateau, without changing its field dependence. To a first approximation, the rounded edges of the plateau will also introduce an additive constant, without significantly affecting the slope. Figure 6 shows the temperature dependence of $\alpha(T)$ (solid symbols). As expected, $\alpha(T)$ decreases with increasing $T$, reflecting the fact that the lock-in angle at fixed $H$ decreases with $T$ due to the reduction of both the line tension and the pinning energy. For a quantitative analysis it is necessary to know the expressions for $\varepsilon_l$ and $\varepsilon_r(T)$. In our experiments the appropriate line tension is that corresponding to in-plane deformations (see pages 1163-1164 in Ref.[@blatter94]), $\varepsilon_l=\left(\varepsilon^2\varepsilon_0/\varepsilon(\Theta)\right)\ln\kappa$, where $\varepsilon_0=\left(\Phi_0/4\pi\lambda\right)^2$, the penetration depth $\lambda$ corresponds to ${\bf H}\parallel$c, the mass anisotropy $\varepsilon \ll 1$ and $\varepsilon^2(\Theta)=\cos^2(\Theta)+\varepsilon^2\sin^2(\Theta)$. The temperature dependence of the superconducting parameters appears in $\varepsilon_l$ through $\lambda(T)$. On the other hand, $\varepsilon_r(T)$ is given by[@blatter94; @nel-vin]
$$\varepsilon_r(T) = \eta \frac{\varepsilon_0}{2}
\ln\left(1+\frac{r^2}{2\xi^2}\right)\times f(x)
\label{eq:pinenergy}$$
where $r \approx 50\AA$ is the radius of the tracks, $\xi$ is the superconducting coherence length, and the dimensionless [*efficiency factor*]{} $\eta \leq 1$ accounts for the experimental fact that the pinning produced by the CD is smaller than the ideal.[@accomodation] Besides the intrinsic temperature dependence of the superconducting parameters, this expression contains an additional temperature dependent factor $f(x)$, known as the [*entropic smearing*]{} function, which accounts for the thermal fluctuations of the flux lines. Here $x=T/T_{dp}$, where $T_{dp}$ is a characteristic field-independent [*depinning temperature*]{}. Combining all these elements, at the track’s direction $\Theta=\Theta_{CD}$ we obtain
$$\alpha(T) \approx \frac{\Phi_0\varepsilon}{8\pi\lambda^2}
\ln\left(1+\frac{r^2}{2\xi^2}\right)\times
\left[\eta \frac{2\ln\kappa}{\varepsilon(\Theta_D)} f(x) \right]^{1/2},
\label{eq:slope2}$$
In the original work of Nelson and Vinokur[@nel-vin], where only a [*short range*]{} pinning potential was considered, the entropic function for $x>1$ was approximately given by $f_{sr}(x) \sim x^2\exp(-2x^2)$. However, according to a further refinement of the model[@blatter94], where the [*long range*]{} nature of the pinning potential was taken into account, this function (for $x>1$) takes the form $f_{lr}(x) \sim \exp(-x)$.
We can now fit the experimentally determined $\alpha(T)$ using eq. (\[eq:slope2\]). To that end we use the long range result $f_{lr}(x)$ and fix the reasonably well known superconducting parameters of the material $\varepsilon \approx 1/5$; $\ln\kappa \approx 4$ and $\xi=15\AA /\sqrt{1-t}$ (where $t=T/T_c$). We also assume the usual two-fluid temperature dependence $\lambda(T)=\lambda_L/2\sqrt{1-t^4}$, where $\lambda_L$ is the zero-temperature London penetration depth. The free parameters are then $T_{dp}$ and the combination $\lambda_L/\eta^{1/4}$. The best fit, shown in figure 6 as a solid line, yields $\lambda_L/\eta^{1/4}=360 \AA$ and $T_{dp}=30K$.
Based on the results of figs. 5 and 6, there are a number of considerations that can be made at this point. The first one is that the Bose-glass scenario contained in eqs. (\[eq:lockin\]) to (\[eq:slope2\]) provides a quite satisfactory description of the lock-in effect over the whole range of temperature and field of our study. In addition, the obtained $T_{dp}$ is smaller, but still reasonably similar to the value $\sim 41K$ that we had previously found for several YBCO crystals using a completely different experimental method.[@accomodation; @gaby] This low $T_{dp}$ (well below the initial theoretical expectations) indicates that the efficiency factor $\eta$ is rather small, what is also consistent with the less than optimum $J_c$ observed here and in several previous studies. For low matching fields as that used in the present work, it was estimated[@accomodation] that $\eta \sim 0.2 - 0.25$.
The exact value of $\eta$ has little influence in our estimate of $\lambda_L$, as it only appears as $\eta^{1/4}$. For $\eta=0.2$ and $\eta=1$ we get $\lambda_L=250 \AA$ and $360 \AA$ respectively, a factor of 4 to 5 smaller than the accepted value $\lambda_L \sim 1400\AA$. Zhukov et al.[@zhukov97b] had reported a similar discrepancy when studying the lock-in effect by both CD and twin boundaries in YBCO. In a previous study in YBCO crystals with CD, we had also found that the misalignment between ${\bf B}$ and ${\bf H}$ at low fields (due to anisotropy effects) was well described using a $\lambda_L$ significantly smaller than the accepted value[@silhanek99a]. Thus, this numerical discrepancy appears to be a common result associated to the study of angular dependencies in YBCO-type superconductors with correlated disorder.
Finally, it is relevant to note that eq. (\[eq:lockin\]) was derived for the [*single vortex pinning*]{} regime, which occurs below a temperature dependent accommodation field[@blatter94; @accomodation] $B^*(T) < B_{\Phi}$, while a large fraction of the data shown in fig. 5 lies above this line, in the [*collective pinning*]{} regime. Unfortunately, to our knowledge there is no available expression for $\varphi_L \left(H,T \right)$ in the collective regime. Blatter et al.[@blatter94] only argued that collective effects should result in a reduction of the lock-in angle. The experimental fact is that eq. (\[eq:lockin\]) satisfactorily describes both the temperature and field dependence of $\varphi_L$. This suggests that, at least to a first approximation, collective effects in the range of our measurements simply result in a different prefactor in eq. (\[eq:lockin\]). Clearly, lock-in effects in the collective regime deserve further theoretical study.
Conclusions
===========
We have measured the irreversible magnetization (${\bf M_i}$) of an ErBa$_{2}$Cu$_{3}$O$_{7-\delta}$ single crystal with columnar defects (CD), using an alternative technique based on sample rotation under a fixed magnetic field. The resulting ${\bf M_i}
(\Theta)$ curves for several temperatures agreed very well with independent hysteresis loop experiments, showing a peak in the CD direction at higher fields, while a very well defined plateau due to the lock-in of the vortices into the CD was observed at lower fields. The lock-in angle satisfactorily follows the field and temperature dependence predicted by the Bose-glass scenario.
acknowledgments
===============
Work partially supported by FAPESP, Brazil, Procs. \#96/01052-7 and \#96/05800-8; ANPCyT, Argentina, PICT 97 No.01120; and CONICET PIP 4207.
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Figure 1. $M_T$ versus $M_L$ polar graph in the Meissner phase for $T=60 K$ and $H=50 Oe$. Open symbols: raw data. Solid symbols: pure Meissner response (the remnant contribution was removed).
Figure 2. (a) Components of the magnetization vector, $M_T$ and $M_L$, as a function of the angle for $T=70 K$ and $H=8 kOe$. The direction of rotation is such that the conditions $\bf H \perp c$; $\bf H \parallel$ CD and $\bf H \parallel$ c proceed in that order. (b) $M_T$ vs $M_L$ polar graph of the same CW rotation of (a) (full symbols) together with the CCW rotation.
Figure 3. Irreversible magnetization $M_i$ as a function of $\Theta$ at $T=70K$ for (a) $H=8 kOe$ together with data obtained from hysteresis loop measurements (b) several fields.
Figure 4. Irreversible magnetization $M_i$ as a function of $\Theta$ in the region of the plateau at $T=50 K$ and $70 K$ for several fields.
Figure 5. Lock-in angle $\varphi_{L}$ versus $1/H$ for several temperatures. The straight lines are fits according to equation (\[eq:lockin\]).
Figure 6. Temperature dependence of the slopes $\alpha(T)=d\varphi_L/d(H^{-1})$ of the linear fits of fig. 5 (full symbols). The solid line is a fit to eq. (\[eq:slope2\]).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We investigate the generalized second law of thermodynamics (GSL) in generalized theories of gravity. We examine the total entropy evolution with time including the horizon entropy, the non-equilibrium entropy production, and the entropy of all matter, field and energy components. We derive a universal condition to protect the generalized second law and study its validity in different gravity theories. In Einstein gravity, (even in the phantom-dominated universe with a Schwarzschild black hole), Lovelock gravity, and braneworld gravity, we show that the condition to keep the GSL can always be satisfied. In $f(R)$ gravity and scalar-tensor gravity, the condition to protect the GSL can also hold because the gravity is always attractive and the effective Newton constant should be approximate constant satisfying the experimental bounds.'
author:
- 'Shao-Feng Wu$^{1}$[^1], Bin Wang$^{2}$[^2], Guo-Hong Yang$^{1}$[^3], and Peng-Ming Zhang$^{3,4}$[^4]'
title: The generalized second law of thermodynamics in generalized gravity theories
---
Introduction
============
Motivated by the black hole physics, it was realized that there is a profound connection between gravity and thermodynamics. In Einstein gravity, the evidence of this connection was first discovered in [@Jacobson] by deriving the Einstein equation from the proportionality of entropy and horizon area together with the first law of thermodynamics in the Rindler spacetime. For a general static spherically symmetric spacetime, Padmanabhan pointed out that Einstein equations at the horizon give rise to the first law of thermodynamics [@Padmanabhan]. Recently the study on the connection between gravity and thermodynamics has been extended to cosmological context. Frolov and Kofman [@Frolov] employed the approach proposed by Jacobson [@Jacobson] to a quasi-de Sitter geometry of inflationary universe, and calculated the energy flux of a background slow-roll scalar through the quasi-de Sitter apparent horizon. By applying the first law of thermodynamics to a cosmological horizon, Danielsson obtained Friedmann equation in the expanding universe [@Danielsson]. In the quintessence dominated accelerating universe, Bousso [@Bousso] showed that the first law of thermodynamics holds at the apparent horizon. The relation between gravity and thermodynamics has been further disclosed in extended gravity theories, including Lovelock gravity [@Akbar1; @Cao], braneworld gravity [@Cao1; @brane], nonlinear gravity [Eling,Akbar,Cao]{}, and scalar-tensor gravity [@Akbar; @Cao] etc. In the nonlinear gravity and scalar-tensor gravity, it was argued that the non-equilibrium thermodynamics instead of the equilibrium thermodynamics should be taken into account to build the relation to gravity [Eling,Akbar,Cao]{}. In our previous work [@Wu1], we have presented a general procedure to build the connection between gravity and thermodynamics. From the Friedmann equations, we have constructed the first law of thermodynamics on the apparent horizon in generalized gravity theories. We found that the non-equilibrium entropy production term arising in non-linear gravity and scalar-tensor gravity is due to the existence of other dynamic fields besides the ordinary matter dominating the cosmological evolution.
It is of great interest to extend our discussion in [@Wu1] to study the generalized second law (GSL) of thermodynamics in the generalized gravity theories. There have been a lot of interest on investigating the GSL in gravity [Babichev,Setare,Pollock,Davies,Mohseni1,Izquierdo,Mohseni2,Wang,Zhou]{}, but all of them concentrate on the Einstein gravity. The modified theory of gravity was argued to be a possible candidate to explain the accelerated expansion of our universe, thus it is interesting to examine the GSL in the extended gravity theories. An attempt to study this problem was carried out in [@Mohseni], where it was found that some additional conditions are needed for validity of GSL. Even for the Einstein gravity, it was found that GSL breaks down in phantom-dominated universe in the presence of Schwarzschild black hole [@Izquierdo], at least in transition epoch [Mohseni2]{}. In our paper we will adopt the formalism proposed in [@Wu1]. We will derive the entropy of the horizon from the first law of thermodynamics constructed in [@Wu1]. We will examine the total entropy evolution with time including the horizon entropy, the non-equilibrium entropy production, and the entropy of all matter, field and energy components. We will derive a universal condition to protect the GSL in generalized gravity theories and examine its validity in the Einstein gravity (even in the presence of Schwarzschild black hole), Lovelock gravity, braneworld gravity, nonlinear gravity and scalar-tensor gravity.
The organization of the paper is as follows: In section 2, we briefly review the generalized first law of thermodynamics in extended theories of gravity. In section 3, we derive the universal condition to protect the GSL and examine its validity in some extended gravity theories. The last section is devoted to summary.
The first law of thermodynamics on the apparent horizon in FRW cosmology
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In this section we briefly go over the general procedure to construct the first law of thermodynamics on the apparent horizon in generalized gravity theory [@Wu1].
The homogenous and isotropic ($n+1$)-dimensional FRW universe is described by$$ds^{2}=h_{ab}dx^{a}dx^{b}+\tilde{r}^{2}d\Omega _{n-1}^{2}, \label{FRW}$$where $h_{ab}=$diag$(-1,\frac{a^{2}}{1-ka^{2}})$, $d\Omega _{n-1}^{2}$ is the $\left( n-1\right) $-dimensional sphere element, and $%
x^{0}=t,\;x^{1}=r,\;\tilde{r}=ar$ is the radius of the sphere and $a$ is the scale factor. For simplicity, we consider the flat space $k=0$ in this paper, however, our discussion can also be generalized to the non-flat cases. It is known that the dynamical apparent horizon, the marginally trapped surface with vanishing expansion, is defined as a sphere situated at $r=r_{A}$ satisfying$$h^{ab}\partial _{a}\tilde{r}\partial _{b}\tilde{r}=0.$$The sphere radius is$$\tilde{r}_{A}\equiv r_{A}a=\frac{1}{H}. \label{Horizon}$$The associated temperature on the apparent horizon is defined by surface gravity $\kappa =\frac{1}{\sqrt{-h}}\partial _{a}\left( \sqrt{-h}%
h^{ab}\partial _{b}\tilde{r}\right) $$$T=\frac{\left\vert \kappa \right\vert }{2\pi }=\frac{1}{2\pi \tilde{r}_{A}}%
\left( 1-\epsilon \right) \label{T}$$where $\epsilon \equiv \frac{\partial _{t}\tilde{r}_{A}}{2H\tilde{r}_{A}}<1$. $\epsilon <1$ ensures that the temperature is positive. Using the definition of the horizon (\[Horizon\]), the positive temperature condition can be written as $\epsilon =-\frac{\dot{H}}{2H^{2}}<1$, i.e.$$\dot{H}>-2H^{2}, \label{postive T}$$which is useful in our later discussion. In our previous work [@Wu1], in order to study the mass-like function, we construct the first law on the assumption $\epsilon \ll 1$ meaning that the apparent horizon radius is approximately fixed thereby the temperature $T=\frac{1}{2\pi \tilde{r}_{A}}$ [@Cai; @Cao]. Here we will drop this assumption.
In Einstein gravity, the entropy is proportional to the horizon area$$S_{E}=\frac{A}{4G},$$where the horizon area $A=n\Omega _{n}\tilde{r}_{A}^{n-1}$. The thermodynamical fluid $\delta Q$ can be written as$$TdS_{E}=\frac{n(n-1)V\tilde{r}_{A}^{-3}d\tilde{r}_{A}}{8\pi G}-\frac{n(n-1)V%
\tilde{r}_{A}^{-3}d\tilde{r}_{A}}{8\pi G}\frac{\partial _{t}\tilde{r}_{A}}{2H%
\tilde{r}_{A}},$$where $V=\Omega _{n}\tilde{r}_{A}^{n}$ is the volume in the horizon. Using the definition of the horizon (\[Horizon\]) and the temperature (\[T\]), we can obtain$$TdS_{E}=\frac{-n(n-1)V}{16\pi G}\frac{dH^{2}}{dt}dt-\frac{n(n-1)V}{16\pi G}%
\frac{\dot{H}^{2}}{H}dt, \label{dr1}$$which is purely a geometric relation.
In all gravity theories, Friedmann equations can be expressed in the form as that in the Einstein gravity$$H^{2}=\frac{16\pi G}{n(n-1)}\rho _{eff} \label{H21}$$$$\dot{H}=-\frac{8\pi G}{(n-1)}(\rho _{eff}+p_{eff}). \label{H22}$$Though we do not know the exact form of $\rho _{eff}$ (and $p_{eff}$), we know that there must be ordinary matter density $\rho $ in $\rho _{eff}$ and also other variables $\rho _{p}$. In some cases, $\rho _{p}$ (or their combination) may describe other matter field $\rho _{f}$ or effective energy component $\rho _{e}$ besides the ordinary matter $\rho $ in $\rho _{eff}$. The first Friedmann equation can be expressed in the form$$H^{2}=H^{2}(\rho ,\;\rho _{1},\cdots \rho _{p},\cdots ).$$Then the relation (\[dr1\]) can be changed as$$TdS_{E}=\frac{-n(n-1)V}{16\pi G}dt(\frac{\partial H^{2}}{\partial \rho }\dot{%
\rho}+\frac{\partial H^{2}}{\partial \rho _{p}}\dot{\rho}_{p})-\frac{n(n-1)V%
}{16\pi G}\frac{\dot{H}^{2}}{H}dt. \label{dr2}$$To construct the first law of thermodynamics, we need to know the energy flux $dE$ or entropy $S$. In the general gravity theory, they are not specified. However, it is known that the energy flux of ordinary matter includes $V\dot{\rho}dt$. Multiplying $\frac{16\pi G}{n(n-1)}\frac{1}{\frac{%
\partial H^{2}}{\partial \rho }}$ on both sides of (\[dr2\]), we can extract it clearly$$\frac{16\pi G}{n(n-1)}\frac{1}{\frac{\partial H^{2}}{\partial \rho }}%
TdS_{E}=-V\dot{\rho}dt-Vdt\frac{1}{\frac{\partial H^{2}}{\partial \rho }}%
\frac{\dot{H}^{2}}{H}-Vdt\frac{1}{\frac{\partial H^{2}}{\partial \rho }}%
\frac{\partial H^{2}}{\partial \rho _{p}}\dot{\rho}_{p}. \label{dr4}$$In the general case, we have the conservation$$\dot{\rho}_{eff}+nH(\rho _{eff}+p_{eff})=0. \label{Con1}$$We furthermore assume that ordinary matter (All matter and energy in this paper are assumed as perfect fluid) has energy exchange with other energy source, described by$$\dot{\rho}+nH(\rho +p)=q, \label{Con2}$$If the gravity theory has matter $\rho _{f}$ and energy content $\rho _{e}$, one may also have similar semi-conserved laws$$\begin{aligned}
\dot{\rho}_{f}+nH(\rho _{f}+p_{f})& =q_{f}, \notag \\
\dot{\rho}_{e}+nH(\rho _{e}+p_{e})& =q_{e}, \notag \\
\dot{\rho}_{t}+nH(\rho _{t}+p_{t})& =q_{t}. \label{qt}\end{aligned}$$In the last equation the total density $\rho _{t}\equiv \rho +\rho _{f}+\rho
_{e}$, total pressure $p_{t}\equiv p+p_{f}+p_{e}$, and total energy exchange $q_{t}\equiv q+q_{f}+q_{e}$ are introduced. However, it should be emphasized that one can not impose total energy fluid $q_{t}=0$, because there may be energy exchange with the horizon. Equations (\[Con1\]) and (\[Con2\]) will be used later to express the first law explicitly. Since we will consider thermodynamical effect with the change of horizon volume, we introduce the work density. Defining $T_{a}^{b}$ as the projection of energy-momentum tensor $T_{\nu }^{\mu }$ of the perfect fluid in the FRW universe in the normal direction of the ($n-1$)-sphere, we have the density $%
W\equiv -\frac{1}{2}T_{a}^{a},$ which may be viewed as the work done by the change of the apparent horizon, as pointed out in [@Hayward].
Consider the entropy change should be an exact form in the first law of thermodynamics. If there is just ordinary matter $\rho $ in the space, $%
\frac{\partial H^{2}}{\partial \rho }$ can be rewritten as a function of $%
\tilde{r}_{A}$. Then a total differential can be obtained by the integration$$S=\int \frac{16\pi G}{n(n-1)}\frac{1}{\frac{\partial H^{2}}{\partial \rho }(%
\tilde{r}_{A})}d\left( S_{E}\right) , \label{S0}$$and the relation (\[dr4\]) can be written as$$TdS=\delta Q, \label{first law1}$$where the thermodynamical fluid$$\delta Q=-V\dot{\rho}dt-Vdt\frac{1}{\frac{\partial H^{2}}{\partial \rho }}%
\frac{\dot{H}^{2}}{H}. \label{dE1}$$Since the gravity is only determined by ordinary matter, we can impose conservation $q=0$. Using the conservation equation (\[Con2\]), one can prove$$-V\frac{1}{\frac{\partial H^{2}}{\partial \rho }}\frac{\dot{H}^{2}}{H}=-%
\frac{1}{2}\left( \rho +p\right) \dot{V}.$$Then the thermodynamical fluid can be rewritten as$$\delta Q=-dE+WdV, \label{dq}$$which includes just the energy flux of ordinary matter $E=\rho V$ and the work done by the change of the apparent horizon. Thus the expression ([S0]{}) should be understood as the entropy to assure the first law$$TdS=-dE+WdV.$$ If the modified gravity theory has other dynamic fields resulting that $%
\frac{\partial H^{2}}{\partial \rho }$ is a function of $\tilde{r}_{A}$ and $%
\rho _{p}$$$\frac{\partial H^{2}}{\partial \rho }=\frac{\partial H^{2}}{\partial \rho }(%
\tilde{r}_{A},\rho _{p}),$$we can not integral the l.h.s in (\[dr4\]) directly. However we can express it as$$T\frac{16\pi G}{n(n-1)}\frac{1}{\frac{\partial H^{2}}{\partial \rho }}%
dS_{E}=Td\left( \frac{16\pi G}{n(n-1)}\frac{1}{\frac{\partial H^{2}}{%
\partial \rho }}S_{E}\right) -T\frac{16\pi G}{n(n-1)}S_{E}d\frac{1}{\frac{%
\partial H^{2}}{\partial \rho }}. \label{first law20}$$It can be rewritten as$$TdS=\delta Q,$$where$$S\equiv \frac{16\pi G}{n(n-1)}\frac{1}{\frac{\partial H^{2}}{\partial \rho }}%
S_{E}, \label{S}$$$$\delta Q\equiv -V\dot{\rho}dt-Vdt\frac{1}{\frac{\partial H^{2}}{\partial
\rho }}\frac{\dot{H}^{2}}{H}-Vdt\frac{1}{\frac{\partial H^{2}}{\partial \rho
}}\frac{\partial H^{2}}{\partial \rho _{p}}\dot{\rho}_{p}+T\frac{16\pi G}{%
n(n-1)}S_{E}d\frac{1}{\frac{\partial H^{2}}{\partial \rho }}. \label{dE}$$To construct the first law, we can write$$\delta Q=-dE+W_{t}dV-Td_{p}S,$$where $E\equiv \rho _{t}V$ is the total intrinsic energy and $W_{t}$ the total work. $d_{p}S$ is defined as$$d_{p}S\equiv -\frac{1}{T}\left( \delta Q+dE-W_{t}dV\right) \text{,}
\label{dis}$$where the subindex p" denotes the term resulted from other dynamical fields. In general, $d_{p}S$ is non-vanishing and can not be written out exactly (We will show $d_{p}S$ clearly in concrete gravity theories). If the first law of generalized gravity theory can be constructed, the entropy term contains the form (\[S\]) together with the entropy production $d_{p}S$ (\[dis\]) following the idea given in [Eling]{}.** **Here we do not consider whether the entropy is developed internally by the system as in [@Eling]. The first law is expressed as $$TdS+Td_{p}S=-dE+W_{t}dV. \label{first law2}$$The exact forms of entropy production $d_{p}S$ and entropy $S$ depend on the concrete gravity theory.
In [@Gong] it was argued that the entropy correction can be absorbed in the mass-like function and the entropy generation in the nonequilibrium can be reinterpreted. Whether one should interpret the thermodynamical equations as representing systems in equilibrium or out of equilibrium seems unclear. This is also under debate in recent works on $f(R)$ gravity [Eling1,Elizalde]{}. In our case, there is work term, since the horizon is not fixed as that in [@Gong] and the form of the mass-like function is not known. The entropy expression, inner energy and the work term in the first law Eq. (\[first law2\]) are all determined. The extra term can not be absorbed into any other terms and it has to be interpreted as the entropy production as done in [@Eling]. Since the mass-like function here is not available, we do not know how to reinterprete this entropy correction as done in [@Gong].
One might argue that there seems ambiguities in entropy expressions (\[S0\]) (\[S\]) since one might add a proper quantity to the expression of entropy, which may vanish in the Einstein gravity, and this extra term could be absorbed into the redefinition of the entropy production. This worry is not necessary. As pointed out in [@Wu1], the known black hole entropy in different gravity theories will strictly restrict the form of the additional quantities in the entropy expressions. The definition of entropy in (\[S0\]) and (\[S\]) can recover the exact expression of the known black hole entropy [@Wu1] such as in Lovelock gravity [@Cai1], nonlinear gravity [@Wald] and scalar-tensor gravity [@Cai2]. Moreover, following the work in [@Gong], the general mass-like function which can be reduced to the corresponding Misner-Sharp mass has been found. If we add other quantities in entropy expressions (\[S0\]) (\[S\]), it seems very difficult to obtain the corresponding mass-like function which can be reduced to the known Misner-Sharp mass. This serves as another restriction on adding additional terms to the entropy expressions.
GSL of extended gravity theories
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Now we start to discuss the GSL in generalized gravity theories. Most discussions on the GSL focus on the Einstein gravity. In generalized gravity theories, from the first law of thermodynamics (\[first law2\]) one can see that an entropy production term appears, which characterizes the non-equilibrium thermodynamical process about the horizon. In describing the GSL one should include this non-equilibrium entropy production term. Besides, in addition to the matter and fields, one should also include the entropy for the effective energy in describing the GSL, since the change of energy is relative to the change of entropy.
Now we are going to use the first law of thermodynamics (\[first law2\]) to find the general condition need to hold the GSL in any gravity theories. From (\[first law2\]), we have the expression of the horizon entropy$$TdS_{h}=-Td_{p}S-dE+W_{t}dV. \label{Sh2}$$ On the other hand, if there is only ordinary matter inside the cosmological horizon of a comoving observer, the entropy of the ordinary matter is relative to the energy $\rho $ and the pressure $p$ via the Gibbs equation$$T_{\rho }dS_{\rho }=d(\rho V)+pdV=Vd\rho +(\rho +p)dV. \label{Sroi}$$As pointed out in [@Pavon], the ordinary matter should be understood as a phenomenological representation of a mixture of fields, each of which may or may not be in a pure state, and therefore entitled to an entropy. $%
T_{\rho }$ refers to the temperature of matter inside the horizon. If there are other matter field and energy component, one can similarly have the Gibbs equation (including all matter, field, and energy contents) $$T_{t}dS_{t}=d(\rho _{t}V)+p_{t}dV=Vd\rho _{t}+(\rho _{t}+p_{t})dV.
\label{St}$$Here the temperature of total energy inside the horizon is denoted as $T_{t}$. It should be noted that the temperature $T_{t}$ is not equal to $T$ in general because there may be energy flow $q_{t}$ (thereby thermodynamical fluid) between the horizon and energy inside the horizon. Since the only temperature scale we have at our disposal is the temperature of the apparent horizon $T$, we assume$$T_{t}=bT,$$where the temperature parameter satisfying $0<b<1$ assures the temperature being positive and smaller than the horizon temperature. This ansatz is similar to the disposal when the horizon is taken as the event horizon [Pavon, Mohseni1]{}, where the only temperature scale is the de Sitter temperature [@Davies].
Now, we propose that GSL should be expressed as$$\dot{S}_{h}+d_{p}\dot{S}+\dot{S}_{t}\geq 0 \label{S00}$$where $d_{p}\dot{S}\equiv \partial _{t}\left( d_{p}S\right) $. Summing up Eqs. (\[Sh2\]) and (\[St\]), the GSL reads$$-b\dot{E}+bW_{t}\dot{V}+V\dot{\rho}_{t}+(\rho _{t}+p_{t})\dot{V}=(1-b)\dot{%
\rho}_{t}V+(1-\frac{b}{2})(\rho _{t}+p_{t})\dot{V}\geq 0. \label{Condition0}$$It should be noted that if there is no energy flow ($q_{t}=0$) between the horizon and energy therein, the horizon and the energy inside it are in thermal equilibrium $b=1$.
One can immediately find that the GSL holds for Einstein gravity. Two Friedmann equations (\[H21\]) (\[H22\]) make the energy flow $q$ in the continuity equation (\[Con2\]) vanish, thereby the thermal flow vanishes. Employing two Friedmann equations (\[H21\]) (\[H22\]) in condition ([Condition0]{}) by replacing $\rho _{t}$ ($p_{t}$) as $\rho $ ($p$), we have $$-\dot{E}+W\dot{V}+V\dot{\rho}+(\rho +p)\dot{V}=\frac{1}{2}(\rho +p)\dot{V}=%
\frac{n\left( n-1\right) }{16\pi G}\Omega _{n}H^{-n-1}\dot{H}^{2}\geq 0.
\label{EinC}$$In the derivation of the above equation we have used the second Friedmann equation $\dot{H}=-\frac{8\pi G}{(n-1)}(\rho +p)$ and $\dot{V}=\partial
_{t}(\Omega _{n}\tilde{r}_{A}^{n})=\partial _{t}(\Omega _{n}H^{-n})=-n\Omega
_{n}H^{-n-1}\dot{H}$. In [@Zhou], the condition (\[EinC\]) without the term $W\dot{V}$ was obtained, which corresponds to the approximation $T=%
\frac{1}{2\pi \tilde{r}_{A}}$.
Now we consider a Schwarzschild black hole inside the apparent horizon in $%
\left( 3+1\right) $-dimensional spacetime, whose mass is assumed to be small enough $M\ll\rho V$ so that the FRW metric remains unchanged (thereby $b=1$). Using the first Friedmann equation (\[H21\]) in $\left( 3+1\right) $-dimensional spacetime, this condition reduces to$$MH\ll\frac{\tilde{r}_{A}^{3}H^{3}}{2}=\frac{1}{2}, \label{MH}$$ where we set $G=1$. In a fluid with the energy density $\rho$ and the pressure $p$, the change rate of the black hole mass was obtained in [Babichev]{}$$\dot{M}=4\alpha r_{h}^{2}(\rho+p)M^{2}=-4\alpha M^{2}\dot{H},$$ where $r_{h}$ is the radius of the black hole horizon and $\alpha\sim O(1)$ is a positive numerical constant. The entropy of the black hole is $%
S_{bl}=4\pi M^{2}$ [@Bekenstein1], therefore$$\dot{S}_{bl}=-32\pi\alpha M^{2}\dot{H}. \label{sbl}$$ To protect the GSL, we require that the sum of the apparent horizon entropy, the ordinary matter entropy and the black hole entropy cannot decrease with time:$$\dot{S}_{h}+\dot{S}_{\rho}+\dot{S}_{bl}\geq0.$$ Employing Eqs. (\[Sh2\]), (\[Sroi\]), (\[sbl\]), together with the Friedmann equation (\[H21\]) and $b=1$, the condition to protect the GSL reads$$2\pi\dot{H}\left( -16\alpha M^{3}+\frac{\dot{H}}{2H^{5}}\frac{1}{1-\epsilon }%
\right) \geq0.$$ If the fluid surrounding the black hole is quintessence $\dot{H}<0$, the GSL holds always. If the fluid is the phantom type, $\dot{H}>0$, the GSL can also hold, because$$\frac{\dot{H}}{H^{2}}\frac{1}{1-\epsilon}\geq32\alpha M^{3}H^{3}\sim0,$$ where the condition (\[MH\]) and the positive temperature has been considered.
In the following, we are going to extend the discussion on the condition (\[Condition0\]) to protect the GSL to generalized gravity theories.
Lovelock gravity
----------------
The Lagrangian of the Lovelock gravity consists of the dimensionally extended Euler densities [@Lovelock]$$L=\sum_{i=1}^{[n/2]}c_{i}L_{i},$$where $c_{i}$ is an arbitrary positive constant and $L_{i}$ is the Euler density of a ($2i$)-dimensional manifold$$L_{i}=2^{-i}\delta _{\alpha _{1}\beta _{1}\cdots \alpha _{ii}\beta
_{i}}^{\mu _{1}\nu _{1}\cdots \mu _{i}\nu _{i}}R_{\mu _{1}\nu _{1}\cdots \mu
_{i}\nu _{i}}^{\alpha _{1}\beta _{1}\cdots \alpha _{ii}\beta _{i}}.$$$L_{1}$ is just the Einstein-Hilbert term, and $L_{2}$ corresponds to the so called Gauss-Bonnet term. Using the FRW metric, we obtain Friedmann equations in ($n+1$)-dimensional spacetime$$\sum_{i=1}^{[n/2]}\hat{c}_{i}\left( H^{2}\right) ^{i}=\frac{16\pi G}{n(n-1)}%
\rho , \label{FM1Lovelock}$$and$$\sum_{i=1}^{[n/2]}\hat{c}_{i}i\left( H^{2}\right) ^{i-1}(\dot{H})=-\frac{%
8\pi G}{(n-1)}(\rho +p), \label{LoveH2}$$where$$\hat{c}_{i}=\frac{(n-2)!}{(n-2i)!}c_{i}$$Since only one dynamic field $\rho $ in the first Friedmann equation ([FM1Lovelock]{}), the first law on the horizon is described by the equilibrium thermodynamics $d_{p}S=0$. Two Friedmann equations (\[FM1Lovelock\]) ([LoveH2]{}) make the energy flow $q=0$, so the horizon and energy are in thermodynamical equilibrium $b=1$. From two Friedmann equations ([FM1Lovelock]{}) (\[LoveH2\]), one can find that the condition ([Condition0]{}) always holds$$\frac{1}{2}(\rho +p)\dot{V}=\frac{n\left( n-1\right) }{16\pi G}\Omega
_{n}H^{-1-n}\sum_{i=1}^{[n/2]}\hat{c}_{i}i\left( H^{2}\right) ^{i-1}\dot{H}%
^{2}\geq 0. \label{GSllovelock}$$The condition to protect the GSL in the Einstein gravity (\[EinC\]) can be recovered when $\hat{c}_{1}=1$, $\hat{c}_{i}=0$ ($i>1$).
Randall-Sundrum braneworld gravity
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We consider a $n$-dimensional brane embedded in a ($n+2$)-dimensional spacetime. Using the junction condition on the brane, we can obtain Friedmann equations$$H^{2}=\frac{1}{4n^{2}}\rho^{2}+\frac{1}{2n^{2}}\lambda\rho
\label{RSFriedmann1}$$$$\dot{H}=-\frac{1}{4n}(\rho+p)(\rho+\lambda) \label{RSFriedmann2}$$ where the Randall-Sundrum fine-turning condition$$\frac{1}{4n^{2}}\lambda^{2}+\frac{2\Lambda_{n+2}}{n(n+1)}=0$$ has been used. Using the junction condition into the $(05)$ component of the field equation we can obtain the conserved equation$$\dot{\rho}+nH(\rho+p)=q=0, \label{RSq3}$$ which results that the horizon and energy inside the horizon are in thermodynamical equilibrium $b=1$.
We see that $\rho $ is the only freedom in the first Friedmann equation ([RSFriedmann1]{}), so we only need to consider the equilibrium thermodynamics about the horizon. From (\[RSFriedmann1\]) and (\[RSFriedmann2\]), one can find that the condition (\[Condition0\]) to protect the GSL always holds$$\frac{1}{2}(\rho +p)\dot{V}=\frac{2n^{2}\Omega _{n}}{\sqrt{\lambda
^{2}+4n^{2}H^{2}}}H^{-n-1}\dot{H}^{2}\geq 0, \label{GSLbrane}$$The condition (\[GSLbrane\]) can be reduced to the condition in Einstein gravity (\[EinC\]) at low energy $\rho \ll \lambda $.
Nonlinear gravity
-----------------
For the nonlinear gravity $f(R)$, the Lagrangian is$$L=\frac{1}{16\pi G}f(R)$$The variational principle gives equations of motion. Using the FRW metric, one can obtain Friedmann equations in ($n+1$)-dimensional space-time$$H^{2}=\frac{16\pi G}{n(n-1)}\frac{1}{f^{\prime }}\left( \rho +\rho
_{c}f^{\prime }\right) \label{FMfr1}$$$$\dot{H}=-\frac{8\pi G}{(n-1)}\frac{1}{f^{\prime }}(\rho +\rho _{c}f^{\prime
}+p+p_{c}f^{\prime }), \label{FMfr2}$$where$$\rho _{c}=\frac{1}{8\pi Gf^{\prime }}\left[ -\frac{f-Rf^{\prime }}{2}%
-nHf^{\prime \prime }\dot{R}\right]$$$$p_{c}=\frac{1}{8\pi Gf^{\prime }}\left[ (f-Rf^{\prime })-f^{\prime \prime }%
\ddot{R}+f^{\prime \prime \prime }\dot{R}^{2}+n(n-1)f^{\prime \prime }\dot{R}%
\right] .$$The prime denotes the derivate respect to $R$. Since $\frac{\partial H^{2}}{%
\partial \rho }$ is determined by dynamic field $f^{\prime }$ while not the horizon radius uniquely, one should consider the horizon described by non-equilibrium thermodynamics. One can select $\rho _{p}$ arbitrarily. For example, we select $\rho _{p}=(f^{\prime },\rho _{c})$. There is not the real matter field besides the ordinary matter $\rho $. It is important to observe that from the Friedmann equations, $\rho _{e}\equiv \rho
_{c}f^{\prime }$ ($p_{e}\equiv p_{c}f^{\prime }$) acts as the density (pressure) of an effective energy component in $f(R)$ gravity.
The condition (\[Condition0\]) reads$$(1-b)\dot{\rho}_{t}V+(1-\frac{b}{2})(\rho _{t}+p_{t})\dot{V}\geq 0,$$where the total density (pressure) is $\rho _{t}\equiv \rho +\rho _{e}$ ($%
p_{t}\equiv p+p_{e}$). Solving $\rho _{t}$ and $p_{t}$ from two Friedmann equations (\[FMfr1\]) and (\[FMfr2\]), and substituting them into the above inequality, we find$$\frac{n(n-1)\Omega _{n}}{16\pi G}H^{-(n+1)}\left[ (1-b)H^{3}\dot{f}^{\prime
}+2(1-b)f^{\prime }H^{2}\dot{H}+(2-b)f^{\prime }\dot{H}^{2}\right] \geq 0.
\label{GSLfr0}$$This inequality is the condition to protect the GSL in the $f(R)$ gravity. Now we like to point out a nontrivial observation that this condition may hold always. From two Friedmann equations (\[FMfr1\]) and (\[FMfr2\]), we find that $\dot{f}^{\prime }$ is related to the total energy flow $q_{t}$$$q_{t}=\frac{n(n-1)}{16\pi G}H^{2}\dot{f}^{\prime }. \label{qtt}$$Since $G/f^{\prime }$ takes role as the effective Newton gravitational constant, the relation (\[qtt\]) means the energy fluid is determined by the evolvement of effective Newton gravitational constant. Consider the temperature of total energy $bT$, which in general can not equal to the temperature of horizon $T$. However, it is known that the experimental bounds acquires the Newton constant should be approximate constant [Ozan]{}. From the relation (\[qtt\]), one can find that the big energy fluid is prohibited to protect the effective Newton constant as approximate constant. Thereby the thermodynamic fluid is small and the temperature of the horizon is very closed with the one of the energy source therein $b\sim 1
$. This is more reasonable if two systems undergo some length of time. So the former two terms in the bracket of condition (\[GSLfr0\]) may be neglected. Moreover, since the gravity is always attractive, we can impose $%
f^{\prime }>0$. Thus the last term in the bracket is positive and the GSL always holds. We also note that when the effective Newton constant is constant indeed $\dot{f}^{\prime }=0$ which leads $q_{t}=0$ then $b=1$, the GSL always holds, recovering the results of Einstein gravity.
It is interesting to show the entropy production $d_{i}S$ clearly. Recalling the conserved equation (\[Con1\])$$\dot{\rho}_{eff}=-nH(\rho _{eff}+p_{eff}) \label{con}$$and using the first Friedmann equation (\[FMfr1\]), one can find that the l.h.s in Eq. (\[con\]) reads$$\begin{aligned}
\dot{\rho}_{eff}& =\frac{n(n-1)}{16\pi G}\left( \frac{\partial H^{2}}{%
\partial \rho }\dot{\rho}+\frac{\partial H^{2}}{\partial \rho _{p}}\dot{\rho}%
_{p}\right) \notag \\
& =\frac{n(n-1)}{16\pi G}\left( \frac{16\pi G}{n(n-1)}\frac{1}{f^{\prime }}%
\dot{\rho}+\frac{\partial H^{2}}{\partial \rho _{p}}\dot{\rho}_{p}\right)
\notag \\
& =\frac{1}{f^{\prime }}\dot{\rho}+\frac{n(n-1)}{16\pi G}\frac{\partial H^{2}%
}{\partial \rho _{p}}\dot{\rho}_{p} \label{r1}\end{aligned}$$while by using the second Friedmann equation (\[FMfr2\]) the r.h.s in Eq. (\[con\]) reads $$-nH(\rho _{eff}+p_{eff})=-nH\left[ \frac{1}{f^{\prime }}\left( \rho
_{t}+p_{t}\right) \right] . \label{r2}$$Employing the continuous equation (\[Con2\]) to Eqs. (\[r1\]) and ([r2]{}), one can find$$\frac{\partial H^{2}}{\partial \rho _{p}}\dot{\rho}_{p}=-\frac{16\pi G}{%
n(n-1)}\frac{1}{f^{\prime }}\left[ nH(\rho _{t}+p_{t}-\rho -p)+q\right] .
\label{ls}$$Using the continuous equation (\[qt\]) and substituting Eq. (\[ls\]) into Eq. (\[dE\]), we have$$\begin{aligned}
\delta Q &=&nVH(\rho +p)dt-Vqdt-Vdt\frac{1}{\frac{\partial H^{2}}{\partial
\rho }}\frac{\partial H^{2}}{\partial \rho _{p}}\dot{\rho}_{p}-V\frac{1}{%
\frac{\partial H^{2}}{\partial \rho }}\frac{\dot{H}^{2}}{H}+T\frac{16\pi G}{%
n(n-1)}S_{E}d\frac{1}{\frac{\partial H^{2}}{\partial \rho }} \notag \\
&=&nVH(\rho _{t}+p_{t})dt-Vdt\frac{1}{\frac{\partial H^{2}}{\partial \rho }}%
\frac{\dot{H}^{2}}{H}+TS_{E}df^{\prime } \notag \\
&=&nVH(\rho _{t}+p_{t})dt-\frac{1}{2}\left( \rho _{t}+p_{t}\right) \dot{V}%
+TS_{E}df^{\prime } \notag \\
&=&-dE+W_{t}dV+Vq_{t}dt+TS_{E}df^{\prime }. \label{dqq}\end{aligned}$$One can find clearly that the energy fluid $Vq_{t}dt$ accounts for some part of the thermodynamical fluid. From Eqs. (\[dis\]) and (\[dqq\]), the entropy production can be obtained $$\begin{aligned}
d_{i}S &=&-\frac{1}{T}Vq_{t}dt-S_{E}df^{\prime } \\
&=&\frac{-n\Omega _{n}H^{-n+1}df^{\prime }\left[ \left( n+1\right) H^{2}+%
\dot{H}\right] }{4G(2H^{2}+\dot{H})}.\end{aligned}$$The second equation has used the two Friedmann equations and Eq. (\[qtt\]). One can find $d_{i}S$ is not an exact form as desired and it is vanishing when the effective Newton gravitational constant $f^{\prime }$ is constant indeed. This is a reasonable result since the Einstein gravity is recovered in this situation. It should be noticed that this entropy production is different with the one given in Ref. [@Eling; @Cao], where they have not considered the energy flux of effective energy $\rho _{e}$. If we really omit the energy flux of effective energy, the condition to protect GSL ([S00]{}) is (for simplicity, we consider the ($3+1$)-dimensional space-time)$$\begin{aligned}
&&\dot{S}_{h}+d_{i}\dot{S}+\dot{S}_{t}=-b\left[ \partial _{t}(\rho V)+W\dot{V%
}\right] +V\dot{\rho}_{t}+(\rho _{t}+p_{t})\dot{V} \\
&=&\frac{\Omega _{3}}{32\pi GH^{4}}\left\{
\begin{array}{c}
12(1+b)H^{3}\dot{f}^{\prime }-3bf\dot{H}+6(4+b)f^{\prime }\dot{H}^{2} \\
+12H^{2}\left[ (2+5b)f^{\prime }\dot{H}-b\ddot{f}^{\prime }\right]
-2bH\left( \dot{f}+9\dot{f}^{\prime }\dot{H}-6f^{\prime }\ddot{H}\right)
\end{array}%
\right\} \geq 0,\end{aligned}$$Obviously, the GSL can not be held always in this situation, which is a strong point to favor the present entropy production (\[dis\]).
Scalar-tensor gravity
---------------------
The general scalar-tensor theory of gravity is described by the Lagrangian$$L=F\left( \phi \right) R-\frac{1}{2}g^{\mu \nu }\partial _{\mu }\phi
\partial _{\nu }\phi -V\left( \phi \right) ,$$where $F(\phi )$ is a positive continuous function of the scalar field $\phi
$ and $V(\phi )$ is its potential. Using the FRW metric, we obtain Friedmann equations in ($n+1$)-dimensional space-time$$H^{2}=\frac{16\pi G}{n(n-1)}\frac{1}{F}\left( \rho +\rho _{f}+\rho
_{c}F\right) \label{FMfai1}$$$$\dot{H}=\frac{8\pi G}{(n-1)}\frac{1}{F}\left( \rho +p+\rho _{f}+p_{f}+\rho
_{c}F+p_{c}F\right) . \label{FMfai2}$$where the density and pressure of scalar field $\phi $ are$$\begin{aligned}
\rho _{f}& =\frac{1}{2}\dot{\phi}^{2}+V\left( \phi \right) \\
p_{f}& =\frac{1}{2}\dot{\phi}^{2}-V\left( \phi \right) ,\end{aligned}$$and $\rho _{e}\equiv \rho _{c}F$ ($p_{e}\equiv p_{c}F$) can be understood as effective density (pressure) of the energy component in scalar-tensor theory: $$\rho _{c}=-\frac{n}{8\pi GF}H\dot{F}$$$$p_{c}=\frac{1}{8\pi GF}\left[ \ddot{F}+\left( n-1\right) H\dot{F}\right] .$$Obviously, we need to consider the non-equilibrium thermodynamics of horizon. We select $\rho _{p}=(\rho _{f},F,\rho _{e})$.
Consider the condition (\[Condition0\]) for GSL$$(1-b)\dot{\rho}_{t}V+(1-\frac{b}{2})(\rho _{t}+p_{t})\dot{V}\geq 0,$$where the total density (pressure) is $\rho _{t}\equiv \rho +\rho _{f}+\rho
_{e}$ ($p_{t}\equiv p+p_{f}+p_{e}$). Solving $\rho _{t}$ and $p_{t}$ from two Friedmann equations (\[FMfai1\]) and (\[FMfai2\]), and substituting them into the above inequality, we find that the inequality can be reduced to,$$\frac{n(n-1)\Omega _{n}}{16\pi G}H^{-(n+1)}\left[ (1-b)H^{3}\dot{F}%
+2(1-b)FH^{2}\dot{H}+(2-b)F\dot{H}^{2}\right] \geq 0 \label{GSLScalar0}$$The effective Newton gravitational constant in the scalar tensor theory is taken as $G/F$. From two Friedmann equations (\[FMfai1\]) and (\[FMfai2\]), the dynamical Newton constant is related to the energy flow$$q_{t}=\frac{n(n-1)}{16\pi G}H^{2}\dot{F}. \label{qt2}$$The condition (\[GSLScalar0\]) and the relation (\[qt2\]) are the same as those in the $f(R)$ gravity. As observed in the $f(R)$ gravity, we can think that the GSL for scalar-tensor theory always holds, since we can impose $F>0$ which means the gravity is always attractive, and the big energy fluid is prohibited to protect the effective Newton constant as approximate constant under experimental bounds, thereby the temperature of the horizon is very closed with the one of the energy source therein $b\sim
1 $. Moreover, it should be noticed that, actually $f(R)$ gravity is a special scalar-tensor theory by introducing the scalar field $\phi =R$ and potential $V=\phi f^{\prime }-f$ and choosing the Brans-Dick parameter $%
\omega =0$ ([@Brans; @Bergmann]—see [@Faraoni] for a review).
Now we will evaluate the entropy production. The process is similar to the one of nonlinear gravity. Recalling the continuous equation (\[Con1\]) and using the first Friedmann equation (\[FMfai1\]), one can find that the l.h.s in Eq. (\[con\]) reads $$\dot{\rho}_{eff}=\frac{1}{F}\dot{\rho}+\frac{n(n-1)}{16\pi G}\frac{\partial
H^{2}}{\partial \rho _{p}}\dot{\rho}_{p}. \label{r3}$$Using the second Friedmann equation (\[FMfai2\]) the r.h.s in Eq. ([con]{}) reads $$-nH(\rho _{eff}+p_{eff})=-nH\frac{1}{F}(\rho _{t}+p_{t}). \label{r4}$$Employing the continuous equation (\[Con2\]) to Eqs. (\[r3\]) and ([r4]{}), one can find$$\frac{\partial H^{2}}{\partial \rho _{p}}\dot{\rho}_{p}=-\frac{16\pi G}{%
n(n-1)}\frac{1}{F}\left[ nH(\rho _{t}+p_{t}-\rho -p)+q\right] . \label{ls1}$$By substituting Eq. (\[ls1\]) into Eq. (\[dE\]), one can obtain $$\delta Q=-dE+W_{t}dV+Vq_{t}dt+TS_{E}dF.$$Thus the entropy production can be obtained from Eqs. (\[dis\]) and ([qt2]{}) $$d_{i}S=\frac{-n\Omega _{n}H^{-n+1}dF\left[ \left( n+1\right) H^{2}+\dot{H}%
\right] }{4G(2H^{2}+\dot{H})}.$$One can find $d_{i}S$ is not an exact form and it is vanishing when the effective Newton gravitational constant $F$ is constant indeed.
Summary
=======
In this paper, we have examined the GSL in generalized theories of gravity. We have adopted the procedure developed in [@Wu1] to obtain the entropy on the horizon. In studying the GSL, we have examined the evolution of the entropy contributed by all matter, field and energy contents. We have derived a universal condition to protect the GSL and examined its validity. In Einstein gravity, (even in the phantom-dominated universe with a Schwarzschild black hole), Lovelock gravity, and braneworld gravity, we show that the condition to keep the GSL can always be satisfied. In $f(R)$ gravity and scalar-tensor gravity, the condition to protect the GSL can also hold under the consideration that the gravity is always attractive and the energy fluid between the horizon and total energy source therein is very small to protect the effective Newton constant as the approximate constant satisfying the experimental bounds. The same requirement for the $f(R)$ gravity to hold the GSL as that of the scalar-tensor gravity shows again that $f(R)$ gravity is a special scalar-tensor theory.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was partially supported by the NSFC, Shanghai Education Commission, Science and Technology Commission. This work was also supported by the NSFC under Grant Nos. 10575068 and 10604024, the Shanghai Research Foundation No. 07dz22020, the CAS Knowledge Innovation Project Nos. KJcx.syw.N2, the Shanghai Education Development Foundation, and the Innovation Foundation of Shanghai University.
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[^1]: Corresponding author. Email: [email protected]; Phone: +86-021-66136202.
[^2]: Email: [email protected]
[^3]: Email: [email protected]
[^4]: Email: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This paper studies $U(1)$-Chern-Simons theory and its relation to a construction of Chris Beasley and Edward Witten ([@bw]). The natural geometric setup here is that of a three-manifold with a Seifert structure. Based on a suggestion of Edward Witten we are led to study the stationary phase approximation of the path integral for $U(1)$-Chern-Simons theory after one of the three components of the gauge field is decoupled. This gives an alternative formulation of the partition function for $U(1)$-Chern-Simons theory that is conjecturally equivalent to the usual $U(1)$-Chern-Simons theory (as in [@m]). The goal of this paper is to establish this conjectural equivalence rigorously through appropriate regularization techniques. This approach leads to some rather surprising results and opens the door to studying hypoelliptic operators and their associated eta-invariants in a new light.'
address:
- 'Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4'
- 'Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4'
author:
- Lisa Jeffrey
- Brendan McLellan
bibliography:
- 'finalthesis.bib'
date: 'January 15, 2010'
title: 'Eta-invariants and Anomalies in $U(1)$-Chern-Simons theory'
---
[^1]
[^2]
\[proposition\][Example]{}
Introduction
============
In [@bw] the authors study the Chern-Simons partition function (see [@bw], (3.1)), $$\label{orgchern}
Z(k)=\frac{1}{\text{Vol}(\mathcal{G})}\left(\frac{k}{4\pi^2}\right)^{\Delta{\mathcal{G}}}\int \mathcal{D}A\,\,\text{exp}\left[i\frac{k}{4\pi}\int_{X}\text{Tr}\left(A\wedge dA+\frac{2}{3}A\wedge A\wedge A\right)\right],$$ where,
- $A\in\mathcal{A}_{P}=\{ A\in (\Omega^{1}(P)\otimes\frak{g})^{G}\,\,|\,\,A(\xi^{\sharp})=\xi, \,\,\forall\,\xi\in\frak{g}\}$ is a connection on a principal $G$-bundle $\pi:P\rightarrow X$[^3] over a closed three-manifold $X$,
- $\frak{g}=\text{Lie{G}}$ and $\xi^{\sharp}\in\Gamma(TX)$ is the vector field on $P$ generated by the infinitesimal action of $\xi$ on $P$,
- $k\in{\mathbb{Z}}$ (thought of as an element of $H^{4}(BG,{\mathbb{Z}})$ that parameterizes the possible Chern-Simons invariants),
- $\mathcal{G}:=\{\psi\in(\text{Diff}(P,P))^{G}\,\,|\,\,\pi\circ\psi=\pi\}$ is the *gauge group*,
- $\Delta(\mathcal{G})$ is formally defined as the dimension of the gauge group.[^4]
In general, the partition function of Eq. \[orgchern\] does not admit a general mathematical interpretation in terms of the cohomology of some classical moduli space of connections, in contrast to Yang-Mills theory for example (cf. [@w2]). The main result of [@bw], however, is that if $X$ is assumed to carry the additional geometric structure of a Seifert manifold, then the partition function of Eq. \[orgchern\] *does* admit a more conventional interpretation in terms of the cohomology of some classical moduli space of connections. Using the additional Seifert structure on $X$, [@bw] decouple one of the components of a gauge field $A$, and introduce a new partition function (cf. [@bw] ; Eq. 3.7), $$\label{newchern}
\bar{Z}(k)=K\cdot\int \mathcal{D}A\mathcal{D}\Phi\,\,\text{exp}\left[i\frac{k}{4\pi}\left(CS(A)-\int_{X}2\kappa\wedge\text{Tr}(\Phi F_{A})+\int_{X}\kappa\wedge d\kappa\,\,\text{Tr}(\Phi^{2})\right)\right],$$ where
- $K:=\frac{1}{\text{Vol}(\mathcal{G})}\frac{1}{\text{Vol}(\mathcal{S})}\left(\frac{k}{4\pi^2}\right)^{\Delta{\mathcal{G}}}$,
- $\kappa\in\Omega^{1}(X,{\mathbb{R}})$ is a contact form associated to the Seifert fibration of $X$ (cf. [@bw] ; §3.2),
- $\Phi\in\Omega^{0}(X,\frak{g})$ is a Lie algebra-valued zero form on $X$,
- $\mathcal{D}\Phi$ is a measure on the space of fields $\Phi$,[^5]
- $\mathcal{S}$ is the space of local *shift symmetries*[^6] that “acts” on the space of connections $\mathcal{A}_{P}$ and the space of fields $\Phi$ (cf. [@bw] ; §3.1),
- $F_{A}\in\Omega^{2}(X,\frak{g})$ is the curvature of $A$, and
- $CS(A):=\int_{X}\text{Tr}\left(A\wedge dA+\frac{2}{3}A\wedge A\wedge A\right)$ is the Chern-Simons action. [^7]
[@bw] then give a heuristic argument showing that the partition function computed using the alternative description of Eq. \[newchern\] should be the same as the Chern-Simons partition function of Eq. \[orgchern\]. In essence, they show $$\label{maint}
Z(k)=\bar{Z}(k),$$ by gauge fixing $\Phi=0$ using the shift symmetry. [@bw] then observe that the $\Phi$ dependence in the integral can be eliminated by simply performing the Gaussian integral over $\Phi$ in Eq. \[newchern\] directly. They obtain the alternative formulation: $$\label{newchern2}
Z(k)=\bar{Z}(k)=K'\cdot\int \mathcal{D}A\,\,\text{exp}\left[i\frac{k}{4\pi}\left(CS(A)-\int_{X}\frac{1}{\kappa\wedge d\kappa}\,\,\text{Tr}\left[(\kappa\wedge F_{A})^{2}\right]\right)\right],$$ where $K':=\frac{1}{\text{Vol}(\mathcal{G})}\frac{1}{\text{Vol}(\mathcal{S})}\left(\frac{-ik}{4\pi^2}\right)^{\Delta{\mathcal{G}}/2}$.[^8]\
\
The objective in this article is to study the partition function for *$U(1)$-Chern-Simons theory* using the analogue of Eq. \[newchern2\] in this case. Thus, we are also assuming here that $X$ is a Seifert manifold with a “compatible” contact structure, $(X,\kappa)$ (cf. [@bw] ; §3.2). Note that any compact, oriented three-manifold possesses a contact structure and one aim of future work is to extend our results to *all* closed three-manifolds using this fact. For now, we restrict ourselves to the case of closed three-manifolds that possess contact compatible Seifert structures (see Definition \[geodef\] for example). We restrict to the gauge group $U(1)$ so that the action is quadratic and hence the stationary phase approximation is exact. A salient point is that the group $U(1)$ is not simple, and therefore may have non-trivial principal bundles associated with it. This makes the $U(1)$-theory very different from the $SU(2)$-theory in that one must now incorporate a sum over bundle classes in a definition of the $U(1)$-partition function. As an analogue of Eq. \[orgchern\], our basic definition of the partition function for $U(1)$-Chern-Simons theory is now $$\label{abelchern1}
Z_{U(1)}(X,k)=\sum_{p\in\text{Tors}H^{2}(X;{\mathbb{Z}})}Z_{U(1)}(X,p,k)$$ where $$\label{abelchern2}
Z_{U(1)}(X,p,k)=\frac{1}{Vol(\mathcal{G}_{P})}\int_{\mathcal{A}_{P}}\mathcal{D}A e^{\pi i k S_{X,P}(A)},$$ recalling that the torsion subgroup $\text{Tors}H^{2}(X;{\mathbb{Z}})< H^{2}(X;{\mathbb{Z}})$[^9] enumerates the $U(1)$-bundle classes that have flat connections. Note that the bundle $P\rightarrow X$ in Eq. \[abelchern2\] is taken to be any representative of a bundle class with first Chern class $c_{1}(P)=p\in \text{Tors}H^{2}(X;{\mathbb{Z}})$. Also note that some care must be taken to define the Chern-Simons action, $S_{X,P}(A)$, in the case that $G=U(1)$. We outline this construction in Appendix \[appen1\].\
\
The main results of this article may be summarized as follows. First, our main objective is the rigorous confirmation of the heuristic result of Eq. \[maint\] in the case where the gauge group is $U(1)$. This statement is certainly non-trivial and involves some fairly deep facts about the “contact operator” as studied by Michel Rumin (cf. [@r]). Recall that this is the second order operator “$D$” that fits into the complex, $$\label{complex}
C^{\infty}(X)\xrightarrow{\text{$d_{H}$}}\Omega^{1}(H)\xrightarrow{\text{$D$}}\Omega^{2}(V)\xrightarrow{\text{$d_{H}$}}\Omega^{3}(X),$$ and is defined by: $$D\alpha=\kappa\wedge [\mathcal{L}_{\xi}+d_{H}\star_{H} d_{H}]\alpha,\,\,\alpha\in\Omega^{1}(H).$$ This operator is elaborated upon in §\[Dsec\] below. A somewhat surprising observation is that this operator shows up quite naturally in $U(1)$-Chern-Simons theory (see Prop. \[prop1\] below), and this leads us to make several conjectures motivated by the rigorous confirmation of the heuristic result of Eq. \[maint\]. Our main result is the following:
\[mprop\] Let $(X,\phi,\xi,\kappa,g)$ be a closed, *quasi-regular K-contact* three manifold. If, $$\label{nz}
\bar{Z}_{U(1)}(X,p,k)=k^{n_X}e^{\pi i k S_{X,P}(A_{0})}e^{\frac{\pi i}{4}\left(\eta(-\star D)+\frac{1}{512}\int_{X}R^{2}\,\,\kappa\wedge d\kappa\right)}\int_{\mathcal{M}_{P}}\,\, (T^{d}_{C})^{1/2}$$ where $R\in C^{\infty}(X)$ $=$ the Tanaka-Webster scalar curvature of $X$, and ([@m]), $$\label{oz}
Z_{U(1)}(X,p,k)=k^{m_X}e^{\pi i k S_{X,P}(A_{0})}e^{\pi i\left(\frac{\eta(-\star d)}{4}+\frac{1}{12}\frac{\text{CS}(A^{g})}{2\pi}\right)}\int_{\mathcal{M}_{P}}\,\, (T^{d}_{RS})^{1/2}$$ then, $$Z_{U(1)}(X,k)=\bar{Z}_{U(1)}(X,k)$$ as topological invariants.
Following [@m], we rigorously define $\bar{Z}_{U(1)}(X,k)$ in §\[partsec\] using the fact that the stationary phase approximation for our path integral should be exact. This necessitates the introduction of the regularized determinant of $D$ in Eq. \[regdet\], which in turn naturally involves the hypoelliptic Laplacian of Eq. \[maxLap\]. The rigorous quantity that we obtain for the integrand of Eq. \[intzeta\] in §\[partsec\] is derived in Prop. \[rigdet\]. Using an observation from §\[gsec\] that identifies the volume of the isotropy subgroup of the gauge group $\mathcal{G}_{P}$, we identify the integrand of Eq. \[intzeta\] with the contact analytic torsion $T^{d}_{C}$ defined in Def. \[torsdef\]. After formally identifying the signature of the contact operator $D$ with the $\eta$-invariant of $D$ in §\[esec\], we obtain our fully rigorous definition of $\bar{Z}_{U(1)}(X,k)$ in Eq. \[newpar\] below, which is repeated in Eq. \[nz\] above.\
\
On the other hand, [@m] provides a rigorous definition of the partition function $Z_{U(1)}(X,k)$ that does not involve an *a priori* choice of a contact structure on $X$. The formula for this is recalled in Eq. \[oldpar\] below, and is the term $Z_{U(1)}(X,p,k)$ in Eq. \[oz\] of Prop. \[mprop\] above.\
\
Our first main step in the proof of Prop. \[mprop\] is confirmation of the fact that the Ray-Singer analytic torsion (cf. [@rsi]) of $X$, $T_{RS}^{d}$, is identically equal to the contact analytic torsion $T^{d}_{C}$.[^10] We observe that this result follows directly from ([@rs] ; Theorem 4.2).\
\
We also observe in Remark \[rmknX\] that the quantities $m_{X}$ and $n_{X}$ that occur in Prop. \[mprop\] are also equal. This leaves us with the main final step in the confirmation of Prop. \[mprop\], which involves a study of the $\eta$-invariants, $\eta(-\star d)$, $\eta(-\star D)$, that naturally show up in $Z_{U(1)}(X,k)$, $\bar{Z}_{U(1)}(X,k)$, respectively. This analysis is carried out in §\[fsec\], where we observe that the work of Biquard, Herzlich, and Rumin ([@bhr]) is our most pertinent reference. Our main observation here is that the quantum anomalies that occur in the computation of $Z_{U(1)}(X,k)$ and $\bar{Z}_{U(1)}(X,k)$ should, in an appropriate sense, be completely equivalent. In our case, these quantum anomalies are made manifest precisely in the failure of the $\eta$-invariants to represent topological invariants. As observed by Witten (cf. [@w3]), this is deeply connected with the fact that in order to *actually* compute the partition function, one needs to make a choice that is tantamount to either a valid gauge choice for representatives of gauge classes of connections, or in some other way by breaking the symmetry of our problem. Such a choice for us is equivalent to a choice of metric, which is encoded in the choice of a quasi-regular K-contact structure on our manifold $X$. Witten observes in [@w3] that the quantum anomaly that is introduced by our choice of metric may be canceled precisely by adding an appropriate “counterterm” to the $\eta$-invariant, $\eta(-\star d)$. This recovers topological invariance and effectively cancels the anomaly.[^11] This counterterm is found by appealing to the Atiyah-Patodi-Singer theorem, and is in fact identified as the gravitational Chern-Simons term $$\text{CS}(A^{g}):=\frac{1}{4\pi}\int_{X}Tr(A^{g}\wedge dA^{g}+\frac{2}{3} A^{g}\wedge A^{g}\wedge A^{g}),$$ where $A^{g}$ is the Levi-Civita connection on the spin bundle of $X$ for the metric, $$g=\kappa\otimes\kappa+d\kappa(\cdot,J\cdot),$$ on our quasi-regular K-contact three manifold, $(X,\phi,\xi,\kappa,g)$. In particular, we use the fact that, $$\label{regg}
\frac{\eta(-\star d)}{4}+\frac{1}{12}\frac{\text{CS}(A^{g})}{2\pi},$$ is a topological invariant of $X$, after choosing the canonical framing. As is discussed in §\[fsec\], this leads us to conjecture that there exists an appropriate counterterm for the $\eta$-invariant associated to the contact operator $D$ that yields the same topological invariant as in Eq. \[regg\]. More precisely, we conjecture that there exists a counterterm, $C_{T}$, such that $$e^{\pi i\left[\frac{\eta(-\star d)}{4}+\frac{1}{12}\frac{\text{CS}(A^{g})}{2\pi}\right]}=e^{\frac{\pi i}{4}\left[\eta(-\star_{H} D^{1})+C_{T}\right]},$$ as topological invariants. We establish the following in Proposition \[lprop\],
$(X,\phi,\xi,\kappa,g)$ closed, quasi-regular K-contact three-manifold. The counterterm, $C_{T}$, such that $e^{\frac{\pi i}{4}\left[\eta(-\star_{H} D^{1})+C_{T}\right]}$ is a topological invariant that is identically equal to the topological invariant $e^{\pi i\left[\frac{\eta(-\star d)}{4}+\frac{1}{12}\frac{\text{CS}(A^{g})}{2\pi}\right]}$ is $$C_{T}=\frac{1}{512}\int_{X}R^{2}\,\,\kappa\wedge d\kappa,$$ where $R\in C^{\infty}(X)$ is the Tanaka-Webster scalar curvature of $X$.
This proposition is proven in §\[fsec\] by appealing to the following result, which is established using a “Kaluza-Klein” dimensional reduction technique for the gravitational Chern-Simons term. This result is modeled after the paper [@gijp], and is listed as Proposition \[mc2\].
([@mcl2]) $(X,\phi,\xi,\kappa,g)$ closed, quasi-regular K-contact three-manifold, $$\xymatrix{{\makeatletter
\xydef@\xymatrixcolsep@{2pc}
\makeatother
}{\makeatletter
\xydef@\xymatrixrowsep@{1pc}
\makeatother
}U(1) \ar@{^{(}->}[r] & X \ar[d]\\
& \Sigma}.$$ Let $g_{\epsilon}:=\epsilon^{-1}\,\kappa\otimes\kappa+\pi^{*}h$. After choosing a framing for $TX\oplus TX$, corresponding to a choice of vielbeins, then, $$CS(A^{g_{\epsilon}})=\left(\frac{\epsilon^{-1}}{2}\right)\int_{\Sigma}r\,\omega+\left(\frac{\epsilon^{-2}}{2}\right)\int_{\Sigma}f^{2}\,\omega$$ where $r\in C^{\infty}_{orb}(\Sigma)$ is the (orbifold) scalar curvature of $(\Sigma,h)$, $\omega\in\Omega^{2}_{orb}(\Sigma)$ is the (orbifold) Hodge form of $(\Sigma,h)$, and $f:=\star_{h}\omega$. In particular, the adiabatic limit of $\text{CS}(A^{g_{\epsilon}})$ vanishes: $$\lim_{\epsilon\rightarrow \infty}\text{CS}(A^{g_{\epsilon}})=0.$$
Finally, as a consequence of these investigations, we are able to compute in Proposition \[cprop\] the $U(1)$-Chern-Simons partition function fairly explicitly.
$(X,\phi,\xi,\kappa,g)$ closed, quasi-regular K-contact three-manifold. Then, $$\begin{aligned}
\eta(-\star d)+\frac{1}{3}\frac{\text{CS}(A^{g})}{2\pi}&=&\eta(-\star D)+\frac{1}{512}\int_{X}R^{2}\,\,\kappa\wedge d\kappa\\
&=&1+\frac{d}{3}+4\sum_{j=1}^{N}s(\alpha_{j},\beta_{j}),\end{aligned}$$ where $d=c_1(X)=n+\sum_{j=1}^{N}\frac{\beta_{j}}{\alpha_j}\in{\mathbb{Q}}$ and $$s(\alpha,\beta):=\frac{1}{4\alpha}\sum_{k=1}^{\alpha-1}cot\left(\frac{\pi k}{\alpha}\right)cot\left(\frac{\pi k\beta}{\alpha}\right)\in{\mathbb{Q}}$$ is the classical Rademacher-Dedekind sum, where $[n; (\alpha_{1},\beta_{1}),\ldots,(\alpha_{N},\beta_{N})]$ (for gcd$(\alpha_{j},\beta_{j})=1$) are the Seifert invariants of $X$. In particular, we have computed the $U(1)$-Chern-Simons partition function as: $$\begin{aligned}
Z_{U(1)}(X,p,k)&=&k^{n_X}e^{\pi i k S_{X,P}(A_{0})}e^{\frac{\pi i}{4}\left(1+\frac{d}{3}+4\sum_{j=1}^{N}s(\alpha_{j},\beta_{j})\right)}\int_{\mathcal{M}_{P}}\,\, (T^{d}_{C})^{1/2},\\
&=&k^{m_X}e^{\pi i k S_{X,P}(A_{0})}e^{\frac{\pi i}{4}\left(1+\frac{d}{3}+4\sum_{j=1}^{N}s(\alpha_{j},\beta_{j})\right)}\int_{\mathcal{M}_{P}}\,\, (T^{d}_{RS})^{1/2}.\end{aligned}$$
Preliminary Results
===================
Our starting point is the analogue of Eq. \[newchern2\] for the $U(1)$-Chern-Simons partition function: $$\label{Anom1}
\bar{Z}_{U(1)}(X,p,k)=\frac{e^{\pi i k S_{X,P}(A_{0})}}{Vol(\mathcal{S})Vol(\mathcal{G}_{P})}\int_{\mathcal{A}_{P}}DA\,\, exp\,\left[\frac{i k}{4\pi}\left(\int_{X} A\wedge dA-\int_{X} \frac{(\kappa\wedge dA)^{2}}{\kappa\wedge d\kappa}\right)\right]$$ where $S_{X,P}(A_{0})$ is the Chern-Simons invariant associated to $P$ for $A_{0}$ a flat connection on $P$. The derivation of Eq. \[Anom1\] can be found in Appendix \[appen1\]. It is obtained by expanding the $U(1)$ analogue of Eq. \[newchern2\] around a critical point $A_{0}$ of the action. Note that the critical points of this action, up to the action of the shift symmetry, are precisely the flat connections ([@bw] ; Eq. 5.3). In our notation, $A\in T_{A_{0}}\mathcal{A}_{P}$. Let us define the notation $$\label{newact}
S(A):=\int_{X} A\wedge dA-\int_{X} \frac{(\kappa\wedge dA)^{2}}{\kappa\wedge d\kappa}$$ for the new action that appears in the partition function. Also, define $$\bar{S}(A):=\int_{X} \frac{(\kappa\wedge dA)^{2}}{\kappa\wedge d\kappa}$$ so that we may write $$S(A)=CS(A)-\bar{S}(A)$$ The primary virtue of Eq. \[Anom1\] above is that it is exactly equal to the original Chern-Simons partition function of Eq. \[abelchern2\] and yet it is expressed in such a way that the action $S(A)$ is invariant under the shift symmetry. This means that $S(A+\sigma\kappa)=S(A)$ for all tangent vectors $A\in T_{A_{0}}(\mathcal{A}_{P})\simeq\Omega^{1}(X)$ and $\sigma\in \Omega^{0}(X)$. We may naturally view $A\in\Omega^1(H)$, the sub-bundle of $\Omega^{1}(X)$ restricted to the contact distribution $H\subset TX$. Equivalently, if $\xi$ denotes the Reeb vector field of $\kappa$, then $\Omega^{1}(H)=\{\omega\in\Omega^{1}(X)\,\,|\,\,\iota_{\xi}\omega=0\}$. The remaining contributions to the partition function come from the orbits of $\mathcal{S}$ in $\mathcal{A}_{P}$, which turn out to give a contributing factor of $Vol(\mathcal{S})$ (cf. [@bw] ; Eq. 3.32). We thus reduce our integral to an integral over $\bar{\mathcal{A}}_{P}:=\mathcal{A}_{P}/\mathcal{S}$ and obtain: $$\begin{aligned}
Z_{U(1)}(X,p,k)&=&\frac{e^{\pi i k S_{X,P}(A_{0})}}{Vol(\mathcal{G}_{P})}\int_{\bar{\mathcal{A}}_{P}}\bar{D}A\,\, exp\,\left[\frac{i k}{4\pi}\left(\int_{X} A\wedge dA-\int_{X} \frac{(\kappa\wedge dA)^{2}}{\kappa\wedge d\kappa}\right)\right]\\
&=&\frac{e^{\pi i k S_{X,P}(A_{0})}}{Vol(\mathcal{G}_{P})}\int_{\bar{\mathcal{A}}_{P}}\bar{D}A\,\, exp\,\left[\frac{i k}{4\pi}S(A)\right]\end{aligned}$$\
where $\bar{D}A$ denotes an appropriate quotient measure on $\bar{\mathcal{A}}_{P}$, and we can now assume that $A\in\Omega^1(H)\simeq T_{A_{0}}\bar{\mathcal{A}}_{P}$.\
\
Contact structures
==================
At this point, we further restrict the structure on our $3$-manifold and assume that the Seifert structure is compatible with a contact metric structure $(\phi,\xi,\kappa,g)$ on $X$. In particular, we restrict to the case of a quasi-regular K-contact manifold. Let us review some standard facts about these structures in the case of dimension three.
Our three manifolds $X$ are assumed to be closed throughout this paper.
\[eq1\] A *K-contact* manifold is a manifold $X$ with a contact metric structure $(\phi,\xi,\kappa,g)$ such that the Reeb field $\xi$ is Killing for the associated metric $g$, $\mathcal{L}_{\xi}g=0$.
where,
- $\kappa\in\Omega^{1}(X)$ contact form, $\xi=$ Reeb vector field.
- $H:=\text{ker}\kappa\subset TX$ denotes the horizontal or contact distribution on $(X,\kappa)$.
- $\phi\in \text{End}(TX)$, $\phi(Y)=JY$ for $Y\in \Gamma(H)$, $\phi(\xi)=0$ where $J\in \text{End}(H)$ complex structure on the contact distribution $H\subset TX$.
- $g=\kappa\otimes\kappa+d\kappa(\cdot, \phi\cdot)$
Note that we will assume that our contact structure is “co-oriented,” meaning that the contact form $\kappa\in\Omega^{1}(X)$ is a global form. Generally, one can take the contact structure to be to be defined only locally by the condition $H:=\text{ker}\,\kappa$, where $\kappa\in\Omega^{1}(U)$ for open subsets $U\in X$ contained in an open cover of $X$.
\[eq2\] The characteristic foliation $\mathcal{F}_{\xi}$ of a contact manifold $(X,\kappa)$ is said to be *quasi-regular* if there is a positive integer $j$ such that each point has a foliated coordinate chart $(U,x)$ such that each leaf of $\mathcal{F}_{\xi}$ passes through $U$ at most $j$ times. If $j=1$ then the foliation is said to be *regular*.
Definitions \[eq1\] and \[eq2\] together define a quasi-regular $K$-contact manifold, $(X,\phi,\xi,\kappa,g)$. Such three-manifolds are necessarily “Seifert” manifolds that fiber over a two dimensional orbifold $\widehat{\Sigma}$ with with some additional structure. Recall:
A *Seifert manifold* is a three manifold $X$ that admits a locally free $U(1)$-action.
Thus, Seifert manifolds are simply $U(1)$-bundles over an orbifold $\widehat{\Sigma}$, $$\xymatrix{{\makeatletter
\xydef@\xymatrixcolsep@{2pc}
\makeatother
}{\makeatletter
\xydef@\xymatrixrowsep@{1pc}
\makeatother
}U(1) \ar@{^{(}->}[r] & X \ar[d]\\
& \widehat{\Sigma}}.$$ We have the following classification result: $X$ is a quasi-regular K-contact three manifold $\iff$
- ([@bg]; Theorem 7.5.1, (i)) $X$ is a $U(1)$-Seifert manifold over a Hodge orbifold surface, $\widehat{\Sigma}$.
- ([@bg]; Theorem 7.5.1, (iii)) $X$ is a $U(1)$-Seifert manifold over a normal projective algebraic variety of real dimension two.
All 3-dimensional Lens spaces, $L(p,q)$ and the Hopf fibration $S^{1}\hookrightarrow S^{3}\rightarrow {\mathbb{C}}\mathbb{P}^{1}$ possess quasi-regular K-contact structures. Note that any trivial $U(1)$-bundle over a Riemann surface $\Sigma_{g}$, $X=U(1)\times \Sigma_{g}$, possesses *no* K-contact structure ([@itoh]), however, and our results do not apply in this case.
Note that in fact our results apply to the class of all closed *Sasakian* three-manifolds. This follows from the observation that every Sasakian three manifold is K-contact (cf. [@b] ; Corollary 6.5), and every K-contact manifold possesses a quasi-regular K-contact structure (cf. [@bg] ; Theorem 7.1.10).
A useful observation for us is that for a quasi-regular K-contact three-manifold, the metric tensor $g$ must take the following form (cf. [@bg] ; Theorem 6.3.6): $$g=\kappa\otimes\kappa+\pi^{*}h$$ where $\pi:X\rightarrow \Sigma$ is our quotient map, and $h$ represents any (orbifold)Kähler metric on $\widehat{\Sigma}$ which is normalized so that the corresponding (orbifold)Kähler form, $\widehat{\omega}\in\Omega^{2}_{orb}(\Sigma,{\mathbb{R}})$, pulls back to $d\kappa$.\
Note that the assumption that the Seifert structure on $X$ comes from a quasi-regular K-contact structure $(\phi,\xi,\kappa,g)$ is equivalent to assuming that $X$ is a $CR$-Seifert manifold (cf. [@bg] ; Prop. 6.4.8). Recall the following
\[geodef\] A *CR-Seifert* manifold is a three-dimensional compact manifold endowed with both a strictly pseudoconvex CR structure $(H,J)$ and a Seifert structure, that are compatible in the sense that the circle action $\psi:U(1)\rightarrow \text{Diff}(X)$ preserves the CR structure and is generated by a Reeb field $\xi$. In particular, given a choice of contact form $\kappa$, the Reeb field is Killing for the associated metric $g=\kappa\otimes\kappa+d\kappa(\cdot,J\cdot)$.
The assumption that $X$ is CR-Seifert (hence quasi-regular K-contact) is sufficient to ensure that the assumption in ([@bw] ; Eq. 3.27), which states that the $U(1)$-action on $X$, $\psi:U(1)\rightarrow \text{Diff}(X)$, acts by isometries, is satisfied.\
\
We now employ the natural Hodge star operator $\star$, induced by the metric $g$ on $X$, that acts on $\Omega^{\bullet}(X)$ taking $k$ forms to $3-k$ forms. As a result of this normalization convention, we have $\star 1=\kappa\wedge d\kappa$ and $\star\kappa=d\kappa$. Now let $$\star_{H}=-\iota_{\xi}\circ\star$$ as in equation (3.30) of [@bw]. This operator then satisfies $$\begin{aligned}
\star_{H}\kappa&=&0\\
\star_{H}(\kappa\wedge d\kappa)&=&0\\
\star_{H} 1&=&-d\kappa\\
(\star_{H})^{2}&=&-1\end{aligned}$$ as is shown in ([@bw] ; pg. 20). We also define a horizontal exterior derivative $d_H$ as the usual exterior derivative $d$ restricted to the space of horizontal forms $\Omega^{\bullet}(H)$.\
\
Our key observation is that the action $S(A)$ may now be expressed in terms of these horizontal quantities. Let us start with the term $\bar{S}(A)$. Firstly, the term $\kappa\wedge dA$ in $\bar{S}(A)$ is equivalent to $\kappa\wedge d_{H}A$ since the vertical part of $dA$ is annihilated by $\kappa$ in the wedge product. The term $\frac{\kappa\wedge dA}{\kappa\wedge d\kappa}$ is equivalent to $\star(\kappa\wedge d_{H}A)$ by the properties of $\star$ above. By the definition of $\star_{H}$, $\star(\kappa\wedge d_{H}A)=\star_{H}d_{H}A$. We then have, $$\begin{aligned}
\bar{S}(A)&=&\int_{X} \frac{(\kappa\wedge dA)^{2}}{\kappa\wedge d\kappa}\\
&=&\int_{X} \star_{H}(d_{H}A)\wedge \kappa \wedge d_{H}A\\
&=&\int_{X} \kappa \wedge [d_{H}A \wedge\star_{H}(d_{H}A)]\\\end{aligned}$$ We claim that $\bar{S}(A)$ is now expressed in terms of an inner product on $\Omega^{2}{H}$. More generally, we define an inner product on $\Omega^{l}(H)$ for $0\leq l\leq 2$:
Define the pairing $\langle\cdot,\cdot\rangle^{l}_{\kappa}:\Omega^{l}{H}\times\Omega^{l}{H}\rightarrow{\mathbb{R}}$ as $$\label{inner}
\langle\alpha,\beta\rangle^{l}_{\kappa}:=(-1)^{l}\int_{X} \kappa \wedge [\alpha\wedge\star_{H}\beta]$$ for any $\alpha, \beta\in \Omega^{l}{H}$, $0\leq l\leq 2$.
The pairing $\langle\cdot,\cdot\rangle^{l}_{\kappa}$ is an inner product on $\Omega^{l}{H}$.
It can be easily checked that this pairing is just the restriction of the usual $L^{2}$-inner product, $\langle \cdot, \cdot \rangle:\Omega^{l}{X}\times\Omega^{l}{X}\rightarrow{\mathbb{R}}$, $$\langle\alpha,\beta\rangle:=\int_{X} \alpha\wedge\star\beta$$ restricted to horizontal forms. i.e. for any $\beta\in \Omega^{l}{H}$, $0\leq l\leq 2$, we have $\star\beta=\kappa\wedge\star_{H}\beta$. We then have $\alpha\wedge\star\beta=(-1)^{l}\kappa \wedge [\alpha\wedge\star_{H}\beta]$ for any $\alpha, \beta\in \Omega^{l}{H}$, $0\leq l\leq 2$. Thus, $\langle\cdot,\cdot\rangle^{l}_{\kappa}=\langle \cdot, \cdot \rangle$ on $\Omega^{l}{H}$ and therefore defines an inner product.
By our definition, we may now write $\bar{S}(A)=\langle d_{H}A, d_{H}A\rangle^{2}_{\kappa}$. We make the following
Define the formal adjoint of $d_{H}$, denoted $d_{H}^{*}$, via: $$\langle d_{H}^{*}\gamma,\phi\rangle^{l-1}_{\kappa}=\langle \gamma,d_{H}\phi\rangle^{l}_{\kappa}$$ for $\gamma\in\Omega^{l}(H)$, $\phi\in\Omega^{l-1}(H)$ where $l=1,2$ and $d_{H}^{*}\gamma=0$ for $\gamma\in\Omega^{0}(H)$.
$d_{H}^{*}=(-1)^{l}\star_{H}d_{H}\star_{H}:\Omega^{l}(H)\rightarrow\Omega^{l-1}(H)$, $0\leq l \leq 2$, where $\Omega^{-1}(H):=0$.
This just follows from the definition of $d^{*}$ relative to the ordinary inner product $\langle \cdot,\cdot\rangle$, and the facts that $\langle \cdot, \cdot\rangle^{l-1}_{\kappa}$ is just this ordinary inner product restricted to horizontal forms and $d^{*}=(-1)^{l}\star d\star$.
Thus, we may now write $\bar{S}(A)=\langle A, d_{H}^{*}d_{H}A\rangle^{1}_{\kappa}$ and identify this piece of the action with the second order operator $d_{H}^{*}d_{H}$ on horizontal forms.\
\
Now we turn our attention to the Chern-Simons part of the action $CS(A)=\int_{X} A\wedge dA$. We would like to reformulate this in terms of horizontal quantities as well. This is straightforward to do; simply observe that $dA=\kappa\wedge\mathcal{L}_{\xi}A+d_{H}A$. Thus, we have: $$\begin{aligned}
CS(A)&=&\int_{X} A\wedge dA\\
&=&\int_{X} A\wedge [\kappa\wedge\mathcal{L}_{\xi}A+d_{H}A]\\
&=&\int_{X} A\wedge [\kappa\wedge\mathcal{L}_{\xi}A]+\int_{X} A\wedge d_{H}A\\
&=&\int_{X} A\wedge [\kappa\wedge\mathcal{L}_{\xi}A]\end{aligned}$$ where the last line follows from the fact that $A\wedge d_{H}A=0$ since both forms are horizontal. Putting this all together, we may now express the total action $S(A)$ in terms of horizontal quantities as follows: $$\begin{aligned}
S(A)&=&CS(A)-\bar{S}(A)\\
&=&\int_{X} A\wedge [\kappa\wedge\mathcal{L}_{\xi}A]+\int_{X} A\wedge [\kappa\wedge d_{H}\star_{H} d_{H} A]\\
&=&\int_{X} A\wedge [\kappa\wedge (\mathcal{L}_{\xi}+d_{H}\star_{H} d_{H})A]\end{aligned}$$
The contact operator $D$ {#Dsec}
========================
A surprising observation is that $\kappa\wedge (\mathcal{L}_{\xi}+d_{H}\star_{H} d_{H})$ turns out to be well known. It is the second order operator “$D$” that fits into the complex, $$\label{complex}
C^{\infty}(X)\xrightarrow{\text{$d_{H}$}}\Omega^{1}(H)\xrightarrow{\text{$D$}}\Omega^{2}(V)\xrightarrow{\text{$d_{H}$}}\Omega^{3}(X)$$ where, $$\Omega^{\bullet}(V):=\{\kappa\wedge\alpha\,\,|\,\,\alpha\in\Omega^{\bullet}(H)\}=\kappa\wedge\Omega^{\bullet}(H)$$ and for $f\in C^{\infty}(X)$, $d_{H}f\in\Omega^{1}(H)$ stands for the restriction of $df$ to $H$ as usual, while $$d_{H}:\Omega^{2}(V)\rightarrow \Omega^{3}(X)$$ is just de Rham’s differential restricted to $\Omega^{2}(V)$ in $\Omega^{2}(X)$. $D$ is defined as follows: since $d$ induces an isomorphism $$d_{0}:\Omega^{1}(V)\rightarrow\Omega^{2}(H),\,\,\text{with}\,\,d_{0}(f\kappa)=fd\kappa|_{\Lambda^{2}(H)}$$ then any $\alpha\in\Omega^{1}(H)$ admits a unique extension $\textit{l}(\alpha)$ in $\Omega^{1}(X)$ such that $d\textit{l}(\alpha)$ belongs to $\Omega^{2}(V)$; i.e. given any initial extension $\bar{\alpha}$ of $\alpha$, one has $$\textit{l}(\alpha)=\bar{\alpha}-d_{0}^{-1}(d\bar{\alpha})|_{\Lambda^{2}(H)}$$ We then define $$D\alpha:=d\textit{l}(\alpha)$$ We then have ([@bhr] ; Eq. 39), $$\label{Ddef1}
D\alpha=\kappa\wedge [\mathcal{L}_{\xi}+d_{H}\star_{H} d_{H}]\alpha$$ for any $\alpha\in\Omega^{1}(H)$. Thus, $$\begin{aligned}
S(A)&=&\int_{X} A\wedge [\kappa\wedge (\mathcal{L}_{\xi}+d_{H}\star_{H} d_{H})A]\\
&=&\int_{X} A\wedge DA\\
&=&\langle A, -\star DA\rangle\end{aligned}$$ where $\langle \cdot, \cdot\rangle$ is the usual $L^{2}$ inner product on $\Omega^{1}(X)$.\
\
Alternatively, we make the following
Let $D^{1}:\Omega^{1}(H)\rightarrow\Omega^{1}(X)$ denote the operator $$\label{Ddef}
D^{1}:=\mathcal{L}_{\xi}+d_{H}\star_{H} d_{H}$$
and observe that we can also write $S(A)=\langle A,-\star_{H}D^{1}A\rangle_{\kappa}^{1}$, identifying $S(A)$ with the operator $-\star_{H}D^{1}$ on $\Omega^{1}(H)$. Thus, we have proven the following
\[prop1\] The new action, $S(A)$, as defined in Eq. \[newact\], for the “shifted” partition function of Eq. \[Anom1\] can be expressed as a quadratic form on the space of horizontal forms $\Omega^{1}(H)$ as follows: $$S(A)= \langle A, -\star DA\rangle$$ or equivalently as, $$S(A)=\langle A,-\star_{H}D^{1}A\rangle_{\kappa}^{1}$$ where $D$ and $D^{1}$ are the second order operators defined in Eq.’s \[Ddef1\] and \[Ddef\], respectively. $\langle \cdot, \cdot\rangle$ is the usual $L^{2}$ inner product on $\Omega^{1}(X)$, and $\langle \cdot, \cdot\rangle^{1}_{\kappa}$ is defined in Eq. \[inner\].
Gauge group and the isotropy subgroup {#gsec}
=====================================
In order to extract anything mathematically meaningful out of this construction we will need to divide out the action of the gauge group $\mathcal{G}_{P}$ on $\mathcal{A}_{P}$. At this point we observe that the gauge group $\mathcal{G}_{P}\simeq \text{Maps}(X\rightarrow U(1))$ naturally descends to a “horizontal” action on $\bar{\mathcal{A}}_{P}$, which infinitesimally can be written as: $$\label{action}
\theta\in \text{Lie}(\mathcal{G}_{P}):A\mapsto A+d_{H}\theta$$ Following [@s2], we let $H_{A}$ denote the isotropy subgroup of $\mathcal{G}_{P}$ at a point $A\in\bar{\mathcal{A}}_{P}$. Note that $H_{A}$ can be canonically identified for every $A\in\bar{\mathcal{A}}_{P}$, and so we simply write $H$ for the isotropy group. The condition for an element of the gauge group $h(x)=e^{i\theta(x)}$ to be in the isotropy group is that $d_{H}\theta=0$, given definition \[action\] above. By ([@r] ; Prop. 12), we see that the condition $d_{H}\theta=0$ implies that $\theta$ is harmonic, and so $\mathcal{L}_{\xi}\theta=0$. Therefore we have $d\theta=0$ since $d=d_{H}+\kappa\wedge \mathcal{L}_{\xi}$. Thus, the group $H$ can be identified with the group of constant maps from $X$ into $U(1)$; hence, is isomorphic to $U(1)$. We let $Vol(H)$ denote the volume of the isotropy subgroup, computed with respect to the metric induced from $\mathcal{G}_{P}$, so that $$\label{volu}
Vol(H)=\left[\int_{X}\kappa\wedge d\kappa\right]^{1/2}=\left[ n+\sum_{j=1}^{N}\frac{\beta_{j}}{\alpha_{j}}\right]^{1/2}$$ where $[n; (\alpha_{1},\beta_{1}),\ldots,(\alpha_{N},\beta_{N})]$ are the Seifert invariants of our Seifert manifold $X$. The last equality in Eq. \[volu\] above follows from Eq. 3.22 of [@bw].
The partition function {#partsec}
======================
We now have $$\begin{aligned}
Z_{U(1)}(X,p,k)&=&\frac{e^{\pi i k S_{X,P}(A_{0})}}{Vol(\mathcal{G}_{P})}\int_{\bar{\mathcal{A}}_{P}}\bar{D}A\,\, e^{\left[\frac{i k}{4\pi}S(A)\right]}\nonumber\\
&=&\frac{Vol(\mathcal{G}_{P})}{Vol(H)}\frac{e^{\pi i k S_{X,P}(A_{0})}}{Vol(\mathcal{G}_{P})}\int_{\bar{\mathcal{A}}_{P}/\mathcal{G}_{P}}\,\, e^{\left[\frac{i k}{4\pi}S(A)\right]}\left[det'(d_{H}^{*}d_{H})\right]^{1/2}\,\,\mu\nonumber\\
&=&\frac{e^{\pi i k S_{X,P}(A_{0})}}{Vol(H)}\int_{\bar{\mathcal{A}}_{P}/\mathcal{G}_{P}}\,\, e^{\left[\frac{i k}{4\pi}S(A)\right]}\left[det'(d_{H}^{*}d_{H})\right]^{1/2}\,\,\mu\label{oscil}\end{aligned}$$ where $\mu$ is the induced measure on the quotient space $\bar{\mathcal{A}}_{P}/\mathcal{G}_{P}$ and $det'$ denotes a regularized determinant to be defined later. Since $S(A)=\langle A,-\star_{H}D^{1}A\rangle_{\kappa}^{1}$ is quadratic in $A$, we may apply the method of stationary phase ([@s1], [@gs]) to evaluate the oscillatory integral (\[oscil\]) exactly. We obtain, $$\begin{aligned}
\label{intzeta}
&&\\
Z_{U(1)}(X,p,k)&=&\frac{e^{\pi i k S_{X,P}(A_{0})}}{Vol(H)}\int_{\mathcal{M}_{P}}\,\, e^{\frac{\pi i}{4}\,\,sgn(-\star_{H}D^{1})}\frac{\left[det'(d_{H}^{*}d_{H})\right]^{1/2}}{\left[det'(-k\star_{H}D^{1})\right]^{1/2}}\,\,\nu\nonumber\end{aligned}$$ where $\mathcal{M}_{P}$ denotes the moduli space of flat connections modulo the gauge group and $\nu$ denotes the induced measure on this space. Note that we have included a factor of $k$ in our regularized determinant since this factor occurs in the exponent multiplying $S(A)$.
Zeta function determinants
==========================
We will use the following to define the regularized determinant of $-k\star_{H}D^{1}$
\[product\][@s2] Let $\mathcal{H}_{0}$, $\mathcal{H}_{1}$ be Hilbert spaces, and $S:\mathcal{H}_{1}\rightarrow \mathcal{H}_{1}$ and $T:\mathcal{H}_{0}\rightarrow \mathcal{H}_{1}$ such that $S^{2}$ and $TT^{*}$ have well defined zeta functions with discrete spectra and meromorphic extensions to ${\mathbb{C}}$ that are regular at 0 (with at most simple poles on some discrete subset). If $ST=0$, and $S^{2}$ is self-adjoint, then $$det'(S^{2}+TT^{*})=det'(S^{2})det'(TT^{*})$$
This equality follows from the facts that $S^{2}TT^{*}=0$ and $TT^{*}S^{2}=0$ (i.e. these operators commute), which both follow from $ST=0$ and the fact that $S^{2}$ and $TT^{*}$ are both self-adjoint.
Following the notation of Eq.’s (3)-(6) in section 2 of [@s2], we set the operators $S=-k\star_{H}D^{1}$ and $T=k d_{H}d_{H}^{*}$ on $\Omega^{1}(H)$ and observe that $ST=0$ since (\[complex\]) is a complex. With Prop. \[product\] as *motivation*, we make the formal definition $$\label{regdet}
det'(-k\star_{H}D^{1}):=C(k,J)\cdot\frac{[det'(S^{2}+TT^{*})]^{1/2}}{[det'(TT^{*})]^{1/2}}$$ where $S^{2}+TT^{*}=k^{2}((D^{1})^{*}D^{1}+(d_{H}d_{H}^{*})^{2})$, $TT^{*}=k^{2}(d_{H}d_{H}^{*})^{2}$ and $$C(k,J):=k^{\left(-\frac{1}{1024}\int_{X}R^{2}\,\kappa\wedge d\kappa\right)}$$ is a function of $R\in C^{\infty}(X)$, the Tanaka-Webster scalar curvature of $X$, which in turn depends only on a choice of a compatible complex structure $J\in \text{End}(H)$. That is, given a choice of contact form $\kappa\in\Omega^{1}(X)$, the choice of complex structure $J\in \text{End}(H)$ determines uniquely an associated metric. We have defined $det'(-k\star_{H}D^{1})$ in this way to eliminate the metric dependence that would otherwise occur in the $k$-dependence of this determinant. The motivation for the definition of the factor $C(k,J)$ comes explicitly from Prop. \[Jdepend\] below.\
\
The operator $$\label{maxLap}
\Delta:=(D^{1})^{*}D^{1}+(d_{H}d_{H}^{*})^{2}$$ is actually equal to the middle degree Laplacian defined in Eq. (10) of [@rs] and has some nice analytic properties. In particular, it is maximally hypoelliptic and invertible in the Heisenberg symbolic calculus (See [@rs] ; §3.1). We define the regularized determinant of $\Delta$ via its zeta function ([@rs] ; Pg. 10) $$\zeta(\Delta)(s):=\sum_{\lambda\in\text{spec}^{*}(\Delta)}\lambda^{-s}$$ Note that our definition agrees with [@rs] up to a constant term $\text{dim}H^{1}(X,D)$, which is finite by hypoellipticity ([@rs] ; Pg. 11). Also, $\zeta(\Delta)(s)$ admits a meromorphic extension to ${\mathbb{C}}$ that is regular at $s=0$ ([@p2] ; §4). Thus, we define the regularized determinant of $\Delta$ as $$det'(\Delta):=e^{-\zeta'(\Delta)(0)}$$ Let $\Delta_{0}:=(d_{H}^{*}d_{H})^{2}$ on $\Omega^{0}(X)$, $\Delta_{1}:=\Delta$ on $\Omega^{1}(H)$ and define $\zeta_{i}(s):=\zeta(\Delta_{i})(s)$. We claim the following\
For any real number $0<c\in{\mathbb{R}}$, $$\label{delta}
det'(c\Delta_{i}):=c^{\zeta_{i}(0)}det'(\Delta_{i})$$ for $i=0,1$.
To prove this claim, recall that $\zeta_{i}(s)=\zeta(\Delta_{i})(s)$ for $i=0,1$, scale as follows: $$\label{unsure}
\zeta(c\Delta_{i})(s)=c^{-s}\zeta(\Delta_{i})(s).$$ From here we simply calculate the scaling of the regularized determinants using the definition $$det'(\Delta_{i}):=e^{-\zeta'(\Delta_{i})(0)}$$ and the claim is proven.
The following will be useful.
\[Jdepend\] For $\Delta_{0}:=(d_{H}^{*}d_{H})^{2}$ on $\Omega^{0}(X)$, $\Delta_{1}:=\Delta$ on $\Omega^{1}(H)$ defined as above and $\zeta_{i}(s):=\zeta(\Delta_{i})(s)$, we have $$\begin{aligned}
&&\\\label{dimen}
\zeta_{0}(0)-\zeta_{1}(0)&=&\left(-\frac{1}{512}\int_{X}R^{2}\,\kappa\wedge d\kappa\right)+\text{dim Ker}\Delta_{1}-\text{dim Ker}\Delta_{0}\nonumber\\
&=&\left(-\frac{1}{512}\int_{X}R^{2}\,\kappa\wedge d\kappa\right)+\text{dim} H^{1}(X,d_{H})-\text{dim} H^{0}(X,d_{H}).\end{aligned}$$ where $R\in C^{\infty}(X)$ is the Tanaka-Webster scalar curvature of $X$ and $\kappa\in\Omega^{1}(X)$ is our chosen contact form as usual.
Let $$\begin{aligned}
\hat{\zeta_{0}}(s)&:=&\text{dim Ker}\Delta_{0}+\zeta_{0}(s)\\
\hat{\zeta_{1}}(s)&:=&\text{dim Ker}\Delta_{1}+\zeta_{1}(s)\end{aligned}$$ denote the zeta functions as defined in [@rs]. From ([@rs] ; Cor. 3.8), one has that $$\hat{\zeta_{1}}(0)=2\hat{\zeta_{0}}(0)$$ for all 3-dimensional contact manifolds. By ([@bhr] ; Theorem 8.8), one knows that on CR-Seifert manifolds that $$\hat{\zeta_{0}}(0)=\hat{\zeta}(\Delta_{0})(0)=\hat{\zeta}(\Delta_{0}^{2})(0)=\frac{1}{512}\int_{X}R^{2}\,\kappa\wedge d\kappa$$ Thus, $$\hat{\zeta_{1}}(0)=\frac{1}{256}\int_{X}R^{2}\,\kappa\wedge d\kappa$$ By our definition of the zeta functions, which differ from that of [@rs] by constant dimensional terms, we therefore have $$\begin{aligned}
\zeta_{0}(0)&=&\frac{1}{512}\int_{X}R^{2}\,\kappa\wedge d\kappa-\text{dim Ker}\Delta_{0}\\
\zeta_{1}(0)&=&\frac{1}{256}\int_{X}R^{2}\,\kappa\wedge d\kappa-\text{dim Ker}\Delta_{1}\end{aligned}$$ Hence, $$\begin{aligned}
\zeta_{0}(0)-\zeta_{1}(0)&=&\left[\frac{1}{512}\int_{X}R^{2}\,\kappa\wedge d\kappa-\text{dim Ker}\Delta_{0}\right]-\left[\frac{1}{256}\int_{X}R^{2}\,\kappa\wedge d\kappa-\text{dim Ker}\Delta_{1}\right]\\
&=&\left(-\frac{1}{512}\int_{X}R^{2}\,\kappa\wedge d\kappa\right)+\text{dim Ker}\Delta_{1}-\text{dim Ker}\Delta_{0}\\
&=&\left(-\frac{1}{512}\int_{X}R^{2}\,\kappa\wedge d\kappa\right)+\text{dim} H^{1}(X,d_{H})-\text{dim} H^{0}(X,d_{H}).\end{aligned}$$ and the result is proven.
We now have the following
\[rigdet\] The term inside of the integral of Eq. \[intzeta\] has the following expression in terms of the hypoelliptic Laplacians, $\Delta_{0}$ and $\Delta_{1}$, as defined in Prop. \[Jdepend\]: $$\frac{\left[det'(d_{H}^{*}d_{H})\right]^{1/2}}{\left[det'(-k\star_{H}D^{1})\right]^{1/2}}=k^{n_{X}}\frac{[det'(\Delta_{0})]^{1/2}}{\left[det'(\Delta_{1})\right]^{1/4}}$$ where $$\label{nX}
n_{X}:=\frac{1}{2}(\text{dim} H^{1}(X,d_{H})-\text{dim} H^{0}(X,d_{H})).$$
$$\begin{aligned}
\label{cal1}
&&\\
\frac{\left[det'(d_{H}^{*}d_{H})\right]^{1/2}}{\left[det'(-k\star_{H}D_{\kappa}^{1})\right]^{1/2}}&=&C(k,J)^{-1}\cdot\frac{\left[det'(d_{H}^{*}d_{H})^{2}\right]^{1/4}\cdot \left[det'k^{2} (d_{H}d_{H}^{*})^{2}\right]^{1/4}}{\left[det'(k^{2}\Delta)\right]^{1/4}}\nonumber\\\label{cal2}
&=&C(k,J)^{-1}\cdot\frac{k^{\zeta_{0}(0)/2}\left[det'(\Delta_{0})\right]^{1/4}\cdot \left[det'(\Delta_{0})\right]^{1/4}}{k^{\zeta_{1}(0)/2}\left[det'(\Delta_{1})\right]^{1/4}}\\
&=&C(k,J)^{-1}\cdot k^{\frac{1}{2}(\zeta_{0}(0)-\zeta_{1}(0))}\frac{[det'(\Delta_{0})]^{1/2}}{\left[det'(\Delta_{1})\right]^{1/4}}\nonumber\\
&=&C(k,J)^{-1}\cdot C(k,J)\cdot k^{n_{X}}\frac{[det'(\Delta_{0})]^{1/2}}{\left[det'(\Delta_{1})\right]^{1/4}},\,\text{Prop. \ref{Jdepend}},\nonumber\\
&=&k^{n_{X}}\frac{[det'(\Delta_{0})]^{1/2}}{\left[det'(\Delta_{1})\right]^{1/4}}\nonumber\end{aligned}$$
where the second last line comes from Eq. \[dimen\]. Also note that $d_{H}^{*}d_{H}$ and $d_{H}d_{H}^{*}$ have the same eigenvalues (by standard arguments), which allows us to proceed to Eq. \[cal2\] from Eq. \[cal1\].
\[rmknX\] Note that by ([@rs] ; Prop. 2.2), the definition of $n_{X}$ (see Eq. \[nX\]) here is exactly equal to the quantity $m_{X}:=\frac{1}{2}(\text{dim} H^{1}(X,d)-\text{dim} H^{0}(X,d))$ of ([@m] ; Eq. 5.18). This shows that our partition function has the same $k$-dependence as that in [@m].
The eta invariant {#esec}
=================
Next we regularize the signature $sgn(-\star_{H}D^{1})$ via the eta-invariant and set $sgn(-\star_{H}D^{1})=\eta(-\star_{H}D^{1})(0):=\eta(-\star_{H}D^{1})$ where $$\eta(-\star_{H}D^{1})(s):=\sum_{\lambda\in\text{spec}^{*}(-\star_{H}D^{1})}(sgn\lambda)|\lambda|^{-s}$$ Finally, we may now write the result for our partition function $$\begin{aligned}
\label{ctorsion}
&&\\
Z_{U(1)}(X,p,k)&=&k^{n_X}e^{\pi i k S_{X,P}(A_{0})}e^{\frac{\pi i}{4}\eta(-\star_{H}D^{1})}\int_{\mathcal{M}_{P}}\,\, \frac{1}{Vol(H)}\frac{[det'(\Delta_{0})]^{1/2}}{\left[det'(\Delta_{1})\right]^{1/4}}\,\,\nu\nonumber\end{aligned}$$ where $n_{X}:=\frac{1}{2}(\text{dim} H^{1}(X,d_{H})-\text{dim} H^{0}(X,d_{H}))$. Note that $\nu$ is a measure on $\mathcal{M}_{P}$ (the moduli space of flat connections modulo the gauge group) relative to the horizontal structure on the tangent space of $\mathcal{M}_{P}$.\
\
Torsion {#tsec}
=======
Now we will study the quantity $\frac{1}{Vol(H)}\frac{[det'(\Delta_{0})]^{1/2}}{\left[det'(\Delta_{1})\right]^{1/4}}\,\,\nu$ inside of the integral in Eq. \[ctorsion\], and in particular how it is related to the analytic contact torsion $T_{C}$. First, recall that ([@rs];Eq. 16) $$\label{torsion}
T_{C}:=\text{exp}\left( \frac{1}{4}\sum_{q=0}^{3}(-1)^{q}w(q)\zeta'(\Delta_{q})(0)\right)$$ where $$w(q)=
\begin{cases} q & \text{if $q\leq 1$,}
\\
q+1 &\text{if $q>1$.}
\end{cases}$$ in the case where $\text{dim}(X)=3$. Note that we have chosen a sign convention that leads to the inverse of the definition of $T_C$ in [@rs]. Recall ([@rs], Eq. 10), $$\label{conlapl}
\Delta_{q}=
\begin{cases} (d_{H}^{*}d_{H}+d_{H}d_{H}^{*})^{2} & \text{if $q = 0,3$,}
\\
D^{*}D+(d_{H}d_{H}^{*})^{2} &\text{if $q=1$.}
\\
DD^{*}+(d_{H}^{*}d_{H})^{2} &\text{if $q=2$.}
\end{cases}$$ We would, however, like to work with torsion when viewed as a density on the determinant line $$\begin{aligned}
|\text{det}H^{\bullet}(X,d_{H})^{*}|&:=&|\text{det}H^{0}(X,d_{H})|\otimes|\text{det}H^{1}(X,d_{H})^{*}|\\
&\otimes&|\text{det}H^{2}(X,d_{H})|\otimes|\text{det}H^{3}(X,d_{H})^{*}|\end{aligned}$$ We follow [@rsi] and [@m] and make the analogous definition.
\[torsdef\] Define the analytic torsion as a density as follows $$T^{d}_{C}:=T_{C}\cdot\delta_{|\text{det}H^{\bullet}(X,d_{H})|}$$ where $T_{C}$ is as defined in Eq. \[torsion\], and $$\delta_{|\text{det}H^{\bullet}(X,d_{H})|}:=\otimes_{q=0}^{dim X}|\nu_{1}^{q}\wedge\cdots\wedge \nu_{b_{q}}^{q}|^{(-1)^{q}}$$ where $\{\nu_{1}^{q},\cdots ,\nu_{b_{q}}^{q}\}$ is an orthonormal basis for the space of harmonic contact forms $\mathcal{H}^{q}(X,d_{H})$ with the inner product defined in Eq. \[inner\]. Note that $\mathcal{H}^{q}(X,d_{H})$ is canonically identified with the cohomology space $H^{q}(X,d_{H})$, and $b_{q}:=\text{dim}(H^{q}(X,d_{H}))$ is the $q^{th}$ contact Betti number.
Let $$\nu^{(q)}:=\nu_{1}^{q}\wedge\cdots\wedge \nu_{b_{q}}^{q}$$ and write the analytic torsion of a compact connected Seifert 3-manifold $X$ as $$\begin{aligned}
T^{d}_{C}=T_{C}\times |\nu^{(0)}|\otimes|\nu^{(1)}|^{-1}\otimes|\nu^{(2)}|\otimes|\nu^{(3)}|^{-1}.\end{aligned}$$ In terms of regularized determinants, we have $$T_{C}=\left[(det'(\Delta_{0}))^{0}\cdot (det'(\Delta_{1}))^{1}\cdot (det'(\Delta_{2}))^{-3}\cdot (det'(\Delta_{3}))^{4}\right]^{1/4}$$ where $\Delta_{q}$, $0\leq q\leq 3$, denotes the Laplacians on the contact complex as defined in ([@rs] ; Eq. 10) and recalled in Eq. \[conlapl\] above. This notation agrees with our notation for $\Delta_{0}$, $\Delta_{1}$ as in Eq. \[delta\]. The Hodge $\star$-operator induces the equivalences $\Delta_{q}\simeq \Delta_{3-q}$ (see [@rs];Theorem 3.4) and allows us to write $$\begin{aligned}
T_{C}&=&\left[(det'(\Delta_{0}))^{0}\cdot (det'(\Delta_{1}))^{1}\cdot (det'(\Delta_{2}))^{-3}\cdot (det'(\Delta_{3}))^{4}\right]^{1/4}\\\label{eq3}
&=&\frac{det'(\Delta_{0})}{(det'(\Delta_{1}))^{1/2}}\end{aligned}$$ Also, from the isomorphisms $H^{q}(X,{\mathbb{R}})\simeq H^{q}(X,d_{H})$ of Prop. 2.2 of [@rs], we have Poincaré duality $H^{q}(X,d_{H})\simeq H^{3-q}(X,d_{H})^{*}$, and therefore $$\label{eqzo}
T^{d}_{C}=T_{C}\times |\nu^{0}|^{\otimes 2}\otimes(|\nu^{1}|^{-1})^{\otimes 2}$$ Moreover, by [@r] ( Prop. 12), $\mathcal{H}^{q}(X,d_{H})=\mathcal{H}^{q}(X,{\mathbb{R}})$, and thus any orthonormal basis $\nu^{(0)}$ of $\mathcal{H}^{0}(X,d_{H})\simeq{\mathbb{R}}$ is a constant such that $$\label{eq1}
|\nu^{(0)}|=\left[\int_{X}\kappa\wedge d\kappa\right]^{-1/2}$$ Also, recall that the tangent space $T_{A}\mathcal{M}_{P}\simeq H^{1}(X,d_{H}) \simeq H^{1}(X,{\mathbb{R}})$, at any point $A\in\mathcal{M}_{P}$. The measure $\nu$ on $\mathcal{M}_{P}$ that occurs in Eq. \[ctorsion\] is defined relative to the metric on $H^{1}(X,d_{H})\simeq\mathcal{H}^{1}(X,d_{H})$, which can be identified with the usual $L^{2}$-metric on forms. Thus the measure $\nu$ may be identified with the inverse of the density $|\nu^{(1)}|$ by dualizing the orthogonal basis $\{\nu_{1}^{1}, \ldots, \nu_{b_{1}}^{1}\}$ for $\mathcal{H}^{1}(X,d_{H})$; i.e. $$\label{eq2}
\nu=|\nu^{(1)}|^{-1}=|\nu_{1}^{1}\wedge \cdots \wedge \nu_{b_{1}}^{1}|^{-1}$$ Putting together equations \[eq3\], \[eq1\], \[eq2\] into equation \[eqzo\], we have $$\begin{aligned}
T^{d}_{C}&=&T_{C}\times |\nu^{0}|^{\otimes 2}\otimes(|\nu^{1}|^{-1})^{\otimes 2}\\
&=&\frac{det'(\Delta_{0})}{(det'(\Delta_{1}))^{1/2}}\cdot\left[\int_{X}\kappa\wedge d\kappa\right]^{-1} \nu^{\otimes 2}\\
&=&\text{Vol}(H)^{-2}\frac{det'(\Delta_{0})}{(det'(\Delta_{1}))^{1/2}}\cdot\nu^{\otimes 2}\end{aligned}$$ We have thus proven the following,
\[tprop\] The contact analytic torsion, when viewed as a density $T^{d}_{C}$ as in definition \[torsdef\], can be identified as follows: $$(T^{d}_{C})^{1/2}=\frac{1}{Vol(H)}\frac{[det'(\Delta_{0})]^{1/2}}{\left[det'(\Delta_{1})\right]^{1/4}}\,\,\nu$$
Our partition function is now $$\label{newpar}
\bar{Z}_{U(1)}(X,p,k)=k^{n_X}e^{\pi i k S_{X,P}(A_{0})}e^{\frac{\pi i}{4}\eta(-\star_{H}D^{1})}\int_{\mathcal{M}_{P}}\,\, (T^{d}_{C})^{1/2}$$ This partition function should be completely equivalent to the partition function defined in ([@m] ; Eq. 7.27): $$\label{oldpar}
Z_{U(1)}(X,p,k)=k^{m_X}e^{\pi i k S_{X,P}(A_{0})}e^{\frac{\pi i}{4}\eta(-\star d)}\int_{\mathcal{M}_{P}}\,\, (T^{d}_{RS})^{1/2}.$$ Our goal in the remainder is to show that this is indeed the case. Our first observation is that $(T^{d}_{C})^{1/2}$ is equal to the Ray-Singer torsion $(T^{d}_{RS})^{1/2}$ that occurs in ([@m] ; Eq. 7.27). This follows directly from ([@rs] ; Theorem 4.2); note that their sign convention makes $T_C$ the inverse of our definition.
Regularizing the eta-invariants {#fsec}
===============================
Since we have seen that our $k$-dependence matches that in [@m] (i.e. $m_{X}=n_{X}$ ; cf. Remark \[rmknX\]), the only thing left to do is to reconcile the eta invariants, $\eta(-\star_{H}D^{1})$ and $\eta(-\star d)$. As observed in [@w3], the correct quantity to compare our eta invariant to would be $$\label{reg1}
\frac{\eta(-\star d)}{4}+\frac{1}{12}\frac{\text{CS}(A^{g})}{2\pi}.$$ where, $$\text{CS}(A^{g})=\frac{1}{4\pi}\int_{X}Tr(A^{g}\wedge dA^{g}+\frac{2}{3} A^{g}\wedge A^{g}\wedge A^{g})$$ is the gravitational Chern-Simons term, with $A^{g}$ the Levi-Civita connection on the spin bundle of $X$ for a given metric $g$ on $X$. See Appendix \[appen2\] for a short exposition on the regularization of $\eta(-\star d)$ in Eq. \[reg1\]. It was noticed in [@w3] that in the quasi-classical limit, quantum anomalies can occur that can break topological invariance. Invariance may be restored in this case only after adding a counterterm to the eta invariant. Our job then is to perform a similar analysis for the eta invariant $\eta(-\star_{H}D^{1})$, which depends on a choice of metric. Of course, our choice of metric is natural in this setting and is adapted to the contact structure. One possible approach is to consider variations over the space of such natural metrics and calculate the corresponding variation of the eta invariant, giving us a local formula for the counterterm that needs to be added. Such a program has already been initiated in [@bhr].\
\
Our starting point is the conjectured equivalence that results from the identification of Eq.’s \[newpar\] and \[oldpar\]: $$\label{etavar}
e^{\pi i\left[\frac{\eta(-\star d)}{4}+\frac{1}{12}\frac{\text{CS}(A^{g})}{2\pi}\right]}\text{``$=$''}e^{\frac{\pi i}{4}\left[\eta(-\star_{H} D^{1})+C_{T}\right]}$$ where $C_{T}$ is some appropriate counterterm that yields an invariant comparable to the left hand of this equation. As noted in Appendix \[appen2\], the left hand side of this equation depends on a choice of $2$-framing on $X$, and since we have a rule (cf. Eq. \[partform\]) for how the partition function transforms when the framing is twisted, we basically have a topological invariant. Alternatively, as also noted in Appendix \[appen2\], one can use the main result of [@at] and fix the canonical $2$-framing on $TX\oplus TX$. We therefore expect the same type of phenomenon for the right hand side of this equation, having at most a ${\mathbb{Z}}$-dependence on the regularization of our eta invariant, along with a rule that tells us how the partition function changes when our discrete invariants are “twisted,” once again yielding a topological invariant.\
\
Let us first make the statement of the conjecture of Eq. \[etavar\] more precise. We should have the following
$(X,\phi,\xi,\kappa,g)$ a closed quasi-regular K-contact three-manifold. Then there exists a counterterm, $C_{T}$, such that $$e^{\frac{\pi i}{4}\left[\eta(-\star_{H} D^{1})+C_{T}\right]}$$ is a topological invariant that is identically equal to the topological invariant $$e^{\pi i\left[\frac{\eta(-\star d)}{4}+\frac{1}{12}\frac{\text{CS}(A^{g})}{2\pi}\right]},$$ where $\text{CS}(A^{g})$ and all relevant operators are defined with respect to the metric $g$ on $X$ and we use the canonical 2-framing [@at].
Our regularization procedure for $\eta(-\star_{H} D^{1})$ will be quite different than that used for $\eta(-\star d)$. Since we are restricted to a class of metrics that are compatible with our contact structure, we are really only concerned with finding appropriate counterterms for $\eta(-\star_{H} D^{1})$ that will eliminate our dependence on the choice of contact form $\kappa$ and complex structure $J\in\text{End}(H)$. In the case of interest, we observe that our regularization may be obtained in one stroke by introducing the *renormalized $\eta$-invariant*, $\eta_{0}(X,\kappa)$, of $X$ that is discussed in ([@bhr] ; §3). Before giving the definition of $\eta_{0}(X,\kappa)$, we require the following
([@bhr] ; Lemma 3.1) Let $(X,J,\kappa)$ be a strictly pseudoconvex pseudohermitian 3-manifold. The $\eta$-invariants of the family of metrics $g_{\epsilon}:=\epsilon^{-1}\kappa\otimes\kappa+d\kappa(\cdot,J\cdot)$ have a decomposition in homogeneous terms: $$\label{etalem}
\eta(g_{\epsilon})=\sum_{i=-2}^{2}\eta_{i}(X,\kappa)\epsilon^{i}.$$ The terms $\eta_{i}$ for $i\neq 0$ are integrals of local pseudohermitian invariants of $(X,\kappa)$, and the $\eta_{i}$ for $i>0$ vanish when the Tanaka-Webster torsion, $\tau$, vanishes.
We then make the following
Let $(X,\kappa)$ be a compact strictly pseudoconvex pseudohermitian 3-dimensional manifold. The *renormalized $\eta$-invariant* $\eta_{0}(X,\kappa)$ of $(X,\kappa)$ is the constant term in the expansion of Eq. \[etalem\] for the $\eta$-invariants of the family of metrics $g_{\epsilon}:=\epsilon^{-1}\kappa\otimes\kappa+d\kappa(\cdot,J\cdot)$.
Our assumption that $X$ is K-contact ensures that the Reeb flow preserves the metric. In this situation, it is known that the Tanaka-Webster torsion necessarily vanishes (cf. [@bhr] ; §3). In the case where the torsion of $(X,\kappa)$ vanishes, the terms $\eta_{i}(X,\kappa)$ in Eq. \[etalem\] vanish for $i>0$, so that when $\epsilon\rightarrow \infty$, one has $$\label{vantor}
\eta_{0}(X,\kappa)=\lim_{\epsilon\rightarrow\infty}\eta(g_{\epsilon}):=\eta_{ad}$$ The limit $\eta_{ad}$ is known as the *adiabatic limit* and has been studied in [@bc] and [@dai], for example. The adiabatic limit is the case where the limit is taken as $\epsilon$ goes to infinity, $$\eta_{ad}:=\lim_{\epsilon\rightarrow\infty}\eta(g_{\epsilon}),$$ while the the renormalized $\eta$-invariant, $\eta_{0}(X,\kappa)$, is naturally interpreted as the constant term in the asymptotic expansion for $(\eta(g_{\epsilon}))$ in powers of $\epsilon$, when $\epsilon$ goes to $0$. This reverse process of taking $\epsilon$ to $0$ is also known as the *diabatic* limit. When torsion vanishes (i.e. when the Reeb flow preserves the metric), Eq. \[vantor\] is the statement that the diabatic and adiabatic limits agree. One of the main challenges for our future work will be to extend beyond the case where torsion vanishes. This will naturally involve the study of the diabatic limit. For now, we are restricted to the case of vanishing torsion. In this case, the main result that we will use is the following
\[eta0thm\]([@bhr] ; Theorem 1.4) Let $X$ be a compact CR-Seifert 3-manifold, with $U(1)$-action generated by the Reeb field of an $U(1)$-invariant contact form $\kappa$. If $R$ is the Tanaka-Webster curvature of $(X,\kappa)$ and $D^{1}$ is the middle degree operator of the contact complex (cf. Eq. \[complex\] and \[Ddef\]), then $$\eta_{0}(X,\kappa)=\eta(-\star_{H}D^{1})+\frac{1}{512}\int_{X}R^{2}\,\,\kappa\wedge d\kappa.$$
Theorem \[eta0thm\] compels us to conjecture that $C_{T}=\frac{1}{512}\int_{X}R^{2}\,\,\kappa\wedge d\kappa$. Our motivation for this comes from the fact that $\eta_{0}(X,\kappa)$ is a topological invariant in our case. We have the following,
\[topthm\]([@bhr] ; cf. Remark 9.6 and Eq. 27) If $X$ is a CR-Seifert manifold, then $\eta_{0}(X,\kappa)$ is a topological invariant and $$\eta_{0}(X,\kappa)=1+\frac{d}{3}+4\sum_{j=1}^{N}s(\alpha_{j},\beta_{j}),$$ where $d\in{\mathbb{Q}}$ is the degree of $X$ as a compact $U(1)$-orbifold bundle and $$s(\alpha,\beta):=\frac{1}{4\alpha}\sum_{k=1}^{\alpha-1}cot\left(\frac{\pi k}{\alpha}\right)cot\left(\frac{\pi k\beta}{\alpha}\right)$$ is the classical Rademacher-Dedekind sum, where $[n; (\alpha_{1},\beta_{1}),\ldots,(\alpha_{N},\beta_{N})]$ (for $(\alpha_{i},\beta_{i})=1$ relatively prime) are the Seifert invariants of $X$.
Thus, we are led to consider the natural topological invariant $e^{\frac{\pi i}{4}\left[\eta_{0}(X,\kappa)\right]}$ and how it compares with the topological invariant $e^{\pi i\left[\frac{\eta(-\star d)}{4}+\frac{1}{12}\frac{\text{CS}(A^{g})}{2\pi}\right]}$. We consider the limit $$\begin{aligned}
\lim_{\epsilon\rightarrow\infty}e^{\pi i\left[\frac{\eta(-\star_{\epsilon} d)}{4}+\frac{1}{12}\frac{\text{CS}(A^{g_{\epsilon}})}{2\pi}\right]}\end{aligned}$$ where $g_{\epsilon}=\epsilon^{-1}\kappa\otimes\kappa+d\kappa(\cdot,J\cdot)$ is the natural metric associated to $X$. On the one hand, since this is a topological invariant, and is independent of the metric, we must have $$\begin{aligned}
\lim_{\epsilon\rightarrow\infty}e^{\pi i\left[\frac{\eta(-\star_{\epsilon} d)}{4}+\frac{1}{12}\frac{\text{CS}(A^{g_{\epsilon}})}{2\pi}\right]}=e^{\pi i\left[\frac{\eta(-\star d)}{4}+\frac{1}{12}\frac{\text{CS}(A^{g})}{2\pi}\right]}.\end{aligned}$$ where we take $g_{1}:=g$ so that $\star_{g_{1}}:=\star$.
On the other hand, since $\eta(g_{\epsilon})=\eta(-\star_{\epsilon} d)$ by definition, and we know that its limit exists as $\epsilon\rightarrow\infty$ (in fact $\eta_{0}(X,\kappa)=\lim_{\epsilon\rightarrow\infty}\eta(g_{\epsilon})$), we have $$\begin{aligned}
\lim_{\epsilon\rightarrow\infty}e^{\pi i\left[\frac{\eta(-\star_{\epsilon} d)}{4}+\frac{1}{12}\frac{\text{CS}(A^{g_{\epsilon}})}{2\pi}\right]}=e^{\pi i\left[\frac{\eta_{0}(X,\kappa)}{4}+\left\{\lim_{\epsilon\rightarrow\infty}\frac{1}{12}\frac{\text{CS}(A^{g_{\epsilon}})}{2\pi}\right\}\right]}.\end{aligned}$$ Thus, we have $$\begin{aligned}
\label{topeq}
e^{\pi i\left[\frac{\eta(-\star d)}{4}+\frac{1}{12}\frac{\text{CS}(A^{g})}{2\pi}\right]}=e^{\pi i\left[\frac{\eta_{0}(X,\kappa)}{4}\right]}e^{\pi i\left\{\lim_{\epsilon\rightarrow\infty}\frac{1}{12}\frac{\text{CS}(A^{g_{\epsilon}})}{2\pi}\right\}}.\end{aligned}$$ We therefore see that if we can understand the limit $\lim_{\epsilon\rightarrow\infty}\frac{1}{12}\frac{\text{CS}(A^{g_{\epsilon}})}{2\pi}$, we will obtain crucial information for our problem. The following has been established using a “Kaluza-Klein” dimensional reduction technique modeled after the paper [@gijp],
\[mc2\]([@mcl2]) $(X,\phi,\xi,\kappa,g)$ quasi-regular K-contact three-manifold, $$\xymatrix{{\makeatletter
\xydef@\xymatrixcolsep@{2pc}
\makeatother
}{\makeatletter
\xydef@\xymatrixrowsep@{1pc}
\makeatother
}U(1) \ar@{^{(}->}[r] & X \ar[d]\\
& \Sigma}.$$ Let $g_{\epsilon}:=\epsilon^{-1}\,\kappa\otimes\kappa+\pi^{*}h$. After choosing a framing for $TX\oplus TX$, corresponding to a choice of vielbeins, then, $$CS(A^{g_{\epsilon}})=\left(\frac{\epsilon^{-1}}{2}\right)\int_{\Sigma}r\,\omega+\left(\frac{\epsilon^{-2}}{2}\right)\int_{\Sigma}f^{2}\,\omega$$ where $r\in C^{\infty}_{orb}(\Sigma)$ is the (orbifold) scalar curvature of $(\Sigma,h)$, $\omega\in\Omega^{2}_{orb}(\Sigma)$ is the (orbifold) Hodge form of $(\Sigma,h)$, and $f:=\star_{h}\omega$. In particular, the adiabatic limit of $\text{CS}(A^{g_{\epsilon}})$ vanishes: $$\lim_{\epsilon\rightarrow \infty}\text{CS}(A^{g_{\epsilon}})=0.$$
Proposition \[mc2\] combined with Eq. \[topeq\] and Theorem \[eta0thm\] gives us the following,
\[lprop\] $(X,\phi,\xi,\kappa,g)$ closed, quasi-regular K-contact three-manifold. The counterterm, $C_{T}$, such that $e^{\frac{\pi i}{4}\left[\eta(-\star_{H} D^{1})+C_{T}\right]}$ is a topological invariant that is identically equal to the topological invariant $e^{\pi i\left[\frac{\eta(-\star d)}{4}+\frac{1}{12}\frac{\text{CS}(A^{g})}{2\pi}\right]}$ is $$C_{T}=\frac{1}{512}\int_{X}R^{2}\,\,\kappa\wedge d\kappa.$$
Given Proposition \[lprop\] and Theorem \[topthm\], we conclude the following as an immediate consequence,
\[cprop\] $(X,\phi,\xi,\kappa,g)$ closed, quasi-regular K-contact three-manifold. Then, $$\begin{aligned}
\eta(-\star d)+\frac{1}{3}\frac{\text{CS}(A^{g})}{2\pi}&=&\eta(-\star D)+\frac{1}{512}\int_{X}R^{2}\,\,\kappa\wedge d\kappa\\
&=&1+\frac{d}{3}+4\sum_{j=1}^{N}s(\alpha_{j},\beta_{j}),\end{aligned}$$ where $d=c_1(X)=n+\sum_{j=1}^{N}\frac{\beta_{j}}{\alpha_j}\in{\mathbb{Q}}$ and $$s(\alpha,\beta):=\frac{1}{4\alpha}\sum_{k=1}^{\alpha-1}cot\left(\frac{\pi k}{\alpha}\right)cot\left(\frac{\pi k\beta}{\alpha}\right)\in{\mathbb{Q}}$$ is the classical Rademacher-Dedekind sum, where $[n; (\alpha_{1},\beta_{1}),\ldots,(\alpha_{N},\beta_{N})]$ (for gcd$(\alpha_{j},\beta_{j})=1$) are the Seifert invariants of $X$. In particular, we have computed the $U(1)$-Chern-Simons partition function as: $$\begin{aligned}
Z_{U(1)}(X,p,k)&=&k^{n_X}e^{\pi i k S_{X,P}(A_{0})}e^{\frac{\pi i}{4}\left(1+\frac{d}{3}+4\sum_{j=1}^{N}s(\alpha_{j},\beta_{j})\right)}\int_{\mathcal{M}_{P}}\,\, (T^{d}_{C})^{1/2},\\
&=&k^{m_X}e^{\pi i k S_{X,P}(A_{0})}e^{\frac{\pi i}{4}\left(1+\frac{d}{3}+4\sum_{j=1}^{N}s(\alpha_{j},\beta_{j})\right)}\int_{\mathcal{M}_{P}}\,\, (T^{d}_{RS})^{1/2}.\end{aligned}$$
Basic construction of $U(1)$-Chern-Simons theory {#appen1}
================================================
Let $X$ be a closed oriented $3$-manifold. For any $U(1)$-connection $A\in\mathcal{A}_{P}$, [@m] defined an induced $SU(2)$-connection $\hat{A}$ on an associated principal $SU(2)$-bundle $\hat{P}=P\times_{U(1)} SU(2)$. i.e. $$\hat{A}|_{[p,g]}=Ad_{g^{-1}}(\rho_{*}pr_{1}^{*}A|_{p})+pr_{2}^{*}\vartheta_{g}$$ where $\rho:U(1)\rightarrow SU(2)$ is the diagonal inclusion, $pr_{1}:P\times SU(2)\rightarrow P$ and $pr_{2}:P\times SU(2)\rightarrow SU(2)$. Since for any $3$-manifold $X$, $\hat{P}$ is trivializable, let $\hat{s}:X\rightarrow \hat{P}$ be a global section. The definition we use for the Chern-Simons action is as follows:
The Chern-Simons action functional of a $U(1)$-connection $A\in\mathcal{A}_{P}$ is defined by: $$\label{act}
S_{X,P}(A)=\int_{X}\hat{s}^{*}\alpha(\hat{A})\,\,\,(mod\,\, {\mathbb{Z}})$$ where $\alpha(\hat{A})\in\Omega^{3}(\hat{P},{\mathbb{R}})$ is the Chern-Simons form of the induced $SU(2)$-connection $\hat{A}\in\mathcal{A}_{\hat{P}}$, $$\alpha(\hat{A})=Tr(\hat{A}\wedge F_{\hat{A}})-\frac{1}{6}Tr(\hat{A}\wedge [\hat{A},\hat{A}])$$
We then define the partition function for $U(1)$-Chern-Simons theory as (as in [@m], [@mpr]): $$Z_{U(1)}(X,k)=\sum_{p\in\text{Tors}H^{2}(X;{\mathbb{Z}})}Z_{U(1)}(X,p,k)$$ where, $$Z_{U(1)}(X,p,k)=\frac{1}{Vol(\mathcal{G}_{P})}\int_{\mathcal{A}_{P}}\mathcal{D}A e^{\pi i k S_{X,P}(A)}$$ and $$S_{X,P}(A)=\int_{X}\hat{s}^{*}\alpha(\hat{A})$$ Then for any principal $U(1)$-bundle $P$ we follow [@bw] and define a new action $$S_{X,P}(A,\Phi):=S_{X,P}(A-\kappa\Phi)$$ where we may view $\Phi\in \Omega^{0}(X)$ and, $$\begin{aligned}
S_{X,P}(A,\Phi)&=&\int_{X}\alpha(\widehat{A-\kappa\Phi})\\
&=&\int_{X}\alpha(\hat{A}-\kappa\hat{\Phi})\\
&=&S_{X,P}(A)-\int_{X}[2\kappa\wedge Tr (\hat{\Phi}\wedge F_{\hat{A}})-\kappa\wedge d\kappa\,\, Tr (\hat{\Phi}^{2})]\end{aligned}$$ where the second equality follows from the definition of $\hat{A}$ and $\hat{\Phi}$ (where $\hat{\Phi}|_{[p,g]}=Ad_{g^{-1}}(\rho_{*}pr_{1}^{*}\Phi|_{p})$) on $\hat{P}=P\times_{U(1)} SU(2)$. The third equality follows from Eq. 3.6 of [@bw]. We then define a new partition function $$\bar{Z}_{U(1)}(X,p,k):=\frac{1}{Vol(S)}\frac{1}{Vol(\mathcal{G}_{P})}\int_{\mathcal{A}(P)}DA\,D\Phi\,\, e^{\pi i k S_{X,P}(A,\Phi)}$$ where $D\Phi$ is defined by the invariant, positive definite quadratic form, $$\label{phiprod}
(\Phi,\Phi)=-\frac{1}{4\pi^{2}}\int_{X}\Phi^{2}\kappa\wedge d\kappa$$ As observed in [@bw], our new partition function is identically equal to our original partition function defined for $U(1)$-Chern-Simons theory. On the one hand, we can fix $\Phi=0$ above using the shift symmetry, $\delta\Phi=\sigma$, which will cancel the pre-factor $Vol(S)$ from the resulting group integral over $S$ and yield exactly our original partition function: $$Z_{U(1)}(X,p,k)=\frac{1}{Vol(\mathcal{G}_{P})}\int_{\mathcal{A}_{P}}\mathcal{D}A\,\, e^{\pi i k S_{X,P}(A)}$$ Thus, we obtain the heuristic result, $$\label{equivpart}
\bar{Z}_{U(1)}(X,p,k)=Z_{U(1)}(X,p,k).$$ On the other hand, we obtain another description of $\bar{Z}_{U(1)}(X,p,k)$ by integrating $\Phi$ out. We will briefly review this computation here. Our starting point is the formula for the shifted partition function $$\bar{Z}_{U(1)}(X,p,k)=\frac{1}{Vol(S)}\frac{1}{Vol(\mathcal{G}_{P})}\int_{\mathcal{A}(P)}DA\,D\Phi\,\, e^{\pi i k S_{X,P}(A,\Phi)}$$ where $$S_{X,P}(A,\Phi)=S_{X,P}(A)-\int_{X}[2\kappa\wedge Tr (\hat{\Phi}\wedge F_{\hat{A}})-\kappa\wedge d\kappa\,\, Tr (\hat{\Phi}^{2})]$$ We formally complete the square with respect to $\hat{\Phi}$ as follows: $$\begin{aligned}
\int_{X}[\kappa\wedge d\kappa\,\, Tr (\hat{\Phi}^{2})&-&2\kappa\wedge Tr (\hat{\Phi}\wedge F_{\hat{A}})]\\
&=&\int_{X}\left[Tr (\hat{\Phi}^{2})-\frac{2\kappa\wedge Tr (\hat{\Phi}\wedge F_{\hat{A}})}{\kappa\wedge d\kappa}\right]\kappa\wedge d\kappa\\
&=&\int_{X}Tr \left(\hat{\Phi}^{2}-\frac{2\kappa\wedge
F_{\hat{A}}}{\kappa\wedge d\kappa}\hat{\Phi}\right)\kappa\wedge d\kappa\\
&=&\int_{X}Tr \left(\left[\hat{\Phi}-\frac{\kappa\wedge F_{\hat{A}}}{\kappa\wedge d\kappa}\right]^{2}-\left[\frac{\kappa\wedge F_{\hat{A}}}{\kappa\wedge d\kappa}\right]^{2}\right)\kappa\wedge d\kappa\end{aligned}$$ We then only need to compute the Gaussian $$\begin{aligned}
\int D\Phi\,\,&\text{exp}&\left[ \pi i k\,\int_{X} \text{Tr} \left(\left[\hat{\Phi}-\frac{\kappa\wedge F_{\hat{A}}}{\kappa\wedge d\kappa}\right]^{2}\right)\kappa\wedge d\kappa\right]\\
&=&\int D\Phi\,\text{exp}\left[ \pi i k\,\, \int_{X} Tr (\hat{\Phi}^{2})\kappa\wedge d\kappa\right]\\
&=&\int D\Phi\,\text{exp}\left[ \frac{i k}{4\pi}\,\, \int_{X}\Phi^{2}\kappa\wedge d\kappa\right]\\
&=&\int D\Phi\,\text{exp}\left[ -\frac{1}{2}\,\, (\Phi, A\Phi) \right]\end{aligned}$$ where we take $A=2\pi i k{\mathbb{I}}$ acting on the space of fields $\Phi$ and the inner product $(\Phi,\Phi)$ is defined as in Eq. \[phiprod\]. We then formally get $$\begin{aligned}
\int D\Phi\,\text{exp}\left[ -\frac{1}{2}\,\, (\Phi, A\Phi) \right]&=&\sqrt{\frac{(2\pi)^{\Delta\mathcal{G}}}{\text{det}A}}\\\label{dimterm}
&=&\left(\frac{-i}{k}\right)^{\Delta\mathcal{G}/2}\end{aligned}$$ where the quantity $\Delta\mathcal{G}$ is formally the dimension of the gauge group $\mathcal{G}$. Note that we have abused notation slightly throughout by writing $\frac{1}{\kappa\wedge d\kappa}$. We have done this with the understanding that since $\kappa\wedge d\kappa$ is non-vanishing, then $\kappa\wedge F_{\hat{A}}=\phi\,\kappa\wedge d\kappa$ for some function $\phi\in 2\pi i \Omega^{0}(X)$, and we identify $\frac{\kappa\wedge F_{\hat{A}}}{\kappa\wedge d\kappa}:=\phi$.\
\
Our new description of the partition function is now, $$\label{part}
\bar{Z}_{U(1)}(X,p,k)=C\int_{\mathcal{A}_{P}}DA\,\, exp\,\left[\pi i k\left(S_{X,P}(A)-\int_{X} \frac{Tr[(\kappa\wedge F_{\hat{A}})^{2}]}{\kappa\wedge d\kappa}\right)\right]$$ where $C=\frac{1}{Vol(S)}\frac{1}{Vol(\mathcal{G}_{P})}\left(\frac{-i}{k}\right)^{\Delta\mathcal{G}/2}$. We may rewrite this partition function after choosing a flat base point $A_{0}$ in $\mathcal{A}_{P}$ so that $F_{A_{0}}=0$ and identifing $\mathcal{A}(P)=A_{0}+2\pi i \Omega^{1}(X)$. We then obtain $$\label{Anom1a}
\bar{Z}_{U(1)}(X,p,k)=C_1\int_{\mathcal{A}_{P}}DA\,\, exp\,\left[\frac{i k}{4\pi}\left(\int_{X} A\wedge dA-\int_{X} \frac{(\kappa\wedge dA)^{2}}{\kappa\wedge d\kappa}\right)\right]$$ where $$C_1 = \frac{e^{\pi i k S_{X,P}(A_{0})}}{Vol(\mathcal{S})Vol(\mathcal{G}_{P})} \left(\frac{-i}{k}\right)^{\Delta\mathcal{G}/2}.$$ We may further simplify Eq. \[Anom1a\] by reducing $\mathcal{A}_{P}$ to its quotient under the shift symmetry $\bar{\mathcal{A}}_{P}:=\mathcal{A}_{P}/\mathcal{S}$, effectively canceling the factor of $Vol(\mathcal{S})$ out front of the integral. We obtain: $$\label{Anom2a}
\bar{Z}_{U(1)}(X,p,k) =
C_2 \int_{\bar{\mathcal{A}}_{P}}\bar{D}A\,\, exp\,\left[\frac{i k}{4\pi}
\left(\int_{X} A\wedge dA-\int_{X} \frac{(\kappa\wedge dA)^{2}}{\kappa\wedge d\kappa}\right)\right]\\$$ where $C_2 = C_1 Vol (S) $.
Note that we are justified in excluding the factor $\left(\frac{-i}{k}\right)^{\Delta\mathcal{G}/2}$ from Eq. \[Anom1\] since this factor would cancel in the stationary phase approximation in any case.
Framing dependence and the gravitational Chern-Simons term {#appen2}
==========================================================
As observed in ([@m] ; Eq. 7.17), Eq. \[abelchern2\] can also be rigorously defined by setting $$Z_{U(1)}(X,p,k)=\frac{e^{\pi i k S_{X,P}(A_{P})}}{\text{Vol}U(1)}\int_{\mathcal{M}_{P}}e^{\frac{\pi i}{4} \text{sgn}(-\star d)}\frac{[\text{det}'(d^{*}d)]^{1/2}]}{[\text{det}'(-k\star d)]^{1/2}]}\nu$$ where $\nu$ is the metric induced on the moduli space of flat connections on $P$, $\mathcal{M}_{P}$. This last expression has rigorous mathematical meaning if the determinants and signatures of the operators are regularized. The signature of the operator $-\star d$ on $\Omega^{1}(X;{\mathbb{R}})$ is regularized via the eta invariant, so that $\text{sgn}(-\star d)=\eta(-\star d)+\frac{1}{3}\frac{\text{CS}(A^{g})}{2\pi}$, where $$\eta(-\star d)=\lim_{s\rightarrow 0}\sum_{\lambda_{j}\neq 0}sign\lambda_{j}|\lambda_{j}|^{-s}$$ and $\lambda_{j}$ are the eigenvalues of $-\star d$, and $$\text{CS}(A^{g})=\frac{1}{4\pi}\int_{X}Tr(A^{g}\wedge dA^{g}+\frac{2}{3} A^{g}\wedge A^{g}\wedge A^{g})$$ is the gravitational Chern-Simons term, with $A^{g}$ the Levi-Civita connection on the spin bundle of $X$. The determinants are regularized as in Remark 7.6 of [@m].\
It is straightforward to see that the the term inside of the integral $$\frac{1}{\text{Vol}U(1)}\frac{[\text{det}'(d^{*}d)]^{1/2}]}{[\text{det}'(-k\star d)]^{1/2}]}$$ may be identified with the Reidemeister torsion of the 3-manifold $X$, $T^{d}_{RS}$ (cf. [@m] ; Eq. 7.22). We obtain, $$Z_{U(1)}(X,p,k)=k^{m_X}e^{\pi i k S_{X,P}(A_{P})}e^{\pi i\left(\frac{\eta(-\star d)}{4}+\frac{1}{12}\frac{\text{CS}(A^{g})}{2\pi}\right)}\int_{\mathcal{M}_{P}}(T^{d}_{RS})^{1/2}$$ where $m_X=\frac{1}{2}(\text{dim}H^{1}(X;{\mathbb{R}})-\text{dim}H^{0}(X;{\mathbb{R}}))$. The Atiyah-Patodi-Singer theorem says that the combination $$\frac{\eta(-\star d)}{4}+\frac{1}{12}\frac{\text{CS}(A^{g})}{2\pi}$$ is a topological invariant depending only on a $2$-framing of $X$. Recall ([@at]) that a 2-framing is choice of a homotopy equivalence class $\pi$ of trivializations of $TX\oplus TX$, twice the tangent bundle of $X$ viewed as a Spin$(6)$ bundle. The possible $2$-framings correspond to ${\mathbb{Z}}$. The identification with ${\mathbb{Z}}$ is given by the signature defect defined by $$\delta(X,\pi)=\text{sign}(M)-\frac{1}{6}p_{1}(2TX,\pi)$$ where $M$ is a $4$-manifold with boundary $X$ and $p_{1}(2TX,\pi)$ is the relative Pontrjagin number associated to the framing $\pi$ of the bundle $TX\oplus TX$. The canonical $2$-framing $\pi^{c}$ corresponds to $\delta(X,\pi^{c})=0$. Either we can choose the canonical framing, and work with this throughout, or we can observe that if the framing of $X$ is twisted by $s$ units, then $CS(A^{g})$ transforms by $$CS(A^{g})\rightarrow CS(A^{g})+2\pi s$$ and so the partition function $Z_{U(1)}(X,k)$ is transformed by $$\label{partform}
Z_{U(1)}(X,k)\rightarrow Z_{U(1)}(X,k)\cdot \text{exp}\left(\frac{2\pi i s}{24}\right)$$ Then $Z_{U(1)}(X,k)$ is a topological invariant of framed, oriented $3$-manifolds, with a transformation law under change of framing. This is tantamount to a topological invariant of oriented $3$-manifolds without a choice of framing.
[^1]: The first author was supported in part by a grant from NSERC
[^2]: We would like to thank John Bland, Eckhard Meinrenken, Raphaël Ponge, Edward Witten and especially Frédéric Rochon and Michel Rumin for helpful advice related to this work.
[^3]: In fact, [@bw] consider only $G$ compact, connected and simple, and for concreteness one may assume $G=SU(2)$.
[^4]: Note that the definition of the Chern-Simons partition function in Eq. \[orgchern\] is completely heuristic. The measure $\mathcal{D}A$ has not been defined, but only assumed to “exist heuristically,” and the volume and dimension of the gauge group, $\text{Vol}(\mathcal{G})$ and $\Delta(\mathcal{G})$, respectively, are at best formally defined.
[^5]: The measure $\mathcal{D}\Phi$ is defined independently of any metric on $X$ and is formally defined by the positive definite quadratic form $$(\Phi,\Phi):=-\int_{X}\kappa\wedge d\kappa\,\,\text{Tr}(\Phi^{2}),$$ which is invariant under the choice of representative for the contact structure $(X,H)$ on $X$, i.e. under the scaling $\kappa\mapsto f\kappa$, $\Phi\mapsto f^{-1}\Phi$, for some non-zero function $f\in\Omega^{0}(X,{\mathbb{R}})$.
[^6]: $\mathcal{S}$ may be identified with $\Omega^{0}(X,\frak{g})$, where the “action” on $\mathcal{A}_P$ is defined as $\delta_{\sigma}(A):=\sigma\kappa$, and on the space of fields $\Phi$ is defined as $\delta_{\sigma}(\Phi):=\sigma$, for $\sigma\in\Omega^{0}(X,\frak{g})$. $\delta_{\sigma}$ denotes the action associated to $\sigma$.
[^7]: Note that the partition functions of Eq.’s \[orgchern\] and \[newchern\] are defined implicitly with respect the pullback of some trivializing section of the principal $G$-bundle $P$. Of course, every principal $G$-bundle over a three-manifold for $G$ compact, connected and simple is trivializable. It is basic fact that the partition functions of Eq.’s \[orgchern\] and \[newchern\] are independent of the choice of such trivializations.
[^8]: Note that we have abused notation slightly by writing $\frac{1}{\kappa\wedge d\kappa}$. We have done this with the understanding that since $\kappa\wedge d\kappa$ is non-vanishing (since $\kappa$ is a contact form), then $\kappa\wedge F_{A}=\phi\,\kappa\wedge d\kappa$ for some function $\phi\in \Omega^{0}(X,\frak{g})$, and we identify $\frac{\kappa\wedge F_{A}}{\kappa\wedge d\kappa}:=\phi$.
[^9]: Recall the definition of the torsion of an abelian group is the collection of those elements which have finite order.
[^10]: We consider the square roots thereof, viewed as densities on the moduli space of flat connections $\mathcal{M}_{X}$.
[^11]: In this case, topological invariance is recovered only up to a choice of two-framing for $X$. Of course, there is a canonical choice of such framing ([@at]), and we assume this choice throughout.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Energy-efficient simultaneous localization and mapping (SLAM) is crucial for mobile robots exploring unknown environments. The mammalian brain solves SLAM via a network of specialized neurons, exhibiting asynchronous computations and event-based communications, with very low energy consumption. We propose a brain-inspired spiking neural network (SNN) architecture that solves the unidimensional SLAM by introducing spike-based reference frame transformation, visual likelihood computation, and Bayesian inference. We integrated our neuromorphic algorithm to Intel’s Loihi neuromorphic processor, a non-Von Neumann hardware that mimics the brain’s computing paradigms. We performed comparative analyses for accuracy and energy-efficiency between our neuromorphic approach and the GMapping algorithm, which is widely used in small environments. Our Loihi-based SNN architecture consumes 100 times less energy than GMapping run on a CPU while having comparable accuracy in head direction localization and map-generation. These results pave the way for scaling our approach towards active-SLAM alternative solutions for Loihi-controlled autonomous robots.'
author:
- 'Guangzhi Tang, Arpit Shah, and Konstantinos P. Michmizos[^1][^2]'
bibliography:
- 'myref.bib'
title: '**Spiking Neural Network on Neuromorphic Hardware for Energy-Efficient Unidimensional SLAM** '
---
Introduction
============
Localization, knowing one’s pose, and mapping, knowing the positions of surrounding landmarks, are essential skills for both humans and robots as they navigate in unknown environments. The main challenge is to produce accurate estimates from noisy, error-prone cues, with robustness, efficiency, and adaptivity. Graph-based [@kummerle2011g; @dellaert2010subgraph] and filter-based approaches [@grisetti2007improved; @strom2011occupancy] have solved the simultaneous localization and mapping (SLAM) problem by either optimizing a constrained graph or performing recursive Bayesian estimation. As they are tackling SLAM in a growing number of real-world applications, these approaches face increasing challenges for minimizing energy consumption.
Interestingly, efficient and highly accurate localization and mapping are “effortless” characteristics of mammalian brains [@poulter2018neurobiology]. Over the last few decades, a number of specialized neurons, including border cells, head direction cells, place cells, grid cells, and speed cells, have been found to be part of a brain network that solves localization and mapping [@grieves2017representation] in an energy-efficient manner [@sengupta2014power].
Large-scale neuromorphic processors [@davies2018loihi; @merolla2014million; @schemmel2010wafer; @furber2014spinnaker] have been proposed as a non-Von Neumann alternative to traditional computing hardware. These processors offer asynchronous event-based parallelism and relatively efficient solutions to many mobile robot applications [@mei2005case; @blum2017neuromorphic; @hwu2017self]. Particularly, Intel’s Loihi processor [@davies2018loihi] supports on-chip synaptic learning, multilayer dendritic trees, and other brain-inspired components such as synaptic delays, homeostasis, and reward-based learning.
To leverage the disruptive potential of neuromorphic computing, we need to develop new algorithms that call for a bottom-up rethinking of our already developed solutions. Neuromorphic processors are designed to run Spiking Neural Networks (SNN), a specialized brain-inspired architecture where simulated neurons emulate the learning and computing principles of their biological counterparts. SNNs can introduce an orthogonal dimension to neural processing by adhering to the structure of the biological networks associated with the targeted tasks. Specifically, the brain’s spatial navigation and sensorimotor systems have inspired the design of SNNs that solved a number of problems in robotics [@bing2018end; @bing2018survey; @hwu2018adaptive] Of particular interest for this study is an SNN inspired by the brain’s navigational system that enables a mobile robot to correct its estimate of pose and map of a simple environment, by periodically using a ground-truth signal [@kreiser2018pose].
In this paper, we present a biologically constrained SNN architecture which solves the unidimensional SLAM problem on Loihi, without depending on the external ground truth information. To do so, our proposed model determines the robot’s heading via spike-based recursive Bayesian inference of multisensory cues, namely visual and odometry information. We validated our implementation in both real-world and simulated environments, by comparing with the GMapping algorithm [@grisetti2007improved]. The SNN generated representations of the robot’s heading and mapped the environment with comparable performance to the baseline while consuming less than 1% of dynamic power.
Method
======
We developed a recursive SNN that suggests a cue-integration connectome performing head direction localization and mapping, and we integrated the network to Loihi. Inspired by the spatial navigation system found in the mammalian brain, the head direction and border cells in our network exhibited biologically realistic activity [@grieves2017representation]. Our model had intrinsic asynchronous parallelism by incorporating spiking neurons, multi-compartmental dendritic trees, and plastic synapses, all of which are supported by Loihi.
Our model had 2 sensory spike rate-encoders and 5 sub-networks (Fig. \[fig: overall\]). The odometry sensor and the RGB-Depth camera signals drove the neural activity of speed cells and sensory neurons encoding the angular speed and the distance to the nearest object, respectively. The Head Direction (HD) network received the input from the speed cells and represented the heading of the robot. The Reference Frame Transformation (RFT) network received the egocentric input from sensory neurons and generated allocentric distance representation in the world reference frame, as defined by the HD network. The Distance Mapping (DM) network learned the allocentric observations from the RFT network and formed the map of the robot’s surrounding environment. The Observation Likelihood (OL) network used the map from the DM network to compute the observation likelihood distribution of the robot’s heading based on the egocentric observation from sensory neurons. The Bayesian Inference (BI) network produced a near-optimal posterior of the robot’s heading and corrected the heading representation within the HD network. To do so, the BI network used the observation likelihood from the OL network and the odometry likelihood from the HD network. Each one of the networks is briefly described below.
Head Direction Network
----------------------
Head direction cells in the HD network changed their spiking activity according to the robot’s heading, as follows. The HD network comprised of 75 neurons, each having a 5-degree resolution. We used the Continuous Attractor Model [@wu2016continuous] to integrate angular speed and form a stable representation of the robot’s heading (Fig. \[fig: SNN\]a). The HD attractor state shifted either clockwise or counter-clockwise, depending on the robot’s rotation, with the help of transition neurons. There were two populations of such neurons to represent the two possible directions of rotation. Each transition neuron had a dendritic tree with one dendrite receiving spikes from the speed cell and the other from its corresponding head direction cell. The neuron fired when both dendrites were activated, thereby changing the HD attractor state.
Reference Frame Transformation Network
--------------------------------------
Border cells in the RFT network represented distance observations in the world reference frame (Fig. \[fig: SNN\]b). The sensory neurons represented discretized distances between the observable objects and the robot, in an egocentric manner. Similarly to our previous work [@tang2018gridbot], the RFT network used the HD activity to create an allocentric representation of the surrounding environment, therefore translating from egocentric observations to mapping. Other spike-based methods exist that perform reference frame transformation [@schneegans2012neural; @blum2017neuromorphic; @bicanski2018neural].
To perform the transformation in the RFT network, we leveraged the concurrent activity of sensory neurons and HD cells, as follows. Sensory neurons encoded the depth signal at the robot’s heading, as represented by HD cells. We created a group of border cells with the same preferred headings as the HD cells, allowing the border cells and HD cells to be on the same reference frame and have a one-to-one correspondence on preferred headings. Each border cell had a dendritic tree receiving spikes from HD cells and sensory neurons. A border cell fired maximally when the HD cells and sensory neurons connected to its dendritic tree were activated at the same time.
![Structure of the SNN architecture. Each block is a sub-network.[]{data-label="fig: overall"}](small_fig2.png)
Distance Mapping Network
------------------------
The spike activity of map neurons in the DM network represented the mapping of the robot’s surrounding environment (Fig. \[fig: SNN\]c). The map was stored in the synapses between a single place cell and all map neurons, using an unsupervised, Hebbian-type rule. When the map (post-synaptic) neuron fired, the synaptic weight, $w$, increased proportionally to the trace of the place cell’s (pre-synaptic) spikes, as follows: $$\delta w = A*x_1*y_0 - B*u_k, \eqno{(1)}$$ where the trace $x_1$ was the convolution of the pre-synaptic neuron’s spikes with a decaying exponential function; $y_0$ changes from 0 to 1 whenever a post-synaptic neuron fires; and $u_k$ is a decay factor which changed from 0 to 1 every k time-steps and prevented overlearning in synapses with inconsistent pre-synaptic activity. That way, the network learned only the obstacles in the map that were observed with high certainty. During learning, map neurons were activated by border cells, and a single place cell was activated by the location of the robot. A winner-take-all (WTA) mechanism was implemented as an inhibition of the nearby map neurons and ensured that a single map neuron would be active at each location.
Observation Likelihood Network
------------------------------
Likelihood neurons in the OL network changed their spike activity based on the encoded distances and formed an observation likelihood distribution (Fig. \[fig: SNN\]d). The network encoded the likelihoods of different headings based on the observed distance pattern. The distribution was multimodal when multiple similar distance patterns existed in the environment. This enabled the robot to estimate its heading without reference to its odometry sensor. The OL network is a spike-based alternative to the previously proposed scan matching methods [@olson2009real; @olson2015m3rsm], which compute observation likelihoods based on visual observations.
To generate the likelihood activity, we computed similarities between the depth signal and the map, by employing asynchronous dendritic processing, as follows. Synaptic connections from map neurons to OL neurons formed spatial windows in the learned representations of the environment. Since this pattern comparison considered only the excited neurons between the observation and the map, it could generate wrong likelihoods. To overcome this, we used a group of inverse sensory neurons to compute the similarity of the inverse distance pattern with the map. This second dendritic branch inhibited the likelihood, since it represented the non-active part of the environment. These two branches of the OL neurons, increased the contrast in inferring the heading.
Bayesian Inference Network
--------------------------
Bayesian neurons in the BI network generated a posterior distribution from the likelihood functions (Fig. \[fig: SNN\]e), as defined in Equation 2: $$p(s|d,o) \propto p(d|s)p(o|s)p(s), \eqno{(2)}$$ where $s$ is the heading of the robot, $d$ is the distance observed, $o$ is the odometry sensing. With a flat prior $p(s)$, the posterior distribution over the robot’s heading is proportional to the product of two likelihood functions, $p(d|s)$ and $p(o|s)$, through Bayes’ theorem.
It is known that multiplying two Gaussian distributions produces another Gaussian distribution. This property enabled us to use dendritic trees to estimate the posterior distribution from likelihood distributions represented by the OL network and the HD network. Specifically, each Bayesian neuron had two dendritic compartments connected with its corresponding OL neuron and HD cell. The *PASS* dendritic operation on Loihi integrated the OL neuron voltage into the Bayesian neuron voltage when the HD cell spiked. Through this operation, the Bayesian neuron estimated the product of activities from the OL neuron and the HD cell.
![(left column) Experimental environments; (middle column) Learned maps as represented by map neurons in our SNN architecture; (right column) Learned maps generated by the GMapping algorithm, with the lowest resolution that gave a stable solution.[]{data-label="fig: map"}](iros19fig4.png)
Neuromorphic Realization in Loihi
---------------------------------
We implemented our SNN architecture in one Loihi research chip. With a mesh layout, Loihi supports 128 neuromorphic cores with 1,024 compartments (primitive spiking neural units) in each core. Overall, a single chip provides up to 128k neurons and 128M synapses for building large-scale SNNs [@davies2018loihi]. Our SNN architecture used 15,162 compartments and 31,935 synapses distributed over 82 neuromorphic cores, slightly more than ten percent of the resources in a single Loihi research chip. When encoding the input from the distance observation, the encoder transformed all values to 3 discrete distance levels. Additionally, all neurons with HD receptive fields had a resolution of 5 degrees. For example, each sensory neuron encoded a single distance level for representing objects observable within 5 degrees.
Experiment and Results
======================
Experimental Setup
------------------
We used a mobile robot equipped with an RGB-Depth camera, in both the real-world and Gazebo simulator, for validating our method. During all experiments, the robot rotated for 120 seconds with only angular velocity commands. We created 1 real-world and 3 simulated environments (Fig. \[fig: map\]). Environments 1 and 2 provided continuous borders with environment 2 simulating the real-world environment. We also considered more simulated scenarios where non-continuous objects (Environments 3 and 4) left gaps between themselves. In the simulated environments, we retrieved the ground truth of the robot’s heading directly from Gazebo model states. In the real-world environment, we used the AprilTag detection system [@olson2011apriltag] and 4 AprilTag tags to determine sufficiently the ground truth values.
![Mean and STD of localization error over 5 experiments for both methods in a) Environment 1 (real-world environment), b) Environment 2, c) Environment 3, and d) Environment 4.[]{data-label="fig: arrcuracy"}](fig4.png)
The Baseline Method
-------------------
We chose the GMapping algorithm [@grisetti2007improved] as the baseline method solving the same unidimensional SLAM problem. To equally compare GMapping with our method, we limited GMapping to the lowest resolution that gave stable results. For the real-world environment, GMapping built the map using a resolution of 0.04 meters and did scan-matching using all distance data from each update with a minimum score parameter of 700. For the simulated environments, GMapping built maps using a resolution of 0.1 meters and did scan-matching using 15 evenly distributed distance observations with a minimum score parameter of 10.
Localization and Mapping
------------------------
We compared the heading from the HD cells with the ground truth values (Fig. \[fig: arrcuracy\]). We conducted 5 experiments in the 4 environments and estimated the average error of headings to less than 15 degrees, for both our method and GMapping. Given the 5 degrees resolution of the HD cells, the error was in practice a 1 to 3 neuron-drift in the attractor model, which had up to 10 active neurons. We observed a higher variance in the errors for environments 3 and 4, which was due to the free space between the objects and the instability in correcting the activity of the attractor model. Indeed, when there was no object observed, the error increased temporarily until an object was within the range of the visual observation. Similarly to any filter-based approach on SLAM, as soon as an object was detected, there was a sharp correction resulting in error decrease (Figs. \[fig: arrcuracy\]c and d).
We decoded the activity of the map neurons into a 20x20 gridmap representing a 4mx4m environment. Environments 1 and 2 had a square shape, and the maps generated by the SNN (Fig. \[fig: map\]a,b) successfully captured the repetitive distance pattern at the corners. Environments 3 and 4 had two objects with different shapes. The maps learned by our method (Fig \[fig: map\]c,d) reflected the differences between the two objects as perceived by the robot. We further show how our SNN can scale to map a 2D environment by using more than one place cell in the DM network (Fig. \[fig: bigmap\]).
![Learned double T maze environment to demonstrate scalability using multiple unidimensional maps. A single place cell in the DM network corresponded to a location in the maze (blue dots). The learned 2D map was constructed by superimposing the maps from all place cells.[]{data-label="fig: bigmap"}](big_map.png)
Observation Likelihood and Bayesian Inference
---------------------------------------------
The activity of OL neurons captured the distinctive patterns in the learned environment. For instance, firing rates of OL neurons in Fig. \[fig: likelihood\]a formed a bimodal distribution representing two possible headings due to the repetitive objects in Environment 4. We evaluated the activity of the Bayesian neurons by decoding the spikes from HD cells and OL neurons within a range of head directions into two Gaussian distributions, $N_1(\mu_1, \sigma^2_1)$ (red) and $N_2(\mu_2, \sigma^2_2)$ (blue) respectively. Equations (3) and (4) give the optimal posterior distribution $N_3(\mu_3, \sigma^2_3)$ (green) from these two likelihood distributions (Fig. \[fig: likelihood\]a): $$\mu_3 = \frac{\sigma^2_2}{\sigma^2_1+\sigma^2_2}\mu_1 + \frac{\sigma^2_1}{\sigma^2_1+\sigma^2_2}\mu_2 \eqno{(3)}$$ $$\sigma^2_3 = \frac{1}{\frac{1}{\sigma^2_1} + \frac{1}{\sigma^2_2}} \eqno{(4)}$$ We also computed the differences of the means and the standard deviations (STDs) between the decoded posterior distribution from Bayesian neurons and the optimal posterior distribution during runtime (Fig. \[fig: likelihood\]b). During the experiments, the difference of the mean and STD was always less than 5 degrees, which is, in fact, the resolution of the head direction in our SNN. The transient increase in the STD differences in Fig \[fig: likelihood\]b was caused by the small resolution, constrained to 2 or 3 neurons, for representing the posterior distribution. Overall, the BI network generated near-optimal posterior distribution by performing spike-based Bayesian inference.
![Spike-based Bayesian inference. a) (upper panel) Neuronal activities within the BI network for Environment 4 and (bottom panel) comparison between the decoded and the optimal Bayesian inference results. b) Mean and STD differences between the decoded and the optimal posterior distribution during a single run in Environment 4.[]{data-label="fig: likelihood"}](iros19fig6.png)
![Power consumption of our SNN architecture ran on Loihi and GMapping ran on CPU, solving the same unidimensional SLAM problem.[]{data-label="fig: power"}](fig7.png)
Energy Efficiency
-----------------
Our Loihi-run SNN was two orders of magnitude more energy efficient compared to the CPU-run GMapping solving the same unidimensional SLAM problem (Fig. \[fig: power\]). We compared the power consumption of the SNN on a Nahuku board, an 8-chip Loihi system, with that of GMapping on an Intel i7-4850HQ CPU. To measure the idle power on Loihi, we set all compartments to non-updating state in multiple 10,000 time-step runs. The idle CPU power was measured by running only the operating system for ten minutes. The running power for both methods was averaged over six such experiments.
Similarly to GMapping, our SNN architecture performed real-time data processing by only using 0.3 seconds for execution per wall-clock second, on average. This allowed us to compare the Loihi power consumption and the CPU power consumption against the same wall-clock time of the running robot (Fig. \[fig: power\]). We computed the dynamic power of each method by subtracting the idle power from the running power. An 8-chip Loihi board uses 4 times less power compared to a quad-core CPU in the idle state and our SNN running on Loihi was 100 times more energy efficient compared to GMapping running on a CPU in terms of dynamic power consumption. Since Loihi is at an early development stage, the power consumption, especially the idle power consumption, can be lowered further to 0.031 Watts in a more customized system [@blouw2018benchmarking] compared to the Nahuku board we utilized.
Discussion and Conclusion
=========================
In this paper, we showed that an SNN architecture inspired by the brain’s spatial navigation system and run on a neuromorphic processor can have similar accuracy and much lower power consumption, compared to a widely used method for solving the unidimensional SLAM problem. Although the error in the sparse environments was larger than GMapping, our proof-of-concept results can be improved by increasing the resolution or the stability of the HD network, to further demonstrate the validity of our proposed method as similarly accurate and much more efficient in terms of power consumption SLAM method. Similar to other solutions running on neuromorphic processors addressing speech recognition [@blouw2018benchmarking] and image processing [@davies2018loihi], our approach currently yields results that are only comparable to the state-of-the-art methods that have been well-tuned to run on traditional Von Neumann CPUs.
For applications such as planetary exploration and disaster rescue, where robots have limited recharging capabilities, minimizing energy consumption is crucial. Our proposed neuromorphic approach provides an energy efficient solution to the SLAM problem, which accounts for a large portion of the computational cost and its energy consumption.
Overall, this work points to the real-time neuromorphic control of robots as a strong alternative, complementing widely used solutions for foundational robotic problems. Although it probably requires a lot more small insights before it can scale to outperform a highly developed technology, the fact that our Loihi-run SNN compares in accuracy to a mainstream method while offering unparalleled energy efficiency, is an indication that end-to-end neuromorphic solutions for fully autonomous systems is a direction worth exploring.
[^1]: \*This work is supported by Intel’s INRC Grant Award
[^2]: GT, AS and KM are with the Computational Brain Lab, Department of Computer Science, Rutgers University, New Jersey, USA. [[email protected]]{}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The Lindblad master equation for an arbitrary [*quadratic*]{} system of $n$ fermions is solved explicitly in terms of diagonalization of a $4n \times 4n$ matrix, provided that all Lindblad bath operators are [*linear*]{} in the fermionic variables. The method is applied to the explicit construction of non-equilibrium steady states and the calculation of asymptotic relaxation rates in the far from equilibrium problem of heat and spin transport in a nearest neighbor Heisenberg $XY$ spin $1/2$ chain in a transverse magnetic field.'
address: 'Department of physics, FMF, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia'
author:
- Tomaž Prosen
title: 'Third quantization: a general method to solve master equations for quadratic open Fermi systems'
---
Introduction
============
Understanding time evolution of an open quantum system of many interacting particles is of primary importance in fundamental problems of quantum physics, such as decoherence [@zurek; @zeh] and closely related quantum measurement problem [@neumann; @schlos], quantum computation [@chuang; @casati], or the problem of computation of [*non-equilibrium steady states*]{} (NESS) in quantum statistical mechanics [@araki; @ruelle; @jaksic]. Even though application of the methods of Hamiltonian dynamical systems and ergodic theory to quantum systems out of equilibrium gives many promising results [@piere2; @lee; @prosenjpa], the field of open quantum systems is still lacking non-trivial explicitly solvable models, as compared to studies of closed (isolated) quantum systems where we know a large body of the so-called [*completely integrable*]{} systems [@korepin; @fadeev]. Examples of explicitly solvable models of master equations for open quantum systems are limited to quite restricted models of a single particle, single spin or harmonic oscillators (see e.g. [@petruccione; @haake; @alicki]).
In this paper we show that the generator of the master equation of a general quadratic system of $n$ interacting fermions which are coupled to a general set of Markovian baths, specified in terms of Lindblad operators which are linear in the fermionic variables - the so called quantum Liouville super-operator (or Liouvillean) - can be explicitly diagonalized in terms of $2n$ [*normal master modes*]{}, i.e. anticommuting super-operators which act on the Fock space of density operators. This can be understood as a complex (non-canonical) version of the Bogoliubov transformation [@lieb] lifted on the operator space, and has very powerful consequences: (i) The NESS of the master equation can be understood as the ‘ground state’ normal mode of the Liouvillean, whereas the long time relaxation rate is given by the eigenmode closest to the real axis. (ii) The covariance matrix of NESS can be computed explicitly in terms of the eigenvectors of $4n\times 4n$ antisymmetric complex matrix. It can be used to completely express physical observables in NESS, such as particle/spin densities, currents, etc. We demonstrate the power of this novel method by applying it to the problem of heat and spin transport far from equilibrium in nearest neighbor Heisenberg XY spin $1/2$ chains subject to a transverse magnetic field. As a result we reproduce [*ballistic transport*]{} in the [*integrable*]{} spatially homogeneous case (see e.g. [@zotos; @saito; @hartmann; @mejia05; @mejia07; @dhar; @prosenjpa] for related recent studies of quantum thermal conductivity in one dimension), and predict [*ideally insulating*]{} behaviour (at all temperatures) in a disordered case of spatially random interactions/field. Apart from obtaining numerical results which go by far beyond what was so far accessible by direct numerical solution of the many-particle Lindblad equation, either directly or by means of quantum trajectories [@petruccione], we also obtain two notable analytical results in the spatially homogeneous (non-disordered) case: (i) We compute the spectral gap of the Liouvillean i.e. the rate of of relaxation to the NESS and show that it scales with the inverse cube of the chain length. (ii) We construct [*evanescent*]{} normal master modes of the Liouvillean, for long chains, by which we explain quantitatively the exponential falloff of energy density or temperature profiles near the bath contacts.
The paper is organized as follows. In section \[sect:method\] we shall outline a general method for the diagonalization of the Liouvillean super-operator for finite quadratic open Fermi systems and an explicit construction of NESS. In section \[sect:trivia\] we illustrate the method by working out a simple example of a single fermion or a two level quantum system in a bath. In section \[sect:nontrivia\] we demonstrate the usefulness of the new method by applying it to quantum transport in XY spin chains. In section \[sect:conc\] we discuss possible alternative applications and generalizations of the method and reach some conclusions.
General method of solution for the Lindblad equation {#sect:method}
====================================================
The general master equation governing time evolution of the density matrix $\rho(t)$ of an open quantum system, preserving trace and positivity of $\rho$, can be written in the Lindblad form [@lindblad; @alicki] as (we set $\hbar=1$) $$\frac{{ {\rm d} }\rho }{{ {\rm d} }t} = {{\hat {\cal L}}}\rho :=
-{ {\rm i} }[H,\rho] + \sum_{\mu} \left(2 L_\mu \rho L_\mu^\dagger - \{L_\mu^\dagger L_\mu,\rho\} \right)
\label{eq:lind}$$ where $H$ is a Hermitian operator (Hamiltonian), $[x,y]:=xy-yx$, $\{x,y\}:=xy+yx$, and $L_\mu$ are arbitrary operators representing couplings to different baths (at possibly different values of thermodynamic potentials). We are now going to describe a general method of explicit solution of (\[eq:lind\]) for a [*quadratic*]{} system of $n$ fermions (or spins $1/2$) with [*linear*]{} bath operators $$\begin{aligned}
H &=& \sum_{j,k=1}^{2n} w_j H_{jk} w_k = {{\underline{w}}} \cdot {{\mathbf{H}}}\, {{\underline{w}}} \label{eq:hamil} \\
L_\mu &=& \sum_{j=1}^{2n} l_{\mu,j} w_j = {{\underline{l}}}_\mu \cdot {{\underline{w}}}\ \label{eq:lindb}\end{aligned}$$ where $w_j$, $j=1,2,\ldots,2n$, are abstract [*Hermitian*]{} Majorana operators satisfying the anti-commutation relations $$\{w_j,w_k\} = 2\delta_{j,k} \qquad j,k =1,2,\ldots, 2n$$ and generate a Clifford algebra. Thus, $2n \times 2n$ matrix ${{\mathbf{H}}}$ can be chosen to be antisymmetric ${{\mathbf{H}}}^T = -{{\mathbf{H}}}$. Throughout this paper ${{\underline{x}}}=(x_1,x_2,\ldots)^T$ will designate a vector (column) of appropriate scalar valued or operator valued symbols $x_k$.
Two notable examples, to which our formalism is immediately applicable, are: (i) canonical fermions $c_m$, $m=1,2,\ldots, n$, $$w_{2m-1}= c_m + c_m^\dagger \qquad
w_{2m} = { {\rm i} }(c_m- c^\dagger_m)
\label{eq:physfermi}$$ or (ii) spins $1/2$ with canonical Pauli operators $\vec{\sigma}_m$, $m=1,2,\ldots, n$, $$w_{2m-1} = \sigma^{{\rm x}}_m \prod_{m'<m} \sigma^{{\rm z}}_{m'} \qquad
w_{2m} = \sigma^{{\rm y}}_m \prod_{m'<m} \sigma^{{\rm z}}_{m'}
\label{eq:jordan}$$
Here we are not concerned with physical criteria for the validity of the so-called Markovian approximation under which eq. (\[eq:lind\]) is derived, so we shall make no assumptions on the smallness of the bath coupling constants $l_{\mu,j}$. We merely consider the Lindblad equation (\[eq:lind\]) as a possible parametrization of an important subset of [*Markovian*]{} completely positive quantum channels and demonstrate its complete solvability for quadratic systems. Note that generalization of our formalism to [*explicitly time dependent*]{} Hamiltonians $H(t)$ and Lindblad operators $L_\mu(t)$, generating more general and possibly non-Markovian open system dynamics, is straightforward. See e.g. [@wolf] for a discussion of [*Markovianity*]{}.
Fock space of operators
-----------------------
We begin by associating a Hilbert space structure $x \to {{\vert x \rangle}}$ to a linear $2^{2n}=4^n$ dimensional space ${\cal K}$ of operators, with a canonical basis ${{\vert P_{{{\underline{\alpha}}}} \rangle}}$ with $$P_{\alpha_1,\alpha_2,\ldots,\alpha_{2n}} := w_1^{\alpha_1}w_2^{\alpha_2}\cdots w_{2n}^{\alpha_{2n}} \qquad
\alpha_j\in\{ 0,1\}$$ [*orthonormal*]{} with respect to an inner product $${\langle x \vert y \rangle} = 2^{-n} \tr x^\dagger y$$ The form of the canonical basis of the operator space ${\cal K}$ suggests that it is just a usual Fock space with an unusual physical interpretation. Namely we can define the following set of $2n$ adjoint [*annihilation linear maps*]{} $\hat{c}_j$ over ${\cal K}$ $${ {\hat c} }_j {{\vert P_{{{\underline{\alpha}}}} \rangle}} = \delta_{\alpha_j,1} {{\vert w_j P_{{{\underline{\alpha}}}} \rangle}}
\label{eq:defanih}$$ and derive the actions of their Hermitian adjoints - the [*creation linear maps*]{} $\hat{c}^\dagger$, ${{\langle P_{{{\underline{\alpha}}}'} \vert}}{ {\hat c} }^\dagger_j{{\vert P_{{{\underline{\alpha}}}} \rangle}} =
{{\langle P_{{{\underline{\alpha}}}} \vert}}{ {\hat c} }_j{{\vert P_{{{\underline{\alpha}}}'} \rangle}}^* =
\delta_{\alpha'_j,1} {\langle P_{{{\underline{\alpha}}}} \vert w_j P_{{{\underline{\alpha}}}'} \rangle}^*=
\delta_{\alpha_j,0} {\langle P_{{{\underline{\alpha}}}'} \vert w_j P_{{{\underline{\alpha}}}} \rangle} $: $${ {\hat c} }^\dagger_j {{\vert P_{{{\underline{\alpha}}}} \rangle}} = \delta_{\alpha_j,0} {{\vert w_j P_{{{\underline{\alpha}}}} \rangle}}$$ Straightforward inspection then shows that they satisfy the canonical anticommutation relations $$\{{ {\hat c} }_j,{ {\hat c} }_k\} = 0 \qquad \{{ {\hat c} }_j,{ {\hat c} }_k^\dagger\} = \delta_{j,k} \qquad j,k=1,2,\ldots, 2n$$ The key is now to realize that the quantum Liouville map ${{\hat {\cal L}}}$ defined by eqs. (\[eq:lind\],\[eq:hamil\],\[eq:lindb\]) can be written as a quadratic form in [*adjoint Fermi maps*]{} ${ {\hat c} }_j,{ {\hat c} }^\dagger_j$ (or for short, [*a-fermions*]{}). [^1]\
\
First, we consider the unitary part of Liouvillean $${{\hat {\cal L}}}_0 \rho := -{ {\rm i} }[H,\rho]
\label{eq:exprunitary}$$ Since ${\cal K}$ is a Lie algebra, one defines the [*adjoint representation*]{} of a Lie derivative for an arbitrary $A\in {\cal K}$ back on ${\cal K}$ as, $\ad A {{\vert B \rangle}} := {{\vert [A,B] \rangle}}$. It is now straightforward to compute the action of a Lie derivative of a product of two Majorana operators on an arbitrary basis element $$\begin{aligned}
\ad w_j w_k {{\vert P_{{\underline{\alpha}}} \rangle}} &=& {{\vert w_j w_k P_{{\underline{\alpha}}} \rangle}} - {{\vert P_{{\underline{\alpha}}} w_j w_k \rangle}} = \nonumber \\
&=& 2 (\delta_{\alpha_j,1}\delta_{\alpha_k,0} + \delta_{\alpha_j,0}\delta_{\alpha_k,1}){{\vert w_j w_k P_{{\underline{\alpha}}} \rangle}} = \nonumber \\
&=& 2 ({ {\hat c} }_j { {\hat c} }^\dagger_k + { {\hat c} }^\dagger_j { {\hat c} }_k) {{\vert P_{{\underline{\alpha}}} \rangle}} =
2 ({ {\hat c} }^\dagger_j { {\hat c} }_k - { {\hat c} }^\dagger_k{ {\hat c} }_j) {{\vert P_{{\underline{\alpha}}} \rangle}}\end{aligned}$$ Extending this relation by linearity to an arbitrary element of ${\cal K}$, it follows that for an arbitrary quadratic Hamiltonian (\[eq:hamil\]) its Lie derivative has a very similar quadratic form in a-Fermi maps $${{\hat {\cal L}}}_0 = -{ {\rm i} }\ad H = -4{ {\rm i} }\sum_{j,k=1}^{2n} { {\hat c} }^\dagger_j H_{jk} { {\hat c} }_k =
-4{ {\rm i} }\, {{\underline{{ {\hat c} }}}}^\dagger\cdot {{\mathbf{H}}}\,{{\underline{{ {\hat c} }}}}
\label{eq:unitary}$$ It is worth stressing here that for an arbitrary (complex) matrix ${{\mathbf{H}}}$, ${{\hat {\cal L}}}_0$ (\[eq:unitary\]) conserves the total number of a-fermions ${{\hat {\cal N}}}:= \sum_{j} { {\hat c} }^\dagger_j { {\hat c} }_j = {{\underline{{ {\hat c} }}}}^\dagger\cdot{{\underline{{ {\hat c} }}}}$, namely $[{{\hat {\cal L}}}_0,{{\hat {\cal N}}}] = 0$.\
\
Second, we consider the action of the Lindblad maps $${{\hat {\cal L}}}_\mu \rho := 2 L_\mu \rho L_\mu^\dagger - \{L_\mu^\dagger L_\mu,\rho\} =
\sum_{j,k=1}^{2n} l_{\mu,j} l_{\mu,k}^* {{\hat {\cal L}}}_{j,k}\rho
\label{eq:exprlindb}$$ where we write ${{\hat {\cal L}}}_{j,k}\rho := 2 w_j \rho w_k - w_k w_j \rho - \rho w_k w_j$. Again we proceed by computing the actions of ${{\hat {\cal L}}}_{j,k}$ on elements of the canonical basis of operator Fock space ${\cal K}$. In order to do so, it is crucial to observe that the question whether $w_j$ commutes or anticommutes with $P_{{{\underline{\alpha}}}}$ depends on the number of a-fermions $|{{\underline{\alpha}}}|:=\sum_{k=1}^{2n} \alpha_k$ in ${{\vert P_{{{\underline{\alpha}}}} \rangle}}$, namely $P_{{{\underline{\alpha}}}}w_j = (-1)^{|{{\underline{\alpha}}}|+\alpha_j} w_j P_{{{\underline{\alpha}}}}$, and hence $${{\hat {\cal L}}}_{j,k} {{\vert P_{{{\underline{\alpha}}}} \rangle}}
= \left[2(-1)^{|{{\underline{\alpha}}}|+\alpha_k} w_j w_k - w_k w_j - (-1)^{\alpha_j+\alpha_k} w_k w_j\right]
{{\vert P_{{{\underline{\alpha}}}} \rangle}}
\label{eq:Ljk1}$$ Observing that $$\begin{aligned}
\phantom{(-1)^{\alpha_j}}{{\vert w_j P_{{{\underline{\alpha}}}} \rangle}} &=& ({ {\hat c} }^\dagger_j + { {\hat c} }_j){{\vert P_{{{\underline{\alpha}}}} \rangle}} \\
(-1)^{\alpha_j} {{\vert w_j P_{{{\underline{\alpha}}}} \rangle}} &=& ({ {\hat c} }^\dagger_j - { {\hat c} }_j){{\vert P_{{{\underline{\alpha}}}} \rangle}} \\
(-1)^{|{{\underline{\alpha}}}|} {{\vert P_{{{\underline{\alpha}}}} \rangle}} &=& \exp({ {\rm i} }\pi{{\hat {\cal N}}}) {{\vert P_{{{\underline{\alpha}}}} \rangle}}\end{aligned}$$ one derives from (\[eq:Ljk1\]) the general expression for ${{\hat {\cal L}}}_{j,k}$ $$\begin{aligned}
{{\hat {\cal L}}}_{j,k} &=& \left(\hat{\one}+\exp({ {\rm i} }\pi{{\hat {\cal N}}})\right)\left(2 { {\hat c} }^\dagger_j { {\hat c} }^\dagger_k - { {\hat c} }^\dagger_j { {\hat c} }_k -
{ {\hat c} }^\dagger_k { {\hat c} }_j\right) \nonumber \\
&+& \left(\hat{\one}-\exp({ {\rm i} }\pi{{\hat {\cal N}}})\right)\left(2 { {\hat c} }_j { {\hat c} }_k - { {\hat c} }_j { {\hat c} }^\dagger_k -
{ {\hat c} }_k { {\hat c} }^\dagger_j\right)\end{aligned}$$ Obviously, the maps ${{\hat {\cal L}}}_{j,k}$, and hence also the total Lindblad part of Liouvillean $\sum_\mu{{\hat {\cal L}}}_\mu$, do not conserve the number of a-fermions. But they conserve its [*parity*]{} i.e. the product of any two creation/annihilation a-Fermi maps commutes with the parity operation ${{\hat {\cal P}}}= \exp({ {\rm i} }\pi{{\hat {\cal N}}})$, with respect to which the operator space can be decomposed into a direct sum ${\cal K} = {\cal K}^+ \oplus {\cal K}^-$, and even/odd operator spaces are orthogonally projected as ${\cal K}^{\pm} = (\hat{\one} \pm \exp({ {\rm i} }\pi{{\hat {\cal N}}})){\cal K}$. Thus the positive parity subspace ${\cal K}^+$ is a linear space spanned by ${{\vert P_{{{\underline{\alpha}}}} \rangle}}$ with [*even*]{} $|{{\underline{\alpha}}}|$. All the maps ${{\hat {\cal L}}}_{j,k}$ now act separately on ${\cal K}^\pm$, ${{\hat {\cal L}}}_{j,k} {\cal K}^\pm
\subseteq {\cal K}^\pm$. For example, the maps defined on even parity subspace are indeed quadratic in a-fermions $${{\hat {\cal L}}}_{j,k}\vert_{{\cal K}^+} = 4 { {\hat c} }^\dagger_j { {\hat c} }^\dagger_k - 2 { {\hat c} }^\dagger_j { {\hat c} }_k -
2 { {\hat c} }^\dagger_k { {\hat c} }_j
\label{eq:lindbpart}$$ In this paper we shall focus on physical observables which are products of an [*even*]{} number of Majorana fermions – operators $w_j$ – so we shall in the following discuss only Liouville dynamics on the subspace ${\cal K}^+$. The extension to the dynamics of [*odd*]{} observables should be straightforward.
Putting the results (\[eq:exprunitary\],\[eq:unitary\],\[eq:exprlindb\],\[eq:lindbpart\]) together we arrive at the final compact quadratic representation of the Liouvillean ${{\hat {\cal L}}}_+ := {{\hat {\cal L}}}\vert_{{\cal K}^+}$ $${{\hat {\cal L}}}_+ = -2\, {{\underline{{ {\hat c} }}}}^\dagger\cdot(2{ {\rm i} }{{\mathbf{H}}} + {{\mathbf{M}}} + {{\mathbf{M}}}^T)\,{{\underline{{ {\hat c} }}}}
+ 2\,{{\underline{{ {\hat c} }}}}^\dagger\cdot ({{\mathbf{M}}}-{{\mathbf{M}}}^T)\,{{\underline{{ {\hat c} }}}}^\dagger
\label{eq:liouv1}$$ where ${{\mathbf{M}}}$ is a complex Hermitian matrix parametrizing the Lindblad operators $$M_{jk} = \sum_{\mu} l_{\mu,j} l_{\mu,k}^*$$
Reduction to normal master modes
--------------------------------
Next we want to show that the representation (\[eq:liouv1\]) allows us to reduce it further by a linear transformation of the set of maps $\{{ {\hat c} }_j,{ {\hat c} }_j^\dagger; j=1,2,\ldots,2n\}$ to [*normal master modes*]{} (NMM) in terms of which the complete spectrum of the Liouvillean, as well as its eigenvectors, can be explicitly constructed; in particular the zero-mode eigenvector which is just the physically relevant NESS.
In fact we proceed in analogy to Lieb [*et al.*]{} [@lieb] and define $4n$ adjoint Hermitian Majorana maps ${ {\hat a} }_r = { {\hat a} }_r^\dagger$, $r=1,2,\ldots,4n$: $${ {\hat a} }_{2j-1} = \frac{1}{\sqrt{2}}({ {\hat c} }_j + { {\hat c} }^\dagger_j) \qquad
{ {\hat a} }_{2j} = \frac{{ {\rm i} }}{\sqrt{2}}({ {\hat c} }_j - { {\hat c} }^\dagger_j)
\label{eq:aexpr}$$ satisfying the anti-commutation relations $$\{{ {\hat a} }_r,{ {\hat a} }_s\} = \delta_{r,s}
\label{eq:clifa}$$ in terms of which the Liouvillean (\[eq:liouv1\]) can be rewritten as $${{\hat {\cal L}}}_+ = {{\underline{{ {\hat a} }}}}\cdot{{\mathbf{A}}}\,{{\underline{{ {\hat a} }}}} - A_0 \hat{\one}
\label{eq:liouv2}$$ where ${{\mathbf{A}}}$ is an antisymmetric complex $4n\times 4n$ matrix with entries $$\begin{aligned}
A_{2j-1,2k-1} &=&-2{ {\rm i} }H_{jk}-M_{jk}+M_{kj} \nonumber \\
A_{2j-1,2k} &=& \phantom{wj}2{ {\rm i} }M_{kj} \nonumber \\
A_{2j,2k-1} &=& -2{ {\rm i} }M_{jk} \nonumber \\
A_{2j,2k} &=&-2{ {\rm i} }H_{jk}+M_{jk}-M_{kj}
\label{eq:explA}\end{aligned}$$ $\hat{\one}$ is an identity map over ${\cal K}$ and $A_0$ is a scalar $$A_0 = 2\sum_{j=1}^{2n} M_{jj} = 2\tr{{\mathbf{M}}}$$ Obviously, the [*bi-linear*]{} Liouvillean (\[eq:liouv2\]) cannot be brought to a normal form with a linear [*canonical*]{} transformation since the matrix ${{\mathbf{A}}}$ – which shall in the following be referred to as a [*shape matrix*]{} of Liouvillean – is not anti-Hermitian like in Hamiltonian systems. So we should proceed in more general terms.
We first recall few facts about complex antisymmetric matrices of even dimension. If ${{\underline{v}}}$ is a [*right*]{} eigenvector ${{\mathbf{A}}}{{\underline{v}}} = \beta {{\underline{v}}}$ with complex eigenvalue $\beta$, then ${{\underline{v}}}$ is also a [*left*]{} eigenvector with eigenvalue $-\beta$, ${{\mathbf{A}}}^T {{\underline{v}}} = -{{\mathbf{A}}} {{\underline{v}}} = -\beta {{\underline{v}}}$. Hence eigenvalues always come in pairs $\beta,-\beta$. Let as assume that ${{\mathbf{A}}}$ can be [*diagonalized*]{}[^2], i.e. there exist $4n$ linearly independent vectors ${{\underline{v}}}_r,r=1,\ldots,4n$ with the corresponding eigenvalues $\beta_1,-\beta_1,\beta_2,-\beta_2,\ldots,\beta_{2n},-\beta_{2n}$, $${{\mathbf{A}}}{{\underline{v}}}_{2j-1} = \beta_j {{\underline{v}}}_{2j-1} \qquad
{{\mathbf{A}}}{{\underline{v}}}_{2j} = -\beta_j {{\underline{v}}}_{2j}
\label{eq:eigv}$$ ordered such that $\re\beta_1\ge \re\beta_2\ge \ldots \ge \re\beta_{2n} \ge 0.$ The $2n$ complex numbers $\beta_j$ shall be referred to as [*rapidities*]{}. It is easy to check that we can always choose and normalize ${{\underline{v}}}_r$ such that[^3] $${{\underline{v}}}_r \cdot {{\underline{v}}}_s = J_{rs} \quad \textrm{where} \quad
{{\mathbf{J}}} := \sigma^{{{\rm x}}}\otimes {{\mathbf{I}}}_{2n} =
\pmatrix{0 & 1 & 0 & 0 & \cdots \cr
1 & 0 & 0 & 0 & \cdots \cr
0 & 0 & 0 & 1 & \cdots \cr
0 & 0 & 1 & 0 & \cdots \cr
\vdots &\vdots & \vdots & \vdots& \ddots \cr}
\label{eq:norm}$$ Let ${{\mathbf{V}}}$ be $4n\times 4n$ matrix whose $r$th row is given by ${{\underline{v}}}_r$, $V_{rs} := v_{r,s}$. Then eqs. (\[eq:eigv\],\[eq:norm\]) rewrite as $$\begin{aligned}
{{\mathbf{A}}}{{\mathbf{V}}}^T &=& {{\mathbf{V}}}^T{{\mathbf{D}}}\quad \textrm{where}\quad {{\mathbf{D}}} := {\rm diag}\{\beta_1,-\beta_1,
\beta_2,-\beta_2,\ldots,\beta_{2n},-\beta_{2n}\} \label{eq:AVVD} \\
{{\mathbf{V}}} {{\mathbf{V}}}^T &=& {{\mathbf{J}}}
\label{eq:VVJ}\end{aligned}$$ Expressing ${{\mathbf{V}}}^T$ in terms of (\[eq:VVJ\]) and plugging the result into eq. (\[eq:AVVD\]) we arrive at a very convenient canonical form of a generic complex antisymmetric matrix ${{\mathbf{A}}}$ $${{\mathbf{A}}} = {{\mathbf{V}}}^T {{\mathbf{\Lambda}}} {{\mathbf{V}}}\quad\textrm{where}\quad
{{\mathbf{\Lambda}}} = {{\mathbf{D}}}{{\mathbf{J}}} =
\pmatrix{0 & \beta_1 & 0 & 0 & \cdots \cr
-\beta_1 & 0 & 0 & 0 & \cdots \cr
0 & 0 & 0 & \beta_2 & \cdots \cr
0 & 0 & -\beta_2 & 0 & \cdots \cr
\vdots &\vdots & \vdots & \vdots& \ddots \cr}
\label{eq:canform}$$
Now we apply decomposition (\[eq:canform\]) to the Liouvillean (\[eq:liouv2\]) $${{\hat {\cal L}}}_+ = {{\underline{{ {\hat a} }}}}\cdot {{\mathbf{V}}}^T {{\mathbf{\Lambda}}}{{\mathbf{V}}}{{\underline{{ {\hat a} }}}} - A_0\hat{\one} =
({{\mathbf{V}}}{{\underline{{ {\hat a} }}}})\cdot{{\mathbf{\Lambda}}}({{\mathbf{V}}}{{\underline{{ {\hat a} }}}}) - A_0\hat{\one}
\label{eq:liouv4}$$ Let us define the NMM maps ${{\underline{{ {\hat b} }}}} := ({ {\hat b} }_1,{ {\hat b} }'_1,{ {\hat b} }_2,{ {\hat b} }'_2,\ldots,{ {\hat b} }_{2n},{ {\hat b} }'_{2n}) :=
{{\mathbf{V}}}{{\underline{{ {\hat a} }}}}$ or $${ {\hat b} }_j = {{\underline{v}}}_{2j-1}\cdot{{\underline{{ {\hat a} }}}} \qquad
{ {\hat b} }'_j = {{\underline{v}}}_{2j}\cdot{{\underline{{ {\hat a} }}}}
\label{eq:bexpr}$$ We note that due to (\[eq:clifa\],\[eq:norm\]) NMM maps satisfy [*almost-canonical*]{} anti-commutation relations $$\{{ {\hat b} }_j,{ {\hat b} }_k\} = 0 \qquad \{{ {\hat b} }_j,{ {\hat b} }'_k\} = \delta_{j,k} \qquad \{{ {\hat b} }'_j,{ {\hat b} }'_k\}=0
\label{acar}$$ namely ${ {\hat b} }_j$ could be interpreted as annihilation map and ${ {\hat b} }'_j$ as a creation map of $j$th NMM, but we should note that ${ {\hat b} }'_j$ is in general [*not*]{} the Hermitian adjoint of ${ {\hat b} }_j$ [@thomas]. In terms of NMM the Liouvillean (\[eq:liouv4\]) now achieves a very convenient normal form $${{\hat {\cal L}}}_+ = -2\sum_{j=1}^{2n} \beta_j { {\hat b} }'_j { {\hat b} }_j - B_0 \hat{\one}
\label{eq:liouv3}$$ where $B_0 = A_0 - \sum_{j=1}^{2n}\beta_j$. We shall later show that the constant $B_0$ is in fact equal to $0$.
Non-equilibrium steady states and a complete spectrum of the Liouvillean
------------------------------------------------------------------------
The Liouvillean can always be represented in terms of a large but finite $4^n \times 4^n$ matrix. We shall now outline the procedure of complete construction of its spectrum in terms of NMM which are easy to calculate in terms of diagonalization of $4n \times 4n$ matrix ${{\mathbf{A}}}$ as described in the previous subsection.
We proceed by constructing the Liouvillean ‘vacuum’. From the representation (\[eq:liouv1\]) it follows immediately that ${{\langle 1 \vert}} = {{\langle P_{(0,0\ldots,0)} \vert}}$ is left-annihilated by ${{\hat {\cal L}}}_+$, ${{\langle 1 \vert}}{{\hat {\cal L}}}_+ = 0$, or equivalently ${{\hat {\cal L}}}_+^\dagger {{\vert 1 \rangle}} = 0$. So we have just shown that $0$ is always an eigenvalue of ${{\hat {\cal L}}}_+$, hence there should also exist the corresponding right eigenvector $\ness$, normalized as ${\langle 1 \vert {\rm NESS} \rangle} = \tr \rho_{\rm NESS} = 1$, which represents physical NESS, i.e. stationary solutions of the Lindblad equation (\[eq:lind\]) $${{\hat {\cal L}}}_+\ness = 0$$ Let us define NMM number maps as ${{\hat {\cal N}}}_j := { {\hat b} }'_j { {\hat b} }_j$. From eqs. (\[acar\]) it follows that ${{\hat {\cal N}}}_j$ satisfy a projection property ${{\hat {\cal N}}}_j^2 = {{\hat {\cal N}}}_j$, so they are diagonalizable since no nontrivial Jordan block could satisfy the projection property. Furthermore, ${{\hat {\cal N}}}_j$ are mutually commuting $[{{\hat {\cal N}}}_j,{{\hat {\cal N}}}_k]=0$, so they can be simultaneously diagonalized and there should exist a vacuum state on which all ${{\hat {\cal N}}}_j$ have value 0. It follows from the stability of completely positive evolution (\[eq:lind\]) that all eigenvalues $\lambda$ of ${{\hat {\cal L}}}_+$ should obey $\re\lambda \le 0$, and since by assumption $\re\beta_j \ge 0$, ${{\langle 1 \vert}}$ and $\ness$ should be the left and right vacua which are simultaneously annihilated by NMM maps $${{\langle 1 \vert}} { {\hat b} }'_j = 0 \qquad { {\hat b} }_j \ness = 0
\label{eq:anihb}$$ and hence also ${{\hat {\cal N}}}_j\ness = 0$. Thus we have also shown that the NMM representation (\[eq:liouv3\]) is only consistent if $B_0=0$ so we find an interesting sum rule for rapidities $$\sum_{j=1}^{2n} \beta_j = 2\tr{{\mathbf{M}}}$$
The complete excitation spectrum and the corresponding left/right eigenvectors of the Liouvillean are given in terms of a sequence of $2n$ binary integers (NMM occupation numbers) ${{\underline{\nu}}}=(\nu_1,\nu_2,\ldots,\nu_{2n})$, $\nu_j\in\{0,1\}$, $${{\langle \Theta^{\rm L}_{{\underline{\nu}}} \vert}}{{\hat {\cal L}}}_+ = \lambda_{{{\underline{\nu}}}} {{\langle \Theta^{\rm L}_{{\underline{\nu}}} \vert}}
\qquad
{{\hat {\cal L}}}_+ {{\vert \Theta^{\rm R}_{{\underline{\nu}}} \rangle}} = \lambda_{{{\underline{\nu}}}} {{\vert \Theta^{\rm R}_{{\underline{\nu}}} \rangle}}$$ namely $$\begin{aligned}
\lambda_{{\underline{\nu}}} &=& -2\sum_{j=1}^{2n} \beta_j \nu_j \label{eq:eval}\\
{{\langle \Theta^{\rm L}_{{\underline{\nu}}} \vert}} &=& {{\langle 1 \vert}}{{ {\hat b} }_{2n}}^{\nu_{2n}}\cdots {{ {\hat b} }_2}^{\nu_2}\,{{ {\hat b} }_1}^{\nu_1}
\qquad
{{\vert \Theta^{\rm R}_{{\underline{\nu}}} \rangle}} = {{ {\hat b} }_1}^{'\nu_1}\,{{ {\hat b} }_2}^{'\nu_2} \cdots {{ {\hat b} }_{2n}}^{' \nu_{2n}} \ness \label{eq:evec}\end{aligned}$$ where by construction, left and right eigenvectors satisfy the bi-orthonormality relation ${\langle \Theta^{\rm L}_{{\underline{\nu'}}} \vert \Theta^{\rm R}_{{\underline{\nu}}} \rangle} = \delta_{{{\underline{\nu}}}',{{\underline{\nu}}}}$.
The main general results: uniqueness of NESS, rate of relaxation to NESS, and expectation values of physical observables
------------------------------------------------------------------------------------------------------------------------
Given a [*physical observable*]{} $X\in {\cal K}^+$ and an [*arbitrary*]{} initial state with a density operator $\rho_0 \in {\cal K}$, the time dependent expectation value of $X$ can be written in terms of the spectral resolution of the Liouvillean, $$\exp(t {{\hat {\cal L}}}_+) = \sum_{{{\underline{\nu}}}} \exp(t \lambda_{{{\underline{\nu}}}})
{{\vert \Theta^{\rm R}_{{{\underline{\nu}}}} \rangle}}{{\langle \Theta^{\rm L}_{{{\underline{\nu}}}} \vert}}
\label{eq:propag}$$ namely $${{\langle X(t)\rangle}} = \tr X\rho(t) = \tr\!\!\left[X\exp(t{{\hat {\cal L}}}_+)\rho_0\right] =
\sum_{{{\underline{\nu}}}} \exp(t\lambda_{{{\underline{\nu}}}}){{\langle \Theta^{\rm L}_{{{\underline{\nu}}}} \vert}}\rho_0 X{{\vert \Theta^{\rm R}_{{{\underline{\nu}}}} \rangle}}
\label{eq:timedeptexp}$$ We remind the reader that ${{\hat {\cal L}}}_+$ correctly represents physical Liouvillean only on the subspace ${\cal K}^+$ of operators with [*even*]{} number of a-fermions. However, since the dynamics is closed on ${\cal K}^+$ and test physical observable $X$ also belongs to ${\cal K}^+$ it follows that the component of $\rho_0$ from ${\cal K}^-$ does not contribute to the expectation value ${{\langle X(t)\rangle}}$.
Given the exact and explicit constructions developed in this section we can now make the following rigorous and useful conclusions, assuming throughout that Liouvillean shape matrix ${{\mathbf{A}}}$ is diagonalizable:\
\
[**Theorem 1:**]{} NESS of Lindblad equation (\[eq:lind\]) is [*unique*]{} if and only if the rapidity spectrum $\{\beta_j\}$ does not contain $0$, in our ordering convention, if $\beta_{2n} \ne 0$. In the opposite case, if we have $d \ge 1$ vanishing rapidities, then we have a $2^d$ dimensional convex set of non-equilibrium steady states which can be spanned with ${{\vert \Theta^{\rm R}_{(0,\ldots,0,\nu_{1},\ldots,\nu_{d})} \rangle}}$.\
\
[**Theorem 2:**]{} An arbitrary initial state with a density operator $\rho_0 \in {\cal K}$ converges with time to NESS if and only if all rapidities have [*strictly positive*]{} real parts, $\re \beta_j > 0$. Then, the rate of exponential relaxation to NESS is given by the [*spectral gap*]{} $\Delta$ of the Liouvillean which equals $\Delta=2\re\beta_{2n}$.\
\
[**Theorem 3:**]{} Assume that the rapidity spectrum does not contain 0, i.e. $\beta_{2n}\ne 0$. Then the expectation value of any quadratic observable $w_j w_k$ in a (unique) NESS can be explicitly computed as $$\begin{aligned}
{{\langle w_j w_k\rangle}}_{\rm NESS} &=& \delta_{j,k} + {{\langle 1 \vert}}{ {\hat c} }_j { {\hat c} }_k \ness = \label{eq:pairwise} \\
&=& \delta_{j,k} + \frac{1}{2}\sum_{m=1}^{2n} \biggl(
v_{2m,2j-1} v_{2m-1,2k-1} - v_{2m,2j} v_{2m-1,2k} \nonumber \\
&& \qquad\qquad- { {\rm i} }v_{2m,2j} v_{2m-1,2k-1} - { {\rm i} }v_{2m,2j-1} v_{2m-1,2k}\biggr)
\label{eq:pairwise2}\end{aligned}$$
The statements of theorems 1 and 2 simply follow from exact and explicit spectral decomposition (\[eq:eval\],\[eq:evec\],\[eq:propag\]).
The proof of theorem 3 is also straightforward: The first expression (\[eq:pairwise\]) follows from the definition of the annihilation maps (\[eq:defanih\]) and the explicit representation of the density operator of NESS, $\rho_{\rm NESS}$, in the canonical basis $P_{{{\underline{\alpha}}}}$. The second, very useful equality (\[eq:pairwise2\]) is then obtained by expressing ${ {\hat c} }_j$ thru (\[eq:aexpr\]) in terms of NMM maps (\[eq:bexpr\]) and using the annihilation relations (\[eq:anihb\]).
The quadratic correlator of theorem 3 covers many physically interesting observables such as densities or currents. However if one needs expectation values of more general observables, e.g. an expectation value of a high order monomial ${{\langle P_{{{\underline{\alpha}}}}\rangle}}_{\rm NESS} =
{{\langle 1 \vert}}{ {\hat c} }^{\alpha_1}_{1}{ {\hat c} }^{\alpha_2}_2\cdots{ {\hat c} }^{\alpha_{2n}}_{2n}\ness
$, then one may use a [*Wick theorem*]{} and rewrite it as a sum of products of pair-wise contractions (\[eq:pairwise\]).
Trivial example: A single fermion in a bath {#sect:trivia}
===========================================
In order to illustrate the method and demonstrate convenience of the results derived in the previous section we first work out a simple example of a single fermion $n=1$ (or equivalently, an arbitrary qubit, a two-level quantum system), in a thermal bath. We take the most general single fermion Hamiltonian $H = -{ {\rm i} }h w_1 w_2 + {\rm const} =
2h c^\dagger c + {\rm const}'$ and the following bath operators (see e.g. [@haake; @wich]) $$L_1 = \frac{1}{2}\sqrt{\Gamma_1} (w_1 - { {\rm i} }w_2) =
\sqrt{\Gamma_1} c \qquad
L_2 = \frac{1}{2}\sqrt{\Gamma_2} (w_1 + { {\rm i} }w_2) =
\sqrt{\Gamma_2} c^\dagger
\label{eq:canlind}$$ where the ratio of coupling constants determine the bath temperature $T$, $\Gamma_2/\Gamma_1 = \exp(-2h/T)$. Leaving out the details of a straightforward calculation, simply following the steps of the previous section, we arrive at the following shape matrix of the Liouvillean (\[eq:liouv2\]) $${{\mathbf{A}}} = -h{{\mathbf{R}}} + {{\mathbf{B}}}_{\Gamma_+,\Gamma_-} \qquad A_0 = \Gamma_+$$ where $${{\mathbf{R}}} := \pmatrix{ 0 & 0 & 1 & 0 \cr
0 & 0 & 0 & 1 \cr
\!\!-1 & 0 & 0 & 0 \cr
0 &\!\!\!-1 & 0 & 0}
\quad
{{\mathbf{B}}}_{\Gamma_+,\Gamma_-} :=
\pmatrix{
0 & \frac{{ {\rm i} }}{2} \Gamma_+ &\!\!\!-\frac{{ {\rm i} }}{2}\Gamma_- & \frac{1}{2}\Gamma_- \cr
\!\!-\frac{{ {\rm i} }}{2}\Gamma_+ & 0 & \frac{1}{2}\Gamma_- & \frac{{ {\rm i} }}{2}\Gamma_- \cr
\frac{{ {\rm i} }}{2}\Gamma_- &\!\!\!-\frac{1}{2}\Gamma_- & 0 & \frac{{ {\rm i} }}{2}\Gamma_+ \cr
\!\!-\frac{1}{2}\Gamma_- &\!\!\!-\frac{{ {\rm i} }}{2}\Gamma_- &\!\!\!-\frac{{ {\rm i} }}{2}\Gamma_+ & 0}
\label{eq:canbath}$$ and $\Gamma_\pm := \Gamma_2 \pm \Gamma_1$. Further, we compute NMM rapidities $\beta_{1,2} = \frac{1}{2}\Gamma_+ \pm { {\rm i} }h$ and the eigenvector matrix $${{\mathbf{V}}} = \pmatrix{
\frac{\Gamma_-}{\Gamma_+} - 1 & { {\rm i} }\frac{\Gamma_-}{\Gamma_+} + { {\rm i} }&
-{ {\rm i} }\frac{\Gamma_-}{\Gamma_+} + { {\rm i} }& \frac{\Gamma_-}{\Gamma_+} + 1 \cr
-\frac{1}{4} & -\frac{{ {\rm i} }}{4} & -\frac{{ {\rm i} }}{4} & \frac{1}{4} \cr
\frac{\Gamma_-}{\Gamma_+} + 1 & { {\rm i} }\frac{\Gamma_-}{\Gamma_+} - { {\rm i} }&
{ {\rm i} }\frac{\Gamma_-}{\Gamma_+} + { {\rm i} }& -\frac{\Gamma_-}{\Gamma_+} + 1 \cr
\frac{1}{4} & \frac{{ {\rm i} }}{4} & -\frac{{ {\rm i} }}{4} & \frac{1}{4}}\qquad$$ Then, using theorem 3 we compute the expectation value of occupation number ${{\langle c^\dagger c\rangle}} = \frac{1}{2} - \frac{{ {\rm i} }}{2} {{\langle w_1 w_2\rangle}} = \Gamma_2/(\Gamma_1 + \Gamma_2)$ which is what we expect in canonical equilibrium.
Non-trivial example: transport in quantum spin chains {#sect:nontrivia}
=====================================================
Here we work out a physically more interesting example, namely we construct NESS for the magnetic and heat transport of a Heisenberg XY spin $1/2$ chain, with arbitrary – either homogeneous or positionally dependent (e.g. disordered) – nearest neighbour interaction $$H =
\sum_{m=1}^{n-1} \left( J^{{\rm x}}_m \sigma^{{\rm x}}_m \sigma^{{\rm x}}_{m+1} + J^{{\rm y}}_m \sigma^{{\rm y}}_m \sigma^{{\rm y}}_{m+1}\right)
+ \sum_{m=1}^n h_m \sigma^{{\rm z}}_m
\label{eq:hamsc}$$ which is coupled to [*two*]{} thermal/magnetic baths [*at the ends*]{} of the chain, generated by two pairs of canonical Lindblad operators [@wich] (similar to (\[eq:canlind\])) $$\begin{aligned}
L_1 &=& \frac{1}{2}\sqrt{\Gamma_1^{\rm L}} \sigma^{-}_1 \qquad
L_3 = \frac{1}{2}\sqrt{\Gamma_1^{\rm R}} \sigma^{-}_n \nonumber \\
L_2 &=& \frac{1}{2}\sqrt{\Gamma_2^{\rm L}} \sigma^{+}_1 \qquad
L_4 = \frac{1}{2}\sqrt{\Gamma_2^{\rm R}} \sigma^{+}_n
\label{eq:bathsc}\end{aligned}$$ where $\sigma^{\pm}_m = \sigma^{{\rm x}}_m \pm { {\rm i} }\sigma^{{\rm y}}_m$ and $\Gamma^{{\rm L},{\rm R}}_{1,2}$ are positive coupling constants related to bath temperatures/magnetizations, for example if spins were non-interacting the bath temperatures $T_{{\rm L},{\rm R}}$ would be given with $\Gamma^{{\rm L},{\rm R}}_2/\Gamma^{{\rm L},{\rm R}}_1 =
\exp(-2h_{\rm 1,n}/T_{{\rm L},{\rm R}})$.
Applying the inverse of Jordan-Wigner transformation (\[eq:jordan\]), $\sigma^{{\rm x}}_m = (-{ {\rm i} })^{m-1}\prod_{j=1}^{2m-1} w_j$, $\sigma^{{\rm y}}_m = (-{ {\rm i} })^{m-1}(\prod_{j=1}^{2m-2} w_j)w_{2m}$, we rewrite (\[eq:hamsc\],\[eq:bathsc\]) in terms of Majorana fermions $$\begin{aligned}
H &=& -{ {\rm i} }\sum_{m=1}^{n-1}\left(J^{{\rm x}}_m w_{2m} w_{2m+1} - J^{{\rm y}}_m w_{2m-1}w_{2m+2}\right)
-{ {\rm i} }\sum_{m=1}^n h_m w_{2m-1}w_{2m} \\
L_1 &=& \frac{1}{2}\sqrt{\Gamma_1^{\rm L}} (w_1 - { {\rm i} }w_2)
\qquad
L_3 = -\frac{(-{ {\rm i} })^n}{2}\sqrt{\Gamma_1^{\rm R}} (w_{2n-1} - { {\rm i} }w_{2n})W \nonumber \\
L_2 &=& \frac{1}{2}\sqrt{\Gamma_2^{\rm L}} (w_1 + { {\rm i} }w_2)
\qquad
L_4 = -\frac{(-{ {\rm i} })^n}{2}\sqrt{\Gamma_2^{\rm R}} (w_{2n-1} + { {\rm i} }w_{2n})W\end{aligned}$$ where $W=w_{1}w_{2}\cdots w_{2n}$ is a Casimir operator which commutes with all the elements of the Clifford algebra generated by $w_j$ and squares to unity $W^2=1$. Therefore, $W$ [*does not affect*]{} the action of bath operators (\[eq:exprlindb\]) where $L_\mu$ enter quadratically, so we find $$\begin{aligned}
{{\hat {\cal L}}}_1 + {{\hat {\cal L}}}_2 &=& -\Gamma_+^{\rm L} ({ {\hat c} }^\dagger_1{ {\hat c} }_1 + { {\hat c} }^\dagger_2{ {\hat c} }_2) -
2{ {\rm i} }\Gamma_-^{\rm L} { {\hat c} }^\dagger_1{ {\hat c} }^\dagger_2 \nonumber \\
{{\hat {\cal L}}}_3 + {{\hat {\cal L}}}_4 &=& -\Gamma_+^{\rm R} ({ {\hat c} }^\dagger_{2n-1}{ {\hat c} }_{2n-1} + { {\hat c} }^\dagger_{2n}{ {\hat c} }_{2n}) -
2{ {\rm i} }\Gamma_-^{\rm R} { {\hat c} }^\dagger_{2n-1}{ {\hat c} }^\dagger_{2n}\end{aligned}$$ leading to the bath shape matrix (\[eq:canbath\]) identical to the single fermion case (\[eq:canlind\]). Again, carefully following the steps of section \[sect:method\], we derive the Liouvillean in the form (\[eq:liouv2\]) with $4n\times 4n$ shape matrix, which we write in a [*block tridiagonal*]{} form in terms of $4\times 4$ matrices as $${{\mathbf{A}}}=
\pmatrix{
{{\mathbf{B}}}_{\rm L} - h_1 {{\mathbf{R}}} & {{\mathbf{R}}}_1 & {{\mathbf{0}}} & \cdots & {{\mathbf{0}}} \cr
-{{\mathbf{R}}}^T_1 & - h_2 {{\mathbf{R}}} & {{\mathbf{R}}}_2 & \ddots & {{\mathbf{0}}} \cr
{{\mathbf{0}}} & -{{\mathbf{R}}}^T_2 & -h_3 {{\mathbf{R}}} & & \vdots \cr
\vdots & \ddots & & \ddots & {{\mathbf{R}}}_{n-1} \cr
{{\mathbf{0}}} & {{\mathbf{0}}} & \cdots & -{{\mathbf{R}}}^T_{n-1} & {{\mathbf{B}}}_{\rm R} - h_n {{\mathbf{R}}} }
\label{eq:bigA}$$ and $A_0= \Gamma_+^{\rm L} + \Gamma_+^{\rm R}$, where ${{\mathbf{B}}}_{\rm L}:={{\mathbf{B}}}_{\Gamma_+^{\rm L},\Gamma_-^{\rm L}}$, ${{\mathbf{B}}}_{\rm R}:={{\mathbf{B}}}_{\Gamma_+^{\rm R},\Gamma_-^{\rm R}}$ (in terms of (\[eq:canbath\])), with $\Gamma_{\pm}^{{\rm L},{\rm R}} := \Gamma_2^{{\rm L},{\rm R}}\pm\Gamma_1^{{\rm L},{\rm R}}$, and $${{\mathbf{R}}}_m := \pmatrix{ 0 & 0 & J^{{\rm y}}_m & 0 \cr
0 & 0 & 0 & J^{{\rm y}}_m \cr
-J_m^{{\rm x}}& 0 & 0 & 0 \cr
0 & -J_m^{{\rm x}}& 0 & 0 }$$ We are not able to perform a complete diagonalization of the antisymmetric matrix ${{\mathbf{A}}}$ (\[eq:bigA\]) of the general XY model analytically. For example, even in the spatially homogeneous case $J^{{{\rm x}},{{\rm y}}}_m \equiv J^{{{\rm x}},{{\rm y}}}, h_m\equiv h$ it is not possible to proceed like in the classical harmonic oscillator chain where the corresponding matrix is a sum of a Toeplitz and a bordered matrix [@rieder]. Namely, in our case ${{\mathbf{A}}}$ is a sum of a [*block Toeplitz*]{} and [*block bordered*]{} matrix and its explicit exact diagonalization remains an open problem. However, we stress that even relying on numerical diagonalization of ${{\mathbf{A}}}$ yielding a set of rapidities $\beta_j$ and properly normalized eigenvector matrix ${{\mathbf{V}}}$, represents a dramatic progress with respect to previously existing numerical methods which needed diagonalization of matrices which were exponentially large in $n$. We shall later derive some exact theoretical and analytical results, explaining results of exact numerical computations, in the special case of a [*homogeneous*]{} transverse Ising chain (subsection \[sect:ti\]), and the case of a [*disordered*]{} XY chain (subsection \[sect:disord\]) for which we shall relate NMM to the problem of Anderson localization in one dimension,
Let us continue by discussing transport observables in the spin chain whose expectation values in NESS are easy to calculate. Note that the bulk Hamiltonian (\[eq:hamsc\]) can be written in terms of the two-body [*energy density*]{} operator $$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
H_m = -{ {\rm i} }J^{{\rm x}}_m w_{2m} w_{2m+1} + { {\rm i} }J^{{\rm y}}_m w_{2m-1}w_{2m+2}
-\frac{{ {\rm i} }h_m}{2}w_{2m-1}w_{2m}-\frac{{ {\rm i} }h_{m+1}}{2}w_{2m+1}w_{2m+2}
\label{eq:hdens}$$ as $H=\sum_m H_m$. One can derive the local [*energy current*]{} $Q_m={ {\rm i} }[H_m,H_{m+1}]$ from the [*continuity equation*]{} $$({ {\rm d} }/{ {\rm d} }t){{\langle H_m\rangle}} = \tr H_m { {\rm d} }\rho/{ {\rm d} }t = {{\langle { {\rm i} }[H,H_m]\rangle}}
= -{{\langle Q_{m}\rangle}} + {{\langle Q_{m-1}\rangle}}
\label{eq:cont}$$ where $Q_m := { {\rm i} }[H_{m},H_{m+1}] $ $$\begin{aligned}
Q_m =
&&2{ {\rm i} }(2J^{{\rm y}}_m J^{{\rm x}}_{m+1} w_{2m-1}w_{2m+3} + 2J^{{\rm x}}_mJ^{{\rm y}}_{m+1} w_{2m}w_{2m+4} -
\nonumber\\
&&-J^{{\rm y}}_m h_{m+1} w_{2m-1}w_{2m+1} - J^{{\rm x}}_m h_{m+1} w_{2m}w_{2m+2}- \nonumber\\
&&-h_{m+1}J^{{\rm x}}_{m+1}w_{2m+1}w_{2m+3}-h_{m+1}J^{{\rm y}}_{m+1}w_{2m+2}w_{2m+4}) \label{eq:hcurr}\end{aligned}$$ The validity of the above continuity equation (\[eq:cont\]) depends on two assumptions only: (i) All Lindblad operators $L_\mu$ [*commute*]{} with the density $H_m$ in the [*bulk*]{}, $2 \le m \le n-2$ (second equality sign), and (ii) $[H_m,H_{m'}] = 0$ if $|m-m'| \ge 2$ (third equality sign).
Using eq. (\[eq:pairwise2\]) of theorem 3 we can now compute NESS expectation values of energy density $H_m$ and energy current $Q_m$, and also of somewhat simpler [*spin density*]{} $$\sigma^{{\rm z}}_m = -{ {\rm i} }w_{2m-1}w_{2m}
\label{eq:sdens}$$ and [*spin current*]{} $$S_m = \sigma^{{\rm x}}_{m}\sigma^{{\rm y}}_{m+1}-\sigma^{{\rm y}}_{m}\sigma^{{\rm x}}_{m+1} = -{ {\rm i} }w_{2m}w_{2m+2} -{ {\rm i} }w_{2m-1}w_{2m+1}
\label{eq:scurr}$$ which are all quadratic in $w_j$. Note, however, that the spin density satisfies continuity equation $({ {\rm d} }/{ {\rm d} }t){{\langle \sigma^{{\rm z}}_m\rangle}} = -{{\langle S_{m}\rangle}} + {{\langle S_{m-1}\rangle}}$ only in the isotropic case, when $J^{{\rm x}}_m \equiv J^{{\rm y}}_m$.
Homogeneous transverse Ising chain {#sect:ti}
----------------------------------
Here we limit our discussion to the spatially homogeneous case $J_n^{{{\rm x}},{{\rm y}}}\equiv J^{{{\rm x}},{{\rm y}}},h_n\equiv h$. We shall show that in this case the eigenvalue problem $${{\mathbf{A}}}{{\underline{v}}} =
\beta{{\underline{v}}}
\label{eq:evA}$$ for (\[eq:bigA\]) can be most easily and elegantly treated if formulated in terms of an abstract inelastic scattering problem in one dimension, with asymptotic solutions given in terms of free normal modes for the infinite translationally invariant chain ${{\underline{v}}}=(\ldots, {{\underline{u}}} \xi^{m-1},{{\underline{u}}} \xi^{m}, {{\underline{u}}} \xi^{m+1},\ldots)^T$, where $\xi$ is a complex [*quasi–momentum*]{} (Bloch) parameter and ${{\underline{u}}}$ is a 4-dimensional amplitude vector satisfying the condition $$(-{{\mathbf{R}}}^T_1 \xi^{-1} - h {{\mathbf{R}}} + {{\mathbf{R}}}_1 \xi - \beta {{\mathbf{I}}}_4) {{\underline{u}}} = 0
\label{eq:freemode}$$ and the baths playing the role of inelastic (absorbing) scatterers at the edges of a finite lattice. The ‘elastic’ (Hamiltonian) version of this trick has been used to compute temporal correlation functions in kicked Ising chain [@ProsenPTPS].
The singularity condition for the free mode equation (\[eq:freemode\]) results, for a general homogeneous $XY$ model, in eight [*master bands*]{} - different values of momenta $\xi$ for each value of the spectral parameter (rapidity) $\beta$. In order to simplify the discussion - which will still get rather involved - we shall in the following restrict ourselves to the transverse Ising model $J^{{\rm x}}= J, J^{{\rm y}}=0$. In this case we find just two master bands with simple dispersion relations $$\xi_{\pm}(\beta) = \frac{h^2 + J^2 + \beta^2 \pm \omega(\beta)}{2 h J}
\quad
\omega(\beta):=\sqrt{(h^2+J^2+\beta^2)^2 - (2 h J)^2}
\label{eq:dispersion}$$ but each band is doubly-degenerate, since the corresponding amplitude problem (\[eq:freemode\]) has two linearly independent solutions $$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
{{\underline{u}}}^\pm_{1}(\beta) = \pmatrix{
-h (h^2 - J^2+\beta^2 \pm \omega) \cr
0 \cr
\beta ( h^2 + J^2 + \beta^2 \pm \omega) \cr
0
} \quad
{{\underline{u}}}^\pm_{2}(\beta) = \pmatrix{
0 \cr
-h (h^2 - J^2+\beta^2 \pm \omega)\cr
0 \cr
\beta ( h^2 + J^2 + \beta^2) \pm \omega)
\label{eq:freemodes}
}$$ Naively speaking, $\xi_-$ represents left moving and $\xi_+$ right moving free modes, each having two possible polarizations. Note that $\xi_- \xi_+ = 1$. For a general complex $\beta$ we shall choose the branch of square root $\omega(\beta)$ (\[eq:dispersion\]) for which $|\xi_-| \le 1$. Let us now write the scattering problem on the [*left*]{} bath in terms of an ansatz $${{\underline{v}}} = \pmatrix{ {{\underline{u}}}_{\rm L} \cr
\psi_1^- {{\underline{u}}}^-_1 + \psi_2^- {{\underline{u}}}^-_2 + \psi_1^+ {{\underline{u}}}^+_1 + \psi_2^+ {{\underline{u}}}^+_2 \cr
(\psi_1^- {{\underline{u}}}^-_1 + \psi_2^- {{\underline{u}}}^-_2)\xi_- + (\psi_1^+ {{\underline{u}}}^+_1 + \psi_2^+ {{\underline{u}}}^+_2)\xi_+ \cr
(\psi_1^- {{\underline{u}}}^-_1 + \psi_2^- {{\underline{u}}}^-_2)\xi_-^2 + (\psi_1^+ {{\underline{u}}}^+_1 + \psi_2^+ {{\underline{u}}}^+_2)\xi_+^2 \cr
\vdots}
\label{eq:scatL}$$ where ${{\underline{u}}}_{\rm L}$ represents a $4$-dimensional vector of left-most eigenvector components, $\psi^-_{1,2}$ are the amplitudes of (known) incident free modes, and $\psi^+_{1,2}$ are the amplitudes of the scattered, outgoing free modes. Plugging the scattering ansatz to the eigenproblem (\[eq:evA\]), the first two rows of ${{\mathbf{A}}}$ (\[eq:bigA\]) result in 6 linearly independent equations for 6 unknowns $\psi^+_{1,2},{{\underline{u}}}_{\rm L}$. Eliminating four variables ${{\underline{u}}}_{\rm L}$ we finally arrive to the non-unitary $2 \times 2$ S-matrix $$\pmatrix{
\psi^+_1 \cr
\psi^+_2} = {{\mathbf{S}}}^{\rm L} \pmatrix{
\psi^-_1 \cr
\psi^-_2} \label{eq:SL} \\$$ with $$\begin{aligned}
S^{\rm L}_{11} &=& \tau^{-1} \beta^2 (-(\GLp)^4+4 (\GLp)^2 (\beta^2-3h^2) -16 h (h J^2 + { {\rm i} }\GLm \omega)) \nonumber \\
S^{\rm L}_{12} &=& \tau^{-1} \beta ( (\GLp)^3 + 8{ {\rm i} }\GLm h \beta + 4 \GLp (h^2 - \beta^2)) (2{ {\rm i} }\omega) \nonumber \\
S^{\rm L}_{21} &=& \tau^{-1} \beta ( (\GLp)^3 - 8{ {\rm i} }\GLm h \beta + 4 \GLp (h^2 - \beta^2)) (-2{ {\rm i} }\omega) \nonumber \\
S^{\rm L}_{22} &=& \tau^{-1} \beta^2 (-(\GLp)^4+4 (\GLp)^2 (\beta^2-3h^2) -16 h (h J^2 - { {\rm i} }\GLm \omega)) \label{eq:Sexplicit} \\
\tau &:=& (\GLp)^4 \beta^2 + 8 \beta^2 (h^4 + (J^2 + \beta^2)(J^2 + \beta^2 - \omega) + h^2 (2 \beta^2 - \omega)) \nonumber \\
&-& 2(\GLp)^2 (h^4 + J^4 + 3\beta^4 + J^2 (2 \beta^2-\omega) - \beta^2\omega + h^2 (\omega - 2 J^2 - 4\beta^2)) \nonumber
\end{aligned}$$ Similarly, one can solve the scattering problem from the [*right*]{} bath with the scattering ansatz $${{\underline{v}}} = \pmatrix{
\vdots \cr
(\psi_1^+ {{\underline{u}}}^+_1 + \psi_2^+ {{\underline{u}}}^+_2)\xi_+^{-2} + (\psi_1^- {{\underline{u}}}^-_1 + \psi_2^- {{\underline{u}}}^-_2)\xi_-^{-2} \cr
(\psi_1^+ {{\underline{u}}}^+_1 + \psi_2^+ {{\underline{u}}}^+_2)\xi_+^{-1} + (\psi_1^- {{\underline{u}}}^-_1 + \psi_2^- {{\underline{u}}}^-_2)\xi_-^{-1} \cr
\psi_1^+ {{\underline{u}}}^+_1 + \psi_2^+ {{\underline{u}}}^-_2 + \psi_1^- {{\underline{u}}}^-_1 + \psi_2^- {{\underline{u}}}^-_2 \cr
{{\underline{u}}}_{\rm R}}$$ defining the right S-matrix $$\pmatrix{
\psi^-_1 \cr
\psi^-_2} = {{\mathbf{S}}}^{\rm R} \pmatrix{
\psi^+_1 \cr
\psi^+_2}$$ Note that since the two directions of free modes (\[eq:freemodes\]) do not have left-right symmetry an explicit expression for $S^{\rm R}$ is considerably more complicated than (\[eq:Sexplicit\]) and shall not be written out here. We shall now show that there exist two qualitatively different types of NMM - complex rapidities $\beta$ solving (\[eq:evA\]) for sufficiently [*large*]{} $n$.
![ Rapidities $\beta_j$ (black dots) for a transverse Ising chain with $J=1.5,h=1$ and bath couplings $\Gamma^{\rm L}_1=1,\Gamma^{\rm L}_2=0.6$, $\Gamma^{\rm R}_1=1,\Gamma^{\rm R}_2=0.3$, for three different sizes $n=6$ (upper), $n=30$ (middle), and $n=150$ (lower panel). Big blue/red dots indicate positions of evanescent rapidities (solutions of eq.(\[eq:4order\])) for the left/right bath. \[fig:rapid\]](rapid.eps)
First, we shall discuss the so called [*evanescent normal master modes*]{}. These are characterized with amplitudes (\[eq:scatL\]) which decay exponentially with the distance from – say the left – bath, so the other – the right boundary condition becomes physically irrelevant in the limit $n\to \infty$. Such solutions $\psi^{+}_{1,2} = 0$ of eq. (\[eq:SL\]) exist exactly when the determinant of S-matrix vanishes $\det S^{\rm L} = 0$. Using (\[eq:Sexplicit\]) the determinant can be written as $\det S^{\rm L} = (\beta/\tau)^2 p^{\rm L}(\beta)$ where[^4] $$\begin{aligned}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!p^{\rm L}(\beta) &=& (\GLp)^8 \beta^2 - 4(\GLp)^6 ((h^2\!-\!J^2)^2 + (2J^2\!-\!4h^2)\beta^2 + 3\beta^4) \nonumber \\
&-& 16 (\GLp)^4(2 h^2 (h^2\!-\!J^2)^2 - (7h^4\!-\!6 h^2J^2\!+\!2 J^4)\beta^2 + 4(h^2\!-\!J^2)\beta^4 - 3 \beta^6) \nonumber \\
&-& 64 (\GLp)^2(h^4 (h^2\!-\!J^2)^2-2h^2 J^4\beta^2 -(2h^4\!+\!4 h^2 J^2\!-\!J^4)\beta^4 + 2 J^2 \beta^6 + \beta^8) \nonumber \\
&+& 256 h^4 J^4 \beta^2 \label{eq:p}\end{aligned}$$ Thus, for sufficiently large spin chains we find at most four NMM whose rapidities are given as the roots of 4-th order polynomial in $\beta^2$ $$p^{\rm L}(\beta_{\rm evan}) = 0 \label{eq:evan}
\label{eq:4order}$$ that are [*not*]{} simultaneously zeroes of $\tau(\beta)$. Clearly, such NMM asymptotically do not depend on the chain size $n$. In addition, we find evanescent NMM corresponding to the right bath simply by replacing $\GLp$ by $\GRp$ in (\[eq:evan\],\[eq:p\]). In fig. \[fig:rapid\] we compare evanescent rapidities computed from eq. (\[eq:evan\]) to numerically calculated spectrum of ${{\mathbf{A}}}$, at several different sizes $n$, for a typical case of transverse Ising chain, $J=1.5,h=1.0$, strongly coupled to two baths at considerably different temperatures, $\Gamma^{\rm L}_1=1,\Gamma^{\rm L}_2=0.6$, $\Gamma^{\rm R}_1=1,\Gamma^{\rm R}_2=0.3$ Note that the same parameter values will be used for numerical demonstrations throughout this subsection.
Second, we shall discuss the other extreme of [*soft normal master modes*]{} with rapidities that are closest to the imaginary axis, and thus determining the spectral gap of the Liouvillean and relaxation time to NESS. Composing the scattering from the two baths with the free propagation along the chain (back and forth) we arrive at the general secular equation for the eigenvalue problem (\[eq:evA\]) in terms of a $2\times 2$ determinant $$\det ( \xi_+^{2(n-3)} {{\mathbf{S}}}^{\rm R} {{\mathbf{S}}}^{\rm L}- {{\mathbf{I}}}_2) = 0
\label{eq:sec}$$ In the absence of the baths, $\Gamma^{\rm L,R}_{\pm} = 0$, the solutions of the above problem exist only for real quasi-momenta, namely $\xi_{\pm} = \exp(\pm{ {\rm i} }\vartheta), \vartheta \in {\mathbb{R}}$. For such [*extended*]{} master modes the local coupling to the baths can be considered as a small perturbation, thus only slightly perturbing the Bloch-like bands $\beta(e^{{ {\rm i} }\vartheta}) = \pm { {\rm i} }\varepsilon(\vartheta)$ with ‘energy’ $$\varepsilon(\vartheta) = \sqrt{h^2 + J^2 - 2 |h J| \cos\vartheta}$$ The softest NMM, namely the one for which the coupling to the baths is expected to be the weakest, should have nearly nodes at the ends of the chain, i.e. $\vartheta \approx \pi/n$, or $\vartheta \approx \pi + \pi/n$, and should thus lie near the band edges $\pm { {\rm i} }|h| \pm { {\rm i} }|J|$ (see fig. \[fig:rapid\]). In the following we shall focus our calculation on the band edge $\beta^* = { {\rm i} }(|h| + |J|)$ which, as can be checked aposteriori by a straightforward but tedious calculation, always gives smaller real part of the rapidity than the lower edge ${ {\rm i} }(|h|-|J|)$, and hence really determines the gap of the Liouvillean. So we write $$\beta = { {\rm i} }(|h| + |J|) + z$$ where $z\in{\mathbb{C}}$ is a small parameter, and expand the S-matrices around the band edge $${{\mathbf{S}}}^{\rm L,R} = -{{\mathbf{I}}}_2 + \frac{4g}{\eta^{\rm L,R}}
{{\mathbf{Z}}}^{\rm L,R} \sqrt{-{ {\rm i} }z} + {\cal O}(|z|)
\label{eq:SZ}$$ where $g:=\sqrt{\frac{|hJ|}{2(|h|+|J|)}}$, $\eta^{\rm L,R}:=(\Gamma^{\rm L,R}_+)^4 + 4 (\Gamma^{\rm L,R}_+)^2(4 h^2 + 2 |h J| + J^2) + 16 h^2 J^2$ and $$\begin{aligned}
Z^{\rm L}_{11} &=& 4 |h|(\GLp)^2 + 16|h|(|h|+|J|)(|J|-{ {\rm i} }\GLm) \nonumber \\
Z^{\rm L}_{12} &=& -2 (\GLp)^3 - 16 \GLm |h|(|h|+|J|)-8(2h^2+2|hJ|+J^2) \nonumber\\
Z^{\rm L}_{21} &=& +2 (\GLp)^3 - 16 \GLm |h|(|h|+|J|)+8(2h^2+2|hJ|+J^2) \nonumber\\
Z^{\rm L}_{22} &=& 4 |h|(\GLp)^2 + 16|h|(|h|+|J|)(|J|+{ {\rm i} }\GLm) \label{eq:ZL}\end{aligned}$$ and $$\begin{aligned}
Z^{\rm R}_{11} &=& (\GRp)^4(2|h|+|J|)+4(\GRp)^2(8|h|^3+9h^2|J|+4|h|J^2+|J|^3) \nonumber \\
&+& 16 h^2|J|(|J|(3|h|+2|J|)-{ {\rm i} }\GRm(|h|+|J|)) \nonumber \\
Z^{\rm R}_{12} &=& -2 (\GRp)^3 - 16 \GRm |h|(|h|+|J|)-8(2h^2+2|hJ|+J^2) \nonumber\\
Z^{\rm R}_{21} &=& +2 (\GRp)^3 - 16 \GRm |h|(|h|+|J|)+8(2h^2+2|hJ|+J^2) \nonumber\\
Z^{\rm R}_{22} &=& (\GRp)^4(2|h|+|J|)+4(\GRp)^2(8|h|^3+9h^2|J|+4|h|J^2+|J|^3) \nonumber \\
&+& 16 h^2|J|(|J|(3|h|+2|J|)+{ {\rm i} }\GRm(|h|+|J|))
\label{eq:ZR}\end{aligned}$$ Next we expand $\xi_+$ (\[eq:dispersion\]) in $z$, yielding $$\xi_+ = -1 - g^{-1} \sqrt{-{ {\rm i} }z} + {\cal O}(|z|)
\label{eq:xi1}$$ and so the [*free propagator*]{} in (\[eq:sec\]) can be written as $$\xi_+^{2(n-3)} = \exp(2n g^{-1}\sqrt{-{ {\rm i} }z}) + {\cal O}(|z|).
\label{eq:xiexp}$$ In eqs. (\[eq:SZ\],\[eq:xi1\],\[eq:xiexp\]) the branch cut along the negative real axis has been chosen for $\sqrt{-{ {\rm i} }z}$. Since the product of S-matrices in (\[eq:sec\]) is near identity, the free propagator should be near one as well, hence $2n g^{-1}\sqrt{-{ {\rm i} }z}$ should be near $2\pi{ {\rm i} }$. Let us define $z_0$ by setting $2n g^{-1}\sqrt{-{ {\rm i} }z_0} = 2\pi{ {\rm i} }$, so $$z_0 = -{ {\rm i} }\pi^2 g^2 n^{-2}$$ and write $z = z_0 (1 + y) $ where $|y| \ll 1$ is another small complex parameter. However, since $z_0$ is purely imaginary, we need to compute a small but non-vanishing $y$ which will, in the leading order in $n$, solve (\[eq:sec\]) since the real part of the soft mode’s rapidity is determined as $$\re \beta = \re z_0 y = \pi^2 g^2 n^{-2} \im y
\label{eq:y}$$
Now, writing $\sqrt{-{ {\rm i} }z} = \sqrt{-{ {\rm i} }z_0}\sqrt{1+y} = { {\rm i} }\pi g n^{-1} (1 + y/2 - y^2/8) + {\cal O}(y^3)$ in (\[eq:SZ\],\[eq:xiexp\]), plugging all that to eq. (\[eq:sec\]) and computing to order ${\cal O}(n^{-2})$, noting that ${\cal O}(|z|)={\cal O}(n^{-2})$, we arrive to a simple quadratic equation for $y$, whose solution, plugged to (\[eq:y\]), gives the final result, namely the sectral gap of Liouvillean $\Delta= 2\re \beta $ $$\begin{aligned}
\Delta &=& \frac{(2\pi h J)^2}{(|h|+|J|)^2} \frac{\Delta_1}{\Delta_2} n^{-3} + {\cal O}(n^{-4}) \label{eq:delta} \\
\Delta_1 &:=& 64 (\GLp + \GRp) h^2 J^2 (2 h^2 + 2|h J| + J^2) \nonumber \\
&+& 16 ((\GLp)^3 + (\GRp)^3) h^2 J^2 \nonumber \\
&+& 16 \GLp \GRp (\GLp + \GRp) (2 h^2 + 2|hJ| + J^2)(4 h^2 + 2|hJ| + J^2) \nonumber \\
&+& 4 \GLp \GRp ((\GLp)^3 + (\GLp)^2 \GRp + \GLp (\GRp)^2 + (\GRp)^3) (2 h^2+ 2|h J| + J^2) \nonumber \\
&+& (\GLp \GRp)^3 (\GLp+\GRp) \nonumber \\
\Delta_2 &:=& ((\GLp)^4 + 4(\GLp)^2 (4 h^2 + 2|hJ| + J^2)+ 16 h^2 J^2 ) \nonumber \\
&\times& ((\GRp)^4 + 4(\GRp)^2 (4 h^2 + 2|hJ| + J^2)+ 16 h^2 J^2) \nonumber
\end{aligned}$$ In fig. (\[fig:delta\]) we compare this analytical result to exact numerical calculations of the eigenvalue of ${{\mathbf{A}}}$ with minimal real part, confirming both, its precise numerical value and that the relative scaling of the next order correction is indeed ${\cal O}(n^{-1})$.
Note that, interestingly, both main analytical results of this subsection, namely evanescent and soft mode rapidities [*do not depend*]{} on $\Gamma^{\rm L,R}_-$. Physically speaking, they only depend on the effective strengths of the bath couplings and not on the temperatures.
![ Spectral gap $\Delta$ times a third power of the chain length $n$ for a transverse Ising chain with $J=1.5,h=1$ and bath couplings $\Gamma^{\rm L}_1=1,\Gamma^{\rm L}_2=0.6$, $\Gamma^{\rm R}_1=1,\Gamma^{\rm R}_2=0.3$. Thin horizontal line indicates the theoretical asymptotic value (\[eq:delta\]). In the inset we show deviation from asymptotic constant value of $\Delta n^3$ in log-log scale and demonstrate that it decays as $\propto n^{-1}$ (thin line). \[fig:delta\]](deltafine.eps)
![ Spectral gap $\Delta$ times a third power of the chain length $n$ for a transverse Ising chain with $J=1.5,h=1$ and bath couplings $\Gamma^{\rm L}_1=1,\Gamma^{\rm L}_2=0.6$, $\Gamma^{\rm R}_1=1,\Gamma^{\rm R}_2=0.3$. Thin horizontal line indicates the theoretical asymptotic value (\[eq:delta\]). In the inset we show deviation from asymptotic constant value of $\Delta n^3$ in log-log scale and demonstrate that it decays as $\propto n^{-1}$ (thin line). \[fig:delta\]](delta.eps)
![ Complete spectrum of $2^{12}$ complex eigenvalues of Liouvillean for a transverse Ising chain with $n=6$ spins and $J=1.5,h=1$ and bath couplings $\Gamma^{\rm L}_1=1,\Gamma^{\rm L}_2=0.6$, $\Gamma^{\rm R}_1=1,\Gamma^{\rm R}_2=0.3$ (the case of the upper panel of fig. \[fig:rapid\]). \[fig:spc\]](spc.eps)
![ Energy current (upper/blue points), and average spin current (lower/red points), versus chain length $n$ for a transverse Ising chain with $J=1.5,h=1$ and bath couplings $\Gamma^{\rm L}_1=1,\Gamma^{\rm L}_2=0.6$, $\Gamma^{\rm R}_1=1,\Gamma^{\rm R}_2=0.3$. \[fig:current\]](current.eps)
![ Energy density profile (lower, blue points), and spin density profile (upper, red points), for a transverse Ising chain of $n=80$ spins with $J=1.5,h=1$ and bath couplings $\Gamma^{\rm L}_1=1,\Gamma^{\rm L}_2=0.6$, $\Gamma^{\rm R}_1=1,\Gamma^{\rm R}_2=0.3$. The insets display logarithm of the difference to the bulk values $\delta H_m := |{{\langle H_m\rangle}} - H_{\rm bulk}|$ (blue points), $\delta \sigma^{{\rm z}}_m := |{{\langle \sigma^{{\rm z}}_m\rangle}} - \sigma^{{\rm z}}_{\rm bulk}|$ (red points) in comparison with $\pm (4 \log \xi_- ) m + {\rm const}$ with quasi-momentum $\xi_- = 0.584692$ corresponding (\[eq:dispersion\]) to the leading evanescent rapidity $\beta_{\rm evan} = 0.438739$ (full lines). \[fig:profile\]](profile1.eps)
![ Energy density profile (lower, blue points), and spin density profile (upper, red points), for a transverse Ising chain of $n=80$ spins with $J=1.5,h=1$ and bath couplings $\Gamma^{\rm L}_1=1,\Gamma^{\rm L}_2=0.6$, $\Gamma^{\rm R}_1=1,\Gamma^{\rm R}_2=0.3$. The insets display logarithm of the difference to the bulk values $\delta H_m := |{{\langle H_m\rangle}} - H_{\rm bulk}|$ (blue points), $\delta \sigma^{{\rm z}}_m := |{{\langle \sigma^{{\rm z}}_m\rangle}} - \sigma^{{\rm z}}_{\rm bulk}|$ (red points) in comparison with $\pm (4 \log \xi_- ) m + {\rm const}$ with quasi-momentum $\xi_- = 0.584692$ corresponding (\[eq:dispersion\]) to the leading evanescent rapidity $\beta_{\rm evan} = 0.438739$ (full lines). \[fig:profile\]](profile2.eps "fig:")![ Energy density profile (lower, blue points), and spin density profile (upper, red points), for a transverse Ising chain of $n=80$ spins with $J=1.5,h=1$ and bath couplings $\Gamma^{\rm L}_1=1,\Gamma^{\rm L}_2=0.6$, $\Gamma^{\rm R}_1=1,\Gamma^{\rm R}_2=0.3$. The insets display logarithm of the difference to the bulk values $\delta H_m := |{{\langle H_m\rangle}} - H_{\rm bulk}|$ (blue points), $\delta \sigma^{{\rm z}}_m := |{{\langle \sigma^{{\rm z}}_m\rangle}} - \sigma^{{\rm z}}_{\rm bulk}|$ (red points) in comparison with $\pm (4 \log \xi_- ) m + {\rm const}$ with quasi-momentum $\xi_- = 0.584692$ corresponding (\[eq:dispersion\]) to the leading evanescent rapidity $\beta_{\rm evan} = 0.438739$ (full lines). \[fig:profile\]](profile3.eps "fig:")
We end this subsection by presenting some further numerical results on heat transport in the open transverse Ising chain in the Lindblad form. In fig. \[fig:spc\] we demonstrate expression (\[eq:eval\]) for constructing the full spectrum of the Liouvillean in terms of a set of rapidities, for a short chain. In fig. \[fig:current\] we demonstrate eq. (\[eq:pairwise2\]) of Theorem 3 by computing the energy current $Q_m$ (\[eq:hcurr\]), and the average spin current $S=\frac{1}{n-1}\sum_{m=1}^{n-1} S_m$ (\[eq:scurr\]) in NESS of a typical transverse Ising chain. Numerical results give a clear indication of [*ballistic transport*]{} ${{\langle Q\rangle}} = {\cal O}(n^0), {{\langle S\rangle}}={\cal O}(n^0)$, however its rigorous proof and analytical calculation of the currents would require full control over the complete set of NMM which is at present not available. In fig. \[fig:profile\] we plot the energy density (\[eq:hdens\]) and spin density (\[eq:sdens\]) profiles in NESS. Again, we note flat profiles in the bulk of the chain, $m,n-m \gg 1$, with exponential falloff due to adjustment to the non-equilibrium bath values. Since the densities can be written, by means of (\[eq:pairwise2\]), as $4-$point functions in NMM components, the leading falloff exponents of the profile $|{{\langle H_m\rangle}} - H_{\rm bulk}| \sim
|\xi_-|^{4m} $ is given by the quasi-momentum $\xi_-$ (\[eq:dispersion\]) corresponding to the maximal evanescent rapidity $\beta_{\rm evan}$ (\[eq:evan\]).
Disordered XY chain {#sect:disord}
-------------------
In this subsection we treat the opposite extreme, a disordered XY chain (\[eq:hamsc\]) where three sets of physical parameters are chosen as [*random uncorrelated*]{} variables from [*uniform*]{} distributions on the intervals, $J^{{\rm x}}_m \in [J^{{\rm x}}_{\rm min},J^{{\rm x}}_{\rm max}]$, $J^{{\rm y}}_m \in [J^{{\rm y}}_{\rm min},J^{{\rm y}}_{\rm max}]$, $h_m \in [h_{\rm min},h_{\rm max}]$. Clearly, the eigenvalue problem (\[eq:evA\]) for the matrix (\[eq:bigA\]) then becomes equivalent to the Anderson tight-binding problem in one dimension for a quantum particle with a $4-$level internal degree of freedom. We do not pursue any theoretical analysis of this problem here, but merely state that numerical investigations indicate existence of exponential localization of [*all*]{} eigenvectors (or normal master modes) for disorder of any strength in anyone of system’s parameters.
![ Average Liouvillean spectral gap ${{\langle \Delta\rangle}}$ versus the chain length $n$ for disordered XY models: (i) $J^{{\rm x}}_m = 0.5, J^{{\rm y}}_m = 0, h_m \in [1,2]$ (transverse Ising with field disorder, blue points), (ii) $J^{{\rm x}}_m \in [0.5,2], J^{{\rm y}}_m = 0, h_m = 1$ (transverse Ising with interaction disorder, red points), (iii) $J^{{\rm x}}_m \in [0.5,1],J^{{\rm y}}_m \in [0.5,1], h_m = 1$ (XY with interaction disorder, golden points), all for bath couplings $\Gamma^{\rm L}_1=1,\Gamma^{\rm L}_2=0.6$, $\Gamma^{\rm R}_1=1,\Gamma^{\rm R}_2=0.3$. Full lines indicate exponential fits to right halves of data. Averaging is performed over $2000$ disorder realizations. \[fig:disdelta\]](disdelta.eps)
![ The scaling of the energy current ${{\langle Q_m\rangle}}$ with chain length $n$ of the disordered XY model in the same regimes/parameters/plot styles as in fig. \[fig:disdelta\]. \[fig:discurrent\]](discurrent.eps)
![ Scaled energy density profile of interaction disordered XY chains (case (iii) of fig. \[fig:disdelta\]) for three chain sizes: $n=20$ (blue points), $n=40$ (red points), $n=60$ (golden points). Averages over 50000 disorder realizations have been performed. \[fig:disprofile\]](disprofile.eps)
With the picture of localization of NMM in mind, the effect of the couplings to the heat baths at the chain’s ends on quantum transport can be predicted by theoretical arguments (see [@livi] for a review of related studies): The spectral gap of the Liouvillean should be exponentially small $\Delta \sim \exp(-n/\ell)$ where $\ell$ is the localization length of NMM which is expected to be proportional to the square of inverse disorder strength. This is demonstrated in fig.\[fig:disdelta\]. If all NMM are exponentially localized, the currents should decrease with the chain size $n$ faster than any power, perhaps exponentially, and the system should behave as an ideal insulator (at all temperatures). This is demonstrated by straightforward numerical calculations of the heat current (\[eq:hcurr\]) in fig. \[fig:discurrent\]. In the final figure \[fig:disprofile\] we plot the energy density profile ${{\langle H_m\rangle}}$ (\[eq:hdens\]) in a typical case of disordered XY chain, versus a scaled spatial coordinate $(m-1)/(n-1) \in [0,1]$, for several different chain lengths $n$, and demonstrate sharping up of energy density profiles with increasing $n$, which is again indicating insulating behaviour.
Discussion and conclusions {#sect:conc}
==========================
The main result of the paper is a general method of explicit solution of master equations describing dynamics of open quantum system, under the condition that the system’s Hamiltonian is [*quadratic*]{} and all Lindblad operators are [*linear*]{} in canonical fermionic operators (which can either represent real physical fermions or any abstract 2-level quantum systems (qubits) thru the Jordan-Wigner transformation). Using a novel concept of Fock space of physical operators (or density operators of physical states), and the adjoint structure of canonical creation and annihilation maps over this space, the problem can be treated in terms of a non-Hamiltonian generalization of the method of Lieb, Schultz and Mattis [@lieb] lifted to an operator space. We have explicitly constructed a non-canonical analog of Bogoliubov transformation of the quantum Liouville map to normal master modes. Related ideas in the Hamiltonian context have been used by the author [@ProsenPTPS; @prosen; @pizorn] in order to approach the problem of real time dynamics and ergodic properties of [*isolated*]{} interacting many-body quantum systems.
As an illustration of the method we have solved far from equilibrium quantum heat and spin transport problem in Heisenberg XY spin 1/2 chains which are coupled to canonical heat baths only at the two ends. Irrespectively of the strength of the coupling to the baths and their temperatures, we have shown a ballistic transport in the spatially homogeneous (non-disordered) case, and an ideally insulating behaviour in the disordered case associated to localization of normal master modes of the quantum Liouville operator. In this context the method can be considered as a simple alternative to the solution of quantum Langevin equations [@dhar].
However, the method should easily be applicable to variety of other physical situations, for example if all fermions are coupled to the baths one could make a solvable model of genuine quantum diffusion, a many-body generalization of the tight-binding model [@piere]. We also expect the method to be applicable to the Redfield type of master equations (see e.g. [@piere]) - which do not conserve positivity for a short initial (slippage) time interval - provided only the system part of the Hamiltonian is [*quadratic*]{} and system-bath couplings are [*linear*]{} in fermionic variables. Furthermore, extension of the method to open [*many-boson*]{} systems should be straightforward, simply by replacing anticommutators by commutators throughout the exposition of section \[sect:method\].
Treating density operators of NESS as elements of a Hilbert space of operators one may also extend the concept of [*entanglement entropy*]{}, with respect to a bipartition of a system of many fermions [@latorre], to NESS which can in our approach be viewed as a kind of ground state of the Liouvillean. Saturation of such [*operator space entanglement entropy*]{} [@pizorn] (which is suggested by numerical experiments [@PZ08]) indicates [*efficient simulability*]{} of NESS by elaborate numerical methods such as [*density matrix renormalization group*]{} [@dmrg], perhaps even for more general, non-solvable quantum systems.
As last we mention a more ambitious extension of the present work: Namely we propose to explore a question, whether more involved algebraic methods of solution of interacting many-body quantum systems, like e.g. Bethe Ansatz or quantum inverse scattering [@korepin], could be generalized to open quantum systems, e.g. by means of the proposed concept of Fock space of operators. Could one discuss completely integrable open quantum systems which go beyond quadratic Liouvilleans?
Acknowledgements {#acknowledgements .unnumbered}
================
I gratefully acknowledge stimulating discussions with Pierre Gaspard, Keiji Saito and Walter Strunz, thank Carlos Mejia-Monasterio and Thomas H. Seligman for reading the manuscript and many useful comments, and Iztok Pižorn and Marko Žnidarič for collaboration on related projects. The work has been supported by the grants P1-0044 and J1-7347 of Slovenian research agency (ARRS). Explicit analytical calculations reported in subsection (\[sect:ti\]) were assisted by [*Mathematica*]{} software package.
References {#references .unnumbered}
==========
[10]{}
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We note a similarity to the formalism of second quantization with non-orthogonal orbitals introduced in: M. Moshinsky and T. H. Seligman, Ann. Phys. (New York) [**66**]{}, 311 (1971).
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[^1]: Throughout this paper Dirac’s bra-ket notation shall be used only for a Hilbert space ${\cal K}$ of physical operators, including density operators, in a sense of GNS construction although here [*all*]{} spaces will be [*finite*]{} dimensional. Symbols with a [*hat*]{} shall designate [*linear maps*]{} over the operator space ${\cal K}$. For instance, we note a key distinction between [*physical fermions*]{} $c_m$ (\[eq:physfermi\]) and a-fermions $\hat{c}_j$ (\[eq:defanih\]).
[^2]: It is not known at present whether explict form (\[eq:explA\]) guarantees diagonalizability of any such ${{\mathbf{A}}}$. Note that one can construct certain types of complex antisymmetric matrices with degenerate eigenvalues which cannot be diagonalized [@semrl].
[^3]: For a non-degenerate rapidity spectrum $\{\beta_j\}$ the proof of this statement is a trivial consequence of antisymmetry ${{\mathbf{A}}}=-{{\mathbf{A}}}^T$, whereas in case of degeneracies it can be shown that one can always choose appropriate linear combinations of eigenvectors.
[^4]: Trivial zero $\beta=0$ of course does not represent a physical solution since then the whole S-matrix (\[eq:Sexplicit\]) vanishes.
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{
"pile_set_name": "ArXiv"
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[**[Summary on Theoretical Aspects [^1]]{}**]{}
1.40cm [**Jacques Soffer**]{} 0.3cm Physics Department, Temple University\
Barton Hall, 1900 N, 13th Street\
Philadelphia, PA 19122-6082, USA 1.0cm [**Abstract**]{}
During the five days of this conference a very dense scientific program has enlighted our research fields, with the presentation of large number of interesting lectures. I will try to summarize the theoretical aspects of some of these new results.
Introduction
============
This meeting has confirmed once more a clear scientific evolution, that is the foundations of elastic scattering and diffraction phenomena at high energy, are now best understood in terms of the first principles of quantum chromodynamics (QCD). Since we are just at the start-up of the LHC, a great deal of the theoretical activity focus on the accessibility to this new energy regime, for the interpretation of relevant aspects of the strong interactions, by means of basic QCD mechanisms. Clearly they have to be confronted with experimental measurements, a very relevant part which will be summarized elsewhere [@JD].\
I will essentially cover the following topics:
- Elastic and Total Cross Section
- Soft Diffraction
- Hard Diffraction and Central Production
Unfortunately, I have left out some important topics, in particular, ultra high energy cosmic rays and heavy-ion physics, because of lack of time and I apologize for that.
Elastic Scattering and Total Cross Section
==========================================
This is a classical subject which was very largely discussed from different viewpoints. Let us first mention a study of the amplitudes $pp$ and $\bar pp$ elastic scattering in the Coulomb-Nuclear interference (CNI) region [@EF; @KFK], using a method based on derivative dispersion relations. The real and imaginary parts of the hadronic amplitude near the forward direction, whose detailed knowledge is needed, are parametrised by a single exponential, with two different hadronic slopes $B_R$ and $B_I$. The analysis of the available data, in the range from $\sqrt{s}$=19GeV to 1800GeV, leads to the conclusion that $B_R > B_I$, although the determination of $B_R$ is far less precise than $B_I$, for obvious reasons. Note that strickly speaking, this concept of hadronic slope is very misleading, since it is known that the derivative of the amplitude with respect to $|t|$ is a slowly decreasing function of $|t|$ as shown in Ref. [@BSW84], an approach where real and imaginary parts of the amplitude are strongly related. The relevance of the measurement of the real part of the $pp$ forward scattering amplitude at the LHC has been also emphasized in Ref. [@BKMSW].\
A model for $pp$ and $\bar pp$ elastic scattering, based on the electromagnetic and gravitational form factors related to a new set of generalized parton distributions (GPD) was used, after unitarization, to fit the data [@OS]. Unfortunately the quality of the fit is rather poor, with a $\chi^2$/pt=6 and it predicts a high value of the total cross section at LHC, $\sigma_{tot}$=146mb. One gets an even higher prediction at $\sqrt{s}$=14TeV, $\sigma_{tot}$=230mb, in another approach, which introduces the concept of reflective elastic scattering at very high energies [@ST]. This picture also predicts that the scattering amplitude at the LHC energy goes beyond the black disk limit.
Another phenomenological investigation of $pp$ and $\bar pp$ elastic scattering was carried out by considering that the proton consists of an outer region of $\bar qq$ condensed ground state, an inner shell of topological baryonic charge and a core where valence quarks are confined [@RL]. It leads to $\sigma_{tot}$=110mb and for the ratio of real to imaginary parts of the forward amplitude $\rho$=0.12 at LHC. The predicted differential cross section $d\sigma/dt$ has a smooth behavior beyond the bump at $|t|\simeq1\mbox{GeV}^2$, with no oscillations and a much larger value, in contrast with other models.
Concerning the specific issue of the value of the $pp$ total cross section at the LHC, a highly non-perturbative quantity which cannot be predicted by QCD, we had a general presentation of different models (double poles, triple poles, cuts, etc...) and their experimental consequences [@JRC]. It was stressed that the theoretical uncertainty is large, as discussed above, and therefore an accurate measurement is badly needed since it will also tell us a lot about the analytic structure of the $pp$ elastic amplitude.
The eikonal approach has been proven to be very useful in describing high energy elastic scattering. Clearly it relies on the knowledge of the impact parameter profile, which can be either more peripheral or central. The analysis of the $pp$ data at the ISR energy $\sqrt{s}$=53GeV led to the conclusion that a peripheral profile is preferred in this case [@VK]. In another presentation [@MVL], the validity of the optical theorem commonly used to extract the total cross section has been questionned.
A new rigorous result on the inelastic cross section was obtained recently [@AM] and it reads $\sigma_{inel}(s)< \pi/4m_{\pi}^2 (\mbox{ln} s)^2$. This bound is four times smaller than the old Froissart bound derived in 1967, $\sigma_{tot}(s)< \pi/m_{\pi}^2 (\mbox{ln} s)^2$ where $\pi/m_{\pi}^2$ = 60mb. This last result can be also improved by a factor two, using some reasonable assumptions and it would be nice to prove it rigorously.
A possible description of high-energy small-angle scattering in QCD can be done by means of two vacuum exchanges with $C=\pm 1$, the Pomeron and the Odderon. Recent developments in this subject, based on the weak/strong duality, relating Yang-Mills theories to string theories in Anti-de Sitter (AdS) space, were presented in some details [@CIT]. If the QCD Pomeron is viewed as a two-component object, soft and hard, a dual description of the Pomeron emerges unambiguously through the AdS/CFT approach and the Odderon is related to the anti-symmetric Kalb-Ramond field. Some aspects of analyticity, unitarity and confinement were also discussed.
Soft Diffraction
================
In an overview of soft diffraction [@AK], several theoretical approaches were considered for a better understanding of the relevant mechanisms of high-energy interactions and making an instructive comparison between s- and t-channel view points. Diffractive production in the s-channel is peripheral in the impact parameter and there is a strong influence of unitarity effects due to multi-pomeron exchanges. The calculation of the survival probability for hard processes is very important, in particular for high mass diffraction, as we will see later, for example, for central Higgs production.
Following the above ideas, a model based on Gribov’s Reggeon calculus was proposed and applied to soft diffraction processes at high energy [@MP]. By giving a special attention to the absorptive corrections, the parameters of the model are determined following from a good description of the existing experimental data on inclusive diffraction in the energy range from ISR up to Tevatron. The model predictions for single and double diffraction at LHC energy are also given later. In another contribution [@EM], one was recalling the method to unitarize the Pomeron for elastic and inclusive scattering, providing as well as a comparison with data, mainly for one-particle inclusive production and some LHC predictions.
Soft scattering theory was re-visited by considering some eikonal models for simplicity and to secure s-channel unitarity [@UM]. After recalling the main features of two specific models [@GLMM; @RMK], the interplay between theory and data analysis led to some LHC predictions, in particular a total cross section of the order of 90mb, in contrast with the prediction $\sigma_{tot}=103.6 \pm 1.1$mb from Ref. [@BSW]. Another important point from Ref. [@BSW] to notice here, is the fact that the ratio $\sigma_{el}/\sigma_{tot}$ rises from the value 0.18 for $\sqrt{s}$=100GeV to 0.30 for $\sqrt{s}$=100TeV, whereas Refs. [@GLMM; @RMK] predict almost no energy dependence in this range.
Some special features of the model of soft interactions of Ref. [@GLMM] mentioned above, were discussed together with the results of the fit to determine the parameters of the model [@EG]. Needless to say that it is very important to estimate the survival probability for central exclusive production of the Higgs boson, which was also compared with the results of the model of Ref. [@RMK].
This question is also related to the notion of color fluctuations in the nucleon in high energy scattering [@MS], so it is legitimate to ask: how strong are fluctuations of the gluon field in the nucleon? A simple dynamical model can explain the ratio of the inelastic to the elastic cross section in vector meson production in $ep$ collisions at HERA and leads to a new sum rule [@FSTW]. However it cannot explain the Tevatron CDF data and it reduces the expected survival probability in central exclusive production.
Hard Diffraction and Central Production
=======================================
The standard QCD mechanism for central exclusive production for heavy systems, using the formalism of collinear generalized parton distributions has been proposed some time ago and applied for Higgs production at the LHC [@KKMR]. In this case also, it is relevant to question a possible violation of QCD factorization and some aspects of analyticity and crossing properties [@OT]. At the phenomenological level, the same mechanism was used to calculate the amplitudes for the central exclusive production of the $\chi_c$ mesons, using different unintegrated gluon distribution functions (UGDF) [@TPS]. The extention of the UGDF to the non-forward case, can be obtained by saturation of positivity constraints. The resulting total cross sections for all charmonium states $\chi_c(0^+,1^+,2^+)$ are compared at Tevatron energy.
The present situation of theoretical predictions for central exclusive production of Higgs bosons and other heavy systems at the LHC was reviewed [@JRC2]. It was shown that the CDF dijet data can be used to reduce the uncertainty on the cross section prediction for the Higgs boson. The claim is that a cross section between 0.3 and 2fb is expected for a standard Higgs of mass 120GeV. Central exclusive production of vector mesons may be used as a discovery channel for the odderon.
Some simple examples of physics beyond the Standard Model (SM), which require an extended Higgs sector, were considered [@SH]. Assuming a central exclusive production mechanism, the sensitivity of the search for the corresponding Higgs was studied, with some experimental aspects like signal and background rates. In another presentation, along the same lines of extending the Higgs sector beyond the SM, the search for the lightest neutral Higgs boson of a model containing triplets was discussed [@KH]. By means of some Monte Carlo simulations, it was found that the central exclusive production mechanism is again a very powerful tool to study this new object.
Deep-inelastic scattering data in the very low-$x$ region is known to be dominated, in the Regge picture, by the Pomeron. By using a discretized version of the BFKL Pomeron, which generates discrete Regge pole solutions, an integrated positive gluon distribution was obtained [@DR]. It allows a good fit of the ZEUS $F_2$ data in the kinematic range $10^{-4}<x<10^{-2}$ and $4.5< Q^2<350\mbox{GeV}^2$ and this gluon distribution must be tested in hadronic collisions at the LHC.
In jet production at LHC, gaps between jets is an important issue which deserves serious theoretical studies, because, it is sensitive to various QCD processes. The phenomenological impact of the Coulomb gluon contributions and super-leading logarithms on the gaps between jets cross section, has been investigated [@SM].\
Acknowledgments
===============
I would like to thank the conference organizers for their warm hospitality at CERN and for making the 13th “Blois Workshop” a very successful meeting. I am also grateful to all the conference speakers for the high quality of their contributions.
[99]{} J. Dainton, Summary on Experimental Aspects, these proceedings. E. Ferreira, these proceedings. A. Kendi Kohara, E. Ferreira and T. Kodama, arXiv:hep-ph/0905.1955 (2009). C. Bourrely, J. Soffer and T.T. Wu, Nucl. Phys. [**B247**]{} 15 (1984). C. Bourrely, N.N. Khuri, A. Martin, J. Soffer and T.T. Wu, arXiv:hep-ph/0511135 (2005). O. Selyugin, these proceedings. S. Troshin, these proceedings, arXiv:hep-ph/0909.3926 (2009). R. Luddy, these proceedings. J.R. Cudell, Total Cross Section at the LHC, these proceedings. V. Kundrat, these proceedings, arXiv:hep-ph/0909.3199(2009). M.V. Lokajicek, these proceedings, arXiv:hep-ph/0906.3961 (2009). A. Martin, these proceedings, Phys. Rev. [**D80**]{} 065013 (2009). Chung-I. Tan, these proceedings. A. Kaidalov, these proceedings. M.G. Poghosyan, these proceedings, arXiv:hep-ph/0909.5156 (2009). E. Martynov, these proceedings. U. Maor, these proceedings, arXiv:hep-ph/0910.1196 (2009). E. Gotsman, E. Levin, U. Maor and J.S. Miller, arXiv:hep-ph/0903.0247 (2009) and references therein. M.G. Ryskin, A.D. Martin and V.A. Khoze, Eur. Phys. J. [**C60**]{} 265 (2009) and references therein. C. Bourrely, J. Soffer and T.T. Wu, Eur. Phys. J. [**C28**]{} 97 (2003) and references therein. E. Gotsman, these proceedings, arXiv:hep-ph/0910.0598 (2009). M. Strikman, these proceedings. L. Frankfurt, M. Strikman, D. Treleani and C. Weiss, Phys. Rev. Lett. [**101**]{} 202003 (2008). A. Kaidalov, V.A. Khoze, A.D. Martin and M.G. Ryskin, Eur. Phys. J. [**C33**]{} 261 (2004) and references therein. O. Teryaev, these proceedings. O. Teryaev, R. Pasechnik and A. Szczurek, these proceedings. J.R. Cudell, Central Exclusive Production, these proceedings. S. Heinemeyer, these proceedings, arXiv:hep-ph/0909.4665 (2009). K. Huitu, these proceedings. D. Ross, these proceedings. S. Marzani, these proceedings.
[^1]: Invited talk at the 13th International Conference on Elastic and Diffractive Scattering, “13th Blois Workshop”, CERN, Geneva, June 29-July 3, 2009.
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---
abstract: |
Chemically peculiar stars present spectral and photometric variability with a single period. In the oblique rotator model, the non homogeneous distribution of elements on the stellar surface is at the origin of the observed variations. As to helium weak stars, it has been suggested that photometric and helium line equivalent width variations are out of phase. To understand the behaviour of helium in CP stars, we have obtained time resolved spectra of the He[i]{}5876 [Å]{} line for a sample of 16 chemically peculiar stars in the spectral range B3 – A1 and belonging to different sub-groups.\
The He[i]{}5876 [Å]{} line is too weak to be measured in the spectra of the stars HD24155, HD41269, and HD220825. No variation of the equivalent width of the selected He line has been revealed in the stars HD22920, HD24587, HD36589, HD49606, and HD209515. The equivalent width variation of the He[i]{}5876 [Å]{} line is in phase with the photometric variability for the stars HD43819, HD171247 and HD176582. On the contrary it is out of phase for the stars HD28843, HD182255 and HD223640. No clear relation has been found for the stars HD26571 and HD177003.
author:
- 'G. Catanzaro, F. Leone'
- 'F. A. Catalano'
date: 'Received June 3, 1998; accepted August 3, 1998'
title: 'Variability of the He[i]{}5876 [Å]{} line in early type chemically peculiar stars'
---
Introduction
============
Among the early type stars of the main sequence various groups of chemically peculiar stars (henceforth CP stars) are found. According to the [*General Catalogue of Ap and Am stars*]{} (Renson et al. 1991), among the 6684 so far known or suspected CP stars, more than half (3427) are Am stars (CP1) the remaining 3257 being Bp or Ap. Among these only 190 stars belong to the He strong (82) or He weak (108) subgroups. Usually helium is underabundant in the coolest CP stars and overabundant in the hottest ones.
CP stars are characterised by spectral and photometric variations with a common period. In the oblique rotator model, proposed by Stibbs (1950), chemical elements are not homogeneously distributed on the stellar surface and the observed variations are due to the stellar rotation. Studying the photometric variability of six helium weak stars, Catalano & Leone (1996) found that the equivalent width of the He[i]{}4026[Å]{} varies out of phase with respect to the photometric variability.
To investigate the behavior of helium and its relation to the spectral and light variability, we have performed time resolved spectroscopy of the He[i]{}5876Åline for a sample of 16 CP stars in the B6-A1 spectral range and belonging to different peculiarity classes.
[rclcrrcccrlll]{} HD & HR & Sp. Type & V & $v_{e} \sin i$ & T$_{\rm eff}$ & log g & N & $\langle$EW$\rangle$ & $\sigma$ & Remarks & P(d) & Reference\
22920 & 1121 & B8 Si & 5.53 & 120 & 13700 & 3.72 & 8 & 125 & 13 & constant & &\
24155 & 1194 & B9 Si & 6.30 & 50 & 13700 & 3.96 & 9 & & & no line & &\
24587 & 1213 & B6 & 4.65 & 40 & 14100 & 4.23 & 18 & 325 & 20 & constant & &\
26571 & 1297 & B8 Si & 6.12 & 29 & 13000 & 3.16 & 18 & 145 & 15 & & 1.0646 & Winzer (1974)\
28843 & 1441 & B9 He wk& 5.81 & 70$^*$ & 15300 & 4.17 & 21 & 70 & 20 & out of phase & 1.373813 & Mathys et al. (1986)\
36589 & 1860 & B7 & 6.18 & 90 & 14000 & 3.97 & 9 & 350 & 8 & constant & &\
41269 & 2139 & B9 Si & 6.20 & 75 & 10800 & 3.82 & 10 & & & no line & &\
43819 & 2258 & B9 Si & 6.30 & 14 & 11100 & 3.66 & 12 & 95 & 30 & in phase & 15.0305 & Adelman (1997)\
49606 & 2519 & B8 HgMnSi& 5.85& 35 & 13100 & 3.83 & 14 & 145 & 7 & constant & &\
171247& 6967 & B8 Si & 6.42 & 60 & 11300 & 3.40 & 23 & 95 & 15 & in phase & 3.9124 & North (1992)\
176582& 7185 & B5 He wk & 6.41 & 145 & 18300 & 4.31 & 12 & 360 & 30 & in phase & 1.58175 & This work\
177003& 7210 & B3 He & 5.37 & 40 & 18700 & 4.11 & 14 & 705 & 25 & & 1.835 & This work\
182255& 7358 & B6 He wk & 5.13 & 45 & 14400 & 4.17 & 18 & 325 & 15 & out of phase & 1.26263 & This work\
209515& 8407 & A0 CrSiMg & 5.60 & 100 & 9600 & 3.73 & 7 & 40 & 5 & constant & &\
220825& 8911 & A1 CrSrEu & 4.94 & 30 & 10400 & 4.46 & 7 & & & no line & &\
223640& 9031 & B9 SiSrCr & 5.19 & 20 & 12400 & 3.62 & 6 & 60 & 15 & out of phase & 3.735239 & North et al. (1992)\
\
Observations and data analysis
==============================
For the chemically peculiar stars listed in Table \[listcp\] echelle spectra were obtained in 1995 at the 2.1 m telescope of the Complejo Astronómico El Leoncito equipped with a Boller & Chivens cassegrain spectrograph and in 1997 at the 91 cm telescope of the Catania Astrophysical Observatory equipped with a Czerney-Turner echelle spectrograph.\
The data were analysed by using IRAF package. The lines of the wavelength calibration lamp show that R=16000 for the 1997 data set and R=13000 for the 1995 data set. The achieved S/N was between 100 and 200. When possible, equivalent widths were measured by a Gaussian fit of spectral lines after having removed possible continuum slope; otherwise a measure of the area between the line profile and the continuum was obtained. Following Leone et al. (1995), we estimated the error in the measured equivalent width with the relation: $$\Delta W = \frac{1}{2} \left(2 \frac{v_e \sin i}{c} \lambda\right)
\frac{1}{S/N}
\label{errorbar}$$ where the quantity in brackets is the total extension of the line as deduced from the rotational broadening. Adopted $v_e \sin i$ values (Table1) are from SIMBAD with the exception of HD28843 whose projected rotational velocity was measured from the unblended Si[ii]{}5865[Å]{} line.\
The initial ephemeris of program stars were taken from Catalano & Renson (1984, 1988, 1997), and Catalano et al. (1991, 1993), and references therein. If necessary, periods were established using our spectral observations and Hipparcos photometry[^1]. A least squares fit of measured EW’s and H$_{\rm p}$ magnitudes has been performed by adopting the function: $$\begin{aligned}
& & A_0 + A_1 \sin (2 \pi (t-t_0)/P + \phi_1) \nonumber \\
& & \qquad \qquad \qquad \qquad + A_2 \sin (4 \pi (t-t_0)/P + \phi_2) \end{aligned}$$ where $t$ is the JD date, $t_0$ is the assumed initial epoch, $P$ is the period in days. A sine wave and its first harmonic appear to be quite adequate functions to describe the light curves and the spectral variations (North 1984, Mathys & Manfroid 1985). The error in the period value has been evaluated according to the relation given in Horne & Baliunas (1986).
As to the coolest CP stars, the effective temperatures and gravities have been determined by means of [*ad hoc*]{} Napiwotzki et al. (1993) relations. As to helium peculiar stars, Hauck & North (1993) found that [*classical*]{} methods are still reliable to determine their effective temperature. Thus, we have used the Moon & Dworetsky (1985) grids as coded by Moon (1985). The source of Strömgren photometry was SIMBAD.
To ascertain if the selected stars present a peculiar helium abundance we have compared the measured equivalent widths of the He[i]{}5876[Å]{} line with the NLTE computations of Leone & Lanzafame (1998) for solar composition stars with $\log$g = 3.5, 4.0 and 4.5 and 9000 K $<$ T$_{\rm eff} <$ 19000 K.
Individual stars
================
HD22920 (= HR1121 = 22 Eri)
---------------------------
According to Maitzen (1976), the silicon star HD22920 has a low value of the photometric peculiarity index $\Delta a$ (= 0.011). Photometric observations have been carried out by Bartholdy (1988) who found this star to be variable with a period of 3.95 d. North (1990 priv. comm.) found two possible periods almost equally probable: 3.96 d, very close to Bartholdy’s (1988), and 1.33 d.
No evidence of variability has been found in our spectra of the He[i]{}5876[Å]{} line. The mean value of the equivalent width is: $\langle$EW$\rangle$ = 125 $\pm$ 13 m[Å]{}. The effective temperature of HD22920 resulting from Napiwotzki et al. (1993) relation is T$_{\rm eff}$ = 13700 K. Figure9 shows that the He[i]{}5876[Å]{} line equivalent width of HD22920 is smaller than expected for a main sequence star of the same effective temperature.
HD24155 (= HR1194 = V766 Tau)
-----------------------------
The UBV photometric variability of HD24155 has been studied by Winzer (1974), who reported a possible period of 2.5352 d. Renson & Manfroid (1981) found P = 2.53465 $\pm$ 0.00015 d. The observed light curves show a quite large amplitude (0.10 mag) with very sharp minima and quite broad maxima, hence this star is the fourth largest amplitude silicon star known, exceeded only by HD215441, CU Vir and HR7058.
Assuming Renson & Manfroid’s (1981) period, our nine spectra are well distributed in phase, but none of them shows a measurable He[i]{}5876 [Å]{} line. Because of its effective temperature of 13700 K, HD24155 is an extremely helium weak star (Fig.9).
HD24587 (= HR1213 = $\tau^8$ Eri)
---------------------------------
HD24587 is listed in the [*General Catalogue of Ap and Am stars*]{} by Renson et al. (1991) as a suspected CP star. Feinstein (1978) used this star as standard for his measurements of hydrogen lines in He weak stars. HD24587 has in fact been considered as a standard for [*uvby*]{} (Garnier 1972 - personal communication to Mathys et al. 1986) and [*$\beta$*]{} photometry (Strauss & Ducati 1981). Mathys et al. (1986) found this star to be a light variable with a period of 1.728 d and concluded that the light curves resemble those of many CP stars. Recently Leone & Catanzaro (1998) have performed a spectroscopic study and concluded that this star presents chemical elements which are slightly underabundant with respect to main sequence stars.
Our measurements of the He[i]{}5876 [Å]{} line do not show any variation of the equivalent width; the mean value is: $\langle$EW$\rangle$ = 325 $\pm$ 20 m[Å]{}. From Moon’s algorithm we find that T$_{\rm eff}$ = 14100 K and the EW of the He[i]{}5876 [Å]{} line is close to the value expected for a main sequence star (Fig.9). These facts confirm Leone & Catanzaro’s (1998) conclusion that HD24587 is not a peculiar star.
HD26571 (= HR1297)
------------------
The peculiarity of HD26571 was first noted by Gulliver (1971) and independently confirmed by Bond (1972). On the basis of his spectra, Gulliver (1971) described this star as a spectrum variable. Photometric observations were obtained by Winzer (1974), who found HD26571 to vary with a period of 1.0646 d.
Figure \[hd26571\] shows the EW variation of the He[i]{}5876 [Å]{} line versus the phase computed assuming the initial epoch coincident with the light maximum as given by Winzer (1974): $$JD(UBV \; max.) = 2441246.81 + 1.0646 E .$$
Winzer (1974) has not published the uncertainty in the period determination, hence we have estimated the error by applying the Horne & Baliunas (1986) relation to our data and have found that equivalent widths are phased with an expected error $\Delta \Phi$ = 0.3. This means that no phase relation can be determined between the photometric and our spectral variations.
HD28843 (= HR1441 = DZ Eri)
---------------------------
HD 28843 was classified as B9IV Si He-wk by Davis (1977) and it is classified as B9 He wk in the [*General catalogue of Ap and Am Stars*]{}. The photometric variability of HD28843 had been detected for the first time by Cousins & Stoy (1966) while its peculiar character had been confirmed by Jaschek et al. (1969). Photometric observations of this star have been carried out by Pedersen & Thomsen (1977) who found variability with a period of 1.374 $\pm$ 0.006 d. This value was improved by Pedersen (1979) to the value 1.37375 $\pm$ 0.00035 d. Manfroid et al. (1984) also used Pedersen & Thomsen’s (1977) data to improve the period, their most probable value being 1.373813 $\pm$ 0.000012 d. Mathys et al. (1986) concluded that the ambiguity in the choice of the best peak in the periodogram could be removed by inclusion of the measurements of Dean (1980), confirming the value obtained by Manfroid et al. (1984). Further photometric observations have been carried out by Waelkens (1985), by the team of the ESO Long-Term Photometry of Variable Project (Manfroid et al. 1994, Sterken et al. 1995), and by the team of Hipparcos (ESA, 1997).\
Our spectroscopic data are plotted in Fig. \[hd28843\], versus the phase computed from the ephemeris elements of Mathys et al. (1986): $$JD(uvby \; max) = 2442777.5 + 1.373813 E
\label{ephe}$$ The amplitude of the equivalent line width variations is of the order of 75 m[Å]{}. From Fig. \[hd28843\] a clear anti-correlation is evident between the He[i]{}5876 [Å]{} equivalent line width and all the Hipparcos and $uvby$ light curves, in the sense that light minima occur at the phase of maximum He[i]{}. Because of the period error determined by Manfroid et al. (1984), the expected phase error in our EW variations is $\Delta\Phi$ = 0.03. EW variations of the He[i]{}5876 [Å]{} line are then out of phase with respect to light variations.\
Even if most of our equivalent widths periodically vary with the ephemeris computed with Eq. (\[ephe\]), we have found several (5 out of 21) spectra where the He[i]{}5876 line is absent (Fig.2).
HD36589 (= HR1860)
------------------
In the [*General Catalogue of Ap and Am stars*]{}, the star HD36589 is a suspected CP star. Bossi & Guerrero (1989) and Hao et al. (1996) have used it as a comparison star for photometric observations. Leone & Catanzaro (1998) derived chemical abundances and found that HD36589 shows nearly solar values and no evidence of spectral variability.
From our spectra we confirm this result: no evidence of variation is found in the He[i]{}5876 [Å]{} equivalent line width. On the hypothesis that HD36589 is not a CP star, we have determined T$_{\rm eff}$ = 14000 K by mean of Moon’s relations and found that the average value of the equivalent widths ($\langle W \rangle$ = 350 $\pm$ 8 m[Å]{}) is very close to that of normal main sequence stars of the same spectral type (Fig.9).
HD41269 (= HR2139)
------------------
This star has been classified as B9p by Cowley et al. (1969) who described it as a mild silicon star. On the basis of a single observing run in UBV, Winzer (1974) found a period of 1.68 d, although he could not rule out the resonance period of 2.47 d, because of the few observed points.\
In our spectra the He[i]{}5876 [Å]{} line is too weak to be measured. According to Napiwotzki et al. (1993) relation, T$_{\rm eff}$ = 10800 K. Figure \[summary\] shows that the helium abundance is lower than the expected value for a main sequence star of this temperature.
HD43819 (= HR2258 = HIP30019 = V 1155 Ori)
------------------------------------------
Cowley (1972) classified this star as B9IIIp Si. Photometric measurements of HD43819 were performed in the UBV system by Winzer (1974) who found a light variation with a period of 1.0785 d. Later on Maitzen (1980) found the light variation to occur with two possible periods: 0.93 d and 1.077 d. A spectroscopic study of this sharp lined star ($v_{e} \sin i =
14$ km s$^{-1}$, Wolff & Preston 1978) was carried out by Lopez-Garcia & Adelman (1994), who found iron peak elements ten times overabundant and rare earths 1000 times overabundant with respect to solar values. From photometric $uvby$ observations Adelman (1997) has deduced a period of 15.0305$\pm$0.0003 d, longer than Winzer’s (1974) and more consistent with the low rotational velocity of this star. This period is also confirmed by the Hipparcos observations (Fig.3).
Our He[i]{}5876 [Å]{} equivalent line widths are plotted in Fig. \[hd43819\] versus the phase computed by means of Adelman’s (1997) ephemeris elements: $$JD(U \; max) = 2441254.16 + 15.0305 E$$ The observed EW variation has an amplitude of the order of 80 m[Å]{}. From Fig. \[hd43819\] we see a clear in-phase correlation between light and spectral variations. This correlation is expected to be real, the phase error being $\Delta\Phi\sim$ 0.01.
HD49606(= HR2519 = 33 Gem)
--------------------------
The star HD49606 is classified as a B8HgMnSi star by Renson et al. (1991). Photometric observations of HD49606 were performed by Chunakova et al. (1981), who found the light variations to occur with a period of 3.099 d, and by Glagolevskii et al. (1985), who found two possible period values, namely 3.3546 d and 1.41864 d.
The He[i]{}5876 [Å]{} equivalent line width observed in our spectra does not show any detectable variation, so that we consider this line does not vary with time. The average equivalent width is $\langle W
\rangle = 145 \pm 7$ m[Å]{}. This result confirms the one obtained by Hubrig & Launhardt (1993) who searched for variations in the equivalent width of helium and some metallic lines and did not find any evidence of variability.\
According to an elemental abundances analysis performed by Adelman & al. (1996) our observations show that helium is underabundant with respect to solar composition (Fig.\[summary\]).
HD171247 (= HR6967 = HIP90971)
------------------------------
The photometric variability of this star was detected by North (1992) who found the period to be 3.9124 d. This value of the period is confirmed by the Hipparcos photometry (1997) which gives an error on the period equal to 0.0004 d applying Horne & Baliunas (1986) formula.
Computing the phase of the measured equivalent widths by means of North’s (1992) ephemeris: $$JD([U]{\rm \, Geneva\, max}) = 2447178.245 + 3.9124 E$$ we find a sinusoidal variation of the He[i]{}5876[Å]{} line strength (Fig.\[hd171247\]) with an amplitude of the order of 45 m[Å]{}.\
Converting the period error to a phase error, we get $\Delta\Phi$ = 0.07. We can thus conclude that H$_{\rm p}$ and EW variability are in phase for the silicon star HD171247.
HD176582 (= HR7185 = HIP93210)
------------------------------
This star is classified as a silicon star (Renson et al. 1991). Spectroscopic observations of the He[i]{}4026 [Å]{} line strength were carried out by Pedersen (1976), who found a variation with the period 0.8143 d. The period is not representative of the variability of Hipparcos photometry and He[i]{}5876 [Å]{} equivalent width.
By using our spectroscopic data and Hipparcos photometry we found a period of 1.5817$\pm$0.0003 d. The observations are plotted in Fig. \[hd176582\] versus the phase computed by means of the ephemeris elements: $$JD(EW \; min.) = 2450624.6410 + 1.5817 E$$ From this figure we see that both curves show a clear evidence of a double-wave variation. The observed EW amplitude is of the order of 40 m[Å]{}. The expected phase error is $\Delta\Phi$ = 0.05, and the Hipparcos photometry appears to vary in phase with the equivalent width variations of the He[i]{}5876 [Å]{} line.
HD177003 (= HR7210 = HIP93299)
------------------------------
Schöneich & Zelwanowa (1984) from their photometric observations in the UV filters found two possible periods: 0.66 d and 2.1 d. From UBVRI photometric observations, Vetö (1993) found this star to be light variable with a period of 0.724 d and amplitudes of about 0.1 mag. in all filters. An analysis of Hipparcos photometric data does not give a clear variability period.
Our spectroscopic data are not consistent with the periods given in the literature. A period search of our data yelds two possible values: 1.835$\pm$0.004 d and 2.186$\pm$0.005 d. The He[i]{}5876[Å]{} line equivalent width variation is plotted in Fig. \[hd177003\] versus the phase computed by means of the ephemeris elements: $$JD(EW \; max.) = 2450629.4099 + 1.835 E$$ where we have adopted the shorter value of the period which has a smaller $\chi^{2}$ value. The variation shown in Fig. \[hd177003\] has an amplitude of the order of 75 m[Å]{}.
The photometric variability is not clear for the H$_{\rm p}$ filter assuming this period (Fig.6) and no conclusion can be drawn concerning a possible phase relation between photometric and spectral variations of the He[i]{}5876[Å]{} line.
HD182255 (= HR7358 = HIP95260 = 3 Vul)
--------------------------------------
According to Hube & Aikman (1991) this star is a nonradial pulsator. It has also been observed by Hipparcos, from whose photometry a period of 1.26239 d has been derived. However this value of the period is not perfectly consistent with our spectroscopic observations; instead, by using both sets of data the most probable value appears to be 1.26263 $\pm$ 0.00005 d. Adopting this period, the measured EW of the He[i]{}5876[Å]{} line are plotted in Fig. \[hd182255\] versus the phase computed by means of the ephemeris elements: $$JD(EW \; max.) = 2450650.4729 + 1.26263 E$$ The observed EW amplitude is of the order of 65 m[Å]{}. Since the period error corresponds to a phase error $\Delta\Phi$ = 0.09, the reported out of phase relation between photometric and helium line variations is expected to be real (Fig.7).
HD209515 (= HR8407 = V1942 Cyg)
-------------------------------
This star was classified as A0p by Osawa (1965) and as A0 IV by Cowley et al. (1969). From his photometric observations, Winzer (1974) found a period of 0.63703 d, concluded that the observed photometric variation is typical for a silicon star and suggested that the correct classification should be A0p Si.\
The equivalent width of the He[i]{}5876[Å]{} line of the cool CP star HD209515 is constant: 40$\pm$5 m[Å]{}. This value of equivalent width is consistent with the helium abundance of a main sequence star (Fig.9).
HD220825 (= HR8911 = $\kappa$ Psc)
----------------------------------
The variability of HD220825 had been detected for the first time by Rakosch (1962) who found a period of 0.5805 d. Recently, Ryabchikova et al. (1996) determined the period to be 1.418 d and magnetic observations performed by Borra & Landstreet (1980) are also consistent with this value.
The He[i]{}$\lambda$5876 [Å]{} line is too weak to be measured. Assuming T$_{\rm eff}$ = 10400 K, Fig.9 shows that helium is underabundant in HD220825 with respect to main sequence stars.
HD223640 (= HR9031 = HIP 117629 = 108 Aqr = ET Aqr)
---------------------------------------------------
The photometric variability of this star has been studied by several authors. Morrison & Wolff (1971) found HD223640 to be variable in the Strömgren system with a period of 3.73 d and noted that light curves show quite the same behaviour in all filters. Spectroscopic observations were carried out by Megessier & Garnier (1972) who found strongly variable the Ti and Sr lines and constant the Fe lines. Moreover the Ti lines correlate with photometric variations in the sense that Ti lines are strongest when the star is brightest. This correlation has been interpreted by Megessier (1974, 1975) in terms of the oblique rotator model taking also into account the sign changes of the magnetic field measurements by Babcock (1958). Photometric observations in the Geneva system have been performed by North et al. (1992), they found a period of 3.735239 $\pm$ 0.000024 d which is consistent with the magnetic data. This period has been confirmed by photometric observations in the uvby system performed by Adelman & Knox (1994) and Adelman (1997).\
According to North et al. (1992), we phased the measured equivalent widths of the He[i]{}5876[Å]{} line by means of the ephemeris elements: $$JD(uvby \; max.) = 2444696.820 + 3.735239 E$$ The period uncertainty corresponds to a phase error $\Delta\Phi$ = 0.004. There is evidence of an anti-correlation between light and spectroscopic curves. The He[i]{} line is strongest in coincidence with the light minimum: the helium distribution on the surface of HD223640 is then not coincident with the Ti distribution.
Conclusion
==========
In this paper we have presented spectroscopic observations of the He[i]{}5876 [Å]{} line in 16 CP stars (see Table \[listcp\]). In the case of HD26571, HD28843, HD43819, HD171247 and HD223640 the literature period values are accurate enough to represent our observations quite well. No variability has been detected in the stars HD22920, HD24587, HD36589, HD49606, and HD209515, while the He[i]{}5876 [Å]{} line has been found to be too weak to be measured in the stars HD24155, HD41269 and HD220825. In the case of the remaining stars, ie. HD176582, 177003, and HD182255, we have refined the value of the period by using both our own and literature data, when available.
In the attempt to study the phase correlations between light and spectral variations, we have calculated the error on phase. From these calculations we can see that three stars, HD28843 (He weak), HD182255 (He weak) and HD223640 (B9SiSrCr), show a clear anti-phase correlation. The equivalent width of the He[i]{}5876[Å]{} line varies in phase with the photometric variations for the stars HD43819 (B9 Si), HD171247 (B8Si) and HD176582 (He weak). As to HD26571 the error on $\Phi$ is too large to draw any conclusion. In the case of HD177003 (B3 He) nothing can be said since the Hipparcos light curve has too large a dispersion and a low amplitude. Hence no unique correlation exists, and this fact is independent of the spectral types of both groups of stars, which are all in the B5-B9 range. This result confirms the one obtained by Catalano & Leone (1996). In an attempt to clarify the nature of the correlation between light and helium lines variations, these authors compared the emerging fluxes of two atmosphere models with the same effective temperature (T$_{eff}$=15000 K) and gravity (log g=4.0) but with different helium abundance. The models were computed by means of the ATLAS9 code (Kurucz 1993) and are characterized by solar and zero helium abundance. By comparing these fluxes, they found no observable magnitude differences and concluded that the photometric variations presented by helium weak stars cannot be entirely ascribed to the non homogeneous distribution of helium on the stellar surface.\
Figure \[summary\] shows the measured average value of the equivalent width of the He[i]{}5876[Å]{} line together with the theoretical behaviour computed in the NLTE approximation by Leone & Lanzafame (1998) for solar composition stars. We conclude the helium abundance is not peculiar for the stars HD36589, HD43819, HD171247, HD177003, HD182255 and HD209515, while helium is underabundant in the remaining stars. It is worthy note, that the equivalent width of the He[i]{}5876[Å]{} line for HD182255 is close to the value of a solar composition main sequence star, even though this star is classified as an helium weak star.
Adelman S. J., 1997, A&AS 127, 421 Adelman S. J., Knox jr J. R., 1994, A&AS 103, 1 Adelman S. J., Philip A. G. D., Adelman C. J., 1996, MNRAS 282, 953 Babcock H. W., 1958, ApJS 3, 141 Bartholdy P., 1988, in: Halbwachs J.-L., Jasniewicz G., Egret D. (eds), Detection et classification des etoiles variables. Comptes Rendus des Journees de Strasbourg, 10me reunion, 21 April 1988, p. 77 Bond H. E., 1972, PASP 84, 446 Borra E. F., Landstreet J. D., 1980, ApJS 42, 421 Bossi M., Guerrero G., 1989, IBVS 3326 Catalano F. A., Leone F., 1996, A&A 311, 230 Catalano F. A., Renson P., 1984, A&AS 55, 371 Catalano F. A., Renson P., 1988, A&AS 72, 1 Catalano F. A., Renson P., 1997, A&AS 121, 57 Catalano F. A., Renson P., Leone F., 1991, A&AS 86, 59 Catalano F. A., Renson P., Leone F., 1993, A&AS 98, 269 Chunakova N. M., Bychov V. D., Glagolevskii Yu. V., 1981, Soobshch. Spetsial’noi Astrofiz. Obs 31, 5 Cousins A. W. J., Stoy R. H., 1966, R. Obs. Bull. No. 121 Cowley A., 1972, AJ 77, 750 Cowley A., Cowley C., Jaschek M., Jaschek C., 1969, AJ 74, 375 Davis R. J., 1977, ApJ 213, 105 Dean J. F., 1980, Mon. Notes Astron. Soc. S. Afr. 39, 13 Feinstein A., 1978, Rev. Mex. Astron. Astrofis. 2, 331 ESA SP-1200, 1997 Glagolevskii Yu. V., Panov K., Chunakova N. M., 1985, Pis’ma AZh 11, 749 Gulliver A. F., 1971, Thesis, University of Toronto Hao J. X., Huang L., Guo Z. H., 1996, A&A 308, 499 Hauck B., North P., 1993, A&A 269, 403 Horne J.H., Baliunas S.L., 1986, ApJ 302, 757 Hube D. P., Aikman G. C. L., 1991, PASP 103, 49 Hubrig S., Launhardt R., 1993, Line Variability in 33 Gem? In: Dworetsky M. M., Castelli F., Faraggiana R. (eds) Proc. IAU Coll. 138, Peculiar versus Normal Phenomena in A-Type and Related Stars. ASP Conf. Series 44, p. 350 Jaschek M., Jaschek C., Arnal M., 1969, PASP 81, 650 Kurucz R. L., 1993, A new opacity-sampling model atmosphere program for arbitrary abundances. In: M. M. Dworetsky, F. Castelli, R. Castelli (eds.) IAU Col. 138, Peculiar versus normal phenomena in A-type and related stars. A.S.P. Conferences Series Vol. 44, p. 87 Leone F., Catanzaro G., 1998, A&A 331, 627 Leone F., Lanzafame A.C., 1998, A&A 330, 306 Leone F., Lanzafame A. C., Pasquini L., 1995, A&A 293, 457 Lopez-Garcia Z., Adelman S. J., 1994 A&AS 107, 353 Maitzen H. M., 1976, A&A 51, 223 Maitzen H. M., 1980, IBVS 1735 Manfroid J., Mathys G., Cousins A. W. J., 1984, IBVS 2625 Manfroid J., Sterken C., Cunow B. et al., 1994, Third Catalogue of Stars Measured in the Long-Term Photometry of Variable Project (1990-1992), ESO Scientific Report No. 14 Mathys G., Manfroid J., 1985, A&AS 60, 17 Mathys G., Manfroid J., Renson P., 1986, A&AS 63, 403 Megessier C., 1974, A&A 34, 53 Megessier C., 1975, A&A 39, 263 Megessier C., Garnier R., 1972, Astrophys. Lett. 11, 113 Moon T. T., 1985 in: Communications from the University of London Observatory No. 78 Moon T. T., Dworetsky M. M., 1985, MNRAS 217, 305 Morrison N. D., Wolff S. C., 1971, PASP 83, 474 Napiwotzki R., Schönberner D., Wenske V., 1993, A&A 268, 653 North P., 1984, A&AS 55, 259 North P., 1992, Photometric Periods of Some Old Si Stars. In: Glagolevskii Yu. V., Romanyuk I. I. (eds) Proc. Int. Meeting on the Problem Physics and Evolution of the Stars. Stellar Magnetism. NAUKA, Sankt-Petersburg, p. 73 North P., Brown D. N., Landstreet J. T., 1992, A&A 258, 389 Osawa K., 1965, Ann. Tokyo Astron. Obs. Ser. 2, 9, 123 Pedersen H., 1976, A&A 49, 217 Pedersen H., 1979, A&AS 35, 313 Pedersen H., Thomsen B., 1977, A&AS 30, 11 Rakosch K. D., 1962, Lowell Obs. Bull. 5, 227 Renson P., Manfroid J., 1981, A&AS 44, 23 Renson P., Gerbaldi M., Catalano F. A., 1991, A&AS 89, 429 Ryabchicova T. A., Pavlova V. M., Davydova E. S., Piskunov N. E., 1996, Astronomy Letters 22, 822 Schöneich W., Zelwanowa E., 1984, in Magnetic Stars ed. V. Khokhlova et al., Proceedings of the 6-th Conference on “Physics and Evolution of Stars”, Riga, April 10-12 1984, p.73 Sterken C., Manfroid J., Beele D., et al., 1995, Fourth Catalogue of Stars Measured in the Long-Term Photometry of Variable Project (1992-1994), ESO Scientific Report No. 16 Stibbs D. W. N., 1950, MNRAS 110, 395 Strauss F. M., Ducati J. R., 1981, Revised Catalogue of Stellar Rotational Velocities Vetö B., 1993, Do Bp Stars have “Flares”? In: Dworetsky M. M., Castelli F., Faraggiana R. (eds) Proc. IAU Coll. 138, Peculiar versus Normal Phenomena in A-Type and Related Stars. A.S.P. Conferences Series 44, p. 340 Waelkens C., 1985, A&AS 61, 127 Winzer J. E., 1974, Thesis, University of Toronto Wolff S. C., Preston G. W., 1978 ApJS, 37, 371
[^1]: The Hipparcos filter, referred to as Hp, extends from 3550 [Å]{} to 8900 [Å]{} with the maximum at 4350 [Å]{}. The typical accuracy of Hipparcos measurements, at the 8th magnitude, is given as 0.0015 mag (ESA, 1997).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We compute the complexity of the classes of operators $\mathfrak{G}_{\xi, \zeta}\cap \mathcal{L}$ and $\mathfrak{M}_{\xi, \zeta}\cap \mathcal{L}$ in the coding of operators between separable Banach spaces. We also prove the non-existence of universal factoring operators for both $\complement \mathfrak{G}_{\xi, \zeta}$ and $\complement \mathfrak{M}_{\xi, \zeta}$. The latter result is an ordinal extension of a result of Johnson and Girardi.'
author:
- 'R.M. Causey'
title: The complexity of some ordinal determined classes of operators
---
Introduction
============
In this work, we investigate the complexity of some recently isolated operator ideals from two different points of view. The first point of view is by the classical search for a universal factoring operator for the complement of the ideal. The second point of view makes use of descriptive set theory and the coding $\mathcal{L}$ of the class of operators between separable Banach spaces first given in [@BF]. The ideals of interest are ordinal-defined classes which are related to three important ideals: The weak Banach-Saks operators $\mathfrak{wBS}$, the completely continuous operators $\mathfrak{V}$, and the class $\mathfrak{DP}$ whose space ideal is the class of spaces with the Dunford-Pettis property. Each of these classes is defined by the behavior of weakly null sequences. Therefore it is natural to use the weakly null hierarchy defined by the Banach-Saks index of a weakly null sequence defined in [@AMT] to define quantified classes. We give the the formal definition of $\xi$-weakly null in Section $2$. Heuristically, given a weakly null sequence $(x_n)_{n=1}^\infty$, an ordinal assignment $\mathcal{BS}((x_n)_{n=1}^\infty)$ is defined which measures how weakly null the sequence $(x_n)_{n=1}^\infty$ is. Sequences with smaller Banach-Saks index are “more” weakly null than sequences with larger index. Given a Banach space $X$, we can define for each $0\leqslant \xi\leqslant \omega_1$ the set $\mathcal{WN}_\xi(X)$ to be the set of all weakly null sequences $(x_n)_{n=1}^\infty$ in $X$ with $\mathcal{BS}((x_n)_{n=1}^\infty)\leqslant \xi$. The properties of these classes relevant to this work are summarized in the following items.
(i) $\mathcal{WN}_0(X)$ consists of the norm null sequences in $X$,
(ii) $\mathcal{WN}_1(X)$ consists of those sequences in $X$ such that every subsequence has a further subsequence whose Cesaro means converge to zero in norm.
(iii) $\mathcal{WN}_{\omega_1}(X)=\cup_{\xi<\omega_1}\mathcal{WN}_\xi(X)$ is the set of all weakly null sequences in $X$.
Let us recall definitions of the classes $\mathfrak{wBS}, \mathfrak{V}$, and $\mathfrak{DP}$ using the notation from the previous paragraph The class $\mathfrak{wBS}$ is the class of all operators $A:X\to Y$ such that for every $(x_n)_{n=1}^\infty\in \mathcal{WN}_{\omega_1}(X)$, $(Ax_n)_{n=1}^\infty\in \mathcal{WN}_1(Y)$. The class $\mathfrak{V}$ is the class of all operators $A:X\to Y$ such that for every $(x_n)_{n=1}^\infty \in\mathcal{WN}_{\omega_1}(X)$, $(Ax_n)_{n=1}^\infty \in \mathcal{WN}_0(Y)$. The class $\mathfrak{DP}$ is the class of all operators $A:X\to Y$ such that for each $(x_n)_{n=1}^\infty\in \mathcal{WN}_{\omega_1}(X)$ and $(y^*_n)_{n=1}^\infty\in \mathcal{WN}_{\omega_1}(Y^*)$, $\lim_n y^*_n(Ax_n)=0$. Now for $0\leqslant \zeta,\xi\leqslant \omega_1$, we let $\mathfrak{G}_{\xi, \zeta}$ denote the class of all operators $A:X\to Y$ such that for each $(x_n)_{n=1}^\infty\in \mathcal{WN}_\xi(X)$, $(Ax_n)_{n=1}^\infty \in \mathcal{WN}_\zeta(Y)$. Then $\mathfrak{wBS}=\mathfrak{G}_{\omega_1, 1}$ and $\mathfrak{V}=\mathfrak{G}_{\omega_1, 0}$. It is easily verified that if $(x_n)_{n=1}^\infty\in \mathcal{WN}_\xi(X)$, then $(Ax_n)_{n=1}^\infty \in \mathcal{WN}_\xi(Y)$. From this it follows that $\mathfrak{G}_{\xi, \zeta}$ is simply the class $\mathfrak{L}$ of all bounded, linear operators when $0\leqslant \xi\leqslant \zeta\leqslant \omega_1$. Therefore we will be interested in these classes only in the non-trivial case $0\leqslant \zeta<\xi\leqslant \omega_1$. The classes $(\mathfrak{G}_{\xi, \zeta})_{0\leqslant \zeta<\xi\leqslant \omega_1}$ are closed, distinct, injective, two-sided ideals which contain all compact operators [@Causey1] and each of which contains the class $\mathfrak{V}$. For $0\leqslant \zeta, \xi\leqslant \omega_1$, we let $\mathfrak{M}_{\xi, \zeta}$ denote the class of all operators $A:X\to Y$ such that for each $(x_n)_{n=1}^\infty\in \mathcal{WN}_\xi(X)$ and $(y^*_n)_{n=1}^\infty\in \mathcal{WN}_\zeta(Y^*)$, $\lim_n y^*_n(Ax_n)=0$. Then $\mathfrak{DP}=\mathfrak{M}_{\omega_1, \omega_1}$. Furthermore, $\mathfrak{M}_{\xi, \zeta}=\mathfrak{L}$ if $\xi=0$ or $\zeta=0$, so we will restrict our attention to the cases $1\leqslant \xi, \zeta\leqslant \omega_1$. The classes $(\mathfrak{M}_{\xi, \zeta})_{1\leqslant \xi, \zeta\leqslant \omega_1}$ are closed, distinct, non-injective, two-sided ideals which contain all compact operators [@Causey1]. One benefit of defining and studying such classes is that results which fail for a set which is too complex may have (sometimes quantitatively weaker) positive results when we restrict our attention to sets with lower complexity. Results of this type using descriptive set theory can be found in [@BC2] and [@BF]. To that end, we show that when restricting to countable ordinals, we obtain strictly lower complexity for the classes in the coding of operators between separable Banach spaces. We also compute complexity of the associated space ideals in the coding $\textbf{SB}$ of separable Banach spaces, which complements recent computations of Kurka of the classes of separable Schur spaces and separable spaces with the Dunford-Pettis property. Kurka’s results are the spatial versions of items $(iii)$ and $(vi)$ of the following theorem.
(i) For $0\leqslant \zeta<\xi<\omega_1$, the class $\mathfrak{G}_{\xi, \zeta}\cap \mathcal{L}$ is $\Pi_1^1$-complete and therefore non-Borel in the coding $\mathcal{L}$ of operators between separable Banach spaces.
(ii) For $0\leqslant \zeta< \xi<\omega_1$, the class $\textsf{\emph{G}}_{\xi, \zeta}\cap \textbf{\emph{SB}}$ of spaces $X$ such that $I_X\in \mathfrak{G}_{\xi, \zeta}$ is $\Pi_1^1$-complete and therefore non-Borel in the coding $\textbf{\emph{SB}}$ of separable Banach spaces.
(iii) For each $0\leqslant \zeta<\omega_1$, the class $\mathfrak{G}_{\omega_1, \zeta}\cap \mathcal{L}$ is $\Pi_2^1$-complete and therefore not $\Sigma_2^1$ in $\mathcal{L}$.
(iv) For $1\leqslant \zeta, \xi<\omega_1$, the class $\mathfrak{M}_{\xi, \zeta}\cap \mathcal{L}$ is $\Pi_1^1$-complete and therefore non-Borel in the coding $\mathcal{L}$ of operators between separable Banach spaces.
(v) For $1\leqslant \zeta, \xi<\omega_1$, the class $\textsf{\emph{M}}_{\xi, \zeta}\cap \textbf{\emph{SB}}$ of spaces $X$ such that $I_X\in \mathfrak{M}_{\xi, \zeta}$ is $\Pi_1^1$-complete and therefore non-Borel in the coding $\textbf{\emph{SB}}$ of separable Banach spaces.
(vi) For each $1\leqslant \zeta, \xi\leqslant \omega_1$ with $\max\{\xi, \zeta\}=\omega_1$, the class $\mathfrak{M}_{\omega_1, \zeta}\cap \mathcal{L}$ is $\Pi_2^1$-complete and therefore not $\Sigma_2^1$ in $\mathcal{L}$.
We also investigate the classes above by searching for one or a class of universal factoring operators for the complement of the ideal. If $\mathfrak{I}$ is an ideal and $U:F\to G$ is a member of the complement $\complement \mathfrak{I}$ which factors through another operator $A:X\to Y$, then $A\in \complement \mathfrak{I}$. This motivates a search for an easily understood class $\mathfrak{U}\subset \complement \mathfrak{I}$ such that for each $A:X\to Y\in \complement \mathfrak{I}$, there exists $U:F\to G\in \mathfrak{U}$ which factors through $A$. The best result of this type would be for $\mathfrak{U}$ to be a singleton. One notable of such results is the universal non-weakly compact operator $\Sigma:\ell_1\to \ell_\infty$ of Lindenstrauss and Pełczński [@LP] which takes the $n^{th}$ member of the canonical $\ell_1$ basis to the sequence $(1, 1, \ldots, 1, 0, 0, \ldots)$, where $1$ appears $n$ times. Another example is Johnson’s universal non-compact operator $J:\ell_1\to \ell_\infty$ [@Johnson] which takes the canonical $\ell_1$ basis to the canonical $c_0$ basis. A simple, universal class for the complement of an ideal can provide a route to investigating that ideal. For example, Bourgain’s result [@Bourgain] that the binary trees of arbitrary, finite height embed with uniformly bounded distortion into any non-superreflexive Banach space uses the fact that the universal non-super-weakly compact operator factors through the identity of such a space. A generalization of this argument was used in [@CD] to prove the analogous operator version of Bourgain’s spatial result. In certain instances, one can show that no universal operator exists for a given class (see, for example, [@GJ] and [@Oikhberg]), or that the existence of a “nice” class of universal factoring operators is impossible (see [@BC], where it was shown by descriptive set theoretic considerations that no Borel subset of $\mathcal{L}$ can be a universal factoring class for $\mathfrak{V}$). As we have quantified classes which depend on ordinals parameters, one can ask for weaker conclusions by, for example, searching for a “nice” subset $\mathfrak{U}$ of $\complement \mathfrak{G}_{\xi+1, \zeta}$ such that each member $A:X\to Y$ of $\complement \mathfrak{G}_{\xi, \zeta}$ factors of member of $\mathfrak{U}$. This complements the negative result of Girardi and Johnson and offers another example of the aforementioned theme within descriptive set theory: Given an ordinal quantification on some class, restricting our attention to subsets whose ordinal quantification does not exceed some fixed, countable bound $\xi$ yields classes for which positive results hold, while the analogous results fail if we consider the entire class without a countable bound. Our negative and positive results regarding universal classes are summarized in the following theorem.
For the following theorem, if $F$ is a Banach space with basis $(f_i)_{i=1}^\infty$, and if $M=\{m_1<m_2<\ldots\}$ is an infinite subset of ${\mathbb{N}}$, we let $F_M$ denote the closed span in $F$ of the subsequence $(f_{m_i})_{i=1}^\infty$ of $(f_i)_{i=1}^\infty$.
(i) For $0\leqslant \zeta<\xi\leqslant \omega_1$, $\complement\mathfrak{G}_{\xi, \zeta}$ does not admit a universal operator.
(ii) For any $0\leqslant \zeta<\xi<\omega_1$, there exist a Banach space $F$ with basis $(f_i)_{i=1}^\infty$ and an operator $U:F\to \ell_\infty$ such that for each $\zeta<\beta<\xi$, each subsequence $(f_{m_i})_{i=1}^\infty$ of the basis $(f_i)_{i=1}^\infty$ and $A:X\to Y\in \complement \mathfrak{G}_{\beta, \zeta}$, $U|_{F_M}\in \complement \mathfrak{G}_{\xi, \zeta}$ and $U|_{F_M}$ factors through $A$.
(iii) For each $1\leqslant \xi, \zeta\leqslant \omega_1$, the class $\mathfrak{M}_{\xi, \zeta}$ does not admit a universal factoring operator.
We remark that, as $\complement\mathfrak{G}_{\beta, \zeta}\subset \complement \mathfrak{G}_{\xi, \zeta}\subset \complement \mathfrak{G}_{\alpha, \zeta}$ whenever $\beta<\xi<\alpha\leqslant \omega_1$, item $(ii)$ is quantitatively the strongest possible result in light of the negative result of $(i)$. That is, for $0\leqslant \zeta<\beta<\omega_1$, we exhibit a fairly simple class $\mathfrak{U}$ of operators in $\complement\mathfrak{G}_{\beta+1, \zeta}$ such that each member of $\complement\mathfrak{G}_{\beta, \zeta}$ factors a member of $\mathfrak{U}$.
Definitions
===========
Throughout, for a subset $M$ of ${\mathbb{N}}$, we let $[M]$ (resp. $[M]^{<{\mathbb{N}}}$) denote the set of all infinite (resp. finite) subsets of $M$. Throughout, we will denote sets as sets as well as strictly increasing sequences in the natural way. We let $E<F$ denote the relation that either $E=\varnothing$, $F=\varnothing$, or $\max E<\min F$. We topologize $\{0, 1\}^{\mathbb{N}}$ with the product topology and endow the power set $2^{\mathbb{N}}$ of ${\mathbb{N}}$ with the topology making the map $2^{\mathbb{N}}\ni E\leftrightarrow 1_E\in \{0,1\}^{\mathbb{N}}$ a homeomorphism. Given two members $(m_i)_{i=1}^k, (n_i)_{i=1}^k\in [{\mathbb{N}}]^{<{\mathbb{N}}}$, we say $(n_i)_{i=1}^k$ is a *spread* of $(m_i)_{i=1}^k$ if $m_i\leqslant n_i $ for all $1\leqslant i\leqslant k$. We say a subset $\mathcal{F}\subset [{\mathbb{N}}]^{<{\mathbb{N}}}$ is *spreading* if it contains all spreads of its members. We say $\mathcal{F}\subset [{\mathbb{N}}]^{<{\mathbb{N}}}$ is *hereditary* if it contains all subsets of its members. We say $\mathcal{F}\subset [{\mathbb{N}}]^{<{\mathbb{N}}}$ is *regular* if it is spreading, hereditary, and compact.
Given two non-empty, regular families $\mathcal{F}, \mathcal{G}$, we let $$\mathcal{G}[\mathcal{F}]=\{\varnothing\}\cup \Bigl\{\bigcup_{i=1}^n E_i: \varnothing \neq E_i\in \mathcal{F}, E_1<\ldots <E_n, (\min E_i)_{i=1}^n\in \mathcal{G}\Bigr\}.$$ Let $$\mathcal{A}_n=\{E: |E|\leqslant n\}$$ and let $$\mathcal{S}=\{\varnothing\}\cup \{E: |E|\leqslant \min E\}.$$ We next recall the Schreier families, defined in [@AMT]. We let $$\mathcal{S}_0=\mathcal{A}_1,$$ if $\mathcal{S}_\xi$ has been defined for $\xi<\omega_1$, $$\mathcal{S}_{\xi+1}= \mathcal{S}[\mathcal{S}_\xi],$$ and if $\xi<\omega_1$ is a limit ordinal, there exists a sequence $(\xi_n)_{n=1}^\infty$ such that $\xi_n\uparrow \xi$, $\mathcal{S}_{\xi_n+1}\subset \mathcal{S}_{\xi_{n+1}}$ for all $n\in{\mathbb{N}}$, and $$\mathcal{S}_\xi=\{\varnothing\}\cup \{E: \varnothing\neq E\in \mathcal{S}_{\xi_{\min E}+1}\}=\{E: \exists n\leqslant E\in \mathcal{S}_{\xi_n+1}\}.$$ We note that the existence of such as sequence was discussed, for example, in [@C2].
For a regular family $\mathcal{F}$, let us note that the set of isolated points of $\mathcal{F}$ is precisely the set of maximal (with respect to inclusion) members of $\mathcal{F}$. Let us denote this set by $MAX(\mathcal{F})$. Then we let $\mathcal{F}'=\mathcal{F}\setminus MAX(\mathcal{F})$. It is easy to see that $\mathcal{F}'$ is also regular. Then the Cantor-Bendixson derivatives are given by $$\mathcal{F}^0=\mathcal{F},$$ $$\mathcal{F}^{\xi+1}=(\mathcal{F}^\xi)',$$ and if $\xi$ is a limit ordinal, $$\mathcal{F}^\xi=\bigcap_{\zeta<\xi}\mathcal{F}^\zeta.$$ We let $CB(\mathcal{F})$ be the minimum ordinal $\xi$ such that $\mathcal{F}^\xi=\varnothing$, noting that such a $\xi$ must exist. Furthermore, we note that for a non-empty, regular family $\mathcal{F}$, $CB(\mathcal{F})$ must be a successor ordinal. For this reason, it is convenient to let $\iota(\mathcal{F})=CB(\mathcal{F})-1$ whenever $\mathcal{F}$ is a non-empty, regular family. We next recall some important facts regarding these notions. A reference for these facts is [@C2]. For what follows, for $N=(n_i)_{i=1}^\infty\in [{\mathbb{N}}]$ and a regular family $\mathcal{F}$, we let $\mathcal{F}(N)=\{(n_i)_{i\in E}: E\in \mathcal{F}\}$.
Let $\mathcal{F}, \mathcal{G}$ be regular families.
(i) For every $\xi<\omega_1$, $\mathcal{S}_\xi$ is regular with $\iota(\mathcal{S}_\xi)=\omega^\xi$.
(ii) For every $n\in{\mathbb{N}}$, $\mathcal{A}_n$ is regular with $\iota(\mathcal{A}_n)=n$.
(iii) The set $\mathcal{F}[\mathcal{G}]$ is regular and $\iota(\mathcal{F}[\mathcal{G}])=\iota(\mathcal{G})+\iota(\mathcal{F})$.
(iv) There exists $N\in[{\mathbb{N}}]$ such that $\mathcal{F}(N)\subset \mathcal{G}$ if and only if for every $M\in[{\mathbb{N}}]$, there exists $N\in[M]$ such that $\mathcal{F}(N)\subset \mathcal{G}$ if and only if $CB(\mathcal{F})\leqslant CB(\mathcal{G})$.
(v) If $\mathcal{G}$ is regular and $(m_n)_{n=1}^\infty\in [{\mathbb{N}}]$, then $\{E\in[{\mathbb{N}}]^{<{\mathbb{N}}}: (m_n)_{n\in E}\in \mathcal{G}\}$ is regular with the same Cantor-Bendixson index as $\mathcal{G}$.
(vi) For any $0\leqslant \zeta\leqslant \xi<\omega_1$, there exists $k\in{\mathbb{N}}$ such that for any $E\in \mathcal{S}_\zeta$ with $k\leqslant E$, $E\in \mathcal{S}_\xi$.
\[gra\]
Given a regular family $\mathcal{F}$, a Banach space $X$, and a sequence $(x_n)_{n=1}^\infty\subset X$, let us say $(x_n)_{n=1}^\infty$ is an $\ell_1^\mathcal{F}+$-*spreading model* if $(x_n)_{n=1}^\infty$ is bounded and $$\inf\{\|x\|:F\in \mathcal{F}, x\in \text{co}(x_n:n\in F)\}>0.$$ We say $(x_n)_{n=1}^\infty$ is an $\ell_1^\mathcal{F}$-spreading model if $(x_n)_{n=1}^\infty$ is bounded and $$\inf\{\|x\|: F\in \mathcal{F}, x=\sum_{n\in F} a_nx_n, \sum_{n\in F}|a_n|=1\}>0.$$ If $\mathcal{F}=\mathcal{S}_\xi$, we write $\ell_1^\xi+$ (resp. $\ell_1^\xi$) in place of $\ell_1^{\mathcal{S}_\xi}+$ (resp. $\ell_1^{\mathcal{S}_\xi}$). For $\xi<\omega_1$, we say the sequence $(x_n)_{n=1}^\infty$ is $\xi$-*weakly null* if it has no subsequence which is an $\ell_1^\xi+$-spreading model. This implies weak nullity by the Mazur lemma.
If $\mathcal{F}$ is a regular family containing all singletons, we define the norm $\|\cdot\|_\mathcal{F}$ on $c_{00}$ by $$\|x\|_\mathcal{F}=\sup\{\|Ex\|_{\ell_1}: E\in \mathcal{F}\}.$$ Here, $E\subset {\mathbb{N}}$ also denotes the projection on $c_{00}$ given by $E\sum_{n=1}^\infty a_ne_n=\sum_{n\in E}a_ne_n$. In the case that $\mathcal{F}=\mathcal{S}_\xi$, we write $\|\cdot\|_\xi$ in place of $\|\cdot\|_{\mathcal{S}_\xi}$. These are the Schreier spaces. We also define the *mixed Schreier spaces*. For a null sequence $(\varpi_n)_{n=1}^\infty\subset (0,1]$ and a sequence $\mathcal{F}_1, \mathcal{F}_2, \ldots$ of regular families such that each $\mathcal{F}_n$ contains all singletons, the completion of $c_{00}$ with respect to the norm $$[x]=\sup \{\varpi_n \|Ex\|_{\ell_1}: n\in{\mathbb{N}}, E\in \mathcal{F}_n\}.$$
The non-existence of universal operators
========================================
A persistent question regarding any class $\mathfrak{I}$ with the ideal property is whether or not there exists an operator $U:X\to Y$ lying in $\complement \mathfrak{I}$ which factors through every member of $\complement\mathfrak{I}$. Important examples of such operators are the universal factoring non-weakly compact operator $\Sigma:\ell_1\to \ell_\infty$ which takes the $\ell_1$ basis to the summing basis of $c_0$ [@LP], universal factoring non-super weakly compact operator $\Sigma_n:(\oplus_{n=1}^\infty \ell_1^n)_{\ell_1}\to (\oplus_{n=1}^\infty \ell_\infty^n)_{\ell_\infty}$ which takes the basis of $\ell_1^n$ to the summing basis of $\ell_\infty^n$, a universal $\ell_p$-singular operator, which is any isomorphic embedding of $\ell_p$ into $\ell_\infty$, and the universal factoring non-super $\ell_p$-singular operator $jP$, where $P:(\oplus_{n=1}^\infty \ell_p^n)_{\ell_1}\to (\oplus_{n=1}^\infty \ell_p^n)_{c_0}$ is the formal inclusion and $j:(\oplus_{n=1}^\infty \ell_p^n)_{c_0}\to \ell_\infty$ is an isomorphic embedding [@Oikhberg]. In this section, we will prove that none of our classes of interest admits a universal factoring operator. We begin with a technical piece for later use.
Fix $0<\eta<\omega_1$ and suppose that $\mathcal{F}_1, \mathcal{F}_2, \ldots$ are regular families which contain all singletons and such that $CB(\mathcal{F}_n)<\omega^\eta$ for all $n\in{\mathbb{N}}$. Fix a sequence $(\varpi_n)_{n=1}^\infty\subset (0, 1]$ of numbers converging to $0$. Define $|\cdot|_n$ on $c_{00}$ by $$|x|_i=\sup\Bigl\{\sum_{i=1}^t \|I_i x\|_{\ell_2}:t\in {\mathbb{N}}, I_1<\ldots <I_t, (\min I_i)_{i=1}^t\in \mathcal{F}_n\Bigr\}.$$ Define $[\cdot]$ on $c_{00}$ by $[x]=\sup_n \varpi_n|x|_n$ and let $Z$ be the completion of $c_{00}$ with respect to this norm. Then the canonical basis of $Z$ is $\eta$-weakly null.
\[btsu\]
In the proof below, we make use of the repeated averages hierarchy, introduced in [@AMT]. As a precise definition of the repeated averages hierarchy is not necessary for the following proof, and the full definition of the hierarchy would be unnecessarily technical, we simply state here the essential facts about the repeated averages hierarchy needed for the following proof. For each $\xi<\omega_1$, each $n\in{\mathbb{N}}$, and $M\in[{\mathbb{N}}]$, $\mathbb{S}^\xi_{M,n}=(\mathbb{S}^\xi_{M,n}(i))_{i=1}^\infty$ is a sequence of non-negative numbers such that $1=\sum_{i=1}^\infty \mathbb{S}^\xi_{M,n}(i)$ and $\{i: \mathbb{S}^\xi_{M,n}(i)\neq 0\}\in \mathcal{S}_\xi$.
Suppose the result is not true. Then there exist a subsequence $(e_{m_i})_{i=1}^\infty$ and $0<{\varepsilon}<1$ such that $${\varepsilon}<\{[x]: F\in \mathcal{S}_\eta, x\in \text{co}(e_{m_n}: n\in F)\}.$$ First choose $k\in{\mathbb{N}}$ such that $1/k<{\varepsilon}^2/16$. Choose $N=(n_i)_{i=1}^\infty\in[{\mathbb{N}}]$ such that $\mathcal{S}_\eta[\mathcal{A}_k](N)\subset \mathcal{S}_\eta$. Such an $N$ exists by Proposition \[gra\]$(iv)$, since $$CB(\mathcal{S}_\eta[\mathcal{A}_k])=k\omega^\eta+1=\omega^\eta+1=CB(\mathcal{S}_\eta).$$ Fix $G_1<G_2<\ldots$ with $|G_i|=k$ and define $x_i=\frac{1}{k}\sum_{j\in G_i} e_{m_{n_j}}$. Note that $\|x_i\|_{\ell_2}=1/k^{1/2}<{\varepsilon}/4$ and $\|x_i\|_{\ell_1}= 1$ for all $i\in{\mathbb{N}}$. Also, by our choice of $N$, it follows that $${\varepsilon}\leqslant \inf\{[x]: F\in \mathcal{S}_\eta, x\in \text{co}(x_i: i\in F)\}.$$ Fix $m\in{\mathbb{N}}$ such that $\varpi_m<{\varepsilon}/2$. Let $\mathcal{F}=\cup_{i=1}^m \mathcal{F}_i$ and note that $CB(\mathcal{F})= \max_{1\leqslant i\leqslant m}CB(\mathcal{F}_i)<\omega^\eta$. For each $i\in{\mathbb{N}}$, let $s_i=\max \text{supp}(x_i)$ and let $$\mathcal{G}=\{E: (s_i)_{i\in E}\in \mathcal{F}\},$$ so $CB(\mathcal{G})=CB(\mathcal{F})<\omega^\eta$ by Proposition \[gra\]$(v)$. By [@CN Lemma $4.3$], there exists $P\in [{\mathbb{N}}]$ such that $$\sup\{\mathbb{S}^\eta_{Q,1}(A): A\in \mathcal{G}, Q\in [P]\}<{\varepsilon}/4.$$ Let $$x=\sum_{i=1}^\infty \mathbb{S}^\eta_{P,1}(i)x_i\in \text{co}(x_i: i\in\text{supp}(\mathbb{S}^\eta_{P,1})).$$ Since $\text{supp}(\mathbb{S}^\eta_{P,1})\in \mathcal{S}_\eta$, $[x]\geqslant {\varepsilon}$. Now fix $n\in{\mathbb{N}}$. If $n>m$, $$\varpi_n|x|_n \leqslant \varpi_n\|x\|_{\ell_1} <{\varepsilon}/2.$$ If $1\leqslant n\leqslant m$, fix $I_1<\ldots <I_t$ such that $(\min I_i)_{i=1}^t\in \mathcal{F}_n$. Let $A$ denote the set of those $i\in \text{supp}(\mathbb{S}^\eta_{N,1})$ such that $I_jx_i\neq 0$ for at least two values of $j$, and let $B=\text{supp}(\mathbb{S}^\eta_{N,1})\setminus A$. For each $i\in A$, let $j_i$ be the minimum $j\in \{1, \ldots, t\}$ such that $I_jx_i\neq 0$ and note that $A\ni i\mapsto j_i$ is an injection of $A$ into $\{1, \ldots, t\}$. Moreover, $(s_i)_{i\in A}$ is a spread of $(\min I_{j_i})_{i\in A}\subset (\min I_j)_{j=1}^t\in \mathcal{F}_n$, whence $(s_i)_{i\in A}\in \mathcal{F}$ and $A\in \mathcal{G}$. Then $$\begin{aligned}
\varpi_n\sum_{j=1}^t\|I_j x\|_{\ell_2} & \leqslant \sum_{i\in A} \mathbb{S}^\eta_{P,1}(i)\sum_{j=1}^\infty \|I_j x_i\|_{\ell_1} + \sum_{i\in B} \mathbb{S}^\eta_{P,1}(i)\|x_i\|_{\ell_2} \\ & \leqslant \sum_{i\in A} \mathbb{S}^\eta_{P,1}(i)\|x_i\|_{\ell_1} + \sum_{i\in B}\mathbb{S}^\eta_{P,1}(i)k^{-1/2} \\ & \leqslant \mathbb{S}^\eta_{P,1}(A)+ k^{-1/2} \leqslant {\varepsilon}/2. \end{aligned}$$ Since this holds for any $I_1<\ldots <I_t$ with $(\min I_i)_{i=1}^t\in \mathcal{F}_n$, it follows that $\varpi_n|x|_n\leqslant {\varepsilon}/2$. Therefore we have shown that $$[x]=\sup_n \varpi_n[x]_n \leqslant {\varepsilon}/2,$$ a contradiction.
We recall that for a sequence $(x_n)_{n=1}^\infty$ in the Banach space $X$ and $\delta>0$, $$\mathfrak{F}_\delta((x_n)_{n=1}^\infty)=\{E\in [{\mathbb{N}}]^{<{\mathbb{N}}}: (\exists x^*\in B_{X^*})(\forall n\in E)(\text{Re\ }x^*(x_n)\geqslant \delta)\}.$$ We will use the following fact.
[@CN Lemma $3.12$] For a Banach space $X$, $0<\eta<\omega_1$, and an $\eta$-weakly null sequence $(x_i)_{i=1}^\infty \subset X$, for every $\delta>0$ and $M\in[{\mathbb{N}}]$, there exists $N\in [M]$ such that $$CB(\mathfrak{F}_\delta((x_i)_{i=1}^\infty)\cap [N]^{<{\mathbb{N}}})<\omega^\eta.$$
\[cb\]
We are now ready to prove the non-existence of a universal operator for $\complement \mathfrak{M}_{\xi, \zeta}$.
For $1\leqslant \zeta, \xi\leqslant \omega_1$, $\complement \mathfrak{M}_{\xi,\zeta}$ does not admit a universal factoring operator.
\[nu1\]
Seeking a contradiction, assume that $U:X\to Y$ is a universal factoring operator for $\complement \mathfrak{M}_{\xi, \zeta}$. This means there exists a sequence $(x_n)_{n=1}^\infty \subset X$ which is $\xi$-weakly null and such that $\inf_n \|Ux_n\|>0$. If $\xi<\omega_1$, let $\eta=\xi$. If $\xi=\omega_1$, fix $\eta<\omega_1$ such that $(x_i)_{i=1}^\infty$ is $\eta$-weakly null. Note that in either case, $0<\eta<\omega_1$. By Lemma \[cb\], we may select $M_1\supset M_2\supset \ldots$ such that for each $n\in{\mathbb{N}}$, $CB(\mathfrak{F}_{3^{-n}}((x_i)_{i=1}^\infty)\cap [M_n]^{<{\mathbb{N}}})<\omega^\eta$. For each $n\in {\mathbb{N}}$, we may fix $\nu_n<\eta$ and $k_n\in{\mathbb{N}}$ such that $CB(\mathfrak{F}_{3^{-n}}((x_i)_{i=1}^\infty)\cap [M_n]^{<{\mathbb{N}}})<\omega^{\nu_n}k_n$. Now fix $m_1<m_2<\ldots$, $m_n\in M_n$, and let $M=(m_n)_{n=1}^\infty$. First note that for any $n\in{\mathbb{N}}$ and $L\in [M]$, there exists $N\in [L]$ such that $CB(\mathfrak{F}_{3^{-n}}((x_i)_{i=1}^\infty)\cap [N]^{<{\mathbb{N}}})<\omega^{\nu_n}k_n$. Indeed, any $N\in [L]$ which is also a subset of the tail set $(m_i)_{i=n}^\infty$ of $M$ has this property. Let $\mathcal{F}_n=\mathcal{A}_{k_n}[\mathcal{S}_{\nu_n}]$, which has Cantor-Bendixson index $\omega^{\nu_n}k_n<\omega^\eta$. Let $Z$ be the space from Lemma \[btsu\] with $\varpi_n=2^{-n}$. More precisely, $$[x]= \sup\{2^{-n} \sum_{i=1}^t \|I_ix\|_{\ell_2}: n\in{\mathbb{N}}, I_1<\ldots <I_t, (\min I_i)_{i=1}^t\in \mathcal{A}_{k_n}[\mathcal{S}_{\nu_n}]\}.$$ By Lemma \[btsu\], the basis of $Z$ is $\eta$-weakly null. The basis of this space is also $\xi$-weakly null, since $\eta\leqslant \xi$. Let $I:Z\to \ell_2$ be the formal inclusion. Since the canonical basis of $\ell_2^*$ is $1$-weakly null and $e^*_n(Ie_n)=1$ for all $n\in{\mathbb{N}}$, $I\in \complement \mathfrak{M}_{\xi, 1}$. This means there exist $R:X\to Z$ and $L:\ell_2\to Y$ such that $U=LIR$. Since $(Rx_i)_{i=1}^\infty$ and $(IRx_i)_{i=1}^\infty$ are weakly null and seminormalized in $Z$ and $\ell_2$, respectively, by a standard perturbation argument, we may fix $N=(n_i)_{i=1}^\infty\in[{\mathbb{N}}]$ and a block sequence $(z_i)_{i=1}^\infty$ with respect to the $c_{00}$ basis such that ${\varepsilon}:=\inf_i \|z_i\|_{\ell_2}>0$ and for all $(a_i)_{i=1}^\infty\in c_{00}$, $$[\sum_{i=1}^n a_iz_i]\leqslant 2[\sum_{i=1}^\infty a_i Rx_{n_i}].$$ Fix $n\in{\mathbb{N}}$ such that $2^n/3^n<2\|R\|/{\varepsilon}$. By our remark above, by replacing $N$ with an infinite subset thereof, we may assume $$CB(\mathfrak{F}_{3^{-n}}((x_i)_{i=1}^\infty)\cap [N]^{<{\mathbb{N}}})<\omega^{\nu_n}k_n.$$ Note that for any $F\in \mathcal{A}_{k_n}[\mathcal{S}_{\nu_n}]$ and non-negative scalars $(a_i)_{i\in F}$ summing to $1$, if $I_i=\text{supp}(z_i)$, then since $(\min \text{supp}(z_i))_{i\in F}$ is a spread of $F$, $$[\sum_{i\in F}a_iz_i]\geqslant 2^{-n}\sum_{j\in F}\|I_j\sum_{i\in F}a_iz_i\|_{\ell_2} \geqslant 2^{-n} {\varepsilon}.$$
Next let us note that by the geometric Hahn-Banach theorem, for any sequence $(y_i)_{i=1}^\infty$ and $\delta>0$, $F\in \mathfrak{F}_\delta((y_i)_{i=1}^\infty)$ if and only if $\min \{\|y\|: y\in \text{co}(y_i: i\in F)\}\geqslant \delta$. By the last inequality from the previous paragraph, it follows that $CB(\mathfrak{F}_{2^{-n}{\varepsilon}}((z_i)_{i=1}^\infty))\geqslant CB(\mathcal{A}_{k_n}[\mathcal{S}_{\nu_n}])=\omega^{\nu_n}k_n+1$. Now for any finite subset $F$ of ${\mathbb{N}}$, $$\min \{\|z\|: z\in \text{co}(z_i:i\in F)\} \leqslant 2\min\{\|Rx\|: x\in \text{co}(x_{n_i}: i\in F)\}\leqslant 2\|R\|\min \{\|x\|: x\in \text{co}(x_{n_i}: i\in F)\}.$$ From this it follows that for any $\delta>0$, $\mathfrak{F}_\delta((z_i)_{i=1}^\infty)\subset \mathfrak{F}_{2\|R\|\delta}((x_{n_i})_{i=1}^\infty)$ and $$CB(\mathfrak{F}_\delta((z_i)_{i=1}^\infty))\leqslant CB(\mathfrak{F}_{2\|R\|\delta}((x_{n_i})_{i=1}^\infty)) \leqslant CB(\mathfrak{F}_{2\|R\|\delta}((x_i)_{i=1}^\infty)\cap [N]^{<{\mathbb{N}}}).$$ Applying this with $\delta= 2^{-n}{\varepsilon}$ and noting that $2\|R\|\delta>3^{-n}$,$$\begin{aligned}
\omega^{\nu_n}k_n & <CB(\mathfrak{F}_{2^{-n}{\varepsilon}/2}((z_i)_{i=1}^\infty)) \leqslant CB(\mathfrak{F}_{2\|R\|2^{-n}{\varepsilon}/2}((x_i)_{i=1}^\infty)\cap [N]^{<{\mathbb{N}}}) \\ & \leqslant CB(\mathfrak{F}_{3^{-n}}((x_i)_{i=1}^\infty)\cap [N]^{<{\mathbb{N}}})<\omega^{\nu_n}k_n.\end{aligned}$$ This contradiction finishes the proof.
For the classes $\mathfrak{G}_{\xi, \zeta}$, we will prove a slightly stronger non-existence result, followed by a parallel positive result. Let $F,G$ be Banach spaces with bases $(f_i)_{i=1}^\infty$, $(g_i)_{i=1}^\infty$ such that the formal inclusion $U:F\to G$ is well-defined. For an infinite subset $M\in[{\mathbb{N}}]$, let $F_M$ (resp. $G_M$) denote the closed span of $(f_{m_i})_{i=1}^\infty$ in $F$ (resp. $(g_{m_i})_{i=1}^\infty$ in $G$). Let $U_M:F_M\to G_M$ be the restriction of the formal inclusion $U$ to $F_M$. Given an ideal $\mathfrak{I}$, let us say that $U$ is *subsequentially universal* for $\complement \mathfrak{I}$ if for any $A:X\to Y\in \complement \mathfrak{I}$ and any $L\in[{\mathbb{N}}]$, there exist $M\in [L]$ and subspaces $X_0$, $Y_0$ of $X$ and $Y$, respectively, such that $A(X_0)\subset Y_0$ and $U_M$ factors through $A|_{X_0}:X_0\to Y_0$.
An operator $U:F\to G$ being subsequentially universal for the class $\complement\mathfrak{I}$ provides a potentially small, easy to understand collection (formal inclusions between subsequences of fixed bases) which can be used to study the class $\mathfrak{I}$. Furthermore, the definition not only requires that we can factor formal inclusions of these subsequences through members of $\complement \mathfrak{I}$, but that the subsequences of this type are fairly abundant.
If $\mathfrak{I}$ is injective (which our classes $\mathfrak{G}_{\xi, \zeta}$ are) and $U:F\to G$ is subsequentially universal for $\complement \mathfrak{I}$, first fix an isometric embedding $j:G\to \ell_\infty$. Then the conclusion that $U_M$ factors through a restriction $A|_{X_0}:X_0\to Y_0$ of $A$ together with injectivity imply the existence of a factorization of $jU_M$ through $A$.
(i) For $0\leqslant \zeta<\xi\leqslant \omega_1$, there do not exist basic sequences $(f_i)_{i=1}^\infty$ and $(g_i)_{i=1}^\infty$ such that $(f_i)_{i=1}^\infty$ is $\xi$-weakly null, $(g_i)_{i=1}^\infty$ is not $\zeta$-weakly null, and the formal identity $U:[f_i:i\in{\mathbb{N}}]\to [g_i:i\in{\mathbb{N}}]$ is well-defined and subsequentially universal for $\complement\mathfrak{G}_{\xi, \zeta}$.
(ii) For any $0\leqslant \zeta<\xi\leqslant \omega_1$, there exists a formal identity operator $I$ between mixed Schreier spaces which lies in $\complement \mathfrak{G}_{\xi, \zeta}$ such that for each $\zeta<\beta<\xi$, $I$ is subsequentially universal for $\complement \mathfrak{G}_{\beta, \zeta}$.
\[big show\]
Given item $(i)$ of Theorem \[big show\], item $(ii)$ Theorem \[big show\] is the best possible quantitative weakening in the search for universal factoring operators.
We will need the following consequence of Lemma \[btsu\].
Suppose $0<\eta<\omega_1$ and $\mathcal{F}_1, \mathcal{F}_2, \ldots$ are regular families containing all singletons and such that for each $j\in {\mathbb{N}}$, $CB(\mathcal{F}_j)<\omega^\eta$. Fix a positive sequence of numbers $(\varpi_n)_{n=1}^\infty$ converging to zero. Let $Z$ be the completion of $c_{00}$ with respect to the norm $$[x]_0= \sup\Bigl\{\varpi_j\|Ex\|_{\ell_1}: j\in {\mathbb{N}}, E\in\mathcal{F}_j\}.$$ Then the basis of $Z_0$ is $\eta$-weakly null.
\[sop\]
Fix $j\in{\mathbb{N}}$ and $E\in \mathcal{F}_j$. Write $E=(n_i)_{i=1}^t$ and let $I_i=(n_i)$. Then $I_1<\ldots <I_t$ and $(\min I_i)_{i=1}^t\in \mathcal{F}_j$. Then if $Z$ is the space from Lemma \[btsu\], $$[x]\geqslant \varpi_j\|Ex\|_{\ell_1}.$$ From this it follows that the formal inclusion $I:Z\to Z_0$ is bounded with norm $1$. Since the canonical $c_{00}$ basis is $\eta$-weakly null in $Z$, its image is $\eta$-weakly null in $Z_0$.
$(i)$ Seeking a contradiction, assume $F$ is the closed span of a $\xi$-weakly null, basic sequence $(f_i)_{i=1}^\infty$, $G$ is the closed span of a basic, $\ell_1^\zeta+$-spreading model $(g_i)_{i=1}^\infty$, and the linear extension of the map taking $f_i$ to $g_i$ extends to a continuous linear operator $U:F\to G$ which is subsequentially universal for $\complement \mathfrak{G}_{\xi, \zeta}$. If $\xi<\omega_1$, let $\eta=\xi$. If $\xi=\omega_1$, let $\eta<\omega_1$ be such that $(f_i)_{i=1}^\infty$ is $\eta$-weakly null. Note that in either case, $0<\eta<\omega_1$. As in the proof of Theorem \[nu1\], we may recursively select $M_1\supset M_2\supset \ldots$, $\nu_n<\eta$, $k_n\in{\mathbb{N}}$ such that for all $n\in{\mathbb{N}}$, $$CB(\mathfrak{F}_{3^{-n}}((f_i)_{i=1}^\infty)\cap [M_n]^{<{\mathbb{N}}})<\omega^{\nu_n}k_n.$$ Let us note that since $(f_i)_{i=1}^\infty$ is $\eta$-weakly null and $(g_i)_{i=1}^\infty=(Uf_i)_{i=1}^\infty$ is not, $\zeta<\eta$. Therefore by replacing $\nu_n$ with $\zeta$ for any $n$ such that $\nu_n<\zeta$, we may assume that $\nu_n\geqslant \zeta$ for all $n\in{\mathbb{N}}$. Let $\varrho_n=\nu_n-\zeta$. That is, $\varrho_n$ is the unique ordinal such that $\zeta+\varrho_n=\nu_n$. For each $n\in{\mathbb{N}}$, let $$\mathcal{F}_n= \mathcal{A}_{k_n}[\mathcal{S}_{\varrho_n}[\mathcal{S}_\zeta]].$$ Note that for each $n\in{\mathbb{N}}$, $CB(\mathcal{F}_n)=\omega^\zeta \omega^{\varrho_n} k_n+1=\omega^{\nu_n}k_n+1<\omega^\eta$. Note that $\mathcal{F}_n\supset \mathcal{S}_\zeta$ for all $n\in{\mathbb{N}}$. Let $Z$ be the completion of $c_{00}$ with respect to the mixed Schreier norm $$[x]=\sup \{2^{-n}\|Ex\|_{\ell_1}: n\in{\mathbb{N}}, E\in\mathcal{F}_n\}.$$ Let $I:Z\to X_\zeta$ be the formal inclusion, which has norm $2$. Moreover, by Proposition \[sop\], the basis of $Z$ is $\eta$-weakly null. But the canonical $X_\zeta$ basis is not $\zeta$-weakly null, so $I\in \complement \mathfrak{G}_{\eta, \zeta}\subset \complement \mathfrak{G}_{\xi, \zeta}$.
Fix $m_1<m_2<\ldots$, $m_n\in M_n$, and let $M=(m_n)_{n=1}^\infty$. By the definition of subsequentially universal, there exists $P\in [M]$ such that the restriction $U_P:F_P\to G_P$ factors through some restriction of $I$. Fix $V_0, W_0$ and $R:F_P\to V_0$, $L:W_0\to G_P$ such that $I(U_0)\subset V_0$ and $LIR=U_P$. Since $(LIRf_n)_{n\in P}$ is an $\ell_1^\zeta+$-spreading model, so is $(IRf_n)_{n\in P}$, and $(IRf_n)_{n\in P}$ has no $\zeta$-weakly null subsequence. By standard perturbation arguments, we may find $N=(n_i)_{i=1}^\infty\in [P]$ and a block sequence $(z_i)_{i=1}^\infty$ with respect to the $c_{00}$ basis such that $(Iz_i)_{i=1}^\infty$ is an $\ell_1^\zeta+$-spreading model in $X_\zeta$ and for all $(a_i)_{i=1}^\infty$, $$[\sum_{i=1}^\infty a_i z_i]\leqslant 2[\sum_{i=1}^\infty a_i Rf_{n_i}].$$ We must consider two cases. If $\zeta=0$, we fix $0<{\varepsilon}<\inf_i \|z_i\|_\zeta=\inf_i \|z_i\|_{c_0}$. If $\zeta>0$, let $\gamma=\max\{\omega^\alpha: \omega^\alpha \leqslant \zeta\}$. In the $\zeta>0$ case, by [@Causey1 Theorem $2.14$], there exists $\beta<\gamma$ such that $\lim\sup \|z_i\|_\beta>0$. In this case, we may pass to a subsequence of $N$, relabel, and assume there exists ${\varepsilon}>0$ such that ${\varepsilon}<\inf_i \|z_i\|_\beta$. This is because if no such ${\varepsilon}$ and $\beta$ exist, $(Iz_i)_{i=1}^\infty$ is $\zeta$-weakly null in $X_\zeta$. Now in either case, fix $n\in{\mathbb{N}}$ so large that $2^n/3^n<2\|R\|/{\varepsilon}$. By passing to a subset of $N$ and relabeling once more, we may assume that $$CB(\mathfrak{F}_{3^{-n}}((f_{n_i})_{i=1}^\infty))\leqslant CB(\mathfrak{F}_{3^{-n}}((f_i)_{i=1}^\infty)\cap [N]^{<{\mathbb{N}}})< \omega^{\nu_n}k_n.$$ As in the proof of Theorem \[nu1\], we will show that $CB(\mathfrak{F}_{2^{-n}{\varepsilon}}((z_i)_{i=1}^\infty))>\omega^{\nu_n}k_n$, and this contradiction will finish $(i)$. In the $\zeta=0$ case, for each $i\in {\mathbb{N}}$, we fix a singleton $E_i=(s_i)\in \text{supp}(z_i)$ such that $\|E_iz_i\|_{\ell_1}>{\varepsilon}$. In the $0<\zeta$ case, we fix $E_i\in \mathcal{S}_\beta$ such that $\|E_iz_i\|_{\ell_1}>{\varepsilon}$. In either case, there exists $T=(t_i)_{i=1}^\infty\in [{\mathbb{N}}]$ such that for any $G\in \mathcal{S}_\zeta$, $\cup_{i\in G}E_{t_i}\in \mathcal{S}_\zeta$. In the $\zeta=0$ case, we may take $T={\mathbb{N}}$, since $G$ and $E_i$ are singletons. For the $\zeta>0$ case, we appeal to [@Causey1 Lemma $2.2$$(i)$] and the fact that $\beta+\zeta=\zeta$ by properties of the additively indecomposable ordinal $\gamma$. In the $\zeta=0$ case, $\varrho_n=\nu_n$ and $\mathcal{F}_n=\mathcal{A}_{k_n}[\mathcal{S}_{\nu_n}]$. Then for any $F\in \mathcal{F}_n$ and non-negative scalars $(a_i)_{i\in F}$ summing to $1$, $$[\sum_{i\in F} a_i z_i]\geqslant 2^{-n}\sum_{j\in F}\|E_j \sum_{i\in F}a_iz_i\|_{\ell_1} \geqslant 2^{-n}{\varepsilon}.$$ By another appeal to the geometric Hahn-Banach theorem, $\mathcal{A}_{k_n}[\mathcal{S}_{\nu_n}]\subset \mathfrak{F}_{2^{-n}{\varepsilon}}((z_i)_{i=1}^\infty)$, which gives the required lower estimate on the Cantor-Bendixson index and finishes the $\zeta=0$ case of the proof. Now assume $\zeta>0$ and let $T$ be as above. Now fix $F\in \mathcal{A}_{k_n}[\mathcal{S}_{\varrho_n}[\mathcal{S}_\zeta]]$, which means we can write $$F=\bigcup_{i=1}^l G_i,$$ $G_1<\ldots <G_i$, $\varnothing\neq G_i\in \mathcal{S}_\zeta$, $(\min G_i)_{i=1}^l\in\mathcal{A}_{k_n}[\mathcal{S}_{\varrho_n}]$. Now for each $1\leqslant i\leqslant l$, let $H_i=\cup_{j\in G_i} E_{t_j}$ and note that this set lies in $ \mathcal{S}_\zeta$ by our choice of $T$. Note that $\min H_i\geqslant \min G_i$, so $(\min H_i)_{i=1}^l$ is a spread of $(\min G_i)_{i=1}^l$, and therefore lies in $\mathcal{A}_{k_n}[\mathcal{S}_{\varrho_n}]$. Therefore $$H:=\bigcup_{i=1}^l H_i\in \mathcal{A}_{k_n}[\mathcal{S}_{\varrho_n}[\mathcal{S}_\zeta]].$$ For any non-negative scalars $(a_i)_{i\in F}$ summing to $1$, $$[\sum_{i\in F} a_i z_{t_i}]\geqslant 2^{-n}\|H\sum_{i\in F}a_iz_{t_i}\|_{\ell_1} \geqslant 2^{-n}{\varepsilon}.$$ One more appeal to the geometric Hahn-Banach theorem yields that $$\{(t_i)_{i\in E}: E\in \mathcal{A}_{k_n}[\mathcal{S}_{\varrho_n}[\mathcal{S}_\zeta]]\}\subset \mathfrak{F}_{2^{-n}{\varepsilon}}((z_i)_{i=1}^\infty).$$ Since $$CB(\{(t_i)_{i\in E}: E\in \mathcal{A}_{k_n}[\mathcal{S}_{\varrho_n}[\mathcal{S}_\zeta]]\})=CB(\mathcal{A}_{k_n}[\mathcal{S}_{\varrho_n}[\mathcal{S}_\zeta]])=\omega^\zeta\omega^{\varrho_n}k_n+1=\omega^{\nu_n}k_n+1,$$ this gives the required lower estimate on the Cantor-Bendixson index and finishes $(i)$.
$(ii)$ We first consider the case in which $\xi$ is a successor, say $\xi=\eta+1$. If $\xi=\zeta+1$, the conclusion is vacuous, as there are no $\beta$ with $\zeta<\beta<\xi$. Therefore we assume $\zeta<\eta$, so $0<\eta<\omega_1$. Let $I:X_\eta\to X_\zeta$ be the formal inclusion, which is bounded by Proposition \[gra\]. Let $X,Y$ be Banach spaces and suppose $A:X\to Y\in \complement \mathfrak{G}_{\eta, \zeta}$. Fix an $\eta$-weakly null sequence $(x_i)_{i=1}^\infty\subset B_X$ such that $(Ax_i)_{i=1}^\infty$ is basic, seminormalized and an $\ell_1^\zeta+$-spreading model. By [@AMT Theorem A], we may assume that $(x_i)_{i=1}^\infty$ and $(Ax_i)_{i=1}^\infty$ are convexly unconditional. This means that for any $\delta>0$, there exists $C(\delta)>0$ such that for any $(a_i)_{i=1}^\infty$ with $\sum_{i=1}^\infty |a_i|\leqslant 1$ and $\|\sum_{i=1}^\infty a_ix_i\|\geqslant \delta$, then $\|\sum_{i=1}^\infty \lambda_i a_ix_i\|\geqslant C(\delta)$ for any scalars $(\lambda_i)_{i=1}^\infty$ with $|\lambda_i|=1$ for all $i\in{\mathbb{N}}$. A similar inequality holds for $(Ax_i)_{i=1}^\infty$. By [@AG Theorem $1.10$], we may assume $(y_i)_{i=1}^\infty$ is $\mathcal{S}_\zeta$-unconditional. This means that there exists a constant $a>0$ such that for any $(a_i)_{i=1}^\infty\in c_{00}$, $$\sup_{F\in \mathcal{S}_\zeta} a\|\sum_{i\in F}a_i Ax_i\|\leqslant \|\sum_{i=1}^\infty a_i Ax_i\|.$$ Since $(Ax_i)_{i=1}^\infty$ is convexly unconditional and an $\ell_1^\zeta+$-spreading model, $(Ax_i)_{i=1}^\infty$ is an $\ell_1^\zeta$-spreading model, which means there exists $b>0$ such that $$\inf\{\|\sum_{i\in F} a_i Ax_i\|: F\in \mathcal{S}_\zeta, \sum_{i\in F}|a_i|=1\}=b.$$ Then for any $(a_i)_{i=1}^\infty \in c_{00}$, $$\|\sum_{i=1}^\infty a_i Ax_i\|\geqslant a\sup \{\|\sum_{i\in F}a_i Ax_i\|: F\in \mathcal{S}_\zeta\} \geqslant ab \sup \{\sum_{i\in F}|a_i|: F\in \mathcal{S}_\zeta\}.$$ This yields that the formal inclusion $J:[Ax_i:i\in{\mathbb{N}}]\to X_\zeta$ given by $JAx_i=e_i$ is well-defined and bounded. Now for $\delta>0$, let $$\mathfrak{H}_\delta=\{E\in [{\mathbb{N}}]^{<{\mathbb{N}}}: (\exists x^*\in B_{X^*})(\forall n\in E)(|x^*(x_i)|\geqslant \delta)\}.$$ Since $(x_i)_{i=1}^\infty$ is $\eta$-weakly null, for every $\delta>0$ and $M\in[{\mathbb{N}}]$, there exists $N\in[M]$ such that $$CB(\mathfrak{F}_\delta((x_i)_{i=1}^\infty)\cap [N]^{<{\mathbb{N}}})<\omega^\eta.$$ By convex unconditionality, this implies that for every $\delta>0$ and $M\in [{\mathbb{N}}]$, there exists $N\in [M]$ such that $CB(\mathfrak{H}_\delta\cap [N]^{<{\mathbb{N}}})<\omega^\eta$. Now let us fix $0<\vartheta<1$. Let $L\in [{\mathbb{N}}]$ be arbitrary and recursively select $L\supset M_1\supset M_2\supset \ldots$ such that for all $n\in{\mathbb{N}}$, either $\mathfrak{H}_{\vartheta^n}\cap [M_n]^{<{\mathbb{N}}}\subset \mathcal{S}_\eta$ or $\mathcal{S}_\eta\cap [M_n]^{<{\mathbb{N}}}\subset \mathfrak{H}_{\vartheta^n}$. We may make these selections by [@Gasparis Theorem $1.1$]. But our remark preceding the fixing of $\vartheta$ yields that the second option cannot hold, and $\mathfrak{H}_{\vartheta^n}\cap [M_n]^{<{\mathbb{N}}}\subset \mathcal{S}_\eta$ for all $n\in{\mathbb{N}}$. Fix $m_1<m_2<\ldots$, $m_n\in M_n$, and let $M=(m_n)_{n=1}^\infty$. Now fix $(a_i)_{i=1}^\infty\in c_{00}$ and $x^*\in B_{X^*}$ such that $$\|\sum_{i=1}^\infty a_ix_{m_i}\|=x^*(\sum_{i=1}^\infty a_i x_{m_i}).$$ For each $n\in{\mathbb{N}}$, let $$I_n=\{i<n: |x^*(x_{m_i})|\in (\vartheta^n, \vartheta^{n-1}]\}$$ and $$J_n=\{i\geqslant n: |x^*(x_{m_i})|\in (\vartheta^n, \vartheta^{n-1}]\}.$$ For each $n\in{\mathbb{N}}$, $$(m_i)_{i\in J_n}\in \mathfrak{H}_{\vartheta^n}\cap [M_n]^{<{\mathbb{N}}}\subset \mathcal{S}_\eta,$$ so $$x^*(\sum_{i\in I_n\cup J_n} a_ix^*_{m_i}) \leqslant \vartheta^{n-1}\Bigl[\|(a_i)_{i=1}^\infty\|_\infty |I_n|+ \sum_{i\in J_n}|a_i|\Bigr] \leqslant n\vartheta^{n-1}\|\sum_{i=1}^\infty a_i e_{m_i}\|_\eta.$$ Therefore $$\|\sum_{i=1}^\infty a_ix_{m_i}\|\leqslant (1-\vartheta)^{-2}\|\sum_{i=1}^\infty a_ie_{m_i}\|_\eta.$$ Thus the maps taking $(e_{m_i})_{i=1}^\infty\subset X_\eta$ to $(x_{m_i})_{i=1}^\infty$ and $(Ax_{m_i})_{i=1}^\infty$ to $(e_{m_i})_{i=1}^\infty \subset X_\zeta$ are bounded. Since $L\in[{\mathbb{N}}]$ was arbitrary, this shows that $I:X_\eta\to X_\zeta$ is subsequentially universal for $\complement \mathfrak{G}_{\eta, \zeta}$. Since for any $\zeta<\beta<\xi$, $\complement \mathfrak{G}_{\beta, \zeta}\subset \complement \mathfrak{G}_{\eta, \zeta}$, this completes the successor case.
Now suppose that $\xi$ is a limit ordinal. Fix $\zeta<\xi_1<\xi_2<\ldots$ with $\xi_n\uparrow \xi$ and a null sequence $(\varpi_n)_{n=1}^\infty\subset (0,1]$ of positive numbers. Let $Z$ be the completion of $c_{00}$ with respect to the norm $$[x]=\sup \{\varpi_n \|Ex\|_{\ell_1}: n\in{\mathbb{N}}, E\in \mathcal{S}_{\xi_n}\}$$ and let $I:Z\to X_\zeta$ be the formal inlusion. Suppose that for some $0\leqslant \zeta<\beta<\xi$, $A:X\to Y\in \complement \mathfrak{G}_{\beta, \zeta}$. Arguing as in the successor case, we may select a sequence $(x_i)_{i=1}^\infty\subset B_X$ which is $\beta$-weakly null and such that the map taking $(Ax_i)_{i=1}^\infty $ to $(e_i)_{i=1}^\infty\subset X_\zeta$ is bounded. Also, for $L\in[{\mathbb{N}}]$, we may select $(m_i)_{i=1}^\infty\in [L]$ such that the map taking $(e_{m_i})_{i=1}^\infty\subset X_\beta$ to $(x_{m_i})_{i=1}^\infty$ is bounded. Now if $n\in{\mathbb{N}}$ is such that $\xi_n>\beta$, the formal inclusion of $X_{\xi_n}$ into $X_\beta$ is bounded by Proposition \[gra\], as is the map taking $(e_{m_i})_{i=1}^\infty \subset X_{\xi_n}$ to $(e_{m_i})_{i=1}^\infty$ into $X_\beta$. Now the map taking $(e_{m_i})_{i=1}^\infty \subset Z$ to $(e_{m_i})_{i=1}^\infty\subset X_{\xi_n}$, and therefore the maps taking $(e_{m_i})_{i=1}^\infty\subset Z$ to $(x_{m_i})_{i=1}^\infty$ and $(Ax_{m_i})_{i=1}^\infty$ to $(e_{m_i})_{i=1}^\infty\subset X_\zeta$ are well-defined and bounded. This yields the appropriate factorization of $I$ through a restriction of $A$ and gives the limit ordinal case.
The following is implicitly contained in $(i)$ of the preceding proof.
For $0\leqslant \zeta<\xi\leqslant \omega_1$, $\complement \mathfrak{G}_{\xi, \zeta}$ does not admit a universal factoring operator.
If $U:F\to G\in \complement \mathfrak{G}_{\xi, \zeta}$ were universal for $\complement \mathfrak{G}_{\xi,\zeta}$, we first fix $(f_i)_{i=1}^\infty\subset X$ which is $\xi$-weakly null and such that $(Uf_i)_{i=1}^\infty=(g_i)_{i=1}^\infty$ is an $\ell_1^\zeta+$-spreading model. Let $\eta=\xi$ if $\xi<\omega_1$ and otherwise let $\eta<\omega_1$ be such that $(f_i)_{i=1}^\infty$ is $\eta$-weakly null. We fix $M_1\supset M_2\supset \ldots$, $\nu_n<\eta$, and $k_n\in{\mathbb{N}}$ such that $$CB(\mathfrak{F}_{3^{-n}}((f_i)_{i=1}^\infty)\cap [M_n]^{<{\mathbb{N}}})<\omega^{\nu_n}k_n.$$ We may assume $\nu_n\geqslant \zeta$ for all $n\in{\mathbb{N}}$ and write $\nu_n=\zeta+\varrho_n$. If $\zeta=0$, let $Z$ be the completion of $c_{00}$ with respect to the mixed Schreier norm $$[x]=\sup\{2^{-n}\|Ex\|_{\ell_1}: n\in{\mathbb{N}}, E\in \mathcal{A}_{k_n}[\mathcal{S}_{\nu_n}]\}.$$ If $\zeta>0$, let $Z$ be the completion of $c_{00}$ with respect to the mixed Schreier norm $$[x]=\sup \{2^{-n}\|Ex\|_{\ell_1}: n\in{\mathbb{N}}, E\in \mathcal{A}_{k_n}[\mathcal{S}_{\varrho_n}[\mathcal{S}_\zeta]]\}.$$ In either case, the formal inclusion $I:Z\to X_\zeta$ lies in $\complement \mathfrak{G}_{\xi, \zeta}$. If $U$ were to factor through $Z$ as $U=LIR$, then arguing as in the proof of Theorem \[big show\]$(i)$, we would be able to find $N\in[M]$, ${\varepsilon}>0$, $n$ such that $2^n/3^n<2\|R\|/{\varepsilon}$, and a block sequence $(z_i)_{i=1}^\infty$ with respect to $c_{00}$ such that $$\begin{aligned}
\omega^{\nu_n}k_n & < CB(\mathfrak{F}_{2^{-n}{\varepsilon}}((z_i)_{i=1}^\infty)) \leqslant CB(\mathfrak{F}_{3^{-n}}((f_i)_{i=1}^\infty)\cap [N]^{<{\mathbb{N}}})<\omega^{\nu_n}k_n. \end{aligned}$$
Codings of $\text{SB}$, $\mathcal{L}$, and dual spaces
======================================================
We first recall some facts and constructions from descriptive set theory. Two references for such facts are the books [@Dodos] and [@Ke].
The following fact is standard. However, since it will be used freely, we isolate it here.
Let $X$ be a Polish space with topology $\tau$. Let $Y_n$ be a sequence of second countable topological spaces and $f_n:X\to Y_n$ a sequence of Borel functions. Then there exists a Polish topology $\tau'$ on $X$ finer than $\tau$ such that the Borel $\sigma$-algebras of $\tau$ and $\tau'$ coincide and for each $n\in{\mathbb{N}}$, $f_n:(X, \tau')\to Y_n$ is continuous.
\[improv\]
For each $n\in{\mathbb{N}}$, let $(U_{m,n})_{m=1}^\infty$ be a countable base for the topology of $Y_n$. Then by [@Ke Lemma $13.3$, Page $82$], there exists a Polish topology $\tau'$ on $X$ finer than $\tau$, generating the same Borel $\sigma$-algebra as $\tau$, such that each of the sets $f^{-1}_n(U_{m,n})$, $m,n\in{\mathbb{N}}$, is clopen in $\tau'$.
Let us recall that a subset $A$ of a Polish space $S$ is $\Sigma_1^1$ if there exist a Polish space $P$, a Borel subset $B$ of $P$, and a Borel function $f:P\to S$ such that $f(B)=A$. A subset $C$ of $S$ is $\Pi_1^1$ if $S\setminus C$ is $\Sigma_1^1$. We say a subset $A$ of a Polish space $S$ is $\Sigma_2^1$ if there exist a Polish space $P$, a $\Pi_1^1$ subset $B$ of $P$, and a Borel function $f:S\to P$ such that $f(B)=A$. If for subset $A,B$ of Polish spaces $S,P$, respectively, and a Borel function $f:S\to P$, $f^{-1}(B)=A$, then we say $A$ is *Borel reducible* to $B$. Given $j\in \{1, 2\}$, we say a set $A\subset S$ is $\Pi^1_j$-*hard* if for any Polish space $P$ and any $\Pi_j^1$ subset $B$ of $P$, $A$ is Borel reducible to $B$. We say $A\subset S$ is $\Pi^1_j$-*complete* if it is $\Pi_j^1$-hard and $\Pi_j^1$.
We let $C(2^{\mathbb{N}})$ be the space of continuous functions on the Cantor set. We endow $F(C(2^{\mathbb{N}}))$, the set of closed subsets of $C(2^{\mathbb{N}})$, with its Effros-Borel $\sigma$-algebra, and recall that this is a standard Borel space. That is, there exists a Polish topology on $F(C(2^{\mathbb{N}}))$ the Borel $\sigma$-algebra of which is the Effros-Borel $\sigma$-algebra. By a result of Kuratowski and Ryll-Nardzewski [@KRN], there exists a sequence $d_n:F(C(2^{\mathbb{N}}))\setminus\{\varnothing\}\to C(2^{\mathbb{N}})$ such that each $d_n$ is Borel and for each $\varnothing\neq F\in F(C(2^{\mathbb{N}}))$, $\{d_n(F): n\in{\mathbb{N}}\}$ is a dense subset of $F$. By standard techniques, we may assume that for each finite subset $F$ of ${\mathbb{N}}$ and all rational numbers $(p_i)_{i\in F}$, there exists $n\in {\mathbb{N}}$ such that $\sum_{i\in F}p_i d_i=d_n$. We let $\textbf{SB}$ denote the subset of $F(C(2^{\mathbb{N}}))$ consisting of those closed subsets of $C(2^{\mathbb{N}})$ which are linear subspaces. This is easily seen to be a Borel subset of $F(C(2^{\mathbb{N}}))$, whence it is also a standard Borel space. From now on, we will treat $\textbf{SB}$ as a Polish space. However, as we are not concerned with the particular Polish topology on $\textbf{SB}$ which generates the Effros-Borel $\sigma$-algebra as its Borel $\sigma$-algebra, we will fix a Polish topology on $\textbf{SB}$ which generates the Effros-Borel $\sigma$-algebra and such that each selector $d_n$ is continuous with respect to this topology. Define $r_n(X)=d_n(X)$ if $\|d_n(X)\|\leqslant 1$ and $r_n(X)=d_n(X)/\|d_n(X)\|$ if $\|d_n(X)\|>1$. Note that each $r_n$ is also continuous and $\|r_n(X)\|\leqslant 1$ for each $n\in{\mathbb{N}}$ and $X\in \textbf{SB}$. We also let $\mathcal{L}$ denote the set of all triples $(X,Y, (y_n)_{n=1}^\infty)\in \textbf{SB}\times \textbf{SB}\times C(2^{\mathbb{N}})^{\mathbb{N}}$ such that $y_n\in Y$ for all $n\in{\mathbb{N}}$ and there exists $k\in {\mathbb{N}}$ such that for all $n\in{\mathbb{N}}$ and scalars $(a_i)_{i=1}^n$, $$\|\sum_{i=1}^n a_i y_i\|\leqslant k\|\sum_{i=1}^n a_id_i(X)\|.$$ It is easy to see that this is a Borel subset of $\textbf{SB}\times \textbf{SB}\times C(2^{\mathbb{N}})^{\mathbb{N}}$, and is also therefore a standard Borel space. We fix a Polish topology on $\mathcal{L}$ stronger than the topology inherited as a subspace of the product $\textbf{SB}\times\textbf{SB}\times C(2^{\mathbb{N}})^{\mathbb{N}}$ and which generates the Effros-Borel $\sigma$-algebra. Note that the functions $(X, Y, (y_n)_{n=1}^\infty)\mapsto d_m(X), r_m(X)$ are still continuous for all $m\in{\mathbb{N}}$. This is the coding of all operators between separable Banach spaces introduced in [@BF]. That is, for any $(X, Y, (y_n)_{n=1}^\infty)\in \mathcal{L}$, the map $A_0:\{d_n(X):n\in{\mathbb{N}}\}\to Y$ given by $A_0d_n(X)=y_n$ is well-defined and extends to a continuous, linear operator $A:X\to Y$. Conversely, if $A:X\to Y$ is a continuous, linear operator for $X,Y\in \textbf{SB}$, then $(X, Y, (Ad_n(X))_{n=1}^\infty)\in \mathcal{L}$.
We also recall the coding of dual spaces. Let $H=[-1,1]^{\mathbb{N}}$, endowed with a Polish topology such that the coordinate functional $(a_i)_{i=1}^\infty\mapsto a_n$ is continuous for each $n\in{\mathbb{N}}$, and the map $(a_i)_{i=1}^\infty\mapsto \|(a_i)_{i=1}^\infty\|_\infty$ is continuous. We can see that such a topology exists by first endowing $H$ with its product topology $\tau$ and then using Proposition \[improv\] to find a finer Polish topology $\tau'$ such that $(a_i)_{i=1}^\infty\mapsto \|(a_i)_{i=1}^\infty\|_\infty$, which is Borel with respect to $\tau$, is continuous with respect to $\tau'$. We leave this topology fixed throughout. Given $X\in \textbf{SB}$ and $x^*\in B_{X^*}$, we define $$H\ni f_{x^*}= (x^*(r_1(X)), x^*(r_2(X)), x^*(r_3(X)), \ldots).$$ We let $K_X=\{f_{x^*}\in H: x^*\in B_{X^*}\}$. We define $D\subset \textbf{SB}\times H$ by $(X,f)\in D\Leftrightarrow f\in K_X$. Then $D$ is a Borel set and the bijective identification $B_{X^*}\ni x^*\leftrightarrow f_{x^*}\in K_X$ is isometric (see properties P10-P12 from [@Dodos Page 12]). More generally, for $x^*_1, \ldots, x^*_n\in K_X$ and scalars $(a_i)_{i=1}^n$, $\|\sum_{i=1}^n a_i x^*_i\|_{X^*}=\|\sum_{i=1}^n a_i f_{x^*_i}\|_\infty$.
We also remark that since $2^{\mathbb{N}}$ with its product topology is compact and the subset $[{\mathbb{N}}]$ of infinite subsets of ${\mathbb{N}}$ is $G_\delta$ in $2^{\mathbb{N}}$, $[{\mathbb{N}}]$ with its inherited topology is a Polish space.
Our first result regarding this is that functional evaluation is Borel.
The set $\mathcal{E}:=\{(Y,y, f_{y^*})\in \textbf{\emph{SB}}\times C(2^{\mathbb{N}})\times D: y\in Y, f_{y^*}\in K_Y\}$ is Borel and the map $(Y, y, f_{y^*})\mapsto y^*(y)$ is Borel from $\mathcal{E}$ to ${\mathbb{R}}$. \[hardest part\]
The map $(Y, y, f_{y^*})\mapsto (Y, f_{y^*})$ is continuous and therefore the set of $(Y, y, f_{y^*})$ such that $f_{y^*}\in Y$ is the set of $(Y, y, f_{y^*})$ such that $(Y, f_{y^*})\in D$ is Borel. It is known that the set of $(Y, y)\in\textbf{SB}\times C(2^{\mathbb{N}})$ such that $y\in Y$ is Borel (see [@Dodos Property (P4), Page 10]). Thus $\mathcal{E}$ is Borel.
To prove that evaluation is Borel, it is sufficient to prove that evaluation is continuous when $\textbf{SB}$ and $H$ are endowed with the topologies we have fixed above. We therefore proceed assuming that for each $n\in{\mathbb{N}}$, $d_n$ and $r_n$ are continuous on $\textbf{SB}$ for each $n\in{\mathbb{N}}$, and $(a_i)_{i=1}^\infty\mapsto a_n$, $(a_i)_{i=1}^\infty\mapsto \|(a_i)_{i=1}^\infty\|_\infty$ are continuous on $H$. Assume $(Y, y, f_{y^*})\in \mathcal{E}$ is the limit of a sequence $((Y_n, y_n, f_{y^*_n}))_{n=1}^\infty\subset \mathcal{E}$.
Define $T:[0, \infty)\to [1, \infty)$ by $T(x)=1$ if $0\leqslant x\leqslant 1$ and $T(x)=x$ for all $x>1$. Note that $T$ is $1$-Lipschitz. Fix ${\varepsilon}>0$ and $j\in{\mathbb{N}}$ such that $\|y-d_j(Y)\|<{\varepsilon}$. Fix $m\in{\mathbb{N}}$ such that for all $n\geqslant m$, $\|y_n-y\|<{\varepsilon}$, $|f_{y^*}(j)-f_{y^*_n}(j)|<{\varepsilon}$, and $\|d_j(Y)-d_j(Y_n)\|<{\varepsilon}$. Now let us note that $d_j(Y)=T(\|d_j(Y)\|) r_j(Y)$, so $$y^*(d_j(Y))= T(\|d_j(Y)\|) f_{y^*}(j).$$ Similarly, for any $n\in{\mathbb{N}}$, $$y^*_n(d_j(Y_n))=T(\|d_j(Y_n)\|)f_{y^*_n}(j).$$ By the triangle inequality, for any $n\geqslant m$, $$\|y_n-d_j(Y_n)\|\leqslant \|y_n-y\|+\|y-d_j(Y)\|+\|d_j(Y)-d_j(Y_n)\|<3{\varepsilon}.$$ Thus $$|y^*(y)-T(\|d_j(Y)\|)f_{y^*}(j)|=|y^*(y)-y^*(d_j(Y))|\leqslant \|y^*\|\|y-d_j(Y)\|\leqslant \|y-d_j(Y)\| <{\varepsilon}$$ and for any $n\geqslant m$, $$|y^*_n(y_n)-T(\|d_j(Y_n)\|)f_{y^*_n}(j)|=|y^*_n(y_n)-y^*_n(d_j(Y_n))|\leqslant \|y^*_n\|\|y_n-d_j(Y_n)\|\leqslant \|y_n-d_j(Y_n)\| <3{\varepsilon}.$$ From this it follows that for any $n\geqslant m$, $$\begin{aligned}
|y^*(y)-y^*_n(y_n)| & \leqslant |y^*(y)- T(\|d_j(Y)\|)f_{y^*}(j)| + |T(\|d_j(Y)\|)f_{y^*}(j)- T(\|d_j(Y)\|)f_{y^*_n}(j)| \\ & + |T(\|d_j(Y)\|)f_{y^*_n}(j) - T(\|d_j(Y_n)\|)f_{y^*_n}(j)| + |T(\|d_j(Y_n)\|)f_{y^*_n}(j) - y^*_n(y_n)| \\ & \leqslant {\varepsilon}+ T(\|d_j(Y)\|){\varepsilon}+ {\varepsilon}+ 3{\varepsilon}\\ & \leqslant (5+\|y\|+{\varepsilon}){\varepsilon}. \end{aligned}$$ Since ${\varepsilon}>0$ was arbitrary, we are done.
For the remainder of this work, when an ideal is denoted by a fraktur letter (with subscripts), the associated subsets of $\mathcal{L}$ and $\textbf{SB}$ will be denoted by the same letter (with the same subscripts) in calligraphic and bold fonts, respectively. That is, for an ideal $\mathfrak{I}$, we let $\mathcal{I}$ denote subset of $\mathcal{L}$ consisting of those members $(X, Y, (y_n)_{n=1}^\infty)$ of $\mathcal{L}$ such that the unique continuous extension of the function $d_n(X)\mapsto y_n$ lies in $\mathfrak{I}$. We let $\textbf{I}$ denote the subset of $\textbf{SB}$ consisting of those $X\in \textbf{SB}$ such that $X\in \textsf{I}$ (equivalently, such that $I_X\in \mathfrak{I}$).
The map $\Phi:\textbf{SB}\to \mathcal{L}$ given by $\Phi(X)=(X,X, (d_n(X))_{n=1}^\infty)$ is Borel. From this it follows that for any ideal $\mathfrak{I}$ of operators, if $\mathcal{I}$ is $\Pi_1^1$ (resp. $\Pi_2^1$), then $\textbf{I}$ is $\Pi_1^1$ (resp. $\Pi_2^1$). Therefore to provide an upper estimate on the complexities of $\mathcal{I}$ and $\textbf{I}$, it is sufficient to provide that upper estimate only for $\mathcal{I}$.
Similarly, in order to show that $\mathcal{I}$ and $\textbf{I}$ are $\Pi_1^1$-hard (resp. $\Pi_2^1$-hard), it is sufficient to show that $\textbf{I}$ is $\Pi_1^1$-hard (resp. $\Pi_2^1$-hard). To see this, if $P$ is a Polish space, $C\subset P$ is a $\Pi_1^1$ subset of $P$, and $\Psi:P\to \textbf{I}$ is a Borel map such that $\Psi^{-1}(\textbf{I})=C$, then $\Phi\circ\Psi:P\to \mathcal{L}$ is a Borel reduction of $\mathcal{I}$ to $C$. A similar statement holds for $\Pi_2^1$-hard sets.
\[oopyal\]
Given $\xi<\omega_1$ and $M\in [{\mathbb{N}}]$, there exists a unique, non-empty, finite initial segment of $M$ which is a maximal member of $\mathcal{S}_\xi$. We denote this initial segment by $M|_\xi$. Given a Banach space $X$, a sequence $(x_n)_{n=1}^\infty\subset X$, $\xi<\omega_1$, and $M\in[{\mathbb{N}}]$, we let $$\Xi_\xi((x_n)_{n=1}^\infty, M)= \min\{\|x\|: x\in \text{co}(x_n: n\in M|_\xi)\}.$$
For a Banach space $X$, $(x_n)_{n=1}^\infty\subset X$, and $M=(m_n)_{n=1}^\infty\in [{\mathbb{N}}]$, let us say that the pair $((x_n)_{n=1}^\infty, M)$ has $D_\xi$ provided that for any $k\in {\mathbb{N}}$ and $N\in [M]$ with $\min N\geqslant m_k$, $\Xi_\xi((x_n)_{n=1}^\infty, N)\leqslant 1/k$.
Let $(x_n)_{n=1}^\infty$ be a sequence in the Banach space $X$. Let $\xi<\omega_1$.
(i) $(x_n)_{n=1}^\infty$ fails to be $\xi$-weakly null if and only if there exist $m\in{\mathbb{N}}$ and $M\in [{\mathbb{N}}]$ such that for all $N\in [M]$, $\Xi_\xi((x_n)_{n=1}^\infty, N)\geqslant 1/m$.
(ii) If $(x_n)_{n=1}^\infty\subset X$ is $\xi$-weakly null, then for any $M_0\in [{\mathbb{N}}]$, there exists $M\in [M_0]$ such that $((x_n)_{n=1}^\infty, M)$ has $D_\xi$.
(iii) If $M=(m_n)_{n=1}^\infty\in [{\mathbb{N}}]$ is such that $((x_n)_{n=1}^\infty, M)$ has $D_\xi$, then $(x_{m_n})_{n=1}^\infty$ is $\xi$-weakly null.
\[fras\]
First, for each $m\in{\mathbb{N}}$, let $\mathcal{V}_m$ denote the set of subsets $M\in [{\mathbb{N}}]$ such that $$\Xi_\xi((x_n)_{n=1}^\infty, M)\geqslant 1/m.$$ Since this is a closed set, the infinite Ramsey theorem yields that for each $m\in{\mathbb{N}}$ and $M\in[{\mathbb{N}}]$, there exists $N\in [M]$ such that either $[N]\subset \mathcal{V}_m$ or $[N]\cap \mathcal{V}_m=\varnothing$. From this and a standard diagonalization, we establish the dichotomy that either there exist $m\in{\mathbb{N}}$ and $M\in[{\mathbb{N}}]$ such that for all $N\in[M]$, $\Xi_\xi((x_n)_{n=1}^\infty, N)\geqslant 1/m$, or for every $M_0\in [{\mathbb{N}}]$, there exists $M\in[{\mathbb{N}}]$ such that $((x_n)_{n=1}^\infty, M)$ has $D_\xi$. We will show that the first of these two conditions is equivalent to $(x_n)_{n=1}^\infty$ failing to be $\xi$-weakly null, which will yield both $(i)$ and $(ii)$.
First suppose that there exist $m\in {\mathbb{N}}$ and $M=(m_i)_{i=1}^\infty\in[{\mathbb{N}}]$ such that for all $N\in [M]$, $\Xi_\xi((x_n)_{n=1}^\infty, N)\geqslant 1/m$. Now fix $F\in \mathcal{S}_\xi$ and let $E=(m_n)_{n\in F}\in \mathcal{S}_\xi$. Let $N$ be any infinite subset of $M$ such that $E$ is an initial segment of $N$. Now let $(a_n)_{n\in F}$ be non-negative numbers summing to $1$ and note that $$\|\sum_{n\in F}a_nx_{m_n}\|\geqslant \Xi_\xi((x_n)_{n=1}^\infty, N)\geqslant 1/m.$$ This yields that $(x_{m_n})_{n=1}^\infty$ is an $\ell_1^\xi+$-spreading model, and $(x_n)_{n=1}^\infty$ is not $\xi$-weakly null.
Now suppose that $(x_n)_{n=1}^\infty$ is not $\xi$-weakly null. If $(x_n)_{n=1}^\infty$ is not weakly null, then there exist $m\in{\mathbb{N}}$ and $M=(m_n)_{n=1}^\infty$ such that $\inf \{\|x\|: x\in \text{co}(x_{m_n}:n\in{\mathbb{N}})\}\geqslant 1/m$. Then $\Xi_\xi((x_n)_{n=1}^\infty, N)\geqslant 1/m$ for all $N\in [M]$. Now suppose that $(x_n)_{n=1}^\infty$ is weakly null but not $\xi$-weakly null. This means there exist ${\varepsilon}>0$ and $r_1<r_2<\ldots$ such that $(x_{r_n})_{n=1}^\infty$ is $2$-basic and for each $F\in \mathcal{S}_\xi$ and $x\in \text{co}(x_{r_n}:n\in F)$, $\|x\|\geqslant {\varepsilon}$. Now choose $1=s_1<s_2<\ldots$ such that for each $n\in{\mathbb{N}}$, $s_{n+1}>r_{s_n}$. Let $m_n=r_{s_n}$ and $M=(m_n)_{n=1}^\infty$. Fix $N\in [M]$ and let $N|_\xi=(r_{s_{t_i}})_{i=1}^l$. Then $F:=(s_{t_i})_{i=2}^l$ is a spread of $(r_{s_{t_i}})_{i=1}^{l-1}$ and therefore lies in $\mathcal{S}_\xi$. Fix non-negative scalars $(a_i)_{i=1}^l$ summing to $1$ and note that $$\begin{aligned}
\|\sum_{i=1}^l a_i x_{r_{s_{t_i}}}\| & \geqslant \frac{1}{3}\max\Bigl\{a_1\|x_{r_{s_{t_1}}} \|, \|\sum_{i=2}^l a_i x_{r_{s_{t_i}}}\|\Bigr\} \geqslant \frac{{\varepsilon}}{3}\max\Bigl\{a_1, \sum_{i=2}^l a_i\Bigr\} \geqslant {\varepsilon}/6. \end{aligned}$$ Thus for any $N\in [M]$, $\Xi_\xi((x_n)_{n=1}^\infty, M)\geqslant {\varepsilon}/6$. Fixing $m>6/{\varepsilon}$, we conclude the stated equivalence. This yields $(i)$ and $(ii)$.
$(iii)$ If $((x_n)_{n=1}^\infty, M)$ has $D_\xi$ and $(x_{m_n})_{n=1}^\infty$ is not $\xi$-weakly null, we may argue as in the previous paragraph to find $m\in{\mathbb{N}}$, $r_1<r_2<\ldots$, and $s_1<s_2<\ldots$ such that for each $N\in[(r_{s_n})_{n=1}^\infty]$, $\Xi_\xi((x_n)_{n=1}^\infty, N)\geqslant 1/m$, with the added condition that $(r_n)_{n=1}^\infty \in [(m_n)_{n=1}^\infty]$. Now if we fix $k>m$ and $N\in [M]$ with $\min N\geqslant k$, these two conditions yield that $$1/m\leqslant \Xi_\xi((x_n)_{n=1}^\infty, N)\leqslant 1/k,$$ which is a contradiction.
The sets $$W=\{(X, (n_i)_{i=1}^\infty)\in \textbf{\emph{SB}}\times {\mathbb{N}}^{\mathbb{N}}: (d_{n_i}(X))_{i=1}^\infty\text{\ is weakly null}\}$$ and $$W^*=\{(Y, (f_i)_{i=1}^\infty)\in \textbf{\emph{SB}}\times H^{\mathbb{N}}: (f_i)_{i=1}^\infty\subset K_Y, (f_i)_{i=1}^\infty\text{\ is weakly null in\ } \ell_\infty\}$$ are $\Pi_1^1$. For $\xi<\omega_1$, the sets $$W_\xi=\{(X, (n_i)_{i=1}^\infty,M)\in \textbf{\emph{SB}}\times{\mathbb{N}}^{\mathbb{N}}\times [{\mathbb{N}}]: ((d_{n_i}(X))_{i=1}^\infty, M) \text{\ has\ }D_\xi\}$$ and $$W_\xi^*=\{(Y, (f_i)_{i=1}^\infty, P)\in \textbf{\emph{SB}}\times H^{\mathbb{N}}\times [{\mathbb{N}}]: (f_i)_{i=1}^\infty\subset K_Y, ((f_i)_{i=1}^\infty, P)\text{\ has\ }D_\xi\text{\ in\ }\ell_\infty\}$$ are Borel.
\[boring\]
First let $C$ denote the set of those $(X, (n_i)_{i=1}^\infty, M, p)\in \textbf{SB}\times{\mathbb{N}}^{\mathbb{N}}\times [{\mathbb{N}}]\times {\mathbb{N}}$ such that for all $k\in{\mathbb{N}}$ and non-negative scalar sequences $(a_i)_{i=1}^k$, $\|\sum_{i=1}^k a_i d_{n_{m_i}}(X)\|\geqslant 1/p$. Here, $M=(m_i)_{i=1}^\infty$. It is evident that $C$ is closed. Let $\pi:\textbf{SB}\times {\mathbb{N}}^{\mathbb{N}}\times [{\mathbb{N}}]\times{\mathbb{N}}\to \textbf{SB}\times {\mathbb{N}}^{\mathbb{N}}$ be the projection and note that, by the Mazur lemma, $(\textbf{SB}\times{\mathbb{N}}^{\mathbb{N}})\setminus \pi(C)$ is the set $W$.
Now let $A$ denote the set of those $(Y, (f_i)_{i=1}^\infty, M, p)\in \textbf{SB}\times H^{\mathbb{N}}\times [{\mathbb{N}}]\times {\mathbb{N}}$ such that for all $i\in{\mathbb{N}}$, $f_i\in K_Y$. Let $B$ denote the set of those $(Y, (f_i)_{i=1}^\infty, M, p)\in \textbf{SB}\times H^{\mathbb{N}}\times [{\mathbb{N}}]\times {\mathbb{N}}$ such that for all $k\in{\mathbb{N}}$ and non-negative scalars $(a_i)_{i=1}^k$, $\|\sum_{i=1}^k a_i f_{m_i}\|_\infty\geqslant 1/p$. Since $\{(Y, f)\in \textbf{SB}\times H: f\in K_Y\}$ is Borel, $A$ is Borel. It is obvious that the set $B$ is closed, as we have assumed a topology on $H$ making the supremum norm continuous. Then $A\cap B$ is Borel. Let $\pi:\textbf{SB}\times H^{\mathbb{N}}\times [{\mathbb{N}}]\times {\mathbb{N}}\to \textbf{SB}\times H^{\mathbb{N}}$ be the projection and note that, by another appeal to the Mazur lemma, $W^*=(\textbf{SB}\times H^{\mathbb{N}})\setminus \pi(A\cap B)$.
We next show that, with our fixed topologies, the set $W_\xi$ is closed. To that end, fix $(X, (n_i)_{i=1}^\infty, M)\in (\textbf{SB}\times {\mathbb{N}}^{\mathbb{N}}\times [{\mathbb{N}}])\setminus W_\xi$. This means there exist ${\varepsilon}>0$, $k\in{\mathbb{N}}$, and $L\in [M]$ such that $\min L\geqslant m_k$ and for all non-negative scalars $(a_i)_{i\in L|_\xi}$ summing to $1$, $$\|\sum_{i\in L|_\xi}a_i d_{n_i}(X)\|>{\varepsilon}+1/k.$$ Here, $L=(l_i)_{i=1}^\infty$ and $M=(m_i)_{i=1}^\infty$. Let $t=\max L|_\xi$. Let $U_1$ denote the set of $Y\in \textbf{SB}$ such that for each $1\leqslant i\leqslant n_t$, $\|d_i(Y)-d_i(X)\|<{\varepsilon}$. By continuity of the selectors, $U_1$ is open in $\textbf{SB}$. Let $U_2$ denote the subset of ${\mathbb{N}}^{\mathbb{N}}$ consisting of those $(p_i)_{i=1}^\infty$ such that $p_i=n_i$ for all $1\leqslant i\leqslant t$, which is open. Let $U_3$ denote the subset of $[{\mathbb{N}}]$ consisting of those $Q\in[{\mathbb{N}}]$ such that for all $1\leqslant i\leqslant \max\{m_k, t\}$, $1_M(i)=1_Q(i)$. This is an open set in $[{\mathbb{N}}]$. Now let $U=U_1\times U_2\times U_3$, which is an open subset of $\textbf{SB}\times {\mathbb{N}}^{\mathbb{N}}\times [{\mathbb{N}}]$ containing $(X, (n_i)_{i=1}^\infty, M)$. We claim that $U\cap W_\xi=\varnothing$. Indeed, suppose $(Y, (p_i)_{i=1}^\infty, Q)\in U$. Let us note that, if $L|_\xi=(l_1, \ldots, l_s)$, $l_s=t$. Since $Y\in U_1$, it follows that $\|d_i(Y)-d_i(X)\|<{\varepsilon}$ for all $1\leqslant i\leqslant n_{l_s}$. Since $(p_i)_{i=1}^\infty\in U_2$, $n_i=p_i$ for all $i\leqslant l_s$. Since $Q\in U_3$, if $j\in{\mathbb{N}}$ is such that $t=m_j$, $q_i=m_i$ for all $i\leqslant j$. The last condition implies that $(l_1, \ldots, l_s)$ is an initial segment of some infinite subset $R$ of $Q$ such that $\min R=\min L\geqslant m_k=q_k$. Moreover, since $(l_1, \ldots, l_s)$ is a maximal member of $\mathcal{S}_\xi$ which is also an initial segment of $R$, $R|_\xi=L|_\xi$. Then for any non-negative scalars $(a_i)_{i\in R|_\xi}=(a_i)_{i\in L|_\xi}$ summing to $1$, $$\|\sum_{i\in R|_\xi}a_i d_{p_i}(Y)\| = \|\sum_{i\in L|_\xi}a_i d_{n_i}(Y)\|\geqslant \|\sum_{i\in L|_\xi}a_i d_{n_i}(X)\|-\sum_{i\in L|_\xi}a_i \|d_{n_i}(X)-d_{n_i}(Y)\|>{\varepsilon}+1/k-{\varepsilon}=1/k.$$ This yields that $(Y, (p_i)_{i=1}^\infty, Q)\notin W_\xi$, and $W_\xi$ is closed.
Now let $A$ denote the set of $(Y, (f_i)_{i=1}^\infty, M)\in \textbf{SB}\times H^{\mathbb{N}}\times [{\mathbb{N}}]$ such that $f_i\in K_Y$ for all $i\in{\mathbb{N}}$ and let $B$ denote the set of $(Y, (f_i)_{i=1}^\infty, M)\in \textbf{SB}\times H^{\mathbb{N}}\times [{\mathbb{N}}]$ such that $((f_i)_{i=1}^\infty, M)$ has $D_\xi$ in $\ell_\infty$. If $(Y, (f_i)_{i=1}^\infty, M)\notin B$, there exist ${\varepsilon}>0$, $k\in{\mathbb{N}}$, $L\in [M]$ such that $\min L\geqslant m_k$ and for all non-negative scalars $(a_i)_{i\in L|_\xi}$ summing to $1$, $$\|\sum_{i\in L_\xi}a_i f_i\|_\infty > {\varepsilon}+1/k.$$ Let $t=\max L|_\xi$. We let $U_1$ be the set of those $(g_i)_{i=1}^\infty\in H^{\mathbb{N}}$ such that $\|f_i-g_i\|_\infty<{\varepsilon}$ for all $1\leqslant i\leqslant t$. Let $U_2$ be the set of those $Q\in [{\mathbb{N}}]$ such that $1_M(i)=1_Q(i)$ for all $i\leqslant \max\{ m_k,t\}$. Then as in the previous paragarph, if $(Z, (g_i)_{i=1}^\infty, Q)\in \textbf{SB}\times U_1\times U_2$, there exists $R\in [Q]$ with $\min L=\min R\geqslant m_k=q_k$ such that $R|_\xi=L|_\xi$ and $$\|\sum_{i\in R|_\xi}a_i g_i\|_\infty =\|\sum_{i\in L|_\xi} a_ig_i\|_\infty \geqslant \|\sum_{i\in L|_\xi}a_if_i\|_\infty-\sum_{i\in L|_\xi}a_i\|f_i-g_i\|_\infty>1/k$$ for all non-negative scalars $(a_i)_{i\in R|_\xi}$ summing to $1$. This yields that $(Z, (g_i)_{i=1}^\infty, Q)\notin B$. This yields that $B$ is closed. Since $A\cap B=W_\xi^*$ and $A$ is Borel, $W_\xi^*$ is Borel.
We are now ready to prove the upper estimates.
(i) For $0\leqslant \zeta<\xi\leqslant \omega_1$, $\mathcal{G}_{\xi, \zeta}$ and $\textbf{\emph{G}}_{\xi, \zeta}$ are $\Pi_2^1$, and $\Pi_1^1$ if $\xi<\omega_1$.
(ii) For $1\leqslant \zeta,\xi\leqslant \omega_1$, $\mathcal{M}_{\xi, \zeta}$ and $\textbf{\emph{M}}_{\xi, \zeta}$ are $\Pi_2^1$, and $\Pi_1^1$ if $\zeta, \xi<\omega_1$.
\[upper\]
$(i)$ It suffices to prove that $\mathcal{G}_{\xi, \zeta}$ is $\Pi_1^1$ if $0\leqslant \zeta<\xi<\omega_1$ and that $\mathcal{G}_{\omega_1, \zeta}$ is $\Pi_2^1$ for any $\zeta<\omega_1$. The desired membership of the classes of spaces then follows from Remark \[oopyal\].
First fix $\xi<\omega_1$. Let $B$ denote the set of $((X, Y, (y_i)_{i=1}^\infty), (n_i)_{i=1}^\infty, M, p)\in \mathcal{L}\times {\mathbb{N}}^{\mathbb{N}}\times [{\mathbb{N}}]\times {\mathbb{N}}$ such that $((d_{n_i}(X))_{i=1}^\infty, M)$ has $D_\xi$ and let $C$ denote the set of $((X, Y, (y_i)_{i=1}^\infty), (n_i)_{i=1}^\infty, M, p)\in \mathcal{L}\times {\mathbb{N}}^{\mathbb{N}}\times [{\mathbb{N}}]\times {\mathbb{N}}$ such that for all $F\in \mathcal{S}_\zeta$ and non-negative scalars $(a_i)_{i\in F}$ summing to $1$, $\|\sum_{i\in F}a_i y_{n_i}\|\geqslant 1/p$. It is obvious that $C$ is Borel (and actually closed with our fixed topologies), and we know that $B$ is Borel by Lemma \[boring\]. Let $\pi:\mathcal{L}\times {\mathbb{N}}^{\mathbb{N}}\times [{\mathbb{N}}]\times {\mathbb{N}}\to \mathcal{L}$ be the projection and note that $\pi(B\cap C)$ is $\Sigma_1^1$. In order to show that $\mathcal{G}_{\xi, \zeta}$ is $\Pi_1^1$, it suffices to show that $\mathcal{L}\setminus \pi(B\cap C)=\mathcal{G}_{\xi, \zeta}$.
If $(X,Y, (y_i)_{i=1}^\infty)\in \mathcal{L}\setminus \mathcal{G}_{\xi, \zeta}$, then there exists a $\xi$-weakly null sequence $(x_i)_{i=1}^\infty\subset X$ whose image under the operator associated with the triple $(X, Y, (y_i)_{i=1}^\infty)$ is an $\ell_1^\zeta+$-spreading model. By perturbing, we may assume $x_i=d_{n_i}(X)$ for some $(n_i)_{i=1}^\infty\in {\mathbb{N}}^{\mathbb{N}}$. By Proposition \[fras\], there exists $M\in[{\mathbb{N}}]$ such that $((d_{n_i}(X))_{i=1}^\infty, M)$ has $D_\xi$. Furthermore, since the image of $d_{n_i}(X)$ under the operator associated with the triple is $y_{n_i}$, $(y_{n_i})_{i=1}^\infty$ is an $\ell_1^\zeta+$-spreading model. This yields the existence of some $p\in{\mathbb{N}}$ such that $((X, Y, (y_i)_{i=1}^\infty), (n_i)_{i=1}^\infty, M, p)\in B\cap C$. Therefore $\mathcal{L}\setminus \pi(B\cap C)\subset \mathcal{G}_{\xi, \zeta}$. For the reverse inclusion, assume that $(X, Y, (y_i)_{i=1}^\infty)\in \pi(B\cap C)$. Fix $(n_i)_{i=1}^\infty \in {\mathbb{N}}^{\mathbb{N}}$, $M\in [{\mathbb{N}}]$, $p\in{\mathbb{N}}$ such that $((X, Y, (y_i)_{i=1}^\infty), (n_i)_{i=1}^\infty, M, p)\in B\cap C$. Then by Proposition \[oopyal\], $(d_{n_{m_i}}(X))_{i=1}^\infty$ is $\xi$-weakly null, while $(y_{n_{m_i}})_{i=1}^\infty$ is an $\ell_1^\zeta+$-spreading model. Thus $(X, Y, (y_i)_{i=1}^\infty)\in \mathcal{L}\setminus \mathcal{G}_{\xi, \zeta}$.
The $\xi=\omega_1$ case is similar. We replace $W_\xi$ with $W$ from Lemma \[boring\].
$(ii)$ Suppose that $\xi, \zeta<\omega_1$. Let $P=\mathcal{L}\times{\mathbb{N}}^{\mathbb{N}}\times H^{\mathbb{N}}\times [{\mathbb{N}}]\times {\mathbb{N}}$. We let $A,B,C$ be the subsets of $P$ consisting of those $((X, Y, (y_i)_{i=1}^\infty), (n_i)_{i=1}^\infty, (f_i)_{i=1}^\infty, M, p)\in P$ such that $$\text{for all\ }i\in{\mathbb{N}}, f_i\in K_Y, \tag{A}$$ $$((d_{n_i}(X))_{i=1}^\infty, M)\text{\ has\ }D_\xi,\tag{B}$$ $$((f_i)_{i=1}^\infty, M)\text{\ has\ }D_\zeta.\tag{C}$$ Note that $A,B,C$ are Borel. Let $E$ denote the subset of $A$ consisting of those $((X, Y, (y_i)_{i=1}^\infty), (n_i)_{i=1}^\infty, M, p)$ such that for all $i\in{\mathbb{N}}$, $f_i(y_{n_i})\geqslant 1/p$. Note that $E$ is a Borel subset of $A$, and is therefore Borel in $P$. Let $\pi:P\to \mathcal{L}$ be the projection. Arguing as in the first paragraph, we deduce that $\mathcal{L}\setminus \pi(E\cap B\cap C)=\mathcal{M}_{\xi, \zeta}$. Thus this set is $\Pi_1^1$. Here we are using the fact that for $f_{y_1^*}, \ldots, f_{y_n^*}\in K_Y$ and scalars $(a_i)_{i=1}^n$, $\|\sum_{i=1}^n a_i f_{y^*_i}\|_\infty=\|\sum_{i=1}^n a_i y^*_i\|_{Y^*}$. Therefore if $f_i\in K_Y$, $f_i=f_{y^*_i}$, and $((f_i)_{i=1}^\infty, M)$ has $D_\zeta$ in $\ell_\infty$, $((y^*_i)_{i=1}^\infty, M)$ has $D_\zeta$ in $Y^*$. Therefore $(y^*_{m_i})_{i=1}^\infty$ is $\zeta$-weakly null.
Now if $\xi=\omega_1$ and $\zeta<\omega_1$, we replace $B$ above with the set of $((X, Y, (y_i)_{i=1}^\infty), (n_i)_{i=1}^\infty, M, p)$ such that $(d_{n_i}(X))_{i=1}^\infty$ is weakly null, which is $\Pi_1^1$. This gives that the resulting set $\mathcal{M}_{\omega_1, \zeta}$ is $\Pi_2^1$. If $\xi<\omega_1$ and $\zeta=\omega_1$, we replace the set $C$ above with the set of $((X, Y, (y_i)_{i=1}^\infty), (n_i)_{i=1}^\infty, M, p)$ such that $(f_i)_{i=1}^\infty$ is weakly null. If $\xi=\zeta=\omega_1$, we make both of these replacements of $B$ and $C$.
Throughout, for a given set $S$, $2^S$ will be topologized with the product topology. Given a set $\Lambda$, we let $\textbf{Tr}(\Lambda)$ denote the subset of $2^{\Lambda^{<{\mathbb{N}}}}$ consisting of those subsets which contain all initial segments of their members. Let $\textbf{Tr}=\textbf{Tr}({\mathbb{N}})$. Let $\textbf{WF}$ and $\textbf{IF}$, respectively, denote the subsets of $\textbf{Tr}$ consisting of well-founded and ill-founded trees. Let us recall that $T$ is *ill-founded* if there exists $(n_i)_{i=1}^\infty\in {\mathbb{N}}^{\mathbb{N}}$ such that $(n_i)_{i=1}^l\in T$ for all $l\in{\mathbb{N}}$, and $T$ is *well-founded* otherwise. Let us also recall that $\textbf{Tr}$ with the topology inherited from $2^{\Lambda^{<{\mathbb{N}}}}$ is a Polish space and $\textbf{WF}$ is a $\Pi_1^1$-complete subset of $\textbf{Tr}$ [@Dodos Theorem A$.9$, page $130$]. Let us note that a regular family, by compactness, is always identified with a well-founded tree on ${\mathbb{N}}$.
For $T\in \text{Tr}(2\times {\mathbb{N}})$ and $\sigma=({\varepsilon}_n)_{n=1}^\infty\in 2^{\mathbb{N}}$, let $T(\sigma)=\varnothing$ if $T=\varnothing$ and otherwise let $$T(\sigma)=\{\varnothing\}\cup \{(n_i)_{i=1}^l\in {\mathbb{N}}^{<{\mathbb{N}}}: ({\varepsilon}_i, n_i)_{i=1}^l\in T\}.$$
Let us define the subset $C$ of $\text{Tr}(2\times {\mathbb{N}})$ by $$C=\{T\in \text{Tr}(2\times {\mathbb{N}}): (\forall \sigma\in 2^{\mathbb{N}})(T(\sigma)\in \textbf{IF})\}.$$ Then $C$ is $\Pi_2^1$-complete [@K Lemma $4.1$].
For a finite sequence $v=(n_1, \ldots, n_k)$ of natural numbers, let $\overline{v}=(n_1, n_1+n_2, \ldots, n_1+\ldots +n_k)$. For an infinite sequence $v=(n_1, n_2, \ldots)$, let $\overline{v}=(n_1, n_1+n_2, \ldots)$. Let $\overline{\varnothing}=\varnothing$. For the following proposition, let us recall that we identify subsets of ${\mathbb{N}}$ with sequences in ${\mathbb{N}}$ in the natural way. A subset is identified with the sequence obtained by listing the members of that subset in strictly increasing order. Therefore a regular family is identified with a tree on ${\mathbb{N}}$.
Suppose $\mathcal{F}$ is a regular family. Define the map $U_\mathcal{F}:\text{\emph{Tr}}(2\times {\mathbb{N}})\to \text{\emph{Tr}}(2\times {\mathbb{N}})$ by letting $\varnothing\in U_\mathcal{F}(T)$ and by letting $({\varepsilon}_i, n_i)_{i=1}^k\in U_\mathcal{F}(T)$ if and only if either $({\varepsilon}_i, n_i)_{i=1}^k\in T$ or $\overline{(n_i)_{i=1}^k}\in \mathcal{F}$. Then $T\mapsto U_\mathcal{F}(T)$ is continuous. Furthermore, $U_\mathcal{F}(T)\in C$ if and only if $T\in C$.
\[crusoe\]
Fix $t\in (2\times {\mathbb{N}})^{<{\mathbb{N}}}$. Fix a sequence $T_n$ of trees on $2\times {\mathbb{N}}$ converging to the tree $T$. If $t=\varnothing$, then for all $n\in{\mathbb{N}}$, $$1_{U_\mathcal{F}(T_n)}(t)=1=1_{U_\mathcal{F}(T)}(t).$$ If $t\neq \varnothing$, write $t=({\varepsilon}_i, n_i)_{i=1}^k$ and let $v=(n_i)_{i=1}^k$. If $\overline{v}\in \mathcal{F}$, then for all $n\in{\mathbb{N}}$, $$1_{U_\mathcal{F}(T_n)}(t)=1=1_{U_\mathcal{F}(T)}(t).$$ Otherwise $$1_{U_\mathcal{F}(T_n)}(t)= 1_{T_n}(t)\to 1_T(t)=1_{U_\mathcal{F}(T)}(t).$$
Now let $I$ denote the set of all trees $T$ on ${\mathbb{N}}$ such that if $(n_i)_{i=1}^k\in T$, then $n_1<\ldots <n_k$. Note that $\mathcal{F}\in I$. Define the map $\Psi:\text{Tr}\to I$ by $\Psi(T)=\{\overline{v}: v\in T\}$ and note that $\Psi$ is a bijection. Note also that $\Psi(T)$ is well-founded if and only if $T$ is. Furthermore, it is well-known that if $S,T$ are two trees, then $S\cup T$ is well-founded if and only if $S,T$ are. From this it follows that for a tree $T$ on ${\mathbb{N}}$, then $T$, $\Psi(T)$, $\Psi(T)\cup \mathcal{F}$, and $\Psi^{-1}(\Psi(T)\cup \mathcal{F})$ are all well-founded, or all ill-founded.
One can describe $U_\mathcal{F}$ by noting that for each $\sigma\in 2^{\mathbb{N}}$ and $V\in \textbf{Tr}(2\times{\mathbb{N}})$, $U_\mathcal{F}(V)(\sigma)= \Psi^{-1}(\Psi(V(\sigma))\cup \mathcal{F})$. Now suppose that $V$ is a tree on $2\times {\mathbb{N}}$. Suppose that $V\in C$. Then for each $\sigma\in 2^{\mathbb{N}}$, by the last paragraph applied with $T=V(\sigma)$, $U_\mathcal{F}(V)(\sigma)=\Psi^{-1}(\Psi(V(\sigma))\cup \mathcal{F})$ is ill-founded. Since this holds for any $\sigma$, $U_\mathcal{F}(V)\in C$. Now if $V\in \textbf{Tr}(2\times {\mathbb{N}})\setminus C$, then there exists $\sigma\in 2^{\mathbb{N}}$ such that $V(\sigma)$ and $\Psi^{-1}(\Psi(V(\sigma))\cup \mathcal{F})$ are well-founded. In this case, $U_\mathcal{F}(V)\in \textbf{Tr}(2\times {\mathbb{N}})\setminus C$.
Let us recall that for a Banach space $R$ and an ordinal $0<\alpha<\omega_1$, a basis $(e_i)_{i=1}^\infty$ for $R$ is said to be *asymptotic* $c_0^\alpha$ (resp. *asymptotic* $\ell_1^\alpha$) in $R$ provided that there exists $a>0$ such that whenever $(x_i)_{i=1}^l$ is a block sequence with respect to $(e_i)_{i=1}^\infty$ such that $(\min \text{supp}(x_i))_{i=1}^l\in \mathcal{S}_\alpha$, $$\|\sum_{i=1}^l x_i\|\leqslant a\max_{1\leqslant i\leqslant l}\|x_i\|$$ $$(\text{resp.\ }\|\sum_{i=1}^l x_i\|\geqslant a\sum_{i=1}^l \|x_i\|).$$ Note that every seminormalized block sequence in a space with a basis which is asymptotic-$\ell_1^\alpha$ in the space is an $\ell_1^\alpha$-spreading model, and is therefore not $\alpha$-weakly null.
For each $0<\alpha<\omega_1$, there exist Borel maps $\mathfrak{S}_\alpha, \mathfrak{S}_\alpha^*:\text{\emph{Tr}}(2\times {\mathbb{N}})\to \textbf{\emph{SB}}$ such that
(i) if $T\in C$, then $\mathfrak{S}_\alpha(T)^*, \mathfrak{S}^*_\alpha(T)$ have the Schur property, and therefore $\mathfrak{S}_\alpha(T), \mathfrak{S}_\alpha^*(T)$ have the Dunford-Pettis property, and
(ii) if $T\notin C$, then $\mathfrak{S}_\alpha(T)$ (resp. $\mathfrak{S}_\alpha^*(T)$) has a complemented, reflexive subspace $R$ with a basis which is asymptotic $c_0^\alpha$ (resp. asymptotic $\ell_1^\alpha$) in $R$.
\[rob\]
For a tree $T$ on ${\mathbb{N}}$, let us recall that $$[T]=\{(n_i)_{i=1}^\infty \in {\mathbb{N}}^{\mathbb{N}}: (\forall k\in{\mathbb{N}})((n_i)_{i=1}^k\in T)\}.$$ define $$\mathcal{M}_T=\{\varnothing\}\cup \{\{n_1, n_1+n_2, \ldots, n_1+\ldots +n_k\}: (n_i)_{i=1}^k\in T\}\cup \{\{n_1, n_1+n_2, \ldots\}: (n_i)_{i=1}^\infty\in [T]\}\in 2^{\mathbb{N}}.$$ Note that $\mathcal{M}_T$ is compact. Given $\sigma=({\varepsilon}_i)_{i=1}^\infty\in 2^{\mathbb{N}}$ and $l\in{\mathbb{N}}$, we let $\sigma|_l=({\varepsilon}_i)_{i=1}^l$. Now for $T\in \textbf{Tr}(2\times {\mathbb{N}})$, we define the space $E_T$ to be the completion of $c_{00}(2^{<{\mathbb{N}}}\setminus \{\varnothing\})$ with respect to the norm $$[\sum_{t\in 2^{<{\mathbb{N}}}\setminus \{\varnothing\}}a_te_t]= \sup_{\sigma \in 2^{\mathbb{N}}} \|\sum_{l=1}^\infty a_{\sigma|_l} e_l\|_{\mathcal{M}_{T(\sigma)}}.$$ Here, for a compact set $\mathcal{M}\subset 2^{\mathbb{N}}$, $\|\cdot\|_\mathcal{M}$ denotes the Tsirelson space $\mathcal{T}^*[\mathcal{M}, 1/2]$ as defined in [@K]. Kurka showed that there exist Borel maps $\mathfrak{S}, \mathfrak{S}^*:\text{Tr}(2\times {\mathbb{N}})\to \textbf{SB}$ such that for each $T\in \text{Tr}(2\times {\mathbb{N}})$, $\mathfrak{S}(T)$ is isometric to $E_T$ and $\mathfrak{S}^*(T)$ is isometric to $E_T^*$. Kurka also showed that if $T\in C$, $E_T^*$ has the Schur property. Moreover, it is easy to see that for any $\sigma=({\varepsilon}_i)_{i=1}^\infty\in 2^{\mathbb{N}}$, $E_T$ contains a complemented copy of the space $\mathcal{T}^*[\mathcal{M}_{T(\sigma)},1/2]$, namely the closed span of the branch $(e_{\sigma|_l})_{l=1}^\infty$.
Now for $0<\alpha<\omega_1$, let us define $\mathfrak{S}_\alpha =\mathfrak{S}\circ U_{\mathcal{S}_\alpha}$ and $\mathfrak{S}^*_\alpha= \mathfrak{S}^*\circ U_{\mathcal{S}_\alpha}$, where $U_{\mathcal{S}_\alpha}$ is as defined in Proposition \[crusoe\]. Thus these maps are Borel. Furthermore, if $T\in C$, then so is $U_{\mathcal{S}_\alpha}(T)$. By Kurka’s result, for $T\in C$, $\mathfrak{S}_\alpha(T)$ is isometric to $E_{U_{\mathcal{S}_\alpha}(T)}$, the dual of which has the Schur property. Similarly, for $T\in C$, $\mathfrak{S}_\alpha(T)^*$ is isometric to $E_{U_{\mathcal{S}_\alpha}(T)}^*$, which has the Schur property. Now if $T\in \text{Tr}(2\times {\mathbb{N}})\setminus C$, there exists $\sigma\in 2^{\mathbb{N}}$ such that $T(\sigma)$, and $U_{\mathcal{S}_\alpha}(T)(\sigma)$, are well-founded. Let $\mathcal{M}=\mathcal{M}_{U_{\mathcal{S}_\alpha}(T)(\sigma)}$. Then since $\mathcal{M}$ contains only finite sets (that is, since $U_{\mathcal{S}_\alpha}(T)(\sigma)$ is well-founded), $[U_{\mathcal{S}_\alpha}(T)(\sigma)]=\varnothing$, it is well-known that $\mathcal{T}^*[\mathcal{M},1/2]$ is reflexive. By construction, $\mathcal{S}_\alpha\subset \mathcal{M}$, whence the basis of $\mathcal{T}^*[\mathcal{M},1/2]$ is asymptotic $c_0^\alpha$ and the basis of the dual space is asymptotic $\ell_1^\alpha$. Then $\mathfrak{S}_\alpha(T)$ contains a complemented copy of the reflexive space $\mathcal{T}^*[\mathcal{M},1/2]$ with asymptotic $c_0^\alpha$ basis. Since $\mathfrak{S}_\alpha^*(T)$ is isometric to $E^*_T$, it contains a complemented copy of the reflexive space $\mathcal{T}[\mathcal{M},1/2]$ with asymptotic $\ell_1^\alpha$ basis.
If $X$ is a Banach space with a complemented, reflexive subspace $R$ having a seminormalized basis which is asymptotic $c_0$ in $R$, then $X$ lies in $\complement\textsf{M}_{1, \omega_1}$. Indeed, since $R$ is complemented in $X$, it is sufficient to show that $R$ itself lies in $\complement \textsf{M}_{1, \omega_1}$. But a normalized, asymptotic $c_0$ basis for a reflexive Banach space is $1$-weakly null and the coordinate functionals to this basis are weakly null. Therefore such a space cannot lie in $\textsf{M}_{1, \omega_1}$.
Similarly, if $X$ is a Banach space with a complemented, reflexive subspace $R$ having a seminormalized basis which is asymptotic $\ell_1$ in $R$, then $X$ lies in $\complement\textsf{M}_{\omega_1,1}$. Indeed, since $R$ is complemented in $X$, it is sufficient to show that $R$ itself lies in $\complement \textsf{M}_{ \omega_1,1}$. Arguing as in the previous paragraph, the basis of $R$ is weakly null and the coordinate functionals to the basis are asymptotic $c_0$ in $R^*$, and therefore $1$-weakly null. Thus $R^*\in \complement \textsf{M}_{\omega_1, 1}$.
Finally, let us note that if $X$ is a Banach space with a reflexive subspace $R$ having a basis which is asymptotic $\ell_1^\alpha$ in $R$, and if $\zeta<\alpha$, then $X\in \complement \textsf{G}_{\omega_1, \zeta}$. Indeed, $R$, and therefore $X$, admits a weakly null, normalized sequence. Since $R$ is asymptotic $\ell_1^\alpha$, a subsequence of this sequence must be an $\ell_1^\alpha$-spreading model, so this sequence cannot be $\zeta$-weakly null.
\[econ\]
We are now ready to prove the lower estimates on complexity in the case that at least one of the ordinals is uncountable.
(i) If $0\leqslant \zeta<\omega_1$, then $\mathcal{G}_{\omega_1, \zeta}$ and $\textbf{\emph{G}}_{\omega_1, \zeta}$ are $\Pi_2^1$-hard.
(ii) If $0<\zeta\leqslant \omega_1$, then $\mathcal{M}_{\omega_1, \zeta}$, $\textbf{\emph{M}}_{\omega_1, \zeta}$, $\mathcal{M}_{\zeta, \omega_1}$, and $\textbf{\emph{M}}_{\zeta, \omega_1}$ are $\Pi_2^1$-hard.
We can deduce that the classes of operators are $\Pi_2^1$-hard if we know that the classes of spaces are $\Pi_2^1$-hard. Therefore it suffices to produce for each class $\textbf{G}_{\omega_1, \zeta}$, $\textbf{M}_{\zeta, \omega_1}$, $\textbf{M}_{\omega_1, \zeta}$ a Borel reduction of the class to $C$.
$(i)$ Let us fix $\zeta<\alpha<\omega_1$ and let $\mathfrak{S}_\alpha^*$ be the map from Corollary \[rob\]. Then if $T\in C$, $\mathfrak{S}^*_\alpha(T)$ has the Schur property and therefore lies in $ \textbf{G}_{\omega_1, \zeta}$. If $T\notin C$, then $\mathfrak{S}^*_\alpha(T)$ has a reflexive subspace $R$ with a basis which is asymptotic $\ell_1^\alpha$ in $R$. We deduce that if $T\notin C$, $\mathcal{S}^*_\alpha(T)\in \textbf{SB}\setminus \textbf{G}_{\omega_1, \zeta}$ by Remark \[econ\].
$(ii)$ Let us fix any $0<\alpha<\omega_1$. If $T\in C$, $\mathfrak{S}_\alpha(T)^*, \mathfrak{S}^*_\alpha(T)$ have the Schur property, so $\mathfrak{S}_\alpha(T), \mathfrak{S}^*_\alpha(T)$ lie in $\textbf{M}_{\omega_1, \omega_1}$ if $T\in C$. Now if $T\notin C$, $\mathfrak{S}_\alpha(T)$ has a complemented, reflexive subspace $R$ with asymptotic $c_0^1$-basis, so $\mathfrak{S}_\alpha(T)\in \textbf{SB}\setminus \textbf{M}_{1, \omega_1}\subset \textbf{SB}\setminus \textbf{M}_{\zeta, \omega_1}$ by Remark \[econ\]. A similar appeal to Remark \[econ\] yields that if $T\notin C$, $\mathfrak{S}^*_\alpha(T)$ has a complemented, reflexive subspace $R$ with asymptotic $\ell_1^1$-basis, so $\mathfrak{S}^*_\alpha(T)\in \textbf{SB}\setminus \textbf{M}_{\omega_1, 1}$.
We now complete the lower estimate on the complexity in the case of two countable ordinals. This follows from a standard tree space construction.
(i) For $0\leqslant \zeta<\xi\leqslant \omega_1$, $\mathcal{G}_{\xi, \zeta}$ and $\textbf{\emph{G}}_{\xi, \zeta}$ are $\Pi_1^1$-hard.
(ii) For $1\leqslant \zeta, \xi\leqslant \omega_1$, $\mathcal{M}_{\xi, \zeta}$ and $\textbf{\emph{M}}_{\xi, \zeta}$ are $\Pi_1^1$-hard.
We prove the result for spaces. We prove $(i)$ and $(ii)$ simultaneously. First fix any space $Y$ with a normalized, bimonotone basis $(y_i)_{i=1}^\infty$ such that $Y$ lies in $\complement\textsf{G}_{\xi, \zeta}$ (resp. $\complement\textsf{M}_{\xi, \zeta}$). Note that $X_\zeta\in \complement\textsf{G}_{\xi, \zeta}$, as the canonical basis is $\zeta+1$-weakly null and not $\zeta$-weakly null, and $\ell_2\in \complement \textsf{M}_{1,1}\subset \complement \textsf{M}_{\xi, \zeta}$. So such a $Y$ exists.
Now let $\mathcal{T}$ denote the finite, non-empty sequences of natural numbers and for such a sequence, let $|t|$ denote the length of $t$. Let us define the relation $\preceq$ on $\mathcal{T}$ by $s\preceq t$ if $s$ is an initial segment of $t$. We say a subset $\mathfrak{s}\subset \mathcal{T}$ is a *segment* if it is of the form $\mathfrak{s}=\{v: u\preceq v\preceq w\}$ for some $u,w\in \mathcal{T}$. Let us say two segments $\mathfrak{s}_0, \mathfrak{s}_1$ are *incomparable* if for $j\in \{0,1\}$, no member of $\mathfrak{s}_j$ is an initial segment of any member of $\mathfrak{s}_{1-j}$. Now let $Z$ denote the completion of $c_{00}(\mathcal{T})$ with respect to the norm $$\|\sum_{t\in \mathcal{T}}a_t e_t\|=\sup\Bigl\{\sum_{i=1}^n \|\sum_{t\in \mathfrak{s}_i} a_t y_{|t|}\|_Y: n\in{\mathbb{N}}, \mathfrak{s}_1, \ldots, \mathfrak{s}_n\subset \mathcal{T}\text{\ pairwise incomparable segments}\Bigr\}.$$ For a tree $T$ on ${\mathbb{N}}$, let $Z(T)$ denote the closed span in $Z$ of $\{e_t: t\in T\setminus\{\varnothing\}\}$. Then by an easy induction on the rank of $T$, if $T$ is well-founded, $Z(T)$ has the Schur property and therefore lies in $\textsf{G}_{\xi, \zeta}$ (resp. $\textsf{M}_{\xi, \zeta}$). If $T$ is ill-founded, $Z(T)$ contains a complemented copy of $Y$, and therefore lies in $\complement \textsf{G}_{\xi, \zeta}$ (resp. $\complement \textsf{M}_{\xi, \zeta}$). Furthermore, by standard techniques (see, for example, [@Bossard]), there exists a map $J:\text{Subs}(Z)\to \textbf{SB}$ mapping the space of closed subspaces of $Z$ into $\textbf{SB}$ such that $T\mapsto J(Z(T))$ is Borel. This is a reduction of $\textbf{WF}$ to $\textbf{G}_{\xi, \zeta}$ (resp. $\textbf{M}_{\xi, \zeta}$).
[HD]{}
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Applying probabilistic techniques we study regularity properties of quantum master equations (QMEs) in the Lindblad form with unbounded coefficients; a density operator is regular if, roughly speaking, it describes a quantum state with finite energy. Using the linear stochastic Schrödinger equation we deduce that solutions of QMEs preserve the regularity of the initial states under a general nonexplosion condition. To this end, we develop the probabilistic representation of QMEs, and we prove the uniqueness of solutions for adjoint quantum master equations. By means of the nonlinear stochastic Schrödinger equation, we obtain the existence of regular stationary solutions for QMEs, under a Lyapunov-type condition.'
address: |
CI$^2$MA, Departamento de Ingeniería Matemática\
Facultad de Ciencias Físicas y Matemáticas\
Universidad de Concepción\
Casilla 160 C, Concepción\
Chile\
\
author:
-
title: 'Regularity of solutions to quantum master equations: A stochastic approach'
---
.
.
Introduction
============
In order to establish the well-posedness of the mean values of quantum observables represented by unbounded operators, we investigate the regularity of solutions of quantum master equations (with unbounded coefficients) in stationary and transient regimes. For this purpose, we use classical stochastic analysis.
Gorini–Kossakowski–Lindblad–Sudarshan equations
-----------------------------------------------
In many open quantum systems, the states of a small quantum system with Hamiltonian $H\dvtx \mathfrak{h}
\rightarrow\mathfrak{h}$ evolve according to the operator equation $$\label{3}
\frac{d}{dt} \rho_{t}( \varrho) = \mathcal{L}_{\ast}(
\rho_{t}( \varrho) ),\qquad \rho_{0}( \varrho) =\varrho,$$ where $
\mathcal{L}_{\ast}( \rho)
=
G \rho+ \rho G^{\ast} +\sum_{k=1}^{\infty}L_{k} \rho L_{k}^{\ast}
$ (see, e.g., [@BreuerPetruccione2002; @GardinerZoller2004; @WisemanMilburn2010]). Here, $( \mathfrak{h}, \langle\cdot,\cdot\rangle)$ is a separable complex Hilbert space, $G, L_{1}, L_{2}, \ldots$ are given linear operators in $\mathfrak{h}$ satisfying $
G=-iH-\frac{1}{2} \sum_{k=1}^{\infty}L_{k}^{\ast}L_{k}
$ on suitable common domain and the unknown density operator $\rho_{t}( \varrho)$ is a nonnegative operator in $\mathfrak{h}$ with unit trace. The operators $L_{1},L_{2},\ldots$ describe the weak interaction between the small quantum system and a heat bath.
The measurable physical quantities of the small quantum system are represented by self-adjoint operators in $\mathfrak{h}$, which are called observables. Very important observables are unbounded, like position, momentum and kinetic energy operators. In the Schrödinger picture, the mean value of the observable $A$ at time $t$ is given by $\operatorname{tr} ( \rho_{t}(\varrho) A )$, the trace of $\rho_{t}( \varrho) A$.
In the Heisenberg picture, the initial density operator $\varrho$ is fixed. Using, for instance, (\[3\]) we obtain the following equation of motion for the observable $A$: $$\label{41}
\frac{d}{dt} \mathcal{T}_{t}( A )
=
\mathcal{L} ( \mathcal{T}_{t}( A ) ),\qquad
\mathcal{T}_{0}( A ) = A,$$ where $
\mathcal{L} ( \mathcal{T}_{t}( A ) )
=
\mathcal{T}_{t}( A ) G
+ G^{\ast} \mathcal{T}_{t}( A ) +\sum_{k=1}^{\infty} L_{k}^{\ast}
\mathcal{T}_{t}( A ) L_{k}
$; see, for example, . The expected value of $A$ at time time $t$ is given by $\operatorname{tr} (\varrho\mathcal{T}_{t}( A ) )$.
Stochastic Schrodinger equations (SSEs)
---------------------------------------
The evolution of the state of a quantum system conditioned on continuous measurement is governed (see, e.g., [@BarchielliGregoratti2009; @Belavkin1989; @WisemanMilburn2010]) by the stochastic evolution equation on $\mathfrak{h}$. $$\label{5}
Y_{t}=Y_{0}+\int_{0}^{t}G( Y_{s}) \,ds+\sum_{k=1}^{\infty}\int
_{0}^{t}L_{k}( Y_{s}) \,dB_{s}^{k}.$$ Here $ G( y) = G y + \sum_{k=1}^{\infty}(\Re\langle y,L_{k} y \rangle
L_{k} y -\frac{1}{2} \Re^{2}\langle y,L_{k} y \rangle y) $, $ L_{k}( y)
=L_{k} y -\break \Re\langle y, L_{k} y \rangle y $ and $ B^{1}, B^{2}, \ldots$ are real valued independent Wiener processes.
\[exmeasurement\] Set $\mathfrak{h} = L^{2}( \mathbb{R},\mathbb{C})$. Let $Q,P\dvtx \mathfrak{h} \rightarrow\mathfrak{h}$ be defined by $ Q f ( x ) = x f( x ) $ and $ P f ( x ) = -i f ^{\prime} ( x ) $. In (\[5\]), take $H = \frac{1}{2m} P^2 + c Q^2$, $ L_{1} = \alpha Q $ and $ L_{2} = \beta P $, with $m>0$, $\alpha, \beta\geq0$ and $c \in\mathbb{R}$. For all $k\geq3$, fix $L_{k}=0$.
Example \[exmeasurement\] with $\alpha, \beta, c >0$ describes the simultaneous monitoring of position and momentum of a linear harmonic oscillator; see, for example, [@GoughSobolev2004; @ScottMilburn2001]. Taking instead $\alpha>0$ and $\beta= c = 0$ we get a well-studied model for the continuous measurement of position of a free particle; see, for example, [@Diosi1988; @GoughSobolev2004; @BassiDurrKolb2010; @Kolokoltsov2000] and references therein.
Our main tool for studying (\[3\]) and (\[41\]) is the following linear SSE on $\mathfrak{h}$: $$\label{2}
X_{t}( \xi) =
\xi+\int_{0}^{t}GX_{s}( \xi) \,ds + \sum_{k=1}^{\infty}\int
_{0}^{t}L_{k}X_{s}( \xi) \,dW_{s}^{k},$$ where $ W^{1}, W^{2}, \ldots$ are real valued independent Wiener processes on a filtered complete probability space $( \Omega,\mathfrak{F},(\mathfrak{F}_{t}) _{t\geq0},\mathbb{P}) $. In fact, the basic assumption of this paper is:
There exists a nonnegative self-adjoint operator $C$ in $\mathfrak{h}$ such that: (i) $G$ is relatively bounded with respect to $C$; and (ii) (\[2\]) has a unique $C$-solution for any initial condition $\xi$ satisfying $ \mathbb{E} ( \Vert C \xi\Vert^{2} + \Vert\xi\Vert^{2} ) <
\infty$.
Here, a strong solution $X ( \xi)$ of (\[2\]) is called $C$-solution if $\mathbb{E}\Vert X_{t}( \xi) \Vert^{2}\leq\mathbb{E}\Vert\xi\Vert^{2}$, and the function $t \mapsto\mathbb{E}\Vert C X_{t}( \xi) \Vert^{2}$ is uniformly bounded on compact time intervals; see Definition \[definicion2\] for details.
The law of $X_{t}( Y_0 ) / \| X_{t}( Y_0 ) \|$ with respect to $\Vert X_{T}( Y_0 ) \Vert^{2}\cdot\mathbb{P}$ coincides with the law of $Y_{t}$ for all $t \in[0, T]$; see [@MoraReAAP2008]. The main technical and conceptual advantage of (\[5\]) over (\[2\]) is that the norm of $Y_t$ is equal to $1$.
Principal objectives {#subsecobjectives}
--------------------
Our main goal is to make progress in the understanding of the evolution of $\operatorname{tr} ( \rho_{t}(\varrho) A )$ when $A$ is unbounded.
Given a self-adjoint nonnegative operator $C$ in $\mathfrak{h}$, we denote by $\mathfrak{L}_{1,C}^{+}( \mathfrak{h}) $ the set of all nonnegative operators $\varrho\dvtx \mathfrak{h}
\rightarrow\mathfrak{h}$ for which, loosely speaking, $C \varrho$ is a trace-class operator; see Definition \[def2\]. From Section \[secprob-rep\] we have that the expected value of $A$ with respect to $\varrho\in\mathfrak{L}_{1,C}^{+}( \mathfrak{h}) $ is well defined whenever $A \in\mathfrak{L}( ( \mathcal{D}( C)$, $\mbox{$\Vert\cdot\Vert_{C}$})
,\mathfrak{h})$, where $\mathfrak{L}( ( \mathcal{D}( C), \mbox{$\Vert\cdot\Vert_{C}$})
,\mathfrak{h})$ is the space of all operators relatively bounded with respect to $C$. Our first objective is:
To prove that the solution $\rho_{t}( \varrho)$ of (\[3\]) belongs to $\mathfrak{L}_{1,C}^{+}(
\mathfrak{h})$ (for all $t > 0$) provided that $C$ satisfies hypothesis (H) and that $ \varrho\in\mathfrak{L}_{1,C}^{+}( \mathfrak{h})$.
The key condition to guarantee the uniqueness of solution of (\[3\]) is the existence of a self-adjoint nonnegative operator $C$ in $\mathfrak{h}$ such that formally $$\label{i2}
\mathcal{L} ( C^2 ) \leq K ( C^2 + I ),$$ where $I$ is the identity operator in $\mathfrak{h}$ and $K \in[ 0,
\infty[$. This condition, introduced by Chebotarev and Fagnola [@ChebFagn1998] (see also [@Chebotarev2000; @Fagnola1999; @GarQue1998]), is a quantum analog of the Lyapunov condition for nonexplosion of classical Markov processes; see [@Chebotarev2000] for heuristic arguments. Since hypothesis (H) holds under a weak version of (\[i2\]) (see [@FagnolaMora2010; @MoraReIDAQP2007] and Remark \[notaSuffCond\]), inequality (\[i2\]) is the underlying assumption of objective (O1). In many physical examples, relevant observables belong to $\mathfrak{L}( ( \mathcal{D}( C), \mbox{$\Vert\cdot\Vert_{C}$}) ,\mathfrak{h})$ for some $C$ satisfying (\[i2\]). In Example \[exmeasurement\], for instance, $C = P^2 + Q^2$ satisfies hypothesis (H) (see, e.g., [@FagnolaMora2010; @MoraReIDAQP2007]), and the position and momentum operators $Q$ and $P$ are $( P^2 + Q^2 )$-bounded.
Previously, the regularity of the solutions to (\[3\]) has been treated in [@ArnoldSparber2004; @ChebGarQue1998; @Davies1977b] using methods from the operator theory. Exploiting the characteristics of a model describing a variable number of neutrons moving in a translation invariant external reservoir of unstable atoms, Davies [@Davies1977b] established that $\rho_{t}(\varrho) \in\mathfrak{L}_{1,C}^{+}( \mathfrak{h})$ whenever $C$ is the particle number operator on an adequate Fermion Fock space. Arnold and Sparber [@ArnoldSparber2004] obtained the same property with $C$ being essentially the energy operator for a linear quantum master equation associated to a diffusion model with Hartree interaction.
The second objective presents the first attempt (to the best of my knowledge) to show the existence of stationary solutions of (\[3\]) with finite energy.
- To prove the existence of a stationary solution of (\[3\]) belonging to $\mathfrak
{L}_{1,D}^{+}( \mathfrak{h})$ provided essentially that:
- There exist two nonnegative self-adjoint operators $C$ and $D$ and a constant $K>0$ such that $\{ x \in\mathfrak{h}\dvtx \| D x \| ^2 + \| x \| ^2 \leq1 \}$ is compact in $\mathfrak{h}$, and $
\mathcal{L} ( C^2 ) \leq-D^2 + K ( 1 + I )
$.
Hypothesis (L) is a quantum version of the Lyapunov criterion for the existence of invariant probability measures for stochastic differential equations, which applies to many open quantum systems; see, for example, [@FagReb2001] and Section \[secoscillator\]. Let $\mathfrak{L}( \mathfrak{h} ) $ be the set of all bounded operators from $\mathfrak{h}$ to $\mathfrak{h}$. In the case where $G$ is the infinitesimal generator of a strongly continuous contraction semigroup on $\mathfrak{h}$ and, loosely speaking, $ \mathcal{T}_{t} ( I ) = I$ for all $t \geq0$, Fagnola and Rebolledo [@FagReb2001] proved that under hypothesis (L), there exists at least one density operator $\varrho_{\infty}$ satisfying $
\operatorname{tr} (\varrho_{\infty} \mathcal{T}_{t}( A ) )
=
\operatorname{tr} (\varrho_{\infty} A )
$ for all $t \geq0$ and $A \in\mathfrak{L} ( \mathfrak{h})$. The main point of objective (O2) is that among such stationary states $\varrho_{\infty} $ we can select a finite-energy density operator belonging to the domain of $ \mathcal{L}_{\ast}$, under the same hypothesis (L).
The third main objective develops the rigorous probabilistic representation of solution of (\[3\]), the key step to achieve objectives (O1) and (O2).
Assume hypothesis (H), and let $
\varrho= \mathbb{E} \vert\xi\rangle\langle\xi\vert
$, where $\xi$ is a $\mathfrak{h}$-valued random variable such that $
\mathbb{E} \Vert\xi\Vert^{2} = 1
$ and $
\mathbb{E} \Vert C \xi\Vert^{2} < \infty
$. We wish to prove that (\[3\]) has a unique solution, which is $$\label{I1}
\rho_{t}( \varrho)
= \mathbb{E} \vert X_{t}( \xi) \rangle\langle X_{t}( \xi) \vert.$$
In Dirac notation, $\vert x\rangle\langle y\vert\dvtx \mathfrak{h} \rightarrow\mathfrak{h}$ is defined by $
\vert x\rangle\langle y\vert( z ) = \langle y,z\rangle x
$, with . Using (\[I1\]) we can assert that $$\label{i9}
\rho_{t}( \varrho) =
\mathbb{E} \vert Y_{t} \rangle\langle Y_{t} \vert$$ with $Y_0 = \xi$ (see [@MoraReAAP2008]). Objective (O3), together with (\[i9\]), shows that physical models based on the stochastic Schrödinger equations are in good agreement with their formulations in terms of quantum master equations.
In the physical literature, the probabilistic representations (\[I1\]) and (\[i9\]) of the density operator at time $t$ have been obtained by means of formal computations; see, for example, [@BarchielliBelavkin1991; @BreuerPetruccione2002; @GisinPercival1992]. Barchielli and Holevo [@BarchielliHolevo1995] established essentially (\[I1\]) and (\[i9\]) in situations where $G,L_{1},L_{2},\ldots$ are bounded.
Approach {#subsecapproach}
--------
In the perspective of the operator theory, methods based on the Hille–Yosida theorem and perturbations of linear operators [@Kato1995; @Pazy1983] present severe limitations for studying linear functionals of the solutions of (\[3\]) and (\[41\]). For example, it is very difficult to decompose $ \mathcal{L}_{\ast}$ into $ \mathcal{L}_{\ast}^1+ \mathcal{L}_{\ast}^2$ for a dissipative operator $ \mathcal{L}_{\ast}^1$ in $\mathfrak{L}_{1}(
\mathfrak{h})$ and an infinitesimal generator $ \mathcal{L}_{\ast}^2$ of a $C_0$ semigroup of contractions on $\mathfrak{L}_{1}( \mathfrak{h})$, which together satisfy $
\Vert\mathcal{L}_{\ast}^1 ( \varrho) \Vert_{1}
\leq
\alpha\Vert\mathcal{L}_{\ast}^2 ( \varrho) \Vert_{1} + K \Vert
\varrho\Vert_{1}
$ whenever $\varrho\in\mathcal{D} ( \mathcal{L}_{\ast}^2 )$. Here, $0 \leq\alpha< 1$, $K\geq0$ and $\mathfrak{L}_{1}( \mathfrak{h})$ is the Banach space of trace-class operators on $\mathfrak{h}$ equipped with the trace norm . Another difficulty is that $ \mathcal{L}$ and $ \mathcal{L}_{\ast}$ are defined formally; indeed $ \mathcal{L}$ and $ \mathcal{L}_{\ast}$ can be interpreted as sesquilinear forms, but without having a priori knowledge about their cores.
When $G$ is the generator of a $C_0$ semigroup of contractions on $\mathfrak{h}$, Davis [@Davies1977a] provided solutions of (\[3\]) by means of semigroups. Modifying Davis’s ideas, Chebotarev constructed a quantum dynamical semigroup $\mathcal{T}^{(\min)}$ that is weak solution of (\[41\]) by generalizing the Chung construction of the minimal solution of Feller–Kolmogorov equations for countable state Markov chains; see Remark \[nota7\]. Under certain conditions involving (\[i2\]) and invariant sets for $\exp(Gt )$, Chebotarev and Fagnola [@ChebFagn1998] proved the uniqueness of $\mathcal{T}^{(\min)}$; see Remark \[nota7\]. This property implies that $ \mathcal{L}_{\ast}$ is the infinitesimal generator of the predual semigroup $\rho^{(\min)}$ of $\mathcal{T}^{(\min)}$, and a core for $ \mathcal{L}_{\ast}$ is formed by the linear span of all $\vert x\rangle\langle y\vert$ with $x,y$ belonging to $\mathcal{D} ( G )$, the domain of $G$; see Remark \[nota8\]. In Remark \[nota8\], we outline how to obtain $\rho^{(\min)} ( \mathfrak{L}_{1,C}^{+}( \mathfrak{h}) ) \subset
\mathfrak{L}_{1,C}^{+}( \mathfrak{h}) $ under various assumptions including $ \exp( G t ) \mathcal{D}( C ) \subset\mathcal{D}( C )$. It is a hard problem, in general, to find $C$ satisfying (\[i2\]) whose domain $ \mathcal{D}( C )$ is invariant under the action of $
\exp( G t ) $.
In contrast to closed quantum systems, solutions of (\[3\]) are not decomposable as dyadic products of solutions of evolution equations in $\mathfrak{h}$. Nevertheless, the solution of (\[3\]) is unraveled into stochastic quantum trajectories; more precisely, objective (O3) establishes that $
\rho_{t}( \varrho)
$ is expressed as the mean value of quadratic functionals of the solutions of SSEs in a general context. This property allows us to achieve objectives (O1) and (O2) by using SSEs, without serious difficulties and without assumptions involving invariant sets for $\exp(Gt )$. Applying (\[I1\]) we also deduce that $
\rho_{t}( \varrho)
$ satisfies (\[3\]) in both sense integral and $\mathfrak{L}_{1}(
\mathfrak{h}) $-weak. This leads to prove rigorously some dynamical properties of $ \rho_{t}( \varrho)$ given in physics; see, for example, Theorem \[teorema5\].
We now focus on objective (O1). By Section \[secprob-rep\], $ \varrho\in\mathfrak{L}_{1,C}^{+}( \mathfrak{h})$ iff there exists a $\mathfrak{h}$-valued random variable $\xi$ satisfying $
\mathbb{E} ( \Vert C \xi\Vert^{2} + \Vert\xi\Vert^{2} ) < \infty
$ and $
\varrho= \mathbb{E} \vert\xi\rangle\langle\xi\vert
$. Therefore (\[I1\]) leads directly to objective (O1) since $
\mathbb{E} \Vert C X_{t}( \xi) \Vert^{2}
+
\mathbb{E} \Vert X_{t}( \xi) \Vert^{2}
< \infty
$. Assumption (\[i2\]) is natural in the context of (\[2\]) because (\[i2\]) is essentially the dissipative condition for (\[2\]).
We turn to objective (O2). Here, hypothesis (L) is a classical Lyapunov condition for (\[2\]). Relation (\[I1\]) suggests us that $ \int_{\mathfrak{h}}\vert x\rangle\langle x\vert\mu( dx)$ is a good candidate for being a stationary solution for (\[3\]) when $\mu$ is an invariant probability measure for (\[2\]) such that $ \int_{\mathfrak{h}} \| x \|^2 \mu( dx) = 1$. This reduces objective (O2) to prove that there exists an invariant probability measure for (\[2\]), different from the Dirac measure at $0$, which is a difficult problem. We instead use (\[i9\]). Under a weak version of hypothesis (L), there exists an invariant probability measure $\Gamma$ for (\[5\]) such that $ \int_{\mathfrak{h}} \| x \|^2 \Gamma( dx) = 1$ and $ \int_{\mathfrak{h}} \| D x \|^2 \Gamma( dx) < \infty$; see [@MoraReAAP2008]. Then, using (\[i9\]) we deduce that $
\varrho_{\infty}=\int_{\mathfrak{h}}\vert x\rangle\langle x\vert
\Gamma( dx)
$ is a stationary solution to (\[3\]) that belongs to $\mathfrak{L}_{1,D}^{+}( \mathfrak{h})$; see Section \[secStatSol\]. This is a step forward in the study of the long time behavior of unbounded observables.
Technical ideas: Unraveling {#subsecexistence}
---------------------------
Fix $ \varrho\in\mathfrak{L}_{1,C}^{+}( \mathfrak{h})$. Then $
\varrho= \mathbb{E} \vert\xi\rangle\langle\xi\vert
$ for some $\mathfrak{h}$-valued random variable $\xi$ satisfying $
\mathbb{E} ( \Vert C \xi\Vert^{2} + \Vert\xi\Vert^{2} ) < \infty
$; see Section \[secprob-rep\]. We can define $$\label{i4}
\rho_{t}( \varrho)
: = \mathbb{E} \vert X_{t}( \xi) \rangle\langle X_{t}( \xi) \vert,$$ because $\rho_{t}( \varrho) $ does not depend on the choice of $\xi$; see Theorem \[teor7\]. We next outline how to establish that $
\rho_{t}( \varrho)
$ is a solution of (\[3\]).
Applying Itô’s formula we obtain $$\begin{aligned}
\langle X_{t}( \xi) , x \rangle X_{t}( \xi)
&=&
\langle\xi, x \rangle\xi
+ \int_{0}^{t} \bigl(
\langle X_{s}( \xi) ,x \rangle GX_{s}( \xi)
+
\langle G X_{s}( \xi) , x \rangle X_{s}( \xi)
\bigr) \,ds
\\
&&{}
+ \sum_{k=1}^{\infty} \int_{0}^{t} \langle L_k X_{s}( \xi) , x
\rangle L_k X_{s}( \xi)
\,ds
+ M_t\end{aligned}$$ with $
M_t
=
\sum_{k = 1}^{\infty} \int_{0}^{t} (
\langle X_{s}( \xi) , x \rangle L_k X_{s}( \xi)
+
\langle L_k X_{s}( \xi) , x \rangle X_{s}( \xi)
) \,dW^{k}_{s}
$. Since $M_t$ is a local martingale, we use stopping times and the dominated convergence theorem to deduce that $$\begin{aligned}
\label{65}
&&\mathbb{E} \langle X_{t}( \xi) , x \rangle X_{t}( \xi)\nonumber\\
&&\qquad=
\mathbb{E} \langle\xi, x \rangle\xi
+ \int_{0}^{t}
\mathbb{E} \langle X_{s}( \xi) ,x \rangle GX_{s}( \xi)
\,ds
\\
&&\qquad\quad{}
+ \int_{0}^{t}
\mathbb{E} \langle G X_{s}( \xi) , x \rangle X_{s}( \xi)
\,ds
+ \sum_{k=1}^{\infty} \int_{0}^{t} \mathbb{E} \langle L_k X_{s}( \xi
) , x \rangle L_k X_{s}( \xi)
\,ds .\nonumber\end{aligned}$$ Define the operator $
\mathcal{L}_{*} ( \xi, s ) \dvtx
\mathfrak{h} \rightarrow\mathfrak{h}
$ to be $$\mathbb{E} \vert GX_{s}( \xi) \rangle\langle X_{s}( \xi) \vert
+
\mathbb{E} \vert X_{s}( \xi) \rangle\langle G X_{s}( \xi) \vert
+
\sum_{k=0}^{\infty}
\mathbb{E} \vert L_k X_{s}( \xi) \rangle\langle L_k X_{s}( \xi)
\vert.$$ We now face the major technical difficulties; we have to prove that $\mathcal{L}_{*} ( \xi, s )$ is a trace-class operator such that: (i) $
\mathcal{L}_{*} ( \xi,t )
=
\mathcal{L}_{\ast}( \rho_{s}( \varrho) )
$; (ii) the function $ s \mapsto
\Vert
\mathcal{L}_{*} ( \xi, s )
\Vert_{1}$ is locally bounded; and (iii) $s \mapsto\mathcal{L}_{*} ( \xi, s )$ is weakly continuous in $\mathfrak{L}_{1}( \mathfrak{h})$. Then, applying (\[65\]) yields $$\label{I5}
\rho_{t}( \varrho)
=
\varrho
+
\int_{0}^{t} \mathcal{L}_{\ast}( \rho_{s}( \varrho) ) \,ds,$$ where we understand the integral of (\[I5\]) in the sense of the Bochner integral in $\mathfrak{L}_{1}( \mathfrak{h})$. Thus, we can deduce that for any $A \in\mathfrak{L}( \mathfrak{h} )$, $$\label{I6}
\frac{d}{dt}\operatorname{tr}( A\rho_{t}( \varrho) )
=
\operatorname{tr}( A \mathcal{L}_{\ast}( \rho_{t}( \varrho) ) ) .\vadjust{\goodbreak}$$
Technical ideas: Uniqueness {#subsecuniqueness}
---------------------------
Recall that $\rho_{t}( \varrho)$ is defined by (\[i4\]) for any $ \varrho\in\mathfrak{L}_{1,C}^{+}( \mathfrak{h})$. In order to establish the uniqueness of the solution of (\[3\]) under hypothesis (H), Theorem \[teor8\] extends $\rho_{t}( \varrho)$ to a strongly continuous semigroup $( \rho_{t})_{t\geq0}$ of bounded operators on $\mathfrak{L}_{1} ( \mathfrak{h})$. Thus, $( \rho_{t})_{t\geq0}$ belongs to the class $\mathcal{S}$ formed by all the locally bounded semigroups $( \widehat{\rho}_{t} )_{t\geq0}$ on $\mathfrak{L}_{1} ( \mathfrak{h})$ such that for any $ x \in\mathcal{D} (C )$: (i) $t \mapsto\widehat{\rho}_{t} ( \vert x \rangle\langle x \vert) $ is weakly continuous in $\mathfrak{L}_{1} ( \mathfrak{h}))$; and (ii) $\widehat{\rho}_{t} ( \vert x \rangle\langle x \vert)$ satisfies (\[I6\]) in $t=0$; see Theorem \[teorema9\] and Lemma \[lema26\] for details. We next outline the proof that $( \rho_{t})_{t\geq0}$ is the unique element in $\mathcal{S}$, and so (\[3\]) has a unique solution (in the semigroup sense).
Let $( \widehat{\rho}_{t}) _{t\geq0} \in\mathcal{S}$. Taking in mind that $\mathfrak{L} ( \mathfrak{h})$ is the dual of $\mathfrak{L}_{1} ( \mathfrak{h})$, we consider the semigroup $ ( \mathcal{T}_{t} ) _{t\geq0} $ on $\mathfrak{L} ( \mathfrak{h})$ which is the adjoint semigroup of $( \widehat{\rho}_{t}) _{t\geq0}$. Using techniques from operator theory we obtain in Lemma \[lema20\] that $ ( \mathcal{T}_{t} ) _{t\geq0} $ is a weak solution of (\[41\]), namely, for all $t \geq0 $, $A \in\mathfrak{L} ( \mathfrak{h})$ and $x \in\mathcal{D} (C ) $ we have $$\label{I7}
\frac{d}{dt} \langle x,\mathcal{T}_{t}( A ) x \rangle
=
\langle x, \mathcal{L} ( \mathcal{T}_{t}( A ) ) x \rangle.$$
Now, we wish to prove that $ ( \mathcal{T}_{t} ) _{t\geq0} $ is the unique weak solution of (\[41\]), which is an important problem itself; see, for example, [@ChebFagn1993; @ChebFagn1998; @ChebGarQue1998; @Chebotarev2000; @Fagnola1999; @MoraJFA2008]. Suppose for a moment that $\mathfrak{h}$ is finite-dimensional and $L_k \ne0$ for only a finite number of $k$. Applying the Itô formula to $
\langle X_{s}( x ) , \mathcal{T}_{t -s}( A ) X_{s}( x ) \rangle
$ we deduce that $$\begin{aligned}
&&
\langle X_{t}( x ) , A X_{t}( x ) \rangle\\
&&\quad=
\langle x , \mathcal{T}_{t}( A ) x \rangle
+
M_t
\\
&&\qquad{}
+
\int_{0}^{t} \biggl(
\langle X_{s}( x ) ,
\mathcal{L} ( \mathcal{T}_{t-s}( A ) ) X_{s}( x ) \rangle
-
\biggl\langle X_{s}( x ) ,
\frac{d \mathcal{T}_r ( A )}{dr} \bigg| _{r=t-s} X_{s}( x ) \biggr\rangle
\biggr) \,ds\end{aligned}$$ with $$M_t
=
\sum_{k=1}^{\infty} \int_{0}^{t}
\bigl(
\langle L_{k} X_{s}( x ) , \mathcal{T}_{t-s}( A ) X_{s}( x ) \rangle
+
\langle X_{s}( x ) , \mathcal{T}_{t-s}( A ) L_{k} X_{s}( x ) \rangle
\bigr)\,
dW_s^{k} .$$ From (\[I7\]) we obtain $
\langle X_{t}( x ) , A X_{t}( x ) \rangle
=
\langle x , \mathcal{T}_{t}( A ) x \rangle
+
M_t
$, and so the martingale property of $M_t$ leads to $
\mathbb{E} \langle X_{t}( x ) , A X_{t}( x ) \rangle
=
\langle x , \mathcal{T}_{t}( A ) x \rangle
$, and hence all the elements in $\mathcal{S}$ are the same semigroups, which implies $ \widehat{\rho} = \rho$.
In the general case, $G$ and $L_k$ are unbounded operators. Therefore $$( s, x )
\mapsto
\frac{d}{ds} \langle x,\mathcal{T}_{t-s}( A ) x \rangle
\bigl( =
\langle x, \mathcal{L} ( \mathcal{T}_{t-s}( A ) ) x \rangle
\bigr)$$ is not continuous on $[ 0, t ] \times\mathfrak{h}$, and consequently we cannot apply directly Itô’s formula to $
\langle X_{s}( x ) , \mathcal{T}_{t -s}( A ) X_{s}( x ) \rangle
$. We overcome this difficulty in Section \[subsecteorema3\] by applying Itô’s formula to a regularized version of $\langle x,\mathcal{T}_{t-s}( A ) x \rangle$; the resulting stochastic integrals (similar to those in $M_t$) are only local martingales, and so we have to use stopping times.
Outline
-------
Section \[secAdQME\] addresses the existence and uniqueness of solutions for the adjoint quantum master equation, as well as its probabilistic representation. Section \[secprob-rep\] deals with the probabilistic interpretations of regular density operators. In Section \[secQME\] we construct Schrödinger evolutions by means of stochastic Schrödinger equations and study the regularity of solutions to (\[3\]). Section \[secStatSol\] focusses on the existence of regular stationary solutions for (\[3\]). In Section \[secoscillator\] we apply our results to a quantum oscillator. Section \[secproofs\] is devoted to proofs.
Notation {#subsecnot}
--------
Throughout this paper, the scalar product $\langle\cdot,\cdot\rangle$ is linear in the second variable and anti-linear in the first one. We write $\mathfrak
{B}( \mathfrak{h}) $ for the Borel $\sigma$-algebra on $\mathfrak{h}$. Suppose that $A$ is a linear operator in $\mathfrak{h}$. Then $A^{\ast}$ denotes the adjoint of $A$. If $A$ has a unique bounded extension to $\mathfrak{h}$, then we continue to write $A$ for the closure of $A$.
Let $\mathfrak{X}$, $\mathfrak{Z}$ be normed spaces. We write $\mathfrak{L}( \mathfrak{X},\mathfrak{Z}) $ for the set of all bounded operators from $\mathfrak{X}$ to $\mathfrak{Z}$ (together with norm ). We abbreviate to , if no misunderstanding is possible, and define $\mathfrak{L}( \mathfrak{X}) = \mathfrak{L}( \mathfrak
{X},\mathfrak{X}) $. By $\mathfrak{L}_{1}^{+}( \mathfrak{h})$ we mean the subset of all nonnegative trace-class operators on $\mathfrak{h}$.
Let $C$ be a self-adjoint positive operator in $\mathfrak{h}$. Then, for any $x,y\in\mathcal{D}( C) $ we set $\langle x,y\rangle
_{C}=\langle x,y\rangle+\langle Cx,Cy\rangle$ and $ \Vert x\Vert
_{C}=\sqrt{\langle x,x\rangle_{C}}$. As usual, $L^{2}( \mathbb{P},\mathfrak{h}) $ stands for the set of all square integrable random variables from $( \Omega,\mathfrak{F},\mathbb
{P})$ to $ ( \mathfrak{h},\mathfrak{B}( \mathfrak{h}) )$. We write $L_{C}^{2}( \mathbb{P},\mathfrak{h}) $ for the set of all $\xi
\in L^{2}( \mathbb{P},\mathfrak{h}) $ satisfying $\xi\in\mathcal{D}(
C) $ a.s. and $\mathbb{E} \Vert\xi\Vert_{C}^{2}<\infty$. The function $\pi_{C}\dvtx\mathfrak{h\rightarrow h}$ is defined by $\pi_{C}( x) = x$ if $x\in\mathcal{D}( C)$ and $\pi_{C}( x) = 0$ whenever $x\notin\mathcal{D}( C)$. In the sequel, the letter $K$ denotes generic constants.
Adjoint quantum master equation {#secAdQME}
===============================
We begin by presenting in detail the notion of $C$-solution to (\[2\]).
\[HipN3\] Suppose that $C$ is a self-adjoint positive operator in $\mathfrak{h}$ such that $\mathcal{D}( C)$ is a subset of the domains of $G, L_{1}, L_{2}, \ldots,$ and the maps $G\circ\pi_{C}, L_{1}\circ\pi_{C}, L_{2}\circ\pi_{C},
\ldots$ are measurable.
\[definicion2\] Let Hypothesis \[HipN3\] hold. Assume that $\mathbb{I}$ is either $[
0,\infty[ $ or $[ 0,T] $, with $T\in\mathbb{R}_{+}$. An $\mathfrak{h}$-valued adapted process $( X_{t}( \xi) ) _{t \in
\mathbb{I}}$ with continuous sample paths is called strong $C$-solution of (\[2\]) on $\mathbb{I}$ with initial datum $\xi$ if and only if for all $t\in\mathbb{I}$:
- $\mathbb{E}\Vert X_{t}( \xi) \Vert^{2}\leq\mathbb{E}\Vert\xi
\Vert^{2}$, $X_{t}( \xi) \in\mathcal{D}( C) $ a.s., $
\sup_{s\in[ 0,t] }\mathbb{E}\Vert C X_{s}( \xi) \Vert^{2} < \infty.
$
- $
X_{t}( \xi) =\xi+\int_{0}^{t}G\pi_{C}( X_{s}( \xi
) ) \,ds+\sum_{k=1}^{\infty}\int_{0}^{t}L_{k}\pi_{C}(
X_{s}( \xi) ) \,dW_{s}^{k}$ $\mathbb{P}$-a.s.
\[notX\] The symbol $X ( \xi)$ will be reserved for the strong $C$-solution of (\[2\]) with initial datum $\xi$.
\[notaMedibilidad\] Suppose that $C$ is a self-adjoint positive operator in $\mathfrak{h}$, together with $A \in\mathfrak{L}( ( \mathcal{D}( C), \mbox{$\Vert\cdot\Vert_{C}$})
,\mathfrak{h})$. Then $A \circ\pi_{C}\dvtx \mathfrak{h} \rightarrow\mathfrak{h}$ is measurable whenever $\mathfrak{h}$ is equipped with its Borel $\sigma
$-algebra (see, e.g., [@FagnolaMora2010] for details).
We now make more precise our basic assumptions, that is hypothesis (H).
\[HipN5\] Suppose that Hypothesis \[HipN3\] holds. In addition, assume:
The operator $G$ belongs to $\mathfrak{L}( ( \mathcal{D}(
C), \mbox{$\Vert\cdot\Vert_{C}$}) ,\mathfrak{h})$.
For all $x\in\mathcal{D}( C) $, $
2\Re\langle x,Gx\rangle+{\sum_{k=1}^{\infty}}\Vert L_{k}x\Vert^{2} = 0
$.
Let $\xi\in L_{C}^{2}( \mathbb{P}, \mathfrak{h} )$ be $\mathfrak
{F}_{0}$-measurable. Then for all $T > 0$, (\[2\]) has a unique strong $C$-solution on $[
0,T]$ with initial datum $\xi$.
\[nota1\] Let $A$ be a closable operator in $\mathfrak{h}$ whose domain is contained in $\mathcal{D}( C)$, where $C$ is a self-adjoint positive operator in $\mathfrak{h}$. Applying the closed graph theorem we obtain $A\in\mathfrak{L}( ( \mathcal{D}( C) ,\mbox{$\Vert
\cdot\Vert_{C}) ,\mathfrak{h}$}) $, which leads to a sufficient condition for (H2.1).
\[nota3\] Let $C$ be a self-adjoint positive operator in $\mathfrak{h}$ such that $\mathcal{D}( C) \subset\mathcal{D}( G)$. Assume that $
2\Re\langle x, Gx\rangle+\sum_{k=1}^{\infty}\Vert L_{k}x \Vert
^{2}\leq0
$ for all $x \in\mathcal{D}( G)$. Then the numerical range of $G$ is contained in the left half-plane of $\mathbb{C}$, and so $G$ is closable. Therefore $G \in\mathfrak{L}( ( \mathcal{D}( C), \Vert\cdot\Vert
_{C}) ,\mathfrak{h})$ by Remark \[nota1\].
Using arguments given in Section \[subsecuniqueness\] we prove the following theorem, establishing the uniqueness of the solution of (\[41\]).
\[definicion3\] Suppose that $A \in\mathfrak{L}( \mathfrak{h})$ and that $C$ is a self-adjoint positive operator in $\mathfrak{h}$. A family of operators $( \mathcal{A}_{t} )_{t \geq0}$ belonging to $\mathfrak{L}( \mathfrak{h})$ is a $C$-solution of (\[41\]) with initial datum $A$ iff $\mathcal{A}_{0} = A$ and for all $t \geq0$:
$
\frac{d}{dt} \langle x, \mathcal{A}_{t} y \rangle= \langle x, \mathcal
{A}_{t} G y \rangle+ \langle G x, \mathcal{A}_{t} y \rangle+ \sum
_{k=1}^{\infty} \langle L_{k} x, \mathcal{A}_{t} L_{k} y \rangle
$ for all $x,y \in\mathcal{D}(C )$.
${\sup_{s \in[ 0,t]}} \Vert\mathcal{A}_{s} \Vert_{\mathfrak{L}(
\mathfrak{h})} < \infty$.
\[teorema3\] Suppose that Hypothesis \[HipN5\] holds. Let $A$ belong to $\mathfrak{L}( \mathfrak{h})$. Then, for every nonnegative real number $t$ there exists a unique $\mathcal{T}_{t}( A ) $ in $\mathfrak{L}( \mathfrak{h})$ such that for all $x,y$ in $\mathcal{D}(C )$, $$\label{42}
\langle x, \mathcal{T}_{t}( A ) y \rangle= \mathbb{E} \langle X_{t} (
x ), A X_{t} ( y ) \rangle.$$ Moreover, any $C$-solution of (\[41\]) with initial datum $A$ coincides with $ \mathcal{T} ( A ) $, and $
\Vert\mathcal{T}_{t}( A ) \Vert_{\mathfrak{L}( \mathfrak{h})}
\leq
\Vert A \Vert_{\mathfrak{L}( \mathfrak{h})}$ for all $t \geq0$.
The proofs fall naturally into Lemmata \[lema41\] and \[lema42\].
As a by-product of our proof of the existence of solutions to (\[3\]), we “construct” a solution to (\[41\]), and so Theorem \[teorema3\] leads to Theorem \[teorema10\].
\[teorema10\] Let Hypothesis \[HipN5\] hold. Suppose that $A \in\mathfrak{L}( \mathfrak{h})$ and that $ \mathcal{T}_{t}( A )$ is as in Theorem \[teorema3\]. Then $( \mathcal{T}_{t}( A ) )_{t \geq0}$ is the unique $C$-solution of (\[41\]) with initial datum $A$.
Lemmata \[lema26\] and \[lema20\] shows that $( \mathcal{T}_{t}( A ) )_{t \geq0}$ is a $C$-solution of (\[41\]) with initial datum $A$. Theorem \[teorema3\] now completes the proof.
In [@MoraJFA2008], C. M. Mora developed the existence and uniqueness of the solution to (\[41\]) with $A$ unbounded, as well as its probabilistic representation. Thus taking $A \in\mathfrak{L}( \mathfrak{h})$, Corollary 14 of [@MoraJFA2008] established the statement of Theorem \[teorema10\] under assumptions including the existence of an orthonormal basis $( e_{n} )_{n \in\mathbb{N}}$ of $\mathfrak{h}$ that satisfies, for example, $G e_n, L_k e_n \in\mathcal{D}( C)$ and ${\sup_{n\in\mathbb{Z}_{+}}}\Vert CP_{n}x\Vert
\leq\Vert C x\Vert$ for all $x \in\mathcal{D}( C)$, where $P_{n}$ is the orthogonal projection of $\mathfrak{h}$ over the linear manifold spanned by $e_{0},\ldots, e_{n}$. In Theorem \[teorema10\] we remove this basis, extending the range of applications.
\[nota7\] Suppose that $
2\Re\langle x,Gx\rangle+{\sum_{k=1}^{\infty}}\Vert L_{k}x\Vert^{2}
\leq0
$ for all $x \in\mathcal{D}(G )$. Let $G$ be the infinitesimal generator of a $C_0$-semigroup of contractions. Define the sequence $( \mathcal{T}^{(n)} )_{n \geq0}$ of linear contractions on $\mathfrak{L}( \mathfrak{h})$ by $$\bigl\langle u, \mathcal{T}_{t}^{(n+1)}( A ) v \bigr\rangle
=
\langle e^{G t } u, A e^{G t } v \rangle
+
\sum_{k=1}^{\infty} \int_{0}^{t} \bigl\langle L_{k} e^{G ( t -s )} u,
\mathcal{T}_{s}^{(n)} ( A ) L_{k} e^{G ( t -s )} v \bigr\rangle
\,ds,$$ where $u,v \in\mathcal{D}(G ) $, $A\in\mathfrak{L}( \mathfrak{h})$, and $\mathcal{T}^{(-1)} = 0$. A. M. Chebotarev proved that Picard’s successive approximations $\mathcal
{T}^{(n)}$ converge as $n \rightarrow\infty$ to a quantum dynamical semigroup $\mathcal{T}^{(\min)}$ which is a weak solution to (\[41\]); see, for example, [@Chebotarev2000; @Fagnola1999]. Holevo [@Holevo1996] developed the probabilistic representation of $\mathcal{T}^{(\min)}$ under restrictions, including that $G$ and $G^{\ast}$ are the infinitesimal generators of $C_0$-semigroup of contractions. From Chebotarev and Fagnola [@ChebFagn1998] we have that $\mathcal{T}^{(\min)}_{t}( I ) = I$ for any $t \geq0$, provided that there exists a self-adjoint positive operator $C$ in $\mathfrak{h}$ and a linear manifold $\mathfrak{D} \subset\mathcal{D}(G )$ which is a core for $C$ such that: (i) The semigroup generated by $G$ leaves invariant $\mathfrak{D}$; and (ii) For some $\gamma>0$, $
2\Re\langle C^{2} x, Gx\rangle+\sum_{k=1}^{\infty}\Vert C L_{k}x
\Vert^{2}\leq\alpha\Vert x\Vert_{C}^{2}
$ for all $x \in( \gamma I - G )^{-1} ( \mathfrak{D})$; see also [@ChebGarQue1998; @Fagnola1999]. This implies the uniqueness (in the semigroup sense) of the solution to (\[41\]) with $A$ bounded; see, for example, [@Fagnola1999].
In addition to its proof, the main novelty of Theorem \[teorema10\] is that we do not assume properties like $G$ are the infinitesimal generators of a semigroup and condition (i), which involves the study of invariant sets for $\exp(Gt )$. The latter is not an easy problem in general.
Probabilistic representations of regular density operators {#secprob-rep}
==========================================================
The following notion of a regular density operator was introduced by Chebotarev, García and Quezada [@ChebGarQue1998] to investigate the identity preserving property of minimal quantum dynamical semigroups.
\[def2\] Let $C$ be a self-adjoint positive operator in $\mathfrak{h}$. An operator $\varrho$ belonging to $\mathfrak{L}_{1}^{+}( \mathfrak{h} )$ is called $C$-regular iff $
\varrho=\sum_{n\in\mathfrak{I}}\lambda_{n}\vert u_{n}\rangle\langle
u_{n}\vert
$ for some countable set $\mathfrak{I}$, summable nonnegative real numbers $( \lambda_{n}) _{n\in\mathfrak{I}}$ and family $( u_{n}) _{n\in
\mathfrak{I}}$ of elements of $\mathcal{D}( C) $, which together satisfy $
\sum_{n\in\mathfrak{I}}\lambda_{n}\Vert Cu_{n}\Vert^{2}<\infty
$. We write $\mathfrak{L}_{1,C}^{+}( \mathfrak{h}) $ for the set of all $C$-regular density operators.
We next formulate the concept of $C$-regular operators in terms of random variables. This characterization of $\mathfrak{L}_{1,C}^{+}( \mathfrak{h}) $ complements those given in [@ChebGarQue1998] using operator theory; see also [@Chebotarev2000].
\[teorema4\] Suppose that $C$ is a self-adjoint positive operator in $\mathfrak{h}$. Let $\varrho$ be a linear operator in $\mathfrak{h}$. Then $\varrho$ is $C$-regular if and only if $\varrho= \mathbb{E}\vert\xi\rangle
\langle\xi\vert$ for some $\xi\in L_{C}^{2}( \mathbb{P},\mathfrak{h}) $. Moreover, $\mathbb{E}\vert\xi\rangle\langle\xi\vert$ can be interpreted as a Bochner integral in both $\mathfrak{L}_{1}( \mathfrak
{h}) $ and $\mathfrak{L}( \mathfrak{h}) $.
The proof is divided into Lemmata \[lema6\] and \[lema53\].
By the following theorem, the mean values of a large number of unbounded observables are well posed when the density operators are $C$-regular. Theorem \[teorema8\] also provides probabilistic interpretations of these expected values.
\[teorema8\] Suppose that $C$ is a self-adjoint positive operator in $\mathfrak{h}$, and fix $\varrho=\mathbb{E}\vert\xi\rangle\langle\xi\vert$ with $\xi\in
L_{C}^{2}( \mathbb{P},\mathfrak{h}) $. Then:
The range of $\varrho$ is contained in $\mathcal{D}( C)$ and $
C\varrho= \mathbb{E}\vert C \xi\rangle\langle\xi\vert
$.
Consider $A \in\mathfrak{L}( ( \mathcal{D}( C) ,\mbox{$\Vert\cdot\Vert_{C}$})
,\mathfrak{h}) $, and let $B$ be a densely defined linear operator in $\mathfrak{h}$ such that $\mathcal{D}( C) \subset\mathcal{D}( B^{\ast})$. Then $A\varrho B$ is densely defined and bounded. The unique bounded extension of $A\varrho B$ belongs to $\mathfrak
{L}_{1}( \mathfrak{h}) $ and is equal to $\mathbb{E}\vert A\xi\rangle
\langle B^{\ast} \xi\vert$, where $\mathbb{E}\vert A\xi\rangle\langle B^{\ast} \xi\vert$ is a well defined Bochner integral in both $\mathfrak{L}_{1}( \mathfrak{h}) $ and $\mathfrak{L}( \mathfrak
{h}) $. Moreover, $$\operatorname{tr}( A\varrho B ) =\mathbb{E}\langle B^{\ast} \xi,A\xi
\rangle.$$
Deferred to Section \[subsecteorema8\].
Quantum master equation {#secQME}
=======================
We first deduce that (\[i4\]) defines a density operator.
\[teor7\] Let Hypothesis \[HipN5\] hold. Then, for every $t\geq0 $ there exists a unique operator $\rho_{t} \in\mathfrak{L} ( \mathfrak{L}_{1}( \mathfrak{h}) ) $ such that for each $C$-regular operator $\varrho$, $$\label{31}
\rho_{t}( \varrho) =\mathbb{E}\vert X_{t}( \xi)
\rangle\langle X_{t}( \xi) \vert,$$ where $\xi$ is an arbitrary random variable in $L_{C}^{2}( \mathbb
{P},\mathfrak{h}) $ satisfying $\varrho=\mathbb{E} \vert\xi\rangle\langle\xi\vert$. Here $ X( \xi) $ is the strong $C$-solution of (\[2\]) with initial datum $\xi$, and we can interpret $
\mathbb{E}\vert X_{t}( \xi) \rangle\langle X_{t}( \xi) \vert
$ as a Bochner integral in $\mathfrak{L}_{1}( \mathfrak{h}) $ as well as in $\mathfrak{L}( \mathfrak{h}) $. Moreover, $
\Vert\rho_{t} \Vert_{\mathfrak{L} ( \mathfrak{L}_{1}( \mathfrak{h})
)} \leq1
$ for all $t \geq0$.
Deferred to Section \[subsecteor7\].
\[notrho\] From now on, $\rho_{t}$ stands for the operator given by (\[31\]).
The next theorem says that the expected value $\mathbb{E}$ commutes with the action of $\rho_{t}$ on random $C$-regular pure density operators.
\[teor9\] Assume that Hypothesis \[HipN5\] holds. Let $\varrho= \mathbb{E}\vert\xi\rangle\langle\xi\vert$, with $\xi\in$ $L_{C}^{2}( \mathbb{P},\mathfrak{h}) $. Then $
\mathbb{E}\rho_{t}( \vert\xi\rangle\langle\xi\vert)
=
\rho_{t}( \varrho)
$ for all $t\geq0$.
Deferred to Section \[subsecteor9\].
We now summarize some relevant properties of the family of linear operators $ ( \rho_{t}
)_{t \geq0}$.
\[teor8\] Adopt Hypothesis \[HipN5\]. Then $( \rho_{t})_{t\geq0}$ is a semigroup of contractions such that $\rho_{t}(
\mathfrak{L}_{1}^{+}( \mathfrak{h}) ) \subset\mathfrak{L}_{1}^{+}(
\mathfrak{h}) $, $\rho_{t}( \mathfrak{L}_{1,C}^{+}( \mathfrak{h}) ) \subset\mathfrak
{L}_{1,C}^{+}( \mathfrak{h}) $, and for all $\varrho\in\mathfrak{L}_{1,C}^{+}( \mathfrak{h}) $, $$\label{310}
{\lim_{s\rightarrow t}\operatorname{tr}}\vert\rho_{s}( \varrho) -\rho
_{t}( \varrho) \vert= 0 .$$
The proof is divided into Lemmata \[lema14\], \[lema24\] and \[lema25\].
The analysis outlined in Section \[subsecexistence\] leads to our first main theorem, which asserts that $\mathbb{E}\vert X_{t}( \xi)
\rangle\langle X_{t}( \xi) \vert$ satisfies (\[3\]) in both senses, integral and $\mathfrak{L}_{1}( \mathfrak{h}) $-weak, whenever $\varrho= \mathbb{E} \vert\xi\rangle\langle\xi\vert$ is $C$-regular.
\[HipN1\] The operators $G, L_{1}, L_{2}, \ldots$ are closable.
\[teor10\] Let Hypotheses \[HipN5\] and \[HipN1\] hold. Suppose that $\varrho$ is $C$-regular. Then for all $t\geq0$, $$\label{311}
\rho_{t}( \varrho) =\varrho+\int_{0}^{t}\Biggl( G\rho_{s}(
\varrho) +\rho_{s}( \varrho) G^{\ast}+\sum_{k=1}^{\infty
}L_{k}\rho_{s}( \varrho) L_{k}^{\ast}\Biggr) \,ds,$$ where we understand the above integral in the sense of the Bochner integral in $\mathfrak{L}_{1}( \mathfrak{h}) $. Moreover, for any $A\in\mathfrak{L}( \mathfrak{h}) $ and $t\geq0$, $$\label{312}
\frac{d}{dt}\operatorname{tr}( A\rho_{t}( \varrho) )
=
\operatorname{tr}\Biggl(
A\Biggl( G\rho_{t}( \varrho) +\rho_{t}( \varrho)G^{\ast}
+\sum_{k=1}^{\infty}L_{k}\rho_{t}( \varrho) L_{k}^{\ast}\Biggr)
\Biggr).$$
Deferred to Section \[subsecteor10\].
Let $G, L_{1}, L_{2}, \ldots$ be densely defined. Then Hypothesis \[HipN1\] is equivalent to saying that $G^{\ast}, L_{1}^{\ast}, L_{2}^{\ast}, \ldots$ are densely defined.
The second main theorem of this paper establishes that under Hypothesis \[HipN5\], $\rho_{t}( \varrho)$ is the unique solution of (\[312\]) in the semigroup sense. Its proof is based on arguments given in Section \[subsecuniqueness\].
\[defSemigroupSol\] A semigroup $( \widehat{\rho}_{t}) _{t\geq0}$ of bounded operators on $\mathfrak{L}_{1}( \mathfrak{h}) $ is called semigroup $C$-solution of (\[3\]) if and only if:
For each nonnegative real number $T$, ${\sup_{t\in[ 0,T]
}}\Vert\widehat{\rho}_{t}\Vert_{\mathfrak{L}( \mathfrak{L}_{1}(
\mathfrak{h}) ) }<\infty$.
For any $x\in\mathcal{D}( C) $ and $A\in\mathfrak{L}(
\mathfrak{h}) $, the function $t\mapsto\operatorname{tr}( \widehat{ \rho}_{t}( \vert
x\rangle\langle x\vert) A) $ is continuous.
$
\lim_{t\rightarrow0+}
(
\operatorname{tr}( A \widehat{\rho}_{t} ( \vert x\rangle\langle x\vert
) )
-
\operatorname{tr}( A \vert x\rangle\langle x\vert)
) / t
=
\langle x, A Gx\rangle+\langle Gx, A x\rangle
+\break
\sum_{k=1}^{\infty}\langle L_{k}x, A L_{k}x\rangle
$ whenever $x\in\mathcal{D}( C) $ and $A\in\mathfrak{L}( \mathfrak{h}) $.
\[teorema9\] Let Hypothesis \[HipN5\] hold. Then $( \rho_{t})_{t\geq0}$ is the unique semigroup $C$-solution of (\[3\]).
Deferred to Section \[subsecteorema9\].
Theorems \[teor10\] and \[teorema9\], together with Theorem \[teorema8\], show that the mean values of the observables with respect to the solutions of the quantum master equations are well posed in many physical situations. Moreover, Theorems \[teorema8\], \[teor10\] and \[teorema9\] allow us to make rigorous some explicit computations concerning the evolution of unbounded observables, like the following Ehrenfest-type theorem.
\[teorema5\] Assume the setting of Example \[exmeasurement\]. Then $( \rho_{t})_{t\geq0}$ is the unique semigroup $( P^2 + Q^2 )$-solution of (\[3\]). If $\varrho\in\mathfrak{L}_{1, P^2 + Q^2}^{+}( L^{2}( \mathbb{R},\mathbb
{C}) ) $, then for all $t \geq0$, $$\label{31n}\quad
\frac{d}{dt}\operatorname{tr}( Q \rho_{t}( \varrho) )
=
\frac{1}{m} \operatorname{tr}( P \rho_{t}( \varrho) ) ,\qquad
\frac{d}{dt} \operatorname{tr}( P \rho_{t}( \varrho) )
=
- 2c \operatorname{tr}( Q \rho_{t}( \varrho) ) .$$
Deferred to Section \[subsecteorema5\].
\[nota8\] A novelty of this paper lies in the use of probabilistic methods for proving Theorems \[teor10\] and \[teorema9\]. In order to adopt a purely Operator Theory viewpoint, we now return to Remark \[nota7\]. Let $( \mathcal{T}_{\ast t} ) _{t\geq0}$ be the semigroup on $\mathfrak
{L}_{1} ( \mathfrak{h})$ whose adjoint semigroup is $( \mathcal{T}^{(\min)}_{t} ) _{t\geq0}$; that is, $\mathcal{T}_{\ast}$ is the predual semigroup of $\mathcal
{T}^{(\min)}$. In case $\mathcal{T}^{(\min)}$ leaves invariant the identity operator, the linear span of $ \{ \vert x\rangle\langle y\vert\dvtx x,y\in\mathcal{D}( G ) \}$ is a core for the infinitesimal generator of $( \mathcal{T}_{\ast t} )
_{t\geq0}$, which is denoted by $\mathcal{L}_{\ast}$ for simplicity of notation; see, for example, Proposition 3.32 of [@Fagnola1999]. Then, under conditions (i) and (ii) given in Remark \[nota7\], $( \mathcal{T}_{\ast t}) _{t\geq0}$ is the unique strongly continuous semigroup on $\mathfrak{L}_{1}( \mathfrak{h}) $ satisfying a version of (\[3\]) for all $\varrho= \vert x \rangle\langle y \vert$ with $x,y \in\mathcal{D}( G) $. In order to establish $ \mathcal{T}_{\ast t}( \mathfrak{L}_{1,C}^{+}( \mathfrak{h}) )
\subset\mathfrak{L}_{1,C}^{+}( \mathfrak{h}) $ as well as the assertions of Theorem \[teor10\], we have to prove first that $\mathfrak{L}_{1,C}^{+}( \mathfrak{h}) \subset\mathcal{D}( \mathcal
{L}_{\ast} ) $. If we are able to do it, then $\mathfrak{L}_{1,C}^{+}( \mathfrak{h}) $ is an invariant set for $
\mathcal{T}_{\ast t} $ provided that $$\label{i10}
\sup_{n \in\mathbb{N}} \bigl| \operatorname{tr}\bigl( \mathcal{T}^{(\min)} \bigl( n(n
+ C^2 ) ^{-1} C^2 \bigr) \varrho\bigr) \bigr|
<
\infty$$ for any $\varrho\in\mathfrak{L}_{1,C}^{+}( \mathfrak{h})$. When $C$ is invertible, (\[i10\]) follows from $$\label{i11} \bigl\| C^{-1} \mathcal{T}^{(\min)} \bigl( n(n + C^2 ) ^{-1}C^2
\bigr) C^{-1} \bigr\| \leq K \| n(n + C^2 ) ^{-1} \| .$$ A careful reading of [@GarQue1998] reveals that for any $A \in\mathfrak{L}( \mathfrak{h})$ we have $$\label{i12}
\bigl\| C^{-1} \mathcal{T}^{(\min)} ( A ) C^{-1} \bigr\|
\leq
K \| C^{-1} A C^{-1} \|$$ under assumptions of type (\[i2\]), together with $ \exp( G t )$, leaves invariant a core of $C$ contained in $\mathcal{D}( G )$; see also [@ChebGarQue1998; @Chebotarev2000]. This gives (\[i11\]), and so (\[i10\]) holds. Under the same assumptions, an alternative is to obtain (\[i12\]) by proving $
\| C^{-1} \mathcal{T}^{(n)} ( A )\times C^{-1} \|
\leq
K \| C^{-1} A C^{-1} \|
$ directly from the definition of $\mathcal{T}^{(n)}$, but with effort. Here $\mathcal{T}^{(n)}$ is as in Remark \[nota7\]. Finally, to establish (\[311\]) and (\[312\]) we have to get that $
\mathcal{L}_{\ast}( \varrho) =G\varrho+\varrho
G^{\ast}+\sum_{k=1}^{\infty}L_{k}\varrho L_{k}^{\ast}
$ for any $\varrho\in\mathfrak{L}_{1,C}^{+}( \mathfrak{h})$.
Regular stationary solutions of quantum master equations {#secStatSol}
========================================================
This section is devoted to objective (O2). In this direction, the next theorem provides the representation of the density operator at time $t$ as the average of all pure states $\vert Y_{t}\rangle\langle Y_{t}\vert$ associated to the nonlinear stochastic Schrödinger equation (\[5\]). This model has a sound physical basis; see, for example, [@BarchielliBelavkin1991; @BarchielliGregoratti2009; @GoughSobolev2004; @ScottMilburn2001].
Let $C$ satisfy Hypothesis \[HipN3\]. Suppose that $\mathbb{I}$ is either $[ 0,+\infty[ $ or $[ 0,T] $ provided $T\in[ 0,+\infty[ $. We say that $( \mathbb{Q}, ( Y_{t})_{t\in\mathbb{I}},( B_{t})_{t\in\mathbb{I}})$ is a $C$-solution of (\[5\]) with initial distribution $\theta$ on $\mathbb{I}$ if and only if:
- $B = ( B^{k} )_{k \in\mathbb{N}}$ is a sequence of real valued independent Brownian motions on the filtered complete probability space $( \Omega,\mathfrak{F},(
\mathfrak{F}_{t}) _{t\in\mathbb{I}},\mathbb{Q}) $.
- $( Y_{t}) _{t\in\mathbb{I}}$ is an $\mathfrak{h}$-valued process with continuous sample paths such that the law of $Y_{0}$ coincides with $\theta$ and $\mathbb{Q}( \Vert Y_{t}\Vert=1$ for all $t\in\mathbb{I}) =1$.
- For every $t\in\mathbb{I}\dvtx Y_{t}\in\mathcal{D}( C)$ $\mathbb{Q}$-a.s. and $\sup_{s\in[ 0,t] }\mathbb{E}_{\mathbb{Q}}\Vert CY_{s}\Vert
^{2}<\infty$.
- $\mathbb{Q}$-a.s., $
Y_{t}=Y_{0}+\int_{0}^{t}G( \pi_{C} (Y_{s})) \,ds+\sum_{k=1}^{\infty}\int
_{0}^{t}L_{k}( \pi_{C} ( Y_{s} ) ) \,dB_{s}^{k}
$ for all $t\in\mathbb{I}$.
\[corolario2\] Suppose that Hypothesis \[HipN5\] holds. Let $
\varrho=\int_{\mathfrak{h}} \vert y\rangle\langle y\vert\theta( dy)
$, with $\theta$ probability measure over $\mathfrak{h}$ satisfying $
\theta( \mathcal{D}( C) \cap\{ x\in\mathfrak{h}\dvtx\Vert
x\Vert=1\} ) =1
$ and $
\int_{\mathfrak{h}}\Vert Cx\Vert^{2}\theta( dx) <\infty
$. Then for all $t \geq0$, $$\rho_{t}( \varrho) =\mathbb{E}_{ \mathbb{Q}} \vert Y_{t}\rangle\langle
Y_{t}\vert,$$ where $\rho_{t}( \varrho)$ is defined by (\[31\]), and $( \mathbb{Q},( Y_{t}) _{t\geq0},( B_{t}) _{t\geq0}) $ is the $C$-solution of (\[5\]) with initial law $\theta$.
Deferred to Section \[subseccorolario2\].
\[nota6\] Let $\theta$ be as in Theorem \[corolario2\]. Suppose that Hypothesis \[HipN5\] holds. Then, we can use the same arguments as in the proof of Theorem 1 of [@MoraReAAP2008] for establishing that (\[5\]) has a unique (in the probabilistic sense) $C$-solution $(\mathbb{Q},( Y_{t}) _{t\geq0},( B_{t})_{t\geq0})$ with initial law $\theta$.
From Theorems \[teorema8\] and \[corolario2\] we obtain that the expected value of $A\in\mathfrak{L}( ( \mathcal{D}( C), \mbox{$\Vert\cdot\Vert_{C}$})
,\mathfrak{h})$ at time $t$ is equal to $
\mathbb{E} \langle Y_{t} , A Y_{t} \rangle
$. This gives theoretical support for the numerical computation of the mean value of an observable $A$ at time $t$ through (\[5\]), which is the principal method for computing efficiently $
\operatorname{tr}( A\rho_{t}( \varrho) )
$; see, for example, [@BreuerPetruccione2002; @MoraAAP2005; @Percival1998].
Our third main theorem deals with the existence of regular stationary states for (\[3\]). This is a step forward in the understanding of the long-time behavior of unbounded observables.
\[Hip2\] Let Hypothesis \[HipN5\] hold. Assume the existence of a probability measure $\Gamma$ on $\mathfrak{B}( \mathfrak{h}) $ such that: $\Gamma( \operatorname{Dom}( C) \cap\{ x\in\mathfrak{h}\dvtx\Vert x\Vert=1\} ) =1$, $\int_{\mathfrak{h}} \Vert C z\Vert^{2}\Gamma( dz) <\infty$ and $$\label{IM1}\Gamma( A) =\int_{\mathfrak{h}}P_{t}( x,A)
\Gamma( dx)$$ for any $t\geq0$ and $A\in\mathfrak{B}( \mathfrak{h}) $. Here $
P_{t}( x,A) = \mathbb{Q}_{x}( Y_{t}^{x}\in A)
$ if $x\in \operatorname{Dom}( C)$ and $
P_{t}( x,A) = \delta_{x}( A)
$ otherwise; the $C$-solution of (\[5\]) with initial data $x\in \operatorname{Dom}( C) $ is denoted by $( \mathbb{Q}_{x},( Y_{t}^{x}) _{t\geq0},(B_{t}^{\cdot,x}) _{t\geq0}) $.
\[teorema7\] Under Hypothesis \[Hip2\], there exists a $C$-regular operator $\varrho_{\infty}$ such that $
\rho_{t}( \varrho_{\infty}) =\varrho_{\infty}
$ for all $t\geq0$.
Deferred to Section \[subsecteorema7\].
Combining the results of Section \[secQME\] with Theorem \[teorema7\] yields the existence of a $C$-regular stationary solution to (\[3\]).
Quantum oscillator {#secoscillator}
==================
In this section we illustrate our general results with the following quantum oscillator.
\[Ejemplo1\] Consider $\mathfrak{h}=l^{2}( \mathbb{Z}_{+}) $, together with its canonical orthonormal basis $( e_{n}) _{n\in\mathbb{Z}_{+}}$. The closed operators $a^{\dagger}$, $a$ are given by: for all $n \in\mathbb{Z}_{+}$ $a^{\dagger}e_{n}= \sqrt{ n+1}e_{n+1}$, $ae_{0} = 0$ and $ae_{n} = \sqrt{n}e_{n-1}$ if $n \in\mathbb{N}$. Define $N=a^{\dagger}a$.
Choose $
H=i\beta_{1}( a^{\dagger}-a) +\beta_{2}N+\beta_{3}( a^{\dagger}) ^{2}a^{2}
$ with $\beta_{1}, \beta_{2}, \beta_{3} \in\mathbb{R}$. Let $L_{1}=\alpha_{1}a$, $L_{2}=\alpha_{2}a^{\dagger}$, $L_{3}=\alpha_{3}N$, $L_{4}=\alpha_{4}a^{2}$, $L_{5}=\alpha_{5}( a^{\dagger}) ^{2}$ and $L_{6}=\alpha_{6}N^{2}$, where $\alpha_{1},\ldots,\alpha_{6} \in\mathbb{C}$. Set $L_{k}=0$ for any $k\geq7$, and so take $
G=-iH- \sum_{k=1}^{6}L_{k}^{\ast}L_{k} /2
$.
Example \[Ejemplo1\] describes a laser-driven quantum oscillator in a Kerr medium that interacts with a thermal bath. In addition, Example \[Ejemplo1\] unifies concrete physical systems such as the following two basic models:
- A mode with natural frequency $\omega$ of a electromagnetic field inside of a cavity is described by $\beta_{2}=\omega$, $\alpha_{1}=\sqrt{A(\nu+1)}$, $\alpha_{2}=\sqrt
{A\nu}$ and $\beta_{1}=\beta_{3}=\alpha_{k }=0$, with $k =3, \ldots, 6$. Here, the mode is damped with rate $\alpha_{1}$ by a thermal reservoir, and $\nu$ is a parametrization of the bath temperature; see, for example, [@BreuerPetruccione2002; @GardinerZoller2004; @WisemanMilburn2010].
- A simple two-photon absorption and emission process is modeled by $\beta_{3}\in\mathbb{R}$, $\alpha_{4}>0$, $\alpha_{5} \geq0$ and $\beta_{1} = \beta_{2} = \alpha_{1} = \alpha_{2} =\alpha_{3} = \alpha
_{6} =0$; see, for example, [@CarboneFagGaQue2008; @FagQue2005] and references therein.
The next theorem characterizes the well-posedness of the mean values of observables formed by a finite composition of $a^{\dagger}$ and $a$ in transient and stationary regimes. Important examples of such observables are $ Q = i( a^{\dagger}+a) /\sqrt{2}$, $P = i( a^{\dagger}-a) /\sqrt{2}$ and $N$.
\[teor14\] Assume the setting of Example \[Ejemplo1\], and let $\rho_{t}( \varrho) $ be as in Theorem \[teor7\]. Suppose that $p$ is a natural number greater than or equal to $4$.=-1
Let $\vert\alpha_{4}\vert\geq\vert\alpha_{5}\vert$ and let $\varrho\in\mathfrak{L}_{1,N^{p}}^{+}( l^{2}( \mathbb{Z}_{+}) ) $. Then $\rho_{t}( \varrho) $ is a $N^{p}$-regular operator that satisfies both (\[311\]) and (\[312\]). Moreover, $( \rho_{t}) _{t\geq0}$ is the unique semigroup $N^{p}$-solution of (\[3\]).
Suppose that either $\vert\alpha_{4}\vert>\vert\alpha_{5}\vert$ or $\vert\alpha_{4}\vert=\vert\alpha_{5}\vert$ with $\vert\alpha_{2}\vert^{2}-\vert\alpha_{1}\vert
^{2}+4( 2p+1) \vert\alpha_{4}\vert^{2}<0$. Then, there exists a $N^{p}$-regular operator $\varrho_{\infty}$ such that $
\rho_{t}( \varrho_{\infty}) =\varrho_{\infty}
$ for any $t\geq0$.
Deferred to Section \[subsecteor14\].
\[notaSuffCond\] In the proof of Theorem \[teor14\] we use the following sufficient condition for condition (H2.3), which is developed in [@FagnolaMora2010].
\[HipN4\] Suppose that $C$ is a self-adjoint positive operator in $\mathfrak{h}$ such that $G, L_{1}, L_{2}, \ldots$ belong to $\mathfrak{L}( ( \mathcal{D}( C), \mbox{$\Vert\cdot\Vert_{C}$})
,\mathfrak{h})$, and $
2\Re\langle x, Gx\rangle+{\sum_{k=1}^{\infty}}\Vert L_{k}x \Vert
^{2}\leq0
$ for any $x$ in a core of $C$. In addition, assume that for any $x$ belonging to a core of $C^{2}$, $
2\Re\langle C^{2} x, Gx\rangle+{\sum_{k=1}^{\infty}}\Vert C L_{k}x
\Vert^{2}
\leq
K ( \Vert x\Vert_{C}^{2} + 1 )
$.
Proofs {#secproofs}
======
Proof of Theorem 2.1 {#subsecteorema3}
--------------------
We first prove that (\[42\]) defines implicitly a bounded operator $\mathcal{T}_{t}( A ) $.
\[lema41\] Adopt the assumptions of Hypothesis \[HipN5\] with the exception of condition . Consider $A \in\mathfrak{L}( \mathfrak{h})$. Then for every $t \geq0$ there exists a unique $\mathcal{T}_{t}( A ) $ belonging to $\mathfrak{L}( \mathfrak{h})$ for which (\[42\]) holds for all $x,y$ in $\mathcal{D}(C )$. Moreover, $\Vert\mathcal{T}_{t}( A ) \Vert\leq\Vert A \Vert$ for any $t \geq0$.
By Definition \[definicion2\], $
\vert\mathbb{E} \langle X_{t} ( x ), A X_{t} ( y ) \rangle\vert
\leq
\Vert A \Vert\Vert x \Vert\Vert y \Vert
$ for all $x,y \in\mathcal{D}( C )$. Hence the sesquilinear form over $\mathcal{D}( C ) \times\mathcal{D}(
C )$ given by $
(x, y ) \mapsto\mathbb{E} \langle X_{t} ( x ), A X_{t} ( y ) \rangle
$ can be extended uniquely to a sesquilinear form $\lbrack\cdot, \cdot
\rbrack$ over $\mathfrak{h} \times\mathfrak{h}$ with the property that $
\vert\lbrack x, y \rbrack\vert
\leq
\Vert A \Vert\Vert x \Vert\Vert y \Vert
$ for any $x,y \in\mathfrak{h}$. There exists a unique bounded operator $\mathcal{T}_{t}( A )$ on $\mathfrak{h}$ such that $ \vert\lbrack x, y \rbrack\vert= \langle
x, \mathcal{T}_{t}( A ) y \rangle$ for all $x,y$ in $\mathfrak{h}$. Furthermore, $\Vert\mathcal{T}_{t}( A ) \Vert\leq\Vert A \Vert$.
Using arguments given in Section \[subsecuniqueness\] we next establish the uniqueness of solutions for the adjoint quantum master equations.
\[lema42\] Let Hypothesis \[HipN5\] hold. Assume that $( \mathcal{A}_{t} )_{t
\geq0}$ is a $C$-solution of (\[41\]) with initial datum $A \in
\mathfrak{L}( \mathfrak{h})$. Then $\mathcal{A}_{t} = \mathcal{T}_{t}(
A )$ for all $t \geq0$, where $\mathcal{T}_{t}( A )$ is as in Therorem \[teorema3\].
Using Itô’s formula we will prove that for all $x,y \in\mathcal{D}( C )$, $$\label{45}
\mathbb{E} \langle X_{t}( x ), A X_{t} ( y ) \rangle
=
\langle x, \mathcal{A}_{t} y \rangle.$$ This, together with Lemma \[lema41\], implies $\mathcal{A}_{t} = \mathcal{T}_{t}( A )$.
Motivated by the fact that $\mathcal{A}_t$ is only a weak solution, we fix an orthonormal basis $( e_{n} )_{n \in\mathbb{N}}$ of $\mathfrak{h}$ and consider the function $
F_{n} \dvtx [0 , t ] \times\mathfrak{h} \times\mathfrak{h}
\rightarrow
\mathbb{C}
$ defined by $$F_{n} ( s, u, v ) = \langle R_{n} \overline{u}, \mathcal{A}_{t-s}
R_{n} v \rangle,$$ where $R_{n} = n (n+C )^{-1}$ and $
\bar{u}
=
\sum_{n \in\mathbb{N}} \overline{\langle e_{n}, u \rangle} e_{n}
$. Since the range of $R_{n}$ is contained in $\mathcal{D}( C )$, condition (a) of Definition \[definicion3\] yield $$\label{46}
\frac{d}{ds} F_{n} ( s, u, v ) = - g ( s, R_{n} \overline{u}, R_{n} v )$$ with $
g ( s, x, y )
=
\langle x, \mathcal{A}_{t-s} G y \rangle
+ \langle G x, \mathcal{A}_{t-s} y \rangle
+ \sum_{k=1}^{\infty} \langle L_{k} x, \mathcal{A}_{t-s} L_{k} y
\rangle
$. According to conditions (a), (b) of Definition \[definicion3\], we have that $t \longmapsto\langle u, \mathcal{A}_{t} v \rangle$ is continuous for all $u,v \in\mathfrak{h}$, and so combining $C R_{n} \in\mathfrak{L}( \mathfrak{h})$ with Hypothesis \[HipN5\] we get the uniform continuity of $
(s, u, v )
\longmapsto
g ( s, R_{n} \overline{u}, R_{n} v )
$ on bounded subsets of $[ 0, t ] \times\mathfrak{h} \times\mathfrak{h}$. Therefore we can apply Itô’s formula to $ F_{n} ( s \wedge\tau_{j}, \overline{X_{s}^{\tau_{j}}( x )},
X_{s}^{\tau_{j}}( y ) ) $, with $
\tau_{j} =
\inf{ \{ t \geq0\dvtx \Vert X_{t}( x ) \Vert+ \Vert X_{t}( y ) \Vert> j
\} }
$.
Fix $x,y \in\mathcal{D}( C )$. Combining Itô’s formula with (\[46\]) we deduce that $$F_{n} \bigl( t \wedge\tau_{j}, \overline{X_{t}^{\tau_{j}}( x )}, X_{t}^{\tau
_{j}}( y ) \bigr)
=
F_{n} ( 0, \overline{X_{0}( x )}, X_{0}( y ) )
+ I_{t \wedge\tau_{j}}^n + M_t .$$ Here for $s \in[ 0, t ]$: $
M_{s}
=
\sum_{k = 1}^{\infty} \int_{0}^{s \wedge\tau_{j}} \langle
R_{n} X_{r} ^{\tau_{j}} ( x ),
\mathcal{A}_{t-r}
R_{n} L_{k} X_{r}^{\tau_{j}} ( y )
\rangle \,dW^{k}_{r}
+\break
\sum_{k = 1}^{\infty} \int_{0}^{s \wedge\tau_{j}} \langle
R_{n} L_k X_{r} ^{\tau_{j}} ( x ),
\mathcal{A}_{t-r}
R_{n} X_{r}^{\tau_{j}} ( y )
\rangle \,dW^{k}_{r}
$ and $$I_s^n
=
\int_{0}^{s}
\bigl(
- g ( r, R_n X_{r} ( x ) , R_n X_{r} ( y ) )
+
g_{n} ( r, X_{r} ( x ) , X_{r} ( y ) )
\bigr) \,dr,$$ the function $
g_{n} ( r, u, v )
$ is equal to $
\langle R_{n} u, \mathcal{A}_{t-r} R_{n} G v \rangle
+ \langle R_{n} G u, \mathcal{A}_{t-r} R_{n} v \rangle
+ \sum_{k=1}^{\infty} \langle R_{n} L_{k} u, \mathcal{A}_{t-r} R_{n}
L_{k} v \rangle$.
We next establish the martingale property of $M_s$. For all $r \in[0, t ]$ we have $$\Vert R_{n} X_{r} ^{\tau_{j}} ( x ) \Vert^2
\Vert\mathcal{A}_{t-r} \Vert^2
\Vert R_{n} L_{k} X_{r}^{\tau_{j}} ( y ) \Vert^2
\leq
j^2 \sup_{s \in[0, t ]} \Vert\mathcal{A}_{s} \Vert^2
\Vert L_{k} X_{r}^{\tau_{j}} ( y ) \Vert^2 .$$ By (H2.1) and (H2.2), $
\mathbb{E}
\int_{0}^{t \wedge\tau_{j}}
\sum_{k = 1}^{\infty}
|
\langle
R_{n} X_{r} ^{\tau_{j}} ( x ),
\mathcal{A}_{t-r}
R_{n} L_{k} X_{r}^{\tau_{j}} ( y )
\rangle
|^{2} \,ds
<
\infty
$. Thus $
(
\sum_{k = 1}^{\infty}
\int_{0}^{s \wedge\tau_{j}} \langle
R_{n} X_{r} ^{\tau_{j}} ( x ),
\mathcal{A}_{t-r}
R_{n} L_{k} X_{r}^{\tau_{j}} ( y )
\rangle \,dW^{k}_{r}
)_{s \in[ 0, t ]}
$ is a martingale. The same conclusion can be drawn for $$\sum_{k = 1}^{\infty}
\int_{0}^{s \wedge\tau_{j}} \langle
R_{n} L_k X_{r} ^{\tau_{j}} ( x ),
\mathcal{A}_{t-r}
R_{n} X_{r}^{\tau_{j}} ( y )
\rangle \,dW^{k}_{r},$$ and so $( M_s )_{s \in[ 0, t ]}$ is a martingale. Hence $$\label{415}
\mathbb{E}
\langle R_{n} X_{ t }^{\tau_{j}}( x ) ,
\mathcal{A}_{t - t \wedge\tau_{j}}
R_{n} X_{t}^{\tau_{j}}( y )
\rangle
=
\langle R_{n} x , \mathcal{ A}_{t} R_{n} y \rangle
+
\mathbb{E} I_{t \wedge\tau_{j}}^n .$$
We will take the limit as $j \rightarrow\infty$ in (\[415\]). Since $
\mathbb{E} ( {\sup_{s \in[0, t ]} }\Vert X_s( \xi) \Vert^{2} )
< \infty
$ for $\xi= x, y$ (see, e.g., Theorem 4.2.5 of [@Prevot2007]), using the dominated convergence theorem, together with the continuity of $t \longmapsto\langle u, \mathcal{A}_{t} v \rangle$, we get $$\mathbb{E} \langle R_{n} X_{t}^{\tau_{j}}( x ), \mathcal{A}_{t- t
\wedge\tau_{j}} R_{n} X_{t}^{\tau_{j}}( y ) \rangle
\longrightarrow_{j \rightarrow\infty}
\mathbb{E} \langle R_{n} X_{t}( x ), A R_{n} X_{t} ( y ) \rangle.$$ Applying again the dominated convergence theorem yields $
\mathbb{E} I_{t \wedge\tau_{j}}^n
\longrightarrow_{j \rightarrow\infty}
\mathbb{E} I_{t }^n
$, and hence letting $j \rightarrow\infty$ in (\[415\]) we deduce that $$\begin{aligned}
\label{414}
&& \mathbb{E} \langle R_{n} X_{t} ( x ), A R_{n} X_{t} ( y ) \rangle
- \langle R_{n} x , \mathcal{ A}_{t} R_{n} y \rangle
\nonumber\\[-8pt]\\[-8pt]
&&\qquad=
\mathbb{E} \int_{0}^{t}
\bigl(
- g ( s, R_n X_{s} ( x ) , R_n X_{s} ( y ) )
+ g_{n} ( s, X_{s} ( x ), X_{s} ( y ) )
\bigr) \,ds.\nonumber\end{aligned}$$
Finally, we take the limit as $n \rightarrow\infty$ in (\[414\]). Since $\Vert R_{n} \Vert\leq1$ and $ R_{n}$ tends pointwise to $I$ as $n \rightarrow\infty$, the dominated convergence theorem yields $$\lim_{n \rightarrow\infty} \mathbb{E} \int_{0}^{t}
g_{n} ( s, X_{s} ( x ), X_{s} ( y ) ) \,ds
=
\mathbb{E} \int_{0}^{t}
g ( s, X_{s} ( x ), X_{s} ( y ) ) \,ds .$$ For any $x \in\mathcal{D}( C )$, $ \lim_{n \rightarrow\infty} C R_{n} x = C x$. By $\Vert C R_{n} x \Vert\leq\Vert C x \Vert$, using the dominated convergence theorem gives $$\lim_{n \rightarrow\infty} \mathbb{E} \int_{0}^{t}
g ( s, R_n X_{s} ( x ) , R_n X_{s} ( y ) )
\,ds
=
\mathbb{E} \int_{0}^{t}
g ( s, X_{s} ( x ), X_{s} ( y ) ) \,ds.$$ Thus, letting $n \rightarrow\infty$ in (\[414\]) we obtain (\[45\]).
Proof of Theorem 3.2 {#subsecteorema8}
--------------------
We begin by examining the properties of the Bochner integral $\mathbb{E}\vert\xi\rangle\langle\chi\vert$ when $\xi, \chi\in L^{2}( \mathbb{P},\mathfrak{h})$.
\[lema51\] Suppose that $\xi$ and $\chi$ belong to $L^{2}( \mathbb{P},\mathfrak
{h})$. Then $\mathbb{E}\vert\xi\rangle\langle\chi\vert$ defines an element of $\mathfrak{L}_{1}( \mathfrak{h}) $, which moreover, is given by $$\label{53}
\langle x, \mathbb{E}\vert\xi\rangle\langle\chi\vert y \rangle
=
\mathbb{E} \langle x, \xi\rangle\langle\chi, y \rangle$$ for all $x,y \in\mathfrak{h}$. Here, $\mathbb{E}\vert\xi\rangle\langle\chi\vert$ is well defined as a Bochner integral with values in both $\mathfrak{L}_{1}( \mathfrak{h}) $ and $\mathfrak{L} ( \mathfrak{h}) $. In addition, $
\operatorname{tr} ( \mathbb{E}\vert\xi\rangle\langle\chi\vert) =
\mathbb{E} \langle\chi, \xi\rangle$.
We first get $\mathbb{E}\vert\xi\rangle\langle\chi\vert
\in
\mathfrak{L}_{1}( \mathfrak{h}) $. Since the image of $\vert\xi\rangle\langle\chi\vert$ lies in the set of all rank-one operators on $\mathfrak{h}$, $\vert\xi\rangle\langle
\chi\vert$ takes values in $ \mathfrak{L}_{1}( \mathfrak{h})$. Applying Parseval’s equality yields $$\label{58}
\operatorname{tr} (A \vert\xi\rangle\langle\chi\vert)
=
\langle\chi, A \xi\rangle.$$ Hence $\vert\xi\rangle\langle\chi\vert$ is $ \mathfrak{B}( \mathfrak
{L}_{1}( \mathfrak{h}) )$-measurable because the dual of $ \mathfrak{L}_{1}( \mathfrak{h})$ is formed by all maps $\varrho\mapsto\operatorname{tr} (A \varrho) $ with $A \in\mathfrak
{L}( \mathfrak{h})$. Let $x,y \in\mathfrak{h}$. The absolute value of the operator $ \vert x \rangle\langle y \vert$ is equal to the operator $
\vert y \rangle\langle y \vert
\Vert x \Vert/ \Vert y \Vert
$ in case $y \neq0$, and coincides with the null operator otherwise. Therefore $$\label{55}
\Vert
\vert x \rangle\langle y \vert
\Vert_{1}
= \frac{\Vert x \Vert}{ \Vert y \Vert} \Vert y \Vert^2
= \Vert x \Vert\Vert y \Vert.$$ Combining $\xi, \chi\in L^{2}( \mathbb{P},\mathfrak{h})$ with (\[55\]) gives $\mathbb{E} \Vert\vert\xi\rangle\langle\chi\vert\Vert_{1} < \infty$, and so the Bochner integral $\mathbb{E}\vert\xi\rangle\langle\chi\vert$ is well defined in the separable Banach space $\mathfrak{L}_{1}
( \mathfrak{h}) $.
We now turn to work in $\mathfrak{L}( \mathfrak{h}) $. The application $( x, y ) \mapsto\vert x \rangle\langle y \vert$ from $\mathfrak{h} \times\mathfrak{h}$ to $\mathfrak{L}( \mathfrak
{h})$ is continuous, and in consequence the measurability of $\xi$ and $\chi$ implies that $\vert\xi\rangle\langle\chi\vert$ is $ \mathfrak{B} ( \mathfrak{L}(
\mathfrak{h}) )$-measurable. Thus using we deduce that $\vert\xi\rangle\langle\chi\vert$ is Bochner $\mathbb{P}$-integrable in $\mathfrak{L}( \mathfrak{h}) $; see, for example, [@Yosida1995] for a treatment of the Bochner integral in Banach spaces which, in general, are not separable. Since $ \mathfrak{L}_{1}( \mathfrak{h})$ is continuously embedded in $\mathfrak{L}( \mathfrak{h}) $, either of the interpretations of $\mathbb{E}\vert\xi\rangle\langle
\chi\vert$ given above refers to the same operator.
Finally, for any $x,y$ belonging to $\mathfrak{h}$, the linear function $A
\mapsto\langle x, A y \rangle$ is continuous as a map from $\mathfrak
{L}( \mathfrak{h}) $ to $\mathbb{C}$. This gives (\[53\]). Similarly, (\[58\]) yields $
\operatorname{tr} ( \mathbb{E}\vert\xi\rangle\langle\chi\vert)
=
\mathbb{E} \operatorname{tr} ( \vert\xi\rangle\langle\chi\vert)
= \mathbb{E} \langle\chi, \xi\rangle
$, because $\operatorname{tr}( \cdot) \in\mathfrak{L}_{1}( \mathfrak
{h}) '$.
Under the assumptions of Lemma \[lema51\], $\mathbb{E}\vert\xi\rangle
\langle\chi\vert$ can also be interpreted as a Bochner integral in the pointwise sense; see, for example, [@DaPratoZabczyk1992].
To prove Theorem \[teorema8\], we need the following lemma.
\[lema52\] Let $C$ be a self-adjoint positive operator in $\mathfrak{h}$. Suppose that $\xi\in L^{2}_{C}( \mathbb{P},\mathfrak{h})$ and $A \in
\mathfrak{L}( ( \mathcal{D}( C) ,\Vert\cdot\Vert_{C}) ,\mathfrak{h}) $. Then $A \xi$ belongs to $L^{2}( \mathbb{P},\mathfrak{h})$.
Since $A \xi= A \pi_{C} ( \xi)$ $\mathbb{P}$-a.s., from Remark \[notaMedibilidad\] we deduce that $A \xi$ is strongly measurable. Thus $A \xi\in L^{2}( \mathbb{P},\mathfrak{h})$.
[Proof of Theorem \[teorema8\]]{} We start by proving statement (a). Let $x \in\mathcal{D}( C)$ and let $y \in\mathfrak{h}$. Using Lemma \[lema51\] yields $$\langle Cx, \varrho y \rangle
=
\mathbb{E} \langle C x, \xi\rangle\langle\xi, y \rangle
=
\mathbb{E} \langle x, C \xi\rangle\langle\xi, y \rangle.$$ In Lemma \[lema52\] we take $A=C$ to obtain $C \xi\in L^{2}( \mathbb
{P},\mathfrak{h})$. Thus, Lemma \[lema51\] implies $
\mathbb{E} \langle x, C \xi\rangle\langle\xi, y \rangle
=
\langle x, \mathbb{E}\vert C \xi\rangle\langle\xi\vert y \rangle
$, and so $
\langle Cx, \varrho y \rangle
=
\langle x, \mathbb{E}\vert C \xi\rangle\langle\xi\vert y \rangle
$. Then $\varrho y \in\mathcal{D}( C^{\ast}) = \mathcal{D}( C ) $ and $
C\varrho y = \mathbb{E}\vert C \xi\rangle\langle\xi\vert y
$, which is our assertion.
Part (a) yields $
\mathcal{D}( B )
=
\mathcal{D}( A \varrho B ) $, and so $A \varrho B$ is densely defined. We next prove that $A \varrho B$ coincides with $ \mathbb{E}\vert A \xi\rangle\langle B^{\ast} \xi\vert$ on $\mathcal{D}( B )$. For this purpose, we approximate $A$ by $A R_{n}$, where $R_{n}$ is the Yosida approximation of $-C$.
Suppose that $x \in\mathfrak{h}$ and $y \in\mathcal{D}( B )$. As in the proof of Lemma \[lema42\] we consider $
R_{n} = n (n+C )^{-1}
$, and so $C R_{n} z \longrightarrow_{n \rightarrow\infty} Cz$ for any $x \in
\mathcal{D}( C )$. Therefore $
\langle x, A R_{n} \varrho B y \rangle
\longrightarrow_{n \rightarrow\infty}
\langle x, A \varrho B y \rangle
$, and hence Lemma \[lema51\] gives $$\label{56}\quad
\langle x, A \varrho B y \rangle
=
\lim_{n \rightarrow\infty} \mathbb{E} \langle( A R_{n} )^{\ast} x,
\xi\rangle\langle\xi, B y \rangle
=
\lim_{n \rightarrow\infty} \mathbb{E} \langle x, A R_{n} \xi\rangle
\langle\xi, B y \rangle.$$ Since $\Vert R_{n} \Vert\leq1$ and $R_{n}$ commutes with $C$, $\Vert A R_{n} z \Vert\leq K \Vert z \Vert_{C}$. Using the dominated convergence theorem we obtain $$\label{57}
\langle x, A \varrho B y \rangle
=
\lim_{n \rightarrow\infty} \mathbb{E} \langle x, A R_{n} \xi\rangle
\langle\xi, B y \rangle
=
\mathbb{E} \langle x, A \xi\rangle\langle B^{\ast}\xi, y \rangle.$$
Since $B$ is densely defined, $B^{\ast}$ is a closed operator. Remark \[nota1\] now shows that $B^{\ast} \in\mathfrak{L}( ( \mathcal{D}( C) ,\mbox{$\Vert\cdot\Vert_{C}$})
,\mathfrak{h})$, and so applying Lemma \[lema52\] gives $
A \xi,\break B^{\ast} \xi\in L^{2}( \mathbb{P},\mathfrak{h})
$. Combining (\[57\]) with Lemma \[lema51\] we get $
\langle x, A \varrho B y \rangle
=\break
\langle x, \mathbb{E}\vert A \xi\rangle\langle B^{\ast} \xi\vert y
\rangle
$. Since the closure of $A \varrho B$ is equal to $\mathbb{E}\vert A \xi
\rangle\langle B^{\ast} \xi\vert$, we complete the proof of statement (b) by using Lemma \[lema51\].
Proof of Theorem 3.1
--------------------
First, we easily construct a random variable that represents a given $C$-regular operator.
\[lema6\] Let $\varrho\in\mathfrak{L}_{1,C}^{+}( \mathfrak{h}) $, with $C$ self-adjoint positive operator in $\mathfrak{h}$. Then there exists $\xi\in L_{C}^{2}( \mathbb{P},\mathfrak{h}) $ such that $\varrho=\mathbb{E}\vert\xi\rangle\langle\xi\vert$ and $\Vert\xi\Vert^{2}=\operatorname{tr}( \varrho) $ a.s.
In case $\varrho=0$, we take $\xi=0$. Otherwise, consider that $\varrho$ is written as in Definition \[def2\]. Then, we choose $\Omega= \mathfrak{I}$, and for any $n \in\mathfrak{I}$ we define $\mathbb{P}( \{ n \} )= \lambda_{n}/\operatorname{tr}( \varrho)$ and $\xi( n ) = \sqrt{\operatorname{tr}( \varrho) }u_{n}$.
Second, we use part (a) of Theorem \[teorema8\], together with Lemma \[lema51\], to establish the sufficient condition of Theorem \[teorema4\].
\[lema53\] Let $C$ be a self-adjoint positive operator in $\mathfrak{h}$. Suppose that $\varrho= \mathbb{E}\vert\xi\rangle\langle\xi\vert$, with $\xi\in L_{C}^{2}( \mathbb{P},\mathfrak{h}) $. Then $\varrho$ is $C$-regular.
Lemma \[lema51\] shows that $ \varrho\in\mathfrak{L}_{1}^{+}(
\mathfrak{h} )$, hence $
\varrho=\sum_{n\in\mathfrak{I}}\lambda_{n}\vert u_{n}\rangle\langle
u_{n}\vert
$, where $\mathfrak{I}$ is a countable set, $( \lambda_{n}) _{n\in\mathfrak
{I}}$ are summable positive real numbers and $( u_{n}) _{n\in\mathfrak
{I}}$ is a orthonormal family of vectors of $\mathfrak{h}$. Using statement (a) of Theorem \[teorema8\] yields $ u_{n} \in\mathcal{D}( C) $ for all $n\in\mathfrak{I}$.
We can extend $( u_{n}) _{n\in\mathfrak{I}}$ to an orthonormal basis $( e_{n}) _{n\in\mathfrak{I}'}$ of $\mathfrak{h}$ formed by elements of $\mathcal{D}( C) $. From Parseval’s equality we obtain $$\sum_{n\in\mathfrak{I}}\lambda_{n}\Vert Cu_{n}\Vert^{2}
=
\sum_{n\in\mathfrak{I}}\sum_{k\in\mathfrak{I}'}\lambda_{n}\vert
\langle Cu_{n},e_{k}\rangle\vert^{2}
=
\sum_{k\in\mathfrak{I}'}\sum_{n\in\mathfrak{I}}\lambda_{n}\langle
Ce_{k},\vert u_{n}\rangle\langle u_{n}\vert Ce_{k}\rangle,$$ and so $
\sum_{n\in\mathfrak{I}}\lambda_{n}\Vert Cu_{n}\Vert^{2}
=
\sum_{k\in\mathfrak{I}'}\langle Ce_{k},\varrho Ce_{k}\rangle
$. Combining Lemma \[lema51\] with Parseval’s equality we now get $$\sum_{n\in\mathfrak{I}}\lambda_{n}\Vert Cu_{n}\Vert^{2}
=
\sum_{k\in\mathfrak{I}'}\mathbb{E}\vert\langle
\xi,Ce_{k}\rangle\vert^{2}
=
\mathbb{E}\sum_{k\in\mathfrak{I}'}\vert\langle C\xi
,e_{n}\rangle\vert^{2}
=
\mathbb{E}\Vert C\xi\Vert^{2}.$$ This gives $\varrho\in\mathfrak{L}_{1,C}^{+}( \mathfrak{h}) $.
Proof of Theorem 4.1 {#subsecteor7}
--------------------
We first establish, in our framework, the well-known relation between Heisenberg and Schrödinger pictures.
\[lema10\] Suppose that Hypothesis \[HipN5\] holds, together with $\xi\in L_{C}^{2}( \mathbb{P},\mathfrak{h}) $. Let $\mathcal{T}_{t}( A) $ be as in Theorem \[teorema3\]. Then for all $A\in\mathfrak{L}( \mathfrak{h}) $, $$\label{35}
\operatorname{tr}( A\mathbb{E}\vert X_{t}( \xi) \rangle\langle X_{t}(
\xi) \vert) =\operatorname{tr}( \mathcal{T}_{t}(
A) \mathbb{E}\vert\xi\rangle\langle\xi\vert).$$
Fix $A\in\mathfrak{L}( \mathfrak{h}) $, and define the function $f_{n} \dvtx \mathfrak{h} \rightarrow\mathbb{C}$ by $
f_{n}( x) = \langle x,Ax\rangle
$ if $\Vert x\Vert\leq n$, and $
f_{n}( x) = 0
$ otherwise. Using the Markov property of $X_t ( \xi)$, which can be obtained by techniques of well-posed martingale problems, we get $$\label{n311}
\mathbb{E}( f_{n}( X_{t}( \xi) ) )
=
\mathbb{E}
\bigl(
( f_{n}( X_{t}( \xi) ) )
{/}\mathfrak{F}_{0}
\bigr)
=
\mathbb{ E}P_{t}f_{n}( \xi),$$ where $P_{t}f_{n}( x) =\mathbb{E}( f_{n}( X_{t}( x) ) ) $ for all $x\in\mathcal{D}( C) $.
We will take the limit as $n \rightarrow\infty$ in (\[n311\]). The dominated convergence theorem leads to $$\label{62}
\lim_{n \rightarrow\infty} \mathbb{E}( f_{n}( X_{t}( \xi) ) )
=
\mathbb{E}\langle X_{t}( \xi) ,AX_{t}( \xi) \rangle.$$ Combining (\[62\]) with (\[42\]) yields $
P_{t}f_{n}( x) \longrightarrow_{n\rightarrow\infty}
\langle x,\mathcal{T}_{t}( A) x\rangle
$ whenever $x\in\mathcal{D}( C) $. Since $\Vert P_{t}f_{n}( x) \Vert\leq\Vert A \Vert\Vert x \Vert^{2}$, according to the dominated convergence theorem, we have $
\mathbb{ E}P_{t}f_{n}( \xi)
\longrightarrow
\mathbb{ E} \langle\xi,\mathcal{T}_{t}( A) \xi\rangle
$ as $
n \rightarrow\infty
$. Then, letting $n \rightarrow\infty$ in (\[n311\]) we get $
\mathbb{E}\langle X_{t}( \xi) ,AX_{t}( \xi)
\rangle=\mathbb{E}\langle\xi,\mathcal{T}_{t}( A)
\xi\rangle
$ by (\[62\]), and so Theorem \[teorema8\] leads to (\[35\]).
We next check that $\rho_{t}( \varrho) $ is well defined by (\[31\]).
\[lema13\] Let Hypothesis \[HipN5\] hold and consider $\xi, \varphi\in L_{C}^{2}( \mathbb{P},\mathfrak{h}) $ such that $\mathbb{E}\vert\xi\rangle\langle\xi\vert=\mathbb{E}\vert
\varphi\rangle\langle\varphi\vert$. Then $
\mathbb{E}\vert X_{t}( \xi) \rangle\langle X_{t}(
\xi) \vert=\mathbb{E}\vert X_{t}( \varphi)
\rangle\langle X_{t}( \varphi) \vert
$.
Let $A\in\mathfrak{L}( \mathfrak{h}) $. Using Lemma \[lema10\] yields $$\operatorname{tr}( A\mathbb{E}\vert X_{t}( \xi) \rangle\langle X_{t}(
\xi) \vert)
=
\operatorname{tr}( \mathcal{T}_{t}( A) \mathbb{E}\vert\xi\rangle
\langle\xi\vert)
=
\operatorname{tr}( A\mathbb{E}\vert X_{t}( \varphi) \rangle\langle
X_{t}( \varphi) \vert) .$$ Hence $\Vert\mathbb{E}\vert X_{t}( \xi) \rangle\langle
X_{t}( \xi) \vert-\mathbb{E}\vert X_{t}(
\varphi) \rangle\langle X_{t}( \varphi) \vert
\Vert_{\mathfrak{L}_{1}( \mathfrak{h}) }=0$; see, for example, Proposition 9.12 of [@Parthasarathy1992].
We now address the contraction property of the restriction of $\rho_{t}$ to $\mathfrak{L}_{1,C}^{+} ( \mathfrak{h}) $.
\[lema11\] Let Hypothesis \[HipN5\] hold. If $\varrho,\widetilde{\varrho}
$ are $C$-regular, then $$\label{n312}
{\operatorname{tr}}\vert\rho_{t}( \varrho) -\rho_{t}( \widetilde{\varrho}) \vert\leq{\operatorname{tr}}\vert\varrho-\widetilde{\varrho}
\vert.\vadjust{\goodbreak}$$
Since $
{\operatorname{tr} }\vert\rho_{t}( \varrho) -
\rho_{t}( \widetilde{\varrho}) \vert
=
\sup_{\Vert A\Vert_{\mathfrak{L}( \mathfrak{h} )} =1}
\vert{\operatorname{tr}}( A\rho_{t}(
\varrho) ) -\operatorname{tr}( A\rho_{t}( \widetilde{\varrho})
) \vert
$, according to Lemma \[lema10\] we have $${\operatorname{tr}}\vert\rho_{t}( \varrho) -\rho_{t}( \widetilde{\varrho
}) \vert
=
\sup_{A\in\mathfrak{L}( \mathfrak{h}) ,\Vert A\Vert=1}
\vert{\operatorname{tr}}( \mathcal{T}_{t}( A) \varrho)
-\operatorname{tr}( \mathcal{T}_{t}( A) \widetilde{\varrho})
\vert.$$ Therefore $
{\operatorname{tr}}\vert\rho_{t}( \varrho) -\rho_{t}( \widetilde{\varrho
}) \vert
\leq
{\operatorname{tr}}\vert\varrho-\widetilde{\varrho}\vert
\sup_{\Vert A\Vert_{\mathfrak{L}( \mathfrak{h})} =1}
\Vert\mathcal{T}_{t}( A) \Vert
$, and so Theorem \[teorema3\] leads to (\[n312\]).
The following lemma helps us to extend $\rho_{t}$ to all $\mathfrak
{L}_{1}( \mathfrak{h}) $.
\[lema12\] Suppose that $C$ is a self-adjoint positive operator in $\mathfrak{h}$. Then $\mathfrak{L}_{1,C}^{+}( \mathfrak{h}) $ is dense in $\mathfrak{L}_{1}^{+}( \mathfrak{h}) $ with respect to the trace norm.
Let $\varrho\in\mathfrak{L}_{1}^{+}( \mathfrak{h}) $. Then there exists a sequence of orthonormal vectors $( u_{j}) _{j\in
\mathbb{N}}$ for which $\varrho=\sum_{j\in\mathbb{N}}\lambda_{j}\vert u_{j}\rangle
\langle u_{j}\vert$, with $\lambda_{j}\geq0$ and $\sum_{j\in\mathbb{N}}\lambda_{j}<\infty$. For any $x,y\in\mathfrak{h}$ we have $${\operatorname{tr}}\bigl\vert
\vert x\rangle\langle x\vert-\vert
y\rangle\langle y\vert
\bigr\vert
=
\sup_{\Vert A\Vert_{\mathfrak{L}( \mathfrak{h})} =1}
\vert\langle
x,Ax\rangle-\langle y,Ay\rangle\vert
\leq
\Vert x-y\Vert^{2}+2\Vert y\Vert\Vert x-y\Vert,$$ and so $\{ \vert x\rangle\langle x\vert\dvtx x\in\mathcal{D}(
C) \} $ is a -dense subset of $\{ \vert
x\rangle\langle x\vert\dvtx x\in\mathfrak{h}\} $ since $\mathcal{D}( C) $ is dense in $\mathfrak{h}$. Now, the lemma follows from $
{\operatorname{tr}}\vert\varrho-\sum_{j=1}^{n}\lambda_{j}\vert
u_{j}\rangle\langle u_{j}\vert\vert
=
\sum_{j=n+1}^{\infty}\lambda_{j}
\longrightarrow_{n\rightarrow\infty} 0
$.
[Proof of Theorem \[teor7\]]{} Combining Theorem \[teorema4\] with Lemma \[lema13\] we obtain that (\[31\]) defines unambiguously a linear operator $\rho_{t}( \varrho) $ for any $\varrho\in\mathfrak{L}_{1,C}^{+}( \mathfrak{h}) $ and $t\geq0$. Lemma \[lema12\] guarantees the uniqueness of the operator belonging to $\mathfrak{L}( \mathfrak{L}_{1}( \mathfrak{h})
) $ for which (\[31\]) holds. We next extend $\rho_{t}$ to a bounded linear operator in $\mathfrak
{L}_{1}( \mathfrak{h}) $ by means of density arguments.
Suppose that $\varrho\in\mathfrak{L}_{1}^{+}( \mathfrak{h}) $. By Lemma \[lema12\], there exists a sequence $(
\varrho_{n}) _{n\in\mathbb{N}}$ of $C$-regular operators for which $
\lim_{n\rightarrow\infty}
\Vert\varrho-\varrho_{n}\Vert_{\mathfrak{L}_{1}( \mathfrak{h}) }\rightarrow0
$. We define $
\rho_{t}( \varrho)
$ to be the limit in $\mathfrak{L}_{1}( \mathfrak{h} ) $ of $\rho_{t}( \varrho_{n}) $ as $n\rightarrow\infty$; according to Lemma \[lema11\] this limit exists and does not depend on the choice of $( \varrho_{n}) _{n\in\mathbb{N}}$. Recall that every $A \in\mathfrak{L}( \mathfrak{h}) $ has a unique decomposition of the form $A=\Re( A) +i$ $\Im( A) $, with $\Re( A) $ and $\Im( A) $ self-adjoint operators in $\mathfrak{h}$. For each $\varrho\in\mathfrak{L}_{1}( \mathfrak{h}) $ we set $$\rho_{t}( \varrho) =\rho_{t}( \Re( \varrho)
_{+}) -\rho_{t}( \Re( \varrho) _{-})
+i\bigl( \rho_{t}( \Im( \varrho) _{+})
-\rho_{t}( \Im( \varrho) _{-}) \bigr) ,$$ where $A_{+}$, $A_{-}$ denotes, respectively, the positive and negative parts of the self-adjoint operator $A$; see, for example, [@BratteliRobinson1987] for details.
We will verify that $\rho_{t} \in\mathfrak{L} ( \mathfrak{L}_{1}( \mathfrak{h}) ) $. Let $\varrho=\varrho_{1}-\varrho_{2}+i( \varrho_{3}-\varrho_{4}) $, with $\varrho_{j}\in\mathfrak{L}_{1,C}^{+}( \mathfrak{h} ) $ for any $j=1,\ldots,4$. Since $\Vert\mathcal{T}_{t}( A) \Vert\leq\Vert A \Vert$, Lemma \[lema10\] yields $${\operatorname{tr}}\vert\rho_{t}( \varrho) \vert
=
{\sup_{\Vert A\Vert_{\mathfrak{L}( \mathfrak{h}) } =1}}
\vert{\operatorname{tr}}( A\rho_{t}( \varrho) ) \vert
=
{\sup_{\Vert A\Vert_{\mathfrak{L}( \mathfrak{h}) } =1}}
\vert{\operatorname{tr}}( \mathcal{T}_{t}( A) \varrho) \vert
\leq
\operatorname{tr}( \vert\varrho\vert) .$$ The construction of $\rho_{t}( \varrho) $ now implies $
\Vert\rho_{t}( \varrho) \Vert_{\mathfrak{L}_{1}( \mathfrak{h}) }\leq
\Vert\varrho\Vert_{\mathfrak{L}_{1}( \mathfrak{h}) }
$ for all $\varrho\in\mathfrak{L}_{1}( \mathfrak{h}) $. Consider two $C$-regular operators $\varrho,\widetilde{\varrho}$ and $\alpha\geq0$. By Definition \[def2\], $\varrho+\alpha\widetilde{\varrho}$ belongs to $\mathfrak{L}_{1,C}^{+}( \mathfrak{h}) $. If $A\in\mathfrak{L}( \mathfrak{h}) $, then applying Lemma \[lema10\] we obtain $$\operatorname{tr}\bigl( \rho_{t}( \varrho+\alpha\widetilde{\varrho}) A\bigr)
=
\operatorname{tr}( \mathcal{T}_{t}( A) \varrho)
+
\alpha\operatorname{tr}( \mathcal{T}_{t}( A) \widetilde{\varrho}) \\
=
\operatorname{tr}\bigl( \bigl( \rho_{t}( \varrho)
+
\alpha
\rho_{t}( \widetilde{\varrho}) \bigr) A\bigr) .$$ Therefore $\Vert\rho_{t}( \varrho+\alpha\widetilde{\varrho}) -\rho_{t}( \varrho) -\alpha\rho_{t}( \widetilde{\varrho}) \Vert_{\mathfrak{L}_{1}( \mathfrak{h}) }=0$, and so Lemma \[lema12\] leads to $\rho_{t}( \varrho+\alpha\widetilde{\varrho}) =\rho_{t}( \varrho)
+\alpha\rho_{t}( \widetilde{\varrho}) $ for any $\varrho,\widetilde{\varrho}\in\mathfrak{L}_{1}^{+}( \mathfrak{h}) $. Careful algebraic manipulations now show the linearity of $\rho_{t}\dvtx\mathfrak{L}_{1}( \mathfrak{h}) \rightarrow\mathfrak{L}_{1}( \mathfrak{h}) $.
Proof of Theorem 4.2 {#subsecteor9}
--------------------
Let us first prove the continuity of the map $
\xi\mapsto\rho_{t}( \mathbb{E}\vert
\xi\rangle\langle\xi\vert)
$.
\[lema16\] Assume that Hypothesis \[HipN5\] holds. Let $\xi$ and $\xi_{n}$, with , be random variables in $L_{C}^{2}( \mathbb{P},\mathfrak{h}) $ satisfying $\mathbb{E}\Vert\xi-\xi_{n}\Vert^{2}\longrightarrow_{n\rightarrow
\infty}0$. Then$\rho_{t}( \mathbb{E}\vert\xi_{n}\rangle\langle\xi_{n}\vert) $ converges in $\mathfrak{L}( \mathfrak{h} ) $ to $\rho_{t}( \mathbb{E}\vert\xi\rangle\langle\xi\vert) $ as $n \rightarrow\infty$.
Let $x \in\mathfrak{h}$. Combining (\[31\]) with the linearity of (\[2\]) we get $$\begin{aligned}
&&
\bigl\Vert\rho_{t}( \mathbb{E}\vert\xi_{n}\rangle\langle\xi_{n}\vert)
x-\rho_{t}( \mathbb{E}\vert\xi
\rangle\langle\xi\vert) x\bigr\Vert
\\
&&\qquad \leq
\mathbb{E}\vert\langle X_{t}( \xi_{n}) ,x\rangle\vert\Vert X_{t}( \xi
_{n}- \xi) \Vert
+
\mathbb{E}\vert\langle X_{t}( \xi- \xi_{n}) ,x\rangle\vert\Vert
X_{t}( \xi) \Vert
\\
&&\qquad \leq
\Vert x\Vert
\bigl(
\mathbb{E}\Vert\xi-\xi_{n}\Vert^{2} +
2 \sqrt{\mathbb{E}\Vert\xi-\xi_{n}\Vert^{2}} \sqrt{ \mathbb{E}\Vert\xi
\Vert^{2}}
\bigr).\end{aligned}$$ In the last inequality we used that $
\mathbb{E}\Vert X_{t}( \eta) \Vert^{2}
\leq
\mathbb{E} \Vert\eta\Vert^{2}
$ for $\eta\in L_{C}^{2}( \mathbb{P},\mathfrak{h}) $.
[Proof of Theorem \[teor9\]]{} There exits a sequence $( \xi_{n}) _{n}$ of -valued random variables with finite ranges such that $\Vert\xi_{n}-\xi\Vert$ converges monotonically to $0$; see, for example, [@DaPratoZabczyk1992]. By Lemma \[lema16\], $
\rho_{t}( \mathbb{E}\vert\xi_{n}\rangle\langle\xi_{n}\vert)
$ converges to $
\rho_{t}( \mathbb{E}\vert\xi\rangle\langle\xi\vert)
$ in $\mathfrak{L}( \mathfrak{h})$. Since $\rho_{t}$ is linear, an easy computation shows that $
\mathbb{E}\rho_{t}( \vert\xi_{n}\rangle\langle\xi_{n}\vert
)
=
\rho_{t}( \mathbb{E}\vert\xi_{n}\rangle
\langle\xi_{n}\vert)
$, hence $$\label{n37}
\mathbb{E}\rho_{t}( \vert\xi_{n}\rangle\langle\xi_{n}\vert
)
\longrightarrow_{n\rightarrow\infty}
\rho_{t}( \mathbb{E}\vert\xi\rangle\langle\xi\vert)
\qquad\mbox{in } \mathfrak{L}( \mathfrak{h}) .$$
We will prove that $\mathbb{E}\rho_{t}( \vert\xi_{n}\rangle\langle\xi_{n}\vert) $ converges to $\mathbb{E}\rho_{t}( \vert\xi\rangle\langle\xi\vert) $ in $\mathfrak
{L}( \mathfrak{h}
) $ as $n \rightarrow\infty$, which together with (\[n37\]) implies $
\rho_{t}( \mathbb{E}\vert\xi\rangle\langle\xi\vert)
=
\mathbb{E} \rho_{t}( \vert\xi\rangle\langle\xi\vert)
$. From Lemma \[lema16\] we obtain $
\Vert\rho_{t}( \vert\xi_{n}\rangle\langle\xi_{n}\vert
) -\rho_{t}( \vert\xi\rangle\langle\xi\vert)
\Vert_{\mathfrak{L}( \mathfrak{h}) }\longrightarrow
_{n\rightarrow\infty}0
$. For any $x,y \in\mathfrak{h}$ we have $
\Vert
\vert x \rangle\langle y \vert
\Vert_{1}
= \Vert x \Vert\Vert y \Vert
$, and so Lemma \[lema11\] yields $$\Vert\rho_{t}( \vert\xi_{n}\rangle\langle\xi_{n}\vert
) \Vert
\leq
\Vert\rho_{t}( \vert\xi_{n}\rangle\langle\xi_{n}\vert
) \Vert_{1}
\leq
\Vert\xi_{n}\Vert^{2}
\leq
2( \Vert\xi_{1}-\xi\Vert^{2}+\Vert
\xi\Vert^{2}) .$$ Therefore $
\mathbb{E}\Vert\rho_{t}( \vert\xi_{n}\rangle\langle\xi
_{n}\vert) -\rho_{t}( \vert\xi\rangle\langle
\xi\vert) \Vert_{\mathfrak{L}( \mathfrak{h})
}\longrightarrow_{n\rightarrow\infty}0.
$
Proof of Theorem 4.3
--------------------
Our proof is divided into three lemmata. The first two deal with the semigroup property of $( \rho_t )_{t \geq
0} $.
\[lema14\] Let Hypothesis \[HipN5\] hold, and let $\varrho$ be $C$-regular. Then for all $t \geq0$, $\rho_{t}( \varrho)$ belongs to $\mathfrak{L}_{1,C}^{+}( \mathfrak{h}) $ and $\rho_{t+s}( \varrho) =\rho_{t}\circ\rho_{s}( \varrho) $ whenever $s\geq0$.
Since $X_{t}( \xi) \in L_{C}^{2}( \mathbb{P},\mathfrak{h}) $, combining Theorem \[teorema4\] with (\[31\]) gives $
\rho_{t}( \mathfrak{L}_{1,C}^{+}( \mathfrak{h}) )
\subset\mathfrak{L}_{1,C}^{+}( \mathfrak{h})
$.
We will establish the semigroup property of the restriction of $ \rho$ to $ \mathfrak{L}_{1,C}^{+}( \mathfrak{h})$. Consider $\xi\in L_{C}^{2}( \mathbb{P},\mathfrak{h}) $ satisfying $\rho=\mathbb{E}\vert\xi\rangle\langle\xi\vert$, and fix $ x,y \in\mathfrak{h}$. For all $z \in\mathfrak{h}$ we define $p_{n}( z )
=
\langle z,x\rangle\langle y,z\rangle
$ if $
\vert\langle z,x\rangle\langle y,z\rangle\vert\leq n
$, and $p_{n}( z ) = 0$ otherwise. Using the Markov property of $
X_{t}( \xi)
$ we deduce that $$\label{38}
\mathbb{E}( p_{n} ( X_{t+s}( \xi) ) )
=
\mathbb{E} \bigl(
( p_{n} ( X_{t+s}( \xi) ) )
{/}\mathfrak{F}_{s}
\bigr)
=
\mathbb{E}P_{t}( p_{n} ) ( X_{s}( \xi)
) ,$$ where for all $z \in\mathcal{D}( C) $, $P_{t}( p_{n} ) ( z) = \mathbb{E}( p_{n} ( X_{t}( z) ) ) $.
Let $z \in\mathcal{D}( C) $. Applying the dominated convergence theorem gives $$\lim_{n\rightarrow\infty} \mathbb{E}( p_{n} ( X_{t}( z) ) )
=
\mathbb{E} \langle X_{t}( z) ,x\rangle\langle y,X_{t}( z) \rangle
=
\langle y,\rho_{t}( \vert z\rangle\langle z\vert) x\rangle,$$ hence $
\lim_{n\rightarrow\infty} P_{t}( p_{n} ) ( z)
=
\langle y,\rho_{t}( \vert z\rangle\langle z\vert) x\rangle
$. Then $
\mathbb{E} P_{t} ( p_{n} ) ( X_{s} ( \xi) )
\longrightarrow_{n\rightarrow\infty}\break
\mathbb{E}
\langle y,\rho_{t}( \vert X_{s}( \xi)
\rangle\langle X_{s}( \xi) \vert) x\rangle
$, and so Theorem \[teor9\] leads to $$\label{331}\quad\qquad
\lim_{n\rightarrow\infty}
\mathbb{E} P_{t} ( p_{n} ) ( X_{s} ( \xi) )
=
\langle y,\rho_{t}( \mathbb{E}\vert X_{s}( \xi)
\rangle\langle X_{s}( \xi) \vert) x\rangle
=
\langle y,\rho_{t} \circ\rho_{s} ( \varrho) x\rangle.$$ By (\[38\]), in (\[331\]) we replace $s$ by $0$ and $t$ by $t+s$ to obtain $$\lim_{n\rightarrow\infty}
\mathbb{E}( p_{n} ( X_{t+s}( \xi) ) )
=
\lim_{n\rightarrow\infty}
\mathbb{E} P_{t+s} ( p_{n} ) ( X_{0} ( \xi) )
=
\langle y,\rho_{t+s}( \varrho) x\rangle.$$ Thus, letting $n\rightarrow\infty$ in (\[38\]), we get $\rho_{t+s}( \varrho) =\rho_{t}\circ\rho_{s}( \varrho) $ by (\[331\]).
\[lema24\] Under Hypothesis \[HipN5\], $( \rho_{t})_{t\geq0}$ is a semigroup of contractions which leaves $\mathfrak{L}_{1}^{+}( \mathfrak{h}) $ invariant.
By Theorem \[teor7\], $\Vert\rho_{t}\Vert_{\mathfrak{L}( \mathfrak{L}_{1}( \mathfrak{h}) )
}\leq1$. Since $\rho_{t}( \varrho) $ is positive whenever $\varrho$ is $C$-regular, using Lemma \[lema12\] yields $\langle x,\rho_{t}( \varrho) x\rangle\geq0$ for any $\varrho\in\mathfrak{L}_{1}^{+}( \mathfrak{h}) $ and $x\in
\mathfrak{h}$.
Suppose that $\varrho=\varrho_{1}-\varrho_{2}+i( \varrho_{3}-\varrho_{4}) $, where $\varrho_{1},\ldots,\varrho_{4}$ are $C$-regular operators. Applying (\[31\]) gives $\rho_{0}( \varrho) =\varrho$, and Lemma \[lema14\] asserts that $\rho_{t+s}( \varrho) =\rho_{t}\circ\rho_{s}( \varrho) $ for any $s,t\geq0$. Then, combining Lemma \[lema12\] with density arguments, we deduce that $( \rho_{t}) _{t\geq0}$ is a semigroup.
We now examine the continuity of the map $t\mapsto$ $\rho_{t}( \varrho) $ when $\varrho$ is $C$-regular.
\[lema25\] Adopt Hypothesis \[HipN5\], together with $\varrho\in\mathfrak{L}_{1,C}^{+}( \mathfrak{h}) $. Then the map $t\mapsto\rho_{t}( \varrho) $ from $[ 0,\infty[ $ to $\mathfrak{L}_{1}( \mathfrak{h}) $ is continuous.
Consider $\xi\in L_{C}^{2}( \mathbb{P},\mathfrak{h}) $ such that $\varrho=\mathbb{E}\vert\xi\rangle\langle\xi\vert$. Theorem \[teorema8\] yields $
\mathbb{E}\Vert X_{t}( \xi) \Vert^{2}
\leq
\mathbb{E}\Vert\xi\Vert^{2}
=
\operatorname{tr}( \varrho)
$ for all $t\geq0$, and so combining Theorem \[teorema8\] with the Cauchy–Schwarz inequality yields $$\begin{aligned}
{\operatorname{tr}}\vert\rho_{t}( \varrho) -\rho_{s}( \varrho)
\vert &=& \sup_{A\in\mathfrak{L}( \mathfrak{h}) ,\Vert
A\Vert=1}\vert\mathbb{E}\langle X_{t}( \xi)
,AX_{t}( \xi) \rangle-\langle X_{s}( \xi)
,AX_{s}( \xi) \rangle\vert\\
&\leq& 2( \operatorname{tr}( \varrho) ) ^{1/2}\bigl( \mathbb{E}\Vert X_{t}( \xi) -X_{s}( \xi) \Vert
^{2}\bigr) ^{1/2}.\end{aligned}$$ Since $
\mathbb{E} ( \sup_{s \in[0, T ]} \Vert X_s( \xi) \Vert^{2} )
< \infty
$ for any $T>0$ (see, e.g., Theorem 4.2.5 of [@Prevot2007]), using the dominated convergence theorem, we get (\[310\]).
Proof of Theorem 4.4 {#subsecteor10}
--------------------
First, we establish the weak continuity of the map $t\mapsto AX_{t}( \xi
) $ when $A$ is relatively bounded by $C$.
\[lema17\] Assume that Hypothesis \[HipN5\] holds. If $\xi$ belongs to $L_{C}^{2}( \mathbb{P},\mathfrak{h}) $ and if $A \in\mathfrak{L}( ( \mathcal{D}( C) ,\mbox{$\Vert\cdot\Vert_{C}$})
,\mathfrak{h}) $, then for all $\psi\in L^{2}( \mathbb{P},\mathfrak{h}) $ and $t\geq0$ we have $$\label{n39}
\lim_{s\rightarrow t}\mathbb{E}\langle\psi,AX_{s}( \xi)
\rangle=\mathbb{E}\langle\psi,AX_{t}( \xi)
\rangle.$$
Let $( s_{n}) _{n}$ be a sequence of nonnegative real numbers converging to $t$. Since $( ( X_{s_{n}}( \xi) ,AX_{s_{n}}( \xi) ,CX_{s_{n}}( \xi) ) )_{n}$ is a bounded sequence in $L^{2}( \mathbb{P}, \mathfrak{h}^{3} ) $ with $\mathfrak{h}^{3} = \mathfrak{h}\times\mathfrak{h}\times
\mathfrak{h}$, there exists a subsequence $( s_{n( k) }) _{k}$ for which $$\label{n38} \bigl( X_{s_{n( k) }}( \xi) ,AX_{s_{n( k)}}( \xi)
,CX_{s_{n( k) }}( \xi) \bigr) \longrightarrow_{k\rightarrow\infty} (
Y,U,V)$$ weakly in $L^{2}( \mathbb{P},\mathfrak{h}^{3})$.
Set $\mathfrak{M}=\{ ( \eta,A\eta,C\eta) \dvtx\eta\in L_{C}^{2}( \mathbb
{P},\mathfrak{h}) \} $. Then $\mathfrak{M}$ is a linear manifold of $L^{2}( \mathbb{P},\mathfrak
{h}^{3}) $ closed with respect to the strong topology. In fact, suppose that $( ( \eta_{n},A\eta_{n}, C\eta_{n}) ) _{n}$ is a sequence of elements of $\mathfrak{M}$ that converges to $( \eta
_{1},\eta_{2},\eta_{3}) $ in $L^{2}(
\mathbb{P},\mathfrak{h}^{3}) $. Hence there exists a subsequence $( ( \eta_{n( j) },A\eta_{n( j)}, C\eta_{n( j) }) ) _{j}$ converging almost surely to $( \eta_{1},\eta_{2},\eta_{3}) $. Therefore $\eta_{1} \in\mathcal{D}( C )$ and $\eta_{3}=C\eta_{1}$ by $C$ is closed. Using $A\in\mathfrak{L}( ( \mathcal{D}( C) ,\mbox{$\Vert\cdot\Vert_{C}$})
,\mathfrak{h}) $ gives $\eta_{2}=A\eta_{1}$.
For any $k \in\mathbb{N}$, $(X_{s_{n( k) }}( \xi) ,AX_{s_{n( k) }}( \xi) ,CX_{s_{n( k) }}( \xi) )
$ belongs to $\mathfrak{M}$. Since $\mathfrak{M}$ is a closed linear manifold of $L^{2}( \mathbb
{P},\mathfrak{h}^{3}) $, (\[n38\]) implies $( Y,U,V) \in\mathfrak{M}$; see, for example, Section III.1.6 of [@Kato1995]. Combining the dominated convergence theorem with $
\mathbb{E} ( \sup_{s \in[0, t+1 ]} \Vert X_s( \xi) \Vert^{2} )
< \infty
$ we get that $
\mathbb{E} \Vert X_{s_{n( k)}}( \xi) -X_{t}( \xi) \Vert^{2}
$ converges to $0$. Thus $Y=X_{t}( \xi) $, and so $U=AX_{t}( \xi) $. Hence $
AX_{s_{n( k) }}( \xi)
$ converges to $AX_{t} ( \xi) $ weakly in $L^{2}( \mathbb{P},\mathfrak{h})$.
Second, we show that the probabilistic representation of the right-hand side of (\[312\]) is continuous as a function from $[
0,+\infty[ $ to $\mathbb{C}$.
\[lema18\] Let Hypothesis \[HipN5\] hold. Fix $\xi\in L_{C}^{2}( \mathbb{P},\mathfrak{h}) $ and $A \in\mathfrak
{L}( \mathfrak{h}) $. Then, the function that maps each $t$ in $[ 0,+\infty[ $ to the complex number $
\mathbb{E}\langle GX_{t}( \xi)$, $AX_{t}( \xi) \rangle
+
\mathbb{E}\langle X_{t}( \xi) ,AG X_{t}( \xi) \rangle
+\sum_{k=1}^{\infty}
\mathbb{E}\langle L_{k}X_{t}( \xi) ,AL_{k}X_{t}( \xi) \rangle
$ is continuous.
Let $( t_{n}) _{n}$ be a sequence of nonnegative real numbers such that $t_{n}$ converges to $t$. Since $
\mathbb{E} ( \sup_{s \in[0, t+1 ]} \Vert X_s( \xi) \Vert^{2} )
< \infty
$ (see, e.g., Theorem 4.2.5 of [@Prevot2007]), $
AX_{t_{n}}( \xi) \longrightarrow_{n\rightarrow\infty}AX_{t}( \xi)
$ in $L^{2}( \mathbb{P},\mathfrak{h})$. Hence Lemma \[lema17\] yields $$\label{138}
\lim_{n\rightarrow\infty} \mathbb{E}\langle GX_{t_{n}}( \xi)
,AX_{t_{n}}( \xi) \rangle
=
\mathbb{E}\langle GX_{t}( \xi) ,AX_{t}( \xi) \rangle;$$ see, for example, Section III.1.7 of [@Kato1995]. By (\[138\]) with $A$ replaced by $A ^{*}$, $t \mapsto\mathbb{E}\langle A ^{*}X_{t}( \xi) ,G X_{t}( \xi) \rangle$ is continuous, then so is $t \mapsto\mathbb{E}\langle X_{t}( \xi) ,AG X_{t}( \xi) \rangle$.
We now focus on $
\sum_{k=1}^{\infty}
\mathbb{E}\langle L_{k}X_{t}( \xi) ,AL_{k}X_{t}( \xi) \rangle
$. Taking $A=I$ in (\[138\]) we get $
\mathbb{E} \Re\langle X_{t_{n}}( \xi) ,GX_{t_{n}}( \xi) \rangle
\rightarrow_{n\rightarrow\infty}
\mathbb{E} \Re\langle X_{t}( \xi) ,GX_{t}( \xi) \rangle$. Thus condition (H2.2) leads to $$\label{313}
\sum_{k=1}^{\infty}
\mathbb{E}\Vert L_{k}X_{t_{n}}( \xi) \Vert^{2}
\longrightarrow_{n\rightarrow\infty}
\sum_{k=1}^{\infty} \mathbb{E}\Vert L_{k}X_{t}( \xi) \Vert^{2}.$$
Using (\[313\]) we will deduce that $L_{k}X_{t_{n}}( \xi) $ converges strongly in $L^{2}( \mathbb
{P},\mathfrak{h})$ to $L_{k}X_{t}( \xi) $ as $n \rightarrow\infty$. Conversely, suppose that for a given $j \in\mathbb{N}$, $$\label{317}
\limsup_{n \rightarrow\infty} \mathbb{E}\Vert L_{j}X_{t_{n}}( \xi)
\Vert^{2}
>
\mathbb{E}\Vert L_{j}X_{t}( \xi) \Vert^{2} .$$ Since $
\mathbb{E}\Vert L_{k}X_{t}( \xi) \Vert^{2}
\leq
\lim\inf_{n\rightarrow\infty}
\mathbb{E}\Vert L_{k}X_{t_{n}}( \xi) \Vert^{2}
$, Fatou’s lemma shows $$\label{318}
\sum_{k \neq j} \mathbb{E}\Vert L_{k}X_{t}( \xi) \Vert^{2}
\leq
\liminf_{n \rightarrow\infty} \sum_{k \neq j} \mathbb{E}\Vert
L_{k}X_{t_{n}}( \xi) \Vert^{2}.$$ According to (\[313\]) and (\[317\]) we have $$\begin{aligned}
\liminf_{n \rightarrow\infty} \sum_{k \neq j} \mathbb{E}\Vert
L_{k}X_{t_{n}}( \xi) \Vert^{2}
& = &
\sum_{k=1}^{\infty} \mathbb{E}\Vert L_{k}X_{t}( \xi) \Vert^{2}
-
\limsup_{n \rightarrow\infty} \mathbb{E}\Vert L_{j}X_{t_{n}}( \xi)
\Vert^{2} \\
& < &
\sum_{k \neq j} \mathbb{E}\Vert L_{k}X_{t}( \xi) \Vert^{2} ,\end{aligned}$$ contrary to (\[318\]), and so $$\label{332}
\limsup_{n \rightarrow\infty} \mathbb{E}\Vert L_{j}X_{t_{n}}( \xi)
\Vert^{2}
\leq
\mathbb{E}\Vert L_{j}X_{t}( \xi) \Vert^{2}
.$$ Applying Lemma \[lema17\] we get that $L_{j}X_{t_{n}}( \xi) $ converges weakly in $L^{2}( \mathbb{P},\mathfrak{h})$ to $L_{j}X_{t}( \xi) $ as $n \rightarrow\infty$, and so (\[332\]) leads to $
L_{k}X_{t_{n}}( \xi)
\longrightarrow_{n \rightarrow\infty}
L_{k}X_{t}( \xi)
$ in $L^{2}( \mathbb{P},\mathfrak{h})$.
From condition (H2.2) it follows that $
\sum_{k=1}^{n}
\mathbb{E}\langle L_{k}X_{t}( \xi) ,AL_{k}X_{t}( \xi) \rangle
$ converges to $
\sum_{k=1}^{\infty}
\mathbb{E}\langle L_{k}X_{t}( \xi) ,AL_{k}X_{t}( \xi) \rangle
$ as $n\rightarrow\infty$ uniformly on any finite interval. Since $
\mathbb{E}\langle L_{k}X_{t_{n}}( \xi) ,AL_{k}X_{t_{n}}( \xi) \rangle
\longrightarrow_{n\rightarrow\infty}
\mathbb{E}\langle L_{k}X_{t}( \xi) ,AL_{k}X_{t}( \xi) \rangle
$, the map $
t \mapsto
\sum_{k=1}^{\infty}
\mathbb{E}\langle L_{k}X_{t}( \xi) ,AL_{k}X_{t}( \xi) \rangle
$ is continuous.
Third, we deal with basic properties of the probabilistic representation of the right-hand side of (\[311\]).
\[lema61\] Let Hypothesis \[HipN5\] hold. For any $\xi\in L_{C}^{2}( \mathbb{P},\mathfrak{h}) $, we define $$\mathcal{L}_{*} ( \xi,t )
=
\mathbb{E} \vert G X_{t}( \xi) \rangle\langle X_{t}( \xi) \vert
+
\mathbb{E} \vert X_{t}( \xi) \rangle\langle GX_{t}( \xi) \vert
+
\sum_{k=1}^{\infty}\mathbb{E} \vert L_{k} X_{t}( \xi) \rangle
\langle L_{k}X_{t}( \xi) \vert
.$$ Then $\mathcal{L}_{*} ( \xi,t )$ is a trace-class operator on $\mathfrak{h}$ whose trace-norm is uniformly bounded with respect to $t$ on bounded time intervals; the series involved in the definition of $\mathcal{L}_{*}$ converges in $\mathfrak{L} _{1}( \mathfrak{h} )$.
By condition (H2.2), using (\[55\]) and Lemma \[lema51\] we get $$\begin{aligned}
&& \Vert\mathbb{E} \vert G X_{t}( \xi) \rangle\langle X_{t}( \xi)
\vert\Vert_{1}
+
\Vert\mathbb{E} \vert X_{t}( \xi) \rangle\langle GX_{t}( \xi)
\vert\Vert_{1}
+
\sum_{k=1}^{\infty}
\Vert\mathbb{E} \vert L_{k} X_{t}( \xi) \rangle
\langle L_{k}X_{t}( \xi) \vert
\Vert_{1}
\\
&&\qquad \leq
4 \mathbb{E} ( \Vert X_{t}( \xi) \Vert\Vert GX_{t}( \xi) \Vert)
\leq
K \sqrt{ \mathbb{E} \Vert\xi\Vert^{2} }
\sqrt{ \mathbb{E} \Vert X_{t}( \xi) \Vert_{C}^{2} },\end{aligned}$$ where the last inequality follows from $G \in\mathfrak{L}( ( \mathcal{D}( C) ,\Vert\cdot\Vert_{C})
,\mathfrak{h}) $.
Applying Lemmata \[lema51\] and \[lema18\] we easily obtain Lemma \[lema61b\].
\[lema61b\] Adopt the assumptions and notation of Lemma \[lema61\], together with $A \in\mathfrak{L} ( \mathfrak{h} )$. Then, the trace of $ A \mathcal{L}_{*} ( \xi,t ) $ is equal to $$\mathbb{E}\langle X_{t}( \xi) ,AG X_{t}( \xi) \rangle
+
\mathbb{E}\langle GX_{t}( \xi) ,AX_{t}( \xi) \rangle
+
\sum_{k=1}^{\infty}
\mathbb{E}\langle L_{k}X_{t}( \xi) ,AL_{k}X_{t}( \xi) \rangle,$$ and $t \mapsto\operatorname{tr} ( A \mathcal{L}_{*} ( \xi,t ) )$ is continuous as a function from $[0, \infty[$ to $ \mathbb{C}$.
We proceed to prove that $\mathbb{E}\vert X_{t}( \xi) \rangle\langle X_{t}( \xi) \vert$ satisfies an integral version of (\[3\]). To this end, we combine the regularity of $X( \xi)$ with Itô’s formula.
\[lema23\] Adopt Hypothesis \[HipN5\] together with $\xi\in L_{C}^{2}( \mathbb
{P},\mathfrak{h}) $. Then $$\label{136}
\rho_{t}( \mathbb{E}\vert\xi\rangle\langle\xi\vert)
=
\mathbb{E}\vert\xi\rangle\langle\xi\vert
+
\int_{0}^{t} \mathcal{L}_{*} ( \xi,s ) \,ds,$$ where $t\geq0$ and $\mathcal{L}_{*} ( \xi,s )$ is as in Lemma \[lema61\]; we understand the above integral in the sense of Bochner integral in $\mathfrak{L}_{1}( \mathfrak{h}) $.
Our proof is based on arguments given in Section \[subsecexistence\]. Fix $x \in\mathfrak{h}$, and choose $
\tau_{n}=\inf\{ s\geq0\dvtx\Vert X_{s}( \xi) \Vert>n\}
$, with $n \in\mathbb{N}$. Applying the complex Itô formula we obtain that $$\label{135}
\langle X_{t\wedge\tau_{n}}( \xi) ,x\rangle X_{t\wedge\tau_{n}}( \xi)
=
\langle\xi,x\rangle\xi+\mathbb{E}\int_{0}^{t\wedge\tau_{n}} L_{x}
(X_{s}( \xi) ) \,ds
+
M_t,$$ where $
M_t
=
\sum_{k=1}^{\infty}\int_{0}^{t\wedge\tau_{n}}( \langle
X_{s}( \xi) ,x\rangle L_{k}X_{s}( \xi)
+\langle L_{k}X_{s}( \xi) ,x\rangle X_{s}(
\xi) ) \,dW_{s}^{k}
$, and $
L_{x} (z )
=
\langle z ,x\rangle G z
+ \langle Gz ,x\rangle z
+\sum_{k=1}^{\infty}\langle L_{k} z ,x\rangle L_{k}z
$ for any $z \in\mathcal{D}( C )$. According to condition (H2.2) we have $$\begin{aligned}
&&
\mathbb{E}\sum_{k=1}^{\infty}\int_{0}^{t\wedge\tau_{n}}\Vert\langle
X_{s}( \xi) ,x\rangle L_{k}X_{s}(
\xi) +\langle L_{k}X_{s}( \xi) ,x\rangle X_{s}( \xi) \Vert^{2}\,ds
\\
&&\qquad \leq
4 n^{3} \Vert x\Vert^{2}
\mathbb{E} \int_{0}^{t\wedge\tau_{n}} \Vert G X_{s} \Vert \,ds.\end{aligned}$$ Therefore $
\mathbb{E} M_t =0
$ by $G$ belongs to $\mathfrak{L}( ( \mathcal{D}( C) ,\Vert\cdot\Vert
_{C}) ,\mathfrak{h}) $, and so (\[135\]) yields $$\label{131}
\mathbb{E} \langle X_{t\wedge\tau_{n}}( \xi) ,x\rangle X_{t\wedge\tau
_{n}}( \xi)
=
\mathbb{E}\langle\xi,x\rangle\xi+\mathbb{E}\int_{0}^{t\wedge\tau
_{n}} L_{x} (X_{s}( \xi) ) \,ds.$$
We will take the limit as $n \rightarrow\infty$ in (\[131\]). Since $ X ( \xi)$ has continuous sample paths, $\tau_{n}\nearrow_{n\rightarrow\infty}\infty$. By (H2.1) and (H2.2), applying the dominated convergence yields $
\lim_{n \rightarrow\infty} \mathbb{E}\int_{0}^{t\wedge\tau_{n}} L_{x}
(X_{s}( \xi) ) \,ds
=
\mathbb{E}\int_{0}^{t} L_{x} (X_{s}( \xi) ) \,ds.
$ Combinig $
\mathbb{E} ( \sup_{s \in[0, t+1 ]} \Vert X_s( \xi) \Vert^{2} )
< \infty
$ with the dominated convergence theorem gives $
\lim_{n \rightarrow\infty} \mathbb{E} \langle X_{t\wedge\tau_{n}}(
\xi) , x\rangle X_{t\wedge\tau_{n}}( \xi)
=
\mathbb{E} \langle X_{t}( \xi) ,x\rangle X_{t}( \xi).
$ Then, letting first $n \rightarrow\infty$ in (\[131\]) and then using Fubini’s theorem, we get $$\label{63}
\mathbb{E}\langle X_{t}( \xi) ,x\rangle X_{t}( \xi)
=
\mathbb{E}\langle\xi,x\rangle\xi
+
\int_{0}^{t} \mathbb{E} L_{x} (X_{s}( \xi) ).$$
By condition (H2.2), the dominated convergence theorem leads to $$\mathbb{E} \sum_{k=1}^{\infty} \langle L_{k} X_{s}( \xi) ,x\rangle
L_{k}X_{s}( \xi)
=
\sum_{k=1}^{\infty}\mathbb{E} \langle L_{k} X_{s}( \xi) ,x\rangle
L_{k}X_{s}( \xi) ,$$ and so Lemma \[lema51\] yields $
\mathbb{E} L_{x} (X_{s}( \xi) )
=
\mathcal{L}_{*} ( \xi,s ) x
$, hence $$\label{61}
\int_{0}^{t} \mathbb{E} L_{x} (X_{s}( \xi) )
=
\int_{0}^{t} \mathcal{L}_{*} ( \xi,s ) x \,ds .$$
Since the dual of $ \mathfrak{L}_{1}( \mathfrak{h})$ consists in all linear maps $\varrho\mapsto\operatorname{tr} ( A \varrho)$ with $A \in\mathfrak
{L}( \mathfrak{h})$, Lemma \[lema61b\] implies that $
t \mapsto\mathcal{L}_{*} ( \xi,t )
$ is measurable as a function from $[0, \infty[$ to $ \mathfrak{L}_{1}( \mathfrak{h})$. Furthermore, using Lemma \[lema61\] we get that $
t \mapsto\mathcal{L}_{*} ( \xi,t )
$ is a Bochner integrable $\mathfrak{L}_{1}( \mathfrak{h}) $-valued function on bounded intervals. Then (\[63\]), together with (\[61\]), gives (\[136\]).
We are in position to show (\[311\]) and (\[312\]) with the help of Hypothesis \[HipN1\].
[Proof of Theorem \[teor10\]]{} By Theorem \[teorema4\], $
\varrho= \mathbb{E}\vert\xi\rangle\langle\xi\vert
$ for certain $\xi\in L_{C}^{2}( \mathbb{P},\mathfrak{h}) $. Theorem \[teorema8\] now gives $
A G\rho_{t}(\varrho) = \mathbb{E} \vert A GX_{t} ( \xi) \rangle
\langle X_{t}( \xi) \vert
$. Applying Hypothesis \[HipN1\] we get that $G^{\ast}, L_{1}^{\ast}, L_{2}^{\ast}, \ldots$ are densely defined and $G^{\ast\ast}$, $L_{1}^{\ast\ast}, \ldots$ coincide with the closures of $G, L_{1}, \ldots,$ respectively; see, for example, Theorem III.5.29 of [@Kato1995]. Theorem \[teorema8\] yields $
A \rho_{t}(\varrho) G^{\ast} = \mathbb{E} \vert AX_{t}( \xi) \rangle
\langle GX_{t}( \xi) \vert
$ and $
A L_{k} \rho_{t}(\varrho) L_{k}^{\ast}
=
\mathbb{E} \vert A L_{k}X_{t}( \xi) \rangle\langle L_{k}X_{t}( \xi)
\vert
$. Therefore $$\label{64}
\mathcal{L}_{*} ( \xi,t )
=
G\rho_{t}( \varrho) + \rho_{t}( \varrho)G^{\ast}
+\sum_{k=1}^{\infty} L_{k}\rho_{t}( \varrho) L_{k}^{\ast} ,$$ where $\mathcal{L}_{*} ( \xi,t )$ is as in Lemma \[lema61\]. Combining (\[64\]) with Lemma \[lema23\] we get (\[311\]), and so $
\operatorname{tr}( A\rho_{t}( \varrho) )
=
\operatorname{tr}( A \varrho)
+
\int_{0}^{t}
\operatorname{tr}(
A \mathcal{L}_{*} ( \xi,s )
) \,ds
$ for all $t \geq0$. Using the continuity of $\mathcal{L}_{*} ( \xi,\cdot)$ we obtain (\[312\]).
Proof of Theorem 4.5 {#subsecteorema9}
--------------------
We first obtain the existence of a solution of (\[3\]) in the semigroup sense, without Hypothesis \[HipN1\].
\[lema26\] Under Hypothesis \[HipN5\], $ \rho$ is a semigroup $C$-solution of (\[3\]).
By Theorem \[teor8\], $( \rho_{t}) _{t\geq0}$ is a semigroup of bounded operators on $\mathfrak{L}_{1}( \mathfrak{h}) $ that satisfies property (i) of Definition \[defSemigroupSol\]. Fix $\varrho= \vert x\rangle\langle x\vert$, with $x\in\mathcal{D}( C) $. Thus $\varrho$ is a $C$-regular operator, and so (\[310\]) leads to property (ii). Finally, using Lemmata \[lema61b\] and \[lema23\] we get property (iii).
We next make it legitimate to use in our context the duality relation between quantum master equations and adjoint quantum master equations.
\[lema20\] Let Hypothesis \[HipN5\] hold. Suppose that $A \in\mathfrak{L}( \mathfrak{h}) $ and that $( \widehat{\rho}_{t}) _{t\geq0}$ is a semigroup $C$-solution of (\[3\]). Then $( \widehat{\rho}_{t}^{\ast}( A ) ) _{t\geq0}$ is a $C$-solution of (\[41\]) with initial datum $A$, where $( \widehat{\rho}_{t}^{\ast} ) _{t\geq0}$ is the adjoint semigroup of $( \widehat{\rho}_{t}) _{t\geq0}$ (see, e.g., [@Pazy1983]), that is, $( \widehat{\rho}_{t}^{\ast})
_{t\geq0} $ is the unique semigroup of bounded operators on $\mathfrak
{L}( \mathfrak{h}) $ such that for all $B\in\mathfrak{L}( \mathfrak{h}) $ and $\varrho\in\mathfrak
{L}_{1}( \mathfrak{h}) $, $$\label{315}
\operatorname{tr}( \widehat{\rho}_{t}( \varrho) B)
=
\operatorname{tr}( \widehat{\rho}_{t}^{\ast}( B) \varrho) .$$
Using (\[315\]) we get that for all vectors $x,y \in\mathfrak{h}$ whose norm is $1$, $$\begin{aligned}
\vert\langle y,\widehat{\rho}_{t}^{\ast}( A) x\rangle\vert
& = &
\vert{\operatorname{tr}}( \widehat{\rho}_{t}^{\ast}( A) \vert x\rangle
\langle y\vert) \vert =
\operatorname{tr}( \vert\widehat{\rho}_{t}( \vert x\rangle\langle
y\vert) A\vert)\\
&\leq&
\Vert A\Vert\Vert\widehat{\rho}_{t}\Vert_{\mathfrak{L}( \mathfrak
{L}_{1}( \mathfrak{h} ) )}
\operatorname{tr}( \vert\vert x\rangle\langle y\vert\vert).\end{aligned}$$ We conclude from (\[55\]) that $ \operatorname{tr}( \vert\vert x\rangle\langle y\vert\vert) =1$, hence that $
\vert\langle y,\widehat{\rho}_{t}^{\ast}( A) x\rangle\vert
\leq
\Vert A\Vert\times\Vert\widehat{\rho}_{t}\Vert_{\mathfrak{L}( \mathfrak
{L}_{1}( \mathfrak{h}) ) }
$, and finally that $$\label{316}
\Vert\widehat{\rho}_{t}^{\ast}( A) \Vert_{\mathfrak{L}( \mathfrak{h}
) }
\leq\Vert A\Vert\Vert\widehat{\rho}_{t}\Vert_{\mathfrak{L}( \mathfrak
{L}_{1}( \mathfrak{h}) ) }.$$ Applying property (i) of Definition \[defSemigroupSol\] gives property (b) of Definition \[definicion3\].
In order to verify property (a), we will prove the continuity of $t\mapsto\langle x,\break\widehat{\rho}_{t}^{\ast}( A) y \rangle$ for any $x ,y \in\mathfrak{h}$. As in the proof of Lemma \[lema42\], we define $R_{n} = n (n+C )^{-1}$ for $n \in\mathbb{N}$. According to (\[315\]) we have $$\langle R_{n}x,\widehat{\rho}_{t}^{\ast}( A) R_{n} x\rangle
=
\operatorname{tr}( \widehat{\rho}_{t}^{\ast} ( A ) \vert R_{n} x \rangle
\langle R_{n} x\vert)
=
\operatorname{tr}( \widehat{\rho}_{t} ( \vert R_{n} x \rangle\langle
R_{n} x\vert)
A) .$$ Since $ R_{n} x\in\mathcal{D}( C) $, property (ii) of Definition \[defSemigroupSol\] implies the continuity of the function $t\mapsto\langle R_{n} x,\widehat{\rho}_{t}^{\ast}( A) R_{n} x\rangle$. By (\[316\]), $$\begin{aligned}
&&
\vert\langle x,\widehat{\rho}_{t}^{\ast}( A) x\rangle-\langle
x,\widehat{\rho}_{s}^{\ast}( A)
x\rangle\vert\\
&&\qquad\leq
\vert\langle R_{n} x,\widehat{\rho}_{t}^{\ast}( A)
R_{n} x\rangle-\langle R_{n} x,\widehat{\rho}_{s}^{\ast}(
A) R_{n} x\rangle\vert
\\
&&\qquad\quad{}+ 2 \Vert A\Vert\bigl( \Vert\widehat{\rho}_{t}\Vert_{\mathfrak{L}(
\mathfrak{L}_{1}( \mathfrak{h}) ) }
+
\Vert\widehat{\rho}_{s}\Vert_{\mathfrak{L} ( \mathfrak{L}_{1}(
\mathfrak{h}) ) }\bigr)
\Vert x\Vert\Vert x - R_{n} x\Vert.\end{aligned}$$ Using $R_{n} x \longrightarrow_{n \rightarrow\infty} x$ we deduce that the map $t\mapsto\langle x,\widehat{\rho}_{t}^{\ast}( A) x\rangle$ is continuous, so is $t\mapsto\langle x,\widehat{\rho}_{t}^{\ast}( A) y \rangle$ by the polarization identity.
Assume that $x \in\mathcal{D}( C )$. By (\[315\]), combining $
\widehat{\rho}_{t+s}^{\ast}( A)
=
\widehat{\rho}_{s}^{\ast}( \widehat{\rho}_{t}^{\ast}( A))
$ with property (iii) of Definition \[defSemigroupSol\] yields $$\begin{aligned}
&& \lim_{s\rightarrow0+}\frac{1}{s}\bigl( \langle x,\widehat{\rho
}_{t+s}^{\ast}( A) x\rangle-\langle x,\widehat{\rho}_{t}^{\ast}( A)
x\rangle\bigr) \\[-2pt]
&&\qquad =
\lim_{s\rightarrow0+}\frac{1}{s}\bigl( \operatorname{tr} ( \widehat{\rho
}_{s}( \vert x\rangle\langle x\vert) \widehat{\rho}_{t}^{\ast}( A) )
-\operatorname{tr}( \vert x\rangle\langle x\vert\widehat{\rho
}_{t}^{\ast}( A) ) \bigr)\\[-2pt]
&&\qquad=
\mathcal{L} ( \widehat{\rho}_{t}^{\ast}( A),x )\end{aligned}$$ with $
\mathcal{L} ( \widehat{\rho}_{t}^{\ast}( A),x )
=
\langle x,\widehat{\rho}_{t}^{\ast}( A)
Gx\rangle+\langle Gx,\widehat{\rho}_{t}^{\ast}( A)
x\rangle+\sum_{k=1}^{\infty}\langle L_{k}x,\widehat{\rho}_{t}^{\ast}( A) L_{k}x\rangle
$. Thus $$\label{325}
\frac{d}{dt}^{+}\langle x,\widehat{\rho}_{t}^{\ast}( A)
x\rangle
=
\mathcal{L} ( \widehat{\rho}_{t}^{\ast}( A),x ).$$ From (\[316\]) and condition (H2.2) we get that $\sum_{k=1}^{\infty}\langle
L_{k}x,\widehat{\rho}_{t}^{\ast}( A) L_{k}x\rangle$ is uniformly convergent on bounded intervals, and so $
t\mapsto\sum_{k=1}^{\infty}\langle L_{k}x,\widehat{\rho}_{t}^{\ast}(
A) L_{k}x\rangle
$ is continuous, and hence the application $t\mapsto\frac{d}{dt}^{+}\langle x,\widehat{\rho}_{t}^{\ast}( A)
x\rangle$ is continuous. Therefore $\langle x,\widehat{\rho}_{t}^{\ast}( A) x\rangle$ is continuously differentiable (see, e.g., Section 2.1 of [@Pazy1983]). Property (a) of Definition \[definicion3\] now follows from (\[325\]).
We are in position to show our second main theorem.
[Proof of Theorem \[teorema9\]]{} Let $( \widehat{\rho}_{t})_{t\geq0}$ be a semigroup $C$-solution of (\[3\]). Consider the adjoint semigroup $ (
\widehat{\rho}_{t}^{\ast} ) _{t\geq0}$ of $( \widehat{\rho}_{t})
_{t\geq0}$, and let $( \mathcal{T}_{t}( A) ) _{t\geq0}$ be given by Theorem \[teorema3\]. Combining Lemma \[lema20\] with Theorem \[teorema3\] we obtain $\widehat{\rho}_{t}^{\ast}( A) =
\mathcal{T}_{t}( A) $ for all $t\geq0$ and $A\in\mathfrak{L}(
\mathfrak{h}) $. If $\varrho\in\mathfrak{L}_{1,C}^{+}( \mathfrak{h}) $ and $A\in\mathfrak{L}( \mathfrak{h}) $, then applying (\[315\]) and Lemma \[lema10\] yields $$\operatorname{tr}( \rho_{t}( \varrho) A)
= \operatorname{tr}( \mathcal{T}_{t}( A) \varrho)
= \operatorname{tr}( \widehat{\rho}_{t}^{\ast}( A) \varrho)
= \operatorname{tr}( \widehat{\rho}_{t}( \varrho) A)$$ and so $\rho_{t}( \varrho)
= \widehat{\rho}_{t}( \varrho) $. Lemma \[lema12\] now implies that $\rho_{t}( \varrho)
= \widehat{\rho}_{t}( \varrho) $ for all $\varrho$ belonging to $\mathfrak{L}_{1}^{+}( \mathfrak{h}) $, hence $\rho_{t} = \widehat{\rho}_{t}$. Finally, Lemma \[lema26\] completes the proof.
Proof of Theorem 4.6 {#subsecteorema5}
--------------------
From [@FagnolaMora2010] we have that Hypothesis \[HipN5\] holds with $C = P^2 + Q^2$. Hence Theorem \[teorema9\] yields our first assertion.
Suppose that $A=P$ or $A=Q$. Using, for instance, the spectral theorem, we deduce the existence of a sequence $A_n$ of bounded self-adjoint operators in $\mathfrak{h}$ such that for all $f \in\mathcal{D} ( A )$ we have $\| A_n f \| \leq\| A f \|$ and $A_n f \longrightarrow_{n \rightarrow\infty} Af$. Applying Theorems \[teorema8\] and \[teor10\] (or better Lemmata \[lema61b\] and \[lema23\]) gives $$\begin{aligned}
\label{32}\quad
&&\operatorname{tr}( A_n \rho_{t}( \varrho) )\nonumber\\
&&\qquad=
\operatorname{tr}( A_n \varrho)
+
\int_{0}^{t}
\Biggl(
\mathbb{E}\langle A_n X_{t}( \xi) , G X_{t}( \xi) \rangle
+
\mathbb{E}\langle GX_{t}( \xi) ,A_nX_{t}( \xi) \rangle\\
&&\qquad\quad\hspace*{65.3pt}\hspace*{52.4pt}{}
+
\sum_{k=1}^{\infty}
\mathbb{E}\langle L_{k}X_{t}( \xi) ,A_nL_{k}X_{t}( \xi) \rangle
\,ds
\Biggr),\nonumber\end{aligned}$$ where $
\varrho= \mathbb{E} \vert\xi\rangle\langle\xi\vert
$ with $
\mathbb{E} ( \Vert C \xi\Vert^{2} + \Vert\xi\Vert^{2} )
<\infty
$. By the dominated convergence theorem, letting $n \rightarrow\infty$ we obtain $$\begin{aligned}
\label{32n}\quad
&&\operatorname{tr}( A \rho_{t}( \varrho) )\nonumber\\
&&\qquad=
\operatorname{tr}( A \varrho)
+
\int_{0}^{t}
\Biggl(
\mathbb{E}\langle A X_{t}( \xi) , G X_{t}( \xi) \rangle
+
\mathbb{E}\langle GX_{t}( \xi) ,A X_{t}( \xi) \rangle
\\
&&\hspace*{140pt}{}+
\sum_{k=1}^{\infty}
\mathbb{E}\langle L_{k}X_{t}( \xi) ,A L_{k}X_{t}( \xi) \rangle
\,ds
\Biggr).\nonumber\end{aligned}$$
Let $f \in C_{c}^{\infty} ( \mathbb{R}, \mathbb{C} )$. Using $
[ P, Q ] = -i I
$ we get that $ \mathcal{L} ( P ) f = i [ H, P ] f $ and $ \mathcal{L} ( Q ) f = i [ H, Q ] f $. Therefore $$\begin{aligned}
\label{34}
&&\langle A f, Gf \rangle
+
\langle Gf, Af \rangle
+\sum_{k=1}^{\infty}\langle L_{k}f, A L_{k}f
\rangle\nonumber\\[-8pt]\\[-8pt]
&&\qquad=
\cases{
\langle f, P f \rangle/ m ,
&\quad
if $A=Q$,\cr
-2c \langle f, Q f \rangle, &\quad
if $A=P$.}\nonumber\end{aligned}$$ Since $C_{c}^{\infty} ( \mathbb{R}, \mathbb{C} )$ is a core for $C = P^2 + Q^2$, combining a limit procedure with, for instance, Lemma 12 of [@FagnolaMora2010] we get that (\[34\]) holds for all $f \in\mathcal{D} ( C )$. Then, (\[32n\]) leads to (\[31n\]).
Proof of Theorem 5.1 {#subseccorolario2}
--------------------
Let $\xi$ be distributed according to $\theta$. Set $\widetilde{\mathbb{Q}} = \Vert X_{T}( \xi) \Vert^{2}\cdot\mathbb{P}$, with $T > 0$. For any $t \in[ 0, T ]$, we choose $ \widetilde{Y}_{t} = X_{t}( \xi) /\Vert X_{t}( \xi) \Vert$ if $X_{t}( \xi) \neq0$ and $ \widetilde{Y}_{t} = 0 $ otherwise; let $
B_{t}^{k}=W_{t}^{k}-\int_{0}^{t}\frac{1}{\Vert X_{s}(
\xi) \Vert^{2}}\,d[ W^{k}, X( \xi) ]_{s}
$ for any $k\in\mathbb{N}$. Proceeding along the same lines as in the proof of Proposition 1 of [@MoraReAAP2008] we obtain that $
( \mathbb{Q},( Y_{t}) _{t\in[ 0,T] },( B_{t}^{k}) _{t\in[ 0,T] }^{k\in
\mathbb{N}})
$ is a $C$-solution of (\[5\]) with initial law $\theta$. By Remark \[nota6\], (\[5\]) has a unique $C$-solution with initial distribution $\theta$. Therefore the distribution of $\widetilde{Y}_{t}$ with respect to $\widetilde{\mathbb{Q}}$ coincides with the distribution of $Y_{t}$ under $\mathbb{Q}$. From [@FagnolaMora2010] we have that $( \Vert X_{t}\Vert^{2})_{t\in[ 0,T] }$ is a martingale, and hence for any $x\in\mathfrak{h}$ and $t \in[ 0, T ]$, $$\mathbb{E}_{\mathbb{Q}}\vert\langle x,Y_{t}\rangle\vert^{2}
=
\mathbb{E}_{\widetilde{ \mathbb{Q}}}\vert\langle x,\widetilde
{Y}_{t}\rangle\vert^{2}
=
\mathbb{E}_{\mathbb{P}}( \vert\langle x,\widetilde{Y}_{t}\rangle\vert
^{2}\Vert X_{t}( \xi)
\Vert^{2})
=
\mathbb{E}_{\mathbb{P}}\vert\langle x,X_{t}( \xi) \rangle\vert^{2}.$$ Applying (\[31\]) and the polarization identity gives $\rho_{t}( \varrho) =\mathbb{E} \vert Y_{t}\rangle\langle
Y_{t}\vert$.
Proof of Theorem 5.2 {#subsecteorema7}
--------------------
Let $( \mathbb{Q},( Y_{t}) _{t\geq0},(
B_{t})_{t\geq0}) $ be the $C$-solution of (\[5\]) with initial distribution $\Gamma$; see Remark \[nota6\]. Choose $\varrho_{\infty}=\mathbb{E}\vert Y_{0}\rangle\langle
Y_{0}\vert$. Then, Theorem \[corolario2\] shows that $
\rho_{t}( \varrho_{\infty})
=
\mathbb{E}\vert Y_{t}\rangle\langle Y_{t}\vert
$ for all $t\geq0$.
As in the proof of Theorem 3 of [@MoraReAAP2008], applying techniques of well-posed martigale problems we obtain the Markov property of the $C$-solutions of (\[5\]) under Hypothesis \[HipN5\]. Hence for any $x\in\mathfrak{h}$ and $t\geq0$ $$\mathbb{E}\bigl( \mathbf{1}_{[ 0,\Vert x\Vert^{2}]
}( \vert\langle x,Y_{t}\rangle\vert^{2})\bigr)=
\mathbb{E}\biggl( \int_{\mathfrak{h}}\mathbf{1}_{[ 0,\Vert
x\Vert^{2}] }( \vert\langle x,y\rangle
\vert^{2}) P_{t}( Y_{0},dy) \biggr).\vadjust{\goodbreak}$$ On the other hand, using (\[IM1\]) we deduce that $$\mathbb{E}\bigl( \mathbf{1}_{[ 0,\Vert x\Vert^{2}]
}( \vert\langle x,Y_{0}\rangle\vert^{2})
\bigr)
=
\int_{\mathfrak{h}}\biggl( \int_{\mathfrak{h}}\mathbf{1}_{[
0,\Vert x\Vert^{2}] }( \vert\langle
x,y\rangle\vert^{2}) P_{t}( z,dy) \biggr)
\Gamma( dz) .$$ Now, combining $ \Vert Y_{t} \Vert= 1 $ with $ \int_{\mathfrak{h}}(
\int_{\mathfrak{h}}\mathbf{1}_{[ 0,\Vert x\Vert^{2}] }( \vert\langle
x,y\rangle\vert^{2}) P_{t}( z,dy) ) \Gamma( dz) =\break \mathbb{E}(
\int_{\mathfrak{h}}\mathbf{1}_{[ 0,\Vert x\Vert^{2}] }( \vert\langle
x,y\rangle \vert^{2}) P_{t}( Y_{0},dy) ) $ we get $
\mathbb{E}\vert\langle x,Y_{0}\rangle\vert ^{2}=\mathbb{E}\vert\langle
x,Y_{t}\rangle\vert ^{2} $. This gives $ \mathbb{E}\vert
Y_{t}\rangle\langle Y_{t}\vert = \mathbb{E}\vert Y_{0}\rangle\langle
Y_{0}\vert $ and so $\rho_{t}( \varrho _{\infty}) =\varrho_{\infty}$.
Proof of Theorem 6.1 {#subsecteor14}
--------------------
Since $\mathcal{D}( G ) = \mathcal{D}( N^{4} )$, from Remark \[nota3\] we have that $G$ is a closable operator satisfying $G
\in\mathfrak{L}( ( \mathcal{D}( N^{p}), \mbox{$\Vert\cdot\Vert_{ N^{p}}$})
,\mathfrak{h})$. Fix $x \in\mathfrak{h}$ such that $x_{n} := \langle
e_{n}, x \rangle$ is equal to $0$ for all $n \in\mathbb{Z}_{+}$ except a finite number. An easy computation shows that $ 2\Re\langle N^{2p} x,
Gx\rangle+\sum_{k=1}^{\infty} \Vert N^{p} L_{k}x \Vert^{2} $ is equal to the sum of $ 4 p ( \vert\alpha_{5} \vert^{2} - \vert\alpha_{4}
\vert^{2} ) \sum _{n = 0}^{\infty} n^{2p+1} \vert x_{n} \vert^{2} $ and $$2 \beta_{1} \sum_{n=1}^{\infty} \sqrt{n+1} \bigl( ( n+1 )^{2p} - n^{2p} \bigr)
\Re( x_{n} \overline{ x_{n+1}} )
+
\sum_{n = 0}^{\infty} f(n ) \vert x_{n} \vert^{2} ,$$ where $f$ is a $2p$-degree polynomial whose coefficients depend on $
\vert\alpha_{k} \vert^{2} $ with $k=1,2,4,5$. Hence $N^{p}$ satisfies Hypothesis \[HipN4\] whenever $\vert\alpha_{4}\vert\geq\vert\alpha_{5}\vert$. From [@FagnolaMora2010] it follows that $N^{p}$ fulfills condition (H2.3) of Hypothesis \[HipN5\], and so Theorems \[teor10\] and \[teorema9\] lead to statement (i).
From Theorem 8 of [@MoraReAAP2008] we have the existence of an invariant probability measure $\Gamma$ for (\[5\]) that satisfies the properties given in Hypothesis \[Hip2\] with $C=N^p$. Using Theorem \[teorema7\] yields statement (ii).
Acknowledgments {#acknowledgments .unnumbered}
===============
The author wishes to express his gratitude to the anonymous referees, whose suggestions and constructive criticisms led to substantial improvements in the presentation. Moreover, I thank Franco Fagnola and Roberto Quezada for helpful comments.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We classify Lagrangian subcategories of the representation category of a twisted quantum double $D^\omega(G)$, where $G$ is a finite group and $\omega$ is a $3$-cocycle on it. In view of results of [@DGNO] this gives a complete description of all braided tensor equivalent pairs of twisted quantum doubles of finite groups. We also establish a canonical bijection between Lagrangian subcategories of ${\mbox{Rep}}(D^\omega(G))$ and module categories over the category ${\mbox{Vec}}_G^\omega$ of twisted $G$-graded vector spaces such that the dual tensor category is pointed. This can be viewed as a quantum version of V. Drinfeld’s characterization of homogeneous spaces of a Poisson-Lie group in terms of Lagrangian subalgebras of the double of its Lie bialgebra [@D]. As a consequence, we obtain that two group-theoretical fusion categories are weakly Morita equivalent if and only if their centers are equivalent as braided tensor categories.'
address:
- 'Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, USA'
- 'Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, USA'
author:
- Deepak Naidu
- Dmitri Nikshych
date: 'June 28, 2007'
title: Lagrangian subcategories and braided tensor equivalences of twisted quantum doubles of finite groups
---
[Introduction]{} Throughout this paper we will work over an algebraically closed field $k$ of characteristic zero. Unless otherwise stated all cocycles appearing in this work will have coefficients in the trivial module $k^\times$. All categories considered in this work are assumed to be $k$-linear and semisimple with finite-dimensional ${\mbox{Hom}}$-spaces and finitely many isomorphism classes of simple objects. All functors are assumed to be additive and $k$-linear.
Let $G$ be a finite group and $\omega$ be a $3$-cocycle on $G$. In [@DPR1; @DPR2] R. Dijkgraaf, V. Pasquier, and P. Roche introduced a quasi-triangular quasi-Hopf algebra $D^\omega(G)$. When $\omega =1$ this quasi-Hopf algebra coincides with the Drinfeld double $D(G)$ of $G$ and so $D^\omega(G)$ is often called a [*twisted quantum double*]{} of $G$. It is well known that the representation category ${\mbox{Rep}}(D^\omega(G))$ of $D^\omega(G)$ is a modular category [@BK; @T] and is braided equivalent to the center [@K] of the tensor category ${\mbox{Vec}}_G^\omega$ of finite-dimensional $G$-graded vector spaces with the associativity constraint defined using $\omega$. The category ${\mbox{Vec}}_G^\omega$ is a typical example of a [*pointed*]{} fusion category, i.e., a finite semisimple tensor category in which every simple object is invertible.
In [@DGNO] a criterion for a modular category ${\mathcal{C}}$ to be braided tensor equivalent to the center of a category of the form ${\mbox{Vec}}_G^\omega$ for some finite group $G$ and $\omega\in Z^3(G,\, k^\times)$ is given. Namely, such a braided equivalence exists if and only if ${\mathcal{C}}$ contains a Lagrangian subcategory, i.e., a maximal isotropic subcategory of dimension $\sqrt{{\mbox{dim}}({\mathcal{C}})}$. More precisely, Lagrangian subcategories of ${\mathcal{C}}$ parameterize the classes of braided equivalences between ${\mathcal{C}}$ and centers of pointed categories, see [@DGNO Section 4]. Note that any Lagrangian subcategory of ${\mathcal{C}}$ is equivalent (as a symmetric tensor category) to the representation category of some group by the result of P. Deligne [@De].
This means that a description of Lagrangian subcategories of ${\mbox{Rep}}(D^\omega(G))$ for all groups $G$ and $3$-cocycles $\omega$ is equivalent to a description of all braided equivalences between representation categories of twisted group doubles. Such equivalences for elementary Abelian and extra special groups were studied in [@MN] and [@GMN]. A motivation for such study comes from a relation between holomorphic orbifolds in the Rational Conformal Field Theory and twisted group doubles observed in [@DVVV], [@DPR2].
A complete classification of Lagrangian subcategories of ${\mbox{Rep}}(D^\omega(G))$ is the principal goal of this paper.
Main results
------------
Let $G$ be a finite group and let $\omega \in Z^3(G,\, k^\times)$ be a $3$-cocycle on $G$.
\[thm 1\] Lagrangian subcategories of the representation category of the Drinfeld double $D(G)$ are classified by pairs $(H, B)$, where $H$ is a normal Abelian subgroup of $G$ and $B$ is an alternating $G$-invariant bicharacter on $H$.
The proof is based on the analysis of modular data (i.e., the $S$- and $T$-matrices) associated to $D(G)$.
Theorem \[thm 1\] gives a simple classification of Lagrangian subcategories for the untwisted double $D(G)$. In the twisted ($\omega \neq 1$) case the notion of a $G$-invariant bicharacter needs to be twisted as well, cf. Definition \[alt omega def\].
\[thm 2\] Lagrangian subcategories of the representation category of the twisted double $D^\omega(G)$ are classified by pairs $(H, B)$, where $H$ is a normal Abelian subgroup of $G$ such that $\omega|_{H\times H\times H}$ is cohomologically trivial and $B: H\times H \to k^\times$ is a $G$-invariant alternating $\omega$-bicharacter in the sense of Definition \[alt omega def\].
Note that bicharacters in the statement of Theorem \[thm 2\] are in bijection with equivalence classes of $G$-invariant cochains $\mu \in C^2(H,\, k^\times)$ such that $\delta^2 \mu =
\omega|_{H\times H\times H}$, see .
Let ${\mathcal{L}}_{(H, B)}$ denote the Lagrangian subcategory of ${\mbox{Rep}}(D^\omega(G))$ corresponding to a pair $(H, B)$ in Theorems \[thm 1\] and \[thm 2\]. Then there is a group $G'$, defined up to an isomorphism, such that ${\mathcal{L}}_{(H, B)}$ is equivalent to ${\mbox{Rep}}(G')$ as a symmetric tensor category, where the braiding of ${\mbox{Rep}}(G')$ is the trivial one [@De]. The group $G'$ can be described explicitly in terms of $G$, $H$, and $B$, see Remark \[G’\]. Note that $G \not\cong G'$ in general.
There is a canonical subcategory ${\mathcal{L}}_{(\{1\},\, 1)} \cong {\mbox{Rep}}(G)$ corresponding to the forgetful functor ${\mbox{Rep}}(D^\omega(G)) \cong {\mathcal{Z}}({\mbox{Vec}}_G^\omega) \to {\mbox{Vec}}_G^\omega$. We have ${\mathcal{L}}_{(\{1\},\, 1)} \cap {\mathcal{L}}_{(H, B)} \cong {\mbox{Rep}}(G/H)$.
In [@N] the first named author classified indecomposable ${\mbox{Vec}}_G^\omega$-module categories ${\mathcal{M}}$ with the property that the dual fusion category $({\mbox{Vec}}_G^\omega)^*_{\mathcal{M}}$ is pointed. Such module categories ${\mathcal{M}}$ can be thought of as categorical analogues of homogeneous spaces. The above property gives rise to an equivalence relation on the set of pairs $(G,\, \omega)$, where $G$ is a finite group and $\omega \in Z^3(G,\, k^\times)$ with $$\label{Deepak's equivalence}
(G,\, \omega) \approx (G',\, \omega') \mbox{ if and only if }
({\mbox{Vec}}_G^\omega)^*_{\mathcal{M}}\cong {\mbox{Vec}}_{G'}^{\omega'} \mbox{ for some } {\mathcal{M}}.$$ In other words, ${\mbox{Vec}}_G^\omega$ and ${\mbox{Vec}}_{G'}^{\omega'}$ are [*weakly Morita equivalent*]{} in the sense of M. Müger [@M].
\[thm 3\] There is a canonical bijection between equivalence classes of indecomposable ${\mbox{Vec}}_G^\omega$-module categories ${\mathcal{M}}$ with respect to which the dual fusion category $({\mbox{Vec}}_G^\omega)^*_{\mathcal{M}}$ is pointed and Lagrangian subcategories of ${\mbox{Rep}}(D^\omega(G))$.
Lagrangian subalgebras of the double of a Lie bialgebra $\mathfrak{g}$ were used by V. Drinfeld in [@D] to describe Poisson homogeneous spaces of the Poisson-Lie group $G$ corresponding to $\mathfrak{g}$. So Theorem \[thm 3\] can perhaps be understood as a quantum version of the correspondence in [@D]. Note that quantization of Poisson homogeneous spaces was studied in [@EK].
Recall that a fusion category is called [*group-theoretical*]{} if it has a pointed dual.
\[thm 4\] Let ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ be group-theoretical fusion categories. Then ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ are weakly Morita equivalent if and only if their centers ${\mathcal{Z}}({\mathcal{C}}_1)$ and ${\mathcal{Z}}({\mathcal{C}}_2)$ are braided equivalent.
If ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ are arbitrary (i.e., not necessarily group-theoretical) weakly Morita equivalent finite tensor categories then it was observed by M. Müger in [@M Remark 3.18] that ${\mathcal{Z}}({\mathcal{C}}_1)$ is braided tensor equivalent to ${\mathcal{Z}}({\mathcal{C}}_2)$. Also, ${\mathcal{Z}}({\mathcal{C}}_1)$ being equivalent to ${\mathcal{Z}}({\mathcal{C}}_2)$ (even in a non-braided way) implies that ${\mathcal{C}}_1\boxtimes {\mathcal{C}}_1^{\text{rev}}$ is weakly Morita equivalent to ${\mathcal{C}}_2\boxtimes {\mathcal{C}}_2^{\text{rev}}$, where ${\mathcal{C}}^{\text{rev}}$ denotes the fusion category obtained by reversing the tensor product in ${\mathcal{C}}$. At the moment of writing we do not know if braided equivalence of centers implies weak Morita equivalence of ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ in a general case.
Note that it was shown by S. Natale [@Nat] that a fusion category ${\mathcal{C}}$ is group-theoretical if and only if ${\mathcal{Z}}({\mathcal{C}})$ is braided equivalent to the representation category of some twisted group double $D^\omega(G)$ (this also follows from [@O2]).
Combining the explicit description of weak Morita equivalence classes of pointed categories from [@N] and correspondence between braided equivalences of centers and Lagrangian subcategories from [@DGNO] one obtains a complete description of braided equivalences of twisted quantum doubles.
Recall that two finite groups $G_1$ and $G_2$ were called [*categorically Morita equivalent*]{} in [@N] if ${\mbox{Vec}}_{G_1}$ and ${\mbox{Vec}}_{G_2}$ are weakly Morita equivalent. Let us write $G_1 \approx G_2$ for such groups. It follows from Theorem \[thm 4\] that $G_1 \approx G_2$ if and only if the corresponding Drinfeld doubles $D(G_1)$ and $D(G_2)$ have braided tensor equivalent representation categories. By Theorems \[thm 1\] and \[thm 3\] groups categorically Morita equivalent to a given group $G$ correspond to pairs $(H, B)$, where $H$ is a normal Abelian subgroup of $G$ and $B$ is an alternating bicharacter on $H$. Note that such pairs $(H, B)$ for which $B$ is, in addition, a [*nondegenerate*]{} bicharacter were used by P. Etingof and S. Gelaki in [@EG] to describe groups [*isocategorical*]{} to $G$, i.e., such groups $G'$ for which ${\mbox{Rep}}(G') \cong {\mbox{Rep}}(G)$ as tensor categories. This non-degeneracy condition in [@EG] is the reason why the categorical Morita equivalence extends isocategorical equivalence. Indeed, since ${\mbox{Rep}}(D(G))$ is determined by the tensor structure of ${\mbox{Rep}}(G)$, it is clear that isocategorical groups are categorically Morita equivalent. On the other hand, the first author constructed in [@N] examples of categorically Morita equivalent but non-isocategorical groups.
For an Abelian group $H$ let $\widehat{H}$ denote the group of linear characters of $H$.
\[thm 5\] Let $G, G'$ be finite groups, $\omega \in Z^3(G,\, k^\times)$, and $\omega' \in Z^3(G',\, k^\times)$. Then the representation categories of twisted doubles $D^\omega(G)$ and $D^{\omega'}(G')$ are equivalent as braided tensor categories if and only if $G$ contains a normal Abelian subgroup $H$ such the following conditions are satisfied:
1. $\omega|_{H \times H \times H}$ is cohomologically trivial,
2. there is a $G$-invariant $($see $)$ $2$-cochain $\mu \in C^2(H, \, k^\times)$ such that that $\delta^2 \mu = \omega|_{H \times H \times H}$, and
3. there is an isomorphism $a: G' \xrightarrow{\sim} \widehat{H} \rtimes_{\nu}
(H \backslash G)$ such that $\varpi \circ (a \times a \times a)$ and $\omega'$ are cohomologically equivalent.
Here $\nu$ is a certain $2$-cocycle in $Z^2(H \backslash G, \, \widehat{H})$ that comes from the $G$-invariance of $\mu$ and $\varpi$ is a certain $3$-cocycle on $\widehat{H} \rtimes_{\nu}
(H \backslash G)$ that depends on $\nu$ and on the exact sequence $1 \to H \to G \to H \backslash G \to 1$ $($see [@N Theorem 5.8] for precise definitions$)$.
Note that in the special case when $\omega =1$ and $\mu$ is a non-degenerate $G$-invariant alternating $2$-cocycle on $H$, our construction of the “dual” group $G'$ in Corollary \[thm 5\] becomes the construction of a group isocategorical to $G$ from [@EG]. This can be seen by comparing [@N 4.2] and [@EG Formula (2)].
Organization of the paper
-------------------------
Section 2 contains necessary preliminary information about fusion categories, module categories, and modular categories. We also recall definitions and results from [@DGNO] concerning Lagrangian subcategories of modular categories.
Section 3 (respectively, Section 4) is devoted to classification of Lagrangian categories of the representation category of the Drinfeld double (respectively, twisted double) of a finite group. The reason we prefer to treat untwisted and twisted cases separately is because our constructions in the former case do not involve rather technical cohomological computations present in the latter. We feel that the reader might get a better understanding of our results by exploring the untwisted case first. Of course when $\omega =1$ the results of Section 4 reduce to those of Section 3.
The Sections 3 and 4 contain proofs of our main results stated above. Theorems \[thm 1\] - \[thm 4\] and Corollary \[thm 5\] correspond to Theorems \[untwisted bijection\], \[bij 1\], \[bijection 1\], \[main 1\], and Corollary \[main 2\].
Section 5 contains examples in which we compute Lagrangian subcategories of Drinfeld doubles of finite symmetry groups. Here we also show that the four non-equivalent non-pointed fusion categories of dimension $8$ with integral dimensions of objects are pairwise weakly Morita non-equivalent, and hence their centers are pairwise non-equivalent as braided tensor categories.
Acknowledgments
---------------
The present paper would not be possible without [@DGNO] and the second named author is happy to thank his collaborators, Vladimir Drinfeld, Shlomo Gelaki, and Victor Ostrik for many useful discussions. The authors thank Nicolas Andruskiewitsch, Pavel Etingof, and Leonid Vainerman for helpful comments. The second author thanks the Université de Caen for hospitality and excellent working conditions during his visit. The authors were supported by the NSF grant DMS-0200202. The research of Dmitri Nikshych was supported by the NSA grant H98230-07-1-0081.
[Preliminaries]{}
[Fusion categories and their module categories]{} A [*fusion category*]{} over $k$ is a $k$-linear semisimple rigid tensor category with finitely many isomorphism classes of simple objects, finite-dimensional Hom-spaces, and simple neutral object.
In this paper we only consider fusion categories with integral Frobenius-Perron dimensions of simple objects. It was shown in [@ENO Propositions 8.23, 8.24] that any such category is equivalent to the representation category of a semisimple quasi-Hopf algebra and has a canonical spherical structure with respect to which the categorical dimension of any object is equal to its Frobenius-Perron dimension. In particular, all categorical dimensions are positive integers. For any object $X$ of a fusion category ${\mathcal{C}}$ let $d(X)$ denote its (Frobenius-Perron) dimension.
By a fusion subcategory of a fusion category we will always mean a full fusion subcategory.
A fusion category is said to be [*pointed*]{} if all its simple objects are invertible. A typical example of a pointed category is ${\mbox{Vec}}_G^{\omega}$ - the category of finite-dimensional vector spaces over $k$ graded by the finite group $G$. The morphisms in this category are linear transformations that respect the grading and the associativity constraint is given by a $3$-cocycle $\omega$ on $G$.
Let ${\mathcal{C}}= ({\mathcal{C}}, \, \otimes, \, 1_{{\mathcal{C}}}, \, \alpha, \, \lambda, \, \rho)$ be a tensor category, where $1_{{\mathcal{C}}}$, $\alpha$, $\lambda$, and $\rho$ are the unit object, the associativity constraint, the left unit constraint, and the right unit constraint, respectively. A right [*module category*]{} over ${\mathcal{C}}$ is a category ${\mathcal{M}}$ together with an exact bifunctor $\otimes: {\mathcal{M}}\times {\mathcal{C}}\to
{\mathcal{M}}$ and natural isomorphisms $\mu_{M, \, X, \, Y}: M \otimes (X \otimes Y) \to (M \otimes X) \otimes Y, \,\,
\tau_M: M \otimes 1_{{\mathcal{C}}} \to M$, for all $M \in {\mathcal{M}}, \, X, Y \in {\mathcal{C}}$ such that the following two equations hold for all $M \in {\mathcal{M}}, \, X, Y, Z \in {\mathcal{C}}$: $$\label{module pentagon}
\mu_{M \otimes X, \, Y, \, Z} \, \circ
\, \mu_{M, \, X, \, Y \otimes Z} \, \circ
\, ({\text{id}}_M \otimes \alpha_{X,Y,Z})
= (\mu_{M, \, X, \, Y}\otimes {\text{id}}_Z) \, \circ
\, \mu_{M, \, X \otimes Y, \, Z},$$ $$\label{module triangle}
(\tau_M \otimes {\text{id}}_Y) \, \circ
\, \mu_{M, \, 1_{\mathcal{C}}, \, Y}
= {\text{id}}_M \otimes \lambda_Y.$$
Note that having a ${\mathcal{C}}$-module structure on a category ${\mathcal{M}}$ is the same as having a tensor functor from ${\mathcal{C}}$ to the (strict) tensor category of endofunctors of ${\mathcal{M}}$. The coherence conditions on a module action follow automatically from those of a tensor functor.
Let $({\mathcal{M}}_1, \, \mu^1, \tau^1)$ and $({\mathcal{M}}_2, \, \mu^2, \tau^2)$ be two right module categories over ${\mathcal{C}}$. A ${\mathcal{C}}$-[*module functor*]{} from ${\mathcal{M}}_1$ to ${\mathcal{M}}_2$ is a functor $F: {\mathcal{M}}_1\to {\mathcal{M}}_2$ together with natural isomorphisms $\gamma_{M, \, X}: F(M \otimes X) \to F(M) \otimes X$, for all $M \in {\mathcal{M}}_1, \, X \in {\mathcal{C}}$ such that the following two equations hold for all $M \in {\mathcal{M}}_1, \, X, Y \in {\mathcal{C}}$: $$\label{module functor pentagon}
(\gamma_{M, \, X} \otimes {\text{id}}_Y) \, \circ
\, \gamma_{M\otimes X, \, Y} \, \circ
\, F(\mu^1_{M, \, X, \, Y})
= \mu^2_{F(M), \, X, \, Y} \, \circ
\, \gamma_{M, X \otimes Y},$$ $$\label{module functor triangle}
\tau^1_{F(M)} \, \circ \, \gamma_{M, \, 1_{\mathcal{C}}} = F(\tau^1_M).$$ Two module categories ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ over ${\mathcal{C}}$ are [*equivalent*]{} if there exists a module functor from ${\mathcal{M}}_1$ to ${\mathcal{M}}_2$ which is an equivalence of categories. For two module categories ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ over a tensor category ${\mathcal{C}}$ their [*direct sum*]{} is the category ${\mathcal{M}}_1 \oplus {\mathcal{M}}_2$ with the obvious module category structure. A module category is [*indecomposable*]{} if it is not equivalent to a direct sum of two non-trivial module categories.
[**Indecomposable module categories over pointed categories**]{}. Let $G$ be a finite group and $\omega \in Z^3(G, \, k^\times)$. Indecomposable right module categories over ${\mbox{Vec}}_G^{\omega}$ correspond to pairs $(H, \, \mu)$, where $H$ is a subgroup of $G$ such that $\omega|_{H \times H \times H}$ is cohomologically trivial and $\mu \in C^2(H, \, k^\times)$ is a $2$-cochain satisfying $\delta^2\mu = \omega|_{H \times H \times H}$, i.e., $$\label{mu omega}
\mu(h_2, \, h_3)
\mu(h_1h_2, \, h_3)^{-1} \mu(h_1, \, h_2h_3)
\mu(h_1, \, h_2)^{-1} = \omega(h_1, \, h_2, \,h_3).$$ for all $h_1, h_2, h_3 \in H$ (see [@O1]). Let ${\mathcal{M}}:= {\mathcal{M}}(H, \, \mu)$ denote the right module category constructed from the pair $(H, \, \mu)$. The simple objects of ${\mathcal{M}}$ are given by the set $H \backslash G$ of right cosets of $H$ in $G$, the action of ${\mbox{Vec}}_G^{\omega}$ on ${\mathcal{M}}$ comes from the action of $G$ on $H \backslash G$, and the module category structure isomorphisms are induced from the $2$-cochain $\mu$. Let $H, H'$ be subgroups of $G$ such that restrictions of $\omega$ are trivial in $H^3(H, \, k^\times)$ and $H^3(H', \, k^\times)$. Two pairs $(H, \, \mu)$ and $(H', \, \mu')$, where $\delta^2 \mu = \omega|_{H \times H \times H}$ and $\delta^2 \mu' = \omega|_{H' \times H' \times H'}$ give rise to equivalent ${\mbox{Vec}}_G^{\omega}$-module categories if and only if there is $g\in G$ such that $H'=gHg^{-1}$ and $\mu$ and the $g$-conjugate of $\mu'$ differ by a coboundary. We will say that two elements of $\{ \mu \in C^2(H, \, k^\times) \, | \,
\delta^2 \mu = \omega|_{H \times H \times H}\}$ are equivalent if they differ by a coboundary. Let $$\label{Omega}
\Omega_{H, \omega} := \text{equivalence classes of }
\left\{ \mu \in C^2(H, \, k^\times) \mid
\delta^2 \mu = \omega|_{H \times H \times H}\right\}.$$ There is an (in general, non-canonical) bijection between $\Omega_{H, \omega}$ and $H^2(H, \, k^\times)$, i.e., $\Omega_{H, \omega}$ is a (non-empty) torsor over $H^2(H, \, k^\times)$. Note that $\Omega_{H, 1} = H^2(H, \, k^\times)$.
Let ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ be two right module categories over a tensor category ${\mathcal{C}}$. Let $(F^1, \, \gamma^1)$ and $(F^2, \, \gamma^2)$ be module functors from ${\mathcal{M}}_1$ to ${\mathcal{M}}_2$. A [*natural module transformation*]{} from $(F^1, \, \gamma^1)$ to $(F^2, \, \gamma^2)$ is a natural transformation $\eta: F^1 \to F^2$ such that the following equation holds for all $M \in {\mathcal{M}}_1$, $X \in {\mathcal{C}}$: $$\label{module trans square}
(\eta_M \otimes {\text{id}}_X) \, \circ \, \gamma_{M, \, X}^1
= \gamma_{M, \, X}^2 \, \circ \, \eta_{M \otimes X}.$$
Let ${\mathcal{C}}$ be a tensor category and let ${\mathcal{M}}$ be a right module category over ${\mathcal{C}}$. The [*dual category*]{} of ${\mathcal{C}}$ with respect to ${\mathcal{M}}$ is the category ${\mathcal{C}}^*_{\mathcal{M}}:=Fun_{\mathcal{C}}({\mathcal{M}},{\mathcal{M}})$ whose objects are ${\mathcal{C}}$-module functors from ${\mathcal{M}}$ to itself and morphisms are natural module transformations. The category ${\mathcal{C}}^*_{\mathcal{M}}$ is a tensor category with tensor product being composition of module functors. It is known that if ${\mathcal{C}}$ is a fusion category and ${\mathcal{M}}$ is semisimple $k$-linear and indecomposable, then ${\mathcal{C}}^*_{\mathcal{M}}$ is a fusion category [@ENO].
Let $G$ be a finite group and $\omega \in Z^3(G, \, k^\times)$. For each $x \in G$, define $\Upsilon_x : G \times G \to k^\times$ by $$\label{Upsilon}
\Upsilon_x(g_1, \, g_2) := \frac{\omega(xg_1x^{-1}, \, xg_2x^{-1}, \, x)
\omega(x, \, g_1, \, g_2)}{\omega(xg_1x^{-1}, \, x, \, g_2)}, \qquad
\mbox{for all } g_1, g_2 \in G.$$ It is straightforward to verify that $\delta^2 \Upsilon_x = \frac{\omega}{\omega^x}$, for all $x \in G$, where $$\omega^x(g_1, \, g_2, \, g_3)
= \omega(xg_1x^{-1}, \, xg_2x^{-1}, \, xg_3x^{-1}),$$ for all $g_1, g_2, g_3 \in G$.
For each $x \in G$, define $\nu_x : G \times G \to k^\times$ by $$\nu_x(g_1, \, g_2) := \frac{\omega(g_1, \, g_2, \, x)
\omega(g_1g_2xg_2^{-1}g_1^{-1}, \, g_1, \, g_2)}
{\omega(g_1, \, g_2xg_2^{-1}, \, g_2)}, \qquad \mbox{ for all }
g_1, g_2 \in G.$$ It is easy to verify that the following relation holds: $$\label{nu upsilon}
\frac{\Upsilon_{x_1x_2}(g_1, \, g_2)}
{\Upsilon_{x_1}(x_2g_1x_2^{-1}, \, x_2g_2x_2^{-1})
\Upsilon_{x_2}(g_1, \, g_2)} =
\frac{\nu_{g_1}(x_1, \, x_2) \nu_{g_2}(x_1, \, x_2)}
{\nu_{g_1g_2}(x_1, \, x_2)}, \quad \mbox{ for all }
x_1, x_2, g_1, g_2 \in G.$$
Let $H$ be a normal subgroup of $G$ such that $\omega|_{H \times H \times H}$ is cohomologically trivial. For any $x \in G$ and $\mu \in C^2(H, \, k^\times)$ such that $\delta^2 \mu =
\omega|_{H \times H \times H}$, define $$\mu \triangleleft x :=
\mu^x \times \Upsilon_x|_{H \times H},$$ where $\mu^x(h_1, \, h_2) = \mu(xh_1x^{-1}, \, xh_2x^{-1})$, for all $h_1, h_2 \in H$. It is easy to verify that $\delta^2(\mu \triangleleft x)
= \omega|_{H \times H \times H}$. This induces an action of $G$ on $\Omega_{H, \omega}$ (defined in ). Indeed, that this is an action follows from . Let $(\Omega_{H, \omega})^G$ denote the set of $G$-invariant elements of $\Omega_{H, \omega}$, i.e., $$\label{Omega 1}
(\Omega_{H, \omega})^G := \left\{\mu
\in \Omega_{H, \omega} \,\, \vline \,\,
\frac{\mu^x}{\mu} \times \Upsilon_x|_{H \times H} \text{ is trivial in } H^2(H, \, k^\times),
\text{ for all } x \in G \right\}.$$
\[pointed mod cats\] [**Module categories over ${\mbox{Vec}}_G^{\omega}$ with pointed duals**]{}. Let $G$ be a finite group and $\omega \in Z^3(G, \, k^\times)$. It is shown in [@N Theorem 3.4] that the set of equivalence classes of indecomposable module categories over ${\mbox{Vec}}_G^{\omega}$ such that the dual is pointed is in bijection with the set of pairs $(H, \, \mu)$, where $H$ is a normal Abelian subgroup of $G$ such that $\omega|_{H \times H \times H}$ is cohomologically trivial and $\mu \in (\Omega_{H, \omega})^G$ (the description in [@N Theorem 3.4] is given in somewhat different but equivalent terms).
Two fusion categories ${\mathcal{C}}$ and ${\mathcal{D}}$ are said to be [*weakly Morita equivalent*]{} if there exists an indecomposable (semisimple $k$-linear) right module category ${\mathcal{M}}$ over ${\mathcal{C}}$ such that the categories ${\mathcal{C}}^*_{{\mathcal{M}}}$ and ${\mathcal{D}}$ are equivalent as fusion categories. It was shown by M. Müger [@Mu] that this is indeed an equivalence relation.
A fusion category ${\mathcal{C}}$ is said to be [*group theoretical*]{} if it is weakly Morita equivalent to a pointed category.
[Modular categories and centralizers]{}
Let ${\mathcal{C}}$ be a modular fusion category with braiding $c$, twist $\theta$, and S-matrix $S$ (see [@BK]). Let ${\mathcal{D}}$ be a full (not necessarily tensor) subcategory of ${\mathcal{C}}$. Its dimension is defined by ${\mbox{dim}}({\mathcal{D}}) := \sum_{X \in {\text{Irr}}({\mathcal{D}})} d(X)^2$, where ${\text{Irr}}({\mathcal{D}})$ is the set of isomorphism classes of simple objects in ${\mathcal{D}}$. In [@M], M. Müger introduced the notion of the [*centralizer*]{} of ${\mathcal{D}}$ in ${\mathcal{C}}$ as the fusion subcategory $${\mathcal{D}}' := \left\{ X \in {\mathcal{C}}\mid c(Y, \, X) \circ c(X, \, Y)
= {\text{id}}_{X \otimes Y}, \mbox{ for all } Y \in {\mathcal{D}}\right\}.$$ It was also shown in [@M] that if ${\mathcal{D}}$ is a fusion subcategory then ${\mathcal{D}}''={\mathcal{D}}$ and $$\label{dimension}
{\mbox{dim}}({\mathcal{D}}) \cdot {\mbox{dim}}({\mathcal{D}}') = {\mbox{dim}}({\mathcal{C}}).$$ Following M. Müger, we will say that two objects $X, Y \in {\mathcal{C}}$ [*centralize*]{} each other if $$c(Y, \, X) \circ c(X, \, Y) = {\text{id}}_{X \otimes Y}.$$ For simple $X$ and $Y$ this condition is equivalent to $S(X, \, Y) = d(X) d(Y)$ [@M Corollary 2.14].
\[square dim\] If ${\mathcal{D}}$ is a full subcategory of ${\mathcal{C}}$ such that all objects in ${\mathcal{D}}$ centralize each other, i.e., ${\mathcal{D}}\subseteq {\mathcal{D}}'$ then ${\mbox{dim}}({\mathcal{D}})^2 \leq {\mbox{dim}}({\mathcal{C}})$. Indeed, we have ${\mbox{dim}}({\mathcal{D}}) \leq {\mbox{dim}}({\mathcal{D}}')$ and so it follows from that ${\mbox{dim}}({\mathcal{D}})^2 \leq {\mbox{dim}}({\mathcal{C}})$. In particular, if ${\mathcal{D}}$ is a symmetric fusion subcategory of ${\mathcal{C}}$, then ${\mbox{dim}}({\mathcal{D}})^2 \leq {\mbox{dim}}({\mathcal{C}})$.
\[lag\] Let ${\mathcal{D}}$ be a full subcategory of ${\mathcal{C}}$ $($which is not [*a priori*]{} assumed to be closed under the tensor product or duality$)$ such that ${\mathcal{D}}\subseteq {\mathcal{D}}'$. Then the fusion subcategory $\tilde{{\mathcal{D}}}
\subseteq {\mathcal{C}}$ generated by ${\mathcal{D}}$ is symmetric.
We may assume that ${\mathcal{D}}$ is closed under taking duals. Indeed, it follows from [@ENO Proposition 2.12] that $X$ centralizes $Y$ if and only if $X$ centralizes $Y^*$ for any two simple objects $X,Y$ in ${\mathcal{C}}$.
Let $Z_1, Z_2$ be simple objects in $\tilde{{\mathcal{D}}}$. There exist simple objects $X_1, X_2, Y_1, Y_2$ in ${\mathcal{D}}$ such that $Z_1$ is contained in $X_1{\otimes}Y_1$ and $Z_2$ is contained in $X_2{\otimes}Y_2$. By [@M Lemma 2.4 (i)], it follows that $Z_1$ centralizes $X_2 \otimes Y_2$, and hence $Z_1, Z_2$ centralize each other.
Let ${\mathcal{D}}$ be a full subcategory of ${\mathcal{C}}$ such that ${\mathcal{D}}\subseteq {\mathcal{D}}'$ and ${\mbox{dim}}({\mathcal{D}})^2 = {\mbox{dim}}({\mathcal{C}})$. Then ${\mathcal{D}}$ is a symmetric fusion subcategory.
[Lagrangian subcategories and braided equivalences of twisted group doubles]{}
Let ${\mathcal{C}}$ be a modular category. Recall that we chose the canonical spherical twist for ${\mathcal{C}}$ with respect to which the categorical dimension of any object of ${\mathcal{C}}$ is equal to its Frobenius-Perron dimension. This is possible by [@ENO Proposition 8.23, 8.24]. Let us recall some definitions and results from [@DGNO].
A fusion subcategory ${\mathcal{D}}\subseteq {\mathcal{C}}$ is said to be [*isotropic*]{} if the twist of ${\mathcal{C}}$ restricts to identity on ${\mathcal{D}}$. An isotropic subcategory ${\mathcal{D}}\subseteq {\mathcal{C}}$ is said to be [*Lagrangian*]{} if $({\mbox{dim}}({\mathcal{D}}))^2 = {\mbox{dim}}({\mathcal{C}})$.
An isotropic subcategory ${\mathcal{D}}$ of ${\mathcal{C}}$ is necessarily symmetric and its objects have positive categorical dimensions. It follows from [@De] that there is a (unique up to an isomorphism) group $G$ such that ${\mathcal{D}}\cong {\mbox{Rep}}(G)$ as a symmetric fusion category, where ${\mbox{Rep}}(G)$ is considered with its trivial braiding.
Consider the set of all braided tensor equivalences $F: {\mathcal{C}}\xrightarrow{\sim} {\mathcal{Z}}({\mathcal{P}})$, where ${\mathcal{P}}$ is a pointed fusion category and ${\mathcal{Z}}({\mathcal{P}})$ denotes its center. There is an equivalence relation on this set defined as follows. We say that $F_1: {\mathcal{C}}\xrightarrow{\sim} {\mathcal{Z}}({\mathcal{P}}_1)$ and $F_2: {\mathcal{C}}\xrightarrow{\sim} {\mathcal{Z}}({\mathcal{P}}_2)$ are equivalent if there exists a tensor equivalence $\iota: {\mathcal{P}}_1
\xrightarrow{\sim} {\mathcal{P}}_2$ such that $\mathcal{F}_2\circ F_2 = \iota\circ \mathcal{F}_1\circ F_1$, where $\mathcal{F}_i : {\mathcal{Z}}({\mathcal{P}}_i) \to
{\mathcal{P}}_i, \, i=1,2$, are the canonical forgetful functors. Let $\mbox{E}({\mathcal{C}})$ be the collection of equivalence classes of such equivalences. Informally, $\mbox{E}({\mathcal{C}})$ is the set of all “different” braided equivalences between ${\mathcal{C}}$ and centers of pointed categories, i.e., representation categories of twisted group doubles.
Let $\mbox{Lagr}({\mathcal{C}})$ be the set of all Lagrangian subcategories of ${\mathcal{C}}$.
In [@DGNO Theorem 4.5] it was proved that there is a bijection $$\label{lagr dgno}
f: \mbox{E}({\mathcal{C}}) \xrightarrow{\sim} \mbox{Lagr}({\mathcal{C}})$$ defined as follows. Note that each braided tensor equivalence $F: {\mathcal{C}}\xrightarrow{\sim} {\mathcal{Z}}({\mathcal{P}})$ gives rise to the Lagrangian subcategory $f(F)$ of ${\mathcal{C}}$ formed by all objects sent to multiples of the unit object ${\mathbf{1}}$ under the forgetful functor ${\mathcal{Z}}({\mathcal{P}})\to {\mathcal{P}}$. This subcategory is clearly the same for all equivalent choices of $F$.
In particular, the center of a fusion category ${\mathcal{D}}$ contains a Lagrangian subcategory if and only if ${\mathcal{D}}$ is group-theoretical [@DGNO].
[The Schur multiplier of an Abelian group.]{} \[schur abelian\]
Let $H$ be a normal Abelian subgroup of a finite group $G$. Let $\Lambda^2H$ denote the Abelian group of alternating bicharacters on $H$, i.e., $$\Lambda^2H := \left\{B: H \times H \to k^\times \,\, \vline \,
\begin{tabular}{l}
$B(h_1h_2, \, h) = B(h_1, h)B(h_2, \, h),$\\
$B(h, \, h_1h_2) = B(h, h_1)B(h, \, h_2), \text{ and }$\\
$B(h, \, h) = 1, \mbox{ for all } h, h_1, h_2 \in H$
\end{tabular}
\right\}.$$
Let $Z^2(H, \, k^\times)$ be the group of $2$-cocycles on $H$. Define a homomorphism $alt: Z^2(H, \, k^\times) \to \Lambda^2H : \mu \to alt(\mu)$ by $$alt(\mu)(h_1, \, h_2) :=
\frac{\mu(h_2, \, h_1)}{\mu(h_1, h_2)}, \quad h_1, h_2 \in H.$$
It is well known that $alt$ induces an isomorphism between the Schur multiplier $H^2(H, \, k^\times)$ of $H$ and $\Lambda^2H$. By abuse of notation we denote this isomorphism also by $alt$: $$\label{alt}
alt: H^2(H, \, k^\times)
\xrightarrow{\sim} \Lambda^2H.$$
Note that both $H^2(H, \, k^\times)$ and $\Lambda^2H$ are right $G$-modules via the conjugation and that $alt$ is $G$-linear.
[Lagrangian subcategories in the untwisted case]{} \[section 3\]
We fix notation for this Section. Let $G$ be a finite group. For any $g \in G$, let $K_g$ denote the conjugacy class of $G$ containing $g$. Let $R$ denote a complete set of representatives of conjugacy classes of $G$. Let ${\mathcal{C}}$ denote the representation category ${\mbox{Rep}}(D(G))$ of the Drinfeld double of the group $G$: $${\mathcal{C}}:= {\mbox{Rep}}(D(G)).$$ The category ${\mathcal{C}}$ is equivalent to ${\mathcal{Z}}({\mbox{Vec}}_G)$, the center of ${\mbox{Vec}}_G$. It is well known that ${\mathcal{C}}$ is a modular category. Let $\Gamma$ denote a complete set of representatives of simple objects of ${\mathcal{C}}$. The set $\Gamma$ is in bijection with the set $\{(a, \, \chi) \mid a \in R \mbox{ and } \chi \mbox{ is an
irreducible character of }
C_G(a) \}$, where $C_G(a)$ is the centralizer of $a$ in $G$ (see [@CGR]). In what follows we will identify $\Gamma$ with the previous set, $$\label{Gamma}
\Gamma := \{(a, \, \chi) \mid a \in R \mbox{ and } \chi \mbox{ is an
irreducible character of } C_G(a) \}.$$ Let $S$ and $\theta$ be (see, e.g. [@BK], [@CGR]) the $S$-matrix and twist, respectively, of ${\mathcal{C}}$. Recall that we take the canonical twist. It is known that the entries of the $S$-matrix lie in a cyclotomic field. Also, the values of characters of a finite group are sums of roots of unity. So we may assume that all scalars appearing herein are complex numbers; in particular, complex conjugation and absolute values make sense. We have the following formulas for the $S$-matrix, twist and dimensions: $$\begin{split}
S((a, \, \chi), \, (b, \, \chi^\prime))
&= \frac{|G|}{|C_G(a)||C_G(b)|} \sum_{g \in G(a, \, b)}
\overline{\chi}(gbg^{-1}) \, \overline{\chi}^\prime(g^{-1}ag),\\
\theta(a, \, \chi)
& = \frac{\chi(a)}{{\text{deg }}\chi},\\
d((a, \, \chi))
& = |K_a| \, {\text{deg }}\chi = \frac{|G|}{|C_G(a)|} \, {\text{deg }}\chi,
\end{split}$$ for all $(a, \, \chi), (b, \, \chi^\prime) \in \Gamma$, where $G(a, \, b) = \{g \in G \mid agbg^{-1} = gbg^{-1}a\}$.
[Classification of Lagrangian subcategories of $\mathbf{{\mbox{Rep}}(D(G))}$]{}
\[centralize\] Two objects $(a, \, \chi), (b, \, \chi^\prime) \in \Gamma$ centralize each other if and only if the following conditions hold:\
(i) The conjugacy classes $K_a, K_b$ commute element-wise,\
(ii) $\chi(gbg^{-1}) \, \chi^\prime(g^{-1}ag) = {\text{deg }}\chi \, {\text{deg }}\chi^\prime$, for all $g \in G$.
By [@M Corollary 2.14] two objects $(a, \, \chi), (b, \, \chi^\prime) \in \Gamma$ centralize each other if and only if $$S((a, \, \chi), (b, \, \chi^\prime)) = {\text{deg }}\chi \, {\text{deg }}\chi^\prime.$$ This is equivalent to the equation $$\label{eqn}
\sum_{g \in G(a, \, b)}
\chi(gbg^{-1}) \, \chi^\prime(g^{-1}ag) = |G| \, {\text{deg }}\chi \, {\text{deg }}\chi^\prime,$$ where $G(a, \, b) = \{g \in G \mid agbg^{-1} = gbg^{-1}a\}$. It is clear that if the two conditions of the Lemma hold, then holds since $G(a, \, b) = G$.
Now suppose that holds. We will show that this implies the two conditions in the statement of the Lemma. We have $$\begin{split}
|G| \, {\text{deg }}\chi \, {\text{deg }}\chi^\prime
&= |\sum_{g \in G(a, \, b)} \chi(gbg^{-1}) \, \chi^\prime(g^{-1}ag)|\\
&\leq \sum_{g \in G(a, \, b)} |\chi(gbg^{-1})| \, |\chi^\prime(g^{-1}ag)|\\
&\leq |G| \, {\text{deg }}\chi \, {\text{deg }}\chi^\prime.
\end{split}$$ So $\sum_{g \in G(a, \, b)} |\chi(gbg^{-1})| \, |\chi^\prime(g^{-1}ag)|
= |G| \, {\text{deg }}\chi \, {\text{deg }}\chi^\prime$. Since $$|G(a, \, b)| \leq |G|,\\
\,\,|\chi(gbg^{-1})| \leq {\text{deg }}\chi, \mbox{ and }
|\chi^\prime(g^{-1}ag)| \leq {\text{deg }}\chi^\prime,$$ we must have $G(a, \, b) = G$, $|\chi(gbg^{-1})| = {\text{deg }}\chi$, and $|\chi^\prime(g^{-1}ag)| = {\text{deg }}\chi^\prime$. The equality $G(a, \, b) = G$ implies that the conjugacy classes $K_a, K_b$ commute element-wise, which is Condition (i) in the statement of the Lemma. Since $|\chi(gbg^{-1})| = {\text{deg }}\chi$, and $|\chi^\prime(g^{-1}ag)| = {\text{deg }}\chi^\prime$, there exist roots of unity $\alpha_g$ and $\beta_g$ such that $\chi(gbg^{-1}) = \alpha_g \, {\text{deg }}\chi$, and $\chi^\prime(g^{-1}ag) = \beta_g \, {\text{deg }}\chi^\prime$, for all $g \in G$. Put this in to get the equation $$\label{a}
\sum_{g \in G} \alpha_g \beta_g = |G|.$$ Note that holds if and only if $\alpha_g \beta_g = 1$, for all $g \in G$. This is equivalent to saying that $\chi(gbg^{-1}) \, \chi^\prime(g^{-1}ag)
= {\text{deg }}\chi \, {\text{deg }}\chi^\prime$, for all $g \in G$ and the Lemma is proved.
\[FR\] Let $E$ be a normal subgroup of a finite group $K$. Let ${\text{Irr}}(K)$ denote the set of irreducible characters of $K$. Let $\rho$ be a $K$-invariant character of $E$ of degree 1. Then $$\sum_{\chi \in {\text{Irr}}(K) : \chi|_E = ({\text{deg }}\chi) \, \rho}
({\text{deg }}\chi)^2 = \frac{|K|}{|E|}.$$
Suppose $\chi$ is any irreducible character of $K$. Since $\rho$ is $K$-invariant, by Clifford’s Theorem, if $\rho$ is an irreducible constituent of $\chi|_E$, then $$\label{chiE}
\chi|_E = ({\text{deg }}\chi) \, \rho.$$ By Frobenius reciprocity, the multiplicity of any irreducible $\chi$ in $\mbox{Hind}_{E}^{K}\rho$ is equal to the multiplicity of $\rho$ in $\chi|_E$. The latter is equal to ${\text{deg }}\chi$ if $\chi$ satisfies and $0$ otherwise. Therefore, $$\sum_{\chi \in {\text{Irr}}(K) : \chi|_E = ({\text{deg }}\chi) \, \rho}
({\text{deg }}\chi)^2 = {\text{deg }}\mbox{Ind}_{E}^{K}\rho = \frac{|K|}{|E|},$$ as required.
Let $H$ be a normal Abelian subgroup of $G$ and let $B$ be a $G$-invariant alternating bicharacter on $H$. Then $H = \bigcup_{a \in H \cap R}K_a$. Let $$\label{L}
\begin{split}
{\mathcal{L}}_{(H, \, B)} :=
&\text{ full Abelian subcategory of } {\mathcal{C}}\text{ generated by } \\
&\left\{(a, \, \chi) \in \Gamma \,\, \vline \,
\begin{tabular}{l}
$a \in H \cap R \text{ and } \chi \text{ is an irreducible character of } C_G(a)$ \\
$\text{ such that } \chi(h) = B(a, \, h) \,{\text{deg }}\chi, \text{ for all } h \in H$
\end{tabular}
\right\}.
\end{split}$$
\[Proposition L\] The subcategory ${\mathcal{L}}_{(H, \, B)} \subseteq {\mbox{Rep}}(D(G))$ is Lagrangian.
We have $$\begin{split}
\chi(gbg^{-1}) \, \chi^\prime(g^{-1}ag)
& = B(a, \, gbg^{-1}) \, {\text{deg }}\chi \, B(b, \, g^{-1}ag)
\, {\text{deg }}\chi^\prime\\
& = B(a, \, gbg^{-1}) \, B(gbg^{-1}, \,a)
\, {\text{deg }}\chi \,{\text{deg }}\chi^\prime\\
& = {\text{deg }}\chi \,{\text{deg }}\chi^\prime,
\end{split}$$ for all $(a, \, \chi), (b, \, \chi^\prime) \in {\mathcal{L}}_{(H, \, B)}
\cap \Gamma, g \in G$. The second equality above is due to $G$-invariance of $B$ and the third equality holds since $B$ is alternating. By Lemma \[centralize\], it follows that objects in ${\mathcal{L}}_{(H, \, B)}$ centralize each other.
Also, we have $\theta_{(a, \, \chi)} = \frac{\chi(a)}{{\text{deg }}\chi}
= \frac{B(a, \, a)}{{\text{deg }}\chi} \, {\text{deg }}\chi = 1$, for all $(a, \, \chi)
\in {\mathcal{L}}_{(H, \, B)} \cap \Gamma$. Therefore, $\theta|_{{\mathcal{L}}_{(H, \, B)}} = {\text{id}}$.
The dimension of ${\mathcal{L}}_{(H, \, B)}$ is equal to $|G|$. Indeed, $$\begin{split}
{\mbox{dim}}({\mathcal{L}}_{(H, \, B)})
&= \sum_{(a, \, \chi) \in {\mathcal{L}}_{(H, \, B)} \cap \Gamma} d(a, \, \chi)^2 \\
&= \sum_{(a, \, \chi) \in {\mathcal{L}}_{(H, \, B)} \cap \Gamma} |K_a|^2 \, ({\text{deg }}\chi)^2\\
&= \sum_{a \in H \cap R} |K_a|^2 \sum_{\chi :
(a, \, \chi) \in {\mathcal{L}}_{(H, \, B)} \cap \Gamma} ({\text{deg }}\chi)^2\\
&= \sum_{a \in H \cap R} |K_a|^2 \frac{|C_G(a)|}{|H|}\\
&= \frac{|G|}{|H|} \sum_{a \in H \cap R} |K_a|\\
&= |G|.
\end{split}$$ The fourth equality above is explained as follows. Fix $a \in H \cap R$. Define $\rho : H \to k^\times$ by $\rho(h) := B(a, \, h)$. Observe that $\rho$ is a $C_G(a)$-invariant character of $H$ of degree $1$ and then apply Lemma \[FR\].
It follows from Lemma \[lag\] that ${\mathcal{L}}_{(H, \, B)}$ is a Lagrangian subcategory of ${\mbox{Rep}}(D(G))$ and the Proposition is proved.
Now, let ${\mathcal{L}}$ be a Lagrangian subcategory of ${\mathcal{C}}$. So, in particular, the two conditions in Lemma \[centralize\] hold for all simple objects in ${\mathcal{L}}$. Define $$\label{H_L}
H_{{\mathcal{L}}} := \bigcup_{a \in R : (a, \, \chi) \in {\mathcal{L}}\text{ for some }\chi} K_a.$$ Note that $H_{{\mathcal{L}}}$ is a normal Abelian subgroup of $G$. Indeed, that $H_{{\mathcal{L}}}$ is a subgroup follows from the fact that ${\mathcal{L}}$ contains the unit object and is closed under tensor products. The subgroup $H_{{\mathcal{L}}}$ is normal in $G$ because it is a union of conjugacy classes of $G$. Finally, that $H_{{\mathcal{L}}}$ is Abelian follows by Condition (i) of Lemma \[centralize\].
For each $a \in H \cap R$, define $\xi_a : H_{{\mathcal{L}}} \to k^\times$ by $$\xi_a(h) := \frac{\chi(h)}{{\text{deg }}\chi},$$ for $h \in H_{{\mathcal{L}}}$, where $\chi$ is any irreducible character of $C_G(a)$ such that $(a, \, \chi) \in {\mathcal{L}}\cap \Gamma$. To see that this definition does not depend on the choice of $\chi$, let $(a, \, \chi), \, (a, \, \chi^\prime), \, (b, \, \chi^{\prime\prime})
\in {\mathcal{L}}\cap \Gamma$ and apply Condition (ii) of Lemma \[centralize\] to pairs $(a, \, \chi), \, (b, \, \chi^{\prime\prime})$ and $(a, \, \chi^\prime), \, (b, \, \chi^{\prime\prime})$ to get $$\frac{\chi(gbg^{-1})}{{\text{deg }}\chi} = {\left(}\frac{\chi^{\prime\prime}(g^{-1}ag)}
{{\text{deg }}\chi^{\prime\prime}} {\right)}^{-1} \quad \mbox{ and } \quad
\frac{\chi^\prime(gbg^{-1})}{{\text{deg }}\chi^\prime} = {\left(}\frac {\chi^{\prime\prime}(g^{-1}ag)}
{{\text{deg }}\chi^{\prime\prime}} {\right)}^{-1},$$ for all $g \in G$. This implies that $\frac{\chi|_{H_{{\mathcal{L}}}}}{{\text{deg }}\chi}
= \frac{\chi^\prime|_{H_{{\mathcal{L}}}}}{{\text{deg }}\chi^\prime}$, for any two pairs $(a, \, \chi), (a, \, \chi^\prime) \in {\mathcal{L}}\cap \Gamma$.
For any $a, b \in H_{{\mathcal{L}}} \cap R$, by Condition (ii) of Lemma \[centralize\], $\xi_a$ and $\xi_b$ satisfy the equation: $$\label{eqn1}
\xi_a(gbg^{-1}) = \xi_b(g^{-1}ag)^{-1}, \quad
\mbox{ for all } g \in G.$$ Define a map $B_{{\mathcal{L}}}: H_{{\mathcal{L}}} \times H_{{\mathcal{L}}} \to k^\times$ by $$\label{B_L}
B_{{\mathcal{L}}}(h_1, \, h_2) := \xi_a(g^{-1}h_2g),$$ where $h_1 = gag^{-1}, g \in G, a \in H_{{\mathcal{L}}} \cap R$.
\[Proposition B\_L\] $B_{{\mathcal{L}}}$ is a well-defined $G$-invariant alternating bicharacter on $H_{{\mathcal{L}}}$.
First, let us show that $B_{{\mathcal{L}}}$ is well-defined. Suppose $gag^{-1} = kak^{-1}$, where $a \in H_{{\mathcal{L}}} \cap R, g, k \in G$. Then $$\begin{split}
B_{{\mathcal{L}}}(gag^{-1}, \, lbl^{-1})
&= \xi_a((g^{-1}l)b(g^{-1}l)^{-1})\\
&= \xi_b((g^{-1}l)^{-1}a(g^{-1}l))^{-1}\\
&= \xi_b(l^{-1}(gag^{-1})l)^{-1}\\
&= \xi_b(l^{-1}(kak^{-1})l)^{-1}\\
&= \xi_a((l^{-1}k)^{-1}b(l^{-1}k))\\
&= \xi_a(k^{-1}(lbl^{-1})k)\\
&= B_{{\mathcal{L}}}(kak^{-1}, \, lbl^{-1}),
\end{split}$$ for all $b \in H_{{\mathcal{L}}} \cap R, l \in G$. The second and the fifth equalities above are due to .
Let $h_1 = kak^{-1}, h_2 \in H_{{\mathcal{L}}}, g \in G$, where $a \in H_{{\mathcal{L}}} \cap R, k \in G$. Then $$\begin{split}
B_{{\mathcal{L}}}(gh_1g^{-1}, \, gh_2g^{-1})
&= B_{{\mathcal{L}}}(gkak^{-1}g^{-1}, \, gh_2g^{-1})\\
&= \xi_a((gk)^{-1}(gh_2g^{-1})(gk))\\
&= \xi_a(k^{-1}h_2k)\\
&= B_{{\mathcal{L}}}(kak^{-1}, \, h_2)\\
&= B_{{\mathcal{L}}}(h_1, \, h_2).
\end{split}$$ So, $B_{{\mathcal{L}}}$ is $G$-invariant.
Now, $$\begin{split}
B_{{\mathcal{L}}}(gag^{-1}, \, gag^{-1})
&= B_{{\mathcal{L}}}(a, \, a)\\
&= \xi_a(a)\\
&= \frac{\chi(a)}{{\text{deg }}\chi}\\
&= \theta_{(a, \, \chi)}\\
&= 1,
\end{split}$$ for all $a \in H_{{\mathcal{L}}} \cap R, g \in G$. The first equality above is due to the $G$-invariance of $B_{{\mathcal{L}}}$. So $B_{{\mathcal{L}}}(h, \, h) = 1$, for all $h \in H_{{\mathcal{L}}}$.
Also, $B_{{\mathcal{L}}}(g_1ag_1^{-1}, \, g_2bg_2^{-1}) B_{{\mathcal{L}}}(g_2bg_2^{-1}, \, g_1ag_1^{-1})
= \xi_a(g_1^{-1}g_2bg_2^{-1}g_1) \xi_b(g_2^{-1}g_1ag_1^{-1}g_2) = 1$, for all $g_1, g_2 \in G, a, b \in H \cap R$. We used in the last equality.
To see that $B_{{\mathcal{L}}}$ is a bicharacter, observe first that $\xi_a$ is a homomorphism, for all $a \in H_{{\mathcal{L}}} \cap R$. We have $$\begin{split}
B_{{\mathcal{L}}}(gag^{-1}, \, h_1) \, B_{{\mathcal{L}}}(gag^{-1}, \, h_2)
&= \xi_a(g^{-1}h_1g) \, \xi_a(g^{-1}h_2g)\\
&= \xi_a(g^{-1}h_1h_2g)\\
&= B_{{\mathcal{L}}}(gag^{-1}, \, h_1h_2),
\end{split}$$ for all $a \in H_{{\mathcal{L}}} \cap R, g \in G, h_1, h_2 \in H_{{\mathcal{L}}}$. We conclude that $B_{{\mathcal{L}}}$ is a $G$-invariant alternating bicharacter on $H_{{\mathcal{L}}}$ and the Proposition is proved.
Recall that ${\mbox{Lagr}}({\mathcal{C}})$ denotes the set of Lagrangian subcategories of a modular category ${\mathcal{C}}$.
\[untwisted bijection\] Lagrangian subcategories of the representation category of the Drinfeld double $D(G)$ are classified by pairs $(H, B)$, where $H$ is a normal Abelian subgroup of $G$ and $B$ is an alternating $G$-invariant bicharacter on $H$.
Let $\mathcal{E} := \{(H, \, B) \mid H \mbox{ is a normal Abelian subgroup
of $G$ and } B \in (\Lambda^2H)^G \}$. Define a map $\Psi : \mathcal{E} \to {\mbox{Lagr}}({\mathcal{C}}) : (H, \, B) \mapsto {\mathcal{L}}_{(H, \, B)}$, where ${\mathcal{C}}= {\mbox{Rep}}(D(G))$ and ${\mathcal{L}}_{(H, \, B)}$ is defined in ([\[L\]]{}). It was shown in Proposition \[Proposition L\] that ${\mathcal{L}}_{(H, \, B)}$ is a Lagrangian subcategory.
To see that $\Psi$ is injective pick any $(H, \, B), (H^\prime, \, B^\prime) \in \mathcal{E}$ and assume that $\Psi((H, \, B)) = \Psi((H^\prime, \, B^\prime))$. So in particular we will have ${\mathcal{L}}_{(H, \, B)} \cap \Gamma
= {\mathcal{L}}_{(H^\prime, \, B^\prime)} \cap \Gamma$. Note that $H = \cup_{(a, \, \chi) \in {\mathcal{L}}_{(H, \, B)}\cap \Gamma}K_a$ and $H^\prime = \cup_{(a, \, \chi) \in {\mathcal{L}}_{(H^\prime, \, B^\prime)}\cap \Gamma}K_a$. Since ${\mathcal{L}}_{(H, \, B)}\cap \Gamma = {\mathcal{L}}_{(H^\prime, \, B^\prime)}\cap \Gamma$, it follows that $H = H^\prime$. Also note that for any $(a, \, \chi) \in {\mathcal{L}}_{(H, \, B)}\cap \Gamma
= {\mathcal{L}}_{(H^\prime, \, B^\prime)}\cap \Gamma$, we have $\chi(h) = B(a, \, h) \, {\text{deg }}\chi
= B^\prime(a, \, h) \, {\text{deg }}\chi$, for all $h \in H = H^\prime$. Since $B, B^\prime$ are $G$-invariant, it follows that $B = B^\prime$. So $\Psi$ is injective.
To see that $\Psi$ is surjective pick any ${\mathcal{L}}\in {\mbox{Lagr}}({\mathcal{C}})$. Consider the pair $(H_{{\mathcal{L}}}, \, B_{{\mathcal{L}}})$, where $H_{{\mathcal{L}}}$ and $B_{{\mathcal{L}}}$ are defined in (\[H\_L\]) and (\[B\_L\]), respectively. Proposition \[Proposition B\_L\] showed that $(H_{{\mathcal{L}}}, \, B_{{\mathcal{L}}})$ belongs to the set $\mathcal{E}$. We contend that $\Psi((H_{{\mathcal{L}}}, \, B_{{\mathcal{L}}})) = {\mathcal{L}}$. It suffices to show that ${\mathcal{L}}\cap \Gamma \subseteq {\mathcal{L}}_{(H_{{\mathcal{L}}}, \, B_{{\mathcal{L}}})}$. But this hold by definition of ${\mathcal{L}}_{(H_{{\mathcal{L}}}, \, B_{{\mathcal{L}}})}$ and the observation that $\frac{\chi|_{H_{{\mathcal{L}}}}}{{\text{deg }}\chi}
= \frac{\chi^\prime|_{H_{{\mathcal{L}}}}}{{\text{deg }}\chi^\prime}$, for any two pairs $(a, \, \chi), (a, \, \chi^\prime) \in {\mathcal{L}}\cap \Gamma$, $a \in H_{{\mathcal{L}}} \cap R$. So $\Psi$ is surjective and the Theorem is proved.
[Bijective correspondence between Lagrangian subcategories and module categories with pointed duals]{}
Let ${\mathcal{D}}$ be a fusion category and let ${\mathcal{M}}$ be an indecomposable ${\mathcal{D}}$-module category. There is a canonical braided tensor equivalence [@Mu; @EO] $$\label{iota}
\iota_{\mathcal{M}}: {\mathcal{Z}}({\mathcal{D}})\xrightarrow{\sim}
{\mathcal{Z}}({\mathcal{D}}^*_{\mathcal{M}})$$ defined by identifying both centers with the category of ${\mathcal{D}}\boxtimes ({\mathcal{D}}^*_{\mathcal{M}})^{{\text{rev}}}$-module endofunctors of ${\mathcal{M}}$.
Let $f: E({\mathcal{C}}) \xrightarrow{\sim} {\mbox{Lagr}}({\mathcal{C}})$ be the bijection between the set of (equivalence classes of) braided tensor equivalences between ${\mathcal{C}}$ and centers of pointed fusion categories and the set of Lagrangian subcategories of ${\mathcal{C}}$ defined in [@DGNO], see .
\[bijection\] The assignment ${\mathcal{M}}\mapsto \iota_{\mathcal{M}}$ restricts to a bijection between the set of equivalence classes of indecomposable ${\mbox{Vec}}_G$-module categories ${\mathcal{M}}$ with respect to which the dual fusion category $({\mbox{Vec}}_G)^*_{\mathcal{M}}$ is pointed and $E( {\mbox{Rep}}(D(G)))$.
Comparing the result of [@N] (see Example \[pointed mod cats\]) and Theorem \[untwisted bijection\] and taking into account that the isomorphism $alt:H^2(H, k^\times) \xrightarrow{\sim} (\Lambda^2H)$ is $G$-linear, we see that the two sets in question have the same cardinality. Thus, to prove the theorem it suffices to check that for ${\mathcal{M}}:={\mathcal{M}}(H, \mu)$ one has $f(\iota_{\mathcal{M}}) \subseteq {\mathcal{L}}_{(H, \, alt(\mu))}$, where ${\mathcal{L}}_{(H, \, alt(\mu))}$ is the Lagrangian subcategory defined in .
By definition, $f(\iota_{\mathcal{M}})$ consists of all objects $Z$ in ${\mathcal{C}}= {\mathcal{Z}}(Vec_G)$ (identified with ${\mbox{Rep}}(D(G))$) such that the ${\mbox{Vec}}_G$-module endofunctor $F_Z: {\mathcal{M}}\to {\mathcal{M}}: M \mapsto M {\otimes}Z$ is isomorphic to a multiple of ${\text{id}}_{\mathcal{M}}$. Note that here we abuse notation and write $Z$ for both object of the center and its forgetful image.
Let us recall the parameterization of simple objects of ${\mathcal{Z}}({\mbox{Vec}}_G)$ in . Suppose that a simple $Z$ corresponds to the conjugacy class $K_a$ represented by $a\in R$ and the character afforded by the irreducible representation $\pi:C_G(a)\to GL(V_\pi)$. Then as a $G$-graded vector space $Z = \oplus_{x\in K_a}\,V_\pi^x$ and the permutation isomorphism $c_{g, Z} : g {\otimes}Z \xrightarrow{\sim}
Z {\otimes}g$ is induced from $\pi$, where we identify simple objects of ${\mbox{Vec}}_G$ with the elements of the group $G$.
It is clear that $F_Z$ is isomorphic to a multiple of ${\text{id}}_{\mathcal{M}}$ as an ordinary functor if and only if $K_a \subseteq H$. Note that this implies that $H \subseteq C_G(a)$. Note that for every ${\mbox{Vec}}_G$-module functor $F:{\mathcal{M}}\to {\mathcal{M}}$ the module functor structure on $F$ is completely determined by the collection of isomorphisms $F(H1 {\otimes}h) \xrightarrow{\sim}
F(H1) {\otimes}h,\, h\in H$, where $H1$ denotes the trivial coset in $H \backslash G = {\text{Irr}}({\mathcal{M}})$.
For $F=F_Z$ the latter isomorphism is given by the composition $$(H1 {\otimes}h){\otimes}Z \xrightarrow{\oplus_{x} \mu(h, x)^{-1} {\text{id}}_{V_\pi^x}}
H1 {\otimes}(h {\otimes}Z) \xrightarrow{ {\text{id}}_{H1}
{\otimes}c_{h, Z}} H1 {\otimes}(Z {\otimes}h)
\xrightarrow{\oplus_{x} \mu(x, h) {\text{id}}_{V_\pi^x}} (H1 {\otimes}Z){\otimes}h.$$ The restriction of $c_{h, Z}$ to $h {\otimes}V_{\pi}^a$ is given by $\pi(h)$, for all $h \in C_G(a)$. If the above composition equals identity, then $\pi(h) = alt(\mu)(a, \, h) \,\, {\text{id}}_{V_{\pi}}$, for all $h \in H$. So $Z \in {\mathcal{L}}_{(H, \, alt(\mu))}$ and, therefore, $f(\iota_{\mathcal{M}}) \subseteq {\mathcal{L}}_{(H, \, alt(\mu))}$, as required.
\[G’\] Let us explicitly describe the subcategory ${\mathcal{L}}_{(H, \, B)}$. By [@De] there is a unique up to an isomorphism group $G'$ such that ${\mathcal{L}}_{(H, \, B)} \cong {\mbox{Rep}}(G')$ as a symmetric category. This group $G'$ is precisely the group of invertible objects in the dual category $({\mbox{Vec}}_G)^*_{{\mathcal{M}}(H, \mu)}$, where $\mu \in Z^2(H,\,k^\times)$ is such that $alt(\mu)=B$. It was shown in [@N] that $G'$ is an extension $$0 \to \widehat{H} \to G' \to H\backslash G \to 0,$$ with the corresponding second cohomology class being the image of the cohomology class of $\mu$ under the canonical homomorphism $H^2(H,\,k^\times)^G \to H^2(H\backslash G,\,
\widehat{H})$, see [@N] for details. Note that in general $G \not\cong G'$, see Section \[exs\] for examples.
[Lagrangian subcategories in the twisted case]{}
In this Section we extend the constructions of the previous Section when the associativity is given by a $3$-cocycle $\omega\in Z^3(G,\,k^\times)$. Note that the results of this Section reduce to the results in Section \[section 3\] when $\omega \equiv 1$.
For this Section we follow the notation fixed at the beginning of Section \[section 3\]. Let $\omega$ be a normalized $3$-cocycle on $G$, i.e., $\omega$ is a map from $G \times G \times G$ to $k^\times$ satisfying: $$\label{3-cocycle}
\omega(g_2, \, g_3, \, g_4)\omega(g_1, \, g_2g_3, \, g_4)\omega(g_1, \, g_2, \, g_3)
= \omega(g_1g_2, \, g_3, \, g_4)\omega(g_1, \, g_2, \, g_3g_4),\\$$ $$\omega(g, \, 1_G, \, l) = 1,$$ for all $g, l, g_1, g_2, g_3, g_4 \in G$.
Let ${\mathcal{C}}$ denote the representation category ${\mbox{Rep}}(D^{\omega}(G))$ of the twisted quantum double of the group $G$ [@DPR1; @DPR2]: $${\mathcal{C}}:= {\mbox{Rep}}(D^{\omega}(G)).$$ The category ${\mathcal{C}}$ is equivalent to ${\mathcal{Z}}({\mbox{Vec}}_G^{\omega})$. It is well known that ${\mathcal{C}}$ is a modular category. Replacing $\omega$ by a cohomologous $3$-cocycle we may assume that the values of $\omega$ are roots of unity.
For all $a, g, h \in G$, define $$\label{beta}
\beta_a(h, g) := \omega(a, \, h, \, g) \omega(h, \, h^{-1}ah, \, g)^{-1}
\omega(h, \, g, \, (hg)^{-1}ahg).$$ The $\beta_a$’s satisfy the following equation: $$\label{beta relation}
\beta_a(x, \, y) \beta_a(xy, \, z) = \beta_a(x, \, yz) \beta_{x^{-1}ax}(y, \, z),
\qquad \mbox{for all } x, y, z \in G.$$ Observe that the restriction of each $\beta_a$ to the centralizer $C_G(a)$ of $a$ in $G$ is a normalized $2$-cocycle. Let $\Gamma$ denote a complete set of representatives of simple objects of ${\mathcal{C}}$. The set $\Gamma$ is in bijection with the set $\{(a, \, \chi) \mid a \in R \mbox{ and } \chi
\mbox{ is an irreducible $\beta_a$-character of } C_G(a) \}$. In what follows we will identify $\Gamma$ with the previous set: $$\label{Gamma 1}
\Gamma := \{(a, \, \chi) \mid a \in R \mbox{ and } \chi \mbox{ is an
irreducible $\beta_a$-character of } C_G(a) \}.$$ Let $S$ and $\theta$ be the $S$-matrix and twist, respectively, of ${\mathcal{C}}$. It is known that the entries of the $S$-matrix lie in a cyclotomic field. Also, the values of $\alpha$-characters of a finite group are sums of roots of unity, so they are algebraic numbers, where $\alpha$ is any $2$-cocycle whose values are roots of unity. So we may assume that all scalars appearing herein are complex numbers; in particular, complex conjugation and absolute values make sense. We have the following formulas for the $S$-matrix, twist, and dimensions (see [@CGR]):\
\
$S((a, \, \chi), \, (b, \, \chi^\prime))$\
$$\begin{split}
&= \sum_{g \in K_a, g^\prime \in K_b \cap C_G(g)}
\overline{{\left(}\frac {\beta_a(x, \, g^\prime) \beta_a(xg^\prime, \, x^{-1})
\beta_b(y, \, g) \beta_b(yg, \, y^{-1})}
{\beta_a(x, \, x^{-1}) \beta_b(y, \, y^{-1})} {\right)}}
\overline{\chi}(xg^{\prime}x^{-1}) \, \overline{\chi}^\prime(ygy^{-1}),\\
\theta(a, \, \chi)
& = \frac{\chi(a)}{{\text{deg }}\chi},\\
d((a, \, \chi))
& = |K_a| \, {\text{deg }}\chi = \frac{|G|}{|C_G(a)|} \, {\text{deg }}\chi,
\end{split}$$ for all $(a, \, \chi), (b, \, \chi^\prime) \in \Gamma$, where $g = x^{-1}ax, g^\prime = y^{-1}by$.
[Classification of Lagrangian subcategories of $\mathbf{{\mbox{Rep}}(D^{\omega}(G))}$]{}
\[proj. chars.\] Let $\rho:K \to GL(V)$ be a finite-dimensional projective representation with $2$-cocycle $\alpha$ on the finite group $K$, i.e., $\rho(xy) = \alpha(x,\,y) \rho(x) \rho(y)$, for all $x,y\in K$. Let $\chi$ be the projective character afforded by $\rho$, i.e., $\chi(x) = \text{Trace}(\rho(x))$, for all $x \in K$. Suppose that the values of $\alpha$ are roots of unity. Then $|\chi(x)| \leq {\text{deg }}\chi$, for all $x \in K$ and we have equality if and only if $\rho(x) \in k^\times \cdot {\text{id}}_V$.\
\[centralize 1\] Two objects $(a, \, \chi), (b, \, \chi^\prime) \in \Gamma$ centralize each other if and only if the following conditions hold:\
(i) The conjugacy classes $K_a, K_b$ commute element-wise,\
(ii) ${\left(}\frac {\beta_a(x, \, y^{-1}by) \beta_a(xy^{-1}by, \, x^{-1})
\beta_b(y, \, x^{-1}ax) \beta_b(yx^{-1}ax, \, y^{-1})}
{\beta_a(x, \, x^{-1}) \beta_b(y, \, y^{-1})} {\right)}\chi(xy^{-1}byx^{-1}) \, \chi^\prime(yx^{-1}axy^{-1})
= {\text{deg }}\chi \, {\text{deg }}\chi^\prime$, for all $x, y \in G$.
Two objects $(a, \, \chi), (b, \, \chi^\prime) \in \Gamma$ centralize each other if and only if $S((a, \, \chi), (b, \, \chi^\prime)) = {\text{deg }}\chi \, {\text{deg }}\chi^\prime$. This is equivalent to the equation: $$\begin{split}
\label{eqn 1}
\sum_{g \in K_a, g^\prime \in K_b \cap C_G(g)}
{\left(}\frac {\beta_a(x, \, g^\prime) \beta_a(xg^\prime, \, x^{-1})
\beta_b(y, \, g) \beta_b(yg, \, y^{-1})}
{\beta_a(x, \, x^{-1}) \beta_b(y, \, y^{-1})} {\right)}\chi(xg^{\prime}x^{-1}) \, \chi^\prime(ygy^{-1}) \\
= |K_a||K_b| \, {\text{deg }}\chi \, {\text{deg }}\chi^\prime,
\end{split}$$ where $g = x^{-1}ax, g^\prime = y^{-1}by$. It is clear that if the two conditions of the Lemma hold, then holds since the set over which the above sum is taken is equal to $K_a \times K_b$.
Now suppose that holds. We will show that this implies the two conditions in the statement of the Lemma. We have $$\begin{split}
&|K_a||K_b| \, {\text{deg }}\chi \, {\text{deg }}\chi^\prime\\
&= \left|\sum_{g \in K_a, g^\prime \in K_b \cap C_G(g)}
{\left(}\frac {\beta_a(x, \, g^\prime) \beta_a(xg^\prime, \, x^{-1})
\beta_b(y, \, g) \beta_b(yg, \, y^{-1})}
{\beta_a(x, \, x^{-1}) \beta_b(y, \, y^{-1})} {\right)}\chi(xg^{\prime}x^{-1}) \, \chi^\prime(ygy^{-1}) \right|\\
&\leq
\sum_{g \in K_a, g^\prime \in K_b \cap C_G(g)}
\left|{\left(}\frac {\beta_a(x, \, g^\prime) \beta_a(xg^\prime, \, x^{-1})
\beta_b(y, \, g) \beta_b(yg, \, y^{-1})}
{\beta_a(x, \, x^{-1}) \beta_b(y, \, y^{-1})} {\right)}\right|
\left|\chi(xg^{\prime}x^{-1})\right| \, \left|\chi^\prime(ygy^{-1})\right|\\
&=
\sum_{g \in K_a, g^\prime \in K_b \cap C_G(g)}
\left|\chi(xg^{\prime}x^{-1})\right| \, \left|\chi^\prime(ygy^{-1})\right|\\
&\leq |K_a||K_b| \, {\text{deg }}\chi \, {\text{deg }}\chi^\prime
\end{split}$$
So $$\sum_{g \in K_a, g^\prime \in K_b \cap C_G(g)}
\left|\chi(xg^{\prime}x^{-1})\right| \, |\chi^\prime(ygy^{-1})|
= |K_a||K_b| \, {\text{deg }}\chi \, {\text{deg }}\chi^\prime.$$
Since $|\{(g, \, g^\prime) \mid g \in K_a, g^\prime \in K_b \cap C_G(g)\}|
\leq |K_a||K_b|$, $|\chi(xg^{\prime}x^{-1})| \leq {\text{deg }}\chi$, and\
$|\chi^\prime(ygy^{-1})| \leq {\text{deg }}\chi^\prime$, we must have $|\{(g, \, g^\prime) \mid g \in K_a, g^\prime \in K_b \cap C_G(g)\}| = |K_a||K_b|$, i.e. $\{(g, \, g^\prime) \mid g \in K_a, g^\prime \in K_b \cap C_G(g)\}
= K_a \times K_b$, $|\chi(xg^{\prime}x^{-1})| = {\text{deg }}\chi$, and $|\chi^\prime(ygy^{-1})| = {\text{deg }}\chi^\prime$. The equality $\{(g, \, g^\prime) \mid g \in K_a, g^\prime \in K_b \cap C_G(g)\}
= K_a \times K_b$ implies that $K_b \subseteq C_G(g)$, for all $g \in K_a$ . This is equivalent to the condition that $K_a, K_b$ commute element-wise which is Condition (i) in the statement of the Lemma. Now, becomes: $$\label{eqn 2}
\sum_{(g, \, g^\prime) \in K_a \times K_b}
{\left(}\frac {\beta_a(x, \, g^\prime) \beta_a(xg^\prime, \, x^{-1})
\beta_b(y, \, g) \beta_b(yg, \, y^{-1})}
{\beta_a(x, \, x^{-1}) \beta_b(y, \, y^{-1})} {\right)}\frac {\chi(xg^{\prime}x^{-1})}{{\text{deg }}\chi} \,
\frac{\chi^\prime(ygy^{-1})}{{\text{deg }}\chi^\prime}
= |K_a||K_b|,$$ where $g = x^{-1}ax, g^\prime = y^{-1}by$. Since $|\chi(xg^{\prime}x^{-1})| = {\text{deg }}\chi$, and $|\chi^\prime(ygy^{-1})| = {\text{deg }}\chi^\prime$, by Remark \[proj. chars.\], $\frac {\chi(xg^{\prime}x^{-1})}{{\text{deg }}\chi}$ and $\frac{\chi^\prime(ygy^{-1})}{{\text{deg }}\chi^\prime}$ are roots of unity. Note that holds if and only if $${\left(}\frac {\beta_a(x, \, g^\prime) \beta_a(xg^\prime, \, x^{-1})
\beta_b(y, \, g) \beta_b(yg, \, y^{-1})}
{\beta_a(x, \, x^{-1}) \beta_b(y, \, y^{-1})} {\right)}\chi(xg^{\prime}x^{-1}) \, \chi^\prime(ygy^{-1}) = {\text{deg }}\chi \,\,
{\text{deg }}\chi^\prime,$$ for all $g \in K_a, g^\prime \in K_b$, where $g = x^{-1}ax, g^\prime = y^{-1}by$. This is equivalent to Condition (ii) in the statement of the Lemma.
Let $E$ be a subgroup of a finite group $K$. Let $\alpha$ be a $2$-cocycle on $K$. Let $\chi$ be a projective $\alpha$-character of $E$. For any $x \in K$, define $\chi^x$ by $$\chi^x(l) := \alpha(lx, \, x^{-1})^{-1} \alpha(x, \, x^{-1}lx)^{-1}
\alpha(x, \, x^{-1}) \, \chi(x^{-1}lx),$$ for all $l \in E$. Then $\chi^x$ is a projective $\alpha$-character of $xEx^{-1}$. Suppose $E$ is normal in $K$. Then $\chi$ is said to be $K$-[*invariant*]{} if $\chi^x = \chi$, for all $x \in K$.
\[FR 1\] Let $E$ be a normal subgroup of a finite group $K$. Let $\alpha$ be a $2$-cocycle on $K$. Let ${\text{Irr}}(K)$ denote the set of irreducible projective $\alpha$-characters of $K$. Let $\rho$ be a $K$-invariant projective $\alpha|_{E\times E}$-character of $E$ of degree $1$. Then $$\sum_{\chi \in {\text{Irr}}(K) : \chi|_E = ({\text{deg }}\chi) \, \rho}
({\text{deg }}\chi)^2 = \frac{|K|}{|E|}.$$
The proof is completely similar to the one given in Lemma \[FR\] except in this case we apply Clifford’s Theorem [@KAR Theorem 8.1] and Frobenius reciprocity [@KAR Proposition 4.8] for projective characters.
Let $H$ be a normal Abelian subgroup of $G$.
Recall that $\omega \in Z^3(G,\, k^\times)$ gives rise to a collection of $2$-cochains $\beta_a,\, a\in G$.
We will say that a map $B: H \times H \to k^\times$ is an [*alternating $\omega$-bicharacter on $H$*]{} if it satisfies the following three conditions: $$\begin{aligned}
\label{1 1}
B(h_1, \, h_2) = B(h_2, \, h_1)^{-1},\\
\label{1 2}
B(h, \, h) = 1,\\
\label{1 3}
\delta^1B_h = \beta_{h}|_{H \times H},\end{aligned}$$ for all $h, h_1, h_2 \in H$, where the map $B_h: H \to k^\times$ is defined by $B_h(h_1) := B(h, \, h_1)$, for all $h, h_1 \in H$.
\[alt omega def\] We will say that an alternating $\omega$-bicharacter $B: H \times H \to k^\times$ on $H$ is [*$G$-invariant*]{} if it satisfies the following condition: $$\label{1 4}
B(x^{-1}ax, \, h) = \frac{\beta_a(x,\, h) \beta_a(xh, \, x^{-1})}
{\beta_a(x, \, x^{-1})}
\, B(a, \, xhx^{-1}), \quad \mbox{ for all }
x \in G, a \in H \cap R, h \in H.$$
Define $$\begin{split}
\label{alt omega}
\Lambda_{\omega}^2H := \{B: H \times H \to k^\times \mid
B \text{ is an alternating } \omega-\text{bicharacter on } H\},
\end{split}$$ and $$\begin{split}
\label{alt omega 1}
(\Lambda_{\omega}^2H)^G := \{B \in \Lambda_{\omega}^2H \mid
B \text{ is $G$-invariant} \}.
\end{split}$$
If $\omega \equiv 1$, then $(\Lambda_{\omega}^2H)^G$ is the Abelian group of $G$-invariant alternating bicharacters on $H$.
If $B$ is an alternating $\omega$-bicharacter on $H$, then the restriction $\omega|_{H \times H \times H}$ must be cohomologically trivial. Indeed, let $\omega_H := \omega|_{H \times H \times H}$. Then $B$ defines a braiding on the fusion category ${\mbox{Vec}}_H^{\omega_H}$. The isomorphism $h_1 {\otimes}h_2 \xrightarrow{\sim} h_2 {\otimes}h_1$ is given by $B(h_1, \, h_2)$, for all $h_1, h_2 \in H$, where we identify simple objects of ${\mbox{Vec}}_H^{\omega_H}$ with elements of $H$. It is known ( see, e.g., [@Q], [@FRS]) that in this case $\omega_H$ is an [*Abelian*]{} $3$-cocycle on $H$. By a classical result of Eilenberg and MacLane [@EM] the third Abelian cohomology group of $H$ is isomorphic to the (multiplicative) group of quadratic forms on $H$. The value of the corresponding quadratic form $q$ on $h\in H$ is given by $q(h) = B(h,h)$. Since $B$ is alternating we have $q\equiv 1$ and so $\omega_H$ must be cohomologically trivial.
Let $B \in (\Lambda^2_{\omega}H)^G$ and define: $$\label{L 1}
\begin{split}
{\mathcal{L}}_{(H, \, B)} :=
&\text{ full Abelian subcategory of } {\mathcal{C}}\text{ generated by } \\
&\left\{(a, \, \chi) \in \Gamma \,\, \vline \,
\begin{tabular}{l}
$a \in H \cap R \text{ and } \chi \text{ is an irreducible $\beta_a$-character of } C_G(a)$ \\
$\text{ such that } \chi(h) = B(a, \, h) \,{\text{deg }}\chi, \text{ for all } h \in H$
\end{tabular}
\right\}
\end{split}$$
\[Proposition L 1\] The subcategory ${\mathcal{L}}_{(H, \, B)} \subseteq {\mbox{Rep}}(D^{\omega}(G))$ is Lagrangian.
Pick any $(a, \, \chi), (b, \, \chi^\prime) \in {\mathcal{L}}_{(H, \, B)}\cap \Gamma$. We have $$\begin{split}
&{\left(}\frac {\beta_a(x, \, y^{-1}by) \beta_a(xy^{-1}by, \, x^{-1})
\beta_b(y, \, x^{-1}ax) \beta_b(yx^{-1}ax, \, y^{-1})}
{\beta_a(x, \, x^{-1}) \beta_b(y, \, y^{-1})} {\right)}\chi(xy^{-1}byx^{-1}) \, \chi^\prime(yx^{-1}axy^{-1})\\
& = \frac {\beta_a(x, \, y^{-1}by) \beta_a(xy^{-1}by, \, x^{-1})}
{\beta_a(x, \, x^{-1})} B(a, \, xy^{-1}byx^{-1}) \\
& \hspace{1.2in} \times \frac {\beta_b(y, \, x^{-1}ax) \beta_b(yx^{-1}ax, \, y^{-1})}
{\beta_b(y, \, y^{-1})} B(b, \, yx^{-1}axy^{-1})
\times {\text{deg }}\chi \, {\text{deg }}\chi^\prime\\
& = B(x^{-1}ax, \, y^{-1}by) \, B(y^{-1}by, \, x^{-1}ax) \, {\text{deg }}\chi
\, {\text{deg }}\chi^{\prime}\\
& = {\text{deg }}\chi \,{\text{deg }}\chi^\prime,
\end{split}$$ for all $x, y \in G$. The second equality above is due to while the third equality is due to . Note that $K_a, K_b$ commute element-wise since $H$ is Abelian. By Lemma \[centralize 1\], it follows that objects in ${\mathcal{L}}_{(H, \, B)}$ centralize each other.
Also, $\theta|_{{\mathcal{L}}_{(H, \, B)}} = {\text{id}}$. The proof of this assertion is exactly the one given in Proposition \[Proposition L\].
Now, fix $a \in H \cap R$ and observe that $B_a$ defines a $C_G(a)$-invariant $\beta_a$-character of $H$ of degree $1$. Indeed, $$\begin{split}
(B_a)^x(h)
&= \frac{\beta_a(x, \, x^{-1})}{\beta_a(hx, \, x^{-1}) \beta_a(x, \, x^{-1}hx)}
B(a, \, x^{-1}hx)\\
&= B(x^{-1}ax, \, x^{-1}hx)^{-1} \, B(a, \, h) \, B(a, \, x^{-1}hx)\\
&= B(a, \, h),
\end{split}$$ for all $x \in C_G(a), h \in H$. The second equality above is due to .
The dimension of ${\mathcal{L}}_{(H, \, B)}$ is equal to $|G|$. The proof of this assertion is exactly the one given in Proposition \[Proposition L\] except we appeal to Lemma \[FR 1\] in this case.
It follows from Lemma \[lag\] that ${\mathcal{L}}_{(H, \, B)}$ is a Lagrangian subcategory of ${\mbox{Rep}}(D^{\omega}(G))$ and the Proposition is proved.
\[coboundary\] Let $H$ be a normal Abelian subgroup of $G$. Let $B:H \times H \to k^\times$ be a map satisfying ,, and . Suppose $\delta^1B_a = \beta_{a}|_{H \times H},
\mbox{ for all } a \in H \cap R$. Then $B \in (\Lambda^2_{\omega}H)^G$.
We only need to verify that holds. We have $$\begin{split}
& (\delta^1B_{x^{-1}ax})(h_1, \, h_2) \\
& = \frac{B(x^{-1}ax, \, h_1) B(x^{-1}ax, \, h_2)}{B(x^{-1}ax, \, h_1h_2)}\\
& = {\left(}\frac{\beta_a(x, \, h_1) \beta_a(xh_1, \, x^{-1})}
{\beta_a(x, \, x^{-1})} {\right)}B(a, \, xh_1x^{-1}) \times
{\left(}\frac{\beta_a(x, \, h_2) \beta_a(xh_2, \, x^{-1})}
{\beta_a(x, \, x^{-1})} {\right)}B(a, \, xh_2x^{-1}) \\
& \hspace{2.3in} \times {\left(}\frac{\beta_a(x, \, x^{-1})}
{\beta_a(x, \, h_1h_2) \beta_a(xh_1h_2, \, x^{-1})}{\right)}B(a, \, xh_1h_2x^{-1})^{-1}\\
& = \frac{\beta_a(x, \, h_1) \beta_a(xh_1, \, x^{-1})
\beta_a(x, \, h_2) \beta_a(xh_2, \, x^{-1}) \beta_a(xh_1x^{-1}, \, xh_2x^{-1})}
{\beta_a(x, \, x^{-1}) \beta_a(x, \, h_1h_2) \beta_a(xh_1h_2, \, x^{-1})} \\
& = \frac{\beta_{x^{-1}ax}(h_1, \, h_2) \beta_a(xh_1, \, x^{-1})
\beta_a(x, \, h_2) \beta_a(xh_2, \, x^{-1}) \beta_a(xh_1x^{-1}, \, xh_2x^{-1})}
{\beta_a(xh_1, \, h_2) \beta_a(x, \, x^{-1}) \beta_a(xh_1h_2, \, x^{-1})}\\
& = \frac{\beta_{x^{-1}ax}(h_1, \, h_2) \beta_a(xh_1, \, h_2x^{-1})
\beta_{x^{-1}ax}(x^{-1}, \, xh_2x^{-1}) \beta_a(x, \, h_2x^{-1})
\beta_{x^{-1}ax}(h_2, \, x^{-1})}
{\beta_a(xh_1, \, h_2) \beta_a(x, \, x^{-1}) \beta_a(xh_1h_2, \, x^{-1})}\\
& = \beta_{x^{-1}ax}(h_1, \, h_2),
\end{split}$$ for all $x \in G, a \in H \cap R, h_1, h_2 \in H$. In the second equality above, we used . In the third equality we used $\delta^1B_a = \beta_{a}|_{H \times H}$. and canceled some factors. In the fourth equality we used with $(x, \, y, \, z) = (x, \, h_1, \, h_2)$. In the fifth equality we used twice with $(x, \, y, \, z) = (x, \, h_2, \, x^{-1}), (xh_1, \, x^{-1}, \, xh_2x^{-1})$. In the last equality we used twice with $(x, \, y, \, z) = (xh_1, \, h_2, \, x^{-1}), (x, \, x^{-1}, \, xh_2x^{-1})$.
Now, let ${\mathcal{L}}$ be a Lagrangian subcategory of ${\mathcal{C}}$. So, in particular, the two conditions in Lemma \[centralize 1\] hold for all objects in ${\mathcal{L}}\cap \Gamma$. Define $$\label{H_L 1}
H_{{\mathcal{L}}} := \bigcup_{a \in R : (a, \, \chi) \in {\mathcal{L}}\cap \Gamma
\text{ for some }\chi} K_a.$$ Note that $H_{{\mathcal{L}}}$ is a normal Abelian subgroup of $G$.
Define a map $B_{{\mathcal{L}}}: H_{{\mathcal{L}}} \times H_{{\mathcal{L}}} \to k^\times$ by $$\label{B_L 1}
B_{{\mathcal{L}}}(h_1, \, h_2) := \frac {\beta_a(x, \, h_2) \beta_a(xh_2, \, x^{-1})}
{\beta_a(x, \, x^{-1})} \times \frac{\chi(xh_2x^{-1})}{{\text{deg }}\chi},$$ where $h_1 = x^{-1}ax, x \in G, a \in H_{{\mathcal{L}}} \cap R$ and $\chi$ is any $\beta_a$-character of $C_G(a)$ such that $(a, \, \chi) \in {\mathcal{L}}\cap \Gamma$. The above definition does not depend on the choice of $\chi$. The proof of this assertion is similar to the proof given for the corresponding assertion in the untwisted case.
\[Proposition B\_L 1\] The map $B_{{\mathcal{L}}}$ defined in is an element of $(\Lambda^2_{\omega}H)^G$.
First, let us show that $B_{{\mathcal{L}}}$ is well-defined. Suppose $x^{-1}ax = z^{-1}az$, where $a \in H_{{\mathcal{L}}} \cap R, x, z \in G$. Then $$\begin{split}
B_{{\mathcal{L}}}(x^{-1}ax, \, y^{-1}by)
&= \frac {\beta_a(x, \, y^{-1}by) \beta_a(xy^{-1}by, \, x^{-1})}
{\beta_a(x, \, x^{-1})} \times \frac{\chi(xy^{-1}byx^{-1})}{{\text{deg }}\chi}\\
&= {\left(}\frac {\beta_b(y, \, x^{-1}ax) \beta_b(yx^{-1}ax, \, y^{-1})}
{\beta_b(y, \, y^{-1})} {\right)}^{-1}
{\left(}\frac{\chi^\prime(yx^{-1}axy^{-1})}{{\text{deg }}\chi^{\prime}} {\right)}^{-1}\\
&= {\left(}\frac {\beta_b(y, \, z^{-1}az) \beta_b(yz^{-1}az, \, y^{-1})}
{\beta_b(y, \, y^{-1})} {\right)}^{-1}
{\left(}\frac{\chi^\prime(yz^{-1}azy^{-1})}{{\text{deg }}\chi^{\prime}} {\right)}^{-1}\\
&= \frac {\beta_a(z, \, y^{-1}by) \beta_a(zy^{-1}by, \, z^{-1})}
{\beta_a(z, \, z^{-1})} \times \frac{\chi(zy^{-1}byz^{-1})}{{\text{deg }}\chi}\\
&= B_{{\mathcal{L}}}(z^{-1}az, \, y^{-1}by),
\end{split}$$ for all $b \in H_{{\mathcal{L}}} \cap R, y \in G$, where $\chi^\prime$ is any irreducible $\beta_b$-character of $C_G(b)$ such that $(b, \, \chi^\prime) \in {\mathcal{L}}\cap \Gamma$. The second and the fourth equalities above are due to Condition (ii) of Lemma \[centralize 1\].
The map $B_{{\mathcal{L}}}$ satisfies because Condition (ii) of Lemma \[centralize 1\] holds. Let us show that holds for $B_{{\mathcal{L}}}$: $$\begin{split}
&B_{{\mathcal{L}}}(x^{-1}ax, \, x^{-1}ax) \\
& = \frac{\beta_a(x, \, x^{-1}ax) \beta_a(ax, \, x^{-1})}{\beta_a(x, \, x^{-1})}
\times \frac{\chi(a)}{{\text{deg }}\chi}\\
& = \omega(a, \, x, \, x^{-1}ax) \times
\frac{\omega(a, \, ax, x^{-1}) \omega(ax, \, x^{-1}, \, a)}
{\omega(ax, \, x^{-1}ax, x^{-1})} \times
\frac{\omega(x, \, x^{-1}ax, \, x^{-1})}
{\omega(a, \, x, \, x^{-1}) \omega(x, \, x^{-1}, \, a)}
\times \theta_{(a, \, \chi)}\\
& = \frac{\omega(a, \, x, \, x^{-1}a) \omega(ax, \, x^{-1}, \, a)}
{\omega(a, \, x, \, x^{-1}) \omega(x, \, x^{-1}, \, a)}\\
& = 1,
\end{split}$$ for all $x \in G, a \in H_{{\mathcal{L}}} \cap R$. In the second equality we used the definition of $\beta_a$. In the third equality we used with $(g_1, \, g_2, \, g_3, \, g_4) = (a, \, x, \, x^{-1}ax,\, x^{-1})$ and used the fact that $\theta_{(a, \, \chi)} = 1$ . In the fourth equality we used with $(g_1, \, g_2, \, g_3, \, g_4) =
(a, \, x, \, x^{-1}, \, a)$.
The map $B_{{\mathcal{L}}}$ satisfies because $B_{{\mathcal{L}}}(a, \, xhx^{-1}) = \frac{\chi(xhx^{-1})}{{\text{deg }}\chi}$, for all $a \in H \cap R, x \in G, h \in H$. We have $B_{{\mathcal{L}}}(a, \, h_1) B_{{\mathcal{L}}}(a, \, h_2) = \frac{\chi(h_1)}{{\text{deg }}\chi}
\frac{\chi(h_2)}{{\text{deg }}\chi} = \beta_a(h_1, \, h_2)\frac{\chi(h_1h_2)}{{\text{deg }}\chi}
= \beta_a(h_1, \, h_2) \, B_{{\mathcal{L}}}(a, \, h_1h_2)$, for all $a \in H \cap R, h_1, h_2 \in H$. The second last equality above is because $H$ acts as scalars on the projective $\beta_a$-representation of $C_G(a)$ whose projective character is $\chi$. By Lemma \[coboundary\] it follows that $B_{{\mathcal{L}}} \in \Lambda^2_{\omega}H$ and the Proposition is proved.
\[bij 1\] Lagrangian subcategories of the representation category of the twisted double $D^\omega(G)$ are classified by pairs $(H, B)$, where $H$ is a normal Abelian subgroup of $G$ such that $\omega|_{H\times H\times H}$ is cohomologically trivial and $B: H\times H \to k^\times$ is a $G$-invariant alternating $\omega$-bicharacter in the sense of Definition \[alt omega def\].
The proof is completely similar to the one given in Theorem \[untwisted bijection\].
[Bijective correspondence between Lagrangian subcategories and module categories with pointed duals]{}
Recall [@N] that equivalence classes of indecomposable module categories over ${\mbox{Vec}}_G^{\omega}$ for which the dual is pointed are in bijection with pairs $(H, \, \mu)$, where $H$ is a normal Abelian subgroup of $G$ such that $\omega|_{H \times H \times H}$ is cohomologically trivial and $\mu \in (\Omega_{H, \omega})^G$ (defined in ).
Theorem \[bij 1\] showed that Lagrangian subcategories of ${\mbox{Rep}}(D^{\omega}(G))$ are in bijection with pairs $(H, \, B)$, where $H$ is a normal Abelian subgroup of $G$ such that $\omega|_{H \times H \times H}$ is cohomologically trivial and $B \in (\Lambda_{\omega}^2H)^G$ (the latter was defined in ).
In this Subsection we will first show that the set of equivalence classes of indecomposable module categories over ${\mbox{Vec}}_G^{\omega}$ such that the dual is pointed is in bijection with the set of Lagrangian subcategories of ${\mbox{Rep}}(D^{\omega}(G))$. Let $H$ be a normal Abelian subgroup of $G$ such that $\omega|_{H \times H \times H}$ is cohomologically trivial. We will establish the aforementioned bijection by showing that there is a bijection between $\Omega_{H, \omega}$ (defined in ) and $\Lambda_{\omega}^2H$ that restricts to a bijection between $(\Omega_{H, \omega})^G$ and $(\Lambda_{\omega}^2H)^G$.
Let $\mu \in C^2(H, \, k^\times)$ be a $2$-cochain satisfying $\delta^2\mu = \omega|_{H \times H \times H}$. Define $alt^\prime(\mu)$ by $$alt^\prime(\mu)(h_1, \, h_2) := \frac{\mu(h_2, \, h_1)}
{\mu(h_1, \, h_2)}, h_1, h_2 \in H.$$
The map $alt^\prime(\mu): H \times H \to k^\times$ defined above is an element of $\Lambda_{\omega}^2H$.
Clearly $alt^\prime(\mu)(h_1, \, h_2) = alt^\prime(\mu)(h_2, \, h_1)^{-1}$ and $alt^\prime(\mu)(h, \, h) = 1$, for all $h, h_1, h_2 \in H$. We have $$\begin{split}
\frac{alt^\prime(\mu)(h, \, h_1) \,\,\, alt^\prime(\mu)(h, \, h_2)}
{alt^\prime(\mu)(h, \, h_1h_2)}
&= \frac{\mu(h_1, \, h)}{\mu(h, \, h_1)}
\times \frac{\mu(h_2, \, h)}{\mu(h, \, h_2)}
\times \frac{\mu(h, \, h_1h_2)}{\mu(h_1h_2, \, h)}\\
&= \frac{\mu(h_1, \, h) \mu(h_2, \, h) \mu(hh_1, \, h_2)}
{\mu(h, \, h_2) \mu(h_1h_2, \, h) \mu(h_1, \, h_2)}
\times \omega(h, \, h_1, \, h_2)\\
&= \frac{\mu(h_1, \, h) \mu(hh_1, \, h_2)}
{\mu(h, \, h_2) \mu(h_1, \, hh_2)}
\times \omega(h, \, h_1, \, h_2)
\times
\omega(h_1, \, h_2, \, h)\\
&= \frac{\omega(h, \, h_1, \, h_2)\omega(h_1, \, h_2, \, h)}
{\omega(h_1, \, h, \, h_2)}\\
&= \beta_h(h_1, \, h_2),
\end{split}$$ for all $h, h_1, h_2 \in H$. In the second, third, and fourth equalities above we used with $(h_1, \, h_2, \, h_3) = (h, \, h_1, \, h_2),
(h_1, \, h_2, \, h),
(h_1, \, h, \, h_2)$, respectively.
The map $alt^\prime$ induces a map between $\Omega_{H, \omega}$ and $\Lambda_{\omega}^2H$. By abuse of notation we denote this map also by $alt^\prime$: $$\label{alt prime}
alt^\prime:
\Omega_{H, \omega} \to \Lambda_{\omega}^2H :
{\mu} \mapsto alt^\prime(\mu).$$
\[bijec\] The map $alt^\prime$ defined above is a bijection.
First note that $alt^\prime$ is well-defined. Fix $\mu_0 \in C^2(H, \, k^\times)$ satisfying $\delta^2\mu_0 = \omega|_{H \times H \times H}$. Let $B_0 := alt^\prime(\mu_0)$. Define bijections $f_1: \Lambda_{\omega}^2H \xrightarrow{\sim} \Lambda^2H : B \mapsto \frac{B}{B_0}$ and $f_2: \Omega_{H, \omega} \xrightarrow{\sim}
H^2(H, \, k^\times): {\mu} \mapsto {{\left(}\frac{\mu}{\mu_0}{\right)}}$. Note that the cardinality of the two sets $\Omega_{H, \omega}$ and $\Lambda_{\omega}^2H$ are equal. Injectivity, and hence bijectivity, of $alt^\prime$ follows from the equality $f_1\circ alt^\prime = alt \circ f_2$.
\[abc\] The following relation holds: $$\frac{\Upsilon_x(h_2, \, h_1)}{\Upsilon_x(h_1, \, h_2)}
= \frac{\beta_{xh_1x^{-1}}(x, \, h_2)\beta_{xh_1x^{-1}}(xh_2, x^{-1})}
{\beta_{xh_1x^{-1}}(x, \, x^{-1})}, \mbox{ for all } x \in G, h_1, h_2 \in H.$$
We have $$\begin{split}
&\frac{\Upsilon_x(h_2, \, h_1)}{\Upsilon_x(h_1, \, h_2)}
\times \frac {\beta_{xh_1x^{-1}}(x, \, x^{-1})}
{\beta_{xh_1x^{-1}}(x, \, h_2)\beta_{xh_1x^{-1}}(xh_2, x^{-1})} \\
&= \frac{\omega(xh_2x^{-1}, \, xh_1x^{-1},\, x)}
{\omega(xh_2x^{-1}, \, x, \, h_1) \omega(xh_1x^{-1}, \, xh_2x^{-1}, \, x)}
\times
\frac{\omega(xh_1x^{-1}, \, x, \, x^{-1}) \omega(x, \, x^{-1}, \, xh_1x^{-1})}
{\omega(x, \, h_1, \, x^{-1})}\\
& \hspace{3in} \times
\frac{\omega(xh_2, \, h_1, \, x^{-1})}
{\omega(xh_1x^{-1}, \, xh_2, \, x^{-1}) \omega(xh_2, \, x^{-1}, \, xh_1x^{-1})}\\
&= \frac {\omega(xh_2x^{-1}, \, xh_1x^{-1}, \, x)
\omega(xh_1x^{-1}, \, x, \, x^{-1})\omega(x, \, x^{-1}, \, xh_1x^{-1})
\omega(xh_2x^{-1}, \, xh_1, \, x^{-1})}
{\omega(xh_1x^{-1}, \, xh_2x^{-1}, \, x) \omega(xh_1x^{-1}, \, xh_2, \, x^{-1})
\omega(xh_2, \, x^{-1}, \, xh_1x^{-1}) \omega(xh_2x^{-1}, \, x, \, h_1x^{-1})}\\
&= \frac{\omega(x, \, x^{-1}, \, xh_1x^{-1})
\omega(xh_1h_2x^{-1}, \, x, \, x^{-1})}
{\omega(xh_1x^{-1}, \, xh_2x^{-1}, \, x)
\omega(xh_1x^{-1}, \, xh_2, \, x^{-1}) \omega(xh_2, \, x^{-1}, \, xh_1x^{-1})
\omega(xh_2x^{-1}, \, x, \, h_1x^{-1})}\\
& = \frac{\omega(x, \, x^{-1}, \, xh_1x^{-1}) \omega(xh_2x^{-1}, \, x, \, x^{-1})}
{\omega(xh_2, \, x^{-1}, \, xh_1x^{-1}) \omega(xh_2x^{-1}, \, x, \, h_1x^{-1})}\\
&= 1,
\end{split}$$ for all $x \in G, h_1, h_2 \in H$. In the first equality above we used the definition of $\Upsilon$ and $\beta$ and canceled some factors. In the second, third, fourth, and fifth equalities we used with $(g_1, \, g_2, \, g_3, \, g_4)
= (xh_2x^{-1}, \, x, \, h_1, \, x^{-1}), \, (xh_2x^{-1}, \, xh_1x^{-1}, \, x, \, x^{-1}), \,
(xh_1x^{-1}, \, xh_2x^{-1}, \, x, \, x^{-1}),$ and $(xh_2x^{-1}, \, x, \, x^{-1}, \, xh_1x^{-1})$, respectively.
The map $alt^\prime$ defined in restricts to a bijection between $(\Omega_{H, \omega})^G$ and $(\Lambda_{\omega}^2H)^G$.
Let us first show that $alt^\prime((\Omega_{H, \omega})^G) \subseteq (\Lambda_{\omega}^2H)^G$. Pick any $\mu \in (\Omega_{H, \omega})^G$.\
So $alt {\left(}\frac{\mu^x}{\mu} \times \Upsilon_x|_{H \times H} {\right)}= 1$, for all $x \in G$. We have $$\begin{split}
alt^\prime(\mu)(x^{-1}ax, \, h) \times
alt^\prime(\mu)(a, \, xhx^{-1})^{-1}
&= \frac{\mu(h, \, x^{-1}ax)}{\mu(x^{-1}ax, \, h)}
\times \frac{\mu(a, \, xhx^{-1})}{\mu(xhx^{-1}, \, a)}\\
&= \frac{\mu^x(x^{-1}ax, \, h)}{\mu(x^{-1}ax, \, h)}
\times \frac{\mu(h, \, x^{-1}ax)}{\mu^x(h, \, x^{-1}ax)}\\
&= alt {\left(}\frac{\mu^x}{\mu} \times \Upsilon_x|_{H \times H} {\right)}(h, \, x^{-1}ax)
\times \frac{\Upsilon_x(h, \, x^{-1}ax)}{\Upsilon_x(x^{-1}ax, \, h)}\\
&= \frac{\Upsilon_x(h, \, x^{-1}ax)}{\Upsilon_x(x^{-1}ax, \, h)}\\
&= \frac{\beta_a(x, \, h) \beta_a(xh, \, x^{-1})}{\beta_a(x, \, x^{-1})},
\end{split}$$ for all $x \in G, a \in H \cap R, h \in H$. In the fourth equality above we used the fact that $alt {\left(}\frac{\mu^x}{\mu} \times \Upsilon_x|_{H \times H} {\right)}= 1$ and in the fifth equality we used Lemma \[abc\]. So $alt^\prime((\Omega_{H, \omega})^G)
\subseteq (\Lambda_{\omega}^2H)^G$, as desired.
Now let us show that $(\Lambda_{\omega}^2H)^G
\subseteq alt^\prime((\Omega_{H, \omega})^G)$. Pick any $\mu \in \Omega_{H, \omega}$ and suppose that $alt^\prime(\mu) \in (\Lambda_{\omega}^2H)^G$. Suffices to show that $alt {\left(}\frac{\mu^x}{\mu} \times \Upsilon_x|_{H \times H} {\right)}= 1$, for all $x \in G$. Let $B := alt^\prime(\mu)$. We have $$\begin{split}
&alt {\left(}\frac{\mu^x}{\mu} \times \Upsilon_x|_{H \times H} {\right)}(h_1, \, h_2)
\times \frac{\Upsilon_x(h_1, \, h_2)}{\Upsilon_x(h_2, \, h_1)}\\
& = B(xh_1x^{-1}, \, xh_2x^{-1}) B(h_1, \, h_2)^{-1}\\
&= B((yx^{-1})^{-1}a(yx^{-1}), \, xh_2x^{-1}) B(y^{-1}ay, \, h_2)^{-1}
\qquad \qquad (\mbox{where $h_1 = y^{-1}ay$})\\
&= \frac{\beta_a(yx^{-1}, \, xh_2x^{-1}) \beta_a(yh_2x^{-1}, \,xy^{-1})}
{\beta_a(yx^{-1}, \, xy^{-1})}
\times
\frac{\beta_a(y, \, y^{-1})}{\beta_a(y, \, h_2) \beta(yh_2, \, y^{-1})}\\
&= \frac{\beta_a(yx^{-1}, \, xh_2) \beta_a(yh_2, \, x^{-1})
\beta_a(yh_2x^{-1}, \, xy^{-1})}
{\beta_{xh_1x^{-1}}(xh_2, \, x^{-1}) \beta_a(y, \, y^{-1})
\beta_a(yx^{-1}, \, xy^{-1}) \beta_a(y, \, h_2) \beta_a(yh_2, \, y^{-1})}\\
&= \frac{\beta_a(yx^{-1}, \, x) \beta_a(yh_2, \, x^{-1})
\beta_a(yh_2x^{-1}, \, xy^{-1}) \beta_a(y, \, y^{-1})}
{\beta_{xh_1x^{-1}}(x, \, h_2) \beta_{xh_1x^{-1}}(xh_2, \, x^{-1})
\beta_a(yx^{-1}, \, xy^{-1}) \beta_a(yh_2, \, y^{-1})}\\
&= \frac{\beta_{xh_1x^{-1}}(x, \, x^{-1}) \beta_a(yh_2, \, x^{-1})
\beta_a(yh_2x^{-1}, \, xy^{-1}) \beta_a(y, \, y^{-1})}
{\beta_{xh_1x^{-1}}(x, \, h_2) \beta_{xh_1x^{-1}}(xh_2, \, x^{-1})
\beta_a(y, \, x^{-1}) \beta_a(yx^{-1}, \, xy^{-1})
\beta_a(yh_2, \, y^{-1})}\\
& = \frac{\Upsilon_x(h_1, \, h_2)}{\Upsilon_x(h_2, \, h_1)}
\times
\frac{\beta_a(y, \, y^{-1}) \beta_{h_1}(x^{-1}, \, xy^{-1})}
{\beta_a(y, \, x^{-1}) \beta_a(yx^{-1}, \, xy^{-1})}\\
& = \frac{\Upsilon_x(h_1, \, h_2)}{\Upsilon_x(h_2, \, h_1)},
\end{split}$$ for all $x \in G, h_1, h_2 \in H$. In the fourth through eight equalities above we used with $(x, \, y, \, z)
= (yx^{-1}, \, xh_2, \, x^{-1}), (yx^{-1}, \, x, \, h_2),
(yx^{-1}, \, x, \, x^{-1}),
(yh_2, \, x^{-1}, \, xy^{-1}),$ and $(y, \, x^{-1}, \, xy^{-1})$, respectively. It follows that $(\Lambda_{\omega}^2H)^G
\subseteq alt^\prime((\Omega_{H, \omega})^G)$ and the Lemma is proved.
Recall that $\mbox{E}({\mathcal{C}})$ denotes the set of (equivalence classes of) braided tensor equivalences between a modular category ${\mathcal{C}}$ and the centers of pointed fusion categories.
\[bijection 1\] The assignment ${\mathcal{M}}\mapsto \iota_{\mathcal{M}}$ $($defined in $)$ restricts to a bijection between equivalence classes of indecomposable ${\mbox{Vec}}_G^\omega$-module categories ${\mathcal{M}}$ with respect to which the dual fusion category $({\mbox{Vec}}_G^\omega)^*_{\mathcal{M}}$ is pointed and $E({\mbox{Rep}}(D^\omega(G)))$.
The proof is completely similar to the one given in Theorem \[bijection\].
The equivalence type of the symmetric category ${\mathcal{L}}_{(H, B)}$ of ${\mbox{Rep}}(D^\omega(G))$ can be explicitly described in a way similar to Remark \[G’\], cf. [@N Theorem 4.5].
\[main 1\] Let ${\mathcal{C}}_1, {\mathcal{C}}_2$ be group-theoretical fusion categories. Then ${\mathcal{C}}_1,{\mathcal{C}}_2$ are weakly Morita equivalent if and only if their centers ${\mathcal{Z}}({\mathcal{C}}_1)$ and ${\mathcal{Z}}({\mathcal{C}}_2)$ are equivalent as braided fusion categories.
That the “if" part is true for all fusion categories was first observed by M. Müger in [@Mu Remark 3.18]. This follows from the definition of weak Morita equivalence and a theorem of P. Schauenburg [@S]. See also [@Nat; @O1; @EO].
For the “only if" part, let $(G_1, \, \omega_1), (G_2, \, \omega_2)$ be two pairs of groups and $3$-cocycles such that ${\mathcal{C}}_1$ is weakly Morita equivalent to ${\mbox{Vec}}_{G_1}^{\omega_1}$ and ${\mathcal{C}}_2$ is weakly Morita equivalent to ${\mbox{Vec}}_{G_2}^{\omega_2}$.
If ${\mathcal{Z}}({\mathcal{C}}_1) \cong {\mathcal{Z}}({\mathcal{C}}_2)$ (as braided fusion categories) then ${\mathcal{Z}}({\mbox{Vec}}_{G_1}^{\omega_1})\cong {\mathcal{Z}}({\mbox{Vec}}_{G_2}^{\omega_2})$ (as braided fusion categories) and therefore, ${\mbox{Vec}}_{G_1}^{\omega_1}$ and ${\mbox{Vec}}_{G_2}^{\omega_2}$ are weakly Morita equivalent by Theorem \[bijection 1\] and hence, ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ are weakly Morita equivalent.
\[main 2\] Let $G, G'$ be finite groups, $\omega \in Z^3(G,\, k^\times)$, and $\omega' \in Z^3(G',\, k^\times)$. Then the representation categories of twisted doubles $D^\omega(G)$ and $D^{\omega'}(G')$ are equivalent as braided tensor categories if and only if $G$ contains a normal Abelian subgroup $H$ such the following conditions are satisfied:
1. $\omega|_{H \times H \times H}$ is cohomologically trivial,
2. there is a $G$-invariant $($see $)$ $2$-cochain $\mu \in C^2(H, \, k^\times)$ such that that $\delta^2 \mu = \omega|_{H \times H \times H}$, and
3. there is an isomorphism $a: G' \xrightarrow{\sim} \widehat{H} \rtimes_{\nu}
(H \backslash G)$ such that $\varpi \circ (a \times a \times a)$ and $\omega'$ are cohomologically equivalent.
Here $\nu$ is a certain $2$-cocycle in $Z^2(H \backslash G, \, \widehat{H})$ coming from the $G$-invariance of $\mu$ and $\varpi$ is a certain $3$-cocycle on $\widehat{H} \rtimes_{\nu}
(H \backslash G)$ depending on $\nu$ and on the exact sequence $1 \to H \to G \to H \backslash G \to 1$ $($see [@N Theorem 5.8] for precise definitions$)$.
[Examples]{} \[exs\]
Lagrangian subcategories of the Drinfeld doubles of finite symmetry groups
--------------------------------------------------------------------------
\[dihedral\] Consider the group of symmetries of a regular $n$-gon, i.e., the dihedral group $D_{2n}= {\langle}r,\, s \mid r^n = s^2= 1,\, rs = sr^{-1} {\rangle},\, n \geq 2$. Let us describe the Lagrangian subcategories of ${\mbox{Rep}}(D(D_{2n})) \cong
{\mathcal{Z}}({\mbox{Vec}}_{D_{2n}})$.
Let $n= 2$. Then $D_4 \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. There are six different Lagrangian subcategories of ${\mbox{Rep}}(D(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}))$, all of them equivalent to ${\mbox{Rep}}(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z})$.
Let $n= 4$. The group $D_8$ has five normal Abelian subgroups which give rise to seven Lagrangian subcategories of ${\mathcal{Z}}({\mbox{Vec}}_{D_8})$. With an exception of the Lagrangian subcategory corresponding to the center ${\langle}r^2{\rangle}$ of $D_8$, all Lagrangian subcategories are equivalent to ${\mbox{Rep}}(D_8)$. The one corresponding to ${\langle}r^2{\rangle}$ is equivalent to ${\mbox{Rep}}(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z})$. Applying [@DGNO] we conclude that for some $3$-cocycle $\omega$ on $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}
\times \mathbb{Z}/2\mathbb{Z}$ there is a braided tensor equivalence ${\mbox{Rep}}(D(D_8)) \cong
{\mbox{Rep}}(D^\omega(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}))$ (existence of such an equivalence is already known to experts, see [@CGR] and [@GMN]).
If $n =3$ or $n \geq 5$ then normal Abelian subgroups of $D_{2n}$ are precisely rotation subgroups ${\langle}r^k{\rangle}$, where $k$ is a divisor of $n$. Each of these subgroups is cyclic and so has a trivial Schur multiplier. The Lagrangian subcategory of ${\mathcal{Z}}({\mbox{Vec}}_{D_{2n}})$ corresponding to ${\langle}r^k{\rangle},\, k|n$, is equivalent to ${\mbox{Rep}}(\mbox{Dih}(\mathbb{Z}/\frac{n}{k}\mathbb{Z} \times
\mathbb{Z}/{k}\mathbb{Z}))$, where for any Abelian group $A$ we denote $\mbox{Dih}(A) = A \rtimes \mathbb{Z}/2\mathbb{Z}$ the generalized dihedral group (the action of $ \mathbb{Z}/2\mathbb{Z}$ on $A$ is by inverting elements). Consequently, there is a $3$-cocycle $\omega $ on $\mbox{Dih}(\mathbb{Z}/\frac{n}{k}\mathbb{Z} \times
\mathbb{Z}/{k}\mathbb{Z})$ such that ${\mbox{Rep}}(D^\omega(\mbox{Dih}(\mathbb{Z}/\frac{n}{k}\mathbb{Z} \times
\mathbb{Z}/{k}\mathbb{Z}))) \cong
{\mbox{Rep}}(D(D_{2n}))$ as braided tensor categories.
Next, consider the symmetry groups of Platonic solids.
\[platonic\] The tetrahedron group $A_4$ has two normal Abelian subgroups: the trivial one and another isomorphic to $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. The Schur multiplier of $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$ with the trivial $A_4$-action. So ${\mbox{Rep}}(D(A_4))$ has three Lagrangian subcategories. All three are equivalent to ${\mbox{Rep}}(A_4)$.
The double of the cube/octahedron group $S_4$ also has three Lagrangian subcategories (corresponding to the same data as in the case of $A_4$). One can check that corresponding three Lagrangian subcategories of ${\mbox{Rep}}(D(S_4))$ are all isomorphic to ${\mbox{Rep}}(S_4)$.
Finally, the group of symmetries of dodecahedron/icosahedron is a simple non-Abelian group $A_5$. It is clear that for any simple non-Abelian group $G$ the category ${\mbox{Rep}}(D(G))$ contains a unique Lagrangian subcategory (corresponding to the trivial subgroup of $G$).
Recall [@N] that a group $G$ is called [*categorically Morita rigid*]{} if ${\mbox{Vec}}_G$ being weakly Morita equivalent to ${\mbox{Vec}}_{G'}^\omega$ implies that groups $G$ and $G'$ are isomorphic. One can check that in this case $\omega$ must be cohomologically trivial. Our results imply that $G$ is categorically Morita rigid if and only if all Lagrangian subcategories of ${\mbox{Rep}}(D(G))$ are equivalent to ${\mbox{Rep}}(G)$ as symmetric categories.
It follows from Examples \[dihedral\] and \[platonic\] that groups $A_4,\, S_4,\, A_5$, as well as groups $D_{2n}$, where $n$ is a square-free integer, are categorically Morita rigid. It is clear that any group $G$ without non-trivial normal Abelian subgroups is categorically Morita rigid.
Finally, let us consider symmetries of vector spaces.
Let $n$ be a positive integer and let $\mathbb{F}_q$ be the finite field with $q$ elements. Let $G= SL(n, \, q)$ denote the [*special linear*]{} group of $n \times n$ matrices with entries from $\mathbb{F}_q$, i.e., matrices having determinant equal to $1$.
Let $PSL(n, \, q) = SL(n, \, q)/Z(SL(n, \, q))$, where $Z(SL(n, \, q))$ is the center of $SL(n, \, q)$. Let us assume that $(n,\, q) \not = (2, \, 2), (2, \, 3)$. It is known that in this case $PSL(n, \, q)$ is simple. Let $d= (n, \, q-1)$ be the greatest common divisor of $n$ and $q-1$. The group $Z(SL(n, \, q)) \cong \mathbb{Z}/d\mathbb{Z}$ is cyclic and any normal subgroup of $SL(n, \, q)$ is contained in $Z(SL(n, \, q))$. Thus, Lagrangian subcategories of ${\mbox{Rep}}(D(SL(n, \, q)))$ correspond to divisors of $d$. One can easily describe the equivalence types of these subcategories. For instance, the Lagrangian subcategory corresponding to $Z(SL(n, \, q)) \cong \mathbb{Z}/d\mathbb{Z}$ is equivalent to ${\mbox{Rep}}( \mathbb{Z}/d\mathbb{Z} \times PSL(n, \, q))$. Therefore, ${\mbox{Rep}}(D(SL(n, \, q))$ is equivalent (as a braided tensor category) to ${\mbox{Rep}}(D^{\omega}(\mathbb{Z}/d\mathbb{Z} \times PSL(n, \, q)))$ for some $3$-cocycle $\omega$.
Non-pointed categories of dimension $8$
---------------------------------------
\[eight\] It is known [@TY] that there are exactly four non-pointed fusion categories of dimension $8$ with integral dimensions of objects: ${\mbox{Rep}}(D_8)$; ${\mbox{Rep}}(Q_8)$, where $Q_8$ is the group of quaternions; $KP$, the representation category of the Kac-Paljutkin Hopf algebra [@KP]; and $TY$, the category of representations of a unique $8$-dimensional quasi-Hopf algebra which is not gauge equivalent to a Hopf algebra [@TY] (equivalently, $TY$ is the unique non-pointed fusion category of dimension $8$ with integral dimensions of objects which does not have a fiber functor).
Let us show that these four categories belong to four different weak Morita equivalence classes and hence, in view of Theorem \[main 1\], their centers are not equivalent as braided tensor categories.
All proper subgroups of $Q_8$ are normal Abelian and have trivial Schur multipliers. Hence all of them produce pointed duals. So ${\mbox{Rep}}(Q_8)$ is the only non-pointed dual of ${\mbox{Vec}}_{Q_8}$.
The only non-normal subgroups of $D_8={\langle}r,\, s \mid r^4 = s^2= 1,\, rs = sr^3 {\rangle}$ are reflection subgroups ${\langle}s{\rangle},\, {\langle}sr{\rangle}$, and their conjugates. Such subgroups appear as factors in exact factorizations of $D_8$ (one can take ${\langle}r{\rangle}$ as another factor). The corresponding dual categories $({\mbox{Vec}}_{D_8})^*_{{\mathcal{M}}({\langle}s{\rangle},\, 1)}$ and $({\mbox{Vec}}_{D_8})^*_{{\mathcal{M}}({\langle}sr{\rangle},\, 1)}$ admit fiber functors by [@O2 Corollary 3.1] and so are representations of semisimple Hopf algebras. But Hopf algebras corresponding to exact factorizations of $D_8$ are known to be either commutative or cocommutative. We conclude that $$({\mbox{Vec}}_{D_8})^*_{{\mathcal{M}}({\langle}s{\rangle},\, 1)} \cong ({\mbox{Vec}}_{D_8})^*_{{\mathcal{M}}({\langle}sr{\rangle},\, 1)}
\cong {\mbox{Rep}}(D_8),$$ and, hence, ${\mbox{Rep}}(D_8)$ is the unique non-pointed dual of ${\mbox{Vec}}_{D_8}$.
Thus, neither ${\mbox{Rep}}(Q_8)$ nor ${\mbox{Rep}}(D_8)$ is weakly Morita equivalent to any other non-pointed category.
It remains to check that the same is true for $TY$. Let $\omega_0$ be a non-trivial $3$-cocycle on $D_8/ {\langle}r^2,\, s{\rangle}\cong \mathbb{Z}/2\mathbb{Z}$ (corresponding to the non-zero element of $H^3(\mathbb{Z}/2\mathbb{Z},\, k^\times) \cong \mathbb{Z}/2\mathbb{Z}$). Let $\pi : D_8 \to D_8/ {\langle}r^2,\, s{\rangle}$ be the canonical projection. Define a $3$-cocycle $\omega$ on $D_8$ by $\omega = \omega_0 \circ(\pi\times \pi\times \pi)$. Then $\omega \equiv 1$ on ${\langle}r^2,\, s{\rangle}$ and the restrictions of $\omega$ on each of the subgroups ${\langle}r{\rangle}$ and ${\langle}sr{\rangle}$ are cohomologically non-trivial. This means that the complete list of equivalence classes of indecomposable module categories over ${\mbox{Vec}}_{D_8}^\omega$ consists of ${\mathcal{M}}(\{ 1\},\, 1)$, ${\mathcal{M}}({\langle}r^2{\rangle},\, 1)$, ${\mathcal{M}}({\langle}s{\rangle},\, 1)$, and ${\mathcal{M}}({\langle}r^2,\, s{\rangle},\, \mu)$, where $\mu\in
H^2(\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z},\,k^\times)$. Therefore, the only non-pointed dual category of ${\mbox{Vec}}_{D_8}^\omega$ corresponds to ${\mathcal{M}}({\langle}s{\rangle},\, 1)$, see Example \[pointed mod cats\]. It follows from the classification of fiber functors on group-theoretical categories obtained in [@O2 Corollary 3.1] that the category $({\mbox{Vec}}_{D_8}^\omega)^*_{{\mathcal{M}}({\langle}s{\rangle},\, 1)}$ does not have a fiber functor and hence $({\mbox{Vec}}_{D_8}^\omega)^*_{{\mathcal{M}}({\langle}s{\rangle},\, 1)} \cong TY$. Since all other duals of ${\mbox{Vec}}_{D_8}^\omega$ are pointed, it follows that $TY$ is not weakly Morita equivalent to any other non-pointed fusion category.
Hence, ${\mbox{Rep}}(D_8)$, ${\mbox{Rep}}(Q_8)$, $KP$, and $TY$ are pairwise weakly Morita non-equivalent fusion categories. Our claim about their centers follows from Theorem \[main 1\].
Let us note that there is another $3$-cocycle $\eta$ on $D_8$ such that $({\mbox{Vec}}_{D_8}^\eta)^*_{{\mathcal{M}}({\langle}s{\rangle},\, 1)} \cong KP$. Up to a conjugation, such $\eta$ must have a trivial restriction on ${\langle}r^2,\, s{\rangle}$ and ${\langle}sr{\rangle}$ (this can be seen from the Kac exact sequence [@Kac]).
The braided tensor equivalence classes of twisted doubles of groups of order $8$ were studied in detail in [@GMN] using the higher Frobenius-Schur indicators. In particular, it was shown that there are precisely $20$ equivalence classes of such [ *non-pointed*]{} doubles. In view of results of the present paper this description can be interpreted in terms of weak Morita equivalence classes of pointed categories of dimension $8$.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We develop an estimator for the correlation function which, in the ensemble average, returns the shape of the correlation function, even for signals that have significant correlations on the scale of the survey region. Our estimator is general and works in any number of dimensions. We develop versions of the estimator for both diffuse and discrete signals. As an application, we examine Monte Carlo simulations of X-ray background measurements. These include a realistic, spatially-inhomogeneous population of spurious detector events. We discuss applying the estimator to the averaging of correlation functions evaluated on several small fields, and to other cosmological applications.'
author:
- 'K. M. Huffenberger, M. Galeazzi, E. Ursino'
bibliography:
- 'notes.bib'
title: Reconstructing the shape of the correlation function
---
Introduction
============
Two-point statistics encode valuable information about the fields that they describe, such as the cosmological matter density traced by galaxies or the intensity of radiation in backgrounds like the Cosmic Microwave Background (CMB), the Cosmic Infrared Background (CIB), or the Diffuse X-ray Background (DXB).
For discrete objects, the two-point, dimensionless correlation function can be defined in terms of the probability of finding a pair of objects in two small cells, with sizes $\delta\Omega_1$ and $\delta\Omega_2$, separated by $\theta_{12}$ [@1980lssu.book.....P §31, 45]: $$\delta P_{12} = {\cal N}^2 \delta \Omega_1 \delta\Omega_2 \left[1+w(\theta_{12}) \right]$$ where ${\cal N}$ is the mean density of sources. For diffuse fields, the equivalent definition for a signal $s$ with mean $\langle s \rangle = \mu$ is $$\langle s_1 s_2 \rangle = \mu^2 \left[ 1 + w(\theta_{12}) \right],$$ where here and throughout $\langle \dots \rangle$ denotes the ensemble average.[^1] We denote the covariance of $s$ as $C(\theta) = \mu^2 w(\theta)$, which we also refer to as the (dimensionful) correlation function. This work mostly deals with the dimensionful correlation function and addresses the bias in its estimation. With similar expressions, we can define correlation functions in any number of dimensions, replacing the angular separation $\theta$ by a linear separation or time interval or whatever is appropriate.
The estimation of the correlation function has been studied extensively in the literature. For galaxy clustering, @1982MNRAS.201..867H, @1983ApJ...267..465D, and @1993ApJ...417...19H suggest different Monte Carlo estimators, but the most common estimator now in use was advocated by @1993ApJ...412...64L, which employs the data in concert with a synthetic, random catalog. Their estimator combines counts of objects pairs within and between the data and random catalogs. This estimator is biased, but for surveys where the correlation length of the objects is much smaller than the survey area, the bias is small [@1994ApJ...424..569B]. Such is the case for modern galaxy surveys like 2dF and SDSS [@2001MNRAS.327.1297P; @2000AJ....120.1579Y]. However, the bias can become significant when structures approach the size of the survey . This bias can be corrected [e.g. @2002ApJ...579...48S], but the correction depends on same correlation function that is being estimated.
For diffuse signals like the CMB, where using the dimensionful correlation function is more common, a typical estimator looks like [@1996ApJ...464L..25H; @2007PhRvD..75b3507C]: $$\tilde C_0(\theta) = \frac{\sum_{ij}\alpha_i\alpha_j(s_i-\tilde \mu)(s_j - \tilde \mu)}{ \sum_{ij} \alpha_i\alpha_j} \label{eq:intro_continuous_C0}$$ where $\alpha_i$ are the weights applied to the pixels or cells (for the purpose of downweighting noisy regions), $\tilde \mu$ is an estimate of the mean, and the sum over $ij$ refers to pixels separated by $\theta$. These estimators suffer the same biases on small fields.
In this paper we introduce a new method to address the biases in these above estimators. Our estimator is also biased, but biased in a particularly convenient way: regardless of the survey geometry or weighting, the shape of the correlation function is preserved on average, and only information about a constant offset is lost. This permits the straightforward averaging of correlation functions from several small patches across the sky. Building upon the estimator in eqn. (\[eq:intro\_continuous\_C0\]), we develop classes of estimators for both diffuse signals and discrete objects.
This work was prompted by our group’s efforts to compute correlation function from observations of the diffuse X-ray background. The signal in that case comes from a diffuse, gaseous source, but arrives and is recorded as individual, discrete X-ray photons, and so can be analyzed with either scheme above. Indeed, for simulations of diffuse X-ray emission from the WHIM, @2011MNRAS.414.2970U found that the @1993ApJ...412...64L estimator gave roughly equivalent results to an estimator of the type in eqn. (\[eq:intro\_continuous\_C0\]). We focused on the correlation function biases because the angular correlation scale of this gas (several arcminutes) is substantial compared to the field-of-view ($\sim 8$ arcminutes) for single-field observations with the Chandra X-ray Observatory.
The paper is organized as follows. In section \[sec:continuous\] we find the bias for the naive estimator (eqn. \[eq:intro\_continuous\_C0\]), verifying our result with Monte Carlo simulations, and introduce a method for correcting it up to a constant offset. In section \[sec:poisson\] we extend this estimate to Poisson-distributed counts, allowing for the possibility of a spatially-varying set of spurious detector events. Finally, we summarize our conclusions in section \[sec:conclusions\]. An appendix contains the detailed derivations of the bias terms.
Correlation function estimator bias {#sec:continuous}
===================================
We begin by defining our signals. Let $s_i$ represent a pixelized, diffuse signal that is statistically homogeneous and isotropic. Let it be described by a mean and covariance as follows: $$\begin{aligned}
\langle s_i \rangle &=& \mu \\ \nonumber
%\end{equation}
%\begin{equation}
\langle (s_i-\mu)(s_j - \mu) \rangle &=& \langle s_is_j \rangle -\mu^2 = C(\theta_{ij} )\end{aligned}$$ where $\theta_{ij}$ represents the separation between cells $i$ and $j$. In our derivations we use $C(\theta)$ rather that $w(\theta)$ because the examination of biases is convenient; $C(\theta)$ also makes sense for diffuse fields where $\mu = 0$. No other special properties of $s$ are required, except that the covariance matrix is positive semi-definite: $ 0 \leq |C(\theta)| \leq C(0)$. In particular, the signal need not be a Gaussian random field: we could define higher-order moments without disrupting our following arguments. Note that by this definition, the correlation function $C(\theta)$ is a property of the probability distribution for our signal $s$, and it is not a descriptive statistic.
With a set of weights on the pixels, $\alpha_i$, we can compute a weighted average to estimate the mean, $$\tilde\mu = \frac{\sum_i \alpha_i s_i}{\sum_i \alpha_i}$$ where the sum is over all pixels. These weights could be chosen to be uniform or to suppress noisy or polluted portions of the measurement. Throughout we mark estimated quantities with tildes. This mean estimate is unbiased, $\langle \tilde\mu \rangle = \mu$. Additionally we define the deviation between the true mean and the estimated mean by $$\delta\tilde\mu = \tilde\mu - \mu$$ with $\langle \delta\tilde\mu \rangle = 0$.
Naive correlation function estimator
------------------------------------
Based on the estimated mean, we make an initial estimate of the correlation function in a bin labeled by $\theta_p$, which we call the naive estimator: $$\tilde C_0(\theta_p) = \frac{ \sum_{ij} d_{ij}(\theta_p) \alpha_i \alpha_j (s_i-\tilde\mu)(s_j - \tilde\mu)}{\sum_{ij} d_{ij}(\theta_p) \alpha_i \alpha_j}% = \frac{\cal N}{\cal D}.
\label{eq:Cestimator}$$ This is just a more explicit rewriting of eqn. (\[eq:intro\_continuous\_C0\]). The function $$d_{ij}(\theta_p) = \left\{
\begin{array}{ll}
1, & \mbox{if $i$ and $j$ are separated by $\theta_p \pm \delta\theta$/2} \\
0, & \mbox{otherwise}
\end{array} \right.$$ chooses the separation bin to which the pixel sum contributes.[^2] Evaluation of the estimator costs ${\cal O}(N^2)$ operations over $N$ pixels. If the true mean $\mu$ replaces the estimated mean $\tilde\mu$ in eqn. (\[eq:Cestimator\]), then this correlation function estimate is unbiased,[^3] and we find $\langle \tilde C_0(\theta_p) \rangle = C(\theta_p)$. However, since we do not know the true mean, our estimate will be biased, because we are forced to use the same (correlated) set of pixels to compute the mean and the correlation function. The smaller the survey compared to the correlation length of the signal, the worse this bias—the “integral constraint”—becomes. (See @1993ApJ...417...19H for further discussion of bias due to the mean error and other approaches to avoid it.) In the appendix, we compute the bias explicitly. We further show that the ensemble average of the naive, biased estimator may be cast as a linear operation applied to the true correlation function: $$\langle \tilde C_0(\theta_p) \rangle = \sum_{q} M_{pq} C(\theta_q),$$ or as a matrix equation, $$\langle \mathbf{\tilde C_0} \rangle = \mathbf{M C}.$$ [Writing it this way is somewhat analogous to the MASTER technique [@2002ApJ...567....2H] for CMB power spectrum estimation on the partial sky.]{}
In the appendix we find that the matrix is $$M_{pq} = \delta_{pq} - 2 \frac{\sum_{ij} d_{ij}(\theta_p) \alpha_i \alpha_jD^{(1)}_{iq}}{\sum_{ij} d_{ij}(\theta_p) \alpha_i \alpha_j} + D_{q}^{(2)} \tag{\ref{eqn:matrix_cont}}$$ where the auxiliary operations $$D_{iq}^{(1)} = \frac{\sum_{k} \alpha_k d_{ik}(\theta_q) }{\sum_k \alpha_k}
\qquad \qquad D_{q}^{(2)} = \frac{\sum_{kl} \alpha_l \alpha_k d_{kl}(\theta_q) }{\left(\sum_k \alpha_k\right)^2} \tag{\ref{eqn:D1}, \ref{eqn:D2}}$$ are functions of the pixel weights. This matrix is composed of three terms. The first term is the identity matrix and the following two terms are responsible for the bias. The matrix costs ${\cal O}(N^2)$ operations to compute, the same as the naive correlation function estimator.
Monte Carlo simulation
----------------------
To test our expression for the bias terms, we performed Monte Carlo simulations of continuous, diffuse fields; later we will include shot noise. The survey size, roughly $7' \times 8'$, mimics an actual observation with Chandra. For the correlation function $C(\theta)$ in the simulation, we use a Gaussian function with correlation length (i.e. standard deviation) of $3.9'$, significant compared to the size of the field. For weights we use the inverse of the exposure for a real set of observations. These downweight the edges of the observations compared to the center (and correspond to inverse-variance pixel weights in the Poisson-noise-dominated limit.)
![The input correlation function (black) was used to create a set of $N_{\rm MC} = 1000$ Monte Carlo realizations of a simulated map (without shot noise). At each angular separation, 95 percent of naive estimates $\tilde C_0(\theta)$ for the correlation function fall within the pink region. The average of the Monte Carlo ensemble of naive estimates is solid blue, and has fluctuations reduced by a factor $\sqrt{N_{\rm MC}} \sim 30$. The sum of the bias terms computed from the input $C(\theta)$ is shown as the dashed red line. The ensemble average minus the bias terms is shown with the dash-dot blue line, and closely matches the input.[]{data-label="fig:Wtheta_MC"}](Wtheta_MC.pdf){width="0.6\columnwidth"}
Figure \[fig:Wtheta\_MC\] shows the input correlation, and the ensemble average (and dispersion) of the naive estimates, which are biased. Compared to the input correlation, the ensemble average is offset and the shape differs. The bias terms capture this difference, but the bias terms depend on the input correlation function, and so when working with data are not directly available. We address this shortcoming in the next section. The matrix $\mathbf{M}$ for our example is depicted in Figure \[fig:biasmatrix\].
In the simulations shown, we generated the diffuse signal $s$ as a Gaussian random field, but obtain the same results with a log-normal random field (constructed with the recipe from @2012ApJ...750...28C to keep the same mean and correlation function). The ensemble average and bias terms are the same in the Gaussian and non-Gaussian cases, however the non-Gaussianities substantially increase the dispersion of the naive estimates.
![Left: the matrix $\mathbf M$ which relates the true correlation function to the ensemble average of the naive estimate. The columns represent the input scale and the rows the output scale. The matrix is dimensionless. Right: Without the identity matrix, we have the biasing terms only.[]{data-label="fig:biasmatrix"}](M.pdf "fig:"){width="0.49\columnwidth"} ![Left: the matrix $\mathbf M$ which relates the true correlation function to the ensemble average of the naive estimate. The columns represent the input scale and the rows the output scale. The matrix is dimensionless. Right: Without the identity matrix, we have the biasing terms only.[]{data-label="fig:biasmatrix"}](M12.pdf "fig:"){width="0.49\columnwidth"}
Correcting the naive estimator {#sec:continuous_svd}
------------------------------
Once we have $\mathbf{M}$, we can define a *reconstructed correlation function* $\tilde C(\theta_q)$ as the solution to the linear equation $$\tilde C_0(\theta_p) = \sum_{q} M_{pq} \tilde C(\theta_q),
\label{eq:reconstructed}$$ where the left-hand-side is the naive estimate we already obtained and the right-hand-side contains our reconstruction.
Unfortunately this equation does not have a unique solution. Explicit computation in the appendix shows that $\mathbf{M}$ maps any constant offset to zero. Thus constant offsets to the correlation function are in the null space of the matrix. In particular this implies that $\mathbf{M}$ is not invertible, ruling out a straightforward solution to the linear equation. However, we can recover the true $C(\theta)$ in the ensemble average up to an unknown constant function.
Since we know this matrix has a non-empty null space, we analyze it by singular value decomposition, factoring it as $$\mathbf{M = U s V}^T$$ where $\mathbf{U}$ and $\mathbf{V}$ are orthogonal and $\mathbf{s}$ is diagonal and contains the singular values. The matrix has one singular value near zero, and the column of $\mathbf{V}$ that corresponds to the singular mode contains the constant function we identified previously as being in the null space.
The upshot of this discussion is that although $\mathbf{M}$ does not have an inverse, we can construct a pseudo-inverse $$\mathbf{M^+ = V s^+ U}^T$$ where $\mathbf{s^+}$ is a diagonal matrix constructed from the reciprocal of the diagonal of $\mathbf{s}$ except at the singular value where it is set to zero. Then the reconstructed correlation function $$\tilde C(\theta_p) = \sum_q M^+_{pq} \tilde C_0(\theta_q)$$ solves equation (\[eq:reconstructed\]). This solution is not unique, however, since adding any constant function also yields a solution. This procedure chooses the solution which minimizes the squared norm of the reconstructed correlation function [e.g. @1992nrca.book.....P] $$\sum_p | \tilde C(\theta_p) |^2.$$
![The ensemble of 1000 realizations made with the input correlation shown in black yields the average naive correlation function shown in blue. Multiplying the ensemble average by $\mathbf M^+$, the pseudo-inverse of the biasing matrix, gives the reconstructed correlation function (in green), which has the same shape as the input spectrum, but has lost the information about the constant offset. It resembles the input spectrum after the input is offset to minimize the square norm.[]{data-label="fig:Wtheta_MC_reconstruction"}](Wtheta_MC_reconstruction.pdf){width="0.6\columnwidth"}
Therefore, in the ensemble average, we can reconstruct the correlation matrix up to a constant offset factor, as shown in Fig. \[fig:Wtheta\_MC\_reconstruction\] for our Monte Carlo simulation. This shows how the incorrect shape of the ensemble average has been repaired in the reconstruction, except for residual fluctuations in the ensemble average.
Thus we have $$\langle \tilde C(\theta_p) \rangle = C(\theta_p) + \mbox{const.}$$ where the constant is unknown. Our estimator is therefore biased. Note however, that the shape is not biased, as we can see from a comparison of the reconstructed correlation function at two separations: $$\langle \tilde C(\theta_p) - \tilde C(\theta_q) \rangle = C(\theta_p) + \mbox{const.} - C(\theta_q) - \mbox{const.} = C(\theta_p) - C(\theta_q)$$ for any scales $\theta_p$ and $\theta_q$ accessible by the survey. Thus we can say that the shape information is preserved in an unbiased way. If we further have theoretical expectations or other constraints, these can help fix the offset for the correlation function.
Poisson shot noise {#sec:poisson}
==================
If the observations have significant shot noise from measuring discrete photons or objects, additional bias terms appear. We use a Poisson model [@1980lssu.book.....P §33] for our computations. Let $N_i$ be the count of events in pixel or cell $i$. This quantity is Poisson-distributed with a mean parameter $\lambda_i$ that is proportional to our diffuse signal. In our X-ray example, $\lambda_i = s_i t_i A$, where $s_i$ is our diffuse signal from before, representing a photon rate per area, time $t_i$ is the duration of the pixel’s exposure, and $A$ is the pixel’s collecting area.[^4] Note $\lambda_i$ is a mean number of counts, and so is dimensionless. The Chandra observations we have studied have a large fraction of counts ($\sim 85$ percent) that are spurious events unrelated to the cosmic signal. We first derive the bias and corrections for the naive estimator neglecting these spurious counts, and then including them.
No spurious contamination {#sec:nobackground}
-------------------------
If all the counts are genuinely related to the cosmic signal, the observed rate ($R$) of signal events is $$R_i = N_i/t_i A$$ which has the same units as $s_i$. The ensemble average of $R_i$ is $$\langle R_i \rangle = \frac{\langle N_i \rangle}{t_i A} = \frac{\langle s_i \rangle t_i A}{t_i A} = \mu.$$ We can estimate the mean of our rate map $$\bar R = \frac{\sum_i \alpha_i R_i}{\sum_i \alpha_i}$$ which is an unbiased estimate: $\langle \bar R \rangle = \mu$. The fluctuation in the map’s mean we call $$\delta \bar R = \bar R - \mu$$ which has $\langle \delta \bar R \rangle = 0$. The covariance of the observed rate map is $$\begin{aligned}
{\rm Cov}(R_i,R_j) %&=& \langle R_i R_j \rangle - \mu^2 \nonumber \\
% &=& \frac{ \langle N_i N_j \rangle}{ t_i t_j A^2} - \mu^2 \nonumber \\
% &=& \frac{\langle s_i \rangle t_i A }{t_i^2 A^2} \delta_{ij} + \frac{\langle s_i s_j \rangle t_i t_j A^2}{t_i t_j A^2} - \mu^2 \nonumber \\
&=& \frac{\mu}{t_iA} \delta_{ij} + C(\theta_{ij}). \label{eq:poissonratecov}\end{aligned}$$ This has an additional shot noise component compared to the covariance of the diffuse signal. The shot noise term can be avoided if the sums over pixel pairs exclude common pixels, at the cost of slightly more complicated pixel accounting. Here we include it in our computations for completeness.
Note that since $C(\theta)$ is a property of the diffuse field’s probability distribution, in the discrete case it is not subject to any particular new constraints compared to the continuous case. The total number of counts (or objects) summed over all pixels is a random variable, and is *not* fixed [@1980lssu.book.....P cf. §31, 33 vs. §32], and there is no specific constraint on the integral of $C(\theta)$.
The field $s$, representing a rate of counts or objects, must be non-negative, which implies that its statistics are non-Gaussian. For the derivation of the estimator biases, this matters little because, as before, the higher-order moments do not appear in our argument. On the other hand, it may matter more when constructing simulations. A Gaussian random field can be a suitable approximation for $s$, but only if the particular realizations do not contain negative pixels, which would lead to negative (and thus ill-defined) expected counts. Otherwise, a log-normal random field, which is positive-definite and which we employ below, provides another useful candidate.
As before we make a naive estimate of the correlation function $$\tilde C_0^R(\theta) = \frac{ \sum_{ij} d_{ij}(\theta) \alpha_i \alpha_j (R_i-\bar R)(R_j - \bar R)}{\sum_{ij} d_{ij}(\theta) \alpha_i \alpha_j} \label{eqn:poisson_naive}.$$
In the appendix, we show that the ensemble average of the naive estimator for the discrete field can be written as a linear function of both the true mean and the true correlation function. $$\langle \tilde C_0^R(\theta_p) \rangle = v^R_p \mu + \sum_q M_{pq} C(\theta_q) \tag{\ref{eqn:bias_poisson}}$$ where $$\begin{gathered}
v^R_p = \frac{ \sum_{ij} d_{ij}(\theta_p) \alpha_i \alpha_j [(1/t_iA)\delta_{ij} - 2 E^{(1)}_i + E^{(2)}]}{\sum_{ij} d_{ij}(\theta_p) \alpha_i \alpha_j} \nonumber\\
E^{(1)}_i = \frac{\alpha_i/t_iA}{\sum_k \alpha_k}
\qquad \qquad
E^{(2)} = \frac{\sum_k \alpha_k^2/t_kA}{(\sum_k \alpha_k)^2}
\tag{\ref{eqn:E1}, \ref{eqn:E2}, \ref{eqn:vR}}\end{gathered}$$ and $\mathbf{M}$ is the same matrix as before.
We can express this relationship in matrix form as $$\left(
\begin{array}{c}
\langle \bar R \rangle \\
\langle \mathbf{ \tilde C}_0^R \rangle
\end{array}
\right)
= \left(
\begin{array}{cc}
1 & (0 \dots 0) \\
\mathbf{v}^R & \mathbf{M}
\end{array}
\right)
\left(
\begin{array}{c}
\mu \\
\mathbf{C}
\end{array}
\right)$$ where we used that $\bar R$ is an unbiased estimator for $\mu$.
Like $\mathbf M$ before, this larger square matrix is amenable to the construction of a pseudo-inverse by singular value decomposition. Analogous to equation (\[eq:reconstructed\]), we can solve the linear equation $$\left(
\begin{array}{c}
\bar R \\
\mathbf{ \tilde C}_0^R
\end{array}
\right)
= \left(
\begin{array}{cc}
1 & (0 \dots 0) \\
\mathbf{v}^R & \mathbf{M}
\end{array}
\right)
\left(
\begin{array}{c}
\tilde \mu \\
\mathbf{ \tilde C}
\end{array}
\right)$$ to reconstruct estimates (on the right-hand side) for the mean (this estimate is unbiased because it just takes the already unbiased $\bar R$ directly) and correlation function, with the same limitation as before: a constant function added to the correlation function is unconstrained. As before, the shape of the reconstructed correlation function in the ensemble average matches the true correlation function.
With spurious contamination {#sec:background}
---------------------------
In the presence of an uncorrelated, but spatially varying, set of spurious counts, the analysis changes slightly, with the spurious counts contributing additional shot noise terms. In the case of Chandra data, these spurious counts are well-characterized in the sense that their mean rate is well-understood. However, counts cannot be classified as signal or spurious on an individual basis.
Now our counts include events from both the signal and the spurious set: $N_i = N_i^s + N_i^{sp}$. Then the ensemble average photon count is $\langle N_i \rangle = \mu t_i A + \lambda^{sp}_i$, where $\lambda_i^{sp}$ is the known spurious mean count for each pixel. We redefine the signal rate map as $$R_i = \frac{N_i - \lambda^{sp}_i}{t_iA} \label{eqn:rate_spur}$$ so that $\langle R_i \rangle = \mu$. Defining the map mean as before yields $\langle \bar R \rangle = \mu$ and the fluctuation from the mean has average $\langle\delta \bar R\rangle = 0$. From here the analysis proceeds much as before. Noting that
$$\langle (N_i - \lambda^{sp}_i)(N_j - \lambda^{sp}_j) \rangle = \langle N_i^s N_j^s \rangle + {\rm Cov}(N_i^{sp},N_j^{sp})$$
we can show that $${\rm Cov}(R_i,R_j) = \left( \frac{\mu}{t_iA} + \frac{\lambda^{sp}_i}{t_i^2A^2} \right)\delta_{ij} + C(\theta_{ij}). \label{eqn:cov_spur}$$ which includes an additional shot noise term compared to the similar eqn. (\[eq:poissonratecov\]).
This allows the ensemble average of the naive estimate to be written as the sum of the spurious-event-free naive estimate and additional shot-noise terms which depend on the known mean spurious rate, $\lambda_i^{sp}$. In the appendix we show that this is: $$\begin{gathered}
\langle \tilde C_0^{R,sp}(\theta) \rangle = \langle \tilde C_0^{R}(\theta) \rangle +
\frac{ \sum_{ij} d_{ij}(\theta) \alpha_i \alpha_j (\lambda_i^{sp}/t_i^2A^2)\delta_{ij} - 2\left(\alpha_i\lambda_i^{sp} /(t_i^2A^2 \sum_k \alpha_k) \right)}{ \sum_{ij} d_{ij}(\theta) \alpha_i \alpha_j} + \frac{\sum_k \alpha_k^2 \lambda_k^{sp}/t_k^2}{A^2\left( \sum_k\alpha_k \right)^2}
\tag{\ref{eqn:ensemble_spur}}\end{gathered}$$ Subtracting away these spurious terms, we can proceed to reconstruct the correlation function as described at the end of section \[sec:nobackground\].
Poisson Monte Carlo simulation
------------------------------
For a set of 5000 Monte Carlo realizations that include shot noise, we show in Fig. \[fig:Wtheta\_obs\_MC\] the ensemble average and dispersion for the naive estimate, and also the analytic computation of the bias terms. The mean rate of photons, $\mu = 4.3 \times 10^{-9}$ counts/s/pixel, was chosen based on a real Chandra observation, and is low enough that a Gaussian random field with this correlation function will have negative pixels. For this reason we used a log-normal random field in this case, which accounts for much of the increase in the dispersion compared to Fig. \[fig:Wtheta\_MC\]. The shot-noise bias terms are large in the first bin of the correlation function, which contains the same-pixel pairs. Elsewhere, they are small because in this application, we have enough photons to make the shot noise contribution to $\delta \bar R$ sub-dominant. The bias terms we computed account for the shot noise well. The dispersion due to shot noise is extreme at $>9'$ separations for two reasons: only the periphery of the map provides these separations, so there are few pixel pairs, and the effective exposure for pixels at the edge of the map is less, so there are many fewer photons than at the center of the field.
![Similar to figure \[fig:Wtheta\_MC\], except including shot noise from signal photons and background events, based on 5000 log-normal random fields. The Poisson bias terms (dashed green and cyan) are very small except in the first bin, which contains common pixel pairs. Accounting for all bias terms, the average closely matches the input, including at the first bin.[]{data-label="fig:Wtheta_obs_MC"}](Wtheta_obs_MC_iv.pdf){width="0.6\columnwidth"}
In Fig. \[fig:Wtheta\_MC\_poisson\_reconstruction\], we demonstrate that the reconstruction of the correlation function by the singular value decomposition method works well to correct the shape distortion in the ensemble average.
![Similar to figure \[fig:Wtheta\_MC\_reconstruction\], reconstructing the correlation function, except including shot noise from signal photons and background events.[]{data-label="fig:Wtheta_MC_poisson_reconstruction"}](Wtheta_MC_poisson_reconstruction_iv.pdf){width="0.6\columnwidth"}
Conclusions {#sec:conclusions}
===========
We have developed an estimator for the correlation function which allows the shape, but not the overall offset, of the correlation function to be estimated properly in the ensemble average. If there are significant signal correlations on the largest scales that the survey region can probe, as with X-ray observations and some other astronomical data sets, the large sample variance will limit the utility of the correlation function shape measurement. However, when $\tilde C(\theta)$’s from multiple fields are averaged, we beat down the noise on the shape, while the average of unknown offsets simply yields a new unknown offset. Put another way, averaging improves our knowledge of the shape but does not worsen our lack of knowledge about the offset.
The estimators written here, although motivated by observations of the diffuse X-ray background, easily generalize to galaxy counts-in-cells (setting $\lambda_i = {\cal N}_i\Delta\Omega$ in section \[sec:poisson\]). The estimator can be trivially adapted for cross-correlations between fields, or extended from angular correlations in two dimensions to linear or time-series correlations in one dimension or spatial correlations in three dimensions.
These estimators may be usefully applied to any situation with correlations on the scale of the observed region. One example is the CMB, which in the $\Lambda$CDM model has significant correlations even between points on the sky separated by $180^\circ$. However, estimates from the COBE and WMAP data [@1996ApJ...464L..25H; @2003ApJS..148..175S; @2007PhRvD..75b3507C; @2009MNRAS.399..295C] show surprisingly little correlations at scales larger than $60^\circ$. These authors have used the biased, naive estimator (equation \[eq:Cestimator\]), but our preliminary tests on WMAP maps and the $\Lambda$CDM CMB correlation function indicate that the bias terms we have computed here are too small to account for this difference.
We have computed the variance of our estimates in Monte Carlo simulations, but not analytically, nor have we tried to find optimal weights to minimize the variance. When sample variance dominates the covariance for the correlation function, it is unlikely that the optimal weighting can be done on a pixel-by-pixel basis, and instead pixel pairs will need to be jointly weighted by the inverse covariance for that pair, accounting for the signal covariance and the signal and spurious shot noise. Compared to the real-space estimators we examine here, @2004MNRAS.348..885E and @2010MNRAS.407.2530E argue that a correlation function estimate built from a maximum likelihood estimate of the harmonic space power spectrum will have lower variance, because it effectively gives pixel pairs closer-to-optimal weights in this way. This task we leave for future work.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Enzo Branchini and the anonymous referee for useful comments on earlier versions of this work. We thank Gabriela Degwitz for help in the preparation of this manuscript. This work was supported by NASA through the Smithsonian Astrophysical Observatory (SAO), award G0112177X, and NASA award NNX11AF80G. KMH also receives support from NASA-JPL subcontract 1363745.
Bias terms: continuous case {#sec:bias_cont}
===========================
In this appendix we compute the bias terms for the continuous signal. Rewriting $ \tilde\mu = \mu + \delta\tilde\mu$, the ensemble average of the numerator of the naive estimator (\[eq:Cestimator\]) is $$% \langle {\cal N} \rangle =
\sum_{ij} d_{ij}(\theta_p) \alpha_i \alpha_j \left[ C(\theta_{ij}) - \langle s_i\delta\tilde\mu \rangle - \langle s_j\delta\tilde\mu \rangle + \langle \delta\tilde\mu^2 \rangle \right]$$ where we have used $\langle \delta\tilde\mu \rangle = 0$. Further we can use the sum’s symmetry between $i$ and $j$ to show that it equals $$% \langle {\cal N} \rangle =
\sum_{ij} d_{ij}(\theta_{p}) \alpha_i \alpha_j \left[ C(\theta_{ij}) - 2\langle s_i\delta\tilde\mu \rangle + \langle \delta\tilde\mu^2 \rangle \right]\label{eqn:numerator_ensemble_Average}.$$ If we had used the true mean, only the $C(\theta_{ij})$ term would be present, and we could pull it out of the sum as $C(\theta_p)$. The sum over weights would cancel the denominator, and we would indeed find that $\langle \tilde C_0(\theta_p) \rangle = C(\theta_p)$. This is not the case here because of the middle and last terms in the brackets, which are responsible for the bias.
We can compute both bias terms from the field’s correlation function. We call the first bias term $B^{(1)}_i$ because it is first order in the mean estimation error $\delta\tilde\mu$, and compute it as $$\begin{aligned}
\nonumber
B^{(1)}_i = \langle s_i\delta\tilde\mu \rangle &=& \langle s_i(\tilde\mu - \mu) \rangle \\ \nonumber
% &=&\left\langle s_i \frac{\sum_k \alpha_ks_k}{\sum_k \alpha_k} - s_i \mu \right\rangle \\ \nonumber
&=& \frac{\sum_k \alpha_k\langle s_i s_k\rangle}{\sum_k \alpha_k} - \mu^2 \\ \nonumber
% &=& \frac{\sum_k \alpha_k [C(\theta_{ik}) + \mu^2]}{\sum_k \alpha_k} - \mu^2 \\
&=& \frac{\sum_k \alpha_k C(\theta_{ik})}{\sum_k \alpha_k}.\end{aligned}$$ The second bias term, $B^{(2)}$, which is second order in the mean’s error, has no dependence on the pixel index. $$\begin{aligned}
\nonumber
B^{(2)} = \langle \delta\tilde\mu^2 \rangle&=& \langle (\tilde\mu - \mu)^2 \rangle \\\nonumber
% &=& \langle \tilde\mu^2 \rangle - \mu^2 \\\nonumber
& = & \left\langle \frac{\sum_k \alpha_ks_k}{\sum_k \alpha_k} \frac{\sum_l \alpha_ls_l}{\sum_l \alpha_l} \right\rangle - \mu^2 \\\nonumber
&=& \frac{\sum_{kl} \alpha_l \alpha_k \langle s_k s_l \rangle}{\left(\sum_k \alpha_k\right)^2} - \mu^2\\\nonumber
% & = &\frac{\sum_{kl} \alpha_l \alpha_k [C(\theta_{kl}) + \mu^2]}{\left(\sum_k \alpha_k\right)^2} - \mu^2 \\
& = &\frac{\sum_{kl} \alpha_l \alpha_k C(\theta_{kl}) }{\left(\sum_k \alpha_k\right)^2}\end{aligned}$$ Because $B^{(2)}$ does not depend on the pixel index, this term too can slip outside the sum over pixel pairs in eqn. (\[eqn:numerator\_ensemble\_Average\]). Therefore, finally, we have $$\langle \tilde C_0(\theta_p) \rangle = C(\theta_p) - 2 \frac{\sum_{ij} d_{ij}(\theta_p) \alpha_i \alpha_jB^{(1)}_i}{\sum_{ij} d_{ij}(\theta_p) \alpha_i \alpha_j} + B^{(2)} \label{eq:naivebias}$$ which states the bias in our estimate explicitly. Each bias term costs ${\cal O}(N^2)$ operations to compute, the same as the correlation function.
Note that our naive estimator has a peculiar reaction to correlation functions such as $C(\theta) = c$ for all separations sampled by our survey.[^5] In this case $\langle \tilde C_0(\theta) \rangle = 0 $, which we show by examining the bias terms. If $C(\theta) = c$, then the constant can be set outside the sums, which cancel the denominators. Therefore bias factors $B_i^{(1)} = c$ and $B^{(2)} = c$, and the middle term of eqn. (\[eq:naivebias\]) is $-2c$. Therefore $\langle \tilde C_0(\theta) \rangle = c - 2c + c = 0$. Thus, if the naive estimator is viewed as a linear operator on the input correlation function, constant functions are in the null space of the operator, since any constant maps to zero. Moreover, the naive estimator loses the information about any constant baseline in the correlation function, although the information about the shape is preserved.
The bias terms depend on $C(\theta)$ only on scales accessible by the survey region, and not on any larger scales. This permits an (imperfect) reconstruction of the correlation function. To proceed, we can rewrite eqn. (\[eq:naivebias\]) as a matrix multiplication: $$\langle \tilde C_0(\theta_p) \rangle = \sum_{q} M_{pq} C(\theta_q)$$ where the sum is over the angular bins. Then we set about finding the matrix $\mathbf{M}$.
To write down $\mathbf{M}$, we make use of the relationship $$C(\theta_{ik}) = \sum_q d_{ik}(\theta_q) C(\theta_q).$$ Note that this sum is over angular bin, not pixel. We rewrite the bias terms more explicitly as linear operations on the vector $C(\theta_q)$. The first bias term is $$B_i^{(1)} = \frac{\sum_{kq} \alpha_k d_{ik}(\theta_q) C(\theta_q)}{\sum_k \alpha_k} = \sum_q D_{iq}^{(1)} C(\theta_q),$$ where we define $$D_{iq}^{(1)} = \frac{\sum_{k} \alpha_k d_{ik}(\theta_q) }{\sum_k \alpha_k}.\label{eqn:D1}$$ Note that the first index refers to pixel and the second to bin. The second bias term is $$B^{(2)} = \frac{\sum_{kl} \alpha_l \alpha_k d_{kl}(\theta_q) C(\theta_q) }{\left(\sum_k \alpha_k\right)^2} = \sum_q D_{q}^{(2)} C(\theta_q),$$ where we define $$D_{q}^{(2)} = \frac{\sum_{kl} \alpha_l \alpha_k d_{kl}(\theta_q) }{\left(\sum_k \alpha_k\right)^2}. \label{eqn:D2}$$ Since $C(\theta_p) = \sum_q \delta_{pq} C(\theta_q)$, we finally have $$M_{pq} = \delta_{pq} - 2 \frac{\sum_{ij} d_{ij}(\theta_p) \alpha_i \alpha_jD^{(1)}_{iq}}{\sum_{ij} d_{ij}(\theta_p) \alpha_i \alpha_j} + D_{q}^{(2)} \label{eqn:matrix_cont}$$
To sum up, in this appendix we have: (1) computed the bias of the naive correlation function estimator; (2) shown that the ensemble average of the naive estimate is a linear operation acting upon the true correlation; (3) computed that linear operator in terms of the pixel weights; and (4) shown that constant offsets are in the null space of that operator. The method to estimate the shape of the correlation function in section \[sec:continuous\_svd\] depends on these results.
Bias terms: discrete case {#sec:bias_discrete}
=========================
No spurious contamination {#no-spurious-contamination}
-------------------------
To compute the bias for the discrete case, we write the numerator of the naive estimator (\[eqn:poisson\_naive\]) in terms of the fluctuation of the mean $\delta \bar R$ and take the ensemble average: $$\begin{aligned}
&& \left\langle\sum_{ij} d_{ij}(\theta) \alpha_i \alpha_j (R_i-\mu - \delta\bar R)(R_j - \mu - \delta \bar R) \right\rangle \nonumber \\
&=& \left\langle \sum_{ij} d_{ij}(\theta) \alpha_i \alpha_j \left[ (R_i-\mu)(R_j-\mu) - 2(R_i - \mu) \delta\bar R + (\delta\bar R)^2 \right] \right\rangle \nonumber \\
&=& \sum_{ij} d_{ij}(\theta) \alpha_i \alpha_j \left[(\mu/t_iA)\delta_{ij} + C(\theta_{ij}) -2 \langle R_i \delta\bar R \rangle + \langle (\delta\bar R)^2 \rangle \right]\end{aligned}$$ Now we examine the last two terms, which are analogous to the bias terms for the diffuse signal. First, $$\begin{aligned}
B^{R(1)}_i = \langle R_i \delta\bar R \rangle &=& \frac{\sum_k \alpha_k \langle R_i R_k \rangle}{\sum_k \alpha_k} - \mu^2 \nonumber \\
&=& \frac{\sum_k \alpha_k [(\mu/t_iA) \delta_{ik} + C(\theta_{ik})]}{\sum_k \alpha_k} \nonumber \\
&=& E^{(1)}_i \mu + B^{(1)}_i \label{eq:BR1}\end{aligned}$$ where we define $$E^{(1)}_i = \frac{\alpha_i/t_iA}{\sum_k \alpha_k}. \label{eqn:E1}$$ This shows that for a signal of discrete photons, this bias term can be written as a sum of a new shot noise term and the old $B^{(1)}$ bias term from the diffuse case.
The final term is $$\begin{aligned}
B^{R(2)} = \langle (\delta\bar R)^2 \rangle &=& \frac{\sum_{kl} \alpha_k \langle R_k R_l \rangle}{\sum_{kl} \alpha_k \alpha_l} - \mu^2 \nonumber \\
&=& \frac{\sum_{kl} \alpha_k \alpha_l [(\mu/t_iA)\delta_{kl} + C(\theta_{kl})]}{(\sum_{k} \alpha_k)^2} \nonumber \\
&=&E^{(2)}\mu + B^{(2)} \label{eq:BR2}\end{aligned}$$ where we define $$E^{(2)} = \frac{\sum_k \alpha_k^2/t_kA}{(\sum_k \alpha_k)^2}. \label{eqn:E2}$$ Again this bias term has a new, shot-noise component added to the old bias term from the diffuse signal. These shot noise bias terms cannot be avoided by excluding $i=j$ from the naive estimator’s pixel sums.
Each of the new shot noise terms is proportional to $\mu$. We can gather those terms together and notice that the remaining terms are just those which appear on the right side of eqn. (\[eq:naivebias\]), so that: $$\langle \tilde C_0^R(\theta_p) \rangle = \left[\frac{ \sum_{ij} d_{ij}(\theta_p) \alpha_i \alpha_j [(1/t_iA)\delta_{ij} - 2 E^{(1)}_i + E^{(2)}]}{\sum_{ij} d_{ij}(\theta_p) \alpha_i \alpha_j} \right] \mu + \langle \tilde C_0(\theta_p) \rangle \label{eqn:vR}$$ Therefore the ensemble average of the naive estimate for the discrete signal equals the ensemble average of the naive estimate for the diffuse signal plus an additional shot noise bias term which is proportional to the mean of the diffuse field.
Thus the ensemble average of the naive estimator for the discrete field can be written as a linear function of the true mean and correlation function. $$\langle \tilde C_0^R(\theta_p) \rangle = v^R_p \mu + \sum_q M_{pq} C(\theta_q). \label{eqn:bias_poisson}$$ This formulation leads to the reconstruction method for the correlation function discussed in section \[sec:nobackground\].
Including spurious contamination
--------------------------------
Starting from equations (\[eqn:rate\_spur\]) and (\[eqn:cov\_spur\]), we find that the two bias terms also have additional shot noise components due to the spurious signal. Instead of eqn. (\[eq:BR1\]) we have $$B_i^{R(1)} = \langle R_i \delta\bar R \rangle = E_i^{(1)}\mu + B_i^{(1)} + \frac{\alpha_i\lambda^{sp}_i}{t_i^2A^2\sum_k \alpha_k},$$ and instead of eqn. (\[eq:BR2\]) we have $$B^{R(2)} = \langle (\delta\bar R)^2 \rangle = E^{(2)}\mu + B^{(2)} + \frac{\sum_k \alpha_k^2 \lambda_k^{sp}/t_k^2}{A^2\left( \sum_k\alpha_k \right)^2}.$$
Thus there are additional terms which can be subtracted away to yield the naive estimator in the contamination-free case. $$\begin{gathered}
\langle \tilde C_0^{R,sp}(\theta) \rangle = \langle \tilde C_0^{R}(\theta) \rangle +
\frac{ \sum_{ij} d_{ij}(\theta) \alpha_i \alpha_j (\lambda_i^{sp}/t_i^2A^2)\delta_{ij} - 2\left(\alpha_i\lambda_i^{sp} /(t_i^2A^2 \sum_k \alpha_k) \right)}{ \sum_{ij} d_{ij}(\theta) \alpha_i \alpha_j} + \frac{\sum_k \alpha_k^2 \lambda_k^{sp}/t_k^2}{A^2\left( \sum_k\alpha_k \right)^2}
\label{eqn:ensemble_spur}\end{gathered}$$
[^1]: If the signal $s$ records the object count in a cell with size $\delta \Omega$, then $\langle s \rangle = \mu={\cal N}\delta\Omega$. If the cells are so small that they contain at most one object, $\langle s_1 s_2 \rangle = \delta P_{12}$, making the correspondence between the two definitions clear.
[^2]: $d_{ij}(\theta)$ is equivalent to the $\Theta_{ij}^\theta$ function defined by @1993ApJ...412...64L. In practice we loop over all pixels and just select which separation bin is appropriate to accumulate the sum.
[^3]: Technically, biases are also introduced by averaging the smooth sky into pixels—this pixel window function is severe if the pixels approach the size of the correlation length—and by binning the smooth correlation function into a stepwise function. These can often be made insignificant by choosing finer discretization schemes, and we do not treat such biases here.
[^4]: These may differ for other applications. For the example of galaxy counts, the galaxy number density plays the role of the signal and the cell volume plays the role of the exposure-weighted area.
[^5]: On scales larger than the survey, this correlation function could vary without changing the discussion.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'To study spacelike surfaces of codimension two in the Lorentz-Minkowski space $\Bbb R^{n+1}_1,$ we construct a pair of maps whose values are in $HS_r:=H_+^n(\textbf v,1)\cap \{x_{n+1}=r\},$ called $\textbf n_r^{\pm}$-Gauss maps. It is showed that they are well-defined and useful to study practically flat as well as umbilic spacelike surfaces of codimension two in $\Bbb R^{n+1}_1.$'
address:
- |
Dang Van Cuong\
Hue Geometry Group\
Departement of Mathematics\
Duy Tan University\
Danang\
Vietnam
- |
Doan The Hieu\
Hue Geometry Group\
Departement of Mathematics\
College of Education, Hue University\
Hue\
Vietnam
author:
- |
\
[^1]
title: '[$HS_{\lowercase {r}}$-valued Gauss maps and umbilic spacelike surfaces of codimension two ]{}'
---
[^2] [^3]
Introduction
============
In classical differential geometry, the Gauss map plays an important role in the study of the behaviour or geometric invariants of surfaces of codimension one. In the case of surfaces of codimension larger than one, Gauss map associated with some arbitrary normal field $\nu$ is considered. By that way, one can consider the second fundamental form associated with $\nu$ and study invariants or properties of surfaces, concerning to the concept of $\nu$-curvatures, that are dependent or independent on $\nu.$
In 1989, Marek Kossowski [@ko] used Gauss maps, whose values are in the lightcone, to study spacelike 2-surfaces in $\Bbb R^4_1,$ followed by Izumiya et. al. (see [@izu2]). In 2004, Izumiya et. al. [@izu1] used Gauss maps associated with a normal field $\nu$ to study $\nu$-umbilicity for spacelike surfaces of codimension two in Lorentz-Minkowski spaces. Long before, in the study of minimal 2-surfaces in $\Bbb R^n,$ it is well-known that the mean curvature vector $\overrightarrow{H}$ does not depend on $\nu$ (see [@os]).
Motivated by these ideas, to study practically spacelike surfaces of codimension two in $\Bbb R_1^{n+1},$ we construct a kind of Gauss map whose values are in a hyperbolic space, called $\textbf n_r^{\pm}$-Gauss maps.
Let $M$ be a spacelike surface of codimension two in $\Bbb R^{n+1}_1.$ The normal plane of $M$ at $p\in M,$ denoted by $N_pM$ is a timelike 2-plane. We identify $N_pM$ with its image under the translation given by the vector $-p.$ Then, the intersection of $N_pM$ and the hyperbolic space with center ${\bf v}=(0,0,\dots,0, -1)$ and radius $1$, $H_+^n({\bf v},1)$, is a hyperbola. For a fixed $r>0$, the hyperplane $\{x_{n+1}=r\}$ meets this hyperbola exactly at two points, denoted by $\textbf n_r^{\pm}(p).$
This gives two differential maps $p\mapsto \textbf n_r^{\pm}(p),$ called $\textbf n_r^{\pm}$-Gauss maps. Their derivatives are self-adjoint, and hence we can define the $\textbf n_r^{\pm}$-Weingarten maps, $\textbf n_r^{\pm}$-Gauss-Kronecker curvatures, $\textbf n_r^{\pm}$-mean curvatures, $\textbf n_r^{\pm}$-principal curvatures, $\textbf n_r^{\pm}$-flat points, $\textbf n_r^{\pm}$-umbilic points ….
We use these maps to study the flatness and umbilicity for spacelike surfaces of codimension two in $\Bbb R^{n+1}_1.$
In this situation, some criteria for a spacelike surface to be flat or umbilic as well as examples of some kinds of flat and umbilic spacelike surfaces of codimension two are established. These examples show that we can use $\textbf n_r^{\pm}$-Gauss maps to study some properties of spacelike surfaces of codimension two practically.
Prelimineries
=============
The Lorentz-Minkowski space $\Bbb R^{n+1}_1$
--------------------------------------------
The Lorentz-Minkowski space $\Bbb R^{n+1}_1$ is the $(n+1)$-dimensional vector space $\Bbb R^{n+1}=\{( x_1, x_2,\ldots, x_{n+1}): x_i\in \Bbb R, i=1,2,\ldots, n+1\}$ endowed the pseudo scalar product $$\langle \textbf x, \textbf y\rangle=\sum_{i=1}^{n}x_iy_i-x_{n+1}y_{n+1},$$ where $\textbf x=( x_1, x_2,\ldots, x_{n+1}), \textbf y=(y_1, y_2, \ldots y_{n+1})\in \Bbb R^{n+1}.$ Since $\langle, \rangle$ is non-positive definite, $\langle \textbf x, \textbf x\rangle$ may be zero or negative. We say a nonzero vector $\textbf x\in \Bbb R^{n+1}_1$ spacelike, lightlike or timelike if $\langle \textbf x, \textbf x\rangle>0$, $\langle \textbf x, \textbf x\rangle=0$ or $\langle \textbf x, \textbf x\rangle<0$, respectively. If $\langle \textbf x, \textbf y\rangle=0,$ we say $\textbf x,\textbf y$ are pseudo-orthogonal.
The norm of a vector $\textbf x\in \Bbb R^{n+1}_1$, denoted by $\|\textbf x\|$, is defined by $\sqrt{|\langle \textbf x,\textbf x\rangle|}.$ For a nonzero vector $\textbf n\in \Bbb R^{n+1}_1$, a hyperplane with the pseudo normal $\textbf n$ is defined as $$HP(\textbf n,c)=\{\textbf x\in\Bbb R^{n+1}_1 : \langle \textbf x,\textbf n\rangle=c,\ c\in\Bbb R\}.$$ The hyperplane is said to be spacelike, lightlike or timelike if $\textbf n$ is timelike, lightlike or spacelike, respectively.
It is easy to see that, $HP(\textbf n,c)$ is spacelike if any vector $\textbf x\in HP(\textbf n,0)$ is spacelike; $HP(\textbf n,c)$ is lightlike if $HP(n,0)$ is tangent to the lightcone and $HP(\textbf n,c)$ is timelike if $HP(\textbf n,0)$ contains timelike vectors.
In $\Bbb R^{n+1}_1,$ we have three kinds of pseudo-hyperspheres
1. $H^{n}(\textbf a,R)=\{\textbf x\in\Bbb R^{n+1}_1\ |\ \langle \textbf x-\textbf a,\textbf x-\textbf a\rangle=-R^2,\ R>0\}:$ the hyperbolic with center $\textbf a$ and radius $R;$
2. $S_1^{n}(\textbf a,R)=\{\textbf x\in\Bbb R^{n+1}_1\ |\ \langle \textbf x-\textbf a,\textbf x-\textbf a\rangle=R^2,\ R>0 \}:$ the de Sitter with center $\textbf a$ and radius $R;$
3. $LC(\textbf a)=\{\textbf x\in\Bbb R^{n+1}_1: \langle \textbf x-\textbf a,\textbf x-\textbf a\rangle=0\}:$ the lightcone with vertex $\textbf a.$
And we call
$$H_+^{n}(\textbf a,R)=\{\textbf x\in H^{n}(\textbf a,R):\ x_{n+1}-a_{n+1}\geq 0 \}$$ the hyperbolic space with center $\textbf a$ and radius $R.$
The $\textbf{n}_r^{\pm}$-Gauss maps
-----------------------------------
In this paper a surface is always spacelike and is of codimension two in $\Bbb R_1^{n+1},$ unless otherwise stated. It is an embedding $\textbf X:U\to\Bbb R^{n+1}_1$ , where $U$ is an open domain in $\Bbb R^{n-1}.$ We often identify $M=\textbf X(U)$ with $\textbf X.$
In this section we introduce two concrete spacelike normal fields on a surface that are useful to study the flatness and umbilicity, practically.
The normal plane of $M$ at $p\in M,$ denoted by $N_pM,$ can be viewed as a timelike 2-plane passing the origin. The intersection of this plane and the hyperbolic space with center $\text{\bf v}=(0,0,\dots,0,-1)$ and radius $1,$ $H_+^n(\text{\bf v},1)$ is a hyperbola. For a fixed $r>0$, the hyperplane $\{x_{n+1}=r\}$ meets this hyperbola exactly at two points, denoted by $\textbf {n}_r^{\pm}(p).$
The following maps $$\begin{aligned} \textbf {n}^{\pm}_r: M&\to HS_r:=H_+^n(\textbf v,1)\cap \{x_{n+1}=r\}\\
p&\mapsto \textbf {n}^{\pm}_r(p).
\end{aligned}$$ are called $\textbf {n}^{\pm}_r$-Gauss maps.
The first property of $\textbf {n}^{\pm}_r$-Gauss maps is
The $\textbf{n}_r^{\pm}$-Gauss maps are smooth.
Locally, $\textbf {n}^{\pm}_r(p)$ are the solutions of the following system of equations $$\begin{cases}
\langle \textbf X_{u_i},\textbf a\rangle&=0,\ \ i=1,2,\ldots, n-1;\\
\langle \textbf a-\textbf v, \textbf a-\textbf v\rangle&=-1;\\
\end{cases}$$ where $\textbf a=(a_1, a_2, \ldots,a_n, r).$
Since $\text{rank} (\textbf X_{u_1}, \textbf X_{u_2},\ldots, \textbf X_{u_{n-1}})=n-1,$ we can assume that $a_1, a_2, \ldots,a_{n-1}$ are linearly expressed in term of $a_n.$ Substituting these to the last equation, we get a quadratic equation in term of $a_n.$ This equation has exactly two solutions and of course they are smooth.
From now on, the symbol “ \* ” means “ + ” or “ - ”, unless otherwise stated. The derivative of $\textbf n_{r}^{*}$ at $p$ $$d\textbf n_r^{*}(p)\ :\ T_pM\rightarrow T_{\textbf n_{r}^{*}(p)}H_+^n(\textbf v,1)\subset T_pM\oplus N_pM;$$ can be writen as $$d\textbf n_r^{*}(p)=d{\textbf n_r^{*}}^T(p)+d{\textbf n_r^{*}}^N(p),$$ where $d{\textbf n_r^{*}}^T$ and $d{\textbf n_r^{*}}^N$ are the tangent and normal components of $d{\textbf n_r^{*}},$ respectively.
We recall some definitions and facts concerning to $\nu$-umbilic (see [@izu1]) but restated for $\textbf n_r^{*}.$ Denoted by
1. $A_p^{\textbf n_r^{*}}:=-d{\textbf n_r^{*}}^T(p),$ the $\textbf n_r^{*}$-Weingarten map of $M$ at $p;$
2. $K_p^{\textbf n_r^{*}}:=\det(A_p^{\textbf n_r^{*}}),$ the $\textbf n_r^{*}$-Gauss-Kronecker curvature of $M$ at $p;$
3. $H^{\textbf n_r^{*}}_p:=\frac{1}{n-1}\text{tr}(A_p^{\textbf n_r^{*}}),$ the $\textbf n_r^{*}$-mean curvature of $M$ at $p;$
4. $k_1^{\textbf n_r^{*}}(p),k_2^{\textbf n_r^{*}}(p),\dots,k_{n-1}^{\textbf n_r^{*}}(p),$ (the eigenvalues of $A_p^{\textbf n_r^{*}}$) the $\textbf n_r^{*}$-principal curvatures of $M$ at $p$.
Of course $$K_p^{\textbf n_r^{*}}=k_1^{\textbf n_r^{*}}(p)k_{2}^{\textbf n_r^{*}}(p)\dots k_{n-1}^{\textbf n_r^{*}}(p),$$ and $$H_p^{\textbf n_r^{*}}=\frac 1{n-1}(k_1^{\textbf n_r^{*}}(p)+k_{2}^{\textbf n_r^{*}}(p)+\dots +k_{n-1}^{\textbf n_r^{*}}(p)).$$
We have some well-known facts.
1. The $\text{\bf n}_r^{*}$-Weingarten map is self-adjoint.
2. The $\text{\bf n}_r^{*}$-principal curvatures $k_i^{\text{\bf n}_r^{*}}(p),i=1,2,\dots,n-1$ of $M$ at $p$ are the solutions of the following equation $$\label{principal} \det(b_{ij}^{\text{\bf n}_r^{*}}(p)-kg_{ij}(p))=0,$$ where $b_{ij}^{\text{\bf n}_r^{*}}(p):=\langle \textbf X_{u_iu_j}(p),\text{\bf n}_r^{*}(p)\rangle,\ i,j=1,2,\dots,n-1 ,$ the coefficients of the $\text{\bf n}_r^{*}$-second fundamental form of $M$ at $p.$
3. $K_p^{\text{\bf n}_r^{*}}={\det(b_{ij}^{\text{\bf n}_r^{*}}(p))}.{\det(g_{ij}(p))}^{-1}.$
<!-- -->
1. A point $p\in M$ is said to be $\textbf n^{*}_r$-umbilic if $k_i^{\textbf n^{*}_r}(p)=k(p),\ i=1,2,\ldots, n-1.$ If $k(p)=0,$ then $p$ is called $\textbf n^{*}_r$-flat.
2. $M$ is said to be $\textbf n^{*}_r$-umbilic ($\textbf n^{*}_r$-flat) if every point $p\in M$ is $\textbf n^{*}_r$-umbilic ($\textbf n^{*}_r$-flat).
3. $M$ is said to be totally umbilic (totally flat) if every point $p\in M$ is $\textbf n^{*}_{r}$-umbilic ($\textbf n^{*}_{r}$-flat) for every $r>0.$
The $\textbf n_r^{*}$- flatness
===============================
We begin with a useful lemma.
\[lem1\] If $(\text{\bf n}_r^*)_{u_i}\in N_pM,$ where $i\in\{1,2,\ldots, n-1\},$ then $(\text{\bf n}_r^*)_{u_i}=0.$
We observe that, the last coordinate of $(\textbf n_r^*)_{u_i}$ is zero because the last coordinate of $\textbf n_r^*$ is constant. Therefore, since $\{\textbf n_r^+, \textbf n_r^-\}$ is a basis of $N_pM,$ we have $$\label{pt1}(\textbf n_r^*)_{u_i}=\lambda (\textbf n_r^+- \textbf n_r^-).$$
An easy calculation shows that $\langle \textbf n_r^*,\textbf n_r^*\rangle=2r.$ Therefore,
$$\langle (\textbf n_r^*)_{u_i},\textbf n_r^*\rangle=\lambda\langle \textbf n_r^+-\textbf n_r^-,\textbf n_r^*\rangle=0.$$
If $\lambda\ne 0,$ then $$\langle \textbf n_r^+,\textbf n_r^+\rangle=\langle \textbf n_r^-,\textbf n_r^-\rangle=\langle \textbf n_r^+,\textbf n_r^-\rangle=2r.$$ And hence, $$\langle \textbf n_r^+-\textbf n_r^-,\textbf n_r^+-\textbf n_r^-\rangle=0,$$ a contradiction, because $\textbf n_r^+-\textbf n_r^-$ is a nonzero spacelike vector. Thus, $\lambda= 0,$ and the lemma is proved.
\[theoflat1\]Let $M$ be a connected surface. The following statements are equivalent
1. there exists an $r>0,$ $M$ is $\text{\bf n}_r^*$-flat;
2. there exists an $r>0,$ $\text{\bf n}_r^*$ is constant;
3. there exists a spacelike vector $\text{\bf a}=(a_1,a_2,\dots,a_n, a_{n+1}), a_{n+1}\ne 0$ and a real number $c$ such that $M\subset HP(\text{\bf a},c).$
($1.\Rightarrow 2.$) Since $M$ is $\textbf n_r^*$-flat, i.e. $A_p^{\textbf n_r^*}=0,$ we have $$\label{eq1}\langle \textbf X_{u_iu_j},\textbf n_r^*\rangle=-\langle \textbf X_{u_i},(\textbf n_r^*)_{u_j}\rangle=0,\ i,j=1,2,\dots,n-1.$$
But (\[eq1\]) means that $(\textbf n_r^*)_{u_i}\in N_pM$ and hence $(\textbf n_r^*)_{u_i}=0, \ i=1,2,\dots,n-1$ by virtue of Lemma \[lem1\].
($2\Rightarrow 1$) Obviously.
($2.\Rightarrow 3.$)If $\textbf n_r^*$ is constant, then $$\frac{\partial}{\partial u_i}\langle\textbf X,\textbf n_r^*\rangle=\langle\textbf X_{u_i},\textbf n_r^*\rangle-\langle\textbf X,(\textbf n_r^*)_{u_i}\rangle=0.$$ Thus $\textbf X\subset H(\textbf n_r^*, c),$ for some constant $c.$
($3.\Rightarrow 2.$) If $M$ is contained in a timelike hyperplane with a unit spacelike normal vector $\textbf a=(a_1, a_2,\dots, a_n,a_{n+1}), a_{n+1}\ne 0$, then it is not hard to check that we can choose the constant vector $\textbf n_r^*=2a_{n+1}\textbf a\in H^n_+(\text{\bf v},1).$
1. The Theorem \[theoflat1\] is a necessary and sufficient condition for a surface belonging to a timelike hyperplane that does not contain the $x_{n+1}$-axis. For the case of surfaces belonging to a timelike hypersurface containing the $x_{n+1}$-axis, see Example \[exflat\].
2. A necessary and sufficient condition for a surface belonging to a lightlike hyperplane based on the totally lightlike flatness was established in [@izu3].
3. A similar result with an assumption on parallelism of the normal field was given ([@izu1 Theorem 4.3]).
Let $M$ be a connected surface and $\text{\bf n}_{r_1}^*\ne\text{\bf n}_{r_2}^*,$ i.e. $r_1\ne r_2$ or $\text{\bf n}_{r_1}^*= \text{\bf n}_{r}^+,\ \ \text{\bf n}_{r_2}^*=\text{\bf n}_{r}^-$ for some fixed $r.$ If $M$ is both $\text{\bf n}_{r_1}^*$- and $\text{\bf n}_{r_2}^*$-flat, then $M$ is a part of a spacelike $(n-1)$-plane. In this cases, $\text{\bf n}_r^{*}$ are constant for every $r>0$ or equivalently, $M$ is totally flat, i.e. $\text{\bf n}_r^{*}$-flat for every $r>0.$
If $M$ is connected and contained in a timelike hyperplane not containing the $x_{n+1}$-axis, then there exists a unique possitive real number $r$ such that $M$ is $\text{\bf n}_r^*$-flat unless $M$ is (or a part of) a spacelike $(n-1)$-plane.
The $\text{\bf n}_r^*$-umbilicity
=================================
In this section, we study the $\text{\bf n}_r^{*}$-umbilicity for spacelike surfaces of codimension two in $\Bbb R_1^{n+1}.$ For a pseudo-hypersphere, we mean a hyperbolic or a de Sitter with center $\textbf a$ and radius $R,$ or a lightcone with vertex $\textbf a.$ Because $\textbf n_{r}^{\pm}$-umbilicity is an invariant under translations, we can assume that $\textbf a$ is the origin in the study of the $\text{\bf n}_r^{*}$-umbilicity for surfaces lying in a pseudo-hypersphere. We begin this section with another useful lemma.
\[funlem\] Suppose that $\nu_1$ and $\nu_2$ are smooth normal fields on $M$ and for every $p\in M, \ \nu_1(p), \nu_2(p)$ are linear independent. If $M$ is both $\nu_1$- and $\nu_2$-umbilic then $M$ is $\nu$-umbilic for every smooth normal field $\nu.$
By the assumption, for every smooth normal field $\nu$ $$\nu=\lambda_1\nu_1+\lambda_2\nu_2$$ where $\lambda_i,\ i=1,2$ are smooth functions on $M.$
Because $d(\lambda_i\nu_i)^T=\lambda_i d(\nu_i)^T,\ i=1,2$ $$A^{\nu}=\lambda_1 A^{\nu_1}+\lambda_2 A^{\nu_2}.$$ Since $A^{\nu_i}=k^{\nu_i}\text{id},\ i=1,2$ $$A^{\nu}=(\lambda_1 k^{\nu_1}+\lambda_2 k^{\nu_2})\text{id}.$$
Because $\textbf n_{r}^+,$ $ \textbf n_{r}^-$ are linear independent by the construction and so are $\textbf n_{r_1}^*,\ \textbf n_{r_2}^*$ if $r_1\ne r_2,$ we have
If $M$ is $\textbf n_{r_1}^*$- and $\textbf n_{r_2}^*$-umbilic, where $\textbf n_{r_1}^*\ne\textbf n_{r_2}^*;$ then $M$ is totally umbilic.
1. By virue of Lemma \[funlem\], a surface is totally umbilic iff it is $\nu$-umbilic for every smooth normal field $\nu.$
2. It is well-known that (see [@izu1 Lemma 4.1]), a surface lying in a pseudo-hypersphere is always $\nu$-umbilic, where $\nu$ is the position vector field. Therefore, Lemma \[funlem\] is useful in the study of the totally umbilicity for surfaces lying in a hyperbolic or a lightcone, because the position vector field and $\textbf n_{r}^*$ are always linear independent. The case of the de Sitter can be studied in a similar way by using the lightcone Gauss maps (see [@izu2], [@ko]...). So for simplicity in statements, we just state for the case of the hyperbolic spaces.
By using of Theorem \[theoflat1\], Lemma \[funlem\] or by a direct computation (see Example \[exflat\]), we have
\[cor54\] If $M$ is contained in the intersection of a hyperbolic space and a hyperplane, then $M$ is totally umbilic.
\[theoum1\] Let $M$ be a spacelike surface of codimension two in $H_+^n(0,R).$ The following statements are equivalent.
1. there exists $r>0,$ $M$ is $\textbf n_r^*$-umbilic;
2. $M$ is totally umbilic;
3. $M$ is contained in a hyperplane.
($1. \Rightarrow 2.$) Because $M$ is contained in $H_+^n(0,R),$ $M$ is umbilic with respect to the position vector field $\textbf X.$ Moreover, because $\textbf X$ is timelike while $\textbf n_r^*$ is spacelike, $M$ is totally umbilic by virtue of Lemma \[funlem\].
($2. \Rightarrow 3.$) Let $$\nu=\frac{\textbf X\wedge \textbf X_{u_1}\wedge \textbf X_{u_2}\wedge \dots \wedge \textbf X_{u_{n-1}}}{\left|\textbf X\wedge \textbf X_{u_1}\wedge \textbf X_{u_2}\wedge \dots \wedge \textbf X_{u_{n-1}}\right|}.$$ Because $$\label{hapdan}\langle \nu,\textbf X\rangle =0,\qquad \langle \nu,\nu\rangle =\pm 1,\qquad \langle \nu,\textbf X_{u_i}\rangle =0,\ i=1,2,...,n-1;$$ we have $$\langle d\nu,\textbf X\rangle = \langle \nu,d\textbf X\rangle =0;\ \langle \nu,d\nu\rangle =0.$$ Since $\{\nu,\textbf X\}$ is a basis of $N_pM,$ $d\nu\in T_pM,$ i.e. $\nu$ is parallel.
By virtue of Lemma 4.2 in [@izu1], $d\nu=\lambda d\textbf X,$ where $\lambda$ is constant and hence $\nu=\lambda \textbf X+ \textbf a,$ where $\textbf a$ is a constant vector. Since $\langle \nu,\textbf X\rangle =0, $ $\langle \textbf X, \textbf a\rangle=-\langle \textbf X,\lambda \textbf X\rangle=-\lambda R=c$ (a constant). Thus, $M\subset HP(\textbf a,c)$.\
($3.\Rightarrow 1.$) follows by Corollary \[cor54\].
\[lempara\] Let $\nu_1,\ \nu_2$ be parallel vector fields on the connected surface $M$ and $\nu=\alpha\nu_1+\beta\nu_2.$ Suppose that for every $p\in M,$ $\nu_1(p), \nu_2(p)$ are linear independent, then $\nu$ is parallel if and only if $\alpha$ and $\beta$ are constants.
The assumption that $\nu, \ \nu_1,\ \nu_2$ are parallel yields $$d\alpha\nu_1+d\beta\nu_2=0.$$ But this implies $d\alpha=d\beta=0$ since $\nu_1, \nu_2$ are linear independent. Conversely, it is obvious that if $\alpha$ and $\beta$ are constants then $\nu$ is parallel.
Among all hyperspheres $HP(\textbf n,c)\cap H_+^{n}(0,R)$ ($\textbf n$ is timelike) of the hyperbolic space $H_+^{n}(0,R),$ the case of right hyperspheres, i.e. $\textbf n=(0,0,\ldots, 1),$ are special. The following theorem give some necessary and sufficient conditions for a surface lying in a hyperbolic space to be a part of a right hypersphere.
\[theoum2\] Let $M$ be a surface contained in $H_+^{n}(0,R).$ The following statements are equivalent:
1. $M$ is contained in a right hypersphere;
2. $\text{\bf n}_r^{*}$ is parallel for any $r>0;$
3. there exists two different parallel normal fields $\text{\bf n}_{r_1}^*,\ \text{\bf n}_{r_2}^*;$
4. there exists $r>0,$ such that $A^{\text{\bf n}_r^*}=-\alpha{\text{id}},$ where $\alpha$ is constant.
($1. \Rightarrow 2.$) It is not hard to see that, because $M\subset\{x_{n+1}=c\}\cap H_+^{n}(0,R),$ for every $r>0,$ $$\label{chinhhang}\textbf n_r^*=\alpha\textbf X+ \beta\textbf v,$$ where $\alpha,\beta$ are constants. Since $\textbf X$ is parallel and $\textbf v=(0,0,\ldots,0, -1)$ is constant, $\textbf n_r^*$ is parallel.
($2. \Rightarrow 3.$) Obviously.
($3. \Rightarrow 1.$)Because $\textbf X$ is a parallel normal field and $\{\text{\bf n}_{r_1}^{*},\text{\bf n}_{r_2}^{*}\}$ is a basis of $N_pM,$ we have the linear expression $$\label{bieuthi}\textbf X=\alpha \textbf n_{r_1}^*+\beta \textbf n_{r_2}^*,$$ where $\alpha,\beta$ are constants by virtue of Lemma \[lempara\]. Since the last coordinates of $\text{\bf n}_{r_1}^{*}$ and $\text{\bf n}_{r_2}^{*}$ are constants, the last coordinate of $\textbf X$ is constant.
($1.\Rightarrow 4.$) The equation (\[chinhhang\]) implies that $$A^{\textbf n_r^*}=-\alpha\text{id}.$$
($4.\Rightarrow 1.$) By the assumption, $M$ is $\textbf n_r^*$-umbilic. By virtue of Theorem \[theoum1\], $M\subset HP(\textbf a,c),$ where $\textbf a$ is a unit vector. Except at most one point, where $\textbf X$ is parallel to $\textbf a,$ $$\textbf n_r^*=\alpha\textbf X+\beta\textbf a,$$ where $\beta$ is a differential function on $M$.
Since $\langle\textbf n_r^*,\textbf n_r^*\rangle=2r,\ \ \langle\textbf X,\textbf X\rangle=-R^2, \ \ \langle\textbf X,\textbf a\rangle=c$ we obtain the following equation $$2r=-\alpha^2 R^2+2\alpha c\beta +\textbf a^2\beta^2 .$$ Thus, $\beta$ is constant and therefore so is the last coordinate of $\textbf X.$
The following theorem give another necessary and sufficient condition for a surface to be a part of a right hypersphere of a hyperbolic space without the assumption of lying in the hyperbolic space.
\[hang\] Let $M$ be a surface in $\Bbb R_1^{n+1}.$ The following statements are equivalent
1. there exists $r>0$ such that $\textbf n_r^*$ is parallel, not constant, and $M$ is $\textbf n_r^*$-umbilic;
2. $M$ is contained in a right hypersphere in a hyperbolic space.
($(1)\Rightarrow (2)$) Since $M$ is $\textbf n_r^* $-umbilic, $\textbf n_r^*$ is parallel and $\langle \textbf n_r^*,\textbf n_r^*\rangle =2r;$ $d\textbf n_r^*=\alpha d\textbf X,\ \alpha=\text{const.}\ne 0,$ by virtue of Lemma 4.2 in [@izu1]. Therefore, $$\label{111} \textbf n_r^*=\alpha\textbf X+\textbf a,$$ where $\textbf a$ is constant.
Let $\textbf v=(0,0,\dots,0,-1).$ From (\[111\]) we have $$\textbf X-\frac{1}{\alpha}(\textbf v-\textbf a)=\frac{1}{\alpha}(\textbf n_r^*-\textbf v).$$ A simple calculation yields $$\langle \text X-\frac{1}{\alpha}(\textbf v-\textbf a),\langle \text X-\frac{1}{\alpha}(\textbf v-\textbf a)\rangle=-\frac{1}{\alpha^2},$$ i.e. $M$ is contained in the hyperbolic space with center $\frac{1}{\alpha}(\textbf v-\textbf a) $ and radius $R=\frac{1}{\alpha},$ and hence contained in a right hypersphere by virtue of Theorem \[theoum2\].
($(2)\Rightarrow (1)$) is obvious by Theorem \[theoum2\].
The following is somewhat similar to the first statement of Lemma 4.2 in [@izu1].
\[theoum4\] Let $M$ be a connected surface in $\Bbb R_1^{n+1}.$ If there exists $r>0,$ such that $M$ is $n_r^*$-umbilic and for every $i, j \in\{1,2,\ldots, n-1\}$ $$\label{dk}[(\textbf n_r^*)^T_{u_i}]_{u_j}=[(\textbf n_r^*)^T_{u_j}]_{u_i}$$ then $A_p^{\textbf n_r^*}=-\alpha\text{id},$ where $\alpha$ is constant.
By the assumption, we have $$(\textbf n_r^+)_{u_i}^T=\alpha \textbf X_{u_i},\ \ i=1,2,\ldots n-1.$$ Therefore, for every $i, j \in\{1,2,\ldots, n-1\}$ $$[(\textbf n_r^*)^T_{u_i}]_{u_j}=\alpha_{u_j}\textbf X_{u_i}+\alpha \textbf X_{u_iu_j},$$ and $$[(\textbf n_r^*)^T_{u_j}]_{u_i}=\alpha_{u_i}\textbf X_{u_j}+\alpha \textbf X_{u_ju_i}.$$ Since $[(\textbf n_r^*)^T_{u_i}]_{u_j}=[(\textbf n_r^*)^T_{u_j}]_{u_i}$ and $\textbf X_{u_iu_j}=\textbf X_{u_ju_i},$ we have $$\alpha_{u_i}\textbf X_{u_j}-\alpha_{u_j}\textbf X_{u_i}=0;$$ and hence $\alpha_{u_i}=\alpha_{u_j}=0$ because $\textbf X_{u_i},\textbf X_{u_j}$ are linear independent; and therefore $\alpha$ is constant because $M$ is connected.
Examples
========
We construct some concrete examples to illustrate the above results.
This example shows that there exists an $\textbf n_r^*$-umbilic surface but not totally umbilic.
Let $M$ be a parametric surface in $\mathbb \Bbb R^4_1$, defined by the parametric equation $$\textbf X(u,v)=\left(\frac{1}{2}u^2,au-\frac{1}{2}u^2,u^2+v^2,u\right),\ v>0,\ u>1,\ a=\sqrt 3-1$$
A direct computation shows that $\textbf X$ is spacelike and $$\textbf n_a^-=\left(1,1,0, a\right),$$ $$\textbf n_a^+=\left(\frac{-a^2+4ua-2u^2}{a^2-2ua+2u^2},\frac{a^2-2u^2}{a^2-2ua+2u^2},0,a \right).$$
Since $\textbf n_a^-$ is constant, $M$ is $\textbf n_a^-$-flat. We can check that $\textbf X\subset HP(\textbf n_a^-, 0).$
Calculating the first and the second fundamental forms (with respect to $\textbf n_a^+$) yields $$(g_{ij})=\left(\begin{matrix}6u^2-2au+a^2-1&4uv\\4uv&4v^2 \end{matrix}\right),$$ and $$(b_{ij}(\textbf n_a^+))=\begin{pmatrix}\frac{-2a^2+4au}{a^2-2au+2u^2}&0\\0&0 \end{pmatrix}.$$ Therefore, the principal curvatures $k_1^{\textbf n_a^-}$ and $k_2^{\textbf n_a^-}$ are the solutions of the following equation $$\label{kho}4v^2\left(2u^2-2au+a^2-1\right)k^2-4v^2\left(\frac{-2a^2+4au}{a^2-2au+2u^2} \right)k =0.$$ It is easy to see that $k_1^{\textbf n_a^+}=0$ and $k_2^{\textbf n_a^+}\ne 0.$ Thus, $M$ is not $\textbf n_a^+$-umbilic.
\[exflat\] This is an example of a totally umbilic surfaces, but the curvature $\lambda$ is not constant. Consider the equidistance hypersurface in $H_+^3(0,1)$ $$M=H_+^3(0,1)\cap \{x_1=0\}=\textbf X(\mathbb R^2)$$ defined by $$\textbf X(u,v)=(0,u,v,\sqrt{u^2+v^2+1});\ (u,v)\in\mathbb R^2.$$
A direct computation yields $$\textbf X_u=\left(0,1,0,\frac{u}{\sqrt{u^2+v^2+1}}\right),\ \textbf X_v=\left(0,0,1,\frac{v}{\sqrt{u^2+v^2+1}}\right);$$ $$g_{11}=\frac{v^2+1}{u^2+v^2+1},\ \ \ g_{12}=g_{21}=\frac{-uv}{u^2+v^2+1},\ \ \ g_{22}=\frac{u^2+1}{u^2+v^2+1};$$
$$\textbf n_r^{\pm}=\left(\pm\sqrt{\frac{r^2}{u^2+v^2+1}+2r},\frac{ru}{\sqrt{u^2+v^2+1}},\frac{rv}{\sqrt{u^2+v^2+1}},r\right);$$
$$\textbf X_{uu}=\left(0,0,0,\frac{v^2+1}{(u^2+v^2+1)^{3/2}}\right),\textbf X_{vv}
=\left(0,0,0,\frac{u^2+1}{(u^2+v^2+1)^{3/2}}\right),$$ $$\textbf X_{uv}=\left(0,0,0,\frac{-uv}{(u^2+v^2+1)^{3/2}}\right);$$
$$(g_{ij})=\frac{1}{u^2+v^2+1}\left(\begin{matrix}v^2+1&-uv\\-uv&u^2+1\end{matrix}\right);\ (g_{ij})^{-1}=\left(\begin{matrix}u^2+1&uv\\uv&v^2+1\end{matrix}\right);$$
$$(b_{ij}^{\textbf n_r^{\pm}})=\frac{-r}{(u^2+v^2+1)^{3/2}}\left(\begin{matrix}v^2+1&-uv\\-uv&u^2+1\end{matrix}\right);$$
$$(a_{ij}^{\textbf n_r^{\pm}})=(b_{ij}^{\textbf n_r^{\pm}})(g_{ij})^{-1}=\frac{-r}{\sqrt{u^2+v^2+1}}\left(\begin{matrix}1&0\\0&1\end{matrix}\right);$$
$$[(\textbf n_r^+)^T_{u}]_{v}=\left(0,\frac{-rv}{\sqrt{(u^2+v^2+1)^3}},0,\frac{-2ruv}{(u^2+v^2+1)^2} \right);$$ $$[(\textbf n_r^+)^T_{v}]_{u}=\left(0,0,\frac{-ru}{\sqrt{(u^2+v^2+1)^3}},\frac{-2ruv}{(u^2+v^2+1)^2}\right).$$
We can see that $M$ is totally umbilic. Moreover, $$k_p^{\textbf n_r^{\pm}}=\frac{-r}{\sqrt{u^2+v^2+1}}$$ is not constant and $[(\textbf n_r^+)^T_{u}]_{v}\ne [(\textbf n_r^+)^T_{v}]_{u}$ (see Theorem \[theoum4\]).
This is an example of a $\nu$-umbilic but neither $\textbf n_r^+$- nor $\textbf n_r^-$-umbilic for any $r\in \Bbb R_+.$ Let $$\textbf X:(0,\frac{\pi}{2})\times(-\frac{\pi}{2},0)\rightarrow \mathbb R_1^4,\ \ \ \ (u,v)\mapsto (u,\sin v,v,\cos u).$$ A direct computation yields $$\textbf X_u=(1,0,0,-\sin u),\ \ \ \textbf X_v=(0,\cos v,1,0);$$ $$\textbf X_{uu}=(0,0,0,-\cos u),\ \ \textbf X_{uv}=\textbf X_{vu}=(0,0,0,0),\ \ \textbf X_{vv}=(0,-\sin v,0,0);$$ $$g_{11}=\langle \textbf X_u,\textbf X_u\rangle=\cos^2u>0,\ \ g_{12}=\langle \textbf X_u,\textbf X_v\rangle=0,\ \ g_{22}=\langle \textbf X_v,\textbf X_v\rangle=1+\cos^2v>0;$$
$$\textbf n_r^+=\left(-r\sin u,-\sqrt{\frac{r^2\cos^2u+2r}{1+\cos^2v}},\cos v\sqrt{\frac{r^2\cos^2u+2r}{1+\cos^2v}},r\right);$$ $$\textbf n_r^-=\left(-r\sin u,\sqrt{\frac{r^2\cos^2u+2r}{1+\cos^2v}},-\cos v\sqrt{\frac{r^2\cos^2u+2r}{1+\cos^2v}},r\right);$$
$$(b_{ij}^{\textbf n_r^{\pm}})=\begin{pmatrix}r\cos u&0\\0&\mp\sin v\sqrt{\frac{r^2\cos^2u+2r}{1+\cos^2v}}\end{pmatrix};$$
$$(g_{ij})=\begin{pmatrix}\cos^2u&0\\0&1+\cos^2v\end{pmatrix};$$
$$\label{mtA} (a_{ij}^{\textbf n_r^{\pm}})
=(b_{ij}^{\textbf n_r^{\pm}}).(g_{ij})^{-1}
=\begin{pmatrix}\frac{r}{\cos u}&0\\0&\mp\sin v\sqrt{\frac{r^2\cos^2u+2r}{(1+\cos^2v)^3}}\end{pmatrix};$$
$$\label{dcc} k_1^{\textbf n_r^{\pm}}(P)=\frac{r}{\cos u},\ k_2^{\textbf n_r^{\pm}}(p)=\mp\sin v\sqrt{\frac{r^2\cos^2u+2r}{(1+\cos^2v)^3}}.$$
At each point $p=x(u,v)\in M,$ let $\nu(p)= \text n_{r_{p}},$ where $r_{p}=\frac{2\sin^2v\cos^2 u}{(1+\cos^2v)^3-\cos^4u\sin^2v}.$ We can see that $\nu$ is a smooth normal vector field on $M$ and $M$ is $\nu$-umbilic but neither $\textbf n_r^+$- nor $\textbf n_r^-$-umbilic for any $r\in \Bbb R_+.$
Let $$\textbf X(u,v)=\left(u,\sin v,\cos v,\sqrt{2+u^2}\right),\quad u\in\mathbb R,\ \ v\in(-\pi/2,\pi/2).$$ Because $\langle \textbf X,\textbf X\rangle =-1,$ $M\subset H_+^4(0,1).$ A direct computation yields $$\textbf X_u=\left(1,0,0,\frac{u}{\sqrt{u^2+2}}\right) ,\qquad \textbf X_v=\left(0,\cos v,-\sin v,0\right);$$ $$g_{11}=\frac{2}{2+u^2},\ g_{12}=g_{21}=0,\ g_{22}=1;$$ $$\textbf n_r^{\pm}=\left(\frac{ru}{\sqrt{u^2+2}},\pm\sin v\sqrt{-\frac{u^2r^2}{u^2+2}+r^2+2r },\pm\cos v\sqrt{-\frac{u^2r^2}{u^2+2}+r^2+2r },r \right);$$ $$\textbf X_{uu}=\left(0,0,0,\frac{2}{(u^2+2)^{3/2}} \right),\ \ \textbf X_{uv}= \textbf X_{vu}= \left(0,0,0,0\right),\ \ \textbf X_{vv}=\left(0,-\sin v,-\cos v,0\right)$$
$$b_{11}^{\textbf n_r^{\pm}}=\frac{-2r}{(u^2+2)^{3/2}},\ b_{12}^{\textbf n_r^{\pm}}=0,\ b_{22}^{\textbf n_r^{\pm}}=\mp\sqrt{\frac{2r(u^2+r+2)}{u^2+2} };$$
$$k_1^{\textbf n_r^{\pm}}=\frac{-r}{\sqrt{u^2+2}},\ \ k_2^{\textbf n_r^{\pm}}=\mp\sqrt{\frac{2r(u^2+r+2)}{u^2+2}}.$$
We can see that $k_1^{\textbf n_r^+}> k_2^{\textbf n_r^+}$ while $k_1^{\textbf n_r^-}< k_2^{\textbf n_r^-}$ for any $r>0.$ Thus, $M$ is not $\nu$-umbilic, for any normal vector field $\nu\ne\textbf X.$
[99]{} , [*Differential Geometry of Curves and Surfaces*]{}, Prentice Hall, 1976. , [*The Gauss map of surfaces in $R\sp{n}$*]{}, J. Differential Geom. 18 (1983), no. 4, 733-754. , [*The lightcone Gauss map and the lightcone developable of a spacelike curve in Minkowski 3-space*]{}, Glasgow. Math. J (42), (2000) 75-89. , [*Singularities of hyperbolic Gauss maps*]{}, Proceedings of the London Mathematical Society (86), (2003) 485-512. , [*Umbilicity of spacelike submanifolds of Minkowski space*]{}, Proceedings of the Royal Society of Edinburgh (134A), (2004) 375-387. , [*The lightlike flat geometry on spacelike submanifolds of codimension two in Minkowski space*]{}, Selecta Math. (N.S.) 13 (2007), no. 1, 23-55. , [*Global properties on spacelike submanifolds of codimension two in Minkowski space*]{}, Proceedings of the Royal Society of Edinburgh 53A40, 53A35, 2007. , [*The $S^2$-valued Gauss maps and split total curvature of a spacelike codimension-2 surface in Minkowski space*]{}, J. London Math. Soc. (2) 40 (1989) 179-192. , [*The Gauss map of surfaces in $R\sp{n}$*]{}, J. Differential Geom. 18 (1983), no. 4, 733-754. , [*A survey of minimal surfaces*]{}, Dover Publ. Inc., New York, 1986.
[[email protected]]{}
[[email protected]]{}
[^1]: The authors is supported in part by the National Foundation for Science and Technology Development, Vietnam (Grant No. 101.01.30.09).
[^2]: 2000 *Mathematics Subject Classification*. Primary 00; Secondary 00.
[^3]: *Key words and phrases*. Lorentz-Minkowski space, $\textbf n_r^{\pm}$-Gauss map, Umbilicity.
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author:
- 'M. Ackermann, M. Ajello, A. Allafort, E. Antolini, W. B. Atwood, M. Axelsson, L. Baldini, J. Ballet, G. Barbiellini, D. Bastieri, K. Bechtol, R. Bellazzini, B. Berenji, R. D. Blandford, E. D. Bloom, E. Bonamente, A. W. Borgland, E. Bottacini, A. Bouvier, J. Bregeon, M. Brigida, P. Bruel, R. Buehler, T. H. Burnett, S. Buson, G. A. Caliandro, R. A. Cameron, P. A. Caraveo, J. M. Casandjian, E. Cavazzuti, C. Cecchi, E. Charles, C. C. Cheung, J. Chiang, S. Ciprini, R. Claus, J. Cohen-Tanugi, J. Conrad, L. Costamante, S. Cutini, A. de Angelis, F. de Palma, C. D. Dermer, S. W. Digel, E. do Couto e Silva, P. S. Drell, R. Dubois, L. Escande, C. Favuzzi, S. J. Fegan, E. C. Ferrara, J. Finke, W. B. Focke, P. Fortin, M. Frailis, Y. Fukazawa, S. Funk, P. Fusco, F. Gargano, D. Gasparrini, N. Gehrels, S. Germani, B. Giebels, N. Giglietto, P. Giommi, F. Giordano, M. Giroletti, T. Glanzman, G. Godfrey, I. A. Grenier, J. E. Grove, S. Guiriec, M. Gustafsson, D. Hadasch, M. Hayashida, E. Hays, S. E. Healey, D. Horan, X. Hou, R. E. Hughes, G. Iafrate, G. Jóhannesson, A. S. Johnson, W. N. Johnson, T. Kamae, H. Katagiri, J. Kataoka, J. Knödlseder, M. Kuss, J. Lande, S. Larsson, L. Latronico, F. Longo, F. Loparco, B. Lott, M. N. Lovellette, P. Lubrano, G. M. Madejski, M. N. Mazziotta, W. McConville, J. E. McEnery, P. F. Michelson, W. Mitthumsiri, T. Mizuno, A. A. Moiseev, C. Monte, M. E. Monzani, E. Moretti, A. Morselli, I. V. Moskalenko, S. Murgia, T. Nakamori, M. Naumann-Godo, P. L. Nolan, J. P. Norris, E. Nuss, M. Ohno, T. Ohsugi, A. Okumura, N. Omodei, M. Orienti, E. Orlando, J. F. Ormes, M. Ozaki, D. Paneque, D. Parent, M. Pesce-Rollins, M. Pierbattista, S. Piranomonte, F. Piron, G. Pivato, T. A. Porter, S. Rainò, R. Rando, M. Razzano, S. Razzaque, A. Reimer, O. Reimer, S. Ritz, L. S. Rochester, R. W. Romani, M. Roth, D.A. Sanchez, C. Sbarra, J. D. Scargle, T. L. Schalk, C. Sgrò, M. S. Shaw, E. J. Siskind, G. Spandre, P. Spinelli, A. W. Strong, D. J. Suson, H. Tajima, H. Takahashi, T. Takahashi, T. Tanaka, J. G. Thayer, J. B. Thayer, D. J. Thompson, L. Tibaldo, M. Tinivella, D. F. Torres, G. Tosti, E. Troja, Y. Uchiyama, J. Vandenbroucke, V. Vasileiou, G. Vianello, V. Vitale, A. P. Waite, E. Wallace, P. Wang, B. L. Winer, D. L. Wood, K. S. Wood, S. Zimmer'
title: 'The Second Catalog of Active Galactic Nuclei Detected by the [*Fermi*]{} Large Area Telescope'
---
\[sec:intro\]Introduction
=========================
This paper presents a catalog of active galaxy nuclei (AGNs) associated through formal probabilities with high-energy $\gamma$-ray sources detected in the first two years of the [*Fermi*]{} Gamma-ray Space Telescope mission by the Large Area Telescope (LAT). This catalog is based on the larger second LAT catalog, 2FGL [@2FGL] and is a follow-up of the first LAT AGN catalog, 1LAC [@1LAC]. The second LAT AGN catalog, 2LAC includes a number of analysis refinements and additional association methods which have substantially increased the number of associations over 1LAC.
The high sensitivity and nearly uniform sky coverage of the LAT make it a powerful tool for investigating the properties of large populations. The first list of bright AGNs detected by the LAT, the LAT Bright AGN Sample [LBAS; @LBAS] included AGNs at high Galactic latitude ($|b|>10\arcdeg$) detected with high significance (Test Statistic[^1], $TS>100$) during the first three months of scientific operation. This list comprised 58 flat-spectrum radio quasars (FSRQs), 42 BL Lacs, two radio galaxies, and four AGNs of unknown type. The next evolution, 1LAC, based on the first 11 months of data included 671 sources detected with $TS > 25$ at high Galactic latitudes ($|b| > 10\arcdeg$). The 1LAC Clean Sample (sources with single associations and not affected by analysis issues) comprised 599 sources: 248 FSRQs, 275 BL Lacs, 26 other AGNs and 50 blazars of unknown type. The main findings of 1LAC, summarized below, were consistent with those found with the LBAS.
1. Only a small number of non-blazar AGNs detected;
2. redshift distributions peaking at $z \approx 1$ for 1LAC FSRQs and at low redshift for 1LAC BL Lacs with known redshifts (only 60% of the total);
3. similar numbers of BL Lacs and FSRQs;
4. high-synchrotron-peaked sources representing the largest subclass among BL Lacs;
5. little evidence for different variability properties for FSRQs and BL Lacs using monthly light curves; a more detailed analysis based on weekly light curves [@Abdo_var] showed that bright FSRQs exhibit larger fractional variability than do BL Lacs.
6. the detected high-synchrotron-peaked sources have harder spectra and lower $\gamma$-ray luminosity than lower synchrotron-peaked sources.
The 1LAC catalog has proven to be an invaluable resource opening the way to numerous studies on the blazar sequence and the BL Lac-FSRQ dichotomy issue [@Ghi11; @Ghi10t; @Bjo10; @Che11; @Tra10], blazar evolution [@Ino10], the comparison of properties of $\gamma$-ray loud and $\gamma$-ray quiet blazars [@Mahony2010; @Linford2011; @Kar10; @Cha11], the contribution of AGNs to the extragalactic diffuse $\gamma$-ray background [@Abdo_EDB; @Sin11; @Ven11], the correlation between AGNs and the sources of ultra high-energy cosmic rays [@Jia10; @Der10; @Nem10; @Kim10], the timing correlations between the activity in the $\gamma$-ray bands and other bands [@Pus10; @Ric10], and the attenuation of $\gamma$-rays by Extragalactic Background Light [@Abdo_EBL; @Rau10]. The release of the 1LAC also triggered TeV observations leading to discoveries of new TeV-emitting blazars [e.g., @RBS0413:ATel2272].
Here we report on the AGNs associated with LAT sources detected after 24 months of scientific operation. The second LAT AGN catalog comprises a total of 1017 sources detected with $TS>25$ at high Galactic latitudes ($|b| > 10\arcdeg$). Due to some analysis issues, some sources were flagged in the 2FGL catalog and 26 sources have two possible associations, so we define a Clean Sample, which includes 886 sources. An additional 104 sources at $|b| < 10\arcdeg$ are also presented here.
In Section 2, the observations by the LAT and the analysis employed to produce the two-year catalog are described. In Section 3, we explain the methods for associating sources with AGN counterparts and present the results of these methods. Section 4 describes the different schemes for classifying 2LAC AGNs. Section 5 provides a brief census of the 2LAC sample. Section 6 summarizes some of the properties of the 2LAC, including the flux distribution, the photon spectral index distribution, the variability properties, the redshift distribution, and the luminosity distribution. In Section 7, we discuss some radio, optical and TeV properties of the 2LAC AGNs. We discuss the implications of the 2LAC results in Section 8 and conclude in Section 9.
In the following, we use a $\Lambda$CDM cosmology with values within $1\sigma$ of the [*Wilkinson Microwave Anisotropy Probe*]{} ([*WMAP*]{}) results [@WMAP11]; in particular, we use $h = 0.70$, $\Omega_m = 0.27$, and $\Omega_\Lambda = 0.73$, where the Hubble constant $H_0=100h$ km s$^{-1}$ Mpc$^{-1}$. We also define the radio spectral indices such that $S(\nu) \propto \nu^{-\alpha}$.
\[sec:obs\]Observations with the Large Area Telescope — Analysis Procedures
===========================================================================
The 2LAC sources are a subset of those in the 2FGL catalog, so we only briefly summarize the analysis here and we refer the reader to the paper describing the 2FGL catalog [@2FGL] for details. The data were collected over the first 24 months of the mission from 2008 August 4 to 2010 August 1, with an overall data-taking efficiency of 74%. Time intervals during which the rocking angle of the LAT was greater than 52$^{\circ}$ were excluded (leading to a reduction in exposure of less than 2%). A cut on the zenith-angle of $\gamma$-rays of 100$^{\circ}$ was applied. The Pass 7\_V6 Source event class [@2FGL] was used, with photon energies between 100 MeV and 100 GeV. In the study of the highest-energy photons detected for each source, presented in §\[sec:hep\], photons belonging to the purest (i.e., with the lowest instrumental background) class (Pass 7\_V6 Ultraclean) were used, without any high-energy cut.
The source detection procedure considered seed sources taken from 1FGL and the results of three point-source detection methods, described in [@1FGL] were used: [*mr\_filter*]{} [@sp98], [*PGWave*]{} [@ciprini07] and the minimal spanning tree method [@Cam08]. With these seeds, an all-sky likelihood analysis produced an “optimized” model, where parameters characterizing the diffuse components[^2] in addition to sources were fitted. The analysis of the residual TS map provided new seeds that were included in the model for a new all-sky likelihood analysis. This iterative procedure yielded 3499 seeds that were then passed on to the maximum likelihood analysis for source characterization.
The analysis was performed with the binned likelihood method implemented in the [*pyLikelihood*]{} library of the Science Tools[^3] (v9r23p0). Different spectral fits were carried out with a single power-law function () and a LogParabola function\
(), where $E_0$ is an arbitrary reference energy adjusted on a source-by-source basis to minimize the correlation between $N_0$ and the other fitted parameters over the whole energy range (0.1 to 100 GeV). Whenever the difference in log(likelihood) between these two fits was greater than 8 [i.e., $TS_{curve}$, defined as twice this difference, see @2FGL was greater than 16] the LogParabola results were retained. The photon spectral index ($\Gamma$) presented in this paper was obtained from the single power-law fit for all sources. A threshold of $TS=25$ was applied to all sources, corresponding to a significance of approximately 4 $\sigma$. At the end of this procedure, 1873 sources survived the cut on TS. Power-law fits were also performed in five different energy bands (0.1-0.3, 0.3-1, 1-3, 3-10 and 10-100 GeV), from which the energy flux was derived. A variability index [[*TS$_{VAR}$*]{}, see @2FGL] was constructed from a likelihood test based on the monthly light curves, with the null (alternative) hypothesis corresponding to the source being steady (variable). A source is identified as being variable at the 99% level if the variability index is equal or greater than 41.6.
Some of the 2FGL sources were flagged as suspicious when certain issues arose during their analysis [see @2FGL for a full list of these flags]. The issues that most strongly affected the 2LAC list are: i) sources moving beyond their 95% error ellipse when changing the model of Galactic diffuse emission, ii) sources with $TS >$ 35 going down to $TS <$ 25 when changing the diffuse model, iii) sources located closer than $\theta_{ref}$ [defined in Table 2 of @2FGL] to a brighter neighbor, iv) source $Spectral\_Fit\_Quality >$ 16.3 ($\chi^{2}$ between spectral model and flux in five energy bands). Therefore, we applied a selection on sources to build a clean sample of AGNs.
Thanks to its large field of view and sky survey mode, the LAT sensitivity is relatively uniform at large Galactic latitudes, although the switch from a rocking angle of 35$\arcdeg$ to 50$\arcdeg$ in September 2009 reduced this uniformity [@2FGL]. A map of the flux limit, calculated for the two-year period covered by this catalog, a $TS=25$ and a photon index of 2.2, is shown in Galactic coordinates in Figure \[fig:sens\]. The 95% error radius [defined as the geometric mean of the semimajor and semiminor axes of the ellipse fitted to the $TS$ map, see @2FGL] is plotted as a function of $TS$ in Figure \[fig:r95\_TS\]. It ranges from about $0\fdg01$ for 3C 454.3, the brightest LAT blazar, to $0\fdg2$ on average for sources just above the detection threshold (similar to 1LAC).
Source Association
==================
The LAT localization accuracy is not precise enough to permit the determination of a lower-energy counterpart based only on positional coincidence. We assert a firm counterpart identification only if the variability detected by the LAT corresponds with variability at other wavelengths. In practice, such identifications have been made only for 28 2FGL AGNs (see Table 5 in @2FGL). For the rest, we use statistical approaches for finding associations between LAT sources and AGNs.
In 1FGL, several sources were flagged as [*affiliated*]{} AGNs (and thus not included in 1LAC) as the methods providing associations were not able to give a quantitative association probability. Moreover some LAT-detected blazars turn out to be fainter in radio than the flux limit of catalogs of flat-spectrum radio sources. In order to improve over the results of 1LAC by including these faint radio sources, the association procedure for building the 2LAC list makes use of three different methods: the Bayesian Method (used in 1FGL/1LAC) and two additional methods, namely the Likelihood Ratio Method and the $\log N - \log S$ Method. These procedures are described respectively in §\[sec:gtsrcid\], \[sec:like\], \[sec:steve\]. For a counterpart to be considered as associated, its association probability must be $>$ 0.8 for at least one method.
The two additional methods improve the association results through the use of physical properties of the candidate counterparts, such as the surface density and the spectral shape in the radio energy band, in addition to the positional coincidence with the $\gamma$-ray source. Considering potential counterparts with lower radio flux enables more high-synchrotron peaked BL Lacs to be selected but the number of FSRQs is also increased. This is achieved through the use of surveys and serendipitous findings, as the available catalogs (used by the Bayesian Method) are not deep enough.
\[sec:gtsrcid\]The Bayesian Association Method
----------------------------------------------
The Bayesian method [@deRuiter77; @ss92], implemented by the [*gtsrcid*]{} tool in the LAT [*ScienceTools*]{}, is similar to that used by @mhr01 to associate EGRET sources with flat-spectrum radio sources. A more complete description is given in the Appendix of [@1FGL] and in [@2FGL], but we provide a basic summary here. The method uses Bayes’ theorem to calculate the posterior probability that a source from a catalog of candidate counterparts is truly an emitter of detected by the LAT. The significance of a spatial coincidence between a candidate counterpart from a catalog $C$ and a LAT-detected source is evaluated by examining the local density of counterparts from $C$ in the vicinity of the LAT source. We can then estimate the likelihood that such a coincidence is due to random chance and establish whether the association is likely to be real. To each catalog $C$, we assign a prior probability, assumed for simplicity to be the same for all sources in $C$, for detection by the LAT. The prior probability for each catalog can be tuned to give the desired number of false positive associations for a given threshold on the posterior probability, above which the associations are considered reliable (see § \[sec:cat\]). The posterior probability threshold for high-confidence associations was set to 80%.
Candidate counterparts were drawn from a number of source catalogs. With respect to 1FGL, all catalogs for which more comprehensive compilations became available have been updated. The catalogs used are the 13th edition of the Veron catalog [@AGNcatalog], version 20 of BZCAT [@bzcat], the 2010 December 5 version of the VLBA Calibrator Source List[^4], and the most recent version of the TeVCat catalog[^5]. We also added new counterpart catalogs, the Australia Telescope 20-GHz Survey (AT20G) [@AT20G_CAT; @Mas11] and the [*Planck*]{} Early Release Catalogs [@Planckcatalog].
\[sec:like\]The Likelihood-Ratio (LR) Association Method
--------------------------------------------------------
The Likelihood Ratio method has been introduced to make use of uniform surveys in the radio and in X-ray bands in order to search for possible counterparts among the faint radio and X-ray sources. The main differences with the Bayesian method are that i) the LR makes use of counterpart densities through the $\log N - \log S$ and therefore the source flux, ii) the LR assumes, in this paper, that the counterpart density is constant over the survey region. An improved version of the LR should take into consideration the local density, which is mandatory in the case of optical counterparts but not for radio and X-ray because of their lower surface densities. We assigned $\gamma$-ray associations and estimate their reliability using a likelihood ratio analysis which has frequently been used to assess identification probabilities for radio, infrared and optical sources [e.g., @deRuiter77; @Pre83; @ss92; @Lon98; @Mas01].
We made use of a number of relatively uniform radio surveys. Almost all radio AGN candidates of possible interest are detected either in the NRAO VLA Sky Survey [NVSS; @NVSScatalog] or the Sydney University Molonglo Sky Survey [SUMSS; @SUMSScatalog]. We added the 4.85 GHz Parkes-MIT-NRAO (PMN) Surveys [@PMNcat; @Gri95; @Wri94; @Wri96], with a typical flux limit of about 40 mJy which varies as a function of declination, as well as the recently released AT20G source catalog [@AT20G_CAT; @Mas11], which contains entries for 5890 sources observed at declination $\delta <$0. In this way we are able to look for counterparts with radio flux down to 5 mJy. To look for additional possible counterparts we cross-correlated the LAT sources with the most sensitive all-sky X-ray survey, the ROSAT All Sky Survey Bright and Faint Source Catalogs [@RASSbright; @RASS_FAINT_CAT]. A source is considered as a likely counterpart of the $\gamma$-ray source if its reliability (see Eq. \[rel\]) is $>$0.8 in at least one survey.
The method, which computes the probability that a suggested association is the ‘true’ counterpart, is outlined as follows. For each candidate counterpart $i$ in the search area neighboring a 2FGL $\gamma$-ray source $j$, we calculate the normalized distance between $\gamma$-ray and radio/X-ray positions: $$r_{ij}=\frac{\Delta}{(\sigma_{a}^{2} + \sigma_{b}^{2})^{1/2}}
\label{rij}$$ where $\Delta$ is the angular distance between the $\gamma$-ray source and its prospective counterpart and $\sigma_{a}$ and $\sigma_{b}$ represent the errors on $\gamma$-ray and counterpart positions respectively.
Given $r_{ij}$, we must now distinguish between two mutually exclusive possibilities: i) the candidate is a confusing background object that happens to lie at distance $r_{ij}$ from the $\gamma$-ray source ii) the candidate is the ‘true’ counterpart that appears at distance $r_{ij}$ owing solely to the $\gamma$-ray and radio/X-ray positional uncertainties. We assume that the $\gamma$-ray and radio/X-ray positions would coincide if these uncertainties were negligibly small [@Mas01].
To distinguish between these cases, we compute the likelihood ratio $LR_{ij}$, defined as: $$LR_{ij}=\frac{e^{-r_{ij}^{2}/2}}{N(>S_{i})~A}
\label{LR}$$ where $N(>S_{i})$ is the surface density of objects brighter than candidate $i$ (i.e., the $\log N - \log S$) and $A$ is the solid angle spanned by the 95% confidence LAT error ellipse. The likelihood ratio $LR_{ij}$ is therefore simply the ratio of the probability of an association (the Rayleigh distribution: $r\exp{(-r^{2}/2)}$), to that of a chance association at $r$. $LR_{ij}$ therefore represents a ‘relative weight’ for each match $ij$, and our aim is to find an optimum cutoff value $LR_{c}$ above which a source is considered to be a reliable candidate.
The value of LR$_c$ can be evaluated using simulations as described in [@Lon98]. We generate a truly random background population with respect to the $\gamma$-ray sources by randomly displacing $\gamma$-ray sources within an annulus with inner and outer radii of $2^{\circ}$ and $10^{\circ}$ respectively around their true positions. In addition to extragalactic sources, 2FGL contains a population of Galactic $\gamma$-ray emitters that follows a rather narrow latitude distribution. We limit the source displacement in Galactic latitude to $ b~\pm~b_{max}$, where $$b_{max}=r_{max}(1-sech^{2}\frac{b}{b_0})$$ $r_{max}=10^{\circ}$, $b$ is the Galactic latitude of the $\gamma$-ray source, and $b_{0}=5^{\circ}$ is the angular scale height above the Galactic plane for which the latitude displacement is reduced. We further require that $b_{max}>0\fdg2$ to allow for a non-zero latitude displacement of sources in the Galactic plane, and require any source to be shifted by at least $r_{min}=2^{\circ}$ away from its original location. The results derived here do not critically depend on the exact values of $r_{max}$, $b_{max}$ and $b_0$ chosen for the simulations.
We generated 100 realizations of this fake $\gamma$-ray sky and for each of the 100 fake $\gamma$-ray catalogs, we calculated the respective LR value for all counterparts. Then we compared the number of associations for ([*true*]{}) $\gamma$-ray source positions with the number of associations found for ([*random*]{}) $\gamma$-ray source positions, which enabled us to determine a critical value LR$_c$ for reliable association. From these distributions, we computed the reliability as a function of LR. $$R(LR_{ij})=1-\frac{N_{random}(LR_{ij})}{N_{random}(LR_{ij})+N_{true}(LR_{ij})}
\label{rel}$$ where $N_{true}$ and $N_{random}$ are the number of associations with $\gamma$-ray sources in the [*true*]{} sky and those in the simulated ([*random*]{}) one respectively. The reliability computed in this way also represents an approximate measure of the association probability for a candidate with given LR.
Figure \[fig:nvss\_LR\] shows the two distributions of [*true*]{} (blue) and [*fake*]{} (red) LR values for the NVSS survey, which we report as an example. In order to obtain $R$ as a function of $LR$ we parametrize the reliability curve with the following function: $$f(LR)=1-a ~ exp(-b ~ LR)$$ The $a$ and $b$ parameters are given in Table \[tbl-ab\] for the different surveys. We use this function to calculate the reliability for each value of LR and select high-confidence counterparts. The values of log (LR$_c$) above which the reliability is greater than $80\%$ are listed in Table \[tbl-ab\] as well for the different surveys.
After having calculated the reliability of the association with the use of the LR based on the $\log N - \log S$ cited above, we look for typical blazar characteristics of a source taking into consideration its optical class and radio spectrum slope. The 2LAC being a list of AGN candidate counterparts for 2FGL sources, we include only AGN-type sources. We therefore looked at their optical spectra through an extensive program of optical follow-up (M. S. Shaw et al., 2011, in preparation and S. Piranomonte et al., 2011, in preparation) and the BZCAT list. Moreover we evaluated their spectral slopes in the radio through a cross-correlation with catalogs of flat-spectrum radio sources.
\[sec:steve\]$\log N - \log S$ Method
-------------------------------------
The $\log N - \log S$ association method is a modified version of the Bayesian method for blazars. The Bayesian method assesses the probability of association between a $\gamma$-ray source and a candidate counterpart using the local density of such candidates; this local density is estimated simply by counting candidates in a nearby region of the sky. The $\log N - \log S$ method differs in one small but important way: the density of “competing” candidates is estimated by using a model of the radio $\log N - \log S$ distribution of the candidate population. Specifically, the density $\rho$ that goes into the Bayesian calculation for a candidate $k$ with radio flux density $S_k$ and radio spectral index $\alpha_k$ is $\rho(S > S_k, \, \alpha < \alpha_k)$, the density of sources that are at least as bright and have spectra at least as flat as source $k$. [This attrition-based approach—considering only those sources that are as “good” as or “better” than the candidate in question—was used in practically the same way by @mat97; @mhr01]. The $\log N - \log S$ method has the distinct advantage of being extensible to radio data not found in any formal catalog. In particular, the method can be applied to new radio observations that explicitly target unassociated LAT sources with no loss of statistical validity.
In order to exploit the size and uniformity of the CRATES catalog and its proven utility as a source of radio/$\gamma$-ray blazar associations, we sought a model of the 8.4 GHz $\log N - \log S$ distribution of the flat-spectrum radio population. For $S \ga 85$ mJy, CRATES itself provides sufficient coverage of this population that the $\log N - \log S$ distribution can be directly examined and modeled. Below this flux density, however, the CRATES coverage declines rapidly. By definition, CRATES only includes sources with 4.85 GHz flux densities of at least 65 mJy, so the faint population is explicitly disfavored. In addition, because of this 4.85 GHz flux density limit, CRATES sources that are faint at 8.4 GHz are far more likely to be steep-spectrum objects.
Because the LAT selects $\gamma$-ray sources with radio counterparts fainter than those in radio catalogs of flat-spectrum radio sources such as CRATES, we required another source of 8.4 GHz data to study the faint end of the $\log N - \log S$ distribution. For this purpose, we looked to the Cosmic Lens All-Sky Survey [CLASS; @class1; @class2]. While CLASS did target sources down to a fainter limit than CRATES, we were able to push to even lower flux densities by considering serendipitous CLASS detections (i.e., sources that were not explicitly targeted by CLASS but which were detected in CLASS pointings). We assembled this sample by taking CLASS detections that were at least 60$\arcsec$ away from any CLASS pointing position in order to ensure that we were not using any component of the “real” CLASS target (e.g., a jet). We also considered only those sources with $S > 10$ mJy at 8.4 GHz to avoid sidelobes or other mapping errors.
Because the serendipitous sources were not intentionally targeted and appear in the CLASS data purely by a coincidence of their locations on the sky, they represent a statistically unbiased sample of the 8.4 GHz population, unaffected by any selection criterion other than their ability to be detected cleanly by the VLA. In order to model just the flat-spectrum members of this population, we computed spectral indices using 1.4 GHz data from NVSS and imposed a spectral index cut of $\alpha < 0.5$ (the same cut as for CRATES). In the end, we had a sample of $\sim$ 300 flat-spectrum sources with flux densities ranging from 10 mJy to $\sim$110 mJy. However, while the shape of the $\log N - \log S$ distribution for this sample could be studied, the sky area of this “survey” was not well defined, so the $\log N - \log S$ was not properly normalized. Fortunately, the flux density range of the CRATES coverage overlapped sufficiently with that of the serendipitous sample to allow us to scale the latter until it agreed with the former in the overlap region. We then had a full characterization of the 8.4 GHz $\log N - \log S$ distribution of the flat-spectrum population from 10 mJy to $\sim$10 Jy (see Figure \[fig:logNlogS\_steve\]). The integral form of the distribution is well modeled piecewise by
$$\begin{aligned}
\log N(>S) = 4.07 - 2.0 \log S \: \mathrm{for} \: \log S > 3.2 \\
\log N(>S) = 2.15 - 1.4 \log S \: \mathrm{for} \: \log S < 3.2\end{aligned}$$
where $N(>S)$ is the number of sources per square degree with flux density greater than $S$ at 8.4 GHz, expressed here in mJy .
With an understanding of the flux density distribution in hand, we turned to the second component of the attrition, the spectral indices. In particular, we sought to characterize how the spectral index distribution varied with increasing flux density. We sorted the radio data into logarithmic bins in flux density centered on $10$ mJy, $10^{1.5}$ mJy, and so on up to $10^4$ mJy, and we examined the spectral index distribution for each bin. In every case, the spectral index distribution was very well approximated by a Gaussian, and as it turned out, the widths of these Gaussians were very nearly the same, never deviating from the mean value of 0.29 by more than 0.01. Since these deviations are statistically insignificant, we adopt 0.29 as the fiducial standard deviation of the $\alpha$ distribution for all flux densities. The centers of the Gaussians increased with increasing flux density; we approximated the flux density dependence of the mean $\alpha$ as
$$\mu_\alpha(S) = 0.527 - 0.187 \log S$$
Thus, for a candidate counterpart $k$ with flux density $S_k$ and spectral index $\alpha_k$, the fraction $F_\alpha$ of competing counterparts that have spectra at least as flat as $k$ is the area to the left of $\alpha_k$ under a Gaussian with $\sigma_\alpha = 0.29$ centered on $\alpha = \mu_\alpha(S)$. The sought-after density of competing counterparts, $\rho(S > S_k, \, \alpha < \alpha_k)$, is then given simply by
$$\rho(S > S_k, \, \alpha < \alpha_k) = F_\alpha \times N(>S)$$
Once the attrition-based value is used for $\rho$, the rest of the Bayesian method is unchanged. The prior probability can be calibrated in exactly the same way; for this approach, we find that a value of 0.0475 gives the proper number of false positives.
Association Results
-------------------
Using three different methods has increased the fraction of formally associated counterparts with respect to the 1LAC work. In total we found that 1095 2FGL sources have been associated with at least one counterpart source at other wavelengths (corresponding to a total of 1120 counterparts). Only 26 2FGL sources have been associated with more than one counterpart. A total of 1017 counterparts are at high Galactic latitude ($|b|>10\arcdeg$), comprising the full 2LAC sample. Of these 1017 sources, 704 sources (69%) are associated by all three methods. We found that 886 2LAC sources have a single counterpart and are free of the analysis issues mentioned in §\[sec:obs\] (103 sources were discarded on these grounds), defining the Clean Sample. We note that 640 sources of the Clean Sample (72%) are associated by all three methods. Table \[tbl-assoc\] compares the performance of the different methods in terms of total number of associations, number of false associations $N_{false}$, calculated as $N_{false}=\sum_i (1-P_i)$ and number of sources solely associated via a given method, $N_S$, for the full and Clean samples. The largest probability from the three methods has been used to evaluate the overall value of $N_{false}$. The contamination is found to be less than 2% in both 2LAC and the Clean Sample. The distribution of separation distance between 2LAC sources and their assigned counterparts is shown in Figure \[fig:separation\].
The probabilities given by the three methods are listed in Tables \[tab:clean\] and \[tab:lowlat\] for the high- and low-latitude sources respectively. Where possible, counterpart names have been chosen to adhere to the NASA/IPAC Extragalactic Database[^6] nomenclature. In these tables, a redshift z=0 means that the redshift could not be evaluated even though an optical spectrum was available, e.g., for BL Lacs without redshifts, while no mentioned redshift means that no optical spectrum was available.
\[sec:classif\]Source Classification
=====================================
The ingredients of the classification procedure are optical spectrum or other blazar characteristics (radio loudness, flat radio spectrum, broad band emission, variability, and polarization). We made use of different surveys, including the VLBA Calibrator Survey [VCS; @Bea02; @Fom03; @Pet05; @Pet06; @Pet08; @Kov07]. PMN-CA [@Wri97] is a simultaneous 4.8 GHz and 8.64 GHz survey of PMN sources in the region $-87\arcdeg <\delta< -38.5\arcdeg$ observed with the Australia Telescope Compact Array. CRATES-Gaps is an extension of the CRATES sample to areas of the sky not covered by CRATES due to a lack of PMN coverage from which to draw targets. It consists of an initial 4.85 GHz finding survey performed with the Effelsberg 100-m telescope and follow-up at 8.4 GHz with the VLA [@Hea09]. FRBA, standing for Finding and Rejecting Blazar Associations, is a VLA survey at 8.4 GHz that explicitly targeted otherwise unidentified 1FGL sources.
- To classify a source optically we made use of, in decreasing order of precedence: optical spectra from our intensive follow-up programs, the BZCAT list (i.e., FSRQs and BL Lacs in this list), spectra available in the literature. The latter information was used only if we found a published spectrum.
- If an optical spectrum was not available, we looked for the evidence of typical blazar characteristics, such as radio loudness, a flat radio spectrum at least between 1.4 GHz and 5 GHz, broad band emission (i.e., detection of the candidate counterpart at a frequency outside the radio band). We did not take into account the optical polarization. In this context we made use of, in decreasing order of precedence: BZCAT (i.e., the BZU objects in this list), detection from high frequency surveys and catalogs (AT20G, VCS, CRATES, FRBA, PMN-CA, CRATES-Gaps, CLASS lists), radio and X-ray coincidence association with probability $\ga$ 0.8.
The classes are the following:
- FSRQ, BL Lac, radio galaxy, steep-spectrum radio quasar (SSRQ), Seyfert, NLS1, starburst galaxy – for sources with well-established classes in literature and/or through an optical spectrum with a good evaluation of emission lines.
- AGU – for sources without a good optical spectrum or without an optical spectrum at all: $a$) BZU objects in the BZCAT list; $b$) sources in AT20G, VCS, CRATES, FRBA, PMN-CA, CRATES-Gaps, or CLASS lists, selected by the $\log N - \log S$ method (see §\[sec:steve\]) and the Likelihood Ratio method (see Sect. \[sec:like\]); $c$) coincident radio and X-ray sources selected by the Likelihood Ratio method (see Sect \[sec:like\]).
- AGN – this class is more generic than AGU. These sources are not confirmed blazars nor blazar candidates (such as AGU). Although they may have had evidence for their flatness in radio emission or broad-band emission, our intensive optical follow-up program did not provide a clear evidence for optical blazar characteristics.
As compared to the 1LAC, the classification scheme in the 2LAC has improved thanks to the two additional association methods, allowing for two more types of AGUs (classes [*b*]{} and [*c*]{} in the above description). With the previous association procedure, only about 50% of the current AGUs would have been included in the 2LAC.
In addition to the optical classifications, sources have also been classified according to their SEDs using the scheme detailed in §\[sec:sedclass\].
\[sec:optclass\] Follow-up Optical Program for Redshift and Optical Classification
----------------------------------------------------------------------------------
A large fraction ($\sim$ 60%) of the redshifts and optical classifications presented in Table \[tab:clean\] are derived from dedicated optical follow-up campaigns and specifically from spectroscopic observations performed with the Marcario Low-Resolution Spectrograph [@lrs] on the 9.2 m Hobby-Eberly Telescope (HET) at McDonald Observatory. Other spectroscopic facilities used for these optical results include the 3.6 m New Technology Telescope at La Silla, the 5 m Hale Telescope at Palomar, the 8.2 m Very Large Telescope at Paranal, the 10 m Keck I Telescope at Mauna Kea and the DOLORES spectrograph at 3.6 m Telescopio Nazionale Galileo at La Palma. Our spectroscopic campaigns first considered all the sources which were statistically associated (probability larger than 90%) with one of the still unclassified $\gamma$-ray sources in the 1LAC which have X-ray, radio and optical counterparts within their error boxes. We then consider all sources with a flat radio spectrum. This work will be detailed in two upcoming publications (M. S. Shaw et al., 2011, in preparation and S. Piranomonte et al., 2011, in preparation). Overall, about 67 1LAC sources have gained a measured redshift between the 1LAC and the 2LAC.
\[sec:sedclass\]SED Classification
----------------------------------
As in 1LAC, we classify blazars also based on the synchrotron peak frequency of the broadband SED [@SEDpaper]. This scheme extends to all blazars the standard classification system introduced by [@pg95] for BL Lacs. We estimate the synchrotron peak frequency $\nu^S_\mathrm{peak}$, using the broadband indices $\alpha_{ro}$ (between 5 GHz and 5000 Å) and $\alpha_{ox}$ (between 5000 Å and 1 keV). The analytic relationship $\nu^S_\mathrm{peak}=f(\alpha_{ro},\alpha_{ox})$ was calibrated with 48 SEDs in @SEDpaper. We use the estimated value of $\nu_\mathrm{peak}^\mathrm{S}$ to classify the source as either a low-synchrotron-peaked blazar (LSP, for sources with $\nu_\mathrm{peak}^\mathrm{S} < 10^{14}$ Hz), an intermediate-synchrotron-peaked blazar (ISP, for $10^{14}$ Hz $< \nu_\mathrm{peak}^\mathrm{S} < 10^{15}$ Hz), or a high-synchrotron-peaked blazar (HSP, if $\nu_\mathrm{peak}^\mathrm{S} > 10^{15}$ Hz).
In this work the broad-band spectral indices are calculated from data in the radio, optical and X-ray bands. The radio flux measurements are obtained mainly from the GB6 [@GB6cat] and PMN catalogs. The optical fluxes are taken mainly from the USNO-B1.0 [@USNOcat] and SDSS [@SDSS] catalogs. For BL Lac objects we applied a correction to the optical flux assuming a giant elliptical galaxy with absolute magnitude M$_r$= $-$23.7 as the host galaxy of the blazar [see @2000Urry]. In the case of FSRQs we neglected the dilution of non-thermal light by the host galaxy. Finally, the X-ray fluxes are derived from the RASS [@RASSbright], [*Swift*]{}-XRT, WGA [@WGAcat], [*XMM*]{} [@XMMcat] and BMW [@BMWcat] catalogs.
We express the value of $\nu_\mathrm{peak}^\mathrm{S}$ in the rest frame. BL Lacs without known redshifts were assigned the median BL Lac redshift, z=0.27. The same redshift was assigned to AGU without measured redshifts, except for those with FSRQ-like properties ($\nu_\mathrm{peak}^\mathrm{S}<10^{15}$ Hz in the observer frame and $\Gamma\ge 2.2$, corresponding to the approximate dividing line between FSRQs and BL Lacs found in 1LAC), which were given the FSRQ redshift median, z=1.12.
We note that the SED classification method assumes that the optical and fluxes come exclusively from non-thermal emission. Recently, using simultaneous [*Planck*]{}, [*Swift*]{} and [*Fermi*]{} data, [@paperPLANCK] found that the optical/UV emission was significantly contaminated by thermal/disk radiation (known as the big blue bump). FSRQs (and the AGUs which we assumed to be FSRQ like) are most affected by this contamination. To account for this, we systematically reduce $\nu_\mathrm{peak}^\mathrm{S}$ by 0.5 in logarithmic space for these sources as suggested by [@paperPLANCK].
The $\nu_\mathrm{peak}^\mathrm{S}$ distributions for FSRQs and BL Lacs are displayed in Figure \[fig:syn\_hist\]. Some individual sources can differ from the general behavior of their class, e.g., 2FGL J0747.7+4501 seems to be an ISP-FSRQ with $\log \nu_\mathrm{peak}^\mathrm{S}= 14.66$. Inspection of the SED reveals that this high peak value is partly due to the blue bump (thermal emission in the optical band). The same feature is found in the other ISP-FSRQs. Indeed, we can conclude that even with the applied corrections this method may lead to a significant overestimation of the position of $\nu_\mathrm{peak}^\mathrm{S}$ for some sources where the thermal components are non-negligible.
However, looking at the whole sample we can see that the two classes of objects have different distributions. For FSRQs, the average $\langle \log \nu^S_\mathrm{peak}\rangle$ obtained in the 2LAC Clean Sample is 13.02 $\pm 0.35 $ while BL Lacs are spread over the whole parameter space from low (LSP) to the highest frequencies (HSP). These results are consistent with those presented in [@1LAC] and in [@paperPLANCK].
Figure \[fig:alpha\_alpha\] displays $\alpha_\mathrm{ro}$ versus $\alpha_\mathrm{ox}$. Some sources, filling the bottom part of the $\alpha_\mathrm{ox} - \alpha_\mathrm{ro}$ plane, have much greater contamination by the host galaxy than the average assumed in our estimate. Other outliers can be found in the upper part of the plane especially for some extreme HSP sources including 2FGL J2343.6+3437, 2FGL J0304.5$-$2836, 2FGL J2139.1$-$2054, 2FGL J0227.3+0203 have a very low value of $\alpha_\mathrm{ox}$. This is probably due their being in high states in the X-ray band during the ROSAT observations. However, the SEDs built from archival data do point to a HSP classification.
The X-ray flux is plotted against the radio flux in Figure \[fig:Fr\_Fx\]. As in 1LAC, we see that the FSRQs (essentially all of the LSP type) and HSPs (all BL Lacs) are clearly divided. This plot supports our method to classify the sources using multifrequency properties to estimate synchrotron peak frequency.
\[sec:cat\] The Second LAT AGN Catalog (2LAC)
==============================================
The 2LAC catalog includes all sources with a significant detection over the two-year time period. Sources with only sporadic activity will be missing if they do not make the $TS>25$ cut as computed over the full time span.
\[sec:census\]2LAC Population Census
------------------------------------
Table \[tab:census\] presents the breakdown of sources by type for the entire 2LAC, the Clean Sample, and the low-latitude sample. The entire 2LAC includes 360 FSRQs, 423 BL Lacs, 204 blazars of unknown type and 30 other AGNs. Of the 373 unassociated 1FGL sources located at $|b|>10\arcdeg$, 107 are now firmly associated with AGNs and listed in the 2LAC. Interestingly, 84 of these were predicted to be AGNs in [@1FGL_un]. In the following only the Clean Sample is considered in tallies and figures. The Clean Sample comprises 886 sources in total, 395 BL Lacs, 310 FSRQs, 157 sources of unknown type, 22 other AGNs, and 2 starburst galaxies. For BL Lacs, 302 (76% of the total) have an SED classification (i.e., 93 sources cannot be classified for lack of archival data), with HSPs representing the largest subclass (53% of SED-classified sources), ISPs the second largest (27%) and LSPs the smallest subclass (20%, see Figure \[fig:syn\_hist\]). FSRQs with SED classification (224/310=72%) are essentially all LSPs (99%).
Figure \[fig:sky\_map\] shows the locations of the 2LAC sources. Some relative voids are present, the most prominent centered on ($l$,$b$)=($-$45$\arcdeg$,$-$45$\arcdeg$) reflecting a relative lack of counterparts in the BZCAT catalog at that location. More generally, the observed anisotropy is mainly governed by the non-uniformity of the counterpart catalogs. A difference in the numbers of sources between the northern and the southern Galactic hemispheres is clearly visible for BL Lacs in Figure \[fig:sky\_map\]. This conclusion is confirmed in Figure \[fig:gal\_lat\] displaying the Galactic-latitude distributions for FSRQs and BL Lacs and blazars of unknown type. While the FSRQs show an approximately isotropic distribution[^7], only 40% of the total number of BL Lacs are found in the southern Galactic hemisphere (152 at $b<-10\arcdeg$, 243 at $b>10\arcdeg$). At least approximately 100 other 2FGL sources at $b<-10\arcdeg$ are thus expected to be BL Lac blazars. Some of them fall into the category blazars of unknown type, which are indeed found to be more numerous at $b<-10\arcdeg$ than at $b>10\arcdeg$ (97 versus 60), but a large fraction of these BL Lacs obviously remain unassociated 2FGL sources.
The comparison of the results inferred from the 1LAC and 2LAC enables the following observations:
- The 2LAC Clean Sample includes 287 more sources than the 1LAC Clean Sample, i.e., a 48% increase. Of these, 234 were not present in 1FGL (58 FSRQs, 65 BL Lacs, 108 blazars of unknown type, 3 non-blazar objects); a total of 116 sources were present in 1FGL but not included in the 1LAC Clean Sample for various reasons (their associations were not firm enough, they had more than one counterpart or were flagged in the analysis).
- The fraction of FSRQs has dropped from 41% to 35% between the 1LAC and the 2LAC. The number of 2LAC Clean-Sample FSRQs has increased by 22% relative to the 1LAC Clean Sample.
- The fraction of BL Lacs has remained about constant ($\sim$45% for both 1LAC and 2LAC). The number of 2LAC Clean-Sample BL Lacs has increased by 42% relative to the 1LAC Clean Sample.
- The fraction of sources with unknown type has increased fairly dramatically between the two catalogs (from 8% to 18%), in part due to the improved association procedure. The number of these sources in the 2LAC Clean Sample has increased by more than a factor of 3 relative to that in the 1LAC Clean Sample.
- The overall fraction of FSRQs and BL Lacs without SED classification has increased from 25% to 32%: 155 sources in the Clean Sample are without optical magnitude while 227 are without X-ray flux.
- Out of 599 sources in the 1LAC Clean Sample, a total of 45 sources (listed in Table \[tab:1LAC\]) are missing in the full 2LAC sample, most of them due to variability effects. A few others are present in 2FGL but with shifted positions, ruling out the association with their former counterparts. The significances reported in the 1LAC for these 45 sources are relatively low (Figure \[fig:TS\_gone\]).
These findings point to a need for more multiwavelength data, in particular in the optical and X-ray bands, enabling better classification and characterization of the $\gamma$-ray loud blazars.
\[sec:RG\] Non-Blazar Objects and Misaligned AGNs
-------------------------------------------------
Non-blazar $\gamma$-ray AGN are those not classified as FSRQs, BL Lacs, or as blazars of unknown/uncertain type, and constituted a small fraction of sources in the 1LAC ($\sim$4$\%$ in the Clean Sample). In the 2LAC, this fraction is similarly small ($\sim$3$\%$). Amongst these AGN are radio galaxies, which have emerged as a $\gamma$-ray source population due to the [*Fermi*]{}-LAT [e.g., @ngc1275; @m87; @magn]. The 2LAC contains in particular two new radio galaxies – Centaurus B and Fornax A, associated with 2FGL J1346.6$-$6027 and 2FGL J0322.4$-$3717, respectively. The LAT detects extended emission from Centaurus A [@cena], and this source is modeled with a extended spatial template in 2FGL. @che07 and @geo08 predicted that the radio lobes of Fornax A might be seen as extended sources in the LAT, though to date no extension has been detected. In this context we also note that the position of the 2FGL source associated with the large radio galaxy NGC 6251 ($\sim1\fdg2$ in angular extent), 2FGL J1629.4$+$8236, is shifted toward the western radio lobe with respect to the 1FGL source position (1FGL J1635.4$+$8228).
The source 2FGL J0316.6+4119 is associated with the head-tail radio galaxy IC 310, whose spectrum extends up to TeV energies and was discovered with the LAT [@ner10] and with MAGIC [@ale10ic]. Missing from the 2LAC/2FGL are three radio galaxies reported previously – 1FGL J0308.3+0403 and 1FGL J0419.0+3811, associated with 3C 78 (NGC 1218) and 3C 111, respectively [@1LAC], and 3C 120 [@magn]. In the cases of 3C 111 and 3C 120 this may be due to the $\gamma$-ray emission being variable [@kat11] and the analysis being complicated by their relatively low Galactic latitudes ($b=-8.8\arcdeg$ and $b=-27.4\arcdeg$ respectively). The 1FGL J0308.3+0403/3C 78 source is confirmed but at a significance level lower than the $TS = 25$ threshold for inclusion in the 2FGL catalog [see Table 7 of @2FGL].
Nearby AGN with dominant $\gamma$-ray emitting starburst components were detected in the first year of LAT observations: M 82 and NGC 253 [@starburst1] and NGC 1068 and NGC 4945 [@starburst2]. A study on star-forming galaxies observed with the LAT has been carried out [@sta11]. The low-probability association of 1FGL J1307.0$-$4030 with the nearby Seyfert galaxy ESO 323$-$G77 is confirmed with 2FGL J1306.9$-$4028, with a probability of 0.8, just above the threshold. The low-probability (65$\%$) association of 1FGL J2038.1+6552 with NGC 6951 in the 1LAC is not confirmed – instead, the $\gamma$-ray source in this vicinity, 2FGL J2036.6+6551, is now associated with the blazar CLASS J2036+6553. Finally, one new Seyfert association of note is NGC 6814 to 2FGL J1942.5$-$1024 with a probability of 0.91 for its radio-$\gamma$-ray match. LAT studies of other nearby Seyfert galaxies have so far resulted only in upper limits [@seyfert]. We conclude that such radio-quiet sources do not emit strongly in $\gamma$-rays.
No new radio-loud narrow-line Seyfert I galaxies beyond those four detected in the first year [@pmnj0948; @rlnlsy1] were found, although such objects can be highly variable in $\gamma$-rays and one such example (SBS 0846+513) has been recently detected while flaring [@don11], though it does not make it into 2FGL/2LAC as it was too faint during the first 24 months of LAT operation.
\[sec:lowlat\]Low-Latitude AGNs
-------------------------------
Diffuse radio emission, Galactic point sources, and heavy optical extinction make the low-latitude sky a difficult region for AGN studies, and catalogs of AGNs and AGN candidates often avoid it partially or entirely. However, we are able to make associations with 104 low-latitude AGNs (while about 210 AGNs would be expected in this region from the high-latitude observations if the LAT sensitivity remained the same); these are presented in Table \[tab:lowlat\]. Although the associations are considered valid, these sources have, in general, been studied much less uniformly and much less thoroughly than the high-latitude sources at virtually all wavelengths, so we do not include them as part of the Clean Sample in order to keep them from skewing any of our analyses of the overall $\gamma$-ray AGN population.
\[sec:indiv\] Notes on Individual Sources
-----------------------------------------
As in the 1LAC, we provide additional notes on selected sources. Associations discussed in the previous subsection (§\[sec:RG\]) on non-blazars and misaligned AGNs are not repeated.
[**2FGL J0319.8+4130:**]{} This is the LAT source associated with the radio galaxy NGC 1275 discovered early in the *Fermi* mission [@ngc1275]. During the first two years of LAT operation, the MeV/GeV emission is variable with significant spectral changes at $>$GeV energies [@Kat10; @Bro11].
[**2FGL J0339.2$-$1734:**]{} As noted in the 1LAC, the optical spectrum of the associated AGN source PKS 0336$-$177 is not easily classified as BL Lac or FSRQ.
[**2FGL J0523.0$-$3628:**]{} The radio source associated with this EGRET $\gamma$-ray source is PKS 0521$-$36, which has historically been classified as a BL Lac object because of its optically variable continuum [@Dan79]. However, its spectrum obtained in our optical follow-up program did not enable a clear classification. It is thus flagged as a generic AGN.
[**2FGL J0627.1$-$3528:**]{} This LAT source was associated with PKS 0625$-$35, classified as a radio galaxy, but with BL Lac characteristics in the optical as discussed in [@magn].
[**2FGL J0840.7+1310:**]{} This LAT source was associated with 3C 207, classified as a SSRQ, and was analyzed in more detail in [@magn].
[**2FGL J0847.0$-$2334:**]{} This source is associated with CRATES J0847$-$2337 and has been classified as a “galaxy” in our optical follow-up program.
[**2FGL J0903.6+4238:**]{} This radio source, S4 0900+42 was selected by @Fan01 in a search for candidate Compact Steep Spectrum radio sources. It was then rejected because – interestingly – deeper observations revealed an extended ($>$40 kpc) low frequency radio structure. In the lack of an optical spectrum, this source could then be considered as a candidate misaligned AGN.
[**2FGL J0904.9$-$5735:**]{} The associated radio source, PKS 0903$-$57, was classified as a Seyfert-I galaxy at $z=0.695$ by [@Tho90]. Its spectrum obtained in our optical follow-up program did not enable a clear classification.
[**2FGL J0942.8$-$7558:**]{} The LAT source was associated with the radio source, PKS 0943$-$76, and studied in [@magn]. The photometric redshift of the radio source is $z=0.26$ and it appears to have an FR II morphology [@Bur06].
[**2FGL J1230.8+1224:**]{} This LAT source is associated with the radio galaxy M87, discovered initially in the first year LAT data [@m87]. No significant variability is observed with the LAT within the first two years of observations [see @M87a].
[**2FGL J1256.5$-$1145:**]{} The associated source is CRATES J1256$-$1146 ($z=0.058$) whose spectrum obtained in our optical follow-up program did not enable a clear classification.
[**2FGL J1329.3$-$0528:**]{} The associated AGN, 1RXS 132928.0$-$053132, is not a known radio emitter (e.g., in the NVSS survey).
[**2FGL J1641.0+1141:**]{} The associated AGN, CRATES J1640+1144, was noted in the 1LAC as simply a “galaxy.” Its spectrum obtained in our optical follow-up program did not enable a clear classification.
[**2FGL J1647.5+4950:**]{} The associated AGN is SBS 1646+499, already noted in the 1LAC as characterized as a nearby ($z=0.047$) late-type galaxy. It is a BZU type in BZCAT. Its spectrum obtained in our optical follow-up program did not enable a clear classification.
[**2FGL J1829.7+4846:**]{} This LAT source was associated with 3C 380, classified as a SSRQ, and was analyzed in more detail in [@magn].
[**2FGL J2250.8$-$2808:**]{} The LAT detected a flare from this object in 2009 March [@Koe09]. The associated flat spectrum radio source, PMN J2250$-$2806, has a redshift $z=0.525$. Its spectrum obtained in our optical follow-up program did not enable a clear classification.
Properties of the 2LAC Sources
==============================
\[sec:z\]Redshift Distributions
-------------------------------
The redshift distributions of the various classes are shown in Figure \[fig:redshift\]. They are very similar to those obtained with 1LAC. The distribution peaks around z=1 for FSRQs (Fig. \[fig:redshift\] top) and extends to z=3.10. This distribution contrasts with that of sources observed in the BAT catalog [@Aje09] where 40% of FSRQs have a redshift greater than 2. The distribution peaks at a lower redshift for BL Lacs (Figure \[fig:redshift\] middle). Note that 56% of the BL Lacs have no measured redshifts. The fraction of BL Lacs having a measured redshift is higher for sources with a SED-based classification. This fraction is essentially constant for the different subclasses, (49%, 49%, 54%) for (LSPs, ISPs, HSPs) respectively. Figure \[fig:redshift\] bottom shows the redshift distributions for the different subclasses of BL Lacs. These distributions gradually extend to lower redshifts as the location of the synchroton peak shifts to higher frequency, i.e., from LSPs to HSPs.
The redshift distributions of FSRQs and BL Lacs are compared in Figure \[fig:redshift\_w\] to the corresponding distributions for the sources obtained by cross correlating the seven-year WMAP catalog [@Gol11] with BZCat, using a correlation radius of 11$\arcmin$ (thus selecting 339 sources of a total of 471). Good agreement is observed for FSRQs. The agreement between the 2LAC and WMAP distributions of BL Lacs is more marginal, but the low number of BL Lacs with measured redshifts in the WMAP sample (29 sources) prevents us from drawing definite conclusions. Note that all BL Lacs in the WMAP catalog are detected by the LAT, while only 50% (130 of 260) of the WMAP FSRQs fulfill this condition.
\[sec:flux\]Flux and Photon Spectral Index Distributions
--------------------------------------------------------
The photon index is plotted versus the mean flux (E$>$100 MeV) in Figure \[fig:index\_flux\], along with an estimate of the flux limit. The flux limit strongly depends on the photon index as harder sources are easier to discriminate against the background,which is due to the narrowing of the point-spread function (PSF) of the LAT with increasing energy and to the relative softness of the diffuse Galactic $\gamma$-ray emission. In contrast, the limit in energy flux above 100 MeV is almost independent of the photon index as illustrated in Figure \[fig:index\_S\].
The photon index distributions are given in Figure \[fig:index\] for the different classes of blazars. The now well-established spectral difference in the LAT energy range between FSRQs and BL Lacs, with a moderate overlap between the distributions [@LBAS; @1LAC] is still present. The index distribution of sources with unknown types spans a wider range than those of FSRQs and BL Lacs separately. Assuming that the class of sources with unknown types is entirely made up of FSRQs and BL Lacs lacking classification, each with the same photon index distributions as the classified sources, FSRQs and BL Lacs would contribute about equally to this component.
The photon index is plotted versus the frequency of the synchrotron peak in Figure \[fig:index\_nu\_syn\]. A relatively strong correlation between these two parameters, again reported earlier [@LBAS; @1LAC] is observed. Strong conclusions regarding the HSP-BL Lac outliers (e.g., 2FGL J1213.2$-$2616/ RBS 1080 and 2FGL J1023.6+2959/RX J1023.6+3001 with $\Gamma$=2.4 and $\Gamma$=1.2 respectively) should not be made as these sources are very faint and are significantly detected at best in only one energy band. In order to make a meaningful comparison between the photon index distributions for different classes, it is advantageous to use the flux-limited sample, i.e., sources with Flux\[E$>$100 MeV\]$>$1.5$\times$10$^{-8}$ , which is free of the bias arising from the photon-index dependence of the flux limit (Figure \[fig:index\_flux\]). The resulting photon index distributions are shown in Figure \[fig:index\_c\]. The distribution mean values and rms are (2.42$\pm$0.17, 2.17$\pm$0.12, 2.13$\pm$0.14, 1.90$\pm$0.17) for (FSRQs, LSP-BL Lacs, ISP-BL Lacs, HSP-BL Lacs) respectively. For orientation, the mean values in the significance-limited sample are (2.39, 2.14, 2.09, 1.81) for (FSRQs, LSP-BL Lacs, ISP-BL Lacs, HSP-BL Lacs). No significant dependence of the photon index on redshift is observed [*if blazar subclasses are considered separately*]{}, as illustrated in Figure \[fig:index\_z\], corroborating the conclusion drawn with 1LAC. Note that the region populated by LSP-BL Lacs in the (redshift, $\Gamma$) plane overlaps but does not strictly coincide with that populated by FSRQs. The FSRQ with z=2.941 and $\Gamma= 1.59\pm0.23$ is 2FGLJ0521.9+0108/CRATES J0522+0113, which, while having a definite classification, exhibits a complex optical spectrum. This source is located in the Orion region, where uncertainties in our knowledge of the Galactic diffuse emission can affect the determination of the source photon spectral index. The three photon index distributions for BL Lacs with z$<$0.5 (mostly HSPs), with z$>$0.5 (mostly LSPs), and for BL Lacs without redshifts are compared in Figure \[fig:index\]. The distribution of BL Lacs without redshifts is markedly different from the two other distributions and thus does not favor any conclusions concerning the actual redshift distributions of these blazars.
The time-averaged, mean flux distributions for FSRQs and BL Lacs are compared in Figure \[fig:flux\_mean\_peak\]a. As suggested by Figure \[fig:index\_flux\], the fluxes of the FSRQs extend to higher values than do BL Lacs, but FSRQs have a higher detection flux limit due to their spectral softness. For sources showing significant variability, the monthly peak-flux distributions are compared in Figure \[fig:flux\_mean\_peak\]b. These distributions are more similar for the two blazar classes. The peak flux is plotted as a function of mean flux in Figure \[fig:flux\_mean\_peak\]c, and the distribution of peak flux over mean flux ratio is given in Figure \[fig:flux\_mean\_peak\]d. Larger flux ratios are observed for FSRQs. Variability is discussed further in §\[sec:var\].
Comparison of 2LAC and 1LAC fluxes
----------------------------------
Photon flux distributions from 1LAC and 2LAC are displayed in Figure \[fig:flux\_11\_24\]. The top two panels show the 1LAC fluxes and 2LAC fluxes for sources present in both 1LAC and 2LAC. As expected the 2LAC distribution is broader than the 1LAC distribution, especially at the low-flux end. The bottom two panels represent the 1LAC flux distribution for the 45 missing 1LAC sources and the 2LAC flux distribution for the 250 newly-detected 2LAC sources in the Clean Sample. The high-flux end of these distributions look alike, which can presumably arise from the facts that a similar pool of sources i) were comparatively bright during the first 11 months and then faded away, or ii) have brightened during the last 13 months spanned by the 2LAC while being faint during the 1LAC period. Of course, the low-flux ends of the two distributions are different as the new 2LAC sources include sources fainter than the 1LAC detection limit.
\[sec:curv\] Energy Spectra
---------------------------
First observed for 3C 454.3 [@Abdo_3C] early in the [*Fermi*]{} mission, a significant curvature in the energy spectra of many bright FSRQs and some bright LSP-/ISP-BL Lacs is now a well-established feature [@spec_an; @1LAC]. The break energy obtained from a broken power-law fit has been found to be remarkably constant as a function of the flux, at least for 3C 454.3 [@Abdo_3C_11]. Several explanations have been proposed to account for this feature, including $\gamma\gamma$ attenuation from He[ii]{} line photons [@Pou10], intrinsic electron spectral breaks [@Abdo_3C], Ly $\alpha$ scattering [@Abdo_3C_10], and hybrid scattering [@Fin10].
Although broken power-law (BPL) functions have been found to better reproduce most curved blazar energy spectra, the LogParabola function (§\[sec:obs\]) has been selected here since it has only one more degree of freedom with respect to a power law, convergence of spectral fits is easier than for BPL and the function decreases more smoothly at high energy than a power law with exponential cutoff form. Physical arguments supporting the use of a LogParabola function have been presented in [@Tra11].
The spectral curvature is characterized by the parameter $Signif\_Curve$, equal to $\sqrt{c \times TS_{curve}}$, where $TS_{curve}$ is defined in §\[sec:obs\] and $c$ is a source-dependent correction factor accounting for systematic effects [see @2FGL for details]. $Signif\_Curve$ is plotted as a function of $TS$ in Figure \[fig:TSCurve\_TS\]. For $TS>$ 1000, most FSRQs have large $Signif\_Curve$, while BL Lacs exhibit a variety of behaviors. As mentioned earlier, LogParabola results were retained for sources with $TS_{curve}$$>$16 (corresponding to $Signif\_Curve \simeq$ 4). The LogParabola parameter $\beta$ is plotted as a function of the flux in Figure \[fig:beta\_flux\] for the 57 FSRQs and 12 BL Lacs in the Clean Sample with $TS_{curve}>$16. The average $\beta$ is significantly lower for BL Lacs than for FSRQs (0.11$\pm$0.02 versus 0.18$\pm$0.02 respectively), possibly due to the fact that different regions of the Inverse-Compton peak (assuming a leptonic scenario) are probed in the LAT energy band.
The 12 BL Lacs comprise 7 LSPs, 3 ISPs, 1 HSP and 1 BL Lac lacking SED classification. The HSP is BZB J1015+4926 (GB 1011+496), the SED of which has a maximum at a few GeV. The flux distributions for these sources are compared to the overall distributions in Figure \[fig:Flux\_curv\], and are seen to confirm the trend observed in Figure \[fig:TSCurve\_TS\].
\[sec:var\]Variability
----------------------
Variability at all time scales is one of the distinctive properties of blazars. Since launch, detections by the [*Fermi*]{}-LAT of $\gamma$-ray activity from 81 flaring blazars have been reported in Astronomer’s Telegrams (ATels). Four of them are not listed in the 2LAC since they did not pass the $TS=25$ cut for inclusion in the 2FGL: SBS 0846+513, PMN J1123$-$6417 (at b=3.0$\arcdeg$), PMN J1913$-$3630, PKS 1915$-$458.
Two-year light curves with monthly binning were obtained as part of the 2FGL catalog. The large bin width leads to a substantial smoothing of the light curves for the brightest blazars, for which peak fluxes may be much higher than the one-month average fluxes reported here. A more extensive analysis using higher-resolution light curves, thus containing richer temporal information will be presented elsewhere. Nevertheless these light curves constitute the largest set ever produced in the $\gamma$-ray band, allowing variability analysis on a wide sample of blazars. In this section we will give an overview of the variability properties for the sources in the 2LAC Clean Sample. This includes the detection of variability via the LAT $\gamma$-ray variability index, a measure of the $\gamma$-ray variability duty cycle and a derivation of population variability characteristics from the Discrete Auto Correlation Function, DACF, first order Structure Function, SF, and from Power Density Spectra, PDS. DACF [see, e.g., @ede88; @huf92], SF [see, e.g., @sim85; @smi93; @lai93; @pal97], and PDS [@Vaugh03] are methods providing insights into fluctuation modes, characteristic timescales and flavors of the variability modes in the $\gamma$-ray monthly-bin light curves. A short description of these three analysis methods are given in @Abdo_var.
The variability index $TS_{var}$, which is described in section \[sec:obs\], is plotted as a function of the relative flux uncertainty in Figure \[fig:varind\_relunc\]. The relative flux uncertainty, computed with a fixed photon index [see section 3.6 of @2FGL], reflects the photon statistics. This parameter allows meaningful comparisons between sources with different fluxes and photon indices. Figure \[fig:varind\_relunc\] illustrates the fact that for a source to be labeled as variable on the basis of its variability index it must be both intrinsically variable and sufficiently bright. All very bright sources, including both FSRQs and BL Lacs are found to be variable at a confidence level greater than 99%, depicted by the line at $TS_{var}>$41.6 in Figure \[fig:varind\_relunc\]. At a given relative flux uncertainty, BL Lacs have on average lower $TS_{var}$ than FSRQs.
A total of 224 FSRQs (out of 310), 91 BL Lacs (out of 395) and 33 sources of unknown type (out of 157) are variable at a confidence level greater than 99%. Thus 348 blazars of the 2LAC Clean Sample fulfill this condition, while there were only 189 in the 1LAC Clean Sample. Figure \[fig:varind\_sync\] shows the variability index versus synchrotron peak position. Only a small fraction of the HSP-BL Lacs detected by the LAT shows significant variability (27 out of 160), substantially less than LSP-BL Lacs (25 out of 61) and ISP-BL Lacs (30 out of 81). The photon indices of variable FSRQs and BL Lacs are shown in Figure \[fig:index\_var\] versus the normalized excess variance [@Vaugh03]. The plot reveals a trend of variability with spectral index. Most variable sources have a photon index greater than 2.2. These sources are observed at energies greater than the peak energies of their SEDS, where the variability amplitude tends to be larger. The harder sources, including all but one (PKS 0301$-$243) of the HSPs and ISPs have normalized excess variance $<$ 0.5. The average normalized excess variance for each of the blazar classes is 0.37 $\pm$0.03 (FSRQs), 0.28$\pm$0.07 (LSP-BL Lacs), 0.19$\pm$0.04 (ISP-BL Lacs) and 0.20$\pm$0.10 (HSP-BL Lacs). Excluding the outlier (PKS 0301$-$243) the value for the HSP-BL Lacs becomes 0.10$\pm$0.03 which implies that even if significant variability is detected only in a fraction of the individual HSPs, they do, as a class, exhibit variability but at a lower level than the other classes. The variability index and normalized excess variance are also plotted against $\gamma$-ray luminosity. These are shown in Figure \[fig:L\_varind\] and \[fig:L\_sig\] respectively. The normalized excess variance does show a gradual increase with $\gamma$-ray luminosity for both BL Lacs and FSRQs. The BL Lac with low luminosity and high normalized excess variance ($>$1.5) is 2FGL J0217.4+0836, which underwent a flare with a Flux\[E$>$100 MeV\]=1.3$\times$10$^{-7}$ flare in January 2010.
The monthly-binned light curves also provide information about the duty cycle of blazars at $\gamma$-ray energies. Sources are in general not detected in all 1-month-bins. This is illustrated in Figure \[fig:coverage\_lc\], which shows the distribution of [*coverage*]{}, i.e., the fraction of months where the source was detected with $TS>$4. Not surprisingly, the coverage distribution is skewed toward low values. We find that 161 FSRQs and 152 BL Lacs have a coverage greater than 0.5. Only these sources will be considered in the variability studies presented below. We define the [*duty cycle*]{} as the fraction of monthly periods N$_b$/N$_{tot}$ where the flux exceeds $<$F$>+1.5 S + \sigma_i$, where $<F>$ is the average flux, $S$ is the total standard deviation and $\sigma_i$ is the flux uncertainty of month $i$ [@Abdo_var]. These duty cycle values are shown as a function of $TS$ in Figure \[fig:duty\_cycle\]. Bright sources with $TS >$ 1000 essentially have all N$_b$/N$_{tot}\ge$ 0.05. Simulations considering the actual $TS$ distributions of both blazar classes were performed and showed that the measurement of N$_b$/N$_{tot}$ for these sources was not significantly affected by measurement noise. The wider distribution in N$_b$/N$_{tot}$ for sources with $TS <$ 1000 is consistent with these sources having similar duty cycle as the brighter ones and only results from a lower signal-to-noise ratio.
DACF and PDS were calculated for all sources with coverage larger than 0.5 and mean flux above 100 MeV exceeding $3 \times 10^{-8}$ (156 FSRQs and 59 BL Lacs), while the SF analysis was applied to the whole Clean Sample. From each DACF a correlation timescale was estimated as the time lag of the first zero crossing of the function, computed by linear interpolation between the lag points. These observer-frame timescale estimates for both FSRQs and BL Lacs are plotted in Figure \[fig:ACF\] as a function of synchrotron peak frequency for the selected sources. The timescale distribution is shown in the inset plot. Interestingly the observation that FSRQs have $\gamma$-ray correlation extending to longer timescales than BL Lacs confirms the trend found for the LBAS sample [@LBAS] using weekly light curves obtained over the first 11 months of observation [@Abdo_var].
The SF, which is equivalent to the PDS of the signal but calculated in the time domain, which makes it less subject to irregular sampling, low significance bins and upper limit problems, was applied to the light curves of the entire 2LAC Clean Sample sources. Results are shown in Figure \[fig:SF\] where the distribution of the PDS power-law indexes evaluated in the time domain ($\beta + 1$, where $\beta$ is the blind power-law slope estimated from the $SF$ of each light curve) are reported for the FSRQs and BL Lacs. The resulting distributions of the power-law indices appear whitened (i.e., closer to white noise with flatter $SF$ power-law indices) because of the short extent of the time lag range investigated (from 1 to 24 months) and of the fact that a consistent subset of the 2LAC Clean Sample showed low-flux, noisy and non-variable monthly-bin light curves, when compared with the same analysis performed on the brightest and better sampled light curves of the LBAS sample [@Abdo_var]. Again the distribution shows FSRQs with slightly more Brownian-like (steeper) and more scattered SF indexes, with respect to the more flicker-like (flatter) ones for BL Lacs in agreement with what was already found for the LBAS sample [@Abdo_var].
In Figure \[fig:PDS\] we have plotted the average PDS for FSRQs and BL Lacs. The power density is normalized to fractional variance per frequency unit (, where I is the average flux) and the PDS points are averaged in logarithmic frequency bins. The white noise level was estimated from the rms of the flux errors and was subtracted for each PDS. The error bars were computed as the standard error of the mean for each frequency bin. The PDS slope (power-law index) is similar for the two groups, $\sim$ 1.15 $\pm$0.10. This is somewhat flatter than was deduced for the very brightest sources in the LBAS sample [@Abdo_var]. The difference in the height of the PDS means that the fractional variability of BL Lacs is lower than that of FSRQs. This is in line with the LBAS results. With the PDS normalization used here, we can compute a normalized excess variance by integrating the PDS over frequency. To limit the effect of statistical noise this integration was done for frequencies up to 0.2 month$^{-1}$, which also contains most of the variance. The resulting normalized excess variance for the different blazar classes is 0.44 $\pm$0.04 (FSRQs), 0.27 $\pm$0.10 (LSP-BL Lacs), 0.19 $\pm$0.04 (ISP-BL Lacs) and 0.14 $\pm$0.07 (HSP-BL Lacs). The trend and values are consistent with the normalized excess variance calculated directly from the light curves as described above.
\[sec:hep\] Highest-energy photons
----------------------------------
Figure \[fig:redshift\_he\] displays, as a function of redshift, the highest energy photon (HEP) detected by the LAT from the 2LAC AGN sample using the Pass 7\_V6 Ultraclean event selection and that is associated with the source within the 68% containment radius. Further work is being carried out to improve the capability to reconstruct event tracks and reject background at high energy [@Baldini]. In comparison to the corresponding sample based on 11 months of LAT operation [@Abdo_EBL] we find about a factor $\sim 2$ more candidate photon events coming from sufficiently high redshift ($z>0.5$) to probe the models of the extragalactic background light (EBL).
Predictions of $\gamma\gamma$ opacity curves, $\tau_{\gamma\gamma} = 1$ (top panel) and $\tau_{\gamma\gamma} = 3$ (bottom panel), for different EBL models are also shown in Figure \[fig:redshift\_he\]. Detection of HEPs above the opacity curve predicted by a given model makes the model less likely. In the new 2LAC AGN sample, we find 30 HEP events from $z>0.5$ sources beyond the $\tau_{\gamma\gamma} = 3$ regime of the [@Ste06] “baseline model”, which is already severely constrained by the LAT 11 month data set [@Abdo_EBL]. Only one event appears beyond $\tau_{\gamma\gamma} = 3$ of the [@Kne04] “best-fit” and “high-UV” models.
None of the HEP events seems to be in strong contradiction with EBL models that are of lower photon density [e.g., @Fran08; @Fin10b; @Gil09]. Note, however, that we don’t have redshift information for more than 50% of the 36 sources with HEPs at energies greater than 100 GeV, which can therefore not be tested against any EBL models. Apparent in Figure \[fig:redshift\] is the clustering of HSPs at low redshifts ($z\le 0.2$) while LSPs cover a broad redshift range up to $z = 3.1$. Because HSPs are intrinsically hard sources, and LSPs intrinsically soft (see Figure \[fig:index\_nu\_syn\]) any systematic trend between redshift and spectral properties (spectral index, HEP) is unlikely to be caused by EBL absorption only. For the $> 500$ events without an assigned source redshift, the HEP is located above $\sim 10$ GeV in more than $\sim 70\%$ of all cases. Interestingly, we found $\sim 4$ FSRQs with HEPs that reach energies $> 100$ GeV [4C +55.17, see @McC11 4C +21.35, PKS 1958$-$179, BZQ J1722+1013] with the latter two (at redshifts $z = 0.652$ and $z = 0.732$ respectively) displaying no significant deviation from a power-law spectrum (with indices $\Gamma \sim 2.4$ and $\Gamma \sim 2.2$, respectively) in the energy range of the LAT. One BL Lac (2FGL J0428.6$-$3756, PKS 0426$-$380) at redshift $z =1.10$ of LSP spectral type has also been detected at $> 100$ GeV.
Luminosity Distributions
------------------------
The $\gamma$-ray luminosity is plotted as a function of redshift in Figure \[fig:L\_redshift\]. A Malmquist bias is readily apparent in this figure as only high-luminosity sources (mostly FSRQs) are detected at large distances. Given their $\gamma$-ray luminosity distribution, most BL Lacs could not be detected if they were located at redshifts greater than 1.
Figure \[fig:index\_L\] shows photon index versus $\gamma$-ray luminosity. This correlation has been discussed in detail in the context of the “blazar divide” [@Ghisellini09]. Note that since the $\gamma$-ray luminosity is derived from the energy flux and that the detection limit in energy flux is essentially independent of the photon index (Figure \[fig:index\_S\]), no significant LAT-related detection bias is expected to affect this correlation. The ISP-BL Lac outlier at L$_\gamma \simeq$ 3$\times$ 10$^{43}$ erg cm$^{-2}$ s$^{-1}$ is 4C 04.77 (2FGL J2204.6+0442) at z=0.027, which was classified as an AGN in 1LAC.
Figure \[fig:index\_L\_2\] shows photon index versus $\gamma$-ray luminosity for FSRQs (top) and BL Lacs (bottom) separately. The Pearson correlation coefficients are $-$0.04 and 0.14 for FSRQs and BL Lacs respectively. For a given class, the correlation is very weak.
Multiwavelength properties of the 2LAC sample
=============================================
In this section, we explore the properties of the 2LAC sample in the radio, optical, X-ray and TeV bands. Table \[tab:prob1\] gives archival fluxes in different bands for these sources. For completeness, Table \[tab:prob2\] provides the corresponding fluxes for the low-latitude sources.
Radio Properties
----------------
The 2LAC sources are associated with a population of radio sources, whose flux density distribution spans the range between a few mJy and several tens of Jy. This is rather typical for blazars, whose radio emission has often been found to be correlated with the $\gamma$-ray activity [@Kovalev2009; @Ghirlanda2010; @Ghirlanda2011; @Mahony2010; @radiogamma]. In particular, @radiogamma have shown a highly significant correlation (chance probability $<10^{-7}$) between the radio and $\gamma$-ray fluxes for both FSRQ and BL Lacs in the 1LAC, although with a large scatter.
In Figure \[fig:fluxhisto\] we plot the radio flux density distributions for sources in the 2LAC, divided according to the optical type. For all sources, we plot the radio flux density at 8 GHz, obtained either using interferometric data from CRATES [@crates or similar surveys, when available], or extrapolated from low frequency (NVSS or SUMSS) measurements assuming $\alpha=0.0$; we also plot the distribution of the radio flux density at higher frequency, i.e., at 20 GHz as obtained from the AT20G survey and at 30 GHz as obtained from the [*Planck*]{} ERCSC [@Planckcatalog]. Since AT20G only covers half of the sky, we multiply the counts by 2 to have a consistent normalization (2LAC and [*Planck*]{} are all-sky surveys).
The distributions for BL Lacs and FSRQs are quite broad, with well separated peaks, FSRQs being on average significantly brighter radio sources. The median flux densities of the two distributions at 8 GHz are 86 and 581 mJy for BL Lacs and FSRQ, respectively. In the highest flux density bins, the various surveys are all basically complete. The distributions are similar for the three frequencies (8 GHz, 20 GHz, 30 GHz), confirming that the 2LAC sources have flat radio spectra. Below 1 Jy, [*Planck*]{} counts drop rapidly owing to sensitivity limits, while AT20G becomes less and less complete below 100 mJy. Interestingly, AT20G shows a deficit of BL Lac sources in the 100–300 mJy range, which cannot be attributed to sensitivity limits; this is most likely to arise from the lack of spectroscopic information for sources in the Southern hemisphere (see Fig. \[fig:sky\_map\]), where the AT20G survey was carried out.
As shown in the 2FGL paper, the radio flux density distribution of the [*Fermi*]{} sources accounts for nearly all the brightest radio sources in CRATES, while a significant fraction of lower flux density sources have not been detected by [*Fermi*]{} so far. One viable possibility is that the $\gamma$-ray duty cycle of FSRQs (which is the dominant population in CRATES) is quite low, so these sources have not yet gone through a phase of activity during the [*Fermi*]{} lifetime; combined with the typically soft $\gamma$-ray spectra of FSRQs and the lower sensitivity and broader PSF of the LAT at low energy, this could account for the lack of such sources.
On the other hand, the BL Lac population extends to lower flux densities (even below the CRATES sensitivity) and is more consistently detected by the LAT. For example, the $\gamma$-ray detection rate in the VIPS survey established with the 1LAC sample, was $\sim 2/3$ for BL Lacs and only 9% (50/529) for the FSRQs [@Linford2011]. In particular, a large number of BL Lacs have now been detected and associated thanks to the extension to lower flux density of the association methods, which is essential for the radio-weak HSP sources, and their more persistent (less dramatically variable) $\gamma$-ray emission.
When combined with the different redshift distributions (see Sect. \[sec:z\]), the different flux density distributions result in markedly distinct radio luminosity distributions, as shown by Figure \[fig:lumhisto\]. The overall luminosity interval spans the range between $10^{40}$ and $10^{45}$ erg s$^{-1}$, with FSRQs more clustered at high luminosity ($\log L_{\rm r, FSRQ} [$ergs$^{-1}] = 44.1 \pm
0.7$), while the BL Lacs span a broader interval, down to lower luminosities ($\log L_{\rm r, BL} [$ergs$^{-1}]= 42.3 \pm 1.1$).
Not unexpectedly, given the large overlap between the two samples, these properties are entirely consistent with those of the sources in the 1LAC. Also the radio spectral index distribution for sources with data at both 8 GHz and $\sim
1$ GHz remains consistent with a flat value, with $\langle \alpha \rangle
= 0.08 \pm 0.30$. This is also suggestive that our extrapolation of the low frequency data is solid, as confirmed by the similar distributions of the 8 GHz, 20 GHz, and 30 GHz flux densities in the range where the three surveys are complete.
Properties in the optical/infrared and hard X-ray bands
--------------------------------------------------------
Optical and infrared bands are important for our understanding of GeV $\gamma$-ray blazars. For LSPs, the peak of the synchrotron emission is located in these bands and significant correlation with the GeV emission has been observed. Both synchrotron and thermal emission components can contribute in these bands, creating a complex spectral-temporal behavior. On the other hand, our limited knowledge about their host galaxy, nucleus and stellar core profiles hamper studies in these bands, as do difficulties in measuring line widths, ratios, and fluxes.
Correlated variability between optical-infrared and $\gamma$-ray variability points to a common population of electrons producing non-thermal emission through synchrotron and Inverse Compton processes. High-quality data (GeV and optical/NIR) obtained on flaring sources thanks to intensive multifrequency campaigns [e.g., @Abdo_1502; @Abdo_3C279], have already revealed the existence of correlated flares, with no true orphan flares [as sometimes observed in the X-ray band, e.g., @Abdo_3C279].
Our 2LAC sample is characterized by different optical spectra, with a number of BL Lac - FSRQ transition objects. Those include BL Lacertae itself, the prototype of the class displaying at times moderately strong, broad lines and a complex SED [@Abdo_BLLacertae], and 3C 279, one of the prototypes of the FSRQ class, which can appear nearly featureless in the optical band in a bright state [@Abdo_3C279]. The four NLS1 sources in 2LAC have flat radio spectra and strong but narrow emission lines, interpreted as the apparent luminosity of the jets compared to the line luminosity being lower, possibly because of lower intrinsic jet power, or slight misalignment of the jet with respect to our line of sight.
Figure \[fig:Magu\] shows the V magnitude reported in SDSS for the FSRQs and BL Lacs of the Clean Sample. The BL Lacs are associated with brighter galaxies relative to the FSRQs, although the sources are all relatively bright. This brightness enables the monitoring of all Clean Sample sources with small optical telescopes to study correlated variability.
Cross-correlating the 2LAC with the [*Swift*]{} BAT 58-month survey [@BATcatalog] yields a total of 47 sources present in both catalogs. The redshift distributions of the FSRQs and BL Lacs from this subset are given in Figure \[fig:BAT\_z\]. All 15 BL Lacs are of the HSP type, except one, which is an ISP. These distributions are very similar to those of the LAT blazars not detected by BAT. The photon spectral index measured in the BAT band is plotted against the photon spectral index in the LAT band in Figure \[fig:BAT\_index\]. A clear anticorrelation is visible in this Figure (Pearson correlation factor=$-$0.73). For the HSP-BL Lacs considered here, BAT probe the high-frequency (falling) part of the $\nu F_{\nu}$ synchrotron peak while the LAT probes the rising side of the Inverse Compton peak (assuming a leptonic scenario). For FSRQs, which are all LSPs, BAT and LAT probe the rising and falling parts of the Inverse-Compton peak respectively. Note that for this subset of sources which are quite distinct in properties, the LAT spectral indices for FSRQs and BL Lacs do not overlap. The Pearson correlation factor is only $-0.15$ and $-0.17$ for FSRQs and BL Lacs considered independently, respectively.
GeV-TeV connection
------------------
At the time of publication of 1LAC [@1LAC], 32 AGNs had been detected in the “TeV” or very high energy (VHE; E$\ge$ 100GeV) regime [@TevCat]. All but four of these (RGBJ0152$+$017, 1ES0347$-$121, PKS0548$-$322 and 1ES0229$+$200) were in 1LAC. Since then, an additional 13 AGNs (14 if we include the unidentified, but likely AGN, VERJ0648$+$152 that is discussed below) have been detected at TeV energies, which brings the total number of TeV AGNs to 45, 39 of which are in 2FGL. Just one of the TeV AGNs, RGBJ0152$+$017, that was not in 1LAC is in 2LAC. The clean 2LAC sample contains 34 of the TeV AGNs, which we will refer to as the GeV-TeV AGNs. The five TeV AGNs that are in 2FGL but not the clean sample are: VERJ0521+211, MAGICJ2001+435 and 1ES2344+514 (due to their low Galactic latitudes) and IC310 and 1RXSJ101015.9$-$311909 (due to their flags[^8]). All of the TeV AGNs that were in 1LAC remained significant LAT sources and are thus in the 2LAC Clean Sample. As can be seen in Table \[TAB:GeVTeV\], the largest subclass in the GeV-TeV AGNs (18) is the HSPs but there also 6 ISPs, 5 LSPs and 5 AGNs whose SED class remains unclassified using the technique described in § \[sec:sedclass\]. The mean photon index of the 2LAC sources associated with the TeV AGNs is $1.87\,\pm\,0.27$, while the mean photon index of the clean 2LAC sample is $2.13\,\pm\,0.30$, indicating that those AGNs which are detected at TeV are, in general, harder than the majority of the 2LAC sources at [*[Fermi]{}*]{}-LAT energies.
Since the launch of [*[Fermi]{}*]{}, 22 AGNs and one TeV source that was classified as unidentified when discovered, VERJ0648$+$152,[^9] have been discovered in the VHE regime. [*[Fermi]{}*]{}-LAT was implicated in the detection of nine of these objects [@ATel2486; @1ES0502+675:ATel2301; @PKS1424+240:ATel2084; @VERJ0521+211:ATel2260; @MAGICJ2001+435:ATel2753; @NGC1275:ATel2916; @4C+21.35:ATel2684; @RBS0413:ATel2272; @APLib:ATel2743], a significant percentage of the entire catalog of TeV AGNs ($20\%$). This demonstrates the close ties between these energy regimes and also the unique capability of the LAT to provide the Cherenkov telescopes with prime TeV candidates, which is especially valuable input for these instruments since they have small fields of view and low duty cycles ($\sim$10%). These sources are flagged with asterisks in Table \[TAB:GeVTeV\].
As discussed in @1LAC, the majority of the GeV-TeV AGNs can be well fit with power-law (PL) spectra in both $\gamma$-ray energy regimes although, as detailed below, sometimes a LogParabola spectrum was the preferred fit in the GeV regime. In many cases, there is a significant difference between the PL spectral indices measured by [*[Fermi]{}*]{} LAT, $\Gamma_{GeV}$, and by the Cherenkov telescopes, $\Gamma_{TeV}$, indicating that the spectrum undergoes a break somewhere in the $\gamma$-ray regime. In the same manner as described in @1LAC, the difference in photon index between that measured by [*[Fermi]{}*]{} LAT and that reported in the TeV regime, $\Delta\Gamma\,\equiv\,\Gamma_{TeV}\,-\,\Gamma_{GeV}$, for the GeV-TeV AGNs with reliable redshifts and reported TeV spectra (flagged in Table \[TAB:GeVTeV\]), are plotted as a function of the redshift in Figure \[fig:GeVTeVEBL\]. It should be noted that the data used to measure the spectral indices in question were not necessarily simultaneous. It can be seen that there is a deficit of distant sources with small values of $\Delta\Gamma$, confirming the trend previously reported [@pg1553; @1LAC]. One possible explanation for this is the effect of the EBL: the $\gamma$-ray photons pair produce with the photons of the EBL, softening the spectrum in the VHE band in a redshift-dependent way.
As can be seen in the 2LAC, most of the GeV-TeV AGNs, 26 out of 34, are best fit with power-law spectra in the [*[Fermi]{}*]{}-LAT band pass. Of these sources, 17 are HSPs, two are ISPs, two are LSPs and five are GeV-TeV AGN that are unclassified. Out of the three SED classes, the HSPs have, by definition, their synchrotron peak frequencies at the highest energies. Thus, in many emission model scenarios, it is expected that their second SED peak would also occur at the highest energies. For sources not subject to significant absorption by the EBL, this means that their spectral turn-over may occur at higher energies than covered by the 2LAC. Following these arguments, it is not surprising then that most of the GeV-TeV sources with power-law spectra in the LAT bandpass are HSPs.
By extension, it would seem likely that at least some of the five GeV-TeV AGN that were not assigned SED classes by the procedure described in § \[sec:sedclass\] are HSPs. An examination of the literature reveals three of them (2FGLJ0416.8$+$0105/1H0413$+$009, 2FGLJ1101$-$2330/1H1100$-$230 and 2FGLJ2009.5$-$4850/PKS2009$-$489) to have been classified as high-frequency peaked BL Lacs (). One of the remaining sources (2FGLJ1325.6$-$4300) is associated with the CentaurusA core, a Fanaroff-Riley Type I galaxy. We note that these are all Southern Hemisphere sources and that, typically, this hemisphere is not as well surveyed at radio and optical wavelengths. This could be a factor in the non-classification of their SEDs. The two ISPs that are best fit by power laws are WComae (2FGLJ1221.4$+$2814; $z=0.103$) and PG1424$+$240 (2FGLJ1427.0+2347; the redshift is unknown but upper limits of $z<1.19$ and $z<0.66$ have been derived by @2010arXiv1006.4401Y and @2010ApJ...708L.100A while @2011arXiv1101.4098P estimate $z=0.24\pm0.05$). The two LSPs that were best fit by power laws are among the closest known GeV-TeV AGN: APLib (2FGLJ1517.7$-$2421; $z=0.048$) and M87 (2FGLJ1230.8$-$1224, z=0.0036), a Fanaroff-Riley Type I galaxy.
Of the eight GeV-TeV whose [*[Fermi]{}*]{} LAT spectra are best fit by a LogParabola, only one, 1H1013$+$498 (2FGLJ1015.1+4925), is classified as an HSP. With a redshift of $z\,=\,0.212$, this object is less distant than many of the other sources (both those best-fit by LogParabolas and by power laws) so the curvature in its spectrum is not likely to be solely attributable to absorption from the photons of the EBL. The remaining GeV-TeV sources with LogParabola spectra, comprise four ISPs and 3 LSPs.
The six TeV AGNs that are not in 2FGL (SHBLJ001355.9$-$185406, 1ES0229+200, 1ES0347$-$121, PKS0548$-$322, 1ES1312$-$423 and HESSJ1943+213[^10]) are all high-frequency-peaked BL Lacs, and are amongst the weakest extragalactic TeV sources detected to date, with fluxes ranging between 0.4% and 2% that of the Crab Nebula in that energy regime. The fact that it is the weakest TeV HBL that remain below the 2LAC detection threshold is compatible with the characteristics of this subclass of AGN, namely, that their second emission peak occurs at high frequencies and that they have low bolometric luminosities (when compared to that of the other blazar subclasses).
Discussion and Summary
======================
The 2FGL catalog contains 1319 sources at $|b| > 10\arcdeg$, of which 1017 sources are associated at high confidence with AGNs. These constitute the 2LAC. The 2LAC Clean Sample consists of 886 sources (see Table \[tab:census\]), and is defined by requiring that sources have only one counterpart each and no analysis flags. It includes 395 BL Lacs, 310 FSRQs, 157 blazars of unknown type, 8 misaligned AGNs, 4 NLSy1 galaxies, 10 AGNs of other types, and 2 starburst galaxies. The 2LAC Clean Sample represents a 48% increase over the 599 high-latitude AGNs in the 1LAC Clean Sample. This reflects not only the increased exposure, but also follow-up campaigns on individual targets and the availability of more extensive catalogs.
Unassociated Sources and Redshift Incompleteness
------------------------------------------------
The observed deficit of BL Lac objects at negative Galactic latitudes compared to positive latitudes (Figure \[fig:gal\_lat\]) is not fully accounted for by blazars of unknown type, suggesting that a significant number of blazars (at least 60) are present in the unassociated sample of 2FGL sources. This deficit results primarily from the greater incompleteness of the current counterpart catalogs at Southern declinations, in particular, the BZCAT [@bzcat], which is biased by the greater number of Northern hemisphere arrays that have better exposure to positive Galactic latitudes. There is furthermore a modest anisotropy in LAT exposure favoring positive Galactic latitudes (Figure \[fig:sens\]). The lack of extensive archival multiwavelength data also leads to an incomplete characterization of the 2LAC Clean Sample. Consequently we find that
1. 157 of the 862 blazars in the 2LAC ($\sim 18$%, referred to as “of unknown type”) lack firm optical classification. Their photon index distribution (Figure \[fig:index\] bottom) suggests that they comprise roughly equal numbers of BL Lacs and FSRQs.
2. 220 of the 395 BL Lac objects ($\sim 55$%) lack measured redshifts, and this fraction is roughly the same for LSP, ISP, and HSP BL Lac objects;
3. 93 of the 395 BL Lac objects ($\sim 23$%), and 86 of the 310 FSRQs ($\sim 28$%), lack SED-based classifications.
Despite the fact that intensive optical follow-up programs are underway (M. S. Shaw et al., 2011, in preparation and S. Piranomonte et al., 2011, in preparation), these limitations, as was also the case for the 1LAC, hamper interpretation.
The smaller error boxes that result from longer exposure fortunately result in fewer multiple associations in 2LAC than in 1LAC. Only 26 2LAC sources have more than one counterpart, whereas 33 sources had more than one counterpart in the 1LAC. Moreover, 2LAC sources have at most two counterparts, while there were cases of three counterparts in 1LAC. Besides the difference in exposure, comparisons between 1LAC and 2LAC must take several other factors into account [a full description is given in @2FGL]: i) the switch from unbinned to binned likelihood analysis; ii) the use of different instrument response functions (“P7\_V6 SOURCE" instead of “P6\_V3 DIFFUSE"); and iii) the use of different association methods. None of these changes is, however, expected to affect the number of overall associated sources by more than $\sim 10$% (the former change leads to a lower 2LAC/1LAC count ratio, while the latter two have the opposite effect).
Comparisons between the properties of BL Lac objects and FSRQs must carefully take into account the redshift incompleteness, given that more than half of the BL Lac objects in the 2LAC lack redshifts. Because the photon spectral-index distribution of blazars of unknown type differs from both those of BL Lac objects or FSRQs (Fig. \[fig:index\]), the sample of blazars lacking redshifts therefore does not, apparently, represent a uniform subsample of any one class of objects with measured redshift. This incompleteness influences any conclusions concerning luminosity or other properties that depend on knowledge of redshift [@1LAC]. For example, strongly beamed emission can overwhelm the atomic line radiation flux and might preferentially arise from high luminosity, high redshift BL Lac objects [@Gio11]. These would then be absent in the spectral index/luminosity diagram (Fig. \[fig:index\_L\]) and skew the correlation. Until the redshift incompleteness, the nature of the unassociated sources in the 2LAC, and underlying biases introduced by using different source catalogs [@Gio99; @Padovani03; @Gio11] are resolved, conclusions about the blazar sequence [@Fossati98; @Ghi98] and the blazar divide [@Ghisellini09] remain tentative.
The GeV spectra of most FSRQs are softer than those of BL Lac objects, suggesting that the strength of the emission lines is connected with and possibly determines the position of the external Compton scattering peak, as would be expected in leptonic scenarios for blazar jets [e.g., @Ghi98; @Boe02]. From their general properties, in particular in the $\gamma$-ray band, LSP BL Lacs appear to be transitional objects between FSRQs and the general BL Lac population, confirming the trend established from their broadband SEDs [@Ghi11]. Clarifying the relationship between the line luminosities and the broadband SEDs of blazars is crucial to determine the evolutionary connection between various classes of $\gamma$-ray emitting blazars, and whether this is reflected in the blazar sequence.
$\log N - \log S$ Distribution
------------------------------
A complete analysis of the $\log N - \log S$ distribution requires a dedicated study ([*Fermi*]{}-LAT Collaboration, in preparation). Assuming, however, that the sources at high Galactic latitude are dominated by blazars, and furthermore neglecting the aforementioned effects of different analysis procedures, then the observed increase in the detected number of $|b| > 10^\circ$ sources between 1FGL and 2FGL is roughly compatible with the extrapolation of the integral $\log N - \log S$ derived from the 1LAC to lower fluxes, which exhibits a slope of $\sim -0.6$ at the low-flux end of the distribution [@Abdo_EDB]. The roles of source confusion, flux limits of the cataloged AGN data used to make AGN associations, and intrinsic AGN variability must be carefully considered, however. With respect to the first issue, approximately 8% of the $|b|>10^\circ$, $TS > 25$ sources were missing because of source confusion in 1FGL [@1FGL], but this fraction went down to $\sim 3.3$% in the 2FGL due to improved analysis techniques [@2FGL]. Source confusion is, of course, even more important for soft sources due to the larger PSF and the lower effective area for detection of lower energy photons that leads to poorer position determination, but should not strongly affect the results presented here.
Regarding the flux limits of the cataloged sources, Figure \[fig:fluxhisto\] shows that BL Lac objects are on average much fainter radio sources, with median 8 GHz fluxes nearly an order of magnitude fainter than for FSRQs ($\sim 80$ mJy for BL Lac objects and $\sim 500$ mJy for FSRQs). Incompleteness in radio catalogs therefore would likely be more important for BL Lac objects and especially the HSP BL Lac objects which, if this selection bias were not present, would further increase the fractional number of BL Lacs compared to FSRQs. Finally, concerning the issue of variability, we note that the averaging of fluxes over 2 years will dilute the presence of blazars with small duty cycles on monthly and yearly timescales.
Threshold sensitivity in terms of photon flux is strongly dependent on source spectral index (Figure \[fig:index\_flux\]), whereas energy flux is not (Figure \[fig:index\_S\]). BL Lacs and FSRQs are both complete to an energy flux of $\sim 5\times 10^{-12}$ erg cm$^{-2}$ s$^{-1}$. The $\log N - \log S$ energy-flux distribution of unassociated 2FGL sources with $|b| > 10^{\circ}$ and $\Gamma>2.2$ that are potential FSRQ candidates is displayed in Figure \[fig:logn\_logse\] (bottom; black histogram). Adding the $\log N - \log S$ distribution for these sources to that for FSRQs results in the magenta histogram, which exhibits a steeper slope at low fluxes than the case with FSRQs alone. Thus we conclude that the unassociated sources are likely to be a mixture of FSRQs and BL Lacs, including possibly other source types.
Aligned and Misaligned Sources
------------------------------
The [*Fermi*]{}-LAT has increased the number of known, high-confidence $\gamma$-ray emitting BL Lacs by a factor of $\sim 20$ over the number detected with EGRET [@3EGcatalog; @mhr01; @Din01; @Sow03; @Sow04]. The number of BL Lacs has increased by 43% (395 versus 275) from the 1LAC to 2LAC Clean Samples, while the number of FSRQs has increased by only $\sim 25$% (310 versus 248). This discrepancy might be even larger due to the evident lack of cataloged southern hemisphere BL Lac objects, as noted above. Yet the number of misaligned AGNs observed at large, $\gtrsim 10^\circ$ angles to the jet axis, remains small—only 11 were reported in the dedicated Fermi paper on these sources [@magn]. Three of these, 3C 78, 3C 111, and 3C 120, are not now in the 2LAC, evidently due to variability (§\[sec:RG\]), illustrating that the jetted component can make a dominant contribution to the $\gamma$-ray emission in radio galaxies. Two other radio galaxies—Centaurus B and Fornax A—are, however, now included.
The LAT-detected Fanaroff-Riley II (FR II) radio galaxies and steep spectrum radio quasars (SSRQs) have $\gamma$-ray luminosities $\sim 10^{45}$ – $10^{46}$ erg s$^{-1}$, and are found at the faint end of the luminosity distribution of FSRQs, which extends upwards to $\gtrsim 10^{49}$ erg s$^{-1}$. In comparison, the LAT-detected Fanaroff-Riley I (FR I) radio galaxies have $\gamma$-ray luminosities 2 – 4 orders of magnitude lower than the lowest typical $\gamma$-ray luminosities, $\sim 10^{44}$ erg s$^{-1}$, of BL Lac objects [see Figure \[fig:index\_L\] and @magn]. Besides the slow increase in numbers, this raises the interesting and possibly related question why the ratio of measured $\gamma$-ray luminosities of FR I galaxies and BL Lac objects span a much larger range than that for FR II galaxies and FSRQs. If SSRQs are FSRQs seen at slightly larger angle to the jet axis, then the low-luminosity range of FSRQs could be a mixture of sources with lower-power jets and those with powerful jets, but with slight misalignment. One possibility is that this could be due to different $\gamma$-ray emission beaming factors, with the emission being more beamed in the latter case due to external Compton scattering [@dermer1995; @Geo2001]. The more rapid fall-off in off-axis flux, combined with the relative paucity of nearby FR II galaxies, could therefore make detection of FR IIs less likely than for the FR Is. Another possibility is that the preferential detection of FR Is over FR IIs reflects the difference in jet structure in FSRQs and BL Lac objects [e.g., @Chiaberge2000; @Mey11], with broader emission cones in BL Lacs that consequently favor the detection of FR Is. Furthermore, extended jet or lobe emission could be present in the FR Is that is missing in FR II galaxies. The situation is further complicated in that some LSP BL Lac objects have properties associated with FR II rather than FR I radio galaxies [@Kollgaard1992].
Variability
-----------
Monthly light curves established for the whole 2LAC have enabled the confirmation of trends obtained over a more limited source sample and shorter time span, namely that:
1. The mean fractional variability on time scales sampled by our data, as given by the normalized excess variance, is higher for FSRQs than for BL Lacs. The normalized excess variance for BL Lacs decreases from LSP to ISP and HSP BL Lac objects.
2. With the definition of duty cycle used in Section \[sec:var\] based on monthly-averaged time bin light curves, bright FSRQs and BL Lac objects both have duty cycles of about 0.05 - 0.10.
3. The Power Density Spectra in the frequency range $\sim (0.033$ – 0.5) month$^{-1}$ for bright FSRQs and bright BL Lacs of all types are each described by a power law with mean index of $\sim 1.2$ (Figure \[fig:PDS\]). The discrete auto-correlation and structure function analyses shows that FSRQs display slightly longer correlation timescales and steeper and more broadly distributed structure function indices than HSP BL Lac sources (Figure \[fig:ACF\] and \[fig:SF\]). Thus the FSRQs tend to be slightly more “Brownian-variable," i.e., driven by longer-memory processes, than HSP objects.
Differences between variability properties of BL Lac objects and FSRQs at GeV energies are important for understanding the jet location and jet radiation mechanisms, considering that rapid variability is more likely to be related to emission sites near the central nucleus, whereas extended ($\gtrsim$ kpc) jets can only make weakly variable or quiescent emission. Earlier analysis of GeV light curves indicate that FSRQs have larger variability amplitudes than BL Lacs [@Abdo_var], and this result is confirmed here by considering the normalized excess variance (Figure \[fig:index\_var\]), which also follows from a comparison of BL Lac and FSRQ light curves with similar photon statistics (Figure \[fig:varind\_relunc\]). The larger variability amplitudes in FSRQs than BL Lacs can be interpreted as a result of shorter cooling timescales of electrons making GeV emission through external Compton processes in FSRQs above the $\nu F_\nu$ peak compared with the longer cooling timescales of the lower-energy electrons making GeV emission through synchrotron self-Compton processes in HSP BL Lac objects at frequencies below the $\nu F_\nu$ peak . This assumes, however, that the jet is long-lived and not subject to adiabatic expansion that would make achromatic variability at all frequencies. Radiation from extended jets in BL Lac objects [@2008ApJ...679L...9B], which might be less important in the relatively younger but more powerful FSRQs, could also make a weakly varying high-energy radiation component, as could cascade emission induced by ultra-high energy cosmic-ray protons [@Ess09; @Ess10; @2011ApJ...731...51E], or the cascade emission from TeV $\gamma$ rays interacting with photons of the extragalactic background light .
EBL and High Redshift AGNs
--------------------------
The number of high-energy ($> 10$ GeV) photons from $z> 0.5$ sources that can constrain EBL models has increased by a factor $\sim 2$ in 2LAC compared with the 11 month data [@Abdo_EBL], due to increased exposure and better background rejection. This should increase further with improvements in our capability to reconstruct event tracks and reject background at high energy [@Baldini]. The detection of 30 photons with $E>10$ GeV and $z>0.5$ in the 2LAC that are also above the $\tau_{\gamma\gamma} = 3$ opacity curve predicted by the [@Ste06] “baseline model” will further constrain this high EBL model. EBL models that produce lower opacity [e.g. @Fran08; @Fin10b; @Gil09] in the $(E,z)$ phase space cannot, however, be constrained in this manner. The detection of 5 photons in 2LAC with $E>100$ GeV and from $z > 0.5$ sources can probe the EBL at much longer wavelengths than was previously possible with [*Fermi*]{}-LAT data.
Remarkably, no source detected in the 2LAC is at higher redshift than in the 1LAC, even though the exposure has more than doubled. The most distant blazar detected is still at $z = 3.10$. Thus the lower flux limits in the 2LAC have helped detect fainter objects at lower redshifts, rather than finding objects with comparable luminosities as those found in the 1LAC but farther away. With the detection limits of the 2LAC, FSRQs with a $\gamma$-ray luminosity of $\sim 10^{48}$ erg s$^{-1}$ (many of which are present in the 2LAC at $z \geq 1$, see Figure \[fig:L\_redshift\]) would have been detected up to $z\sim 6$, and up to $z\sim 4$ for luminosities as low as $10^{47.5}$ erg s$^{-1}$. Thus the lack of high-redshift objects is not due to luminosity selection. A change of SED properties for blazars at high redshift is suggested by comparing the overlapping sources from the BAT survey in the hard X-ray band with LAT samples (7 above $z=2$ and $|b|>10^\circ$, among 14 and 30 sources in the BAT 58-month and 2LAC samples, respectively) and the fact that none of the more than 50 known luminous FSRQs above $z = 3.10$ in the BZCAT [@bzcat] is detected in the 2LAC. These are likely characterized by a much lower $\nu_{peak}$ frequency of the SED [see, e.g., @2010MNRAS.402..497G], with the $\gamma$-ray peak near 1 MeV rather than at $\sim 10$ – 100 MeV. A source with this type of SED would be very difficult to detect with [*Fermi*]{}, since the LAT band would be sampling the $\gamma$-ray cutoff of the SED, but should be easily detectable in the hard X-ray band with upcoming missions like NuSTAR [@2010SPIE.7732E..21H] and Astro-H [@2010SPIE.7732E..27T].
Summary of Results
------------------
The 2LAC represents a significant advance with respect to the 1LAC, including many more sources and reduced uncertainties thanks to the doubling of exposure and refinement of the analysis. This has resulted in an $\sim 52$% (1017 versus 671) increase in the number of associated sources, better localization, more accurate time-averaged spectra, and more detailed light curves and characterization of variability patterns. Despite the problems outlined above concerning the incomplete classification of the 2LAC Clean Sample, the following results—most of which were already found in 1LAC—can be stated with confidence:
1. $\gamma$-ray AGNs are almost exclusively blazars, with $\gtrsim 95$% of the 2LAC sources associated with members of this class. The number of non-blazar sources in the Clean Sample has dropped from 26 to 24 between 1LAC and 2LAC, though part of this reduction is due to variability of sources previously classified as radio galaxies. There is no compelling evidence for $\gamma$-ray emission from radio-quiet AGNs.
2. BL Lacs outnumber FSRQs. BL Lacs, with generally harder spectra, can be detected more easily with the [*Fermi*]{}-LAT than FSRQs at a given significance limit with increased exposure (as was also the case in the LBAS and 1LAC samples).
3. A strong correlation is found between spectral index and blazar class for sources with measured redshift. This effect is most clearly visible in the flux-limited sample shown in Figure \[fig:index\_c\]. For that sample, the average photon spectral index $\langle \Gamma \rangle$ continuously shifts to lower values (i.e., harder spectra) as the class varies from FSRQs ($\langle \Gamma \rangle = 2.42$) to LSP-BL Lacs ($\langle \Gamma \rangle = 2.17$), ISP-BL Lacs ($\langle \Gamma \rangle = 2.14$), and HSP-BL Lacs ($\langle \Gamma \rangle = 1.90$). These values are systematically slightly lower, by $\sim$ 0.06 units, than those found in 1LAC.
4. Among BL Lacs, HSP sources dominate over ISPs and LSPs. The percentages, $\sim 20$% 27%, 53% for LSPs, ISPs, HSPs, respectively, are essentially the same as for the 1LAC.
5. Due to the flattening of the $\log N - \log S$ distribution for FSRQs (Figure \[fig:logn\_logse\]), increased exposure should yield only a modest addition to the number of such sources.
6. BL Lac objects and FSRQs display significantly different variability properties. The differences are weaker than those found in the bright LBAS sample [@Abdo_var], probably due to the use of coarser time binning (one month instead of one week) and the inclusion in the larger 2LAC sample of fainter or less variable sources.
7. Most of the 45 TeV AGN have now been detected with [*Fermi*]{}. Of these, 39 are in the 2FGL and 34 of these are in the 2LAC Clean Sample. The six that have not been detected with [*Fermi*]{} are HSPs. The increase in the break between the spectral index measured by [*[Fermi]{}*]{} and that reported in the TeV regime as a function of the redshift of the AGNs [@FermiTeV] has been confirmed with this larger sample of GeV-TeV AGNs.
The fact that many sources lack proper classification or a measured redshift calls for a large multiwavelength effort by the blazar community, emphasizing optical spectroscopy when the jet activity is low and the emission line flux is not hidden by nonthermal jet radiation. The general trends identified in the 1LAC, many of them already apparent in the LBAS, are confirmed. Overall, the 2LAC should allow for a deeper understanding of the blazar phenomenon and the relations between blazar classes.
Acknowledgments
===============
The *Fermi* LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis. These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat à l’Energie Atomique and the Centre National de la Recherche Scientifique / Institut National de Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the Swedish National Space Board in Sweden. Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d’Études Spatiales in France.
This work is partly based on optical spectroscopy observations performed at Telescopio Nazionale Galileo, La Palma, Canary Islands (proposal AOT20/09B and AOT21/10A). Part of this work is based on archival data, software or on-line services provided by the ASI Science Data Center (ASDC). 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.
[*Facilities:*]{} .
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[lccc]{} NVSS & 0.162 $\pm$ 0.001 & 0.744 $\pm$0.004 & $-$0.28\
SUMSS & 0.50 $\pm$ 0.03 & 0.88 $\pm$ 0.02 & 0.79\
RASS & 0.70 $\pm$ 0.03 & 0.79 $\pm$0.02 & 1.71\
PMN & 0.59 $\pm$ 0.03 & 0.88 $\pm$0.02 & 1.36\
AT20G & 0.59 $\pm$ 0.07 & 0.25 $\pm$0.02 & 2.91\
[lccccccccccc]{} All& 1017& 16.3& 846& 12.5& 2 & 1007& 27.4&113 & 763& 22.7& 6\
Clean Sample& 886& 11.7&754& 9.1& 2 & 877& 21.0 & 82 & 691& 19.1 & 5\
[llrrcccccccccc]{} J0000.9$-$0748\*&PMN J0001$-$0746&0.32502&$-$7.77411&0.099&0.181&BL Lac&ISP&0&2.39$\pm$0.14&0.98&0.83&0.97&0.81\
J0001.7$-$4159\*&1RXS J000135.5$-$41551&0.38794&$-$41.92392&0.082&0.118&AGU&HSP&0&2.14$\pm$0.19&&&0.81&0.89\
J0004.7$-$4736\*&PKS 0002$-$478&1.14842&$-$47.60567&0.022&0.104&FSRQ&LSP&0.88&2.45$\pm$0.09&1.00&1.00&0.99&0.95\
J0006.1+3821\*&S4 0003+38&1.48810&38.33754&0.032&0.133&FSRQ&LSP&0.229&2.60$\pm$0.08&1.00&1.00&0.99&\
J0007.8+4713\*&MG4 J000800+4712&1.99986&47.20213&0.033&0.058&BL Lac&LSP&0.28&2.10$\pm$0.06&1.00&0.98&0.98&0.96\
J0008.7$-$2344&RBS 0016&2.14734&$-$23.65775&0.090&0.174&BL Lac&&0.147&1.62$\pm$0.25&0.99&&0.92&\
J0008.7$-$2344$-$&PKS 0005$-$239&2.00159&$-$23.65512&0.196&0.174&FSRQ&&1.412&1.62$\pm$0.25&&&0.96&\
J0009.0+0632$-$&GB6 J0009+0625&2.32097&6.43164&0.125&0.126&AGU&&&2.40$\pm$0.16&&&0.96&\
J0009.0+0632&CRATES J0009+0628&2.26701&6.47266&0.070&0.126&BL Lac&LSP&0&2.40$\pm$0.16&0.99&0.97&0.98&0.91\
J0009.1+5030\*&NVSS J000922+503028&2.34475&50.50801&0.034&0.050&AGU&&&1.85$\pm$0.06&&0.88&&\
J0009.9$-$3206&IC 1531&2.39901&$-$32.27696&0.180&0.147&AGU&LSP&0.025&2.17$\pm$0.16&&&0.97&\
J0011.3+0054&PMN J0011+0058&2.87641&0.96429&0.078&0.199&FSRQ&LSP&1.4934&2.43$\pm$0.13&0.99&0.99&0.96&\
J0012.9$-$3954\*&PKS 0010$-$401&3.24980&$-$39.90718&0.007&0.107&BL Lac&&0&2.16$\pm$0.16&1.00&1.00&0.99&\
J0013.8+1907\*&GB6 J0013+1910&3.48510&19.17825&0.056&0.158&BL Lac&&0.473&2.06$\pm$0.19&0.99&1.00&0.97&\
J0017.4$-$0018\*&S3 0013$-$00&4.04574&$-$0.25404&0.322&0.280&FSRQ&LSP&1.574&2.60$\pm$0.14&&&0.97&\
J0017.6$-$0510\*&PMN J0017$-$0512&4.39900&$-$5.21179&0.030&0.071&FSRQ&LSP&0.226&2.44$\pm$0.07&1.00&1.00&0.99&0.97\
J0018.5+2945\*&RBS 0042&4.61563&29.79174&0.035&0.098&BL Lac&HSP&0&1.24$\pm$0.28&1.00&&0.95&0.99\
J0018.8$-$8154\*&PMN J0019$-$8152&4.84104&$-$81.88083&0.028&0.134&AGU&HSP&&2.14$\pm$0.12&&0.87&0.93&0.96\
J0019.4$-$5645\*&PMN J0019$-$5641&4.86058&$-$56.69525&0.061&0.174&AGU&&&2.66$\pm$0.28&0.98&0.88&0.89&\
J0021.6$-$2551\*&CRATES J0021$-$2550&5.38552&$-$25.84700&0.024&0.079&BL Lac&ISP&0&1.98$\pm$0.11&1.00&0.91&0.98&\
J0022.2$-$1853\*&1RXS 002209.2$-$185333&5.53816&$-$18.89249&0.020&0.063&AGU&HSP&&1.53$\pm$0.12&&0.95&0.97&0.96\
J0022.3$-$5141\*&1RXS 002159.2$-$514028&5.49937&$-$51.67408&0.062&0.150&AGU&HSP&&2.22$\pm$0.17&&&0.85&0.97\
J0022.5+0607\*&PKS 0019+058&5.63526&6.13457&0.013&0.059&BL Lac&LSP&0&2.09$\pm$0.06&1.00&1.00&0.99&\
J0023.2+4454\*&B3 0020+446&5.89755&44.94339&0.069&0.107&FSRQ&&1.062&2.36$\pm$0.12&1.00&1.00&0.97&\
J0024.5+0346\*&GB6 J0024+0349&6.18826&3.81761&0.055&0.166&FSRQ&&0.545&2.24$\pm$0.16&&0.97&0.91&\
\[tab:clean\]
[llrrcccccccccc]{} J0010.5+6556&GB6 J0011+6603&2.91238&66.06075&0.168&0.190&AGU&&&2.41$\pm$0.23&0.87&&0.91&\
J0035.8+5951&1ES 0033+595&8.96930&59.83486&0.019&0.040&BL Lac&HSP&0&1.87$\pm$0.07&1.00&&0.99&1.00\
J0047.2+5657&GB6 J0047+5657&11.75224&56.96170&0.031&0.064&BL Lac&&0&2.06$\pm$0.07&1.00&1.00&0.99&\
J0102.7+5827&TXS 0059+581&15.69076&58.40321&0.059&0.059&FSRQ&LSP&0.644&2.28$\pm$0.05&0.99&1.00&0.99&\
J0103.5+5336&1RXS 010325.9+533721&15.85868&53.62000&0.026&0.067&AGU&HSP&&1.75$\pm$0.16&&&0.97&0.99\
J0109.9+6132&TXS 0106+612&17.44394&61.55816&0.026&0.044&FSRQ&LSP&0.785&2.19$\pm$0.06&1.00&1.00&0.99&\
J0110.3+6805&4C +67.04&17.55254&68.09483&0.011&0.052&AGU&ISP&&2.13$\pm$0.08&1.00&1.00&1.00&0.98\
J0131.1+6121&1RXS 013106.4+612035&22.77986&61.34246&0.014&0.041&AGU&HSP&&1.91$\pm$0.08&&&0.98&1.00\
J0137.7+5811&1RXS 013748.0+581422&24.45948&58.23698&0.039&0.094&AGU&HSP&&2.33$\pm$0.12&&&0.98&0.99\
J0241.3+6548&NVSS J024121+654311&40.34080&65.71981&0.089&0.071&AGU&HSP&&1.97$\pm$0.16&&&0.97&0.96\
J0250.7+5631&NVSS J025047+562935&42.69858&56.49304&0.033&0.108&AGU&&&2.25$\pm$0.13&&&0.95&0.97\
J0253.5+5107&NVSS J025357+510256&43.48992&51.04909&0.096&0.087&FSRQ&&1.732&2.44$\pm$0.07&0.93&0.86&0.98&\
J0303.5+4713&4C +47.08&45.89702&47.27117&0.054&0.061&BL Lac&LSP&0&2.24$\pm$0.07&1.00&0.99&1.00&0.95\
J0303.5+6822&TXS 0259+681&46.09134&68.36020&0.076&0.138&AGU&&&2.77$\pm$0.11&0.98&0.99&0.99&0.91\
J0334.3+6538&TXS 0329+654&53.48632&65.61562&0.046&0.074&AGU&ISP&&1.82$\pm$0.14&0.99&0.98&0.99&0.96\
J0359.1+6003&TXS 0354+599&59.76081&60.08954&0.035&0.103&FSRQ&ISP&0.455&2.30$\pm$0.08&0.99&1.00&0.99&0.97\
J0423.8+4149&4C +41.11&65.98325&41.83412&0.023&0.036&BL Lac&&0&1.80$\pm$0.06&1.00&1.00&1.00&\
J0503.3+4517&1RXS 050339.8+451715&75.91498&45.28299&0.048&0.089&AGU&&&1.85$\pm$0.14&&&0.95&0.98\
J0512.9+4040&B3 0509+406&78.21907&40.69547&0.031&0.102&AGU&&&1.89$\pm$0.12&0.99&1.00&0.99&0.96\
J0517.0+4532&4C +45.08&79.36892&45.61742&0.111&0.127&FSRQ&LSP&0.839&2.13$\pm$0.11&0.93&0.93&0.99&\
J0521.7+2113&VER J0521+211&80.44167&21.21429&0.009&0.023&BL Lac&ISP&0&1.93$\pm$0.03&1.00&1.00&1.00&1.00\
J0533.0+4823&TXS 0529+483&83.31617&48.38132&0.039&0.058&FSRQ&LSP&1.16&2.31$\pm$0.05&1.00&1.00&0.99&0.95\
J0622.9+3326&B2 0619+33&95.71749&33.43628&0.026&0.043&AGU&&&2.13$\pm$0.04&1.00&0.99&0.99&\
J0643.2+0858&PMN J0643+0857&100.86013&8.96074&0.049&0.069&FSRQ&&0.882&2.49$\pm$0.09&0.98&0.99&0.99&\
\[tab:lowlat\]
[lccc]{} &[**1017**]{}&[**886**]{}&[**104**]{}\
\
[**FSRQ**]{}&[**360**]{}&[**310**]{}&[**19**]{}\
[…]{}LSP&246&221&7\
[…]{}ISP&4&3&2\
[…]{}HSP&2&0&0\
[…]{}no classification &108&86&10\
\
[**BL Lac**]{}&[**423**]{}&[**395**]{}&[**16**]{}\
[…]{}LSP&65&61&3\
[…]{}ISP&82&81&3\
[…]{}HSP&174&160&5\
[…]{}no classification &102&93&5\
\
[**Blazar of Unknown type**]{}&[**204**]{}&[**157**]{}&[**67**]{}\
[…]{}LSP&24&19&10\
[…]{}ISP&13&11&3\
[…]{}HSP&65&53&13\
[…]{}no classification &102&74&41\
\
[**Other AGN**]{}&[**30**]{}&[**24**]{}&[**2**]{}\
[llrrccccccccccccc]{} J0013.7$-$5022 & BZB J0014$-$5022 & 3.54675 & $-$50.37575 & BLL & HSP & & S & Y & 1.00& C\
J0019.3+2017 & PKS 0017+200 & 4.90771 & 20.36267 & BLL & LSP & & S & Y & 0.99 & C\
J0041.9+2318 & PKS 0039+230 & 10.51896 & 23.33367 & FSRQ & & 1.426 & S & Y & 0.98 & C\
J0202.1+0849 & RX J0202.4+0849 & 30.61000 & 8.82028 & BLL & LSP & & S & Y & 0.99 & C\
J0208.6+3522 & BZB J0208+3523 & 32.15913 & 35.38686 & BLL & HSP & 0.318 & S & Y & 1.00& C\
J0305.0$-$0601 & CRATES J0305$-$0607 & 46.25238 & $-$6.12819 & BLL & & & S & Y & 0.95 & NC, V\
J0308.3+0403 & NGC 1218 & 47.10927 & 4.11092 & AGN & & 0.029 & S & Y & 0.98 & C\
J0343.4$-$2536 & PKS 0341$-$256 & 55.83138 &$-$25.50480 & FSRQ & LSP & 1.419 & S & Y & 0.97 & C\
J0422.1+0211 & PKS 0420+022 & 65.71754 & 2.32414 & FSRQ & LSP & 2.277 & S & Y & 0.86 & NC, V\
J0457.9+0649 & 4C +06.21 & 74.28212 & 6.75203 & FSRQ & LSP & 0.405 & S & Y & 0.84 & UnA\
J0622.3$-$2604 & CRATES J0622-2606 & 95.59888 & $-$26.10767 & & & & S & Y & 0.99 & S\
J0625.9$-$5430 & CGRaBS J0625$-$5438 & 96.46771 & $-$54.64739 & FSRQ & LSP & 2.051 & S & Y & 0.99 & BC\
J0626.6$-$4254 & CRATES J0626$-$4253 & 96.53292 & $-$42.89219 & & & & S & Y & 0.89 & CC\
J0645.5+6033 & BZU J0645+6024 & 101.25571 & 60.41175 & AGN & & 0.832 & S & Y & 0.87 & UnA\
J0722.3+5837 & BZB J0723+5841 & 110.80817 & 58.68844 & BLL & HSP & & S & Y & 0.95 & NC, V\
J0809.4+3455 & B2 0806+35 & 122.41204 & 34.92700 & BLL & HSP & 0.082 & S & Y & 0.99 & C\
J0835.4+0936 & CRATES J0835+0937 & 128.93008 & 9.62167 & BLL & & & S & Y & 0.96 & NC, V\
J0842.2+0251 & BZB J0842+0252 & 130.6063 & 2.88131 & BLL & HSP & 0.425 & S & Y & 0.99 & BC\
J0850.2+3457 & RX J0850.6+3455 & 132.65083 & 34.92305 & BLL & ISP & 0.149 & S & Y & 0.99 & C\
J0952.2+3926 & BZB J0952+3936 & 148.06129 & 39.60442 & BLL & HSP & & S & Y & 0.82 & NC, V\
J1007.0+3454 & BZB J1006+3454 & 151.73527 & 34.91255 & BLL & HSP & & S & Y & 1.00& NC, V\
J1119.5$-$3044 & BZB J1119$-$3047 & 169.91458 & $-$30.78894 & BLL & HSP & 0.412 & S & Y & 1.00& C\
J1220.2+3432 & CGRaBS J1220+3431 & 185.03454 & 34.52269 & BLL & ISP & & S & Y & 1.00& C\
J1226.8+0638 & BZB J1226+0638 & 186.68428 & 6.64811 & BLL & HSP & & S & Y & 0.99 & C\
J1253.7+0326 & CRATES J1253+0326 & 193.44588 & 3.44178 & BLL & HSP & 0.065 & S & Y & 0.99 & C\
J1331.0+5202 & CGRaBS J1330+5202 & 202.67750 & 52.03761 & AGN & & 0.688 & S & Y & 0.99 & C\
J1341.3+3951 & BZB J1341+3959 & 205.27127 & 39.99595 & BLL & HSP & 0.172 & S & Y & 0.93 & C\
J1422.2+5757 & 1ES 1421+582 & 215.66206 & 58.03208 & BLL & HSP & & S & Y & 0.95 & C\
J1422.7+3743 & CLASS J1423+3737 & 215.76921 & 37.62516 & BLL & & & S & Y & 0.90 & S\
J1442.1+4348 & CLASS J1442+4348 & 220.52979 & 43.81020 & BLL & & & S & Y & 0.99 & CC\
J1503.3+4759 & CLASS J1503+4759 & 225.94999 & 47.99195 & BLL & LSP & & S & Y & 0.96 & UnA\
J1531.8+3018 & BZU J1532+3016 & 233.00929 & 30.27468 & BLL & HSP & 0.065 & S & Y & 0.99 & C\
J1536.6+8200 & CLASS J1537+8154 & 234.25036 & 81.90862 & & & & S & Y & 0.82 & CC\
J1616.1+4637 & CRATES J1616+4632 & 244.01571 & 46.54033 & FSRQ & & 0.95 & S & Y & 0.96 & C\
J1624.7$-$0642 & 4C $-$06.46 & 246.13717 & $-$6.83047 & & & & S & Y & 0.94 & NC\
J1635.4+8228 & NGC 6251 & 248.13325 & 82.53789 & AGN & & 0.025 & S & Y & 0.88 & O\
J1735.4$-$1118 & CRATES J1735$-$1117 & 263.86325 & $-$11.29292 & & & & S & Y & 1.00& C\
J1804.1+0336 & CRATES J1803+0341 & 270.9845 & 3.68544 & FSRQ & & 1.42 & S & Y & 0.95 & BC\
J1925.1$-$1018 & CRATES J1925$-$1018 & 291.26333 & $-$10.30344 & BLL & & & S & Y & 1.00& S\
J2006.6$-$2302 & CRATES J2005$-$2310 & 301.48579 & $-$23.17417 & FSRQ & LSP & 0.833 & S & Y & 0.91 & UnA\
J2008.6$-$0419 & 3C 407 & 302.10161 & $-$4.30814 & AGN & & 0.589 & S & Y & 0.99 & NC, V\
J2025.9$-$2852 & CGRaBS J2025$-$2845 & 306.47337 & $-$28.76353 & & LSP & & S & Y & 0.97 & C\
J2117.8+0016 & CRATES J2118+0013 & 319.57250 & 0.22133 & FSRQ & & 0.463 & S & Y & 0.91 & C\
J2126.1$-$4603 & PKS 2123$-$463 & 321.62846 & $-$46.09633 & FSRQ & & 1.67 & S & Y & 0.98 & S\
J2322.3$-$0153 & PKS 2320$-$021 & 350.76929 & $-$1.84669 & FSRQ & & 1.774 & S & Y & 0.84 & C\
\[tab:1LAC\]
[llrrccrrr]{} J0000.9$-$0748\*&PMN J0001$-$0746&46&209&8.10&17.61&&1.33&0.53\
J0001.7$-$4159\*&1RXS J000135.5$-$41551&45&12&11.01&18.97&&1.12&0.17\
J0004.7$-$4736\*&PKS 0002$-$478&173&995&14.70&17.30&&1.12&0.67\
J0006.1+3821\*&S4 0003+38&164&573&14.10&17.72&&1.20&0.65\
J0007.8+4713\*&MG4 J000800+4712&326&61&14.80&18.28&&1.08&0.51\
J0008.7$-$2344&RBS 0016&25&36&&16.64&&&0.42\
J0008.7$-$2344$-$&PKS 0005$-$239&25&375&&16.51&&&0.55\
J0009.0+0632$-$&GB6 J0009+0625&43&180&&19.49&19.19&&\
J0009.0+0632&CRATES J0009+0628&43&247&13.00&18.70&18.10&1.17&0.63\
J0009.1+5030\*&NVSS J000922+503028&310&12&&19.52&&&\
J0009.9$-$3206&IC 1531&35&389&5.01&8.91&&2.78&$-$0.09\
J0011.3+0054&PMN J0011+0058&49&167&5.41&20.17&20.40&0.86&0.78\
J0012.9$-$3954\*&PKS 0010$-$401&50&495&&18.09&&&0.74\
J0013.8+1907\*&GB6 J0013+1910&25&161&&18.41&&&0.61\
J0017.4$-$0018\*&S3 0013$-$00&38&1086&3.19&19.99&19.17&1.11&0.84\
J0017.6$-$0510\*&PMN J0017$-$0512&185&178&17.40&18.09&&1.10&0.63\
J0018.5+2945\*&RBS 0042&31&34&143.00&17.47&&0.90&0.36\
J0018.8$-$8154\*&PMN J0019$-$8152&69&83&29.70&16.35&&1.32&0.30\
J0019.4$-$5645\*&PMN J0019$-$5641&37&61&&20.36&&&\
J0021.6$-$2551\*&CRATES J0021$-$2550&116&69&1.72&17.22&&1.63&0.49\
J0022.2$-$1853\*&1RXS 002209.2$-$185333&141&23&10.90&17.45&&1.34&0.32\
J0022.3$-$5141\*&1RXS 002159.2$-$514028&36&20&50.30&16.58&&1.15&0.23\
J0022.5+0607\*&PKS 0019+058&391&340&2.45&19.51&&1.04&0.82\
J0023.2+4454\*&B3 0020+446&76&141&&&&&\
J0024.5+0346\*&GB6 J0024+0349&32&22&&19.81&&&0.61\
\[tab:prob1\]
[llrrccrrr]{} J0010.5+6556&GB6 J0011+6603&71&64&&19.70&&&\
J0035.8+5951&1ES 0033+595&243&148&318.00&18.21&&1.01&0.37\
J0047.2+5657&GB6 J0047+5657&201&190&&19.58&&&0.62\
J0102.7+5827&TXS 0059+581&298&849&3.83&18.06&&1.61&0.64\
J0103.5+5336&1RXS 010325.9+533721&44&31&63.70&16.09&&1.43&0.15\
J0109.9+6132&TXS 0106+612&1102&305&2.60&19.10&&1.70&0.48\
J0110.3+6805&4C +67.04&145&1707&23.20&17.13&&1.69&0.42\
J0131.1+6121&1RXS 013106.4+612035&276&20&471.00&19.29&&1.03&0.15\
J0137.7+5811&1RXS 013748.0+581422&65&171&252.00&18.40&17.04&1.23&0.29\
J0241.3+6548&NVSS J024121+654311&70&191&41.60&19.43&&1.22&0.44\
J0250.7+5631&NVSS J025047+562935&41&36&34.30&&&&\
J0253.5+5107&NVSS J025357+510256&141&430&&20.24&&&0.71\
J0303.5+4713&4C +47.08&218&964&3.59&17.45&&1.63&0.68\
J0303.5+6822&TXS 0259+681&81&1208&&&&&\
J0334.3+6538&TXS 0329+654&51&288&16.60&18.57&&1.41&0.45\
J0359.1+6003&TXS 0354+599&90&953&38.80&17.25&&1.46&0.48\
J0423.8+4149&4C +41.11&335&1756&&19.78&&&0.72\
J0503.3+4517&1RXS 050339.8+451715&45&35&75.20&&&&\
J0512.9+4040&B3 0509+406&35&877&&15.81&&&\
J0517.0+4532&4C +45.08&42&1336&1.55&20.04&&1.54&0.70\
J0521.7+2113&VER J0521+211&1542&530&60.20&16.29&&1.52&0.37\
J0533.0+4823&TXS 0529+483&400&435&10.80&19.18&&1.16&0.66\
J0622.9+3326&B2 0619+33&566&240&&&&&\
J0643.2+0858&PMN J0643+0857&267&543&&&17.85&&0.46\
\[tab:prob2\]
[lcccccccccccccccc]{} RGBJ0152$+$017 & J0152.6$+$0148 & BL Lac & HSP & 0.08$\dagger$ & PL & —\
3C 66A & J0222.6$+$4302 & BL Lac & ISP & — & LP & Y\
RBS0413$^*$ & J0319.6$+$1849 & BL Lac & HSP & 0.19 & PL & Y\
NGC1275$^*$ & J0319.8$+$4130 & Radio Gal & ISP & 0.018 & LP & Y\
1ES0414$+$009 & J0416.8$+$0105 & BL Lac & — & 0.287 & PL & Y\
PKS0447$-$439 & J0449.4$-$4350 & BL Lac & — & 0.205 & PL & Y\
1ES0502$+$675$^*$ & J0508.0$+$6737 & BL Lac & HSP & 0.416 & PL & Y\
RGBJ0710$+$591 & J0710.5$+$5908 & BL Lac & HSP & 0.125$\dagger$ & PL & Y\
S50716$+$714 & J0721.9$+$7120 & BL Lac & ISP & 0.31$^c$$\dagger$ & LP & Y\
1ES0806$+$524 & J0809.8$+$5218 & BL Lac & HSP & 0.137097$\dagger$ & PL & Y\
1ES1011$+$496 & J1015.1$+$4925 & BL Lac & HSP & 0.212$\dagger$ & LP & Y\
1ES1101$-$232 & J1103.4$-$2330 & BL Lac & — & 0.186$\dagger$ & PL & Y\
Markarian421 & J1104.4$+$3812 & BL Lac & HSP & 0.031$\dagger$ & PL & Y\
Markarian180 & J1136.7$+$7009 & BL Lac & HSP & 0.046$\dagger$ & PL & Y\
1ES1215$+$303 & J1217.8$+$3006 & BL Lac & HSP & 0.13 & PL & Y\
1ES1218$+$304 & J1221.3$+$3010 & BL Lac & HSP & 0.18365$\dagger$ & PL & Y\
WComae & J1221.4$+$2814 & BL Lac & ISP & 0.102891$\dagger$ & PL & Y\
4C$+$21.35$^*$ & J1224.9$+$2122 & FSRQ & LSP & 0.433507 & LP & Y\
M87 & J1230.8$+$1224 & Radio Gal & LSP & 0.0036$\dagger$ & PL & Y\
3C279 & J1256.1$-$0547 & FSRQ & LSP & 0.536 & LP & Y\
CentaurusA & J1325.6$-$4300 & Radio Gal & — & 0.0008$^d$$\dagger$ & PL & Y\
PKS1424$+$240$^*$ & J1427.0$+$2347 & BL Lac & ISP & — & PL & Y\
H1426$+$428 & J1428.6$+$4240 & BL Lac & HSP & 0.129172$\dagger$ & PL & Y\
1ES1440$+$122 & J1442.7$+$1159 & BL Lac & HSP & 0.16309 & PL & Y\
PKS1510$-$089 & J1512.8$-$0906 & FSRQ & LSP & 0.36 & LP & Y\
APLib$^*$ & J1517.7$-$2421 & BL Lac & LSP & 0.048 & PL & Y\
PG1553$+$113 & J1555.7$+$1111 & BL Lac & HSP & — & PL & Y\
Markarian501 & J1653.9$+$3945 & BL Lac & HSP & 0.0337$\dagger$ & PL & Y\
1ES1959$+$650 & J2000.0$+$6509 & BL Lac & HSP & 0.047$\dagger$ & PL & Y\
PKS2005$-$489 & J2009.5$-$4850 & BL Lac & — & 0.071$\dagger$ & PL & Y\
PKS2155$-$304 & J2158.8$-$3013 & BL Lac & HSP & 0.116$\dagger$ & PL & Y\
BLLacertae & J2202.8$+$4216 & BL Lac & ISP & 0.0686$\dagger$ & LP & Y\
B32247$+$381 & J2250.0$+$3825 & BL Lac & HSP & 0.119 & PL & Y\
H2356$-$309 & J2359.0$-$3037 & BL Lac & HSP & 0.165$\dagger$ & PL & Y\
IC310 & J0316.6$+$4119 & Radio Gal & HSP & 0.018849 & PL & —\
VERJ0521$+$211$^*$ & J0521.7$+$2113 & BL Lac & ISP & — & PL & L\
VERJ0648$+$152$^{*,\star}$ & J0648.9$+$1516 & AGU & HSP & — & PL & —\
1RXSJ101015.9$-$311909 & J1009.7$-$3123 & BL Lac & HSP & 0.143 & PL & —\
MAGICJ2001$+$435$^*$ & J2001.1$+$4352 & BL Lac & ISP & — & PL & L\
1ES2344$+$514 & J2347.0$+$5142 & BL Lac & HSP & 0.044$\dagger$ & PL & L\
$^*$ Sources for which [*[Fermi]{}*]{} LAT data motivated the observations leading to their discovery at TeV energies. $^\dagger$ The sources used to make Figure \[fig:GeVTeVEBL\]. $^\star$ VERJ0648$+$152 is listed as an unidentified source in TeVCat. It is spatially consistent with the 2LAC AGN, 2FGLJ0648$+$1516. $^a$ The shape of the best fit spectrum: power-law (PL); LogParabola(LP). $^b$ Sources that are flagged with a “Y” were in the 1LAC Clean Sample; those with an “L” were in 1FGL but not in 1LAC due to their low Galactic latitude . All others were not in 1LAC. $^c$ The redshift assumed for this source is uncertain at $z = 0.31 \pm 0.08$ and is therefore not listed in 2LAC [@2009ApJ...704L.129A]. $^d$ The redshift is not in the 2LAC table because, as a member of the local group, the redshift does not provide a reliable estimate of its distance. @2007ApJ...654..186F used Cepheid variables to calculate its distance and derived a value of $3.42 \pm 0.18$ (random) $\pm 0.25$ (systematic) Mpc, which we converted to redshift of $z = 0.0008$, with the tool at this URL: http://www.astro.ucla.edu/$\sim$wright/CosmoCalc.html assuming the cosmological values quoted in § \[sec:intro\]. \[TAB:GeVTeV\]
[^1]: The Test Statistic is defined as $TS$ = 2(log $\mathcal{L}$(source)- log $\mathcal{L}$(nosource)), where $\mathcal{L}$ represents the likelihood of the data given the model with or without a source present at a given position on the sky,
[^2]: The Galactic diffuse model and isotropic background model (including the $\gamma$-ray diffuse and residual instrumental backgrounds) are described in [@2FGL]. Alternative Galactic diffuse models were tested as well.
[^3]: http://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/Cicerone/
[^4]: The VLBA Calibrator Source List can be downloaded from http://www.vlba.nrao.edu/astro/calib/vlbaCalib.txt.
[^5]: http://tevcat.uchicago.edu
[^6]: http://ned.ipac.caltech.edu/
[^7]: Although a relative deficit exists at intermediate northern Galactic latitudes, this is somewhat offset by blazars of unknown type.
[^8]: IC310 has two flags indicating that its $TS$ changed from $TS$ $>$ 35 to $TS$ $<$ 25 when the diffuse model was changed and that it lies on top of an interstellar gas clump or small-scale defect in the model of the diffuse emission. 1RXSJ101015.9$-$311909 has one flag indicating that when the diffuse model was changed, its position moved beyond the 95% error ellipse; see @2FGL for more details on flagged sources.
[^9]: VERJ0648$+$152 is spatially coincident with 1FGLJ0648.8$+$1516 and 2FGLJ0648.9$+$1516, and seems likely to be an AGN. It is not in the 2LAC Clean Sample due to its low Galactic latitude.
[^10]: The subclass of this source has not been confirmed but all available observations favor its classification as a HBL
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Classes of $(p,q)$-deformations of the Jaynes-Cummings model in the rotating wave approximation are considered. Diagonalization of the Hamiltonian is performed exactly, leading to useful spectral decompositions of a series of relevant operators. The latter include ladder operators acting between adjacent energy eigenstates within two separate infinite discrete towers, except for a singleton state. These ladder operators allow for the construction of $(p,q)$-deformed vector coherent states. Using $(p,q)$-arithmetics, explicit and exact solutions to the associated moment problem are displayed, providing new classes of coherent states for such models. Finally, in the limit of decoupled spin sectors, our analysis translates into $(p,q)$-deformations of the supersymmetric harmonic oscillator, such that the two supersymmetric sectors get intertwined through the action of the ladder operators as well as in the associated coherent states.'
---
ICMPA-MPA/2006/20\
CP3-06-13\
[ ]{}
[**$(p,q)$-Deformations and $(p,q)$-Vector Coherent States**]{}
[**of the Jaynes-Cummings Model**]{}
[**in the Rotating Wave Approximation**]{}
Joseph Ben Geloun$^{\dag}$, Jan Govaerts$^{\ddag,\dag,}$[^1] and M. Norbert Hounkonnou$^{\dag}$
$^{\dag}$[*International Chair in Mathematical Physics and Applications (ICMPA-UNESCO)*]{}\
[*072 B.P. 50 Cotonou, Republic of Benin*]{}\
[*E-mail: [[email protected], [email protected]]{}*]{}
$^{\ddag}$[*Department of Theoretical Physics, School of Physics*]{}\
[*The University of New South Wales, Sydney NSW 2052, Australia*]{}\
[*E-mail: [[email protected]]{}*]{}
Introduction {#Sect1}
============
In recent years, quantum algebras and groups [@maj] which appear as a generalization of the symmetry concept [@wes] and the basics of so-called noncommutative theories, have been the subject of intensive research interest in both mathematics and physics. The $q$- and more generally $(p,q)$-deformation of a pre-defined algebraic structure [@ar; @far; @cha] proves to be a powerful tool widely used in the representation theory of quantum groups. The field of “$q$-mathematics" has a long history [@jac; @ram] dating back to over 150 years, and includes several famous names such as Cauchy, Jacobi and Heine to mention just a few. Its possible relation to physics has been considerably reinforced during the last thirty years [@ar; @hex]. In particular, great attention has been devoted to deformations of the bosonic Fock-Heisenberg algebra. The most commonly studied deformed bosons, with annihilation and creation operators $a$ and $a^\dagger$, respectively, satisfy the $q$-commutation relation [@ar] (also called quommutation) $$aa^{\dagger} - q a^{\dagger}a=\mathbb{I},
\label{eq:q-def1}$$ or some variant forms of such a relation [@far; @vin]. Still more general deformations, which include in specific limits the above standard $q$-deformed case and which also provides consistent extensions of the harmonic oscillator algebra, proceed from the two parameter deformation of the Fock algebra introduced by Chakrabarty and Jagannathan [@cha], namely the so-called $(p,q)$-oscillator quantum algebras generated by three operators $a$, $a^{\dagger}$ and $N$ which obey [@cha; @vin2] $$\begin{aligned}
[N,a]=-a,\quad
[N,a^{\dag}]=a^{\dag}, \quad
aa^{\dag}-qa^{\dag}a=p^{-N},\quad
aa^{\dag}-p^{-1}a^{\dag}a=q^{N}.
\label{eq:pq-def1}\end{aligned}$$ Here, $p$ and $q$ are free parameters, which henceforth are chosen to be both real and such that $p>1$, $0<q<1$ and $pq<1$. Clearly, one recovers the ordinary Fock algebra of the harmonic oscillator algebra in the double limit $p,q \to 1$, with then $[a,a^\dagger]=\mathbb{I}$ and $N=a^\dagger a$. Furthermore, these $q$- and $(p,q)$-deformed algebras have found a number of relevant applications and provide algebraic interpretations of various $q$- and $(p,q)$-special functions [@vin; @vin2; @koe].
The harmonic oscillator algebra is central in the construction of a number of models in physics, among which the Jaynes–Cummings model (${\cal JC}m$) plays a significant role. Indeed ever since Jaynes and Cummings’ historical work [@jc], the ${\cal JC}m$ has been at the basis of many investigations. This system belongs to a class of physically relevant models widely used in atomic physics and quantum optics. As far as we know, a great deal of analytically solvable models of this type have been studied in the rotating wave approximation (r.w.a.) within the framework of non-deformed commutative theories (see [@jc]–[@hus4] and references therein). The ${\cal JC}m$ has also been considered in the context of generalized intensity dependent oscillator algebras including nonlinear dynamical supersymmetry [@daou] or using shape invariance techniques [@bal1; @bal2]. Comparatively, much fewer papers have dealt with generalizations of these models including deformations. Among the latter and mainly based on the generalized intensity-dependent coupling of Buck and Sukumar [@buk], one may mention, on the one hand, the work by Chaichan [*et al.*]{} [@chai], and on the other hand, that by Chang [@chan], both dealing with a generalized $q$-deformed intensity-dependent interaction Hamiltonian of the ${\cal JC}m$ given by the Holstein-Primakoff $su_q(1,1)$ or $su_q(2)$ quantum algebra realizations of the Hamiltonian field operators and the related Peremolov, Glauber or Barut-Girardello group theoretical construction of coherent states. In the same vein, the paper by Naderi [*et al.*]{} considers the dynamical properties of a two-level atom in three variants of the two-photon $q$-deformed ${\cal JC}m$ [@nad]. In this latter work, the authors focused their attention onto the time evolution of atomic properties including population inversion and quantum fluctuations of the atomic dipole variables. However, it is not clear to us how the main issues related to the moment problem as well as the mathematical foundation of the coherent and squeezed states which they use and on which a great part of their analysis rests in a crucial way, are solved.
In a recent publication [@hus1], Hussin and Nieto have performed an interesting systematic search of different types of ladder operators for the ${\cal JC}m$ model in the r.w.a. and constructed associated coherent states. In the present work, and in line with that investigation, we provide a generalization of that analysis to $(p,q)$-deformations of the same model.
The outline of the paper is the following. In Section \[Sect2\], we briefly recall the main results relevant to the ${\cal JC}m$ in the r.w.a. in the non-deformed situation [@hus1]. Section \[Sect3\] then introduces $(p,q)$-deformations of the same model. By providing an explicit diagonalization of the $(p,q)$-deformed Hamiltonian, the spectrum and its eigenstates are exactly identified. As in the non-deformed case [@hus1], except for a singleton state, all other energy eigenstates are organized into two separate discrete towers, for which ladder operators transforming states into one another within each tower separately may be introduced. Using properties of these ladder operators, in Section \[Sect4\] we introduce general classes of $(p,q)$-deformed vector coherent states. The freedom afforded in their construction is fixed from two alternative points of view, discussed in Section \[Sect5\], which in the ordinary case of the non-deformed Fock algebra coincide. However at all stages of our discussion, the double limit $p,q\to 1$ reproduces the corresponding results of [@hus1]. Section \[Sect5\] also briefly considers the situation in the uncoupled limit of the ${\cal JC}m$, while Section \[Sect6\] presents some concluding remarks. An Appendix collects useful facts in connection with properties of $(p,q)$-deformed algebras and related functions.
The Ordinary ${\cal JC}m$ in the Rotating Wave Approximation {#Sect2}
============================================================
The ${\cal JC}m$ describes the interaction between one mode of the quantized electromagnetic field and a two-level model of an atomic system [@jc; @hus1]–[@hus3]. It has proved to be a theoretical laboratory of great relevance to many topics in atomic physics and quantum optics, as well as in the study of ion traps, cavity QED theory and quantum information processing [@mew; @hus1]. Furthermore, the spin-orbit interaction term which appears in the ${\cal JC}m$ is essentially the so-called Dresselhaus spin-orbit term [@dr]. The model is thus also widely used in condensed matter physics for its relevance in spintronics [@js] which exploits the electron spin rather than its charge to develop a new generation of electronic devices [@qui1; @qui2]. The solution of the complete ${\cal JC}m$ is not yet known in a closed form [@hus1]. However, in the r.w.a., although the Hamiltonian remains nonlinear, the model becomes exactly solvable in closed form with explicit expressions for its eigenenergy states. In this Section, we briefly recall, in a streamlined presentation, the main results in the non-deformed case (see [@hus1; @hus2] and references therein) of relevance to our analysis of $(p,q)$-deformations hereafter.
In the r.w.a., the reduced dimensionless ${\cal JC}m$ Hamiltonian reads [@hus2] $$\begin{aligned}
\label{h}
{\cal H}^{\rm red}=\frac{1}{\hbar \omega_0}{\cal H}=
\left(1 + \epsilon\right)\left(a^\dagger a + \frac{1}{2}\right) +
\frac{1}{2}\sigma_{3}+\lambda\left(a^\dagger\sigma_{-} +a \sigma_{+}\right),\end{aligned}$$ where $a$ and $a^\dagger$ are the usual photon annihilation and creation operators, respectively, obeying the ordinary Fock algebra, and $(\sigma_1,\sigma_2,\sigma_3)$ are the Pauli matrices with $\sigma_\pm= \sigma_1 \pm i\sigma_2$. The r.w.a. is related to the detuning parameter $\epsilon$ which is such that $|\epsilon|\ll 1$, with $\omega_0$ being the fixed atomic frequency and $\omega=\omega_0(1+\epsilon)$ the actual field mode frequency. The r.w.a. is reliable provided $|\omega - \omega_0|\ll \omega,\omega_0$. Finally, $\lambda$ is the reduced spin-orbit coupling modelling the interaction strength between the radiation field and the atom.
The Hilbert space ${\cal V}$ of the system is the tensor product of the Fock space representation of the Fock algebra $(a,a^\dagger)$ and the 2-dimensional representation of the SU(2) algebra associated to the Pauli matrices. A basis of the former is provided by the number operator, $N=a^\dagger a$, orthonormalized eigenstates $|n\rangle=(1/\sqrt{n!})(a^\dagger)^n|0\rangle$ ($n=0,1,2,\cdots$), with $a|n\rangle=\sqrt{n}|n-1\rangle$, $a^\dagger|n\rangle=\sqrt{n+1}|n+1\rangle$ and $N|n\rangle=n|n\rangle$, while a basis of the latter spin sector is the orthonormalized set $\{|+\rangle,|-\rangle\}$ such that $\sigma_3|\pm\rangle=\pm|\pm\rangle$. The tensor product space is thus spanned by the states $|n,\pm\rangle=|n\rangle\otimes|\pm\rangle$.
The diagonalization of the Hamiltonian (\[h\]) is readily achieved. The orthonormalized energy eigenspectrum consists of a “singleton" state $|E_*\rangle$, $${\cal H}^{\rm red}|E_*\rangle= E_*|E_*\rangle,$$ with $$E_*=\frac{1}{2}\epsilon,\qquad
|E_*\rangle=|0,-\rangle ,$$ and two infinite discrete towers of states $|E_n^\pm\rangle$ such that ${\cal H}^{\rm red}|E^\pm_n\rangle=E^\pm_n|E^\pm_n\rangle$ for all $n=0,1,2,\cdots$, expressed as [@hus1] $$\begin{aligned}
\label{ei}
|E^+_{n}\rangle&=&\sin\vartheta(n)\, |n,+\rangle + \cos\vartheta(n)\, |n+1,-\rangle ,\\
|E^-_{n}\rangle&=&\cos\vartheta(n)\, |n,+\rangle - \sin\vartheta(n)\, |n+1,-\rangle,\end{aligned}$$ where, given $Q(n+1)=\sqrt{\epsilon^{2}/4 + \lambda^2(n+1)}$, the mixing angle $\vartheta(n)$ is such that $$\begin{aligned}
\label{de}
\sin\vartheta(n)= {\rm sign}(\lambda)
\sqrt{\frac{Q(n+1) - \epsilon/2}{2Q(n+1)}},\qquad
\cos\vartheta(n)= \sqrt{\frac{Q(n+1) + \epsilon/2}{2Q(n+1)}},\end{aligned}$$ while the energy eigenvalues are $$\begin{aligned}
\label{ev}
E^{\pm}_{n} = (1+ \epsilon)(n+1) \pm Q(n+1).\end{aligned}$$ Consequently, one has the spectral decomposition of the reduced Hamiltonian (\[h\]), $${\cal H}^{\rm red}=|E_*\rangle\,E_*\,\langle E_*|\ +\
\sum_{n=0,\pm}^\infty\,|E^\pm_n\rangle\,E^\pm_n\,\langle E^\pm_n| .$$
It proves useful to introduce the following notations. Let ${\cal V}_0$ be the (complex) one-dimensional subspace of the Hilbert space ${\cal V}$ spanned by the state $|0,-\rangle=|E_*\rangle$, and $\overline{\cal V}$ be its complement in the Hilbert space ${\cal V}$, spanned by $\{|E^\pm_n\rangle, n\in\mathbb{N}\}$. We thus have ${\cal V}={\cal V}_0\oplus\overline{\cal V}$.
Furthermore let us introduce [@hus1] operators ${\cal U}$ and ${\cal U}^\dagger$ defined through their action on the above two sets of basis vectors, for all $n\in\mathbb{N}$, $${\cal U}|n,\pm\rangle = |E^\pm_n\rangle ;\qquad
{\cal U}^\dagger|E_*\rangle=0,\quad
{\cal U}^\dagger|E^\pm_n\rangle=|n,\pm\rangle,$$ namely $${\cal U}=\sum_{n=0,\pm}^\infty |E^\pm_n\rangle\langle n,\pm|,\qquad
{\cal U}^\dagger=\sum_{n=0,\pm}^\infty |n,\pm\rangle\langle E^\pm_n|.$$ Clearly we have $${\cal U}\,{\cal V}=\overline{\cal V};\qquad
{\cal U}^\dagger\,{\cal V}={\cal V},\quad
{\cal U}^\dagger\,\overline{\cal V}={\cal V}.$$ Note that even though neither ${\cal U}$ nor ${\cal U}^\dagger$ is unitary on the full Hilbert space ${\cal V}$, they are the adjoint of one another, hence the notation.
It is of interest to apply these operators onto the quantum Hamiltonian (\[h\]). One obtains $$\mathbb{H}^{\rm red}={\cal U}^\dagger\,{\cal H}^{\rm red}\,{\cal U}=
\sum_{n=0,\pm}^\infty\,|n,\pm\rangle\,E^\pm_n\,\langle n,\pm|,$$ and conversely, $${\cal U}\,\mathbb{H}^{\rm red}\,{\cal U}^\dagger=
\sum_{n=0,\pm}^\infty |E^\pm_n\rangle\,E^\pm_n\,\langle E^\pm_n|=
{\cal H}^{\rm red}\,-\,|E_*\rangle\,E_*\,\langle E_*|.$$
The energy eigenstates spanning $\overline{\cal V}$ may be organized into two subspaces referred to as “towers", namely $\left\{|E^+_n\rangle, n\in\mathbb{N}\right\}$ and $\left\{|E^-_n\rangle, n\in\mathbb{N}\right\}$. The states in the tower $\left\{|E^+_n\rangle, n\in\mathbb{N}\right\}$ are associated to strictly increasing eigenvalues so that they constitute a nondegenerate set of eigenstates. The second group does not necessarily possess the same feature depending on the values for the parameters $\lambda$ and $\epsilon$. It is possible [@hus3] to identify a range of values for these parameters such that $\left\{|E^-_n\rangle, n\in\mathbb{N}\right\}$ only contains nondegenerate states of strictly increasing eigenvalues with $n$. Some of the considerations discussed hereafter may require a nondegenerate spectrum, which may always be achieved by properly “detuning" the parameters $\lambda$ and $\epsilon$ away from a degenerate case, but not necessarily a strictly increasing spectrum in the label $n\in\mathbb{N}$. Whatever the case may be though, bounded from below spectra such that $E^\pm_n>E^\pm_0$ for $n=1,2,\cdots$ are always assumed implicitly.
It is possible to consider ladder operators acting between successive energy eigenstates within each of the above two towers, irrespective of whether the spectral values are strictly increasing or not[^2]. Namely, let us first consider operators $\mathbb{M}^-$ and $\mathbb{M}^+$ given as $$\mathbb{M}^-=\sum_{n=0,\pm}^\infty|n-1,\pm\rangle\,K_\pm(n)\,\langle n,\pm|;\qquad
\mathbb{M}^+=\sum_{n=0,\pm}^\infty|n+1,\pm\rangle\,K^*_\pm(n+1)\,\langle n,\pm|,$$ where $K_\pm(n)$ are, at this stage, arbitrary complex coefficients such that $K_\pm(0)=0$. Then, introduce the ladder operators $${\cal M}^-={\cal U}\,\mathbb{M}^-\,{\cal U}^\dagger=
\sum_{n=0,\pm}^{\infty}|E^\pm_{n-1}\rangle\,K_\pm(n)\,\langle E^\pm_n|;\quad
{\cal M}^+={\cal U}\,\mathbb{M}^+\,{\cal U}^\dagger=
\sum_{n=0,\pm}^{\infty}|E^\pm_{n+1}\rangle\,K^*_\pm(n+1)\,\langle E^\pm_n|,$$ which are thus such that, for all $n=0,1,2,\cdots$, $${\cal M}^-|E_*\rangle=0,\quad
{\cal M}^-|E^\pm_n\rangle=K_\pm(n)|E^\pm_{n-1}\rangle;\quad
{\cal M}^+|E_*\rangle=0,\quad
{\cal M}^+|E^\pm_n\rangle=K^*_\pm(n+1)|E^\pm_{n+1}\rangle .
\label{eq:ladder1}$$ Note that ${\cal M}^-$ and ${\cal M}^+$ are adjoint of one another but in effect only act on the subspace $\overline{\cal V}$.
General vector coherent states (VCS) may then be introduced [@al]–[@jp2] on the space $\overline{\cal V}$ as eigenstates of the lowering operator ${\cal M}^-$ with as eigenvalue an arbitrary complex number $z\in\mathbb{C}$. Furthermore, these VCS are also parametrized by two real quantities $\tau_\pm$ which account for their stability under time evolution generated by the operator $\exp\left\{-i\omega_0 t\,{\cal H}^{\rm red}\right\}$, as well as the two spherical coordinates $(\theta,\phi)\in[0,\pi]\times[0,2\pi[$ parametrizing a unit vector in the 2-sphere $S_2$ (hence the name of “vector" coherent states). Explicitly, one has [@hus1] $$\begin{aligned}
\label{cohp}
|z;\tau_\pm;\theta,\phi\rangle&=&
\ \ \ N^+(|z|)\cos\theta \sum_{n=0}^\infty
\frac{z^n}{K_+(n)!} e^{-i \omega_0\tau_+ E^+_n}\,|E^+_n\rangle \cr
& & + \ N^-(|z|)\, e^{i\phi}\sin\theta \sum_{n=0}^\infty
\frac{z^n}{K_-(n)!} e^{-i \omega_0\tau_- E^-_n}\,|E^-_n\rangle ,\end{aligned}$$ where $K_\pm(n)!=\prod_{k=1}^n K_\pm(k)$ (with, by convention, $K_\pm(0)!=1$), while the normalization factors are defined as $$N^{\pm}(|z|)= \left[\sum_{n=0}^\infty
\frac{|z|^{2n}}{|K_\pm(n)!|^2}\right]^{-1/2}$$ in order that the VCS be of unit norm. The smallest value, $R$, of the two convergence radii of these two series in $|z|$ also defines the disk $D_R$ in $z\in\mathbb{C}$ for which these VCS are well defined. These states are clearly such that $${\cal M}^-|z;\tau_\pm;\theta,\phi\rangle = z\, |z;\tau_\pm;\theta,\phi\rangle ,\quad
e^{-i\omega_0 t\,{\cal H}^{\rm red}}\,|z;\tau_\pm;\theta,\phi\rangle=|z;t+\tau_\pm;\theta,\phi\rangle.$$
Further restrictions are necessary to finally specify in a unique fashion the factors $K_\pm(n)$, and then solve the moment problem implied by the requirement of overcompleteness over $\overline{\cal V}$ for the VCS (\[cohp\]) given a choice of a SU(2) matrix-valued integration measure over $\mathbb{C}\times S_2$ [@kl]-[@jp2]. Different choices are available [@hus1], each leading to a different set of VCS. Furthermore, taking the limit case $\lambda \to 0$ or the zero-detuning limit (resonance case) $\epsilon \to 0$, different models arise with their associated VCS.
For the sake of illustration, let us consider one such choice explicitly [@hus1]. The factors $K_\pm(n)$ may be restricted for example by requiring that the ladder operators ${\cal M}^-$ and ${\cal M}^+$ obey the usual Fock algebra of annihilation and creation operators on the space $\overline{\cal V}$, $$\left[{\cal M}^-,{\cal M}^+\right]=
{\cal M}^-\,{\cal M}^+\,-\,{\cal M}^+\,{\cal M}^-\,=\,\mathbb{I}_{\overline{\cal V}}
=\sum_{n=0,\pm}^\infty\,|E^\pm_n\rangle\,\langle E^\pm_n|.$$ From the expressions in (\[eq:ladder1\]) and the initial conditions $K_\pm(0)=0$, it follows that the quantities $K_\pm(n)$ are now determined up to arbitrary phase factors $\varphi_\pm(n)$ as $$K_\pm(n)=e^{i\varphi_\pm(n)}\,\sqrt{n},\qquad n=0,1,2,\cdots .$$ Consequently, one has $N^\pm(|z|)=e^{-|z|^2/2}$, which is well-defined for all $z\in\mathbb{C}$. Hence so are then all the VCS $|z;\tau_\pm;\theta,\phi\rangle$.
The $(p,q)$-Deformed ${\cal JC}m$ in the Rotating Wave Approximation {#Sect3}
====================================================================
Let us now introduce a $(p,q)$-deformation of the ${\cal JC}m$ Hamiltonian (\[h\]), namely $(p,q)$-${\cal JC}m$ models. The eigenstates and spectrum are first identified, before considering the construction of ladder operators following the same rationale as in Section \[Sect2\]. A study of the associated VCS and examples of exactly solvable reduced models is differed to Section \[Sect4\].
Energy spectrum and eigenstates {#Subsect3.1}
-------------------------------
Given the $(p,q)$-deformation (\[eq:pq-def1\]) of the ordinary Fock algebra (see the Appendix for further details and identities pertaining to such deformations), we now consider $(p,q)$-deformations of the Hamiltonian (\[h\]) of the form[^3] $$\begin{aligned}
{\cal H}^{red}= \left(1+\epsilon\right)\left\{h(p,q)[N] + \frac{1}{2} \right\}
+ \frac{1}{2}\sigma_3 + \lambda\left(a^\dagger\sigma_- + a \sigma_+\right),
\label{eq:hq}\end{aligned}$$ where $[N]=(p^{-N}-q^N)/(p^{-1}-q)$, and $h(p,q)$ is some arbitrary positive function of the real parameters $p>1$ and $0<q<1$ (with $pq<1$) such that $\lim_{p,q\to 1}h(p,q)=1$ in order to recover (\[h\]) in the non-deformed case.
The Hilbert space ${\cal V}$ of quantum states of the model is again the tensor product of the $(p,q)$-deformed Fock space spanned by the states[^4] $|n\rangle$ ($n\in\mathbb{N}$) such as $a|n\rangle=\sqrt{[n]}|n-1\rangle$ and $a^\dagger|n\rangle=\sqrt{[n+1]}|n+1\rangle$ (see the Appendix), with the 2-dimensional representation of the SU(2) algebra associated to the Pauli matrices $\sigma_i$ ($i=1,2,3$). Hence the diagonalization of (\[eq:hq\]) is readily achieved in the same way as in the non-deformed case, on the basis $|n,\pm\rangle=|n\rangle\otimes|\pm\rangle$ of ${\cal V}$.
For any $n\in\mathbb{N}$, let us introduce the following quantities, $${\cal E}([n+1])=\left(1+\epsilon\right)h(p,q)\Big([n+1]-[n]\Big)-1,\quad
Q([n+1])=\sqrt{\frac{1}{4}{\cal E}^2([n+1])\,+\,\lambda^2\,[n+1]},
\label{eq:EQ}$$ as well as the mixing angles $\vartheta([n])$ defined by $$\sin\vartheta([n])={\rm sign}(\lambda)
\sqrt{\frac{Q([n+1])-{\cal E}([n+1])/2}{2Q([n+1])}},\quad
\cos\vartheta([n])=
\sqrt{\frac{Q([n+1])+{\cal E}([n+1])/2}{2Q([n+1])}}.
\label{eq:mixang}$$ The energy eigenspectrum of (\[eq:hq\]) is then obtained as follows. First, there exists a singleton state $|E_*\rangle=|0,-\rangle$ such that $${\cal H}^{\rm red}\,|E_*\rangle=E_*\,|E_*\rangle,\qquad
E_*=\frac{1}{2}\epsilon,$$ with an eigenvalue which is thus independent of the deformation parameters $p$ and $q$. Next, one also finds two infinite discrete towers of states for all $n\in\mathbb{N}$ such that $$\begin{aligned}
|E^+_n\rangle &=& \sin\vartheta([n])\,|n,+\rangle\,+\,\cos\vartheta([n])\,|n+1,-\rangle ,\\
|E^-_n\rangle &=& \cos\vartheta([n])\,|n,+\rangle\,-\,\sin\vartheta([n])\,|n+1,-\rangle ,\end{aligned}$$ with $${\cal H}^{\rm red}\,|E^\pm_n\rangle= E^\pm_n\,|E^\pm_n\rangle ,\quad
E^\pm_n=\frac{1}{2}\left(1+\epsilon\right)
\Big\{h(p,q)\Big([n+1]+[n]\Big)+1\Big\}\,\pm\,Q([n+1]).
\label{eq:pqev}$$ Note that the energy spectrum of these states is deformed by the parameters $p$ and $q$ as compared to the ordinary case. In particular, the Zeeman spin splitting $\Delta E_n=E^+_n-E^-_n=2Q([n+1])$, proportional to the Rabi frequency, is function of the values for $p$ and $q$. In terms of these results, the reduced Hamiltonian (\[eq:hq\]) possesses the spectral resolution $${\cal H}^{\rm red}=|E_*\rangle\,E_*\,\langle E_*|\ +\,
\sum_{n=0,\pm}^\infty\,|E^\pm_n\rangle\,E^\pm_n\,\langle E^\pm_n| .$$
Let us again introduce the following notations and operators. Let ${\cal V}_0$ denote the subspace of the Hilbert space ${\cal V}$ spanned by the singleton state $|E_*\rangle=|0,-\rangle$, and $\overline{\cal V}$ its complement in ${\cal V}$, namely the subspace spanned by $\{|E^\pm_n\rangle,n\in\mathbb{N}\}$, with of course ${\cal V}={\cal V}_0\oplus\overline{\cal V}$. Acting on these spaces, let us consider the operators $${\cal U}=\sum_{n=0,\pm}^\infty\,|E^\pm_n\rangle\,\langle n,\pm|;\qquad
{\cal U}^\dagger=\sum_{n=0,\pm}^\infty\,|n,\pm\rangle\,\langle E^\pm_n|,$$ such that, for all $n=0,1,2,\cdots$, $${\cal U}|n,\pm\rangle=|E^\pm_n\rangle ;\quad
{\cal U}^\dagger|E_*\rangle=0,\quad
{\cal U}^\dagger|E^\pm_n\rangle=|n,\pm\rangle,$$ and thus $${\cal U}\,{\cal V}=\overline{\cal V};\quad
{\cal U}^\dagger\,{\cal V}={\cal V},\quad
{\cal U}^\dagger\,\overline{\cal V}={\cal V}.$$ Hence once again the operators ${\cal U}$ and ${\cal U}^\dagger$, even though non unitary on ${\cal V}$, are adjoint of one another. More specifically, one has $${\cal U}^\dagger\,{\cal U}=\sum_{n=0,\pm}^\infty\,|n,\pm\rangle\,\langle n,\pm|=\mathbb{I}_{\cal V},\qquad
{\cal U}\,{\cal U}^\dagger=\sum_{n=0,\pm}^\infty\,|E^\pm_n\rangle\,\langle E^\pm_n|=\mathbb{I}_{\overline{\cal V}}.$$
Applying these operators to the reduced Hamiltonian, one finds $$\mathbb{H}^{\rm red}={\cal U}^\dagger\,{\cal H}^{\rm red}\,{\cal U}=
\sum_{n=0,\pm}^\infty\,|n,\pm\rangle\,E^\pm_n\,\langle n,\pm|,
\label{eq:hpq}$$ and conversely, $${\cal U}\,\mathbb{H}^{\rm red}\,{\cal U}^\dagger=
\sum_{n=0,\pm}^\infty\,|E^\pm_n\rangle\,E^\pm_n\,\langle E^\pm_n|=
{\cal H}^{\rm red}\,-\,|E_*\rangle\,E_*\,\langle E_*| .$$
Some remarks on the spectrum are in order. First, as in the ordinary ${\cal JC}m$, except for the singleton state $|E_*\rangle=|0,-\rangle$, the spectrum is the direct sum of two towers of states $\{|E^\pm_n\rangle, n\in\mathbb{N}\}$. However, in contradistinction to the non-deformed case or even the $q$-deformation with $p=1$, the $(p,q)$-basic numbers $[n]=[n]_{(p,q)}$ are not strictly increasing as a function of $n \in {\mathbb{N}}$ when $p>1$, $0<q<1$ and $pq<1$. There always exists a finite positive value $n_0\in\mathbb{N}$ such that $[n]$ decreases once $n>n_0$. Hence, depending on the values for the parameters $\lambda$ and $\epsilon$ as well as the positive function $h(p,q)$, parts of the spectrum $E^\pm_n$ may turn negative or present some degeneracies (as in [@hus3]). Without exploring this issue any further in the present work, henceforth we shall assume that parameter values are such that no degeneracies occur and that the spectrum $E^-_n$ remains bounded from below ($E^+_n$ is obviously positive). The definition of the ladder operators to be considered next does not require a strictly increasing spectrum, while it is only for one of possible choices leading to vector coherent states to be discussed hereafter that the condition of non degeneracy in $E^\pm_n>E^\pm_0$, for $n\ge 1$, becomes relevant. Since it has been shown [@hus3] that such conditions may be met in the non-deformed case for appropriate ranges of values for the available parameters, through an argument of continuity in the deformation parameters $p$ and $q$, similar ranges ought to exist also for the $(p,q)$-deformed realizations of the ${\cal JC}m$ model.
Another feature of potential interest related to these facts, and which will also not be pursued here, is the possibility that through the $(p,q)$-deformation of the ${\cal JC}m$, the levels $E^+_n$ and $E^-_{n+1}$ cross one another. Such a property may lead to effects similar to the phenomenon of resonant spin-Hall conductance at the Fermi level recently observed in spintronics [@qui1; @qui2]. Note that this $(p,q)$-dependent crossing phenomenon is expected since the Zeeman splitting $\Delta E_n$ is also modified as a function of $p$ and $q$. This remark is also in line with the recent suggestion [@Scholtz1; @Scholtz2; @JBG1] that $(p,q)$-deformed or space noncommutative realizations of exactly solvable systems may provide useful model approximations to more realistic complex interacting dynamics of collective phenomena.
Ladder operators {#Subsect3.2}
----------------
In order to construct ladder operators mapping each of the successive states $|E^\pm_n\rangle$ into one another separately within each of the towers, let us first introduce the following operators acting on ${\cal V}$, $$\mathbb{A}^-=\sum_{n=0,\pm}^\infty\,|n-1,\pm\rangle\,K_\pm([n])\,
\langle n,\pm|;\quad
\mathbb{A}^+=\sum_{n=0,\pm}^\infty\,|n+1,\pm\rangle\,K^*_\pm([n+1])\,
\langle n,\pm|,$$ where $K_\pm([n])$ are arbitrary complex quantities such that $K_\pm([0])=K_\pm(0)=0$. Note that $\mathbb{A}^-$ and $\mathbb{A}^+$ are adjoint of one another on ${\cal V}$.
Then the relevant ladder operators are obtained as $${\cal A}^-={\cal U}\,\mathbb{A}^-\,{\cal U}^\dagger=
\sum_{n=0,\pm}^\infty\,|E^\pm_{n-1}\rangle\,K_\pm([n])\,\langle E^\pm_n|;\quad
{\cal A}^+={\cal U}\,\mathbb{A}^+\,{\cal U}^\dagger=
\sum_{n=0,\pm}^\infty\,|E^\pm_{n+1}\rangle\,K^*_\pm([n+1])\,\langle E^\pm_n|.$$ Consequently, we have indeed, for all $n\in\mathbb{N}$, $${\cal A}^-|E_*\rangle=0,\quad
{\cal A}^-|E^\pm_n\rangle=K_\pm([n])\,|E^\pm_{n-1}\rangle;\qquad
{\cal A}^+|E_*\rangle=0,\quad
{\cal A}^+|E^\pm_n\rangle=K^*_\pm([n+1])\,|E^\pm_{n+1}\rangle.$$ Note that ${\cal A}^-$ and ${\cal A}^+$ are adjoint of one another, but that in effect they act only on the subspace $\overline{\cal V}$.
It is of course possible to express these ladder operators in the $|n,\pm\rangle$ basis. In the case of the lowering operator, one finds $$\begin{array}{rclcl}
{\cal A}^-&=& \ \ \ \ \sum_{n=0}^\infty\,|n,+\rangle\,{\cal A}^-_{++}(n)\,\langle n+1,+| \ &+&\
\sum_{n=0}^\infty\,|n,+\rangle\,{\cal A}^-_{+-}(n)\,\langle n+2,-| \\
&& && \\
&& +\ \sum_{n=0}^\infty\,|n,-\rangle\,{\cal A}^-_{-+}(n)\,\langle n,+| \ &+&\
\sum_{n=0}^\infty\,|n,-\rangle\,{\cal A}^-_{--}(n)\,\langle n+1,-|
\end{array}$$ where $$\begin{aligned}
{\cal A}^-_{++}(n)&=&\sin\vartheta([n])\,\sin\vartheta([n+1])\,K_+([n+1])\,+\,
\cos\vartheta([n])\,\cos\vartheta([n+1])\,K_-([n+1]),\cr
& & \cr
{\cal A}^-_{+-}(n)&=&\sin\vartheta([n])\,\cos\vartheta([n+1])\,K_+([n+1])\,-\,
\cos\vartheta([n])\,\sin\vartheta([n+1])\,K_-([n+1]),\cr
& & \cr
{\cal A}^-_{-+}(n)&=&\cos\vartheta([n-1])\,\sin\vartheta([n])\,K_+([n])\,-\,
\sin\vartheta([n-1])\,\cos\vartheta([n])\,K_-([n]),\cr
& & \cr
{\cal A}^-_{--}(n)&=&\cos\vartheta([n-1])\,\cos\vartheta([n])\,K_+([n])\,+\,
\sin\vartheta([n-1])\,\sin\vartheta([n])\,K_-([n]).\end{aligned}$$ Likewise for the raising operator, $$\begin{array}{rclcl}
{\cal A}^+&=& \ \ \ \ \sum_{n=0}^\infty\,|n+1,+\rangle\,\left({\cal A}^{-}_{++}(n)\right)^*\,\langle n,+| \ &+&\
\sum_{n=0}^\infty\,|n,+\rangle\,\left({\cal A}^{-}_{-+}(n)\right)^*\,\langle n,-| \\
&& && \\
&& +\ \sum_{n=0}^\infty\,|n+2,-\rangle\,\left({\cal A}^{-}_{+-}(n)\right)^*\,\langle n,+| \ &+&\
\sum_{n=0}^\infty\,|n+1,-\rangle\,\left({\cal A}^{-}_{--}(n)\right)^*\,\langle n,-|.
\end{array}$$ Note that we have ${\cal A}^-_{-+}(0)=0={\cal A}^-_{--}(0)$, since $K_\pm([0])=0$.
The quantities $K_\pm([n])$ parametrize the freedom available in the choice of such ladder operators. Further restrictions arise when considering first the possible existence of vector coherent states meeting a series of general conditions charateristic of such states [@kl]-[@jp2], starting with one involving the lowering operator ${\cal A}^-$ itself.
$(p,q)$-Vector Coherent States for the $(p,q)$-${\cal JC}m$ {#Sect4}
===========================================================
By considering the action of the lowering operator ${\cal A}^-$, we are able to construct an overcomplete set of vectors in $\overline{\cal V}$, so-called vector coherent states [@kl]-[@jp2] for the $(p,q)$-${\cal JC}m$. Since these states are associated to unit vectors in the 2-sphere $S_2$ [@al], they are referred to as $(p,q)$-vector coherent states ($(p,q)$-VCS). As in Section \[Sect2\], these $(p,q)$-VCS are parametrized by a complex variable $z\in\mathbb{C}$, two real parameters $\tau_\pm$ to track a stable time evolution of the $(p,q)$-VCS, and finally the spherical angle coordinates $(\theta,\phi)$ on $S_2$, $|z;\tau_\pm;\theta,\phi\rangle$. In the double limit that $p,q\to 1$, these $(p,q)$-VCS reduce to those of [@hus1] discussed in Section \[Sect2\]. The dependence of the $(p,q)$-VCS on all these quantities is introduced as follows, according to the discussion in [@kl].
Identifying $(p,q)$-VCS {#Subsect4.1}
-----------------------
As a slight extension of the analysis so far, given two real parameters $\mu$ and $\nu$, let us consider the operator $$\mathbb{Q}_{\cal V}=|E_*\rangle\,\langle E_*|\ +\
\sum_{n=0,\pm}^\infty\,|E^\pm_n\rangle\,
\left(\frac{q^{\mu}}{p^{\nu}}\right)^{n}
\,\langle E^\pm_n|.$$ Hence, the energy eigenstates of the $(p,q)$-${\cal JC}m$ are also eigenstates of this operator $\mathbb{Q}_{\cal V}$, with eigenvalues given through the above spectral decomposition.
We are now in a position to successively identify the dependence of the $(p,q)$-VCS to be constructed on each of the parameters of which they are functions, first $z$, then $\tau_\pm$, and finally, $\theta$ and $\phi$. Having defined both the operators ${\cal A}^-$ and $\mathbb{Q}_{\cal V}$, let us consider the following eigenvalue problem in $z$ for the $(p,q)$-VCS, $$\begin{aligned}
\label{coh}
{\cal A}^-|z;\tau_\pm;\theta,\phi\rangle =
z\,\mathbb{Q}_{\cal V}\,|z;\tau_\pm;\theta,\phi\rangle\end{aligned}$$ which generalizes to a two-level system the definition of coherent states as advocated in [@kl]-[@jp2]. The particular case $\mu=0=\nu$ yields also a consistent definition of $(p,q)$-VCS viewed as the limit $\mu,\nu \to 0$ of the present definition (note that their domain of definition in $z$, required for the convergence of the infinite series to be considered hereafter, may have to be adapted accordingly).
By expanding the $(p,q)$-VCS in the Hamiltonian eigenstate basis as $$\begin{aligned}
\label{ser}
|z;\tau_\pm;\theta,\phi\rangle=
C_*(z)|E_*\rangle + \sum_{n=0,\pm}^\infty\,C^\pm_n(z)|E^\pm_n\rangle,\end{aligned}$$ where $C_*(z)$ and $C^\pm_n(z)$ are complex continuous functions of $z$ to be specified presently, the condition (\[coh\]) then requires, for all $n\in\mathbb{N}$, $$\begin{aligned}
\label{cc}
C_*(z)=0,\qquad
C^\pm_{n+1}(z)K_\pm([n+1]) = z\,\frac{q^{\mu n}}{p^{\nu n}}\,C^\pm_n(z),\end{aligned}$$ of which the solution is $$\begin{aligned}
C^\pm_n(z)= \left(\frac{q^{\mu}}{p^{\nu}}\right)^{n(n-1)/2}\,
\frac{z^n}{K_\pm([n])!}\,C^\pm_0(z),\end{aligned}$$ where $C^\pm_0(z)$ are arbitrary complex functions of $z$, while we defined $K_\pm([n])!=\prod_{k=1}^n K_\pm([k])$ with, by convention, $K_\pm([0])!=1$. Hence, the general solution to (\[coh\]) defines states lying only within the subspace $\overline{\cal V}$, of the form $$\begin{aligned}
\label{coh2}
|z;\tau_\pm;\theta,\phi\rangle=\sum_{n=0,\pm}^\infty\,
\left(\frac{q^{\mu}}{p^{\nu}}\right)^{n(n-1)/2}\,
\frac{z^n}{K_\pm([n])!}\,C^\pm_0(z)\,|E^\pm_n\rangle.\end{aligned}$$ Note that the eigenvalue problem (\[coh\]) is singular at the particular value $z=0$, since its solution is an arbitrary superposition of the three states $|E_*\rangle$ and $|E^\pm_0\rangle$. Nevertheless, we shall consider the $(p,q)$-VCS associated to $z=0$, $|z=0;\tau_\pm;\theta,\phi\rangle$, as being defined through the continuous limit in $z\to 0$ of the construction in (\[coh2\]), namely $|z=0;\tau_\pm;\theta,\phi\rangle=C^+_0(0)|E^+_0\rangle +C^-_0(0)|E^-_0\rangle$.
Let us now turn to the issue of the stability of the $(p,q)$-VCS under time evolution generated by the Hamiltonian (\[eq:hq\]). Namely, we now require furthermore that $(p,q)$-VCS are transformed into one another under time evolution according to the following dependence on the real parameters $\tau_\pm$, for all $t\in\mathbb{R}$, $$\begin{aligned}
\label{evol}
e^{-i\omega_0 t\,{\cal H}^{\rm red}}\,|z; \tau_\pm;\theta,\phi\rangle
= |z; t+\tau_\pm;\theta,\phi\rangle.\end{aligned}$$ Since one has, for all $n\in\mathbb{N}$, $$e^{-i\omega_0 t\,{\cal H}^{\rm red}}\,|E^\pm_n\rangle=
e^{-i\omega_0 t\,E^\pm_n}\,|E^\pm_n\rangle,$$ one needs to factor out their complex phases from the quantities $K_\pm([n])$, $$\begin{aligned}
K_\pm([n]) = e^{i\varphi_\pm([n])}K^0_\pm([n]),\end{aligned}$$ where $K^0_\pm([n])>0$ are now real positive scalars. The stability condition (\[evol\]) is then solved by choosing, for all $n=1,2,\cdots$, $$\varphi_\pm([n])=\omega_0\tau_\pm\left[E^\pm_n-E^\pm_{n-1}\right],$$ and redefining $$C^\pm_0(z)={\cal C}^\pm_0(z)\,e^{-i\omega_0\tau_\pm E^\pm_0},$$ where ${\cal C}^\pm_0(z)$ are new complex functions of $z$. Hence, $$|z;\tau_\pm;\theta,\phi\rangle = \sum_{n=0,\pm}^\infty\,
\left(\frac{q^{\mu}}{p^{\nu}}\right)^{n(n-1)/2}\,
\frac{z^n}{K^0_\pm([n])!}\,{\cal C}^\pm_0(z)\,
e^{-i\omega_0\tau_\pm\,E^\pm_n}\,|E^\pm_n\rangle.
\label{coh3}$$
Having identified both the $z$ and $\tau_\pm$ dependences of the coherent states, finally let us account for their $(\theta,\phi)$ dependence and $S_2$ vector character implicit so far through the two functions ${\cal C}^\pm_0(z)$. The latter are now chosen to be given as $${\cal C}^+_0(z)=N^+(|z|)\,\cos\theta,\qquad
{\cal C}^-_0(z)=N^-(|z|)\,e^{i\phi}\,\sin\theta,$$ $N^\pm(|z|)$ being factors such that the constructed $(p,q)$-VCS be of unit norm, $$N^\pm(|z|)=\left\{\sum_{n=0}^\infty\,\left(\frac{q^{\mu}}{p^{\nu}}\right)^{n(n-1)}\,
\frac{|z|^{2n}}
{\left(K^0_\pm([n])!\right)^2}\right\}^{-1/2}.$$ The convergence radii $R_\pm$ of these two series in $z$, $$R_\pm=\lim_{n\to\infty}\left\{(q^{\mu}p^{-\nu})^{-(n-1)}\,K^0_\pm([n])\right\},$$ depend on the choice of functions $K^0_\pm([n])$ as well as on $(\mu,\nu)$ possibly. Specific cases are considered hereafter.
Consequently, the $(p,q)$-VCS constructed here are properly defined provided $z\in D_R$ where $D_R$ denotes the disk in the complex plane centered at $z=0$ and of radius $R={\rm min}\left(R_+,R_-\right)$. Their general structure is thus of the form $$\begin{aligned}
\label{coh5}
|z;\tau_\pm;\theta,\phi\rangle &=& \ \ \ N^+(|z|)\,\cos\theta\,
\sum_{n=0}^\infty \left(\frac{q^{\mu}}{p^{\nu}}\right)^{n(n-1)/2}\,
\frac{z^n}{K^0_+([n])!}\,
e^{-i\omega_0\tau_+\,E^+_n}\,|E^+_n\rangle \cr
&& +\ N^-(|z|)\,e^{i\phi}\,\sin\theta\,
\sum_{n=0}^\infty \left(\frac{q^{\mu}}{p^{\nu}}\right)^{n(n-1)/2}\,
\frac{z^{n}}{K^0_-([n])!}\,
e^{-i\omega_0\tau_-\,E^-_n}\,|E^-_n\rangle.\end{aligned}$$ Only the real positive functions $K^0_\pm([n])$ still need to be specified. They parametrize the remaining freedom in the construction. Particular examples will be considered hereafter by imposing further requirements on these $(p,q)$-VCS. Note that the double limit $p,q\to 1$ yields the VCS of the non-deformed ${\cal JC}m$ as obtained by Hussin and Nieto [@hus1], briefly described in Section \[Sect2\].
Some expectation values {#Sect4.2}
-----------------------
Before dealing with further requirements on the family of $(p,q)$-VCS, among which their overcompleteness in the space $\overline{\cal V}$, let us consider some relevant expectation values for these states. Given (\[coh5\]), the mean value of ${\cal H}^{\rm red}$ for any of the $(p,q)$-VCS is simply $$\begin{aligned}
\label{min}
\langle{\cal H}^{\rm red}\rangle
&=&\ \ \ |N^+(|z|)|^2\,\cos^2\theta\,
\sum_{n=0}^\infty\, \left(\frac{q^{\mu}}{p^{\nu}}\right)^{n(n-1)}
\frac{|z|^{2n}}{\left(K^0_+([n])!\right)^2}\,E^+_n\cr
&& +\ |N^-(|z|)|^2\,\sin^2\theta\,
\sum_{n=0}^\infty\, \left(\frac{q^{\mu}}{p^{\nu}}\right)^{n(n-1)}
\frac{|z|^{2n}}{\left(K^0_-([n])!\right)^2}\,E^-_n.\end{aligned}$$ Likewise for the “number" operator associated to the ladder operators ${\cal A}^-$ and ${\cal A}^+$, one finds the expectation value $$\begin{aligned}
\langle{\cal A}^+\,{\cal A}^-\rangle
&=& |z|^{2}\left\{\ \ |N^+(|z|)|^2\,\cos^2\theta\,
\sum_{n=0}^\infty\, \left(\frac{q^{\mu}}{p^{\nu}}\right)^{n(n+1)}
\frac{|z|^{2n}}{\left(K^0_+([n])!\right)^2}\right.\,\cr
&&\ \ \ \ \ \ \ +\left. |N^-(|z|)|^2\,\sin^2\theta\,
\sum_{n=0}^\infty\, \left(\frac{q^{\mu}}{p^{\nu}}\right)^{n(n+1)}
\frac{|z|^{2n}}{\left(K^0_-([n])!\right)^2}\right\}.\end{aligned}$$
Finally, the average atomic spin time evolution $\langle\sigma_3(t)\rangle=\langle U^{-1}(t)\sigma_3 U(t)\rangle$, with $U(t)=exp\{-i\omega_0 t\,{\cal H}^{\rm red}\}$ being the time evolution operator, has the form $$\begin{aligned}
&&\langle\sigma_3(t)\rangle =
\frac{1}{2}\sum_{n=0}^\infty \left(\frac{q^{\mu}}{p^{\nu}}\right)^{n(n-1)}|z|^{2n}
\frac{{\cal E}([n+1])}{Q([n+1])}
\left\{-\frac{|N^+(|z|)|^2}{\left(K^0_+([n])!\right)^2}\cos^2\theta
+\frac{|N^-(|z|)|^2}{\left(K^0_-([n])!\right)^2}\sin^2\theta\right\}\cr
&&+ \lambda N^+(|z|)\,N^-(|z|)\sin 2\theta
\sum_{n=0}^\infty \left(\frac{q^{\mu}}{p^{\nu}}\right)^{n(n-1)}
\frac{|z|^{2n}}{K^0_+([n])! K^0_-([n])!}
\frac{[n+1]}{Q([n+1])}\cos\Psi_n(t),
\label{ato}\end{aligned}$$ with $$\Psi_n(t)=\omega_0\left[\left(t+\tau_+\right)E^+_n\,-\,
\left(t+\tau_-\right)E^-_n\right]\,+\,\phi=
\omega_0\Delta E_n\,t\,+\,\omega_0\left[\tau_+ E^+_n - \tau_- E^-_n\right]+\phi.$$ As is the case in the non-deformed model, the explicit time dependence which arises for the atomic inversion $\langle\sigma_3(t)\rangle$ is due to the mixed state sector, namely the fact that the mixed-spin matrix elements of the Heisenberg picture operator $\sigma_3(t)$ do not vanish when $\lambda\ne 0$. Hence, the proposition which states that the time dependence of atomic inversion consists of Rabi oscillations when a system is prepared in a coherent state of the radiation field [@hus4] extends to $(p,q)$-VCS. However, in the limit where $\lambda\to 0$, no such oscillations occur. Let us also point out that the time dependence of $\langle\sigma_3(t)\rangle$ diplays chaotic behaviour for appropriate values of the model parameters, as was previously mentioned for the $q$-deformation of the model, with $0<q<1$, in the work by Naderi [*et al.*]{} [@nad].
Overcompleteness and the moment problem {#Sect4.3}
---------------------------------------
An important property that coherent states ought to meet is that of overcompleteness in the space over which they are defined [@kl]. In the present case, this means that the $(p,q)$-VCS in (\[coh5\]) must also provide a resolution of the identity operator over the subspace $\overline{\cal V}$, namely $$\mathbb{I}_{\cal V}=\mathbb{I}_{{\cal V}_0}\,+\,
\mathbb{I}_{\overline{\cal V}}=|E_*\rangle\,\langle E_*|\,+\,
\mathbb{I}_{\overline{\cal V}},$$ while $$\mathbb{I}_{\overline{\cal V}}=
\sum_{n=0,\pm}^\infty\,|E^\pm_n\rangle\,\langle E^\pm_n|=
\int_{D_R\times S_2}\,d\mu(z;\theta,\phi)\,
|z;\tau_\pm;\theta,\phi\rangle\,\langle z;\tau_\pm;\theta,\phi|,
\label{res}$$ where $d\mu(z;\theta,\phi)$ is some SU(2) matrix-valued integration measure over $D_R\times S_2$ to be determined from the above requirement.
Let us thus consider the following parametrization of that measure, $$d\mu(z;\theta,\phi)=d^2z\,d\theta\,\sin\theta\,d\phi\,
\left\{{\cal W}^+(|z|)\sum_{n=0}^\infty|E^+_n\rangle\langle E^+_n|\,+\,
{\cal W}^-(|z|)\sum_{n=0}^\infty|E^-_n\rangle\langle E^-_n|\right\},
\label{eq:weight}$$ in terms of real weight functions ${\cal W}^\pm(|z|)$ to be identified. Using the radial parametrization $z=r\,e^{i\varphi}$ and $d^2z=dr\,r\,d\varphi$ where $r\in[0,\infty[$ and $\varphi\in[0,2\pi[$, a direct substitution in (\[res\]) leads to the moment problem associated to the overcompleteness relation (\[res\]). In terms of the functions $h^\pm(r^2)$ defined through $$h^+(r^2)=\frac{4\pi^2}{3}\,|N^+(r)|^2\,{\cal W}^+(r),\qquad
h^-(r^2)=\frac{8\pi^2}{3}\,|N^-(r)|^2\,{\cal W}^-(r),
\label{eq:hfunctions}$$ the following two infinite sets of moment identities must be met, for all $n\in\mathbb{N}$, $$\int_0^{R^2} du\,u^n\,h^\pm(u)= \left(\frac{q^{\mu}}{p^{\nu}}\right)^{-n(n-1)}\,
\left(K^0_\pm([n])!\right)^2.
\label{eq:moment}$$
In conclusion, the resolution of the identity operator over $\overline{\cal V}$ in terms of the $(p,q)$-VCS is achieved provided the Stieljes moment problem (\[eq:moment\]) can be solved [@hs; @sim]. This requires a choice of functions $K^0_\pm([n])>0$ such that not only the conditions (\[eq:moment\]) may all be met, but also such that the normalization factors $N^\pm(|z|)$ converge in a non-empty disc of the complex plane.
As a result of this analysis, [*a priori*]{} there may exist a large number of sets of $(p,q)$-VCS which fulfill all the above properties, namely continuity in the complex parameter $z$, temporal stability through a simple additive time dependence in the real parameters $\tau_\pm$, a unit vector valued characterization on the sphere $S_2$ in terms of the spherical coordinates $\theta$ and $\phi$, and the completeness property of a resolution of the unit operator with a SU(2) matrix-valued integration measure over these spaces. These sets of $(p,q)$-VCS are distinguished from one another by different choices of real positive weight factors $K^0_\pm([n])$, in agreement with the considerations developed in [@kl; @kl2]. The above construction of $(p,q)$-VCS is general, but can admit explicit exact solutions to the moment problem (\[eq:moment\]) for particular cases. Concrete examples are discussed in Section \[Sect5\]..
Action-angle variables {#Sect4.4}
----------------------
One of the useful properties that general coherent states constructed according to the arguments of [@kl2] possess, is that action-angle variables are readily identified in relation to the continuous parameters ensuring stability of the coherent states under time evolution. In the present case, canonical reduced action-angle variables $(J_\pm(t),\tau_\pm(t))$ are such that for the previously evaluated expectation values of the reduced Hamiltonian (\[eq:hq\]) in the $(p,q)$-VCS, one has $$\langle{\cal H}^{\rm red}\rangle=J_+\,\omega_+\ +\ J_-\,\omega_-
=\sum_\pm\,J_\pm\,\omega_\pm,$$ in relation to the action-angle variational principle of the form $$\int\,dt\sum_\pm\,\left[\frac{d\tau_\pm}{dt}\,J_\pm\,-\,\omega_\pm\,J_\pm\right]\
\longleftrightarrow\
\int\,dt\left[\langle \frac{i}{\omega_0}\frac{d}{dt}\rangle\,-\,
\langle{\cal H}^{\rm red}\rangle\right] ,$$ where $\omega_\pm$ are two constant factors to be chosen appropriately. Consequently $$\frac{d\tau_\pm}{dt}
=\frac{\partial\langle{\cal H}^{\rm red}\rangle}{\partial J_\pm}
=\omega_\pm,\qquad
\frac{dJ_\pm}{dt}
=-\frac{\partial\langle{\cal H}^{\rm red}\rangle}{\partial\tau_\pm}
=0.$$ Given the time evolution, $\tau_\pm(t)=t+\tau_\pm(0)$, one simply finds $\omega_\pm =1$. From the expression in (\[min\]), one then has the identifications $$\begin{aligned}
J_+&=&|N^+(|z|)|^2\,\cos^2\theta\,
\sum_{n=0}^\infty\, \left(\frac{q^{\mu}}{p^{\nu}}\right)^{n(n-1)}\,\frac{|z|^{2n}}
{\left(K^0_+([n])!\right)^2}\,E^+_n, \cr
J_-&=&|N^-(|z|)|^2\,\sin^2\theta\,
\sum_{n=0}^\infty\, \left(\frac{q^{\mu}}{p^{\nu}}\right)^{n(n-1)}\,\frac{|z|^{2n}}
{\left(K^0_-([n])!\right)^2}\,E^-_n.
\label{eq:J}\end{aligned}$$
As a final remark, let us mention that the saturated Heisenberg uncertainty relations which are obeyed by $q$- and $(p,q)$-coherent states are also well-known in $q$-mechanics (see for instance [@kemp]). Such minimal uncertainties may be characterized through small corrections to canonical commutation relations defined in [@kemp; @pens]. Such properties in the case of the $(p,q)$-VCS constructed here are deferred to a later study.
Explicit Solutions {#Sect5}
==================
In order to completely specify the quantities $K^0_\pm([n])$, one last set of conditions needs to be implemented. In the present Section, two such choices are discussed, one of which allows for an exact and explicit solution to the moment problem, hence the construction of a set of $(p,q)$-VCS. First, in line with the illustrative example of Section \[Sect2\], we consider restricting the algebra of the ladder operators ${\cal A}^\pm$. Then as a second and independent possibility, we apply a final additional criterion developed in [@kl] in order to uniquely characterize a set of coherent states which meet already all the requirements considered heretofore and having led to the representation (\[coh5\]), even though the moment problem remains unsolved for that choice.
Constraining the ladder operator algebra {#Sect5.1}
----------------------------------------
In order to uniquely identify the set of functions $K^0_\pm([n])>0$, let us consider the possibility that this may be achieved by restricting the algebraic properties of the ladder operators. In line with the general $(p,q)$-deformations of the Fock algebra in (\[eq:pq-def1\]), let us constrain the algebra of the operators $\mathbb{A}^\pm$ acting on ${\cal V}$ to be such that $$\begin{aligned}
\mathbb{A}^-\,\mathbb{A}^+\,-\,q_0\,\mathbb{A}^+\,\mathbb{A}^-=p^{-N}_0
&=&\sum_{n=0,\pm}^\infty|n,\pm\rangle\,p^{-n}_0\,\langle n,\pm|,\cr
\mathbb{A}^-\,\mathbb{A}^+\,-\,p^{-1}_0\,\mathbb{A}^+\,\mathbb{A}^-=q^{N}_0
&=&\sum_{n=0,\pm}^\infty\,|n,\pm\rangle\,q^n_0\,\langle n,\pm|,
\label{eq:const1}\end{aligned}$$ where $p_0$ and $q_0$ are again two real parameters such that $p_0>1$, $0<q_0<1$ and $p_0 q_0<1$, which may or may not be identical to $p$ and $q$. For instance, we could have $p_0=1$ and $q_0=1$ thus corresponding to an ordinary Fock algebra, or else $p_0=p$ and $q_0=q$, but also more generally $p_0=p^\alpha$ and $q_0=q^\alpha$, $\alpha$ being some real constant. As a matter of fact, exact solutions to the moment problem are presented hereafter in all these situations.
In terms of the ladder operators ${\cal A}^\pm={\cal U}\,\mathbb{A}^\pm\,{\cal U}^\dagger$ acting on the subspace $\overline{\cal V}$, the associated algebraic constraint reads $$\begin{aligned}
{\cal A}^-\,{\cal A}^+\,-\,q_0\,{\cal A}^+\,{\cal A}^-
&=&\sum_{n=0,\pm}^\infty|E^\pm_n\rangle\,p^{-n}_0\,\langle E^\pm_n|,\cr
{\cal A}^-\,{\cal A}^+\,-\,p^{-1}_0\,{\cal A}^+\,{\cal A}^-
&=&\sum_{n=0,\pm}^\infty\,|E^\pm_n\rangle\,q^n_0\,\langle E^\pm_n|.
\label{eq:const2}\end{aligned}$$ Whether in terms of (\[eq:const1\]) or (\[eq:const2\]), these algebraic constraints translate into the following identities, for all $n\in\mathbb{N}$, $$\label{eq:recrel}
\left(K^0_\pm([n+1])\right)^2\,-\,q_0\,\left(K^0_\pm([n])\right)^2=p^{-n}_0,\quad
\left(K^0_\pm([n+1])\right)^2\,-\,p^{-1}_0\,\left(K^0_\pm([n])\right)^2=q^n_0.$$ Given the initial values $K^0_\pm([0])=0$, the solution to these recursion relations is simply $$K^0_\pm([n])=\sqrt{[n]_{(p_0,q_0)}}=
\sqrt{[n]_{(q^{-1}_0,p^{-1}_0)}},
\label{eq:sol1K0}$$ where[^5] $$[n]_{(p_0,q_0)}=\frac{p^{-n}_0-q^n_0}{p^{-1}_0-q_0}=
\frac{\left(q^{-1}_0\right)^{-n}-\left(p^{-1}_0\right)^n}
{\left(q^{-1}_0\right)^{-1}-\left(p^{-1}_0\right)}=[n]_{(q^{-1}_0,p^{-1}_0)}.$$
Given this solution, the normalization factors are defined by the series $$|N^\pm(|z|)|^{-2}=\sum_{n=0}^\infty\,\left(\frac{q^{\mu}}{p^{\nu}}\right)^{n(n-1)}\,
\frac{|z|^{2n}}{[n]_{(p_0,q_0)}!},$$ of which the convergence radius is $$R=\lim_{n\to\infty}\left[\left(\frac{q^{\mu}}{p^{\nu}}\right)^{-2(n-1)}
\frac{p^{-n}_0-q^n_0}{p^{-1}_0-q_0}\right]^{1/2}
=\lim_{n\to\infty}
\left[\left(p_0p^{-2\nu}q^{2\mu}\right)^{-(n-1)}\,\frac{1-(p_0q_0)^n}{1-(p_0q_0)}\right]^{1/2}.$$ Provided $p_0p^{-2\nu}q^{2\mu}<1$, a condition which we shall henceforth assume to be satisfied[^6], this radius of convergence is infinite, $R=\infty$, and the moment problem (\[eq:moment\]) then becomes, for all $n\in\mathbb{N}$, $$\int_0^\infty\,du\,u^n\,h^\pm(u)=\left(\frac{q^{\mu}}{p^{\nu}}\right)^{-n(n-1)}\,
\left([n]_{(p_0,q_0)}!\right).
\label{eq:moment2}$$ In order to solve these equations, the Ramanujan integral (\[eq:App-Ramanujan\]) discussed in the Appendix suggests itself quite naturally, through a simple but appropriate rescaling of its arguments in the form of (\[eq:App-Ramanujan2\]).
After a little moment’s thought one comes to the conclusion that a solution to (\[eq:moment2\]) based on (\[eq:App-Ramanujan2\]) is possible for the following choice of parameters, $$\mu=\frac{1}{2},\qquad \nu=0,\qquad p_0=p,\qquad q_0=q,$$ in which case $p_0 p^{-2\nu} q^{2\mu}=pq<1$, hence corresponding indeed to an infinite radius of convergence. For this choice, one has (for definitions of the $(p,q)$-exponential functions appearing in these expressions, see the Appendix), $$h^\pm\left(|z|^2\right)=\frac{\left(p^{-1}-q\right)}{q\log\left(1/pq\right)}\,
e_{(p,q)}\left(-|z|^2\,p^{-1/2} q^{-1}\left(p^{-1}-q\right)\right),
\label{eq:sol-mom}$$ as well as[^7] $$\left(K^0_\pm([n])\right)^2=[n],\qquad
|N^\pm(|z|)|^{-2}
={\cal E}^{(1/2,0)}_{(p,q)}\left(|z|^2q^{-1/2}\left(p^{-1}-q\right)\right),$$ with for the weight functions ${\cal W}^\pm(|z|)$ in the integration measure (\[eq:weight\]) of the overcompleteness relation (\[res\]), $${\cal W}^+\left(|z|\right)=\frac{3}{4\pi^2}\,|N^+\left(|z|\right)|^{-2}\,h^+\left(|z|^2\right),\qquad
{\cal W}^-\left(|z|\right)=\frac{3}{8\pi^2}\,|N^-\left(|z|\right)|^{-2}\,h^-\left(|z|^2\right).
\label{eq:weight2}$$ Explicit expressions for all previously computed quantities readily follow, beginning with the definition of the associated $(p,q)$-VCS which then meet all the necessary requirements expected of coherent states. Note that up to the coefficients $3/(2\pi)$ and $3/(4\pi)$, the reduced weights obtained are compatible with that of the $q$-shape invariant harmonic oscillator [@bal2]. Furthermore, (\[eq:sol-mom\]) is a $(p,q)$-generalization of the $q$-harmonic oscillator coherent state moment problem solution constructed in [@quesne]. Finally, in the double limit $p,q\to 1$, the results of [@hus1] are recovered.
The functions (\[eq:sol-mom\]) thus provide a complete and explicit solution to the moment problem of the $(p,q)$-VCS for the $(p,q)$-${\cal JC}m$ such that the ladder operators ${\cal A}^\pm$ obey the same $(p,q)$-Fock algebra as the original modes $a$ and $a^\dagger$ of the initial Hamiltonian (\[eq:hq\]), namely with the choice $p_0=p$ and $q_0=q$. It is also possible to construct an explicit solution when the ladder operators ${\cal A}^\pm$ are constrained to rather obey the ordinary non-deformed Fock algebra on $\overline{\cal V}$, corresponding to the choice $p_0=1$ and $q_0=1$. One then has to consider[^8], for all $n\in\mathbb{N}$, $$K^0_\pm([n])=\sqrt{n},\qquad
\int_0^\infty\,du\,u^n\,h^\pm(u)=
\left(\frac{q^\mu}{p^\nu}\right)^{-n(n-1)}\,\left(n!\right),\qquad
p^{-\nu}q^\mu\le 1.$$ An obvious solution to this moment problem is obtained when $\mu=0=\nu$, in which case the condition for an infinite radius of convergence is saturated. One then has $$h^\pm\left(|z|^2\right)=e^{-|z|^2},\qquad
|N^\pm\left(|z|\right)|^{-2}=e^{|z|^2},\qquad
{\cal W}^+\left(|z|\right)=\frac{3}{4\pi^2},\qquad
{\cal W}^-\left(|z|\right)=\frac{3}{8\pi^2}.$$
In fact, the above two explicit solutions belong to a general class of solutions obtained by taking $(p_0,q_0)=(p^\alpha,q^\alpha)$ with $\alpha$ a positive real parameter, $\alpha>0$, such that $p^{\alpha-2\nu}q^{2\mu}<1$ in order to ensure an infinite radius of convergence[^9] in $z\in\mathbb{C}$. Once again based on (\[eq:App-Ramanujan2\]), an explicit solution to the moment problem (\[eq:moment2\]) is achieved for the following choice of parameters, $$\mu=\frac{1}{2}\alpha,\qquad \nu=0,\qquad p_0=p^\alpha,\qquad q_0=q^\alpha,$$ for which the radius of convergence is indeed infinite, $p^{\alpha-2\nu}q^{2\mu}=(pq)^\alpha<1$. One then has $$h^\pm\left(|z|^2\right)=\frac{\left(p^{-\alpha}-q^\alpha\right)}
{q^\alpha\log\left(1/p^\alpha q^\alpha\right)}\,
e_{(p^\alpha,q^\alpha)}\left(-|z|^2\,p^{-\alpha/2} q^{-\alpha}\left(p^{-\alpha}-q^\alpha\right)\right),
\label{eq:sol-momalpha}$$ with $$|N^\pm(|z|)|^{-2}
={\cal E}^{(1/2,0)}_{(p^\alpha,q^\alpha)}\left(|z|^2q^{-\alpha/2}\left(p^{-\alpha}-q^\alpha\right)\right),$$ leading finally to the weight functions ${\cal W}^\pm(|z|)$ given in terms of the latter two quantities through the same relations as in (\[eq:weight2\]). In the limits that $\alpha\to 1$ or $\alpha\to 0$, the previous two explicit solutions are then recovered as particular cases.
The action identity constraint {#Sect5.2}
------------------------------
An alternative to fixing the factors $K^0_\pm([n])$ through conditions on the algebra of ladder operators, is to consider the action identity constraint discussed in [@kl] as the one last requirement which singles out coherent states uniquely. In the case of the ordinary Fock algebra, this action identity constraint is equivalent to requiring that the ladder operators obey themselves the Fock algebra as well. We shall establish that this is not the case for the $(p,q)$-VCS of the $(p,q)$-${\cal JC}m$ constructed above.
Given the relations (\[eq:J\]), in the present model the action identity constraint is of the form $$J_+=\cos^2\theta \left(|z|^2 + E^+_0\right),\qquad
J_-=\sin^2\theta \left(|z|^2 + E^-_0\right).
\label{eq:J2}$$ By direct substitution into these constraints of the relations (\[eq:J\]), the identification of the successive powers in $|z|^2$ leads to the following solution for the factors $K^0_\pm([n])$, $$K^0_\pm([n])=\left(\frac{q^\mu}{p^\nu}\right)^{(n-1)}\,
\sqrt{E^\pm_n\,-\,E^\pm_0}.$$ These positive real quantities are thus well-defined provided one has $E^\pm_n>E^\pm_0$ for all $n\ge 1$, as is implicitly assumed. It is noteworthy that, as $(p,q)\to (1^+,1^-)$, these factors reduce to exactly those obtained in [@hus3] by the factorization method. On the other hand, since the present solution for $K^0_\pm([n])$ cannot be brought into the form of (\[eq:sol1K0\]) for some choice of constants $p_0$ and $q_0$ meeting our assumptions for these quantities, it follows indeed that for the $(p,q)$-${\cal JC}m$ the action identity constraint is not equivalent to requiring an algebraic constraint on the ladder operators of the $(p_0,q_0)$-deformed Fock algebra type.
This choice also allows for the factorization of the Hamiltonian in (\[eq:hpq\]) in the form $$\mathbb{H}^{\rm red}=\mathbb{A}^+\,\left(\frac{q^\mu}{p^\nu}\right)^{-2N}\,\mathbb{A}
\ +\ \sum_{n=0,\pm}^\infty\,|n,\pm\rangle\,E^\pm_0\,\langle n,\pm|,$$ extending a similar expression in [@hus1].
Given this solution for the factors $K^0_\pm([n])$, the general moment problem (\[eq:moment\]) reduces to the following conditions, $$\int_0^{R^2}du\,\,h^\pm(u)=1;\qquad
\int_0^{R^2}du\,u^n\,h^\pm(u)=
\prod_{k=1}^n \left(E^\pm_k-E^\pm_0\right),\quad n=1,2,3,\cdots,$$ where the radius of convergence $R$ is given as $$R={\rm min}\,\left(R_+,R_-\right),\qquad
R_\pm=\lim_{n\to+\infty}\sqrt{E^\pm_n-E^\pm_0}.$$ In the absence of a detailed analysis of the energy spectra $E^\pm_n$ as functions of the parameters $p$, $q$, $\lambda$ and $\epsilon$ and the function $h(p,q)$, nothing more explicit may be said concerning this moment problem. Since when $p>1$ the quantities $[n]$ always possess a turn-around behaviour as functions of $n$ for $n$ sufficiently large, it is to be expected generally that the radius of convergence $R$, hence the moment problem as well, are associated to a finite disk $D_R$ in the complex plane. Nevertheless, one conclusion of the present discussion is that indeed for the $(p,q)$-VCS considered in this work, the action identity constraint leads to coherent states different from those constructed in Section \[Sect5.1\] and for which explicit solutions to the moment problem have been given.
The spin decoupled limit $\lambda=0$ {#Sect5.3}
------------------------------------
In the limit that $\lambda=0$, the two spin sectors of the model are decoupled, and the $(p,q)$-${\cal JC}m$ reduces to the supersymmetric harmonic oscillator [@arag; @org; @daou] with a $(p,q)$-deformation. Diagonalization of the reduced Hamiltonian (\[eq:hq\]) is then of course straightforward in the $\sigma_3$-eigenbasis, with, for $n=0,1,2,\cdots$, $${\cal H}^{\rm red}_{\lambda=0}\,|n,\pm\rangle=\epsilon^\pm_n\,|n,\pm\rangle,\qquad
\epsilon^\pm_n=(1+\epsilon)h(p,q)[n]\,+\,\frac{1}{2}(1+\epsilon)\pm\frac{1}{2}.
\label{eq:lambda0}$$
From that point of view, one thus has two decoupled $(p,q)$-deformed Fock bases, for which one could consider the usual $(p,q)$-coherent states in each spin sector separately. However, such coherent states do not coincide with any of those constructed in this paper and obtained in the limit $\lambda=0$, because of the distinguished role played by the singleton state $|E_*\rangle=|0,-\rangle$ and the $S_2$ unit vector character of the $(p,q)$-VCS. In particular the ladder operators ${\cal A}^\pm$ acting within each of the towers $|E^\pm_n\rangle$ do not coincide with the annihilation and creation operators $a$ and $a^\dagger$ defining the Hamiltonian (\[eq:hq\]), even in the decoupled limit $\lambda=0$. As a matter of fact, the action of the ladder operators ${\cal A}^\pm$ may switch between the two spin sectors as a function of $n$ depending on the sign of the quantity ${\cal E}([n+1])$.
More specifically, let us introduce the notation $$s_n={\rm sign}\,{\cal E}([n+1]),\qquad n\in\mathbb{N}.$$ In the limit that $\lambda=0$, one has $Q([n+1])=|{\cal E}([n+1])|/2$, so that the mixing angle $\theta([n])$ is now such that, for all $n\in\mathbb{N}$, $$\lambda=0:\quad
\sin\theta([n])=\frac{1}{2}(1-s_n)\,({\rm sign}\,\lambda),\quad
\cos\theta([n])=\frac{1}{2}(1+s_n).$$ Consequently, the towers of energy eigenstates $|E^\pm_n\rangle$ are then given as follows, for all $n\in\mathbb{N}$, $$\begin{array}{rll}
{\rm If}\ s_n=+1: &\quad |E^+_n\rangle_{\lambda=0}=|n+1,-\rangle,\quad
&|E^-_n\rangle_{\lambda=0}=|n,+\rangle ;\\
& & \\
{\rm If}\ s_n=-1: &\quad
|E^+_n\rangle_{\lambda=0}=({\rm sign}\,\lambda)\,|n,+\rangle,\quad
&|E^-_n\rangle_{\lambda=0}=-({\rm sign}\,\lambda)\,|n+1,-\rangle,
\end{array}$$ while the energy eigenvalues are given as $$\begin{array}{rl}
{\rm If}\ s_n=+1: &\quad
E^+_n(\lambda=0)=(1+\epsilon)h(p,q)[n+1]\,+\,\frac{1}{2}(1+\epsilon)\,-\,\frac{1}{2}, \\
& \\
&\quad E^-_n(\lambda=0)=(1+\epsilon)h(p,q)[n]\,+\,\frac{1}{2}(1+\epsilon)\,+\,\frac{1}{2}; \\
& \\
{\rm If}\ s_n=-1: &\quad
E^+_n(\lambda=0)=(1+\epsilon)h(p,q)[n]\,+\,\frac{1}{2}(1+\epsilon)\,+\,\frac{1}{2}, \\
& \\
&\quad E^-_n(\lambda=0)=(1+\epsilon)h(p,q)[n+1]\,+\,\frac{1}{2}(1+\epsilon)\,-\,\frac{1}{2}.
\end{array}$$ These spectra do indeed coincide with those in (\[eq:lambda0\]), once the singleton state $|E_*\rangle=|0,-\rangle$ with $E_*=\epsilon/2$ is included as well.
These expressions show how, even in the decoupled spin limit $\lambda=0$, the $(p,q)$-VCS constructed here are not simply the juxtaposition of two separate $(p,q)$-coherent states of the $(p,q)$-deformed Fock algebra in each of the two spin sectors. Since the spectrum of the system is discrete infinite, by leaving aside the singleton state $|0,-\rangle$, all the remaining states still allow for similar types of constructions of coherent states, but in such a way that different spin sectors are getting superposed, leading to the SU(2) vector coherent states of the type studied here. All the expressions detailed in the previous sections for the $(p,q)$-VCS may readily be particularized to the limit $\lambda\to 0$.
Conclusion {#Sect6}
==========
In this work, we considered $(p,q)$-deformations of the Jaynes-Cummings model in the rotating wave approximation, extending recent developments on this topic in the non-deformed case [@hus1]. Having introduced $(p,q)$-deformed versions of the model, first its energy eigenspectrum has been identified, enabling the definition of different relevant operators acting on Hilbert space and the characterization of the spectrum in terms of two separate infinite discrete towers and a singleton state. Among these operators, ladder operators acting within each of the two towers separately may be considered, defined up to some arbitrary normalization factors.
Such a structure sets the stage for the introduction of vector coherent states for the $(p,q)$-deformed Jaynes-Cummings model, following the approach of [@hus1] and the rationale outlined in [@kl]. These $(p,q)$-VCS are parametrized by elements of $\mathbb{C}\times S_2$, and enjoy temporal stability through a further action-angle identification. The moment problem associated to the overcompleteness property of these $(p,q)$-VCS involves SU(2)-valued matrix weight functions. Using $(p,q)$-arithmetic techniques, some explicit and exact solutions to the moment problem have been displayed, hence characterizing specific classes of such $(p,q)$-VCS. All these solutions provide $(p,q)$-extensions to the non-deformed vector coherent states of the ${\cal JC}m$ considered in [@hus1]. These explicit solutions are obtained by requiring that specific algebraic constraints of the $(p,q)$-deformed Fock algebra type be obeyed by the ladder operators. However, in contradistinction to [@hus1], we have not been able to display an explicit and exact solution to the moment problem in the generic case by imposing an action identity constraint.
Finally, the spin decoupled limit of these models was considered, corresponding to a $(p,q)$-supersymmetric oscillator of which the two sectors are intertwined in a manner depending on the sign of the energy level spacing between the two decoupled spin sectors as function of the excitation level. In the non-deformed limit $(p,q)=(1,1)$, this feature disappears, reproducing the ordinary supersymmetric oscillator. Our results thus provide new classes of generalized versions of the ${\cal JC}m$ in the rotating wave approximation [@bal2; @daou]. Finally, the $(p,q)$-VCS built here extend the $q$-coherent states obtained by other techniques involving supersymmetric shape invariance and self-similar potential formalisms applied to the harmonic oscillator [@bal2; @coo].
Acknowledgements {#acknowledgements .unnumbered}
================
J. B. G. is grateful to the Abdus Salam International Centre for Theoretical Physics (ICTP, Trieste, Italy) for a Ph.D. fellowship under the grant . M. N. H. is particularly indebted to V. Hussin for discussions relating to the ${\cal JC}m$ as well as for provided references during his stay at the Centre de Recherches Mathématiques, Université de Montréal, Canada. The ICMPA is in partnership with the Daniel Iagoniltzer Foundation (DIF), France.
J. G. acknowledges a visiting appointment as Visiting Professor in the School of Physics (Faculty of Science) at the University of New South Wales. He is grateful to Prof. Chris Hamer and the School of Physics for their hospitality during his sabbatical leave, and for financial support through a Fellowship of the Gordon Godfrey Fund. His stay in Australia is also supported in part by the Belgian National Fund for Scientific Research (F.N.R.S.) through a travel grant.
J. G. acknowledges the Abdus Salam International Centre for Theoretical Physics (ICTP, Trieste, Italy) Visiting Scholar Programme in support of a Visiting Professorship at the ICMPA. His work is also supported by the Belgian Federal Office for Scientific, Technical and Cultural Affairs through the Interuniversity Attraction Pole (IAP) P5/27.
Appendix {#App .unnumbered}
========
This appendix lists some useful facts related to the $(p,q)$-boson algebra and associated functions. The $(p,q)$-deformed oscillator algebra introduced in [@cha] is generated by operators $a$, $a^{\dag}$ and $N$ obeying the relations $$\begin{aligned}
\label{pqalg}
& [N,a]=-a,&\quad [N,a^\dagger]=a^\dagger,\cr
& aa^\dagger-qa^\dagger a=p^{-N},&\quad
aa^\dagger-p^{-1}a^\dagger a=q^{N}.\end{aligned}$$ Throughout the text, we assume the real parameters $p$ and $q$ are such that $p>1$, $0<q<1$ and $pq<1$. The limit $p\to 1^{+}$ yields the $q$-oscillator of Arik and Coon [@ar] while $p=q$ gives the $q$-deformed oscillator algebra of Biedenharn and MacFarlane [@far]. Finally, the algebra (\[pqalg\]) reduces to the ordinary harmonic oscillator Fock algebra as $q\to 1$ for $p=1^{+}$ or $p=q$. At any stage of the discussion, the $(p,q)$-deformed model readily reduces to its usual counterpart as $(p,q) \to (1,1)$.
The associated $(p,q)$-deformed Fock-Hilbert space representation is spanned by the vacuum $|0\rangle$ annihilated by $a$ and the orthonormalized states $|n\rangle$, such that $$\begin{aligned}
\label{pqrep}
&&
a|0\rangle=0,\quad \langle 0|0\rangle=1,\quad
|n\rangle = \frac{1}{\sqrt{[n]_{(p,q)}!}}\left(a^\dagger\right)^n|0\rangle,\cr
&&
a|n\rangle=\sqrt{[n]_{(p,q)}}|n-1\rangle,\quad
a^\dagger |n\rangle = \sqrt{[n+1]_{(p,q)}}|n+1\rangle,\quad
N|n\rangle =n|n\rangle,\end{aligned}$$ where the symbol $[n]_{(p,q)}=\left(p^{-n}-q^{n}\right)/\left(p^{-1}-q\right)$ is called $(p,q)$-basic number with, by convention, $[0]_{(p,q)}=0$, and its $(p,q)$-factorial is defined through $[n]_{(p,q)}!=[n]_{(p,q)}\left([n-1]_{(p,q)}!\right)$ and the convention $[0]_{(p,q)}!=1$. There exists a formal $(p,q)$-number operator denoted by $[N]_{(p,q)}$, or simply by $[N]$ when no confusion arises. As a matter of fact, from the second pair of relations in (\[pqalg\]), it follows that $[N]=a^\dagger a$ as well as $[N+1]=a a^\dagger$. One has of course $[N]|n\rangle= [n]|n\rangle$. Hence, (\[pqrep\]) provides a well defined Fock-Hilbert representation space of the algebra (\[pqalg\]).
The following relations hold for any function $f\equiv f(N)$ and consequently for any function of $[N]$, $$\begin{aligned}
\label{afa}
a f(N -1)= f(N)a,\quad a^\dagger f(N) = f(N -1)a^\dagger.\end{aligned}$$
Let us define $q$-shifted products and factorials and their $(p,q)$-analogues. Using the notations of [@ga], for any quantity $x$, $(x;q)_{\alpha}$ is constructed as follows, $$\begin{aligned}
(x;q)_0=1,\qquad(x;q)_\alpha=\frac{(x;q)_\infty}{(xq^\alpha;q)_\infty},
\qquad (x;q)_\infty= \prod_{n=0}^\infty\left(1-xq^n \right).\end{aligned}$$ Furthermore, in the notations of [@vin2], $(p,q)$-shifted products and factorials are defined as follows, for any real quantities $a$ and $b$ such that $a\neq 0$, $$\begin{aligned}
[a,b;p,q]_{0}=1,\qquad
[a,b;p,q]_\alpha=\frac{[a,b;p,q]_\infty}{[ap^\alpha,bq^\alpha;p,q]_\infty},\qquad
[a,b;p,q]_\infty= \prod_{n=0}^\infty\left(\frac{1}{ap^n}-bq^n \right).\end{aligned}$$ For $\alpha=n \in \mathbb{N}$, we have $$\begin{aligned}
\label{pqprod}
[p^\mu,q^\nu;p,q]_n&=&\left(\frac{1}{p^\mu}-q^\nu \right)
\left(\frac{1}{p^{\mu+1}}-q^{\nu+1}\right)\dots
\left(\frac{1}{p^{\mu+n-1}}-q^{\nu+n-1}\right)\cr
&=&p^{-\mu n - n(n-1)/2}(p^\mu q^\nu;pq)_n.\end{aligned}$$ This identity is a central formula since it defines a bridge between $q$- and $(p,q)$-analogue quantities and functions.
Let us now introduce $q$-analogues of the ordinary exponential funtion. There exist many types of $q$-deformations of the exponential function $e^{z}$, $z\in \mathbb{C}$ (see, for instance, [@vin]). For any $(z,\mu) \in \mathbb{C}\times\mathbb{R}$, the $(\mu,q)$-exponential is the complex function [@vin] $$\begin{aligned}
\label{muex}
E_{q}^{(\mu)}(z)= \sum_{n=0}^{\infty}\frac{q^{\mu n^{2}}}
{(q;q)_{n}}z^{n}.\end{aligned}$$ This series has an infinite radius of convergence for $\mu>0$. For $\mu=0$ its domain of definition reduces to the unit disk, $|z|<1$, while it is nowhere convergent in $\mathbb{C}$ for $\mu<0$. Rescaling $z \to z(1-q)$ and taking the limit $\lim_{q\to 1}E_{q}^{\mu}(z(1-q))$, one recovers $e^{z}$. For some specific values of $\mu$, (\[muex\]) reproduces some standard $q$-exponentials [@vin; @koe], $$\begin{aligned}
\label{eq}
E_{q}^{(0)}(z)&=& e_{q}(z)=\frac{1}{(z;q)_{\infty}}=
\sum_{n=0}^{\infty}\frac{z^{n}}{(q;q)_{n}},\qquad |z|<1,\\
\label{edemi}
E_{q}^{(1/2)}(z) &=& E_{q}(q^{1/2}z)=(-q^{1/2}z;q)_{\infty},
\qquad z\in \mathbb{C},\end{aligned}$$ where $$\begin{aligned}
\label{eqjac}
E_{q}(z)= \sum_{n=0}^{\infty}\frac{q^{n(n-1)/2} z^{n}}{(q;q)_{n}},
\qquad z\in\mathbb{C},\end{aligned}$$ is known as the Jackson $q$-exponential [@jac]. Note that whereas $E_{q}^{(\mu)}(z)$ is defined in the entire complex plane, $|z|<\infty$, for any $\mu > 0 $, its reduction $e_{q}(z)$ is only defined on the unit disc. Finally, it is also well established that [@koe] $$\begin{aligned}
\label{invert}
E_{q}(-z)e_{q}(z)=1.\end{aligned}$$
$(p,q)$-analogues of the usual exponential function $e^{z}$, $z\in \mathbb{C}$ may also be introduced (see, for instance, [@vin2]). Given any $(z,\mu,\nu) \in \mathbb{C}\times\mathbb{R}\times\mathbb{R}$, consider the $(\mu,\nu,p,q)$-exponential function $$\begin{aligned}
\label{pgex}
{\cal E}_{(p,q)}^{(\mu,\nu)}(z)= \sum_{n=0}^{\infty}
\left( \frac{q^{\mu}}{p^{\nu}} \right)^{n^{2}}
\frac{z^{n}}{[p,q;p,q]_{n}}.\end{aligned}$$ Keeping in mind the condition $pq < 1$, the radius of convergence $R$ of this series is such that $$\begin{aligned}
\label{rad}
R_1= \left\{\begin{array}{ll}
\infty, & \qquad {\rm if}\ q^{2\mu} p^{1-2\nu} < 1; \\
p^{\nu-1}q^{-\mu}, & \qquad {\rm if}\ q^{2\mu}p^{1-2\nu} = 1; \\
0, & \qquad {\rm if}\ q^{2\mu}p^{1-2\nu} > 1.
\end{array}\right.\end{aligned}$$ Thus the function ${\cal E}^{(\mu,\nu)}_{(p,q)}(z)$ exists only provided $q^{2\mu}p^{1-2\nu}\le 1$.
In order to recover the usual exponential function, one has to rescale $z \to z(p^{-1}-q)$, for example, and then take the limit $\lim_{(p,q)\to (1,1)}{\cal E}_{(p,q)}^{\mu,\nu}(z(p^{-1}-q))=e^{z}$. For particular values of the parameters $\mu$ and $\nu$, (\[pgex\]) reproduces known $(p,q)$-exponentials, $$\begin{aligned}
\label{pgex2}
{\cal E}_{(p,q)}^{(1/2,1/2)}(z)&=&
E_{(p,q)}\left(\left(\frac{q}{p}\right)^{1/2}z\right)=
\sum_{n=0}^{\infty} \left(\frac{q}{p}\right)^{n^{2}/2}
\frac{z^{n}}{[p,q;p,q]_{n}},\end{aligned}$$ where $$\begin{aligned}
\label{epq}
E_{(p,q)}(z)&=&\sum_{n=0}^{\infty}
\left(\frac{q}{p}\right)^{n(n-1)/2}
\frac{z^{n}}{[p,q;p,q]_{n}}.\end{aligned}$$ The function $E_{(p,q)}$ may be found in [@vin2]. Note that (\[epq\]) coincides with (\[eqjac\]) as $p\to 1$. In the same limit, (\[pgex\]) reproduces the $(\mu,q)$-deformed exponential map $E^{(\mu)}_q(z)$ [@vin]. If $\mu=0=\nu$ the series (\[pgex\]) is not defined since then $R=0$, unless one has taken $p=1$ in which case the radius of convergence is unity. A $(p,q)$-analogue of (\[eq\]) is given by $$\begin{aligned}
\label{egex}
e_{(p,q)}(z) =\sum_{n=0}^{\infty}
\frac{1}{p^{n^{2}/2}}\frac{z^{n}}{[p,q;p,q]_{n}},\qquad |z|<p^{-1/2},\end{aligned}$$ which reproduces exactly $e_{q}(z)$ converging in the unit disc as $p\to 1^{+}$. Furthermore, we have from (\[pqprod\]) $$\begin{aligned}
\label{inv}
e_{(p,q)}(z)=\sum_{n=0}^{\infty}
\frac{(p^{1/2}z)^{n}}{(pq;pq)_{n}} =
e_{pq}(p^{1/2}z) .\end{aligned}$$ Using (\[pqprod\]) and (\[eqjac\]), we may also write $$\begin{aligned}
\label{epqder}
E_{(p,q)}(z)&=& \sum_{n=0}^{\infty}
\left(\frac{q}{p}\right)^{n(n-1)/2}
\frac{z^{n}}{p^{-n(n+1)/2}(pq;pq)_{n}}
\cr
&=&
\sum_{n=0}^{\infty}
q^{n(n-1)/2} \frac{(zp)^{n}}{(pq;pq)_{n}}=E_{pq}(pz).\end{aligned}$$ Then taking into account (\[invert\]), (\[inv\]) and (\[epqder\]), a $(p,q)$-analogue of (\[invert\]) is given by $$E_{pq}(-pz)e_{pq}(pz)= E_{(p,q)}(-z)e_{(p,q)}(p^{1/2}z)=1.$$ Finally, consider $$\begin{aligned}
\label{pqpens}
\mathfrak{e}_{(p,q)}^{(\mu,\nu)}(z)=
\sum_{n=0}^{\infty}\left(\frac{q^{\mu}}{p^{\nu}}\right)^{n^{2}}
\frac{z^{n}}{n!}.\end{aligned}$$ Therefore, $\mathfrak{e}_{(p,q)}^{(\mu,\nu)}(z)$, which converges to $e^{z}$ as $(p,q)\to (1,1)$, provides a $(p,q)$-deformed exponential analogue to the $q$-function used by Penson and Solomon [@pens2] which coincides with $\mathfrak{e}_{(1,q)}^{(1,\nu)}(q^{-1/2}z)$. The radius of convergence of (\[pqpens\]) is given as $$\begin{aligned}
\label{rad2}
R_2= \left\{\begin{array}{ll}
\infty, & \qquad\ {\rm if}\ q^{\mu}p^{-\nu} \leq 1; \\
0, & \qquad\ {\rm if}\ q^{\mu}p^{-\nu} > 1.
\end{array}\right.\end{aligned}$$
Finally, consider the Ramanujan integral [@ram; @bal1], valid for any integer $n\in\mathbb{N}$, $$\begin{aligned}
\label{eq:App-Ramanujan}
\int_{0}^{\infty}dt\,t^n\,e_{q}(-t)
= - \frac{(q;q)_{n}}{q^{n(n+1)/2}} \log q.\end{aligned}$$ Through the change of variables $$\begin{aligned}
q \to pq ,\qquad t \to \lambda_0\,p^{-1/2}\,t,\qquad \lambda_0 >0,\end{aligned}$$ and using once again (\[pqprod\]), the following identity is obtained, for any $n\in\mathbb{N}$, $$\begin{aligned}
\int_{0}^{\infty} dt\,t^n\,e_{(p,q)}\left(-\lambda_0 p^{-1/2}t\right)
= \frac{[p,q;p,q]_{n}}{\lambda^{n+1}_0\,q^{n(n+1)/2}} \log\left(\frac{1}{pq}\right).
\label{eq:App-Ramanujan2}\end{aligned}$$ This result is indeed a $(p,q)$-analogue of the Ramanujan integral (\[eq:App-Ramanujan\]).
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[^1]: On sabbatical leave from the Center for Particle Physics and Phenomenology (CP3), Institute of Nuclear Physics, Catholic University of Louvain, 2, Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium.
[^2]: We differ on this point with [@hus1], where strictly increasing energy spectra in each tower are required.
[^3]: Make no mistake that henceforth, all quantities correspond to the $(p,q)$-deformed analysis even though the notations used coincide with those of Section \[Sect2\] and do not make explicit the fact that all expressions correspond now to the deformed case. When wanting to make the difference explicit, notations such as for instance $[N]\equiv [N]_{(p,q)}=(p^{-N}-q^N)/(p^{-1}-q)$ and $[n]\equiv [n]_{(p,q)}=(p^{-n}-q^n)/(p^{-1}-q)$ are used.
[^4]: Once again, the states $|n\rangle=|n\rangle_{(p,q)}$ are not to be confused with the number operator eigenstates of the ordinary Fock algebra as in Section \[Sect2\], in spite of an identical notation.
[^5]: Incidentally, it is because of this identity, corresponding to the exchange $p_0\leftrightarrow q^{-1}_0$, that the two solutions to the above two recursion relations are consistent, as are the two algebraic restrictions in (\[eq:const1\]) and (\[eq:const2\]).
[^6]: If $p_0p^{-2\nu}q^{2\mu}=1$, the radius of convergence is finite with $R=(1-p_0q_0)^{-1/2}$, while when $p_0p^{-2\nu}q^{2\mu}>1$ the radius of convergence vanishes, implying that $(p,q)$-VCS cannot be constructed in such a case.
[^7]: Restricting to $p_0=p$ and $q_0=q$ but keeping $\mu$ and $\nu$ arbitrary such that $p^{1-2\nu} q^{2\mu}<1$ in order to retain an infinite radius of convergence, one has $\left(K^0_\pm([n])\right)^2=[n]$ and $|N^\pm\left(|z|\right)|^{-2}={\cal E}^{(\mu,\nu)}_{(p,q)}
\left(|z|^2\,p^\nu\,q^{-\mu}\left(p^{-1}-q\right)\right)$, hence also all other previous expressions given accordingly.
[^8]: Leading to $|N^\pm\left(|z|\right)|^{-2}=
\mathfrak{e}^{(\mu,\nu)}_{(p,q)}\left(|z|^2\,p^\nu\,q^{-\mu}\right)$, which converges for all $|z|<\infty$ provided $p^{-\nu}q^\mu\le 1$.
[^9]: Leading to $|N^\pm\left(|z|\right)|^{-2}=
{\cal E}^{(\mu/\alpha,\nu/\alpha)}_{(p^\alpha,q^\alpha)}\left(|z|^2p^\nu q^{-\mu}(p^{-\alpha}-q^\alpha)\right)$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The purpose of this article is to extend the wavelet transform to quaternion algebra using the kernel of the two-sided quaternion Fourier transform (QFT). We study some fundamental properties of this extension such as scaling, translation, rotation, Parseval’s identity, inversion theorem, and a reproducing kernel, then we derive the associated Heisenberg-Pauli-Weyl uncertainty principle UP. Finally, using the quaternion Fourier representation of the CQWT we generalize the logarithmic UP and Hardy’s UP to the CQWT domain.'
---
.2cm
**The Continuous quaternion Algebra-Valued Wavelet Transform and the Associated Uncertainty Principle**
**Youssef El Haoui$^{1,}$, Said Fahlaoui$^1$**
$^1$Department of Mathematics and Computer Sciences, Faculty of Sciences, University Moulay Ismail, Meknes 11201, Morocco\
E-MAIL: [email protected], [email protected]
**Key words:** quaternion algebra; quaternion Fourier transform; Admissible quaternion wavelet; Uncertainty principle.
Introduction
============
The wavelet transform (WT) is of great importance due to its applications in different disciplines including: signal analysis, image processing and denoising, pattern recognition, quantum mechanics, astronomy, sampling theory and other fields. The WT was introduced in the classical case, at first for one dimension by Grassman and Morlet [@GM84], Later Murenzzi [@MU89] generalized the WT to more than one dimension. Thereafter Brackx and Sommen extended the classical wavelets to Clifford algebra [@BS00; @BS01]. Considering quaternion algebra as a special case of Clifford algebra, the generalization of the WT to the quaternion framework came quite naturally [@TR01], [@ZP01]. In Ref. [@TR01], Traversoni proposed a discrete quaternion wavelet transform using the (two-sided) quaternion Fourier transform (QFT).\
Our contribution to these developments is that we introduce the two-dimensional continuous quaternion wavelet transform by means of the kernel of the two-sided QFT, and using the similitude group of the plane. We thoroughly study this generalization of the continuous wavelet transform to quaternion algebra which we call the two-sided continuous quaternion wavelet transform CQWT.\
To the best of our knowledge, the study of a CQWT from the similitude groupe $SIM(2)$ using the kernel of the two-sided QFT, has not been carried out yet. Int this regard, the novelty in the present work can be staded as follows: following the same processus as in Clifford case [@HI09], and CQWT case based on the kernel of the right sided QFT [@BAV11], we construct our new transform and investigate its important properties such as linearity, scaling, rotation, inversion formula, reproducing kernel.., we show that these properties of the two-sided CQWT can be established whenever the quaternion wavelet satisfies a particular admissibility condition. Even our generalization does not verify the spectral QFT representation property[^1]used on the case of the two-dimensional CQWT based on the kernel of the (right-sided) QFT [@BAV11] which was the key to the demonstration of the Heisenberg principle. However, one could establish the Heisenberg-Pauli-Weyl UP related for the two-sided CQWT using the UP and derivative theorem of the two-sided QFT.\
The hope is that such a transform could be useful in signal processing and optics.\
The manuscript is structured as follows: The remainder of the section 2 briefly reviews quaternions and the two-sided QFT. In section 3, we discuss the basis ideas for the construction of a CQWT based on the two-sided QFT, and derive some important properties, We then, in section 4, prove the Heisenberg-Pauli-Weyl inequality related to the CQWT, and extend the corresponding results of logarithmic UP and Hardy’s UP to the CQWT domain respectively. Finally, a conclusion is given in section 5.\
Preliminaries
=============
The quaternion algebra ${\mathbb{H}}$ over ${\mathbb{R}}$, is a special Clifford algebra $Cl_{0,2} $, it is an associative non-commutative four-dimensional algebra, its basis : $e_0,e_1,e_2,e_3$ satisfies Hamilton’s multiplication rules
$e^2_1=e^2_2=-1, e_1e_2=e_3, e_1e_2=-e_2e_1.$
Let $q=\sum^{k=3}_{k=0}{q_k}e_k, q'=\sum^{k=3}_{k=0}{q^{'}_k}e_k\in {\mathbb{H}}.$
Then the product $qq'$ is given by
$qq' = (q_0q^{'}_0-q_1q^{'}_1-q_2q^{'}_2-q_3q^{'}_3)e_0+(q_1q^{'}_0+q_0q^{'}_1-q_3q^{'}_2+q_2q^{'}_3)e_1+(q_2q^{'}_0+q_3q^{'}_1+q_0q^{'}_2-q_1q^{'}_3)e_2+(q_3q^{'}_0-q_2q^{'}_1+q_1q^{'}_2+q_0q^{'}_3)e_3.$
We define the conjugation of q $\in {\mathbb{H}}$ by :
$$\overline{q}=q_0e_0-\sum^{k=3}_{k=1}{q_k}e_k.$$
The quaternion conjugation is a linear anti-involution\
$$\label{conj}
\overline{qp}= \overline{p}\ \overline{q} ,\ \overline{p+q}= \overline{p}+\overline{q},\ \overline{\overline{p}}=p.$$
The modulus of a quaternion q is defined by:
$$|q|=\sqrt{q\overline{q}}=\sqrt{{q_0}^2{ +}\ {q_1}^2{ +}{q_2}^2+{q_3}^2}.$$
It is easy to verify that:
$$|pq|=|p||q|.$$
And $0\ne q \in {\mathbb{H}}$ implies :
$$q^{-1}=\frac{\overline{q}}{{{|q|}}^2}.$$ This means that ${\mathbb{H}}$ is a normed division algebra.
A quaternion module $L^2({{\mathbb{R}}}^2,{\mathbb{H}})$ is given by\
$$L^2\left({\mathbb{R}}^2,{\mathbb{H}}\right)=\{f=\sum^{k=3}_{k=0}{f_k}e_k~:{\mathbb{R}}^2\to {\mathbb{H}}, f_k\in L^2\left({\mathbb{R}}^2,{\mathbb{R}}\right)\ k=0,1,2,3\},$$ Let the inner product of $f,g\in L^2({\mathbb{R}}^2,{\mathbb{H}})$ be defined by $$\label{in_product}
(f,g):= \int_{{\mathbb{R}}^2}{f}(x)\ \overline{g(x)}dx.$$ If $f=g,$ we get the associated norm:
$${\left\|f\right\|}_{L^2\left({\mathbb{R}}^2, {\mathbb{H}}\right)}=\sqrt{(f,f)} =\sqrt{\int_{{\mathbb{R}}^2}{{|f(x)|}^2dx}},$$ From , we obtain the quaternion Schwartz’s inequality
$$\forall f,g\in L^2\left({{\mathbb{R}}}^2,{\mathbb{H}}\right):\ \ \ \ \ {\left|\int_{{{\mathbb{R}}}^2}{f(x)}\overline{g(x)}dx\right|}^2\le \int_{{\mathbb{R}}^2}{{\left|f(x)\right|}^2 dx}\int_{{\mathbb{R}}^2}{{\left|g(x)\right|}^2dx}.$$ We denote by ${\mathcal S}({\mathbb{R}}^2,{\mathbb{H}})$, the quaternion Schwartz space of $C^{\infty }$- functions $f$, from ${{ {\mathbb{R}}}}^2$ to ${\mathbb{H}}$, that for all $m,n \in \mathbb N$\
$${sup}_{t\in {\mathbb{R}}^2,{{\alpha}_1+{\alpha}_2\le n}}
{({\left(1+\left|t\right|\right)}^m{\left|\frac{{\partial }^{{\alpha }_1+{\alpha }_2}}{{{\partial t}_1}^{{\alpha }_1}{{\partial t}_2}^{{\alpha }_2}}f(t)\right|}_Q)}<\infty, where \ ({\alpha }_1,{\alpha }_1)\in \mathbb N^2.$$ Let $SIM(2)$denote the similitude group, a subgroup of the affine group of ${{\mathbb{R}}}^2,$ which is given by $$\textit{SIM(2)}=({\mathbb{R}}^{*}_+\times SO(2)) \otimes {\mathbb{R}}^2,$$ $=\{(a,r_{\theta },b), a>0,\ \theta \in [0,2\pi[, \ b \in {\mathbb{R}}^2\}.$
Where $SO(2),$ is the special orhtogonal group of ${{\mathbb{R}}}^2.$\
The group law of $SIM(2)$ is given by
$\{x,b\}\{x',b'\}= \{ xx',\ xb'+b\},$ where $x =ar,\ r\in SO(2)$.
The rotation operator $r_\theta \in SO(2)$ acts on $x=(x_1,x_2) \in {{\mathbb{R}}}^2$ as usual,
$$\label{rot}
r_{\theta }\left(x\right)\ =\ \left( \begin{array}{cc}
{ cos}(\theta ) & sin(\theta ) \\
-sin(\theta ) & { -cos}(\theta ) \end{array}
\right)\left( \begin{array}{c}
x_1 \\
x_2 \end{array}
\right),\ \ \ 0\le \theta < 2\pi.$$
We define $L^2\left(SIM\left(2\right),{\mathbb{H}}\right)$ as follows:\
$L^2\left(SIM\left(2\right),{\mathbb{H}}\right)=\{f\left(a,\theta ,b\right):\ \int^{+\infty }_0 {\int_{SO(2)}{ \int_{{{\mathbb{R}}}^2} {{{|f\left(a,\theta ,b\right)|}^2}d\mu(a,\theta) db<+\infty }}}\},$\
where $d\mu(a,\theta)db$ is the left Haar measure on $\textit{SIM(2)}$ with $d\mu(a,\theta)=a^{-3} dad\theta$ and $d\theta$ is the Haar measure on $SO(2).$\
Let $L^{\infty }(SIM(2),{\mathbb{H}})$ be the collection of essentially bounded measurable functions $f$ with the norm ${\left\|f\right\|}_{L^{\infty }({ SIM}(2),{\mathbb{H}})}= ess\ {sup}_{\left(a,r_\theta ,b\right)\ \in\ SIM(2)}\left|f\left(a,\theta ,b\right)\right|.$\
If $f\in \ L^{\infty }(SIM(2),{\mathbb{H}})$ is continuous, then ${\left\|f\right\|}_{L^{\infty }({ SIM}(2),{\mathbb{H}})}={{ sup}}_{\left(a,r_\theta ,b\right)\in SIM(2)}$.
For the sake of simplicity, We write $L^2\left(SIM\left(2\right),{\mathbb{H}}\right)$ as ${L}^2(SIM\left(2\right),{\mathbb{H}},dad\theta db$) and $d\mu(a,\theta)=d\mu$ and write the element $f(a,\theta,b)$ of $L^2\left(SIM\left(2\right),{\mathbb{H}}\right)$ as $f(a,r_\theta,b)$.
We introduce an inner product for $f,g:\ SIM\left(2\right)\to {\mathbb{H}}$ as follows :
$$\label{SIM_2}
<f,g> =\int_{SIM\left(2\right)}{f\left(a,\theta ,b\right)\overline{g\left(a,\theta ,b\right)}} d\mu db,$$
and we obtain $L^2\left(SIM\left(2\right),{\mathbb{H}}\right)-\ norm$
$$\label{norm_SIM_2}
{\left\|f\right\|}^2_{L^2\left(SIM\left(2\right),{\mathbb{H}}\right)}=\int_{SIM\left(2\right)}{{\left|f\left(a,\theta ,b\right)\right|}^2} d\mu db.$$
(the two-sided QFT)\
The two-sided QFT with respect to $e_1,e_2$ [@HS13], is defined by:\
For $f$ in $L^1\left({{\mathbb{R}}}^2,{\mathbb{H}}\right),$\
$$\label{QFT}
{\mathcal F}^{e_1, e_2 }\{f\}(u) =\hat f(u)=\int_{{{\mathbb{R}}}^2}{e^{-2\pi e_1 {{ u}}_1t_1}}\ f(t) e^{-2\pi e_2 {{ u}}_2t_2}dt,\ \ where ~t, u \in {{\mathbb{R}}}^2.$$
[Inverse QFT]{} ([@BU99], Thm. 2.5)\
For $f,\hat{f}\in \ L^1\left({\mathbb{R}}^2,{\mathbb{H}}\right)$, the inverse transform for the QFT is given by ${\mathcal F}^{-e_1,-e_2}$
$$\label{inverseQFT}
f(t)={\mathcal F}^{-e_1,-e_2}\left\{\hat{f}\left(\xi \right)\right\}(t)=\int_{{\mathbb{R}}^2} e^{2\pi e_1 \xi _1t_1}\ \hat{f}(\xi ) e^{2\pi e_2 \xi_2 t_2 d^2} d\xi.$$
[Derivative theorem (QFT)]{} ([@BU99], Thm. 2.10)
If $f, \frac{{\partial }^{m+n}}{{\partial }^m_{x_1}{\partial }^n_{x_2}}f\in L^2\left({\mathbb{R}}^2,{\mathbb{H}}\right)$ for $m,n\in {\mathbb N},$
Then\
$$\label{derQFT}
{\mathcal{F}^{e_1,e_2}}\{\frac{{\partial }^{m+n}}{{\partial }^m_{x_1}{\partial }^n_{x_2}}f(x)\}\left(\xi \right){ =}{(2\pi )}^{m+n}{(e_1{\xi }_1)}^m{\mathcal{F}^{e_1,e_2}}\{f(x)\}\left(\xi \right){(e_2{\xi }_2)}^n.$$
[QFT of laplacian]{}\[QFT\_laplacian\]
Let $f, \frac{{\partial }^2}{{\partial }^2_{{{x}}_{1}}}f \in $ ${{L}}^2\left({{{\mathbb{R}}}}^2,{{\mathbb{H}}}\right),\ l=1,2$
One has
$${\mathcal{F}^{e_1,e_2}}\{\triangle {f}{\}}\left(\xi \right){=-}{\left({2}\pi \right)}^2{\left|\xi \right|}^2{\mathcal{F}^{e_1,e_2}}\{{f}\}\left(\xi \right),$$
where $\triangle$ stands the Laplace operator $\triangle = \sum^{{l=2}}_{{l=1}}{\frac{{\partial }^2}{{\partial }^2_{{{x}}_{{l}}}}}.$
proof. by linearity of ${\mathcal{F}^{e_1,e_2}}$, and using we get $$\begin{aligned}
{\mathcal{F}^{e_1,e_2}}\{\triangle {f}\left({x}\right)\}(\xi)&=& {\mathcal{F}^{e_1,e_2}}\{\frac{{\partial }^2}{{\partial }^2_{{{x}}_{1}}}{f}\left({x}\right)\}\left(\xi \right)+{\mathcal{F}^{e_1,e_2}}\{\frac{{\partial }^2}{{\partial }^2_{{{x}}_2}}{f}\left({x}\right)\}\left(\xi \right)\\
&=& {(2\pi )}^2{({e}_{1}{\xi }_{1})}^2{\mathcal{F}^{e_1,e_2}}\{{f}({x})\}\left(\xi \right){+}{(2\pi )}^2{\mathcal{F}^{e_1,e_2}}\{{f}({x})\}\left(\xi \right){({e}_2{\xi }_2)}^2\\
&=& -{\left({2}\pi \right)}^2\left({{\xi }_{1}}^2{+}{\xi }^2\right){\mathcal{F}^{e_1,e_2}}\{{f}\left({x}\right)\}\left(\xi \right)\\
&=& {-}{(2\pi )}^2{\left|\xi \right|}^2{\mathcal{F}^{e_1,e_2}}\{{f}({x})\}\left(\xi \right).\hspace*{3cm} \square\end{aligned}$$
The following lemma states the Plancherel’s formula, specific to the two-sided QFT ,
[Plancherel theorem for (QFT)]{} ([@CKL15], Thm 3.2)
For $f,g \in L^2\left({\mathbb{R}}^2,{\mathbb{H}}\right):$\
$$\label{Planch}(f,g)=({\mathcal{F}^{e_1,e_2}}{f},{\mathcal{F}^{e_1,e_2}}{g}).$$
In particular , if $f=g$, we find Parseval’s formula,\
$$\label{Pars}
{\left\|f\right\|}^2_{L^2({\mathbb{R}}^2,{\mathbb{H}})}={\left\|{\mathcal{F}^{e_1,e_2}}\right\|}^2_{L^2({{\mathbb{R}}}^2,{\mathbb{H}})}.$$
The next lemma states that the QFT of a Gaussian function, is also a Gaussian function.
[QFT of a Gaussian]{}([@EF17], Lem. 3.5) $$\label{GaussQFT}
{{\mathcal{F}^{e_1,e_2}}}\left\{e^{-\pi {\left|x\right|}^2}\right\}\left(y\right)=e^{-\pi {\left|y\right|}^2},$$ where $x,y \in {{\mathbb{R}}}^2.$
[$SO({\mathbb{R}}^2)$ transformation of the QFT]{}\[SO2QFT\]
The QFT of a signal $f\in L^2({\mathbb{R}}^2,{\mathbb{H}}),\ $ with a $SO({\mathbb{R}}^2)$ transformation $A =\left( \begin{array}{cc}
{ cos}(\theta ) & -{ sin}(\theta ) \\
{ sin}(\theta ) & { cos}(\theta ) \end{array}
\right)$,\
is given by
$$\label{SO2_QFT}
{\mathcal{F}^{e_1,e_2}}\{\ f(Ax)\}(\xi )= \frac{1}{2}\{ \hat{f}( A\xi )+ \hat{f}(A^{-1}\xi )+e_1[\hat{f}\left(A\xi \right)-\hat{f}(A^{-1}\xi )]e_2\}.$$
Proof. The proof was first given by Thomas Bülow ([@BU99], Thm. 2.12) for a real 2D signal $f\in L^2({\mathbb{R}}^2,{\mathbb{R}})$. After, the result was generalized by Eckhard Hitzer (([@HI07], Thm. 2.6) for $f \in L^2({\mathbb{R}}^2,{\mathbb{H}})$.
Construction of The quaternion Algebra- Valued Wavelet Transform
================================================================
Admissible quaternion Wavelet
-----------------------------
**Definition(admissibility condition)**
A two-sided admissible quaternion wavelet is a function $\varphi \in L^2\left({\mathbb{R}}^2,\ {\mathbb{H}}\right),$ not identically zero, satisfying $$\label{ad_cond}
0 <C_{\varphi }= \int_{{\mathbb{R}}^+}{\int_{SO(2)}{{ {\left|{\mathcal{F}^{e_1,e_2}}\left\{\varphi (r_{-\theta }\left(.\right))\right\}\left(a\xi \right)\right|}^2\ }d\theta a^{-1}da}}<+\infty.$$
is called the admissibility condition, it is given in order to have the two-sided CQWT satisfy a Parseval formula.
Let AQW =$\{ \varphi \in L^2\left({\mathbb{R}}^2,\ {\mathbb{H}}\right),$ satisfying }
The inner product of AQW is given by
$${<{\varphi }_1,{\varphi }_2>}_{AQW}=\int_{{\mathbb{R}}^2}{{\widehat{{\varphi }_1}(\xi )\overline{\widehat{{\varphi }_2}}(\xi )|\xi |}^{-2}d\xi }.$$
As a consequence, AQW is a left ${\mathbb{H}}$-module.
Using the same technique as in the case of the classical wavelets, the two-dimensional quaternion wavelet ${\varphi }_{(a,\theta ,b)}\ $can be obtained from a mother $\varphi \in L^2\left({ {\mathbb{R}}}^2{ ,{\mathbb{H}}}\right)$ by the combination of dilation, translation and rotation as $$\label{wav}
{\varphi }_{(a,\theta ,b)}\left(x\right)=\frac{1}{a}\varphi (r_{-\theta }\left(\frac{x-b}{a}\right)),$$ Where $a\in {\mathbb{R}}^+, b\in {{\mathbb{R}}}^2,r_{\theta }$ is the rotation given by .
We note that if $\varphi \in L^2\left({ {\mathbb{R}}}^2{ ,{\mathbb{H}}}\right),\ $ then ${\varphi }_{(a,\theta ,b)}\in L^2\left({ {\mathbb{R}}}^2{ ,{\mathbb{H}}}\right).$
Indeed
$${\left\|{\varphi }_{(a,\theta ,b)}\right\|}^2_{ L^2\left({ {\mathbb{R}}}^2; {\mathbb{H}}\right)}\mathop{(12)}_{=}\frac{1}{{a}^2}\int_{{ {\mathbb{R}}}^2}{{\left|\varphi (r_{-\theta }\left(\frac{x-b}{a}\right))\right|}^2}dx$$ =$\int_{{ {\mathbb{R}}}^2}{{\left|\varphi (y)\right|}^2}dy\ ={\left\|\varphi \right\|}^2_{ L^2\left({ {\mathbb{R}}}^2{ ,{\mathbb{H}}}\right)}.$
Generally $<\varphi_1, \varphi_2>_{AQW}$ is quaternion-valued, so it cannot be taken out of inner product, but if $\varphi_1=\varphi_2$, then $<\varphi_1, \varphi_2>_{AQW} = C_{\varphi_1}$ is real-valued.
$$\label{Fourier_wav}
{\widehat{\varphi }}_{a,\theta ,b}\left(\xi \right)= a\ e^{-2\pi {\xi }_1b_1e_1}{\mathcal{F}^{e_1,e_2}}\{ \varphi (r_{-\theta }(.)){\}}(a\xi )e^{-2\pi {{\xi }_2b_2e_2}}.$$
Proof. $${\widehat{\varphi }}_{a,\theta ,b}\left(\xi \right)=\frac{1}{a}\int_{{\mathbb{R}}^2}{e^{-2\pi {{\xi }_1x_1e}_1}}\ \varphi (r_{-\theta }\left(\frac{x-b}{a}\right))\ e^{-2\pi {\xi }_2x_2e_2}dx$$ =$\ a\int_{{\mathbb{R}}^2}{e^{-2\pi e_1{\xi }_1(ay_1+b_1)}}\ \varphi (r_{-\theta }(y))\ e^{-2\pi e_2{\xi }_2{(ay}_2+b_2)}dx$\
= $ae^{-2\pi {\xi }_1b_1e_1}\int_{{\mathbb{R}}^2}{e^{-2\pi {a\xi }_1y_1e_1}}\ \varphi (r_{-\theta }(y))\ e^{-2\pi {{a\xi }_2y_2e}_2}dy e^{-2\pi {{\xi }_2b_2e}_2}$\
= $\ a e^{-2\pi {\xi }_1b_1e_1}{\mathcal{F}^{e_1,e_2}}\left\{ \varphi (r_{-\theta }\left(.\right))\right\}\left(a\xi \right)e^{-2\pi {{\xi }_2b_2e_2}}.\hfill \square$
\[ex\_wav\] Let $f\left(t\right)=e^{-\pi {|t|}^2}$, yields ${\mathcal{F}^{e_1,e_2}}\{-\frac{1}{4{\pi }^2}\triangle f\}(\xi)= {\left({2}\pi \right)}^2{\left|\xi \right|}^2e^{-\pi {|\xi |}^2}.$
Now, we take
$$\varphi \left(t\right)=-\frac{1}{4{\pi}^2}\triangle f\left(t\right),\ t\in {\mathbb{R}}^2.$$
Let’s prove that $\varphi $ belongs to AQW.
While it is obvious that $\varphi \in L^2\left({\mathbb{R}}^2,{\mathbb{H}}\right),\ $it has yet to be shown that
$$C_{\varphi}=\int_{{\mathbb{R}}^+}{\int_{SO(2)}{{\left|{\mathcal{F}^{e_1,e_2}}{\{}\varphi (r_{-\theta }(.))\}(\ a\ \xi )\right|}^2}}d\theta a^{-1}da<+\infty .$$
Applying lemma \[SO2QFT\], we have that for all $r_{\theta }\in SO\left(2\right),\ a>0$
$$\begin{aligned}
{\mathcal{F}^{e_1,e_2}}\left\{\ \varphi \left(r_{-\theta }\left(.\right)\right)\right\}\left(\ a\ \xi \right)&=&\frac{1}{2}\{ \widehat{\varphi } (a\ r_{-\theta }(\xi ))+ \widehat{\varphi } (a\ r_{\theta }(\xi ))+ e_1(\widehat{\varphi }{\ (}a\ r_{-\theta }(\xi ){)-}\widehat{\varphi }{\ (}a\ r_{\theta }(\xi ))e_2\}\\
&=& a^2{\left|\xi \right|}^2e^{-\pi a^2{|\xi |}^2}.\end{aligned}$$
Where in the last equality we applied $|r_{\theta }(\xi )|$=$\ |r_{\theta }(\xi )|$=$\ |\xi |$.
Thus we have $$\begin{aligned}
C_{\varphi }&=&\int_{{\mathbb{R}}^+}{\int_{SO(2)}{{\left|{\mathcal{F}^{e_1,e_2}}{\{}\ \varphi (r_{-\theta }(.))\}(\ a\ \xi )\right|}^2}}d\theta a^{-1}da\\
&=& 2\pi {\left|\xi \right|}^{{4}}\int^{+\infty }_0{a^3}e^{-2\pi a^2{\left|\xi \right|}^2}da<+\infty.\ \ \ \ \ \ \ \hfill \square\end{aligned}$$
As ${\mathcal{F}^{e_1,e_2}}\left\{{\ }\varphi \left({{r}}_{{-}\theta }\left({.}\right)\right)\right\}\left({\ a\ }\xi \right){\ }$in example \[ex\_wav\], is real-valued function, then it commutes with $e_2$, hence one see that the wavelet in this example satisfy the assumption of the main theorems of this article.
Furthemore\
$\widehat{\varphi }\left(0\right){=0,}$ Then $\int_{{{\mathbb{R}}}^2}{\sum^{{k=3}}_{{k=0}}{{\varphi }_{{k}}{(x)}}{e}_{{k}}}{\ dx}$=0,
That is $\int_{{{\mathbb{R}}}^2}{{\varphi }_{{k}}{(x)}}$=0, k=0,1,2,3.,
which means, similar to classical wavelets, that the integral of every component ${\varphi }_{{k}}$ is zero.
Two- Dimensional Continuous quaternion Wavelet Transform
--------------------------------------------------------
Let $\varphi \in $ AQW, the two-sided CQWT $:T_{\varphi }$ is defined by
$T_{\varphi }:\ L^2\left({\mathbb{R}}^2,{\mathbb{H}}\right)\to L^2\left(SIM(2),{\mathbb{H}}\right)$
$$\label{CQWT}
f\rightarrowtail T_{\varphi }f:\left(a,\theta ,b\right)\rightarrowtail T_{\varphi }f\left(a,\theta ,b\right) = (f,{\varphi }_{a,\theta ,b})=\ \int_{{\mathbb{R}}^2}{f}(x)\ \overline{\frac{1}{a}\varphi (\frac{r_{-\theta }(x-b)}{a})}dx.$$
We now investigate some basic properties of the CQWT:
Its Properties
--------------
Let$\ \varphi ,\psi $ are quaternion admissible wavelets.
If $f,g\ \in L^2({\mathbb{R}}^2,{\mathbb{H}})$, then
1. Left linearity:$\ T_{\varphi }(\lambda f+\mu g)\left(a,\theta ,b\right) = \lambda T_{\varphi }f\left(a,\theta ,b\right)$+${\mu T}_{\varphi }g\left(a,\theta ,b\right).\ $ For arbitrary quaternion constants $\lambda , \mu \in {\mathbb{H}}.$
2. Anti-linearity: $T_{\lambda \varphi +\mu \psi }f\left(a,\theta ,b\right)=T_{\varphi }f\left(a,\theta ,b\right)\overline{\lambda }+T_{\psi }f\left(a,\theta ,b\right) \overline{\mu } .$ Where $\lambda$ and $\ \mu$ are constants in$\ {\mathbb{H}}.$
3. Scaling: $T_{\varphi }f\left(c.\right)\left(a,\theta ,b\right)=\frac{1}{c}T_{\varphi }f\left(c.\right)\left(ca,\theta ,cb\right):\ \ c\ \in {\mathbb{R}}^*$.
4. Translation: $T_{\varphi }{\tau }_cf\left(a,\theta ,b\right)=T_{\varphi }f\left(a,\theta ,b-c\right).$ where $c\ \in {\mathbb{R}}^2$, and ${\tau }_c\ $ is the translation operator given by ${\tau }_c f\left(.\right)=f(.-c).$
5. Rotation: $T_{\varphi }f(r_{\omega }.)\left(a,\theta ,b\right)\ =T_{\varphi }f\left(a,\theta +\omega ,r_{\omega }\left(b\right)\right)$, where $r_{\omega }\ $is a rotation.
Proof.
1. For $\lambda ,\mu \in {\mathbb{H}}$, we have $T_{\varphi }\left(\lambda f+\mu g\right)\left(a,\theta ,b\right)=(\lambda f+\mu g,{\varphi }_{a,\theta ,b})$\
$=\lambda \left(f,{\varphi }_{a,\theta ,b}\right)+\mu (g,{\varphi }_{a,\theta ,b})$\
=$\lambda T_{\varphi }f\left(a,\theta ,b\right)$+${\mu T}_{\varphi }g\left(a,\theta ,b\right).$
2. For $\lambda , \mu \in {\mathbb{H}}$, we have $T_{\lambda \varphi +\mu \psi }f\left(a,\theta ,b\right)=\int_{{\mathbb{R}}^2}{f}$*(*$t$*)*$\ \overline{\frac{1}{a}(\lambda \varphi +\mu \psi )(r_{-\theta }(\frac{t-b}{a})}dt$\
$=\int_{{\mathbb{R}}^2}{f}$*(*$t$*)*$\ \overline{\frac{1}{a}\varphi (r_{-\theta }(\frac{t-b}{a})}dt\overline{\lambda }$*+*$\int_{{\mathbb{R}}^2}{f}$*(*$t$*)*$\ \overline{\frac{1}{a}\psi (r_{-\theta }(\frac{t-b}{a})}dt\overline{\mu }$\
${=T}_{\varphi }f\left(a,\theta ,b\right)\overline{\lambda }+T_{\varphi }f\left(a,\theta ,b\right)\ \overline{\mu } .$
3. For$\ c \in {\mathbb{R}}^*,$ we have $T_{\varphi }f\left(c.\right)\left(a,\theta ,b\right)=(f\left(c.\right),{\varphi }_{a,\theta ,b})$\
$=\int_{{\mathbb{R}}^2}{f}(ct) \overline{\frac{1}{a}\varphi (r_{-\theta }(\frac{t-b}{a})}dt$\
=$ \frac{1}{c}\int_{{\mathbb{R}}^2}{f}(u) \overline{\frac{1}{a}\varphi (r_{-\theta }(\frac{u-bc}{ac})}du$\
=$\frac{1}{c}T_{\varphi }f\left(c.\right)\left(ca,\theta ,cb\right).$
4. For$\ c \in {\mathbb{R}}^2,\ { we\ have}\ $ $T_{\varphi }{\tau }_cf\left(a,\theta ,b\right)=({\tau }_cf,{\varphi }_{a,\theta ,b})$\
=$ \int_{{\mathbb{R}}^2}{f}(t-c) \overline{\frac{1}{a}\varphi (r_{-\theta }(\frac{t-b}{a})}dt$\
=$\int_{{\mathbb{R}}^2}{f}(u)\ \overline{\frac{1}{a}\varphi (r_{-\theta }(\frac{u-(b-c)}{a})}du$ (By change of variable $t-c=u$)\
=$T_{\varphi }f\left(a,\theta ,b-c\right).$
5. Applying and using the fact that $r^{-1}_{\omega }=r_{-\omega }$, and $r_{\theta }r_{\omega }=r_{\left(\theta +\omega \right),}$ we obtain
$ T_{\varphi }f(r_{\omega }.)\left(a,\theta ,b\right)=\int_{{\mathbb{R}}^2}{f}(r_{\omega }t)\overline{\frac{1}{a}\varphi (r_{-\theta }\left(\frac{t-b}{a}\right))}dt$\
$=\int_{{\mathbb{R}}^2}{f}(u) \overline{\frac{1}{a}\varphi (r_{-\theta }\left(\frac{r_{-\omega }u-b}{a}\right))}\ \ {det}^{-1}(r_{\omega })du$\
$=\int_{{\mathbb{R}}^2}{f}(u) \overline{\frac{1}{a}\varphi (r_{-\theta }r_{-\omega }\left(\frac{u-r_{\omega }\left(b\right)}{a}\right))}du$\
$=\int_{{\mathbb{R}}^2}{f}(u)\overline{\frac{1}{a}\varphi (r_{-\left(\theta +\omega \right)}\left(\frac{u-r_{\omega }\left(b\right)}{a}\right))}du$\
=$T_{\varphi }f\left(a,\theta +\omega ,r_{\omega }\left(b\right)\right)\hfill \square.$
\[Pars\_CQWT\](Parseval’s identity for the CQWT).
Suppose that $\varphi \in AQW$ be a quaternion admissible wavelet, and assume that ${\mathcal{F}^{e_1,e_2}}\left\{\varphi \left({{r}}_{-\theta}\left(.\right)\right)\right\}$ commute with $e_2$, then for every $f,\ g\in L^2({\mathbb{R}}^2, {\mathbb{H}}),$ we have
$$\label{Pars_CQWT}
<T_{\varphi }f,T_{\varphi}g>\ = C_{\varphi }(f,g).$$
Proof. $\ <T_{\varphi }f,T_{\varphi }g>\ =\int_{SIM\left(2\right)}{T_{\varphi }f}{\left(a,\theta ,b\right)\overline{T_{\varphi }g\left(a,\theta ,b\right)}}d\mu db$\
=$\int_{SIM\left(2\right)}{[\int_{{\mathbb{R}}^2}{f}(x)\ \overline{{\varphi }_{a,\theta ,b}(x)}dx\ \ \ ]}{\overline{[\int_{{\mathbb{R}}^2}{g}(y)\ \overline{{\varphi }_{a,\theta ,b}(x)}dy]}}d\mu db$ $$\mathop{\eqref{Planch}}_=\ \int_{SIM\left(2\right)}{[\int_{{\mathbb{R}}^2}{\hat{f}}(\xi )\ \overline{\widehat{{\varphi }_{a,\theta ,b}}(\xi )}d\xi ]}{\overline{[\int_{{\mathbb{R}}^2}{\hat{g}}(\eta )\ \overline{\widehat{{\varphi }_{a,\theta ,b}}(\eta )}d\eta ]}}d\mu db$$ $=\int_{SIM\left(2\right)}{[\int_{{\mathbb{R}}^2}{\hat{f}}(\xi )e^{2\pi{{\xi }_2b_2e}_2}\overline{{\mathcal{F}^{e_1,e_2}}\left\{\varphi {(r}_{-\theta }\left(.\right))\right\}(a\xi )}e^{2\pi {{\xi }_1b_1e}_1}d\xi]}$\
$\times {[\int_{{\mathbb{R}}^2}{e^{-2\pi {{\eta }_1b_1e}_1}{\mathcal{F}^{e_1,e_2}}\left\{\varphi {(r}_{-\theta }\left(.\right))\right\}(a\eta )e^{-2\pi {{\eta }_2b_2e}_2}}\overline{\hat{g}(\eta )}d\eta ]]a}^{-1}dad\theta db$\
(By and ).\
Since ${\mathcal{F}^{e_1,e_2}}\left\{\varphi \left({{r}}_{-\theta}\left(.\right)\right)\right\}$ commute with $e_2,$ we have\
$<T_{\varphi }f,T_{\varphi }g>\ = \int^{+\infty }_0{\int_{SO\left(2\right)}{\int_{{\mathbb{R}}^2}{\int_{{\mathbb{R}}^2}{\hat{f}\left(\xi \right)\left[\int_{{\mathbb{R}}^2}{{e^{2\pi b_2\left({\xi }_2-{\eta }_2\right)e_2}e}^{2\pi b_1\left({\xi }_1-{\eta }_1\right)e_1}}db\right]}}}}\ \overline{{\mathcal{F}^{e_1,e_2}}\left\{\varphi (r_{-\theta }\left(.\right))\right\}\left(a\xi \right)}$\
$\times {\mathcal{F}^{e_1,e_2}}\left\{\varphi {(r}_{-\theta }\left(.\right))\right\}\left(a\eta \right)\overline{\hat{g}\left(\eta \right)} d\eta d\xi d\theta a^{-1}da$\
$=\int^{+\infty }_0{\int_{SO\left(2\right)}{\int_{{\mathbb{R}}^2}{\int_{{\mathbb{R}}^2}{\hat{f}\left(\xi \right)\delta (\xi { -}\eta )}}}}\ \overline{{\mathcal{F}^{e_1,e_2}}\left\{\varphi {(r}_{-\theta }\left(.\right))\right\}\left(a\xi \right)}{\mathcal{F}^{e_1,e_2}}\left\{\varphi (r_{-\theta }\left(.\right))\right\}\left(a\eta \right)\overline{\hat{g}\left(\eta \right)}d\eta d\xi d\theta a^{-1}da,$ In the second equality we applied the orthogonality of harmonic exponential functions.\
Furthermore, we get\
$ <T_{\varphi }f,T_{\varphi }g> =\int^{+\infty }_0{\int_{SO\left(2\right)}{\int_{{\mathbb{R}}^2}{\hat{f}\left(\xi \right)}}} \overline{{\mathcal{F}^{e_1,e_2}}\left\{\varphi {(r}_{-\theta }\left(.\right))\right\}\left(a\xi\right)}{\mathcal{F}^{e_1,e_2}}\left\{\varphi ({(r}_{-\theta }\left(.\right))\right\}\left(a\xi \right)\overline{\hat{g}\left(\xi \right)}d\xi a^{-1}d\theta da$ $$\mathop{{ Fubini}}_{=}\int_{{\mathbb{R}}^2}{\hat{f}\left(\xi \right){ [}\int_{{\mathbb{R}}^+}{\int_{SO(2)}{{\ \ \ {\left|{\mathcal{F}^{e_1,e_2}}\left\{\varphi {(r}_{-\theta }\left(.\right))\right\}\left(a\xi \right)\right|}^2} \ d\theta a^{-1}da]\overline{\widehat{g}\left(\xi \right)}}}}d\xi$$ $=(\hat{f}C_{\varphi },\hat{g}$) (by and and )\
$=C_{\varphi }$ ($\hat{f},\hat{g}$) ($C_{\varphi }$ is a real constant)\
$=C_{\varphi } (f,g).$ (by using ) .
Hence the theorem follows. $\hfill \square$
Theorem \[Pars\_CQWT\] could be interpreted as preservation of energy by CQWT.
The following corollary follows directly from Theorem \[Pars\_CQWT\].
(Plancherel’s Formula for CQWT).\
Suppose that $\varphi $ is quaternion admissible wavelet with ${\mathcal{F}^{e_1,e_2}}\left\{\varphi \left({{r}}_{-\theta}\left(.\right)\right)\right\}$ commute with $e_2$,\
then for every $f\in L^2({\mathbb{R}}^2,{\mathbb{H}})$ we have\
$$\label{Planch_CQWT}
{\left\|T_{\varphi }f\right\|}^2_{L^2(SIM(2),{\mathbb{H}})} = {C_{\varphi }\left\|f\right\|}^2_{L^2({\mathbb{R}}^2,{\mathbb{H}})}.$$ Thus, exept for the factor $C_{\varphi },$ CQWT is an isometry from $L^2({\mathbb{R}}^2,{\mathbb{H}})$ to $L^2(SIM(2),{\mathbb{H}}).$
The inversion formula for the CQWT is given by the following theorem .
(Inversion theorem for CQWT).\
If $f\in L^2({\mathbb{R}}^2,{\mathbb{H}})$, then $f$ can be reconstructed by the formula\
$$\label{inverse_CQWT}
f\left(t\right)=\frac{1}{C_{\varphi }}\int_{SIM(2)}{T_{\varphi }f\left(a,\theta ,b\right){\varphi }_{a,\theta ,b}} d\mu db.$$
Proof. Applying theorem \[Pars\_CQWT\], we obtain for every $g\in L^2({\mathbb{R}}^2,{\mathbb{H}})$,\
$C_{\varphi }(f,\ g) =\int_{SIM\left(2\right)}{T_{\varphi }f}{\left(a,\theta ,b\right)\overline{T_{\varphi }g\left(a,\theta ,b\right)}}d\mu db$\
=$\int_{SIM\left(2\right)}{T_{\varphi }f}{\left(a,\theta ,b\right)\int_{{\mathbb{R}}^2}{{\varphi }_{a,\theta ,b}(t)}\overline{g(t)}}dtd\mu db$\
=$\int_{{\mathbb{R}}^2}{[\int_{SIM\left(2\right)}{T_{\varphi }f}}\left(a,\theta ,b\right){\varphi }_{a,\theta ,b}(t)d\mu db]\overline{g\left(t\right)}dt$ $$\label{aide}=(\int_{SIM\left(2\right)}{T_{\varphi }f\left(a,\theta ,b\right){\varphi }_{a,\theta ,b}\left(t\right)d\mu db,g}).$$ Where in the third equality we applied Fubini’s theorem to interchange the order of integrations.\
Since holds for every $g\in L^2({\mathbb{R}}^2,{\mathbb{H}})$, it follows, therefore\
$f(t) =\frac{1}{C\varphi}\int_{SIM\left(2\right)}{T_{\varphi }f\left(a,\theta ,b\right){\varphi }_{a,\theta ,b}\left(t\right)d\mu db}.\hfill \square$\
Next, let’s establish the Reproducing kernel theorem of CQWT
(Reproducing Kernel).\
Let$\ \varphi \ be\ $ a quaternion admissible wavelet.
We have\
$$\label{rep}
T_{\varphi'}f(a',\theta',b')= \frac{1}{C_{\varphi }}\int_{SIM\left(2\right)}{T_{\varphi }f\left(a,\theta ,b\right)({\varphi }_{a,\theta ,b},{{\varphi' }}_{a',\theta ',b'})}d\mu db.$$
Proof. $T_{\varphi'}f(a',\theta',b')=\int_{{\mathbb{R}}^2}{f}(t)\overline{{\varphi'}_{a',\theta ',b'}(t)}dt$(by )\
=$\int_{{\mathbb{R}}^2}{[\frac{1}{C_{\varphi}}\int_{SIM\left(2\right)}{T_{\varphi }f\left(a,\theta ,b\right){\varphi }_{a,\theta ,b}\left(t\right)d\mu db}}]\overline{{{\varphi' }}_{a',\theta',b'}(t)}db$ (by applying )\
=$\frac{1}{C_{\varphi }}\int_{SIM\left(2\right)}{T_{\varphi }f\left(a,\theta ,b\right)[\int_{{\mathbb{R}}^2}{{\varphi }_{a,\theta ,b}\left(t\right)\overline{{{\varphi' }}_{a',\theta ',b'}(t)}}dt]} d\mu db$\
=$\frac{1}{C_{\varphi }}\int_{SIM\left(2\right)}{T_{\varphi }f\left(a,\theta ,b\right)({\varphi }_{a,\theta ,b},{{\varphi' }}_{a',\theta ',b'})}d\mu db.$\
which completes the proof. $\hfill \square$
\[Haus\_CQWT\]
Let $\varphi$ be a quaternion admissible wavelet.\
For every ${f\in L}^2({\mathbb{R}}^2,{\mathbb{H}})$, we have $T_{\varphi }f\in L^p\left({\mathbb{R}}^2,{\mathbb{H}}\right),2 \le p\le \infty .$\
And the following inequality holds $${\left\|T_{\varphi }f\right\|}_{L^p\left({ SIM(2),{\mathbb{H}}}\right)}\le C^{\frac{ 1}p}_{\varphi }{{\left\|\varphi \right\|}^{{ 1-}\frac2p}_{L^2\left({\mathbb{R}}^2,{\mathbb{H}}\right)}\left\|f\right\|}_{L^2\left({\mathbb{R}}^2,{\mathbb{H}}\right)}.$$
Proof. For $p=2$, by (17) we have\
${\left\|T_{\varphi }f\right\|}_{L^2\left(SIM(2),{\mathbb{H}}\right)}=\ C^{\frac{1}{2}}_{\varphi }{\left\|f\right\|}_{L^2\left({\mathbb{R}}^2,{\mathbb{H}}\right)}.$\
For $ p=\infty $, the theorem is obtained by Hôlder’s inequality, indeed\
$|T_{\varphi }f\left(a,\theta ,b\right)|=|\int_{{\mathbb{R}}^2}{f}(x)\ \overline{{\varphi }_{a,\theta ,b}(x)}dx|$\
$\le {{\left\|\varphi \right\|}_{L^2\left({\mathbb{R}}^2,{\mathbb{H}}\right)}\ \left\|f\right\|}_{L^2\left({\mathbb{R}}^2,{\mathbb{H}}\right)}.$\
Thus ${\left\|T_{\varphi }f\right\|}_{L^{\infty }\left(SIM(2),{\mathbb{H}}\right)}\le \ {{\left\|\varphi \right\|}_{L^2\left({\mathbb{R}}^2,{\mathbb{H}}\right)}\ \left\|f\right\|}_{L^2\left({\mathbb{R}}^2,{\mathbb{H}}\right)}.$\
Let us show the theorem for the case $2<p<\infty ,$\
From the two previous cases we have $T_{\varphi }$ is a bounded linear operator of type (2,2) with norm $C^{\frac{ 1}{2}}_{\varphi }$,\
and it is of type (2,$\infty $) with norm bounded by ${\left\|\varphi \right\|}_{L^2\left({\mathbb{R}}^2,{\mathbb{H}}\right)}$,\
Therefore Riesz-Thorin interpolation theorem ([@LP04], Thm. 2.1) guarantees that\
$T_{\varphi }$ is bounded from $L^{p_{\alpha }}\left({\mathbb{R}}^2,{\mathbb{H}}\right)$) in $L^{{q}_{\alpha }}\left(SIM(2),{\mathbb{H}}\right)$ with norm $M_{\alpha }$ such that $M_{\alpha }$ $\le {(C^{\frac{ 1}2}_{\varphi })}^{1-\alpha }{\left\|\varphi \right\|}^{\alpha }_{L^2\left({\mathbb{R}}^2,{\mathbb{H}}\right)},$\
whith $\frac{ 1}{p_{\alpha }}$=$\frac{{ 1-}\alpha }2+\frac{\alpha }2=\frac{ 1}2,\ \ \ \frac{ 1}{{q}_{\alpha }}=\frac{{ 1-}\alpha }2+\frac{\alpha }{\infty }=\frac{{ 1-}\alpha }2,\ \ \ \ \ \ 0<\alpha <1$,\
Thus $p_{\alpha }{ =}$2, and by taking $p=q_{\alpha }{,\ i.e}\alpha { =1-}\frac2p$\
we get $T_{\varphi }$ is of type $(2,p),\ \ \ 2{ <}p<\infty ,$ with norm $M_{\alpha }$ bounded by ${C^{\frac{ 1}p}_{\varphi }}{\left\|\varphi \right\|}^{{ (1-}\frac2p)}_{L^2\left({\mathbb{R}}^2,{\mathbb{H}}\right)}.\hfill \square$
Uncertainty Principles for the Two- Dimensional Continuous quaternion Wavelet Transform
========================================================================================
In this section we will prove the famous Heisenberg-Weyl’sUP, and its logarihmic version for the CQWT. Moreover, we establish Hardy’s theorem in the setting of the CQWT.
Heisenberg-Weyl’s uncertainty principle
---------------------------------------
It is known that Heisenberg-Weyl’s UP for the two-sided QFT states that a nonzero quatenion algebra-valued function and its QFT cannot both be sharply localized [@CKL15].
In what follows, we will extend the validity of the Heisenberg-Weyl’s inequality for the CQWT.
\[Heisenberg1\] Let Let $\varphi $ be a quaternion admissible wavelet, with ${\mathcal{F}^{e_1,e_2}}\left\{\varphi \left({{r}}_{-\theta}\left(.\right)\right)\right\} \in {\mathbb{R}}+{\mathbb{R}}e_2,$ If $f,\frac{{\partial }^2}{{\partial }^2_{b_1}}f\in L^2({\mathbb{R}}^2,{\mathbb{H}})$
Then
$$\label{Heis_CQWT}
\int_{SO(2)}{\int_{{\mathbb{R}}^+}{{\left\|{ {\xi }_l{\mathcal{F}^{e_1,e_2}}\{T}_{\varphi }f\left(a,\theta ,.\right)\}\right\|}^2_{L^2({\mathbb{R}}^2,{\mathbb{H}})}d\mu }} = \ C_{\varphi } {\left\|{\xi }_l\hat{f}\right\|}^2_{L^2({\mathbb{R}}^2,{\mathbb{H}})}.\ \ \l=1,2,$$
Proof.
For $l=1,$ we have
$\int_{SO(2)}{\int_{{\mathbb{R}}^+}{{\left\|{{\xi }_1{\mathcal{F}^{e_1,e_2}}\{T}_{\varphi }f\left(a,\theta ,.\right)\}\right\|}^2_{L^2({\mathbb{R}}^2,{\mathbb{H}})}d\mu }}$\
$$\label{Heis_CQWT2}
\int_{SO(2)}{\int_{{\mathbb{R}}^+}{{ {\int_{{\mathbb{R}}^2}{{\xi }^2_1}{\mathcal{F}^{e_1,e_2}}}\{T}_{\varphi }f}{\left(a,\theta ,b)\}(\xi \right)\overline{{{\mathcal{F}^{e_1,e_2}}\{T}_{\varphi }f\left(a,\theta ,b\right)\}\left(\xi \right)}}d\xi d\mu }$$ By using , we obtain
${\mathcal{F}^{e_1,e_2}}\{\frac{{\partial }^2}{{\partial }^2_{b_1}}f\left(a,\theta ,b\right)\}\left(\xi \right)={-\left(2\pi \right)}^2{\xi }^2_1{\mathcal{F}^{e_1,e_2}}\{f\left(a,\theta ,b\right)\}\left(\xi \right)$,
Then becomes
$\int_{SO\left(2\right)}{\int_{{\mathbb{R}}^+}{{\left\|{{\xi }_1{\mathcal{F}^{e_1,e_2}}\{T}_{\varphi }f\left(a,\theta ,.\right)\}\right\|}^2_{L^2\left({\mathbb{R}}^2,{\mathbb{H}}\right)}d\mu }}$\
$=\int_{SO(2)}{\int_{{\mathbb{R}}^+}{\int_{{\mathbb{R}}^2}{{\mathcal{F}^{e_1,e_2}}\{}-\frac{1}{{\left(2\pi \right)}^2}\frac{{\partial }^2}{{\partial }^2_{b_1}}T_{\varphi }f}{\left(a,\theta ,b)\}(\xi \right)\overline{{{\mathcal{F}^{e_1,e_2}}\{T}_{\varphi }f\left(a,\theta ,b\right)\}\left(\xi \right)}}d\xi d\mu }$
=$\int_{SO\left(2\right)}{\int_{{\mathbb{R}}^+}{\int_{{\mathbb{R}}^2}{-\frac{1}{{\left(2\pi \right)}^2}\frac{{\partial }^2}{{\partial }^2_{b_1}}}T_{\varphi }f}{\left(a,\theta ,b\right)\overline{T_{\varphi }f\left(a,\theta ,b\right)}}dbd\mu }$ (Using )\
=$\int_{SO\left(2\right)}{\int_{{\mathbb{R}}^+}{\int_{{\mathbb{R}}^2}{-\frac{1}{{\left(2\pi \right)}^2}\frac{{\partial }^2}{{\partial }^2_{b_1}}}[\int_{{\mathbb{R}}^2}{f}(x)\ \overline{{\varphi }_{a,\theta ,b}(x)}dx\ \ \ ]}\overline{[\int_{{\mathbb{R}}^2}{f}(y)\ \overline{{\varphi }_{a,\theta ,b}(y)}dy]}dbd\mu}$\
(By )\
=$\int_{SO\left(2\right)}{\int_{{\mathbb{R}}^+}{\int_{{\mathbb{R}}^2}{-\frac{1}{{\left(2\pi \right)}^2}\frac{{\partial }^2}{{\partial }^2_{b_1}}}[\int_{{\mathbb{R}}^2}{\hat{f}}(\xi )\ \overline{\widehat{{\varphi }_{a,\theta ,b}}(\xi )}d\xi \ \ ]}\overline{[\int_{{\mathbb{R}}^2}{\hat{f}}(\eta )\ \overline{\widehat{{\varphi }_{a,\theta ,b}}(\eta )}d\eta ]}dbd\mu}$\
(by using again)\
$=a^2\int_{SO(2)}{\int_{{\mathbb{R}}^+}{\int_{{\mathbb{R}}^2}{-\frac{1}{{\left(2\pi \right)}^2}}\int_{{\mathbb{R}}^2}{\hat{f}}(\xi )\frac{{\partial }^2}{{\partial }^2_{b_1}}\ [\overline{ae^{-2\pi {\xi }_1b_1e_1}{\mathcal{F}^{e_1,e_2}}\left\{\varphi {(r}_{-\theta }\left(.\right))\right\}\left(a\xi \right)e^{-2\pi {{\xi }_2b_2e}_2}}]}d\xi}$\
$\times\int_{{\mathbb{R}}^2}{e^{-2\pi {{\eta }_1b_1e}_1}{{\mathcal{F}^{e_1,e_2}}\left\{\varphi {(r}_{-\theta }\left(.\right))\right\}\left(a\eta \right)e}^{-2\pi{{\eta }_2b_2e}_2}}\overline{\hat{f}(\eta )}d\eta ]dbd\mu $\
(by applying )\
= $a^2\int_{SO\left(2\right)}{\int_{{\mathbb{R}}^+}{\int_{{\mathbb{R}}^2}{-\frac{1}{{\left(2\pi \right)}^2}}[\int_{{\mathbb{R}}^2}{\hat{f}}(\xi )e^{2\pi {{\xi }_2b_2e}_2}\ \overline{{\mathcal{F}^{e_1,e_2}}\left\{\varphi {(r}_{-\theta }\left(.\right))\right\}\left(a\xi \right)}{\frac{{\partial }^2}{{\partial }^2_{b_1}}[e}^{2\pi {\xi }_1b_1e_1}]d\xi }}$\
$\times\int_{{\mathbb{R}}^2}{e^{-2\pi {{\eta }_1b_1e}_1}{\mathcal{F}^{e_1,e_2}}\left\{\varphi {(r}_{-\theta }\left(.\right))\right\}\left(a\eta \right)e^{-2\pi {{\eta }_2b_2e}_2}}\overline{\hat{f}(\eta )}d\eta ]dbd\mu$\
=$\int_{SO(2)}{\int_{{\mathbb{R}}^+}{\int_{{\mathbb{R}}^2}{{\xi }^2_1}[\int_{{\mathbb{R}}^2}{\hat{f}}(x)e^{2\pi {{\xi }_2b_2e}_2}\overline{{\mathcal{F}^{e_1,e_2}}\left\{\varphi {(r}_{-\theta }\left(.\right))\right\}\left(a\xi \right)}e^{2\pi {\xi }_1b_1e_1}d\xi }}$\
$\times\int_{{\mathbb{R}}^2}{e^{-2\pi {{\eta }_1b_1e}_1}{\mathcal{F}^{e_1,e_2}}\left\{\varphi {(r}_{-\theta }\left(.\right))\right\}\left(a\eta \right)e^{-2\pi {{\eta }_2b_2e}_2}}\overline{\hat{f}(\eta )}d\eta ]db\ a^{-1}dad\theta $\
=$\int_{SO(2)}{\int_{{\mathbb{R}}^+}{\int_{{\mathbb{R}}^2}{{\int_{{\mathbb{R}}^2}{}\xi }^2_1\hat{f}(\xi )}\overline{{\mathcal{F}^{e_1,e_2}}\left\{\varphi {(r}_{-\theta }\left(.\right))\right\}\left(a\xi \right)}[}\int_{\mathbb{R}}{e^{2\pi {{(\xi }_1{-{\eta }_1)b}_1e}_1}{db}_1\int_{\mathbb{R}}{e^{2\pi {{(\xi }_2{-{\eta }_2)b}_2e}_2}}}db_2]}$\
$\times{\mathcal{F}^{e_1,e_2}}\left\{\varphi {(r}_{-\theta }\left(.\right))\right\}\left(a\eta \right)\overline{\hat{f}(\eta )}d\eta d\xi a^{-1}dad\theta. $\
Where we assume that ${\mathcal{F}^{e_1,e_2}}\left\{\varphi \left({{r}}_{-\theta}\left(.\right)\right)\right\} \in {\mathbb{R}}+{\mathbb{R}}e_2$, i.e ${\mathcal{F}^{e_1,e_2}}\left\{\varphi \left({{r}}_{-\theta}\left(.\right)\right)\right\}$ commute with $e_2$.
Moreover, as $\int_{{\mathbb{R}}}{e^{2\pi {{(\xi }_l{-{\eta }_l)b}_le}_l}db_l}={\delta }({\xi }_l-{\eta }_l)$, for l=1,2.\
We obtain $\int_{SO\left(2\right)}{\int_{{\mathbb{R}}^+}{{\left\|{{\xi }_1{\mathcal{F}^{e_1,e_2}}\{T}_{\varphi }f\left(a,\theta ,.\right)\}\right\|}^2_{L^2\left({\mathbb{R}}^2,{\mathbb{H}}\right)}d\mu }}$ $$\mathop{{ Fubini}}_{=}\int_{{\mathbb{R}}^2}{{\left|{\xi }_1\hat{f}(\xi )\right|}^2{ [}\int_{{\mathbb{R}}^+}{\int_{SO(2)}{{{\left|{\mathcal{F}^{e_1,e_2}}\left\{\varphi {(r}_{-\theta }\left(.\right))\right\}\left(a\xi \right)\right|}^2}d\theta a^{-1}da]}}}d\xi$$ $=C_{\varphi } {|\ {\xi }_1\hat{f}(\xi )|}^2_{L^2({\mathbb{R}}^2,{\mathbb{H}})}.$ ($C_{\varphi }$ is a real constant)\
For $l=2,$ the argument is similar to the one used for $l=1$.\
The proof is complete.$\square$
\[Heisenberg2\] Under the same conditions as in Lemma \[Heisenberg1\], one has $$\int_{SO(2)}{\int_{{\mathbb{R}}^+}{{\left\|{\left|\xi \right|{\mathcal{F}^{e_1,e_2}}\{T}_{\varphi }f\left(a,\theta ,.\right)\}\right\|}^2_{L^2({\mathbb{R}}^2,{\mathbb{H}})}d\mu }}=C_{\varphi } {\left\|\left|\xi \right|\hat{f}\right\|}^2_{L^2({\mathbb{R}}^2,{\mathbb{H}})}.$$
Proof. We have\
$\int_{SO(2)}{\int_{{\mathbb{R}}^+}{{\left\|{\left|\xi \right|{\mathcal{F}^{e_1,e_2}}\{T}_{\varphi }f\left(a,\theta ,.\right)\}\right\|}^2_{L^2({\mathbb{R}}^2,{\mathbb{H}})}d\mu }}=\int_{SO(2)}{\int_{{\mathbb{R}}^+}{\int_{{\mathbb{R}}^2}{{{|\xi |}^2\left|{{\mathcal{F}^{e_1,e_2}}\{T}_{\varphi }f\left(a,\theta ,.\right)\right\}(\xi )|}^2d\xi }d\mu }}$\
=$\int_{SO\left(2\right)}{\int_{{\mathbb{R}}^+}{\int_{{\mathbb{R}}^2}{{({\xi }^2_1{ +}{\xi }^2_2)\left|{{\mathcal{F}^{e_1,e_2}}\{T}_{\varphi }f\left(a,\theta ,.\right)\right\}(\xi )|}^2d\xi }d\mu }}$\
=$\ C_{\varphi }\int_{{\mathbb{R}}^2}{{\xi }^2_1}{|\hat{f}(\xi )|}^2d\xi + C_{\varphi }\int_{{\mathbb{R}}^2}{{\xi }^2_2}{|\hat{f}(\xi )|}^2d\xi $\
(by linearity of the integral, and (21))\
= $\ C_{\varphi }\int_{{\mathbb{R}}^2}{{\left|\xi \right|}^2}{\left|\hat{f}\left(\xi \right)\right|}^2d\xi $\
$=\ C_{\varphi } {\left\|\left|\xi \right|\hat{f}\right\|}^2_{L^2({\mathbb{R}}^2,{\mathbb{H}})}.\hfill \square$
The following proposition is Heisenberg–Weyl’s inequality related to the two-sided QFT, it is a generalization of the inequality obtained, in the remark on page 12, in [@CKL15], for $f \in {\mathcal S}({\mathbb{R}}^2,{\mathbb{H}})$ with $\|f\|_{{\mathbb{R}}^2,{\mathbb{H}}}=1$ .\
The proof is quite similar to the one of Thm. 4.1 in [@EF17] and will be omitted.
([[@CKL15], Heisenberg-Weyl’s UP associated with two-sided QFT]{}\
Let $f \in {\mathcal S}({\mathbb{R}}^2,{\mathbb{H}})$, we have\
$$\label{Heis_CQWT3}{\left\|\left|t\right|f\right\|}^2_{L^2({\mathbb{R}}^2,{\mathbb{H}})} \ {\left\|\left|\xi\right|\hat{f}\right\|}^2_{L^2({\mathbb{R}}^2,{\mathbb{H}})} \ge \frac{1}{16{\pi }^2} {\left\|f\right\|}^4_{L^2({\mathbb{R}}^2,{\mathbb{H}})}.$$
\[Heisenberg\_CQWT\][(Heisenberg-Weyl’s UP associated with CQWT)]{}\
For $\varphi \in {\mathcal S}({\mathbb{R}}^2,{\mathbb{H}})$ satisfying the admissibility condition , with ${\mathcal{F}^{e_1,e_2}}\left\{\varphi \left({{r}}_{-\theta}\left(.\right)\right)\right\} \in {\mathbb{R}}+{\mathbb{R}}e_2$.\
For every $f \in {\mathcal S}({\mathbb{R}}^2,{\mathbb{H}}),$ such as $\frac{{\partial }^2}{{\partial }^2_{b_1}}f\in L^2({\mathbb{R}}^2,{\mathbb{H}}).$ We have the following inequality: $${\left\|{\left|b\right|\ T}_{\varphi }f\right\|}_{L^2(SIM(2),{\mathbb{H}})}{\left\|\left|\xi \right|\ \hat{f}\right\|}_{L^2({\mathbb{R}}^2,{\mathbb{H}})}\ge \frac{1}{\sqrt{C_{\varphi }}4\pi } {\left\| T_{\varphi }f\right\|}^2_{L^2(SIM(2),{\mathbb{H}})}.$$
Proof. Firstly we note that $f, \varphi \in {\mathcal S}({\mathbb{R}}^2,{\mathbb{H}})$ implies that $T_{\varphi }f\left(a,\theta ,.\right)\in{\mathcal S}({\mathbb{R}}^2,{\mathbb{H}})$.\
Replacing $f$ by $T_{\varphi }f\left(a,\theta ,.\right)$ in , we get $$[\int_{{\mathbb{R}}^2}{{\left|b\right|}^2{\left|T_{\varphi }f\left(a,\theta ,b\right)\right|}^2}db]^\frac{1}{2} \ [\int_{{\mathbb{R}}^2}{{\left|\xi \right|}^2{\left|{\mathcal{F}^{e_1,e_2}}\{T_{\varphi }f\left(a,\theta ,b\right)\} \left(\xi \right)\right|}^2}d\xi]^\frac{1}{2} \ge \frac{1}{4\pi } \int_{{\mathbb{R}}^2}|T_{\varphi }f\left(a,\theta ,b\right)|^2 db,$$ Integrating both sides of the above inequality with respect to the Haar measure $d\mu $, we have $$\int_{SO\left(2\right)}{\int_{{\mathbb{R}}^+}{[\int_{{\mathbb{R}}^2}{{\left|b\right|}^2{\left|T_{\varphi }f\left(a,\theta ,b\right)\right|}^2}db]^\frac{1}{2}\ [\int_{{\mathbb{R}}^2}{{\left|\xi \right|}^2{\left|{\mathcal{F}^{e_1,e_2}}\{T_{\varphi }f\left(a,\theta ,b\right)\}\left(\xi \right)\right|}^2}d\xi]^ \frac{1}{2}}d\mu }\ge \frac{1}{4\pi } {\left\| T_{\varphi }f\right\|}^2_{L^2(SIM(2),{\mathbb{H}})} ,$$ Using the quaternion Scwartz’s inequality, we can write,\
$[\int_{SO\left(2\right)}{\int_{{\mathbb{R}}^+}{\int_{{\mathbb{R}}^2}{{\left|b\right|}^2{\left|T_{\varphi }f\left(a,\theta,b\right)\right|}^2}db }d\mu}]^\frac{1}{2} \ [\int_{SO\left(2\right)}{\int_{{\mathbb{R}}^+}{\int_{{\mathbb{R}}^2}{{\left|\xi \right|}^2{\left|{\mathcal{F}^{e_1,e_2}}\{T_{\varphi }f\left(a,\theta ,b\right)\}\left(\xi \right)\right|}^2}d\xi }d\mu }]^\frac{1}{2} $\
$\ge \frac{1}{4\pi } {\left\| T_{\varphi }f\right\|}^2_{L^2(SIM(2),{\mathbb{H}})},$\
By applying lemma \[Heisenberg2\]\
$[\int_{SO\left(2\right)}{\int_{{\mathbb{R}}^+}{\int_{{\mathbb{R}}^2}{{\left|b\right|}^2{\left|\ T_{\varphi }f\left(a,\theta ,b\right)\right|}^2}db }d\mu }]^\frac{1}{2}\ \sqrt{C_{\varphi }}{\left\|\left|\xi \right|\ \hat{f}\right\|}_{L^2({\mathbb{R}}^2,{\mathbb{H}})}\ge \frac{1}{4\pi } {\left\| T_{\varphi }f\right\|}^2_{L^2(SIM(2),{\mathbb{H}})},$\
As the first term in the above expression is $L^2(SIM(2),{\mathbb{H}})$-norm, we finally obtain $${\left\|{\left|b\right|\ T}_{\varphi }f\right\|}_{L^2(SIM(2),{\mathbb{H}})}{\left\|\left|\xi \right|\ \hat{f}\right\|}_{L^2({\mathbb{R}}^2,{\mathbb{H}})}\ge \frac{1}{\sqrt{C_{\varphi }}4\pi } {\left\| T_{\varphi }f\right\|}^2_{L^2(SIM(2),{\mathbb{H}})}.$$
The proof of theorem \[Heisenberg\_CQWT\] is complete.$\hfill \square$
Logarithmic uncertainty principle
---------------------------------
Based on the classical Pitt’s inequality, Beckner [@BE95] proved the logarithmic version of Heisenberg’s UP. Recently this principle has been carried out for different two-dimensional time-frequency domain transforms [@CKL15; @BR16; @TR01]. Here, we derive the logarithmic inequality in CQWT domains.
[(Quaternion Fourier representation of the CQWT)]{}\
Let $ \varphi $ be a quaternion admissible wavelet, and $f \in L^2\left({\mathbb{R}}^2,{\mathbb{H}}\right).$ If we assume that $\hat{f}$ is ${\mathbb{R}}+{\mathbb{R}}e_2$-valued, one has\
$$\label{QF_rep_CQWT}
{\mathcal{F}^{e_1,e_2}}\{\overline{T_{\varphi }f\left(a,\theta ,-b\right)}\}\left(\xi \right)=a{\mathcal{F}^{e_1,e_2}}\{\varphi (r_{-\theta }(.)\} \left(a \xi \right)\overline{\hat{f}}\left(\xi \right).$$
Proof. We have by , and\
$T_{\varphi }f\left(a,\theta ,b\right)=\int_{{\mathbb{R}}^2}{\hat{f}}(\xi )\ \overline{\widehat{{\varphi }_{a,\theta ,b}}(\xi )}d\xi$\
= $a \int_{{\mathbb{R}}^2}{\hat{f}}(\xi )\ e^{2\pi {{\xi }_2b_2e}_2}\ \overline{{\mathcal{F}^{e_1,e_2}}\left\{\varphi {(r}_{-\theta }\left(.\right))\right\}(a\xi )}e^{2\pi {{\xi }_1b_1e}_1}d\xi.$\
Hence by , we have\
$\overline{T_{\varphi }f\left(a,\theta ,b\right)}=a\int_{{\mathbb{R}}^2}{e^{-2\pi {{\xi }_1b_1e}_1}}{\mathcal{F}^{e_1,e_2}}\left\{\varphi {(r}_{-\theta }\left(.\right))\right\}(a\xi )\ e^{-2\pi {{\xi }_2b_2e}_2}\ \hat{f}(\xi )d\xi.$\
Using the assumption, that $\hat{f}\in {\mathbb{R}}+{\mathbb{R}}e_2$ which means that $\ \hat{f}$ commute with $e_2$, we obtain by $\overline{T_{\varphi }f\left(a,\theta ,-b\right)}=a{\mathcal F}^{-e_1,-e_2}\left\{{\mathcal{F}^{e_1,e_2}}\left\{\varphi {(r}_{-\theta }\left(.\right))\right\}(a\xi )\hat{f}\left(\xi \right)\right\}(b).$\
*Due to the* Linearity of ${\mathcal{F}^{e_1,e_2}}$, and (7), we get\
${\mathcal{F}^{e_1,e_2}}\{\overline{T_{\varphi }f\left(a,\theta ,-b\right) }\}\left(\xi \right)=a{\mathcal{F}^{e_1,e_2}}{\{}\varphi (r_{-\theta }(.)\} \left(a \xi \right)\overline{\hat{f}}\left(\xi \right).\hfill \square$
[@CKL15], For $f \in {\mathcal S}\left({\mathbb{R}}^2,{\mathbb{H}}\right),$ we have\
$$\label{log_CQWT}
\int_{{\mathbb{R}}^2}{ ln( \left|y\right|)}\ {\left| {\mathcal{F}^{e_1,e_2}}\left\{f\right\}(y)\right|}^2dy\ +\int_{{\mathbb{R}}^2}{ \ln \left(\left|t\right|\right) }{\left|f(t)\right|}^2dt\ge A \int_{{\mathbb{R}}^2}{\left|f(t)\right|}^2 dt.$$ With $A= -{\ln \left(\pi \right)}+{\Gamma { '}\left(1\right)}/{\Gamma (1)},$ and $\Gamma { (.)}$ is the Gamma function.
Let $\varphi \in {\mathcal S}\left({\mathbb{R}}^2,{\mathbb{H}}\right)$ satisfying the admissibility condition given by , and suppose that ${\mathcal{F}^{e_1,e_2}}\left\{\varphi \left({{r}}_{-\theta}\left(.\right)\right)\right\}$ commute with $e_2,$ and let $f \in {\mathcal S}\left({\mathbb{R}}^2,{\mathbb{H}}\right).$\
If we assume that $\hat{f}$ is ${\mathbb{R}}+{\mathbb{R}}e_2$, then we have\
$ C_{\varphi }\int_{{\mathbb{R}}^2}{ln( \left|y\right|){\left|\hat{f}\left(y\right)\right|}^2dy}+\int_{SIM(2)}{{\ln \left(\left|b\right|\right)}}{\left|T_{\varphi }f\left(a,\theta ,b\right)\ \right|}^2 d\mu db \ge A\ C_{\varphi }{\left\|f\right\|}^2_{L^2\left({\mathbb{R}}^2,{\mathbb{H}}\right)}$.\
Proof. Replacing $f$ by $\overline{T_{\varphi }f\left(a,\theta ,-.\right)}$ in , we get\
$\int_{{\mathbb{R}}^2}{ln( \left|y\right|)}{\left|{\mathcal{F}^{e_1,e_2}}\left\{\overline{T_{\varphi }f\left(a,\theta ,-b\right)}\right\}(y)\right|}^2dy\ $+ $\int_{{\mathbb{R}}^2}{{\ln \left(\left|b\right|\right)}}{\left|\overline{T_{\varphi }f\left(a,\theta ,-b\right)}\right|}^2db\ge \ A \int_{{\mathbb{R}}^2}{\left|\overline{T_{\varphi }f\left(a,\theta ,-b\right)}\right|}^2 db,$\
As $\left|\overline{q}\right|=\left|q\right|$, for any ${ q}$ in ${\mathbb{H}},$ and by change of variable in the second term on the left-hand side , and the term on the right-hand side of the above inequality, we have\
$\int_{{\mathbb{R}}^2} {ln(\left|y\right|) }{\left|{\mathcal{F}^{e_1,e_2}}\left\{\overline{T_{\varphi }f\left(a,\theta ,-b\right)} \right\}(y)\right|}^2dy\ $+ $\int_{{\mathbb{R}}^2}{{\ln \left(\left|b\right|\right)}}{\left|T_{\varphi }f\left(a,\theta ,b\right)\ \right|}^2db\ge A\int_{{\mathbb{R}}^2}{\left|T_{\varphi }f\left(a,\theta ,-b\right)\ \right|}^2db.$\
We now integrate both sides of this inequality with respect to the measure $d\mu $, we obtain\
$\int_{SO(2)}{\int_{{\mathbb{R}}^+}{\int_{{\mathbb{R}}^2}{{ln(\left|y\right|)}}{\left|{\mathcal{F}^{e_1,e_2}}\left\{\overline{T_{\varphi }f\left(a,\theta ,-b\right)}\right\}(y)\right|}^2dyd\mu }}+\int_{SIM(2)}{{{\ {\ln \left(\left|b\right|\right)\ }\ }{\left|T_{\varphi }f\left(a,\theta ,b\right)\ \right|}^2d\mu db }}$\
$$\label{log_CQWT2}
\ge A\int_{SIM(2)}\ {\left|T_{\varphi }f\left(a,\theta ,b\right)\ \right|}^2d\mu db.$$ We will show the following assumption\
$$\label{log_CQWT3}
\int_{SO\left(2\right)}{\int_{{\mathbb{R}}^+}{\int_{{\mathbb{R}}^2}{ln( \left|y\right|)}{\left|{\mathcal{F}^{e_1,e_2}}\left\{\overline{T_{\varphi }f\left(a,\theta ,-b\right)}\right\}\left(y\right)\right|}^2 d\mu }}dy = C_{\varphi }\int_{{\mathbb{R}}^2}{ln( \left|y\right|){\left|\hat{f}\left(y\right)\right|}^2dy}.$$ Using , and straightforward computations show that\
$\int_{SO\left(2\right)}{\int_{{\mathbb{R}}^+}{\int_{{\mathbb{R}}^2}{ln(\left|y\right|)}{\left|{\mathcal{F}^{e_1,e_2}}\left\{\overline{T_{\varphi}f\left(a,\theta,-b\right)}\right\}\left(y\right)\right|}^2d\mu }}dy = \int_{SO\left(2\right)}{\int_{{\mathbb{R}}^+}{\int_{{\mathbb{R}}^2}ln( \left|y\right|){|a{\mathcal{F}^{e_1,e_2}}\{\varphi (r_{-\theta }(.)\}\left(ay\right)\overline{\hat{f}}\left(y\right)|}^2d\mu }}dy$\
=$\int_{{\mathbb{R}}^+}{\int_{SO(2)}{\int_{{\mathbb{R}}^2}{a^2ln( \left|y\right|)}{|{\mathcal{F}^{e_1,e_2}}\{\varphi (r_{-\theta }(.)\}\left(ay\right)\overline{\hat{f}}\left(y\right)|}^2\ a^{-3}dad\theta }}dy$\
=$\int_{{\mathbb{R}}^+}{\int_{SO(2)}{{{\left|{\mathcal{F}^{e_1,e_2}}\{\varphi (r_{-\theta }(.)\}\left(ay\right)\right|}^2}d\theta a^{-1}da}}\ \int_{{\mathbb{R}}^2}{{ln( \left|y\right|)}{\left|\hat{f}\left(y\right)\right|}^2dy}$\
=$C_{\varphi }\int_{{\mathbb{R}}^2}{{ln( \left|y\right|)}{\left|\hat{f}\left(y\right)\right|}^2dy}.$\
In the last equality we used . The proof of is completed.\
Inserting into the first integral of , we obtain\
$C_{\varphi }\int_{{\mathbb{R}}^2}{{ln( \left|y\right|)}{\left|\hat{f}\left(y\right)\right|}^2dy}+\int_{SIM(2)}{{\ln \left(\left|b\right|\right)}}{\left|T_{\varphi }f\left(a,\theta ,b\right)\ \right|}^2d\mu db \ge A\int_{SIM(2)}{{{\left|T_{\varphi }f\left(a,\theta ,b\right)\ \right|}^2}} d\mu db, $\
Using gives\
$C_{\varphi }\int_{{\mathbb{R}}^2}{{ln(\left|y\right|)}{\left|\hat{f}\left(y\right)\right|}^2dy}+\int_{SIM(2)}{{\ln \left(\left|b\right|\right)}}{\left|T_{\varphi }f\left(a,\theta ,b\right)\ \right|}^2d\mu db\ge A\ C_{\varphi }{\left\|f\right\|}^2_{L^2\left({\mathbb{R}}^2,{\mathbb{H}}\right)}.$\
The result follows.$\hfill \square$
Hardy’s uncertainty principle
-----------------------------
We remind that Hary’s UP associated with the QFT asserts that is impossible for a non-zero function and its QFT to both decrease very rapidly as it was given in [@EF17]. Actually, we establish Hardy’s UP for two-dimensional continuous quaternion wavelet transform.\
Let us review Hardy’s UP in the quaternion Fourier transform domain as follows.
\[Elhaoui\]([@EF17], Thm. 5.3)
Let $\alpha \ $ and $\beta $ are positive constants. Suppose $f\in L^2({\mathbb R}^2,{\mathbb H})$ with\
${|f\left(x\right)|}_Q={\mathcal O}(e^{-\alpha {\left|x\right|}^2}), \ x\in {\mathbb R}^2.$\
${|\hat{f} \left(y\right)|}_Q={\mathcal O}(e^{-\beta {\left|y\right|}^2}),\ y\in {\mathbb R}^2.$\
Then, three cases can occur :
1. If $\alpha \beta >{\pi }^2$, then $f=0$.
2. If $\alpha \beta ={\pi }^2$, then $\ f=A e^{-\alpha {\left|x\right|}^2}$ ,where A is a quaternion constant.
3. If $\alpha \beta <{\pi }^2,$ then there are infinitely many such functions $f$.
Based on Proposition \[Elhaoui\], we derive the corresponding Hardy’s UP for the CQWT.
Let $\alpha \ $ and $\beta $ are positive constants, and Let $\varphi$ be a quaternion admissible wavelet,\
Suppose $f\in L^2({\mathbb{R}}^2;{\mathbb{H}})$ such that $\hat{f}$ is ${\mathbb{R}}+ {\mathbb{R}}e_2$-valued, we assume that\
$$\label{Hardy_CQWT}
\left|T_{\varphi }f\left(a,\theta ,b\right)\ \right|= {\mathcal O}(e^{-\alpha {\left|b\right|}^2}), b\in {\mathbb{R}}^2$$ and\
$$\label{Hardy_CQWT2}
\left|\hat{f}\left(\xi \right)\right|={\mathcal O}(e^{-\beta {\left|\xi \right|}^2}),\ \xi \in {\mathbb{R}}^2$$ for $\alpha ,\ \beta >0.$\
Then,
1. If $\alpha \beta >{\pi }^2$, then $T_{\varphi }f(a,\theta ,.)= 0 $.
2. If $\alpha \beta ={\pi }^2,$ then $T_{\varphi }f\left(a,\theta ,b\right)=A{{\rm e}}^{-\alpha {\left|b\right|}^2}$, where $A$ is a quaternion constant.
3. If $\alpha \beta <{\pi }^2,$ then there are infinitely many $T_{\varphi }f$.
Proof. Since $f$ and $ \varphi $ are in $L^2({\mathbb{R}}^2;{\mathbb{H}})$ we have $T_{\varphi }f\left(a,\theta,.\right){\in L}^2({\mathbb{R}}^2;{\mathbb{H}})$ by the use of Theorem \[Haus\_CQWT\].\
Then it follows from that\
$\left|{\mathcal{F}^{e_1,e_2}}\{\overline{T_{\varphi }f\left(a,\theta ,-b\right) }\}\left(\xi \right) \right| =a \left|{\mathcal{F}^{e_1,e_2}}\{\varphi (r_{-\theta }(.)\}\left(\xi \right)\right|\left|\overline{\hat{f}}\left(\xi \right)\right|=a\ \left|{\mathcal{F}^{e_1,e_2}}{\rm \{}\varphi (r_{-\theta }(.)\}\left(\xi \right)\right|\left|\hat{f}\left(\xi \right)\right|\ $\
As the quaternion Fourier transform is isometry on $L^2({\mathbb{R}}^2;{\mathbb{H}}))$ ([@CKL15], Thm. 3.2),\
We have $\widehat{\varphi }\in L^2({\mathbb{R}}^2;{\mathbb{H}})$ therefore\
$\left|{\mathcal{F}^{e_1,e_2}}{\rm \{}\overline{T_{\varphi }f\left(a,\theta ,-b\right)\ }\}\left(\xi \right)\ \right|=\ O(e^{-\beta {\left|\xi \right|}^2}),$
On the other hand, by assumption we obtain
$$\left|\overline{T_{\varphi }f\left(a,\theta ,-b\right)\ }\ \right|= \left|T_{\varphi }f\left(a,\theta ,-b\right)\ \right|= {O(e}^{-\alpha {\left|b\right|}^2}),$$
Hence, it follows from Proposition \[Elhaoui\] that that\
if $\alpha \beta ={\pi }^2,$ then $\overline{T_{\varphi }f\left(a,\theta ,-b\right)\}\ }=Be^{-\alpha {\left|b\right|}^2}$, whit B is a quaternion constant. That is $T_{\varphi }f\left(a,\theta ,b\right)=\overline {B}e^{-\alpha {\left|b\right|}^2}.$\
If $\alpha \beta >{\pi }^2,$ then $T_{\varphi }f\left(a,\theta ,.\right)=0\ $on ${\mathbb{R}}^2$.\
If $\alpha \beta <{\pi }^2,$ then there are infinitely many such functions $T_{\varphi }f\left(a,\theta ,.\right)$, that verify and .
This completes the proof.$\hfill \square$
Conclusion
==========
In this paper, we developed the definition of CQWT using the Kernel of the two-sided QFT, and the similitude group. The various important properties of the CQWT such as scaling, translation, rotation, Parseval’s identity, inversion theorem, and a reproducing kernel are established.\
Using the derivative theorem, and Heisenberg-Weyl’ UP related to the two-sided QFT, we derived the Heisenberg-Weyl’s UP associated with the CQWT. Finally, due to the spectral QFT representation of CQWT and based on the logarithmic UP and Hardy’s UP for the two-sided QFT, the forms associated with these UPs have been proved in the CQWT Domain.\
With the help of this paper, we hope to introduce an extension of the wavelet transform to Clifford algebra by means of the kernel of Two-sided Clifford Fourier transform defined by Hitzer in [@HI14]. The investigation on this topic will be reported in a forthcoming paper.\
\
\
[|p[1.2in]{}|p[1.3in]{}|p[2in]{}|]{}\
Left linearity TranslationScalingRotationPlancherelParseval & $(\lambda f+\mu g )\left(a,\theta ,b\right)$ $f\left(a,\theta ,b-c\right)$$f\left(\alpha .\right)\left(a,\theta ,b\right) $ $f(r_{\omega }.)\left(a,\theta ,b\right)$$ C_{\varphi }{(f,g)}_{L^2\left({\mathbb{R}}^2,{\mathbb{H}}\right)}$${\sqrt{{{ C}}_{\varphi }}\left\|f\right\|}_{L^2({\mathbb{R}}^2,{\mathbb{H}})}$& $\lambda T_{\varphi }f\left(a,\theta ,b\right)$+${\mu T}_{\varphi }g\left(a,\theta ,b\right)$$T_{\varphi }f\left(a,\theta ,b-c\right)$$\frac{1}{\alpha }T_{\varphi }f\left(\alpha .\right)\left(\alpha a,\theta ,\alpha b\right)$$T_{\varphi }f\left(a,\theta +\omega ,r_{\omega }\left(b\right)\right)$${<T_{\varphi }f,T_{\varphi }g>}_{L^2\left(SIM\left(2\right),{\mathbb{H}}\right)}$${\left\|T_{\varphi }f\right\|}_{L^2(SIM(2),{\mathbb{H}})}$\
\
Some important properties of the CQWT are summarized in Table 1.
[9]{} M. Bahri, R. Ashino and R. Vaillancourt, Two-dimensional quaternion wavelet transform, Applied Mathematics and Computation, 218 (2011), pp. 10-21. M. Bahri, R. Ashino, Logarithmic Uncertainty Principle for quaternion linear canonical , , Proceedings of the 2016 International Conference on Wavelet Analysis and Pattern Recognition, Jeju, South Korea, pp. 10-13 July. W. Beckner, Pitt’s inequality and the uncertainty principle. Proc. Amer. Math. Soc. 123(6), pp. 1897-1905 (1995). F. Brackx, F. Sommen, Clifford-Hermite wavelets in Euclidean space, J. Fourier Anal. Appl., 2000, 6(3): pp. 209-310. F. Brackx, F. Sommen, The continuous wavelet transform in Clifford analysis. Clifford analysis and its applications, Prague, 2000, 9-26, NATO Sci. Ser II Math Phys Chem. Dordrecht: Kluwer Acad Publ, 2001, 25. T. Bülow, Hypercomplex spectral signal representations for the processing and analysis of images, Ph.D. Thesis, 1999, Institut für Informatik und Praktische Mathematik, University of Kiel, Germany.
L.P. Chen, K.I. Kou, M.S. Liu, Pitt’s inequality and the uncertainty principle associated with the quaternion Fourier transform, J. Math.Anal.Appl.423(2015), pp. 681-700.
Y. El Haoui, S. Fahlaoui, The Uncertainty principle for the two-sided quaternion Fourier transform, Mediterr. J. Math. (2017) doi:10.1007/s00009-017-1024-5. Y. El Haoui, S. Fahlaoui, Generalized Uncertainty Principles associated with the Quaternionic Offset Linear Canonical Transform, https://arxiv.org/abs/1807.04068, 2018.
T.A. Ell, Quaternion-Fourier transfotms for analysis of two-dimensional linear time-invariant partial differential systems. In: Proceeding of the 32nd Conference on Decision and Control, San Antonio, Texas, pp. 1830-1841, 1993.
A. Grossman, J. Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J Math Anal, 1984, 15: pp. 723-736. W. Heisenberg, Uber den anschaulichen inhalt der quanten theoretischen kinematik und mechanik. Zeitschrift für Physik 43, pp. 172-198, 1927. E. Hitzer, Quaternion Fourier transform on quaternion fields and generalizations, Advances in Applied Clifford Algebras, 17 (3) (2007), pp. 497-517. E. Hitzer, Clifford (Geometric) Algebra Wavelet Transform, in V. Skala and D. Hildenbrand (eds.), roc. of GraVisMa 2009, 02-04 Sep. 2009, Plzen, Czech Republic, pp. 94-101 (2009). Preprint: http://arxiv.org/abs/1306.1620. E. Hitzer, S. J. Sangwine, The Orthogonal 2D Planes Split of Quaternions and Steerable Quaternion Fourier Transformations, in E. Hitzer, S.J. Sangwine (eds.), “Quaternion and Clifford Fourier transforms and wavelets”, Trends in Mathematics 27, Birkhauser, Basel, 2013, pp. 15-39. $DOI: 10.1007/978-3-0348-0603-9_2$, Preprint: http://arxiv.org/abs/1306.2157. E. Hitzer, Two-Sided Clifford Fourier Transform with Two Square Roots of 1 in Cl(p, q). Adv. Appl. Clifford Algebras 24, pp. 313-332 (2014). doi: 10.1007/s00006-014-0441-9, preprint: http://arxiv.org/abs/1306.2092 F. Linares, G. Ponce, Introduction to Nonlinear Dispersive Equations, Publicações Matemáticas, IMPA, Rio de Janeiro, Brazil, 2004. R. Murenzi, Wavelet transforms associated to the n-dimensional Euclidean group with dilations, In: Combes J, ed. Wavelet, Time-Frequency Methods and Phase Space. Boston-London: Jones and Bartlett Publishers, 1989, pp. 239–246.
L. Traversoni, Imaging analysis using quaternion wavelet, in geometric algebra with applications, in: E.B. Corrochano, G. Sobczyk (Eds.), Science and Engineering, Birkhäuser, Boston, 2001. J. Zhao, L. Peng, Quaternion-valued admissible wavelets associated with the 2-dimensional Euclidean group with dilations, J. Nat. Geom. 20 (1), pp. 21-32, 2001.
[^1]: unless the two-sided QFT of a function commute with the quaternion basis vector $e_2\ $, see lemma 4.5 in the present paper.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider the wreath product of two symmetric groups as a group of blocks permutations and we study its conjugacy classes. We give a polynomiality property for the structure coefficients of the center of the wreath product of symmetric groups algebra. This allows us to recover an old result of Farahat and Higman about the polynomiality of the structure coefficients of the center of the symmetric group algebra and to generalize our recent result about the polynomiality property of the structure coefficients of the center of the hyperoctahedral group algebra. A particular attention is paid to the cases when the blocks contain two or three elements.'
address: 'Instytut Matematyczny, Polska Akademia Nauk, ul. Śniadeckich 8, 00-656 Warszawa, Poland'
author:
- Omar Tout
bibliography:
- 'biblio.bib'
title: |
The center of the wreath product of\
symmetric groups algebra
---
Introduction
============
The conjugacy classes of the symmetric group $\mathcal{S}_n$ can be indexed by partitions of $n.$ The conjugacy class associated to a partition $\lambda$ is the set of all permutations with cycle-type $\lambda.$ The center of the symmetric group algebra is the algebra over $\mathbb{C}$ generated by the conjugacy classes of the symmetric group. Its structure coefficients have nice combinatorial properties. In [@FaharatHigman1959], Farahat and Higman, gave a polynomiality property for the structure coefficients of the center of the symmetric group algebra. By introducing partial permutations in [@Ivanov1999], Ivanov and Kerov, gave a combinatorial proof to this result.\
We introduce in this paper the group $\mathcal{B}_{kn}^k$ which permutes $n$ blocks of $k$ elements each. The permutation of the $k$ elements in each block is allowed. This group is the symmetric group $\mathcal{S}_n$ if $k=1$ and in case $k=2$ it is the hyperoctahedral group $\mathcal{H}_n$ on $2n$ elements. In general, we show that the group $\mathcal{B}_{kn}^k$ is isomorphic to the wreath product $\mathcal{S}_k\sim \mathcal{S}_n$ of the symmetric group $\mathcal{S}_k$ by the symmetric group $\mathcal{S}_n.$ It is well known that the conjugacy classes of $\mathcal{H}_n$ are indexed by pairs of partitions $(\lambda,\delta)$ verifying $|\lambda|+|\delta|=n,$ see [@geissinger1978representations] and [@stembridge1992projective]. We show that in general, for any fixed integer $k,$ the conjugacy classes of the group $\mathcal{B}_{kn}^k$ are indexed by families of partitions $\lambda=(\lambda(\rho))_{\rho\vdash k}$ indexed by the set of partitions of $k$ such that the sum of the sizes of all $\lambda(\rho)$ equals $n.$ This comes with no surprise since it was shown by Specht in [@specht1932verallgemeinerung] that the conjugay classes of $\mathcal{S}_k \sim \mathcal{S}_n$ are indexed by families of partitions indexed by the set of partitions of $k,$ see [@kerber2006representations] and [@McDo] for more information about this fact.\
Recently, in [@Tout2017], we developed a framework in which the polynomiality property for double-class algebras, and subsequently centers of groups algebra, holds. In particular, we showed that our framework contains the sequence of the symmetric groups and that of the hyperoctahedral groups. Thus we obtained again the result of Farahat and Higman for the structure coefficients of the center of the symmetric group algebra. In addition we gave a polynomiality property for the structure coefficients of the center of the hyperoctahedral group algebra in [@Tout2017 Section 6.2].
In this paper we show that the general framework we gave in [@Tout2017] contains the sequence of groups $(\mathcal{B}_{kn}^k)_n$ when $k$ is a fixed integer. Thus, we will be able to give a polynomiality property for the structure coefficients of the center of the group $\mathcal{B}_{kn}^k$ algebra. A particular attention to the cases $k=2$ (hyperoctahedral group) and $k=3$ is given. The question whether the partial permutation concept of Ivanov and Kerov can be generalized to obtain a combinatorial proof to this result arises directly. We will try to answer this question in another paper.\
The paper is organized as follows. In Section \[sec\_2\], we review all necessary definitions of partitions and we describe the conjugacy classes of the symmetric group. Then, we define explicitly the group $\mathcal{B}_{kn}^k$ in section \[sec\_3\] and we study in details its conjugacy classes. Then, we show that it is isomorphic to the wreath product $\mathcal{S}_k\sim \mathcal{S}_n.$ In sections \[sec\_hyp\] and \[seck=3\] a special treatment is given for the cases $k=2$ and $k=3$ respectively. The last section contains our main result, that is a polynomiality property for the structure coefficients of the center of the group $\mathcal{B}_{kn}^k$ algebra. In addition some examples are given for the cases $k=2$ and $k=3.$
Partitions and conjugacy classes of the Symmetric group {#sec_2}
=======================================================
If $n$ is a positive integer, we denote by $\mathcal{S}_n$ the symmetric group of permutations on the set $[n]:=\lbrace 1,2,\cdots,n\rbrace.$ A *partition* $\lambda$ is a list of integers $(\lambda_1,\ldots,\lambda_l)$ where $\lambda_1\geq \lambda_2\geq\ldots \lambda_l\geq 1.$ The $\lambda_i$ are called the *parts* of $\lambda$; the *size* of $\lambda$, denoted by $|\lambda|$, is the sum of all of its parts. If $|\lambda|=n$, we say that $\lambda$ is a partition of $n$ and we write $\lambda\vdash n$. The number of parts of $\lambda$ is denoted by $l(\lambda)$. We will also use the exponential notation $\lambda=(1^{m_1(\lambda)},2^{m_2(\lambda)},3^{m_3(\lambda)},\ldots),$ where $m_i(\lambda)$ is the number of parts equal to $i$ in the partition $\lambda.$ In case there is no confusion, we will omit $\lambda$ from $m_i(\lambda)$ to simplify our notation. If $\lambda=(1^{m_1(\lambda)},2^{m_2(\lambda)},3^{m_3(\lambda)},\ldots,n^{m_n(\lambda)})$ is a partition of $n$ then $\sum_{i=1}^n im_i(\lambda)=n.$ We will dismiss $i^{m_i(\lambda)}$ from $\lambda$ when $m_i(\lambda)=0.$ For example, we will write $\lambda=(1^2,3,6^2)$ instead of $\lambda=(1^2,2^0,3,4^0,5^0,6^2,7^0).$ If $\lambda$ and $\delta$ are two partitions we define the *union* $\lambda \cup \delta$ and subtraction $\lambda \setminus \delta$ (if exists) as the following partitions: $$\lambda \cup \delta=(1^{m_1(\lambda)+m_1(\delta)},2^{m_2(\lambda)+m_2(\delta)},3^{m_3(\lambda)+m_3(\delta)},\ldots).$$ $$\lambda \setminus \delta=(1^{m_1(\lambda)-m_1(\delta)},2^{m_2(\lambda)-m_2(\delta)},3^{m_3(\lambda)-m_3(\delta)},\ldots) \text{ if $m_i(\lambda)\geq m_i(\delta)$ for any $i.$ }$$ A partition is called *proper* if it does not have any part equal to 1. The proper partition associated to a partition $\lambda$ is the partition $\bar{\lambda}:=\lambda \setminus (1^{m_1(\lambda)})=(2^{m_2(\lambda)},3^{m_3(\lambda)},\ldots).$
The *cycle-type* of a permutation of $\mathcal{S}_n$ is the partition of $n$ obtained from the lengths of the cycles that appear in its decomposition into product of disjoint cycles. For example, the permutation $(2,4,1,6)(3,8,10,12)(5)(7,9,11)$ of $\mathcal{S}_{12}$ has cycle-type $(1,3,4^2).$ In this paper we will denote the cycle-type of a permutation $\omega$ by $\operatorname{ct}(\omega).$ It is well known that two permutations of $\mathcal{S}_n$ belong to the same conjugacy class if and only if they have the same cycle-type. Thus the conjugacy classes of the symmetric group $\mathcal{S}_n$ can be indexed by partitions of $n.$ If $\lambda=(1^{m_1(\lambda)},2^{m_2(\lambda)},3^{m_3(\lambda)},\ldots,n^{m_n(\lambda)})$ is a partition of $n,$ we will denote by $C_\lambda$ the conjugacy class of $\mathcal{S}_n$ associated to $\lambda:$ $$C_\lambda:=\lbrace \sigma\in \mathcal{S}_n \text{ $\mid$ } \operatorname{ct}(\sigma)=\lambda \rbrace.$$ The cardinal of $C_\lambda$ is given by: $$|C_\lambda|=\frac{n!}{z_\lambda},$$ where $$z_\lambda:=1^{m_1(\lambda)}m_1(\lambda)!2^{m_2(\lambda)}m_2(\lambda)!\cdots n^{m_n(\lambda)}m_n(\lambda)!.$$
The family $(\mathcal{C}_\rho)_{\rho\vdash n}$, indexed by partitions of $n$ where $\mathcal{C}_\lambda$ is the sum of permutations of $\mathcal{S}_n$ with cycle-type $\rho$ $$\mathcal{C}_\rho=\sum_{\sigma\in C_\rho}\sigma$$ forms a basis for the center of the symmetric group algebra. If $\lambda$ and $\delta$ are two partitions of $n,$ the product $\mathcal{C}_\lambda \mathcal{C}_\delta$ can be written as a linear combination of the elements $(\mathcal{C}_\rho)_{\rho\vdash n}$ as follows:
$$\mathcal{C}_\lambda \mathcal{C}_\delta=\sum_{\rho\vdash n}c_{\lambda\delta}^{\rho}\mathcal{C}_\rho.$$
The coefficients $c_{\lambda\delta}^{\rho}$ are called the *structure coefficients* of the center of the Symmetric group algebra. It was shown in [@FaharatHigman1959] and in [@Ivanov1999] by a more combinatorial way that the coefficients $c_{\bar{\lambda}\bar{\delta}}^{\bar{\rho}}$ are polynomials in $n.$
The conjugacy classes of the group $\mathcal{B}_{kn}^{k}$ {#sec_3}
=========================================================
In this section we define the group $\mathcal{B}_{kn}^{k}$ then we study its conjugacy classes. We show in Proposition \[class\_conj\_classes\] that these latter are indexed by families of partitions indexed by all partitions of $k.$ Then in Proposition \[Prop\_size\_conj\] we give an explicit formula for the size of any of its conjugacy classes. The group $\mathcal{B}_{kn}^{k}$ is isomorphic to the wreath product $\mathcal{S}_k\sim \mathcal{S}_n$ as will be shown in Proposition \[isom\]. However, we decided to work with this copy of the wreath product in this paper since it seems very natural to present our main result.\
If $i$ and $k$ are two positive integers, we denote by $p_{k}(i)$ the following set of size $k:$ $$p_k(i):=\lbrace (i-1)k+1, (i-1)k+2, \cdots , ik\rbrace.$$
The above set $p_k(i)$ will be called a $k$-tuple in this paper. We define the group $\mathcal{B}_{kn}^{k}$ to be the subgroup of $\mathcal{S}_{kn}$ formed by permutations that send each set of the form $p_{k}(i)$ to another set with the same form: $$\mathcal{B}_{kn}^{k}:=\lbrace w \in \mathcal{S}_{kn}; \ \forall \ 1 \leq r \leq n, \ \exists \ 1 \leq r' \leq n \text{ such that } w(p_{k}(r))=p_{k}(r')\rbrace.$$
$\begin{pmatrix}
1&2&3&&4&5&6\\
1&3&2&&6&5&4
\end{pmatrix}\in \mathcal{B}_{6}^{3}$ but $\begin{pmatrix}
1&2&3&&4&5&6\\
1&3&6&&2&4&6
\end{pmatrix}\notin \mathcal{B}_{6}^{3}.$
The group $\mathcal{B}_{kn}^{k}$ as it is defined here appears in [@Tout2017 Section 7.4]. When $k=1,$ it is clear that $\mathcal{B}_{n}^1$ is the symmetric group $\mathcal{S}_n.$ When $k=2,$ the group $\mathcal{B}_{2n}^2$ is the hyperoctahedral group $\mathcal{H}_n$ on $2n$ elements, see [@toutejc] where the author treats $\mathcal{H}_n$ as being the group $\mathcal{B}_{2n}^2.$ It would be clear to see that the order of the group $\mathcal{B}_{kn}^{k}$ is $$|\mathcal{B}_{kn}^{k}|=(k!)^nn!$$
The decomposition of a permutation $\omega\in \mathcal{B}_{kn}^{k}$ into product of disjoint cycles has remarkable patterns. To see this, fix a $k$-tuple $p_{k}(i)$ and a partition $\rho=(\rho_1,\rho_2,\cdots,\rho_l)$ of $k.$ Suppose now that while writing $\omega$ as product of disjoint cycles we get among the cycles the following pattern: $$\label{canonical_decom}
\mathcal{C}_1=(a_1,\cdots,a_2,\cdots,a_{\rho_1},\cdots),$$ $$\mathcal{C}_2=(a_{\rho_1+1},\cdots,a_{\rho_1+2},\cdots,a_{\rho_1+\rho_2},\cdots),$$ $$\vdots$$ $$\mathcal{C}_l=(a_{\rho_1+\cdots+\rho_{l-1}+1},\cdots,a_{\rho_1+\cdots+\rho_{l-1}+2},\cdots,a_{\rho_1+\cdots+\rho_{l-1}+\rho_l},\cdots),$$ where $$\lbrace a_1,a_2,\cdots,a_{\rho_1+1},\cdots,a_{\rho_1+\cdots+\rho_{l-1}+\rho_l}\rbrace=p_k(i) \text{ for a certain $i\in [n]$}.$$ We should remark that since $\omega\in \mathcal{B}_{kn}^{k},$ if we consider $b_j=\omega(a_j)$ for any $1\leq j\leq |\rho|$ then there exists $r\in [n]$ such that: $$p_{k}(r)=\lbrace b_1,b_2,\cdots,b_{\rho_1+1},\cdots,b_{\rho_1+\cdots+\rho_{l-1}+\rho_l}\rbrace.$$ This can be redone till we reach $a_2$ which implies that in cycle $\mathcal{C}_1,$ we have the same number of elements between $a_i$ and $a_{i+1}$ for any $1\leq i\leq \rho_1-1.$ Thus the size of the cycle $\mathcal{C}_1$ is a multiple of $\rho_1,$ say $|\mathcal{C}_1|=m\rho_1.$ The same can be done for all the other cycles $\mathcal{C}_i$ and in fact for any $1\leq i\leq l,$ $|\mathcal{C}_i|=m\rho_i.$ In addition, if we take the set of all the elements that figure in the cycles $\mathcal{C}_i$ we will get a disjoint union of $m$ $k$-tuples. That means: $$\sum_{j=1}^{l}|\mathcal{C}_j|=\sum_{j=1}^{l} m\rho_j=mk.$$ This integer $m$ can be obtained from any of the cycles $\mathcal{C}_j$ by looking to the number of elements that separate two consecutive elements of the same $k$-tuple. Now construct the partition $\omega(\rho)$ by grouping all the integers $m$ as above.
We should pay the reader’s attention to two important remarks after this construction. First of all, for any partition $\rho$ of $k$ there are $rk$ elements involved when adding the part $r$ to $\omega(\rho).$ That means: $$\sum_{\rho\vdash k}\sum_{r\geq 1}krm_r(\omega(\rho))=kn \text{ which implies } \sum_{\rho\vdash k}\sum_{r\geq 1}rm_r(\omega(\rho))=n.$$ Now let $p_\omega$ denotes the blocks permutation of $n$ associated to $\omega.$ That is $p_\omega(i)=j$ whenever $\omega(p_k(i))=p_k(j).$ Adding a part $r$ to one of the partitions $\omega(\rho)$ implies that $r$ is the length of one of the cycles of $p_\omega.$ Thus we have: $$\bigcup_{\rho\vdash k}\omega(\rho)=\operatorname{ct}(p_\omega).$$ In other words, one may first find the cycle-type of $p_\omega$ and then distribute all its parts between the partitions $\omega(\rho)$ according to the above construction.
\[k=3,equivex\] Consider the following permutation, written in one-line notation, $\omega$ of $\mathcal{B}^3_{24}$
$$\omega=12~~10~~11\,\,\,\, 20~~21~~19\,\,\,\, 8~~7~~9\,\,\,\, 1~~2~~3\,\,\,\, 16~~18~~17 \,\,\,\,15~~14~~13 \,\,\,\, 5~~4~~6 \,\,\,\, 22~~23~~24$$ Its decomposition into product of disjoint cycles is: $$\omega=\color{red}(1,12,3,11,2,10)\color{blue}(4,20)(5,21,6,19)\color{green}(7,8)(9)\color{brown}(13,16,15,17,14,18)\color{black}(22)(23)(24).$$ The first red cycle contains all the elements of $p_3(1)$ thus it contributes to $\omega(3).$ In it, there are two elements between $1$ and $3$ thus we should add $2$ to the partition $\omega(3).$ The brown cycle contributes to $\omega(3)$ by $2$ also, thus $\omega(3)=(2,2).$ By looking to the blue cycles we see that $5$ and $6$ belong to the same cycle while $4$ belongs to the other, thus these cycles will contribute to $\omega(2,1).$ The distance between $5$ and $6$ is two (the same can be done by looking to the cycle containing $4$ but in this case since $4$ is the only element of $p_3(3)$ in this cycle we count the distance between $4$ and $4$ which is $2$) which means that $\omega(2,1)$ contains a cycle of length $2.$ The green cycle will add $1$ to $\omega(2,1)$ to become the partition $(2,1).$ The black cycles give $\omega(1,1,1)=(1).$ The reader should remark that $p_\omega$ is the permutation $(1,4)(2,7)(3)(5,6)(8)$ of $8$ and that: $$\omega(3)\cup \omega(2,1)\cup \omega(1^3)=\operatorname{ct}(p_\omega)=(1^2,2^3).$$
If $\omega\in \mathcal{B}_{kn}^k,$ define $\operatorname{type}(\omega)$ to be the following family of partitions indexed by partitions of $k$ $$\operatorname{type}(\omega):=(\omega(\rho))_{\rho\vdash k}.$$
\[class\_conj\_classes\] Two permutations $\alpha$ and $\beta$ of $\mathcal{B}_{kn}^k$ are in the same conjugacy class if and only if they both have the same type.
Suppose $\alpha=\gamma\beta\gamma^{-1}$ for some $\gamma\in \mathcal{B}_{kn}^k$ and fix a partition $\rho$ of $k.$ Suppose that the cycles $\mathcal{C}_1,\cdots,\mathcal{C}_l$ of Equation (\[canonical\_decom\]) contribute to $\beta(\rho)$ then the cycles $$\mathcal{C}^{'}_1=(\gamma(a_1),\cdots,\gamma(a_2),\cdots,\gamma(a_{\rho_1}),\cdots),$$ $$\mathcal{C}^{'}_2=(\gamma(a_{\rho_1+1}),\cdots,\gamma(a_{\rho_1+2}),\cdots,\gamma(a_{\rho_1+\rho_2}),\cdots),$$ $$\vdots$$ $$\mathcal{C}^{'}_l=(\gamma(a_{\rho_1+\cdots+\rho_{l-1}+1}),\cdots,\gamma(a_{\rho_1+\cdots+\rho_{l-1}+2}),\cdots,\gamma(a_{\rho_1+\cdots+\rho_{l-1}+\rho_l}),\cdots),$$ will contribute to $\alpha(\rho)$ and they have respectively the same lengths as $\mathcal{C}_1,\cdots,\mathcal{C}_2.$ This proves the first implication.
Conversely, $\operatorname{type}(\alpha)=\operatorname{type}(\beta)$ means that $\alpha(\rho)=\beta(\rho)$ for any partition $\rho$ of $k.$ In order to simplify, we will look at the elements of two fixed $k$-tuples, say $p_k(1)=\lbrace 1,2,\cdots, k\rbrace$ for $\alpha$ and $p_k(2)=\lbrace k+1,k+2,\cdots ,2k\rbrace$ for $\beta,$ such that the distribution of the elements of $p_k(1)$ among the cycle decomposition in $\alpha$ is similar to that of the elements of $p_k(2)$ in $\beta.$ In other words, the cycle decompositions of $\alpha$ and $\beta$ are as follows: $$\alpha=(1,\cdots,2,\cdots,\rho_1,\cdots)(\rho_1+1,\cdots,\rho_1+2,\cdots,\rho_1+\rho_2,\cdots)\cdots$$ $$\cdots(\rho_1+\cdots+\rho_{l-1}+1,\cdots,\rho_1+\cdots+\rho_{l-1}+2,\cdots,\rho_1+\cdots+\rho_{l-1}+\rho_l,\cdots)$$ and $$\beta=(k+1,\cdots,k+2,\cdots,k+\rho_1,\cdots)(k+\rho_1+1,\cdots,k+\rho_1+2,\cdots,k+\rho_1+\rho_2,\cdots)\cdots$$ $$\cdots(k+\rho_1+\cdots+\rho_{l-1}+1,\cdots,k+\rho_1+\cdots+\rho_{l-1}+2,\cdots,k+\rho_1+\cdots+\rho_{l-1}+\rho_l,\cdots)$$ Construct $\gamma$ to be the permutation that orderly takes each element of the above cycles of $\alpha$ to each element with the same order in $\beta,$ that is: $$\gamma(1)=k+1, \gamma(\alpha(1))=\beta(k+1),\cdots \gamma(\rho_1)=k+\rho_1,\cdots$$ It is then clear that $\alpha=\gamma^{-1}\beta\gamma$ and $\gamma\in \mathcal{B}_{kn}^k.$
\[Prop\_size\_conj\] Let $\omega$ be a permutation of $\mathcal{B}_{kn}^k$ then the size of the conjugacy class of $\omega$ is given by: $$\frac{n!(k!)^n}{\displaystyle \prod_{\rho\vdash k}z_{\omega(\rho)}z_\rho^{l(\omega(\rho))}}.$$
Suppose that $\operatorname{ct}(p_\omega)=\lambda,$ where $\lambda$ is a partition of $n.$ We will explain how to obtain all the elements $\gamma\in \mathcal{B}_{kn}^k$ that have the same type as $\omega.$ First of all, $p_\gamma$ must be decomposed in cycles with sizes equivalent to those of $p_\omega.$ In other words, $p_\gamma$ should have the same cycle-type as $p_\omega$ and there are $$\frac{n!}{z_{\lambda}}$$ choices to make $p_\gamma.$
Now suppose that $m_r(\lambda)\neq 0$ for some $r\in [n].$ That means that there are exactly $m_r(\lambda)$ cycles of length $r$ distributed in the family $(\omega(\rho))_{\rho\vdash k}.$ Suppose they all appear in the partitions $\omega(\rho_1),\cdots, \omega(\rho_m).$ In order to have $\operatorname{type}(\gamma)=\operatorname{type}(\omega),$ we should make $\gamma(\rho_1)=\omega(\rho_1),$ $\gamma(\rho_2)=\omega(\rho_2)$ and so on. That is $\gamma(\rho_1)$ should have parts equal to $r$ as many as $\omega(\rho_1)$ has and so on. The total number of $\gamma$ verifying these conditions is $$\frac{m_r(\lambda)!}{\displaystyle \prod_{1\leqslant j\leqslant m}m_r(\omega(\rho_j))!}=\frac{m_r(\lambda)!}{\displaystyle \prod_{\rho\vdash k}{m_r(\omega(\rho))!}}$$
It remains to see how many $\gamma\in \mathcal{B}_{kn}^k$ one can make when he nows all $\gamma(\rho).$ For this fix a partition $\rho_1$ of $k$ such that $\gamma(\rho_1)\neq \emptyset$ and suppose $m_1$ is a part of $\gamma(\rho_1).$ There will be $m_1$ $k$-tuples involved in the construction of $\gamma$ now. Choose one of them then distribute its elements according to $\rho_1$ in $\frac{k!}{z_{\rho_1}}$ ways. To complete $\gamma,$ there will be $(k!)^{m_1-1}$ choices for the elements between any two consecutive elements of the fixed $k$-tuples in any chosen cycle.
By bringing together all the above arguments, the size of the conjugacy class of $\omega$ is:
$$\frac{n!}{z_{\lambda}}\displaystyle \prod_{r,m_r(\lambda)\neq 0} \frac{m_r(\lambda)!}{\displaystyle \prod_{\rho\vdash k}{m_r(\omega(\rho))!}}\displaystyle \frac{(k!)^{n}}{z_{\rho}^{l(\omega(\rho))}}=\frac{n!(k!)^n}{\displaystyle \prod_{\rho\vdash k}z_{\omega(\rho)}z_\rho^{l(\omega(\rho))}}.$$ This equality is due to the fact that $$l(\lambda)=\sum_{r\geq 1}m_r(\lambda)=\sum_{r\geq 1}\sum_{\rho\vdash k}m_r(\omega(\rho)).$$
We turn now to show that the group $\mathcal{B}_{kn}^k$ is isomorphic to the wreath product $\mathcal{S}_k \sim \mathcal{S}_n.$ The wreath product $\mathcal{S}_k \sim \mathcal{S}_n$ is the group with underlying set $\mathcal{S}_k^n\times \mathcal{S}_n$ and product defined as follows: $$((\sigma_1,\sigma_2,\cdots ,\sigma_n); p).((\epsilon_1,\epsilon_2,\cdots ,\epsilon_n); q)=((\sigma_1\epsilon_{p^{-1}(1)},\sigma_2\epsilon_{p^{-1}(2)},\cdots ,\sigma_n\epsilon_{p^{-1}(n)});pq),$$ for any $((\sigma_1,\sigma_2,\cdots ,\sigma_n); p),((\epsilon_1,\epsilon_2,\cdots ,\epsilon_n); q)\in \mathcal{S}_k^n\times \mathcal{S}_n.$ The identity in this group is $(1;1):=((1_k,1_k,\cdots ,1_k); 1_n),$ where $1_i$ denotes the identity function of $\mathcal{S}_i.$ The inverse of an element $((\sigma_1,\sigma_2,\cdots ,\sigma_n); p)\in \mathcal{S}_k \sim \mathcal{S}_n$ is given by $$((\sigma_1,\sigma_2,\cdots ,\sigma_n); p)^{-1}:=((\sigma^{-1}_{p(1)},\sigma^{-1}_{p(2)},\cdots ,\sigma^{-1}_{p(n)}); p^{-1}).$$
For each permutation $\omega\in \mathcal{B}^k_{kn}$ and each integer $i\in [n],$ define $\omega_i$ to be the normalized restriction of $\omega$ on the block $p_\omega^{-1}(i).$ That is:
$$\begin{array}{ccccc}
\omega_i & : &[k] & \to & [k] \\
& & b & \mapsto & \omega_i(b):=\omega\big(k(p_\omega^{-1}(i)-1)+b\big)\%k,\\
\end{array}$$ where $\%$ means that the integer is taken modulo $k$ (for a multiple of $k$ we use $k$ instead of $0$).
\[ex\_psi\] Consider the following permutation $\alpha$ of $\mathcal{B}^3_{18}:$ $$\alpha=\begin{pmatrix}
1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18\\
12&10&11&5&6&4&8&7&9&15&13&14&16&18&17&3&2&1
\end{pmatrix}.$$ The blocks permutation associated to $\alpha$ is $p_\alpha=(1,4,5,6)(2)(3).$ In addition, we have: $$\alpha_1=(1,3),~~\alpha_2=(1,2,3),~~\alpha_3=(1,2),~~\alpha_4=(1,3,2),~~\alpha_5=(1,3,2) \text{ and }\alpha_6=(2,3).$$
\[isom\] The application $$\begin{array}{ccccc}
\psi & : &\mathcal{B}^k_{kn} & \to & \mathcal{S}_k\sim \mathcal{S}_n \\
& & \omega & \mapsto & \psi(\omega):=((\omega_1,\cdots ,\omega_n);p_\omega),\\
\end{array}$$ is a group isomorphism.
$\psi$ is clearly a bijection with inverse given by:
$$\begin{array}{ccccc}
\phi & : &\mathcal{S}_k\sim \mathcal{S}_n & \to & \mathcal{B}^k_{kn} \\
& & ((\sigma_1,\sigma_2,\cdots ,\sigma_n); p) & \mapsto & \sigma,\\
\end{array}$$ where $\sigma\big(k(a-1)+b\big)=k(p(a)-1)+\sigma_{p(a)}(b)$ for any $a\in [n]$ and any $b\in [k].$ It remains to show that if $x=((\sigma_1,\sigma_2,\cdots ,\sigma_n); p)$ and $y=((\epsilon_1,\epsilon_2,\cdots ,\epsilon_n); q)$ are two elements of $\mathcal{S}_k\sim \mathcal{S}_n$ then $\phi(x.y)=\phi(x)\phi(y).$ To prove this let $a\in [n]$ and $b\in [k].$ On the right hand we have: $$\phi(x)\phi(y)\big(k(a-1)+b\big)=\phi(x)\big(k(q(a)-1)+\epsilon_{q(a)}(b)\big)=k(p(q(a))-1)+\sigma_{p(q(a))}(\epsilon_{q(a)}(b))$$ and on the left hand we have: $$\phi(xy)\big(k(a-1)+b\big)=k((pq)(a)-1)+\sigma_{(pq)(a)}\epsilon_{p^{-1}((pq)(a))}(b).$$ This shows that both functions $\phi(x.y)$ and $\phi(x)\phi(y)$ are equal which finishes the proof.
Recall the permutation $\alpha\in \mathcal{B}^3_{18}$ of Example \[ex\_psi\] and consider the following permutation $\beta$ of the same group: $$\beta=\begin{pmatrix}
1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18\\
4&5&6&18&17&16&8&9&7&1&2&3&12&11&10&15&14&13
\end{pmatrix}.$$ We have: $$\alpha\beta=\begin{pmatrix}
1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18\\
5&6&4&1&2&3&7&9&8&12&10&11&14&13&15&17&18&16
\end{pmatrix},$$ $$\psi(\alpha)=\Big(\big( (1,3),(1,2,3),(1,2),(1,3,2),(1,3,2),(2,3)\big);(1,4,5,6)(2)(3)\Big),$$ $$\psi(\beta)=\Big(\big( 1,1,(1,2,3),(1,3),(1,3),(1,3)\big);(1,2,6,5,4)(3)\Big)$$ and $$\psi(\alpha\beta)=\Big(\big( 1,(1,2,3),(2,3),(1,3,2),(1,2),(1,2,3)\big);(1,2)(3)(4)(5)(6)\Big).$$ Now we can easily verify that $\psi(\alpha\beta)=\psi(\alpha).\psi(\beta).$
Special cases
=============
In this section we will treat two special cases, the case of $k=2$ and that of $k=3.$ We will see that when $k=2,$ the group $\mathcal{B}_{2n}^2$ is the hyperoctahedral group on $2n$ elements. There are many papers in the literature, see [@stembridge1992projective] and [@geissinger1978representations] for examples, that study the representations of the hyperoctahedral group. Especially we are interested here in studying its conjugacy classes. We will recover some of its nice properties using our approach.
Case $k=2:$ The hyperoctahedral group {#sec_hyp}
-------------------------------------
When $k=2$ there are only two types of partitions of $2,$ mainly $\lambda_1=(1^2)$ and $\lambda_2=(2).$ Now we are going to see how things go in the context of $\mathcal{B}^2_{2n}.$ We use the same arguments given in [@Tout2017 Section 6.2]. If $a\in p_2(i)$, we shall denote by $\overline{a}$ the element of the set $p_2(i)\setminus \lbrace a\rbrace.$ Therefore, we have, $\overline{\overline{a}}=a$ for any $a\in [2n].$ As seen in our general construction, the cycle decomposition of a permutation of $\mathcal{B}^2_{2n}$ will have two types of cycles. To see this, suppose that $\omega$ is a permutation of $\mathcal{B}^2_{2n}$ and take the following cycle $\mathcal{C}$ of its decomposition: $$\mathcal{C}=(a_1,\cdots ,a_{l}).$$ We distinguish two cases :
1. $\overline{a_1}$ appears in the cycle $\mathcal{C},$ for example $a_j=\overline{a_1}.$ Since $\omega\in\mathcal{B}^2_{2n}$ and $\omega(a_1)=a_2,$ we have $\omega(\overline{a_1})=\overline{a_2}=\omega(a_j).$ Likewise, since $\omega(a_{j-1})=\overline{a_1},$ we have $\omega(\overline{a_{j-1}})=a_1$ which means that $a_{l}=\overline{a_{j-1}}.$ Therefore, $$\mathcal{C}=(a_1,\cdots a_{j-1},\overline{a_1},\cdots,\overline{a_{j-1}})$$ and $l=2(j-1)$ is even. We will denote such a cycle by $(\mathcal{O},\overline{\mathcal{O}}).$
2. $\overline{a_1}$ does not appear in the cycle $\mathcal{C}.$ Take the cycle $\overline{\mathcal{C}}$ which contains $\overline{a_1}.$ Since $\omega(a_1)=a_2$ and $\omega\in \mathcal{B}^2_{2n}$, we have $\omega(\overline{a_1})=\overline{a_2}$ and so on. That means that the cycle $\overline{\mathcal{C}}$ has the following form, $$\overline{\mathcal{C}}=(\overline{a_1},\overline{a_2},\cdots ,\overline{a_{l}})$$ and that $\mathcal{C}$ and $\overline{\mathcal{C}}$ appear in the cycle decomposition of $\omega.$
The cycles of the first case will contribute to $\omega(\lambda_2)$ while those of the second will contribute to $\omega(\lambda_1).$ Suppose now that the cycle decomposition of a permutation $\omega$ of $\mathcal{B}^2_{2n}$ is as follows: $$\omega=\mathcal{C}_1\overline{\mathcal{C}_1}\mathcal{C}_2\overline{\mathcal{C}_2}\cdots \mathcal{C}_k\overline{\mathcal{C}_k}(\mathcal{O}^1,\overline{\mathcal{O}^1})(\mathcal{O}^2,\overline{\mathcal{O}^2})\cdots (\mathcal{O}^l,\overline{\mathcal{O}^l}),$$ where the cycles $\mathcal{C}_i$ (resp. $(\mathcal{O}^j,\overline{\mathcal{O}^j})$) are written decreasingly according to their sizes. From this decomposition, we obtain that the parts of the partition $\omega(\lambda_1)$ are the sizes of the sets $\mathcal{C}_j,$ while the parts of the partition $\omega(\lambda_2)$ are the sizes of the sets $\mathcal{O}^i:$ $$\omega(\lambda_1)=(|\mathcal{C}_1|,\cdots,|\mathcal{C}_k|),~~\omega(\lambda_2)=(|\mathcal{O}^1|,\cdots,|\mathcal{O}^l|) \text{ and }|\omega(\lambda_1)|+|\omega(\lambda_2)|=n.$$
Consider the following permutation $$\omega=\begin{pmatrix}
1&2&&3&4&&5&6&&7&8&&9&10&&11&12&&13&14&&15&16\\
14&13&&1&2&&16&15&&7&8&&12&11&&10&9&&4&3&&5&6
\end{pmatrix}\in \mathcal{B}^2_{16}.$$ Its decomposition into product of disjoint cycles is as follows: $$\omega=(1,14,3)(2,13,4)(7)(8)(9,12)(10,11)(5,16,6,15).$$ Then $\omega(\lambda_1)=(3,2,1)$ and $\omega(\lambda_2)=(2).$
Apply Proposition \[Prop\_size\_conj\] to obtain the following result.
\[size\_conj\_k=2\] The size of the conjugacy class of a permutation $\omega\in \mathcal{B}^2_{2n}$ is: $$\frac{2^nn!}{2^{l(\omega(\lambda_1))+l(\omega(\lambda_2))}z_{\omega(\lambda_1)}z_{\omega(\lambda_2)}}.$$
The above Corollary \[size\_conj\_k=2\] is a well known formula for the sizes of the conjugacy classes of the hyperoctahedral group, see [@stembridge1992projective] and [@geissinger1978representations].
Case $k=3$ {#seck=3}
----------
In a way similar to that of case $k=2,$ the fact that there are only three partitions of $3$ suggests that there are three types of cycles in the decomposition into product of disjoint cycles of a permutation $\omega\in \mathcal{B}^3_{3n}.$
Let $\mathcal{C}$ be a cycle of $\omega\in \mathcal{B}^3_{3n},$ we distinguish the following three cases:
1. first case: all three elements of a certain $p_3(s)$ belong to $\mathcal{C}.$ For simplicity, suppose: $$\mathcal{C}=(a_1=1,a_2,a_3,\cdots, a_j=2,a_{j+1},\cdots, a_l=3,a_{l+1},\cdots a_k).$$ Since $\omega\in \mathcal{B}^3_{3n},$ the sets $\lbrace a_2,a_{j+1},a_{l+1}\rbrace,$ $\lbrace a_3,a_{j+2},a_{l+2}\rbrace,\cdots,$ $\lbrace a_{j-1},a_{l-1},a_k\rbrace$ all have the form $p_3(m)$ and thus $\mathcal{C}$ is a cycle of length $3(j-1)$ that contains a union of sets of the form $p_3(r).$
2. second case: two and only two elements of a certain $p_3(s)$ belong to $\mathcal{C}.$ Say, $$\mathcal{C}=(a_1=1,a_2,a_3,\cdots, a_j=2,a_{j+1},\cdots , a_k).$$ Since $\omega\in \mathcal{B}^3_{3n},$ there exists integers $b_i,$ $1\leq i\leq j-1,$ such that $$\lbrace a_1,a_{j}\rbrace=p_3(b_1)\setminus \lbrace c_1\rbrace,$$ $$\lbrace a_2,a_{j+1}\rbrace=p_3(b_2)\setminus \lbrace c_2\rbrace,$$ $$\vdots$$ $$\lbrace a_{j-1},a_k\rbrace=p_3(b_{j-1})\setminus \lbrace c_{j-1}\rbrace,$$ and another cycle $(c_1,c_2,\cdots ,c_{j-1})$ should thus appear in the decomposition of $\omega.$
3. third case: all three elements of a certain $p_3(s)$ belong to different cycles. In this case, all three cycles will have the same lengths and each one of them will contain elements belonging to different triplets.
Now for any permutation $\omega\in \mathcal{B}^3_{3n},$ define $\gamma_\omega$ to be the partition obtained from the lengths divided by three of the cycles of the first case, $\beta_\omega$ the partition obtained from the lengths of the cycles of the second case divided by two and $\alpha_\omega$ the partition obtained from the lengths of the cycles of the third case. It would be clear that $$\omega(\lambda_1)=\alpha_\omega,~~ \omega(\lambda_2)=\beta_\omega,~~ \omega(\lambda_3)=\gamma_\omega \text{ and } |\gamma_\omega|+|\beta_\omega|+|\alpha_\omega|=n.$$ For example, for the permutation $\omega$ of Example \[k=3,equivex\], we have: $$\gamma_\omega=(2,2),~~\beta_\omega=(2,1) \text{ and } \alpha_\omega=(1).$$
Using Proposition \[Prop\_size\_conj\], we obtain the following result.
\[size\_con\_k=3\] The size of the conjugacy class of $\omega\in \mathcal{B}_{3n}^3$ is:
$$\frac{(3!)^nn!}{(3!)^{l(\alpha_\omega)}z_{\alpha_\omega}2^{l(\beta_\omega)}z_{\beta_\omega}3^{l(\gamma_\omega)}z_{\gamma_\omega}}=2^{n-l(\alpha_\omega)-l(\beta_\omega)}3^{n-l(\alpha_\omega)-l(\gamma_\omega)}.\frac{n!}{z_{\alpha_\omega}z_{\beta_\omega}z_{\gamma_\omega}}.$$
The center of the group $\mathcal{B}_{kn}^k$ algebra {#sec_6}
====================================================
In this section, we present in Theorem \[main\_the\] a polynomiality property for the structure coefficients of the center of the group $\mathcal{B}_{kn}^k$ algebra. This can be seen as a generalisation of the Farahat and Higman result in [@FaharatHigman1959] and our result in [@Tout2017] that gave polynomiality properties for the structure coefficients of the center of the symmetric group and the hyperoctahedral group algebras respectively. A special treatment for the cases $k=2$ and $k=3$ is given.
The algebra $Z(\mathbb{C}[\mathcal{B}_{kn}^k])$
-----------------------------------------------
The center of the group $\mathcal{B}_{kn}^k$ algebra will be denoted $Z(\mathbb{C}[\mathcal{B}_{kn}^k]).$ It is the algebra over $\mathbb{C}$ spanned by the “formal sum of elements of the” conjugacy classes of $\mathcal{B}_{kn}^k.$ According to Proposition \[class\_conj\_classes\], these are indexed by families of partitions $x=(x(\lambda))_{\lambda\vdash k}$ satisfying the property: $$\label{co}
|x|:=\sum_{\lambda\vdash k}|x(\lambda)|=n$$ and for each such a family its associated conjugacy class $C_{x}$ is $$C_{x}=\lbrace t\in \mathcal{B}_{kn}^k \text{ such that }\operatorname{type}(t)=x\rbrace$$ while its formal sum of elements is $$\mathbf{C}_x:=\sum_{t\in C_x}t.$$ From now on and unless stated otherwise, $x$ is a family of partition would mean $x=(x(\lambda))_{\lambda\vdash k}$ with the condition \[co\].
Let $x$ and $y$ be two family of partitions. In the algebra $Z(\mathbb{C}[\mathcal{B}_{kn}^k]),$ the product $\mathbf{C}_x\mathbf{C}_y$ can be written as a linear combination as following $$\label{struc_coef}
\mathbf{C}_x\mathbf{C}_y=\sum_{z}c_{xy}^z \mathbf{C}_z$$ where $z$ runs through all the families of partitions. The coefficients $c_{xy}^z$ that appear in this equation are called the structure coefficients of the center of the group $\mathcal{B}_{kn}^k$ algebra.
Polynomiality of the structure coefficients of $Z(\mathbb{C}[\mathcal{B}_{kn}^k])$
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When $k=1,$ Farahat and Higman were the first to give a polynomiality property for the structure coefficients of the center of the symmetric group algebra in [@FaharatHigman1959]. In the case of the center of the hyperoctahedral group algebra, we gave a polynomiality property for its structure coefficients in [@Tout2017]. The goal of this section is to generalize these results and show that the structure coefficients of $Z(\mathbb{C}[\mathcal{B}_{kn}^k])$ have a polynomiality property for any fixed $k.$\
The most natural way to see a permutation $\omega\in \mathcal{B}_{kn}^k$ as an element of $\omega\in \mathcal{B}_{k(n+1)}^k$ is by extending it by identity. By doing so, the new permutation will have the same type as $\omega$ except that the partition $\omega(1^k)$ will become $\omega(1^k)\cup (1).$
\[def\_propre\] A family of partitions $x=(x(\lambda))_{\lambda\vdash k}$ is said to be proper if and only if the partition $x(1^k)$ is proper. If $x=(x(\lambda))_{\lambda\vdash k}$ is a proper family of partitions such that $|x|<n,$ we define $C_{x}(n)$ to be the set of elements $t\in \mathcal{B}_{kn}^k$ that have type equals to $x$ except that $x(1^k)$ is replaced by $x(1^k)\cup (1^{n-|x|}).$
If $x=(x(\lambda))_{\lambda\vdash k}$ is a proper family of partitions such that $|x|=n_0$ then for any $n>n_0$ we have by Proposition \[Prop\_size\_conj\] the following result:
$$\begin{aligned}
|C_x(n)|&=&\frac{n!(k!)^n}{z_{x(1^k)\cup (1^{n-n_0})}(k!)^{l(x(1^k))+n-n_0}\displaystyle \prod_{\lambda\vdash k, \lambda\neq (1^k)}z_{x(\lambda)}z_\lambda^{l(x(\lambda))}}\\
&=&\frac{n!(k!)^{n_0-l(x(1^k))}}{z_{x(1^k)}(n-n_0)!\displaystyle \prod_{\lambda\vdash k, \lambda\neq (1^k)}z_{x(\lambda)}z_\lambda^{l(x(\lambda))}}.\end{aligned}$$
Take $x$ and $y$ to be two proper families of partitions. For any integer $n>|x|,|y|$ we have the following equation in $Z(\mathbb{C}[\mathcal{B}_{kn}^k])$ $$\label{eq_str_coe}
\mathbf{C}_x(n)\mathbf{C}_y(n)=\sum_{z}c_{xy}^h(n) \mathbf{C}_z(n),$$ where $h$ runs through all proper families of partitions verifying $|z|\leq n.$
In [@Tout2017], under some conditions, a formula describing the form of the structure coefficients of centers of finite group algebras is given. We show below that the sequence $(\mathcal{B}_{kn}^k)_n$ satisfies these conditions. This will allow us to use [@Tout2017 Corollary 6.3] in order to give a polynomiality property for the structure coefficients $c_{xy}^h(n)$ described in Equation (\[eq\_str\_coe\]).
We will show below the conditions required in [@Tout2017] for our sequence of groups $(\mathcal{B}_{kn}^k)_n.$ To avoid repetitions and confusing notations and since the integer $k$ will be fixed, we will use simply $G_n$ to denote the group $\mathcal{B}_{kn}^k.$
**Hypothesis 1:** For any integer $1\leq r\leq n,$ there exists a group $G_n^r$ isomorphic to $G_{n-r}.$ Set $$G_n^r:=\lbrace \omega\in G_n \text{ such that } \omega(i)=i \text{ for any $1\leq i\leq kr$}\rbrace.$$ for this reason.
**Hypothesis 2:** The elements of $G_n^r$ and $G_r$ commute between each other which is normal since the permutations in these groups act on disjoint sets.
**Hypothesis 3:** $G_{n+1}^r\cap G_n=G_n^r$ which is obvious.
**Hypothesis 4:** For any $z\in G_n,$ $\mathrm{k}(G_n^{r_1} zG_n^{r_2}):=\min\lbrace s|G_n^{r_1} zG_n^{r_2}\cap G_s\neq \emptyset\rbrace\leq r_1+r_2.$ To prove this, remark first that the size of the set $\lbrace 1,\cdots,kr_1\rbrace \cap \lbrace z(1),\cdots,z(kr_2)\rbrace$ is a multiple of $k$ since $z\in G_n,$ say it is $km.$ Suppose that $$\lbrace h_1,\cdots, h_{kr_1-km}\rbrace =\lbrace1,\cdots,kr_1\rbrace\setminus \lbrace z(1),\cdots,z(kr_2)\rbrace.$$ We can find a permutation of the following form $$\begin{matrix}
1 & 2 & \cdots & kr_2 & kr_2+1 & \cdots & kr_1+kr_2-km & kr_1+kr_2-km+1 & \cdots & kn \\
z(1) & z(2) & \cdots & z(kr_2) & h_1 & \cdots & h_{kr_1-km} & * & \cdots & *
\end{matrix}$$ in $zG_n^{r_2}$ since it contains permutations that fixes the first $kr_2$ images of $z.$ The stars are used to say that the images may not be fixed. Since the multiplication by an element of $G_n^{r_1}$ to the left permutes the elements greater than $kr_1$ in the second line defining this permutation, the set $G_n^{r_1} zG_n^{r_2}$ contains thus a permutation of the following form $$\begin{matrix}
1 & 2 & \cdots & kr_2 & kr_2+1 & \cdots & kr_1+kr_2-km & kr_1+kr_2-km+1 & \cdots & kn \\
* & * & \cdots & * & h_1 & \cdots & h_{kr_1-km} & kr_1+kr_2-km+1 & \cdots & kn
\end{matrix}$$ This permutation is also in $G_{r_1+r_2-m}$ which ends the proof. **Hypothesis 5:** If $z\in G_n$ then we have $zG_n^{r_1}z^{-1}\cap G_n^{r_2}= G_n^{r(z)}$ where $$\begin{aligned}
r(z)&=&|\lbrace z(1),z(2),\cdots , z(kr_1),1,\cdots ,kr_2\rbrace| \\
&=&kr_1+kr_2-|\lbrace z(1),z(2),\cdots , z(kr_1)\rbrace\cap\lbrace 1,\cdots ,kr_2\rbrace|.\end{aligned}$$ To prove this, let $a=zbz^{-1}$ be an element of $G_n$ which fixes the $kr_2$ first elements while $b$ fixes the $kr_1$ first elements. Then $a$ also fixes the elements $z(1),\cdots, z(kr_1)$ which proves that $zG_n^{r_1}z^{-1}\cap G_n^{r_2}\subset G_n^{r(z)}.$ In the opposite direction, if $p$ is a permutation of $n$ which fixes the elements of the set $\lbrace z(1),z(2),\cdots , z(kr_1),1,\cdots ,kr_2\rbrace$ then $p$ is in $G_n^{r_2}$ and in addition $z^{-1}pz$ is in $G_n^{r_1}$ which implies that $p=zz^{-1}pzz^{-1}$ is in $zG_n^{r_1}z^{-1}.$\
Now with all the necessary hypotheses verified we can apply the main result in [@Tout2017] to get the following theorem.
\[main\_the\] Let $x,$ $y$ and $h$ be three proper families of partitions. For any $n>|x|,|y|,|h|,$ the coefficients $c_{xy}^h(n)$ defined in Equation \[eq\_str\_coe\] are polynomials in $n$ with non-negative rational coefficients. In addition, $$\deg(c_{xy}^h(n))<|x|+|y|-|h|.$$
By [@Tout2017 Corollary 6.3], if $n>|x|,|y|,|h|$ then $$c_{xy}^h(n)=\frac{|C_x(n)||C_y(n)||\mathcal{B}_{k(n-|x|)}^k||\mathcal{B}_{k(n-|y|)}^k|}{|\mathcal{B}_{kn}^k||C_h(n)|}\sum_{|h|\leq r \leq |x|+|y|}\frac{a_{xy}^h(r)}{|\mathcal{B}_{k(n-r)}^k|}$$ where the $a_{xy}^h(r)$ are positive, rational and independent numbers of $n.$ Since all the cardinals involved in this formula are known, we get after simplification the following formula for $c_{xy}^h(n)$ $$c_{xy}^h(n)=(k!)^{l(h(1^k))-l(x(1^k))-l(y(1^k))}\frac{z_{h(1^k)}c_h}{z_{x(1^k)}z_{y(1^k)}c_xc_y}\sum_{|h|\leq r \leq |x|+|y|}\frac{(k!)^{r-|h|}a_{xy}^h(r)(n-|h|)!}{(n-r)!}$$ where $c_x$ denotes $\prod_{\lambda\vdash k, \lambda\neq (1^k)}z_{x(\lambda)}z_\lambda^{l(x(\lambda))}.$ The result follows.
Special cases
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In this section we revisit the two already known results of polynomiality for the structure coefficients of the center of the symmetric group ($k=1$) algebra and the center of the hyperoctahedral ($k=2$) group algebra. In addition, as an application of our main theorem, we give a polynomiality property in the case $k=3.$ In these three cases, we give explicit expressions of products of conjugacy classes in the associated center algebra in order to see our results.
### k=1: The symmetric group
As seen in Section \[sec\_2\], the conjugacy classes of the symmetric group $\mathcal{S}_n$ are indexed by partitions of $n.$ If $\lambda$ is a partition of $n$ the size of its associated conjugacy class is $$|C_\lambda|=\frac{n!}{z_\lambda}.$$
If $\lambda$ is a proper partition with $|\lambda|<n,$ we define $\underline{\lambda}_n$ to be the partition $\lambda\cup (1^{n-|\lambda|}).$ Now let $\lambda$ and $\delta$ be two proper partitions with $|\lambda|,|\delta|<n.$ In the center of the symmetric group algebra we have the following equation: $$\label{Eq_str_sym}
\textbf{C}_{\underline{\lambda}_n}\textbf{C}_{\underline{\delta}_n}=\sum_{\gamma}c_{\lambda\delta}^\gamma (n)\textbf{C}_{\underline{\gamma}_n}$$ where the sum runs through all proper partitions $\gamma$ satisfying $|\gamma|\leq |\lambda|+|\delta|.$ If we apply Theorem \[main\_the\], we re-obtain the following result of Farahat and Higman in [@FaharatHigman1959 Theorem 2.2].
Let $\lambda,$ $\delta$ and $\gamma$ be three proper partitions and let $n\geq |\lambda|,|\delta|,|\gamma|$ be an integer. The structure coefficient $c_{\lambda\delta}^{\gamma}(n)$ of the center of the symmetric group algebra defined by Equation (\[Eq\_str\_sym\]) is a polynomial in $n$ with non-negative coefficients and $$\deg(c_{\lambda\delta}^{\gamma}(n))\leq |\lambda|+|\delta|-|\gamma|.$$
See [@touPhd14] for more information about the computation of the following two complete expressions in the center of the symmetric group $\mathcal{S}_n,$ $$\mathbf{C}_{\underline{(2)}_n}\mathbf{C}_{\underline{(2)}_n}=\frac{n(n-1)}{2} \mathbf{C}_{\underline{\emptyset}_n}+3\mathbf{C}_{\underline{(3)}_n}+2\mathbf{C}_{\underline{(2,2)}_n} \text{ for any $n\geq 4$},$$ and $$\mathbf{C}_{\underline{(2)}_n}\mathbf{C}_{\underline{(3)}_n}=2(n-2) \mathbf{C}_{\underline{(2)}_n}+4\mathbf{C}_{\underline{(4)}_n}+\mathbf{C}_{\underline{(2,3)}_n} \text{ for any $n\geq 5$}.$$ $$\mathbf{C}_{(1^{n-2},2)}\mathbf{C}_{(1^{n-2},2)}=\frac{n(n-1)}{2} \mathbf{C}_{(1^{n})}+3\mathbf{C}_{(1^{n-3},3)}+2\mathbf{C}_{(1^{n-4},2^2)} \text{ for any $n\geq 4$},$$ and $$\mathbf{C}_{(1^{n-2},2)}\mathbf{C}_{(1^{n-3},3)}=2(n-2) \mathbf{C}_{(1^{n-2},2)}+4\mathbf{C}_{(1^{n-4},4)}+\mathbf{C}_{(1^{n-5},2,3)} \text{ for any $n\geq 5$}.$$
### k=2: The hyperoctahedral group
In Section \[sec\_hyp\], we showed that the conjugacy classes of the hyperoctahedral group are indexed by pairs of partitions $(\lambda,\delta)$ such that $|\lambda|+|\delta|=n.$ The partition $\lambda$ is that associated to the partition $(1,1)$ of $2$ while $\delta$ is associated to the partition $(2).$ The size of the class $C_{(\lambda,\delta)}$ is given in Corollary \[size\_conj\_k=2\]
$$|C_{(\lambda,\delta)}|=\frac{2^nn!}{2^{l(\lambda)+l(\delta)}z_{\lambda}z_{\delta}}.$$
By Definition \[def\_propre\], the pair $(\lambda,\delta)$ is proper if and only if the partition $\lambda$ is proper. For a proper pair $(\lambda,\delta)$ of partitions and for any integer $n>|\lambda|+|\delta|,$ we define the following pair of partitions: $$\underline{(\lambda,\delta)}_n:=(\lambda\cup (1^{n-|\lambda|-|\delta|}),\delta).$$ Let $(\lambda_1,\delta_1)$ and $(\lambda_2,\delta_2)$ be two proper pairs of partitions. We have the following equation in the center of the hyperoctahedral group algebra for any integer $n$ greater than $|\lambda_1|+|\delta_1|, |\lambda_2|+|\delta_2|,$
$$\label{Eq_str_hyp}
\textbf{C}_{\underline{(\lambda_1,\delta_1)}_n}\textbf{C}_{\underline{(\lambda_2,\delta_2)}_n}=\sum_{(\lambda_3,\delta_3)}c_{(\lambda_1,\delta_1)(\lambda_2,\delta_2)}^{(\lambda_3,\delta_3)} (n)\textbf{C}_{\underline{(\lambda_3,\delta_3)}_n}$$
where the sum runs over all the proper pairs of partitions $(\lambda_3,\delta_3)$ satisfying $|\lambda_3|+|\delta_3|\leq |\lambda_1|+|\delta_1|+\lambda_2|+|\delta_2|.$ As an application of Theorem \[main\_the\], we re-obtain the following result in [@Tout2017 Corollary 6.11].
Let $(\lambda_1,\delta_1), (\lambda_2,\delta_2)$ and $(\lambda_3,\delta_3)$ be three proper pairs of partitions, then for any $n\geq |\lambda_1|+|\delta_1|,|\lambda_2|+|\delta_2|,|\lambda_3|+|\delta_3|$ the structure coefficient $c_{(\lambda_1,\delta_1)(\lambda_2,\delta_2)}^{(\lambda_3,\delta_3)}(n)$ of the center of the hyperoctahedral group algebra defined in Equation (\[Eq\_str\_hyp\]) is a polynomial in $n$ with non-negative coefficients and we have $$\deg(c_{(\lambda_1,\delta_1)(\lambda_2,\delta_2)}^{(\lambda_3,\delta_3)}(n))\leq |\lambda_1|+|\delta_1|+|\lambda_2|+|\delta_2|-|\lambda_3|-|\delta_3|.$$
We give in this example the complete product of the class $C_{((1^{n-2}),(2))}$ by itself whenever $n\geq 4:$ $$\mathbf{C}_{\underline{(\emptyset,(2))}_n}\mathbf{C}_{\underline{(\emptyset,(2))}_n}=n(n-1)\mathbf{C}_{\underline{(\emptyset,\emptyset)}_n}+2\mathbf{C}_{\underline{(\emptyset,(2^2))}_n}+2\mathbf{C}_{\underline{(\emptyset,(1^2))}_n}+3\mathbf{C}_{\underline{((3),\emptyset)}_n}.$$ $$\mathbf{C}_{((1^{n-2}),(2))}\mathbf{C}_{((1^{n-2}),(2))}=n(n-1)\mathbf{C}_{((1^{n}),\emptyset)}+2\mathbf{C}_{((1^{n-4}),(2^2))}+2\mathbf{C}_{((1^{n-2}),(1^2))}+3\mathbf{C}_{((1^{n-3},3),\emptyset)}.$$
${C}_{((1^{n}),\emptyset)}$ is the identity class and since any element in ${C}_{((1^{n-2}),(2))}$ has its inverse in ${C}_{((1^{n-2}),(2))},$ the coefficient of $\mathbf{C}_{((1^{n}),\emptyset)}$ is the size of the conjugacy class ${C}_{((1^{n-2}),(2))}$ which is $n(n-1).$ The coefficient of $\mathbf{C}_{((1^{n-4}),(2^2))}$ is $2$ since if we fix a permutation of ${C}_{((1^{n-4}),(2^2))},$ say $$(1,3,2,4)(5,7,6,8)(9)(10)\cdots (2n),$$ then there exists only two pairs $(\alpha;\beta)\in {C}_{((1^{n-2}),(2))}\times {C}_{((1^{n-2}),(2))}$ such that $$\alpha\beta=(1,3,2,4)(5,7,6,8)(9)(10)\cdots (2n).$$ Mainly: $$(\alpha,\beta)=((1,3,2,4)(5)\cdots (2n);(1)(2)(3)(4)(5,7,6,8)(9)\cdots (2n))$$ or $$(\alpha,\beta)=((1)(2)(3)(4)(5,7,6,8)(9)\cdots (2n);(1,3,2,4)(5)\cdots (2n)).$$ There are only two permutations in $\mathcal{B}_4^2,$ namely $\alpha=(1,3,2,4)$ and $\beta=(1,4,2,3),$ such that $\alpha,\beta\in {C}_{(\emptyset,(2))}$ and $\alpha^2=\beta^2=(12)(34).$ Thus the coefficient of $\mathbf{C}_{((1^{n-2}),(1^2))}$ is $2.$ The last coefficient can be obtained by identifying both sides.
### k=3: the group $\mathcal{B}_{3k}^3$
Moving to the case $k=3,$ we showed in Section \[seck=3\] that the conjugacy classes of the group $\mathcal{B}_{3n}^3$ are indexed by triplet of partitions $(\alpha,\beta,\gamma)$ such that $|\alpha|+|\beta|+|\gamma|=n.$ For a given triplet $(\alpha,\beta,\gamma),$ the size of its associated conjugacy class is given in Corollary \[size\_con\_k=3\],
$$|C_{(\alpha,\beta,\gamma)}|=2^{n-l(\alpha)-l(\beta)}3^{n-l(\alpha)-l(\gamma)}.\frac{n!}{z_{\alpha}z_{\beta}z_{\gamma}}.$$
By Definition \[def\_propre\], the triplet $(\alpha,\beta,\gamma)$ is proper if and only if the partition $\alpha$ is proper. Fix three proper triplet of partitions, $(\alpha_1,\beta_1,\gamma_1),$ $(\alpha_2,\beta_2,\gamma_2)$ and $(\alpha_3,\beta_3,\gamma_3),$ the structure coefficient associated to these three triplet has the following form according to the proof of Theorem \[main\_the\],
$$c_{(\alpha_1,\beta_1,\gamma_1)(\alpha_2,\beta_2,\gamma_2)}^{(\alpha_3,\beta_3,\gamma_3)}(n)=(3!)^{l(\alpha_3)-l(\alpha_1)-l(\alpha_2)}\frac{z_{\alpha_3}z_{\beta_3}2^{l(\beta_3)}z_{\gamma_3}3^{l(\gamma_3)}}{z_{\alpha_1}z_{\alpha_2}z_{\beta_1}2^{l(\beta_1)}z_{\gamma_1}3^{l(\gamma_1)}z_{\beta_2}2^{l(\beta_2)}z_{\gamma_2}3^{l(\gamma_2)}}\times$$ $$\sum_{r}\frac{(3!)^{r-|\alpha_3|-|\beta_3|-|\gamma_3|}a_{(\alpha_1,\beta_1,\gamma_1)(\alpha_2,\beta_2,\gamma_2)}^{(\alpha_3,\beta_3,\gamma_3)}(r)(n-|\alpha_3|-|\beta_3|-|\gamma_3|)!}{(n-r)!}$$ where the sum runs through all integers $r$ with $$|\alpha_3|+|\beta_3|+|\gamma_3|\leq r \leq |\alpha_1|+|\beta_1|+|\gamma_1|+|\alpha_2|+|\beta_2|+|\gamma_2|.$$
Let $(\alpha_1,\beta_1,\gamma_1),$ $(\alpha_2,\beta_2,\gamma_2)$ and $(\alpha_3,\beta_3,\gamma_3)$ be three proper triplet of partitions, then for any $n\geq |\alpha_1|+|\beta_1|+|\gamma_1|,|\alpha_2|+|\beta_2|+|\gamma_2|,|\alpha_3|+|\beta_3|+|\gamma_3|$ the structure coefficient $c_{(\alpha_1,\beta_1,\gamma_1)(\alpha_2,\beta_2,\gamma_2)}^{(\alpha_3,\beta_3,\gamma_3)}(n)$ of the center of the group $\mathcal{B}_{3n}^3$ algebra is a polynomial in $n$ with non-negative coefficients and we have $$\deg(c_{(\alpha_1,\beta_1,\gamma_1)(\alpha_2,\beta_2,\gamma_2)}^{(\alpha_3,\beta_3,\gamma_3)}(n))\leq |\alpha_1|+|\beta_1|+|\gamma_1|+|\alpha_2|+|\beta_2|+|\gamma_2|-|\alpha_3|-|\beta_3|-|\gamma_3|.$$
For $n\geq 3,$ we leave it to the reader to verify the following two complete expressions in $Z(\mathbb{C}[\mathcal{B}_{3n}^3]):$
$$\mathbf{C}_{(\emptyset,(1),(1))}(n)\mathbf{C}_{(\emptyset,\emptyset,(1))}(n)=2\mathbf{C}_{(\emptyset,(1),(1^2))}(n)+2(n-1)\mathbf{C}_{(\emptyset,(1),\emptyset)}(n)+3\mathbf{C}_{(\emptyset,(1),(1))}(n)$$ and $$\mathbf{C}_{(\emptyset,(1),(1))}(n)\mathbf{C}_{(\emptyset,(1),\emptyset)}(n)=2\mathbf{C}_{(\emptyset,(1^2),(1))}(n)+3(n-1)\mathbf{C}_{(\emptyset,\emptyset,(1))}(n)+4\mathbf{C}_{(\emptyset,(1^2),\emptyset)}(n)+6\mathbf{C}_{(\emptyset,\emptyset,(1^2))}(n).$$ $$\mathbf{C}_{((1^{n-2}),(1),(1))}\mathbf{C}_{((1^{n-1}),\emptyset,(1))}=2\mathbf{C}_{((1^{n-3}),(1),(1^2))}+2(n-1)\mathbf{C}_{((1^{n-1}),(1),\emptyset)}+3\mathbf{C}_{((1^{n-2}),(1),(1))}$$ and $$\mathbf{C}_{((1^{n-2}),(1),(1))}\mathbf{C}_{((1^{n-1}),(1),\emptyset)}=2\mathbf{C}_{((1^{n-3}),(1^2),(1))}+3(n-1)\mathbf{C}_{((1^{n-1}),\emptyset,(1))}+4\mathbf{C}_{((1^{n-2}),(1^2),\emptyset)}$$$$+6\mathbf{C}_{((1^{n-2}),\emptyset,(1^2))}.$$
We give in this example the complete product of the class $C_{(\emptyset,(1),(1))}(n)$ by itself whenever $n\geq 12:$ $$C_{(\emptyset,(1),(1))}(n)^2=6n(n-1)C_{(\emptyset,\emptyset,\emptyset)}(n)+3(n-1)C_{(\emptyset,\emptyset,(1))}(n)+4nC_{(\emptyset,(1^2),\emptyset)}(n)$$ $$+6(n-2)C_{(\emptyset,\emptyset,(1^2))}(n)+4C_{(\emptyset,(1^2),(1^2))}(n)+2C_{(\emptyset,(1^2),(1))}(n)+18C_{(\emptyset,\emptyset,(1^3))}(n).$$
The algebra of $k$-partial permutations
=======================================
Suppose we have a set $d$ that is a disjoint union of some $k$-tuples $$d=\bigcup_{i=1}^rp_k(a_i),$$ we define the group $\mathcal{B}_d^k$ to be the following group of permutations: $$\mathcal{B}_d^k=\lbrace \omega\in \mathcal{S}_d \mid \forall 1\leq i \leq r, \exists 1\leq j\leq r \text{ with }\omega(p_k(a_i))=p_k(a_j) \rbrace.$$ In other words, the group $\mathcal{B}_d^k$ consists of permutations that permute the blocks of the set $d.$
A $k$-partial permutation of $n$ is a pair $(d,\omega)$ where $d\subset [kn]$ is a disjoint union of some $k$-tuples and $\omega$ is a permutation of the group $\mathcal{B}_d^k.$
The concept of $k$-partial permutation can be seen as a generalization of the concept of partial permutation defined by Ivanov and Kerov in [@Ivanov1999]. In fact when $k=1,$ a $1$-partial permutation is a partial permutation as defined in [@Ivanov1999].\
We will denote by $\mathcal{P}_{kn}^k$ the set of all $k$-partial permutations of $n.$ It would be clear that the cardinal of the set $\mathcal{P}_{kn}^k$ is $$|\mathcal{P}_{kn}^k|=\sum_{r=1}^n {n \choose r}(k!)^rr!=\sum_{r=1}^n (n)_r(k!)^r,$$ where $(n)_r:=n(n-1)\cdots (n-r+1).$
\[def\_supp\_ext\] If $(d,\omega)$ is a $k$-partial permutation of $n,$ we define:
1. $\operatorname{supp}(\omega)$ to be the support of $\omega.$ That is the minimal union of $k$-tuples of $d$ on which $\omega$ does not act like the identity.
2. $\underline{\omega}_n$ to be the natural extension of $\omega$ to the set $[kn].$
Consider the $3$-partial permutation $(d,\omega),$ where $d=p_3(1)\cup p_3(2)\cup p_3(4)\cup p_3(6)$ and $$\omega=
\begin{pmatrix}
1&2&3&4&5&6&10&11&12&16&17&18\\
12&10&11&4&5&6&16&18&17&1&2&3
\end{pmatrix}.$$ We have $\operatorname{supp}(\omega)=p_3(1)\cup p_3(4)\cup p_3(6)$ and $$\underline{\omega}_6=
\begin{pmatrix}
1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18\\
12&10&11&4&5&6&7&8&9&16&18&17&13&14&15&1&2&3
\end{pmatrix}.$$
The notion of $\operatorname{type}$ defined for the permutations of $\mathcal{B}_{kn}^n$ can be extended to the $k$-partial permutations of $n.$ If $(d,\omega)$ is a $k$-partial permutation of $n,$ we define its type $\lambda=(\lambda(\rho))_{\rho\vdash k}$ to be the type of its permutation $\omega.$ For example the cycle decomposition of the $3$-partial permutation given in the above example is $$(1,12,17,2,10,16)(3,11,18)(4)(5)(6)$$ and its type is formed by $\omega(2,1)=(3)$ and $\omega(1^3)=(1).$
If $(d_1,\omega_1)$ and $(d_2,\omega_2)$ are two $k$-partial permutations of $n,$ we define their product as follows: $$(d_1,\omega_1)(d_2,\omega_2)=(d_1\cup d_2,\omega_1\omega_2).$$
It is clear that the set $\mathcal{P}_{kn}^k$ equipped with the above product of $k$-partial permutations is a semi-group. That is the product is associative with identity element the $k$-partial permutation $(\emptyset, 1_\emptyset)$ where $1_\emptyset$ is the trivial permutation of the empty set.
The group $\mathcal{B}_{kn}^k$ acts on the set $\mathcal{P}_{kn}^k$ by the following action: $$\sigma . (d,\omega)=(\sigma(d),\sigma \omega \sigma^{-1}),$$ for any $\sigma\in \mathcal{B}_{kn}^k$ and $(d,\omega)\in \mathcal{P}_{kn}^k.$ We will use the term conjugacy class to denote an orbit of this action and we will say that two elements of $\mathcal{P}_{kn}^k$ are conjugate if they belong to the same orbit. Two $k$-partial permutations $(d_1,\omega_1)$ and $(d_2,\omega_2)$ of $n$ are in the same conjugacy class if and only if there exists a permutation $\sigma\in \mathcal{B}_{kn}^k$ such that $(d_2,\omega_2)=(\sigma(d_1),\sigma \omega_1 \sigma^{-1}).$ That is $|d_2|=|d_1|$ and $\operatorname{type}(\omega_2)=\operatorname{type}(\omega_1).$ Thus the conjugacy classes of the action of the group $\mathcal{B}_{kn}^k$ on the set of $k$-partial permutations $\mathcal{P}_{kn}^k$ can be indexed by families $\lambda=(\lambda(\rho))_{\rho\vdash k}$ with $|\lambda|<n$ and for such a family, its associated conjugacy class is: $$C_{\lambda;n}:=\lbrace (d,\omega)\in \mathcal{P}_{kn}^k \text{ such that } |d|=k|\lambda| \text{ and } \operatorname{type}(\omega)=\lambda\rbrace.$$
If $\lambda$ is a proper family of partitions with $|\lambda|<n,$ we define $\underline{\lambda}_n$ to be the family of partitions $\lambda$ except that $\lambda(1^k)$ is replaced by $\lambda(1^k)\cup (1^{n-|\lambda|}).$ Consider now the following surjective homomorphism $\psi$ that extends $k$-partial permutations of $n$ to elements of $\mathcal{B}_{kn}^k:$
$$\begin{array}{ccccc}
\psi & : &\mathcal{P}^k_{kn} & \to & \mathcal{B}_{kn}^k \\
& & (d,\omega) & \mapsto & \underline{\omega}_n,\\
\end{array}$$
where $\underline{\omega}_n$ is defined in Definition \[def\_supp\_ext\]. Now let $\lambda$ be a family of partitions with $|\lambda|<n$ and fix a permutation $x\in C_{\underline{\lambda}_n},$ where $C_{\underline{\lambda}_n}$ is the conjugacy class in $\mathcal{B}_{kn}^k$ associated to the family $\underline{\lambda}_n.$ The inverse image of $x$ by $\psi$ is formed by all the $k$-partial permutations $(d,\omega)$ of $n$ that satisfy the two conditions: $d\supset \operatorname{supp}(\omega)$ and $\omega$ coincide with $x$ on $d.$ Since $|\operatorname{supp}(\omega)|=|\lambda|-m_1(\lambda(1^k)),$ there are $${n-|\lambda|+m_1(\lambda(1^k))\choose m_1(\lambda(1^k))}$$ elements in $\psi^{-1}(x).$ All of these elements are in $C_{\lambda;n}$ and we recover of its elements when $x$ runs through all the elements of $C_{\underline{\lambda}_n}.$ Thus we get the following proposition.
\[prop\_rel\_conj\] If $\lambda$ is a family of partitions with $|\lambda|<n$ then: $$|C_{\lambda;n}|={n-|\lambda|+m_1(\lambda(1^k))\choose m_1(\lambda(1^k))}|C_{\underline{\lambda}_n}|.$$
The action of the group $\mathcal{B}_{kn}^k$ on the set $\mathcal{P}_{kn}^k$ of $k$-partial permutations can be extended linearly to an action of $\mathcal{B}_{kn}^k$ on the algebra $\mathbb{C}[\mathcal{P}_{kn}^k].$ The homomorphism $\psi$ can be also extended by linearity to become a surjective homomorphism between the algebras $\mathbb{C}[\mathcal{B}_{kn}^k]$ and $\mathbb{C}[\mathcal{P}_{kn}^k].$ For any $\sigma\in \mathcal{B}_{kn}^k$ and $(d,\omega)\in \mathcal{P}_{kn}^k$ we have:
$$\psi(\sigma.(d,\omega))=\psi(\sigma(d),\sigma\omega\sigma^{-1})=\underline{\sigma\omega\sigma^{-1}}_n=\sigma\underline{\omega}_n\sigma^{-1}=\sigma.\underline{\omega}_n=\sigma.\psi(d,\omega).$$
Let $\mathcal{I}_{kn}^k$ be the sub-algebra of $\mathbb{C}[\mathcal{P}_{kn}^k]$ generated by “the formal sum of” the conjugacy classes $C_{\lambda;n},$ then we have $\psi(\mathcal{I}_{kn}^k)=Z(\mathbb{C}[\mathcal{B}_{kn}^k]),$ where $Z(\mathbb{C}[\mathcal{B}_{kn}^k])$ is the center of the group algebra $\mathbb{C}[\mathcal{B}_{kn}^k]$ and by Proposition \[prop\_rel\_conj\], $$\psi(\mathbf{C}_{\lambda;n})={n-|\lambda|+m_1(\lambda(1^k))\choose m_1(\lambda(1^k))}\mathbf{C}_{\underline{\lambda}_n},$$ for any family of partitions $\lambda$ with $|\lambda|<n.$
Let $\mathbb{C}[\mathcal{P}_\infty^k]$ denote the algebra generated by all the $k$-partial permutations on finite support. An element $a$ of $\mathbb{C}[\mathcal{P}_\infty^k]$ can be canonically written as follows: $$\label{can_form}
a=\sum_{r=0}^\infty \sum_{d}\sum_{\omega \in \mathcal{B}_d^k}a_{d,\omega}(d,\omega),$$ where the second sum runs through all the set $d$ that are unions of $r$ $k$-tuples and $a_{d,\omega}\in \mathbb{C}$ for any $(d,\omega).$ Denote by $\operatorname{Proj}_n$ the natural projection homomorphism of $\mathbb{C}[\mathcal{P}_\infty^k]$ on $\mathbb{C}[\mathcal{P}_{kn}^k],$ that is if $a\in \mathbb{C}[\mathcal{P}_\infty^k]$ is canonically written as in \[can\_form\], then $$\operatorname{Proj}_n(a)=\sum_{r=0}^n \sum_{d}\sum_{\omega \in \mathcal{B}_d^k}a_{d,\omega}(d,\omega),$$ where the second sum is now taken over all the sets $d\subset [kn]$ that are unions of $r$ $k$-tuples.
Let $\mathcal{B}_\infty^k$ denote the infinite group of permutations permuting $k$-tuples. A permutation $x\in \mathcal{B}_\infty^k$ is a permutation that permutes finitely $k$-tuples, i.e. it has a finite support. The action defined for $\mathcal{B}_{kn}$ on $\mathbb{C}[\mathcal{P}_{kn}^n]$ can be generalized to an action of $\mathcal{B}_\infty^k$ on the algebra $\mathbb{C}[\mathcal{P}_\infty^k].$ In concordance with our notations, let us denote $\mathcal{I}_\infty^k$ the sub-algebra generated by the conjugacy classes of this action. It is generated by the elements $\mathbf{C}_\lambda,$ indexed by families of partitions, where
$$\mathbf{C}_\lambda=\sum_{(d,\omega)}(d,\omega),$$ where the sum runs through all $k$-partial permutations $(d,\omega)\in \mathcal{P}_{\infty}^k$ such that $d$ is a union of $|\lambda|$ $k$-tuples and $\omega$ has type $\lambda.$ It would be clear that if $|\lambda|<n,$
$$\operatorname{Proj}_n(\mathbf{C}_\lambda)=\mathbf{C}_{\lambda;n}.$$
Now $\lambda$ and $\delta$ be two proper families of partitions with $|\lambda|,|\delta|<n.$ In the algebra $\mathcal{I}_\infty^k,$ we can write the product $\mathbf{C}_\lambda\mathbf{C}_\delta$ as a linear combination of the basis elements, that is $$\label{str_coe_I_infty}
\mathbf{C}_\lambda\mathbf{C}_\delta=\sum_{\gamma}c_{\lambda\delta}^\gamma\mathbf{C}_\gamma,$$ where $\gamma$ runs through all the families of partitions and $c_{\lambda\delta}^\gamma$ are non-negative integers independent of $n.$ If we apply $\operatorname{Proj}_n$ to this equality we get the following identity in $\mathcal{I}_{kn}^k:$ $$\mathbf{C}_{\lambda;n}\mathbf{C}_{\delta;n}=\sum_{\gamma}c_{\lambda\delta}^\gamma\mathbf{C}_{\gamma;n}.$$
Since $\lambda$ and $\delta$ are proper, by applying $\psi$ to this equality we obtain using Proposition \[prop\_rel\_conj\] the following identity in the center of the group $\mathcal{B}_{kn}^k$ algebra:
$$\mathbf{C}_{\underline{\lambda}_n}\mathbf{C}_{\underline{\delta}_n}=\sum_{\gamma}c_{\lambda\delta}^\gamma{n-|\gamma|+m_1(\gamma(1^k))\choose m_1(\gamma(1^k))}\mathbf{C}_{\underline{\gamma}_n}.$$
The sum over all the families of partitions in the above equation can be turned into a sum over all the proper families of partitions if we sum up all the partitions that give $\mathbf{C}_{\underline{\gamma}_n}.$ Explicitly, we have
$$\mathbf{C}_{\underline{\lambda}_n}\mathbf{C}_{\underline{\delta}_n}=\sum_{\gamma}\Big(\sum_{r=1}^{n-|\gamma|}c_{\lambda\delta}^{\underline{\gamma}_{|\gamma|+r}}{n-|\gamma|\choose r}\Big)\mathbf{C}_{\underline{\gamma}_n},$$ where the sum now runs over all proper families of partitions. The sums over $\gamma$ in the above all equations are not infinite sums. That means that there will be a limited number of family partitions $\gamma$ appearing in each equation. To see this, one needs to understand what are the families of partitions $\gamma$ that may appear in Equation (\[str\_coe\_I\_infty\]).
For a fixed three families of partitions, $\lambda,$ $\delta$ and $\gamma,$ the coefficient $c_{\lambda\delta}^{\gamma}$ in Equation (\[str\_coe\_I\_infty\]) counts the number of pairs of $k$-partial permutations $\big( (d_1,\omega_1),(d_2,\omega_2)\big)\in C_\lambda\times C_\delta$ such that $$(d_1,\omega_1).(d_2,\omega_2)=(d,\omega)$$ where $(d,\omega)$ is a fixed $k$-partial permutation of $C_\gamma.$ We should remark that when multiplying $(d_1,\omega_1)$ by $(d_2,\omega_2),$ the permutation $\omega_1\omega_2$ acts on maximum $k|\lambda|+k|\delta|$ elements when $d_1$ and $d_2$ are disjoint sets. This means that each family of partitions $\gamma$ that appears in the sum of Equation (\[str\_coe\_I\_infty\]) must verify the following condition: $$\max(|\lambda|,|\delta|)\leq |\gamma|\leq |\lambda|+|\delta|.$$ In other words, we have showed that the function $\deg:\mathcal{I}_\infty^k \rightarrow \mathbb{N}$ defined on the basis elements of $\mathcal{I}_\infty^k$ by $\deg(\mathbf{C}_\lambda)=|\lambda|$ is a filtration on $\mathcal{I}_\infty^k.$
We can reduce the number of $\gamma$ by showing that .... are filtrations on the algebra $\mathcal{I}_\infty^k.$\
We use [@McDo Appendix B] The irreducible characters of the group $\mathcal{B}_{kn}^{k}$ are indexed by families of partitions. Let $\lambda$ be a family of partition, the associated irreducible character of $\lambda$ is $\mathcal{X}^\lambda$ defined by:
$$\mathcal{X}^\lambda=\prod_{\gamma\vdash k}\mathcal{X}^{\lambda(\gamma)}(\gamma).$$
Now, for a family of partitions $\lambda$ let us compute the following composition of morphisms $F^\lambda:=\frac{\mathcal{X}^{\lambda}}{\dim \lambda}\circ \psi \circ \operatorname{Proj}_{|\lambda|}.$ To see what the value of this morphism is, let us apply it on $\mathbf{C}_\delta$ for some family of partitions $\delta.$ First of all, it would be clear that if $|\lambda|<|\delta|$ then $F^\lambda(\mathbf{C}_\delta)=0.$ Suppose now that $|\lambda|\geq |\delta|,$ then we have: $$\begin{aligned}
F^{\lambda}(\mathbf{C}_\delta)&=&\big( \frac{\mathcal{X}^{\lambda}}{\dim \lambda}\circ \psi \circ \operatorname{Proj}_{|\lambda|} \big)(\mathbf{C}_\delta)\\
&=& \big( \frac{\mathcal{X}^{\lambda}}{\dim \lambda}\circ \psi \big) (\mathbf{C_{\underline{\delta}_{|\lambda|}}})\\
&=& {n-|\delta|+m_1(\delta(1^k))\choose m_1(\delta(1^k))}\frac{|\lambda|!(k!)^{|\lambda|}}{Z_{\underline{\delta}_{|\lambda|}}}\frac{\mathcal{X}^\lambda}{\dim \lambda}\\
&=&{n-|\delta|+m_1(\delta(1^k))\choose m_1(\delta(1^k))}\frac{|\lambda|!(k!)^{|\lambda|}}{\frac{(n-|\delta|+m_1(\delta(1^k)))!}{m_1(\delta(1^k))!}((k!)^{n-|\delta|}) Z_{\delta}}\frac{\mathcal{X}^\lambda_{\underline{\delta}_{|\lambda|}}}{\dim \lambda}\\
&=&\frac{|\lambda|!(k!)^{|\delta|}}{(|\lambda|-|\delta|)!}\frac{\mathcal{X}^\lambda_{\underline{\delta}_{|\lambda|}}}{Z_{\delta} \dim \lambda}\\
&=&(k!)^{|\delta|}\frac{P_\delta(\lambda)}{Z_\delta}. ???\end{aligned}$$
Acknowledgement {#acknowledgement .unnumbered}
===============
I would like to thank Professor Piotr Śniady for many interesting discussions about the topics presented in this paper.
This research is supported by Narodowe Centrum Nauki, grant number 2017/26/A/ST1/00189.
\[sec:biblio\]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We consider the stochastic analysis of information ranking algorithms of large interconnected data sets, e.g. Google’s PageRank algorithm for ranking pages on the World Wide Web. The stochastic formulation of the problem results in an equation of the form $$R \stackrel{\mathcal{D}}{=} Q + \sum_{i=1}^N C_i R_i,$$ where $N, Q, \{R_i\}_{i\ge 1}, \{C,C_i\}_{i \geq 1}$ are independent non-negative random variables, $\{C,C_i\}_{i \geq 1}$ are identically distributed, and $\{R_i\}_{ i \geq 1}$ are independent copies of $R$; $\stackrel{\mathcal{D}}{=}$ stands for equality in distribution. We study the asymptotic properties of the distribution of $R$ that, in the context of PageRank, represents the frequencies of highly ranked pages. The preceding equation is interesting in its own right since it belongs to a more general class of weighted branching processes that have been found useful in the analysis of many other algorithms.
Our first main result shows that if $E N E[C^\alpha]=1, \alpha >0$ and $Q, N$ satisfy additional moment conditions, then $R$ has a power law distribution of index $\alpha$. This result is obtained using a new approach based on an extension of Goldie’s (1991) implicit renewal theorem. Furthermore, when $N$ is regularly varying of index $\alpha>1$, $E N E[C^\alpha]<1$ and $Q, C$ have higher moments than $\alpha$, then the distributions of $R$ and $N$ are tail equivalent. The latter result is derived via a novel sample path large deviation method for recursive random sums. Similarly, we characterize the situation when the distribution of $R$ is determined by the tail of $Q$. The preceding approaches may be of independent interest, as they can be used for analyzing other functionals on trees. We also briefly discuss the engineering implications of our results.
title: Information Ranking and Power Laws on Trees
---
Introduction
============
We consider a problem of ranking large interconnected information (data) sets, e.g., ranking pages on the World Wide Web (Web). A solution to the preceding problem is given by Google’s PageRank algorithm, the details of which are presented in Section \[ss:pr\]. Given the large scale of these information sets, we adopt a stochastic approach to the page ranking problem, e.g. Google’s PageRank algorithm. The stochastic formulation naturally results in an equation of the form $$\label{eq:GeneralPR}
R \stackrel{\mathcal{D}}{=} Q + \sum_{i=1}^N C_i R_i ,$$ where $N, Q, \{R_i\}_{i\ge 1}, \{C,C_i\}_{i \geq 1}$ are independent non-negative random variables, $P(Q > 0) > 0$, $\{C,C_i\}_{i \geq 1}$ are identically distributed, and $\{R_i\}_{ i \geq 1}$ are independent copies of $R$; $\stackrel{\mathcal{D}}{=}$ stands for equality in distribution. We study the asymptotic properties of the distribution of $R$ that, in the context of PageRank, represents the frequencies of highly ranked pages. In somewhat smaller generality, the preceding stochastic setup was first introduced and analyzed in [@Volk_Litv_Dona_07] for the PageRank algorithm; the formulation given in was later studied in [@Volk_Litv_08].
The canonical representation given by recursion is also of independent interest since it belongs to a more general class of weighted branching processes (WBPs) [@Rosler_93; @Liu_98; @Kuhl_04]; the connection to WBPs is discussed in more detail in Section \[SS.RelatedProcesses\]. With a small abuse of notation, we also refer to our more restrictive processes as WBPs. These processes have been found useful in the average-case analysis of many algorithms [@Ros_Rus_01], e.g. quicksort algorithm [@Fill_Jan_01], and thus, our study of recursion (\[eq:GeneralPR\]) may be useful in these types of applications. Furthermore, when $Q=1, C_i \equiv 1$, the steady state solution to (\[eq:GeneralPR\]) represents the total number of individuals born in an ordinary branching process. Also, by letting $N$ be a Poisson random variable and fixing $Q = 1, C_i \equiv 1$, equation (\[eq:GeneralPR\]) reduces to the recursion that is satisfied by the busy period of an M/G/1 queue. Similarly, selecting $N=1$ yields the fixed point equation satisfied by the first order autoregressive process; see Section \[SS.RelatedProcesses\] for a more thorough discussion on related processes.
In Section \[S.ModelDescription\] we connect the iterations of recursion to an explicit construction of a WBP on a tree, such that the sum of all the weights of the first $n$ generations of the tree are directly related to the $n$th iteration of the recursion. Then, in Section \[S.Moments\] we present explicit estimates for the moments of the total weight, $W_n$, of the $n$th generation in the corresponding WBP. Using these moment estimates and the WBP representation, we show in Section \[SS.Convergence\] that under mild conditions the iterations of converge in distribution to a unique and finite steady state random variable $R$. Hence, under the stated assumptions, this limiting distribution $P(R\le x)$ is the unique solution to (\[eq:GeneralPR\]). The steady state variable $R$ represents the sum of all the weights in the corresponding branching tree.
Studying the asymptotic tail properties of the constructed steady state solution $R$ to (\[eq:GeneralPR\]) represents the main focus of this paper. In particular, we study the possible causes that can result in power tail asymptotics for $P(R > x)$. We discover that the tail behavior of $R$ can be determined/dominated by the statistical properties of any of the three variables $C, N$ and $Q$. The corresponding results are presented in Sections \[S.C\_dominates\], \[S.NDominates\] and \[S.QDominates\], respectively. Our emphasis on power law asymptotics is motivated by the well established empirical fact that the number of pages that point to a specific page (in-degree) on the Web, represented by $N$ in recursion (\[eq:GeneralPR\]), follows a power law distribution; other complex data sets, e.g. citations, are found to posses similar power law properties as well.
Our first main result on the tail behavior of $P(R>x)$ is presented in Theorem \[T.GoldieApplication\], showing that if $E N E[C^\alpha]=1, \alpha >0$ and $Q, N$ satisfy additional moment conditions, then $R$ has a power law distribution of index $\alpha$, with an explicitly characterized constant of proportionality. In particular, when $\alpha$ is an integer, the constant of proportionality of the power law distribution is explicitly computable, see Corollary \[C.explicit\]. This result is obtained by an extension of Goldie’s (1991) implicit renewal theorem that we present in Theorem \[T.Goldie\]. This extension may be of independent interest since $R$ and $C$ in the statement of Theorem \[T.Goldie\] can be any two independent random variables that may satisfy a different recursion. In the context of the broader literature on WBPs, our results are related to the studies in [@Rosler_93] (see Theorem 6), and more recently in [@Alsm_Rosl_05], both of which study recursion using stable law methods when $Q$, $\{C_i\}$ are deterministic constants. However, these deterministic assumptions fall outside of the scope of this paper; for more details see the discussion in Section \[SS.RelatedProcesses\] and the remarks after Theorem \[T.GoldieApplication\]. Outside of these results, the majority of the work on WBPs considers the homogeneous equation ($Q \equiv 0$), e.g. in [@Liu_98] the behavior of the distribution of $R$ was characterized using stable-law distributions for $0 < \alpha \leq 1$. Also, related results for the homogeneous case ($Q \equiv 0$) and $\alpha > 1$ can be found in Theorem 2.2 of [@Liu_00] and Proposition 7 of [@Iksanov_04]. Interestingly, our approach for the nonhomogeneous case ($P(Q>0) > 0$) shows that the distribution of $R$ can have a uniform treatment for any $\alpha>0$. For additional comments on results related to our Theorem \[T.GoldieApplication\] see the remarks following its statement. Furthermore, this result may provide a new explanation of why power laws are so commonly found in the distribution of wealth since weighted branching processes appear to be reasonable models for the total wealth of a family tree.
Section \[S.NDominates\] studies the case when $N$ is power law and dominates the tail behavior of $R$. This is the case that more closely relates to the original formulation of PageRank and the structure of the Web graph since the in-degree $N$ is well accepted to be a power law. Our main result in this case, stated in Theorem \[T.Main\_N\], shows that, when $N$ is regularly varying of index $\alpha>1$, $E N E[C^\alpha]<1$ and $Q, C$ have higher moments than $\alpha$, then the distribution of $R$ is tail equivalent to that of $N$. Our approach in deriving this result is based on a new sample path heavy-tailed large deviation method for weighted recursions on trees. The key technical result is given by Proposition \[P.UniformBound\] that provides a uniform bound (in $n$ and $x$) on the distribution of the total weight of the $n$th generation $P(W_n>x)$. We would also like to point out that Proposition \[P.UniformBound\] resembles to some extent a classical result by Kesten (see Lemma 7 on p. 149 of [@Ath_McD_Ney_78]), which provides a uniform bound for the sum of heavy-tailed (subexponential) random variables. The main difference between the latter result and our uniform bound is that $n$ refers to the depth of the recursion in our case, while in Lemma 7 of [@Ath_McD_Ney_78], $n$ is the number of terms in the sum. This makes the derivation of Proposition \[P.UniformBound\] considerably more complicated, and perhaps implausible, if it were not for the fact that we restrict our attention to regularly varying distributions, as opposed to the general subexponential class.
Section \[S.QDominates\] investigates a third possible source of heavy tails for $R$, the one that arises from the innovation, $Q$, being power law; see Theorem \[T.MainQ\]. For $N=1$, this result is consistent with a corresponding result for the first order autoregressive process in Lemma A.3 of [@Mik_Sam_00]. The proofs of more technical results are postponed to Section \[S.Proofs\].
Finally, from a mathematical perspective, we would like to emphasize that our sample path large deviation approach as well as the extension of the implicit renewal theory, provide a new set of tools that can be of potential use in other applications, as well as in studying the broader class of recursions on trees, e.g., one can readily characterize the asymptotic behavior of the distribution that solves $R = Q + \max_{1\leq i\leq N} C_i R_i$. Furthermore, from an engineering perspective, our Theorem \[T.Main\_N\] shows that for highly ranked pages, the PageRank algorithm basically reflects the popularity vote given by the number of references $N$, implying that overly inflated referencing may be advantageous. A more detailed discussion on the engineering implications of the performance and design of ranking algorithms, e.g. PageRank, can be found at the end of Section \[S.NDominates\].
Google’s algorithm: PageRank {#ss:pr}
----------------------------
PageRank is an algorithm trademarked by Google, the Web search engine, to assign to each page a numerical weight that measures its relative importance with respect to other pages. We think of the Web as a very large interconnected graph where nodes correspond to pages. The Google trademarked algorithm PageRank defines the page rank as: $$\label{eq:PR}
R(p_i) = \frac{1-d}{n} + d \sum_{p_j \in M(p_i)} \frac{R(p_j)}{L(p_j)},$$ where, using Google’s notation, $p_1, p_2,\dots, p_n$ are the pages under consideration, $M(p_i)$ is the set of pages that link to $p_i$, $L(p_j)$ is the number of outbound links on page $p_j$, $n$ is the total number of pages on the Web, and $d$ is a damping factor, usually $d=0.85$. As noted in the original paper by Brin and Page (1998) [@Bri_Pag_98] PageRank “can be calculated using a simple iterative algorithm, and corresponds to the principal eigenvector of the normalized link matrix of the Web. Also, a PageRank for 26 million web pages can be computed in a few hours on a medium size workstation." Other link-based ranking algorithms for web pages include the HITS algorithm, developed by Kleinberg [@Klein_98], and the TrustRank algorithm [@Gyo_Gar_Ped_04].
While in principle the solution to reduces to the solution of a large system (possibly billions) of linear equations, we believe that finding page ranks in such a way is unlikely to be insightful. Specifically, if one obtains the principal eigenvector of the normalized link matrix, it is hard to obtain from the solution qualitative insights about the relationship between highly ranked pages and the in-degree/out-degree statistical properties of the graph.
In particular, the division by the out-degree, $L(p_j)$ in equation , was meant to decrease the contribution of pages with highly inflated referencing, i.e., those pages that basically point/reference possibly indiscriminately to other documents. However, the stochastic approach (to be described in the following sections) reveals that highly ranked pages are essentially insensitive to the parameters of the out-degree distribution, implying that the PageRank algorithm may not reduce the effects of overly inflated referencing (citations, voting) as originally intended, i.e., it may lead to possibly unjustifiable highly ranked pages. An analytical explanation as to why the tail of the rank distribution is dominated by $N$ was first given in [@Volk_Litv_Dona_07] and [@Volk_Litv_08]. More discussions on this topic are provided at the end of Section \[S.NDominates\].
A stochastic approach to analyze is to consider the recursion $$\label{eq:StochPR}
R \stackrel{\mathcal{D}}{=} \gamma + c \sum_{i=1}^N \frac{R_i}{D_i} ,$$ where $\gamma, c > 0$ are constants, $cE[1/D] < 1$, $N$ is a random variable independent of the $R_i$’s and $D_i$’s, the $D_i$’s are iid random variables satisfying $D_i \geq 1$, and the $R_i$’s are iid random variables having the same distribution as $R$. In terms of recursion , $R$ is the rank of a random page, $N$ corresponds to the in-degree of that node, the $R_i$’s are the ranks of the pages pointing to it, and the $D_i$’s correspond to the out-degrees of each of these pages. The experimental justification of these independence assumptions can be found in [@Volkovich2009]. This stochastic setup was first introduced in [@Volk_Litv_Dona_07], where the process resulting after a finite number of iterations of was analyzed. More recently, in a follow up paper [@Volk_Litv_08], the more general recursion $$R \stackrel{\mathcal{D}}{=} Q + \sum_{i=1}^N C_i R_i$$ was analyzed via tauberian theorems for the cases when $N$ or $Q$ dominate. In [@Volk_Litv_08], dependancy between $N$ and $Q$ is allowed, but additional moment conditions are imposed. Recall that in the setup considered here $N, Q, \{R_i\}_{i\ge 1}, \{C,C_i\}_{i \geq 1}$ are independent non-negative random variables, $P(Q > 0) > 0$, $\{C,C_i\}_{i \geq 1}$ are identically distributed, and $\{R_i\}_{ i \geq 1}$ are independent copies of $R$.
Model Description {#S.ModelDescription}
=================
As outlined above, we study the sequence of random variables that are obtained by iterating . Specifically, we consider for $n \geq 0$ $$\label{eq:Rstar_Rec}
R_{n+1}^* = Q_{n} + \sum_{i=1}^{N_n} C_{n,i} R_{n,i}^*,$$ where $\{ R_{n,i}^* \}_{i \geq 1}$ are iid copies of $R_n^*$ from the previous iteration, and $\{N_n\}, \{ C_{n,i} \}, \{ Q_{n}\}$ are mutually independent iid sequences of random variables; for $n = 0$, $R_{0,i}^*$ are iid copies of the initial value $R_0^*$.
In this section we will discuss the weak convergence of $R_n^*$ to a finite random variable $R$, independently of the initial condition $R_0^*$. In other words, $R$ is the unique solution to under the assumptions of Lemma \[L.Convergence\]. In particular, we will construct a process $R^{(n)}$ on a tree that converges a.s. to $R$. These convergence results may be of practical interest as well since ranking algorithms are implemented recursively. The actual proofs are postponed until Section \[SS.Convergence\].
Construction of $R$ on a Tree {#SS.TreeConstruction}
-----------------------------
To better understand the dynamics of our recursion, we give below a sample path construction of the random variable $R$ on a tree. First we construct a random tree $T$. We use the notation $\emptyset$ to denote the root node of $T$, and $A_n$, $n \geq 0$, to denote the set of all individuals in the $n$th generation of $T$, $A_0 = \{\emptyset\}$. Let $Z_n$ be the number of individuals in the $n$th generation, that is, $Z_n = |A_n|$, where $| \cdot |$ denotes the cardinality of a set; in particular, $Z_0 = 1$. We iteratively construct the tree as follows. Let $N^{(0)}$ be the number of individuals born to the root node $\emptyset$ and let $N^{(0)}, \{N^{(n)}_{i_1,\dots, i_n} \}_{n \geq 1}$ be iid copies of $N$. Define now $$A_1 = \{ i: 1 \leq i \leq N^{(0)} \}, \quad A_n = \{ (i_1, i_2, \dots, i_n): (i_1, \dots, i_{n-1}) \in A_{n-1}, 1 \leq i_n \leq N^{(n-1)}_{i_1, \dots, i_{n-1}} \},$$ and then the number of individuals $Z_n = |A_n|$ in the $n$th generation, $n \geq 1$, satisfies the following branching recursion $$Z_{n} = \sum_{(i_1, \dots, i_{n-1}) \in A_{n-1}} N^{(n-1)}_{i_1,\dots, i_{n-1}}.$$
Suppose now that individual $(i_1,\dots,i_n)$ in the tree has a weight ${\bf C}_{i_1,\dots,i_n}^{(n)}$ defined via the recursion $${\bf C}_{i_1}^{(1)} =C_{i_1}^{(1)}, \qquad {\bf C}_{i_1,\dots, i_n}^{(n)} = C_{i_1,\dots, i_{n-1}, i_n}^{(n)} {\bf C}_{i_1,\dots,i_{n-1}}^{(n-1)}, \quad n \geq 2,$$ where ${\bf C}^{(0)} =1$ is the weight of the root node and the random variables $\{C_{i_1,\dots, i_n}^{(n)}: n \geq 0, i_k \geq 1\}$ are iid with the same distribution as $C$. Note that ${\bf C}_{i_1,\dots, i_n}^{(n)}$ is equal to the product of all the weights $C_\cdot^{(\cdot)}$ along the branch leading to node $(i_1, \dots, i_n)$, as depicted on the figure below. Define now the process $$W_n = \sum_{(i_1,\dots, i_n) \in A_n} Q_{i_1,\dots, i_n}^{(n)} {\bf C}_{i_1, \dots, i_n}^{(n)}, \qquad n \geq 0,$$ where $A_n$ is the set of all individuals in the $n$th generation and $\{Q_{i_1,\dots,i_n}^{(n)}\}_{n \geq 0}$ is a sequence of iid random variables having the same distribution as $Q$ (see Figure \[F.Tree\]), and independent of ${\bf C}_{\cdot}^{(\cdot)}$.
(430,160)(0,0) (0,0) (136,150)[${\bf C}^{(0)}$]{} (65,83)[${\bf C}_{1}^{(1)}$]{} (127,83)[${\bf C}_{2}^{(1)}$]{} (215,83)[${\bf C}_{3}^{(1)}$]{} (22,17)[${\bf C}_{1,1}^{(2)}$]{} (78,17)[${\bf C}_{1,2}^{(2)}$]{} (126,17)[${\bf C}_{2,1}^{(2)}$]{} (162,17)[${\bf C}_{3,1}^{(2)}$]{} (213,17)[${\bf C}_{3,2}^{(2)}$]{} (268,17)[${\bf C}_{3,3}^{(2)}$]{} (350,150)[$W_0 = Q^{(0)} {\bf C}^{(0)}$]{} (350,135)[$Z_0 = 1$]{} (350,83)[$W_1 = \sum_{i} Q_{i}^{(1)} {\bf C}_{i}^{(1)}$]{} (350,68)[$Z_1 = 3$]{} (350,17)[$W_2 = \sum_{i,j} Q_{i,j}^{(2)} {\bf C}_{i,j}^{(2)}$]{} (350,2)[$Z_2 = 6$]{}
Observe that when $C_\cdot^{(\cdot)} \equiv 1$ and $Q_\cdot^{(\cdot)} \equiv 1$, $W_n$ is equal to the number of individuals in the $n$th generation of the corresponding branching process, and in particular $Z_n = W_n$. Otherwise, $W_n$ represents the sum of the weights of all the individuals in the $n$th generation. Related processes known as weighted branching processes have been considered in the existing literature [@Rosler_93; @Liu_98; @Kuhl_04] and are discussed in more detail in Section \[SS.RelatedProcesses\]. With a small abuse of notation we also refer to our more restrictive processes as WBPs.
Define the process $\{R^{(n)}\}_{n \geq 0}$ according to $$R^{(n)} = \sum_{k=0}^n W_k , \qquad n \geq 0,$$ that is, $R^{(n)}$ is the sum of the weights of all the individuals on the tree. Clearly, when $Q_\cdot \equiv 1$ and $C_\cdot^{(\cdot)} \equiv 1$, $R^{(n)}$ is simply the number of individuals in a branching process up to the $n$th generation. We define the random variable $R$ according to $$\label{eq:R_Def}
R \triangleq \lim_{n \to \infty} R^{(n)} = \sum_{k=0}^\infty W_k.$$ Furthermore, it is not hard to see that $R^{(n)}$ satisfies the recursion $$\label{eq:MainRec}
R^{(n)} = \sum_{j=1}^{N^{(0)}} C_{j}^{(1)} R^{(n-1)}_{j} + Q^{(0)},$$ for $n \geq 1$, where $\{R_{j}^{(n-1)}\}$ are independent copies of $R^{(n-1)}$ corresponding to the tree starting with individual $j$ in the first generation and ending on the $n$th generation; note that $R_j^{(0)} = Q_j^{(1)}$.
Moreover, since the tree structure repeats itself after the first generation, $W_n$ satisfies $$\begin{aligned}
W_n &= \sum_{(i_1,\dots,i_n) \in A_n} Q_{i_1,\dots,i_n} ^{(n)} {\bf C}_{i_1,\dots,i_n}^{(n)} \\
&= \sum_{k = 1}^{N^{(0)}} C_{k}^{(1)} \sum_{(k,\dots,i_n) \in A_n} Q_{k,\dots,i_n} ^{(n)} {\bf C}_{k,\dots,i_n}^{(2,n)} \\
&\stackrel{\mathcal{D}}{=} \sum_{k=1}^N C_k W_{n-1,k},\end{aligned}$$ where $N$, $C_k$, $W_{n-1,k}$ are independent of each other and of all other random variables, $ W_{n-1,k}$ has the same distribution as $W_{n-1}$, and ${\bf C}_{k,\dots,i_n}^{(2,n)}=\prod_{j=2}^n C_{k, i_2,\dots, i_j}^{(j)} $, i.e., if $C_{k}^{(1)} >0$, then ${\bf C}_{k,\dots,i_n}^{(2,n)}={\bf C}_{k,\dots,i_n}^{(n)} / C_{k}^{(1)}$.
Connection between $R_n^*$ and $R^{(n)}$ {#SS.Connection}
----------------------------------------
We now connect the two processes $R_n^*$ and $R^{(n)}$, the one obtained by iterating and the one obtained from the tree construction, respectively. To do this define $$W_n(R_0^*) = \sum_{(i_1,\dots, i_n) \in A_n} R^*_{0,(i_1,\dots, i_n)} {\bf C}_{i_1,\dots, i_n}^{(n)},$$ where $R^*_{0, (\cdot)}$ are iid copies of the initial condition $R_0^*$, independent of ${\bf C}_\cdot^{(n)}$, and the weights ${\bf C}^{(n)}_{\cdot}$ are the ones defined in Section \[SS.TreeConstruction\]. In words, $W_n(R_0)$ is the sum of all the weights in the $n$th generation of the tree with the coefficients $Q_{\cdot}^{(n)}$ substituted by the corresponding $R^*_{0,(\cdot)}$. We claim that $$R_n^* \stackrel{\mathcal{D}}{=} R^{(n-1)} + W_n(R_0^*).$$ To see this note that for $n = 1$, $$R_1^* = Q_0 + \sum_{i=1}^{N_0} C_{0,i} R_{0,i}^* \stackrel{\mathcal{D}}{=} Q^{(0)} {\bf C}^{(0)} + \sum_{i=1}^{N^{(0)}} {\bf C}_i^{(1)} R_{0,i}^* = R^{(0)} + W_1(R_0^*) \qquad \text{(recall ${\bf C}^{(0)} = 1$)},$$ and by induction in $n$, $$\begin{aligned}
R_{n+1}^* &= Q_{n} + \sum_{i=1}^{N_n} C_{n,i} R_{n,i}^* \\
&\stackrel{\mathcal{D}}{=} Q_n + \sum_{i=1}^{N_n} C_{n,i} (R^{(n-1)}_{i} + W_{n,i}(R_0^*)) \qquad \text{(by induction)} \\
&\stackrel{\mathcal{D}}{=} Q^{(0)} + \sum_{i=1}^{N^{(0)}} C_i^{(1)} \left( R_i^{(n-1)} + \sum_{(i,i_1,\dots,i_n) \in A_{n+1}}
R_{0,(i,i_1,\dots,i_n)}^* {\bf C}_{i,i_1,\dots,i_n}^{(2,n+1)} \right) \\
&= R^{(n)} + \sum_{i=1}^{N^{(0)}} \sum_{(i,i_1,\dots,i_n) \in A_{n+1}} R_{0,(i,i_1,\dots,i_n)}^* {\bf C}_{i,i_1,\dots,i_n}^{(n+1)} \\
&= R^{(n)} + W_{n+1}(R_0^*) ,\end{aligned}$$ where $R_i^{(n-1)}$ corresponds to the process $R^{(n-1)}$ obtained from the tree starting with individual $i$ in the first generation (a descendent of the root) and ending on the $n$th generation, and $(R^{(n-1)}_i, W_{n,i}(R_0^*))$ are iid copies of $(R^{(n-1)}, W_n(R_0^*))$. Since $R^{(n-1)} \to R$ a.s., it will follow from Slutsky’s Theorem (see Theorem 1, p. 254 in [@ChowTeich1988]) that if $W_{n}(R_0^*) \Rightarrow 0$, then $$R_n^* \Rightarrow R,$$ where $\Rightarrow$ denotes convergence in distribution. The proof of this convergence and that of the finiteness of $R$ are given in Section \[SS.Convergence\]. Understanding the asymptotic properties of the distribution of $R$, as defined by , is the main objective of this paper.
Related Processes {#SS.RelatedProcesses}
-----------------
As we mentioned earlier, the stochastic equation defined in leads to the analysis of a process known in the literature as a weighted branching process (WBP). WBPs were introduced by Rösler [@Rosler_93] in a construction that is more general than ours. More precisely, each individual in the tree has potentially an infinite number of offsprings, and each offspring inherits a certain (nonnegative) weight from its parent and multiplies it by a factor $T_i$, where the index $i$ refers to his birth order (i.e., a first born multiplies his inheritance by $T_1$, a second born by $T_2$, etc.). Each individual branches independently, using an independent copy of the sequence $T_1, T_2, \dots$. However, within the sequence, $T_1, T_2, \dots$ can be dependent. Only individuals whose weights are different than zero are considered to be alive. The construction we give in this paper would correspond to having $$T_i = C_i 1_{(N \geq i)}.$$ The definition of a WBP described above leads to the following stochastic recursion for the total weight of the $n$th generation, $$\label{eq:RoslerHomogeneous}
W_n \stackrel{\mathcal{D}}{=} \sum_{i=1}^\infty T_i W_{n-1,i}$$ and a corresponding nonhomogeneous fixed point equation of the form $$\label{eq:RoslerNonHomogeneous}
R \stackrel{\mathcal{D}}{=} \sum_{i=1}^\infty T_i R_i + Q.$$ In the construction given in [@Rosler_93], the $\{T_i\}$ and $Q$ are allowed to be dependent as well.
We now briefly describe some of the existing literature on WBPs, most of which considers the homogeneous equation, i.e. $Q \equiv 0$. The nonhomogeneous equation has only been studied for the special case when $Q$ and the $\{T_i\}$ are deterministic constants. In particular, Theorem 5 of [@Rosler_93] analyzes the solutions to when $Q$ and the $\{T_i\}$ are nonnegative deterministic constants, which implies that $T_i \leq 1$ for all $i$ and $\sum_{i} T_i^\alpha \log T_i \leq 0$ for all $\alpha > 0$, falling outside of the scope of this paper. More results about the solutions to for the case when $Q$ and the $T_i$’s are real valued deterministic constants were derived in [@Alsm_Rosl_05].
Regarding the homogeneous equation, in [@Rosler_93], the martingale structure of $W_n/m^n$ ($m = E[\sum_i T_i ]$) was used to point out the existence of $W = \lim_{n\to \infty} W_n / m^n$, and it was shown that positive stable distributions with $\alpha \in (0,2)$ arise when $E\left[ \sum_i T_i^\alpha \right] = 1$ and some additional moment conditions are satisfied. Furthermore, for a detailed analysis of the case when $W$ follows a positive stable distribution $(0 < \alpha \leq 1)$ see [@Liu_98]. The convergence of $W_n/ m^n$ to $W$ was studied in [@Ros_Top_Vat_00], and conditions for $W$ to belong to the domain of attraction of an $\alpha$-stable law $(1 < \alpha < 2)$ were given in [@Ros_Top_Vat_00], along with an analysis of the rate of convergence. A generalization of the WBP described in [@Rosler_93] to a random environment was given in [@Kuhl_04], where necessary and sufficient conditions for $W$ to be nondegenerate were derived. The existence of moments of $W$ was studied in [@Alsm_Kuhl_07]. The power law tail of $W$ for the critical case $E\left[ \sum_{i=1}^N C_i\right] = 1$ and $\alpha > 1$ was derived in Theorem 2.2 of [@Liu_00] and Proposition 7 of [@Iksanov_04]. For an even longer list of references to WBPs and related work see [@Kuhl_04] and [@Alsm_Rosl_05].
From the discussion above it is clear that the prior literature on WBPs is extensive, but we point out that the more specific structure of our model, given by , as well as our novel analysis via implicit renewal theory, allow us to characterize the asymptotic power law behavior of the distribution of $R$ for all $\alpha > 0$ when the $\{C_i\}$ dominate the tail. In addition, we study the nonhomogeneous equation , while the preceding work primarily focuses on the homogeneous case . The case when $N$ dominates the tail, which is important for the page ranking problem, has not been considered until very recently in [@Volk_Litv_Dona_07] and [@Volk_Litv_08]. In reference to the latter work, our analysis is based on a new sample path approach, while the studies in [@Volk_Litv_Dona_07; @Volk_Litv_08] use transforms and tauberian theorems as well as somewhat different assumptions. We will provide more details on these connections throughout the paper in remarks after the corresponding theorems.
From a different mathematical perspective, our model also constitutes a generalization of several important types of processes. For instance, by setting $N \equiv 1$, reduces to an autoregressive process of order one. Also, by letting $N$ be a Poisson random variable and fixing $C_i \equiv 1$, $Q \equiv 1$, becomes the recursion that the number of customers in a busy period of an M/G/1 queue satisfies. Recursion and its connection to the busy period when the weights $D_i$ are equal to a deterministic constant was exploited in [@Lit_Sch_Volk_07].
It is worth noting that probabilistic sample path approaches for the busy period ($C_i \equiv 1$, $Q \equiv 1$) were developed in [@Zwart_01; @Jel_Mom_04; @Bal_Dal_Klu_04]; the work in [@Zwart_01; @Jel_Mom_04] is also relying on the theory of cycle maximum [@Asm_98]. However, for our more general model (random $C_i$’s) it is not clear if there is a tractable way of generalizing this analysis. Instead of pursuing the preceding directions, we develop a direct sample path large deviation analysis for recursive random sums that provides greater generality.
Moments of $W_n$ {#S.Moments}
================
In this section we provide explicit estimates for the moments of the total weight, $W_n$, of the $n$th generation that will be used throughout the paper. In particular, we apply these estimates in Section \[SS.Convergence\] to prove that $R_n^* \Rightarrow R$ where $R < \infty$ a.s. Our estimates may be of independent interest due to their explicit nature.
A simple calculation shows that provided $E[N], E[Q], E[C] < \infty$, then $E[W_n] < \infty$ and is given by $$E[W_n] = E[N] E[C] E[W_{n-1}] = (E[N] E[C])^{n} E[W_0] = (E[N] E[C])^{n} E[Q].$$ We give below upper bounds on the general moments of $W_n$.
Throughout the paper we will use $K$ to denote a large positive constant that may be different in different places, say $K = K/2$, $K = K^2$, etc.
\[L.MomentSmaller\_1\] Suppose $E[Q^\beta] E[N] E[C^\beta] < \infty$ for $0 < \beta \leq 1$, then $$E[ W_n^\beta ] \leq (E[C^\beta] E[N])^{n} E[Q^\beta]$$ for all $n \geq 0$.
Simply note that $$\begin{aligned}
E[W_n^\beta] &= E\left[ \left( \sum_{i=1}^N C_i W_{n-1,i} \right)^\beta \right] \end{aligned}$$ and use the well known inequality $\left( \sum_{i=1}^k y_i \right)^\beta \leq \sum_{i=1}^k y_i^\beta$ for $0 < \beta \leq 1$, $y_i \geq 0$ (see e.g., Exercise 4.2.1, p. 102, in [@ChowTeich1988]).
The lemma for moments greater than one is given below.
\[L.GeneralMoment\] Suppose $E[Q^\beta]< \infty$, $E[N^\beta] < \infty$, and $E[N] \max\{ E[C^\beta], E[C] \} < 1$ for some $\beta > 1$. Then, there exists a constant $K_\beta > 0$ such that $$E[ W_n^\beta ] \leq K_\beta (E[N] \max\{ E[C^\beta], E[C]\} )^{n}$$ for all $n \geq 0$.
The proof of Lemma \[L.GeneralMoment\] is given in Section \[SS.Moments\_Proofs\].
[Remark:]{} Recall that when $C \equiv 1$ and $Q \equiv 1$ then $E[W_n^\beta]$ is the $\beta$-moment of a subcritical branching process $Z_n$ and our result reduces to $E[Z_n^\beta] \leq K_\beta (E[N])^n$, which is in agreement with the classical results from branching processes, e.g. see Corollary 1 on p. 18 of [@Athreya_Ney_2004]. Moreover, from the proof of the integer $\beta$ case (given in Section \[SS.Moments\_Proofs\]), it is clear that $E[W_n^\beta]$ scales as $\rho^{\beta n}$ if $\rho^\beta > \rho_\beta$ and as $\rho_\beta^n$ if $\rho^\beta < \rho_\beta$, where $\rho = E[N] E[C]$ and $\rho_\beta = E[N] E[C^\beta]$. Note that this is not quite the same as our upper bounds, and the reason we choose the geometric term $(\rho \vee \rho_\beta)^n$ instead is that it makes the proofs simpler and is sufficient for our purposes. Similar techniques to those used in proving the preceding lemmas can yield, with some additional work, lower bounds for the $\beta$-moments of $W_n$, showing that the correct leading term is $(\rho^\beta \vee \rho_\beta)^n$.
More technical results dealing with the existence of the $\beta$-moments of $W \triangleq \lim_{n \to \infty} W_n/ \rho^n$ can be found in [@Alsm_Kuhl_07]. There, necessary and sufficient conditions are given for the finiteness of $E[W^\beta L(W)]$ when $\beta \geq 1$ and $L(\cdot)$ is slowly varying (see Theorems 1.2 and 1.3). In particular, the approach the authors take is to first normalize the process so that $\rho = E[W_1] = 1$, and then impose a condition that in our case reduces to $\rho_\beta = E[N] E[C^\beta] < 1$, that is, they preclude the situation where $W_n^\beta$ might scale as $\rho_\beta^n$ when $\rho^\beta < \rho_\beta$. An example where $E[W_n^\beta]$ scales as $\rho_\beta^n$ is when $N \equiv 1$, since then $W_n^\beta \stackrel{\mathcal{D}}{=} Q^\beta \prod_{i=1}^n C_i^\beta$.
Furthermore, observe that when $\rho = 1$ and $\rho_\beta < 1$ for $\beta > 1$, our proof of the lemma shows that $\limsup_{n \to \infty} E[W_n^\beta ] < \infty$, but it does not converge to zero, which is in agreement with [@Alsm_Kuhl_07]. However, since we study $R$, it is necessary to have $\rho < 1$ for the finiteness of $E[R^\beta]$. Otherwise, if $\rho =1$, $\rho_\beta < 1$, $\beta > 1$, then $E[R^{(n)}] = n E[Q]$ which by monotone convergence and implies that $E[R] = \infty$, and therefore, by convexity, $E[R^\beta] = \infty$.
Convergence of $R_n^*$ and finiteness of $R$ {#SS.Convergence}
--------------------------------------------
As discussed in Section \[SS.Connection\], there are two issues regarding the process $R_n^*$ that remain to be addressed. One, is the proof that $$R_n^* \Rightarrow R = \sum_{k=0}^\infty W_k$$ for any initial condition $R_0^*$; the other one is the finiteness of $R$. The lemma below shows that $R< \infty$ a.s.
\[L.Stability\] Suppose that $E[Q^\beta] < \infty$, $E[N^\beta] < \infty$, and either $E[N] E[C^\beta] < 1$ for some $0 < \beta < 1$, or $E[N]\max\{ E[C], E[C^\beta]\} < 1$ for some $\beta \geq 1$. Then, $E[R^\gamma] < \infty$ for all $0 < \gamma \leq \beta$, and in particular, $R < \infty$ a.s. Moreover, if $\beta \geq 1$, $R^{(n)} \stackrel{L_\beta}{\to} R$, where $L_\beta$ stands for convergence in $(E|\cdot|^\beta)^{1/\beta}$ norm.
Let $$\eta = \begin{cases} E[N] E[C^\beta], & \text{ if }\beta < 1 \\ E[N] \max\{E[C], E[C^\beta] \}, & \text{ if } \beta \geq 1. \end{cases}$$ Then by Lemmas \[L.MomentSmaller\_1\] and \[L.GeneralMoment\], $$\label{eq:EW_n}
E[W_n^\beta] \leq K \eta^n$$ for some $K > 0$. Suppose $\beta \geq 1$, then, by monotone convergence and Minkowski’s inequality, $$\begin{aligned}
E[R^\beta] &= E\left[ \lim_{n\to\infty} \left(\sum_{k=0}^n W_k \right)^\beta \right] = \lim_{n\to \infty} E\left[ \left(\sum_{k=0}^n W_k\right)^\beta \right] \\
&\leq \lim_{n\to\infty} \left( \sum_{k=0}^n E[W_k^\beta]^{1/\beta} \right)^\beta \leq K \left( \sum_{k=0}^\infty \eta^{k/\beta} \right)^\beta < \infty.\end{aligned}$$ This implies that $R < \infty$ a.s. When $0 < \beta < 1$ use the inequality $\left( \sum_{k=0}^n y_k \right)^\beta \leq \sum_{k=0}^n y_k^\beta$ for any $y_i \geq 0$ instead of Minkowski’s inequality. Furthermore, for any $0 < \gamma \leq \beta$, $$E[R^\gamma] = E\left[ (R^\beta)^{\gamma/\beta}\right] \leq \left(E[R^\beta] \right)^{\gamma/\beta} < \infty.$$ That $R^{(n)} \stackrel{L_\beta}{\to} R$ whenever $\beta \geq 1$ follows from noting that $E[|R^{(n)} - R|^\beta] = E\left[ \left( \sum_{k = n+1}^\infty W_k \right)^\beta \right]$ and applying the same arguments used above to obtain the bound $E[|R^{(n)} - R|^\beta] \leq K \eta^{n+1}/(1 - \eta^{1/\beta})^\beta$.
Next, by monotone convergence in equation it can be verified that $R$ must solve $$R \stackrel{\mathcal{D}}{=} \sum_{i=1}^N C_i R_i + Q,$$ where $\{R_i\}_{i \geq 1}$ are iid copies of $R$, independent of $N$, $Q$, and $\{C_i\}$.
We now turn our attention to the proof of the convergence of $R_n^*$ to $R$. Recall from Section \[SS.Connection\] that $$\label{eq:connection}
R_n^* \stackrel{\mathcal{D}}{=} R^{(n-1)} + W_n(R_0^*),$$ where $$W_n(R_0^*) = \sum_{(i_1,\dots,i_n) \in A_n} R_{0,(i_1,\dots,i_n)}^* {\bf C}_{i_1,\dots,i_n}^{(n)}.$$ The following lemma shows that $R_n^* \Rightarrow R$ for any initial condition $R_0^*$ satisfying a moment assumption.
\[L.Convergence\] For any $R_0^* \geq 0$, if $E[Q^\beta]< \infty$, $E[(R_0^*)^\beta] < \infty$ and $E[N] E[C^\beta] < 1$ for some $0 < \beta \leq 1$, then $$R_n^* \Rightarrow R,$$ with $E[R^\beta] < \infty$. Furthermore, under these assumptions, the distribution of $R$ is the unique solution with finite $\beta$ moment to recursion .
In view of , and since $R^{(n)} \to R$ a.s., the result will follow from Slutsky’s Theorem (see Theorem 1, p. 254 in [@ChowTeich1988]) once we show that $W_n(R_0^*) \Rightarrow 0$. Recall that $W_n(R_0^*)$ is the same as $W_n$ if we substitute the $Q_{i_1, \dots, i_n}$ by the $R_{0,(i_1,\dots,i_n)}^*$. Fix $\epsilon > 0$, then $$\begin{aligned}
P( W_n(R_0^*) > \epsilon) &\leq \epsilon^{-\beta} E[ W_n(R_0^*)^\beta] \\
&\leq \epsilon^{-\beta} (E[C^\beta] E[N])^n E[(R_0^*)^\beta] \qquad \text{(by Lemma \ref{L.MomentSmaller_1})} .\end{aligned}$$ Since by assumption the right hand side converges to zero as $n \to \infty$, then $R_n^* \Rightarrow R$. Furthermore, $E[R^\beta] < \infty$ by Lemma \[L.Stability\]. Clearly, the distribution of $R$ represents the unique solution with finite $\beta$-moment to , since any other possible solution would have to converge to the same limit.
[Remarks:]{} (i) Note that when $E[N] < 1$, then the branching tree is a.s. finite and no conditions on the $C$’s are necessary for $R < \infty$ a.s. This corresponds to the second condition in Theorem 1 of [@Brandt_86]. (ii) In view of the same theorem from [@Brandt_86], one could possibly establish the convergence of $R_n^* \Rightarrow R < \infty$ under milder conditions. However, since the conditions that we will impose on $N$, $Q$ and $C$ in the main theorems will be stronger, this lemma is not restrictive. Furthermore, the initial values, $R_0^*$, are typically small (e.g. constant in applications), and thus the polynomial moment condition imposed on $R_0^*$ is general enough.
The case when the $C$’s dominate: Implicit renewal theory {#S.C_dominates}
=========================================================
In this section we study the power law phenomenon that arises from the multiplicative effects of the weights $\{C_i\}$ in .
Implicit Renewal Theorem on Trees {#S.Renewal}
---------------------------------
One observation that will help gain some intuition about is to consider the case when $N \equiv 1$. The process $\{R^{(n)}\}$ then reduces to a (random coefficient) autoregressive process of order one, whose steady state solution satisfies $$R \stackrel{\mathcal{D}}{=} Q + C R,$$ where $R$ is independent of $C$ and $Q$. This is precisely one of the stochastic recursions considered in [@Goldie_91] (see also [@Kesten_73]), where it is shown that under the assumption that $E[C^\alpha] = 1$ and some other technical conditions on the distribution of $C$ and $Q$, we have that $$\label{eq:PowerLaw}
P(R > x) \sim H x^{-\alpha}$$ for some (computable) constant $H > 0$ (see Theorem 4.1 in [@Goldie_91]). The fact that the index of the power law depends on the distribution of the weights is already promising in terms of our goal of identifying other sources of power law behavior.
Informally speaking, the recursions studied in [@Goldie_91] are basically multiplicative away from the boundary. However, always has an additive component given by $\sum_{i=1}^N C_i R_i$ regardless of how far from the boundary one may be. Fortunately, due to the heavy-tailed nature of $R$, our intuition says that it is only one of the additive $C_iR_i$ components that determines the behavior of , thus the sum will behave as the maximum term, simplifying to $$\label{eq:heuristic}
P\left( Q + \sum_{i=1}^N C_i R_i > x \right) \sim E[N] P(C R > x),$$ assuming that $Q$ has a light enough tail. This heuristic suggests the following generalization of Theorem 2.3 from [@Goldie_91].
Here, we would like to emphasize that $R$ and $C$ in the following theorem can be any two independent random variables that satisfy the stated conditions, i.e., they do not have to be related by recursion . Hence, the theorem may be of potential use in other applications. Note that we prove the theorem for a general constant $m$, that in our application refers to $E[N]$, as suggested by .
\[T.Goldie\] Suppose $C \geq 0$ a.s., $0 < E[C^\alpha \log C] < \infty$ for some $\alpha > 0$, and that the conditional distribution of $\log C$ given $C \neq 0$ is nonarithmetic. Suppose further that $R$ is independent of $C$, $m E[C^\alpha] = 1$, and that $E[R^\beta] < \infty$ for any $0 < \beta < \alpha$. If $$\label{eq:Goldie_condition}
\int_0^\infty \left| P(R > t) - m P(CR > t) \right| t^{\alpha-1} dt < \infty,$$ then $$P(R > t) \sim H t^{-\alpha}, \qquad t \to \infty,$$ where $H \geq 0$ is given by $$H = \frac{1}{m E[C^\alpha \log C]} \int_{0}^\infty v^{\alpha -1} (P(R > v) - mP(C R > v)) \, dv.$$
The proof of this theorem follows the same steps as Theorem 2.3 from [@Goldie_91], and is presented in Section \[SS.CDominates\_Proofs\].
[Remarks:]{} (i) As pointed out in [@Goldie_91], the statement of the theorem has content only when $R$ has infinite moment of order $\alpha$, since otherwise the constant $H = (\alpha E[N] E[C^\alpha\log C])^{-1} (E[R^\alpha] - E[N] E[(CR)^\alpha])$ will be zero by independence of $R$ and $C$. (ii) Note that some of the assumptions of Theorem \[T.Goldie\] are different than the corresponding ones from Theorem 2.3 in [@Goldie_91]. In particular, it is no longer the case that convexity implies $E[C^\alpha \log C] > 0$ whenever $\alpha$ solves $m E[C^\alpha] = 1$ and $E[C^\alpha \log C] < \infty$, since if $m > 1$ it is possible to construct counterexamples, hence the need to include this as an assumption. Another difference is our requirement that $E[R^\beta] < \infty$ for any $0 < \beta < \alpha$. In the case of applying Theorem \[T.Goldie\] to equation , the condition on $E[R^\beta]$ is not restrictive since we readily obtain the moments of $R$ for $0<\beta <\alpha$ from the computed moments of $W_n$ from Section \[S.Moments\]. (iii) A similar result for the case when $\log C$ is lattice valued can be derived using the corresponding renewal theorem.
In what follows we will use the preceding theorem to derive the asymptotic behavior of $P(R > x)$ where $R$, as given by , satisfies . Here, the main difficulty will be to show that condition holds. For brevity we use $x \vee y$ to denote $\max\{x, y\}$ and $x \wedge y$ to denote $\min\{x, y\}$.
\[T.GoldieApplication\] Suppose that $0 < E[C^\alpha \log C] < \infty$ for some $\alpha > 0$, the conditional distribution of $\log C$ given $C \neq 0$ is nonarithmetic, and that $C$ and $R$ are independent, where $R$ is defined by . Assume that $E[N] E[C^\alpha] = 1$, $0 < E[Q^{\alpha}] < \infty$ and $E[N^{\alpha \vee (1+\epsilon)}] < \infty$ for some $0 < \epsilon < 1$; if $\alpha > 1$ assume further that $E[N] E[C] < 1$. Then, $$P(R > t) \sim H t^{-\alpha}, \qquad t \to \infty,$$ where $$\begin{aligned}
H &= \frac{1}{E[N] E[C^\alpha \log C]} \int_{0}^\infty v^{\alpha -1} (P(R > v) - E[N]P(C R > v)) \, dv \\
&= \frac{E\left[ \left( \sum_{i=1}^N C_i R_i +Q \right)^\alpha - \sum_{i=1}^N (C_i R_i )^\alpha \right]}{\alpha E[N]E[C^\alpha\log C]}.\end{aligned}$$
[Remarks:]{} (i) Note that the second expression for $H$ is more suitable for actually computing it, especially in the case of $\alpha$ being an integer, as will be stated in the forthcoming corollary. (ii) When $\alpha$ is not an integer we can derive an explicit bound on $H$ by using the forthcoming Lemma \[L.Max\_Approx\] and . (ii) For the homogeneous equation ($Q \equiv 0$) and $\alpha > 1$, closely related results to our theorem can be found in Theorem 2.2 of [@Liu_00] and Proposition 7 of [@Iksanov_04]. The approach from [@Liu_00] transforms the recursion $W \stackrel{\mathcal{D}}{=} \sum_{i=1}^N C_i W_i$ for the critical case $E[W] = 1$, $E\left[ \sum_{i=1}^N C_i \right] = 1$ to a first order difference (autoregressive) equation on a different probability space, see Lemma 4.1 in [@Liu_00]. Note that the tail behavior of $W$ does not imply that of $R$. Furthermore, it appears that the method from [@Liu_00] does not extend to the nonhomogeneous case since the proof of Lemma 4.1 in [@Liu_00] critically depends on having both $E[W] = 1$ and $E\left[ \sum_{i=1}^N C_i \right] = 1$, which is only possible when $Q \equiv 0$. For $0 < \alpha \leq 1$, the homogeneous equation was studied in [@Liu_98] using stable laws. (iii) Related results for the nonhomogeneous equation with deterministic constants $Q, \{C_i\}$, $N = \infty$, have been considered in [@Rosler_93] (see Theorem 5), and more recently in [@Alsm_Rosl_05], also using stable laws. (iv) Moreover, the results obtained in the references cited above appear to be less explicit in the expression for $H$ than the statement of Theorem \[T.GoldieApplication\], as Corollary \[C.explicit\] below illustrates. (v) Furthermore, Theorem \[T.Goldie\] and the preceding technique of Theorem \[T.GoldieApplication\] can be adapted to analyze other, possibly non-linear, recursions on trees, e.g., one can characterize the asymptotic behavior of $P(R > x)$ that solves $$R = Q + \max_{1 \leq i \leq N} C_i R_i.$$
We also want to point out that one can obtain the logarithmic asymptotics of $R$, that is, the behavior of $\log P(R > x)$, much easier and under less restrictive conditions, e.g. $\log C_i$ needs not be nonarithmetic (this condition is required because of the use of the Renewal Theorem). An upper bound can be obtained from Lemma \[L.Stability\] and Markov’s inequality. For the lower bound, using the notation from Section \[SS.TreeConstruction\], we obtain $$\begin{aligned}
P(R > x) &\geq P(W_n > x) \geq P\left( \max_{1 \leq i \leq N} C_i W_{n-1,i} > x \right) \\
&= E\left[ (1 - P(C W_{n-1} \leq x)^N) \right] \\
&\geq E\left[ N P(C W_{n-1} \leq x )^N \right] P(C W_{n-1} > x),\end{aligned}$$ where in the last step we used the relation $1-t^m \geq m t^m (1-t)$ for $0 \leq t \leq 1$. Now we use the fact that $P(C W_{n-1} \leq x) \geq P(R \leq x)$, for all $x$, to show that $$\begin{aligned}
P(R > x) &\geq E\left[ N P(R \leq x )^N \right] P(C W_{n-1} > x) \\
&\geq E\left[ N P(R \leq x )^N \right] P\left( C_1 \max_{1\leq i\leq N} C_{2,i} W_{n-2,i} > x \right) \\
&\geq E\left[ N P(R \leq x )^N \right] E\left[ N P(C_1 C_2 W_{n-2} \leq x )^N \right] P(C_1 C_2 W_{n-2} > x),\end{aligned}$$ which, by using $P(C_1 C_2 W_{n-2} \leq x) \geq P(R \leq x)$, for all $x$, yields $$\begin{aligned}
P(R > x) &\geq \left(E\left[ N P(R \leq x )^N \right] \right)^2 P(C_1 C_2 W_{n-2} > x). \end{aligned}$$ Next, by continuing this inductive argument we obtain $$\begin{aligned}
P(R > x) &\geq \left(E\left[ N P(R \leq x )^N \right] \right)^n P\left( Q \prod_{i=1}^n C_i > x \right). \end{aligned}$$ Finally, for any $0 < \epsilon < 1$, we can choose $x_0$ such that $E\left[ N P(R \leq x_0 )^N \right]\geq(1-\epsilon)E [N]$, implying that for all $n \geq 0$ and $x \geq x_0$, $$P(R > x) \geq (1-\epsilon)^n (E[N])^n P\left( Q \prod_{i=1}^n C_i > x \right) \geq P(Q > 1/\log x) (1-\epsilon)^n (E[N])^n P\left( \prod_{i=1}^n C_i > x \log x \right) .$$ Now define $S_n = \log C_1 + \dots + \log C_n$, $\kappa(\theta) = \log E[ C^\theta]$, and choose $\alpha$ to be the solution to $\kappa(\alpha) = -\log E[N]$ (i.e. $E[N]E[C^\alpha] = 1$). The proof can be completed by choosing $n = n(x) = \log (x\log x)/\mu_\alpha$, where $\mu_\alpha = \kappa'(\alpha) = E[C^\alpha \log C]/E[C^\alpha] > \kappa'(0) = E[\log C]$ by convexity of $\kappa(\cdot)$. Then, by Theorem 2.1 in Chapter XIII in [@Asm2003], $$\liminf_{x \to \infty} \frac{\log P(R > x)}{\log x} \geq \frac{\log((1-\epsilon) E[N])}{\mu_\alpha} + \liminf_{x \to \infty} \frac{\log P(S_{n(x)} > \mu_\alpha n(x))}{\mu_\alpha n(x)} = \frac{\log(1-\epsilon)}{\mu_\alpha} -\alpha.$$ Hence, one can derive with a considerably smaller effort the following theorem.
Suppose that $0 < E[C^\alpha \log C] < \infty$ for some $\alpha > 0$, and that $R$ is given by . Assume that $E[N] E[C^\alpha] = 1$, $0 < E[Q^{\alpha}] E[N^\alpha] < \infty$; if $\alpha > 1$ assume further that $E[N] E[C] < 1$. Then, $$\log P(R > t) \sim -\alpha \log t, \qquad t \to \infty.$$
Therefore, the majority of the work in proving Theorem \[T.GoldieApplication\] goes into the derivation of the exact asymptotic. Furthermore, it is worth noting that the logarithmic approach, although less precise, can be obtained in a more general setting. For example, one can have $C_\cdot^{(\cdot)}$ to be dependent across different generations, as in the so called WBP in a random environment. Here, one could derive the asymptotics of $\log P(R > x)$ if $E\left[ \left( \prod_{i=1}^n C_{(1,1,\dots,1)}^{(n)} \right)^\alpha \right]$ satisfies the polynomial type Gärtner-Ellis conditions that were recently considered in [@Jel_Tan_07].
\[C.explicit\] For integer $\alpha \geq 1$, and under the same assumptions of Theorem \[T.GoldieApplication\], the constant $H$ can be explicitly computed as a function of $E[R^k], E[C^k], E[Q^k]$, $0 \leq k \leq \alpha-1$. In particular, for $\alpha = 1$, $$H = \frac{E[Q]}{E[N] E[C \log C]},$$ and for $\alpha = 2$, $$H = \frac{E[Q^2] + 2 E[Q] E[C] E[N] E[R] + E[N(N-1)] (E[C] E[R])^2 }{2 E[N] E[C^2 \log C]}, \qquad E[R] = \frac{E[Q]}{1-E[N]E[C]}.$$
The proof follows directly from multinomial expansions of the second expression for $H$ in Theorem \[T.GoldieApplication\].
Before giving the proof of Theorem \[T.GoldieApplication\] we state the following three preliminary lemmas. The proof of Lemma \[L.Alpha\_Moments\] is given in Section \[SS.Moments\_Proofs\] and the proof of Lemma \[L.Max\_Approx\] is given in Section \[SS.CDominates\_Proofs\].
\[L.Moments\_R\] Suppose that $0 < E[C^\alpha \log C] < \infty$ for some $\alpha > 0$ and $E[N]E[C^\alpha] = 1$; if $\alpha > 1$ suppose further that $E[N] E[C] < 1$. Assume also that $E[Q^\alpha] < \infty$, $E[N^{\alpha \vee 1}] < \infty$. Then, $$E[R^\beta] < \infty$$ for all $0 < \beta < \alpha$.
The derivative condition $0 < E[C^\alpha \log C] < \infty$ and $E[N] E[C^\alpha] = 1$ imply that $E[N] E[C^\beta] < 1$ for all $\beta < \alpha$ that are sufficiently close to $\alpha$. Hence, the conclusion of the result follows from Lemma \[L.Stability\].
\[L.Alpha\_Moments\] Let $\beta > 1$ and $p = \lceil \beta \rceil \in \{2, 3, 4, \dots\}$. For any sequence of nonnegative iid random variables $\{Y, Y_i\}_{i\geq 1}$ and any $k \in \{1,2,3,\dots\}$ we have $$E\left[ \left( \sum_{i=1}^k Y_i \right)^\beta - \sum_{i=1}^k Y_i^\beta \right] \leq k^\beta E[ Y^{p-1} ]^{\beta/(p-1)}.$$
\[L.Max\_Approx\] Suppose $\{C, C_i\}$ and $\{R, R_i\}$ are iid sequences of nonnegative random variables independent of each other and of $N$. Assume that $E[C^\alpha] < \infty$, $E[N^{1 + \epsilon}] < \infty$ for some $0 < \epsilon < 1$, and $E[R^\beta]< \infty$ for any $0 < \beta < \alpha$. Then, $$0 \leq \int_{0}^\infty \left( E[N] P(C R > t) - P\left( \max_{1\leq i \leq N} C_i R_i > t \right) \right) t^{\alpha -1} \, dt = \frac{1}{\alpha} E \left[ \sum_{i=1}^N \left(C_i R_i \right)^\alpha - \left( \max_{1\leq i \leq N} C_i R_i \right)^\alpha \right]
< \infty.$$
By Lemma \[L.Moments\_R\] we know that $E[R^\beta] < \infty$ for any $0 < \beta < \alpha$. The statement of the theorem with the first expression for $H$ will follow from Theorem \[T.Goldie\] once we prove condition for $m = E[N]$. Define $$R^* = \sum_{i=1}^N C_i R_i + Q.$$ Then, $$\begin{aligned}
\left| P(R>t) - E[N] P(CR > t) \right| &\leq \left| P(R > t) - P\left( \max_{1\leq i \leq N} C_i R_i > t \right) \right| \\
&\hspace{5mm} + \left| P\left( \max_{1\leq i \leq N} C_i R_i > t \right) - E[N] P(CR > t) \right|.\end{aligned}$$ Since $R \stackrel{\mathcal{D}}{=} R^* \geq \max_{1\leq i\leq N} C_i R_i$, the first absolute value disappears. For the second one note that by the union bound $$\begin{aligned}
E[N] P(CR > t) - P\left( \max_{1\leq i \leq N} C_i R_i > t \right) &= E\left[ N P(CR > t) - 1 + P(CR \leq t)^N \right] \geq 0.\end{aligned}$$ It follows that $$\begin{aligned}
\left| P(R > t) - E[N] P(CR > t) \right| &\leq P(R > t) - P\left( \max_{1\leq i \leq N} C_i R_i > t \right) \notag \\
&\hspace{5mm} + E[N] P(CR > t) - P\left( \max_{1\leq i \leq N} C_i R_i > t \right) . \notag \end{aligned}$$ Note that we only need to verify that $$\int_0^\infty \left( P(R > t) - P\left( \max_{1\leq i \leq N} C_i R_i > t \right) \right) t^{\alpha-1} \, dt < \infty,$$ since $$\int_0^\infty \left( E[N]P(CR > t) - P\left( \max_{1\leq i \leq N} C_i R_i > t \right) \right) t^{\alpha-1} dt < \infty$$ by Lemma \[L.Max\_Approx\]. To see this note that $R \stackrel{\mathcal{D}}{=} R^*$ and $1_{(R^* > t)} - 1_{(\max_{1\leq i\leq N} C_i R_i > t)} \geq 0$, thus, by Fubini’s Theorem, we have $$\label{eq:Fubini}
\int_0^\infty \left( P(R > t) - P\left( \max_{1\leq i \leq N} C_i R_i > t \right) \right) t^{\alpha-1} \, dt = \frac{1}{\alpha} E \left[ (R^*)^\alpha - \left( \max_{1\leq i \leq N} C_i R_i \right)^\alpha \right].$$ If $0 < \alpha \leq 1$ we apply the inequality $\left( \sum_{i=1}^k x_i \right)^\beta \leq \sum_{i=1}^k x_i^\beta$ for $0 < \beta \leq 1$, $x_i \geq 0$, to obtain $$E \left[ (R^*)^\alpha - \left( \max_{1\leq i \leq N} C_i R_i \right)^\alpha \right] \leq E \left[ Q^\alpha + \sum_{i=1}^N (C_iR_i)^\alpha - \left( \max_{1\leq i \leq N} C_i R_i \right)^\alpha \right],$$ which is finite by Lemma \[L.Max\_Approx\] and the assumption $E[Q^\alpha] < \infty$. If $\alpha > 1$ we use the well known inequality $\left(\sum_{i=1}^k x_i \right)^\alpha \geq \sum_{i=1}^k x_i^\alpha$, $x_i\geq 0$ (see Exercise 4.2.1, p. 102, in [@ChowTeich1988]) to split the expectation as follows $$\begin{aligned}
E \left[ (R^*)^\alpha - \left( \max_{1\leq i \leq N} C_i R_i \right)^\alpha \right] &= E \left[ (R^*)^\alpha - \sum_{i=1}^N \left(C_i R_i \right)^\alpha \right] + E \left[ \sum_{i=1}^N \left(C_i R_i \right)^\alpha - \left( \max_{1\leq i \leq N} C_i R_i \right)^\alpha \right],\end{aligned}$$ which can be done since both expressions inside the expectations on the right hand side are nonnegative. The second expectation is again finite by Lemma \[L.Max\_Approx\]. To see that the first expectation is finite let $S = \sum_{i=1}^N C_i R_i$ and note that $R^* = S + Q$, where $S$ and $Q$ are independent. Let $p = \lceil \alpha \rceil$ and note that $1 \leq p-1 < \alpha$. Then, by Lemma \[L.Alpha\_Moments\], $$\begin{aligned}
E \left[ (R^*)^\alpha - \sum_{i=1}^N \left(C_i R_i \right)^\alpha \right] &= E\left[ (S+Q)^\alpha - S^\alpha \right] + E\left[ \left( \sum_{i=1}^N C_i R_i \right)^\alpha - \sum_{i=1}^N (C_iR_i)^\alpha \right] \\
&\leq E\left[ (S+Q)^\alpha - S^\alpha \right] + E\left[ N^\alpha \right] (E[(CR)^{p-1}])^{\alpha/(p-1)} .\end{aligned}$$ The second expectation is finite since by Lemma \[L.Stability\], $E[R^\beta] < \infty$ for any $0 <\beta < \alpha$. For the first expectation we use the inequality $$(x+t)^\kappa \leq \begin{cases}
x^\kappa + t^\kappa, & 0 < \kappa \leq 1, \\
x^\kappa + \kappa (x+t)^{\kappa-1} t, & \kappa > 1,
\end{cases}$$ for any $x,t \geq 0$. We apply the second expression $p-1$ times and then the first one to obtain $$(x+t)^\alpha \leq x^\alpha + \alpha (x+t)^{\alpha-1} t \leq \dots \leq x^\alpha + \sum_{i=1}^{p-2} \alpha^i x^{\alpha-i} t^i + \alpha^{p-1} (x+t)^{\alpha-p+1} t^{p-1} \leq x^\alpha + \alpha^p t^\alpha + \alpha^p \sum_{i=1}^{p-1} x^{\alpha-i} t^i.$$ We conclude that $$\label{eq:Alpha_diff}
E\left[(S+Q)^\alpha - S^\alpha\right] \leq \alpha^p E[Q^\alpha] + \alpha^p \sum_{i=1}^{p-1} E[S^{\alpha-i}] E[Q^i],$$ where $E[S^{\alpha-i}] \leq E[(R^*)^{\alpha-i}] < \infty$ for any $1\leq i\leq p-1$ by Lemma \[L.Stability\].
Finally, applying Theorem \[T.Goldie\] gives $$P(R > t) \sim H t^{-\alpha},$$ where $H = (E[N] E[C^\alpha \log C])^{-1} \int_0^\infty v^{\alpha-1} (P(R > v) - E[N] P(CR > v)) \, dv$.
To obtain the second expression for $H$ note that $$\begin{aligned}
&\int_0^\infty v^{\alpha-1} (P(R > v) - E[N] P(CR > v)) \, dv \notag \\
&= \int_0^\infty v^{\alpha-1} \left( E\left[1_{(\sum_{i=1}^N C_i R_i + Q > v)}\right] - E\left[ \sum_{i=1}^N 1_{(C_iR_i > v)} \right] \right) \, dv \notag \\
&= E \left[ \int_0^\infty v^{\alpha-1} \left( 1_{(\sum_{i=1}^N C_i R_i + Q > v)} - \sum_{i=1}^N 1_{(C_iR_i > v)} \right) dv \right] \label{eq:Fubini} \\
&= E \left[ \int_0^{\sum_{i=1}^N C_i R_i + Q} v^{\alpha-1} dv - \sum_{i=1}^N \int_0^{C_i R_i} v^{\alpha-1} dv \right] \label{eq:DiffIntegrals} \\
&= \frac{1}{\alpha} E\left[ \left( \sum_{i=1}^N C_i R_i + Q \right)^\alpha - \sum_{i=1}^N (C_i R_i)^\alpha \right] , \notag\end{aligned}$$ where is justified by Fubini’s Theorem and the absolute integrability of $v^{\alpha-1} (P(R > v) - E[N] P(CR > v))$, and is justified from the observation that $$v^{\alpha-1} 1_{(\sum_{i=1}^N C_i R_i + Q > v)} \qquad \text{and} \qquad v^{\alpha-1} \sum_{i=1}^N 1_{(C_iR_i > v)}$$ are each almost surely absolutely integrable as well. This completes the proof.
The case when $N$ dominates {#S.NDominates}
===========================
We now turn our attention to the distributional properties of $R^{(n)}$ and $R$ when $N$ has a heavy-tailed distribution (in particular, regularly varying) that is heavier than the potential power law effect arising from the multiplicative weights $\{C_i\}$. This case is particularly important for understanding the behavior of Google’s PageRank algorithm since the $C_i$’s are smaller than one and the in-degree distribution of the Web graph is well accepted to be a power law. We start this section by stating the corresponding lemma that describes the asymptotic behavior of $R^{(n)}$. The main technical difficulty of extending this lemma to steady state ($R = R^{(\infty)}$) is to develop a uniform bound for $R-R^{(n)}$, which is enabled by our main technical result of this section, Proposition \[P.UniformBound\]. The following lemmas are proved in Section \[SS.NDominates\_Proofs\].
Before stating the lemmas, let us recall that a function $L: [0, \infty) \to (0, \infty)$ is slowly varying if $L(\lambda x)/L(x) \to 1$ as $x \to \infty$ for any $\lambda > 0$. We then say that the function $x^{-\alpha} L(x)$ is regularly varying with index $\alpha$.
\[L.Finite\_n\] Suppose that $P(N > x) = x^{-\alpha} L(x)$ with $L(\cdot)$ slowly varying, $\alpha > 1$, and $E[Q^{\alpha+\epsilon}] < \infty$, $E[C^{\alpha+\epsilon}] < \infty$ for some $\epsilon > 0$. Let $\rho = E[N] E[C]$ and $\rho_\alpha = E[N] E[C^\alpha]$. Then, for any fixed $n \in \{1, 2, 3,\dots\}$, $$\label{eq:Asymp_R_n}
P(R^{(n)} > x) \sim \frac{(E[C]E[Q])^\alpha}{(1-\rho)^\alpha} \sum_{k=0}^n \rho_\alpha^k (1-\rho^{n-k})^\alpha P(N > x)$$ as $x \to \infty$, where $R^{(n)}$ was defined in Section \[SS.TreeConstruction\].
\[L.W\_n\_Finite\_n\] Suppose that $P(N > x) = x^{-\alpha} L(x)$ with $L(\cdot)$ slowly varying, $\alpha > 1$, and $E[Q^{\alpha+\epsilon}] < \infty$, $E[C^{\alpha+\epsilon}] < \infty$ for some $\epsilon > 0$. Let $\rho = E[N] E[C]$ and $\rho_\alpha = E[N] E[C^\alpha]$. Then, for any fixed $n \in \{1, 2, 3,\dots\}$, $$P(W_n > x) \sim (E[C] E[Q])^\alpha \sum_{k=0}^{n-1} \rho_\alpha^k \rho^{(n-1-k)\alpha} P(N > x)$$ as $x \to \infty$, where $W_n$ was defined in Section \[SS.TreeConstruction\].
From this result, provided $\rho \vee \rho_\alpha < 1$, it is to be expected that a bound of the form $$P(W_n > x) \leq K \eta^n P(N > x)$$ might hold for all $n$ and $x \geq 1$, for some $\rho \vee \rho_\alpha < \eta < 1$. Such a bound will provide the necessary tools to ensure that $R-R^{(n)}$ is negligible for large enough $n$, allowing the exchange of limits in Lemma \[L.Finite\_n\]. Proving this result is the main technical contribution of this section; the actual proof is given in Section \[SS.NDominates\_Proofs\]. This bound may be of independent interest for computing the distributional properties of other recursions on branching trees, e.g. it is straightforward to apply our method to study the solution to $$R = Q + \max_{1 \leq i \leq N} C_i R_i,$$ and similar recursions.
\[P.UniformBound\] Suppose $P(N > x) = x^{-\alpha} L(x)$, with $L(\cdot)$ slowly varying and $\alpha > 1$, $E[C^{\alpha+\nu}] < \infty$, $E[Q^{\alpha+\nu}] < \infty$ for some $\nu > 0$, and let $E[N] \max\{E[C^{\alpha}], E[C]\} < \eta < 1$. Then, there exists a constant $K = K(\eta,\nu) > 0$ such that for all $n \geq 1$ and all $x \geq 1$, $$\label{eq:UniformBound}
P(W_n > x) \leq K \eta^n P(N > x).$$
We would also like to point out that a bound of type resembles a classical result by Kesten (see Lemma 7 on p. 149 of [@Ath_McD_Ney_78]) stating that the sum of heavy-tailed (subexponential) random variables satisfies $$P( X_1 + \dots + X_n > x) \leq K (1+\epsilon)^n P(X_1 > x),$$ uniformly for all $n$ and $x$, for any $\epsilon > 0$ (see also [@Den_Foss_Kor_09] for more recent work). The main difference between this result and is that while $n$ above refers to the number of terms in the sum, in it refers to the depth of the recursion. This makes the derivation of considerably more complicated, and perhaps implausible if it were not for the fact that we restrict our attention to regularly varying distributions, as opposed to the general subexponential class.
In view of , we can now prove the main theorem of this section.
\[T.Main\_N\] Suppose $P(N > x) = x^{-\alpha} L(x)$, with $L(\cdot)$ slowly varying and $\alpha > 1$. Let $\rho = E[N] E[C]$ and $\rho_\alpha = E[N] E[C^\alpha]$. Assume $\rho \vee \rho_\alpha < 1$, and $E[C^{\alpha+\epsilon}] < \infty$, $E[Q^{\alpha+\epsilon}] < \infty$ for some $\epsilon > 0$. Then, $$P(R > x) \sim \frac{(E[C]E[Q])^\alpha}{(1-\rho)^\alpha(1-\rho_\alpha)} P(N > x)$$ as $x \to \infty$, where $R$ was defined by .
[Remarks:]{} (i) A related result that also allows $Q$ and $N$ to be dependent was derived very recently in [@Volk_Litv_08] using transform methods and tauberian theorems under the moment conditions $E[Q]< 1$, $E[C] = (1-E[Q])/E[N]$. (ii) Note that this result implies the classical result on the busy period of an M/G/1 queue derived in [@Mey_Teug_80]. Specifically, the total number of customers in a busy period $B$ satisfies the recursion $B \stackrel{\mathcal{D}}{=} 1 + \sum_{i=1}^{N(S)} B_i$, where the $B_i$’s are iid copies of $B$, $N(t)$ is a Poisson process of rate $\lambda$ and $S$ is the service distribution; $\{B_i\}$, $N(t)$ and $S$ are mutually independent and $\rho = E[N(S)] < 1$. Now, the recursion for $B$ is obtained from our theorem by setting $C \equiv 1$ and $Q \equiv 1$, implying that $P(B > x) \sim P(N(S) > x)/ (1-\rho)^{\alpha+1}$. Next, one can obtain the asymptotics for the length of the busy period $P$ by using the identity $B = N(P)$. This can be easily derived, in spite of the fact that $N(t)$ and $P$ are correlated, since $N(t)$ is highly concentrated around its mean. For recent work on the power law asymptotics of the GI/GI/1 busy period see [@Zwart_01]. (iii) In view of Lemma \[L.Finite\_n\], the theorem shows that the limits $\lim_{x \to \infty} \lim_{n \to \infty} P(R^{(n)} > x)/ P(N > x)$ are interchangeable.
Fix $0< \delta <1$ and $n_0 \geq 1$. Choose $\rho \vee \rho_\alpha < \eta < 1$ and use Proposition \[P.UniformBound\] to obtain that for some constant $K_0 > 0$, $$P(W_n > x) \leq K_0 \eta^n P(N > x)$$ for all $n \geq 1$ and all $x \geq 1$. Let $H_\alpha^{(n)} = (E[C]E[Q])^\alpha (1-\rho)^{-\alpha} \sum_{k=0}^n \rho_\alpha^k (1-\rho^{n-k})^\alpha$ and $H_\alpha = H_\alpha^{(\infty)}$. Then, $$\begin{aligned}
&\left| P(R > x) - H_\alpha P(N > x) \right| \notag \\
&\leq \left| P(R > x) - P(R^{(n_0)} > x) \right| \label{eq:Tail} \\
&\hspace{5mm} + \left| P(R^{(n_0)} > x) - H_\alpha^{(n_0)} P(N > x) \right| \label{eq:FiniteIterations} \\
&\hspace{5mm} + \left| H_\alpha^{(n_0)} - H_\alpha \right| P(N > x). \label{eq:FiniteError}\end{aligned}$$ By Lemma \[L.Finite\_n\], there exists a function $\varphi(x) \downarrow 0$ as $x \to \infty$ such that $$\left| P(R^{(n_0)} > x) - H_\alpha^{(n_0)} P(N > x) \right| \leq \varphi(x) H_\alpha P(N > x).$$ To bound let $\beta = \eta^{1/(2\alpha+2)} < 1$ and note that $$\begin{aligned}
\left| P(R > x) - P(R^{(n_0)} > x) \right| &\leq P\left(R^{(n_0)} + (R-R^{(n_0)}) > x, \, R-R^{(n_0)} \leq \delta x\right) - P(R^{(n_0)} > x) \\
&\hspace{5mm} + P\left(R-R^{(n_0)} > \delta x\right) \\
&\leq P(R^{(n_0)} > (1-\delta) x) - P(R^{(n_0)} > x) + P\left( \sum_{n = n_0+1}^\infty W_n > \delta x \right) \\
&\leq P(R^{(n_0)} > (1-\delta) x) - H_\alpha^{(n_0)} P(N > (1-\delta) x) \\
&\hspace{5mm} + H_\alpha^{(n_0)} P(N > x) - P(R^{(n_0)} > x) \\
&\hspace{5mm} + H_\alpha^{(n_0)} P(N > (1-\delta) x) - H_\alpha^{(n_0)} P(N > x) \\
&\hspace{5mm} + \sum_{n=n_0+1}^\infty P\left( W_n > \delta x
(1-\beta) \beta^{n-n_0-1} \right) \\
&\leq \left\{ 2\varphi((1-\delta)x) \frac{P(N > (1-\delta) x)}{P(N > x)} \right. \\
&\hspace{5mm} + \left. \left( \frac{P(N > (1-\delta)x)}{P(N>x)} - 1 \right) \right\} H_\alpha P(N > x) \\
&\hspace{5mm} + \sum_{n = n_0+1}^\infty K_0 \eta^n P\left( N > \delta x
(1-\beta) \beta^{n-n_0-1} \right) ,\end{aligned}$$ where in the last inequality we applied the uniform bound from Proposition \[P.UniformBound\]. The expression in curly brackets is bounded by $$2 \varphi((1-\delta)x) (1-\delta)^{-\alpha} \frac{L((1-\delta)x)}{L(x)} + \left( (1-\delta)^{-\alpha} \frac{L((1-\delta)x)}{L(x)} -1 \right) \to (1-\delta)^{-\alpha} - 1$$ as $x \to \infty$. By Potter’s Theorem (see Theorem 1.5.6 (ii) on p. 25 in [@BiGoTe1987]), there exists a constant $A = A(1) > 1$ such that $$\begin{aligned}
&\sum_{n = n_0+1}^\infty K_0 \eta^n P\left( N > \delta x
(1-\beta) \beta^{n-n_0-1} \right) \\
&\leq K_0 A \sum_{n = n_0+1}^\infty \eta^n \left( \delta
(1-\beta) \beta^{n-n_0-1} \right)^{-\alpha-1} P(N > x) \\
&= K_0 A (\delta(1-\beta))^{-\alpha-1} (1-\eta^{1/2})^{-1} \eta^{n_0+1} P(N > x) \\
&\leq K \delta^{-\alpha-1} \eta^{n_0} P(N > x) .\end{aligned}$$ Next, for simply note that $$\begin{aligned}
&\frac{1}{H_\alpha} \left| H_\alpha^{(n_0)} - H_\alpha \right| \\
&= (1-\rho_\alpha) \left( \sum_{k=0}^\infty \rho_\alpha^k - \sum_{k=0}^{n_0} \rho_\alpha^k (1-\rho^{n_0-k})^\alpha \right) \\
&= (1-\rho_\alpha) \sum_{k=0}^{n_0} \rho_\alpha^k (1 - (1-\rho^{n_0-k})^\alpha ) + (1-\rho_\alpha) \sum_{k=n_0+1}^\infty \rho_\alpha^k \\
&\leq (1-\rho_\alpha) \sum_{k=0}^{n_0} \rho_\alpha^k \alpha \rho^{n_0-k} + \rho_\alpha^{n_0+1} \\
&\leq [\alpha (1-\rho_\alpha) (n_0+1) + \rho_\alpha] (\rho_\alpha \vee \rho)^{n_0} \\
&\leq K \eta^{n_0} .\end{aligned}$$ Finally, by replacing the preceding estimates in - , we obtain $$\begin{aligned}
\lim_{x \to \infty} \left| \frac{P(R > x)}{H_\alpha P(N > x)} - 1 \right| &\leq (1-\delta)^{-\alpha} - 1 + K \delta^{-\alpha-1} \eta^{n_0} .\end{aligned}$$ Since the right hand side can be made arbitrarily small by first letting $n_0 \to \infty$ and then $\delta \downarrow 0$, the result of the theorem follows.
[**Engineering implications.**]{} Recall that for Google’s PageRank algorithm the weights are given by $C_i = c/D_i < 1$, where $0 < c < 1$ is a constant related to the damping factor and the number of nodes in the Web graph, and $D_i$ corresponds to the out-degree of a page. We point out that dividing the ranks of neighboring pages by their out-degree has the purpose of decreasing the contribution of pages with highly inflated referencing. However, Theorem \[T.Main\_N\] reveals that the page rank is essentially insensitive to the parameters of the out-degree distribution, which means that PageRank basically reflects the popularity vote given by the number of references $N$. This same observation was previously made in [@Volk_Litv_08].
Furthermore, Theorem \[T.Goldie\] clearly shows that the choice of weights $C_i$ in the ranking algorithm can determine the distribution of $R$ as well. Note that for the PageRank algorithm the weights $C_i = c/D_i < 1$ can never dominate the asymptotic behavior of $R$ when $N$ is a power law. Therefore, Theorem \[T.Goldie\] suggests a potential development of new ranking algorithms where the ranks will be much more sensitive to the weights.
The case when $Q$ dominates {#S.QDominates}
===========================
This section of the paper treats the case when the heavy-tailed behavior of $R$ arises from the $\{Q_i\}$, known in the autoregressive processes literature as innovations. The results presented here are very similar to those in Section \[S.NDominates\], and so are their proofs. We will therefore only present the statements of the results and skip most of the proofs. We start with the equivalent of Lemmas \[L.Finite\_n\] and \[L.W\_n\_Finite\_n\] in this context; their proofs are given in Section \[SS.QDominate\_Proofs\].
\[L.Finite\_nQ\] Suppose $P(Q > x) = x^{-\alpha} L(x)$ with $L(\cdot)$ slowly varying, $\alpha > 1$, and $E[N^{\alpha+\epsilon}] < \infty$, $E[C^{\alpha+\epsilon}] < \infty$ for some $\epsilon > 0$; let $\rho_\alpha = E[N] E[C^\alpha]$ . Then, for any fixed $n \in \{1,2,3,\dots\}$, $$P(R^{(n)} > x) \sim \sum_{k=0}^n \rho_\alpha^k \, P(Q > x)$$ as $x \to \infty$, where $R^{(n)}$ was defined in Section \[SS.TreeConstruction\].
As for the case when $N$ dominates the asymptotic behavior of $R$, we can here expect that $$P(R > x) \sim (1-\rho_\alpha)^{-1} P(Q > x),$$ and the technical difficulty is justifying the exchange of limits. The same techniques used in Section \[S.NDominates\] can be used in this case as well. Therefore, we give a sketch of the arguments in Section \[SS.QDominate\_Proofs\] but omit the proof. The following is the equivalent of Lemma \[L.W\_n\_Finite\_n\].
\[L.W\_n\_Finite\_nQ\] Suppose $P(Q > x) = x^{-\alpha} L(x)$ with $L(\cdot)$ slowly varying, $\alpha > 1$, and $E[N^{\alpha+\epsilon}] < \infty$, $E[C^{\alpha+\epsilon}] < \infty$ for some $\epsilon > 0$; let $\rho_\alpha = E[N] E[C^\alpha]$. Then, for any fixed $n \in \{1, 2, 3,\dots\}$, $$P(W_n > x) \sim \rho_\alpha^n P(Q > x)$$ as $x \to \infty$, where $W_n$ was defined in Section \[SS.TreeConstruction\].
The corresponding version of Proposition \[P.UniformBound\] is given below.
\[P.UniformBoundQ\] Suppose $P(Q > x) = x^{-\alpha} L(x)$, with $L(\cdot)$ slowly varying and $\alpha > 1$, $E[C^{\alpha + \nu}] < \infty$, $E[N^{\alpha+\nu}] < \infty$ for some $\nu > 0$, and let $E[N] \max\{ E[C^{\alpha}] , E[C] \} < \eta < 1$. Then, there exists a constant $K = K(\eta,\nu) > 0$ such that for all $n \geq 1$ and all $x \geq 1$, $$P(W_n > x) \leq K \eta^n P(Q > x).$$
A sketch of the proof can be found in Section \[SS.QDominate\_Proofs\].
And finally, the main theorem of this section. The proof again greatly resembles that of Theorem \[T.Main\_N\] and is therefore omitted.
\[T.MainQ\] Suppose $P(Q > x) = x^{-\alpha} L(x)$, with $L(\cdot)$ slowly varying and $\alpha > 1$. Let $\rho = E[N] E[C]$ and $\rho_\alpha = E[N] E[C^\alpha]$. Assume $\rho \vee \rho_\alpha < 1$, and $E[C^{\alpha+\epsilon}] < \infty$, $E[N^{\alpha+\epsilon}] < \infty$ for some $\epsilon > 0$. Then, $$P(R > x) \sim (1-\rho_\alpha)^{-1} P(Q > x)$$ as $x \to \infty$, where $R$ was defined in .
Compare this result with Lemma A.3 in [@Mik_Sam_00], where the autoregressive process of order one with regularly varying innovations is shown to be tail-equivalent to $Q$. In particular, if we set $N \equiv 1$ in Theorem \[T.MainQ\] and let $A_k = \prod_{i=1}^{k-1} C_i$, our result reduces to $$P\left( \sum_{k=0}^\infty A_k Q_k > x \right) \sim \sum_{k=0}^\infty E[A_k^\alpha] P(Q > x),$$ which is in line with the commonly accepted intuition about heavy-tailed large deviations where large sums are due to one large summand $Q_k$.
Proofs {#S.Proofs}
======
This section contains the proofs to most of the technical results presented in the paper, together with some auxiliary lemmas that are needed along the way. The section is divided into four subsections, each corresponding to the content of Sections \[S.Moments\], \[S.C\_dominates\], \[S.NDominates\], and \[S.QDominates\], respectively.
Moments of $W_n$ {#SS.Moments_Proofs}
----------------
Here we give the proof of the moment bound for the $\beta$-moment, $\beta > 1$, of the sum of the weights, $W_n$ of the $n$th generation. As an intermediate step, we present a lemma for the integer moments of $W_n$, but first we give the proof of Lemma \[L.Alpha\_Moments\].
Let $p = \lceil \beta \rceil \in \{2,3,\dots\}$ and $\gamma = \beta/p \in (0, 1]$. Define $A_p(k) = \{ (j_1, \dots, j_k) \in \mathbb{Z}^k: j_1 + \dots + j_k = p, 0 \leq j_i < p\}$. Then, $$\begin{aligned}
\left( \sum_{i=1}^k y_i \right)^\beta &= \left( \sum_{i=1}^k y_i \right)^{p \gamma} \notag \\
&= \left( \sum_{i=1}^k y_i^p + \sum_{(j_1,\dots,j_k) \in A_p(k)} \binom{p}{j_1,\dots,j_k} y_1^{j_1} \cdots y_k^{j_k} \right)^\gamma \notag \\
&\leq \sum_{i=1}^k y_i^{p\gamma} + \left( \sum_{(j_1,\dots,j_k) \in A_p(k)} \binom{p}{j_1,\dots,j_k} y_1^{j_1} \cdots y_k^{j_k} \right)^\gamma, \notag\end{aligned}$$ where for the last step we used the well known inequality $\left( \sum_{i=1}^k x_i \right)^\gamma \leq \sum_{i=1}^k x_i^\gamma$ for $0 < \gamma \leq 1$ and $x_i \geq 0$ (see the proof of Lemma \[L.MomentSmaller\_1\]). We now use Jensen’s inequality to obtain $$\begin{aligned}
E\left[ \left( \sum_{i=1}^k Y_i \right)^\beta - \sum_{i=1}^k Y_i^{\beta} \right] &\leq E\left[ \left( \sum_{(j_1,\dots,j_k) \in A_p(k)} \binom{p}{j_1,\dots,j_k} Y_1^{j_1} \cdots Y_k^{j_k} \right)^\gamma \right] \\
&\leq \left( E\left[ \sum_{(j_1,\dots,j_k) \in A_p(k)} \binom{p}{j_1,\dots,j_k} Y_1^{j_1} \cdots Y_k^{j_k} \right] \right)^\gamma \\
&= \left( \sum_{(j_1,\dots,j_k) \in A_p(k)} \binom{p}{j_1,\dots,j_k} E\left[ Y_1^{j_1} \cdots Y_k^{j_k} \right] \right)^\gamma.\end{aligned}$$ Since the $\{Y_i\}$ are iid, we have $$E\left[ Y_1^{j_1} \cdots Y_k^{j_k} \right] = || Y||_{j_1}^{j_1} \cdots ||Y||_{j_k}^{j_k},$$ where $|| Y||_\kappa = E[|Y|^\kappa]^{1/\kappa}$ for $\kappa \geq 1$ and $|| Y ||_0 \triangleq 1$. Since $|| Y||_\kappa$ is increasing for $\kappa \geq 1$ it follows that $|| Y ||_{j_i}^{j_i} \leq || Y ||_{p-1}^{j_1}$. It follows that $$|| Y||_{j_1}^{j_1} \cdots ||Y||_{j_k}^{j_k} \leq || Y ||_{p-1}^p,$$ which in turn implies that $$\begin{aligned}
E\left[ \left( \sum_{i=1}^k Y_i \right)^\beta - \sum_{i=1}^k Y_i^{\beta} \right] &\leq \left( \sum_{(j_1,\dots,j_k) \in A_p(k)} \binom{p}{j_1,\dots,j_k} || Y ||_{p-1}^p \right)^\gamma \\
&= || Y||_{p-1}^\beta (k^p - k)^\gamma \\
&\leq || Y ||_{p-1}^\beta k^\beta.\end{aligned}$$
\[L.IntegerMoment\] Suppose $E[Q^p]< \infty$, $E[N^p] < \infty$, and $E[N] \max\{ E[C^p], E[C] \} < 1$ for some $p \in \{2,3,\dots\}$. Then, there exists a constant $K_p > 0$ such that $$E[ W_n^p ] \leq K_p \left( E[N] \max\{E[C], E[C^p]\} \right)^n$$ for all $n \geq 0$.
Let $Y = C W_{n-1}$, where $C$ is independent of $W_{n-1}$ and let $\{Y_i\}$ be independent copies of $Y$. We will give an induction proof in $p$. For $p = 2$ we have $$\begin{aligned}
E[W_n^2] &= E\left[ \left( \sum_{i=1}^N Y_i \right)^2 \right] \\
&= E[N] E[Y^2] + E[N(N-1)] (E[Y])^2 \\
&= E[N] E[C^2] E[W_{n-1}^2] + E[N(N-1)] (E[C] E[W_{n-1}])^2 .\end{aligned}$$ Using the preceding recursion, letting $\rho = E[N] E[C]$, $\rho_2 = E[N] E[C^2]$, and noting that, $$E[W_{n-1}] = \rho^{n-1} E[Q],$$ we obtain $$\label{eq:2_recur}
E[W_n^2] = \rho_2 E[W_{n-1}^2] + K \rho^{2(n-1)},$$ where $K = E[N(N-1)] (E[C] E[Q])^2$. Now, iterating gives $$\begin{aligned}
E[W_n^2] &= \rho_2 \left( \rho_2 E[W_{n-2}^2] + K \rho^{2(n-2)} \right) + K \rho^{2(n-1)} \\
&= \rho_2^{n-1} \left( \rho_2 E[W_{0}^2] + K \right) + K \sum_{i=0}^{n-2} \rho_2^i \, \rho^{2(n-1-i)} \\
&= \rho_2^n E[Q^2] + K \sum_{i=0}^{n-1} \rho_2^i \, \rho^{2(n-1-i)} \\
&\leq (\rho_2 \vee \rho)^n E[Q^2] + K (\rho_2 \vee \rho)^n \sum_{i=0}^{n-1} (\rho_2 \vee \rho)^{n-2 - i } \\
&\leq \left( E[Q^2] + \frac{K}{\rho_2 \vee \rho} \sum_{j=0}^{\infty} (\rho_2 \vee \rho)^{j} \right) (\rho_2 \vee \rho)^n \\
&= K_2 (\rho_2 \vee \rho)^n .\end{aligned}$$ Next, for any $p \in \{2, 3, \dots\}$ let $\rho_p = E[N] E[C^p]$. Suppose now that there exists a constant $K_{p-1} > 0$ such that $$\label{eq:Induction_p}
E[W_n^{p-1}] \leq K_{p-1} \left( \rho_{p-1} \vee \rho \right)^n$$ for all $n \geq 0$. By Lemma \[L.Alpha\_Moments\] we have $$\begin{aligned}
E[W_n^p] &= \sum_{k=1}^\infty E\left[ \left( \sum_{i=1}^k Y_i \right)^p \right] P(N = k) \\
&\leq \sum_{k=1}^\infty \left( k E\left[ Y^p \right] + k^p (E[ Y^{p-1}])^{p/(p-1)} \right) P(N = k) \\
&= E[N] E[ C^p] E[W_{n-1}^p] + E[N^p] (E[C^{p-1}])^{p/(p-1)} (E[W_{n-1}^{p-1}])^{p/(p-1)} \\
&\leq \rho_p E[W_{n-1}^p] + E[N^p] (E[C^{p-1}])^{p/(p-1)} (K_{p-1})^{p/(p-1)} (\rho_{p-1} \vee \rho)^{(n-1)p/(p-1)},\end{aligned}$$ where the last inequality corresponds to the induction hypothesis. We then obtain the recursion $$\label{eq:p_recur}
E[W_n^p] \leq \rho_p E[W_{n-1}^p] + K (\rho_{p-1} \vee \rho)^{\frac{(n-1)p}{p-1}},$$ where $K = E[N^p] (E[C^{p-1}])^{p/(p-1)} (K_{p-1})^{p/(p-1)}$. Iterating as for the case $p=2$ gives $$\begin{aligned}
E[W_n^p] &\leq \rho_p^n E[Q^p] + K \sum_{i=0}^{n-1} \rho_p^i \, (\rho_{p-1} \vee \rho)^{\frac{(n-1-i)p}{p-1}} \\
&\leq (\rho_p \vee \rho)^n E[Q^p] + K \sum_{i=0}^{n-1} (\rho_p \vee \rho)^{\frac{(n-1)p -i}{p-1} } \\
&= (\rho_p \vee \rho)^n E[Q^p] + K (\rho_p \vee \rho)^n \sum_{i=0}^{n-1} (\rho_p \vee \rho)^{\frac{ n- i - p }{p-1} } \\
&\leq \left( E[Q^p] + K (\rho_p \vee \rho)^{-1} \sum_{j=0}^{\infty} (\rho_p \vee \rho)^{\frac{j}{p-1}} \right) (\rho_p \vee \rho)^n \\
&= K_p (\rho_p \vee \rho)^n.\end{aligned}$$
The proof for the general $\beta$-moment, $\beta > 1$, is given below.
Set $p = \lceil \beta \rceil \geq \beta > 1$. Since the result when $p = \beta$ follows from Lemma \[L.IntegerMoment\], we assume that $p > \beta$. Let $Y = C W_{n-1}$, where $C$ is independent of $W_{n-1}$ and $\{Y_i\}$ are independent copies of $Y$. Also, recall that $\rho = E[N] E[C]$ and $\rho_\beta = E[N] E[C^\beta]$. Then, by Lemma \[L.Alpha\_Moments\], $$\begin{aligned}
E[W_n^\beta] &= E\left[ \left( \sum_{i=1}^N Y_i \right)^\beta \right] \\
&= \sum_{k=1}^\infty E\left[ \left( \sum_{i=1}^k Y_i \right)^{\beta} \right] P(N = k) \\
&= \sum_{k=1}^\infty \left( E\left[ \left( \sum_{i=1}^k Y_i \right)^{\beta} - \sum_{i=1}^k Y_i^\beta \right] + E\left[ \sum_{i=1}^k Y_i^\beta \right] \right) P(N = k) \\
&\leq \sum_{k=1}^\infty \left( k^\beta E[Y^{p-1}]^{\beta/(p-1)} + k E\left[ Y^\beta \right] \right) P(N = k) \\
&= E[N^\beta] (E[C^{p-1}])^{\beta/(p-1)} (E[W_{n-1}^{p-1}])^{\beta/(p-1)} + \rho_\beta E[ W_{n-1}^\beta] .\end{aligned}$$ Then, by Lemma \[L.IntegerMoment\], $$\begin{aligned}
E[W_n^\beta] &\leq \rho_\beta E[ W_{n-1}^\beta] + E[N^\beta] (E[C^{p-1}])^{\beta/(p-1)} (K_{p-1} (\rho_{p-1} \vee \rho)^{n-1})^{\beta/(p-1)} \\
&= \rho_\beta E[ W_{n-1}^\beta] + K (\rho_{p-1} \vee \rho)^{(n-1)\gamma},\end{aligned}$$ where $\gamma = \beta/(p-1) > 1$. Finally, iterating the preceding bound $n-1$ times gives $$\begin{aligned}
E[W_n^\beta] &\leq \rho_\beta^n E[W_0^\beta] + K \sum_{i=0}^{n-1} \rho_\beta^i (\rho \vee \rho_{p-1})^{\gamma(n-1-i)} \\
&\leq E[W_0^\beta] (\rho \vee \rho_\beta)^n + K \sum_{i=0}^{n-1} (\rho \vee \rho_\beta)^{\gamma(n-1-i) + i} \\
&= E[Q^\beta] (\rho \vee \rho_\beta)^n + K (\rho \vee \rho_\beta)^{n-1} \sum_{i=0}^{n-1} (\rho \vee \rho_\beta)^{(\gamma-1) i} \\
&\leq K_\beta (\rho \vee \rho_\beta)^n .\end{aligned}$$ This completes the proof.
The case when the $C$’s dominate: Implicit renewal theory {#SS.CDominates_Proofs}
---------------------------------------------------------
In this section we state a lemma that is used in the proof of Theorem \[T.Goldie\] and we give the proofs to Theorem \[T.Goldie\] and Lemma \[L.Max\_Approx\].
\[L.Derivative\] Let $\alpha, \beta > 0$ and $H \geq 0$. Suppose $\int_0^t v^{\alpha+\beta-1} P(R > v) dv \sim H t^{\beta}/\beta$ as $t \to \infty$. Then, $$P(R > t) \sim H t^{-\alpha}, \qquad t \to \infty.$$
This lemma is a special case of the Monotone Density Theorem, see Theorem 1.7.5 (also Exercise 1.11.14) in [@BiGoTe1987]. However, for completeness, we give a direct proof here, similar to the one of Lemma 9.3 in [@Goldie_91]. By assumption, for any $b > 1$, $\epsilon \in (0,1)$, and $t$ sufficiently large, $$\begin{aligned}
P(R > t) t^{\alpha +\beta} \cdot \frac{b^{\alpha+\beta}-1}{\alpha+\beta} &\geq \int_{t}^{b t} v^{\alpha+\beta-1} P(R > v) \, dv \geq \frac{(H-\epsilon)}{\beta} (b t)^\beta - \frac{(H+\epsilon)}{\beta} t^\beta \\
&\geq \frac{t^\beta}{\beta} \left( H (b^\beta-1) -\epsilon (1 + b^\beta) \right).\end{aligned}$$ Since $\epsilon$ was arbitrary, we can take the limit as $\epsilon \to 0$ and obtain $$\liminf_{t \to \infty} P(R > t) t^{\alpha} \geq \frac{H (\alpha+\beta) (b^\beta-1)}{\beta(b^{\alpha+\beta} - 1)} \to H, \qquad b \downarrow 1.$$ Similarly, one can prove that $\limsup_{t \to \infty} P(R > t) t^\alpha \leq H$ starting from $\int_{bt}^t v^{\alpha+\beta -1} P(R > v) \, dv$ with $0 < b < 1$.
For any $k \in \mathbb{N}$ define $\Pi_k = \prod_{i=1}^k C_i$ and $V_k = \sum_{i=1}^k \log C_i$, with $\Pi_0 = 1$ and $V_0 = 0$, where the $C_i$’s are independent copies of $C$. Then, for any $t \in \mathbb{R}$, $$\begin{aligned}
P(R > e^t) &= \sum_{k=1}^n \left( m^{k-1} P(\Pi_{k-1} R > e^t) - m^k P(\Pi_k R > e^t) \right) + m^n P( \Pi_n R > e^t) \\
&= \sum_{k=1}^n \left( m^{k-1} P(e^{V_{k-1}} R > e^t) - m^k P( e^{V_{k-1}} C_k R > e^t) \right) + m^n P( e^{V_n} R > e^t) \\
&= \sum_{k=0}^{n-1} m^{k} \int_{-\infty}^\infty \left( P( R > e^{t-v}) - mP( C R > e^{t-v}) \right) P(V_{k} \in dv) + m^n P( e^{V_n} R > e^t).\end{aligned}$$ Next, define $$\nu_n(dt) = e^{\alpha t} \sum_{k=0}^n m^{k} P(V_k \in dt), \qquad g(t) = e^{\alpha t} (P(R > e^t) - m P(CR > e^t)),$$ $$r(t) = e^{\alpha t} P(R > e^t) \qquad \text{and} \qquad \delta_n(t) = m^n P( e^{V_n} R > e^t).$$ Then, for any $t \in \mathbb{R}$ and $n \in \mathbb{N}$, $$r(t) = (g*\nu_{n-1})(t) + \delta_n(t).$$ Next, for any $\beta > 0$, define the smoothing operator $$\breve{f}(t) = \int_{-\infty}^t e^{-\beta(t-u)} f(u) \, du$$ and note that $$\begin{aligned}
\breve{r}(t) &= \int_{-\infty}^t e^{-\beta(t-u)} (g*\nu_{n-1})(u) \, du + \breve{\delta}_n (t) \notag \\
&= \int_{-\infty}^t e^{-\beta(t-u)} \int_{-\infty}^\infty g(u-v) \nu_{n-1}(dv) \, du + \breve{\delta}_n (t) \notag \\
&= \int_{-\infty}^\infty \int_{-\infty}^t e^{-\beta(t-u)} g(u-v) \, du \, \nu_{n-1}(dv) + \breve{\delta}_n (t) \notag \\
&= \int_{-\infty}^\infty \breve{g}(t-v) \, \nu_{n-1}(dv) + \breve{\delta}_n (t) \notag \\
&= (\breve{g}* \nu_{n-1})(t) + \breve{\delta}_n(t) . \label{eq:SmoothOperator}\end{aligned}$$
Next, we will show that one can pass $n \to \infty$ in the preceding identity. To this end, let $\eta(du) = e^{\alpha u} m P(\log C \in du)$, and note that this measure places no mass at $-\infty$. Also, by assumption, $\eta(\cdot)$ is a nonarithmetic measure on $\mathbb{R}$. Moreover, $$\int_{-\infty}^\infty \eta(du) = m E[ e^{\alpha \log C}] = m E[ C^\alpha] = 1$$ and $$\int_{-\infty}^\infty u\, \eta(du) = m E[ e^{\alpha \log C} \log C] = m E[C^\alpha \log C] = m\mu$$ imply that $\eta(\cdot)$ is a probability measure with mean $0 < m\mu < \infty$. Furthermore, $$\nu(dt) = \sum_{k=0}^\infty m^k e^{\alpha t} P(V_k \in dt)$$ is its renewal measure since $\nu(dt) = \sum_{n=0}^\infty \eta^{*n}(dt)$. Since $m\mu > 0$, then $(|f|*\nu)(t) < \infty$ for all $t$ whenever $f$ is directly Riemann integrable. From we know that $g \in L^1$, so by Lemma 9.2 from [@Goldie_91], $\breve{g}$ is directly Riemann integrable, resulting in $(|\breve{g}|*\nu)(t) < \infty$ for all $t$. Thus $(|\breve{g}|*\nu)(t) = E\left[ \sum_{k=0}^\infty m^k e^{\alpha V_k} | \breve{g}(t - V_k) | \right] < \infty$. By Fubini’s Theorem, $E\left[ \sum_{k=0}^\infty m^k e^{\alpha V_k} \breve{g}(t - V_k) \right] $ exists and $$(\breve{g}*\nu)(t) = E\left[ \sum_{k=0}^\infty m^k e^{\alpha V_k} \breve{g}(t - V_k) \right] = \sum_{k=0}^\infty E\left[ m^k e^{\alpha V_k} \breve{g}(t - V_k) \right] = \lim_{n\to \infty} (\breve{g}*\nu_n)(t).$$ Now, by assumption, we can choose $\beta$ in the definition of the smoothing operator such that $0 < \beta < \alpha$ and $m E[C^\beta] < 1$. We show below that for such $\beta$ we have $\breve{\delta}_n(t) \to 0$ as $n \to \infty$ for all fixed $t$, since $$\begin{aligned}
\breve{\delta}_n(t) &= \int_{-\infty}^t e^{-\beta(t-u)} m^n P(e^{\beta V_n} R^\beta > e^{\beta u}) \, du \\
&= \frac{e^{-\beta t} m^n}{\beta} \int_0^{e^{\beta t}} P( e^{\beta V_n} R^\beta > v) \, dv \ \\
&\leq \frac{e^{-\beta t}}{\beta} E[R^\beta] (m E[C^\beta])^n \to 0 \end{aligned}$$ as $n \to \infty$. Hence, the preceding arguments allow us to pass $n \to \infty$ in , and obtain $$\breve{r}(t) = (\breve{g}*\nu)(t).$$
Now, by the key renewal theorem for two-sided random walks in [@Ath_McD_Ney_78], $$e^{-\beta t} \int_{0}^{e^t} v^{\alpha+\beta-1} P(R > v) \, dv = \breve{r}(t) \to \frac{1}{m\mu} \int_{-\infty}^\infty \breve{g}(u) \, du \triangleq \frac{H}{\beta}, \qquad t \to \infty.$$ Clearly, $H \geq 0$ since the left-hand side of the preceding equation is positive, and thus, by Lemma \[L.Derivative\] $$P(R > t) \sim H t^{-\alpha}, \qquad t \to \infty.$$ Finally, $$\begin{aligned}
H &= \frac{\beta}{m\mu} \int_{-\infty}^\infty \int_{-\infty}^u e^{-\beta (u-t)} g(t) \, dt \, du \\
&= \frac{1}{m\mu} \int_{-\infty}^\infty g(t) \, dt \\
&= \frac{1}{m\mu} \int_{0}^\infty v^{\alpha -1} (P(R > v) - mP(C R > v)) \, dv.\end{aligned}$$
We end this section with the proof of Lemma \[L.Max\_Approx\].
That the integral is positive follows from the union bound. That $$\int_0^\infty \left( E[N] P(CR > t) - P\left( \max_{1\leq i\leq N} C_i R_i > t \right) \right) t^{\alpha-1} dt = \frac{1}{\alpha} E\left[ \sum_{i=1}^N (C_iR_i)^\alpha - \left( \max_{1\leq i\leq N} C_iR_i \right)^\alpha \right]$$ follows from similar arguments to those used to derive the alternative expression for $H$ in the proof of Theorem \[T.GoldieApplication\]. The rest of the proof shows that the integral is finite.
Clearly $$\begin{aligned}
\int_{0}^{1} \left( E[N] P(C R > t) - P\left( \max_{1\leq i \leq N} C_i R_i > t \right) \right) t^{\alpha -1} \, dt &\leq E[N] \int_0^{1} t^{\alpha-1} \, dt < \infty.\end{aligned}$$ Hence, it remains to prove that the remaining part of the integral $\left( \int_{1}^\infty \cdots dt \right)$ is finite. To do this, we start by letting $Y = CR$ and $F(y) = P(Y \leq y)$. Then $$\begin{aligned}
E[N] P(CR > t) - P\left( \max_{1\leq i \leq N} C_i R_i > t \right)&= \sum_{k=1}^\infty \left( F(t)^k -1 + k \overline{F}(t) \right) P(N = k) \\
&= E\left[ (1-\overline{F}(t))^N - 1 + N \overline{F}(t) \right] .\end{aligned}$$ Use the inequality $1 - x \leq e^{-x}$ for $x \geq 0$ to obtain $$E\left[ (1-\overline{F}(t))^N - 1 + N \overline{F}(t) \right] \leq E\left[ e^{-\overline{F}(t) N} - 1 + N \overline{F}(t) \right].$$ Choose $0 < \delta < \alpha\epsilon/(1+\epsilon)$ (recall that $0 < \epsilon < 1$) and let $\beta = \alpha-\delta$. By Markov’s inequality and Lemma \[L.Stability\] $$\overline{F}(t) \leq t^{-\beta} E[ Y^\beta] = t^{-\beta} E[R^\beta] E[C^\beta] \triangleq c t^{-\beta} < \infty$$ for any $t > 0$. Note that the function $h(x) = e^{-x} - 1 + x$ is increasing on $[0,\infty)$, so $h(N \overline{F}(t)) \leq h(cN t^{-\beta})$. Thus, by Fubini’s Theorem (the integrand is nonnegative), $$\int_{1}^\infty \left( E[N] P(C R > t) - P\left( \max_{1\leq i \leq N} C_i R_i > t \right) \right) t^{\alpha -1} \, dt \leq E\left[ \int_{1}^\infty \left( e^{-cN t^{-\beta}} - 1 + cN t^{-\beta} \right) t^{\alpha -1} \, dt \right].$$ Using the change of variables $u = c N t^{-\beta}$ gives $$\begin{aligned}
\int_{1}^\infty \left( e^{-cN t^{-\beta}} - 1 + c N t^{-\beta} \right) t^{\alpha -1} \, dt &= \frac{(cN)^{\alpha/\beta}}{\beta} \int_0^{cN} \left( e^{-u} - 1 + u \right) u^{-\alpha/\beta -1} \, du \\
&\leq \frac{(cN)^{\alpha/\beta}}{\beta} \int_0^\infty \left( e^{-u} - 1 + u \right) u^{-\alpha/\beta -1} \, du.\end{aligned}$$ Our choice of $\beta = \alpha-\delta$ guarantees that $1 < \alpha/\beta < 1+\epsilon$, so $E[(cN)^{\alpha/\beta}] < \infty$. It only remains to show that the last (non-random) integral is finite. To see this note that $e^{-x} -1 + x \leq x^2/2$ and $e^{-x} - 1 \leq 0$ for any $x \geq 0$, so $$\begin{aligned}
\int_0^\infty \left( e^{-u} - 1 + u \right) u^{-\alpha/\beta -1} \, du &\leq \frac{1}{2} \int_0^1 u^{1-\alpha/\beta} \, du + \int_1^\infty u^{-\alpha/\beta } \, du \\
&= \frac{1}{2(2-\alpha/\beta)} + \frac{1}{\alpha/\beta-1} < \infty.\end{aligned}$$ This completes the proof.
The case when $N$ dominates {#SS.NDominates_Proofs}
---------------------------
This section contains the proofs of Lemma \[L.Finite\_n\] and Proposition \[P.UniformBound\]; the proof of Lemma \[L.W\_n\_Finite\_n\] is omitted since it is basically the same as that of Lemma \[L.Finite\_n\]. We also present in Lemma \[L.TruncBound\] a result for sums of iid truncated random variables that may be of independent interest in the context of heavy-tailed asymptotics, since it provides bounds that do not depend on the distribution of the summands. Most of the work involved in the proof of Proposition \[P.UniformBound\] goes into obtaining a bound for one iteration of the recursion satisfied by $W_n$, and for the convenience of the reader it is presented separately in Lemma \[L.Bound1Iter\].
We proceed by induction in $n$. For $n = 1$ fix $\alpha/(\alpha+\epsilon) < \delta < 1$ and note that $$\begin{aligned}
P(R^{(1)} > x) &= P\left( \sum_{i=1}^{N^{(0)}} C_i^{(1)} R_{i}^{(0)} + Q^{(0)} > x \right) \\
&= P\left( \sum_{i=1}^N C_i Q_i > x - Q, \, Q \leq x^\delta \right) + P\left( Q > x^\delta \right) \\
&\sim P\left( \sum_{i=1}^N C_i Q_i > x \right) + O\left( x^{-\delta(\alpha+\epsilon)} \right) \\
&\sim P( N > x/ E[CQ] ) + o(P(N > x)) \\
&\sim (E[C] E[Q])^\alpha P(N > x),\end{aligned}$$ where $N, \{C_i\},$ and $Q$ are independent and the fourth step is justified by Lemma 3.7(2) from [@Jess_Miko_06]. Now suppose that we have $$P(R^{(n)} > x) \sim \frac{(E[C]E[Q])^\alpha}{(1-\rho)^\alpha} \sum_{k=0}^n \rho_\alpha^k (1-\rho^{n-k})^\alpha P(N > x).$$ Note that since $E[C^{\alpha + \epsilon}] < \infty$, then by Lemma 4.2 from [@Jess_Miko_06], for $C$ independent of $R^{(n)}$, $$P(C R^{(n)} > x) \sim E[C^\alpha] P(R^{(n)} > x).$$ Let $c^{-1} = E[C^\alpha] (E[C]E[Q])^\alpha (1-\rho)^{-\alpha} \sum_{k=0}^n \rho_\alpha^k (1-\rho^{n-k})^\alpha$, then $$P(N > x) \sim c P(C R^{(n)} > x),$$ and by Lemma 3.7(5) from [@Jess_Miko_06] we have $$\begin{aligned}
P(R^{(n+1)} > x) &= P\left( \sum_{i=1}^{N^{(0)}} C_i^{(1)} R_{i}^{(n)} + Q^{(0)} > x \right) \\
&\sim P\left( \sum_{i=1}^{N^{(0)}} C_i^{(1)} R_{i}^{(n)} > x \right) \\
&\sim (E[N] + c (E[C R^{(n)}])^\alpha) P( C R^{(n)} > x) \\
&\sim (E[N] + c (E[C R^{(n)}])^\alpha) c^{-1} P( N > x) .\end{aligned}$$ Next, observing that $E[R^{(n)}] = \sum_{i=0}^n E[W_i] = E[Q] \sum_{i=0}^n \rho^i = E[Q] (1-\rho^{n+1})/(1-\rho)$, we obtain $$\begin{aligned}
(E[N] + c (E[C R^{(n)}])^\alpha) c^{-1} &= \left(\rho_\alpha+ \frac{ E[R^{(n)}]^\alpha (1-\rho)^\alpha}{ E[Q]^\alpha \sum_{k=0}^n \rho_\alpha^k (1-\rho^{n-k})^\alpha} \right) \frac{(E[C]E[Q])^\alpha}{(1-\rho)^\alpha} \sum_{k=0}^n \rho_\alpha^k (1-\rho^{n-k})^\alpha \\
&= \left(\rho_\alpha \sum_{k=0}^n \rho_\alpha^k (1-\rho^{n-k})^\alpha + (1-\rho^{n+1})^\alpha \right) \frac{(E[C]E[Q])^\alpha}{(1-\rho)^\alpha} \\
&= \frac{(E[C]E[Q])^\alpha}{(1-\rho)^\alpha} \sum_{k=0}^{n+1} \rho_\alpha^k (1-\rho^{n+1-k})^\alpha.\end{aligned}$$ This completes the proof.
Lemma \[L.TruncBound\] below is based on traditional heavy-tailed techniques based on Chernoff’s inequality for truncated random variables, such as those used in [@Nag82] and [@Bor00], to name some references. The reason why we cannot simply use existing results is our need to guarantee that the bounds do not depend on the distribution of the summands, which will be key when we apply them to $W_n$. Hence special care goes into accounting for the constants explicitly. The corollary that we obtain from this lemma will be used in the proof of Lemma \[L.Bound1Iter\].
\[L.TruncBound\] Suppose that $Y_1, Y_2, \dots$ are nonnegative iid random variables with the same distribution as $Y$, where $E[Y^\beta] < \infty$ for some $\beta > 0$. Fix $0 < \epsilon < 1$. Then,
1. for $0 < \beta < 1$, $1 \leq k \leq x^\beta/ E[Y^\beta]$, and $x \geq e^{(Ke)^{1/(1-\beta)}}$, $$P\left( \sum_{i=1}^k Y_i > x, \, \max_{1\leq i \leq k} Y_i \leq x/\log x \right) \leq e^{-(1-\beta)(\log x)(\log\log x) \left( 1 - \frac{\log (eK)}{(1-\beta)\log\log x} \right) },$$
2. for $\beta > 1$, $1 \leq k \leq (1-\epsilon) x/(E[Y] \vee E[Y^\beta])$, and $x \geq e \vee (Ke/\epsilon)^{2/(\beta-1)}$, $$P\left( \sum_{i=1}^k Y_i > x, \, \max_{1\leq i \leq k} Y_i \leq x/\log x \right) \leq e^{-\epsilon (\beta-1) (\log x)^2 \left( 1 - \frac{\log\log x}{\log x} - \frac{\log(Ke/\epsilon)}{(\beta-1)\log x} \right) + e^5 (\beta-1)^2},$$
where $K = K(\beta) > 1$ is a constant that does not depend on $\epsilon$, $k$ or the distribution of $Y$.
Let $F(t) = P(Y \leq t)$, set $y = x/\log x$ and note that $$\begin{aligned}
P\left( \sum_{i=1}^k Y_i > x, \max_{1\leq i \leq k} Y_i \leq y \right) = P\left( \sum_{i=1}^k Y_i^{(y)} > x \right) F(y)^k ,\end{aligned}$$ where $P( Y^{(y)} \leq t) = F(t \wedge y)/F(y)$. Fix $\theta \geq 1/y$ and use the standard Chernoff’s bound method for truncated heavy tailed sums (see, e.g. [@Nag82; @Bor00]) to obtain $$\begin{aligned}
P\left( \sum_{i=1}^k Y_i^{(y)} > x \right) &\leq e^{-\theta x} E\left[ e^{\theta Y_1^{(y)}} \right]^k = e^{-\theta x} E \left[ e^{\theta Y} 1_{(Y \leq y)} \right]^k F(y)^{-k}.\end{aligned}$$ From where it follows that $$P\left( \sum_{i=1}^k Y_i^{(y)} > x \right) F(y)^k \leq e^{-\theta x} E \left[ e^{\theta Y} 1_{(Y \leq y)} \right]^k.$$ To analyze the preceding truncated exponential moment suppose first that $\beta > 1$. Then, by using the identity $$\label{eq:EtaMoment}
E[Y^\eta] = \int_0^\infty \eta t^{\eta-1} \overline{F}(t) dt$$ we obtain $$\begin{aligned}
E \left[ e^{\theta Y} 1_{(Y \leq y)} \right] &= \overline{F}(0) - e^{\theta y} \overline{F}(y) + \theta \int_0^y e^{\theta t} \overline{F}(t) \, dt \notag \\
&\leq 1 + \theta \int_0^{1/\theta} \overline{F}(t) \, dt + \theta \int_0^{1/\theta} (e^{\theta t} - 1) \overline{F}(t) \, dt + \theta \int_{1/\theta}^y e^{\theta t} \overline{F}(t) \, dt \notag \\
&\leq 1 + \theta E[Y] + e \theta^2 \int_0^{1/\theta} t \overline{F}(t) \, dt + \theta \int_{1/\theta}^y e^{\theta t} \overline{F}(t) \, dt \notag \\
&\leq 1 + \theta E[Y] + \frac{e \theta^{ 2\wedge \beta}}{2 \wedge \beta} E[Y^{2\wedge \beta}] + \theta \int_{1/\theta}^y e^{\theta t} \overline{F}(t) \, dt , \label{eq:SecondOrd}\end{aligned}$$ where in the second inequality we use $e^x - 1 \leq x e^x$, $x \geq 0$, and in the last inequality we use $t^{2 - (2\wedge \beta)} \leq \theta^{-2 + (2\wedge \beta)}$ and with $\eta = 2 \wedge \beta$. Similarly, if $0 < \beta \leq 1$, then $$\begin{aligned}
E \left[ e^{\theta Y} 1_{(Y \leq y)} \right] &\leq 1 + \theta \int_0^{1/\theta} e^{\theta t} \overline{F}(t) \, dt + \theta \int_{1/\theta}^y e^{\theta t} \overline{F}(t) \, dt \notag \\
&\leq 1 + \frac{e \theta^{\beta}}{\beta} E[Y^\beta] + \theta \int_{1/\theta}^y e^{\theta t} \overline{F}(t) \, dt . \label{eq:FirstOrd}\end{aligned}$$ Next, by Markov’s inequality we have $$\overline{F}(t) \leq E[Y^\beta] t^{-\beta},$$ which, in combination with and , gives $$\label{eq:Cases}
E \left[ e^{\theta Y} 1_{(Y \leq y)} \right] \leq \begin{cases}
1 + \theta E[Y] + \frac{e \theta^{2}}{2} E[Y^{2}] + E[Y^\beta] \theta \int_{1/\theta}^y e^{\theta t} t^{-\beta} dt , & \beta > 2, \\
1 + \theta E[Y] + \frac{e \theta^\beta}{\beta} E[Y^\beta] + E[Y^\beta] \theta \int_{1/\theta}^y e^{\theta t} t^{-\beta} dt , & 1 < \beta \leq 2, \\
1 + \frac{e \theta^\beta}{\beta} E[Y^\beta] + E[Y^\beta] \theta \int_{1/\theta}^y e^{\theta t} t^{-\beta} dt, & 0 < \beta \leq 1.
\end{cases}$$ To analyze the remaining integral we split it as follows, for $\beta > 0$, $$\begin{aligned}
\theta \int_{1/\theta}^y e^{\theta t} t^{-\beta} dt &\leq \theta^{1+\beta} \int_{1/\theta}^{y/2} e^{\theta t} dt + \theta \int_{y/2}^y e^{\theta t} t^{-\beta} \, dt \\
&\leq \theta^\beta e^{\theta y/2} + \theta y^{1-\beta} \int_{1/2}^1 e^{\theta y u} u^{-\beta} \, du \\
&\leq \theta^\beta e^{\theta y/2} + \theta y^{1-\beta} 2^{\beta} \int_{1/2}^1 e^{\theta y u} \, du \\
&\leq \theta^\beta e^{\theta y/2} + 2^{\beta} e^{\theta y} y^{-\beta} ,\end{aligned}$$ from where it follows that $$\begin{aligned}
&2e \theta^\beta E[Y^\beta] + E[Y^\beta] \theta \int_{1/\theta}^y e^{\theta t} t^{-\beta} dt \\
&\leq 2e \theta^\beta E[Y^\beta] + E[Y^\beta] \theta^\beta e^{\theta y/2} + E[Y^\beta] 2^{\beta} e^{\theta y} y^{-\beta} \\
&\leq 2^{\beta} E[Y^\beta] e^{\theta y} y^{-\beta} \left( 1 + e 2^{1-\beta} (\theta y)^\beta e^{-\theta y} + 2^{-\beta} (\theta y)^\beta e^{-\theta y/2} \right) \\
&\leq 2^{\beta} E[Y^\beta] e^{\theta y} y^{-\beta} \left( 1 + 2e \sup_{t \geq 1} t^\beta e^{-t} + \sup_{t \geq 1/2} t^{\beta} e^{-t} \right).\end{aligned}$$ Hence, we have shown that $$2e \theta^\beta E[Y^\beta] + E[Y^\beta] \theta \int_{1/\theta}^y e^{\theta t} t^{-\beta} dt \leq K E[Y^\beta] e^{\theta y} y^{-\beta},$$ where $K = 2^\beta \left(1 + (2e+1) \sup_{t \geq 1/2} t^\beta e^{-t} \right)$ does not depend on $\theta$ or the distribution of $Y$. Replacing the preceding inequality in and using $1+t \leq e^t$ give, $$\label{eq:TripleBound}
e^{-\theta x} E\left[ e^{\theta Y} 1_{(Y \leq y)} \right]^k \leq \begin{cases}
e^{-\theta (x - k E[Y]) + e k \theta^2 E[Y^2] + K k E[Y^\beta] e^{\theta y} y^{-\beta} } , & \beta > 2, \\
e^{-\theta (x - k E[Y]) + K k E[Y^\beta] e^{\theta y} y^{-\beta} } , & 1 < \beta \leq 2, \\
e^{-\theta x + K k E[Y^\beta] e^{\theta y} y^{-\beta}}, & 0 < \beta \leq 1.
\end{cases}$$ Now, to complete the proof, we optimize the choice of $\theta$ in the preceding bounds. For $0 < \beta < 1$, choose $\theta = \frac{1}{y} \log \left( \frac{x}{K k E[Y^\beta] y^{1-\beta}} \right)$ and note that for all $1 \leq k \leq x^\beta/E[Y^\beta]$ and $x \geq e^{(Ke)^{1/(1-\beta)}}$, $$\theta y \geq \log\left( \frac{(\log x)^{1-\beta}}{K} \right) \geq 1.$$ Then, $$\begin{aligned}
e^{-\theta x + K k E[Y^\beta] e^{\theta y} y^{-\beta}} &= e^{-(\log x) \log \left( \frac{ x^\beta (\log x)^{1-\beta}}{K e k E[Y^\beta]} \right) } \\
&\leq e^{-(\log x) \log \left( \frac{(\log x)^{1-\beta}}{K e} \right) } \\
&= e^{-(1-\beta)(\log x)(\log\log x) \left( 1 - \frac{\log (eK)}{(1-\beta)\log\log x} \right) } .\end{aligned}$$
Now, for $\beta > 1$, set $\theta = \frac{1}{y} \log\left( \frac{(x-kE[Y]) y^{\beta-1}}{K x} \right)$ and note that for and $x \geq e \vee (Ke/\epsilon)^{2/(\beta-1)}$, $$\theta y \geq \log\left( \frac{\epsilon y^{\beta-1}}{K } \right) \geq \log\left( \frac{\epsilon x^{(\beta-1)/2}}{K } \right) \geq 1.$$ Then, for $1 < \beta \leq 2$ and all $1 \leq k \leq (1-\epsilon) x/(E[Y] \vee E[Y^\beta])$, $$\begin{aligned}
e^{-\theta (x-kE[Y]) + K k E[Y^\beta] e^{\theta y} y^{-\beta}} &= e^{- \frac{(x-kE[Y])}{y} \log\left( \frac{(x-kE[Y]) y^{\beta-1}}{Kx} \right) + k E[Y^\beta] \frac{(x-kE[Y])}{xy} }\\
&\leq e^{- \frac{(x-kE[Y])}{y} \log\left( \frac{\epsilon y^{\beta-1}}{Ke } \right) } \\
&\leq e^{-\epsilon (\beta-1) (\log x)^2 \left( 1 - \frac{\log\log x}{\log x} - \frac{\log(Ke/\epsilon)}{(\beta-1)\log x} \right) } .\end{aligned}$$ In addition, for $\beta > 2$ note that $$\sup_{x \geq e} e k \theta^2 E[Y^2] \leq \sup_{x\geq e} \frac{e x}{y^2} \left( \log\left( \frac{ y^{\beta-1}}{K} \right) \right)^2 \leq \sup_{x \geq e} \frac{e (\beta-1)^2 (\log x)^4}{x} \leq e^5 (\beta-1)^2.$$ Finally, by combining the preceding two bounds with the first two inequalities in , we derive $$\begin{aligned}
P\left( \sum_{i=1}^k Y_i > x, \, \max_{1\leq i \leq k} Y_i \leq y \right) &\leq e^{-\epsilon (\beta-1) (\log x)^2 \left( 1 - \frac{\log\log x}{\log x} - \frac{\log(Ke/\epsilon)}{(\beta-1)\log x} \right) + e^5 (\beta-1)^2}\end{aligned}$$ for any $\beta > 1$.
As an immediate corollary to the preceding lemma we obtain:
\[C.SimpleTrunc\] Suppose that $Y_1, Y_2, \dots$ are nonnegative iid random variables with the same distribution as $Y$, where $E[Y^\beta] < \infty$ for some $\beta > 0$. Then, for any $\kappa > 0$ there exists a constant $x_0 > 0$ that does not depend on the distribution of $Y$ such that $$\sup_{1 \leq k \leq m_\beta(x)} P\left( \sum_{i=1}^k Y_i > x, \, \max_{1\leq i \leq k} Y_i \leq x/\log x \right) \leq x^{-\kappa}$$ for all $x \geq x_0$, where $$m_\beta(x) = \begin{cases}
\frac{x^\beta}{E[Y^\beta]}, & 0 < \beta < 1, \\
\frac{(1-\epsilon) x}{E[Y] \vee E[Y^\beta]}, & \beta > 1, 0 < \epsilon < 1.
\end{cases}$$
Lemma \[L.Bound1Iter\] below gives a bound for the distribution of $W_{n+1}$ in terms of that of $W_{n}$. This lemma can also be used to prove the corresponding uniform bound for $W_n$ in the case when $Q$ dominates recursion . In the statement of the lemma we assume that $1/L(x)$ is locally bounded on $[1, \infty)$.
\[L.Bound1Iter\] Suppose that $P(N > x) \leq x^{-\alpha} L(x)$, with $\alpha > 1$ and $L(\cdot)$ slowly varying, and $E[N] \max\{E[C^{\alpha}], E[C]\} < \eta < 1$. Then, for any $c > 0$, $0 < \epsilon < 1$, and $0 < \delta < 1 \wedge (\alpha-1)/2$, there exist constants $K = K(\delta,\epsilon,c,\eta) > 0$ and $x_0 = x_0(\delta,\epsilon,c,\eta) > 0$, that do not depend on $n$, such that for all $1 \leq n \leq c\log x/|\log\eta|$ and all $x \geq x_0$, $$P(W_{n+1} > x) \leq K \eta^{(2\wedge (\alpha-\delta)) n} x^{-\alpha} L(x) + E[N] P(C W_{n} > (1-\epsilon) x) ,$$ where $C$ and $W_n$ are independent.
[Remark:]{} Note that the condition $E[N] \max\{E[C^\alpha], E[C]\} < 1$ is natural since it is needed for the finiteness of $E[R^\beta]$ for any $\beta < \alpha$. It is also in agreement with Lemma \[L.Finite\_n\] in the sense that it is a necessary condition for the convergence (as $n \to \infty$) of the sum appearing in . The choice of $\eta$ is also suggested by the fact that for $\beta < \alpha$ one can obtain a weaker uniform bound by applying the moment estimate on $E[W_n^\beta]$ from Lemma \[L.GeneralMoment\], i.e., $P(W_n > x) \leq E[W_n^\beta] x^{-\beta} \leq K_\beta (E[N]\max\{ E[C], E[C^\beta] \})^n x^{-\beta}$.
Before going into the proof, we would like to emphasize that special care goes into making sure that $K$ and $x_0$ in the statement of the lemma do not depend on $n$. This is important since Lemma \[L.Bound1Iter\] will be applied iteratively in the proof of Proposition \[P.UniformBound\], where one does not want $K$ and $x_0$ to grow from one iteration to the next.
By convexity of $f(\theta) = E[C^\theta]$, $\max\{E[C^\alpha], E[C]\} \geq \max\{ E[C^{\alpha-\delta}], E[C] \}$, implying $$\varepsilon \triangleq \frac{\eta}{E[N] \max\{E[C^{\alpha-\delta}], E[C]\}} - 1 > 0.$$ Next, recall that $W_{n+1} \stackrel{\mathcal{D}}{=} \sum_{i=1}^N C_i W_{n,i}$ where $W_{n,i}$ are iid copies of $W_n$, let $Y \stackrel{\mathcal{D}}{=} Y_i = C_i W_{n,i}$ and $\beta = \alpha- \delta > 1$. Note that by Lemma \[L.GeneralMoment\] there exists a constant $K_1 > 0$ (that does not depend on $n$) such that, $$\begin{aligned}
E[Y^\beta] &= E[C^\beta] E[W_n^\beta] \notag \\
&\leq K_1 (E[N] \max\{E[C^{\alpha-\delta}], E[C]\})^n \notag \\
&= K_1 (1+\varepsilon)^{-n} \eta^n, \label{eq:NewMomentBound}\end{aligned}$$ where the last equality comes from the definition of $\varepsilon$. And since $E[Y] = E[Q] (E[N] E[C])^n$ $\leq E[Q] (E[N] \max\{ E[C^{\alpha-\delta}], E[C]\})^n$, then $$\label{eq:OtherMomentBound}
E[Y^\beta] \vee E[Y] \leq K_2 (1+\varepsilon)^{-n} \eta^{n}$$ for some constant $K_2 > 0$ that does not depend on $n$. With the intent of applying Corollary \[C.SimpleTrunc\], we define $$y \triangleq \epsilon x \qquad \text{and} \qquad m_\beta(x) \triangleq \lfloor \epsilon^2 x/(E[Y^\beta] \vee E[Y]) \rfloor.$$ Let $M_k^{(i)}$ is the $i$th order statistic of $\{Y_1, \dots, Y_k \}$, with $M_k^{(k)}$ being the largest. Then, $$\begin{aligned}
P\left( W_{n+1} > x \right) &= P\left( \sum_{i=1}^N Y_i > x \right) \notag \\
&\leq P\left( \sum_{i=1}^N Y_i > x, \, N \leq m_\beta(x) \right) + P\left( N > m_\beta(x) \right) \notag \\
&\leq P\left( \sum_{i=1}^N Y_i > x, \, M_N^{(N)} \leq (1-\epsilon) x, \, N \leq m_\beta(x) \right) \notag \\
&\hspace{5mm} + P\left( M_N^{(N)} > (1-\epsilon) x, \, N \leq m_\beta(x) \right) + P\left( N > m_\beta(x) \right) \notag \\
&\leq P\left( \sum_{i=1}^N Y_i > x, \, M_N^{(N)} \leq (1-\epsilon) x, \, M_N^{(N-1)} \leq y/\log y, \, N \leq m_\beta(x) \right) \label{eq:FirstIneq} \\
&\hspace{5mm} + P\left( M_N^{(N-1)} > y /\log y, \, N \leq m_\beta(x) \right) \label{eq:SecondIneq} \\
&\hspace{5mm} + P\left( M_N^{(N)} > (1-\epsilon) x, \, N \leq m_\beta(x) \right) + P\left(N > m_\beta(x)\right) . \label{eq:ThirdIneq}\end{aligned}$$ Note that the term in can be bounded as follows $$\begin{aligned}
&P\left( \sum_{i=1}^N Y_i > x, \, M_N^{(N)} \leq (1-\epsilon) x, \, M_N^{(N-1)} \leq y/\log y, \, N \leq m_\beta(x) \right) \\
&\leq P\left( \sum_{i=1}^N Y_i - M_N^{(N)} > y, \, M_N^{(N-1)} \leq y/\log y, \, N \leq m_\beta(x) \right) \\
&\leq P\left( \sum_{i=1}^{N} Y_i > y, \, M_{N}^{(N)} \leq y/\log y, \, N \leq m_\beta(x) \right) \\
&\leq P\left( \sum_{i=1}^{m_\beta(x)} Y_i > y, \, \max_{1\leq i < m_\beta(x)} Y_i \leq y/\log y \right).\end{aligned}$$ Fix $\nu = \alpha + \delta + c(\alpha-\delta)$, then, by Corollary \[C.SimpleTrunc\], there exists a constant $x_1 \geq e$, that does not depend on the distribution of $Y$ (and therefore, does not depend on $n$), such that $$\begin{aligned}
P\left( \sum_{i=1}^{m_\beta(x)} Y_i > y, \, \max_{1\leq i < m_\beta(x)} Y_i \leq y/\log y \right) &\leq y^{-\nu} = \epsilon^{-\nu} \eta^{\frac{c(\alpha-\delta)}{|\log\eta|} \cdot \log x} x^{-\alpha-\delta} \\
&\leq \epsilon^{-\nu} \eta^{(\alpha-\delta) n} x^{-\alpha-\delta} = \epsilon^{-\nu} \frac{x^{-\delta}}{L(x)} \,\eta^{\beta n} x^{-\alpha} L(x) \\
&\leq \epsilon^{-\nu} \sup_{t \geq 1} \frac{t^{-\delta}}{L(t)} \, \eta^{\beta n} x^{-\alpha} L(x)\end{aligned}$$ for all $y \geq x_1$, where the second inequality follows from the assumption $n \leq c\log x/|\log\eta|$, and in the second equality we use the definition $\beta = \alpha-\delta$. To bound , we condition on $N$, $$\begin{aligned}
P\left( M_N^{(N-1)} > y /\log y, \, N \leq m_\beta(x) \right) &= \sum_{k=1}^{m_\beta(x)} P\left( M_k^{(k-1)} > y /\log y \right) P(N = k) \\
&\leq \sum_{k=1}^{m_\beta(x)} \binom{k}{2} P(Y > y/\log y)^2 P(N = k) \\
&\leq E\left[ N^2 1_{(N \leq m_\beta(x))} \right] P(Y > y/\log y)^2 \\
&\leq E\left[ N^{2 \wedge \beta} \right] m_\beta(x)^{(2-\beta)^+} P(Y > y/\log y)^2 ,\end{aligned}$$ where in the last inequality we use $N \leq m_\beta(x)$ in case $N$ does not have a second moment. Now, by Markov’s inequality and the definition of $m_\beta(x)$, $$\begin{aligned}
m_\beta(x)^{(2-\beta)^+} P(Y > y/\log y)^2 &\leq m_\beta(x)^{(2-\beta)^+} \left( \frac{E[Y^{\beta}] (\log y)^\beta}{y^\beta} \right)^2 \\
&\leq \left( \frac{ E[Y^\beta] }{E[Y^\beta] \vee E[Y]} \right)^{(2-\beta)^+} \frac{\epsilon^{(2-\beta)^+} E[Y^\beta]^{2 \wedge \beta} (\log y)^{2\beta} }{y^{2\beta \wedge (3\beta-2)}} \\
&\leq \frac{\epsilon^{(2-\beta)^+} E[Y^\beta]^{2 \wedge \beta} (\log y)^{2\beta} }{y^{2\beta \wedge (3\beta-2)}} \\
&\leq \frac{\epsilon^{(2-\beta)^+} (K_1 (1+\varepsilon)^{-n} \eta^{n})^{2 \wedge \beta} (\log y)^{2\beta} }{y^{2\beta \wedge (3\beta-2)}} \qquad \text{(by \eqref{eq:NewMomentBound})}.\end{aligned}$$ Our choice of $\delta$ guarantees that $2\beta \wedge (3\beta -2) > \alpha + \delta$ and $\beta = \alpha-\delta > 1$, and therefore, $$\begin{aligned}
P\left( M_N^{(N-1)} > y /\log y, \, N \leq m_\beta(x) \right) &\leq K_3 \, \frac{\eta^{(2\wedge\beta)n}}{(1+\varepsilon)^{(2\wedge \beta)n}} x^{-\alpha-\delta} \\
&\leq K_3 \, \frac{x^{-\delta}}{L(x)} \eta^{(2\wedge\beta)n} x^{-\alpha} L(x) \\
&\leq K_3 \sup_{t \geq 1} \frac{t^{-\delta}}{L(t)} \, \eta^{(2\wedge\beta)n} x^{-\alpha} L(x)\end{aligned}$$ for all $x \geq x_2 = \epsilon^{-1} e$, where $$K_3 = K_3(\epsilon,\delta) = E\left[ N^{2 \wedge \beta} \right] \epsilon^{(2-\beta)^+-\alpha-\delta} K_1^{2\wedge \beta} \, \sup_{t \geq e} \frac{(\log t)^{2\beta} }{t^{2\beta \wedge (3\beta-2) -\alpha-\delta}}.$$ To bound the second term in , we first note that by Potter’s Theorem (see Theorem 1.5.6 (ii) on p. 25 in [@BiGoTe1987]), there exists a constant $x_3 = x_3(\varepsilon, \delta)$ such that for all $x \geq x_3$ $$\begin{aligned}
P(N > m_\beta(x)) &\leq \frac{(m_\beta(x))^{-\alpha} L(m_\beta(x))}{x^{-\alpha} L(x)} \cdot x^{-\alpha} L(x) \\
&\leq (1+\varepsilon) \max\left\{ \left( \frac{m_\beta(x)}{x} \right)^{-\alpha+\delta}, \, \left( \frac{m_\beta(x)}{x} \right)^{-\alpha-\delta} \right\} x^{-\alpha} L(x) \\
&= (1+\varepsilon) \max\left\{ \left( \frac{E[Y^\beta] \vee E[Y]}{\epsilon^2} \right)^{\alpha-\delta}, \, \left( \frac{E[Y^\beta] \vee E[Y]}{\epsilon^2} \right)^{\alpha+\delta} \right\} x^{-\alpha} L(x) \\
&\leq \frac{(1+\varepsilon)}{\epsilon^{2(\alpha+\delta)}} (E[Y^\beta] \vee E[Y])^\beta x^{-\alpha} L(x) \\
&\leq \frac{K_2^\beta }{\epsilon^{2(\alpha+\delta)}} \cdot \frac{\eta^{\beta n}}{(1+\varepsilon)^{\beta n - 1}} \cdot x^{-\alpha} L(x) \qquad \text{(by \eqref{eq:OtherMomentBound})} \\
&\leq K_4 \eta^{\beta n} x^{-\alpha} L(x).\end{aligned}$$ Finally, for the first term in , $$\begin{aligned}
P\left( M_N^{(N)} > (1-\epsilon) x, \, N \leq m_\beta(x) \right) &\leq P\left( M_N^{(N)} > (1-\epsilon) x \right) \\
&\leq E[N] P(Y > (1-\epsilon) x).\end{aligned}$$ Combining the preceding bounds for - and setting $x_0 = \max\{x_1, x_2, x_3\}$ and $K = (\epsilon^{-\nu} + K_3) \sup_{t \geq 1} \frac{t^{-\delta}}{L(t)} + K_4$ completes the proof.
Finally, we give the proof of Proposition \[P.UniformBound\], the main technical contribution of Section \[S.NDominates\].
Note that it is enough to prove the proposition for all $x \geq x_0$ for some $x_0 = x_0(\eta,\nu) > 1$, since for all $1 \leq x \leq x_0$ and $n \geq 1$, $$\begin{aligned}
P(W_n > x) &= \frac{P(W_n > x)}{ \eta^n P(N > x)} \, \eta^n P(N > x)\\
&\leq \frac{E[Q] (E[N]E[C])^n x^{-1}}{\eta^n P(N > x)} \, \eta^n P(N > x) \qquad \text{(by Markov's inequality)} \\
&\leq \sup_{1 \leq t \leq x_0} \frac{E[Q]}{t P(N > t)} \, \cdot \eta^n P(N > x) .\end{aligned}$$ Next, choose $0 < \epsilon < 1$ such that $$\label{eq:EpsilonChoice}
E[N] E[C^\alpha] \left( (1-\epsilon)^{-\alpha-1} + 2\epsilon \right) \leq \eta,$$ define $c = \nu/2$, $$\gamma = \frac{1}{|\log\eta|} \log\left( \frac{\eta}{E[N] \max\{E[C^{\alpha}], E[C]\}} \right),$$ and select $0 < \delta < \min\{1, (\alpha-1)/2, c\gamma \}$. Now, by Lemma \[L.Bound1Iter\], there exist constants $K_1, x_1 > 0$ (that do not depend on $n$) such that $$P(W_{n+1} > x) \leq K_1 \eta^{(2\wedge (\alpha-\delta)) n} P(N > x) + E[N] P(C W_{n} > (1-\epsilon) x)$$ for all $x \geq x_1$. Hence, by defining $n_0 = (2 \wedge(\alpha-\delta)-1)^{-1} (\log\eta)^{-1} \log(\epsilon E[N]E[C^\alpha])$, we obtain $$\label{eq:oneIter}
P(W_{n+1} > x) \leq K_1 E[N]E[C^\alpha] \epsilon \eta^{n} P(N > x) + E[N] P(C W_{n} > (1-\epsilon) x)$$ for all $n \geq n_0$, and all $x \geq x_1$.
Next, in order to derive an explicit bound for $P(W_n > x)$, we need the following two estimates and . In this regard, choose $x_0 \geq 1 \vee x_1$ such that $$\label{eq:boundForCN}
P(C N > (1-\epsilon) x) \leq E[C^\alpha] (1-\epsilon)^{-\alpha-1} P(N > x)$$ for all $x \geq x_0$. This is possible since by Lemma 4.2 from [@Jess_Miko_06] $P(C N > (1-\epsilon) x) \sim E[C^\alpha] (1-\epsilon)^{-\alpha} P(N > x)$. Also, by Markov’s inequality, we have that for all $1\leq n \leq c\log x/|\log\eta|$, $$\begin{aligned}
P(C > (1-\epsilon) x/x_0) &\leq E[C^{\alpha+\nu}] (1-\epsilon)^{-\alpha-\nu} x_0^{\alpha+\nu} x^{-\alpha-\nu} \notag \\
&= \frac{E[C^{\alpha+\nu}] x_0^{\alpha+\nu} }{(1-\epsilon)^{\alpha+\nu} x^{\nu/2} L(x)} x^{-\nu/2} P(N > x) \notag \\
&\leq \frac{E[C^{\alpha+\nu}] x_0^{\alpha+\nu}}{(1-\epsilon)^{\alpha+\nu} x^{\nu/2}L(x)} \, \eta^n P(N > x) ,\label{eq:boundForC}\end{aligned}$$ where in the second inequality we use $x^{-\nu/2} = x^{-c} = \eta^{\frac{c\log x}{|\log\eta|}} \leq \eta^n$. Now, define $$K_2 = \max\left\{1, \, K_1, \, \sup_{x \geq x_0} \frac{E[C^{\alpha+\nu}] x_0^{\alpha+\nu}}{\epsilon E[C^\alpha] (1-\epsilon)^{\alpha+\nu} x^{\nu/2}L(x)} \right\}.$$
Now we proceed to derive bounds for $P(W_n > x)$ for different ranges of $n$. For all $1\leq n \leq n_0$ and all $x \geq x_0$, by Lemma \[L.Finite\_n\], there exists a constant $K_0 \geq K_2$ such that $$\label{eq:InductionHyp}
P(W_n > x) \leq K_0 \, \eta^n P(N > x).$$
Next, for the values $n_0 \leq n \leq c \log x/|\log \eta|$ we proceed by induction using . To this end, suppose holds for some $n$ in the specified range. Then, note that by and the induction hypothesis , we have for all $x \geq x_0$, $$\begin{aligned}
P(C W_n > (1-\epsilon) x) &\leq P(C W_n > (1-\epsilon) x, C \leq (1-\epsilon)x/x_0) + P(C > (1-\epsilon)x/x_0) \\
&\leq \int_0^{(1-\epsilon)x/x_0} P(W_n > (1-\epsilon) x/y) P(C \in dy) + K_2 E[C^\alpha] \epsilon \eta^n P(N > x) \\
&\leq K_0 \eta^n \int_0^{\infty} P(N > (1-\epsilon) x/y) P(C \in dy) + K_2 E[C^\alpha] \epsilon \eta^n P(N > x) \\
&= K_0 \eta^n P(CN > (1-\epsilon) x) + K_2 E[C^\alpha] \epsilon \eta^n P(N > x) \\
&\leq K_0 E[C^\alpha] \left( (1-\epsilon)^{-\alpha-1} + \epsilon \right) \eta^n P(N > x) ,\end{aligned}$$ where in the last inequality we used and $K_0 \geq K_2$. Then, by replacing the preceding bound in and using , we derive $$\begin{aligned}
P(W_{n+1} > x) &\leq K_0 E[N] E[C^\alpha] \left( (1-\epsilon)^{-\alpha-1} + 2\epsilon \right) \eta^n P(N > x) \\
&\leq K_0 \eta^{n+1} P(N > x)\end{aligned}$$ for all $x \geq x_0$ and all $1 \leq n \leq c\log x/|\log\eta|$.
Finally, for $n \geq c \log x/|\log\eta|$, we follow a different approach that comes from our moment estimates for $W_n$. Let $$\varepsilon = \frac{\eta}{E[N] \max\{E[C^{\alpha}], E[C]\}} - 1 > 0$$ and note that by convexity $$E[N] \max\{E[C^{\alpha-\delta}], E[C]\} \leq E[N] \max\{E[C^\alpha], E[C]\} = (1+ \varepsilon)^{-1} \eta.$$ Then, by Markov’s inequality and Lemma \[L.GeneralMoment\], we have $$\begin{aligned}
P(W_n > x) &\leq E[W_n^{\alpha-\delta}] x^{-\alpha+\delta} \notag \\
&\leq K_{\alpha-\delta} (E[N] \max\{E[C^{\alpha-\delta}], E[C]\})^n x^{-\alpha+\delta} \notag \\
&= K_{\alpha-\delta} (1+\varepsilon)^{- n} \eta^n x^{-\alpha+\delta} \notag \\
&\leq K_{\alpha-\delta} x^{- \log(1+\varepsilon) c /|\log\eta|} \eta^n x^{-\alpha+\delta} \label{eq:boundForW_n}\end{aligned}$$ for all $x > 0$. Note that the preceding bound, $$\begin{aligned}
\frac{\log (1+\varepsilon)}{|\log\eta|} &= \frac{1}{|\log\eta|} \log\left( \frac{\eta}{E[N] \max\{E[C^{\alpha}], E[C]\}} \right) = \gamma,\end{aligned}$$ and yield $$\begin{aligned}
P(W_n > x) &\leq K_{\alpha-\delta} \eta^n x^{-c\gamma -\alpha+\delta} \\
&\leq K_{\alpha-\delta} \eta^n x^{-\alpha+\delta -c\gamma } = K_{\alpha-\delta} \eta^n \frac{x^{\delta -c\gamma }}{L(x)} P(N > x) \\
&\leq K_{\alpha-\delta} \sup_{t \geq 1} \frac{t^{\delta-c\gamma}}{ L(t)} \, \eta^n P(N > x) \end{aligned}$$ for all $x \geq 1$; recall that $\delta < c\gamma$. Thus, setting $K = \max\{K_0, \, K_{\alpha-\delta} \sup_{t \geq 1} t^{\delta-c\gamma} (L(t))^{-1} \}$ completes the proof.
The case when $Q$ dominates {#SS.QDominate_Proofs}
---------------------------
We end the paper with the proof of Lemma \[L.Finite\_nQ\] (the proof of Lemma \[L.W\_n\_Finite\_nQ\] is basically the same) and a sketch of the proof of Proposition \[P.UniformBoundQ\]. As mentioned before, the proofs of the other results presented in Section \[S.QDominates\] have been omitted since they are very similar to those from Section \[S.NDominates\].
We proceed by induction in $n$. By Lemma 4.2 from [@Jess_Miko_06], $$P(C Q > x) \sim E[C^\alpha] P(Q > x),$$ by Lemma 3.7(1) from the same source, $$P\left( \sum_{i=1}^{N^{(0)}} C_i^{(1)} Q^{(1)}_i > x \right) \sim E[N] P(C Q > x) \sim E[N] E[C^\alpha] P(Q> x),$$ and by Lemma 3.1, again from the same source, we have $$\begin{aligned}
P(R^{(1)} > x) &= P\left( \sum_{i=1}^{N^{(0)}} C_i^{(1)} Q_i^{(1)} + Q^{(0)} > x \right) \\
&\sim P\left( \sum_{i=1}^{N^{(0)}} C_i^{(1)} Q_i^{(1)} > x \right) + P(Q > x) \\
&\sim (\rho_\alpha + 1) P(Q > x).\end{aligned}$$ Now suppose that we have $$P(R^{(n)} > x) \sim \sum_{k=0}^n \rho_\alpha^k \, P(Q > x).$$ Then, $$\begin{aligned}
P(R^{(n+1)} > x) &= P\left( \sum_{i=1}^{N^{(0)}} C_i^{(1)} R_{i}^{(n)} + Q^{(0)} > x \right) \\
&\sim P\left( \sum_{i=1}^{N^{(0)}} C_i^{(1)} R_{i}^{(n)} > x \right) + P(Q > x) \\
&\sim E[N] E[C^\alpha] P(R^{(n)} > x) + P(Q > x) \\
&\sim \left( \rho_\alpha \sum_{k=0}^n \rho_\alpha^k + 1 \right) P(Q > x) \\
&= \sum_{k=0}^{n+1} \rho_\alpha^k \, P(Q > x).\end{aligned}$$
By Markov’s inequality $$P(N > x) \leq E[N^{\alpha+\nu}] x^{-\alpha-\nu}$$ for all $x > 0$. Use Lemma \[L.Bound1Iter\] to obtain $$P(W_{n+1} > x) \leq K_1 E[N] E[C^\alpha] \epsilon \eta^n P(Q > x) + E[N] P(C W_n > (1-\epsilon) x)$$ for all $n_0 \leq n \leq \kappa \log x$ and all $x \geq x_1$ (for suitably chosen constants $\epsilon, n_0, \kappa$). Choose $x_0 \geq 1 \vee x_1$ such that $$P(CQ > (1-\epsilon) x) \leq E[C^\alpha] (1-\epsilon)^{-\alpha-1} P(Q > x).$$ The rest of the proof continues as in Proposition \[P.UniformBound\] with some modifications.
The authors are grateful to Professor Charles Goldie for pointing out a reference, and also to an anonymous referee for his or her helpful comments.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'To properly describe heating in weakly collisional turbulent plasmas such as the solar wind, inter-particle collisions should be taken into account. Collisions can convert ordered energy into heat by means of irreversible relaxation towards the thermal equilibrium. Recently, Pezzi et al. ([*Phys. Rev. Lett.*]{}, vol. 116, 2016, p. 145001) showed that the plasma collisionality is enhanced by the presence of fine structures in velocity space. Here, the analysis is extended by directly comparing the effects of the fully nonlinear Landau operator and a linearized Landau operator. By focusing on the relaxation towards the equilibrium of an out of equilibrium distribution function in a homogeneous force-free plasma, here it is pointed out that it is significant to retain nonlinearities in the collisional operator to quantify the importance of collisional effects. Although the presence of several characteristic times associated with the dissipation of different phase space structures is recovered in both the cases of the nonlinear and the linearized operators, the influence of these times is different in the two cases. In the linearized operator case, the recovered characteristic times are systematically larger than in the fully nonlinear operator case, this suggesting that fine velocity structures are dissipated slower if nonlinearities are neglected in the collisional operator.'
author:
- 'Oreste Pezzi[^1]'
date: '?; revised ?; accepted ?. - To be entered by editorial office'
title: Solar wind collisional heating
---
Authors should not enter PACS codes directly on the manuscript, as these must be chosen during the online submission process and will then be added during the typesetting process (see http://www.aip.org/pacs/ for the full list of PACS codes)
Introduction {#sec:intro}
============
Since the beginning of the last Century, many theoretical efforts have been performed to model natural and laboratory plasmas. One of the first attempts to describe the interplanetary medium and its interaction with the planetary magnetospheres was conducted by S. Chapman and V.C.A. Ferraro [@chapman30; @chapman31], widely considered the fathers of the Magnetohydrodynamics (MHD) theory. Their main intuition was to treat plasmas, approximated by neutral conducting fluids, as self-consistent media. One of the basic assumptions of this framework is that inter-particle collisions are sufficiently strong to maintain a local thermodynamical equilibrium, e.g. the particle velocity distribution function (VDF) is close to the equilibrium Maxwellian shape. This approach is still widely adopted to analyze plasma dynamics at large scales and many models have been developed to study the features of the MHD turbulence [@elsasser50; @chandrasekhar56; @iroshnikov64; @kraichnan65; @moffatt78; @parker79; @dobrowolny80a; @dobrowolny80b; @NgBhattacharjee; @MatthaeusEA99; @verdini09; @bruno13; @howes13; @pezzi16b]. One of the most studied natural plasmas is the solar wind, which is the high temperature, low density, supersonic flow emitted from the solar atmosphere. The solar wind is a strongly turbulent flow: the typical Reynolds number is about $Re\approx 10^{5}$ [@matthaeus05]; fluctuations are broadband and often exhibit a power-law spectra; several indicators of intermittency are routinely observed [@bruno13; @matthaeus15]. Despite the solar wind is usually approached in terms of MHD turbulence, spacecraft [ *in-situ*]{} measurements reveal much complex features, which go beyond the fluid MHD approach. Once the energy is transferred by turbulence towards smaller scales close to the ion inertial scales, kinetic physics signatures are often observed [@SahraouiEA07; @GaryEA10; @AlexandrovaEA08; @bruno13]. The particle VDF often displays a distorted out-of-equilibrium shape characterized by the presence of non-Maxwellian features such as temperature anisotropies, particle beams along the local magnetic field direction, rings-like structures [@marsch06; @kasper08; @maruca11; @maruca13; @he15]. The principal models to take into account kinetic effects are based on the assumption that the plasma is [*collisionless*]{}, e.g. collisions are far too weak to produce any significant effect on the plasma dynamics [@daughton09; @ParasharPP09; @camporeale11; @valentini11; @servidio12; @greco12; @perrone12; @valentini14; @franci15; @servidio15; @valentini16].
We would point out that, in order to comprehend the heating mechanisms of the solar wind, collisional effects should be considered. Indeed collisions are the unique mechanism able to produce irreversible heating from a thermodynamic point of view. Furthermore, to show that collisions can be neglected, the shape of the particle VDF is usually assumed to be close to the equilibrium Maxwellian [@spitzer56; @hernandez85; @maruca13]. This approximation may result problematic for weakly-collisional turbulent plasmas, where kinetic physics strongly distorts the particle VDFs and produces fine structure in velocity space. Collisional effects, which explicitly depend on gradients in velocity space, may be enhanced by the presence of these small scale structures in velocity space [@pezzi16a] (here after Paper I). Indeed, in Paper I we showed that the the collisional thermalization of fine velocity structures occurs on much smaller times with respect to the usual Spitzer-Harm time [@spitzer56] $\nu_{SH}^{-1}$ (being $\nu_{SH}\simeq 8 \times (0.714 \pi n e^4
\ln \Lambda)/(m^{0.5} (3 k_B T)^{3/2})$, where $n$, $e$, $\ln \Lambda$, $m$, $k_B$ and $T$ are respectively the particle number density, the unit electric charge, the Coulombian logarithm, the Boltzmann constant and the plasma temperature). The smallest characteristic times may be comparable with the characteristic times of other physical processes. Therefore, collisions could play a significant role into the dissipation of strong gradients in the VDF, thus contributing to the plasma heating.
In this paper we focus on the importance of retaining nonlinearities in the collisional operators. In particular, by means of numerical simulations of a homogeneous force-free plasma, we describe the collisional relaxation towards the equilibrium of an initial VDF which exhibits strong non-Maxwellian signatures. Collisions among particles of the same species are here modeled through the fully nonlinear Landau operator and a linearized Landau operator. A detailed comparison concerning the effects of the two operator indicates that retaining nonlinearities in the collisional integral is crucial to give the proper importance to collisional effects. Indeed, both operators are able to highlight the presence of several characteristic times associated with the dissipation of fine velocity structures. However, the magnitude of these times is different if nonlinearities are neglected: in the linearized operator case, the characteristic times are systematically larger compared to the case of the fully nonlinear operator. This indicates that, when nonlinearities are not taken into account in the mathematical form of the collisional operator, fine velocity structures are dissipated much slower. Results here described support the idea that to properly quantify the enhancement of collisional effects and, hence, to correctly compare collisional times with other dynamical times, it is important to adopt nonlinear collisional operators.
We would remark that, since the Landau operator is demanding from a computational perspective, self-consistent high-resolution simulations cannot be currently afforded and we are forced to restrict to the case of a force-free homogeneous plasma, where both force and advection terms have been neglected. This approximation represents a caveat of the work here presented and future studies will be devoted to the generalization of the results here shown to the self-consistent case.
The paper is organized as follows: in Sec. \[sec:SWheat\] the solar wind heating problem is revisited in order to address and motivate our work. Then, in Sec. \[sec:numres\] we give a brief description of the numerical codes and the adopted methods of analysis. Numerical results of our simulations are also reported and discussed in detail. Finally, in Sec. \[sec:concl\] we conclude and summarize.
Solar wind heating: a huge problem {#sec:SWheat}
==================================
As introduced above, the solar wind is a weakly collisional, strongly turbulent medium [@bruno13]. Several observations indicate that the solar wind is incessantly heated during its travel through the heliosphere: the temperature decay along the radial distance is indeed much slower than the decay expected within adiabatic models of the wind expansion [@marsch82; @goldstein96; @marino08; @cranmer09]. Therefore, some local heating mechanisms must play a significant role to supply the energy needed to heat the plasma. Numerous scenarios have been proposed to understand the plasma heating and a long-standing debate about which processes are preferred is still waiting for a clear and definitive answer \[See @bruno13 and references therein\]. Among these processes, it is widely known that the turbulence efficiently contributes to the local heating of solar wind [@sorriso07; @marino08; @sahraoui09], since it can efficiently transfer a significant amount of energy towards smaller scales, where dissipative mechanisms are at work. In fact, in a turbulent flow, much more energy is transferred towards smaller scales with respect to a laminar flow: the ratio between the energy transfer flux due to turbulence at a certain scale with respect to the heating production due to dissipation at the same scale is proportional to the Reynolds number $Re$, thus indicating that the energy transfer towards smaller scales gets more efficient as the flow becomes more turbulent.
In the simple neutral fluid scenario, the cascade is arrested once that the dissipative scale is reached [@frisch95]. On the other hand, the cascade evolves in a more complex way in a plasma: the presence of other processes (for example dispersion and kinetic effects) strongly modify the cascade before reaching the dissipative scale. A relatively wide agreement has been achieved about the importance of turbulence for transferring energy towards smaller scales. Instead, many scenarios have been proposed to explain the transition from the inertial range towards the kinetic scales and the nature of dissipative processes. These scenarios are often based on the “collisionless” assumption, that is justified by the fact that the Spitzer-Harm collisional time [@spitzer56] is much larger than other dynamical times. We would remark that two important caveats should be considered.
First, any mechanism which does not consider collisions is not able to describe the last part of the heating process, namely the heat production due to the irreversible dissipation of phase space structures and the approach towards the thermal equilibrium. For example, several mechanisms (e.g. nonlinear waves) can indeed increase the particle temperature, evaluated as the second order moment of the particle distribution function, by producing non-Maxwellian features as beams of trapped particles. However, this temperature growth due to the beam production does not represent a temperature growth in the thermodynamic sense, because the beam presence makes the system out of equilibrium. The particles beam can be instead interpreted as a form of free energy stored into the VDF. This energy is not in general converted into heat by means of irreversible processes but it can be also transformed in other forms of ordered energy (e.g. through micro-instabilities) [@lesur14]. Collisions are the unique mechanism able to degrade this information into heat by approaching the thermal equilibrium, thus producing heating in the general thermodynamic and irreversible sense.
Second, the evaluation of the Spitzer-Harm collisional time strictly assumes that the VDF shape is close to the equilibrium Maxwellian. This assumption may not be held in the solar wind [@marsch06; @servidio15], where VDFs shape is strongly perturbed by kinetic turbulence. In this direction, by focusing on the collisional relaxation in a homogeneous force-free plasma where collisions are modeled with the fully nonlinear Landau operator [@landau36], we recently showed that fine velocity structures are dissipated much faster than global non-thermal features such as temperature anisotropy (Paper I). The entropy production due to the relaxation of the VDF towards the equilibrium occurs on several characteristic times. These characteristic times are associated with the dissipation of particular velocity space structures and can be much smaller than the Spitzer-Harm time [@spitzer56], this indicating that collisions could effectively compete with other processes (e.g. micro-instabilities). In this perspective, high-resolution measurements of the particle VDF in the solar wind are crucial for a proper description of the heating problem [@vaivads16].
In principle the combination of the turbulent nature of the solar wind with its weakly collisionality may constitute a new scenario to describe the solar wind heating. In fact, turbulence is able to transfer energy towards smaller scales. Then, when kinetic scales are reached, since the plasma is weakly collisional, the VDF becomes strongly distorted and exhibits non-Maxwellian features, such as beams, anisotropies, ring-like structures [@belmont08; @chust09; @servidio12; @servidio15]. The presence of strong gradients in velocity space tends to naturally enhance the effect of collisions, which - ultimately - may become efficient for dissipating these structures and for producing heat.
Based on these last considerations, numerous studies have been recently conducted in order to take into account collisional effects in a weakly collisional plasma such as the solar wind [@filbet02; @bobylev13; @pezzi13; @pezzi14a; @escande15; @pezzi15b; @pezzi16a; @banonnavarro16; @hirvijoki16; @tigik16], where collisions are usually introduced by means of a collisional operator at the right hand-side of the Vlasov equation. The choice of the [*proper*]{} collisional operator remains an open problem. Several derivations from first principles (e.g. Liouville equation) indicate that the most general collisional operators for plasmas are the Lenard-Balescu operator [@lenard60; @balescu60] or the Landau operator [@landau36; @akhiezer86]. Both operators are nonlinear “Fokker-Planck”-like operators which involve velocity space derivatives and three-dimensional integrals. The Landau operator introduces an upper cut-off of the integrals at the Debye length to avoid the divergence for large impact parameters, while the Balescu-Lenard operator solves this divergence in a more consistent way through the dispersion equation. Therefore, the Balescu-Lenard operator is more general compared to the Landau operator from this point of view. However, both operators are derived by assuming that the plasma is not extremely far from the thermal equilibrium. Hence, both operators could lack the description of inter-particle collisions in a strongly turbulent system. The numerical approach of operators is also much more difficult for the Balescu-Lenard operator with respect to the Landau operator, because it involves the evaluation of dispersion function. Finally, we would also point out that, as far as we know, an explicit derivation of the Boltzmann operator for plasmas starting by the Liouville equation does not exist [@villani02]. Despite the adoption of the Boltzmann operator for describing collisional effects in plasmas is questionable from a theoretical perspective, it still represents a valid options since Boltzmann and Fokker-Planck like operators such as the Landau one are intrinsically similar [@landau36; @bobylev13].
The computational cost to evaluate both the Landau and Balescu-Lenard operators numerically is huge: for $N$ gridpoints along each direction of the $3D$–$3V$ numerical phase space ($3D$ in physical space and $3D$ in velocity space), the computation for the Landau operator would require about $N^9$ operations at each time step. In fact, for each point of the six-dimensional grid, a three-dimensional integral must be computed. To avoid this numerical complexity, several simplified operators have been proposed. We may distinguish these simpler operators in two classes. The first type of operators, as the Bathanar-Gross-Krook [@bgk54; @livi86] and the Dougherty operators [@dougherty64; @dougherty67; @pezzi15a], models collisions in the realistic three-dimensional velocity space by adopting a simpler structure of the operator. The second class of collisional operators works instead in a reduced, one-dimensional velocity space assuming that the dynamics mainly occur in one direction. Although this approach is quite “unphysical” (collisions naturally act in three dimensions), these operators can satisfactorily model collisions in laboratory plasmas devices, such as the Penning-Malmberg traps, where the plasma is confined into a long and thin column and the dynamics occurs mainly along a single direction [@anderson07a; @anderson07b; @pezzi13].
Numerical approach and simulation results {#sec:numres}
=========================================
As described above, to highlight the importance of nonlinearities present in the collisional operator, here we compare the effects of the fully Landau operator with a model of linearized Landau operator, obtained by simplifying the structures of the Landau operator coefficients. We restrict to the case of a force-free homogeneous plasma and we just model collisions between particles of the same species. Our interest is in fact to understand how collisional effects change when the mathematical kernel of the collisional operator is modified. Based on these assumptions, we numerically integrate the following dimensionless collisional evolution equations for the particle distribution function $f({\mathbf{v}},t)$: $$\begin{aligned}
\frac{{\partial}f({\mathbf{v}},t)}{{\partial}t} & = & \pi \left(\frac{3}{2}\right)^{\frac{3}{2}}\frac{{\partial}}{{\partial}v_{i}} \int d^3v' \ U_{ij}
(\mathbf{u}) \left[ f({\mathbf{v}}',t)\frac{{\partial}f({\mathbf{v}},t)}{{\partial}v_{j}} - f({\mathbf{v}},t) \frac{{\partial}f({\mathbf{v}}',t)}{{\partial}v'_{j}} \right] \ ,
\label{eq:lanNL} \\
\frac{{\partial}f({\mathbf{v}},t)}{{\partial}t} & = & \pi \left(\frac{3}{2}\right)^{\frac{3}{2}}\frac{{\partial}}{{\partial}v_{i}} \int d^3v' \ U_{ij}
(\mathbf{u}) \left[ f_0({\mathbf{v}}')\frac{{\partial}f({\mathbf{v}},t)}{{\partial}v_{j}} - f({\mathbf{v}},t) \frac{{\partial}f_0({\mathbf{v}}')}{{\partial}v'_{j}} \right] \ .
\label{eq:lanLIN}\end{aligned}$$ being $f$ normalized such that $\int d^3v f({\mathbf{v}}) =n=1$ and $U_{ij}(\mathbf{u})$ $$U_{ij}(\mathbf{u}) = \frac{\delta_{ij}u^2 - u_i u_j}{u^3} \ ,
\label{lanproj}$$
where $\mathbf{u}={\mathbf{v}}-{\mathbf{v}}'$, $u=|\mathbf{u}|$ and the Einstein notation is introduced. In Eqs. (\[eq:lanNL\]–\[eq:lanLIN\]), and from now on, time is scaled to the inverse Spitzer-Harm frequency $\nu_{SH}^{-1}$ [@spitzer56] and velocity to the particle thermal speed $v_{th}$. Details about the numerical solution of Eqs. (\[eq:lanNL\]–\[eq:lanLIN\]) can be found in Refs. [@pezzi15a; @pezzi16a]. In Eq. (\[eq:lanLIN\]), $f_0({\mathbf{v}})$ is the three-dimensional Maxwellian distribution function associated with the initial condition of our simulations $f({\mathbf{v}},t=0)$ and built in such a way that density, bulk velocity and temperature of the two distributions $f({\mathbf{v}},t=0)$ and $f_0({\mathbf{v}})$ are equal. The two equations clearly differ because Eq. (\[eq:lanLIN\]) is a linearized model of Eq. (\[eq:lanNL\]). The operator described in Eq. (\[eq:lanLIN\]) has been in fact obtained by linearizing the coefficients of the Landau operator. Although this linear operator does not represent the exact linearization of the Landau operator, the procedure here adopted for linearizing the operator (i.e. simplifying only the Fokker-Planck coefficients) is commonly adopted. In the following, we will note that the simulations performed with the linearized operator thermalize to the same final VDF and produce also the same total entropy growth of the nonlinear simulations. This suggests that the term which is not included in the form collisional operator (whose Fokker-Planck coefficients depend on $(f-f_0)({\mathbf{v}}')$) is not extremely relevant in the global thermalization of the system. This approximation corresponds to retain the gradients related to the out-of-equilibrium structures but to neglect their contribute to the integral in the ${\mathbf{v}}'$ space. In other words, here we locally consider gradients but we neglect their contribute to the global Fokker-Planck coefficients.
When simulations are completed, we perform the following multi-exponential fit [@curtis70; @pezzi16a] of the entropy growth $\Delta S$ to point out the presence of several characteristic times: $$\Delta S (t) = \sum_{i=1}^{K} \Delta S_i \left( 1 - e^{-t/\tau_i} \right) \ ,
\label{eqfit}$$ $\tau_i$ being the $i$–th characteristic time, $\Delta S_i$ the growth of entropy related to the characteristic time $\tau_i$ and $K$ is evaluated through a recursive procedure. This procedure has been already adopted in Paper I to highlight the importance of fine velocity structures in the entropy growth. In the following subsections, we report and describe the results of the simulations performed with two different initial distribution functions, already adopted in Paper I. The first initial condition concerns the presence of non-Maxwellian signatures due to a strongly nonlinear wave - an Electron Acoustic Wave (EAW) [@holloway91; @kabantsev06; @valentini06; @anderegg09a; @anderegg09b; @johnston09; @valentini12] - in the core of the distribution function. The EA waves here excited are quite different from another type of electron acoustic fluctuations which occur in a plasma composed by two components at different temperature [@watanabe77] and can be also observed in the Earth’s magnetosphere [@tokar84; @lu05]. The EAWs here excited are undamped waves whose phase speed is close to the thermal speed. It has been shown that, in the usual theory of the equilibrium Maxwellian plasma, these waves are strongly damped; while they can survive if the distribution function is locally modified (and exhibits a flat region) around the wave phase velocity. To generate the nonlinearity in the distribution function and let these waves survive, external drivers are usually adopted to force the plasma. EAWs are also characterized by the presence of phase space Bernstein-Green-Kruskal (BGK) structures [@bgk57] in the core of the electron distribution function, associated with trapped particle populations. The second initial distribution is instead a typical VDF recovered in hybrid Vlasov-Maxwell simulations of solar wind decaying turbulence [@servidio12; @valentini14; @servidio15]. The two simulations from which we selected our initial VDF are quite different. Indeed, in the first case, the out-of-equilibrium structures present in the initial VDF are due to the wave-particle interaction with the EAW, which is an almost monochromatic (few excited wavenumbers), electrostatic wave. On the other hand, in the second case, the initial distribution function has been strongly distorted due to the presence of an electromagnetic, turbulent cascade.
First case study: wave-particle interactions and collisions
-----------------------------------------------------------
=
The first initial condition here adopted, which is a three-dimensional VDF that evolves according to Eqs. (\[eq:lanNL\]–\[eq:lanLIN\]) in the three-dimensional velocity space, has been designed as follows. We separately performed a $1D$–$1V$ Vlasov-Poisson simulation of a electrostatic plasma composed by kinetic electrons and motionless protons whose resolution, in the $z-v_z$ phase space domain, is $N_z=256$, $N_{v_z}=1601$. In order to excite a large amplitude EAW, we forced the system with an external sinusoidal electric field, which has been adiabatically turned on and off to properly trigger the wave. Fig. \[fig:fd0\](a) reports the power spectral density of the electric energy $E_E(k_z)$ as a function of the wavenumber $k_z$, evaluated at the final time instant of the Vlasov-Poisson simulation (where the EAW is fully developed). Few wavenumbers are significantly excited and the EAW is almost monochromatic. The features of the electric fluctuations spectrum are reflected into the shape of the distribution function, which is locally distorted around the phase speed and present a clear BGK hole, as reported in Fig. \[fig:fd0\](b). Since the gridsize in velocity space is quite small in the current simulation, relatively small velocity scales are dynamically generated during the simulation by wave-particle interaction.
Then, we selected the spatial point $z_0$ in the numerical domain \[red vertical line in Fig. \[fig:fd0\](b)\], where this BGK-like phase space structure displays its maximum velocity width, and we considered the velocity profile $\hat{f}_e(v_z)=f_e(z_0,v_z)$, whose shape as a function of $v_z$ is reported in Fig. \[fig:fd0\](c). $\hat{f}_e$ is highly distorted due to nonlinear wave-particle interactions and exhibits sharp velocity gradients (bumps, holes, spikes around the resonant speed). Finally, by evaluating the density $n_e$, the bulk speed $V_e$ and the temperature $T_e$ of $\hat{f}_e$, we built up the three-dimensional VDF $f(v_x,v_y,v_z)=f_{M}(v_x)f_{M}(v_y)\hat{f}_e(v_z)$, which represents our initial condition, being $f_M$ the one-dimensional Maxwellian associated with $\hat{f}_e$. We remark that this VDF does not exhibit any temperature anisotropy but it still exhibits strong non-Maxwellian deformations along $v_z$, due to the presence of trapped particles, which make the system far from thermal equilibrium. The three-dimensional velocity domain is here discretized by $N_{v_x}=N_{v_y}=51$ and $N_{v_z}=1601$ gridpoints in the region $v_i=[-v_{max},v_{max}]$, being $v_{max}=6v_{th}$ and $i=x,y,z$, while boundary conditions assume that the distribution function is set equal to zero for $|v_j|>v_{max}$.
Since no temperature anisotropies are present, the evolutions of the total temperature and of the temperatures along each direction are trivial, the total temperature is preserved. On the other hand, the evolution of the entropy variation $\Delta S=S(t)-S(0)$ ($S=-\int
f\ln{f} d^3v$) gives information about the approach towards equilibrium. The time history of $\Delta S$ obtained with the nonlinear Landau operator (black) and with the linearized Landau operator (red) is showed in Fig. \[fig:entr\]. Since the initial condition and the equilibrium Maxwellian reached under the effect of collisions is the same for both operators, the total entropy growth $\Delta S$ is the same in the two cases. In other words, the free energy contained in the out-of-equilibrium structures in the initial VDF produces the same entropy growth in absolute terms but the growth occurs on different time scales in the two cases. Indeed, in the nonlinear operator case the entropy grows much faster ($1\div2 \nu_{SH}^{-1}$) compared to the linearized operator case ($4\div 5 \nu_{SH}^{-1}$).
=
To quantify the presence of several characteristic times, we perform the multi exponential fit of Eq. (\[eqfit\]) on the entropy growth curves reported in Fig. \[fig:entr\]. The analysis of the growth recovered in the fully nonlinear Landau operator indicates that three different characteristic times are recovered in the entropy growth:
- $\tau^{nl}_1 = 3.5 \cdot 10^{-3}\, \nu_{SH}^{-1} \rightarrow \Delta S^{nl}_1 / \Delta S_{tot} = 13 \% $
- $\tau^{nl}_2 = 1.3 \cdot 10^{-1}\, \nu_{SH}^{-1} \rightarrow \Delta S^{nl}_2 / \Delta S_{tot} = 42 \% $
- $\tau^{nl}_3 = 4.9 \cdot 10^{-1}\, \nu_{SH}^{-1} \rightarrow \Delta S^{nl}_3 / \Delta S_{tot} = 40 \% $
As discussed in Paper I, the presence of several characteristic times is associated with the dissipation of different velocity space structures. Fig. \[fig:fdNL\] reports $f(v_x=v_y=0,v_z)$ as a function of $v_z$ at the time instants $t=T_{nl,1}=\tau^{nl}_1$ (a), $t=T_{nl,2}=\tau^{nl}_1+\tau^{nl}_2$ (b), $t=T_{nl,3}=\tau^{nl}_1+\tau^{nl}_2+\tau^{nl}_3$ (c) and $t=t_{fin}$ (d), These time instants are displayed in Fig. \[fig:entr\] with blue diamonds. After the time $t=T_{nl,1}=\tau^{nl}_1$ (a), steep spikes visible in Fig. \[fig:fd0\](b) are almost completely smoothed out; then, at time $t=T_{nl,2}=\tau^{nl}_1+\tau^{nl}_2$ (b), the remaining plateau region is significantly rounded off, only a gentle shoulder being left; finally, after a time $t=T_{nl,3}=\tau^{nl}_1+\tau^{nl}_2+\tau^{nl}_3$ (c), the collisional relaxation to equilibrium is completed for the most part. A small percentage $\simeq 5\%$ of the total entropy growth is finally recovered for larger times and corresponds to the final approach to the equilibrium Maxwellian (d).
By performing the same analysis for the linearized Landau operator case, three characteristic times are also recovered:
- $\tau^{lin}_1 = 1.1 \cdot 10^{-2}\, \nu_{SH}^{-1} \rightarrow \Delta S^{lin}_1 / \Delta S_{tot} = 11 \% $
- $\tau^{lin}_2 = 2.7 \cdot 10^{-1}\, \nu_{SH}^{-1} \rightarrow \Delta S^{lin}_2 / \Delta S_{tot} = 23 \% $
- $\tau^{lin}_3 = 1.5 \;\;\;\;\;\;\;\;\;\;\; \nu_{SH}^{-1} \rightarrow \Delta S^{lin}_3 / \Delta S_{tot} = 63 \% $
These characteristic times are systematically larger than the times recovered in the nonlinear operator case. The shape of the distribution function after each characteristic time (not shown here) is quite similar to the shape recovered in the case of the fully nonlinear operator evolution. The process of dissipation of fine velocity structure is, hence, qualitatively similar if one adopts nonlinear or linearized operators. However, significant quantitative differences occur: similar profiles in velocity space are indeed reached at very different times, being the characteristic times recovered in the linearized case significantly larger (about $4\div5$ times) than the times recovered in the nonlinear operator case.
Therefore, from a qualitative point of view, both operators are able to recover the fact that fine velocity space structures are dissipated faster as their scale gets finer (i.e. as the velocity space gradients become stronger). However, fine velocity structures are dissipated slower by linearizing the collisional operator. Moreover, it is also worth mentioning that the amount of entropy growth associated with each characteristic time slightly changes by ignoring nonlinearities. For example, in the case of the fully nonlinear Landau operator, about $55\%$ of the total entropy growth is produced when the initial spikes and the successive flat plateau are dissipated. On the other hand, in the linearized operator case, only about the $30\%$ of the total entropy growth is associated with these processes.
=5.5cm
Second case study: kinetic turbulence and collisions
----------------------------------------------------
To support the scenario described in the previous section, here we focus on a second initial condition. This initial VDF has been selected from a $2D$–$3V$ hybrid Vlasov-Maxwell numerical simulation of decaying turbulence in solar wind like conditions [@valentini07; @valentini14]. The hybrid Vlasov-Maxwell simulation, whose resolution is $N_x=N_y=512$ and $N_{v_x}=N_{v_y}=N_{v_z}=51$, is initialized with a out of the plane background magnetic field. Then, magnetic and bulk speed perturbations at large, MHD scales are introduced. As a result of nonlinear couplings among the fluctuations, the energy cascades towards smaller kinetic scales. Hence, the particle VDF strongly departs from the thermal equilibrium due to the presence of kinetic turbulence and exhibits a potato-like shape similar to the solar wind [*in-situ*]{} observations [@marsch06]. The omni-directional power spectral densities of the magnetic (black) and electric energy (line), evaluated at the time instant where the turbulent activity is maximum, are reported in Fig. \[fig:TURBfd0\](a): clearly a broadband spectrum is recovered. The iso-contour of the initial VDF, selected where non-Maxwellian effects are strongest [@servidio15], is shown in Fig. \[fig:TURBfd0\](b). The VDF exhibits a hole-like structure in the upper part of the box and a thin ring-like structure in the bottom part of the box, while the VDF is clearly elongated on the $v_z$ direction. Compared to the first case study, the current distribution function reflects the presence of a spectrum of excited wavenumbers and it contains several kinds of distortions, not only concentrated around the resonant speed as in the previous case.
In Paper I we showed that, once velocity space gradients are artificially smoothed out through a fitting procedure, the presence of several characteristic times associated with the dissipation of fine velocity structures is definitively lost. Here, we instead compare the evolution towards the equilibrium of this initial condition under the effect of the fully nonlinear Landau operator \[Eq. (\[eq:lanNL\])\] and the linearized Landau operator \[Eq. (\[eq:lanLIN\])\]. The velocity domain is here discretized with $N_{v_x}=N_{v_y}=N_{v_z}=51$ points. Note that, compared to the first case study, the resolution is here smaller and it cannot be incremented due to the computational cost of the hybrid Vlasov-Maxwell code. Therefore, the quite small velocity scales recovered in the first case study (spikes around the resonant speed etc. etc.) are not present in this case.
Figure \[fig:TURBentr\] reports the entropy growth obtained with the fully nonlinear Landau operator (black) and with its linearized version (red). The entropy growth is, also here, slower in the linearized operator case compared to the fully nonlinear operator case. To quantify the different evolution observed in Fig. \[fig:TURBentr\], we perform the multi-exponential fit [@curtis70; @pezzi16a] described in Eq. (\[eqfit\]).
The analysis performed in the case of the fully nonlinear operator indicates that the entropy grows with two characteristic times:
- $\tau^{nl}_1 = 0.20\, \nu_{SH}^{-1} \rightarrow \Delta S^{nl}_1 / \Delta S_{tot} = 26 \% $
- $\tau^{nl}_2 = 0.82 \, \nu_{SH}^{-1} \rightarrow \Delta S^{nl}_2 / \Delta S_{tot} = 74 \% $
As in the case described in the previous section, each characteristic time is associated with the dissipation of a different out-of-equilibrium features. Figure \[fig:TURBVDFNL\] reports the iso-surface of the particle VDF at the time $t=T_{nl,1}=\tau^{nl}_1$ (left) and at the time $t=T_{nl,2}=\tau^{nl}_1+ \tau^{nl}_2$. At $t=T_{nl,1}$, the initial hole-like structure and the slight ring-like signature has been significantly smoothed out. Then, at $t=T_{nl,2}$, the approach towards the equilibrium is almost complete, being the VDF shape almost Maxwellian. Only a slight temperature anisotropy, which is finally thermalized in the late stage of the simulation, is still recovered. The approach towards the equilibrium confirms that small scale gradients are dissipated quite faster, while the final approach towards the equilibrium - concerning also the thermalization of temperature anisotropy - occurs on larger characteristic times.
In the linearized operator case, two characteristic times are also recovered:
- $\tau^{lin}_1 = 0.54 \, \nu_{SH}^{-1} \rightarrow \Delta S^{nl}_1 / \Delta S_{tot} = 16 \% $
- $\tau^{lin}_2 = 2.60 \, \nu_{SH}^{-1} \rightarrow \Delta S^{nl}_2 / \Delta S_{tot} = 84 \% $
As described for the first case study, these recovered characteristic times are systematically larger (about three times) compared to the times recovered in the fully nonlinear operator case. The amount of entropy growth associated with each characteristic time is also different, a smaller amount of entropy growth is indeed associated with the fastest characteristic time when nonlinearities are neglected. The results here described confirm the insights described in the previous section. The evolution obtained with the two operators is qualitatively similar: in both cases, several characteristic times are recovered in the entropy growth and these characteristic times are associated with the dissipation of different velocity space structures. However, the observed evolutions are different from a quantitative point of view: the recovered characteristic times are much different in the two cases, being significantly larger in the case of the linearized operator.
As described in Paper I, in the first case study, much smaller characteristic times are in general recovered compared to the second case study, probably since the numerical resolution in the second case study is about $30$ times smaller compared to the first case study and the sharp velocity gradients present in the first case study \[Fig. \[fig:fd0\](c)\] are not accessible in the second case study \[Fig. \[fig:TURBfd0\](b)\]. The presence of finer velocity structures in the first case compared to the second case introduces smaller characteristic times.
Conclusion {#sec:concl}
==========
To summarize, here we discussed in detail the importance of considering collisions in the description of the weakly collisional plasmas. Collisions are enhanced by the presence of fine velocity space structures, such as the ones naturally generated by kinetic turbulence in the solar wind; therefore, they could play a role into the conversion of VDFs free energy into heat, by means of irreversible processes.
In particular, we focused on the importance of retaining nonlinearities in the collisional operator by performing a comparative analysis of the collisional relaxation of a out-of-equilibrium initial VDF. Collisions have been modeled by means of two collisional operators: the fully nonlinear Landau operator and a linearized Landau operator. Due to the demanding computational cost of the collisional integral, we restricted to the collisional relaxation in a force-free homogeneous plasma. Our results must be clearly extended to the more general, self-consistent case; however, performing a high-resolution collisional simulation cannot be currently afforded.
The cases of study here analyzed indicate that both nonlinear and linearized collisional operators are able to detect the presence of several time scales associated with the collisional dissipation of small velocity scales in the particle VDF. A possible explanation of this behavior is that also the linearized operator involves gradients in its structure while it does not describe the “second-order” gradients related to the Fokker-Planck coefficients of the operator; therefore it is able to recover the presence of several characteristic times. The general message given in Paper I, namely the presence of sharp velocity space gradients speeds up the entropy growth of the system, is confirmed also in the case of the linearized operator: indeed, the fastest recovered characteristic times are significantly smaller than the common Spitzer-Harm collisional time [@spitzer56].
However, we would point out that the importance of the fine velocity structures is weakened if nonlinearities are ignored in the collisional operator. In the case of a linearized collisional operator, slower characteristic times are systematically recovered with respect to the nonlinear operator case. This indicates that, when one neglects the nonlinearities of the collisional integral, fine velocity structures are dissipated slower. Therefore, to properly address the role of collisions and to attribute them the correct relevance with respect to other physical processes [@matthaeus14; @gary93; @tigik16], nonlinearities should be explicitly considered.
Acknowledgements {#acknowledgements .unnumbered}
================
Dr. O. Pezzi would sincerely thank Prof. P. Veltri, Dr. F. Valentini and Dr. D. Perrone for the fruitful discussions which significantly contributed to the construction of this work. Dr. O. Pezzi would also thank the anonymous Referees for their suggestions which improved the quality of this work. Numerical simulations here discussed have been run on the Fermi parallel machine at Cineca (Italy), within the Iscra–C project IsC26–COLTURBO and on the Newton parallel machine at University of Calabria (Rende, Italy). This work has been supported by the Agenzia Spaziale Italiana underthe Contract No. ASI-INAF 2015-039-R.O “Missione M4 di ESA: Partecipazione Italiana alla fase di assessment della missione THOR”.
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[^1]: Email address for correspondence: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Donaldson conjectured[@Dona96] that the space of Kähler metrics is geodesic convex by smooth geodesic and that it is a metric space. Following Donaldson’s program, we verify the second part of Donaldson’s conjecture completely and verify his first part partially. We also prove that the constant scalar curvature metric is unique in [**each**]{} Kähler class if the first Chern class is either strictly negative or 0. Furthermore, if $C_1 \leq 0,$ the constant scalar curvature metric realizes the global minimum of Mabuchi energy functional; thus it provides a new obstruction for the existence of constant curvature metric: if the infimum of Mabuchi energy (taken over all metrics in a fixed Kähler class) isn’t bounded from below, then there doesn’t exist a constant curvature metric. This extends the work of Mabuchi and Bando[@Bando87]: they showed that Mabuchi energy bounded from below is a necessary condition for the existence of Kähler-Einstein metrics in the first Chern class.'
author:
- 'Xiuxiong Chen[^1]'
bibliography:
- 'test.bib'
date: 'Revised on July, 1999'
nocite:
- '[@Bourg97]'
- '[@Bourg85]'
- '[@Semmes93]'
- '[@Semmes95]'
- '[@GuanB98]'
- '[@GuanP971]'
- '[@GuanP972]'
- '[@Roch84]'
- '[@Slod96]'
title: The Space of Kähler metrics
---
Introduction to the problem
===========================
Brief introduction to the classical problems in Kähler geometry
---------------------------------------------------------------
Let $V$ be a Kähler manifold. E. Calabi conjectured in 1954 that any (1,1) form which represents $C_1(V)$ (the first Chern class) is the Ricci form of some Kähler metrics on $V.\;$ Yau [@Yau78], in 1978, proved this Calabi’s conjecture. Around the same time, Aubin [@Au76] and Yau proved independently the existence of a Kähler-Einstein metric on a Kähler manifold with negative first Chern class (also a conjecture of E. Calabi). G. Tian [@Tian87], in 1987, proved the existence of Kähler-Einstein metric in a canonical Kähler class on complex surfaces if the first Chern class is positive and the group of automorphism is reductive. For further references on this subject, see [@Tian96] and [@Tian97]. An important conjecture by Yau [@Yau92] relates the existence of Kähler-Einstein metrics to the stability in the sense of Hilbert schemes and Geometric invariant theory.\
Kähler-Einstein metrics could be treated as a special kind of extremal Kähler metrics. The question of extremal kähler metric was first raised by E. Calabi in his paper[@calabi82] : he considered $L^2$ norm of curvature as a functional from a given Kähler class; a critical point of this functional is called an “extremal Kähler metric.” He showed that any extremal Kähler metric must be symmetric under a maximal compact subgroup of the holomorphic transformation group. Using this structure theorem of Calabi, Marc Levine [@Levin85] was able to construct a Kähler surface on which there is no extremal Kähler metric. In 1992, D. Burns and P. de Bartolomeis [@BurnsBa92] also produced an example of non-existence of extremal Kähler metric; their example suggests some new obstruction for the existence of extremal metrics which is related to some borderline semi-stability of hermitian vector bundle. LeBrun[@LeBrun95] also demonstrated that the existence of critical Kähler metrics might be tied up with the stability of corresponding vector bundles. Donaldson[@Dona96] thought that Yau’s conjecture [@Yau92] should be extend over to the general extremal Kähler metrics. For further references in the subject of extremal metrics, please see [@LeBrun-Simanca94], [@LeBrun95] [@futaki88] and references therein.\
Futaki[@futaki83] in 1983 introduced an analytic invariant for any Kähler manifold with positive first Chern class. The vanishing of this invariant is a necessary condition for the existence of a Kähler-Einstein metric on the manifold. Later Futaki and Calabi [@calabi85] generalized the invariant to any compact Kähler class. This generalized Futaki invariant, i.e., Calabi-Futaki invariant, is an analytic obstruction to the existence of constant scalar curvature metric in a Kähler manifold. In the same paper, Calabi also shows that constant scalar curvature metric and extremal Kähler metric with non-constant scalar curvature do not co-exist in a single Kähler class.\
For the uniqueness, the known results are as follows: 1)in 1950s, E. Calabi showed the uniqueness of Kähler-Einstein metrics if $C_1 \leq 0.\;$ 2)in 1987, Mabuchi and S. Bando [@Bando87] showed the uniqueness of Kähler - Einstein metrics up to holomorphic transformation if the first Chern class is positive. Recently, Tian and X.H. Zhu [@TianZhu98] proved the uniqueness of Kähler-Ricci Soliton with respect to a fixed holomorphic vector field on any Kähler manifolds with positive first Chern class. Although very little was known about the uniqueness of general extremal Kähler metrics, most experts in Kähler geometry expect that the extremal Kähler metric is unique in each Kähler class up to holomorphic transformation. In [@chen943] (also see [@chen981] for further references), we demonstrated two degenerate extremal Kähler metrics in the same Kähler class with different energy levels and different symmetry groups: one example is due to Calabi, the other is due to the author. To my knowledge, it appears that this is the only non-uniqueness example known today.\
[**Main results.**]{} Mabuchi ([@Ma87])[^2] in 1987 defined a Riemannian metric on the space of Kähler metrics, under which it becomes (formally) a non-positive curved infinite dimensional symmetric space. Apparently unaware of Mabuchi’s work, Semmes [@Semmes92] and Donaldson [@Dona96] re-discover this same metric again from different angles. In [@Semmes92], Semmes S. first pointed out that the geodesic equation is a homogeneous complex Monge-Ampere equation on a manifold of one dimension higher. In [@Dona96], Donaldson further conjectured that the space is geodesically convex and it is a genuine metric space. We prove that it is at least convex by $C^{1,1}$[^3] geodesics, and from which we conclude that the space is indeed a metric space, thus verifying the second part of Donaldson’s conjecture. Moreover, this $C^{1,1}$ geodesic realizes the absolute minimum of length over all possible paths connecting the end points; thus the metric aforementioned is a genuine one. Using these results, we are able to show that the constant curvature metric is unique in each Kähler class if $C_1 < 0$ or $C_1 = 0.\;$ Furthermore, if $C_1 \leq 0,$ we show that constant scalar metric (if exists) realizes the global minimum of Mabuchi energy, which gives an affirmative answer to a question raised by Gang Tian [@TianPrivate2] in this special case. This last statement also extends the work of Mabuchi and Bando[@Bando87]: they showed that Mabuchi energy bounded from below is a necessary condition for the existence of Kähler-Einstein metrics in the first Chern class. In the light of Tian’s work[@Tian97] in which he shows that in Khler manifold with positive first Chern class and no non-trivial holomorphic fields, the Khler-Einstein metric exists if and only if the Mabuchi functional is proper (he actually uses an equivalent functional instead of the Mabuchi functional)[^4]. One would like to ask: is this still true for constant scalar curvature metrics[^5]?\
[**Organization**]{}: In section 2, we first summarize the different approaches taken by Mabuchi, Semmes and Donaldson independently in the space of Kähler metrics; then we introduce this Riemannian metric on this infinite dimensional space and prove that it has non-positive sectional curvature in the formal sense. Then we introduce Donaldson’s two conjectures and reduce the 1st conjecture to the existence problem for the complex homogeneous Monge-Ampere equation with Drichelet boundary data. Readers are alerted that material in section 2.3-2.5 is essentially a re-presentation of Donaldson’s work [@Dona96], included here for the convenience of readers. In section 3, we prove that this geodesic(CHMA) equation always has a $C^{1,1}$ solution. In section 4, we prove that a continuous solution to the geodesic(CHMA) equation in some appropriate weak sense is unique. In section 5, we show that the geodesic distance defined by the length of $C^{1,1}$ geodesic satisfies the triangular inequality. Using this, we prove the space of Kähler metrics is a metric space. In section 6 we show that extremal Kähler metric is unique in each Kähler class if either $C_1(V) < 0$ or $C_1(V) = 0.\;$\
[**Acknowledgment**]{}: The author is very grateful for Simon Donaldson who not only introduced him to this problem, but also spent many long sessions with him. He is also grateful for the constant encouragement and support of E. Calabi, L. Simon and R. Schoen during this work. he also want to thank Professor E. Stein for his help in soblev functions and embedding theorems. The author wish to thank S. Semmes for points out some important references in this problem. Thanks also goes to W.Y. Ding and his colleagues, and P. Guan for pointing out some errors in an earlier version of this paper.
Space of Kähler metrics
=======================
Mabuchi and S. Semmes’ Ideas
----------------------------
Shortly after introducing the now famous Mabuchi functional, Mabuchi [@Ma87] set out and defined a Riemannian metric in the space of Kähler metrics. Besides showing formally it is a locally symmetric space with non-positive sectional curvature, he also pointed out that the Mabuchi energy is formally convex in this infinite dimensional space (in the sense that the Hessian is semi-positive definite). Perhaps, this is his original motivation for introducing such a metrics. Unaware of Mabuchi’s work, in a remarkable paper [@Semmes92], S. Semmes studied the geometry of solution of complex Homogeneous Monge-Ampere equation (CHMA). He observed that in some special domain $\Omega \times D$ where $\Omega$ is n-dimensional domain in $C^n$ and $D$ is a domain in complex plane, the solution to CHMA is some sort of geodesic equation if the data is rotationally symmetric when restricted to $D.\;$ He then considered the space of pluri-subharmonic functions in $\Omega$ and defined a Riemannian metric in this space according to this geodesic equation. It turns out that this space becomes non-positively curved (locally) symmetric space in some formal sense. He also went on to study the variational problem of finding a geodesic. It seems that he is mainly motivated from providing a proper geometric meaning to solution of CHMA with right domain setting. Unlike real homogeneous Monge-Ampere equation (RHMA) whose solution always has proper geometric meaning, solution of a CHMA equation doesn’t have a preferred geometric interpretation. Without a proper geometry interpretation, it is very hard to work on this subject. Of course, great progress has been made since the famous work of L. Caffarelli, L. Nirenberg and J. Spruck[@CNS84] and later their joint work with J. Kohn [@CKNS85]. For instance, Lempert L. [@Lempt83], E. Bedford and B.A. Taylor [@Bedford76]; leong P.[@Lelong86] and important work of Krylov [@Krylov87] and Evans [@Evans82] $\cdots, $etc.. This is by no means a complete list of papers in complex Monge-Ampere equation since the author is quite new to this important field. For a complete and updated references, please see S. Kolodziej [@Kolo98]. Donaldson’s recent work certainly makes Mabuchi and Semmes’s original work all the more remarkable.\
Brief summary of Donaldson’s theory on space of Kähler metrics
---------------------------------------------------------------
Motivated from complete different reasons, S. K. Donaldson [@Dona96] re-discovered this metric. More importantly, he outlined a strategy in [@Dona96] to relate this geometry of infinite dimensional space to the existence problems in Kähler geometry. In particular, he explains how one can uses this extra structure in the infinite dimensional space to solve the problems of the existence and uniqueness of extremal Kähler metrics. In general, the later are intractable problems from traditional means. He regards the space of Kähler metrics in a fixed Kähler class as an infinite dimensional symplectic manifold with the automorphism group $SDiff(V)$ (symplectic diffeomorphism group of $V$ into itself). In [@Dona97], he pointed out that scalar curvature is the moment map $\mu$ from this infinite dimension symplectic manifold to the dual space of the Lie algebra of its automorphism group[^6]. Thus, to find an extremal Kähler metric in a fixed Kähler class in classical Kähler geometry could be re-interpreted as to find a pre-image of $0$ of the moment map $\mu$ in this symplectic setting. This acute observation sheds new light into the otherwise intractable problem of the existence of extremal Kähler metrics in a Kähler manifold; at least conceptually, the picture looks much clear. He then proposed several conjectures whose ultimate resolution will lead to a better understanding of extremal Kähler metric, and for that matter, better understanding of Kähler geometry as well. The most fundamental one among his conjectures is the so called geodesic conjecture: any two Kähler metrics in the same class is connected by a smooth geodesic. A second conjecture by him is that this space of Kähler metric is a metric space under this metric. If the geodesic conjecture is true, this second conjecture will be a direct consequence (since this space of Kähler metrics in a fixed Kähler class is non-positively curved in the formal sense.). He went on to show that the uniqueness of extremal Kähler metric is a consequence of this geodesic conjecture as well.\
Riemannian metrics in the infinite dimensional space.
------------------------------------------------------
Now we introduce this metric here. Readers are referred to Mabuchi, S. Semmes and Donaldson’s original writing for details. Consider the space of Kähler potentials in a fixed Kähler class as: $${\cal{H}} = \{ \varphi \in C^{\infty}(V) : \omega_{\varphi} = \omega_{0} + \sqrt{-1} \partial \overline{\partial} \varphi > 0 \; {\rm on}\; V\}.$$ Clearly, the tangent space $T \cal {H} $ is $C^{\infty} (V).\;$ Each Kähler potential $\phi \in \cal {H}$ defines a measure $d\,\mu_{\phi} = {1\over {n!}} \omega_{\phi}^n.\; $ Now we define a Riemannian metric on the infinite dimensional manifold $\cal {H}$ using the $L^2 $ norm provided by these measures. A tangent vector in $\cal {H}$ is just a function in $V.\;$ For any vector $\psi \in T_{\varphi} \cal {H}, $ we define the length of this vector as $$\|\psi\|^2_{\varphi} =\int_{V}\psi^2\;d\;\mu_{\varphi}.$$
For a path $\varphi(t) \in {\cal {H}} (0\leq t \leq 1),$ the length is given by $$\int_0^1 \sqrt{\int_V {\varphi(t)'}^2 d\,\mu_{\varphi(t)}} \; d\,t$$
and the geodesic equation is $$\varphi(t)'' - {1\over 2} |\nabla \varphi'(t)|^2_{ \varphi(t)} = 0,
\label{geodesic}$$ where the derivative and norm in the 2nd term of the left hand side are taken with respect to the metric $\omega_{\varphi(t)}.\;$\
This geodesic equation shows us how to define a connection on the tangent bundle of ${\cal H}$. The notation is simplest if one thinks of such a connection as a way of differentiating vector fields along paths. Thus, if $\phi(t)$ is any path in ${\cal H}$ and $\psi(t)$ is a field of tangent vectors along the path (that is, a function on $V \times [0,1]$), we define the covariant derivative along the path to be $$D_{t}\psi = \frac{\partial\psi}{\partial t} - {1\over 2} (\nabla \psi, \nabla
\phi')_{\phi}.$$ This connection is torsion-free because in the canonical co-ordinate chart”, which represents ${\cal H}$ as an open subset of $C^{\infty}(V)$, the Christoffel symbol” $$\Gamma: C^{\infty}(V) \times C^{\infty}(V) \rightarrow C^{\infty}(V)$$ at $\phi$ is just $$\Gamma(\psi_{1}, \psi_{2}) = - {1\over 2} (\nabla \psi_{1},
\nabla\psi_{2})_{\phi}$$ which is symmetric in $\psi_{1},\psi_{2}$. The connection is metric-compatible because $$\begin{array}{lrl} {1\over 2} \frac{d}{dt} \Vert \psi\Vert^{2}_{\phi} & = &\frac{d}{dt}\int_{V}\psi^{2}
d\mu_{\phi}\\
&= & \int_{V} \frac{\partial \psi}{\partial t} \psi + {1\over 2} \psi^{2} \Delta
(\phi') d\mu_{\phi}\\
&= & \int_{V} \frac{\partial \psi}{\partial t} \psi - {1\over 4}(\nabla(\psi^{2}),
\nabla \phi')_{\phi}\ d\mu_{\phi}\\
&= & \int_{V} (\frac{\partial \psi}{\partial t} - {1\over 2} (\nabla\psi, \nabla
\phi')_{\phi}\ ) \psi d\mu_{\phi}\\
&= & \langle D_{t} \psi, \psi\rangle.\end{array}$$ Here $\triangle$ is complex Laplacian operator. The main theorem proved in [@Ma87](and later reproved in [@Semmes92] and [@Dona96]) is:\
[**Theorem A**]{} [*The Riemannian manifold $\cal {H} $ is an infinite dimensional symmetric space; it admits a Levi-Civita connection whose curvature is covariant constant. At a point $\phi\in{\cal {H}}$ the curvature is given by $$R_{\phi}(\delta_{1}\phi, \delta_{2}\phi) \delta_{3}\phi=
- {1\over 4} \{ \{ \delta_{1}\phi, \delta_{2}\phi\}_{\phi},
\delta_{3}\phi\}_{\phi},$$ where $\{\ ,\ \}_{\phi}$ is the Poisson bracket on $C^{\infty}(V)$ of the symplectic form $\omega_{\phi}$; and $\delta_1 \phi, \delta_2 \phi \in T_{\phi} {\cal H}.$* ]{}
(Recall that in infinite dimensions the usual argument gives the uniqueness of a Levi-Civita \[i.e. torsion-free, metric-compatible\] connection, but not the existence in general.) The formula for the curvature of ${\cal {H}}$ entails that the sectional curvature is non-positive, given by $$K_{\phi}(\delta_{1}\phi, \delta_{2}\phi) = - {1\over 4} \Vert \{ \delta_{1}
\phi , \delta_{2}\phi\}_{\phi}\Vert_{\phi}^2.$$
Different proofs of this theorem have been appeared in [@Ma87], [@Semmes92] and [@Dona96]. We will skip the proof here, interested readers are referred to these papers if they are interested in the proof.\
The expression for the curvature tensor in terms of Poisson brackets shows that $R$ is invariant under the action of the symplectic-morphism group. Since the connection on $T{\cal H}$ is induced from an ${\rm SDiff}$-connection, it follows that $R$ is covariant constant, and hence ${\cal H}$ is indeed an infinite-dimensional symmetric space.
Splitting of ${\cal H}$
-----------------------
There is obviously a decomposition of the tangent space: $$T_{\phi}{\cal H} = \{ \psi : \int_{V} \psi d\mu_{\phi}=0 \} \oplus {\bf R}.$$ We claim that this corresponds to a Riemannian decomposition $${\cal H} = {\cal H}_{0}\times {\bf R}.$$ We are interested to see this Riemannian splitting more explicitly, partly because we see the appearance of a functional $I$ on the space of Kähler potentials, which is well-known in the literature, see [@Au84], [@Tian97] for example. The decomposition of tangent space of $\cal H$ gives a $1$-form $\alpha$ on ${\cal H}$ with $$\alpha_{\phi}(\psi) = \int_{V} \psi d\mu_{\phi},$$ and it is straightforward to verify that this $1$-form is [*closed*]{}. Indeed
$$(d\alpha)_{\phi}(\psi,\tilde{\psi}) = \int_{V} \left(\tilde{\psi}\Delta\psi- \psi
\Delta\tilde{\psi}\right) = 0.\;$$ This means that there is a function $I:{\cal H}\rightarrow {\bf R}$ with $I(0)=0$ and $dI=\alpha$, and it is this function which gives rise to the corresponding Riemannian decomposition. We call a Kähler potential $\phi$ [*normalized*]{} if $I(\phi)=0$. Then any Kähler metric has a unique normalized potential, and the restriction of our metric on ${\cal H}$ to $I^{-1}(0)$ endows the space ${\cal H}_{0}$ of Kähler metrics with a Riemannian structure; this is independent of the choice of base point $\omega_{0}$ and clearly makes ${\cal H}_{0}$ into a symmetric space. The functional $I$ can be written more explicitly by integrating $\alpha$ along lines in ${\cal H} $ to give the formula $$I(\phi) =\sum_{p=0}^{n} \frac{1}{(p+1)! (n-p)!} \int_{V} \omega_{0}^{n-p}
(\partial \overline{\partial}\phi)^{p}\ \phi
.$$
Donaldson’ Conjectures
----------------------
We will now study the geodesic equation in $\cal {H}$ in more detail, and interpret the solutions geometrically. Suppose $\phi_{t},\ t\in [0,1]$, is a path in $\cal {H}$. We can view this as a function on $V\times [0,1]$ and in turn as a function on $V\times [0,1]\times S^{1}$, with trivial dependence on the $S^{1}$ factor; that is, we define $$\Phi(v,t,e^{is}) = \phi_{t}(v).$$ We regard the cylinder ${\bf R}=[0,1]\times S^{1}$ as a Riemann surface with boundary in the standard way—so $t+is$ is a local complex co-ordinate. Let $\Omega_{0}$ be the pull-back of $\omega_{0}$ to $V\times {\bf R}$ under the projection map and put $\Omega_{\Phi} = \Omega_{0}+ \partial \overline{\partial} \Phi$, a $(1,1)$-form on $V\times {\bf R}$. Then we have:\
The path $\phi_{t}$ satisfies the geodesic equation (\[geodesic\]) if and only if $\Omega_{\Phi}^{n+1} =0$ on $V\times {\bf R}$.
[**Proof:**]{} Denote the metric defined by $\omega_0, \omega_{\phi}$ as $g, g'.\;$ Then $${1\over {n!}} \omega_{\phi}^n = \det\, g';\qquad {1\over {n!}} \omega_{0}^n = \det\, g.$$ Then geodesic equation is equivalent to the following (if $det \;g' \neq 0$) $$(\phi'' - {1\over 2} \mid \nabla \phi'\mid_{g'}^2 ) \;det\; g' = 0.$$ The last equation is equivalent to $$det \left( \begin{array}{cc} g' & \left( \begin{array} {c} {{\partial \phi'} \over {\partial z_1 }} \\
{{\partial \phi'} \over {\partial z_2 }}\\
\downarrow\\{{\partial \phi'} \over {\partial z_n }} \end{array}\right)\\
\left( \begin{array} {cccc} {{\partial \phi'} \over {\partial \overline{z_1} }} &
{{\partial \phi'} \over {\partial \overline{z_2} }} &
\cdots & {{\partial \phi'} \over {\partial \overline {z_n} }} \end{array}\right) & \phi'' \end{array} \right) = 0.$$ Let $w = t + \sqrt{-1} s,$ then $t = Re(w).\; $ The above equation could be re-written as $$det \left( \begin{array}{cc} (g +{{\partial^2 \phi}\over{\partial {z_{\alpha}} \partial {\overline z_{\beta}}}} )_{n\,n} & \left( \begin{array} {c} {{\partial^2 \phi} \over {\partial z_1 \partial {\overline w} }} \\
{{\partial^2 \phi} \over {\partial z_2 \partial {\overline w} }}\\
\downarrow\\{{\partial^2 \phi} \over {\partial z_n \partial {\overline w}}} \end{array}\right)\\
\left( \begin{array} {cccc} {{\partial^2 \phi} \over {\partial \overline{z_1} \partial {w} }} &
{{\partial^2 \phi} \over {\partial \overline{z_2} \partial {w} }} &
\cdots & {{\partial^2 \phi} \over {\partial \overline {z_n} \partial {w}}} \end{array}\right) & {{\partial^2 \phi}\over{\partial {w} \partial {\overline w}}} \end{array} \right) = 0.$$ This is just $\Omega_{\Phi}^{n+1} =0.\;$ The proposition is then proved. $\qquad QED.$\
Given boundary data —a real value function $\rho\in C^{\infty} (\partial
(V \times {\bf R})),$ we consider the set of functions $\Phi$ on $V\times {\bf R}$ which agree with $\rho$ on the boundary. Then we define the variation of $I_{\rho}$ on this set by $$\delta I_{\rho} = \frac{1}{(n+1)!}\int_{V\times {\bf R}} \delta \Phi \
\Omega_{\Phi}^{n+1} ,$$ where the variation $\delta \Phi$ vanishes on the boundary by hypothesis. This boundary condition means that we can show easily that this formula defines a functional $I_{\rho}.\;$ To prove this, one only need to show that the second derivatives of $I_{\rho}$ with respect to two infinitesimal variation $\delta_1 \Phi$ and $ \delta_2 \Phi $ is symmetric. The second derivatives is: $${1\over 2} \cdot \frac{1}{(n+1)!}
\int_{V} \delta_1 \Phi \; \triangle \;\delta_2 \Phi \;\Omega_{\Phi}^{n+1}$$ which is clearly symmetric on $\delta_1 \Phi$ and $ \delta_2 \Phi.\; $ Here $\triangle$ is the Laplacian operator of $\Omega_{\Phi}$ on $V \times {\bf R}.\;$\
This functional $I_{\rho}$ reduces to the energy functional on paths, by an integration by parts, in the case when ${\bf R}$ is the cylinder and we restrict to $S^{1}$-invariant data. Suppose $\phi(t) (0\leq t \leq 1)$ is a path in $\cal H, \;$ and $\delta \phi$ represents the infinestimal variation of $\phi$ while keep value of $\phi$ fixed when $t=0, 1.\;$ Thus, the variation of $I_{\rho}$ in $\delta \phi$ direction is (follow notations in the proof of previous proposition): $$\delta I_{\rho} = \frac{1}{(n+1)!}\int_{V\times R} \delta \phi\;
\Omega_{\Phi}^{n+1} = \frac{1}{(n+1)!}\int_{t=0}^{1} \,\int_{V} \delta \phi
(\phi''- {1\over 2} \mid \nabla \phi'\mid_{g'}^2 ) \;det\; g' d\,t.$$ On the other hand, the variation of energy functional along this path is: $$\delta E = \int_{t=0}^{1}\; \int_V \delta \phi\;
(\phi''- {1\over 2} \mid \nabla \phi'\mid_{g'}^2 ) \;det\; g' d\,t$$ where $E = \int_{t=0}^{1}\; \int_V \phi'(t)^2 \;det\; g' d\,t.\;$ Thus, in case when ${\bf R}$ is the cylinder and we restrict to $S^{1}$-invariant data, $I_{\rho}$ equal to the energy functional on the path up to a multiple of constant.\
The following is the first conjecture by Donaldson in [@Dona96]:
(Donaldson) Let ${\bf R}$ be a compact Riemann surface with boundary and $\rho:V\times
\partial R\rightarrow {\bf R}$ be a function such that $\omega_{0}- {\sqrt{-1}\;\overline{\partial}\partial} \rho$ is a strictly positive (1,1) form on each slice $V\times \{z\}$ for each fixed $z\in \partial R$. Let $\cal{S}_{\rho}$ be the set of functions $\Phi$ on $V\times
R$ equal to $\rho$ over the boundary and such that $\omega_{0} -{\sqrt{-1}\;\overline{\partial}\partial}\Phi$ is strictly positive on every slice $V\times \{w\}, w\in R$. Then there is a unique solution of the Monge-Ampere equation $(\Omega_{0} -{\sqrt{-1}\;\overline{\partial}\partial}\Phi)^{n+1}=0$ in $\cal{S}_{\rho}$, and this solution realizes the absolute minimum of the functional $I_{\rho}$.
This question is a version of the Dirichlet problem for the complete degenerate Monge-Ampere equation, a topic around which there is a substantial literature; see [@Au84],[@Klimek91] for example. Note that regularity questions are very important in this theory, since the equation is not elliptic.\
In the case of the geodesic problem, when the functional can be rewritten as the energy of a path; if these infimum are strictly positive, for all choices of fixed, distinct, end points, they make ${\cal H}$ into a metric space, in the usual fashion. In this connection, Donaldson proposes the following conjecture (after verifying that it will be satisfied by a smooth geodesic):
(Donaldson) If $\phi\in{\cal {H}}_{0}$ is normalized and $\tilde{\phi}_{t},\ t\in[0,1]$ is [ any]{} path from $0$ to $\phi$ in ${\cal {H}}$ then $$\int_{0}^{1} \int_{V} \left(\frac{d\tilde{\phi}}{dt}\right)^{2}
d\mu_{\tilde{\phi}_{t}} dt \geq M^{-1} \left( \max( \int_{\phi>0} \phi
d\mu_{\phi}, -\int_{\phi<0} \phi d\mu_{0}) \right)^{2}. \label{eq:lowerbound}$$
The restriction to normalized potentials $\phi$ is not important since we know that ${\cal {H}}$ splits as a product, and we could immediately write down a corresponding inequality, involving $I(\phi)$, for any $\phi\in{\cal {H}}$. If this conjecture and the geodesic conjecture are proved, then $\cal H$ is a metric space.
we want to use continuous method to treat this existence problem of geodesic between any two points in $\cal H$.
Existence of $ C^{1,1} $ solution
=================================
Let $V$ be a $n-$ dimensional Kähler manifold without boundary, ${\bf R}$ be a Riemann surface with boundary. The case we concerned most is when ${\bf R}$ is a cylinder. Suppose $g = g_{\alpha \overline{\beta}} dz_{\alpha} d\,\overline{z_{\beta}} (1\leq \alpha,\beta \leq n)$ is a given Kähler metric in $V.\;$ Then $ \tilde{g} = g_{\alpha \overline{\beta}} dz_{\alpha} d\,\overline{z}_{\beta} + dw\, \overline{dw}$ is a Kähler metric in $V\times {\bf R},\;$ and $\tilde{\varphi} = \varphi - |w|^2.\;$ For convenience, we still denote $\tilde{g}$ as $g$, and $\tilde{\varphi}$ as $\varphi$ when there is no confusion arisen. Also, let $z_{n+1}=w.\;$ Then $z= (z_1,z_2,\cdots,z_n,z_{n+1})$ is a point in $V\times {\bf R} $ and $z'=(z_1, z_2,\cdots z_n)$ is a point in $V.\;$ Let $\varphi(z) = \varphi(z', w)$ be a function in $V\times {\bf R}$ such that $g + \partial_{z'} \overline {\partial_{z'}}
\varphi(z',w)$ is a Kähler metric in $V$ for each $w \in {\bf R}.\;$ We want to solve the degenerated Monge-Ampere equation: $$det\;(g + {{\partial^2 \varphi}\over{\partial z_{\alpha} \partial \overline{z}_{\beta}}})_{(n+1)(n+1)} =0 \;\; {\rm in}\; V\times {\bf R}; \qquad {\rm and}\;\;
\varphi = \varphi_0 \;{\rm in}\;\; \partial (V\times {\bf R}).
\label{eq:euler1}$$
We want to use the continue method to solve this equation. Consider the continuous equation $0 \leq t \leq 1.\;$ $$det\;(g + {{\partial^2 \varphi}\over{\partial z_{\alpha} \partial \overline{z}_{\beta}}}) = t \; det\;(g + {{\partial^2 \varphi_0}\over{\partial z_{\alpha} \partial \overline{z}_{\beta}}}), \;\; {\rm in}\; V\times {\bf R}; \qquad {\rm and}\;\;
\varphi = \varphi_0 \;{\rm in}\;\; \partial (V\times {\bf R}).
\label{eq:euler2}$$ Suppose $\varphi_0$ is a solution to (\[eq:euler2\]) at $t=1$ such that $\displaystyle \sum_{\alpha,\beta = 1}^{n+1} (g_{\alpha \overline{\beta}} + {{\partial^2 \varphi_0}\over{\partial z_{\alpha} \partial \overline{z}_{\beta}}}) dz_{\alpha} d\,\overline{z}_{\beta} $ is strictly positive Kähler metric in $V \times {\bf R}$[^7]. Denote $f = det\;(g + {{\partial^2 \varphi_0}\over{\partial z_{\alpha} \partial \overline{z}_{\beta}}}) (det \;g)^{-1} > 0.\;$ Then equation (\[eq:euler2\]) can be re-written in a better form $$det\;(g + {{\partial^2 \varphi}\over{\partial z_{\alpha} \partial \overline{z}_{\beta}}}) = t \cdot f\cdot det\;(g) \;\; {\rm in}\; V\times {\bf R}; \qquad {\rm and}\;\;
\varphi = \varphi_0 \;{\rm in}\;\; \partial (V\times {\bf R}).
\label{eq:euler3}$$ Clearly, $\varphi_0$ is the unique solution to this equation at $t=1.\;$ Since the equation is elliptic, this equation can be uniquely solved for $t $ sufficiently closed to $1 $(the kernal of linearized operator is zero for any $t > 0$). Let $t_0$ be such that (\[eq:euler3\]) has a unique smooth solution for every $t \in (t_0,1].\;$ We want to show that $t_0=0 $ in this section. Observe that equation (\[eq:euler3\]) is elliptic for every $t > 0.\; $ Hence, the solution will be as smooth as the boundary value once we show that 2nd derivatives of $\varphi$ is uniformly bounded. Let $h$ be a super harmonic function on $V\times {\bf R}$ with respect to $g$ such that $\triangle_g h + n + 1 = 0.$ and $h = \varphi_0$ in $ \partial (V\times {\bf R} ).\;$ Then for any solution of equation (\[eq:euler3\]) for $t < 1$, we have $C^0$ bound of the solution:
If $\varphi$ is a solution of equation (\[eq:euler3\]) at $0<t <1,$ then $\varphi$ has the following a priori $C^0 $ estimate due to maximum principal: $$\varphi_0 \leq \varphi \leq h,\qquad {\rm in } \; V\times {\bf R}.$$
For $C^2$ estimate, we follow Yau’s famous work in Calabi’s conjecture. Essentially, we reduce it to a boundary estimate since we have $C^0$ estimate:
(Yau) If $\varphi$ is a solution of equation (\[eq:euler3\]) at $0<t <1,$ then $\varphi$ has the following a priori $C^2 $ estimate: $$\begin{array}{lcl}
\triangle' ( e^{- C\varphi} (n+1 + \triangle \varphi)) & \geq & e^{-C\varphi} (\triangle \ln f - (n+1)^2 \displaystyle \inf_{i\neq l} (R_{i\overline{i} l\overline{l}})) - C e^{-C\varphi} (n+1) (n+1 + \triangle \varphi) \\ & &
+ (C +\displaystyle \inf_{i\neq l} (R_{i\overline{i} l\overline{l}})) e^{-C\varphi} (n+1 + \triangle \varphi)^{ 1 + {1\over n}} ( t f)^{-1}.
\end{array}$$ where $C + \displaystyle \inf_{i\neq l} (R_{i\overline{i} l\overline{l}}) > 1,\;$ $\triangle $ is the Laplacian operator with respect to to $g,$ while $\triangle' $ is the Laplacian operator with respect to to $g' = g + {{\partial^2 \varphi}\over{\partial z_{\alpha} \partial \overline{z}_{\beta}}} d\,z_{\alpha}\; \overline{d\, z_{\beta}} $ and $R_{i\overline{i} l\overline{l}}$ is the Riemannian curvature of $g.\;$
From the a priori estimate in Lemma 2, either $e^{- C\varphi} (n+1 + \triangle \varphi)$ is uniformly bounded in $V \times {\bf R}$ or it achieves maximum value at $\partial (V \times {\bf R}).\;$ Lemma 1 asserts that $\varphi$ is uniformly bounded from above and below, then
There exists a constant $C$ which depends only on $(V\times {\bf R}, g) $ such that $$\displaystyle \max_{V \times {\bf R}}\; (n+1 + \triangle \varphi)
\leq C ( 1 + \displaystyle \max_{\partial (V \times {\bf R})}\; (n+1 + \triangle \varphi)).$$
If $\varphi$ is a solution of equation (\[eq:euler3\]) at $0<t <1,$ then there exists a constant $C$ which depends only on $(V\times {\bf R},g)$ such that: $$\displaystyle \max_{V \times {\bf R}}\; (n+1 + \triangle \varphi)
\leq C \displaystyle \max_{V \times {\bf R}} \;(|\nabla \varphi|_g^2+1).$$
In light of Corollary 1, we only need to prove the inequality (6) on the boundary, i.e,, $$\displaystyle \max_{\partial (V \times {\bf R})}\; (n+1 + \triangle \varphi)
\leq C \; \displaystyle \max_{V \times {\bf R}} \;(|\nabla \varphi|_g^2 + 1).$$ We will prove this inequality in the next subsection.
If $\varphi_i (i = 1, 2, \cdots)$ are solutions of equation (\[eq:euler3\]) at $0<t_i <1,$ and the inequality (6) holds uniformly for all these solutions $\{\varphi_i, i \in {\bf N}\}$, then there exists a constant $C_1$ independent of $i$ such that $$\displaystyle \max_{V \times {\bf R}}\; (n+1 + \triangle \varphi)
\leq C \; \displaystyle \max_{V \times {\bf R}} \;(|\nabla \varphi|_g^2 + 1)
< C_1.$$
This is proved via a blowing up argument. We will show this in subsection 3.2.
By now it is standard estimate of Monge-Ampere equations, that if $$\displaystyle \max_{V \times {\bf R}}\; (n+1 + \triangle \varphi)
\leq C \; \displaystyle \max_{V \times {\bf R})} \;(|\nabla \varphi|_g^2 + 1)
< C_1$$ then equation (\[eq:euler3\]) for $ t_1, t_2 ,\cdots$ is a sequence of uniform elliptic equations. The higher derivative of the solution $\varphi_i$ has a uniform bound as long as $\displaystyle \liminf_{i \rightarrow \infty} t_i > 0.\;$
There exists a $C^{1,1} (V\times {\bf R})
$ function which solves equation (\[eq:euler1\]) weakly. In other words, for any two points $\varphi_0, \varphi_1 \in \cal H,$ there exists a geodesic path $\varphi(t): [0,1] \rightarrow \overline{\cal H} $ and a uniform constant $C$ such that the following holds: $$0 \leq \left( g_{i \overline{j}}\; + \; {{\partial^2 \varphi} \over
{\partial z_i \partial \overline{z_j}}} \right)_{(n+1) (n+1)} \leq C \left({\tilde{g}}_{i \overline{j}}\right)_{(n+1)(n+1)}.$$ Here $z_1, z_2 \cdots, z_n$ are local coordinates in $V$ and $ t = Re \;(z_{n+1}).\;$ And $\tilde{g} = g_{\alpha \overline{\beta}} dz_{\alpha} d\,\overline{z}_{\beta} + dw\, \overline{dw} $ is a fixed product metric in $V \times {\bf R}.\;$
Following notations in theorem 2, we want to show that $t_0 =
\displaystyle \liminf_{i \rightarrow \infty} t_i =0.\; $ Otherwise, assume $t_0>0.\;$ Then equation (\[eq:euler3\]) has a unique smooth solution for $1 \geq t > t_0.\;$ Following from theorem 2, then we have uniform upper bound for $\triangle \varphi + (n+1)$ for all $t_i > t_0 > 0.\;$ Then equation (\[eq:euler3\]) implies that $g'_i = g + {{\partial^2 \varphi_i}\over{\partial z_{\alpha} \partial \overline{z}_{\beta}}} d\,z_{\alpha}\; \overline{d\, z_{\beta}} $ is bounded uniformly from below by a uniform positive constant (this positive low bound approaches 0 when $t \rightarrow 0)$. Thus, from equation (\[eq:euler3\]), we obtain uniform higher derivative estimates for solution $\varphi_i.\;$ Therefore these solution converge to a regular solution at $t_0 > 0.\;$ Again, since equation (\[eq:euler3\]) at $t_0$ is an elliptic equation and the kernal of the linearized operator is zero, it can then be solved for any $t$ sufficiently closed to $t_0.\;$ But this contradicts to the definition of $t_0.\;$ Thus $t_0 = 0.\;$ We can choose a subsequence of $t_i \rightarrow 0$ such that $\varphi_i$ converge weakly in $ C^{1,1} (V \times {\bf R}) $ where $\Omega$ is relative compact subset of $ V \times {\bf R}.$ Again via maximum principal, we can show this limit is unique and define a weak solution of equation (\[eq:euler1\]).
Boundary estimate
-----------------
We want to estimate $ \triangle \varphi $ at any point in the boundary $ \partial (V \times {\bf R}) = V \times \partial {\bf R}.\;$ Let $p$ be a generic point in $\partial (V \times {\bf R}).\;$ Now choose a small neighborhood $U$ of $p$ in $V\times {\bf R}$ (this will be a half geodesic ball since $ p \in \partial ( V\times {\bf R}))$ and a local coordinate chart such that $ g_{\alpha \overline{\beta}}(p) = \delta_{\alpha \overline{\beta}}$ and $p = (z=0)$ $${1\over 2} \delta_{\alpha \overline{\beta}} \leq g_{\alpha \overline{\beta}}(q) \leq 2 \delta_{\alpha \overline{\beta}} ,\qquad \forall q\; \in\; U.\;$$ Since $ \displaystyle \sum_{\alpha,\beta = 1}^{n+1}(g_{\alpha \overline{\beta}} + {{\partial^2 \varphi_0}\over{\partial z_{\alpha} \partial \overline{z}_{\beta}}}) d\,z_{\alpha} d\,\overline{z}_{\beta} $ is a positive Kähler metric in $V\times {\bf R},$ there exists a constant $\epsilon > 0$ such that $$g_{\alpha \overline{\beta}} + {{\partial^2 \varphi_0}\over{\partial z_{\alpha} \partial \overline{z}_{\beta}}} > 2\;\epsilon \cdot g_{\alpha \overline{\beta}}, \qquad {\rm in}\; \; V \times {\bf R}.$$ In the neighborhood $U$ of $p,$ we have $$g_{\alpha \overline{\beta}} + {{\partial^2 \varphi_0}\over{\partial z_{\alpha} \partial \overline{z}_{\beta}}} > \epsilon \cdot \delta_{\alpha \overline{\beta}}\qquad {\rm in}\; \; V \times {\bf R}.$$ We have the trivial estimates in $\partial (V \times {\bf R})$: $${{ \partial (\varphi - \varphi_0)}\over {\partial z_{\alpha}}} = 0,\qquad
{{\partial^2 (\varphi-\varphi_0)}\over
{\partial z_{\alpha} \partial \overline{z}_{\beta}}} = 0,\qquad \forall
\; 1\leq \alpha, \beta \leq n.$$ In order to estimate $\triangle \varphi = \displaystyle \sum_{\alpha,\beta=1 }^{n+1}g^{\alpha \overline{\beta}} \; {{\partial^2 \varphi}\over
{\partial z_{\alpha} \partial \overline{z}_{\beta}}} $ in $\partial (V \times {\bf R}),$ we only need to estimate ${{\partial^2 (\varphi-\varphi_0)}\over
{\partial z_{\alpha} \partial \overline{z}_{\beta}}} $ when either $\alpha $ or $\beta$ is $n+1.\;$ We will estimate ${{\partial^2 (\varphi-\varphi_0)}\over
{\partial z_{\alpha} \partial \overline{z}_{n+1}}} (\alpha \leq n)$ first, then use equation (\[eq:euler3\]) to derive estimate for ${{\partial^2 (\varphi-\varphi_0)}\over
{\partial z_{n+1} \partial \overline{z}_{n+1}}}.\; $\
Now we set up some conventions: $$z_{\alpha} = x_{\alpha} + \sqrt{-1}\; y_{\alpha},\;\; \forall \; 1\leq \alpha \leq n;\qquad z_{n+1} = x + \sqrt{-1} \;y$$ where $ {\bf R} $ near $\partial {\bf R}$ is given by $ x\geq 0$.
There exists a constant $C$ which depends only on $(V\times {\bf R}, g) $ such that $$| {{\partial^2 \varphi }\over
{\partial z_{\alpha} \partial \overline{z}_{n+1}}}(p)| \leq C ( \displaystyle
\max_{V\times {\bf R}} \; |\nabla \varphi|_g + 1) .$$
[**Proof of theorem 1**]{}: At point $p$, equation (\[eq:euler3\]) reduces to $$det ( \delta_{\alpha \overline{\beta}} + {{\partial^2 \varphi}\over
{\partial z_{\alpha} \partial \overline{z}_{\beta}}}) = t \cdot f.$$ In other words, $${{\partial^2 \varphi}\over
{\partial z_{n+1} \partial \overline{z}_{n+1}}} = t \cdot f - {{\partial^2 \varphi}\over
{\partial z_{\alpha} \partial \overline{z}_{n+1}}}\cdot {{\partial^2 \varphi}\over
{\partial \overline{z}_{\alpha} \partial {z}_{n+1}}}.$$ Lemma 3 then implies that $$| {{\partial^2 \varphi}\over
{\partial z_{n+1} \partial \overline{z}_{n+1}}}| \leq C (\displaystyle
\max_{V\times {\bf R}}|\nabla \varphi|_g^2 + 1 ).$$ Then, $$|\triangle \varphi(p)| = |\displaystyle \sum_{\alpha,\beta=1 }^{n+1}g^{\alpha \overline{\beta}} \; {{\partial^2 \varphi}\over
{\partial z_{\alpha} \partial \overline{z}_{\beta}}}(p)| \leq C (\displaystyle
\max_{V\times {\bf R}}|\nabla \varphi|_g^2 + 1 ).$$ Since $p$ is a generic point in $\partial (V \times {\bf R}),$ then theorem 2 holds true. QED.\
Let $D$ be any constant linear 1st order operator near the boundary ( for instance $ D = \pm {\partial \over {\partial x_{\alpha}}},\; \pm
{\partial \over {\partial y_{\alpha}}}$ for any $1 \leq \alpha \leq n).\;$ Notice $D$ is just defined locally. Define a new operator $\cal {L}$ as ( $\phi$ is any test function): $${\cal {L} } \phi = \displaystyle \sum_{\alpha,\beta = 1}^{n+1}
\;g'^{\alpha \overline{\beta}} {{\partial^2 \phi}\over
{\partial z_{\alpha} \partial \overline{z}_{\beta}}}$$ where $(g'^{\alpha \overline{\beta}}) = (g'_{\alpha \overline{\beta}})^{-1} =
\left( g_{\alpha \overline{\beta}} + {{\partial^2 \varphi}\over
{\partial z_{\alpha} \partial \overline{z}_{\beta}}}\right)^{-1}.\;$ Differentiating both side of equation (\[eq:euler3\]) by $D,$ we get $${\cal {L} }\; D \varphi = D \ln f + \displaystyle \sum_{\alpha,\beta = 1}^{n+1} g'^{\alpha \overline{\beta}} D g_{\alpha \overline{\beta}}.$$ Thus there exists a constant $C$ which depends only on $(V \times {\bf R},g)$ such that $${\cal {L} } D ( \varphi - \varphi_0) \leq C ( 1 + \displaystyle \sum_{\alpha=1}^{n+1} g'^{\alpha \overline {\alpha}})$$
We will now employ a barrier function of the form $$\nu = (\varphi - \varphi_0) + s\; ( h - \varphi_0) - N \cdot x^2$$ near the boundary point, and $s, N$ are positive constants to be determined. We may take $\delta $ small enough so that $x$ is small in $\Omega_{\delta} = (V\times {\bf R}) \cap B_{\delta} (0).\;$ The main essence of the proof is:
For $N$ sufficiently large and $s, \delta$ sufficiently small, we have $${\cal {L}}\; \nu \leq -{\epsilon \over 4} ( 1 + \displaystyle \sum_{\alpha=1}^{n+1}
g'^{\alpha \overline {\alpha}}) \;\;{\rm in}\;\; \Omega_{\delta},\;\;
\nu \geq 0\;\; {\rm on} \; \partial \Omega_{\delta}.$$
[**Proof**]{} Since $ g_{\alpha \overline{\beta}} + {{\partial^2 \varphi_0}\over
{\partial z_{\alpha} \partial \overline{z}_{\beta}}} \geq \epsilon \delta_{\alpha \overline{\beta}},$ we have $${\cal {L}} (\varphi - \varphi_0) = \displaystyle \sum_{\alpha,\beta=1}^{n+1}
g'^{\alpha \overline{\beta}} [ (g_{\alpha \overline{\beta}} + {{\partial^2 \varphi}\over
{\partial z_{\alpha} \partial \overline{z}_{\beta}}}) -( g_{\alpha \overline{\beta}} + {{\partial^2 \varphi_0}\over
{\partial z_{\alpha} \partial \overline{z}_{\beta}}}) ] \leq n+1 - \epsilon \;\displaystyle \sum_{\alpha=1}^{n+1}
g'^{\alpha \overline {\alpha}}$$ and $${\cal {L}} (h - \varphi_0) \leq C_1 ( 1 + \displaystyle \sum_{\alpha=1}^{n+1}
g'^{\alpha \overline {\alpha}})$$ for some constant $C_1.\;$ Furthermore, ${\cal {L}}\; x^2 = 2 g'^{(n+1)\overline{n+1}}.\;$ Thus $$\begin{array}{lcl} {\cal {L}} \nu & = & {\cal {L}} (\varphi-\varphi_0) +
s \cdot {\cal {L}} (h - \varphi_0) - 2\cdot N\cdot g'^{(n+1)\overline{n+1}}\\
& \leq & n+1 - \epsilon \displaystyle \sum_{\alpha=1}^{n+1}
g'^{\alpha \overline {\alpha}} + s C_1 + s C_1 \displaystyle \sum_{\alpha=1}^{n+1}
g'^{\alpha \overline {\alpha}} - 2 N g'^{(n+1)\overline{n+1}}.
\end{array}$$ Suppose $ 0 < \lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_{n+1}$ are eigenvalues of $ (g'_{\alpha \overline{\beta}})_{(n+1)(n+1)}.\;$ Thus $$\displaystyle \sum_{\alpha=1}^{n+1} g'^{\alpha \overline {\alpha}}
=\displaystyle \sum_{\alpha=1}^{n+1} {\lambda_{\alpha}}^{-1},\qquad g'^{(n+1)\overline{n+1}} \geq \lambda_n^{-1}.$$ Thus, $$\begin{array}{lcl} {\epsilon \over 4} \displaystyle \sum_{\alpha=1}^{n+1} g'^{\alpha \overline {\alpha}} + N g'^{(n+1)\overline{n+1}} &\geq & {\epsilon \over 4} \displaystyle \sum_{\alpha=1}^{n} {\lambda_{\alpha}}^{-1} + (N +{\epsilon \over 4}) {\lambda_{n+1}}^{-1} \\
& \geq & (n+1) {\epsilon \over 4} N^{{1\over {n+1}}} (\lambda_1 \cdot \lambda_2 \cdots \lambda_{n+1})^{-{1 \over {n+1}}} = C_2 N^{ {1 \over {n+1}}}.
\end{array}$$ Choose $N$ large enough so that $$-C_2 N^{ {1 \over {n+1}}} + (n+1) + s C_1 < - {\epsilon \over 4}.$$ Choose $s$ small enough so that $ s \cdot C_1 \leq {\epsilon \over 4}.\;$ Then $${\cal {L}} \nu \leq - {\epsilon \over 4} ( 1 + \displaystyle \sum_{\alpha=1}^{n+1} g'^{\alpha \overline {\alpha}})).$$ From now on we fix $N.\;$ Observe that $\triangle (h -\varphi_0) < - 2\epsilon,$ then there exists a constant $C_0$ which depends only on $g$ such that $ h - \varphi_0 > C_0\; x$ near $\partial (V \times {\bf R}).\;$ Choose $\delta $ small enough so that $$s (h -\varphi_0) - N x^2 \geq (s C_0 - N \delta)x \geq 0.$$ Then $\nu \geq 0$ in $\partial \Omega_\delta.\;$ QED.\
[**Proof of Lemma 3**]{}: Let $M = \displaystyle \max (|\nabla \varphi|_g + 1).$ Choose $ A \gg B \gg C,C_1.\;$ In additional, choose $A, B$ as a big multiple of $M.\;$ Notice that $ |D \varphi | \leq 2 M $ in $\Omega_{\delta}.\;$ For $\delta$ fixed as in Lemma 4, we have $B \delta^2 - | D(\varphi - \varphi_0)| > 0.\;$ Consider $w = A\; \nu + B \;|z|^2 + D(\varphi-\varphi_0).\;$ Then $w \geq 0$ in $\partial \Omega_{\delta}$ and $ w(0) = 0.\;$ Moreover, $${\cal {L}} w \leq ( -{{\epsilon A }\over {4}} + 2 B + C) ( 1 + \displaystyle \sum_{\alpha=1}^{n+1} g'^{\alpha \overline {\alpha}}) < 0.$$ Maximal Principal implies that $w \geq 0$ in $\Omega_{\delta}.\;$ Since $w(0) = 0,$ then $ {{\partial w}\over {\partial x}} \geq 0.\;$ In other words, $${{\partial} \over {\partial x}} D \varphi (0) < C_3 \cdot M$$ for some uniform constant $C_3.\; $ Since $D$ is any 1st order constant operator near $\partial (V \times {\bf R}).\;$ Replace $D$ with $-D,$ we get $$- {{\partial} \over {\partial x}} D \varphi (0) < C_3 \cdot M$$ On the other hand, since $\partial {\bf R}$ is given by $x=0$ in our special case, we then have the trivial estimate: $${{\partial} \over {\partial y}} D (\varphi-\varphi_0) (0) = 0.$$ Therefore, $$| {{\partial} \over {\partial z_{n+1}}} D \varphi (0) | < C_3 \cdot M$$ Lemma 3 follows from here directly. QED.\
Blowing up analysis
-------------------
Any bounded weakly sub-harmonic function in two dimensional plane is a constant.
This is a standard fact in geometry analysis, we will omit the proof here. Notice this lemma is false if dimension is no less than 3.
The essence of blowing up analysis is to use “micro-scope” to analyze what happen in a small neighborhood via rescaling. Hence it doesn’t make any difference what the global structure of background metric is, or what the metric is. Under rescaling, everything become Euclidean anyway. We may as well view the manifold as a domain in Euclidean space. we will use variable $x$ to denote position in $V \times {\bf R}.\;$
[**Proof of theorem 2**]{}: Suppose ${1\over \epsilon_i} =
\displaystyle \max_{V \times {\bf R}} |\nabla \varphi_i|_g \rightarrow \infty.\;$ We want to draw a contradiction from this statement.
Suppose $ |\nabla \varphi_i|_g(x_i) = {1\over \epsilon_i}.\;$ By theorem 1, we have $\displaystyle \max_{V \times {\bf R}} \triangle \varphi_i \leq
{1\over \epsilon_i^2}.\;$ Choose a convergent subsequence of $x_i$ such that $x_i \rightarrow \underline{x}.\;$ Choose a tiny neighborhood $B_{\delta}(\underline{x})$ of $\underline{x} $ so that $g_{\alpha\overline{\beta}}(\underline{x}) = \delta_{\alpha\overline{\beta}}$ and $g $ is essentially an identical matrix in $B_{\delta}(\underline{x}).\;$For simplicity, let us pretend that $g$ is an Euclidean metric in $B_{\delta}(\underline{x}).\;$ There are two cases to consider: the first case is when $\underline{x} \in \partial (V\times {\bf R})$ and the 2nd case is when $\underline{x}$ is in the interior of $V\times {\bf R}.\;$\
We define the blowing up sequence as $$\tilde{\varphi}_i(x) = \varphi_i (x_i +\epsilon_i x), \forall x \in B_{{\delta\over \epsilon_i}}(0).$$
Then $ |\nabla \tilde{\varphi}_i(0) =1 $ and $$\displaystyle \max_{B_{{\delta\over \epsilon_i}}(0)} |\nabla \tilde{\varphi}_i | \leq 1, \qquad {\rm and}\; \displaystyle \max_{B_{{\delta\over \epsilon_i}}(0)}
|\triangle \tilde{\varphi}_i | \leq C.$$
Observe $\varphi_0 \leq \varphi_i \leq h \;(\;\forall i).\;$ Re-scale $\varphi_0$ and $h$ accordingly: $$\tilde{\varphi}_0 (x) = \varphi_0 (x_i + \epsilon_i x),\qquad
\tilde{h}(x) = h(x_i + \epsilon_i x),\;\; \forall \; x\; \in B_{{\delta\over \epsilon_i}}(0).$$
Thus $\displaystyle \lim_{i \rightarrow \infty} \tilde{\varphi}_0 (x)= \varphi_0(\underline{x})$ and $ \displaystyle \lim_{i \rightarrow \infty} \tilde{h} (x)= h(\underline{x}).\;$ Moreover, $$\tilde{\varphi}_0 \leq \tilde{\varphi}_i \leq \tilde{h},\qquad \forall i = 1, 2,\cdots. \label{eq:c0bound}$$
There exists a subsequence of $\tilde{\varphi}_i$ and a limit function $ \tilde{\varphi}$ in $C^{n+1}$ (or half plane in case $\underline{x}$ in the boundary) such that in any fixed ball of $B_{l}(0)$(or half ball if $\underline{x}$ is in the boundary) we have $\tilde{\varphi}_i \rightarrow \tilde{\varphi}$ in $C^{1,\eta}$ in the ball $ B_{l}(0)$ (or half ball) for any $0 < \eta < 1.$ This implies $$|\nabla \tilde{\varphi}(0)| = 1.
\label{eq:gradient=1}$$
In additional, inequality (\[eq:c0bound\]) holds in the limit: $$\varphi_0(\underline{x}) \leq \tilde{\varphi}(x) \leq h(\underline{x}),\qquad \forall\; x.
\label{eq:limitc0bound}$$
Case 1: Suppose $\underline{x} \in \partial (V\times {\bf R}).\;$ Then $h(\underline{x}) = \varphi_0(\underline{x}).\;$ Inequality (\[eq:limitc0bound\] ) implies that $\tilde{\varphi}$ is a constant function in its domain. In particular, we have $|\nabla \tilde{\varphi}(x)| \equiv 0.\;$ This contradicts our assertion (\[eq:gradient=1\]). Thus the theorem is proved in this case.\
Case 2: Suppose $\underline{x}$ is in the interior of $ V \times {\bf R}.\;$ Then $\tilde{\varphi}(x)$ is a well defined $C^{1,\eta}$ and bounded function in $C^{n+1}.\;$ We claim that this function is weakly sub-harmonic in any complex line through origin. If this claim is true, then Lemma 5 says it must be constant for any complex line through origin. Therefore, the function itself must be a constant as well. Thus $|\nabla \tilde{\varphi}|
\equiv 0.\;$ It again contradicts with our assertion (\[eq:gradient=1\]). Thus the theorem is proved also, provided we can prove this claim.\
Without loss of generality, we consider the complex line $T$ is $$z_2 = z_3 =\cdots =z_{n+1} =0.$$
Observed that (near $\underline{x}$) the following holds $$0 < (\delta_{\alpha\overline{\beta}} + {{\partial ^2 \varphi_i}\over{\partial z_{\alpha} \partial \overline{z}_\beta}})_{(n+1) (n+1)} < { C \over {\epsilon_i^2}} ( \delta_{\alpha\overline{\beta}} )_{(n+1) (n+1)}, \forall \;i.$$ After rescaling, we have $$0 < \epsilon_i^2 \cdot (\delta_{\alpha\overline{\beta}})_{(n+1) (n+1)} + ({{\partial ^2 \tilde{\varphi}_i}\over{\partial z_{\alpha} \partial \overline{z}_\beta}})_{(n+1) (n+1)} < C \cdot ( \delta_{\alpha\overline{\beta}} )_{(n+1) (n+1)}.$$ Restricting this to a complex line $T,$ we have $$0 < \epsilon_i^2 + {{\partial ^2 \tilde{\varphi}_i}\over{\partial z_{1} \partial \overline{z}_1}} < C$$ Thus one can choose a subsequence of $\tilde{\varphi}_i$ which converges $C^{1,\eta} (0 < \eta < 1)$ locally in $T$ to some function $\psi.\;$ Since the convergence is in $C^{1,\eta},$ thus $\psi = \tilde{\varphi}|_{T};$ i.e., $\psi $ is the restriction of $\tilde{\varphi}$ in this complex line $T.\;$ By taking weak limit in inequality (11), then $\tilde{\varphi}_i |_{T} $ weakly converge to $\psi $ in $H^{2,p}_{loc}$ topology for any $p > 1.\;$ Therefore, $\psi $ is a weakly sub-harmonic function by taking weak limit in inequality (11). Therefore $\psi = \tilde{\varphi}|_{T}$ is a constant by Lemma 5. Our claim is then proved. QED.\
Uniqueness of weak $C^0 $ geodesic
==================================
Notation follows from previous section.
A function $\varphi$ is [*generalized-pluri-subharmonic*]{} in $V\times {\bf R}$ if $\displaystyle \sum_{\alpha,\beta = 1}^{n+1} (g_{\alpha \overline{\beta}} + {{\partial^2 \varphi}\over{\partial z_{\alpha} \partial \overline{z}_{\beta}}}) dz_{\alpha} d\,\overline{z}_{\beta} $ defines a strictly positive Kähler metric in $V \times {\bf R}.\;$
A continuous function $\varphi$ in $V\times {\bf R}$ is a weak $C^0$ solution to degenerated Monge-Ampere equation (\[eq:euler1\]) with prescribing boundary data $\varphi_0$ if the following statement is true: $\forall \; \epsilon > 0,$ there exists a pluri subharmonic function $\tilde{\varphi}$ in $V\times {\bf R}$ such that $ |\varphi -\tilde{ \varphi}| < \epsilon$ and $\tilde{\varphi}$ solves equation (\[eq:euler3\]) with some positive function $0 < f < \epsilon $ at $t=1,\; $ and with the same boundary data $\varphi_0.\;$
Clearly, the solution we obtain through continuous method is a weak $C^0$ solution of equation (\[eq:euler1\]).
Suppose $\varphi_1,\varphi_2$ are two $C^0$ weak solutions to the degenerated Monge-Ampere equation with prescribing boundary condition $h_1, h_2.\;$ Then $$\displaystyle \max_{V\times {\bf R}}\; |\varphi_1 -\varphi_2| \leq
\max_{\partial (V\times {\bf R})} |h_1 -h_2|.$$
The solution to degenerated Monge-Ampere equation is unique as soon as the boundary data is fixed.
[**Proof**]{}: Suppose $\phi_1,\phi_2$ are two approximate [*generalized-pluri-subharmonic*]{} solutions of $\varphi_1,\varphi_2$ in the sense of definition 2. In other words $$det\;(g + {{\partial^2 \phi_i}\over{\partial z_{\alpha} \partial \overline{z}_{\beta}}}) = f_i\cdot det\;(g) > 0 \;\; {\rm in}\; V\times {\bf R}; \qquad {\rm and}\;\;
\phi_i = h_i \;{\rm in}\;\; \partial (V\times {\bf R}),\;\; i=1,2$$ such that $ \displaystyle \max_{V\times {\bf R}}
(\;|\varphi_1 - \phi_1| + f_1)$ and $\displaystyle \max_{V\times {\bf R}} (\;|\varphi_2 - \phi_2| + f_2)$ could be made as small as we wanted.\
$\forall \epsilon > 0,$ we want to show $$\displaystyle \max_{V\times {\bf R}}\; ( \varphi_1 - \varphi_2) \leq \displaystyle \max_{V\times {\bf R}}\;(h_1 -h_2) + 2 \epsilon.$$ Choose $f_1$ such that $0 < f_1 < \epsilon$ and $ \displaystyle \max_{V\times {\bf R}} \;|\varphi_1 - \phi_1| < \epsilon.\;$ Choose $f_2$ such that $0 < f_2 \leq {1\over 2} \displaystyle \min_{V\times {\bf R}}\; f_1 < \epsilon $ and $ \displaystyle \max_{V\times {\bf R}} \;|\varphi_2 - \phi_2| < \epsilon.\;$ Then $\phi_1$ is a sub-solution to $\phi_2$ (thus $\phi_1 < \phi_2$) if $h_1 = h_2.\;$ In general, we have $$\displaystyle \max_{V\times {\bf R}}\; (\phi_1 - \phi_2) \leq
\displaystyle \max_{\partial (V\times {\bf R})}\;(h_1 - h_2).$$ Thus $$\begin{array} {lcl} \displaystyle \max_{V\times {\bf R}}\; (\varphi_1 -\varphi_2) & = &\displaystyle \max_{V\times {\bf R}}\; (\varphi_1 -\phi_1) +\displaystyle \max_{V\times {\bf R}}\; (\phi_1 -\phi_2) + \displaystyle \max_{V\times {\bf R}}\;(\phi_2 - \varphi_2) \\
& \leq & \epsilon + \displaystyle \max_{\partial (V\times {\bf R})}\;(h_1 - h_2) +\epsilon \\
& = & \displaystyle \max_{\partial (V\times {\bf R})}\;(h_1 - h_2) + 2 \epsilon.
\end{array}$$ Change the role of $\varphi_1$ and $\varphi_2,$ we obtain $$\displaystyle \max_{V\times {\bf R}}\; (\varphi_2 -\varphi_1) \leq \displaystyle \max_{\partial (V\times {\bf R})}\;(h_2 - h_1) + 2 \epsilon.$$ Thus $$\displaystyle \max_{V\times {\bf R}}\; |\varphi_1 -\varphi_2| \leq \displaystyle \max_{\partial (V\times {\bf R})}\;|h_1 - h_2| + 2 \epsilon.$$ Let $\epsilon \rightarrow 0, $ we obtain the desired result. QED.
The space of Kähler metric is a metric space—Triangular inequality
==================================================================
In this section, we want to prove that the space of Kähler metric is a metric space and the $C^{1,1}$ geodesic between any two points realizes the global minimal length over all possible paths. To prove this claim, one inevitably need to take derivatives of lengths for a family of $C^{1,1}$ geodesics. However, the length for a $C^{1,1}$ geodesic is just barely defined (the integrand is in $L^p$ space). In general, one can not take derivatives. Therefore, we must find ways to circumvent this trouble.
A path $\varphi(t) (0< t < 1)$ in the space of Kähler metrics is a convex path if $\varphi(t)$ is a [*generalized-pluri-subharmonic*]{} function in $V\times (I\times S^1)$ (see definition 1).
Suppose $vol(t) (0\leq t \leq 1)$ is a family of strictly positive volume form in $V$ such that $$\displaystyle \int_{V} vol(t) = \displaystyle \int_{V} det\; g.$$ The notion of $\epsilon$-approximate geodesic is defined with respect to such a volume form:
A convex path $\varphi(t)$ in the space of Kähler metrics is called $\epsilon$-approximate geodesic if the following holds: $$(\varphi'' - |\nabla \varphi'|_{g(t)}^2)\; det\; g(t) = \epsilon \cdot vol(t)$$ where $g(t)_{\alpha\overline{\beta}} = g_{\alpha\overline{\beta}} + {{\partial^2 \varphi}\over{\partial z_{\alpha}\partial \overline{z}_{\beta}}}\; (1\leq \alpha,\beta\leq n).\;$
The definition is really independent of these volume forms since we only care what happens when $\epsilon$ is really small. For convenience, sometimes we choose $vol(t) \equiv det\; g $ (a volume form independent of $t$).
Suppose $\varphi(t) (0\leq t \leq 1)$ is an $\epsilon$-approximate geodesics. Define the energy element as $E(t) = \displaystyle \int_{V} \varphi'(t)^2 d\;g(t).\;$ Then $$\displaystyle \max_{t} |{{d \, E}\over {d\,t}}| \leq 2 \;\epsilon \cdot
\displaystyle \max_{V\times I}\; |\varphi'(t)| \cdot M$$ where $ M = \displaystyle \int_{V} det \,g$ is the total volume of $V$ which depends only on the Kähler class.
[**Proof**]{}: $$\begin{array} {lcl} |{{d \, E}\over {d\,t}}| & = & |\displaystyle \int_V ( 2\varphi''\varphi' +\varphi'^2 \triangle_{g(t)} \varphi') \; d\, g(t)| \\
& = & 2 \; |\displaystyle \int_V \varphi'( \varphi'' - {1\over 2} |\nabla \varphi'|_{g(t)}^2) det \, g(t)|\\ & = & 2\;|\displaystyle \int_V \varphi'\, \epsilon\, vol(t)|
\leq 2 \; \epsilon \cdot \displaystyle \max_{V\times I}\; |\varphi'(t)| \cdot M. \end{array}\qquad {\rm QED}.$$
Suppose $\varphi(t)$ is a $C^{1,1}$ geodesic in $\cal H$ from $0$ to $\varphi$ and $I(\varphi)=0.\;$ Then the following inequality holds $$\int_{0}^{1} \sqrt{\int_{V} \varphi'^{2}
d\mu_{{\varphi}_{t}} } dt \geq M^{-1} \left( \max( \int_{\varphi>0} \varphi
d\mu_{\varphi}, -\int_{\varphi<0} \varphi d\mu_{0}) \right).$$ In other words, the length of any $C^{1,1}$ geodesic is strictly positive.
[**Proof**]{}: As in definition (4), suppose $\varphi(\epsilon,t)$ is a $\epsilon-$ approximated geodesic between $0$ and $\varphi.\;$ (We will drop the dependence of $\epsilon$ in this proof since no confusion shall arise from this omition). First of all, from definition of $\epsilon-$approximated geodesic, we have
$$\varphi''- {1\over 2} |\nabla \varphi'|^2_{g(t)} > 0.$$ In particular, we have $\varphi''(t) \geq 0.\;$ Thus $$\varphi'(0) \leq \varphi \leq \varphi'(1).
\label{eq:convex1}$$ Consider $f(t) = I(t\varphi), t\in [0,1].\;$ Then $f'(t) = \displaystyle \int_V\; \varphi\; d\mu_{t\varphi}$ and $$f''(t) = \displaystyle \int_V \varphi\; \triangle_{g(t\varphi)}\; \varphi\; d\mu_{t\varphi} \leq 0.$$ Thus, we have $f'(0) \geq { {f(1)-f(0)}\over {1-0}} \geq f'(1).\;$ In other words, we have $$\displaystyle \int_V \varphi\; d\;\mu_0 \geq I(\varphi) \geq \int_V \varphi\; d\; \mu_{\varphi}.$$
Since we assume $I(\varphi)=0,$ and $\varphi$ not identically zero, then it must take both positive and negative values. Then the length (or energy) of the geodesic is given by $$E=\int_{V} {\varphi'}^{2} d\mu_{\varphi_{t}},$$ for any $t\in[0,1]$. In particular, taking $t=1$, $$\sqrt{E(1)} \geq M^{-1/2} \int_{V} \vert \varphi'(1) \vert d\mu_{\varphi} > M^{-
1/2} \int_{\varphi'(1)>0} \varphi'(1) d\mu_{\varphi},$$ where $M$ is the volume of $V$ (which is of course the same for all metrics in ${\cal H}$). It follows from inequality (\[eq:convex1\]) that $$\int_{ \varphi'(1)>0} \varphi' \; d\mu_{\varphi}\geq \int_{\varphi>0} \varphi\;
d\mu_{\varphi},$$ where the last term is strictly positive by the remarks above, and depends only on $\varphi$ and not on the geodesic. A similar argument gives $$\sqrt {E(0)}> -M^{-1/2} \int_{\varphi<0} \varphi \;d\mu_{0}.$$ The previous lemma implies that for any $t_1,t_2 \in [0,1],$ we have $$|E(t_1) - E(t_2)| < C\cdot \epsilon$$ for some constant $C$ independent of $\epsilon.\;$ Thus $$\sqrt{E(t)} \geq M^{-
1/2} \displaystyle \max (\int_{\varphi>0} \varphi d\mu_{\varphi}, - \int_{\varphi<0} \varphi d\mu_{0}) - C\cdot \epsilon.$$ Now integrating from $t=0$ to $1$ and let $\epsilon \rightarrow 0.\;$Then $$\int_{0}^1 \sqrt{\displaystyle \int_V \varphi'^2 d\mu_{\varphi}}\;d\;t
\geq M^{-
1/2} \displaystyle \max (\int_{\varphi>0} \varphi d\mu_{\varphi}, - \int_{\varphi<0} \varphi d\mu_{0}).$$ Then this proposition is proved. $QED.$
This proposition verifies Donaldson’s 2nd conjecture [^8]. However, it will not imply $\cal H$ is a metric space automatically since the geodesic is not sufficiently differentiable. However, one can easily verifies that $C^{1,1}$ geodesic minimizes length over all possible convex curves between the two end points. To show that it minimizes length over all possible curves, not just convex ones, we need to prove that the triangular inequality is satisfied by the geodesic distance (see definition below).
Let $\varphi_1, \varphi_2$ be two distinct points in the space of metrics. According to theorem 3 and Corollary 2, there exists a unique geodesic connecting these two points. Define the geodesic distance as the length of this geodesic. Denoted as $d(\varphi_1,\varphi_2).\;$
Suppose $C: \varphi(s): [0,1] \rightarrow {\cal {H}}$ is a smooth curve in $\cal {H}.\;$ Suppose $p$ is a base point of $\cal {H}.\;$ For any $s,$ the geodesic distance from $p$ to $\varphi(s)$ is no greater than the sum of geodesic distance from $p$ to $\varphi(0)$ and the length from $\varphi(0)$ to $\varphi(s)$ along this curve $C.\;$ In particular, if $C: \varphi(s): [0,1] \rightarrow {\cal {H}}$ is a geodesic, then the geodesic distance satisfies: $$d(0,\varphi(1)) \leq d(0,\varphi(0)) + d(\varphi(0),\varphi(1)).$$
(Geodesic approximation lemma): Suppose $C_i: \varphi_i(s):
[0,1] \rightarrow {\cal {H}} (i = 1, 2)$ are two smooth curves in ${\cal {H}}.\;$ For $\epsilon_0$ small enough, there exist two parameters smooth families of curves $ C(s,\epsilon): \phi(t,s,\epsilon): [0,1]\times [0,1]\times (0,\epsilon_0] ( 0\leq t, s \leq 1, 0 < \epsilon \leq \epsilon_0) $ such that the following properties hold:
1. For any fixed $s $ and $\epsilon, C(s,\epsilon)$ is a $\epsilon$-approximate geodesic from $\varphi_1(s)$ to $\varphi_2(s).\;$ More precisely, $\phi(z,t,s,\epsilon) $ solves the corresponding Monge-Ampere equation: $$det\;(g + {{\partial^2 \phi}\over{\partial z_{\alpha} \partial \overline{z}_{\beta}}}) = \epsilon \cdot det\;(g) \;\; {\rm in}\; V\times {\bf R}; \qquad {\rm and}\;\;
\phi(z',0,s,\epsilon) = \varphi_1(z',s),\;\phi(z',1,s,\epsilon) = \varphi_2(z',s).$$ Here we follows notation in section 3, and $z_{n+1} = t + \sqrt{-1}\, \theta$ where depends of $\phi$ on $\theta $ is trivial.
2. There exists a uniform constant $C$ (which depends only on $\varphi_1, \varphi_2$ such that $$|\phi| + |{{\partial \phi}\over {\partial s}} | + |{{\partial \phi}\over {\partial t}} | < C; \qquad 0 \leq {{\partial^2 \phi}\over {\partial t^2}} < C,
\qquad {{\partial^2 \phi}\over {\partial s^2}} < C.$$
3. For fixed $s,$ let $\epsilon \rightarrow 0,$ the convex curve $ C(s,\epsilon)$ converges to the unique geodesic between $\varphi_1(s)$ and $\varphi_2(s)$ in weak $C^{1,1}$ topology.
4. Define energy element along $C(s,\epsilon)$ by $$E(t,s,\epsilon ) = \displaystyle \int_{V} |{{\partial \phi}\over {\partial t}}|^2 d\;g(t,s,\epsilon)$$ where $g(t,s,\epsilon)$ is the corresponding Kähler metric define by $\phi(t,s,\epsilon).\;$ Then there exists a uniform constant $C$ such that $$\max_{t,s} |{{\partial \, E}\over {\partial\,t}}| \leq \epsilon \cdot C \cdot M.$$ In other words, the energy/length element converges to a constant along each convex curve if $\epsilon \rightarrow 0.\;$
[**Proof** ]{}: Everything follows from theorem 3 and 4 and lemma 6 except the bound on $ |{{\partial \phi}\over {\partial s}} |$ and a upbound on $ {{\partial^2 \phi}\over {\partial s^2}}$ which follow from maximal principal directly since $${\cal {L}} ({{\partial \phi}\over {\partial s}} ) = 0$$ and $${\cal {L}} ({{\partial^2 \phi}\over {\partial s^2}} ) = tr_{g'} \{ Hess {{\partial \phi}\over {\partial s}}\cdot Hess {{\partial \phi}\over {\partial s}}\}
\geq 0. \qquad {\rm QED.}$$
[**Proof of theorem 5**]{}: Apply geodesic approximation lemma with special case that $\varphi_1(s)\equiv p.\;$ We follow notations in the previous lemma. For $\epsilon_0$ small enough, there exist two parameters smooth families of curves $ C(s,\epsilon): \phi(t,s,\epsilon): [0,1]\times [0,1]\times (0,\epsilon_0] ( 0\leq t, s \leq 1, 0 < \epsilon \leq \epsilon_0) $ such that $$det\;(g + {{\partial^2 \phi}\over{\partial z_{\alpha} \partial \overline{z}_{\beta}}}) = \epsilon \cdot det\;(g), \;\; {\rm in}\; V\times {\bf R}; \qquad {\rm and}\;\;
\phi(z',0,s,\epsilon) = 0,\;\phi(z',1,s,\epsilon) = \varphi(z',s).$$ Denote the length of the curve $\phi(t,s,\epsilon)$ from $p$ to $\varphi(s)$ as $L(s,\epsilon),$ denote the geodesic distance between $p$ and $\varphi(s) $ as $L(s),\;$ and denote the length from $\varphi(0)$ to $\varphi(s)$ along curve $C$ as $l(s).\;$ Clearly, $l(s) = \displaystyle \int_0^s \sqrt{\displaystyle \int_{V} |{{\partial \varphi}\over {\partial \tau}}|^2 d\;g(\tau) }\; d\,\tau$ where $g(\tau)$ is the Kähler metric defined by $\varphi(\tau), $ and $$L(s,\epsilon) = \displaystyle \int_0^1 \sqrt{E(t,s,\epsilon)} \,d\,t = \displaystyle \int_0^1 \sqrt{\displaystyle \int_{V} |{{\partial \phi}\over {\partial t}}|^2 d\;g(t,s,\epsilon) }\; d\,t,\;{\rm and}\; \displaystyle \lim_{\epsilon\rightarrow 0} L(s,\epsilon) = L(s).$$ Define $F(s,\epsilon) = L(s,\epsilon) + l(s)$ and $F(s) = L(s) + l(s).\;$ What we need to prove is : $F(1) \geq F(0).\;$ This will be done if we can show that $F'(s) \geq 0,\;\forall \;s\;\in [0,1].\;$ The last statement would be straightforward if the deformation of geodesics is $C^1.\;$ Since we don’t have it, we need to take derivatives on $F(s,\epsilon)$ for $\epsilon > 0 $ instead. Notice ${{\partial \phi}\over {\partial s}} = 0 $ at $t= 0$ in the following deduction: $$\begin{array}{lcl} {{d\, L(s,\epsilon)}\over {d\, s}} & =
& \displaystyle \int_0^1 {1\over 2} \; E(t,s,\epsilon)^{-{1\over 2}} \displaystyle \displaystyle \int_V \left( 2 {{\partial \phi}\over {\partial t}} {{\partial^2 \phi}\over {\partial t \partial s}} + ({{\partial \phi}\over {\partial t}})^2 \triangle_{g(t,s,\epsilon)} {{\partial \phi}\over {\partial s}} \right) d g(t,s,\epsilon)\; d\,t \\
& = & \displaystyle \int_0^1 \; E(t,s,\epsilon)^{-{1\over 2}} \{{{\partial} \over {\partial
t }}\; \left( \displaystyle \int_V
{{\partial \phi}\over {\partial t}}\;{{\partial \phi}\over {\partial s}}\; d\,
g(t,s,\epsilon)\right) - \displaystyle \int_V \;{{\partial \phi}\over {\partial s}}\; ({{\partial^2 \phi}\over {\partial t^2}}- {1\over 2} |\nabla {{\partial \phi}\over {\partial t}} |^2 ) d\,g(t,s,\epsilon) \}\; d\,t\\
& = & \{E(t,s,\epsilon)^{-{1\over 2}}\;\displaystyle \int_V
{{\partial \phi}\over {\partial t}}\;{{\partial \phi}\over {\partial s}}\; d\,
g(t,s,\epsilon))\}|_0^1 \;
- \; \displaystyle \int_0^1 \{ E(t,s,\epsilon)^{-{1\over 2}} \displaystyle \int_V \;{{\partial \phi}\over {\partial s}}\; ({{\partial^2 \phi}\over {\partial t^2}}- {1\over 2} |\nabla {{\partial \phi}\over {\partial t}} |^2 ) d\,g(t,s,\epsilon)\} d\,t\\
& & \qquad + \; \displaystyle \int_0^1 \{ E(t,s,\epsilon)^{-{3\over 2}}\; \displaystyle \int_V
{{\partial \phi}\over {\partial t}}\;{{\partial \phi}\over {\partial s}}\; d\,
g(t,s,\epsilon))\cdot \displaystyle \int_V \;{{\partial \phi}\over {\partial t}}\; ({{\partial^2 \phi}\over {\partial t^2}}- {1\over 2} |\nabla {{\partial \phi}\over {\partial t}} |^2 ) d\,g(t,s,\epsilon)\} \;d\,t\\
& = & \displaystyle \int_V
{{\partial \phi(1,s,\epsilon)}\over {\partial t}}\;{{d\, \varphi}\over {d\, s}}\; d\,
g(s) \cdot \{\displaystyle \int_V
|{{\partial \phi(1,s,\epsilon)}\over {\partial t}}|^2 d\,g(s)\}^{-{1\over 2}}
- \; \displaystyle \int_0^1 \{ E(t,s,\epsilon)^{-{1\over 2}} \; \displaystyle \int_V \;{{\partial \phi}\over {\partial s}}\; \epsilon \cdot det\; g \}\; d\,t\\
& & \qquad \qquad
+ \displaystyle \int_0^1 \{ E(t,s,\epsilon)^{-{3\over 2}}\; \displaystyle \int_V
{{\partial \phi}\over {\partial t}}\;{{\partial \phi}\over {\partial s}}\; d\,
g(t,s,\epsilon))\cdot \displaystyle \int_V \;{{\partial \phi}\over {\partial t}}\; \epsilon \cdot det\; g\}\; d\,t.
\end{array}$$ Observe that By Schwartz inequality, we have $${{d\, l(s)}\over {d\,s}} = \sqrt{\displaystyle \int_{V} |{{\partial \varphi}\over {\partial s}}|^2 d\;g(s) } \geq -\; \displaystyle \int_V
{{\partial \phi(1,s,\epsilon)}\over {\partial t}}\;{{d\, \varphi}\over {d\, s}}\; d\,
g(s) \cdot \{\displaystyle \int_V
|{{\partial \phi(1,s,\epsilon)}\over {\partial t}}|^2 d\,g(s)\}^{-{1\over 2}}.$$ Observe that $F(s,\epsilon) = L(s,\epsilon) + l(s).\;$ Thus $${{d\, F(s,\epsilon)}\over {d\,s}} \geq
- \displaystyle \int_0^1 \{ E(t,s,\epsilon)^{-{1\over 2}} \; \displaystyle \int_V \;{{\partial \phi}\over {\partial s}}\; \epsilon \cdot det\; g \}\; d\,t
+ \displaystyle \int_0^1 \{ E(t,s,\epsilon)^{-{3\over 2}}\; \displaystyle \int_V
{{\partial \phi}\over {\partial t}}\;{{\partial \phi}\over {\partial s}}\; d\,
g(t,s,\epsilon))\cdot \displaystyle \int_V \;{{\partial \phi}\over {\partial t}}\; \epsilon \cdot det\; g\}\; d\,t.$$ Integrating from $0$ to $s \in (0,1],$ we obtain $$\begin{array}{lcl}
F(s,\epsilon) - F(0,\epsilon) & \geq & - \displaystyle \int_0^s \displaystyle \int_0^1 \{ E(t,\tau,\epsilon)^{-{1\over 2}} \; \displaystyle \int_V \;{{\partial \phi}\over {\partial \tau}}\; \epsilon \cdot det\; g \}\; d\,t\; d\,\tau \\
& & \qquad + \displaystyle \int_0^s \displaystyle \int_0^1 \{ E(t,\tau,\epsilon)^{-{3\over 2}}\; \displaystyle \int_V
{{\partial \phi}\over {\partial t}}\;{{\partial \phi}\over {\partial \tau}}\; d\,
g(t,\tau,\epsilon))\cdot \displaystyle \int_V \;{{\partial \phi}\over {\partial t}}\; \epsilon \cdot det\; g\}\; d\,t \; d\,\tau\\
&\geq & - C \epsilon
\end{array}$$ for some big constant $C$ depends only on $(V\times {\bf R}, g)$ and the initial curve $C:\varphi(s):[0,1] \rightarrow {\cal {H}}.\;$ Now take limit as $\epsilon \rightarrow 0,$ we have $F(s) \geq F(0).\;$ In other words, the geodesic distance from $p$ to $\varphi(s)$ is no greater than the sum of geodesic distance from $p$ to $\varphi(0)$ and the length from $\varphi(0)$ to $\varphi(s)$ along this curve $C.\;$ QED.
The geodesic distance between any two metrics realize the absolute minimum of the lengths over all possible paths.
[**Proof**]{}: For any smooth curve $C: \varphi(s): [0,1] \rightarrow {\cal {H}},$ we want to show that the geodesic distance between the two end points $\varphi(0)$ and $\varphi(1)$ is no greater than the length of $C.\;$ However, this follows directly from Theorem 5 by taking $p = \varphi(1)$ and $s=1.\;$ QED.
For any two Kähler potentials $\varphi_1, \varphi_2,$ the minimal length $d(\varphi_1,\varphi_2)$ over all possible paths which connect these two Kähler potentials is strictly positive, as long as $\varphi_1 \neq \varphi_2.\;$ In other words, $ ({\cal {H}},d)$ is a metric space. Moreover, the distance function is at least $C^1.\;$
[**Proof**]{} Immediately from Corollary 3 and proposition 2, we imply that $({\cal H}, d)$ is a metric space. Now we want to prove the differentiability of distance function. From the Proof of theorem 5, we have $$\begin{array}{lcl} {{d\, L(s,\epsilon)}\over {d\, s}} & = &
\displaystyle \int_V
{{\partial \phi(1,s,\epsilon)}\over {\partial t}}\;{{d\, \varphi}\over {d\, s}}\; d\,
g(s) \cdot \{\displaystyle \int_V
|{{\partial \phi(1,s,\epsilon)}\over {\partial t}}|^2 d\,g(s)\}^{-{1\over 2}}
- \; \displaystyle \int_0^1 \{ E(t,s,\epsilon)^{-{1\over 2}} \; \displaystyle \int_V \;{{\partial \phi}\over {\partial s}}\; \epsilon \cdot det\; g \}\; d\,t\\
& & \qquad \qquad + \displaystyle \int_0^1 \{ E(t,s,\epsilon)^{-{3\over 2}}\; \displaystyle \int_V
{{\partial \phi}\over {\partial t}}\;{{\partial \phi}\over {\partial s}}\; d\,
g(t,s,\epsilon))\cdot \displaystyle \int_V \;{{\partial \phi}\over {\partial t}}\; \epsilon \cdot det\; g\}\; d\,t.
\end{array}$$
Integrating this from $s_1$ to $s_2$ and divide by $s_2-s_1,$ we have $$\begin{array}{l}
|{{L(s_2,\epsilon)-L(s_1,\epsilon)} \over {s_2 - s_1}} - {1\over{s_2-s_1}}
\displaystyle \int_{s_1}^{s_2} \displaystyle \int_V
{{\partial \phi(1,s,\epsilon)}\over {\partial t}}\;{{d\, \varphi}\over {d\, s}}\; d\,
g(s) \cdot \{\displaystyle \int_V
|{{\partial \phi(1,s,\epsilon)}\over {\partial t}}|^2 d\,g(s)\}^{-{1\over 2}}\;d\,s| \\
\leq {1\over {s_2-s_1}} \displaystyle \int_{s_1}^{s_2} \displaystyle \int_0^1 \{ E(t,s,\epsilon)^{-{1\over 2}}\;\displaystyle \int_V \;|{{\partial \phi}\over {\partial s}}|\; \epsilon \cdot det\; g \}\; d\,t\; d\,s \\
\\
\qquad \qquad + {1\over {s_2-s_1}} \displaystyle \int_{s_1}^{s_2} \displaystyle \int_0^1 \{ E(t,s,\epsilon)^{-{3\over 2}}\; \displaystyle \int_V
|{{\partial \phi}\over {\partial t}}|\;|{{\partial \phi}\over {\partial s}}|\; d\,
g(t,s,\epsilon))\cdot \displaystyle \int_V \;|{{\partial \phi}\over {\partial t}}|\; \epsilon \cdot det\; g\}\; d\,t\; d\,s \\
\leq C \epsilon.
\end{array}$$ Let $\epsilon \rightarrow 0,$ and then let $s_2 \rightarrow s_1$ we have $$\begin{array}{lcl}
\displaystyle \lim_{s_2 \rightarrow s_1} {{L(s_2)-L(s_1)} \over {s_2 - s_1}} & = & \displaystyle \lim_{s_2 \rightarrow s_1} {1\over{s_2-s_1}}
\displaystyle \int_{s_1}^{s_2} \displaystyle \int_V
{{\partial \phi(1,s)}\over {\partial t}}\;{{d\, \varphi}\over {d\, s}}\; d\,
g(s) \cdot \{\displaystyle \int_V
|{{\partial \phi(1,s)}\over {\partial t}}|^2 d\,g(s)\}^{-{1\over 2}}\;d\,s \\
& = & \displaystyle \int_V
{{\partial \phi(1,s)}\over {\partial t}}\;{{d\, \varphi}\over {d\, s}}\; d\,
g(s) \cdot \{\displaystyle \int_V
|{{\partial \phi(1,s)}\over {\partial t}}|^2 d\,g(s)\}^{-{1\over 2}}.
\end{array}$$ The distance function $L$ is then a differentiable function. QED.
Application: Uniqueness of Extremal Kähler metrics if $C_1 (V) < 0$ and $C_1(V) = 0 $
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In this section, we want to show that if $C_1 (V) < 0,$ or if $C_1 (V)= 0,$ then extremal Kähler metric is unique in any Kähler class. Furthermore, if $C_1(V) \leq 0,$ extremal Kähler metric (if existed) realizes the global minimum of Mabuchi energy functional in any Kähler class, thus gave a affirmative answer to a question raised by Tian Gang in this special case.
Uniqueness of c.s.c metric when $C_1(V)=0$ and the lower bound of Mabuchi energy for $C_1(V) \leq 0$
----------------------------------------------------------------------------------------------------
We should now introduce an important operator— Lichernowicz operator $\cal D.\;$ For any function $h,$ ${\cal D} h = h_{,\alpha \beta} dz^{\alpha} \otimes dz^{\beta}.\;$ If ${\cal D} h = 0,$ then $ \uparrow \overline{\partial} h = g^{\alpha \overline{\beta}} {{\partial h}\over {\partial \overline{\beta}}} {{\partial }\over {\partial {z_{\alpha}}}}
$ is a holomorphic vector field. Now let us introduce Mabuchi functional. Like $I_{\rho}, I,$ it is again defined by its derivatives and one should check it is well defined by verifying the second derivatives is symmetric (we will leave this to the reader). Let $R$ be the scalar curvature of metric $g= g_0 + \sqrt{-1} \partial \overline{\partial} \varphi $ and $\underline{R}$ be the average scalar curvature in the cohomology class, let $\psi \in T_{\varphi} {\cal H}.\;$ Then the variation of Mabuchi energy of $g$ at direction $\psi$ is: $$\delta_{\psi} E = - \int_{V} \; (R - \underline{R})\cdot \psi \; det\; g.$$ Along any smooth geodesic $\varphi(t) \in {\cal H}, $ S. Donaldson shows $${{d^2 E}\over{d\,t^2}} = \int_V |{\cal D} \varphi'(t)|^2_g \;det\; g.$$ Using this, Donaldson shows that constant curvature metric is unique in each Kähler class if the smooth geodesic conjecture is true. Now we want to prove the uniqueness of constant curvature metric in each Kähler class when $C_1(V) < 0$ or $C_1(V) = 0,$ despite the fact we have not proven the smooth geodesic conjecture yet.
If either $C_1(V) < 0$ or $C_1(V) = 0,$ then the constant curvature metric (if existed) in any Kähler class must be unique.
[**Proof**]{}: Notations follow from section 5. Suppose $\varphi(t) $ is a $\epsilon-$ approximate geodesic. Then $$det \;g\; (\varphi'' - {1\over 2} |\nabla \varphi'|_g^2 ) = \epsilon \cdot det\; h$$ where $h$ is a given metrics in the Kähler class such that $ Ric(h) < -c \; h$ if $ C_1(V) < 0$ and $ Ric(h) \equiv 0 $ if $ C_1(V) = 0.\;$ Let $f = \varphi'' - {1\over 2} |\nabla \varphi'|_g^2 \geq 0.\;$ Then $$\nabla \ln { {det\, g}\over {det\, h}} = - \nabla \ln\, f
\label{eq:Mabuchi1}$$
and $${{d}\over {d\, t}} (\int_{V} \varphi'(t) det\,g ) = \int_V\; f\cdot det g =
\epsilon \cdot \int_V \;det\, h.
\label{eq:Mabuchi2}$$ Let $E$ denote the Mabuchi energy functional. Then $${{d E}\over{d\,t}} = - \int_{V} \; (R - \underline{R})\cdot \psi \; det\; g$$ A direct calculation yields $${{d^2 E}\over{d\,t^2}} = \int_V |{\cal D} \varphi'(t)|^2_g \;det\; g
- \int_V (\varphi'' - {1\over 2} |\nabla \varphi'|_g^2 ) \cdot R\; det \, g +
\epsilon \cdot \underline{R} \cdot \int_V \;det\, h.
\label{eq:Mabuchi3}$$ where we already use equation (\[eq:Mabuchi2\]). Now the 2nd term in the right hand side of above equation is: $$\begin{array}{ccl} - \int_V \, R \cdot f \, det g & = & \int_V {\Delta_g} \ln det \, g \cdot f\cdot det\, g\\
& = & \int_V \;{\Delta_g} {{\ln det \, g}\over{\ln\, det h}} \cdot f\cdot det\, g + \int_V \; {\Delta_g} \ln det \, h \cdot f\cdot det\, g\\
& = & - \int_V \nabla_g {{\ln det \, g}\over{\ln\, det h}} \cdot \nabla \ln f\, det g \;-\; \int_V tr_g (Ric(h)) \; f\, det\, g \\
& = & \int_V |\nabla f|_g^2 {1\over f} det\,g - \; \int_V tr_g (Ric(h)) \; f\, det\, g.
\end{array}$$
Thus integrate from $t=0$ to $1,$ $$\int_{V\times I} | {\cal D} \varphi'|_g^2\, det\, g\, d\,t + \int_{V\times I}
{{|\nabla f|^2}\over{f}} \, det\, g\, d\,t - \int_{V\times I} tr_g (Ric(h)) \;f\, det\, g\, d\,t = {{d E}\over{d\,t}} \mid_{0}^{1} - \epsilon \underline{R} \cdot \int_V\, det\, h d\,t.$$ If $\varphi(0) $ and $\varphi(1)$ are both of constant scalar curvature metrics, then ${{d E}\over{d\,t}} \mid_{0}^{1} = 0$ and $$\int_{V\times I} | {\cal D} \varphi'|_g^2\, det\, g \,d\,t + \int_{V\times I}
{{|\nabla f|_g^2}\over{f}} \, det\, g \,d\,t- \int_{V\times I} tr_g (Ric(h)) \;f \, det\, g\, d\, t = - \epsilon \underline{R} \cdot \int_V\, det\, h\, d\,t.$$ Observe that $f \det \, g = \epsilon \cdot det h .\;$ We then imply from previous equation $$\int_{V\times I} {{ | {\cal D} \varphi'|_g^2}\over f}\, det\, h + \int_{V\times I}
{|\nabla \ln f|_g^2} \, det\, h - \int_{V\times I} tr_g (Ric(h)) \, det\, h = - \underline{R} \int_V\, det\, h.$$ Clearly, if $C_1(V) = 0,$ then $\underline{R} = 0.\; $ Thus $$\int_{V\times I} {{| {\cal D} \varphi'|_g^2}\over f}\, det\, h d\,t + \int_{V\times I}
{|\nabla \ln f|_g^2} \, det\, h
d\,t = 0$$ This easily implies that $ {\cal D} \varphi(t) \equiv 0 $ and $ \uparrow \overline{\partial} \varphi'(t) $ is a holomorphic vector field. Since $C_1 = 0,$ the only holomorphic vector field is constant vector field. Thus $\varphi'(t)$ is constant on $V$ direction. In other words, $\varphi'(t) $ is a functional of $t$ only. Hence, there exist at most one constant scalar curvature metric in each Kähler class when $C_1 = 0.\;$ We postpone the proof of case $C_1 < 0$ to the next subsection.
If $C_1(V) \leq 0, $ then constant scalar curvature metric, if existed, realizes the global minimum of Mabuchi energy functional in each Kähler class. In other words, if Mabuchi energy doesn’t have a lower bound, then there exists no constant curvature metric in that cohomology class.
[**Proof**]{} Suppose $\varphi_0 \in \cal H $ is a metric of constant curvature, then $${{d E}\over{d\,t}}|_{\varphi_0} = - \int_{V} \; (R - \underline{R})\cdot \psi \; det\; g = 0$$ For any metric $\varphi(1),$ let $\varphi(t) (0\leq t \leq 1)$ is a path in $\cal H$ which connects between $\varphi(0)$ and $\varphi(1).\;$ In additional, let us assume this is a $\epsilon-$ approximate geodesic where $\epsilon > 0 $ may be chosen arbitrary small. From equation (17), we have $$\begin{array} {ccl}
{{d^2 E}\over{d\,t^2}} & = & \int_V |{\cal D} \varphi'(t)|^2_g \;det\; g
- \int_V (\varphi'' - {1\over 2} |\nabla \varphi'|_g^2 ) \cdot R\; det \, g +
\epsilon \cdot \underline{R} \cdot \int_V \;det\, h\\
& = & \int_V |{\cal D} \varphi'(t)|^2_g \;det\; g + \int_V |\nabla f|_g^2 {1\over f} det\,g - \; \int_V tr_g (Ric(h)) \; f\, det\, g + \epsilon \cdot \underline{R} \cdot \int_V \;det\, h\\
& > & - C\epsilon.
\end{array}$$ The last inequality holds since the average of scalar curvature is a topological invariant. Thus $$E(t) -E(0) \geq -C\epsilon {{t^2}\over 2}, \qquad \forall t \in [0,1].$$ In particular, this holds for $t=1$ $$E(\varphi(1)) - E(\varphi(0)) = E(1) - E(0) \geq - {{ C\cdot \epsilon}\over 2}.$$ Now let $\epsilon \rightarrow 0,$ we have $$E(\varphi(1)) \geq E(\varphi(0)).$$ Thus the theorem is proved since $\varphi(1)$ is arbitrary chosen.
Uniqueness of c.s.c. metric when $C_1 < 0$
------------------------------------------
Now we turn our attentions to the case $C_1 < 0.\;$ By initial assumption, $Ric(h) < -c h$ for some positive constant $c > 0.\;$ Thus $$\int_{V\times I} {{| {\cal D} \varphi'|_g^2}\over{f}} \, det\, h + \int_{V\times I}
{|\nabla \ln f|_g^2} \, det\, h + c \cdot \int_{V\times I} tr_g (h) \, det\, h \leq C (= -\underline{R} \int_V\, det\, h).
\label{eq:patch0}$$ We want to show that in the limit as $\epsilon \rightarrow 0,$ we still have ${\cal D}\varphi'(t) = 0.\;$ Let us first get an integral estimate on $f^{q\over {2-q}} (1 < q < 2)$ with respect to measure $det\,h\; d\,t:$ $$\begin{array}{ccl}
\displaystyle \int_{V\times I} f^{q\over {2-q}} det\, h \; d\, t & \leq &
C\cdot \displaystyle \int_{V\times I} f det\, h \; d\, t \\
&\leq & C\cdot \displaystyle \int_{V\times I} \{ f \cdot { {det\, g}\over {det\, h}} \}^{1\over n} \cdot \{{ {det\, h}\over {det\, g}} \}^{1\over n}
det\, h \; d\, t \\
&\leq & {\epsilon}^{1\over n} \displaystyle \int_{V\times I} \{{ {det\, h}\over {det\, g}} \}^{1\over n}
det\, h \; d\, t \\
& \leq & C \cdot {\epsilon}^{1\over n} \displaystyle \int_{V\times I} tr_g (h)
det\, h \; d\, t \rightarrow 0.
\end{array}$$
Let $X = \uparrow \overline{\partial} \varphi'(t) = g^{\alpha \overline{\beta}} {{\partial \varphi'}\over{\partial \overline{z_{\beta}}}} {{\partial}\over{\partial z_{\alpha}}}.\; $ Then we want to show that $X$ is uniformly in $L^2$ with respect to measure $h + d\,t^2.\;$ $$\begin{array} {ccl}\int_{V\times I} |X|_h^2 \;det\, h \;d\,t & = & \displaystyle \int_{V\times I} \displaystyle \sum_{\alpha,\beta} h_{\alpha \overline{\beta}} X^{\alpha} \overline{X^{\beta}} \;det\, h \;d\,t \\
& = & \displaystyle \int_{V\times I}\; \displaystyle \sum_{\alpha,\beta,\gamma,\delta} h_{\alpha \overline{\beta}}
g^{\alpha \overline{\gamma}} {{\partial \varphi'}\over{\partial \overline{z_{\gamma}}}} \overline{\{g^{\beta \overline{\delta}} {{\partial \varphi'}\over{\partial \overline{z_{\delta}}}}\}}\; det\, h \;d\,t\\
& = & \displaystyle \int_{V\times I}\;\displaystyle \sum_{\alpha,\beta,\gamma,\delta} h_{\alpha \overline{\beta}} g^{\alpha \overline{\gamma}} g^{\delta \overline{\beta}} {{\partial \varphi'}\over{\partial \overline{ z_{\gamma}}}} {{\partial \varphi'}\over{\partial {z_{\delta}}}} \; det\, h \;d\,t\\
& \leq & \displaystyle \int_{V\times I} tr_g (h) |\nabla \varphi'|^2_g \; det\,h\\
& \leq & C \cdot \displaystyle \int_{V\times I} tr_g (h) \; det\,h \;d\,t\leq C.
\end{array}$$ The second to last inequality holds since $f = \varphi'' - {1\over 2} |\nabla \varphi'|_g^2 \geq 0$ and $\varphi'' < C.\;$ Thus $X \in L^2( V \times I)$ has a uniform upbound for the $L^2$ norm.
Consider $|D\varphi'|_g $ as a function in $L^2 (V \times I). \;$ First of all, it has a weak limit in $L^2 (V\times I);$ secondly, its $L^{q} (1 < q < 2) $ norm tends to $0$ as $\epsilon \rightarrow 0.$ $$\begin{array}{ccl}
\displaystyle \int_{V\times I} {|{\cal D}\varphi'|_g}^q \; det\, h \; d\, t
& = & \cdot \displaystyle \int_{V\times I} { {|{\cal D}\varphi'|_g}^q \over {f^{l}}} \cdot f^l \; det\, h \; d\, t \\
& \leq & \cdot ( \displaystyle \int_{V\times I} { {|{\cal D}\varphi'|_g}^{s q} \over {f^{l s}}} \; det\, h \; d\, t)^{1\over s} \cdot ( \displaystyle \int_{V\times I} f^{l \tau } \; det\, h \; d\, t)^{1\over \tau} \qquad ({\rm where}\; {1\over s} + {1\over \tau} = 1).
\end{array}$$ Now $l$ is some number we should choose appropriately: $$l s = 1; q s = 2; {1\over s} + {1\over \tau } = 1.$$ Thus for any $q < 2,$ we have $$s= {2 \over q}; l = {q\over 2}; \tau = {2 \over {2-q}}.$$
Thus the above inequality reduce to $$\displaystyle \int_{V\times I} {|{\cal D}\varphi'|_g}^q \; det\, h \; d\, t \leq
C \cdot (\displaystyle \int_{V\times I} { {|{\cal D}\varphi'|_g}^{2} \over {f^{}}} \; det\, h \; d\, t)^{2 \over q} \cdot (\displaystyle \int_{V\times I} f^{q\over{2-q}} \; det\, h \; d\, t)^{(2-q)\over 2 } \rightarrow 0.$$ For any vector $Y \in T^{1,0} (V \times I)$ (i.e., $Y = \displaystyle \sum_{i=1}^n \; Y^{i} {{\partial }\over {\partial z_i}}\;$ where $z_1, z_2, \cdots z_n$ are all of the coordinate functions in a local chart in $V$. We use ${{\partial Y} \over {\partial \overline{z}}}$ to denote the vector valued (0,1) form $ \displaystyle \sum_{i,j=1}^n {{\partial Y^i} \over {\partial \overline {z_j}}} \;{{\partial }\over {\partial z_i}}\otimes
d\;\overline {z_j}.)\;$ For a scalar function $\psi$ in $V\times I,$ denote $ {{\partial \psi} \over {\partial \overline{z}}}$ as $ \displaystyle \sum_{j=1}^n {{\partial \psi } \over {\partial \overline {z_j}}} d\;\overline {z_j}.\;$Now the norm of ${{\partial Y} \over {\partial \overline{z}}}$ and $ {{\partial \psi} \over {\partial \overline{z}}}$ in terms of the metric $h$ are: $$\mid{{\partial Y}\over {\partial \overline{z}}}\mid_h^2 = \displaystyle \sum_{\alpha,\beta,r,\delta=1}^n\; h_{\alpha \overline{r}} h^{\overline{\beta} \delta} {{\partial Y^{\alpha}}\over {\partial \overline{z_{\beta}}}} \overline{ \left({{\partial Y^{r}}\over {\partial \overline{z_{\delta}}}}\right)}$$ and $$\mid{{\partial \psi}\over {\partial \overline{z}}}\mid_h^2 = \displaystyle \sum_{\alpha,\beta=1}^n\;h^{\alpha \overline{\beta}} {{\partial \psi } \over {\partial {z_{\alpha}}}} {{\partial \psi } \over {\partial \overline {z_\beta}}}.$$
We claim the following inequality holds (for some uniform constant $C$): $$\mid{{\partial X}\over {\partial \overline{z}}}\mid_h \leq \sqrt{\displaystyle \sum_{\alpha,\beta,r,\delta=1}^n\;h_{\alpha \overline{r}} h^{\overline{\beta} \delta} {{\partial X^{\alpha}}\over {\partial \overline{z_{\beta}}}} \overline{ \left({{\partial X^{r}}\over {\partial \overline{z_{\delta}}}}\right)} }\leq C \;\sqrt{tr_g (h)}\; |D\varphi'|_g.
\label{eq:patch1}$$ This could be proven by choosing a preferred coordinate, where $h_{i \overline{j}} = \delta_{i \overline{j}} (1 \leq i, j \leq n) $ while $g_{i \overline{j}} = \lambda_i \delta_{i \overline{j}} (1 \leq i, j \leq n)\;$ in an arbitrary point $O.\;$ Here $\lambda_i$ are eigenvalues of metric $g$ in terms of metric $h.\;$ These $\lambda_i$’s are uniformly bounded from above since $g$ is uniformly bounded from above. We want to verify the above inequality in this point $O.\;$ $$\begin{array}{lcl}
\mid{{\partial X}\over {\partial \overline{z}}}\mid_h^2 & = &
\displaystyle \sum_{\alpha,\beta,a,b=1}^n\;
{{\partial X^{\alpha} } \over {\partial z_{\overline{\beta}}}}
{{\partial X^{\overline{a}} } \over {\partial z_{b}}} h_{\alpha \overline{a}} h^{\overline{\beta}\, b} \\
& = & \displaystyle \sum_{\alpha,\beta,a,b,c,d=1}^n\;h_{\alpha \overline{a}} h^{\overline{\beta} \,b} g^{\alpha \overline{c}} {\varphi'}_{,\overline{c} \overline{\beta}} {\varphi'}_{,d\,b}
g^{\overline{a} d} = \displaystyle \sum_{\alpha,\beta=1}^n\;\delta_{\alpha \overline{a}} \delta^{\overline{\beta} \,b} {1\over {\lambda_{\alpha}}} \delta^{\alpha \overline{c}} {\varphi'}_{,\overline{c} \overline{\beta}} {\varphi'}_{,d\,b}
{1\over {\lambda_{a}}} \delta^{\overline{a} d} \\
& = & \displaystyle \sum_{\alpha, \beta=1}^n\;{1\over {\lambda_{\alpha}^2}} {\varphi'}_{,\overline{\alpha} \overline{\beta}} {\varphi'}_{,\alpha \,\beta}
\leq \left(\displaystyle \sum_{\alpha, \beta=1}^n\; {{\lambda_{\beta}} \over {\lambda_{\alpha}}} \right) \;
\displaystyle \sum_{\alpha, \beta=1}^n\;{1\over {\lambda_{\alpha} \lambda_{\beta}}} {\varphi'}_{,\overline{\alpha} \overline{\beta}} {\varphi'}_{,\alpha \,\beta}
\\& \leq & C \cdot tr_g(h) \cdot |D\varphi'|_g^2.
\end{array}$$ Here $C$ is a uniform constant. From inequality (\[eq:patch0\]) and the fact $g$ is bounded from above, we have $$\int_{V\times I} \mid \nabla \;log {{det \,g} \over {det\,h}}\mid_h^2
= \int_{V\times I} \mid \nabla \;log f \mid_h^2 \leq C \cdot \int_{V\times I} \mid \nabla log f \mid_g^2 det \,g d\,t \leq C.$$ and $$\int_{V\times I} \left({{det \;h}\over {det\;g}}\right)^{1\over n} \; det \;h \leq \int_{V\times I} \; tr_g (h) \;det\;h \leq C.$$ From now on, all of the norm, inner product and integration are taken w.r.t. metric $h + d\,t^2$ unless otherwise specified. Now define a new vector field $Y$ as $$Y = X \cdot {{det \;g}\over {det\;h}}.$$ Then $$|Y|_h = |X|_h {{det \;g}\over {det\;h}} \leq C.$$ In other words, $Y$ has uniform $L^{\infty}$ bound. This implies that $Y\cdot {{\partial \ln {{det \;g}\over { det\; h}}} \over {\partial \overline{z}}}$ has uniform $L^q$ bound for any $1<q < 2.\;$ Moreover, for any $1< q < 2,$ we have $$\begin{array} {lcl} \displaystyle \int_{V\times I} \mid {{\partial Y}\over {\partial \overline{z}}} - Y \cdot {{\partial \ln {{det \;g}\over { det\; h}}} \over {\partial \overline{z}}}\mid_h^q & = & \displaystyle \int_{V\times I} \left( \mid
{{\partial X}\over {\partial \overline{z}}} \mid_h {{det \;g}\over {det\;h}}\right)^q \\ & \leq & \displaystyle \int_{V\times I} \;(\sqrt{tr_g (h)} {{det \;g}\over {det\;h}})^q\; |D\varphi'|_g^q
\\ &\leq & \displaystyle \int_{V\times I} C \;|D\varphi'|_g^q \rightarrow 0.
\end{array}$$ This immediately implies that ${{\partial Y}\over {\partial \overline{z}}}
$ are uniformly bounded in $L^q$ for any $1<q < 2.\;$\
Now, all of these quantities, $X, Y, {{\partial Y}\over {\partial \overline{z}}},\;$ and $ {{det \;g}\over {det\;h}},\cdots$ are geometrical quantities which depend on $\epsilon.\;$ Since their respective soblev norms are uniformly controlled, we can take weak limits of these quantities in some appropriate sense. Denote the corresponding weak limits (when $\epsilon \rightarrow 0$) as $X, Y, {{det \;g}\over {det\;h}}, \cdots.\;$ Then $X(\epsilon) \rightharpoonup X$ weakly in $L^2(V \times I),\; Y (\epsilon) \rightharpoonup X$ weakly in $L^{\infty}(V\times I)\;$ and $ {{\det\;g} \over {det\;h}} (\epsilon)
\rightharpoonup {{det\;g} \over {det\;h}}$ weakly in $L^{\infty} (V\times I),\cdots.\;$\
Consider $u = \ln {{det h}\over {det\;g}}.\; $ For simplicity, assume $ u > 0$ (otherwise $u > -c$ for some positive constants). Then the following two equations holds in the limit $${{\partial Y}\over {\partial \overline{z}}} + Y \cdot {{\partial u} \over {\partial \overline{z}}} = 0,\qquad {\rm and}\qquad Y= X \;e^{-u}$$ in the sense of $L^q (V \times I)$ for any $1 < q < 2.\;$ Moreover, we have the following estimates: $$\int_{V \times I} e^{{1\over n} u} \leq C; \qquad \int_{V \times I} |{{\partial u}\over{\partial \overline{z}}}|^2 \leq C;\qquad {\rm and}\; \int_{V \times I} |X|^2 \leq C.$$
Now define a new sequence of vectors $X_{,k} (k =1,2,\cdots )$ as $X_{,k} = Y \displaystyle \sum_{i=0}^k {{u^i}\over {i!}}.\; $ This is well defined since $u$ is in $L^p (V \times I) $ for any $p>1.\;$ Then $$\begin{array} {lcl}
|X_{,k}| & = & |Y| \displaystyle \sum_{i=0}^k {{u^i}\over {i!}}\\
& \leq & (|X| e^{-u}) e^{u} \leq |X|.
\end{array}$$ The equality holds in the last inequality whenever $e^{-u} \neq 0.\;$ Thus $$\displaystyle \int_{V \times I} \; |X_{,k}|^2 \leq \int_{V \times I} |X|^2 \leq C.$$ By definition, it is clear $\| X_{,k} \|_{L^2 (V \times I)} \leq
\| X_{,m} \|_{L^2 (V \times I)}$ whenever $ k \leq m.\;$ Thus, there exists a positive number $A \leq \| X \|_{L^2 (V \times I)} $ such that $\displaystyle \lim_{ k \rightarrow \infty} \|X_{,k}\|_{L^2 (V \times I)} = A.\; $ For $m > k$, we have $$\begin{array} {lcl}
\| X_{,m} \|_{L^2 (V \times I)}^2 & = & \displaystyle \int_{V \times I}\; |X_{,m}|^2 \\
& = & \displaystyle \int_{V \times I} \; |Y|^2 (\displaystyle \sum_{i=0}^m {{u^i}\over {i!}})^2 \geq
\displaystyle \int_{V \times I} \; |Y|^2 \left( (\displaystyle \sum_{i=0}^k {{u^i}\over {i!}})^2 + (\displaystyle \sum_{i=k+1}^m {{u^i}\over {i!}})^2 \right) \\
& = & \displaystyle \int_{V\times I}\; (|X_{,k}|^2 + |X_{,m} - X_{,k}|^2) = \| X_{,k} \|_{L^2 (V \times I)}^2 + \| X_{,m} - X_{,k} \|_{L^2 (V \times I)}^2.
\end{array}$$ Taking limits as $m,k \rightarrow \infty,$ we have $ \| X_{,m} - X_{,k} \|_{L^2 (V \times I)}^2 \rightarrow 0.\;$ Thus $X_{,k}\; (k =1,2, \cdots )$ is a Cauchy sequence in $L^2 (V \times I)\;$ and there exists a strong limit $X_{,\infty} $ in $L^2 (V \times I).\;$ By definition, we know that $X_{,\infty} = X$ almost everywhere in $V \times I$ [^9]. We want to show that $X_{,\infty}$ is weakly holomorphic in $V-$ direction. A straightforward calculation yields $$\begin{array}{lcl} {{\partial X_{,k}}\over {\partial \overline{z}}} & = & {{\partial Y}\over {\partial \overline{z}}} \displaystyle \sum_{i=0}^k {{u^i}\over {i!}}
+ Y {{\partial }\over {\partial \overline{z}}} \left(\displaystyle \sum_{i=0}^k {{u^i}\over {i!}} \right) \\
& = & - Y {{\partial u}\over {\partial \overline{z}}} \displaystyle \sum_{i=0}^k {{u^i}\over {i!}} + Y \left(\displaystyle \sum_{i=0}^{k-1} {{u^i}\over {i!}}\right) \;
{{\partial u}\over {\partial \overline{z}}} \\
& = & - ( X_{,k} - X_{,k-1}) {{\partial u}\over {\partial \overline{z}}}.
\end{array}$$ We want to show that ${{\partial X_{,\infty}}\over {\partial \overline{z}}} =0$ in the sense of distribution. We just need to show it in any open set $U \times I$ where $U$ is a coordinate chart in $V.\;$ Denote $(z_1, z_2, \cdots z_n)$ as coordinate variable in $U.\;$ Then, for any vector valued smooth function $\psi= (\psi^1,\psi^2,\cdots \psi^n)$ which vanish in $\partial (U\times I),\;$ and for any $1\leq j\leq n.\;$ we have $$\begin{array} {lcl}
\mid \displaystyle \int_{V \times I} \; X_{,k} \cdot {{\partial \overline{\psi}}\over {\partial \overline{z_j}}} \mid & = & \mid - \displaystyle \int_{V \times I} \;{{\partial X_{,k}}\over {\partial \overline{z_j}}}\cdot \overline{\psi} \mid \\
& = & \mid \displaystyle \int_{V \times I} \;( X_{,k} - X_{,k-1}) {{\partial u}\over {\partial \overline{z_j}}} \overline{\psi} \mid \\ & \leq & C \cdot \| X_{,k} -X_{,k-1}\|_{L^2 (V\times I)} \cdot \sqrt {\displaystyle \int_{V\times I} |\nabla u|^2} \\ & \leq & C \| X_{,k} -X_{,k-1}\|_{L^2 (V\times I)}.
\end{array}$$ Now, taking limit as $k \rightarrow \infty,$ we have $$\displaystyle \int_{V \times I} \; X_{,\infty} \cdot {{\partial \overline{\psi}}\over {\partial \overline{z_j}}} = 0,\qquad {\rm for \; any \;} j =1,2,\cdots n$$ and for any smooth vector valued function $\psi= (\psi^1,\psi^2,\cdots \psi^n)$ which vanish in $\partial (U \times I).\;$ Thus, $X_{,\infty}$ is a weak holomorphic vector field in $V$ direction for almost all $t$.
Now recalls that $$\int_{V\times I} |X_{,\infty}|_h^2 \;det\, h \;d\,t < C.$$ This implies that $X_{,\infty}$ is in $L^2(V \times \{t\})$ for almost all $t \in [0,1].\;$ Since $X_{,\infty}$ is weakly holomorphic in $V \times \{t\}$ for all $t,$ thus $X_{,\infty}$ must be holomorphic for those $t$ where $X_{,\infty}$ is in $L^2(V\times \{t\}).\;$ However, there is no holomorphic vector field in $V$ since $C_1 < 0.\;$ Thus $X_{,\infty} \equiv 0$ for all of those $t$ where $X_{,\infty}$ is in $L^2(V\times \{t\}).\;$ This implies that $X_{,\infty} = 0$ in $V\times I.\;$ Thus $X=0$ since $X = X_{,\infty}$ in the sense of $L^q(V\times I)$ for any $1 < q < 2.\;$ Recall $${{\partial \varphi'(t)}\over {\partial z_{\alpha}}} = \displaystyle \sum_{\beta=1}^n\; g_{\alpha \overline{\beta}} {{X}}^{\overline{\beta}} = \displaystyle \sum_{\beta=1}^n\; g_{\alpha \overline{\beta}} {{X_{,\infty}}}^{\overline{\beta}} = 0.$$ In other words, $\varphi'(t)$ is trivial in $V-$direction and it is a function of $t$ only for all $t \in [0,1].\; $ Thus, $\varphi(0) $ and $\varphi(1)$ differ only by a constant in $V$ direction. Therefore they represent same metric in each Kähler class.
[^1]: Research was supported partially by NSF postdoctoral fellowship.
[^2]: Around the same time with Mabuchi’s work, Bourguignon J. P. has worked on something similar in a related subject [@Bourg85].
[^3]: Here we mean the mixed second derivatives is uniformly bounded. See theorem 3 in section 3 for details.
[^4]: The sufficient part of this result was proved in [@DingTian93].
[^5]: Tian inform us that he[@tian98] has conjectured that constant scalar curvature metrics exist if and only if Mabuchi functional is proper.
[^6]: The [*moment map*]{} point of view here was also observed by A. Fujiki[@fujiki92].
[^7]: By definition, for any $\varphi_0 \in {\cal H},$ $\displaystyle \sum_{\alpha,\beta = 1}^{n+1} (g_{\alpha \overline{\beta}} + {{\partial^2 \varphi_0}\over{\partial z_{\alpha} \partial \overline{z}_{\beta}}}) dz_{\alpha} d\,\overline{z}_{\beta} $ is strictly positive Kähler metric in each $V-$ slice $V \times \{w\}.\;$ Let $\Psi$ be a strictly convex function of $w$ which vanishes on $\partial {\bf R}.\;$ Then for large enough constants $m , \displaystyle \sum_{\alpha,\beta = 1}^{n+1} (g_{\alpha \overline{\beta}} + {{\partial^2 (\varphi_0 + m \Psi) }\over{\partial z_{\alpha} \partial \overline{z}_{\beta}}}) dz_{\alpha} d\,\overline{z}_{\beta} $ is a strictly positive Kähler metric in $V \times {\bf R}.\;$
[^8]: In [@Dona96], Donaldson provided a formal proof to this proposition after assuming the existence of a smooth geodesic between any two metrics. Our proof follows his idea closely.
[^9]: It is easy to prove that $X_{,\infty} = X$ in the sense of $L^q (V\times I) $ for any $ (1<q<2).$
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Cross-modal food retrieval is an important task to perform analysis of food-related information, such as food images and cooking recipes. The goal is to learn an embedding of images and recipes in a common feature space, so that precise matching can be realized. Compared with existing cross-modal retrieval approaches, two major challenges in this specific problem are: 1) the large intra-class variance across cross-modal food data; and 2) the difficulties in obtaining discriminative recipe representations. To address these problems, we propose *Semantic-Consistent and Attention-based Networks* (SCAN), which regularize the embeddings of the two modalities by aligning output semantic probabilities. In addition, we exploit self-attention mechanism to improve the embedding of recipes. We evaluate the performance of the proposed method on the large-scale Recipe1M dataset, and the result shows that it outperforms the state-of-the-art.'
author:
- |
Hao Wang$^{\flat,}$[^1] Doyen Sahoo$^{\ddag,*}$ Chenghao Liu$^\dag$ Ke Shu$^\dag$ Palakorn Achananuparp$^\dag$\
Ee-peng Lim$^\dag$ Steven C. H. Hoi$^{\dag,\ddag}$\
$^\flat$Nanyang Technological University$^\dag$Singapore Management University$^\ddag$Salesforce Research Asia\
[[email protected] [email protected] {chliu, keshu, palakorna, eplim, chhoi}@smu.edu.sg]{}
bibliography:
- 'egbib.bib'
title: 'Cross-Modal Food Retrieval: Learning a Joint Embedding of Food Images and Recipes with Semantic Consistency and Attention Mechanism'
---
=1
Introduction
============
In recent years, cross-modal retrieval has drawn much attention with the rapid growth of multimodal data. In this paper we address the problem of cross-modal food retrieval based on large-amount of heterogeneous food dataset [@salvador2017learning], i.e. take cooking recipes (ingredients & cooking instructions) as the query to retrieve the food images, and vice versa. There have been many works [@salvador2017learning; @carvalho2018cross; @wang2019learning] on cross-modal food retrieval. Despite those efforts, cross-modal food retrieval remains challenging mainly due to the following two reasons: 1) the large intra-class variance across food data pairs; and 2) the difficulties of obtaining discriminative recipe representation.
In cross-modal food data, given a recipe, we have many food images that are cooked by different chefs. Besides, the images from different recipes can look very similar because they have similar ingredients. Hence, the data representation from the same food can be different, but different food may have similar data representations. This leads to large intra-class variance but small inter-class variance in food data. Existing studies [@carvalho2018cross; @chen2018deep; @chen2017cross] only addressed the small inter-class variance problem by utilizing triplet loss to measure the similarities among cross-modal data. Specifically, the objective of this triplet loss is to make inter-class feature distance larger than intra-class feature distance by a predefined margin [@cheng2016person]. Therefore, cross-modal instances from the same class may form a loose cluster with large average intra-class distance. As a consequence, it eventually results in less-than-optimal ranking, i.e., irrelevant images are closer to the queried recipe than relevant images. See Figure \[fig:motivation\] for an example.
Besides, many recipes share common ingredients in different food. For instance *fruit salad* has ingredients of *apple*, *orange* and *sugar* etc., where *apple* and *orange* are the main ingredients, while *sugar* is one of the ingredients in many other foods. If the embeddings of ingredients are treated equally during training, the features learned by the model may not be discriminative enough. The cooking instructions crawled from cooking websites tend to be noisy, some instructions turn out irrelevant to cooking e.g. *’Enjoy!’*, which convey no information for cooking instruction features but degrade the performance of cross-modal retrieval task. In order to find the attended ingredients, Chen et al. [@chen2017cross] apply a two-layer deep attention mechanism, which learns joint features by locating the visual food regions that correspond to ingredients. However, this method relies on high-quality food images and essentially increases the computational complexity.
To resolve those issues, we propose a novel unified framework of **S**emantic-**C**onsistent and **A**ttention-Based **N**etwork (SCAN) to improve the cross-modal food retrieval performance. The pipeline of the framework is shown in Figure \[fig:pipeline\]. To reduce the intra-class variance, we introduce a semantic consistency loss, which imposes Kullback-Leibler (KL) Divergence to minimize the distance between the output semantic probabilities of paired image and recipe. In order to obtain discriminative recipe representations, we combine self-attention mechanism [@vaswani2017attention] with LSTM to find the key ingredients and cooking instructions for each recipe. Without requiring food images or adding extra layers, we can learn better discriminative recipe embeddings, compared to that trained with plain LSTM.
Our work makes two major contributions as follows:
- We introduce a semantic consistency loss to cross-modal food retrieval task and the result shows that it can align cross-modal matching pairs and reduce the intra-class variance of food data representations.
- We integrate self-attention mechanism with LSTM, and learn discriminative recipe features without requiring the food images. It is useful to discriminate samples of similar recipes.
We perform an extensive experimental analysis on Recipe1M, which is the largest cross-modal food dataset and available in public. We find that our proposed cross-modal food retrieval approach SCAN outperforms state-of-the-art methods. Finally, we show some visualizations of the retrieved results.
Related Work
============
Our work is closely related to cross-modal retrieval. Canonical Correlation Analysis (CCA) [@hotelling1936relations] is a representative work in cross-modal data representation learning, it utilizes global alignment to allow the mapping of different modalities which are semantically similar to be close in the common space. Many recent works [@srivastava2012multimodal; @feng2014cross] utilize deep architectures for cross-modal retrieval, which have the advantage of capturing complex non-linear cross-modal correlations. With the advent of generative adversarial networks (GANs) [@goodfellow2014generative], which is helpful in modeling the data distributions, some adversarial training methods [@wang2019learning; @ghasedi2018unsupervised] are frequently used for modality fusion.
For cross-modal food retrieval, [@chen2016deep; @min2017being] are early works in cross-modal food retrieval. [@chen2016deep] uses vanilla attention mechanism with extra learnable layers, and only test their model in a small-scale dataset. [@min2017being] utilize a multi-modal Deep Boltzmann Machine for recipe-image retrieval.
[@chen2017cross; @chen2018deep] both integrate attention mechanism into cross-modal retrieval, [@chen2017cross] introduce a stacked attention network (SAN) to learn joint space from images and recipes for cross-modal retrieval. Consequently, [@chen2018deep] make full use of the ingredient, cooking instruction and title (category label) information of Recipe1M, and concatenate the three types of features above to construct the recipe embeddings. Compared with the self-attention mechanism we adopt in our model, the above two works increase the computational complexity by depending on the food images or adding some extra learnable layers and weights.
In order to have a better regularization on the shared representation space learning, some researchers corporate the semantic labels with the joint training. [@salvador2017learning] develop a hybrid neural network architecture with a cosine embedding loss for retrieval learning and an auxiliary cross-entropy loss for classification, so that a joint common space for image and recipe embeddings can be learned for cross-modal retrieval. [@carvalho2018cross] is an extended version of [@salvador2017learning], by providing a double-triplet strategy to express both the retrieval loss and the classification loss. [@wang2019learning] preserve the semantic information on the learned cross-modal features with generative approaches. Here we include the KL loss into the training, and preserve well the semantic information across image-recipe pairs. We also adopt the self-attention mechanism for discriminative feature representations and achieve better performance.
Proposed Methods
================
In this section, we introduce our proposed model, where we utilize food image-recipe paired data to learn cross-modal embeddings as shown in Figure \[fig:pipeline\].
Overview
--------
We formulate the proposed cross-modal food retrieval with three networks, i.e. one convolutional neural network (CNN) for food image embeddings, and two LSTMs to encode ingredients and cooking instructions respectively. The food image representations $I$ can be obtained from the output of CNN directly, while the recipe representations $R$ come from the concatenation of the ingredient features $f_{ingredient}$ and instruction features $f_{instruction}$. Specifically, for obtaining discriminative ingredient and instruction embeddings, we integrate the self-attention mechanism [@vaswani2017attention] into the LSTM embedding. Triplet loss is used as the main loss function $L_{Ret}$ to map cross-modal data to the common space, and semantic consistency loss $L_{SC}$ is utilized to align cross-modal matching pairs for retrieval task, reducing the intra-class variance of food data. The overall objective function of the proposed SCAN is given as: $$\begin{aligned}
L = L_{Ret} + \lambda L_{SC},
\label{total_eq}
\end{aligned}$$
Recipe Embedding
----------------
We use two LSTMs to get ingredient and instruction representations $f_{ingredient}$, $f_{instruction}$, concatenate them and pass through a fully-connected layer to give a 1024-dimensional feature vector, as the recipe representation $R$.
### Ingredient Representation Learning
Instead of word-level word2vec representations, ingredient-level word2vec representations are used in ingredient embedding. To be specific, *ground ginger* is regarded as a single word vector, instead of two separate word vectors of *ground* and *ginger*.
We integrate self-attention mechanism with LSTM output to construct recipe embeddings. The purpose of applying self-attention model lies in assigning higher weights to main ingredients for different food items, making the attended ingredients contribute more to the ingredient embedding, while reducing the effect of common ingredients.
Given an ingredient input $\{z_1, z_2, ..., z_n\}$, we first encode it with pretrained embeddings from word2vec algorithm to obtain the ingredient representation $Z_t$. Then $\{Z_1, Z_2, ..., Z_n\}$ will be fed into the one-layer bidirectional LSTM as a sequence step by step. For each step $t$, the recurrent network takes in the ingredient vector $Z_t$ and the output of previous step $h_{t-1}$ as the input, and produces the current step output $h_t$ by a non-linear transformation, as follow:
$$\begin{aligned}
h_t = \mathrm{tanh}(\textbf{W}{Z_t} + \textbf{U}{h_{t-1}} + b),
\end{aligned}$$
The bidirectional LSTM consists of a forward hidden state $\overrightarrow{h_t}$ which processes ingredients from $Z_1$ to $Z_n$ and a backward hidden state $\overleftarrow{h_t}$ which processes ingredients from $Z_n$ to $Z_1$. We obtain the representation $h_t$ of each ingredient $z_t$ by concatenating $\overrightarrow{h_t}$ and $\overleftarrow{h_t}$, i.e. $h_t=[\overrightarrow{h_t}, \overleftarrow{h_t}]$, so that the representation of the ingredient list of each food item is $H = \{h_1, h_2, ..., h_n\}$.
We further measure the importance of ingredients in the recipe with the self-attention mechanism which has been studied in Transformer [@vaswani2017attention], where the input comes from queries $Q$ and keys $K$ of dimension $d_k$, and values $V$ of dimension $d_v$ (the definition of $Q$, $K$ and $V$ can be referred in [@vaswani2017attention]), we compute the attention output as:
$$\begin{aligned}
\mathrm{Attention}(Q, K, V) = \mathrm{softmax}(\frac{QK^T}{\sqrt{d_k}})V,
\end{aligned}$$
Different from the earlier attention-based methods [@chen2018deep], we use self-attention mechanism where all of the keys, values and queries come from the same ingredient representation $H$. Therefore, the computational complexity is reduced since it is not necessary to add extra layers to train attention weights. The ingredient attention output $H_{attn}$ can be formulated as:
$$\begin{aligned}
H_{attn} = \mathrm{Attention}(H, H, H) \\
= \mathrm{softmax}(\frac{HH^T}{\sqrt{d_h}})H,
\end{aligned}$$
where $d_h$ is the dimension of $H$. In order to enable unimpeded information flow for recipe embedding, skip connections are used in the attention model. Layer normalization [@ba2016layer] is also used since it is effective in stabilizing the hidden state dynamics in recurrent network. The final ingredient representation $f_{ingredient}$ is generated from summation of $H$ and $H_{attn}$, which can be defined as:
$$\begin{aligned}
f_{ingredient} = \mathrm{LayerNorm}(H_{attn} + H),
\end{aligned}$$
### Instruction Representation Learning
Considering that cooking instructions are composed of a sequence of variable-form and lengthy sentences, we compute the instruction embedding with a two-stage LSTM model. For the first stage, we apply the same approach as [@salvador2017learning] to obtain the representations of each instruction sentence, in which it uses skip-instructions [@salvador2017learning] with the technique of skip-thoughts [@kiros2015skip].
The next stage is similar to the ingredient representation learning. We feed the pre-computed fixed-length instruction sentence representation into the LSTM model to generate the hidden representation of each cooking instruction sentence. Based on that, we can obtain the self-attention representation. The final instruction feature $f_{instruction}$ is generated from the layer normalization function on the previous two representations, as we formulate in the last section. By doing so, we are able to find the key sentences in cooking instruction. Some visualizations on attended ingredients and cooking instructions can be found in Section \[attention\_vis\].
Image Embedding
---------------
We use ResNet-50 [@he2016deep] pretrained on ImageNet to encode food images. We empirically remove the last layer, which is used for ImageNet classification, and add an extra FC layer to obtain the final food image features $I$. The dimension of features $I$ is 1024.
Cross-modal Food Retrieval Learning
-----------------------------------
Triplet loss is utilized to do retrieval learning, the objective function is:
$$\begin{aligned}
L_{Ret} = & \sum_V\left[d(I_{a}, R_{p})-d(I_{a}, R_{n})+\alpha\right]_+ \; \\
& + \sum_R\left[d(R_{a}, I_{p})-d(R_{a}, I_{n})+\alpha\right]_+,
\end{aligned}$$
where $d(\bullet)$ is the Euclidean distance, subscripts $a,p$ and $n$ refer to anchor, positive and negative samples respectively and $\alpha$ is the margin. To improve the effectiveness of training, we adopt the *BatchHard* idea proposed in [@hermans2017defense]. Specifically in a mini-batch, given an anchor sample, we simply select the most distant positive instance and the closest negative instance, to construct the triplet.
Semantic Consistency
--------------------
Given the pairs of food image and recipe representations $I$, $R$, we first transform $I$ and $R$ into $I'$ and $R'$ for classification with an extra FC layer. The dimension of the output is same as the number of categories $N$. The probabilities of food category $i$ can be computed by a softmax activation as:
$$\begin{aligned}
p_{i}^{img} = \frac{exp(I_{i}')}{\sum_{i=1}^{N} exp(I_{i}')},
\end{aligned}$$
$$\begin{aligned}
p_{i}^{rec} = \frac{exp(R_{i}')}{\sum_{i=1}^{N} exp(R_{i}')},
\end{aligned}$$
where $N$ represents the total number of food categories, and it is also the dimension of $I'$ and $R'$. Given $i \in \{1, 2, ..., N\}$, with the probabilities of $p_{i}^{img}$ and $p_{i}^{rec}$ for each food category, the predicted label $l^{img}$ and $l^{rec}$ can be obtained. We formulate the classification (cross-entropy) loss as $L_{cls}(p^{img}, p^{rec}, c^{img}, c^{rec})$, where $c^{img}$, $c^{rec}$ are the ground-truth class label for food image and recipe respectively.
In the previous work [@salvador2017learning], $L_{cls}(p^{img}, p^{rec}, c^{img}, c^{rec})$ consists of $L^{img}_{cls}$ and $L^{rec}_{cls}$, which are treated as two independent classifiers, focusing on the regularization on the embeddings from food images and recipes separately. However, food image and recipe embeddings come from heterogeneous modalities, the output probabilities of each category can be significantly different. As a result, the distance of intra-class features remains large. To improve image-recipe matching and make the probabilities predicted by different classifiers consistent, we minimize Kullback-Leibler (KL) Divergence between the probabilities $p_{i}^{img}$ and $p_{i}^{rec}$ of paired cross-modal data, which can be formulated as:
$$\begin{aligned}
L_{KL}({p}^{img}\|{p}^{rec}) = \sum_{i=1}^{N} \ p_i^{img} \log \frac{p_i^{img}}{p_i^{rec}},
\end{aligned}$$
$$\begin{aligned}
L_{KL}({p}^{rec}\|{p}^{img}) = \sum_{i=1}^{N} \ p_i^{rec} \log \frac{p_i^{rec}}{p_i^{img}},
\end{aligned}$$
The overall semantic consistency loss $L_{SC}$ is defined as:
$$\begin{aligned}
L_{SC} = \{( L^{img}_{cls} + L_{KL}({p}^{rec}\|{p}^{img})) \; \\
+ ( L^{rec}_{cls}+ L_{KL}({p}^{img}\|{p}^{rec}))\} / 2.
\end{aligned}$$
\[tab:results\]
Experiments
===========
Dataset {#Dataset}
-------
We conduct extensive experiments to evaluate the performance of our proposed methods in Recipe1M dataset [@salvador2017learning], the largest cooking dataset with recipe and food image pairs available to the public. Recipe1M was scraped from over 24 popular cooking websites and it not only contains the image-recipe paired labels but also more than half amount of the food data with semantic category labels extracted from food titles on the websites. The category labels provide semantic information for cross-modal retrieval task, making it fit in our proposed method well. The paired labels and category labels construct the hierarchical relationships among the food. One food category (e.g. *fruit salads*) may contain hundreds of different food pairs, since there are a number of recipes of different *fruit salads*.
We perform cross-modal the food retrieval task based on food data pairs, i.e. when we take the recipes as the query to do retrieval, the ground truth will be the food images in food data pairs, and vice versa. We use the original Recipe1M data split [@salvador2017learning], containing 238,999 image-recipe pairs for training, 51,119 and 51,303 pairs for validation and test, respectively. In total, the dataset has 1,047 categories.
Evaluation Protocol
-------------------
We evaluate our proposed model with the same metrics used in prior works [@salvador2017learning; @chen2018deep; @carvalho2018cross; @wang2019learning]. To be specific, median retrieval rank (MedR) and recall at top K (R@K) are used. MedR measures the median rank position among where true positives are returned. Therefore, higher performance comes with a lower MedR score. Given a food image, R@K calculates the fraction of times that the correct recipe is found within the top-K retrieved candidates, and vice versa. Different from MedR, the performance is directly proportional to the score of R@K. In the test phase, we first sample 10 different subsets of 1,000 pairs (1k setup), and 10 different subsets of 10,000 (10k setup) pairs. It is the same setting as in [@salvador2017learning]. We then consider each item from food image modality in subset as a query, and rank samples from recipe modality according to L2 distance between the embedding of image and that of recipe, which is served as image-to-recipe retrieval, and vice versa for recipe-to-image retrieval.
Implementation Details
----------------------
We set the trade-off parameter $\lambda$ in Eq. based on empirical observations, where we tried a range of values and evaluated the performance on the validation set. We set the $\lambda$ as 0.05. The model was trained using Adam optimizer [@kingma2014adam] with the batch size of 64 in all our experiments. The initial learning rate is set as 0.0001, and the learning rate decreases 0.1 in the 30th epoch. Note that we update the two sub-networks, i.e. image encoder $E_I$ and recipe encoder $E_R$, alternatively. It only takes 40 epochs to get the best performance with our proposed methods, while [@salvador2017learning] requires 220 epochs to converge. Our training records can be viewed in Figure \[fig:training\]. We do our experiments on a single Tesla V100 GPU, which costs about 16 hours to finish the training.
Baselines
---------
We compare the performance of our proposed methods with several state-of-the-art baselines, and the results are shown in Table \[tab:results\].
**CCA [@hotelling1936relations]:** Canonical Correlation Analysis (CCA) is one of the most widely-used classic models for learning a common embedding from different feature spaces. CCA learns two linear projections for mapping text and image features to a common space that maximizes their feature correlation.
**SAN [@chen2017cross]:** Stacked Attention Network (SAN) considers ingredients only (and ignores recipe instructions), and learns the feature space between ingredient and image features via a two-layer deep attention mechanism.
**JE [@salvador2017learning]:** They [@salvador2017learning] use pairwise cosine embedding loss to find a joint embedding (JE) between the different modalities. To impose regularization, they add classifiers to the cross-modal embeddings which predict the category of a given food data.
**AM [@chen2018deep]:** Attention mechanism (AM) over the recipe is adopted in [@chen2018deep], applied at different parts of a recipe (title, ingredients and instructions). They use an extra transformation matrix and context vector in the attention model.
**AdaMine [@carvalho2018cross]:** A double triplet loss is used, where triplet loss is applied to both the joint embedding learning and the auxiliary classification task of categorizing the embedding into an appropriate category. They also integrate the adaptive learning schema (AdaMine) into the training phase, which performs an adaptive mining for significant triplets.
**ACME [@wang2019learning]:** Adversarial training methods are utilized in ACME for modality alignment, to make the feature distributions from different modalities to be similar. In order to further preserve the semantic information in the cross-modal food data representation, Wang et al. introduce a translation consistency component.
In summary, our proposed model SCAN is lightweight and effective and outperforms all of earlier methods by a margin, as is shown in Table \[tab:results\].
\[tab:ablation\]
{width="\textwidth"}
Ablation Studies
----------------
Extensive ablation studies are conducted to evaluate the effectiveness of each component of our proposed model. Table \[tab:ablation\] illustrates the contributions of self-attention model (**SA**), semantic consistency loss (**SC**) and their combination on improving the image to recipe retrieval performance.
**TL** serves as a baseline for **SA**, which adopts the *BatchHard* [@hermans2017defense] training strategy. We then add **SA** and **SC** incrementally, and significant improvements can be found in both of the two components. To be specific, integrating **SA** into **TL** helps improve the performance of the image-to-recipe retrieval more than 4% in R@1, illustrating the effectiveness of self-attention mechanism to learn discriminative recipe representations. The model trained with triplet loss and classification loss (**cls**) used in [@salvador2017learning] is another baseline for **SC**. It shows that our proposed semantic consistency loss improves the performance in R@1 and R@10 by more than 2%, which suggests that reducing intra-class variance can be helpful in cross-modal retrieval task.
We show the training records in Figure \[fig:training\], in the left figure, we can see that for the first 20 epochs, the performance gap between **TL** and **TL+SA** gets larger, while the performance of **TL+cls** and **TL+SC** keeps to be similar, which is shown in the middle figure. But for the last 20 training epochs, the performance of **TL+SC** improves significantly, which indicts that for those hard samples whose intra-variance can hardly be reduced by **TL+cls**, **TL+SC** contributes further to the alignment of paired cross-modal data.
In conclusion, we observe that each of the proposed components improves the cross-modal embedding model, and the combination of those components yields better performance overall.
Recipe-to-Image Retrieval Results
---------------------------------
We show three recipe-to-image retrieval results in Figure \[fig:re2im\]. In the top row, we select a recipe query *dessert* from Recipe1M dataset, which has the ground truth for retrieved food images. Images with the green box are the correctly retrieved ones, which come from the retrieved results by SCAN and TL+SA. But we can see that the model trained only with semantic consistency loss (TL+SC) has a reasonable retrieved result as well, which is relevant to the recipe query.
In the middle and bottom row, we remove some ingredients and the corresponding cooking instruction sentences in the recipe, and then construct the new recipe embeddings for the recipe-to-image retrieval. In the bottom row where we remove the *walnuts*, we can see that all of the retrieved images have no *walnuts*. However, only the image retrieved by our proposed SCAN reflects the richest recipe information. For instance, the image from SCAN remains visible ingredients of *frozen whipped topping*, while images from TL+SC and TL+SA have no *frozen whipped toppings*.
The recipe-to-image retrieval results might indicate an interesting way to satisfy users’ needs to find the corresponding food images for their customized recipes.
![Visualizations of image-to-recipe retrieval. We show the retrieved recipes of the given food images, along with the attended ingredients and cooking instruction sentences.[]{data-label="fig:attn_vis"}](fig/attn_vis.pdf){width="50.00000%"}
Image-to-Recipe Retrieval Results & Effect of Self-Attention model {#attention_vis}
------------------------------------------------------------------
In this section, we show some of the image-to-recipe retrieval results in Figure \[fig:attn\_vis\] and then focus on analyzing the effect of our self-attention model. Given images from *cheese cake*, *meat loaf* and *salad*, we show the retrieved recipe results by SCAN, which are all correct. We visualize the attended ingredients and instructions for the retrieved recipes with the yellow background, where we choose the ingredients and cooking instruction sentences of the top 2 attention weights as the attended ones. We can see that some frequently used ingredients like *water*, *milk*, *salt*, etc. are not attended with high weights, since they are not visible and shared by many kinds of food, which cannot provide enough discriminative information for cross-modal food retrieval. This is an intuitive explanation for the effectiveness of our self-attention model.
Another advantage of using self-attention mechanism is that the image quality cannot affect the attended outputs. Obviously, the top two rows of food images *cheese cake* and *meat loaf* do not possess good image quality, while our self-attention model still outputs reasonable attended results. This suggests that our proposed attention model has good capabilities to capture informative and reasonable parts for recipe embedding.
Effect of Semantic Consistency
------------------------------
In order to have a concrete understanding of the ability of our proposed semantic consistency loss on reducing the mean intra-class feature distance (intra-class variance) between paired food image and recipe representations, we show the difference of the intra-class feature distance on cross-modal data trained with and without semantic consistency loss, i.e. SCAN and TL, in Figure \[fig:SCC\_vis\]. In the test set, we select the recipe and food image data from *chocolate chip*, which in total has 425 pairs. We obtain the food data representations from models trained with two different methods, then we compute the Euclidean distance between paired cross-modal data to obtain the mean intra-class feature distance. We adopt t-SNE [@maaten2008visualizing] to do dimensionality reduction to visualize the food data.
It can be observed that cross-modal food data which is trained with semantic consistency loss (SCAN) has smaller intra-class variance than that trained without semantic consistency loss (TL). This means that semantic consistency loss is able to correlate paired cross-modal data representations effectively by reducing the intra-class feature distance, and also our experiment results suggest its efficacy.
![The difference on the intra-class feature distance of cross-modal paired data trained with and without semantic consistency loss. The food data is selected from the same category, *chocolate chip*. **SCAN** obtains closer image-recipe feature distance than **TL**. (Best viewed in color.)[]{data-label="fig:SCC_vis"}](fig/SCC_vis_new.pdf){width="50.00000%"}
Conclusion
==========
In conclusion, we propose SCAN, a lightweight and effective training framework, for cross-modal food retrieval. It introduces a novel semantic consistency loss and employs a self-attention mechanism to learn the joint embedding between food images and recipes for the first time. To be specific, we apply semantic consistency loss on cross-modal food data pairs to reduce the intra-class variance, and utilize self-attention mechanism to find the important parts in the recipes to construct discriminative recipe representations. SCAN is easy to implement and can extend to other general cross-modal datasets. We have conducted extensive experiments and ablation studies. We achieved state-of-the-art results in Recipe1M dataset.
[^1]: Work done in Singapore Management University
|
{
"pile_set_name": "ArXiv"
}
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---
author:
-
bibliography:
- 'IEEEabrv.bib'
- 'bibliography.bib'
title: Deep Reinforcement Learning Autoencoder with Noisy Feedback
---
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We present a homogeneous kinematic analysis of red giant branch stars within 18 of the 28 Andromeda dwarf spheroidal (dSph) galaxies, obtained using the Keck I LRIS and Keck II DEIMOS spectrographs. Based on their $g-i$ colors (taken with the CFHT MegaCam imager), physical positions on the sky, and radial velocities, we assign probabilities of dSph membership to each observed star. Using this information, the velocity dispersions, central masses and central densities of the dark matter halos are calculated for these objects, and compared with the properties of the Milky Way dSph population. We also measure the average metallicity (\[Fe/H\]) from the co-added spectra of member stars for each M31 dSph and find that they are consistent with the trend of decreasing \[Fe/H\] with luminosity observed in the Milky Way population. We find that three of our studied M31 dSphs appear as significant outliers in terms of their central velocity dispersion, And XIX, XXI and XXV, all of which have large half-light radii ($\gta700$pc) and low velocity dispersions ($\sigma_v<5\kms$). In addition, And XXV has a mass-to-light ratio within its half-light radius of just $[M/L]_{\rm
half}=10.3^{+7.0}_{-6.7}$, making it consistent with a simple stellar system with no appreciable dark matter component within its $1\sigma$ uncertainties. We suggest that the structure of the dark matter halos of these outliers have been significantly altered by tides.
author:
- 'Michelle L. M. Collins, Scott C. Chapman, R. Michael Rich, Rodrigo A. Ibata, Nicolas F. Martin, Michael J. Irwin, Nicholas F. Bate, Geraint F. Lewis, Jorge Peñarrubia, Nobuo Arimoto, Caitlin M. Casey, Annette M. N. Ferguson, Andreas Koch, Alan W. McConnachie, Nial Tanvir'
title: A kinematic study of the Andromeda dwarf spheroidal system
---
Introduction
============
The underlying nature of the dark matter halos of dwarf spheroidal galaxies (dSphs) has garnered significant attention from the scientific community over the past decade. The goal of recent observational studies of these objects has been to make critical tests of structure formation scenarios, particularly focusing on the viability of the canonical $\Lambda$CDM model. There is the long standing issue of the relative dearth of these faintest of galaxies observed surrounding nearby galaxies when compared with the number of dark matter subhalos produced in $N-$body simulations, which is referred to as the “missing satellite” problem [@klypin99; @moore99]. The extent to which this mismatch is considered problematic has decreased over recent years as both theorists and observers have sought to reconcile the simulated and observable Universe. From a modelling point of view, one does not expect stars to be able to form within all dark matter subhalos seen in simulations, and at a certain mass limit ($V_{\rm max}\lta15\kms$, see @penarrubia08a [@koposov09]), star formation is unable to proceed. Thus, there is a lower limit placed on galaxy formation. This mass limit is also tied to feedback processes that can remove the baryonic reservoirs required for star formation (e.g., @bullock00 [@somerville02; @kravtsov10; @bullock10; @nickerson11; @kazantzidis11]). This would imply that only the most massive subhalos seen in simulations are able to form and retain luminous populations. Observers have also attempted to quantify current survey completeness and radial selection effects to account for the number of satellites we are not currently able to detect (e.g., @koposov08 [@tollerud08; @walsh09]). These studies suggest that there are of order a few hundred satellites within the Milky Way’s virial radius that we have yet to detect.
The high dark matter dominance of dSph galaxies also singles them out as objects of interest. With total dynamic mass-to-light ratios of $[M/L]\sim10-1000$s and half-light radii of $r_{\rm half}\sim10-1000$ pc, they are ideal systems with which to probe the inner density profiles of dark matter halos. Recent imaging and spectroscopic observations of these objects within the Local Group have shown that, despite spanning approximately 5 decades in luminosity, the dSphs of the MW share a common mass scale and a universal density profile [@strigari08; @walker09b; @wolf10]. With a kinematic resolution of a few 10s of parsecs, these objects allow us to start addressing the question of whether the central regions of these halos follow cuspy density profiles as predicted by cosmological simulations [@navarro97], or constant density cores, similar to what is observed in low surface brightness galaxies [@blaise01; @deblok02; @deblok03; @deblok05; @swaters03; @kassin06; @spano08]. From studies of brighter dSphs, such as Sculptor and Fornax [@walker11; @amorisco12; @jardel12] it appears that their halo density profiles are also inconsistent with hosting central cusps. It is possible that these objects originally formed with cuspy density profiles, and that these have been subsequently modified by baryonic feedback. If this is truly the case, one should be able to observe cuspy profiles in the fainter dSph population (@zolotov12) as these do not contain enough baryons to drive this change in the dark matter density profile. Perhaps the only way to gain further insight into this contentious issue is by measuring the kinematics for large numbers of stellar tracers within these objects and analysing them with detailed models that do not make [*ab initio*]{} assumptions about the underlying density profiles, or the velocity anisotropy of both the dark matter and stars, such as those employed by @walker11 and @jardel12.
To date, the majority of studies involving the detailed kinematics of dSphs have revolved largely around those belonging to the Milky Way, as these are nearby enough that we can measure the velocities of their member stars to a high degree of accuracy. However, there are currently only $\sim25$ known MW dSphs, with luminosities ranging from $10^2-10^7\lsun$. For the very faintest, some controversy remains as to whether they are massively dark-matter dominated (see e.g., @niederste09 [@simon11]), but almost all of them have been shown to be consistent with the universal mass profiles of @walker09a and @wolf10. One notable exception to this is the Hercules object [@aden09], which some have argued is currently undergoing significant tidal disruption [@martin10]. Andromeda represents the only other system for which comparable kinematic analyses can be performed. M31 now has 28 dSph companions known, whose luminosities range from $\sim10^4 - 10^8\lsun$, the majority of which have been discovered by the CFHT Pan-Andromeda Archaeological Survey (PAndAS @martin06 [@ibata07; @irwin08; @mcconnachie08; @martin09; @richardson11]). The relatively brighter lower bound for the luminosities of M31 dSphs compared to the MW is a detection limit issue, rather than a sign of differing stellar populations (Martin et al. 2013, in prep). It has been noted by a number of authors (e.g., @mcconnachie06a [@tollerud12; @mcconnachie12]) that for the brighter dSphs ($M_V<-8$), those belonging to M31 are 2–3 times more extended in terms of their half-light ($r_{\rm half}$) and tidal ($r_t$) radii compared with the MW. In these papers, the underlying cause of this discrepancy was not identified, but it has been argued that it could be an effect of environment, with the mass distribution of the host playing an important role. Subsequent work by @brasseur11b, who included the fainter, non-classic M31 dSphs for the first time, showed that statistically, the relationship between size and luminosity for dSphs in the MW and Andromeda are actually largely consistent with one another, however there remain a number of significantly extended outliers within the Andromedean system (e.g., And II, And XIX, $r_{\rm
half}\sim1.2\kpc$ and $1.5\kpc$ respectively), and the scatter in this relationship is large (up to an order of magnitude at a given luminosity, @mcconnachie12).
Working from the @mcconnachie06a results, @penarrubia08a modelled the expected velocity dispersions for the M31 dSphs, assuming that all dSph galaxies are embedded within similar mass dark matter halos. A robust prediction of their modeling was that, given the larger radial extents, the dSphs of M31 should be [*kinematically hotter*]{} than their MW counterparts by a factor of $\sim2$. At the time of writing, they had only 2 measured velocity dispersions for the M31 dSphs, those of And II and And IX [@cote99; @chapman05]. New studies of the kinematics of M31 dSphs [@collins10; @collins11b; @kalirai10; @tollerud12; @chapman12] have dramatically increased the number of systems with a measured velocity dispersion, and have shown that instead of being kinematically hotter, these systems are either very similar to, or in a number of cases (e.g., And II, And XII, And XIV, And XV and And XXII), [*colder*]{} than their MW counterparts. In particular, a significant recent kinematic study of 15 M31 dSph companions using the Keck II DEIMOS spectrograph by the Spectroscopic and Photometric Landscape of Andromeda’s Stellar Halo (SPLASH, @tollerud12) concluded that the M31 dSph system largely obeys very similar mass-size-luminosity scalings as those of the MW. However, they also identified 3 outliers (And XIV, XV and XVI) that appear to possess much lower velocity dispersions, and hence maximum circular velocities, than would be expected for these systems. Such a result suggests that there are significant differences in the formation and/or evolution of the M31 and MW dSph systems.
To investigate this further, our group has been systematically surveying the known dSphs of M31 with the Keck I LRIS and Keck II DEIMOS spectrographs, and have obtained kinematic data for 18 of the 28 galaxies. In this paper, we present new spectroscopic analysis for the 11 dSphs, Andromeda (And) XVII, And XVIII, And XIX, And XX, And XXI, And XXIII, And XXIV, And XXV, And XXVI, the tidally disrupting And XXVII, And XXVIII and And XXX (Cassiopeia II) using an algorithm we have developed that implements a probabilistic method of determining membership for each galaxy. In addition we re-analyze the kinematics of 6 dSphs that our group has previously observed (And V, VI, XI, XII, XIII, XXII) using this method with the aim of providing a homogeneous analysis of all dSphs observed by our group to date. We also provide the individual stellar velocities and properties for every star observed in our dSphs survey, allowing us to present a large catalog of stellar kinematics that will be of interest to those studying dSph systems and Milky Way-like galaxies, whether observationally or theoretically.
The outline of this paper is as follows. In § 2 we discuss the relevant observations, data reduction techniques. In § 3 we outline our algorithm for the classification of member stars. In § 4 we present an analysis of our new kinematic datasets. In § 5 we report on the masses and densities of the dark matter halos of the M31 dSphs, comparing them to those of the MW dSphs. In §6 we report on the metallicities, \[Fe/H\], of the M31 dSphs as measured from the co-added spectra of their member stars. Finally, we conclude in § 7.
Observations {#sect:obs}
============
Photometry and target selection {#sect:photobs}
-------------------------------
The PAndAS survey [@mcconnachie09], conducted using the 3.6 metre Canada France Hawaii Telescope (CFHT), maps out the stellar density of the disc and halo regions of the M31–M33 system over a projected area of $\sim350\rm{deg}^2$ ($\sim55,000\kpc^2$), resolving individual stars to depths of $g=26.5$ and $i=25.5$ with a signal to noise ratio of 10, making this survey the deepest, highest resolution, contiguous map of the majority of the extended stellar halo of an L$_*$ galaxy to date. Each of the $411$ fields in this survey ($0.96\times0.94$ deg$^2$) has been observed for at least 1350s in both MegaCam $g$ and $i$ filters, in $<0.8^{\prime\prime}$ seeing. This survey was initiated following two precursor surveys of the M31 system, the first of which surveyed the central $\sim40$ deg$^2$ conducted with the 2.5 metre Isaac Newton Telescope [@ferguson02; @irwin05], and revealed a wealth of substructure in the Andromeda stellar halo, including the giant southern stream [@ibata01c]. To better understand this feature, and to probe deeper into the M31 (and M33) stellar halo, a survey of the south west quadrant of the M31 halo was initiated using the CFHT [@ibata07], and revealed yet more substructure, including the arc like stream Cp and Cr [@chapman08] and a number of dwarf spheroidal satellites [@martin06]. This CFHT survey was then extended into the full PAndAS project. For details of the processing and reduction of these data, see @richardson11. This survey has introduced us to a wealth of stellar substructure, debris and globular clusters within the Andromeda–Triangulum system. In addition, it has led to the discovery of 17 dSphs. These objects were detected in the PAndAS survey maps as over-densities in matched-filter surface density maps of metal poor red giant branch (RGB) stars and were presented in @martin06 [@ibata07; @irwin08; @mcconnachie08; @martin09] and @richardson11. We briefly summarise the photometric properties of all dSphs discussed within this paper in Table \[tab:photobs\].
[lccccc]{} And V & 01:10:17.1 & +47:37:41.0 & -10.2 & 302$\pm44$ & 742$^{+21}_{-22}$\
And VI & 23:51:39.0 & +24:35:42.0 & -10.6 & 524$\pm49$ & 783$\pm28$\
And XI & 00:46:20.0 & +33:48:05.0 & -6.9 & 158$^{+9}_{-23}$ & 763$^{+29}_{-106}$\
And XII & 00:47:27.0 & +34:22:29.0 & -6.4 & 324$^{+56}_{-72}$ & 928$^{+40}_{-136}$\
And XIII & 00:51:51.0 & +33:00:16.0 & -6.7 & 172$^{+34}_{-39}$ & 760$^{+126}_{-154}$\
And XVII & 00:37:07.0 & +44:19:20.0 & -8.5 & 262$^{+53}_{-46}$ & 727$^{+39}_{-25}$\
And XVIII & 00:02:14.5 & +45:05:20.0 & -9.7 & 325$\pm24$ & 1214$^{+40}_{-43}$\
And XIX & 00:19:32.1 & +35:02:37.1 & -9.6 & 1481$^{+62}_{-268}$& 821$^{+32}_{-148}$\
And XX & 00:07:30.7 & +35:07:56.4 & -6.3 & 114$^{+31}_{-12}$ & 741$^{+42}_{-52}$\
And XXI & 23:54:47.7 & +42:28:15.0 & -9.8 & 842$\pm77$ & 827$^{+23}_{-25}$\
And XXII & 01:27:40.0 & +28:05:25.0 & -6.5 & 252$^{+28}_{-47}$ & 920$^{+32}_{-139}$\
And XXIII & 01:29:21.8 & +38:43:08.0 & -10.2 & 1001$^{+53}_{-52}$& 748$^{+31}_{-21}$\
And XXIV & 01:18:30.0 & +46:21:58.0 & -7.6 & 548$^{+31}_{-37}$ & 898$^{+28}_{-42}$\
And XXV & 00:30:08.9 & +46:51:07.0 & -9.7 & 642$^{+47}_{-74}$ & 736$^{+23}_{-69}$\
And XXVI & 00:23:45.6 & +47:54:58.0 & -7.1 & 219$^{+67}_{-52}$ & 754$^{+218}_{-164}$\
And XXVII & 00:37:27.2 & +45:23:13.0 & -7.9 & 657$^{+112}_{-271}$ &1255$^{+42}_{-474}$\
And XXVIII & 22:32:41.2 & +31:12:51.2 & -8.5 & 210$^{+60}_{-50}$ &650$^{+150}_{-80}$\
And XXX (Cass II) & 00:36:34.9 & +49:38:48.0 & -8.0 & 267$^{+23}_{-36}$ & 681$^{+32}_{-78}$\
For the majority of these objects, the PAndAS dataset formed the basis for our spectroscopic target selection. Using the color selection boxes presented in @mcconnachie08 [@martin09] and @richardson11, we isolated the RGBs of each dSph, then prioritised each star on this sequence depending on their color, $i$-band magnitude, and distance from the centre of the dSph. Stars lying directly on the RGB, with $20.3<i_0<22.5$ and distance, $d< 4r_{\rm half}$ (where $r_{\rm half}$ is the half-light radius, measured on the semi-major axis of the dSph) were highly prioritised (priority A), followed by stars on the RGB within the same distance from the centre with $22.5<i_0<23.5$ (priority B). The remainder of the mask was then filled with stars in the field with $20.3<i_0<23.5$ and $0.5<g-i< 4$ (priority C). In general, it is the brighter, higher priority A stars that ultimately show the highest probability of membership. We also use the PAndAS photometry to help us determine membership of the dSph (discussed in § \[sect:membership\]).
And XXVIII is the one dSph in our sample that is not covered by the PAndAS survey. For this object, target selection followed an identical methodology tot hat detailed above, but used photometry from the 8th data release of the Sloan Digital Sky Survey (SDSS-III). Details of the observations and analysis of this photometry can be found in @slater11.
Keck Spectroscopic Observations {#sect:specobs}
-------------------------------
The DEep-Imaging Multi-Object Spectrograph (DEIMOS), situated on the Nasmyth focus of the Keck II telescope is an ideal instrument for obtaining medium resolution (R$\sim1.4$Å) spectra of multiple, faint stellar targets in the M31 dSphs. The data for the dSphs within this work were taken between Sept 2004 and Sept 2012 in photometric conditions, with typical seeing between $0.5-1^{\prime\prime}$. Our chosen instrumental setting covered the observed wavelength range from 5600–9800Å and employed exposure times of 3x20 minute integrations. The majority of observations employed the 1200 line/mm grating, although for 4 dSphs (And XI, XII, XIII and XXIV) the lower resolution 600 line/mm grating ($R\sim3.8$Å FWHM) was used. The spectra from both setups typically possess signal-to-noise (S:N) ratios of $>3$Å$^{-1}$ for our bright targets ($i\lta22.0$), though some of our fainter targets fall below this level. Information regarding the spectroscopic setup and observations for each dSph are displayed in Table \[tab:specobs\].
The resulting science spectra are reduced using a custom built pipeline, as described in @ibata11. Briefly, the pipeline identifies and removes cosmic rays, corrects for scattered light, performs flat-fielding to correct for pixel-to-pixel variations, corrects for illumination, slit function and fringing, wavelength calibrates each pixel using arc-lamp exposures, performs a 2-dimensional sky subtraction, and finally extracts each spectra – without resampling – in a small spatial region around the target. This results in a large set of pixels for each target, each of which carries a flux and wavelength (with associated uncertainties) plus the value of the target spatial profile at that pixel. We then derive velocities for all our stars with a Bayesian approach, using the Ca II Triplet absorption feature. Located at rest wavelengths of 8498, 8542 and 8662 Å, these strong features are ideal for determining the velocities of our observed stars. We determine the velocities by using an Markov Chain Monte Carlo procedure where a template Ca II spectrum was cross-correlated with the non-resampled data, generating a most-likely velocity for each star, and a likely uncertainty based on the posterior distribution that incorporates all the uncertainties for each pixel. Typically our velocity uncertainties lie in the range of $5-15\kms$. Finally, we also correct these velocities to the heliocentric frame.
[lcccccccccc]{} And V & 16 Aug 2009 & LRIS &831/8200&3.0Å& 01:10:18.21 & +47:37:53.3 & 0$\deg$ & 3600 & 50&15\
And VI & 17–19 Sept 2009& DEIMOS & 1200 & 1.3Å& 23:51:51.49 & +24:34:57.0 & 0$\deg$ & 5400 & 113& 43\
And XI & 23 Sept 2006 & DEIMOS & 600 & 3.5Å& 00:46:28.08 & +33:46:28.8 & 0$\deg$ & 3600 &33 &5\
And XII & 21–23 Sept 2006& DEIMOS & 600 & 3.5Å& 00:47:32.89 & +34:22:28.6 & 0$\deg$ & 3600 &49 &8\
And XIII & 23 Sept 2006 & DEIMOS & 600 & 3.5Å& 00:52:00.22 &+32:59:16.2 & 0$\deg$ & 3600 & 46& 4\
And XVII & 26 Sept 2011 & DEIMOS & 1200 & 1.3Å& 00:37:51.09 & +44:17:51.9 & 280$\deg$ & 3600 &149& 8\
And XVIII & 9 Sept 2010 & DEIMOS & 1200 & 1.3Å& 00:02:14.50 &+45:05:20.0&90$\deg$& 3600 & 73 & 4\
And XIXa & 26 Sept 2011 & DEIMOS & 600 & 3.5Å& 00:19:45.04 & +35:05:28.8 & 270$\deg$& 3600 & 107&15\
And XIXb & 26 Sept 2011 & DEIMOS & 1200 & 1.3Å& 00:19:30.88 & +35:07:34.1 & 0$\deg$ & 3600 & 108& 9\
And XX & 9 Sept 2010 & DEIMOS & 1200 & 1.3Å& 00:07:30.69 & +35:08:02.4 & 90$\deg$ &3600 & 85 &4\
And XXI & 26 Sept 2011 & DEIMOS & 1200 & 1.3Å& 23:54:47.70 & +42:28:33.6 & 180$\deg$ &3600& 157&32\
And XXIIa & 23 Sept 2009 & DEIMOS & 1200 & 1.3Å& 01:27:52.37 &+28:05:22.3 & 90$\deg$&1200 &93 &4\
And XXIIb & 10 Sept 2010 & DEIMOS & 1200 & 1.3Å& 01:27:52.37 & +28:05:22.3 & 0$\deg$&3600& 73&6\
And XXIIIa & 9 Sept 2010 & DEIMOS & 1200 & 1.3Å& 01:29:18.18 & +38:43:50.4 & 315$\deg$ & 3600 &196 & 24\
And XXIIIb & 10 Sept 2010 & DEIMOS & 1200 & 1.3Å& 01:29:21.87 & +38:44:58.7 & 245$\deg$ & 3600 & 189&18\
And XXIVa & 9 Sept 2010 & DEIMOS & 1200 & 1.3Å& 01:18:32.90 & +46:22:50.0 & 30$\deg$ &3600 & 192 & 1\
And XXIVb & 31 May 2011 & DEIMOS & 600 & 3.5Å& 01:18:32.90 & +46:22:50.0 & 0$\deg$ &2700 & 115 & 11\
And XXV & 10 Sept 2010 & DEIMOS & 1200 & 1.3Å& 00:30:01.88 &+46:50:31.0 & 90$\deg$ &3600& 183& 26\
And XXVI & 10 Sept 2010 & DEIMOS & 1200 & 1.3Å& 00:23:41.42 & +47:54:56.8 & 90$\deg$ &3600 & 179 &6\
And XXVII & 10 Sept 2010 & DEIMOS & 1200 & 1.3Å& 00:37:31.93 & +45:23:55.4 & 45$\deg$ & 3600 & 131&8\
And XXVIII& 21 Sept 2012 & DEIMOS & 1200 & 1.3Å& 22:32:32.71 & +31:13:13.2 & 140$\deg$ & 3600 & 102&17\
Cass II & 28 Sept 2011 & DEIMOS & 1200 & 1.3Å& 00:37:00.84 & +49:39:12.0 & 270$\deg$ & 3600 & 156& 8\
### Telluric velocity corrections {#sect:telluric}
With slit-spectroscopy, systematic velocity errors can be introduced if stars are not well aligned within the centre of their slits. Such misalignments can result from astrometric uncertainties or a slight offset in the position angle of the mask on the sky. For our astrometry, we take the positions of stars from PAndAS photometry, which have an internal accuracy of $\sim0.1^{\prime\prime}$ and a global accuracy of $\sim0.^{\prime\prime}25$ [@segall07]. This can translate to velocity uncertainties of up to $\sim15\kms$ for our DEIMOS setup. In previous studies, authors have tried to correct for this effect by cross correlating their observed spectra with telluric absorption features (e.g., @sohn07 [@simon07; @kalirai10; @collins10; @tollerud12]). These atmospheric absorption lines are superimposed on each science spectrum, and should always be observed at their rest-frame wavelengths. Thus, if one is able to determine the offset of these features, shifts caused by misalignment of the science star within the slit can be corrected for, and this can be applied on a slit by slit basis. The strongest of these features is the Fraunhofer A-band, located between 7595–7630Å. An example of this feature is shown in the left panel of Fig. \[fig:telluric\].
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While we believe this correction is robust in the high S:N regime, we argue against applying this correction in studies of Andromedean satellites, where S:N is often quite low (typically less than 8–10Å$^{-1})$ for 1 hour observations of faint ($i\gta21$) RGB stars. We find that when we compute this offset for all stars within our sample, those with high S:N tend to cluster within a few $\kms$ of the average telluric correction found for the mask. As the S:N decreases below about 10Å$^{-1}$, the scatter about this mean value increases dramatically, as do the uncertainties computed for each individual correction. This is because the telluric feature is a single, very broad and asymmetric feature. It is therefore easy in the noisy regime for the cross correlation routine to misalign the template and science spectrum whilst still producing a high confidence cross-correlation maximum. We show this effect explicitly in the right hand panel of Fig. \[fig:telluric\] where we plot the deviation of the telluric correction for every star within our sample from the average correction determined for the spectroscopic mask it was observed with ($v_{tel}-<v_{tel}>$) as a function of S:N. The cyan points represent the individual data, and the large black points represent the median of all points within 1Å bins in S:N. The error bars represent the dispersion within each bin. It is plainly seen that the median value for each bin is consistent with zero (i.e., the median), and that the dispersion increases with decreasing S:N. If we were to apply these velocity corrections to all our stars, it is probable that we would merely increase the velocity uncertainties rather than reducing them.
For this reason, we take a different approach. Using solely the telluric velocity corrections of stars from each observed mask whose spectra have S:N$>7$, we measure (a) the average telluric correction for the mask and (b) the evolution of the telluric correction as a function of mask position. In this way, we can track any gradient in our measurements that could be caused by e.g., rotational offsets in our mask. In general, we find these corrections to be slight. The average measured offset across all our masks is $3.8\kms$ (ranging from between $-3.4\kms$ and $+10.6\kms$). The measured gradients are very slight, resulting in an average end-to-end mask difference of 2.6$\kms$, with a range of $0.1-7.2\kms$, typically within our measured velocity uncertainties.
A probabilistic determination of membership {#sect:membership}
===========================================
Determining membership for Andromedean dwarf spheroidals is notoriously difficult in the best of cases. We only possess information about the velocity, CMD position, distance from the centre of the dSph and spectroscopic metallicity (although this carries large uncertainties of $>0.3$ dex for individual stars). Depending on the systemic velocity of the dSph, we must try to use these properties to distinguish the likely members from either Milky Way halo K-dwarfs ($v_{hel}\ga-150\kms$) or M31 halo giants ($v_{halo}\approx-300\kms$, $\sigma_{v,halo}\approx90\kms$, @chapman06). In the case of Galactic contamination, our spectra also cover the region of the Na I doublets ($\sim 8100$Å). As this feature is dependent on the stellar surface gravity, it is typically stronger in dwarf stars than in giants. However, there is a significant overlap between the two, especially in the CMD color region of interest for Andromedean RGB stars. In the past, groups have focused on making hard cuts on likely members in an attempt to weed out likely contaminants (e.g., based on their distance from the centre of the galaxy or their velocity, @chapman05 [@collins10; @collins11b; @kalirai10]), but such ‘by eye’ techniques are not particularly robust. @tollerud12 recently presented an analysis of a number of M31 dSphs where they used a more statistical method to ascertain likely membership, using the distance from the centre of, and position in the CMD of stars targeted within their DEIMOS masks. Here we employ a similar technique that will assess the probability of stars being members of a dSph based on (i) their position on the CMD; (ii) their distance from the centre of the dSph and, in addition; (iii) their position in velocity space, giving the likelihood of membership as:
$$\label{eqn:probtotal}
P_i\propto P_{CMD}\times P_{dist}\times P_{vel}$$
Below, we fully outline our method, and implement a series of tests to check it can robustly recover the kinematics of the M31 dSphs.
Probability based on CMD position, $P_{CMD}$
--------------------------------------------
For the first term, $P_{CMD}$, we are interested in where a given star observed in our mask falls with respect to the RGB of the observed dSph. @tollerud12 use the distance of a given star from a fiducial isochrone fit to the dwarf photometry to measure this probability. Here, we determine this value from the data itself, rather than using isochrones. Using the PAndAS CFHT photometry, we construct a normalised Hess diagram for the central region (i.e., within $2\times r_{\rm half}$) of the dSph, and one of a surrounding ‘field’ comparison region. By combining these two Hess diagrams, we can then map both the color distribution of the dSph and that of our contaminating populations. We use both directly as probability maps, where the densest region would have a value of 1.
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So we are not dominated by shot noise of sparsely populated regions of the CMD, we use only the region of the RGB. We do this by assigning a generous bounding box around the RGB as seen in the left hand panel of Fig. \[hess\] where we display the PAndAS CMD for the well populated And XXI RGB. We have zoomed in on the region for which DEIMOS observations with reliable velocities can be obtained, e.g., $i<23.5$. The bounding box is shown with red dashed lines. Anything that falls outside this region is therefore assigned a probability of $P_{CMD}=0$. The resulting probability map for And XXI is shown in the right hand panel. Red points represent all DEIMOS stars that have $P_{CMD}>10^{-6}$, while the blue stars show stars from the DEIMOS mask that are far removed from the And XXI RGB, and thus not considered to be members.
Probability based on distance position, $P_{dist}$
--------------------------------------------------
The second term in our probability function, $P_{dist}$ can be easily determined from the known radial profile of the dSphs. The half-light radii of all these objects are known and can be found in @mcconnachie06b [@zucker04; @zucker07; @mcconnachie08; @martin09; @collins10; @collins11b; @richardson11]. We also know that their density profiles are well represented by a Plummer profile with a scale radius of $r_p\equiv r_{\rm half}$. Therefore, we can define the probability function as a normalised Plummer profile [@plummer11], i.e.,:
$$P_{dist}=\frac{1}{\pi r_p^2[1+(r/r_p)^{2}]^2}$$
The above equation assumes that the systems we are studying are perfectly spherical. While the majority of these systems are not significantly elliptical, it is important to consider the effect of any observed deviations from sphericity. We therefore modify $r_p$ based on a given stars angular position with respect to the dwarfs major axis, $\theta_i$, such that:
$$r_p=\frac{r_{\rm half}(1-\epsilon)}{1+\epsilon{\rm cos}\theta_i}$$
where $\epsilon$ is the measured ellipticity of the dSph as taken from @mcconnachie12.
Probability based on velocity, $P_{vel}$ {#sect:velprob}
----------------------------------------
The final term, $P_{vel}$, contains information about the likelihood of a given star belonging to a kinematic substructure that is not well described by the velocity profiles of either the MW halo dwarfs or the Andromeda halo giants, both of which are determined empirically from our DEIMOS database of $>20000$ stars. From an analysis of these stars (selecting out any obvious substructures and the Andromedean disc) we find that the M31 halo is well approximated by a single Gaussian with a systemic velocity of $v_{r,halo}=-308.8\kms$ and $\sigma_{v,halo}=96.3\kms$, giving a probability density function for a given star with a velocity $v_i$ and velocity uncertainty of $v_{err,i}$ of:
$$\begin{aligned}
P_{halo}=\frac{1}{\sqrt{2\pi(\sigma_{v,halo}^2+v_{err,i}^2)}}\times\\{\rm exp}\Big[-\frac{1}{2}\left(\frac{v_{r,halo}-v_{r,i}}{\sqrt{\sigma_{v,halo}^2+v_{err,i}^2}}\right)^2\Big] \end{aligned}$$
The MW halo population is well approximated by 2 Gaussians with $v_{r,MW 1}=-81.2\kms$, $\sigma_{v,MW 1}=36.5\kms$ and $v_{r,MW 2}=-40.2\kms$ and $\sigma_{v,MW 2}=48.5\kms$, resulting in a probability density function for a given star with a velocity $v_i$ and velocity uncertainty of $v_{err,i}$ of: $$\begin{aligned}
P_{MW}=\frac{R}{\sqrt{2\pi(\sigma_{v,MW
1}^2+v_{err,i}^2)}}\times\\{\rm exp}\Big[-\frac{1}{2}\left(\frac{v_{r,MW
1}-v_{r,i}}{\sqrt{\sigma_{v,MW1}^2+v_{err,i}^2}}\right)^2\Big] \\+ \frac{(1-R)}{\sqrt{2\pi(\sigma_{v,MW
2}^2+v_{err,i}^2)}}\times\\{\rm exp}\Big[-\frac{1}{2}\left(\frac{v_{r,MW 2}-v_{r,i}}{\sqrt{\sigma_{v,MW
2}^2+v_{err,i}^2}}\right)^2\Big]
\end{aligned}$$
where $R$ is the fraction of stars in the first MW peak, and $(1-R)$ is the fraction of stars in the second peak. The value of $R$ is determined empirically from our DEIMOS data set.
A strong kinematic peak outside of these two populations can then be searched for using a maximum likelihood technique, based on the approach of @martin07. We search for the maximum in the likelihood function that incorporates the two contamination populations plus an additional Gaussian structure with systemic velocity $v_{r,substr}$ and a dispersion of $\sigma_{v,substr}$, defined as:
$$\begin{aligned}
\label{eq:mlpeta}
P_{substr}=\frac{1}{\sqrt{2\pi([\eta\sigma_{v,substr}]^2+v_{err,i}^2)}}\times\\{\rm exp}\Big[-\frac{1}{2}\left(\frac{v_{r,substr}-v_{r,i}}{\sqrt{[\eta\sigma_{v,substr}]^2+v_{err,i}^2}})]\right)^2\Big]
\end{aligned}$$
Here, to ensure we haven’t biased our $P_{substr}$ strongly against stars that lie within the wings of the Gaussian distribution of dwarf spheroidal velocities, we have included a multiplicative free parameter, $\eta$, to our derived value of $\sigma_v$. To determine the ideal value of $\eta$, we ran our algorithm over all our datasets, changing the value of $\eta$ from $0.5-10.5$ to see its effect on the final derived systemic velocities and velocity dispersion. We find that in all cases, the solutions converge at values of $\eta\sim2-4$. For dSphs whose kinematics are well separated from contaminants, the derived kinematics can remain stable up to much larger values of $\eta$, however for those with systemic velocities within the velocity regime of the Milky Way, the solution quickly destabilizes as more contaminants are included as probable members. We show this implicitly in Fig. \[fig:etatest\], where we present the effect of modifying $\eta$ on 6 dSphs, And XII, XIX, XXI, XXII, XXIII and XXV. These objects were selected as they nicely probe our datasets with low numbers of probable member stars ($\sim8$), to those where we have 10s of probable members, as well as sampling dSphs from highly contaminated to well isolated kinematic regimes. The value of $\eta$ is therefore independently determined for each dataset separately, and we report its final value in Table \[tab:kprops\].
$$\begin{aligned}
\label{eq:mlp}
P_{substr}=\frac{1}{\sqrt{2\pi(\sigma_{v,substr}^2+v_{err,i}^2)}}\times\\{\rm exp}\Big[-\frac{1}{2}\left(\frac{v_{r,substr}-v_{r,i}}{\sqrt{\sigma_{v,substr}^2+v_{err,i}^2}})]\right)^2\Big]
\end{aligned}$$
The likelihood function can then be simply written as:
$$\rm{log}[{\mathcal{L}}(v_r,\sigma_v)]=\sum_{i=1}^{N}\rm{log}\Big(
\alpha P_{i,{\rm halo}}+\beta P_{i,{\rm MW}}+\gamma P_{i,{\rm substr}}\Big)$$
where $\alpha$, $\beta$ and $\gamma$ represent the Bayesian priors, i.e., the expected fraction of stars to reside in each population. These are determined by starting with arbitrary fractions (for example, 0.2, 0.5 and 0.3 respectively) and are then adjusted to the posterior distribution until priors and posteriors match. This technique will therefore identify an additional kinematic peak, independent of the MW and M31 halo populations, if it exists. We stress that these are not the final systemic velocity and dispersion of the dSph, but merely indicate a region in velocity space in which an excess of stars above the two contaminant populations is seen. In Fig. \[fig:vtest\], we show the result of this process for the And XXI dSph. Here, the substructure is clearly visible as a cold spike at $\sim-400\kms$.
![A velocity histogram for all observed stars in the field of And XXI. Our empirical Gaussian fits to the full Keck II data set for the M31 and MW halos are overlaid in blue and green respectively. A cold, kinematic peak at $-400\kms$ is also seen, and this is the likely signature of the dSph. Our coarse, initial ML procedure identifies this peak, and the values of $v_{sys}$ and $\sigma_v$ it measures are used to derive our kinematic probability, $P_{vel}$.[]{data-label="fig:vtest"}](fig2.eps){width="0.85\hsize"}
Now that we have a velocity profile for our three components (MW, M31 halo and the dSph), we can assign probabilities for each star within our sample belonging to each population using simple Bayesian techniques, i.e., the probability that a given star belongs to the substructure, $P_{vel}$, is:
$$P_{vel}=\frac{\gamma P_{substr}}{\alpha P_{halo}+\beta P_{MW}+\gamma P_{substr}}$$
and the probability of being a contaminant is:
$$P_{nvel}=\frac{\alpha P_{halo}+\beta P_{MW}}{\alpha P_{halo}+\beta P_{MW}+\gamma P_{substr}}$$
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Measuring $v_r$ and $\sigma_v$
------------------------------
Upon applying this to our data, we can identify the most probable members of each dSph, without having to apply any additional constraints or cuts.
Having established the membership probability for each observed star (as detailed above) we now calculate the kinematic properties of each dSph; namely their systemic velocities ($v_r$) and velocity dispersions ($\sigma_v$). We use the maximum likelihood technique of @martin07, modified to include our probability weights for each star. We sample a coarse grid in $(v_r,\sigma_v)$ space and determining the parameter values that maximise the likelihood function (ML), defined as:
$$\label{eq:ml}
{\rm log}[{\mathcal{L}}(v_r,\sigma_v)]=-\frac{1}{2}\sum_{i=1}^{N}\Big[P_i{\rm
log}(\sigma_{\mathrm{tot}}^2)+P_i\frac{v_r-v_{r,i}}{\sigma_{\mathrm{tot}}}^2\\+P_i{\rm
log}(2\pi)\Big]
$$
where $N$ is the number of stars in the sample, $v_{r,i}$ is the radial velocity measured for the $i^\mathrm{th}$ star, $v_{err,i}$ is the corresponding uncertainty and $\sigma_{\mathrm{tot}}=\sqrt{\sigma_v^2+v_{err,i}^2}$. In this way, we aim to separate the intrinsic dispersion of a system from the dispersion introduced by the measurement uncertainties.
Testing our probabilistic determination of membership and calculations of kinematic properties {#sect:test}
----------------------------------------------------------------------------------------------
Having developed the above technique, it is important for us to rigorously test that it is robust enough to accurately determine the global kinematic properties for each of our datasets. In Appendix A, we examine in detail a number of potential issues that could cause our algorithm to return biased or incorrect results. These are the inclusion of a velocity dependent term in our probability calculation, the effect of including low S:N data (S:N$<5$Å) in our analysis and the effect of small sample sizes ($N_*<8$) on our measurements of kinematic properties. We briefly summarize our findings here, and refer the reader to Appendix A for a more detailed description.
This work has introduced the concept of assigning a probability of membership for a given star to a dSph based on the prior knowledge of the velocity profiles of our expected contaminant populations, $P_{vel}$, a technique that has not previously been used in the study of M31 dSphs. To test that this is not biasing our results, we can simply remove this term from Eqn. \[eqn:probtotal\], and follow the technique of T12, where they use only $P_{CMD}$ and $P_{dist}$ terms and then cut all stars with $P_{i}<0.1$ and velocities that lie greater than $3\sigma$ from the mean of this sample from their final analysis. We find that both techniques produce very similar results, however our algorithm is more robust in regimes where the systemic velocity of the dSph is close to that of the MW, and in dSphs where our number of probable member stars is low ($N_*<10$).
To test the effect of low S:N data on our calculations of $v_r$ and $\sigma_v$, we use our datasets for And XXI, XXIII and XXV, all of which have $\geq25$ associated members. For each dataset, we apply a series of cuts to the sample based on S:N (at levels of S:N$>2,3,4$ and 5Å) and rerun our algorithm. In all cases we find that the derived probabilities do not significantly differ when the low S:N data are included, justifying our inclusion of all stars for which velocities are calculated by our pipeline.
We also test the ability of our algorithm to measure $v_r$ and $\sigma_v$ in the small $N_*$ regime. For some of our datasets, we are only able to identify a handful of stars as probable members. In theory, one can calculate velocity dispersions accurately from only 3 stars if one is confident of ones measurement uncertainties, as is demonstrated by @aaronson83 measurement of the velocity dispersion for Draco from only 3 stars, which remains consistent with modern day measurements from significantly larger datasets [@walker09b]. We can test if our results are similarly robust using our larger datasets (such as And XXI, XXIII and XXV) by randomly selecting 4, 6, 8, 10, 15, 20 and 25 stars from these datasets and rerunning our algorithm to determine $v_r$ and $\sigma_v$ from these subsets. We repeat this exercise 1000 times for each sample size, and examine the mean and standard deviations for the computed quantities. We find that on average, for all sample sizes, our routine measures systemic velocities and velocity dispersions that are entirely consistent with those measured from the full sample, with a spread that is very comparable to typical errors produced by our ML routine in these low $N_*$ regimes. As such, we conclude that our technique is able to place sensible limits on these values, even when dealing with as few as four member stars.
Finally, as the individual positions, velocities and velocity uncertainties for all the stars analyzed in T12 are publicly available (with the data from the non-member stars having been kindly passed on to us by the SPLASH team), we can check that our algorithm is able to reproduce the values they measure for their M13 dSph sample. We find that, in all cases, we calculate systemic velocities and velocity dispersions that agree with their measured values to well within their $1\sigma$ uncertainties.
These tests demonstrate that our method is robust enough to accurately determine the global kinematic properties of M31 dSphs across a wide range of sample sizes and data quality. We therefore proceed to apply it to the datasets of all of the dSphs for which our group has acquired Keck II DEIMOS observations to date.
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The kinematics of M31 dSphs {#sect:results2}
===========================
With this vast dataset of dSph kinematics now in hand, we can begin to statistically probe their structures more fully. In Fig. \[fig:vels\] we display a summary of the velocities and positions of 26 of the 28 dSphs for which kinematic data are available, where the values are taken from this work, T12 and @kalirai10. In the following sections, we will discuss the individual stellar kinematics, masses and chemistries of the dSphs analyzed within this work.
Andromeda XVII {#sect:and17}
--------------
And XVII was discovered by @irwin08, and it is located at a projected distance of $\sim40$ kpc to the North West of Andromeda. A detailed study of deep imaging obtained with the Large Binocular Camera on the LBT was also performed by @brasseur11a, and throughout, we use the structural properties as determined from this work. It is a faint, compact galaxy ($M_V=-8.61, r_{\rm half}=1.24'$ or 262$^{+53}_{-46}$ pc). In the left panel of Fig. \[fig:And17\] we display the PAndAS color magnitude diagram for And XVII. Over-plotted we show the observed DEIMOS stars, color-coded by their probabilities of membership. The open symbols represent stars for which $P_i<10^{-6}$. We employ this cut solely to make clarify which stars have the highest probability of belonging to And XVII. In the right hand panel, we display the basic kinematic information for And XVII. In the top panel of this subplot, we show a velocity histogram for all stars observed within the LRIS mask, and stars with $P_{i}>10^{-6}$ are highlighted with a filled red histogram. The centre panel shows the velocities as a function of distance from the centre of And XVII (and the red dashed lines indicate $1,2,3$ and $4\times r_{\rm half}$), and the lower panel shows the photometric metallicities for all stars, as determined using @dart08 CFHT isochrones. Again, all points are color-coded by their probability of membership. Finally, the two lower panels show the resulting, one dimensional, probability weighted, marginalized maximum likelihood distributions for $v_r$ and $\sigma_v$ for this data set. From the kinematics presented in Fig. \[fig:And17\], which represent the first spectroscopic observations of this object, we see the signature of the dwarf galaxy as a cold spike at $v_r\sim-250\kms$. From the lower panels of this figure and the accompanying CMD we see that there is a cluster of 7 stars sitting within this spike that are centrally concentrated and are consistent with the RGB of the dwarf itself, leading us to believe that our algorithm has cleanly detected the signature of the galaxy. Interestingly, we also see 3 stars that, kinematically, are indistinguishable from the stars that have been dubbed as probable members in our analysis. However, they all sit at large distances from the centre of And XVII, equivalent to greater than 6 times the half-light radius of the dwarf, and hence the routine has classified them as likely members of the M31 halo rather than And XVII members. But, given their tight correlation in velocity to the systemic velocity of And XVII the possibility exists that these are extra-tidal stars of And XVII. No sign of extra-tidal features were cited in either the discovery paper of And XVII or the LBT followup, but given its position in the north M31 halo where contamination from the MW becomes increasingly problematic, and its relatively low luminosity ($M_V=-8.5$), such features may be difficult to see within the imaging. However, at present they are considered unlikely members by our routine, and do not factor into our calculation of global kinematic properties for this object. We find $v_r=-251.6^{+1.8}_{-2.0}\kms$ and $\sigma_v=2.9^{+2.2}_{-1.9}\kms$.
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Andromeda XVIII {#sect:and18}
---------------
Andromeda XVIII (And XVIII) was detected by @mcconnachie08 in the PAndAS CFHT maps. Located at a projected distance of $\sim110$ kpc to the North-West of M31, it is one of the most distant of its satellites, sitting $\sim600$ kpc behind the galaxy, making spectroscopic observations of its individual RGB stars taxing, as they are all relatively faint ($i\gta22.2$). Thus, from our 1 hour DEIMOS observation, we were only able to confirm 4 stars as probable members (see Fig. \[fig:And18\]). We determine the global systemic velocity to be $v_r=-346.8\pm2.0\kms$, and we are unable to resolve a velocity dispersion, finding $\sigma_v=0.0^{+2.7}\kms$ where the upper bound is determined from the formal $1\sigma$ confidence interval produced by our maximum likelihood analysis. This suggests that the 4 stars we are able to confirm as members do not adequately sample the underlying velocity profile. The systemic velocity we measure is different to that presented in T12 of $v_r=-332.1\pm2.7$ at a level of 3.4$\sigma$. Our 1$\sigma$ limit of 2.7$\kms$ is also at odds with the dispersion determined by T12 ($\sigma_v=9.7\pm2.3$). They were able to measure velocities for significantly more probable member stars (22 vs. 4) owing to their longer integration of 3 hours. The faintness of our targets and shorter exposure time could mean that strong sky absorption lines have systematically skewed our velocity measurements for these stars, and this could explain the discrepancy of our measurements with respect to those of T12. To check against this, we again perform weighted cross-correlations to each of the 3 Ca II lines individually. We find that the values we obtain, and their average, are fully consistent with that derived from the technique described in § \[sect:specobs\], differing by less than $3\kms$ from those values. The true systemic velocity of And XVIII therefore remains unclear. However, given their larger sample size, the T12 systemic properties are more statistically robust than those we present here.
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Andromeda XIX {#sect:and19}
-------------
Andromeda XIX (And XIX) was first reported in @mcconnachie08, and is a relatively bright, very extended ($M_V=-9.3$, $r_{\rm half}=1.5$ kpc) dSph, located at a projected distance of $\sim180$ kpc to the south west of M31. Its unusual morphology, very low surface brightness $\Sigma_v=30.2$mag/arcsec$^2$, and evidence in the photometry for a possible link to the major axis substructure reported in @ibata07 caused @mcconnachie08 to question whether And XIX was truly a dynamically relaxed system, or whether it had experienced a significant tidal interaction. Here, we present the first spectroscopic observations of the And XIX satellite in Fig. \[fig:And19\] from two DEIMOS masks placed at different position angles. These data allow us to comment on its dark matter content, and on the likelihood of a tidal origin for its unique structure. We identify 27 stars where $P_i>10^{-6}$ within the system. These measurements were made increasingly challenging as the systemic velocity we measure is $v_r=-111.6^{+1.6}_{-1.4}\kms$, placing it within the regime of Galactic contamination. However, we are confident that our algorithm is robust to this unfortunate location of And XIX in velocity space (see discussion in § \[sect:test\] and Appendix A). As a further check that none of the stars we define as probable members are actually foreground contaminants, we measure the equivalent widths of the Na I doublet lines (located at $\sim8100$Å). These gravity-sensitive absorption lines are typically significantly stronger in foreground dwarf stars than M31 RGB stars, although there is some overlap between the two populations. For the stars tagged as probable members by our algorithm, we find no evidence of strong absorption in the region of the Na I doublet, indicating that we are not selecting foreground stars as members. We measure a relatively cold velocity dispersion for this object of $\sigma_v=4.7^{+1.6}_{-1.4}\kms$, which is surprising given the radial extent of this galaxy. This result will be discussed further in § \[sect:mass\], and in a follow-up paper (Collins et al. in prep).
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Andromeda XX {#sect:and20}
------------
And XX was the third of three dSphs discovered by @mcconnachie08, and is notable for being one of the faintest dSph companions detected surrounding Andromeda thus far. With $M_V=-6.3$ and $r_{\rm half}=114^{+31}_{-12}$ pc, it is a challenging object to study spectroscopically as there are very few stars available to target on its RGB, as shown in the top left subplot in Fig. \[fig:And20\]. As a result, our algorithm is only able to find 4 stars for which $P_i>10^{-6}$. These are found to cluster around $v_r=-456.2^{+3.1}_{-3.6}\kms$, with a dispersion of $\sigma_v=7.1^{+3.9}_{-2.5}\kms$. Despite the low number of stars, we are confident in this detection, as the systemic velocity places it in the outer wings of the velocity profile of the M31 halo. And XX is also located at a large projected distance from M31 of $\sim130$ kpc, where we expect the density of the M31 halo to be very low. As such, seeing 4 halo stars so tightly correlated in velocity in the wings of the halo velocity profile within such a small area of the sky (all stars are within 1 arcmin of the centre of And XX) is highly unlikely. We caution the reader that, while we are confident that our algorithm is able to measure velocity dispersions for sample sizes as small as 4 stars, as we are not probing the full velocity profile of this object this measurement ideally needs to be confirmed with larger numbers of member stars.
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Andromeda XXI {#sect:and21}
-------------
Andromeda XXI (And XXI) was identified within the PAndAS imaging maps by @martin09. It is a relatively bright dSph ($M_V=-9.9$), located at a projected distance of $\sim150$ kpc from M31, and it has a half-light radius of $r_{\rm half}=842\pm77$ pc. We present our spectroscopic observations for this object in Fig. \[fig:And21\], and in the top right subplot, we can clearly see the signature of And XXI as a cold spike in velocity with 32 probable member stars, located at $v_r=-362.5\pm0.9\kms$, with a curiously low velocity dispersion of only $\sigma_v=4.5^{+1.2}_{-1.0}\kms$. These results are completely consistent with those of T12, where they measured $v_r=-361.4\pm5.8\kms$ and $\sigma_v=7.2\pm5.5\kms$. As their sample contained only 6 likely members compared with the 29 we identify here, ours constitute a more statistically robust measurement of the global kinematics for this object than those presented in T12.
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Andromeda XXII {#sect:and22}
--------------
Andromeda XXII (And XXII) was identified within the PAndAS imaging maps by @martin09, and is a relatively faint dSph, with $M_V=-6.5$. Its physical position in the halo, located at a distance of 224 kpc in projection from M31, but only 42 kpc in projection from M33, led the authors to postulate that it could be the first known dSph satellite of M33. Subsequent work analysing the kinematics of And XXII by T12 measured a systemic velocity for And XXII of $-126.8\pm3.1\kms$ from 7 stars, more compatible with the systemic velocity of M33 ($-178\kms$, @mateo98) than that of M31. Another study by @chapman12 using the same data and the same method we present here concluded the same, measuring a systemic velocity for the satellite of $-129.8\pm2.0$ from 12 probable member stars, consistent with the T12 value. @chapman12 also compare the position and kinematics of And XXII with a suite of $N-$body simulations of the M31-M33 system, concluding that And XXII was a probable M33 satellite.
In Fig. \[fig:And22\], we present the same data as analyzed by @chapman12 for completeness. The velocity dispersion of And XXII is just resolved at $\sigma_v=2.8^{+1.9}_{-1.4}\kms$, completely consistent with the value of $\sigma_v=3.5^{+4.2}_{-2.5}\kms$ from T12. As our values are calculated from a 50% greater sample size, we posit that they are the more statistically robust.
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Andromeda XXIII {#sect:and23}
---------------
Andromeda XXIII (And XXIII) was the first of five M31 dSphs identified by @richardson11. Located at a projected distance of $\sim130$ kpc to the east of Andromeda, it is relatively bright, with $M_V=-10.2$, and extended, with $r_{\rm half}=1001^{+53}_{-52}$ pc. Our routine clearly detects a strong cold kinematic peak for And XXIII located around $-230\kms$ and calculates a systemic velocity of $v_r=-237.7\pm1.2\kms$, and a velocity dispersion of $\sigma_v=7.1\pm1.0\kms$ from 40 probable member stars, as show in Fig. \[fig:And23\]. This small, positive velocity relative to M31, combined with its large projected distance from the host suggests that And XXIII is not far past the apocentre of its orbit, heading back towards M31.
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Andromeda XXIV {#sect:and24}
--------------
And XXIV was also first reported in @richardson11. Relatively faint and compact ($M_V=-7.6$, $r_{\rm half}=548^{+31}_{-37}$ pc), spatially it is located $\sim200\kpc$ from M31, along its northern major axis. And XXIV was observed on two separate occasions as detailed in Table \[tab:specobs\]. For the first mask, there was an error in target selection, and as a result, only one star that lay on the RGB of And XXIV was observed. The second mask was observed in May 2011, however owing to target visibility, only a short integration of 45 minutes was obtained, which resulted in higher velocity uncertainties than typically expected ($\sim8\kms$ vs. $\sim5\kms$). For this reason, we have only included stars from this mask with $i<22.0$, as the spectra for fainter stars were too noisy to determine reliable velocities from. The systemic velocity of And XXIV also unsatisfactorily coincides with that of the MW halo contamination, as can be seen in Fig. \[fig:and24\]. As for And XIX, we check the strength of the Na I doublet of all the stars classified as potential members for And XXIV, and find no significant absorption, making them unlikely foreground contaminants. But, given the lower quality of this dataset, this check is far from perfect, and it is possible that we have included contaminants from the MW within our sample. Owing to the larger velocity uncertainties of the And XXIV dataset, and the overlap of And XXIV with the MW, the determination of probability of membership for stars within this dataset is based largely on their position in the color magnitude diagram (e.g., location on the RGB) and their distance from the centre of And XXIV.
When we run our machinery over the data acquired from both masks, we identify only 3 probable members and determine a systemic velocity of $v_r=-128.2\pm5.2\kms$ and we resolve a velocity dispersion of $\sigma_v=0.0^{+7.3}\kms$. Given the lower quality of this dataset in comparison to the remainder of those we present in this work, and the overlap of And XXIV in velocity space with contamination from the MW, a robust kinematic detection and characterisation of this galaxy is made incredibly challenging. As such, we present these results as a tentative identification of And XXIV, and do not include its measured properties in the remainder of our analysis. Further kinematic follow up of And XXIV is required to understand this system. We present the velocities of all bright stars for which velocity measurements were possible in Table \[tab:members\] so that they may be helpful for any future kinematic analysis of this system.
[lccccccccc]{} And V & 9 & 1:10:2.38 &47:37:48.5 & 23.640 & 22.402 & -371.270 & 5.080 & 1.700 & 0.014\
And V & 12 & 1:10:5.540 & 47:36:41.6 & 23.100 & 21.530 & -406.500 & 3.600& 2.000 & 0.111\
...&...&...&...&...&...&...&...&...&...\
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Andromeda XXV {#sect:and25}
-------------
And XXV was identified in @richardson11 as a relatively bright ($M_V=-9.7$), extended ($r_{\rm half}=642^{+47}_{-74}$ pc) dwarf spheroidal, located at a projected distance of $\sim90$ kpc to the north west of M31. As with And XXIII, we present here a kinematic analysis of And XXV. The results are displayed in Fig. \[fig:And25\]. We see that the systemic velocity of And XXV ($v_r=-107.8\pm1.0\kms$), places it in the regime of the Galactic foreground. However, given the strong over-density of stars with this velocity relative to the expected contribution of MW stars, we are confident that our routine has detected 25 likely members for this object. We check the strength of the Na I doublet in these likely members, and find no significant absorption, making them unlikely foreground contaminants. As for And XIX and XXI, we find that And XXV has a curiously low velocity dispersion for its size, with $\sigma_v=3.0^{+1.2}_{-1.1}\kms$. We discuss the significance of this further in § \[sect:mass\] and Collins et al (2013, in prep).
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Andromeda XXVI {#sect:and26}
--------------
And XXVI is a relatively faint ($M_V=-7.1$) dSph with $r_{\rm half}=219^{+67}_{-52}$ pc, also first reported in @richardson11. Its low luminosity makes observing large numbers of member stars difficult, owing to the paucity of viable targets on the RGB that can be observed with DEIMOS. As a result, our routine has identified only 6 stars as potential members, highlighted in Fig. \[fig:And26\]. The dwarf has a systemic velocity of $v_r=-261.6^{+3.0}_{-2.8}\kms$, and a fairly typical velocity dispersion of $\sigma_v=8.6^{+2.8}_{-2.2}\kms$. As with And XX, while we believe our routine can robustly measure the velocity dispersions of systems with only 6 confirmed members, to be truly confident of this value, follow up of And XXVI to increase the number of likely members is required.
In @conn12, from an analysis of the photometry of And XXVI, they determined a distance modulus to the object of $(m-M)_0=24.39^{+0.55}_{-0.53}$ from a Markov-Chain-Monte-Carlo analysis of the PAndAS photometry of And XXVI. This value corresponds to an $i-$band magnitude for the TRGB of And XXVI of $m_{i,0}=21.1^{+0.55}_{-0.53}$. Our CMDs for the dwarfs are not extinction corrected, but using the extinction values from @richardson11 of $E(B-V)=0.110$ [@schlegel98], this would correspond to an $i-$band magnitude of $m_{i,TRGB}=21.3^{+0.55}_{-0.53}$. Three targets were observed with magnitudes and colors that should be consistent with their belonging to And XXVI. However. we find that all these objects have velocities that are consistent with being Galactic foreground contaminants. Given the position of And XXVI in the northern M31 halo, where contamination from the MW increases, this is not unexpected. The brightest star we observe that is likely associated with And XXVI has $m_i=21.9$ ($m_{i,0}=21.7$). Assuming that this brightest confirmed member star of And XXVI sits at the RGB tip, the distance estimate becomes 1.1 Mpc. While this value is higher than that of @conn11, it is still within their upper distance estimate. Additionally, as not every star brighter than the member with $m_{i,0}=21.7$, with colors consistent with the And XXVI RGB was observed, this value merely represents an upper limit, on the distance to And XXVI and highlights the difficulty of calculating distances to these faint galaxies where RGB stars are sparse.
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Andromeda XXVII {#sect:and27}
---------------
Andromeda XXVII (And XXVII) is a somewhat unusual object as it is currently undergoing tidal disruption, spreading its constituent stars into a large stellar stream, named the northwestern arc, discovered in the PAndAS survey by @richardson11. As such, it is unlikely to be in virial equilibrium, if it remains bound at all.
When determining the kinematics of And XXVII, we find the results somewhat unsatisfactory. Our routine determines $v_r=-539.6^{+4.7}_{-4.5}\kms$ and $\sigma_v=14.8^{+4.3}_{-3.1}\kms$ from 11 stars. However, from an inspection of Fig. \[fig:And27\], we see that there is significant substructure around $v_{hel}\sim-500\kms$, much of which is considered to be unassociated with And XXVII in this analysis as it does not fall within a cold, well-defined Gaussian velocity peak. Given the disrupting nature of And XXVII, it is likely that a different analysis is required for this object, and we shall discuss this further in a future analysis, where the kinematics of the northwestern arc itself are also addressed. From this first pass however, it would appear that And XXVII may no longer be a gravitationally bound system.
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Andromeda XXVIII {#sect:and28}
----------------
And XXVIII was recently discovered in the 8th data release of the SDSS survey [@slater11]. It has $M_v=-8.5$ and $r_{\rm half}=210^{+60}_{-50}$ pc. It is also potentially one of Andromeda’s most distant satellites, with a host-satellite projected separation of $365^{+17}_{-1}$ kpc. The And XXVIII satellite is not covered by the PAndAS footprint, so we must instead use the original SDSS photometry for our analysis. A CMD with the SDSS $i-$band and $r-i$ colors for And XXVIII is shown in Fig. \[fig:And28\], where all targets brighter that $i\sim23.5$ within $3r_{\rm half}$ are shown. The photometry here do not show an RGB that is as convincing as those from the PAndAS survey, so to guide the eye, we also overplot an isochrone from @dart08 with a metallicity of $\feh=-2.0$, corrected for the distance of And XXVIII as reported in @slater11.
In a recent paper, @tollerud13 discussed the kinematics of this object as derived from 18 members stars. They find $v_r=-328.0\pm2.3\kms$ and $\sigma_v=8.1\pm1.8\kms$ from their full sample. They then remove two stars that they categorize as outliers based on their distance from the centre of And XXVIII, which alters their measurements to $v_r=-331.1\pm1.8\kms$ and $\sigma_v=4.9\pm1.6\kms$. Analyzing our own DEIMOS dataset for this object, we find $v_r=-326.1\pm2.7\kms$ and $\sigma_v=6.6^{+2.9}_{-2.2}\kms$ based on 17 probable members. This is fully consistent with the results from the full sample in @tollerud13. However, the systemic velocity we measure is offset at a level of $\sim1\sigma$ from their final value (calculated after excluding 2 outliers). This offset is small, and is probably attributable to our differing methodologies for classifying stars as members. As we believe our method is more robust (as discussed in § \[sect:test\] and Appendix A), we will use our derived parameters for this object in the remainder of our analysis.
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And XXX/Cassiopeia II {#sect:and30}
---------------------
And XXX – also known as Cass II owing to its spatial location, overlapping the Cassiopeia constellation – is a recently discovered dSph from the PAndAS survey (Irwin et al. in prep). It has $M_v=-8.0$ and $r_{\rm
half}=267^{+23}_{-36}$ pc. Located to the north west of Andromeda, it sits within 60 kpc of the two close dwarf elliptical M31 companions, NGC 147 and NGC 185. With these 3 objects found so close together in physical space, it is tempting to suppose them a bound system within their own right, but this can only be borne out by comparing their kinematics.
Conspiring to confound us, we find that Cass II has kinematics that place it well within the regime of Galactic foreground, as can be seen in Fig. \[fig:Cass2\]. However, our analysis is able to detect the dSph as a cold spike consisting of 8 likely members. As for And XIX, we check the strength of the Na I doublet in these likely members, and find no significant absorption, making them unlikely foreground contaminants. We measure $v_r=-139.8^{+6.0}_{-6.6}\kms$, and a fairly typical velocity dispersion of $\sigma_v=11.8^{+7.7}_{-4.7}\kms$.
The systemic velocity of Cass II ($v_r=-139.8^{+6.0}_{-6.6}\kms$) puts it within $\sim50\kms$ of those of NGC 147 and NGC 185 ($v_r=-193\pm3\kms$ and $v_r=-210\pm7\kms$, @mateo98), lending further credence to the notion that these 3 systems are associated with one another. This will be discussed in more detail in Irwin et al. (2013, in prep).
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A note on previous work
-----------------------
Finally, we also use our new algorithm to reanalyze all our previously published M31 dSph datasets. These include And V, VI [@collins11b], XI, XII and XIII [@chapman07; @collins10]. Details of the results of this reanalysis can be found in Appendix B. In summary, we find that our algorithm measures systemic velocities and velocity dispersions that are fully consistent with our previous work. We present these results in Table \[tab:kprops\]. And V, XI, XII and XIII are also analyzed by T12, so we compare our findings with theirs. For And XI and XII, our results are based on two and four times the number of stars respectively, and as such, supercede those presented in T12. In the case of And V and XIII, the T12 measurements are based sample sizes with four times the number of stars as our datasets, making their findings more robust.
In previous studies by our group [@chapman05; @letarte09; @collins10], we also published kinematic analyses for three additional M31 dSphs; And IX, And XV and And XVI. In T12, it was noted that the values presented in these works for systemic velocities and velocity dispersions were not consistent with those measured in their analyses. We revisited these datasets in light of this discrepancy, to see if our new technique could resolve this issue. We found that these discrepancies remained. For And IX, we measure a systemic velocity of $v_r=-204.8\pm2.1\kms$ cf. $v_r=-209.4\pm2.5\kms$ in T12 and a velocity dispersion of $\sigma_v=2.0^{+2.7}_{-2.0}\kms$ cf. $\sigma_v=10.9\pm2.0\kms$. Not only are their measurements determined from 4 times the member stars that we possess, we also experienced problems with our radial velocity measurements for the stars observed with this mask, due to the use of the minislitlet approach pioneered by @ibata05. This setup resulted in poor sky subtraction for many of the science spectra, lowering the quality of our radial velocity measurements. As such, the T12 results supercede those of our previous work [@chapman05; @collins10].
For And XV and XVI, we measure a systemic velocities of $v_r=-354.6\pm4.9\kms$ and $v_r=-374.1\pm6.8\kms$ cf. $v_r=-323.0\pm1.4\kms$ and $v_r=-367.3\pm2.8\kms$ from T12. We also note offsets in our velocity dispersions for And XV and XVI, where we measure $\sigma_v=9.6^{+4.1}_{-2.6}\kms$ and $\sigma_v=17.3^{+6.4}_{-4.4}\kms$ cf. $\sigma_v=3.8\pm2.9\kms$. In this instance, the data for both And XV and XVI were taken in poor conditions, with variable seeing that averaged at $1.8^{\prime\prime}$ and patchy cirrus. These conditions significantly deteriorated the quality of our spectra, and made the measurement of reliable radial velocities extremely difficult. Again, this leads us to conclude that the measurements made in T12 supercede those presented by our group in @letarte09.
The masses and dark matter content of M31 dSphs
===============================================
Measuring the masses and mass-to-light ratios of our sample {#sect:mass}
-----------------------------------------------------------
As dSph galaxies are predominantly dispersion supported systems, we can use their internal velocity dispersions to measure masses for these systems, allowing us to infer how dark matter dominated they are. There are several methods in the literature for this (e.g., @illingworth76 [@richstone86]), but recent work by @walker09a has shown that the mass contained within the half-light radius ($M_{\rm half}$)of these objects can be reliably estimated using the following formula:
$$M_{\rm half}=\mu r_{\rm half}\sigma_{v,{\rm half}}^2$$
where $\mu=580\msun{\rm pc}^{-1}$km$^{-2}$s$^2$, $r_{\rm half}$ is the spherical half-light radius in pc and $\sigma_{v,{\rm half}}$ is the luminosity- averaged velocity dispersion. This mass estimator is independent of the (unknown) velocity anisotropy of the tracer population, however, it is sensitive to the embeddedness of the stellar component within the DM halo. Particularly, the mass tends to be slightly over-estimated the more embedded the stars are [@walker11], especially if the dark matter halo follows a cored density profile.
As numerous authors have shown that the velocity dispersion profiles of dSphs are constant with radius (e.g., @walker07 [@walker09b]), we assume our measured values of $\sigma_v$ are representative of the luminosity-averaged velocity dispersion ($\sigma_{v,{\rm half}}$) used by @walker09b. However, if it transpired that the velocity dispersion profiles of the Andromedean dSphs were not flat, but declined or increased with radius, this would no longer true. We see no evidence for this behaviour in our dataset, although low-number statistics means we are unable to completely rule out this possibility. We calculate this for all our observed dSphs (including those we reanalyzed from previous works, see Appendix B) using results from the Keck LRIS and DEIMOS dataset, and report their masses within $r_{\rm half}$ ($M_{\rm half}$) in Table \[tab:kprops\].
[lcccccc]{} And V & 2.0 & $-391.5\pm2.7$ & $12.2^{+2.5}_{-1.9}$ & $2.6^{+0.66}_{-0.56}$& $88.4^{+22.3}_{-18.9}$ & $-2.0\pm0.1$\
And VI & 2.5 & $-339.8\pm1.8$ & $12.4^{+1.5}_{-1.3}$ & $4.7\pm0.7$& $27.5^{+4.2}_{-3.9}$ & $-1.5\pm0.1$\
And XI & 2.5 & $-427.5^{+3.5}_{-3.4}$ & $7.6^{+4.0(*)}_{-2.8}$ & $0.53^{+0.28}_{-0.21}$& $216^{+115}_{-87}$ & $-1.8\pm0.1$\
And XII & 2.5 & $-557.1\pm1.7$ & $0.0^{+4.0}$ & $0.0^{+0.3}$ & $0.0^{+194}$ & $-2.2\pm0.2$\
And XIII & 2.5 & $-204.8\pm4.9$ & $0.0^{+8.1(*)}$ & $0.0^{+0.7}$ & $0.0^{+330}$ & $-1.7\pm0.3$\
And XVII & 2.5 & $-251.6^{+1.8}_{-2.0}$ & $2.9^{+2.2}_{-1.9}$ & $0.13^{+0.22}_{-0.13}$& $12^{+22}_{-12}$& $-1.7\pm0.2$\
And XVIII & 2.5 & $-346.8\pm2.0$ & $0.0^{+2.7}$ & $0.0^{+0.14}$ & $0^{+5}$&$-1.4\pm0.3$\
And XIX & 2.0 & $-111.6^{+1.6}_{-1.4}$ & $4.7^{+1.6}_{-1.4}$ &$1.9^{+0.65}_{-0.66}$& $84.3^{+37}_{-38}$ & $-1.8\pm0.3$\
And XX & 2.5 & $-456.2^{+3.1}_{-3.6}$ & $7.1^{+3.9(*)}_{-2.5}$ & $0.33^{+0.20}_{-0.12}$& $238.1^{+147.6}_{-90.2}$& $-2.2\pm0.4$\
And XXI & 5.0 & $-362.5\pm0.9$ & $4.5^{+1.2}_{-1.0}$ & $0.99^{+0.28}_{-0.24}$ & $25.4^{+9.4}_{-8.7}$& $-1.8\pm0.1$\
And XXII & 2.0 & $-129.8\pm2.0$ & $2.8^{+1.9}_{-1.4}$ & $0.11^{+0.08}_{-0.06}$& $76.4^{+58.4}_{-48.1}$& $-1.8\pm0.6$\
And XXIII & 4.0 & $-237.7\pm1.2$ & $7.1\pm1.0$ & $2.9\pm4.4$& $58.5\pm36.2$& $-2.2\pm0.3$\
And XXIV & 1.5 & $-128.2\pm5.2$ & $0.0^{+7.3(*)}$ & $0.4^{+0.7}_{-0.4}$ & $82^{+157}_{-82}$ & $-1.8\pm0.3$\
And XXV & 2.5 & $-107.8\pm1.0$ & $3.0^{+1.2}_{-1.1}$ & $0.34^{+0.14}_{-0.12}$& $10.3^{+7.0}_{-6.7}$ & $-1.9\pm0.1$\
And XXVI & 3.0 & $-261.6^{+3.0}_{-2.8}$ & $8.6^{+2.8(*)}_{-2.2}$ & $0.96^{+0.43}_{-0.34}$ & $325^{+243}_{-225}$ & $-1.8\pm0.5$\
And XXVII & 1.5 & $-539.6^{+4.7}_{-4.5}$ & $14.8^{+4.3}_{-3.1}$ & $8.3^{+2.8}_{-3.9}$& $1391^{+1039}_{-1128}$ & $-2.1\pm0.5$\
And XXVIII & 2.5 & $-326.2\pm2.7$ & $6.6^{+2.9}_{-2.1}$ & $0.53^{+0.28}_{-0.21}$& $51^{+30}_{-25}$ & $-2.1\pm0.3$\
And XXX (Cass II) & 2.0 & $-139.8^{+6.0}_{-6.6}$ & $11.8^{+7.7}_{-4.7}$ & $2.2^{+1.4}_{-0.9}$ & $308^{+269}_{-219}$ & $-1.7\pm0.4$\
From these masses, it is trivial to estimate the dynamical central mass-to-light ratios for the objects, $[M/L]_{\rm half}$. We list these values for each dSph in Table \[tab:kprops\], where the associated uncertainties also take into account those from the measured luminosities and distances to these dSphs [@mcconnachie12; @conn12], as well as those on the masses measured in this work.
Comparing the mass-to-light ratios of M31 and Milky Way dSphs
-------------------------------------------------------------
By combining our measurements of the kinematic of M31 dSphs in this work with those from T12 and @tollerud13, we find ourselves with a set of kinematic properties as measured for 27 of the 28 Andromeda dSphs (owing to the difficulties experienced with the And XXIV dataset, we do not include this object in our subsequent analysis). This near-complete sample allows us to fully compare the masses and mass-to-light ratios for the M31 satellite system with those measured in the Milky Way satellites. Before beginning this analysis, we compile Table \[tab:summary\] which presents the kinematics for the full M31 satellite system, which combines the results from this work, T12, @kalirai10, and @tollerud13. In cases where two measurements for a dSph exist, we use those that were calculated from larger numbers of likely members, as these are the more robust. We begin by comparing the mass-to-light ratios (which indicate the relative dark matter dominance of these objects) of the two populations as a function of luminosity. In Fig \[fig:ml\] we show these values for all MW (red triangles, with values taken from @walker09b), and M31 (blue circles) dSphs as a function of their luminosity. We can see that all these objects are clearly dark matter dominated, excluding And XII and And XVII where we are unable to resolve the mass with current datasets. We also see that they follow the trend of increasing $[M/L]_{\rm half}$ with decreasing luminosity, as is seen in their MW counterparts.
The one tentative exception to this is the And XXV dSph. From our dataset, we measure a value of $[M/L]_{\rm half}=10.3^{+7.0}_{-6.7}$ for this object, making it consistent with a stellar population with no dark matter within its $1\sigma$ uncertainties. This result is surprising and would be of enormous importance if confirmed with a larger dataset than our catalog of 26 likely members as it would be the first dSph to be observed with a negligible dark matter component. And XXV is also one of the members of the recently discovered thin plane of satellites in Andromeda [@ibata13], and so the presence or absence of dark matter in And XXV might tell us more about the origins of this plane which are currently poorly understood.
![Dynamical mass-to-light ratio within the half-light radius, $[M/L]_{\rm half}$, as a function of half-light radius for all M31 (blue circles), MW (red triangles) and isolated dSphs (cyan squares).[]{data-label="fig:ml"}](fig19.eps){width="0.95\hsize"}
Comparing the masses of M31 and Milky Way dSphs
-----------------------------------------------
Finally, we discuss how the masses for the full sample of Andromeda dSphs for which kinematic data are available compare with those of the MW dSphs. For the M31 dSph population, we again use our compilation of kinematic properties assembled in Table \[tab:summary\]. We plot the velocity dispersions, mass within the half-light radius, and central densities for all M31 (blue circles) and MW (red triangles, @walker09b [@aden09; @koposov11; @simon11]) dSphs as a function of radius. We then overplot the best-fit NFW and cored mass profiles for the MW, taken from @walker09b. In general, we see that the M31 and MW objects are similarly consistent with these profiles, an agreement that was also noted by T12. However, there are 3 objects which are clear outliers to these relations. These are And XIX, XXI and XXV, with velocity dispersions of $\sigma_v=4.7^{+1.6}_{-1.4}\kms,
\sigma_v=4.5^{+1.2}_{-1.0}\kms$ and $\sigma_v=3.0^{+1.2}_{-1.1}\kms$, as derived in this work. Given their half-light radii ($r_{\rm
half}=$1481$^{+62}_{-268}$ pc, $r_{\rm half}=$842$\pm77$ pc and $r_{\rm
half}=$642$^{+47}_{-74}$ pc), one would expect them to have dispersions of closer to $9\kms$ in order to be consistent with the MW mass profile. As they stand, these three objects are outliers at a statistical significance of $2.5\sigma$, $3.0\sigma$ and $3.4\sigma$ (calculated directly from their likelihood distributions as presented in Figs. \[fig:And19\], \[fig:And21\] and \[fig:And25\]). Similarly, in T12 they noted that And XXII and And XIV were outliers in the same respect as And XIX, XXI and XXV, albeit at a lower significance. These difference can also be observed in terms of the enclosed masses and densities within $r_{\rm half}$.
{width="0.9\hsize"}
In @collins11b we argued that the low velocity dispersion seen in some Andromeda dwarfs were a result of tidal forces exerted on their halos by the host over the course of their evolution, and that this effect was predominantly seen in dSphs where their half-light radii were more extended for a given luminosity than expected, such is the case our three outliers, And XIX, XXI and XXV. This result therefore adds weight to the trend presented in that work. A number of recent works trying to account for the lower than predicted central masses of dSph galaxies within the Local Group also support this notion. For example, @penarrubia10 demonstrated that the presence of a massive stellar disk in the host galaxy (such as those of the MW and M31) can significantly reduce the total masses of its associated satellites. In addition, recent, papers by @zolotov12 and @brooks12, where the effect of baryons within dark matter only simulations was measured also find that tidal forces exerted by host galaxies where a massive disk is present will serve to reduce the masses of its satellite population at a far greater rate than hosts without baryons. And XIX, XXI and XXV may thus represent a population of dSph satellites whose orbital histories about M31 have resulted in substantial fractions of their central mass being removed by tides. It should be noted, however, that tides not only reduce the central masses and densities of dSph halos, they also reduce the spatial size of the luminous component [@penarrubia08b; @penarrubia10], albeit at a slower rate. The tidal scenario is therefore slightly difficult to reconcile with these outlying M31 dSphs having the largest sizes, unless they were both more massive and spatially larger in the past.
Other recent theoretical works have also shown that the removal of baryons from the very centres of dark matter halos by baryonic feedback (from star formation and supernovae, for example) can also help to lower the central masses and densities of satellite galaxies (e.g., @pontzen12 [@zolotov12; @brooks12]). For this method to work effectively, however, very large ‘blow outs’ of gas are required, of the order $\sim10^8-10^9\msun$, equivalent to $\sim40000$ SNe. This would require a minimum initial satellite luminosity of $M_V<-12$ [@zolotov12; @garrison13], significantly brighter than the current luminosities of our outliers ($M_V\sim-10$). Therefore, if feedback has indeed played a role in the shaping of the dark matter halos of And XIX, XXI and XXV, one assumes it would have to have operated in tandem with tidal stripping. We will discuss the implications and interpretation of these result further in a companion paper (Collins et al. in prep).
Metallicities {#sect:metals}
=============
Our observational setup was such that we cover the calcium triplet region (Ca II) of all our observed stars. This strong, absorption feature is useful not only for calculating velocities for each star, but also metallicities. For RGB stars, such as we have observed, there is a well known relation between the equivalent widths (EWs) of the Ca II lines, and the iron abundance, $\feh$, of the object. The calibration between these two values has been studied and tested by numerous authors, using both globular clusters and dSphs, and is valid down to metallicities as low as $\feh\sim-4$ (see e.g., @battaglia08 [@starkenburg10]). Following the @starkenburg10 method, which extends the sensitivity of this method down to as low as $\feh\sim-4$, we fit Gaussian functions to the three Ca II peaks to estimate their equivalent widths (EWs), and calculate \[Fe/H\] using equation \[eqn:cat\]:
$$\begin{aligned}
\feh=-2.87+0.195(V_{RGB}-V_{HB})+0.48\Sigma \rm{Ca}\\-0.913\Sigma
\rm{Ca}^{-1.5}+0.0155\Sigma \rm{Ca}(V_{RGB}-V_{HB})
\end{aligned}
\label{eqn:cat}$$
where $\Sigma$Ca=0.5EW$_{8498}$+EW$_{8542}$+0.6EW$_{8662}$, $V_{RGB}$ is the magnitude (or, if using a composite spectrum, the average, S:N weighted magnitude) of the RGB star, and $V_{HB}$ is the mean $V$-magnitude of the horizontal branch (HB). Using $V_{HB}-V_{RGB}$ removes any strong dependence on distance or reddening in the calculated value of \[Fe/H\], and gives the Ca II line strength at the level of the HB. For M31, we set this value to be $V_{HB}=$25.17 [@holland96][^1]. As the dSphs do not all sit at the same distance as M31, assuming this introduces a small error into our calculations, but it is at a far lower significance than the dominant uncertainty introduced by the noise within the spectra themselves. For individual stars, these measurements carry large uncertainties ($\gta0.4$ dex), but these are significantly reduced when stacking the spectra into a composite in order to measure an average metallicity for a given population.
Uncertainties on the individual measurements of \[Fe/H\] from our stellar spectra are typically large ($\ge0.5$ dex), so for a more robust determination of the average metallicities we co-add the spectra for each dSph (weighting by the S:N of each individual stellar spectrum, which is required to be a minimum of 2.5Å$^{-1}$) and measure the resulting EWs. In a few cases, not all 3 Ca II lines are well resolved. For And V, IX, XVII, XVIII XXVI and XXVIII, the third Ca II line is significantly affected by skylines, whilst for And XXIV, the first Ca II line is distorted. In the case of And XIII, only the second Ca II line appears well resolved. In these cases, we neglect the affected lines in our estimate of \[Fe/H\], and derive reduced equivalent widths from the unaffected lines. Where the third line is affected, this gives $\Sigma$Ca=1.5EW$_{8498}$+EW$_{8542}$. Where the second line is affected, we find $\Sigma$Ca=EW$_{8542}$+EW$_{8662}$. Finally, where only the second line seems reliable we use $\Sigma$Ca=1.7EW$_{8542}$. These coefficients are derived empirically from high S:N spectra where the absolute values of \[Fe/H\] are well known. We test these variations of $\Sigma$Ca by applying them to our high S:N co-added spectra where all three lines are well resolved, such as And XXI and XXV, and we find that all three formulae produce consistent values of \[Fe/H\]. The composite spectra for each satellite are shown in Figs. \[fig:sumspec\] and \[fig:sumspec2\]. In all cases, we find that our results are consistent with photometric metallicities derived in previous works.
{width="0.45\hsize"} {width="0.45\hsize"} {width="0.45\hsize"} {width="0.45\hsize"} {width="0.45\hsize"} {width="0.45\hsize"} {width="0.45\hsize"} {width="0.45\hsize"}
{width="0.45\hsize"} {width="0.45\hsize"} {width="0.45\hsize"} {width="0.45\hsize"} {width="0.45\hsize"}
In the MW, it has been observed that the average metallicities of the dSph population decrease with decreasing luminosity (e.g., @kirby08 [@kirby11]). In Fig. \[fig:mvfeh\], we plot the spectroscopic metallicities of the M31 dSphs (blue dSphs) as a function of absolute magnitude. We plot the MW dSphs as red triangles (@martin07 [@kirby08; @kirby11; @belokurov09; @koch09]). We also include those M31 dSphs for which only photometric measurements of \[Fe/H\] are available (And I, II, III, VII and XIV, @kalirai10 [@tollerud12]), and these are highlighted as encircled blue points. The dashed line represents the best-fit to the MW dSph population from @kirby11. The three relatively metal rich (\[Fe/H\]$\sim-1.5$ to $-2.0$), faint ($L\sim1000\lsun$) MW points are the three ultra faint dSphs Willman I, Boötes II and Segue 2, and these three were not included in the @kirby11 analysis, where the best fit MW relation was determined. We see that the metallicities for a given luminosity in the M31 dSphs also loosely define a relationship of decreasing metallicity with decreasing luminosity, and they agree with that defined by their MW counterparts within their associated uncertainties. However, it is also noteworthy that for dSphs with $L<10^6\lsun$, the Andromeda satellites are also consistent with having a constant metallicity of $\sim-1.8$. The same levelling off of average metallicity at lower luminosities was noted by @mcconnachie12, where they note that this break occurs at the same luminosity as a break in the luminosity-surface brightness relation for faint galaxies. As such, it could imply that the denisty of baryons in these systems, rather than the total number of baryons, could be the most important facor in determining their chemical evolution. The error bars we present here are still significant, so it is hard to fully interpret this result, but the hint of a metallicity floor in these lower luminosity systems is intriguing.
In Fig. \[fig:mvfeh\], we highlight the positions of our three kinematic outliers, And XIX, XXI and XXV, and we see that they fall almost exactly on the MW relation. In this figure, systems that have experienced extreme tidal stripping would move horizontally to the left, as their luminosity would gradually decrease as stars are stripped, but their chemistry would remain unaffected. One would expect to see such behaviour only after the stellar component began to be removed in earnest, after the majority of the dark matter halo had been removed. If their central densities were lowered by some active feedback mechanism, such as SNII explosions (e.g., @zolotov12), one would expect the objects to become more enriched and perhaps brighter, moving them up and to the right, potentially allowing them to remain on the MW relation. To confirm that this was the case for And XIX, XXI and XXV, we would require more information on the abundances of these objects and their star formation histories, which we do not currently possess.
![Spectroscopically derived \[Fe/H\] vs. luminosity for all MW (red triangles, taken from @kirby11, with additional measurements taken from @martin07 [@belokurov09; @koch09]) and M31 dSphs (blue circles, this work). The solid line represents the best fit relationship between these two parameters as taken from @kirby11. The dashed lines represent the $1\sigma$ scatter about this relationship. We see that the M31 dSphs follow this relationship very well within their associated uncertainties. As we discuss in § \[sect:metals\], those galaxies with $L<10^6\lsun$ are also consistent with having a constant metallicity, which could indicate a metallicity floor in these fainter systems.[]{data-label="fig:mvfeh"}](fig22.eps){width="0.95\hsize"}
[lccccccccccc]{} And I& -11.8 & 80 & -1.45$\pm$0.37 & N/A &-376.3$\pm2.2$ & 10.2$\pm1.9$&$656^{+68}_{-67}$ & 727$^{+18}_{-17}$ &(1),(2),(3)\
And II& -12.6 & 95 &-1.64$\pm$0.34 & N/A &-193.6$\pm1.0$ & 7.3$\pm$ 0.8& 1136$\pm46$ & 630$\pm15$ &(1),(3),(4)\
And III& -10.2 & 43 & -1.78$\pm$0.27 & N/A &-344.3$\pm1.7$ & 9.3$\pm1.4$& $463^{+44}_{-45}$ & 723$^{+18}_{-24}$&(1),(2),(3)\
And V& -9.6 & 85 & -1.6$\pm0.3$ & -1.8$\pm$ 0.2& -397.3$\pm1.5^{(b)}$ & 10.5$\pm1.1^{(b)}$& $302\pm44$ & 742$^{+21}_{-22}$ &(2),(3),(5)\
And VI& -11.5 & 38 & -1.3$\pm0.3$ & -1.5$\pm$ 0.3& -339.8$\pm1.9$ & 12.4$^{+1.5}_{-1.3}$& $524\pm49$ & 783$\pm28$ & (3),(5),(6)\
And VII& -13.3 & 18 & -1.4$\pm$0.3 & N/A &-307.2$\pm1.3$ & 13.0$\pm$ 1.0& $776\pm42$ & $762\pm35$ &(1),(2),(3)\
And IX& -8.1 & 32 & -2.2$\pm0.2$ & -1.9$\pm$ 0.6&-209.4$\pm2.5^{(b)}$ & 10.9$\pm2.0^{(b)}$& $436^{+68}_{-24}$ &$600^{+91}_{-23}$ & (1),(2),(3)\
And X& -8.1 & 22 &-1.93$\pm$0.48 & N/A & -164.1$\pm1.7$ & 6.4$\pm1.4$& $253^{+21}_{-65}$ &$670^{+24}_{-39}$ & (1),(2),(3)\
And XI& -6.9 & 5 &-2.0$\pm0.2$ & -2.0$\pm$ 0.3 & -427.5$^{+3.4}_{-3.5}$ & 7.6$^{+4.0}_{-2.8}$&$158^{+9}_{-23}$ & $763^{+29}_{-106}$& (3),(6),(7)\
And XII&-6.4 & 8 & -1.9$\pm0.2$ & -2.0$\pm$ 0.3 & -557.1$\pm1.7$ & 0.0$^{+4.0}$&$324^{+56}_{-72}$ & $928^{+40}_{-136}$ & (3),(6),(7)\
And XIII& -6.7 & 12 & -2.0$\pm0.2$ & -1.9$\pm$ 0.7 & -185.4$\pm2.4^{(b)}$ &5.8$\pm2.0^{(b)}$ & $172^{+34}_{-39}$ & $760^{+126}_{-154}$ &(2),(3),(7)\
And XIV & -8.3 & 38 & -2.26$\pm$0.3 & N/A & -480.6$\pm1.2$ &5.3$\pm$ 1.0 & $392^{+185}_{-205}$ & $793^{+23}_{-179}$ &(1),(2),(3)\
And XV & -9.4 & 29 & -1.1 & N/A &-323$\pm1.4^{(b)}$ & 4.0$\pm1.4^{(b)}$ & $220^{+29}_{-15}$ & $626^{+79}_{-35}$& (1),(2),(3)\
And XVI & -9.4 & 7 & -1.7 & -2.0$\pm$0.5 &-367.3$\pm2.8^{(b)}$ & 3.8$\pm2.9^{(b)}$& $123^{+13}_{-10}$ &$476^{+44}_{-29}$ & (1),(2),(3)\
And XVII& -8.5 & 7 & -1.9 &-1.7$\pm0.3$ & -251.6$^{+1.8}_{-2.0}$ & 2.9$^{+2.2}_{-1.9}$ &$262^{+53}_{-46}$ &$727^{+39}_{-25}$ & (1),(3),(6)\
And XVIII&-9.7 & 22 & -1.8$\pm0.5$ & N/A &-332.1$\pm2.7^{(b)}$ & 9.7$\pm2.3^{(b)}$ &$325\pm24$ & $1214^{+40}_{-43}$ & (1),(2),(3)\
And XIX & -9.3& 27 & -1.9$\pm0.4$ & -1.9$\pm0.6$ & -111.6$^{+1.6}_{-1.4}$ & 4.7$^{+1.6}_{-1.4}$ &$1481^{+62}_{-268}$ & $821^{+32}_{-148}$ & (1),(3),(6)\
And XX& -6.3 & 4 & -1.5$\pm0.5$ & -2.3$\pm0.8$ & -456.2$^{+3.1}_{-3.6}$ &7.1$^{+3.9}_{-2.5}$ &$114^{+31}_{-12}$ & $741^{+42}_{-52}$&(1),(3),(6)\
And XXI & -9.9 & 32 & -1.8 & -1.8$\pm0.4$ &-362.5$\pm0.9$ & 4.5$^{+1.2}_{-1.0}$ & $842\pm77$ & $827^{+23}_{-25}$ &(1),(3),(6)\
And XXII& -6.5 & 12 & -1.8 &-1.85$\pm$ 0.1&-129.8$\pm2.0$& 2.8$^{+1.9}_{-1.4}$ & $252^{+28}_{-47}$ &$920^{+32}_{-139}$ & (1),(2),(3),(8)\
And XXIII &-10.2 & 42&-1.8$\pm0.2$ & -2.3$\pm0.7$ & -237.7$\pm1.2$ & 7.1$\pm1.0$&$1001^{+53}_{-52}$ & $748^{+31}_{-21}$ & (1),(3),(6)\
AndXXIV& -7.6 & 3 &-1.8$\pm0.2$ & -1.8$\pm0.3$ & $-128.2\pm5.2^{(c)}$ &$0.0^{+7.3(c)}$ & $548^{+31}_{-37}$ & $898^{+28}_{42}$ &(1),(3),(6)\
And XXV & -9.7 & 25 &-1.8$\pm0.2$ &-2.1$\pm0.2$ &-107.8$\pm1.0$ & 3.0$^{+1.2}_{-1.1}$& $642^{+47}_{-74}$ &$736^{+23}_{-69}$ & (1),(3),(6)\
And XXVI& -7.1 & 6&-1.9$\pm0.2$ & -1.8$\pm0.5$ & -261.6$^{+3.0}_{-2.8}$ & 8.6$^{+2.8}_{-2.2}$&$219^{+67}_{-52}$ & $754^{+218}_{-164}$ & (1),(3),(6)\
AndXXVII & -7.9 & 11 &-1.7$\pm0.2$ & -1.5$\pm0.28$ & -539.6$^{+4.7}_{-4.5}$ & 14.8$^{+4.3}_{-3.1}$& $657^{+112}_{-271}$ &$1255^{+42}_{-474}$ & (1),(3),(6)\
AndXXVIII & -8.5 & 17 &-2.0$\pm0.2$ & -2.1$\pm0.3$ & -326.2$\pm2.7$ & 6.6$^{+2.9}_{-2.1}$& $210^{+60}_{-50}$ &$650^{+150}_{-80}$ & (6),(9)\
AndXXIX & -8.3 & 24 &-1.8$\pm0.2$ & N/A & $-194.4\pm1.5$ & $5.7\pm1.2$& $360\pm60$ &$730\pm75$ & (10),(11)\
And XXX (Cass II)& -8.0 & 8 &-1.6$\pm0.4$ & -2.2$\pm0.4$ &-139.8$^{+6.0}_{-6.6}$ & 11.8$^{+7.7}_{-4.7}$& $267^{+23}_{-36}$ & $681^{+32}_{-78}$ & (3),(6),(12)\
Conclusions {#sect:conc}
===========
Using new and existing spectroscopic data from the Keck I LRIS and Keck II DEIMOS spectrographs, we have homogeneously derived kinematic properties for 18 of the 28 known Andromeda dSph galaxies. Using a combination of their $g-i$ colors, positions on the sky and radial velocities, we determine the likelihood of each observed star belonging to a given dSph, thus filtering out MW foreground or M31 halo contaminants. We have measured both their systemic velocities and their velocity dispersions, with the latter allowing us to constrain the mass and densities within their half-light radii. For the first time, we confirm that And XVII, XIX, XX, XXIII, XXVI and Cass II are dark matter dominated objects, with dynamical mass-to-light ratios within the half-light radius of $[M/L]_{\rm half}>10\msun/\lsun$.
For And XXV, a bright M31 dSph ($M_V=-9.7$) we measure a mass-to-light ratio of only $[M/L]_{\rm half}=10.3^{+7.0}_{-6.7}\msun/\lsun$ from a sample of 26 stars, meaning that it is consistent with a simple stellar system with no appreciable dark matter component within its $1\sigma$ uncertainties. If this were confirmed with larger datasets, it would prove to be a very important object for our understanding of the formation and evolution of galaxies.
We compare our computed velocity dispersions and mass estimates with those measured for MW dSphs, and find that the majority of the M31 dSphs have very similar mass-size scalings to those of the MW. However, we note 3 significant outliers to these scalings, namely And XIX, XXI and XXV, who possess significantly lower velocity dispersions than expected for their size. These results builds on the identification of three potential outliers in the @tollerud12 dataset (And XIV, XV and XVI). We suggest that the lower densities of the dark matter halos for these outliers could be an indication that they have encountered greater tidal stresses from their host over the course of their evolution, decreasing their masses. However, these bright systems still fall on the luminosity-metallicity relation established for the dSph galaxies of the Local Group. If these objects had undergone significant tidal disruption, we would expect them to lie above this relation. As such, this remains puzzling, and requires dedicated follow up studies to fully map out the kinematics of these unusual systems.
We measure the metallicities of all 18 dSphs from their co-added spectra and find that they are consistent with the established MW trend of decreasing metallicity with decreasing luminosity.
This work represents a significant step forward in understanding the mass profiles of dwarf spheroidal galaxies. Far from residing in dark matter halos with identical mass profiles, we show that the halos of these objects are complex, and differ from one to the next, with their environment and tidal evolution imprinting themselves upon the dynamics of their stellar populations. The Andromeda system of dSphs presents us with an opportunity to better understand these processes, and our future work will further illuminate the evolutionary paths taken by these smallest of galaxies.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Hans-Walter Rix for helpful discussions regarding this manuscript. We are also grateful to the referee for their helpful and detailed suggestions for improving this work. We thank the SPLASH collaboration for providing us with details of their observations of dSphs as presented in T12.
Most of the data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation.
Based in part on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institute National des Sciences de l’Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii.
Based in part on data collected at Subaru Telescope, which is operated by the National Astronomical Observatory of Japan.
The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/.
SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, University of Cambridge, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.
R.I. gratefully acknowledges support from the Agence Nationale de la Recherche though the grant POMMME (ANR 09-BLAN-0228).
G.F.L. gratefully acknowledges financial support for his ARC Future Fellowship (FT100100268) and through the award of an ARC Discovery Project (DP110100678).
N.B. gratefully acknowledges financial support through the award of an ARC Discovery Project (DP110100678).
A.K. thanks the Deutsche Forschungsgemeinschaft for funding from Emmy-Noether grant Ko 4161/1.
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Testing the membership probability algorithm
============================================
The inclusion of a velocity term in the calculation of $P_i$ {#sect:vptest}
------------------------------------------------------------
In T12, the authors do not impose a velocity probability criterion for their membership calculations. Instead they require all member stars to have a total probability, based on their positions and colors, of $P_{member}>0.1$, and then apply a $3\sigma$ clipping to this final sample to prevent any outliers from significantly inflating their calculated velocity dispersions. In our analysis, we have avoided making any hard cuts to our sample by also utilising prior information on the velocities of our expected contaminant populations and member stars. In § \[sect:velprob\], we tested our velocity probability criteria was not overly biasing our final measurements of velocity dispersion to artificially lower values with the introduction of an extra parameter, $\eta$, that allows us to add additional weight to stars in the tails of the Gaussian velocity distribution. However, we can further test our velocity criterion by removing it entirely from the probabilistic determination, and instead implementing the same cuts presented by T12. This involves cutting stars where $P_i<0.1$ (as determined from $P_{CMD}$ and $P_{dist}$), and also by iteratively removing all stars that have velocities that do not lie within $3\sigma$ of the mean of the remaining sample. In Table \[tab:vpcuts\], we present the results of this on our measured values of $v_r$ and $\sigma_v$ for our full sample of dSphs. For all objects (bar And XXVII, which is a unique case, as described in § \[sect:and27\]) the systemic velocities derived are within $\sim2-3\kms$ of one another. The velocity dispersions we measure from our full algorithm tend to be slightly higher on average, and this is to be expected as we do not cut any stars from our analysis, and therefore outliers in the velocity profile may be assigned non-negligible membership probabilities that will allow them to increase this measurement. By and large, these differences are not significant, with the final values agreeing to well within their $1\sigma$ uncertainties.
It is interesting that our algorithm appears to perform better when dealing with dSphs where the number of member stars is low. This is best demonstrated by And XI (see Fig. \[fig:And11\]). Our algorithm identifies 5 stars with non-negligible probabilities of membership, clustered around $v_r\sim-430\kms$. Our full algorithm measures a systemic velocity of $v_r=-427.5^{+3.5}_{-3.4}\kms$ and a velocity dispersion of $\sigma_v=7.5^{+4.0}_{-2.8}\kms$. One of these stars is slightly offset from the other 4 with a more negative velocity of $v_r=-456.8\kms$. Although this star has a reasonably high probability of being a member based on its distance from the centre of And XI, and its position in the CMD, it does not survive the $3\sigma$ velocity clipping procedure of T12. As the number of member stars is so low, cutting one star from the sample can have a significant effect, and as such, while the T12 procedure determines a very similar systemic velocity of $v_r=-425.0\pm3.1\kms$ it is unable to resolve a velocity dispersion. This effect is also seen other systems (such as And XIII, XVII, XXII and XXVI), although it is typically less pronounced.
Another regime where our algorithm performs better than that of T12 is where the systemic velocity of the system in question is within the regime of the contaminating Milky Way K-dwarfs. An example of this is the unusual system, And XIX, where our algorithm measures a systemic velocity of $v_r=-111.6^{+1.6}_{-1.4}\kms$ and a velocity dispersion of $\sigma_v=4.7^{+1.6}_{-1.4}$. However, the procedure of T12 is less able to resolve the kinematics of the system, measuring $v_r=-109.3\pm5.3\kms$ and a velocity dispersion of $\sigma_v=1.8^{+9.1}_{-1.8}$. The much larger uncertainty on the dispersion is a result of including Milky Way contaminants in the sample which can be difficult to cut out without applying prior knowledge of the velocity profile of this population. $3\sigma$ clipping allows outliers to contribute more significantly to the measured profile in this instance, increasing the uncertainty. A similar effect is seen in the And XXIV and And XXX (Cass II) objects, which also have systemic velocities in the Milky Way contamination regime.
These results lead us to conclude that the inclusion of a $P_{vel}$ term in our analysis allows us to more effectively determine the true kinematics of the systems we are studying. Further, as no cuts to the sample are required using this method, it allows for a more unbiased study of the kinematics of dSphs than that of T12.
{width="0.45\hsize"} {width="0.45\hsize"} {width="0.9\hsize"}
[lcccc]{} And V & $-391.1\pm2.9$ & $10.8^{+3.0}_{-2.3}$ & $-391.5\pm2.7$ & $12.2^{+2.5}_{-1.9}$\
And VI & $-339.0\pm3.0$ & $11.9^{+2.9}_{-2.3}$ & $-339.8\pm1.8$ & $ 12.4^{+1.5}_{-1.3}$\
And XI & $-425.0\pm3.1$ & $0^{+3.5}$ & $-427.5^{+3.5}_{-3.4}$ & $7.6^{+4.0}_{-2.8}$\
And XII & $-558.8\pm3.7$ & $0^{+6.8}$ & $-557.1\pm1.7$ & $0^{+4.0}$\
And XIII & $-203.8\pm8.4$ & $0^{+16.2}$ & $-204.8\pm4.9$& $0.0^{+8.1}$\
And XVII & $-260.0^{+8.0}_{-7.8}$ & $1.8^{+9.1}_{-1.8}$ &$-254.3^{+3.3}_{-3.7}$ &$2.9^{+5.0}_{-2.9}$\
And XVIII &$-345.1\pm3.3$ & $0^{+4.4}$ & $-346.8\pm2.0$& $0^{+2.7}$\
And XIX & $-109.3\pm5.3$ & $1.5^{+6.8}_{-1.5}$ & $-111.6^{+1.6}_{-1.4}$& $4.7^{+1.6}_{-1.4}$\
And XX & $-454.6^{+4.6}_{-5.7}$ & $7.7^{+8.4}_{-3.9}$ &$-456.2^{+3.1}_{-3.6}$ &$7.1^{+3.9}_{-2.5}$\
And XXI &$-363.4^{+2.0}_{-1.8}$ & $3.2^{+2.3}_{-2.1}$ & $-362.5\pm0.9$&$4.5^{+1.2}_{-1.0}$\
And XXII &$-131.4\pm2.7$ & $0^{+3.1}$ &$-129.8\pm2.0$ &$2.8^{+1.9}_{-1.4}$\
And XXIII & $-236.9\pm2.1$ & $8.4^{+1.9}_{-1.5}$ &$-237.7\pm1.2$ &$7.1\pm1.0$\
And XXIV &$-129.2\pm3.6$ & $0^{+6.1}$ & $-129.9^{+4.3}_{-4.4}$& $3.5^{+6.6}_{-3.5}$\
And XXV & $-107.7^{+1.9}_{-1.8}$ & $3.3^{+2.2}_{-1.8}$ &$-107.8\pm1.0$ & $3.0^{+1.2}_{-1.1}$\
And XXVI & $-264.1\pm4.5$ & $0^{+4.8}$ & $-261.7^{+3.1}_{-2.8}$& $8.7^{+2.9}_{-2.3}$\
And XXVII & $-517.6^{+42.8}_{-43.2}$ & $19.3^{+17}_{-19}$ & $-539.6^{+4.7}_{-4.5}$& $14.8^{+4.3}_{-3.1}$\
And XXX (CassII) & $-140.1^{+8.6}_{-9.3}$ & $14.1^{+12.9}_{-6.1}$ &$-139.8^{+6.0}_{-6.6}$ &$11.8^{+7.7}_{-4.7}$\
The effect of low signal-to-noise data on measuring $v_r$ and $\sigma_v$
------------------------------------------------------------------------
For our brightest targets ($i\lta22.5$) the S:N of our spectra is typically $>3$Å$^{-1}$. However, as our targets become fainter, so too their S:N falls. For spectra with S:N$\gta1.5$Å$^{-1}$, our pipeline is still able to measure velocities based on the Ca II triplet, with reasonable measurement uncertainties. However, it is prudent to check whether the inclusion of these velocities, calculated from significantly noisier spectra, has a detrimental effect on our ability to measure the kinematic properties of our dSph sample.
Such a test is straightforward to implement. We have a number of dSphs within our sample for which our probabilistic analysis identifies $\sim30$ likely members (such as And XXI, XXIII and XXV). We can therefore use these samples to impose S:N cuts on our data to see the effect of this on our measurements of $v_r$ and $\sigma_v$. We present the results of this test in Table \[tab:sntest\], and our finding is that, as the level of our imposed S:N cut increases (and so the number of included stars decreases), the systemic velocity remains more or less constant. The measured velocity dispersion, however, shows some variation. In the case of And XXI and XXV, the dispersion increases with increased S:N, however not significantly. In both cases the dispersion calculated from the higher S:N data lies well within $1\sigma$ of that calculated from the lower S:N data. Intuitively, this makes sense as the spectra with higher S:N are likely to have lower velocity uncertainties, and so our maximum-likelihood analysis will attribute more of the spread in measured velocities to an intrinsic dispersion, rather than to our measurement errors. In the case of And XXIII, we find the opposite to be true. As our S:N cut increases, we find that our measured dispersion decreases. This may be because the number of member stars in subsequent quality cuts drops off more rapidly for And XXIII than And XXI and XXV. This suggests that we should be extra cautious when interpreting our measured velocity dispersions for dSphs where both the average S:N of member stars, and the number of member stars, is low.
[lccccccccc]{} S:N$>2$ & 20 & $-362.9\pm0.9$& $3.5^{+0.9}_{-0.7}$ & 22 & $-238.0\pm1.2$ & $6.6\pm1.1$ & 27 & $-107.7\pm0.9$ & 2.7$\pm1.1$\
S:N$>3$ & 11 & $-364.2\pm0.9$ & $3.1\pm0.8$ & 10 & $-238.3\pm1.4$ & $5.1^{+1.4}_{-1.2}$ & 24 & $-107.7\pm1.0$ & $3.0\pm1.2$\
S:N$>4$ & 5 & $-363.9\pm1.5$ & $4.0^{+1.5}_{-1.1}$ & 5 & $-239.4\pm1.1$ & $5.7^{+1.5}_{-1.3}$ & 16 & $-108.2\pm1.2$ & $3.0^{+1.4}_{-1.2}$\
S:N$>5$ & 3 & $-362.6^{+2.2}_{-2.3}$ & $4.5^{+2.6}_{-1.5}$ & 2 & $-239.5\pm1.9$ & $0.0^{+4.9}$ & 13 & $-109.0\pm1.2$ & $2.8^{+1.5}_{-1.3}$\
The effect of small sample sizes on determining kinematic properties of dwarf galaxies
--------------------------------------------------------------------------------------
Obtaining reliable velocities for member stars of faint and distant systems is a difficult task that can only be achieved with the largest optical telescopes, such as Keck. Given the demand for facilities such as this, any observing time awarded must be used as effectively as possible, and this often means compromising between deep pointings for a few objects, and shallower pointings for a number of objects. With longer or multiple exposures on a single target, one can build up impressive samples of member stars for an individual dSph. For example, the SPLASH collaboration observed a total of 95 members in And II, one of the brightest M31 dSph companions [@kalirai10] by taking 2 separate exposure fields over this large object. However, multiple exposures such as these produce diminishing returns as you move down the luminosity scale to fainter, more compact dSphs. This is both because of their smaller size with respect to the DEIMOS field of view, and the fewer number of bright stars available on the RGB to target. In this case, the only way to identify more members is by integrating for longer, but given the paucity of stars, the trade-off between time spent exposing and additional members observed can be quite expensive. Such difficulties inevitably lead to the inference of dynamical properties for an entire system from a handful of stars. It is important for us to understand the effect this bias has on our results, and how reliable the quoted values are. We test this using our datasets for which we identify $>25$ member stars, namely And XIX, XXI, XXIII and XXV using the following method. We select 4, 6, 8, 10, 15, 20, 25 and 30 stars at random from each dataset and then measure the systemic velocity and velocity dispersion using our probability algorithm. This was repeated 1000 times for each sample size. In cases where the algorithm is unable to resolve a velocity dispersion, we throw out the result and resimulate, as null results here will affect our averages and will not inform us whether the instances in which we are able to resolve a velocity dispersion from small numbers of stars are producing valid, reliable result. We display the resulting values in Table \[tab:sampsz\], with the true value recovered from the full sample shown in bold in the final row for comparison. We show that in all these cases, the systemic velocity and velocity dispersion are recovered well within the scatter of the 1000 simulations even when dealing with sample sizes as small as 4 stars, so long as the measurement is resolved. In cases where we are unable to resolve a dispersion, we find that our resulting uncertainties are not meaningful. This is shown explicitly in the case of And XVIII, where we can compare our upper limit for the velocity dispersion as determined from our algorithm with the dispersion calculated in T12 from a much larger dataset. We see that our uncertainty is not consistent with their result. As such, we advise that in all cases where we calculate velocity dispersions from small samples ($N_*<8$, And XI, XII, XX XXIV and XXVI), the dispersion measurements should be treated as indications of the likely dispersion, and need to be confirmed with follow-up studies.
[ccccccc]{} 4 & $-362.6\pm3.1$ & $5.0\pm3.3$ & $-237.2\pm3.8$ & $7.5\pm3.6$ & $-108.8\pm2.4$ & 5.4$\pm4.3$\
6 & $-363.0\pm2.0$ & $4.0\pm1.9$ & $-237.3\pm3.0$ & $6.9\pm2.6$& $-108.0\pm2.7$ & $4.0\pm2.3$\
8 & $-362.9\pm2.1$ & $4.2\pm2.0$ & $-236.8\pm2.8$ & $6.9\pm2.3$ & $-107.6\pm2.0$ & $2.7\pm1.1$\
10 & $-362.7\pm1.8$ & $4.1\pm1.8$ & $-237.3\pm2.5$ & $6.5\pm2.1$ & $-107.9\pm1.5$ & $3.1\pm1.0$\
15 & $-362.9\pm1.5$ & $4.3\pm1.2$ & $-237.4\pm2.0$ & $6.8\pm1.4$ & $-107.9\pm1.5$ & $3.1\pm1.0$\
20 & $-362.9\pm1.3$ & $4.3\pm1.2$ & $-237.6\pm1.7$ & $6.9\pm1.4$ & $-107.8\pm1.2$ & $3.0\pm0.9$\
25 & $-363.0\pm1.0$ & $4.4\pm1.0$ & $-237.4\pm1.4$ & $7.1\pm1.7$ & $-107.8\pm1.0$ & $3.1\pm0.8$\
30 & $-362.8\pm1.1$ & $4.4\pm1.1$ & $-237.7\pm1.3$ & $7.2\pm1.4$ & – & –\
[**Full sample**]{} & $\boldsymbol{-362.5\pm0.9}$ &$\boldsymbol{4.5^{+1.2}_{-1.1}}$ & $\boldsymbol{-237.7\pm1.2}$ & $\boldsymbol{7.1\pm1.0}$ & $\boldsymbol{-107.8\pm1.0}$& $\boldsymbol{3.0^{+1.2}_{-1.1}}$\
Testing our algorithm on the SPLASH sample of M31 dSphs
-------------------------------------------------------
In T12, the authors reported on the kinematic properties of 15 M31 dSphs, And I, III, V, VII, IX, X, XI, XII, XIII, XIV, XV, XVI, XVIII, XXI and XXII, and the positions, and measured velocities (plus uncertainties) for each likely member stars were published as part of that work. The authors were kind enough to also give us access to these properties for their non-member stars so that we might run our algorithm over the full samples to see if we reproduce their results. Our technique for assigning membership probability differs from theirs in that we use the velocities of stars as an additional criterion for membership, whereas they use a cut on both the resulting probability ($P({\mathrm member}>0.1$), and a $3\sigma$ clipping on the velocity. In addition, where we use PAndAS CFHT MegaCam $g-$ and $i-$band photometry for our membership analysis,the SPLASH team use their own Washington-DDO51 filter photometric dataset, in membership classification. As such, small differences might be expected, but if our technique is robust our results should well mirror those of T12. In Table \[tab:splashcomp\] we compare our calculated values of $v_r$ and $\sigma_v$ to those published in T12. As the measurements made in T12 for And XI and XII are made from only 2 stars, we do not include these in this test. In general, the results from both analyses agree to within $1\sigma$ of one another, with the majority of them being well within this bound. Typically, we find that our procedure measures slightly larger values for $\sigma_v$ than that of T12 (with the exception of And IX, And XVIII and And XXII). This is to be expected, as we do not cut stars from our analysis based on their velocity, instead we down-weight their probability of membership. As such, those stars considered as outliers would naturally inflate our dispersions above those measured by T12, but the effect is marginal. These results demonstrate that our technique for assigning probability of membership of individual stars within M31 dSphs based on their photometric properties and velocities is robust, and comparable to that of T12. However, as discussed in § \[sect:vptest\], we find that our technique is superior as it requires no cuts to the final dataset to be made, reducing the bias in these measurements.
[lcccc]{} And I & -376.3$\pm2.2$ & 10.2$\pm1.9$ & $-376.3\pm1.3$ & $10.6\pm1.0$\
And III & -344.3$\pm1.7$ & 9.3$\pm1.4$& $-344.2\pm1.2$ & $10.1\pm1.9$\
And V & -397.3$\pm1.5$ & 10.5$\pm1.1$ & $-396.0\pm1.0$ & $11.4\pm1.2$\
And VII & -307.2$\pm1.3$ & 13.0$\pm$ 1.0& $-307.1\pm1.1$ & $13.1\pm0.9$\
And IX & -209.4$\pm2.5$ & 10.9$\pm2.0$ & $-210.3\pm1.9$ & $10.2^{+1.9}_{-1.7}$\
And X & -164.1$\pm1.7$ & 6.4$\pm1.4$ & $-165.3\pm1.5$ & $6.0^{+1.3}_{-1.2}$\
And XIII& -185.4$\pm2.4$ & 5.8$\pm2.0$ & $-183.0^{+2.4}_{-2.3}$ & $8.6^{+2.1}_{-1.7}$\
And XV & -323$\pm1.4$ & 4.0$\pm1.4$ & $-322.6\pm1.1$ & $6.0^{+2.0}_{-1.8}$\
And XVI & -367.3$\pm2.8$ & 3.8$\pm2.9$& $-366.1^{+4.0}_{-3.1}$ & $4.2^{+4.8}_{-4.2}$\
And XVIII &-332.1$\pm2.7$ & 9.7$\pm2.3$ & $-330.7^{+3.9}_{-4.1}$ & $7.5^{+4.5}_{-3.1}$\
And XXI & $-361.4\pm5.8$ & $7.2\pm5.5$ & $-358.9^{+5.1}_{-5.6}$ & $8.5^{+6.3}_{-5.1}$\
And XXII & $-126.8\pm3.1$ & $3.54^{+4.16}_{-2.49}$ & $-124.2^{+4.6}_{-4.5}$ & $0.0^{+5.7}$\
Reanalyzing our previously published results
============================================
To ensure our analysis of the global properties of the M31 dSph population in § \[sect:mass\] is homogenous, we reanalyzed our previously published datasets using our new probability algorithm. Here we briefly summarize the results of this analysis and compare the new results to the published works. The objects we discuss here are And V, VI (first published in @collins11b), XI, XII and XIII (published in @chapman05 [@chapman07; @collins10]). And V was observed with the LRIS instrument, while the remaining objects were observed with the DEIMOS instrument.
Andromeda V {#sect:and5}
-----------
Andromeda V (And V) was observed using the LRIS instrument on Keck I rather than the DEIMOS instrument on Keck II. LRIS has a lower resolution than DEIMOS, and a smaller field of view, which lowers the accuracy of velocity measurement and limits us to only $\sim50$ targets within a mask compared with $100-200$ for a DEIMOS mask. The raw data were also not reduced using our standard pipeline, owing to problems with the arc-lamp calibrations, and were instead analyzed using the NOAO.ONEDSPEC and NOAO.TWODSPEC packages in IRAF. The results from this reduction, plus an analysis of the data using hard cuts in velocity, distance and color to determine membership were first published in @collins11b. Our full probabilistic analysis identifies 17 stars with a non-negligible probability of belonging to And V. Our technique determines a systemic velocity of $v_r=-391.5\pm2.7\kms$ and $\sigma_v=12.2^{+2.5}_{-1.9}\kms$. Comparing these values to our previously published results ($v_r=-393.1\pm4.2\kms$ and $\sigma_v=11.5^{+5.3}_{-4.3}\kms$, @collins11b) we find them to be consistent within the quoted uncertainties. We also compare our results to those of T12, who measured $v_r=−397.3\pm1.5\kms$ and $\sigma_v=10.5\pm1.1\kms$ from a larger sample of stars (85 members cf. 17) using the higher resolution DEIMOS spectrograph. The velocity dispersions of both are consistent within their $1\sigma$ uncertainties, as are the systemic velocities. Given the difference of a factor of 5 in number of probable member stars between our study and that of T12, this consistency is reassuring, and demonstrates the ability of our technique to accurately determine the kinematics of M31 dSph galaxies from small sample sizes.
Andromeda VI {#sect:and6}
------------
As And VI sits at a large projected distance from the centre of M31 ($\sim270$ kpc), it was not observed as part of the PAndAS survey and we were unable to use CFHT data for our $P_{CMD}$ determination. Instead, we use Subaru Suprime-cam data (PI. N. Arimoto, see @collins11b for a full discussion of this data). Using our full probabilistic analysis, we identify 45 stars with $P_i>10^{-6}$. Our technique determines a most-likely $v_r=-339.8\pm1.8\kms$, with $\sigma_v=12.4^{+1.5}_{-1.3}\kms$. Comparing these values with the results of @collins11b who measured $v_r=-344.8\pm2.5\kms$ and $\sigma_v=9.4^{+3.2}_{-2.4}\kms$, we see that both the systemic velocities and the velocity dispersions are consistent within quoted uncertainties. The slight differences between our previous study and this work are simply a result of the application of our new technique.
Andromeda XI {#sect:and11}
------------
The kinematic properties for Andromeda XI as measured from this DEIMOS data set were first published in @collins10. Here we identify 5 stars as probable members. We determine most-likely parameters of $v_r=-427.5^{+3.5}_{-3.4}\kms$ and $\sigma_v=7.6^{+4.0}_{-2.8}\kms$. In @collins10 we measured $v_r=-419.4^{+4.4}_{-3.8}\kms$ and we were unable to resolve the velocity dispersion for the dSph, measuring $\sigma_v=0.0^{+4.6}\kms$ (where the upper bound represents the formal 1$\sigma$ uncertainty on the unresolved dispersion), which implied a higher systemic velocity and lower velocity dispersion for And XI. However, in that analysis, one star with a velocity of $\sim-440\kms$ was considered to be an outlier based on its velocity, and thus excluded from the kinematic analysis. Here, our algorithm gives this star a non-zero probability of membership, which likely decreases the systemic velocity and increases the dispersion. T12 also presented observations for And XI, but they were not able to cleanly detect the galaxy. They identified 2 stars with highly negative velocities ($\sim-460\kms$), which are offset from our systemic velocity by $\sim30\kms$. Given the very negative velocities of their 2 stars, the probability of them both being M31 contaminants seems low, and some other explanation may be more suitable. Between observations, there is one star common to both ($\alpha=$00:46:19.10,$\delta=$+33:48:4.1), for which we measure a velocity of $-427.16\kms$ compared with $-461.6\kms$. This amounts to a statistical difference at the level of $5\sigma$. One obvious avenue to check is that there has been no velocity offset introduced by a rogue skyline that falls within the region of one of the three Ca II lines. We have carefully checked the spectra of each of our 5 probable members to see if this has been the case. We also rederive the velocity based on cross-correlations with each of the three lines individually, rather than with the full triplet. We find these results, and their average to be entirely consistent within the associated errors from the velocities derived using the technique discussed in § \[sect:specobs\]. This large discrepancy is puzzling, particularly as the methods used to measure velocities in this work are almost identical to those of T12. Without further data it is not possible for either team to pin down the exact issue, or which of the datasets gives the true systemic velocity. This argues for taking further observations within these faint systems, in the hopes of better understanding both the systems themselves, and any systematics introduced by the DEIMOS instrument. It is comforting that, in all cases where this offset is observed, the velocity dispersions measured by each team are consistent with one another, suggesting the problem affects all observations identically.
In this case, we have identified a greater number of potential members stars than in the work of T12, the spectra for all of which have relatively high S:N (S:N$\sim5-10$Å$^{-1}$). Therefore, our results should be considered as more robust than those of T12.
Andromeda XII {#sect:and12}
-------------
Kinematic properties for Andromeda XII as determined from the DEIMOS data set presented here were previously published in @chapman07 and @collins10, and in both cases, membership was largely determined using hard cuts in velocity. This object has been of particular interest as it possesses an extremely negative systemic velocity with respect to Andromeda, suggesting that it is on its first infall into the local group (see @chapman07 for a full discussion). Using our new algorithm, we measure $v_r=-557.1\kms$ and an unresolved velocity dispersion of $\sigma_v=0.0^{+4.0}\kms$, where the upper bound represents the formal $1\sigma$ uncertainty on the measurement. As the dispersion is unresolved, the lower error bound is undefined, as having a negative velocity dispersion is unphysical. These values are completely consistent with the results of @chapman07 and @collins10 ($v_r=-558.4\pm3.2\kms$ and $\sigma_v=2.6^{+5.1}_{-2.6}\kms$). T12 also presented observations for And XII, but as for And XI, they were not able to cleanly detect the galaxy. They identified 2 stars with highly negative velocities ($\sim-530\kms$), which are offset from our systemic velocity by $\sim30\kms$. In this instance, both the stars they observed overlap with two of our likely members, situated at $\alpha$=00:47:28.63,$\delta$=+34:22:43.1 and $\alpha=$00:47:24.69,$\delta=$+34:22:23.9, and these velocities are offset from those that we measure at a statistical level of $3.8\sigma$. This suggests that the self-same calibration effect that causes an offset between our results for And XI and those of T12, is present here also. In the case of And XII, our mask was observed over two separate nights, giving us two velocity measurements for each star (as discussed in @chapman07 and @collins10), and we saw no evidence for systematic offsets of this magnitude in the night to night velocities, making a calibration error within our dataset seem unlikely, though not impossible. We therefore conclude that, owing to our larger sample of members and repeat observations, our measurements for the kinematic properties of And XII are more robust than those of T12.
Andromeda XIII {#sect:and13}
--------------
The kinematic properties of Andromeda XIII were also presented in @collins10, And XIII sits at a large projected distance from Andromeda $(~\sim120$ kpc) in the southern M31 halo, so we expect contamination from the Milky Way and Andromeda halo to be low. It is surprising then, that we see significant structure within our DEIMOS field. This is also seen within the 3 fields observed by T12, who attribute this over-density of stars located at $v_r\sim-120\kms$ to an association with the TriAnd over-density within the Galactic halo. We too see a number of stars between $-140\kms$ and $-100\kms$. From their positions within the CMD of And XIII, they appear more consistent with MW foreground K-dwarfs than M31 RGB stars. As such, these are also likely associated to this MW substructure.
Using our full probabilistic analysis, we identify the most probable And XIII stars as those 4 that cluster around $v\sim-200\kms$, and we determine $v_r=-204.8\pm4.9\kms$ and are unable to resolve a velocity dispersion, with $\sigma_v=0.0^{+8.1}$, where the upper limit indicates the formal $1\sigma$ uncertainty on the likelihood distribution. Given the very large uncertainties on these values (mostly a factor of the low number of member stars) it comes as no surprise perhaps that these results are consistent with the results in @collins10 ($v_r=-195.0^{+7.4}_{-8.4}\kms$ and $\sigma_v=9.7^{+8.9}_{-4.5}$), although not with those of T12 ($v_r=-185.4\pm2.4\kms$ and $\sigma_v=5.6\pm2.0\kms$), where they derived parameters from three times the number of member stars that we present here. Given the highly substructured nature of the And XIII field, and the fact that our detection is at a very low significance (only 4 stars), we find that the T12 measurements for the kinematics of And XIII are more robust than ours.
[^1]: This assumed value is sensitive to age and metallicity effects, see @chen09 for a discussion, however owing to the large distance of M31, small differences in this value within the M31 system will have a negligible effect on metallicity calculations
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We introduce a new parametrization for the two-parameter species sampling model with [*finite*]{} but [*random*]{} number of different species recently introduced in Gnedin (2010a). We show the reparametrization yields a representation in terms of generalized Waring mixture of Fisher species sampling models and derive the structural distribution of the model.\
[*Keywords*]{}: Exchangeable Gibbs partitions, Generalized Waring distribution, Gnedin-Fisher model, Poisson-Dirichlet model, Species Sampling model.
author:
- |
<span style="font-variant:small-caps;">Annalisa Cerquetti</span>[^1]\
*[Department of Methods and Models for Economics, Territory and Finance]{}\
*[Sapienza University of Rome, Italy ]{}**
title: '[****]{}[^2] '
---
Introduction
============
Gnedin (2010a) introduces a two parameter family of exchangeable partition models belonging to the Gibbs class of genius $\alpha =-1$ (Gnedin and Pitman, 2006), by suitably mixing Fisher’s (1943) $(-1, \xi)$ partitions over the fixed number of boxes. The resulting $(\gamma, \zeta)$ [*Gnedin-Fisher*]{} species sampling model has exchangeable partition probability function (EPPF) of the form $$\label{gnedintwo}
p_{\gamma, \zeta}(n_1, \dots, n_k)=\frac{(\gamma)_{n-k} \prod_{i=1}^{k-1}(i^2 -\gamma i +\zeta)}{\prod_{l=1}^{n-1} (l^2 +\gamma l +\zeta)} \prod_{j=1}^k n_j!$$ obtained by sequential construction with [*one-step*]{} allocation rules $$\mbox{(O)}: p_{j}({\bf n}) :=\frac{(n-k+\gamma)(n_j +1)}{n^2 +\gamma n +\zeta} \mbox{ \hspace{0.5cm} for $j=1,\dots, k$ \hspace{0.5cm} and \hspace{0.5cm} } \ \mbox{(N)}: p_0({\bf n}):=\frac{k^2 -\gamma k +\zeta}{n^2 +\gamma n +\zeta},$$ for $\gamma \geq 0$ and (i) either $i^2 -\gamma i +\zeta$ (strictly) positive for all $i \in \mathbb{N}$ or (ii) the quadratic is positive for $i \in \{1,\dots, i_0-1\}$ and has a root at $i_0$. (See Pitman, 2006, Hansen and Pitman, 2000, for background on exchangeable partitions and sequential constructions). As from Theorem 1. in Gnedin (2010a) model (\[gnedintwo\]) arises by mixing Poisson-Dirichlet $(-1, \xi)$ models (Pitman and Yor, 1997), for $\xi=1,2,3,\dots,$ over $\xi$ with $$\label{mix}
\mathbb{P}_{\gamma, \zeta} (\Xi=\xi)=\frac{\Gamma(z_1+1)\Gamma(z_2 +1)}{\Gamma(\gamma)}\frac{\prod_{i=1}^{\xi -1}(i^2 -\gamma i +\zeta)}{\xi!(\xi-1)!},$$ for some complex $z_1$ and $z_2$. For $\zeta =0$ then $\gamma \in (0,1)$ (cfr. Gnedin, 2010a, Sect. 6) and (\[gnedintwo\]) reduces to $$\label{gnedin}
p_{\gamma}(n_1, \dots, n_k)= \frac{(k-1)!}{(n-1)!}\frac{(1-\gamma)_{k-1} (\gamma)_{n-k}}{(1+\gamma)_{n-1}} \prod_{j=1}^k n_j!.$$ The sequence of the number of occupied boxes $K_n$ for both model (\[gnedin\]) and (\[gnedintwo\]) is a nondecreasing Markov chain with $0-1$ increments and transition probabilities determined by the specific rule (N), whose distribution follows by the general formula for Gibbs partitions of genius $\alpha \in (-\infty, 1)$ $$\label{nblocks}
\mathbb{P}(K_n=k)= V_{n,k} S_{n,k}^{-1, -\alpha},$$ for $V_{n,k}$ the general Gibbs weights satisfying the backward recursion $V_{n,k}=(n-k\alpha)V_{n+1, k}+ V_{n+1, k+1}$ and $S_{n,k}^{-1, \alpha}$ generalized Stirling numbers. For $\alpha=-1$ those reduce to Lah numbers (see e.g. Charalambides, 2005) $S_{n,k}^{-1,1}= {n-1 \choose k-1} \frac{n!}{k!}$ hence, e.g. for the one-parameter model $$\label{number}
\mathbb{P}_{\gamma}(K_n=k)= {n \choose k} \frac{(1-\gamma)_{k-1} (\gamma)_{n-k}}{(1+\gamma)_{n-1}}.$$ As from Gnedin (2010a, cfr. eq. (9) and (10)) the mixing law yielding the one-parameter model (\[gnedin\]) arises from (\[number\]) for $n \rightarrow \infty$ by the standard asymptotics $\Gamma(n+\alpha)/\Gamma(n+b) \sim n^{a-b}$ and corresponds to $$\label{priorone}
\mathbb{P}_\gamma(\Xi=\xi)=\frac{\gamma(1-\gamma)_{\xi-1}}{\xi!},$$ for $\xi=1,2,\dots,$ and $\gamma \in (0,1)$, while, for $1 \leq k \leq n$, a [*posterior*]{} distribution for $\Xi$ results $$\label{postgamma}
\mathbb{P}_\gamma(\Xi=\xi |K_n=k)=\frac{(n-1)!}{(k-1)!} \frac{\Gamma(\gamma +n)}{\Gamma(\gamma +n -k)} \frac{(k- \gamma)_{\xi} \Gamma(k+\xi)}{\Gamma(\xi +1)\Gamma(k+\xi+n)}.$$
The new parametrization
=======================
Gnedin (2010a, cfr. Sect. 2) points out that the Gibbs weights of model (\[gnedintwo\]) can be split in linear factors by factoring the quadratics as $$x^2+\gamma x +\zeta =(x+z_1)(x+z_2), \mbox{ and } x^2 -\gamma x + \zeta=(x+s_1)(x+ s_2)$$ thus providing the alternative [*five*]{} parameters representation $$\label{essezeta}
V_{n,k}^{\gamma, \zeta}= \frac{(\gamma)_{n-k}(s_1+1)_{k-1} (s_2 +1)_{k-1}}{(z_1 +1)_{n-1}(z_2 +1)_{n-1}}$$ for some complex $z_1,z_2,s_1,s_2$, such that $z_1+z_2=\gamma$, $z_1z_2=\zeta$, $s_1 +s_2=-\gamma$, $s_1s_2=\zeta$.\
Here we show those constraints limit admissible values for the four parameters $s_1, s_2, z_1$ and $z_2$ yielding an interesting alternative [*two*]{} parameter representation of the weights of the Gnedin-Fisher model.\
\
[**Theorem 1.**]{} [*For $\psi \in [0,1)$ and $0 <\gamma < \psi +1$ the EPPF of the two-parameter $(\gamma, \zeta)$-Gnedin-Fisher species sampling model*]{} (\[gnedintwo\]) [*admits the following alternative representation $$\label{miogned}
p_{\gamma, \psi}(n_1, \dots, n_k)= \frac{(\gamma)_{n-k} (1- \psi)_{k-1} (1 -\gamma +\psi)_{k-1}}{(1+\psi)_{n-1}(1 +\gamma -\psi)_{n-1}} \prod_{j=1}^k n_j!.$$ For $\psi=0$, then $\gamma \in (0,1)$ and*]{} (\[miogned\]) [*yields the one-parameter Gnedin-Fisher model*]{} (\[gnedin\]).\
\
[*Proof:*]{} For $z_1+z_2=\gamma$, $z_1z_2=\zeta$, $s_1 +s_2=-\gamma$ and $s_1s_2=\zeta$ vectors $(z_1, z_2)$ and $(s_1, s_2)$ must be the roots (complex or real) of the following quadratic polynomials $$z_1^2 -\gamma z_1 +\zeta=0 \mbox{ \hspace{0.8cm} and \hspace{0.8cm} }
z_2^2 -\gamma z_2 +\zeta=0,$$ $$s_1^2 +\gamma s_1 +\zeta=0 \mbox{ \hspace{0.8cm} and \hspace{0.8cm} }
s_2^2 +\gamma s_2+\zeta=0.$$ For $\gamma^2 -4\zeta>0$ admissible [*real*]{} solutions are $$z_1= \frac{\gamma \pm \sqrt{\gamma^2 -4 \zeta}}{2} \mbox{ \hspace{0.8cm} and \hspace{0.8cm} } z_2= \frac{\gamma \pm \sqrt{\gamma^2 -4 \zeta}}{2},$$ $$s_1= \frac{-\gamma \pm \sqrt{\gamma^2 -4\zeta}}{2} \mbox{\hspace{0.8cm} and \hspace{0.8cm}} s_2 =\frac{ -\gamma \pm \sqrt{\gamma^2 -4\zeta}}{2}.$$ For $\gamma^2 -4\zeta<0$ admissible [*complex*]{} solutions are $$z_1= \frac{\gamma \pm i \sqrt{4 \zeta-\gamma^2}}{2} \mbox { \hspace{0.8cm} and \hspace{0.8cm} } z_2= \frac{\gamma \pm i \sqrt{4 \zeta-\gamma^2}}{2},$$ $$s_1= \frac{-\gamma \pm i \sqrt{4 \zeta-\gamma^2}}{2} \mbox{ \hspace{0.8cm} and \hspace{0.8cm}} s_2 =\frac{ -\gamma \pm i \sqrt{4 \zeta-\gamma^2}}{2}.$$ Now let indifferently $A=i\sqrt{4 \zeta -\gamma^2}/2$ or $A= \sqrt{\gamma^2 -4\zeta}/2$. Then, regardless of the solutions being real or complex, possible vectors satisfying the constraints $z_1+z_2=\gamma$, $s_1+s_2=-\gamma$, $z_1z_2= \zeta$ and $s_1s_2=\zeta$ must be as follows $$(z_1, z_2)=\left(\frac{\gamma}{2}+A, \frac{\gamma}{2}-A \right) \mbox{ or } \left(\frac{\gamma}{2}-A, \frac{\gamma}{2}+A \right)$$ and $$(s_1, s_2)=\left(-\frac{\gamma}{2}+A, -\frac{\gamma}{2}-A \right) \mbox{ or } \left(-\frac{\gamma}{2}-A, -\frac{\gamma}{2}+A \right),$$ which shows admissible solutions reduce to $z_1=-s_1$ and $z_2=-s_2$ or $z_1=-s_2$ and $z_2=-s_1$. Since (\[essezeta\]) is invariant to permutations of $(z_1, z_2)$ and $(s_1, s_2)$ the five parameters in (\[essezeta\]) reduce to $\psi$ and $\gamma$ for $z_1=\psi$ and $z_2= \gamma -\psi$, and $s_1=-\psi$, $s_2=\psi -\gamma$ thus yielding (\[miogned\]). Moreover the positiveness of the numerator in (\[miogned\]) implies $1-\psi>0$ and $1-\gamma +\psi>0$. For $\psi=0$, $0 < \gamma < 1$ and (\[miogned\]) reduces to (\[gnedin\]) by standard combinatorial calculus. $\square$\
\
[**Remark 2.**]{} We stress that, as from Lemma 5.2 in Gnedin (2010b), beside the extended two-parameter Poisson-Dirichlet $(\alpha, \theta)$ family of partitions models, with $\alpha \in (0,1)$ and $\theta > -\alpha$ or $\alpha < 0$ and $\theta= |\alpha|\xi$, $\xi=1, 2,\dots$, the two-parameter Gnedin-Fisher models is the unique class of exchangeable random partitions with Gibbs weights in the nice multiplicative form $$\label{niceform}
V_{n,k}= \frac{\prod_{i=0}^{n-k-1}g_0(i) \prod_{j=1}^{k-1} g_1(j)}{\prod_{l=1}^{n-1} g(l)}$$ for $g_0(\cdot)$, $g_1(\cdot)$ and $g(\cdot): \mathbb{N} \rightarrow \mathbb{R}$ satisfying the identity $$(n- \alpha k)g_0(n-k)+g_1(k)= g(n), \hspace{0.5cm} 1 \leq k \leq n, n \in \mathbb{N}.$$ In terms of equation (\[niceform\]) the weights in (\[miogned\]) may be written as $$V_{n,k}^{\gamma, \psi}= \frac{\prod_{i=0}^{n-k-1}(i+\gamma) \prod_{j=1}^{k-1} (j-\psi)(j-\gamma +\psi)}{\prod_{l=1}^{n-1} (l +\psi)(l +\gamma -\psi)}$$ for $g_0(i)=(i+\gamma)$, $g_1(j)=(j-\psi)(j -\gamma +\psi)$ and $g(l)=(l +\psi)(l +\gamma -\psi)$.\
\
For the reparametrized model (\[miogned\]) [*multistep allocation rules*]{} may be derived specializing the general form for Gibbs partitions of genius $\alpha \in (-\infty, 1)$ introduced in Cerquetti (2008) as follows. Start with box $B_{1,1}$, containing a single ball 1. At step $n$ the allocation of $n$ balls is a certain random partition $\Pi_n=(B_{n,1},\dots, B_{n,K_n})$ of the set of balls $[n]$. Given the number of boxes is $K_n=k$ and the occupancy counts are $(n_1,\dots, n_k)$, the partition of $[n+m]$ at step $n+m$ is obtained by randomly placing the additional $m$ balls\
\
(AO): in $k$ [*old*]{} boxes in configuration $(m_1,\dots, m_k)$, for $m_j \geq 0$, $\sum_{j=1}^k m_j=m$, with probability $$\label{gibbsallold}
p_{{\bf m}}({\bf n}):=\frac{(\gamma +n -k)_{m}}{(\psi +n)_m(\gamma -\psi +n)_m} \prod_{j=1}^k (n_j+1)_{m_j\uparrow},$$ (AN): in $k^*$ [*new*]{} boxes in configuration $(s_1, \dots, s_{k^{*}})$, for $\sum_{j=1}^{k^*} s_j =m$, $1 \leq k^* \leq m$, $s_j \geq 1$, with probability $$\label{gibbsallnew}
p_{{\bf s}}({\bf n}):=\frac{(\gamma +n -k)_{m-k^*} (k- \psi)_{k^*} (k-\gamma +\psi)_{k^*}}{(\psi +n)_m(\gamma -\psi +n)_m}\prod_{j=1}^{k^*} s_j!,\\$$ (ON): $s < m$ balls in $k^*$ [*new*]{} boxes in configuration $(s_1,\dots,s_{k^*})$ and the remaining $m-s$ balls in the $k$ [*old*]{} boxes in configuration $(m_1,\dots, m_k)$ for $\sum_{j=1}^{k} m_j= m-s$, $1 \leq s \leq m$, $\sum_{j=1}^{k^*} s_j=s$, $m_j \geq 0$, $s_j \geq 1$ with probability $$\label{oldenew}
p_{{s, m}}({\bf n}):=\frac{(\gamma +n -k)_{m-k^*} (k- \psi)_{k^*} (k-\gamma +\psi)_{k^*}}{(\psi +n)_m(\gamma -\psi +n)_m}\prod_{j=1}^k (n_j+1)_{m_j\uparrow}\prod_{j=1}^{k^*}s_j.\\\\$$ Corresponding (O) and (N) one-step allocation rules under the new $(\gamma, \psi)$ parametrization arise respectively from (\[gibbsallold\]) for $m=1$, $m_j=1$ and $m_l=0$ for $l \neq j$, and from (\[gibbsallnew\]) for $m=1$, $k^*=1$ and $s_1=1$, and are given by $$\mbox{(O)}: p_{j}({\bf n}): =\frac{(n-k+\gamma)(n_j +1)}{n^2 +n\gamma +\psi(\gamma -\psi)} \mbox{ \hspace{0.2cm} for $j=1,\dots, k$ \hspace{0.0cm} and \hspace{0.0cm} } \mbox{(N)}: p_0({\bf n}):=\frac{k^2 -k\gamma +\psi(\gamma -\psi)}{n^2 +n\gamma +\psi(\gamma -\psi)}.$$\
[**Remark 3.**]{} Notice that (\[oldenew\]), which is obtained specializing (19) in Cerquetti (2008), provides, once the notation is made consistent, the explicit form for $p_{\bf b}(n_1, \dots, n_{\xi})$ in Sect. 7 of Gnedin (2010a) for the two-parameter $(\gamma, \psi)$ model. This kind of conditional distributions play a significant role in a Bayesian nonparametric approach to the treatment of species sampling problems under Gibbs priors, (see e.g. Favaro et al. 2009, Lijoi et al. 2007, 2008). Here we don’t deal with this kind of applications. Some results in this perspective for the one parameter $(\gamma)$ Gnedin-Fisher model are in Cerquetti (2010).
Mixture representation and the number of occupied boxes
=======================================================
The fundamental result in Gnedin and Pitman (2006, cfr. Th. 12) establishes that the EPPF of each Gibbs partition of genius $\alpha \in (-\infty, 1)$ corresponds to a mixture of extreme partitions probability function, which differ for $\alpha \in (-\infty, 0)$, $\alpha =0$ and $\alpha \in (0,1)$. Gnedin (2010a) provides the mixing law (\[mix\]) over Poisson-Dirichlet $(-1, \xi)$ (i.e. $\alpha=-1$) extreme partitions that corresponds to the parametrization (\[gnedintwo\]). Here we derive the mixing law for the reparametrization introduced in Theorem 1. as the limit distribution of the number of blocks following the approach in Gnedin (2010a). Additionally, by an application of Bayes theorem, we provide a direct proof of the weights in model (\[miogned\]) actually arising by mixing over $\xi$ the extreme $PD(-1, \xi)$ weights.\
\
First notice that both the [*prior*]{} (\[priorone\]) and the [*posterior*]{} (\[postgamma\]) for the number of blocks of the one-parameter $(\gamma)$-Gnedin-Fisher model may be rewritten respectively as $$\label{priGW}
\mathbb{P}_\gamma(\Xi=\xi)=\frac{\gamma(1-\gamma)_{\xi-1}}{\xi!}=\frac{(1)_{\xi -1}}{\Gamma(\xi)} \frac{(1-\gamma)_{\xi-1} (\gamma)_1}{(1)_{\xi}}$$ and $$\label{postGW}
\mathbb{P}_\gamma(\Xi=\xi|K_n=k)= \frac{(k)_{\xi-1}}{\Gamma(\xi)} \frac{(k-\gamma)_{\xi-1} (n+\gamma -k)_k}{(n)_{k+\xi-1}},$$ for $\xi=1,2,\dots$, which shows that both belong to the class of [*shifted*]{} univariate [*generalized Waring distributions*]{}, (also known as inverse Markov-Polya). This is a family of distributions on $\mathbb{N} \cup 0$ (Irwin, 1975; Xekalaki, 1983; see also Johnson et al. 2005), whose probability mass function is given by $$\mathbb{P}(N=i)=\frac{(\rho)_\eta}{i!} \frac{(a)_i (\eta)_i}{(a +\rho)_{\eta+i}}$$ for $i=0,1,2, \dots,$ for parameter $a, \eta, \rho$ positive reals, which arises by $Beta(\rho, a)$ mixture of a Negative Binomial distribution $(\eta, p)$. Hence equations (\[priGW\]) and (\[postGW\]) correspond respectively to $NB(1, p)$ and $Be(\gamma, 1-\gamma)$ and $NB(k, p)$ and $Be(n +\gamma -k, k-\gamma )$. The probability generating function is, except for the constant, the Gaussian hypergeometric function $$_2F_1(a,\eta; a+\eta+\rho;z)=\sum_{i=0}^{\infty} \frac{(a)_i(\eta)_i}{(a +\eta +\rho)_i}\frac{z^i}{i!}$$ which implies the generalized Waring distribution is overdispersed and characterized by heavy tail effect. Moreover $E(X^k)< \infty$ if and only if $\rho >k$.\
\
The following result shows the reparametrization introduced in Theorem 1. yields even for the two-parameter $(\gamma, \psi)$ Gnedin-Fisher model a representation in terms of shifted generalized Waring mixture of Fisher $(-1, \xi)$ models.\
\
[**Theorem 4.**]{} [*The EPPF in*]{} (\[miogned\]) [*arises by mixing the family of $PD(-1, \xi)$ partition models $$\label{alfa1}
p_{\xi, -1}(n_1, \dots, n_k)=\frac{(\xi-1)_{k-1\uparrow -1}}{(\xi+1)_{n-1}}\prod_{j=1}^{k} {n_{j}},$$ over $\xi$, with a shifted generalized Waring distribution of parameters $a= 1-\gamma +\psi$, $\eta= 1-\psi$ and $\rho=\gamma$*]{}, $$\label{mixlaw}
\mathbb{P}_{\gamma, \psi}(\Xi=\xi)= \frac{(1 -\psi)_{\xi-1}(1-\gamma +\psi)_{\xi-1}(\gamma)_{1-\psi}}{\Gamma(\xi) (1+\psi)_{\xi -\psi}},$$ for $\psi \in [0,1)$ and $\gamma \in (0, \psi+1)$.\
\
[*Proof.*]{} By the general formula (\[nblocks\]) for the law of the number of blocks for Gibbs partitions of genius $\alpha$, and exploting the definition of Lah numbers, the analogous of (\[number\]) for the two-parameter model is given by $$\mathbb{P}_{\gamma, \psi}(K_n=k)= {n-1 \choose k-1} \frac{n!}{k!} \frac{(\gamma)_{n-k} (1- \psi)_{k-1} (1 -\gamma +\psi)_{k-1}}{(1+\psi)_{n-1}(1 +\gamma -\psi)_{n-1}}.$$ Rewriting in terms of Gamma functions yields $$\mathbb{P}_{\gamma, \psi}(K_n=k)=\frac{\Gamma(1+\psi) \Gamma(1+\gamma -\psi) (1-\psi)_{k-1} (1- \gamma +\psi)_{k-1}}{\Gamma(k+1) \Gamma(k) \Gamma(\gamma) } \frac{\Gamma(n+1) \Gamma(n) \Gamma(\gamma +n -k)}{\Gamma(n -k +1) \Gamma(\psi +n) \Gamma(\gamma -\psi +n)}$$ and, by Stirling approximations, for $n \rightarrow \infty$ reduces to $$\label{mixing2}
\mathbb{P}_{\gamma, \psi}(\Xi=\xi)= \frac{(1 -\psi)_{\xi-1}(1-\gamma +\psi)_{\xi-1}(\gamma)_{1-\psi}}{\Gamma(\xi) (1+\psi)_{ \xi-\psi }}$$ which provides the analogous of Eq. (5) in Gnedin (2010a) for the new parametrization.\
\
As $\xi \rightarrow \infty$ the power-like decay of the masses in (\[mixlaw\]) (cfr. Gnedin, 2010a, Sect. 3) is rewritten as $$\mathbb{P}_{\gamma, \psi} (\Xi = \xi ) \sim \frac{c}{\xi^{1+\gamma}} \mbox{ with } c=\frac{\Gamma(1+ \gamma -\psi) \Gamma(1 +\psi)}{\Gamma(1 -\psi) \Gamma(1- \gamma +\psi) \Gamma(\gamma)}.$$\
To show that the weights in (\[miogned\]) actually arise by mixing the weights of the extreme Poisson-Dirichlet $(-1, \xi)$ partitions over $\xi$ with (\[mixing2\]) we apply Bayes theorem. The [*posterior*]{} distribution for $\Xi$ for the reparametrized model may be obtained by the general form for Gibbs models of the posterior of the number of new blocks arising in a new sample of dimension $m$ (cfr. Cerquetti (2008, eq. (32) , see also Lijoi et al. 2007, eq. (4)) which expressed in terms of non-central generalized Stirling numbers is given by $$\label{genpost}
\mathbb{P}(K^*_m=k^*| K_n=k)= \frac{V_{n+m, k+k^*}}{{V_{n,k}}}S_{m, k^*}^{-1,-\alpha, -(n -\alpha k)}.$$ For $\alpha=-1$, inserting the specific weights in (\[miogned\]), and exploiting the definition of non-central Lah numbers $S_{n,k}^{-1, 1, r}= \frac{n!}{k!} {n -r-1 \choose n-k}$, (\[genpost\]) yields $$\label{postfinite}
\mathbb{P}_{\gamma, \psi} (K_m=k^*|K_n=k) = {m \choose k^*} (\gamma +n -k)_{m-k^*} (n+k+k^*)_{m-k^*} \frac{(k - \psi)_{k^*}(k -\gamma +\psi)_{k^*}}{(n +\psi)_{m} (n +\gamma -\psi)_m},$$ and for $m \rightarrow \infty$, by standard Stirling approximations, the [*posterior*]{} for the number of blocks of the two-parameter $(\gamma, \psi)$ model results $$\label{posttwo}
\mathbb{P}_{\psi, \gamma}(\Xi=\xi |K_n=k)=\frac{(k-\psi)_{\xi-1} (k -\gamma +\psi)_{\xi-1}(n+\gamma -k)_{k-\psi}}{\Gamma(\xi) (n+\psi)_{k-\psi+\xi-1}},$$ which is still in the class of shifted univariate Waring distributions for parameters $a= k-\gamma +\psi$, $\eta= k-\psi$ and $\rho= n+\gamma -k$, for $\gamma < k < n +\gamma$. Now, by Bayes theorem, $$\mathbb{P}_{\psi,\gamma}(\Xi=\xi) V_{n,k}^{-1, \xi}=\mathbb{P}_{\psi, \gamma}(\Xi=\xi |K_n=k) V_{n,k}^{\psi, \gamma},$$ therefore, exploiting (\[posttwo\]) $$V_{n,k}^{\psi, \gamma}=\frac{(1 -\psi)_{\xi-1}(1-\gamma +\psi)_{\xi-1}(\gamma)_{1-\psi}}{\Gamma(\xi) (1+\psi)_{1-\psi +\xi -1}} \frac{(\xi-1)_{k-1\uparrow -1}}{(\xi+1)_{n-1}}\frac{\Gamma(\xi) (n+\psi)_{k-\psi+\xi-1}}{(k-\psi)_{\xi-1} (k -\gamma +\psi)_{\xi-1}(n+\gamma -k)_{k-\psi}}=$$ and with the substitution $\xi -k=y$, $$=\frac{(1 -\psi)_{ y+k-1}(1-\gamma +\psi)_{y+k-1}(\gamma)_{1-\psi}}{\Gamma(y+k) (1+\psi)_{y +k -\psi }} \frac{(y+k-1)_{k-1\uparrow -1}}{(y+k+1)_{n-1}}\frac{\Gamma(y+k) (n+\psi)_{k-\psi+y+k-1}}{(k-\psi)_{y+k-1} (k -\gamma +\psi)_{y+k-1}(n+\gamma -k)_{k-\psi}}.$$ Then, by the multiplicative property of rising factorials $(x)_{a+b}=(x)_a (x+a)_b$, the last expression easily simplifies to $$V_{n,k}^{\psi, \gamma}=\frac{(\gamma)_{n-k}(1 -\gamma +\psi)_{k-1} (1 -\psi)_{k-1}}{(1 +\psi)_{n-1} (1 +\gamma -\psi)_{n-1}},$$ and the proof is complete. $\square$\
\
We provide an additional result for the two-parameter $(\gamma, \psi)$ Gnedin-Fisher model by exploiting the mixture representation introduced in Theorem 4. to obtain the [*structural distribution*]{}, the law of the frequency of Box 1, $$\tilde{P}_{1}:= \lim_{n \to \infty } \frac{\#(B_1 \cap [n])}{n}.$$\
[**Proposition 5.**]{} [*The frequency $\tilde{P_1}$ of box $B_1$ of the $(\gamma, \psi)$ Gnedin-Fisher model has distribution $$\label{lawP_1}
\mathbb{P}_{\gamma, \psi}(\tilde{P}_1 \in dy)= \gamma_{1-\psi} \Gamma(1+\psi) \left[\delta_1(dy)+(1-\gamma +\psi)(1-\psi) _2F_1 (2-\psi, 2-\gamma+\psi, 2; 1-y)
\right]dy$$ for $_2F_1(a, b, c; x)$ the Gaussian hypergeometric function.*]{}\
\
[*Proof:*]{} By the mixture representation of Theorem 4. $$\mathbb{P}_{\gamma, \psi}(\tilde{P}_1 \in dy)= \sum_{\xi=1}^\infty \mathbb{} \mathbb{P}_{\gamma, \psi}(\Xi=\xi) \mathbb{P}(\tilde{P}_{\xi, 1} \in dy).$$ By the theory of the symmetric Dirichlet model, it is known that $\tilde{P}_{\xi, i}\stackrel{d}=Beta(2, \xi-i)$, therefore, since $Be(2, 0)=\delta_1(dy)$ $$\mathbb{P}_{\gamma, \psi}(\tilde{P}_1 \in dy)=(\gamma)_{1 -\psi} \Gamma(1+\psi)\delta_1 (dy) +\sum_{\xi=2}^\infty \frac{(1 -\psi)_{\xi-1}(1-\gamma +\psi)_{\xi-1}(\gamma)_{1-\psi}}{\Gamma(\xi) (1+\psi)_{\xi -\psi}} \frac{\Gamma(\xi +1)}{\Gamma(\xi -1)}y (1-y)^{\xi -2}.$$ By the change of variable $\xi-2=z$ $$\mathbb{P}_{\gamma, \psi}(\tilde{P}_1 \in dy)=(\gamma)_{1 -\psi} \Gamma(1+\psi)\delta_1(dy) +\sum_{z=0}^\infty
\frac{(1-\psi)_{z+1}(1-\gamma +\psi)_{z+1} (\gamma)_{1-\psi}\Gamma(z+3)}{\Gamma(z+1)\Gamma(z+2)(1+\psi)_{z+2-\psi}} y(1-y)^{z}=$$ and by standard combinatorial calculus $$=(\gamma)_{1 -\psi} \Gamma(1+\psi)\left[\delta_1(dy) + (1-\psi)(1 -\gamma +\psi)\sum_{z=0}^\infty
\frac{(2-\psi)_{z}(2-\gamma +\psi)_{z} }{\Gamma(z+1)\Gamma(z+2)} y(1-y)^{z}\right]$$ and the result follows.$\square$\
\
[**Remark 6.**]{} For $\psi=0$ (\[lawP\_1\]) yields the result in Gnedin (2010a, Sect. 6) for the distribution of the frequency $\tilde{P}_1$ of box $B_1$ for the one parameter $(\gamma)$ Gnedin-Fisher model. In fact $$_2F_1(2, 2-\gamma, 2; 1-y)= \sum_{z=0}^\infty \frac{(2-\gamma)_z}{\Gamma(z+1) (2)_z}(1-y)^z,$$ and multiplying and dividing by $y^{1-\gamma}$ and exploiting the probability mass function of the Negative Binomial $(2-\gamma, y)$ yields $$\sum_{z=0}^\infty \frac{(2-\gamma)_z}{\Gamma(z+1) (2)_z}y(1-y)^z = y^{\gamma -1},$$ hence for $y \in (0,1]$ $$\mathbb{P}_\gamma(\tilde{P}_1 \in dy)= \gamma \delta_1(dy)+ \gamma(1 -\gamma) y^{\gamma-1}dy.$$
References {#references .unnumbered}
==========
[ ]{}
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<span style="font-variant:small-caps;">Cerquetti, A.</span> (2008) Generalized chinese restaurant construction of exchangeable Gibbs partitions and related results. [arXiv:0805.3853v1 \[math.PR\]]{}
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<span style="font-variant:small-caps;">Cerquetti, A.</span> (2010) Bayesian nonparametric analysis of a species sampling model with finitely many types. [arXiv:1001.0245v1 \[math.PR\]]{}
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<span style="font-variant:small-caps;">Charalambides, C. A.</span> (2005) [*Combinatorial Methods in Discrete Distributions*]{}. Wiley, Hoboken NJ.
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<span style="font-variant:small-caps;">Favaro, S., Lijoi, A., Mena, R. and Prünster, I.</span> (2009) Bayesian non-parametric inference for species variety with a two-parameter Poisson-Dirichlet process prior. [*JRSS-B*]{}, 71, 993-1008.
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<span style="font-variant:small-caps;">Gnedin, A.</span> (2010b) Boundaries from inhomogeneous Bernoulli trials. [arXiv:0909.4933 \[math.PR\]]{}
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<span style="font-variant:small-caps;">Lijoi, A., Mena, R.H. and Prünster, I.</span> (2007) Bayesian nonparametric estimation of the probability of discovering new species. [*Biometrika*]{}, 94, 769–786.
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[^1]: Corresponding author, SAPIENZA University of Rome, Via del Castro Laurenziano, 9, 00161 Rome, Italy. E-mail: [[email protected]]{}
[^2]: [*AMS (2000) subject classification*]{}. Primary: 60G58. Secondary: 60G09.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Andrew M. Goldsborough$^{1,2}$'
- 'Rudolf A. Römer$^{1}$'
title: 'Using entanglement to discern phases in the disordered one-dimensional Bose-Hubbard model'
---
Introduction
============
The study of bosons in one dimension has been of great interest in both theoretical and experimental physics for many years due in part to the existence of a quantum phase transition from a superfluid to insulator at zero temperature [@GreMEH02]. The introduction of disorder causes a further phase transition into a localised *Bose glass* phase, which is insulating but remains compressible [@FisWGF89]. The experimental study of phase transitions in bosonic systems is possible using Helium in porous media [@CroHST83; @CroVR97], Josephson junction arrays [@VanEGM96], thin films [@HavLG89; @Pan89] and, more recently, optical lattices [@GreMEH02; @BakGPF09]. It is now possible to introduce disorder in a controlled manner to optical lattices using speckle potentials [@HorC98; @BilJZB08] to study these transitions directly [@WhiPMZ09; @PasMWD10].
Analytical results even for clean systems are limited. There is an approximate Bethe-ansatz solution [@Kra91], where the maximum number of bosons per site is set to two. For disordered systems Giamarchi and Schulz used renormalization group (RG) techniques to determine the weak disorder physics given the Luttinger parameter $K$ [@GiaS87; @RisPLG12]. There are further real-space RG results for the case of strong disorder [@AltKPR10; @PieA13]. Numerical approaches provide some of the most effective means of garnering information. Quantum Monte Carlo has been employed in 1, 2 and 3 dimensions [@ScaBZ91; @ProS98; @KraTC91; @SoyKPS11; @GurPPS09; @Pol13], but these methods become difficult in the limit of zero temperature. An ideal method for analysing one dimensional systems is the density-matrix RG (DMRG) [@Whi92]. It has been applied with great success to a number of physical systems from quantum chemistry [@ShaC12] to quantum information[@PerVWC07], including the disordered Bose-Hubbard model [@PaiPKR96; @RapSZ99]. The phase diagrams obtained using these methods for the one dimensional case [@ProS98; @RapSZ99], whilst qualitatively agreeing, are quantitatively quite different. This difference could be down to the choice of different observables and the difficulties each has with finite size effects.
In recent years the use of entanglement properties as a means of deciphering phase has become commonplace [@LiH08; @EisCP10; @PolTBO10; @DenS11; @DenCOM13; @KjaBP14]. Entanglement is a measurement of a wavefunction’s non-locality and as such it is an ideal means of analysing various phases. Modern numerical techniques such as *tensor networks* and DMRG obtain entanglement information as part of the update algorithms, so large amounts of information about the phase is gathered automatically in the course of the RG iterations [@Sch11]. In this paper we perform a DMRG simulation of the disordered Bose-Hubbard model in the form of a variational update of a matrix product state (MPS) [@OstR95; @Sch11] implemented in the [ITensor]{} libraries [@Itensor023]. The disordered Bose-Hubbard model is made up of bosonic creation, $b_{i}^{\dagger}$, and annihilation operators, $b_{i}$, on sites of a linear lattice. The Hamiltonian is [@RapSZ99] $$H= - \sum_{i}^{L-1} \frac{t}{2} (b^{\dagger}_{i} b_{i+1} + \mbox{h.c.} ) + \sum_{i}^{L} \frac{U}{2} n_{i}(n_{i}-1) + \mu_{i} n_{i},$$ where $n_{i}=b^{\dagger}_{i}b_{i}$ is the local occupation or *number operator* that gives the number of bosons on site $i$. The potential disorder is modelled via uniformly distributed random chemical potentials $\mu_i\in [-\Delta \mu/2, \Delta \mu/2]$. For ease of comparison, we have adopted prior conventions [@RapSZ99] for hopping $t$ and interaction $U$ and throughout the rest of the analysis $t=1$.
Observables
===========
The Mott insulator can be differentiated from the Bose glass phase by the existence of the *Mott gap*, $E_{g}$, between the ground and first excited state. While DMRG ordinarily finds the ground state of the system, low lying excited states have to be constructed iteratively by orthogonalising with respect to the lower lying states [@Sch11]. For the Bose-Hubbard chain it is numerically more convenient to use the fact that the energy of the excited state is equal to the difference in energy between the chemical potential for particle, $\mu_{p} = E_{N+1}-E_{N}$, and hole, $
\mu_{h} = E_{N}-E_{N-1}$, excitations [@RapSZ99]. This means that $E_{g}$ can be found by calculating the energies $E_{N+1}$, $E_{N}$ and $E_{N-1}$ of the $N+1$, $N$ and $N-1$ particle sectors, respectively, as $$E_{g} = E_{N+1}-2E_{N}+E_{N-1}.$$ Hence the determination of $E_{g}$ requires a DMRG run for each of the three different particle numbers and each set of parameters.
The superfluid phase is determined by a non-zero *superfluid fraction* $\rho_{s}$. This is defined as the difference between the ground state energies of a chain with periodic boundaries and anti-periodic boundaries, $$\rho_{s} = \frac{2L^{2}}{\pi^{2}N} \left( E_{N}^{\mathrm{anti-periodic}} - E_{N}^{\mathrm{periodic}} \right),$$ where $L$ is the chain length and $N$ the number of bosons [@RapSZ99]. For Mott insulator and Bose glass phases, we have $\rho_{s}=0$, so a finite $\rho_{s}$ indicates superfluidity in the phase diagram. From a computational point of view, $\rho_{s}$ is not an easy quantity to determine as it requires the use of periodic boundaries, which are well-known to converge slower and be less accurate than for open systems when using DMRG [@Sch11]. Furthermore, as $\rho_{s}$ is the difference between two energies, two such periodic DMRG calculations have to be performed for each set of parameters.
The two-point correlation function $\langle b_{i}^{\dagger} b_{j} \rangle$ provides information regarding the localisation of the wavefunction. For the Bose glass and Mott insulating phases the correlation function decays exponentially, $\langle \langle b_{i}^{\dagger} b_{j} \rangle \rangle \propto e^{-|i-j|/\xi}$, where $\xi$ is the correlation length and $\langle \langle \dots \rangle \rangle$ denotes the expectation value when averaged over all pairs of sites separated by $|i-j|$ and all disorder realisations [@RouBMK08]. Extended phases like the superfluid are not localised so $\xi$ diverges in the thermodynamic limit. In the absence of disorder the superfluid phase will be described by Luttinger liquid theory [@Voi95], hence the correlation function will admit a power law decay $$\langle \langle b_{i}^{\dagger} b_{j} \rangle \rangle \propto |i-j|^{-1/2K},
\label{eq-pld-poly}$$ where $K$ is the Luttinger parameter. $K$ takes the value $2$ for a Kosterlitz-Thouless (KT) transition from superfluid to Mott insulator [@Gia97; @KuhWM00]. By utilizing an RG approach, Giamarchi and Schulz [@GiaS87] showed that disorder scales to zero in the weak disorder regime when $K>3/2$, giving a superfluid phase. On the other hand, disorder grows for $K<3/2$ signifying a Bose glass. This was later extended [@RisPLG12] to the *medium disorder* case ($U \sim \Delta\mu$). Instead of the infinite-system size result we use the conformal field theory (CFT) expression [@RouBMK08] for an open chain of size $L$, $$\langle b_{i}^{\dagger} b_{j} \rangle \propto \left[ \frac{\pi}{2L} \frac{ \sqrt{ \left| \sin \left( \frac{\pi i}{L} \right) \right| \left| \sin \left( \frac{ \pi j}{L} \right) \right| }}{ \left| \sin \frac{ \pi (i+j)}{2L} \right| \left| \sin \frac{ \pi (i-j)}{2L} \right|} \right]^{{1}/{2K}}.$$ The expression has to be averaged over all $i$ and $j$ with separation $|i-j|$ and, of course, also averaged over disorder realisations. Calculating correlation functions does not require multiple DMRG runs, but requires the calculation of an expectation value for each combination of $i$ and $j$, of which there are $L(L-1)/2$. Furthermore, the accuracy of locating the KT transition from correlation functions for the Bose-Hubbard model has previously been questioned [@KuhM98; @RouBMK08].
In each DMRG run, bipartitioning the chain into *system* and *environment* blocks is done routinely to compute singular-value decompositions [@Sch11]. These singular values, $s_{a}$, can themselves be used to obtain information regarding the phase [@PolTBO10; @DenS11; @DenCOM13] without the need for multiple DMRG runs, thus saving substantial numerical costs. The most common such measure is the *entanglement entropy* or *Von Neumann entropy* defined as $$S_{\mathrm{A}|\mathrm{B}} = -\mathrm{Tr} \rho_{\mathrm{A}} \log_{2} \rho_{\mathrm{A}} = - \sum_{a=1} s_{a}^{2} \log_{2} s_{a}^{2},
\label{eq:entropy}$$ which gives the entanglement between regions A and B [@Sch11]. The reduced density matrix, $\rho_{\mathrm{A}}$, for region A is obtained from the density matrix by tracing over degrees of freedom from region B. Its eigenvalues are given as squares of the $s_a$’s. Hence $S_{\mathrm{A}|\mathrm{B}}$ is a measure of the spread of the $s_a$ values. If there is one non-zero singular value then the regions are in a product state of the two regions. The other extreme is if all singular values are equal, in which case the subsystems are *maximally entangled*. In the subsequent analysis we shall average the entanglement entropy over all possible bipartitions along the chain. This averaged entanglement entropy can distinguish between phases with high and low entanglement, for example the superfluid and Mott insulating phases.
(a){width="30.00000%"} (b){width="30.00000%"} (c){width="30.00000%"}
Deng et. al. [@DenCOM13] used the *entanglement spectral parameter*, $\zeta$, to obtain the phase diagram for an extended Bose-Hubbard model. The $\zeta$ parameter is defined as the sum of the difference between the first and second, and third and fourth, respectively, eigenvalues $s_a^2$ of $\rho_{\mathrm{A}}$ when averaged over all bipartition positions such that $L_{\mathrm{A}}+L_{\mathrm{B}} = L$, i.e.$$\zeta = \overline{\lambda}_{1} - \overline{\lambda}_{2} + \overline{\lambda}_{3} - \overline{\lambda}_{4},
\label{eq:zeta}$$ with $
\overline{\lambda}_{a}
%= \frac{1}{L-1} \sum_{L_{A}=1}^{L-1} \lambda_{i}(L_{A})
\equiv \sum_{L_{\mathrm{A}}=1}^{L-1} s^{2}_{a}(L_{\mathrm{A}})/({L-1})$, $a= 1, 2, 3, 4$, the bipartition-averaged $a$-th eigenvalue. In fig. \[fig:eespecs\] we show the typical behaviour of the four lowest $s_{a}$ values in the superfluid, fig. \[fig:eespecs\](a), Mott insulator, fig. \[fig:eespecs\](b), and Bose glass, fig. \[fig:eespecs\](c), regimes We see that the entanglement spectrum of the superfluid phase is somewhat noisy with the four singular values being of the same order of magnitude. Therefore the resulting $S_{\mathrm{A}|\mathrm{B}}$ is large, $\zeta$ is small, but neither have too much variation along $L_{\mathrm{A}}$. One of the striking features of the entanglement spectrum for the Mott insulator regime is that $s_{1} \approx 1$ while $s_{2} \approx s_{3} \approx s_{4} \approx 0$ for *all* bipartitions, even in the presence of disorder. This means that the average $S_{\mathrm{A}|\mathrm{B}}$ will be low and $\zeta \approx 1$, with a negligible deviation. Last, entanglement spectra of the Bose-glass show pronounced localised regions separated by areas of low entanglement. This results in a much larger variation of $\zeta$ and $S_{\mathrm{A}|\mathrm{B}}$ than in the superfluid phase, with $\zeta$ and average $S_{\mathrm{A}|\mathrm{B}}$, between the other phases. These findings suggest that the average and spatial variations of $S_{\mathrm{A}|\mathrm{B}}$ and $\zeta$ might also be used to distinguish the phases of the disordered Bose-Hubbard model.
Results
=======
We use $S_{\mathrm{A}|\mathrm{B}}$ and $\zeta$ to create qualitative phase diagrams for a modest size of $L=50$, disorder-averaged over $100$ samples using as our DMRG implementation the [ITensor]{} libraries [@Itensor023]. The finite-size scaling (FSS) behaviour of $S_{\mathrm{A}|\mathrm{B}}$ and $\zeta$ is currently not well understood, thus to find the phase boundaries for $L\rightarrow\infty$, we perform scaling with $K$ and $E_g$ from estimates up to $L=200$. This is a numerically more expensive procedure, so we concentrate on a small number of points with positions motivated by the phase diagram from the entanglement properties. For disordered systems, getting stuck in local minima is particularly problematic, so we use a relatively large bond dimension $\chi=200$ and perform $20$ DMRG sweeps of the chain for each sample. Our truncation error is less than $10^{-10}$. We also introduce a small noise term for the first few sweeps; this perturbs a perhaps bad initial wavefunction, allowing faster convergence into the ground state.
Bosons do not obey the Pauli exclusion principle and hence can condense onto a single site. In order to capture such a behaviour, the one-site basis dimension should to be as large as the number of particles in the system. This is numerically infeasible and it is necessary to introduce a finite maximum number of bosons that can occupy each site. We use $\max (n_i) = 5$, consistent with ref. [@KuhWM00] who find that a higher particle number does not effect the results appreciably for $U>0$ [@RouBMK08; @PaiPKR96; @RapSZ99].[^1]
Density = 1
-----------
For particle density ${N}/{L} = 1$, in the clean case, the system is in a superfluid phase for small $U$ but transitions into a Mott insulating phase at a critical $U_c$. Introducing disorder enables the existence of a localized *Bose glass* phase [@FisWGF89]. The possibility of a direct transition from superfluid to Mott insulator has been discussed extensively (see references in [@PolPST09]). In one dimension it was shown [@Svi96] that the transition necessarily goes via the Bose glass phase. This is now also the accepted picture for any dimension [@PolPST09].
We show our results based on $\zeta$ and $S_{\mathrm{A}|\mathrm{B}}$ for $L=50$ in fig. \[fig:eepds\], (a) and (b), respectively.
The superfluid, small $U\lesssim 1.5$, and the Mott insulator, $U \gtrsim 2$, are clearly distinguishable in both panels. The boundary of the superfluid to the Bose glass is less well defined and it is not clear that there is a Bose glass region between the Mott insulator and the superfluid — very different wavefunctions give similar average entanglement entropy. Following on from our prior discussion of fig. \[fig:eespecs\], we also plot in fig. \[fig:eepds\](c) the standard error[^2] of $\zeta$, $\Delta \zeta$, and, similarly, (d) $\Delta S_{\mathrm{A}|\mathrm{B}}$. In these plots the phases become clear and their boundaries are consistent with earlier work [@RapSZ99]. In particular, a Bose glass phase can be easily identified between Mott insulator and superfluid. Furthermore, we see that the contours for $\zeta$ and $S_{\mathrm{A}|\mathrm{B}}$ in fig. \[fig:eepds\] are qualitatively similar, just as those for $\Delta \zeta$ and $\Delta S_{\mathrm{A}|\mathrm{B}}$. We emphasize that for the entanglement-based measures present here, it is in fact possible to discern all of the phases with just a *single* DMRG run for each $(U, \Delta \mu$, disorder realisation) data point. This is a clear advantage in terms of numerical costs when compared to calculations based on $E_g$, $\rho_s$ or $K$.
In order to augment the finite-size phases identified in fig. \[fig:eepds\], we now perform runs with larger $L$ and employ FSS. To find the superfluid-Bose glass transition in the thermodynamic limit we calculate $K$ for various points along the boundary for system sizes $L = 30$, $50$, $100$ and $150$. The transition is of KT type at $K=3/2$. The corresponding points in $(U, \Delta\mu)$ which are shown as filled circles in fig. \[fig:eepds\]. For reference, we also plot the points where $K=3/2$ for $L=50$ (stars). Similarly, the superfluid-Mott insulator transition point $U_c$ is the point on the zero disorder axis where $K=2$. We estimate it value as $U_{c} = 1.634 \pm 0.002$. We also calculate $E_g$ for the same system sizes and use FSS to find the Mott insulator-Bose glass boundary indicated as squares in fig. \[fig:eepds\]. The superfluid region we find is significantly smaller than that of ref. [@RapSZ99] but matches ref. [@ProS98]; the position of Mott insulator-Bose glass boundary is very similar to [@RapSZ99] and different to [@ProS98]. The RG analysis of Refs. [@GiaS87] and [@RisPLG12] suggests that there may be a further *Anderson glass* phase in the low $U < \Delta\mu / 2$ region of the Bose glass phase highlighted by the dashed line in fig. \[fig:eepds\]. This would imply a critical point along the superfluid boundary at which point $K$ at the transition becomes disorder dependent. Our entanglement analysis shows no sign of such a transition either within the Bose glass phase or on the boundary with the superfluid. However, when $U \ll \Delta\mu$ the truncation of the basis, i.e. $\max (n_i) \leq 5$, becomes more problematic so we cannot rule out the existence of another phase in this region.
Density = 1/2
-------------
The clean case for $N/L=1/2$ remains a superfluid for all values of $U$ [@FisWGF89]. When $\Delta\mu$ is increased, our entanglement measures indicate the eventual emergence of a Bose glass phase as shown in fig. \[fig:eepds\_half\](a+b).
\[fig:eepds\_half\] \[fig:eepds\_two\]
Still, the superfluid phase for $L=50$ seems to extend up to $\Delta\mu \lesssim 1$ for $U \lesssim 5$ as shown by all four entanglement measures. The Giamarchi-Schulz criterion [@GiaS87; @RapSZ99] implies that the Bose-Hubbard model should be in a Bose glass phase for $K<3/2$. In fig. \[fig:eepds\_half\](a+b) we show that the resulting boundaries indicate that the superfluid phase extends as far as $U_{K=3/2}=3.5\pm 0.1$, i.e. it ends somewhat earlier for low $\Delta\mu$ than suggested by our entanglement measures. In order to explore this region further, we have also calculated $\rho_{s}$ for fixed $\Delta\mu = 0.5$ and sizes $L=50$, $100$, $150$, and $200$ as shown in fig. \[fig:stiffness\_half\](a). The results for $\rho_{s}$ have been computed for increased bond dimension $\chi=400$ with $40$ DMRG sweeps and $20$ disorder configuration to offset the reduction in precision due to periodic boundaries. The figure shows that for $U\gtrsim 3$, $\rho_{s}$ decreases when increasing $L$ as expected in the Bose glass phase. However, the decrease is very slow and, for the system sizes attainable by us, even seems to saturate at non-zero values. These results suggest that for finite systems, the $K=3/2$ criterion significantly underestimates the extent of the superfluid phase, while our four entanglement measures and $\rho_{s}$ predict a much larger region. Performing a FSS analysis for $K$ as shown in fig. \[fig:stiffness\_half\](a) we find the $U$ values, for which $K = 3/2$ in the limit $L\rightarrow\infty$, converge towards a limiting value of $U_{c} = 3.09 \pm 0.01$ (see also fig. \[fig:eepds\_half\]). This again indicates that in an infinite system, we expect the superfluid to Bose glass transition to take place at much lower values of $U_c$ than observed for $L=50$. The relevance of this result is of course that experimental realizations of the Bose-Hubbard model are typically in cold atom systems, which are limited to finite system sizes, currently a typical lattice dimension is $\sim 50-100$ [@GreMEH02].
For values of $\Delta\mu\gtrsim 1$, the situation is less severe and we see in fig. \[fig:eepds\_half\](a+b) that our entanglement-based measures again qualitatively agree with the Giamarchi-Schulz criterion, both for $L=50$ and estimated via FSS at $L\rightarrow\infty$.
Density = 2
-----------
To the best of our knowledge, the phases for $N/L = 2$ have not been shown before in the literature. Due to our numerical restriction of five bosons per site, this regime is close to the limit of what can be studied reliably, particularly for small $U$ where the occupancy per site should be large. For large $U$, one might expect that we will have a Mott insulator of boson *pairs*, while a superfluid of boson pairs emerges for small $U$ and small $\Delta\mu$. As before, we envisage a disordered Bose glass phase for large $\Delta\mu$. With more particles per site than in the $N/L=1$ case, we could furthermore expect that onset of the Mott transition at $\Delta\mu=0$ is at larger values of $U$, since there is a larger energy penalty to pay for a doubly occupied site [@FisWGF89]. Similarly, as the cost for two boson pairs to go onto the same site is $2U$, we expect the $2 U = \Delta\mu/2$ line to characterize the superfluid phase as in the $N/L=1$ case. In addition, one might conjecture to see a remnant of the $U = \Delta\mu/2$ condition.
In fig. \[fig:eepds\_two\](c+d), we show that our expectations are largely validated. In particular, a double lobe shape for the superfluid phase emerges and allows a possible re-entrant behaviour given a suitable cut across parameter space. The gradient of the Mott insulating phase boundary is shallower ($\sim 4/3$) when compared to $N/L=1$. Furthermore, both the $\zeta$ and $S_{\mathrm{A}|\mathrm{B}}$ based entanglement measures, as well as their errors, $\Delta\zeta$ and $\Delta S_{\mathrm{A}|\mathrm{B}}$, capture the phases equally well and agree with the $K$ and $E_g$ estimates. Note that for $N/L=2$, the KT superfluid-to-Mott transition at $\Delta\mu=0$ corresponds to $K=2$ and we finite-size scale the Luttinger parameter to find $U_{c} = 2.75 \pm 0.03$. We emphasize that the points for small $U$, see top left of the phase diagram in fig. \[fig:eepds\_two\](c+d), should be viewed with caution as the basis truncation will affect the results. The grey line in fig. \[fig:eepds\_two\] — as in fig. \[fig:eepds\] — indicates the points at which the probability of obtaining a site with $\langle n_{i} \rangle \ge 4.9$ reaches $10^{-3}$. This clearly shows that for the Bose glass with small $U$ all wavefunctions are beginning to reach the limit five of bosons per site, however in the bulk of the phase diagram the results are not affected.
(b)![(Color online) (a) Superfluid fraction $\rho_{s}(U)$ for $N/L = 1/2$ with $\Delta\mu = 0.5$ for lengths $50$–$200$. The vertical line indicates $U_c= 3.09$. The inverted triangles give the finite-size scaled $L\rightarrow\infty$ limit with the dashed line a guide to the eye. (b) Luttinger parameter $K$ for various lengths $30$–$150$ at $N/L=1/2$. The horizontal line highlights $K = 3/2$. The inset shows the FSS analysis. []{data-label="fig:stiffness_half"}](./lutt_clean+finite_half.pdf "fig:"){width="22.00000%"}
Conclusion
==========
We have analysed the phase diagrams of the disordered Bose-Hubbard model for fillings $N/L = 1/2$, $1$ and $2$ using the entanglement-based measures $\zeta$, $S_{\mathrm{A}|\mathrm{B}}$, $\Delta\zeta$ and $\Delta S_{\mathrm{A}|\mathrm{B}}$. We find that despite success in ref. [@DenCOM13], $\zeta$ or $S_{\mathrm{A}|\mathrm{B}}$ alone do not always faithfully reproduce the phase diagrams. The error-based measures, $\Delta\zeta$ and $\Delta S_{\mathrm{A}|\mathrm{B}}$, provide a much clearer picture — the distributions of the values contain more information regarding the nature of the phase than the mean values alone. These measures are an excellent means of quickly identifying the different phases of the system while removing the need for multiple DMRG runs per measurement and special boundary conditions. Unfortunately, they do not seem to exhibit a simple FSS behavior, at least for the system size up to $L=200$ used here. While $\zeta$ and $S_{\mathrm{A}|\mathrm{B}}$ and, in particular, $\Delta\zeta$ and $\Delta S_{\mathrm{A}|\mathrm{B}}$ provide a numerically convenient, qualitative outline of the phase boundaries, it seems still necessary to apply FSS to $K$ and $E_g$ for estimates of the boundaries in the $L\rightarrow\infty$ limit. For $N/L = 1$ our phase diagram is found to complement the results of refs. [@ProS98; @RapSZ99]. For $N/L = 1/2$ the diagram shows strong finite-size effects and the critical $U$ defined by the Giamarchi-Schulz criterion is not apparent for these finite systems. Finally, for $N/L = 2$ the superfluid phase has a *double-lobed* appearance giving rise to re-entrance phenomena.
We would like to thank Miles Stoudenmire for help with [ITensor]{}. We gratefully acknowledge discussions with Kai Bongs and Nadine Meyer. We are grateful to the EPSRC for financial support (EP/J003476/1) and provision of computing resources through the MidPlus Regional HPC Centre (EP/K000128/1).
The supporting data for this research is openly available from the University of Warwick research archive portal at http://wrap.warwick.ac.uk/71189/.
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version: 0.2.3. <http://itensor.org/>
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[^1]: Five is the current hard limit in the [ITensor]{} code.
[^2]: We note that for larger system sizes the variance or standard deviation may be better measures of distribution width as they do not approach zero in the infinite system limit.
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{
"pile_set_name": "ArXiv"
}
|
[ Naoyuki Haba and Toshifumi Yamada ]{}
[ *Graduate School of Science and Engineering, Shimane University, Matsue 690-8504, Japan* ]{}
We investigate the scenario where the Standard Model is extended with classical scale invariance, which is broken by chiral symmetry breaking and confinement in a new strongly-coupled gauge theory that resembles QCD. The Standard Model Higgs field emerges as a result of the mixing of a scalar meson in the new strong dynamics and a massless elementary scalar field. The mass and scalar decay constant of that scalar meson, which are generated dynamically in the new gauge theory, give rise to the Higgs field mass term, automatically possessing the correct negative sign by the bosonic seesaw mechanism. Using analogy with QCD, we evaluate the dynamical scale of the new gauge theory and further make quantitative predictions for light pseudo-Nambu-Goldstone bosons associated with the spontaneous breaking of axial symmetry along chiral symmetry breaking in the new gauge theory. A prominent consequence of the scenario is that there should be a Standard Model gauge singlet pseudo-Nambu-Goldstone boson with mass below 220 GeV, which couples to two electroweak gauge bosons through the Wess-Zumino-Witten term, whose strength is thus determined by the dynamical scale of the new gauge theory. Other pseudo-Nambu-Goldstone bosons, charged under the electroweak gauge groups, also appear. Concerning the theoretical aspects, it is shown that the scalar quartic coupling can vanish at the Planck scale with the top quark pole mass as large as 172.5 GeV, realizing the flatland scenario without being in tension with the current experimental data.
Introduction
============
Now that the Standard Model (SM) picture of electroweak symmetry breaking is soundly supported by the Higgs particle measurement, the next theoretical challenge is to derive the Higgs potential, which has been added to the SM *ad hoc* to trigger electroweak symmetry breaking, from an underlying theory behind the SM. An appealing scenario is that a new strongly-coupled gauge theory generates the Higgs field mass term through dimensional transmutation, as it is capable of addressing the origin of the electroweak scale. Such scenarios should come along with a symmetry or an assumption that forbids a tree-level Higgs field mass, like classical scale invariance [@csi], since otherwise the Higgs mass would arise as a genuine parameter of the fundamental Lagrangian, not as a consequence of dynamics. In this paper, we aim at a derivation of the SM Higgs field mass term from strong dynamics of a new gauge theory in a classically scale invariant setup. To this end, we consider the minimal model described below: Besides the SM fermions and gauge fields, our model includes an elementary scalar field with the same electroweak charge as the SM Higgs field, which however is massless due to classical scale invariance. The model further contains a new $SU(N)_T$ gauge group and 3 flavors of massless Dirac fermions that are charged under both the $SU(N)_T$ and electroweak gauge groups and have a Yukawa-type coupling with the elementary scalar field. At infrared scales, the new gauge theory becomes strongly-coupled and triggers chiral symmetry breaking and confinement, analogously to QCD. The strong dynamics gives rise to a massive scalar meson as a bound state of the new fermions, which mixes with the elementary scalar through the Yukawa-type coupling. One of the mass eigenstates necessarily has a negative mass squared term due to the “bosonic seesaw” mechanism [@bosonicseesaw], at a scale given by the product of the Yukawa-type coupling and the dynamical scale of the new gauge theory. It is this state that we identify with the SM Higgs field. We thereby attribute the scale and negative sign of the SM Higgs mass term to dynamics of the new gauge theory. Since the principal motivation of this study is to link the Higgs field mass to a new dynamical scale, we do not adhere to “naturalness” of the electroweak scale; we take into account all situations where the dynamical scale is arbitrarily high, as long as the SM Higgs field mass term is reproduced at the correct scale.
Concerning experimental signatures of the model, we intensively study pseudo-Nambu-Goldstone (pNG) bosons associated with the spontaneous breaking of $SU(3)$ axial symmetry along chiral symmetry breaking in the $SU(N)_T$ gauge theory. The pNG bosons acquire mass from the electroweak interactions and the Yukawa-type interaction that explicitly violate the $SU(3)$ axial symmetry, which are the only source for their mass because the current mass is zero due to classical scale invariance. A prominent prediction of the model is that there should be a SM gauge singlet pNG boson with mass below 220 GeV, which couples to the SM $W$, $Z$ bosons and photon through the Wess-Zumino-Witten term [@wz; @w]. The mass prediction is insensitive to model parameters because this pNG boson is massive due only to the Yukawa-type interaction, and hence its mass is given by the dynamical scale times the coupling constant for the Yukawa-type interaction. Since this is also the scale at which the Higgs field mass term is generated, the pNG boson necessarily has mass at the electroweak scale. The coupling with the electroweak gauge bosons is, however, suppressed by the unknown dynamical scale of the new gauge theory, but searching for a particle with $O(100)$ GeV mass that interacts feebly with electroweak gauge bosons inspires new experimental techniques and is intriguing in its own light. There also appear pNG bosons with electroweak charges, which gain mass from the electroweak interactions at a scale of the dynamical scale times the electroweak gauge couplings. Their masses are therefore enhanced accordingly with the dynamical scale and these pNG bosons may not be kinematically accessible at colliders.
The model has an implication for physics at the Planck scale. The vanishing of the scalar quartic coupling at the Planck scale in a classically scale invariant model, or equivalently, a flat scalar potential at the Planck scale, has been pursued in Ref. [@flatland] in the hope of finding hints for quantum gravity. In previous models, the vanishing of the Higgs quartic coupling is achieved only with the top quark pole mass well below 172 GeV, and hence those models are in tension with top quark mass measurements; note that the ATLAS Collaboration has recently reported the top quark mass to be $172.84\pm0.70$ GeV [@topatlas]. Our model, in contrast, can realize the vanishing of the scalar quartic coupling with the top quark pole mass as large as 172.5 GeV. This is possible owing to contributions of the new fermions to the renormalization group (RG) running of the weak gauge coupling, which enhance this gauge coupling at ultraviolet scales and eventually lift the scalar quartic coupling at the Planck scale through the RG evolution as compared to the SM.
Models where strong dynamics supplies the SM Higgs field mass term in a classically scale invariant setup have been proposed in Ref. [@csi-chsb], which however cannot address the origin of its tachyonic feature. A model similar to ours has been studied in Refs. [@similar1; @similar2; @similar3]. Our model is more attractive in that the field content is minimal and thus a more definite prediction for the pNG boson mass spectrum is feasible.
The content of the paper is as follows: In Secion 2, we describe the model and expound how the SM Higgs field mass term is generated from strong dynamics of the $SU(N)_T$ gauge theory in the classically scale invariant setup. We further evaluate the dynamical scale of the gauge theory based on analogy with QCD. In Section 3, we study phenomenology of the pNG bosons by deriving their mass spectrum and interaction strengths, again relying on analogy with QCD. In Section 4, we demonstrate that the model can achieve the vanishing of the scalar quartic coupling at the Planck scale with the top quark pole mass as large as 172.5 GeV. Section 5 is for the summary and discussions.\
Model
=====
Model description and the origin of the Standard Model Higgs field mass
-----------------------------------------------------------------------
The gauge symmetry of the model is $SU(N)_T\times SU(3)_C\times SU(2)_W\times U(1)_Y$, where $SU(3)_C$, $SU(2)_W$ and $U(1)_Y$ are the SM color, weak and hypercharge gauge groups, respectively, and $SU(N)_T$ is a newly introduced gauge group. We require $N\geq3$. The model includes the same matter fermions as the SM and further contains massless Dirac fermions, $\chi,\psi$, which are in the fundamental representation of the $SU(N)_T$ gauge group and are also charged under $SU(2)_W\times U(1)_Y$ gauge group; $\chi$ is an isospin doublet with hypercharge $Y=b/2$ and $\psi$ is an isospin singlet with hypercharge $Y=(b-1)/2$, with $b$ being an arbitrary number. There also is a massless elementary scalar field, $H$, which possesses the same electroweak charge as the SM Higgs field and is a singlet under the $SU(N)_T$ gauge group. The charges of the fields $\chi,\psi$ and $H$ are summarized in Table \[fields\].
---------------------------------------------------------------------------------------------------
Field Lorentz $SO(1,3)$ $SU(N)_T$ $SU(2)_W$ $U(1)_Y$
------- -------------------------------------------------------- ----------- ----------- ----------
$H$ **[1]{} & **[1]{} & **[2]{} & $1/2$\
$\chi$ & **[(2, 2)]{} & **[N]{} & **[2]{} & $b/2$\
$\psi$ & **[(2, 2)]{} & **[N]{} & **[1]{} & $(b-1)/2$\
******************
---------------------------------------------------------------------------------------------------
: The charges of the fields $\chi,\psi$ and $H$. Representations in the Lorentz group, the $SU(N)_T$ gauge group and the SM electroweak gauge groups $SU(2)_W\times U(1)_Y$ are displayed. $b$ is an arbitrary number.[]{data-label="fields"}
Classical scale invariance forbids Dirac mass terms for $\chi$ and $\psi$ and a mass term for $H$, while $\chi$ and $\psi$ are allowed to have a Yukawa-type coupling with $H$. We further assume invariance of the theory under the parity transformation on $\chi$ and $\psi$, $\chi \to \gamma_0\chi$, and $\psi \to \gamma_0\psi$. This assumption is required to fully use analogy with QCD. The Lagrangian for $\chi,\psi$ and $H$ fields thus reads $$\begin{aligned}
{\cal L}_{{\rm ewsb}} &= \left\vert \left( \partial_{\mu} + i g_W W^a_{\mu} \tau^a + i \frac{1}{2} g_Y B_{\mu} \right) H \right\vert^2
\nonumber \\
&+ i\bar{\chi} \gamma^{\mu} \left( \partial_{\mu} + i g_T X_{\mu}^\alpha T^\alpha + i g_W W_{\mu}^a \tau^a + i \frac{b}{2} g_Y B_{\mu} \right) \chi
+ i\bar{\psi} \gamma^{\mu} \left( \partial_{\mu} + i g_T X_{\mu}^\alpha T^\alpha + i \frac{b-1}{2} g_Y B_{\mu} \right) \psi
\nonumber \\
&- y \, \bar{\chi} \psi H - y \, H^{\dagger} \bar{\psi} \chi
\label{lagrangian1} \\
&- \lambda \, (H^{\dagger}H)^2 -Y_u \, \bar{q}u \, \epsilon H^* - Y_d \, \bar{q}d \, H - Y_e \, \bar{\ell}e \, H - {\rm H.c.},
\label{lagrangian2}\end{aligned}$$ where $y$ is a Yukawa coupling constant for $H$, $\chi$ and $\psi$ fields, which is taken to be real by phase redefinition of $\psi$ field. Here, $g_T$, $g_W$ and $g_Y$ denote the $SU(N)_T$, SM weak and hypercharge gauge coupling constants, $X_{\mu}^\alpha$, $W_{\mu}^a$ and $B_{\mu}$ denote the corresponding gauge fields, and $T^\alpha \, (\alpha=1,2,...,N^2-1)$ and $\tau^a \, (a=1,2,3)$ are generators of the $SU(N)_T$ and weak gauge groups, respectively. $q$, $u$, $d$, $\ell$ and $e$ respectively denote SM isospin doublet quarks, singlet up-type quarks, down-type quarks, doublet leptons and singlet leptons, $Y_u$, $Y_d$ and $Y_e$ are coupling constants proportional to the SM Yukawa couplings (flavor indices are omitted), and $\epsilon$ denotes the antisymmetric tensor in the isospin space.
The $SU(N)_T$ gauge theory is assumed to become strongly-coupled at infrared scales and induce chiral symmetry breaking and confinement of $\chi$,$\psi$ fields in the same way as QCD. We infer the pattern of chiral symmetry breaking from the most attractive channel hypothesis. When applied to weak and hypercharge gauge boson exchange forces [@mac], the hypothesis argues that composite operators with the smallest values of the quadratic Casimir operators of $SU(2)_W$ and $U(1)_Y$ groups form condensates, while when applied to the exchange of the elementary scalar $H$ [@scalarmac], those with the largest values of the quadratic Casimir operators are expected to form condensates. In the current model, we assume $y$ to be sufficiently small that the $\bar{\chi}\psi H$ Yukawa interaction is subdominant compared to the weak and hypercharge gauge interactions, which gives that operators with the smallest quadratic Casimir operators go into chiral condensation, namely, we have $$\begin{aligned}
\langle 0 \vert \bar{\chi}\chi \vert 0 \rangle &\neq 0, \ \ \ \ \ \langle 0 \vert \bar{\chi}\tau^a\chi \vert 0 \rangle = 0 \ (a=1,2,3), \ \ \ \ \ \langle 0 \vert \bar{\psi}\psi \vert 0 \rangle \neq 0, \ \ \ \ \ \langle 0 \vert \bar{\chi}\psi \vert 0 \rangle = 0.
\label{chsb}\end{aligned}$$ Note that the electroweak symmetry is maintained at this stage, unlike in the technicolor model [@tc]. This owes to the fact that $\chi,\psi$ fields are vector-like with respect to the weak and hypercharge gauge groups.
In the mass spectrum below the confinement scale, there exists a scalar meson, $\Theta$, that corresponds to a scalar bound state of $\bar{\psi}\chi$. We write the mass of $\Theta$ meson as $M_\Theta$, and further define the scalar decay constant for $\Theta$ meson, $F_\Theta$, in the following fashion: $$\begin{aligned}
\hat{y} F_\Theta M_\Theta &\equiv \langle 0 \vert y \, \bar{\psi}\chi(0) \vert \Theta \rangle,
\label{decayconstant}\end{aligned}$$ where $\vert \Theta \rangle$ denotes the state with one $\Theta$ meson, $y \, \bar{\psi}\chi$ denotes a scalar current accompanied by the coupling constant $y$, and $\hat{y}$ is a quantity proportional to $y$ that is RG-invariant in the $SU(N)_T$ gauge theory. Note that inclusion of the coupling constant $y$ in the definition of the scalar current is advantageous in that the current becomes independent of the wavefunction renormalization in the $SU(N)_T$ gauge theory. Below the confinement scale, $\bar{\psi}\chi$ term in the Lagrangian Eq. (\[lagrangian1\]) asymptotes to $\Theta$ meson term, and accordingly, $y \, H \bar{\chi}\psi$ term becomes a mixing term for $H$ and $\Theta$ as $$\begin{aligned}
y \, \bar{\chi}\psi \, H \ &\Rightarrow \ (\hat{y} F_\Theta M_\Theta) \, \Theta^{\dagger}H.\end{aligned}$$ The mass matrix for $H$ and $\Theta$ is thus found to be $$\begin{aligned}
-{\cal L}_{{\rm ewsb}} &\supset \left(
\begin{array}{cc}
H^{\dagger} & \Theta^{\dagger}
\end{array}
\right)
\left(
\begin{array}{cc}
0 & \hat{y} F_\Theta M_\Theta \\
\hat{y} F_\Theta M_\Theta & M_\Theta^2
\end{array}
\right)
\left(
\begin{array}{c}
H \\
\Theta
\end{array}
\right).
\label{hthetamass}\end{aligned}$$ We hereafter concentrate on the limit with small $\hat{y}$ that leads to $\hat{y} F_\Theta \ll M_\Theta$. Upon diagonalization, we obtain the following mass terms for mass eigenstate scalar fields $H_1$ and $H_2$: $$\begin{aligned}
-{\cal L}_{{\rm ewsb}} &\supset -(\hat{y} F_\Theta)^2 \, H_1^\dagger H_1 + M_\Theta^2 \, H_2^\dagger H_2,
\nonumber \\
&\left(
\begin{array}{c}
H \\
\Theta
\end{array}
\right)=
\left(
\begin{array}{cc}
c_H & s_H \\
-s_H & c_H
\end{array}
\right)
\left(
\begin{array}{c}
H_1 \\
H_2
\end{array}
\right), \ \ \ s_H = \frac{\hat{y}F_\Theta}{M_\Theta}, \
c_H = 1 - \frac{1}{2} \frac{(\hat{y}F_\Theta)^2}{M_\Theta^2}.
\label{diag}\end{aligned}$$ Since $H_1$ has a negative mass squared term, it develops a non-zero vacuum expectation value (VEV) and triggers electroweak symmetry breaking. Hence, we identify $H_1$ with the SM Higgs field. The $H_1$ field has a quartic coupling and Yukawa couplings induced from Eq. (\[lagrangian2\]) as $$\begin{aligned}
-{\cal L}_{{\rm ewsb}} &\supset \lambda \, c_H^4 \, (H_1^\dagger H_1)^2 + Y_u \, c_H \, \bar{q}u \, \epsilon H_1^* + Y_d \, c_H \, \bar{q}d \, H_1 + Y_e \, c_H \, \bar{\ell}e \, H_1 + {\rm H.c.}
\nonumber \\ &{\rm with \ } c_H = 1 - \frac{1}{2} \frac{(\hat{y}F_\Theta)^2}{M_\Theta^2}.\end{aligned}$$ The mass term, quartic coupling and Yukawa couplings of $H_1$ field should agree with those of the SM Higgs field at the energy scale of the $H_2$ mass, that is, $M_\Theta$. These requirements are encapsulated in the following matching conditions: $$\begin{aligned}
-\frac{1}{2}m_h^2 = ({\rm SM \ Higgs \ field \ mass}) &= -(\hat{y}F_\Theta)^2,
\label{matching1}
\\
\lambda^{{\rm SM}}(M_\Theta) &= \lambda(M_\Theta) \, c_H^4 = \lambda(M_\Theta) \, \left(1-2\frac{(\hat{y}F_\Theta)^2}{M_\Theta^2} \right),
\label{matching2}
\\
Y_k^{{\rm SM}}(M_\Theta) &= Y_k(M_\Theta) \, c_H = Y_k(M_\Theta) \, \left(1-\frac{1}{2}\frac{(\hat{y}F_\Theta)^2}{M_\Theta^2} \right) \ \ \ (k=u,d,e), \nonumber \\
\label{matching3}\end{aligned}$$ where $m_h$ denotes the pole mass of the SM Higgs particle, and $\lambda^{{\rm SM}}(M_\Theta)$ and $Y_u^{{\rm SM}}(M_\Theta)$, $Y_d^{{\rm SM}}(M_\Theta)$, $Y_e^{{\rm SM}}(M_\Theta)$ respectively represent the SM Higgs quartic coupling and SM Yukawa couplings at the energy scale $M_\Theta$.
In this way, the Higgs field mass is generated at the scale determined by the $\Theta$ meson decay constant $F_\Theta$ and the $\bar{\chi}\psi H$ Yukawa coupling constant $y$, with the correct negative sign owing to the negative determinant of the mass matrix Eq. (\[hthetamass\]). The scale and sign of the SM Higgs mass term are thus attributed to strong dynamics of the $SU(N)_T$ gauge theory and its coupling with the elementary scalar field.\
We comment in passing that the elementary scalar $H$ does not have a mixing term with any pNG boson associated with the chiral symmetry breaking, due to the assumption that the theory is invariant under the parity transformation on $\chi$ and $\psi$ that makes the Yukawa coupling appear in the form Eq. (\[lagrangian1\]). To prove this, we employ chiral perturbation theory: First we define the fields, $$\begin{aligned}
\Psi_L &\equiv \frac{1-\gamma_5}{2}
\left(
\begin{array}{c}
\chi \\
\psi
\end{array}
\right),
\ \ \
\Psi_R \equiv \frac{1+\gamma_5}{2}
\left(
\begin{array}{c}
\chi \\
\psi
\end{array}
\right),\end{aligned}$$ in terms of which the Yukawa interaction of Eq. (\[lagrangian1\]) is expressed as [^1] $$\begin{aligned}
-{\cal L} &\supset \bar{\Psi}_L
\left(
\begin{array}{cc}
0 & yH \\
y^*H^\dagger & 0
\end{array}
\right)
\Psi_R
+ {\rm h.c.}\end{aligned}$$ The $SU(N)_T$ gauge theory possesses approximate $U(3)_L \times U(3)_R$ symmetry under the transformation $\Psi_L \to U_L \Psi_L$, $\Psi_R \to U_R \Psi_R$, with $U_L$, $U_R$ being unitary matrices. Along the chiral symmetry breaking, the axial part $U(3)_A$ is spontaneously broken and there appear 9 pNG bosons (one of which gains mass at the dynamical scale from instantons). We parametrize them as $$\begin{aligned}
U(x) &= \exp\left( \ 2i \frac{\Pi^j(x)}{f_\Pi}\frac{\lambda^j}{2} + \sqrt{2}i \frac{\Pi(x)}{f_\Pi} \ \right),\end{aligned}$$ where $\Pi^j(x)$ $(j=1,2,...,8)$ and $\Pi(x)$ denote the pNG boson fields, $\lambda^j$’s are the Gell-Mann matrices, and $f_\Pi$ is the NG boson decay constant. $U(x)$ transforms under $U(3)_L \times U(3)_R$ as $$\begin{aligned}
U(x) &\to U_R U(x) U_L^\dagger.\end{aligned}$$ Since we can assign to the Yukawa coupling and $H$ the following spurious transformation property, $$\begin{aligned}
\left(
\begin{array}{cc}
0 & yH \\
y^*H^\dagger & 0
\end{array}
\right)
&\to
U_L
\left(
\begin{array}{cc}
0 & yH \\
y^*H^\dagger & 0
\end{array}
\right)
U_R^\dagger,\end{aligned}$$ the effective Lagrangian reads $$\begin{aligned}
{\cal L}_{eff} &= \frac{f_\Pi^2}{4} {\rm tr}\left[ \left(D_\mu U(x)\right)^\dagger D^\mu U(x) \right]
- B_0 \ {\rm tr}\left[U(x)
\left(
\begin{array}{cc}
0 & yH \\
y^*H^\dagger & 0
\end{array}
\right)
\right]
- {\rm h.c.},
\label{efflag}\end{aligned}$$ where $B_0$ is a constant and $D_\mu$ is a covariant derivative. Based on analogy with QCD, we expect $B_0$ to be real. Then the second term of Eq. (\[efflag\]) is recast into the form, $$\begin{aligned}
-{\cal L}_{eff} &\supset B_0 \ {\rm tr}\left[ \ \left\{ U(x)+U(x)^\dagger \right\} \
\left(
\begin{array}{cc}
0 & yH \\
y^*H^\dagger & 0
\end{array}
\right)
\right].\end{aligned}$$ Since the above interaction term is symmetric under the transformation $\Pi^j \to -\Pi^j$ or $\Pi \to -\Pi$, the couplings between $H$ and the pNG bosons contain even numbers of pNG bosons and hence no mixing term exists between them.\
Evaluation of the dynamical scale of the $SU(N)_T$ gauge theory
---------------------------------------------------------------
We evaluate the dynamical scale of the $SU(N)_T$ gauge theory from the matching condition Eq. (\[matching1\]) under analogy between the $SU(N)_T$ gauge theory and QCD. This is done by expressing the $\Theta$ meson mass, total width and decay constant in terms of corresponding quantities in QCD, rescaled by the ratio of the dynamical scales of the $SU(N)_T$ gauge theory and QCD, $r$, defined by $$\begin{aligned}
r &\equiv \frac{\Lambda_T}{\Lambda_{QCD}},\end{aligned}$$ where $\Lambda_T$ and $\Lambda_{QCD}$ denote the dynamical scales of the $SU(N)_T$ gauge theory and QCD, respectively.
Along with $r$, a factor depending on $N/N_c$ ($N_c=3$ is the number of colors in QCD) appears when one relates some quantities in the $SU(N)_T$ gauge theory to their QCD counterparts. In this paper, we make the two approximations below:\
($\alpha$) The Casimir operator $C_F$ is approximated as $C_F=(N^2-1)/(2N)\simeq N/2$.
($\beta$) The contribution of $\chi,\psi$ fermion loop to the gauge field propagator is subdued compared to those of gauge field loop and ghost loop, and is hence negligible. For example, the one-loop correction to the gauge field propagator is proportional to $11N-2n_f$ with $n_f$ being the number of flavors which equals $3$ in our model, and this is approximated as $11N-2n_f=11N-6\simeq 11N$.\
($\alpha$) and ($\beta$) give that the coefficient for $O(\alpha_T^n)$ correction term is proportional to $N^n$. It immediately follows that the $SU(N)_T$ gauge coupling $g_T$ and the QCD gauge coupling $g_s$ satisfy $g_T^2(r\mu)\simeq(N_c/N)g_s^2(\mu)$. Also, correlation functions of $SU(N)_T$-singlet operators, with all operators connected, are obtained by rescaling corresponding correlation functions in QCD by $N/N_c$, *e.g.*, we have $$\begin{aligned}
\langle 0 \vert T\{ \bar{f}\Gamma_1 f(x_1) \, \bar{f}\Gamma_2 f(x_2) \, ... \, \bar{f}_n\Gamma_n f_n(x_n) \} \vert 0 \rangle &=
\frac{N}{N_c} r^{3n} \, \langle 0 \vert T\{ \bar{q}\Gamma_1 q(x_1) \, \bar{q}\Gamma_2 q(x_2) \, ... \, \bar{q}\Gamma_n q(x_n) \} \vert 0 \rangle,
\label{scaling}\end{aligned}$$ where $\bar{f}\Gamma_i f$ and $\bar{q} \Gamma_i q$ ($i=1,2,...,n$) represent $SU(N)_T$-singlet and QCD-singlet bilinear operators of massless Dirac fermions in the fundamental representation of the $SU(N)_T$ and QCD gauge groups, respectively, with $\Gamma_i$ being a combination of the gamma matrices. As a corollary, the wavefunction overlap of a one-meson-state with a $SU(N)_T$-singlet current operator roughly scales by $\sqrt{N/N_c}$, because the two-point self-correlation function of that current operator scales by $N/N_c$. In particular, the decay constant for a Nambu-Goldstone (NG) boson is proportional to $\sqrt{N/N_c}$. Moreover, the decay amplitude for a meson decaying into two mesons through the $SU(N)_T$ gauge interaction scales by $(N/N_c)^{-1/2}$. This is because a correlation function for three $SU(N)_T$-singlet current operators scales by $N/N_c$, and when the current operators asymptote to one-meson creation operators, a factor $(N/N_c)^{-1/2}$ appears for each current, which leads to an overall factor of $(N/N_c)\{(N/N_c)^{-1/2}\}^3=(N/N_c)^{-1/2}$ for the decay amplitude.\
### $\Theta$ meson mass $M_\Theta$
We evaluate the $\Theta$ meson mass $M_\Theta$ by regarding the $K_0^*(1430)$ meson as a QCD analog of the $\Theta$ meson for two reasons [^2]:\
(i)It is likely that $K_0^*(1430)$ belongs to a nonet of quark-anti-quark bound states whereas $K_0^*(800)$ belongs a nonet of diquark-anti-diquark bound states, notably because the $K_0^*(800)$ mass is considerably smaller than the $a_0(980)$ mass, which suggests that $K_0^*(800)$ is mainly a bound state of $(s,q,\bar{q},\bar{q}')$ and $a_0(980)$ is that of $(s,q,\bar{s},\bar{q}')$ ($q,q'$ denote up and down quarks and $s$ denotes strange quark). Hence, $K_0^*(1430)$ meson corresponds to the lightest $\bar{\chi}\psi$ scalar bound state, *i.e.*, $\Theta$ meson.
(ii)$K_0^*(1430)$ meson and $\Theta$ meson do not mix with glueball state and its $SU(N)_T$-gauge-theory counterpart due to non-zero strangeness and electroweak charge, respectively.\
Analogy between $\Theta$ meson and $K_0^*(1430)$ meson enables us to express the $\Theta$ meson mass $M_\Theta$ in terms of the $K_0^*(1430)$ mass, $m_{K_0^*(1430)}$, as $$\begin{aligned}
M_\Theta &= r \, m_{K_0^*(1430)}.
\label{mtheta}\end{aligned}$$ The experimental central value [@pdg] gives $M_\Theta = r\cdot 1.425$ GeV.\
### $\Theta$ meson total width $\Gamma_\Theta$
The $\Theta$ meson total width, $\Gamma_\Theta$, can be written with the $K_0^*(1430)$ meson total width, $\Gamma_{K_0^*(1430)}$. In doing so, we remind that about 90% of $K_0^*(1430)$ decays as $K_0^*(1430) \to K \pi$ [@pdg], in which process the $K,\pi$ meson masses modify the phase space, while the corresponding pNG bosons in the $SU(N)_T$ gauge theory are nearly massless due to zero current mass. Hence, we rescale the $K_0^*(1430)$ total width by the phase space ratio, together with the dynamical scale ratio $r$ and the factor $(N/N_c)^{-1}$, to evaluate $\Gamma_\Theta$. It is thus found to be $$\begin{aligned}
\Gamma_\Theta &= r \, \frac{N_c}{N} \, \frac{1}{ \sqrt{1-2\frac{m_\pi^2+m_K^2}{m_{K_0^*(1430)}^2}+\frac{(m_\pi^2-m_K^2)^2}{m_{K_0^*(1430)}^4}} } \, \Gamma_{K_0^*(1430)},
\label{gammatheta}\end{aligned}$$ where $m_\pi$ and $m_K$ denote the $\pi$ and $K$ masses, respectively. The experimental central values [@pdg] give $\Gamma_\Theta = r(N_c/N)\cdot 0.311$ GeV.\
### $\Theta$ meson scalar decay constant $F_\Theta$
Since a QCD analog of the $\Theta$ meson scalar decay constant $F_\Theta=\langle 0 \vert y \, \bar{\psi}\chi \vert \Theta \rangle/(\hat{y}M_\Theta)$ has not been measured, we evaluate it by the aid of explicit calculations. To this end, we confront the spectral density of states that couple to the scalar current $y \, \bar{\chi}\psi$, with the two-point self-correlation function of that scalar current for large space-like momenta; the former spectral density contains the term $\hat{y}F_\Theta M_\Theta$, as it quantifies the coupling of the one-$\Theta$-meson state with the scalar current. The latter correlation function can be described with perturbative calculation in the $SU(N)_T$ gauge theory and with empirical vacuum condensates. To embody the idea, we formulate the correlation function, $\Pi_{y\bar{\chi}\psi}(q^2)$, and the spectral density, $\rho_s(s)$, in the following fashion [^3]: $$\begin{aligned}
\Pi_{y\bar{\chi}\psi}(q^2) &\equiv \frac{1}{2}i\int{\rm d}^4x \, e^{iqx} \langle 0 \vert T\left\{ y \, \bar{\chi}(x)\psi(x) \ y \, \bar{\psi}(0)\chi(0) \right\} \vert 0 \rangle,
\label{correlatordef} \\
\rho_s(s) &\equiv {\rm Im}\Pi_{y\bar{\chi}\psi}(s) \ \ \ \ \ {\rm for} \ s\geq0.
\label{spectraldef}\end{aligned}$$ We relate the spectral density to the correlation function for space-like momenta using the Cauchy’s theorem. Since no one-massless-particle state couples to the scalar current, we have $$\begin{aligned}
\lim_{\vert q^2\vert\to0} \, q^2\Pi_{y\bar{\chi}\psi}(q^2) &= 0\end{aligned}$$ for complex $q^2$, which allows us to use the Cauchy’s theorem to obtain $$\begin{aligned}
\int_0^{s_0}{\rm d}s \, \{P(s)-P(s_0)\} \, \frac{1}{\pi}{\rm Im}\Pi_{y\bar{\chi}\psi}(s) = -\frac{1}{2\pi i}\oint_{\vert q^2 \vert=s_0}{\rm d}q^2 \, \{P(q^2)-P(s_0)\} \, \Pi_{y\bar{\chi}\psi}(q^2),
\label{cauchy}\end{aligned}$$ where $s_0$ can take any real positive value and $P(s)$ can be any analytic function. Here, on the right-hand side, the correlation function for complex $q^2$ off the real positive axis can be derived through analytic continuation from space-like momenta, while the discontinuity of the correlation function at $q^2=s_0$ is avoided as the function $P(q^2)-P(s_0)$ vanishes there. The left-hand side includes $\hat{y}F_\Theta M_\Theta$, and the right-hand side is calculable by perturbation theory and operator product expansion in the $SU(N)_T$ gauge theory if $s_0$ is taken sufficiently large that these calculations are valid. Therefore, the equality Eq. (\[cauchy\]) enables us to evaluate $\hat{y}F_\Theta M_\Theta$ through analytic calculations. The use of Eq. (\[cauchy\]) for such $s_0$ constitutes the basis for the finite energy sum rules [@fesr].
First we express the spectral density $\rho_s(s)$ in terms of $\hat{y}F_\Theta M_\Theta$ based on the following two assumptions on quantum states that couple to the scalar current $y \, \bar{\chi}\psi$:\
(i) In the range $M_\Theta^2-r^2\cdot1~{\rm GeV}^2 \lesssim s \lesssim M_\Theta^2+r^2\cdot1~{\rm GeV}^2$, the spectral density $\rho_s(s)$ is dominated by contributions from the $\Theta$ meson resonance.
\(ii) The $\Theta$ meson resonance is well approximated by the relativistic Breit-Wigner function.\
These ansaetze find no support from hadron physics experiments, even given analogy between $\Theta$ meson and $K_0^*(1430)$ meson, because an elementary scalar current that couples to light quarks has not been measured. Nevertheless, $S$-wave $K\pi$ scattering data from the LASS experiment [@lass] hint us that the $K_0^*(1430)$ meson resonance may dominantly couple to $\bar{d}s$ scalar current ($d$ and $s$ respectively denote down and strange quarks) for invariant mass $m_{K_0^*(1430)}^2-1~{\rm GeV}^2 \lesssim s \lesssim m_{K_0^*(1430)}^2+1~{\rm GeV}^2$, which is rendered into the assumption (i) through the rescaling. The same data also suggest that the $K_0^*(1430)$ meson resonance can be fit with the relativistic Breit-Wigner function. Once one accepts (i) and (ii), the spectral density satisfies $$\begin{aligned}
{\rm for} \ &M_\Theta^2-r^2\cdot1~{\rm GeV}^2 \lesssim s \lesssim M_\Theta^2+r^2\cdot1~{\rm GeV}^2,
\nonumber \\
\rho_s(s) &= \frac{1}{\pi}{\rm Im}\Pi_{y\bar{\chi}\psi}(s)
\nonumber \\
&= \frac{1}{\pi}{\rm Im}\left[ \, \langle 0 \vert y \, \bar{\psi}\chi \vert \Theta \rangle \frac{1}{s-M_\Theta^2+iM_\Theta\Gamma_\Theta} \langle \Theta \vert y \, \bar{\chi}\psi \vert 0 \rangle \, \right]
\nonumber \\
&= (\hat{y} F_\Theta M_\Theta)^2 \, \frac{1}{\pi}\frac{M_\Theta \Gamma_\Theta}{(s-M_\Theta^2)^2+M_\Theta^2 \Gamma_\Theta^2}.
\label{spectral}\end{aligned}$$ Note here that since $\Theta$ meson decays into two pNG bosons, which are nearly massless due to the absence of current mass, the Breit-Wigner function reduces to the form as appears in Eq. (\[spectral\]).
Next we present the scalar current correlation function for space-like momenta $q^2 < 0$ as a perturbative series of the $SU(N)_T$ gauge coupling and an operator product expansion involving vacuum condensates, following analogous calculations for QCD in Refs. [@chetyrkin; @chetyrkin2; @qcdsumrules; @bagan]. The scalar current correlation function is given, for $q^2 < 0$, by $$\begin{aligned}
&\Pi_{y\bar{\chi}\psi}(q^2) = A \, q^2 + y^2(\mu^2) \frac{N}{8\pi^2}(-q^2) \log\left(\frac{-q^2}{\mu^2}\right) \left\{ \, 1
+ \sum_{k=1}^\infty \, \left(\frac{\alpha_T(\mu^2)}{\pi}\right)^k \, \sum_{l=0}^k \, d_{k,l}(C_F,C_A) \, \log^l\left(\frac{-q^2}{\mu^2}\right) \, \right\}
\nonumber \\
&+ \frac{1}{8\pi}\frac{1}{(-q^2)} \, y^2(\mu^2) \, \langle 0 \vert \alpha_T X_{\mu\nu}^\alpha X^{\alpha\mu\nu} \vert 0 \rangle
\nonumber \\
&+ 2\pi\frac{1}{(-q^2)^2} y^2(\mu^2) \langle 0 \vert \alpha_T \{ \, \bar{\chi}\sigma_{\mu\nu} T^\alpha \psi \, \bar{\psi}\sigma^{\mu\nu} T^\alpha \chi + \frac{1}{3}(\bar{\chi}\gamma_\mu T^\alpha \chi+2\bar{\psi}\gamma_\mu T^\alpha \psi)(\bar{\chi}\gamma^\mu T^\alpha \chi+\bar{\psi}\gamma^\mu T^\alpha \psi) \, \} \vert 0 \rangle
\nonumber \\
&+ ({\rm other \ vacuum \ condensates}) + ( \ O(\alpha_T(\mu^2)/\pi){\rm \ corrections \ to \ vacuum \ condensates} \ ).
\label{correlator}\end{aligned}$$ Here, $A$ is a constant of no interest, $\alpha_T\equiv g_T^2/(4\pi)$, $C_F=(N^2-1)/(2N)$, $C_A=N$, $T^\alpha$ denotes $SU(N)_T$ group generators, and $X_{\mu\nu}^\alpha$ denotes the $SU(N)_T$ gauge field strength. $\mu$ denotes the renormalization scale in the $\overline{MS}$ scheme for the correlation function itself, the running coupling constant $y(\mu^2)$ and the running gauge coupling constant $\alpha_T(\mu^2)$, which is taken to be common for simplicity. $d_{k,l}(C_F,C_A)$’s are coefficients that depend on $C_F,C_A$, whose explicit forms are found in Ref. [@chetyrkin]. The first term in the last line represents vacuum condensates of other operators, which are known to be subdominant compared to the field strength condensate and four-fermion condensate for $-q^2 \gtrsim r^2\cdot1$ GeV$^2$. We stress that fermion bilinear condensates $\langle 0 \vert \bar{f}f \vert 0 \rangle$ ($f=\chi,\psi$) do not enter into Eq. (\[correlator\]) due to the absence of $\chi,\psi$ current mass.
Finally, we associate the spectral density Eq. (\[spectral\]) with the correlation function Eq. (\[correlator\]) through the Cauchy’s theorem Eq. (\[cauchy\]). For an arbitrary analytic function $P(s)$, we choose $$\begin{aligned}
P(s) &= \exp\left[ \, -\frac{(s-M_\Theta^2)^2}{\Delta s^2} \, \right]
\ \
{\rm with \ \ } \Delta s = r^2 \cdot 1~{\rm GeV}^2,
\label{windowfunc}\end{aligned}$$ whose shape is depicted in Figure \[pfunc\].
![ The function $P(s)$ Eq. (\[windowfunc\]) in the range $0\leq s \leq r^2\cdot4~{\rm GeV}^2$. The solid vertical line corresponds to $s=M_\Theta^2$ and dashed lines to $s=M_\Theta^2\pm r^2\cdot1~{\rm GeV}^2$. []{data-label="pfunc"}](window.eps){width="120mm"}
The function $P(s)$ as given in Eq. (\[windowfunc\]) is beneficial for extracting the $\Theta$ meson contribution to the spectral density, because $P(s)$ is sizable only in the range $M_\Theta^2-r^2\cdot1~{\rm GeV}^2 \lesssim s \lesssim M_\Theta^2+r^2\cdot1~{\rm GeV}^2$, where the $\Theta$ meson resonance is assumed to dominate the spectral density, while contributions from the spectral density outside the range are exponentially suppressed. We set the common renormalization scale for $\Pi_{y\bar{\chi}\psi}(q^2)$ as $\mu^2=s_0$. We thus obtain $$\begin{aligned}
&\int_0^{s_0}{\rm d}s \, \{P(s)-P(s_0)\} \, \rho_s(s) = -\frac{1}{2\pi i}\oint_{\vert q^2 \vert=s_0}{\rm d}q^2 \, \{P(q^2)-P(s_0)\} \, \Pi_{y\bar{\chi}\psi}(q^2)
\nonumber \\
&\Longleftrightarrow
\nonumber \\
&(\hat{y} F_\Theta M_\Theta)^2 \, \int_0^{s_0} {\rm d}s \, \{P(s)-P(s_0)\} \, \frac{1}{\pi} \frac{M_\Theta \Gamma_\Theta}{(s-M_\Theta^2)^2+M_\Theta^2 \Gamma_\Theta^2}
\nonumber \\
&= y^2(s_0) \frac{N}{8\pi^2} \, s_0^2 \, \left\{ \, B_0 + \sum_{k=1}^\infty \, \left(\frac{\alpha_T(s_0)}{\pi}\right)^k \, \sum_{l=0}^k \, d_{k,l}(C_F,C_A) \, B_{l+1} \, \right\}
\nonumber \\
&+ \frac{1}{8\pi} \, y^2(s_0) \, C_{1} \, \langle 0 \vert \alpha_T X_{\mu\nu}^\alpha X^{\alpha\mu\nu} \vert 0 \rangle
\nonumber \\
&+ 2\pi y^2(s_0) \, C_{2} \, \frac{1}{s_0} \langle 0 \vert \alpha_T \{ \, \bar{\chi}\sigma_{\mu\nu} T^\alpha \psi \, \bar{\psi}\sigma^{\mu\nu} T^\alpha \chi + \frac{1}{3}(\bar{\chi}\gamma_\mu T^\alpha \chi+2\bar{\psi}\gamma_\mu T^\alpha \psi)(\bar{\chi}\gamma^\mu T^\alpha \chi+\bar{\psi}\gamma^\mu T^\alpha \psi) \, \} \vert 0 \rangle
\nonumber \\
&+ ({\rm other \ vacuum \ condensates}) + ( \ O(\alpha_T(s_0)/\pi){\rm \ corrections \ to \ vacuum \ condensates} \ ),
\label{sumrule}\end{aligned}$$ where $B_l$’s and $C_{l}$’s are numbers defined as $$\begin{aligned}
B_l &\equiv -\frac{1}{s_0^2} \, \frac{1}{2\pi i}\oint_{\vert q^2 \vert=s_0}{\rm d}q^2 \, \{ P(q^2)-P(s_0) \} \, (-q^2)\log^{l+1}\left(\frac{-q^2}{s_0}\right)
\nonumber \\
&= \frac{1}{2\pi i}\int_0^{2\pi}{\rm d}\theta \, i \, e^{2i\theta} \, \left\{ \, \exp\left[ \, -\frac{(s_0 e^{i\theta}-M_\Theta^2)^2}{2M^4} \, \right] - \exp\left[ \, -\frac{(s_0-M_\Theta^2)^2}{2M^4} \, \right] \, \right\} \, \{i(\theta-\pi)\}^{l+1},
\nonumber \\
C_{l} &\equiv -s_0^{l-1} \, \frac{1}{2\pi i}\oint_{\vert q^2 \vert=s_0}{\rm d}q^2 \, \{ P(q^2)-P(s_0) \} \, \frac{1}{(-q^2)^{l}}
\nonumber \\
&= \frac{(-1)^{l+1}}{2\pi i}\int_0^{2\pi}{\rm d}\theta \, i \, e^{-i \theta (l-1)} \, \left\{ \, \exp\left[ \, -\frac{(s_0 e^{i\theta}-M_\Theta^2)^2}{2M^4} \, \right] - \exp\left[ \, -\frac{(s_0-M_\Theta^2)^2}{2M^4} \, \right] \, \right\}.\end{aligned}$$ One can derive $(\hat{y} F_\Theta M_\Theta)^2$ from the equality Eq. (\[sumrule\]). The derived value would not depend on $s_0$ if the true spectral density were used and the correlation function were calculated to all orders in perturbation theory and in operator product expansion. In reality, it does exhibit a $s_0$ dependence due to the error of spectral density assumed and the truncation of perturbative series and operator product expansion. Nevertheless, we argue that a value of $(\hat{y} F_\Theta M_\Theta)^2$ that is most stable against variations of $s_0$ is a physically meaningful estimate for $(\hat{y} F_\Theta M_\Theta)^2$. To search for such a value, we numerically evaluate both sides of Eq. (\[sumrule\]) to express $(\hat{y} F_\Theta M_\Theta)^2$ as a function of $s_0$\
For the left-hand side of Eq. (\[sumrule\]), we numerically integrate the relativistic Breit-Wigner function Eq. (\[spectral\]) multiplied by the function $P(s)-P(s_0)$ Eq. (\[windowfunc\]), substituting the previously evaluated values of $\Theta$ meson mass $M_\Theta$ and total width $\Gamma_\Theta$. Contributions from states other than the $\Theta$ meson resonance are ignored, as they are exponentially suppressed by the function $P(s)$.
In the right-hand side of Eq. (\[sumrule\]), we evaluate the perturbative series to the order of $\alpha_T^4$. We quote the analytic expressions of $d_{1,l}$, $d_{2,l}$ and $d_{3,l}$ from Ref. [@chetyrkin]. As for $d_{4,0}$, only the formula for $N=3$ is available in the literature. Therefore, we estimate $d_{4,0}$ for general $N$ by rescaling the analytic expression of $d_{4,0}$ for $N=3$ in Ref. [@chetyrkin2] by the following rule, based on the approximation with $C_F=(N^2-1)/(2N)\simeq N/2$; the $n_f^0$ term is rescaled by $(N/3)^4$; the $n_f^1$ term is rescaled by $(N/3)^3$; the $n_f^2$ term is rescaled by $(N/3)^2$; the $n_f^3$ term is rescaled by $N/3$, with $n_f$ denoting the number of quark flavors. $d_{4,1},d_{4,2},d_{4,3},d_{4,4}$ are calculated from terms of order $\alpha_T^3$ or below through RG equations. We separately evaluate $\alpha_T(s_0)$ and $y(s_0)$ by solving RG equations; to obtain $\alpha_T(s_0)$, we employ the $O(\alpha_s^4)$ RG equation for the QCD gauge coupling with the replacement of $C_F,C_A$ with those for the $SU(N)_T$ gauge group, and impose an initial condition at the scale $\mu=r\cdot1.777$ GeV. The equations are given below: $$\begin{aligned}
\frac{{\rm d}\alpha_T(\mu^2)}{{\rm d}\log\mu^2} &= -\beta_1(C_F) \, \alpha_T^2(\mu^2) -\beta_2(C_F,C_A) \, \alpha_T^3(\mu^2) -\beta_3(C_F,C_A) \, \alpha_T^4(\mu^2),
\nonumber \\
\alpha_T(\mu^2) &= \frac{N_c}{N} \cdot 0.33 \ \ {\rm at} \ \ \mu=r\cdot1.777~{\rm GeV},\end{aligned}$$ where the forms of $\beta_1,\beta_2,\beta_3$ are found in Ref. [@qcd3loop]. Here, the initial condition is based on the QCD gauge coupling constant at the $\tau$ lepton mass scale $m_\tau = 1.777$ GeV studied in Ref. [@pich], which is rescaled by the factor $N_c/N$ under the approximations $(\alpha),(\beta)$ in Section 2.2. To gain $y(s_0)$, we exploit the $O(\alpha_s^4)$ RG equation for the running current quark mass in Ref. [@larin] by exchanging $C_F,C_A$ with those for the $SU(N)_T$ gauge group. This is justifiable because the coupling constant $y$ and the current quark mass obey the same RG equation in the $SU(N)_T$ gauge theory and QCD. Since the overall scale of the coupling constant $y$ is a free parameter of the model, we express $y(s_0)$ as normalized by the value at one particular scale. For later convenience, we express $y(s_0)$ in terms of $y$ evaluated at $\mu=r\cdot2$ GeV scale in the $\overline{MS}$ scheme, which we denote by $y_{r2}\equiv y(\mu^2=r^2\cdot2^2~{\rm GeV}^2)$.
The vacuum condensates are assessed by analogy with QCD; using the values of QCD gluon condensate and four-quark condensate obtained from $e^+e^-$ collisions, heavy quarkonia and $\tau$ lepton decays in Ref. [@narison], we evaluate them as $$\begin{aligned}
&\frac{1}{8\pi} \langle 0 \vert \alpha_T X_{\mu\nu}^\alpha X^{\alpha\mu\nu} \vert 0 \rangle
\simeq \frac{1}{8\pi} \, \frac{N}{N_c} \, r^4 \, \langle 0 \vert \alpha_s G_{\mu\nu}^aG^{a\mu\nu} \vert 0 \rangle = \frac{1}{8\pi} \, \frac{N}{N_c} \, r^4 \cdot 6.8\times10^{-2} \, {\rm GeV}^4,
\label{ggcondensate} \\
&2\pi\langle 0 \vert \alpha_T \{ \, \bar{\chi}\sigma_{\mu\nu}T^\alpha \psi \, \bar{\psi}\sigma^{\mu\nu}T^\alpha \chi + \frac{1}{3}(\bar{\chi}\gamma_\mu T^\alpha \chi + 2\bar{\psi}\gamma_\mu T^\alpha \psi)(\bar{\chi}\gamma^\mu T^\alpha \chi + \bar{\psi}\gamma^\mu T^\alpha \psi) \, \} \vert 0 \rangle
\nonumber \\
&\simeq - \frac{22\pi}{3}\frac{N^2-1}{N^2} \, \rho \alpha_T(\langle 0 \vert \bar{f}f \vert 0 \rangle)^2
\simeq -\frac{22\pi}{3}\frac{N^2-1}{N^2}\frac{N}{N_c} \, r^6 \, \rho\alpha_s(\langle 0 \vert \bar{q}q \vert 0 \rangle)^2
\nonumber \\
&= -\frac{22\pi}{3}\frac{N^2-1}{N^2}\frac{N}{N_c} \, r^6 \cdot 4.5\times10^{-4} \, {\rm GeV}^6,
\label{4fcondensate}\end{aligned}$$ where in Eq. (\[4fcondensate\]), $f$ represents one massless Dirac fermion in the fundamental representation of the $SU(N)_T$ gauge group, and $\rho$ denotes the ratio of four-quark condensate and the square of quark bilinear condensate. The factor $N/N_c$ ensues from the rescaling of the correlation functions of operators in the adjoint representation, which scale by $N^2/N_c^2$, and that of the gauge coupling, which scales by $N_c/N$.
As a reference, we numerically express the perturbative series and the two vacuum condensates for $N=3,6$ and for $s_0=r^2\cdot3^2~{\rm GeV}^2, \, r^2\cdot5^2~{\rm GeV}^2$. Also shown is the value of the gauge coupling $a\equiv (N/3)\alpha_T(s_0)/\pi$ for each $s_0$ and $N$. We comment that for $s_0<r^2\cdot3^2~{\rm GeV}^2$, the perturbative series does not show good convergence, while for $s_0>r^2\cdot5^2~{\rm GeV}^2$, numerical calculation of the contour integral is computationally expensive due to a rapid oscillation of the function $P(q^2)$ for complex $q^2$. $$\begin{aligned}
&{\rm For \ } N=3 {\rm \ and \ } s_0=r^2\cdot3^2~{\rm GeV}^2,
\nonumber \\
&-\frac{1}{2\pi i}\oint_{\vert q^2 \vert=s_0}{\rm d}q^2 \, \{P(q^2)-P(s_0)\} \, \Pi_{y\bar{\chi}\psi}(q^2)
\nonumber \\
&= y^2(s_0) \frac{N}{8\pi^2} \, s_0^2 \, ( \, 0.0444 + 0.378 \, a + 3.82 \, a^2 + 35.0 \, a^3 + 332 a^4 \, )
\nonumber \\
&+0.0162 \, \frac{y^2(s_0)}{8\pi} \frac{N}{3} \, r^4\cdot6.8\times10^{-2} \, {\rm GeV}^4
+0.592 \, \frac{y^2(s_0)}{s_0}\frac{22\pi}{3}\frac{N^2-1}{N^2}\frac{N}{3} \, r^6\cdot4.5\times10^{-4} \, {\rm GeV}^6;
\nonumber \\
&a = \frac{N}{3}\frac{\alpha_T(s_0)}{\pi} = \frac{\alpha_T(s_0)}{\pi} = 0.0810.
\label{3r3} \end{aligned}$$ $$\begin{aligned}
&{\rm For \ } N=3 {\rm \ and \ } s_0=r^2\cdot5^2~{\rm GeV}^2,
\nonumber \\
&-\frac{1}{2\pi i}\oint_{\vert q^2 \vert=s_0}{\rm d}q^2 \, \{P(q^2)-P(s_0)\} \, \Pi_{y\bar{\chi}\psi}(q^2)
\nonumber \\
&= y^2(s_0) \frac{N}{8\pi^2} \, s_0^2 \, ( \, 0.00576 + 0.0608 \, a + 0.779 \, a^2 + 9.54 \, a^3 + 95.7 \, a^4 \, )
\nonumber \\
&+0.0162 \, \frac{y^2(s_0)}{8\pi} \frac{N}{3} \, r^4\cdot6.8\times10^{-2} \, {\rm GeV}^4
+1.64 \, \frac{y^2(s_0)}{s_0}\frac{22\pi}{3}\frac{N^2-1}{N^2}\frac{N}{3} \, r^6\cdot4.5\times10^{-4} \, {\rm GeV}^6;
\nonumber \\
&a = \frac{N}{3}\frac{\alpha_T(s_0)}{\pi} = \frac{\alpha_T(s_0)}{\pi} = 0.0666.
\label{3r5}\end{aligned}$$ $$\begin{aligned}
&{\rm For \ } N=6 {\rm \ and \ } s_0=r^2\cdot3^2~{\rm GeV}^2,
\nonumber \\
&-\frac{1}{2\pi i}\oint_{\vert q^2 \vert=s_0}{\rm d}q^2 \, \{P(q^2)-P(s_0)\} \, \Pi_{y\bar{\chi}\psi}(q^2)
\nonumber \\
&= y^2(s_0) \frac{N}{8\pi^2} \, s_0^2 \, ( \, 0.0444 + 0.414 \, a + 4.56 \, a^2 + 47.2 \, a^3 + 467 \, a^4 \, )
\nonumber \\
&+0.0162 \, \frac{y^2(s_0)}{8\pi} \frac{N}{3} \, r^4\cdot6.8\times10^{-2} \, {\rm GeV}^4
+0.592 \, \frac{y^2(s_0)}{s_0}\frac{22\pi}{3}\frac{N^2-1}{N^2}\frac{N}{3} \, r^6\cdot4.5\times10^{-4} \, {\rm GeV}^6;
\nonumber \\
&a = \frac{N}{3}\frac{\alpha_T(s_0)}{\pi} = 2 \, \frac{\alpha_T(s_0)}{\pi} = 0.0784.
\label{6r3}\end{aligned}$$ $$\begin{aligned}
&{\rm For \ } N=6 {\rm \ and \ } s_0=r^2\cdot5^2~{\rm GeV}^2,
\nonumber \\
&-\frac{1}{2\pi i}\oint_{\vert q^2 \vert=s_0}{\rm d}q^2 \, \{P(q^2)-P(s_0)\} \, \Pi_{y\bar{\chi}\psi}(q^2)
\nonumber \\
&= y^2(s_0) \frac{N}{8\pi^2} \, s_0^2 \, ( \, 0.00576 + 0.0665 \, a + 0.931 \, a^2 + 12.7 \, a^3 + 139 \, a^4 \, )
\nonumber \\
&+0.0162 \, \frac{y^2(s_0)}{8\pi}\frac{N}{3} \, r^4\cdot6.8\times10^{-2} \, {\rm GeV}^4
+1.64 \, \frac{y^2(s_0)}{s_0}\frac{22\pi}{3}\frac{N^2-1}{N^2}\frac{N}{3} \, r^6\cdot4.5\times10^{-4} \, {\rm GeV}^6;
\nonumber \\
&a = \frac{N}{3}\frac{\alpha_T(s_0)}{\pi} = 2 \, \frac{\alpha_T(s_0)}{\pi} = 0.0634.
\label{6r5}\end{aligned}$$ It is observed that the perturbative series converges at a rate about $0.7^k$ for all cases above. We also find that for all $N$, the field-strength condensate $\langle 0 \vert \alpha_T X_{\mu\nu}^\alpha X^{\alpha\mu\nu} \vert 0 \rangle$ contributes to the right-hand side of Eq. (\[sumrule\]) by less than 0.03% for $s_0=r^2\cdot3^2~{\rm GeV}^2$ and by less than 0.004% for $s_0=r^2\cdot5^2~{\rm GeV}^2$, while the four-fermion condensate contributes by less than 1% for $s_0=r^2\cdot3^2~{\rm GeV}^2$ and by less than 0.05% for $s_0=r^2\cdot5^2~{\rm GeV}^2$. Since vacuum condensates have minor or negligible impact on the evaluation of $(\hat{y} F_\Theta M_\Theta)^2$, we ignore all vacuum condensates except the two displayed in Eq. (\[sumrule\]), and further discard $O(\alpha_T)$ corrections to vacuum condensates.
We plot in Figure \[s0dep\] the values of $(\hat{y} F_\Theta M_\Theta)^2$ derived from Eq. (\[sumrule\]) in the range $r^2\cdot3^2~{\rm GeV}^2 \leq s_0 \leq r^2\cdot5^2~{\rm GeV}^2$ for $N=3,4,5,6$. We normalize $(\hat{y} F_\Theta M_\Theta)^2$ by $N/3$, as it is roughly linearly proportional to $N$. It is further normalized by $y_{r2}$, the value of the coupling constant $y$ at $\mu=2$ GeV scale in the $\overline{MS}$ scheme.
![ $(\hat{y} F_\Theta M_\Theta)^2$ as derived from Eq. (\[sumrule\]) with various values of $s_0$. The upper, middle-upper, middle-lower and lower lines respectively correspond to the cases with $N=3,4,5,6$. The value of $(\hat{y} F_\Theta M_\Theta)^2$ is normalized by $N/3$ and $y_{r2}$, which is the value of the coupling constant $y$ at $\mu=$2 GeV scale in the $\overline{MS}$ scheme. []{data-label="s0dep"}](s0dep.eps){width="100mm"}
The decrease of $(\hat{y} F_\Theta M_\Theta)^2$ with $s_0$ is totally ascribed to the truncation of perturbative series at order $\alpha_T^4$, because the integral on the left-hand side of Eq. (\[sumrule\]), *i.e.*, the integral of the Breit-Wigner function times the function $P(s)-P(s_0)$, is virtually constant for $s_0 \geq r^2\cdot3^2~{\rm GeV}^2$ due to exponential suppression in $P(s)$. Since the variation of $(\hat{y} F_\Theta M_\Theta)^2$ with $s_0$, namely, steepness of the curves in Figure \[s0dep\], falls off as $s_0$ approaches to $r^2\cdot5^2~{\rm GeV}^2$, we consider that the value corresponding to $s_0 = r^2\cdot5^2~{\rm GeV}^2$ is most close to the true physical value. Hence, we conclude with the following estimate for $(\hat{y} F_\Theta M_\Theta)^2$ derived by setting $s_0 = r^2\cdot5^2~{\rm GeV}^2$ in Eq. (\[sumrule\]): $$\begin{aligned}
(\hat{y} F_\Theta M_\Theta)^2 &= \frac{N}{3} \, y_{r2}^2 \, r^4 \cdot 0.437~{\rm GeV}^4 \ \ \ \ \ {\rm for} \ N=3,
\nonumber \\
(\hat{y} F_\Theta M_\Theta)^2 &= \frac{N}{3} \, y_{r2}^2 \, r^4 \cdot 0.403~{\rm GeV}^4 \ \ \ \ \ {\rm for} \ N=4,
\nonumber \\
(\hat{y} F_\Theta M_\Theta)^2 &= \frac{N}{3} \, y_{r2}^2 \, r^4 \cdot 0.381~{\rm GeV}^4 \ \ \ \ \ {\rm for} \ N=5,
\nonumber \\
(\hat{y} F_\Theta M_\Theta)^2 &= \frac{N}{3} \, y_{r2}^2 \, r^4 \cdot 0.367~{\rm GeV}^4 \ \ \ \ \ {\rm for} \ N=6.
\label{ftheta}\end{aligned}$$ The uncertainty of $(\hat{y} F_\Theta M_\Theta)^2$ due to the truncation of perturbative series is estimated to be $0.7^4\simeq25$%, as the perturbative series in Eqs. (\[3r3\]), (\[3r5\]), (\[6r3\]), (\[6r5\]) converge at a rate about 0.7$^k$.\
### Ratio of the dynamical scale of the $SU(N)_T$ gauge theory and that of QCD
From $M_\Theta$ and $\hat{y}F_\Theta$ evaluated in Eqs. (\[mtheta\]), (\[ftheta\]) and the matching condition Eq. (\[matching1\]), we arrive at the following values of $r$: $$\begin{aligned}
r &= \sqrt{\frac{3}{N}} \, \frac{1}{y_{r2}} \, 191 \ \ \ \ \ {\rm for \ }N=3,
\ \ \ \ \ \ \
r = \sqrt{\frac{3}{N}} \, \frac{1}{y_{r2}} \, 199 \ \ \ \ \ {\rm for \ }N=4,
\nonumber \\
r &= \sqrt{\frac{3}{N}} \, \frac{1}{y_{r2}} \, 204 \ \ \ \ \ {\rm for \ }N=5,
\ \ \ \ \ \ \
r = \sqrt{\frac{3}{N}} \, \frac{1}{y_{r2}} \, 208 \ \ \ \ \ {\rm for \ }N=6,
\label{ratio}\end{aligned}$$ with $y_{r2}$ denoting the $\bar{\chi}\psi H$ coupling constant $y$ at the scale $\mu=r\cdot2$ GeV in the $\overline{MS}$ scheme. Here, the experimental value [@pdg] $m_h=125.09$ GeV is used. As a reference, we have $\Lambda_T \simeq $4.4 TeV for $y_{r2}=10^{-2}$ and $N=3$, with $\Lambda_{QCD}=0.23$ GeV.
Given the relation Eq. (\[ratio\]), the only model parameters are $y_{r2}$ and $N$. Some may wonder that because $y_{r2}$ corresponds to the value of $y$ at the scale $r\cdot2$ GeV, which by itself contains $r$, Eq. (\[ratio\]) might contain self-inconsistency. As a matter of fact, the value of $y$ at one particular scale is involved in Eq. (\[ratio\]), and only after $r$ is determined from Eq. (\[ratio\]), can one calculate $y$ at different scales using the RG equation with the initial condition $y(\mu=r\cdot2~{\rm GeV})=y_{r2}$.\
Phenomenology of pseudo-Nambu-Goldstone bosons
==============================================
We are concerned with phenomenology of the pNG bosons that arise due to chiral symmetry breaking in the $SU(N)_T$ gauge theory. They are associated with the spontaneous breaking of the axial $SU(3)_A\times U(1)_A$ symmetry, under which $\chi$ and $\psi$ transform as $$\begin{aligned}
\left(
\begin{array}{c}
\chi \\
\psi
\end{array}
\right)
&\to \exp\left( i\theta_A^j \frac{\lambda^j}{2} \gamma_5 + i\theta_A \gamma_5 \right) \left(
\begin{array}{c}
\chi \\
\psi
\end{array}
\right),\end{aligned}$$ where $\lambda^i$ $(i=1,2,...,8)$ are the Gell-Mann matrices, and $\theta_A^i$ $(i=1,2,...,8)$ and $\theta_A$ represent parameters of $SU(3)_A$ and $U(1)_A$ transformations. Since the $U(1)_A$ symmetry is anomalous in the $SU(N)_T$ gauge theory, the corresponding pNG boson gains mass from instantons at the scale $4\pi \Lambda_T$, *i.e.*, the dynamical scale multiplied by $4\pi$. On the other hand, the $SU(3)_A$ symmetry is explicitly broken by the SM weak and hypercharge gauge interactions and the $\bar{\chi}\psi H$ Yukawa interaction. Thus, the corresponding 8 pNG bosons are rendered massive by radiative corrections involving electroweak gauge bosons and elementary scalar $H$ as well as by the VEV of $H$, whose masses are at the scale $g_W \Lambda_T$ or $y \Lambda_T$. Note that these masses are generated by the same mechanism as the charged pion-neutral pion mass difference, which stems from the electromagnetic interaction of quarks. We eventually find that order $4\pi/g_W \sim 20$ mass hierarchy exists between the pNG boson of the $U(1)_A$ symmetry and those of the $SU(3)_A$ symmetry. In the ensuing study of phenomenology, therefore, we ignore the former pNG boson and its mixing with the latter.\
Mass spectrum of pseudo-Nambu-Goldstone bosons
----------------------------------------------
The mass matrix for the 8 pNG bosons of the $SU(3)_A$ symmetry is calculated with the Dashen’s formula [@dashen] in the leading order of the electroweak gauge couplings and the $\bar{\chi}\psi H$ Yukawa coupling as $$\begin{aligned}
M^2_{ij} &= \frac{1}{f_\Pi^2} \langle 0 \vert [ Q_A^i, [Q_A^j, {\cal H}_{{\rm break}}]] \vert 0 \rangle \ \ \ (i,j=1,2,...,8),
\label{massmatrix}\end{aligned}$$ where $i,j$ label Gell-Mann matrices to which the pNG bosons correspond, $Q_A^i$ denotes the charge for a $SU(3)_A$ symmetry current $(\bar{\chi},\bar{\psi}) \gamma^\mu \gamma_5 \dfrac{\lambda^i}{2} \left(
\begin{array}{c}
\chi \\
\psi
\end{array}
\right)$, and $f_\Pi$ denotes the NG boson decay constant, which is approximated to be common for the 8 pNG bosons. ${\cal H}_{{\rm break}}$ is the effective Hamiltonian density that explicitly breaks the $SU(3)_A$ symmetry, which comprises two parts; one is obtained from the Lagrangian Eq.(\[lagrangian1\]) by contracting $W$, $Z$, photon and scalar fields with free field propagators; the other comes from the electroweak symmetry breaking VEV of $H$, $c_H v$. In the calculation of ${\cal H}_{{\rm break}}$, we impose the unitary gauge for $W$ and $Z$ fields. Also, we make the approximation that $c_H =1,s_H=0$. Then only the physical Higgs field $h$ and $W,Z$ and photon fields contribute to ${\cal H}_{{\rm break}}$, and it is expressed as $$\begin{aligned}
&{\cal H}_{{\rm break}} = y \, \frac{v}{\sqrt{2}} \, (\bar{\chi}_2\psi + \bar{\psi}\chi_2)
\nonumber \\
&-\frac{i}{2} \, \frac{g_W^2}{2} \int{\rm d}^4x \, D^W_{\mu\nu}(x) \, \{ \bar{\chi}_1(x)\gamma^{\mu}\chi_2(x) \ \bar{\chi}_2(0)\gamma^{\nu}\chi_1(0) + \bar{\chi}_2(x)\gamma^{\mu}\chi_1(x) \ \bar{\chi}_1(0)\gamma^{\nu}\chi_2(0) \}
\nonumber \\
&- \frac{i}{2} \frac{g_Z^2}{4} \int{\rm d}^4x D^Z_{\mu\nu}(x)
\{ (c_W^2-bs_W^2)\bar{\chi}_1(x)\gamma^{\mu}\chi_1(x) - (c_W^2+bs_W^2)\bar{\chi}_2(x)\gamma^{\mu}\chi_2(x) + (1-b)s_W^2\bar{\psi}(x)\gamma^{\mu}\psi(x)\}
\nonumber \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times \{ (c_W^2-bs_W^2) \, \bar{\chi}_1(0)\gamma^{\nu}\chi_1(0) - (c_W^2+bs_W^2)\, \bar{\chi}_2(0)\gamma^{\nu}\chi_2(0) + (1-b)s_W^2 \, \bar{\psi}(0)\gamma^{\nu}\psi(0)\}
\nonumber \\
&- \frac{i}{2} \, \frac{e^2}{4} \int{\rm d}^4x \, D^\gamma_{\mu\nu}(x) \,
\{ (1+b) \, \bar{\chi}_1(x)\gamma^{\mu}\chi_1(x) - (1-b) \, \bar{\chi}_2(x)\gamma^{\mu}\chi_2(x) - (1-b) \, \bar{\psi}(x)\gamma^{\mu}\psi(x)\}
\nonumber \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times \{ (1+b) \, \bar{\chi}_1(0)\gamma^{\nu}\chi_1(0) - (1-b) \, \bar{\chi}_2(0)\gamma^{\nu}\chi_2(0) - (1-b) \, \bar{\psi}(0)\gamma^{\nu}\psi(0)\}
\nonumber \\
&- \frac{i}{2} \, \frac{y^2}{2} \int{\rm d}^4x \, D^h(x) \, \{ \bar{\chi}_2(x)\psi(x) + \bar{\psi}(x)\chi_2(x) \}\{ \bar{\chi}_2(0)\psi(0) + \bar{\psi}(0)\chi_2(0) \}
\label{effh}\end{aligned}$$ where $D^{W}_{\mu\nu}(x)$, $D^{Z}_{\mu\nu}(x)$, $D^\gamma_{\mu\nu}(x)$ and $D^h(x)$ denote the free field propagators for $W$, $Z$, photon and the physical Higgs field $h$, respectively, and in particular, the unitary gauge is chosen for $D^{W}_{\mu\nu}(x)$ and $D^{Z}_{\mu\nu}(x)$. $c_W,s_W$ respectively denote the cosine and sine of the Weinberg angle. By substituting ${\cal H}_{{\rm break}}$ into the Dashen’s formula Eq. (\[massmatrix\]), the non-vanishing components of the mass matrix are computed as $$\begin{aligned}
M^2_{11} = M^2_{22} &= \frac{1}{f_\Pi^2} \left( \, g_W^2 \, {\cal C}^W + c_W^4g_Z^2 \, {\cal C}^Z + e^2 \, {\cal C}^\gamma + \frac{1}{4} \, {\cal F}_{12} \, \right),
\nonumber \\
M^2_{33} &= \frac{1}{f_\Pi^2} \left( \, 2g_W^2 \, {\cal C}^W + \frac{1}{4} \, {\cal F}_{38} \, \right),
\nonumber \\
M^2_{44} = M^2_{55} &= \frac{1}{f_\Pi^2} \left( \, \frac{1}{2}g_W^2 \, {\cal C}^W + \frac{(c_W^2-s_W^2)^2}{4}g_Z^2 \, {\cal C}^Z + e^2 \, {\cal C}^\gamma + \frac{1}{4} \, {\cal F}_{45} \, \right),
\nonumber \\
M^2_{66} = M^2_{77} &= \frac{1}{f_\Pi^2} \left( \, \frac{1}{2}g_W^2 \, {\cal C}^W + \frac{1}{4}g_Z^2 \, {\cal C}^Z + {\cal F}_{67} \, \right),
\nonumber \\
M^2_{88} &= \frac{1}{f_\Pi^2} \, \frac{1}{12} \, {\cal F}_{38},
\nonumber \\
M^2_{38} &= \frac{1}{f_\Pi^2} \, \frac{1}{4\sqrt{3}} \, {\cal F}_{38},
\nonumber \\
M^2_{14} = M^2_{25} = -M^2_{36} = \sqrt{3}M^2_{68} &= -\frac{1}{f_\Pi^2} \, \frac{v}{\sqrt{2}} \, \langle 0 \vert y \, \bar{f}f \vert 0 \rangle.
\label{pngmass}\end{aligned}$$ Here, ${\cal C}^W,{\cal C}^Z,{\cal C}^\gamma$ are quantities defined with one massless Dirac fermion $f$ in the fundamental representation of the $SU(N)_T$ gauge group as $$\begin{aligned}
{\cal C}^k &\equiv \frac{i}{2}\int{\rm d}^4x \, D^k_{\mu\nu}(x) \, \left[ \, \langle 0 \vert T \{ \bar{f}(x) \gamma^\mu f(x) \, \bar{f}(0) \gamma^\nu f(0) \} \vert 0 \rangle
- \langle 0 \vert T \{ \bar{f}(x) \gamma^\mu \gamma_5 f(x) \, \bar{f}(0) \gamma^\nu \gamma_5 f(0) \} \vert 0 \rangle \, \right]
\nonumber \\
&(k=W,Z,\gamma).
\label{cdef}\end{aligned}$$ ${\cal F}_{12}$, ${\cal F}_{38}$, ${\cal F}_{45}$, ${\cal F}_{67}$ are defined as $$\begin{aligned}
{\cal F}_{12} &\equiv i\int{\rm d}^4x \, D^h(x) \, \left[ \, \langle 0 \vert T \{ y \, \bar{\chi}_2(x)\psi(x) \ y \, \bar{\psi}(0) \chi_2(0) \} \vert 0 \rangle - \langle 0 \vert T \{ y \, \bar{\chi}_1(x) i\gamma_5 \psi(x) \ y \, \bar{\psi}(0) i\gamma_5 \chi_1(0) \} \vert 0 \rangle \, \right],
\label{ddef12}
\\
{\cal F}_{38} &\equiv i\int{\rm d}^4x \, D^h(x) \, \left[ \, \langle 0 \vert T \{ y \, \bar{\chi}_2(x)\psi(x) \ y \, \bar{\psi}(0) \chi_2(0) \} \vert 0 \rangle - \langle 0 \vert T \{ y \, \bar{\chi}_2(x) i\gamma_5 \psi(x) \ y \, \bar{\psi}(0) i\gamma_5 \chi_2(0) \} \vert 0 \rangle \, \right],
\label{ddef3}
\\
{\cal F}_{45} &\equiv i\int{\rm d}^4x \, D^h(x) \, \left[ \, \langle 0 \vert T \{ y \, \bar{\chi}_2(x)\psi(x) \ y \, \bar{\psi}(0) \chi_2(0) \} \vert 0 \rangle - \langle 0 \vert T \{ y \, \bar{\chi}_2(x) i\gamma_5 \chi_1(x) \ y \, \bar{\chi}_1(0) i\gamma_5 \chi_2(0) \} \vert 0 \rangle \, \right],
\label{ddef45}
\\
{\cal F}_{67} &\equiv i\int{\rm d}^4x \, D^h(x) \, \left[ \, \langle 0 \vert T \{ y \, \bar{\chi}_2(x) \psi(x) \ y \, \bar{\psi}(0) \chi_2(0) \} \vert 0 \rangle \right.
\nonumber \\
&- \left. \frac{1}{2}\langle 0 \vert T \{ y \, \bar{\chi}_2(x) i\gamma_5 \chi_2(x) \ y \, \bar{\chi}_2(0) i\gamma_5 \chi_2(0) \} \vert 0 \rangle
- \frac{1}{2}\langle 0 \vert T \{ y \, \bar{\psi}(x) i\gamma_5 \psi(x) \ y \, \bar{\psi}(0) i\gamma_5 \psi(0) \} \vert 0 \rangle \, \right].
\label{ddef67}\end{aligned}$$ Finally, in the last line of Eq. (\[pngmass\]), $\langle 0 \vert y \, \bar{f}f \vert 0 \rangle$ collectively denotes the fermion bilinear condensate with the coupling constant $y$, $\langle 0 \vert y \, \bar{\chi}_1\chi_1 \vert 0 \rangle=\langle 0 \vert y \, \bar{\chi}_2\chi_2 \vert 0 \rangle=\langle 0 \vert y \, \bar{\psi}\psi \vert 0 \rangle$.
In the rest of the subsection, we evaluate the pNG boson masses Eq. (\[pngmass\]) by analogy with QCD. This is done by equating ${\cal C}^W,{\cal C}^Z,{\cal C}^\gamma,{\cal F}_1,{\cal F}_2$ and the pNG boson decay constant $f_\Pi$ with certain quantities in QCD, rescaled by the dynamical scale ratio $r$ as well as $N/N_c$. Then $r$ is rendered into the coupling constant $y_{r2}$ and $N$ through Eq. (\[ratio\]), by which the pNG boson masses are expressed solely in terms of $y_{r2}$ and $N$.\
### Evaluation of $f_\Pi$
$f_\Pi$ is evaluated from the pion decay constant in the chiral-limit QCD, $f_\pi^{{\rm chiral}}$, obtained in Ref. [@durr] by fitting a lattice simulation with the next-to-leading order chiral perturbation theory [@leutwyler]. It is given by $$\begin{aligned}
f_\pi^{{\rm chiral}} &= 0.08678~{\rm GeV}.
\label{lattice}\end{aligned}$$ Reminding that the NG boson decay constant scales by $\sqrt{N/N_c}$ as well as $r$, we obtain the following estimate: $$\begin{aligned}
f_\Pi &= \sqrt{\frac{N}{N_c}} \, r \, f_\pi^{{\rm chiral}} = \sqrt{\frac{N}{N_c}} \, r \cdot 0.08676~{\rm GeV}.\end{aligned}$$\
### Evaluation of ${\cal C}^W,{\cal C}^Z,{\cal C}^\gamma$
${\cal C}^W,{\cal C}^Z,{\cal C}^\gamma$ in Eq. (\[cdef\]) can be estimated from the mass difference between the charged and neutral pions in QCD. Recall that in QCD, this mass difference stems mainly from the electromagnetic interaction. Substituting the effective Hamiltonian density for the electromagnetic interaction of quarks into the Dashen’s formula, we compute the pion mass difference $m_{\pi^\pm}^2-m_{\pi^0}^2$ to be $$\begin{aligned}
&m_{\pi^\pm}^2 - m_{\pi^0}^2
\nonumber \\
&= \frac{e^2}{f_\pi^2} \frac{i}{2}\int{\rm d}^4x \, D_{\mu\nu}^\gamma(x) \left[ \, \langle 0 \vert T \{ \bar{q}(x) \gamma^\mu q(x) \, \bar{q}(0) \gamma^\nu q(0) \} \vert 0 \rangle
- \langle 0 \vert T \{ \bar{q}(x) \gamma^\mu \gamma_5 q(x) \, \bar{q}(0) \gamma^\nu \gamma_5 q(0) \} \vert 0 \rangle \, \right],
\label{pionmassdifference}\end{aligned}$$ where $q$ denotes up and down quarks, and $f_\pi$ denotes the pion decay constant in real QCD. Analogy between QCD and the $SU(N)_T$ gauge theory gives that the integral on the right hand side of Eq. (\[pionmassdifference\]) is $(N_c/N){\cal C}^\gamma/r^4$, where the factor $N/N_c$ enters because the correlation function scales by $N/N_c$. We thus find the following relation between ${\cal C}^\gamma$ and the pion mass difference: $$\begin{aligned}
{\cal C}^\gamma &= r^4 \, \frac{N}{N_c} \frac{1}{e^2} f_\pi^2(m_{\pi^\pm}^2 - m_{\pi^0}^2).
\label{cestimate}\end{aligned}$$ Using experimental central values [@pdg] $e^2=4\pi/137.0$, $m_{\pi^\pm}=0.13957018$ GeV, $m_{\pi^0}=0.1349766$ GeV and $f_\pi=0.921$ GeV, we obtain $$\begin{aligned}
{\cal C}^\gamma &= \frac{N}{N_c} \, r^4 \, (0.104~{\rm GeV})^4. \label{cnumestimate}\end{aligned}$$
Regarding ${\cal C}^W$ and ${\cal C}^Z$, notice that they are independent of the gauge choice for $W$ and $Z$ fields, because the correlation function for the axial current $\bar{f} \gamma^\mu \gamma_5 f$, as with the vector current, is proportional to $g_{\mu\nu}p^2 - p_\mu p_\nu$ ($p^\mu$ denotes the momentum of the correlation function) in the zeroth order of the electroweak gauge couplings and $\bar{\chi}\psi H$ Yukawa coupling due to the absence of $\chi,\psi$ current mass. Also, in the limit with $y_{r2} \ll 1$, the dynamical scale of the $SU(N)_T$ gauge theory is much larger than the electroweak scale and hence the $W$ and $Z$ boson masses can be ignored in the calculation of ${\cal C}^W$ and ${\cal C}^Z$. Therefore, ${\cal C}^W$, ${\cal C}^Z$ and ${\cal C}^\gamma$ can be calculated with the identical free field propagator for the gauge field and we find $$\begin{aligned}
{\cal C}^W &= {\cal C}^Z = {\cal C}^\gamma.\end{aligned}$$\
### Evaluation of ${\cal F}_{12},{\cal F}_{38},{\cal F}_{45},{\cal F}_{67}$
Since ${\cal F}_{12}$, ${\cal F}_{38}$, ${\cal F}_{45}$ and ${\cal F}_{67}$ have no corresponding quantities in QCD, we calculate them explicitly. We commence the calculation from ${\cal F}_{38}$. It can be recast in the following form: $$\begin{aligned}
{\cal F}_{38} &= \frac{i}{(2\pi)^4}\int{\rm d}^4p \, \frac{1}{p^2-m_h^2}\left\{ \, \Pi_{y\bar{\chi}_2\psi}(p^2) - \Pi_{y\bar{\chi}_2i\gamma_5\psi}(p^2) \, \right\}
\nonumber \\
&= \frac{1}{16\pi^2}\int_0^\infty{\rm d}(p_E^2) \, \frac{p_E^2}{p_E^2+m_h^2}\left\{ \, \Pi_{y\bar{\chi}_2\psi}(-p_E^2) - \Pi_{y\bar{\chi}_2i\gamma_5\psi}(-p_E^2) \, \right\},
\label{f3calc}\end{aligned}$$ where $p_E$ is the Euclidean momentum with $p^0=ip_E^0, p^{1,2,3}=p_E^{1,2,3}$, and the scalar and pseudoscalar current correlation functions $\Pi_{y\bar{\chi}_2\psi}$ and $\Pi_{y\bar{\chi}_2i\gamma_5\psi}$ are given by $$\begin{aligned}
\Pi_{y\bar{\chi}_2\psi}(p^2) &= i\int{\rm d}^4x \, e^{ipx} \langle 0 \vert T\left\{ y \, \bar{\chi}_2(x)\psi(x) \ y \, \bar{\psi}(0)\chi_2(0) \right\} \vert 0 \rangle,
\nonumber \\
\Pi_{y\bar{\chi}_2i\gamma_5\psi}(p^2) &= i\int{\rm d}^4x \, e^{ipx} \langle 0 \vert T\left\{ y \, \bar{\chi}_2(x)i\gamma_5\psi(x) \ y \, \bar{\psi}(0)i\gamma_5\chi_2(0) \right\} \vert 0 \rangle.\end{aligned}$$ ${\cal F}_{38}$ is thus calculated from the difference between the scalar and pseudoscalar correlation functions for space-like momenta $p^2\leq0$. The correlation functions for space-like momenta are connected to their imaginary parts for time-like momenta by the dispersion relation. As the operator product expansion [@qcdsumrules] yields $\Pi_{y\bar{\chi}_2\psi}(-p_E^2) - \Pi_{y\bar{\chi}_2i\gamma_5\psi}(-p_E^2) = O(1/p_E^4)$ for $p_E^2 \to \infty$ (recall that fermion bilinear condensate does not enter into the correlation function due to the absence of $\chi,\psi$ current mass), no subtraction term is needed for the correlation function difference and we find $$\begin{aligned}
\Pi_{y\bar{\chi}_2\psi}(-p_E^2) - \Pi_{y\bar{\chi}_2i\gamma_5\psi}(-p_E^2) &= \frac{1}{\pi}\int_0^\infty{\rm d}s \, \frac{{\rm Im}\Pi_{y\bar{\chi}_2\psi}(s) - {\rm Im}\Pi_{y\bar{\chi}_2i\gamma_5\psi}(s)}{s+p_E^2}.
\label{dispersion2}\end{aligned}$$
The imaginary part of the correlation function difference, ${\rm Im}\Pi_{y\bar{\chi}_2\psi}(s)-{\rm Im}\Pi_{y\bar{\chi}_2i\gamma_5\psi}(s)$, corresponds to the difference between the spectral densities of scalar and pseudoscalar bound states. We infer the form of the spectral density difference from the fact that chiral symmetry is restored for large momenta $s \to \infty$. This implies that the scalar and pseudoscalar spectral densities coincide for heavy bound states and their difference is described with several light bound states. We thus assume that the spectral density difference is approximated by contributions from the scalar meson $\Theta$, the pNG boson $\frac{1}{\sqrt{2}}(\Pi^6 \pm i\Pi^7)$ and one heavier pseudo-scalar meson, which we denote by $\Pi'$. [^4] It is further assumed that $\Theta$ and $\Pi'$ meson resonances are described by the relativistic Breit-Wigner function, whereas the pNG boson contribution is represented by a delta function at $s=0$ because it is massless in the zeroth order of the electroweak gauge couplings and the Yukawa coupling $y$. Given these approximations, we arrive at the following spectral density difference: $$\begin{aligned}
&\frac{1}{\pi}\left\{ {\rm Im}\Pi_{y\bar{\chi}_2\psi}(s)-{\rm Im}\Pi_{y\bar{\chi}_2i\gamma_5\psi}(s) \right\}
\nonumber \\
&=
(\hat{y} F_\Theta M_\Theta)^2 \, \frac{1}{\pi}\frac{M_\Theta \Gamma_\Theta}{(s-M_\Theta^2)^2+M_\Theta^2 \Gamma_\Theta^2}
- \hat{y}^2 G_{\Pi}^2 \, \delta(s) - \hat{y}^2 G_{\Pi'}^2 \, \frac{1}{\pi}\frac{M_{\Pi'} \Gamma_{\Pi'}}{(s-M_{\Pi'}^2)^2+M_{\Pi'}^2 \Gamma_{\Pi'}^2}
\label{spsspectral}\end{aligned}$$ where $M_{\Pi'}$ and $\Gamma_{\Pi'}$ respectively denote the mass and width of $\Pi'$ meson, and $\hat{y}G_{\Pi}$ and $\hat{y}G_{\Pi'}$ are defined as $$\begin{aligned}
\langle 0 \vert y \, \bar{\chi}_2(x) i\gamma_5 \psi(x) \vert \, \frac{1}{\sqrt{2}}(\Pi^6+i\Pi^7)(p) \, \rangle &\equiv \hat{y} G_{\Pi} \, e^{-ipx},
\label{gpipre} \\
\langle 0 \vert y \, \bar{\chi}_2(x) i\gamma_5 \psi(x) \vert \Pi'(p) \rangle &\equiv \hat{y} G_{\Pi'} \, e^{-ipx},\end{aligned}$$ with $\hat{y}$ denoting the renormalization group invariant quantity made from the coupling constant $y$.
Substituting the spectral density difference Eq. (\[spsspectral\]) into the dispersion relation Eq. (\[dispersion2\]), we compute the correlation function difference to be $$\begin{aligned}
&\Pi_{y\bar{\chi}_2\psi}(-p_E^2) - \Pi_{y\bar{\chi}_2i\gamma_5\psi}(-p_E^2)
\nonumber \\
&=
(\hat{y} F_\Theta M_\Theta)^2 \, \int_0^\infty {\rm d}s \, \frac{1}{s+p_E^2}\frac{1}{\pi}\frac{M_\Theta \Gamma_\Theta}{(s-M_\Theta^2)^2+M_\Theta^2 \Gamma_\Theta^2} - \hat{y}^2 G_{\Pi}^2 \frac{1}{p_E^2}
\nonumber \\
&- \hat{y}^2 G_{\Pi'}^2 \int_0^\infty {\rm d}s \, \frac{1}{s+p_E^2}\frac{1}{\pi}\frac{M_{\Pi'} \Gamma_{\Pi'}}{(s-M_{\Pi'}^2)^2+M_{\Pi'}^2 \Gamma_{\Pi'}^2}.
\label{spscorrelator}\end{aligned}$$ ${\cal F}_{38}$ is calculated from the above correlation function difference through Eq. (\[f3calc\]). Before that, we remind that in the limit with $p_E^2\to \infty$, the correlation function difference Eq. (\[spscorrelator\]) asymptotes to the VEV of the operator product expansion calculated in the $SU(N)_T$ gauge theory, as the expansion is reliable in this limit. The calculation in Ref. [@qcdsumrules] tells us that $O(1)$ and $O(1/p_E^2)$ terms in $\Pi_{y\bar{\chi}_1\psi}(-p_E^2) - \Pi_{y\bar{\chi}_1i\gamma_5\psi}(-p_E^2)$ vanish for $p_E^2\to \infty$ (note that fermion bilinear condensate does not appear in our case), yielding the following constraint: $$\begin{aligned}
&\frac{1}{\pi}\int_0^\infty{\rm d}s \, \left\{ \, (\hat{y} F_\Theta M_\Theta)^2 \frac{M_\Theta \Gamma_\Theta}{(s-M_\Theta^2)^2+M_\Theta^2 \Gamma_\Theta^2} - \hat{y}^2 G_{\Pi'}^2 \frac{M_{\Pi'} \Gamma_{\Pi'}}{(s-M_{\Pi'}^2)^2+M_{\Pi'}^2 \Gamma_{\Pi'}^2} \, \right\} - \hat{y}^2 G_{\Pi}^2 = 0.
\label{gpipconstraint}\end{aligned}$$ By virtue of the above constraint, the integral over $p_E^2$ in Eq. (\[f3calc\]) is convergent and we obtain $$\begin{aligned}
{\cal F}_{38} &= -\frac{1}{16\pi^2}\int_0^\infty{\rm d}s \, \frac{s}{s-m_h^2}\log\left(\frac{s}{m_h^2}\right) \frac{1}{\pi}
\nonumber \\
&\times \left\{ \,
(\hat{y} F_\Theta M_\Theta)^2 \frac{M_\Theta \Gamma_\Theta}{(s-M_\Theta^2)^2+M_\Theta^2 \Gamma_\Theta^2} - \hat{y}^2 G_{\Pi'}^2 \frac{M_{\Pi'} \Gamma_{\Pi'}}{(s-M_{\Pi'}^2)^2+M_{\Pi'}^2 \Gamma_{\Pi'}^2} \, \right\}.
\label{f1}\end{aligned}$$ We evaluate ${\cal F}_{38}$ by making the following two assumptions: First, we assume that $M_{\Pi'}$ is located at the same scale as $M_\Theta$, namely, $M_{\Pi'}\sim M_\Theta$ is assumed. Second, the ratio of the total width over the mass for $\Pi'$ is similar to that for $\Theta$, *i.e.*, the $\Pi'$ resonance is as narrow as the $\Theta$ resonance. Then, since $\Gamma_{\Pi'}/M_{\Pi'} \simeq \Gamma_\Theta/M_\Theta \simeq 0.2$, the integrand has a significant peak in the region with $s \sim M_\Theta \sim M_{\Pi'}$ and the integral in Eq. (\[f1\]) is dominated by contributions from this region. It follows that in the limit with $y_{r2} \ll 1$, we have $M_{\Pi'} \sim M_\Theta \gg m_h$ and $\vert \log(M_\Theta^2/M_{\Pi'}^2) \vert \ll \log(M_\Theta^2/m_h^2)$, and are thus allowed to make the approximations $s/(s-m_h^2) = 1$ and $\log(s/m_h^2) = \log(M_\Theta^2/m_h^2)$ in the integrand. Under these approximations, ${\cal F}_{38}$ is evaluated as $$\begin{aligned}
{\cal F}_{38} &= -\frac{1}{16\pi^2}\log\left(\frac{M_\Theta^2}{m_h^2}\right) \frac{1}{\pi} \int_0^\infty{\rm d}s \, \left\{ \,
(\hat{y} F_\Theta M_\Theta)^2 \frac{M_\Theta \Gamma_\Theta}{(s-M_\Theta^2)^2+M_\Theta^2 \Gamma_\Theta^2} - \hat{y}^2 G_{\Pi'}^2 \frac{M_{\Pi'} \Gamma_{\Pi'}}{(s-M_{\Pi'}^2)^2+M_{\Pi'}^2 \Gamma_{\Pi'}^2} \, \right\}
\label{f3app}\end{aligned}$$ Again making use of the constraint Eq. (\[gpipconstraint\]), we arrive at $$\begin{aligned}
{\cal F}_{38} &= -\frac{1}{16\pi^2}\log\left(\frac{M_\Theta^2}{m_h^2}\right) \, \hat{y}^2 G_{\Pi}^2.
\label{f3app2}\end{aligned}$$
We will express the term $\hat{y}G_\Pi$ as a combination of certain quantities in QCD multiplied by the dynamical scale ratio $r$ and $N$. As a first step, we prove that $\hat{y}G_{\Pi}$ is linked to the fermion bilinear condensate and the NG boson decay constant in the following manner: $$\begin{aligned}
\hat{y} G_{\Pi} &= \langle 0 \vert y \, \bar{\chi}_1(x) i\gamma_5 \psi(x) \vert \, \frac{1}{\sqrt{2}}(\Pi^6+i\Pi^7)(p) \, \rangle \, e^{ipx}
\nonumber \\
&= -\sqrt{2} \, \frac{\langle 0 \vert y \, \bar{f}f \vert 0 \rangle }{f_\Pi},
\label{gpi}\end{aligned}$$ with $f$ representing one massless Dirac fermion in the fundamental representation of the $SU(N)_T$ gauge group. To prove Eq. (\[gpi\]), consider adding an infinitesimal current mass $y \delta v$ to the $\psi$ field as $-\delta{\cal L}=y\delta v \bar{\psi}\psi$, with $\delta v$ being an infinitesimal RG invariant constant. Its contribution to the $\Pi^6,\Pi^7$ mass is computed as $$\begin{aligned}
\delta M_{66}^2 = \delta M_{77}^2 &= -\frac{1}{f_\Pi^2} \langle 0 \vert y\delta v \, \bar{f}f \vert 0 \rangle.
\label{infinitesimalmass}\end{aligned}$$ On the other hand, the NG boson low-energy theorem relates the axial vector current and the one-NG-boson state in the following way: $$\begin{aligned}
\langle 0 \vert \frac{1}{\sqrt{2}}\bar{\chi}_2(x) \gamma_\mu \gamma_5 \psi(x) \vert \, \frac{1}{\sqrt{2}}(\Pi^6+i\Pi^7)(p) \, \rangle &= if_\Pi \, p_\mu \, e^{-ipx}.
\label{low}\end{aligned}$$ Taking total derivative on both sides of Eq. (\[low\]), and utilizing the equation of motion for $\psi$ field operator with the electroweak and $\bar{\chi}\psi H$ Yukawa interactions ignored, which reads $(i\partial_\mu \gamma^\mu - y \, \delta v)\psi=0$, we obtain $$\begin{aligned}
\langle 0 \vert \frac{y \, \delta v}{\sqrt{2}}\bar{\chi}_2(x) i\gamma_5 \psi(x) \vert \, \frac{1}{\sqrt{2}}(\Pi^6+i\Pi^7)(p) \, \rangle &= f_\Pi \, p^2 \, e^{-ipx} = f_\Pi \, \delta M_{66}^2 \, e^{-ipx}.
\label{low2}\end{aligned}$$ Substituting the mass formula Eq. (\[infinitesimalmass\]) into the right-hand side of Eq. (\[low2\]) and then taking a derivative with $\delta v$ on both sides, we find $$\begin{aligned}
\langle 0 \vert \frac{y}{\sqrt{2}}\bar{\chi}_2(x) i\gamma_5 \psi(x) \vert \, \frac{1}{\sqrt{2}}(\Pi^6+i\Pi^7)(p) \, \rangle &= -\frac{1}{f_\Pi} \langle 0 \vert y \, \bar{f}f \vert 0 \rangle \, e^{-ipx},
\label{low3}\end{aligned}$$ reproducing Eq. (\[gpi\]) as desired. As a second step, we compare the fermion bilinear condensate $\langle 0 \vert y \, \bar{f}f \vert 0 \rangle$ and the NG boson decay constant $f_\Pi$ to corresponding quantities in QCD, for which we employ the pion decay constant and the quark bilinear condensate in the chiral-limit QCD obtained in Ref. [@durr] by fitting a lattice simulation with the next-to-leading order chiral perturbation theory [@leutwyler], which read $$\begin{aligned}
f_\pi^{{\rm chiral}} &= 0.08678~{\rm GeV},
\label{lattice1}
\\
-\frac{ \langle 0 \vert (m_u+m_d) \bar{q}q \vert 0 \rangle^{{\rm chiral}} }{ (f_\pi^{{\rm chiral}})^2 } &= 0.01861~{\rm GeV}^2.
\label{lattice2}\end{aligned}$$ Here, the quark bilinear condensate in Eq. (\[lattice2\]) is defined in terms of the product of a quark bilinear operator and the current quark mass, which is the sum of up and down quark current mass $(m_u+m_d)$. This product is independent of the wavefunction renormalization in QCD. Eq. (\[lattice2\]) is computed by first tuning $(m_u+m_d)$ to reproduce the real pion mass-pion decay constant ratio $m_{\pi^0}/f_{\pi}$. For this value of $(m_u+m_d)$, the lattice spacing is determined from the experimental value of pion mass. Then the quark bilinear condensate is extrapolated for $(m_u+m_d)=0$. To translate $\langle 0 \vert (m_u+m_d) \bar{q}q \vert 0 \rangle^{{\rm chiral}}$ into $\langle 0 \vert y \, \bar{f}f \vert 0 \rangle$, we exploit the fact that the current quark mass $(m_u+m_d)$ in QCD and the coupling constant $y$ in the $SU(N)_T$ gauge theory are renormalized in the same way with respect to the QCD and $SU(N)_T$ gauge coupling constants. For $N=3$, the correspondence is exact and we have $(m_u+m_d)(\mu')/(m_u+m_d)(\mu) = y(r \mu')/y(r \mu)$ for any two scales $\mu,\mu'$. For $N\geq4$, we still have an approximate relation $(m_u+m_d)(\mu')/(m_u+m_d)(\mu) \simeq y(r \mu')/y(r \mu)$. This is because the gauge coupling roughly scales as $\alpha_T(r\mu) \simeq (N_c/N)\alpha_s(\mu)$ whereas the Casimir operator $C_F$ scales as $C_F(N)\simeq (N/N_c)C_F(N_c)$ and $C_A$ scales as $C_A(N)=(N/N_c)C_A(N_c)$. The $\chi,\psi$ loop contribution to the gauge field propagator does not scale in this way, but its impact is subdominant compared to contributions from the gauge field and ghost field loops. Therefore, the factor $N/N_c$ mostly cancels and the radiative corrections to $(m_u+m_d)$ and $y$ are similar in numerical values. Accordingly, $\langle 0 \vert y \, \bar{f}f \vert 0 \rangle$ is estimated from $\langle 0 \vert (m_u+m_d) \bar{q}q \vert 0 \rangle^{{\rm chiral}}$ as $$\begin{aligned}
\langle 0 \vert y \, \bar{f}f \vert 0 \rangle &\simeq r^3 \, \frac{N}{N_c} \, y(r \mu) \frac{ \langle 0 \vert (m_u+m_d) \bar{q}q \vert 0 \rangle^{{\rm chiral}} }{(m_u+m_d)(\mu)},
\label{yrelation}\end{aligned}$$ where the factor $N/N_c$ is because the fermion bilinear condensate scales in the same manner as a correlation function for $SU(N)_T$-singlet operators. We quote the numerical value of $(m_u+m_d)(\mu)$ for $\mu=2$ GeV in the $\overline{MS}$ scheme obtained by a lattice simulation in Ref. [@durr2], which reads $$\begin{aligned}
\frac{1}{2}(m_u+m_d)(\mu=2~{\rm GeV}) &= 0.003469~{\rm GeV}.
\label{lattice3}\end{aligned}$$ Assembling the lattice results Eqs. (\[lattice1\]), (\[lattice2\]), (\[lattice3\]), the relation Eq. (\[yrelation\]) and the value of $f_\Pi$ evaluated in Section 3.1.1, we numerically evaluate ${\cal F}_{38}$ Eq. (\[f3app2\]) as $$\begin{aligned}
{\cal F}_{38} &= -\frac{1}{16\pi^2}\log\left(\frac{M_\Theta^2}{m_h^2}\right) \, \left(-\sqrt{2}\frac{\langle 0 \vert y \, \bar{f}f \vert 0 \rangle }{f_\Pi}\right)^2
\nonumber \\
&= -\frac{1}{16\pi^2}\log\left(\frac{r^2 \, m_{K_0^*(1430)}^2}{m_h^2}\right) \, \left(-\sqrt{2}y(r\mu)\frac{\langle 0 \vert (m_u+m_d)\bar{q}q \vert 0 \rangle^{{\rm chiral}} }{(m_u+m_d)(\mu)} \, \frac{1}{f_\pi^{{\rm chiral}}}\right)^2 \, \frac{N}{N_c}r^4
\nonumber \\
&= -\frac{N}{N_c} \, y_{r2}^2 \, r^4 \, \log(0.000130r^2) \, (0.162~{\rm GeV})^4, $$ where $y_{r2}$ is the coupling constant $y$ evaluated at $\mu=r\cdot2$ GeV scale in the $\overline{MS}$ scheme, which coincides with what has appeared in Section 2.2.
${\cal F}_{12}$ and ${\cal F}_{45}$ can be derived by the same reasoning as ${\cal F}_{38}$, expect that the pNG boson $(\Pi^6\pm i\Pi^7)$ should be exchanged with $(\Pi^1\pm i\Pi^2)$ or $(\Pi^4\pm i\Pi^5)$. Since these pNG bosons can all be represented by a delta function at $s=0$, the calculations are identical and we obtain $$\begin{aligned}
{\cal F}_{12} &={\cal F}_{45}={\cal F}_{38}.\end{aligned}$$ On the other hand, the evaluation of ${\cal F}_{67}$ does not proceed in the same way as ${\cal F}_{38}$, because ${\cal F}_{67}$ involves the pNG boson associated with an anomalous axial current $\bar{\psi} i \gamma_\mu \gamma_5 \psi + \bar{\chi}_2 i \gamma_\mu \gamma_5 \chi_2$. We speculate ${\cal F}_{67} \sim \frac{N}{N_c} \, y_{2r}^2 \, \Lambda_T^4$ and postpone its evaluation for future works.\
### Evaluation of the pseudo-Nambu-Goldstone boson mass
Now that $f_\Pi,{\cal C}^\gamma,{\cal C}^W,{\cal C}^Z,{\cal F}_{38},{\cal F}_{12},{\cal F}_{45}$ are written with the dynamical scale ratio $r$ as well as $N$, we are in position to express the pNG boson mass matrix Eq. (\[pngmass\]) solely in terms of $y_{r2}$ and $N$, using Eq. (\[ratio\]). We restrict ourselves to the limit with $y_{r2} \ll g_W$. Also, we ignore the tiny numerical difference in the evaluation of $r$ for $N=3,4,5,6$ and approximate it as $$\begin{aligned}
r &= \sqrt{\frac{3}{N}} \, \frac{1}{y_{r2}} \, 2.0\times10^2 \ \ \ \ \ {\rm for \ all \ }N.\end{aligned}$$
When $y_{r2} \ll g_W$, we have $g_W^2{\cal C}^W \gg {\cal F}_{38} = {\cal F}_{12} = {\cal F}_{45}$, $g_W^2{\cal C}^W \gg {\cal F}_{67}$ and $g_W^2{\cal C}^W \gg v \hat{y} \langle 0 \vert \bar{f}f \vert 0 \rangle$, which gives that off-diagonal mass terms $M^2_{ij} \, (1\leq i \neq j \leq 8)$ can be ignored in comparion to the diagonal ones [^5] and the terms ${\cal F}_{12}, \, {\cal F}_{38}, \, {\cal F}_{45}$ and ${\cal F}_{67}$ in $M^2_{11}, M^2_{22},...,M^2_{77}$ can further be discarded. The pNG boson masses are thus found to be $$\begin{aligned}
M^2_{11} = M^2_{22} = M^2_{33} = \frac{1}{f_\Pi^2} 2g_W^2 \, {\cal C}^\gamma &= \frac{3}{N}\frac{2g_W^2}{y^2_{r2}} \, (25~{\rm GeV})^2, \nonumber \\
M^2_{44} = M^2_{55} = M^2_{66} = M^2_{77} = \frac{1}{f_\Pi^2} \left(\frac{g_W^2}{2} + \frac{g_Z^2}{4} \right) {\cal C}^\gamma &= \frac{3}{N}\frac{1}{y^2_{r2}} \left(\frac{g_W^2}{2} + \frac{g_Z^2}{4} \right) \, (25~{\rm GeV})^2,
\nonumber \\
M^2_{88} = \frac{1}{f_\Pi^2} \, \frac{1}{12} \, {\cal F}_{38} &= -\frac{3}{N} \log\left(5.2 \, \frac{3}{N} \frac{1}{y^2_{r2}}\right) \, (17~{\rm GeV})^2. \label{pngmass2}\end{aligned}$$ Note that ${\cal F}_{38}$ and hence $M^2_{88}$ are negative, in contrast with ${\cal C}^\gamma$ being positive. A physical interpretation for this is that because scalar exchange force is always attractive, the energy of a system containing a scalar interaction ($\Pi^8$ meson in this case) diminishes, which manifests itself as a negative radiative correction to the mass. On the other hand, vector exchange force between the same charge is repulsive and thus contributes positively to the energy, and hence to the mass, of a system containing this type of interaction.
Since $M^2_{88}$ is negative, the $\Pi^8$ field acquires a VEV. The vacuum can be stabilized by a quartic coupling for the $\Pi^8$ field, which is generated radiatively from the $\bar{\chi}\psi H$ Yukawa interaction that explicitly violates the $\lambda^8$ component of the $SU(3)_A$ symmetry. The quartic coupling is proportional to $$\begin{aligned}
\int{\rm d}^4x \, D^h(x) \, \langle 0 \vert \, y \bar{\chi}_2\psi(x) \, y \bar{\psi}\chi_2(0) \, \vert \Pi^8 \Pi^8 \Pi^8 \Pi^8 \rangle
\label{quartic}\end{aligned}$$ in the leading order of the coupling constant $y$. The quartic coupling is naïvely estimated as $(N_c/N)y_{r2}^2$, giving rise to the term $$\begin{aligned}
-{\cal L} &\supset c \, \frac{N_c}{N}y_{r2}^2 \, (\Pi^8)^4,\end{aligned}$$ where $c$ is a $O(1)$ constant, and the factor $N_c/N$ originates from the fact that when the $\Pi^8$-meson creation operator asymptotes to a current operator, a factor $\sqrt{N_c/N}$ appears, and the correlation function for $SU(N)_T$-singlet current operators scales by $N/N_c$. If positive, the quartic coupling is responsible for vacuum stabilization and the VEV of $\Pi^8$ satisfies $$\begin{aligned}
\langle \Pi^8 \rangle &\sim \sqrt{ \frac{-M^2_{88}}{\frac{N_c}{N}y_{r2}^2} } = \frac{1}{y_{r2}} \sqrt{\log\left(5.2 \, \frac{3}{N} \frac{1}{y^2_{r2}}\right)} \, 17~{\rm GeV}.
\label{pi8vev}\end{aligned}$$ Phenomenologically, the VEV breaks parity ($P$) and charge-conjugation-parity ($CP$) symmetries, and brings about a $CP$-violating mixing of the physical Higgs field $h$ and the pNG boson $\Pi^6$, $$\begin{aligned}
-{\cal L} &\supset h \, \Pi^6 \, \langle \, \Pi^6 \, \vert ( \, y \bar{\chi}_2\psi(0)+ y \bar{\psi}\chi_2(0) \, ) \vert \, \Pi^8 \, \rangle \, \langle \Pi^8 \rangle.\end{aligned}$$ Since $\Pi^6$ does not couple to SM fermions, the above mixing should be detected as a suppression on the coupling of the observed Higgs particle to SM fermions as compared to the SM. To gain information on the mixing angle for $h$ and $\Pi^6$, we estimate the mixing term based on its mass dimension and $y$ dependence as $$\begin{aligned}
-{\cal L} &\supset \sim \ h \, \Pi^6 \, y_{r2} \Lambda_T \, \langle \Pi^8 \rangle,\end{aligned}$$ which yields the following mass matrix for $h$ and $\Pi^6$: $$\begin{aligned}
&-{\cal L}\supset \sim \ \frac{1}{2}\left(
\begin{array}{cc}
\Pi^6 & h
\end{array}
\right)
\left(
\begin{array}{cc}
M_{66}^2 & y_{r2} \Lambda_T \, \langle \Pi^8 \rangle \\
y_{r2} \Lambda_T \, \langle \Pi^8 \rangle & m_h^2
\end{array}
\right)
\left(
\begin{array}{c}
\Pi^6 \\
h
\end{array}
\right)
\nonumber \\
&= \frac{1}{2}\left(
\begin{array}{cc}
\Pi^6 & h
\end{array}
\right)
\left(
\begin{array}{cc}
\frac{3}{N}\frac{1}{y^2_{r2}} \left(\frac{g_W^2}{2} + \frac{g_Z^2}{4} \right) \, (25~{\rm GeV})^2 & \frac{1}{y_{r2}} \sqrt{\frac{3}{N}\log\left(5.2 \, \frac{3}{N} \frac{1}{y^2_{r2}}\right)} \, 46\cdot17~{\rm GeV}^2
\nonumber \\
\frac{1}{y_{r2}} \sqrt{\frac{3}{N}\log\left(5.2 \, \frac{3}{N} \frac{1}{y^2_{r2}}\right)} \, 46\cdot17~{\rm GeV}^2 & m_h^2
\end{array}
\right)
\left(
\begin{array}{c}
\Pi^6 \\
h
\end{array}
\right),\end{aligned}$$ where we have used the estimate for $\langle \Pi^8 \rangle$ in Eq. (\[pi8vev\]) and the relation $y_{r2} \Lambda_T = \sqrt{3/N} (2.0\times10^2) \Lambda_{QCD}$ as well as $\Lambda_{QCD}=0.23$ GeV. We find that in the limit with $y_{r2} \ll g_W$, the mixing angle for $h$ and $\Pi^6$ decreases with $y_{r2}$ and hence does not lead to a tension with the current Higgs particle measurement.
When the $\Pi^8$ quartic coupling is positive and stabilizes the vacuum, the physical mode of the $\Pi^8$ field, $\Pi^8_{{\rm phys}}=\Pi^8 - \langle \Pi^8 \rangle$, gains the following mass term: $$\begin{aligned}
-{\cal L} &\supset \frac{1}{2} \, (-2 M^2_{88}) \, (\Pi^8_{{\rm phys}})^2 \equiv \frac{1}{2} \, M_{\Pi^8_{{\rm phys}}}^2 \, (\Pi^8_{{\rm phys}})^2.\end{aligned}$$ Fortunately, the mass of the physical mode $M_{\Pi^8_{{\rm phys}}}$ is determined solely by $M^2_{88}$, with no dependence on the value of the quartic coupling. We further stress that the mass of $\Pi^8_{{\rm phys}}$ is rather insensitive to $y_{r2}$; when $y_{r2}$ varies from $10^{-1}$ to $10^{-17}$, with $y_{r2}=10^{-17}$ corresponding to the case when $\Lambda_T$ is above the Planck scale $\Lambda_T \gtrsim 2.44\times10^{18}$ GeV, the mass changes in the following range: $$\begin{aligned}
62~{\rm GeV} &< M_{\Pi^8_{{\rm phys}}} < 220~{\rm GeV} \ \ \ \ \ \ {\rm for \ } N=3 {\rm \ and \ } 10^{-1} < y_{r2} < 10^{-17},
\nonumber \\
41~{\rm GeV} &< M_{\Pi^8_{{\rm phys}}} < 155~{\rm GeV} \ \ \ \ \ \ {\rm for \ } N=6 {\rm \ and \ } 10^{-1} < y_{r2} < 10^{-17}.
\label{pi8physmass}\end{aligned}$$ The insensitivity to $y_{r2}$ is because the mass term for $\Pi^8$ stems only from the $\bar{\chi}\psi H$ Yukawa interaction and is thus of the order of $y_{r2} \Lambda_T$. However, the scale of $\Lambda_T$ is about the SM Higgs field mass divided by $y_{r2}$, *i.e.*, $\Lambda_T \sim m_h/y_{r2}$, and therefore the $\Pi^8$ mass is of the same order as the SM Higgs field mass. The mild logarithmic dependence on $y_{r2}$ is due to the fact that the SM Higgs particle mass functions as an infrared cutoff for the momentum integral in the ${\cal F}_{38}$ formula Eq. (\[f3calc\]).
It is convenient to organize pNG bosons in electroweak charge eigenstates as $\Pi = (\Pi^+,~\Pi^0,~\Pi^-) \equiv (\frac{1}{\sqrt{2}}(\Pi^1+i\Pi^2),~\Pi^3,~\frac{1}{\sqrt{2}}(\Pi^1-i\Pi^2))$, $\Sigma = (\Sigma^+,~\Sigma^0) \equiv (\frac{1}{\sqrt{2}}(\Pi^4+i\Pi^5),~\frac{1}{\sqrt{2}}(\Pi^6+i\Pi^7))$, $\bar{\Sigma} = (\bar{\Sigma}^0,~-\Sigma^-) = (\frac{1}{\sqrt{2}}(\Pi^6-i\Pi^7),~-\frac{1}{\sqrt{2}}(\Pi^4-i\Pi^5))$. $\Pi$ is an isospin triplet with hypercharge $Y=0$, $\Sigma$ is an isospin doublet with hypercharge $Y=-1/2$, and $\Pi^8_{{\rm phys}}$ has no gauge charge. Note that unlike neutral kaons, $\Sigma^0$ and $\bar{\Sigma}^0$ do not mix because of the absence of $\chi,\psi$ current mass. The mass and electroweak charges of $\Pi$, $\Sigma$ and $\Pi^8_{{\rm phys}}$ are summarized in Table \[png\].
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$\Pi$ $\Sigma$ $\Pi^8_{{\rm phys}}$
------------------------- --------------------------------------------------------- -------------------------------------------------------------------- -------------------------------------------------------------------------------------------------
Mass $\sqrt{\dfrac{3}{N}}\dfrac{\sqrt{2}g_W}{y_{r2}}$ 25 GeV $\sqrt{\dfrac{3}{N}}\dfrac{\sqrt{2g_W^2 + g_Z^2}}{2y_{r2}}$ 25 GeV $ \sqrt{\dfrac{3}{N}} \sqrt{\log\left( 5.2 \, \dfrac{3}{N} \dfrac{1}{y^2_{r2}} \right)}$ 25 GeV
$SU(2)_W \times U(1)_Y$ **[3]{}$_{0}$ & **[2]{}$_{1/2}$ & **[1]{}$_{0}$\
******
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Mass and electroweak charges of the pNG bosons $\Pi$, $\Sigma$ and $\Pi^8_{{\rm phys}}$ in the limit with $y_{r2} \ll g_W$, where electroweak symmetry breaking is negligible. That 25 GeV appears both in the $\Pi$, $\Sigma$ masses and in the $\Pi^8_{{\rm phys}}$ mass is merely accidental. []{data-label="png"}
We find that $\Pi$ is always heavier than $\Sigma$, but the $\Pi$ mass does not exceed twice the $\Sigma$ mass. $\Pi^8_{{\rm phys}}$ is the lightest pNG boson in any case.\
Interactions, decay pattern and production channels of the pseudo-Nambu-Goldstone bosons
----------------------------------------------------------------------------------------
The $\Pi$, $\Sigma$ and $\Pi^8$ fields couple to the physical Higgs field $h$, whose strength is determined by the following correlation functions: $$\begin{aligned}
&\langle \, \Sigma \, \vert ( \, y \bar{\chi}_2\psi(0)+ y\bar{\psi}\chi_2(0) \, ) \vert \, \Pi \, \rangle, \ \ \ \ \ \langle \, \Pi^8 \, \vert ( \, y \bar{\chi}_2\psi(0) + y \bar{\psi}\chi_2(0) \, ) \vert \, \Sigma \, \rangle.\end{aligned}$$ Evaluation of the above correlation functions is left for future works. There is another type of interaction described by the Wess-Zumino-Witten (WZW) term [@wz; @w] in chiral perturbation theory for the pNG bosons $\Pi$, $\Sigma$ and $\Pi^8$. This is particularly important for phenomenology of the $\Pi^8_{{\rm phys}}$ particle, as it is the principal source for the production and decay channels of the $\Pi^8_{{\rm phys}}$ particle. We below extract phenomenologically relevant part of the WZW term [@krs] which involves only $\Pi^8$ and electroweak gauge fields with the number of fields limited to 3 or 4: $$\begin{aligned}
-{\cal L}_{WZW} &\supset \frac{N}{48\pi^2} \epsilon^{\mu\nu\rho\sigma} \frac{1}{f_\Pi} \, \left\{ \, 2\sqrt{3}g_W^2 \, \Pi^8 \partial_\mu W^+_\nu \partial_\rho W^-_\sigma + \sqrt{3}(1-2s_W^2+2b s_W^4)g_Z^2 \, \Pi^8 \partial_\mu Z_\nu \partial_\rho Z_\sigma \right.
\nonumber \\
&+ 2\sqrt{3}(1-2b s_W^2)g_Z \, e \, \Pi^8 \partial_\mu Z_\nu \partial_\rho A_\sigma + 2\sqrt{3}b \, e^2 \, \Pi^8 \partial_\mu A_\nu \partial_\rho A_\sigma
\nonumber \\
&\left. -i\frac{1}{\sqrt{3}}g_W^3 \, W^+_\mu W^-_\nu (c_W Z_\rho + s_W A_\rho) \partial_\sigma \Pi^8 \, \right\},
\label{wzw}\end{aligned}$$ where $A_\mu$ denotes photon and $b/2$ is the hypercharge for the $\chi$ field. We have the non-vanishing WZW term above because the axial current corresponding to $\Pi^8$ is anomalous with respect to the $SU(2)_W$ weak gauge group, and when $b\neq1/2$, also to the $U(1)_Y$ hypercharge gauge group.
The decay pattern of the pNG bosons is as follows. For the $\Pi$ particle, since the $\Pi$ mass is below twice the $\Sigma$ mass, $\Pi \to \Sigma \bar{\Sigma}$ decay through the $SU(N)_T$ gauge interaction is kinematically forbidden. The $SU(2)_W$ and $U(1)_Y$ current conservation prohibits $\Pi \to (W \, {\rm or} \, Z) + \Sigma$ and $\Pi \to (W \, {\rm or} \, Z) + \Pi^8_{{\rm phys}}$ decays. Therefore, the main decay channels of the $\Pi$ particle are $\Pi^{\pm} \to h + \Sigma^{\pm}$, $\Pi^0 \to h + \Sigma^0$ and $\Pi^0 \to h + \bar{\Sigma}^0$ through the $\bar{\chi}\psi H$ Yukawa interaction, whose decay amplitudes are $$\begin{aligned}
{\cal A}(\Pi^+(p+q) \to h(q) \, \Sigma^+(p)) &= \frac{1}{\sqrt{2}} \, \langle \, \Sigma^+(p) \, \vert y \, \bar{\chi}_2(0)\psi(0) \vert \, \Pi^+(p+q) \, \rangle,
\label{pidecay1} \\
{\cal A}(\Pi^0(p+q) \to h(q) \, \bar{\Sigma}^0(p)) &= \frac{1}{\sqrt{2}} \, \langle \, \bar{\Sigma}^0(p) \, \vert y \, \bar{\chi}_2(0)\psi(0) \vert \, \Pi^0(p+q) \, \rangle,
\label{pidecay2} \\
\left\vert {\cal A}(\Pi^-(p+q) \to h(q) \, \Sigma^-(p)) \right\vert &= \left\vert {\cal A}(\Pi^+(p+q) \to h(q) \, \Sigma^+(p)) \right\vert,
\nonumber \\
\left\vert {\cal A}(\Pi^0(p+q) \to h(q) \, \bar{\Sigma}^0(p)) \right\vert &= \left\vert {\cal A}(\Pi^0(p+q) \to h(q) \, \Sigma^0(p)) \right\vert,
\label{cp}\end{aligned}$$ where $p+q$ denotes the $\Pi$ meson momentum, $p_2$ denotes the final-state $\Sigma$ meson momentum, and $q$ satisfies $q^2=m_h^2$. In addition to the above decay channel, $\Pi^{\pm} \to W^{\pm} \Pi^0$ or $\Pi^{0} \to W^{\mp} \Pi^{\pm}$ decay through the $SU(2)_W$ interaction is possible depending on the mass splitting in the $\Pi$ triplet, which we have omitted in the derivation of the mass spectrum.
For the $\Sigma$ particle, the $SU(2)_W$ and $U(1)_Y$ current conservation again forbids $\Sigma \to (W \, {\rm or} \, Z) + \Pi^8_{{\rm phys}}$ decays. Hence, the $\Sigma^0$ and $\bar{\Sigma}^0$ particles dominantly decay as $\Sigma^0 \to h + \Pi^8_{{\rm phys}}$ and $\bar{\Sigma}^0 \to h + \Pi^8_{{\rm phys}}$ through the $\bar{\chi}\psi H$ Yukawa interaction, with the decay amplitudes below: $$\begin{aligned}
{\cal A}(\Sigma^0(p+q) \to h(q) \, \Pi^8_{{\rm phys}}(p)) &= \frac{1}{\sqrt{2}} \, \langle \, \Pi^8(p) \, \vert y \, \bar{\psi}(0)\chi_2(0) \vert \, \Sigma^0(p+q) \, \rangle,
\label{sigmadecay} \\
\left\vert {\cal A}(\bar{\Sigma}^0(p+q) \to h(q) \, \Pi^8_{{\rm phys}}(p)) \right\vert &= \left\vert {\cal A}(\Sigma^0(p+q) \to h(q) \, \Pi^8_{{\rm phys}}(p)) \right\vert.
\label{cp2}\end{aligned}$$ On the other hand, the only tree-level decay channel of the $\Sigma^{\pm}$ particle is $\Sigma^+ \to W^+ \bar{\Sigma}^0$ and $\Sigma^- \to W^- \Sigma^0$ through the $SU(2)_W$ interaction provided the mass splitting in the $\Sigma$ doublet allows it kinematically. Otherwise, the main tree-level decay channel is $\Sigma^+ \to W^+ h \Pi^8_{{\rm phys}}$ $\Sigma^- \to W^- h \Pi^8_{{\rm phys}}$ via off-shell $\Sigma^0$ and $\bar{\Sigma}^0$ fields.
For the $\Pi^8_{{\rm phys}}$ particle, the decay proceeds through the WZW term Eq. (\[wzw\]). The partial widths are calculated from Eq. (\[wzw\]) and are found to be (the latter three modes are valid only when kinematically allowed) $$\begin{aligned}
\Gamma(\Pi^8_{{\rm phys}} \to \gamma \gamma) &= \frac{b^2 \, e^4}{3072\pi^5} \, N^2 \, \frac{M_{\Pi^8_{{\rm phys}}}^3}{f_\Pi^2}
\nonumber \\
\Gamma(\Pi^8_{{\rm phys}} \to \gamma Z) &= \frac{e^2g_Z^2(1-2b \, s_W^2)^2}{6144\pi^5} \, N^2 \, \frac{M_{\Pi^8_{{\rm phys}}}^3}{f_\Pi^2} \left(1-\frac{M_Z^2}{M_{\Pi^8_{{\rm phys}}}^2}\right)^3
\nonumber \\
&= \Gamma(\Pi^8_{{\rm phys}} \to \gamma \gamma) \, \frac{(1-2b \, s_W^2)^2}{2b^2 \, s_W^2 c_W^2}\left(1-\frac{M_Z^2}{M_{\Pi^8_{{\rm phys}}}^2}\right)^3,
\nonumber \\
\Gamma(\Pi^8_{{\rm phys}} \to Z Z) &= \frac{g_Z^4(1-2s_W^2 + 2b \, s_W^4)^2}{12288\pi^5} \, N^2 \, \frac{M_{\Pi^8_{{\rm phys}}}^3}{f_\Pi^2} \left(1-\frac{4M_Z^2}{M_{\Pi^8_{{\rm phys}}}^2}\right)^{3/2}
\nonumber \\
&= \Gamma(\Pi^8_{{\rm phys}} \to \gamma \gamma) \, \frac{(1-2s_W^2 + 2b \, s_W^4)^2}{4b^2 \, s_W^4 c_W^4}\left(1-\frac{4M_Z^2}{M_{\Pi^8_{{\rm phys}}}^2}\right)^{3/2},
\nonumber \\
\Gamma(\Pi^8_{{\rm phys}} \to W^+ W^-) &= \frac{g_W^4}{6144\pi^5} \, N^2 \, \frac{M_{\Pi^8_{{\rm phys}}}^3}{f_\Pi^2} \left(1-\frac{4M_W^2}{M_{\Pi^8_{{\rm phys}}}^2}\right)^{3/2}
\nonumber \\
&= \Gamma(\Pi^8_{{\rm phys}} \to \gamma \gamma) \, \frac{1}{2b^2 \, s_W^4}\left(1-\frac{4M_W^2}{M_{\Pi^8_{{\rm phys}}}^2}\right)^{3/2}.\end{aligned}$$ Numerically, we have $$\begin{aligned}
\frac{1}{\Gamma(\Pi^8_{{\rm phys}} \to \gamma \gamma)} &= \frac{1}{b^2 N^2} \left(\frac{N}{N_c}\right)^{3/2} \frac{1}{y_{r2}^2} \frac{1}{\log^{3/2}\left( 5.2 \, \dfrac{3}{N} \dfrac{1}{y^2_{r2}} \right)} \, 1.48 \times 10^{-18} \ {\rm s},
\nonumber \\\end{aligned}$$ Additionally, the $\Pi^8_{{\rm phys}} \to W^+ W^- \gamma$ decay is possible depending on the $\Pi^8_{{\rm phys}}$ mass, but the partial width is suppressed by the factor $e^2/(2\pi)^2$ compared to the $\Pi^8_{{\rm phys}} \to W^+ W^-$ decay.
Note that the pNG bosons cannot decay into two SM fermions via a $s$-channel gauge boson, because $\chi$ and $\psi$ fields couple to the electroweak gauge fields through vector currents [^6], while the pNG bosons couple to axial-vector currents.
We briefly discuss the model’s signatures at the LHC. Since the $\Pi^8_{{\rm phys}}$ particle is predicted to have mass below 220 GeV, kinematically it is accessible at the LHC. The main production channels are gauge-boson-associated productions through the WZW term Eq. (\[wzw\]) described as $$\begin{aligned}
q\bar{q}' &\to W^* \to \Pi^8_{{\rm phys}} + W, \ \ \ \ \ q\bar{q} \to Z^*/\gamma^* \to \Pi^8_{{\rm phys}} + Z, \ \ \ \ q\bar{q} \to Z^{(*)}/\gamma^* \to \Pi^8_{{\rm phys}} + \gamma,
\nonumber \\
(&q, \, q' = u, \, d),\end{aligned}$$ where the intermediate gauge fields are off-shell except for the $\Pi^8_{{\rm phys}} + \gamma$ production, in which $Z$ boson can be on-shell. Note that the vector boson fusion process does not enjoy collinear enhancement proportional to $\log(\hat{s}/M_{\Pi^8_{{\rm phys}}}^2)$ ($\hat{s}$ denotes the parton center-of-mass energy), because the $\Pi^8$ field couples to the transverse polarization modes, not the longitudinal mode, of $W$ and $Z$ bosons. Therefore, the vector-boson-fusion cross sections are simply suppressed by the electroweak gauge couplings and a phase space factor compared to those for the associated productions. At hadron colliders, decay channels practically usable for detecting $\Pi^8_{{\rm phys}}$ particle signals are those in which the $W$ or $Z$ boson associated with $\Pi^8_{{\rm phys}}$ decays into electron or muon, or the associated photon is not soft, namely, $$\begin{aligned}
q\bar{q}' &\to W^* \to \Pi^8_{{\rm phys}} + W(\to \ell \nu),
\ \ \
q\bar{q} \to Z^*/\gamma^* \to \Pi^8_{{\rm phys}} + Z(\to \ell \bar{\ell}),
\ \ \
q\bar{q} \to Z^{(*)}/\gamma^* \to \Pi^8_{{\rm phys}} + \gamma_{{\rm hard}}
\nonumber \\
(&\ell = e, \, \mu; \ \ \ \gamma_{{\rm hard}} {\rm \ denotes \ a \ photon \ sufficiently \ hard \ to \ be \ detected}).
\label{lhc}\end{aligned}$$ This is because the presence of a muon, electron or photon associated with $\Pi^8_{{\rm phys}}$ decay products is indispensable for reducing SM backgrounds. To make a crude estimate for the upper bound on $y_{r2}$ at the 13 TeV LHC, we present in Figures \[w\], \[z\], \[gam\] the cross section times branching ratio for each production and decay process in Eq. (\[lhc\]) in 13 TeV $pp$ collisions. In the calculation, the leading order MSTW 2008 parton distribution function [@mstw] is employed, with the factorization scale set at $\mu_F=M_{\Pi^8_{{\rm phys}}}+M_W$ for the $\Pi^8_{{\rm phys}} + W$ channel, $\mu_F=M_{\Pi^8_{{\rm phys}}}+M_Z$ for the $\Pi^8_{{\rm phys}} + Z$ channel, and $\mu_F=M_{\Pi^8_{{\rm phys}}}$ for the $\Pi^8_{{\rm phys}} + \gamma$ channel. Remind that since the $\Pi^8_{{\rm phys}}$ particle mass and the WZW term are entirely determined by the coupling constant $y_{r2}$ and the gauge group size $N$, the cross sections depend only on these variables. In Figure \[gam\], the energy of the associated photon in the parton center-of-mass frame is required to be above 20 GeV.
![ Cross section for the production of a $\Pi^8_{{\rm phys}}$ particle associated with a $W$ boson in 13 TeV $pp$ collisions, multiplied by the $W$ boson branching ratio into electron or muon. The horizontal axis is in the logarithm of $y_{r2}$, $\log_{10} y_{r2}$, and the solid and dashed lines respectively correspond to the cases with $N=3$ and $N=6$. []{data-label="w"}](w.eps){width="80mm"}
![ Cross section for the production of a $\Pi^8_{{\rm phys}}$ particle associated with a $Z$ boson in 13 TeV $pp$ collisions, multiplied by the $Z$ boson branching ratio into electrons or muons. The left plot is for $b=0$, with the hypercharge of $\chi$ being 0, and the right plot is for $b=1$, with the hypercharge of $\chi$ being $1/2$. The horizontal axis is in the logarithm of $y_{r2}$, $\log_{10} y_{r2}$, and the solid and dashed lines respectively correspond to the cases with $N=3$ and $N=6$. []{data-label="z"}](zb0.eps "fig:"){width="80mm"} ![ Cross section for the production of a $\Pi^8_{{\rm phys}}$ particle associated with a $Z$ boson in 13 TeV $pp$ collisions, multiplied by the $Z$ boson branching ratio into electrons or muons. The left plot is for $b=0$, with the hypercharge of $\chi$ being 0, and the right plot is for $b=1$, with the hypercharge of $\chi$ being $1/2$. The horizontal axis is in the logarithm of $y_{r2}$, $\log_{10} y_{r2}$, and the solid and dashed lines respectively correspond to the cases with $N=3$ and $N=6$. []{data-label="z"}](zb1.eps "fig:"){width="80mm"}
![ Cross section for the production of a $\Pi^8_{{\rm phys}}$ particle associated with a photon in 13 TeV $pp$ collisions, where the photon energy in the parton center-of-mass frame is above 20 GeV. The left plot is for $b=0$, with the hypercharge of $\chi$ being 0, and the right plot is for $b=1$, with the hypercharge of $\chi$ being $1/2$. The horizontal axis is in the logarithm of $y_{r2}$, $\log_{10} y_{r2}$, and the solid and dashed lines respectively correspond to the cases with $N=3$ and $N=6$. []{data-label="gam"}](gamb0.eps "fig:"){width="80mm"} ![ Cross section for the production of a $\Pi^8_{{\rm phys}}$ particle associated with a photon in 13 TeV $pp$ collisions, where the photon energy in the parton center-of-mass frame is above 20 GeV. The left plot is for $b=0$, with the hypercharge of $\chi$ being 0, and the right plot is for $b=1$, with the hypercharge of $\chi$ being $1/2$. The horizontal axis is in the logarithm of $y_{r2}$, $\log_{10} y_{r2}$, and the solid and dashed lines respectively correspond to the cases with $N=3$ and $N=6$. []{data-label="gam"}](gamb1.eps "fig:"){width="80mm"}
In the range $10^{-2.4} < y_{r2} < 10^{-1}$ displayed above, the $\Pi^8_{{\rm phys}}$ particle mass varies as 88 GeV$> M_{\Pi^8_{{\rm phys}}} >$62 GeV for $N=3$ case, and 60 GeV$> M_{\Pi^8_{{\rm phys}}} >$41 GeV for $N=6$ case. It follows that for $b\neq 0$, $\Pi^8_{{\rm phys}}$ almost exclusively decays into two photons and hence can be reconstructed as a diphoton resonance. For $b=0$, the term $\Pi^8 \partial^\mu A^\nu \partial^\rho A^\sigma$ in the WZW term is zero and the decay into diphoton is absent, in which case $\Pi^8_{{\rm phys}}$ mainly decays into a photon and a pair of SM fermions via an off-shell $Z$ boson. When the off-shell $Z$ boson decays into quarks, neutrinos or tau leptons, the reconstruction of a $\Pi^8_{{\rm phys}}$ particle is challenging. Although derivation of the precise bound on $y_{r2}$ calls for a collider simulation with rapidity and transverse momentum selection cuts, efficiency factor for the $\Pi^8_{{\rm phys}}$ particle reconstruction and examination of SM backgrounds, it is evident that the region with $y_{r2}\lesssim10^{-1.6}$ for $N=3$ case and that with $y_{r2}\lesssim10^{-2.4}$ for $N=6$ case would not be excluded even with about 10 fb$^{-1}$ of data at the 13 TeV LHC.
We finally comment on accessibility of the other pNG bosons, $\Pi$ and $\Sigma$, at the LHC. Since the $\Pi$ and $\Sigma$ particles are charged under the electroweak gauge groups, they can be pair-produced through the Drell-Yan process. When the upper bound on $y_{r2}$ is saturated with $N=3$ and $y_{r2} = 10^{-1.6}$, the $\Pi$ and $\Sigma$ masses are found to be $M_\Pi = 920$ GeV and $M_\Sigma=590$ GeV, in which case $\Pi$ and $\Sigma$ particles may be detected at the 14 TeV LHC with 300 fb$^{-1}$ of data. For $N=6$ and $y_{r2} = 10^{-2.4}$, the masses are found to be $M_\Pi = 4.1$ TeV and $M_\Sigma=2.6$ TeV, and $\Pi$ and $\Sigma$ particles are out of reach of the 14 TeV LHC.\
Vanishing of the scalar quartic coupling at the Planck scale
============================================================
We demonstrate that the model can realize, with the top quark pole mass as large as 172 GeV, the vanishing of the scalar quartic coupling at the Planck scale $M_P=2.44\times10^{18}$ GeV, namely, a flat scalar potential at the Planck scale. For this purpose, we evaluate the scalar quartic coupling at the Planck scale by numerically solving the RG equations for the quartic coupling of the elementary scalar field $H$, the SM gauge couplings and top quark Yukawa coupling. We vary the coupling constant $y_{r2}$ and the top quark pole mass, considering that the top quark pole mass is subject to sizable experimental and theoretical uncertainties. We concentrate on the case with $N=3$ and the range $10^{-1.6} \geq y_{r2} \geq 10^{-8}$, where the upper limit is based on a rough experimental bound at the 13 TeV LHC estimated in Section 3.2. Since $y_{r2}$ is that small, the $\bar{\chi}\psi H$ Yukawa interaction plays no role in the RG evolution of the scalar quartic coupling. Still, it is an important parameter that determines at which scale the confinement occurs and the particle content changes from (SM particles+light pNG bosons) to (SM particles+fermionic particles made of $\chi,\psi$ fields).
To obtain the RG equations for our model, we alter the SM two-loop RG equations in Ref. [@rge] by adding contributions of new particles. We make the approximation that the particle content changes from (SM particles+light pNG bosons) to (SM particles+fermionic particles made of $\chi,\psi$ fields) precisely at the energy scale $M_\Theta$ and ignore loop-level threshold corrections. With this simplification, the new particle contributions are included in the following way: Below the scale $\mu=M_\Theta$, the pNG bosons with electroweak charges, $\Pi$ and $\Sigma$, contribute to the RG runnings of the weak and hypercharge gauge couplings. At the scale $\mu=M_\Theta$, the SM Higgs quartic coupling $\lambda^{SM}$ and top quark Yukawa coupling $y_t^{SM}$ are matched to the $H$ quartic coupling $\lambda$ and the Yukawa coupling $(Y_u)_{33}$ in Eq. (\[lagrangian2\]) by the conditions Eqs. (\[matching2\]), (\[matching3\]). Above the scale $\mu=M_\Theta$, fermionic particles comprising $\chi,\psi$ fields contribute to the RG runnings of the weak and hypercharge gauge couplings. Remember that $M_\Theta$ is linked to the coupling constant $y_{r2}$ as $M_\Theta = r \, m_{K_0^*(1430)} \simeq (191/y_{r2}) \, m_{K_0^*(1430)}$ through Eqs. (\[mtheta\]), (\[ratio\]). We fix SM parameters as $M_W=80.384$ GeV, $\alpha_s(M_Z)=0.1184$ and $m_h=125.09$ GeV. Also, the hypercharge of the $\chi$ field is set as $b/2=1/2$. However, we have separately confirmed that $O(1)$ variation of $b$ does not affect the main result.
We present in Figure \[lambda@planck\] a contour plot of the value of the $H$ quartic coupling at the Planck scale, $\lambda(M_P)$, on the plane spanned by the coupling constant $y_{r2}$ and top quark pole mass $m_t^{{\rm pole}}$. Additionally shown are the central value and the 2$\sigma$ lower bound for the top quark mass reported by the ATLAS Collaboration [@topatlas], which gives the top quark mass to be $m_t=172.84\pm0.70$ GeV.
![ Contour plot of the quartic coupling for the elementary scalar field $H$ at the Planck scale, $\lambda(M_P)$. The $\bar{\chi}\psi H$ Yukawa coupling constant varies in the range $10^{-1.6} \geq y_{r2} \geq 10^{-8}$, and the top quark pole mass in the range $173~{\rm GeV} \geq m_t^{{\rm pole}} \geq 170~{\rm GeV}$. The thick curve corresponds to $\lambda(M_P)=0$, while the upper and lower dashed curves respectively correspond to $\lambda(M_P)=-0.01$ and $\lambda(M_P)=0.01$. Also shown are the central value (horizontal solid line) and the 2$\sigma$ lower bound (horizontal dashed line) for the top quark mass reported by the ATLAS Collaboration [@topatlas]. []{data-label="lambda@planck"}](lambdaplanck.eps){width="100mm"}
We find that $\lambda(M_P)=0$ can be achieved for the top quark pole mass as large as 172.5 GeV, which is quite consistent with the top quark mass reported by the ATLAS Collaboration. This is a remarkable progress from previous models of classical scale invariance with $\lambda(M_P)=0$, where the top quark pole mass needs to be well below 172 GeV and hence outside the 2$\sigma$ bound. The realization of $\lambda(M_P)=0$ with $m_t^{{\rm pole}}\simeq172.5$ GeV owes to the fact that fermionic particles made of $\chi$ field contribute positively to the RG running of the weak gauge coupling $g_W$ and thereby enhance it at high scales. The weak gauge coupling then raises the $H$ quartic coupling through RG evolutions as compared to the SM, so that $\lambda(M_P)$ can reach 0 even when the top quark Yukawa coupling is large. The above situation is illustrated in Figure \[rge\], where we contrast the RG runnings of the scalar quartic coupling and the weak gauge coupling in our model with $y_{r2}=10^{-1.6}$ and $m_t^{{\rm pole}}=172.55$ GeV, with those of the Higgs quartic coupling and weak gauge coupling in the SM with the same top quark pole mass. Here, the RG running of the $H_1$ quartic coupling is shown below the scale $M_\Theta$, and that of $H$ is shown above the scale $M_\Theta$.
![ The RG evolutions of the $H_1$ and $H$ quartic coupling and weak gauge coupling $g_W$ in our model with $y_{r2}=10^{-1.6}$ and $m_t^{{\rm pole}}=172.55$ GeV, contrasted with those of the Higgs quartic coupling and weak gauge coupling in the SM with the same top quark pole mass. The scale $\mu$ moves from $\mu=10^{2.3}$ GeV$\simeq m_t^{{\rm pole}}$ to $\mu=10^{18.4}$ GeV$\simeq M_P$. The vertical line indicates the confinement scale $M_\Theta$. The quartic coupling of $H_1$ is shown below the scale $M_\Theta$, and that of $H$ is shown above that scale. The solid black line starting from about 1.2 corresponds to the $H_1$ and $H$ quartic coupling, rescaled by $\times 10$ for visibility, and the solid red line corresponds to the weak gauge coupling. The nearby dashed lines describe the RG evolutions of the Higgs quartic coupling and weak gauge coupling in the SM. []{data-label="rge"}](rge.eps){width="100mm"}
As $y_{r2}$ decreases, realization of $\lambda(M_P)=0$ requires smaller top quark pole mass. This is because the pNG bosons, as being bosonic, contribute negatively to the RG evolution of the weak gauge coupling, in contrast to fermionic particles made of $\chi$ field. Hence, the higher the confinement scale $M_\Theta$ is, the less the new particles enhance the weak gauge coupling and the scalar quartic coupling at ultraviolet scales. Therefore, the top quark Yukawa coupling needs to be smaller to have $\lambda(M_P)=0$.\
Summary and discussions
=======================
We have studied a classically scale invariant extension of the Standard Model, in which chiral symmetry breaking and confinement in a new $SU(N)_T$ gauge theory break the scale invariance. The SM Higgs field emerges through the mixing of a scalar meson resulting from the confinement and an elementary scalar field. The SM Higgs field mass is dynamically generated at the scale given by $\Lambda_T$, the dynamical scale of the $SU(N)_T$ gauge theory, times $y$, the Yukawa coupling constant for $SU(N)_T$-charged fermions and the elementary scalar field, automatically with the correct negative sign. Concerning phenomenological signatures of the model, we have investigated the mass and interactions of pseudo-Nambu-Goldstone bosons associated with the spontaneous breaking of the $SU(3)_A$ axial symmetry along chiral symmetry breaking in the $SU(N)_T$ gauge theory. We have found that the model predicts the existence of a Standard Model gauge singlet pseudoscalar particle with mass below 220 GeV, which couples to two electroweak gauge bosons through the Wess-Zumino-Witten term, with the strength thus proportional to $1/\Lambda_T$. Regarding the theoretical aspects, we have shown that the model can realize the vanishing of the scalar quartic coupling at the Planck scale with the top quark pole mass as large as 172.5 GeV, which is consistent with the current top quark mass measurement.
If $y_{r2}$ were as large as $0.3$, the gap between the Standard Model Higgs quartic coupling and the quartic coupling of $H$ as appears in Eq. (\[matching2\]) would be sizable and could lead to the simultaneous vanishing of the scalar quartic coupling and its beta function at the Planck scale, realizing so-called “multiple-point principle” conjectured in Ref. [@mpp]. In the current model, however, $y_{r2}\simeq 0.3$ would incur too large cross section for the $\Pi^8_{{\rm phys}}$ particle production associated with a hard photon, which contradicts with the null result for new physics searches at the LHC.
We comment in passing that the $SU(N)_T$ gauge theory gives rise to baryonic states composed of $\chi,\psi$ fields. The lightest baryon is stable due to the conservation of the $U(1)$ vector charge for $\chi,\psi$ fields. We expect the lightest baryon to be a Standard Model gauge singlet, or at least neutral with respect to the electromagnetic interaction, because the electroweak interactions lift the baryon mass. It can therefore be a dark matter candidate.\
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors thank Nobuchika Okada (University of Alabama), Noriaki Kitazawa (Tokyo Metropolitan University) and Masayasu Harada (Nagoya University) for useful comments. This work is partially supported by Scientific Grants by the Ministry of Education, Culture, Sports, Science and Technology of Japan (Nos. 24540272, 26247038, 15H01037, 16H00871, and 16H02189).\
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[^1]: We here refrain from the phase redefinition of $H$ that makes $y$ real.
[^2]: For studies on scalar mesons, see Ref. [@scalarmesons] and references therein.
[^3]: We insert the factor $1/2$ in Eq. (\[correlatordef\]) to cancel the $SU(2)_W$ degree of freedom of $\chi$.
[^4]: $\Pi'$ may correspond to the unestablished $K(1460)$ meson.
[^5]: $M^2_{38}$ can be ignored despite $M^2_{88}$ being smaller than $M^2_{38}$, because $M^2_{33} \gg M^2_{38}$.
[^6]: In the current model, $\chi$ and $\psi$ must be vector-like with respect to the electroweak gauge groups in order to avoid electroweak symmetry breaking along chiral symmetry breaking.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'An elliptic curve may be immersed in $\P^{N-1}$ as a degree $N$ curve using level $N$ structure. In the case where $N$ is odd, there are well known classical results dating back to Bianchi and Klein. In this paper we study the case of even $N$ in some detail. In particular, over the complex number field, we define an immersion using suitably chosen theta functions, and study the quadratic equations satisfied by them.'
address:
- 'Faculty of Mathematics, Kyushu University, Motooka 744, Nishi-ku, Fukuoka 819-0395, Japan'
- 'Faculty of Economics, Chuo University, 742-1 Higashinakano, Hachioji-shi, Tokyo 192-0393, Japan'
author:
- Masanobu Kaneko
- Masato Kuwata
date: 'May 26, 2020'
title: Elliptic normal curves of even degree and theta functions
---
Introduction
============
An elliptic normal curve of degree $N$ is an elliptic curve $E$ with an immersion in a projective space $\P^{N-1}$ as a degree $N$ curve that is contained in no hyperplane. Suppose $N\ge 4$, and the base field $K$ is a field of characteristic not dividing $N$. By the Riemann-Roch theorem, $E/K$ can be realized as an elliptic normal curve of degree $N$ in $\P^{N-1}_{K}$ by means of a complete linear system $|D|$ with any effective divisor $D$ of degree $N$. In particular, we may take $D=N\cdot O$, where $O$ is the origin of the group structure of $E$. Moreover, it is known that such an $E$ is defined by a system of $N(N-3)/2$ quadratic equations.
Classically, over the complex number field $\C$, Bianchi [@Bianchi] first wrote down equations for the case $N=5$, where elliptic quintic in $\P^{4}$ is given by a set of five quadratic equations: $$\left\{\renewcommand{\arraystretch}{1.33}\begin{array}{l}
x_{0}^{2} + \phi x_{2}x_{3}-\phi^{-1}x_{1}x_{4} = 0, \\
x_{1}^{2} + \phi x_{3}x_{4}-\phi^{-1}x_{2}x_{0} = 0, \\
x_{2}^{2} + \phi x_{4}x_{0}-\phi^{-1}x_{3}x_{1} = 0, \\
x_{3}^{2} + \phi x_{0}x_{1}-\phi^{-1}x_{4}x_{2} = 0, \\
x_{4}^{2} + \phi x_{1}x_{2}-\phi^{-1}x_{0}x_{3} = 0.
\end{array}\right.$$ Here, $\phi$ is a parametrizing modular function of level 5. Klein [@Klein] generalized it to obtain such quadratic equations for general odd integers $N$, using Riemann’s sigma functions. What is significant is that their formulas describe the universal family of elliptic curves with level $N$ structure. In fact, Vélu [@Velu:1978] realized it as schemes over $\operatorname{Spec}\Z[1/N]$. By looking at the origin $O\in E$, we also obtain a model of modular curve associated with the principal congruence subgroup $\Gamma(N)$.
In the case where $N$ is even, Hurwitz [@Hurwitz] obtained similar formulas following Klein’s method, although it does not give the same kind of universal family. The purpose of this paper is to study this case in detail. We take a slightly different immersion from that of Hurwitz, and obtain the universal family of elliptic curves associated with a certain congruence subgroup of level $2N$.
Let us describe it in more detail. Consider the immersion $E\to \P^{N-1}$ using the complete linear system $|N\cdot O|$. For a $N$-torsion point $T\in E[N]$, let us denote by $\tau_{T}$ the translation-by-$T$ map $P\mapsto P+T$. It is easy to see that $\tau_{T}$ can be lifted to an automorphism of the ambient space $\P^{N-1}$. By choosing a basis $(S,T)$ of $E[N]$, we can find a coordinate system such that the translations $\tau_{S}$ and $\tau_{T}$ are expressed in simple forms similar to Bianchi’s equations above (see Proposition \[prop:proj-coord-odd\] for detail).
In case $N$ is odd, the choice of such a coordinate system is unique once we fix a primitive $N$th root of unity $\zeta_{N}$, and the pair $(S,T)$ such that the Weil pairing $e_{N}(S,T)=\zeta_{N}$. Thus, we obtain an immersion of the family of elliptic curves with level $N$ structure to a single $\P^{N-1}$, and by associating the origin of the elliptic curves, we obtain a morphism from the modular curve $X(N)$ to $\P^{N-1}$.
In case $N$ is even, the situation is more complicated. In this case, the coordinate system mentioned above is not unique; we need to specify a further structure than a level $N$ structure. As a consequence we will obtain the universal elliptic curve corresponding to a slightly smaller group than $\Gamma(N)$, which we will denote by $\Gamma^{(N)}(2N)$. (See §3 for detail.) In addition, we write down the quadratic equations satisfied by the image of $E$ in this particular coordinate system in the case of $K=\C$ using a certain type of theta functions. More precisely, we define the theta functions denoted by $\theta_{k}^{(N)}(z,\tau)$ (see Definition \[def:theta-N\]) that become the coordinate functions of the coordinate system described above, and we obtain quadratic equations of the image using the relations between theta functions coming from Jacobi’s identity or . It should be noted that it is essentially the addition formula of the elliptic curve.
In §§\[sec:level-4\]–\[sec:level-8\] we work out in detail the cases $N=4,6$, and $8$. There we show the explicit equations of the modular curve and the universal curve over it.
Acknowledgements {#acknowledgements .unnumbered}
----------------
Kaneko was supported in part by JSPS KAKENHI Grant Numbers JP23340010, JP15K13428, JP16H06336. Kuwata was supported by JSPS KAKENHI Grant Numbers JP23540028, JP26400023, and by the Chuo University Grant for Special Research.
Preliminaries
=============
Let $E$ be an elliptic curve defined over $K$. Throughout this section we fix a positive integer $N$ greater than $3$, and suppose that the base field $K$ is a field of characteristic not dividing $N$. Let $K_{s}$ be a separable closure of $K$, and let $\Gamma_{K}={\operatorname{Gal}}(K_{s}/K)$. Denote by $E(K)$ the group of $K$-rational points of $E$, and $E[N]=\{P\in E(K_{s})\mid NP=O\}$ the subgroup of $N$-torsion points.
In this section we give a summary of necessary facts about the Weil pairings and central extensions of $E[N]$. We follow Vélu [@Velu:1978] faithfully, but we included here for the convenience of the reader.
The Weil pairing
----------------
We first fix some notation.
- $K(E)^{\times}$ : the multiplicative group of the function field $K(E)$ of $E$.
- ${\operatorname{Div}}_{K}(E)$ : the group of $K$-divisors of $E$; i.e., the group of formal $\Z$-linear combination $D=\sum_{P\in E(K_{s})} n_{P}\{P\}$[^1] such that $n_{P}\in \Z$, $n_{P}=0$ for all but finitely many $P\in E(K_{s})$, and $n_{\, {{}^{\sigma}\!}\!P}=n_{P}$ for all $\sigma\in \Gamma_{K}={\operatorname{Gal}}(K_{s}/K)$.
- ${\operatorname{Div}}_{K}^{0}(E)$ : the kernel of $\deg:{\operatorname{Div}}_{K}(E)\to \Z$.
- ${\operatorname{Nul}}_{K}(E)$ : the kernel of the homomorphism ${\operatornamewithlimits{\mathrm{Sum}}}:{\operatorname{Div}}_{K}(E)\to E(K)$ defined by $\sum_{P}n_{P}\{P\}\mapsto{\operatornamewithlimits{\mathrm{Sum}}}\limits_{P}\,n_{P}P$, where ${\operatornamewithlimits{\mathrm{Sum}}}$ means the addition in $E$.
- $P_{K}(E) = {\operatorname{Div}}_{K}^{0}(E)\cap {\operatorname{Nul}}_{K}(E)$.
We have an obvious homomorphism ${\operatorname{div}}:K(E)^{\times}\to P_{K}(E)$, and the following is fundamental.
\[th:able\] The following sequence is exact. $$\label{eq:Abel}
1 \longrightarrow K^{\times} \longrightarrow K(E)^{\times}
\overset{{\operatorname{div}}}{\longrightarrow} P_{K}(E) \longrightarrow 0.
\qedhere$$
Let $G\subset E[N](K)$ be a cyclic group of order $N$. We consider as the exact sequence of $G$-modules. Define an action of $G$ on various groups as follows:
- trivially on $K^{\times}$, $\Z$, and $E(K)$.
- by translation on $K(E)$; i.e., if $f\in K(E)$ and $T\in G$, define ${{}^{T}\!}\!f(X)=f(X-T)$, where $X$ is a generic point of $E$.
- by translation on ${\operatorname{Div}}_{K}(E)$; i.e., if ${D}=\sum_{P}n_{P}\{P\}$ and $T\in G$, define $${{}^{T}\!}\!{D}=\sum_{P}n_{P}\{P+T\}.$$
Clearly, the action of $G$ on ${\operatorname{Div}}_{K}(E)$ induces actions on ${\operatorname{Div}}_{K}^{0}(E)$ and $P_{K}(E)$. The Weil pairing is obtained essentially as the connecting homomorphism $\delta$ of the long exact sequence of the group cohomology obtained from : $$\label{eq:long-exact-seq}
0 \longrightarrow {K(E)^{\times}}^{G} \overset{{\operatorname{div}}}\longrightarrow P_{K}(E)^{G}
\overset{\delta}{\longrightarrow} H^{1}(G,K^{\times}).$$
\[thm:weil-pairing\]
1. The homomorphism $\delta$ in induces a $\Gamma_{K}$-isomorphism $\Psi:G'=E[N]/G \overset{\simeq}{\longrightarrow}{\operatorname{Hom}}(G,\mu_{N})$.
2. For $S'\in G'$ and $T\in G$, define $e_{G}(S',T)$ by $$e_{G}(S',T)=\Psi(S')(T).$$ Then $e_{G}$ is a $\Gamma_{K}$ compatible nondegenerate bilinear form $G'\times G\to \mu_{N}$.
3. For $S'\in G'$, choose $S\in E[N]$ such that $S\bmod G=S'$, and define ${D}_{S}=\{S\}-\{O\}\in {\operatorname{Div}}_{K_{s}}^{0}(E)$. Choose $f_{S}\in K_{s}(E)^{\times}$ such that $${\operatorname{div}}f_{S} = \sum_{T\in G}{{}^{T}\!}\!{D}_{S}
= \sum_{T\in G}\bigl(\{S+T\}-\{T\}\big).$$ Then, we have$$e_{G}(S',T)={{}^{T}\!}\!f_{S}/f_{S}=f_{S}(X-T)/f_{S}(X).$$
Since $G\subset E[N](K)$ by assumption, we have $\mu_{N}\in K^{\times}$ and thus $H^{1}(G,K^{\times})\simeq {\operatorname{Hom}}(G,\mu_{N})$. The proof is a standard diagram chasing. See Vélu [@Velu:1978 Ch. 1] for detail.
The usual Weil pairing $e_{N}:E[N]\times E[N]\to \mu_{N}$ is defined in a similar manner. The relation between $e_{G}$ and $e_{N}$ is given by $$e_{N}(S,T)=e_{G}(\phi(S),T) \quad \text{for $S, T\in E[N]$},$$ where $\phi$ is the isogeny $E\to E/G$.
Central extension $E[N]({D})$
-----------------------------
A central extension of a group $G$ is a short exact sequence of groups $1\to A\to H\to G\to 1$ such that $A$ is in the center of the group $H$. Here, we consider central extensions of $E[N]$ by $K^{\times}$, that is, short exact sequences of groups $$1 \longrightarrow K^{\times} \longrightarrow H
\longrightarrow E[N] \longrightarrow 0.$$ (Note that the operation of $K^{\times}$ is noted multiplicatively, while that of $E[N]$ is noted additively.)
For a divisor ${D}\in{\operatorname{Div}}_{K}(E)$, we can construct a central extension.
\[def:extension\] Let ${D}\in {\operatorname{Div}}_{K}(E)$ be a divisor of degree divisible by $N$. Define $$E[N]({D})=\{(T,f)\in E[N]\times K(E)^{\times}\mid {\operatorname{div}}f ={{}^{T}\!}\!{D}-{D}\},$$ with a group operation on $E[N]({D})$ given by $$(S,g)(T,f)=(S+T,g\cdot{{}^{S}\!}\!f).$$
Indeed, we have a natural inclusion $K^{\times}\to E[N]({D})$ given by $c\mapsto (O,c)$, and an exact sequence $$1 \longrightarrow K^{\times} \longrightarrow E[N]({D})
\longrightarrow E[N] \longrightarrow 0.$$ From now on, we identify $(O,c)\in E[N]({D})$ with $c\in K^{\times}$ and consider $K^{\times}$ as a subgroup in $E[N]({D})$ (to ease the notation).
\[rem:Heisenberg\] If $N$ is odd, $E[N]({D})$ is essentially the Heisenberg group attached to $E[N]$.
\[lem:lin-equiv\] If two divisors ${D}_{1}$ and ${D}_{2}$ are linearly equivalent, then the extensions $E[N]({D}_{1})$ and $E[N]({D}_{2})$ are isomorphic.
Straightforward.
In general $E[N]({D})$ is not commutative and each element may be of infinite order. For $S,T\in E[N]$, we denote the commutator and the $N$th power by $$\begin{aligned}
\label{eq:<T,T'>}
&\<S,T\> =(S,g)(T,f)(S,g)^{-1}(T,f)^{-1}
=(g/{{}^{T}\!}\!g)\cdot({{}^{S}\!}\!f/f)
\\
&v(T)=(T,f)^{N}=f\cdot{{}^{T}\!}f\cdot{{}^{2T}\!}f\cdot\dots\cdot{{}^{(N-1)T}\!}f.\end{aligned}$$ Recall that we identify $(O,c)$ with $c$. It is easy to see that $\<S,T\>$ and $v(T)$ are independent of the choice of $f$ and $g$.
\[prop:<T,T’>\]
Let ${D}$ be a divisor of degree divisible by $N$, and let $S,T\in E[N]$. Then, we have
1. $\<S,T\>$ is a bilinear form on $E[N]$ with its value in $\mu_{N}$, and $$\<S,T\> = e_{N}(T,S)^{\frac{\deg{D}}{N}}.$$
2. $v(T)$ is determined modulo ${K^{\times}}^{N}$
Vélu [@Velu:1978 Def. 2.2 and Prop. 2.3].
The case where ${D}=N\{O\}$ is of particular interest. In this case, the results depend on the parity of $N$.
\[prop:E\[N\](NO)\] Let ${D}$ be a divisor linearly equivalent to $N\{O\}$. Let $T$ be in $E[N]$, $\ph$ a function in $K_{s}(E)$ such that ${\operatorname{div}}\ph=N\{T\}-N\{O\}$.
1. If $N$ is odd, $v$ is trivial; i.e., $v(T)\equiv 1 \bmod {K^{\times}}^{N}$.
2. Let $\tilde T$ be a point in $E[2N]$ such that $2\tilde T=T$. Then, we have $v(T)\equiv \ph(\tilde T)^{N} \bmod {K^{\times}}^{N}$. As a consequence, the homomorphism $v^{2}$ is trivial.
Vélu [@Velu:1978 Prop. 2.6].
Projective immersion of an elliptic curve associated with a cyclic subgroup
===========================================================================
In this section we assume $E[N]\subset E(K)$. Let ${D}\in {\operatorname{Div}}_{K}(E)$ be a divisor. If $\deg {D}\ge 3$, it is well known that the complete linear system $|{D}|$ gives an immersion $E\hookrightarrow |{D}|$ in a projective space. Here, we take a closer look at this fact.
\[def:L(D)\] For ${D}\in {\operatorname{Div}}_{K}(E)$ and nonnegative integer $d$, define $$\begin{aligned}
&{\mathscr{L}}^{d}({D}) =\{ h\in K(E)^{\times}\mid {\operatorname{div}}h + d{D}\geq 0\} \cup \{0\}
\\
&{\mathscr{L}}^{{*}}({D})=\textstyle{\bigoplus_{d\ge0}}\,{\mathscr{L}}^{d}({D}),
\\
&{\mathscr{S}}^{{*}}({D})=\operatorname{Sym}^{{*}}{\mathscr{L}}^{1}({D}).\end{aligned}$$
By the Riemann-Roch theorem, ${\mathscr{L}}^{d}({D})$ is a $K$-vector space of dimension $d\deg D$, and the space ${\mathscr{L}}^{{*}}({D})$ is equipped naturally with a structure of graded algebra by the multiplication of functions. There is a canonical homomorphism of graded algebras ${\mathscr{S}}^{{*}}({D})\to {\mathscr{L}}^{{*}}({D})$, and let ${\mathscr{I}}^{{*}}({D})$ be its kernel. We have the exact sequence $$0\longrightarrow {\mathscr{I}}^{{*}}({D}) \longrightarrow {\mathscr{S}}^{{*}}({D})
\longrightarrow {\mathscr{L}}^{{*}}({D}).$$
\[prop:proj-normality\] Let $D$ be an effective divisor of degree $N\geq 4$.
1. There is an exact sequence $$0\longrightarrow {\mathscr{I}}^{{*}}({D}) \longrightarrow {\mathscr{S}}^{{*}}({D})
\longrightarrow {\mathscr{L}}^{{*}}({D}) \longrightarrow 0.$$
2. $\dim{\mathscr{I}}^{2}({D})=N(N-3)/2$.
3. ${\mathscr{I}}^{{*}}({D})$ is generated by ${\mathscr{I}}^{2}({D})$, i.e., ${\mathscr{I}}^{{*}}({D})={\mathscr{I}}^{2}({D})\cdot {\mathscr{S}}^{{*}}({D})$.
4. $E\hookrightarrow {\operatorname{Proj}}{\mathscr{S}}^{{*}}({D})\simeq \P^{N-1}$ is the scheme-theoretic intersection of the quadrics which contain it.
\(1) The right exactness of this sequence is one of the equivalent definition of projective normality. For proof see Vélu [@Velu:1978 Th. 3.3]. See also Hartshorne [@Hartshorne:AG Ex. IV.4.2] and Mumford [@Mumford:1970a p. 55]. (2) follows immediately from (1) as $\dim {\mathscr{S}}^{2}({D})=N(N+1)/2$ and $\dim {\mathscr{L}}^{2}({D})=2N$.
\(3) Vélu [@Velu:1978 Th. 3.9]. (4) follows from (3).
The central extension $E[N]({D})$ has a representation in ${\mathscr{L}}^{{*}}({D})$. More precisely, we have
\[prop:L(D)-rep\]
1. For $h\in {\mathscr{L}}^{d}({D})$ and $(T,f)\in E[N]({D})$, the function ${{}^{T}\!}h\cdot f^{d}$ is in ${\mathscr{L}}^{d}({D})$.
2. If we denote by $\tau_{d}(T,f)$ the automorphism $h\mapsto {{}^{T}\!}h\cdot f^{d}$ of ${\mathscr{L}}^{d}({D})$, the map $\tau_{d}:(T,f)\mapsto \tau_{d}(T,f)$ is a representation of $E[N]({D})$ in ${\mathscr{L}}^{d}({D})$. In addition, the representations $\tau_{d}$ extend to a representation $\tau$ of $E[N]({D})$ in the group of automorphism of the graded ring ${\mathscr{L}}^{{*}}({D})$.
3. If ${D}$ is an effective divisor of degree $\geq 2$ and $d\geq 1$, the kernel of the representation $\tau_{d}$ consists of the elements $(O,f)$, where $f$ is a constant satisfying $f^{d}=1$. In particular, $\tau_{1}$ is faithful.
Vélu [@Velu:1978 Prop. 2.13].
We call $\tau$ the canonical representation of $E[N]({D})$.
\[lem:G(dd)-lin-equiv\] If ${D}_{1}$ and ${D}_{2}$ are two linearly equivalent effective divisors, the canonical representations of the extensions $E[N]({D}_{1})$ and $E[N]({D}_{2})$ are isomorphic. Moreover, if $\sigma:E[N]\to E[N]({D}_{1})$ is a section as sets of $E[N]({D}_{1})\to E[N]$, then the map $\mu\sigma$ is also a section of $E[N]({D}_{2})\to E[N]$, and the associated $2$-cocycles are equal.
Let $\ph$ be a function such that ${\operatorname{div}}\ph={D}_{2}-{D}_{1}$. Define the maps $
\nu: E[N]({D}_{1})\to E[N]({D}_{2})$ and $\mu: {\mathscr{L}}^{{*}}({D}_{1})\to {\mathscr{L}}^{{*}}({D}_{2}),
$ by $$\nu\bigl((T,f)\bigr)=\bigl(T,f\cdot{{}^{T}\!}\!\ph/\ph\bigr)
\quad\text{ and }\quad
\mu(h)=h/\ph^{d}$$ for any $(T,f)\in E[N]({D})$ and $h\in {\mathscr{L}}^{d}({D})$. Then it is easy to show that they are isomorphisms, and we have $$\tau_{d}\bigl(\nu\bigl((T,f)\bigr)\bigr)\mu(h)
=\mu\bigl(\tau_{d}(T,f)h\bigr).$$ The last statement is also straight forward.
Let $C\subset E[N]$ be a cyclic subgroup of order $N$. Suppose ${D}$ is an effective divisor of degree $N$ invariant under the translations by $C$. Then, we may choose a particular coordinate system of $\P^{N-1}$ so that the immersion $E\hookrightarrow |{D}|\simeq \P^{N-1}$ is expressed in a simple way. To do so, we consider the decomposition of ${\mathscr{L}}^{1}({D})$ into eigenspaces.
\[def:L(D,chi)\] For any character $\chi\in {\operatorname{Hom}}(C,\mu_{N})$ and a positive integer $d$, define $${\mathscr{L}}^{d}({D},\chi) = \{h\in {\mathscr{L}}^{d}({D})\mid
{{}^{U}\!}h=\chi(U)\cdot h \text{ for all $U$ in $C$} \}.$$ Then, define $${\mathscr{L}}^{d}({D}) = \textstyle{\bigoplus_{\chi}}{\mathscr{L}}^{d}({D},\chi),
\quad
\text{and}
\quad
{\mathscr{L}}^{{*}}({D})=\textstyle{\bigoplus_{d,\chi}}{\mathscr{L}}^{d}({D},\chi).$$
If $h\in {\mathscr{L}}^{d}({D},\chi)$ and $h'\in {\mathscr{L}}^{d'}({D},\chi')$, then the function $hh'$ belongs to ${\mathscr{L}}^{d+d'}\bigl({D},\chi\chi')$. As a consequence, ${\mathscr{L}}^{{*}}({D})$ has a structure of graded algebra by $\N\times {\operatorname{Hom}}(C,\mu_{N})$.
\[def:char\_S\] Let $C=\<T\>\subset E[N]$ be the cyclic subgroup generated by $T$. For any $S\in E[N]$, define the character $\chi_{S}\in {\operatorname{Hom}}(C,\mu_{N})$ by $$\chi_{S}(U)=e_{N}(S,U) \quad\text{for all $U\in C$}.$$
\[prop:L(D)\]
1. The space ${\mathscr{L}}^{1}({D},\chi)$ is $1$-dimensional.
2. If $(S,f_{S})$ is in $E[N]({D})$, the function $f_{S}$ is a base of ${\mathscr{L}}^{1}({D},\chi_{S})$.
3. The map $\tau_{d}(S,f_{S})$ from ${\mathscr{L}}^{d}({D},\chi)$ to ${\mathscr{L}}^{d}({D},\chi\chi_{S}^{d})$ is an isomorphism for all $d\ge 0$.
Let $(S,f_{S})$ be in $E[N]({D})$. Since ${\operatorname{div}}f_{S} + {D}={{}^{S}\!}{D}\ge0$, $f_{S}$ is in ${\mathscr{L}}^{1}({D})$. For $U\in C$, we have $(U,1)\in E[N]({D})$. By , we have $\<U,S\>={{{}^{U}\!}\!f_{S}}/{f_{S}}$, and by Proposition \[prop:<T,T’>\], we have $\<U,S\>=e_{N}(S,U)=\chi_{S}(U)$. Thus, we have ${{}^{U}\!}\!f_{S}=\chi_{S}(U)f_{S}$, which shows $g\in {\mathscr{L}}({D},\chi_{S})$.
If $h\in {\mathscr{L}}^{d}({D},\chi)$, then $${{}^{U}\!}({{}^{S}\!}h\cdot f_{S}^{d}) = {{}^{U+S}\!}h\cdot({{}^{U}\!}\!f_{S})^{d}=\chi(U)\cdot{{}^{S}\!}h\cdot({{}^{U}\!}\!f_{S})^{d}
=\chi(U)\chi_{S}(U)^{d}({{}^{S}\!}h\cdot f_{S}^{d}).$$ This implies $\tau_{d}(S,f_{S})$ sends ${\mathscr{L}}^{d}({D},\chi)$ to ${\mathscr{L}}^{d}({D},\chi\chi_{S}^{d})$. As a consequence, $\tau_{d}$ permutes among the spaces ${\mathscr{L}}^{d}({D},\chi)$ transitively as long as $d$ is relatively prime to $N$. In that case, the spaces ${\mathscr{L}}^{d}({D},\chi)$ have the same dimension independent of $\chi$.
By the Riemann-Roch theorem, $\dim {\mathscr{L}}^{1}({D})=\deg{D}=N$. On the other hand $$\begin{aligned}
\dim {\mathscr{L}}^{1}({D}) &= \sum_{\chi}\dim {\mathscr{L}}^{1}({D},\chi)
= \#{\operatorname{Hom}}(C,\mu_{N}) \cdot\dim {\mathscr{L}}^{1}({D},\chi_{0}) \\
&= N \dim {\mathscr{L}}^{1}({D},\chi_{0}),\end{aligned}$$ where $\chi_{0}$ is the trivial character. Thus, we conclude that $\dim {\mathscr{L}}^{1}({D},\chi)=1$ for any $\chi$, and any nonzero function $h$ in ${\mathscr{L}}^{1}({D},\chi_{S})$ is its base.
\[prop:I\^2\] For any character $\chi\in {\operatorname{Hom}}(C,\mu_{N})$, the sequence $$0 \longrightarrow {\mathscr{I}}^{2}({D},\chi)
\longrightarrow {\mathscr{S}}^{2}({D},\chi)
\longrightarrow {\mathscr{L}}^{2}({D},\chi)
\longrightarrow 0$$ is exact. Moreover, we have $$\dim {\mathscr{I}}^{2}({D},\chi) = \frac{N-4+r_{\chi}}{2},$$ where $r_{\chi}$ is a number of $\chi'\in {\operatorname{Hom}}(C,\mu_{N})$ such that $\chi'^{2}=\chi$.
Vélu [@Velu:1978 Cor. 3.6].
Canonical coordinate system
===========================
In this section we consider the projective immersion $E\hookrightarrow \bigl|N\{O\}\bigr|\simeq \P^{N-1}$. We show that, by fixing a certain level structure on $E$, we can choose a unique coordinate system with prescribed properties. The situation differs depending on the parity of $N$. The odd case is well known, but we include here for the comparison with the even case.
Odd case {#subsec:4.1}
--------
Throughout this subsection we assume $N$ to be a positive odd integer. This section serves as a prototype for the even case, and all the material in this section is written in Vélue [@Velu:1978]. See also Fisher [@Fisher:thesis], [@Fisher:2001] for geometric treatment.
Consider the action on the pair of points in $E[N]$ by an element $\sltwo(a,b;c,d)\in M_{2}(\Z)$ given by $$(S',T')=(S,T)\sltwo(a,b;c,d)=(aS+cT,bS+dT).$$ Recall that if the pair $(S, T)$ is a basis of $E[N]$ satisfying $e_{N}(S,T)=\zeta$, then the pair $(S',T')=(aS+cT,bS+dT)$ is once again a basis of $E[N]$ satisfying $e_{N}(S',T')=\zeta$ if and only if $\sltwo(a,b;c,d)\in SL_{2}(\Z)$. The kernel of this action is given by $$\Gamma(N) = \left\{\left.
\sltwo(a,b;c,d)\in SL_{2}(\Z) \,\right|\,
\sltwo(a,b;c,d)\equiv \sltwo(1,0;0,1)\bmod N\right\},$$ which is the principal congruence subgroup of level $N$ in $SL_{2}(\Z)$.
\[def:level-Gamma(N)\] Fix a primitive $N$-th root of unity $\zeta$. A *$\Gamma(N)$-structure* on an elliptic curve $E$ is a pair $(S, T)$ of points in $E[N]$ satisfying $e_{N}(S,T)=\zeta$.
From now on we fix a primitive $N$th root of unity $\zeta$ in $K_{s}$, and a $\Gamma(N)$-structure $(S,T)$. Define the divisor ${D}_{T}$ by $${D}_{T}=\sum_{n=0}^{N-1}\{nT\}.$$ Clearly, ${D}_{T}$ is invariant under the translation by the cyclic group $C=\<T\>$.
Since ${\operatornamewithlimits{\mathrm{Sum}}}_{n=0}^{N-1}nT=O$, ${D}_{T}$ is linearly equivalent to $N\{O\}$. Choose $f_{S,T}$ such that $${\operatorname{div}}f_{S,T}={{}^{S}\!}{D}_{T}-{D}_{T}.$$ Since $(S,f_{S,T})^{N}\equiv 1 \bmod {K^{\times}}^{N}$ by Proposition \[prop:E\[N\](NO)\] a), by multiplying a suitable constant if necessary, we may assume $(S,f_{S,T})^{N}=1$. Define functions $X_{0},X_{1},\dots,$ $X_{N-1}\in K(E)$ indexed by the elements of $k\in\Z/N\Z$ by $$\begin{aligned}
&X_{0} = 1,
\\
&X_{k}
=\tau(S,f_{S,T})X_{k-1}
\quad \text{for $1\le k \le N-1$.}\end{aligned}$$
\[prop:X\_k-odd\]
1. $X_{k}$ is a basis of ${\mathscr{L}}^{1}({D}_{T},\chi_{S}^{k})$ for $k\in \Z/N\Z$.
2. The actions of $(T,1)\in E[N]({D}_{T})$ is given by $$\tau(T,1)X_{k}={{}^{T}\!}X_{k}=e_{N}(S,T)^{k}X_{k}.$$
3. $X_{k}(-P)=X_{-k}(P)$ for any $P\in E$.
Interpreting the above proposition geometrically, we have
\[prop:proj-coord-odd\] Let $E$ be an elliptic curve, and $E\hookrightarrow\P^{N-1}$ be an immersion as an elliptic normal curve of degree $N$ via the complete linear system $|N\{O\}|$. Choose a primitive $N$th root of unity $\zeta$ and a $\Gamma(N)$-structure $(S,T)$. Then, there exists a unique coordinate system of $\P^{N-1}$ such that the translation maps $\tau_{S}$, $\tau_{T}$, and the multiplication-by-$(-1)$ map $[-1]$ are given by the following elements $M_{T}$, $M_{S}$ and $M_{[-1]}$ in $PGL_{N}(K(\zeta))$, respectively. $$\label{eq:matrices}
\begin{gathered}
M_{S}=
\setlength{\arraycolsep}{3pt}
\renewcommand{\arraystretch}{0.9}
\left[\begin{array}{*5c|c}
0 & & \cdots & & 0 & 1\\\hline
1 & & & & & 0 \\
& 1 & & & & \\
& & \ddots & & & \vdots \\
& & & 1 & & \\
& & & & 1 & 0
\end{array}\right],
\quad
M_{T}=
\renewcommand{\arraystretch}{1.1}
\left[\begin{array}{*5c}
1 & & & & \\
& \zeta & & & \\
& & \zeta^{2} & & \\
& & & \ddots & \\
& & & & \zeta^{N-1}
\end{array}\right],
\\
\setlength{\arraycolsep}{3pt}
\renewcommand{\arraystretch}{0.9}
M_{[-1]}=\left[\begin{array}{c|*5c}
1 & 0 & & \cdots & & 0 \\\hline
0 & & & & & 1 \\
& & & & 1 & \\
\vdots & & & \udots & & \\
& & 1 & & & \\
0 & 1 & & & &
\end{array}\right]
\end{gathered}$$
When we use the square bracket $[\quad ]$ for matrix, we mean it is a class in $PGL$ or $PSL$.
\[prop:quad-eqs-odd\] The notation being as above, we have the following. $$\begin{aligned}
&{\mathscr{I}}^{2}({D}_{T})
= \textstyle{\bigoplus_{k\in\Z/N\Z}}{\mathscr{I}}^{2}({D}_{T},\chi_{S}^{k}),\\
&{\mathscr{I}}^{2}({D}_{T},\chi_{S}^{k})={\mathscr{I}}^{2}({D}_{T})\cap \<X_{i}X_{j}\mid i+j\equiv k \mod N\>,
\\
&\dim {\mathscr{I}}^{2}({D}_{T},\chi_{S}^{k}) =(N-3)/2, \quad
\dim {\mathscr{I}}^{2}({D}_{T}) =N(N-3)/2\end{aligned}$$ In other words, $E\hookrightarrow \P^{N-1}$ is defined by $N(N-3)/2$ quadratic equations.
Even case {#subsec:4.2}
---------
In this subsection we assume $N$ to be a positive even integer. A critical difference in this case is that ${\operatornamewithlimits{\mathrm{Sum}}}_{n=0}^{N-1}nT=\frac{N}{2}T\neq O$, and thus ${D}_{T}=\sum_{n=0}^{N-1}\{nT\}$ is *not* linearly equivalent to $N\{O\}$. To define a divisor linearly equivalent to $N\{O\}$, we use a $\tilde T$ such that $2\tilde T=T$.
First we fix a primitive $N$-th root of unity $\zeta$ and a $\Gamma(N)$-structure $(S,T)$, then we consider $\Gamma(2N)$-structures on $E$ in relation to this fixed $\Gamma(N)$-structure. To do so, we first choose a primitive $2N$th root of unity $\tilde\zeta$ such that $\tilde\zeta^{2}=\zeta$.
\[def:level-2N\] Suppose $(S,T)$ is a $\Gamma(N)$-structure on $E$ with $e_{N}(S,T)=\zeta=\tilde \zeta^{2}$.
1. We say that a $\Gamma(2N)$-structure $(\tilde S,\tilde T)$ on $E$ is *above* $(S,T)$ if $$2\tilde S=S, \quad 2\tilde T=T, \quad\text{and}\quad
e_{2N}(\tilde S,\tilde T)=\tilde\zeta.$$
2. We say that two $\Gamma(2N)$-structures $(\tilde S,\tilde T)$ and $(\tilde S',\tilde T')$ are *similar* if $(\tilde S',\tilde T')$ equals either $$(\tilde S,\tilde T) \quad\text{or}\quad
(\tilde S,\tilde T)\sltwo(1+N,0;0,1+N).$$
\[lem:level-N-2N\] For a given $\Gamma(N)$-structure $(S,T)$, there are eight $\Gamma(2N)$-structures $(\tilde S,\tilde T)$ above $(S,T)$. Up to similarity, these eight are classified into four classes. If $(\tilde S,\tilde T)$ is one of them, then the following four represent the four different classes: $$(\tilde S,\tilde T), \quad
(\tilde S,\tilde T)\sltwo(1,N;0,1),\quad
(\tilde S,\tilde T)\sltwo(1,0;N,1),\quad
(\tilde S,\tilde T)\sltwo(1,N;N,1).$$
There are sixteen pairs $(\tilde S,\tilde T)$ satisfying $2\tilde S=S$ and $2\tilde T=T$. Choose one pair $(\tilde S,\tilde T)$. Replacing $\tilde S$ by $\tilde S+\frac{N}{2}S$ if necessary, we may assume $e_{2N}(\tilde S,\tilde T)=\tilde\zeta$. Thus, we obtain at least one $\Gamma(2N)$-structure above $(S,T)$. All the pairs $(\tilde S',\tilde T')$ satisfying $2\tilde S'=S$ and $2\tilde T'=T$ are given by $$(\tilde S,\tilde T) +(\tfrac{N}{2}S,\tfrac{N}{2}T)\sltwo(\epsilon_{1},\epsilon_{3};\epsilon_{2},\epsilon_{4}),
\quad \epsilon_{j}=0, \text{or } 1 \ (j=1,\dots,4).$$ Now, using properties of the Weil pairings, we have $$\begin{gathered}
e_{2N}(\tilde S+\epsilon_{1}\tfrac{N}{2}S+\epsilon_{2}\tfrac{N}{2}T,
\tilde T+\epsilon_{3}\tfrac{N}{2}S+\epsilon_{4}\tfrac{N}{2}T)
\\
=e_{2N}(\tilde S,\tilde T) e_{2N}(\tilde S,\tfrac{N}{2}T)^{\epsilon_{4}}e_{2N}(\tfrac{N}{2}S,\tilde T)^{\epsilon_{1}}
=(-1)^{\epsilon_{1}+\epsilon_{4}}e_{2N}(\tilde S,\tilde T).\end{gathered}$$ Thus, the Weil pairing coincide if and only if $\epsilon_{1}=\epsilon_{4}$. This implies that there are eight different $\Gamma(2N)$-structures above $(S,T)$. It is also easy to see that the four pairs above represent the different classes up to similarity.
In view of Lemma \[lem:level-N-2N\], we define a subgroup $\Gamma^{(N)}(2N)$ of $\Gamma(N)$ by $$\Gamma^{(N)}(2N) =\left\{\left.\sltwo(a,b;c,d) \in SL_{2}(\Z)\,\right|\,
a\equiv d\equiv 1 \bmod N, b\equiv c\equiv 0\bmod 2N\right\}.$$ It is easy to see that $$\Gamma(N)\vartriangleright\Gamma^{(N)}(2N)\vartriangleright\Gamma(2N).$$ Since $N$ is even, we have $$\begin{gathered}
\Gamma(N)/\Gamma^{(N)}(2N)
=\left\<\sltwo(1,N;0,1), \sltwo(1,0;N,1)\right\>
\simeq \Z/2\Z\times\Z/2\Z,
\\
\Gamma^{(N)}(2N)/\Gamma(2N)
=\left\<\sltwo(1+N,0;0,1+N)\right\>
\simeq \Z/2\Z.\end{gathered}$$
If $N$ is odd, we have $\Gamma^{(N)}(2N)$ conincide with $\Gamma(2N)$, and $[\Gamma(N):\Gamma^{(N)}(2N)]=6$.
\[def:Gamma\^(N)(2N)-structure\] Fix a primitive $N$th root of unity $\zeta$, and choose $\tilde \zeta$ such that $\tilde\zeta^{2}=\zeta$. Let $(S,T)$ be a $\Gamma(N)$-structure. By a *$\Gamma^{(N)}(2N)$-structure* we mean an equivalence class of $\Gamma(2N)$-structure $(\tilde S,\tilde T)$ above $(S,T)$ modulo similarity.
We now fix a $\Gamma^{(N)}(2N)$-structure $(\tilde S, \tilde T)$ above $(S,T)$. Define $${D}_{\tilde T}=\sum\limits_{n=0}^{N-1}\bigl\{\tilde T+nT\bigr\}.$$
\[lem:D-tilde-T\]
1. The divisor ${D}_{\tilde T}$ is linearly equivalent to $N\{O\}$, and there is a function $\ph_{\tilde T}$ such that ${\operatorname{div}}\ph_{\tilde T}={D}_{\tilde T}-N\{O\}$.
2. Suppose $\tilde T'$ is another point satisfying $2\tilde T'=T$, and let $T_{2}=\tilde T'-\tilde T\in E[2]$. Then we have $${D}_{\tilde T'}={{}^{T_{2}}\!}{D}_{\tilde T}.$$
\(1) Since ${\operatornamewithlimits{\mathrm{Sum}}}_{E}{D}_{\tilde T}=N\tilde T+\frac{N}{2}T=O$, the assertion follows immediately from Able’s Theorem (Theorem \[th:able\]). (2) follows immediately from the definition.
We may still use ${D}_{T}=\sum_{n=1}^{N-1}nT$ to obtain an immersion to $\P^{N-1}$ even though ${\operatornamewithlimits{\mathrm{Sum}}}_{E}{D}_{T}\neq O$. This is how Hurwitz [@Hurwitz] and Stevens [@Stevens] studied.
For a given $\Gamma^{(N)}(2N)$-structure $(\tilde S, \tilde T)$, there is a unique function $f_{\tilde S,\tilde T}$ satisfies the condition $${\operatorname{div}}f_{\tilde S,\tilde T}={{}^{S}\!}\!{D}_{\tilde T} - {D}_{\tilde T},
\quad f_{\tilde S,\tilde T}(\tilde S)=1.$$ In other words, $(S,f_{\tilde S,\tilde T})$ is in $E[N]({D}_{\tilde T})$.
\[def:X\_k-even\] Define $X^{(\tilde S,\tilde T)}_{0},X^{(\tilde S,\tilde T)}_{1},\dots,X^{(\tilde S,\tilde T)}_{N-1}\in {\mathscr{L}}(D_{\tilde T})$ indexed by the elements of $\Z/N\Z$ by $$\begin{aligned}
&X^{(\tilde S,\tilde T)}_{0} = 1,
\\
&X^{(\tilde S,\tilde T)}_{k}
=\tau(S,f_{\tilde S,\tilde T})X^{(\tilde S,\tilde T)}_{k-1}
\quad \text{for $1\le k \le N-1$.}\end{aligned}$$ If $(\tilde S,\tilde T)$ is understood, we write $X^{(\tilde S,\tilde T)}_{k}=X_{k}$ for simplicity.
\[prop:X\_k-even\]
1. $X_{k}$ is well-defined, i.e., $$\tau(S,f_{\tilde S,\tilde T})X_{N-1} = X_{0}$$
2. $X_{k}$ is a basis of ${\mathscr{L}}^{1}({D}_{\tilde T},\chi_{S}^{k})$ for $k\in \Z/N\Z$.
3. The actions of $(T,1)\in E[N]({D}_{\tilde T})$ is given by $$\tau(T,1)X_{k}=e_{N}(S,T)^{k}X_{k}.$$
\(1) follows from the fact $(S,f_{\tilde S,\tilde T})^{N}=1$, which is by Proposition \[prop:E\[N\](NO)\] b).
\(2) and (3) follow immediately form Proposition \[prop:L(D)\] (1) and (3)
\[prop:X\_k-2N-str\]
1. If $\tilde T'$ is another point such that $2\tilde T'=T$, then $$X_{k}^{(\tilde S,\tilde T')}
=\begin{cases}
X_{k}^{(\tilde S,\tilde T)} & \text{ if $e_{2}(\tilde T'-\tilde T,\frac{N}{2}T)=1$,}\\[\medskipamount]
X_{k+\frac{N}{2}}^{(\tilde S,\tilde T)}& \text{ if $e_{2}(\tilde T'-\tilde T,\frac{N}{2}T)=-1$.}
\end{cases}$$
2. If $\tilde S'$ is another point such that $2\tilde S'=S$, then $$X_{k}^{(\tilde S',\tilde T)}
=\begin{cases}
X_{k}^{(\tilde S,\tilde T)} & \text{ if $e_{2}(\tilde S'-\tilde S,\frac{N}{2}S)=1$,}\\[\smallskipamount]
(-1)^{k}X_{k}^{(\tilde S,\tilde T)} & \text{ if $e_{2}(\tilde S'-\tilde S,\frac{N}{2}S)=-1$.}
\end{cases}$$
See Vélu [@Velu:1978 Lemma 2.7].
\[prop:level-2N\] Let $(\tilde S,\tilde T)$ be a $\Gamma^{(N)}(2N)$-structure above a $\Gamma(N)$-structure $(S,T)$, and let $X_{k}^{(\tilde S,\tilde T)}$ $(k=0,\dots,N)$ be the functions defined in Definition \[def:X\_k-even\]. The functions $X_{k}^{(\tilde S,\tilde T)}$ are uniquly determined by the $\Gamma^{(N)}(2N)$-structure. More precisely, if $(\tilde S',\tilde T')$ is another $\Gamma(2N)$-structure above $(S,T)$, then $X_{k}^{(\tilde S',\tilde T')}=X_{k}^{(\tilde S,\tilde T)}$ for all $k$ if and only if $(\tilde S',\tilde T')$ is similar to $(\tilde S,\tilde T)$.
This is an immediate consequence of Proposition \[prop:X\_k-2N-str\].
\[prop:proj-coord-even\] Let $E$ be an elliptic curve, and $E\hookrightarrow\P^{N-1}$ be an immersion as an elliptic normal curve of degree $N$ via the complete linear system $|N\{O\}|$. Choose a primitive $N$th root of unity $\zeta$, and $\tilde \zeta$ such that $\tilde\zeta^{2}=\zeta$. Let $(\tilde S,\tilde T)$ be a $\Gamma^{(N)}(2N)$-structure above a $\Gamma(N)$-structure $(S,T)$. Then, there exists a unique coordinate system of $\P^{N-1}$ such that the translation maps $\tau_{S}$, $\tau_{T}$, and the multiplication-by-$(-1)$ map $[-1]$ are given by the matrices $M_{T}$, $M_{S}$ and $M_{[-1]}$ in , respectively.
For a given $\Gamma(N)$-structure $(S,T)$, there are four different choices of such coordinate systems related by the change of coordinates of $\P^{N-1}$ given by the transition matrices generated by the following two: $$M_{T}^{\frac{N}{2}}=
\left[
\begin{array}{*5r}
1 & 0 & & & \\
0 & -1& & O & \\
& & \ddots & & \\
& O & & 1 & 0 \\
& & & 0& -1
\end{array}
\right],
\quad
\setlength{\arraycolsep}{3pt}
\renewcommand{\arraystretch}{0.8}
M_{S}^{\frac{N}{2}}=
\left[
\begin{array}{*3c|*3c}
& & & & &\\
& O & & & I_{\frac{N}{2}} & \\
& & & & &\\
\hline
& & & & &\\
& I_{\frac{N}{2}} & & & O \\
& & & & &
\end{array}
\right],$$
The existence of such a coordinate system follows from Proposition \[prop:X\_k-even\], and the uniqueness follows from Proposition \[prop:level-2N\]. The last part follows from Proposition \[prop:X\_k-2N-str\].
\[prop:quad-eqs-even\] The notation being as above, we have the following. $$\begin{aligned}
&{\mathscr{I}}^{2}({D}_{\tilde T})
= \textstyle{\bigoplus_{k\in\Z/N\Z}}{\mathscr{I}}^{2}({D}_{\tilde T},\chi_{S}^{k}),\\
&{\mathscr{I}}^{2}({D}_{\tilde T},\chi_{S}^{k})={\mathscr{I}}^{2}({D}_{\tilde T})\cap \<X_{i}X_{j}\mid i+j\equiv k \mod N\>,
\\
&\dim {\mathscr{I}}^{2}({D}_{\tilde T},\chi_{S}^{k}) =
\begin{cases}
(N-2)/2 & \text{if $k\equiv 0\bmod 2$,}\\(N-4)/2 & \text{if $k\equiv 1\bmod 2$,}
\end{cases}
\\
&\dim {\mathscr{I}}^{2}({D}_{\tilde T}) =N(N-3)/2.\end{aligned}$$
To find a basis of ${\mathscr{I}}^{2}({D}_{\tilde T})$, we need to find a basis of ${\mathscr{I}}^{2}({D}_{\tilde T},\chi_{0})$ and ${\mathscr{I}}^{2}({D}_{\tilde T},\chi_{S})$, which we will do for the case $K=\C$, the field of complex numbers using theta functions.
Modular curves
==============
Suppose $N$ is even. Fix a primitive $N$th root of unity $\zeta$, and fix $\tilde \zeta$ such that $\tilde\zeta^{2}=\zeta$. Let $(E, (S,T), (\tilde S, \tilde T))$ be a triple such that $E$ is an elliptic curve, $(S,T)$ a $\Gamma(N)$-structure on $E$, and $(\tilde S,\tilde T)$ a $\Gamma^{(N)}(2N)$-structure above $(S,T)$. Such triples are parametrized by the modular curve corresponding to the modular group $\Gamma^{(N)}(2N)$. We denote it by $Y^{(N)}(2N)$, and its compactification by $X^{(N)}(2N)$.
As we have seen in §2.3, such a triple $(E, (S,T), (\tilde S, \tilde T))$ determines a unique immersion $\iota:E \hookrightarrow \P^{N-1}$. If we choose two different $\Gamma^{(N)}(2N)$ structures $(\tilde S,\tilde T)$ and $(\tilde S',\tilde T')$ related by $$(\tilde S',\tilde T')=(\tilde S,\tilde T)\sltwo(a,b;c,d),
\quad \sltwo(a,b;c,d)\in SL_{2}(\Z),$$ two different coordinate systems are related by $\rho\bigl(\sltwo(a,b;c,d)\bigr)$ of the change of coordinates of $\P^{N-1}$. We thus have a representation $$\rho: SL_{2}(\Z) \to PGL_{n}(\bar K); \quad {\textstyle \sltwo(a,b;c,d) \mapsto \rho\left(\sltwo(a,b;c,d)\right)}.$$ In the coordinate system through the immersion using $(\tilde S',\tilde T')$, the translations $\tau_{S}$ and $\tau_{T}$ are represented by the same matrices $M_{S}$ and $M_{T}$ as before. Since $u\tilde S+v\tilde T=(du-bv)\tilde S'+(-cu+av)\tilde T'$, we have the following relation $$\rho\begin{pmatrix} a & b \\c & d\end{pmatrix}^{-1}
M_{S}^{u}M_{T}^{v}
\,\rho\sltwo(a,b;c,d)
=M_{S}^{du-bv}M_{T}^{-cu+av} \quad\text{ in } PGL_{N-1}(\bar K).$$ Recall that $SL_{2}(\Z)$ is generated by $$A=\sltwo(0,-1;1,0)
\quad \text{and} \quad
B=\sltwo(1,1;0,1),$$ and it is characterized by the relations among them: $$SL_{2}(\Z)=\<A,B\mid (AB)^{3}=A^{2}, B^{4}=I\>.$$ The lemma below shows the candidates of $\rho(A)$ and $\rho(B)$.
\[lem:rep\] [(1)]{} Let $R$ be an element of $PGL_{N-1}(\bar K)$ satisfying the relations $$R^{-1}M_{S}R = M_{T}^{-1}, \quad
R^{-1}M_{T}R = M_{S}, \quad
R^{-1}M_{[-1]}R = M_{[-1]},$$ then $R$ is one of the following. $$\begin{aligned}
&A_{0}=\left[\renewcommand{\arraystretch}{1.25}
\begin{matrix}
1 & 1 & 1 &\cdots & 1 \\
1 & \zeta^{1} & \zeta^{2} &\cdots & \zeta^{(N-1)}\\
1 & \zeta^{2} & \zeta^{4} &\cdots & \zeta^{2(N-1)}\\
\vdots & \vdots & \vdots & \ddots &\vdots\\
1 & \zeta^{(N-1)} & \zeta^{2(N-1)} & \cdots & \zeta^{(N-1)^{2}}
\end{matrix}\right]
=\bigl[\zeta^{ij}\bigr]_{i,j=0}^{N-1},
\\
&A_{0}M_{S}^{\frac{N}{2}}, \quad
A_{0}M_{T}^{\frac{N}{2}}, \quad
A_{0}M_{S}^{\frac{N}{2}}M_{T}^{\frac{N}{2}}.\end{aligned}$$
[(2)]{} Let $R$ be an element of $PGL_{N}(\bar K)$ satisfying the relations $$R^{-1}M_{P}R = M_{P}, \quad
R^{-1}M_{Q}R = M_{P}^{-1}M_{Q}, \quad
R^{-1}M_{[-1]}R = M_{[-1]},$$ then $R$ is one of the following. $$\begin{aligned}
&B_{0}=\setlength{\arraycolsep}{2pt}
\left[\begin{array}{cclclc}
1 & & & & & \\
& \tilde\zeta^{1(N-1)} & & & & \\
& & \ddots & & & \\
& & & \tilde\zeta^{i(N-i)} & & \\
& & & & \ddots & \\
& & & & & \tilde\zeta^{(N-1)\cdot1}
\end{array}\right]
=\operatorname{Diag}\bigl[\tilde\zeta^{i(N-i)}\bigr]_{i=0}^{N-1},
\\
&B_{0}M_{S}^{\frac{N}{2}}, \quad
B_{0}M_{T}^{\frac{N}{2}}, \quad
B_{0}M_{S}^{\frac{N}{2}}M_{T}^{\frac{N}{2}}.\end{aligned}$$ where $\tilde\zeta$ is a primitive $2N$th root of unity with $\tilde\zeta^2=\zeta$. In particular, $R$ is an element of order $2N$.
\[prop:rep\] There is a representation $\rho:SL_{2}(\Z)\to PGL_{N-1}(K(\tilde\zeta))$ satisfying $$\rho\sltwo(0,-1;1,0) = A_{0}
\quad \text{and} \quad
\rho\sltwo(1,1;0,1) = B_{0},$$ where $A_{0}$ and $B_{0}$ are in Lemma \[lem:rep\]. Its kernel equals $\Gamma^{(N)}(2N)$.
Since $SL_{2}(\Z)$ is characterized by the relation $(AB)^{3}=A^{2}$ and $A^{4}=I$, it suffices to verify that $(A_{0}A_{0})^{3}=A_{0}^{2}$ and $A_{0}^{4}=I$. This can be done by straightforward calculations.
Since $\ord(B_{0})=2N$, the kernel of $\rho$ contains $\Gamma(2N)$ but does not contain $\Gamma(N)$. Also, we have the relation $$\sltwo(1,N;0,1)
\sltwo(1,0;N-1,1)
\sltwo(1,N;0,1)
\sltwo(1,0;1,1)
\equiv
\sltwo(1+N,0;0,1+N) \bmod 2N,$$ and a simple calculation shows that $$B_{0}^{N}(A_{0}^{-1}B_{0}^{-1}A_{0})B_{0}^{N}(A_{0}^{N-1}B_{0}A_{0})
=I_{N}.$$ This implies $\ker \rho=\Gamma^{(N)}(2N)$.
Let $\iota$ be the immersion associated with a triple $(E, (S,T), (\tilde S, \tilde T))$. Then, associating $\iota(O)\in \P^{N-1}$ to the triple $(E, (S,T), (\tilde S, \tilde T))$, we obtain a map $Y^{(N)}(2N)\to \P^{N-1}$. Since $\iota(O)$ is fixed by $M_{[-1]}$, it is contained in the linear subspace $H$ of $\P^{N-1}$ fixed by $M_{[-1]}$. $H$ is given by the equations $X_{k}=X_{N-k}$, $k=1,\dots, \tfrac{N}{2}-1$. Thus the coordinate of $\iota(O)$ is of the form $$\label{eq:iota(O)}
(a_{0}:a_{1}:\dots:a_{\frac{N}{2}-1}:a_{\frac{N}{2}}:a_{\frac{N}{2}-1}:\dots:a_{2}:a_{1}).$$ Let $E_{Y^{(N)}(2N)}$ be the universal elliptic curve over $Y^{(N)}(2N)$, and $o$ the $0$-section $Y^{(N)}(2N)\to E_{Y^{(N)}(2N)}$. We have the diagram $$\begin{CD}
E_{Y^{(N)}(2N)} @>>> \P^{N-1} \\
@A o AA @AAA\\
Y^{(N)}(2N) @>>> H \simeq \P^{\frac{N}{2}}.
\end{CD}$$
The linear space $H$ is stable under the representation $\rho:SL_{2}(\Z)\to PGL_{N-1}(K(\tilde\zeta))$ in Proposition \[prop:rep\]. Moreover, $\rho(-I_{2})=M_{[-1]}$ acts trivially on $H$, and $\ker\rho=\Gamma^{(N)}(2N)$. Thus, $\rho$ induces a representation $$\bar\rho: SL_{2}(\Z)/(\pm\Gamma^{(N)}(2N)) \to PGL_{\frac{N}{2}+1}(K(\tilde\zeta)).$$ Define the coordinates $(\bar X_{0}:\bar X_{1}:\dots:\bar X_{\frac{N}{2}})$ of $H$ by $$\renewcommand{\arraystretch}{1.2}
\left\{
\begin{array}{l}
\bar X_{0} = X_{0}, \\
\bar X_{1} = X_{1}+X_{N-1}, \\
\bar X_{2} = X_{2}+X_{N-2}, \\
\qquad \dots \\
\bar X_{\frac{N}{2}-1} = X_{\frac{N}{2}-1}+X_{\frac{N}{2}+1},\\
\bar X_{\frac{N}{2}} = X_{\frac{N}{2}}.
\end{array}\right.$$ Then, $\bar\rho$ is given by $$\begin{gathered}
\bar\rho\sltwo(0,-1;1,0)
=\left[\renewcommand{\arraystretch}{1.25}
\setlength{\arraycolsep}{3pt}
\begin{matrix}
1 & 1 & 1 &\cdots & 1 \\
2 & \zeta+\zeta^{-1} & \zeta^{2}+\zeta^{-2} & \cdots & -2\\
2 & \zeta^{2}+\zeta^{-2} & \zeta^{4}+\zeta^{-4} &\cdots & -2\\
\vdots & \vdots & \vdots & \ddots &\vdots\\
1 & -1 & 1 & \cdots & (-1)^{\frac{N}{2}}
\end{matrix}\right], \\
\bar\rho\sltwo(1,1;0,1)
=\setlength{\arraycolsep}{1pt}
\left[\begin{array}{*5c}
1 & & & & \\
& \tilde\zeta^{1(N-1)} & & & \\
& & \tilde\zeta^{2(N-2)} & & \\
& & & \ddots & \\
& & & & \tilde\zeta^{N^{2}/4}\\
\end{array}\right].\end{gathered}$$ In particular, we have $$\begin{gathered}
\bar\rho\sltwo(1,N;0,1)
=
\left[\addtolength{\arraycolsep}{0.5pt}\begin{array}{*5c}
1 & & & &\\
& -1 & & \\
& & 1 & & \\
& & & \ddots &\\
& & & &(-1)^{\frac{N}{2}}
\end{array}\right],
\\ \bar\rho\sltwo(1,0;N,1)
=\addtolength{\arraycolsep}{4pt}
\left[\begin{array}{*5c}
& & & & 1 \\
& & & 1 & \\
& & \udots & & \\
& 1 & & & \\
1 & & & &
\end{array}\right].\end{gathered}$$
Later we will show that the coordinates of $\iota(O)$ satisfies a set of quartic equations. In §§8–10, we will study these equations in more detail.
Theta functions
===============
From now on, we assume that the base field is the field of complex numbers $\C$. We construct functions $X_{0},X_{1},\dots,X_{N-1}$ in §4 using theta functions. Then, we find quadratic relations among them from the relations among theta functions.
\[def:theta\] For a pair of real numbers $(p,q)$, we define the theta function $\theta_{(p,q)}(z,\tau)$ with characteristic $(p,q)$ by $$\theta_{(p,q)}(z,\tau):=\sum_{n\in\Z}
{\mathbf{e}}\bigl({\tfrac{1}{2}(n+p)^2\tau + (n+p)(z+q)}\bigr),$$ where $z\in\C$ and $\tau\in\H=\{\tau\in\C\mid \Im\tau>0\}$, and ${\mathbf{e}}(x)=e^{2\pi i x}$.
This conforms with the definition in Mumford [@Mumford:Tata-I Ch. I. §3]. We have the following fundamental formulas.
\[prop:theta-basic\] Suppose $p,q,r,s\in \R$, and $l,m\in\Z$. Then, we have
1. $\theta_{(p,q)}(z+s,\tau) = \theta_{(p,q+s)}(z,\tau)$.
2. $\theta_{(p,q)}(z+r\tau,\tau)=
{\mathbf{e}}(-\frac{1}{2}r^{2}\tau-rz-rq)\theta_{(p+r,q)}(z,\tau)$.
3. $\theta_{(p+l,q+m)}(z,\tau)
= {\mathbf{e}}(pm)\,\theta_{(p,q)}(z,\tau)$.
Let $N$ be a positive integer. Although we are interested mainly in the case where $N$ is even, we do not restrict ourselves to even $N$ throughout this section.
\[def:theta-N\] Let $N$ be a positive integer. For an integer or a half-integer $k$, define $$\begin{aligned}
\theta^{(N)}_k(z,\tau)
&:=\theta_{(\frac{1}{2}-\frac{k}{N},\frac{N}{2})}(Nz,N\tau)
\\
&\phantom{:}
=\sum_{n\in\Z}{\mathbf{e}}\Bigl(
{\tfrac{1}{2}N\bigl(n-\tfrac{k}{N}+\tfrac1{2}\bigr)^2\tau
+N\bigl(n-\tfrac{k}{N}+\tfrac1{2}\bigr)\bigl(z+\tfrac12\bigr)}\Bigr).\end{aligned}$$
It is easy to verify that $\theta^{(N)}_{k+N}(z,\tau)=
\theta^{(N)}_k(z,\tau)$, and thus $\theta^{(N)}_k(z,\tau)$ depends only on the class $k\bmod N$.
Basic properties of $\theta^{(N)}_{k}(z,\tau)$ as a function of $z$
-------------------------------------------------------------------
First, we fix a positive integer $N$ and a point $\tau\in \H$.
\[prop:theta-basic2\] For any positive integer $N\in \N$, and any integer or half-integer $k\in \frac{1}{2}\Z$, the following relations hold.
1. $\theta^{(N)}_k(z+1,\tau) = (-1)^{N+2k}\,\theta^{(N)}_k(z,\tau)$.
2. $\theta^{(N)}_k(z+\tau,\tau)
= (-1)^{N}{\mathbf{e}}\bigl(-\frac{N}{2}\tau - Nz\bigr)\,\theta^{(N)}_k(z,\tau)$.
3. $\theta^{(N)}_k\bigl(z+\tfrac{1}{N},\tau\bigr)
= -{\mathbf{e}}\bigl(-\frac{k}{N}\bigr)\theta^{(N)}_k(z,\tau)$.
4. $\theta^{(N)}_k\bigl(z+\tfrac{\tau}{N},\tau\bigr)
= -{\mathbf{e}}\bigl(-\frac{\tau}{2N} -z\bigr)\,\theta^{(N)}_{k-1}(z,\tau)$.
5. $\theta^{(N)}_k\bigl(z+\tfrac{\tau}{2N},\tau\bigr)
= {\mathbf{e}}\bigl(-\tfrac{\tau}{8N} -\tfrac{z}{2}-\tfrac{1}{4}\bigr)\,\theta^{(N)}_{k-\frac{1}{2}}(z,\tau)$.
6. $\theta^{(N)}_k(-z,\tau) = (-1)^{N+2k}\theta^{(N)}_{-k}(z,\tau)$.
These formulas follow easily from the definition and Proposition \[prop:theta-basic\]. See also [@Mumford:Tata-I] for details.
From now on, we denote $\theta^{(N)}_k(z,\tau)$ simply by $\theta_k(z)$.
\[lem:zeros\] Let $Z^{(N)}_{k}$ be the set of zeros of $\theta_{k}(z)$.
1. If $N$ is odd, then we have $Z^{(N)}_{k}=\left\{\left.\frac{m}{N}+\frac{k}{N}\tau+n\tau\,\right|\, m,\,n\in\Z\right\}$. $$\includegraphics[scale=0.8]{zeros-n-odd.eps}$$
2. If $N$ is even, then $Z^{(N)}_{k}=\left\{\left.\frac{1}{2N}+\frac{m}{N}+\frac{k}{N}\tau+n\tau\,\right|\, m,\,n\in\Z\right\}$. $$\includegraphics[scale=0.8]{zeros-n-even.eps}$$
It follows easily from the fact that the set of zeros of $\theta_{(p,q)}(z,\tau)$ is given by $\{(l+\frac{1}{2}-p)\tau+
(m+\frac{1}{2}-q)\mid l,m\in\Z\}$.
\[lem:lin-indep\] $\theta_{0}(z), \theta_{1}(z),\dots,\theta_{N-1}(z)$ are linearly independent over $\C$.
This is easily seen by looking at the series expansions of $\theta_i(z)$’s.
\[def:S-T-in-C\] For $\tau\in\H$, let $\Lambda_{\tau}=\<1,\tau\>$ be the lattice in $\C$ spanned by $1$ and $\tau$, and let $E_{\tau}=\C/\Lambda_{\tau}$ be the elliptic curve with modulus $\tau$. For $N\in \N$, define points $S,T\in E_{\tau}[N]$, and $ \tilde S,\tilde T \in E_{\tau}[2N]$ by $$\begin{array}{ll}
S=\frac{\tau}{N} \bmod \Lambda_{\tau},&T=\frac{1}{N} \bmod \Lambda_{\tau},
\\[\medskipamount]
\tilde S=\frac{\tau}{2N} \bmod \Lambda_{\tau}, \quad &\tilde T=\frac{1}{2N} \bmod \Lambda_{\tau}.
\end{array}$$
\[lem:weil-pairing-in-C\] Let $\zeta_{N}={\mathbf{e}}\bigl(\tfrac{1}{N})=e^{2\pi i/N}$ and $\tilde \zeta_{N}=\zeta_{2N}={\mathbf{e}}\bigl(\tfrac{1}{2N})=e^{2\pi i/2N}$. Then $(S,T)$ is a level $N$ structure with $e_{N}(S,T)=\zeta_{N}$ and $(\tilde S,\tilde T)$ is a level $2N$ structure above $(S,T)$ with $\tilde \zeta_{N}^{2}=\zeta_{N}$.
Define the function $f(z)$ on $\C$ by $$f(z)=
\frac{\theta_{1}(z)}
{\theta_{0}(z)}
\quad \text{if $N$ is odd,}
\quad
f(z)=\frac{\theta_{1}(z-\frac{1}{2N})}
{\theta_{0}(z-\frac{1}{2N})}
\quad\text{if $N$ is even.}$$ Then, the divisor of $f$ is given by $$\operatorname{div} f = \sum_{j=0}^{N-1}
\left(\bigl\{\tfrac{\tau}{N}+\tfrac{j}{N}\bigr\}
- \bigl\{\tfrac{j}{N}\bigr\}\right).$$ Thus, by Theorem \[thm:weil-pairing\] and Proposition \[prop:theta-basic2\] (3), we have $$e_{N}(S,T)
= e_{N}\left(\tfrac{\tau}{N}\bmod \Lambda_{\tau},
\tfrac{1}{N}\bmod \Lambda_{\tau}\right)
=\frac{{{}^{T}\!}\!f(z)}{f(z)}=\frac{f\left(z-\frac{1}{N}\right)}{f(z)}
= \zeta_{N}.$$ By the same token we have $e_{2N}(\tilde S,\tilde T)=\zeta_{2N}=\tilde\zeta_{N}$, which implies $(\tilde S,\tilde T)$ is above $(S,T)$.
Projective immersion via theta functions
----------------------------------------
\[prop:immersion\] Let $N\in \N$ and $\tau\in\H$, and let $\Lambda_{\tau}=\<1,\tau\>$ be the lattice in $\C$ spanned by $1$ and $\tau$. The map $$z \longmapsto \bigl(\theta_{0}(z),\theta_{1}(z),
\dots,\theta_{N-1}(z)\bigr) \in \C^{N}.$$ induces an immersion $\Theta$ of the elliptic curve $E_{\tau}=\C/\Lambda_{\tau}$ into the projective space $\P^{N-1}$[:]{} $$\begin{array}{rcccc}
\Theta:&E_{\tau}=\hskip-\arraycolsep&\C/\Lambda_{\tau} & \longrightarrow & \P^{N-1}
\\
&&z & \longmapsto & \bigl(\theta_{0}(z):\theta_{1}(z):
\dots:\theta_{N-1}(z)\bigr).
\end{array}$$ The image $E_{\Theta}$ of $\Theta$ is a curve of degree $N$ defined as the intersection of $N(N-3)/2$ quadrics in $\P^{N-1}$.
Let $S,T,\tilde S$, and $\tilde T$ be as in Definition \[def:S-T-in-C\].
If $N$ is odd, let ${D}_{T}=\sum_{k=0}^{N-1}\{kT\}=\sum_{k=0}^{N-1}\{\frac{k}{N}\}$. Then, for $k\in\Z/N\Z$, we have ${\operatorname{div}}\theta_{k}(z)/\theta_{0}(z)={{}^{kS}\!}{D}_{T}-{D}_{T}$, and it has the same divisor as $X_{k}$ in §\[subsec:4.1\]. This shows that $\Theta$ is an immersion to $\P^{N-1}$ and has the properties in the statement.
If $N$ is even, let $D_{\tilde T}=\sum_{k=0}^{N-1}\{\tilde T+kT\}=\sum_{k=0}^{N-1}\{\frac{1}{2N}+\frac{k}{N}\}$. Then, for $k\in\Z/N\Z$, we have ${\operatorname{div}}\theta_{k}(z)/\theta_{0}(z)={{}^{kS}\!}{D}_{\tilde T}-{D}_{\tilde T}$, and it has the same divisor as $X^{(\tilde S,\tilde T)}_{k}$ in Definition \[def:X\_k-even\]. Again, this shows that $\Theta$ has the required properties in the statement.
\[thm:theta-immersion\] Let $N\in \N$ and $\tau\in\H$ be as in Proposition \[prop:immersion\]. Let $E_{\Theta}$ be the image of $\Theta$ in Proposition \[prop:immersion\]. Then, $E_{\Theta}$ is an elliptic curve whose group operation is such that $\Theta$ is a homomorphism. By an abuse of notatin, we denote $\Theta\bigl(\frac{\tau}{N}\bigr)$ and $Q=\Theta\bigl(\frac{1}{N}\bigr)$ also by $S$ and $T$. They are points of order $N$ in $E_{\Theta}$, and we have $$\begin{aligned}
&\bigl(\theta_{0}(z):\theta_{1}(z):
\dots:\theta_{N-1}(z)\bigr) + S
= \bigl(\theta_{N-1}(z):\theta_{0}(z):\theta_{1}(z):
\dots:\theta_{N-2}(z)\bigr),
\\
&\bigl(\theta_{0}(z):\theta_{1}(z):
\dots:\theta_{N-1}(z)\bigr) + T
= \bigl(\theta_{0}(z):\zeta_{N}^{-1}\theta_{1}(z):
\dots:\zeta_{N}^{-N+1}\theta_{N-1}(z)\bigr).\end{aligned}$$ In other words, the translations of $E_{\Theta}$ by $S$ and $T$ are given by the matrices $M_{S}$ and $M_{T}$ in , respectively. Moreover, the map $[-1]$ on $E_{\Theta}$ is given by the matrix $M_{[-1]}$ in . If $N$ is odd, the coordinate system of $E_{\Theta}\in \P^{N-1}$ given by $\Theta$ is exactly the one described in Propositions \[prop:proj-coord-odd\]. If $N$ is even, it is exactly the one described in Propositions \[prop:proj-coord-even\] with the level $2N$ structure $(\tilde S,\tilde T)=\bigl((\Theta\bigl(\frac{\tau}{2N}\bigr),\Theta\bigl(\frac{1}{2N}\bigr)\big)$.
The formulas for the translations by $S$ and $T$, and the inversion $[-1]$ are consequences of Proposition \[prop:theta-basic2\] (3)(4)(5), respectively. The latter part of the statement follows from Proposition \[prop:immersion\].
Transformation formula for $\theta^{(N)}_{k}(z,\tau)$
-----------------------------------------------------
We now consider $\theta^{(N)}_{k}(z,\tau)$ as a function of $\tau$. The matrix $\sltwo(a,b;c,d)\in SL_{2}(\Z)$ acts on the upper half plane by $\tau\mapsto \frac{a\tau+b}{c\tau+d}$. We would like to know how this $SL_{2}(\Z)$ action affects the immersion $\Theta:E_{\tau}\to\P^{N-1}$.
The following transformation formula for the theta function is classical and fundamental.
\[lem:tr\_formula\] Let $M=\sltwo(a,b;c,d)\in SL_{2}(\Z)$. Then we have $$\label{eq:tr_formula}
\theta_{(p,q)}\Bigl(\frac{z}{c\tau+d},\frac{a\tau+b}{c\tau+d}\Bigr)
=\kappa(M){\mathbf{e}}(\phi_{M}(p,q)){\mathbf{e}}\bigl(\tfrac{cz^{2}}{2(cz+d)}\bigr)\sqrt{c\tau+d}\,
\theta_{(p',q')}(z,\tau),$$ where $\kappa(M)$ is an eighth root of unity that does not depend on neither $\tau$ nor $(p,q)$, $$\begin{aligned}
\phi_{M}(p,q) &= -\frac{1}{2}\bigl(abp^{2} + 2bcpq + cdq^{2}
- bd(ap+cq)\bigr), \\
(p',q')&=\bigl(ap+cq-\frac{1}{2}ac, bp+dq-\frac{1}{2}bd\bigr),\end{aligned}$$ and $\sqrt{c\tau+d}$ is the principal value.
\[prop:tr\_formula\] The following transformation formulas hold. $$\begin{aligned}
&\theta_k^{(N)}(z,\tau+1)=c\, {\mathbf{e}}\bigl(-\tfrac{k(N-k)}{2N}\bigr)\, \theta_k^{(N)}(z,\tau),
\tag 1
\\
&\theta_k^{(N)}\Bigl(\frac{z}{\tau},-\frac1{\tau}\Bigr)
=c'\,{\mathbf{e}}\bigl(\tfrac{z}{2}\bigr)\sqrt{\tfrac{\tau}{N}}\,
\sum_{j=0}^{N-1} \zeta_{N}^{-jk} \theta_j^{(N)}(z,\tau).
\tag 2\end{aligned}$$ Here, $c$ and $c'$ are constants independent of $k$, and $\zeta_N={\mathbf{e}}(\tfrac1N)$.
\(1) Since $\theta_k^{(N)}(z,\tau+1)
=\theta_{(\frac{1}{2}-\frac{k}{N},\frac{N}{2})}(Nz,N\tau+N)$, we use the transformation formula with $M=\sltwo(1,N;0,1)$. Then we have $$\begin{aligned}
\theta_k^{(N)}(z,\tau+1)&=\theta_{(\frac{1}{2}-\frac{k}{N},\frac{N}{2})}(Nz,N\tau+N)
\\
&=\kappa(M){\mathbf{e}}\bigl(\tfrac{N}{2}(\tfrac{1}{4}-\tfrac{k^{2}}{N^{2}})\bigr)
\,\theta_{(\frac{1}{2}-\frac{k}{N},\frac{N}{2}-k)}(Nz,N\tau)
\\
&=\kappa(M){\mathbf{e}}\bigl(\tfrac{N}{2}(\tfrac{1}{4}-\tfrac{k^{2}}{N^{2}}\bigr)
{\mathbf{e}}\bigl(-(\tfrac{1}{2}-\tfrac{k}{N})k\bigr)
\,\theta_{(\frac{1}{2}-\frac{k}{N},\frac{N}{2})}(Nz,N\tau)
\\
&=\kappa(M){\mathbf{e}}\bigl(\tfrac{N}{8}-\tfrac{k(N-k)}{2N}\bigr)
\,\theta_{k}^{(N)}(z,\tau)
\\
&=c\,{\mathbf{e}}\bigl(-\tfrac{k(N-k)}{2N}\bigr)\, \theta_{k}^{(N)}(z,\tau).\end{aligned}$$ (2) Since $\theta_k^{(N)}(\frac{z}{\tau},-\frac{1}{\tau})
=\theta_{(\frac{1}{2}-\frac{k}{N},\frac{N}{2})}(\frac{z}{\tau/N},-\frac{1}{\tau/N})$, we let $\tau'=\tau/N$ and we use the transformation formula with $M=\sltwo(0,-1;1,0)$. Then, we have $\phi_{M}(p,q)=\frac{N}{4}-\frac{k}{2}$ and $(p',q')=\left(\frac{N}{2},-\frac{1}{2}+\frac{k}{N}\right)$, and $$\begin{aligned}
\theta_k^{(N)}\Bigl(\frac{z}{\tau},\frac{-1}{\tau}\Bigr)
&=\theta_{(\frac{1}{2}-\frac{k}{N},\frac{N}{2})}
\Bigl(\frac{z}{\tau'},\frac{-1}{\tau'}\Bigr) \\
&=\kappa(M){\mathbf{e}}\bigl(\tfrac{N}{4}-\tfrac{k}{2}\bigr){\mathbf{e}}(\tfrac{z}{2})\sqrt{\tau'}
\,\theta_{(\frac{N}{2},-\frac{1}{2}+\frac{k}{N})}(z,\tau').\end{aligned}$$ We compute the series of $\theta_{(\frac{N}{2},-\frac{1}{2}+\frac{k}{N})}(z,\tau')$ by splitting it into $N$ parts: $$\begin{gathered}
\theta_{(\frac{N}{2},-\frac{1}{2}+\frac{k}{N})}(z,\tau')
\\
\begin{aligned}
&=\sum_{n\in\Z}{\mathbf{e}}\Bigl(
{\tfrac{1}{2}\bigl(n+\tfrac{N}{2}\bigr)^2\tfrac{\tau}{N}
+\bigl(n+\tfrac{N}{2}\bigr)\bigl(z-\tfrac{1}{2}+\tfrac{k}{N}\bigr)}\Bigr)
\\
&=\sum_{j=0}^{N-1}\biggl(
\sum_{n+j\equiv 0\bmod N}{\mathbf{e}}\Bigl(
{\tfrac{1}{2}\bigl(n+\tfrac{N}{2}\bigr)^2\tfrac{\tau}{N}
+\bigl(n+\tfrac{N}{2}\bigr)\bigl(z-\tfrac{1}{2}+\tfrac{k}{N}\bigr)}\Bigr)
\biggr).
\end{aligned}\end{gathered}$$ Since $n+j\equiv 0\bmod N$ if and only if $n=Nm-j$ for some $m\in\Z$, we have $$\begin{aligned}
&\sum_{n+j\equiv 0\bmod N}{\mathbf{e}}\Bigl(
{\tfrac{1}{2}\bigl(n+\tfrac{N}{2}\bigr)^2\tfrac{\tau}{N}
+\bigl(n+\tfrac{N}{2}\bigr)\bigl(z-\tfrac{1}{2}+\tfrac{k}{N}\bigr)}\Bigr)
\\
&\qquad=\sum_{m\in\Z}{\mathbf{e}}\Bigl(
{\tfrac{1}{2}\bigl((Nm-j)+\tfrac{N}{2}\bigr)^2\tfrac{\tau}{N}
+\bigl((Nm-j)+\tfrac{N}{2}\bigr)\bigl(z-\tfrac{1}{2}+\tfrac{k}{N}\bigr)}\Bigr)
\\
&\qquad=\sum_{m\in\Z}{\mathbf{e}}\Bigl(
\tfrac{1}{2}N\bigl(m-\tfrac{j}{N}+\tfrac{1}{2}\bigr)^2\tau
+N\bigl(m-\tfrac{j}{N}+\tfrac{1}{2}\bigr)\bigl(z+\tfrac{1}{2}\bigr)
\\
\noalign{\hfill$
+\bigl(Nm-j+\tfrac{N}{2}\bigr)(\tfrac{k}{N}-1\bigr)\Bigr)$
\quad\null}
&\qquad=\sum_{m\in\Z}{\mathbf{e}}\Bigl(
{\tfrac{1}{2}N\bigl(n-\tfrac{j}{N}+\tfrac{1}{2}\bigr)^2\tau
+N\bigl(n-\tfrac{j}{N}+\tfrac{1}{2}\bigr)\bigl(z+\tfrac{1}{2}\bigr)}\Bigr)
{\mathbf{e}}(-\tfrac{jk}{N}+\tfrac{k-N}{2})
\\
&\qquad ={\mathbf{e}}(\tfrac{k-N}{2})\zeta_{N}^{-jk}\theta_{j}^{(N)}(z,\tau)\end{aligned}$$ Therefore, $$\begin{aligned}
\theta_k^{(N)}\Bigl(\frac{z}{\tau},\frac{-1}{\tau}\Bigr)
&=\kappa(M){\mathbf{e}}\bigl(\tfrac{N}{4}-\tfrac{k}{2}\bigr){\mathbf{e}}(\tfrac{z}{2})\sqrt{\tfrac{\tau}{N}}
\,\theta_{(\frac{N}{2},-\frac{1}{2}+\frac{k}{N})}(z,\tau')
\\
&=\kappa(M){\mathbf{e}}\bigl(-\tfrac{N}{4}\bigr){\mathbf{e}}\bigl(\tfrac{z}{2}\bigr)\sqrt{\tfrac{\tau}{N}}\,
\sum_{j=0}^{N-1} \zeta_{N}^{-jk}\theta_{j}^{(N)}(z,\tau).
\qedhere\end{aligned}$$
The action of $SL_{2}(\Z)$ on the upper half plane induces coordinate changes of $\P^{N-1}$ via the immersion $\Theta$ by theta functions. As a consequence we have a representation $\rho_{\theta}:SL_{2}(\Z)\to PGL_{N-1}(\C)$.
\[thm:sl2z-action\] The images of $\rho_{\theta}$ of the generators $S=\sltwo(0,-1;1,0)$ and $T=\sltwo(1,1;0,1)$ of $SL_{2}(\Z)$ are given by $$\rho_{\theta}\sltwo(0,-1;1,0) = \bigl(\zeta^{ij}\bigr)_{i,j=0}^{n-1}
\quad \text{and} \quad
\rho_{\theta}\sltwo(1,\ 1;0,1)
= \operatorname{Diag}\bigl(\zeta^{-i^{2}/2}\bigr)_{i=0}^{n-1}.$$ The kernel of $\rho_{\theta}$ is equal to $\Gamma^{(N)}(2N)$.
Immediate from Proposition \[prop:tr\_formula\].
Finally, by using Lemma \[lem:tr\_formula\] (theta transformation formula), we can describe the transformation properties of the “Theta Null Werte" $$a_k^{(N)}(\tau):=\theta_k^{(N)}(0,\tau)=\theta_{(\frac12-\frac{k}N,\frac{N}2)}(0,N\tau)\quad (k\in\Z)$$ under the action of the group $\Gamma(N)$.
By Proposition \[prop:theta-basic2\] (5), we see $$a_{k}^{(N)}(\tau) = (-1)^N a_{-k}^{(N)}(\tau) = (-1)^Na_{N-k}^{(N)}(\tau).$$ Thus, $a_0^{(M)}(\tau)=0$ if $N$ is odd, and we use $a_{k}^{(N)}(\tau)$ for $k=1,\ldots, (N-1)/2$ to embed the modular curve $\H/\Gamma(N)$ into $\P^{(N-3)/2}$ when $N$ is odd. On the other hand, when $N$ is even, we use $a_{k}^{(N)}(\tau)$ for $k=0,\ldots, N/2$ for an immersion of $\H/\Gamma^{(N)}(2N)$, [*not*]{} for $\H/\Gamma(N)$, as the following Proposition suggests. Recall that the principal congruence subgroup $\Gamma(N)$ is generated by the subgroup $\Gamma^{(N)}(2N)$ and the elements $\sltwo(1,N;0,1)$ and $\sltwo(1,0;N,1)$. The difference between $N$ being odd or even is also clear here.
Let $N$ be a positive integer. Write $a^{(N)}_{k}(\tau)=a_{k}$ for short.
1. If $N$ is odd, then the ratio $(a_{1}:\dots:a_{\frac{N-1}{2}})$ is invariant under the action of $\Gamma(N)$.
2. If $N$ is even, then the ratio $(a_{0}:a_{1}:\dots:a_{\frac{N}{2}})$ is invariant under the action of $\Gamma^{(N)}(2N)$. Moreover, the following hold.
1. The action $\tau\mapsto \tsltwo(1,N;0,1)\tau=\tau + N$ induces the action $$(a_{0}:a_{1}:\dots:a_{k}:\dots:a_{\frac{N}{2}}) \mapsto
(a_{0}:-a_{1}:\dots:(-1)^{k}a_{k}:\dots:(-1)^{\frac{N}{2}}a_{\frac{N}{2}}).$$
2. The action $\tau\mapsto \tsltwo(1,0;N,1)\tau=\frac{\tau}{N\tau + 1}$ induces the action $$(a_{0}:a_{1}:\dots:a_{k}:\dots:a_{\frac{N}{2}}) \mapsto
(a_{\frac{N}{2}}:a_{\frac{N}{2}-1}:\dots:a_{\frac{N}{2}-k}:\dots:a_{0}).$$
Straightforward calculation using Lemma \[lem:tr\_formula\].
Quadratic equations satisfied by theta functions
================================================
Let $E$ be an elliptic curve $E$ over a filed $K\hookrightarrow\C$. If we embed $E$ via the complete linear system $|N\{O_{E}\}|$ for some integer $N$, there exists a coordinate system $(X_{0}:X_{1}:\cdots:X_{N-1})\in\P^{N-1}$ described in Proposition \[prop:proj-coord-odd\]. Then, by a suitable choice of $\tau\in\H$, we have a map $\C/\Lambda_{\tau}\to E\subset\P^{N-1}$ given by $X_{k}=\theta_{k}^{(N)}(z,\tau)$. In order to describe $N(N-3)/2$ quadratic equations satisfied by $E$ explicitly, we would like to find relations satisfied among $\theta_{k}^{(N)}(z,\tau)$. It turns out that the situation is quite different depending on the parity of $N$.
We use the method of Jacobi [@Jacobi:Theta], which is completely elementary and algebraic (no function theory is used).
\[def:Jacobi-basic\] Jacobi’s basic theta functions $\vartheta_i(z)$ ($i=0,1,2,3$) are defined by the following formulas: $$\begin{aligned}
{2}
\vartheta_0(z)&=\theta_{(0,\frac12)}(z,\tau)
&&=\sum_{n\in\Z} {\mathbf{e}}\bigl({\tfrac{1}{2}n^2\tau + n(z+\tfrac12)}\bigr),\\
\vartheta_1(z)&=\theta_{(\frac12,\frac12)}(z,\tau)
&&=\sum_{n\in\Z} {\mathbf{e}}\bigl({\tfrac{1}{2}(n+\tfrac12)^2\tau + (n+\tfrac12)(z+\tfrac12)}\bigr),\\
\vartheta_2(z)&=\theta_{(\frac12,0)}(z,\tau)
&&=\sum_{n\in\Z} {\mathbf{e}}\bigl({\tfrac{1}{2}(n+\tfrac12)^2\tau + (n+\tfrac12)z}\bigr),\\
\vartheta_3(z)&=\theta_{(0,0)}(z,\tau)
&&=\sum_{n\in\Z} {\mathbf{e}}\bigl({\tfrac{1}{2}n^2\tau + nz}\bigr).\end{aligned}$$
Note that our definition differs slightly from Jacobi’s original notation by some rescaling and sign. Although we should write $\vartheta_{i}(z,\tau)$ instead of $\vartheta_{i}(z)$, we omit $\tau$ for simplicity.
The function $\vartheta_1(z)$ is an odd function and the others are even functions: $$\label{eq:theta-odd-even}
\begin{array}{ll}
\vartheta_0(-z)=\vartheta_1(z), \ &
\vartheta_1(-z)=-\vartheta_1(z), \\
\vartheta_2(-z)=\vartheta_2(z), \ &
\vartheta_3(-z)=\vartheta_3(z),
\end{array}$$ The following formulas are immediate from the definition. $$\label{jacobitheta}
\begin{aligned}
&\vartheta_0(z+\tfrac12)=\vartheta_3(z),\quad
\vartheta_1(z+\tfrac12)=-\vartheta_2(z),\\
&\vartheta_2(z+\tfrac12)=\vartheta_1(z),\quad
\vartheta_3(z+\tfrac12)=\vartheta_0(z).
\end{aligned}$$ The starting point is the identity (A)-(1) in Jacobi [@Jacobi:Theta p.507].
For independent variables $w,x,y,z$, define variables $w',x',y',z'$ by $$\label{eq:xyzw}
\left\{\begin{aligned}
w' &=\tfrac{1}{2}(w+x+y+z), \\
x' &=\tfrac{1}{2}(w+x-y-z), \\
y' &=\tfrac{1}{2}(w-x+y-z), \\
z' &=\tfrac{1}{2}(w-x-y+z),
\end{aligned}\right.
\quad\text{or}\quad
\setlength{\arraycolsep}{2pt}
\left(\begin{array}{c}
w' \\ x' \\ y' \\ z'
\end{array}\right)
=
\frac{1}{2}
\left(\begin{array}{rrrr}
1 & 1 & 1 & 1 \\
1 & 1 & -1 & -1 \\
1 & -1 & 1 & -1 \\
1 & -1 & -1 & 1
\end{array}\right)
\left(\begin{array}{c}
w \\ x \\ y \\ z
\end{array}\right).$$ Denote by $A$ the $4\times 4$ matrix above. In other words, define $A$ such that ${}^{t}(w',x',y',z')=A\,{}^{t}(w,x,y,z)$. Then, we have $A\in O(4)$, $A^{2}=I$, and $\det A=-1$. This shows that the transformation given by induces an involution on the set $$\{(w,x,y,z)\in \Z^4\,\mid\,w\equiv x\equiv y\equiv z\!\pmod2\}$$ and preserving the norm $w^2+x^2+y^2+z^2$.
\[prop:Jacobi-A-1\] Let $w,x,y,z$ be independent variables, and let $w',x',y',z'$ be the variables defined by . Then, the following identity holds: $$\label{eq:Jacobi4}
\begin{aligned}
\vartheta_{3}(w)\vartheta_{3}(x)&\vartheta_{3}(y)\vartheta_{3}(z)
+\vartheta_2(w)\vartheta_2(x)\vartheta_2(y)\vartheta_2(z)
\\
&= \vartheta_{3}(w')\vartheta_{3}(x')\vartheta_{3}(y')\vartheta_{3}(z')
+\vartheta_2(w')\vartheta_2(x')\vartheta_2(y')\vartheta_2(z'),
\end{aligned}$$
Jacobi’s identity follows from the properties of . The reader is encouraged to consult the beautiful, original article of Jacobi [@Jacobi:Theta].
\[th:three-term\] Let $w$, $x$, $y$, $z$ be independent variables. As in Proposition \[prop:Jacobi-A-1\], define $w'$, $x'$, $y'$, $z'$ by . Furthermore, define variables $w'',x'',y'',z''$ by $$\label{eq:xyzw3}
{}^{t}\!(w'', x'', y'', z'') = A\, {}^{t}\!(w,x,y,-z).$$ Then, the following three-term identity holds: $$\begin{gathered}
\label{eq:threeterm}
\vartheta_1(w)\vartheta_1(x)\vartheta_1(y)\vartheta_1(z)
\\
+\vartheta_{0}(w')\vartheta_{0}(x')\vartheta_{0}(y')\vartheta_{0}(z')
\\
-\vartheta_0(w'')\vartheta_0(x'')\vartheta_0(y'')\vartheta_0(z'')=0.\end{gathered}$$
Replacing $w$ by $w+1$ in and using , we obtain $$\begin{gathered}
\label{eq:Jacobi2}
\vartheta_{3}(w)\vartheta_{3}(x)\vartheta_{3}(y)\vartheta_{3}(z)
-\vartheta_2(w)\vartheta_2(x)\vartheta_2(y)\vartheta_2(z)
\\
= \vartheta_{0}(w')\vartheta_{0}(x')\vartheta_{0}(y')\vartheta_{0}(z')
+\vartheta_1(w')\vartheta_1(x')\vartheta_1(y')\vartheta_1(z').\end{gathered}$$ Adding and subtracting and , we obtain $$\begin{gathered}
2\vartheta_3(w)\vartheta_3(x)\vartheta_3(y)\vartheta_3(z)\\
=\vartheta_{3}(w')\vartheta_{3}(x')\vartheta_{3}(y')\vartheta_{3}(z')
+\vartheta_2(w')\vartheta_2(x')\vartheta_2(y')\vartheta_2(z')\\
+\vartheta_{0}(w')\vartheta_{0}(x')\vartheta_{0}(y')\vartheta_{0}(z')
+\vartheta_1(w')\vartheta_1(x')\vartheta_1(y')\vartheta_1(z'),\end{gathered}$$ and $$\begin{gathered}
2\vartheta_2(w)\vartheta_2(x)\vartheta_2(y)\vartheta_2(z)\\
=\vartheta_{3}(w')\vartheta_{3}(x')\vartheta_{3}(y')\vartheta_{3}(z')
+\vartheta_2(w')\vartheta_2(x')\vartheta_2(y')\vartheta_2(z')\\
-\vartheta_{0}(w')\vartheta_{0}(x')\vartheta_{0}(y')\vartheta_{0}(z')
-\vartheta_1(w')\vartheta_1(x')\vartheta_1(y')\vartheta_1(z').\end{gathered}$$ Replace $w,x,y,z$ by $w+\tfrac12,x+\tfrac12,y+\tfrac12,z+\tfrac12$ in these two identities. Then $w'$ becomes $w'+1$, and $x',y',z'$ unchanged. By we obtain $$\begin{gathered}
\label{eq:th0}
2\vartheta_0(w)\vartheta_0(x)\vartheta_0(y)\vartheta_0(z)\\
=\vartheta_{0}(w')\vartheta_{0}(x')\vartheta_{0}(y')\vartheta_{0}(z')
-\vartheta_1(w')\vartheta_1(x')\vartheta_1(y')\vartheta_1(z')\\
-\vartheta_2(w')\vartheta_2(x')\vartheta_2(y')\vartheta_2(z')
+\vartheta_{3}(w')\vartheta_{3}(x')\vartheta_{3}(y')\vartheta_{3}(z'),\end{gathered}$$ and $$\begin{gathered}
\label{eq:th1}
2\vartheta_1(w)\vartheta_1(x)\vartheta_1(y)\vartheta_1(z)\\
=-\vartheta_{0}(w')\vartheta_{0}(x')\vartheta_{0}(y')\vartheta_{0}(z')
+\vartheta_1(w')\vartheta_1(x')\vartheta_1(y')\vartheta_1(z')\\
-\vartheta_2(w')\vartheta_2(x')\vartheta_2(y')\vartheta_2(z')
+\vartheta_{3}(w')\vartheta_{3}(x')\vartheta_{3}(y')\vartheta_{3}(z').\end{gathered}$$
Since the relation between $w,x,y,z$ and $w',x',y',z'$ are symmetric, we have $$\begin{gathered}
\label{eq:th0-inv}
2\vartheta_0(w')\vartheta_0(x')\vartheta_0(y')\vartheta_0(z')\\
=\vartheta_{0}(w)\vartheta_{0}(x)\vartheta_{0}(y)\vartheta_{0}(z)
-\vartheta_1(w)\vartheta_1(x)\vartheta_1(y)\vartheta_1(z)\\
-\vartheta_2(w)\vartheta_2(x)\vartheta_2(y)\vartheta_2(z)
+\vartheta_{3}(w)\vartheta_{3}(x)\vartheta_{3}(y)\vartheta_{3}(z),\end{gathered}$$ By the definition of the transformation , the identity can be translated to $$\begin{gathered}
\label{eq:th0-inv-2}
2\vartheta_0(w'')\vartheta_0(x'')\vartheta_0(y'')\vartheta_0(z'')\\
=\vartheta_{0}(w)\vartheta_{0}(x)\vartheta_{0}(y)\vartheta_{0}(-z)
-\vartheta_1(w)\vartheta_1(x)\vartheta_1(y)\vartheta_1(-z)\\
-\vartheta_2(w)\vartheta_2(x)\vartheta_2(y)\vartheta_2(-z)
+\vartheta_{3}(w)\vartheta_{3}(x)\vartheta_{3}(y)\vartheta_{3}(-z).\end{gathered}$$ Now, calculating $+$ $-$ , and using , we obtain the relation .
Odd case {#odd-case}
--------
Assume $N$ is odd. From our definition of $\theta^{(N)}_{k}(z,\tau)$ and Proposition \[prop:theta-basic\], we have the relations $$\theta^{(N)}_{0}(z,\tau)= (-1)^{\frac{N-1}{2}}\vartheta_{1}(Nz,N\tau)\quad\text{and}\quad
\theta^{(N)}_{\frac{N}{2}}(z,\tau)= \vartheta_{0}(Nz,N\tau).$$ Dropping the superscript “$(N)$”, translates to $$\begin{gathered}
\label{eq:finalthreeterm-odd}
\theta_0(w)\theta_0(x)\theta_0(y)\theta_0(z)
\\
+\theta_{\frac{N}2}(w')\theta_{\frac{N}2}(x')\theta_{\frac{N}2}(y')\theta_{\frac{N}2}(z')
-\theta_{\frac{N}2}(w'')\theta_{\frac{N}2}(x'')\theta_{\frac{N}2}(y'')\theta_{\frac{N}2}(z'')=0.\end{gathered}$$
We use this equation to obtain our quadratic equations.
\[th:eq-odd\] Suppose $N\in \N$ is odd. Let $E_{\Theta}$ be the image of $\Theta$ in Proposition \[prop:immersion\]. Let $V=H^{0}(\P^{N-1},\mathcal{I}_{E_{\Theta}}(2))$, and $V_{k}=V\cap \<X_{i}X_{j}\mid i+j\equiv k \mod N\>$, where $X_{i}$ $(i=0,\dots,N-1)$ are the coordinate functions of $\P^{N-1}$ in Proposition \[prop:proj-coord-odd\]. Let $a_{i}=\theta^{(N)}_{i}(0,\tau)$. Then, the quadratic forms $$\label{eq:deg2-N-odd}
a_{j+1}a_{N-j}X_{0}^{2}
-a_{\frac{N-1}{2}-j}a_{\frac{N+1}{2}+j}
X_{\frac{N+1}{2}}X_{\frac{N-1}{2}}
+a_{\frac{N-1}{2}}a_{\frac{N+1}{2}}X_{\frac{N-1}{2}-j}
X_{\frac{N+1}{2}+j}$$ with $j=1,\ldots,\tfrac{N-3}2$ forms a basis of the vector space $V_{0}$.
First, we show that if we replace $X_{i}$ by $\theta^{(N)}_{i}(z,\tau)$, the quadratic forms in vanishes. In , let $$(w,x,y,z) =\bigl(z,z,\tfrac{j\tau}{N},-\tfrac{(j+1)\tau}{N}\bigr).$$ Then, we have $$\begin{aligned}
(w',x',y',z') &= \bigl(z-\tfrac{\tau}{2N},z+\tfrac{\tau}{2N},\tfrac{(2j+1)\tau}{2N},-\tfrac{(2j+1)\tau}{2N}\bigr),
\\
(w'',x'',y'',z'') &=\bigl(z+\tfrac{(2j+1)\tau}{2N},z-\tfrac{(2j+1)\tau}{2N},-\tfrac{\tau}{2N},\tfrac{\tau}{2N}\bigr).\end{aligned}$$ Then, use Proposition \[prop:theta-basic2\] (4) to see that is satisfied by $X_{i}=\theta^{(N)}_{i}(z,\tau)$ and $a_{i}=\theta^{(N)}_{i}(0,\tau)$. Recall that $a_k$ depends only on $k \bmod N$, and we have $a_0=0$ and $a_k=-a_{N-k}$. Thus, we may restrict the range of $j$ to $1\le j\le (N-3)/2$. ($j=(N-1)/2$ gives the trivial relation.)
Next we show that these $(N-3)/2$ relations are independent. Indeed, the coefficients of the terms $X_{\frac{N-1}{2}-j}X_{\frac{N+1}{2}+j}$ for $1\le j\le (N-3)/2$ are all equal to $a_{\frac{N-1}{2}}a_{\frac{N+1}{2}}$, which is nonzero, and the terms $X_{0}^{2}, X_{\frac{N+1}{2}}X_{\frac{N-1}{2}}$ are different from $X_{\frac{N-1}{2}-j}X_{\frac{N+1}{2}+j}$ for $1\le j\le (N-3)/2$. Therefore those $(N-3)/2$ quadratic forms are independent.
We briefly describe below the classical examples $N=5$ and 7.
When $N=5$, yields $(5-3)/2=1$ quadratic form, which is $$a_1a_2X_0^2-a_1^2X_{2}X_{3}+a_2^2X_{1}X_{4}.$$ $V=H^{0}(\P^{N-1},\mathcal{I}_{E_{\Theta}}(2))$ is generated by this form and its various permutations by $M_{Q}$ in Proposition \[prop:proj-coord-odd\]. If we define $\phi(\tau)$ by $$\phi(\tau)=-\frac{a_1(\tau)}{a_2(\tau)}
=q^{\frac{1}{5}} - q^{\frac{6}{5}} + q^{\frac{11}{5}} - q^{\frac{21}{5}} + q^{\frac{26}{5}} - q^{\frac{31}{5}} + \cdots\quad
(q={\mathbf{e}}(\tau)=e^{2\pi i \tau}),$$ we have the equations $$X_i^2+\phi(\tau)X_{i+2}X_{i-2}-\frac1{\phi(\tau)}X_{i+1}X_{i-1}=0
\quad (i=0,\ldots,4).$$ This set of equations is the well-known Bianchi normal form [@Bianchi].
When $N=7$, quadratic forms give two equations $$\begin{aligned}
&a_1a_2X_0^2-a_2^2X_{3}X_{4}+a_3^2 X_{2}X_{5}=0,\\
&a_2a_3X_0^2-a_1^2X_{3}X_{4}+a_3^2 X_{1}X_{6}=0.\end{aligned}$$ By the specialization $z=0$ in $X_i$ ($X_i\mapsto a_i$), we obtain the unique relation $$a_1^3a_2=a_2^3a_3+a_1a_3^3.$$ This is the renowned Klein’s quartic, which is a model of the modular curve $\H/\Gamma(7)$ of genus 3.
Even case {#even-case}
---------
Next suppose $N$ is even. Again we start with Jacobi’s , which in terms of our theta’s is written as $$\label{eq:start-even}
\begin{aligned}
\theta_{\frac{N}{2}}(w)\theta_{\frac{N}{2}}(x)&\theta_{\frac{N}{2}}(y)\theta_{\frac{N}{2}}(z)
+\theta_{0}(w)\theta_{0}(x)\theta_{0}(y)\theta_{0}(z)
\\
&= \theta_{\frac{N}{2}}(w')\theta_{\frac{N}{2}}(x')\theta_{\frac{N}{2}}(y')\theta_{\frac{N}{2}}(z')
+\theta_{0}(w')\theta_{0}(x')\theta_{0}(y')\theta_{0}(z').
\end{aligned}$$ We can deduce our quadratic relations directly from this with the following specializations. For integers $i$ and $j$, consider the four substitutions
$$\begin{aligned}
(w,x,y,z)=
&\left(z,-\tfrac{j\tau}{N}, -\tfrac{j\tau}{N}, z\right),\quad
\left(-\tfrac{2j\tau}{N},0, z, z\right),\\
&\left(z-\tfrac{\tau}{N},-\tfrac{j\tau}{N}, -\tfrac{(j+1)\tau}{N}, z\right),\quad
\left(-\tfrac{(2j+1)\tau}{N},0,z, z-\tfrac{\tau}{N}\right),\end{aligned}$$
which, respectively, yield $$\begin{gathered}
(w',x',y',z')=
\left(z-\tfrac{j\tau}{N},0,0, z+\tfrac{j\tau}{N}\right),\
\left(z-\tfrac{j\tau}{N},-z-\tfrac{j\tau}{N}, -\tfrac{2j\tau}{N},-\tfrac{2j\tau}{N}\right),\\
\left(z-\tfrac{(j+1)\tau}{N},0, -\tfrac{\tau}{N}, z+\tfrac{j\tau}{N}\right),\
\left(z-\tfrac{(j+1)\tau}{N},-z-\tfrac{j\tau}{N}, -\tfrac{j\tau}{N},-\tfrac{(j+1)\tau}{N},\right).\end{gathered}$$ Applying these to and using Proposition \[prop:theta-basic2\], we obtain the following set of equations.
$$\begin{gathered}
\left\{
\begin{array}{rcrcrcr}
a_{j}^{2}\,X_{0}^{2} & + & a_{\frac{N}{2}+j}^{2}\,X_{\frac{N}{2}}^{2} & = &
a_{0}^{2}\,X_{j}X_{N-j} & + & a_{\frac{N}{2}}^{2}\,X_{\frac{N}{2}+j}X_{\frac{N}{2}-j},
\\
a_{0}a_{2j}\,X_{0}^{2} & + & a_{\frac{N}{2}}a_{\frac{N}{2}+2j}\,X_{\frac{N}{2}}^{2} & = &
a_{j}^{2}\,X_{j}X_{N-j} & + & a_{\frac{N}{2}+j}^{2}\,X_{\frac{N}{2}+j}X_{\frac{N}{2}-j},
\end{array} \label{eq:515}
\right.
\\[\smallskipamount]
\def\+{\hskip 4pt+\hskip 4pt}
\left\{
\begin{array}{rcl}
a_{j}a_{j+1}\,X_{0}X_{1} & + & a_{\frac{N}{2}+j}a_{\frac{N}{2}+j+1}\,X_{\frac{N}{2}}X_{\frac{N}{2}+1}
\\
& = & a_{0}a_{1}\ X_{j+1}X_{N-j} \+ a_{\frac{N}{2}}a_{\frac{N}{2}+1} X_{\frac{N}{2}+j+1}X_{\frac{N}{2}-j},
\\
a_{0}a_{2j+1}\,X_{0}X_{1} & + & a_{\frac{N}{2}}a_{\frac{N}{2}+2j+1}\,X_{\frac{N}{2}}X_{\frac{N}{2}+1}
\\
& = & a_{j}a_{j+1}\,X_{j+1}X_{N-j} \+ a_{\frac{N}{2}+j}a_{\frac{N}{2}+j+1}\,X_{\frac{N}{2}+j+1}X_{\frac{N}{2}-j}.
\end{array} \label{eq:516}
\right.\end{gathered}$$ Here, as before, the indices are considered modulo $N$. Equations (resp. ) belongs to the space $V_{0}$ (resp. $V_{1}$) (see §2.3). Because of the relation $a_j=a_{N-j}$, it is easy to see that we may restrict ourselves to the case $0\le j\le \frac{N}2$. Furthermore, $j=0\text{ and }\frac{N}2$ give trivial relations, and becomes also trivial for $j=\frac{N}2-1$. And finally, the second equation of (resp. ) is unchanged if we replace $j$ by $\frac{N}2-j$ (resp. $\frac{N}2-j-1$). Out of these equations, first equations of and give $N-3$ independent equation.
\[prop:deg2-N-even\] The equations $$\label{eq:V2i}
a_{j}^{2}\,X_{0}^{2} + a_{\frac{N}{2}+j}^{2}\,X_{\frac{N}{2}}^{2} =
a_{0}^{2}\,X_{j}X_{N-j} + a_{\frac{N}{2}}^{2}\,X_{\frac{N}{2}+j}X_{\frac{N}{2}-j}$$ with $j=1,\ldots,\frac{N}{2}-1$ form a basis of $V_{0}$, and the equations $$\label{eq:V2ip1}
\begin{aligned}
a_{j}a_{j+1}\,X_{0}X_{1} &+ a_{\frac{N}{2}+j}a_{\frac{N}{2}+j+1}\,X_{\frac{N}{2}}X_{\frac{N}{2}+1} \\
& = a_{0}a_{1}\ X_{j+1}X_{N-j} + a_{\frac{N}{2}}a_{\frac{N}{2}+1} X_{\frac{N}{2}+j+1}X_{\frac{N}{2}-j}
\end{aligned}$$ with $j=1,\ldots,\frac{N}{2}-2$ form a basis of $V_{1}$.
Looking at the indices of $X$, only possible dependency occurs between equations with $j=j_0$ and $j=\frac{N}2-j_0$. But then the determinant $a_0^4-a_{\frac{N}2}^4$ of the two by two matrix of coefficients of $X_{j_0}X_{-j_0}$ and $X_{\frac{N}2+j_0}X_{\frac{N}2-j_0}$ is non-zero, as is seen by looking at its Fourier series. The same argument applies for equations in .
As will be seen in the examples in the following sections, the whole set of equations and are needed to obtain relations among $a_i$’s.
Other equations for $N$ even
----------------------------
If $N$ is even, our $\theta_0(z)=\theta^{(N)}_{0}(z,\tau)$ and $\theta_{\frac{N}2}(z)=\theta^{(N)}_{\frac{N}2}(z,\tau)$ are related to the Jacobi theta functions in the following way: $$\theta_{0}(z)=(-1)^{\frac{N}{2}}\vartheta_2(Nz,N\tau)\quad
\text{and}\quad
\theta_{\frac{N}{2}}(z)=\vartheta_3(Nz,N\tau).$$ Since we have $$(-1)^{\frac{N}{2}}\theta_{0}\left(z+\tfrac{1}{2N}\right)
=\vartheta_{1}(Nz,N\tau), \quad
\theta_{\frac{N}{2}}\left(z+\tfrac{1}{2N}\right)
=\vartheta_{0}(Nz,N\tau),$$ the relation translates to $$\begin{gathered}
\label{eq:finalthreeterm-even}
\theta_{0}\left(w+\tfrac{1}{2N}\right)\theta_0\left(x+\tfrac{1}{2N}\right)\theta_0\left(y+\tfrac{1}{2N}\right)\theta_0\left(z+\tfrac{1}{2N}\right)\\
-\theta_{\frac{N}2}\left(w'+\tfrac{1}{2N}\right)\theta_{\frac{N}2}\left(x'
+\tfrac{1}{2N}\right)\theta_{\frac{N}2}\left(y'+\tfrac{1}{2N}\right)\theta_{\frac{N}2}\left(z'+\tfrac{1}{2N}\right)\\
+\theta_{\frac{N}2}\left(w''+\tfrac{1}{2N}\right)\theta_{\frac{N}2}\left(x''+\tfrac{1}{2N}\right)\theta_{\frac{N}2}\left(y''+\tfrac{1}{2N}\right)\theta_{\frac{N}2}\left(z''+\tfrac{1}{2N}\right)=0.\end{gathered}$$
\[th:eq-even-1\] Suppose $N\in \N$ is even. Let $E_{\Theta}$ be the image of $\Theta$ in Proposition \[prop:immersion\]. Let $V=H^{0}(\P^{N-1},\mathcal{I}_{E_{\Theta}}(2))$, and $V_{k}=V\cap \<X_{i}X_{j}\mid i+j\equiv k \mod N\>$, where $X_{i}$ $(i=0,\dots,N-1)$ are the coordinate functions of $\P^{N-1}$ in Definition \[def:X\_k-even\]. Let $$s_{i}=\theta^{(N)}_{i}(\tfrac{1}{2N},\tau).$$ Then, the quadratic forms $$\label{eq:deg2-N-even-0}
s_{j}s_{N-j}X_{0}^{2}
+s_{\frac{N}{2}-j}s_{\frac{N}{2}+j}
X_{\frac{N}{2}}^{2}
-s_{\frac{N}{2}}^{2}X_{\frac{N}{2}-j}
X_{\frac{N}{2}+j}$$ with $j=1,\ldots,\tfrac{N}2-1$ forms a basis of the vector space $V_{0}$, and $$\label{eq:deg2-N-even-1}
s_{j+1}s_{N-j}X_{0}X_{1}
+s_{\frac{N}{2}-j}s_{\frac{N}{2}+j+1}
X_{\frac{N}{2}}X_{\frac{N}{2}+1}
-s_{\frac{N}{2}}s_{\frac{N}{2}+1}X_{\frac{N}{2}-j}
X_{\frac{N}{2}+j+1}$$ with $j=1,\ldots,\tfrac{N}2-2$ forms a basis of the vector space $V_{1}$
If we let $$(w,x,y,z) = \bigl(z-\tfrac{1}{2N},
z-\tfrac{1}{2N},\tfrac{j\tau}{N},-\tfrac{j\tau}{N}\bigr),$$ then, we have $$\begin{aligned}
(w',x',y',z') &= \bigl(z-\tfrac{1}{2N},
z-\tfrac{1}{2N},\tfrac{j\tau}{N},-\tfrac{j\tau}{N}\bigr)
\\
(w'',x'',y'',z'') &=\bigl(z+\tfrac{j\tau}{N}-\tfrac{1}{2N},
z-\tfrac{j\tau}{N}-\tfrac{1}{2N},0,0\bigr)\end{aligned}$$ Substituting these in , and using Proposition \[prop:theta-basic2\], we obtain .
If we let $$(w,x,y,z) = \bigl(z-\tfrac{\tau}{N}-\tfrac{1}{2N},
z-\tfrac{1}{2N},\tfrac{j\tau}{N},-\tfrac{(j+1)\tau}{N}\bigr),$$ then we have $$\begin{aligned}
(w',x',y',z') &= \bigl(z-\tfrac{\tau}{N}-\tfrac{1}{2N},
z-\tfrac{1}{2N},\tfrac{j\tau}{N},-\tfrac{(j+1)\tau}{N}\bigr)
\\
(w'',x'',y'',z'') &= \bigl(z-\tfrac{j\tau}{N}-\tfrac{1}{2N},
z-\tfrac{(j+1)\tau}{N}-\tfrac{1}{2N},-\tfrac{\tau}{N},0\bigr).\end{aligned}$$ Substituting these in , and using Proposition \[prop:theta-basic2\], we obtain .
Level 4 {#sec:level-4}
=======
Consider the case $N=4$. When $i$ and $j$ run from $0$ to $2$ in the formulas in Proposition \[prop:deg2-N-even\] the only nontrivial relations we obtain are
$$\begin{gathered}
V_{0}:\left\{\begin{array}{*{2}rc}
a_{1}^{2}(X_{0}^{2} + X_{2}^{2}) &\ =\ &
(a_{0}^{2}+a_{2}^{2})X_{1}X_{3},
\\
a_{0}a_{2}(X_{0}^{2} + X_{2}^{2}) &\ =\ &
2a_{1}^{2}X_{1}X_{3},
\end{array}\right.
\\
V_{2}:\left\{\begin{array}{*{2}rc}
a_{1}^{2}(X_{1}^{2} + X_{3}^{2}) &\ =\ &
(a_{0}^{2}+a_{2}^{2})X_{0}X_{2},
\\
a_{0}a_{2}(X_{1}^{2} + X_{3}^{2}) &\ =\ &
2a_{1}^{2}X_{0}X_{2}.
\end{array}\right.\end{gathered}$$
Note that $V_{1}=V_{3}=0$ in this case. If we let $z=0$, then $X_{i}$ becomes $a_{i}$. Considering the fact $a_{1}=a_{3}$, we obtain a unique nontrivial relation from the above equations $$\label{eq:X4_8}
a_{0}a_{2}(a_{0}^{2} + a_{2}^{2}) = 2a_{1}^{4}.$$ This is a nonsingular curve in $\P^{2}$, which is a genus $3$ curve. On the other hand, the congruence subgroup $\Gamma^{(4)}(8)$ is the subgroup of $\Gamma(1)$ of index $96$ and genus $3$. Thus, the curve is nothing but $X^{(4)}(8)$.
Using the relation we see that the two relations in $V_{0}$ and the two relations in $V_{1}$ are equivalent, respectively. Thus, the following two equations determine the universal elliptic curve $\mathscr{E}^{(4)}(8)$ over $X^{(4)}(8)$: $$\label{eq:E4_8}
\renewcommand{{1.4}}{1.4}
\setlength{\arraycolsep}{2pt}
\mathscr{E}^{(4)}(8) : \
\left\{
\begin{array}{*3c}
a_{0}a_{2}(X_{0}^{2} + X_{2}^{2}) &= & 2a_{1}^{2}\, X_{1}X_{3},
\\
2a_{1}^{2}\, X_{0}X_{2} &= & a_{0}a_{2}(X_{1}^{2} + X_{3}^{2}).
\end{array}\right.$$
\[prop:E4\_8\] The curve $\mathscr{E}^{(4)}(8)$ is isomorphic over $K$ to $$\label{eq:Weier_E4_8}
Y^{2}=X\bigl(X-(a_{0}-a_{2})^4\bigr)\bigl(X-(a_{0}+a_{2})^4\bigr).$$ The points on $X^{(4)}(8)$ at which the fiber of $\mathscr{E}^{(4)}(8)$ becomes a Néron polygon of four sides are $$(a_{0}:a_{1}:a_{2})
= (0:0:1), (1:0:0), (\pm\zeta_{8}^{2}:0:1), (1:\zeta_{8}^{k}:1) \ (k=0,\dots,7),$$ where $\zeta_{8}$ is a primitive $8$th root of unity.
From two quadrics , eliminate $X_{1}$ to obtain a quartic equation. Then, using the rational point $(X_{0}:X_{1}:X_{2}:X_{3})=(a_{0}:a_{1}:a_{2}:a_{1})$, we obtain a Weierstrass equation. After some simplification, we obtain with the change of coordinates $$\label{eq:chang_E4_8}
\left\{\begin{aligned}
X &=
(a_{0}^{2}-a_{2}^{2})^2\frac{
(a_{1}^{2} X_{0} X_{2}+a_{0}a_{2} X_{1}X_{3}) }
{(a_{1}^{2}X_{0} X_{2}-a_{0}a_{2} X_{1} X_{3}) },
\\
Y &=
4a_{1}^{2}(a_{0}^{2}-a_{2}^{2})^2\frac{
(X_{1}+X_{3})(X_{0}+X_{2})(a_{0}a_{2}X_{0}X_{2}-a_{1}^{2} X_{1} X_{3})}
{(X_{1}-X_{3})(X_{0}-X_{2})(a_{1}^{2} X_{0} X_{2}-a_{0}a_{2} X_{1} X_{3})}.
\end{aligned}\right.$$
The locus of degenerate fibers are where the right hand side of has a multiple root.
The action of $SL_{2}(\Z)/\Gamma^{(4)}(8)$ on $\mathscr{E}^{(4)}(8)$ and $X^{(4)}(8)$ are given as follows $$\begin{aligned}
{2}
\rho\tsltwo(0,-1;1,0)
&=\left[\begin{array}{rcrc}
1 & 1 & 1 & 1 \\
1 &\zeta_{8}^{2} & -1 & -\zeta_{8}^{2} \\
1 & -1 & 1 & -1 \\
1 &-\zeta_{8}^{2} & -1 & \zeta_{8}^{2}
\end{array}\right],
\quad &
\rho\tsltwo(1,1;0,1)
&=\left[\begin{array}{rcrc}
1 & 0 & 0 & 0 \\
0 &\zeta_{8}^{3} & 0 &0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & \zeta_{8}^{3}
\end{array}\right],
\\
\bar\rho\tsltwo(0,-1;1,0)
&=\left[\begin{array}{rcr}
1 & 2 & 1 \\
1 & 0 & -1\\
1 & -2 & 1 \\
\end{array}\right],
\quad &
\bar\rho\tsltwo(1,1;0,1)
&=\left[\begin{array}{rcrc}
1 & 0 & 0\\
0 &\zeta_{8}^{3} & 0 \\
0 & 0 & -1 \\
\end{array}\right].\end{aligned}$$ In particular, we have $$\begin{aligned}
\rho\tsltwo(1,4;0,1)&:
(X_{0}:X_{1}:X_{2}:X_{3}) \mapsto
(X_{0}:-X_{1}:X_{2}:-X_{3}),
\\
\rho\tsltwo(1,0;4,1)&:
(X_{0}:X_{1}:X_{2}:X_{3}) \mapsto
(X_{2}:X_{3}:X_{0}:X_{1}),
\\
\bar\rho\tsltwo(1,4;0,1)&:
(a_{0}:a_{1}:a_{2}) \mapsto
(a_{0}:-a_{1}:a_{2}),
\\
\bar\rho\tsltwo(1,0;4,1)&:
(a_{0}:a_{1}:a_{2}) \mapsto
(a_{2}:a_{1}:a_{0}).\end{aligned}$$ Let $G=\Gamma(4)/\Gamma^{(4)}(8)=\<\tsltwo(1,4;0,1)
\Gamma^{(4)}(8),\tsltwo(1,0;4,1)
\Gamma^{(4)}(8)\>\simeq \Z/2\Z\times \Z/2\Z$. In order to obtain a model of $\mathscr{E}(4)\to X(4)$, we take the quotient of $\mathscr{E}^{(4)}(8)\to X^{(4)}(8)$ by the action of $G$. To do so, we first dehomogenize the equations by letting $$x_{0}=\frac{X_{0}}{X_{3}}, \quad
x_{1}=\frac{X_{1}}{X_{3}}, \quad
x_{2}=\frac{X_{2}}{X_{3}}, \quad
\alpha_{0}=\frac{a_{0}}{a_{1}}, \quad
\alpha_{2}=\frac{a_{2}}{a_{1}},$$ and consider the function field $K(\alpha_{0},\alpha_{2},x_{0},x_{1},x_{2})$. Then, these variables satisfy the following relations: $$\begin{gathered}
\label{eq:X4_8-a}
\alpha_{0}\alpha_{2}(\alpha_{0}^{2} + \alpha_{2}^{2}) = 2, \\
\label{eq:E4_8-a}
\left\{
\begin{aligned}
& \alpha_{0}\alpha_{2}(x_{0}^{2} + x_{2}^{2}) = 2 x_{1},\\
& 2 x_{0}x_{2} = \alpha_{0}\alpha_{2}(x_{1}^{2} + 1).
\end{aligned}\right.\end{gathered}$$ Let $\sigma_{1}=\tsltwo(1,4;0,1)\Gamma^{(4)}(8)$ and $\sigma_{2}=\tsltwo(1,0;4,1))\Gamma^{(4)}(8)$ be the generators of $G$. Then, they act as follows: $$\begin{aligned}
\sigma_{1}&:
(\alpha_{0},\alpha_{2},x_{0},x_{1},x_{2}) \mapsto
(-\alpha_{0},-\alpha_{2},-x_{0},x_{1},-x_{2}),
\\
\sigma_{2}&:
(\alpha_{0},\alpha_{2},x_{0},x_{1},x_{2}) \mapsto
\left(\alpha_{2},\alpha_{0},\frac{x_{2}}{x_{1}},\frac{1}{x_{1}},\frac{x_{0}}{x_{1}}\right).\end{aligned}$$ It is easy to see that $K(\alpha_{0},\alpha_{2})^{G}=K(\alpha_{0}\alpha_{2},\alpha_{0}^{2}+\alpha_{2}^{2})$. Considering , we see that $K(\alpha_{0},\alpha_{2})^{G}=K(\lambda)$ with $\lambda=\alpha_{0}\alpha_{2}$. This shows that $X(4)\simeq \P^{1}_{\lambda}$. Also, we have $$\begin{aligned}
\lambda(\tau) &= \frac{a_{0}(\tau)a_{2}(\tau)}{a_{1}(\tau)^{2}} =
2\left(\frac{\eta(\tau)\eta(4\tau)^2}{\eta(2\tau)^3}\right)^2\\
&=2q^{\frac14} -4q^{\frac54}+10q^{\frac94}-20q^{\frac{13}4}+36q^{\frac{17}4}+\cdots
\quad (q=e^{2\pi i \tau}),\end{aligned}$$ where $\eta(\tau)$ is the Dedekind eta function. Further calculations show that the fixed field $K(\alpha_{0},\alpha_{2},x_{0},x_{1},x_{2})^{G}$ is generated by $$\lambda=\alpha_{0}\alpha_{2}, \quad
\xi_{0}=\frac{x_{0}x_{2}}{x_{1}}, \quad
\xi_{1}=x_{1}+\frac{1}{x_{1}}, \quad
\xi_{2}=\frac{(x_{1}+1)(x_{0}+x_{2})}{(x_{1}-1)(x_{0}-x_{2})}.$$ Now, can be written in terms of $\lambda$, $\xi_{0}$ and $\xi_{1}$: $$X=\frac{4(1-\lambda^4)}{\lambda^{2}}
\frac{(\xi_{0}+\lambda)}{(\xi_{0}-\lambda)}, \quad
Y=\frac{16(1-\lambda^4)}{\lambda^{2}}
\frac{\xi_{2}(1-\lambda \xi_{0})}
{(\xi_{0}-\lambda)}.$$ If we let $X'=\lambda^{2}X/4$ and $Y'=\lambda^{2}Y/8$, then the relation above becomes $$\mathscr{E}(4): Y'^{2} = X'\bigl(X'-(\lambda^2-1)^{2}\bigr)
\bigl(X'-(\lambda^2+1)^{2}\bigr).$$ The action of $PSL_{2}(\Z/4\Z)$ is given by $$\begin{aligned}
\left[\begin{smallmatrix} 0 & -1 \\ 1 & 0\end{smallmatrix}\right]:
(\lambda,X',Y')&\mapsto
\left(\frac{-\lambda+1}{\lambda+1},
-\frac{4(X'+(\lambda^2+1)^2)}{(\lambda+1)^4},
-\frac{8\zeta_{8}^{2}Y'}{(\lambda+1)^6}\right),\\
\left[\begin{smallmatrix} 1 & 1 \\ 0 & 1\end{smallmatrix}\right]:
(\lambda,X',Y')&\mapsto ( -\zeta_{8}^{2}\lambda, X',Y').\end{aligned}$$
Level 6 {#sec:level-6}
=======
In this section we study the case $N=6$ in detail. In this case, the nontrivial relations obtained from Proposition \[prop:deg2-N-even\] are as follows By letting $z=0$ (i.e. $X_{i}=a_{i}$), we obtain quartic relations among $a_i$’s. Noting $a_5=a_1$ and $a_4=a_2$, we obtain only two relations: $$\label{eq:level-6-homogeneous}
\left\{
\begin{gathered}
a_{1}^{4}+a_{2}^{4}=a_{0}^{3}a_{2}+a_{1}a_{3}^{3} \\
a_{0}a_{3} \left( a_{0}a_{1}+a_{2}a_{3} \right)
=2\,a_{1}^{2}a_{2}^{2}
\end{gathered}
\right.$$ Using these relations we see that the equations are not linearly independent, and we have $\dim V_{0}=2$, $\dim V_{1}=1$.
The curve in $\P^{3}$ defined by turns out to be reducible. Computations using Groebner basis reveal that there are five irreducible components; four of them are lines $$\begin{gathered}
a_{1}=a_{2}=0, \quad
a_{0}-a_{2}=a_{1}-a_{3}=0, \\
a_{0}-\omega a_{2}=\omega a_{1}-a_{3}=0, \quad
a_{0}-\omega^{2} a_{2}=\omega^{2} a_{1}-a_{3}= 0, \end{gathered}$$ where $\omega$ is third root of unity, and the other component is an irreducible curve of genus $13$. On the other hand, the congruence subgroup $\Gamma^{(6)}(12)$ is the subgroup of $\Gamma(1)$ of index $288$ and genus $13$. Thus, the last irreducible component is the modular curve $X^{(6)}(12)$ associated with the group $\Gamma^{(6)}(12)$.
$$\begin{gathered}
\bar\rho\tsltwo(0,-1;1,0)
=\left[\begin{array}{rcrc}
1 & 2 & 2 & 1 \\
1 & 1 & -1 & -1\\
1 & -1 & -1 & 1\\
1 & -2 & 2 & -1
\end{array}\right],
\quad
\bar\rho\tsltwo(1,1;0,1)
=\left[\begin{array}{*4c}
1 & 0 & 0 & 0\\
0 &\zeta_{12}^{5} & 0 & 0\\
0 & 0 &\zeta_{12}^{8} & 0 \\
0 & 0 & 0 &\zeta_{12}^{9}
\end{array}\right].\end{gathered}$$
In particular, we have $$\begin{aligned}
\bar\rho\tsltwo(1,6;0,1)&:
(a_{0}:a_{1}:a_{2}:a_{3}) \mapsto
(a_{0}:-a_{1}:a_{2}:-a_{3}),
\\
\bar\rho\tsltwo(1,0;6,1)&:
(a_{0}:a_{1}:a_{2}:a_{3}) \mapsto
(a_{3}:a_{2}:a_{1}:a_{0}).\end{aligned}$$
Let us find a model of $X(6)$ by taking the quotient of $X^{(6)}(12)$ by $G=\Gamma(6)/\Gamma^{(6)}(12)\simeq \Z/2\Z\times \Z/2\Z$. Dehomogenize the coordinates by setting $$\alpha_{1}=\frac{a_{1}}{a_{0}}, \quad \alpha_{2}=\frac{a_{2}}{a_{0}}, \quad
\alpha_{3}=\frac{a_{3}}{a_{0}}.$$ Now the equation becomes $$\label{eq:inhomogeneous}\left\{
\begin{gathered}
\alpha_{1}^{4}+\alpha_{2}^{4}=\alpha_{2}+\alpha_{1}\alpha_{3}^{3}, \\
\alpha_{3}(\alpha_{1}+\alpha_{2}\alpha_{3})=2\alpha_{1}^{2}\alpha_{2}^{2}.
\end{gathered}
\right.$$ Let $\sigma_{1}$ and $\sigma_{2}$ denote the automorphisms induced on the function field $k(\alpha_{1},\alpha_{2},\alpha_{3})$ by $\bar\rho\tsltwo(1,6;0,1)$ and $\bar\rho\tsltwo(1,0;6,1)$, respectively. We have $$\sigma_{1} : (\alpha_{1},\alpha_{2},\alpha_{3}) \mapsto (-\alpha_{1},\alpha_{2},-\alpha_{3}), \quad
\sigma_{2} : (\alpha_{1},\alpha_{2},\alpha_{3}) \mapsto
\Bigl(\frac{\alpha_{2}}{\alpha_{3}},\frac{\alpha_{1}}{\alpha_{3}},\frac{1}{\alpha_{3}}\Bigr).$$ The fixed subfields by $\sigma_{1}$ and $\sigma_{2}$ are $$\begin{aligned}
&k(\alpha_{1},\alpha_{2},\alpha_{3})^{\sigma_{1}}
=k\Bigl(\alpha_{1}\alpha_{3},\frac{\alpha_{1}}{\alpha_{3}},\alpha_{2}\Bigr), \\
&k(\alpha_{1},\alpha_{2},\alpha_{3})^{\sigma_{2}}
=\Bigl(\alpha_{1}+\frac{\alpha_{2}}{\alpha_{3}},\alpha_{2}+\frac{\alpha_{1}}{\alpha_{3}},\alpha_{3}+\frac{1}{\alpha_{3}}\Bigr).\end{aligned}$$ Finally, define $$b_{1}=\alpha_{1}\alpha_{3}+\frac{\alpha_{2}}{\alpha_{3}^{2}}, \quad
b_{2}=\alpha_{2}+\frac{\alpha_{1}}{\alpha_{3}}, \quad
b_{3}=\alpha_{3}^{2}+\frac{1}{\alpha_{3}^{2}}.$$ Clearly, $b_{1}$, $b_{2}$, and $b_{3}$ are fixed by $G=\<\sigma_{1},\sigma_{2}\>$. It is easy to show that the fixed field by the group $G$ is given by $$k(\alpha_{1},\alpha_{2},\alpha_{3})^{G} = k(b_{1},b_{2},b_{3}).$$ The system of equations becomes $$\left\{\begin{gathered}
b_{3}(b_{1}^2+b_{2}^2-b_{1}b_{2}b_{3})^2+b_{2}(2b_{1}-b_{2}b_{3})(b_{1}^2-b_{2}^2)(b_{3}^2-4)=b_{1}(b_{3}^{2}-4)^2
\\
b_{2}(b_{3}^{2}-4)^2 = 2(b_{1}^2+b_{2}^2-b_{1}b_{2}b_{3})^2
\end{gathered}\right.$$ Using calculations based on Groebner basis, we find that if we let $$X = \frac{-2(b_{1}^2+b_{2}^2-b_{1}b_{2}b_{3})}{b_{3}^2-4},
\quad
Y = \frac{b_{2}(2b_{1}-b_{2}b_{3})}{b_{1}^2-b_{2}^2},$$ they satisfy $$Y^{2} = X^{3} + 1,$$ which is well-known model of the modular curve $X(6)$. Furthermore, we have $$b_{1} = \frac{X^2(Y^2-3)}{4Y},\quad
b_{2} = \frac{1}{2}X^2,\quad
b_{3} = \frac{Y^4-6X^2-3}{4Y},$$ and thus $k(b_{1},b_{2},b_{3})=k(X,Y)$.
On the curve $Y^{2}=X^{3}+1$, the actions of $\bar\rho\sltwo(0,1;-1,0)$ and $\bar\rho\sltwo(1,1;0,1)$ are given by $$\begin{aligned}
\bar\rho\tsltwo(0,1;-1,0)
&:(X,Y) \mapsto (2,-3) - (X,Y),
\\
\bar\rho\tsltwo(1,1;0,1)
&:(X,Y) \mapsto [-\omega](X,Y)=(\omega X,-Y),\end{aligned}$$ where the operation “$-$” in the first map is the group operation of $Y^{2}=X^{3}+1$, and the map $[\omega]$ is the complex multiplication of $Y^{2}=X^{3}+1$. In terms of $a_{i}(\tau)=\theta_{i}^{(N)}(0,\tau)$, $X$ and $Y$ are expressed as follows: $$X = \frac{2a_{1}(\tau)a_{2}(\tau)}{a_{0}(\tau)a_{3}(\tau)}, \quad
Y = \frac{a_{0}(\tau)^{2}a_{1}(\tau)^{2}
-a_{2}(\tau)^{2}a_{3}(\tau)^{2}}
{a_{0}(\tau)^{2}a_{2}(\tau)^{2}
-a_{1}(\tau)^{2}a_{3}(\tau)^{2}}.$$ Incidentally, $X$ and $Y$ (which are modular functions on $\Gamma(6)$) can also be written by using the Dedekind eta function as $$X=\frac{\eta(2\tau)\eta(3\tau)^3}{\eta(\tau)\eta(6\tau)^3}=q^{-\frac13} + q^{\frac{2}{3}} + q^{\frac{5}{3}}
- q^{\frac{8}{3}} - q^{\frac{11}{3}} + q^{\frac{17}{3}} + 2\, q^{\frac{20}{3}} -\cdots$$ and $$Y=\frac{\eta(2\tau)^4\eta(3\tau)^2}{\eta(\tau)^2\eta(6\tau)^4}=q^{-\frac12} + 2 q^{\frac{1}{2}} + q^{\frac{3}{2}}
- 2 q^{\frac{7}{2}} - 2 q^{\frac{9}{2}} + 2 q^{\frac{11}{2}} + 4 q^{\frac{13}{2}} + \cdots,$$ where $q=e^{2\pi i \tau}$. The function $Y$ has a simpler expression in terms of $a_i(\tau)=\theta_i^{(6)}(0,\tau)$ as $$Y=\frac{a_0(\frac{\tau}{3})a_3(\frac{\tau}{3})}{a_0(\tau)a_3(\tau)}.$$
At the end of this section, we remark the connection to the “Hesse cubic”, that is the elliptic normal curve of degree 3. Put $$\begin{aligned}
3\mu&=X^2-\frac{2}{X} =\frac{Y^2-3}{X}\\
&=q^{-\frac23}+5q^{\frac43}-7q^{\frac{10}3}+3q^{\frac{16}3}+15q^{\frac{22}3}-32q^{\frac{28}3}
+9q^{\frac{34}3}+\cdots. \end{aligned}$$ We can check that $\mu(\tau/2)$ is a modular function for the group $\Gamma(3)$, and we have the relation $$X_0^3+X_2^3+X_4^3=3\mu X_0 X_2 X_4.$$ Actually, our theta functions $\theta_k^{(3)}(z,\tau)\ (k=0,1,2)$ in the case of $N=3$ can be obtained from those for $N=6$ ($k=0,2,4$) by changing the variables $z\to z/2-1/4, \,\tau\to\tau/2$. We refer the reader to [@KKNT] for the derivation of the Hesse cubic in the same line of the current paper.
Level 8 {#sec:level-8}
=======
In this section we study the case $N=8$ in detail.
For $N=8$, the quadratic equations obtained in §3 are as follows. $$\begin{gathered}
\setlength{\arraycolsep}{2pt}
V_{0}: \left\{
\begin{array}{*9c}
a_{1}^{2}X_{0}^{2} &+& a_{3}^{2}X_{4}^{2}
&=& a_{0}^{2}X_{1}X_{7}& & &+& a_{4}^{2}X_{3}X_{5}, \\
a_{2}^{2}X_{0}^{2} &+& a_{2}^{2}X_{4}^{2}
&=& & & (a_{0}^{2}+a_{4}^{2})X_{2}X_{6}, &
\\
a_{3}^{2}X_{0}^{2} &+& a_{1}^{2}X_{4}^{2}
&=& a_{4}^{2}X_{1}X_{7} & & &+& a_{0}^{2}X_{3}X_{5},
\\
a_{0}a_{2}X_{0}^{2} &+& a_{2}a_{4}X_{4}^{2}
&=& a_{1}^{2}X_{1}X_{7} & & &+& a_{3}^{2}X_{3}X_{5},
\\
a_{0}a_{4} X_{0}^{2} &+& a_{0}a_{4}X_{4}^{2}
&=& & &2a_{2}^{2}X_{2}X_{6},
\end{array}\right.
\\
V_{1}: \left\{
\begin{array}{*9c}
a_{1}a_{2}X_{0}X_{1} &-& a_{0}a_{1}X_{2}X_{7}
&-& a_{3}a_{4}X_{3}X_{6} &+& a_{2}a_{3}X_{4}X_{5} =0,
\\
a_{2}a_{3}X_{0}X_{1} &-& a_{3}a_{4}X_{2}X_{7}
&-& a_{0}a_{1}X_{3}X_{6} &+& a_{1}a_{2}X_{4}X_{5} =0,
\\
a_{0}a_{3}X_{0}X_{1} &-& a_{1}a_{2}X_{2}X_{7}
&-& a_{2}a_{3}X_{3}X_{6} &+& a_{1}a_{4}X_{4}X_{5}=0.
\end{array}\right.\end{gathered}$$ By letting $z=j\tau/N$, $j=0,1,\dots,$ in these formula, we obtain the following relations among $a_{i}$: $$\label{eq:level-8-homogeneous}
\left\{
\begin{aligned}
&a_{0}a_{4}(a_{0}^{2}+a_{4}^{2}) = 2a_{2}^{4}
\\
&a_{0}a_{4}(a_{1}^{2}+a_{3}^{2}) = 2a_{1}a_{3}a_{2}^{2}
\\
&a_{0}a_{2}a_{4}(a_{0}+a_{4})=2a_{1}^{2}a_{3}^{2}
\\
&a_{2}^{3}(a_{0}+a_{4}) = a_{1}a_{3}(a_{1}^{2}+a_{3}^{2})
\\
&a_{1}a_{3}(a_{0}^{2}+a_{4}^{2}) = a_{2}^{2}(a_{1}^{2}+a_{3}^{2})
\\
&a_{2}(a_{0}^{3} + a_{4}^{3}) = a_{1}^{4} + a_{3}^{4}
\end{aligned}
\right.$$ The curve in $\P^{4}$ defined by turns out to be an irreducible, and this is a model of $X^{(8)}(16)$.
Dehomogenize by letting $$\alpha_{0}=\frac{a_{0}}{a_{2}}, \quad \alpha_{1}=\frac{a_{1}}{a_{2}}, \quad
\alpha_{3}=\frac{a_{3}}{a_{2}}, \quad \alpha_{4}=\frac{a_{4}}{a_{2}}.$$ It turns out that, in the function field $k(\alpha_{0},\alpha_{1},\alpha_{3},\alpha_{4})$, the following three equations are enough to generate the above six equations $$\label{eq:level8-rel}
\alpha_{0}\alpha_{4}(\alpha_{0}^{2}+\alpha_{4}^{2}) = 2,\quad
\alpha_{0}+\alpha_{4} = \alpha_{1}\alpha_{3}(\alpha_{1}^{2}+\alpha_{3}^{2}),\quad
\alpha_{1}\alpha_{3}(\alpha_{0}^{2}+\alpha_{4}^{2}) = \alpha_{1}^{2}+\alpha_{3}^{2}.$$
Let $\sigma_{1}$ and $\sigma_{2}$ denote the automorphisms induced on the function field $k(\alpha_{0},\alpha_{1},\alpha_{3},\alpha_{4})$ by $\bar\rho\tsltwo(1,8;0,1)$ and $\bar\rho\tsltwo(1,0;8,1)$, respectively. We have $$\begin{aligned}
& \sigma_{1} : (\alpha_{0},\alpha_{1},\alpha_{3},\alpha_{4}) \longmapsto
(\alpha_{0},-\alpha_{1},-\alpha_{3},\alpha_{4}), \\
&\sigma_{2} : (\alpha_{0},\alpha_{1},\alpha_{3},\alpha_{4}) \longmapsto
(\alpha_{4},\alpha_{3},\alpha_{1},\alpha_{0}).\end{aligned}$$ It is easy to see that $\alpha$ and $\beta$ commute, and thus the group of automorphism $\<\alpha,\beta\>$ induced by $\alpha$ and $\beta$ is isomorphic to $(\Z/2\Z)^{2}$. The fixed subfields by $\alpha$ and $\beta$ are $$\begin{aligned}
k(\alpha_{0},\alpha_{1},\alpha_{3},\alpha_{4})^{\sigma_{1}}
&=k\Bigl(\alpha_{0},\alpha_{1}\alpha_{3},\frac{\alpha_{1}}{\alpha_{3}},\alpha_{4}\Bigr), \\
k(\alpha_{0},\alpha_{1},\alpha_{3},\alpha_{4})^{\sigma_{2}}
&=k\left(\alpha_{0}+\alpha_{4},\alpha_{1}+\alpha_{3}, (\alpha_{1}-\alpha_{3})(\alpha_{0}-\alpha_{4}),\frac{\alpha_{1}-\alpha_{3}}{\alpha_{0}-\alpha_{4}}\right).\end{aligned}$$ Finally, define $$b_{0}=\alpha_{0}+\alpha_{4},\quad
b_{1}=\alpha_{1}\alpha_{3},\quad
b_{3}=\frac{\alpha_{1}}{\alpha_{3}}+\frac{\alpha_{3}}{\alpha_{1}},\quad
b_{4}=\frac{\alpha_{1}\alpha_{3}(\alpha_{0}-\alpha_{4})}{(\alpha_{1}+\alpha_{3})(\alpha_{1}-\alpha_{3})}.$$ Then, $b_{1}$, $b_{2}$, and $b_{3}$ are fixed by $G=\<\sigma_{1},\sigma_{2}\>$. It is easy to show that the fixed field by the group $G$ is given by
$$k(\alpha_{0},\alpha_{1},\alpha_{3},\alpha_{4})^{G}
=k(b_{0},b_{1},b_{3},b_{4}).$$ Further calculations show that the relations translates to the relations $$b_{0}=b_{1}b_{3}, \quad b_{3}=\frac{1}{b_{4}^{2}}, \quad
b_{1}^{4}=4b_{4}^{6}+b_{4}^{2}.$$ This shows that $$k(\alpha_{0},\alpha_{1},\alpha_{3},\alpha_{4})^{G}
=k(b_{1},b_{4}), \quad \text{with } b_{1}^{4}=4b_{4}^{6}+b_{4}^{2}$$ It is easy to see that $b_{1}^{4}=4b_{4}^{6}+b_{4}^{2}$ defines a curve of genus $5$. Thus, an affine model of $X(8)$ is given by $$X(8): b_{4}^{2}(1+4b_{4}^{2})=b_{1}^{4}.$$ Recall that an affine equation of $X^{(4)}(8)$ is given by $\alpha_{2}(1 + \alpha_{2}^{2}) = 2\alpha_{1}^{4}$ (cf. ). Thus, there is a two-to-one map $X(8)\to X^{(4)}(8)$ given by $
(\alpha_{1},\alpha_{2})=(b_{1},2b_{4}^{2})$.
The modular functions $b_1$ and $b_4$ (on $\Gamma(8)$) can be written in terms of the Dedekind eta function as $$b_1=\frac{\eta(2\tau)^4\eta(8\tau)^2}{\eta(\tau)\eta(4\tau)^5}=q^{\frac{1}{8}} + q^{\frac{9}{8}} - 2 q^{\frac{17}{8}} - q^{\frac{25}{8}}
+ 4 q^{\frac{33}{8}} + 2 q^{\frac{41}{8}} - 7 q^{\frac{49}{8}} -\cdots$$ and $$b_4=-\frac{\eta(2\tau)\eta(8\tau)^2}{\eta(4\tau)^3}=q^{\frac{1}{4}} - q^{\frac{9}{4}} + 2 q^{\frac{17}{4}} - 3 q^{\frac{25}{4}} + 4 q^{\frac{33}{4}} -
6 q^{\frac{41}{4}} + 9 q^{\frac{49}{4}}- \cdots.$$ They also have neat expressions in terms of our theta series as $$b_1=\frac{a_2(\frac{\tau}{2})a_2(2\tau)}{a_2(\tau)^2},\quad b_4=\frac{a_2(2\tau)}{a_2(\tau)}.$$
\#1[arXiv:[\#1](http://arXiv.org/abs/#1)]{} \[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{}
[10]{}
Luigi Bianchi, *Ueber die [N]{}ormalformen dritter und fünfter [S]{}tufe des elliptischen [I]{}ntegrals erster [G]{}attung*, Math. Ann. **17** (1880), no. 2, 234–262.
Tom A. Fisher, *On $5$ and $7$ descents for elliptic curves*, Ph.D. thesis, Cambridge University, 2000.
, *[Some examples of 5 and 7 descent for elliptic curves over Q]{}*, Journal of the European Mathematical Society **3** (2001), no. 2, 169–201.
Robin Hartshorne, *Algebraic geometry*, Springer-Verlag, New York-Heidelberg, 1977.
Adolf Hurwitz, *Ueber endliche [G]{}ruppen linearer [S]{}ubstitutionen, welche in der [T]{}heorie der elliptischen [T]{}ranscendenten auftreten*, Math. Ann. **27** (1886), no. 2, 183–233.
Jun-Ichi Igusa, *Theta functions*, Die Grundlehren der mathematischen Wissenschaften, Band 194., Springer-Verlag, New York-Heidelberg, 1972.
C. G. J. Jacobi, *Theorie der elliptischen funktionen aus den eigenschaften der thetareihen abgeleitet*, Gesammelte werke, bd. 1, s. 497, Leipzig, (1881)., 1927 (German).
Kenji Kajiwara, Masanobu Kaneko, Atsushi Nobe, and Teruhisa Tsuda, *Ultradiscretization of a solvable two-dimensional chaotic map associated with the [H]{}esse cubic curve*, Kyushu Journal of Mathematics **63** (2009), no. 2, 315–338.
Felix Klein, *[[Ü]{}ber die elliptischen Normalkurven der n-ten Ordnung (1885)]{}*, Gesammelte Mathematische Abhandlungen, 1923.
David Mumford, *Varieties defined by quadratic equations*, Questions on [A]{}lgebraic [V]{}arieties ([C]{}.[I]{}.[M]{}.[E]{}., [III]{} [C]{}iclo, [V]{}arenna, 1969), Edizioni Cremonese, Rome, 1970, pp. 29–100.
, *[Tata lectures on theta. I]{}*, Progress in Mathematics, vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1983.
Jan Stevens, *Degenerations of elliptic curves and equations for cusp singularities*, Math. Ann. **311** (1998), no. 2, 199–222.
Jacques Vélu, *Courbes elliptiques munies d’un sous-groupe [${\bf
Z}/n{\bf Z}\times {\bf \mu }_{n}$]{}*, Bull. Soc. Math. France Mém. (1978), no. 57, 5–152.
[^1]: For a point $P\in E(K_{s})$, we denote by $\{P\}$ the base of the formal sums associated with $P$. Thus, the sum $\{P\}+\{Q\}$ is a formal sum, while the sum in $\{P+Q\}$ means the addition in the elliptic curve $E$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider the late-time tails of spherical waves propagating on even-dimensional Minkowski spacetime under the influence of a long range radial potential. We show that in six and higher even dimensions there exist exceptional potentials for which the tail has an anomalously small amplitude and fast decay. Along the way we clarify and amend some confounding arguments and statements in the literature of the subject.'
author:
- Piotr Bizoń
- Tadeusz Chmaj
- Andrzej Rostworowski
title: Anomalously small wave tails in higher dimensions
---
Introduction
============
It is well known that sharp propagation of free waves along light cones in even-dimensional flat spacetimes, known as Huygens’ property, is blurred by the presence of a potential. Physically, the spreading of waves inside the light cone is caused by the backscattering off the potential. If the potential falls off exponentially or faster at spatial infinity, then the backscattered waves decay exponentially in time, while the long range potentials with an algebraic fall-off give rise to tails which decay polynomially in $1/t$. The precise description of these tails is an important issue in scattering theory. There are two main approaches to this problem in the literature. On the one hand, there are mathematical results in the form of various decay estimates. These results are rigorous, however they rarely give optimal decay rates inside the light cone and provide very poor information about the amplitudes of tails. The notable exception is the work of Strauss and Tsutaya [@st] (recently strengthened by Szpak [@sz]) where the optimal pointwise decay estimate for the tail was proved in four dimensions. Unfortunately, to the best of our knowledge, there is no analogous result in higher dimensions.
On the other hand, there are non-rigorous results in the physics literature based on perturbation theory. The most complete work in this category was done by Ching *et al.* [@ching] who derived first-order approximations of the tails for radial potentials. Although these results were originally formulated for partial waves in four dimensions, they can be easily translated to spherical waves in higher dimensions. Ching *et al.* noticed that there are exceptional potentials for which the first-order tail vanishes, however they did not pursue their analysis to the second order, apart from giving some dimensional arguments. The main purpose of this paper is to analyze the tails for such exceptional potentials in more detail.
One of the physical motivations behind our work stems from the fact that this kind of potentials arise in the study of linearized perturbations of higher even-dimensional Schwarzschild black holes. The behavior of tails on the Schwarzschild background in well known in four dimensions (see [@p], [@l], [@ching], [@gpp], [@b], [@dr]), but not in higher even dimensions (despite statements to the contrary in the literature [@car]). Although our analysis is restricted to the flat background, it sheds some light on the problem of tails on the black hole background because the properties of tails are to some extent independent of what happens in the central region.
The rest of the paper is organized as follows. In section 2 we construct the iterative scheme for the perturbation expansion of a spherically symmetric solution of the linear wave equation with a potential. This scheme is applied in section 3 to derive the first and second-order approximations of the tails for radial potentials which fall off as pure inverse-power at infinity. In section 4 we discuss the modifications caused by subleading terms in the potential. Section 5 contains numerical evidence confirming the analytic formulae from sections 3 and 4. Finally, in section 6 we give a heuristic argument to predict the behavior of tails outside Schwarzschild black holes in higher even dimensions. Technical details of most calculations are given in the appendix.
Throughout the paper we use the succinct notation and summation technics from the excellent book by Graham *et al.* [@gkp]. In particular, we shall frequently use the following abbreviations $$\begin{aligned}
x^{\underline{0}} := 1, &\qquad& x^{\underline{k}} := x \cdot (x-1)
\cdot \dots \cdot (x-(k-1)), \quad k>0\,,
\\
x^{\overline{0}} := 1, &\qquad& x^{\overline{k}} := x \cdot (x+1)
\cdot \dots \cdot (x+(k-1)), \quad k>0\,.\end{aligned}$$
Iterative scheme
================
We consider the wave equation with a potential in even-dimensional Minkowski spacetime $R^{d+1}$ $$\label{eqm}
\partial_t^2 \phi -\Delta \phi + \lambda V \phi =0\,.$$ The prefactor $\lambda$ is introduced for convenience - throughout the paper we assume that $\lambda$ is small which allows us to use it as the perturbation parameter. The precise assumptions about the fall-off of the potential will be formulated below. We restrict attention to spherical symmetry, i.e., we assume that $\phi=\phi(t,r)$ and $V=V(r)$. Then, equation (\[eqm\]) becomes $$\label{eqs}
\mathcal{L} \phi + \lambda V(r) \phi =0\,,\qquad \mathcal{L}
:=\partial_t^2 -\partial_r^2 -\frac{d-1}{r} \partial_r\,.$$ We are interested in the late-time behavior of $\phi(t,r)$ for smooth compactly supported (or exponentially localized) initial data. $$\label{id}
\phi(0,r)=f(r),\qquad \partial_t \phi(0,r)=g(r)\,.$$ To determine the asymptotic behavior of solutions we define the perturbative expansion (Born series) $$\label{pert}
\phi=\sum_{n=0} \lambda^n \phi_n\,,$$ where $\phi_0$ satisfies initial data (\[id\]) and all $\phi_n$ with $n>0$ have zero initial data. Substituting this expansion into equation (\[eqs\]) we get the iterative scheme $$\label{scheme}
\mathcal{L} \phi_n = -V \phi_{n-1}\,, \qquad \phi_{-1}=0\,,$$ which can be solved recursively. The zeroth-order solution is given by the general regular solution of the free radial wave equation which is a superposition of outgoing and ingoing waves [@k] $$\label{f0}
\phi_0(t,r)=\phi_0^{ret}(t,r)+\phi_0^{adv}(t,r)\,,$$ where $$\label{f1}
\phi_0^{ret}(t,r)= \frac{1}{r^{l+1}}\,\sum_{k=0}^{l} \frac {(2l-k)!} {k!(l-k)!} \frac
{a^{(k)}(u)}{(v-u)^{l-k}}\,, \qquad \phi_0^{adv}(t,r) =
\frac{1}{r^{l+1}}\,\sum_{k=0} ^{l} (-1)^{k+1} \frac {(2l-k)!}
{k!(l-k)!} \frac {a^{(k)}(v)}{(v-u)^{l-k}}\,,$$ and $u=t-r$, $v=t+r$ are the retarded and advanced times, respectively. Here and in the following, instead of $d$, we use the index $l$ defined by $d=2l+3$ (remember that we consider only *odd* space dimensions $d$). Note that for compactly supported initial data the generating function $a(x)$ can be chosen to have compact support as well (this condition determines $a(x)$ uniquely).
To solve equation (\[scheme\]) for the higher-order perturbations we use the Duhamel representation for the solution of the inhomogeneous equation $\mathcal{L} \phi = N(t,r)$ with zero initial data $$\label{a2}
\phi(t,r)= \frac{1}{2 r^{l+1}}
\int\limits_{0}^{t} d\tau \int\limits_{|t-r-\tau|}^{t+r-\tau} \rho^{l+1} P_l(\mu)
N(\tau,\rho) d\rho\,,$$ where $P_l(\mu)$ are Legendre polynomials of degree $l$ and $\mu=(r^2+\rho^2-(t-\tau)^2)/2r\rho$ (note that $-1\leq \mu \leq 1$ within the integration range). This formula can be readily obtained by integrating out the angular variables in the standard formula $\phi=G^{ret} * N$ where $G^{ret}(t,x)=(2\pi^{l+1})^{-1}\Theta(t)
\delta^{(l)}(t^2-|x|^2)$ is the retarded Green’s function of the wave operator in $d+1$ dimensions (see, for example, [@ls]). It is convenient to express (\[a2\]) in terms of null coordinates $\eta=\tau-\rho$ and $\xi=\tau+\rho$ $$\label{duh1}
\phi(t,r)= \frac{1}{2^{l+3} r^{l+1}}
\int\limits_{|t-r|}^{t+r} d\xi \int\limits_{-\xi}^{t-r} (\xi-\eta)^{l+1} P_l(\mu)
N(\eta,\xi) d\eta\,,$$ where now $ \mu=(r^2+(\xi-t)(t-\eta))/r(\xi-\eta)$. Using this representation we can rewrite the iterative scheme (\[scheme\]) in the integral form $$\label{iter}
\phi_{n}(t,r)= -\frac{1}{2^{l+3} r^{l+1}}
\int\limits_{|t-r|}^{t+r} d\xi \int\limits_{-\xi}^{t-r}
(\xi-\eta)^{l+1} P_l(\mu) V(\rho(\eta,\xi))\phi_{n-1}(\eta,\xi)
d\eta\,.$$ This “master” equation will be applied below to evaluate the first two iterates for a special class of potentials. It is natural to expect that for sufficiently small $\lambda$ these iterates provide good approximations of the true solution.
Pure inverse-power potentials at infinity
=========================================
In this section we consider the simple case (below referred to as type I) when the potential is *exactly* $V(r)= r^{-\alpha}$ for $r$ greater than some $r_0>0$. We assume that $\alpha>2$. The modifications caused by subleading corrections to the pure inverse-power decay of the potential will be discussed in section 4.
Generic case
------------
We wish to evaluate the first iterate $\phi_1(t,r)$ near timelike infinity, i.e, for $r=const$ and $t\rightarrow \infty$. Thanks to the fact that $\phi_0(\eta,\xi)$ has compact support we may interchange the order of integration in (\[iter\]) and drop the advanced part of $\phi_0(\eta,\xi)$ to obtain $$\label{iter1}
\phi_{1}(t,r) = -\frac{2^{\alpha}}{2^{l+3} r^{l+1}}
\int\limits_{-\infty}^{\infty} d\eta \int\limits_{t-r}^{t+r} (\xi-\eta)^{l+1-\alpha}
P_l(\mu) \phi_0^{ret}(\eta,\xi)
d\xi\,,$$ where we have substituted $V= 2^{\alpha} (\xi-\eta)^{-\alpha}$. Plugging (\[f1\]) into (\[iter1\]), after a long calculation (see appendix A for the technical details), we get $$\begin{aligned}
\phi_1(t,r) &=& - 2^{\alpha+3l-1} \left( \frac {\alpha-3} {2}
\right)^{\underline{l}} \left( \frac {\alpha} {2}
\right)^{\overline{l}} \int \limits_{-\infty}^{+\infty} d\eta \,
a(\eta) \, \frac {(t-\eta)^{\alpha-2}} {\left[ (t-\eta)^2 - r^2 \right]^{\alpha-1+l}}
\nonumber\\
&\times& \sum_{0 \leq n \leq \lfloor (\alpha-2) / 2 \rfloor} (-1)^{n} \frac {2^{2n}(l+n)!} {n! (2l+2n+1)!}
\left( - \frac {\alpha-2} {2} - l - 1 \right)^{\underline{n}} \left( \frac {\alpha-1} {2} - l - 1 \right)^{\underline{n}}
\nonumber\\
&\times& \sum_{n \leq m \leq n+l} (-1)^{m} \left( \begin{array}{c} l \\ m-n \end{array} \right)
\frac {\left( - \frac {\alpha} {2} + 1 \right)^{\overline{m}}} {\left( \frac {\alpha} {2} \right)^{\overline{m}}} \left( \frac {r} {t-\eta} \right)^{2m}.
\label{tailf1}\end{aligned}$$ Asymptotic expansion of (\[tailf1\]) near timelike infinity yields the following first-order approximation of the tail $$\label{tail1}
\phi(t,r) \approx \lambda \phi_1(t,r) = \lambda \, \frac {C(l,\alpha)} {t^{\alpha+2l}} \left[ A + (\alpha+2l)\frac{B}{t} \, + \mathcal{O} \left( \frac{1}{t^2}
\right) \right] \, ,$$ where $$\label{B}
C(l,\alpha) = - \frac {2^{\alpha+2l-1}}{(2l+1)!!} \left( \frac {\alpha-3} {2} \right)^{\underline{l}}
\left( \frac {\alpha} {2} \right)^{\overline{l}}\,,$$ and $$\label{ab}
A=\int \limits_{-\infty}^{+\infty} a(\eta)\, d\eta\,,\qquad B=\int \limits_{-\infty}^{+\infty}
a(\eta)\,\eta\,d\eta\,.$$ In general $A\neq 0$ and the tail decays as $t^{-\alpha-2l}$, however there are nongeneric initial data for which $A=0$ and then the tail decays as $t^{-\alpha-2l-1}$; in particular this happens for time symmetric initial data for which $a(x)$ is an odd function. 0.2cm *Remark 1.* It is easy to check that if the function $\phi(t,r)$ satisfies equation (\[eqs\]), then the function $\psi=r^{l+1}\phi$ satisfies the radial wave equation for the $l$th multipole $$\label{a3}
(\partial_t^2-\partial_r^2+l(l+1)/r^2)\psi + \lambda V(r)\psi =0\,.$$ The late-time tails for this equation were studied by Ching *et al.* [@ching] who derived the formula equivalent to (\[tail1\]) via the Fourier transform methods.
Exceptional case
----------------
It follows from (\[tail1\]) that if $\alpha$ is an odd integer satisfying $3\leq \alpha \leq 2l+1$, then $\phi_1(t,r)$ vanishes identically due to factor $\left( \frac {\alpha-3} {2} \right)^{\underline{l}}$ in (\[B\]) and there is no (polynomial) tail whatsoever in the first order. Thus, in order to compute the tail in this exceptional case we need to go the second order of the perturbation expansion.
Using (\[iter\]) and proceeding as above we get the second iterate $$\label{iter2}
\phi_{2}(t,r) = -\frac{2^{\alpha}}{2^{l+3} r^{l+1}}
\int\limits_{-\infty}^{\infty} d\eta \int\limits_{t-r}^{t+r}
(\xi-\eta)^{l+1-\alpha} P_l(\mu) \phi_1^{ret}(\eta,\xi)
d\xi\,,$$ where $\phi_1^{ret}$ is the outgoing solution of the inhomogeneous equation $$\label{NH}
\mathcal{L} \phi_1=-V \phi_0\,.$$ In general $\phi_1$ is a sum of the solution of the homogeneous equation and the particular solution of the inhomogeneous equation. The homogeneous part has the form (\[f1\]) (with a different generating function than $a$, but still compactly supported), thus for the same reason as above it gives no contribution to the tail. The particular solution of the inhomogeneous equation (\[NH\]) reads $$\label{f1nul} \phi_l^{NH} = \frac {1}{2 (\alpha-1) r^{\alpha +l}}
\sum_{q=0}^{l-\alpha/2+1/2} (l-\alpha/2+1/2)^{\underline{q}} \;
\frac {2^q \left(\alpha/2 \right)^{\overline{q}}}
{\alpha^{\overline{q}}} \frac {\phi_{l-1-q}^{H}} {r^q} \, ,$$ where $\phi_{l-1-q}^{H}$ denotes the solution of the homogeneous equation with $d=2(l-1-q)+3$ and the same generating function $a$ as in $\phi_0$ (see (\[f1\])). The formula (\[f1nul\]) can be easily derived by the method of undetermined coefficients (we emphasize that this formula is valid *only* for odd $\alpha$ satisfying $3\leq \alpha \leq 2l+1$). Substituting (\[f1nul\]) into (\[iter2\]), after a long calculation (see appendix A for the technical details), we obtain the following asymptotic behavior near timelike infinity $$\label{tail2}
\phi(t,r) \approx \lambda^2 \, \phi_2(t,r) = \lambda^2 \frac {D(l,\alpha)} {t^{2(\alpha+l-1)}}
\left[ A + 2(\alpha+l-1)\frac{B}{t} + \mathcal{O} \left( \frac{1}{t^2} \right) \right]\,,$$ where the coefficients $A$ and $B$ are defined in (\[ab\]) and $$\label{c2}
D(l,\alpha)= \frac {2^{2(\alpha+l-2)}}{(2l+1)!!} \cdot \frac{(2\alpha-3)}{2(\alpha-1)}
\left( \alpha - \frac {5} {2} \right)^{\underline{l-1}} \left( \alpha - 2 + l
\right)^{\underline{l-1}}\,
F\left( \left.
\begin{array}{c} -l+\alpha/2-1/2,\, \alpha/2,\, 2\alpha-2,\, 1 \\ \alpha,\, \alpha,\,
\alpha - l-1/2 \end{array} \right| 1
\right)\,.$$ Here $F$ stands for the generalized hypergeometric function $$\label{F}
F\left( \left.
\begin{array}{c}
a_1,\, \dots,\, a_m
\\
b_1,\, \dots,\, b_n
\end{array}
\right| z \right) = \sum _{k \geq 0} \frac {a_1^{\overline{k}},\,
\dots,\, a_m^{\overline{k}}}
{b_1^{\overline{k}},\, \dots,\, b_n^{\overline{k}}}
\frac{z^k}{k!}\,.$$
[|c||\*[4]{}[c|]{}]{} & & & &\
$1$ & $4$ & & &\
$2$ & $-8/5$ & $2240/3$ & &\
$3$ & $96/35$ & $1792$ & $2523136/5$ &\
$4$ & $-64/7$ & $-17920/9$ & $16580608/5$ & $4638965760/7$\
We remark that the behavior $\mathcal{O}(\lambda^2)\,
t^{-2(l+\alpha-1)}$ of the tail (\[tail2\]) was conjectured before by Ching *et al.* [@ching] on the basis of dimensional analysis.
General polynomially decaying potentials
========================================
In this section we analyze how the presence of subleading corrections to the pure inverse-power asymptotic behavior of the potential affects the results obtained in section 3. We restrict ourselves to the most interesting and common case (below referred to as type II) when near infinity $$\label{case2}
V(r) = \frac{1}{r^{\alpha}} \left(1+\frac{\beta}{r^{\gamma}}\right)
+o\left(\frac{1}{r^{\alpha+\gamma}}\right),\,\qquad
\gamma>0\,.$$ If $C(\alpha,l)\neq 0$, then the dominant behavior of the tail is of course the same as in (\[tail1\]): $$\label{gtail1}
\phi(t,r) \sim \lambda \, A \, C(l,\alpha)\,
t^{-(\alpha+2l)}\,.$$ However, in the exceptional case, when $C(\alpha,l)=0$, the situation is more delicate. As we showed above, in this case there is the second-order contribution to the tail given by (\[tail2\]) $$\label{gtail2}
\phi_2(t,r) \sim A \, D(l,\alpha)\,
t^{-2(\alpha+l-1)}\,.$$ In contrast to the type I case where the first-order tail vanishes identically, in the type II case the subleading term in the potential produces the first-order contribution which is given by (\[tail1\]) with $\alpha$ replaced by $\alpha+\gamma$: $$\label{tailf1p}
\phi_1(t,r) \sim \beta \, A\, C(l,\alpha+\gamma)\,
t^{-(\alpha+\gamma+2l)}\,,$$ assuming that $\alpha+\gamma$ is not an odd integer $\leq
d-2$ (otherwise one has to repeat the analysis for the next subleading term in the potential).
Now, comparing the decay rates in (\[gtail2\]) and (\[tailf1p\]) we conclude that the leading asymptotics of the tail is given by the first-order term $\lambda \phi_1(t,r)$ if $\gamma\leq \alpha-2$ (we call it subtype IIa), but otherwise, *i.e.* for $\gamma
> \alpha-2$ (subtype IIb), the second-order term $\lambda^2 \phi_2(t,r)$ is dominant for $t\rightarrow \infty$. 0.2cm *Remark 2.* In the context of equation (\[a3\]) a formula analogous to (\[tailf1p\]) was obtained by Hod who studied tails in the presence of subleading terms in the potential (see subgroup IIIb in [@hod]). However, Hod’s analysis, restricted to the first-order approximation, was inconclusive because, as we just have shown, without the second-order formula (\[gtail2\]) one is not in position to make assertions about the dominant behavior of the tail. 0.2cm
Numerics
========
In order to verify the above analytic predictions we solved numerically the initial value problem (4-5) for various potentials and initial data. Our numerical algorithm is based on the method of lines with finite differencing in space and explicit fourth-order accurate Runge-Kutta time stepping. As was pointed out in [@ching], a reliable numerical computation of tails requires high-order finite-difference schemes, since otherwise the ghost potentials generated by discretization errors produce artificial tails which might mask the genuine behavior. The minimal order of spatial finite-difference operators depends on the fall-off of the potential – for the cases presented below the fourth-order accuracy was sufficient, but for the faster decaying potentials a higher-order accuracy is needed. To eliminate high-frequency numerical instabilities we added a small amount of Kreiss-Oliger artificial dissipation All computations were performed using quadruple precision which was essential in suppressing round-off errors at late times.
The numerical results presented here were produced for initial data of the form $$\label{idn}
\phi(0,r) = \exp(-r^2),\qquad
\partial_t \phi(0,r)= \exp(-r^2)\,.$$ As follows from (\[f1\]) the generating function for these data is $$\label{a}
a(x)=2^{-(l+2)} (1-2x) \exp(-x^2)\,,\quad \mbox{hence}\quad
A=\int_{-\infty}^{+\infty} a(x) dx=\sqrt{\pi}/2^{l+2}\,.$$ We considered the following potentials
[V(r)=]{} & )\
(1+) & ,
for various values of $\alpha$ and $\gamma$. The regularizing factor $\tanh(r)$ introduces exponentially decaying corrections to the pure inverse-power behavior at infinity but such corrections do not affect the polynomial tails. The numerical verification of the formulae (\[tail1\]), (\[tailf1p\]), and (\[tail2\]) is shown in tables II and III. The observed decay rates agree perfectly with analytic predictions, while small errors in the amplitudes are due to (neglected) higher-order terms in the perturbation expansion.
-- ----------- --------- ---------- ---------- ---------- --------- ----------
Theory Numerics Theory Numerics Theory Numerics
Exponent 2.5 2.499 3.01 3.009 4 4.00002
Amplitude -0.1253 -0.0881 -0.1785 -0.1518 -0.3545 -0.3320
Exponent 4.5 4.501 5.01 5.0101 6 5.9999
Amplitude 0.0261 0.0235 -0.00089 -0.00085 -0.2363 -0.2318
Exponent 6.5 6.501 7.01 7.01 8 7.9999
Amplitude -0.0294 -0.0276 0.00089 0.00087 0.1418 0.1404
-- ----------- --------- ---------- ---------- ---------- --------- ----------
: The generic case: numerical verification of the analytic formula (\[tail1\]) for the potential (31a) ($\lambda=0.1$) and initial data (\[idn\]). Comparing the second column of this table (corresponding to $\alpha=3.01$) with the last column of table III one can see the discontinuity of the decay rate at $\alpha=3$ (for $d=5$ and $7$).[]{data-label="table 3"}
-- ----------- --------- ---------- ---------- ---------- ---------- ----------
Theory Numerics Theory Numerics Theory Numerics
Exponent 5.5 5.4993 6 6.002 6 6.0000
Amplitude -0.0731 -0.0696 0.00886 0.00862 0.00886 0.00843
Exponent 7.5 7.4998 8 8.0003 8 7.9999
Amplitude 0.0603 0.0579 -0.00177 -0.00175 -0.00177 -0.00172
Exponent 9.5 9.4999 10 9.9957 10 9.9997
Amplitude -0.1131 -0.1115 0.00152 0.00145 0.00152 0.00149
-- ----------- --------- ---------- ---------- ---------- ---------- ----------
: The exceptional case: comparison of analytic and numerical parameters of the tails for the potential (31b) (the first two columns) and (31a) (the third column) with $\alpha=3$, $\lambda=0.1$, and initial data (\[idn\]). The analytic results are given by the formula (\[tailf1p\]) for the subtype IIa potential, and by the formula (\[gtail2\]) for the type I and IIb potentials. Note that although the dominant tails for the type I and the subtype IIb potentials are theoretically the same, in the case IIb there is an additional first order error due to the subdominant term $\mathcal{O}(\lambda)
t^{-(2l+\alpha+\gamma)}$ which accounts for a slight difference in numerical accuracy between these two cases.[]{data-label="table 3"}
Schwarzschild background
========================
Consider the evolution of the massless scalar field outside the $d+1$ dimensional Schwarzschild black hole $$\label{sch}
ds^2=-\left(1-\frac{1}{r^{d-2}}\right) dt^2 +
\left(1-\frac{1}{r^{d-2}}\right)^{-1}
dr^2 + r^2
d\Omega_{d-1}^2\,,$$ where $d\Omega_{d-1}^2$ is the round metric on the unit sphere $S^{d-1}$ and $d\geq 5$ is odd. Here we use units in which the horizon radius is at $r=1$. Introducing the tortoise coordinate $x$, defined by $dr/dx=1-1/r^{d-2}$, and decomposing the scalar field into multipoles, one obtains the following reduced wave equation for the $j$th multipole [@ik] $$\label{eqsch}
\partial_t^2\psi -\partial_x^2 \psi +U(x) \psi=0, \qquad
U=\left(1-\frac{1}{r^{d-2}}\right) \left(\frac{(2j+d-3)(2j+d-1)}{4r^2}+\frac{(d-1)^2}{4
r^d}\right)\,.$$ Note that (\[eqsch\]) is the $1+1$ dimensional wave equation on the whole axis $-\infty<x<\infty$. For large positive $x$ we have $$\label{exp2}
r =
x+\frac{1}{d-3}\frac{1}{x^{d-3}}-\frac{d-2}{(2d-5)(d-3)}\frac{1}{x^{2d-5}} +\mathcal{O} \left( \frac{1}{x^{3d-7}}\right)\,,$$ which implies that $$\label{expv}
U(x) =
\frac{(2j+d-3)(2j+d-1)}{4x^2}+V(x)\,,\qquad V(x)=
\frac{a}{x^d}+\frac{b}{x^{2d-2}} +\mathcal{O} \left( \frac{1}{x^{3d-4}}\right) \quad \mbox{as}\quad x\rightarrow
\infty\,,$$ with $$\label{wsp} a = - \frac {(d-1) j (j+d-2)} {d-3} \qquad \mbox{and}
\qquad b = - \frac {(2d - 3) ((d-3)(d-2)^2(d-1) - 4 j
(j+d-2)(1+d(d-3)))} {4(2d-5)(d-3)^2}\,.$$ For large negative $x$ (near the horizon) the potential is exponentially small, so one expects that the backscattering off the left edge of the potential can be neglected. If so, the decay rate (but not the amplitude!) should follow from the analysis of section 4. Comparing equation (\[eqsch\]) for large positive $x$ to equation (\[a3\]) with the potential (\[case2\]) and using (\[expv\]) we find that $l=j+(d-3)/2$ and the potential $V$ is of the subtype IIa with $\alpha=d$ and $\gamma=d-2$. Thus, applying (\[tailf1p\]) we get the first-order tail $$\label{tails}
\psi(t,x) \sim t^{-(2j+3d-5)}\,.$$ *Remark 3.* Late-time tails outside higher dimensional Schwarzschild black holes were studied in [@car], however in the even-dimensional case the reasoning presented there is not correct, even though the result agrees with (\[tails\]). The reason is that the analysis of [@car] is based on the application of Ching *et al.* conjecture about the decay of the second-order tail $t^{-(2l+2\alpha-2)}$ which for $l=j+(d-3)/2$ and $\alpha=d$ gives $t^{-(2j+3d-5)}$. Unfortunately, this conjecture does not apply to the problem at hand. For $j=0$ this is evident because the leading term in $V$ (proportional to $x^{-d}$) vanishes (since by (\[wsp\]) $a=0$), while the subleading term (proportional to $x^{-(2d-2)}$) is of generic type. For $j>0$ this follows from the fact that the potential is of the subtype IIa. Thus, for all $j\geq
0$ the dominant (first-order) contribution to the tail comes from the subleading term in the potential. The agreement of the decay rate obtained in [@car] with (\[tails\]) is accidental and due to the fact that the subdominant term in (\[expv\]) (not considered in [@car]) is on a borderline between subtypes IIa and IIb. Admittedly, the handwaving argument leading to (\[tails\]) is far from satisfactory. Unfortunately, we have not been able to carry over the analysis from sections 2-4 in the case of equation (\[eqsch\]). There are two difficulties in this respect. First, in contrast to the spherical case, Huygens’ principle is not valid for the free wave equation in $1+1$ dimensions. Second, there is no natural small parameter in the problem. In the impressive tour de force work [@b] Barack showed how to overcome these difficulties for a restricted class of initial data in four dimensions. It would be interesting to generalize Barack’s approach to higher even-dimensional Schwarzschild spacetimes.
0.3cm **Acknowledgments:** PB thanks Nikodem Szpak for helpful discussions and Leor Barack for clarifying some details of the paper [@b]. AR thanks Prof. Bernd Brügmann for hospitality in his group at FSU Jena, where a part of this work was done. This research was supported in part by the MNII grant 1PO3B01229 and grant 189/6. PR UE/2007/7.
Throughout the appendix we use the notation of [@gkp] in which the square bracket around a logical expression returns a value $1$ if the expression is true and a value $0$ if the expression is false: $$[condition] = \left\{ \begin{array}{ccl} 1 & \mbox{if} & condition =
\mbox{true} \\ 0 & \mbox{if} & condition = \mbox{false} \end{array}
\right.\nonumber$$ In order to derive the asymptotic behavior of the iterates (\[iter1\]) and (\[iter2\]) near timelike infinity (fixed $r$ and $t\rightarrow\infty$) we need to evaluate the following expression $$\mathcal{F} (t,r;\,\beta,\,L) = - \frac {2^{\beta}}{4 r^{l+1}}
\sum_{k=0}^L c_{L,k} \, \int \limits_{-\infty}^{+\infty} d\eta \,
\int \limits_{t-r}^{t+r} d\xi \, \frac {P_l (\mu)}
{(\xi-\eta)^{\beta+L-k}} a^{(k)}(\eta), \label{master}$$ where $$c_{L,k} = \frac {(2 L - k)!} {k! (L-k)!}$$ and $$\label{mu} \mu = \frac {(\xi-t)(t-\eta)+r^2}{r(\xi-\eta)}\, .$$ From (\[f1\]) and (\[iter1\]) we have $$\phi_{1}(t,r) = \mathcal{F} (t,r;\,\alpha,\,l),$$ and from (\[iter2\]) and (\[f1nul\]) we have $$\label{a5}
\phi_{2}(t,r) = \frac {1}{2 (\alpha-1) r^{\alpha +l}}
\sum_{q=0}^{l-\alpha/2+1/2} (l-\alpha/2+1/2)^{\underline{q}} \cdot
\frac {2^q \left(\alpha/2 \right)^{\overline{q}}}
{\alpha^{\overline{q}}} \, \mathcal{F} (t,r;\,2 \alpha - 1 +
q,\,l-1-q).$$ Since $a(\eta)$ has compact support, it is advantageous to begin with integration by parts $$\int \limits_{-\infty}^{+\infty} d\eta \, \frac {P_l (\mu)}
{(\xi-\eta)^{\beta+L-k}} a^{(k)}(\eta)
= \int \limits_{-\infty}^{+\infty} d\eta \, (-1)^k \frac {d^k} {d\eta^k} \left( \frac {P_l (\mu)} {(\xi-\eta)^{\beta+L-k}} \right)
a(\eta)\,.\nonumber$$ For $\mu$ as defined in (\[mu\]) and for any function $g(\mu)$ the following identity holds $$\frac {d^k} {d\eta^k} \left( \frac {g(\mu)} {(\xi-\eta)^{\beta}}
\right) = \sum_{j=0}^k \left( \begin{array}{c} k \\ j
\end{array} \right) (\beta+k-1)^{\underline{k-j}} \left( \frac
{r^2-(t-\xi)^2} {r} \right)^j \frac {g^{(j)}(\mu)}
{(\xi-\eta)^{\beta+k+j}}\,,$$ hence $$\mathcal{F} (t,r;\,\beta,\,L) = - \frac {2^{\beta}}{4 r^{l+1}} \int
\limits_{-\infty}^{+\infty} d\eta \, a(\eta) \, \sum_{0\leq j\leq k
\leq L} (-1)^k \left( \begin{array}{c} k \\ j \end{array} \right)
c_{L,k} (\beta+L-1)^{\underline{k-j}} \frac {1} {r^j} \int
\limits_{t-r}^{t+r} d\xi \, \frac {\left( r^2-(t-\xi)^2 \right)^j}
{(\xi-\eta)^{\beta+L+j}} P^{(j)}_{l} (\mu)\,. \label{master(2)}$$ The sum over $k$ can be evaluated explicitly $$\label{k-sum} \sum_{k=j}^L (-1)^k \left( \begin{array}{c} k
\\ j \end{array} \right) \frac {(2 L - k)!}
{k! (L-k)!} (\beta+L-1)^{\underline{k-j}} = (-1)^L \left( \begin{array}{c} L \\ j \end{array} \right) (\beta-2)^{\underline{L-j}}\,.$$ Let us define $$\label{I}
\mathcal{I} := \frac {1} {r^j} \int \limits_{t-r}^{t+r} d\xi \,
\frac {\left( r^2-(t-\xi)^2 \right)^j} {(\xi-\eta)^{\beta+L+j}}
P^{(j)}_{l} (\mu) \,.$$ Changing the integration variable from $\xi$ to $\mu$ and integrating by parts, we get $$\mathcal{I} = (-1)^j \frac {r^{j+1} (t-\eta)^{\beta-2+L-j}} {\left[
(t-\eta)^2-r^2 \right]^{\beta-1+L}} \int \limits_{-1}^{+1} d\mu \,
P_{l} (\mu) \frac {d^j} {d\mu^j} \left[ (1 - \mu^2)^j \left( 1 -
\frac {r} {t-\eta} \mu \right)^{\beta-2+L-j} \right]. \label{I}$$ Using the identity [@wolfram] $$\label{mu-to-k} \mu^k = \sum_{l=k,k-2,k-4,\dots} \frac {(2l+1) k!}
{2^{(k-l)/2} \left( \frac {k-l} {2} \right)! (k+l+1)!!}\, P_l
(\mu)\,,$$ and expanding $\dfrac {d^j} {d\mu^j} \left[ (1 - \mu^2)^j \left( 1 -
\frac {r} {t-\eta} \mu \right)^{\beta-2+L-j} \right]$ in Taylor series we get $$\begin{aligned}
\mathcal{I} &=& (-1)^j \frac {r^{j+1} (t-\eta)^{\beta-2+L-j}}
{\left[ (t-\eta)^2-r^2 \right]^{\beta-1+L}} \,
\sum_{n=0}^{\beta-2+L} (j+n)^{\underline{j}} \, \int
\limits_{-1}^{+1} d\mu \, P_{l} (\mu) \mu^{n}
\nonumber\\
&\times& \sum_{m=0}^{\lfloor (j+n)/2 \rfloor} \left(
\begin{array}{c} j \\ m \end{array} \right) \left( \begin{array}{c}
\beta-2+L-j \\ j+n-2m \end{array} \right) (-1)^{j+n+m} \left( \frac
{r} {t-\eta} \right)^{j+n-2m}
\nonumber\\
&=& \frac {r^{l+1} (t-\eta)^{\beta-2+L-l}} {\left[ (t-\eta)^2 - r^2
\right]^{\beta-1+L}} \, \sum_{n=0}^{\lfloor (\beta-2+L-l) / 2
\rfloor} (j+l+2n)^{\underline{j}} \, \int \limits_{-1}^{+1} d\mu \,
P_{l} (\mu) \mu^{l+2n}
\nonumber\\
&\times& \sum_{m=0}^{\lfloor (j+l+2n)/2 \rfloor} \left(
\begin{array}{c} j \\ m \end{array} \right) \left(
\begin{array}{c} \beta-2+L-j \\ j+l+2n-2m \end{array} \right)
(-1)^{l+m} \left( \frac {r} {t-\eta} \right)^{2j+2n-2m}
\nonumber\\
&=& \frac {r^{l+1} (t-\eta)^{\beta-2+L-l}} {\left[ (t-\eta)^2 - r^2
\right]^{\beta-1+L}} \, \sum_{n=0}^{\lfloor (\beta-2+L-l) / 2
\rfloor} (j+l+2n)^{\underline{j}} \, \, 2^{l+1} \frac {(l+2n)!
(l+n)!} {n! (2l+2n+1)!}
\nonumber\\
&\times& \sum_{m=0}^{\lfloor (j+l+2n)/2 \rfloor} \left(
\begin{array}{c} j \\ m \end{array} \right) \left(
\begin{array}{c} \beta-2+L-j \\ j+l+2n-2m \end{array} \right)
(-1)^{l+m} \left( \frac {r} {t-\eta} \right)^{2j+2n-2m}
\label{I(2)}\,.\end{aligned}$$ Collecting the results of (\[k-sum\], \[I\], \[I(2)\]) and plugging them into (\[master(2)\]) we get $$\mathcal{F} (t,r;\,\beta,\,L)
= - \frac {2^{\beta+l+1}}{4} \int \limits_{-\infty}^{+\infty}
d\eta \, a(\eta) \, \frac {(t-\eta)^{\beta-2+L-l}} {\left[
(t-\eta)^2 - r^2 \right]^{\beta-1+L}} \sum_{n=0}^{\lfloor
(\beta-2+L-l) / 2 \rfloor} \frac {(l+2n)! (l+n)!} {n! (2l+2n+1)!}
(-1)^{L+l} L! \, S (\beta, L), \label{master(3)}$$ where $$\!\!S (\beta, L) = \sum_{j=0}^{L} \left( \begin{array}{c}
\beta-2 \\ L-j \end{array} \right) \left( \begin{array}{c} j+l+2n \\
j \end{array} \right) \sum_{m=0}^{\lfloor (j+l+2n)/2 \rfloor}\!\!\!
(-1)^{m} \left( \begin{array}{c} j
\\ m \end{array} \right) \left( \begin{array}{c} \beta-2+L-j \\
j+l+2n-2m \end{array} \right) \left( \frac {r} {t-\eta}
\right)^{2j+2n-2m} \,.$$
First-order approximation
-------------------------
To evaluate the first iterate $\phi_1(t,r)$ we apply the formula (\[master(3)\]) with $\beta=\alpha$ and $L=l$. Then $$S(\alpha,l) = \sum_{j=0}^{l} \left( \begin{array}{c} \alpha-2
\\ l-j \end{array} \right) \left( \begin{array}{c} l+2n+j
\\ j \end{array} \right) \sum_{m=(j-l)/2}^{j+n} \!\!\!(-1)^{j+n-m} \left(
\begin{array}{c} j \\ j+n-m \end{array} \right) \left(
\begin{array}{c} \alpha-2+l-j \\ l-j+2m \end{array} \right) \left(
\frac {r} {t-\eta} \right)^{2m}\,, \label{SI}$$ where we shifted the summation index $m\rightarrow j+n-m$. Next, we interchange the order of summation according to $$\begin{aligned}
&& [ 0 \leq j ] [ j \leq l ] [ m-n \leq j ] [ j \leq 2m+l ]
\nonumber\\
&\Leftrightarrow& [ -\frac {l}{2} \leq m < 0 ] [ 0 \leq j \leq l+2m
] \, + \, [ 0 \leq m < n ] [ 0 \leq j \leq l ] \, + \, [ n \leq m
\leq l+n ] [ m-n \leq j \leq l ]\,,\nonumber\end{aligned}$$ and convert the sum over $j$ into a generalized hypergeometric function [@gkp]. Defining $$t_j = (-1)^{j+n-m} \left( \begin{array}{c} \alpha-2 \\ l-j
\end{array} \right) \left( \begin{array}{c} l+2n+j \\ j \end{array}
\right) \left( \begin{array}{c} j \\ j+n-m \end{array} \right)
\left( \begin{array}{c} \alpha-2+l-j \\ l-j+2m \end{array}
\right)\,,$$ we see that $t_0\neq 0$ iff $n=m$, thus the sums for $[ -\frac
{l}{2} \leq m < 0 ]$ and $[ 0 \leq m < n ]$ do not contribute to (\[SI\]) and we are left with $$\begin{aligned}
S (\alpha, l)&=& \sum_{m=n}^{n+l} \left( \frac {r}
{t-\eta} \right)^{2m} \sum_ {j=0}^{l+n-m} (-1)^{j} \left(
\begin{array}{c} \alpha-2 \\ l+n-m-j \end{array} \right) \left(
\begin{array}{c} l+n+m+j \\ j+m-n \end{array} \right)
\nonumber\\
& \times & \left( \begin{array}{c} j+m-n \\ j \end{array} \right)
\left(
\begin{array}{c} \alpha-2+l+n-m-j \\ l+n+m-j \end{array} \right),\end{aligned}$$ where we shifted the summation index $j \rightarrow j+m-n$. Defining $$\tilde{t}_j = (-1)^{j} \left( \begin{array}{c} \alpha-2 \\ l+n-m-j
\end{array} \right) \left( \begin{array}{c} l+n+m+j \\ j+m-n
\end{array} \right) \left( \begin{array}{c} j+m-n \\ j \end{array}
\right) \left( \begin{array}{c} \alpha-2+l+n-m-j \\ l+n+m-j
\end{array} \right)$$, we see that $$\begin{aligned}
\tilde{t}_0 &=& \frac {(\alpha-2)^{\underline{l+n-m}}} {(l+n-m)!}
\cdot \frac {(l+n+m)!} {(m-n)! (l+2n)!} \cdot \frac
{(\alpha-2+l+n-m)^{\underline{l+n+m}}} {(l+n+m)!}\nonumber\end{aligned}$$ and $$\frac {\tilde{t}_{j+1}} {\tilde{t}_j} = \frac {(j - (l+n-m)) (j -
(l+n+m)) (j + (l+n+m+1)) } {(j + ((\alpha-1) - (l+n-m))) (j +
(-(\alpha-2) - (l+n-m)))
(j +1)}\,,$$ hence $$\begin{aligned}
S (\alpha, l) &=& \sum_{m=n}^{n+l} \left( \frac {r} {t-\eta}
\right)^{2m} \frac {(\alpha-2)^{\underline{l+n-m}}} {(l+n-m)!} \cdot
\frac {(\alpha-2+l+n-m)^{\underline{l+n+m}}} {(m-n)! (l+2n)!}
\nonumber\\
&\times& F\left( \left. \begin{array}{c} - (l+n-m),\, - (l+n+m),\,
(l+n+m+1) \\ (\alpha-1) - (l+n-m),\, -(\alpha-2) - (l+n-m)
\end{array} \right| 1 \right)
\nonumber\\
&=& \sum_{m=n}^{n+l} \left( \frac {r} {t-\eta} \right)^{2m} 2^{1 +
2(l+n-m)} \pi \,\frac {(\alpha-2)^{\underline{l+n-m}}} {(l+n-m)!}
\cdot \frac {(\alpha-2+l+n-m)^{\underline{l+n+m}}} {(m-n)! (l+2n)!}
\nonumber\\
&\times& \frac {\Gamma(-(\alpha-2) - (l+n-m)) \Gamma((\alpha-1) -
(l+n-m))} {\Gamma \left( - \frac {\alpha-3} {2} + m \right) \Gamma
\left( - \frac {\alpha-2} {2} - (l+n) \right) \Gamma \left( \frac
{\alpha} {2} + m \right) \Gamma \left( \frac {\alpha-1} {2} - (l+n)
\right)}\,, \label{SI(2)}\end{aligned}$$ where in the last equation we used the identity $$F\left( \left. \begin{array}{c} a+1,\, -a,\, (b+c-1)/2 \\ b,\, c
\end{array} \right| 1 \right) = 2^{2-(b+c)} \pi \frac {\Gamma(b)
\Gamma(c)} {\Gamma \left( \frac {b-a} {2} \right) \Gamma \left(
\frac {c-a} {2} \right)
\Gamma \left( \frac {1+b+a} {2} \right) \Gamma \left( \frac {1+c+a} {2}
\right)}\,.$$ Substituting $$(\alpha-2)^{\underline{l+n-m}} \Gamma((\alpha-1) - (l+n-m)) =
\Gamma(\alpha-1),\nonumber$$ and $$(\alpha-2+l+n-m)^{\underline{l+n+m}} \Gamma(-(\alpha-2) - (l+n-m)) =
(-1)^{l+n+m} \Gamma(-\alpha + 2 + 2m)\nonumber$$ into (\[SI(2)\]) we get $$\begin{aligned}
\!S (\alpha, l) \! &\!=\!&\! \sum_{m=n}^{n+l} \left( \frac {r}
{t-\eta} \right)^{2m}\! \!\frac {(-1)^{l+n+m} 2^{1 + 2(l+n-m)} \pi}
{(l+n-m)! (m-n)! (l+2n)!} \frac {\Gamma(\alpha-1) \Gamma(-\alpha + 2
+ 2m)} {\Gamma \left( \frac {\alpha} {2} + m \right) \Gamma \left(
- \frac {\alpha-3} {2} + m \right) \Gamma \left( - \frac {\alpha-2}
{2} - (l+n) \right) \Gamma \left( \frac {\alpha-1} {2} - (l+n)
\right)}\,. \nonumber \label{SI(3)}\end{aligned}$$ The last equation can be still simplified due to the identity $$\label{GammasId} \frac {\Gamma(\alpha -1) \Gamma(-\alpha + 2)}
{\Gamma \left( \frac {\alpha} {2} \right) \Gamma \left( - \frac
{\alpha-3} {2} \right) \Gamma \left( - \frac {\alpha-2} {2} - l
\right) \Gamma \left( \frac {\alpha-1} {2} - l \right)} = \frac
{(-1)^l} {2 \pi} \left( \frac {\alpha-3} {2} \right)^{\underline{l}}
\left( \frac {\alpha} {2} \right)^{\overline{l}}.$$ We have $$\begin{aligned}
\label{Gamma1} \Gamma(-\alpha + 2 + 2m) &=& (-\alpha +
2)^{\overline{2m}} \Gamma(-\alpha + 2),\nonumber\\
\label{Gamma2} \Gamma \left( - \frac {\alpha-3} {2} + m \right)& =&
\left( - \frac {\alpha-3} {2} \right)^{\overline{m}} \Gamma \left( -
\frac {\alpha-3} {2} \right),\nonumber\\
\label{Gamma3} \Gamma \left( - \frac {\alpha-2} {2} - l - n \right)
&=& \frac {\Gamma \left( - \frac {\alpha-2} {2} - l \right)} {\left(
- \frac {\alpha-2} {2} - l - 1 \right)^{\underline{n}}}\,,\nonumber \\
\label{Gamma4} \Gamma \left( \frac {\alpha-1} {2} - l - n \right)
&=& \frac {\Gamma \left( \frac {\alpha-1} {2} - l \right)} {\left(
\frac {\alpha-1} {2} - l - 1 \right)^{\underline{n}}}\,,\nonumber\\
\label{Gamma5} \Gamma \left( \frac {\alpha} {2} + m \right) &= &
\left( \frac {\alpha} {2} \right)^{\overline{m}} \Gamma \left( \frac
{\alpha} {2} \right),\nonumber\end{aligned}$$ and $$\frac {(-\alpha + 2)^{\overline{2m}}} {\left( - \frac {\alpha-3}
{2} \right)^{\overline{m}}} = 2^{2m} \left( - \frac {\alpha} {2} + 1
\right)^{\overline{m}},\nonumber$$ so finally $$\begin{aligned}
S (\alpha, l) &=& \sum_{m=n}^{n+l} \left( \frac {r} {t-\eta}
\right)^{2m} \frac {(-1)^{n+m} 2^{2(l+n)}} {(l+n-m)! (m-n)! (l+2n)!}
\left( \frac {\alpha-3} {2} \right)^{\underline{l}} \left( \frac
{\alpha} {2} \right)^{\overline{l}}
\nonumber\\
&\times& \frac {\left( - \frac {\alpha} {2} + 1
\right)^{\overline{m}} \left( - \frac {\alpha-2} {2} - l - 1
\right)^{\underline{n}} \left( \frac {\alpha-1} {2} - l - 1
\right)^{\underline{n}}} {\left( \frac {\alpha} {2}
\right)^{\overline{m}}}\,. \label{SI(4)}\end{aligned}$$ Plugging (\[SI(4)\]) into (\[master(3)\]) with $\beta=\alpha$ and $L=l$ we get the expression (\[tailf1\]).
Second-order approximation
--------------------------
The calculation in the second order ($\beta=2\alpha-1+q$ and $L=l-1-q$) is only a slight modification of what we have already done in the first order. Following the same steps which led us from (\[SI\]) to (\[SI(3)\]) we get $$\begin{aligned}
S (\beta, L)
&=& \sum_{m=n}^{n+L} \left( \frac {r} {t-\eta}
\right)^{2m} \frac {(-1)^{l+n+m} 2^{1 + 2(L+n-m)} \pi} {(L+n-m)!
(m-n)! (l+2n)!}
\nonumber\\
&\times& \frac {\Gamma(\beta-1) \Gamma(-\beta + 2 + l-L + 2m)}
{\Gamma \left( \frac {\beta} {2} + \frac{l-L}{2} + m \right) \Gamma
\left( - \frac {\beta-3} {2} + \frac{l-L}{2} + m \right) \Gamma
\left( - \frac {\beta-2} {2} - \left(\frac{l+L}{2}+n\right)
\right)\Gamma \left( \frac {\beta-1} {2} -
\left(\frac{l+L}{2}+n\right) \right)}\,. \label{j-sum(3)II}
\label{SII(3)}\end{aligned}$$ The last equation can be simplified due to the identity $$\begin{aligned}
&& \frac {\Gamma(\beta - 1) \Gamma(-\beta + 2 + l-L)} {\Gamma \left(
\frac {\beta} {2} + \frac{l-L}{2} \right) \Gamma \left( - \frac
{\beta-3} {2} + \frac{l-L}{2} \right) \Gamma \left( - \frac
{\beta-2} {2} - \frac{l+L}{2} \right) \Gamma \left( \frac {\beta-1}
{2} - \frac{l+L}{2} \right)}
\nonumber\\
&=& \frac {(-1)^l} {2 \pi} \left( \frac {\beta-3-(l-L)} {2}
\right)^{\underline{L}} \left( \frac {\beta + l-L} {2}
\right)^{\overline{L}} (\beta-2)^{\underline{l-L}}\,,\end{aligned}$$ which for $L=l$ reduces to (\[GammasId\]). We have $$\begin{aligned}
\label{Gamma1II} \Gamma(-\beta + 2 + l-L + 2m) &=& (-\beta + 2 +
l-L)^{\overline{2m}} \,\Gamma(-\beta + 2 + l-L)\,,\nonumber\\
\label{Gamma2II} \Gamma \left( - \frac {\beta-3} {2} + \frac{l-L}{2}
+ m \right) &=& \left( - \frac {\beta-3} {2} + \frac{l-L}{2}
\right)^{\overline{m}} \Gamma \left( - \frac {\beta-3} {2} +
\frac{l-L}{2} \right)\,,\nonumber\\
\label{Gamma3II} \Gamma \left( - \frac {\beta-2} {2} - \frac{l+L}{2}
- n \right) &=& \frac {\Gamma \left( - \frac {\beta-2} {2} -
\frac{l+L}{2} \right)} {\left( - \frac {\beta-2} {2} - \frac{l+L}{2}
- 1 \right)^{\underline{n}}}\,,\nonumber\\
\label{Gamma4II} \Gamma \left( \frac {\beta-1} {2} - \frac{l+L}{2} -
n \right) &=& \frac {\Gamma \left( \frac {\beta-1} {2} -
\frac{l+L}{2} \right)} {\left( \frac {\beta-1} {2} - \frac{l+L}{2} -
1 \right)^{\underline{n}}}\,,\nonumber\\
\label{Gamma5II} \Gamma \left( \frac {\beta} {2} + \frac{l-L}{2} + m
\right) &=& \left( \frac {\beta} {2} + \frac{l-L}{2}
\right)^{\overline{m}} \Gamma \left( \frac {\beta} {2} +
\frac{l-L}{2} \right)\,,\nonumber\end{aligned}$$ and $$\frac {(-\beta + 2 + l-L)^{\overline{2m}}} {\left( - \frac
{\beta-3} {2} + \frac{l-L}{2} \right)^{\overline{m}}} = 2^{2m}
\left( - \frac {\beta} {2} + \frac{l-L}{2} + 1
\right)^{\overline{m}},\nonumber$$ hence $$\begin{aligned}
S (\beta, L) &\!=\!& \sum_{m=n}^{n+L} \left( \frac {r} {t-\eta}
\right)^{2m} \frac {(-1)^{n+m} 2^{2(L+n)}} {(L+n-m)! (m-n)! (l+2n)!} \label{SII(4)}\\
&\times& \left( \frac {\beta-3-(l-L)} {2} \right)^{\underline{L}}
\left( \frac {\beta + l-L} {2} \right)^{\overline{L}}
(\beta-2)^{\underline{l-L}} \frac {\left( - \frac {\beta} {2} +
\frac{l-L}{2} + 1 \right)^{\overline{m}} \left( - \frac {\beta-2}
{2} - \frac{l+L}{2} - 1 \right)^{\underline{n}} \left( \frac
{\beta-1} {2} - \frac{l+L}{2} - 1 \right)^{\underline{n}}} {\left(
\frac {\beta} {2} + \frac{l-L}{2} \right)^{\overline{m}}}.\nonumber\end{aligned}$$ Plugging (\[SII(4)\]) into (\[master(3)\]) we get $$\begin{aligned}
&& \mathcal{F} (t,r;\,2\alpha-1+q,\,l-1-q) = (-1)^{q} \,\frac
{2^{2\alpha+3l-2-q}}{4} \left( \alpha - \frac {5} {2}
\right)^{\underline{l-1-q}} \left( \alpha - 2 + l
\right)^{\underline{l-1-q}} (2\alpha-3)^{\overline{1+q}}
\nonumber\\
&\times& \int \limits_{-\infty}^{+\infty} d\eta \, a(\eta) \, \frac
{(t-\eta)^{2\alpha-4}} {\left[ (t-\eta)^2 - r^2
\right]^{2\alpha-3+l}}
\\
&\times& \sum_{n=0}^{\alpha-2}\! (-1)^{n} \frac {2^{2n}(l+n)!} {n!
(2l+2n+1)!} \left( -\alpha+1-l \right)^{\underline{n}} \left( \alpha
- \frac {3} {2} - l + q \right)^{\underline{n}} \, \sum_{m=n}^{
n+l-1-q} \!(-1)^{m} \left(
\begin{array}{c} l-1-q \\ m-n \end{array} \right) \frac {\left(
-\alpha + 2 \right)^{\overline{m}}} {\left( \alpha + q
\right)^{\overline{m}}} \left( \frac {r} {t-\eta}
\right)^{2m}.\nonumber\end{aligned}$$ Substituting this into (\[a5\]) and expanding in $1/t$ we have $$\begin{aligned}
\phi_{2}(t,r) &=& \frac {1} {2(\alpha-1)} \cdot \frac
{2^{2\alpha+2l-2}}{4 (2l+1)!!} \cdot \frac {1} {t^{2\alpha+2l-2}}
\left[ A + 2(\alpha+l-1) \frac{B}{t} \, + \mathcal{O} \left(
\frac{1}{t^2} \right) \right]
\nonumber\\
&\times& \left( \sum_{q=0}^{l-(\alpha-1)/2} (-1)^{q}
(l-p)^{\underline{q}} \,\, \frac {2^q \left( \alpha/2
\right)^{\overline{q}}} {\alpha^{\overline{q}}} \left( \alpha -
\frac {5} {2} \right)^{\underline{l-1-q}} \left( \alpha - 2 + l
\right)^{\underline{l-1-q}} (2\alpha-3)^{\overline{1+q}} \right),\end{aligned}$$ with $A$ and $B$ defined in (\[ab\]). Converting the sum over $q$ into the generalized hypergeometric function we get (\[tail2\]).
[10]{}
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R. L. Graham, D. E. Knuth and O. Patashnik, *Concrete Mathematics* (Reading, Massachusetts: Addison-Wesley, 1994).
http://mathworld.wolfram.com/
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We have investigated the electrical transport through strained $p-Si/Si_{1-x}Ge_x$ double-barrier resonant tunnelling diodes. The confinement shift for diodes with different well width, the shift due to a central potential spike in a well, and magnetotunnelling spectroscopy demonstrate that the first two resonances are due to tunnelling through heavy hole levels, whereas there is no sign of tunnelling through the first light hole state. This demonstrates for the first time the conservation of the total angular momentum in valence band resonant tunnelling. It is also shown that conduction through light hole states is possible in many structures due to tunnelling of carriers from bulk emitter states.'
author:
- 'U. Gennser,$^{(1)}$ M. Scheinert,$^{(2)}$ L. Diehl,$^{(2, 3)}$ S. Tsujino,$^{(2)}$ A. Borak,$^{(2)}$ C. V. Falub,$^{(2)}$ D. Grützmacher,$^{(2)}$ A. Weber,$^{(2)}$ D. K. Maude,$^{(4)}$ G. Scalari,$^{(5)}$ Y. Campidelli,$^{(6)}$ O. Kermarrec,$^{(6)}$ and D. Bensahel$^{(6)}$'
title: Total Angular Momentum Conservation During Tunnelling through Semiconductor Barriers
---
The challenge of introducing spin as an additional degree of freedom in semiconductor devices has lately attracted great attention.[@Prinz; @Wolf] One approach to couple the spin to the carrier motion is through the spin-orbit interaction; one suggestion is to use it in conjunction with resonant tunnelling devices (RTDs) for injection and detection of spin currents.[@Hall; @Glasov] Whereas the spin-orbit coupling in the conduction band, mediated by the Dresselhaus mechanism [@Dresselhaus; @Malcher] or the Rashba mechanism,[@Bychkov] is generally rather weak, the interaction is strong in the valence band. Since this band is made up from p-orbitals, the interaction term $V_{so} \sim {\bf L}\cdot {\bf S}$ is non-zero, and there is a strong coupling between the orbital angular momentum ${\bf L}$ and the spin ${\bf S}$, so that the total angular momentum $ J = L + S$ is a proper eigenvalue at the band edge. [@luttinger] In order to examine the feasibility of such devices for spintronics applications, one may therefore already consider spin (or $J$) detection in p-RTDs. It is then rather disconcerting to find, that in all previous investigations, tunnelling has been observed from heavy hole states (HH; with $(J, m_{J}) = (3/2, \pm 3/2)$ at $k = 0$) to light hole states (LH; $(3/2, \pm 1/2)$) or split-off states (SO, $(1/2, \pm 1/2)$).[@Mendez; @Liu; @Lewis; @Hayden; @Gennser1] It has been proposed that this non-conservation of the total angular momentum $(J, m_{J})$ in resonant tunnelling is due to either the band mixing at finite in-plane momentum $k_{p}$, or because of interface roughness scattering. However, especially in strained quantum wells, the non-parabolicity and band mixing for the lowest states is quite small. This suggests that scattering plays a large role even in systems with interfaces known for their good quality. In our present study, we show the absence of resonances in the $I-V$ characteristics from heavy holes tunnelling through the first light-hole state in a double barrier p-type quantum well. This demonstrates conclusively that there is ${\bf J}$ conservation during resonant tunnelling. Furthermore, by investigating specially designed RTDs, we are able to show that the emitter structure away from the barrier interface may explain an apparent mixing of $J$ in the tunnelling process.
The samples were grown by molecular beam epitaxy on fully relaxed Si$_{0.5}$Ge$_{0.5}$ pseudosubstrates, of which the top 2 $\mu$m is p-doped, $p = 1\times10^{19}cm^{-3}$. The active part of the initial structures consist of $40$Å$ $ barriers surrounding a single Si$_{0.2}$Ge$_{0.8}$ quantum well (QW) of width $W$ ($W = $$25$, $35$, or $45$Å$ $ for three different samples). Symmetrically on either side of the active structure are $150$Å$ $ thick SiGe emitter layers that are linearly graded, from $80$% Ge closest to the barriers to $50$% Ge away from the barriers. These emitter layers consists of an undoped spacer ($100$Å, closest to the barriers) and a doped part ($50$Å, $p = 2\times10^{18}cm^{-3}$). A $2000$Å, $p = 2\times10^{18}cm^{-3}$ Si$_{0.5}$Ge$_{0.5}$ top contact layer terminates the structure. The corresponding structure for the $35$Å$ $ QW is schematically shown in Fig. \[struc\]. For clarity, the graded emitter region is also shown, and the lowest energy levels in the quantum well at $k_p = 0$ are indicated. Due to the strain splitting, only HH states will be populated in the emitter closest to the barrier. Since the LH and SO bands are coupled even for zero in-plane momentum $k_p$, we have denoted these states as LHSO; however, the LHSO1 level is in fact predominately LH.
The diodes were processed into mesas with diameters varying between $10\mu m$ and $300\mu m$. All measurements were performed at $T \le 4$K, using separate voltage and current leads connected to both diode contacts, unless otherwise stated. However, we found that the resonance voltages changed by less than $10$% $ $ between 4K and 77K.
In Fig. \[IVpic\]a the 77K current versus voltage characteristics of the three different RTDs are plotted. They show up to three resonances, with a maximum peak-to-valley current ratio of $5:1$ at $4$K. These characteristics are comparable to the best p-type RTDs in any material system, and indicate a good interface quality minimizing interface-roughness assisted tunnelling. In the following we will focus on the two lowest resonances.
The first evidence for assigning the resonances comes from the confinement shift clearly evident in the $I-V$ characteristics. The shift, obtained from measurements of the smallest diodes, is unaffected by the contact layer resistance from the substrate, as verified by the dependence of the current on the mesa size. In Fig. \[IVpic\]b the resonance voltages vs. well width are plotted. On the right hand scale, these are compared with the calculated energies. The scales can be directly compared by assuming a linear voltage drop across the double barriers, the quantum well and the undoped part of the structure. Including the Stark changes the energies much less than the measurement uncertainties. Because of the graded nature of the emitter, the zero bias emitter states lie $\approx$30 meV higher then than the quantum well edge. This is included as an energy offset between the two scales. The so-called ’lever arm’ - i.e. the ratio between the energy drop between the emitter and the centre of the quantum well and the applied voltage - is in good agreement with what can be expected from geometrical considerations, and the energies are consistent with those obtained from intersubband absorption measurements.[@Diehl] A good agreement between theory and experiment is found if the first two resonances correspond to tunnelling through the HH1 and HH2 states, respectively. Moreover, the difference between the first two resonances increases with decreasing well width. This effect is only obtained for states with different index, such as HH1 and HH2. In view of the simplicity of the model - e.g. neither depletion width nor carrier accumulation in the structure is taken into account - this result alone can only be taken as an indication of the nature of the resonances. However, support for the model is found through magnetocurrent oscillations. For low, fixed $V$ and with a magnetic field $B$ applied parallel to the current, it is possible to observe weak oscillations periodic in $1/B$ (period $B_{f}$). They are due to Landau levels passing through the quasi-Fermi energy in the emitter accumulation layer, the two-dimensional charge density of which is $p_{e} = 2eB_{f}/h$. Unlike similar oscillations in GaAs/AlAs p-type RTDs [@Hayden2], no decrease in $B_{f}$ is found as $V$ passes through the resonances, from which we conclude that the charge density in the quantum wells is negligible. Furthermore, the electric field $F = ep_{e}/\epsilon \epsilon_{0}$ over the QW structure is in reasonable agreement with the simple lever arm model.
Further, conclusive evidence that the above assignment of the resonances is correct can be found in experiments where the resonances are shifted by a central potential spike. Even symmetry states (HH1, LHSO1), with a wave function maximum in the middle of the quantum well, are much more affected by a central, repulsive potential spike than odd symmetry states (HH2, LHSO2).[@JYM] In our samples, the spike has been approximated by a thin Si layer; a $35$Å$ $ QW with a $5$Å$ $ spike in the middle was investigated and compared to the initial $35$Å$ $ structure (Fig. \[spike\]a). The plotted wavefunctions in the figure give a clear picture of the described effect. An example of the $I-V$ characteristics measured at 77K is displayed in Fig. \[spike\]b) and clearly demonstrate the predicted behaviour for the HH1 and HH2 states. We find shifts for the first and second resonances equal to $(0.31 \pm 0.03) V$ and $(0.06 \pm 0.08) V$, respectively, where the uncertainty is due to the natural scatter of the measured resonances for different diodes of the same structure. The values compares well with the calculated values (using the model described above, including Stark shifts) of $0.21 V$ and $0.06 V$ for the HH1 and HH2 resonance respectively.[@HH1] In contrast, assuming a lever arm compatible with the second resonance due to LHSO1 tunnelling, the expected shift would be $\ge 0.2 V$.
Having shown that it is possible to observe $J$-conservation in these tunnelling experiments, we now try to understand the difference between the present samples and those of previous studies, where tunnelling through LHSO states was observed. One important contrast is the higher strain used in the present study. For example, in previous studies of Si/SiGe RTDs on Si substrates, the Ge content was around $20 - 25 \%$. [@Liu; @Gennser1]¥ One consequence is that the HH and LHSO states in the emitter were less decoupled in these samples, with a separation between the HH and LH potentials $\le 45$ meV, whereas for the present samples it is $\approx 85$ meV. To study the role of the emitter, a structure with a $25$Å$ $ QW and an emitter region with a grading from $50$% $ $ to $65$% $ $ was investigated (See Fig. \[LHpic\](a)). Two resonances, at $\approx$100 mV and $\approx$470 mV, are observed in this ’emitter ramp’ sample. The second resonance voltage is compatible with the estimated resonance voltage of the tunneling through HH2 but the first resonance is likely due to the tunneling through LHSO1: the tunneling through HH1 is prohibited by design. Also the 370 mV separation between the two resonances is more than a factor of 2 smaller than the separation between the HH 1 and the HH2 resonances of $80$% $ $ emitter sample (Fig.2).
To further compare these resonances, we use magnetotunnelling spectroscopy with $B$ up to 23 T. A magnetic field $B_{\perp}$ applied perpendicular to the current $I$ accelerates the carriers in the direction perpendicular to both $B_{\perp}$ and $I$, so that they tunnel through the quantum well levels at a non-zero in-plane momentum, centered around $\Delta k_{p} =
q \Delta s B_{\perp}/\hbar$ where $\Delta s$ is the tunnelling distance.[@Hayden] The in-plane dispersion relations can then be mapped out and compared with the calculated dispersion $E(k_{p})$.[@Hayden; @Hayden2; @Gennser1; @Gassot] All the HH1 resonances of the three regular structures show a parabolic behaviour. The corresponding effective masses (0.04 $m_0$, 0.15 $m_0$, and 0.13 $m_0$ for the $25$Å, $35$Å$ $ and $45$Å$ $ wells, respectively) are in reasonable agreement with the calculated dispersions (0.17 $m_0$, 0.155 $m_0$, and 0.144 $m_0$) though quantitative comparisons are difficult to make. [@Hayden2] In Fig. \[LHpic\](b) we compare the $B_{\perp}$ shift for the first resonance of the sample with a $25$Å$ $ QW and an 80$\%$ emitter and of the emitter ramp sample. In contrast to the 80$\%$ emitter resonance, the first resonance of the emitter ramp sample shows a very distinct linear behaviour. A log plot clearly demonstrates these dependences (Fig. \[LHpic\]c). This indicates that it is the first resonance rather than the second that is not due to tunnelling through one of the HH states. Furthermore, a magnetic field $B_{\perp}$ cannot lead to a linear energy shift of the valence band QW states or the emitter states next to the barrier. The Zeeman effect is given by $E_{Z} = \kappa \mu _{B}¥{\bf J} \cdot {\bf B}$ (plus a small term proportional to $B^{2}$) [@Luttinger], which is only a small perturbation since the direction of $J$ is frozen in the direction of the confinement and the strain. Since the well thickness is much smaller than the cyclotron orbit even for the highest fields, Landau level formation can also be excluded. Neither can the linear shift in Fig. \[LHpic\] be explained by the acceleration in k-space, since the levels are quite parabolic, and never linear in $k_p$. In fact, we find that only an unstrained valence band bulk state can give rise to the observed linear shift. We propose that there are two reservoires of holes in the emitter: states confined close to the Si barrier and states in the unstrained ’bulk’ part of the emitter. The latter, tunnelling through the LHSO1 state, are responsible for the first resonance of the ramp emitter sample. In the bulk the $\bf{J}$ vector is free to turn along the B-field axis, and with $\bf{J}$ perpendicular to the growth axis, the quantum well state will ’see’ a mixed HH-LHSO state coming from the emitter. Because of the lower Ge content in this emitter, the barrier for the holes from the bulk is smaller, making it possible for them either to tunnel directly into the quantum well states, or to form hybrid states with the emitter states in the HH emitter well. The Landau level separation in the Si$_{0.5}$Ge$_{0.5}$ bulk is $\approx$ 0.6 meV/T, and the Zeeman energy a factor of 2-10 smaller.[@Winkler] This compares reasonably well with the measured slope of 4.1 mV/T $\approx$ 1.4 meV/T. It seems plausible that the apparent tunnelling from HH to LHSO states in other p-type RTDs may be due to the inevitable bulk part of the emitter, as well as band mixing in the well states. A similar linear behaviour has indeed been observed in a Si/Si$_{0.75}$Ge$_{0.25}$ RTD with the strain fully in the SiGe layer. [@Gassot]
Concerning the third resonance of the $35$Å$ $ and $45$Å$ $ sample, the fit with the energy levels in Fig. \[IVpic\] indicates that it corresponds to tunnelling through the second LH-like state (LHSO3 in the figure), and this also agrees with the observed shift in the sample with a central Si spike. This state is much less parabolic than the three lower states, and one would therefore expect a larger amount of band mixing. However, further experiments are necessary to confirm this.
We have demonstrated that the total angular momentum is conserved during resonant tunnelling in a system with strong spin-orbit coupling. This does not necessarily imply that the same holds true for the case of weakly coupled spin, but is certainly an encouraging sign. However, it may also have direct implications for the field of spintronics, since in order to inject spin in a semiconductor, a possible path is through the growth of magnetic semiconductors as electrodes. Much of the work has been focused on GaMnAs alloys, where the Mn not only provides the ferromagnetic properties, but also is a p-dopant.[@Dietl; @Mattana] Finally, it should also be noted that these results may have an additional relevance for the development of a Si/SiGe based quantum cascade laser, in they exclude one of the possible non-radiative conduction paths for the HH carriers in these structures.[@Gennser2]
We would like to thank J. Faist for help with this work. It has been partly financially supported by the Swiss National Science Foundation, the EC Contract Si-GeNET, “Région Ile de France”, “Conseil Général de l’Essonne” and the program Nano2008.
\[References\]
G. Prinz, Science [**282**]{}, 1660 (1998). S. A. Wolf, Science $et\; al.$, [**294**]{}, 1488 (2001). K. C. Hall, $et\; al.$, Appl. Phys. Lett. [**83**]{}, 2973 (2003). M. M. Glasov, $et\;al$, cond-mat 0406191 (2004). G. Dresselhaus, Phys. Rev. [**100**]{}. 580 (1955). F. Malcher, G. Lommer, and U. Rœssler, Superlatt. Microstr. [**2**]{}, 267 (1986). Y. Bychkov and E. Rashba, JETP Lett. [**39**]{}, 78 (1984). J. M. Luttinger and W. Kohn, Phys. Rev. [**97**]{}, 869 (1965). E. E. Mendez, W. I. Wang, B. Ricco, and L. Esaki, Appl. Phys. Lett. [**47**]{}, 415 (1985). H. C. Liu, D. Landheer, M. Buchanan, and D. C. Houghton, Appl. Phys. Lett. [**52**]{}, 1809 (1988). R. M. Lewis, H. P. Wei, S. Y. Lin, and J. F. Klem, Appl. Phys. Lett. [**77**]{}, 2722 (2000). R.K. Hayden, $et\;al$, Phys. Rev. Lett. [**66**]{}, 1749 (1991). U. Gennser, $et\;al$, Phys. Rev. Lett. [**67**]{}, 3828 (1991). L. Diehl, $et\;al$, Appl. Phys. Lett. [**80**]{}, 3274 (2002). R.K. Hayden, $et\;al$, Semicond. Sci. Technol. [**7**]{}, B413 (1992). J.-Y. Marzin and J.-M. G' erard, Phys. Rev. Lett. [**62**]{}, 2172 (1989). The larger discrepancy for the HH1 is expected since it is more sensitive to the exact thickness of the Si spike. P. Gassot, $et\;al$, Physica E [**2**]{}, 758 (1998). J. M. Luttinger, Phys. Rev. [**102**]{}, 1030 (1956). The value depends on the interpolation used between Si and Ge. See R. Winkler, M. Merkler, T. Darnhofer, and U. Rœssler, Phys. Rev. [**B 53**]{}, 10858 (1996). T. Dietl, $et\;al$, Science [**287**]{}, 1019 (2000). R. Mattana, $et\;al$, Phys. Rev. Lett. [**90**]{}, 166601 (2003). U. Gennser, $et\;al$, in ’Future Trends in Microelectronics’, Ed. S. Lyuri, J. Xu, and A. Zaslavsky (Wiley-IEEE Press, 2004).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Image posterization is converting images with a large number of tones into synthetic images with distinct flat areas and a fewer number of tones. In this technical report, we present the implementation and results of using fuzzy logic in order to generate a posterized image in a simple and fast way. The image filter is based on fuzzy logic and bilateral filtering; where, the given image is blurred to remove small details. Then, the fuzzy logic is used to classify each pixel into one of three specific categories in order to reduce the number of colors. This filter was developed during building the Specs on Face dataset in order to add a new level of difficulty to the original face images in the dataset. This filter does not hurt the human detection performance; however, it is considered a hindrance evading the face detection process. This filter can be used generally for posterizing images, especially those have a high contrast to get images with vivid colors.'
author:
- |
Mahmoud Afifi\
IT Dept., Assiut University, Egypt\
email: [email protected]
date: '20 Dec, 2016'
title: Image Posterization Using Fuzzy Logic and Bilateral Filter
---
Introduction
============
Image posterization is converting an image that has a large number of tones into an image with distinct flat areas and a fewer number of tones. In this technical report, we present the implementation and results of image posterization using fuzzy logic and bilateral filtering. This image filter is considered a simple way to obtain a posterized image in low computational time. Figure 1 shows the result of the image filter. The image posterization filter is based on pre-smoothing of the original image using bilateral filtering [@bilateral] that smooths the original image without affecting the edges in the blurred image. The reason of smoothing the image is to remove small details before the quantization process. Next, an image quantization process is applied to the blurred image to generate the final synthetic image. In this process, fuzzy logic is used to classify each pixel in each channel of the color model into one of three categories which are: 1) bright, 2) gray, or 3) dark [@IP]. After the classification process, each pixel is assigned to a specific intensity value, that for each color channel, based on its category. The synthetic image filter gives vivid color images compared with Adobe Photoshop Poster Edges filter. If this filter is applied to a face image, it does not hurt the human detection performance; however, this filter is considered a hindrance evading the face detection process. For that reason, this filter was developed within constructing the Specs on Face dataset [@AIFIF] in order to add a new level of difficulty to the original face images in the dataset.
{width="0.9\linewidth"}
Image Posterization
===================
For generating a posterized image, two main stages are performed. The first stage is to remove small details of the given image. To that end, the bilateral filter is used for preserving the edges in the filtered image [@bilateral]. The bilateral filter closes to Gaussian smoothing by using a weighted average of the pixels. However, the bilateral filter considers two pixels close together not only relying on the spatial coordinates of them, but they must close enough in their intensity. Thus, the bilateral filter is considered edge-preserving filter. Figure 2 shows the blurred image after using the bilateral filter. The enlargement of the ground region shows the result of removing small objects.
{width="0.9\linewidth"}
\
After preparing the given image to the image quantization process, fuzzy logic plays the role of classifying each pixel into one of three categories which are: 1) bright, 2) gray, or 3) dark [@IP]. Fuzzy logic solves the problem of crisp sets using membership functions that determine the element’s degree of belonging to each set. In other words, the membership function $\upsilon$ maps element $x$ to degree of membership in the fuzzy set $A$, i.e., $\upsilon(x)=Degree(x\in A)$. Where, $A=\{x,\upsilon(x)\}$. Thus, the element $x$ may be a full member of the set ($\upsilon(x)=1$), not a member of the set ($\upsilon(x)=0$), or has a partial membership in the set ($0<\upsilon(x)<1$) \[2\]. By using linguistic labels, each element has its membership to one or more labels using a set of (IF-THEN) rules. Scaler inputs are converted into fuzzy sets using the fuzzification process. After that, combining rules using (ANDs or ORs) are performed to establish a robust rule. After determining the fuzzy output, the defuzzification process is performed to generate the final crisp output.\
Fuzzy logic is used in this context to determine the category (the linguistics label) of each pixel in the given image. There are three linguistics labels which are: 1) dark, 2) gray, or 3) bright. For each pixel in each channel of the color model, the fuzzification process is performed and the membership of the three categories is calculated. As shown in Figure 3, the dark membership function is represented using the following sigma function:
{width="0.5\linewidth"}
$$\upsilon_{dr}(p) = \left\{
\begin{array}{lr}
1-\frac{a_{dr}-p}{b_{dr}} \texttt{\hspace{2 mm} if\hspace{1 mm}} a_{dr}\leq p \leq a_{dr}+b_{dr} \\\\
1-\frac{(p-a_{dr})}{c_{dr}} \texttt{\hspace{1 mm} if\hspace{1 mm}} p < a_{dr} \\\\
0 \texttt{\hspace{13 mm} otherwise}, \\\\
\end{array}
\right.\$$ Gray membership function is represented using the following triangular function:\
$$\upsilon_{g}(p) = \left\{
\begin{array}{lr}
1-\frac{a_{g}-p}{b_{g}} \texttt{\hspace{2 mm} if\hspace{1 mm}} a_{g}-b_{g} \leq p < a_{g} \\\\
1-\frac{(p-a_{g})}{b_{g}} \texttt{\hspace{1 mm} if\hspace{1 mm}} a_{g}\leq p \leq a_{g}+b_{g} \\\\
0 \texttt{\hspace{13 mm} otherwise}, \\\\
\end{array}
\right.\$$ Bright membership function is represented using the following sigma function: $$\upsilon_{br}(p) = \left\{
\begin{array}{lr}
1-\frac{a_{br}-p}{b_{br}} \texttt{\hspace{2 mm} if\hspace{1 mm}} a_{br}-b_{br} \leq p \leq a_{br} \\\\
1-\frac{(p-a_{br})}{c_{br}} \texttt{\hspace{1 mm} if\hspace{1 mm}} a_{br} < p \\\\
0 \texttt{\hspace{13 mm} otherwise}, \\\\
\end{array}
\right.\$$ where $a_K$, $b_K$, and $c_K$ are the parameters of the membership functions. $\upsilon_{K}(p)$ represents the membership’s degree of the intensity of the current pixel $p$ where, $K \in \{dr,g,br\}$. $dr$, $g$, and $br$ denote dark, gray, and bright linguistics, respectively.\
The defuzzification process is performed for determining the crisp output of the current pixel. By determining the center of gravity which is given by: $$\label{s}
v_{0}=\frac{\sum_{v=1}^{N}vQ(v)}{\sum_{v=1}^{N}Q(v)},$$ where $Q$ denotes the fuzzy output, $v$ is the output variable, and $N$ is the possible values of $Q$. In our case, the output of the membership functions are constants, so equ. \[s\] can be represented as: $$v_{0}=\frac{\upsilon_{dr}(p) \texttimes v_{dr} + \upsilon_{g}(p) \texttimes v_g + \upsilon_{br}(p) \texttimes v_{br}}{\upsilon_{dr}(p) + \upsilon_{g}(p) + \upsilon_{br}(p)},$$ where $v_{dr}$, $v_g$, and $v_{br}$ are the output variables of dark, gray, and bright membership functions, respectively.\
By getting the final value, the final crisp output is given by finding the minimum between $v_0$ and the output variables of the membership functions: $$min(\textbar v_0 - v_{dr} \textbar, \textbar v_0 - v_g \textbar, \textbar v_0 - v_{br} \textbar).$$ The minimum value refers to the category of the pixel, either dark, gray, or bright. By Applying the following rules, the new intensity of the pixel is found:\
IF $p$ is dark, THEN make it $v_{dr}$.\
IF $p$ is gray, THEN make it $v_g$.\
IF $p$ is bright, THEN make it $v_{br}$.\
To get several effects, a series of posterized images can be generated by:
- $max(I,O)$,
- $min(I,O)$,
- $\alpha$ $O$+ (1-$\alpha$) $I$,
where $I$ is the original image, $O$ is the raw output of the posterization process, and $\alpha$ is the blending weight.\
\
\
Implementation
==============
This filter is implemented using both Matlab and OpenCV. In the next lines, the pseudocode of the image posterization filter is presented:
-----------------------------------------
Pseudocode
-----------------------------------------
Function F = Posterization(I)
-- Posterization Filter
I=Bilateral(I)
if I is colored image
for i=1 to 3
F.ColorChannel(i)=...
Fuzzy(I.ColorChannel(i))
end
else
F=Fuzzy(I)
end
Return F
////////////////////////////////
Function F = Fuzzy(I)
-- Fuzzy logic
foreach pixel p in I
vDR=darkkMembFunc(I(p))
vG=grayMembFunc(I(p))
vBR=brightMembFunc(I(p))
v=(vDR*vd+vG*vg+vBR*vb)/...
(vDR+vG+vBR)
index=min([abs(v-vd),...
abs(v-vg),abs(v-vb)])
if index=1
F(p)=vd
else
if index=2
F(p)=vg
else
F(p)=vb;
end
end
return F
/////////////////////////////////
Function vBR=brightMembFunc(z)
-- Bright Membership function
if a-b<=z<=a
vBR=1-(a-z)/b
else
if z>a
vBR=1
else
vBR=0
end
end
return vBR
/////////////////////////////////
Function vG=grayMembFunc(z)
-- Gray Membership function
if a-b<=z<=a
vG=1-(a-z)/b
else
if a<=z<a+b
vG=1-(z-a)/b
else
vG=0
end
end
return vG
/////////////////////////////////
Function vDR=darkkMembFunc(z)
-- Dark Membership function
if a<=z<=a+b
vDR=1-(a+b-z)/a
else
if z<a
vDR=1
else
vDR=0
end
end
return vDR
////////////////////////////////
{width="\linewidth"}
![A comparison between the image posterization using fuzzy logic and the pixelated image abstraction method [@Pixelated]. (a) The original image. (b) The result of [@Pixelated] using six colors. The results of the posterization using fuzzy logic using (c) $\alpha=0.3$, (d) $\alpha=0.5$, (e) $\alpha=0.7$, (f) $\alpha=1.0$, (g) $min (I,O)$, and (h) $max (I,O)$.](obama){width="\linewidth"}
You can download the Matlab source code [^1] and the OpenCV source code [^2] from the links below.
Results
=======
Figure 5 shows the results of the synthetic image filter compared to the Adobe Photoshop Poster Edges filter. Membership function’s variables are $v_{dr}=0$, $v_g=127$, $v_{br}=255$, $a_{dr}=73$, $b_{dr}=50$, $a_g=127$, $b_{g}=50$, $a_{br}=177$, and $b_{br}=50$. The Adobe Photshop filter’s parameters are $edge thickness=0$, $edge intensity=0$ and $posterization=0, 1,$ and $2$. As shown, the posterization image filter generates images with vivid colors compared with the results of the Adobe Photoshop Poster Edges filter. Figure 6 shows the effect of using three different weights (0.3, 0.5, 0.7) and the min/max operations. Again, the image posterization filter gives images that have more contrast in comparison with the pixelated image abstraction method [@Pixelated].
[9]{} Paris, Sylvain, et al. *A gentle introduction to bilateral filtering and its applications*. ACM SIGGRAPH 2007 courses. ACM, 2007. Gonzalez, Rafael C., and Richard E. Woods. *Digital image processing*. Pearson/Prentice Hall, 2008. Mahmoud Afifi and Abdelrahman. *AFIF$^4$: Deep gender classification based on an AdaBoost-based fusion of isolated facial features and foggy faces*. arXiv preprint arXiv:1706.04277, 2017. Gerstner, Timothy, et al. *Pixelated image abstraction with integrated user constraints*. Computers & Graphics 37.5, 333-347, 2013.
[^1]: https://www.dropbox.com/s/mlt7ks4p3irq9bk/Image%20Posterization%20Filter%20Matlab.rar?dl=0
[^2]: https://www.dropbox.com/s/cdaqluqvyhtvn98/Image%20Posterization%20Filter.rar?dl=0
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Modifying Rudin’s original construction of the Rudin–Shapiro sequence, we derive a new substitution-based sequence with purely absolutely continuous diffraction spectrum.'
address: 'School of Mathematics and Statistics, The Open University, Milton Keynes MK7 6AA, UK'
author:
- Lax Chan and Uwe Grimm
title: 'Substitution-based sequences with absolutely continuous diffraction'
---
Introduction
============
Substitution dynamical systems are widely used as toy models for aperiodic order in one dimension [@PF02]. The binary Rudin–Shapiro sequence is a paradigm for a substitution-based structure with (in its balanced weight case) purely absolutely continuous diffraction spectrum; we refer to [@BG13] for details and background. Indeed, this deterministic sequence has the stronger property that its two-point correlations vanish for *any* non-zero distance, as it would be the case for a random sequence [@BG09]. While currently only very few examples of substitution-based sequences with this property are known, a systematic generalisation in terms of Hadamard matrices [@Fra03] allows the construction of one-dimensional as well as higher-dimensional examples.
We start by reviewing Rudin’s original construction [@Rud59], and modify it to obtain a new example of a substitution-based structure with absolutely continuous diffraction spectrum. Our approach differs from the construction in [@Fra03]. We then use recent work of Bartlett [@BG13] to investigate the properties of the new system in more detail.
Original construction of the Rudin–Shapiro sequence
===================================================
In 1958, Salem [@Rud59] asked the following question: is it possible to find a sequence $(\varepsilon_{n})_{n\in{{\mathbb{N}}}}\in\{\pm
1\}^{{{\mathbb{N}}}}$ such that there is a constant $C>0$ for which $$\label{equation:0}
\sup_{\theta\in\mathbb{R}}\,\biggl|\sum_{n<N}\varepsilon_{n}
\, e^{2\pi in\theta}\biggr|\, \leq\, C\sqrt{N}$$ holds for any positive integer $N$? This is known as the ‘root $N$’ property, which implies absolutely continuous diffraction of the sequence $(\varepsilon_{n})_{n\in{{\mathbb{N}}}}$ by an application of [@MQ10 Prop. 4.9]. Rudin [@Rud59] and Shapiro [@Sha51] independently gave positive answers to the question by constructing what is now known as the Rudin–Shapiro sequence. We briefly review Rudin’s original construction to obtain the substitution dynamical system and its balanced weight version (with weights in $\pm 1$).
We start by defining polynomials $P_{k}(x)$ and $Q_{k}(x)$ of degree $2^{k}$ for $k\in\mathbb{N}_{0}$ recursively by $$\label{equation:1}
\begin{split}
P_{k+1}(x)\, =\, P_{k}(x)+x^{2^{k}}Q_{k}(x),\\
Q_{k+1}(x)\, =\, P_{k}(x)-x^{2^{k}}Q_{k}(x),
\end{split}$$ with $P^{}_{0}(x)=Q^{}_{0}(x)=x$. Note that from Eq. it is clear that the first $2^k$ terms of $P_{k+1}(x)$ and of $Q_{k+1}(x)$ coincide with those of $P_{k}(x)$, and the remaining terms differ by a sign. By construction, $P_{k}(x)$ is of the form $$\label{equation:1b}
P_{k}(x)\, =\,\sum_{n=1}^{2^{k}}\,\varepsilon_{n}\,x^n,$$ so we can define a binary sequence $(\varepsilon_{n})^{}_{n\in\mathbb{N}}\in\{\pm 1\}^{\mathbb{N}}$ from the corresponding coefficients. This is the binary Rudin–Shapiro sequence. For example, $P_3(x)=x+x^{2}+x^{3}-x^{4}+x^{4}(x+x^{2}-x^{3}+x^{4})$, from which we read off the sequence $111\overline{1}11\overline{1}1$ with $\overline{1}=-1$. The main ingredient in the proof of property for this sequence is the parallelogram law, $$|\alpha+\beta|^{2}+|\alpha-\beta|^{2}
\, =\, 2|\alpha|^{2}+2|\beta|^{2},$$ for $\alpha,\beta\in{{\mathbb{C}}}$; see [@Rud59] for details. This can be used to establish the bound on $P_{k}(e^{2\pi
i\theta})$.
Often, the Rudin–Shapiro sequence is defined by a four-letter substitution rule and a subsequent reduction map from four to two letters. The underlying four-letter substitution on the alphabet $\{A,B,C,D\}$ can be read off from the recursion , noting that the recursion implies the concatenation of the sequences corresponding to $P_{k}$ and $Q_{k}$. Associating letters $A$ and $B$ to the coefficients in $P$ and $Q$ and the letters $C$ and $D$ to those of $-Q$ and $-P$, respectively, this gives rise to the four-letter substitution rule $$\varrho\!:\quad A\mapsto AB, \quad B\mapsto AC, \quad C\mapsto DB,
\quad D\mapsto DC,$$ which corresponds to the standard four-letter Rudin–Shapiro subsitution. Iterating the sequence on the initial seed $A$ gives $$A\mapsto AB\mapsto ABAC\mapsto ABACABDB\mapsto \dots$$ which, under the mapping $A\mapsto 1$, $B\mapsto 1$, $C\mapsto -1$, $D\mapsto -1$ (which corresponds to the choice of signs above) produces the binary Rudin–Shapiro sequence $(\varepsilon_{n})^{}_{n\in\mathbb{N}}$.
Generalising Rudin’s construction
=================================
Let us now modify Rudin’s original argument by considering the following system $$\label{equation:2}
\begin{split}
P_{k+1}(x)\, =\, P_{k}(x)+(-1)^{k}x^{2^{k}}Q_{k}(x),\\
Q_{k+1}(x)\, =\, P_{k}(x)-(-1)^{k}x^{2^{k}}Q_{k}(x),
\end{split}$$ starting again from $P^{}_{0}(x)=Q^{}_{0}(x)=x$. Since from now on we shall only look at this set of recursion relations, we use the same notation as above. What has changed in comparison with Eq. is that we swap the sign in the recursion relation in each step. Since the sign does not affect the argument used in Rudin’s proof [@Rud59], it is straightforward to show that the parallelogram law can again be used to prove that the corresponding *new* sequence of coefficients $(\varepsilon_{n})^{}_{n\in\mathbb{N}}$, defined as above via Eq. , also satisfies the root $N$ property.
Can we read off the corresponding substitution rule as for the Rudin–Shapiro case? Indeed this is possible, but it is slightly more complicated, because the recursion relations in Eq. alternate. One way to deal with that is to look at two consecutive steps together, $$\label{equation:2b}
\begin{split}
P_{k+2}(x)\, =\, P_{k}(x) + (-1)^{k}x^{2^{k}}Q_{k}(x) +
(-1)^{k+1}x^{2\cdot 2^{k}} P_{k}(x)
+ x^{3\cdot 2^{k}}Q_{k}(x),\\
Q_{k+2}(x)\, =\, P_{k}(x) + (-1)^{k}x^{2^{k}}Q_{k}(x) -
(-1)^{k+1}x^{2\cdot 2^{k}} P_{k}(x)
- x^{3\cdot 2^{k}}Q_{k}(x).
\end{split}$$ Choosing $k$ to be even, say, and associating again four letters $A,B,C,D$ to the sequences corresponding to $P,Q,-Q,-P$, we can read off the substitution $$\label{equation:3}
\sigma\!:\quad
A\mapsto ABDB,\quad B\mapsto ABAC,\quad C\mapsto DCDB,\quad D\mapsto DCAC,$$ which is now a constant length substitution of length four, because we used a double step of the recursion. Therefore Eq. corresponds to concatenation of four sets of coefficients. As before, the binary sequence is obtained from iterating the substitution on the initial letter $A$; $$\label{equation:3b}
A\mapsto ABDB\mapsto ABDBABACDCACABAC\mapsto \cdots.$$ By mapping $A$ and $B$ to $1$ and mapping $C$ and $D$ to $\overline{1}=-1$, we obtain $11\overline{1}1111\overline{1}\overline{1}
\overline{1}1\overline{1}111\overline{1}\ldots$ as the initial part of our new binary sequence.
Clearly, we can generate infinitely many such examples by changing the signs in the original recursion relation in more complicated ways. Any finite sequence of signs, when used periodically, will give rise to a substitution-based system. However, the length of the substitution will increase with the length of the sign sequence. There neither appears to be an obvious relation between the different Rudin–Shapiro type sequences obtained from this construction, nor between these sequences and those derived from Frank’s construction [@Fra03]. In the latter case, the number of letters in the alphabet increases with the size of the Hadamard matrix, whereas all our sequences will only use four letters. Having said that, it would be possible to use more letters rather than consider multiple recursion steps, so the relation between these two approaches still needs to be analysed in more detail.
Spectral properties
===================
Since we have Rudin’s original argument at our disposal, we know that this new binary sequence satisfies the root $N$ property , and hence has (in the balanced weight case with weights $\pm 1$) purely absolutely continuous diffraction. Nevertheless, we would like to use the remainder of this paper to apply Bartlett’s algorithm [@AB14] to our new (four-letter substitution) sequence, and to verify that (in the balanced weight case) it indeed has absolutely continuous diffraction spectrum only. Due to space constraints, we cannot introduce all quantities here, and instead refer to [@AB14] for definitions and further details.
The balanced weight sequence derived from the substitution rule $\sigma$ of Eq. has purely absolutely continuous diffraction spectrum.
The four instruction matrices $R_{i}$ and the substitution matrix $M$ can be read off of the substitution rule . They are given by $$R_0=
\begin{pmatrix}
1 & 1 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 1 & 1
\end{pmatrix},\quad
R_1=
\begin{pmatrix}
0 & 0 & 0 & 0\\
1 & 1 &0 &0\\
0& 0& 1&1\\
0& 0&0& 0
\end{pmatrix}, \quad
R_2=
\begin{pmatrix}
0& 1 & 0 &1 \\
0& 0&0&0\\
0&0&0&0\\
1&0&1&0
\end{pmatrix},\quad
R_3=
\begin{pmatrix}
0&0&0&0\\
1&0&1&0\\
0&1&0&1\\
0&0&0&0
\end{pmatrix},$$ with $M=R_{0}+R_{1}+R_{2}+R_{3}$. As $M^{2}$ has positive entries only, the substitution is primitive. The second iterate of the seed $A$ obtained in Eq. shows that the letter $A$ can be preceded by either $B$ or $C$. Hence $A$ has two distinct neighbourhoods and, by Pansiot’s lemma [@AB14 Lem. 3.6], the substitution is aperiodic.
Since we are dealing with a constant-length substitution, the leading (Perron–Frobenius) eigenvalue [@BG13 Thm. 2.2] of $M$ is $\lambda_{\textnormal{PF}}=4$ with (statistically normalised) eigenvector and $u=\frac{1}{4}(1,1,1,1)$. By applying [@AB14 Thm. 4.3], we obtain $\widehat{\Sigma}(0)=\frac{1}{4}\sum_{\alpha}e_{\alpha\alpha}$, where $e_{\alpha\alpha}$ denotes the unit vector corresponding to the word $\alpha\alpha$. As $\sigma$ is a length four substitution, we have $\Delta_{1}(1)=\{3\}$. Using [@AB14 Thm. 4.3], we find $$\widehat{\Sigma}(1)\,=\,
\frac{1}{8}\left(0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0\right)\,=\,
\widehat{\Sigma}(3),$$ and $$\widehat{\Sigma}(2)\,=\,
\frac{1}{8}\left(1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1\right)\,=\,
\widehat{\Sigma}(4).$$ One can then verify the following recursive relations $$\label{equation:4}
\widehat{\Sigma}(4n)\,=\,\widehat{\Sigma}(4),\quad
\widehat{\Sigma}(4n+1)\,=\,\widehat{\Sigma}(1),\quad
\widehat{\Sigma}(4n+2)\,=\,\widehat{\Sigma}(2), \quad
\widehat{\Sigma}(4n+3)\,=\,\widehat{\Sigma}(3).$$ Using [@AB14 Prop. 2.2], we calculate the ergodic decomposition of the bi-substitution $\sigma\otimes\sigma$, and obtain $E_{1}=\{AA,BB,CC,DD\}$ and $E_{2}=\{AD,DA,BC, CB\}$ as the two ergodic classes and $T=\{AB,AC,BA,BD,CA,CD,DB,DC\}$ as the transient part. From [@AB14 Lem. 4.7], it then follows that $$v\, =\, w^{}_1
\begin{pmatrix}
1&0&0&0\\
0&1&0&0\\
0&0&1&0\\
0&0&0&1\end{pmatrix}+w^{}_2
\begin{pmatrix}
0&0&0&1\\
0&0&1&0\\
0&1&0&0\\
1&0&0&0
\end{pmatrix}+\frac{w^{}_1+w^{}_2}{2}
\begin{pmatrix}
0&1&1&0\\
1&0&0&1\\
1&0&0&1\\
0&1&1&0
\end{pmatrix}.$$ Diagonalising the matrix $v$, we obtain $$v^{}_{d}=
\begin{pmatrix}
2(w^{}_1+w^{}_2)&0&0&0\\
0&w^{}_1-w^{}_2&0&0\\
0&0&w^{}_1-w^{}_2&0\\
0&0&0&0
\end{pmatrix}.$$ Setting $w^{}_1=1$, strong semi-positivity is equivalent to $w^{}_2$ satisfying $-1\leq w^{}_2\leq 1$. The extreme points are then given by $(w^{}_1,w^{}_2)=(1,1)$ and $(w^{}_1,w^{}_2)=(1,-1)$. Thus the extreme rays are $$\begin{split}
v_1=&(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1),\\
v_2=&(1,0,0,-1,0,1,-1,0,0,-1,1,0,-1,0,0,1).
\end{split}$$ As usual, $\lambda_{v^{}_1}=\delta^{}_{0}$, which gives rise to the pure point component of the spectrum. Using Eq. , one checks that $\widehat{{\lambda}_{v^{}_2}}(k)=0$ for all $k\neq 0$, which gives us the absolutely continuous component. Thus, we have a purely absolute continuous spectrum in the balanced weight case, in which the pure point component is extinguished.
The authors would like to thank Ian Short for many helpful comments. L.C. would like to thank the organising committee for a young scientist award to attend ICQ13 in Kathmandu.
References {#references .unnumbered}
==========
[1]{}
Pytheas Fogg N 2002 *Substitutions in Dynamics, Arithmetics and Combinatorics* (Lecture Notes in Mathematics vol 1794) (Springer, Berlin)
Rudin W 1959 Some theorems on Fourier coefficients *Proc. Amer. Math. Soc.* **10** 855–859
Shapiro H 1951 *Extremal Problems for Polynomials and Power Series* Masters thesis (MIT, Boston)
Baake M and Grimm U 2013 *Aperiodic Order. Vol. 1. A Mathematical Invitation* (Encyclopedia of Mathematics and its Applications vol 149) (Cambridge University Press, Cambridge)
Baake M and Grimm U 2009 Kinematic diffraction is insufficient to distinguish order from disorder *Phys. Rev. B* **79** 020203 and **80** 029903 (Erratum)
Frank N 2003 Substitution sequences in $\mathbb{Z}^{d}$ with a non-simple Lebesgue component in the spectrum *Ergod. Th. & Dynam. Syst.* **23** 519–532
Queff[é]{}lec M 2010 *Substitution Dynamical Systems—Spectral Analysis* 2nd ed (Lecture Notes in Mathematics vol 1294) (Springer, Berlin)
Bartlett A 2014 Spectral theory of $\mathbb{Z}^{d}$ substitutions *Ergod. Th. & Dynam. Syst.* (to appear, preprint arXiv:1410.8106)
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- Yusuf Hafidh
- Rizki Kurniawan
- Suhadi Saputro
- Rinovia Simanjuntak
- Steven Tanujaya
- Saladin Uttunggadewa
subtitle: '...'
title: Multiset Dimensions of Trees
---
Introduction
============
Let $G$ be a simple and connected graph with vertex set $V(G)$. The *distance $d(u,v)$ between two vertices $u,v \in V(G)$* is the length of a shortest path between them. The *eccentricity of a vertex $v$*, $ecc(v)$ is the maximum distance from $v$ to other vertices in $G$. The *radius of $G$* is $rad(G):=\min\{ecc(v):v\in G\}$. A *center of $G$* is a vertex with the smallest eccentricity (equal to the radius) and the set of centers of $G$ is denoted by $C(G)$. The diameter of $G$ is $diam(G):=\max\{ecc(v):v\in G\}$ and an *end-vertex* is a vertex with the highest eccentricity (equal to the diameter).
The concept of multiset dimension was introduced in [@SSV] as a natural variation of metric dimension. In both concepts, the location of a vertex in a graph is uniquely identified by utilising the distance from that vertex to a set of “landmarks”. Each vertex is then allocated a distinct “coordinate”: in metric dimension, the coordinates are vectors, while in mustiset dimension, the coordinates are instead multisets.
For an ordered set of $k$ vertices $W = \{ w_1, w_2, \dots , w_k \}$, the *representation of a vertex $v$ with respect to $W$* is the ordered $k$-tuple $$r(v|W) = ( d(v,w_1), d(v,w_2), \dots , d(v,w_z)).$$ $W$ is a *resolving set of $G$* if every two vertices of $G$ have distinct representations. A resolving set with minimum cardinality is called a *basis* and the number of vertices in a basis is called the *metric dimension*, denoted by $dim(G)$.
Now suppose that $W'$ is an unordered subset of $V(G)$. The *representation multiset of $v$ with respect to $W'$*, $r_m (v|W')$, is defined as a multiset of distances between $v$ and the vertices in $W'$. If $r_m (u|W') \neq r_m(v|W')$ for every pair of distinct vertices $u$ and $v$, then $W'$ is called an *m-resolving set* of $G$. If $G$ has an m-resolving set, then the cardinality of a smallest m-resolving set is called the *multiset dimension of $G$*, denoted by $md(G)$; otherwise, we say that $md(G) = \infty$.
In [@SSV], a few basic results of multiset dimension were proved. Here we list those connected to the results of this paper.
[@SSV]
1. The multiset dimension of a graph $G$ is one if and only if $G$ is a path. \[path\]
2. No graph has multiset dimension $2$. \[not2\]
3. Let $G$ be a graph other than a path. Then $md(G) \ge 3$. \[bound1\]
4. If $G$ is a non-path graph of diameter at most $2$, then $md(G) = \infty$. \[diam2\]
Another basic property needs the definition of twin vertices as follow. Two vertices $u$ and $v$ are said to be *twins* if $N(u)\setminus\{v\}=N(v)\setminus\{u\}$. We define a relation $\sim$ where $u \sim v$ if and only if $u=v$ or $u$ and $v$ are twins. It is quite obvious that $\sim$ is an equivalence relation on $V(G)$ and we denote by $v^*$ the equivalence class containing the vertex $v$. The fact that a pair of twins have the same distance to every other vertex gives rise to a necessary condition for graphs having finite multiset dimension.
\[twins\] [@SSV; @KI18] If $G$ contains a vertex $v$ with $|v^*|\geq 3$, then $md(G) = \infty$.
In general, the necessary conditions for the finiteness of multiset dimension in Lemmas \[diam2\] and \[twins\] are not sufficient, so it is interesting to find such conditions for trees, as proposed in [@SSV].
[@SSV] \[chartree\] Characterize all trees having finite multiset dimension. Give exact values of the multiset dimension of trees if it is finite.
It is obvious that if a graph $G$ on $n$ vertices has finite multiset dimension, then $md(G)\leq n$. However it was proposed that the natural upper bound is not sharp as stated in the following conjecture.
[@SSV] \[n-1\] If $G$ is a graph on $n$ vertices having finite multiset dimension, then $md(G)\leq n-1$.
In this paper, we show that Conjecture \[n-1\] is true for trees in Section \[s\_bounds\]. Additionally, we provide necessary and sufficient conditions for caterpillars and lobsters having finite multiset dimension in Section \[s\_catlob\] which partially settled Open Problem \[chartree\].
To prove the aforementioned results, we shall utilise the following properties of trees, which follow from the fact that in a tree, there exists a unique path connecting every pair of distinct vertices.
Let $T$ be a tree.
1. $rad(T)=\left\lceil \frac{diam(T)}{2} \right\rceil$.
2. If $T$ has even diameter then $|C(T)|=1$ and if $T$ has odd diameter then $C(T)=\{u,v\}$, where $u,v$ are two adjacent vertices.
3. For every $v\in V(T)$, $ecc(v)=rad(T)+d(v,C(T))$. \[ecc\]
From now on, we shall denote by $P_n$ the path on $n$ vertices and $S_n$ the star on $n$ vertices.
Upper bound for multiset dimensions of trees {#s_bounds}
============================================
In this section, we provide partial proof for Conjecture \[n-1\], that is the conjecture is true for trees. We start with trees of small diameter.
\[diam3\] If $T$ is a tree of order $n$ and diameter 3 then $md(T)\leq n-2$.
Let $u$ and $v$ be the centers of $T$, by Lemma \[twins\], $deg(u)\leq 3$ and $deg(v)\leq 3$. Since $diam(T)=3$, the centers are not leaves, and so $2 \leq deg(u), deg(v) \leq 3$.
We shall consider 3 cases:
**(1) $deg(u)=deg(v)=2$**:
: Thus $T\approx P_4$ and $md(T)=1=n-3$.
**(2) $deg(u)=2$ and $deg(v)=3$:**
: Let $N(u)=\{v,u_1\}$ and $N(v)=\{u,v_1,v_2\}$. Thus we have $R=\{u_1,u,v_1\}$ is an m-resolving set for $T$, which means $md(T)=3=n-2$.
**(3) $deg(u)=deg(v)=3$:**
: Let $N(u)=\{v,u_1,u_2\}$ and $N(v)=\{u,v_1,v_2\}$. Therefore $R=\{u_1,u,v_1\}$ is an m-resolving set for $T$, or $md(T)=3=n-3$.
\[bounds\] Let $T$ be a tree of order $n$ and diameter at least $2$. If $md(T) < \infty$, then $md(T) \leq n-2$.
Let $T$ be a tree with finite multiset dimention. If $diam(T)=2$, by Lemma \[path\] and \[diam2\], $T\approx P_3$ and the result follows.
Now let $diam(T)\geq 3$. First we prove $md(T)\leq n-1$. Let $R$ be arbitrary m-resolving set of $T$. Since if $R \ne V(T)$ we already have the desired result, consider $R=V(T)$. We claim that $R'=V(T)-C(T)$ is also an m-resolving set for $T$. Consider two cases based on the parity of the diameter of $T$.
**Case 1: $\mathbf{diam(T)}$ is even.** Recall that there exists a unique center. To the contrary, suppose that $R'$ is not an m-resolving set, then there exist two vertices $u$ and $v$ where $r_m(u|R')=r_m(v|R')$. The maximum element in $r_m(u|R')$ is $ecc_T(u)$, thus $ecc_T(u)=ecc_T(v)$. By (\[ecc\]), $d(u,C(T))=ecc_T(u)-rad(T)=ecc_T(v)-rad(T)=d(v,C(T))$, and so $r_m(u|R)=r_m(u|R')\cup \{d(u,C(T))\} = r_m(v|R') \cup \{d(v,C(T))\} = r_m(v|R)$, which contradict the fact that $V(T)$ is a resolving set.
**Case 2: $\mathbf{diam(T)}$ is odd.** Let $C(T)=\{c_1,c_2\}$, where $c_1$ and $c_2$ are adjacent. Similar to the first case, it can be showed that $r_m(u|R)=r_m(u|R')\cup\{d(u,c_1),d(u,c_2)\}=r_m(u|R')\cup\{ecc_T(u)-rad(T),ecc_T(u)-rad(T)+1\}$, a contradiction.
Now we are ready to prove that $md(T)\leq n-2$. If $diam(T)=3$, the result follows from Lemma \[diam3\]. Let $diam(T)\geq 4$ and let $R$ be an m-resolving set for $T$ with $|R| \leq n-1$. It is only necessary to consider the case when $R=V(T)-\{x\}$, since otherwise the desired result holds. Let $T'$ be the minimum induced subgraph of $T$ containing $R$, that is $T'=T-x$ if $x$ is a leaf and $T'=T$ otherwise. Note that $diam(T') \geq diam(T)-1 \geq 3$. We shall consider two cases, based on whether $x$ is a center of $T'$.
**Case 1: $\mathbf{x \notin C(T')}$.** We claim that $R'=R-C(T')=V(T)-\{x\}-C(T')$ is also an m-resolving set. By the construction of $T'$, the maximum element in $r_m(u|R')$ is $ecc_{T'}(u)$, for any $u\in V(T')$. Since $x\notin R'$ then $0\notin r_m(x|R')$, and $x\notin C(T')$ implies that $\max(r_m(x|R'))=rad(T')+d(x,C(T'))>rad(T')$. The only vertex $u$ other than $x$ with $0\notin r_m(u|R')$ is $u\in C(T')$, but if $u\in C(T')$ then $\max(r_m(x|R'))=rad(T')+d(x,C(T'))>rad(T')=ecc_{T'}(u)=\max(r_m(u|R'))$. Therefore $r_m(x|R') \ne r_m(u|R')$ for all $u\in V(T)-\{x\}$. Let $u,v$ be two vertices in $V(T)-\{x\}$. If $r_m(u|R')=r_m(v|R')$, then $r_m(u|R)=r_m(v|R)$, a contradiction. These show that the multiset representation of every vertex is distinct.
**Case 2 : $\mathbf{x\in C(T')}$.** Here $x$ is not a leaf and $T'=T$. Since $diam(T)\geq 4$, then neither a center nor a neighbor of a center is an end-vertex. We again separate our observation into two subcases, based on the diameter of $T$.
**Case 2.1:**
: **$\mathbf{diam(T)}$ is even.** In this case $C(T)=\{x\}$. Let $N=N(x)$ be the set of neighbors of the center or the set of vertices with eccentricity $rad(T)+1$. Let $R'=R-N=V(T)-\{x\}-N$, we claim that $R'$ is also an m-resolving set. Since $diam(T)\geq 4$ the vertices in $N$ are not end-vertices, and so the maximum element in $r_m(u|R')$ is still $ecc_T(u)$. Since $x\notin R'$ and $x\in C(T)$, then $0\notin r_m(x|R')$ and $\max(r_m(x|R'))=ecc_T(x)=rad(T)$. If $u$ is a vertex other than $x$ with $0\notin r_m(u|R')$, then $u\in N$ and $\max(r_m(x|R')) = rad(T) \ne rad(T) + 1 = \max(r_m(u|R'))$. Therefore $r_m(x|R')\ne r_m(u|R')$ for all $u\in V(T)-\{x\}$.
Now let $k=|N|$ and consider a vertex $v\in V(T)-\{x\}$. Thus, $$\begin{aligned}
r_m(v|R) &=r_m(v|R') \cup \{d(v,y):y\in N\}\\
&=r_m(v|R') \cup \{d(v,x)-1,(d(v,x)+1)^{k-1}\}\\
&=r_m(v|R') \cup \{ecc_T(v)-rad(T)-1,(ecc_T(v)-rad(T)+1)^{k-1}\}.
\end{aligned}$$ Since the maximum element in $r_m(u|R')$ is $ecc_{T}(u)$, if $r_m(u|R')=r_m(v|R')$, we will obtain $r_m(u|R)=r_m(v|R)$, a contradiction.
**Case 2.2:**
: **$\mathbf{diam(T)}$ is odd.** Since $|C(T)|=2$, let $C(T)=\{x,y\}$.
First we consider the case when $|T|$ is odd. Obviously, $|T-C(T)|$ is also odd. Consider $R'=R-\{y\}=V(T)-\{x,y\}$ and $T_x$ and $T_y$ are the components of $T-xy$ containing $x$ and $y$, respectively. Let $u$ and $v$ be two vertices in $T$, with $r_m(u|R')=r_m(v|R')$. This means $ecc(u)=ecc(v)$. If both $u$ and $v$ is in either $T_x$ or $T_y$, then they have the same distance to $y$ and $r_m(u|R)=r_m(v|R)$, a contradiction. If $u$ in $T_x$ and $v$ in $T_y$, then $d(u,x)=d(u,C(T))=ecc(u)-rad(T)=ecc(v)-rad(T)=d(v,C(T))=d(v,y)$, and so $d(u,v)$ is odd. Now we count the number of vertices in $R'$ with odd and even distance to $u$, and denote them with $o$ and $\epsilon$, respectively. Since $o + \epsilon = |R'|$ is odd, then $o \ne \epsilon$. However $d(u,v)$ is odd, and so, by the uniqueness of path between two vertices, a vertex with odd distance to $u$ has even distance to $v$ and vice versa. This means the number of vertices in $R'$ with odd distance to $v$ is $\epsilon$ which is not equal to the number of vertices with odd distance to $u$, a contradiction.
Now we consider the case when $|T|$ is even. Our idea is to construct a new m-resolving set of odd cardinality by removing some neighbors of the center. By doing so, two vertices in $T_x$ (or $T_y$) will have the same distance to the removed vertices; while a vertex in $T_x$ and a vertex in $T_y$ will have different number of vertices in $R'$ with odd distance. As we already established in the previous cases, this will guarantee that the newly constructed set is m-resolving. Our construction depends on the degrees of $x$ and $y$. If either $deg(x)$ or $deg(y)$ is odd, then either $|N|=|N(x)-\{y\}|$ or $|N|=|N(y)-\{x\}|$ is even. Choose the $N$ with even cardinality. Since $|T|$ is even then $|R|$ is odd and $|R'|=|R-N|$ is also odd. If both $deg(x)$ and $deg(y)$ is even, let $N_1=N(x)-\{y\}$ and $N_2=N(y)-\{x\}$. Thus $R'=V(T)-N_1-N_2$ is the required new m-resolving set.
To study whether the upper bound in Theorem \[bounds\] is sharp, we conducted exhaustive search for multiset basis for all trees of order up to 10, which were generated by using McKay’s geng software [@MP13]. We found that there are only two trees of order $n$ with multiset dimension $n-2$. They are the path on 3 vertices and the tree on 5 vertices constructed from the star on 4 vertices by subdividing exactly one of its edges. The complete statistics for trees of order $n$, $6 \leq n \leq 10$ can be seen in Table \[trees\].
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: Multiset dimensions of trees of order up to 10.[]{data-label="trees"}
Based on the result of our computer search, we would like to propose the following.
\[upperbound\] Let $T$ be a tree. If $md(T) < \infty$, then $md(T) \leq n-diam(T)+1$ and the bound is sharp.
If Conjecture \[upperbound\] is true, an example for the sharpness of the bound is the tree on $n$ vertices constructed from the star on 4 vertices by subdividing exactly one of its edges $n-4$ times. This tree has multiset dimension 3 which is equal to $n-(n-2)+1$, where $n-2$ is the diameter of the tree.
Caterpillars and lobsters with finite multiset dimensions {#s_catlob}
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In this paper we define caterpillars and lobsters by using the notion of a $k$-center-path. Let $G$ be a connected graph, a subgraph $P$ of $G$ is called a *$k$-center-path of $G$* if $P$ is a path and $d(u,P)\leq k$ for every vertex $u$ in $G$. A *minimum $k$-center-path* is a $k$-center path with minimum length. A *caterpillar* is a tree containing a $1$-center-path and a *lobster* is a tree containing a $2$-center-path. Let $T$ be a rooted tree with $v$ as its root, a *separation of $T$*, denoted by $[T]$, is a graph obtained by subdividing all edge attached to $v$ and then deleting $v$. Note that the number of components in $[H]$ is the degree of $v$ in $H$.
We start by characterising all lobsters having finite multiset dimension.
\[Lobster\] Let $G$ be a lobster. If $P$ is the minimum $2$-center path of $G$, then the following are equivalent.
- $G$ has finite multiset dimension.
- The only component of $G-E(P)$ with infinite multiset dimension is an $S_4$.
- If $H$ is a component of $G-E(P)$ then $[H]$ has at most $4$ components which are either a $P_2$, a $P_3$, or an $S_4$, with at most two $P_2$s and two $S_4$s.
We will prove $(2) \Rightarrow (3) \Rightarrow (1) \Rightarrow (2)$.
- Let $H$ be a component of $G-E(P)$ and $v$ the root of $H$ in $P$. If $H$ has infinite multiset dimension then $H$ is an $S_4$ which fullfils (3). Now consider that $H$ has finite multiset dimension.
Let $u$ be a neighbor of $v$ in $H$. Since $G$ is a lobster then all neighbors of $u$ other than $v$ are leaves, and by Lemma \[twins\], $deg(u)\leq 3$. This means the component in $[H]$ containing $u$ is either a $P_2$, a $P_3$, or an $S_4$.
Let $R$ be an m-resolving set of $H$, if $[H]$ contains an $S_4$ then exactly one of the leaf of $S_4$ is in $R$. If there are more than two $S_4$s in $[H]$, then there exist two $S_4$s with either both centers are in $R$ or not in $R$, which results on both centers having the same multiset representations with respect to $R$, a contradiction. The fact that $[H]$ can not have more than two $P_2$s is due to Lemma \[twins\].
Thus we are left to prove that $[H]$ has at most $4$ components. Since for each component of $[H]$, there are at most one vertex in $R$ with distance two from a vertex $v$ in $H$, we shall catagorize each component of $[H]$ into the following 4 types:
- *type 0*: The component has no vertex in $R$,
- *type 1*: The component has one vertex in $R$ with distance 1 to $v$,
- *type 2*: The component has one vertex in $R$ with distance 2 to $v$, or
- *type 12*: The component has two vertices in $R$, each with distance 1 or 2 to $v$.
Suppose that $[H]$ has more than $4$ components. Thus there are two components with the same type, and so the two neighbors of $v$ in those two components will have the same multiset representation with respect to $R$, a contradiction.
- Let $G$ be a graph that satisfies (3), $P=p_0,p_1,\ldots,p_n$, $H_i$ be the component of $G-E(P)$ containing $p_i$, and $d_i$ be the degree of $p_i$ in $H_i$. For $i=0,1,\ldots, n$, let $R(H_i)$ be a the resolving set of $H_i$ with maximum cardinality which does not contain the root. We shall construct an m-resolving set for $G$. If $d_i=0$, define $R(H_i)=\emptyset$, otherwise do the folowing steps.
1. Define $H_i^{(1)},H_i^{(2)},\ldots, H_i^{(d_i)}$ a non-increasing sequence of components of $H_i$ with component ordering: $S_4 > P_3 > P_2$.
2. If $H_i^{(1)} \not\approx P_2$, choose exactly two vertices in $H_i^{(1)}$, each with distance $1$ and $2$ to $p_i$, as members of $R(H_i)$. If $H_i^{(1)} \approx P_2$, define $H_i^{(d_i+1)}\approx P_2$.
3. If $H_i^{(2)}$ exists and is not a $P_2$, choose exactly one vertex in $H_i^{(2)}$ with distance $2$ to $p_i$ as a member of $R(H_i)$. If $H_i^{(2)} \approx P_2$, define $H_i^{(d_i+e_i)}\approx P_2$ with $e_i=1$ if $H_i^{(1)} \not\approx P_2$ and $e_i=2$ otherwise.
4. If $H_i^{(3)}$ exists, choose the vertex in $H_i^{(3)}$ with distance $1$ to $p_i$ as a member of $R(H_i)$.
Since $P$ is a minimum path and $R(H_i)$ a maximum resolving set, for $i=0$ and $i=n$, there exists a vertex in $R(H_i)$ with distance two to $P$. This means the maximum element in $r_m\left(v|\cup_{i=0}^n R(H_i)\right)$ is $ecc(v)$.
Now we shall propose two constructions of an m-resolving set for $G$, depending on the diameter of $G$.
**Construction 1**: for odd $diam(G)$.
: If $\sum_{i=0}^{n} |R(H_i)|$ is odd, define $R=\cup_{i=0}^n R(H_i)$, otherwise $R=p_0\cup\left(\cup_{i=0}^n R(H_i)\right)$. To prove that $R$ is a resolving set, we only need to show that vertices with the same eccentricity have different representations. In Figure \[diamodd\], the boxed vertices have the same eccentricity. Here we define sides as components of $G-c_1c_2$, where $c_1$ and $c_2$ are the centers.
\[h\]
(-5.6,4.4) – (-4.4,4.4) – (-4.4,3.6) – (-5.6,3.6) – cycle; (-7.8,2.4) – (-6.2,2.4) – (-6.2,1.6) – (-7.8,1.6) – cycle; (-9.4,0.4) – (-8.6,0.4) – (-8.6,-0.4) – (-9.4,-0.4) – cycle; (8.6,0.4) – (9.4,0.4) – (9.4,-0.4) – (8.6,-0.4) – cycle; (6.2,2.4) – (7.8,2.4) – (7.8,1.6) – (6.2,1.6) – cycle; (4.6,4.4) – (5.4,4.4) – (5.4,3.6) – (4.6,3.6) – cycle; (-9.0,0.0) – (-9.0,2.0) – (-9.0,4.0) (-11.0,0.0)– (-10.5,2.0) (-11.0,0.0)– (-11.0,2.0) (-11.0,0.0)– (-11.5,2.0) (-11.0,0.0)– (-9.0,0.0) (-11.0,2.0)– (-11.0,4.0) (-11.5,2.0)– (-11.25,4.0) (-11.5,2.0)– (-11.75,4.0) (-7.0,0.0)– (-7.5,2.0) (-7.0,0.0)– (-6.5,2.0) (-6.5,2.0)– (-6.5,4.0) (-7.5,2.0)– (-7.25,4.0) (-7.5,2.0)– (-7.8,4.0) (-5.0,0.0)– (-5.0,2.0) (-5.0,2.0)– (-5.25,4.0) (-5.0,2.0)– (-4.75,4.0) (-3.0,0.0)– (-3.0,2.0) (-1.0,0.0)– (-1.0,2.0) (-1.0,2.0)– (-1.0,4.0) (1.0,0.0)– (1.0,2.0) (1.0,2.0)– (1.0,4.0) (3.0,0.0)– (3.0,2.0) (3.0,2.0)– (3.0,4.0) (5.0,0.0)– (5.0,2.0) (5.0,2.0)– (5.0,4.0) (7.5,2.0)– (7.0,0.0) (9.0,0.0)– (9.5,2.0) (11.0,0.0)– (11.0,2.0) (11.0,2.0)– (11.0,4.0) (9.0,0.0)– (8.5,2.0) (9.0,0.0)– (9.0,2.0) (9.0,2.0)– (9.0,4.0) (7.0,0.0)– (6.5,2.0) (-11.0,0.0)– (11.0,0.0); (-1.0,0.0) circle (0.1) (1.0,0.0) circle (0.1) (3.0,0.0) circle (0.1) (5.0,0.0) circle (0.1) (7.0,0.0) circle (0.1) (-3.0,0.0) circle (0.1) (-5.0,0.0) circle (0.1) (-7.0,0.0) circle (0.1) (-9.0,0.0) circle (0.1) (9.0,0.0) circle (0.1) (11.0,0.0) circle (0.1) (-11.0,0.0) circle (0.1) (-11.5,2.0) circle (0.1) (-11.0,2.0) circle (0.1) (-10.5,2.0) circle (0.1) (-11.0,4.0) circle (0.1) (-11.25,4.0) circle (0.1) (-11.75,4.0) circle (0.1) (-9.0,2.0) circle (0.1) (-9.0,4.0) circle (0.1) (-7.5,2.0) circle (0.1) (-6.5,2.0) circle (0.1) (-6.5,4.0) circle (0.1) (-7.25,4.0) circle (0.1) (-7.8,4.0) circle (0.1) (-5.0,2.0) circle (0.1) (-5.25,4.0) circle (0.1) (-4.75,4.0) circle (0.1) (-3.0,2.0) circle (0.1) (-1.0,2.0) circle (0.1) (-1.0,4.0) circle (0.1) (1.0,2.0) circle (0.1) (1.0,4.0) circle (0.1) (3.0,2.0) circle (0.1) (3.0,4.0) circle (0.1) (5.0,2.0) circle (0.1) (5.0,4.0) circle (0.1) (7.5,2.0) circle (0.1) (9.5,2.0) circle (0.1) (11.0,2.0) circle (0.1) (11.0,4.0) circle (0.1) (8.5,2.0) circle (0.1) (9.0,2.0) circle (0.1) (9.0,4.0) circle (0.1) (6.5,2.0) circle (0.1); (0,6)–(0,-1); (-6,5.2)node[Side $1$]{}; (6,5.2)node[Side $2$]{}; (-9,-0.5)node\[below\][$p_{i-2}$]{}; (-7,-0.5)node\[below\][$p_{i-1}$]{}; (-5,-0.5)node\[below\][$p_{i}$]{}; (-1,-0.5)node\[below\][$c_1$]{}; (1,-0.5)node\[below\][$c_2$]{}; (9,-0.5)node\[below\][$p_{n-i+2}$]{}; (7,-0.5)node\[below\][$p_{n-i+1}$]{}; (5,-0.5)node\[below\][$p_{n-i}$]{};
The vertices in the same box will have different representation with respect to $R$, because $R(H_i)$ is a resolving set for $H_i$. Now consider vertices in different boxes. Vertices in a box in Side 1 are with the same distance to vertices in $M:=\cup_{k=i+1}^n R(H_k)$, but they have distinct multiset representations with respect to $N:=R-M$, since the maximum distance to vertices in $N$ is distinct. This means that their multiset representations with respect to $R$ is also distinct. Similar argument can be applied to vertices in a box in Side 2.
Now let $u$ be a vertex in a box in Side $1$ and $v$ be a vertex in a box in Side $2$. Since $diam(G)$ is odd, $d(u,v)$ is also odd. Let $o$ and $\epsilon$ be the number of vertices in $R$ with odd and even distances to $u$, respectively. Since $o + \epsilon = |R|$ is odd, then $o \ne \epsilon$. Since $d(u,v)$ is odd, a vertex with odd distance to $u$ will have even distance to $v$ and vice versa. This means the number of vertices in $R$ with odd distance to $v$ is $\epsilon$ which is not equal to the number of vertices with odd distance to $u$. Therefore $r_m(u|R) \ne r_m(v|R)$.
**Construction 2**: for even $diam(G)$.
: Here $H_{n/2}$ will be exactly in the middle and thus will not be included in any side. For $i=0,1,2$ we define $$a_i:=|\{w\in \cup_{i=0}^n R(H_i)|w\ \text{in Side 1 and }ecc(w)=diam(G)-i\}|$$ and $$b_i:=|\{w\in \cup_{i=0}^n R(H_i)|w\ \text{in Side 2 and }ecc(w)=diam(G)-i\}|.$$ By considering the vectors $(a_0,a_1,a_2)$ and $(b_0,b_1,b_2)$; the side with larger vector (lexicographically) will be named the dominant side, and if $(a_0,a_1,a_2)=(b_0,b_1,b_2)$ we name Side $1$ as the dominant side.
Assume that $n>2$. If Side $1$ is dominant, define $R=\{p_0,p_{n/2+1}\}\cup_{i=0}^n R(H_i)$, otherwise define $R=\{p_n,p_{n/2-1}\}\cup_{i=0}^n R(H_i)$. The proof that $R$ is an m-resolving set will only be given for the case when Side $1$ is dominant, the other case can be proved similarly.
(-5.6,4.4) – (-4.4,4.4) – (-4.4,3.6) – (-5.6,3.6) – cycle; (-7.8,2.4) – (-6.2,2.4) – (-6.2,1.6) – (-7.8,1.6) – cycle; (-9.4,0.4) – (-8.6,0.4) – (-8.6,-0.4) – (-9.4,-0.4) – cycle; (6.6,0.4) – (7.4,0.4) – (7.4,-0.4) – (6.6,-0.4) – cycle; (4.6,2.4) – (5.4,2.4) – (5.4,1.6) – (4.6,1.6) – cycle; (2.6,4.4) – (3.4,4.4) – (3.4,3.6) – (2.6,3.6) – cycle; (-9.0,0.0) – (-9.0,2.0) – (-9.0,4.0) (-11.0,0.0)– (-10.5,2.0) (-11.0,0.0)– (-11.0,2.0) (-11.0,0.0)– (-11.5,2.0) (-11.0,0.0)– (-9.0,0.0) (-11.0,2.0)– (-11.0,4.0) (-11.5,2.0)– (-11.25,4.0) (-11.5,2.0)– (-11.75,4.0) (-7.0,0.0)– (-7.5,2.0) (-7.0,0.0)– (-6.5,2.0) (-6.5,2.0)– (-6.5,4.0) (-7.5,2.0)– (-7.25,4.0) (-7.5,2.0)– (-7.8,4.0) (-5.0,0.0)– (-5.0,2.0) (-5.0,2.0)– (-5.25,4.0) (-5.0,2.0)– (-4.75,4.0) (-3.0,0.0)– (-3.0,2.0) (-1.0,0.0)– (-1.0,2.0) (-1.0,2.0)– (-1.0,4.0) (1.0,0.0)– (1.0,2.0) (1.0,2.0)– (1.0,4.0) (3.0,0.0)– (3.0,2.0) (3.0,2.0)– (3.0,4.0) (5.0,0.0)– (5.0,2.0) (5.0,2.0)– (5.0,4.0) (7.5,2.0)– (7.0,0.0) (9.0,0.0)– (9.5,2.0) (9.0,0.0)– (8.5,2.0) (9.0,0.0)– (9.0,2.0) (9.0,2.0)– (9.0,4.0) (7.0,0.0)– (6.5,2.0) (-11.0,0.0)– (9.0,0.0); (-1.0,0.0) circle (0.1) (1.0,0.0) circle (0.1) (3.0,0.0) circle (0.1) (5.0,0.0) circle (0.1) (7.0,0.0) circle (0.1) (-3.0,0.0) circle (0.1) (-5.0,0.0) circle (0.1) (-7.0,0.0) circle (0.1) (-9.0,0.0) circle (0.1) (9.0,0.0) circle (0.1) (-11.0,0.0) circle (0.1) (-11.5,2.0) circle (0.1) (-11.0,2.0) circle (0.1) (-10.5,2.0) circle (0.1) (-11.0,4.0) circle (0.1) (-11.25,4.0) circle (0.1) (-11.75,4.0) circle (0.1) (-9.0,2.0) circle (0.1) (-9.0,4.0) circle (0.1) (-7.5,2.0) circle (0.1) (-6.5,2.0) circle (0.1) (-6.5,4.0) circle (0.1) (-7.25,4.0) circle (0.1) (-7.8,4.0) circle (0.1) (-5.0,2.0) circle (0.1) (-5.25,4.0) circle (0.1) (-4.75,4.0) circle (0.1) (-3.0,2.0) circle (0.1) (-1.0,2.0) circle (0.1) (-1.0,4.0) circle (0.1) (1.0,2.0) circle (0.1) (1.0,4.0) circle (0.1) (3.0,2.0) circle (0.1) (3.0,4.0) circle (0.1) (5.0,2.0) circle (0.1) (5.0,4.0) circle (0.1) (7.5,2.0) circle (0.1) (9.5,2.0) circle (0.1) (8.5,2.0) circle (0.1) (9.0,2.0) circle (0.1) (9.0,4.0) circle (0.1) (6.5,2.0) circle (0.1); (0,6)–(0,-1); (-2,6)–(-2,-1); (-6,5.2)node[Side $1$]{}; (4,5.2)node[Side $2$]{}; (-11,-0.5)node\[below\][$p_0$]{}; (-1,-0.5)node\[below\][$p_{n/2}$]{}; (1,-0.5)node\[below\][$p_{n/2+1}$]{}; (9,-0.5)node\[below\][$p_{n}$]{};
Similar argument from Construction 1 can be applied to prove that the vertices in the same side have distinct representations. Let $u$ and $v$ be vertices with the same eccentricity in Side $1$ and side $2$, respectively. For $i=0,1,\ldots,ecc(v)$, we define $m_v(i)$ as the multiplicity of $i$ in $r_m(v|R)$. If $v$ is in $H_{n/2+1}$, then $m_v:=(m_v(ecc(v)),m_v(ecc(v)-1),m_v(ecc(v)-2))=(a_0,a_1,a_2+b_0+1)$, otherwise $m_v=(a_0,a_1,a_2+1)$. If $u$ is in $H_{n/2-1}$, then $m_u=(b_0,b_1,b_2+a_0)$, otherwise $m_u=(b_0,b_1,b_2)$.
In cases other than $u\in V(H_{n/2-1})$ and $v\notin V(H_{n/2}+1)$, we have $m_v > m_u$, and so $r_m(u|R)\ne r_m(v|R)$. Therefore we only need to prove the case when $u \in H_{n/2-1}$ and $v\notin H_{n/2+1}$. Consider the lobster in Figure \[uv\], where the black vertices indicate the members of $R$.
\[h\]
(-3.5,4.4)–(-2.5,4.4)–(-2.5,3.6)–(-3.5,3.6)–cycle; (3.25,2.4)–(4.25,2.4)–(4.25,1.6)–(3.25,1.6)–cycle; (-5.0,0.0)– (-5.0,2.0); (-3.0,0.0)– (-3.5,2.0); (-3.0,0.0)– (-2.5,2.0); (-1.0,0.0)– (-1.0,2.0); (-1.0,2.0)– (-1.0,4.0); (1.0,0.0)– (1.0,2.0); (1.0,2.0)– (1.0,4.0); (3.0,0.0)– (3.5,2.0); (3.5,2.0)– (3.5,4.0); (-3.5,2.0)– (-3.25,4.0); (-3.5,2.0)– (-3.75,4.0); (-2.5,2.0)– (-2.25,4.0); (-2.5,2.0)– (-2.75,4.0); (2.5,2.0)– (2.75,4.0); (2.5,2.0)– (2.25,4.0); (2.5,2.0)– (3.0,0.0); (3.0,0.0)– (3.0,2.0); (3.0,0.0)– (4.0,2.0); (-9.0,0.0)– (7.0,0.0); (5.0,0.0)– (5.0,2.0); (5.0,2.0)– (5.0,4.0); (5.0,0.0)– (5.5,2.0); (5.0,0.0)– (4.5,2.0); (-1.0,0.0) circle (0.15); (1.0,0.0) circle (0.15); (3.0,0.0) circle (0.15); (5.0,0.0) circle (0.15); (-3.0,0.0) circle (0.15); (-5.0,0.0) circle (0.15); (-5.0,2.0) circle (0.15); (-3.5,2.0) circle (0.15); (-2.5,2.0) circle (0.15); (-1.0,2.0) circle (0.15); (-1.0,4.0) circle (0.15); (1.0,2.0) circle (0.15); (1.0,4.0) circle (0.15); (3.5,2.0) circle (0.15); (3.5,4.0) circle (0.15); (-3.75,4.0) circle (0.15); (-2.75,4.0) circle (0.15); (-2.25,4.0) circle (0.15); (-3.25,4.0) circle (0.15); (2.5,2.0) circle (0.15); (2.75,4.0) circle (0.15); (2.25,4.0) circle (0.15); (3.0,2.0) circle (0.15); (4.0,2.0) circle (0.15); (5.0,2.0) circle (0.15); (5.0,4.0) circle (0.15); (5.5,2.0) circle (0.15); (4.5,2.0) circle (0.15); (-7.0,0.0) circle (0.15); (-5,5.2)node[Side $1$]{}; (3,5.2)node[Side $2$]{}; (0,6)–(0,-1); (-2,6)–(-2,-1); (-1,-0.5)node\[below\][$p_{n/2}$]{}; (1,-0.5)node\[below\][$p_{n/2+1}$]{}; (-3,-0.5)node\[below\][$p_{n/2-1}$]{}; (-3,4.3)node\[above\][$u$]{}; (3.95,2.3)node\[above\][$v$]{};
Since $p_{n/2+1} \in R$, then $2\in r_m(v|R)$. In this case, $u$ is a leaf of an $S_4$ component in $[H_{n/2-1}]$ and $u\notin R$. If $v\notin R$, then by the maximality of $R(H_{n/2+1})$, there exists a component in $[H_{n/2+1}]$ with type 1 or type 12, and so $m_v(2)\geq 2$. However $m_u(2)\leq 1$, which leads to $rm(u|R)\ne rm(v|R)$.
In both constructions, we we prove that $R$ is an m-resolving set for $G$, and therefore $md(G)$ is finite.
- Let $R$ be an m-resolving set of $G$ and suppose that $G-E(P)$ has a component $H \not\approx S_4$ with infinite multiset dimension. Let $v$ be the vertex in $H$ which is also in $P$. We will prove that there are two vertices $x$ and $y$ in $H$ with the same distance to $v$ and thus $r_m(x|R_H)=r_m(y|R_H)$.
Let $N_H(v)=\{v_1,v_2,\cdots,v_{deg(v)}\}$. If there exists a vertex $v_i$ in $N_H(v)$ with degree at least $4$ then there is at least $3$ leaves attached to $u$, and by Lemma \[twins\], there exists two vertices with the same representation. Now assume that $deg(v_i)\leq 3$ for all $i$. By the assumption of $H$, we have $deg(v)\geq2$. If $[H]$ contains an $S_4$ with two leaves other than $v$, then both are either in $R$ or not in $R$, and so the two leaves will have the same representation. Now the only case to consider is when the components of $[H]$ are a $P_2$, a $P_3$, or an $S_4$ with exactly one leaf (other than $v$) in $R$. If all of the components of $[H]$ has different types then we already established that $md(H)$ is finite. So there are components with the same type which means the neighbors of $v$ in those components will have the same representation.
Let $x$ and $y$ be the two vertices in $H$ with the same distance to $v$ with $r_m(x|R_H)=r_m(y|R_H)$. The path from $x$ or $y$ to any vertex $r$ in $R-R_H$ goes through $v$, and so $d(r,x)=d(r,y)$. Thus, $r_m(x|R)=r_m(y|R)$, a contradiction. We conclude that $H$ is either with finite multiset dimension or is an $S_4$.
Note that if we let $P=p_0p_1\dots p_n$ to be any $2$-center path, not necessarily the minimum, then it is possible that the components of $G-E(P)$ satisfy (3) but not (1). One example is when $H_0=K_1$ and $H_1=P_3$, since $p_1$ has 3 leaves as neighbors.
If either $P$ is a path not containing an end-vertex; or $H_0$ and $H_n$ are either a $P_2$ or a $P_3$; or $[H_1]$ and $[H_{n-1}]$ only have at most $3$ components which are either a $P_2$, a $P_3$, or an $S_4$, with at most one $S_4$ and two $P_2$s, then Theorem \[Lobster\] still hold. These assumptions are redundant for characterizing lobsters since they are equivalent with (3) in the theorem. However they could be used to characterise caterpillar with finite multiset dimension as stated in the following.
\[Catterpillar\] Let $G$ be a caterpillar with $P$ its minimum $1$-center-path. $md(G)$ is finite if and only if every vertex in $P$ has at most $2$ neighbors in $G-P$.
In Theorem \[Catterpillar\], it is necessary for $P=p_0p_1\cdots p_n$ to be a minimum $1$-center path. An example to show its necessity is when $p_0$ is a leaf and $p_1$ have two neighbors in $G-P$.
Theorems \[Catterpillar\] and \[Lobster\] partially answered the question proposed in Problem \[chartree\]. However, to characterize all trees with finite multiset dimension, we might have to use a different approach. We believe that applying our argument inductively to the size of the minimum center-path of a tree will be difficult to prove.
This research was partially supported by Penelitian Dasar Unggulan Perguruan Tinggi 2017-2019, funded by Indonesian Ministry of Research, Technology and Higher Education.
[99]{}
F. Harary and R. A. Melter, On the metric dimension of a graph, Ars Combin., 2 (1976), 191-195.
V. Khemmani and S. Isariyapalakul, The multiresolving sets of graphs with prescribed multisimilar equivalence classes, Int. J. Math. Math. Sci., 2018 (2018), ID 8978193.
B. D. McKay and A. Piperno, Practical graph isomorphism, II, J. Symbolic Computation, 60 (2013), 94–112.
R. Simanjuntak, P. Siagian, and T. Vetrík, On the multiset dimension of a graph, arXiv:1711.00225.
P. J. Slater, Leaves of trees, Congr. Numer., 14 (1975), 549-559.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this work, we aim to study the thermal properties of materials using classical molecular dynamics simulations and specialized numerical methods. We focus primarily on the thermal conductivity $\kappa$ using non-equilibrium molecular dynamics to study the response of a crystalline solid, namely hematite ($\alpha\textrm{-Fe}_2\textrm{O}_3$), to an imposed heat flux as is the case in real life applications. We present a methodology for the calculation of $\kappa$ as well as an adapted potential for hematite. Taking into account the size of the simulation box, we show that not only the longitudinal size (in the direction of the heat flux) but also the transverse size plays a role in the determination of $\kappa$ and should be converged properly in order to have reliable results. Moreover we propose a comparison of thermal conductivity calculations in two different crystallographic directions to highlight the spatial anisotropy and we investigate the non-linear temperature behavior typically observed in NEMD methods.'
author:
- Jonathan Severin
- Philippe Jund
bibliography:
- 'biblio.bib'
title: 'Thermal conductivity calculation in anisotropic crystals by molecular dynamics : application to $\alpha\textrm{-Fe}_2\textrm{O}_3$'
---
Introduction {#sec:intro}
============
From [*ab initio*]{} calculations of material properties to finite-element models of macroscopic systems, numerical studies of the thermal transport are motivated by the wide range of technological applications. Examples of such applications are found in nanoelectronics [@sinha2005review], aerospace [@lee2012large], automotive [@yang2005potential] and building sector [@jelle2011traditional]. Therefore, being able to model the heat transfer and predict thermal properties presents a definite advantage for designing or improving materials and industrial processes.\
NEMD methods have been applied to thermal conductivity calculations since the beginning of the 1980s [@evans1982homogeneous] and a number of different algorithms have been developed [@schelling2002comparison]. Two main approaches can be identified: imposing the temperature gradient or imposing the heat flux. The latter approach has the advantage of a faster convergence [@heino2003thermal] and two of its algorithms are now commonly applied [@schelling2002comparison]: the particle velocity exchange presented in [@muller1997simple] and the heat source and heat sink method described in [@jund1999molecular], [@ikeshoji1994non] and [@aubry2004robust]. Here we make use of the source and sink method which was first developed for amorphous materials and simple Lennard-Jones (LJ) liquids and has been generalized to simple crystals, e.g. FCC Cu, Ag, Au, etc. in [@bracht2014thermal; @ge2013vibrational; @zhang2013thermal; @zhou2007phonon] and, in some cases, to more complex structures such as tetragonal ZrO$_2$[@schelling2001mechanism] or zeolitic imidazolate frameworks in [@zhang2013zeolitic].\
The work reported here is part of a wider effort which aims to predict the material-lubricant interactions involved in the operation of a car engine by determining numerically the thermal conductivity of the different parts. As such, we focused in a first step on developing a methodology that we then applied to iron oxide (hematite) as a model material for the surface of the solid parts of the engine and we considered a temperature range between 300 and 500 K. In addition to its industrial and technological importance [@dzade2014density], in recent years hematite has been subject to a renewed interest due to discoveries concerning the geological structure and mineral properties of Mars [@christensen2000detection; @morris2006mossbauer; @fraeman2013hematite]. Bulk hematite ($\alpha\textrm{-Fe}_2\textrm{O}_3$), of space group $R\overline{3}c (167)$ , has a crystal structure that can be indexed as hexagonal with a unit cell consisting of 6 formula units in which the oxygen ions lie approximately in a hexagonal close-packed framework while the iron ions are positioned symetrically in two-thirds of the octahedral interstices [@morrish1994canted; @nolze2014exploring]. We chose to model this material using an empirical interatomic potential that we adapted and tested.\
The main objective of the study was to put together a set of methods and tools for the calculation of the thermal conductivity of a crystal based on a previous work on amorphous materials by non equilibrium molecular dynamics (NEMD) [@jund1999molecular]. We considered the influence of sample size, temperature and crystallographic orientation and we present a step by step methodology.\
The paper is divided into three sections. In the first section we describe the NEMD algorithm and methodology, the empirical potential used to model the material and the numerical methods applied to circumvent sample size effects. In the second part we present and discuss our results on the thermal conductivity of hematite single crystals, its temperature dependence, the spatial anisotropy and the nonlinear behavior observed in specific regions of the simulation box. Finally, we draw the major conclusions in the last section.
Numerical methods
=================
Molecular dynamics and NEMD
---------------------------
Classical molecular dynamics (MD) is a very popular method providing atomic-scale information on material properties and processes. Based on the integration of the Newton equations of motion at the atomic level, it allows to use relatively large simulation boxes compared to first-principles calculations. In non equilibrium molecular dynamics (NEMD) one or several internal variables are constrained to keep the system out of the equilibrium state. In our case we use the standard velocity-Verlet algorithm for time integration with a time step of 0.6 fs and we apply the method described in [@jund1999molecular] for the determination of the thermal conductivity $\kappa$ . The first step is choosing the direction of the heat flux. Here we will start by considering the z direction, parallel to the \[001\] crystallographic direction in the hexagonal lattice of hematite. Then we define two plates positioned at 1/4 and 3/4 of the simulation box length and orthogonal to the z axis. Periodic boundary conditions (PBC) are applied in every direction. At each time step, one of these plates receives a fixed amount of energy $\Delta \epsilon$ while the same amount is subtracted from the other. This is done by constantly rescaling the velocities of the atoms within the plates. This effectively results in imposing a heat flux across the box and parallel to the z direction. Care is taken to prevent a drift of the center of mass caused by the velocity rescaling. For a particle $i$ in the hot plate $P_+$, the rescaling can be expressed as
$$\label{eq:rescal}
\mathbf{\overline{v}_i} = \mathbf{v_G} + \alpha (\mathbf{v_i} - \mathbf{v_G})$$
where $\mathbf{\overline{v}_i}$ is the updated velocity, $\mathbf{v_G}$ the velocity of the center of mass of the ensemble of particles in $P_+$ and
$$\alpha = \sqrt{1 + \frac{\Delta \epsilon}{E_{P_+}}}.$$
Of course, in the case of the cold plate $P_-$, $\Delta \epsilon$ should be subtracted. The non-translational kinetic energy $E_{P_+}$ is given by
$$E_{P_+} = \frac{1}{2} \sum_i m_i \mathbf{v_i}^2 - \frac{1}{2} \sum_i m_i \mathbf{v_G}^2.$$
![Diagram of the NEMD method. The temperature slices are represented along with the heat source and sink (gray slabs). Adapted with permission from [@jund1999molecular]. Copyrighted by the American Physical Society.[]{data-label="method"}](method4.eps){width="8cm"}
To keep track of the temperature profile along the z axis, the box is divided into a set of slices in which the local temperature is computed, as summarized in figure \[method\]. The width $a$ of those slices, and therefore the number of atoms in each of them, affects the temperature calculation. A larger number of atoms provides better statistics and so allows to reduce the time over which the values need to be averaged. But to obtain a detailed temperature profile the number of slices has to be large enough and therefore their width is limited. In order to eliminate this constraint we introduce several sets of overlapping bins, as described in figure \[overlap\]. This allows to increase the temperature profile resolution without decreasing the number of atoms per slice. We find that 4 sets of 12 slices is a good compromise between precision and computational effort.
![Diagram of the NEMD method with the addition of a second set of temperature slices (dotted lines) translated by a/2 from the first.[]{data-label="overlap"}](overlap3.eps){width="10cm"}
The initial state of the simulated system is obtained by replicating a smaller cell previously optimized by energy minimization. After a first run of relaxation in a (NPT) ensemble at the desired temperature, the heat flux is switched on. We have checked that small variations of the amount of energy added to or subtracted from the plates don’t significantly affect the value of $\kappa$. Nevertheless, large values of $\Delta \epsilon$ will cause a large temperature difference between the source and the sink. A value of 1.5 % of $k_B T$ seems reasonable for a simulation box containing 60000 atoms. Once the heat flux is switched on, it is necessary to let the dynamics run long enough to produce a steady state. Typically, this stationarization is of the order of 1 ns as reported in [@schelling2002comparison; @maiti1997dynamical] for Stillinger-Weber silicon. We observe an average time of 0.8 ns before stable temperatures are observed in the source and sink plates. Then we start collecting and averaging the temperature values in every bin. Depending on the size of the system, we find that the averaging time necessary to remove the numerical noise and obtain a smooth temperature profile, on which a linear function can be fitted, varies from 2 to 6 ns. Except for the first relaxation run, all the calculations are performed in a (NVE) microcanonical ensemble. Finally, the temperature gradient is calculated from the time-averaged temperature profile and the Fourier’s law (\[eq:Fourier\]) is used to obtain $\kappa$. Recently, an adapted version of the heat exchange algorithm was proposed by Wirnsberger et al. [@wirnsberger2015enhanced] to correct a large drift of the total energy of the system observed in long-term simulations. However, in the present work we observe a moderate difference between the initial and final values of the total energy (of the order of $10^{-4}$ %). This is due to the length of our simulations which is less than 10 ns and to the relatively small value of the time step chosen to ensure energy conservation.
$$\label{eq:Fourier}
\vec{J} = - \kappa \vec{\nabla}T$$
The application of such a method takes its toll on numerical performance. Indeed the permanent temperature rescaling in the heat source and heat sink and the local temperature computation in each slice are time consuming. And with the constraints of stationarization and averaging, the time cost is a primary issue. Some of the simulations performed as part of this work required more than 100000 CPU hours individually. All molecular dynamics simulations were conducted using a modified version of the LAMMPS package [@plimpton1995fast] (based on version may 2015).
Interaction potential {#sec:potential}
---------------------
To perform molecular dynamics on an ionic crystal such as hematite one needs to apply an interatomic potential defined to model the structure of the material as well as its physical properties. As mentioned in section \[sec:intro\], we consider a hexagonal unit cell with 6 formula units (30 atoms) and the following lattice constants, respectively a, b and c: $5.039$, $5.039$, $13.77$ Å[@sadykov1996effect]. We use a modified version of the potential $U(r)$ published in 2006 by Pedone et al. [@pedone2006self] to model our material. Three terms contribute to the equation of $U$: a short-range Morse potential, a short-range $r^{-12}$ repulsion and a long range Coulomb interaction. A cutoff distance of 7 Å was applied to the Morse and short-range repulsion terms and the Coulomb interaction was implemented by way of an Ewald summation.
$$\begin{aligned}
\label{eq:potential}
U(r) = D_{ij} ([1 - e^{-a_{ij} (r - r_0)}]^2 - 1) + \frac{C_{ij}}{r^{12}} + \frac{z_i z_j e^2}{r}\end{aligned}$$
The parameters of the potential were initially fitted on the experimental lattice constants, atomic positions and elastic constants using free energy minimization [@catlow1997computer] and other empirical fitting methods implemented in the GULP package [@gale1997gulp; @gale1996empirical; @richard1994self]. These methods allow to fit the potential on a structure relaxed at a finite temperature taking into acount quantities such as mechanical and dielectric properties. Nevertheless the experimental lattice constants used for this fit are for some reason different from the values obtained in most of the studies [@sadykov1996effect; @sorescu2004hydrothermal; @sawada1996electron; @barinov1995effects; @morrish1994canted; @maslen1994synchrotron]. A fine-tuning of the Morse equilibrium parameter ($r_0$) for the Fe-O and the O-O interactions allowed us to accurately fit the “correct” experimental values. Indeed differences between lattice constants and interatomic distances experimentally observed and obtained with the modified potential are well under 1%, as presented in table \[table:cellTab\]. The new values of $r_0$ are 2.41810 Å and 3.65455 Å respectively for the Fe-O and O-O interactions.
Cell parameters. Experimental (Å) Calculated.[^1] (Å) Calculated[^2] (Å) Difference (%)
------------------ ------------------ --------------------- -------------------- ----------------
a 5.039 4.95 5.066 0.54
b 5.039 4.95 5.066 0.54
c 13.77 13.42 13.74 0.23
c/a 2.73 2.71 2.71 0.77
\[table:cellTab\]
In addition to the structural properties, the bulk modulus was investigated in two different ways in order to validate further the modified potential function. First the elastic constants were calculated by deforming the hexagonal box in the 6 degrees of freedom and observing the change in the stress tensor at zero temperature. The Voigt-Reuss approximation applied to the calculated elastic tensor gives a value of 217 GPa for the bulk modulus. Additionnally, we performed single-point energy calculations at different volumes around the equilibrium volume of the cell and fitted the curve E=f(V) with the Vinet equation of state [@vinet1987compressibility]. We obtained this way a bulk modulus of 219 GPa. These calculated results fall within the measured values of 203 [@liebermann1968elastic] and 230 GPa [@catti1995theoretical] available in the literature. The small difference between the two calculated values (less than 1%) may be used as an indication of the precision of this kind of calculations.
Finite size effects {#finiteSize}
-------------------
With a band gap of 2.1 eV [@morrish1994canted], the heat transport in hematite is primarily described by the lattice thermal conductivity which, in turn, is determined by the phonon transport. When performing NEMD on a finite simulation box, the phonon mean free path (MFP), which is the average distance a phonon travels before being scattered, is of particular importance. Phonons with different MFPs contribute differently to the thermal transport and in some materials the MFP of the contributing phonons can take values larger than several hundred nanometers [@sellan2010size; @hermet2016lattice]. The NEMD method, as described in the previous sections, requires that two plates be defined at one quarter and three quarters of the box length to serve as heat source and sink. It would therefore be necessary to have a simulation box twice as long as the length of the largest contributing MFP. These scales are very difficult to handle in molecular dynamics simulations since the simulations would require enormous computing resources and very long execution times. We studied the effect of finite box dimensions in the direction perpendicular to the heat flux and we applied the method proposed by Schelling et al. in [@schelling2002comparison] to the finite size effects parallel to the heat flux. For the latter, several simulations with increasing box sizes $L$ are performed and their results are combined with the use of the so-called linear extrapolation procedure. As the size of the simulation box grows, more phonon mean free paths are taken into account which increases the calculated value of the thermal conductivity. As described in [@sellan2010size], within a first order approximation a linear dependence can be expected between $1/\kappa$ and $1/L$. As a consequence, $\kappa_{\infty}$ can be evaluated by plotting $1/\kappa$ against $1/L$ and extrapolating the curve to $1/L_{\infty} = 0$ with a linear function. To obtain a reasonable precision of the curve to be fitted, a large number of individual simulations has to be performed. Nevertheless, the computational effort needed is considerably lower than for a real-size simulation with $L$ larger than the largest phonon mean free path. This method has also the advantage of providing a thermal conductivity value that relies on a significant number of different simulations with different initial states, thus reducing the statistical errors that could be attributed to an individual simulation.\
Sellan et al. [@sellan2010size] mentioned a number of limitations of this method, due primarily to the first order approximation. For example, when considering a large collection of different box sizes, they reported good results for the application of the extrapolation method to Lennard-Jones argon, while an underestimation was observed in the case of Stillinger-Weber silicon. In the latter case, the calculated values and the extrapolated curve would diverge for the largest sizes. To address this issue, Hu et al. [@hu2011one] suggested a relation between the divergence and the aspect ratio of the simulation box (length/width). Studying Lennard-Jones solid argon, Lennard-Jones WSe$_2$ and graphite, their first conclusion is that a divergence can be observed at very high aspect ratios (200 to 300) for an elastically isotropic material such as LJ argon. Nevertheless these authors add that such high limits don’t have practical consequences since a converged value of $\kappa$ can be computed well below those limits. Their second conclusion is that a clear divergent behavior is observed for low aspect ratios ($\sim$ 30) for elastically anisotropic materials such as LJ WSe$_2$ and even earlier for graphite. With the smallest lateral size considered for the initial tests, we reached a maximum aspect ratio of 133. With the lateral size used in most of our simulations the largest aspect ratios range from 83 to 116. Since no divergence can be seen in the results presented hereafter, we can conclude that our material is probably elastically isotropic. Moreover, Hu et al. proposed a criterion to predict if the thermal conductivity of a material can be modeled by NEMD methods or not. This criterion is based on the ratio of the elastic constants along two crystallographic directions ($\frac{C_{xx}}{C_{zz}}$) which should be reasonably low (< 5). We computed the 6x6 matrix of the elastic constants as described in section \[sec:potential\] and obtained the following ratio: $$\frac{C_{xx}}{C_{zz}} = \frac{362}{326} \sim 1.1$$ which shows that our material is perfectly suited for NEMD methods and is not prone to the divergence issues described in [@hu2011one].
Results and discussion
======================
Size-dependent simulations {#sec:simulations}
--------------------------
As explained in the previous section, the calculation of $\kappa$ requires two steps. First a number of size-dependent simulations, then a size-independent extrapolation. Figure \[typical\] shows the typical result for an individual simulation for a given box size $L$. The time-averaged temperature profile is presented along with the position and width of the slabs where heat is added or subtracted (heat source and heat sink). In the central part of the simulation box, the temperature exhibits a linear profile which can be fitted to calculate the thermal gradient. Non-linear areas are noticeable close to the heat source and sink.
![Typical simulation result with the time-averaged temperature profile (blue dots) as a function of the position along the the z axis, parallel to the heat flux. The gray slabs are the areas corresponding to the heat sink and source. The dotted vertical lines are the limits of the linear domain where the slope of the curve is calculated with a fitting procedure.[]{data-label="typical"}](fitted_profile_big_slurm-2213.eps){width="10cm"}
In figure \[300size\] the thermal conductivity at 300 K is shown as a function of the box size in the directions both orthogonal and parallel to the heat flux. One can observe a quick convergence of $\kappa$ as a function of the lateral size, and we fixed this size to 30 Å for the rest of the study. As for the evolution of $\kappa$ as a function of $L$, the size parallel to the heat flux (horizontal axis in figure \[300size\]), it requires the application of the extrapolation method discussed in section \[finiteSize\].
![Thermal conductivity at 300 K for different lateral box sizes as a function of the box size in the direction of the heat flux.[]{data-label="300size"}](kappa_vs_lz.eps){width="10cm"}
A plot of $1/\kappa$ against $1/L$ is presented in figure \[extrapolated\] showing the calculated values and the extrapolated curve with a logscale x axis. The corresponding value of $\kappa$ is calculated to be 16 W.m$^{-1}$.K$^{-1}$ for a pure single crystal of hematite at 300 K in the \[001\] crystallographic direction. Several thermal conductivity measurements of polycrystalline hematite at room temperature are reported in the litterature. The values presented in [@diment1988thermal; @clark1966handbook; @horai1971thermal] range between 11 and 13 W.m$^{-1}$.K$^{-1}$ while Akiyama et al. report a much larger value of 17 W.m$^{-1}$.K$^{-1}$ in [@akiyama1992measurement]. However, experimental values of $\kappa$ for single-crystal hematite samples are less common. In [@clauser1995thermal], references are made to a 1974 work [@dreyer1974properties] where a value of 12.1 W.m$^{-1}$.K$^{-1}$ was reported for a single crystal in the \[001\] direction and 14.7 W.m$^{-1}$.K$^{-1}$ (22 % more) for a second direction orthogonal to \[001\].
![The inverse of the thermal conductivity as a function of the inverse of the box size with a decimal logscale x-axis (blue dots). The line is the result of the extrapolation procedure.[]{data-label="extrapolated"}](extrapolated.eps){width="10cm"}
Temperature dependence
----------------------
In order to determine the temperature dependence of $\kappa$ in the range 300 to 500 K, it is in principle necessary to apply the previously described procedure for several temperatures. But taking into account the computational cost of such a brute force method, we propose an optimized approach where a full study is conducted at 300 K and 500 K but only a partial analysis is done in between. Applying the full extrapolation procedure at the limits of the range provides a validation of its applicability. Assuming that the thermal conductivity follows the same behavior at the intermediate temperatures, we perform only a limited number of simulations at the lowest and highest box sizes for those temperatures providing thus the start and end points for the extrapolation. The results are coherent as can be observed in figure \[temperatures\] and validate the approach. We estimate that proceeding in this way required 30 to 40 % less computational time than a full study.
![Thermal conductivity as a function of box size for different temperatures along with the corresponding extrapolations (lines).[]{data-label="temperatures"}](kappa_vs_lz_temp.eps){width="10cm"}
From the infinite box size extrapolations, the temperature dependence of the thermal conductivity can be assessed. The results are presented in figure \[kappaTemp\] along with a comparison with measurements made on polycristalline hematite in [@akiyama1992measurement]. As expected, $\kappa$ decreases with the temperature in the investigated range. In this temperature range it is reasonable to think that Umklapp processes are dominant and thus $\kappa$ should decrease like 1/T: this behavior is not obvious from the calculated values of figure \[kappaTemp\]. However, we observed that the individual, size-dependent, conductivity values such as the ones shown in figure \[300size\] may vary by up to 7.5% when differences are introduced in the initial state of the simulations (e.g. random initial velocity distributions). And even though the extrapolated, size-independent, values of $\kappa$ rely on several different simulations, the lack of precision makes it difficult to assess a precise mathematical function from the curve of figure \[kappaTemp\].
![Thermal conductivity as a function of temperature. The black squares are experimental values for polycristalline hematite from [@akiyama1992measurement]. These values were fitted with a function of the form $a/T+b$ to highlight the $1/T$ dependence (green line).[]{data-label="kappaTemp"}](kappa_vs_temp_fit.eps){width="10cm"}
Spatial Anisotropy
------------------
The results presented in the previous sections were obtained for a heat flux in the \[001\] crystallographic direction. To investigate the spatial anisotropy of $\kappa$, we applied the same methods to the \[100\] direction at 300 K. Figure \[extrapolated100\] shows the evolution of $\kappa$ as a function of system size together with the extrapolated curve. A spatial anisotropy is observed for hematite since we obtained a value of 20 W.m$^{-1}$.K$^{-1}$ for $\kappa$ with the heat flux orientated in the \[100\] crystallographic direction. This is 25 % more than in the \[001\] direction which is consistent with the experimental values mentioned earlier in section \[sec:simulations\].
![Thermal conductivity in the \[100\] direction as a function of box size (blue squares). The line is the result of the extrapolation procedure.[]{data-label="extrapolated100"}](kappa_vs_lz_100.eps){width="10cm"}
Non-linearity
-------------
As can be observed in figure \[typical\], the temperature profile exhibits a nonlinear behavior near the source and the sink. This behavior has been partially explained by phonon scattering in previous studies [@schelling2002comparison; @sellan2010size]. In order to calculate the temperature gradient and apply Fourier’s law, it is thus necessary to consider the profile far enough from those non-linear areas. Our experience led us to choose an “exclusion” distance equal to the width $a$ of the temperature bins (heat source and sink included). Indeed, we found that this specific choice was appropriate in most of the simulations performed in this work, independently of the actual value of $a$. Moreover, investigating the evolution of the non-linearity as a function of time, we find that in the case of long enough simulations this non-linearity decreases significantly even after the system is considered to have reached the steady state. Figure \[prof\_vs\_temp\] shows the evolution of the time-averaged temperature profile in a 140 nm long simulation box made of a 6x6x105 supercell (113000 atoms) during simulations lasting up to 7.8 ns.
![Evolution of the time-averaged temperature profile for a 140 nm long simulation box made of 3780 hematite unit cells (113000 atoms) during an 7.8 ns long simulation. The yellow-colored areas between the temperature curve and the linear regression are computed to quantify the non-linearity.[]{data-label="prof_vs_temp"}](temp_profile_vs_time.eps){width="16cm"}
The area between the temperature curve and the linear regression (colored area in figure \[prof\_vs\_temp\]) was calculated to quantify the non-linearity. Figure \[nonLinear\] shows the evolution of this quantity, starting after the stationarization. The value peaks around 3 ns and decreases afterwards, until a minimum value is reached. We have observed this evolution with time for different system sizes. Thus it appears that the non-linearity of the temperature gradient close to the source and the sink can partly be characterized as a slow transient phenomenon.
![Size of the non-linearity areas as a function of time from the end of the stationarization up to 8 ns.[]{data-label="nonLinear"}](non-linearity_vs_time.eps){width="10cm"}
Conclusion
----------
A detailed methodology has been presented for the determination of the thermal conductivity of crystals and applied to pure single-crystalline hematite. Calculated values for different temperatures are in reasonable agreement with available experimental data on polycrystals with values ranging from 16 to 10 W.m$^{-1}$.K$^{-1}$ between 300 and 500 K. Moreover, an investigation of the spatial anisotropy has been undertaken and shows, at 300 K, a 25% increase of the thermal conductivity in the \[100\] direction with respect to the \[001\] direction in agreement with measurements on single crystals. Finally, some specific elements of the calculation procedure, such as the width of the temperature bins or the nature of the nonlinear behavior, have been analyzed highlighting new aspects in the application of the NEMD scheme for the determination of the thermal conductivity of crystalline solids.
This work was granted access to the HPC resources of CINES under the allocation 2016-c2016097598 made by GENCI. We gratefully acknowledge Total S.A. and Total M.S. for their financial support and we thank Sophie Loehlé and Vincent Lacour for fruitful discussions.
[^1]: calculated values from [@pedone2006self]
[^2]: this work
|
{
"pile_set_name": "ArXiv"
}
|
---
address: ' Canadian Institute for Advanced Research and Physics Department, University of British Columbia, Vancouver, BC, V6T 1Z1, Canada '
author:
- Ian Affleck
title: 'Conformal Field Theory Approach to the Kondo Effect[@Zakopane]'
---
OUTLINE
.5cm I. Renormalization Group and Fermi Liquid Approaches to the Kondo Effect
A\) Introduction to The Kondo Effect
B\) Renormalization Group Approach
C\) Mapping to a One Dimensional Model
D\) Fermi Liquid Approach at Low T\
\
II. Conformal Field Theory (“Luttinger Liquid”) Techniques: Separation of Charge and Spin Degrees of Freedom, Current Algebra, “Gluing Conditions”, Finite-Size Spectrum\
\
III. Conformal Field Theory Approach to the Kondo Effect: “Completing the Square”
A\) Leading Irrelevant Operator, Specific Heat, Susceptibility, Wilson Ratio, Resistivity at $T>0$\
\
IV. Introduction to the Multi-Channel Kondo Effect: Underscreening and Overscreening
A\) Large-k Limit
B\) Current Algebra Approach\
\
V. Boundary Conformal Field Theory\
\
VI. Boundary Conformal Field Theory Results on the Multi-Channel Kondo Effect:
A\) Fusion and the Finite-Size Spectrum
B\) Impurity Entropy
C\) Boundary Green’s Functions: Two-Point Functions, T=0 Resistivity
D\) Four-Point Boundary Green’s Functions, Spin-Density Green’s Function
E\) Boundary Operator Content and Leading Irrelevant Operator:
Specific Heat, Susceptibility, Wilson Ratio, Resistivity at $T>0$
Renormalization Group and Fermi Liquid Approaches to the Kondo Effect
=====================================================================
Introduction to the Kondo Effect
--------------------------------
Most mechanisms contributing to the resistivity of metals, $\rho(T)$, give either $\rho(T)$ decreasing to $0 $, as $T\rightarrow 0$ (phonons or electron-electron interactions), or $\rho(T)\rightarrow$ constant, as $T\rightarrow 0$ (non-magnetric impurities). However, metals containing magnetic impurities show a $\rho(T)$ which increases as $T \rightarrow 0$. This was explained by Kondo[@Kondo] in 1964 using a simple Hamiltonian: $$H=\sum_{\vec{k}\alpha}\psi^{\dagger\alpha}_{\vec{k}}
\psi_{\vec{k}\alpha}\epsilon(k)+\lambda\vec{S}\cdot\sum_{\vec{k}\vec{k'}}
\psi^{\dagger}_{\vec k} \frac{\vec{\sigma}}{2}\psi_{\vec{k'}}$$ where $\psi_{\vec{k}\alpha}$’s are conduction electron annihilation operators, (of momentum $\vec{k}$, spin $\alpha$) and $\vec{S}$ represents the spin of the magnetic impurity with $$[S^a, S^b]=i\epsilon^{abc}S^c.$$ The interaction term represents an impurity spin interacting with the electron spin at $\vec{x}=0$.
With the above Hamiltonian, the Born approximation gives: $\rho(T)\sim\lambda^2$, independent of $T$. The next order term has a divergent coefficient at $T=0$: $$\rho(T)\sim[\lambda+\nu\lambda^2 \ln \frac{D}{T}+...]^2$$ Here $D$ is the band-width, $\nu$ the density of states. This result stimulated an enormous amount of theoretical work. As Nozières put it, “Theorists ‘diverged’ on their own, leaving the experiment realities way behind”.[@Nozieres1] What happens at low $T$, i.e. $T \sim
T_K=De^{-\frac{1}{\upsilon\lambda}}$? In that case the $O(\lambda^2)$ term will be as big as the term of $O(\lambda)$. What about the $O(\lambda^3)$ term? Such questions helped lead to the development of the renormalization group needed to understand the problem.
In particle physics such a growth of a coupling constant at low energies explains quark confinement (1973) and “asymptotic freedom" at $E\rightarrow\infty$. To solve these problems, Wilson[@Wilson] developed a very powerful numerical renormalization group approach. The Kondo model was also “solved" by the Bethe ansatz[@Andrei; @Weigmann] which gives the specific heat and magnetization. Nozières,[@Nozieres1; @Nozieres2] following ideas of Anderson[@Anderson] and Wilson,[@Wilson] developed a very simple, and in a sense exact, picture of the low $T$ behaviour. With A. Ludwig, I have generalized and reformulated Nozières’ approach[@Affleck1; @Affleck2; @Affleck3; @Affleck4; @Ludwig1; @Affleck5; @Affleck6; @Ludwig2] using recent results in conformal field theory. The latter approach is very general and can be applied to a number of other problems including multi-channel and higher spin Kondo effect, [@Affleck1; @Affleck2; @Affleck3; @Affleck4; @Ludwig1; @Affleck5; @Affleck6; @Ludwig2] two (or more)-impurity Kondo effects,[@Affleck7; @Affleck8] impurity assisted tunneling,[@Ralph1] impurities in one-dimensional conductors (“quantum wires")[@Wong] or 1D antiferromagnets,[@Eggert; @Sorensen] baryon-monopole interaction,[@Affleck9]. Some of these problems, including the multi-channel Kondo effect, exhibit non-Fermi liquid behaviour. These are among the very few exactly solved problems that do this (the others are 1D Luttinger liquids). It has been suggested that this may be connected with exotic behaviour of certain compounds, including high-$T_c$ superconductors.[@Cox; @Ruckenstein]
Renormalization Group
---------------------
We could integrate out $\psi(k)$ for $k$ far from $k_{F}$, the Fermi wave-vector, and successively reduce the band-width $D$ to obtain a new effective interaction. \[See Figure (\[fig:red\]).\] This is hard to do exactly. At weak coupling one can do it perturbatively in $\lambda$. With the simplest approach, real-time, time-ordered perturbation theory, we expand $$T\exp
\left[-i\lambda\int\vec{S}(t)\cdot\psi^\dagger\frac{\vec{\sigma}}{2}
\psi(\vec{0},t)\right],$$ where the fields are in the interaction picture.
=10 cm
As $\vec{S}(t)$ is independent of $t$, we simply multiply powers of $\vec{S}$ using $$[S^a,S^b]=i\epsilon^{abc}S^c,~~~~\vec{S}^2=s(s+1).$$ We must time-order $\vec{S}$’s which don’t commute. The first few diagrams are shown in Figure (\[fig:pert\]). In 2nd order in $\lambda$, we have: $$-\frac{\lambda^2}{2}\int dt\ dt'T(S^a(t)S^b(t'))\cdot
T[\psi^\dagger (t)\frac{\sigma^a}{2}\psi (t)\psi^\dagger
(t')\frac{\sigma^b}{2} \psi (t')],$$\
which can be reduced, using Wick’s theorem, to: $$\begin{aligned}
&&-\frac{1}{2}\lambda^2\int
dt\ dt'\psi^\dagger \left[
\frac{\sigma^a}{2},\frac{\sigma^b}{2}\right] \psi
T\langle\psi(t)\psi^\dagger (t')\rangle (\theta(t-t')S^a
S^b+\theta(t'-t)S^b S^a)\nonumber\\ =&&\frac{\lambda^2}{2}\int dt\
dt'\psi^\dagger \frac{\vec{\sigma}}{2}\psi\cdot\vec{S}\
\hbox{sn}(t-t') \langle\psi(t)\psi^\dagger(t')\rangle
,\end{aligned}$$ where sn $(t-t')$ is the sign-function which arises from $T$-ordering spins.
=10 cm
We see that the integral $$\int dt
\epsilon(t)G(t)=-i\int\frac{dt}{|t|}$$ is divergent in the infrared limit: $t\rightarrow\infty$, where $G(t)=\langle\psi(t)\psi^\dagger(0)\rangle.$ But we only integrate out electrons with $D'<k<D$ which gives $\ln D/D'$. To do it explicitly we use the Fourier transformed form: $$\begin{aligned}
&&\int\frac{d^3k}{(2\pi)^3}\int\frac{d\omega }{2\pi}\left[
\frac{1}{i\omega +\delta} + \frac{1}{i\omega -\delta}\right]
\frac{i}{\omega-\epsilon_k+i\delta \hbox{sn}(\epsilon_k)}\\ &&
=\int\frac{d^3\vec{k}}{(2\pi)^3}\frac{1}{|\epsilon_k|}\approx
2\nu\int ^D_{D'}\frac{d\epsilon}{\epsilon}=2\nu\ln\frac{D}{D'}
.\end{aligned}$$ Thus $$\delta\lambda=\nu\lambda^2 \ln
\frac{D}{D'},$$ and $$\frac{d\lambda}{d\ln D}=-\nu\lambda^2.$$ We see that lowering the band cut-off increases $\lambda$ or, defining a length-dependent cut-off, $l\sim v_F/D$, $$\frac{d\lambda}{d\ln l}=\nu\lambda^2.$$ Integrating the equation (equivalent to performing an infinite sum of diagrams), gives: $$\lambda_{\hbox{eff}}(D)=\frac{\lambda_0}{1-\nu\lambda_0
\ln \frac{D_0}{D}}.$$ If $\lambda_0>0$ (antiferromagnetic), then $\lambda_{\hbox{eff}}(D)$ diverges at $D\sim T_k \sim D_0e^{-\frac{1}{\nu\lambda_0}}$, If $\lambda_0<0$ (ferromagnetic), $\lambda_{\hbox{eff}}(D)\rightarrow 0$. See Figure (\[fig:flowex\]).
=10 cm
The behaviour at temperature $T$ is determined by $\lambda_{\hbox{eff}}(T)$: $\rho(T)\rightarrow 0$ as $T\rightarrow
0$ for the ferromagnetic case. What happens for the antiferromagnetic case?
Mapping to a One-Dimensional Model
----------------------------------
The above discussion can be simplified if we map the model into a one dimensional one. We assume a spherically symmetric $\epsilon(\vec k)$, $$\epsilon(k)=\frac{k^2}{2m}-\epsilon_F\approx v_F(k-k_F),$$ and a $\delta-$function Kondo interaction. There is only s-wave scattering, i.e. $$\begin{aligned}
\psi(\vec{k})&=&\frac{1}{\sqrt{4\pi}k}\psi_0(k)+\mbox{higher
harmonics}, \nonumber\\ H_0&=&\int dk \psi_{0 k}^\dagger\psi_{0
k}\epsilon(k)+ \mbox{higher harmonics}, \nonumber\\
H_{\hbox{INT}}&=&\lambda v_F\nu\int dkdk'\psi_{0,k}^\dagger{\vec
\sigma \over 2}\psi_{0,k'} \cdot \vec S, \end{aligned}$$ where $\nu=k_F^2/2\pi^2v_F$ is the density of states per spin. This can also be written in terms of radial co-ordinate. We eliminate all modes except for a band width 2D: $|k-k_F|<D$. Defining left and right movers (incoming and outgoing waves), $$\Psi_{L,R}(r)\equiv\int^\wedge_{-\wedge} dke^{\pm
ikr}~~\psi_0(k+k_F), ~~~\Rightarrow~~~ \psi_L(0)=\psi_R(0),$$ we have $$\begin{aligned}
H_0&=&\frac{v_F}{2\pi}\int^{\infty}_0 dr(\psi^\dagger
_Li\frac{d}{dr}\psi_L-\psi^ \dagger _Ri \frac{d}{dr}\psi_R)~~~
\mbox{(note the unconventional normalization)},\nonumber\\
H_{\hbox{INT}}&=&v_F\lambda\psi_L(0)^\dagger
\frac{\vec{\sigma}}{2}\psi_L(0)\cdot\vec{S}. \label{HINT}\end{aligned}$$ Here we have redefined a dimensionless Kondo coupling, $\lambda \rightarrow \lambda \nu$. Using the notation $$\psi_L=\psi_L(x,\tau)=\psi_L(z=\tau+ix),~~
\psi_R(x,\tau)=\psi_R(z^*=\tau-ix) ,$$ where $\tau$ is imaginary time and $x=r$, (and we set $v_F=1$) we have $$\langle\psi_L(z)\psi_{L}^+(0)\rangle=\frac{1}{z},~~
\langle\psi_R(z^*)\psi_R^\dagger(0)\rangle=\frac{1}{z^*}.$$ Alternatively, since $$\psi_L(0,\tau)=\psi_R(0,\tau)~~~~~\psi_L=\psi_L(z),~~\psi_R=\psi_R(z^*),$$ we may consider $\psi_R$ to be the continuation of $\psi_L$ to the negative $r$-axis: $$\psi_R(x,
\tau)\equiv\psi_L(-x,\tau).$$ Now we obtain a relativistic $(1+1)$ dimensional field theory ( a “chiral” one, containing left-movers only) interacting with the impurity at $x=0$ with $$H_0={v_F\over
2\pi}\int_{-\infty}^\infty dx\psi^\dagger_Li {d \over
dx}\psi_L\label{Hochi}$$ and $H_{\hbox{INT}}$ as in Eq. (\[HINT\]). See Figure (\[fig:1DLR\]).
=10 cm
Fermi Liquid Approach at Low T
------------------------------
What is the $T\rightarrow 0 $ behavior of the antiferromagetic Kondo model? The simplest assumption is $\lambda_{\hbox{eff}}\rightarrow\infty$. But what does that really mean? Consider the strong coupling limit of a lattice model,[@Nozieres1] for convenience, in spatial dimension $D=1$. ($D$ doesn’t really matter since we can always reduce the model to $D=1$.) $$H=t\sum_{i}(\psi^\dagger_i\psi_{i+1}+\psi^\dagger_{i+1}
\psi_i)+\lambda\vec{S}\cdot \psi^\dagger_0
\frac{\vec\sigma}{2}\psi_0\label{lattice}$$ Consider the limit $\lambda>>|t|$. The groundstate of the interaction term will be the following configuration: one electron at the site $0$ forms a singlet with the impurity: $|\Uparrow\downarrow\rangle-|\Downarrow\uparrow\rangle$. (We assume $S_{IMP}=1/2$). Now we do perturbation theory in $t$. We have the following low energy states: an arbitary electron configuration occurs on all other sites-but other electrons or holes are forbidden to enter the site-$0$, since that would destroy the singlet state, costing an energy, $\Delta E \sim \lambda >>t$. Thus we simply form free electron Bloch states with the boundary condition $\phi(0)=0$, where $\phi(i)$ is the single-electron wave-function. Note that at zero Kondo coupling, the parity even single particle wave-functions are of the form $\phi (i)=\cos ki$ and the parity odd ones are of the form $\phi (i)=\sin ki$. On the other hand, at $\lambda\rightarrow \infty$ the parity even wave-functions become $\phi (i)=|\sin ki|$, while the parity odd ones are unaffected.
The behaviour of the parity even channel corresponds to a $\pi/2$ phase shift in the s-wave channel. $$\phi_j\sim
e^{-ik|j|}+e^{+2i\delta}e^{ik|j|}, ~~\delta=\pi/2.$$ In terms of left and right movers on $r>0$ we have changed the boundary condition, $$\begin{aligned}
\psi_L(0)&=&\psi_R(0),
{}~~~~\lambda=0, \nonumber\\ \psi_L(0)&=&-\psi_R(0),~~~~
\lambda=\infty. \end{aligned}$$ The strong coupling fixed point is the same as the weak coupling fixed point except for a change in boundary conditions (and the removal of the impurity). In terms of the left-moving description of the $P$-even sector, the phase of the left-mover is shifted by $\pi$ as it passes the origin. Imposing another boundary condition a distance $l$ away quantizes $k$: $$\begin{aligned}
\psi(l)=\psi_L(l)+\psi_R(l)=\psi_L(l)+\psi_L(-l)&=&0,\nonumber\\
\lambda&=&0:~~~~~~k=\frac{\pi}{l}(n+1/2)\nonumber\\
\lambda&=&\infty :~~~~~~k=\frac{\pi n}{l} \end{aligned}$$
Near the Fermi surface the energies are linearly spaced. Assuming particle-hole symmetry, the Fermi energy lies midway between levels or on a level. \[See Figures (\[fig:anti\]) and (\[fig:per\]).\] The two situations switch with the phase shift. Wilson’s numerical RG scheme[@Wilson] involves calculating the low-lying spectrum numerically and looking for this shift. This indicates that $\lambda$ renormalizes to $\infty$ even if it is initially small. However, now we expect the screening to take place over a longer length scale $$\xi
\sim\frac{v_F}{T_K}\sim\frac{v_F}{D}e^{1/\nu\lambda}
.$$ In other words, the wave function of the screening electron has this scale. We get low energy Bloch states of free electrons only for $|k-k_F|<<1/\xi$ (so we must take $l>>\xi$). \[See Figure (\[fig:cloud\]).\] The free electron theory with a phase shift corresponds to a universal stable low energy fixed point for the Kondo problem. This observation determines the $T=0$ resistivity for an array of Kondo impurities at random locations of low density $n_i$. It is the same as for non-magnetic s-wave scatterers with a $\pi /2$ phase shift at the Fermi energy. $\delta = \pi /2$ gives the so-called unitary limit resistivity: $$\rho_{\hbox{u}} = {3n_i\over \pi\nu^2
v_F^2e^2}.$$
=10 cm
=10 cm
=10 cm
The low-$T$ behaviour, so far, seems trivial. Much of the interesting behaviour comes from the leading irrelevant operator. The impurity spin has disappeared (screened) from the description of the low -$T$ physics. However certain interactions between electrons are generated (at the impurity site only) in the process of eliminating the impurity spin. We can determine these by simply writing the lowest dimension operators allowed by symmetry.
It is simplest to work in the 1D formulation, with left-movers only. We write the interaction in terms of $\psi_L$, obeying the new boundary condition (but the impurity spin). The dimension of the operator is determined as in 1D field theory $$H=\int dx\psi^\dagger_L i\frac{d}{dx}\psi_L +....$$ The length and time dimensions are equivalent (we convert with $v_F$), $$[H]=E\Rightarrow[\psi]=E^{\frac{1}{2}}.$$ The interactions are local $$\delta H=\sum_{i}\lambda_iO_i(x=0),
{}~~~~[\lambda_i]+[O_i]=1.$$ So $\lambda_i$ has negative energy dimension if $[O_i]>1$, implying that it is irrelevant. In RG theory one usually defines a dimensionless coupling constant by multiplying powers of the cut-off $D$, if $$[\lambda_i]=E^{-a},
{}~~~~\tilde \lambda_i\equiv \lambda_iD^a,$$ $\tilde \lambda_i$ decreases as we lower $D$: $$\frac{d\tilde
\lambda_i}{dlnD}=a\tilde \lambda_i.$$ Such a coupling, with $a>0$, produces no infrared divergences in perturbation theory. The ultraviolet ones are cancelled by the expicit factors of the ultraviolet cut-off, $D$, appearing in the Lagrangian, $\tilde \lambda O/D^a$. What are the lowest dimension operators allowed by symmetry? Consider $\psi^{+\alpha}(0)\psi_\alpha(0)$. This has $d=1$. However, it is not allowed because it breaks particle-hole symmetry. If particle-hole symmetry is broken then we do get this, a potential scattering term; it adds a term to the phase shift. Consider another term, $$i\psi^{\dagger
\alpha}\frac{d}{dx}\psi_\alpha (0)- i{d\over dx}\psi^{\dagger
\alpha}\psi_\alpha (0).$$ This has d=2 . This term produces a $k$-dependent phase shift. The only other term with $d\leq 2$ is $\psi^{\dagger \uparrow}\psi_{\uparrow}\psi^{\dagger
\downarrow}\psi_{\downarrow} $. This term represents the electron-electron interaction induced by an impurity spin-flip. The first electron flips the impurity spin. This makes it possible for the second electron to flip it back if the electron spin is correct. These are the only $d\leq 2$ operators. There are no relevant ($d\leq 1$) operators, implying the stability of the low energy fixed point. Note that, by contrast, the high energy, zero Kondo coupling, fixed point is unstable. The dimension 1 operator (the Kondo interaction) can occur there because of the presence of the impurity spin.
We can’t calculate these two coupling constants exactly except by using complicated methods: Wilson’s numerical melthod or the Bethe ansatz. They both have dimension $E^{-1}$. We expect them to be $O[1/{T_K}]$ by a standard scaling argument. That is, functions of the cut-off $D$ and coupling constant $\lambda$ can be replaced by functions of the reduced cut-off, $D'$ and the renormalized coupling constant, $\lambda_{\hbox{eff}}(D')$: $f[D,\lambda]=f[D',\lambda_{\hbox{eff}}(D')]$. We can lower the cut-off down to $T_K$ where $\lambda_{\hbox{eff}}$ is $O(1)$ so $f=f(T_K,1)=f(T_K)$. This is a characteristic scale introduced by the infrared divergences of perturbation theory. For $T_K<<D
(\lambda<<1)$, Nozières argued that the two irrelevant coupling constants have a universal ratio. So there is only one unknown parameter (“the Wilson number”). Essentially all low-temperature information is given by this irrelevant coupling constant, if it was not already determined by the $\pi/2$ phase shift at $k_F$. I will give a different derivation of the ratio of the two coupling constants later using conformal field theory. We now simply do perturbation theory in the irrelevant coupling constant $\sim1/{T_K}$. We can determine powers of $T$ by dimensional analysis. For the specific heat we find: $$C \sim
\frac{\pi}{3v_F}lT+a\frac{T}{T_K}.$$ This is the specific heat for the one-dimensional system with a single impurity at the origin. Note that the first term is simply the specific heat of the free system, proportional to system length. The second term is independent of length and is the impurity specific heat. It is the result of first order perturbation theory in the irrelevant coupling constant, of $O(1/T_K)$. The (linear) power of $T$ can be fixed by dimensional analysis; $a$ is a pure number. Note that while this is formally an “irrelevant” contribution, it in fact gives the leading impurity specific heat at low $T$. To obtain the specific heat for the three dimensional system we simply multiply the first term by the ratio $\nu V/(l/2\pi v_F)$ i.e. the ratio of densities of states per unit energy. For a dilute random array the last term gets multiplied by the number of impurities. At high T we get, approximately, the entropy for a decoupled s=1/2 impurity: $$S(T) = {\pi l\over 3v_F}T + \ln 2.$$ At low $T$, the impurity entropy decreases to 0: $$S(T) = {\pi l\over 3v_F}T+ {aT\over T_K}.$$ In general we may write: $$S(T) - {\pi l\over 3v_F}T \equiv
S_{\hbox{imp}}= g(T/T_K),$$ where $g$ is a scaling function which is universal for weak bare coupling. See Figure (\[fig:entropy\]). The behaviour of $g(x)$ for small arguments is determined by RG-improved weak coupling perturbation theory. It’s behaviour at low T is determined from the theory of the low energy fixed point. Its behaviour at arbitrary $T/T_K$ is a property of the universal crossover between fixed points. It has been found from the Bethe ansatz.
=10 cm
Similarly, the susceptibility, at $T=0$, is given by: $$\chi \sim \frac{l}{2\pi v_F}+ \frac{b}{T_K}.$$ The ratio $b/a$ is universal since the coupling constant ($1/T_K$) drops out. This is known as the Wilson Ratio.
At high $T$, we must (for weak bare coupling) obtain approximately the results for a free spin: $$\chi \sim {l\over
2\pi v_F} + {1\over 4T}.$$ At lower $T$, using RG improved perturbation theory this becomes: $$\chi
\sim {l\over 2\pi v_F} + {1\over 4T}\left[1-{1\over \ln (T/T_K)}+
...\right].$$ In general, we may write: $$\chi - {l\over 2\pi v_F} \equiv \chi_{\hbox{imp}}=
{1\over T}f(T/T_K),$$ where $f(T/T_K)$ is another universal scaling function. See Figure (\[fig:chi\]).
=10 cm
The temperature dependent part of the low $T$ resistivity for the dilute random array is $2nd$ order in perturbation theory, $$\rho=\rho_{\hbox{u}}[1-d\left(
\frac{T}{T_K}\right)^2],$$ where $d$ is another dimensionless constant. The second term comes from second order perturbation theory in the irrelevant coupling constant. Another universal ratio can be formed. I will discuss this in Sec. III, in some detail, using the CFT approach.
We now expect a scaling behaviour: $$\rho (T) = n_if(T/T_K),$$ sketched in Figure (\[fig:resex\]).
=10 cm
Conformal Field Theory Techniques
=================================
It is very useful to bosonize free fermions to understand the Kondo effect. This allows separation of spin and charge degrees of freedom which greatly simplifies the problem.
We start by considering a left-moving spinless fermion field with Hamiltonian density: $${\cal
H}=\frac{1}{2\pi}\psi_L^{\dagger}i\frac{d}{dx}\psi_L.$$
Define the current (=density) operator, $$\begin{aligned}
J_L(x-t)
&=&:\psi_L^+\psi_L:(x,t)\nonumber\\
&=&\lim_{\epsilon\rightarrow 0}[\psi_L(x)\psi_L
(x+\epsilon)-\langle 0|\psi_L(x)\psi_L(x+\epsilon)|0\rangle ]\end{aligned}$$ (Henceforth we generally drop the subscripts “L”.) We will reformulate the theory in terms of currents (key to bosonization). Consider: $$\begin{aligned}
&&J(x)~~J(x+\epsilon)~~~~~\mbox{as}~~~~\epsilon\rightarrow
0\nonumber\\ =&& :\psi^\dagger (x)\psi (x)\psi^\dagger (x+\epsilon
)\psi (x+\epsilon ):\nonumber \\ +&&[: \psi^\dagger
(x)\psi(x+\epsilon):+:\psi (x)\psi^\dagger
(x+\epsilon):]G(\epsilon)+ G(\epsilon)^2\nonumber\\ G(\epsilon)= &&
\langle 0|\psi(x)\psi^\dagger (x+\epsilon)|0\rangle
=\frac{1}{-i\epsilon} .\end{aligned}$$ By Fermi statistics the 4-Fermi term vanishes as $\epsilon\rightarrow 0$ $$:\psi^\dagger (x)\psi(x)\psi^\dagger (x)\psi(x):\ =-:\psi^\dagger
(x)\psi^\dagger (x)\psi(x)\psi(x):\ =0 .$$ The second term becomes a derivative, $$\begin{aligned}
\lim_{\epsilon
\rightarrow 0}[J(x)J(x+\epsilon )+\frac{1}{\epsilon^2}]
&=&\lim_{\epsilon\rightarrow 0}\frac{1}{-i\epsilon}[:\psi^\dagger
(x)\psi(x+\epsilon):- :\psi^\dagger
(x+\epsilon)~~\psi(x):]\nonumber\\ &=&2i:\psi^\dagger
\frac{d}{dx}\psi:\nonumber\\ {\cal H}&=&\frac{1}{4\pi}
J(x)^2+\mbox{constant} .\end{aligned}$$ Now consider the commutator, $[J(x),J(y)]$. The quartic and quadratic terms cancel. We must be careful about the divergent c-number part, $$\begin{aligned}
[J(x),J(y)]&=&
-\frac{1}{(x-y-i\delta)^2}+\frac{1}{(x-y+i\delta)^2}\ \
(\delta\rightarrow 0^+)\nonumber\\ &=&\frac{d}{dx}\left[
\frac{1}{x-y-i\delta}-\frac{1}{x-y+i\delta}\right] \nonumber\\
&=&2\pi i \frac{d}{dx} \delta(x-y) \end{aligned}$$
Now consider the free massless boson theory with Hamiltonian density (setting $v_F=1$): $${\cal
H}=\frac{1}{2}\left( \frac{\partial\phi}{\partial
t}\right)^2+\frac{1}{2}\left( \frac{\partial\phi}{\partial
x}\right)^2,
~~~~[\phi(x), \frac{\partial}{\partial t}\phi (y)]=i\delta(x-y)$$
We can again decompose it into the left and right-moving parts, $$\begin{aligned}
({\partial_t}^2-{\partial_x}^2)\phi&=&
(\partial_t+\partial_x)(\partial_t-\partial_x)\phi \nonumber\\
\phi(x,t)&=&\phi_L(x+t)+\phi_R(x-t)\nonumber\\
(\partial_t-\partial_x)\phi_L&\equiv&\partial_-\phi_L=0,
{}~~\partial_+\phi_R=0\nonumber\\
H&=&\frac{1}{4}(\partial_-\phi)^2+\frac{1}{4}(\partial_+\phi)^2=
\frac{1}{4}(\partial_-\phi_R)^2+\frac{1}{4}(\partial_+\phi_L)^2\end{aligned}$$
Consider the Hamiltonian density for a left-moving boson field: $$\begin{aligned}
{\cal H}&=&\frac{1}{4}(\partial_+\phi_L)^2
\nonumber\\ {[}\partial_+\phi_L(x), \partial_+\phi_L(y)]&=&
[\dot\phi+\phi',\dot\phi+\phi']=2i\frac{d}{dx}\delta(x-y)\end{aligned}$$
Comparing to the Fermionic case, we see that: $$J_L=\sqrt{\pi}\partial_+\phi_L=\sqrt{\pi}\partial_+\phi,$$ since the commutation relations and Hamiltonian are the same. That means the operators are the same with appropriate boundary conditions.
Let’s compare the spectra. For the Fermionic case, choose boundary condition: $$\psi(l)=-\psi(-l)~~~~(i.e.~~\psi_L(l)+\psi_R(l)=0),~~~~
k=\frac{\pi}{l}(n+\frac{1}{2}),~~ n=0,\pm 1,\pm 2...$$ \[See Figure (\[fig:anti\]). Note that we have shifted $k$ by $k_F$.\] Consider the minimum energy state of charge $Q$ (relative to the ground state). See Figure (\[fig:Q\]). We have the single Fermion energy: $$E=v_Fk,$$ so:$$E(Q)=v_F\frac{\pi}{l}\sum^{Q-1}_{n=0}(n+\frac{1}{2})=\frac{v_F
\pi}{2l}Q^2 .\label{E(Q)}$$ Now consider particle hole excitations relative to the $Q$-ground state: The most general particle-hole excitation is obtained by raising $n_m$ electrons by $m$ levels, then $n_{m-1}$ electrons by $m-1$ levels, etc. \[See Figure (\[fig:ph\]).\] $$E=\frac{\pi v_F}{l}(\frac{1}{2}Q^2+\sum^\infty_{m=1}n_m\cdot m)
\label{1spectrum}$$
=10 cm
=10 cm
Now consider the bosonic spectrum. What are the boundary conditions? Try the periodic one, $$\phi(l)=\phi(-l)\Rightarrow k=\frac{\pi m}{l}$$ The $m^{th}$ single particle level has $E_m=v_Fk_m$. The total energy is $$E=\frac{\pi v_F}{l}(\sum^{\infty}_1 n_m \cdot
m),~~~~ n_m= \mbox{occupation~~number}:~0,1,2,...$$ Where does the $Q^2$ term in Eq. (\[1spectrum\]) come from? We need more general boundary condition on the boson field. Let $\phi$ be an angular variable: $$\begin{aligned}
&&\phi_L(-l)=\phi_L(l)+\sqrt{\pi}Q,~~~~~ Q=0,\pm
1,\pm2,...\nonumber\\ \Rightarrow
&&\phi_L(x+t)=\frac{\sqrt{\pi}}{2}\frac{Q}{l}\cdot(x+t)+\sum^\infty_{m=1}
{_\frac{1}{\sqrt{4\pi m}} (e^{-i\frac{\pi m}{l}(x+t)} a_m+h.c.)}
,\end{aligned}$$ where $a_n$’s are the annihilation operators and $Q$ is the winding number, $$E=\int^l_{-l}
dx[\frac{1}{2}\left( \frac{\partial\phi}{\partial
t}\right)^2+\frac{1}{2}\left( \frac{\partial\phi}{\partial
x}\right)^2]=\frac{\pi}{l}[\frac{1}{2}Q^2+...].$$ Here we have set $v_F=1$. We have the following correspondence:\
soliton $\leftrightarrow$ electron, oscillator $\leftrightarrow$ particle-hole pair.\
It is also possible to represent fermion operators in terms of the boson, $$\psi_L\sim
e^{i\sqrt{4\pi}\phi_L},$$ which gives the correct Green’s function and implies the same angular definition of $\phi_L$.
For the Kondo effect we are also interested in the phase-shifted boundary condition: \[See Figure (\[fig:per\]).\] $$\begin{aligned}
\psi_L(l)&=& + \psi_L(-l),~~~~~~~~~k=\frac{\pi}{l}n,~~~ \mbox{(for
fermions)} \nonumber\\ E&=&\frac{\pi v_F}{l}\left[
\frac{Q(Q-1)}{2}+\sum^\infty_1 n_m m\right] . \end{aligned}$$ We have the degenerate ground state, $Q=0$ or $1$, which correspond to an anti-periodic boundary condition on $\phi$, $$\begin{aligned}
\phi(l)&=&\phi(-l)+\sqrt{\pi}(Q-\frac{1}{2})\nonumber\\
E&=&\frac{\pi}{l}\frac{1}{2}(Q-\frac{1}{2})^2+...=\frac{\pi}{l}
(\frac{1}{2}Q(Q-1)+\mbox{const.} +...) \end{aligned}$$
Now we include spin, i.e. we have $2$-component electrons, $$H_0=iv_F\psi^{\alpha
\dagger}\frac{d}{dx}\psi_\alpha,~~~~~~~(\alpha=1,2,~~~\mbox{summed}).$$ Now we have charge and spin currents (or densities). We can write H in a manifestly $SU(2)$ invariant way, quadratic in charge and spin currents: $$J=:\psi^{\alpha\dagger}\psi_{\alpha}:~~,~~~~\vec{J}=\psi^{\dagger\alpha}
\frac{\vec{\sigma}_\alpha^\beta}{2}\psi_{\beta}$$ Using: $$\begin{aligned}
\vec{\sigma}_\alpha^\beta\cdot\vec{\sigma}_\gamma^{\delta}
&=&2\delta^{\beta}_{\gamma}\delta^{\delta}_{\alpha}-
\delta^\alpha_\beta\delta^\gamma_\delta\\
{\vec{J}}^2&=&-\frac{3}{4}:\psi^{\dagger\alpha}\psi_{\alpha}
\psi^{\dagger\beta}\psi_{\beta}:+\frac{3i}{2}\psi^{\alpha
+}\frac{d}{dx}\psi_{\alpha}+c\mbox{-number},\nonumber\\
J^2&=&:\psi^{\dagger
\alpha}\psi_{\alpha}\psi^{\dagger\beta}\psi_{\beta}:+2i\psi^{\alpha
+}\frac{d}{dx}\psi_{\alpha}+ c\mbox{-number},\nonumber\\ {\cal
H}&=&\frac{1}{8\pi}J^{2}+\frac{1}{6\pi}{\vec{J}}^2, \end{aligned}$$ we have the following commutation relations, $$\begin{aligned}
[J(x),J(y)]&=&4\pi i\delta '(x-y), \mbox{(twice the result for the
spinless case)} \nonumber\\
{[}J(x),J^z(y)]&=&\frac{1}{2}[J_{\uparrow}+J_{\downarrow},J_{\uparrow}-
J_{\downarrow}]=0 .\end{aligned}$$ From $[J,\vec{J}]=0$, we see that $H$ is sum of commuting charge and spin parts. $$\begin{aligned}
[J^a(x),J^b(y)]&=&2\pi\psi^\dagger
[{\frac{\sigma}{2}}^a,{\frac{\sigma}{2}}^b]
{\psi}\cdot\delta(x-y)+tr[{\frac{\sigma}{2}}^a,{\frac{\sigma}{2}}^b]2\pi
i \frac{d}{dx}\delta(x-y)\nonumber\\ &=&2\pi i
\epsilon^{abc}J^c(x)\cdot\delta(x-y)+\pi i\delta^{ab}
\frac{d}{dx}\delta(x-y) .\end{aligned}$$ We obtain the Kac-Moody algebra of central charge $k=1$. More generally the coefficient of the second term is multiplied by an integer $k$.\
Fourier transforming, $$\vec J_n
\equiv\frac{1}{2\pi}\int^l_{-l} dx e
^{in\frac{\pi}{l}x}\vec{J}(x),~~~~
[J^a_n,J^b_m]=i\epsilon^{abc}J^c_{n+m}+\frac{1}{2}n\delta^{ab}\delta_{n,-m}$$ we have an $\infty$-dimensional generalization of the ordinary $SU(2)$ Lie algebra. The spin part of the Hamiltonian is $$H_{s}=\frac{\pi}{l}\frac{1}{3}\sum^{\infty}_{n=-\infty}:\vec J_{-n}
\cdot\vec J_{n}:$$ The spectrum of $H_{s}$ is again determined by the algebra obeyed by the $\vec{J}_{n}$’s together with boundary conditions. The construction is similar to building representations of $SU(2)$ from commutation relations, i.e. constructing raising operator, etc.
In the $k=1$ case we are considering here it is simplest to use: $$\begin{aligned}
\vec{J}(x)^2 &=&3(J^z(x))^2\nonumber\\
{\cal H} &=&\frac{1}{8\pi}J^2+\frac{1}{2\pi}(J^z)^2\nonumber\\
&=&\frac{1}{4\pi}(J^2_{\uparrow}+J^2_{\downarrow})\nonumber\\
&=&\frac{1}{4}((\partial_+\phi_\uparrow)^2+(\partial_+\phi_\downarrow)^2)
\nonumber\\
&=&\frac{1}{4}[(\partial_+(\frac{\phi_\uparrow+\phi_\downarrow}{\sqrt{2}}))^2+
(\partial_+(\frac{\phi_\uparrow-\phi_\downarrow}{\sqrt{2}}))^2]\nonumber\\
&=&\frac{1}{4}({(\partial_+\phi_c)}^2+{(\partial_+\phi_s)}^2)\end{aligned}$$ Now we have introduced two commuting charge and spin free massless bosons. SU(2) symmetry is now concealed but boundary condition on $\phi_s$ must respect it. Consider the spectrum of fermion theory with boundary condition: $\psi(l)=-\psi(-l)$, $$E=\frac{\pi V}{l}\left[
{\frac{Q_\uparrow}{2}}^2+{\frac{Q_\downarrow}{2}}^2+
\sum^\infty_{m=-\infty}m(n^\uparrow_m+n^\downarrow_m)\right] .$$ Change over to $\phi_c$ and $\phi_s$. We can relabel occupation numbers, $$\begin{aligned}
n^\uparrow_m,~n^\downarrow_m~&\longrightarrow&~n^c_m,~n^s_m\nonumber\\
Q&=&Q_\uparrow +Q_\downarrow\nonumber\\
S^z&=&\frac{1}{2}(Q_\uparrow-Q_\downarrow)\nonumber\\ E&=&\frac{\pi
v_F}{l}[\frac{1}{4}Q^2+{(S^z)}^2+\sum^\infty_1
mn^c_m+\sum^\infty_1mn_m^s]\label{EcEs}\\ &=&E_c+E_s\nonumber\\
\phi_c&=&\frac{\sqrt{\pi}}{2\sqrt{2}}\frac{Q}{l}(x+t)+...\nonumber\\
\phi_s&=&\frac{\pi}{\sqrt{2}}{\frac{S^z}{l}}(x+t)+...\end{aligned}$$ Actually charge and spin bosons are not completely decoupled; we must require $Q=2S^z$ (mod $2$), to correctly reproduce the free fermion spectrum. We see that the boundary conditions on $\phi_c$ and $\phi_c$ are coupled. Now consider the phase-shifted case. $$E=\frac{\pi
v_F}{l}[\frac{1}{4}(Q-1)^2+(S^z)^2+...]$$ Redefine $Q-1\rightarrow Q$ so $$E=\frac{\pi
v_F}{l}[\frac{1}{4}Q^2+(S^z)^2+....] ,$$ the same as before the phase shift, Eq. (\[EcEs\]). One of the $0$-energy single-particle states is filled, for $Q=0$ and there are 4 groundstates, $$(Q,S^z)=(0,\pm \frac{1}{2}),~~(\pm
1,0) .$$ Now $Q=2S^z+1$ (mod $2$); i.e. we “glue" together charge and spin excitations in two different ways, either $$\begin{aligned}
\hbox{(even, integer)}&\oplus& \hbox{(odd,
half-integer)}\nonumber \\
\hbox{or\ \ (even, half-integer)} &\oplus& \hbox{(odd,
integer)},\end{aligned}$$ depending on the boundary conditions. The $\frac{\pi}{2}$ phase shift simply reverses these “gluing conditions”.
The set of all integer spin states form a “conformal tower". They can be constructed from the Kac-Moody algebra by applying the raising operators $\vec J_{-n}$ to the lowest (singlet) state, with all spacings $\frac{\pi v_F}{l}\cdot$(integer). Likewise for all half-integer spin states, $(s^z)^2=\frac{1}{4}+$integer. Likewise for even and odd charge states. The K-M algebra determines uniquely conformal towers but boundary conditions determine which conformal towers occur in the spectrum and in which spin-charge combinations.
Conformal Field Theory Approach to The Kondo Effect
===================================================
The chiral one-dimensional Hamiltonian density of Eq. (\[Hochi\]) and (\[HINT\]) is: $${\cal
H}=\frac{d}{dx}\psi_{L\alpha}+
\lambda\psi^{\dagger\alpha}_L\frac{\vec{\sigma}_\alpha^\beta}{2}
\psi_{L\beta}\cdot\vec{S}~\delta(x)~~~ \mbox{(left-movers only)}$$ We rewrite it in terms of spin and charge currents only, $${\cal
H}=\frac{1}{8\pi}J^2+\frac{1}{6\pi}(\vec{J})^2+\lambda\vec{J}\cdot\vec{S}
{}~\delta(x) .$$ The Kondo interaction involves spin fields only, not charge fields: $H=H_s+H_c.$ Henceforth we only consider the spin part. In Fourier transformed form, $$\begin{aligned}
H_s&=&\frac{\pi}{l}(\frac{1}{3}\sum^\infty_{n=-\infty}\vec
J_{-n}\cdot \vec J_n+\lambda\sum^\infty_{n=-\infty}\vec
J_n\cdot\vec{S})\nonumber\\ {[}J^a_n,
J^b_m]&=&i\epsilon^{abc}J^{c}_{n+m}+\frac{n}{2}\delta^{ab}\delta_{n,-m}
\nonumber\\ {[} S^{a}, S^{b} ]
&=&i \epsilon^{abc}S^c \nonumber\\ {[} S^a , J^{b}_{n} ] &=&0\end{aligned}$$ From calculating Green’s functions for $\vec{J}(x)$ we could again reproduce perturbation theory $\frac{d\lambda}{d
lnD}=-\lambda^2+\cdots $. That is a small $\lambda >0$ grows. What is the infrared stable fixed point? Consider $\lambda=\frac{2}{3}$, where we may “complete the square". $$\begin{aligned}
H&=&\frac{\pi V}{3l}\sum^{\infty}_{n=-\infty}[(\vec J_{-n}+\vec{S})
\cdot(\vec J_n+\vec{S})-\frac{3}{4}]\nonumber\\ {[}J^a_n+S^a,
J^b_m+S^b]&=&i\epsilon^{abc}(J^c_{n+m}+S^c)+
\frac{n}{2}\delta^{ab}\delta_{n,-m} .\end{aligned}$$ $H$ is quadratic in the new currents, $\vec{{\cal J}}_n\equiv\vec
J_n+\vec{S}$, which obey the same Kac-Moody algebra! What is the spectrum of $H(\lambda=\frac{2}{3})?$ We must get back to Kac-Moody conformal towers for integer and half-integer spin. This follows from the KM algebra and the form of $H$ (i.e. starting from the lowest state we produce the entire tower by applying the raising operators, $\vec{J}_{-n}$).
Thus we find that [*the strong-coupling fixed point is the same as the weak-coupling fixed point*]{}. However, the total spin operator is now $\vec{\cal J}_0=\vec J_0+\vec{S}$. We consider impurity spin magnitude, s=1/2. Any integer-spin state becomes a 1/2-integer spin state and vice versa. $$\hbox{Integer}\leftrightarrow \hbox{1/2-Integer}.$$ Presumably $\lambda=2/3$ is the strong coupling fixed point in this formulation of the problem. $\infty$-coupling can become finite coupling under a redefinition, eg. $$\lambda_{\hbox{Lattice}}=\frac{\lambda_{KM}}{1-\frac{3}{2}\lambda_{KM}}.$$ We expect the low-energy, large $l$ spectrum to be KM conformal towers for any $\lambda$. The effect of the Kondo interaction is to interchange the two conformal towers, Integer $\Leftrightarrow ~~ \frac{1}{2}$-Integer. See Figure (\[fig:square\]). This is equivalent to a $\frac{\pi}{2}$ phase-shift, $$\begin{aligned}
\hbox{(even,
integer)}&\oplus&\hbox{(odd,$\frac{1}{2}$-integer)} \nonumber \\
&\Updownarrow&\nonumber \\ \hbox{(even,
$\frac{1}{2}$-integer)}&\oplus&\hbox{(odd,integer)}\end{aligned}$$
Leading Irrelevant Operator:Specific Heat, Susceptibility, Wilson Ratio, Resistivity at $T>0$
---------------------------------------------------------------------------------------------
At the stable fixed point $\vec{S}$ has disappeared; i.e. it is absorbed into $\vec{\cal J}$, $$\vec{\cal
J}(x)=\vec{J}(x)+2\pi\vec{S}~\delta(x).$$ What interactions could be generated in $H_{eff}$? These only involve $\vec{\cal J}$, not $\vec{S}$. $$H_s=\frac{1}{6\pi}\vec{\cal J}(x)^2+\lambda_1\vec{\cal
J}(0)^2\delta(x).$$ This is the only dimension-2 rotationally invariant operator in the spin sector. We have succeeded in reducing two dimension-2 operators to one. The other one is the charge-operator $\lambda_2J(0)^2\delta(x)$, $\lambda_2=0$ because there is no interaction in the charge sector (with other regularization we expect $\lambda_1\sim\frac{1}{T_K}$, $\lambda_2\sim\frac{1}{D}<<\lambda_1$).
=10 cm
Now we calculate the specific heat and susceptibility to $1st$ order in $ \lambda_1$.\
Susceptibility of left-moving free fermions: $$\begin{aligned}
0\mbox{-th order}~~~~
M&=&\frac{1}{2}(n_{\uparrow}-n_{\downarrow}) =l\int d\epsilon\
\nu(\epsilon)[n(\epsilon+\frac{h}{2})-n(\epsilon-\frac{h}{2})]\nonumber\\
\chi&=&\frac{l}{2\pi}~~~~(for~~~~ T<<D)\nonumber\\ 1\mbox{st
order}~~~~ \chi&=&\frac{1}{3T}\langle[\int dx\vec{\cal
J}(x)]^2\rangle_{\lambda_1} \nonumber\\
&=&\chi_0-\frac{\lambda_1}{3T^2}\langle[\int dx \vec{\cal
J}(x)]^2\vec{\cal J}(0)^2\rangle+... \end{aligned}$$ A simplifying trick is to replace: $$\delta {\cal
H}=\lambda_1\vec{\cal J}^2(0)\delta(x)\longrightarrow
\frac{\lambda_1}{2l}\vec{\cal J}^2(x),$$ which gives the same result to first order in $\lambda$ (only) by translational invariance of $H$ at $\lambda=0$. Now the Hamiltonian density changes into $${\cal
H}\rightarrow(\frac{1}{6\pi}+\frac{\lambda_1}{2l})~~\vec{\cal
J}^2(x) .$$ We simply rescale $H$ by a factor $$H\rightarrow(1+\frac{3\pi\lambda_1}{l})H
.$$ Equivalently in a thermal average, $$T\rightarrow\frac{T}{1+\frac{3\pi\lambda_1}{l}}
\equiv T(\lambda_1)$$ $$\begin{aligned}
\chi(\lambda_1,T)&=&\frac{1}{3T}<(\int\vec{\cal
J})^2>_{T(\lambda_1)} \nonumber\\
&=&\frac{1}{1+3\pi\lambda_1/l}\chi(0,T(\lambda_1))\nonumber\\
&\approx& [1-\frac{3\pi\lambda_1}{l}]\chi_0\nonumber\\
&=&\frac{l}{2\pi}-{3\lambda_1\over 2}, \end{aligned}$$ where in the last equality the first term represents the bulk part and the second one, of order $\sim\frac{1}{T_K}$, comes from the impurity part. Specific Heat:\
$$0\mbox{-th order}~~~~
C=C_c+C_s,~~~ C_c=C_s=\frac{\pi l T}{3}.$$ Each free left-moving boson makes an identical contribution. $$\begin{aligned}
1\mbox{st order in } \lambda_1 ~~~~
C_s(\lambda_1,T)&=&\frac{\partial}{\partial
T}<H(\lambda_1)>_{\lambda_1}\nonumber\\
&=&C_s(0,T(\lambda_1))\nonumber\\ &=&\frac{\pi
l}{3}\frac{T}{1+3\pi\lambda_1/l}\nonumber\\ &\approx&\frac{\pi l
T}{3}-\pi^2\lambda_1 T \end{aligned}$$ $$\frac{\delta
C_s}{C_s}=-\frac{3\pi\lambda_1}{l}=2\frac{\delta C_s}{C}$$ The Wilson Ratio: $$R_w\equiv\frac{\delta\chi/\chi}{\delta C/C}=2=\frac{C}{C_s}$$ measues the fraction of $C$ coming from the spin degrees of freedom.
Doing more work, we can calculate the resistivity to $O(\lambda^2)$.[@Nozieres2; @Affleck6] First we get the electron lifetime from the self-energy. The change in the 3D Green’s function comes only from the 1D s-wave part: $$\begin{aligned}
&&G_3(\vec{r_1},\vec{r_2})-G^0_3(|\vec{r_1}-\vec{r_2}|)\nonumber \\
=&&\frac{1}{8\pi^2r_1r_2}[e^{-ik_F(r_1+r_2)}(G_{LR}(r_1,r_2)-G_{LR,0}(r_1,r_2))
+h.c.]\nonumber\\ =&&G^0_3(r_1)\Sigma G^0_3(r_2). \end{aligned}$$ The self-energy $\Sigma$ depends only on the frequency. It gets multiplied by the impurity concentation for a finite density (in the dilute limit). We must calculate the 1D Green’s function $G_{LR}(r_1,r_2,\omega) $ perturbatively in $\lambda$ $$\begin{aligned}
O(\lambda^0_1):~~~~ G_{LR}(r_1,r_2)
&=&-G^0_{LL}(r_1,-r_2)\nonumber\\ &=&-G^0_{LL}(r_1+r_2)\nonumber\\
&=&-G^0_{LR}(r_1,r_2) ,\end{aligned}$$ where the ($-$) sign comes from the change in boundary conditions, $$G_{LR}-G^0_{LR}=-2 G^0_{LR}+O(\lambda_1)$$ To calculate to higher orders it is convenient to write the interaction as: $$~~ {\vec{\cal
J}}^2=-\frac{3}{4}:\psi^{\dagger\alpha}
\psi_{\alpha}\psi^{\dagger\beta}\psi_{\beta}:
+\frac{3i}{4}(\psi^{\dagger\alpha}\frac{d}{dx}\psi_{\alpha}
-\frac{d\psi^{\dagger \alpha}}{dx}\psi_{\alpha})$$ To second order in $\lambda_1$, we have the Feynman diagrams shown in Figure (\[fig:stpert\]), giving: $$\Sigma^R(\omega)=\frac{-in}{2\pi\nu}[2+3\pi
i\lambda_1\omega-\frac{1}{2}(3\pi\lambda_1)^2\omega^2
-\frac{1}{4}(3\pi\lambda_1)^2(\omega^2+\pi^2T^2)] .$$ The first three terms give a phase-shift and the last term represents inelastic scattering. $$\begin{aligned}
\Sigma^R(\omega)&=&\frac{-in_i}{2\pi\nu}[1-e^{2i\delta(\omega)}]+
\Sigma^R_{inel}(\omega)\nonumber\\
\delta&=&\frac{\pi}{2}+\frac{3\pi\lambda_1\omega}{2}+...\nonumber\\
\frac{1}{\tau}&=&-2 I_m\Sigma_R(\omega)\nonumber\\
\frac{1}{\tau(\omega)}&=&\frac{n_i}{\pi\nu}[2-\frac{1}{2}(3\pi\lambda_1)^2
\omega^2-\frac{1}{4}(3\pi\lambda_1)^2(\omega^2+\pi^2T^2)]\end{aligned}$$ The leading $\lambda _1$ dependence is $O(\lambda^2_1)$ in this case. The $O(\lambda_1)$ term in $\Sigma^R$ is real. We calculate the conductivity from the Kubo formula. (There is no contribution from the scattering vertex for pure s-wave scattering.) $$\begin{aligned}
\sigma(T)&=&\frac{2
e^2}{3m^3}\int\frac{d^3\vec{k}}{(2\pi)^3}\left[-\frac{\partial
n}{\partial\epsilon_k}\right]{\vec{k}}^2\tau(\epsilon_k)\nonumber\\
\tau(\epsilon_k)&\approx&\frac{\pi\nu}{2n_i}[1+\frac{1}{4}(3\pi{\lambda_1}^2)
\epsilon_k^2+\frac{1}{8}(3\pi\lambda_1)^2(\epsilon_k^2+(\pi^2T^2)]
\nonumber\\ \rho(T)&
=&\frac{1}{\sigma(T)}=\frac{3n_i}{\pi(ev_F\nu)^2}[1-\frac{9}{4}\pi^4
\lambda_1^2T^2] \end{aligned}$$ All low-temperature properties are determined in terms of one unknown coupling constant $\lambda_1\sim\frac{1}{T_K}$. Numerical or Bethe ansatz methods are needed to find the precise value of $\lambda_1(D,\lambda)\propto\frac{1}{D}e^{1/\lambda}$.
=10 cm
Multi-Channel Kondo Effect
==========================
Normally there are several “channels" of electrons -e.g. different d-shell orbitals. A very simple and symmetric model is: $$H=\sum_{\vec
k,\alpha , i=1,2,...k}\epsilon_{\vec k}\psi^{\dagger\alpha
i}_{\vec k} \psi_{\vec k \alpha
i}+\lambda\vec{S}\cdot\sum_{\vec k,\vec k'\\ \alpha,\beta
i}\psi^{\dagger\alpha i}_{\vec
k}\vec{\sigma}_\alpha^{\beta}\psi_{\vec k' \beta i}.$$ This model has $SU(2)\times SU(k)\times U(1)$ symmetry. Realistic systems do not have this full symmetry. To understand the potential applicability of this model we need to analyse the relevance of various types of symmetry breaking.[@Affleck5] An interesting possible experimental application of the model was proposed by Ralph, Ludwig, von Delft and Buhrman.[@Ralph1] In general, we let the impurity have an arbitrary spin, s, as well.
Perturbation theory in $\lambda$ is similar to the result mentioned before: $$\begin{aligned}
\frac{d\lambda}{d~ln
D}&=&-\nu\lambda^2+\frac{k}{2}\nu^2\lambda^3+O[ks(s+1)\lambda^4]\nonumber\\
{\vec{S}}^2&=&s(s+1) .\end{aligned}$$ Does $\lambda\rightarrow\infty$ as $T\rightarrow 0$? Let’s suppose it does and check consistency. What is the groundstate for the lattice model of Eq. (\[lattice\]), generalized to arbitrary $k$ and $s$, at $\lambda/t \rightarrow\infty$? In the limit we just consider the single-site model: $$H=\lambda\vec{S}\cdot\psi^\dagger_0\frac{\vec{\sigma}}{2}\psi_0
,$$ for $\lambda>0$ (antiferromagnetic case) the minimum energy state has maximum spin for electrons at $\vec{0}$ i.e. spin$=k/2$. Coupling this spin-k/2 to a spin-s, we don’t get a singlet if $s\neq k/2$, but rather an effective spin of size $|s-k/2|$. \[See Figure (\[fig:seff\]).\] The impurity is underscreened $(k/2<s)$ or overscreened $(k/2>s)$.
=10 cm
Now let $\frac{t}{\lambda}<<1 $ be finite. Electrons on site $\pm
1$ can exchange an electron with $0$. This gives an effective Kondo interaction: $$\lambda_{\hbox{eff}}\sim{\frac{t}{\lambda}}^2<<1$$ See Figure (\[fig:exeff\]). What is the sign of $\lambda_{\hbox{eff}}$? The coupling of the electron spins is antiferromagnetic: $\lambda_{\hbox{eff}}\vec S_{e1,0}\cdot \vec S_{e1,1}$, with $\lambda_{\hbox{eff}}>0$ (as in the Hubbard model). But we must combine spins $$\vec S_{\hbox{eff}}=\vec{S}+\vec
S_{el,0}.$$ For $\frac{k}{2}<s, \vec{S}_{eff}\
||-\vec{S}_{el,0}$ but, for $\frac{k}{2}>s, \vec{S}_{eff} \
||+\vec{S}_{el,0}$. So, ultimately, $\lambda_{\hbox{eff}}<0$ in the underscreened case and $\lambda_{\hbox{eff}}>0$ in the overscreened case. In the first (underscreened) case, the assumption $\lambda\rightarrow\infty$ was consistent since a ferromagnetic $\lambda_{\hbox{eff}}\rightarrow 0$ under renormalizaton and this implies $\lambda\rightarrow\infty$, since $\lambda_{\hbox{eff}}\sim-{\frac{t}{\lambda}}^2$. In this case we expect the infrared fixed point to correspond to a decoupled spin of size $s_{\hbox{eff}}=s-k/2$ and free electrons with a $\pi /2$ phase shift. In the second (overscreened) case the $\infty$-coupling fixed point is not consistent. Hence the fixed point occurs at intermediate coupling: This fixed point does not correspond to a simple boundary condition on electrons, instead it is a Non-Fermi-Liquid Fixed Point. See Figure (\[fig:flowov\]).
=10 cm
=10 cm
For the k=2, s=1/2 case we may think of the electrons (one from each channel) in the first layer around the impurity as aligning antiferromagnetically with the impurity. This overscreens it, leaving an effective s=1/2 impurity. The electrons in the next layer then overscreen this effective impurity, etc. At each stage we have an effective s=1/2 impurity. \[See Figure (\[fig:s12\]).\] Note in this special case that there is a duality between the weak and strong coupling unstable fixed points: they both contain an s=1/2 impurity.
=10 cm
Large-k Limit
-------------
The $\beta$-function is: $$\beta = \lambda^2-{k\over 2}\lambda^3+
O(\lambda^4).$$ If we only consider the first two terms, there is a fixed point at: $$\lambda_c
\approx 2/k.$$ At this (small) value of $\lambda$ the quartic term, and all higher terms are $O(1/k^4)$, whereas the quadratic and cubic terms are $O(1/k^2)$. Thus we may ignore all terms but the quadratic and cubic ones, for large k. The slope of the $\beta$-function at the critical point is: $$\left. {d\beta \over dk}\right|_{\lambda_c}=2\lambda_c -{3\over
2}\lambda^2_c=-{2\over k}.$$ This implies that the leading irrelevant coupling constant at the non-trivial (infrared) fixed point has dimension 2/k at large k, so that $(\lambda-\lambda_c)$ scales as $\Lambda^{2/k}$. Thus the leading irrelevant operator has dimension (1+2/k). This is not an integer! This implies that this critical point is not a Fermi liquid.
Current Algebra Approach
------------------------
We can gain some insight into the nature of the non-trivial critical point using the current algebra approach discussed in the previous section for the k=1 case. It is now convenient to use a form of bosonization which separates spin, charge and [*flavour*]{} (i.e. channel) degrees of freedom. This representation is known as a conformal embedding. We introduce charge ($J$), spin ($\vec J$) and flavour ($J^A$) currents. $A$ runs over the $k^2-1$ generators of $SU(k)$. The corresponding elements of the algebra are written $T^A$. These are traceless Hermitean matrices normalized so that: $$\hbox{tr}T^AT^B = {1\over 2} \delta^{AB},$$ and obeying the completeness relation: $$\sum_A
(T^A)_a^b(T^A)^d_c = {1\over 2}\left[ \delta^b_c\delta^d_a -{1\over
k}\delta^b_a\delta^d_c\right] ,\label{complete}$$ and the commutation relations: $$[T^A,T^B] =
if^{ABC}T^C,$$ where the $f^{ABC}$ are the $SU(k)$ structure constants. Thus the currents are: $$\begin{aligned}
J &\equiv& :\psi^{\dagger
i \alpha}\psi_{i \alpha}:\nonumber \\ \vec J &\equiv& \psi^{\dagger
i \alpha}{\vec \sigma_\alpha^\beta \over 2}\psi_{i \beta}\nonumber
\\ J^A &\equiv& \psi^{\dagger i \alpha}(T^A)_i^j\psi_{j
\alpha}.\end{aligned}$$ (All repeated indices are summed.) It can be seen using Eq. (\[complete\]) that the free fermion Hamiltonian can be written in terms of these currents as: $${\cal H} = {1\over 8\pi k}J^2+{1\over 2\pi
(k+2)}\vec J^2+{1\over 2\pi (k+2)}J^AJ^A.$$ The currents $\vec J$ obey the $SU(2)$ Kac-Moody algebra with central charge $k$ and the currents $J^A$ obey the $SU(k)$ Kac-Moody algebra with central charge 2: $$[J^A_n,J^B_m]=if^{ABC}J^C_{n+m}+n\delta^{AB}\delta_{n,-m}.$$ The three types of currents commute with each of the other two types, as do the three parts of the Hamiltonian. Thus we have succeeded in expressing the Hamiltonian in terms of these three types of excitations: charge, spin and flavour. The Virasoro central charge c (proportional to the specific heat) for a Hamiltonian quadratic in currents of a general group $G$ at level $k$ is:[@Knizhnik] $$c_{G,k} =
{\hbox{Dim}(G)\cdot k\over C_V(G)+k},$$ where Dim ($G$) is the dimension of the group and $C_V(G)$ is the quadratic Casimir in the fundamental representation. For $SU(k)$ this has the value: $$C_V(SU(k)) = k.$$ Thus the total value of the central charge, c, is: $$c_{\hbox{TOT}}=1+{3\cdot k\over k+2}+{(k^2-1)\cdot 2\over
k+2}=2k,$$ the correct value for 2k species of free fermions. Complicated “gluing conditions” must be imposed to correctly reproduce the free fermion spectra, with various boundary conditions. These were worked out in general by Altshuler, Bauer and Itzykson.[@Altshuler] The $SU(2)_k$ sector consists of $k+1$ conformal towers, labelled by the spin of the lowest energy (“highest weight”) state: $s=0,1/2,1,...k/2$.[@Zamolodchikov1; @Gepner]
We may now treat the Kondo interaction much as in the single channel case. It only involves the spin sector which now becomes: $${\cal H}_s = {1\over 2\pi (k+2)}\vec J^2 + \lambda
\vec J\cdot \vec S \delta (x).$$ We see that we can always “complete the square” at a special value of $\lambda$: $$\lambda_c={2\over 2+k},$$ where the Hamiltonian reduces to its free form after a shift of the current operators by $\vec S$ which preserves the KM algebra. We note that at large $k$ this special value of $\lambda$ reduces to the one corresponding to the critical point: $\lambda_c\to 2/k.$
While this observation is tantalizing, it leaves many open questions. We might expect that some rearranging of the $(k+1)$ $SU(2)_k$ conformal towers takes place at the critical point but precisely what is it? Does it correspond to some sort of boundary condition? If so what? How can we calculate thermodynamic quantities and Green’s functions? To answer these questions we need to understand some more technical aspects of CFT in the presence of boundaries.
Boundary Conformal Field Theory
===============================
We will assume that the critical point corresponds to a conformally invariant boundary condition on the free theory. Using the general theory of conformally invariant boundary conditions developed by Cardy[@Cardy1] we can completely solve for the critical properties of the model. Why assume that the critical point corresponds to such a boundary condition? It is convenient to work in the space-(imaginary) time picture. The impurity then sits at the boundary, $r=0$ of the half-plane $r>0$ on which the Kondo effect is defined. If we consider calculating a two-point Green’s function when both points are taken very far from the boundary (with their separation held fixed) then we expect to obtain bulk behaviour, unaffected by the boundary. \[See Figure (\[fig:2ptbulk\]).\] This, at long distances and times is the conformally invariant behaviour of the free fermion system. Very close to the boundary, we certainly do not expect the behaviour to be scale invariant (let alone conformally invariant) because various microscopic scales become important. The longest of these scales is presumably the Kondo scale, $\xi_K\approx v_F/T_L\approx
ae^{1/\nu \lambda}$. Beyond this distance, it is reasonable to expect scale-invariant behaviour. However, if the two points are far from each other compared to their distance from the boundary \[Figure (\[fig:2ptbound\])\] then the behaviour is still influenced by the boundary even when both points are far from it. We have a sort of boundary-dependent termination of the bulk conformally invariant behaviour. The dependence on the details of the boundary (such as the value of $\xi_K$) drops out. We may think of various types of boundaries as falling into universality classes, each corresponding to a type of conformally invariant behaviour. Rather remarkably, the above statements hold true whether we are dealing with a 2-dimensional classical statistical system with some boundary condition imposed, or dealing with a (1+1)-dimensional quantum system with some dynamical degrees of freedom living on the boundary. In fact, we already saw an example of this in the single-channel Kondo problem. The dynamical impurity drops out of the description of the low-energy physics and is replaced by a simple, scale-invariant boundary condition, $\psi_L=-\psi_R$.
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Precisely what is meant by a conformally invariant boundary condition? Without boundaries, conformal transformations are analytic mappings of the complex plane: $$z\equiv
\tau+ix,$$ into itself: $$z\to
w(z).$$ (Henceforth, we set the Fermi velocity, $v_F=1$.) We may Taylor expand an arbitrary conformal transformation around the origin: $$w(z) = \sum_0^\infty
a_nz^n,\label{CTexp}$$ where the $a_n$’s are arbitrary complex coefficients. They label the various generators of the conformal group. It is the fact that there is an infinite number of generators (i.e. coefficients) which makes conformal invariance so powerful in (1+1) dimensions. Now suppose that we have a boundary at $x=0$, the real axis. At best, we might hope to have invariance under all transformations which leave the boundary fixed. This implies the condition: $$w(\tau )^* = w(\tau ).$$ We see that there is still an infinite number of generators, corresponding to the $a_n$’s of Eq. (\[CTexp\]) except that now we must impose the conditions: $$a_n^* = a_n.$$ We have reduced the (still $\infty$) number of generators by a factor of 1/2. The fact that there is still an $\infty$ number of generators, even in the presence of a boundary, means that this boundary conformal symmetry remains extremely powerful.
To exploit this symmetry, following Cardy, it is very convenient to consider a conformally invariant system defined on a cylinder of circumference $\beta$ in the $\tau$-direction and length $l$ in the $x$ direction, with conformally invariant boundary conditions $A$ and $B$ at the two ends. \[See Figure (\[fig:cyl\]).\] From the quantum mechanical point of view, this corresponds to a finite temperature, $T=1/\beta$. The partition function for this system is: $$Z_{AB} = \hbox{tr}e^{-\beta
H^l_{AB}},\label{ZAB1}$$ where we are careful to label the Hamiltonian by the boundary conditions as well as the length of the spatial interval, both of which help to determine the spectrum. Alternatively, we may make a modular transformation, $\tau \leftrightarrow x$. Now the spatial interval, of length, $\beta$, is periodic. We write the corresponding Hamiltonian as $H^\beta_P$. The system propagates for a time interval $l$ between initial and final states $A$ and $B$. Thus we may equally well write: $$Z_{AB} =
<A|e^{-lH^\beta_P}|B>.\label{ZAB2}$$ Equating these two expressions, Eq. (\[ZAB1\]) and (\[ZAB2\]) gives powerful constraints which allow us to determine the conformally invariant boundary conditions.
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To proceed, we make a further weak assumption about the boundary conditions of interest. We assume that the momentum density operator, $T-\bar T$ vanishes at the boundary. This amounts to a type of unitarity condition. In the free fermion theory this becomes: $$\psi^{\dagger \alpha i}_L\psi_{L \alpha i}
(t,0)-\psi^{\dagger \alpha i}_R\psi_{R \alpha
i}(t,0)=0.$$ Note that this is consistent with both boundary conditions that occured in the one-channel Kondo problem: $\psi_L=\pm \psi_R$.
Since $T(t,x)=T(t+x)$ and $\bar T(t,x)=\bar T(t-x)$, it follows that $$\bar T(t,x)=T(t,-x).$$ i.e. we may regard $\bar T$ as the analytic continuation of $T$ to the negative axis. Thus, as in our previous discussion, instead of working with left and right movers on the half-line we may work with left-movers only on the entire line. Basically, the energy momentum density, $T$ is unaware of the boundary condition. Hence, in calculating the spectrum of the system with boundary conditions $A$ and $B$ introduced above, we may regard the system as being defined periodically on a torus of length $2l$ with left-movers only. The conformal towers of $T$ are unaffected by the boundary conditions, $A$, $B$. However, which conformal towers occur [*does*]{} depend on these boundary conditions. We introduce the characters of the Virasoro algebra, for the various conformal towers: $$\chi_a (e^{-\pi \beta /l})\equiv \sum_ie^{-\beta
E^a_i(2l)},$$ where $E^a_i(2l)$ are the energies in the $a^{\hbox{th}}$ conformal tower for length $2l$. i.e.: $$E^a_i(2l) = {\pi \over l}x_i^a-{\pi c\over 24
l},$$ where the $x_i^a$’s correspond to the (left) scaling dimensions of the operators in the theory and $c$ is the conformal anomaly. The spectrum of $H^l_{AB}$ can only consist of some combination of these conformal towers. i.e.: $$Z_{AB}=\sum_an^a_{AB}\chi_a(e^{-\pi \beta
/l}),\label{naAB}$$ where the $n^a_{AB}$ are some non-negative integers giving the multiplicity with which the various conformal towers occur. Importantly, only these multiplicities depend on the boundary conditions, not the characters, which are a property of the bulk left-moving system. Thus, a specification of all possible multiplicities, $n^a_{AB}$ amounts to a specification of all possible boundary conditions $A$. The problem of specifying conformally invariant boundary conditions has been reduced to determining sets of integers, $n^a_{AB}$. For rational conformal field theories, where the number of conformal towers is finite, only a finite number of integers needs to be specified.
Now let us focus on the boundary states, $|A>$. These must obey the operator condition: $$[T(x)-\bar T(x)]|A>=0 \ \
(\forall x).$$ Fourier transforming with respect to $x$, this becomes: $$[L_n-\bar
L_n]|A>=0.$$ This implies that all boundary states, $|A>$ must be linear combinations of the “Ishibashi states”:[@Ishibashi] $$|a> \equiv
\sum_m|a;m>\otimes \overline{|a;m>}.\label{Ishibashi}$$ Here $m$ labels all states in the $a^{th}$ conformal tower. The first and second factors in Eq. (\[Ishibashi\]) refer to the left and right-moving sectors of the Hilbert Space. Thus we may write: $$|A> = \sum_a|a><a0|A>.$$ Here, $$|a0>\equiv |a;0>\otimes
\overline{|a;0>}.$$ (Note that while the states, $|a;m>\otimes \overline{|b;n>}$ form a complete orthonormal set, the Ishibashi states, $|a>$ do not have finite norm.) Thus, specification of boundary states is reduced to determining the matrix elements, $<a0|A>$. (For rational conformal field theories, there is a finite number of such matrix elements.) Thus the partition function becomes: $$Z_{AB} =
\sum_a<A|a0><a0|B><a|e^{-lH^\beta_P}|a>.$$ From the definition of the Ishibashi state, $|a>$ we see that: $$<a|e^{-lH^\beta_P}|a>=\sum_me^{-2lE_m^a(\beta )
},$$ the factor of 2 in the exponent arising from the equal contribution to the energy from $T$ and $\bar T$. This can be written in terms of the characters: $$<a|e^{-lH^\beta_P}|a>=\chi_a (e^{-4\pi l/\beta}).$$
We are now in a position to equate these two expressions for $Z_{AB}$: $$Z_{AB} =
\sum_a<A|a0><a0|B>\chi_a(e^{-4\pi
l/\beta})=\sum_an^a_{AB}\chi_a(e^{-\pi \beta
/l}).\label{ZAB12}$$ This equation must be true for all values of $l/\beta$. It is very convenient to use the modular transformation of the characters:[@Kac; @Cardy2] $$\chi_a(e^{-\pi \beta /l})=\sum_bS^b_a\chi_b(e^{-4\pi
l/\beta}).$$ Here $S^a_b$ is known as the “modular S-matrix”. (This name is rather unfortunate since this matrix has no connection with the scattering-matrix.) We thus obtain a set of equations relating the multiplicities, $n^a_{AB}$ which determine the spectrum for a pair of boundary conditions and the matrix elements $<a0|A>$ determining the boundary states: $$\sum_bS^a_bn^b_{AB} = <A|a0><a0|B>.\label{Cardy}$$ We refer to these as Cardy’s equations. They basically allow a determination of the boundary states and spectrum.
How do we go about constructing boundary states and multiplicities which satisfy these equations? Generally, boundary states corresponding to trivial boundary conditions can be found by inspection. i.e., given $n^b_{AA}$ we can find $<a|A>$. We can then generate new (sometimes non-trivial) boundary states by [*fusion*]{}. i.e. given any conformal tower, $c$, we can obtain a new boundary state $|B>$ and new spectrum $n^a_{AB}$ from the “fusion rule coefficients”, $N^c_{ab}$. These non-negative integers are defined by the operator product expansion (OPE) for (chiral) primary operators, $\phi_a$. In general the (OPE) of $\phi_a$ with $\phi_b$ contains the operator $\phi_c$ $N^c_{ab}$ times. In simple cases, such as occur in the Kondo problem, the $N^c_{ab}$’s are all 0 or 1. In the case of $SU(2)_k$, which will be relevant for the Kondo problem, the OPE is:[@Zamolodchikov1; @Gepner] $$j \otimes j' =
|j-j'|, |j-j'|+1, |j-j'|+2, \ldots ,\hbox{min} \{ j+j', k-j-j'\}
.$$ Note that this generalizes the ordinary angular momentum addition rules in a way which is consistent with the conformal tower structure of the theories (i.e. the fact that primaries only exist with $j\leq
k/2$). Thus, $$\begin{aligned}
N^{j''}_{jj'}&=&1\ \ (|j-j'|\leq j'' \leq
\hbox{min} \{ j+j', k-j-j'\} \nonumber \\ &=&0\ \
\hbox{otherwise}.\label{KMfusion}\end{aligned}$$
The new boundary state, $|B>$, and multiplicities obtained by fusion with the conformal tower $c$ are given by: $$\begin{aligned}
<a0|B> &=& <a0|A>{S^a_c\over S^a_0}\nonumber \\ n^a_{AB} &=&
\sum_bN^a_{bc}n^b_{AA}.\label{fusion}\end{aligned}$$ Here $0$ labels the conformal tower of the identity operator. Importantly, the new boundary state and multiplicities so obtained, obey Cardy’s equation. The right-hand side of Eq. (\[Cardy\]) becomes: $$<A|a0><a0|B> = <A|a0><a0|A>{S^a_c\over
S^a_0}.$$ The left-hand side becomes: $$\sum_bS^a_bn^b_{AB} = \sum_{b,d}S^a_bN^b_{dc}n^d_{AA}.$$ We now use a remarkable identity relating the modular S-matrix to the fusion rule coefficients, known as the Verlinde formula:[@Verlinde] $$\sum_bS^a_bN^b_{dc}={S^a_dS^a_c\over S^a_0}.$$ This gives: $$\sum_bS^a_bn^b_{AB} = {S^a_c\over
S^a_0}\sum_dS^a_dn^d_{AA}={S^a_c\over
S^a_0}<A|a0><a0|A>=<A|a0><a0|B>,$$ proving that fusion does indeed give a new solution of Cardy’s equations. The multiplicities, $n^a_{BB}$ are given by double fusion: $$n^a_{BB} =
\sum_{b,d}N^a_{bc}N^b_{dc}n^d_{AA}.$$ \[Recall that $|B>$ is obtained from $|A>$ by fusion with the primary operator $c$.\] It can be checked that the Cardy equation with $A=B$ is then obeyed. It is expected that, in general, we can generate a complete set of boundary states from an appropriate reference state by fusion with all possible conformal towers.
Boundary Conformal Field Theory Results on the Multi-Channel Kondo Effect
=========================================================================
Fusion and the Finite-Size Spectrum
-----------------------------------
We are now in a position to bring to bear the full power of boundary conformal field theory on the Kondo problem. By the arguments at the beginning of Sec. V, we expect that the infrared fixed points describing the low-T properties of the Kondo Hamiltonian correspond to conformally invariant boundary conditions on free fermions. We might also expect that we could determine these boundary conditions and corresponding boundary states by fusion with appropriate operators beginning from some convenient, trivial, reference state.
We actually already saw a simple example of this in Sec. III in the single channel, $s=1/2$, Kondo problem. There we observed that the free fermion spectrum, with convenient boundary conditions could be written: $$(0,\hbox{even})\oplus
(1/2,\hbox{odd}).$$ Here $0$ and $1/2$ label the $SU(2)_1$ KM conformal towers in the spin sector, while “even” and “odd” label the conformal towers in the charge sector. We argued that, after screening of the impurity spin, the infrared fixed point was described by free fermions with a $\pi /2$ phase shift, corresponding to a spectrum: $$(1/2,\hbox{even})\oplus (0,\hbox{odd}).$$ The change in the spectrum corresponds to the interchange of $SU(2)_1$ conformal towers: $$0 \leftrightarrow 1/2.$$ This indeed corresponds to fusion, with the spin-1/2 primary field of the WZW model. To see this note that the fusion rules for $SU(2)_1$ are simply \[from Eq. (\[KMfusion\])\]: $$\begin{aligned}
0\otimes
{1\over 2} &=& {1\over 2}\nonumber \\ {1\over 2}\otimes {1\over 2}
&=& 0.\end{aligned}$$ Thus for an $s=1/2$ impurity, the infrared fixed point is given by fusion with the $j=1/2$ primary. This is related to our completing the square argument. The new currents at the infrared fixed point, $\vec {\cal J}$, are related to the old ones, $\vec J$, by: $$\vec {\cal J}_n = \vec J_n +
\vec S.\label{square}$$ If $\vec J$ and $\vec {\cal J}$ were ordinary spin operators, then the new spectrum would be given by the ordinary angular momentum addition rules. In the case at hand, where $\vec J$ and $\vec {\cal J}$ are KM current operators, it is plausible that the spectrum is given by fusion with the spin-s representation, generalizing the ordinary angular momentum addition rules in a way which is consistent with the structure of the KM CFT. In particular, Eq. (\[square\]) implies, for half-integer $s$, that states of integer total spin are mapped into states of half-integer total spin, and vice versa, a property which follows from fusion with $j=1/2$.
This immediately suggests a way of determining the boundary condition for arbitrary number of channels, k and impurity spin magnitude, s: fusion with spin-s. Actually, while this is possible for $s\leq k/2$, corresponding to exact or overscreening, it is not possible in the underscreened case since there is no spin-s primary with which to fuse for $s>k/2$. Instead, in the underscreened case, we assume fusion with the maximal possible spin, namely $k/2$. This seems to correspond to the (in this case stable) strong coupling fixed point described in Sec. III. $k/2$ electrons partially screen the impurity. The fact that further screening is not possible is related to Fermi statistics. The maximal possible conduction electron spin state at the origin, for k channels is k/2. This is also essentially the reason why there are no primaries with larger spin, as can be seen from the corresponding bosonization of free fermions. We reiterate this essential point: [*The infrared fixed point in the k-channel spin-s Kondo problem is given by fusion with the spin-s primary for $s\leq k/2$ or with the spin $k/2$ primary for $s>k/2$.*]{} We have referred to this as the “fusion rules hypothesis”. If the general assumption that the infrared fixed point should be described by a conformally invariant boundary condition is accepted, then this hypothesis starts to seem very plausible. The general method for generating new boundary conditions is by fusion. Since the Kondo interaction appears entirely in the spin sector of the theory we should expect that the fusion occurs in that sector. The current redefinition $\vec J \to
\vec {\cal J}$ and various self-consistency checks all point towards this particular set of fusions.
An immediate way of checking the fusion rule hypothesis, and more generally the applicability of the boundary CFT framework to this problem, is to work out in detail the finite size spectrum for a few values of k and s and compare with spectra obtained by numerical methods.
Let us first consider the exactly screened and underscreened cases, $s\geq k/2$, where fusion occurs with the spin k/2 primary. In this case the fusion rules are simply: $$j\otimes {k\over
2} = {k\over 2}-j.$$ Each conformal tower is mapped into a unique conformal tower. It can be shown that this gives the free fermion spectrum with a $\pi /2$ phase shift.[@Affleck2]
We demonstrate the case k=2, s=1 in Tables (\[tab:antiper\]) and (\[tab:per\]). Let us start with antiperiodic boundary conditions in the left-moving formalism: $$\psi_L(l)
= -\psi_L(-l).$$ Let us express this free fermion spectrum, for 2 spin components and 2 channels, in terms of products of conformal towers in the charge, spin and flavour sectors. In this case, the flavour sector corresponds to $SU(2)_2$ as does the flavour sector. We will refer to the corresponding quantum numbers as j for ordinary spin and $j_f$ for flavour (or “pseudo-spin”). We need the energy of the “highest weight state” (i.e. groundstate) of each conformal tower. For the Kac-Moody conformal towers, the highest weight state transforming under the representation R of the group G at level k has energy:[@Knizhnik] $$E_R={\pi \over l}{C_R\over
k+C_A},$$ where $C_R$ is the quadratic Casimir in the R representation and $A$ refers to the fundamental representation. For the case of SU(2) the representations are labelled by their spin, j and the Casimirs are: $$C_j =
j(j+1).$$ We also need the energy for the charge sector. These can be worked out by generalizing the method used in the k=1 case in Sec. II. The energy for the lowest charge $Q$ excitation, for each species of fermion is: $$E={\pi
\over l}{Q^2\over 2},$$ as shown in Eq. (\[1spectrum\]). Altogether we obtain 4 terms like this for the 4 species of fermions (2 spin $\times$ 2 flavours). We can express the total energy in terms of the total charge: $$Q\equiv Q_{11}+Q_{12}+Q_{21}+Q_{22},$$ and various difference variables. This gives: $$E={\pi \over
l}{Q^2\over 8} + ...$$ This gives the energy of the charge $Q$ primary. In addition to the energy of the primary state we obtain additional terms in the energy corresponding to the excitation level in the charge, spin and flavour conformal towers: $n_Q$, $n_s$ and $n_f$. These are non-negative integers. Altogether, we may write the energy of any state as: $$E={\pi \over l}\left[ {Q^2\over 8}+{j(j+1)\over 4}
+ {j_f(j_f+1)\over 4}+ n_Q+n_s+n_f\right]
.\label{Ek=2}$$ For primary states, $n_Q=n_s=n_f=0$. $Q$ must be integer and $j$ and $j_f$ must be integer or half-integer. The allowed combinations of $Q$, $j$ and $j_f$ are what we refer to as “gluing conditions”. They depend on the boundary conditions. For antiperiodic boundary conditions, the allowed fermion momenta are $$k = \pi
(n+1/2)/l.$$ The corresponding energy levels are drawn in Figure (\[fig:anti\]). Note that the groundstate is unique. It has $j=j_f=Q=0$. Thus we must include the corresponding product of conformal towers in the spectrum. The single particle or single hole excitation has $j=j_f=1/2$ and $Q=\pm 1$. The energy is: $$E={\pi \over l}\left[{1\over 8} + {3/4\over
4}+{3/4\over 4}\right]={\pi \over l}{1\over 2},$$ the right value. 2-particle excitations have $lE/\pi = 1$, $Q=2$ and either have $j=1$, $j_f=0$ or $j=0$, $j_f=1$, due to Fermi statistics. Again the energy is given correctly by Eq. (\[Ek=2\]). The lowest energy particle-hole excitations, with $lE/\pi = 1$, have $Q=0$ and various values of $j$ and $j_f$. In particular, they can have $j=j_f=1$. It turns out that these excitations are Kac-Moody primaries. Again the energy is given correctly by Eq. (\[Ek=2\]). It can be shown[@Altshuler] that these are all conformal towers that occur, except that we must allow arbitrary values of $Q$, mod 4. These conformal towers are summarized in Table (\[tab:antiper\]).
------------- ----- ------- --------------------------
$Q$ (mod 4) $j$ $j_f$ $(El/\pi )_{\hbox{min}}$
0 0 0 0
0 1 1 1
$\pm$1 1/2 1/2 1/2
2 0 1 1
2 1 0 1
------------- ----- ------- --------------------------
: Conformal towers appearing in the $k=2$ free fermion spectrum with anti-periodic boundary conditions.[]{data-label="tab:antiper"}
Now consider fusion with the $j=1$ primary. This has the effect of shuffling the spin conformal towers in the following way: $$\begin{aligned}
0 &\rightarrow & 1\nonumber \\ 1/2 & \rightarrow
&1/2 \nonumber \\ 1 & \rightarrow 0.\end{aligned}$$ The spectrum of Table (\[tab:antiper\]) goes into that of Table (\[tab:per\]) under this shuffling. It can be checked that this corresponds to free fermions with a $\pi /2$ phase shift, i.e. periodic boundary conditions, or a shift of the Fermi energy by 1/2 a level spacing, drawn in Figure (\[fig:per\]). Now note that the groundstate is $2^4=16$-fold degenerate, since the zero-energy level may be filled or empty for each species of fermion. The charge $,Q$, in Table (\[tab:per\]) is now measured relative to the symmetric case where 2 of these levels are filled and 2 are empty. Also note that if make the replacement: $$Q \to Q-2,$$ we get back the previous spectrum of Table (\[tab:antiper\]). Making this replacement in Eq. (\[Ek=2\]), we obtain: $${ El\over \pi} \to {El\over \pi}+{Q\over
2}$$ (ignoring a constant). This corresponds to shifting the Fermi energy by 1/2-spacing; i.e. a $\pi /2$ phase shift.
------------- ----- ------- --------------------------
$Q$ (mod 4) $j$ $j_f$ $(El/\pi )_{\hbox{min}}$
0 1 0 1/2
0 0 1 1/2
$\pm$1 1/2 1/2 1/2
2 1 1 3/2
2 0 0 1/2
------------- ----- ------- --------------------------
: Conformal towers appearing in the $k=2$ free fermion spectrum with periodic boundary conditions, obtained from the Table I by fusion with $j=1$.[]{data-label="tab:per"}
In the overscreened case the fusion rules are more interesting. They lead to spectra which cannot be obtained by applying any simple linear boundary conditions to the free fermions. Thus we may refer to these as non-Fermi liquid fixed points. It might be possible to find some kind of non-linear description of the boundary conditions in this case. But note that a non-linear boundary condition effectively introduces an interaction into the theory at the boundary. A boundary condition quadratic in the fermion fields might induce an additional condition quartic in fields, etc. Thus specification of non-linear boundary conditions could be very difficult. Cardy’s formalism cleverly sidesteps this problem by the device of focussing on the boundary [*states*]{} instead of boundary conditions, and providing a method (fusion) for producing these boundary states. As was stated above, and we will continue to see in what follows, knowledge of the boundary states will determine all physical properties of the theory so nothing is lost by using this abstract description of the boundary condition.
For the $k=2$, $s=1/2$ example, the fusion rules give: $$\begin{aligned}
0 &\to & 1/2 \nonumber \\ 1/2 &\to & 0 \oplus 1
\nonumber \\ 1 &\to & 1/2.\end{aligned}$$ Now we get a larger number of conformal towers in the spectrum with this boundary condition, shown in Table (\[tab:1/2fus\]). We have shifted the groundstate energy to 0, in this Table. Note that energies $El/\pi = 1/8$ and $5/8$ now occur. These do not correspond to any possible linear boundary conditions on free fermions. Note in particular that, due to particle-hole symmetry, only phase shifts of 0 or $\pi /2$ are allowed. These give half-integer energies, as we saw above and in Tables (\[tab:antiper\]) and (\[tab:per\]).
------------- ----- ------- --------------------------
$Q$ (mod 4) $j$ $j_f$ $(El/\pi )_{\hbox{min}}$
0 1/2 0 0
0 1/2 1 1/2
$\pm$ 1 0 1/2 1/8
$\pm$ 1 1 1/2 5/8
2 1/2 0 1/2
2 1/2 1 1
------------- ----- ------- --------------------------
: Conformal towers appearing in the $k=2$ free fermion spectrum after fusion with $j=1/2$.[]{data-label="tab:1/2fus"}
This spectrum was compared with numerical work on the $k=2$, $s=1/2$ Kondo effect and the agreement was excellent (to within 5% for several of the lowest energy states).[@Affleck5] This provides evidence that the fusion rule hypothesis is correct in the overscreened case.
Impurity Entropy
----------------
We define the impurity entropy as: $$S_{\hbox{imp}}(T) \equiv lim_{l\to
\infty}[S(l,T)-S_0(l,T)],\label{Simpdef}$$ where $S_0(l,T)$ is the free fermion entropy, proportional to $l$, in the absence of the impurity. We will find an interesting, non-zero value for $S_{\hbox{imp}}(0)$. Note that, for zero Kondo coupling, $S_{\hbox{imp}}=\ln [s(s+1)],$ simply reflecting the groundstate degeneracy of the free spin. In the case of exact screening, ($k=2s$), $S_{\hbox{imp}}(0)=0$. For underscreening, $$S_{\hbox{imp}}(0) = \ln [s'(s'+1)],$$ where $s'\equiv s-k/2$. What happens for overscreening? Surprisingly, we will obtain, in general, the log of a non-integer, implying a sort of “non-integer groundstate degeneracy”.
To proceed, we show how to calculate $S_{\hbox{imp}}(0)$ from the boundary state. All calculations are done in the scaling limit, ignoring irrelevant operators, so that $S_{\hbox{imp}}(T)$ is a constant, independent of $T$, and characterizing the particular boundary condition. It is important, however, that we take the limit $l\to \infty$ first, as specified in Eq. (\[Simpdef\]), at fixed, non-zero $T$. i.e. we are interested in the limit, $l/\beta
\to \infty$. Thus it is convenient to use the first expression for the partition function, $Z_{AB}$ in Eq. (\[ZAB2\]): $$Z_{AB} = \sum_a<A|a0><a0|B>\chi_a(e^{-4\pi
l/\beta})\to e^{\pi lc/6\beta}<A|00><00|B>.$$ Here $|00>$ labels the groundstate in the conformal tower of the identity operator. $c$ is the conformal anomaly. Thus the free energy is: $$F_{AB} = -\pi cT^2l/6-T\ln
<A|00><00|B>.$$ The first term gives the specific heat: $$C=\pi cTl/3$$ and the second gives the impurity entropy: $$S_{\hbox{imp}} = \ln
<A|00><00|B>.$$ This is a sum of contributions from the two boundaries, $$S_{\hbox{imp}}=S_A+S_B.$$ Thus we see that the “groundstate degeneracy” $g_A$, associated with boundary condition A is: $$\exp [S_{\hbox{imp}A}] =
<A|00>\equiv g_A.$$ Here we have used our freedom to choose the phase of the boundary state so that $g_A>0$. For our original, anti-periodic, boundary condition, $g=0$. For the Kondo problem we expect the low T impurity entropy to be given by the value at the infrared fixed point. Since this is obtained by fusion with the spin-s (or k/2) operator, we obtain from Eq. (\[fusion\]), $$g = {S^0_s\over
S^0_0}.$$ The modular S-matrix for $SU(2)_k$ is:[@Altshuler; @Kac] $$S^j_{j'} (k) =
\sqrt{2\over 2+k}\sin \left[{\pi (2j+1)(2j'+1)\over
2+k}\right],$$ so $$g(s,k) = {\sin [\pi
(2s+1)/(2+k)]\over \sin [\pi /(2+k)]}.
\label{Smatrix}$$ This formula agrees exactly with the Bethe ansatz result.[@Tsvelik] This formula has various interesting properties. Recall that in the case of exact or underscreening ($s\geq k/2$) we must replace $s$ by $k/2$ in this formula, in which case it reduces to 1. Thus the groundstate degeneracy is 1 for exact screening. For underscreening we must multiply $g$ by $(2s'+1)$ to account for the decoupled, partially screened impurity. Note that, in the overscreened case, where $s<
k/2$, we have: $${1\over 2+k}<{2s+1\over
2+k}<1-{1\over 2+k},$$ so $g>1$. In the case $k\to
\infty$ with s held fixed, $g\to 2s+1$, i.e. the entropy of the impurity spin is hardly reduced at all by the Kondo interaction, corresponding to the fact that the critical point occurs at weak coupling. In general, for underscreening: $$1<g<2s+1.$$ i.e. the free spin entropy is somewhat reduced, but not completely eliminated. Furthermore, $g$ is not, in general, an integer. For instance, for $k=2$ and $s=1/2$, $g=\sqrt{2}$. Thus we may say that there is a non-integer “groundstate degeneracy”. Note that in all cases the groundstate degeneracy is reduced under renormalization from the zero Kondo coupling fixed point to the infrared stable fixed point. This is a special case of what we believe to be a general result: [*the groundstate degeneracy always decreases under renormalization.*]{} This appears to be related to Zamolodchikov’s c-theorem[@Zamolodchikov2] which states that the conformal anomaly parameter, c, always decreases under renormalization. The intuitive explanation of the c-theorem is that, as we probe lower energy scales, degrees of freedom which appeared approximately massless start to exhibit a mass. This freezes out their contribution to the specific heat, the slope of which can be taken as the definition of c. In the case of the “g-theorem” the intuitive explanation is that, as we probe lower energy scales, approximately degenerate levels of impurities exhibit small splittings, reducing the degeneracy.
So far, only a perturbative proof of the g-theorem has been given.[@Affleck6] It is completely analogous to a perturbative proof of the c-theorem given by Cardy and Ludwig,[@Ludwig3] independently of Zamolodchikov’s more general proof. For the g-theorem proof, we consider perturbing around a boundary CFT fixed point with a barely relevant boundary operator. i.e. the action is: $$S = S_0-\lambda \int_0^\beta d\tau \phi (0,\tau
),$$ where $\phi$ has dimension $1-y$ with $0<y<<1$. The $\beta$-function has the form: $$\beta =
y\lambda -b\lambda^2,$$ for some constant, b. There is a nearby fixed point at: $$\lambda_c=y/b.$$ It is possible to calculate the small change in $g$ using renormalization group improved perturbation theory. This gives: $$\delta g/g = -\pi^2y^3/3b^2
<0.$$
Boundary Green’s Functions: Two-Point Functions, T=0 Resistivity
----------------------------------------------------------------
In this sub-section we explain the basic concepts for calculation of Green’s functions in the presence of a conformally invariant boundary condition.[@Cardy3] We then work out the case of two-point funtions in detail. Finally we show how this gives information about the Kondo problem.
The most important point is the consequence of the identification of left and right-moving sectors, discussed in Sec. V. In general, in the bulk theory, a typical local operator is a product of left and right-moving factors: $$\phi (x) = \phi_L(x)\bar
\phi_R(x).$$ Here $x$ is the spatial co-ordinate; we suppress the time-dependence. However, in the presence of a boundary, we use: $$\bar \phi_R(x) = \bar
\phi_L(-x).$$ Thus a local operator with left and right-moving factors becomes a bilocal operator with only left-moving factors: $$\phi (x) \to \phi_L(x)\bar
\phi_L(-x).$$ \[See Figure (\[fig:1to2\]).\] Thus a one-point function becomes a two-point function, two-point becomes four-point etc.
=10 cm
Henceforth, the number of points in the Green’s function will refer to the larger number after this doubling due to the identification of left with right. In the remainder of this sub-section we show how to calculate boundary two-point functions. Four point functions are discussed in the next sub-section.
Our bulk operators are normally defined so that $<\phi
(x)>_{\hbox{bulk}}=0$. For a semi-infinite plane with a boundary, this one-point function essentially becomes a 2-point function which may have a non-zero value: $$<\phi
(x)>_A=<\phi_L(x)\bar \phi_L(-x)>_A = {C_A\over (2x)^{2d}}.
\label{2pt}$$ Here $d$ is the scaling dimension of $\phi_L$, which does not depend on the boundary condition, A. On the other hand, the coefficient, $C_A$ does depend on the boundary condition. Following Cardy and Lewellen,[@Cardy2] we may calculate $C_A$ in terms of the boundary state $|A>$. We assume that $\phi$ is a primary field.
This is done by making a conformal mapping from the semi-infinite cylinder to the semi-infinite plane: $$z=i\tanh
{\pi w\over \beta}.\label{cylinder}$$ Writing: $$\begin{aligned}
z&=&\tau + ix \nonumber \\ w&=& \tau
'+ix',\end{aligned}$$ we see that, as $x'$ goes from $-\beta /2$ to $\beta /2$, $\tau $ goes from $-\infty$ to $\infty$. \[See Figure (\[fig:pltocyl\]).\] The semi-infinite plane is $x>0$ and the semi-infinite cylinder is $\tau '>0$. Note that we are regarding $x'$ (space) as the periodic variable on the cylinder. i.e. we are calculating imaginary-time propagation from the boundary state $|A>$. We will calculate the one-point function on the cylinder, first directly, then by obtaining it from the half-plane by conformal mapping. To obtain the correlation function on the [*infinite*]{} half-cylinder, it is convenient to take the limit of a finite cylinder, with boundary state $|00>$ (the highest weight state of the identity conformal tower) at the other end. Thus, on the infinite cylinder, $$<\phi (\tau ',0)>_A =
lim_{T\to \infty}{<00|e^{-(T-\tau ' )H^\beta_P}\phi (0,0)e^{-\tau '
H_P^\beta}|A>\over <00|e^{-TH_P^\beta}|A>}.$$ Now we insert a complete set of states between $\phi$ and $|A>$. Since $\phi$ is a primary field, $\phi |00>$ gives only a sum of states in the conformal tower of $\phi$. For convenience, we also consider the limit, $\tau '\to \infty$, so that only the highest weight state, $|\phi 0>$ survives. Thus, $$<\phi
(\tau ')>_A\to_{\tau '\to \infty} {<00|\phi |\phi 0><\phi 0|A>\over
<00|A>}e^{-(2\pi /\beta )2d\tau '}.$$ Note that $$E_\phi = (2\pi /\beta )2d.$$ We need the matrix element $<00|\phi |\phi 0>$, for the periodic Hamiltonian $H^\beta_P$. This can be obtained from the Green’s function $<00|\phi (\tau_1')\phi (\tau_2')|00>$, arising from a calculation on the infinite cylinder of radius $\beta$.[@Cardy2] This can be obtained by a conformal mapping from the infinite plane, giving: $$\begin{aligned}
<00|\phi
(\tau_1')\phi (\tau_2')|00>&=&\left[ {\beta \over \pi}\sinh {\pi
\over \beta}(\tau_1'- \tau_2' )\right]^{-4d}\nonumber \\
&\to&_{\tau_1' - \tau_2'\to \infty}\left({2\pi \over \beta
}\right)^{4d}e^{-4\pi d(\tau_1'-\tau_2')/\beta} \nonumber \\
&=&|<00|\phi |\phi 0>|^2e^{-E_\phi
(\tau_1'-\tau_2')}.\end{aligned}$$ Thus we obtain: $$<00|\phi |\phi >=\left({2\pi \over
\beta}\right)^{2d}.$$ Thus the desired one-point function, on the half-cylinder is: $$<\phi (\tau '
)>_A \to_{\tau '\to \infty} \left({2\pi \over
\beta}\right)^{2d}{<\phi 0|A>\over <00|A>}e^{-(2\pi /\beta )2d\tau
'}.\label{1pthalf1}$$
=10 cm
Now let us repeat this calculation, by conformal transformation from the semi-infinite plane. As argued above, the result on the half-plane takes the form: $$<\phi (x)>_A={C_A\over
(2x)^{2d}}.$$ We now obtain the result on the half-cylinder by the conformal transformation of Eq. (\[cylinder\]). This gives: $$\begin{aligned}
<\phi (\tau
)>_A^{\hbox{1/2-cylinder}}&=&|{dz\over dw}|^{2d}<\phi
(x)>_{A}^{\hbox{plane}}\nonumber \\ &=&\left({\pi \over
\beta}\hbox{sech}^2{\pi \tau '\over \beta}\right)^{2d}{C_A\over
(2x)^{2d}}\nonumber \\ &=&C_A\left({\pi \over
\beta}{\hbox{sech}^2\pi \tau '/\beta \over 2\tanh \pi \tau
'/\beta}\right)^{2d}\nonumber \\ &\to & C_A \left({2\pi \over
\beta}e^{-2\pi \tau '
/\beta}\right)^{2d}.\label{1pthalf2}\end{aligned}$$ Thus, equating Eq. (\[1pthalf1\]) and (\[1pthalf2\]), we finally obtain the desired formula for the coefficient $C_A$, defined in Eq. (\[2pt\]): $$C_A = {<\phi 0|A >\over
<00|A>}.$$ If $|A>$ is obtained by fusion with primary operator $c$ from some reference state $|F>$ (standing for free), then: $$C_A=C_A^{\hbox{free}}{S^\phi_c/S^\phi_0\over
S^0_c/S^0_0}.\label{C_Afusion}$$ Eq. (\[2pt\]) can be immediately generalized to the case where the left and right factors occur at different points in the upper half-plane: $$<\phi_L(z)\bar \phi_R(z')>_A={C_A\over [(\tau
-\tau ')+i(x+x')]^{2d}}.$$
An important application of this formula in the Kondo problem is to the single fermion Green’s function, $<\psi_L(z)^\dagger\psi_R(z')>$. In the case of periodic boundary conditions: $$\begin{aligned}
<\psi^{\dagger
i\alpha}_L(x)\psi_{Rj\beta}(x)>_{\hbox{free}} &= &
<\psi^{\dagger i\alpha}_L(x)\psi_{Lj\beta}(-x)>\nonumber \\
&=&{\delta^i_j\delta ^\alpha_\beta \over 2x}.\end{aligned}$$ We may obtain the one-point function at the Kondo fixed point by fusion. The fermion operator can be written as a product of spin flavour and charge operators: $$\psi_{Lj\alpha} \propto
g_\alpha h_je^{i\sqrt{2\pi /k}\phi}. \label{bosonizek}$$ Here $g$ and $h$ are the left moving factors of the primary fields of the WZW models, transforming under the fundamental representation. i.e. $g$ has s=1/2. We may use Eq. (\[C\_Afusion\]). The Kondo boundary condition is obtained by fusion with the spin-s conformal tower, where s is the spin of the impurity. Thus: $$\begin{aligned}
<\psi^{\dagger
i\alpha}_L(x)\psi_{Rj\beta}(x)>^{\hbox{Kondo}}_{s}
&=&{\delta^i_j\delta ^\alpha_\beta \over
2x}{S^{1/2}_s/S^{1/2}_0\over S^0_s/S^0_0}\nonumber \\ &=&{\cos [\pi
(2s+1)/(2+k)]\over \cos [\pi /(2+k)]} {\delta^i_j\delta
^\alpha_\beta \over 2x}\label{1pGF}\end{aligned}$$ Here we have used the SU(2) modular S-matrix given in Eq. (\[Smatrix\]) . There are several interesting points to notice about this formula. First of all, we see that for the case of exact or underscreening, where $s=k/2$, the cosine in the numerator in Eq. (\[1pGF\]) becomes: $$\cos \pi (k+1)/(k+2) = -\cos
\pi/(k+2).$$ Thus: $$<\psi^\dagger_L\psi_R>^{\hbox{Kondo}}_{k/2} = -
<\psi^\dagger_L\psi_R>_{\hbox{free}}.$$ As expected, this corresponds to a $\pi /2$ phase shift: $$\psi_R(0) = e^{2i\delta}\psi_L(0)=-\psi_L(0).$$ In general, we may define a one-particle into one-particle S-matrix element , $S^{(1)}$ by: $$<\psi^\dagger_L\psi_R> =
S^{(1)} <\psi^\dagger_L\psi_R>_{\hbox{free}},$$ with: $$S^{(1)} = {\cos [\pi (2s+1)/(2+k)]\over \cos [\pi
/(2+k)]}.\label{S(1)}$$ We see that in the overscreened case, $|S^{(1)}|<1$. $S^{(1)}$ is the matrix element, at the Fermi energy, for a single electron to scatter off the impurity into a single electron. In the case where $|S^{(1)}|=1$, we see, by unitarity, that there is zero probability for a single electron to scatter into anything but a single electron at the Fermi energy, at zero temperature. This is precisely the starting point for Landau’s Fermi liquid theory. Thus we refer to such cases as Fermi liquid boundary conditions. In the overscreened case, where $|S^{(1)}|<1$, this inelastic scattering probability is non-zero so we have non-Fermi liquid boundary conditions. It is interesting to note that in the large k limit (with s held fixed), $S^{(1)}\to
1$, corresponding to the fixed point occuring at weak coupling. We also note that, for $k=2$, $s=1/2$, $S^{(1)}=0$. This is related to the symmetry between the zero coupling ($S^{(1)}=1$) and infinite coupling ($S^{(1)}=-1$) fixed point, mentioned above. In this case the one particle to one particle scattering rate vanishes!
From the electron self-energy we can obtain the lifetime and hence the resistivity for a dilute array of impurities by the Kubo formula[@Affleck6] giving a $T=0$ resistivity: $$\rho (0) = {3n_i\over k\pi (e\nu v_F)^2}\left[ {1-S^{(1)}\over
2}\right] .$$ Here $n_i$ is the impurity density. The first factor is the “unitary limit”. i.e. this is the largest possible resistivity that can occur for a dilute array of non-magnetic impurities. It is only realised in the Kondo problem in the Fermi liquid case (exact or overscreened). Otherwise the resistivity is reduced by the factor $\left[ {1-S^{(1)}\over
2}\right]$, which goes to zero for large k.
Four Point Boundary Green’s Functions, Spin-Density Green’s Function
--------------------------------------------------------------------
In this sub-section we sketch how four-point functions of chiral operators (or two-point functions of non-chiral ones) are calculated in the presence of a boundary, using the particular example of the fermion four-point function with the Kondo boundary condition. For more details, see Ref. (). Similarly to the case of the two-point function, we will find that the four-point function is determined by bulk properties up to a constant which can be expressed in terms of matrix elements involving the boundary state.
We consider the Green’s function: $$G\equiv
<\psi_{L\alpha i}(z_1)\psi_L^{\dagger \bar \beta \bar
j}(z_2)\psi_{R\beta j}(z_3)\psi_R^{\dagger \bar \alpha \bar
i}(z_4)>.$$ We suppress spin and flavour indices in what follows, but it must be understood that we are dealing with tensors throughout. The first step is to regard the right-moving fields as reflected left-moving ones: $$\psi_R(z) \to
\psi_L(z^*).$$ Then we use Eq. (\[bosonizek\]) to express $G$ as a product of charge, spin and flavour Green’s functions: $$G = G_cG_sG_f.$$ The form of $G_c$ is unique up to a multiplicative constant, and is unaffected by the boundary. $G_s$ and $G_f$ are partially determined by the bulk conformal field theories. i.e. they must be solutions of the linear Knizhnik-Zamolodchikov (KZ) equations.[@Knizhnik] These equations have two solutions in both cases. We label them $G_s(p)$ and $G_f(q)$ where $p$ and $q$ take values 0 and 1. They are tensors in spin and flavour space; we suppress these indices. Thus we may schematically write the solution as : $$G=\sum_{p,q=0}^1a_{p,q}G_cG_s(p)G_f(q).\label{4ptgen}$$ The four constants, $a_{p,q}$ remain to be determined. It turns out that three of them can be determined from general considerations, with only one ($a_{1,1}$) depending on the boundary conditions.
To see this, and to understand better how the solutions $G_s(p)$ and $G_f(q)$ are defined, it is convenient to consider the limit $z_1\to
z_2$ and $z_3\to z_4$, illustrated in Figure (\[fig:4ptbulk\]). In the limit we may use the operator product expansion of $\psi_L
(z_1)$ with $\psi^\dagger_L(z_2)$. Since these points are at a fixed distance from the boundary as they approach each other, the bulk OPE applies. This is simply the trivial OPE of free fermions: $$\psi (z_1)\psi^\dagger (z_2) \sim {1\over z_1-z_2}
+ J + \vec J + J^A + {\cal O}_s^{ad}{\cal O}^{ad}_f +
...$$ Here we have used the fact that the spin and flavour currents are bilinear in the fermion fields. ${\cal
O}_s^{ad}$ and ${\cal O}^{ad}_f$ are the primary fields in the adjoint representation of the spin and flavour groups. These have scaling dimension $2/(2+k)$ and $k/(2+k)$ respectively, so their product has scaling dimension $1$ and corresponds to the product of fermion fields with spin and flavour traces subtracted. Note, importantly, that neither adjoint primary can appear by itself, since it has a fractional scaling dimension that does not occur in the free fermion OPE.
=10 cm
Now let us consider the bosonized expression for the fermion fields, of Eq. (\[bosonizek\]). Consider the OPE of each factor separately: $$\begin{aligned}
e^{i\sqrt{2\pi
/k}\phi}(z_1)e^{-i\sqrt{2\pi /k}\phi}(z_2) &\sim&
(z_1-z_2)^{-1/2k}+...\nonumber \\ g(z_1) g^\dagger (z_2) &\sim&
(z_1-z_2)^{-3/[2(2+k)]}+(z_1-z_2)^{1/[2(2+k)]}{\cal O}_s^{ad}
+...\nonumber \\ h(z_1)h^\dagger (z_2) &\sim &
(z_1-z_2)^{-(k^2-1)/[k(2+k)]}+ (z_1-z_2)^{1/[k(2+k)]}{\cal
O}_f^{ad}+...\end{aligned}$$ A general solution of the KZ equations for the four-point spin and flavour Green’s functions, $G_s$ and $G_f$ will have singularities corresponding to both the singlet and adjoint terms in the OPE shown above. The two independent solutions, $G_s(p)$ and $G_f(q)$ can be chosen so that only the singlet appears for $p=0$ and $q=0$ and only the adjoint appears for $p=1$ and $q=1$. It is now clear that, in order for $G$ to reproduce free fermion behaviour in the limit $z_1\to z_2$, we must demand that the coefficients $a_{1,0}$ and $a_{0,1}$ vanish in Eq. (\[4ptgen\]) so that the spin or flavour adjoint field doesn’t occur without being multiplied by the other adjoint field. Similarly, requiring the correct normalization for the singlet-singlet singularity in this limit: $$G \to
{1\over (z_1-z_2)} {1\over (z_3^*-z_4^*)},$$ determines the non-zero value of the coefficient $a_{0,0}$. i.e. it has the same value as in the free fermion case, independent of the boundary conditions. Thus only $a_{1,1}$ remains to be determined by detailed consideration of the boundary condition.
From considering the limit $z_1\to z_2$, $z_3\to z_4$, we see that: $$a_{1,1} \propto <[{\cal O}^{ad}_s{\cal
O}^{ad}_f](z_1) [{\cal O}^{ad}_s{\cal
O}^{ad}_f](z_3^*)>_A.$$ This is a two-point Green’s function with the two points staddling the boundary. Thus its normalization [*does*]{} depend on the particular boundary condition, A. In fact this normalization is precisely the coefficient $C_A$ which was calcuated in the previous sub-section, Eq. (\[C\_Afusion\]). Thus we obtain, upon identifying $\phi$ in Eq. (\[C\_Afusion\]) with the adjoint (spin 1) primary in the spin sector, for the Kondo boundary condition: $$a_{1,1}
= {S^1_s/S^1_0\over
S^0_s/S^0_0}a_{1,1}^{\hbox{free}}.$$ This result completes the determination of the four point function. The general form of $G$ is rather complicated, involving hypergeometric functions. We only give explicit results here for the simplest non-trivial case, $k=2$, $s=1/2$. We consider only the most singular ($2k_F$ part) of the spin density Green’s function. Writing:$${\cal S}^a(\vec r,\tau ) \equiv
e^{2ik_Fr}\psi_L^{\dagger}{\sigma^a\over 2}\psi_R +
h.c.,$$ this becomes: $$<S^a(\vec
r_1,\tau_1 )S^b(\vec r_2,\tau_2)> = {\delta^{ab}\over
8\pi^4r^2_1r_2^2}{\eta^{-1/2}\over |z_1-z_2^*|^2}\left[ 2\cos
2k_F(r_1+r_2) + \left(2+{\eta \over 1-\eta }\right)\cos
2k_F(r_1-r_2)\right]+ ...$$ Here the $...$ represents less singular terms and $\eta$ is the cross-ratio: $$\eta \equiv -{(z_1-z_1^*)(z_2-z_2^*)\over
(z_1-z_2^*)(z_2-z_1^*)}={4r_1r_2\over
(\tau_1-\tau_2)^2+(r_1+r_2)^2}.$$
Note that in the bulk limit, $|z_1-z_2|<<r_1,r_2$, \[Figure (\[fig:4ptbulk\]) with $z_1=z_3$ and $z_2=z_4$\] $\eta \to 1$ and the Green’s function reduces to its bulk form: $$<S^a(\vec r_1,\tau_1 )S^b(\vec r_2,\tau_2)>\to {\delta^{ab}\over
8\pi^4r^2_1r_2^2}{\cos 2k_F(r_1-r_2)\over
(\tau_1-\tau_2)^2+(r_1-r_2)^2},$$ the free fermion result. On the other hand, in the boundary limit, \[Figure (\[fig:4ptbound\])\] $r_1,r_2<<|\tau_1-\tau_2|$, $\eta \to 0$ and $$<S^a(\vec r_1,\tau_1 )S^b(\vec r_2,\tau_2)>\to
{\delta^{ab}\cos 2k_Fr_1\cos 2k_Fr_2\over
4\pi^4(r_1r_2)^{3/2}|\tau_1-\tau_2|}.$$ The $\tau$-dependence implies that $S^a$ has scaling dimension 1/2 at the boundary, although it has dimension 1 in the bulk, corresponding to the spin current operator.
=10 cm
This change at the boundary can be understood as follows. As explained in the previous sub-section, in the presence of a boundary, the spin density becomes a bilocal operator, at the points $z$ and $z^*$. As $r\to 0$ we use the O.P.E. to express it as a sum of local operators. In general, any operator in the O.P.E. of the two left-moving operators ${\cal O}_L$ and $\bar
{\cal O}_L$ may appear. The coefficients of the various terms in the O.P.E. (some of which may be zero) depend on the particular boundary condition. This is very natural since we are taking an O.P.E. for two points which straddle the boundary. \[See Figure (\[fig:1to2\]).\] In this case we are obtaining the spin adjoint primary field in the O.P.E., unaccompanied by the flavour adjoint, as it is in the bulk O.P.E. We give a general prescription for determining the boundary operator content in the next sub-section.
Boundary Operator Content and Leading Irrelevant Operator: Specific Heat, Susceptibility, Wilson Ratio, Resistivity at $T>0$
----------------------------------------------------------------------------------------------------------------------------
As we saw in Secs. I and III, the leading irrelevant operator plays a very important role in the Kondo problem, determining the low temperature behaviour of the specific heat, susceptibility and resistivity. One of the novel features of a non-Fermi liquid boundary condition is that boundary operators may appear which do not occur in the bulk theory. As explained in the previous sub-sections, this is a consequence of the bilocal nature of operators in the presence of a boundary and the fact that the O.P.E. coefficients depend on the particular boundary condition. In this sub-section we derive a general formula which gives all boundary operators that occur with a particular boundary condition. Then we analyse the effect of the leading irrelevant operator in the overscreened Kondo problem.
We can identify the boundary operator content from a general relationship between the finite-size spectrum and operator content. This is established by a conformal mapping from the semi-infinite plane to the infinite strip. We consider a correlation function for some primary operator, ${\cal O}$, on the semi-infinite plane: $$<{\cal O}(\tau_1){\cal O}^\dagger
(\tau_2)>_A.$$ \[See Figure (\[fig:plstrip\]).\] Now we make the conformal mapping: $$z = le^{\pi
w/l}.$$ Here $w$ is on the strip: $$-l/2< \hbox {Im}w <l/2.$$ We define: $$\begin{aligned}
z &=& \tau + ix \nonumber \\ w&=&u+iv.\end{aligned}$$ It is convenient to assume $\tau_1, \tau_2 >0$ so that $v_1=v_2=0$, as shown in Figure (\[fig:plstrip\]). The correlation function on the infinite plane has the form: $$<{\cal
O}(\tau_1){\cal O}^\dagger (\tau_2)>_A={1\over
(\tau_1-\tau_2)^{2x}}.$$ From this we obtain the correlation function on the strip: $$\begin{aligned}
<{\cal
O}(u_1){\cal O}^\dagger (u_2)>_{AA} &=&\left\{{ {\partial z\over
\partial w}(u_1){\partial z\over \partial w}(u_2)\over
[z(u_1)-z(u_2)]^2}\right\}^x\nonumber \\ &=& \left[{2l\over
\pi}\sinh {\pi \over 2l}(u_1-u_2)\right]^{2x}.\end{aligned}$$ Here $AA$ denotes the correlation function on the strip with boundary condition $A$ on both sides. As $u_2-u_1\to \infty$ this approaches $$<{\cal O}(\tau_1){\cal O}^\dagger
(\tau_2)>_A \to \left({\pi \over l}\right)^{2x}e^{-\pi
x(u_2-u_1)/l}.$$ Alternatively, we may evaluate the correlation function on the strip by inserting a complete set of states: $$<{\cal O}(u_1){\cal O}^\dagger (u_2)>_{AA}=
\sum_n|<0|{\cal O}|n>_{AA}|^2e^{-E_n(u_2-u_1)}.$$ Here we get the eigenstates on the strip with boundary conditions of type ‘$A$’ on both sides. As $u_2-u_1\to \infty$, the lowest energy state created from the groundstate by ${\cal O}$ dominates. This is simply the highest weight state corresponding to the primary field ${\cal O}$. Clearly, the corresponding energy is: $$E_n = \pi x/l.$$ Thus we see that the primary boundary operators with boundary condition $A$, are in one-to correspondance with the conformal towers appearing in the spectrum with boundary conditions $AA$.
=10 cm
Actually we have made an assumption here, that the operator ${\cal
O}$ does not change the boundary condition. This appears to be reasonable for the Kondo problem, but in some situations, such as the X-ray edge singularity, it is neccessary to consider more general boundary condition changing operators.[@Affleck10]
Thus to obtain the boundary operator content with a boundary condition $A$, we simply need to know the finite size spectrum with the boundary condition $A$ at both sides. In the case of the Kondo boundary condition, this is obtained by “double fusion”. i.e. beginning with the free fermion spectrum, fusing once with the spin-$s$ primary gives the spectrum with free fermion boundary conditions at one end and Kondo boundary conditions at the other. Fusing a second time with the spin-$s$ primary gives the spectrum with Kondo boundary conditions at both ends, as mentioned in Sec. III. The fact that double fusion gives the spectrum with Kondo boundary conditions at both ends can also be seen by a “completing the square” argument. i.e., if we have Kondo impurities at $x=0$ and $x=l$, then at a special value of both Kondo couplings, $\lambda_1=\lambda_2=2/3$, the Hamiltonian returns to its free form if we redefine the $SU(2)$ currents by: $$\vec {\cal
J}_n \equiv \vec J_n + \vec S_1+(-1)^n\vec S_2.$$ Double fusion represents a generalization of ordinary angular momentum addition rules, for the case of two impurity spins, which is consistent with the conformal tower structure of the WZW model. With free fermion boundary conditions, the boundary operators are simply the bulk free fermion operators. After double fusion, new boundary operators, not present in the free fermion theory, are generally present. However, an exception occurs in the case $s=k/2$ corresonding to exact or underscreening. In this case under double fusion the jth conformal tower is mapped into itself: $$j \to k/2-j \to j.$$ Thus the free fermion operator content is recovered, corresponding to a Fermi liquid boundary condition.
On the other hand, in the overscreened case, $k>2s$, non-trivial operators are always obtained. In particular, the spin $j=1$, charge $Q=0$ flavour singlet operator, $\vec \phi$, always occurs. This follows from double fusion with the identity operator: $$\begin{aligned}
(Q=0, j=0,\hbox{flavour singlet}) &\to & (Q=0,
j=s,\hbox{flavour singlet}) \nonumber \\ &\to& (Q=0,
j=0,1,...\hbox{min}(2s,k-2s),\hbox{flavour singlet}).\end{aligned}$$ For overscreening, min$(2s, k-2s)\geq 1$, so $\vec \phi$ (with $j=1$) always occurs. $\vec \phi$ itself, cannot appear in the effective Hamiltonian, since it is not s spin singlet. However, its first descendent, $\vec J_{-1}\cdot \vec \phi$ [*can*]{} appear since it is invariant under all the symmetries of the underlying Kondo Hamiltonian. It has dimension, $$x= 1 +
{2\over 2+k}.$$ This dimension is $>1$ meaning that the operator is irrelevant. On the other hand, the dimension is $<2$ so it is more relevant than $\vec {\cal J}^2$, the leading irrelevant operator in the Fermi liquid case. It can be seen to be the leading irrelevant operator in all overscreened cases.[@Affleck3]
We now discuss the calculation of various quantities to lowest non-vanishing order in the leading irrelevant term in the effective Hamiltonian: $$\delta H = \lambda_1 \vec J_{-1}\cdot
\vec \phi .$$ We begin with the specific heat. The simple change of variables trick that we used in Sec. III for the Fermi liquid case doesn’t work here. In fact, since $\vec
J_{-1}\cdot \vec \phi$ is a Virasoro primary operator (although a Kac-Moody descendant) its one-point function continues to vanish even at finite temperature. Consequently, the leading contribution to the specific heat is second order in $\lambda_1$. The contribution to the impurity free energy of $O(\lambda_1^2)$ is: $$-\beta f_{\hbox{imp}} = {\lambda_1^2\over
2}\int_{-\beta /2}^{\beta /2}d\tau_1\int_{-\beta /2}^{\beta
/2}d\tau_2T<\vec J_{-1}\cdot \vec \phi (\tau_1)\vec J_{-1}\cdot
\vec \phi (\tau_2)>.\label{cimp}$$ Since $\vec
J_{-1}\cdot \vec \phi$ is a Virasoro primary, this two-point function is given by: $$\begin{aligned}
<\vec J_{-1}\cdot \vec \phi
(\tau_1)\vec J_{-1}\cdot \vec \phi (\tau_2)>&=&{3(2+k/2)\over
|\tau_1-\tau_2|^{2(1+\Delta )}}\nonumber \\ &\to &{3(2+k/2)\over
|{\beta \over \pi}\sin {\pi \over
\beta}(\tau_1-\tau_2)|^{2(1+\Delta )}}. \end{aligned}$$ Here $$\Delta \equiv {2\over 2+k},$$ and we have assumed that $\vec \phi$ has unit normalization: $$<\phi^a(\tau_1)\phi^b(\tau_2)> = {\delta^{ab}\over
|\tau_1-\tau_2|^{2\Delta}}.$$ The normalization of $\vec J_{-1}\cdot \vec \phi$ can then be fixed using the Kac-Moody algebra. A naive rescaling argument implies that: $$f_{\hbox{imp}} \propto
T^{1+2\Delta}.$$ Actually the integral in Eq. (\[cimp\]) requires an ultraviolet cut-off. However, this only leads to additional terms in $f_{\hbox{imp}}$ that vanish more rapidly as $T\to 0$. Evaluating the universal term in Eq. (\[cimp\]) explicitly, we obtain: $$\begin{aligned}
C_{\hbox{imp}}&=&-T{\partial ^2f_{\hbox{imp}}\over \partial T^2}
=\lambda_1^2\left[T^{2\Delta}\pi^{1+2\Delta}(2\Delta)^23({k\over
2}+2) {1\over 2}{\Gamma (1/2-\Delta )\Gamma (1/2)\over \Gamma
(1-\Delta )}+ O(T)\right] + O(\lambda_1^3T^{3\Delta}) + ...
\nonumber \\ &&(\hbox{for}\ k>2)\end{aligned}$$ Here $\Gamma (x)$ is Euler’s Gamma function. Note that this result is independent of the magnitude of the impurity spin, $s$. Also note that since, for $k>2$, $\Delta =2/(2+k)<1/2$, so the impurity specific heat is more singular than in the Fermi liquid case. In the particular case $k=2$, where $\Delta =1/2$, we obtain from Eq. (\[cimp\]): $$C_{\hbox{imp}}=\lambda_1^2\pi^29T\ln (T_K/T) +
...$$ Note that this is only more singular by a log than in the Fermi liquid case.
The impurity susceptibility is also second order in $\lambda_1$, $$\chi_{\hbox{imp}}\propto \beta \lambda_1^2\int
d\tau_1d\tau_2dx_1dx_2 <\vec J_{-1}\cdot \vec \phi (\tau_1) \vec
J_{-1}\cdot \vec \phi
(\tau_2)J^3(0,x_1)J^3(0,x_2)>.\label{chiimp}$$ Again, by a scaling argument we see that: $$\chi_{\hbox{imp}} \propto \lambda_1^2T^{2\Delta -1}.$$ By evaluating the integral in Eq. (\[chiimp\]) explicitly, we can caluculate the Wilson ratio, $R_W$ for the overscreened Kondo problem: $${\chi_{\hbox{imp}}\over
C_{\hbox{imp}}}{C_{\hbox{bulk}}\over \chi_{\hbox{bulk}}}\equiv R_W
= {(2+k/2)(2+k)^2\over 18}.$$
We can calculate the Wilson ratio in the exactly screened, Fermi liquid, case, much more simply. In this case, the simple rescaling argument of Sec. III always works giving: $${C_{\hbox{imp}}/C_s\over \chi_{\hbox{imp}}/\chi}=1.$$ Here $C_s$ is the part of the bulk specific heat coming from the spin degrees of freedom. Hence the Wilson ratio is given by $$R_W={C\over C_s}.$$ It simply measures the ratio of the total bulk specific heat to that coming from the spin degrees of freedom. Using: $$C = {\pi
\over 3}cT,$$ where $c$ is the conformal anomaly, and the fact that $c=2k$ for 2k species of free fermions and $c=3k/(2+k)$ for the level k SU(2) WZW model, we obtain: $$R_W={2(2+k)\over 3},$$ for the exactly screened Kondo problem.
Finally we consider the T-dependence of the impurity resistivity. For more details, see Ref. (). We may write the electron Green’s function in the one-dimensional theory with the impurity at the origin in the form: $$G(r_1,-r_2,\omega_n)-G_0(r_1+r_2,\omega_n)=G_0(r_1,\omega_n)
\Sigma_1(\omega_n)G_0(r_2,\omega_n).$$ In the three dimensional theory with a dilute random array of Kondo impurities, of density $n_i$ the self-energy becomes: $$\Sigma_3={n_i\over \nu}\Sigma_1.$$ The electron life-time is given by: $${1\over \tau}=-2\hbox{Im}
\Sigma_3^R,$$ where the superscript $R$ denotes the retarded Green’s function, obtained by analytic continuation from the Matsubara Green’s function. The conductivity is then expressed in terms of the lifetime in the standard way: $$\sigma = {2e^2\over 3m^2}\int{d^3p\over (2\pi )^3}\left[{-dn_F\over
d\epsilon_p}\right]\vec p^2\tau (\epsilon_p).$$ Thus, we just need to calculate the one-dimensional self-energy, $\Sigma_1$. This was already done, at T=0, in Subsection (VIC). There we showed that: $$G=G_0S^{(1)},$$ where $S^{(1)}$ is the scattering matrix element for 1-electron to go into 1-electron, given explicitly in Eq. (\[S(1)\]). This gave the resistivity: $$\rho = {1-S^{(1)}\over
2}\rho_{\hbox{u}},$$ where $\rho_{\hbox{u}}$ is the unitary limit resistivity. To obtain the leading T-dependence, we do perturbation theory in the leading irrelevant coupling constant, $\lambda_1$. In this case, there is a non-zero first order contribution: $$\delta \Sigma_1 \propto \lambda_1
<\psi \vec J_{-1}\cdot \vec \phi \psi^\dagger >.$$ Since $\lambda_1$ has scaling dimension $\Delta$, it follows that: $$\rho = \rho_{\hbox{u}}{1-S^{(1)}\over 2}\left[1+
\alpha \lambda_1 T^{\Delta}+...\right],$$ where $\alpha
$ is a dimensionless constant which can be obtained by explicit evaluation of first order perturbation theory. After a rather long calculation, $\alpha$ can be expressed in terms of an integral over hypergeometric functions. We evaluated this integral numerically for the case $k=2$, where $\Delta = 1/2$, obtaining: $$\alpha =4\sqrt{\pi}.$$ Thus we can form another universal ratio involving the square of the temperature-dependent term in the resistivity, and the specific heat coefficient.
Note that in the Fermi liquid case the resistivity had the form: $$\rho = \rho_{\hbox{u}}[1-\alpha
\lambda_1^2T^2].$$ The leading T-dependence is second order in $\lambda_1$ and $T$. In the non-Fermi liquid case the sign of the leading T-dependence depends on the sign of the irrelevant coupling constant, $\lambda_1$. As we increase the Kondo coupling constant, so as to pass through the non-trivial critical point, at Kondo couplings of $O(1)$, the leading irrelevant coupling constant, $\lambda_1$, should change sign. Thus the sign of the leading T-dependent term in the resistivity actually switches at the critical point. We can’t determine the value of $\lambda_1$ from our methods. However, if we make the plausible assumption that $\lambda_1 <0$ for weak Kondo coupling and hence $\lambda_1 >0$ for strong Kondo coupling, then the resistivity behaves, at low T as shown in Figure (\[fig:resov\]) (since the constant, $\alpha >0$). For weak Kondo coupling the resistivity decreases with T, exhibiting a power law singularity at $T=0$. On the other hand, for strong Kondo coupling, the resistivity initially [*increases*]{} with $T$ \[Figure (\[fig:resov\])\]. This is quite reasonable for very strong Kondo coupling where, at high $T$ the effective renormalized Kondo coupling is close to the unstable fixed point at $\infty$. Hence, we expect to obtain the unitary limit resistivity at high $T$ in this case. This requires $\rho$ to increase with $T$.
=10 cm
This $\sqrt{T}$ behaviour of the resistivity, and other scaling behaviour predicted by the present approach for the case $k=2$, $s=1/2$, were observed in recent experiments.[@Ralph1] However, the interpretation of these experiments in terms of the 2-channel Kondo problem is controversial at present.[@Wingreen; @Ralph2; @Moustakas]
I would especially like to thank my main collaborator in this work, Andreas Ludwig. I would also like to thank Chandra Varma for interesting me in this problem, Nathan Seiberg for suggesting the “fusion rule hypothesis” and John Cardy and Dan Cox for their collaboration and helpful suggestions. I am very grateful to Ming Yu who produced a TeXversion, from my transparencies, of some lectures which I gave in Beijing in June, 1991. This essentially became Sections I-III. This research was supported in part by NSERC of Canada.
Lectures given at the XXXVth Cracow School of Theoretical Physics, Zakopane, Poland, June, 1995. These lecture notes assume a certain familiarity with conformal field theory. Useful background information may be found in the lecture notes of Jean-Bernard Zuber in this series. J. Kondo, Prog. Theor. Phys. [**32**]{}, 37 (1964). P. Nozières, Proc. of 14th Int. Conf. on Low Temp. Phys. \[ Ed. M. Krusius and M. Vuorio \] V.5, P.339, (1975). K.G. Wilson, Rev. Mod. Phys. [**47**]{}, 773 (1975). N. Andrei, Phys. Rev. Lett. [**45**]{}, 379 (1980). P.B. Weigmann, Sov. Phys. J.E.T.P. Lett. [**31**]{}, 392 (1980). P.W. Anderson, J. Phys. [**C3**]{}, 2346 (1970). P. Nozières, J. Low Temp. Phys. [**17**]{}, 31 (1974). P. Nozières and A. Blandin, J. de Physique, [**41**]{}, 193 (1980). I. Affleck, Nucl. Phys. [**B336**]{}, 517, 1990. I. Affleck and A.W.W. Ludwig, Nucl. Phys. [**B352**]{}, 849(1991) . I. Affleck and A.W.W. Ludwig, Nucl. Phys. [**B360**]{}, 641(1991) . I. Affleck and A.W.W. Ludwig, Phys. Rev. Lett. [**67**]{}, 161(1991) . A.W.W. Ludwig and I. Affleck, Phys. Rev, Lett. [**67**]{}, 3160(1991) . I. Affleck, A.W.W. Ludwig, H-B. Pang and D. L. Cox, Phys. Rev. [**B45**]{}, 7918 (1992). I. Affleck and A.W.W. Ludwig, Phys. Rev. [**B48**]{}, 7297 (1993). A.W.W. Ludwig and I. Affleck, Nucl. Phys. [**B428**]{}, 545 (1994). I. Affleck and A.W.W. Ludwig, Phys. Rev. Lett. [**68**]{}, 1046 (1992). I. Affleck, A.W.W. Ludwig and B.A. Jones, Phys. Rev. [**B52**]{}, 9528 (1995). D.C. Ralph, A.W.W. Ludwig, J. von Delft and R.A. Buhrman, Phys. Rev. Lett. [**72**]{}, 1064 (1994). E. Wong and I. Affleck, Nucl. Phys. [**B417**]{}, 403 (1994). S. Eggert and I. Affleck, Phys. Rev. [**B46**]{}, 10866 (1992). E.S. Sørensen, S. Eggert and I. Affleck, J. Phys. [**A26**]{}, 6756 (1993). I. Affleck and J. Sagi, Nuc. Phys. [**B417**]{}, 374 (1994). D.L. Cox, Phys. Rev. Lett. [**59**]{}, 1240 (1987). C. Sire, C.M. Varma, A.E. Ruckenstein and T. Giamarchi, Phys. Rev. Lett. [**72**]{}, 2478 (1994). V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. [**B247**]{}, 83 (1984). D. Altschüler, M. Bauer and C. Itzykson, Comm. Math. Phys. [**132**]{}, 349 (1990). J.L. Cardy, Nuc. Phys. [**B324**]{}, 581 (1989). N. Ishibashi, Mod. Phys. Lett. [**A4**]{}, 251 (1989); T. Onogi and N. Ishibashi, Nuc. Phys. [**B318**]{}, 239 (1989). V.G. Kac and M. Wakimoto, Adv. in Math. [**70**]{}, 156 (1988). J.L. Cardy, Nucl. Phys. [**B270**]{}, 186 (1986). A.B. Zamolodchikov and V.A. Fateev, Sov. Jour. Nuc. Phys. [**43**]{}, 657 (1986). D. Gepner and E. Witten, Nucl. Phys. [**B278**]{}, 493 (1986). E. Verlinde, Nucl. Phys. [**B300**]{}, 360 (1988). J.L. Cardy and D. Lewellen, Phys. Lett. [**B259**]{}, 274 (1991). A.M. Tsvelick, J. Phys. [**C18**]{}, 159 (1985). A.B. Zamolodchikov, Pis’ma Zh. Eksp. Teor. Fiz. [**43**]{}, 565 (1986) \[J.E.T.P. Lett. [**43**]{}, 730 (1986)\]. A.W.W. Ludwig and J.L. Cardy, Nucl. Phys. [**B285**]{}, 687 (1987). I. Affleck and A.W.W. Ludwig, J. Phys. [**A27**]{}, 5375 (1994). N.S. Wingreen and B.L. Altshuler, Phys. Rev. Lett. [**75**]{}, 769 (1995). D.C. Ralph, A.W.W. Ludwig, J. von Delft and R.A. Buhrman, Phys. Rev. Lett. [**75**]{}, 770 (1995). A.L. Moustakas and D.S. Fisher, preprint cond-mat/9508011.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We establish a combinatorial connection between the real geometry and the K-theory of complex *Schubert curves* $S(\lambda_\bullet)$, which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In [@bib:Levinson], it was shown that the real geometry of these curves is described by the orbits of a map $\omega$ on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of $\mathbb{RP}^1$, with $\omega$ as the monodromy operator.
We provide a fast, local algorithm for computing $\omega$ without rectifying the skew tableau, and show that certain steps in our algorithm are in bijective correspondence with Pechenik and Yong’s *genomic tableaux* [@bib:Pechenik], which enumerate the K-theoretic Littlewood-Richardson coefficient associated to the Schubert curve. We then give purely combinatorial proofs of several numerical results involving the K-theory and real geometry of $S(\lambda_\bullet)$.
address:
- |
Mathematics Department\
University of California\
Davis, CA
- |
Mathematics Department\
University of Michigan\
Ann Arbor, MI
author:
- Maria Monks Gillespie
- Jake Levinson
title: |
Monodromy and K-theory of Schubert curves\
via generalized jeu de taquin
---
Introduction {#sec:introduction}
============
In this paper, we study the real and complex geometry of certain one-dimensional intersections $S$ of Schubert varieties defined with respect to ‘osculating’ flags. To define $S$, recall first that the *rational normal curve* is the image of the Veronese embedding $\mathbb{P}^1 \hookrightarrow \mathbb{P}^{n-1} = \mathbb{P}(\mathbb{C}^n)$, defined by $$t \mapsto [1 : t : t^2 : \cdots : t^{n-1}].$$ Let $\mathcal{F}_t$ be the *osculating* or *maximally tangent flag* to this curve at $t \in \mathbb{P}^1$, i.e. the complete flag in $\mathbb{C}^n$ formed by the iterated derivatives of this map. The $i$-th part of the flag is spanned by the top $i$ rows of the matrix $$\label{eqn:flag-matrix}
\begin{bmatrix}
\big(\frac{\mathrm{d}}{\mathrm{d}t}\big)^{i-1}(t^{j-1})
\end{bmatrix}
=
\begin{bmatrix}
1 & t & t^2 & \cdots & t^{n-1} \\
0 & 1 & 2t & \cdots & (n-1) t^{n-2} \\
0 & 0 & 2 & \cdots & (n-1)(n-2) t^{n-3} \\
\vdots & \vdots & \vdots &\ddots & \vdots \\
0 & 0 & 0 & \cdots & (n-1)!
\end{bmatrix}.$$ Let $G(k,\mathbb{C}^n)$ be the Grassmannian, and $\mathrm{\Omega}(\lambda,\mathcal{F}_t)$ the Schubert variety for the condition $\lambda$ with respect to $\mathcal{F}_t$. The [***Schubert curve***]{} is the intersection $$S = S(\lambda^{(1)}, \ldots, \lambda^{(r)}) = \mathrm{\Omega}(\lambda^{(1)}, \mathcal{F}_{t_1}) \cap \cdots \cap \mathrm{\Omega}(\lambda^{(r)}, \mathcal{F}_{t_r}),$$ where the osculation points $t_i$ are real numbers with $0 = t_1 < t_2 < \cdots < t_r = \infty$, and $\lambda^{(1)},\ldots,\lambda^{(r)}$ are partitions for which $\sum |\lambda^{(i)}|=k(n-k)-1$. For simplicity, we always consider intersections of only three Schubert varieties, though the results of this paper (in particular, Theorems \[thm:intro-2\], \[thm:MainResult2\] and \[thm:intro-parity\]) extend to the general case without difficulty. With this in mind, we consider a triple of partitions $\alpha, \beta, \gamma$ with $|\alpha| + |\beta| + |\gamma| = k(n-k) - 1$, and we study the Schubert curve $$S(\alpha,\beta,\gamma) = \mathrm{\Omega}(\alpha,\mathcal{F}_0) \cap \mathrm{\Omega}(\beta,\mathcal{F}_1) \cap \mathrm{\Omega}(\gamma,\mathcal{F}_\infty).$$
Schubert varieties with respect to osculating flags have been studied extensively in the context of degenerations of curves [@bib:Chan] [@bib:EH86] [@bib:Oss06], Schubert calculus and the Shapiro-Shapiro Conjecture [@bib:MTV09] [@bib:Pur13] [@bib:Sot10], and the geometry of the moduli space $\overline{M_{0,r}}(\mathbb{R})$ [@bib:Speyer]. They satisfy unusually strong transversality properties, particularly under the hypothesis that the osculation points $t$ are real [@bib:EH86] [@bib:MTV09]; in particular, $S$ is known to be one-dimensional (if nonempty) and reduced [@bib:Levinson]. Moreover, intersections of such Schubert varieties in dimensions zero and one have been found to have remarkable topological descriptions in terms of Young tableau combinatorics. [@bib:Chan] [@bib:Levinson] [@bib:Pur10] [@bib:Speyer]
The Schubert curve is no exception: recent work [@bib:Levinson] has shown that its *real* connected components can be described by combinatorial operations, related to jeu de taquin and Schützenberger’s promotion and evacuation, on chains of skew Young tableaux. Recall that a skew semistandard Young tableau is *Littlewood-Richardson* if its reading word is *ballot*, meaning that every suffix of the reading word has partition content.
\[def:chains\] We write $\mathrm{LR}(\lambda^{(1)},\ldots,\lambda^{(r)})$ to denote the set of sequences $(T_1, \ldots, T_r)$ of skew Littlewood-Richardson tableaux, filling a $k\times (n-k)$ rectangle, such that the shape of $T_i$ extends that of $T_{i-1}$ and $T_i$ has content $\lambda^{(i)}$ for all $i$. (The tableaux $T_1$ and $T_r$ are uniquely determined and may be omitted.)
The theorem below describes the topology of $S(\alpha,\beta,\gamma)(\mathbb{R})$ in terms of tableaux:
\[thm:intro-2\] There is a map $S \to \mathbb{P}^1$ that makes the real locus $S(\mathbb{R})$ a smooth covering of the circle $\mathbb{RP}^1$ (Figure \[fig:covering-space\]). The fibers over $0$ and $\infty$ are in canonical bijection with, respectively, ${{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$ and ${{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}$. Under this identification, the arcs of $S(\mathbb{R})$ covering $\mathbb{R}_-$ induce the *jeu de taquin bijection* $${\mathrm{sh}}: {{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}\to {{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)},$$ and the arcs covering $\mathbb{R}_+$ induce a different bijection ${\mathrm{esh}}$, called *evacuation-shuffling*. The monodromy operator $\omega$ is, therefore, given by $\omega = {\mathrm{sh}}\circ {\mathrm{esh}}.$
![An example of the covering space of Theorem \[thm:intro-2\]. The fibers over $0$ and $\infty$ are indexed by chains of tableaux, with ${\boxtimes}$ denoting the single box. The dashed arcs correspond to sliding the ${\boxtimes}$ through the tableau using jeu de taquin. The monodromy operator is $\omega = {\mathrm{sh}}\circ {\mathrm{esh}}$.[]{data-label="fig:covering-space"}](newest-covering5)
The operators ${\mathrm{esh}}$ and $\omega$ are our objects of study. In [@bib:Levinson], the second author described ${\mathrm{esh}}$ as the conjugation of jeu de taquin *promotion* by *rectification* (see Section \[sec:background\] for a precise definition). Variants of this operation have appeared elsewhere in [@bib:BerKir], [@bib:HenrKam], [@bib:KirBer].
We prove two main theorems. The first is a shorter, ‘local’ combinatorial description of the map ${\mathrm{esh}}$, which no longer requires rectifying or otherwise modifying the skew shape. We call our algorithm *local evacuation shuffling*. Local evacuation-shuffling resembles jeu de taquin: it consists of successively moving the ${\boxtimes}$ through $T$ through a weakly increasing sequence of squares. Unlike jeu de taquin, the path is in general disconnected. (See Section \[sec:local-esh\] for the definition, and Figure \[fig:antidiagonal\] for a visual description of the path of the ${\boxtimes}$.)
![The path of the ${\boxtimes}$ in a local evacuation-shuffle. The black and gray squares are the initial and final locations of the ${\boxtimes}$; the algorithm switched from “Phase 1” to “Phase 2” at the square marked by a $*$. There is an *antidiagonal symmetry*: the Phase 1 path forms a vertical strip, while the Phase 2 path forms a horizontal strip. We characterize this symmetry precisely in Corollary \[cor:antidiag-evacu-path\].[]{data-label="fig:antidiagonal"}](antidiagonal)
\[thm:MainResult1\] The map ${\mathrm{esh}}$ agrees with local evacuation shuffling. In particular, $\omega = {\mathrm{sh}}\circ {\mathrm{local\text{-}esh}}$.
Our second main result is related to K-theory and the orbit structure of $\omega$. We first recall a key consequence of Theorem \[thm:intro-2\]:
\[prop:numerics\] Let $S$ have $\iota(S)$ irreducible components and let $S(\mathbb{R})$ have $\eta(S)$ connected components. Let $\chi(\mathcal{O}_S)$ be the holomorphic Euler characteristic. Then $$\begin{aligned}
\eta(S) &\geq \iota(S) \geq \chi(\mathcal{O}_S) \text{ and } \\
\eta(S) &\equiv \chi(O_S) \pmod 2.\end{aligned}$$
We note that $\eta(S)$ is the number of orbits of $\omega$, viewed as a permutation of ${{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$. The numerical consequences above are most interesting in the context of K-theoretic Schubert calculus, which expresses $\chi(\mathcal{O}_S)$ in terms of both ordinary and K-theoretic [***genomic tableaux***]{}, namely $$\chi(\mathcal{O}_S) = |{{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}| - |{K(\gamma^c/\alpha; \beta)}|.$$ See Section \[sec:K-theory\] for the definition of $K(\gamma^c/\alpha; \beta)$ due to Pechenik-Yong [@bib:Pechenik]. In particular, we see that $$\begin{aligned}
\label{eqn:ktheory-ineq-A}
|{K(\gamma^c/\alpha; \beta)}| &\geq |{{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}| - |\mathrm{orbits}(\omega)|, \text{ and} \\
\label{eqn:ktheory-mod2-A}
|{K(\gamma^c/\alpha; \beta)}| &\equiv |{{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}| - |\mathrm{orbits}(\omega)| \pmod 2.\end{aligned}$$ The following reformulation is instructive: we recall that the *reflection length* of a permutation $\sigma \in S_N$ is the minimum length of a factorization of $\sigma$ into arbitrary (not necessarily adjacent) transpositions. We have $$\mathrm{rlength}(\sigma) = \sum_{\mathcal{O} \in \mathrm{orbits}(\sigma)}(|\mathcal{O}| - 1) = N - |\mathrm{orbits}(\sigma)|.$$ We also recall that the *sign* of a permutation is the parity of the reflection length: $$\mathrm{sgn}(\sigma) \equiv \mathrm{rlength}(\sigma)\pmod 2.$$ where we use the convention that the sign of a permutation is $0$ or $1$ (rather than $\pm 1$). Applying these relations to equations and , we see that $$\begin{aligned}
\label{eqn:ktheory-ineq}
|{K(\gamma^c/\alpha; \beta)}| &\geq \mathrm{rlength}(\omega), \text{ and} \\
\label{eqn:ktheory-mod2}
|{K(\gamma^c/\alpha; \beta)}| &\equiv \mathrm{sgn}(\omega) \pmod 2.\end{aligned}$$
For the case where $\beta$ is a horizontal strip, a combinatorial interpretation of these facts was given in [@bib:Levinson], indexing all but one step of an orbit by genomic tableaux. Our second main result generalizes this combinatorial interpretation, showing that certain steps of local evacuation-shuffling correspond bijectively to the genomic tableaux ${K(\gamma^c/\alpha; \beta)}$:
\[thm:MainResult2\] As $T$ ranges over ${{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$, for either phase of the local description of ${\mathrm{esh}}(T)$, the gaps in the ${\boxtimes}$ path are in bijection with the set $K(\gamma^c/\alpha;\beta)$.
Using the bijections of Theorem \[thm:MainResult2\], we give an independent, purely combinatorial proof of the relations and , by factoring $\omega$ into auxiliary operators $\omega_i$, which roughly correspond to the individual steps of local evacuation-shuffling, applied in isolation. If $\beta$ has $\ell(\beta)$ parts, we have the following:
\[thm:intro-parity\] There is a factorization of $\omega$ as a composition $\omega_{\ell(\beta)} \cdots \omega_{1}$, such that for every $i$, and every orbit $\mathcal{O}$ of $\omega_i$, the bijections of Theorem \[thm:MainResult2\] yield *exactly* $|\mathcal{O}|-1$ distinct genomic tableaux.
By summing over the orbits of the $\omega_i$’s, we deduce $$\begin{aligned}
\mathrm{rlength}(\omega) \leq \sum_i \mathrm{rlength}(\omega_i) = \sum_{i,\mathcal{O}} (|\mathcal{O}|-1) =|{K(\gamma^c/\alpha; \beta)}|,\end{aligned}$$ by the subadditivity of reflection length. The sign computation is analogous.
Finally, we conjecture that the inequality ‘applies orbit-by-orbit on $\omega$’, in the following sense:
\[conj:orbit-by-orbit-intro\] Using the bijections of Theorem \[thm:MainResult2\], each orbit $\mathcal{O}$ of $\omega$ generates *at least* $|\mathcal{O}| - 1$ genomic tableaux.
Conjecture \[conj:orbit-by-orbit-intro\] implies the inequality , by summing over the orbits of $\omega$. In section \[sec:omega-orbits\], we prove this conjecture in certain special cases and give computational evidence that it holds in general.
The paper is organized as follows. In Section \[sec:background\], we briefly recall the necessary background and definitions from the theory of tableaux and dual equivalence. In Section \[sec:local-esh\], we define ${\mathrm{local\text{-}esh}}$ and establish its basic properties. In Section \[sec:main-result\], we prove Theorem \[thm:MainResult1\] that ${\mathrm{local\text{-}esh}}$ agrees with ${\mathrm{esh}}$. We also establish certain symmetries of the algorithm under rotation and transposition of the tableau. Section \[sec:K-theory\] contains the link to K-theory, and the proofs of Theorems \[thm:MainResult2\] and \[thm:intro-parity\].
The remaining sections explore some consequences of the main results, including new geometric facts about Schubert curves. Section \[sec:omega-orbits\] contains the results on orbits of $\omega$, including a characterization of its fixed points. In Section \[sec:constructions\], we construct examples of Schubert curves with ‘extremal’ geometric properties. In Section \[sec:conjectures\] we state some remaining combinatorial and geometric conjectures.
Acknowledgments
---------------
We especially thank Oliver Pechenik for his help with testing our conjectures using Sage [@sage], and for several helpful discussions about tableaux combinatorics. Computations in Sage [@sage] were very helpful for testing conjectures and verifying our results throughout this work. We also thank Mark Haiman and David Speyer for their guidance. Finally, we are grateful to Bryan Gillespie, Nic Ford, Gabriel Frieden, Rachel Karpman, Greg Muller and David Speyer for comments on earlier drafts of this paper.
Background and Notation {#sec:background}
=======================
Partitions and tableaux
-----------------------
Let $\lambda=(\lambda_1\ge \cdots \ge \lambda_k)$ be a partition. We will refer to the partition $\lambda$ and its [***Young diagram***]{} interchangeably throughout, where we use the English convention for Young diagrams in which there are $\lambda_i$ squares placed in the $i$th row from the top. The [***corners***]{} of $\lambda$ are the squares which, if removed, leave a smaller partition behind. The [***co-corners***]{} are, dually, the exterior squares which, if added to $\lambda$, give a larger partition. The [***transpose***]{} of $\lambda$ is the partition $\lambda^\ast$ obtained by transposing its Young diagram. The [***length***]{} of $\lambda$ is $\ell(\lambda) = k$.
If $\mu=(\mu_1\ge \cdots \ge \mu_r)$ is a partition with $r\le k$ and $\mu_i\le \lambda_i$ for all $i$, then the [***skew shape***]{} $\lambda/\mu$ is the diagram formed by deleting the squares of $\mu$ from that of $\lambda$. The [***size***]{} of $\lambda/\mu$, denoted $|\lambda/\mu|$, is the number of squares that remain in the diagram.
We will occasionally refer to (co-)corners of a skew shape $\lambda/\mu$. The [***inner***]{} (respectively, [***outer***]{}) [***corners***]{} of $\lambda/\mu$ are the corners of $\lambda$ (respectively, the co-corners of $\mu$). These are the squares which, if deleted, leave a smaller skew shape. Similarly, the [***inner***]{} (resp. [***outer***]{}) [***co-corners***]{} are the co-corners of $\lambda$ (resp. the corners of $\mu$): the exterior squares which can be added to obtain a larger skew shape (Figure \[fig:corners\]).[^1]
$$\lambda/\mu \ =\ \young(::\hfil\hfil\hfil,:\hfil\hfil\hfil\hfil,\hfil\hfil\hfil) \qquad \young(::\square\hfil\hfil,:\square\hfil\hfil\circ,\square\hfil\circ) \hspace{0.85cm} \raisebox{-7mm}{\young(:\blacksquare\hfil\hfil\hfil\bullet,\blacksquare\hfil\hfil\hfil\hfil,\hfil\hfil\hfil\bullet,\bullet)}$$
We write ${{\scalebox{.3}{\yng(3,3)}}}=((n-k)^k)$ to denote a fixed rectangular shape of size $k\times (n-k)$, and we will always work with skew shapes that fit inside ${{\scalebox{.3}{\yng(3,3)}}}$. We define the [***complement***]{} of a partition $\lambda\subset {{\scalebox{.3}{\yng(3,3)}}}$, denoted $\lambda^c$, to be the partition $(n-k-\lambda_k,n-k-\lambda_{k-1},\ldots,n-k-\lambda_1)$. Note that $\lambda^c$ can be formed by rotating $\lambda$ by $180^\circ$ about the center of ${{\scalebox{.3}{\yng(3,3)}}}$ and then removing it from ${{\scalebox{.3}{\yng(3,3)}}}$.
A [***semistandard Young tableau***]{} (SSYT) of skew shape $\lambda/\mu$ is a filling of the boxes of the Young diagram of $\lambda/\mu$ with positive integers such that the entries in are weakly increasing to the right across each row and strictly increasing down each column. The [***content***]{} of a semistandard Young tableau is the tuple $\beta=(\beta_1,\ldots,\beta_t)$ where $\beta_i$ is the number of times the number $i$ appears in the filling. The [***reading word***]{} is the sequence formed by reading the rows from bottom to top, and left to right within a row.
$$\young(:::11,::122,::3,12,2) \qquad \qquad \young(:::34,::278,::9,16,5)$$
An SSYT $S$ is [***standard***]{} if the numbers $1,\ldots,|S|$ each appear exactly once as entries in $S$. The [***standardization***]{} of an SSYT $T$ is the tableau formed by replacing the entries of $T$ with the numbers $1,\ldots,|T|$ in the unique way that preserves the relative ordering of the entries, where ties are broken according to left-to-right ordering in the reading word (Figure \[fig:reading-word\]).
The [***suffix***]{} of an entry $m$ of $T$ is the suffix of the reading word consisting of the letters *strictly* after $m$. The [***weak suffix***]{} is the suffix including that letter and those after it. A suffix is [***ballot for $(i,i+1)$***]{} if it contains at least as many $i$’s as $i+1$’s, and is [***tied***]{} if it has the same number of $i$’s as $i+1$’s. Finally, a semistandard Young tableau $T$ is [***ballot***]{} or [***Littlewood-Richardson***]{} (also known as *Yamanouchi* or *lattice*) if every weak suffix of its reading word is ballot for $(i,i+1)$, for all $i$. We write ${\mathrm{LR}}_\mu^\lambda(\beta)$ for the set of (semistandard) Littlewood-Richardson tableaux of shape $\lambda/\mu$ and content $\beta$.
A tableau of shape $\lambda/\mu$ is [***straight shape***]{}, or [***shape $\lambda$***]{}, if $\mu={\varnothing}$ is the empty partition. The [***highest weight tableau***]{} of straight shape $\lambda$ is the tableau in which the $i$th row from the top is filled with all $i$’s. It is easily verified that this tableau is the only Littlewood-Richardson tableau of straight shape $\lambda$.
### Jeu de taquin rectification and shuffling
An [***inward (resp. outward) jeu de taquin***]{} slide of a semistandard skew tableau $T$ is the operation of starting with an inner (resp. outer) co-corner of $T$ as the [***empty square***]{}, and at each step sliding either the entry below or to the right (resp. above or to the left) of the empty square into that square in such a way that the resulting tableau is still semistandard. This condition uniquely determines the choice of slide. The former position of the moved entry is the new empty square, and the process continues until the empty square is an outer (resp. inner) co-corner of the remaining tableau. An example is shown below. $$\young(:13,23)\hspace{0.2cm} \longrightarrow \hspace{0.2cm}\young(1\cdot 3,23) \hspace{0.2cm}\longrightarrow\hspace{0.2cm} \young(133,2)$$ See [@bib:Fulton] for a more detailed introduction to jeu de taquin.
The [***rectification***]{} of a skew tableau $T$, denoted ${\mathrm{rect}}(T)$, is defined to be the straight shape tableau formed by any sequence of inwards jeu de taquin slides. It is well known (often called the “fundamental theorem of jeu de taquin”) that any sequence of slides results in the same rectified tableau.
Let $S,T$ be semistandard skew tableaux so that the shape of $T$ extends the shape of $S$, that is, $T$ can be formed by successively adding outer co-corners starting from $S$. We define the (jeu de taquin) [***shuffle***]{} of $(S,T)$ to be the pair of tableaux $(T',S')$, where $S'$ is obtained from $S$ by performing outwards jeu de taquin slides in the order specified by the standardization of $T$, and $T'$ is obtained from $T$ by performing reverse slides in the order specified by the standardization of $S$.
Equivalently, $T'$ records the squares vacated by $S$ as $S$ slides outwards, and $S'$ records the squares vacated by $T$ as $T$ slides inwards. We then say $S$ and $S'$ are [***slide equivalent***]{}, and likewise for $T, T'$.
Shuffling is an involution.
Shuffling can computed by growth diagrams (see [@bib:StanleyEC2], appendix A1.2), with the input on the left and top sides, and the output on the bottom and right sides. The transpose of a growth diagram is again a growth diagram.
Dual equivalence
----------------
We will use the theory of dual equivalence, particularly Lemmas \[lem:upper-shuffle\] and \[lem:outer-esh\], to prove Theorem \[thm:MainResult1\] on the correctness of our local algorithm for the monodromy operator $\omega$. Dual equivalence is not used outside of Section \[sec:main-result\].
Let $S,S'$ be skew standard tableaux of the same shape. Following the conventions of [@bib:Haiman], we say $S$ is [***dual equivalent***]{} to $S'$ if the following is always true: let $T$ be a skew standard tableau whose shape extends, or is extended by, that of $S$. Let $\widetilde{T}, \widetilde{T}'$ be the results of shuffling $T$ with $S$ and with $S'$. Then $\widetilde{T} = \widetilde{T}'$.
In other words, $S$ and $S'$ are dual equivalent if they have the same shape, and they transform *other* tableaux the same way under jeu de taquin. Therefore, the fact that rectification of skew tableaux is well-defined, regardless of the rectification order, can be phrased in terms of dual equivalence as follows.
\[thm-dual-jdt\] Any two tableaux of the same straight shape are dual equivalent.
We will write $D_\beta$ for the unique dual equivalence class of straight shape $\beta$.
It is also known [@bib:Haiman] that $S$ and $S'$ are dual equivalent if *their own* shapes evolve the same way under any sequence of slides. We state this in the following lemma.
\[lem-dual-def2\] Let $S,S'$ be skew standard tableaux of the same shape. Then $S$ is dual equivalent to $S'$ if and only if the following is always true:
- Let $T$ be a tableau whose shape extends, or is extended by, that of $S$. Let $\widetilde{S}$ and $\widetilde{S'}$ be the results of shuffling $S,S'$ with $T$. Then $\widetilde{S}$ and $\widetilde{S}'$ have the same shape.
Additionally, in this case $\widetilde{S}$ and $\widetilde{S}'$ are also dual equivalent.
We can extend the definition of shuffling to dual equivalence classes, using the following result. [@bib:Haiman]
Let $S,T$ be skew tableaux, with $T$’s shape extending that of $S$, and let $(S,T)$ shuffle to $(\widetilde{T},\widetilde{S})$. The dual equivalence classes of $\widetilde{T}$ and $\widetilde{S}$ depend only on the dual equivalence classes of $S$ and $T$.
So we may use any tableau of straight shape $\mu$ to rectify a skew tableau $S$ of shape $\lambda/\mu$. Thus we may speak of the [***rectification tableau***]{} of a slide equivalence class. Similarly, by the above facts, we may speak of the [***rectification shape of a dual equivalence class***]{} ${\mathrm{rsh}}(D)$. This is the shape of any rectification of any representative of the class $D$.
\[slide-dual\] Let ${\mathcal{D}},{\mathcal{S}}$ be a dual equivalence class and a slide equivalence class, with ${\mathrm{rsh}}({\mathcal{D}}) = {\mathrm{rsh}}({\mathcal{S}})$. There is a unique tableau in $\mathcal{D} \cap \mathcal{S}$.
Uniqueness is clear. To produce the tableau, pick any $T_{\mathcal{D}}\in {\mathcal{D}}$. Rectify $T_{\mathcal{D}}$ using an arbitrary tableau $X$, so $(X,T_{\mathcal{D}})$ shuffles to $(\widetilde{T_{\mathcal{D}}},\widetilde{X})$ (and $X$ and $ \widetilde{T_{\mathcal{D}}}$ are of straight shape). Replace $\widetilde{T_{\mathcal{D}}}$ by the rectification tableau $R_{\mathcal{S}}$ for the class ${\mathcal{S}}$, and let $(R_{\mathcal{S}},\widetilde{X})$ shuffle back to $(X,T)$. Then $T$ and $R_{\mathcal{S}}$ are slide equivalent, and by Theorem \[thm-dual-jdt\] and Lemma \[lem-dual-def2\], $T$ and $T_{\mathcal{D}}$ are dual equivalent.
The dual equivalence classes of a given skew shape and rectification shape are counted by a Littlewood-Richardson coefficient.
\[lem:dual-LRcoeff\] Let $\lambda/\mu$ be a skew shape and let $${\mathrm{DE}}_\mu^\lambda(\beta) = \{\text{dual equivalence classes } D \text{ with } {\mathrm{sh}}(D) = \lambda/\mu \text{ and } {\mathrm{rsh}}(D) = \beta \}.$$ Then $|{\mathrm{DE}}_\mu^\lambda(\beta)| = c_{\mu \beta}^\lambda.$
It is well-known that $c_{\mu \beta}^\lambda$ counts tableaux $T$ of shape $\lambda/\mu$ whose rectification is the standardization of the highest-weight tableau of shape $\beta$. This specifies the slide equivalence class of $T$; by Lemma \[slide-dual\], such tableaux are in bijection with ${\mathrm{DE}}_\mu^\lambda(\beta)$.
### Connection to Littlewood-Richardson tableaux
As noted in the proof of Lemma \[lem:dual-LRcoeff\], we know by Lemma \[slide-dual\] that a dual equivalence class $D$ of rectification shape $\beta$ has a unique [***highest-weight representative***]{}, that is, the unique tableau $T$ dual equivalent to $D$ and slide equivalent to the standardization of the highest weight tableau of shape $\beta$. By the fundamental theorem of jeu de taquin, if $S,T$ are highest-weight skew tableaux, the shuffles $(T',S')$ are also of highest weight. We wish to work with Littlewood-Richardson tableaux, which are in bijection with these highest-weight representatives:
\[lem:DE-LR\] A semistandard skew tableau $T$ is Littlewood-Richardson (of content $\beta$) if and only if its standardization is the highest weight representative of its dual equivalence class $D$ (and ${\mathrm{rsh}}(D)=\beta$).
This is well-known; see e.g. [@bib:Fulton]. A consequence of this lemma is that there is a canonical bijection $${\mathrm{DE}}_\mu^\lambda(\beta)\cong {\mathrm{LR}}_\mu^\lambda(\beta).$$ If $T$ is a highest weight representative for $D$ and $\beta$ is understood, we often write $${\mathrm{LR}}(D)=T \text{ and } {\mathrm{DE}}(T)=D.$$
### Transposing and rotating dual equivalence classes {#sec:rotate-transpose-DE}
Let $T$ be a standard tableau of skew shape $\alpha/\beta$, and write $T^R$ for the tableau of shape $\beta^c/\alpha^c$ obtained by rotating $T$ by $180^\circ$, then reversing the numbering of its entries. Rotating commutes with jeu de taquin shuffling, so the dual equivalence class of $T^R$ depends only on the dual equivalence class of $T$. This gives an involution of dual equivalence classes $$D \mapsto D^R : {\mathrm{DE}}_\mu^\lambda(\beta) \to {\mathrm{DE}}^{\mu^c}_{\lambda^c}(\beta).$$ In particular, any tableaux $T, T'$ of ‘anti-straight-shape’ ${{\scalebox{.3}{\yng(3,3)}}}/\lambda^c$ are dual equivalent, and their rectifications have shape $\lambda$. The same remarks apply to transposing standard tableaux, so we may speak of transposing a dual equivalence class: $$D \mapsto D^\ast : {\mathrm{DE}}_\mu^\lambda(\beta) \to {\mathrm{DE}}_{\mu^\ast}^{\lambda^\ast}(\beta^\ast).$$ We note that these operations do not correspond to simple operations on the Littlewood-Richardson tableau $LR(D)$. The combination, however, is straightforward:[^2]
\[lem:highwt-lowwt\] Let $D \in {\mathrm{DE}}_\mu^\lambda(\beta)$. Let $\tilde{D} = (D^R)^\ast$ be obtained by rotating *and* transposing $D$.
Then $\tilde{T} = {\mathrm{LR}}(\tilde{D})$ is obtained from $T = {\mathrm{LR}}(D)$ as follows: for each $j = 1, \ldots, \beta_1$, let $V_j$ be the vertical strip containing the $j$-th-from-last instance of each entry $i$ in $T$. The squares obtained by rotating and transposing $V_j$ contain the entry $j$ in $\tilde{T}$.
We defer the proof to Section \[sec:main-result\], where we prove a stronger statement (Lemma \[lem:5-facts\]).
Chains of dual equivalence classes and tableaux
-----------------------------------------------
Following the conventions of [@bib:Levinson], we define a [***chain of dual equivalence classes***]{} to be a sequence $(D_1, \ldots, D_r)$ of dual equivalence classes, such that the shape of $D_{i+1}$ extends that of $D_i$ for each $i$ (Figure \[fig:DualChain\]). We say the chain has [***type***]{} $(\lambda^{(1)}, \ldots, \lambda^{(r)})$ if for each $i$, ${\mathrm{rsh}}(D_i) = \lambda_i$.
\[lem:dual-multiLRcoeff\] Let ${\mathrm{DE}}_\mu^\nu(\lambda^{(1)}, \ldots, \lambda^{(r)})$ denote the set of chains of dual equivalence classes of type $(\lambda^{(1)}, \ldots, \lambda^{(r)})$, such that $D_1$’s shape extends $\mu$ and $\nu$ is the outer shape of $D_r$. This has cardinality equal to the Littlewood-Richardson coefficient $c_{\mu, \lambda^{(1)}, \ldots, \lambda^{(r)}}^\nu$.
![\[fig:DualChain\] At left, a chain of dual equivalence classes that extend each other to fill a $k\times (n-k)$ rectangle, with rectification shapes $\lambda^{(1)},\ldots,\lambda^{(4)}$. At right, a Littlewood-Richardson tableau with content $\lambda^{(i)}$ is given for the $i$th skew shape for $i=1,\ldots,4$. Each dual equivalence class $D_\lambda$ of skew shape $\nu/\mu$ is represented by a unique Littlewood-Richardson tableau.](DualChain.pdf)
By Lemma \[lem:DE-LR\], we can work with Littlewood-Richardson tableaux in place of dual equivalence classes. Define a [***chain of Littlewood-Richardson tableaux***]{} to be a sequence $(T_1, \ldots, T_r)$ of Littlewood-Richardson tableaux, such that the shape of $T_{i+1}$ extends that of $T_i$ for each $i$. We say the chain has [***type***]{} $(\lambda^{(1)}, \ldots, \lambda^{(r)})$ if $T_i$ has content $\lambda^{(i)}$ for each $i$.
\[lem:LRchain\] Let ${\mathrm{LR}}_\mu^\nu(\lambda^{(1)}, \ldots, \lambda^{(r)})$ denote the set of chains of Littlewood-Richardson tableaux of type $(\lambda^{(1)}, \ldots, \lambda^{(r)})$, such that $T_1$’s shape extends $\mu$, and $\nu$ is the outer shape of $T_r$. There is a natural bijection $${\mathrm{LR}}_\mu^\nu(\lambda^{(1)}, \ldots, \lambda^{(r)})\cong {\mathrm{DE}}_\mu^\nu(\lambda^{(1)},\ldots,\lambda^{(r)}).$$
When $\mu={\varnothing}$ and $\nu={{\scalebox{.3}{\yng(3,3)}}}$, we simply write ${\mathrm{LR}}(\lambda^{(1)},\ldots,\lambda^{(r)})$ and ${\mathrm{DE}}(\lambda^{(1)},\ldots,\lambda^{(r)})$ in place of ${\mathrm{LR}}_\mu^\nu(\lambda^{(1)},\ldots,\lambda^{(r)})$ and ${\mathrm{DE}}_\mu^\nu(\lambda^{(1)},\ldots,\lambda^{(r)})$ respectively.
### Operations on chains {#sec:shuffling-ops}
We define the [***shuffling***]{} operations $${\mathrm{sh}}_i : {\mathrm{DE}}_\mu^\nu(\lambda^{(1)}, \ldots, \lambda^{(i)}, \lambda^{(i+1)}, \cdots \lambda^{(r)}) \to {\mathrm{DE}}_\mu^\nu(\lambda^{(1)}, \ldots, \lambda^{(i+1)}, \lambda^{(i)}, \cdots \lambda^{(r)})$$ $${\mathrm{sh}}_i : {\mathrm{LR}}_\mu^\nu(\lambda^{(1)}, \ldots, \lambda^{(i)}, \lambda^{(i+1)}, \cdots \lambda^{(r)}) \to {\mathrm{LR}}_\mu^\nu(\lambda^{(1)}, \ldots, \lambda^{(i+1)}, \lambda^{(i)}, \cdots \lambda^{(r)})$$ by shuffling $(D_i,D_{i+1})$ or $(T_i,T_{i+1})$ respectively. The shuffling operations commute with the correspondence between ${\mathrm{DE}}$ and ${\mathrm{LR}}$ of Lemma \[lem:LRchain\]. They satisfy the relations ${\mathrm{sh}}_i^2 = \mathrm{id}$ and ${\mathrm{sh}}_i {\mathrm{sh}}_j = {\mathrm{sh}}_j {\mathrm{sh}}_i$ when $|i-j| > 1$. Note, however, that ${\mathrm{sh}}_i {\mathrm{sh}}_{i+1} {\mathrm{sh}}_i \ne {\mathrm{sh}}_{i+1} {\mathrm{sh}}_i {\mathrm{sh}}_{i+1}$ in general.
We next define the $i$-th [***evacuation***]{} operations $${\mathrm{ev}}_i : {\mathrm{DE}}_\mu^\nu(\lambda^{(1)}, \ldots, \lambda^{(r)}) \to {\mathrm{DE}}_\alpha^\beta(\lambda^{(i)}, \ldots, \lambda^{(1)}, \lambda^{(i+1)}, \ldots, \lambda^{(r)})$$ $${\mathrm{ev}}_i : {\mathrm{LR}}_\mu^\nu(\lambda^{(1)}, \ldots, \lambda^{(r)}) \to {\mathrm{LR}}_\alpha^\beta(\lambda^{(i)}, \ldots, \lambda^{(1)}, \lambda^{(i+1)}, \ldots, \lambda^{(r)})$$ by ${\mathrm{ev}}_i = {\mathrm{sh}}_1 ({\mathrm{sh}}_2 {\mathrm{sh}}_1) \cdots ({\mathrm{sh}}_{i-2} \cdots {\mathrm{sh}}_1) ({\mathrm{sh}}_{i-1} \cdots {\mathrm{sh}}_1)$. This results in reversing the first $i$ parts of the chain’s type, by first shuffling $D_1$ (or $T_1$) outwards past $D_i$, then shuffling the $D_2'$ (now the first element of the chain) out past $D_i'$, and so on.
In the case where $\mu = {\varnothing}$ and $\lambda^{(i)} = {\scalebox{.5}{\yng(1)}}$ for all $i$, the operation ${\mathrm{ev}}_i$ reduces to evacuation of the standard tableau formed by the first $i$ entries. In general, ${\mathrm{ev}}_i$ is an involution:
\[evac-involution\] The operation ${\mathrm{ev}}_i$ is an involution.
By definition, ${\mathrm{ev}}_i = {\mathrm{ev}}_{i-1} ({\mathrm{sh}}_{i-1} \cdots {\mathrm{sh}}_1)$. On the other hand, observe that $({\mathrm{sh}}_{i-1} \cdots {\mathrm{sh}}_1){\mathrm{ev}}_i = {\mathrm{ev}}_{i-1}$. (Each extra ${\mathrm{sh}}_j$ cancels the leftmost instance of ${\mathrm{sh}}_j$ in ${\mathrm{ev}}_i$.) Thus we have $${\mathrm{ev}}_i^2 = {\mathrm{ev}}_{i-1}({\mathrm{sh}}_{i-1} \cdots {\mathrm{sh}}_1) {\mathrm{ev}}_i = {\mathrm{ev}}_{i-1}^2,$$ and the claim follows by induction.
Finally, we define the $i$-th [***evacuation-shuffle***]{} operations $${\mathrm{esh}}_i : {\mathrm{DE}}_{\varnothing}^{{{\scalebox{.3}{\yng(3,3)}}}}(\lambda^{(1)}, \ldots, \lambda^{(i)}, \lambda^{(i+1)}, \cdots \lambda^{(r)}) \to {\mathrm{DE}}_{\varnothing}^{{{\scalebox{.3}{\yng(3,3)}}}}(\lambda^{(1)}, \ldots, \lambda^{(i+1)}, \lambda^{(i)}, \cdots \lambda^{(r)})$$ $${\mathrm{esh}}_i : {\mathrm{LR}}_{\varnothing}^{{{\scalebox{.3}{\yng(3,3)}}}}(\lambda^{(1)}, \ldots, \lambda^{(i)}, \lambda^{(i+1)}, \cdots \lambda^{(r)}) \to {\mathrm{LR}}_{\varnothing}^{{{\scalebox{.3}{\yng(3,3)}}}}(\lambda^{(1)}, \ldots, \lambda^{(i+1)}, \lambda^{(i)}, \cdots \lambda^{(r)})$$ by $${\mathrm{esh}}_i = {\mathrm{ev}}_{i+1}^{-1} {\mathrm{sh}}_1 {\mathrm{ev}}_{i+1}.$$ This operation is simpler than it appears: it only affects the $i$-th and $(i+1)$-th entries of the chain, and its effect is local (it depends only on the $i$-th and $(i+1)$-th entries). We have the following:
\[lem:upper-shuffle\] Let ${\bf D} = (D_1,\ldots, D_r) \in {\mathrm{DE}}_{\varnothing}^{{{\scalebox{.3}{\yng(3,3)}}}}(\lambda^{(1)}, \ldots, \lambda^{(r)})$ and write $${\mathrm{esh}}_i({\bf D}) = (D_1', \ldots, D'_{i+1}, D'_i, \ldots, D_r').$$ Then:
- We have $D_j = D_j'$ for all $j \ne i,i+1$.
- The remaining two classes $D_i', D_{i+1}'$ are computed as follows: let $D_1 \sqcup \cdots \sqcup D_{i-1} = D_\tau$ be the concatenation of the first $i-1$ classes (i.e. the unique class of straight-shape $\tau$, the outer shape of $D_{i-1}$). Let $\sigma$ be the outer shape of $D_{i+1}$. Consider $\overline{\bf D} = (D_\tau, D_{i}, D_{i+1}) \in {\mathrm{DE}}_{\varnothing}^\sigma(\tau, \lambda^{(i)}, \lambda^{(i+1)})$. Then $${\mathrm{esh}}_2(\overline{\bf D}) = {\mathrm{sh}}_1 {\mathrm{sh}}_2 \circ {\mathrm{sh}}_1 \circ {\mathrm{sh}}_2 {\mathrm{sh}}_1(\overline{\bf D}) = (D_\tau, D_{i+1}', D_i').$$
In other words, evacuation-shuffling a pair of consecutive tableaux $(S,T)$ in a Littlewood-Richardson chain consists of rectifying $(S,T)$ together, then shuffling them, then un-rectifying.
We may also compute ${\mathrm{esh}}_i$ by anti-rectifying into the lower right corner of the rectangle instead of rectifying:
\[lem:outer-esh\] Let ${\bf D} = (D_1, D_2, D_3, D_4) \in {\mathrm{DE}}_{\varnothing}^{{\scalebox{.3}{\yng(3,3)}}}(\lambda^{(1)}, \lambda^{(2)}, \lambda^{(3)}, \lambda^{(4)})$. Then $${\mathrm{esh}}_2({\bf D}) = {\mathrm{sh}}_3 {\mathrm{sh}}_2 \circ {\mathrm{sh}}_3 \circ {\mathrm{sh}}_2 {\mathrm{sh}}_3({\bf D}).$$
Rotating dual equivalence classes, as in Section \[sec:rotate-transpose-DE\], $$\mathrm{rev} : (D_1, D_2, D_3, D_4) \mapsto (D_4^R, D_3^R, D_2^R, D_1^R),$$ corresponds to the word $$\mathrm{rev} = {\mathrm{sh}}_1 \circ {\mathrm{sh}}_2{\mathrm{sh}}_1 \circ {\mathrm{sh}}_3{\mathrm{sh}}_2{\mathrm{sh}}_1.$$ (See [@bib:Speyer] for a proof via dual equivalence growth diagrams.) We have $$\mathrm{rev} \circ {\mathrm{sh}}_i = {\mathrm{sh}}_{4-i} \circ \mathrm{rev}, \ \ \text{and so}\ \ \mathrm{rev} \circ {\mathrm{esh}}_2 \circ \mathrm{rev} = {\mathrm{sh}}_3 {\mathrm{sh}}_2 \circ {\mathrm{sh}}_3 \circ {\mathrm{sh}}_2 {\mathrm{sh}}_3.$$ On the other hand, we see directly, by simplifying the corresponding words, that $$\mathrm{rev} \circ {\mathrm{esh}}_2 \circ \mathrm{rev} = {\mathrm{esh}}_2$$ and the proof is complete.
We remark that neither of Lemmas \[lem:upper-shuffle\] or \[lem:outer-esh\] is easy to prove directly for ballot semistandard tableaux. We will use them in the proof of Theorem \[thm:main-theorem\].
The case of interest and the operator w {#sec:case-of-interest}
---------------------------------------
The geometry of Schubert curves (see Section \[sec:introduction\]) suggests studying sets of the form $${\mathrm{DE}}_{\varnothing}^{{\scalebox{.3}{\yng(3,3)}}}(\lambda^{(1)}, {\scalebox{.5}{\yng(1)}}, \lambda^{(2)}, \ldots, \lambda^{(r)}),$$ where ${{\scalebox{.3}{\yng(3,3)}}}$ is a $k\times(n-k)$ rectangle and $1 + \sum |\lambda^{(i)}| = k(n-k)$, with the composition of shuffles and evacu-shuffles $$\omega = {\mathrm{sh}}_2 \circ \cdots \circ {\mathrm{sh}}_{r-1} \circ {\mathrm{esh}}_{r-1} \circ \cdots \circ {\mathrm{esh}}_2.$$ In general, $\omega$ describes the monodromy and real connected components of the Schubert curve $$S(\lambda^{(1)}, \ldots, \lambda^{(r)}) = \mathrm{\Omega}(\lambda^{(1)}, \mathcal{F}_{t_1}) \cap \cdots \cap \mathrm{\Omega}(\lambda^{(r)}, \mathcal{F}_{t_r}),$$ where the osculation points $t_i$ are real numbers with $0 = t_1 < t_2 < \cdots < t_r = \infty.$ (See [@bib:Levinson], Corollary 4.9.) Our local description of ${\mathrm{esh}}$ will apply to each of the above ${\mathrm{esh}}_i$ operations, by Lemma \[lem:upper-shuffle\]. Therefore, our main results, in the case of three marked points, generalize without difficulty to this general case. We leave these extensions to the interested reader.
Thus, for simplicity, we restrict for the remainder of the paper to the case of three partitions $\alpha, \beta, \gamma$, i.e. we study the operator $$\omega = {\mathrm{sh}}_2 \circ {\mathrm{esh}}_2$$ on the sets $${{\mathrm{DE}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}\text{ and } {{\mathrm{DE}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)},$$ or equivalently $${{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}\text{ and } {{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}.$$
Since we mostly work only with ${\mathrm{sh}}_2$ and ${\mathrm{esh}}_2$, we often simply abbreviate them as ${\mathrm{sh}}$ and ${\mathrm{esh}}$, as in Section \[sec:introduction\].
Since the straight shape $\alpha$ and anti straight shape $\gamma^c$ each have only one dual equivalence class, an element of ${{\mathrm{DE}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$ can be thought of as a pair $({\boxtimes},D)$, with $D$ a dual equivalence class of rectification shape $\beta$, and ${\boxtimes}$ an inner co-corner of $D$, such that the shape of ${\boxtimes}\sqcup D$ is $\gamma^c/\alpha$. We represent elements of ${{\mathrm{DE}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}$ similarly, with ${\boxtimes}$ as an outer co-corner.
We will occasionally refer to the element as $D$ if the position of the ${\boxtimes}$ is understood. Similar remarks apply to ${{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$ and ${{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}$, and we write $({\boxtimes},T)$ or $(T,{\boxtimes})$ (or simply $T$) to denote elements of these sets.
### Connection to tableau promotion
Combinatorially, $\omega$ can be thought of as a commutator of well-known operations on Young tableaux. Computing ${\mathrm{esh}}({\boxtimes},T)$ is equivalent to the following steps:
- **Rectification.** Treat the ${\boxtimes}$ as having value $0$ and being part of a semistandard tableau $\widetilde{T}={\boxtimes}\sqcup T$. Rectify, i.e. shuffle $(S,\widetilde{T})$ to $(\widetilde{T}',S')$, where $S$ is an arbitrary straight-shape tableau.
- **Promotion (see [@bib:StanleyEC2]).** Delete the $0$ of $\widetilde{T}'$ and rectify the remaining tableau. Label the resulting empty outer corner with the number $\ell(\beta)+1$.
- **Un-rectification.** Un-rectify the new tableau by shuffling once more with $S'$. Replace the $\ell(\beta)+1$ by ${\boxtimes}$.
Note that the promotion step corresponds to shuffling the ${\boxtimes}$ past the rest of the rectified tableau. Thus, evacuation-shuffling corresponds to conjugating the promotion operator (on skew tableaux) by rectifying the tableau. Likewise, $\omega$ is the *commutator* of promotion and rectification. [^3]
A local algorithm for evacuation-shuffling {#sec:local-esh}
==========================================
We will now define [***local evacuation-shuffling***]{}, a local rule for computing ${\mathrm{esh}}$. This section is devoted to the definition of the algorithm and proofs of its elementary properties. In Section \[sec:main-result\], we will prove that local evacuation-shuffling is the same as ${\mathrm{esh}}$.
The base case of the algorithm is the [***Pieri case***]{}, where $\beta$ is a one-row partition. In this case, ${\mathrm{esh}}$ was computed in Theorem 5.10 of [@bib:Levinson], and we recall it here. We will give an alternative proof of the Pieri case in Section 4, in part because the complete algorithm relies heavily on our understanding of it.
\[thm:Pieri\] Let $\beta$ be a one-row partition. Then ${\mathrm{esh}}({\boxtimes},T)$ exchanges ${\boxtimes}$ with the nearest $1 \in T$ *prior to it* in reading order, if possible. If, instead, the ${\boxtimes}$ precedes all $1$’s in reading order, ${\mathrm{esh}}$ exchanges ${\boxtimes}$ with the *last* $1$ in reading order (a [***special jump***]{}).
We give two examples, illustrating the possible actions of ${\mathrm{esh}}$ and the more familiar ${\mathrm{sh}}$.
1. If the skew shape contains a (necessarily unique) vertical domino: $${\small \young(::{\times}11,:11,1)} \hspace{0.2cm}\stackrel{\xrightarrow{{\mathrm{esh}}}}{\xleftarrow[\,{\mathrm{sh}}\,]{}} \hspace{0.2cm}{\small \young(::111,:1{\times},1)}$$
2. Otherwise, the action of ${\mathrm{esh}}\circ {\mathrm{sh}}$ cycles the ${\boxtimes}$ through the rows of $\gamma^c/\alpha$: $${\small\young(:::{\times}11,:11,1)} \hspace{0.2cm}\xrightarrow{{\mathrm{esh}}} \hspace{0.2cm}{\small\young(:::111,:1{\times},1)}
\hspace{0.2cm}\xrightarrow{{\mathrm{sh}}} \hspace{0.2cm} {\small\young(:::111,:{\times}1,1)}$$ $${\small\young(:::111,:11,{\times})} \hspace{0.2cm}\xrightarrow{{\mathrm{esh}}} \hspace{0.2cm}{\small\young(:::11{\times},:11,1)}
\hspace{0.2cm}\xrightarrow{{\mathrm{sh}}} \hspace{0.2cm} {\small\young(:::{\times}11,:11,1)}$$
The algorithm
-------------
We now give the definition of the local algorithm.
\[def:algorithm\] Let $({\boxtimes}, T) \in {{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}$. We define *local evacuation-shuffling*, $${\mathrm{local\text{-}esh}}: {{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}\to {{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)},$$ by the following algorithm.
- **Phase 1.** If the ${\boxtimes}$ does not precede all of the $i$’s in reading order, switch ${\boxtimes}$ with the nearest $i$ *prior* to it in reading order. Then increment $i$ by $1$ and repeat Phase 1.
If, instead, the ${\boxtimes}$ precedes all of the $i$’s in reading order, go to Phase 2.
- **Phase 2.** If the suffix from ${\boxtimes}$ is not tied for $(i,i+1)$, switch ${\boxtimes}$ with the nearest $i$ *after it* in reading order whose suffix is tied for $(i,i+1)$. Either way, increment $i$ by $1$ and repeat Phase 2 until $i=\ell(\beta)+1$.
\[rmk:alternate-phase-2\] We will sometimes use the following equivalent description of Phase 2, which we call the [***step-by-step***]{} version of Phase 2:
- [**Phase 2$'$ (step-by-step).**]{} If the suffix from ${\boxtimes}$ is not tied for $(i,i+1)$, switch ${\boxtimes}$ with the nearest $i$ after it in reading order. Repeat this step until the suffix becomes tied for $(i,i+1)$. Then increment $i$ and repeat Phase 2$'$.
Phase 1 is identical to the Pieri case *unless* the Pieri case calls for a special jump.
Note that in Phase 2, it is not obvious that we can find any $i$ with suffix tied for $(i,i+1)$. We show below, however, that $T$ remains ballot (and semistandard) throughout the algorithm. Consequently, the topmost $i$ is such a square (or ${\boxtimes}$ itself, if ${\boxtimes}$ is above this $i$).
In Phase 1, ${\boxtimes}$ moves down and to the left; in Phase 2 (or 2$'$), ${\boxtimes}$ instead moves to the right and up. We refer to the squares occupied by the box during the step-by-step algorithm as the [***evacu-shuffle path***]{}. See Figures \[fig:antidiagonal\] and \[fig:antidiag-evacu-path\] for examples.
Non-local evacuation-shuffling, as defined in Section \[sec:case-of-interest\], has running time $O(|\alpha| \cdot b)$, where $b = \ell(\beta) + \ell(\beta^*)$. The local algorithm does not involve the $\alpha$ shape directly and is faster, with running time $O(b)$. Computing the entire orbit decomposition of $\omega$ on ${{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$, using the local algorithm, therefore takes $O(b \cdot c_{\alpha,{\boxtimes},\beta,\gamma}^{{\scalebox{.3}{\yng(3,3)}}})$ steps. See Corollary \[cor:running-time\].
We use the following terminology for the $i$-th step of ${\mathrm{local\text{-}esh}}$:\
${{\bf \mathrm{Pieri}}}_i$ – a [***regular Pieri jump***]{}, a Phase 1 move in which the ${\boxtimes}$ moves down-and-left.
${{\bf \mathrm{Vert}}}_i$ – a [***vertical slide***]{}, a Phase 1 move in which the ${\boxtimes}$ moves strictly down.
${{\bf \mathrm{Jump}}}_i$ – a [***Phase 2 jump***]{}, a move in Phase $2$ involving the $(i,i+1)$ suffixes.\
When using Phase 2$'$, we will index moves by their position along the evacu-shuffle path. We write:\
${{\bf \mathrm{CPieri}}}_j$ – a [***conjugate Pieri jump***]{}, a Phase 2 move in which the ${\boxtimes}$ moves up-and-right.
${{\bf \mathrm{Horiz}}}_j$ – a [***horizontal slide***]{}, a Phase 2 move in which the ${\boxtimes}$ moves strictly right.\
Thus a Phase 2 jump consists of, in general, a possibly empty sequence of conjugate Pieri moves and horizontal slides.
We also say that $s$ is the [***transition step***]{} if the algorithm switches to Phase 2 while $i = s$. If the algorithm remains in Phase 1 throughout, we say the transition step is $s=\ell(\beta)+1$.
Examples
--------
We give two examples of our algorithm. For an online animation, see [@bib:Gillespie].
\[exa:first-evacu-shuffle\] Let $$T=\young(::::::111,:::{\times}1122,:::1223,:::334,::44,235).$$ We compute ${\mathrm{local\text{-}esh}}({\boxtimes},T)$. We start in Phase 1 with $i=1$, and do a vertical slide past the $1$’s, then a regular Pieri jump past the $2$’s:
$$\young(::::::111,:::{\times}1122,:::1223,:::334,::44,235)\xrightarrow{{{\bf \mathrm{Vert}}}_1}
\young(::::::111,:::11122,:::{\times}223,:::334,::44,235)\xrightarrow{{{\bf \mathrm{Pieri}}}_2}
\young(::::::111,:::11122,:::2223,:::334,::44,{\times}35)$$
Since the ${\boxtimes}$ now precedes all the $3$’s in reading order, we transition to Phase 2. We look for the first $3$ after the ${\boxtimes}$ (or ${\boxtimes}$ itself) whose $(3,4)$-suffix is tied. We interchange the ${\boxtimes}$ with that $3$. We repeat for $4$ (interchanging the ${\boxtimes}$ with the last $4$, in this case). For $5$, the $(5,6)$-suffix of the ${\boxtimes}$ is already tied, since the ${\boxtimes}$ is past all the $5$’s. Thus the ${\boxtimes}$ does not move further. Phase 2 is as follows:
$$\young(::::::111,:::11122,:::2223,:::334,::44,{\times}35)\xrightarrow{{{\bf \mathrm{Jump}}}_3}
\young(::::::111,:::11122,:::2223,:::334,::44,3{\times}5)\xrightarrow{{{\bf \mathrm{Jump}}}_4}
\young(::::::111,:::11122,:::2223,:::33{\times},::44,345) = {\mathrm{local\text{-}esh}}(T)$$
Note that ${{\bf \mathrm{Jump}}}_3$ corresponds in the step-by-step algorithm to ${{\bf \mathrm{Horiz}}}_3$, and the portion of the evacu-shuffle path corresponding to ${{\bf \mathrm{Jump}}}_4$ is the sequence of moves ${{\bf \mathrm{CPieri}}}_4,{{\bf \mathrm{Horiz}}}_5, {{\bf \mathrm{CPieri}}}_6$.
We will see later (Corollary \[cor:transition-step\]) that the transition step of $s=3$ indicates that the partition $\beta = (6,5,4,3,1)$ has an outer co-corner in its third row, and that the evacu-shuffle path formed by the step-by-step algorithm therefore has $s + \beta_s = 7$ boxes, including both endpoints.
As another example, we illustrate the action of $\omega = {\mathrm{sh}}\circ {\mathrm{local\text{-}esh}}$ in the transpose of the Pieri case, where the skew shape is a vertical strip and $\beta=(1,1,\ldots,1)$ is a single column.
Let $$T=\young(::1,:2,:3,{\times},4).$$ The tableau is already in Phase $2$ at the step $i=1$. Since the $(1,2)$-suffix and the $(2,3)$-suffix of the ${\boxtimes}$ are already tied, the next step in the evacu-shuffle path is a ${{\bf \mathrm{CPieri}}}$ move that interchanges the ${\boxtimes}$ with the $3$. At this point all higher suffixes are tied, and we are done. For the shuffle step, the box then slides up the second column via jeu de taquin. We find: $$\omega(T)=\young(::1,:{\times},:2,3,4).$$ The box continues moving from one column to the next in the until it reaches the top. For the final tableau, the evacuation shuffle consists only of Phase 1 moves and returns to $T$. The $\omega$-orbit of $T$ is therefore:
$$\young(::1,:2,:3,{\times},4)\xrightarrow{\ \omega\ } \young(::1,:{\times},:2,3,4) \xrightarrow{\ \omega\ } \young(::{\times},:1,:2,3,4)\xrightarrow{\ \omega\ } \young(::1,:2,:3,{\times},4).$$
Properties preserved by local evacuation shuffling
--------------------------------------------------
We will require the fact that the tableau remains semistandard and ballot during local evacuation-shuffling, and moves past the strip of $i$’s at the $i$th step of the default algorithm.
\[thm:ballotness\] Let $T$, including the ${\boxtimes}$, be a tableau that appears in the step-by-step (Phase 2$'$) computation of ${\mathrm{local\text{-}esh}}({\boxtimes},T_1)$ for some pair $({\boxtimes},T_1)\in {{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$. Then:
- Omitting the ${\boxtimes}$, the reading word of $T$ is ballot.
- Omitting the ${\boxtimes}$, the rows (resp. columns) of $T$ are weakly (resp. strictly) increasing.
- If $T=T_i$ appears just before the $i$-th step of the *default* (not step-by-step) algorithm, then the ${\boxtimes}$ is an outer co-corner of the collection of squares in $T$ having entries $1,\ldots,i-1$, and an inner co-corner of the squares in $T$ having entries $i,\ldots,t$.
We first show that the conditions hold for the tableaux occurring via the default algorithm. Let $T_i$ be the tableau before the $i$-th move, using the default description of Phase 2. Conditions (1)-(3) are clearly satisfied in the starting tableau $T_1$. Now let $i\ge 1$ and suppose $T=T_{i+1}$. Assume for induction that the conditions are satisfied for $T_{i}$.
**Case 1:** Suppose $T_i$ is in Phase 1, i.e., a Phase 1 move is applied to $T_i$ to get $T_{i+1}$.
We first check that $T_{i+1}$ satisfies (2) and (3). Since the move from $T_i$ to $T_{i+1}$ was a vertical slide or Pieri move that switches the ${\boxtimes}$ with the next $i$ in reverse reading order, the old position of the ${\boxtimes}$ is now filled with an $i$. This position must satisfy (2) in $T_{i+1}$, since $T_i$ satisfied condition (3) and the only way an $i$ could be below this square in $T_i$ is if a vertical slide occurs (in which case it’s no longer there in $T_{i+1}$). All other rows and columns clearly still satisfy (2), and by the definition of the Phase 1 moves we see that $T_{i+1}$ satisfies (3) as well.
We now check that $T_{i+1}$ satisfies (1). The effect of the move on the reading word is to move a single $i$ entry later in the word, so we need only check that the $(i-1,i)$-subword is still ballot after the move. This is vacuous if $i=1$, so assume $i \geq 2$.
Let $x$ and $z$ be the positions of ${\boxtimes}$ in $T_i$ and $T_{i+1}$ respectively. The only suffixes affected by the Phase 1 move are the suffixes of squares $y$ that occur weakly after $x$ and strictly before $z$ in reading order. Let $y$ be such a square. Since $i\ge 2$, we know $x$ contained an $i-1$ in $T_{i-1}$, and that this $i-1$ moved later in the reading word to form $T_i$. Since the suffix of $y$ was ballot in $T_{i-1}$, it follows that in $T_i$ the suffix of $y$ has at least one more $i-1$ than $i$. Thus the suffix of $y$ formed by replacing $x$ by $i$ is ballot as well.
**Case 2:** Suppose $T_i$ is in Phase 2, i.e., a Phase 2 move will be applied to $T_i$ to get $T_{i+1}$.
We first show (2). If the ${\boxtimes}$ moves, the condition (3) on $T_i$ shows that the old location, say $x$, of ${\boxtimes}$ becomes semistandard when filled with $i$ in $T_{i+1}$, except possibly if the square just below $x$ is also filled with $i$. If the previous move was Phase 1 or if $i=1$, then this is impossible since then we would stay in Phase 1 using a vertical slide.
Otherwise, if the previous move was Phase 2, assume for contradiction that the square below $x$ contains $i$. Then it contained $i$ in $T_{i-1}$ and $T_i$ as well. Consider the leftmost $i-1$ in $x$’s row in $T_i$, or ${\boxtimes}$ if there are no other $i$’s. Let $y$ be the square below it, demonstrated with $i=2$ below: $$\young(11\cdots 1{\times},y2\cdots 22)$$ We have $y=i$ since the tableau is semistandard. By definition, the suffix from ${\boxtimes}$ in $T_{i}$ is tied for $(i-1,i)$. Hence, the *weak* suffix starting at $y$ is not ballot for $(i-1,i)$. This contradicts ballotness of $T_{i-1}$. Thus $T_{i+1}$ satisfies (2).
To check (3), we wish to show that the new position of ${\boxtimes}$ in $T_{i+1}$, when filled with $i$, was an outer corner of the strip of $i$’s in $T_i$. Indeed, if not then since the $i$’s form a horizontal strip it must be directly to the left of another $i$, which contradicts ballotness of $T_i$ (since the weak suffix of the ${\boxtimes}$ is already tied for $(i,i+1)$, and so the suffix of the $i$ to the right would not be ballot). Since $T_i$ is semistandard, the ${\boxtimes}$ is then also an inner co-corner of the entries larger than $i$ in $T_{i+1}$.
Finally, we check (1), that $T_{i+1}$ is ballot. If the reading word is unchanged by the move, we are done. Otherwise, it has moved a single $i$ earlier in the word. In the latter case we only need to check that the $(i,i+1)$-subword is still ballot after the move.
By definition, we switch the ${\boxtimes}$ (say in position $x$) with the first $i$ after it whose $(i,i+1)$-suffix is tied (say in position $z$). This does not affect any suffix starting before $x$ or weakly after $z$, so let $y$ be a square between $x$ and $z$ in reading order, possibly equal to $x$. If $y$ contains an $i$ in $T_i$, its suffix is not tied before the move, hence has strictly more $i$’s than $i+1$’s. Thus the suffix remains ballot after losing an $i$. Otherwise, let $y'$ be the closest square containing $i$ prior to $y$ in the reading word. Since $T_i$ is semistandard, the suffix from $y$ contains as many $i$’s, and at most as many $i+1$’s, as the suffix from $y'$. Since the latter remains ballot, the former does as well.
This completes Case 2.
Finally, to deduce properties (1) and (2) for the step-by-step algorithm, consider that ${{\bf \mathrm{Jump}}}_i$ corresponds to moving the ${\boxtimes}$ past a portion of the horizontal strip of $i$’s. Since the tableaux before and after the jump are semistandard and ballot, it’s easy to check that each intermediate tableau (arising in Phase 2$'$) is semistandard and ballot as well.
Reversing the algorithm
-----------------------
We now give an algorithm that undoes ${\mathrm{local\text{-}esh}}$.
\[def:reverse-algorithm\] We define the *reverse (local) evacuation-shuffle* of $(T',{\boxtimes}) \in {{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}$ to be the pair $({\boxtimes},T)$ of the same total shape, defined by the following algorithm.
- Set $i=t$.
- **Reverse Phase 2.** If the suffix of the ${\boxtimes}$ has strictly more $i$’s than $i+1$’s, go to Reverse Phase 1. Otherwise, choose the first $i$ (or ${\boxtimes}$) *prior* to the ${\boxtimes}$ in reading order whose weak suffix (including itself) has exactly as many $i-1$’s as $i$’s. If no such entry exists, choose the very first $i$ in reading order. Interchange this choice of $i$ (or ${\boxtimes}$) with the ${\boxtimes}$. Decrement $i$ and repeat this step.
- **Reverse Phase 1.** Switch ${\boxtimes}$ with the nearest $i$ *after* it in reading order. Decrement $i$ and repeat this step until $i=0$.
\[thm:reverse-algorithm\] Reverse local evacuation shuffling is the inverse of local evacuation shuffling.
Let $({\boxtimes},T) \in {{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}$ and put ${\mathrm{local\text{-}esh}}({\boxtimes},T) = (T',{\boxtimes})$. We show that the reverse evacuation shuffle of $(T',{\boxtimes})$ is equal to $({\boxtimes}, T)$. Since ${\mathrm{local\text{-}esh}}$ is a function between sets of the same cardinality, we will be done.
Let $\beta=(\beta_1,\ldots,\beta_t)$ be the content of $T$. Suppose the local evacuation shuffle of $({\boxtimes},T)$ consists of $k$ moves in Phase 1 and $t-k$ in Phase 2. If $k=t$ then the last step is still in Phase 1, coming from a Pieri move across the horizontal strip of $t$’s. Then after this move, the $(t,t+1)$ suffix is not tied because there are no $t+1$’s and there is at least one $t$ after the ${\boxtimes}$. Thus there is no Reverse Phase 2 when applying the reverse algorithm; it starts immediately in Reverse Phase 1.
Otherwise, if $k<t$, the local evacuation shuffle ended with a sequence of ${{\bf \mathrm{Jump}}}$ moves. We show inductively that each Reverse Phase 2 step undoes a Phase 2 step in succession. Suppose that reverse-shuffling past the $t,t-1,\ldots,t-i+1$ leaves us at the end of step $t-i$ of ${\mathrm{local\text{-}esh}}$, and that step $t-i$ was a ${{\bf \mathrm{Jump}}}$ move.
In what remains, let $r=t-i$. Let $T_{r}$ and $T_{r+1}$ be the respective tableaux before and after the ${{\bf \mathrm{Jump}}}_{r}$ step, and let $s$ and $s'$ denote the squares that contain the ${\boxtimes}$ in $T_{r}$ and $T_{r+1}$ respectively. Then $T_{r+1}$ is formed by switching the ${\boxtimes}$ (from position $s$) with the first $r$ after it (in position $s'$) whose $(r,r+1)$ suffix is tied. The Reverse Phase 2 step, backwards past the $r$ strip, would take the ${\boxtimes}$ and switch it with either the first $r$ to the left whose weak $(r-1,r)$ suffix is tied, or the very first $r$ in reading order. We wish to show that this $r$ is in location $s$ in $T_{r+1}$.
First suppose that the $(r-1)$st step was also a ${{\bf \mathrm{Jump}}}$ move. Then in $T_{r}$, the $(r-1,r)$-suffix of the ${\boxtimes}$ is tied. So, in $T_{r+1}$, the *weak* suffix starting at square $s$ is tied for $(r-1,r)$ as well. Assume for contradiction that there were another $r$ between $s$ and $s'$ in reading order whose weak $(r-1,r)$ suffix is tied in $T_{r+1}$. Then in $T_{r}$, that suffix would have strictly more $r$’s than $r-1$, contradicting ballotness of $T_{r}$ (see Lemma \[thm:ballotness\]). Thus the $r$ in square $s$ is the first $r$ to the left of the ${\boxtimes}$ in reading order in $T_{r+1}$ whose weak $(r-1,r)$ suffix is tied, and so the reverse process moves the ${\boxtimes}$ back to square $s$.
Otherwise, if the $(r-1)$st step was a Pieri (Phase 1) move, then in $T_{r}$, the $(r-1,r)$-suffix of the ${\boxtimes}$ cannot be tied, since $T_{r-1}$ is ballot and we replaced the ${\boxtimes}$ with another $r-1$, which adds to that suffix. Notice also that since the $(r)$th step is the first step in Phase 2, the ${\boxtimes}$ must precede all $r$’s in reading order in $T_{r}$. Thus square $s$ is the leftmost $r$ in reading order in $T_{r+1}$, and no other $r$ can have weakly tied $(r-1,r)$ suffix by the same ballotness argument as above. It follows that the reverse move does switch the ${\boxtimes}$ with the $r$ in square $s$ in this case as well.
This shows that the ${{\bf \mathrm{Jump}}}$ moves are undone by the Reverse Phase 2 moves, and that the reverse algorithm switches to Reverse Phase 1 exactly after undoing all the forward Phase 2 moves. It is easy to see that a Reverse Phase 1 move is the inverse of a forward Phase 1 move as well, so this algorithm reverses the local evacuation shuffling algorithm.
The algorithm in Definition \[def:reverse-algorithm\] reverses the ordinary (not step-by-step) algorithm. To reverse the step-by-step algorithm, we simply break each Reverse Phase 2 jump into smaller steps, interchanging ${\boxtimes}$ with each $i$ that precedes it in succession until it reaches the first $i$ whose suffix had exactly as many $i$’s as $i-1$’s (before switching it with ${\boxtimes}$).
Proof of local algorithm {#sec:main-result}
========================
![An example of a rectified tableau $R$ with transition step $s=3$. The rectification path of the box is down to row $s$ and then directly right.[]{data-label="fig:R-diagram"}](Rdiag.pdf)
In this section we prove the following:
\[thm:main-theorem\] Local evacuation-shuffling is the same map as evacuation-shuffling, that is, for any $({\boxtimes},T)\in {{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}$, $${\mathrm{local\text{-}esh}}({\boxtimes},T) = {\mathrm{esh}}({\boxtimes},T).$$
The main idea is as follows. In computing ${\mathrm{esh}}$, when we first rectify $({\boxtimes},T)$, we obtain a tableau $R$ of the form shown in Figure \[fig:R-diagram\]. In particular, the ${\boxtimes}$ is in the inner corner and the total shape of ${\boxtimes}\sqcup R$ is formed by adding an outer co-corner to $\beta$ in some row $s$. When shuffling the ${\boxtimes}$ past $R$, the ${\boxtimes}$ follows a path directly down to row $s$ and then directly over to the end of row $s$, as shown. It turns out that this corresponds to a more refined process in which we shuffle the ${\boxtimes}$ past rows $1,2,\ldots,s-1$, then shuffle it past the $\beta_s$ vertical strips formed by greedily taking vertical strips from the right of the bottom $l(\beta)-s+1$ rows of the tableau. We call this decomposition into horizontal and vertical strips the [***$s$-decomposition***]{}, as illustrated in Example \[ex:s-decompositions\].
We show that each step of Phase 1 of ${\mathrm{local\text{-}esh}}$ corresponds to a single move of the ${\boxtimes}$ past a horizontal strip, and that the transition step is $s$. We then show, using the *antidiagonal symmetry* suggested by Figure \[fig:antidiagonal\], that the movements of the ${\boxtimes}$ during Phase 2 correspond similarly to shuffles past each of the $s$-decomposition’s vertical strips.
\[def:conjugate-pieri\] Let $V$ be a vertical strip, i.e., no row of $V$ contains more than one entry. Let ${\boxtimes}$ be an inner co-corner of $V$. Then we define the [***conjugate move***]{} to be the action of switching the location of the ${\boxtimes}$ with the square of $V$ that comes directly *after* it in reading order.
s-decompositions
----------------
We formalize the notion of an $s$-decomposition and extend it to an arbitrary Littlewood-Richardson tableau as follows.
Let $1 \leq s \leq \ell(\beta)+1$.
1. Let $\beta'$ be obtained by deleting the first $s-1$ rows of $\beta$. Let $r_1, \ldots, r_{s-1}$ be one-row partitions with lengths the first $s-1$ rows of $\beta$, and let $c_s, \ldots, c_t$ be one-column partitions of lengths given by the columns of $\beta'$ in reverse order. (Here $t = \beta_s + s - 1$.) We say that $(r_1,\ldots,r_{s-1},c_s,\ldots,c_t)$ is the [***$s$-decomposition***]{} of the shape $\beta$.
2. Let $T \in {\mathrm{LR}}_\mu^\lambda(\beta)$ be a ballot SSYT. The $s$[***-decomposition of***]{} $T$ is the decomposition of $T$ into its first $s-1$ horizontal strips $H_1,\ldots,H_{s-1}$ where $H_i$ consists of the entries labeled $i$ in $T$, followed by $\beta_s$ vertical strips $V_s, \ldots, V_t$, where $V_{t+1-i}$ contains the $i$-th-from-last instance (when possible), in reading order, of each of the entries $j \geq s$.
The $s$-decomposition of the highest weight filling of $\beta$ will be of particular importance.
\[ex:s-decompositions\] The $3$-decomposition of the tableau $T$ used in Example \[exa:first-evacu-shuffle\] is shown in Figure \[fig:s-decomposition-color\] (note that $3$ is the transition point for the initial position of the ${\boxtimes}$ in that example). Notice that this corresponds to the $s$-decomposition of the rectified tableau of shape $\beta$ shown in Figure \[fig:rectified-s-decomposition\].
![\[fig:s-decomposition-color\] The $3$-decomposition into horizontal and vertical strips of the tableau discussed in Example \[ex:s-decompositions\].](skew-s-decompositions-6-color.pdf){width="15.8cm"}
![\[fig:rectified-s-decomposition\] At left, the $3$-decomposition of $\beta$, where $\beta$ is the rectification shape of the tableau $T$ from Example \[ex:s-decompositions\]. The $3$-decomposition of $T$ is color-coded at right.](s-decomposition.pdf "fig:"){height="2.6cm"} ![\[fig:rectified-s-decomposition\] At left, the $3$-decomposition of $\beta$, where $\beta$ is the rectification shape of the tableau $T$ from Example \[ex:s-decompositions\]. The $3$-decomposition of $T$ is color-coded at right.](skew-s-decomposition.pdf "fig:"){height="2.6cm"}
\[lem:5-facts\] Let $({\boxtimes},T)\in {{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$, and let $H_1,\ldots,H_{s-1},V_s,\ldots,V_{t}$ be its $s$-decomposition. Then
1. $H_1,\ldots,H_{s-1}$ are horizontal strips with $H_i$ extending $H_{i-1}$ for all $i$.
2. $V_s,\ldots,V_{t}$ are vertical strips, with $V_s$ extending $H_{s-1}$ and $V_j$ extending $V_{j-1}$ for all $j$.
3. For any $i$, $H_i$ rectifies to the $i$th row in ${\mathrm{rect}}(T)$.
4. For any $i$, $V_{t-i+1}$ rectifies to the $i$th vertical strip in the $s$-decomposition of ${\mathrm{rect}}(T)$.
To prove (i) and (iii), note that $H_i$ rectifies to the $i$th row of the highest weight filling of $\beta$ since it is filled with all $i$’s in $T$. They form a horizontal strip because $T$ is semistandard.
To prove (iv), let $j\ge s$. If we order the $j$’s in the highest weight filling of $\beta$ in reading order, then they must occur in that order in $T$ as well, since the reading word of $T$ is Knuth equivalent to that of its rectification and Knuth moves cannot switch equal-valued entries. (See [@bib:Fulton] for an introduction to Knuth equivalence.) The vertical strip $\beta^{(t-i+1)}$ in the rectified picture consists of the $i$th copies from the right of each such entry $j$, and so in $T$ the entry $j$ occurring in $V_{t-i+1}$ is still the $i$th from the end.
For (ii), since the reading word of $T$ is ballot, the $i$th-to-last copy of $j$ must occur strictly after the $i$th-to-last copy of $j+1$ for any $j$, and since the tableau is semistandard, this $j+1$ cannot appear strictly to the left of the $j$. It follows that the $j+1$ in $V_{t-i+1}$ appears in a row strictly below the $j$ in $V_{t-i+1}$ for each $j$. Therefore, $V_{t-i+1}$ is a vertical strip for all $i$. Since each of the entries $j\ge s$ appears in $V_k$ before $V_{k+1}$ for all $k$, the strips must extend each other as well. Finally, $V_s$ extends $H_{s-1}$ because it consists of entries larger than $s-1$ and is the first of each of those entries in its row.
Lemma \[lem:highwt-lowwt\] follows from Lemma \[lem:5-facts\] in the case $s=1$. To see this, observe that the $s$-decomposition is, in particular, preserved by jeu de taquin slides applied to $T$. If we *anti-rectify* $T$ to a tableau of shape ${{\scalebox{.3}{\yng(3,3)}}}/\beta^c$, the explicit description of the entries of $V_{t-i+1}$ shows that it forms precisely the $i$-th-rightmost column of ${{\scalebox{.3}{\yng(3,3)}}}/\beta^c$.
Lemma \[lem:5-facts\] allows us to factor Littlewood-Richardson chains into longer chains based on the $s$-decomposition. In particular, for a horizontal strip $H_i$ or vertical strip $V_j$ in an $s$-decomposition, let $\overline{H_i}$ and $\overline{V_j}$ be the corresponding Littlewood-Richardson tableaux of content $r_i$ and $c_j$ respectively formed by decreasing the entries appropriately. We have the following map.
We write $$\iota_s:{{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}\to {\mathrm{LR}}(\alpha,{\boxtimes},r_1,\ldots,r_{s-1},c_s,\ldots,c_t,\gamma)$$ by $$\iota_s(T_\alpha,{\boxtimes},T,T_\beta)=(T_\alpha,{\boxtimes}, \overline{H_1},\ldots,\overline{H_{s-1}},\overline{V_s},\ldots,\overline{V_{t}},T_\gamma)$$ where $(H_i,V_j)$ is the $s$-decomposition of $T$. We define $$\iota_s:{{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}\to {\mathrm{LR}}(\alpha,r_1,\ldots,r_{s-1},c_s,\ldots,c_t,{\boxtimes},\gamma)$$ in a similar fashion.
Note that $\iota_s$ is injective, because the process of reducing the strips into Littlewood-Richardson tableau can be reversed by increasing the entries of each $\overline{H_i}$ by $i-1$ and increasing those of $\overline{V_j}$ by $s-1$. We also claim that shuffling any tableau with $T$ is the same as shuffling past each of the $H_i$ and $V_j$ in sequence. This is proven in the two lemmas that follow.
In these lemmas it is helpful to use the language of dual equivalence classes in place of Littlewood-Richardson tableaux (note that $s$-decompositions and the map $\iota_S$ can be similarly defined on dual equivalence classes, by taking the associated classes of the tableaux at each step.)
\[lem:factor-horiz\] Let $\lambda/\mu$ be a skew shape and $\beta = (\beta_1, \ldots, \beta_r)$ a partition. Let $\beta' = (\beta_2, \ldots, \beta_r)$. Consider the concatenation map on dual equivalence classes, $${\mathrm{DE}}_\mu^\lambda(\beta_1, \beta') \to \bigsqcup_{\tau} {\mathrm{DE}}_\mu^\lambda(\tau), \qquad (D_1, D') \mapsto D_1 \sqcup D',$$ where the union is over $\tau \subseteq \beta'$ with $\tau/\beta'$ a horizontal strip of length $\beta_1$.
There is a unique ‘factorization’ injection, a right inverse to concatenation, $$\iota_H : {\mathrm{DE}}_\mu^\lambda(\beta) \hookrightarrow {\mathrm{DE}}_\mu^\lambda(\beta_1, \beta').$$ It is ‘compatible with shuffling’ in the sense that the following diagram commutes, for any partition $\pi$: $$\xymatrix{
{\mathrm{DE}}_\mu^\lambda(\beta, \pi) \ar[r]^-{\iota_H} \ar[d]_{{\mathrm{sh}}_1} & {\mathrm{DE}}_\mu^\lambda(\beta_1, \beta', \pi) \ar[d]^{{\mathrm{sh}}_1 {\mathrm{sh}}_2} \\
{\mathrm{DE}}_\mu^\lambda(\pi, \beta) \ar[r]^-{\iota_H} & {\mathrm{DE}}_\mu^\lambda(\pi, \beta_1, \beta').
}$$
We think of $\iota_H$ as ‘extracting the highest-weight horizontal strip’ from the inner edge of the shape.
Let $D \in {\mathrm{DE}}_\mu^\lambda(\beta)$. By the Pieri rule, at least one pair $(D_1, D') \in {\mathrm{DE}}_\mu^\lambda(\beta_1, \beta')$ has $D_1 \sqcup D' = D$. We wish to define $i_H(D) := D'$.
Suppose $(E_1, E')$ is another such pair. Let $D_\mu$ be the unique dual equivalence class of straight shape $\mu$. Perform shuffles to obtain $$\begin{aligned}
{\mathrm{sh}}_2{\mathrm{sh}}_1(D_\mu, D_1, D') &= (D_{\beta_1}, \tilde{D'}, \tilde{D_\mu}), \\
{\mathrm{sh}}_2{\mathrm{sh}}_1(D_\mu, E_1, E') &= (D_{\beta_1}, \tilde{E'}, \tilde{E_\mu}).\end{aligned}$$ Concatenation is compatible with shuffling, so $\tilde{D_\mu} = \tilde{E_\mu}$, as both correspond to shuffling $D_\mu$ with $D$. Moreover, we have $\tilde{E'}, \tilde{D'} \in {\mathrm{DE}}_{\beta_1}^\beta(\beta')$, which is a singleton set. (Note that $\beta / \beta_1$ is effectively a straight shape.) So $\tilde{E'} = \tilde{D'}$ and so, after shuffling once more with $\tilde{D_\mu}$, we conclude $(E_1,E') = (D_1,D')$. Finally, $\iota_H$ is compatible with shuffling because concatenation is (and $\iota_H$ is a right inverse to concatenation).
\[lem:factor-other\] Let $c$ be the first column of $\beta$, and let $\beta''$ be $\beta$ with $c$ deleted. There are injections $$\begin{aligned}
\iota_H^\ast : {\mathrm{DE}}_\mu^\lambda(\beta) &\hookrightarrow {\mathrm{DE}}_\mu^\lambda(\beta', \beta_1), \\
\iota_V : {\mathrm{DE}}_\mu^\lambda(\beta) &\hookrightarrow {\mathrm{DE}}_\mu^\lambda(c, \beta''), \\
\iota_V^\ast : {\mathrm{DE}}_\mu^\lambda(\beta) &\hookrightarrow {\mathrm{DE}}_\mu^\lambda(\beta'', c),\end{aligned}$$ where $\iota_H^\ast$ corresponds to extracting the maximal horizontal strip along the outer (southeast) edge of the shape, and $\iota_V, \iota_V^\ast$ extract maximal *vertical* strips from the inner and outer edges, respectively. Each of these is a right inverse to concatenation and is compatible with shuffling.
We obtain $\iota_H^\ast$ by rotating tableaux, that is, $\iota_H^\ast(D) = \iota_H(D^R)^R.$ Similarly, we obtain $\iota_V$ by transposing, and $\iota_V^\ast$ by rotating and transposing.
Notice that rotating and transposing $D$ exchanges $\iota_H$ with $\iota_V^*$. By Lemma \[lem:highwt-lowwt\], it follows that the maximal outer vertical strips extracted by $\iota_V^\ast$ are the same as those of the $1$-decomposition of $\beta$. More generally, $\iota_s$ is the composition of several applications of $\iota_H$ and $\iota_V^\ast$: if $D = {\mathrm{DE}}(T)$, we have $$\iota_s(T) = {\mathrm{LR}}\circ (\iota_V^\ast)^{\beta_s} (\iota_H)^{s-1}(D).$$
We now refine evacuation-shuffling by factoring ${\mathrm{esh}}$ into a sequence of operations $e_1, \ldots, e_{s-1+\beta_s}$, corresponding to an $s$-decomposition.
For a fixed $s$, and for $1 \leq i \leq t = s-1+\beta_s$, we define the [***partial evacuation shuffle***]{} $$e_i : {\mathrm{LR}}(\alpha, r_1, \ldots, {\boxtimes}, r_i, \ldots, c_t, \gamma) \to {\mathrm{LR}}(\alpha, r_1, \ldots, r_i, {\boxtimes}, \ldots, c_t, \gamma)$$ by the composition $$e_i=({\mathrm{sh}}_1{\mathrm{sh}}_2\cdots {\mathrm{sh}}_{i+1}){\mathrm{sh}}_i({\mathrm{sh}}_{i+1}\cdots {\mathrm{sh}}_2{\mathrm{sh}}_1).$$ (If $i \geq s$, the $r_i$ in the definition above should be replaced by $c_i$.)
Combinatorially, $e_i$ is a modified version of evacuation shuffling, where:
1. We rectify the first $i-1$ strips, obtaining a straight shape tableau $B$;
2. We then perform a “relative” evacuation-shuffle on ${\boxtimes}$ and the $i$-th strip: we rectify them only up to the outer boundary of $B$, then shuffle and un-rectify.
\[lem:partial-esh\] For any $T\in {{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$, and any $s$, we have $$\iota_{s}({\mathrm{esh}}(T))=e_t\cdots e_1(\iota_{s}(T)).$$
Recall that ${\mathrm{esh}}:{{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}\to {{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}$ is the composition $$\xymatrix{{{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}\ar[r]^{{\mathrm{sh}}_2{\mathrm{sh}}_1} & {\mathrm{LR}}({\scalebox{.5}{\yng(1)}},\beta,\alpha,\gamma) \ar[r]^{{\mathrm{sh}}_1} &
{\mathrm{LR}}(\beta,{\scalebox{.5}{\yng(1)}},\alpha,\gamma) \ar[r]^{{\mathrm{sh}}_1{\mathrm{sh}}_2}
& {{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}}$$
The maps $\iota_H$ and $\iota_V^\ast$ respect shuffling (in the sense stated in Lemmas \[lem:factor-horiz\] and \[lem:factor-other\], translated from dual equivalence classes to the corresponding Littlewood-Richardson tableaux). We thus write $$\xymatrix{
{{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}\ar[r]^-{\iota_{s}} \ar[d]^{{\mathrm{sh}}_2{\mathrm{sh}}_1} &
{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},r_1, \ldots, c_t,\gamma)
\ar[d]^{{\mathrm{sh}}_{t+1}\cdots{\mathrm{sh}}_2{\mathrm{sh}}_1} \\
{\mathrm{LR}}({\scalebox{.5}{\yng(1)}},\beta,\alpha,\gamma)
\ar[r]^-{\iota_{s}} \ar[d]^{{\mathrm{sh}}_1} &
{\mathrm{LR}}({\scalebox{.5}{\yng(1)}},r_1, \ldots, c_t,\alpha,\gamma)
\ar[d]^{{\mathrm{sh}}_t\cdots{\mathrm{sh}}_1} \\
{\mathrm{LR}}(\beta,{\scalebox{.5}{\yng(1)}},\alpha,\gamma)
\ar[r]^-{\iota_{s}} \ar[d]^{{\mathrm{sh}}_1{\mathrm{sh}}_2} &
{\mathrm{LR}}(r_1, \ldots, c_t,{\scalebox{.5}{\yng(1)}},\alpha,\gamma)
\ar[d]^{{\mathrm{sh}}_1{\mathrm{sh}}_2\cdots{\mathrm{sh}}_{t+1}}\\
{{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}\ar[r]^-{\iota_{s}} &
{\mathrm{LR}}(\alpha,r_1, \ldots, c_t,{\scalebox{.5}{\yng(1)}},\gamma)
}$$
Thus we have $$\iota_{s} \circ {\mathrm{esh}}=({\mathrm{sh}}_1{\mathrm{sh}}_2\cdots {\mathrm{sh}}_{t+1})({\mathrm{sh}}_{t}\cdots{\mathrm{sh}}_1)({\mathrm{sh}}_{t+1}\cdots {\mathrm{sh}}_2{\mathrm{sh}}_1)\circ \iota_{s}.$$
We now write out the composition of the $e_i$’s as the reverse-ordered product $$e_t\cdots e_1=\prod_{i=t}^1({\mathrm{sh}}_1\cdots {\mathrm{sh}}_{i+1}){\mathrm{sh}}_i({\mathrm{sh}}_{i+1}\cdots{\mathrm{sh}}_1).$$
Notice that, since the shuffles are all involutions, the right-hand term of the $i$-th factor mostly cancels with the left-hand term of the $(i-1)$-st factor. After all such cancellations, we are left with the product $$({\mathrm{sh}}_1\cdots{\mathrm{sh}}_{t+1})({\mathrm{sh}}_{t}{\mathrm{sh}}_{t+1})({\mathrm{sh}}_{t-1}{\mathrm{sh}}_{t})\cdots ({\mathrm{sh}}_3{\mathrm{sh}}_4)({\mathrm{sh}}_{2}{\mathrm{sh}}_{3})({\mathrm{sh}}_{1}{\mathrm{sh}}_2){\mathrm{sh}}_1.$$
Recall that ${\mathrm{sh}}_i$ commutes with ${\mathrm{sh}}_j$ whenever $|i-j|\ge 2$. Thus we can move the rightmost ${\mathrm{sh}}_3$ past the ${\mathrm{sh}}_1$ next to it, then move the rightmost ${\mathrm{sh}}_4$ past the ${\mathrm{sh}}_2$ and ${\mathrm{sh}}_1$ to its right, and so on. We obtain the product $$({\mathrm{sh}}_1\cdots {\mathrm{sh}}_{t+1}){\mathrm{sh}}_t\cdots {\mathrm{sh}}_1 ({\mathrm{sh}}_{t+1}\cdots {\mathrm{sh}}_1).$$ This matches our expression for ${\mathrm{esh}}$ above.
We emphasize that, for each choice of $s$, we have a *distinct* factorization of ${\mathrm{esh}}$ into partial evacuation-shuffles as above. In our proof of Theorem \[thm:main-theorem\], we cannot use the same choice of $s$ for all $({\boxtimes},T) \in {{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$. Our proof relies on a careful choice of $s$ depending on $({\boxtimes},T)$, which will make the partial steps $e_i$ correspond to the steps of *local* evacuation-shuffling for the particular pair $({\boxtimes},T)$.\
The Pieri Case.
---------------
We now give the proof of Theorem \[thm:Pieri\], the Pieri case. We give a more detailed statement:
Let $\beta=(m)$ be a one-row partition.
1. Suppose $\gamma^c/\alpha$ is *not* a horizontal strip. Then $\gamma^c/\alpha$ contains a unique vertical domino; otherwise there is no semistandard filling of $\gamma^c/\alpha$ using a ${\boxtimes}$ and $1$’s.
In this case, ${{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$ and ${{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}$ have one element each, since the ${\boxtimes}$ must be at the top or bottom of the domino. Then ${\mathrm{esh}}$ slides the $\boxtimes$ down.
2. Suppose $\gamma^c/\alpha$ is a horizontal strip having $r$ nonempty rows. There is a natural ordering[^4] of the tableaux $${{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}= \{L_1, \ldots, L_r\},$$ where $L_i$ is the tableau having $\boxtimes$ at the left end of the $i$th row of $\gamma^c/\alpha$. Likewise, $${{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}= \{R_1, \ldots, R_r\},$$ where $R_i$ is the tableau having $\boxtimes$ at the right end of the $i$th row of $\gamma^c/\alpha$.
We have the following: $$\begin{aligned}
{\mathrm{esh}}(L_i) &= R_{i+1} \pmod{r}\end{aligned}$$ We will say that ${\mathrm{esh}}(L_r) = R_1$ is a [***special jump***]{}, and any other application of ${\mathrm{esh}}$ to $L_i$ for $i\neq 1$ is [***non-special***]{}.
Part 1 is clear because ${\mathrm{esh}}$ is a bijection between two one-element sets.
For Part 2, it is clear that these are the only fillings. So, it suffices to show that ${\mathrm{esh}}(L_i)=R_{i+1}$ for any $i$, where the indices are taken modulo $r$. We will show this by induction on the size of $\alpha$.
For the base case, if $\alpha=\emptyset$, then since we are in the case of Part 2, the total shape of the ${\boxtimes}$ and the tableau is a single row of length $m+1$. (The other possibility is that the total partition shape is $(m,1)$, and the ${\boxtimes}$ slides up and down between the two squares of the first column, which is in Case 1.) So, ${{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$ and ${{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}$ both have one element, $L_1$ and $R_1$ respectively, and so $L_1$ must go to $R_1$ under ${\mathrm{esh}}$ and we are done. Notice that this base case is a special jump.
Now, suppose the theorem holds for a given $\alpha$, and we wish to show it holds for a partition $\alpha'$ formed by adding an outer co-corner to $\alpha$. Let $T'\in {\mathrm{LR}}(\alpha',{\scalebox{.5}{\yng(1)}},\beta,\gamma')$ for some $\beta$ and $\gamma'$, and let $T\in {{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$ be the tableau formed by the first jeu de taquin slide on $T'$ in the rectification of $({\boxtimes},T')$ in the evacuation-shuffle, where we start with the unique outer co-corner of $\alpha$ that is contained in $\alpha'$. Here $\gamma$ is formed from $\gamma'$ by adding the unique corner determined by this slide.
Note that $T'=L_i'$ for some $i$, where $L_i'$ is the tableau having the ${\boxtimes}$ in the $i$th row from the top in $(\gamma')^c/\alpha'$. Defining $R_i'$ similarly, we wish to show ${\mathrm{esh}}(L_i')=R_{i+1}'$ with the indices mod $r$.
Recall that ${\mathrm{esh}}$ is the procedure of rectifying $({\boxtimes},T')$, shuffling the box past the rectified tableau, and then unrectifying both using the reverse sequence of slides. Let $S\in {{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}$ be the tableau preceding the last unrectification step in forming $S'={\mathrm{esh}}(T')$. These steps necessarily involve the same inner and outer co-corners, and so $S$ and $T$ have the same shape. Furthermore, ${\mathrm{esh}}(T)=S$ by the definition of ${\mathrm{esh}}$, and so by the induction hypothesis $S$ is formed from $T$ by one of the two Pieri rules.
We will use this to show that $S'$ is formed from $T'$ by the same rules, by considering the rectification/unrectification step that relates them to $S$ and $T$ respectively. We consider the cases of a special jump and a non-special jump separately. Let $r$ be the number of nonempty rows of $(\gamma')^c/\alpha'$.
**Case 1:** Suppose $T'=L_i'$ for some $i\neq r$. The tableau $T$ is formed by a single inwards jeu de taquin slide on $T'$, which can either be on the inner co-corner just to the left of the ${\boxtimes}$ or not.
If the inner co-corner we start at is to the left of the ${\boxtimes}$ in $T'$, then since we assumed our shape has no vertical domino, the entire row containing the ${\boxtimes}$, say row $r$, simply slides to the left to form $T$. Then by the induction hypothesis, $S$ has the ${\boxtimes}$ at the end of the next row down, either just below the ${\boxtimes}$ in $T$ (the vertical domino case) or to its left. Clearly $S'$ is formed by sliding the new contents of row $r$ back to the right, and so $S'=R'_{i+1}$ as desired.
Otherwise, if the inner co-corner we start at is not to the left of the ${\boxtimes}$ in $T'$, the inwards slide consists of either (a) sliding a horizontal row of $1$’s to the left, if the co-corner is to the left of but not above a $1$, (b) sliding a $1$ on an outer corner up by one row, if the co-corner is just above this $1$.
In the subcase (a), the number of rows remains unchanged and $T=L_i$. Thus $S=R_{i+1}'$ by the induction hypothesis and we are done. For (b), the number of rows either remains the same and we are done again, or the $1$ that we moved up forms a new row. If the new row is above the ${\boxtimes}$, then $T=L_{i+1}$, by the induction hypothesis $S=R_{i+2}$, and $S'$ is formed by moving the $1$ back down and therefore $S'=R_{i+1}'$, as desired. Otherwise, if the new row is below the ${\boxtimes}$, we have $T=L_i$ and $S=R_{i+1}$, keeping in mind that if the ${\boxtimes}$ is in row $i$ in $T$ then row $i+1$ is the new row and the ${\boxtimes}$ is in this new square in $S$. Therefore we again have $S'=R_{i+1}'$, and we are done.
**Case 2:** Suppose $T'=L_r'$. Then the ${\boxtimes}$ is weakly below and strictly to the left of all other entries. Notice that any inwards jeu de taquin slide does not change this property; hence $T=L_q$ where $q$ is the bottom row of $T$. Then $S=R_1$ by the induction hypothesis, and by the same argument, any outwards jeu de taquin slide doesn’t change the property of the ${\boxtimes}$ being weakly above and strictly to the right of the rest of the entries in $S$. Hence $S'=R'_1$, as desired.
For use in Section \[sec:main-result\], we describe how to determine the outcome of the Pieri case based on the location of the ${\boxtimes}$ in *either* the original skew tableau *or* its rectification:
\[prop:pieri-criteria\] Let $\beta=(m)$. The following are equivalent for $T \in {{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$:
- Applying ${\mathrm{esh}}$ results in a special jump;
- The ${\boxtimes}$ precedes the rest of $T$ in reading order;
- The rectification of $T$, including the ${\boxtimes}$, forms a horizontal strip.
This follows immediately from the proof of the Pieri case.
The proof of Theorem \[thm:main-theorem\]
-----------------------------------------
[**Step 1**]{}. Fix $({\boxtimes},T) \in {{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}$. We choose $s = s({\boxtimes},T)$ as follows: consider ${\mathrm{sh}}_1 ({\mathrm{sh}}_2 {\mathrm{sh}}_1({\boxtimes},T))$, the tableau obtained by rectifying, then shuffling ${\boxtimes}$ past $T$. Let $s$ be the row containing ${\boxtimes}$. We will use the $s$-decomposition with this choice of $s$, and compute the effect of $e_t \cdots e_1$ on $({\boxtimes},\iota_s(T))$. We write $$\iota_s(T) = (H_1,\ldots, H_{s-1}, V_s, \ldots, V_t).$$
We note that, if we rectify and shuffle the ${\boxtimes}$ past the entirety of $\iota_s(T)$, the shuffle path of the ${\boxtimes}$ through the rectification of $\iota_s(T)$ is to move (one square at a time) down to row $s$, then over to the right. (See Figure \[fig:R-diagram\].)\
[**Step 2**]{}. We show that ${\mathrm{esh}}$ and ${\mathrm{local\text{-}esh}}$ agree up to Phase 1.
\[lem:phase1-agrees\] Suppose $s > 1$ and let $1 \leq i \leq s-1$. Then $T_i = e_i \cdots e_1({\boxtimes},\iota_s(T))$ agrees with the result of applying $i$ Phase 1 local evacuation-shuffle moves to $({\boxtimes},T)$.
Assume the statement holds for $i-1$ (this is vacuous for $i=1$) and write $$T_{i-1} = e_{i-1} \cdots e_1({\boxtimes},\iota_s(T)) = (H'_1, \ldots, H'_{i-1}, {\boxtimes}, H_i, \ldots, H_{s-1}, V_s, \ldots, V_t.$$ In $T_i$, the ${\boxtimes}$ lies between $H_{i-1}$ and $H_i$.
We compute $e_i(T_{i-1})$. For simplicity, let $H'$ be the concatenation of $H'_1, \ldots, H'_{i-1}$. We are effectively computing $$\xymatrix{
(T_\alpha,H',{\boxtimes},H_i, \cdots) \ar@{|->}[d]_-{{\mathrm{sh}}_3{\mathrm{sh}}_2{\mathrm{sh}}_1} \\
(H'',{\boxtimes},H_i',T'_\alpha, \cdots) \ar@{|->}[d]_-{{\mathrm{sh}}_2} \\
(H'',H''_i,{\boxtimes},T'_\alpha, \cdots) \ar@{|->}[d]_-{{\mathrm{sh}}_1{\mathrm{sh}}_2{\mathrm{sh}}_3} \\ (T_\alpha,H''',H'''_i,{\boxtimes}, \cdots). }$$ By our definition of $s$, in the partial rectification $H'' \sqcup {\boxtimes}\sqcup H'_i$, the ${\mathrm{sh}}_2$ step causes the box to move down to row $i+1$ (since $i \leq s-1$).
In particular, we see that $S = {\boxtimes}\sqcup H'_i$ forms a straight shape in the partial rectification. Shuffling ${\boxtimes}$ and $H'_i$ does not change the overall (trivial) dual equivalence class of $S$; consequently, $e_i$ has no effect on the dual equivalence classes of $T_i$ other than the individual classes of ${\boxtimes}$ and $H_i$.
Moreover, since $e_i$ rectifies $({\boxtimes},H_i)$ to a straight shape, then shuffles and un-rectifies, it must have the same effect as simply applying ${\mathrm{esh}}$ to the pair $({\boxtimes},H_i)$, i.e. the Pieri Case. Moreover, in the rectification, the ${\boxtimes}$ shuffled downward rather than right, so by Proposition \[prop:pieri-criteria\], we see that *prior* to rectifying, there was an $i$ below the ${\boxtimes}$. Thus we are in the non-special Pieri case, which agrees with the $i$-th (Phase 1) step of ${\mathrm{local\text{-}esh}}$.
The transition step of ${\mathrm{local\text{-}esh}}({\boxtimes},T)$ is $s$.
By a similar argument to the previous lemma, we see that, had we used the $(s+1)$-decomposition rather than the $s$-decomposition, then applying $e_s$ would, after rectifying, slide the ${\boxtimes}$ all the way to the right through the $s$-th row. This is the ‘special jump’ of the Pieri case (which would not agree with the behavior of ${\mathrm{local\text{-}esh}}$). By Proposition \[prop:pieri-criteria\], this occurs only when, *prior* to rectifying, there are no $s$’s below the ${\boxtimes}$. This is precisely the condition for ${\mathrm{local\text{-}esh}}$ to transition at step $s$.
[**Step 3**]{}. We have shown that ${\mathrm{local\text{-}esh}}$ and ${\mathrm{esh}}$ agree up to the transition point of ${\mathrm{local\text{-}esh}}$, and that this corresponds to the bend in the shuffle path of the ${\boxtimes}$ through the rectification of $\iota_s(T)$. We are left with determining the effects of $e_s, \ldots, e_t$.
\[lem:antidiag\] For $i \geq s$, $e_i$ corresponds to a *conjugate move* across the vertical strip $V_i$, as in Definition \[def:conjugate-pieri\].
We will prove this for all remaining steps simultaneously. Put $M = e_{s-1} \cdots e_1(T)$. We have $$M = (T_\alpha, H'_1, \ldots, H'_{s-1},{\boxtimes},V_s, \ldots, V_t, T_\gamma^R),$$ where each $H'_i$ is a horizontal strip and each $V_i$ a vertical strip. Let $M_H, M_V$ be the concatenations of the $H'_i$’s and of the $V_j$’s. (We note that $M_H$ is the union of the first $s-1$ strips of $T$ at the transition point of ${\mathrm{local\text{-}esh}}$, and that $M_V$ is simply the rest of the tableau.)
The remainder of the computation corresponds to partial-evacuation-shuffling the ${\boxtimes}$ past $M_V$, $${\mathrm{esh}}(T) = {\mathrm{sh}}_1{\mathrm{sh}}_2{\mathrm{sh}}_3\circ {\mathrm{sh}}_2 \circ {\mathrm{sh}}_3{\mathrm{sh}}_2{\mathrm{sh}}_1(T_\alpha, M_H, {\boxtimes}, M_V, T_\gamma^R).$$
We know that, in ${\mathrm{sh}}_3{\mathrm{sh}}_2{\mathrm{sh}}_1(M)$, the ${\boxtimes}$ and $M_V$ class form a straight shape. Thus, by similar reasoning to the proof of Lemma \[lem:phase1-agrees\], the remaining computation is the same as *ordinary* – not partial – evacuation-shuffling the pair $({\boxtimes},M_V)$. Note that in this (smaller) computation, the ${\boxtimes}$ slides right after rectifying, i.e. ${\mathrm{local\text{-}esh}}({\boxtimes},M_V)$ begins in Phase 2, so our earlier results do not apply. However, by Lemma \[lem:outer-esh\], we may instead write $${\mathrm{esh}}({\boxtimes},M_V) = {\mathrm{sh}}_3{\mathrm{sh}}_4\circ {\mathrm{sh}}_3 \circ {\mathrm{sh}}_4{\mathrm{sh}}_3({\boxtimes},M_V),$$ i.e. we may instead anti-rectify outwards, then shuffle and return. To simplify the situation, we ‘rotate and transpose’, obtaining $$(M_V',{\boxtimes}) = {\mathrm{LR}}\big( {\mathrm{DE}}\big({\boxtimes},M_V\big){}^{R*} \big),$$ Note that the vertical strips of $M_V$ correspond to the horizontal strips of $M_V'$ after this transformation. That is, $M_V'$ has entries $i$ in the squares of the antidiagonal reflection of the strip $V_{t+1-i}$. In the rectification of $({\boxtimes},M_V)$, the ${\boxtimes}$ was to the left of one square from each $V_i$. So the anti-rectification of $M$ has the ${\boxtimes}$ in the leftmost corner, and so (by reflecting over the antidiagonal) the rectification of $(M_V',{\boxtimes})$ has the ${\boxtimes}$ at the bottom of the first column:
We set $({\boxtimes},N') = {\mathrm{esh}}(M_V',{\boxtimes})$ and we compare ${\mathrm{esh}}({\boxtimes},N')$ to ${\mathrm{local\text{-}esh}}({\boxtimes},N')$. By the above observation, ${\mathrm{esh}}({\boxtimes},N')$ corresponds to a local evacuation-shuffle that stays entirely in Phase 1. Thus, by our existing Lemmas on evacuation-shuffling in Phase 1, we see that the *partial* evacuation-shuffles of ${\boxtimes}$ through $N'$ correspond to non-special Phase 1 moves applied to the skew tableau. Reflecting back to our original setting $({\boxtimes},M_V)$, we deduce that the remaining Phase 2 *partial* evacuation shuffles result in successive non-special conjugate moves of the ${\boxtimes}$ through the strips $V_s, \ldots, V_t$.
[**Step 4**]{}. Finally, we prove that the description of $e_s, \ldots, e_t$ corresponding to conjugate Pieri moves produces the same ${\boxtimes}$ movements as Phase 2 of local evacuation-shuffling. Note that this step involves only ballot tableaux, not dual equivalence classes.
Conjugate moves correspond to nontrivial movements of the ${\boxtimes}$, in Phase 2, through its evacu-shuffle path.
First, notice that the Phase 2 algorithm, as described in Remark \[rmk:alternate-phase-2\], can be stated as follows. Starting with $i=s$, at each step choose the smallest $k\ge i$ for which the $(k,k+1)$ suffix of the ${\boxtimes}$ is *not* tied, and then switch the ${\boxtimes}$ with the first $k$ that occurs after it in reading order, incrementing $i$ to $k+1$ and repeating. We will show that shuffling past the $V_j$’s using conjugate moves does the same thing.
Suppose we are moving the ${\boxtimes}$ across the strip $V_{j+1}$. Then either on the previous move it switched places with an element $i$ in $V_j$, or $i=s$ and it is at the start of Phase 2. We first show that the ${\boxtimes}$ switches with an element $k\ge i$ by considering these two cases separately.
**Case 1.** If $i=s$ and it is the start of Phase 2, the next move will switch with an element $k\ge i=s$ because the vertical strips all have entries of size at least $s$.
**Case 2.** If the ${\boxtimes}$ just switched with an $i$ in $V_j$, then there exists an $i$ in $V_{j+1}$ because the $V_j$’s weakly increase in length as $j$ increases by the definition of $s$-decomposition. Furthermore, the $i$ in $V_{j+1}$ occurs after the $i$ in $V_j$ in reading order, so the ${\boxtimes}$ switches with an entry $k$ in $V_j$ which weakly precedes the $i$ in reading order. We must therefore have $k\ge i$ since $V_j$’s entries increase down the strip.
Finally, in either case, suppose $k'$ is an index with $i\le k'<k$. Then the $k'$ and $k'+1$ in $V_{j+1}$ both occur later than ${\boxtimes}$ in the reading word before the move, and by the definition of $s$-decomposition this means that the $k'$ and $k'+1$ in each later strip $V_{j'}$ also occur after the ${\boxtimes}$. Hence the $(k',k'+1)$ suffix of ${\boxtimes}$ is tied prior to the move. Since the $k+1$ in $V_{j+1}$ precedes the ${\boxtimes}$, the $(k,k+1)$ suffix is not tied.
So indeed, the ${\boxtimes}$ switches with the smallest $k\ge i$ for which the $(k,k+1)$ suffix is tied.
This completes the proof of Theorem \[thm:main-theorem\].
Since we work with semistandard tableaux, a natural question is to ask what happens if we use only horizontal strips to factor ${\mathrm{esh}}$ (i.e. if we attempt to use the $(\ell(\beta)+1)$-decomposition for all $T$) rather than the appropriate $s$-decomposition. In fact, an earlier version of our algorithm used this approach; its proof relied on the ‘un-rectification’ method, as demonstrated for the Pieri Case. However, the proof is more difficult, and there are drawbacks to the resulting local algorithm. The most notable are that it does not preserve ballotness at the intermediate steps, and, after Phase 1, it consists of 3-cycles rather than simple transpositions switching the ${\boxtimes}$ with one other entry. These drawbacks make the applications to K-theory (see Section \[sec:K-theory\]) harder or impossible to deduce.
Corollaries on Evacuation-Shuffling {#sec:corollaries-evacu-shuffling}
-----------------------------------
For each of the following corollaries, let $({\boxtimes},T) \in {{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$.
\[cor:transition-step\] Suppose the transition step of ${\mathrm{local\text{-}esh}}({\boxtimes},T)$ is $s$. Then $\beta$ has an outer co-corner in row $s$, and the evacu-shuffle path of the ${\boxtimes}$ has length exactly $s+\beta_s$, including the initial and final locations of the ${\boxtimes}$.
From the proof of Theorem \[thm:main-theorem\], the ${\boxtimes}$ ends up in the square $(s,\beta_s+1)$ after rectifying and shuffling past $T$. Thus, this square is an outer co-corner of $\beta$.
From the local description of ${\mathrm{esh}}$, the box moves through a total of $s-1$ squares in Phase 1. From the description of Phase 2 in terms of conjugate moves, the ${\boxtimes}$ moves through $\beta_s$ squares in Phase 2.
\[cor:antidiag-evacu-path\] Define $(T^{R\ast},{\boxtimes})$ by rotating and transposing $({\boxtimes},{\mathrm{DE}}(T))$, then taking its highest-weight representative. Similarly, for $(S,{\boxtimes}) \in {{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}$, define $({\boxtimes},S^{R\ast})$ the same way. Then: $${\mathrm{local\text{-}esh}}({\boxtimes},T) = (S,{\boxtimes}) \qquad \text{ iff } \qquad (T^{R\ast},{\boxtimes}) = {\mathrm{local\text{-}esh}}({\boxtimes},S^{R\ast}).$$ Moreover, the evacu-shuffle path of the ${\boxtimes}$ for ${\mathrm{local\text{-}esh}}({\boxtimes},S^{R\ast})$ is the antidiagonal reflection of the evacu-shuffle path of the ${\boxtimes}$ for ${\mathrm{local\text{-}esh}}({\boxtimes},T)$.
See Figure \[fig:antidiag-evacu-path\] for an example of this phenomenon.
As a map on dual equivalence classes, ${\mathrm{esh}}$ is its own inverse, and it commutes with transposing and rotating (since shuffling does). However, since the ${\boxtimes}$ is then on the opposite side of the tableau, ${\mathrm{esh}}$ corresponds by our main theorem to ${\mathrm{local\text{-}esh}}^{-1}$. Thus ${\mathrm{local\text{-}esh}}({\boxtimes},S^{R\ast}) = (T^{R\ast},{\boxtimes})$.
To see that the evacu-shuffle paths are the same, we compare $s$-decompositions. Observe that rotating and transposing interchanges the functions $\iota_H$ and $\iota_V^\ast$ of Lemma \[lem:factor-other\]. So, the $s$-decomposition of $T^{R\ast}$ corresponds to a ‘dual’ $s$-decomposition of $T$, $$\iota^\ast_s(T) = (\iota_H)^{s-1} \circ (\iota_V^\ast)^{\beta_s}(T) \ \big(= \iota_s(T^{R\ast})^{R\ast}\big),$$ where we extract the $\beta_s$ vertical strips first, *then* extract the $s-1$ horizontal strips.
Consider rectifying and shuffling $({\boxtimes},T)$. It is easy to see that the shuffle path is the same for both $\iota_s(T)$ and $\iota_s^\ast(T)$. The proof of Theorem \[thm:main-theorem\] then shows that the partial evacuation-shuffles corresponding to $\iota_s^\ast$ give the same Pieri and conjugate-Pieri moves as those corresponding to $\iota_s$.
The following are equivalent:
- The transition step of ${\mathrm{local\text{-}esh}}({\boxtimes},T)$ is $s$.
- Let $({\boxtimes},T')$ be the ‘transposed class’, obtained by transposing $({\boxtimes},{\mathrm{DE}}(T))$, then taking the highest-weight representative. Then the transition step of ${\mathrm{local\text{-}esh}}$ on $({\boxtimes},T')$ is $\beta_s+1$.
- Let $(T'',{\boxtimes})$ be the ‘rotated class’, obtained by rotating $({\boxtimes},{\mathrm{DE}}(T))$, then taking the highest-weight representative. Then the transition step of ${\mathrm{local\text{-}esh}}^{-1}$ on $(T'',{\boxtimes})$ is $s$.
To see that (i) implies (ii), note that shuffling commutes with transposing dual equivalence classes, so in Step 1 of the proof of Theorem \[thm:main-theorem\], we find that the ${\boxtimes}$ is in the square $(\beta_s+1,s)$ after rectifying and shuffling. This means the transition step of $({\boxtimes},T')$ will be $\beta_s+1$. The same reasoning with $T$ and $T'$ exchanged shows (ii) implies (i).
To see that (ii) implies (iii), we use the previous corollary. Transposing *and* rotating exchanges the Phase 1 and Phase 2 portions of the evacu-shuffle path. But the length of the Phase 1 portion of the path is exactly the value of the transition step. As above, the same reasoning with $({\boxtimes},T)$ and $(T'',{\boxtimes})$ exchanged shows (iii) implies (ii).
Finally, we briefly consider the running time of ${\mathrm{local\text{-}esh}}$. We assume the set ${{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$ is given along with, for each $({\boxtimes}, T)$, the $1$-decomposition of $T$ into vertical strips. (Computing this decomposition in advance does not increase the asymptotic running time of computing ${{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$, since it can be obtained by simply labeling each $i$ with its distance from the end of its horizontal strip as the tableau is generated.)
\[cor:running-time\] Given ${{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$ as above, computing ${\mathrm{local\text{-}esh}}$ takes $O(b)$ steps, where $b = \ell(\beta) + \ell(\beta^\ast)$. Computing the entire orbit decomposition of $\omega$ takes $O(b \cdot |{{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}|)$ steps.
We compute ${\mathrm{local\text{-}esh}}({\boxtimes},T)$ directly, for any transition step $s$, updating the $1$-decomposition at the same time. Note that during a Phase 1 move, the $i$ that switches with ${\boxtimes}$ remains part of the same *vertical* strip, since its position among the $i$’s in reading order is unchanged. Thus, we apply Phase 1 moves until the transition step, updating the vertical strips accordingly.
For Phase 2, note that the $s$-decomposition is simply the $1$-decomposition with all squares of value less than $s$ deleted. We may thus compute conjugate Pieri moves using the 1-decomposition.
Note that there are at most $\ell(\beta) + \ell(\beta^\ast)$ steps in all.
[m[1cm]{} @[$=\ $]{} m[3.5cm]{} m[2.5cm]{} @[$=\ $]{} m[3cm]{}]{} $\ \ \ \ T $ & ![An example of antidiagonal reflection. The dual equivalence classes of (the standardizations of) $T$ and $T^{R\ast}$ are antidiagonal reflections of one another, as are those of $S$ and $S^{R\ast}$. By Corollary \[cor:antidiag-evacu-path\], their evacu-shuffle paths are likewise antidiagonally-reflected. []{data-label="fig:antidiag-evacu-path"}](T.pdf "fig:") & $\ \xrightarrow{{\mathrm{local\text{-}esh}}} \ \ \ S $ & ![An example of antidiagonal reflection. The dual equivalence classes of (the standardizations of) $T$ and $T^{R\ast}$ are antidiagonal reflections of one another, as are those of $S$ and $S^{R\ast}$. By Corollary \[cor:antidiag-evacu-path\], their evacu-shuffle paths are likewise antidiagonally-reflected. []{data-label="fig:antidiag-evacu-path"}](S.pdf "fig:")\
$\ T^{R\ast} $ & ![An example of antidiagonal reflection. The dual equivalence classes of (the standardizations of) $T$ and $T^{R\ast}$ are antidiagonal reflections of one another, as are those of $S$ and $S^{R\ast}$. By Corollary \[cor:antidiag-evacu-path\], their evacu-shuffle paths are likewise antidiagonally-reflected. []{data-label="fig:antidiag-evacu-path"}](TR.pdf "fig:") & $\ \xleftarrow{{\mathrm{local\text{-}esh}}}\ \ S^{R\ast} $ & ![An example of antidiagonal reflection. The dual equivalence classes of (the standardizations of) $T$ and $T^{R\ast}$ are antidiagonal reflections of one another, as are those of $S$ and $S^{R\ast}$. By Corollary \[cor:antidiag-evacu-path\], their evacu-shuffle paths are likewise antidiagonally-reflected. []{data-label="fig:antidiag-evacu-path"}](SR.pdf "fig:")
Connections to K-theory {#sec:K-theory}
=======================
Background on K-theoretic (genomic) tableaux
--------------------------------------------
We recall the results we need on increasing tableaux and K-theory. The structure sheaves $\mathcal{O}_\lambda$ of Schubert varieties in $Gr(k,\mathbb{C}^n)$ form an additive basis for the K-theory ring $K(Gr(k,\mathbb{C}^n))$, and they have a product formula $$[\mathcal{O}_\mu] \cdot [\mathcal{O}_\nu] = \sum_{|\lambda| \geq |\mu| + |\nu|} (-1)^{|\lambda| - |\mu| - |\nu|}k_{\mu \nu}^\lambda [\mathcal{O}_\lambda],$$ for certain nonnegative integer coefficients $k_{\mu \nu}^\lambda$. These coefficients enumerate certain tableaux, which we now discuss.
In [@bib:ThomasYong], Thomas and Yong have defined a K-theoretic jeu de taquin for [***increasing tableaux***]{}, i.e. tableaux that are both row- and column-strict; the tableaux analogous to highest-weight standard tableaux are those whose K-rectification is superstandard. When the K-rectification shape is a single row $\beta = (d)$, these are the *Pieri strips of max-content $d$*:
Let $\lambda/\mu$ be a horizontal strip, (no two squares are in the same column). We say a tableau $T$ of shape $\lambda/\mu$ is a [***Pieri strip***]{} if:
- the rows of $T$ are strictly increasing,
- the reading word of $T$ is weakly increasing and does not omit any value $1, \ldots, \max(T)$.
We say the [***max-content***]{} of $T$ is $\max(T)$.
For the shape $\lambda/\mu = {\tiny \young(:::\hfil\hfil,:\hfil\hfil,\hfil)}$, there is one Pieri strip of max-content $5$, two of max-content $4$ and one of max-content $3$. These are, respectively: $$\young(:::45,:23,1) \qquad \young(:::34,:12,1) \qquad \young(:::34,:23,1) \qquad \young(:::23,:12,1).$$
For general shapes, there is an analogous theory of ‘(ballot) semistandard increasing tableaux’. These are the [***genomic tableaux***]{} defined by Pechenik in [@bib:Pechenik], whose entries are subscripted integers $i_j$, which we now define.
Let $T$ be a genomic tableau with entries $i_j$. We call $i$ the *gene family* and $j$ the *gene*. First, we say $T$ is [***semistandard***]{} if:
- The tableau $T_{ss}$ obtained by forgetting the genes is semistandard (that is, each gene family forms a horizontal strip);
- Within each gene family, the genes form a Pieri strip.
We say the [***$K$-theoretic content***]{} of $T$ is $(c_1, \ldots, c_r)$ if $c_i$ is the max-content of the Pieri strip of genes in the $i$-th gene family. Finally, we say $T$ is [***ballot***]{} if it is semistandard and has the following property:
- Let $T'$ be any genomic tableau obtained by deleting, within each gene family of $T$, all but one of every repeated gene. Let $T'_{ss}$ be the tableau obtained by deleting the corresponding entries of $T_{ss}$. Then the reading word of $T'_{ss}$ is ballot.
We write $K(\lambda/\mu;\nu)$ for the set of ballot genomic tableaux of shape $\lambda/\mu$ and $K$-theoretic content $\nu$.
We have $k_{\mu \nu}^\lambda = |K(\lambda/\mu;\nu)|$.
We are most concerned with the case of partitions $\alpha, \beta, \gamma$ with $|\alpha| + |\beta| + |\gamma| = k(n-k)-1$. In this case there will only be one repeated gene, in one gene family. Let $K(\gamma^c/\alpha;\beta)(i)$ be the set of increasing tableaux in which $i$ is the repeated gene family. For convenience, we state the following simpler characterization of this set:
\[lem:genomic-criterion\] Let $T$ be an (ordinary) semistandard tableau of shape $\gamma^c/\alpha$ and content equal to $\beta$ except for a single extra $i$. Let $\{{\boxtimes}_1, {\boxtimes}_2\}$ be two squares of $T$, such that
- The squares are non-adjacent and contain $i$,
- There are no $i$’s between ${\boxtimes}_1$ and ${\boxtimes}_2$ in the reading word of $T$,
- For $k = 1,2,$ the word obtained by deleting ${\boxtimes}_k$ from the reading word of $T$ is ballot.
There is a unique ballot genomic tableau $T' \in K(\gamma^c/\alpha; \beta)(i)$ corresponding to the data $(T,\{{\boxtimes}_1,{\boxtimes}_2\})$. Conversely, each $T'$ corresponds to a unique $(T,\{{\boxtimes}_1,{\boxtimes}_2\})$.
The gene families of $T'$ are the entries of $T$. For $j \ne i$, the $j$-th gene family of $T'$ has all distinct genes, obtained by standardizing the $j$-th horizontal strip of $T$. For the $i$-th gene family, there are exactly two repeated genes, in the squares ${\boxtimes}_1, {\boxtimes}_2$. This uniquely determines the Pieri strip. Ballotness of $T'$ is then equivalent to (iii).
Generating genomic tableaux
---------------------------
We now establish connections between local evacuation-shuffling and genomic tableaux. We first describe how the tableaux $K(\gamma^c/\alpha;\beta)$ arise from evacuation-shuffling. In fact, each tableau arises once during some step of Phase 1 and once during Phase 2, for some $T_1, T_2 \in {{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$.
Let $({\boxtimes},T) \in {{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$. Suppose the evacu-shuffle path for ${\mathrm{local\text{-}esh}}({\boxtimes},T)$ is not connected. This occurs whenever ${\mathrm{local\text{-}esh}}$ applies a Pieri or conjugate Pieri move. Let ${\boxtimes}_1, {\boxtimes}_2$ be two successive non-adjacent squares in the path, and suppose the ${\boxtimes}$ switched with an $i$ during this move (that is, the movement occurred during ${{\bf \mathrm{Pieri}}}_i$ or ${{\bf \mathrm{Jump}}}_i$). Let $T_i$ be the tableau before this step, with the ${\boxtimes}$ replaced by $i$. We will show using Lemma \[lem:genomic-criterion\] that $(T_i,\{{\boxtimes}_1, {\boxtimes}_2\})$ corresponds to a genomic tableau. See Figure \[fig:genomic-bijection\] for an example.
$$\left(\ \raisebox{-.45\height}{\includegraphics[scale=0.7]{local-esh2.pdf}}
\xrightarrow{{{\bf \mathrm{Pieri}}}_2}
\raisebox{-.45\height}{\includegraphics[scale=0.7]{local-esh3.pdf}}\ \right)\
\xmapsto{\ \varphi_1\ }
\raisebox{-.45\height}{\includegraphics[scale=0.7]{K-gen1.pdf}}$$
\[thm:generating-ktheory\] The data $(T_i,{\boxtimes}_1,{\boxtimes}_2)$ corresponds to a ballot genomic tableau, as in Lemma \[lem:genomic-criterion\]. Moreover, as $T$ ranges over ${{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$, every tableau $T_K \in K(\gamma^c/\alpha;\beta)(i)$ arises exactly once this way in Phase 1 and once more in Phase 2. This gives two bijections:
--------------- ---------------------------------------------------------------------------------------- -------------------------------------
$\varphi_1$ : $\big\{{{\bf \mathrm{Pieri}}}_i \text{ moves} \big\}$ $\to\ K(\gamma^c/\alpha;\beta)(i),$
$\varphi_2$ : $\big\{{{\bf \mathrm{CPieri}}}_j \text{ moves during } {{\bf \mathrm{Jump}}}_i \big\}$ $\to\ K(\gamma^c/\alpha;\beta)(i)$.
--------------- ---------------------------------------------------------------------------------------- -------------------------------------
By construction, the squares are non-adjacent. From the definition of local evacuation-shuffling, there is no $i$ between ${\boxtimes}_1$ and ${\boxtimes}_2$ in the reading word of $T$. We need only check that after deleting either one of ${\boxtimes}_1, {\boxtimes}_2$, the reading word of $T_i$ is ballot. This follows from Theorem \[thm:ballotness\].
We show that $\varphi_1$ is bijective. It is clearly injective. Next, given a genomic tableau $T_K$, let $(T,\{{\boxtimes}_1,{\boxtimes}_2\})$ be as defined in Lemma \[lem:genomic-criterion\]. Replace either ${\boxtimes}$ entry with $i$, and leave the other as ${\boxtimes}$. This gives a pair of tableaux $T',T''$, which differ by an ordinary, non-vertical Pieri move. An argument similar to that of Theorem \[thm:ballotness\] then shows that applying Reverse Phase 1 moves yields an element $T \in {{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$. (It is important that *both* tableaux $T',T''$ are ballot.)
The proof for $\varphi_2$ is identical, only we elect to think of $T', T''$ as differing by a movement of the ${\boxtimes}$ in Phase 2. We again work backward to get to $T$. Note that the tableaux $T', T''$ occur in the opposite order when we think of them as arising during Phase 2.
\[exa:ktheory-pieri\] Suppose $\beta$ has only one row, and $\gamma^c/\alpha$ is a horizontal strip with $r$ nonempty rows. With notation as in Theorem \[thm:Pieri\], we have $${{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}= \{L_1, \ldots, L_r\}, \qquad \omega(L_i) = L_{i+1 \pmod r}.$$ In this case, the corresponding genomic tableaux are the Pieri strips on $\gamma^c/\alpha$ of $K$-theoretic content $\beta$. Let $G_{i,i+1}$ be the tableau in which the two equal entries are at the beginning of the $i$-th row and the end of the $(i+1)$-st.
In Phase 1, the ordinary step $\omega(L_i) = L_{i+1}$ generates $G_{i,i+1}$ (for $1 \leq i < r$), while the special jump does not correspond to a genomic tableau.
In Phase 2, the ordinary steps $\omega(L_i) = L_{i+1}$ do not correspond to genomic tableaux, while the special jump generates all of them at once.
The sign and reflection length of w via genomic tableaux
--------------------------------------------------------
We recall the statements about $\omega$ known from geometry: $$\begin{aligned}
|{K(\gamma^c/\alpha; \beta)}| &\geq \mathrm{rlength}(\omega), \label{eqn:recall-ineq} \\
|{K(\gamma^c/\alpha; \beta)}| &\equiv \mathrm{sgn}(\omega) \pmod 2. \label{eqn:recall-parity}\end{aligned}$$ where $\mathrm{sgn}(\omega)$ is $0$ or $1$ when $\omega$ is even or odd respectively, and $\mathrm{rlength}(\omega)$ denotes the *reflection length*, the minimum length of a factorization of $\omega$ as a product of transpositions (permutations consisting of a single $2$-cycle). Note that the right-hand sides of equations and are the same, mod $2$.
We now give enumerative proofs of these statements, using the bijection $\varphi_1$ of Theorem \[thm:generating-ktheory\] to count ballot genomic tableaux. The key idea is to break down the steps of the local algorithm and thereby factor $\omega$ into simpler permutations.
Let $X_i$ be the set of all tableaux arising in between steps $i-1$ and $i$ of ${\mathrm{local\text{-}esh}}$. Let $X'_i$ be the set of all tableaux arising during ${\mathrm{sh}}$, when the ${\boxtimes}$ is between the $(i-1)$-st and $i$-th strips. Then $X_i = X'_i$.
Both sets consist of ‘punctured’ semistandard tableaux of content $\beta$ and shape $\gamma^c / \alpha$, with ballot reading word, and where the ${\boxtimes}$ is between the $(i-1)$-st and $i$-th horizontal strips. (It is well-known that ballotness is preserved by jeu de taquin slides. Ballotness is also preserved during ${\mathrm{local\text{-}esh}}$ by Theorem \[thm:ballotness\].) Both shuffling and evacuation-shuffling are invertible, so every such tableau arises in $X_i$ and $X'_i$.
We have $X_1 = {{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$ and we write $X_{t+1} = {{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}$, where $t$ is the length of $\beta$.\
For $1 \leq i \leq t$, let $\ell_i : X_i \to X_{i+1}$ be the $i$-th step of ${\mathrm{local\text{-}esh}}$. Let $s_i : X_{i+1} \to X_i$ be the jeu de taquin shuffle. We have the diagram
$$\xymatrix{
X_1 \ar@/_10pt/[r]_-{\ell_1} &
X_2 \ar@/_10pt/[r]_-{\ell_2} \ar@/_10pt/[l]_-{s_1}&
X_3 \ar@/_10pt/[r]_-{\ell_3} \ar@/_10pt/[l]_-{s_2} &
\cdots \ar@/_10pt/[r]_-{\ell_t} \ar@/_10pt/[l]_-{s_3} &
X_{t+1} \ar@/_10pt/[l]_-{s_t}.
}$$ By definition, $$\omega = {\mathrm{sh}}\circ {\mathrm{local\text{-}esh}}= s_1 \cdots s_t \circ \ell_t \cdots \ell_1.$$
For $i=1,\ldots,t$, we define $$\omega_i=s_1s_2\cdots s_{i-1} (s_i \ell_i) s_{i-1}^{-1}\cdots s_2^{-1} s_1^{-1}.$$ Note that we may factor $\omega$ as $$\omega=\omega_t \omega_{t-1}\cdots \omega_2\omega_1.$$
Hence we have $$\begin{aligned}
\mathrm{sgn}(\omega) &\equiv \sum_{i=1}^t \mathrm{sgn}(\omega_i)\pmod{2}, \label{eqn:omega-i-inequality} \\
\mathrm{rlength}(\omega) &\leq \sum_{i=1}^t \mathrm{rlength}(\omega_i). \label{eqn:omega-i-sign}\end{aligned}$$ It now suffices to determine the orbits of $\omega_i$, a computation interesting in its own right:
\[thm:little-orbits\] Let $\mathrm{orb}_i$ be the set of orbits of $\omega_i$. Then: $$\sum_{\mathcal{O} \in \mathrm{orb}_i} (|\mathcal{O}| - 1) = |K(\gamma^c/\alpha; \beta)(i)|.$$ In particular, $\mathrm{rlength}(\omega_i)=|K(\gamma^c/\alpha; \beta)(i)|$ and $\mathrm{sgn}(\omega_i) \equiv |K(\gamma^c/\alpha; \beta)(i)| \pmod 2$.
We use the bijection $\varphi_1$ of Theorem \[thm:generating-ktheory\] to generate genomic tableaux. Let $T \in X_i$.
First, suppose $\ell_i$ applies a Phase 1 vertical slide, or a Phase 2 ${{\bf \mathrm{Jump}}}$ move consisting of all ${{\bf \mathrm{Horiz}}}$ steps. Both of these steps are equivalent to jeu de taquin slides, so in this case $\ell_i(T) = s_i^{-1}(T)$. Thus $T$ is a fixed point and does not contribute to the sum; it also does not generate a genomic tableau.
Next, it is easy to see that $\ell_i$ applies a Phase 1 move if and only if the following conditions hold:
- The suffix from ${\boxtimes}$ in $T$ is not tied for $(i-1,i)$, and
- There is an $i$ before the ${\boxtimes}$ in the reading word of $T$.
The first condition implies that the $(i-1)$-st step of ${\mathrm{local\text{-}esh}}$ was in Phase 1; the second rules out the transition to Phase 2.
We now analyze the orbits of $s_i \circ \ell_i$. If either of the above conditions fails, $\ell_i$ moves the ${\boxtimes}$ to the first $i$ after it in reading order for which the $(i,i+1)$ suffix is tied; then $s_i$ moves it to the start of that row of $i$’s. Otherwise, $s_i \circ \ell_i$ applies a Pieri move on the horizontal strip of $i$’s. Thus the ${\boxtimes}$ moves downwards in this strip, one row at a time, until one of the conditions fails.
Since $\omega_i$ is a bijection, and the only two possible types of moves are moving down one row at a time on the $i$th strip or jumping upwards some number of rows, it follows that the orbit consists of a cycle, containing exactly one “special jump” up to the top of the cycle, and the rest downward Pieri moves.
Thus every orbit has a form similar to that of the Pieri case (Example \[exa:ktheory-pieri\]): one step does not generate a genomic tableau; all other steps generate exactly one each. Thus during each orbit $\mathcal{O} \in \mathrm{orb}_i$, we generate $|\mathcal{O}| - 1$ genomic tableaux. Since every tableau of $K(\gamma^c/\alpha; \beta)(i)$ arises once in Phase 1, we are done.
Equations and now follow from Theorem \[thm:little-orbits\] and Equations and .
Orbits of w {#sec:omega-orbits}
===========
A stronger conjectured inequality
---------------------------------
For the first statement, numerical evidence suggests that, using either $\varphi_1$ or $\varphi_2$, the inequality in fact holds orbit-by-orbit (see Figure \[fig:numerical-evidence\]):
\[conj:orbit-by-orbit\] Let $\mathcal{O} \subseteq {{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$ be an orbit of $\omega$. Let $K_1(\mathcal{O}), K_2(\mathcal{O})$ denote the sets of genomic tableaux occuring in this orbit in Phases 1 and 2 via the bijections $\varphi_1, \varphi_2$. Then $$|K_i(\mathcal{O})| \geq |\mathcal{O}| - 1 \qquad \text{for } i = 1, 2.$$ Note that, by Corollary \[cor:antidiag-evacu-path\], it is sufficient to prove this for $\varphi_1$.
We have verified Conjecture \[conj:orbit-by-orbit\] for $n$ up to size $10$ and all $k$, $\alpha$, $\beta$, and $\gamma$. Below, we prove the conjecture in two special cases.
[m[2cm]{} m[2cm]{} m[2cm]{} m[2cm]{}|c|c|c]{} & $|\mathcal{O}|$ & $K_1(\mathcal{O})$ & $K_2(\mathcal{O})$\
& & & & 38 & 52 & 51\
&&&& 23 & 31 & 28\
&&&& 10 & 9 & 13\
& & & & 1 & 0 & 0\
&&&& 1 & 0 & 0
The inequalities of equation and Conjecture \[conj:orbit-by-orbit\] are tight bounds, since equality holds in the Pieri case and in several others. Indeed, in the Pieri case $\omega$ has only one orbit and $|K| = |\mathcal{O}| - 1$. Geometrically, this implies that the Schubert curve $S(\alpha,\beta,\gamma)$ is integral and has $\chi(\mathcal{O}_S) = 1$, so $S \cong \mathbb{P}^1$.
Fixed points of w
-----------------
As a base case of Conjecture \[conj:orbit-by-orbit\], we characterize the fixed points of $\omega$.
\[prop:fixed-points\] Let $T \in {{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$. The following are equivalent:
- $\omega(T) = T$.
- In the computation of ${\mathrm{local\text{-}esh}}(T)$, neither bijection $\varphi_1, \varphi_2$ generates a genomic tableau.
- The evacu-shuffle path of the ${\boxtimes}$ is connected.
It is easy to see that (ii) and (iii) are equivalent. Moreover, if (iii) holds then the movements of the ${\boxtimes}$ are equivalent to jeu de taquin slides, so $\omega(T) = T$. Thus (iii) implies (i).
To show (i) implies (ii), suppose first that the computation of ${\mathrm{local\text{-}esh}}(T)$ involves a Pieri jump in Phase 1. Let $i$ be the index of the jump; the effect is that a single $i$ moved strictly up *and* to the right. Since the horizontal strip of $i$’s is unaffected by the remaining steps of ${\mathrm{local\text{-}esh}}$, the movement must be undone by the ${\mathrm{sh}}$. But the jeu de taquin slides can only either move a single $i$ down one row, or move a strip of $i$’s to the right. Neither is enough to undo the movement, so we conclude $\omega(T) \ne T$.
We have shown that if Phase 1 generates a genomic tableau, then $T$ is not a fixed point. By a similar argument, or by antidiagonal symmetry (Corollary \[cor:antidiag-evacu-path\]), if Phase 2 generates a genomic tableau, then $\omega(T) \ne T$. This completes the proof.
One immediate corollary of this result is the following *geometric* fact:
\[cor:w=id\] Suppose $\omega$ acts on ${{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$ as the identity. Then ${K(\gamma^c/\alpha; \beta)}= {\varnothing}$; it follows that the curve $S(\alpha, \beta, \gamma)$ is a disjoint union of $\mathbb{P}^1$’s, and the map $S \to \mathbb{P}^1$ is locally an isomorphism.
In general, a morphism of real algebraic curves $C \to D$, which is a covering map on real points, may have trivial real monodromy but be algebraically nontrivial (i.e. not a local isomorphism). Corollary \[cor:w=id\] shows that this cannot occur for Schubert curves.
If $\omega$ is the identity, Proposition \[prop:fixed-points\] and Theorem \[thm:generating-ktheory\] imply $|{K(\gamma^c/\alpha; \beta)}|=0$. Therefore $$\chi(\mathcal{O}_S) = |{{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}| - |{K(\gamma^c/\alpha; \beta)}| = |{{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}|.$$ There are, moreover, exactly $|{{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}|$ real connected components. It follows that $S$ has the desired form: using the notation of Proposition \[prop:numerics\], the inequalities $$\eta(S)\geq\iota(S)\geq\dim_\mathbb{C} H^0(\mathcal{O}_S)\geq\chi(\mathcal{O}_S)$$ become equalities. Note that $\dim_\mathbb{C} H^0(\mathcal{O}_S)$ is the number of $\mathbb{C}$-connected components of $S$. In particular each $\mathbb{C}$-connected component is irreducible, and of genus 0 because $\dim H^1(\mathcal{O}_S)=0$.
We also obtain a weaker form of the Orbits Conjecture:
\[cor:order2-orbits\] For any orbit $\mathcal{O}$ of $\omega$, $$|K_1(\mathcal{O})| + |K_2(\mathcal{O})| \geq |\mathcal{O}| - 1,$$ and if $|\mathcal{O}| \ne 1$ the inequality is strict.
This follows from Proposition \[prop:fixed-points\], since in each $\omega$-orbit that is not a fixed point, every step involves at least one genomic tableau generated in either Phase 1 or Phase 2.
We think of this as an ‘order-2 approximation’, since summing over the orbits gives $$2 \cdot |{K(\gamma^c/\alpha; \beta)}| \geq |{{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}| - |\mathrm{orbits}(\omega)| = \mathrm{rlength}(\omega),$$ a weaker version of our Theorem \[thm:MainResult2\].
When the rectification shape has two rows
-----------------------------------------
In this section, we prove Conjecture \[conj:orbit-by-orbit\] for $K_1(\mathcal{O})$ when $\beta$ has two rows. We note that the case where $\beta$ has one row (the Pieri Case) is trivial: equality holds for the (unique) orbit. See Example \[exa:ktheory-pieri\].
\[thm:tworows\] Let $\beta$ have two rows. For an $\omega$-orbit $\mathcal{O} \subset {{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}$, let $K_1(\mathcal{O})$ be the set of ballot genomic tableaux occurring in $\mathcal{O}$ during Phase 1. Then $$\label{eqn:orbit-by-orbit-ineq}
|K_1(\mathcal{O})| \geq |\mathcal{O}| - 1.$$
If the skew shape $\gamma^c/\alpha$ contains a column of height 3, then $\omega$ is the identity and $k=0$. For the remainder of this section, we assume every column of $\gamma^c/\alpha$ has height at most 2.
We use the following idea: consider the sub-shape of $\gamma^c/\alpha$ consisting of only its height-one columns. This shape consists of a disjoint union of row shapes. For a tableau $T \in {{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$ or ${{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}$, we will call the fillings of these row shapes the [***words***]{} of $T$.
\[def:exceptional\] Let $(T,{\boxtimes}) \in {{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}$. We say that $T$ is [***exceptional***]{} if the following holds:
- Every square of $T$ strictly above ${\boxtimes}$ contains a $1$.
- From top to bottom, the words of $T$ are a sequence of all-$1$ words, followed by at most one ‘mixed’ word containing $1$’s, $2$’s and/or ${\boxtimes}$, followed by a sequence of all-$2$ words.
The following tableaux are exceptional: $$T_1: \young(:::::::::::::1,::::::::11111:,:::::11122{\times},::1122,222) \qquad T_2 : \young(:::::::::::1,:::::::::11,::::1112{\times},::1222,222)$$ From top to bottom, the words of $T_1$ are $(1,11,11,12,22)$ and the words of $T_2$ are $(1,11,12{\boxtimes},2,22)$.
Note that ${{\mathrm{LR}}(\alpha,\beta,{\scalebox{.5}{\yng(1)}}, \gamma)}$, if nonempty, contains exactly one exceptional tableau (the second condition determines the words and the first determines placement of the ${\boxtimes}$).
As $\beta$ has two rows, ${\mathrm{local\text{-}esh}}$ takes two steps. If both are Phase 1 Pieri moves, we have ‘gained’ a Pieri move. If neither is, we have ‘lost’ one. All other possibilities contribute 1 element to both $\mathcal{O}$ and $K_1(\mathcal{O})$, hence have no effect on the inequality. We will show that, in almost all cases, we ‘gain’ a Pieri move between two successive ‘losses’.
If $|\mathcal{O}| = 1$, we are done by Proposition \[prop:fixed-points\]. Henceforth we assume $|\mathcal{O}| > 1$.
We divide the orbit into (disjoint) segments $T, \omega(T), \ldots, \omega^n(T)$, such that $\omega^{-1}(T) \to T$ and $\omega^{n-1}(T) \to \omega^n(T)$ have transition step $s < 3$, but the intermediate steps have $s=3$ (i.e. they remain in Phase 1). We will show that among all such segments, at most one contributes ${-1}$ to the inequality. The others contribute nonnegatively.
Within a segment, each intermediate ${\mathrm{local\text{-}esh}}$ remains in Phase 1, hence involves at least one regular Pieri move (since the tableau is not fixed). If the last one does as well, or if some intermediate step involves two Pieri moves, then the entire segment contributes nonnegatively to the inequality. If not, we show:
Suppose $\omega^{n-1}(T) \to \omega^n(T)$ does not involve a Pieri move, and every intermediate step involves exactly one. Then ${\mathrm{sh}}^{-1}(T)$ is exceptional.
Theorem \[thm:tworows\] will follow since only one segment can begin with an exceptional tableau.
By our hypotheses, every intermediate ${\mathrm{local\text{-}esh}}$ step must consist of either ${{\bf \mathrm{Vert}}}_1, {{\bf \mathrm{Pieri}}}_2$ or ${{\bf \mathrm{Pieri}}}_1, {{\bf \mathrm{Vert}}}_2$.
First, we claim that if $\omega^i(T) \to \omega^{i+1}(T)$ consists of ${{\bf \mathrm{Vert}}}_1, {{\bf \mathrm{Pieri}}}_2$, then every earlier step is also of this form, and every *word* weakly above the ${\boxtimes}$ in $\omega^i(T)$ consists only of $1$’s. On the other hand, if $\omega^i(T) \to \omega^{i+1}(T)$ consists of ${{\bf \mathrm{Pieri}}}_1, {{\bf \mathrm{Vert}}}_2$, we claim that every subsequent step is of this form, and every word strictly below ${\boxtimes}$ in $\omega^i(T)$ consists entirely of $2$’s.
For the first claim, we work backwards from $\omega^i(T)$ to $\omega^{i-1}(T)$. During the ${\mathrm{sh}}^{-1}$ step, the ${\boxtimes}$ slides one square down, then right; there must then be a $1$ directly above ${\boxtimes}$. If some row above ${\boxtimes}$ contains a $2$, ${\mathrm{local\text{-}esh}}^{-1}$ must begin in Reverse Phase 1. (By construction, this will be the case as long as $i > 0$.) Hence ${\mathrm{local\text{-}esh}}^{-1}$ consists of (Reverse) ${{\bf \mathrm{Pieri}}}_2$ and ${{\bf \mathrm{Vert}}}_1$, as desired. For the claim about words, note that the (Reverse) ${{\bf \mathrm{Pieri}}}_2$ move will only move the ${\boxtimes}$ past words containing all $1$’s. Finally, if $i=0$, then ${\mathrm{local\text{-}esh}}^{-1}$ begins in Reverse Phase 2 because there are no 2’s in any word (in fact, any row) above ${\boxtimes}$ in ${\mathrm{sh}}^{-1}(T)$.
For the second claim, the argument is similar, only we work forward. The computation of ${\mathrm{local\text{-}esh}}(\omega^i(T))$ terminates with ${\boxtimes}$ below a $2$; any words passed over by the ${\boxtimes}$ contain only $2$’s. During ${\mathrm{sh}}$, the ${\boxtimes}$ slides up and left, so it is above a $2$ in $\omega^{i+1}(T)$. If $i+1 < n-1$, then ${\mathrm{local\text{-}esh}}$ will again have the form ${{\bf \mathrm{Pieri}}}_1, {{\bf \mathrm{Vert}}}_2$. Finally, if $i+1 = n-1$, then ${\mathrm{local\text{-}esh}}(\omega^{n-1}(T))$ must begin in Phase 2 (it can’t begin with ${{\bf \mathrm{Vert}}}_1$ since ${\boxtimes}$ is above a 2, and we have assumed it does not involve a regular Pieri move). Thus every row below the ${\boxtimes}$ contains only $2$’s.
We thus divide the segment into a first part, where ${\mathrm{local\text{-}esh}}$ consists of ${{\bf \mathrm{Vert}}}_1, {{\bf \mathrm{Pieri}}}_2$, and a second part, where ${\mathrm{local\text{-}esh}}$ consists of ${{\bf \mathrm{Pieri}}}_1, {{\bf \mathrm{Vert}}}_2$. Note that there can be a single ‘mixed’ word in the tableau (if the second part begins with $\omega^i(T)$, this is the word to the right of the ${\boxtimes}$ in $\omega^i(T)$; in fact the ${\boxtimes}$ slides through this word during the ${\mathrm{sh}}$ step linking the two parts). We see, moreover, that all the non-mixed words remain unchanged from ${\mathrm{sh}}^{-1}(T)$ to $\omega^{n-1}(T)$.
Thus, from top to bottom, the words of ${\mathrm{sh}}^{-1}(T)$ are a (possibly empty) sequence of all-1 words, a single (possibly) ‘mixed’ word containing $1$’s, $2$’s and/or the ${\boxtimes}$, followed by a (possibly empty) sequence of all-2 words. Thus ${\mathrm{sh}}^{-1}(T)$ is exceptional.
In fact, our proof shows something slightly stronger: an orbit $\mathcal{O}$ for which $|K_1(\mathcal{O})| = |\mathcal{O}|-1$ is either a single fixed point, or is the unique orbit containing the exceptional tableau. All other orbits in fact satisfy $|K_1(\mathcal{O})| \geq |\mathcal{O}|$.
Geometrical constructions {#sec:constructions}
=========================
We now give several families of values of $\alpha$, $\beta$, and $\gamma$ for which the Schubert curve $S(\alpha, \beta, \gamma)$ exhibits ‘extremal’ numerical and geometrical properties.
Schubert curves of high genus
-----------------------------
Recall that the arithmetic genus of a (connected) variety $S$ can be defined as $$g_a(S)=(-1)^{\dim S}(1 - \chi(\mathcal{O}_S)).$$ If $S$ is an integral curve, this is just $\dim_{\mathbb{C}} H^1(\mathcal{O}_S)$. (If $S$ is smooth, this is the usual genus of $S(\mathbb{C})$ as a topological space.)
In this section we construct a sequence of Schubert curves $S_t$, $t \geq 2$, for which $\omega$ has only one orbit, and so (by Proposition \[prop:numerics\]) $S_t$ is integral. Moreover, we show that as $t \to \infty$, $g_a(S_t) \to \infty$ as well. In [@bib:Levinson], the second author asked if Schubert curves are always smooth. K-theory does not in general detect singularities, but either possibility is interesting: that $S_t$ gives examples of singular Schubert curves for $t \gg 0$, or that it gives smooth Schubert curves of arbitrarily high genus.
As mentioned in the introduction, for our Schubert curves $S=S(\alpha,\beta,\gamma)$, we also have $$\chi(\mathcal{O}_S)=|{{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}|-|K(\gamma^c/\alpha;\beta)|.$$ Therefore, if $S$ is connected (which is true if $\omega$ has one orbit), we have $$\begin{aligned}
|{{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}|-|K(\gamma^c/\alpha;\beta)| &=& \dim_{\mathbb{C}} H^0(\mathcal{O}_S)-\dim_{\mathbb{C}} H^1(\mathcal{O}_S)\\
&=& 1-g_a(S).\end{aligned}$$ and so $$\label{eqn:connected} g_a(S)=|K(\gamma^c/\alpha;\beta)|-|{{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}|+1.$$
We can now construct our family of high genus curves. Let $t \geq 3$ be a positive integer, and let $$\begin{aligned}
\alpha = \gamma &= (t,t-1,t-2,\ldots,2,1), \\
\beta &= (t+1,2,1^{t-2}).\end{aligned}$$ We work in the Grassmannian $G(t+1,\mathbb{C}^{2t+3})$, so ${{\scalebox{.3}{\yng(3,3)}}}$ has size $(t+1) \times (t+2)$, and $\gamma^c/\alpha$ is a staircase ribbon shape. (See Example \[exa:high-genus\].) We will call $\gamma^c/\alpha$ the [***staircase ribbon of size $t$***]{}.
\[exa:high-genus\] For $t=5$, two of the elements of ${{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$ are $$\young(:::::11,::::12,:::13,::14,:12,{\times}5)\hspace{0.5cm}\text{ and }\hspace{0.5cm} \young(:::::11,::::12,:::23,::{\times}4,:11,15).$$ Each of these will be referred to as illustrations in our proof below.
\[prop:high-genus\] With notation as above, $\omega$ has only one orbit. In particular, $S_t = S(\alpha,\beta,\gamma)$ is integral, and $g_a(S)=(t-1)(t-2).$
We break the proof of Proposition \[prop:high-genus\], into several intermediate lemmas. We first compute the cardinalities in question.
\[lem:count-LR\] With notation as above, $$|{{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}|=2t(t-1).$$
We sort the tableaux into two types: those for which the inner corners are all $1$ or ${\boxtimes}$, as in the first example in Example \[exa:high-genus\], and those for which there is an inner corner whose entry is greater than $1$, as in the second example. We will refer to these as Type A and Type B tableaux.
In a Type A tableau, the topmost outer corner must be a $1$ since the tableau is ballot. Since there are a total of $t$ entries greater than $1$ and exactly $t+1$ outer corners, the remaining outer corners must be filled with $2,2,3,4,\ldots,t$, and all but the second $2$ must occur in that order. There are $t-1$ possibilities for the position of the second $2$, and the remaining outer corners are determined. The ${\boxtimes}$ can then be in any of the $t+1$ inner corners, and the remaining entries then must be filled with $1$’s. This gives a total of $(t-1)(t+1)$ Type A tableaux.
In a Type B tableau, ballotness forces exactly one inner corner to contain a $2$; among the outer corners, the topmost and one other contain $1$’s. The ${\boxtimes}$ must be above this second $1$; the remaining entries are determined. If the $2$ is in the lowest inner corner, there are $(t-1)$ choices for the ${\boxtimes}$. Each of the $(t-1)$ other placements of the $2$ gives $(t-2)$ choices for the ${\boxtimes}$, for a total of $(t-1)+(t-1)(t-2)=(t-1)^2$ Type B tableaux.
\[lem:count-K\] With notation as above, $$|K(\gamma^c/\alpha;\beta)|=3t^2-5t+1.$$
We count the ballot genomic tableaux having an extra $i$ for each $i$ separately. We use the description from Lemma \[lem:genomic-criterion\].
For $i=1$, the tableau must contain $(t+2)$ $1$’s. By semistandardness, we cannot have a $1$ in an outer corner besides the topmost outer corner. Thus the entries *larger* than $1$ fill all the outer corners except the topmost. There are $t-1$ ways to place the second $2$, and all other entries are then determined by ballotness. For each of these tableaux, there are then $t$ pairs of consecutive inner corners to mark as our chosen repeated $1$’s, and each of these satisfy the ballot condition on removal. We therefore have $t(t-1)$ ballot genomic tableaux in this case.
For $i=2$, we wish to count for semistandard genomic tableaux with content $\beta''=(t+1,3,1,1,\ldots,1)$ and two marked $2$’s as above. By semistandardness and ballotness, the topmost $2$ must be in the outer corner in the second row. If the topmost $2$ is in the marked pair of $2$’s, then in order for the word to be ballot upon removing it, the next $2$ (necessarily the other marked $2$) must occur before the $3$. The next $2$ therefore occurs in the third outer corner from the top, and by semistandardness and ballotness all entries larger than $2$ fill in the remaining outer corners, with the third $2$ in one of $t-1$ possible inner corners. This gives $t-1$ genomic tableaux in this case.
If the topmost $2$ is not in the marked pair, then the other two $2$’s must be in an inner and outer corner respectively which are not adjacent. There are $(t-1)$ positions for the $2$ in the outer corner and then $(t-2)$ valid positions for the other $2$ for each of these choices, for a total of $(t-1)(t-2)$ possibilities in this case. Thus we have a total of $(t-1)^2$ ballot genomic tableaux with two marked $2$’s.
Finally, if $i\ge 3$, it is easy to see by the semistandard and ballot conditions that the repeated $i$’s must be in the consecutive outer corners in the $i$th and $i+1$st rows from the top. For each $i$ there are then $t$ inner corners in which the second $2$ can be placed, and all other entries are determined. It follows that there are a total of $t(t-2)$ ballot genomic tableaux in the case $i\ge 3$.
All in all, there are $t(t-1)+(t-1)^2+t(t-2)=3t^2-5t+1$ tableaux.
\[lem:one-w-orbit\] With notation as above, $\omega:{{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}\to {{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$ has only one orbit.
By Lemma \[lem:count-LR\], it suffices to find an orbit of size $2t(t-1)$.
We first introduce some new notation that will clarify the steps in our proof. Let $A_{p,q}$ be the unique tableau having the ${\boxtimes}$ in the inner corner in the $p$th row from the top ($1 \leq p \leq t+1$) and with the $2$’s in the outer corners in the $2$nd and $q$th rows ($3 \leq q \leq t+1$). Let $B_{p,q}$ be the tableau having the ${\boxtimes}$ in the $p$th row and the $2$’s in rows $2$ and $q$, but with the $2$ in the inner corner of row $q$. We have $2 \leq q \leq t+1$ and $1 \leq p \leq t$, and $q \ne p, p+1$. (These are the Type A and Type B tableaux from Lemma \[lem:count-LR\].)
We will show that, for any $q$ with $4\le q \le t+1$, we have $$\label{eqn:q>3}\omega^{2t} A_{t+1,q}=A_{t+1,q-1},$$ and for $q=3$ we have $$\label{eqn:q=3}\omega^{2t}A_{t+1,3}=A_{t+1,t+1}.$$ These facts together will show that the $\omega$-orbit of $A_{t+1,t+1}$ has length $2t(t-1)$. To prove equations (\[eqn:q>3\]) and (\[eqn:q=3\]), let $q\in \mathbb{Z}$ such that $3\le q\le t+1$. Starting with $A_{t+1,q}$, the first application of $\omega$ according to local evacuation shuffling and JDT consists of a single ${{\bf \mathrm{Jump}}}_1$ move to the very top row, followed by a JDT back to the inner corner. Thus $\omega A_{t+1,q}=A_{1,q}$.
Now, if $q$ is sufficiently large then ${\mathrm{local\text{-}esh}}(A_{1,q})$ starts with ${{\bf \mathrm{Pieri}}}_1$ and ${{\bf \mathrm{Pieri}}}_2$, which results in the ${\boxtimes}$ being in row $q$ and the $2$ in the inner corner of row $2$. There is then a single ${{\bf \mathrm{CPieri}}}$ move and an upwards JDT slide. Thus we have $$\omega^2A_{t+1,q}=\omega A_{1,q}=B_{q-2,2}.$$ The next move, to compute $\omega(B_{q-2,2})$, is ${{\bf \mathrm{Vert}}}_1$ followed by a ${{\bf \mathrm{CPieri}}}$ move to the $2$ in the inner corner and a ${{\bf \mathrm{Horiz}}}$ move that is undone by JDT to form $A_{2,q-1}$. This pattern continues, with the next steps in the $\omega$-orbit being $$B_{q-3,3},A_{3,q-2},B_{q-4,4},A_{4,q-3},\ldots$$ until we reach $A_{r,q+1-r}$ where $r$ is such that $r$ and $q+1-r$ differ by either $2$ or $3$. At this point, ${\mathrm{local\text{-}esh}}(A_{r,q+1-r})$ starts with ${{\bf \mathrm{Pieri}}}_1$ and ${{\bf \mathrm{Pieri}}}_2$ as usual, but then the ${{\bf \mathrm{CPieri}}}$ leaves the ${\boxtimes}$ adjacent to the $2$, and the ${\boxtimes}$ and $2$ then switch via JDT. Thus $$\omega A_{r,q+1-r}=A_{r+1,q+1-(r+1)}.$$
After this special step with two consecutive Type A tableaux, the orbit resumes alternating between $A$’s and $B$’s with the first subscript of the $A$’s increasing by $1$ each time and the first subscript of the $B$’s decreasing, starting with $B_{q-r-2,r+2}$, and continuing until we reach $B_{1,q-1}$. At this point we have applied $\omega$ exactly $2(q-2)$ times.
Now, ${\mathrm{local\text{-}esh}}(B_{1,q-1})$ consists of a single ${{\bf \mathrm{Vert}}}_1$ followed by a long sequence of ${{\bf \mathrm{Pieri}}}$ moves, and the upwards JDT slide then results in the tableau $B_{t,q-1}$. The orbit then alternates between $A$’s and $B$’s again in its usual manner until we reach $A_{v,q+t-v}$ where $v$ is such that $v$ and $q+t-v$ differ by either $2$ or $3$. By the same reasoning as above, this maps to $A_{v+1,q+t-(v+1)}$ and the alternation pattern starts again, and continues until we reach $B_{q-2,t+1}$. We have now applied $\omega$ an extra $2(t-q+2)-1$ times, for a total of $2t-1$ times.
Finally, if $q\ge 4$ then $\omega B_{q-2,t+1}=A_{t+1,q-1}$ by the same reasoning as before, and so $\omega^{2t} A_{t+1,q}=A_{t+1,q-1}$. If $q=3$, though, $\omega B_{q-2,t+1}=\omega B_{1,t+1}$, and so before the application of $\omega$ the ${\boxtimes}$ is in the top row and above a $1$, with the topmost $2$ in the row below that. It follows that the local evacuation shuffle consists of a long sequence of ${{\bf \mathrm{Pieri}}}$ moves, and the JDT slide leaves us with $A_{t+1,t+1}$, as desired.
We now finish the proof of Proposition \[prop:high-genus\].
By Lemma \[lem:one-w-orbit\] and Proposition \[prop:numerics\], $S_t = S(\alpha,\beta,\gamma)$ is integral. It follows from Equation \[eqn:connected\] and Lemmas \[lem:count-LR\] and \[lem:count-K\] that $$\begin{aligned}
g(S)&=|K(\gamma^c/\alpha;\beta)|-|{{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}|+1 \\
&= 3t^2-5t+1-2t(t-1)+1\\
&= (t-1)(t-2).
\end{aligned}$$ as desired.
Curves with many connected components
-------------------------------------
We next exhibit a sequence of Schubert curves $S(\alpha,\beta,\gamma)$ having arbitrarily many (complex) connected components. We use Corollary \[cor:w=id\], since in the case that $\omega$ is the identity map we know that the curve must consist of a disjoint union of $\mathbb{P}^1$’s. So, it suffices to find shapes $\alpha$, $\beta$, and $\gamma$ for which $\omega$ is the identity map and ${{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$ has many elements.
\[prop:2x2hook\] Suppose $\beta=(m,1,1,\ldots,1)$ is a hook shape and $\gamma^c/\alpha$ contains a $2\times 2$ square. Then $\omega$ is the identity.
Since the Littlewood-Richardson tableau are semistandard and ballot, the ${\boxtimes}$ must be in the upper left corner of the (necessarily unique) $2\times 2$ square in any tableau in ${{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$. Moreover, there is a unique copy of each entry greater than $1$ and so these entries form a vertical strip. Therefore, the entry just below the ${\boxtimes}$ must be a $1$, and so the $2\times 2$ square looks like $$\young({\times}a,1b)$$ for some $a$ and $b$. We also have $a<b$ since the tableau is semistandard, and so in particular $b>1$.
Now, we wish to show that any such filling maps to itself under $\omega$. The first step in ${\mathrm{local\text{-}esh}}$ must be ${{\bf \mathrm{Vert}}}_1$. At this step, since $b>1$ and the reading word is ballot, the unique $2$ in the tableau must occur after ${\boxtimes}$ in the reading word, and so the transition step is $s=2$.
At this step, since the entries greater than $1$ appear in reverse reading order by ballotness and each occur exactly once, the smallest $k$ for which the $(k,k+1)$ suffix not tied is $k=b$. It follows that the ${\boxtimes}$ switches with the $b$ as its only Phase 2 move; after this point the remaining $(i,i+1)$-suffixes for $i\ge b$ are empty and therefore tied.
$$\raisebox{-5pt}{\young({\times}a,1b)}\xrightarrow{{{\bf \mathrm{Vert}}}_1} \raisebox{-5pt}{\young(1a,{\times}b)} \xrightarrow{{{\bf \mathrm{Horiz}}}_2} \raisebox{-5pt}{\young(1a,b{\times})}\xrightarrow{\text{JDT}} \raisebox{-5pt}{\young({\times}a,1b)}$$
Finally, we perform a JDT slide to move the ${\boxtimes}$ past the tableau, and we see that all entries are restored to their original position, as shown above.
We will now construct our curve in the Grassmannian $\mathrm{Gr}(m+1,\mathbb{C}^{2m+2})$ so that our shapes fill an $(m+1)\times (m+1)$ rectangle.
Let $m$ be a positive integer. Let $\beta=(m,1,1)$, let $\alpha=(m,m-1,m-2,\ldots,2)$, and let $\gamma^c=(m+1,m,m-1\ldots,4,3,2,2)$. Then $S(\alpha,\beta,\gamma)$ consists of a disjoint union of exactly $m-1$ copies of $\mathbb{P}^1$.
The shape $\gamma^c/\alpha$ consists of a single $2\times 2$ square in the lower left corner plus $m-1$ disconnected boxes to the northeast. Thus we have $\omega=\operatorname{id}$ by Proposition \[prop:2x2hook\], and by Corollary \[cor:w=id\], it follows that $S(\alpha,\beta,\gamma)$ is a disjoint union of exactly $|{{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}|$ copies of $\mathbb{P}^1$.
We claim that $|{{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}|=m-1$. Indeed, since $\beta=(m,1,1)$, we wish to count ballot fillings that have one $2$, one $3$, and the rest $1$’s. Since the $2\times 2$ box is at the lower left corner, the $3$ must be in the lower right corner of the $2\times 2$ box by the ballot and semistandard conditions. It is easy to check that the $2$ can be in any of the remaining squares except the top row or in the leftmost column of the skew shape. The positions of the $2$ and $3$ determine the tableau, so we have a total of $m-1$ Littlewood-Richardson tableaux.
Conjectures {#sec:conjectures}
===========
We recall the conjectural ‘orbit-by-orbit’ inequality:
Let $\mathcal{O} \subseteq {{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$ be an orbit of $\omega$. Let $K_1(\mathcal{O}), K_2(\mathcal{O})$ denote the sets of genomic tableaux occuring in this orbit in Phases 1 and 2 (via the bijections $\varphi_1, \varphi_2$). Then $$|K_i(\mathcal{O})| \geq |\mathcal{O}| - 1 \qquad (\text{for } i = 1, 2).$$ Note that, by Corollary \[cor:antidiag-evacu-path\], it is sufficient to prove this for $\varphi_1$.
We have proven Conjecture \[conj:orbit-by-orbit\] in certain cases, but do not know a proof in general. This conjecture suggests that there is additional combinatorial structure in the complex curve $S(\mathbb{C})$ – in particular its irreducible decomposition and, for each irreducible component $S' \subset S(\mathbb{C})$, the number of real connected components of $S'(\mathbb{R})$. We have in mind the following observation:
\[prop:ram-pts\] Suppose $S$ is smooth and let $R = R(\alpha,\beta,\gamma) \subset S(\alpha,\beta,\gamma)$ be the ramification locus of the map $f: S \to \mathbb{P}^1$ of Theorem \[thm:intro-2\]. Then $R$ is a union of complex conjugate pairs of points and, counted with multiplicity, $$\tfrac{1}{2}|R(\alpha,\beta,\gamma)| = |{K(\gamma^c/\alpha; \beta)}|.$$
The quantity $\tfrac{1}{2}R$ is the number (with multiplicity) of complex conjugate pairs of ramification points because $f$ is defined over $\mathbb{R}$ but none of its ramification points are real. The equation then follows from the Riemann-Hurwitz formula, which states $$\chi(\mathcal{O}_S) = (\deg f) \cdot \chi(\mathcal{O}_{\mathbb{P}^1}) - \tfrac{1}{2} \deg R.$$ Note that $\deg f = |{{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}|$, that $\chi(\mathcal{O}_{\mathbb{P}^1}) = 1$, and that $\chi(\mathcal{O}_S) = |{{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}| - |{K(\gamma^c/\alpha; \beta)}|$.
Proposition \[prop:ram-pts\] suggests that genomic tableaux be used to index *complex conjugate pairs* of ramification points.
Is it possible to assign, to each complex conjugate pair of ramification points in $R(\alpha,\beta,\gamma)$, a genomic tableau from $K(\gamma^c/\alpha;\beta)$?
Conjecture \[conj:orbit-by-orbit\] then suggests assigning to each ramification point $p \in R$ an arc on some component of $S(\mathbb{R})$ – ideally on the same irreducible component as $p$ – compatibly with the labeling by genomic tableaux and the bijections $\varphi_i$. Such an assignment would further relate the real and complex topology of $S$. For instance:
Suppose $T \in {{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$ is an $\omega$-fixed point. Let $S' \subseteq S$ be the irreducible component containing $T$. Must $S'$ be a copy of $\mathbb{P}^1$, mapping (via $f$) to $\mathbb{P}^1$ with degree 1?
The converse is true: if some component $S'$ maps isomorphically to $\mathbb{P}^1$, then $S'(\mathbb{R}) \cap f^{-1}(0)$ corresponds to an $\omega$-fixed point under the identification of $f^{-1}(0)$ with ${{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$. On the other hand, we have shown (Proposition \[prop:fixed-points\]) that if *every* $T$ is a fixed point, that is, $\omega$ is the identity, then $S$ is indeed a disjoint union of $\mathbb{P}^1$’s, each mapping isomorphically under $f$.
Let $\mathcal{O} \subseteq {{\mathrm{LR}}(\alpha,{\scalebox{.5}{\yng(1)}},\beta, \gamma)}$ be an orbit such that $K_i(\mathcal{O}) = |\mathcal{O}| - 1$ for $i=1,2$. Let $S' \subseteq S$ be the irreducible component containing $\mathcal{O}$. Must $S'$ be a copy of $\mathbb{P}^1$, mapping to $\mathbb{P}^1$ with degree $|\mathcal{O}|$?
If the global inequality is replaced by an equality (and is then true of every orbit), it is possible to show that this is true, i.e. that $S$ is a disjoint union of $\mathbb{P}^1$’s, each mapping to $\mathbb{P}^1$ with the appropriate degree – in particular, in the Pieri Case. On the other hand, if a single irreducible component $S'$ contains a number of ramification points equal to $(\deg f|_{S'}) - 1$, then the Riemann-Hurwitz formula implies that $g(S') = 0$, i.e. $S' \cong \mathbb{P}^1$ and $S'(\mathbb{R})$ has only one connected component.
Finally, although we have only defined *local* evacuation-shuffling for Littlewood-Richardson tableaux, the evacuation-shuffle ${\mathrm{esh}}$ is defined on *all* tableaux $({\boxtimes},T)$ as the conjugation of shuffling by rectification. Our results do yield local algorithms for certain other classes of tableaux, such as *lowest*-weight semistandard tableaux, via straightforward alterations to ${\mathrm{local\text{-}esh}}$. (For lowest-weight semistandard tableaux, the local algorithm resembles a rotated version of ${\mathrm{local\text{-}esh}}^{-1}$.) It would be interesting to understand the actions of ${\mathrm{esh}}$ and $\omega$ on arbitrary representatives of dual equivalence classes, and on semistandard tableaux in general. We may be more precise:
Let $T$ be **any** (semi)standard skew tableau and ${\boxtimes}$ an inner co-corner of $T$. There exists a local algorithm for computing ${\mathrm{esh}}({\boxtimes},T)$, which does not require rectifying the tableau, such that:
- Each step consists of exchanging the ${\boxtimes}$ with an entry of $T$, of weakly increasing value.
- The slide equivalence class of $T$ is preserved throughout the algorithm.
- The algorithm specializes to jeu de taquin (if $T$ is of straight shape) and ${\mathrm{local\text{-}esh}}$ (if $T$ is ballot).
Each step should correspond (by conjugating with rectification) to a jeu de taquin slide of ${\boxtimes}$ through the rectification ${\mathrm{rect}}({\boxtimes},T)$.
It would also be interesting to investigate how such algorithms might relate to K-theoretic Schubert calculus.
For a *straight-shape* tableau $T$ that is *not* highest-weight, the shuffle path of the ${\boxtimes}$ is just the path given by jeu de taquin slides through $T$. It would be interesting to find a generalization of the $s$-decomposition that describes this shuffle path, and that gives rise to a local algorithm on any skew tableau $T'$ whose rectification is $T$.
We may also ask analogous questions for computing ${\mathrm{esh}}(S,T)$ locally, where both $S$ and $T$ may have more than one box.
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[^1]: Note that the definition of corner of $\lambda/\mu$ depends on the pair of partitions $\lambda$ and $\mu$, not just the squares that make up the skew shape. The same square may be both an inner and outer corner; likewise for co-corners.
[^2]: This phenomenon reflects the fact that both transformations encode the ‘Fundamental Symmetry’ of Young tableau bijections, in the sense of Pak and Vallejo’s work in [@bib:PakVallejo]. Consequently, the composition *does not* encode this deep symmetry, hence is easier to compute.
[^3]: We note, however, that $\omega$ is not a commutator in the sense of group theory, since it involves maps between two different sets. In particular, as computed in Theorem \[thm:intro-parity\], $\omega$ need not be an even permutation.
[^4]: Our ordering is the reverse of the ordering used in [@bib:Levinson].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We have analyzed the $M_{1+}^{(3/2)}$ and $E_{1+}^{(3/2)}$ multipole amplitudes of pion photoproduction in the framework of fixed-$t$ dispersion relations. Applying the speed plot technique to our results for these multipoles, we have determined the position and the residues of the $\Delta$ (1232) resonance pole. The pole is found at total $c.m.$ energy $W = (1211 - 50i)$ MeV on the second Riemann sheet, and the ratio of the electric and magnetic residues is $R_{\Delta} = - 0.035 - 0.046 i$, resulting in an E2/M1 ratio for the “dressed” delta resonance of $- 3.5 \%$.'
address: 'Institut für Kernphysik, Universität Mainz, 55099 Mainz, Germany'
author:
- 'O. Hanstein, D. Drechsel and L. Tiator'
title: |
The position and the residues of the delta resonance pole\
in pion photoproduction
---
Introduction
============
The determination of the quadrupole excitation strength $E_{1+}^{(3/2)}$ in the region of the $\Delta (1232)$ resonance has been the aim of considerable experimental and theoretical activities. Within the harmonic oscillator quark model, the $\Delta$ and the nucleon are both members of the symmetrical 56-plet of $SU(6)$ with orbital momentum $L = 0$, positive parity and a Gaussian wave function in space. In this approximation the $\Delta$ may only be excited by a magnetic dipole transition $M_{1+}^{(3/2)}$ [@Bec65]. However, in analogy with the atomic hyperfine interaction or the forces between nucleons, also the interactions between the quarks contain a tensor component due to the exchange of gluons. This hyperfine interaction admixes higher states to the nucleon and $\Delta$ wave functions, in particular $d$-state components with $L = 2$, resulting in a small electric quadrupole transition $E_{1+}^{(3/2)}$ between nucleon and $\Delta$ [@Kon80; @Ger81; @Isg82], and a quadrupole moment of the $\Delta$ $Q_{\Delta} \approx -.09 fm^2$ [@Dre84]. Therefore an accurate measurement of $E_{1+}^{(3/2)}$ is of great importance in testing the forces between the quarks and, quite generally, models of nucleons and isobars.
The ratio $$\label{R_EM}
R_{EM} = \frac{\mbox{Re}\left[E_{1+}^{(3/2)}M_{1+}^{(3/2)\ast}\right]}
{\mid M_{1+}^{(3/2)}\mid^{2}}$$ has been predicted to be in the range $- 2\%\le R_{EM} < 0\%$ in the framework of constituent quark [@Kon80; @Isg82; @Dre84], relativized quark [@Bie87; @Cap90; @Cap92] and chiral bag models [@Kae83; @Ber88]. Considerably larger values have been obtained in Skyrme models [@Wir87]. A first lattice QCD calculation resulted in a small value with large error bars $(- 6 \% \le R_{EM} \le 12\%)$ [@Lei92]. However, the connection of the model calculations with the experimental data is not evident. Clearly, the $\Delta$ resonance is coupled to the pion-nucleon continuum and final-state interactions will lead to strong background terms seen in the experimental data, particularly in case of the small $E_{1+}$ amplitude. The question of how to ”correct” the experimental data to extract the properties of the resonance has been the topic of many theoretical investigations. As some typical examples we refer the reader to the work of Olsson [@Ols74], Koch et al. [@Koc84] and Laget [@Lag88]. Unfortunately it turns out that the analysis of the small $E_{1+}$ amplitude is quite sensitive to details of the models, e.g. nonrelativistic vs. relativistic resonance denominators, constant or energy-dependent widths and masses of the resonance, sizes of the form factor included in the width etc. In other words, by changing these definitions the meaning of resonance vs. background changes, too. More recently, Nozawa et al. [@Noz90] have included final-state interactions in a dynamical model with quark cores and pions, and Davidson et al. [@Dav86] have analyzed photoproduction in terms of effective Lagrangians, taking account of final-state interactions implicitly through unitarization. By fitting the parameters of the models to older sets of data, Ref. [@Noz90] obtained a ratio $R_{EM}^{\mbox{\scriptsize bare }\Delta} = - 3.1 \%$ for the bare $\gamma N\Delta$ coupling, while Ref. [@Dav86] deduced a value of $R_{EM}^{\mbox{\scriptsize res}} = - 1.4 \%$, including some ”dressing” from final-state interaction. A detailed discussion of these models and a comparison to the data is given in Ref. [@Kha93]. In an extension of the work of Nozawa et al., Bernstein et al. [@Ber93] have decomposed the multipole contributions into resonant and background terms, and compared their analysis to previous investigations. As a result they obtained $R_{EM}^{\mbox{\scriptsize bare }\Delta} = -(3.1 \pm 1.3)\%$ for the ”bare $\Delta$” amplitude and $R_{EM}^{\mbox{\scriptsize res}} = - 2.2\%$ for the ”dressed $\Delta$”. In very recent relativistic and unitarized pion photoproduction calculations, for the ”bare $\Delta$” ratios of $R_{EM}^{\mbox{\scriptsize bare }\Delta} = -1.43\%$ [@Van95] and $R_{EM}^{\mbox{\scriptsize bare }\Delta} = -1.46\%$ [@Sur95] are found.
In order to study the $\Delta$ deformation, pion photoproduction on the proton has recently been measured by the LEGS collaboration [@Bla92] at Brookhaven and by the A2 collaboration [@Bec95] at MAMI using transversely polarized photons, i.e. by measuring the polarized photon asymmetry $\Sigma$. In particular, the cross section $d\sigma_{\parallel}$ for photon polarization in the reaction plane turns out to be very sensitive to the small $E_{1+}$ amplitude. Assuming for simplicity that only the $P$-wave multipoles contribute, the differential cross section is $$\frac{d\sigma_{\parallel}}{d\Omega} = \frac{q}{k} (A_{\parallel} + B_{\parallel}
\cos \Theta_{\pi} + C_{\parallel} \cos^2 \Theta_{\pi}),$$ where $q$ and $k$ are the pion and photon momenta and $\Theta_{\pi}$ is the pion emission angle in the $c.m.$ frame. Neglecting the (small) contributions of the Roper multipole $M_{1-}$, one obtains [@Bec95] $$C_{\parallel}/A_{\parallel} \approx 12 R_{EM} ,$$ because the isospin $\frac{3}{2}$ amplitudes strongly dominate the cross section for $\pi^{0}$ production. In the meantime new precision data have been obtained by the A2 collaboration at MAMI with polarized photons for both charged and neutral pion production over the energy range 270 MeV $\le E_{\gamma} \le$ 420 MeV [@Bec95; @Kra96]. These data will make it possible to determine the partial wave amplitudes over the full region of the $\Delta$ resonance. The preliminary data for $\pi^{0}$ production are in good agreement with the ratio $d\sigma_{\parallel} / d\sigma_{\perp}$ measured by the LEGS collaboration, at the lower energies.
Dispersion relations at fixed [[$t$]{}]{}
=========================================
Starting from fixed-$t$ dispersion relations for the invariant amplitudes of pion photoproduction, the projection of the multipole amplitudes leads to a well known system of integral equations, $$\label{inteq}
\mbox{Re}{\cal M}_{l}(W) = {\cal M}_{l}^{\mbox{\scriptsize P}}(W)
+ \frac{1}{\pi}\sum_{l'}{\cal P}\int_{W_{\mbox{\scriptsize thr}}}^{\infty}
K_{ll'}(W,W')\mbox{Im}{\cal M}_{l'}(W')dW',$$ where ${\cal M}_l$ stands for any of the multipoles $E_{l\pm}, M_{l\pm},$ and ${\cal M}_{l}^{\mbox{\scriptsize P}}$ for the corresponding (nucleon) pole term. The kernels $K_{ll'}$ are known, and the real and imaginary parts of the amplitudes are related by unitarity. In the energy region below two-pion threshold, unitarity is expressed by the final state theorem of Watson [@Wat54], $$\label{watson}
\cal{M}_{l}^{I} (W) = \mid \cal{M}_{l}^{I} (W)\mid e^{i(\delta_{l}^{I} (W)
+ n\pi)},$$ where $\delta_{l}^{I}$ is the corresponding $\pi N$ phase shift and $n$ an integer. We have essentially followed the method of Schwela et al [@Sch69; @Pfe72] to solve Eqs. (\[inteq\]) with the constraint (\[watson\]). In addition we have taken account of the coupling to some higher states neglected in that earlier reference. At the energies above two-pion threshold up to $W = 2$ GeV, Eq. (\[watson\]) has been replaced by an ansatz based on unitarity [@Sch69]. Finally, the contribution of the dispersive integrals from $2$ GeV to infinity has been replaced by $t$-channel exchange, parametrized by certain fractions of $\rho$- and $\omega$-exchange. Furthermore, we have to allow for the addition of solutions of the homogeneous equations to the coupled system of Eq. (\[inteq\]). The whole procedure introduces 9 free parameters, which have to be determined by a fit to the data. In our data base we have included the recent MAMI experiments for $\pi^{\circ}$ and $\pi^{+}$ production off the proton in the energy range from 160 MeV to 420 MeV [@Fuc96; @Kra96; @Hae96], both older and more recent data from Bonn for $\pi^+$ production off the proton [@Men77; @Bue94; @Zuc95], and older Frascati [@Car73] and more recent TRIUMF data [@Bag88] on $\pi^{-}$ production off the neutron. As shown in Fig. \[fig:legs\], the predicted cross sections are in perfect agreement with the ratio $d\sigma_{\parallel}/d\sigma_{\perp}$ measured by the LEGS collaboration [@Bla92; @San96] whose data have not been included in our fit. In Fig. \[fig:e\_over\_m\] we show our result for the ratio $R_{EM}$ which is in general agreement with the analysis of the Virginia group [@Arn95].
The resonance pole parameters as determined by the speed plot
=============================================================
The analytic continuation of a resonant partial wave as function of energy into the second Riemann sheet should generally lead to a pole in the lower half-plane. A pronounced narrow peak reflects a time-delay in the scattering process due to the existence of an unstable excited state. This time-delay is related to the speed $SP$ of the scattering amplitude $T$, defined by [@Hoe92; @Hoe93] $$SP(W) = \left\vert \frac{dT(W)}{dW}\right\vert ,$$ where $W$ is the total $c.m.$ energy. In the vicinity of the resonance pole, the energy dependence of the full amplitude $T = T_{B} + T_{R}$ is determined by the resonance contribution, $$\label{T_res}
T_{R} (W) = \frac{r\Gamma_{R} e^{i\phi}}{M_{R}- W- i\Gamma_{R}/2}\,\,,$$ while the background contribution $T_B$ should be a smooth function of energy, ideally a constant. We note in particular that $W_R = M_{R}- i\Gamma_{R}/2$ indicates the position of the resonance pole in the complex plane, i.e. $M_{R}$ and $\Gamma_{R}$ are constants and differ from the energy-dependent widths, and possibly masses, derived from fitting certain resonance shapes to the data[@RPP94]. If the energy dependence of $T_{B}$ is negligible, the speed is $$SP(W) = r\Gamma_{R}\frac{\{[(M_{R}-W)^{2}-\Gamma_{R}^{2}/4]^2
+\Gamma_{R}^2(M_{R}-W)^{2}\}^\frac{1}{2}}
{\{(M_{R}-W)^{2}+\Gamma_{R}^{2}/4\}^{2}}.$$ Obviously the speed has its maximum at $W = M_R$, $SP(M_{R}) = 4r/\Gamma_{R} = H$, and the half-maximum values are $SP(M_{R} \pm \Gamma_{R}/2) = H/2$. This determines the parameters $M_{R}$ and $\Gamma_{R}$ as well as the absolute value $r$ of the residue. The phase $\phi$ of the complex residue at the pole may be determined from an Argand plot of the speed vector $dT/dW$.
It should be noted that the speed plot technique requires a reasonably smooth representation of the amplitude in order to differentiate it in a meaningful way. As has been shown by Höhler [@Hoe92; @Hoe93] in the case of $\pi N$ scattering, dispersion relations are particularly well suited for this purpose. If we apply the method to the partial waves obtained by solving Eqs. (\[inteq\]), the results for the $\Delta$ multipole clearly show a resonant peak, but the asymmetry with respect to the maximum indicates an energy dependence of the background (see Fig. \[fig:multi\_sp\]). This effect may be traced back to the nucleon pole terms. After subtracting these well-defined terms from the amplitudes, the speed of both $E_{1+}$ and $M_{1+}$ can be well described according to Eq. (\[T\_res\]). Fig. \[fig:sp\_arg\] shows a comparison of this procedure to the ideal shape of a resonance pole and the resulting Argand diagrams for the speed vector. Except for the threshold region, the differences are almost invisible. Having determined all the resonance parameters, we can now decompose the full amplitudes into contributions of the resonance pole and background terms. As may be seen in Fig. \[fig:uni\_m\], the background is a relatively smooth function of energy without any structure around the resonance. However, the background is quite large, in particular in the case of the $E_{1+}$ amplitude.
The resonance parameters derived from our analysis are shown in Table \[tab:res\_par\]. It is seen that the pole position $W_{R} = M_{R} - i \Gamma_{R}/2
= (1211 - 50 i)$ MeV is in excellent agreement with the results obtained from $\pi N$ scattering, $M_{R}= (1210\pm1)$ MeV and $\Gamma_{R} = 100$ MeV [@Hoe92; @Hoe93; @RPP94] . This agreement may not be very surprising in the case of the largely resonant $M_{1+}$ amplitude, for which there exist earlier investigations to determine the pole position [@Cam76; @Mir79]. However, it is much less obvious that the interference pattern of Re $E_{1+}$ of Fig. \[fig:uni\_m\] should lead to the same answer. The excellent agreement in that case, too, is indeed very satisfactory and shows that the speed plot technique is quite reliable for the extraction of resonance properties. The table also shows the absolute values and the phases of the resonance residues. Because of the different backgrounds in the two amplitudes, $\phi_M$ and $\phi_E$ are different, and the ratio of the resonance amplitudes is complex. The fact that the two ”apple shaped” structures in Fig. \[fig:sp\_arg\] are essentially oriented in opposite direction is, however, related to the negative value of $R_{EM}$ for the full (experimental) amplitude. Concerning the resonance pole contributions alone, we obtain $$R_{\Delta} = \frac{r_{E} e^{i\phi^{E}}}{r_{M} e^{i\phi_{M}}}
= - 0.035 - 0.046 i.$$
The ratio of the heights of the speed plots is $H_{E}/H_{M} =
r_{E}/r_{M} = 5.8\%$. We hasten to add, however, that the experimental observable is related to the real part of the ratio (see Eq. \[R\_EM\]), i.e. the (unphysical) case of the resonance without background would lead to $R_{EM}^{\mbox{\scriptsize res}}=$Re$(R_{\Delta}) = - 3.5\%.$
As has been mentioned before, the ratio for the full (experimental) amplitudes is real below two-pion threshold due to the Watson theorem. As may be seen from Fig. \[fig:e\_over\_m\], this ratio $R_{EM}(W)$ is strongly dependent on energy, and increases with energy from negative to positive values, e.g. $R_{EM}
(M_{R} - \Gamma_{R}/2) = -10.4\%$, $R_{EM} (M_{R}) = -4.3\%$, $R_{EM} (M_{R}
+\Gamma_{R}/2) =0.1\%$. The resonance pole in the complex plane, $M_R - i \Gamma_{R}/2$, and the nonresonant background lead to a $\pi N$ phase shift $\delta_{1+} = 90^{\circ}$ at $W = M_{\Delta} = 1232$ MeV. Due to the Watson theorem, both $E_{1+}^{(3/2)}$ and $M_{1+}^{(3/2)}$ are completely imaginary at this point, and the ratio can be determined from the experimental data as $R_{EM}(M_{\Delta})=$Im$ E_{1+}^{(3/2)} (M_{\Delta})/$Im$M_{1+}^{(3/2)}
(M_{\Delta})$. The recent, nearly model-independent value of the Mainz group at $W = M_{\Delta}$ is $(-2.5 \pm 0.2)\%$ [@Bec95; @Kra96].
Conclusion
==========
It has been shown that the method of speed plots can be well applied to analyze the pion photoproduction amplitudes. The resonance pole position of the $\Delta (1232)$ is obtained from these amplitudes in excellent agreement with the results from pion-nucleon scattering. The complex residues of the resonance pole terms give information on those parts of the full amplitude that have a resonance–like behaviour. Whether such a contribution originates from a ”bare” resonance or, e.g. from a nonresonant pion production followed by rescattering into a resonant state, is model-dependent [@Wil96] and cannot be answered by an analysis of the data but only within the framework of a specific model.
In the future it will be interesting to analyze double polarization variables, e.g. both photon and recoil or target polarization, because some of these observables turn out to be very sensitive to electric quadrupole radiation, too. It is also worthwhile pointing out that reactions like $e\vec{p} \rightarrow e'p\pi^{\circ}$ yield a longitudinal-transverse interference term (”fifth structure function”), which is sensitive to the imaginary part of the interference between the resonant and background multipoles.
We would like to thank Prof. G. Höhler for very fruitful discussions and the members of the A2 collaboration at Mainz for providing us with their preliminary data, in particular R. Beck, F. Härter and H.-P. Krahn. This work was supported by the Deutsche Forschungsgemeinschaft (SFB 201).
[99]{} C. M. Becchi and G. Morpurgo, Phys. Lett. 17 (1965) 352. R. Koniuk and N. Isgur, Phys. Rev. D 21 (1980) 1868. S. S. Gershteyn and G. V. Dzhikiya, Sov. J. Nucl. Phys. 34 (1981) 870. N. Isgur, G. Karl and R. Koniuk, Phys. Rev. D 25 (1982) 2394. D. Drechsel and M. M. Giannini, Phys. Lett. 143 B (1984) 329. J. Bienkowska, Z. Dziembowski and H. J. Weber, Phys. Rev. Lett. 59 (1987) 624. S. Capstick and G. Karl, Phys. Rev. D 41 (1990) 2767. S. Capstick, Phys. Rev. D 46 (1992) 2864. G. Kälbermann and J. J. Eisenberg, Phys. Rev. D 28 (1983) 71 and D 29 (1984) 517. K. Bermuth, D. Drechsel, L. Tiator and J. B. Seaborn, Phys. Rev. D 37 (1988) 89. A. Wirzba and W. Weise, Phys. Lett. B 188 (1987) 6. D. B. Leinweber, T. Draper and R. Woloshyn, Contribution to Baryons ’92, p. 29 (1992). M. G. Olsson, Nucl. Phys. B 78 (1974) 55. J. H. Koch, E. J. Moniz and N. Ohtsuka, Ann. Phys. (N. Y.) 154 (1984) 99. J. M. Laget, Nucl. Phys. A 481 (1988) 765. S. Nozawa, B. Blankleider and T.–S. Lee, Nucl. Phys. A 513 (1990) 513. R. M. Davidson, N. C. Mukhopadhyay and R. Wittman, Phys. Rev. Lett. 56 (1986) 804 and Phys. Rev. D 43 (1991) 71. M. A. Khandaker et al., $\pi N$ Newsletter 8 (1993) 114. A. M. Bernstein, S. Nozawa and M. A. Moinester, Phys. Rev. C 47 (1993) 1274. M. Vanderhaeghen, K. Heyde, J. Ryckebusch, and M. Waroquier, preprint SSF95-05-01, Gent (1995). Y. Surya and F. Gross, preprint CEBAF-TH-95-04 (1995). G. S. Blanpied et al., Phys. Rev. Lett. 69 (1992) 1880. R. Beck, Proc. Int. Conf. “Baryons ’95”, Santa Fé, (1995). H.–P. Krahn, Ph.D. thesis, Mainz (1996). K. M. Watson, Phys. Rev. 95 (1954) 228. D. Schwela and R. Weizel, Z. Physik 221 (1969) 71. W. Pfeil and D. Schwela, Nucl. Phys. B 45 (1972) 379. M. Fuchs et al., Phys. Lett. B 368 (1996) 20. F. Härter, PhD. thesis, Mainz (1996). D. Menze, W. Pfeil and R. Wilcke, Compilation of pion photoproduction data, Bonn (1977). K. Buechler et al., Nucl. Phys. A 570 (1994) 580. H. Dutz, PhD. thesis, Bonn (1993),\
D. Krämer, PhD. thesis, Bonn (1993),\
B. Zucht, PhD. thesis, Bonn (1995). F. Carbonara et al., Nuovo Cim. 13 A (1973) 59. A. Bagheri et al., Phys. Rev. C 38 (1988) 875. A. Sandorfi, private communication (1996), release L7a8.0. R. A. Arndt, I. I. Strakovsky and R. L. Workman, Phys. Rev. C 53 (1996) 430. G. Höhler and A. Schulte, $\pi N$ Newsletter 7 (1992) 94. G. Höhler, $\pi N$ Newsletter 9 (1993) 1. Review of Particle Properties, Phys. Rev. D 50 (1994). R. R. Campbell, G. L. Shaw and J. S. Ball, Phys. Rev. D 14 (1976) 2431. I. I. Miroshnichenko, V. I. Nikiforov, V. M. Sanin, P. V. Sorokin, and S. V. Shalatskii, Sov. J. Nucl. Phys. 29 (1979) 94. P. Wilhelm, Th. Wilbois and H. Arenhövel, preprint MKPH-T-96-1, Mainz (1996).
$r$ $[10^{-3}\mathrm{MeV}/m_{\pi}]$ $\phi$ \[$^{\circ}$\] $M_{R}$ \[MeV\] $\Gamma_{R}$ \[MeV\]
--- ------------------------------------- ----------------------- ----------------- ----------------------
E 1.23 -154.7 1211$\pm$1 102$\pm$2
M 21.16 -27.5 1212$\pm$1 99$\pm$2
: Resonance pole parameters determined by applying of the speed plot technique to our results for $E_{1+}^{(\frac{3}{2})}$ and $M_{1+}^{(\frac{3}{2})}$.
\[tab:res\_par\]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We report the possible optical identification of the companion to the eclipsing millisecond pulsar PSR J1740$-$5340 in the globular cluster NGC 6397. A bright variable star with an anomalous red colour and optical variability which nicely correlates to the orbital period of the pulsar has been found close to the pulsar position. If confirmed, the optical light curve, reminiscent of tidal distorsions similar to those observed in detached and contact binaries, support the idea that this is the first case of a Roche lobe filling companion to a millisecond pulsar.'
author:
- 'Francesco R. Ferraro, Andrea Possenti, Nichi D’Amico, Elena Sabbi'
nocite:
- '[@dpms+01]'
- '[@b+83]'
- '[@bi89]'
- '[@ale99]'
- '[@ghemc2001]'
- '[@ghem2001]'
- '[@dpms+01]'
- '[@svdm98]'
- '[@v00]'
- '[@cvr95]'
- '[@svlk99]'
- '[@ts99]'
- '[@s70]'
- '[@rst89]'
- '[@bdm+01]'
title: 'The bright optical companion to the eclipsing millisecond pulsar in NGC 6397[^1]'
---
Introduction {#sec:intro}
============
The millisecond pulsar (MSP) PSR J1740$-$5340 was discovered during a systematic search of the globular cluster (GC) system for millisecond pulsars, carried out with the Parkes radiotelescope (D’Amico [et al. ]{}2001a, 2001b). The pulsar, associated with the globular cluster NGC6397, is member of a binary system with a relatively wide orbit of period $\simeq$ 1.35 days, and it is eclipsed for about 40% of the orbital phase at 1.4 GHz. In a companion paper, D’Amico et al. (2001c) provide strong evidences that the companion star could be an unusual object, and give a precise position for the pulsar.
We here present the results of a deep search of the optical companion to PSR J1740$-$5340 in the HST archive.
Observations and data analysis {#sec:obs}
==============================
The photometric data consists of a series of public HST exposures taken on March 1996 and April 1999, retrieved from the [*ESO/ST-ECF Science Archive*]{}. The 1999 observations consist of 116 WFPC2 exposures with filters F555W, F675W, F814W and F656N (referred here as $V_{555}, R_{675}, I_{814}$ and H$\alpha$), spanning about 1.8 days. The 1996 observations consist of 55 exposures with filters F336W and F439W (referred here as $U_{336}$ and $B_{439}$), spanning 0.4 and 0.2 days respectively.
From the accurate timing position (RAJ 17$^{\rm h}$ 40$^{\rm m}$ 44589; DECJ $-$53$^{o}$ 40$\arcmin$ 40$\farcs9$) the MSP in NGC6397 turns out to be approximately at $29''E $ and $16''S$ from the cluster center (Djorgovski & Meylan 1993). We used the STSDAS program $metric$, to roughly locate the MSP in the retrieved WFPC2 images, and it turns out to be within the field of view of the WF4 chip in both the data set.
Astrometry
----------
The exact location of the MSP in the HST images was obtained by searching an astrometric solution for a wide field CCD image of a region around NGC 6397. We retrieved from the [*ESO Science Archive*]{} an image obtained on May 1999 with the Wide Field Imager (WFI) at the ESO 2.2m telescope (at European Souther Observatory, La Silla, Chile). The entire image consists of a mosaic of 8 chips (each with a field of view of $8'\times 16'$) giving a global field of view of $33'\times 34'$. Only the chip containing the cluster center was used. The new astrometric [*Guide Star Catalog*]{} ($GSCII$) recently released and now available on the WEB, was used to search for astrometric standard lying in the WFI image field of view: several hundreds astrometric $GSCII$ reference stars have been found, allowing an accurate absolute positioning of the image.
In order to derive an astrometric solution for the WFI image we used an appropriate procedure developed at the Bologna Observatory. The resulting rms residuals were of the order of $\sim 0.3''$ both in RA and Dec. By using this astrometry we were able to accurately locate the nominal position of the MSP in the WFI image and in the HST-WFPC2 images.
Fig. 1 shows an enlargement of a $7''\times 7''$ region of the WF4 chip, centered on the MSP position. The $3\sigma$ error circle, which takes into account the global error in the absolute positioning of the MSP, is shown. The global error is fully dominated by the uncertainty due to the astrometric procedure. One relatively bright star (A) has been found within the error box of the MSP. Two additional objects (B, C) are just out-side the error circle. In order to investigate the nature of these objects we performed accurate photometric analysis of the entire HST-WFPC2 data-set retrieved from the Archive.
Photometry
----------
The photometric reductions have been carried out using ROMAFOT (Buonanno et al. 1983), a package developed to perform accurate photometry in crowded fields and specifically optimized to handle under-sampled point spread function (PSF) as in the case of the HST-WF chips (Buonanno & Iannicola 1989). The standard procedure described in Ferraro et al. (1997) was adopted in order to derive PSF-fitting instrumental magnitudes, which were finally calibrated using zero-points listed by Holtzman et al. (1995). In particular we used a sample of median-combined images to construct reference Color Magnitude Diagrams (CMDs). Figure 2 shows multiband CMDs for stars detected in a region of $400\times400$ pixels (corresponding to $40''\times40''$): the location of the three objects are indicated.
The result of the PSF fitting procedure for these stars have been carefully examined by visual inspection. From this accurate photometric analysis we found that the bright object A lying within the error box of the MSP has an anomalous position in the CMD since it is located at the luminosity of the TO region but it has an anomalous red colour; the other two objects (B, C) are normal Main Sequence stars. Individual images were instead used to check the variability of the objects. Objects B and C show no significant time variability compared to the measurements uncertainties. On the other hand, object A shows a remarkable time modulation ($\sim 0.2-0.3$ mag) on a scale of several hours. This object is the variable star WF4-1 proposed by Taylor [et al. ]{}(2001) as a BY Draconis star.
Time series
-----------
In order to check the association of the time variability of [Star A ]{}to the pulsar binary motion, we have carried out a period search analysis. The data available consist of four time series in the H$\alpha$, $R_{675}, I_{814}$, and $V_{555}$ bands taken in 1999, spanning $\sim$ 1.8 days, and two time series in the $B_{439}$ and $U_{336}$ bands taken in 1996, spanning 0.2 and 0.4 days, respectively. The periodicity search was carried out using GRATIS (GRaphycal Analyzer of TIme Series), a software package developed at the Bologna Astronomical Observatory (see Clementini et al. 2000, 2001). Periods and amplitudes were derived for the H$\alpha$-band time series using GRATIS $\chi^{2}$ Fourier fitting routine. This algorithm is almost equivalent to the Lomb-Scargle periodogram (Scargle 1982), but has the advantage to have sensitivity also to periodicities whose light curve is not strictly sinusoidal, and it is more reliable when the data span is of length comparable to the time scale of the stellar variation (Faulkner 1977). As already mentioned, the data span of the H$\alpha$-band time series is $\sim$ 1.8 days, and the searched periodicity is 1.35 days, so the $\chi^{2}$ fitting method is largely preferable.
0.3truecm [ ]{}
Fig. 3 shows the reduced $\chi^{2}$ resulting from the Fourier fitting of the H$\alpha$ data as a function of the modulation period. The most significant feature is indeed a periodicity around the predicted period of 1.35 days, with substantial power also near the 2$^{nd}$ harmonic. The confidence level peak (99.6%) corresponds to a period $P$=1.37$\pm$0.05 days (consistent, within the uncertainties, with the period quoted by Taylor [et al. ]{}(2001) for the WF4-1 variable). The quoted uncertainty corresponds to the period range for which the confidence level is larger than 99%, and it is dominated by the relatively short ($\sim$ 1.8 days) data span available.
Using another option of the GRATIS package, we have then fixed the period $P$ and the reference epoch $T_{0}$ to the radio ephemeris values, and have fitted the same H$\alpha$ data for the best spectral amplitudes of the 1$^{st}$ and 2$^{nd}$ harmonics. The best-fit light curve, shown in the small panel in Fig. 3, is not exactly what we would expect on the basis of a simple pulsar-irradiation model, but as we will discuss in the next section, it could be understood in term of tidal distorsion effects occurring in the companion star to PSR J1740-5340. The time variability observed in the R, I, and V-bands at the same epochs follows a similar pattern to that observed in the H$\alpha$-band, and the period search analysis produces similar results. Fig. 4 shows the same 1999 H$\alpha$-band data and the $U_{336}$-band data taken on 1996, phased using the accurate radio ephemeris. Remarkably, they show the minimum at the same orbital phase, giving further evidence that the optical modulation is indeed associated to the pulsar binary motion.
0.5truecm [ ]{}
Is [Star A ]{}the pulsar companion?
-----------------------------------
Can we claim that [Star A ]{}(WF4-1) is not a BY Draconis system but is indeed the optical companion to the MSP? There is no doubt that [Star A ]{}is variable, and that the modulation period is compatible to the radio orbital period. However is difficult to estimate the chance occurrence probability to find a variable star with a period of $\sim$ 1.3$-$1.4 days in such a small error circle. HST observations (Taylor [et al. ]{}2001) have discovered several variables in a WFPC2/HST field of view. Also, according to Taylor [et al. ]{}(2001), a modulation period of the order of $\sim$day is typical of most BY Dra systems.
On the other hand, in two observations taken three years a part, we find the minima exactly at the same phase with respect to the precise radio orbital period, whilst the light curve shape of the BY Dra systems are expected to change, according to variations in the configuration of the spotted regions (Alekseev 1999) of their convective envelopes. Also, the position of [Star A ]{} in the CMD is anomalous for a BY Dra or whichever other kind of binary system comprising two MS stars. Further support to the proposed association derives from the detection of PSR J1740-5340 in a [*Chandra*]{} pointing of NGC 6397 (Grindlay [et al. ]{}2001b): its X-ray luminosity and color appear similar to those of the MSPs seen in 47 Tuc (Grindlay [et al. ]{}2001a) and its positional coincidence with [Star A ]{}is consistent with the [*Chandra*]{} astrometric uncertainties.
Observed properties of the companion star
=========================================
In the radio timing paper, D’Amico [et al. ]{}(2001c) demonstrate that the companion star can not be the typical WD found in most binary MSPs. They propose that the companion can be a MS star acquired by exchange interaction in the cluster core or alternatively the same star that spun up the MSP and that would be still overflowing its Roche lobe. Assuming that [Star A ]{}is the pulsar companion, we here discuss these two hypotheses, comparing them with observed optical properties of [Star A ]{}.
In order to get some quantitative hints on the effective temperature $T_{eff}$ and the radius $R_c$ of [Star A ]{}, we used the recent set of isochrones by Silvestri [et al. ]{}(1998) and by Vanderberg (2000). By comparing the CMDs in Fig. 2 with those isochrones, for metallicity \[Fe/H\]=$-$2.00 and ages of $t=12-14$ Gyr, (compatible with the values measured for NGC 6397), we derive $R_{c}\sim 1.3-1.8~{\rm R_\odot}$ and $T_{eff}\sim 5500-5800$ K for [Star A ]{}.
The peculiar nature of [Star A ]{}can be unveiled inspecting the amplitude and the shape of its light curves (Figure 3, 4). There are 2 other eclipsing MSPs (PSR B1957+20 (Callanan, van Paradijs & Rengelink 1995) and PSR J2051-0827 (Stappers [et al. ]{}1999) ), both in the Galactic field, whose optical companion displays strong modulations, interpreted as due to the heated side of the companion entering in and out of view according to the orbital motion. Similar trend, though with a much smaller degree of modulation, is seen in 47 Tuc U$_{opt}$, the first identified MSP companion in a GC (Edmonds, [et al. ]{}2001).
The light curves of [Star A ]{}are completely different. We locate the phase 0.0 at the ascending node of the MSP orbit; thus at the phase 0.75 we see the side of the companion facing the pulsar. In contrast to the other known variable MSP companions, the light curves of [Star A ]{}display there a minimum instead of a maximum (see Figure 4). Within the limits in the orbital period coverage of our photometry, the best-fit light curve of Figure 3 shows two maxima and two minima during each binary orbit: thus, tidal distorsions appear the more natural responsible for this shape. They have been already invoked for explaining the light curves of the optical companions to black-hole candidates (van der Klis, [et al. ]{}1985) and NSs (Zurita, [et al. ]{}2000). In this scenario the maxima correspond to quadratures (phases 0.0 and 0.50), when the distorted star presents the longest axis of its ellipsoid to the observer, the minima to the conjunctions. It is easy to recognize this trend in the insert of Figure 3.
Whether this is the correct interpretation, it turns out in severe constraints on the mass and the nature of Star A. The degree of ellipsoidal variations depends roughly on [@r45] $\Delta m=k_\lambda (M_{MSP}/M_c)(R_L/a)^3F^3\sin^2i,$ where $M_{MSP}$ and $M_c$ are the masses of the MSP and its companion, $R_L$ is the Roche lobe radius of the companion, $a$ is the orbital separation, $F$ is the ratio between the average radius of the star and the Roche lobe radius and $i$ is the inclination of the orbit. The term $k_\lambda=2.6$ accounts for limb and gravity darkening for H$_\alpha$-radiation from our source [@l67]. Given the orbital parameters of PSR J1740-5340 [@dpms+01], it follows $(M_{MSP}/M_c)(R_L/a)^3\sim 0.07$ for all the possible companions, and thus $\Delta m_{H_\alpha}\la 0.2F^3\sin^2i$. As the observed modulation (Figure 4) is just $\sim 0.2$ mag, we argue that only a companion almost filling its Roche lobe ($F\sim1$) and nearly edge-on ($i\sim 90^o$) can reproduce that. Remarkably, these two requirements are contemporary accomplished by a companion of mass $\la 0.25~{\rm M_\odot}$, whose Roche lobe radius just matches the lower limit inferred for the observed radius $R_c$ of [Star A ]{}.
The light curve of a star affected only by tidally distorsion would have the minimum at phase 0.25 less deep than that at phase 0.75. The reversal of this rule in the case of [Star A ]{}can result from the overheating of the side facing the pulsar. In contrast to 47 Tuc $U_{opt}$ (having $F\sim 0.17$), and to the companions to PSR J2051-0827 and to PSR B1957+20 (for which $F=0.5$ and $F=0.9$), [Star A ]{}fills up its Roche lobe allowing ellipsoidal variations to dominate over the thermal modulation. A direct detection of both the minima would allow a measurement of the fraction of the impinging power from the pulsar which goes in heating of the surface of [Star A ]{}, a very interesting value for understanding the composition of the pulsar energetic flux.
Discussion
==========
In summary, PSR J1740-5340 appears as the first example of a MSP orbiting a Roche lobe filling companion, whose brightness would allow unprecedented detailed investigations, for example about the origin of this system.
A first hypothesis is that [Star A ]{}is a MS star perturbed by the energetic flux emitted from the MSP. The so-called [*illumination*]{} mechanism [@d95] predicts that if the heating luminosity $L_h\la(1/4)(R_{*}/a)^2L_{irr}$ (where $R_*$ is the star radius and $L_{irr}$ the MSP luminosity) is large enough, the star inflates and increases the effective temperature, thus modifying its photometric characteristics [@p91]. The rotational energy loss from the MSP is $L_{irr}\sim 1.4\times 10^{35}~{\rm erg/s}$ [@dpms+01]. At the distance $a\sim 6.5~{\rm R_\odot}$, this corresponds to a characteristic temperature for the heating bath in which the star is immersed $T_h=[1/(16\sigma\pi)L_{irr}/a^2]^{1/4}\la 4000$ K where $\sigma$ is the constant of Stefan-Boltzmann. We expect that the MSP flux significantly affects the companion only if $T_h\ga T_*$ (where $T_*$ is the effective temperature of an unperturbed MS star) and $T_*\la 4000$ K implies $\la 0.4~{\rm M_\odot}.$ As $T_h$ does not depend on $R_{*}$, it seems energetically difficult to explain an increasing of $\sim 40\%$ of the effective temperature; however, only detailed simulations (Burderi, D’Antona & Burgay 2001, in preparation) of the system will allow to assess if such a low mass MS star of radius $\sim 0.2-0.4~{\rm R_\odot}$ can indeed be bloated up to $\la 1.3~{\rm R_\odot}$ and heated from $\sim 4000$ K to $\sim 5500$ K by the energetic flux of the MSP.
Another fascinating possibility is that PSR J1740$-$5340 is a new-born MSP, the first one observed just after the end of the process of recycling. In this case [Star A ]{}could have been originally a MS star of $1-2~{\rm M_\odot}$, whose evolution triggered mass transfer towards the compact companion, spinning it up to millisecond periods (Alpar, [et al. ]{}1982). Irregularities in the mass transfer rate $\dot M_{c}$ are common in the evolution of these systems (e.g. Tauris & Savonije 1999): even a short decreasing of $\dot M_{c}$ can have easily allowed PSR J1740$-$5340 (having a magnetic field $\sim 8\times 10^8$ G and a rotational period $\la 3.65$ ms) to became source of relativistic particles and magnetodipole emission, whose pressure [*(i)*]{} first swept the environment of the NS, allowing coherent radio emission to be switched on (Shvartsman 1970) and [*(ii)*]{} then kept on expelling the matter overflowing from the Roche lobe of [Star A ]{}(Ruderman, Shaham & Tavani 1989). For a wide enough binary system (as is the case of PSR J1740$-$5340), once the radio pulsar has been switched on, any subsequent restoration of the original $\dot M_{c}$ cannot quench the radioemission (Burderi [et al. ]{}2001). In this case we have now a donor star still losing matter from its Roche lobe at $\dot M_{c}\ga 5\times 10^{-11}~{\rm M_\odot/yr}$, [@dpms+01] (a high mass loss rate, difficult to explain in the model of a bloated star). At the same time, accretion on the NS is inhibited due to the pressure exerted by the pulsar on the infalling matter. This strong interaction between the MSP flux and the plasma wind would explain also the irregularities seen in the radio signals from PSR J1740$-$5340, sometimes showing the presence of ionized matter along the line of sight even when the pulsar is between [Star A ]{}and the observer. The characteristic age of PSR J1740$-$5340 ($\sim 3.5\times 10^8$ yr) seems indicating it is a young MSP, further supporting this scenario.
If [Star A ]{}will continue releasing matter at the present rate $\dot M_{c}$, PSR J1740$-$5340 is not a candidate for becoming an isolated pulsar. When [Star A ]{}will have shrunk well inside its Roche lobe, the system will probably end up as MSP+WD (or a light non degenerate companion). If [Star A ]{}will undergo a significant increasing of $\dot M_{c}$, the condition for the accretion could be re-established and PSR J1740$-$5340 would probably appear again as a Low Mass X-ray Binary or as a Soft X-ray Transient (Campana, [et al. ]{}1998).
[We thank P. Montegriffo for assistance with the astrometry procedure, G. Clementini, L. Burderi and F.D’Antona for useful discussions and Elena Pancino for providing the WFI image. Financial support to this research is provided by the Agenzia Spaziale Italiana (ASI) and the [*Ministero della Università e della Ricerca Scientifica e Tecnologica*]{} (MURST). The GSCII catalog was produced by the Space Telescope Science Institute and the Osservatorio Astronomico di Torino. The ESO/ST-ECF Science Archive facility is a joint collaboration of the European Souther Observatory and the Space Telescope - European Coordinating Facility.]{}
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[^1]: Based on observations with the NASA/ESA HST, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS5-26555
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A geometric transition is a continuous path of geometric structures that changes type, meaning that the model geometry, i.e. the homogeneous space on which the structures are modeled, abruptly changes. In order to rigorously study transitions, one must define a notion of geometric limit at the level of homogeneous spaces, describing the basic process by which one homogeneous geometry may transform into another. We develop a general framework to describe transitions in the context that both geometries involved are represented as sub-geometries of a larger ambient geometry. Specializing to the setting of real projective geometry, we classify the geometric limits of any sub-geometry whose structure group is a symmetric subgroup of the projective general linear group. As an application, we classify all limits of three-dimensional hyperbolic geometry inside of projective geometry, finding Euclidean, Nil, and Sol geometry among the limits. We prove, however, that the other Thurston geometries, in particular $\HH^2 \times \RR$ and $\widetilde{\SL_2\RR}$, do not embed in any limit of hyperbolic geometry in this sense.'
author:
- 'Daryl Cooper, Jeffrey Danciger, and Anna Wienhard'
bibliography:
- 'refs.bib'
title: Limits of Geometries
---
Introduction
============
Following Felix Klein’s Erlangen Program, a geometry is given by a pair $(Y,H)$ of a Lie group $H$ acting transitively by diffeomorphisms on a manifold $Y$. Given a manifold of the same dimension as $Y$, a geometric structure modeled on $(Y,H)$ is a system of local coordinates in $Y$ with transition maps in $H$. The study of deformation spaces of geometric structures on manifolds is a very rich mathematical subject, with a long history going back to Klein and Ehresmann, and more recently Thurston. In this article we are concerned with *geometric transition*, an idea that was recently promoted by Kerckhoff, and studied by Danciger in his thesis [@danciger1; @danciger2]. A geometric transition is a continuous path of geometric structures for which the model geometry $(Y,H)$ abruptly changes to a different geometry $(Y',H')$. The process involves the limiting of the two different geometric structures to a common transitional geometry which, in some sense, interpolates the geometric features of the two geometries. For this to make sense, one must define a notion of geometric limit at the level of homogeneous spaces which describes the basic process by which one homogeneous geometry may transform or limit to another. In this article we develop a general framework to describe such geometric transitions, focusing on the special situation in which both geometries involved are sub-geometries of a larger ambient geometry $(X,G)$. Working in this framework, we then study transitions between certain sub-geometries of real projective geometry, giving an explicit classification in some cases.
The best-known examples of geometric transition arise in the context of Thurston’s geometrization program. For example, the transition between hyperbolic and spherical geometry, passing through Euclidean geometry, was studied by Hodgson [@Hodgson] and Porti [@Porti-98] and plays an important role in the proof of the orbifold theorem by Cooper, Hodgson, Kerckhoff [@Cooper_Hodgson_Kerckhoff], and Boileau, Leeb, Porti [@Boileau_Leeb_Porti_orbi]. More recently, a transition going from hyperbolic geometry to its Lorentzian analogue, anti de Sitter (AdS) geometry, was introduced by Danciger [@danciger1] and studied in the context of cone-manifold structures on Seifert-fibered three-manifolds. The hyperbolic-AdS transition plays an important role in very recent work by Danciger, Maloni, and Schlenker [@Danciger_Maloni_Schlenker] on the classical subject of combinatorics of polyhedra in three-space, which characterizes the combinatorics of polyhedra inscribed in the one-sheeted hyperboloid, generalizing Rivin’s famous characterization of polyhedra inscribed in the sphere. The transition between constant curvature Lorentzian geometries very recently found applications in the setting of affine geometry. One of the most striking features differentiating affine geometry from Euclidean geometry is the existence of properly discontinuous actions by non-abelian free groups. In three-dimensions, such proper actions of free groups preserve a flat Lorentzian metric and their quotients are called Margulis space-times. In [@Danciger_Gueritaud_Kassel_topology], Danciger, Guéritaud, and Kassel study Margulis space-times as limits of collapsing complete AdS three-manifolds, giving related characterizations of the geometry and topology of both types of geometric structures. In particular, they give a proof of the tameness conjecture for Margulis space-times [@Danciger_Gueritaud_Kassel_topology] (also proved using a different approach by Choi and Goldman [@Choi_Goldman]). Each of the results mentioned here involves the construction of a geometric transition in some specialized geometric setting. This paper seeks to broaden the scope of transitional geometry by initiating a classification program for geometric transitions within a very general framework. We expect our results to be useful in a wide array of problems, for example the study of boundaries or compactifications of deformation spaces of geometric structures, and the construction of interesting proper affine actions in higher dimensions.
Let us now illustrate the general context in which the construction of a geometric transition is desirable. Consider a sequence $\mathscr Y_n$ of $(Y,H)$ structures on a manifold $M$, and suppose that as $n \to \infty$, the structures $\mathscr Y_n$ fail to converge, meaning that the charts fail to converge as local diffeomorphisms, even after adjusting by diffeomorphisms of $M$ and coordinate changes in $H$. Of central interest here is the case that the $\mathscr Y_n$ *collapse*: The charts converge to local submersions onto a lower dimensional sub-manifold of $Y$ and the transition maps converge into the subgroup of $H$ that preserves this sub-manifold. Next suppose that $(Y,H)$ is a sub-geometry of $(X,G)$; this means that $Y$ is an open sub-manifold of $X$ and $H$ is a closed subgroup of $G$. The sequence $\mathscr Y_n$ of collapsing $(Y,H)$ structures need not collapse as $(X,G)$ structures, because the larger group $G$ of coordinate changes could be used to prevent collapse. In certain cases, one may find a sequence $(c_n) \subset G$, so that the conjugate structures $c_n \mathscr Y_n$ converge to a (non-collapsed) $(X,G)$ structure $\mathscr Y_\infty$. This limiting structure $\mathscr Y_\infty$ is modeled on a new sub-geometry $(Z,L)$ which is, in a sense to be defined presently, a geometric limit of $(Y,H)$.
Consider two sub-geometries $(Y,H)$ and $(Z,L)$ of $(X,G)$. First, at the level of structure groups, we say $L$ is a *limit* of $H$, if there exists a sequence $(c_n)$ in $G$ so that the conjugates $c_n H c_n^{-1}$ converge to $L$ in the *Chabauty topology* [@chabauty] on closed subgroups (i.e. $c_n H c_n^{-1}$ converges to $L$ in the Hausdorff topology in every compact neighborhood of $G$). If in addition there exists $z \in Z \subset X$ so that $z \in c_n Y$ for all $n$ sufficiently large, then we say $(X,G)$ is a *geometric limit* of $(Y,H)$ as sub-geometries of $(X,G)$. See Section \[sec:limits-of-geometries\]. The description of limit groups and limit geometries in general is a difficult problem. We give a complete classification in certain special cases.
Symmetric subgroups {#symmetric-subgroups .unnumbered}
-------------------
Let $G$ be a semi-simple Lie group of noncompact type with finite center. A subgroup $H \subset G$ is called *symmetric* if $H = G^\sigma$ is the fixed point set of an involution $\sigma: G \to G$, or more generally $G^\sigma_0 \subset H \subset G^\sigma$, where $G^\sigma_0$ denotes the identity component of $G^\sigma$. The coset space $G/H$ is called an affine symmetric space. Affine symmetric spaces have a rich structure theory, generalizing the structure theory of Riemannian symmetric spaces. In particular there is a Cartan involution $\theta: G \to G$ which commutes with $\sigma$. Let $K = G^{\theta}$ and ${\mathfrak{g}}= {\mathfrak{k}}\oplus {\mathfrak{p}}$ the corresponding Cartan decomposition. The involution $\sigma$ analogously defines a decomposition of ${\mathfrak{g}}= {\mathfrak{h}}\oplus {\mathfrak{q}}$ into $(\pm 1)$–eigenspaces. An important tool in determining the limits $L$ of symmetric subgroups $H$ is the following well-known factorization result: Let ${\mathfrak{b}}$ be a maximal abelian subalgebra of ${\mathfrak{p}}\cap {\mathfrak{q}}$. Then any $g \in G$ can be written as $g = kbh$ with $k \in K$, $b\in B = \operatorname{Exp}({\mathfrak{b}})$, and $h \in H$. Moreover $b$ is unique up to conjugation by the Weyl group $W_{H \cap K} := N_{H\cap K} ({{\mathfrak{b}}})/ Z_{H\cap K}( {{\mathfrak{b}}})$. Using this factorization theorem we can characterize all limits of symmetric subgroups as follows (see Section \[sec:symmetric-subgroups\] for a more precise version).
\[thm:symmetric-limits\] Let $H$ be a symmetric subgroup of a semi-simple Lie group $G$ with finite center. Then any limit $L'$ of $H$ in $G$ is the limit under conjugacy by a one parameter subgroup. More precisely, there exists an $X \in \mathfrak{b}$ such that the limit $L = \lim_{t \to \infty} \exp(tX) H \exp(-tX)$ is conjugate to $L'$. Furthermore, $$L = Z_H(X) \ltimes N_+(X),$$ where $Z_H(X)$ is the centralizer in $H$ of $X$, and $N_+(X)$ is the connected nilpotent subgroup $$N_+(X) := \{g \in G : \lim_{t\to \infty} \exp(tX)^{-1} g \exp(tX) = 1 \}.$$
In the special case when $H = K$ (in other words $\sigma$ is a Cartan involution), the limit groups are determined by Guivarc’h–Ji–Taylor [@GJT] and also by Haettel [@Haettel2]. Moreover in these articles the Chabauty-compactification of $G/K$ is shown to be isomorphic to the maximal Satake-Furstenberg compactification. The analysis leading to Theorem \[thm:symmetric-limits\] bears a lot of similarity with the analysis in [@Gorodnik_Oh_Shah], where the maximal Satake-Furstenberg compactification for affine symmetric spaces $G/H$ is defined. We would like to raise the question whether the Chabauty-compactification of $G/H$ is homeomorphic to its maximal Satake-Furstenberg compactification.
The groups $G = {\operatorname{PGL}}_n \KK$ with $\KK = \RR$ or $\CC$ are of particular interest, since they are the structure group for projective geometry. In this case, Theorem \[thm:symmetric-limits\] implies that the limits of symmetric subgroups have a nice block matrix form. Let $H \subset {\operatorname{PGL}}_{n}\KK$ be a symmetric subgroup and let $L$ be a limit of $H$. Then, there is a decomposition $\KK^n = E_0 \oplus \cdots \oplus E_k$ of $\KK^n$ with respect to which $L$ has the following block form: $$\left(\begin{array}{ccccc}
A_1 & 0 & 0 &\cdots & 0\\
* & A_2 & 0 &\cdots & 0\\
* & * & A_3&\cdots& 0\\
\cdots &\cdots &\cdots &\cdots &\cdots \\
* & * & * &\cdots & A_k
\end{array}\right).$$ Here, the blocks denoted $*$ are arbitrary, and the diagonal part $\operatorname{diag}(A_1,\ldots,A_k)$ is an element of $H$. The groups $\PO_n\CC$, ${\operatorname{P}}(\GL_p\CC \times \GL_q\CC)$, ${\operatorname{P}}{\operatorname{Sp}}(2m,\CC)$, where $n= 2m$, ${\operatorname{PGL}}_n\RR$, $\PU(p,q)$, where $n = p+q$, and $\SL(m,\mathbb{H})$, where $n= 2m$ are some of the symmetric subgroups of ${\operatorname{PGL}}_n\CC$. The symmetric subgroups of ${\operatorname{PGL}}_{n} \RR$ are ${\operatorname{P}}(\GL_p\RR \times \GL_q\RR)$, $\PO(p,q)$, where $p+q = n$, or ${\operatorname{P}}{\operatorname{Sp}}(2m, \RR)$ and ${\operatorname{P}}(\GL(m,{\mathbb C}))$, where $n = 2m$. See Section \[sec:symmetric-subgroups\] for a full characterization of the limit groups in each of these cases.
Limits of constant curvature semi-Riemannian geometries. {#limits-of-constant-curvature-semi-riemannian-geometries. .unnumbered}
--------------------------------------------------------
Let $\beta$ denote a quadratic form on $\RR^n$ of signature $(p,q)$, meaning $\beta\sim-I_p\oplus I_q$. The group ${\operatorname{P}}\Isom(\beta) = \PO(p,q)$ acts transitively on the domain $$\XX(p,q) = \{ [x] \in \RP^{n-1} : \beta(x) < 0 \} \subset \RP^{n-1},$$ with point stabilizer isomorphic to ${\operatorname{O}}(p-1,q)$. The geometry $(\XX(p,q), \PO(p,q))$ is the projective model for semi-Riemannian geometry of constant curvature, of dimension $p+q-1$ and of signature $(p-1,q)$. In the cases $(p,q)$ is $(n,0), (1,n-1), (n-1,1)$ or $(2,n-2)$ we obtain spherical geometry, hyperbolic geometry, de Sitter geometry, and anti de Sitter geometry respectively.
By applying Theorem \[thm:symmetric-limits\], we characterize the limits of these constant curvature semi-Riemannian geometries inside real projective geometry. Here is a brief description of the limit geometries. A [*partial flag*]{} ${{\mathcal F}}=\{V_0,V_1,\cdots V_{k+1}\}$ of $\RR^n$ is a descending chain of vector subspaces $$\RR^n=V_0\supset V_1\supset\cdots \supset V_k\supset V_{k+1}=\{0\}.$$ A [*partial flag of quadratic forms*]{} ${{\boldsymbol \beta}}=(\beta_0,\cdots,\beta_k)$ on ${{\mathcal F}}$ is a collection of non-degenerate quadratic forms $\beta_i$ defined on each quotient $V_i/V_{i+1}$ of the partial flag. Define $\Isom({{\boldsymbol \beta}},{{\mathcal F}})$ to be the group of linear transformations which preserve ${\mathcal F}$ and induce an isometry of each $\beta_i$, and denote its image in ${\operatorname{PGL}}_n \RR$ by ${\operatorname{P}}\Isom({{\boldsymbol \beta}}, {\mathcal F})$. Define the domain $\XX({{\boldsymbol \beta}}) \subset \RP^{n-1}$ by $$\XX({{\boldsymbol \beta}}) := \{ [x] \in\RP^{n-1} : \beta_0(x) < 0\}.$$ Then ${\operatorname{P}}\Isom({\mathcal F}, {{\boldsymbol \beta}})$ acts transitively on $\XX({{\boldsymbol \beta}})$. When the flag and quadratic forms are adapted to the standard basis, we denote $\XX({{\boldsymbol \beta}})$ and $\Isom({{\boldsymbol \beta}})$ by $$\begin{aligned}
\label{PFG-notation}
\XX({{\boldsymbol \beta}}) &=: \XX((p_0,q_0),\ldots,(p_k,q_k)),\\
\Isom({{\boldsymbol \beta}}) &=: {\operatorname{O}}((p_0,q_0),\ldots,(p_k,q_k))\\
&= \left(\begin{array}{ccccc}
{\operatorname{O}}(p_0,q_0) & 0 & 0 &\cdots & 0\\
* & {\operatorname{O}}(p_1,q_1) & 0 &\cdots & 0\\
* & * & {\operatorname{O}}(p_2,q_2) &\cdots& 0\\
\cdots &\cdots &\cdots &\cdots &\cdots \\
* & * & * &\cdots & {\operatorname{O}}(p_k,q_k)
\end{array}\right),\end{aligned}$$ where $*$ denotes an arbitrary block. Note that $\XX({{\boldsymbol \beta}})$ is non-empty if and only if $p_0 > 0$. As a set, the space $\XX((p_0,q_0)\ldots (p_k,q_k))$ depends only on the first signature $(p_0,q_0)$ and the dimension $n = \sum_i (p_i + q_i)$. However, we include all $k$ signatures in the notation as a reminder of the structure determined by $\PO((p_0,q_0),\ldots,(p_k,q_k))$.
\[thm:limits-Hpq\_intro\] The limits of the constant curvature semi-Riemannian geometries $(\XX(p,q), \PO(p,q))$ inside $(\RP^{n-1}, {\operatorname{PGL}}_n \RR)$ are all of the form $(\XX({{\boldsymbol \beta}}), {\operatorname{P}}\Isom({\mathcal F}, {{\boldsymbol \beta}}))$. Further, $\XX({{\boldsymbol \beta}})$ is a limit of $\XX(p,q)$ if and only if $p_0 \neq 0$, and the signatures $((p_0,q_0),\ldots,(p_k,q_k))$ of ${{\boldsymbol \beta}}$ partition the signature $(p,q)$ in the sense that $$p_0 + \cdots + p_k = p \ \ \text{ and } \ \ q_0 + \cdots + q_k = q,$$ after exchanging $(p_i, q_i)$ with $(q_i, p_i)$ for some collection of indices $i$ in $\{1,\ldots,k\}$ (the first signature $(p_0,q_0)$ must *not* be reversed).
See Section \[sec:Hpq\] for a detailed discussion of these partial flag geometries.
The Thurston geometries {#the-thurston-geometries .unnumbered}
-----------------------
One motivation for our work is the study of transitions between the eight three-dimensional *Thurston geometries*, homogeneous Riemannian geometries which play an essential role in the classification of compact three-manifolds. Since each of the eight geometries (almost) admits a representation in real projective geometry [@molnar], [@thiel], it is natural to study transitions between them in the projective setting.
From the point of view of Thurston’s picture of hyperbolic Dehn surgery space, one expects to find five of the eight Thurston geometries as limits of hyperbolic geometry. There are various methods, due to Porti and collaborators [@Porti-98; @Porti-02; @HPS], to realize Euclidean, Nil, and Sol geometry structures as limits of certain families of collapsing hyperbolic structures. However, efforts to realize the Thurston geometries which fiber over the hyperbolic plane, namely $\HH^2 \times \RR$ and $\widetilde{\SL_2 \RR}$, as limits of three-dimensional hyperbolic geometry have so far proved fruitless. Many Seifert-fibered three-manifolds which admit a structure modeled on either $\HH^2\times \RR$ or $\widetilde{\SL_2 \RR}$ also admit hyperbolic cone-manifold structures with cone angles arbitrarily close to $2\pi$. Commonly in examples, such structures are found to collapse down to a hyperbolic surface (the base of the Seifert fibration) as the cone angle increases to $2\pi$. Recent work of Danciger [@danciger1; @danciger2] shows that in this context, the most natural sequence of conjugacies in projective space yields a non-metric geometry called *half-pipe* geometry as limit. However, Danciger’s construction does not rule out the possibility that some other clever sequence of conjugacies could produce $\HH^2 \times \RR$ or $\widetilde{\SL_2 \RR}$ geometry as limit. As an application of Theorem \[thm:limits-Hpq\_intro\], we enumerate the limits of hyperbolic geometry inside of projective geometry and prove:
\[cor:Thurston-geoms\_intro\] The Thurston geometries which locally embed in limits of hyperbolic geometry (within real projective geometry) are : ${\mathbb E}^3$, Solv geometry, and Nil geometry. In particular, neither $\HH^2 \times \RR$ nor $\widetilde{\SL_2 \RR}$ locally embed into any limit of hyperbolic geometry.
In future work we intend to give a complete description of the possible transitions between the eight Thurston geometries.
Structure of the paper {#structure-of-the-paper .unnumbered}
----------------------
In Section \[sec:limits-of-geometries\] we introduce the notion of (geometric) limits of groups and limits of geometries and revisit the transition from hyperbolic geometry through Euclidean geometry to spherical geometry (within real projective geometry). In Section \[sec:limit-groups\] we describe several notions of limits of groups, illustrate their differences, show when they agree, and describe some basic properties of geometric limits of real algebraic Lie groups. In Section \[sec:symmetric-subgroups\] we recall the basic structure theory of affine symmetric spaces, and prove (a more descriptive version of) Theorem \[thm:symmetric-limits\], determining the limit groups of symmetric subgroups. We then apply the theorem to give explicit descriptions of the limits of symmetric subgroups of the general linear group. The limit geometries of the semi-Riemannian real hyperbolic geometries in terms of partial flags geometries are described in Section \[sec:pfqf\]. In the end of that section we discuss the applications to Thurston geometries.
Acknowledgements {#acknowledgements .unnumbered}
----------------
This work grew out of discussions at an AMS Mathematical Research Community at Snowbird in 2011, and we thank the [*AMS*]{} and [*NSF*]{} for their supporting this program. Some of this work occured during the trimester “Geometry and Analysis of Surface Groups” at [*IHP*]{}, and we gratefully appreciate their support and the stimulating environment there. We thank the research network “Geometric Structures and Representation Varieties" (GEAR) , funded by the NSF under the grant numbers DMS 1107452, 1107263, and 1107367, for supporting reciprocal visits to Princeton, Santa Barbara and Heidelberg.
We thank [ Marc Burger]{} for interesting discussions, and in particular for suggesting to work in the context of affine symmetric spaces, and pointing us to the relevant structure theorems. We also profited from discussions with [Thomas Haettel]{}. We thank Benedictus Margeaux for providing a proof of Lemma \[lem:centralizers\] on math overflow.
D.C. was partially supported by NSF grants DMS-0706887, 1207068 and 1045292. J.D. was partially supported by the National Science Foundation under the grant DMS 1103939. A. W. was partially supported by the National Science Foundation under agreement No. DMS-1065919 and 0846408, by the Sloan Foundation, by the Deutsche Forschungsgemeinschaft, and by the ERCEA under ERC-Consolidator grant no. 614733.
Limits of Geometries {#sec:limits-of-geometries}
====================
A [*geometry*]{} is a pair $(X,G)$ where $G$ is a Lie group acting transitively by analytic maps on a connected, smooth manifold $X$.
The requirement in the definition that the action be transitive implies that $X$ identifies with $G/G_x$, where $G_x$ denotes the stabilizer of a point $x \in X$. Note that we do not require the point stabilizer $G_x$ to be compact. Some examples of geometries are
1. Euclidean geometry $\mathbb E^n$: The space $X = \RR^n$ and the structure group $G$ is the semi-direct product of the orthogonal group ${\operatorname{O}}(n)$ and the translation group $\RR^n$.
2. Spherical geometry $\mathbb S^n$: The space $X$ is the unit sphere in $\RR^{n+1}$ and the group $G$ is the orthogonal group ${\operatorname{O}}(n+1)$.
3. Hyperbolic geometry $\mathbb H^n$: The space $X = \{ [\mathbf{x}] \in \RP^{n}: -x_{n+1}^2 + x_1^2+ \cdots + x_n^2 < 0\}$ is the set of negative lines with respect to the standard quadratic form of signature $(1,n)$. The structure group $G = \PO(1,n)$ is the group of projective transformations preserving $X$.
4. Real projective geometry: $X = \RP^n$, $G = {\operatorname{PGL}}_{n+1} \RR$.
Given geometries $(X,G)$ and $(X', G')$ a *morphism* $(X,G) \to (X',G')$ is a Lie group homomorphism $\Phi: G \to G'$ such that for some (and hence any) $x \in X$ there is an $x' \in X'$ such that $\Phi(G_x) \subset G_{x'}'$. The map $\Phi$ induces an analytic map $F: X \to X'$ defined by $F(x) = x'$ and the property that $F$ is $\Phi$-equivariant: $$F(g\cdot y) = \Phi(g) F(y).$$ The map $\Phi$ defines an *isomorphism* of geometries if $\Phi$ is an isomorphism of Lie groups and $\Phi(G_x) = G_{x'}'$. If $\Phi$ is surjective, we say $(X,G)$ *fibers* over $(X', G')$.
Recall that a local homomorphism $\varphi: G \dashrightarrow G'$ of Lie groups is a map $\varphi: V \to G$, defined on a neighborhood $V$ of the identity in $G$, such that $\varphi(gh) = \varphi(g)\varphi(h)$ and $\varphi(g)^{-1} = \varphi(g^{-1})$ whenever all terms are defined. A local homomorphism $\varphi$ is a local isomorphism if $\varphi$ is locally injective, meaning injective when restricted to some small neighborhood of the identity in $G$, and locally surjective, meaning has image containing a small neighborhood of the identity in $G'$. Note that if ${\mathfrak{g}}$ and ${\mathfrak{g}}'$ denote the Lie algebras of $G$ and $G'$, then the differential $\varphi_*: {\mathfrak{g}}\to {\mathfrak{g}}'$ of a local homomorphism of Lie groups $\varphi$ is a homomorphism of Lie algebras $\varphi_*: {\mathfrak{g}}\to {\mathfrak{g}}'$ and conversely any homomorphism of Lie algebras is the differential of a local homomorphism of Lie groups as above.
A *local morphism of geometries* $(X,G) \dashrightarrow (X', G')$ is a local homomorphism $\varphi : G \dashrightarrow G'$ such that for some (and hence any) $x \in X$, there is an $x' \in X'$ with the property that the restriction of $\varphi$ to $G_x$ has image in $G_{x'}'$. The local homomorphism $\varphi$ induces a local analytic map $f: X \dashrightarrow X'$, defined on a neighborhood of $x$, which is locally $\varphi$-equivariant, meaning $f(g \cdot y) = \varphi(g)f(y)$ whenever all terms are defined. The local morphism is a *local isomorphism* if $\varphi$ is a local isomorphism $G \dashrightarrow G'$ and $\varphi$ restricts to a local isomorphism $G_x \dashrightarrow G_{x'}'$.
Note that given $\varphi$ as in the definition, the differential $\varphi_*: {\mathfrak{g}}\to {\mathfrak{g}}'$ satisfies that $\varphi_*({\mathfrak{g}}_x) \subset {\mathfrak{g}}_{x'}'$, where ${\mathfrak{g}}_x$ and ${\mathfrak{g}}_{x'}'$ are the Lie algebras of the point stabilizers $G_x$ and $G_{x'}'$. Conversely a Lie algebra homomorphism ${\mathfrak{g}}\to {\mathfrak{g}}'$ taking infinitesimal point stabilizers into infinitesimal point stabilizers (which could be called an infinitesimal morphism of geometries) determines a local morphism of geometries. The map $\varphi: G \dashrightarrow G'$ determines a local isomorphism if and only if $\varphi_*$ is an isomorphism and $\varphi_*({\mathfrak{g}}_x) = {\mathfrak{g}}_{x'}'$.
If $G$ is connected then the [*universal cover*]{} $(\tilde X,\tilde G)\longrightarrow(X,G)$ of a geometry $(X,G)$ is defined as follows: Let $G_x \subset G$ be the stabilizer of a point $x \in X$ so that $X = G/G_x$. Then $\tilde G \to G$ is the universal covering Lie group of $G$, and $\tilde X = \tilde G /\tilde G_x$, where $\tilde G_x \subset \tilde G$ is the identity component of the inverse image of $G_x$. Indeed $\tilde X \to X$ is the universal cover of $X$. Note that the action of $\tilde G$ on $\tilde X$ might not be faithful, even if the action of $G$ on $X$ was faithful. In this case, one may replace $\tilde G$ with its quotient by the kernel of the action. Every geometry is locally isomorphic to its universal cover. A local morphism (resp. local isomorphism) of geometries induces a morphism (resp. isomorphism) of the universal covering geometries.
The geometry $(Y,H)$ is a [*subgeometry*]{} of $(X,G)$, written $(Y,H)\subset(X,G)$, if $H$ is a closed subgroup of $G$ and $Y$ is an open subset of $X$ on which $H$ acts transitively. We say that a geometry $(Y,H)$ *locally embeds* in $(X,G)$ if $(Y,H)$ is locally isomorphic to a subgeometry $(Y',H') \subset (X,G)$.
For example, both hyperbolic and Euclidean geometry, in the forms described above, are subgeometries of real projective geometry. Spherical geometry is a two fold covering of a subgeometry of projective geometry, and therefore spherical geometry locally embeds in projective geometry but it is not a subgeometry. Similarly, the Thurston geometry known as $\widetilde{\SL_2{\mathbb R}}$ is not a subgeometry of projective geometry, but it does locally embed.
In this article we are concerned with limits of geometries, in particular with limits of subgeometries of a given geometry $(X,G)$. First we introduce the notion of limit of closed subgroups.
\[def:limit-subgroups\]
1. A sequence $H_n$ of closed subgroups of a Lie group $G$ *converges geometrically* to a closed subgroup $L$ if every $g \in L$ is the limit of some sequence $h_n \in H_n$, and if every accumulation point of every sequence $h_n \in H_n$ lies in $L$. We also say that $L$ is the *geometric limit* of the sequence $H_n$. Note that $L$ is the geometric limit of $H_n$ if and only if $H_n$ converges to $L$ in the Chabauty topology on closed subgroups.
2. We say $L$ is a *conjugacy limit* (or just *limit*) of $H$ if there exists a sequence $c_n \in G$ so that the conjugate groups $H_n = c_n H c_n^{-1}$ converge geometrically to $L$.
The set ${\mathcal C} (G)$ of closed subgroups with the Chabauty topology is a compact space [@chabauty], [@bridson], [@delaharpe], so for every sequence of closed subgroups there is some subsequence which has a geometric limit. We on the set of all closed subsets of a non-compact space is commonly defined to be the topology of Hausdorff convergence in compact neighborhoods. The Chabauty topology is simply the subspace topology on the set of closed subgroups.
\[def:limit-subgeom\]
1. A sequence of subgeometries $(Y_n,H_n)\subset (X,G)$ [*converges*]{} to the subgeometry $(Z, L)\subset (X,G)$ if $H_n$ converges geometrically to $L$ and there exists $z\in Z \subset X$ such that for all $n$ sufficiently large $z\in Y_n$.
2. We say that a subgeometry $(Z,L)$ is a *conjugacy limit* (or just *limit*) of $(Y,H)$ if there exists a sequence $g_n \in G$ so that the sequence of conjugate subgeometries $(g_n Y, g_n H g_n^{-1})$ converges to $(Z,L)$.
The motivating situation, as described in the introduction, is that of collapsing $(Y,H)$ structures on a manifold: structures for which each chart (or alternatively the developing map) collapses to a local submersion onto a lower-dimensional subset $N$ of $Y$ and each transition map (alternatively the holonomy representations) converges into some smaller subgroup $P \subset H$ that preserves $N$. The goal is to conjugate the $(Y,H)$-structures inside of $(X,G)$, so that the charts (developing maps) no longer collapse and the transition maps (holonomy representations) converge into some limit group $L$ of $H$ which contains $P$. Then, setting $Z = L \cdot N$, the geometry $(Z, L)$ is a limit of $(Y,H)$ in the sense of Definition \[def:limit-subgeom\] and the limiting geometric structure is a $(Z,L)$ structure.
The transition from spherical to Euclidean to hyperbolic {#sec:spherical-Euclidean}
--------------------------------------------------------
Let us now illustrate the definitions in a familiar example.
Consider the path of quadratic forms $\beta_t= -x_{n+1}^2- t(x_1^2+\cdots+x_n^2)$ on ${\mathbb R}^{n+1}$, and assume $t \geq 0$. These quadratic forms define sub-geometries $(\XX(\beta_t), \PO(\beta_t))$ of projective geometry where $$\XX(\beta_t) = \{ [x] \in \RP^n : \beta_t(x) < 0\},$$ $$\PO(\beta_t) = \operatorname{P}\left\{ A \in \GL(n+1) : A^* \beta_t = \beta_t \right\}.$$ For all $t > 0$, $\PO(\beta_t)$ is conjugate to $\PO(\beta_1)$ which is the standard copy of $\PO(n+1)$, and the geometry $(\XX(\beta_t), \PO(\beta_t))$ is conjugate to the standard realization (found at $t=1$) of spherical geometry $\mathbb S^n$ as a (covering) sub-geometry of projective geometry. The element $c_t \in {\operatorname{PGL}}(n+1)$ conjugating $(\XX(\beta_1), \PO(\beta_1))$ to $(\XX(\beta_t), \PO(\beta_t))$ is the diagonal matrix $c_t = \operatorname{diag}(1/\sqrt{t}, \ldots,1/\sqrt{t}, 1)$ with the first $n$ diagonal entries equal to $1/\sqrt{t}$ and the final diagonal entry equal to one (note $c_t^*\beta_t = \beta_1$). This corresponds to scaling the plane $\RR^n \subset \RR^{n+1}$ spanned by the first $n$ coordinate directions.
At time $t = 0$, the quadratic form becomes degenerate. The group preserving $\beta_0$ is simply the group of matrices that preserve the last coordinate $|x_{n+1}|$; this is a copy of the affine group $$\operatorname{Aff}(n)=\left\{\left(\begin{matrix}
A & b\\
0 & 1
\end{matrix}\right)\right\}\subset {\operatorname{PGL}}(n+1,{\mathbb R}).$$ However, the affine group is *not* the limit of the groups $\PO(\beta_t)$ as $t \to 0^+$. In fact, the limit of the conjugate subgroups $\PO(\beta_t))$ as $t \to 0^+$ is the group of Euclidean isometries. In order to simplify the discussion, let us demonstrate this at the level of Lie algebras. The Lie algebra $\mathfrak{so}(\beta_t)$ of $\PO(\beta_t)$ is conjugate to the Lie algebra $\mathfrak{so}(\beta_1) = \mathfrak{so}(n+1)$:
$$\begin{aligned}
\mathfrak{so}(\beta_t) &=
\left(
\begin{matrix}
\sqrt{t}^{-1}& & & \\
& \sqrt{t}^{-1} & & \\
& & \ddots & \\
& & & 1
\end{matrix}
\right)
\left(\begin{matrix}
0 & & a_{ij} & \vdots\\
& \ddots & & b\\
-a_{ji}& & \ddots & \vdots\\
\hdots & -b^T &\ldots & 0
\end{matrix}\right)
\left(
\begin{matrix}
\sqrt{t} & & & \\
& \sqrt{t} & & \\
& & \ddots & \\
& & & 1
\end{matrix}
\right)\\
&=
\left(\begin{matrix}
0 & & a_{ij} & \vdots\\
& \ddots & & \sqrt{t}^{-1}b\\
-a_{ji}& & \ddots & \vdots\\
\hdots & \sqrt{t} b^T &\ldots & 0
\end{matrix}\right).\end{aligned}$$
It is easy to read off the limit Lie algebra via the following heuristic reasoning. To find an element of the limit Lie algebra, we are allowed to vary the entries $a_{ij}, b$ of the matrix as $t \to 0^+$ in any way that produces a limit. Since $\sqrt{t}^{-1} \to \infty$, it follows that we must have $b = O(\sqrt{t})$. Thus the limit Lie algebra has the form:
$$\lim_{t \to0^+} \mathfrak{so}(\beta_t) =
\left(\begin{matrix}
0 & & a_{ij} & \vdots\\
& \ddots & & b'\\
-a_{ji}& & \ddots & \vdots\\
\hdots & 0 &\ldots & 0
\end{matrix}\right) = \mathfrak{isom}(\EE^n)$$ where $b'$ can be any column vector. We recognize this limit as the Lie algebra of the subgroup of the affine group preserving the standard Euclidean metric on the affine patch $x_{n+1} \neq 0$. In fact, it is only slightly more difficult to show that the limit of the Lie groups $\PO(\beta_t)$ is indeed the group of Euclidean isometries $$\Isom(\EE^n) = \operatorname{P}\left( \begin{pmatrix} {\operatorname{O}}(n) & \\ & \pm1 \end{pmatrix} \ltimes \begin{pmatrix} I_n &\RR^n \\ &1 \end{pmatrix}\right).$$ To determine the limit of the homogeneous spaces $\XX(\beta_t)$, we use Definition \[def:limit-subgeom\]. Consider any point $z$ in the affine patch $\EE^n = \{ [x] : x_{n+1} \neq 0\}$. Then, of course $z$ is in $\XX(\beta_t) = \RP^n$ for all $t > 0$. Note that if we choose $z$ to be the usual origin of $\EE^n$, then $z$ is fixed by $c_t$. The notion that the limit of a constant sequence of spaces ($\RP^n$) could be anything other ($\EE^n$) than that space again may seem counter-intuitive. However, the important thing to realize is that the orbit of $z$ under the groups $\PO(\beta_t)$ is $\RP^n$ while the orbit of $z$ under the limit group $\Isom(\EE^n)$ is now the smaller space $\EE^n$. This is indeed the relevant notion of limit in the context of geometric structures.
Next, for $t < 0$, the $\beta_t$ have signature $(1,n)$ and the corresponding sub-geometries $(\XX(\beta_t), \PO(\beta_t))$ are all conjugate to the standard copy $(\XX(\beta_{-1}), \PO(\beta_{-1}))$ of the projective model for hyperbolic space $\HH^n$. The reasoning above applies similarly in this case to show that the limit of $\PO(\beta_t)$ as $t \to 0^-$ is again the group $\Isom(\EE^n)$ of Euclidean isometries. In this case the spaces $\XX(\beta_t)$ are expanding balls in $\RP^n$ which eventually engulf any point $z$ in the affine patch $\EE^n = \{ [x] : x_{n+1} \neq 0\}$. Thus $(\EE^n, \Isom(\EE^n)$ is the limit of $(\XX(\beta_t), \PO(\beta_t))$ as $t \to 0^-$.
It is worth noting that this transition of homogeneous spaces can be seen nicely in terms of certain quadric hyper-surfaces in $\RR^{n+1}$. For, the level sets of the quadratic forms $\beta_t$ are either ellipsoids if $t > 0$, or hyperboloids of two sheets if $t < 0$. Any such level set $\beta_t = -k$ is preserved by the lift ${\operatorname{O}}(\beta_t)$ of $\PO(\beta_t)$ and projects (two to one) to $\XX(\beta_t)$ in projective space; hence it gives a nice model for the same geometry. In the case $t=1$, the level set $\beta_t = -1$ is the unit sphere and describes the standard model for spherical geometry, while in the case $t= -1$, the level set $\beta_t = -1$ is the standard hyperboloid model for hyperbolic geometry. Note that for all $t$, the level set $\beta_t = -1$ contains the two points $(0,\ldots,0,\pm1)$. As $t \to 0$ (from either direction), the limit of the surfaces $\beta_t = -1$, in the topology of Hausdorff convergence on compact sets, is the surface $\beta_0 = -1$, which is two parallel affine hyper-planes $x_{n+1} = \pm 1$. See Figure \[fig:ellipsoids\]. If one wishes, one may define invariant metrics on the $\XX(\beta_t)$ which transition from uniform positive curvature to uniform negative curvature as $t$ changes from positive to negative. However, its important to note that there is no natural such continuous path of metrics defined from the ambient geometry. In particular, the natural metric on the surfaces $\beta_t = -1$ induced by the quadratic forms $\beta_t$ have curvature $+1$ when $t > 0$ and curvature $-1$ when $t < 0$ (and of course, $\beta_0$ itself does not define any metric). Hence the projective geometry formulation of the transition from spherical to Euclidean to hyperbolic is independent of any metric formulation.
Limits of Groups {#sec:limit-groups}
================
Classifying the limits of closed subgroups $H$ of $G$ is the central problem when classifying the limits of a sub-geometry $(Y,H)$ in $(X,G)$. Thus in the next two sections, we restrict our attention to limits of Lie groups and momentarily forget about homogeneous spaces.
We are mainly interested in *geometric limits* of Lie subgroups (Definition \[def:limit-subgroups\]). However, there are other inequivalent definitions of limit of a group, and it is helpful to understand how they differ and when they coincide. We explore a few of these alternative notions in Section \[sec:other-notions\] and illustrate them through examples in Section \[sec:examples\]. In Section \[sec:limit-props\] we derive some basic properties of geometric limits of linear algebraic Lie groups.
Various notions of limit {#sec:other-notions}
------------------------
Let $H_n$ be a sequence of closed Lie subgroups, of constant dimension, of the Lie group $G$. We introduced the [*geometric limit*]{} of $H_n$ in Definition \[def:limit-subgroups\]. Here are several related notions of limit.
1. The [*connected geometric limit*]{} $\lim_0H_n$ is the connected component of the identity in the geometric limit. In general this is different than the geometric limit of the connected component of the identity.
2. In the specific case that $H_n = c_n H c_n^{-1}$ are all conjugate, we may define the [*local geometric limit*]{}, denoted $\operatorname{local}$-$\lim H_n$, as the union of the geometric limits of conjugates $c_n C c_n^{-1}$ of compact neighborhoods $C \subset H$ of the identity. It might have smaller dimension than $H$. This limit is contained in the geometric limit, but excludes conjugates of elements moving to infinity. One may also define a notion of local geometric limit with respect to a subgroup $P \subset H$; this means the union of geometric limits of neighborhoods of the form $P \cdot C$ for $C$ a compact neighborhood of the identity.
3. Very much related to the local geometric limit is the notion of expansive limit. Again, we work in the case that $H_n = c_n H c_n^{-1}$ are all conjugate and we consider a subgroup $P \subset H$, such that $c_n P c_n^{-1} = P$. A (local) geometric limit is called an *expansive limit*, if every element of the limit group $L$ is of the form $\ell = \lim c_n h_n c_n^{-1}$, for some sequence $h_n \in H$ with $h_n \to h_\infty \in P$. Intuitively, an expansive limit is a limit obtained by blowing up an infinitesimal neighborhood of $P$. Expansive limits are often the relevant limits to study in the context of collapsing geometric structures and geometric transitions. See discussion following Definition \[def:limit-subgeom\]. In Section \[sec:symmetric-subgroups\], we will demonstrate that all limits of symmetric subgroups of semi-simple Lie groups are expansive.
4. The [*Lie algebra limit*]{} is the Lie sub-algebra of ${\mathfrak{g}}= \operatorname{Lie}(G)$ obtained from the limit of the sequence of Lie subalgebras ${\mathfrak h}_n\subset {\mathfrak g}$ of the subgroups $H_n$. As the $\mathfrak{h}_n$ are vector sub-spaces of ${\mathfrak{g}}$, we may (up to subsequence) extract a limit $\mathfrak{l}$, which is a vector subspace of the same dimension as the $\mathfrak{h}_n$ and in fact a Lie sub-algebra. Then $\mathfrak{l}$ defines a local group near the identity in $G$ and generates a subgroup that we call ${\mathfrak{a}}$-$\lim H_n$. It has the same dimension as the $H_n$. Note, however, that this subgroup might not be closed. We call the closure of ${\mathfrak{a}}$-$\lim H_n$ the [*algebraic limit*]{}, denoted $a$-$\lim H_n$. It is a connected Lie subgroup which might have larger dimension than the $H_n$.
5. Let us mention a related, but distinct notion of limit at the level of Lie algebras. The Lie algebra ${\mathfrak{h}}$ is determined by its Lie bracket $[\ ,\ ]:{\mathfrak{h}}\times {\mathfrak{h}}\to {\mathfrak{h}}$. This is a bilinear map and so is determined by its values on a basis, and thus by a finite collection of [*structure constants*]{}. One may continuously change these constants, taking care to ensure they still determine a Lie algebra, and in this way pass between the Lie algebras of different geometries. A path corresponding to change of bases leads to the theory, due to Inönü-Wigner, of *contractions* of Lie algebras which is useful in some physics contexts [@burde], [@Wigner]. This notion is independent of any embedding of ${\mathfrak{h}}$ into a larger Lie algebra.
6. Although we do not pursue it here, there is also a notion of geometric limit of a Lie group that does not involve conjugacy inside a larger group. It is based on the idea that an arbitrarily large compact subset of the limit is almost isomorphic to a (possibly small) subset of the original group. A Lie group $L$ is an [*intrinsic limit*]{} of a Lie group $G$ if for every compact subset $C\subset L$ and $\epsilon>0$ there is an open set $U \subset G$ and an immersion $f:U\longrightarrow L$ such that $f(U)\supset C$ and $f$ is $\epsilon$-close to an isomorphism in the sense that if $a,b,ab\in C$ there are $\alpha,\beta,\alpha\beta\in U$ with $f(\alpha)=a$, $f(\beta)=b$ and $d_L(f(\alpha\beta),ab)<\epsilon$. Here $d_L$ is a metric on $L$. It is easy to see that if $L$ is a geometric limit of a closed subgroup $H$ of a Lie group $G$ then $L$ is an intrinsic limit of $H$ in this sense. Moreover, this definition extends in an obvious way to pairs $(H,K)$ with $K$ a closed subgroup of $H$ and gives a notion of limit of the homogeneous geometry $H/K$ independent of an ambient geometry $G/R$.
Examples of limits of groups {#sec:examples}
----------------------------
We now give some examples to demonstrate various possible behavior of limits. In what follows we make use of three $1$-parameter subgroups of $SL(2,{\mathbb R})$:
$$D(t) =\left(\begin{array}{cc} \cosh(t) & \sinh(t)\\ \sinh(t) & \cosh(t) \end{array}\right),
\qquad\qquad
R(t)=\left(\begin{array}{cc} \cos(t) & -\sin(t)\\ \sin(t) & \cos(t)\end{array}\right),
\qquad\qquad
P(t)=\left(\begin{array}{cc} 1 & t\\ 0 & 1\end{array}\right),$$ All groups in this section will be groups of matrices, described in terms of at most three parameters $t,s,u \in \RR$. Whenever one of these parameters is present, the reader is meant to take the union over such matrices for all possible values of the parameters. We will abuse notation as such throughout, because it is cumbersome to write the definitions properly using set notation.
1. Let $c_n = \begin{pmatrix} n & 0 \\ 0 & n^{-1} \end{pmatrix}$. Then as $n \to \infty$, the sequence of conjugates $c_n D(t) c_n^{-1} \subset \SL(2,{\mathbb R})$ converges (for all notions of limits introduced above) to the parabolic subgroup $P(t)$.
2. In $\SL(2,{\mathbb R})$ the sequence of conjugates $c_n R(t)c_n^{-1}$, with $c_n$ as in (1), converges geometrically to the parabolic subgroup $\pm P(t)$ with two connected components. This illustrates that the [*geometric limit*]{} of a connected group might not be connected.
3. A subgroup of $\SL(3,{\mathbb R})$ containing non-diagonalizable elements may have a [*geometric limit*]{} containing only diagonal elements: $$\lim_{n\to \infty}\left(\begin{array}{ccc}
1/n & 0 & 0\\
0 & n & 0\\
0 & 0 & 1\end{array}\right)
\left(\begin{array}{ccc}
1 & t & 0\\
0 & 1 & 0\\
0 & 0 & e^{-2t}\end{array}\right)
\left(\begin{array}{ccc}
1/n & 0 & 0\\
0 & n & 0\\
0 & 0 & 1\end{array}\right)^{-1}=
\left(\begin{array}{ccc}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & e^{-2t}\end{array}\right)$$
4. Next, we give an example of a one-dimensional group with a two-dimensional conjugacy limit. Let $H$ be the 1-parameter closed subgroup of $\SL(4,{\mathbb R})$ defined by $$H=\left(\begin{array}{cc}
P(t) & 0\\
0 & R(t)
\end{array}\right)$$ The geometric limit under conjugacy by $c_n=\operatorname{diag}(n^{-1},n,1,1)$ is two dimensional: $$\lim_{n\to\infty}c_n H c_n^{-1}=\left(\begin{array}{cc}
P(s) & 0 \\
0 & R(t)
\end{array}\right),$$ where $t,s$ are independent parameters. The group $H$ is a one-parameter subgroup of ${\mathbb R}\times S^1$ that looks like a helix, and conjugating by $c_n$ coils the helix more tightly. The [*algebraic limit*]{} is the one-dimensional group $${a\text{-}\lim_{k\to\infty}}P_k H P_k^{-1}=\left(\begin{array}{cc}
I & 0 \\
0 & R(t)
\end{array}\right)$$ because the limit of Lie algebras is described by: $$\begin{pmatrix} 0 & n^{-1} t & & \\ 0 & 0 & & \\ & & 0 & -t \\ & & t & 0\end{pmatrix} \xrightarrow[n \to \infty]{} \begin{pmatrix} 0 & 0 & & \\ 0 & 0 & & \\ & & 0 & -t \\ & & t & 0\end{pmatrix}.$$ In fact, the [*local geometric limit*]{} is also strictly smaller than the geometric limit; it coincides with the algebraic limit. This is because every element with non-trivial entries in the $P(s)$ block of the geometric limit comes from a sequence of elements of $H$ which go to infinity.
5. To construct a limit of a connected group with infinitely many components, consider again the group from (4): $$H=\left(\begin{array}{cc}
P(t) & 0\\
0 & R(t) \end{array}\right).$$ The [*geometric limit*]{} under conjugacy by the sequence $c_n = \operatorname{diag}(1,1,n,n^{-1})$ is $$L=\left\{\left(\begin{array}{cc}
P(N \pi) & 0\\
0 & (-1)^N P(t) \\
\end{array}\right):\quad N\in{\mathbb Z},\quad t\in{\mathbb R}\right\}$$ and this has countably many components.
6. Next, here is an example where the conjugacy limit of a group is a proper subgroup of itself. Consider the group $L$ from (5). Now conjugate $L$ by the sequence $c_n=\operatorname{diag}(n,n^{-1},1,1)$. The limit is: $$L' =\left(\begin{array}{cc}
I_2 & 0\\
0 & P(t)\end{array}\right).$$
7. The following subgroup of $\GL(6,\RR)$ has infinitely many non-conjugate [*geometric limits*]{}. Fix $\alpha$ and define $$H=H(s,t)=\left(\begin{array}{ccccc}
e^s & 0 & 0 & 0 & 0\\
0 & e^t & 0 & 0 & 0\\
0 & 0 & 1 & s & t\\
0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 1 \end{array}\right)\qquad
c_n=\left(\begin{array}{ccccc}
1 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & n & \alpha n\\
0 & 0 & 0 & n & n^{-1}+\alpha n \end{array}\right)$$ Then $$c_n H c_n^{-1}=
\left(\begin{array}{ccccc}
e^s & 0 & 0 & 0 & 0\\
0 & e^t & 0 & 0 & 0\\
0 & 0 & 1 & n^{-1}+n(\alpha s-t) & -n(\alpha s-t)\\
0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 &1 \end{array}\right)$$ The limit as $n\to\infty$ is the two-dimensional group $$L_{\alpha}=L_{\alpha}(s,u)=\left(\begin{array}{ccccc}
e^s & 0 & 0 & 0 & 0\\
0 & e^{\alpha s} & 0 & 0 & 0\\
0 & 0 & 1 & u & -u\\
0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 &1 \end{array}\right)$$ Consideration of the character shows that if $\alpha\ne\beta$ then $L_{\alpha}$ is not conjugate to $L_{\beta}$.
8. We modify the previous example to obtain algebraic limit groups. For $\beta\in{\mathbb R}$ define a two-dimensional representation of $\RR^2$ by $$\sigma_{\beta}:{\mathbb R}^2\longrightarrow \SL(2,{\mathbb R})\qquad \sigma_{\beta}(x,y)=\left
(\begin{array}{cc} 1 & \beta x-y \\
0 & 1 \\
\end{array}\right),$$ and define also the two-dimensional representation of $\RR^2$, by $$\tau(s,t) = \begin{pmatrix} 1 & s & t\\ 0 & 1 & 0\\ 0 & 0 & 1\end{pmatrix}.$$ Now define a three-parameter algebraic subgroup $H(r,s,t)$ of $\SL(11,\RR)$ as a direct sum of representations $$H(r,s,t)=\sigma_0(r,s)\oplus\sigma_1(r,s)\oplus\sigma_2(r,s)\oplus \sigma_1(t,r)\oplus\tau(s,t).$$
Let $c_n = Id_9\oplus b_n$, where $b_n = \begin{pmatrix} n & \alpha n\\ n & n^{-1} + \alpha n \end{pmatrix}$. Then the limit of $H$ under conjugacy by $c_n$ is the algebraic group $$L_{\alpha}(r,s,u)= \sigma_0(r,s)\oplus\sigma_1(r,s)\oplus\sigma_2(r,s)\oplus \sigma_1(\alpha s,r)\oplus\tau(u,-u).$$ We claim that the function which sends $\alpha\in{\mathbb R}$ to the conjugacy class of $L_{\alpha}$ is finite to one. This requires an invariant.
Given a unipotent representation $\rho:{\mathbb R}^N\longrightarrow \SL(k,{\mathbb R})$ and $0\ne x\in{\mathbb R}^N$, the nullity of $(\rho(x)-Id)$ only depends on $[x]\in{\mathbb R}P^{N-1}$. This defines a function $\mathcal N_{\rho}:{\mathbb R}P^{N-1}\longrightarrow \{0,1,\cdots,k\}$. If $\rho$ and $\rho'$ have conjugate images there is a projective transformation $T\in {\operatorname{PGL}}(N,{\mathbb R})$ such that $\mathcal N_{\rho'}=\mathcal N_{\rho}\circ T$.
Thinking of $L_\alpha$, in the above example, as a representation of $\RR^3$ in terms of the parameters $r,s,u$, we have that $\mathcal N_{L_{\alpha}}([r:s:u])=8$ iff $u=0$ and $[r:s]$ is one of four points $$[1:0],\ [1:1],\ [1:2],\ [1:\alpha]$$ on the projective line $[*:*:0]$ in ${\mathbb R}P^2$. The cross ratio of $\{0,1,2,\alpha\}$, up to the action of the finite group that permutes these points, provides an invariant which shows there is a continuum of non-conjugate $L_\alpha$.
Properties of limits {#sec:limit-props}
--------------------
As we saw in the previous section, it is possible for the dimension of a subgroup to increase under taking limits. However, in the algebraic setting, this does not happen. Therefore, in this setting, the connected component of the geometric limit (the connected geometric limit) is equal to the Lie algebra limit (the group generated by taking the limit at the Lie algebra level first and then exponentiating).
\[prop:limit-dimension\] Let $G$ be an algebraic group (defined over $\CC$ or $\RR$). Suppose that $H$ is an algebraic subgroup and $L$ a conjugacy limit of $H$. Then $L$ is algebraic and $\dim L=\dim H$.
Suppose $c_n\in G$ and $H_n=c_n^{-1}Hc_n$ converges in the Chabauty topology to a subgroup $L$. Assume, for contradiction, that $\dim L>\dim H$. Then for every neighborhood $U$ of the identity in $G$ the number of connected components of $U\cap H_n$ goes to infinity as $n \to \infty$. Let $V$ be a variety of dimension $\dim G-\dim H$ passing through the identity, and smooth there. Choose $V$ so that it is transverse to each $H_n$ (for $n$ sufficiently large) in a fixed neighborhood $U$ of the identity. The set $V\cap H_n \cap U$ is a finite set of points, with cardinality going to infinity as $n$ goes to infinity. However, this is impossible, because the degree of the variety $V$ is constant and the degree of the varieties $H_n$ is also constant equal to that of $H$. Therefore the degree of $V \cap H_n$ is bounded and the cardinality of the finite sets $V \cap H_n \cap U$ must also be bounded.
In the case $G$ is defined over $\CC$, the fact that the limit group $L$ is algebraic follows from a theorem of Tworzewski and Winiarski [@Tworzewski] using work of Bishop [@bishop]. They showed that the set of all pure dimensional algebraic subsets of ${\mathbb C}^n$ of bounded degree is compact in the topology of local uniform convergence. From this one may deduce the case $G$ is defined over $\RR$ by complexification.
Note that Proposition \[prop:limit-dimension\] applies also when $G$ is the connected component of an algebraic group. In particular Proposition \[prop:limit-dimension\] holds when $G$ is the general or special linear group over $\RR$ or $\CC$, and also when $G = {\operatorname{PGL}}(n,\RR)$, which are the main cases of interest in this article.
Next, we investigate the behavior of multiple limits taken in succession, with the goal of showing that the relation “$L$ is a limit of $H$" induces a partial order of the space of closed subgroups of $G$. To begin, we study the behavior of the normalizer. Given a closed subgroup $H$ of $G$, let $H_0$ denote its identity component. The normalizer of $H_0$ in $G$ will be denoted $N_G(H_0)$.
\[prop:normalizer-dim\]Let $G$ be an algebraic Lie group (defined over $\CC$ or $\RR$), let $H$ be an algebraic subgroup and let $L$ be any limit of $H$. Then $\dim N_G(H_0)\le\dim N_G(L_0)$ with equality if and only if $L$ and $H$ are conjugate.
Let ${\mathfrak{h}}$ and ${\mathfrak{l}}$ denote the Lie algebras of $H$ and $L$ respectively. Then the normalizers $N_G(H_0)$ and $N_G(L_0)$ are equal to the normalizers of the respective Lie algebras $N_G({\mathfrak{h}})$ and $N_G({\mathfrak{l}})$. By Proposition \[prop:limit-dimension\], $\dim {\mathfrak{h}}= \dim {\mathfrak{l}}=: k$. So ${\mathfrak{h}}$ and ${\mathfrak{l}}$ define points in the projectivization $\mathbb P V$ of the $k^{th}$ exterior power $V = \Lambda^k {\mathfrak{g}}$. The orbit $G \cdot {\mathfrak{h}}$ under the adjoint action of $G$ is a smooth subset of $\mathbb P V$ corresponding to the Lie algebras of conjugates of $H$. The closure (in the classical topology) $\overline{G \cdot {\mathfrak{h}}}$, which contains ${\mathfrak{l}}$, is a union of $G \cdot {\mathfrak{h}}$ and orbits of strictly smaller dimension (in the case that $G$ is defined over $\RR$, this follows because the orbit ${G \cdot {\mathfrak{h}}}$ is semi-algebraic by the Tarski-Seidenberg theorem). If ${\mathfrak{l}}\in G \cdot {\mathfrak{h}}$, then $L$ is conjugate to $H$ and, of course, $N_G(L_0)$ is conjugate to $N_G(H_0)$. Assume then that ${\mathfrak{l}}\in \overline{G \cdot {\mathfrak{h}}} \setminus G \cdot {\mathfrak{h}}$, so that $\dim G \cdot {\mathfrak{l}}< \dim G \cdot {\mathfrak{h}}$. Then $$\begin{aligned}
\dim N_G({\mathfrak{l}}) &= \dim G - \dim G \cdot {\mathfrak{l}}\\
& > \dim G - \dim G \cdot {\mathfrak{h}}= \dim N_G({\mathfrak{h}}).\end{aligned}$$
Let ${\mathsf{Grp}_0}(G)$ denote the set of conjugacy classes of connected, algebraic, Lie subgroups of an algebraic Lie group $G$. If $L$ is the connected geometric limit of $H$ under some sequence of conjugacies (so $L$ is the identity component of a limit of $H$), then we write $H\rightarrow_0 L$.
\[thm:partial-order\] Let $G$ be an algebraic Lie group. The relation of being a connected geometric limit induces a partial order on the connected, algebraic, sub-groups ${\mathsf{Grp}_0}(G)$. Moreover the length of every chain is at most $\dim G$.
It follows from Proposition \[prop:normalizer-dim\] that $H\rightarrow_0 L$ and $L\rightarrow_0 H$ implies $L$ is conjugate to $H$. It remains to show transitivity. Suppose $\lim H_n= L$ and $K=\lim L_m$ with $H_n=a_nHa_n^{-1}$ and $L_m=b_mLb_m^{-1}$. Let $\mathcal C(G)$ denote the closed subgroups of $G$, equipped with the Chabauty topology (i.e. the topology of Hausdorff convergence in compact sets). The map $\theta_m:{\mathcal C}(G)\longrightarrow{\mathcal C}(G)$ given by $\theta_m(P)=b_mPb_m^{-1}$ is continuous. Hence $\lim_{n\to\infty}\theta_m(H_n)=L_m$ in ${\mathcal C}(G)$. It follows there is a sequence $\theta_{m_n}(H_n)$ which converges to $K$ in ${\mathcal C}(G)$.
Now we restrict our attention to the case when $G $ is locally isomorphic to $\GL_n \RR$. If $L$ is a limit of $H$, the eigenvalues of elements of $L$ are related to those of $H$. This leads to an obstruction to $L$ being a limit of $H$. The idea is that under degeneration eigenvalues either are unchanged or degenerate.
An element $A$ of ${\mathfrak{gl}}_N = \operatorname{End}(\RR^n)$ has a well defined characteristic polynomial, denoted ${\operatorname{char}}(A)$. Given a Lie sub-algebra ${\mathfrak{h}}$ of ${\mathfrak{gl}}$, we denote by ${\operatorname{Char}}({\mathfrak{h}})$ the closure of the subset of ${\mathbb R}[x]$ consisting of characteristic polynomials of all elements in $\mathfrak{h}$. Thus ${\operatorname{Char}}({\mathfrak{h}})$ is closed and invariant under conjugation of ${\mathfrak{h}}$.
\[prop:charpoly\] Suppose $H$ is a closed algebraic subgroup of $\GL_n \RR$, and $L$ is a conjugacy limit of $H$. Then ${\operatorname{Char}}({\mathfrak{l}})\subset {\operatorname{Char}}({\mathfrak{h}})$, where ${\mathfrak{h}}, {\mathfrak{l}}\subset {\mathfrak{gl}}(n)$ denote the Lie algebras of $H$ and $L$ respectively.
Suppose $p(x) = {\operatorname{char}}(\ell)$ for some $\ell \in {\mathfrak{l}}$. By assumption we have that $c_k H c_k^{-1} \to L$ and since $H$ is an algebraic subgroup, we have convergence at the Lie algebra level as well: ${\mathrm{Ad}}_{c_k} {\mathfrak{h}}\to {\mathfrak{l}}$. Hence, there exists a sequence $h_{k} \in {\mathfrak{h}}$ so that ${\mathrm{Ad}}_{c_k} h_{k} \to \ell$ as $k \to \infty$. The characteristic polynomials ${\operatorname{char}}(h_{k}) = {\operatorname{char}}({\mathrm{Ad}}_{c_k} h_{k})$ then converge to ${\operatorname{char}}(\ell)$ and therefore ${\operatorname{char}}(\ell) \in {\operatorname{Char}}({\mathfrak{h}})$ since ${\operatorname{Char}}({\mathfrak{h}})$ is closed.
For example, if $P(t)$ and $R(t)$ are the one parameter subgroups of $\GL_2\RR$ described in Section \[sec:examples\], and $\mathfrak{p}$ and $\mathfrak{r}$ denote their respective Lie algebras, then ${\operatorname{Char}}(\mathfrak{r})=\{x^2 + \theta^2: \theta \in \RR\}$ while ${\operatorname{Char}}({\mathfrak{p}}) = \{x^2\}$. Proposition \[prop:charpoly\] implies that $R(t)$ is not a limit of $P(t)$. This also follows from Theorem \[thm:partial-order\] because $P(t)$ is a (connected) limit of $R(t)$ and they are not conjugate (see example (2) from Section \[sec:examples\]).
We conclude this section by applying Proposition \[prop:charpoly\] to prove that $\HH^2 \times \RR$ geometry is not contained in any limit of hyperbolic geometry, when both are considered as sub-geometries of projective geometry. A more general statement on which Thurston geometries can arise as limits of hyperbolic geometry will be given in Theorem \[cor:Thurston-geoms\_intro\].
\[prop:H2timesR\] $\Isom_+({\mathbb H}^2\times{\mathbb R})$ is not a subgroup of a limit of $\PO(3,1)$ in $\GL(4,{\mathbb R})$ and therefore $\HH^2 \times \RR$ is not a sub-geometry of any limit of hyperbolic geometry inside of projective geometry.
The (almost) embedding of $\HH^2 \times \RR$ geometry in $\RP^3$ geometry represents the isometries of $\HH^2 \times \RR$ which preserve the orientation of the $\RR$ direction as $$\Isom_+({\mathbb H}^2\times{\mathbb R})=\left\{\left(\begin{array}{cc} e^t A & 0\\ 0 & e^{-3t}
\end{array}\right):\quad A\in SO(2,1), t \in \RR\right\}.$$ The Lie algebra is described by $$\mathfrak{isom}_+(\HH^2 \times \RR) = \left\{\left(\begin{array}{cc} t I_3 + a & 0\\ 0 & -3t \end{array}\right):\quad a\in {\mathfrak{so}}(2,1), t \in \RR \right\}.$$ The eigenvalues of elements of this Lie sub-algebra of ${\mathfrak{gl}}(4)$ are of the form $t,t+\lambda,t-\lambda,-3t$ with $\lambda^2\in{\mathbb R}$. The set of characteristic polynomials is: $${\operatorname{Char}}( \mathfrak{isom}_+(\HH^2 \times \RR)) = \{ (x-t)((x-t)^2-\lambda^2)(x+3t):t,\lambda^2\in{\mathbb R} \}.$$ On the other hand, the isometries of $\HH^3$ in the projective model are $\PO(3,1)$ and $${\operatorname{Char}}({\mathfrak{so}}(3,1))=\{(x^2-\lambda^2)(x^2+\theta^2):\ \lambda,\theta\in{\mathbb R}\}.$$ Inspection of these two sets and an application of Proposition \[prop:charpoly\] proves the claim.
Symmetric subgroups {#sec:symmetric-subgroups}
===================
We turn now to a special class of Lie groups and their subgroups, namely semi-simple Lie groups $G$ with finite center and their symmetric subgroups $H$. We will give an explicit description of the conjugacy limits of such symmetric subgroups and a more descriptive version of Theroem \[thm:symmetric-limits\] in Section \[sec:symm\_general\]. Then, Sections \[sec:symmetric-PGL\]–\[sec:Opq\] are dedicated to symmetric subgroups of the (projective) general linear group and their limits.
Symmetric subgroups in a semi-simple Lie group {#sec:symm_general}
----------------------------------------------
Let $G$ be a connected semi-simple Lie group of non-compact type and with finite center. Let $\sigma: G \to G$ be an involutive automorphism, i.e. $\sigma$ is a continuous automorphism with $\sigma^2 = 1$. The subset of fixed points $$G^\sigma =\{g\in G\ :\ \sigma(g)=g\ \}$$ is a closed subgroup of $G$. A closed group $H$ with $G^\sigma_0 \subset H \subset G^\sigma$, where $G^\sigma_0$ denotes the connected component of the identity, is called a *symmetric subgroup* of $G$. The quotient space $G/H$ is an *affine symmetric* space. Let us give some examples of affine symmetric spaces:
- When $H = K$ is a maximal compact subgroup, then $G/K$ is a Riemannian symmetric space.
- When $G = L\times L$ and $H = \mathrm{Diag}(L)$ is the diagonal, then $G/H\cong L$, via $(l_1,l_2)\mapsto {l_1l_2^{-1}}$.
- Let $G = \PO(p,q)$ and $H = {\operatorname{O}}(p-1,q)$. Then the affine symmetric space $$\XX(p,q) = \PO(p,q)/{\operatorname{O}}(p-1, q)$$ is a model space for semi-Riemannian manifolds of signature $(p-1,q)$ and of constant curvature $-1$. The geometries $(\XX(p,q), \PO(p,q))$ are in fact subgeometries of real projective geometry $(\RP^{p+q-1}, {\operatorname{PGL}}_{p+q} \RR)$, and we will describe their limiting geometries explicitly in Section \[sec:pfqf\].
- The symmetric subgroups of ${\operatorname{PGL}}_n\RR$ are $\operatorname{P}(\GL_p\RR\oplus \GL_q\RR)$ and $\PO(p,q)$ where $p+q = n$, or $\operatorname{P}(\GL_m\CC)$ and $\operatorname{P}({\operatorname{Sp}}(2m,\RR))$ where $n = 2m$, where for a subgroup $H' \subset \GL_n \RR$, $\operatorname{P}(H')$ denotes the image of $H'$ under the projection $\GL_n \RR \to {\operatorname{PGL}}_n \RR$. See Section \[sec:symmetric-GLn\].
In order to describe the conjugacy limits of symmetric subgroups, we will make use of the rich structure theory of affine symmetric spaces and symmetric subgroups. In order to keep the presentation concise we recall only the necessary details of the structure theory and refer the reader for more details and proofs to [@Schlichtkrull; @Rossmann; @nomizu; @MB]. We denote by ${\mathfrak{g}}$ and ${\mathfrak{h}}$ the Lie algebra of $G$ and $H$ respectively, and let the differential of $\sigma$, an involution of ${\mathfrak{g}}$, be again denoted by $\sigma: {\mathfrak{g}}\to {\mathfrak{g}}$. Then ${\mathfrak{h}}$ is the $+1$ eigenspace of $\sigma$ and we denote the $-1$ eigenspace by ${\mathfrak{q}}$. This gives the orthogonal decomposition $${\mathfrak{g}}= {\mathfrak{h}}\oplus {\mathfrak{q}}.$$ Note also that $[{\mathfrak{h}}, {\mathfrak{q}}] \subset {\mathfrak{q}}$, $[{\mathfrak{q}},{\mathfrak{q}}] \subset {\mathfrak{h}}$.
There exists a Cartan involution $\theta: {\mathfrak{g}}\to {\mathfrak{g}}$, which commutes with $\sigma$. We denote by $K = G^\theta$ the maximal compact subgroup of $G$ given by the fixed points of $\theta$ and we let ${\mathfrak{g}}= {\mathfrak{k}}\oplus {\mathfrak{p}}$ be the corresponding Cartan decomposition of ${\mathfrak{g}}$. Since the involutions commute, all four subspace are preserved by both involutions and so is the following decomposition: $${\mathfrak{g}}= {\mathfrak{k}}\cap {\mathfrak{h}}\oplus {\mathfrak{k}}\cap {\mathfrak{q}}\oplus {\mathfrak{p}}\cap {\mathfrak{h}}\oplus {\mathfrak{p}}\cap {\mathfrak{q}}.$$
Next, we may choose a maximal abelian sub-algebra ${\mathfrak{a}}\subset {\mathfrak{p}}$ so that the intersection ${\mathfrak{b}}= {\mathfrak{a}}\cap {\mathfrak{q}}$ is a maximal abelian sub-algebra of ${\mathfrak{p}}\cap {\mathfrak{q}}$. Note that ${\mathfrak{b}}$ is unique up to the action of $H \cap K$. We let $A := \exp ({\mathfrak{a}})$ be the corresponding connected subgroup of $G$ and $B := \exp({\mathfrak{b}}) \subset A$. Note that in some cases ${\mathfrak{b}}= {\mathfrak{a}}$ while in others the containment is strict. The following is well-known (see for example Proposition 7.1.3 of [@Schlichtkrull]):
\[thm:KAH\] For any $g \in G$ there exists $k \in K$, $b\in B$ and $h \in H$, such that $g = kbh$. Moreover $b$ is unique up to conjugation by the Weyl group $W_{H \cap K} := N_{H\cap K} ({\mathfrak{b}})/ Z_{H\cap K}( {\mathfrak{b}})$.
This factorization theorem will be our main tool in determining the limits of $H$. In the case $G = {\operatorname{PGL}}_n \RR$, this is equivalent to a matrix decomposition theorem for $\GL_n \RR$. Furthermore, in this case we may conjugate so that $A$ is a subgroup of diagonal matrices. Hence every conjugacy limit of a symmetric subgroup $H < {\operatorname{PGL}}_n \RR$ is conjugate to a conjugacy limit by a sequence of diagonal matrices.
Using Theorem \[thm:KAH\], we prove Theorem \[thm:symmetric-limits\] from the introduction, restated here for convenience:
[thm:symmetric-limits]{} Let $H$ be a symmetric subgroup of a semi-simple Lie group $G$ with finite center. Then for any geometric limit $L'$ of $H$ in $G$, there exists $X \in {\mathfrak{b}}$, so that $L'$ is conjugate to the limit group $L= \lim_{t \to \infty} \exp(tX) H \exp(-tX)$ obtained by conjugation by the one parameter group generated by $X$. Furthermore, $$L = Z_H(X) \ltimes N_+(X),$$ where $Z_H(X)$ is the centralizer in $H$ of $X$, and $N_+(X)$ is the connected nilpotent subgroup $$N_+(X) := \{g \in G : \lim_{t\to \infty} \exp(tX)^{-1} g \exp(tX) = 1 \}.$$
Let $(c_n) \subset G$ be a sequence and $L = \lim_{n \to \infty} c_n H c_n^{-1}$. By Theorem \[thm:KAH\], we may factorize $c_n = k_n b_n h_n$, where $k_n \in K, b_n \in B, h_n \in H$. Then, $$\begin{aligned}
c_n H c_n^{-1} &= k_n b_n h_n H h_n^{-1} b_n^{-1} k_n^{-1} \\
&= k_n b_n H b_n^{-1} k_n^{-1}\end{aligned}$$ We may assume, after passing to a subsequence, that $k_n \to k \in K$, so it follows that $c_n H c_n^{-1}$ converges if and only if $b_n H b_n^{-1}$ converges and their limits are conjugate by $k$. Hence we assume that $k_n = 1 = h_n$ and consider only conjugacies by sequences $(b_n) \in B$.
Consider the set of roots $$\Sigma({\mathfrak{g}}, {\mathfrak{a}}):=\{ \alpha \in {\mathfrak{a}}^*\, |\, \text{ there exists non-zero } Z\in {\mathfrak{g}}\text{ with } {\mathrm{ad}}(X)(Z) = \alpha(X) Z , \, \text{ for all } X \in {\mathfrak{a}}\}.$$ Denote by ${\mathfrak{g}}_\alpha:= \{Z\in {\mathfrak{g}}\, |\, {\mathrm{ad}}(X)(Z) = \alpha(X) Z , \, \text{ for all } X \}$ the root spaces and let ${\mathfrak{g}}= \sum_{\alpha \in \Sigma({\mathfrak{g}}, {\mathfrak{a}})} {\mathfrak{g}}_\alpha$ be the corresponding root space decomposition of ${\mathfrak{g}}$. Choose a basis for ${\mathfrak{g}}$ compatible with this root space decomposition. We work with the adjoint representation ${\mathrm{Ad}}: G \to \GL({\mathfrak{g}}) \cong \GL(N)$, expressed in this basis, which takes the subgroup $B$ to a subgroup of diagonal matrices in $\GL(N)$. The diagonal entries ${\mathrm{Ad}}(b_n)_{ii}$ of ${\mathrm{Ad}}(b_n)$ are positive, and we may assume they are arranged in increasing order: $ {\mathrm{Ad}}(b_n)_{jj} \geq {\mathrm{Ad}}(b_n)_{ii}$ for all $n$ and $j > i$. Further, we may assume that for consecutive indices $j = i+1$, the diagonal entries of ${\mathrm{Ad}}(b_n)$ satisfy exactly one of the following:
- Either ${\mathrm{Ad}}(b_n)_{jj} = {\mathrm{Ad}}(b_n)_{ii}$ holds for all $n$, or
- ${\mathrm{Ad}}(b_n)_{ii} < {\mathrm{Ad}}(b_n)_{jj}$ holds for all $n$, and ${\mathrm{Ad}}(b_n)_{jj}/{\mathrm{Ad}}(b_n)_{ii} \to \infty$.
For if ${\mathrm{Ad}}(b_n)_{jj}/{\mathrm{Ad}}(b_n)_{ii}$ remains bounded we may multiply $(b_n)$ by a sequence $(b_n') \subset B$ which remains in a compact subset of $B$ so that the above holds; the resulting limit differs only by conjugation (by the limit of $b_n'$).
Now consider a sequence $(h_n) \subset H$, so that $b_n h_n b_n^{-1} \to \ell \in L$. Then, the matrix entries $${\mathrm{Ad}}(b_n h_n b_n^{-1})_{ji} = \frac{{\mathrm{Ad}}(b_n)_{jj}}{{\mathrm{Ad}}(b_n)_{ii}} {\mathrm{Ad}}(h_n)_{ji}$$ converge to ${\mathrm{Ad}}(\ell)_{ji}$ as $n \to \infty$. We therefore have that, for $i \leq j$, $$\left \{ \begin{array}{ll}
{\mathrm{Ad}}(h_n)_{ji} \longrightarrow {\mathrm{Ad}}(\ell)_{ji} & \text{ if ${\mathrm{Ad}}(b_n)_{ii} = {\mathrm{Ad}}(b_n)_{jj}$, or}\\
{\mathrm{Ad}}(h_n)_{ji} \longrightarrow 0 & \text{ if $\frac{ {\mathrm{Ad}}(b_n)_{jj}}{{\mathrm{Ad}}(b_n)_{ii} }\to \infty$ as $ n \to \infty$.}
\end{array}\right .$$ This gives us information about the block lower diagonal entries of ${\mathrm{Ad}}(h_n)$. We use the involution $\sigma$ to obtain control over the block upper diagonal matrix entries. Note first that the involution $\sigma$ satisfies ${\mathrm{Ad}}(\sigma(b_n h_n b_n^{-1}))_{ji} = {\mathrm{Ad}}(b_n^{-1} h_n b_n)_{ji}$. Now $\sigma(b_n h_n b_n^{-1}) \to \sigma(\ell)$ as $n \to \infty$ and therefore the matrix entries $$\begin{aligned}
{\mathrm{Ad}}(\sigma(b_n h_n b_n^{-1}))_{ji} &= {\mathrm{Ad}}(b_n^{-1} h_n b_n)_{ji}\\
&= \frac{{\mathrm{Ad}}(b_n)_{ii}}{{\mathrm{Ad}}(b_n)_{jj}} {\mathrm{Ad}}(h_n)_{ji}\end{aligned}$$ also converge. Then, similar to the above, we conclude that, for $i \geq j$: $$\left \{ \begin{array}{ll}
{\mathrm{Ad}}(h_n)_{ji} \longrightarrow {\mathrm{Ad}}(\ell)_{ji} & \text{ if ${\mathrm{Ad}}(b_n)_{ii} = {\mathrm{Ad}}(b_n)_{jj}$, or}\\
{\mathrm{Ad}}(h_n)_{ji} \longrightarrow 0 & \text{ if $\frac{ {\mathrm{Ad}}(b_n)_{jj}}{{\mathrm{Ad}}(b_n)_{ii} }\to 0 $ as $ n \to \infty$.}
\end{array}\right .$$ Therefore, we conclude that ${\mathrm{Ad}}(h_n)$ converges, and because $G \to {\mathrm{Ad}}(G)$ is proper (because $Z(G)$ is finite), we may take a subsequence so that $h_n \to h_\infty$ as $n \to \infty$, where $$\left \{ \begin{array}{ll}
{\mathrm{Ad}}(h_\infty)_{ji} = {\mathrm{Ad}}(\ell)_{ji} & \text{ if ${\mathrm{Ad}}(b_n)_{ii} = {\mathrm{Ad}}(b_n)_{jj}$.}\\
{\mathrm{Ad}}(h_\infty)_{ji} = 0 & \text{ if ${\mathrm{Ad}}(b_n)_{ii} \neq {\mathrm{Ad}}(b_n)_{jj}$.}
\end{array}\right .$$ In other words, ${\mathrm{Ad}}(h_\infty)$ belongs to the centralizer $Z_{\GL(N)}({\mathrm{Ad}}(b_n))$. It follows that $$h_\infty \in Z_H(b_n).$$ This is because the commutator of $h_{\infty}$ with $b_n$ is in the kernel of ${\mathrm{Ad}}$. However $b_n$ lies in a one parameter group that commutes with $h_{\infty}$ so their commutator lies in a connected, hence trivial, subgroup of the center of $G$. In the above expressions, note that $Z_{\GL(N)}({\mathrm{Ad}}(b_n))$ and $Z_H(b_n)$ are independent of $n$ by our assumptions on the sequence $(b_n)$.
Observe that $Z_H(b_n) \subset L$. In fact, we have shown that $L$ is an *expansive limit* of $H$ about the subgroup $Z_H(b_n)$ (see Section \[sec:other-notions\]). From the above arguments, we may also conclude that $$\begin{aligned}
{\mathrm{Ad}}(\ell)_{ji} &= \lim_{n \to \infty} \frac{{\mathrm{Ad}}(b_n)_{jj}}{{\mathrm{Ad}}(b_n)_{ii}} {\mathrm{Ad}}(h_n) = 0\end{aligned}$$ in the case that ${\mathrm{Ad}}(b_n)_{jj}/{\mathrm{Ad}}(b_n)_{ii} \to 0$. It then follows that ${\mathrm{Ad}}(b_n^{-1} \ell b_n) \to {\mathrm{Ad}}(h_\infty)$ because $${\mathrm{Ad}}(b_n^{-1}\ell b_n)_{ji} = \frac{{\mathrm{Ad}}(b_n)_{ii}}{{\mathrm{Ad}}(b_n)_{jj}}{\mathrm{Ad}}(\ell)_{ji}
\left \{ \begin{array}{ll}
= {\mathrm{Ad}}(h_\infty)_{ji} & \text{ if ${\mathrm{Ad}}(b_n)_{ii} = {\mathrm{Ad}}(b_n)_{jj}$}\\
= 0 = {\mathrm{Ad}}(h_\infty)_{ji} & \text{ if ${\mathrm{Ad}}(b_n)_{jj} < {\mathrm{Ad}}(b_n)_{ii}$}\\
\longrightarrow 0 = {\mathrm{Ad}}(h_\infty)_{ji} & \text{ if ${\mathrm{Ad}}(b_n)_{jj} > {\mathrm{Ad}}(b_n)_{ii}$}
\end{array}\right .$$
Hence $b_n^{-1} \ell b_n \to h_\infty' \in Z_H(b_n)$ (in fact, $h_\infty' = h_\infty$, but we have only shown ${\mathrm{Ad}}(h_\infty) = {\mathrm{Ad}}(h_\infty')$). Therefore $b_n^{-1} (h_\infty')^{-1} \ell b_n \to 1$ as $n \to \infty$ and therefore $(h_\infty')^{-1} \ell$ lies in the group $$N_+ := \{ g \in G: b_n^{-1} g b_n \to 1 \text{ as } n \to \infty \}.$$ It follows that $$\label{eqn:contained}
L \subset \langle Z_H(b_n), N_+ \rangle = Z_H(b_n) \ltimes N_+.$$ We next show equality. We have that $L = Z_H(b_n) \ltimes N'$ where $N' = N_+ \cap L \subset N_+$ is a closed subgroup. We also have that $N_+$ is connected because $N_+$ is preserved by conjugation by $B$, and conjugation by $b_n^{-1}$, for large $n$, brings elements arbitrarily close to the identity; hence all elements are in the identity component. So equality in will follow by showing that $\dim N_+ + \dim Z_H((b_n)) = \dim H$. We show this at the Lie algebra level. Observe that $\mathfrak g=\mathfrak z +\mathfrak n_++\mathfrak n_-$ where $$\begin{aligned}
\mathfrak z &= \{ Y \in {\mathfrak{g}}: {\mathrm{Ad}}(b_n) Y = Y \text{ for all } n\}\\
\mathfrak n_+ &= \{ Y \in {\mathfrak{g}}: {\mathrm{Ad}}(b_n^{-1})Y \to 0 \text{ as } n \to \infty\}\\
\mathfrak n_- &= \{Y \in {\mathfrak{g}}: {\mathrm{Ad}}(b_n)Y \to 0 \text{ as } n \to \infty\}\end{aligned}$$
The involution $\sigma$ fixes $\mathfrak z$, and exchanges $\mathfrak n_+$ and $\mathfrak n_-$. Therefore $\mathfrak n_+$ and $\mathfrak n_-$ have the same dimension and the dimension of the $+1$ eigenspace ${\mathfrak{h}}$ of $\sigma$ is equal to $\dim \mathfrak z + \dim \mathfrak n_+$ while the dimension of the $-1$ eigenspace is $\dim \mathfrak n_+$. Since $\mathfrak z$ is the Lie algebra of $Z_H(b_n)$ and $\mathfrak n_+$ is the Lie algebra of $N_+$, we conclude that $L = Z_H(b_n) \ltimes N_+$. Observe that ${\mathrm{Ad}}(N_+)$ is upper triangular and therefore $N_+$ is nilpotent.
The proof now concludes by observing that any sequence $(b_n) \subset B$ whose eigenvalues satisfy our above assumptions will produce exactly the same limit. In particular, let $X = \log(b_m)$ for some $m$. Then conjugating $H$ by the sequence $b_n' = \exp(t_n X)$, also produces $L$ as the limit for any sequence of reals $(t_n)$ such that $t_n \to \infty$. In this case $Z_H(\exp(t_n X)) = Z_H(X)$ and this implies the claim.
The proof of Theorem \[thm:symmetric-limits\] shows that the limit is determined up to conjagacy by an ordered partition of the numbers $\{1,\ldots, N\}$. Hence there are finitely many limits, up to conjugacy, of a symmetric subgroup $H$ of $G$. We now introduce some notation in order to enumerate these limits. Consider the root system $\Sigma({\mathfrak{g}}, {\mathfrak{b}}) \subset {\mathfrak{b}}^*$ defined by the adjoint action of ${\mathfrak{b}}$ on ${\mathfrak{g}}$. The Weyl group $W = N_K({\mathfrak{b}})/Z_K({\mathfrak{b}})$ acts on ${\mathfrak{b}}$; it is the group generated by reflections in the hyperplanes determined by the roots. We may similarly consider the adjoint action of ${\mathfrak{b}}$ on the smaller Lie algebra ${\mathfrak{g}}_{\tau} := {\mathfrak{h}}\cap {\mathfrak{k}}\oplus {\mathfrak{p}}\cap {\mathfrak{q}}$; this is the Lie algebra of the symmetric subgroup defined by the involution $\tau = \sigma \theta$. Then the corresponding root system $\Sigma({\mathfrak{g}}_{\tau}, {\mathfrak{b}})$ is contained in $\Sigma({\mathfrak{g}}, {\mathfrak{b}})$, and the corresponding Weyl group $W_{H \cap K} := N_{H \cap K}({\mathfrak{b}})/Z_{H \cap K}({\mathfrak{b}})$ is a subgroup of $W$. Let $\Sigma^+ \subset \Sigma({\mathfrak{g}}, {\mathfrak{b}})$ denote a system of positive roots, and let $\Delta \subset \Sigma^+$ be a choice of simple roots. Define: $$\begin{aligned}
{\mathfrak{b}}^+ &= \{ Y \in {\mathfrak{b}}: \alpha(Y) > 0 \text{ for all } \alpha \in \Delta\}, & B^+ &= \exp{{\mathfrak{b}}^+},\\
\overline{{\mathfrak{b}}^+} &= \text{ the closure of } {\mathfrak{b}}^+, & \overline{B^+} &= \exp{\overline{{\mathfrak{b}}^+}}.
\end{aligned}$$ Then $\overline{{\mathfrak{b}}^+}$ is the closed Weyl chamber corresponding to $\Sigma^+$; it is (the closure of) a fundamental domain for the action of $W$. A closed Weyl chamber for the action of $W_{H \cap K}$ is then given by a union $\mathcal W \cdot \overline{{\mathfrak{b}}^+}$ of translates of $\overline{{\mathfrak{b}}^+}$ by a set $\mathcal W$ of coset representatives for the quotient $W/W_{H \cap K}$.
By Theorem \[thm:KAH\], any element $g \in G$ may be decomposed as $g = kbh$, where $k \in K, b\in B, h \in H$. There exists $w_1 \in W_{H \cap K}$ so that the conjugate $w_1^{-1} b w_1$ lies in the exponential image of the closed Weyl chamber $\mathcal W \cdot \overline{{\mathfrak{b}}^+}$. Therefore $w_1^{-1} b w_1 = w^{-1} b^+ w$ where $b^+ \in \overline{B^+}$ and $w \in \mathcal W$. Therefore, we may write $$g = k'b^+wh'$$ where $k' = k w_1 w^{-1}$ and $h' = w_1^{-1} h$. Here $b^+ \in \overline{B^+}$ and $w \in \mathcal W$ are uniquely determined.
It follows that any limit of $H$ may be obtained by conjugating by a sequence in $\overline{B^+}w$ for some $w \in \mathcal W$. We next apply these observations in combination with Theorem \[thm:symmetric-limits\] to obtain an enumeration of the limit groups in terms of the element $w$ and the behavior of the sequence in $\overline{B^+}$. Let $I \subset \Delta$ be a subset, and $\Sigma_{I}^+$ the span of $\Delta - I$ in $\Sigma^+$. We set $$\begin{aligned}
{{\mathfrak{b}}}_I &= \bigcap_{\alpha \in I} \ker(\alpha) \subset {{\mathfrak{b}}}, & B_I &= \exp({{\mathfrak{b}}}_I),\\
{\mathfrak{b}}_I^+ &= \bigcap_{\alpha \in I} \ker(\alpha) \cap \bigcap_{\alpha \in \Delta - I} \{ X \in {{\mathfrak{b}}} \,|\, \alpha (X) >0 \}, & B_I^+ &= \exp({\mathfrak{b}}_I^+).
\end{aligned}$$
\[thm:limit\_nice\] Let $H$ be a symmetric subgroup of a semi-simple Lie group $G$ with finite center. Let $L$ be a limit of $H$ under conjugacy in $G$. Then $L$ is conjugate to a subgroup of the form $$L_{I,w} = Z_{H_w}({\mathfrak{b}}_I) \ltimes N_I$$ where $w \in \mathcal W$ and $I \subset \Delta$ is a subset of the set of simple roots $\Delta \subset \Sigma({\mathfrak{g}}, {\mathfrak{b}})$. Here, $H_w := wHw^{-1}$ and $N_I$ is the connected subgroup of $G$ with Lie algebra $
{\mathfrak{n}}_I = \sum_{\alpha \in \Sigma_{I}^+ } {\mathfrak{g}}_{\alpha}$. Further, any $L_{I,w}$ is achieved as a limit.
By Theorem \[thm:symmetric-limits\], we may assume (after conjugating) that $L = Z_H(X) \ltimes N_+(X)$ for some $X \in {\mathfrak{b}}$. Define $$I = \{ \alpha \in \Delta: \alpha(X) = 0\}$$ and let $u \in W_{H \cap K}$ and $w \in \mathcal W$ be such that $X' := {\mathrm{Ad}}(w u)X$ lies in $\overline{{\mathfrak{b}}^+}$. Note that, in fact, $X'$ lies in ${\mathfrak{b}}_I^+$. Then, we have $$\begin{aligned}
Z_H(X) &= Z_H({\mathrm{Ad}}(u^{-1} w^{-1}) X') &N_+(X) &= N_+({\mathrm{Ad}}(u^{-1}w^{-1}X') \\ &= u^{-1} w^{-1} Z_{H_{wu}}(X') wu & &= u^{-1}w^{-1} N_+(X') wu\\ &= (wu)^{-1} Z_{H_{w}}(X') wu & &= (wu)^{-1} N_+(X') wu.
\end{aligned}$$ where in the last step $H_{wu} = H_w$ because $u \in H$. Therefore $L$ is conjugate to $Z_{H_w}(X') \ltimes N_+(X')$. Now, its clear that $N_+(X') = N_I$ because their Lie algebras agree. The Lie algebra of $N_+(X')$ consists of all elements $Y \in {\mathfrak{g}}$ such that ${\mathrm{Ad}}(\exp(-tX'))Y \to 0$ as $t \to \infty$; this is exactly the span of the root spaces ${\mathfrak{g}}_\alpha$ for which $\alpha(X') > 0$, in other words $\alpha \in \Sigma_I^+$. It remains to show that $Z_{H_w}(X') = Z_{H_w}({\mathfrak{b}}_I^+)$. The following Lemma will complete the proof.
\[lem:centralizers\] Let $Y_1,Y_2 \in {\mathfrak{b}}_I^+$. Then $Z_G(Y_1) = Z_G(Y_2)$.
By replacing $G$ by ${\mathrm{Ad}}(G) \cong G/Z(G)$, we may assume $G$ is an algebraic subgroup of $\GL(N)$. Let $S_1$ and $S_2$ denote the Zariski closures of the one parameter subgroups generated by $Y_1$ and $Y_2$ respectively. Then $S_1$ and $S_2$ are $\RR$-tori. Therefore $Z_G(Y_1) = Z_G(S_1)$ and $Z_G(Y_2) = Z_G(S_2)$ are Zariski connected closed subgroups of $G$. It follows that $Z_G(Y_1) = Z_G(Y_2)$ since both groups have the same Lie algebra.
\[cor:indexlimits\] Let $\Delta$ be a set of simple positive roots in $\Sigma({\mathfrak{g}}, {\mathfrak{b}})$, let $S$ denote the power set of $\Delta$ and let $\mathcal{W}$ denote a set of representatives of $W/W_{H\cap K}$. Let us denote by $\mathcal{L}(H)$ the conjugacy classes of limits of $H$ in $G$. Then there is a surjection from $S \times \mathcal{W}$ onto $\mathcal{L}(H)$. In particular, there are up to conjugacy only finitely many limits of $H$ in $G$.
\[rem:finer-list\] The map of $S \times \mathcal{W}$ onto $\mathcal{L}(H)$ is in general not injective. For example if $I = \emptyset$, then $L_{I,w}$ is conjugate to $L_{I, w'}$ for any $w,w' \in \mathcal W$. The conjugacy classes of limits can be labeled, with less redundancy, in the following way. First, two subsets $I_1, I_2 \subset \Delta$ give rise to conjugate limit groups $L_{I_1,1}$, $L_{I_2,1}$ if and only if $I_1 = I_2$. Next, consider a fixed subset $I \subset \Delta$. Let $W_I \subset W$ be the subgroup of the Weyl group which acts trivially on ${\mathfrak{b}}_I$. Then for any $w \in \mathcal{W}$ and $u \in W_I$, the limit groups $L_{I,w}$ and $L_{I,u w}$ are conjugate. Let $\mathcal{W}_I$ be a set of representatives of the double cosets $W_I \backslash W /W_{H\cap K}$. For $I = \emptyset$, $W_I$ is trivial, so $\mathcal{W}^{I}= \mathcal{W}$. Then the pairs $(I, w)$, where $I \in S$ and $w \in \mathcal W_I$ give all possible limits $L_{I,w}$ of $H$. We note that this finer enumeration may still have some redundancy; see Remark \[rem:SOpq-redundancy\].
Taking a different point of view, a more detailed analysis of the limits under conjugation can lead to a description of the Chabauty compactification of the affine symmetric space $X= G/H$. The Chabauty compactification is defined as follows. Consider the continuous map $\phi: X = G/H \longrightarrow \mathcal{C}(G)$, which sends the left coset $gH$ to the closed subgroup $gHg^{-1}$. Since $\mathcal{C}(G)$, endowed with the Chabauty topology is compact, the closure of $\phi(X)$ defines a compactification of $X$, called the Chabauty compactification [@chabauty].
In the case when $H=K$ is a maximal compact subgroup the Chabauty compactification of $G/K$ was determined by Guivárch–Ji–Taylor [@GJT] and Haettel [@Haettel2]. They also show that the Chabauty compactification of $G/K$ is homeomorphic to the maximal Furstenberg compactification.
The similarities of the above analysis with the definition of the maximal Satake-compactification of $G/H$ as defined in [@Gorodnik_Oh_Shah Theorem 4.10] suggests that such a homeomorphism might also hold for affine symmetric spaces.
Limits of symmetric subgroups of $G = {\operatorname{PGL}}_n\KK$ and $G = \SL_n\KK$ {#sec:symmetric-PGL}
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We are mainly interested in classifying the limits of sub-geometries of projective geometry whose structure group $H$ is a symmetric subgroup of $G = {\operatorname{PGL}}_n\KK$, with $\KK= \RR$ or $\CC$. So we now apply Theorems \[thm:limit\_nice\] to the setting of symmetric subgroups in the projective linear group. However, we note that everything described in this section can be easily adapted to the (very similar) case $G = \SL_n \KK$. We may choose coordinates so that the Cartan involution $\theta$ that commutes with $\sigma$ is the standard one, i.e. $\theta(X) = X^{-T}$ for ${\operatorname{PGL}}_n\RR$ and $\theta(X) = \overline{X}^{-T}$ for ${\operatorname{PGL}}_n\CC$. Then, thinking of the Lie algebra $\mathfrak g$ as trace-less $n\times n$ matrices, we may choose the Cartan sub-algebra ${\mathfrak{a}}$ to be the trace-less (real) diagonal matrices and so the algebra ${\mathfrak{b}}$ from the previous section is a sub-space of trace-less diagonal matrices. These coordinates are particular nice for calculating limit groups.
Let $X \in {\mathfrak{b}}$, and let $E_0, \ldots, E_k$ be the eigenspaces of $X$, listed in order of increasing eigenvalue. Consider the partial flag ${\mathcal F}= {\mathcal F}(X)$, defined to be the chain of subspaces $V_0 \supset V_1 \supset \ldots \supset V_k$, where $$V_j = E_j \oplus \cdots \oplus E_k$$ and by convention, we take $V_{k+1} = \{0\}$. We define the *partial flag group* ${\operatorname{PGL}}({\mathcal F})$ to be the subgroup of ${\operatorname{PGL}}_n \RR$ which stabilizes ${\mathcal F}$. There is a natural surjection $$\pi_{{\mathcal F}}:{\operatorname{PGL}}({\mathcal F}) \longrightarrow \operatorname{P} \big(\bigoplus_{i=0}^k \GL(V_i/V_{i+1})\big),$$ where on the right-hand side, $\operatorname{P}$ denotes the projection $\GL_n \to {\operatorname{PGL}}_n$. We call the group $U({{\mathcal F}})=\ker(\pi_{{\mathcal F}})$ the [*flag unipotent subgroup*]{}. It is connected and unipotent, however in general it is not maximal among unipotent subgroups of ${\operatorname{PGL}}({{\mathcal F}})$. Here is a simple corollary of Theorem \[thm:symmetric-limits\]:
\[thm:symmetric-PGL\] Let $L = Z_{H}(X) \ltimes N_+(X)$ be any limit of ${H} \subset {\operatorname{PGL}}_n \RR $ as in Theorem \[thm:symmetric-limits\], where $X \in {\mathfrak{b}}$. Then $L \subset {\operatorname{PGL}}({\mathcal F})$. Further, the group $Z_{H}(X)$ consists of all elements of $H$ which preserve the decomposition $\RR^n = E_0 \oplus \cdots \oplus E_k$, while $N_+(X) = U_{\mathcal F}$. So, in a basis respecting the decomposition $\RR^n = E_0 \oplus \cdots \oplus E_k$, every element of $L$ has the form: $$\left(\begin{array}{ccccc}
A_0 & 0 & 0 &\cdots & 0\\
* & A_1 & 0 &\cdots & 0\\
* & * & A_2&\cdots& 0\\
\cdots &\cdots &\cdots &\cdots &\cdots \\
* & * & * &\cdots & A_k
\end{array}\right)$$ where $\operatorname{diag}(A_0,\ldots, A_k) \in H \cap \operatorname{P}(\GL(E_0)\oplus \cdots \oplus \GL(E_k))$ and each $*$ denotes a block which may take arbitrary values.
That $Z_H(X)$ consists of all elements of $H$ that preserve the eigenspaces of $X$ is clear. Next, we examine the action by conjugation of $\exp(t X)$ on ${\operatorname{PGL}}_n$. Writing an element $g \in {\operatorname{PGL}}_n \KK$ in block form with respect to the decomposition $\KK^n = E_0 \oplus \cdots \oplus E_k$, we see that conjugation by $\exp(-t X)$ multiplies the $(i,j)$ block by the scalar $e^{-t(d_i - d_j)}$, where $d_i$ is the $i^{th}$ eigenvalue of $X$ with eigenspace $E_i$. Therefore, $\exp(-tX)g\exp(tX) \to 1$ as $t \to \infty$ if and only if $g \in {\operatorname{PGL}}({\mathcal F})$ and $\pi_{\mathcal F}(g) = 1$.
Symmetric subgroups of $G = \GL_n \RR$ {#sec:symmetric-GLn}
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Although the general linear group $\GL_n \RR$ is not semi-simple, we abuse terminology and call a subgroup $H < \GL_n \RR$ symmetric if there exists an involution $\sigma$, commuting with the Cartan involution $\theta$, such that $H$ is the set of fixed points of $\sigma$. It is sometimes more convenient to work with symmetric subgroups in $\GL_n \RR$ than with symmetric subgroups in ${\operatorname{PGL}}_n \RR$ or $\SL_n \RR$. Theorems \[thm:symmetric-limits\] and \[thm:limit\_nice\] do not directly apply in this setting. However, one may determine the limits of a symmetric subgroup $H$ of $\GL_n \RR$ by applying the theorems to either the image $\operatorname{P}H$ under the projection $\operatorname{P}: \GL_n \to {\operatorname{PGL}}_n$, or to $H \cap \SL_n \RR$. This strategy will be employed in the following sections.
We now list the symmetric subgroups of $\GL_n \RR$. Any involution of $\GL_n \RR$ commutes with a Cartan involution $\theta$. So, we take $\theta$ to be the standard Cartan involution, given by $\theta(X) = X^{-t}$ and give a list of all involutions that commute with $\theta$.
- First, there are inner involutions of the form $\sigma(X) = J X J^{-1}$, where $J = -I_p \oplus I_q$ for some $p +q = n$. Note that $J^2 = Id$ and $J \in K$, so $\sigma$ commutes with $\theta$. In this case the symmetric subgroup of fixed points of $\sigma$ is: $$H_\sigma = Z_{\GL_n\RR}(J) = \GL_p \RR\oplus \GL_q\RR.$$
- For $n=2m$ even, let $J$ be the complex structure on ${\mathbb R}^{2m}$ given by $m$ copies of the standard complex structure on $\RR^2$: $$J=\bigoplus_m \left[\begin{array}{cc}0 & -1 \\1 & 0\end{array}\right].$$ Again, $J$ is orthogonal, so the involution $\sigma$ defined by $\sigma(X) = JXJ^{-1}$ commutes with $\theta$. Then, $$H_{\sigma}= Z_{\GL_n\RR}(J) =: \GL_m{\mathbb C}.$$
- Of course there is the Cartan involution $\theta$ itself. In this case $H_\theta = {\operatorname{O}}(n)$.
- Let $\phi$ be the involution defined by $\phi(X)=|\det X|^{-2/n}X$. Then $H_\phi =\SL^\pm_n \mathbb R := \{ A \in \GL_n \RR: \det A = \pm 1\}.$
- Assume $n= 2m$ is even. Let $\varphi$ be the involution that is the identity on matrices of positive determinant and multiplication by $-1$ on matrices of negative determinant. Then $H_{\varphi} = \GL^+_n \RR =: \{A \in \GL_n \RR : \det A > 0\}$ is the identity component of $\GL_n \RR$.
- In fact, $\theta, \phi$ and $\varphi$ commute with each other and with any inner involution $\sigma$ of type (1a) or (1b). So if $\epsilon_1, \epsilon_2, \epsilon_3 \in \{0,1\}$ and $\sigma$ is any inner involution of type (1a) or (1b), then $\tau = \sigma \theta^{\epsilon_1}\phi^{\epsilon_2}\varphi^{\epsilon_3}$ defines an involution.
Since the involutions of types (3) and (4) are not so interesting, we will be most interested in the inner involutions $\sigma$ of type (1a) and (1b) and their products $\tau = \sigma \theta$ with the Cartan involution. If $\sigma$ is an inner involution of type (1a), then $H_\tau$ is the orthogonal group: $$H_\tau = {\operatorname{O}}(p,q).$$ If $\sigma$ is an inner involution of type (1b), then $H_\tau$ is the symplectic group: $$H_\tau = {\operatorname{Sp}}(2m,\RR).$$
Every continuous involution of $\GL_n \mathbb R$ is conjugate to one listed above.
The outer (continuous) automorphism group of $\SL_n{\mathbb R}$ is cyclic of order $2$ generated by the Cartan involution $\theta$ for $n\ge3$ and trivial otherwise (see Theorem 4.5 of [@delp]). Since $\GL_n \RR\cong\SL_n^{\pm}\times\RR$, it follows that the outer automorphisms of the identity component $\GL^+_n \RR$ are $\operatorname{Out}(\GL^+_n \RR)\cong{\mathbb Z}_2\times \operatorname{Aut}({\mathbb R})$, generated by $\theta$ and $\phi$. If $n$ is even, then there is exactly one nontrivial outer involution of $\GL_n \RR$ which fixes the identity component, namely $\varphi$. If $n$ is odd, there are no such outer involutions.
Every inner involution is given by conjugation by some $J$ with $J^2$ central. We may assume $\det J=\pm1$. Then $J$ is conjugate to one of the matrices listed.
We next apply Theorem \[thm:limit\_nice\] to determine the limits in the most interesting cases.
Limits of $\GL_m{\mathbb C}$ in $\GL_{2m}{\mathbb R}$
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Consider the involution $\sigma$ of $\GL_{2m} \RR$ defined by $\sigma(X) = JXJ^{-1}$ where $$J = {\begin{pmatrix}}0 & I_m \\ -I_m & 0 {\end{pmatrix}}.$$ Then the fixed point set of $\sigma$ naturally identifies with the group $\GL_m \CC$. The standard basis $e_1, \ldots, e_m, e_{m+1}, \ldots, e_{2m}$ is compatible with the complex structure $J$ in the sense that $e_{j+m} = Je_j$ and $e_j = -J e_{j+m}$. Note that the center $Z(\GL_{2m} \RR)$ is contained in $\GL_m \CC$. Therefore, to determine the limits of $\GL_m \CC$ inside of $\GL_{2m} \RR$, we pass to the projective general linear group via the quotient map $\operatorname{P}: \GL_{2m}\RR \to {\operatorname{PGL}}_{2m} \RR$. The image of $\GL_m \CC$ is the subgroup $H = \operatorname{P}( \GL_m \CC) \cong \GL_m \CC / \RR^*$. Note that $\sigma$ is well-defined on ${\operatorname{PGL}}_{2m} \RR$ and that $H$ is precisely the fixed point set of $\sigma$. So we may apply Theorem \[thm:limit\_nice\] to determine the limits of $H$ inside of $G = {\operatorname{PGL}}_{2m} \RR$. All limits of $\GL_m \CC$ in $\GL_{2m} \RR$ are of the form $\operatorname{P}^{-1} L$ where $L \subset {\operatorname{PGL}}_{2m} \RR$ is a limit of $H$.
Note that $\sigma$ commutes with the standard Cartan involution $\theta$. The $-1$ eigenspace ${\mathfrak{q}}$ of $\sigma$ is given by matrices of the form $${\begin{pmatrix}}A & B\\ B & -A {\end{pmatrix}}$$ A maximal abelian subgroup ${\mathfrak{b}}$ of ${\mathfrak{p}}\cap {\mathfrak{q}}$ is given by elements $X$ of the form $$X = {\begin{pmatrix}}D & 0 \\ 0 & -D {\end{pmatrix}}$$ where $D = \operatorname{diag}(d_1,\ldots,d_m)$ is a diagonal $m \times m$ matrix. The system of positive simple roots of ${\mathfrak{g}}{\mathfrak{l}}_{2m} \RR$ with respect to $\frak b$ can be chosen to be $$\Delta = \left\{ d_{i+1} - d_{i} \right\}_{i=1}^{m-1} \cup \{ 2 d_m \}.$$ In this case the inclusion $W_{H \cap K} \hookrightarrow W$ is an isomorphism; both Weyl groups simply permute the diagonal entries of $D$ and also the signs. Therefore, $\mathcal W = \{1\}$ and we may take $\overline{{\mathfrak{b}}^+}$ to be the collection of diagonal matrices $X$ as above, where $0 \leq d_1 \leq \ldots \leq d_m $. Then, by Theorem \[thm:limit\_nice\], the conjugacy classes of limits of $H$ in $G$ are enumerated by subsets $I \subset \Delta$. For a given subset $I \subset \Delta$, the corresponding limit group $$L_I = Z_H({\mathfrak{b}}_I) \ltimes N_I = Z_H(X) \ltimes N_+(X)$$ is the limit under conjugacy by $\exp(t X)$ as $t \to \infty$, where $X \in {\mathfrak{b}}_I^{+}$. Let $$-\lambda_k < -\lambda_{k-1} < \cdots < \lambda_0 = 0 < \lambda_1 < \cdots < \lambda_k$$ denote the eigenvalues of $X$ (symmetric under negation). Note that either $\lambda_j$ or $- \lambda_j$ is a diagonal entry of $D$. Note also that the eigenspaces $E_{\lambda_j}$ satisfy $E_{\lambda_j} = J E_{-\lambda_j}$, and in particular the zero eigenspace $E_0$ is a complex subspace, invariant under $J$. Hence any element $g \in Z_H(X)$ preserves the eigenspace decomposition $\RR^{2m} = E_0 \bigoplus_{j=1}^k E_{\lambda_j} \oplus E_{-\lambda_j}$. The action of $g$ on $E_0$ is $J$-linear. The action of $g$ on $E_{\lambda_j} \oplus E_{-\lambda_j}$ is also $J$-linear and further preserves the real and imaginary parts $E_{\lambda_j}$ and $E_{-\lambda_j} = J E_{\lambda_j}$. Therefore, the matrix for the action of $g$ on $E_{\lambda_j} \oplus E_{-\lambda_j}$ has the form $${\begin{pmatrix}}A_j & 0 \\ 0 & A_j {\end{pmatrix}}$$ in a basis which is the union of a basis for $E_{\lambda_j}$ and $J$ times that basis (which is a basis for $E_{-\lambda_j}$). This characterizes $Z_H(X)$. Next, the flag ${\mathcal F}$ defining the unipotent part $U({\mathcal F}) = N_I = N_+(X)$ of $L$ (see Section \[sec:symmetric-PGL\]) is given by the sub-spaces $$V_{-k} \supset V_{-(k-1)} \supset \cdots V_0 \supset V_{1} \supset \cdots \supset V_{k}$$ where $V_j = E_{\lambda_j} \oplus \cdots \oplus E_{\lambda_{k}},$ where $\lambda_{-j} := -\lambda_j$.
We may explicitly describe the corresponding limit $\operatorname{P}^{-1} L$ of $\GL_m \CC$ in $\GL_{2m} \RR$. In a basis respecting the ordered eigen-space decomposition $$\RR^{2m} := E_{\lambda_k} \oplus \cdots \oplus E_0 \oplus \cdots E_{-\lambda_k}$$ and such that the basis elements for $E_{-\lambda_j}$ are $J$ times the basis elements for $E_{\lambda_j}$ (where $\lambda_j > 0$), the elements of the limit group $\operatorname{P}^{-1} L$ are exactly the matrices of the form:
$${\begin{pmatrix}}A_k & & & & & & & &\\ * & A_{k-1} & & & & & & & \\ \vdots & \vdots & \ddots & & & & & & \\ * & * & \cdots & A_{1} & & & & & \\ * & * & \cdots & * & {\begin{matrix}}A_0 & B_0\\ -B_0 & A_0 {\end{matrix}}& & &\\
* & * & \cdots & * & * & A_{1} & &\\ \vdots & \vdots & \cdots & \vdots & \vdots & \vdots & \ddots &\\ * & * & \cdots & * & * & * & \cdots & A_{k-1}\\ * & * & \cdots & * & * & * & \cdots & * & A_k {\end{pmatrix}}$$
where each $A_j$ is a square matrix with dimension $\operatorname{dim}(E_j)$, which is equal to the multiplicity of the eigenvalue $\lambda_j$ of $X$. Each $*$ is an arbitrary block matrix of the appropriate dimensions, and the (blank) upper diagonal entries are all zero. The central block ${\begin{pmatrix}}A_0 & B_0\\ -B_0 & A_0 {\end{pmatrix}}$ describes the $J$-linear transformations of the $0$-eigenspace $E_0$ of $X$; this block only appears if the root $2 d_m \in I$. The group $\operatorname{P}^{-1}( Z_H({\mathfrak{b}}_I))$ is the subgroup for which all $*$ blocks are zero. The group $\operatorname{P}^{-1} N_I \cong N_I$ is the subgroup for which all diagonal blocks $A_j$ are the identity (and $B_0 = 0$).
In the case that $D = \operatorname{diag}(1,1,\ldots,1)$, we have only two eigenspaces $E_1$ and $JE_1$. The limit group is then the centralizer of the matrix ${\begin{pmatrix}}0 & 0 \\ I_m & 0 {\end{pmatrix}}$ which we should think of as a degeneration of the complex structure. The limit group is isomorphic to the general linear group in dimension $m$ over the ring $\RR[\epsilon]/(\epsilon^2)$.
Limits of ${\operatorname{Sp}}(2m)$ in $\GL_{2m} \RR$
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Consider $H = {\operatorname{Sp}}(2m, \RR)$ inside of $\GL_{2m} \RR$. Note that $H \subset \SL_{2m} \RR$. To determine the conjugacy classes of limits of $H$ in $\GL_{2m} \RR$ it suffices to determine the limits of $H$ in $G = \SL_{2m} \RR$. We apply Theorem \[thm:limit\_nice\]. The defining involution for $H$, at the Lie algebra level, is $\tau(X) = \sigma(\theta(X)) = -JX^TJ^{-1}$ where $\theta$ is the standard Cartan involution, and $\sigma$ is conjugation by a complex structure $J$ fixed by $\theta$. In this case, it will be more convenient to take $$J = {\begin{pmatrix}}J_0 &0 &\cdots & 0 \\ 0 & J_0 & & 0\\ 0 & 0 & \ddots & & \\ 0 & 0 & \cdots & J_0 {\end{pmatrix}},$$ where $J_0 = {\begin{pmatrix}}0 & 1\\ -1 & 0{\end{pmatrix}}$. The $-1$ eigenspace ${\mathfrak{q}}$ of $\tau$ is given by matrices of the form $(A_{jk})_{j,k=1}^m$ where each $A_{jk}$ is a $2 \times 2$ block with $A_{jk} = -J_0 A_{kj}^T J_0$. The diagonal blocks of elements of ${\mathfrak{q}}$ have the form $A_{jj} = {\begin{pmatrix}}d_j & 0\\ 0 & d_j {\end{pmatrix}}$. A maximal abelian sub-algebra ${\mathfrak{b}}$ of ${\mathfrak{p}}\cap {\mathfrak{q}}$ is given by matrices of the form $$D = {\begin{pmatrix}}D_1 &0 &\cdots & 0 \\ 0 & D_2 & & 0\\ 0 & 0 & \ddots & & \\ 0 & 0 & \cdots & D_m {\end{pmatrix}},$$ where $D_j = {\begin{pmatrix}}d_j & 0\\ 0 & d_j {\end{pmatrix}}$ and $d_1 + \cdots + d_m = 0$. The system of positive simple roots can be chosen to be $$\Delta = \left\{ d_{i+1} - d_{i} \right\}_{i=1}^{m-1}.$$ In this case the inclusion $W_{H \cap K} \hookrightarrow W$ is an isomorphism; both Weyl groups simply permute the block diagonal entries of $D$. Therefore, $\mathcal W = \{1\}$ and $\overline{{\mathfrak{b}}^+}$ is the collection of diagonal matrices $X$ as above, where $d_1 \leq \ldots \leq d_m$. Then, by Theorem \[thm:limit\_nice\], the conjugacy classes of limits of $H$ in $G$ are enumerated by subsets $I \subset \Delta$. For a given subset $I \subset \Delta$, the corresponding limit group $$L_I = Z_H({\mathfrak{b}}_I) \ltimes N_I = Z_H(X) \ltimes N_+(X)$$ is the limit under conjugacy by $\exp(t X)$ as $t \to \infty$, where $X \in {\mathfrak{b}}_I^{+}$. Here, $Z_H({\mathfrak{b}}_I) = Z_H(X)$ has block form: $$Z_H({\mathfrak{b}}_I) = {\begin{pmatrix}}{\operatorname{Sp}}(2 m_1) & 0 & \cdots & 0 \\ 0 & {\operatorname{Sp}}(2 m_2) & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & {\operatorname{Sp}}(2m_k) {\end{pmatrix}},$$ where each block ${\operatorname{Sp}}(2m_j)$ is the symplectic group of dimension $2 m_j$ consisting of those elements of ${\operatorname{Sp}}(2m)$ which preserve the $j^{th}$ eigenspace of $X$ and act as the identity on the other eigenspaces of $X$. The flag ${\mathcal F}$ preserved by $L$ is given by $V_0 \supset \ldots \supset V_k$, where $V_j = E_j \oplus \cdots \oplus E_k$ is the direct sum of the last $k-j+1$ eigenspaces $E_i$ of $X$, where the $E_i$ are indexed in order of increasing eigenvalue. The flag unipotent subgroup $N_I = U({\mathcal F})$ (see Section \[sec:symmetric-PGL\]) has block structure: $$N_I = {\begin{pmatrix}}I_{m_1}& 0 & \cdots & 0 \\ * & I_{m_2} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ * & * & \cdots & I_{m_k} {\end{pmatrix}},$$ where all lower diagonal blocks are labeled $*$ to denote that the entries are arbitrary.
Limits of $\GL_p \oplus \GL_q$ in $\GL_{p+q} \RR$
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Consider the involution $\sigma$ defined by $\sigma(X) = JXJ^{-1}$ where $J = {\begin{pmatrix}}-I_p & 0 \\ 0 & I_q {\end{pmatrix}}.$ We assume without loss of generality that $q \geq p$ and set $r = q -p$. The fixed set of $\sigma$ naturally identifies with $\GL_p \RR \oplus \GL_q \RR$. Note that the center $Z(\GL_{p+q})$ is contained in $\GL_p \oplus \GL_q$. Therefore the involution $\sigma$ is well-defined on the quotient ${\operatorname{PGL}}_{p+q} \RR$. The image $H = \operatorname{P}(\GL_p \oplus \GL_q)$ under the projection $\operatorname{P}: \GL_{p+q} \to {\operatorname{PGL}}_{p+q}$ is therefore a symmetric subgroup of ${\operatorname{PGL}}_{p+q} \RR$ and we may apply Theorem \[thm:limit\_nice\] to determine the limit groups $L$ of $H$. Then any limit of $\GL_p \RR \oplus \GL_q \RR$ in $\GL_{p+q} \RR$ is conjugate to $\operatorname{P}^{-1} L$ where $L$ is some limit of $H$.
Note that $\sigma$ commutes with the standard Cartan involution $\theta$. The $-1$ eigenspace ${\mathfrak{q}}$ of $\sigma$ is given by matrices of the form $${\begin{pmatrix}}0_{p \times p} & B \\ C & 0_{q \times q} {\end{pmatrix}}$$ where $B$ is $p \times q$ and $C$ is $q \times p$. A maximal abelian sub-algebra ${\mathfrak{b}}$ of ${\mathfrak{p}}\cap {\mathfrak{q}}$ is given by matrices of the form $$X = {\begin{pmatrix}}0_{p \times p} & D & \\ D & 0_{p\times p} & \\ & & 0_{r \times r} {\end{pmatrix}}$$ where $D = \operatorname{diag}(d_1, \ldots, d_p)$ is a $p \times p$ diagonal matrix. The system of positive simple roots can be chosen to be $$\Delta = \left\{ d_{i+1} - d_{i} \right\}_{i=1}^{p-1} \cup \{ 2 d_p \}.$$ In this case the inclusion $W_{H \cap K} \hookrightarrow W$ is an isomorphism; both Weyl groups simply permute the diagonal entries of $D$ and also the signs. Therefore, $\mathcal W = \{1\}$ and $\overline{{\mathfrak{b}}^{+}}$ is the collection of matrices $X$ as above, where $0 \leq d_1 \leq \ldots \leq d_p$. Then, by Theorem \[thm:limit\_nice\], the conjugacy classes of limits of $H$ in $G$ are enumerated by subsets $I \subset \Delta$. For a given subset $I \subset \Delta$, the corresponding limit group $$L_I = Z_H({\mathfrak{b}}_I) \ltimes N_I = Z_H(X) \ltimes N_+(X)$$ is the limit under conjugacy by $\exp(t X)$ as $t \to \infty$, where $X \in {\mathfrak{b}}_I^{+}$.
Let’s see more explicitly what this group $L_I$ looks like. The eigenvalues of $X$ are $$-\lambda_k < -\lambda_{k-1} < \cdots < \lambda_0 = 0 < \lambda_1 < \cdots < \lambda_k$$ where if $j \neq 0$, $\lambda_j = 2 d_{i_j}$ is twice one of the diagonal elements of $D$. The multiplicity $m_j$ of $\lambda_j$ is determined by the subset $I$. The eigenvalue $\lambda_0 = 0$ has multiplicity equal to $r + 2m_0$ where $m_0$ is the number of the $d_i$ which are zero. Note also that $E_{\lambda_j} = J E_{-\lambda_j}$. Hence an element $h \in Z_H({\mathfrak{b}}_I)$ preserves both $E_{\lambda_j}$ and $E_{-\lambda_j}$ and has identical matrix on both subspaces, when the basis for $E_{-\lambda_j}$ is taken to be $J$ times the basis for $E_{\lambda_j}$; we work in such a basis. Also, the zero eigenspace $E_0$ is invariant under $J$; the elements of $H$ which preserve $E_0$ form a copy of $\GL(r+m_0)\oplus \GL(m_0)$.
Next, the flag ${\mathcal F}$ defining the unipotent part $U({\mathcal F}) = N_I = N_+(X)$ of $L$ (see Section \[sec:symmetric-PGL\]) is given by the sub-spaces $V_{-k} \supset V_{-(k-1)} \supset \cdots V_0 \supset V_{1} \supset \cdots \supset V_{k}$ where $V_j = E_{\lambda_j} \oplus \cdots \oplus E_{\lambda_{k}},$ and where $\lambda_{-j} := -\lambda_j$. Therefore $N_I = U({\mathcal F})$ is the unipotent group which is block lower diagonal in a basis respecting the ordered decomposition $\RR^n = E_{-\lambda_k} \oplus \cdots \oplus E_0 \oplus \cdots \oplus E_{\lambda_k}$ into eigenspaces. Therefore, in such a basis, the elements of the corresponding limit $\operatorname{P}^{-1} L_I$ of $\GL_p \oplus \GL_q$ in $\GL_{p+q}$ have matrix form
$${\begin{pmatrix}}A_k & & & & & & & &\\ * & A_{k-1} & & & & & & & \\ \vdots & \vdots & \ddots & & & & & & \\ * & * & \cdots & A_{1} & & & & & \\ * & * & \cdots & * & {\begin{matrix}}A_0 & 0\\ 0 & B_0 {\end{matrix}}& & &\\
* & * & \cdots & * & * & A_{1} & &\\ \vdots & \vdots & \cdots & \vdots & \vdots & \vdots & \ddots &\\ * & * & \cdots & * & * & * & \cdots & A_{k-1}\\ * & * & \cdots & * & * & * & \cdots & * & A_k {\end{pmatrix}}$$
where for $j \neq 0$, the matrix $A_j$ is the square $m_j \times m_j$ matrix representing both the action on $E_{\lambda_j}$ and on $E_{-\lambda_j} = J E_{\lambda_j}$, and the block matrix ${\begin{pmatrix}}A_0 & 0 \\ 0 & B_0 {\end{pmatrix}}$ represents an element of $\GL(r+m_0)\oplus \GL(m_0)$ corresponding to the action on $E_0$. As always, the $*$ blocks are arbitrary.
Limits of ${\operatorname{O}}(p,q)$ in $\GL_{p+q}\RR$ {#sec:Opq}
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Set $p+q = n$. Let $\tau$ be the involution of $\GL_{p+q}$ defined by $\tau(g) = Jg^{-T}J^{-1}$ where $J$ is the matrix $$J = {\begin{pmatrix}}-I_p & \\ & I_q{\end{pmatrix}}.$$ The group ${\operatorname{O}}(p,q)$ is the fixed point set of $\tau$. Consider the image $\PO(p,q)$ of ${\operatorname{O}}(p,q)$ under the projection $\operatorname{P}: \GL_{n} \RR \to {\operatorname{PGL}}_{n} \RR$. To determine the limits of ${\operatorname{O}}(p,q)$ in $\GL_{n} \RR$, it suffices to determine the limits of $H = \PO(p,q)$ in $G = {\operatorname{PGL}}_{n} \RR$. For, ${\operatorname{O}}(p,q)$ is the intersection of $\operatorname{P}^{-1}( \PO(p,q))$ with the subgroup $\SL^\pm$ of matrices of determinant $\pm 1$. Therefore, all limits of ${\operatorname{O}}(p,q)$ are of the form $\operatorname{P}^{-1}(L) \cap \SL^\pm$ where $L$ is a limit of $H$ in $G$. Note also that $\tau$ preserves the center $Z(\GL_{n})$, therefore $\tau$ descends to a well-defined involution of the projective general linear group; its fixed point set is exactly $H = \PO(p,q)$. So we may apply Theorem \[thm:limit\_nice\] to determine the limits of $H$.
At the Lie algebra level, our involution has the form $\tau(X) = -JX^TJ^{-1}.$ A maximal abelian sub-algebra ${\mathfrak{b}}$ of ${\mathfrak{p}}\cap {\mathfrak{q}}$ is in this case given by the full Cartan sub-algebra ${\mathfrak{a}}$ of traceless diagonal matrices $$D = {\begin{pmatrix}}d_1 & & & \\ & d_2 & & \\ & & \ddots & \\ & & & d_n {\end{pmatrix}}.$$ The system of positive simple roots can be chosen to be $$\Delta = \left\{ d_{i+1} - d_{i} \right\}_{i=1}^{n-1}.$$ In this case, the Weyl group $W$ is the full symmetric group $S_{p+q}$ permuting the standard basis of $\RR^{p+q}$. The closed Weyl chamber $\overline{{\mathfrak{b}}^+}$ corresponding to $\Delta$ consists of the diagonal matrices $X$, as above, such that $d_1 \leq d_2 \leq \cdots \leq d_{n}$. The Weyl group $W_{H \cap K}$ for $\Sigma({\mathfrak{g}}_{\tau \theta}, {\mathfrak{b}})$ is given by the permutations $S_p \times S_q$ of the standard basis which preserve the signature; in other words $W_{H \cap K}$ permutes the first $p$ basis vectors and the last $q$ coordinate directions independently. A closed Weyl chamber $\mathcal W \cdot \overline{{\mathfrak{b}}^+}$ for $\Sigma({\mathfrak{g}}_{\tau \theta}, {\mathfrak{b}})$ is given by the diagonal matrices $\operatorname{diag}(d_1, \ldots, d_n)$ such that $d_i \leq d_{i+1}$ if $i = 1,\ldots, p-1$ or if $i = p+1,\ldots, p+q = n$. Then $\mathcal W$ consists of permutations $\varpi$ of the following form. For some $1 \leq k \leq p$ (assuming $p \leq q$), and two sets of $k$ indices $1 \leq i_1 < \cdots < i_k \leq p$ and $p+1 \leq j_1 < \cdots < j_k \leq p+q = n$, we have that $\varpi(i_r) = p+r$ and $\varpi(j_r) = p -k +r$ and the remaining $p-k$ indices between $1$ and $p$ are mapped in order to the first (smallest) $p-k$ indices, while the remaining $q -k$ indices between $p+1$ and $n$ are mapped in order to last (largest) $q-k$ indices. In fact, this specific form of $\mathcal W$ is not important; we may work with any collection $\mathcal W$ of coset representatives of $W/W_{H \cap K}$.
By Theorem \[thm:limit\_nice\], the conjugacy classes of limits of $H$ in $G$ are enumerated (with redundancy) by subsets $I \subset \Delta$ and elements $w \in \mathcal W$. For a given subset $I \subset \Delta$ and $w \in \mathcal W$, the corresponding limit group $$L_{I ,w}= Z_{H_w}({\mathfrak{b}}_I) \ltimes N_I = Z_{H_w}(X) \times N_+(X)$$ is the limit of $H_w = wHw^{-1}$ under conjugacy by $\exp(t X)$ as $t \to \infty$, where $X \in {\mathfrak{b}}_I^{+}$. Let $E_0, E_1, \ldots, E_k$ be the eigenspaces of $X$ listed in order of increasing eigenvalue. In our chosen coordinates, each $E_i$ is a span of consecutive coordinate directions. Now, $H_w$ is the fixed point set of the involution $\tau_w$ defined by $\tau_w(g) = J_w g^{-T} J_w^{-1}$, where $J_w = w J w^{-1}$ is a diagonal form of signature $(p,q)$ but with the $p$ $(-1)$’s and $q$ $(+1)$’s arranged in a (possibly) different order. Let $(p_i, q_i)$ denote the signature of $J_w$ when restricted to $E_i$. Then $p_1 + \cdots + p_k = p$ and $q_1 + \cdots + q_k = q$, and $Z_{H_w}({\mathfrak{b}}_I)$ is seen to have the block diagonal form $$Z_{H_w}({\mathfrak{b}}_I) = \operatorname{P} {\begin{pmatrix}}{\operatorname{O}}(p_1,q_1) & & & \\ & {\operatorname{O}}(p_2,q_2) & & \\ & & \ \ \ddots \ \ & \\ & & & {\operatorname{O}}(p_k,q_k) {\end{pmatrix}}.$$ The full limit group $L_{I,w} \subset {\operatorname{PGL}}({\mathcal F})$ preserves the flag ${\mathcal F}$ consisting of subspaces $V_0 \supset \cdots \supset V_k$, where $V_j = E_j \oplus \cdots \oplus E_k$. The unipotent part $N_I = U({\mathcal F})$ has the form $$N_I = \operatorname{P} {\begin{pmatrix}}I_{p_1+q_1} & & & \\ * & I_{p_2+q_2} & & \\ \vdots & \vdots & \ddots & \\ * & * & \cdots & I_{p_k+q_k} {\end{pmatrix}}$$ where the upper diagonal blocks, denoted by $*$, are arbitrary.
\[rem:SOpq-redundancy\] In this example, we may see explicitly that the finer enumeration of limit groups described by Remark \[rem:finer-list\] may still have redundancy. For consider the case $p=2$, $q=2$, and consider $I = \{ d_2 - d_1, d_4 - d_3\}$. Then $W_I$ consists of four permutations, namely the group generated by transposing the first and second basis vector and transposing the third and fourth basis vector; so that $W_I = W_{H \cap K}$. One easily computes that $| W_I \backslash W / W_{H \cap K}| = 3$; representatives are given by the permutation $w_1$ that transposes the second and third basis vector, a permutation $w_2$ that exchanges the first and second basis vectors with the third and fourth, and the identity permutation $e$. However, there are only two conjugacy classes of limit group, because $L_{I,w_2} = L_{I,e}$.
### Partial flag of quadratic forms
Finally, we note that if $L_{I,w}$ is as above, then the corresponding limit groups $\operatorname{P}^{-1}(L_{I,w}) \cap \SL^\pm$ of ${\operatorname{O}}(p,q)$ in $\GL_n \RR$ are easily described. We will investigate these limit groups and their corresponding geometries in depth in the next section. Here we introduce some notation to give an invariant description of the limit groups. Let ${\mathcal F}$ be the partial flag formed the chain of subspaces $V_0 \supset V_1 \supset \ldots \supset V_k$. A [*partial flag of quadratic forms*]{} ${{\boldsymbol \beta}}=(\beta_0,\cdots,\beta_k)$ on the partial flag ${\mathcal F}$ is a collection of quadratic forms $\beta_i$ defined on each quotient $V_i/V_{i+1}$ of the partial flag ${\mathcal F}$. We denote the linear transformations which preserve ${\mathcal F}$ and induce an isometry of each $\beta_i$ by $\Isom({{\boldsymbol \beta}},{{\mathcal F}})$. The [*signature*]{} of a non-degenerate quadratic form $\beta$ is $\eps(\beta)=(n_-,n_+)$, where $n_-$ (resp. $n_+$) is the dimension of the largest subspace on which $\beta$ is negative (resp. positive) definite. Two quadratic forms have the same isometry group iff they are scalar multiples of each other thus ${\operatorname{O}}(p,q)\cong {\operatorname{O}}(p',q')$ iff $\{p,q\}=\{p',q'\}$. The [*signature*]{} of a partial flag of quadratic forms ${{\boldsymbol \beta}}=(\beta_0,\cdots,\beta_k)$ is $$\eps({{\boldsymbol \beta}})=(\eps(\beta_0),\cdots,\eps(\beta_k))=((p_0,q_0)\cdots (p_k,q_k)).$$ The signature $\eps({{\boldsymbol \beta}})$ determines $\Isom ({{\boldsymbol \beta}}, {\mathcal F})$ up to conjugation. When ${\mathcal F}$ is adapted to the standard basis and all $\beta_i$ are diagonal, we will use the notation $$\begin{aligned}
\Isom({{\boldsymbol \beta}},{\mathcal F}) &=: {\operatorname{O}}((p_0,q_0),\cdots,(p_k,q_k))\\ &= {\begin{pmatrix}}{\operatorname{O}}(p_0,q_0) & & \\ & \ \ \ddots \ \ & \\ & & {\operatorname{O}}(p_k,q_k) {\end{pmatrix}}\ltimes {\begin{pmatrix}}I_{p_0+q_0} & & & \\ * & I_{p_1+q_1} & & \\ \vdots & \vdots & \ddots & \\ * & * & \cdots & I_{p_k+q_k} {\end{pmatrix}}.\end{aligned}$$ The conjugacy class of this group is unchanged by scaling some $\beta_i$. In fact ${\operatorname{O}}((p_0,q_0),\cdots,(p_k,q_k))$ is conjugate to ${\operatorname{O}}((p_0',q_0'),\cdots,(p_k',q_k'))$ if and only if for all $i =0,\ldots,k$, $(p_i,q_i) = (p_i',q_i')$ or $(p_i,q_i) = (q_i',p_i')$. As a special case observe that when ${\mathcal F}$ is a full flag, then $\Isom({{\boldsymbol \beta}}, {\mathcal F})$ is conjugate to ${\operatorname{O}}((1,0),\cdots,(1,0))$, which is the group of lower triangular matrices with diagonal entries $\pm1$. We will adopt the convention that the signature $(p,0)$ can be denoted by $(p)$ so this group is also written as ${\operatorname{O}}((1),(1),\cdots,(1))$. This is in agreement with denoting ${\operatorname{O}}(n,0)$ by ${\operatorname{O}}(n)$. The application of Theorem \[thm:limit\_nice\] above shows:
\[thm:limits-Opq\] The limits of ${\operatorname{O}}(p,q)$ (resp. $\PO(p,q)$) inside of $\GL_{p+q} \RR$ are all of the form $\Isom({{\boldsymbol \beta}}, {\mathcal F})$ (resp. ${\operatorname{P}}\Isom({{\boldsymbol \beta}}, {\mathcal F})$). Further $\Isom({\mathcal F}, {{\boldsymbol \beta}})$ (resp. ${\operatorname{P}}\Isom({{\boldsymbol \beta}}, {\mathcal F})$) is a limit of ${\operatorname{O}}(p,q)$ (resp. $\PO(p,q)$) if and only if the signature $((p_0,q_0),\ldots,(p_k,q_k))$ of ${{\boldsymbol \beta}}$ satisfies $$p_0 + \cdots + p_k = p \ \text{ and } \ q_0 + \cdots + q_k = q,$$ after exchanging $(p_i , q_i)$ with $(q_i,p_i)$ for some collection of indices $i$ in $\{0,\ldots,k\}$.
The groups ${\operatorname{P}}\Isom({{\boldsymbol \beta}}, {\mathcal F})$ are the structure groups for many interesting geometries, to be described in Section \[sec:pfqf\]. As a corollary to Theorem \[thm:limits-Opq\], we characterize all limits of these groups.
\[cor:limits-pfqf\] Every conjugacy limit of $\Isom({\mathcal F}, {{\boldsymbol \beta}})$ (resp. ${\operatorname{P}}\Isom({\mathcal F}, {{\boldsymbol \beta}})$) is of the form $\Isom({\mathcal F}', {{\boldsymbol \beta}}')$ (resp. ${\operatorname{P}}\Isom({\mathcal F}', {{\boldsymbol \beta}}')$). Further, up to conjugation, the flag ${\mathcal F}' = \{V'_j\}$ is a refinement of ${\mathcal F}= \{ V_i\}$ and the signature $\eps({{\boldsymbol \beta}}')$ is a refinement of the signature $\eps({{\boldsymbol \beta}})$ in the sense that $$\begin{aligned}
\label{eqn:signature-condition}
p_i &= \sum_{j: V_i \supset V_j' \supsetneq V_{i+1}} p_j' & q_i = \sum_{j: V_i \supset V_j' \supsetneq V_{i+1}} q_j'\end{aligned}$$ after exchanging $(p_j', q_j')$ with $(q_j', p_j')$ for some collection of indices $j$. Any $\Isom({\mathcal F}', {{\boldsymbol \beta}}')$ as above is realized as a limit of $\Isom({\mathcal F}, {{\boldsymbol \beta}})$ under some sequence of conjugacies.
Let $H = \Isom({\mathcal F}, {{\boldsymbol \beta}})$. Consider a conjugacy limit $L = \lim_{n \to \infty} c_n H c_n^{-1}$. The space of flags having the same type as ${\mathcal F}$ is compact. Thus, we may assume that $c_n \in {\operatorname{PGL}}({\mathcal F})$ for all $n$, and therefore that $L \subset \GL({\mathcal F})$. Note that $U({\mathcal F})$ is preserved by conjugation by $c_n$. Therefore $U({\mathcal F}) \subset L$. It remains to determine the projections $\pi_i(L)$ where $\pi_i: \GL({\mathcal F}) \to \GL(V_i/V_{i+1})$ is the natural projection map. Now, $\pi_i(H) = \Isom(\beta_i) \cong {\operatorname{O}}(p_i,q_i)$. The projection $\pi_i(L)$ is the limit of $\pi_i(H)$ under conjugation by the projections $\pi_i(c_n)$. Hence, Theorem \[thm:limits-Opq\] implies that $\pi_i(L) = \Isom({\mathcal F}^{(i)}, {{\boldsymbol \beta}}^{(i)})$, where $({\mathcal F}^{(i)}, {{\boldsymbol \beta}}^{(i)})$ is a partial flag of quadratic forms for $V_i/V_{i+1}$. Then, let ${\mathcal F}'$ be the flag of all lifts $\pi_i^{-1}(V^{(i)}_j)$ of subspaces $V^{(i)}_j$ of each flag ${\mathcal F}^{(i)}$. Let ${{\boldsymbol \beta}}'$ be the flag of quadratic forms $\pi_i^*\beta^{(i)}_j$ on those subspaces determined by pullback. Then, $L = \Isom({\mathcal F}', {{\boldsymbol \beta}}')$ is as in the statement of the Corollary.
Next, to see that any $\Isom({{\boldsymbol \beta}}')$ as in the Corollary is achieved as a limit, note that if the condition (\[eqn:signature-condition\]) is satisfied, then both $\Isom({{\boldsymbol \beta}})$ and $\Isom({{\boldsymbol \beta}}')$ are limits of some ${\operatorname{O}}(p,q)$. In fact, the elements $X,X' \in {\mathfrak{b}}$ determining the respective limits $\Isom({{\boldsymbol \beta}})$ and $\Isom({{\boldsymbol \beta}}')$ of ${\operatorname{O}}(p,q)$ have the property that any eigenspace of $X'$ is contained in an eigenspace of $X$. Further if $E_{\lambda_1}', E_{\lambda_2}'$ are eigenspaces of $X'$ corresponding to eigenvalues $\lambda_1 < \lambda_2$, then $E_{\lambda_1}' \subset E_{\mu_1}$ and $E_{\lambda_2}' \subset E_{\mu_2}$, where $E_{\mu_1}, E_{\mu_2}$ are eigenspaces of $X$ corresponding to eigenvalues $\mu_1 \leq \mu_2$. It is then easy to see that $$\lim_{t \to \infty} \exp(t X') \Isom({{\boldsymbol \beta}}) \exp(-t X') = \Isom({{\boldsymbol \beta}}').$$
The geometry of a partial flag of quadratic forms {#sec:pfqf}
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In Section \[sec:Opq\], we showed that the geometric limits of ${\operatorname{O}}(p,q)$ inside of $\GL_{p+q} \RR$ are the groups $\Isom({{\boldsymbol \beta}}, {\mathcal F})$ preserving a partial flag of quadratic forms. In this section we will investigate the corresponding limit geometries.
$\XX(p,q)$ geometry and its limits {#sec:Hpq}
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Let $\beta$ denote a quadratic form on $\RR^n$ of signature $(p,q)$. We assume that $p > 0$. Then $\operatorname{P}\Isom(\beta) \cong \PO(p,q)$ acts transitively on the space $$\XX(p,q) := \{ [x] \in \RP^{n-1} : \beta(x) < 0 \}$$ with stabilizer isomorphic to ${\operatorname{O}}(p-1,q)$. Therefore $\XX(p,q)$ is a semi-Riemannian space of dimension $n-1 = p+q-1$ and signature $(p-1, q)$. We list some familiar cases:
- $(\XX(n,0), \PO(n))$ is doubly covered by spherical geometry $\mathbb S^{n-1}$.
- $(\XX(1,n-1), \PO(1,n-1))$ is the projective model for hyperbolic geometry $\HH^{n-1}$.
- $(\XX(2,n-2), \PO(2,n-2))$ is the projective model for anti de Sitter (AdS) geometry $\AdS^{n-1}$.
- $(\XX(n-1,1), \PO(n-1,1))$ is the projective model for de Sitter (dS) geometry $\dS^{n-1}$.
We now describe the possible limits of $(\XX(p,q), \PO(p,q))$ as a sub-geometry of $(\RP^{n-1}, {\operatorname{PGL}}_n\RR)$. Consider a partial flag ${\mathcal F}$ equipped with a flag of quadratic forms ${{\boldsymbol \beta}}= (\beta_0, \ldots, \beta_k)$ as in Section \[sec:Opq\]. Let $(p_i,q_i)$ be the signature of $\beta_i$. Define the domain $\XX({{\boldsymbol \beta}}) \subset \RP^{n-1}$ by $$\XX({{\boldsymbol \beta}}) := \{ [x] \in\RP^{n-1} : \beta_0(x) < 0\}.$$ Then ${\operatorname{P}}\Isom({\mathcal F}, {{\boldsymbol \beta}})$ acts transitively on $\XX({{\boldsymbol \beta}})$. When the flag and quadratic forms are adapted the standard basis, we denote $\XX({{\boldsymbol \beta}})$ by $$\label{PFG-notation}
\XX({{\boldsymbol \beta}}) = \XX((p_0,q_0),\ldots,(p_k,q_k)).$$ Note that $\XX({{\boldsymbol \beta}})$ is non-empty if and only if $p_0 > 0$ and that as a set, the space $\XX((p_0,q_0)\ldots (p_k,q_k))$ depends only on the first signature $(p_0,q_0)$ and the dimension $n = \sum_i (p_i + q_i)$. However, we include all $k$ signatures in the notation as a reminder of the structure determined by $\PO((p_0,q_0),\ldots,(p_k,q_k))$.
\[thm:limits-Hpq\] The conjugacy limits of $(\XX(p,q), \PO(p,q))$ inside $(\RP^{n-1}, {\operatorname{PGL}}_n)$ are all of the form $(\XX({{\boldsymbol \beta}}), {\operatorname{P}}\Isom({\mathcal F}, {{\boldsymbol \beta}}))$. Further, $\XX({{\boldsymbol \beta}})$ is a limit of $\XX(p,q)$ if and only if $p_0 \neq 0$, and the signatures $((p_0,q_0),\ldots,(p_k,q_k))$ of ${{\boldsymbol \beta}}$ partition the signature $(p,q)$ in the sense that $$p_0 + \cdots + p_k = p \ \ \text{ and } \ \ q_0 + \cdots + q_k = q,$$ after exchanging $(p_i, q_i)$ with $(q_i, p_i)$ for some collection of indices $i$ in $\{1,\ldots,k\}$ (the first signature $(p_0,q_0)$ must *not* be reversed).
More generally:
\[thm:limits-Xbeta\] Every conjugacy limit of $(\XX({{\boldsymbol \beta}}), {\operatorname{P}}\Isom({\mathcal F}, {{\boldsymbol \beta}}))$ is of the form $(\XX({{\boldsymbol \beta}}'), {\operatorname{P}}\Isom({\mathcal F}', {{\boldsymbol \beta}}'))$. Further, up to conjugation, the flag ${\mathcal F}' = \{V'_j\}$ is a refinement of ${\mathcal F}= \{ V_i\}$ and the signature $\eps({{\boldsymbol \beta}}')$ is a refinement of the signature $\eps({{\boldsymbol \beta}})$ in the sense that $$\begin{aligned}
p_i &= \sum_{j: V_i \supset V_j' \supsetneq V_{i+1}} p_j' & q_i = \sum_{j: V_i \supset V_j' \supsetneq V_{i+1}} q_j'\end{aligned}$$ after exchanging $(p_j', q_j')$ with $(q_j', p_j')$ for some collection of indices $j$ excluding $j=0$ (the first signature $(p_0',q_0')$ must not be reversed). Any such geometry $(\XX({{\boldsymbol \beta}}'), {\operatorname{P}}\Isom({\mathcal F}', {{\boldsymbol \beta}}'))$ is realized as a limit of $(\XX({{\boldsymbol \beta}}), {\operatorname{P}}\Isom({\mathcal F}, {{\boldsymbol \beta}}))$ under some sequence of conjugacies (provided $p_0' > 0$).
Let $L = {\operatorname{P}}\Isom({{\boldsymbol \beta}}', {\mathcal F}')$ be a limit of ${\operatorname{P}}\Isom({{\boldsymbol \beta}}, {\mathcal F})$ under some conjugating sequence $(c_n)$ as in Corollary \[cor:limits-pfqf\]. Suppose that $(\XX({{\boldsymbol \beta}}), {\operatorname{P}}\Isom({{\boldsymbol \beta}}, {\mathcal F}))$ limits, under conjugation by $(c_n)$ to $(Y,L)$ (in the sense of Definition \[def:limit-subgeom\]). Then $Y \subset \RP^n$ is an open orbit of $L$. There are at most two such orbits. $\XX({{\boldsymbol \beta}}')$ is an open orbit of $L$, non-empty if and only if $p_0' > 0$. The set of positive lines, $\XX^+$, with respect to $\beta_0'$ is the other open orbit of $L$, non-empty if and only if $q_0' > 0$. Let us now show that $Y = \XX({{\boldsymbol \beta}}')$.
By definition, there is some $y_\infty \in Y$ such that for all $n$ sufficiently large, $y_\infty \in c_n \XX({{\boldsymbol \beta}})$; in other words, $\beta_0(c_n^{-1} y_\infty) < 0$. It is easy to see, from the proof of Theorem \[thm:symmetric-limits\], that $\beta_0'$ is the limit of $\lambda_n c_n^* \beta_0$ where $\lambda_n > 0$ is a sequence of positive scalars. Therefore, $$\beta_0'(y_\infty) = \lim_{n \to \infty} \lambda_n \beta_0(c_n^{-1} y_\infty) \leq 0.$$ It follows that $Y\neq \XX^+$, so we must have $Y = \XX({{\boldsymbol \beta}}')$ as desired.
Next we show that any $\XX({{\boldsymbol \beta}}')$ as in the Theorem is achieved. As in the proof of Corollary \[cor:limits-pfqf\], we note that $\XX({{\boldsymbol \beta}}')$ and $\XX({{\boldsymbol \beta}})$ are both limits of some $\XX(\beta) = \XX(p,q)$ under conjugation by the one parameter groups $\exp(t X)$ resp. $ \exp(t X')$ and that the groups satisfy ${\operatorname{P}}\Isom({{\boldsymbol \beta}}') = \lim_{t \to \infty} \exp(t X') {\operatorname{P}}\Isom({{\boldsymbol \beta}}) \exp(-t X')$. Further, every eigen-space of $X'$ is contained in an eigenspace of $X$ and the eigenspace $E_0'$ corresponding to the smallest eigenvalue of $X'$ is contained in the eigenspace $E_0$ corresponding to the smallest eigenvalue of $X$. Then $\beta_0'$ agrees with $\beta_0$ on $E_0'$ and is zero on all other eigenspaces of $X'$. Let $y_\infty \in \XX({{\boldsymbol \beta}}') \cap \mathbb P E_0'$. Then, since $\beta_0'(y_\infty) < 0$ we have that $\beta_0(y_\infty) < 0$ and so $y_\infty \in \XX({{\boldsymbol \beta}})$. Since $\exp(t X') y_\infty = y_\infty$, we have shown that the conjugate sub-geometries $(\exp(t X') \XX({{\boldsymbol \beta}}), \exp(t X') {\operatorname{P}}\Isom({{\boldsymbol \beta}}) \exp(-t X'))$ limit to the sub-geometry $(\XX({{\boldsymbol \beta}}'), {\operatorname{P}}\Isom({{\boldsymbol \beta}}'))$.
The geometry of $\XX({{\boldsymbol \beta}})$
--------------------------------------------
Let us now describe the geometry of $(\XX({{\boldsymbol \beta}}), {\operatorname{P}}\Isom({\mathcal F}, {{\boldsymbol \beta}}))$. We assume that the flag ${\mathcal F}$ and quadratic forms ${{\boldsymbol \beta}}$ are adapted to the standard basis so that we may use the notation (\[PFG-notation\]). Let $j \in \{1,\ldots,k\}$. The action of $\Isom({\mathcal F}, {{\boldsymbol \beta}})$ preserves the flag of quotient spaces $$V_0/V_j \supset V_1/V_j \supset \cdots \supset V_{j-1}/V_j,$$ which we denote ${\mathcal F}/V_j$, as well as the induced flag of quadratic forms $\beta_0, \ldots, \beta_{j-1}$, which we denote by ${{\boldsymbol \beta}}/V_j$. Then $\XX({{\boldsymbol \beta}}/V_j)$ identifies with $\XX((p_0,q_0)\ldots(p_{j-1},q_{j-1}))$ and $\Isom({\mathcal F}, {{\boldsymbol \beta}})$ acts on $\XX({{\boldsymbol \beta}}/V_j)$ by transformations of $\Isom({\mathcal F}/V_j, {{\boldsymbol \beta}}/V_j) \cong {\operatorname{O}}((p_0,q_0),\ldots,(p_{j-1},q_{j-1}))$.
There is a projection map $\pi_{j}$: $$\xymatrix{ V_{j}/V_{j+1} \ar[r] & \XX({{\boldsymbol \beta}}/V_{j+1}) \ar[d]^{\pi_{j}}\\
& \XX({{\boldsymbol \beta}}/V_{j})}$$ which is the restriction of the natural projection map $\mathbb P(V_0/V_{j+1}) \setminus \mathbb P(V_{j}/V_{j+1}) \to \mathbb P(V_0/V_{j}).$ The fiber $\pi_j^{-1}([x + V_{j}]) = \{[x + v + V_{j+1}] : v \in V_{j}/V_{j+1}\}$ identifies with $V_{j}/V_{j+1}$; the identification depends on the choice of representative $x$. Via the identification, $\beta_{j}$ induces an affine pseudo-metric $\varrho_{j}$ of signature $(p_{j},q_{j})$ on each fiber, defined by: $$\varrho^2_{j}(x + v, x+w) := \beta_{j}(v-w).$$ The metric is well-defined, provided that the representative $x + V_j$ is always chosen to satisfy $\beta_0(x+V_j) = -1$ (there are two such choices). The action of $\Isom({\mathcal F}, {{\boldsymbol \beta}})$ on $\XX({{\boldsymbol \beta}}/V_{j+1})$ preserves this fibration by affine spaces and preserves the affine $(p_{j},q_{j})$ metric $\varrho_j^2$ on the fibers.
Alternatively, we will use the notation $$\xymatrix{ \mathbb A^{p_{j},q_{j}}\ar[r] & \XX((p_0,q_0),\ldots,(p_{j},q_{j}))\ar[d]^{\pi_{j}}\\
& \XX((p_0,q_0),\ldots, (p_{j-1},q_{j-1})) }$$ where the notation $\mathbb A^{p_{j},q_{j}}$ indicates that the fibers are affine spaces equipped with an affine pseudo-metric of signature $(p_{j},q_{j})$. Combining this information for all possible values of $j$, we see that $\XX({{\boldsymbol \beta}})$ is equipped with an *iterated affine bundle* structure (see Figure \[fig:iterated-bundle\]), and the fibers at each iteration are equipped with an invariant affine pseudo-metric:
$$\label{tower}
\xymatrix{ \mathbb A^{p_{k},q_{k}}\ar[r] & \XX((p_0,q_0),\ldots,(p_{k},q_{k}))\ar[d]^{\pi_{k}}\\
\mathbb A^{p_{k-1},q_{k-1}}\ar[r] & \XX((p_0,q_0),\ldots, (p_{k-1},q_{k-1}))\ar[d]^{\pi_{k-1}}\\
& \vdots\ar[d]^{\pi_2} \\
\mathbb A^{p_{1},q_{1}}\ar[r] & \XX((p_0,q_0), (p_{1},q_{1}))\ar[d]^{\pi_1}\\
& \XX(p_0,q_0) }$$
We note that in the case $(p_0, q_0) = (1,0)$ the base of the tower of fibrations is a point and the next space up $\XX((1,0)(p_1,q_1))$ is an affine space $\mathbb A^{p_1,q_1}$ equipped with an invariant affine pseudo-metric of signature $(p_1,q_1)$. In the case that $(p_1,q_1)$ is $(n,0)$ or $(0,n)$, then $\mathbb A^{p_1,q_1}$ identifies with the Euclidean space $\mathbb E^n$.
In the context of Theorem \[thm:limits-Hpq\], the different levels of the tower of fibrations (\[tower\]) correspond to different rates of collapse of $\XX(p,q)$. The initial projection $\pi_k$ should be thought of as a collapse map which collapses the directions in $\XX(p,q)$ most distorted by the conjugation action.
\[rem:double-cover\] It is sometimes easier to work in the double cover $\widetilde \XX({{\boldsymbol \beta}})$ of $\XX({{\boldsymbol \beta}})$, which is naturally described by the hyperboloid $\beta_0 = -1$. In this case, each projection map $\pi_j$ is just the restriction of the quotient map $V_0/V_{j+1} \to V_0/V_j$ to the hyperboloid $\beta_0 = -1$ in $V_0/V_{j+1}$. It is natural to call $\widetilde \XX({{\boldsymbol \beta}})$ the *hyperboloid model*.
The action of the unipotent part $U({\mathcal F})$ preserves each affine fiber of each fibration in . The action on each fiber is simply a translation. However, the amount and direction of translation may vary from fiber to fiber (with respect to some chosen trivialization), so that the fibers appear to shear with respect to one another.
In the following sections we apply Theorem \[thm:limits-Hpq\] to several cases of interest, including the classical two-dimensional geometries, and three-dimensional hyperbolic and AdS geometry. Along the way we will discuss further the geometry of those $(\XX({{\boldsymbol \beta}}), \Isom({{\boldsymbol \beta}}))$ which arise as limits in these cases.
The classical two-dimensional geometries
----------------------------------------
The two-dimensional Riemannian model geometries of constant curvature may each be realized as subgeometries of $(\RP^2, {\operatorname{PGL}}_3 \RR)$. In fact, each is defined by a partial flag of quadratic forms (Section \[sec:pfqf\]). We use the notation of Section \[sec:Hpq\], and for brevity we will only refer to the space $X$ of the geometry $(X,G)$ when the group $G$ is clear from context:
- Spherical geometry is (the double cover of) $\XX(3,0)$.
- Hyperbolic geometry is $\XX(1,2)$.
- Euclidean geometry is $\XX((1,0)(2))$
The following chart depicts all possible limits of geometries given by a partial flag of quadratic forms in dimension two. The completeness/accuracy of the chart is easy to verify using the calculus of Theorem \[thm:limits-Xbeta\]. $$\xymatrix{ \XX(3,0) \ar[d] \ar[drr] & \XX(1,2) \ar[dl]\ar[d] & \XX(2,1)\ar[dl]\ar[d]\ar[dr] & \\ \XX((1,0)(2))\ar[dr] & \XX((1,1)(1))\ar[d] & \XX((2,0)(1))\ar[dl] & \XX((1,0)(1,1))\ar[dll] \\ & \XX((1,0)(1)(1)) & & }$$
The limits of spherical, hyperbolic, and Euclidean geometry may be read off from the chart:
The limits of spherical, hyperbolic, and Euclidean geometry, considered as sub-geometries of projective geometry are the following PFQF geometries:
- limits of spherical: $\XX(3,0)$ (no degeneration), $\XX((1,0)(2)) = $ Euclidean, and $\XX((1,0)(1)(1)$.
- limits of hyperbolic: $\XX(1,2)$ (no degeneration), $\XX((1,0)(2)) = $ Euclidean, and $\XX((1,0)(1)(1))$.
- limits of Eucldiean: $\XX((1,0)(2))$ (no degeneration), and $\XX((1,0)(1)(1))$.
We give a brief description of the most degenerate two-dimensional partial flag of quadratic forms geometry $\XX((1,0)(1)(1))$ which is a limit of all three classical two-dimensional geometries. First, the group $${\operatorname{O}}((1,0)(1)(1)) = \begin{pmatrix} \pm 1 & 0 & 0\\ * & \pm 1 & 0 \\ * & * & \pm 1 \end{pmatrix}$$ preserves a full flag $\RR^3 = V_0 \supset V_1 \supset V_2 \supset V_3 = \{0\}$ in $\RR^{3}$. In this case, the space $\XX((1,0)(1)(1)) \cong \mathbb A^2$ is an affine plane, though note that ${\operatorname{O}}((1,0)(1)(2))$ is not the full group of affine transformations. The iterated affine bundle structure is:
$$\xymatrix{ \mathbb A^{1,0} \ar[r] & \XX((1,0)(1)(1)) \cong \mathbb A^2 \ar[d]^{\pi_2} \\
\mathbb A^{1,0} \ar[r] & \XX((1,0)(1)) = \mathbb A^1\ar[d]^{\pi_1} \\
& \XX(1,0) = \{pt\} }$$
Hence $\XX((1,0)(1)(1))$ is an affine two-space, equipped with a translation invariant fibration in Euclidean lines $\mathbb A^{1,0}$ over a base which is also a Euclidean line $\mathbb A^{1,0}$. The group ${\operatorname{O}}((1,0)(1)(1))$ is the group of affine transformations which preserve the fibration as well as the metric on the fibers and the base. In the standard basis, $\XX((1,0)(1)(1)) = \{ x_1 \neq 0\}/\RR^*$, which we identify with the affine plane $x_1 = 1$ in $V_0 = \RR^3$. The lines of the foliation are the lines of constant $x_2$ and the (square of the) affine metric on these lines is given by: $$\varrho^2_2\left(\begin{pmatrix} 1 \\ x_2 \\ x_3 \end{pmatrix} , \begin{pmatrix} 1 \\ x_2 \\ x_3' \end{pmatrix} \right) = (x_3 - x_3')^2.$$
Limits of three-dimensional hyperbolic geometry
-----------------------------------------------
Theorem \[thm:limits-Hpq\] gives the limits of three-dimensional hyperbolic geometry $\HH^3 = \XX(1,3)$ as a subgeometry of projective geometry. The results are summarized in the following diagram: $$\label{list-H3}
\xymatrix{ & \XX(1,3)\ar[dl]\ar[d]\ar[dr] & \\ \XX((1,0)(3))\ar[d]\ar[dr] & \XX((1,2)(1))\ar[d]\ar[dr] & \XX((1,1)(2))\ar[dll]\ar[d] \\ \XX((1,0)(1)(2))\ar[dr] & \XX((1,0)(2)(1))\ar[d] & \XX((1,1)(1)(1))\ar[dl] \\ & \XX((1,0)(1)(1)(1)) &}$$
We now describe some of the geometries appearing in this list, and their relationships to the Thurston geometries.
0.2cm
The geometry $\XX((1,0)(3))$ is Euclidean geometry.
0.2cm
The geometry $\XX((1,2)(1))$ is *half-pipe* geometry, defined by Danciger in [@danciger1] and used to construct examples of geometric structures (cone-manifolds) transitioning from hyperbolic geometry to anti de Sitter (AdS) geometry. That $\XX((1,2)(1))$ is a limit of three dimensional AdS geometry also follows from Theorem \[thm:limits-Hpq\], since the projective model for $\operatorname{AdS}^3$ is, in our terminology, $\XX(2,2)$.
0.2cm Next, consider the geometry $\XX((1,0)(2)(1))$, which is also a limit of spherical geometry. The iterated affine bundle structure is: $$\xymatrix{ \mathbb E^{1} \ar[r] & \XX((1,0)(2)(1)) \cong \mathbb A^3 \ar[d]^{\pi_2} \\
\mathbb E^{2} \ar[r] & \XX((1,0)(2)) \cong \mathbb E^{2} \ar[d]^{\pi_1} \\
& \XX(1,0) = \{*\} }$$ Hence, $\XX((1,0)(2)(1)$ fibers in Euclidean lines over the Euclidean plane. Let $(1\ x\ y\ z)^T$ be coordinates for $\XX((1,0)(2)(1)) = \mathbb A^3$, let $(1\ x'\ y')^T$ be coordinates for $\XX((1,0)(2)) = \mathbb E^2$, and let the projection $\pi_2$ be given by $x' = x, y' = y$. Then, consider the contact form $\alpha$ on $\mathbb A^3$ defined by $$\alpha = dz +x dy - y dx.$$ Note that $d \alpha = 2 \ \pi_2^* dA$, where $dA$ is the area form on the Euclidean plane $\mathbb E^{2}$. Consider the following Riemannian metric $g_{\mathrm{Nil}}$ on $\XX((1,0)(2)(1))$: The fibers of $\pi_2$ are defined to be $g_{\mathrm{Nil}}$ orthogonal to $\ker \alpha$, and $g_{\mathrm{Nil}}$ is defined to be the pull-back by $\pi_2$ of the Euclidean metric on $\ker \alpha$, while $g _{\mathrm{Nil}}(X,X) = \alpha(X)^2$ for $X$ tangent to the $\pi_2$ fiber direction. The Riemannian metric $g_{\mathrm{Nil}}$ makes $\XX((1,0)(2)(1))$ into the model space for *Nil geometry*. Of course, ${\operatorname{O}}((1,0)(2)(1))$ does not preserve $\alpha$, nor the metric $g_{\mathrm{Nil}}$. However one may check that the isometries $\Isom(g_{\mathrm{Nil}})$ are a proper subgroup (up to finite index) of ${\operatorname{O}}((1,0)(2)(1))$, so Nil geometry locally embeds into $\XX((1,0)(2)(1))$. In coordinates, $$\Isom_0(g_{\mathrm{Nil}}) = \begin{pmatrix} \pm 1 & & \\ & {{\operatorname{O}}(2)} & \\ & & \pm 1\end{pmatrix} \ltimes \begin{pmatrix} 1 & & & \\ a & 1 & & \\ b & 0 & 1 & \\ c & b & -a & 1\end{pmatrix} \subset {\operatorname{O}}((1,0)(2)(1)),$$ where $a,b,c \in \RR$ are arbitrary numbers. That Nil geometry appears, in this context, as a (sub-geometry of a) limit of hyperbolic geometry is not surprising. Porti [@Porti-02] proved that a Nil orbifold with ramification locus transverse to the $\pi_2$ fibration is (metrically) the limit of collapsing hyperbolic cone-manifold structures (after appropriate modification of the collapsing metric).
0.2cm Next consider the geometry $\XX((1,1)(2))$. The iterated affine bundle structure is just one bundle: $$\xymatrix{ \mathbb E^{2} \ar[r] & \XX((1,1)(2)) \ar[d]^{\pi_1} \\
& \XX((1,1)) \cong \mathbb H^{1}. }$$ In a basis that respects the partial flag, the structure group has the form $${\operatorname{O}}((1,1)(2)) = \begin{pmatrix} {\operatorname{O}}(1,1) & \\ & {\operatorname{O}}(2) \end{pmatrix} \ltimes \begin{pmatrix} I_2 & \\ \begin{matrix} * & *\\ * & * \end{matrix} & I_2 \end{pmatrix},$$ where $I_2$ is the $2 \times 2$ identity matrix. In fact, there is a copy of the group $\mathrm{Sol} \cong \SO(1,1) \ltimes \RR^2$ inside of ${\operatorname{O}}((1,1)(2))$, which is described in coordinates as follows: $$\mathrm{Sol} = \begin{pmatrix} \begin{matrix} \cosh z & \sinh z\\ \sinh z & \cosh z \end{matrix} & \\ & I_2 \end{pmatrix} \ltimes \begin{pmatrix} I_2 & \\ \begin{matrix} x & x\\ y & -y \end{matrix} & I_2 \end{pmatrix}.$$ In fact, $\mathrm{Sol}$ acts simply transitively on $\XX((1,1)(2))$. Hence $\XX((1,1)(2))$ is a model for Sol geometry. Four of the eight components of $\Isom \mathrm{Sol}$ lie inside ${\operatorname{O}}((1,1)(2))$, corresponding to multiplying the diagonal blocks of $\mathrm{Sol}$ by $\pm 1$. The missing four components are achieved by adding the block diagonal matrix $\operatorname{diag}\left( \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}, I_2 \right)$. We may also embed $\mathrm{Sol}$ geometry, in the exact same way, as a sub-geometry of $\XX((1,1)(1)(1))$ (see Figure \[fig:iterated-bundle\] for an illustration of the iterated affine bundle structure of $\XX((1,1)(1)(1))$). Let $M$ be a torus bundle over the circle with Anosov mondromy. In [@HPS], Huesener–Porti–Suárez showed that the natural Sol geometry structure on $M$ is realized as a limit of hyperbolic cone manifold structures, in the sense that the hyperbolic metrics converge to the Sol metric after appropriate (non-isotropic) modification. It is possible to recast their construction in the context of projective geometry using the theory developed here. Recently, Kozai [@Kozai-13] used this projective geometry approach to generalize the work of Huesener–Porti–Suárez to the setting of three-manifolds $M$ which fiber as a surface bundle over the circle. Kozai shows (under some assumptions) that the natural singular Sol structure on $M$ may be deformed to nearby singular hyperbolic structures by first deforming from Sol to half-pipe geometry $\XX((1,2)(1))$ and then from half-pipe to hyperbolic geometry.
Thurston geometries as limits of hyperbolic geometry
----------------------------------------------------
All eight Thurston geometries locally embed as sub-geometries of real projective geometry (in fact, each embeds up to finite index and coverings). We have now demonstrated that Euclidean geometry is a limit and both $\mathrm{Nil}$ geometry and $\mathrm{Sol}$ geometry locally embed in limits of hyperbolic geometry. We now prove Theorem \[cor:Thurston-geoms\_intro\], which says that these are the only Thurston geometries that appear in this way.
We show that the projective geometry realizations of the four remaining Thurston geometries, which are $\mathbb S^3$ , $\HH^2 \times \RR$, $\widetilde{\SL_2\RR}$, and $\mathbb S^2 \times \RR$, do not locally the embed in any of the geometries listed in (\[list-H3\]).
Consider spherical geometry $\mathbb S^3$. Up to conjugacy, the local embedding of spherical geometry of projective geometry is unique. It is clear that the structure group ${\operatorname{O}}(4)$ does not locally embed as a subgroup in any of the partial flag isometry groups for the geometries appearing in (\[list-H3\]).
Next, consider $\mathbb S^2 \times \RR$. The isometry group is (up to finite index) a product $\SO(3) \times \RR$. The only geometry in the list (\[list-H3\]) whose structure group contains a subgroup locally isomorphic to $\SO(3)$ is Euclidean geometry $\XX((1,0)(3))$. Of course $\mathbb S^2 \times \RR$ does not locally embed in Euclidean geometry.
The geometry $\widetilde{\SL_2 \RR}$ is locally isomorphic to (in fact an infinite cyclic cover of) the following subgeometry of projective geometry. The space $\PSL(2,\RR)$ embeds in $\RP^3$ by considering the entries of a $2 \times 2$ matrix as coordinates, and the identity component of the isometry group is given by the linear action of $\PSL_2 \RR \times \PSO(2)$ where the $\PSL_2 \RR$ factor acts on the left and the $\PSO(2)$ factor acts on the right. We show that even the stiffening of this geometry obtained by restricting the structure group to the subgroup $\PSL(2,\RR)$ (acting on the left) does not locally embed in any limit of hyperbolic geometry. For, the only geometry appearing in (\[list-H3\]) whose isometry group contains a subgroup locally isomorphic to $\PSL(2,\RR) \cong \SO_0(2,1)$ is half-pipe geometry $\XX((1,2)(1))$. Any such subgroup is conjugate into the block diagonal subgroup $\operatorname{diag}({\operatorname{O}}(2,1), 1)$ and it is then easy to see that such a subgroup does not act transitively on $\XX((1,2)(1))$, but rather preserves a totally geodesic subspace (a copy of the hyperbolic plane, see [@danciger1]). Therefore, since the left action of $\PSL_2R$ is transitive, we have shown that $\widetilde{\SL_2 \RR}$ does not locally embed in half-pipe geometry.
Finally, consider $\HH^2 \times \RR$ geometry. The isometry group is (up to finite index) the product $\SO(2,1) \times \RR$. Again, the only limit of hyperbolic geometry whose isometry group contains a subgroup $H$ locally isomorphic to $\SO(2,1)$ is $\XX((1,2)(1))$, half-pipe geometry. However, the centralizer of the smallest such subgroup $H = \SO_0(2,1) \times \{1\} $ in ${\operatorname{O}}((1,2)(1))$ is $\operatorname{diag}(\pm I_3, \pm 1)$, so in particular there is no subgroup locally isomorphic to $\SO(2,1) \times \RR$ inside of ${\operatorname{O}}((1,2)(1))$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We calculate static properties of non-rotating neutron stars (NS’s) using a microscopic equation of state (EOS) for asymmetric nuclear matter. The EOS is computed in the framework of the Brueckner–Bethe–Goldstone many–body theory. We introduce three-body forces in order to reproduce the correct saturation point of nuclear matter. A microscopic well behaved EOS is derived. We obtain a maximum mass configuration with $M_{max} = 1.8
M_\odot$, a radius $R = 9.7$ km and a central density $n_c =
1.34~fm^{-3}$. We find the proton fraction exceeds the critical value $x^{Urca}$, for the onset of direct Urca processes, at densities $n \geq
0.45~fm^{-3}$. Therefore, in our model, NS’s with masses above $M^{Urca}
= 0.96 M_\odot$ can undergo very rapid cooling depending on whether or not nucleon superfluidity in the interior of the NS takes place. A comparison with other microscopic models for the EOS is done, and neutron star structure is calculated for these models too.
address:
- ' *Dipartimento di Fisica, Universitá di Catania and I.N.F.N. Sezione di Catania, c.so Italia 57, I-95129 Catania, Italy*'
- ' *Dipartimento di Fisica, Universitá di Pisa and I.N.F.N. Sezione di Pisa, Piazza Torricelli 2, I-56100 Pisa, Italy*'
author:
- 'M. Baldo, G.F. Burgio'
- 'and I. Bombaci'
title: '**Microscopic nuclear equation of state with three-body forces and neutron star structure**'
---
[PACS numbers: 97.60.Jd, 21.65.+f ]{}
In the next few years it is expected that a large amount of novel informations on neutron stars (NS’s) will be available from the new generation of X–ray and $\gamma$–ray satellites. Therefore, a great interest is devoted presently to the study of NS’s and to the prediction of their structure on the basis of the properties of dense matter. The equation of state (EOS) of NS matter covers a wide density range, from $\sim 10$ g/cm$^3$ in the surface to several times nuclear matter saturation density ($\rho_0 \sim 2.8~10^{14}$ g/cm$^3$) in the center of the star[@sha]. The interior part (core) of a NS is made by asymmetric nuclear matter with a certain lepton fraction. At ultra–high density, matter might suffer a transition to other exotic hadronic components (like hyperons, a $K^-$ condensate or a deconfined phase of quark matter). The possible appearance of such an exotic core has enormous consequences for the neutron star and black hole formation mechanism[@bomb]. Unfortunately large uncertainities are still present in the theoretical treatment of this ultra–dense regime[@gle; @pra+]. Therefore, in the present work, we consider a more conventional picture assuming the NS core is composed only by an uncharged mixture of neutrons, protons, electrons and muons in equilibrium with respect to the weak interaction ($\beta$–stable matter). Even in this picture, the determination of the EOS of asymmetric nuclear matter to describe the core of the NS, remains a formidable theoretical problem[@mart].
Any “realistic” EOS must satisfy several requirements : i) It must display the correct saturation point for symmetric nuclear matter (SNM); ii) it must give a symmetry energy compatible with nuclear phenomenology and well behaved at high densities; iii) for SNM the incompressibility at saturation must be compatible with phenomenology on monopole nuclear oscillations[@swi]; iv) both for neutron matter (NEM) and SNM the speed of sound must not exceed the speed of light (causality condition), at least up to the relevant densities; the latter condition is automatically satisfied only in fully relativistic theory.
In this letter we present results for some NS properties obtained on the basis of a microscopic EOS, recently developed[@tbf], which satisfies requirements i-iv, and compare them with the predictions of other microscopic EOS’s. The Brueckner-Hartree-Fock (BHF) approximation for the EOS in SNM, within the continuous choice [@bhf], reproduces closely the Brueckner–Bethe–Goldstone (BBG) results which include up to four hole line diagram contributions[@Day], as well as the variational calculations [@var], at least up to few times the saturation density. Non–relativistic calculations, based on purely two–body interactions, fail to reproduce the correct saturation point for SNM. This well known deficiency is commonly corrected introducing three-body forces (TBF). Unfortunately, it seems not possible to reproduce the experimental binding energies of light nuclei and the correct saturation point accurately with one simple set of TBF [@var]. Relevant progress has been made in the theory of nucleon TBF, but a complete theory is not yet available. In ref.[@var] a set of simple TBF has been introduced within the variational approach. We introduced [@tbf] similar TBF within the BHF approach, and we have adjusted the parameters in order to reproduce closely the correct saturation point of SNM, since for NS studies this is an essential requirement, and there is no reason to believe that TBF be the same as in light nuclei. The corresponding EOS (termed BHF3) is depicted in Fig. 1, in comparison with the EOS obtained in BHF approximation without three-body forces (BHF2), but using the same two-body force, i.e. the Argonne $v_{14}$ ($Av_{14}$) potential [@arg]). In the same figure, we show the variational EOS (WFF) of ref.[@var] for the $Av_{14} + TBF$ Hamiltonian, and the EOS from a recent Dirac-Brueckner calculation (DBHF) [@dbhf] with the Bonn–A two–body force. The BHF3 EOS saturates at $n_o = 0.18~fm^{-3}, E = -15.88~MeV$, and is characterized by an incompressibility $K_{\infty} = 240~MeV$, very close to the recent phenomenological estimate of ref.[@swi]. In the low density region ($n < 0.4~fm^{-3}$), BHF3 and DBHF equations of state are very similar. At higher density, however, the DBHF is stiffer than the BHF3. The discrepancy between these two models for the EOS can be easily understood by noticing that the DBHF treatment is equivalent [@tao] to introduce in the non-relativistic BHF2 the three-body force corresponding to the excitation of a nucleon-antinucleon pair, the so-called Z-diagram[@zeta]. The latter is repulsive at all densities. In BHF3 treatment, on the contrary, both attractive and repulsive three-body forces are introduced [@var], and therefore a softer EOS can be expected.
Fractional polynomial fits to each one of these EOS’s allow to compute the corresponding pressure and speed of sound $c_s$ to compare with the speed of light $c$. The ratio $c_s/c$ for all four EOS’s as a function of the number density is reported in Fig. 2. WFF model (circles) violates the causality condition at densities encountered in the core of stars near the maximum mass configuration for that model (see fig. 4b) and tab. I). The DBHF calculations need an extrapolation to slightly higher densities than the largest one considered in ref.[@dbhf]. The extrapolation was done in such a way to keep the causality condition fulfilled. The same procedure was followed for BHF3. In the latter case the BHF procedure was well converging up to densities $n = 0.76~ fm^{-3}$ for SNM and $n =
0.912~ fm^{-3}$ for NEM. For DBHF the causality condition was fulfilled in the extrapolated region only if particular choices of the fitting parameters were used, while for the BHF3 the results were insensitive to a wide range of variation of the parameters [@bbb].
It has to be stressed that the $\beta$-stable matter EOS is strongly dependent on the nuclear symmetry energy, which in turns affects the proton concentration [@asym]. The latter quantity is crucial for the onset of direct Urca processes [@urca], whose occurrence enhances neutron star cooling rates. In our approach, from the difference of the energy per particle $E/A$ in NEM and SNM the symmetry energy $E_{sym}$ can be extracted assuming a parabolic dependence on the asymmetry parameter $\beta = {{(n_n - n_p)} / {(n_n + n_p)}}$, being $n_n$ and $n_p$ respectively the neutron and proton number density. This procedure turns out to be quite reliable[@asym]. The values of $E_{sym}$ for the different EOS’s are reported in Fig. 3, together with the corresponding proton fraction $x = (1 - \beta)/2$. We notice that in both relativistic and non–relativistic Brueckner–type calculations, the proton fraction can exceed the “critical” value $x^{Urca} = (11-15)\%$ needed for the occurrence of direct Urca processes [@urca]. This is at variance with the WFF variational calculation (Fig. 3, circles), which predicts a low absolute value both for the simmetry energy and the proton fraction with a slight bend over. For BHF3 model we find $x^{Urca} = 13.6\%$, which correspond to a critical density $n^{Urca} = 0.447~fm^{-3}$. Therefore, BHF3 neutron stars with a central density higher than $n^{Urca}$ develop inner cores in which direct Urca processes are allowed.
The EOS for $\beta$–stable matter can be used in the Tolman–Oppenheimer–Volkoff [@tov] equations to compute the neutron star mass and radius as a function of the central density. For the outer part of the neutron star we have used the equations of state by Feynman-Metropolis-Teller [@fmt] and Baym-Pethick-Sutherland [@bps], and for the middle-density regime ($0.001~fm^{-3}<n<0.08~fm^{-3}$) we use the results of Negele and Vautherin [@nv]. In the high-density part ($n > 0.08~fm^{-3}$) we use alternatively the three EOS’s discussed above. The results are reported in Fig. 4. We display the gravitational mass $M_G$, in units of solar mass $M_{\odot}$ ($M_{\odot} = 1.99~10^{33}$ g), as a function of the radius R (panel (a)) and the central number density $n_c$ (panel (b)). As expected, the stiffest EOS (DBHF) we used in the present calculation gives higher maximum mass and lower central density with respect to the non-relativistic Brueckner models. The maximum NS mass for the BHF3 is intermediate between BHF2 and DBHF, but closer to the latter. The difference between BHF3 and WFF neutron stars reflects the discrepancy already noticed for the EOS and mainly for the symmetry energy. This point will be discussed in more details in a forthcoming paper[@bbb]. Table I summarizes maximum mass configuration properties, for the different EOS’s used in the present work.
In conclusion, we computed some properties of NS’s on the basis of a microscopic EOS obtained in the framework of BBG many–body theory with two– and three–body nuclear interactions. BHF3 EOS satisfies the general physical requirements (points i–iv) discussed in the introduction. This is the main feature which distinguishes our BHF3 EOS with respect to other microscopic non–relativistic EOS[@var; @nor]. The calculated maximum mass is in agreement with observed NS masses[@vanK]. We found that the neutron star core is “proton rich”. In fact, the proton fraction at the center of the maximum mass configuration in the BHF3 model is $x =
26\%$. Our BHF3 neutron stars with mass above the critical value $M^{Urca}
\equiv M_G(n^{Urca}) = 0.96 M_\odot$ develop inner cores in which direct Urca processes can take place. These stars cool very rapidly or not depending on the properties of nuclear superluidity (values of the superfluid gaps, critical temperatures, density ranges for the superfluid transition)[@page; @bald+]. Our EOS offers the possibility for a selfconsistent microscopic calculation for both the neutron star structure, and nuclear superfluid properties within the same many–body approach and with the same nuclear interaction.
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EOS $M_G/M_{\odot}$ R(km) $n_c(fm^{-3})$
------ ----------------- ------- ----------------
DBHF 2.063 10.39 1.13
BHF3 1.794 9.74 1.34
BHF2 1.59 7.96 1.95
WFF 2.13 9.4 1.25
: Parameters of the maximum mass configuration: the ratio $M_G/M_{\odot}$ is shown for several EOS’s vs. the corresponding radius R and central number density $n_c$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Radiology reports are an important means of communication between radiologists and other physicians. These reports express a radiologist’s interpretation of a medical imaging examination and are critical in establishing a diagnosis and formulating a treatment plan. In this paper, we propose a Bi-directional convolutional neural network (Bi-CNN) model for the interpretation and classification of mammograms based on breast density and chest radiographic radiology reports based on the basis of chest pathology. The proposed approach is a part of an auditing system that evaluates radiology reports to decrease incorrect diagnoses. Our study revealed that the proposed Bi-CNN outperforms convolutional neural network, random forest and support vector machine methods.'
author:
- |
Hojjat Salehinejad, Shahrokh Valaee, , Aren Mnatzakanian, Tim Dowdell,\
Joseph Barfett, and Errol Colak [^1][^2][^3] [^4]
bibliography:
- 'CTLIEEEtrans.bib'
- 'mybibfile.bib'
title: Interpretation of Mammogram and Chest Radiograph Reports Using Deep Neural Networks
---
Breast density, chest radiograph, convolutional neural networks, mammography, radiology reports classification.
Introduction {#sec:introduction}
============
Radiology reports are an important means of communication between radiologists and other physicians [@naik2001radiology]. These reports express a radiologist’s interpretation of a medical imaging examination and are critical in establishing a diagnosis and formulating a treatment plan.
Radiology is among the medical specialties with the highest rate of malpractice claims [@pinto2010]. These claims can arise from a failure to communicate important findings, a failure of perception, lack of knowledge, and misjudgment. A failure to detect an abnormality on a medical imaging examination can lead to significant medical consequences for patients such as a delayed diagnosis. With a delayed or incorrect diagnosis, patients can present later with worsened symptoms and more advanced disease that may require more aggressive treatment or may be untreatable.
In this paper, we propose an auditing system for radiologists that has two main components: a natural language processing (NLP) model to process and interpret a radiology report and a machine vision model that interprets the medical imaging examination. This auditing system reviews the radiologist’s report and compares it with the interpretation of a machine vision model. The proposed system would notify the radiologist if there is a discrepancy between the two interpretations. Many investigators have aimed to develop machine vision models for the interpretation of medical imaging examinations, such as chest radiographs [@cicero2017training], mammograms [@wang2017detecting], and head computerized tomography (CT) scans [@havaei2017brain]. However, fewer attempts have been made for the NLP of radiology reports [@aronow1999ad; @wilcox2000automated; @shin2017classification]. The focus of this paper is on the design and performance evaluation of the NLP component.
We propose a bi-directional convolutional neural network (Bi-CNN) for the NLP of radiology reports. The Bi-CNN will be trained independently for two distinct report data types: breast mammograms and chest radiographs. In particular, this model will classify the degree of breast density and the type of chest pathology based on the radiology report content. Our proposed NLP model differs from a keyword search algorithm. It is capable of interpreting and classifying a radiology report. The proposed Bi-CNN has two independent input channels, where the order of non-padded input to one channel is the reverse of the non-padded input to the other channel. Performance of the proposed model is compared with a single channel convolutional neural network (CNN), random forest (RF), and support vector machine (SVM) models and a comparative study is conducted.
{width="70.00000%"}
Related Work {#sec:relatedworks}
============
The NLP models used for the classification and interpretation of radiology reports can be categorized into rule-based and machine learning models. Rule-based systems are usually built upon a series of “if-then” rules, which require an exhaustive search through documents. Such rules are typically designed by human experts. By contrast machine learning models do not require explicit rules. Instead, they try to train themselves iteratively and extract features from data.
Rule-based Approaches
---------------------
A radiology report mining system generally has three main components. These include a medical finding extractor, a report and image retriever, and a text-assisted image feature extractor [@gong2008text]. For example, a set of hand-crafted semantic rules are presented in [@gong2008text] for mining brain CT radiology reports. A natural language parser for medical reports is proposed in [@taira2007field] that uses four-gram and higher order systems to assign a stability metric of a reference word within a given sentence. This model is inspired from Field theory and uses a dependency tree that represents the global minimum energy state of the system of words for a given sentence in radiology reports [@taira2007field]. A classifier using extracted keywords from clinical chest radiograph reports is developed in [@cooper1998using]. In [@chapman2003creating], a method is developed to identify patients with mediastinal findings related to inhalational anthrax. These methods are applied on different case studies with different sets of rules. Such methods are not flexible enough to be applied to case studies for which they were not designed, and hence are not transferable, and their performance comparison requires specific implementation and fine tuning of rules.
Machine Learning Approaches
---------------------------
A NLP study for a large database of radiology reports was conducted in [@dang2009use] based on two classes: positive and negative radiology findings. These classes have different patient attributes such as age groups, gender, and clinical indications [@dang2009use]. A collection of 99 musculoskeletal radiology examinations were studied using a machine learning model versus a naive Bayes classification algorithm, followed by a SVM to detect limb fractures from radiology reports [@zuccon2013automatic]. A CNN was used in [@shin2015interleaved] as a supervised tool to map from images to label spaces. This system has an interactive component between supervised and unsupervised machine learning models to extract and mine the semantic interactions of radiology images and reports. Recurrent neural networks (RNN) along with the CNN have been used to interpret chest x-ray reports for a limited number of classes [@shin2016learning]. This CNN model learns from the annotated sequences produced by the RNN model. The RNN model with long short term memory (LSTM) is capable of learning long term dependencies between the annotations [@salehinejad2016learning]. The model proposed in [@lai2015recurrent] uses a recurrent design to extract features from word representations. Compared with window-based neural networks, which uses a rolling window to extract features, this model generates less noise and is more stable. The max-pooling method distinguishes the words that have a major impact in the report. Machine learning models have also been developed for understanding medical reports in other languages, Korean being one example, using tools such as Naive Bayes classifiers, maximum entropy, and SVMs [@oh2011extracting].
Learning Vector Representations of Words
----------------------------------------
Word2vec is a well-known model used for learning vector space representations of words. It produces a vector space from a large corpus of text with reduced dimensionality. This model assigns a word vector to each unique word in the corpus. Vectors with closer contexts are positioned in closer proximity in this space [@mikolov2013distributed]. GloVe approach provides a global vector space representation of words. It uses global matrix factorization and local context windowing methods to construct a global log-bilinear regression model [@pennington2014glove]. The word-word co-occurrence matrix of the words is a sparse matrix by nature. However, GloVe only uses the non-zero elements for training and does not consider individual context windows in a large corpus [@pennington2014glove]. CNNs have been shown to work well on $n$-gram representations of data [@majumder2017deep]. For example, a convolution layer can perform feature extraction from various $n$ values of a $n$-gram model and perform personality detection from documents [@majumder2017deep]. Dynamic k-Max pooling can operate as a global operation over linear sequences. Such dynamic CNNs can perform semantic modelling of sentences [@kalchbrenner2014convolutional]. The model receives variable dimension size input sentences and induces a feature graph over the sentence that is capable of explicitly capturing short and long-range relations [@kalchbrenner2014convolutional].
CNN models are also utilized for learning representations, opinion sentiment understanding, and analysis of products. For example, a CNN with a single output softmax layer can classify a vector representation of text with high accuracy [@kim2014convolutional]. The design of output layers in CNNs varies depending on the application of the model. A collaboration of RNNs and CNNs for feature extraction of sentences has been examined, where a CNN learns features from an input sentence and then a gated RNN model discourses the information [@ren2017neural]. In such system, a bi-directional long short term memory (Bi-LSTM) RNNs can sequentially learn words from question-answer sentences. The trained network can select an answer sentence for a question and present the corresponding likelihood for correctness [@Wang2015ALS].
{width="70.00000%"}
A Brief Review on Convolutional Neural Networks {#sec:cnn}
===============================================
As NLP for radiology reports is an interdisciplinary research effort, a brief review on CNN is provided in this section. A CNN is a machine learning model inspired by the visual cortex of cats [@hubel1968receptive]. A CNN has at least one layer of convolution and sub-sampling, in which the number of layers can increase in depth, generally followed by a fully connected multi-layer perceptron (MLP) network. A consecutive arrangement of convolutional layers followed by sub-sampling build a pyramid-shape model, where the number of feature maps increases as the spatial resolution decreases. Instead of hand-designing feature extractors, the convolutional layers extract features from raw data and the MLP network classifies the features [@lecun1995convolutional]. A CNN typically has three main pieces which are local receptive fields, shared weights, and spatial and/or temporal sub-sampling.
Local Receptive Fields
----------------------
Local receptive fields are made from artificial neurons, which observe and extract features from data such as edges in images. Let us consider an image as input to a CNN such as in Figure \[fig:simplecnn\]. A rectangular kernel with size $M\times N$ scans the input matrix at every single element (i.e., a pixel) and performs convolution such as $$h_{u,v} = \sigma(\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}(I_{u+m,v+n}\cdot w_{m,n})+b_{u,v})$$ where the output is the state of the neurons at element $I_{u,v}$, $\sigma(\cdot)$ is a non-linear function, $\textbf{W}$ is the shared weights matrix, and $\textbf{b}$ is the bias vector.
Shared Weights
--------------
The term “shared weights" refers to the weight matrix $\textbf{W}$ which appears repeatedly in the convolution operation for every element of $I$. Weight sharing reduces the number of free parameters of the model and hence improves generalization of the model [@lecun1995convolutional].
For a general machine vision task, such as classification of natural images, the convolution kernel uses the correlation among spatial or temporal elements of the image to extract local features. Since the image is stationary (i.e., statistics of one part of the image are similar to any other part), the learned shared weights using one sub-region of the image can also be used at another sub-region to extract different features.
Sub-sampling
------------
Passing the whole bag of extracted features after the convolution operation to a classifier is computationally expensive [@boureau2010theoretical]. The feature pooling task (i.e. sub-sampling) generalizes the network by reducing the resolution of the dimensionality of intermediate representations (i.e. feature maps) as well as the sensitivity of the output to shifts and distortions [@lecun1995convolutional]. The two most popular subsampling methods are mean-pooling and max-pooling. By dividing the feature map into a number of non-overlapping rectangle-shape sub-regions, the mean-pooling computes the average of features sitting inside a sub-region as the pooled feature of that sub-region. Max-pooling performs the same task as mean-pooling but with a maximum operator. A study on these two operations is provided in [@boureau2010theoretical].
{width="80.00000%"}
![The order of feature vectors and padding for the two input channels of bi-directional convolutional neural network (Bi-CNN). Each channel has two kernels (green and purple frames) of size three. The order of feature vectors in Channel 2 is the reverse order of feature vectors in Channel 1.[]{data-label="fig:reverse"}](reverse.eps){width="30.00000%"}
The Proposed Method {#sec:methods}
===================
An auditing system for radiology is presented in Figure \[fig:audiotory\]. The main components of the system are a machine vision model to interpret the input image and a NLP component to interpret the corresponding radiology report. The system notifies the radiologist if there is a discrepancy between the two components of the system. In this paper, our focus is on the design and evaluation of the NLP component.
Preprocessing
-------------
In general, radiology reports include major sections such as “Indication", “Findings", “Impression", and the name of the reporting radiologist. However, there are variations in report formatting due to “radiologist style" or institutional requirements. Despite brute-force search approaches, the deep learning models have the advantage of requiring the least amount of data cleaning and preprocessing due to their natural adaptation to the data and non-linear feature extraction ability.
In the preprocessing step, the model extracts the “Findings" and “Impression" sections of the report and performs tokenization and string cleaning. Some of the major tasks include converting upper-case characters to lower-case, removing unnecessary punctuation and symbols, and separating the remainder from the attached string. We add every unique word to a vocabulary dictionary. Each vocabulary has a unique associated index which represents it in the sentence vector. For example, given the sentence *“The breasts show scattered fibroglandular tissue."*, the entire vocabulary dictionary would consist of 6 words (breasts, show, scattered, fibroglandular, tissue, the), which is represented as a list of integers such as $[1,...,6]$. A vector to the length of the longest report in terms of words count in the dataset represents the report. Padding fills up the gap for shorter reports.
In the character embedding step, each integer is mapped to a high dimensional (i.e., $D$) vector with a uniform random distribution. In such a high dimensional space, due to the “curse of dimensionality", the vectors are considered independent [@weber1998quantitative], [@domingos2012few]. Character embedding vector generation for medical vocabularies based on the correlation among the vectors (i.e., similar to Word2Vec) is an interesting topic for further investigation.
Bi-CNN Model Architecture
-------------------------
An input report with $N$ words can be represented as a sequence $\textbf{X}= [\textbf{x}_{1},... ,\textbf{x}_{N}]$, where each word is a vector $\textbf{x}_{n}\in \mathbb{R}^{D}$, [@salehinejad2017convolutional]. As Figure \[fig:cnn\] shows, the proposed architecture has two input channels followed by two independent convolution layers. Both channels have an identical design, except the order of non-padded input to one channel is the reverse of the non-padded input to the other channel. In the example in Figure \[fig:reverse\], the dependencies between feature vectors ($\textbf{x}_{3},\textbf{x}_{4},\textbf{x}_{5}$) are visible to the kernels in both channels. However, the dependencies between feature vectors ($\textbf{x}_{1},\textbf{x}_{0},0$) are only visible to the kernels in Channel 2, while these dependencies are not visible to the kernels in Channel 1. Other examples are the feature vectors ($\textbf{x}_{0},0,0$) in Channel 2 and feature vectors ($\textbf{x}_{4},\textbf{x}_{5},0$) and ($\textbf{x}_{5},0,0$) in Channel 1, which are not visible in the corresponding other channel.
Each channel in the proposed model in Figure \[fig:cnn\] has three filters with varying window sizes $K\in\{3,4,5\}$ that slide across the input layer [@kim2014convolutional], [@salehinejad2017convolutional]. The filters extract features from the input layer to construct feature maps of size $(N-K+1)\times F$, where $F$ is the number of feature maps for each filter. Each feature map $\textbf{h}_{(N-K+1)}$ has its own shared weight $\textbf{W}_{K\times D}$ and bias $\textbf{b}_{ N-K+1}$. The value of a hidden neuron $m$ is
$$h_{m}=\sigma(\sum_{k=1}^{K}\sum_{d=1}^{D} x_{m+k-1,d} \cdot w_{k,d} +b_{m}).$$
The window word is $\textbf{x}_{m:m+K-1}$ and $\sigma(\cdot)$ is a rectified linear unit (ReLU) activation function defined as $$\sigma(z)=max(0,z)$$ where $z\in\mathbb{R}$. The max pooling [@collobert2011natural] extracts the most significant feature from the feature map of a filter as $$\hat{h}_{l}=max(\{h_{1},..,h_{N-K+1}\}).$$ The output of the max-pooling layer contains the max-pooled features $\hat{\textbf{h}}=[\hat{h}_{1},...,\hat{h}_{L}]$ where the length of the feature vectors is $L=C\cdot F$ and $C$ is the number of classes. For example, for the breast density classification with five classes we have $L=5F$ (see Figure \[fig:cnn\]). The features from the input channels are concatenated and passed to a fully connected perceptron network. The output of the softmax layer is the probability distribution over all the labels $P$ (e.g., for mammograms the breast density classes). The value of output unit $p$ is
$$y_{p} = \phi(\sum_{l=1}^{L} (\hat{h}_{1,l} \cdot r_{1,l})\cdot w_{l,p}^{ho} + b_{p}^{o})$$
where $w_{l,p}^{ho}$ is the weight of connection from the hidden unit $l$ in the hidden layer $h$ to the output unit $p$ in the output layer $o$, $ b_{p}^{o}$ is the bias of the output unit $p$, and $\phi(\textbf{z})$ is the softmax activation function defined as $$\phi(z_{p})=\frac{e^{z_{p}}}{\sum_{j=1}^{P}e^{z_{j}}} \:\: \textrm{for} \:\: p=[1,...,P].$$
Training
--------
Adam optimizer is a first-order gradient-based optimization method with integrated momentum functionality [@kingma2014adam]. During training, the momentum helps to diminish the fluctuations in weight changes over consecutive iterations. The drop-out regularization method randomly drops units along with their connections from the concatenated layer (i.e. the input layer to classifier) to the output layer using a binary *mask* vector $\textbf{r}_{1\times L}$ with Bernoulli random distribution [@srivastava2014dropout].
Evaluation Scheme
-----------------
Since we are dealing with discrete categories, we use cross-entropy between a predicted value $y^{(i)}$ from the network and the real label $t^{(i)}$ to measure the loss of networks such as $$\mathcal{\hat{L}}(Y, T) = \frac{-1}{R}\sum_{i=1}^{R}t^{(i)}ln(y^{(i)})+(1-t^{(i)})ln(1-y^{(i)})$$ where $R$ is the number of reports, $Y$ is the set of network predictions, and $T$ is the set of targets to predict. Adding the weight decay term (i.e., $L_{2}$ regularization) to the loss function helps the network to avoid over-fitting while training such as $$\mathcal{L}(Y, T) = \mathcal{\hat{L}}(Y, T) + \eta({\left\lVert\mathbf{W}^{ho}\right\rVert}_{2})$$ where $\mathbf{W}^{ho}$ is the weight matrix of connections between the hidden layer $h$ and the output layer $o$ and $\eta$ is the regularization control parameter.
[width=0.48]{}
------ ------- ------- ------ -------- ------ ------- -------- ------- ------ -------- ------
Mean Median Std Mean Median Std Mean Median Std
MRD 4,080 1,691 3.21 3.00 1.27 30.87 27.00 10.44 9.61 9.00 8.22
CRRD 1,030 772 2.01 2.00 1.07 21.46 19.00 9.21 8.21 8.00 7.67
------ ------- ------- ------ -------- ------ ------- -------- ------- ------ -------- ------
: Summary of statistics for the mammogram reports dataset (MRD) and chest radiograph reports dataset (CRRD) for the “findings" and/or “impression" sections. NR: number of reports; VS: vocabulary size; ANS: average number of sentences per report; ANW: average number of words per report; ASL: average sentence length per report; Mean: sample mean; Med: sample median; Std: sample standard deviation.
\[T:stats\]
[0.120]{} ![Four classes of breast density. a) Almost entirely fatty; b) Scattered areas of fibroglandular density; c) Heterogeneously dense; d) Extremely dense.[]{data-label="fig:breast_images"}](grade1.png "fig:"){width="90.00000%"}
[0.120]{} ![Four classes of breast density. a) Almost entirely fatty; b) Scattered areas of fibroglandular density; c) Heterogeneously dense; d) Extremely dense.[]{data-label="fig:breast_images"}](grade2.png "fig:"){width="90.00000%"}
[0.120]{} ![Four classes of breast density. a) Almost entirely fatty; b) Scattered areas of fibroglandular density; c) Heterogeneously dense; d) Extremely dense.[]{data-label="fig:breast_images"}](grade3.png "fig:"){width="90.00000%"}
[0.120]{} ![Four classes of breast density. a) Almost entirely fatty; b) Scattered areas of fibroglandular density; c) Heterogeneously dense; d) Extremely dense.[]{data-label="fig:breast_images"}](grade4.png "fig:"){width="90.00000%"}
[0.12]{} ![Example chest radiographs for each categories. a) Normal; b) Cardiomegaly; c) Consolidation; d) Pulmonary Edema; e) Lung Nodules; f) Lung Mass; g) Pleural Effusion; h) Widened Mediastinum; i) Vertebral Fractures; j) Clavicular Fracture; k) Pneumothorax.[]{data-label="fig:chest_images"}](Normalh.png "fig:"){width="90.00000%"}
[0.12]{} ![Example chest radiographs for each categories. a) Normal; b) Cardiomegaly; c) Consolidation; d) Pulmonary Edema; e) Lung Nodules; f) Lung Mass; g) Pleural Effusion; h) Widened Mediastinum; i) Vertebral Fractures; j) Clavicular Fracture; k) Pneumothorax.[]{data-label="fig:chest_images"}](Cardiomegalyh.png "fig:"){width="90.00000%"}
[0.12]{} ![Example chest radiographs for each categories. a) Normal; b) Cardiomegaly; c) Consolidation; d) Pulmonary Edema; e) Lung Nodules; f) Lung Mass; g) Pleural Effusion; h) Widened Mediastinum; i) Vertebral Fractures; j) Clavicular Fracture; k) Pneumothorax.[]{data-label="fig:chest_images"}](Consolidationh.png "fig:"){width="90.00000%"}
[0.12]{} ![Example chest radiographs for each categories. a) Normal; b) Cardiomegaly; c) Consolidation; d) Pulmonary Edema; e) Lung Nodules; f) Lung Mass; g) Pleural Effusion; h) Widened Mediastinum; i) Vertebral Fractures; j) Clavicular Fracture; k) Pneumothorax.[]{data-label="fig:chest_images"}](Pulmonary_Edemah.png "fig:"){width="90.00000%"}
[0.12]{} ![Example chest radiographs for each categories. a) Normal; b) Cardiomegaly; c) Consolidation; d) Pulmonary Edema; e) Lung Nodules; f) Lung Mass; g) Pleural Effusion; h) Widened Mediastinum; i) Vertebral Fractures; j) Clavicular Fracture; k) Pneumothorax.[]{data-label="fig:chest_images"}](Lung_Noduleh.png "fig:"){width="90.00000%"}
[0.12]{} ![Example chest radiographs for each categories. a) Normal; b) Cardiomegaly; c) Consolidation; d) Pulmonary Edema; e) Lung Nodules; f) Lung Mass; g) Pleural Effusion; h) Widened Mediastinum; i) Vertebral Fractures; j) Clavicular Fracture; k) Pneumothorax.[]{data-label="fig:chest_images"}](Lung_Massh.png "fig:"){width="90.00000%"}
[0.12]{} ![Example chest radiographs for each categories. a) Normal; b) Cardiomegaly; c) Consolidation; d) Pulmonary Edema; e) Lung Nodules; f) Lung Mass; g) Pleural Effusion; h) Widened Mediastinum; i) Vertebral Fractures; j) Clavicular Fracture; k) Pneumothorax.[]{data-label="fig:chest_images"}](Pleural_Effusionh.png "fig:"){width="90.00000%"}
[0.12]{} ![Example chest radiographs for each categories. a) Normal; b) Cardiomegaly; c) Consolidation; d) Pulmonary Edema; e) Lung Nodules; f) Lung Mass; g) Pleural Effusion; h) Widened Mediastinum; i) Vertebral Fractures; j) Clavicular Fracture; k) Pneumothorax.[]{data-label="fig:chest_images"}](Widened_Mediastinumh.png "fig:"){width="90.00000%"}
[0.12]{} ![Example chest radiographs for each categories. a) Normal; b) Cardiomegaly; c) Consolidation; d) Pulmonary Edema; e) Lung Nodules; f) Lung Mass; g) Pleural Effusion; h) Widened Mediastinum; i) Vertebral Fractures; j) Clavicular Fracture; k) Pneumothorax.[]{data-label="fig:chest_images"}](Vertebral_Fracturesh.png "fig:"){width="90.00000%"}
[0.12]{} ![Example chest radiographs for each categories. a) Normal; b) Cardiomegaly; c) Consolidation; d) Pulmonary Edema; e) Lung Nodules; f) Lung Mass; g) Pleural Effusion; h) Widened Mediastinum; i) Vertebral Fractures; j) Clavicular Fracture; k) Pneumothorax.[]{data-label="fig:chest_images"}](p_Fracturesh.png "fig:"){width="90.00000%"}
[0.12]{} ![Example chest radiographs for each categories. a) Normal; b) Cardiomegaly; c) Consolidation; d) Pulmonary Edema; e) Lung Nodules; f) Lung Mass; g) Pleural Effusion; h) Widened Mediastinum; i) Vertebral Fractures; j) Clavicular Fracture; k) Pneumothorax.[]{data-label="fig:chest_images"}](pneumothoraxh.png "fig:"){width="90.00000%"}
\[T:general\_consensus\_mamo\]
[width=1,center=]{}
Label Count Breast Density
------------------------------------------- ------- ------- ---------------- -----------------------------------------------------------------------------------------
Almost entirely fatty 0 342 $<25\%$ fat; fatty; includes variations of mainly fatty; predominantly fatty; predominantly fat
Scattered areas of fibroglandular density 1 1546 $25-50\%$ scattered
Heterogeneously dense 2 1510 $51-75\%$ heterogenous; heterogeneously
Extremely dense 3 436 $>75\%$ dense; extremely dense; very dense
Ambiguous 4 246 - mild dense; mildly dense; overlap in describing categories with label 1 and 2
\[T:breast\_consensus\]
\[T:breast\_sample\]
[width=1,center=]{}
[|l|l|]{} &\
Almost entirely fatty &
------------------------------------------------------------------------------------------------------------------------------------------------
The breasts are almost entirely fatty bilaterally. No suspicious calcification, dominant mass or architectural distortion. No interval change.
------------------------------------------------------------------------------------------------------------------------------------------------
\
Scattered areas of fibroglandular density &The breasts show scattered fibroglandular densities. There are no dominant nodules or suspicious calcification seen in either breast.\
Heterogeneously dense & The breast parenchyma is heterogeneously dense. No suspicious masses, calcification or architectural distortion seen on either side.\
Extremely dense & The breast parenchyma is extremely dense. No suspicious masses, calcifications or architectural distortion seen on either side.\
\[T:general\_consensus\_mamo\]
[width=1,center=]{}
Label Count
--------------------- ------- ------- -------------------------------------------------------------------------------------------------------------------
Normal 0 116 No acute findings; unremarkable study; the cardiopericardial silhouette and hilar anatomy is within normal limits
Cardiomegaly 1 106 Mild cardiomegaly/pericardial effusion; enlarged cardiac silhouette; cardiopericardial silhouette is enlarged
Consolidation 2 81 Resolving consolidation in the lower lungs bilaterally, with overall improved aeration in the lower lungs
Pulmonary Edema 3 104 Mild to moderate interstitial pulmonary edema; airspace edema; right pulmonary edema
Lung Nodules 4 118 Calcified granulomas present; nodules have developed; multiple faint bilateral pulmonary nodules
Lung Mass 5 89 Bilateral pulmonary masses; mass in the right upper lobe; middle mediastinal mass
Pleural Effusion 6 144 Persistent bilateral pleural effusions; bilateral pleural effusions; persistent loculated right pleural effusion
Widened Mediastinum 7 50 Widening of the superior mediastinum; mediastinum appears widened; mediastinum is slightly widened
Vertebral Fractures 8 67 Vertebral fracture; several old vertebral compression injuries; thoracolumbar vertebral compression injuries
Clavicular Fracture 9 73 Fracture deformity of the right lateral clavicle; right lateral clavicle fracture
Pneumothorax 10 82 Partial right upper lobe collapse; chronic collapse of the right middle lobe; right apical pneumothorax
\[T:chest\_consencus\]
\[T:chest\_sample\]
[width=1,center=]{}
[|l|l|]{} &\
Normal &
------------------------------------------------------------------------------------------------------
Cardiomediastinal contours are within normal limits. The hila are unremarkable. The lungs are clear.
------------------------------------------------------------------------------------------------------
\
Cardiomegaly &
-------------------------------------------------------------------------------------------------------------------------------------------------------
The cardiopericardial silhouette is enlarged with LV prominence. Aortic valve prosthesis in situ. There is unfolding of the aorta with calcification.
-------------------------------------------------------------------------------------------------------------------------------------------------------
\
Consolidation &
-----------------------------------------------------------------------------------------------------------------------------
The lungs remain overinflated and show mild chronic parenchymal changes. Inhomogeneous airspace consolidation has developed
in the basal segments of the left lower lobe.
-----------------------------------------------------------------------------------------------------------------------------
\
Pulmonary Edema&
--------------------------------------------------------------------------------------------------------------
Significant fluid has developed in the right minor fissure. Pulmonary venous markings are severely elevated.
--------------------------------------------------------------------------------------------------------------
\
Lung Nodules &
----------------------------------------------------------------------------------------------------------
The left pulmonary nodular density superimposed on the posterior seventh rib is smaller.
There is mild stable bilateral upper lobe pleuropulmonary scarring and a stable right upper lobe nodule.
----------------------------------------------------------------------------------------------------------
\
Lung Mass&
---------------------------------------------------------------------------------------------------------------
Left upper lobe perihilar mass. Prominence of the left hila as well as the left superior mediastinal margins.
---------------------------------------------------------------------------------------------------------------
\
Pleural Effusion&
---------------------------------------------------------------------------------------------------------------------
Midline sternotomy wires appear stable in position. Small bilateral pleural effusions unchanged from previous exam.
---------------------------------------------------------------------------------------------------------------------
\
Widened Mediastinum&
------------------------------------------------------------------------------------------------------------
Superior mediastinal widening and vascular engorgement relate to patient positioning. The lungs are clear.
------------------------------------------------------------------------------------------------------------
\
Vertebral Fractures &
-----------------------------------------------------------------------------------------------------------------
The cardiac silhouette is normal in size and shape with aortic unfolding. The lungs are overinflated but clear.
The pleural spaces, mediastinum and diaphragm appear normal.
-----------------------------------------------------------------------------------------------------------------
\
Clavicular Fracture&
-----------------------------------------------------------------------------
Degenerative changes in the spine and an old right mid clavicular fracture.
-----------------------------------------------------------------------------
\
Pneumothorax &
--------------------------------------------------------------------------------------------------------------------------
Right-sided pleural drain in situ. There is very subtle residual right pneumothorax with lateral pleural edge displaced.
--------------------------------------------------------------------------------------------------------------------------
\
The Datasets {#sec:data}
============
Our institutional review board approved this single-center retrospective study with a waiver for informed consent. The Research Ethics Boards (REB) number is 17-167. A search of our Radiology Information System (RIS) (Syngo; Siemens Medical Solutions USA Inc, Malvern, PA) was preformed for mammography and chest radiograph reports using Montage Search and Analytics (Montage Healthcare Solutions, Philadelphia, PA). This search identified 4,080 mammography reports and 1,030 chest radiographic reports (see Table \[T:stats\]). These reports were exported from the RIS and removed of any patient identifying information.
Mammogram Reports Dataset
-------------------------
The mammography reports dataset (MRD) contained 4,080 reports. Breast density was classified in each report according to the American College of Radiology Breast Imaging Reporting and Data System (BI-RADS) classification system [@taco1998american] (Figure \[fig:breast\_images\]). Breast density reflects the relative composition of fat and fibroglandular tissue. Almost entirely fatty refers to breasts with less than $25\%$ areas of fibroglandular tissue. Scattered ares of fibroglandular density is used for breasts composed of $25-50\%$ fibroglandular tissue. Heterogeneously dense breasts are composed of $50-75\%$ fibroglandular tissue and the extremely dense breasts have greater than $75\%$ of fibroglandular tissue density. Descriptive statistics of this dataset are presented in Table \[T:breast\_consensus\] and sample reports are presented in Table \[T:breast\_sample\].
Chest Radiographs Reports Dataset
---------------------------------
The chest radiograph dataset (CRRD) consisted of 1,030 reports. These reports were classified into 11 categories which includes normal examinations and ten pathologic states (Figure \[fig:chest\_images\]). Descriptive statistics of this dataset are presented in Table \[T:chest\_consencus\] and sample reports are presented in Table \[T:chest\_sample\]. This dataset has more categories, fewer reports per category, and is less imbalanced when compared with the MRD.
[0.33]{} {width="105.00000%"}
[0.33]{} {width="105.00000%"}
[0.33]{} {width="105.00000%"}
Experiments and Results Analysis {#sec:results}
================================
In this section, we compare performance of the proposed Bi-CNN model with the random forest (RF), SVM, and CNN models. The experiments were conducted on both the mammogram and chest radiograph datasets.
The CNN and Bi-CNN models are implemented in TensorFlow [@abadi2016tensorflow] and the RF and SVM models are implemented using the classifiers in scikit-learn [@pedregosa2011scikit]. The experiments are conducted on a DevBox with an Intel Core i7-5930K 6 Core 3.5GHz desktop processor, 64 GB DDR4 RAM, and two TITAN X GPUs with 12GB of memory per GPU.
Parameters Setting
------------------
Unless stated, the CNN and Bi-CNN models are trained with a mini-batch size of 64, drop-out probability of 0.5, filter sizes {3, 4, 5} and 120 feature maps per filter size. The number of training iterations is set to 50 and the initial learning rate for Adam optimizer is set to 0.001 [@kingma2014adam]. An exponential decay adaptive learning rate is applied. The weights at output layers are initialized using the Xavier method [@glorot2010understanding], the weights in the convolutional layer are selected based on normal distribution with standard deviation of 0.1, and biases are set to 0.1. The activation function before the max-pooling layer is ReLU [@nair2010rectified]. The $L_{2}$ regularization is set to $1.0 \times 10^{-4}$ and early-stopping is applied.
For all the experiments, $70\%$, $15\%$, and $15\%$ of data is allocated for training, validation, and test, respectively. The data is shuffled before splitting. In each experiment, the model is cross-validated over 30 independent experiments and the results after statistical testing are reported.
Performance Evaluation and Analysis
-----------------------------------
We experimented with various configurations of the RF [@breiman2001random], SVM [@hearst1998support], and CNN [@kim2014convolutional] models.
- **RF**: Instead of using a bag-of-the-words technique, which uses the frequency of the words in a query as the features, a $n$-gram model with a lower boundary of 1 and upper boundary $n\in\{1,2,3\}$ performs the feature extraction of reports. We use an implementation of RF [@breiman2001random] which combines classifiers by averaging their probabilistic predictions. The number of estimators is set to 10.
- **SVM**: Similar to RF, the SVM model uses $n$-gram feature extraction. The SVM model is deployed for two different kernels, a “sigmoid" and a “polynomial" with degree three.
- **CNN**: The CNN implementation proposed in [@kim2014convolutional] for text classification and later used in [@shin2017classification] for the classification of radiology head CT reports is used. This model has a single input channel and the order of input words is similar to the order in the report. The model has a convolution layer followed by a max-pooling layer and a “softmax" classifier. The studies are for kernel sizes $k\in\{1,2,3\}$.
- **Bi-CNN**: The proposed Bi-CNN with two input channels. The settings are similar to CNN.
The experiments are for breast density classification and chest radiograph classifications based on chest pathology using the MRD and CRRD, respectively. The results of performance comparisons between the RF, SVM, CNN, and Bi-CNN models are presented in Table \[T:performance\]. As $n$ increases in $n$-gram for RF, it considers the dependency between a greater number of words (i.e. $n$). The RF with 3-gram model has better performance than the 2-gram and 1-gram RF models. The SVM with polynomial of degree three (i.e., “poly") and “sigmoid" kernels almost have the same performance. The number of kernels in the CNN models have a minor impact on accuracy. However, the CNN models with 3-kernel have better performance than models with 1-kernel and 2-kernel. The proposed Bi-CNN with 3-kernel has the best performance compared to the other models.
\[T:performance\]
MRD CRRD
------------------- ----------- -----------
RF (1-gram) 69.73 67.32
RF (2-gram) 73.76 69.27
RF (3-gram) 80.53 77.59
SVM (sigmoid) 82.74 80.24
SVM (poly) 82.97 81.73
CNN (1-kernel) 85.72 85.27
CNN (2-kernel) 86.52 86.01
CNN (3-kernel) 86.91 86.05
Bi-CNN (1-kernel) 89.60 89.16
Bi-CNN (2-kernel) 90.88 89.91
Bi-CNN (3-kernel) **92.94** **91.34**
: Classification accuracy (in $\%$) for MRD and CRRD using RF, SVM, CNN, Bi-CNN. The best results are in boldface.
[width=0.48]{}
----------- ------- ----- ------- ----- ----------- --------- ----------- ---------
$\lambda$ Acc. C Acc. C Acc. C Acc. C
0.1 82.54 36 76.42 43 **90.03** **23** **84.04** **26**
0.01 83.27 100 82.78 100 **89.98** **100** **87.85** **100**
0.001 86.91 100 86.05 100 **92.94** **100** **91.34** **100**
----------- ------- ----- ------- ----- ----------- --------- ----------- ---------
: Accuracy (Acc. in $\%$) and convergence iteration (C) of CNN and Bi-CNN models for learning rate $\lambda$ on MRD and CRRD datasets. Best results are in boldface.
\[T:lr\_analysis\]
The convergence behaviour of CNN and Bi-CNN over training iterations for the validation dataset is presented in Figure \[fig:acc\]. For a large learning rate value (i.e., $\lambda=0.1$), the models converge rapidly to a local solution. However, for smaller learning rates, the models converge more slowly but with greater stability towards a better solution. As the results in Table \[T:lr\_analysis\] show, the CNN model converges with a lower accuracy for three different learning rates $\lambda\in\{0.1,0.01,0.001\}$. For $\lambda=0.001$, the best performance is achieved by Bi-CNN for both MRD and CRRD. One of the advantages of Bi-CNN is adding diversity into the model through the integration of two different representations of feature vectors into the model. This diversity helps the CNN avoid convergence to local solutions with low accuracy.
Conclusion {#sec:conclusion}
==========
Radiology reports contain a radiologist’s interpretation of a medical imaging examination. Vast numbers of such reports are digitally stored in health care facilities. Text mining and knowledge extraction from such data may potentially be used for double checking radiologist interpretations as part of an audit system. Furthermore, these methods can be used to audit utilization of limited health care resources and to enhance the study of patient treatment plans over time.
In this paper, we have proposed a Bi-CNN for the interpretation of radiology reports. We have collected and anonymized two datasets for our experiments: a mammogram reports dataset and a chest radiograph reports dataset. The Bi-CNN has two input channels where one channel represents the features of the report (including zero-padding) and the other channel represents the reverse non-padded features of the report. The combination of forward and reverse order of feature vectors represents more information about dependencies between feature vectors. Our comparative studies demonstrate that the Bi-CNN model outperforms the CNN, RF, and SVM methods.
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors thank the support of NVIDIA Corporation with the donation of the Titan X GPUs used for this project.
[^1]: H. Salehinejad is with the Edward S. Rogers Sr. Department of Electrical & Computer Engineering, University of Toronto, Toronto, Canada, and Department of Medical Imaging, St. Michael’s Hospital, University of Toronto, Toronto, Canada, e-mail: [email protected].
[^2]: S. Valaee is with the Edward S. Rogers Sr. Department of Electrical & Computer Engineering, University of Toronto, Toronto, Canada, e-mail: [email protected].
[^3]: A. Mnatzakanian, T. Dowdell, J. Barfett, and E. Colak are with the Department of Medical Imaging, St. Michael’s Hospital, University of Toronto, Toronto, Canada, e-mail: {mnatzakaniana,dowdellt,barfettj,colake}@smh.ca.
[^4]:
|
{
"pile_set_name": "ArXiv"
}
|
[**ON THE NATURE OF THE GAUSSIAN APPROXIMATIONS**]{}\
[**IN PHASE ORDERING KINETICS**]{}\
\
\
\
S. De Siena and M. Zannetti\
[*Dipartimento di Fisica, Università di Salerno\
84081 Baronissi (SA), Italy*]{}
\
[**Abstract**]{}
> The structure of the gaussian auxiliary field approximation in the theory of phase ordering kinetics is analysed with the aim of placing the method within the context of a systematic theory. While we are unable to do this for systems with a scalar order parameter, where the approximation remains uncontrolled, a systematic development about the gaussian approximation can be outlined for systems with a vector order parameter in terms of a suitably defined $1/N$-expansion.
\
05.70.Fh 64.60.Cn 64.60.My 64.75.+g
1 - Introduction {#introduction .unnumbered}
================
Although much progress has been made in the understanding of phase ordering kinetics$^{1,2}$, from the point of view of the theorist basically this remains an unsolved problem. The reason is that in the case of a scalar order parameter, which is the most relevant for experiments, no systematic scheme for the development of a perturbation theory is available. This requires the existence of a soluble zero order approximation which accounts, at least qualitatively, for the relevant physical features of the problem and of a well defined procedure for the calculation, at least in principle, of the higher order corrections. The case of a system with continous symmetry is in better shape since the $1/N$-expansion meets these requirements, at least in the case of a non conserved order parameter. For a conserved order parameter there are indications that the large-N limit might be singular$^{3}$.
Despite this very unsatisfactory situation, recently much progress has been made in the development of analytical methods for the computation of the structure factor, through extensive use of approaches based on the introduction of a gaussian auxiliary field$^{4,5}$ (GAF) which improve on the original idea of Ohta, Jasnow and Kawasaki$^{6}$ (OJK). An exaustive critical account of these theories has been given by Yeung, Oono and Shinozaki$^{5}$. The success of this approach, which for the moment is mostly limited to non conserved order parameter, amounts to the very accurate reproduction of the scaling function for scalar order parameter$^{4,5,6}$ as known from experiments or numerical simulations, and to the prediction of power law tails in the case of vector order parameter$^{7,8}$. Thus, these theories seem to incorporate those basic ingredients that a real theory of phase ordering kinetics should have. The shortcoming is that the assumption that the auxiliary field obeys gaussian statistics is totally uncontrolled. More than theories for the moment these are sophisticated computational prescriptions which are justified a posteriori.
A substantial progress toward a systematic theory then would be made if it were possible to identify a scheme within which a GAF approximation plays the role of the zero order approximation together with the expansion parameter which generates the higher order corrections. This type of project is illustrated in the paper of Bray and Humayun$^{9}$.
Motivated by these considerations here we overview the GAF approximations with the aim of exposing those features which help to put into focus what is required for the eventual development of a systematic theory. For pedagogical reasons we begin with a detailed discussion of the one particle problem which is exactly soluble and therefore allows to illustrate clearly what is involved in a GAF-type approximation. The same pattern of analysis than will be applied to the field theory case.
2 - One Particle {#one-particle .unnumbered}
================
Let us consider one particle in a double well potential and in contact with a thermal reservoir. The decay process from the instability point of this system has been thouroughly studied in the literature$^{10}$. In the limit of zero temperature the equation of motion for the position $\phi(t)$ is given by $${\dot \phi} = r\phi-g\phi^3$$ with $r>0 \,,\, g>0$. In order to study the quench of this system from high temperature to zero temperature, let us consider a gaussian probability distribution for the initial condition $\phi_{0}=\phi(t_{0})$ $$P_{0}(\phi_{0}) = {1 \over {\sqrt{2 \pi \Delta}}}
e^{-{\phi_{0}^2 \over {2 \Delta}}}.$$ Due to the symmetry of the problem the average position of the particle vanishes identically $<\phi(t)> \equiv 0$ and we concentrate on the behaviour of the fluctuations $$S(t) = <\phi^2(t)> = \int_{-\infty}^{+\infty} d\phi P(\phi,t) \phi^2$$ where $P(\phi,t)$ is the probability that the particle occupies the position $\phi$ at the time $t$. This quantity can be computed exactly since the equation of motion (2.1) can be solved $$\phi(t) = f(t-t_{0},\phi_{0}) = {{\tau \phi_{0}}
\over {[1+(g/r)\phi_{0}^2(\tau^2-1)]^{1/2}}}$$ with $\tau=e^{r(t-t_{0})}$. Thus, for the probability density we find $$\begin{aligned}
&& P(\phi,t)=P_{0} \bigl(f^{-1}(t-t_{0},\phi) \bigr) {{df^{-1}(t-t_{0},\phi)}
\over {d\phi}} \nonumber \\
&& \\
&&={1 \over {\sqrt{2 \pi \Delta \tau^2}}} {{\exp \{-{1 \over {2 \Delta \tau^2}}
{\phi^2 \over {[1-(g/r)\phi^2(1-\tau^{-2})]}}\} } \over
{[1-(g/r) \phi^2 (1-\tau^{-2})]^{3/2}}} \nonumber\end{aligned}$$ where $f^{-1}(t-t_{0},\phi_{0})$ is the inverse of $f(t-t_{0},\phi_{0})$. Inserting into (2.3) $$S(t)=\frac{r}{g}\frac{\tau^2}{(\tau^2-1)} \bigl\{1-{\sqrt{\pi x}}e^{x}
[1-erf({\sqrt{x}})] \bigr\}$$ where $x=r/[2 \Delta g (\tau^2-1)]$ and $erf(z)$ is the error function. For short time (2.6) yields exponential growth of the fluctuations $$S(t) \sim \Delta \tau^2$$ due to the initial instability, while for large time we get $$S(t) \sim \frac{r}{g}\frac{\tau^2}{(\tau^2-1)}[1-{\sqrt{\pi x}}]$$ which describes the saturation toward the finite equilibrium value $S(\infty)=\phi^{2}_{eq}=r/g$ due to the fact that eventually the particle sits at the bottom of one of the two potential wells and the probability density (2.5) develops two narrow peaks centered about the equilibrium values $\phi_{eq}=\pm \sqrt{r/g}$.
If the exact solution of the problem was not available, this type of behaviour could not have been obtained via a straightforward perturbation expansion in the nonlinear coupling $g$. Zero order amounts to take the gaussian approximation in (2.5) which describes only the regime of exponential growth (2.7). Hence, the zero order theory does not reproduce the qualitative picture of the process, neither any improvement is obtained by taking into account corrections of finite order. The saturation to a finite final equilibrium value is obtained within a perturbative scheme, as shown by Suzuki$^{10}$, by resorting to the infinite resummation of the most divergent terms in the series.
However, rather than following this route, let us use the method which in the following will be generalized to the field theory case. The idea is to introduce a new auxiliary variable $m(t)$ through a transformation $$\phi(t)=\sigma(m(t))$$ which takes care of the basic non-linear features of the problem in such a way that the behaviour of $m(t)$ can be treated by straightforward perturbation theory. Namely, the transformation must be such that while $m(t)$ is allowed to grow indefinitely the saturation of $\phi(t)$ to a finite value is induced by $\sigma$.
Substituting (2.9) into (2.1) one obtains the equation of motion for $m(t)$ $${\dot m}={\sigma(m) \over \sigma^{'}(m)}[r-g \sigma^2 (m)]$$ with the transformed probability density of initial conditions $$P_{m_{0}}(m_{0})=P_{0}(\sigma(m_{0})|R) \sigma^{\prime}(m_{0})$$ where $m_{0}=m(t_{0})$ and $P_{0}(\phi_{0}|R)$ is the probability density (2.2) conditioned to $\phi_{0}$ belonging to the range of values $R$ for which (2.9) is invertible. Thus, in terms of $m(t)$ we cannot quite get the fluctuations (2.3), but fluctuations conditioned to $\phi \in R$ $$S(t|R)=\int d\phi P(\phi,t|R) \phi^{2}=\int dm P_{m}(m,t) \sigma^{2}(m)$$ where $R$ is the domain $$R=(\phi^{2} \leq r/g)$$ and $P_{m}(m,t)$ is the probability density of $m$ at the time $t$. How important this restriction is depends on what is the statistical weight of trajectories lying outside $R$ and this in turn is related to the size of the variance $\Delta$ of the initial probability density (2.2) compared to the size $r/g$ of the domain $R$. In the following we shall neglect the distinction between $S(t|R)$ and $S(t)$ by assuming $\Delta \ll r/g$.
Now, if the transformation $\sigma$ is such that (2.10) can be solved, at least in perturbation theory, denoting the solution by $m(t)=h(t-t_{0},m_{0})$ we have $$\begin{aligned}
&& P_{m}(m,t)=P_{m_{0}} \bigl(h^{-1}(t-t_{0},m)\bigr)
{{dh^{-1}(t-t_{0},m)} \over {dm}}= \nonumber \\
&& \\
&& P_{0} \bigl(\sigma\bigl(h^{-1}(t-t_{0},m) \bigr)|R \bigr)
{{dh^{-1}(t-t_{0},m)} \over {dm}}
\sigma^{\prime} \bigl(h^{-1}(t-t_{0},m) \bigr) \nonumber\end{aligned}$$ which formally solves the problem since it gives an explicit expression for $P_{m}(m,t)$ in terms of the known quantities $P_{0}$,$\sigma$ and $h$.
In order to see how this works in practice let us go back to the equation of motion (2.10) for $m(t)$ and let us look for the transformation $\sigma$ which simplifies as much as possible the behaviour of $m(t)$. The first attempt is for an outright linearization of (2.10). If this was not possible then, as stated above, $\sigma$ ought to be such that (2.10) can be solved in perturbation theory. However, in this case linearization can be achieved. Imposing $${\sigma(m) \over \sigma^{\prime}(m)}[r-g\sigma^{2}(m)]=rm$$ one finds $$\phi=\sigma(m)={m \over [1+(g/r)m^{2}]^{1 \over 2}}$$ and $$m(t)=h(t-t_{0},m_{0})=\tau m_{0}.$$ Indeed, we have that while $m(t)$ grows exponentially $\phi(t)$ eventually saturates via (2.16) to the final equilibrium value $\phi_{eq}=\pm \sqrt{r/g}$. Namely, the transformation $\sigma$ accounts for the nonperturbative features of the problem.
Putting together (2.14),(2.16) and (2.17) we have the exact solution of the problem in terms of the auxiliary variable $m(t)$. The motivation for going to this form of the solution is that in more complicated cases where $h$ and therefore $P_{m}(m,t)$ cannot be explicitely obtained, the consideration that the auxiliary variable $m(t)$ should not be much affected by the nonlinear nature of the problem authorizes to attempt a gaussian ansatz for $P_{m}(m,t)$. This will be the crucial step of the GAF approximation in the phase ordering problem. The difficulty with an ansatz however is that it may not be possible to control the corrections to it. In any case it should be clear that a gaussian ansatz does not amount to an overall linearization of the problem, since in (2.12) the ansatz amounts to use a gaussian form for $P_{m}(m,t)$ while the nonlinearity remains through the explicit factor $\sigma^{2}(m)$. To be more specific, since $P_{0}$ is gaussian it is evident from (2.14) that $P_{m}(m,t)$ is gaussian if $\sigma$ and $h$ are linear. Thus, should it be possible to find an expansion parameter $\lambda$ such that $\sigma$ and $h$ become linear for $\lambda \rightarrow 0$, the gaussian approximation amounts to take this limit inside $P_{m}(m,t)$ in (2.12) [*but not*]{} in the explicit factor $\sigma^{2}(m)$.
Let us see how this works in the one particle context. Since in this case $h(t-t_{0},m)$ is already linear, in order to make $P_{m}(m,t)$ gaussian we need to linearize only $\sigma$ in (2.14). From (2.16) we may write $$\sigma^{2}(\tau^{-1}m)=\tau^{-2} m^{2}-\frac{g}{r}{{\tau^{-4}m^{4}}
\over {[1+(g/r)\tau^{-2}m^{2}]}}$$ and using this in (2.14) we obtain $$P_{m}(m,t)=P^{(0)}_{m}(m,t)K(\tau^{-1}m,g/r)$$ with $$P^{(0)}_{m}(m,t)={1 \over \sqrt{2 \pi \Delta \tau^{2}}}
\exp \{-{m^{2} \over {2 \Delta \tau^{2}}} \}$$ and $$K(\tau^{-1}m,g/r)=\frac{\exp \{\frac{g}{r}\frac{\tau^{-4}m^{4} }
{[1+(g/r)\tau^{-2}m^{2}]} \} }{[1+(g/r)\tau^{-2}m^{2}]^{\frac{3}{2}}}.$$ Thus, in this case it is possible to identify the nonlinear coupling $g$ with the expansion parameter $\lambda$ which generates gaussian statistics for $m$ in the limit $\lambda \rightarrow 0$. Then, following the previous discussion, the lowest order is obtained by letting $g \rightarrow 0$ in (2.19) but not in the explicit factor $\sigma^{2}(m)$. From (2.12) then we get $$S^{(0)}(t)=\int_{-\infty}^{\infty}
dm P^{(0)}(m,t) \sigma^{2}(m)=
\frac{r}{g} \bigl\{1-\sqrt{\pi y}e^{y}[1-erf(\sqrt{y})] \bigr\}$$ with $y=r/(2\Delta \tau^{2} g)$ which gives $S^{(0)}(t) \sim
\Delta \tau^{2}$ at short time as in (2.7) and $S^{(0)}(t)
\sim (r/g) [1-\sqrt{\pi y}]$ for long time. The qualitative effect of the saturation is correctly reproduced, altough there is a quantitative discrepancy with (2.8) in the law of approach to equilibrium. In conclusion, in the one particle case the gaussian approximation can be identified with the zero order step in a systematic development where higher order corrections are generated by expanding $K(\tau ^{-1}m,g/r)$ in powers of $g$.
3 - Phase Ordering Dynamics {#phase-ordering-dynamics .unnumbered}
===========================
Let us now turn to the field theory case. The phase ordering dynamics following the quench from high temperature to zero temperature of a system with a non conserved order parameter is described by the equation of motion $$\frac{\partial \phi(\vec x,t)}{\partial t}=\nabla^{2} \phi(\vec x,t)
-V^{\prime} \bigl(\phi(\vec x,t) \bigr)$$ with a gaussian initial state which generalizes (2.2) $$P_{0} \bigl[\phi_{0}(\vec x) \bigr]={1 \over {Z_{0}}}
e^{-{1 \over {2 \Delta}} \int d^{d}x \phi_{0}^{2}(\vec x)}$$ and where $V(\phi)$ is a potential of the double well type.
Again, due to the symmetry of the problem the average order parameter vanishes identically $<\phi(\vec x,t)> \equiv 0$ and the observable of interest is the equal time correlation function $$G(\vec u,t)=<\phi(\vec x_{1},t) \phi(\vec x_{2},t)>=
\int d\phi_{1} d\phi_{2} P(\phi_{1},{\vec x_{1}} t;\phi_{2},{\vec x_{2}} t)
\phi_{1} \phi_{2}$$ or the structure factor $$C(\vec{k},t)=\int d^{d}x e^{i\vec{k} \cdot \vec{u}}
G(\vec{u},t)$$ where $\vec{u}=\vec{x}_{1}-\vec{x}_{2}$. In (3.3) $P(\phi_{1},\vec{x}_{1}t;
\phi_{2},\vec{x}_{2}t)$ is the joint probability density that $\phi(\vec{x},t)$ takes the value $\phi_{1}$ at the space-time point $(\vec{x}_{1},t)$ and the value $\phi_{2}$ at the space-time point $(\vec{x}_{2},t)$.
It has been well established$^{1,2}$, both from experiment and numerical simulations, that in the late stage of the dynamics these quantities obey scaling $$G(\vec{u},t) \sim f(u/L(t))$$ $$C(\vec{k},t) \sim L^{d}(t)g(kL(t))$$ where $L(t)$ is the basic length in the problem which is related to the average size of domains and obeys the growth law $L(t) \sim t^{1/2}$, while $f(x)$ and $g(x)$ are scaling functions. The origin of scaling is that in the late stage the order parameter reaches local equilibrium and forms domains of the ordered phases which evolve according to self-similar patterns. From the existence of sharp interfaces separating domains one can deduce$^{2}$ the short distance behaviour of $f(x)$ or the long wavelength behaviour of $g(x)$ (Porod’s law) $$f(u/L)=1-2u/L+...\,\,\,\, for \,\, u/L<<1$$ $$g(kL) \sim (kL)^{-(d+1)} \,\,\,\, for \,\, kL>>1$$ as well as the saturation law$^{4}$ of the order parameter $$S(t)= G(\vec{u}=0,t)=\phi_{eq}^{2} \bigl[1-\frac{a}{L(t)}+O(L^{-2}) \bigr]$$ where $\phi_{eq}$ is the value of the order parameter in the final equilibrium state. Eq.s from (3.5) to (3.9) contain the minimal phenomenological information that a theory of phase ordering dynamics should account for.
At this point it is important to emphasize that the scaling behaviour described above applies to the late stage of the process where domains are close to saturation and grow through the motion of the interfaces. This stage of the dynamics is dominated by the nonlinear nature of the problem and much as in the one particle case it cannot be obtained through any straightforward perturbation expansion. The great difference with the one particle case is that Eq. (3.1) cannot be solved for any realistic potential. Therefore, in order to make analytic progress, we turn to the generalization of the auxiliary variable method.
4 - Auxiliary Field Method {#auxiliary-field-method .unnumbered}
==========================
Following the idea illustrated above we now introduce an auxiliary field through a local nonlinear transformation $$\phi (\vec x,t)=\sigma (m(\vec x,t))$$ which in general is defined through a relation involving the potential $$K[\sigma(m)]=V^{\prime}(\sigma).$$ We note that such a transformation cannot be a linearising transformation as it was in the one particle case. In fact in that case $m(\vec x,t)$ ought to be the free field and the relation between the free field and the interacting field is certainly nonlocal, as it can be easily seen generating the formal solution of (3.1) by iteration. Thus, the transformation (4.1) is introduced in order to take care at least of the gross nonlinear effect which is the saturation of the order parameter to the finite final equilibrium value $\phi_{eq}$, leaving the rest, possibly, to perturbation theory. Accordingly, for large time the transformation must go over to the form $$\sigma (m(\vec x,t))=\phi_{eq} \,\, sign(m(\vec x,t)).$$
The equation of motion of the auxiliary field is obtained from (3.1) $$\frac{\partial m}{\partial t}=\nabla^{2}m+\frac{1}{\sigma^{\prime}}
[\sigma^{\prime \prime}(m)(\nabla m)^{2}-V^{\prime}(\sigma)]$$ with the transformed initial condition $$P_{m_{0}}[m_{0}(\vec x)]=P_{0}[\sigma(m_{0}(\vec x))]J(\phi_{0},m_{0})$$ where $J(\phi_{0},m_{0})$ is the Jacobian of the transformation (4.1) at the initial time. Representing the solution of (4.4) as a functional of the initial configuration labeled by $\vec x$ and $t$ $$m(\vec x,t)=h(\vec x,t-t_{0};[m_{0}({\vec x}^{\prime})])$$ the probability of a configuration $[m(\vec x)]$ at the time $t$ can be obtained in terms of the initial probability density (3.2) $$\begin{aligned}
&& P_{m}[m(\vec x),t]=P_{m_{0}} \bigl[h^{-1}(\vec x,t-t_{0},
[m({\vec x}^{\prime})]) \bigr]J(m_{0},m) \nonumber \\
&& \\
&& =P_{0} \bigl[\sigma
\bigl(h^{-1}(\vec x,t-t_{0},[m({\vec x}^{\prime})]\bigr) \bigr) \bigr]
J(\phi_{0},m_{0})J(m_{0},m) \nonumber\end{aligned}$$ where $h^{-1}$ is the inverse of (4.6) and $J(m_{0},m)$ is the Jacobian of this transformation. The above result is the analogue of (2.14) and specifies the statistics of the auxiliary field $m(\vec x,t)$ in terms of $\sigma , h$ and the statistical properties of the initial condition.
Neglecting for simplicity considerations pertaining to the restriction of averages to domains of configurations where (4.1) is invertible, the correlation function (3.3) may be rewritten as $$G(\vec u,t)=\int dm_{1} dm_{2}
P_{m}(m_{1},{\vec x}_{1} t;m_{2},{\vec x}_{2} t)
\sigma(m_{1}) \sigma(m_{2})$$ where the joint probability of $m$ is related to (4.7) by $$P_{m}(m_{1},{\vec x}_{1} t;m_{2},{\vec x}_{2} t)=
\int d[m(\vec x)] P_{m}[m(\vec x),t]
\delta(m_{1}-m({\vec x}_{1})) \delta (m_{2}-m({\vec x}_{2})).$$ The above form (4.8) for the correlation function makes a progress over (3.3) if the joint probability of $m$ is available. This requires that the transformation $\sigma$ is such that Eq.(4.4) for $m$ is soluble. Short of this, as explained in section 2, one resorts to the GAF approximation through the linearization of $\sigma$ and $h$ inside $P_{0}$.
Let us now review the predictions of the GAF approximations. If $m(\vec x,t)$ is gaussian the probability densities are of the form $$P_{m}^{(0)}(m_{1},{\vec x}_{1} t;m_{2},{\vec x}_{2} t)=
\frac{1}{Z_{m}} \exp \{-\frac{1}{2(1-\gamma^2)S_{0}(t)}
[m_{1}^{2}+m_{2}^{2}-2\gamma m_{1} m_{2}] \}$$ and $$P_{m}^{(0)}(m,{\vec x} t)=\frac{1}{\sqrt{2 \pi S_{0}(t)}}
\exp \{-\frac{m^{2}}{2S_{0}(t)} \}$$ with $$\begin{aligned}
S_{0}(t)=<m^{2}(\vec x,t)>_{0} \,\, & , & \,\,
G_{0}(\vec u,t)=<m({\vec x}_{1},t)m({\vec x}_{2},t)>_{0}
\nonumber \\
& & \\
\gamma=\gamma(\vec u,t)=\frac{G_{0}(\vec u,t)}{S_{0}(t)}
\,\, & , & \,\,
Z_{m}=2 \pi S_{0}(t) \sqrt{1-\gamma^{2}} \nonumber\end{aligned}$$ and where $<\cdot>_{0}$ denotes averages with respect to $P_{m}^{(0)}$. Hence, for the fluctuations of the order parameter one has $$S(t)=\int dm P_{m}^{(0)}(m, \vec x,t)\sigma^{2}(m)$$ and for the scaling function $$f \bigl(u/L(t) \bigr)=\int dm_{1} dm_{2}
P_{m}^{(0)}(m_{1},{\vec x}_{1} t;m_{2},{\vec x}_{2} t)
sign(m_{1}) sign(m_{2})=\frac{2}{\pi} {sin}^{-1}(\gamma).$$ Within this approach the problem is reduced to the computation of $G_{0}(\vec u,t)$.
For $h$ to be linear Eq. (4.4) must be of the form $$\frac{\partial m}{\partial t}=\nabla^{2}m+a(t)m$$ where $a(t)$ is some function of time to be determined. Upon linearizing $\sigma$ the initial probability density (4.5) becomes $$P_{m_{0}}[m_{0}(\vec x)]=P_{0}[c m_{0}(\vec x)]$$ where $c$ is a constant. Solving (4.15) by Fourier transform and averaging over initial conditions with (4.16) one finds $$C_{0}(\vec k,t)=S_{0}(t)L^{d}(t)g_{0} \bigl(kL(t) \bigr)$$ $$G_{0}(\vec u,t)=S_{0}(t) \gamma \bigl(|\vec u|/L(t) \bigr)$$ with $$\left \{ \begin{array}{ll}
L(t)=t^{\frac{1}{2}} \\
\\
g_{0}(kL)=\exp (-2(kL)^{2}) \\
\\
\gamma(u/L)=\exp (-\frac{u^{2}}{8 L^{2}}) \\
\\
S_{0}(t)=\frac{\Delta}{c^{2}L^{d}} \exp (2b(t)) \\
\\
b(t)=\int_{0}^{t} d{t^{\prime}} a(t^{\prime}).
\end{array}
\right.$$
Inserting the above expression for $\gamma$ in (4.14) one obtains the Ohta-Jasnow-Kawasaki$^{6}$ result for the scaling function which correctly reproduces the behaviours (3.7) (3.8). It is then matter of studying the behaviour of $S(t)$ and for this we must go over to the specific implementations of the method.
[***On site linearization***]{}
Making a direct extension to the field theory case of the procedure adopted for one particle, let us look for a transformation $\sigma$ which linearizes the on site potential in (4.4) $$-\frac{V^{\prime}(\sigma)}{\sigma^{\prime}(m)}=rm$$ where $r$ is a constant. This yields $$\frac{\partial m}{\partial t}=\nabla^{2}m+rm-Q(m)(\nabla m)^{2}$$ where $$Q(m)=-\frac{\sigma^{\prime \prime}(m)}{\sigma^{\prime}(m)}.$$ With the double well potential of the form $$V(\phi)=-\frac{r}{2} \phi^{2}+\frac{g}{4} \phi^{4}$$ (4.20) reduces to (2.15) yielding as in (2.16) $$\sigma(m)=\frac{m}{{[1+(g/r) m^{2}]}^{\frac{1}{2}}}$$ and $$Q(m)=3 \frac{g}{r} \frac{m}{[1+(g/r) m^{2}]}.$$
However, contrary to what happens for one particle, even though the transformation (4.24) manages to account for the saturation of the order parameter, yet it is not sufficient to linearize the equation of motion. This is done by introducing an approximation, which is optimized by the mean field prescription$^{11}$ $$Q(m)(\nabla m)^{2} \rightarrow 3 \frac{g}{r} <(\nabla m)^{2}>
< \frac{m^{2}}{1+(g/r) m^{2}} > \frac{m}{<m^{2}>}$$ where averages must be computed self-consistenly. Note that although (4.26) yields the best linear approximation to the equation of motion, it remains an uncontrolled approximation since no small parameter emerges which allows to compute corrections to it. According to the general discussion made above the implementation the GAF approximation requires, besides the linearization of the equation of motion, also the linearization of $\sigma$. Setting $\sigma(m)=m$ in (4.24), the initial condition is given by (4.16) with $c=1$.
With (4.26) the equation of motion is of the form (4.15) with $$a(t)=r-3 \frac{g}{r} D_{0}(t) \frac{S(t)}{S_{0}(t)}$$ where $$D_{0}(t)=<(\nabla m)^{2}>=\int_{\vec k} k^{2} C_{0}(\vec k,t).$$ Next, using (4.13) $$S(t)=\frac{r}{g} \bigl\{1-\sqrt{\frac{\pi r}{2gS_{0}(t)}}
e^{\frac{r}{2gS_{0}(t)}}
\bigl[1-erf \bigl( \sqrt{\frac{r}{2gS_{0}(t)}} \bigr) \bigr] \bigr\}$$ and making the assumption to be verified a posteriori that $S_{0}(t)$ grows with time, asymptotically we have $$S(t)=\frac{r}{g} \bigl\{1-\sqrt{\frac{\pi r}{2gS_{0}(t)}}+
O \bigl(\frac{1}{S_{0}(t)} \bigr) \bigr \}.$$ Inserting into (4.27), to dominant order we get $${\dot b}(t)=r-3 \frac{D_{0}(t)}{S_{0}(t)}=r+O(t^{-1})$$ which gives $b(t)=rt$. Next, using (4.19) we find $$S_{0}(t)=\Delta \frac{\exp \{2rt \}}{t^{\frac{d}{2}}}$$ which is consistent with the assumption made about $S_{0}(t)$. Finally, inserting the above result into (4.30) we obtain that $S(t)$ saturates exponentially fast to the equilibrium value $\phi_{eq}^{2}=r/g$, rather than according to a power law as expected from (3.9).
[***KYG-theory***]{}
The behaviour of $S(t)$ obtained above is what one finds resumming the singular perturbation series of Kawasaki,Yalabik and Gunton$^{12}$ (KYG). The KYG theory is contained in the above treatment as a particular case. If in addition to the mean field approximation one makes also an expansion in the nonlinear coupling $g$, to lowest order $Q(m) \equiv 0$ and the equation of motion becomes $$\frac{\partial m}{\partial t}=\nabla^{2} m+rm$$ namely the auxiliary field coincides with the free field. The transformation (4.24) together with (4.33) corresponds exactly to the KYG theory, which therefore in the present context amounts to the statement that all the important nonlinear features of the problem are adequately taken care of by the transformation (4.24).
[***BH-theory***]{}
If in (4.26) we keep the first order in $g$ the equation of motion becomes $$\frac{\partial m}{\partial t}=\nabla^{2} m
+[r-3 \frac{g}{r} <(\nabla m)^{2}>]m$$ which is of the type of the equation obtained by Bray and Humayun$^{9}$ (BH) starting from an [*ad hoc*]{} potential and which leads to the correct behaviour for $S(t)$. In fact in this case (4.27) reduces to $${\dot b}(t)=r-3 \frac{g}{r} S_{0}(t) L^{d}(t)
\int_{k} k^{2} e^{-2 k^{2} t}$$ and setting to zero the left hand side for large time $$S_{0}(t) \sim L^{2}(t).$$ Inserting this result in (4.30) the behaviour (3.9) of $S(t)$ is recovered. This is due to the cancellation of $S_{0}(t)$ in the denominator of (4.27), which occurs only in first order in $g$. Notice that from the above result for $S_{0}(t)$ and the definition (4.19) one obtains $a(t) \sim (d+2)/4t$ which coincides with the form for $a(t)$ introduced by Oono and Puri$^{13}$ in their improvement of the OJK theory.
In summary, the GAF approximation obtained via the linearization of the on site potential i) does not describe correctly the saturation law of the order parameter, except for the very special case where the BH-theory applies, and ii) it is an uncontrolled approximation since there is not a systematic expansion scheme within which it plays the role of the zero order theory.
[***Mazenko transformation***]{}
Let us now go to a different way of introducing the auxiliary field due to Mazenko$^{4}$ where equation (4.2) is chosen in such a way that $\sigma(m)$ reproduces the profile of the static interface $$\sigma^{\prime \prime}(m)=V^{\prime}(\sigma).$$ In this case $m(\vec x,t)$ has the physical interpretation of the distance to the nearest interface. Using (4.37) in (4.4) we obtain $$\frac{\partial m}{\partial t}=\nabla^{2} m+(1-(\nabla m)^{2})Q(m)$$ where $Q(m)$ is still given by (4.22). Note that since $V^{\prime}(\sigma)$ is an odd function from (4.37) and (4.22) follows that also $Q(m)$ is an odd function. Thus, the mean field linearization of (4.38) yields $$\frac{\partial m}{\partial t}=\nabla^{2} m
+[1-<(\nabla m)^{2}>]H(t)m$$ where $H(t)$ is some function of time whose explicit form is not important. Hence Eq. (4.27) now gives $${\dot b}(t)=[1-S_{0}(t)L^{d} \int_{k} k^{2} e^{-2k^{2}t}]H(t)$$ which, apart for the overall factor $H(t)$, is identical to (4.35) and therefore leads to the same result (4.36) for $S_{0}(t)$ which yields the correct behaviour of $S(t)$. Comparing (4.34) with (4.39) we see that the BH-theory is a particular case arising with $H(t)$ constant. Thus, the GAF approximation obtained within the static interface approach yields correct results, but for the same reasons pointed out above it remains an uncontrolled approximation.
[***Generalization of the transformation***]{}
We end up this section by considering a generalization of the transformation obtained by allowing for an explicit time dependence. The idea is to see if in so doing one may get closer to the linearization of the equation for $m(\vec x,t)$, as suggested by recent work of Puri and Bray$^{14}$. Replacing (4.1) by $$\phi(\vec x,t)=\sigma(t,m(\vec x,t))$$ the equation for $m$ becomes $$\frac{\partial m}{\partial t}=\nabla^{2}m+
\frac{1}{{\partial \sigma}/{\partial m}}
\bigl[\frac{\partial^{2} \sigma}{\partial m^{2}}(\nabla m)^{2}
-\frac{\partial \sigma}{\partial t}-V^{\prime}(\sigma) \bigr].$$ Let us then determine the explicit dependence of $\sigma$ on $t$ by imposing $$\frac{\partial \sigma}{\partial t}=-V^{\prime}(\sigma)$$ which yields $$\frac{\partial m}{\partial t}=\nabla^{2}m-Q(t,m)(\nabla m)^{2}$$ with $$Q(t,m)=-\frac{{\partial^{2} \sigma}/{\partial m^{2}}}
{{\partial \sigma}/{\partial m}}.$$ Eq. (4.43) is nothing but the one particle equation of motion which, using the potential (4.23), yields the solution (2.4) i.e. $$\sigma(t,m)=\frac{\tau \sigma(0,m)}
{\bigl[1+\frac{g}{r}\sigma^{2}(0,m)(\tau^{2}-1) \bigr]^{\frac{1}{2}}}$$ and $$Q(t,m)=\frac{\frac{g}{r} \tau^2
\bigl[\sigma^{\prime \prime}\sigma^{2}
-3\sigma(\sigma^{\prime})^{2}\bigr]
+\bigl[\sigma^{\prime \prime}
-\frac{g}{r}
(\sigma^{\prime \prime}\sigma^{2}
-3\sigma(\sigma^{\prime})^{2}) \bigr]}
{\sigma^{\prime} \bigl[\frac{g}{r} \sigma^{2}
-\frac{g}{r} \tau^{2} \sigma^{2}-1 \bigr]}$$ where the sigma’s on the right hand side stand for $\sigma(0,m)$ and the primes denote derivatives with respect to $m$.
Imposing $\sigma^{\prime \prime}\sigma^{2}-3\sigma(\sigma^{\prime})^{2}=0$ we find $\sigma(0,m)=\pm (m)^{-1/2}$ and inserting into (4.47) and (4.46) eventually we have $$\sigma(t,m)=\pm \bigl[\frac{\tau^{2} m}
{1+(g/r)m(\tau^{2}-1)} \bigr]^{\frac{1}{2}}$$ and $$\frac{\partial m}{\partial t}=\nabla^{2}m
-\frac{3}{2}
\frac{\tau^{-2}}{\bigl[\tau^{-2}m-\frac{g}{r}(\tau^{-2}-1) \bigr]}
(\nabla m)^{2}.$$ Indeed, the equation of motion for $m$ is “almost” linear since the nonlinear term vanishes exponentially fast, but the scheme it is not of much use in generating a GAF approximation, since (4.48) cannot be linearized.
5 - Vector Fields {#vector-fields .unnumbered}
=================
Let us now consider the case of a vector order parameter with $N$-components ${\vec \phi}(\vec x)=(\phi_{1}(\vec x),...,\phi_{N}(\vec x))$. In this case a systematic expansion scheme about the GAF approximation can be outlined, although its practical implementation remains to be explored.
Phenomenological expectations in this case are a power law tail in the scaling function of the structure factor$^{7,8}$ $$g(x) \sim x^{-(d+N)}$$ which generalizes Porod’s law and the saturation law$^{8}$ $$S(t)=\phi_{eq}^{2} \bigl[1-\frac{b}{L^{2}(t)}+O(L^{-3}) \bigr]$$ in place of (3.9). Considering the equation of motion $$\frac{\partial \phi_{\alpha}(\vec x,t)}{\partial t}=
\nabla^{2}\phi_{\alpha}(\vec x,t)
-\frac{\partial}{\partial \phi_{\alpha}}V({\vec \phi}(\vec x,t))$$ with the potential $$V(\vec \phi)=-\frac{r}{2}{\vec \phi}^{2}
+\frac{g}{4N}({\vec \phi}^{2})^{2}$$ the auxiliary field ${\vec m}(\vec x,t)$ is introduced by generalizing to the vector case the transformation (4.24) $$\sigma_{\alpha}(\vec m)=
\frac{m_{\alpha}}
{\bigl[1+\frac{g}{rN} {\vec m}^{2} \bigr]^{\frac{1}{2}}}$$ which yields the equation of motion for $\vec m$ $$\begin{aligned}
&& \frac{\partial m_{\alpha}}{\partial t}=
\nabla^{2}m_{\alpha}+rm_{\alpha}-\frac{g}{rN}
\bigl \{ m_{\alpha} \sum_{\gamma}
( \nabla m_{\gamma})^{2} \nonumber \\
&& \\
&& +\frac{\bigl[2 \nabla m_{\alpha} \cdot
\sum_{\gamma} (m_{\gamma} \nabla m_{\gamma})
-\frac{g}{rN}m_{\alpha}(m_{\gamma} \nabla m_{\gamma})^{2} \bigr]}
{\bigl[1+\frac{g}{rN} \sum_{\beta} m_{\beta}^{2} \bigr]} \bigr \}. \nonumber\end{aligned}$$ For the equal time correlation function $G(\vec u,t)
=<\phi_{\alpha}({\vec x}_{1},t)\phi_{\alpha}({\vec x}_{2},t)>$, which is independent of $\alpha$ due to the rotational symmetry of the potential, we have $$G(\vec u,t)=\int d{\vec m}_{1} d{\vec m}_{2}
P_{m}({\vec m}_{1},{\vec x}_{1} t;{\vec m}_{2},{\vec x}_{2} t)
\sigma_{\alpha}({\vec m}_{1}) \sigma_{\alpha}({\vec m}_{2})$$ where $P_{m}$, which is related to the initial probability density $P_{0}$ through the analogues of (4.7) and (4.9), depends explicitely on $N$ through $\sigma$ and $h$. As previously stated $P_{m}$ becomes gaussian upon linearizing $\sigma$ and $h$. We now show that this is achieved by taking the large-$N$ limit. The major difference with the scalar case than is that now there emerges $\lambda=1/N$ as the natural parameter which yields the gaussian approximation in the limit $\lambda \rightarrow 0$.
Taking the limit $N \rightarrow \infty$ terms of the type $\frac{1}{N} \sum_{\alpha} q_{\alpha}$ in (5.5) and (5.6) are replaced by the average $<q_{\alpha}>$ yielding the linear equations $$\sigma_{\alpha}(\vec m)=
\frac{m_{\alpha}}{\bigl[1+\frac{g}{r}S_{0}(t) \bigl]^{\frac{1}{2}}}$$ and $$\frac{\partial m_{\alpha}}{\partial t}=
\nabla^{2}m_{\alpha}+\bigl[r-\frac{g}{r}
<(\nabla m_{\alpha})^{2}> \bigr]m_{\alpha}$$ since $<m_{\gamma} {\vec \nabla}m_{\gamma}>$ vanishes. Hence, as anticipated, in the large-$N$ limit the auxiliary field $m$ is gaussian. Furthermore Eq. (5.9) is of the BH-type yielding (4.36) for $S_{0}(t)=<m_{\alpha}^{2}(t)>$.
It is important to realize that the large-$N$ limit we are considering here is quite different from the usual large-$N$ limit$^{15}$ performed on the equation of motion (5.3) for $\vec \phi$. The latter one is recovered in the present context by taking the large-$N$ limit, namely using (5.8), also in the explicit $\sigma_{\alpha}$’s appearing in (5.7) and eventually obtaining $$G(\vec u,t)=\frac{r}{g} \bigl[1-\frac{r}{gL^{2}} \bigr]
\exp (-\frac{u}{8L^{2}}).$$ Instead, according to the general structure of the GAF approximation which we have repeatedly illustrated above, $N$ must be kept fixed to whatever value it has been originally specified in the explicit $\sigma_{\alpha}$’s in (5.7), while the $N \rightarrow \infty$ limit is taken inside $P_{m}$. In so doing from (5.5) and (5.7) we obtain the Bray, Puri and Toyoki $^{7}$ (BPT) result for the scaling function $$\begin{aligned}
&& f \bigl(\frac{u}{L(t)})=<\hat{m}({\vec x}_{1},t) \cdot
\hat{m}({\vec x}_{2},t)> \nonumber \\
&& \\
&& =\frac{N \gamma}{2 \pi}
\left [ B \bigl(\frac{N+1}{2},\frac{1}{2} \bigr)^{2}
F \bigl(\frac{1}{2},\frac{1}{2};\frac{N+2}{2},\gamma^{2}
\bigr) \right ] \nonumber\end{aligned}$$ where$B(x,y)$ is the beta function, $F(a,b;c;z)$ the hypergeometric function and $\gamma(u/L)$ is given by (4.19). From the above result follows the power law tail (5.1). Furthermore, from $S(t)=<\sigma_{\alpha}^{2}>$ we obtain $$S(t)=\frac{1}{N} \int \frac{d \vec{m}}{(2\pi S_{0})^{N/2}}
\frac{m^{2}}{\bigl [1+\frac{gm^{2}}{rN} \bigr ]} e^{-\frac{m^{2}}
{2S_{0}}}$$ and carrying out the integral $$S(t)=\frac{Nr}{2g} \bigl (\frac{Nr}{2gS_{0}} \bigr )^{N/2}
e^{\frac{Nr}{2gS_{0}}} \Gamma (-\frac{N}{2},\frac{Nr}{2gS_{0}})$$ where $\Gamma (x,y)$ is the incomplete gamma function. Expanding up to first order in $1/S_{0}$ we obtain $$S(t) \sim \left \{ \begin{array}{ll} \phi_{eq}^{2}
\bigl[1-\frac{N}{(N-2)}\frac{r}{gS_{0}(t)} \bigr] & \mbox{for $N>2$} \\
\\
\phi_{eq}^{2} \bigl [1+\frac{r}{2gS_{0}(t)} \bigr ] & \mbox{for $N=2$}
\end{array}
\right.$$ which yields the power law behaviour (5.2), contrary to the exponential saturation which one obtains in the BTP approach.
These results show that the expected phenomenological behaviour is obtained at zero order within the $1/N$-expansion of the probability density of the auxiliary field. In principle, systematic corrections could be obtained via the higher order terms in the $1/N$-expansion of $P_{m}$, although we do not expect that such a scheme of computation might be easily implemented in practice. It is worth pointing out that the scheme for the systematic improvement of the GAF approximation for vector fields presented here is conceptually different from that proposed by BH in two respects: i) while we use the standard quartic potential (5.4) BH need to invoke an [*ad hoc*]{} potential which cannot even be written in closed form and ii) the expansion is made in $1/N$ where here $N$ is the number of components of the order parameter rather than the number of components of an additional internal color index. Finally, the comparison between (5.10) on one side and (5.11) (5.14) on the other clearly shows the difference between the standard $1/N$-expansion and the one we have presented here. The most important point is that while there are no localized defects in lowest order in the usual $1/N$-expansion since the correlation function (5.10) decays exponentially, the power law tail (5.1) implied by (5.11) shows that our reformulation of the $1/N$-expansion describes defects in lowest order.
6 - Concluding remarks {#concluding-remarks .unnumbered}
======================
In conclusion, in this paper we have analysed the sequence of steps which must be taken within the framework of a first principles theory in order to generate GAF approximations. The analysis has been restricted to systems with non conserved order parameter. The idea was to look for the systematic expansion scheme which allows to control the corrections to the GAF approximations. A project of this type is suggested by the physical motivation behind the introduction of the auxiliary field. This being more smooth and less non linear than the order parameter field, hopefully should be tractable in perturbation theory. Our results are negative for the scalar case, in the sense that we are unable to come up with the expansion scheme within which the GAF approximation can be identified with the zero order approximation. It should be mentioned that there are indications$^{16,9}$ that the GAF approximation becomes exact in the limit of infinite space dimensionality, suggesting the $1/d$-expansion as a possible systematic expansion scheme. This is an interesting line of research worth to be further investigated.
The outlook is somewhat better in the case of a vector order parameter. In this case one can set up the theory in such a way that the large-$N$ limit yields the GAF approximation. Consequently one can expect that there exists a $1/N$-expansion where corrections to the GAF approximation are generated systematically. Finally, approximations which go beyond the GAF approximation have been introduced recently by Mazenko$^{17}$. In future work we plan to look for the connection between that work and the point of view developed here.
\
\
[**References**]{} \
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3. A.J.Bray and K.Humayun, [*Phys. Rev. Lett.*]{} [**68**]{}, 1559, (1992)
4. G.F.Mazenko, [*Phys. Rev. B*]{} [**42**]{}, 4487, (1990) and [*ibidem*]{} [**43**]{}, 5747, (1991)
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H.Toyoki, [*Phys. Rev. B*]{} [**45**]{}, 1965, (1992)
8. F.Liu and G.F.Mazenko, [*Phys. Rev. B*]{} [**45**]{}, 6989, (1992)
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F.de Pasquale andP.Tombesi, [*Phys. Lett.*]{} [**72A**]{}, 7,(1979)
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17. G.F.Mazenko, [*Post gaussian approximations in phase ordering kinetics*]{}, preprint, 1994
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We define a pair of symplectic Dirac operators $(D^+,D^-)$ in an algebraic setting motivated by the analogy with the algebraic orthogonal Dirac operators in representation theory. We work in the settings of $\mathbb Z/2$-graded quadratic Lie algebras $\gg=\kk+\ppp$ and of graded affine Hecke algebras $\mathbb H$.'
author:
- |
Dan Ciubotaru\
`[email protected]`,
- |
Marcelo De Martino\
`[email protected]`
- |
Philippe Meyer\
`[email protected]`
date: |
Mathematical Institute\
University of Oxford\
Oxford, OX2 6GG, UK
title: Symplectic Dirac operators for Lie algebras and graded Hecke algebras
---
Introduction
============
The idea of symplectic spin geometry and symplectic Dirac operators originated with the work of Kostant [@Ko], and it was developed substantially by K. Halbermann and her collaborators (see for example the textbook [@HH]) who studied geometric symplectic Dirac operators for symplectic manifolds in analogy to the classical Riemannian Dirac operator.
In this paper, we introduce certain symplectic Dirac operators in an algebraic setting with a view towards applications to representation theory. We are also motivated by the analogy with the algebraic orthogonal Dirac operators in representation theory (e.g., [@Pa], [@HP], [@BCT]). We define pairs for symplectic Dirac elements $(D^+,D^-)$ in the setting of $\mathbb Z/2$-graded quadratic Lie algebras $\gg=\kk+\ppp$ and for graded Hecke algebras $\mathbb H$. In the Lie algebra case, the symplectic Dirac elements live in $U(\gg)\otimes {{{{\mathcal W}}}}(\ppp+\ppp^*,\omega)$, where $U(\gg)$ is the universal enveloping algebra of $\gg$, while ${{{{\mathcal W}}}}={{{{\mathcal W}}}}(\ppp+\ppp^*,\omega)$ is the Weyl algebra defined with respect to the symplectic form $\omega$ on $\ppp+\ppp^*$, $\omega(\alpha,Z)=\alpha(Z)$, for $\alpha\in\ppp^*$, $Z\in\ppp$. The graded Hecke algebra $\mathbb H$ is a deformation of $\mathbb C[W]\# S(V)$, where $W$ is a finite Weyl group in the orthogonal group of its reflection representation $V$ with respect to a nondegenerate symmetric form $B$. The symplectic Dirac operators $D^+,D^-$ live in $\mathbb H\otimes {{{{\mathcal W}}}}(V+V^*,\omega)$, where $\omega$ is again the natural symplectic form on $V+V^*$.
We compute formulas for the commutator $[D^+,D^-]$ (Theorems \[t:comm\] and \[p:comm\]), which could be viewed as analogues of Partasarathy’s formula for the square of the orthogonal Dirac operator. For Lie algebras, this commutator is expressed in terms of the Casimir element $\Omega(\gg)$ of $\gg$ and the Casimir element $\Omega(\kk)$ of $\kk$, but unlike the case of the square of the orthogonal Dirac operator, here, $\Omega(\kk)$ occurs in all three possible ways: in $U(\gg)\otimes 1\subset U(\gg)\otimes {{{{\mathcal W}}}}$, diagonally in $U(\gg)\otimes {{{{\mathcal W}}}}$, and also in $1\otimes {{{{\mathcal W}}}}$, via the map $\nu: \kk\to \mathfrak{so}(\ppp)\hookrightarrow {{{{\mathcal W}}}}$. The image $\nu(\Omega(\kk))$ is not a scalar in ${{{{\mathcal W}}}}$. The Weyl algebra has a canonical linear isomorphism $\eta$ onto $S(\ppp+\ppp^*)$ and by Kostant’s result [@Ko3 Proposition 2.6], $\eta(\nu(\Omega(\kk)))$ has only degree $4$ and degree $0$ parts. We study $ \eta(\nu(\Omega(\kk)))$ and, in particular, show that the degree $4$ part is nonzero, while the degree $0$ part admits an interesting formula, Corollary \[c:k-0\], related to the strange Freudenthal-de Vries formula.
In the case of the Hecke algebra $\mathbb H$, the analogous formula for $[D^+,D^-]$ leads to two interesting central elements $\Omega_W$ and $\Omega'_W$ in the group algebra of $W$. The element $\Omega_W$ appeared also in the theory of the orthogonal Dirac operator for graded Hecke algebras [@BCT]. In this paper, we explicitly compute it and its action on irreducible representations for the classical root systems, see section \[s:4.3\]. The other element, $$\Omega'_W=\frac {1}{16}\sum_{{{\alpha}},\beta>0,~s_{{\alpha}}(\beta)<0}k_{{\alpha}}k_\beta B({{\alpha}}^\vee,\beta^\vee)^{-1} [s_{{\alpha}}, s_\beta]^2,$$ is new. Here $s_{{\alpha}}$ is the reflection corresponding to the positive root ${{\alpha}}$ and $k_{{\alpha}}$ are the parameters of ${{\mathbb H}}$. In the formula for $[D^+,D^-]$, we are led naturally to consider $\nu'(\Omega'_W)$, the image in ${{{{\mathcal W}}}}(V+V^*,\omega)$ under the embedding $\mathfrak{so}(V,B)\subset {{{{\mathcal W}}}}$, where we regard $[s_{{\alpha}},s_\beta]$ as an element of $\mathfrak{so}(V,B)$. (In fact, $\{[s_{{\alpha}},s_\beta]:{{\alpha}},\beta>0\}$ spans $\mathfrak{so}(V,B)$, see Proposition \[p:span\].) We show that $\nu'(\Omega'_W)$ is an $O(V,B)$-invariant element of ${{{{\mathcal W}}}}$ (Proposition \[p:O-inv\]), and therefore, by the well-known theory of dual pairs, recalled in section \[s:1\], it must equal a constant plus a multiple of the image of the Casimir element of $\mathfrak{sl}(2)$. In section \[s:4.4\], we determine precisely the relation between $\nu'(\Omega'_W)$ and $\Omega(\mathfrak{sl}(2))$ inside ${{{{\mathcal W}}}}$.
As an application, we look at the actions of $D^+,D^-$ on representations, particularly unitary representations (in the settings of admissible $(\gg,K)$-modules and of finite dimensional $\mathbb H$-modules) and find certain generalisations of the classical Casimir inequality.
One would expect that the constructions in this paper could be formalised in the setting of a cocommutative Hopf algebra acting on a symplectic module, similarly to the general setting for an algebraic theory of the orthogonal Dirac operator from [@Fl].
Preliminaries: Weyl algebra {#s:1}
===========================
Let $k$ be a field of characteristic $0$ and let $({{\mathcal V}},\omega)$ be a finite-dimensional symplectic vector space. Consider the Weyl algebra ${{{{\mathcal W}}}}({{\mathcal V}},\omega)$ defined by ${{{{\mathcal W}}}}({{\mathcal V}},\omega)=T({{\mathcal V}})/I$ where $I$ is the two-sided ideal of the tensor algebra $T({{\mathcal V}})$ generated by the elements of the form $$v\otimes w-w\otimes v-\omega(v,w)\cdot1\qquad \forall v,w\in {{\mathcal V}}.$$ Define $\epsilon:{{\mathcal V}}\rightarrow {\rm End}(S({{\mathcal V}}))$ by $\epsilon(v)(P)=v\cdot P$ and define $i:{{\mathcal V}}\rightarrow {\rm End}(S({{\mathcal V}}))$ to be unique derivation of degree $-1$ such that $i(v)(w)=\omega(v,w)$.
The linear map $\gamma:{{\mathcal V}}\rightarrow {\rm End}(S({{\mathcal V}}))$ given by $\gamma(v)=\epsilon(v)+\frac{1}{2}i(v)$ extends to a homomorphism of associative algebras $\gamma:{{{{\mathcal W}}}}({{\mathcal V}},\omega)\rightarrow {\rm End}(S({{\mathcal V}}))$ that yields a linear isomorphism $\eta: {{{{\mathcal W}}}}({{\mathcal V}},\omega)\rightarrow S({{\mathcal V}})$ given by $$\label{e:eta}
\eta(x) = \gamma(x)(1),\qquad\forall x\in{{{{\mathcal W}}}}.$$ For each $x\in {{{{\mathcal W}}}}$, we shall write $(x)_d$ for the degree $d$ component of $\eta(x) \in S({{\mathcal V}}) = \oplus_{d\in {{\mathbb N}}} S^d({{\mathcal V}})$. The inverse map of $\eta$ is the quantization map $Q$ that satisfies $Q(v_1\cdot\ldots\cdot v_n)=\tfrac{1}{n!}\sum_{\sigma\in S_n}v_{\sigma(1)}\ldots v_{\sigma(n)}.$ Occasionally, we shall denote the product in the symmetric algebra by $x\cdot y$ while the product in the Weyl algebra product will be denoted by juxtaposition. Furthermore, the map $\mu : S^2({{\mathcal V}})\rightarrow \sp({{\mathcal V}},\omega)$ defined by $$\label{e:moment}
\mu(u\cdot v)(w)=\omega(u,w)v+\omega(v,w)u \qquad \forall u,v,w\in {{\mathcal V}}$$ is an isomorphism of Lie algebras, where the bracket on $S^2({{\mathcal V}})$ is the Weyl commutator. The inverse map of $\mu$ satisfies $$\mu^{-1}(f)=\frac{1}{2}\sum\limits_i f(v^i)\cdot v_i$$ where $\lbrace v_i\rbrace$ is a basis of ${{\mathcal V}}$ and $\lbrace v^i\rbrace$ is the basis dual to the basis $\lbrace v_i\rbrace$ is the sense that $\omega(v_i,v^j)=\delta_{ij}$.
If $V,V'$ are maximal isotropic subspaces of ${{\mathcal V}}$ such that ${{\mathcal V}}=V\oplus V'$, define $m:{{\mathcal W}}({{\mathcal V}},\omega)\rightarrow {\rm End}(S(V))$ by $$\label{e:polyaction}
m(v)(P)=v\cdot P, \qquad m(\alpha)(P)=i(\alpha)(P) \qquad \forall v\in V, ~ \forall \alpha\in V', ~ \forall P\in S(V).$$
Suppose now that $k$ is ${{\mathbb R}}$ or ${{\mathbb C}}$, that $V=k^n$ is endowed with a nondegenerate (positive definite when $k={{\mathbb R}}$) symmetric bilinear form $B$ and that ${{\mathcal V}}=V\oplus V^*$. For calculations, it may be convenient to work with coordinates. Let $\{v_i\}$ be an orthonormal basis of $(V,B)$. To distinguish between the elements in $V$ and those in ${{\mathcal W}}(V\oplus V^*)$, it is sometimes convenient to denote by $e_i$ the image of $v_i$ in ${{{{\mathcal W}}}}$. Let $\{v_i^*\}$ be the dual basis in $V^*$ and denote the image of $v_i^*$ in ${{{{\mathcal W}}}}$ by $f_i$. In ${{{{\mathcal W}}}}(V\oplus V^*)$, the convention is $[f_j,e_i]=\delta_{ij}$.
Introduce the linear isomorphism $$\iota: V\oplus V^*\to V\oplus V^*, \quad \iota(v_i)=v_i^*,\ \iota(v_i^*)=v_i.$$ The symplectic Lie algebra $\sp(V\oplus V^*,\omega)$ is isomorphic to $\mathfrak{sp}(2n,k)$ where $\mathfrak{sp}(2n,k)$ is the subalgebra of matrices $X\in \mathfrak{gl}(2n,k)$ such that $X^tJ+JX=0$, where $J=\left(\begin{matrix} 0& -I_n\\ I_n&0\end{matrix}\right)$. Hence $X=\left(\begin{matrix} A &B\\C &-A^t\end{matrix}\right),$ where $A,B,C$ are $n\times n$ matrices satisfying $B=B^t$ and $C=C^t$. Denote by $E_{ij}$ the $(i,j)$-elementary matrix. Under the isomorphisms $\mu:S^2(V\oplus V^*)\rightarrow\sp(V\oplus V^*,\omega)$ and $Q:S(V\oplus V^*)\rightarrow {{{{\mathcal W}}}}(V\oplus V^*,\omega)$, the canonical basis of $\mathfrak{sp}(2n,k)$ corresponds to $$\begin{aligned}
e_{ij}-e_{n+j,n+i}&\mapsto E_{ij}+\frac 12\delta_{ij}\\
e_{i,j+n}+e_{j,i+n}&\mapsto M_{ij}\\
-e_{i+n,j}-e_{j+n,i}&\mapsto \Delta_{ij} ,\quad 1\le i,j\le n\end{aligned}$$ where $$\begin{aligned}
\Delta_{ij}=f_if_j,\quad M_{ij}=e_ie_j,\quad E_{ij}=e_i f_j.
\end{aligned}$$ Define $\ss:=Q\circ\mu^{-1}(\sp(V\oplus V^*,\omega))$. The Lie algebra $\mathfrak{gl}(n,k)$ is embedded diagonally into $\mathfrak{sp}(2n,k)$ by $A\mapsto \left(\begin{matrix} A &0\\0 &-A^t\end{matrix}\right)$ and so define $\mathfrak l:=Q\circ\mu^{-1}(\gl(n,k))$.
The map $\iota$ induces the transpose on $\mathfrak l$: $$\iota: \mathfrak l\to \mathfrak l,\quad \iota(E_{ij})=E_{ji}.$$ Let $\mathfrak k\subset \mathfrak l$ be the $(-1)$-eigenspace of $\iota$. Then $\mathfrak k$ is isomorphic to $\mathfrak{so}(n,k)$.
Define $$\label{sl2}
\begin{aligned}
\Delta&=\sum_{j=1}^n \Delta_{jj}, &\quad& X&&=-\frac 12\Delta,\\
E&=\sum_{i=1}^n E_{ii},&\quad& H&&=-E-\frac n2\\
r^2&=\sum_{j=1}^n M_{jj},&\quad& Y&&=\frac 12 r^2.
\end{aligned}$$ It is easy to check (see [@GW (5.86),Theorem 5.6.9]) that $\lbrace X,H,Y\rbrace$ is a Lie triple and that $\mathfrak t=\text{span}\{X,H,Y\}\cong \mathfrak{sl}(2,k)$ commutes with $\mathfrak k\cong\mathfrak{so}(n,k)$ in ${{\mathcal W}}$.
Let ${{\mathcal P}}$ denote the space of polynomials on $V^*$, equivalently ${{\mathcal P}}=S(V)$. As before, this is a module for ${{\mathcal W}}$ via the action $m:{{\mathcal W}}\to{\operatorname{End}}({{\mathcal P}})$ of (\[e:polyaction\]) where $m(e)$ is the multiplication by $e\in V$ operator and $m(f)$ is the directional derivative for $f\in V^*$. Let ${{\mathcal P}}^\ell$ denote the subspace of homogeneous polynomials of degree $\ell\ge 0$.
As an $\mathfrak{so}(n,k)\times \mathfrak {sl}(2,k)$-module, ${{\mathcal P}}$ decomposes $${{\mathcal P}}=\bigoplus_{\ell\ge 0} {{{{\mathcal H}}}}^\ell(V)\otimes M_{-(\ell+\frac n2)},$$ where ${{{{\mathcal H}}}}^\ell(V)=\{p\in {{\mathcal P}}^\ell\mid \Delta(p)=0\}$ is the space of spherical harmonic polynomials of degree $\ell$ and $M_{-(\ell+\frac n2)}$ is the Verma module of $\mathfrak t$ with highest weight $-(\ell+\frac n2)$.
It is well-known that the Casimir element $\Omega(\mathfrak{sl}(2)) = H^2 + 2(XY + YX)\in {{\mathcal W}}$ satisfies $\eta( \Omega(\mathfrak{sl}(2)))\in S^4(V\oplus V^*)\oplus S^0(V\oplus V^*)$, where $\eta$ is the linear isomorphism ${{\mathcal W}}\to S(V\oplus V^*)$ of (\[e:eta\]). For convenience, we recall the values of its components.
\[p:sl2Cas\] We have $\eta(\Omega(\mathfrak{sl}(2))) = (\eta(H)^2 -\eta(\Delta)\eta(r^2))-3n/4$.
From [@Ko2 Proposition 2.6], we know that $\eta(H^2)\in S^4(V\oplus V^*)\oplus S^0(V\oplus V^*)$. A straightforward computation gives $\eta(H^2)=\gamma(H^2)(1) = \eta(H)^2 - \tfrac{n}{4}$. On the other hand, one computes $$\begin{aligned}
4\eta(XY)=4\gamma(X)\gamma(Y)(1) &= 2\gamma(X)(\eta(r^2))\\
&= -\sum_j \gamma(f_j)(f_j\cdot \eta(r^2) + e_j) \\
&= -\eta(\Delta)\eta(r^2) - 2\sum_j f_j\cdot e_j - \tfrac{n}{2},\end{aligned}$$ as $i(f_j)(e_k)= \delta_{jk}$. Since $\Omega(\mathfrak{sl}(2)) = H^2 - 2H + 4XY$ and $\eta(H)=-\sum_j f_j\cdot e_j$, the claim follows.
Lie algebras
============
Symplectic Dirac elements
-------------------------
Let $k$ be a field of characteristic $0$. Let $(\gg,B)$ be a finite-dimensional quadratic Lie algebra and suppose that we have a $\mathbb{Z}_2$-gradation $\gg=\kk\oplus \ppp$ such that $\kk$ and $\ppp$ are $B$-orthogonal. Let $\lbrace w_k\rbrace$ be a basis of $\kk$, let $\lbrace w^k\rbrace$ be its dual basis, let $\lbrace z_i \rbrace$ be a basis of $\ppp$ and let $\lbrace z^i\rbrace$ be its dual basis.
We have ${\rm End}(\ppp)\cong \ppp\otimes\ppp^*$ and the identity corresponds to $$D^-=\sum\limits_i z_i\otimes z_i^*.$$ Using $B$ we have ${\rm End}(\ppp)\cong \ppp\otimes\ppp$ and the identity corresponds to $$D^+=\sum\limits_i z_i\otimes z^i.$$ In particular, $D^+$ and $D^-$ are independent of the choice of the basis $\lbrace z_i \rbrace$. Let $\omega$ be the symplectic form on $\ppp\oplus\ppp^*$ given by $\omega(\ppp,\ppp)=\omega(\ppp^*,\ppp^*)=\lbrace 0 \rbrace$ and $$\omega(\alpha,v)=\alpha(v)\qquad \forall \alpha \in \ppp^*,~\forall v\in \ppp$$ In the Weyl algebra ${{{{\mathcal W}}}}(\ppp\oplus\ppp^*,\omega)$ we have $[z_i^*,z_j]=\delta_{ij}$. There is a Lie algebra morphism $\nu':\so(\ppp,B)\rightarrow {{{{\mathcal W}}}}(\ppp\oplus\ppp^*,\omega)$ given by $$\nu'(f)=\sum\limits_if(z_i)z_i^*\qquad\forall f\in \so(\ppp,B).$$
If we extend $f\in\so(\ppp,B)$ to $f\in {\rm End}(S(\ppp))$ then we have $$m(\nu'(f))=f \qquad \forall f\in\so(\ppp,B).$$
Hence, we have a Lie algebra morphism $\nu:\kk\rightarrow {{{{\mathcal W}}}}(\ppp\oplus\ppp^*,\omega)$ given by $\nu=\nu'\circ {\operatorname{ad}}|_{\ppp}$ and a Lie algebra morphism $\Delta:\kk\rightarrow U(\kk)\otimes {{{{\mathcal W}}}}(\ppp\oplus\ppp^*,\omega)$ given by $$\Delta(x)=x\otimes 1+1\otimes \nu(x)\qquad\forall x\in \kk.$$
We have $$[D^+,\Delta(x)]=[D^-,\Delta(x)]=0 \qquad \forall x\in \kk.$$
\[t:comm\] In $U(\gg)\otimes {{{{\mathcal W}}}}(\ppp\oplus\ppp^*,\omega)$, we have $$\begin{aligned}
[D^+,D^-]&=\Big(-\Omega(\gg)+\Omega(\kk)\Big)\otimes 1-\sum_k w_k\otimes\nu(w^k)\\
&=\Big(-\Omega(\gg)+\frac{3}{2}\Omega(\kk)\Big)\otimes 1-\frac{1}{2}\Delta(\Omega(\kk))+\frac{1}{2}\cdot 1\otimes \nu(\Omega(\kk)).
\end{aligned}$$
We have $$\begin{aligned}
[D^+,D^-]&=\sum\limits_{i,j}\Big(z_iz_j\otimes z^iz_j^*-z_jz_i\otimes z_j^*z^i\Big)\\
&=\sum\limits_{i,j}\Big(z_iz_j\otimes z^iz_j^*-z_jz_i\otimes (z^iz_j^*+\omega(z_j^*,z^i))\Big)\\
&=\sum\limits_{i,j}\Big([z_i,z_j]\otimes z^iz_j^*-z_jz_iz_j^*(z^i)\otimes 1\Big).\end{aligned}$$ We have $$\sum\limits_{i,j}z_jz_iz_j^*(z^i)=\sum_iz^iz_i=\Omega(\gg)-\Omega(\kk).$$ On the other hand $$\begin{aligned}
\sum\limits_{i,j}[z_i,z_j]\otimes z^iz_j^*&=\sum\limits_{i,j,k}B([z_i,z_j],w^k)w_k\otimes z^iz_j^*=\sum\limits_{k}w_k\otimes\sum\limits_{i,j}B([z_i,z_j],w^k) z^iz_j^*\\
&=-\sum\limits_{k}w_k\otimes\sum\limits_{i,j}B([w^k,z_j],z_i) z^iz_j^*
=-\sum\limits_{k}w_k\otimes\sum\limits_{j}[w^k,z_j]z_j^*\\
&=-\sum\limits_{k}w_k\otimes\nu(w^k)
\end{aligned}$$ and so $$[D^+,D^-]=-\sum\limits_{k}w_k\otimes\nu(w^k)-\Big(\Omega(\gg)-\Omega(\kk)\Big)\otimes 1.$$ Since $$\begin{aligned}
\Delta(\Omega(\kk))&=\sum\limits_{k}\Big(w_k\otimes 1+1\otimes \nu(w_k)\Big)\Big(w^k\otimes 1+1\otimes \nu(w^k)\Big)\\
&=\Omega(\kk)\otimes 1+2\sum\limits_{k}w_k\otimes\nu(w^k)+1\otimes \nu(\Omega(\kk)) \end{aligned}$$ then $$[D^+,D^-]=\Big(-\Omega(\gg)+\frac{3}{2}\Omega(\kk)\Big)\otimes 1-\frac{1}{2}\Delta(\Omega(\kk))+\frac{1}{2}\cdot 1\otimes \nu(\Omega(\kk)).$$
The embedding of the Casimir element in the Weyl algebra
--------------------------------------------------------
We now study the element $\nu(\Omega(\kk))$. Remark that from [@Ko2 Proposition 2.6] $$\eta(\nu(\Omega(\kk)))\in S^4(\ppp\oplus\ppp^*)\oplus S^0(\ppp\oplus\ppp^*).$$
We have $$\Big(\eta(\nu(\Omega(\kk)))\Big)_0=\frac{1}{12}Tr\Big(ad_{\kk}(\Omega(\kk))-ad_{\gg}(\Omega(\gg))\Big).$$
Using Equation , We have $$\begin{aligned}
\Big(\eta(\nu(\Omega(\kk)))\Big)_0&=\sum_k \Big(\gamma([w^k,z_i]z_i^*[w_k,z_j]z_j^*)(1)\Big)_0=\frac{1}{4}\sum_{i,j,k} \omega(z_i^*,[w_k,z_j])\omega([w^k,z_i],z_j^*)\\
&=-\frac{1}{4}\sum_{i,j,k}B(z^i,[w_k,z_j])B(z^j,[w^k,z_i])=-\frac{1}{4}\sum_{j}B(z^j,ad(\Omega(\kk))(z_j)).\end{aligned}$$ On the other hand, we have $$\begin{aligned}
Tr\Big(ad_{\gg}(\Omega(\gg))-&ad_{\kk}(\Omega(\kk))\Big)\\
&=\sum_{k}B(w^k,ad(\Omega(\gg))(w_k))+\sum_i B(z^i,ad(\Omega(\gg))(z_i))-\sum_k B(w^k,ad(\Omega(\kk))(w_k))\\
&=\sum_{k}B(w^k,ad(\sum_i z^iz_i)(w_k))+\sum_i B(z^i,ad(\Omega(\kk))(z_i))+\sum_i B(z^i,ad(\sum_j z^jz_j)(z_i)).\end{aligned}$$ Since $$\begin{aligned}
\sum_{k}B(w^k,ad(\sum_i z^iz_i)(w_k))&=\sum_{i,k}B([w^k,z^i],[z_i,w_k])=-\sum_{i,k}B([[w^k,z^i],w_k],z_i)\\
&=\sum_{i,k}B([w_k,[w^k,z^i]],z_i)=\sum_{i,k}B(z_i,ad(\Omega(\kk))(z^i))\end{aligned}$$ and $$\begin{aligned}
\sum_{i}B(z_i,ad(\sum_j z^jz_j)(z^i))&=\sum_{i,j}B(z_i,[z^j,[z_j,z^i]])=\sum_{i,j}B(z_i,[[z^i,z_j],z^j])\\
&=\sum_{i,j}B([z^i,z_j],[z^j,z_i])=\sum_{i,j,k}B(w^k,[z^i,z_j])B([z^j,z_i],w_k)\\
&=-\sum_{i,j,k}B([w^k,z^i],z_j])B(z_i,[z^j,w_k])=-\sum_{j,k}B([w^k,[z^j,w_k]],z_j])\\
&=\sum_jB(z_j,ad(\Omega(\kk))(z^j)),\end{aligned}$$ we obtain $$Tr\Big(ad_{\gg}(\Omega(\gg))-ad_{\kk}(\Omega(\kk))\Big)=3\sum_i B(z^i,ad(\Omega(\kk))(z_i))$$ and so $$\Big(\eta(\nu(\Omega(\kk)))\Big)_0=\frac{1}{12}Tr\Big(ad_{\kk}(\Omega(\kk))-ad_{\gg}(\Omega(\gg))\Big).$$
Using the strange Freudenthal-de Vries formula [@FdV] we can express this constant for reductive Lie algebras as follows:
\[c:k-0\] Suppose that $k$ is algebraically closed and suppose that $\kk$ and $\gg$ are reductive. We have $$\Big(\eta(\nu(\Omega(\kk)))\Big)_0=2\Big(B(\rho_{\kk},\rho_{\kk})-B(\rho_{\gg},\rho_{\gg})\Big),$$ where $\rho_{\gg}$ (resp. $\rho_{\kk}$) is the Weyl vector of $\gg$ (resp. $\kk$) with respect to a Cartan subalgebra $\hh_{\gg}$ (resp. $\hh_{\kk}$) of $\gg$ (resp. $\kk$) and a choice of a full set of positive roots for the action of $ad(\hh_{\gg})$ (resp. $ad(\hh_{\kk})$).
Let $(\uu,B_{\uu})$ be a quadratic reductive Lie algebra. Let $\hh_{\uu}$ be a Cartan subalgebra of $\uu$ and let $\rho_{\uu}$ be the Weyl vector of $\uu$ with respect to a choice of a full set of positive roots for the action of $ad(\hh_{\uu})$. We have $$Tr\Big(ad(\Omega(\uu))\Big)=24B_{\uu^*}(\rho_{\uu},\rho_{\uu})$$ which is a general formulation of the strange Freudenthal-de Vries formula, see [@Ko2 Proposition 1.84].
Recall from Kostant [@Ko3] the following result:
\[p:Kostant\] Let $\rho : \hh \rightarrow \sp(V,\omega)$ be a finite-dimensional symplectic representation of a finite-dimensional quadratic Lie algebra $(\hh,B)$ and let $\mu : S^2(V)\rightarrow\hh$ be the map given by $$B(x,\mu(v,w))=\omega(\rho(x)(v),w)\qquad \forall x\in \hh, ~ \forall v,w\in V.$$ Consider the Lie algebra morphism $\nu : \hh \rightarrow {{{{\mathcal W}}}}(V,\omega)$.
1. If there exists a quadratic Lie superalgebra structure on $(\hh\oplus V,B\perp \omega)$ extending the bracket of $\hh$ and the action of $\hh$ on $V$, then $$\lbrace v,w \rbrace=\mu(v,w) \qquad \forall v,w\in V.$$
2. Let $\hht:=\hh\oplus V$, let $B_{\hht}:=B\perp \omega$ and let $\lbrace \phantom{v},\phantom{v} \rbrace : \hht\times\hht\rightarrow\hht$ be the unique $\mathbb{Z}_2$-graded super-antisymmetric bilinear map which extends the bracket of $\hh$, the action of $\hh$ on $V$ and such that $$\lbrace v,w\rbrace=\mu(v,w) \qquad \forall v,w \in V.$$ Then the following are equivalent:
1. $(\hht,B_{\hht},\lbrace \phantom{v},\phantom{v} \rbrace)$ is a quadratic Lie superalgebra.
2. $\Big(\eta(\nu(\Omega(\hh)))\Big)_4=0\in S^4(V)$.
It follows from this proposition that the element $(\eta(\nu(\Omega(\kk))))_4\in S^4(\ppp\oplus\ppp^*)$ is the obstruction to have a quadratic Lie superalgebra structure on $\kk\oplus(\ppp\oplus\ppp^*)$ which extends the bracket of $\kk$ and the action of $\kk$ on $\ppp\oplus\ppp^*$. In particular:
\[p:Casimirdegree4nonzero\] If $B([\ppp,\ppp],[\ppp,\ppp])\neq \lbrace 0\rbrace$ then we have $$\Big(\eta(\nu(\Omega(\kk)))\Big)_4\neq 0\in S^4(\ppp\oplus \ppp^*).$$
Consider the map $\mu:S^2(\ppp\oplus\ppp^*)\rightarrow \kk$ given by $$B(x,\mu(v,w))=\omega(x(v),w)\qquad \forall x\in \kk,~ \forall v,w\in \ppp\oplus\ppp^*.$$
Let $v,w\in \ppp,~\alpha,\beta \in \ppp^*$ and let $a\in \ppp$ be the unique element such that $\alpha=B(a,\phantom{v})$. We have $$\mu(v,w)=\mu(\alpha,\beta)=0, \qquad \mu(v,\alpha)=[a,v].$$
Since the vector space $\ppp$ is stable under the action of $\kk$ and isotropic for the symplectic form $\omega$, then we have $\mu(v,w)=0$. Similarly we have $\mu(\alpha,\beta)=0$. For $x\in \kk$ we have $$\begin{aligned}
B(x,\mu(v,\alpha))=\omega(x(v),\alpha)=-\alpha(x(v))=-B(a,x(v))=B(x(a),v)=B(x,[a,v])\end{aligned}$$ and so $\mu(v,\alpha)=[a,v]$.
By Proposition \[p:Kostant\] we have $(\eta(\nu(\Omega(\kk))))_4=0$ if and only if $$\label{e:proofdegree4nonzero}
\mu(u,v)(w)+\mu(v,w)(u)+\mu(w,u)(v)=0 \qquad \forall u,v,w\in \ppp\oplus\ppp^*.$$ Let $a,v\in \ppp$ and $\alpha:=B(a,\phantom{v})\in \ppp^*$. We have $$B\Big(\mu(v,\alpha)(v)+\mu(\alpha,v)(v)+\mu(v,v)(\alpha),a\Big)=2B([[a,v],v],a)=-2B([a,v],[a,v]).$$ Hence, if holds, then $$B([a,v],[a,v])=0 \qquad \forall a,v\in\ppp$$ and by polarisation its implies $$B([\ppp,\ppp],[\ppp,\ppp])=\lbrace 0\rbrace$$ which is a contradiction.
Suppose that $\kk=\so(\ppp)$ and $\kk\rightarrow \so(\ppp)$ is the natural representation. Hence $\gg\cong\so(\ppp\oplus L)$ where $L$ is a one-dimensional quadratic vector space.
Using Propositions \[p:Kostant\] and \[p:Casimirdegree4nonzero\], there is no quadratic Lie superalgebra structure on $\so(\ppp)\oplus(\ppp\oplus\ppp^*)$ which extends the bracket of $\so(\ppp)$ and the action of $\so(\ppp)$ on $\ppp\oplus\ppp^*$. However, the Lie algebra $\sl(2,k)$ acts on $\ppp\oplus\ppp^*$ by $$H(z_i)=-z_i,\quad H(z_i^*)=z_i^*,\quad X(z_i)=z_i^*,\quad X(z_i^*)=0,\quad Y(z_i)=0,\quad Y(z_i^*)=z_i$$ where $\lbrace X,H,Y\rbrace$ is the standard basis of $\sl(2,k)$ and there is a quadratic Lie superalgebra structure on $$\so(\ppp)\oplus\sl(2,k)\oplus(\ppp\oplus\ppp^*).$$ This Lie superalgebra is isomorphic to the orthosymplectic Lie superalgebra $\osp((\ppp,B|_{\ppp})\perp(k^2,\omega_{k^2}))$ where $\omega_{k^2}$ is the canonical symplectic form over $k^2$. In other words, the natural representation of $\so(\ppp)$ is orthogonal special in the sense of [@Mey]. In particular, using Proposition \[p:Kostant\], we have that $$\Big(\eta(\nu(\Omega(\so(\ppp))))\Big)_4=-\Big(\eta(\nu(\Omega(\sl(2,k))))\Big)_4\in {{{{\mathcal W}}}}(\ppp\oplus\ppp^*,\omega).$$
Unitary structures
------------------
Suppose now that $k=\mathbb C$ and that $\mathfrak g=\mathfrak k\oplus\mathfrak p$ is the complexification of a real Lie algebra $\mathfrak g_0=\mathfrak k_0\oplus \mathfrak p_0$. Let $\overline\ $ denote the complex-conjugation on $\mathfrak g$ whose real points is $\mathfrak g_0$. Define a star operation (conjugate-linear anti-involution) on ${{{{\mathcal W}}}}={{{{\mathcal W}}}}(\ppp+\ppp^*)$ by $$v^*=\iota(v),\quad \text{for all }v\in \ppp+\ppp^*,$$ and extended as an anti-homomorphism. Then ${{{{\mathcal P}}}}=S(\ppp)$ is naturally a (pre)unitary module for ${{{{\mathcal W}}}}$ with respect to $*$ with the hermitian pairing: $$\langle P_1,P_2\rangle_{{{{\mathcal P}}}}=\partial_{P_2}(\overline P_1), \text{ for all }P_1,P_2\in{{{{\mathcal P}}}},$$ where $\partial_{P_2}$ is the partial differential operator defined by $P_2$.
Define a star operation on $U(\gg)$ by extending as a conjugate-linear anti-homomorphism the assignment $$x^*=-\overline x,\text{ for all }x\in \gg.$$
In $U(\gg)\otimes{{{{\mathcal W}}}}$, $(D^\pm)^*=-D^{\mp}$.
Straightforward.
Another important star operation (the “compact” star operation) on $U(\gg)$ was considered in [@ALTV] in the study of unitarisable Harish-Chandra modules. This is defined on $\gg$ as follows: $$x^c=-\overline x,\ \text{ for }x\in \kk,\quad z^c=\overline z,\ \text{ for } z\in \ppp.$$ If we extend this to a star operation $c$ of $U(\gg)\otimes {{{{\mathcal W}}}}$ (by $c$ on $U(g)$ and the same $*$ as before on ${{{{\mathcal W}}}}$), then it is immediate that $$(D^\pm)^c=D^\mp.$$ The discussion below with respect to the classical $*$ can be easily modified for $c$ as well.
We can consider $D^\pm$ as operators on $M\otimes {{{{\mathcal P}}}}$ for every $U(\gg_{{\mathbb C}})$-module $(\pi, M)$.
Suppose $(\pi,M)$ is an admissible $(\gg_{{\mathbb C}},K)$-module which admits a nondegenerate hermitian form $\langle~,~\rangle_M$ invariant under $*$. Define the product form $$\langle~,~\rangle_{M\otimes{{{{\mathcal P}}}}}=\langle~,~\rangle_{M}\langle~,~\rangle_{{{{{\mathcal P}}}}}.$$
\[p:diff\] Suppose that $M$ admits an infinitesimal character $\chi_M$. Let $\sigma$ be a simple finite-dimensional $\kk$-module and let $(M\otimes{{{{\mathcal P}}}})(\sigma)$ denote the $\Delta(\kk)$-isotypic component of $\sigma$. If $x\in (M\otimes {{\mathcal P}})(\sigma)$, then $$\begin{aligned}
\langle D^+ x, D^+ x\rangle_{M\otimes{{{{\mathcal P}}}}}-\langle D^- x, D^- x\rangle_{M\otimes{{{{\mathcal P}}}}}&=\left(-\chi_M(\Omega(\gg))-\frac 12 \sigma(\Omega(\kk))\right)\langle x,x\rangle_{M\otimes{{{{\mathcal P}}}}}\\
&+\frac 32 \langle (\pi(\Omega(\kk))\otimes 1)x,x\rangle_{M\otimes{{{{\mathcal P}}}}}+\frac 12 \langle x, (1\otimes\nu(\Omega(\kk)))x \rangle_{M\otimes{{{{\mathcal P}}}}}.\end{aligned}$$
This follows immediately from Theorem \[t:comm\] using the adjointness property $({{\mathcal D}}^{\pm})^*=-{{\mathcal D}}^{\mp}.$
As a particular example, notice that when $x\in M(\sigma)\otimes S^0(\ppp)=M(\sigma)\otimes 1$, then $D^- x=0=(1\otimes\nu(\Omega_\kk)x$, hence we recover the well-known “Casimir inequality” for unitary modules:
\[c:Casimir\] For all $x\in M(\sigma)\otimes 1$: $$\langle {{\mathcal D}}^+ x, {{\mathcal D}}^+ x\rangle_{M\otimes{{\mathcal P}}}=(-\chi_M(\Omega(\gg))+\sigma(\Omega(\kk)))\langle x,x\rangle_{M\otimes {{\mathcal P}}}.$$ Moreover, if $M$ is $*$-unitary then $\chi_M(\Omega(\gg))\le \sigma(\Omega(\kk))$, for all simple finite dimensional $\kk$-modules $\sigma$ such that $M(\sigma)\neq 0$.
The inequality follows by taking $x\neq 0$ in $M(\sigma)\otimes 1$ and using that $\langle {{\mathcal D}}^+ x, {{\mathcal D}}^+ x\rangle_{M\otimes{{\mathcal P}}}\ge 0$ for unitary modules. Notice that $(\pi(\Omega(\kk))\otimes 1)x=(\sigma(\Omega(\kk)))x$ in this case.
Suppose $p\in S(\ppp_0)$. Notice that for all $w\in \kk$, $$\langle \nu(w)p,p\rangle_{{{{\mathcal P}}}}=0.$$ This is because $$\begin{aligned}
\langle\nu(w)p,p\rangle_{{{{\mathcal P}}}}&=\sum_{i,j} B([w,z_i],z_j)\langle z^i z_j^*p,p\rangle=\sum_{i,j} B([w,z_i],z_j)\langle z_j^*p,z_i^* p\rangle_{{{{\mathcal P}}}}\\&=\sum_i B([w,z_i],z_i)\langle z_i^*p,z_i^* p\rangle_{{{{\mathcal P}}}}=0,\end{aligned}$$ using that $ \langle z_j^*p,z_i^* p\rangle_{{{{\mathcal P}}}}=0$ if $i\neq j.$ Therefore $\langle \pi(w)\otimes \nu(w) x,x\rangle_{M\otimes {{{{\mathcal P}}}}}=0$ as well for every simple tensor $x\in M\otimes S(\ppp_0)$. Using Theorem \[t:comm\], we obtain immediately:
Suppose $x\in M(\sigma)\otimes S(\ppp_0)$ is a simple tensor. Then $$\langle D^+ x, D^+ x\rangle_{M\otimes{{{{\mathcal P}}}}}-\langle D^- x, D^- x\rangle_{M\otimes{{{{\mathcal P}}}}}=\left(-\chi_M(\Omega(\gg))+\sigma(\Omega(\kk))\right)\langle x,x\rangle_{M\otimes{{{{\mathcal P}}}}}.$$
The structure of $S(\ppp)$ as a $\kk$-module is well known by the theorem of Kostant and Rallis [@KR]. Suppose $G$ is a complex linear algebraic group with Lie algebra $\gg$ and let $\theta:G\to G$ be a regular involution such that $K=G^\theta$ has Lie algebra $\kk$. Then $\ppp$ is the $(-1)$-eigenspace of the differential of $\theta$ on $\gg$. Let $\mathfrak a$ be a Cartan subspace of $\ppp$ and set $M=Z_K(\mathfrak a)$. The subspace of $K$-invariants $S(\ppp)^K$ is a polynomial ring, and denote by $S(\ppp)^K_+$ the subspace of $K$-invariant polynomials with zero constant term. Let $I(\ppp)$ denote the ideal of $S(\ppp)$ generated by $S(\ppp)^K_+$. For every degree $\ell\ge 0$, $I(\ppp)\cap S^\ell(\ppp)$, being $K$-invariant, has a unique $K$-invariant complement in $S^\ell(\ppp)$, denoted $\mathcal H^\ell(\ppp)$. Set $\mathcal H(\ppp)=\oplus_{\ell\ge 0} \mathcal H^\ell(\ppp)$. This is the space of $K$-harmonic polynomials in $S(\ppp)$. Then (cf. [@GW §12.4.1]):
1. $S(\ppp)$ is a free module over $S(\ppp)^K$ and $S(\ppp)\cong S(\ppp)^K\otimes \mathcal H(\ppp)$;
2. As a $K$-representation, $\mathcal H(\ppp)\cong \operatorname{Ind}_M^K(1)$.
$\operatorname{Ind}_M^K(1)$ is the restriction to $K$ of the spherical minimal principal series $G$-representation.
An example: $(\gg,K)$-modules of $SL(2,{{\mathbb R}})$
------------------------------------------------------
Suppose that $\gg$ is $\sl(2,{{\mathbb C}})$ and that $B(x,y)=\frac{1}{2}Tr(xy)$ for all $x,y\in\sl(2,{{\mathbb C}})$. Let $\lbrace X,H,Y \rbrace$ be a $\sl(2,{{\mathbb C}})$-triple and consider the $B$-orthogonal $\mathbb{Z}_2$-gradation $\sl(2,{{\mathbb C}})=\kk\oplus\ppp$ where $\kk={\rm Span}\langle H\rangle$ and $\ppp={\rm Span}\langle X,Y\rangle$. The Dirac operators $D^+,D^-\in U(\sl(2,{{\mathbb C}}))\otimes {{{{\mathcal W}}}}(\ppp\oplus\ppp^*,\omega)$ satisfy $$\begin{aligned}
D^+&=2X\otimes Y+2Y\otimes X,\\
D^-&=X\otimes X^*+Y\otimes Y^*.\end{aligned}$$ The Casimir elements associated to $(\kk,B|_{\kk})$ and $(\sl(2,{{\mathbb C}}),B)$ satisfy $$\begin{aligned}
\Omega(\kk)&=H^2\in U(\kk),\\
\Omega(\sl(2,{{\mathbb C}}))&=H^2+2(XY+YX)\in U(\sl(2,{{\mathbb C}})).\end{aligned}$$
We have $$\nu(\Omega(\kk))(X^kY^l)=4(k-l)^2X^kY^l \qquad \forall X^kY^l \in S(\ppp).$$
Let $\lbrace H,X,Y \rbrace$ be the $\sl(2)$-triple defined by $$H=\begin{pmatrix}
0 & -i\\
i & 0
\end{pmatrix},\qquad X=\frac{1}{2}\begin{pmatrix}
1 & i\\
i & -1
\end{pmatrix},\qquad Y=\begin{pmatrix}
1 & -i\\
i & -1
\end{pmatrix}.$$ Let $\lambda\in\mathbb{C}$ and $\epsilon\in\lbrace 0,1\rbrace$. Let $V_{\lambda,\epsilon}$ be the minimal principal series module [@Vo], defined as vector space with a basis $\lbrace W_n ~ | ~ \forall n\in \mathbb{Z}, ~ n\equiv \epsilon \mod2 \rbrace$ and with an action of $\sl(2,{{\mathbb C}})$ on $V_{\lambda,\epsilon}$ by $$\begin{aligned}
\pi(H)(W_n)&=nW_n,\\
\pi(X)(W_n)&=\frac{1}{2}(\lambda+n+1) W_{n+2},\\
\pi(Y)(W_n)&=\frac{1}{2}(\lambda-n+1) W_{n-2}.\end{aligned}$$ We have $$\Omega(\kk)(W_n)=n^2W_n, \qquad \Omega(\sl(2,{{\mathbb C}}))(W_n)=(\lambda^2-1)W_n.$$
Let $X^kY^l \in S(\ppp)$. We have $$\Delta(\Omega(\kk))(W_n\otimes X^kY^l)=\left(n^2+4(k-l)^2+4n(k-l)\right)W_n\otimes X^kY^l.$$
\[p:ps\] Suppose that $\lambda+1$ is not an integer congruent to $\epsilon$ modulo $2$. Consider the representation $\pi\otimes m:\sl(2,{{\mathbb C}})\times {{\mathcal W}}\rightarrow {\rm End}(V_{\lambda,\epsilon}\otimes S(\ppp))$. We have $$\begin{aligned}
Ker(\pi\otimes m(D^+))&=\lbrace 0 \rbrace,\\
Ker(\pi\otimes m(D^-))&={\rm Span}\langle\sum\limits_{i=0}^{l}(-1)^i {l\choose i} \prod\limits_{j=0}^{i-1}x(n+4j)\prod\limits_{j=i+1}^{l}y(n+4j) W_{n+4i}\otimes X^{l-i}Y^i ~ | ~ \forall l\geq 0, ~ \forall n\in \mathbb{Z}\rangle,\end{aligned}$$ where $x(n)=\frac{1}{2}(\lambda+n+1)$ and $y(n)=\frac{1}{2}(\lambda-n+1)$.
Let $v=\sum\limits_n W_n\otimes P_n\in V_{\lambda,\epsilon}\otimes S^l(\ppp)$.
We first show that $D^+$ is injective. We have $$\begin{aligned}
D^+(v)&=2\sum\limits_n (x(n)W_{n+2}\otimes YP_n+y(n)W_{n-2}\otimes XP_n)\\
&=2\sum\limits_n W_n\otimes(x(n-2)YP_{n-2}+y(n+2)XP_{n+2}).\end{aligned}$$ We have $P_k=\sum\limits_{j=0}^{l}a_{k,j}X^{l-j}Y^{j}$ and so $$D^+(v)=2\sum\limits_n W_{n}\otimes\Big(a_{n+2,0}y(n+2)X^{l+1}+\sum\limits_{j=1}^l(a_{n-2,j-1}x(n-2)+a_{n+2,j}y(n+2))X^{l-j+1}Y^j+a_{n-2,l}x(n-2)Y^{l+1}\Big).$$ If $D^+(v)=0$, then, for all $n$, $a_{n,0}=a_{n,l}=0$ and $$\label{e:D^+inj}
a_{n-2,j-1}x(n-2)+a_{n+2,j}y(n+2)=0 \qquad \forall j\in \llbracket 1,l \rrbracket.$$ Hence, by induction and using we obtain $a_{n,j}=0$ for all $n,j$ and so $v=0$.
We now calculate the kernel of $D^-$. We have $$D^-(v)=\sum\limits_n W_n\otimes\Big(x(n-2)\partial_X(P_{n-2})+y(n+2)\partial_Y(P_{n+2})\Big).$$ We have $P_k=\sum\limits_{j=0}^{l}a_{k,j}X^{l-j}Y^{j}$ and so $$D^-(v)=\sum\limits_n W_n\otimes\sum\limits_{j=0}^{l-1}\Big(a_{n-2,j}x(n-2)(l-j)+a_{n+2,j+1}y(n+2)(j+1)\Big) X^{l-j-1}Y^{j}.$$ If $D^-(v)=0$, then, for all $n$, we have $$\label{e:D^-kernel}
a_{n,j}x(n)(l-j)+a_{n+4,j+1}y(n+4)(j+1) \qquad \forall j\in \llbracket 0,l-1 \rrbracket.$$ Hence, by induction and using we obtain $$a_{n+4i,i}=(-1)^i {l\choose i}\frac{\prod\limits_{j=0}^{i-1}x(n+4j)}{\prod\limits_{j=1}^{i}y(n+4j)}.$$
Suppose that $\lambda=k-1$ and let $v=W_k\otimes Y$. We have $v\in Ker(\pi\otimes m(D^-))$ and $$[D^+,D^-](v)=4(\lambda+1)v.$$
Let $n\in\mathbb{N}$ and let $V_n$ be the $(n+1)$-dimensional module of $\sl(2,{{\mathbb C}})$, i.e., the vector space with a basis $\lbrace W_k ~ | ~ \forall k\in\llbracket 0,n \rrbracket\rbrace$ and define an action of $\sl(2,{{\mathbb C}})$ on $V_n$ by $$\begin{aligned}
\pi(H)(W_k)&=(-n+2k)W_k,\\
\pi(X)(W_k)&=(n-k) W_{k+1},\\
\pi(Y)(W_k)&=k W_{k-1}.\end{aligned}$$
Consider the representation $\pi\otimes m:\sl(2,{{\mathbb C}})\times {{\mathcal W}}\rightarrow {\rm End}(V_n\otimes S(\ppp))$.
1. If $n$ is odd, then $\pi\otimes m(D^+)$ is injective. If $n$ is even, then the kernel of $\pi\otimes m(D^+)$ is generated by elements of the form $$\sum\limits_{i=0}^{\frac{n}{2}}(-1)^i {\frac{n}{2}\choose l+i} \prod\limits_{j=0}^{i-1}(n-2j)\prod\limits_{j=i+1}^{\frac{n}{2}}(2j) W_{2i}\otimes X^{k-i}Y^{l+i}$$ where $k,l\geq \frac{n}{2}$.
2. The kernel of $\pi\otimes m(D^-)$ is generated by elements of the form $$\sum\limits_{i=0}^{min(n',m-l)}(-1)^i {m\choose l+i} \prod\limits_{j=0}^{i-1}(n-2j-k)\prod\limits_{j=i+1}^{m-l}(2j+k) W_{k+2i}\otimes X^{m-l-i}Y^{l+i}$$ where $m\geq 0$, $k\in\llbracket 0,n \rrbracket$, $l\in\llbracket 0,m \rrbracket$ such that $k=0$ or $l=0$ and $n-k=2n'+\epsilon ~ (\epsilon\in\lbrace 0,1\rbrace)$.
Let $v=\sum\limits_{i=0}^n W_i\otimes P_i\in V_n\otimes S^l(\ppp)$.
$a)$ We have $$\label{e:D+ sl2 finite dim rep}
D^+(v)=2W_0\otimes XP_1+2W_n\otimes YP_{n-1}+2\sum\limits_{i=1}^{n-1}W_i\otimes\Big((n-i+1)YP_{i-1}+(i+1)XP_{i+1}\Big).$$ Suppose that $D^+(v)=0$. In particular $P_1=0$ and, using , this implies $P_i=0$ for all $i$ odd. Similarly, if $n$ is odd, then $P_{n-1}=0$ implies that $P_i=0$ for all $i$ even and so $v=0$. If $n$ is even, the formula follows from a computation similar to Proposition \[p:ps\].
$b)$ We have $$\label{e:D- sl2 finite dim rep}
D^-(v)=W_0\otimes \partial_Y(P_1)+W_n\otimes \partial_X(P_{n-1})+\sum\limits_{i=1}^{n-1}W_i\otimes\Big((n-i+1)\partial_X(P_{i-1})+(i+1)\partial_Y(P_{i+1})\Big).$$ Suppose $D^-(v)=0$ and $P_i=\sum\limits_{j=0}^{l}a_{i,j}X^{l-j}Y^{j}$. We have $\partial_Y(P_1)=\partial_X(P_{n-1})=0$ and $$\sum\limits_{j=1}^{l} (i+1)ja_{i+1,j}+(n-i+1)(l-j-1)a_{i-1,j-1}=0,$$ and the formula follows from a computation similar to Proposition \[p:ps\].
Graded Hecke algebras
=====================
Drinfeld’s degenerate Hecke algebra {#s:DrinfeldA}
-----------------------------------
Let $k$ be a field of characteristic $0$. (As before, the interesting cases for us will be $k={{\mathbb R}}$ or ${{\mathbb C}}$.) Let $V$ be a finite-dimensional vector space with a nondegenerate symmetric bilinear form $B$. Let $\Gamma$ be finite subgroup of $SO(V,B)$. Suppose that we have a family of skew-symmetric forms on $V$, $(a_\gamma)$, $\gamma\in \Gamma$. Define the associative unital algebra $\mathbf H$ as the quotient of the smash-product algebra $T(V)\# k[\Gamma]$ by the relations $$[v_1,v_2]=\sum_{\gamma\in\Gamma}a_\gamma(v_1,v_2) \gamma,\quad v_1,v_2\in V.$$ Of interest are those algebras $\mathbf H$ which have a PBW property, i.e., the associated graded algebra with respect to the filtration where $V$ gets degree $1$ and $\Gamma$ gets degree $0$ is naturally isomorphic to the algebra $S(V)\# k[\Gamma]$.
As before, let ${{{{\mathcal W}}}}={{{{\mathcal W}}}}(V\oplus V^*,\omega)$ be the Weyl algebra and define the symplectic Dirac elements in $\mathbf H\otimes {{{{\mathcal W}}}}$: $$D^-=\sum_i v_i\otimes v_i^*,\quad D^+=\sum_i v_i\otimes v^i,$$ where $\{v_i\}$ is a basis of $V$, $\{v^i\}$ is the $B$-dual basis of $V$, and $\{v_i^*\}$ is the dual basis in $V^*$ to $\{v_i\}$.
Let $\psi: ({\bigwedge}^2V)^*\to {\bigwedge}^2V$ be the identification given by $B$. Under this, $a_\gamma$ corresponds to $$a_\gamma\mapsto \sum_{i,j} a_\gamma(v_i,v_j) v^i\wedge v^j.$$ Next, we may identify $\mu': {\bigwedge}^2V\to \mathfrak{so}(V,B)$ by $$\mu'(u\wedge v)(w)=B(u,w)v-B(v,w)u.$$ Finally, recall the Lie algebra morphism $\nu': \mathfrak{so}(V,B)\to {{{{\mathcal W}}}}(V\oplus V^*,\omega)$, $\nu'(f)=\sum_i f(v_i) v_i^*$. Set $$\tau(\gamma)=\nu'\circ\mu'\circ \psi(a_\gamma).$$ We compute $\tau_\gamma$ explicitly. $$\begin{aligned}
\tau(\gamma)&=\nu'\circ\mu' (\sum_{i,j} a_\gamma(v_i,v_j) v^i\wedge v^j)= \sum_{i,j} a_\gamma(v_i,v_j) \nu'(\mu'(v^i\wedge v^j))\\
&=\sum_{i,j} a_\gamma(v_i,v_j) \sum_\ell \mu'(v^i\wedge v^j)(v_\ell) v_\ell^*=\sum_{i,j} a_\gamma(v_i,v_j) \sum_\ell (B(v^i,v_\ell) v^j v_\ell^*- B(v^j,v_\ell) v^i v_\ell^*)\\
&=\sum_{i,j} a_\gamma(v_i,v_j) (v^j v_i^*-v^i v_j^*)\end{aligned}$$ Hence $$\label{e:tau}
\tau(\gamma)=-2\sum_{i,j} a_\gamma(v_i,v_j) v^i v_j^*.$$
\[p:Drinfeld\] In $\mathbf H\otimes{{{{\mathcal W}}}}$, $$[D^+,D^-]=-\Omega_V\otimes 1-\frac 12\sum_{\gamma\in \Gamma} \gamma\otimes \tau(\gamma),$$ where $\Omega_V=\sum_i v^i v_i\in \mathbf H^\Gamma.$
The calculation is similar to that in the Lie algebra case. $$\begin{aligned}
[D^+,D^-]&=\sum_{i,j} (v_i v_j\otimes v^i v_j^*-v_j v_i\otimes v_j^* v^i)\\
&=\sum_{i,j} [v_i,v_j]\otimes v^i v_j^*-\sum_{i,j} v_j v_i v_j^*(v^i)\otimes 1\\
&=-\sum_i v^i v_i +\sum_\gamma \gamma\otimes \sum_{i,j} a_\gamma(v_i,v_j) v^i v_j^*,\end{aligned}$$ and the claim follows from (\[e:tau\]).
The graded affine Hecke algebra
-------------------------------
Let $(V_0^*,\Phi,V_0,\Phi^\vee)$ be a real root system, where $V_0$ is a finite-dimensional real vector space, $V_0^*$ its dual, $\Phi\subset V_0^*$ the set of roots, and $\Phi^\vee\subset V_0$ the set of coroots. Let $W$ be the finite Weyl group and $B$ be a positive-definite $W$-invariant symmetric bilinear form on $V_0$. Denote $V={{\mathbb C}}\otimes_{{\mathbb R}}V_0$ and $V^*={{\mathbb C}}\otimes_{{\mathbb R}}V_0^*$ the complexified vector spaces and extend $B$ to a symmetric bilinear form on $V$. By abuse of notation, denote also by $B$ the dual form on $V_0^*$ and similarly the bilinear extension to $V$.
Fix a choice of positive roots $\Phi^+$ and let $\Pi$ be the corresponding set of simple roots. Let $s_{{\alpha}}\in W$ denote the reflection corresponding to ${{\alpha}}$. Let $T(V)$ denote the tensor algebra of $V$.
The graded affine Hecke algebra ${{\mathbb H}}={{\mathbb H}}(V,\Phi,k)$ attached to this root system and to the $W$-invariant parameter function $k:\Phi\to{{\mathbb C}}$ is the associative unital algebra which is the quotient of the smash product algebra $T(V)\rtimes{{\mathbb C}}[W]$ by the relations: $$v\cdot s_{{\alpha}}-s_{{\alpha}}\cdot s_{{\alpha}}(v)=k_{{\alpha}}{{\alpha}}(v),\text{ for all }v\in V,\ {{\alpha}}\in\Pi.$$ For every $v\in V$, define $$T_v=\frac 12\sum_{{{\alpha}}>0}k_{{\alpha}}{{\alpha}}(v) s_{{\alpha}},\quad {{\widetilde {v}}}=v-T_v.$$ The presentation of ${{\mathbb H}}$ as a Drinfeld Hecke algebra is with the generators $w\in W$ and ${{\widetilde {v}}}$, $v\in V$, via: $$[{{\widetilde {v}}}_i,{{\widetilde {v}}}_j]=[T_{v_j},T_{v_i}]=\frac 14\sum_{\alpha,\beta>0}k_\alpha k_\beta (\alpha(v_j) \beta(v_i) -{{\alpha}}(v_i)\beta(v_j)) s_{{\alpha}}s_\beta$$ and $$w{{\widetilde {v}}} w^{-1}={{\widetilde {w(v)}}}.$$ In particular, the skew-symmetric forms $a_w$ are $0$ unless $w$ is a product of two distinct reflections and in that case $$a_w({{\widetilde {v}}}_i,{{\widetilde {v}}}_j)=\frac 14\sum_{{{\alpha}},\beta>0,~w=s_{{\alpha}}s_\beta} k_{{\alpha}}k_\beta (\alpha(v_j) \beta(v_i) -{{\alpha}}(v_i)\beta(v_j)).$$ The symplectic Dirac elements are $$D^-=\sum_i {{\widetilde {v}}}_i\otimes v_i^*,\quad D^+=\sum_i {{\widetilde {v}}}_i \otimes v^i.$$ Notice that since $(s_{{\alpha}}s_\beta)^{-1}=s_\beta s_{{\alpha}}$ in $SO(V,B)$, the commutator $[s_{{\alpha}},s_\beta]\in {{\mathbb C}}[W]$ lies in fact in $\mathfrak{so}(V,B)$. Moreover, these commutators span $\mathfrak{so}(V,B)$.
\[p:span\] The set $S = \{[s_\alpha,s_\beta]\mid \alpha,\beta \in \Phi\}\subseteq \mathfrak{so}(V,B)$ spans $\mathfrak{so}(V,B)$.
There is an identification of $\mathfrak{so}(V,B)$ with the irreducible $W$-representation $\bigwedge^2(V)$ under which the action of $W$ on $\mathfrak{so}(V,B)$ is via conjugation. Thus, the set $S$ spans a non-zero $W$-invariant subspace of $\mathfrak{so}(V,B)$ which must be the whole space, by irreducibility.
We can then compute the image under $\nu': \mathfrak{so}(V,B)\to{{{{\mathcal W}}}}$ of the commutators.
\[l:comm\] Regarding $[s_{{\alpha}},s_\beta]$ in $\mathfrak{so}(V,B)$, we have $$\begin{aligned}
\nu'([s_{{\alpha}},s_\beta])&={{\alpha}}(\beta^\vee)\beta{{\alpha}}^\vee-\beta({{\alpha}}^\vee) {{\alpha}}\beta^\vee\\
&={{\alpha}}(\beta^\vee){{\alpha}}^\vee\beta-\beta({{\alpha}}^\vee) \beta^\vee{{\alpha}}\end{aligned}$$ in ${{{{\mathcal W}}}}(V\oplus V^*,\omega).$
For every $v\in V$, notice that $[s_{{\alpha}},s_\beta](v)={{\alpha}}(\beta^\vee) \beta(v) {{\alpha}}^\vee-\beta({{\alpha}}^\vee){{\alpha}}(v) \beta^\vee.$ Then $\nu'([s_{{\alpha}},s_\beta])=\sum_i ({{\alpha}}(\beta^\vee) \beta(v_i) {{\alpha}}^\vee-\beta({{\alpha}}^\vee){{\alpha}}(v_i) \beta^\vee)v_i^*$ and the first formula follows. The second is immediate from the commutation relations in ${{{{\mathcal W}}}}$.
It is also straight-forward to compute $$\tau(w)=-\frac 12\sum_{{{\alpha}},\beta>0~w=s_{{\alpha}}s_\beta} k_{{\alpha}}k_\beta (\iota(\beta){{\alpha}}-\iota({{\alpha}})\beta),$$ where $\iota(f)$ is defined via $f(v) = B(\iota(f),v)$ for all $v\in V$. In particular, for any root $\alpha$, $${{\alpha}}^\vee = \frac 2{B({{\alpha}},{{\alpha}})}\iota({{\alpha}}).$$
From Lemma \[l:comm\], (and the fact that $B(\alpha,\beta) = B(\iota(\alpha),\iota(\beta))$, it follows that the element $$B({{\alpha}}^\vee,\beta^\vee)^{-1}\nu'([s_\alpha,s_\beta]) =(\iota({{\alpha}})\beta-\iota(\beta){{\alpha}})\in {{{{\mathcal W}}}}$$ is well-defined and non-zero, even if $B({{\alpha}}^\vee,\beta^\vee)=0$.
Stretching the analogy with the Lie algebra case, to each pair of positive roots, we denote by $\Delta([s_\alpha,s_\beta]) =
[s_\alpha,s_\beta]\otimes 1 + 1 \otimes \nu'([s_\alpha,s_\beta]) \in {{\mathbb H}}\otimes {{{{\mathcal W}}}}$. Note that $$\Delta([s_\alpha,s_\beta]^2) = [s_\alpha,s_\beta]^2\otimes 1
+2[s_\alpha,s_\beta]\otimes\nu'([s_\alpha,s_\beta])
+1\otimes \nu'([s_\alpha,s_\beta])^2.$$ Let also $\Phi_2^+=\{\{{{\alpha}},\beta\}\mid {{\alpha}},\beta\in\Phi^+, {{\alpha}}\neq\beta\}.$ We emphasise that these are unordered pairs of positive roots.
\[p:comm\] In ${{\mathbb H}}\otimes {{{{\mathcal W}}}}$, $$\begin{aligned}
[D^+,D^-]&=-\Omega_{{\mathbb H}}\otimes 1 +\Omega_W\otimes 1 -\frac 14\sum_{\{{{\alpha}},\beta\}\in \Phi_2^+} k_{{\alpha}}k_\beta \frac 1{B({{\alpha}}^\vee,\beta^\vee)}[s_{{\alpha}}, s_\beta]\otimes \nu'([s_{{\alpha}},s_\beta])\\
&= \left(-\Omega_{{\mathbb H}}+\Omega_W + \Omega'_W\right)\otimes 1 - \Delta(\Omega'_W) + 1\otimes\nu'(\Omega'_W),\end{aligned}$$ where $\Omega_{{\mathbb H}}=\sum_i v_i v^i\in Z({{\mathbb H}})$ and $\Omega_W,\Omega'_W\in {{\mathbb C}}[W]^W$ are given by $$\begin{aligned}
\Omega_W &=\frac 14\sum_{{{\alpha}},\beta>0}k_{{\alpha}}k_\beta B({{\alpha}},\beta) s_{{\alpha}}s_\beta\\
\Omega'_W&=\frac {1}{16}\sum_{{{\alpha}},\beta>0}k_{{\alpha}}k_\beta \frac{1}{B({{\alpha}}^\vee,\beta^\vee)} [s_{{\alpha}}, s_\beta]^2.\end{aligned}$$
As noted before, we can write $$\nu'([s_{{\alpha}},s_\beta])=B({{\alpha}}^\vee,\beta^\vee) (\iota({{\alpha}})\beta-\iota(\beta){{\alpha}}).$$ From Proposition \[p:Drinfeld\], $$\begin{aligned}
[D^+,D^-]&=-\sum_{i}{{\widetilde {v}}}_i {{\widetilde {v^i}}} +\frac 14\sum_{{{\alpha}},\beta>0} k_{{\alpha}}k_\beta s_{{\alpha}}s_\beta\otimes (\iota(\beta){{\alpha}}-\iota({{\alpha}})\beta)\\
&=(-\Omega_{{\mathbb H}}+\Omega_W)\otimes 1+\frac 14\sum_{\{{{\alpha}},\beta\}\in\Phi^+_2} k_{{\alpha}}k_\beta [s_{{\alpha}},s_\beta]\otimes(\iota(\beta){{\alpha}}- \iota({{\alpha}})\beta)\quad \text{(using \cite{BCT}) }\\
&=(-\Omega_{{\mathbb H}}+\Omega_W)\otimes 1-\frac 14\sum_{\{{{\alpha}},\beta\}\in\Phi^+_2} k_{{\alpha}}k_\beta
\frac 1{B({{\alpha}}^\vee,\beta^\vee)}[s_{{\alpha}},s_\beta]\otimes\nu'([s_{{\alpha}},s_\beta]).\end{aligned}$$ The second identity then follows from $$\Delta(\Omega'_W) = \Omega'_W\otimes 1 + \sum_{\{{{\alpha}},\beta\}\in\Phi^+_2} \frac14 k_{{\alpha}}k_\beta \frac 1{B({{\alpha}}^\vee,\beta^\vee)}[s_{{\alpha}},s_\beta]\otimes\nu'([s_{{\alpha}},s_\beta]) +
1\otimes\nu'(\Omega'_W),$$ and we are done.
Let ${{\mathbb H}}(A_1)$ be the Hecke algebra of type $A_1$. Here $V_0={{\mathbb R}}{{\alpha}}^\vee$, $V_0^*={{\mathbb R}}{{\alpha}}$, where ${{\alpha}}$ is the unique simple root. The bilinear form is such that $({{\alpha}}^\vee,{{\alpha}}^\vee)=2$. The relation is $$v\cdot s+s\cdot v=2k,$$ where $v\in V$, $s=s_{{\alpha}}$, and $k=k_{{\alpha}}$. Then ${{\widetilde {v}}}= v-k s$ and $D^+={{\widetilde {v}}}\otimes f$, $D^-={{\widetilde {v}}}\otimes e$. It follows that $$[D^+,D^-]=-\Omega_{{\mathbb H}}\otimes 1 + k^2 (1\otimes 1),$$ where $\Omega_{{\mathbb H}}= \frac 12 ({{\alpha}}^\vee)^2$.
Consider the root system of type $A_2$ with simple roots $\{{{\alpha}},\beta\}$. The bilinear form is normalised such that $B({{\alpha}},{{\alpha}})=B(\beta,\beta)=2$. The third positive root is $\gamma={{\alpha}}+\beta$. A direct calculation shows that $$[D^+,D^-]=-\Omega_{{\mathbb H}}\otimes 1+ \frac {3 k^2}2(1\otimes 1)+\frac {k^2}4 (s_{{\alpha}}s_\beta+s_\beta s_{{\alpha}})\otimes 1-\frac {k^2}4 (s_{{\alpha}}s_\beta-s_\beta s_{{\alpha}})\otimes (\beta{{\alpha}}^\vee-{{\alpha}}\beta^\vee).$$
In ${{\mathbb H}}\otimes {{{{\mathcal W}}}}$: $$[1\otimes \Delta,D^-]=2 D^+,\quad [1\otimes \Delta,D^+]=0,\quad [1\otimes \Delta,[D^+,D^-]]=0.$$
Straightforward.
Suppose $(\pi,M)$ is an ${{\mathbb H}}$-module. Then the actions of $D^\pm$ give rise to the symplectic Dirac operators: $$D^\pm: M\otimes {{\mathcal P}}\to M\otimes{{\mathcal P}}.$$ Notice that $D^+$ maps $M\otimes S^j(V)$ to $M\otimes S^{j+1}(V)$, while $D^-$ maps $S^j(V)$ to $S^{j-1}(V)$.
If $\sigma$ is an irreducible $W$-representation, denote by $M(\sigma)$ the $\sigma$-isotypic component of $M$. Suppose $M$ has a central character $\chi_M$ (e.g., $M$ is a simple module). The central characters of ${{\mathbb H}}$-modules are parameterised by $W$-orbits in $V^*$, and we will think implicitly of $\chi_M$ as an element (or a $W$-orbit) in $V^*$. Then $\Omega$ acts on $M$ by a scalar multiple of the identity, where the scalar is $$\pi(\Omega)=B(\chi_M,\chi_M),$$ see [@BCT] for example. On the other hand, $\Omega_W$ acts on $M(\sigma)$ by a scalar multiple of the identity, where the scalar is $$\sigma(\Omega_W)=\frac 14\sum_{{{\alpha}},\beta>0}k_{{\alpha}}k_\beta B({{\alpha}},\beta) \frac{{\operatorname{tr}}\sigma(s_{{\alpha}}s_\beta)}{{\operatorname{tr}}\sigma(1)}.$$
If $\sigma={\mathsf{triv}}$ is the trivial $W$-representation, then $${\mathsf{triv}}(\Omega_W)=B(\rho_k,\rho_k),\text{ where }\rho_k=\frac 12\sum_{{{\alpha}}>0} k_{{\alpha}}{{\alpha}}.$$ Notice that $\rho_k$ is exactly the central character of the trivial ${{\mathbb H}}$-module.
From Proposition \[p:comm\], it follows that, when $M$ has a central character, if $x\in M(\sigma)\otimes {{\mathcal P}}$, then $$\begin{aligned}
\label{e:iso}
[{{\mathcal D}}^+,{{\mathcal D}}^-]x&=(-\pi(\Omega)+\sigma(\Omega_W))x - {{\mathcal E}}_{\sigma,P} x,\qquad\text{ where}\\
{{\mathcal E}}_{\sigma,P}&=\frac 14\sum_{\{{{\alpha}},\beta\}\in\Phi^+_2} k_{{\alpha}}k_\beta \sigma([s_{{\alpha}},s_\beta])\otimes {{\mathcal X}}_{{{\alpha}},\beta},\text{ with } {{\mathcal X}}_{{{\alpha}},\beta}=\iota({{\alpha}})\partial_\beta-\iota(\beta)\partial_{{\alpha}}.\end{aligned}$$
Notice that every ${{\mathcal X}}_{{{\alpha}},\beta}$ is a differential operator on ${{\mathcal P}}$ preserving the degree.
The element $\Omega_W$ {#s:4.3}
----------------------
We look in more detail at the element $\Omega_W=\frac 14\sum_{{{\alpha}},\beta>0}k_\alpha k_\beta B(\alpha,\beta) s_\alpha s_\beta$. It is easy to see that this can be rewritten as (see also [@BCT]): $$\Omega_W=\frac 14\sum_{({{\alpha}},\beta)\in (\Phi^+)^2,\ s_\alpha(\beta)<0}k_\alpha k_\beta B(\alpha,\beta) s_\alpha s_\beta.$$
Suppose $W=S_n$ and without loss of generality, assume that $k_\alpha=1$ for all $\alpha$. Assume the bilinear form is such that $B(\alpha,\alpha)=2$.
\[l:Sn-1\] When $W=S_n$, we have:
1. $\Omega_{S_n}=\frac 14 (n(n-1)+e_{(123)})$, where $e_{(123)}$ is the sum of $3$-cycles in $\mathbb C[S_n]$.
2. $\Omega_{S_n}=\frac 14 (\frac {n(n-1)}2+\sum_{i=1}^n T_i^2)$, where $T_i=(1,i)+(2,i)+\dots+ (i-1,i)$, $1\le i\le n$, are the Jucys-Murphy elements.
3. If $\lambda$ is a partition of $n$ and $\sigma_\lambda$ is the corresponding irreducible $S_n$-representation, then $$\sigma_\lambda(\Omega_{S_n})=\frac 14(\frac {n(n-1)}2+\Sigma_2(\lambda)),$$ where $\Sigma_k(\lambda)$ is the sum of $k$-powers of contents in $\lambda$ viewed as a left-justified decreasing Young diagram.
Parts (a) and (b) are immediate by direct calculation. Part (c) follows from (b) via the known properties of Jucys-Murphy elements, or equivalently, by Frobenius character formula applied in the case of $3$-cycles.
Now suppose $W=W_n$ is the Weyl group of type $B_n$. Let $k_l$ be the parameter on the long roots $\epsilon_i\pm \epsilon_j$ and $k_s$ the parameter on the short roots $\epsilon_i$. Let $B$ be the standard bilinear form $B(\epsilon_i,\epsilon_j)=\delta_{ij}$. The irreducible $W_n$-representations are parameterised by pairs of partitions $(\lambda,\mu)$, where $\lambda$ is a partition of $a$ and $\mu$ is a partition of $b$, such that $a+b=n$. If we denote the corresponding character by $\chi_{(\lambda,\mu)}$, then we also have $$\chi_{(\lambda,\mu)}=\operatorname{Ind}_{W_a\times W_b}^{W_n}(\chi_{(\lambda,0)}\otimes \chi_{(0,\mu)}),$$ where $\chi_{(\lambda,0)}$ is the character of the representation $\sigma_\lambda$ of $S_n$ inflated so that the short reflections act by the identity, and $\chi_{(0,\mu)}$ is the character of the representation $\sigma_\mu$ of $S_n$ inflated so that the short reflections act by the negative of the identity.
\[l:Bn-1\] In the case of the Weyl group of type $B_n$:
1. $$\Omega_{W_n}=\frac 14 (2 n(n-1) k_l^2+n k_s^2)+\frac 14 k_l^2 e_{A_2}+\frac 12 k_l k_s e_{B_2},$$ where $e_{A_2}$ and $e_{B_2}$ are the sums of $W_n$-conjugates of the Coxeter elements of type $A_2$ and $B_2$, respectively.
2. The scalar by which $\Omega_{W_n}$ acts in the irreducible representation labeled by $(\lambda,\mu)$ is $$\chi_{(\lambda,\mu)}(\Omega_{W_n})=\frac 14 (2 n(n-1) k_l^2+n k_s^2)+2 k_l^2 (\sigma_\lambda(e_{(123)})+\sigma_\mu(e_{(123)})))+k_l k_s (\sigma_\lambda(e_{(12)})-\sigma_\mu(e_{(12)})).$$ Moreover, $\sigma_\lambda(e_{(123)})=-\frac {a(a-1)}2+\Sigma_2(\lambda)$, $\sigma_\lambda(e_{(12)})=\Sigma_1(\lambda)$ and similarly for $\mu$.
The pairs of distinct positive roots $(\alpha,\beta)$ with $s_\alpha(\beta)<0$ that contribute are:
- $(\epsilon_i-\epsilon_j,\epsilon_i-\epsilon_k)$, $(\epsilon_i-\epsilon_j,\epsilon_k-\epsilon_j)$, $i<k<j$;
- $(\epsilon_i+\epsilon_j,\epsilon_i-\epsilon_k)$, $i<j$, $i<k$;
- $(\epsilon_i+\epsilon_j,\epsilon_i+\epsilon_k)$, $i<j<k$;
- $(\epsilon_i+\epsilon_j,\epsilon_i)$ and $(\epsilon_i+\epsilon_j,\epsilon_j)$, $i<j$;
- $(\epsilon_i,\epsilon_i+\epsilon_j)$, $i<j$;
- $(\epsilon_i,\epsilon_i-\epsilon_j)$, $i<j$.
From this, we see that the only pairs of distinct roots $(\alpha,\beta)$ that contribute are the ones that form subroot systems of type $A_2$ or $B_2$. So $\Omega_{W_n}=a+b e_{A_2}+ c e_{B_2}$ for some constants $a,b,c$. We compute $a=\sum_{\alpha>0} k_\alpha^2$. For $b$, we only need to know how many times a representative of $e_{A_2}$ appears and this is a calculation we’ve already done for type $A$. For $c$, it is the same calculation for a representative of type $B_2$, and here we see that each such representative can be obtain from $(\epsilon_i,\epsilon_i-\epsilon_j)$ but also $(\epsilon_i+\epsilon_j,\epsilon_i)$ since $s_{\epsilon_i}s_{\epsilon_i-\epsilon_j}=s_{\epsilon_i+\epsilon_j} s_{\epsilon_i}$. Claim (a) follows.
For (b), we use the character formula for induced representations: $$\chi_{(\lambda,\mu)}(w)=\frac 1{|W_a| |W_b|}\sum_{s^{-1}ws^\in W_a\times W_b}(\chi_{(\lambda,0)}(s^{-1} w s)\chi_{(0,\mu)}(s^{-1} w s)).$$ Let $w=(123)$. The number of $s$ such that $s^{-1}w s\in W_a$ is $2^n a(a-1)(a-2)\cdot (n-3)!$. It follows that: $$\chi_{(\lambda,\mu)}((123))=\frac 1{2^n a! b!} ( 2^n a (a-1) (a-2) (n-3)! \sigma_\lambda((123)) \sigma_\mu(1)+2^n b (b-1) (b-2) (n-3)! \sigma_\lambda(1) \sigma_\mu((123)).$$ Noting that $\chi_{(\lambda,\mu)}=\frac{n!}{a! b!} \sigma_\lambda(1)\sigma_\mu(1)$ and that the size of the conjugacy class of $(123)$ in $W_n$ is $\frac 83 n(n-1)(n-2)$, it follows immediately that $$\chi_{(\lambda,\mu)}(e_{(123)})=8 (\sigma_\lambda(e_{(123)})+\sigma_\mu(e_{(123)})).$$ Completely similarly, we deduce that $$\chi_{(\lambda,\mu)}(e_{B_2})=2 (\sigma_\lambda(e_{(12)})-\sigma_\mu(e_{(12)})),$$ using that $\chi_{(\lambda,0)}(w(B_2))=\sigma_\lambda({(12)})$ and $\chi_{(0,\mu)}(w(B_2))=-\sigma_\mu({(12)})$, if $w(B_2)$ is a representative of the conjugacy class of type $B_2$ in $B_n$.
The element $\Omega_W'$ {#s:4.4}
-----------------------
We now look closer at the element $\Omega_W'=\tfrac {1}{16}\sum_{{{\alpha}},\beta>0}k_{{\alpha}}k_\beta B({{\alpha}}^\vee,\beta^\vee)^{-1} [s_{{\alpha}}, s_\beta]^2\in{{\mathbb C}}W^W.$ Firstly, remark that, just as for $\Omega_W$, the element $\Omega_W'$ can be rewritten as $$\Omega_W'=\frac 1{16} \sum_{({{\alpha}},\beta)\in(\Phi^+)^2,~s_\alpha(\beta)<0} k_{{\alpha}}k_\beta B({{\alpha}}^\vee,\beta^\vee)^{-1} [s_{{\alpha}}, s_\beta]^2.$$ This is because the terms corresponding to the pairs $(\alpha,\beta)$ and $(\alpha,\gamma)$, if $\gamma=s_\alpha(\beta)>0$, cancel out.
The image of $\Omega_W'$ in the Weyl algebra satisfy the following properties.
\[p:O-inv\] The element $\nu'(\Omega'_W)$ is an $O(V,B)$-invariant element of ${{{{\mathcal W}}}}$.
It suffices to show that under the faithful representation $m:{{\mathcal W}}({{\mathcal V}},\omega)\rightarrow {\rm End}(S(V))$ the element $m(\Omega'_W)$ commutes with $m(\nu'(X))$, for all $X\in\mathfrak{so}(V,B)$. Indeed, if that is the case, then $\Omega'_W$ will be an $SO(V,B)$-invariant element of ${{{{\mathcal W}}}}$ which also commutes with some reflections $s\in O(V,B)\setminus SO(V,B)$. This then implies that $\Omega'_W$ is $O(V,B)$-invariant.
Now, to each pair of positive roots $(\alpha,\beta)\in \Phi^+\times\Phi^+$, let $X_{\alpha\beta}:=m(\nu'([s_\alpha,s_\beta]))$. Then, we claim that for every $\xi = v_1\cdot v_2\cdot\ldots\cdot v_m\in S^m(V)$ we have $$\label{e:diagsq}
m(\Omega_W')(\xi) = \sum_{j=1}^m \sum_{{{\alpha}},\beta > 0}\frac{k_\alpha k_\beta}{16}\frac1{B(\alpha^\vee,\beta^\vee)} ( v_1\cdot\ldots\cdot X_{\alpha\beta}^2(v_j)\cdot\ldots\cdot v_m) = \sum_{j=1}^m v_1\cdot\ldots\cdot \Omega'_W(v_j)\cdot\ldots\cdot v_m.$$ Indeed, the composition $X_{\alpha\beta}^2 = X_{\alpha\beta}\circ X_{\alpha\beta}$ acts on $S(V)$ via $$\begin{gathered}
X_{\alpha\beta}^2(\xi) = \sum_{j=1}^m v_1\cdot\ldots\cdot X^2_{\alpha\beta}(v_j)\cdot\ldots\cdot v_m + \\
\sum_{j=1}^m\left(\sum_{i<j} v_1 \cdot\ldots\cdot X_{\alpha\beta}(v_i) \cdot\ldots\cdot X_{\alpha\beta}(v_j) \cdot\ldots\cdot v_m +
\sum_{j<k} v_1\cdots X_{\alpha\beta}(v_j) \cdot\ldots\cdot X_{\alpha\beta}(v_k) \cdot\ldots\cdot v_m \right).\end{gathered}$$ However, on each copy of $S^2(V)$ occurring in between parenthesis, the element $X_{\alpha\beta}(v_i)X_{\alpha\beta}(v_j)$ corresponds to the action of $[s_\alpha, s_\beta]\in{{\mathbb C}}W$ on $v_iv_j\in S^2(V)$. Hence, $$\sum_{{{\alpha}},\beta > 0} \frac{k_\alpha k_\beta}{16} \frac1{B(\alpha^\vee,\beta^\vee)} X_{\alpha\beta}(v_i)X_{\alpha\beta}(v_j) =
\sum_{{{\alpha}},\beta > 0} \frac{k_\alpha k_\beta}{16} \frac1{B(\alpha^\vee,\beta^\vee)} [s_\alpha,s_\beta](v_iv_j) = 0,$$ since $[s_\alpha,s_\beta]\in {{\mathbb C}}W$ is anti-symmetric on $\alpha,\beta$, settling the claim. Thus, for all $X\in \mathfrak{so}(V,B)$ and $\xi = v_1v_2\cdots v_m\in S^mV$, using (\[e:diagsq\]) we have $$m(X)m(\Omega_W')(\xi) = m(X)\left(\sum_{j=1}^m v_1 \cdot\ldots\cdot \Omega'_W(v_j) \cdot\ldots\cdot v_m\right) = m(\Omega_W')m(X)(\xi)$$ since $X$ commutes with $\Omega_W'$ on $S^1(V) = V$.
\[p:Casimirs\] The elements $\nu'(\Omega_W')$ and $\Omega(\mathfrak{sl}(2))$ satisfy the following linear relation $$\nu'(\Omega_W') = \left(\frac{(n-4)(\nu'(\Omega_W'))_0}{n-1}\right) + \left(\frac{-4(\nu'(\Omega_W'))_0}{n(n-1)}\right)\Omega(\mathfrak{sl}(2)).$$
By Proposition \[p:O-inv\], $\nu'(\Omega'_W)$ is $O(V,B)$-invariant in ${{{{\mathcal W}}}}$, and therefore, it must lie in $\nu(U(\mathfrak{sl}(2)))$ by the dual pair argument. In addition, $\nu'(\Omega'_W)$ commutes with $\mathfrak{sl}(2)$ itself as well, since it is built of elements of $\mathfrak{so}(V,B)$. Hence $\nu'(\Omega'_W)$ is in the centre of $\nu(U(\mathfrak{sl}(2)))$. It follows thus that we can write $$\label{e:eq1}
\nu'(\Omega_W') = a + b\Omega(\mathfrak{sl}(2)).$$ for some constants $a,b\in {{\mathbb C}}$. Thus, taking the zero degree components we get, using Proposition \[p:sl2Cas\] $$\label{e:eq2}
(\nu'(\Omega_W'))_0 = a -(\tfrac{3n}{4})b.$$ Furthermore, acting with (\[e:eq1\]) on $S^0(V)$, we get $$\label{e:eq3}
0 = a + \frac{n(n-4)}{4}b.$$ Solving for $a$ and $b$ in (\[e:eq2\]) and (\[e:eq3\]) yields the claim.
Therefore, to obtain a precise relation between $\nu'(\Omega'_W)$ and $\Omega(\mathfrak{sl}(2))$ of ${{\mathcal W}}$, it suffices to compute the degree zero component of $\eta(\nu'(\Omega'_W))$.
\[p:0-term\] We have $$\Big(\nu'(\Omega'_W)\Big)_0 = \tfrac{1}{2}B(\rho_k,\rho_k) - \tfrac{1}{8}\sum_{\alpha,\beta>0}k_{{\alpha}}k_\beta\frac{B(\alpha,\beta)^3}{B(\alpha,\alpha)B(\beta,\beta)},$$ where $\rho_k = \tfrac{1}{2}\sum_{\alpha>0}k_{{\alpha}}{{\alpha}}\in V^*$.
Let $\textup{ev}_0:S(V\oplus V^*)\to {{\mathbb C}}$ be the evaluation at $0$ map. We have $$\Big(\nu'(\Omega'_W)\Big)_0 = \textup{ev}_0(\gamma(\nu'(\Omega'_W))(1)).$$ We recall that for $x\in V\oplus V^*$ we have $\gamma(x) = \epsilon(x)+\tfrac{1}{2}i(x)$ so that $\gamma(x)(P)=x\cdot P+\tfrac{1}{2}i(x)(P)$. Moreover, $i(x)(y)=\omega(x,y)$, for all $y\in V\oplus V^*$. Thus, if $v\in V,\lambda\in V^*$ and $P\in {{{{\mathcal W}}}}$ we note that $$\label{e:gtrick}
\gamma(v)(\gamma(\lambda)(P)) = v\cdot \gamma(\lambda)(P) + \tfrac{1}{2}i(v)(\lambda P)
+ \tfrac{1}{4}i(v)(i(\lambda)(P)).$$ Now, for each pair of roots $\alpha,\beta$, we let $X_{{{\alpha}}\beta} = \nu'([s_\alpha,s_\beta])=B({{\alpha}}^\vee,\beta^\vee)(\iota({{\alpha}})\beta-\iota(\beta){{\alpha}})$ and hence $$\gamma(X_{{{\alpha}}\beta})(1) = B({{\alpha}}^\vee,\beta^\vee)(\gamma(\iota({{\alpha}}))(\gamma(\beta)(1))-\gamma(\iota(\beta))(\gamma({{\alpha}})(1))) = B({{\alpha}}^\vee,\beta^\vee)(\iota({{\alpha}})\cdot\beta-\iota(\beta)\cdot{{\alpha}}).$$ Thus, using (\[e:gtrick\]) and applying $\textup{ev}_0$ we obtain that $$\begin{aligned}
\Big(\nu'([s_\alpha,s_\beta]^2\Big)_0 &= B({{\alpha}}^\vee,\beta^\vee)\textup{ev}_0(\gamma(X_{{{\alpha}}\beta})(\iota({{\alpha}})\cdot\beta-\iota(\beta)\cdot{{\alpha}})\\
&=
\tfrac{1}{4}B({{\alpha}}^\vee,\beta^\vee)(i(\iota(\alpha))(i(\beta)(X_{\alpha\beta})) -
i(\iota(\beta))(i(\alpha)(X_{\alpha\beta}))\\
&=\tfrac{1}{4}B({{\alpha}}^\vee,\beta^\vee)^2\Big(2\omega(\beta,\iota(\beta))\omega(\alpha,\iota(\alpha))-\omega(\beta,\iota({{\alpha}}))^2-\omega(\alpha,\iota(\beta))^2\Big)\\
&=\tfrac{1}{2}B({{\alpha}}^\vee,\beta^\vee)^2(B(\beta,\beta)B(\alpha,\alpha)-B(\alpha,\beta)^2)\\
&=2B({{\alpha}}^\vee,\beta^\vee)B({{\alpha}},\beta)\left(1-\frac{B(\alpha,\beta)^2}{B({{\alpha}},{{\alpha}})B(\beta,\beta)}\right).\end{aligned}$$ Hence, as $\Omega_W'=\tfrac {1}{16}\sum_{{{\alpha}},\beta>0}k_{{\alpha}}k_\beta B({{\alpha}}^\vee,\beta^\vee)^{-1} [s_{{\alpha}}, s_\beta]^2$ we get $$\Big(\nu'(\Omega'_W)\Big)_0 = \tfrac{1}{2}B(\rho_k,\rho_k) - \tfrac{1}{8}\sum_{\alpha,\beta>0}k_{{\alpha}}k_\beta\frac{B(\alpha,\beta)^3}{B(\alpha,\alpha)B(\beta,\beta)}$$ as required.
\[c:deg0vals\] For crystallographic root systems, following the normalization conventions of [@Bou], we have: $$\begin{aligned}
\textup{Type } A_n&\qquad (\nu'(\Omega'_W))_0 = \frac{k^2}{32}(n+1)n(n-1)\\
\textup{Type } B_n&\qquad (\nu'(\Omega'_W))_0 = \frac{1}{8}k_l n(n-1)((n-2)k_l + k_s)\\
\textup{Type } C_n&\qquad (\nu'(\Omega'_W))_0 = \frac{1}{8}k_s n(n-1)((n-2)k_s + 2k_l)\\
\textup{Type } D_n&\qquad (\nu'(\Omega'_W))_0 = \frac{k^2}{8}n(n-1)(n-2)\\
\textup{Type } E_6&\qquad (\nu'(\Omega'_W))_0 = \frac{45}{2} k^2\\
\textup{Type } E_7&\qquad (\nu'(\Omega'_W))_0 = 63 k^2\\
\textup{Type } E_8&\qquad (\nu'(\Omega'_W))_0 = 210 k^2\\
\textup{Type } F_4&\qquad (\nu'(\Omega'_W))_0 = \frac{3}{2}(k_s+2k_l)(k_s+k_l)\\
\textup{Type } G_2&\qquad (\nu'(\Omega'_W))_0 = \frac{3}{16}(k_s+3k_l)(k_s+k_l),\end{aligned}$$ where $k_s$ and $k_l$ are the parameters for the short and long roots, respectively.
Suppose first that $\Phi$ is a simply laced root system. Assume that $k=1$. We claim that $\left(\nu'(\Omega'_W)\right)_0=\frac 38 \left(B(\rho,\rho)-\frac N 2\right)$, where $N$ is the number of positive roots. Indeed, notice that in this case $B(\alpha,\beta)^3=B(\alpha,\beta)$ for all positive roots $\alpha\neq \beta$. Then, from Proposition \[p:0-term\] we get that $$\begin{aligned}
\left(\nu'(\Omega'_W)\right)_0&=\frac 12 B(\rho,\rho)-\frac 18 \sum_{\alpha>0} B(\alpha,\alpha)-\frac 1{32} \sum_{\alpha\neq\beta>0} B(\alpha,\beta)\\
&=\frac 12 B(\rho,\rho)-\frac N 4+\frac 1{32} \sum_{\alpha>0}B(\alpha,\alpha)-\frac 1{32} \sum_{\alpha,\beta>0} B(\alpha,\beta)\\
&=\frac 12 B(\rho,\rho)-\frac 3{16} N-\frac 18 B(\rho,\rho)=\frac 38 B(\rho,\rho)-\frac 3{16} N.\end{aligned}$$ For type $A_n$, we substitute $B(\rho,\rho)=\tfrac{1}{12}n(n+1)(n+2)$ and $N=n(n+1)/2$. For type $D_n$, we substitute $B(\rho,\rho)=\tfrac{1}{6}n(n-1)(2n-1)$ and $N=n(n-1)$. For types $E_6, E_7$ and $E_8$, we substitute $B(\rho,\rho)=78, 399/2$ and $620$; $N=36, 63$ and $120$, respectively.
For type $B_n$, note that we also have $B(\alpha,\beta)^3=B(\alpha,\beta)$ whenever $\alpha\neq \beta$ are positive roots. Proceeding similarly to the symply-laced case, and using $\alpha^\vee = 2B(\alpha,\alpha)^{-1}\iota(\alpha)$, we get $$\begin{aligned}
\left(\nu'(\Omega'_W)\right)_0&=\frac 12 B(\rho_k,\rho_k)-\frac 18 \sum_{\alpha>0} k_\alpha^2B(\alpha,\alpha)-\frac 1{32} \sum_{\alpha\neq\beta>0} k_\alpha k_\beta B(\alpha^\vee,\beta^\vee)\\
&=\frac 12 B(\rho_k,\rho_k)- \frac 18 B(\rho^\vee_k,\rho^\vee_k)- \frac{3N_l k_l^2}{16},\end{aligned}$$ where $N_l$ is the number of long positive roots. Substituting $B(\rho_k,\rho_k) = \tfrac{1}{12}(3nk_s^2 + 6n(n-1)k_sk_l + 2n(n-1)(2n-1)k_l^2)$, $B(\rho^\vee_k,\rho^\vee_k) = \tfrac{1}{6}(6nk_s^2 + 6n(n-1)k_sk_l + n(n-1)(2n-1)k_l^2)$ and $N_l = n(n-1)$ yields the claim. Type $C_n$ is obtained from type $B_n$ by setting $k'_s = k_l$, $k'_l = 2k_s$, where $k'_s,k'_l$ are the parameters of $C_n$ and $k_s,k_l$ are the ones of type $B_n$.
Types $F_4$ and $G_2$ are obtained by direct computation using the conventions in Planche VIII and Planche IX of [@Bou].
In the case where the root system is crystallographic, $n>1$ and $k_\alpha = 1$ for all positive roots, we obtain from Corollary \[c:deg0vals\] and Proposition \[p:Casimirs\], that $$(\nu'(\Omega'_W))_4 = -c(\Omega(\mathfrak{sl}(2)))_4$$ for a rational constant $c>0$.
Finally, we compute $\Omega'_W\in \mathbb C[W]$ in two examples.
If $W=S_n$, then $\Omega_{S_n}'=\frac 18(e_{(123)}-|C_{(123)}|)$, and it acts in an irreducible $S_n$-representation $\sigma_\lambda$ by $\sigma_\lambda(\Omega_{S_n})=\frac 18(\Sigma_2(\lambda)-\Sigma_2((n))).$
It is easy to see that $\Omega_{S_n}'=\frac 1{16} 2\sum_{i<k<j} ((ikj)-(ijk))^2=\frac 18(-2\sum_{i<k<j}1+e_{(123)})=\frac 18(e_{(123)}-|C_{(123)}|).$ The second claim follows from the action of $e_{(123)}$ as in Lemma \[l:Sn-1\].
If $W=W_n$ (type $B_n$), then $\Omega_{W_n}'=\frac 1{16} k_l^2(e_{A_2}-|C_{A_2}|)+\frac 14 k_s k_l (e_{2\widetilde A_1}-|C_{2\widetilde A_1}|)$, where $2 \widetilde A_1$ denotes the conjugacy class of products of two commuting reflections in the short roots.
The proof is similar to that of Lemma \[l:Bn-1\](a). First we notice that $[s_\alpha,s_\beta]^2=s_\beta s_\gamma +s_\gamma s_\beta-2$, where $\gamma=-s_\alpha(\beta)>0$. Then from the list of pairs of roots that can contribute, we see that $\Omega_{W_n}'$ must be of the form $\Omega_{W_n}'=a+ b e_{A_2} + c e_{2\widetilde A_1}$. Since every commutator vanishes on the trivial representation, we see that in fact $\Omega_{W_n}'=b (e_{A_2}-|C_{A_2}|) + c (e_{2\widetilde A_1}-|C_{2\widetilde A_1}|)$. To determine $b$ and $c$, we proceed as for Lemma \[l:Bn-1\](a) and count the number of occurrences of a representative of type $A_2$ and of type $2\widetilde A_1$. This is a direct calculation for the root system of type $B_2$.
Unitary structures
------------------
As before ${{\mathcal P}}$ can be endowed with a positive definite hermitian form $\langle~,~\rangle_{{{\mathcal P}}}$. For the Hecke algebra ${{\mathbb H}}$, there exist two natural star operations $*$ and $\bullet$ defined on generators as follows: $$\begin{aligned}
w^*=w^{-1},\quad {{\widetilde {v}}}^*=-{{\widetilde {v}}},\\
w^\bullet=w^{-1},\quad {{\widetilde {v}}}^\bullet={{\widetilde {v}}},
\end{aligned}$$ for all $w\in W$, $v\in V_0$.
Define two star operations on ${{\mathbb H}}\otimes{{\mathcal W}}$ as well: $$(h\otimes \nu)^*=h^*\otimes \nu^*,\quad (h\otimes \nu)^\bullet=h^\bullet\otimes\nu^*,$$ for $h\in{{\mathbb H}}$ and $\nu\in{{\mathcal W}}$.
In ${{\mathbb H}}\otimes{{\mathcal W}}$, $(D^\pm)^*=-D^{\mp}$ and $(D^\pm)^\bullet=D^{\mp}.$
Straightforward.
Let us assume that $M$ has a nondegenerate $*$-invariant hermitian form $\langle~,~\rangle_M$. Define the product form $$\langle~,~\rangle_{M\otimes{{\mathcal P}}}=\langle~,~\rangle_M\langle~,~\rangle_{{{\mathcal P}}}$$ on $M\otimes{{\mathcal P}}$, so that this becomes a $*$-invariant hermitian form on the ${{\mathbb H}}\otimes{{\mathcal W}}$-module $M\otimes{{\mathcal P}}$.
\[l:diff\] With the notation as above, if $x\in M(\sigma)\otimes {{\mathcal P}}$, then $$\langle {{\mathcal D}}^+ x, {{\mathcal D}}^+ x\rangle_{M\otimes{{\mathcal P}}}-\langle {{\mathcal D}}^- x, {{\mathcal D}}^- x\rangle_{M\otimes{{\mathcal P}}}=(-\pi(\Omega)+\sigma(\Omega_W))\langle x,x\rangle_{M\otimes {{\mathcal P}}}+\langle {{\mathcal E}}_{\sigma,{{\mathcal P}}}x,x\rangle_{M\otimes {{\mathcal P}}}.$$
This follows immediately from (\[e:iso\]) using the adjointness property $({{\mathcal D}}^{\pm})^*=-{{\mathcal D}}^{\mp}.$
As a particular example, notice that when $x\in M(\sigma)\otimes S^0(V^*)=M(\sigma)\otimes 1$, then ${{\mathcal D}}^- x=0={{\mathcal E}}_{\sigma,{{\mathcal P}}}x$, hence we have:
\[p:Casimir\] For all $x\in M(\sigma)\otimes 1$: $$\langle {{\mathcal D}}^+ x, {{\mathcal D}}^+ x\rangle_{M\otimes{{\mathcal P}}}=(\sigma(\Omega_W)-\pi(\Omega))\langle x,x\rangle_{M\otimes {{\mathcal P}}}.$$ Moreover, if $M$ is $*$-unitary then $\pi(\Omega)=B(\chi_M,\chi_M)\le \sigma(\Omega_W)$, for all irreducible $W$-representations $\sigma$ such that $M(\sigma)\neq 0$.
In particular, if $M$ is $W$-spherical, i.e., $M({\mathsf{triv}})\neq 0$, and $*$-unitary, then $B(\chi_M,\chi_M)\le B(\rho_k,\rho_k)$.
The inequality follows by taking $x\neq 0$ and using that $\langle {{\mathcal D}}^+ x, {{\mathcal D}}^+ x\rangle_{M\otimes{{\mathcal P}}}\ge 0$, since $M\otimes{{\mathcal P}}$ is a $*$-unitary ${{\mathbb H}}\otimes{{\mathcal W}}$-module.
This is of course the analogue of the Casimir inequality for Lie algebras (and it has already been known in this setting by [@BCT]).
Now suppose more generally that $x\in M(\sigma)\otimes{{\mathcal P}}$ is a simple tensor: $x=m\otimes p$, $m\in M$, $p\in {{{{\mathcal P}}}}$. Then: $$\begin{aligned}
\langle {{\mathcal E}}_{\sigma,{{\mathcal P}}}x,x\rangle_{M\otimes {{\mathcal P}}}&=\frac 14\sum_{\{{{\alpha}},\beta\}\in\Phi_2^+}k_{{\alpha}}k_\beta \langle \sigma([s_{{\alpha}}, s_\beta]) m\otimes (\iota({{\alpha}})\partial_\beta-\iota(\beta)\partial_{{\alpha}})p,m\otimes p\rangle _{M\otimes {{\mathcal P}}}\\
&=\frac 14\sum_{\{{{\alpha}},\beta\}\in\Phi_2^+}k_{{\alpha}}k_\beta\langle \sigma([s_{{\alpha}},s_\beta])m,m\rangle_M \langle (\iota({{\alpha}})\partial_\beta-\iota(\beta)\partial_{{\alpha}})p,p\rangle_{{{\mathcal P}}}.
\end{aligned}$$ But $\langle (\iota({{\alpha}})\partial_\beta-\iota(\beta)\partial_\alpha)p,p\rangle_{{{\mathcal P}}}=\langle\partial_\beta(p),\partial_\alpha (p)\rangle_{{{\mathcal P}}}-\langle\partial_{{\alpha}}(p),\partial_{\beta}(p)\rangle_{{{\mathcal P}}}=0$, when $p\in S(V_0)$ (i.e., $p$ is a real polynomial). Notice that we used in this calculation that multiplication by $\iota({{\alpha}})$ is adjoint to $\partial_{{\alpha}}$ in the inner product on ${{\mathcal P}}$. In conclusion, $$\label{e:zero}
\langle {{\mathcal E}}_{\sigma,{{\mathcal P}}}x,x\rangle_{M\otimes {{\mathcal P}}}=0,\text{ for all } x=m\otimes p,\ p\in S(V_0^*).$$
Suppose $M$ is $*$-hermitian as above and $\sigma$ is an irreducible $W$-representation such that $M(\sigma)\neq 0.$ If $x=m\otimes p\in M(\sigma)\otimes S(V_0)$ is a simple tensor, then $$\langle {{\mathcal D}}^+ x, {{\mathcal D}}^+ x\rangle_{M\otimes{{\mathcal P}}}-\langle {{\mathcal D}}^- x, {{\mathcal D}}^- x\rangle_{M\otimes{{\mathcal P}}}=(-\pi(\Omega)+\sigma(\Omega_W))\langle x,x\rangle_{M\otimes {{\mathcal P}}}.$$
The formula follows at once from Lemma \[l:diff\] and (\[e:zero\]).
[30]{}
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a complete evaluation for the prompt $\eta_c$ production at the LHC at next-to-leading order in $\alpha_s$ in nonrelativistic QCD. By assuming heavy quark spin symmetry, the recently observed $\eta_c$ production data by LHCb results in a very strong constraint on the upper bound of the color-octet long distance matrix element $\mopa$ of $J/\psi$. We find this upper bound is consistent with our previous study of the $J/\psi$ yield and polarization and can give good descriptions for the measurements, but inconsistent with some other theoretical estimates. This may provide important information for understanding the nonrelativistic QCD factorization formulism.'
author:
- 'Hao Han$^a$, Yan-Qing Ma$^{b,c}$, Ce Meng$^a$, Hua-Sheng Shao$^{a,d}$, Kuang-Ta Chao$^{a,c,e}$'
title: '$\eta_c$ production at LHC and implications for the understanding of $J/\psi$ production'
---
[*Introduction.*]{}— Significant improvements for understanding the heavy quarkonium production mechanism have been achieved in recent years. While abundant data of prompt heavy quarkonium production are accumulated at the LHC, one of the main theoretical improvements is to understand the yields and polarizations by including the higher order QCD effects in the framework of nonrelativistic QCD (NRQCD) factorization [@Bodwin:1994jh], where the inclusive production cross section of a quakonium state $\mathcal{Q}$ in $pp$ collisions can be expressed as \[NRQCD\] d\_[pp+X]{}&=&\_[n]{}d\_[ppQ|[Q]{}\[n\]+X]{}\^(n). Here $d{\hat{\sigma}}_{pp\to Q\bar{Q}[n]+X}$ are the short-distance coefficients (SDCs) for producing a heavy quark pair $Q\bar{Q}$ with quantum number $n$, and $\langle\mathcal{O}^{\mathcal{Q}}(n)\rangle$ are the long-distance matrix elements (LDMEs) for $\mathcal{Q}$. The SDCs can be computed in perturbative QCD as partonic cross sections convoluted with parton distributions. At large transverse momentum $p_T$, they behave roughly as some powers of $1/p_T$. The nonperturbative LDMEs can be arranged as a series in powers of $v$ (the relative velocity of the heavy quark $Q$ and antiquark $\bar{Q}$ )[@Bodwin:1994jh]. E.g., for the $J/\psi$ production the sum over $n$ can be truncated at order $v^4$, and the LDMEs are $\moss, \mopa,
\mopb$ and $\mopc$. In the past few years, complete next-to-leading order (NLO) QCD corrections for the $\jpsi$ hadroproduction have been calculated by three groups independently[@Ma:2010yw; @Butenschoen:2010rq; @Gong:2012ug]. Though the three groups obtained consistent SDCs, they had different philosophies on extracting the color-octet (CO) LDMEs, then got different CO LDMEs, and gave completely different predictions/descripctions for the polarization of prompt $\jpsi$. Specifically, our group found that the $\jpsi$ polarization can be explained by NLO NRQCD [@Chao:2012iv], whereas the other two groups concluded that NRQCD can not explain the polarization data[@Butenschoen:2012px; @Gong:2012ug]. More recently, two other groups performed independent fits for the CO LDMEs[@Bodwin:2014gia; @Faccioli:2014cqa], and concluded that the $\jpsi$ polarization can be understood by a $\sps$ channel dominance mechanism, which was first proposed as one possibility to explain the $\jpsi$ polarization in Ref.[@Ma:2010yw] and reemphasized in Ref.[@Chao:2012iv].
To further test the mechanism of quarkonium production it is crucial to have more measurements. Recently, for the first time the LHCb Collaboration measured the $p_T$ differential cross section of prompt $\eta_c$ production via $\eta_c\to p\bar p$ [@Aaij:2014bga]. This measurement is not only significant for $\eta_c$ production, but also provides important information for $\jpsi$ production via heavy quark spin symmetry (HQSS) [@Bodwin:1994jh]. Leading order study of $\eta_c$ hadroproduction can be found in Refs. [@Mathews:1998nk; @Likhoded:2014fta] and references therein. In this letter, we will study the $\eta_c$ hadroproduction at NLO in $\alpha_s$ within the framework of NRQCD factorization. With HQSS, we find that the $\eta_c$ production data are compatible with our previous studies[@Ma:2010yw; @Chao:2012iv] and may provide a further constraint on the possible values of $\jpsi$ LDMEs.
[*Relationship between $\jpsi$ and $\eta_c$ production.*]{}— Let’s first explain why $\eta_c$ production can provide important clues to $\jpsi$ production.
For the $\jpsi$ production, with a relative large $p_T$ cutoff ($p_T>7~$GeV), our group found that [@Ma:2010yw] only two linear combinations of the three CO LDMEs can be well constrained by fitting the CDF data [@Acosta:2004yw] of the yields of $J/\psi$ production, which gives $$\begin{aligned}
\label{M0M1}
M_0 &=& \langle\mathcal{O}^{J/\psi}(^1\!S_0^{[8]})\rangle + r_0 \langle\mathcal{O}^{J/\psi}(^3\!P_0^{[8]})\rangle/m_c^2,\nonumber\\
M_1 &=& \langle\mathcal{O}^{J/\psi}(^3\!S_1^{[8]})\rangle + r_1 \langle\mathcal{O}^{J/\psi}(^3\!P_0^{[8]})\rangle/m_c^2,\end{aligned}$$ where $r_0 = 3.9$ and $r_1 = -0.56$ for the CDF window, and the corresponding values are $M_0=(7.4\pm1.9)\times 10^{-2}~\mbox{GeV}^3$ and $M_1=(0.05\pm0.02)\times 10^{-2}~\mbox{GeV}^3$. Roughly speaking, the SDCs for the LDMEs $M_0$ and $M_1$ defined in (\[M0M1\]) have mainly $p_T^{-6}$ and $p_T^{-4}$ behaviors [@Ma:2010yw], respectively. These two $p_T$ behaviors dominate the $\jpsi$ production in the region $p_T>7~$GeV. The coefficients $r_0$ and $r_1$ change slightly with the rapidity interval but almost not change with the center-of-mass energy $\sqrt{S}$ (see Table I in Ref. [@Ma:2010jj]). Thus, the CMS yield data [@Chatrchyan:2011kc] for $J/\psi$ production can be also well described by the same LDMEs in Eq. (\[M0M1\]) [@Ma:2010yw; @Shao:2014yta]. Importantly, we further found that [@Chao:2012iv] the transversely polarized cross section for direct $J/\psi$ production at NLO is almost proportional to the combined LDME $$\begin{aligned}
\label{M1prime}
M_1^{\prime}=\langle\mathcal{O}^{J/\psi}(^3\!S_1^{[8]})\rangle - 0.52\langle\mathcal{O}^{J/\psi}(^3\!P_0^{[8]})\rangle/m_c^2\end{aligned}$$ for the CDF and CMS window, which is very close to the $M_1$ in (\[M0M1\]). Since the value of $M_1$ is extremely small, much smaller than that of $M_0$ in Eq. (\[M0M1\]), one can expect that the polarizations will be dominated by $M_0$ at least in the intermediate $p_T$ region, which tends to give unpolarized results [@Chao:2012iv]. We emphasize here that the above expectation is independent of the exact values of the three CO LDMEs of $J/\psi$, as long as $M_0$ and $M_1$ are fixed by Eq. (\[M0M1\]). This can be seen from the fact that, by varying the value of $\mopa$ in Table I of Ref. [@Chao:2012iv], the resulted polarizations are similar and basically unpolarized [@Chao:2012iv]. Based on Eq. (\[M0M1\]), assuming all CO LDMEs to be positive, we updated our results for the polarization of direct $J/\psi$ production together with that of the feeddown contributions from $\chi_c$ and $\psi(2S)$ in Ref. [@Shao:2014yta], which are roughly consistent with the LHC data.
The cross section for $\eta_c$ production can also be expressed as Eq. (\[NRQCD\]). Similar to the case for $J/\psi$, four LDMEs are needed up to relative order $v^4$ for the direct $\eta_c$ production, which are $\mossetac,\mopaetac,\mopbetac$ and $\mopcetac$. The dominant feeddown contribution through $h_c\to\eta_c\gamma$ introduces two other LDMEs at relative order $v^2$: $\mopshc$ and $\mosohc$. Superficially, it appears that six channels would be involved in the fit to data, but in fact, some of them are not important. The relative importance of these channels should depend on the power counting both in $v$ and in $\delta=m_{\mathcal{Q}}/p_T$, where $m_{\mathcal{Q}}$ is the mass of the charmonium. The powers of $v$ can be estimated by the velocity scaling rules [@Bodwin:1994jh]. The powers of $\delta$ can be determined by QCD factorization for quarkonium production [@Kang:2014tta], which shows that all channels have a leading power (LP), $p_T^{-4}$, component at the current order in $\alpha_s$. However, because of the relative importance of next-to-leading power (NLP), $p_T^{-6}$, contributions for some channels [@Ma:2014svb], they will behave almost as $p_T^{-6}$ within a large range of $p_T$.
[[ccccccc]{}]{} * $n=$ & $^1\hspace{-1mm}S_0^{[1]}$ & $^3\hspace{-1mm}S_1^{[8]}$ & $^1\hspace{-1mm}P_1^{[8]}$ & $^1\hspace{-1mm}S_0^{[8]}$ & $^1\hspace{-1mm}S_0^{[8]}(h_c)$ & $^1\hspace{-1mm}P_1^{[1]}(h_c)$ \
& $v^0\delta^6$ & $v^{3}\delta^4$ & $v^4\delta^6$ & $v^4\delta^6$ & $v^2\delta^6$ & $v^2\delta^6$\
*
Especially, for the LHCb window, e.g., $6.5~\mbox{GeV}<p_T<14~\mbox{GeV}$ [@Aaij:2014bga], effectively only the $\so$ channel behaves as $p_T^{-4}$, while all other channels behave as $p_T^{-6}$, as shown in Table \[PCRs\]. As a result, only the $^1\hspace{-1mm}S_0^{[1]}$ and the $^3\hspace{-1mm}S_1^{[8]}$ channels give the leading contributions in the combined power counting. By further applying the HQSS relation [@Bodwin:1994jh] $$\begin{aligned}
\label{HQSS-2}
\mopaetac &\approx& \mopa,\end{aligned}$$ which is valid up to $O(v^2)$ corrections, one may expect that the LDME $\mopa$ will be determined by the study of $\eta_c$ production. This will give the third independent constraint on the three CO LDMEs for $J/\psi$ production other than those given in Eq. (\[M0M1\]).
[*The $\eta_c$ production*]{}—Let us proceed to study the $\eta_c$ production numerically. We use the CTEQ6M PDFs [@Whalley:2005nh] for NLO calculations, and use HELAC-Onia[@Shao:2012iz] to calculate the hard non-collinear part of real correction. The charm-quark mass is set to be $m_c = 1.5$ GeV, the renormalization, factorization, and NRQCD scales are $\mu_r = \mu_f = \sqrt{p_T^2+4m_c^2}$ and $\mu_{\Lambda} = m_c$. Thanks to HQSS, the color-singlet (CS) LDMEs for both $\jpsi$ and $\eta_c$ can be estimated by the potential model [@Eichten:1995ch], $$\begin{aligned}
\label{HQSS-1}
\mossetac &=& \moss/3=0.39~\mbox{GeV}^3.\end{aligned}$$ The theoretical uncertainties by varying $m_c$, $\mu_f$ and $\mu_r$ have been studied thoroughly in our earlier publications [@Ma:2010vd; @Ma:2010yw; @Ma:2010jj; @Chao:2012iv], where one found that the uncertainties can be estimated by a systematical error of about 30%.
As mentioned above, only the channels $^1\hspace{-1mm}S_0^{[1]}$ and $^3\hspace{-1mm}S_1^{[8]}$ are essential to account for the $\eta_c$ production in the LHCb window. However, with the fixed value of $\mossetac$ in Eq. (\[HQSS-1\]), we find that the LHCb data are almost saturated by the contribution from the CS channel, which is denoted by the solid lines in Fig. \[eatc-total\]. Similar results have been found in Ref. [@Likhoded:2014fta] with only the LO SDCs and a relative smaller CS LDME. The similarity is mainly caused by that the NLO calculation gives only a modest correction factor for $^1\hspace{-1mm}S_0^{[1]}$ channel. Therefore, the saturation can hardly be avoided if one choose the CS LDME as large as that in Eq. (\[HQSS-1\]).
However, the above result does not mean that there is no contribution from the $^3\hspace{-1mm}S_1^{[8]}$ channel. On the one hand, although there are large uncertainties of the data, one can roughly find in Fig. \[eatc-total\] that the slope of data is different from the contribution of $^1\hspace{-1mm}S_0^{[1]}$ channel itself. On the other hand, the value in Eq. (\[HQSS-1\]) is not exact, but with at least an uncertainty of order $v^2\sim0.3$ because of modeling of potential, relativistic corrections, HQSS broken, and so on. These uncertainties may leave some room for $^3\hspace{-1mm}S_1^{[8]}$ channel to contribute.
Unfortunately, it is very hard at present to determine the exact value of $\mopaetac$ due to the large uncertainties from both data and theory. But we may give a safe upper bound for $\mopaetac$ by letting the data be saturated by the $^3\hspace{-1mm}S_1^{[8]}$ channel only, which gives $\mopaetac=(1.46\pm0.20)\times10^{-2}~\mbox{GeV}^3$. Since the value of $\mopaetac$ is sufficient amplified, we choose the central value as the upper bound for the LDME. To give a lower bound, we assume the $\mopaetac$ to be positive [@Shao:2014yta], which should be acceptable due to the following reason. Since the renormalization dependence of LDME $\mopaetac$ is at higher order in $v^2$, $\mopaetac$ can be approximated as the probability for the $c\bar{c}$ pair in $^3\hspace{-1mm}S_1^{[8]}$ configuration to evolve into $\eta_c$, which should be positive in general. We then get \[UpperLimit\] 0<<1.4610\^[-2]{} \^3. By using the HQSS relation , the result in Eq. (\[UpperLimit\]) can be viewed as another constraint on the three CO LDMEs for $J/\psi$ other than Eq. (\[M0M1\]). We thus constrain all three CO LDMEs of $J/\psi$ into a finite range.
As a feedback, the other two CO LDMEs for direct $\eta_c$ production can be estimated by the HQSS relations [@Bodwin:1994jh]: $\mopbetac = \mopb/3$ and $\mopcetac = 3\mopc$. As for the feeddown contribution through $h_c\to\eta_c\gamma$, the two relevant LDMEs can be estimated again by the HQSS relations: $\mosohc = 3\mosochic$ and $\mopshc = 3\mopschic$, where the LDMEs for $\chi_{c0}$ have been determined in Ref. [@Ma:2010vd; @Shao:2014fca]. Combining the LDMEs estimated by the relations and the the SDCs calculated up to NLO in $\alpha_s$, we show the sizes of the contributions from these channels in Fig. \[eatc-total\], all of which are smaller than that for the CS channel by about one or two orders of magnitude as expected. Thus, the upper bound of the value of $\mopaetac$ given in Eq. (\[UpperLimit\]) will not be changed even these new contributions are taken into account. In addition, to provide an order of magnitude estimation of the contributions from the $^3\hspace{-1mm}S_1^{[8]}$ channel, we use a half of the upper bound of $\mopaetac$ as its input, and the results are shown as the middle-width dashed lines in Fig. \[eatc-total\]. The theoretical errors, which are indicated by the blue band in Fig. \[eatc-total\], are mainly from the uncertainties of the LDME $\mopaetac$ in Eq. (\[UpperLimit\]).
![The differential cross sections of prompt $\eta_c$ production at $\sqrt{S}=7$ TeV (left) and 8 TeV (right) for the LHCb window. The data are taken from Ref. [@Aaij:2014bga]. See text for details. []{data-label="eatc-total"}](LHCbgrid.eps)
[*Indications on the $\jpsi$ production.*]{}—Let us go back to the problem of the $J/\psi$ production. Since the three CO LDMEs for $\jpsi$ can be constrained better by Eqs. (\[M0M1\]) and (\[UpperLimit\]) using the HQSS relation Eq. (\[HQSS-2\]), we update our predictions for both yields and polarizations of $J/\psi$ prompt production, which are shown in Fig. \[fig:polarjpsi\]. The details for these calculations have been explained in Ref. [@Shao:2014yta]. Compared with the old results given in Ref. [@Shao:2014yta], the new predictions for the CMS window are almost unchanged. This is because, for the CMS window, the prediction for yield is only sensitive to the LDMEs $M_0$ and $M_1$ defined in Eq. (\[M0M1\]), and that for polarizations is only sensitive to $M_1^{\prime}$, which is given in (\[M1prime\]) and very close to $M_1$ as mentioned above. Thus, these predictions can hardly be influenced by the extra constraint in Eqs. (\[HQSS-2\]) and (\[UpperLimit\]). On the other hand, since $r_1$ in the forward rapidity interval is smaller than that in the central rapidity interval [@Ma:2010jj], the relative large and positive $\mopc$, which is indicated by Eqs. (\[M0M1\]), (\[HQSS-2\]) and (\[UpperLimit\]), will imply that the transversely polarized component of the cross section should be further reduced in the forward rapidity interval comparing with that in the central one. As a result, our new prediction of the polarization for the LHCb window with the new constraint Eqs. (\[HQSS-2\]) and (\[UpperLimit\]) tends to be more longitudinally polarized. This slightly improves the consistency between the theoretical predictions and the experimental measurements compared with that in Ref. [@Shao:2014yta]. We list the $\chi^2/d.o.f.$ values for the polarization data: 13/10 and 22/10 for the CMS data with $0<|y|<0.6$ and $0.6<|y|<1.2$ respectively and 1.2/2 for the LHCb data. Although the agreement between our predictions and the CMS polarization data in Fig. \[fig:polarjpsi\] is not very good, it is tolerable considering the large experimental and theoretical uncertainties in this stage. In particular, we note that the current CMS data still suffer from large statistical fluctuations, such as in the last bins in |y| < 0.6 and 0.6 < |y| < 1.2.
![Predictions for prompt $\jpsi$ production. Theoretical parameters are constrained by $\jpsi$ yield data at CDF [@Acosta:2004yw] as well as $\eta_c$ yield data at LHCb [@Aaij:2014bga] along with HQSS. Data are taken from CMS [@Chatrchyan:2011kc; @Chatrchyan:2013cla], LHCb [@Aaij:2011jh; @Aaij:2013nlm] and ALICE [@Abelev:2011md].[]{data-label="fig:polarjpsi"}](promptjpsiyields.eps "fig:") ![Predictions for prompt $\jpsi$ production. Theoretical parameters are constrained by $\jpsi$ yield data at CDF [@Acosta:2004yw] as well as $\eta_c$ yield data at LHCb [@Aaij:2014bga] along with HQSS. Data are taken from CMS [@Chatrchyan:2011kc; @Chatrchyan:2013cla], LHCb [@Aaij:2011jh; @Aaij:2013nlm] and ALICE [@Abelev:2011md].[]{data-label="fig:polarjpsi"}](jpsipol-CMS06.eps "fig:") ![Predictions for prompt $\jpsi$ production. Theoretical parameters are constrained by $\jpsi$ yield data at CDF [@Acosta:2004yw] as well as $\eta_c$ yield data at LHCb [@Aaij:2014bga] along with HQSS. Data are taken from CMS [@Chatrchyan:2011kc; @Chatrchyan:2013cla], LHCb [@Aaij:2011jh; @Aaij:2013nlm] and ALICE [@Abelev:2011md].[]{data-label="fig:polarjpsi"}](jpsipol-CMS612.eps "fig:") ![Predictions for prompt $\jpsi$ production. Theoretical parameters are constrained by $\jpsi$ yield data at CDF [@Acosta:2004yw] as well as $\eta_c$ yield data at LHCb [@Aaij:2014bga] along with HQSS. Data are taken from CMS [@Chatrchyan:2011kc; @Chatrchyan:2013cla], LHCb [@Aaij:2011jh; @Aaij:2013nlm] and ALICE [@Abelev:2011md].[]{data-label="fig:polarjpsi"}](jpsipol-LHCb-ALICE.eps "fig:")
The above calculations and analysis indicate that the new constraint Eq. (\[UpperLimit\]) can hardly change our previous conclusions on the $\jpsi$ production [@Ma:2010yw; @Chao:2012iv; @Shao:2014yta]. One should note that the HQSS in Eq. (\[HQSS-2\]) could be violated up to relative order $v^2$. But the violation at this level will not change our conclusion qualitatively.
However, the upper bound of $\mopa$ obtained in Eq. along with HQSS disagree with many other NLO NRQCD fits in the literature [@Butenschoen:2010rq; @Gong:2012ug; @Bodwin:2014gia; @Faccioli:2014cqa]. In Refs. [@Butenschoen:2010rq; @Gong:2012ug; @Bodwin:2014gia], the $\mopa$ is found to be well constrained, with the value $0.0304\pm0.0035~\gev^3$, $0.097\pm0.009~\gev^3$ and $0.099\pm0.022~\gev^3$, respectively. While in Ref. [@Faccioli:2014cqa], the authors argued that $\sps$ will dominate the $\jpsi$ production, and thus their $\mopa$ should be at least larger than $0.07~\gev^3$. As we discussed above, Eq. gives a very safe upper bound for $\mopa$, so the contradiction with these NLO NRQCD fits may indicate that either HQSS is essentially broken or there are still some theoretical problems to be clarified, if the LHCb data[@Aaij:2014bga] are reliable.
Though both Refs. [@Butenschoen:2010rq; @Gong:2012ug; @Faccioli:2014cqa] and our works [@Ma:2010yw; @Ma:2010jj; @Shao:2014yta] are based on complete NLO NRQCD calculations, there are many differences in the fit procedures. E.g., the lower $p_T$ cutoff for experimental data is chosen to be $1\gev$ in Ref. [@Butenschoen:2010rq] (for the photoproduction data), $7\gev$ in Refs. [@Ma:2010yw; @Gong:2012ug], and $3$ times mass of $\jpsi$ in Ref. [@Faccioli:2014cqa]; feeddown contributions are considered in Refs. [@Ma:2010yw; @Gong:2012ug], but not considered in Ref. [@Butenschoen:2010rq]. So further studies are needed to uncover the deep reason for the discrepancies of these fits and to explore the best value of lower $p_T$ cutoff for experimental data. It is interesting to compare the work of Ref. [@Bodwin:2014gia] with ours. In addition to our complete NLO NRQCD results, the crucial element that Ref. [@Bodwin:2014gia] includes is a partial LP contribution at next-to-next-to-leading order (NNLO) in $\alpha_s$. We conjecture that it is mainly this extra LP contribution that changes the theoretical curve of $\pj$ channel, and results in a $\sps$ dominance conclusion in Ref. [@Bodwin:2014gia]. If HQSS is good, a natural way to solve the contradiction could be that the NNLO correction for NLP contribution is also significant. It is needed to perform complete calculations for both the LP and NLP contributions to the same order in $\alpha_s$. Based on QCD factorization up to NLP [@Kang:2014tta] and the method to calculate the partonic hard part at NLP [@Kang:2014pya], the NNLO correction for NLP contribution may be achieved soon. Then the validation of HQSS for charmonium production will be tested on a more rigorous base.
[*Summary.*]{}—Within NLO NRQCD, we demonstrate that only $^1\hspace{-1mm}S_0^{[1]}$ and $^3\hspace{-1mm}S_1^{[8]}$ channels are essential for the $\eta_c$ production at LHC. By comparing with the LHCb data [@Aaij:2014bga], we find the $\eta_c$ production tends to be saturated by contributions from the CS channel. This strongly constrains the CO LDME $\mopaetac$ for $\eta_c$, which can be related to $\mopa$ for $J/\psi$ by the HQSS relation in Eq. (\[HQSS-2\]). With the help of this new information, all three CO LDMEs of $\jpsi$ can be well constrained into a finite range. We conclude that the prompt production of $\eta_c$ and $J/\psi$ can be understood in the same theoretical framework. Moreover, we find some previous works[@Butenschoen:2010rq; @Gong:2012ug; @Bodwin:2014gia; @Faccioli:2014cqa] may overestimate the value of $\mopa$ unless HQSS is broken. All these studies on $\eta_c$ and $J/\psi$ may provide important information for understanding the mechanism of charmonium production.
[*Note added.*]{}—When our calculation was finished and the manuscript was being prepared for publication, an independent study of $\eta_c$ production in NLO NRQCD was reported in Ref. [@Butenschoen:2014dra]. These authors conclude that with HQSS the $\eta_c$ data conflict with all NLO NRQCD fits to $\jpsi$ production data. This conclusion differs from ours, and is due to their using the values of the first row of Table I in Ref. [@Chao:2012iv] but not that of the second and third rows of the same Table. Indeed, we emphasized in Ref.[@Chao:2012iv] that “As the yield and polarization share a common parameter space, and the yield can only constrain two linear combinations of CO LDMEs, the combined fit of both yield and polarization may also not constrain three independent CO LDMEs stringently. In fact we find for a wide range of given $\mopa$, one can fit both yield and polarization reasonably well" and we showed in the second row of Table I that a possible value for $\mopa$ can be even as small as zero. We repeated the conclusion again in our most recent paper [@Shao:2014yta] that only the two linear combinations of CO LDMEs in Eq. can be well constrained.
[*Acknowledgments.*]{}—We thank V. Belyaev, Z. Yang and S. Barsuk for helpful discussions on the experiments of $\eta_c$ production at the LHCb. This work was supported in part by the National Natural Science Foundation of China (No 11075002, No 11475005) and the National Key Basic Research Program of China (No 2015CB856700). Y.Q.M. is supported by the U.S. Department of Energy Office of Science, Office of Nuclear Physics under Award Number DE-FG02-93ER-40762.
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---
abstract: 'A privacy-preserving Support Vector Machine (SVM) computing scheme is proposed in this paper. Cloud computing has been spreading in many fields. However, the cloud computing has some serious issues for end users, such as unauthorized use and leak of data, and privacy compromise. We focus on templates protected by using a random unitary transformation, and consider some properties of the protected templates for secure SVM computing, where templates mean features extracted from data. The proposed scheme enables us not only to protect templates, but also to have the same performance as that of unprotected templates under some useful kernel functions. Moreover, it can be directly carried out by using well-known SVM algorithms, without preparing any algorithms specialized for secure SVM computing. In the experiments, the proposed scheme is applied to a face-based authentication algorithm with SVM classifiers to confirm the effectiveness.'
author:
- '\'
title: 'Privacy-Preserving SVM Computing by Using Random Unitary Transformation'
---
Support Vector Machine, Privacy-preserving, random unitary transformation
Introduction {#sec:intro}
============
Cloud computing and edge computing have been spreading in many fields, with the development of cloud services. However, the computing environment has some serious issues for end users, such as unauthorized use and leak of data, and privacy compromise, due to unreliability of providers and some accidents. While, a lot of studies on secure, efficient and flexible communications, storage and computation have been reported [@Huang; @Lazzeretti; @Barni]. For securing data, full encryption with provable security (like RSA, AES, etc) is the most secure option. However, many multimedia applications have been seeking a trade-off in security to enable other requirements, e.g., low processing demands, retaining bitstream compliance, and flexible processing in the encrypted domain, so that a lot of perceptual encryption schemes have been studied as one of the schemes for achieving a trade-off [@Lagendij; @Ito1; @Chuman2; @Zhou; @Kurihara_1; @Kurihara; @Chu_Kuri_1; @Chu_Kuri_2; @Chu_Iida_1; @Chu_Kuri_3]
In the recent years, considerable efforts have been made in the fields of fully homomorphic encryption and multi-party computation [@Araki1; @Araki2; @Lu; @Toshinori]. However, these schemes can not be applied yet to SVM algorithms, although it is possible to carry out some statistical analysis of categorical and ordinal data. Moreover, the schemes have to prepare algorithms specialized for computing encrypted data.
Because of such a situation, we propose a privacy-preserving SVM computing scheme in this paper . We focus on templates protected by using a random unitary transformation, which have been studied as one of methods for cancelable biometrics [@Rathgeb; @Nandakumar; @Rane; @Wright; @Nakamura1; @Nakamura2; @Georghiades], and then consider some properties of the protected templates for secure SVM computing, where templates mean features extracted from data. As a result, the proposed scheme enables us not only to protect templates, but also to have the same performance as that of unprotected templates under some useful kernel functions as isotropic stationary kernels. Moreover, it can be directly carried out by using well-known SVM algorithms, without preparing any algorithms specialized for secure SVM computing. In the experiments, the proposed scheme is applied to a face recognition algorithm with SVM classifiers to confirm the effectiveness.
preparation
===========
Support Vector Machine
----------------------
Support Vector Machine (SVM) is a supervised machine learning algorithm which can be used for both classification or regression challenges, but it is mostly used in classification problems. In SVM, we input a feature vector to the discriminant function as $$\label{eq:eq_sign}
\begin{split}
y = \mathrm{sign}(\vector{\omega}^T\vector{x}+b)\\
\lefteqn{\hspace{-60mm}with}\\
\mathrm{sign}(u)=\begin{cases}
1 & (u>1) \\
-1 & (u\leq0)
\end{cases},
\end{split}$$ where $\vector{\omega}$ is a weight parameter, and $b$ is a bias.
SVM also has a technique called the kernel trick, which is a function that takes low dimensional input space and transform it to a higher dimensional space. These functions are called kernels. The kernel trick could be applied to Eq. (\[eq:eq\_sign\]) to map an input vector on further high dimension feature space, and then to linearly classify it on that space as $$\label{eq:eq_kernel_sign}
y = \mathrm{sign}(\vector{\omega}^T\phi(\vector{x})+b).$$ The function $\phi(\vector{x}):\mathbb{R}^d\to\mathcal{F}$ maps an input vector $\vector{x}$ on high dimension feature space $\mathcal{F}$, where $d$ is the number of the dimensions of features. In this case, feature space $\mathcal{F}$ includes parameter $\vector{\omega}$ ($\vector{\omega}\in\mathcal{F}$). The kernel function of two vectors $\vector{x}_i$, $\vector{x}_j$ is defined as $$K(\vector{x}_i,\vector{x}_j)=\langle \phi(\vector{x}_i), \phi(\vector{x}_j)\rangle,$$ where $\langle\cdot, \cdot\rangle$ is an inner product. There are various kernel functions. For example, Radial Basis Function(RBF) kernel is given by $$\label{eq:rbf}
K(\vector{x}_i,\vector{x}_j)=\exp(-\varUpsilon \| \vector{x}_i - \vector{x}_j \|^2 )$$ and polynomial kernel is provided by $$K(\vector{x}_i,\vector{x}_j)=(1+\vector{x}_i^T\vector{x}_j)^l,$$ where $\varUpsilon$ is a high parameter to decide the complexity of boundary determination, $l$ is a parameter to decide the degree of the polynomial, and $T$ indicates transpose.
This paper aims to propose a new framework to carry out SVM with protected vectors.
Scenario
--------
Figure \[fig:architecture\] illustrates the scenario used in this paper. In the enrollment, client $i$, $i \in \{1,2,...,N\}$, prepares training samples $\mathrm{g}_{i,j}, j \in \{1,2,...,M\}$ such as images, and a feature set ${\vector{\mathrm{f}}_{i,j}}$, called a template, is extracted from the samples. Next the client creates a protected template set $\hat{\vector{\mathrm{f}}}_{i,j}$ by a secret key $p_i$ and sends the set to a cloud server. The server stores it and implements learning with the protected templates for a classification problem.
In the authentication, Client $i$ creates a protected template as a query and sends it to the server. The server carries out a classification problem with a learning model prepared in advance, and then returns the result to Client $i$.
Note that the cloud server has no secret keys and the classification problem can be directly carried out by using well-known SVM algorithms. In the other words, the server does not have to prepare any algorithms specialized for the classification in the encrypted domain.
Proposed framework
==================
In this section, protected templates generated by using a random unitary matrix are conducted, and a SVM computation scheme with the protected templates is proposed under some kernel functions.
Template Protection {#sec:Uniprotect}
-------------------
Template protection schemes based on unitary transformations have been studied as one of methods for cancelable biometrics[@Rathgeb; @Wright; @Nakamura1; @Nakamura2; @Nandakumar; @Rane]. This paper has been inspired by those studies.
A template $\vector{\mathrm{f}_{i,j}} \in \mathbb{R}^d$ is protected by a unitary matrix having randomness with a key $p_i$, $\vector{\mathrm{Q}}_{p_i} \in \mathbb{C}^{N\times N}$ as, $$\label{eq:trans}
\hat{\vector{\mathrm{f}}}_{i,j}=T(\vector{\mathrm{f}}_{i,j},{p_i})=\vector{\mathrm{Q}}_{p_i}\vector{\mathrm{f}}_{i,j},$$ where $\hat{\vector{\mathrm{f}}}_{i,j}$ is the protected template. Various generation schemes of $\vector{\mathrm{Q}}_{p_i}$ have been studied to generate unitary or orthogonal random matrices such as Gram-Schmidt method, random permutation matrices and random phase matrices[@Nakamura2; @Nakamura1]. For example, the Gram-Schmidt method can be applied to a pseudo-random matrix to generate $\vector{\mathrm{Q}}_{p_i}$. Security analysis of the protection schemes have been also considered in terms of brute-force attacks, diversity and irreversibility.
SVM with protected templates {#sec:SVMpro}
----------------------------
### Properties
Protected templates generated according to Eq. (\[eq:trans\]) have the following properties under $p_i=p_s$[@Nakamura2].
Property 1 : Conservation of the Euclidean distances:\
$$\| \vector{\mathrm{f}}_{i,j} - \vector{\mathrm{f}}_{s,t} \|^2 = \| \hat{\vector{\mathrm{f}}}_{i,j} - \hat{\vector{\mathrm{f}}}_{s,t} \|^2. \nonumber$$\
Property 2 : Conservation of inner products:\
$$\langle \vector{\mathrm{f}}_{i,j},\vector{\mathrm{f}}_{s,t}\rangle=\langle\hat{\vector{\mathrm{f}}}_{i,j},\hat{\vector{\mathrm{f}}}_{s,t}\rangle, \nonumber$$\
Property 3 : Conservation of correlation coefficients:\
$$\frac{\langle \vector{\mathrm{f}}_{i,j},\vector{\mathrm{f}}_{s,t}\rangle}{\sqrt{\langle \vector{\mathrm{f}}_{i,j},\vector{\mathrm{f}}_{s,t}\rangle}\sqrt{\langle \vector{\mathrm{f}}_{i,j},\vector{\mathrm{f}}_{s,t}\rangle}}=\frac{\langle\hat{\vector{\mathrm{f}}}_{i,j},\hat{\vector{\mathrm{f}}}_{s,t}\rangle}{\sqrt{\langle\hat{\vector{\mathrm{f}}}_{i,j},\hat{\vector{\mathrm{f}}}_{s,t}\rangle}\sqrt{\langle\hat{\vector{\mathrm{f}}}_{i,j},\hat{\vector{\mathrm{f}}}_{s,t}\rangle}}. \nonumber$$\
where $\vector{\mathrm{f}}_{s,t}$ is a template of another client $s, s \in \{1,2,...,N\}$, who has M training samples $\mathrm{g}_{s,t}, t \in \{1,2,...,M\}$.
### Classes of kernels
We consider applying the protected templates to a kernel function. In the case of using RBF kernel, the following relation is satisfied from property 1 and Eq.(\[eq:rbf\]) $$\begin{aligned}
\label{eq:propKernel}
K(\hat{\vector{\mathrm{f}}}_{i,j},\hat{\vector{\mathrm{f}}}_{s,t})
&=&\exp(- \varUpsilon \| \hat{\vector{\mathrm{f}}}_{i,j} - \hat{\vector{\mathrm{f}}}_{s,t} \|^2 )\nonumber\\
&=&K(\vector{\mathrm{f}}_{i,j},\vector{\mathrm{f}}_{s,t})\end{aligned}$$ A stationary kernel $K_S(\vector{x}_i - \vector{x}_j)$ is one which is translation invariant: $$K(\vector{x}_i, \vector{x}_j) = K_S(\vector{x}_i - \vector{x}_j),$$ that is, it depends only on the lag vector separating the two vectors $\vector{x}_i$ and $\vector{x}_j$. Moreover, when a stationary kernel depends only on the norm of the lag vectors between two vectors, the kernel $K_I( \| \vector{x}_i - \vector{x}_j \|)$ is said to be isotropic (or homogeneous)[@Genton], and is thus only a function of distance: $$K(\vector{x}_i, \vector{x}_j) = K_I( \| \vector{x}_i - \vector{x}_j \|).$$ For examples, RBF, WAVE and Rational quadratic kernels belong to this class, i.e, isotropic stationary kernel, called kernel class 1 in this paper. If kernels are isotropic, the propose scheme is useful under the kernels.
Besides, from property 3, we can also use a kernel $K_{In}( \langle\vector{x}_i, \vector{x}_j\rangle)$ that depends only on the inner products between two vectors given as $$K(\vector{x}_i, \vector{x}_j) = K_{In}( \langle\vector{x}_i, \vector{x}_j\rangle).$$ Polynomial kernel and linear kernel are in this class, referred to as class 2.
Some kernels such as Fisher and p-spectrum ones, to which the protected templates can not be applied, belong to other classes. We focus on using kernel class 1 and class 2.
### Dual problem
Next, we consider binary classification that is the task of classifying the elements of a given set. A dual problem to implement a SVM classifier with protected templates is expressed as $$\label{eq:eq_dual_kernel}
\begin{split}
&\max_\alpha\ \left(-\frac{1}{2}\sum_{\substack{i,s \in N\\j,t \in M}}\alpha_{i,j} \alpha_{s,t} y_{i,j} y_{s,t} \langle \phi(\hat{\vector{\mathrm{f}}}_{i,j}), \phi(\hat{\vector{\mathrm{f}}}_{s,t})\rangle + \sum_{\substack{i \in N\\j \in M}}\alpha_{i,j}
\right)\\
&s.t.\ \sum_{\substack{i \in N \\ j \in M}}\alpha_{i,j} y_{i,j} = 0, 0\leq\alpha_{i,j}\leq C,
\end{split}$$ where $y_{i,j}$ and $y_{s,t}$$\in\{+1,-1\}$ are correct labels for each training data, $\alpha_{i,j}$ and $\alpha_{s,t}$ are dual variables and C is a regular coefficient. If we use kernel class 1 or class 2 described above, the inner product $\langle\phi(\hat{\vector{\mathrm{f}}}_{i,j}), \phi(\hat{\vector{\mathrm{f}}}_{s,t})\rangle$ is equal to $K(\vector{\mathrm{f}}_{i,j},\vector{\mathrm{f}}_{s,t})$. Therefore,
even in the case of using protected templates, the dual problem with protected templates is reduced to the same problem as that of the original templates. This conclusion means that the use of the proposed templates gives no effect to the performance of the SVM classifier under kernel class 1 and class 2.
Relation among keys {#sec:key}
-------------------
As shown in Fig \[fig:architecture\], a protected template $\hat{\vector{\mathrm{f}}}_{i,j}$ is generated from training data $\mathrm{g}_{i,j}$ by using a key $p_i$. Two relations among keys are summarized, here.
### Key condition 1: $p_1=p_2=...=p_N$
The first key choice is to use a common key in all clients, namely, $p_1=p_2=...=p_N$. In this case, all protected templates satisfy the properties described in \[sec:SVMpro\], so the SVM classifier has the same performance as that of using the original templates.
### Key condition 2: $p_1 \neq p_2 \neq .. .\neq p_N$
The second key choice is to use a different key in each client, namely $p_1 \neq p_2 \neq .. .\neq p_N$. In this case, the three properties are satisfied only among templates with a common key. This key condition allows us to enhance the robustness of the security against various attacks as discussed later.
Experimental Results
====================
The propose scheme was applied to face recognition experiments which were carried out as a dual problem.
Data Set
--------
We used Extended Yale Face Database B[@Georghiades] that consists of 2432 frontal facial images with $192\times168$-pixels of $N=38$ persons like Fig \[fig:db\]. 64 images for each person were divided into half randomly for training data samples and queries. We used random permutation matrices as unitary matrices to produce protected templates. Besides, RBF kernel and linear kernel were used, where they belong to kernel class 1 and class 2, respectively. The protection was applied to templates with 1216 dimensions generated by the down-sampling method[@Wright]. The down-sampling method divides an image into non-overlapped blocks and then calculates the mean value in each block. Figure \[fig:imageProtected\] shows the examples of an original template and the protected one.
-- -- -- --
-- -- -- --
--------------- ----------------
\(a) template \(b) protected
--------------- ----------------
Results and Discussion
----------------------
In face recognition with SVM classifiers, one classifier is created for each enrollee. The classifier outputs a predicted class label and a classification score for each query template $\hat{\vector{\mathrm{f}}}_{q}$, where $\hat{\vector{\mathrm{f}}}_{q}$ is a protected template generated from the template of a query, $\vector{\mathrm{f}}_{q}$. The classification score is the distance from the query to the boundary ranging. The relation between the classification score $S_q$ and a threshold $\tau$ for the positive label of $\vector{\mathrm{f}}_{q}$ is given as $$if \ S_q \geq \tau \ then\ accept;\ else\ reject.$$ In the experiment, False Reject Rate(FRR), False Accept Rate(FAR), and Equal Error Rate(EER) at which FAR is equal to FRR were used to evaluate the performance.
### $p_1=p_2=...=p_N$
Figure \[fig:result\_1\] shows results in the case of using key condition 1. The results demonstrate that SVM classifiers with protected templates (protected in Fig \[fig:result\_1\]) had the same performances as those fo SVM classifiers with the original templates (not protected in Fig \[fig:result\_1\]). From the results, it is confirmed that the proposed framework gives no effect to the performance of SVM classifiers under key condition 1.
### $p_1 \neq p_2 \neq .. .\neq p_N$
Figure \[fig:result\_2\] shows results in the case of using key condition 2. In this condition, it is expected that a query will be authenticated only when it meets two requirements, i.e. the same key and the same person, although only the same person is required under key condition1. Therefore, the performances in Fig. \[fig:result\_2\] were slightly different from those in Fig. \[fig:result\_1\], so the FAR performances for key condition 2 were better due to the strict requirements.
### Unauthorized outflow ($p_1 \neq p_2 \neq .. .\neq p_N$)
Figure \[fig:key\_leak\] shows the FAR performance in the case that a key $p_i$ leaks out. In this situation, other clients could use the key $p_i$ without any authorization as spoofing attacks. As shown in Fig.\[fig:key\_leak\], the FAR (key leaked in Fig.\[fig:key\_leak\]) still had low vales due to two requirements, although it was slightly degraded, compared to Fig.\[fig:result\_2\].
Figure \[fig:template\_leak\] is the FAR performance in the case that a template $\vector{\mathrm{f}}_{i,j}$ leaks out. It is confirmed that the FAR (template leaked in Fig.\[fig:template\_leak\]) still had low vales as well as in Fig.\[fig:key\_leak\].
From these results, the use of key condition 2 enhances the robustness of the security against spoofing attacks.
[![FAR and FFR ($p_1=p_2=...=p_N$)[]{data-label="fig:result_1"}](result_linear_common.pdf "fig:"){width="7.5cm"}]{}
[![FAR and FFR ($p_1=p_2=...=p_N$)[]{data-label="fig:result_1"}](result_rbf_common.pdf "fig:"){width="7.5cm"}]{}
conclusion
==========
In this paper, we proposed a privacy-preserving SVM computing scheme with protected templates. It was shown that templates protected by a unitary transform has some useful properties, and the properties allow us to securely compute SVM algorithms without any degradation of the performances. Besides, two key conditions were considered to enhance the robustness of the security against various attacks. Some face-based authentication experiments using SVM classifiers were also demonstrated to experimentally confirm the effectiveness of the proposed framework.
Acknowledgements {#acknowledgements .unnumbered}
----------------
This work was partially supported by Grant-in-Aid for Scientific Research(B), No.17H03267, from the Japan Society for the Promotion Science.\
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"pile_set_name": "ArXiv"
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---
abstract: 'The generalized amplitude damping channel (GADC) is one of the sources of noise in superconducting-circuit-based quantum computing. It can be viewed as the qubit analogue of the bosonic thermal channel, and it thus can be used to model lossy processes in the presence of background noise for low-temperature systems. In this work, we provide an information-theoretic study of the GADC. We first determine the parameter range for which the GADC is entanglement breaking and the range for which it is anti-degradable. We then establish several upper bounds on its classical, quantum, and private capacities. These bounds are based on data-processing inequalities and the uniform continuity of information-theoretic quantities, as well as other techniques. Our upper bounds on the quantum capacity of the GADC are tighter than the known upper bound reported recently in \[Rosati *et al*., Nat. Commun. 9, 4339 (2018)\] for the entire parameter range of the GADC, thus reducing the gap between the lower and upper bounds. We also establish upper bounds on the two-way assisted quantum and private capacities of the GADC. These bounds are based on the squashed entanglement, and they are established by constructing particular squashing channels. We compare these bounds with the max-Rains information bound, the mutual information bound, and another bound based on approximate covariance. For all capacities considered, we find that a large variety of techniques are useful in establishing bounds.'
author:
- Sumeet Khatri
- Kunal Sharma
- 'Mark M. Wilde'
bibliography:
- 'Ref.bib'
title: 'Information-theoretic aspects of the generalized amplitude damping channel'
---
Introduction
============
One of the main goals of quantum information theory is to determine the optimal rate of sending information (classical or quantum) through quantum channels [@H13book; @MH06; @W17; @Wat18]. Quantum channels model the noisy evolution that quantum states undergo when they are transmitted via some physical medium.
Depending on the message and the availability of resources, communication protocols over quantum channels can be divided into different categories. In particular, classical communication, entanglement-assisted classical communication, private classical communication, and quantum communication are some of the communication protocols that have been studied in the last few decades (see [@H13book; @MH06; @W17; @Wat18] for reviews). The notion of the capacity of a channel defined by Shannon [@Shannon1948] can be extended to the quantum domain for these different communication protocols (see Sec. \[sec:capacity-of-qc\] for formal definitions).
The optimal rate (capacity) of any communication protocol depends on the properties of the quantum channel. In general, the best characterization of the capacities of a quantum channel is given by an optimization over regularized information quantities over an unbounded number of copies of the channel. Hence, it appears to be generally difficult to calculate the quantum and private capacities of quantum channels [@CEMOGS15; @ES15] except for a special class of quantum channels that are degradable (see definitions in Sec. \[sec:prelim\]), in which case the regularized quantities reduce to simpler formulas that are functions of only one copy of the channel [@DS04; @GS08]. Recently, however, it was shown that one can calculate quantum capacity for some channels that are not degradable [@FW07; @GJL18]. Furthermore, recent progress in estimating and understanding the quantum capacity of low-noise and some other channels has been reported in [@LDS18; @LLS18; @LLSb18; @BL18].
Remarkably, even in the qubit case, very little is known when it comes to exact, computable expressions for the communication capacities of quantum channels. For example, two of the most widely considered noise models in quantum information and communication are the depolarizing channel and the amplitude damping channel. The classical capacity of the qubit depolarizing channel is known [@King02; @K03], but its quantum capacity (for its entire parameter range) is not. Similarly, the quantum capacity of the amplitude damping channel is known [@GF05], but its classical capacity (for its entire parameter range) is not. These are two of the most significant open problems in quantum Shannon theory.
In general, the difficulty in obtaining exact expressions for the communication capacities of quantum channels has led to a wide body of work on obtaining lower and upper bounds on these quantities. With the recent developments in quantum communication technologies, it is important to study different physically motivated noisy communication processes (quantum channels) and to establish lower and upper bounds on their communication capacities in terms of the channel parameters. Moreover, these communication rates also play a critical role in the context of distributed quantum computing between remote locations and in benchmarking the performance of quantum key distribution and quantum networks.
In this work, we provide an information-theoretic study of the generalized amplitude damping channel (GADC). As the name suggests, the GADC is indeed a generalization of the amplitude damping channel. Specifically, the GADC is a qubit-to-qubit channel, and it models the dynamics of a two-level system in contact with a thermal bath at non-zero temperature. It can be used to describe the $T_1$ relaxation process due to the coupling of spins to a system that is in thermal equilibrium at a temperature higher than the spin temperature [@NC10; @MKTSKIMW00; @PhysRevA.62.053807]. The GADC is also one of the sources of noise in superconducting-circuit-based quantum computing [@CB08]. It can additionally be used to characterize losses in linear optical systems in the presence of low-temperature background noise [@Pan17]. In the case that the thermal bath is at zero temperature, the GADC reduces to the amplitude damping channel, which arises naturally as a noise model in spin chains [@Bose03; @GF05].
The GADC can be thought of as the qubit analogue of the bosonic thermal channel, which is used to model loss in quantum optical systems and is particularly relevant in the context of communication through optical fibers or free space [@YS78; @S09; @RGRKVRHWE17]. Moreover, in the context of private communication, tampering by an eavesdropper can be modeled as the excess noise realized by a thermal channel [@NH04; @LDTG05]. A lower bound on the quantum capacity of a bosonic Gaussian thermal channel was proposed in [@HW01]. Recently, several upper bounds on the energy-constrained quantum and private capacities of a thermal channel have been established in [@SWAT] (see also [@NAJ18] in the context of lower and upper bounds on the energy-constrained quantum capacity). Moreover, the unconstrained quantum capacity of a thermal channel has been studied in [@PLOB17; @NAJ18; @RMG18; @WTB17; @SWAT]. However, the communication capacities of a qubit thermal channel, i.e., the GADC, have not been studied extensively.
Some prior works have established bounds on the various capacities of the GADC. Since it is not a degradable channel for nearly all parameter values, determining its quantum capacity exactly appears to be a difficult task. It is worth noting, however, that it is degradable in the special case that it reduces to the amplitude damping channel, and thus the quantum and private capacities of the amplitude damping channel are simply given by its coherent information [@GF05], due to the additivity of the coherent and private information for degradable channels [@DS04; @GS08]. An upper bound on the quantum capacity of the GADC in general was established in [@RMG18] by using the notion of weak degradability. Furthermore, lower and upper bounds on the classical capacity of the GADC have been established in [@F18] (see also [@FFK18]). In [@LM07], the mutual information of the GADC was calculated, thus establishing its entanglement-assisted classical capacity [@BSST99; @ieee2002bennett; @Hol01a], which is in turn an upper bound on its unassisted classical capacity. In general, half the mutual information of a quantum channel is an upper bound on its two-way assisted quantum and private capacities [@TGW14a; @TGW14b; @GEW16]. Thus, one can infer from [@LM07] and [@TGW14a; @TGW14b; @GEW16] an upper bound on the two-way assisted quantum and private capacities of the GADC.
Summary of Results
==================
In this paper, we study the GADC in detail by first deriving its intrinsic information-theoretic properties, such as necessary and sufficient conditions for entanglement breakability [@HSR03] and anti-degradability [@CG06].
We then establish several upper bounds on the classical capacity of the GADC. A first upper bound, known as $C_\beta$, is based on the no-signalling and PPT-preserving codes for classical communication over a quantum channel [@WXD18]. In particular, we find an analytical expression for $C_\beta$ of the GADC that depends only on the channel parameters. Another upper bound from [@WXD18] on the classical capacity of any quantum channel is the quantity $C_{\zeta}$. We prove that $C_{\zeta}=C_{\beta}$ for the GADC. Two other upper bounds on the classical capacity of the GADC are established by using the notion of $\varepsilon$-entanglement-breakability and $\varepsilon$-covariance [@LKDW18]. We also compare these upper bounds with the entanglement-assisted classical capacity upper bound for the GADC [@LM07].
We employ a variety of techniques to establish upper bounds on the quantum and private capacities of the GADC. The first four upper bounds are established, related to the approach of [@WP07; @SS08], by decomposing any GADC into a serial concatenation of two amplitude damping channels. Since the quantum capacity of an amplitude damping channel is known [@GF05], upper bounds on the quantum capacity of the GADC follow from the data processing property [@SN96] of the coherent information of a quantum channel. We call these bounds the “data-processing bounds.” Three other upper bounds are established using the notion of approximate degradability and anti-degradability, recently developed in [@SSWR17]. We call these bounds the “$\varepsilon$-degradable bound”, “$\varepsilon$-close-degradable bound,” and “$\varepsilon$-anti-degradable bound.” We finally employ the Rains information strong converse upper bound from [@TWW17] and the relative entropy of entanglement strong converse upper bound from [@WTB17] in order to bound the quantum and private capacities of the GADC, respectively.
We compare these upper bounds on the quantum capacity of the GADC with the known coherent information lower bound, and we find that for certain parameter values, the gap between the lower bound and the upper bounds is relatively small. Moreover, we compare these upper bounds with the upper bound established in [@RMG18], and we find that two of our data-processing upper bounds are tighter than the bound in [@RMG18] for all parameter values of the channel. Furthermore, the strong converse bounds from [@TWW17; @WTB17] can be even tighter for certain parameter values.
We also establish four different upper bounds on the two-way assisted (i.e., feedback-assisted) quantum and private capacities of the GADC. The first two upper bounds are based on the fact that the squashed entanglement of a quantum channel is an upper bound on the two-way assisted quantum and private capacities of any channel [@TGW14a; @TGW14b; @MMW16]. We establish a third upper bound by employing the max-Rains information [@WD16; @WFD18] and the max-relative entropy of entanglement [@CMH17], which are known to be upper bounds on the two-way assisted quantum [@BW18] and private [@CMH17] capacities, respectively, for any quantum channel. In fact, for this third upper bound, we have found an analytical expression that establishes that the max-Rains information and max-relative entropy of entanglement are equal for the GADC. We found this analytical expression by analytically solving the semi-definite programs associated to max-Rains information and max-relative entropy of entanglement. The fourth upper bound is based on the notion of approximate covariance. A comparison of these three upper bounds with the mutual information upper bound leads to the conclusion that all three upper bounds established in our work are significantly tighter than the mutual information upper bound. The rest of the paper is structured as follows. We begin by summarizing relevant definitions and prior results in Sec. \[sec:prelim\]. We derive necessary and sufficient conditions for entanglement breakability and anti-degradability of the GADC in Sec. \[sec:ent-break\] and Sec. \[sec:anti-deg\], respectively. We then establish several upper bounds on the classical capacity and the quantum capacity of the GADC in Sec. \[sec:c-cap-bounds\] and Sec. \[sec:q-cap-bounds\], respectively. In Sec. \[sec:two-way-q-cap-bounds\], we establish several upper bounds on the two-way assisted quantum and private capacities of the GADC. Finally, we summarize our results and conclude in Sec. \[sec:conclusion\].
All codes in Mathematica, Matlab, and Python used to assist with the analytical derivations, numerical computations, and the creation of plots are available as ancillary files with the arXiv posting of this paper. The Mathematica files contain the code used in the proofs of , Proposition \[prop-C\_beta\], Proposition \[prop-GADC\_Emax\], and . The Matlab and Python files have been used to compute all the bounds stated in the paper, and the plots have been generated in the included Jupyter notebooks using Python.
Preliminaries {#sec:prelim}
=============
In this section, we review some definitions and prior results relevant for the rest of the paper. We point readers to [@MH06; @H13book; @W17; @Wat18] for details and further background.
Let $\H$ denote a finite-dimensional Hilbert space. The tensor product of two Hilbert spaces $\H_A$ and $\H_B$ corresponding to the quantum systems $A$ and $B$ is denoted by $\H_{AB}\equiv\H_A \otimes \H_B$. We let $d_A$ denote the dimension of $\mathcal{H}_A$. Let $D(\H)$ denote the set of density operators (positive semi-definite operators with unit trace) acting on a Hilbert space $\H$. An extension of a state $\rho_A \in D(\H_A)$ is some state $\rho_{RA} \in D(\H_R \otimes \H_A)$ such that ${\operatorname{Tr}}_R[\rho_{RA}] = \rho_A$. Similarly, a purification of a state $\rho_A\in D(\H_A)$ is some pure state ${| \phi \rangle}_{RA} \in \H_R \otimes \H_A$ such that ${\operatorname{Tr}}_R[{| \phi \rangle} {\langle \phi |}_{RA}] = \rho_A$.
The quantum entropy of a quantum state $\rho \in D(\H)$ is defined as $H(\rho)\equiv -{\operatorname{Tr}}[\rho \log_2\rho]$. The binary entropy $h_2(x)$ is defined for $x \in [0, 1]$ as $$\begin{aligned}
h_2(x) \equiv - x \log_2(x) - (1-x)\log_2(1-x).
\end{aligned}$$ Moreover, throughout the paper we use the bosonic entropy $g(x)$ for $x\geq0$: $$\begin{aligned}
g(x) &\equiv (1+x)\log_2(1+x) - x\log_2x\\
&= (1+x)h_2\left(\frac{x}{1+x}\right).
\end{aligned}$$ The quantum mutual information of a bipartite state $\rho_{AB} \in D(\H_A \otimes \H_B)$ is defined as $$\begin{aligned}
I(A;B)_{\rho} \equiv H(\rho_A) + H(\rho_B) - H(\rho_{AB}).
\end{aligned}$$
Let $L(\H)$ denote the space of linear operators acting on $\H$. Quantum channels are completely positive and trace preserving maps from $L(\H_A)$ to $L(\H_B)$ and denoted by $\N_{A\to B}$. An isometric extension or Stinespring dilation $U: \H_A \to \H_B \otimes \H_E$ of a quantum channel $\N_{A\to B}$ is a linear isometry such that for all $\rho_A \in L(\H_A)$, the following holds: ${\operatorname{Tr}}_E[U \rho_A U^{\dagger}]= \N(\rho_A)$. A complementary channel $\N^c_{A\to E}$ of $\N_{A\to B}$ is defined as $\N^c_{A\to E}(\rho_A) = {\operatorname{Tr}}_B[U \rho_A U^{\dagger}]$. The Choi state of a quantum channel $\N_{A\to B}$ is given by $$\begin{aligned}
\label{eq:choi-state}
\rho_{AB}^{\N} \equiv ({\operatorname{id}}_A \otimes \N_{A'\to B}) \left( \Phi_{AA'}^+ \right),
\end{aligned}$$ where $\Phi_{AA'}^+$ denotes the maximally entangled state, i.e., $$\Phi_{AA'}^+ \equiv \frac{1}{d_A}\sum_{i,i'=1}^{d_A} {| i \rangle}{\langle i^{\prime} |}_A \otimes {| i \rangle}{\langle i^{\prime} |}_{A'}.$$ We let $$\Gamma_{AB}^{\mathcal{N}}\equiv d_A\rho_{AB}^{\mathcal{N}}$$ denote the Choi matrix of the channel $\mathcal{N}$.
According to the Choi-Kraus theorem, the action of a quantum channel $\N_{A\to B}$ on any $X_A \in L(\H_A)$ can be represented in the following way: $$\begin{aligned}
\N_{A\to B}(X_A) = \sum_{i=1}^r V_i X_A V_i^{\dagger}~,
\end{aligned}$$ where the so-called Kraus operators $V_i:\H_A\to\H_B$, $i \in \{1, \dots, r \}$, satisfy $\sum_{i=1}^r V^{\dagger}_i V_i = \mathbbm{1}_A$, and $r$ need not exceed $d_Ad_B$, with a minimal choice being $r = \text{rank}(\Gamma_{AB}^{\mathcal{N}})$.
A quantum channel $\N_{A\to B}$ is entanglement breaking if the Choi state as in of the channel is separable [@HSR03].
A quantum channel $\mathcal{N}_{A\to B}$ is called *degradable* if there exists a channel $\mathcal{D}_{B\to E}$ such that $$(\mathcal{D}_{B\to E}\circ\mathcal{N}_{A\to B})(X_A ) =\mathcal{N}^c_{A \to E}(X_A),$$ for all $X_A \in L(\H_A)$ [@DS04]. A channel $\mathcal{N}_{A\to B}$ is called *anti-degradable* if its complementary channel $\mathcal{N}^c_{A\to E}$ is degradable, i.e., if there exists a channel $\mathcal{E}_{E \to B}$ such that $$\label{eq-anti_degrade}
(\mathcal{E}_{E\to B}\circ\mathcal{N}^c_{A\to E})(X_A)=\mathcal{N}_{A\to B}(X_A)$$ for all $X_A \in L(\H_A)$ [@CG06].
For any Hermiticity-preserving map $\mathcal{M}_{A\to B}$, its diamond norm ${\lVert\mathcal{M}\rVert}_{\diamond}$ is defined as [@Kit97] $$\label{eq-diamond_norm}
{\lVert\mathcal{M}\rVert}_{\diamond}=\max_{\psi_{RA}}{\lVert\mathcal{M}_{A\to B}(\psi_{RA})\rVert}_1,$$ where the optimization is over all pure states $\psi_{RA}$, with the dimension of the reference system $R$ equal to the dimension of $A$, and ${\lVertX\rVert}_1$ denotes the trace norm of the matrix $X$, which is defined as the sum of the singular values of $X$.
Capacities of quantum channels {#sec:capacity-of-qc}
------------------------------
For any quantum channel $\mathcal{N}$, its classical capacity $C(\mathcal{N})$ is defined to be the highest rate at which classical information can be sent over many uses of the channel with an error probability that converges to zero as the number of channel uses increases. It holds that [@Hol73; @SW97; @Hol98] $$\label{eq-classical_capacity}
C(\mathcal{N})=\lim_{n\to\infty}\frac{1}{n}\chi(\mathcal{N}^{\otimes n}),$$ where $\chi(\mathcal{N})$ is the Holevo information of the channel $\mathcal{N}$, which is defined as $$\chi(\mathcal{N})=\max_{\rho_{XA}}I(X;B)_{\omega},
\label{eq:Holevo-info-channel}$$ where $\omega_{XB}=\mathcal{N}_{A\to B}(\rho_{XA})$, and the maximization is with respect to all classical-quantum states, i.e., states of the form $$\label{eq-cq_state}
\rho_{XA}\equiv\sum_xp_X(x){| x \rangle}{\langle x |}_X\otimes\rho_A^x.$$
For any quantum channel $\mathcal{N}$, its quantum capacity $Q(\mathcal{N})$ is defined to be the highest rate at which quantum information can be sent over many uses of the channel with a fidelity that converges to one as the number of channel uses increases. It has been shown [@BS96; @SN96; @BNS98; @BKN20; @Llo97; @capacity2002shor; @Dev05] that $$Q(\mathcal{N})=\lim_{n\to\infty}\frac{1}{n}I_{\text{c}}(\mathcal{N}^{\otimes n}),$$ where the function $I_{\text{c}}$ is the channel coherent information, which is defined for any quantum channel $\mathcal{N}$ as $$\label{eq:coherent-info-q-chan}
I_{\text{c}}(\mathcal{N})\equiv\max_{\rho}I_{\text{c}}(\rho,\mathcal{N}),$$ where $\rho \in D(\H)$, and $$I_{\text{c}}(\rho,\mathcal{N})\equiv H(\mathcal{N}(\rho))-H(\mathcal{N}^c(\rho)).$$ If the channel $\mathcal{N}$ is anti-degradable [@CG06], then its coherent information in vanishes, which means that anti-degradable channels have zero quantum capacity.
The private capacity $P(\N)$ of a quantum channel $\N$ is defined to be the maximum rate at which a sender can reliably communicate classical messages to a receiver by using the channel many times, such that the environment of the channel obtains negligible information about the transmitted message. The private capacity $P(\N)$ is equal to the regularized private information of the channel $\N$ [@Dev05; @CWY04], i.e., $$\begin{aligned}
P(\N) = \lim_{n\to\infty}\frac{1}{n} P^{(1)}(\N^{\otimes n})~,
\end{aligned}$$ where the private information of the channel is defined as $$\begin{aligned}
\label{eq-priv_inf}
P^{(1)}(\N) \equiv \max_{\rho_{XA}}\bigg[ I(X;B)_{\omega} - I(X;E)_{\omega}\bigg].
\end{aligned}$$ The maximization here is with respect to all states $\rho_{XA}$ as in , and $\omega_{XABE}=\mathcal{U}^{\mathcal{N}}_{A\to BE}(\rho_{XA})$, with $\mathcal{U}^{\mathcal{N}}_{A\to BE}$ being an isometric channel extending $\mathcal{N}$.
In general, the quantum and private capacities of a channel $\mathcal{N}$ are related as follows [@Dev05]: $$\label{eq-quant_priv_cap_ineq}
Q(\mathcal{N})\leq P(\mathcal{N}).$$ For degradable channels $\N$ and $\M$, the coherent information is known to be additive [@DS04] in the following sense: $$\begin{aligned}
I_{\text{c}}(\N \otimes \M) = I_{\text{c}} (\N)+I_{\text{c}} (\M).
\end{aligned}$$ Moreover, the private information of a degradable channel is equal to its coherent information [@GS08]. Therefore, both the quantum and private capacities of a degradable channel are given by its coherent information.
Bounds on the capacities of quantum channels {#sec-bounds_QCap}
--------------------------------------------
In this section, we recall several different techniques for placing upper bounds on the communication capacities of a quantum channel that we use throughout the rest of the paper.
### Data-processing upper bounds
Let $\N \circ \M$ denote the serial concatenation of two quantum channels $\N$ and $\M$. Upper bounds on the quantum capacity of the channel $\N\circ \M$ can be established as follows [@WP07; @SS08]: $$\begin{aligned}
Q(\mathcal{N}\circ\mathcal{M})&\leq Q(\mathcal{M}),\label{eq-QCap_data_proc_1}\\
Q(\mathcal{N}\circ\mathcal{M})&\leq Q(\mathcal{N}).\label{eq-QCap_data_proc_2}
\end{aligned}$$ The first inequality follows from definitions and the quantum data processing inequality. The second inequality is a consequence of the following argument: consider an arbitrary encoding and decoding scheme for quantum communication over the channel $\N\circ \M$. Then this encoding, followed by many uses of the channel $\M$, can be considered as an encoding for the channel $\N$. Since the quantum capacity of the channel $\N$ involves an optimization over all such encodings, the desired inequality follows.
By similar reasoning as above, we can conclude analogous data-processing upper bounds for the private capacity and the classical capacity: $$\begin{aligned}
P(\mathcal{N}\circ\mathcal{M})&\leq P(\mathcal{M}),\label{eq-PCap_data_proc_1}\\
P(\mathcal{N}\circ\mathcal{M})&\leq P(\mathcal{N})\label{eq-PCap_data_proc_2},\\
C(\mathcal{N}\circ\mathcal{M})&\leq C(\mathcal{M}),\\
C(\mathcal{N}\circ\mathcal{M})&\leq C(\mathcal{N}).
\end{aligned}$$
### Classical capacity upper bounds via approximate entanglement breakability and approximate covariance {#subsubsec-CCap_UB_EB_cov}
Upper bounds on the classical capacity of any quantum channel have been obtained using the notions of approximate entanglement-breakability and approximate covariance of channels [@LKDW18]. We now summarize these results. All of these results, as well as their proofs, can be found in [@LKDW18].
A quantum channel $\mathcal{N}$ is called $\varepsilon$-entanglement-breaking if there exists an entanglement-breaking channel $\mathcal{M}$ such that $\frac{1}{2}{\lVert\mathcal{N}-\mathcal{M}\rVert}_{\diamond}\leq\varepsilon$. We let $$\label{eq-approx_EB}
\varepsilon_{\text{EB}}(\mathcal{N})\equiv\min_{\mathcal{M}}\left\{\frac{1}{2}{\lVert\mathcal{N}-\mathcal{M}\rVert}_{\diamond}:\mathcal{M}\text{ entanglement breaking}\right\}$$ denote the smallest $\varepsilon$ such that $\mathcal{N}$ is $\varepsilon$-entanglement-breaking. For qubit-to-qubit channels, the entanglement-breaking parameter $\varepsilon_{\text{EB}}(\mathcal{N})$ can be calculated by means of a semi-definite program [@LKDW18 Lemma III.8]. We suppress the channel dependence on $\varepsilon_{\text{EB}}$ if the channel is understood from the context.
For any $\varepsilon$-entanglement-breaking channel $\mathcal{N}$, the following upper bound on the classical capacity $C(\mathcal{N})$ holds [@LKDW18 Corollary III.7]: $$\label{eq-CCap_UB_approx_EB}
C(\mathcal{N})\leq\chi(\mathcal{M})+2\varepsilon\log_2d_B+g(\varepsilon),$$ where $\mathcal{M}$ is the entanglement-breaking channel such that $\varepsilon=\frac{1}{2}{\lVert\mathcal{N}-\mathcal{M}\rVert}_{\diamond}$.
We now define the notion of approximate covariance of a quantum channel $\mathcal{N}_{A\to B}$. Let $G$ be a finite group with a unitary representation $\{U_A(g)\}_{g\in G}$ on the input system $A$ and a unitary representation $\{V_B(g)\}_{g\in G}$ on the output system $B$. The so-called twirled channel $\mathcal{N}_{A\to B}^G$ is defined as $$\mathcal{N}_{A\to B}^G(\cdot)\equiv \frac{1}{|G|}\sum_{g\in G} V_B(g)^\dagger\mathcal{N}_{A\to B}(U_A(g)(\cdot)U_A(g)^\dagger)V_B(g). \label{eq:twirled-channel}$$ Note that the twirled channel $\mathcal{N}_{A\to B}^G$ can be realized by means of a generalized teleportation protocol [@KW17 Appendix B]. By construction, this channel is covariant with respect to the representations $\{U_A(g)\}_{g\in G}$ and $\{V_B(g)\}_{g\in G}$, meaning that $$\mathcal{N}_{A\to B}^G(U_A(g)\rho_AU_A(g)^\dagger)=V_B(g)\mathcal{N}_{A\to B}^G(\rho_A)V_B(g)^\dagger$$ for all states $\rho_A$ and all $g\in G$. We call $\mathcal{N}$ $\varepsilon$-covariant with respect to the representations $\{U_A(g)\}_{g\in G}$, $\{V_B(g)\}_{g\in G}$ if $\frac{1}{2}{\lVert\mathcal{N}-\mathcal{N}^G\rVert}_{\diamond}\leq\varepsilon$. We let $$\varepsilon_{\text{cov}}(\mathcal{N})\equiv\frac{1}{2}{\lVert\mathcal{N}-\mathcal{N}^G\rVert}_{\diamond}$$ denote the smallest $\varepsilon$ such that $\mathcal{N}$ is $\varepsilon$-covariant. The covariance parameter $\varepsilon_{\text{cov}}(\mathcal{N})$ can be computed by means of a semi-definite program, as observed in [@LKDW18], due to the fact that the diamond norm can be computed by a semi-definite program [@W13]. We suppress the dependence of the covariance parameter on both the group and its representations for simplicity, and if it is clear from the context, we also suppress the dependence on the channel.
Let $\mathcal{N}$ be a qubit-to-qubit channel, and let $G=\mathbb{Z}_2\times\mathbb{Z}_2$, with $\mathbb{Z}_2$ the group consisting of the set $\{0,1\}$ with addition modulo two. This group has the (projective) unitary representation consisting of the Pauli operators $\{\mathbbm{1},\sigma_x,\sigma_y,\sigma_z\}$. With this group and this representation, if $\mathcal{N}$ is $\varepsilon$-covariant, then [@LKDW18 Corollary III.5] $$\label{eq-CCap_UB_approx_cov}
C(\mathcal{N})\leq\chi(\mathcal{N}^G)+2\varepsilon+g(\varepsilon).$$
### Quantum and private capacity upper bounds via approximate degradability and approximate anti-degradability
We now recall techniques to obtain upper bounds on the quantum and private capacities of a quantum channel using the concepts of approximate degradability and approximate anti-degradability. These concepts were developed in [@SSWR17]. All of the results stated in this subsection, as well as their proofs, can be found in [@SSWR17].
A channel $\mathcal{N}$ is called *$\varepsilon$-degradable* if there exists a channel $\mathcal{D}$ such that $\frac{1}{2}{\lVert\mathcal{N}^c-\mathcal{D}\circ\mathcal{N}\rVert}_{\diamond}\leq\varepsilon$. We let $$\label{eq-approx_deg_parameter}
\varepsilon_{\text{deg}}(\mathcal{N})\coloneqq\min_{\mathcal{D}}\left\{\frac{1}{2}{\lVert\mathcal{N}^c-\mathcal{D}\circ\mathcal{N}\rVert}_{\diamond}:\mathcal{D}\text{ is a channel}\right\}$$ denote the smallest $\varepsilon$ such that $\mathcal{N}$ is $\varepsilon$-degradable. We suppress the dependence of this quantity on the channel if it is clear from the context. Note that $\varepsilon_{\text{deg}}(\mathcal{N})$ can be calculated via a semi-definite program.
For an $\varepsilon$-degradable channel $\mathcal{N}$ with corresponding (approximate) degrading channel $\mathcal{D}$, it holds that [@SSWR17 Theorem 7] $$\begin{aligned}
Q(\mathcal{N})&\leq U_{\mathcal{D}}(\mathcal{N})+2\varepsilon\log_2d_E+g(\varepsilon)\label{eq-approx_deg_UB},
\end{aligned}$$ where the quantity $U_{\mathcal{D}}(\mathcal{N})$ is defined as $$U_{\mathcal{D}}(\mathcal{N})\equiv \max_{\rho}\{H(F|\tilde{E})_{\omega}:\omega_{\tilde{E}FE}=(W\otimes\mathbbm{1}_E)V\rho_A V^\dagger(W\otimes\mathbbm{1}_E)^\dagger\},$$ with $V:\mathcal{H}_A\to\mathcal{H}_B\otimes\mathcal{H}_E$ and $W:\mathcal{H}_B\to\mathcal{H}_{\tilde{E}}\otimes\mathcal{H}_F$ being isometric extensions of channels $\mathcal{N}$ and $\mathcal{D}$, respectively. Moreover, the following bound was established on the private capacity of an $\varepsilon$-degradable channel $\N$ in [@SWAT Theorem 13]: $$\begin{aligned}
P(\mathcal{N})&\leq U_{\mathcal{D}}(\mathcal{N})+6\varepsilon\log_2d_E+3g(\varepsilon).\label{eq-approx_deg_UB_PCap}
\end{aligned}$$
Another upper bound on the quantum capacity of a quantum channel $\N$ can be established using the notion of $\varepsilon$-close degradability. A channel $\mathcal{N}$ is called *$\varepsilon$-close-degradable* if there exists a degradable channel $\mathcal{M}$ such that $\frac{1}{2}{\lVert\mathcal{N}-\mathcal{M}\rVert}_{\diamond}\leq\varepsilon$. If $\mathcal{N}$ is an $\varepsilon$-close-degradable channel, then the following bounds hold [@SSWR17 Proposition A2]: $$\begin{aligned}
Q(\mathcal{N})&\leq I_{\text{c}}(\mathcal{M})+2\varepsilon\log_2d_B+2g(\varepsilon),\label{eq-eps_approx_deg_UB}\\
P(\mathcal{N})&\leq I_{\text{c}}(\mathcal{N})+4\varepsilon\log_2d_B+4g(\varepsilon).\label{eq-eps_approx_deg_UB_PCap}
\end{aligned}$$
A channel $\mathcal{N}$ is called an *$\varepsilon$-anti-degradable* channel if there exists a channel $\mathcal{E}$ such that $\frac{1}{2}{\lVert\mathcal{N}-\mathcal{E}\circ\mathcal{N}^c\rVert}_{\diamond}\leq\varepsilon$. We let $$\label{eq-approx_adeg_parameter}
\varepsilon_{\text{a-deg}}(\mathcal{N})\equiv\min_{\mathcal{E}}\left\{\frac{1}{2}{\lVert\mathcal{N}-\mathcal{E}\circ\mathcal{N}^c\rVert}_{\diamond}:\mathcal{E}\text{ is a channel}\right\}$$ denote the smallest $\varepsilon$ such that $\mathcal{N}$ is $\varepsilon$-anti-degradable. We suppress the dependence of this quantity on the channel if it is clear from the context. Note that $\varepsilon_{\text{a-deg}}(\mathcal{N})$ can be calculated via a semi-definite program.
For any $\varepsilon$-anti-degradable channel $\mathcal{N}$, it holds that [@SSWR17 Theorem 11] $$\begin{aligned}
Q(\mathcal{N})&\leq P(\mathcal{N})\leq \varepsilon\log_2(d_B-1)+2\varepsilon\log_2d_B\nonumber\\
&\qquad\qquad\qquad+h_2(\varepsilon)+g(\varepsilon).\label{eq-eps_anti_degrade_upper_bound}
\end{aligned}$$
### Rains information upper bound on quantum capacity and relative entropy of entanglement upper bound on private capacity
The Rains information of a quantum channel is an upper bound on its quantum capacity [@TWW17], and a channel’s relative entropy of entanglement is an upper bound on its private capacity [@WTB17]. Here we briefly recall these results.
The Rains relative entropy $R(A;B)_{\rho}$ [@R01; @AMVW02] and the relative entropy of entanglement $E_R(A;B)_{\rho}$ [@VP98] of a bipartite state $\rho_{AB}$ are defined as $$\begin{aligned}
R(A;B)_{\rho}&\equiv\min_{\sigma_{AB}\in\text{PPT}'(A:B)}D(\rho_{AB}\Vert\sigma_{AB}), \label{eq:Rains-state}\\
E_R(A;B)_{\rho}&\equiv\min_{\sigma_{AB}\in\text{SEP}(A:B)}D(\rho_{AB}\Vert\sigma_{AB}), \label{eq:REE-state}
\end{aligned}$$ where $D(\rho_{AB}\Vert\sigma_{AB})$ is the quantum relative entropy of $\rho_{AB}$ and $\sigma_{AB}$ [@Ume62]. We have $D(\rho_{AB}\Vert\sigma_{AB})={\operatorname{Tr}}[\rho(\log_2\rho-\log_2\sigma)]$ if $\text{supp}(\rho_{AB})\subset\text{supp}(\sigma_{AB})$, and $D(\rho_{AB}\Vert\sigma_{AB})=+\infty$ otherwise. Also, $\text{PPT}'(A\!:\!B)$ denotes the set $\{ \sigma_{AB} : \sigma_{AB} \geq 0, \Vert \sigma_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B} \Vert_1 \leq 1\}$ [@AMVW02], and $\text{SEP}(A\!:\!B)$ denotes the set of separable states acting on $\mathcal{H}_A\otimes\mathcal{H}_B$ [@W89]. Note that one can efficiently calculate the Rains relative entropy by employing convex programming methods [@FF18; @FWTD17; @Wilde2018], due to the fact that the constraints $\sigma_{AB} \geq 0$ and $ \Vert \sigma_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B} \Vert_1 \leq 1$ are semi-definite constraints.
For any channel $\mathcal{N}_{A'\to B}$, we define its Rains relative entropy $R(\mathcal{N})$ and its relative entropy of entanglement $E_R(\mathcal{N})$ as follows: $$\begin{aligned}
R(\N)& \equiv \max_{\phi_{AA'}} R(A;B)_{\rho}, \label{eq:Rains-channel}\\
E_R(\N)& \equiv \max_{\phi_{AA'}} E_R(A;B)_{\rho}, \label{eq:REE-channel}\end{aligned}$$ where $\rho_{AB} \equiv \N_{A'\to B}(\phi_{AA'})$ and the optimization is with respect to all pure bipartite input states $\phi_{AA'}$, with the dimension of $A$ equal to the dimension of the input system $A'$ of the channel $\mathcal{N}$. As stated above, the information measures $R(\mathcal{N})$ and $E_R(\mathcal{N})$ are useful because they bound the quantum and private capacities, respectively, of the channel $\mathcal{N}$: $$\begin{aligned}
Q(\N) & \leq R(\N), \label{eq:Rains-q-cap-bound}\\
P(\N) & \leq E_R(\N). \label{eq:REE-p-cap-bound}\end{aligned}$$
By following an approach similar to that given in [@TWW17 Proposition 2], it follows that the maximizations in and are concave in the reduced density operator ${\operatorname{Tr}}_A[\phi_{AA'}]$:
\[prop:concavity-Rains\] Let $\mathcal{N}_{A^{\prime}\rightarrow B}$ be a quantum channel, $\rho_{A^{\prime}}$ a state, $\phi_{AA^{\prime}}^{\rho}$ a purification of $\rho_{A^{\prime}}$, and $\omega_{AB}\equiv\mathcal{N}_{A^{\prime}\rightarrow
B}(\phi_{AA^{\prime}}^{\rho})$. Then, the functions $\rho_{A^{\prime}}\mapsto R(A;B)_{\omega}$ and $\rho_{A^{\prime}}\mapsto E_{R}(A;B)_{\omega}$ are concave in the reduced state $\operatorname{Tr}_{A}[\phi_{AA^{\prime}}^{\rho}]=\rho_{A^{\prime}}$, regardless of which purification $\phi_{AA^{\prime}}^{\rho}$ of $\rho_{A^{\prime}}$ is chosen.
We give a proof of Proposition \[prop:concavity-Rains\] in Appendix \[proof-prop:concavity-Rains\]. Proposition \[prop:concavity-Rains\], combined with the results of [@FF18; @FWTD17; @Wilde2018], implies that $R(\N)$ can be computed efficiently by convex programming techniques. One can effectively use convex programming techniques to calculate $E_R(\N)$, but it will not be efficient to do so in general since it is well known that optimizing over the set of separable states is difficult [@G03; @L07; @G10].
For qubit-qubit systems $AB$, it is known that $R(A;B)_{\rho} = E_R(A;B)_{\rho}$ [@AS08], which is related to the fact that the positive partial transposition criterion is necessary and sufficient for separability for such low-dimensional systems [@Peres96; @HHH96]. (However, note that the analysis in [@AS08] goes well beyond this observation in order to establish the aforementioned equality.) This equality in turn implies that $R(\N)= E_R(\N)$ for qubit-to-qubit channels, which is useful for our purposes here since our focus is the qubit-to-qubit generalized amplitude damping channel.
The generalized amplitude damping channel
-----------------------------------------
The generalized amplitude damping channel (GADC) $\mathcal{A}_{\gamma,N}$ is a qubit-to-qubit channel with the following four Kraus operators (in the standard basis): $$\begin{aligned}
A_1&=\sqrt{1-N}\left({| 0 \rangle}{\langle 0 |}+\sqrt{1-\gamma}{| 1 \rangle}{\langle 1 |}\right),\\
A_2&=\sqrt{\gamma(1-N)}{| 0 \rangle}{\langle 1 |},\\
A_3&=\sqrt{N}\left(\sqrt{1-\gamma}{| 0 \rangle}{\langle 0 |}+{| 1 \rangle}{\langle 1 |}\right),\\
A_4&=\sqrt{\gamma N}{| 1 \rangle}{\langle 0 |}.
\end{aligned}$$ It is completely positive and trace preserving for all $\gamma, N \in [0, 1]$. If we set $N=0$, then the GADC reduces to the ordinary amplitude damping channel $\mathcal{A}_\gamma$ with two Kraus operators. The GADC also has only two Kraus operators for $N=1$, in which case the channel behaves as an amplification process, driving the signal toward the state ${| 1 \rangle}{\langle 1 |}$.
Let $\rho$ denote a single-qubit density operator: $$\rho=\frac{1}{2}(\mathbbm{1}+r_x\sigma_x+r_y\sigma_y+r_z\sigma_z),$$ where $\vec{r}\equiv(r_x,r_y,r_z)\in\mathbb{R}^3$ is the Bloch vector, which satisfies $r_x^2+r_y^2+r_z^2\leq 1$. The action of the GADC $\A_{\gamma, N}$ on $\rho$ is given by the action of $\A_{\gamma, N}$ on the Pauli operators $\sigma_x,\sigma_y,\sigma_z$. We have that $$\begin{aligned}
\mathcal{A}_{\gamma,N}(\sigma_x)&=\sqrt{1-\gamma}\sigma_x,\label{eq-GADC_Pauli_action_1}\\
\mathcal{A}_{\gamma,N}(\sigma_y)&=\sqrt{1-\gamma}\sigma_y,\label{eq-GADC_Pauli_action_2}\\
\mathcal{A}_{\gamma,N}(\sigma_z)&=(1-\gamma)\sigma_z,\label{eq-GADC_Pauli_action_3}\\
\mathcal{A}_{\gamma,N}(\mathbbm{1})&=\mathbbm{1}+\gamma(1-2N)\sigma_z\label{eq-GADC_Pauli_action_4}
\end{aligned}$$ for all $\gamma,N\in[0,1]$. This implies that the vector $\vec{r}$ of the initial state $\rho$ gets transformed as $$\label{eq-transformed_vec}
\vec{r}\mapsto(r_x\sqrt{1-\gamma},r_y\sqrt{1-\gamma},r_z(1-\gamma)+\gamma(1-2N))\equiv \vec{R},\nonumber$$ where $\vec{R}\equiv (R_x, R_y, R_z)$. In particular, for any state $\rho$, we get $$\begin{gathered}
\left(\frac{R_x}{\sqrt{1-\gamma}}\right)^2+\left(\frac{R_y}{\sqrt{1-\gamma}}\right)^2+\left(\frac{R_z-\gamma(1-2N)}{1-\gamma}\right)^2 \\
=r_x^2+r_y^2+r_z^2\leq 1,
\end{gathered}$$ which implies that the initial Bloch sphere gets transformed to an ellipsoid centered at $(0,0,\gamma(1-2N))$ with $x$-, $y$- and $z$-axes $\sqrt{1-\gamma}$, $\sqrt{1-\gamma}$, $1-\gamma$, respectively. Note that all pure initial states, which satisfy $r_x^2+r_y^2+r_z^2=1$, get mapped to the surface of the ellipsoid.
The relations – also imply that the GADC is covariant with respect to the Pauli-$z$ operator, i.e., $$\label{eq-GADC_Z_covariant}
\mathcal{A}_{\gamma,N}(\sigma_z\rho\sigma_z)=\sigma_z\mathcal{A}_{\gamma,N}(\rho)\sigma_z$$ for all states $\rho$ and all $\gamma,N\in[0,1]$. More generally, the GADC is covariant with respect to the operator ${\mathrm{e}}^{\mathrm{i} a\hat{n}}$, where $$\label{eq-num_operator}
\hat{n}\equiv{| 1 \rangle}{\langle 1 |}$$ is the number operator, i.e., $$\mathcal{A}_{\gamma,N}({\mathrm{e}}^{\mathrm{i} a\hat{n}}\rho{\mathrm{e}}^{-\mathrm{i} a\hat{n}})={\mathrm{e}}^{\mathrm{i} a\hat{n}}\mathcal{A}_{\gamma,N}(\rho){\mathrm{e}}^{-\mathrm{i} a\hat{n}}$$ for all states $\rho$, all $a\in\mathbb{R}$, and all $\gamma,N\in[0,1]$.
We also have that $$\label{eq-GADC_N_symmetry}
\mathcal{A}_{\gamma,N}(\rho)=\sigma_x\mathcal{A}_{\gamma,1-N}(\sigma_x\rho\sigma_x)\sigma_x$$ for all states $\rho$ and all $\gamma,N\in[0,1]$. In other words, the GADC $\mathcal{A}_{\gamma,N}$ is related to the GADC $\mathcal{A}_{\gamma,1-N}$ via a simple pre- and post-processing by the unitary $\sigma_x$. The information-theoretic aspects of the GADC are thus invariant under the interchange $N\leftrightarrow 1-N$, which means that we can, without loss of generality, restrict the parameter $N$ to the interval $\left[0,1/2\right]$.
We now recall the following well-known decomposition theorems for an arbitrary generalized amplitude damping channel $\A_{\gamma, N}$:
1. Let $\gamma\in[0,1]$ and $N\in[0,1]$. Then any generalized amplitude damping channel $\A_{\gamma, N}$ can be decomposed as a convex combination of $\A_{\gamma, 0}$ and $\A_{\gamma, 1}$, i.e., $$\begin{aligned}
\label{eq-GADC_decomp_convex}
\mathcal{A}_{\gamma,N}=(1-N)\mathcal{A}_{\gamma,0}+N\mathcal{A}_{\gamma,1}.
\end{aligned}$$
2. Let $\gamma_1,\gamma_2\in[0,1]$ and $N_1,N_2\in[0,1]$. Then, any generalized amplitude damping channel $\A_{\gamma, N}$ can be decomposed as the concatenation of two generalized amplitude damping channels $\A_{\gamma_1, N_1}$ and $\A_{\gamma_2, N_2}$: $$\label{eq-GADC_decomp_gen}
\mathcal{A}_{\gamma,N}=\mathcal{A}_{\gamma_2,N_2}\circ\mathcal{A}_{\gamma_1,N_1}$$ where $\gamma = \gamma_1+\gamma_2-\gamma_1\gamma_2$ and $N= \frac{\gamma_1(1-\gamma_2)N_1+\gamma_2N_2}{\gamma_1+\gamma_2-\gamma_1\gamma_2}$.
A consequence of is that, for all $\gamma,N\in[0,1]$, $$\begin{aligned}
\mathcal{A}_{\gamma,N}&=\mathcal{A}_{\gamma N,1}\circ\mathcal{A}_{\frac{\gamma(1-N)}{1-\gamma N},0},\label{eq-GADC_decomp_spec_1} \\
\mathcal{A}_{\gamma,N}&=\mathcal{A}_{\gamma(1-N),0}\circ\mathcal{A}_{\frac{\gamma N}{1-\gamma(1-N)},1}.\label{eq-GADC_decomp_spec_2}
\end{aligned}$$
We define $$\label{eq-GADC_comp}
\mathcal{A}_{\gamma,N}^c(\rho_A)\equiv{\operatorname{Tr}}_B[V_{A\to BE}^{\gamma,N}\rho_A(V_{A\to BE}^{\gamma,N})^\dagger]$$ to be a channel complementary to $\mathcal{A}_{\gamma,N}$, where $V_{A\to BE}^{\gamma,N}$ is an isometric extension of $\mathcal{A}_{\gamma,N}$, which we take to be $$\label{eq-GADC_iso_ext}
V_{A\to BE}^{\gamma,N}\equiv A_1\otimes{| 0 \rangle}_E+A_2\otimes{| 1 \rangle}_E+A_3\otimes{| 2 \rangle}_E+A_4\otimes{| 3 \rangle}_E.$$
The qubit thermal channel {#subsec-qubit_thermal_chan}
-------------------------
A qubit thermal channel is defined by analogy with the bosonic thermal channel [@AS17] as the interaction of two qubit systems $A$ and $E$ via a unitary channel, given by the unitary $U^\eta$, followed by discarding the system $E$ [@GF05]. See Fig. \[fig-qubit\_thermal\_noise\] for an illustration. The unitary $U^\eta$ is defined as $$\begin{aligned}
\label{eq-BS_unitary}
U^\eta=\begin{pmatrix} 1&0&0&0\\0&\sqrt{\eta}&\sqrt{1-\eta}&0\\0&-\sqrt{1-\eta}&\sqrt{\eta}&0\\0&0&0&1\end{pmatrix}.
\end{aligned}$$ This unitary is analogous to the unitary transformation induced by an optical beamsplitter with transmissivity $\eta\in[0,1]$. Such an optical beamsplitter is defined such that if one of the input arms contains no light, then the fraction $\eta$ of the light is transmitted unaltered, while the remaining fraction is reflected into the other output arm. The unitary transformation for the optical beamsplitter can be written as ${\mathrm{e}}^{{\text{i}}\theta H_{\text{BS}}}$, where $H_{\text{BS}}={\text{i}}(\hat{a}^\dagger \hat{b}-\hat{b}^\dagger \hat{a})$ and $\theta=\arccos(\sqrt{\eta})$ (see, e.g., [@KMN+07]). Here, $\hat{a}$ and $\hat{b}$ are the bosonic annihilation operators corresponding to the two input arms of the beamsplitter. The unitary $U^\eta$ for the qubit thermal channel can be written in the same form ${\mathrm{e}}^{{\text{i}}\theta H_{\text{BS}}}$ by replacing the bosonic annihilation operator $\hat{a}$ in $H_{\text{BS}}$ with $\sigma_-\otimes\mathbbm{1}$ and the operator $\hat{b}$ with $\mathbbm{1}\otimes\sigma_-$, where $\sigma_-\equiv {| 0 \rangle}{\langle 1 |}$ can be thought of as the qubit analogue of the annihilation operator.
![The qubit thermal channel is defined by analogy with the bosonic thermal channel as the interaction of a system $A$ in the state $\rho_A$ with an environment in the state $\theta_E^N$ (see ) at a “beamsplitter” of transmissivity $\eta$, which is a unitary channel defined by the unitary $U^\eta$ in . The state of the environment is then discarded to obtain the output $\mathcal{L}_{\eta,N}(\rho_A)$.[]{data-label="fig-qubit_thermal_noise"}](qubit_thermalNoise.pdf)
Let $\rho_A$ denote the state of the input system $A$, and let the initial state of the system $E$ be $$\begin{aligned}
\label{eq-thermal_state}
\theta^N_E \equiv (1-N){| 0 \rangle}{\langle 0 |}_E+N{| 1 \rangle}{\langle 1 |}_E.
\end{aligned}$$ Then, the qubit thermal channel $\L_{\eta, N}$ is defined as $$\begin{aligned}
\mathcal{L}_{\eta,N}(\rho_A)&\equiv{\operatorname{Tr}}_E[U_{AE\to BE}^\eta(\rho_A\otimes\theta_E^N)(U_{AB\to AE}^\eta)^\dagger]\label{eq-GADC_unitary_rep}\\
&={\operatorname{Tr}}_{EE'}[(U_{AE\to BE}^\eta\otimes\mathbbm{1}_{E'})(\rho_A\otimes{| \theta^N \rangle}{\langle \theta^N |}_{EE'})\nonumber\\
&\qquad\qquad\qquad\times (U_{AE\to BE}^\eta\otimes\mathbbm{1}_{E'})^\dagger]\label{eq-GADC_unitary_rep_2},
\end{aligned}$$ where $${| \theta^N \rangle}_{EE'}\equiv \sqrt{1-N}{| 0,0 \rangle}_{EE'}+\sqrt{N}{| 1,1 \rangle}_{EE'}.$$ When $N=0$, we call the qubit thermal channel $\mathcal{L}_{\eta,0}$ the qubit pure-loss channel.
The qubit thermal channel as defined in has exactly the same form as the bosonic thermal channel, the latter having the unitary $U^{\eta}$ defined in replaced by ${\mathrm{e}}^{{\text{i}}\theta H_{\text{BS}}}$. In particular, the initial state $\theta_E^N$ of the system $E$ can be thought of as the qubit analogue of the bosonic thermal state ${\mathrm{e}}^{-\beta\hat{a}^\dagger\hat{a}}/{{\operatorname{Tr}}[{\mathrm{e}}^{-\beta\hat{a}^\dagger\hat{a}}]}$ [@AS17], and the parameter $N\in[0,1]$ can be thought of as the mean number of photons. Indeed, if we replace $\hat{a}$ with $\sigma_-$ in the definition of the bosonic thermal state, observe using the definition of the number operator $\hat{n}$ in that $\sigma_-^\dagger\sigma_-=\hat{n}$, and let $\beta=\ln\left(\frac{1-N}{N}\right)$, then we obtain $$\begin{aligned}
\frac{{\mathrm{e}}^{-\beta\sigma_-^\dagger\sigma_-}}{{\operatorname{Tr}}[{\mathrm{e}}^{-\beta\sigma_-^\dagger\sigma_-}]}&=\frac{1}{1+{\mathrm{e}}^{-\beta}}{| 0 \rangle}{\langle 0 |}+\frac{{\mathrm{e}}^{-\beta}}{1+{\mathrm{e}}^{-\beta}}{| 1 \rangle}{\langle 1 |}\\
&=(1-N){| 0 \rangle}{\langle 0 |}+N{| 1 \rangle}{\langle 1 |}\\
&=\theta^N.
\end{aligned}$$
There is a simple connection between the qubit thermal channel and the generalized amplitude damping channel that is straightforward to prove: for all $\gamma\in[0,1]$ and $N \in [0, 1]$, $$\label{eq-GADC_to_thermalnoise}
\mathcal{A}_{\gamma,N}=\mathcal{L}_{1-\gamma,N}.$$
We take a channel complementary to the qubit thermal channel to be $$\begin{aligned}
\mathcal{L}_{\eta,N}^c(\rho_A)&\equiv {\operatorname{Tr}}_B[(U_{AE\to BE}^{\eta}\otimes\mathbbm{1}_{E'})(\rho_A\otimes{| \theta^N \rangle}{\langle \theta^N |}_{EE'})\\
&\qquad\qquad\qquad\times(U_{AE\to BE}^{\eta}\otimes\mathbbm{1}_{E'})^\dagger],
\end{aligned}$$ and we define a *weakly complementary channel* [@CG06] to be $$\label{eq-QTN_weak_complement}
\widetilde{\mathcal{L}}_{\eta,N}^c(\rho_A)\equiv{\operatorname{Tr}}_B[U_{AE\to BE}^{\eta}(\rho_A\otimes\theta^N)(U_{AE\to BE}^\eta)^\dagger].$$
Entanglement breakability of the GADC {#sec:ent-break}
=====================================
Having defined the GADC, we now proceed to examine its properties. We start by determining when the channel is entanglement breaking.
For any two-qubit quantum state $\rho_{AB}$, the condition $$\label{eq-two_qubit_sep}
\det(\rho_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B})\geq 0$$ is necessary and sufficient for the separability of $\rho_{AB}$ [@ADH08].
Since a channel is entanglement breaking if and only if its Choi state is separable [@HSR03], to determine when the GADC $\mathcal{A}_{\gamma,N}$ is entanglement breaking, we can apply the condition in to its Choi state $\rho_{AB}^{\gamma,N}\equiv\rho_{AB}^{\mathcal{A}_{\gamma,N}}$ as defined by . We have $$\begin{aligned}
\rho_{AB}^{\gamma,N}&=\frac{1}{2}\bigg((1-\gamma N){| 0,0 \rangle}{\langle 0,0 |}_{AB}+\sqrt{1-\gamma}{| 0,0 \rangle}{\langle 1,1 |}_{AB}\nonumber\\
&\qquad+\gamma N{| 0,1 \rangle}{\langle 0,1 |}_{AB} +\gamma(1-N){| 1,0 \rangle}{\langle 1,0 |}_{AB}\nonumber\\
&\qquad+\sqrt{1-\gamma}{| 1,1 \rangle}{\langle 0,0 |}_{AB}\nonumber\\
&\qquad+(1-\gamma(1-N)){| 1,1 \rangle}{\langle 1,1 |}_{AB}\bigg)\label{eq-GADC_Choi_state}\\
& = \frac{1}{2}\begin{pmatrix}
1-\gamma N & 0 & 0 & \sqrt{1-\gamma}\\
0 & \gamma N & 0 & 0\\
0 & 0 & \gamma\left( 1-N\right) & 0\\
\sqrt{1-\gamma} & 0 & 0 & 1-\gamma\left( 1-N\right)
\end{pmatrix}.
\end{aligned}$$ Then, $$\det\left(\left(\rho_{AB}^{\gamma,N}\right)^{{{\scriptscriptstyle\mathsf{T}}}_B}\right)=\frac{-1+2\gamma-\gamma^2+\gamma^4(1-N)^2N^2}{16},$$ so that $\det\left(\left(\rho_{AB}^{\gamma,N}\right)^{{{\scriptscriptstyle\mathsf{T}}}_B}\right)\geq 0$ leads to the following necessary and sufficient condition for the GADC to be entanglement breaking: $$\label{eq-GADC_ent_break}
\begin{aligned}
2(\!\!\sqrt{2}-1)&\leq\gamma\leq 1,\\
\frac{1}{2}\left(1-\sqrt{\frac{\gamma^2+4\gamma-4}{\gamma^2}}\right)&\leq N\leq\frac{1}{2}\left(1+\sqrt{\frac{\gamma^2+4\gamma-4}{\gamma^2}}\right).
\end{aligned}$$ Note that $2(\!\!\sqrt{2}-1)\approx 0.8284$. See Fig. \[eq-ent\_break\_region\] for a plot of this region of parameters. It is worth remarking that while the GADC has many parallels with the bosonic thermal channel, as outlined in Section \[subsec-qubit\_thermal\_chan\], the entanglement-breakability condition obtained here is starkly different from the corresponding condition in the bosonic case. In particular, entanglement breakability of the bosonic thermal channel is given by the relatively simple condition $\eta\leq \frac{N}{N+1}$ [@Hol08].
![Region of parameters, indicated in blue as per , for which the GADC is entanglement breaking.[]{data-label="eq-ent_break_region"}](GADC_ent_break_region.pdf)
Degradability and anti-degradability of the GADC {#sec:anti-deg}
================================================
Degradability of the GADC
-------------------------
It is known that the GADC is degradable for all $\gamma\in [0,1/2]$ when $N=0$ or $N=1$ [@GF05]. For $N\in(0,1)$ and $\gamma\in(0,1]$, it follows from [@CRS08 Theorem 4] that the GADC is not degradable.
In the case $N=0$, it can be shown that [@GF05] $$\mathcal{A}_{\gamma,0}^c=\mathcal{A}_{1-\gamma,0}.$$ Then, using , it follows from the condition $\mathcal{D}_{\gamma,0}\circ\mathcal{A}_{\gamma,0}=\mathcal{A}_{\gamma,0}^c=\mathcal{A}_{1-\gamma,0}$ that a degrading channel $\mathcal{D}_{\gamma,0}$ is simply $$\mathcal{D}_{\gamma,0}=\mathcal{A}_{\frac{1-2\gamma}{1-\gamma},0}.$$ In other words, $$\label{eq-GADC_deg}
\mathcal{A}_{\frac{1-2\gamma}{1-\gamma},0}\circ\mathcal{A}_{\gamma,0}=\mathcal{A}_{\gamma,0}^c$$ for all $\gamma\in[0,1/2)$. In terms of the qubit thermal channel, we use the correspondence in to write the condition as $$\mathcal{L}_{\frac{1-\eta}{\eta},0}\circ\mathcal{L}_{\eta,0}=\mathcal{L}_{\eta,0}^c.$$ for all $\eta\in(1/2,1]$.
Although the qubit thermal channel is not degradable for $N>0$, it is *weakly degradable*, meaning that there exists a channel $\widetilde{\mathcal{D}}_{\eta,N}$ such that $$\widetilde{\mathcal{D}}_{\eta,N}\circ\mathcal{L}_{\eta,N}=\widetilde{\mathcal{L}}_{\eta,N}^c.$$ In particular, one possible weakly degrading channel $\widetilde{\mathcal{D}}_{\eta,N}$ is [@RMG18] $$\label{eq-QTN_weak_deg_channel}
\widetilde{\mathcal{D}}_{\eta,N}=\mathcal{P}_{1-2N}\circ\mathcal{L}_{\frac{1-\eta}{\eta},N},$$ where $\mathcal{P}_{\mu}$ denotes the phase damping channel, which is defined via its Kraus operators $$\begin{pmatrix} 1 & 0 \\ 0 & \sqrt{\mu} \end{pmatrix}\quad\text{and}\quad \begin{pmatrix} 0 & 0 \\ 0 & \sqrt{1-\mu}\end{pmatrix}.$$
Anti-degradability of the GADC
------------------------------
To determine the anti-degradability of the GADC, we use the fact that a channel is anti-degradable if and only if its Choi state is two-extendable [@MyhrThesis].
For all $N\in[0,1]$, the condition $$\gamma\geq\frac{1}{2}$$ is necessary and sufficient for the anti-degradability of the GADC $\mathcal{A}_{\gamma,N}$.
Since the GADC is a qubit-to-qubit channel, its Choi state $\rho_{AB}^{\gamma,N}$ is a two-qubit state. For any two-qubit state $\rho_{AB}$, the inequality $$\label{eq-2ext_cond}
{\operatorname{Tr}}[\rho_{AB}^2]-{\operatorname{Tr}}[\rho_B^2]\leq 4\sqrt{\det(\rho_{AB})}$$ is necessary and sufficient for $\rho_{AB}$ to be two-extendable [@ML09; @CJK+14]. For the Choi state $\rho_{AB}^{\gamma,N}$, we find that $$\begin{aligned}
{\operatorname{Tr}}\left[\left(\rho_{AB}^{\gamma,N}\right)^2\right]&=\gamma^2N^2-\gamma^2N+\frac{1}{2}\gamma^2-\gamma+1,\\
{\operatorname{Tr}}\left[\left(\rho_B^{\gamma,N}\right)^2\right]&=2\gamma^2N^2-2\gamma^2N+\frac{1}{2}\gamma^2+\frac{1}{2},\\
\det\left(\rho_{AB}^{\gamma,N}\right)&=\frac{\gamma^4N^2(1-N)^2}{16}.
\end{aligned}$$ Substituting these quantities into the inequality in and simplifying leads to $\gamma\geq\frac{1}{2}$ as the necessary and sufficient condition for two-extendability of the Choi state of the GADC, and hence for anti-degradability of the GADC.
It is interesting to note that the condition for anti-degradability of the GADC has no dependence on $N$, even though, intuitively, the noise of the channel increases with $N$. This is another way in which the GADC is in contrast with the bosonic thermal channel, since for the bosonic thermal channel the anti-degradability condition depends on $N$ and is given by $\eta\leq\frac{N+1/2}{N+1}$ [@dp2006 Eq. (4.6)].
When the GADC is anti-degradable, there exists a simple anti-degrading channel $\mathcal{E}$ satisfying , the form of which follows immediately from the following lemma.
\[lem-anti\_degrad\_chan\_bd\] Define the channel $\mathcal{E}_N^*$ by the Kraus operators $$\begin{aligned}
E_0&={| 0 \rangle}_B{\langle 0 |}_E+{| 1 \rangle}_B{\langle 1 |}_E,\\
E_1&={| 0 \rangle}_B{\langle 3 |}_E+{| 1 \rangle}_B{\langle 2 |}_E,
\end{aligned}$$ which acts on the four-dimensional output space of the complementary channel $\mathcal{A}_{\gamma,N}^c$ defined in . Then, $$\label{eq-anti_degrad_chan_bd}
\mathcal{E}_N^*\circ\mathcal{A}_{\gamma,N}^c=\mathcal{A}_{1-\gamma,N}$$ for all $N\in[0,1]$ and all $\gamma\in[0,1]$.
See Appendix \[app-GADC\_anti\_degrade\].
It follows that the channel $\mathcal{E}_N^*$ defined in Lemma \[lem-anti\_degrad\_chan\_bd\] is an anti-degrading channel at the boundary $\gamma=\frac{1}{2}$ for all $N\in[0,1]$. To find an anti-degrading channel for $\gamma>\frac{1}{2}$, we use to obtain the following.
For all $N\in[0,1]$ and all $\gamma\geq\frac{1}{2}$, the channel $$\mathcal{E}_{\gamma,N}\equiv\mathcal{A}_{\frac{2\gamma-1}{\gamma},N}\circ\mathcal{E}_N^*$$ is an anti-degrading channel for the GADC, meaning that $\mathcal{E}_{\gamma,N}\circ\mathcal{A}_{\gamma,N}^c=\mathcal{A}_{\gamma,N}$.
The decomposition in implies that $$\mathcal{A}_{\frac{2\gamma-1}{\gamma},N}\circ\mathcal{A}_{1-\gamma,N}=\mathcal{A}_{\gamma,N}.$$ Combining this with , we find that $$\mathcal{A}_{\frac{2\gamma-1}{\gamma},N}\circ\mathcal{E}_N^*\circ\mathcal{A}_{\gamma,N}^c=\mathcal{A}_{\gamma,N}$$ for all $N\in[0,1]$ and all $\gamma\geq\frac{1}{2}$. The result then follows.
Bounds on the classical capacity of the GADC {#sec:c-cap-bounds}
============================================
We now consider the communication capacities of the GADC, starting with the classical capacity. In general, the Holevo information recalled in is a lower bound on the classical capacity of any channel. Then, as implied by the formula in , determining the classical capacity of a quantum channel essentially reduces to determining the additivity of the Holevo information for that channel. Remarkably, even in the case $N=0$, in which case the GADC reduces to the amplitude damping channel, determining the additivity of the Holevo information remains an important open problem. In the case $N=\frac{1}{2}$, however, we observe from that the GADC is unital, i.e., $\mathcal{A}_{\gamma,\frac{1}{2}}(\mathbbm{1})=\mathbbm{1}$ for all $\gamma\in[0,1]$. The Holevo information is additive for unital qubit channels [@King02], i.e., $$\chi(\mathcal{N}\otimes\mathcal{M})=\chi(\mathcal{N})+\chi(\mathcal{M})$$ for any unital qubit channel $\mathcal{N}$ and for any channel $\mathcal{M}$. This implies that the classical capacity of any unital qubit channel is equal to its Holevo information. In particular, for the GADC, we obtain $$\label{eq-GADC_CCap_N05}
C(\mathcal{A}_{\gamma,\frac{1}{2}})=\chi(\mathcal{A}_{\gamma,\frac{1}{2}}).$$ Furthermore, the Holevo information for unital qubit channels is directly related to its minimum output entropy [@KR01; @Cort02; @Cort04] (see also [@H13book Example 8.10]), such that for the GADC with $N=\frac{1}{2}$ we obtain $$\chi(\mathcal{A}_{\gamma,\frac{1}{2}})=1-h_2\left(\frac{1-\sqrt{1-\gamma}}{2}\right).$$
The Holevo information is also known to be additive for entanglement breaking channels [@Shor02]. Therefore, using the result in , we obtain $$C(\mathcal{A}_{\gamma,N})=\chi(\mathcal{A}_{\gamma,N}),$$ for all $\gamma$ and $N$ satisfying $$\begin{aligned}
2(\sqrt{2}-1)&\leq\gamma\leq 1,\\
\frac{1}{2}\left(1-\sqrt{\frac{\gamma^2+4\gamma-4}{\gamma^2}}\right)&\leq N\leq \frac{1}{2}\left(1+\sqrt{\frac{\gamma^2+4\gamma-4}{\gamma^2}}\right).
\end{aligned}$$
Using the techniques from [@Cort02; @Ber05], it has been shown in [@LM07b] that the Holevo information of the GADC for its entire parameter range is given by $$\label{eq-GADC_Hol_inf}
\chi(\mathcal{A}_{\gamma,N})=\frac{1}{2}(f(r^*)-\log_2(1-q^2)-qf'(q)),$$ where $$\begin{aligned}
f(x)&\equiv(1+x)\log_2(1+x)+(1-x)\log_2(1-x),\\
f'(x)&=\frac{\text{d}}{\text{d}x}f(x)=\log_2\left(\frac{1+x}{1-x}\right),\\
r^*&\equiv\sqrt{1-\gamma-\frac{(q-\gamma(1-2N))^2}{1-\gamma}+q^2},
\end{aligned}$$ and $q$ is determined as the solution to the equation $$\begin{gathered}
(\gamma q-\gamma^2(1-2N)-\gamma(1-\gamma)(1-2N))f'(r^*)\\=-r^*(1-\gamma)f'(q).
\end{gathered}$$
Let us now compare the Holevo information lower bound with two upper bounds based on the concepts of $\varepsilon$-entanglement-breakability and $\varepsilon$-covariance.
\[prop-GADC\_CCap\_UB\_EB\_cov\] For all $\gamma,N\in(0,1)$ it holds that $$\begin{aligned}
C(\mathcal{A}_{\gamma,N})&\leq \chi(\mathcal{M}_{\gamma,N})+2\varepsilon_1+g(\varepsilon_1)\equiv C_{\operatorname{EB}}^{\operatorname{UB}}(\gamma,N),\label{eq-CCap_UB1_EB}\\
C(\mathcal{A}_{\gamma,N})&\leq \chi(\mathcal{A}_{\gamma,\frac{1}{2}})+2\varepsilon_2+g(\varepsilon_2)\equiv C_{\operatorname{cov}}^{\operatorname{UB}}(\gamma,N),\label{eq-CCap_UB2_cov}
\end{aligned}$$ where $\varepsilon_1=\varepsilon_{\operatorname{EB}}(\mathcal{A}_{\gamma,N})=\frac{1}{2}{\lVert\mathcal{A}_{\gamma,N}-\mathcal{M}_{\gamma,N}\rVert}_{\diamond}$ and $\varepsilon_2=\varepsilon_{\operatorname{cov}}(\mathcal{A}_{\gamma,N})=\gamma\left|N-\frac{1}{2}\right|$.
To obtain , we use and the fact that $d_B=2$ for the GADC. Furthermore, we note here again that since the GADC is a qubit-to-qubit channel, the entanglement-breaking parameter $\varepsilon_{\text{EB}}(\mathcal{A}_{\gamma,N})$ defined in can be calculated via an SDP [@LKDW18 Lemma III.8] due to the fact that, for two-qubit states, the set of separable states is equal to the set of states with positive partial transpose [@Peres96; @HHH96].
For the bound in , we make use of . Let us first show that the channel $\mathcal{A}_{\gamma,N}^G$ obtained by twirling with the Pauli operators $\{\mathbbm{1},\sigma_x,\sigma_y,\sigma_z\}$ is equal to $\mathcal{A}_{\gamma,\frac{1}{2}}$. We start by recalling the convex decomposition of the GADC as stated in : $$\label{eq-GADC_decomp_convex_2}
\mathcal{A}_{\gamma,N}=(1-N)\mathcal{A}_{\gamma,0}+N\mathcal{A}_{\gamma,1}.$$ Thus, by linearity of the twirling channel, we have that $\mathcal{A}_{\gamma,N}^G=(1-N)\mathcal{A}_{\gamma,0}^{G}+N\mathcal{A}_{\gamma,1}^G$. Next, we recall and , respectively: $$\begin{aligned}
\mathcal{A}_{\gamma,0}(\cdot)&=\sigma_z\mathcal{A}_{\gamma,0}(\sigma_z(\cdot)\sigma_z)\sigma_z,\\
\mathcal{A}_{\gamma,1}(\cdot)&=\sigma_x\mathcal{A}_{\gamma,0}(\sigma_x(\cdot)\sigma_x)\sigma_x.
\end{aligned}$$ Using these relations, and the fact that $\sigma_y={\text{i}}\sigma_x\sigma_z$, we obtain $$\begin{aligned}
\mathcal{A}_{\gamma,0}^G&=\frac{1}{2}\mathcal{A}_{\gamma,0}+\frac{1}{2}\mathcal{A}_{\gamma,1}=\mathcal{A}_{\gamma,\frac{1}{2}},\\
\mathcal{A}_{\gamma,1}^G&=\frac{1}{2}\mathcal{A}_{\gamma,0}+\frac{1}{2}\mathcal{A}_{\gamma,1}=\mathcal{A}_{\gamma,\frac{1}{2}},
\end{aligned}$$ where to obtain the last equality in both equations we used . Therefore, $$\label{eq-GADC_twirl}
\mathcal{A}_{\gamma,N}^G=\mathcal{A}_{\gamma,\frac{1}{2}}$$ for all $\gamma,N\in[0,1]$. The final step is to show that $\varepsilon_{\text{cov}}(\mathcal{A}_{\gamma,N})=\frac{1}{2}{\lVert\mathcal{A}_{\gamma,N}-\mathcal{A}_{\gamma,N}^G\rVert}_{\diamond}=\frac{1}{2}{\lVert\mathcal{A}_{\gamma,N}-\mathcal{A}_{\gamma,\frac{1}{2}}\rVert}_{\diamond}=\gamma\left|N-\frac{1}{2}\right|$, which we do in Appendix \[app-GADC\_cov\_parameter\].
We now compare the upper bounds obtained above with two strong converse upper bounds on the classical capacity that hold for any quantum channel $\mathcal{N}$ [@WXD18]. The first upper bound is $$\label{eq-C_beta}
C(\mathcal{N})\leq C_\beta(\mathcal{N})\equiv \log_2\beta(\mathcal{N}),$$ where $$\label{eq-C_beta_primal}
\beta(\mathcal{N})\equiv\left\{\begin{array}{l l}\text{min.} & {\operatorname{Tr}}[S_{B}]\\
\text{subject to} & -R_{AB}\leq \left(\Gamma_{AB}^{\mathcal{N}}\right)^{{{\scriptscriptstyle\mathsf{T}}}_B}\leq R_{AB},\\
& -\mathbbm{1}_A\otimes S_B\leq R_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B}\leq\mathbbm{1}_A\otimes S_B.
\end{array}\right.$$ Note that the optimization is with respect to the operators $S_B$ and $R_{AB}$. We also observe that the optimization problem is a semi-definite program (SDP).
The second upper bound from [@WXD18], which is also given by an SDP, is the following: $$\label{eq-C_zeta}
C(\mathcal{N})\leq C_{\zeta}(\mathcal{N})\equiv \log_2\zeta(\mathcal{N}),$$ where $$\label{eq-C_zeta_primal}
\zeta(\mathcal{N})=\left\{\begin{array}{l l} \text{min.} & {\operatorname{Tr}}[S_B] \\
\text{subject to} & V_{AB}\geq\Gamma_{AB}^{\mathcal{N}},\\
& -\mathbbm{1}_A\otimes S_B\leq V_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B}\leq \mathbbm{1}_A\otimes S_B. \end{array}\right.$$
By considering the dual of the SDPs in and , we obtain analytic expressions for $C_\beta(\mathcal{A}_{\gamma,N})$ and $C_{\zeta}(\mathcal{A}_{\gamma,N})$ for all values of $\gamma$ and $N$, and we find that $C_{\zeta}(\mathcal{A}_{\gamma,N})=C_{\beta}(\mathcal{A}_{\gamma,N})$ for all values of $\gamma$ and $N$.
\[prop-C\_beta\] For all $\gamma,N\in[0,1]$, $$\label{eq-C_beta_GADC}
C_\beta(\mathcal{A}_{\gamma,N})=C_{\zeta}(\mathcal{A}_{\gamma,N})=\log_2(1+\sqrt{1-\gamma}).$$
See Appendix \[app-C\_beta\].
Let us now compare the Holevo information lower bound and the upper bounds in Proposition \[prop-GADC\_CCap\_UB\_EB\_cov\], , and to the upper bound obtained in [@F18]. This bound is obtained using a technique developed in [@FFK18], which is based on a decomposition of the channel of interest in terms of a unital channel (for which we know the classical capacity, as mentioned above). When applied to the GADC, the technique leads to the following upper bound [@F18 Eq. (35)]: $$\begin{aligned}
C(\mathcal{A}_{\gamma,N})&\leq C_{\text{Fil}}^{\text{UB}}\nonumber\\
&\equiv 1-h_2\left(\frac{1}{2}\left(1-\frac{\sqrt{1-\gamma}}{f(\gamma,N)}\right)\right)+\log_2f(\gamma,N)\nonumber\\
&\qquad +\frac{1}{2}\log_2\frac{N}{1-N},\label{eq-GADC_CCap_UB_Fil}
\end{aligned}$$ where $$\begin{gathered}
f(\gamma,N)\equiv \gamma\sqrt{N(1-N)}\\
+\sqrt{N+(1-N)(1-\gamma)}\sqrt{1-N+N(1-\gamma)}.
\end{gathered}$$
Finally, we consider the entanglement-assisted classical capacity as another upper bound on the classical capacity of the GADC. The entanglement-assisted classical capacity of a quantum channel $\mathcal{N}$, denoted by $C_E(\mathcal{N})$, is defined as the maximum rate at which classical information can be sent over the channel in the asymptotic limit, with the assistance of entanglement between the sender and the receiver. It is known [@BSST99; @Hol01a; @ieee2002bennett] that $C_E(\mathcal{N})$ is given simply by the mutual information $I(\mathcal{N})$ of the channel, i.e., $$C_E(\mathcal{N})=I(\mathcal{N})\equiv\max_{\phi_{AA'}}I(A;B)_{\rho},$$ where $\rho_{AB}=\mathcal{N}_{A'\to B}(\phi_{AA'})$ and the dimension of $A$ is equal to the dimension of the input system $A'$ of the channel $\mathcal{N}$. For the GADC, by using its Pauli-$z$ covariance, as well the concavity of the function $\rho_{A'}\mapsto I(A;B)_{\omega}$, where $\omega_{AA'}=\mathcal{N}_{A'\to B}(\phi_{AA'}^{\rho})$ and $\phi_{AA'}^{\rho}$ is any purification of $\rho_{A'}$, it has been shown [@LM07] that $$\label{eq-GADC_mut_inf}
I(\mathcal{A}_{\gamma,N})=\max_{z\in[-1,1]}F(\gamma,N,z)$$ for all $\gamma,N\in(0,1)$, where $$\begin{gathered}
F(\gamma,N,z)\\\equiv -\sum_{i=1}^2\lambda_i\log_2\lambda_i-\sum_{i=1}^2\lambda_i'\log_2\lambda_i'+\sum_{i=1}^4\lambda_i''\log_2\lambda_i''
\end{gathered}$$ and $$\begin{aligned}
\lambda_1&=\frac{1}{2}(1+z),\\
\lambda_2&=\frac{1}{2}(1-z),\\
\lambda_1'&=\frac{1}{2}\left(1+((2N-1)\gamma-(1-\gamma)z)\right),\\
\lambda_2'&=\frac{1}{2}\left(1-((2N-1)\gamma-(1-\gamma)z)\right),\\
\lambda_1''&=\frac{1}{2}(1-N)\gamma(1-z),\\
\lambda_2''&=\frac{1}{2}N\gamma(1+z),\\
\lambda_3''&=\frac{1}{4}\left(2-(1+(2N-1)z)\gamma\right.\nonumber\\
&\quad\left.+\sqrt{4-4(1+z(2N-1))\gamma+(2N-1+z)^2\gamma^2}\right),\\
\lambda_4''&=\frac{1}{4}\left(2-(1+(2N-1)z)\gamma\right.\nonumber\\
&\quad-\left.\sqrt{4-4(1+z(2N-1))\gamma+(1-2p+z)^2\gamma^2}\right).
\end{aligned}$$
In Fig \[fig-CCap\_bounds\], we plot the Holevo information lower bound as well as the $C_{\beta}$ upper bound, the upper bound $C_{\text{EB}}^{\text{UB}}$ based on approximate entanglement breakability, the upper bound $C_{\text{cov}}^{\text{UB}}$ based on approximate covariance, the bound $C_{\text{Fil}}^{\text{UB}}$ defined in , and the entanglement-assisted classical capacity $C_E$. We find that the $C_{\beta}$ upper bound is close to the Holevo information lower bound for low values of $\gamma$ and $N$. For higher values of $\gamma$, the entanglement-assisted classical capacity provides a tighter upper bound than $C_{\beta}$. For values of $N$ close to $\frac{1}{2}$, as one might expect, the approximate covariance upper bound $C_{\text{cov}}^{\text{UB}}$ is tighter than both $C_{\beta}$ and $C_E$, at least for low to intermediate values of $\gamma$. In this same regime for $N$, the bound $C_{\text{Fil}}^{\text{UB}}$ is the tightest for small intervals of $\gamma$ close to $\gamma=0.6$. For $N=\frac{1}{2}$, we know from that the classical capacity of the GADC is given by the Holevo information. Accordingly, the Holevo information and the upper bounds $C_{\text{cov}}^{\text{UB}}$ and $C_{\text{Fil}}^{\text{UB}}$ coincide. Also, as expected, the approximate entanglement-breaking bound $C_{\text{EB}}^{\text{UB}}$ is tight, matching the lower bound, whenever the GADC is entanglement breaking. For values of $\gamma$ and $N$ close to the entanglement breaking region, this upper bound is also the tightest among all of the other upper bounds.
Bounds on the quantum and private capacities of the GADC {#sec:q-cap-bounds}
========================================================
We now consider the quantum and private capacities of the GADC and provide upper bounds using the data-processing bounds, the approximate degradability and approximate anti-degradability bounds, and the Rains information and relative entropy of entanglement bounds defined in Sec. \[sec-bounds\_QCap\].
We start with the decompositions of the GADC in and : $$\begin{aligned}
\mathcal{A}_{\gamma,N}&=\mathcal{A}_{\gamma N,1}\circ\mathcal{A}_{\frac{\gamma(1-N)}{1-\gamma N},0},\label{eq-GADC_decomp_spec_1a}\\
\mathcal{A}_{\gamma,N}&=\mathcal{A}_{\gamma(1-N),0}\circ\mathcal{A}_{\frac{\gamma N}{1-\gamma(1-N)},1}.\label{eq-GADC_decomp_spec_2a}
\end{aligned}$$ These decompositions of the GADC involve the amplitude damping channels $\mathcal{A}_{\frac{\gamma(1-N)}{1-\gamma N},0}$ and $\mathcal{A}_{\gamma(1-N),0}$. Moreover, these decompositions are similar in spirit to the ones used in [@dp2006; @dp2012; @SWAT; @NAJ18; @RMG18] in the context of bosonic Gaussian thermal channels.
Unlike the classical capacity, the quantum capacity of the amplitude damping channel has a known closed-form expression and is given by [@GF05] $$\label{eq-AD_QCap}
Q(\mathcal{A}_{\gamma,0})=\max_{p\in[0,1]}\bigg[h_2((1-\gamma)p)-h_2(\gamma p)\bigg].$$ for $\gamma \in [0,1/2)$, and $Q(\mathcal{A}_{\gamma,0}) = 0$ for $\gamma \in [1/2,1]$. The quantum capacity can be determined easily in this case since the amplitude damping channel is degradable for all $\gamma\in[0,1/2]$, which implies that the coherent information of the channel is additive. The relation between $\mathcal{A}_{\gamma,0}$ and $\mathcal{A}_{\gamma,1}$ given by implies that the quantum capacity of the channel $\mathcal{A}_{\gamma,1}$ is equal to the quantum capacity of the amplitude damping channel, i.e., $Q(\mathcal{A}_{\gamma,1})=Q(\mathcal{A}_{\gamma,0})$. Furthermore, since the private and quantum capacities are equal to each other for degradable channels, we have that $P(\mathcal{A}_{\gamma,0})=Q(\mathcal{A}_{\gamma,0})$.
\[prop-QCap\_data\_processing\] For all $\gamma,N\in(0,1)$, it holds that $$\begin{aligned}
Q(\mathcal{A}_{\gamma,N})&\leq P(\mathcal{A}_{\gamma,N})\leq Q\left(\mathcal{A}_{\frac{\gamma(1-N)}{1-\gamma N},0}\right)\equiv Q_1^{\operatorname{UB}}(\gamma,N),\label{eq-GACD_Qcap_UB_DP_1}\\
Q(\mathcal{A}_{\gamma,N})&\leq P(\mathcal{A}_{\gamma,N})\leq Q(\mathcal{A}_{\gamma(1-N),0})\equiv Q_2^{\operatorname{UB}}(\gamma,N),\label{eq-GACD_Qcap_UB_DP_2}\\
Q(\mathcal{A}_{\gamma,N})&\leq P(\mathcal{A}_{\gamma,N})\leq Q(\mathcal{A}_{\gamma N,1})\equiv Q_3^{\operatorname{UB}}(\gamma,N), \label{eq-GACD_Qcap_UB_DP_3}\\
Q(\mathcal{A}_{\gamma,N})&\leq P(\mathcal{A}_{\gamma,N})\leq Q\left(\mathcal{A}_{\frac{\gamma N}{1-\gamma(1-N)},1}\right)\equiv Q_4^{\operatorname{UB}}(\gamma,N).\label{eq-GACD_Qcap_UB_DP_4}
\end{aligned}$$
All of these inequalities follow from the relation between the quantum and private capacities in , the decompositions of the GADC in and , and the general data processing upper bounds given in and for the quantum capacity and and for the private capacity. In particular, for the bounds on the private capacity, we make use of the fact that the amplitude damping channel is degradable, which means that its private capacity is equal to its quantum capacity, as given in .
We obtain more upper bounds using the concepts of $\varepsilon$-degradability, $\varepsilon$-close-degradability, and $\varepsilon$-anti-degradability.
\[prop-approx\_deg\_adeg\] For all $\gamma\in(0,1/2)$ and all $N\in(0,1)$, it holds that $$\begin{aligned}
Q(\mathcal{A}_{\gamma,N})&\leq Q_5^{\operatorname{UB}}(\gamma,N)\equiv U_{\mathcal{D}}(\mathcal{A}_{\gamma,N})+4\varepsilon_1+g(\varepsilon_1),\label{eq-GACD_Qcap_UB_DP_5}\\
P(\mathcal{A}_{\gamma,N})&\leq U_{\mathcal{D}}(\mathcal{A}_{\gamma,N})+12\varepsilon_1+3g(\varepsilon),\label{eq-GACD_Pcap_UB_DP_5}\\
Q(\mathcal{A}_{\gamma,N})&\leq Q_6^{\operatorname{UB}}(\gamma,N) \equiv Q(\mathcal{A}_{\gamma,0})+2\varepsilon_2+2g(\varepsilon_2),\label{eq-GACD_Qcap_UB_DP_6}\\
P(\mathcal{A}_{\gamma,N})&\leq Q(\mathcal{A}_{\gamma,0})+4\varepsilon_2+4g(\varepsilon_2),\label{eq-GACD_Pcap_UB_DP_6}\\
Q(\mathcal{A}_{\gamma,N})&\leq Q_7^{\operatorname{UB}}(\gamma,N)\equiv 2\varepsilon_3+h_2(\varepsilon_3)+g(\varepsilon_3) ,\label{eq-GADC_QCap_UB7_Adeg}\\
P(\mathcal{A}_{\gamma,N})&\leq 2\varepsilon_3+h_2(\varepsilon_3)+g(\varepsilon_3),\label{eq-GADC_PCap_UB7_Adeg}
\end{aligned}$$ where $\varepsilon_1=\varepsilon_{\operatorname{deg}}(\mathcal{A}_{\gamma,N})$, $\varepsilon_2=\frac{1}{2}{\lVert\mathcal{A}_{\gamma,N}-\mathcal{A}_{\gamma,0}\rVert}_{\diamond}$, and $\varepsilon_3=\varepsilon_{\textnormal{a-deg}}(\mathcal{A}_{\gamma,N})$.
We start with the bounds in and . For the GADC, we have $d_E=4$, since the channel has four Kraus operators (assuming $N\neq 0$ and $N\neq 1$). Therefore, by determining the approximate-degradability parameter $\varepsilon_{\text{deg}}(\mathcal{A}_{\gamma,N})$, we immediately obtain the bounds in and .
Similarly, we obtain the bounds in and using and , respectively, as follows. Since the channel $\mathcal{A}_{\gamma,0}$ is degradable for all $\gamma\in[0,1/2]$, we can take that to be our $\varepsilon$-close-degradable channel to $\mathcal{A}_{\gamma,N}$. Then, since $I_{\text{c}}(\mathcal{A}_{\gamma,0})$ is simply the quantum capacity of $\mathcal{A}_{\gamma,0}$ (as given by ), we obtain .
Finally, we use the bounds in arising from $\varepsilon$-anti-degradability. Since $d_B=2$, after calculating the anti-degradability parameter $\varepsilon_{\text{a-deg}}(\mathcal{A}_{\gamma,N})$, we obtain and .
We obtain another upper bound on the private and quantum capacities of the GADC by employing the Rains information of the GADC, as given in , , and : $$\label{eq-GADC_QCap_UB8}
Q(\mathcal{A}_{\gamma,N}) \leq P(\mathcal{A}_{\gamma,N}) \leq R(\mathcal{A}_{\gamma,N}) \equiv Q_8^{\text{UB}}(\gamma,N),$$ which follows from the fact that, as stated previously, the Rains information $R(\mathcal{A}_{\gamma,N})$ is equal to the channel’s relative entropy of entanglement $E_R(\mathcal{A}_{\gamma,N})$ for qubit-to-qubit channels, due to [@AS08]. To compute the latter, we can perform the minimization over PPT states, due to [@Peres96; @HHH96]. Furthermore, due to the $\sigma_z$ covariance of the GADC, we can make several simplifications to the task of computing the Rains information $R(\mathcal{A}_{\gamma,N})$, which speed it up significantly. First, due to the $\sigma_z$ covariance and concavity of Rains information in the input state, as presented in Proposition \[prop:concavity-Rains\], it suffices to perform the maximization over input states with respect to the one-parameter family of states ${| \theta^p \rangle}_{AA'} = \sqrt{1-p} {| 0,0 \rangle}_{AA'} + \sqrt{p} {| 1,1 \rangle}_{AA'}$. Second, the minimization in the Rains relative entropy in the definition in can be performed over PPT states having the following form: $$\sigma_{AB} = \frac{1}{2}
\begin{pmatrix}
\alpha & 0 & 0 & \xi {\mathrm{e}}^{i \phi} \\
0 & \beta & 0 & 0 \\
0 & 0 & \gamma & 0 \\
\xi {\mathrm{e}}^{-i \phi} & 0 & 0 & \delta
\end{pmatrix},$$ where $\alpha, \beta, \gamma, \delta \geq 0$, $\alpha+ \beta+ \gamma+ \delta =2$, $0 \leq \xi \leq \min\{\sqrt{\alpha \delta},\sqrt{\beta \gamma}\}$, $\phi\in[0,2\pi)$. This latter simplification follows from the same argument given in [@RKB+18 Appendix B].
See Fig. \[fig-GADC\_QCap\_bounds\] for a plot of the upper bounds $Q_1^{\text{UB}}$ to $Q_8^{\text{UB}}$. To get a sense for how good these upper bounds are, it is worth comparing them to a lower bound. The coherent information $I_{\text{c}}(\mathcal{A}_{\gamma,N})$ provides a lower bound on the quantum capacity of the GADC. It can be shown that [@GPLS09] $$\label{eq-GADC_LB1}
I_{\text{c}}(\mathcal{A}_{\gamma,N})=\max_{p\in[0,1]}I_{\text{c}}\left(\begin{pmatrix} 1-p&0\\0&p\end{pmatrix},\mathcal{A}_{\gamma,N}\right)\equiv Q^{\text{LB}}_{\text{CI}}(\gamma,N).$$ By plotting in Fig. \[fig-GADC\_QCap\_bounds\] the coherent information lower bound alongside the upper bounds $Q_1^{\text{UB}}$ to $Q_8^{\text{UB}}$, we find that the gap between the upper bounds and the lower bound is smallest when both $\gamma$ and $N$ are small. We also find that, as expected, the upper bound $Q_5^{\text{UB}}$ based on $\varepsilon$-degradability is a tighter bound for $\gamma$ close to zero, since $\gamma=0$ is the point at which the GADC is close to an identity channel. We note here that the generic behavior of the $\varepsilon$-degradable bound being tangent to the lower bound for low noise quantum channels was studied in detail in [@LLS18]. On the other hand, the upper bound $Q_6^{\text{UB}}$ based on $\varepsilon$-close-degradability is relatively poor for large values of $N$. Similarly, we observe that the upper bound $Q_7^{\text{UB}}$ based on $\varepsilon$-anti-degradability is relatively poor except for values of $\gamma$ close to $\gamma=\frac{1}{2}$, where, as expected, the bound is tighter, since $\gamma=\frac{1}{2}$ is the point beyond which the GADC is anti-degradable. Moreover, the upper bound $Q_7^{\text{UB}}$ is tighter than all other upper bounds for both $\gamma$ and $N$ close to $\frac{1}{2}$. While the Rains information upper bound $Q_8^{\text{UB}}$ is worse than two of the data-processing upper bounds for all values of $\gamma$ when $N$ is close to zero, it is tighter than all four data-processing upper bounds for all values of $\gamma$ when $N$ is close to $\frac{1}{2}$. In this region of $N$ close to $\frac{1}{2}$, it is also tighter than the bounds $Q_5^{\text{UB}}$ and $Q_7^{\text{UB}}$ for values of $\gamma$ roughly between $0.15$ and $0.49$.
Comparison with prior work
--------------------------
Let us now compare the bounds obtained here with those from prior work.
In [@RMG18], in order to obtain an upper bound on the quantum capacity of the qubit thermal channel, the authors consider the “extended” channel $$\begin{aligned}
\widehat{\mathcal{L}}_{\eta,N}(\rho_A)&\equiv {\operatorname{Tr}}_E[(U_{AE\to BE}^\eta\otimes\mathbbm{1}_{E'})(\rho_A\otimes{| \theta^N \rangle}{\langle \theta^N |}_{EE'})\\
&\qquad\qquad\qquad\times (U_{AE\to BE}^{\eta}\otimes\mathbbm{1}_{E'})^\dagger].
\end{aligned}$$ Note that $$\label{eq-QTN_ext_2}
\mathcal{L}_{\eta,N}={\operatorname{Tr}}_{E'}\circ\widehat{\mathcal{L}}_{\eta,N},$$ which implies, via , that $$\label{eq-QTN_QCap_UB_RMG}
Q(\mathcal{L}_{\eta,N})\leq Q(\widehat{\mathcal{L}}_{\eta,N})\equiv Q_{\text{RMG}}^{\text{UB}}(\eta,N).$$
![Comparison of the upper bounds $Q_1^{\text{UB}}$, $Q_2^{\text{UB}}$ and $Q_5^{\text{UB}}$ defined in , and , respectively, with the upper bound $Q_{\text{RMG}}^{\text{UB}}$ obtained in [@RMG18] and defined in . Also shown is the coherent information lower bound $Q_{\text{CI}}^{\text{LB}}$ defined in . The quantum capacity lies within the shaded region.[]{data-label="fig-compRMG"}](QTN_QCap_N001_compRMG.pdf "fig:"){width="\columnwidth"}\
![Comparison of the upper bounds $Q_1^{\text{UB}}$, $Q_2^{\text{UB}}$ and $Q_5^{\text{UB}}$ defined in , and , respectively, with the upper bound $Q_{\text{RMG}}^{\text{UB}}$ obtained in [@RMG18] and defined in . Also shown is the coherent information lower bound $Q_{\text{CI}}^{\text{LB}}$ defined in . The quantum capacity lies within the shaded region.[]{data-label="fig-compRMG"}](QTN_QCap_N01_compRMG.pdf "fig:"){width="\columnwidth"}
As explained in [@RMG18], to compute the upper bound $Q(\widehat{\mathcal{L}}_{\eta,N})$, we observe that by defining a channel complementary to $\widehat{\mathcal{L}}_{\eta,N}$ as $$\begin{aligned}
\widehat{\mathcal{L}}_{\eta,N}^{c}(\rho_A)&\equiv {\operatorname{Tr}}_{BE'}[(U_{AE\to BE}^\eta\otimes\mathbbm{1}_{E'})(\rho_A\otimes{| \theta^N \rangle}{\langle \theta^N |}_{EE'})\\
&\qquad\qquad\qquad\times (U_{AE\to BE}^{\eta}\otimes\mathbbm{1}_{E'})^\dagger],
\end{aligned}$$ we get $$\widehat{\mathcal{L}}_{\eta,N}^{c}=\widetilde{\mathcal{L}}_{\eta,N}^c$$ for all $\eta,N\in[0,1]$, where $\widetilde{\mathcal{L}}_{\eta,N}$ is the channel weakly complementary to $\mathcal{L}_{\eta,N}$ defined in . This implies that whenever the qubit thermal channel is weakly degradable, the extended channel is degradable. Indeed, for all $N>0$ and all $\eta\in[0,1]$, the channel $\widehat{\mathcal{D}}_{\eta,N}\equiv \mathcal{P}_{1-2N}\circ\mathcal{L}_{\frac{1-\eta}{\eta},N}\circ{\operatorname{Tr}}_{E'}$ satisfies $$\begin{aligned}
\widehat{\mathcal{D}}_{\eta,N}\circ\widehat{\mathcal{L}}_{\eta,N}&=\mathcal{P}_{1-2N}\circ\mathcal{L}_{\frac{1-\eta}{\eta},N}\circ{\operatorname{Tr}}_{E'}\circ\widehat{\mathcal{L}}_{\eta,N}\\
&=\mathcal{P}_{1-2N}\circ\mathcal{L}_{\frac{1-\eta}{\eta},N}\circ\mathcal{L}_{\eta,N}\\
&=\widetilde{\mathcal{L}}_{\eta,N}^c\\
&=\widehat{\mathcal{L}}_{\eta,N}^c,
\end{aligned}$$ where to obtain the second equality we used and to obtain the third equality we used . The quantum capacity of the extended channel is therefore given by its coherent information. In other words, $$\begin{aligned}
Q(\widehat{\mathcal{L}}_{\eta,N})&=\max_{\rho}\left(H(\widehat{\mathcal{L}}_{\eta,N}(\rho))-H(\widehat{\mathcal{L}}_{\eta,N}^c(\rho))\right)\\
&=\max_{p\in[0,1]} I_{\text{c}}\left(\begin{pmatrix} 1-p & 0 \\ 0 & p\end{pmatrix},\widehat{\mathcal{L}}_{\eta,N}\right)
\end{aligned}$$ for all $N>0$ and $\eta\in[0,1]$, where the last equality holds due to the fact $\widehat{\mathcal{L}}_{\eta,N}(\sigma_z\rho_A\sigma_z)=(\sigma_z\otimes\mathbbm{1}_{E'})\widehat{\mathcal{L}}_{\eta,N}(\rho)(\sigma_z\otimes\mathbbm{1}_{E'})$ and the fact that the coherent information is concave in the input state of the channel whenever the channel is degradable [@YHD08].
![Comparison between the data-processing upper bounds $Q_1^{\text{UB}}$, $Q_2^{\text{UB}}$ and $Q_8^{\text{UB}}$ and the upper bound $Q_{\text{RMG}}^{\text{UB}}$ obtained in [@RMG18]. Also shown is the coherent information lower bound $Q_{\text{CI}}^{\text{LB}}$ defined in . The quantum capacity lies within the shaded region.[]{data-label="fig-compRMG_all"}](QTN_QCap_compRMG_all.pdf){width="\columnwidth"}
See Fig. \[fig-compRMG\] for a comparison of the upper bounds obtained in this paper and the upper bound obtained in [@RMG18] for $N=0.01$ and $N=0.1$. We find that the upper bound $Q_5^{\text{UB}}$ based on approximate degradability is tighter than $Q_{\text{RMG}}^{\text{UB}}$ beyond roughly $\eta=0.56$ for both $N=0.01$ and $N=0.1$, while the data-processing upper bounds $Q_1^{\text{UB}}$ and $Q_2^{\text{UB}}$ are tighter than $Q_{\text{RMG}}^{\text{UB}}$ for all values of $\eta$. In fact, as shown in Fig. \[fig-compRMG\_all\], these data-processing bounds are tighter for all values of $N$. The data-processing upper bounds are thus tighter than the bound in [@RMG18] for the entire parameter range of the qubit thermal channel/GADC. For values of $N$ close to $\frac{1}{2}$, the Rains information upper bound $Q_8^{\text{UB}}$ is tighter than both data-processing upper bounds for all values of $\eta$.
Bounds on the two-way-assisted quantum and private capacities {#sec:two-way-q-cap-bounds}
=============================================================
In this section, we consider the two-way assisted quantum and private capacities $Q^{\leftrightarrow}(\mathcal{A}_{\gamma,N})$ and $P^{\leftrightarrow}(\mathcal{A}_{\gamma,N})$, respectively, of the GADC.
Two-way assisted communication capacities are defined as the highest achievable rate of communication for protocols involving local operations by the sender and receiver and classical communication in both directions between the sender and receiver [@BGPSSW96; @BDSW] (see also [@TGW14a]). As in the unassisted case, we have that $Q^{\leftrightarrow}(\mathcal{N})\leq P^{\leftrightarrow}(\mathcal{N})$ for all quantum channels $\mathcal{N}$.
Since any one-way, or unassisted, communication protocol is a special case of a two-way assisted communication protocol, we immediately have the lower bound $Q^{\leftrightarrow}(\mathcal{N})\geq I_{\text{c}}(\mathcal{N})$. Another known lower bound is the reverse coherent information [@HHH00; @DW05; @DJKR06], which is defined as $$I_{\text{rc}}(\mathcal{N})\equiv \max_{\rho} I_{\text{rc}}(\rho,\mathcal{N}),\label{eq:reverse-coh-inf}$$ where $$I_{\text{rc}}(\rho,\mathcal{N})\equiv H(\rho)-H(\mathcal{N}^c(\rho)).$$
The reverse coherent information as in was defined in [@HHH00] and shown in [@HHH00; @DW05] to be a lower bound on the two-way assisted quantum capacity. It was proven to be additive in [@DJKR06], and concavity in the input state $\rho$ was shown in [@MH06 Eq. (8.48)].
Squashed entanglement upper bounds
----------------------------------
The squashed entanglement [@CW04] (see also [@RT99; @RT02]) of a bipartite state $\rho_{AB}$ is defined as $$E_{\textnormal{sq}}(A;B)_\rho=\frac{1}{2}\inf\{I(A;B|E)_\omega:{\operatorname{Tr}}_E[\omega_{ABE}]=\rho_{AB}\},
\label{eq:squashed-E}$$ where $$\label{eq-QCMI}
\begin{aligned}
I(A;B|E)&\equiv H(A|E)+H(B|E)-H(AB|E)\\
&=H(AE)+H(BE)-H(E)-H(ABE)
\end{aligned}$$ is the quantum conditional mutual information. Whether the infimum in can be replaced with a minimum is one of the outstanding challenges in quantum information theory.
An alternative way of writing the squashed entanglement is to use the fact that for any extension $\omega_{ABE}$ of a state $\rho_{AB}$ there exists a channel $\mathcal{S}$ acting on a purification ${| \psi \rangle}_{ABE'}$ such that $\mathcal{S}_{E'\to E}({| \psi \rangle}{\langle \psi |}_{ABE'})=\omega_{ABE}$. This leads to the following alternative expression for $E_{\text{sq}}(A;B)_\rho$: $$\begin{gathered}
E_{\text{sq}}(A;B)_\rho\\=\frac{1}{2}\inf_{\mathcal{S}}\{I(A;B|E)_\omega:\omega_{ABE}=\mathcal{S}_{E'\to E}({| \psi \rangle}{\langle \psi |}_{ABE'})\},
\end{gathered}$$ where ${| \psi \rangle}_{ABE'}$ is a purification of $\rho_{AB}$. The channels $\mathcal{S}$ over which we optimize are called *squashing channels*.
The squashed entanglement of a channel $\mathcal{N}$ [@TGW14a; @TGW14b] is defined as $$E_{\textnormal{sq}}(\mathcal{N})\equiv\max_{\phi_{AA'}}E_{\textnormal{sq}}(A;B)_{\rho},$$ where $\rho_{AB}=\mathcal{N}_{A'\to B}(\phi_{AA'})$ and where the optimization is over all pure states $\phi_{AA'}$, with $A$ having the same dimension as the dimension of the input system $A'$ of the channel $\mathcal{N}$.
For any channel $\mathcal{N}$, the following bounds hold [@TGW14a; @TGW14b] (see also [@MMW16] for ): $$\begin{aligned}
Q^{\leftrightarrow}(\mathcal{N})&\leq E_{\text{sq}}(\mathcal{N}),\\
P^{\leftrightarrow}(\mathcal{N})&\leq E_{\text{sq}}(\mathcal{N}). \label{eq:2-way-bound-private-cap}
\end{aligned}$$
By taking the identity squashing channel, and using the fact that $I(A;B|E)_\psi=I(A;B)_\rho$ for any pure state $\psi_{ABE}$, where $\rho_{AB}={\operatorname{Tr}}_E[{| \psi \rangle}{\langle \psi |}_{ABE}]$, we get that $E_{\text{sq}}(A;B)_\rho\leq\frac{1}{2}I(A;B)_\rho$ for all states $\rho_{AB}$. This implies that $E_{\text{sq}}(\mathcal{N})\leq\frac{1}{2}\max_{\phi_{AA'}}I(A;B)_{\rho}=\frac{1}{2}I(\mathcal{N})$, where $\rho_{AB}=\mathcal{N}_{A'\to B}(\phi_{AA'})$. In other words, the squashed entanglement of any channel is always bounded from above by half the mutual information of the channel. Therefore, we have $Q^{\leftrightarrow}(\mathcal{N})\leq \frac{1}{2}I(\mathcal{N})$ for all channels $\mathcal{N}$ [@TGW14a; @TGW14b; @GEW16].
Using the expression for the mutual information of the GADC in , we get $$\label{eq-GADC_Q2cap_UB1}
Q^{\leftrightarrow}(\mathcal{A}_{\gamma,N})\leq \frac{1}{2}\max_{z\in[-1,1]}F(\gamma,N,z)\equiv Q_1^{\leftrightarrow,\text{UB}}(\gamma,N)$$ for all $\gamma,N\in(0,1)$.
A potentially better upper bound on the two-way quantum capacity of the GADC than the one in can be obtained by a different choice of squashing channel. In particular, we make use of the decompositions in and to obtain the following result. Our approach is related to the constructions in [@GEW16; @DSW18].
For all $\gamma,N\in(0,1)$, it holds that $$\begin{aligned}
Q^{\leftrightarrow}(\mathcal{A}_{\gamma,N})&\leq P^{\leftrightarrow}(\mathcal{A}_{\gamma,N})\nonumber\\
&\leq \frac{1}{2}\max_{p\in[0,1]}I(A;B|E_1E_2)_{\tau^p}\equiv Q_2^{\leftrightarrow,\textnormal{UB}}(\gamma,N),\label{eq-GADC_Esq_UB1}
\end{aligned}$$ where $$\label{eq-GADC_Esq_UB_pf3}
\tau_{ABE_1E_2}^p=({\operatorname{id}}_{AB}\otimes\mathcal{A}_{\frac{1}{2},0}\otimes\mathcal{A}_{\frac{1}{2},0})({| \psi_p \rangle}{\langle \psi_p |}_{ABE_1'E_2'}),$$ with ${| \psi_p \rangle}_{ABE_1'E_2'}=V_{B'\to BE_2'}^{\gamma N,1}V_{A'\to B'E_1'}^{\frac{\gamma(1-N)}{1-\gamma N},0}{| \theta^p \rangle}_{AA'}$ and ${| \theta^p \rangle}_{AA'}=\sqrt{1-p}{| 0,0 \rangle}_{AA'}+\sqrt{p}{| 1,1 \rangle}_{AA'}$.
Also, $$\begin{aligned}
Q^{\leftrightarrow}(\mathcal{A}_{\gamma,N})&\leq P^{\leftrightarrow}(\mathcal{A}_{\gamma,N})\nonumber\\
&\leq \frac{1}{2}\max_{p\in[0,1]}I(A;B|E_1E_2)_{\tilde{\tau}^p}\equiv Q_3^{\leftrightarrow,\textnormal{UB}}(\gamma,N),\label{eq-GADC_Esq_UB2}
\end{aligned}$$ where $$\tilde{\tau}_{ABE_1E_2}^p=({\operatorname{id}}_{AB}\otimes\mathcal{A}_{\frac{1}{2},0}\otimes\mathcal{A}_{\frac{1}{2},0})({| \tilde{\psi}_p \rangle}{\langle \tilde{\psi}_p |}_{ABE_1'E_2'}),$$ with ${| \tilde{\psi}_p \rangle}_{ABE_1'E_2'}=V_{B'\to BE_2'}^{\gamma(1-N),0}V_{A'\to B'E_1'}^{\frac{\gamma N}{1-\gamma(1-N)},1}{| \theta^p \rangle}_{AA'}$.
We use the fact that $Q^{\leftrightarrow}(\mathcal{A}_{\gamma,N})\leq E_{\text{sq}}(\mathcal{A}_{\gamma,N})$, where $$\label{eq-GADC_Esq_UB_pf1}
E_{\text{sq}}(\mathcal{A}_{\gamma,N})=\frac{1}{2}\max_{\phi_{AA'}}\inf_{\mathcal{S}_{E'\to E}}I(A;B|E)_{\omega},$$ where $\omega_{ABE}=\mathcal{S}_{E'\to E}({| \psi \rangle}{\langle \psi |}_{ABE'})$ and ${| \psi \rangle}_{ABE'}$ is a purification of the state $({\operatorname{id}}_A\otimes\mathcal{A}_{\gamma,N})({| \phi \rangle}{\langle \phi |}_{AA'})$.
To obtain the first upper bound in , we use the fact that $\mathcal{A}_{\gamma,N}$ can be decomposed as $\mathcal{A}_{\gamma,N}=\mathcal{A}_{\gamma N,1}\circ\mathcal{A}_{\frac{\gamma(1-N)}{1-\gamma N},0}$. This means that, for any pure state ${| \phi \rangle}_{AA'}$, a purification of the state $\rho_{AB}\equiv({\operatorname{id}}_A\otimes\mathcal{A}_{\gamma,N})({| \phi \rangle}{\langle \phi |}_{AA'})$ can be written as $$\label{eq-Esq_rho_AB_purif}
{| \psi \rangle}_{ABE_1'E_2'}\equiv V_{B'\to BE_2'}^{\gamma N,1}V_{A'\to B'E_1'}^{\frac{\gamma(1-N)}{1-\gamma N},0}{| \phi \rangle}_{AA'}$$ As the squashing channels, which act on $E_1'$ and $E_2'$, we take the channels $\mathcal{A}_{\gamma_1,N_1}$ and $\mathcal{A}_{\gamma_2,N_2}$, respectively. The state $\omega_{ABE_1E_2}$ on which the quantum conditional mutual information in is evaluated is then $$\label{eq-GADC_Esq_ext}
\omega_{ABE_1E_2}\equiv ({\operatorname{id}}_{AB}\otimes\mathcal{A}_{\gamma_1,N_1}\otimes\mathcal{A}_{\gamma_2,N_2})({| \psi \rangle}{\langle \psi |}_{ABE_1'E_2'}).$$ We can optimize over the open parameters $\gamma_1,N_1,\gamma_2,N_2\in[0,1]$ such that the squashed entanglement of $\rho_{AB}$ can be bounded from above as $$\label{eq-Esq_UB1_opt}
E_{\text{sq}}(A;B)_\rho\leq \frac{1}{2}\min_{\gamma_1,\gamma_2,N_1,N_1}I(A;B|E_1E_2)_\omega,$$ where the state $\omega_{ABE_1E_2}$ is given in . This means that $$E_{\text{sq}}(\mathcal{A}_{\gamma,N})\leq\frac{1}{2}\max_{\phi_{AA'}}\min_{\gamma_1,\gamma_2,N_1,N_2}I(A;B|E_1E_2)_{\omega}.$$ Now, numerical evidence suggests that $\gamma_1=\frac{1}{2}=\gamma_2$ and $N_1=0=N_2$ is optimal. The corresponding squashing channel can be viewed as qubit pure-loss channels with beamsplitters of transmissivity $\frac{1}{2}$, analogous to the construction in [@GEW16; @DSW18]. So we have $$E_{\text{sq}}(\mathcal{A}_{\gamma,N})\leq\frac{1}{2}\max_{\phi_{AA'}}I(A;B|E_1E_2)_{\tau},$$ where $\tau_{ABE_1E_2}=({\operatorname{id}}_{AB}\otimes\mathcal{A}_{\frac{1}{2},0}\otimes\mathcal{A}_{\frac{1}{2},0})({| \psi \rangle}{\langle \psi |}_{ABE_1'E_2'})$. Finally, due to the covariance of the GADC with respect to the Pauli-$z$ operator, it suffices to optimize over pure states ${| \phi \rangle}_{AA'}={| \theta^p \rangle}_{AA'}=\sqrt{1-p}{| 0,0 \rangle}_{AA'}+\sqrt{p}{| 1,1 \rangle}_{AA'}$, where $p\in[0,1]$. In other words, the following equality holds: $$\label{eq-GADC_Esq_UB_pf2}
\frac{1}{2}\max_{\phi_{AA'}}I(A;B|E_1E_2)_{\tau}=\frac{1}{2}\max_{p\in[0,1]}I(A;B|E_1E_2)_{\tau^p},$$ where $\tau_{ABE_1E_2}^p$ is defined in . See Appendix \[app-GADC\_Esq\_UB\_pf\] for a proof. We thus obtain the bound in .
We obtain the second upper bound in using the decomposition $\mathcal{A}_{\gamma,N}=\mathcal{A}_{\gamma(1-N),0}\circ\mathcal{A}_{\frac{\gamma N}{1-\gamma(1-N)},1}$. In this case, we take a purification of the state $\rho_{AB}=({\operatorname{id}}_A\otimes\mathcal{A}_{\gamma,N})({| \phi \rangle}{\langle \phi |}_{AA'})$ to be $${| \tilde{\psi} \rangle}_{ABE_1'E_2'}\equiv V_{B'\to BE_2}^{\gamma(1-N),0}V_{A'\to B'E_1}^{\frac{\gamma N}{1-\gamma(1-N)},1}{| \phi \rangle}_{AA'}.$$ Then, letting $$\tilde{\omega}_{ABE_1E_2}\equiv ({\operatorname{id}}_{AB}\otimes\mathcal{A}_{\gamma_1,N_1}\otimes\mathcal{A}_{\gamma_2,N_2})({| \tilde{\psi} \rangle}{\langle \tilde{\psi} |}_{ABE_1'E_2'})$$ and performing the optimization $\min_{\gamma_1,\gamma_2,N_1,N_2}I(A;B|E_1E_2)_{\tilde{\omega}}$ analogous to the one in , we find numerically that $\gamma_1=\frac{1}{2}=\gamma_2$ and $N_1=0=N_2$ gives the optimal value. Therefore, we get $$E_{\text{sq}}(\mathcal{A}_{\gamma,N})\leq\frac{1}{2}\max_{p\in[0,1]} I(A;B|E_1E_2)_{\tilde{\tau}^p},$$ as required. As with the first upper bound, it suffices to optimize over pure states ${| \theta^p \rangle}_{AA'}$ due to the covariance of the GADC with respect to the Pauli-$z$ operator, and the proof is analogous to the one presented in Appendix \[app-GADC\_Esq\_UB\_pf\] for the first upper bound.
See Fig. \[fig-2Way\_QCap\_bounds\_all\] for a plot of the squashed entanglement upper bounds in and along with the mutual information upper bound $E_{\text{sq}}(\mathcal{A}_{\gamma,N})\leq\frac{1}{2}I(\mathcal{A}_{\gamma,N})$, with $I(\mathcal{A}_{\gamma,N})$ given in . We also plot the reverse coherent information $I_{\text{rc}}(\mathcal{A}_{\gamma,N})$ lower bound. Due to Pauli-$z$ covariance and concavity of the reverse coherent information, $I_{\text{rc}}(\mathcal{A}_{\gamma, N})$ can be obtained by optimizing over diagonal input states, i.e., $$\label{eq-GADC_RCI}
I_{\text{rc}}(\mathcal{A}_{\gamma,N})=\max_{p\in[0,1]}I_{\text{rc}}\left(\begin{pmatrix}1-p&0\\0&p\end{pmatrix},\mathcal{A}_{\gamma,N}\right)\equiv Q_{\text{RCI}}^{\leftrightarrow,\text{LB}}(\gamma,N).$$
Max-Rains and max-relative entropy of entanglement upper bounds
---------------------------------------------------------------
The max-Rains relative entropy of a bipartite state $\rho_{AB}$ is defined as [@WD16pra] (see also [@TWW17]) $$R_{\max}(A;B)_\rho\equiv\min_{\sigma_{AB}\in\text{PPT}'(A:B)}D_{\max}(\rho_{AB}\Vert\sigma_{AB}),$$ where, as stated before, the set $\text{PPT}'(A\!:\!B)$ is defined as [@AMVW02] $$\text{PPT}'(A\!:\!B)\equiv \{\sigma_{AB}:\sigma_{AB}\geq 0,~{\lVert\sigma_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B}\rVert}_1\leq 1\},$$ and the max-relative entropy $D_{\max}(\rho_{AB}\Vert\sigma_{AB})$ is defined as [@Datta08] $$\label{eq-D_max}
D_{\max}(\rho_{AB}\Vert\sigma_{AB})=\log_2\min_t\{t:\rho_{AB}\leq t\sigma_{AB}\}.$$ The max-Rains information $R_{\max}(\mathcal{N})$ of a channel $\mathcal{N}$ is defined as [@WFD18] (see also [@TWW17]) $$\label{eq-max_Rains_channel}
R_{\max}(\mathcal{N})\equiv \max_{\phi_{AA'}}R_{\max}(A;B)_{\rho},$$ where $\rho_{AB}=\mathcal{N}_{A'\to B}(\phi_{AA'})$, and the optimization is over pure states $\phi_{AA'}$, with the dimension of $A$ the same as that of the input system $A'$ of the channel $\mathcal{N}$. It satisfies [@BW18] $$\label{eq-max_Rains_bound}
Q^{\leftrightarrow}(\mathcal{N})\leq R_{\max}(\mathcal{N}).$$ Furthermore, it is a strong converse rate. As shown in [@WFD18], it holds that $$\label{eq-R_max}
\begin{aligned}
R_{\max}(\mathcal{N})&=\log_2\Delta(\mathcal{N}),\\
\Delta(\mathcal{N})&=\left\{\begin{array}{l l} \text{min.} & {\lVert{\operatorname{Tr}}_B[V_{AB}+Y_{AB}]\rVert}_\infty \\ \text{subject to} & Y_{AB}\geq 0, V_{AB}\geq 0,\\ & (V_{AB}-Y_{AB})^{{{\scriptscriptstyle\mathsf{T}}}_B}\geq \Gamma_{AB}^{\mathcal{N}}, \end{array}\right.
\end{aligned}$$ where $\Gamma_{AB}^{\mathcal{N}}$ is the Choi matrix of the channel $\mathcal{N}$, and ${\lVertX\rVert}_{\infty}$ denotes the spectral norm of the matrix $X$, which is defined as the largest singular value of $X$. In particular, the quantity $\Delta(\mathcal{N})$ is given by an SDP.
For the two-way assisted private capacity $P^{\leftrightarrow}(\mathcal{A}_{\gamma,N})$, we consider the following general strong converse upper bound [@CMH17]: $$P^{\leftrightarrow}(\mathcal{N})\leq E_{\text{max}}(\mathcal{N}),$$ which holds for any channel $\mathcal{N}$. The quantity $E_{\max}(\mathcal{N})$ is the max-relative entropy of entanglement of $\mathcal{N}$, which is defined as [@CMH17] $$E_{\max}(\mathcal{N})\equiv \max_{\phi_{AA'}}E_{\max}(A;B)_{\rho},$$ where $\rho_{AB}=\mathcal{N}_{A'\to B}(\phi_{AA'})$, and the optimization is over pure states $\phi_{AA'}$, with the dimension of $A$ equal to the dimension of the input system $A'$ of the channel $\mathcal{N}$. The max-relative entropy of entanglement $E_{\max}(A;B)_{\rho}$ of any bipartite state $\rho_{AB}$ is defined as [@Datta08] $$E_{\max}(A;B)_{\rho}\equiv \min_{\sigma_{AB}\in\text{SEP}(A:B)}D_{\max}(\rho_{AB}\Vert\sigma_{AB}),$$ where $\text{SEP}(A\!:\!B)$ is the set of separable states acting on the space $\mathcal{H}_A\otimes\mathcal{H}_B$. It has been shown in [@BW18] that, for qubit-to-qubit channels, the quantity $E_{\max}(\mathcal{N})$ can be written as the solution to an SDP as follows: $$\label{eq-E_max_SDP_primal}
\begin{aligned}
E_{\max}(\mathcal{N})&=\log_2 \Sigma(\mathcal{N}),\\
\Sigma(\mathcal{N})&=\left\{\begin{array}{l l}\text{min}. & {\lVert{\operatorname{Tr}}_B[Y_{AB}]\rVert}_{\infty} \\
\text{subject to} & \Gamma_{AB}^{\mathcal{N}}\leq Y_{AB},\\[0.1cm]
& Y_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B}\geq 0. \end{array}\right.
\end{aligned}$$ Using the fact that $\text{PPT}\subset\text{PPT}'$, we obtain $R_{\max}(A;B)_{\rho}\leq E_{\max}(A;B)_{\rho}$ for all states $\rho_{AB}$, which implies that $$\label{eq-Rmax_Emax_ineq}
R_{\max}(\mathcal{N})\leq E_{\max}(\mathcal{N})$$ for any quantum channel $\mathcal{N}$.
For the amplitude damping channel $\mathcal{A}_{\gamma,0}$, it has been shown in [@RKB+18 Proposition 2] that $$E_{\max}(\mathcal{A}_{\gamma,0})=\log_2(2-\gamma). \label{eq:E_max-GADC}$$ We now generalize this formula to all values of $\gamma,N$ for the GADC. We also prove that the inequality opposite to the one in holds for the GADC. As stated, this result generalizes the equality in , and the proof that we give is arguably simpler than that given for [@RKB+18 Proposition 2].
\[prop-GADC\_Emax\] For all $\gamma,N$ such that the GADC $\mathcal{A}_{\gamma,N}$ is not entanglement breaking, it holds that $$\begin{gathered}
\label{eq-GADC_Emax_Rmax}
E_{\max}(\mathcal{A}_{\gamma,N})=R_{\max}(\mathcal{A}_{\gamma,N})\\=\log_2\left(1-\frac{\gamma}{2}+\frac{1}{2}\sqrt{(\gamma(2N-1))^2+4(1-\gamma)}\right).
\end{gathered}$$ If the GADC is entanglement breaking, as given by , then $E_{\max}(\mathcal{A}_{\gamma,N})=R_{\max}(\mathcal{A}_{\gamma,N})=0$.
See Appendix \[app-GADC\_Emax\].
By , and using Proposition \[prop-GADC\_Emax\], we have that $$\begin{gathered}
Q^{\leftrightarrow}(\mathcal{A}_{\gamma,N}),P^{\leftrightarrow}(\mathcal{A}_{\gamma,N})\leq Q_4^{\leftrightarrow,\text{UB}}\\
\equiv \log_2\left(1-\frac{\gamma}{2}+\frac{1}{2}\sqrt{(\gamma(2N-1))^2+4(1-\gamma)}\right).
\end{gathered}$$ for all $\gamma,N\in[0,1]$. In Fig. \[fig-2Way\_QCap\_bounds\_all\], we compare this max-Rains upper bound with the squashed entanglement upper bounds from the previous subsection. We observe that the max-Rains upper bound is tight when the channel is entanglement breaking. This is due to the fact that the state $\rho_{AB}$ for which $R_{\text{max}}(A;B)_{\rho}$ is evaluated in is separable whenever the channel is entanglement breaking, and the fact that any separable state is in the set $\text{PPT}'$.
Approximate covariance upper bounds
-----------------------------------
In [@KW17], the following bounds on the two-way assisted capacities were established for a channel $\mathcal{N}$ that is $\varepsilon$-approximately covariant (see Sec. \[subsubsec-CCap\_UB\_EB\_cov\] for the definition): $$\begin{aligned}
Q^{\leftrightarrow}(\mathcal{N}) & \leq R(A;B)_{\rho}+2\varepsilon\log_{2}d_{B}+g(\varepsilon),\\
P^{\leftrightarrow}(\mathcal{N}) & \leq E_{R}(A;B)_{\rho}+2\varepsilon\log_{2}d_{B}+g(\varepsilon),
\end{aligned}$$ where $\rho_{AB}=\mathcal{N}_{A'\rightarrow B}^{G}(\Phi_{AA'}^+)$ and the twirled channel $\mathcal{N}_{A'\rightarrow B}^{G}$ is defined in . Applying these bounds to the GADC, recalling from that $\mathcal{A}_{\gamma,N}^G=\mathcal{A}_{\gamma,\frac{1}{2}}$, and using the fact that the quantity $R(A;B)_{\rho}$ coincides with $E_R(A;B)_{\rho}$ for qubit-qubit states $\rho_{AB}$ [@AS08 Section III], these bounds reduce to the following: $$\begin{aligned}
Q^{\leftrightarrow}(\mathcal{A}^{\gamma,N})&, P^{\leftrightarrow}(\mathcal{A}^{\gamma,N})\leq Q_5^{\leftrightarrow,\text{UB}}(\gamma,N)\nonumber\\
&\equiv E_{R}(A;B)_{\rho}+2\varepsilon_{\text{cov}}+g(\varepsilon_{\text{cov}})\label{eq:approx-tele-sim-bnd},
\end{aligned}$$ where $\varepsilon_{\text{cov}}\equiv\varepsilon_{\text{cov}}(\mathcal{A}_{\gamma,N})=\gamma\left|N-\frac{1}{2}\right|$ and $$\begin{aligned}
\rho_{AB}^{\gamma}&\equiv\mathcal{A}_{\gamma,\frac{1}{2}}(\Phi_{AA'}^+)\\
& =\frac{1}{2} \begin{pmatrix}
1-\frac{\gamma}{2} & 0 & 0 & \sqrt{1-\gamma}\\
0 & \frac{\gamma}{2} & 0 & 0\\
0 & 0 & \frac{\gamma}{2} & 0\\
\sqrt{1-\gamma} & 0 & 0 & 1-\frac{\gamma}{2} \end{pmatrix}.
\end{aligned}$$ Note that, due to , $\rho_{AB}^{\gamma}$ is entangled only when $0\leq\gamma<2(\!\sqrt{2}-1)$. In this case, it is a Bell-diagonal state of the form: $$\rho_{AB}^{\gamma}=\sum_{i,j=0}^1 r_{i,j}{| \Phi_{i,j} \rangle}{\langle \Phi
_{i,j} |}_{AB},$$ with ${| \Phi_{i,j} \rangle}_{AB}\equiv\left(\mathbbm{1}_{A}\otimes \sigma_x^{i}\sigma_z^{j}\right){| \Phi^+ \rangle}_{AB}$ and $$\begin{aligned}
r_{0,0} & =\frac{1}{4}\left( 2+2\sqrt{1-\gamma}-\gamma\right) \\
r_{0,1} & =\frac{1}{4}\left( 2-2\sqrt{1-\gamma}-\gamma\right) ,\\
r_{1,0} & =r_{11}=\frac{\gamma}{4}.
\end{aligned}$$ The closest separable state for such a Bell-diagonal state with $r_{0,0}\geq\frac{1}{2}$ is well known to have the form [@VPRK97] (see also [@AS08]) $$\begin{aligned}
\sigma_{AB}& =\frac{1}{2}{| \Phi_{0,0} \rangle}{\langle \Phi_{0,0} |}_{AB}\nonumber\\
&\qquad +\frac
{1}{2(1-r_{0,0})}\sum_{i,j\neq\left(0,0\right)}r_{i,j}{| \Phi_{i,j} \rangle}{\langle \Phi_{i,j} |}_{AB}\\
& = \begin{pmatrix}
\frac{1}{2}-x & 0 & 0 & x\\
0 & x & 0 & 0\\
0 & 0 & x & 0\\
x & 0 & 0 & \frac{1}{2}-x
\end{pmatrix},
\end{aligned}$$ where $$x=\frac{\gamma}{2\left( 2-2\sqrt{1-\gamma}+\gamma\right) }.$$ We then find that $$\begin{gathered}
\label{eq-GADC_REE}
E_{R}(A;B)_{\rho}\\
=\sum_{i,j=0}^1 r_{i,j}\log_{2}r_{i,j}+1-\frac{\gamma}{2}\log_{2}\left(\frac{\gamma}{2-2\sqrt{1-\gamma}+\gamma}\right) \\
+\frac{\gamma-2+2\sqrt{1-\gamma}}{4}\log_{2}\left( \frac{4-\gamma-4\sqrt{1-\gamma}}{8+\gamma}\right) ,
\end{gathered}$$ which completes the analytic form of the bound in . Note that this formula for $E_R(A;B)_{\rho}$ holds only for $\gamma\in [0, 2(\!\sqrt{2}-1))$; otherwise, $\rho_{AB}^{\gamma}$ is separable, which means that $E_R(A;B)_{\rho}=0$.
In Fig. \[fig-2Way\_QCap\_bounds\_all\], we plot the bound $Q_5^{\leftrightarrow,\text{UB}}$ in . While the bound is relatively poor for small values of $N$, for values of $N$ close to $\frac{1}{2}$ we find that it is tighter than the other upper bounds for some values of $\gamma$. Notably, at $N=\frac{1}{2}$, this upper bound is the tightest among the other upper bounds, and by a significant margin as well.
Conclusion {#sec:conclusion}
==========
In this work, we provided an information-theoretic study of the generalized amplitude damping channel (GADC), which is a generalized form of the well-known amplitude damping channel and can be thought of as the qubit analogue of the bosonic thermal channel. We first determined the range of parameters for which the channel is entanglement breaking, as well as the range of parameters for which it is anti-degradable.
We then established several upper bounds on the classical capacity of the GADC. We used the concepts of approximate covariance and approximate entanglement-breakability [@LKDW18] to obtain upper bounds. We compared these upper bounds with known SDP-based upper bounds [@WXD18], for which we proved an analytical formula for the GADC, as well as the known entanglement-assisted classical capacity upper bound [@LM07].
We also provided several upper bounds on the quantum and private capacities of the GADC. We exploited the two decompositions of the GADC in and in terms of amplitude damping channels in order to obtain data-processing upper bounds, and we used the concepts of approximate degradability and approximate anti-degradability [@SSWR17] to obtain further upper bounds. We found that one of the data-processing upper bounds is tighter than the recently obtained upper bound from [@RMG18] for all parameter values of the GADC, and that the Rains information upper bound is tighter than the upper bound from [@RMG18] for certain parameter regimes.
We also considered the two-way assisted quantum and private capacities of the GADC. We determined upper bounds on these capacities using the squashed entanglement [@TGW14a; @TGW14b], the max-Rains information [@BW18], and the max-relative entropy of entanglement [@CMH17]. The squashed entanglement upper bounds exploited the decompositions of the GADC in and , as well as a particular choice of squashing channel. This allowed us to obtain upper bounds that are better than the mutual information bound that can be obtained via the identity squashing channel. We also obtained upper bounds using the concept of approximate covariance. Along the way, we also determined an analytic form for both the max-Rains information $R_{\max}$ and the max-relative entropy of entanglement $E_{\max}$ of the GADC, and we found that for the GADC both quantities are equal to each other. In light of the latter result, it is worth exploring whether the equality $R_{\max}(\mathcal{N})=E_{\max}(\mathcal{N})$ holds for all qubit-to-qubit channels $\mathcal{N}$.
Obtaining the communication capacities of the GADC for its entire parameter range remains a challenging open problem. This work has applied many state-of-the-art techniques to obtain upper bounds, and it is clear that obtaining tighter upper bounds, or even to obtain an exact expression for the capacity, will require new techniques. To this end, some directions for future work include: employing a different squashing channel than the one used here to obtain a better upper bound on the two-way assisted quantum and private capacities of the GADC. Another method to reduce the gap between lower and upper bounds for any communication scenario is to look at improving current lower bounds rather than upper bounds, via potential superadditivity effects.
All authors acknowledge support from the National Science Foundation. Also, SK acknowledges the NSERC PGS-D, and MMW the Office of Naval Research.
Proof of Proposition \[prop:concavity-Rains\] {#proof-prop:concavity-Rains}
=============================================
The proof is similar in spirit to [@TWW17 Proposition 2], and in fact implies it for the relative entropy. Let $\psi_{AA^{\prime}}^{0}$ and $\psi_{AA^{\prime}}^{1}$ be pure states and define$$\psi_{A^{\prime}}^{\lambda}\equiv\left( 1-\lambda\right) \psi_{A^{\prime}}^{0}+\lambda\psi_{A^{\prime}}^{1},$$ for $\lambda\in\left[ 0,1\right] $. A purification of $\psi_{A^{\prime}}^{\lambda}$ is given by$${| \psi \rangle}_{PAA'}^{\lambda}\equiv\sqrt{1-\lambda}|0\rangle_{P}|\psi^{0}\rangle_{AA^{\prime}}+\sqrt{\lambda}|1\rangle_{P}|\psi^{1}\rangle_{AA^{\prime}}.$$ This purification is related to another purification $\phi_{AA^{\prime}}^{\lambda}$ by an isometric channel $\mathcal{U}_{A\rightarrow PA}$: $\psi_{PAA^{\prime}}^{\lambda}=\mathcal{U}_{A\rightarrow PA}(\phi_{AA^{\prime
}}^{\lambda})$. Let $\sigma_{AB}^{\lambda}\in\text{PPT}'(A\!:\!B)$ be the operator such that $R(\mathcal{N}_{A'\to B}(\phi_{AA'}^{\lambda}))\equiv R(A;B)_{\rho^{\lambda}}=D(\mathcal{N}_{A'\to B}(\phi_{AA'}^{\lambda})\Vert\sigma_{AB}^{\lambda})$, where $\rho_{AB}^{\lambda}=\mathcal{N}_{A'\to B}(\phi_{AA'}^{\lambda})$, and define $\xi_{PAB}^{\lambda}=\mathcal{U}_{A\rightarrow PA}(\sigma_{AB}^{\lambda})$. Observe that $\xi_{PAB}^{\lambda}\in\text{PPT}'(PA\!:\!B)$. Let $$\overline{\Delta}_{P}(\xi_{PAB}^{\lambda})=q|0\rangle\langle0|_{P}\otimes\tau_{AB}^{0}+\left( 1-q\right) |1\rangle\langle1|_{P}\otimes\tau_{AB}^{1},$$ where $\overline{\Delta}_{P}$ is a completely dephasing channel, defined as$$\begin{aligned}
\overline{\Delta}_{P}(\cdot) & \equiv|0\rangle\langle0|_{P}(\cdot)|0\rangle\langle
0|_{P}+|1\rangle\langle1|_{P}(\cdot)|1\rangle\langle1|_{P},\\
q & \equiv\operatorname{Tr}[\left( |0\rangle\langle0|_{P}\otimes
\mathbbm{1}_{AB}\right) \xi_{PAB}^{\lambda}],\\
\tau_{AB}^{0} & \equiv\frac{1}{q}\operatorname{Tr}_{P}[\left( |0\rangle
\langle0|_{P}\otimes \mathbbm{1}_{AB}\right) \xi_{PAB}^{\lambda}],\\
\tau_{AB}^{1} & \equiv\frac{1}{1-q}\operatorname{Tr}_{P}[\left(
|1\rangle\langle1|_{P}\otimes \mathbbm{1}_{AB}\right) \xi_{PAB}^{\lambda}].\end{aligned}$$ Note that the states $\tau_{AB}^{0}$ and $\tau_{AB}^{1}$ are in the set PPT$^{\prime}(A\!:\!B)$ since $\xi_{PAB}^{\lambda}$ is in $\text{PPT}'(PA\!:\!B)$. Then we have that$$\begin{aligned}
& R(\mathcal{N}_{A^{\prime}\rightarrow B}(\phi_{AA^{\prime}}^{\lambda
}))\nonumber\\
& =D(\mathcal{N}_{A^{\prime}\rightarrow B}(\phi_{AA^{\prime}}^{\lambda})\Vert\sigma_{AB}^{\lambda})\\
& =D(\mathcal{N}_{A^{\prime}\rightarrow B}(\psi_{PAA^{\prime}}^{\lambda})\Vert\xi_{PAB}^{\lambda})\\
& \geq D(\overline{\Delta}_{P}(\mathcal{N}_{A^{\prime}\rightarrow B}(\psi_{PAA^{\prime}}^{\lambda}))\Vert\overline{\Delta}_P(\xi_{PAB}^{\lambda}))\\
& =D(\mathcal{N}_{A^{\prime}\rightarrow B}(\overline{\Delta}_{P}(\psi_{PAA^{\prime}}^{\lambda}))\Vert\overline{\Delta}_{P}(\xi_{PAB}^{\lambda}))\\
& =\left( 1-\lambda\right) D(\mathcal{N}_{A^{\prime}\rightarrow B}(\psi_{AA^{\prime}}^{0})\Vert\tau_{AB}^{0}) +\lambda D(\mathcal{N}_{A^{\prime}\rightarrow B}(\psi_{AA^{\prime}}^{1})\Vert\tau_{AB}^{1})\notag \\
& \qquad +D(\left\{ 1-\lambda,\lambda\right\} \Vert\left\{
q,1-q\right\} )\\
& \geq\left( 1-\lambda\right) D(\mathcal{N}_{A^{\prime}\rightarrow B}(\psi_{AA^{\prime}}^{0})\Vert\tau_{AB}^{0})\nonumber\\
& \qquad +\lambda D(\mathcal{N}_{A^{\prime}\rightarrow B}(\psi_{AA^{\prime}}^{1})\Vert\tau_{AB}^{1})\\
& \geq\left( 1-\lambda\right) R(\mathcal{N}_{A^{\prime}\rightarrow B}(\psi_{AA^{\prime}}^{0}))+\lambda R(\mathcal{N}_{A^{\prime}\rightarrow B}(\psi_{AA^{\prime}}^{1})).\end{aligned}$$ The second equality follows from the isometric invariance of the relative entropy. The first inequality follows from the data processing property of relative entropy. The fourth equality follows from the identity [@W17 Exercise 11.8.8]$$D(\rho_{XB}\Vert\sigma_{XB})=\sum_{x}p(x)D(\rho_{B}^{x}\Vert\sigma_{B}^{x})+D(p\Vert r),$$ holding for classical-quantum states$$\begin{aligned}
\rho_{XB} & =\sum_{x}p(x)|x\rangle\langle x|_{X}\otimes\rho_{B}^{x},\\
\sigma_{XB} & =\sum_{x}r(x)|x\rangle\langle x|_{X}\otimes\sigma_{B}^{x}.\end{aligned}$$ Note that $D(p\Vert r)$ denotes the classical relative entropy of the probability distributions $p$ and $r$. For binary probability distributions such that $p(0)=1-\lambda$, $p(1)=\lambda$, $r(0)=1-q$, $r(1)=q$, we let $D(\left\{1-\lambda,\lambda\right\}\Vert\left\{1-q,q\right\})\equiv D(p\Vert r)$. The second inequality follows from the non-negativity of the relative entropy. The final inequality follows because the Rains relative entropy involves a minimization over all states in PPT$^{\prime}(A:B)$.
A proof for the concavity statement for the relative entropy of entanglement $E_{R}(A;B)_{\omega}$ is identical, except replacing PPT$^{\prime}(A\!:\!B)$ with SEP$(A\!:\!B)$.
Proof of Lemma \[lem-anti\_degrad\_chan\_bd\] {#app-GADC_anti_degrade}
=============================================
Let $\E^*$, $E_0$, and $E_1$ be as defined in the statement of Lemma \[lem-anti\_degrad\_chan\_bd\]. Let $V_{A\to BE}^{\gamma,N}$ be the isometric extension of the GADC defined in , and define the pure state $$\begin{aligned}
{| \psi \rangle}_{ABE}^{\gamma,N}&\equiv (\mathbbm{1}_A\otimes V_{A'\to BE}^{\gamma,N}){| \Phi^+ \rangle}_{AA'}\\
&=\frac{1}{\sqrt{2}}\left(\sqrt{1-N}{| 0,0,0 \rangle}_{ABE}+\sqrt{N(1-\gamma)}{| 0,0,2 \rangle}_{ABE}\right.\nonumber\\
&\quad \left.+\sqrt{N\gamma}{| 0,1,3 \rangle}_{ABE}+\sqrt{(1-\gamma)(1-N)}{| 1,1,0 \rangle}_{ABE}\right.\nonumber\\
&\quad \left.+\sqrt{N}{| 1,1,2 \rangle}_{ABE}+\sqrt{\gamma(1-N)}{| 1,0,1 \rangle}_{ABE}\right)
\end{aligned}$$ Then, $\rho_{AB}^{\gamma,N}\equiv{\operatorname{Tr}}_E[{| \psi \rangle}{\langle \psi |}_{ABE}^{\gamma,N}]$ is the Choi state of the GADC $\mathcal{A}_{\gamma,N}$, while $\rho_{AE}^{\gamma,N}\equiv {\operatorname{Tr}}_B[{| \psi \rangle}{\langle \psi |}_{ABE}^{\gamma,N}]$ is the Choi state of the complementary channel $\mathcal{A}_{\gamma,N}^c$ as defined in . In order to prove that $\mathcal{E}_N^*\circ\mathcal{A}_{\gamma,N}^c=\mathcal{A}_{1-\gamma,N}$, it suffices to show that $(\mathcal{E}_N^*)_{E\to B'}(\rho_{AE}^{\gamma,N})=\rho_{AB}^{1-\gamma,N}$. In other words, it suffices to show that the Choi state of the complementary channel $\mathcal{A}_{\gamma,N}^c$ is mapped to the Choi state of the channel $\mathcal{A}_{1-\gamma,N}$.
We have $$\begin{aligned}
\rho_{AB}^{\gamma,N}&=\frac{1}{2}\left((1-\gamma N){| 0,0 \rangle}{\langle 0,0 |}_{AB}+\sqrt{1-\gamma}{| 0,0 \rangle}{\langle 1,1 |}_{AB}\right.\\
&\qquad\left.+\sqrt{1-\gamma}{| 1,1 \rangle}{\langle 0,0 |}_{AB}+\gamma N{| 0,1 \rangle}{\langle 0,1 |}_{AB}\right.\\
&\qquad\left.+\gamma(1-N){| 1,0 \rangle}{\langle 1,0 |}_{AB}\right.\\
&\qquad\left.+(1-\gamma(1-N)){| 1,1 \rangle}{\langle 1,1 |}_{AB}\right).
\end{aligned}$$ Let an isometric extension of the channel $\mathcal{E}_N^*$ be $$W^{\mathcal{E}_N^*}_{E\to B'E'}=E_0\otimes{| 0 \rangle}_{E'}+E_1\otimes{| 1 \rangle}_{E'}.$$ Then, $$\begin{aligned}
{| \phi \rangle}_{ABB'E'}^{\gamma,N}&\equiv W^{\mathcal{E}_N^*}_{E\to B'E'}{| \psi \rangle}_{ABE}^{\gamma,N}\\
&=\frac{1}{\sqrt{2}}\left(\sqrt{1-N}{| 0,0,0,0 \rangle}_{ABB'E'}\right.\\
&\qquad\left.+\sqrt{N(1-\gamma)}{| 0,0,1,1 \rangle}_{ABB'E}\right.\\
&\qquad\left.+\sqrt{N\gamma}{| 0,1,0,1 \rangle}_{ABB'E'}\right.\\
&\qquad\left.+\sqrt{(1-\gamma)(1-N)}{| 1,1,0,0 \rangle}_{ABB'E'}\right.\\
&\qquad\left.+\sqrt{N}{| 1,1,1,1 \rangle}_{ABB'E'}\right.\\
&\qquad\left.+\sqrt{\gamma(1-N)}{| 1,0,1,0 \rangle}_{ABB'E'}\right).
\end{aligned}$$ Then, $$\begin{aligned}
&{\operatorname{Tr}}_{BE'}[{| \phi \rangle}{\langle \phi |}_{ABB'E'}^{\gamma,N}]=(\mathcal{E}_N^*)_{E\to B'}(\rho_{AE}^{\gamma,N})\\
&=\frac{1}{2}\left((1-(1-\gamma)N){| 0,0 \rangle}{\langle 0,0 |}_{AB'}+\sqrt{\gamma}{| 0,0 \rangle}{\langle 1,1 |}_{AB'}\right.\\
&\qquad\left.+\sqrt{\gamma}{| 1,1 \rangle}{\langle 0,0 |}_{AB'}+N(1-\gamma){| 0,1 \rangle}{\langle 0,1 |}_{AB'}\right.\\
&\qquad\left.+(1-\gamma)(1-N){| 1,0 \rangle}{\langle 1,0 |}_{AB'}\right.\\
&\qquad\left.+(N+\gamma(1-N)){| 1,1 \rangle}{\langle 1,1 |}_{AB'}\right)\\
&=\rho_{AB'}^{1-\gamma,N},
\end{aligned}$$ as required.
Proof of Proposition \[prop-C\_beta\] {#app-C_beta}
=====================================
We start by recalling the convex decomposition of the GADC as stated in : $$\mathcal{A}_{\gamma,N}=(1-N)\mathcal{A}_{\gamma,0}+N\mathcal{A}_{\gamma,1}$$ for all $\gamma,N\in[0,1]$. We also recall from that $$\label{eq-GADC_N_symmetry_2}
\mathcal{A}_{\gamma,1}(\rho)=\sigma_x\mathcal{A}_{\gamma,0}(\sigma_x\rho\sigma_x)\sigma_x$$ for all $\gamma\in[0,1]$. Next, note that it follows from that the quantity $\beta(\mathcal{N})$ in the definition of $C_\beta(\mathcal{N})$ is convex in the channel $\N$: for any two channels $\mathcal{N}_1$ and $\mathcal{N}_2$ and any $\lambda\in[0,1]$, $$\beta(\lambda\mathcal{N}_1+(1-\lambda)\mathcal{N}_2)\leq \lambda\beta(\mathcal{N}_1)+(1-\lambda)\beta(\mathcal{N}_2).$$ Furthermore, $\beta(\mathcal{N})$ is invariant under pre- and post-processing of the channel $\mathcal{N}$ by unitaries. Therefore, $$\begin{aligned}
C_\beta(\mathcal{A}_{\gamma,N})&=C_{\beta}((1-N)\mathcal{A}_{\gamma,0}+N\mathcal{A}_{\gamma,1})\\
&\leq (1-N)C_{\beta}(\mathcal{A}_{\gamma,0})+NC_{\beta}(\mathcal{A}_{\gamma,1})\\
&=C_{\beta}(\mathcal{A}_{\gamma,0}),
\end{aligned}$$ where to obtain the last line we used and the invariance of $C_{\beta}$ under pre- and post-processing of the given channel by unitaries to find that $C_{\beta}(\mathcal{A}_{\gamma,1}) = C_{\beta}(\mathcal{A}_{\gamma,0})$.
Given the facts above, our proof strategy is as follows. First, we provide an upper bound of $1+\sqrt{1-\gamma}$ for the SDP in in the case $N=0$, i.e., for the amplitude damping channel, which establishes that $C_{\beta}(\mathcal{A}_{\gamma,N})\leq\log_2(1+\sqrt{1-\gamma})$. Next, we consider the SDP dual to the one in and prove that $1+\sqrt{1-\gamma}$ is a lower bound on it. By strong duality, it follows that $C_{\beta}(\mathcal{A}_{\gamma,N})=\log_2(1+\sqrt{1-\gamma})$ for all $\gamma,N\in\left[0,1\right]$.
We first recall from that $$\label{eq:c_beta_quantity}
\beta(\mathcal{N})=\left\{\begin{array}{l l}\text{min.} & {\operatorname{Tr}}[S_B]\\
\text{subject to} & -R_{AB}\leq \left(\Gamma_{AB}^{\mathcal{N}}\right)^{{{\scriptscriptstyle\mathsf{T}}}_B}\leq R_{AB},\\
& -\mathbbm{1}_A\otimes S_B\leq R_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B}\leq\mathbbm{1}_A\otimes S_B,
\end{array}\right.$$ where the optimization is with respect to the Hermitian operators $S_{B}$ and $R_{AB}$. Note that it follows from the above constraints that $S_{B},R_{AB}\geq 0$.
As a matrix in the standard basis, the Choi matrix for the amplitude damping channel is (see ) $$\Gamma_{AB}^{\gamma,0}=2\rho_{AB}^{\gamma,0}= \begin{pmatrix}
1 & 0 & 0 & \sqrt{1-\gamma}\\
0 & 0 & 0 & 0\\
0 & 0 & \gamma & 0\\
\sqrt{1-\gamma} & 0 & 0 & 1-\gamma
\end{pmatrix},$$ so that the partial transpose is given by $$(\Gamma_{AB}^{\gamma,0})^{{{\scriptscriptstyle\mathsf{T}}}_B}= \begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 0 & \sqrt{1-\gamma} & 0\\
0 & \sqrt{1-\gamma} & \gamma & 0\\
0 & 0 & 0 & 1-\gamma
\end{pmatrix}.$$ Let us choose the operators $R_{AB}$ and $S_B$ to be $$\begin{aligned}
R_{AB} & = \begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 1-\gamma+a & a & 0\\
0 & a & 1+a & 0\\
0 & 0 & 0 & 1-\gamma
\end{pmatrix},\\
S_{B} & = \begin{pmatrix}
1+a & 0\\
0 & 1-\gamma+a
\end{pmatrix},
\end{aligned}$$ where $a=\frac{1}{2}(\sqrt{1-\gamma}-\left( 1-\gamma\right))$. We first check that the constraint $-R_{AB}\leq \left(\Gamma_{AB}^{\gamma,0}\right)^{{{\scriptscriptstyle\mathsf{T}}}_B}\leq R_{AB}$ is satisfied. Consider that $$R_{AB}-\left(\Gamma_{RB}^{\gamma,0}\right)^{{{\scriptscriptstyle\mathsf{T}}}_B}= \begin{pmatrix}
0 & 0 & 0 & 0\\
0 & b & -b & 0\\
0 & -b & b & 0\\
0 & 0 & 0 & 0
\end{pmatrix},$$ where $$b=\frac{1}{2}\left(\sqrt{1-\gamma}+\left( 1-\gamma\right)\right).$$ Due to the inequality $b\geq 0$ for all $\gamma\in\left[0,1\right] $ and the fact that $\begin{pmatrix}1 & -1\\-1 & 1 \end{pmatrix} \geq 0$, it follows that $R_{AB}-\left(\Gamma_{RB}^{\gamma,0}\right)^{{{\scriptscriptstyle\mathsf{T}}}_B}\geq 0$. We also have that $$\label{eq-C_beta_pf}
R_{AB}+\left(\Gamma_{RB}^{\gamma,0}\right)^{{{\scriptscriptstyle\mathsf{T}}}_B}= \begin{pmatrix}
2 & 0 & 0 & 0\\
0 & b & \sqrt{1-\gamma}+a & 0\\
0 & \sqrt{1-\gamma}+a & 1+\gamma+a & 0\\
0 & 0 & 0 & 2\left( 1-\gamma\right)
\end{pmatrix}.$$ To determine whether $R_{AB}+\left(\Gamma_{RB}^{\gamma,0}\right)^{{{\scriptscriptstyle\mathsf{T}}}_B}\geq0$, it is clear that we can focus on the inner $2\times 2$ matrix. For the cases $\gamma=0$ or $\gamma=1$, one can directly confirm the condition $R_{AB}+\left(\Gamma_{RB}^{\gamma,0}\right)^{{{\scriptscriptstyle\mathsf{T}}}_B}\geq 0$. A general $2\times 2$ matrix is positive definite if and only its trace and determinant are strictly positive. The trace of the inner $2\times 2$ matrix in is $$\sqrt{1-\gamma}+1+\gamma>0$$ for all $\gamma\in(0,1)$, and its determinant is $$\left( 2-\gamma\right) \left( \sqrt{1-\gamma}-(1-\gamma)\right)>0$$ for all $\gamma\in(0,1)$. It thus follows that $R_{AB}+\left(\Gamma_{AB}^{\gamma,0}\right)^{{{\scriptscriptstyle\mathsf{T}}}_B}>0$ for all $\gamma \in\left( 0,1\right) $.
We now check the conditions $-\mathbbm{1}_{A}\otimes S_{B}\leq R_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B}\leq \mathbbm{1}_{A}\otimes S_{B}$. Consider that $$\begin{aligned}
\mathbbm{1}_{A}\otimes S_{B} & = \begin{pmatrix}
1+a & 0 & 0 & 0\\
0 & 1-\gamma+a & 0 & 0\\
0 & 0 & 1+a & 0\\
0 & 0 & 0 & 1-\gamma+a
\end{pmatrix},\\
R_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B} & = \begin{pmatrix}
1 & 0 & 0 & a\\
0 & 1-\gamma+a & 0 & 0\\
0 & 0 & 1+a & 0\\
a & 0 & 0 & 1-\gamma
\end{pmatrix}.
\end{aligned}$$ Then $$\mathbbm{1}_{R}\otimes S_{B}-R_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B}= \begin{pmatrix}
a & 0 & 0 & -a\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
-a & 0 & 0 & a
\end{pmatrix}.$$ Due to the fact that $a\geq 0$ for all $\gamma\in\left[0,1\right] $, it follows that $\mathbbm{1}_{A}\otimes S_{B}-R_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B}\geq0$. We also need to consider $$\begin{gathered}
\mathbbm{1}_{R}\otimes S_{B}+R_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B}\\= \begin{pmatrix}
a+2 & 0 & 0 & a\\
0 & 2a+2\left( 1-\gamma\right) & 0 & 0\\
0 & 0 & 2a+2 & 0\\
a & 0 & 0 & a+2\left( 1-\gamma\right)
\end{pmatrix}.
\end{gathered}$$ We have that $2a+2\left( 1-\gamma\right) \geq0$ and $2a+2\geq0$ for all $\gamma\in\left[ 0,1\right] $. Thus, to determine whether $\mathbbm{1}_{A}\otimes S_{B}+R_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B}\geq 0$, it is clear that we can focus on the “corners” $2\times 2$ submatrix: $$ \begin{pmatrix}
a+2 & a\\
a & a+2\left( 1-\gamma\right)
\end{pmatrix}.$$ For $\gamma=0$ or $\gamma=1$, one can directly confirm that this corners submatrix is positive semi-definite. For $\gamma\in\left(0,1\right)$, the trace of the corners submatrix is $$3-\gamma+\sqrt{1-\gamma}>0,$$ and its determinant is given by $$\left( 1-\gamma\right) \left( 2+\gamma\right) +\left( 2-\gamma\right) \sqrt{1-\gamma}>0$$ for all $\gamma\in(0,1)$. It thus follows that $\mathbbm{1}_{A}\otimes S_{B}+R_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B}>0$ for all $\gamma \in\left( 0,1\right) $. Thus, the proposed operators $R_{AB}$ and $S_{B}$ satisfy the given constraints in , and we conclude that $$\begin{aligned}
\beta(\mathcal{A}_{\gamma,0}) & \leq{\operatorname{Tr}}[S_{B}]\\
& =1+a+1-\gamma+a\\
& =2+2a-\gamma\\
& =2+2\frac{1}{2}\left(\sqrt{1-\gamma}-\left( 1-\gamma\right)\right)-\gamma\\
& =1+\sqrt{1-\gamma}.
\end{aligned}$$ By the arguments presented at the beginning of the proof, we thus conclude that $$\label{eq:c_beta-GADC-lower-bnd}
C_{\beta}(\mathcal{A}_{\gamma,N})\leq\log_{2}(1+\sqrt{1-\gamma})$$ for all $\gamma,N\in\left[ 0,1\right] $.
The SDP dual to the one in is given by $$\label{eq:c-beta-dual}
\hat{\beta}(\mathcal{N})\equiv\left\{\begin{array}{l l} \text{max.} & {\operatorname{Tr}}[\Gamma_{AB}^{\mathcal{N}}(K_{AB}-M_{AB})^{{{\scriptscriptstyle\mathsf{T}}}_B}]\\
\text{subject to} & K_{AB}+M_{AB}\leq (E_{AB}-F_{AB})^{{{\scriptscriptstyle\mathsf{T}}}_B},\\
& E_B+F_B\leq\mathbbm{1}_B,\\
& K_{AB},M_{AB},E_{AB},F_{AB}\geq 0. \end{array}\right.$$ From we have that the Choi matrix for the GADC is $$\Gamma_{AB}^{\gamma,N}= \begin{pmatrix}
1-\gamma N & 0 & 0 & \sqrt{1-\gamma}\\
0 & \gamma N & 0 & 0\\
0 & 0 & \gamma\left( 1-N\right) & 0\\
\sqrt{1-\gamma} & 0 & 0 & 1-\gamma\left( 1-N\right)
\end{pmatrix}.$$ Let us now make the following choice for the operators $K_{AB},M_{AB},E_{AB},F_{AB}$: $$\begin{aligned}
K_{AB} & =\frac{1}{2} \begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 1 & 0\\
0 & 1 & 1 & 0\\
0 & 0 & 0 & 1
\end{pmatrix},\qquad M_{AB}=0,\\
E_{AB} & =\frac{1}{2} \begin{pmatrix}
1 & 0 & 0 & 1\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
1 & 0 & 0 & 1
\end{pmatrix},\qquad F_{AB}=0.
\end{aligned}$$ We find that $K_{AB}=E_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B}$ and $E_{B}=\mathbbm{1}_{B}$, so that the constraints in are satisfied and $$\begin{aligned}
{\operatorname{Tr}}[\Gamma_{AB}^{\gamma,N}(K_{AB}-M_{AB})^{{{\scriptscriptstyle\mathsf{T}}}_B}] & ={\operatorname{Tr}}[\Gamma_{AB}^{\gamma,N}K_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B}]\\
& ={\operatorname{Tr}}[\Gamma_{AB}^{\gamma,N}E_{AB}].
\end{aligned}$$ We find that
$$\begin{aligned}
\Gamma_{AB}^{\gamma,N}E_{AB}=\frac{1}{2} \begin{pmatrix}
\sqrt{1-\gamma}-N\gamma+1 & 0 & 0 & \sqrt{1-\gamma}-N\gamma+1\\
0 & N\gamma & 0 & 0\\
0 & 0 & -\gamma\left( N-1\right) & 0\\
\sqrt{1-\gamma}+\gamma\left( N-1\right) +1 & 0 & 0 & \sqrt{1-\gamma}+\gamma\left( N-1\right) +1
\end{pmatrix},
\end{aligned}$$
so that $${\operatorname{Tr}}[\Gamma_{AB}^{\gamma,N}E_{AB}]=1+\sqrt{1-\gamma}.$$ This implies that $C_{\hat{\beta}}(\mathcal{A}_{\gamma,N})\equiv\log_2\hat{\beta}(\mathcal{A}_{\gamma,N})\geq\log_{2}(1+\sqrt{1-\gamma})$. By strong duality, it holds that $C_{\beta}(\mathcal{A}_{\gamma,N})=C_{\hat{\beta}}(\mathcal{A}_{\gamma,N})$. Therefore, $$\label{eq:c_beta-GADC-upper-bnd}
C_{\beta}(\mathcal{A}_{\gamma,N})\geq\log_{2}(1+\sqrt{1-\gamma}),$$ for all $\gamma, N \in [0,1]$. Putting together and , we obtain $C_{\beta}(\mathcal{A}_{\gamma,N})=\log_2(1+\sqrt{1-\gamma})$, as required.
Let us now show that $C_{\zeta}(\mathcal{A}_{\gamma,N})=\log_2(1+\sqrt{1-\gamma})$ for all $\gamma,N\in[0,1]$. Recall from that $$\label{eq-C_zeta_primal_2}
\zeta(\mathcal{N})=\left\{\begin{array}{l l} \text{min.} & {\operatorname{Tr}}[S_B] \\
\text{subject to} & V_{AB}\geq\Gamma_{AB}^{\mathcal{N}},\\
& -\mathbbm{1}_A\otimes S_B\leq V_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B}\leq \mathbbm{1}_A\otimes S_B. \end{array}\right.$$
The inequality $C_{\zeta}(\mathcal{A}_{\gamma,0})\leq\log_2(1+\sqrt{1-\gamma})$ has been proven in [@WXD18 Theorem 14]. By inspecting the SDP in , it is clear that the quantity $\zeta(\mathcal{N})$ is convex in the channel $\mathcal{N}$. Furthermore, it is invariant under unitary pre- and post-processing. Thus, proceeding in a way similar to the proof of the upper bound $C_{\beta}(\mathcal{A}_{\gamma,N})\leq\log_2(1+\sqrt{1-\gamma})$ above, we find that $$\begin{aligned}
\zeta(\mathcal{A}_{\gamma,N})&=\zeta((1-N)\mathcal{A}_{\gamma,0}+N\mathcal{A}_{\gamma,1})\\
&\leq (1-N)\zeta(\mathcal{A}_{\gamma,0})+N\zeta(\mathcal{A}_{\gamma,1})\\
&=(1-N)\zeta(\mathcal{A}_{\gamma,0})+N\zeta(\mathcal{A}_{\gamma,0})\\
&=\zeta(\mathcal{A}_{\gamma,0})\\
&\leq 1+\sqrt{1-\gamma},
\end{aligned}$$ from which we conclude that $$\label{eq-C_zeta_pf}
C_{\zeta}(\mathcal{A}_{\gamma,N})\leq\log_2(1+\sqrt{1-\gamma})$$ for all $\gamma,N\in[0,1]$.
To arrive at the opposite inequality, consider that the SDP dual to the one in is given by $$\hat{\zeta}(\mathcal{N})\equiv\left\{\begin{array}{l l} \text{max.} & {\operatorname{Tr}}[K_{AB}\Gamma_{AB}^{\mathcal{N}}] \\
\text{subject to} & {\operatorname{Tr}}_A[E_{AB}+F_{AB}]\leq\mathbbm{1}_B,\\
& K_{AB}\leq (E_{AB}-F_{AB})^{{{\scriptscriptstyle\mathsf{T}}}_B},\\
& K_{AB},E_{AB},F_{AB}\geq 0,\end{array}\right.$$ where the optimization is with respect to the operators $K_{AB},E_{AB},F_{AB}$. Now, for the GADC, let us make the following choice for $K_{AB}$, $E_{AB}$, $F_{AB}$: $$\begin{aligned}
K_{AB}&=\frac{1}{2}\begin{pmatrix} 1 & 0 & 0 & 1\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \end{pmatrix},\\
E_{AB}&=\frac{1}{2}\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix},\\
F_{AB}&=0.
\end{aligned}$$ Then, we find that the conditions ${\operatorname{Tr}}_A[E_{AB}+F_{AB}]\leq\mathbbm{1}_B$ and $K_{AB}\leq (E_{AB}-F_{AB})^{{{\scriptscriptstyle\mathsf{T}}}_B}$ are satisfied with equality. Now,
$$K_{AB}\Gamma_{AB}^{\gamma,N}=\frac{1}{2}\begin{pmatrix} \sqrt{1-\gamma}-N\gamma+1 & 0 & 0 & \sqrt{1-\gamma}-\gamma(1-N)+1 \\ 0 & N\gamma & 0 & 0 \\ 0 & 0 & \gamma(1-N) & 0 \\ \sqrt{1-\gamma}-N\gamma+1 & 0 & 0 & \sqrt{1-\gamma}-\gamma(1-N)+1 \end{pmatrix},$$
so that taking the trace yields $${\operatorname{Tr}}[K_{AB}\Gamma_{AB}^{\gamma,N}]=1+\sqrt{1-\gamma}.$$ We thus conclude that $$\begin{aligned}
C_{\hat{\zeta}}(\mathcal{A}_{\gamma,N})&\equiv \log_2\hat{\zeta}(\mathcal{A}_{\gamma,N})\\
&\geq \log_2(1+\sqrt{1-\gamma}).
\end{aligned}$$ By strong duality, it holds that $C_{\zeta}(\mathcal{A}_{\gamma,N})=C_{\hat{\zeta}}(\mathcal{A}_{\gamma,N})$ for all $\gamma,N\in[0,1]$. Therefore, we have that $C_{\zeta}(\mathcal{A}_{\gamma,N})\geq \log_2(1+\sqrt{1-\gamma})$, and combining this with means that $C_{\zeta}(\mathcal{A}_{\gamma,N})=\log_2(1+\sqrt{1-\gamma})$, as required.
Covariance parameter for the GADC {#app-GADC_cov_parameter}
=================================
Using the definition of the diamond norm in , we can write the quantity $\varepsilon_{\text{cov}}(\mathcal{A}_{\gamma,N})$ as $$\varepsilon_{\text{cov}}(\mathcal{A}_{\gamma,N})=\frac{1}{2}\max_{\psi_{RA}}{\left\lVert(\mathcal{A}_{\gamma,N}-\mathcal{A}_{\gamma,\frac{1}{2}})(\psi_{RA})\right\rVert}_1.$$
We first show that the maximum is achieved by taking ${| \psi \rangle}_{RA}$ to be the maximally entangled state, i.e., taking ${| \psi \rangle}_{RA}={| \Phi^+ \rangle}_{RA}=\frac{1}{\sqrt{2}}({| 0,0 \rangle}_{RA}+{| 1,1 \rangle}_{RA})$. We do this by making use of [@LKDW18 Lemma II.3]. Let ${| \psi \rangle}_{RA}$ be an arbitrary pure state, and let $\rho_A\coloneqq{\operatorname{Tr}}_R[\psi_{RA}]$. We take the group $G=\mathbb{Z}_2\times\mathbb{Z}_2$ and the Pauli operators $\{\mathbbm{1},\sigma_x,\sigma_y,\sigma_z\}$ and note that $$\overline{\rho}_A\coloneqq\frac{1}{4}(\rho_A+\sigma_x\rho_A\sigma_x+\sigma_y\rho_A\sigma_y+\sigma_z\rho_A\sigma_z)=\frac{\mathbbm{1}_A}{2}.$$ Due to this fact, one purification of $\overline{\rho}$ is the maximally entangled state ${| \Phi^+ \rangle}_{RA}$. Therefore, by applying [@LKDW18 Lemma II.3] (with the generalized divergence therein taken to be the trace distance), we obtain $$\begin{aligned}
&{\left\lVert\mathcal{A}_{\gamma,N}(\Phi_{RA}^+)-\mathcal{A}_{\gamma,\frac{1}{2}}(\Phi_{RA}^+)\right\rVert}_{1}\nonumber\\
&\quad \geq \left\lVert\frac{1}{4}\sum_{g\in G}{| g \rangle}{\langle g |}_P\otimes\mathcal{A}_{\gamma,N}^g(\psi_{RA})\right.\nonumber\\
&\quad\qquad\left.-\frac{1}{4}\sum_{g\in G}{| g \rangle}{\langle g |}_P\otimes\mathcal{A}_{\gamma,\frac{1}{2}}^g(\psi_{RA})\right\rVert_1,
\end{aligned}$$ where $\mathcal{A}_{\gamma,N}^g\coloneqq \mathcal{S}_g\circ\mathcal{A}_{\gamma,N}\circ\mathcal{S}_g$, with $\mathcal{S}_g(\cdot)=S_g(\cdot)S_g$ and $S_g\in\{\mathbbm{1},\sigma_x,\sigma_y,\sigma_z\}$. Then, recalling that $$\begin{aligned}
\sigma_x\mathcal{A}_{\gamma,\frac{1}{2}}(\sigma_x(\cdot)\sigma_x)\sigma_x&=\mathcal{A}_{\gamma,\frac{1}{2}}(\cdot),\\
\sigma_z\mathcal{A}_{\gamma,\frac{1}{2}}(\sigma_z(\cdot)\sigma_z)\sigma_z&=\mathcal{A}_{\gamma,\frac{1}{2}}(\cdot),\\
\Rightarrow \sigma_y\mathcal{A}_{\gamma,\frac{1}{2}}(\sigma_y(\cdot)\sigma_y)\sigma_y&=\mathcal{A}_{\gamma,\frac{1}{2}}(\cdot),
\end{aligned}$$ we get that $\mathcal{A}_{\gamma,\frac{1}{2}}^g=\mathcal{A}_{\gamma,\frac{1}{2}}$ for all $g\in G$. Therefore, $$\begin{aligned}
&{\left\lVert\mathcal{A}_{\gamma,N}(\Phi_{RA}^+)-\mathcal{A}_{\gamma,\frac{1}{2}}(\Phi_{RA}^+)\right\rVert}_{1}\\
&\quad\geq {\left\lVert\frac{1}{4}\sum_{g\in G}{| g \rangle}{\langle g |}_P\otimes (\mathcal{A}_{\gamma,N}^g-\mathcal{A}_{\gamma,\frac{1}{2}})(\psi_{RA})\right\rVert}_1\label{eq-GADC_cov_param_pf1}\\
&\quad =\frac{1}{4}\sum_{g\in G}{\left\lVert(\mathcal{A}_{\gamma,N}^g-\mathcal{A}_{\gamma,\frac{1}{2}})(\psi_{RA})\right\rVert}_1,
\end{aligned}$$ where to obtain the last line we used the fact that all of the operators in the sum in are supported on orthogonal spaces. Then, using and , which together imply that $\sigma_y\mathcal{A}_{\gamma,N}(\sigma_y(\cdot)\sigma_y)\sigma_y=\mathcal{A}_{\gamma,1-N}(\cdot)$, we get $$\begin{aligned}
&{\left\lVert\mathcal{A}_{\gamma,N}(\Phi_{RA}^+)-\mathcal{A}_{\gamma,\frac{1}{2}}(\Phi_{RA}^+)\right\rVert}_{1}\\
&\quad\geq \frac{1}{2}{\left\lVert(\mathcal{A}_{\gamma,1-N}-\mathcal{A}_{\gamma,\frac{1}{2}})(\psi_{RA})\right\rVert}_1\\
&\qquad + \frac{1}{2}{\left\lVert(\mathcal{A}_{\gamma,N}-\mathcal{A}_{\gamma,\frac{1}{2}})(\psi_{RA})\right\rVert}_1.
\end{aligned}$$ Next, we use the fact that $\mathcal{A}_{\gamma,N}=(1-N)\mathcal{A}_{\gamma,0}+N\mathcal{A}_{\gamma,1}$ to get that $$\begin{aligned}
& {\left\lVert(\mathcal{A}_{\gamma,N}-\mathcal{A}_{\gamma,\frac{1}{2}})(\psi_{RA})\right\rVert}_1\\
&\quad = {\left\lVert\left(\left(\frac{1}{2}-N\right)\mathcal{A}_{\gamma,0}-\left(N-\frac{1}{2}\right)\mathcal{A}_{\gamma,1}\right)(\psi_{RA})\right\rVert}_1\\
&\quad = \left|N-\frac{1}{2}\right|{\left\lVert(\mathcal{A}_{\gamma,0}-\mathcal{A}_{\gamma,1})(\psi_{RA})\right\rVert}_1,\label{eq-GADC_cov_param_pf3}
\end{aligned}$$ and $$\begin{aligned}
& {\left\lVert(\mathcal{A}_{\gamma,1-N}-\mathcal{A}_{\gamma,\frac{1}{2}})(\psi_{RA})\right\rVert}_1\\
&\quad = {\left\lVert\left(\left(N-\frac{1}{2}\right)\mathcal{A}_{\gamma,0}-\left(\frac{1}{2}-N\right)\mathcal{A}_{\gamma,1}\right)(\psi_{RA})\right\rVert}_1\\
&\quad = \left|N-\frac{1}{2}\right|{\left\lVert(\mathcal{A}_{\gamma,0}-\mathcal{A}_{\gamma,1})(\psi_{RA})\right\rVert}_1\\
&\quad ={\left\lVert(\mathcal{A}_{\gamma,N}-\mathcal{A}_{\gamma,\frac{1}{2}})(\psi_{RA})\right\rVert}_1
\end{aligned}$$ Therefore, $$\begin{aligned}
&{\left\lVert\mathcal{A}_{\gamma,N}(\Phi_{RA}^+)-\mathcal{A}_{\gamma,\frac{1}{2}}(\Phi_{RA}^+)\right\rVert}_{1}\\
&\quad \geq {\left\lVert(\mathcal{A}_{\gamma,N}-\mathcal{A}_{\gamma,\frac{1}{2}})(\psi_{RA})\right\rVert}_1
\end{aligned}$$ for all pure states $\psi_{RA}$, which implies that $$\max_{\psi_{RA}}{\left\lVert(\mathcal{A}_{\gamma,N}-\mathcal{A}_{\gamma,\frac{1}{2}})(\psi_{RA})\right\rVert}_1\leq{\left\lVert\mathcal{A}_{\gamma,N}(\Phi_{RA}^+)-\mathcal{A}_{\gamma,\frac{1}{2}}(\Phi_{RA}^+)\right\rVert}_1.$$ Combined with the inequality $$\max_{\psi_{RA}}{\left\lVert(\mathcal{A}_{\gamma,N}-\mathcal{A}_{\gamma,\frac{1}{2}})(\psi_{RA})\right\rVert}_1\geq {\left\lVert(\mathcal{A}_{\gamma,N}-\mathcal{A}_{\gamma,\frac{1}{2}})(\Phi_{RA}^+)\right\rVert}_1,$$ which holds simply by restricting the maximization to the state $\Phi_{RA}^+$, we obtain $$\label{eq-GADC_cov_param_pf2}
\varepsilon_{\text{cov}}(\mathcal{A}_{\gamma,N})=\frac{1}{2}{\left\lVert(\mathcal{A}_{\gamma,N}-\mathcal{A}_{\gamma,\frac{1}{2}})(\Phi_{RA}^+)\right\rVert}_1$$ for all $\gamma,N\in[0,1]$.
Finally, to calculate the right-hand side of , we observe using that $$\begin{aligned}
&{\left\lVert(\mathcal{A}_{\gamma,N}-\mathcal{A}_{\gamma,\frac{1}{2}})(\Phi_{RA}^+)\right\rVert}_1\\
&\quad = \left|N-\frac{1}{2}\right|{\left\lVert(\mathcal{A}_{\gamma,0}-\mathcal{A}_{\gamma,1})(\Phi_{RA}^+)\right\rVert}_1\\
&\quad = \left|N-\frac{1}{2}\right|{\left\lVert(\mathcal{A}_{\gamma,1}+\mathcal{A}_{\gamma,0}-2\mathcal{A}_{\gamma,0})(\Phi_{RA}^+)\right\rVert}_1\\
&\quad = |2N-1|{\left\lVert\left(\frac{1}{2}\mathcal{A}_{\gamma,1}+\frac{1}{2}\mathcal{A}_{\gamma,0}-\mathcal{A}_{\gamma,0}\right)(\Phi_{RA}^+)\right\rVert}_1\\
&\quad = |2N-1|{\left\lVert(\mathcal{A}_{\gamma,\frac{1}{2}}-\mathcal{A}_{\gamma,0})(\Phi_{RA}^+)\right\rVert}_1\\
&\quad = 2|2N-1|\varepsilon_{\text{cov}}(\mathcal{A}_{\gamma,0}).
\end{aligned}$$ Now, it has been shown in [@LKDW18 Appendix C] that $\varepsilon_{\text{cov}}(\mathcal{A}_{\gamma,0})=\frac{\gamma}{2}$. Therefore, $$\varepsilon_{\text{cov}}(\mathcal{A}_{\gamma,N})=\frac{1}{2}\gamma|2N-1|=\gamma\left|N-\frac{1}{2}\right|,$$ as required.
Proof of Eq. (\[eq-GADC\_Esq\_UB\_pf2\]) {#app-GADC_Esq_UB_pf}
========================================
By restricting the optimization on the right-hand side of to pure states ${| \theta^p \rangle}_{AA'}=\sqrt{1-p}{| 0,0 \rangle}_{AA'}+\sqrt{p}{| 1,1 \rangle}_{AA'}$, we obtain $$\begin{aligned}
\frac{1}{2}\max_{\phi_{AA'}}I(A;B|E_1E_2)_{\tau}&\geq\frac{1}{2}\max_{\theta^p_{AA'}}I(A;B|E_1E_2)_{\tau^p}\\
&=\frac{1}{2}\max_{p\in[0,1]}I(A;B|E_1E_2)_{\tau^p}.\label{eq-Esq_bd_pf}
\end{aligned}$$ The remainder of the proof is dedicated to proving the reverse inequality.
Let $\phi_{AA'}$ be an arbitrary pure state, and let $\rho_{A'}\coloneqq{\operatorname{Tr}}_A[\phi_{AA'}]$. The state $\tau$ on which we evaluate the conditional mutual information on the left-hand side of is given by $$\tau_{ABE_1E_2}=({\operatorname{id}}_{AB}\otimes\mathcal{A}_{\frac{1}{2},0}\otimes\mathcal{A}_{\frac{1}{2},0})({| \psi \rangle}{\langle \psi |}_{ABE_1'E_2'}),$$ where $$\label{eq-Esq_bd_pf9}
{| \psi \rangle}_{ABE_1'E_2'}=V_{B'\to BE_2'}^{\gamma N,1}V_{A'\to B'E_1'}^{\frac{\gamma(1-N)}{1-\gamma N},0}{| \phi \rangle}_{AA'}.$$ Note that the GADC has only two Kraus operators when the second parameter is either zero or one. Consequently, for any $\gamma'\in[0,1]$, we can take the isometric extensions in to be of the following form: $$\begin{aligned}
V^{\gamma',0}&=A_1\otimes{| 0 \rangle}+A_2\otimes{| 1 \rangle},\\
V^{\gamma',1}&=A_3\otimes{| 0 \rangle}+A_4\otimes{| 1 \rangle}.
\end{aligned}$$ By using an isometric extension of the same form for the channel $\mathcal{A}_{\frac{1}{2},0}$, we can write $\tau_{ABE_1E_2}$ explicitly as $$\label{eq-Esq_bd_pf1}
\begin{aligned}
&\tau_{ABE_1E_2}={\operatorname{Tr}}_{F_1F_2}[{| \varphi \rangle}{\langle \varphi |}_{ABE_1E_2F_1F_2}],\\
&{| \varphi \rangle}_{ABE_1E_2F_1F_2}\\
&\quad =\left(V_{E_1'\to E_1F_1}^{\frac{1}{2},0}\otimes V_{E_2'\to E_2F_2}^{\frac{1}{2},0}\right)V_{B'\to BE_2'}^{\gamma N,1}V_{A'\to B'E_1'}^{\frac{\gamma(1-N)}{1-\gamma N},0}{| \phi \rangle}_{AA'},
\end{aligned}$$
Now, the Pauli-$z$ covariance of the GADC is equivalent to the relations $A_1\sigma_z=\sigma_zA_1$, $A_2\sigma_z=-\sigma_zA_2$, $A_3\sigma_z=\sigma_zA_3$, and $A_4\sigma_z=-\sigma_zA_4$. Therefore, writing $V^{\gamma',0}$ as $V^{\gamma',0}=A_1\otimes\sigma_z{| 0 \rangle}-A_2\otimes\sigma_z{| 1 \rangle}$, for any state ${| \psi \rangle}$, we obtain $$\begin{aligned}
V^{\gamma',0}\sigma_z{| \psi \rangle}&=A_1\sigma_z{| \psi \rangle}\otimes\sigma_z{| 0 \rangle}-A_2\sigma_z{| \psi \rangle}\otimes\sigma_z{| 1 \rangle}\\
&=\sigma_zA_1{| \psi \rangle}\otimes\sigma_z{| 0 \rangle}+\sigma_zA_2{| \psi \rangle}\otimes\sigma_z{| 1 \rangle}\\
&=(\sigma_z\otimes\sigma_z)(A_1{| \psi \rangle}\otimes{| 0 \rangle}+A_2{| \psi \rangle}\otimes{| 1 \rangle})\\
&=(\sigma_z\otimes\sigma_z)V^{\gamma',0}.\label{eq-Esq_bd_pf7}
\end{aligned}$$ Similarly, we have $$\label{eq-Esq_bd_pf8}
V^{\gamma',1}\sigma_z{| \psi \rangle}=(\sigma_z\otimes\sigma_z)V^{\gamma',1}{| \psi \rangle}$$ for all states ${| \psi \rangle}$.
Next, we observe that by using the definition of the conditional mutual information in , along with the definition of the conditional entropy, we can write $I(A;B|E_1E_2)_{\tau}$ as $$\begin{aligned}
I(A;B|E_1E_2)_{\tau}&=H(B|E_1E_2)_{\tau}-H(B|E_1E_2A)_{\tau}\\
&=H(B|E_1E_2)_{\varphi}+H(B|F_1F_2)_{\varphi},\label{eq-Esq_bd_pf2}
\end{aligned}$$ where to obtain the last line we used the fact that the state ${| \varphi \rangle}_{ABE_1E_2F_1F_2}$ in is pure; in particular, $$\begin{aligned}
H(B|E_1E_2A)_{\tau}&=H(ABE_1E_2)_{\tau}-H(E_1E_2A)_{\tau}\\
&=H(F_1F_2)_{\varphi}-H(BF_1F_2)_{\varphi}\\
&=-H(B|F_1F_2)_{\varphi}.
\end{aligned}$$ Now, since the right-hand side of does not contain the $A$ system, the quantity is a function solely of the state $\rho_{A'}$. For convenience, let us define a function $F$ by $$F(\rho_{A'})=I(A;B|E_1E_2)_{\tau}=H(B|E_1E_2)_{\varphi}+H(B|F_1F_2)_{\varphi},$$ where
$$\begin{aligned}
\varphi_{BE_1E_2F_1F_2}&\equiv\varphi_{BE_1E_2F_1F_2}(\rho_{A'})\\
&=\left(V^{\frac{1}{2},0}_{E_1'\to E_1F_1}\otimes V^{\frac{1}{2},0}_{E_2'\to E_2F_2}\right)V_{B'\to BE_2'}^{\gamma N,1}V_{A'\to B'E_1'}^{\frac{\gamma(1-N)}{1-\gamma N}}\rho_{A'}\left(V_{A'\to B'E_1'}^{\frac{\gamma(1-N)}{1-\gamma N}}\right)^\dagger \left(V_{B'\to BE_2'}^{\gamma N,1}\right)^\dagger\left(V_{E_1'\to E_1F_1}^{\frac{1}{2},0}\otimes V_{E_2'\to E_2F_2}^{\frac{1}{2},0}\right)^\dagger
\end{aligned}$$
Using the relations in and , we get $$\varphi_{BE_1E_2F_1F_2}(\sigma_z\rho_{A'}\sigma_z)=\sigma_z^{\otimes 5}\varphi_{BE_1E_2F_1F_2}(\rho_{A'})\sigma_z^{\otimes 5},$$ which implies that $F(\sigma_z\rho_{A'}\sigma_z)=F(\rho_{A'})$. Furthermore, since the conditional entropy is concave, so is the function $F$. We thus obtain $$\begin{aligned}
F\left(\frac{1}{2}\rho_{A'}+\frac{1}{2}\sigma_z\rho_{A'}\sigma_z\right)&\geq \frac{1}{2}F(\rho_{A'})+\frac{1}{2}F(\sigma_z\rho_{A'}\sigma_z)\\
&=F(\rho_{A'}).
\end{aligned}$$ Now, observe that the state $\frac{1}{2}\rho_{A'}+\frac{1}{2}\sigma_z\rho_{A'}\sigma_z$ is diagonal in the standard basis, meaning that it has a purification of the form ${| \theta^p \rangle}_{AA'}=\sqrt{1-p}{| 0,0 \rangle}_{AA'}+\sqrt{p}{| 1,1 \rangle}_{AA'}$ for some $p\in[0,1]$, say $p^*$. Therefore, by restricting the optimization $\frac{1}{2}\max_{p\in[0,1]}I(A;B|E_1E_2)_{\tau^p}=\frac{1}{2}\max_{\theta_{AA'}^p}I(A;B|E_1E_2)_{\tau^p}$ to $p^*$, we get $$\begin{aligned}
\frac{1}{2}\max_{p\in[0,1]}I(A;B|E_1E_2)_{\tau^p}&\geq F\left(\frac{1}{2}\rho_{A'}+\frac{1}{2}\sigma_z\rho_{A'}\sigma_z\right)\\
&\geq \frac{1}{2}F(\rho_{A'})\\
&=\frac{1}{2}I(A;B|E_1E_2)_{\tau}.
\end{aligned}$$ Since the state $\rho_{A'}$ was arbitrary, we get that $$\frac{1}{2}\max_{p\in[0,1]}I(A;B|E_1E_2)_{\tau^p}\geq\frac{1}{2}\max_{\phi_{AA'}}I(A;B|E_1E_2)_{\tau}.$$ Combining with the inequality in , we get $$\frac{1}{2}\max_{\phi_{AA'}}I(A;B|E_1E_2)_{\tau}=\frac{1}{2}\max_{p\in[0,1]}I(A;B|E_1E_2)_{\tau^p},$$ as required.
Proof of Proposition \[prop-GADC\_Emax\] {#app-GADC_Emax}
========================================
We start by showing that $$\begin{gathered}
E_{\max}(\mathcal{A}_{\gamma,N})\\=\log_2\left(1-\frac{\gamma}{2}+\frac{1}{2}\sqrt{(\gamma(2N-1))^2+4(1-\gamma)}\right)
\label{eq:E-max-GADC-1}
\end{gathered}$$ for all $\gamma,N$ such that the GADC $\mathcal{A}_{\gamma,N}$ is not entanglement breaking. If the channel $\mathcal{A}_{\gamma,N}$ is entanglement breaking, then the Choi matrix $\Gamma_{AB}^{\gamma,N}$ is separable and PPT, so that we can pick the variable $Y_{AB}$ in the SDP to be $\Gamma_{AB}^{\gamma,N}$, for which we have ${\lVert{\operatorname{Tr}}_B[Y_{AB}]\rVert}_{\infty}=1$. This means that $E_{\max}(\mathcal{A}_{\gamma,N})=0$ in this case. In what follows, we thus assume that $\mathcal{A}_{\gamma,N}$ is not entanglement breaking.
We first establish an upper bound on $\Sigma(\mathcal{A}_{\gamma,N})$ by employing the SDP in . To determine an ansatz for the variable $Y_{AB}$ therein, we first consider the positive partial transpose of the Choi matrix $\Gamma_{AB}^{\gamma,N}$ from : $$\left(\Gamma_{AB}^{\gamma,N}\right)^{{{\scriptscriptstyle\mathsf{T}}}_B}=\begin{pmatrix} 1-\gamma N & 0 & 0 & 0 \\ 0 & \gamma N & \sqrt{1-\gamma} & 0 \\ 0 & \sqrt{1-\gamma} & \gamma(1-N) & 0 \\ 0 & 0 & 0 & 1-\gamma(1-N) \end{pmatrix}.$$ To determine the positive semi-definiteness of this matrix, it suffices to focus on the inner $2\times 2$ matrix, given that $1-\gamma N\geq 0$ and $1-\gamma(1-N)\geq 0$ for all $\gamma,N\in[0,1]$. The eigenvalues of the inner $2\times 2$ matrix are given by $$\lambda_{\pm}\equiv \frac{1}{2}\left(\gamma\pm\sqrt{(\gamma(2N-1))^2+4(1-\gamma)}\right).$$ We have that $\lambda_+\geq 0$ for all $\gamma,N\in[0,1]$. The condition $\lambda_-\leq 0$ is equivalent to the channel not being entanglement breaking. If we add $-\lambda_-\mathbbm{1}$ to the inner $2\times 2$ matrix, then it becomes positive semi-definite. This leads to the following ansatz for the matrix $Y_{AB}$: $$\begin{aligned}
Y_{AB}&=\Gamma_{AB}^{\gamma,N}-\begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & \lambda_- & 0 & 0 \\ 0 & 0 & \lambda_- & 0 \\ 0 & 0 & 0 & 0\end{pmatrix}\\
&=\begin{pmatrix} 1-\gamma N & 0 & 0 & \sqrt{1-\gamma} \\ 0 & \gamma N-\lambda_- & 0 & 0 \\ 0 & 0 & \gamma(1-N)-\lambda_- & 0 \\ \sqrt{1-\gamma} & 0 & 0 & 1-\gamma(1-N) \end{pmatrix}.
\end{aligned}$$ By construction, we have that $$\begin{aligned}
Y_{AB}-\Gamma_{AB}^{\gamma,N}&\geq 0,\\
Y_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B}& \geq 0,
\end{aligned}$$ so that $Y_{AB}$ satisfies the constraints of the SDP in . Now, computing ${\operatorname{Tr}}_B[Y_{AB}]$ gives $${\operatorname{Tr}}_B[Y_{AB}]=\begin{pmatrix} 1-\lambda_- & 0 \\ 0 & 1-\lambda_- \end{pmatrix},$$ which implies that ${\lVert{\operatorname{Tr}}_{B}[Y_{AB}]\rVert}_{\infty}=1-\lambda_-$. Therefore, $$\Sigma(\mathcal{A}_{\gamma,N})\leq \frac{1}{2}\left(2-\gamma+\sqrt{(\gamma(2N-1))^2+4(1-\gamma)}\right).$$
We now establish a lower bound on $\Sigma(\mathcal{A}_{\gamma,N})$ by considering the SDP dual to the one in , namely, $$\label{eq-E_max_SDP_dual}
\hat{\Sigma}(\mathcal{N})\equiv\left\{\begin{array}{l l} \text{max}. & {\operatorname{Tr}}[\Gamma_{AB}^{\mathcal{N}}P_{AB}] \\
\text{subject to} & P_{AB},Q_{AB}\geq 0,\\
& P_{AB}+Q_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B}\leq \rho_A\otimes\mathbbm{1}_B,\\
& \rho_A\geq 0,\\
& {\operatorname{Tr}}[\rho_A]\leq 1.\end{array}\right.$$ By strong duality, it follows that these optimization problems have equal solutions, i.e., $\hat{\Sigma}(\mathcal{N})=\Sigma(\mathcal{N})$ for all quantum channels $\mathcal{N}$.
Now, let $$\begin{aligned}
a&\equiv \sqrt{(\gamma(2N-1))^2+4(1-\gamma)},\\
b&\equiv \frac{a-(2N-1)\gamma}{2a}.
\end{aligned}$$ Note that $b\in[0,1]$ for all $\gamma,N\in[0,1]$. Then, let $$\begin{aligned}
\rho_A&=\begin{pmatrix} b & 0 \\ 0 & 1-b \end{pmatrix},\\
P_{AB}&=\begin{pmatrix} b & 0 & 0 & \frac{1}{a}\sqrt{1-\gamma} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \frac{1}{a}\sqrt{1-\gamma} & 0 & 0 & 1-b \end{pmatrix},\\
Q_{AB}&=\begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & b & -\frac{1}{a}\sqrt{1-\gamma} & 0 \\ 0 & -\frac{1}{a}\sqrt{1-\gamma} & 1-b & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}.
\end{aligned}$$ We have that $\rho_A\geq 0$ and ${\operatorname{Tr}}[\rho_A]=1$ for all $\gamma,N\in[0,1]$. Also, for all $\gamma,N\in[0,1]$, the eigenvalues of the corners submatrix of $P_{AB}$ are equal to zero and one, implying that $P_{AB}\geq 0$. Similarly, for all $\gamma,N\in[0,1]$, the eigenvalues of the inner submatrix of $Q_{AB}$ are equal to zero and one, implying that $Q_{AB}\geq 0$. Furthermore, we have that $$\begin{aligned}
Q_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B}&=\begin{pmatrix} 0 & 0 & 0 & -\frac{1}{a}\sqrt{1-\gamma} \\ 0 & b & 0 & 0 \\ 0 & 0 & 1-b & 0 \\ -\frac{1}{a}\sqrt{1-\gamma} & 0 & 0 & 0 \end{pmatrix},\\
\rho_A\otimes\mathbbm{1}_B&=\begin{pmatrix} b & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ 0 & 0 & 1-b & 0 \\ 0 & 0 & 0 & 1-b \end{pmatrix},
\end{aligned}$$ and so we have that $P_{AB}+Q_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B}\leq\rho_A\otimes\mathbbm{1}_B$ (in fact, this inequality is saturated). Thus, all the constraints in are satisfied. Then, since $$\begin{gathered}
{\operatorname{Tr}}[\Gamma_{AB}^{\gamma,N}P_{AB}]\\=\frac{1}{2}\left(2-\gamma+\sqrt{(\gamma(2N-1))^2+4(1-\gamma)}\right),
\end{gathered}$$ we have that $$\hat{\Sigma}(\mathcal{A}_{\gamma,N})\geq \frac{1}{2}\left(2-\gamma+\sqrt{(\gamma(2N-1))^2+4(1-\gamma)}\right).$$ This means that $$\Sigma(\mathcal{A}_{\gamma,N})=1-\frac{\gamma}{2}+\frac{1}{2}\sqrt{(\gamma(2N-1))^2+4(1-\gamma)},$$ thus establishing .
We now show that $$\begin{gathered}
R_{\max}(\mathcal{A}_{\gamma,N})\\
=\log_2\left(1-\frac{\gamma}{2}+\frac{1}{2}\sqrt{(\gamma(2N-1))^2+4(1-\gamma)}\right).
\end{gathered}$$ Due to the inequality in , namely, $R_{\max}(\mathcal{A}_{\gamma,N})\leq E_{\max}(\mathcal{A}_{\gamma,N})$, it suffices to show that $$R_{\max}(\mathcal{A}_{\gamma,N})\geq \log_2\left(1-\frac{\gamma}{2}+\frac{1}{2}\sqrt{(\gamma(2N-1))^2+4(1-\gamma)}\right)$$ when $\mathcal{A}_{\gamma,N}$ is not entanglement breaking.
When the channel $\mathcal{A}_{\gamma,N}$ is entanglement breaking, then the Choi matrix $\Gamma_{AB}^{\gamma,N}$ is separable and PPT. This means that we can pick $V_{AB}=(\Gamma_{AB}^{\gamma,N})^{{{\scriptscriptstyle\mathsf{T}}}_B}$ and $Y_{AB}=0$ in , for which ${\lVert{\operatorname{Tr}}_B[V_{AB}+Y_{AB}]\rVert}_{\infty}={\lVert{\operatorname{Tr}}_B[V_{AB}]\rVert}_{\infty}=1$, implying that $R_{\max}(\mathcal{A}_{\gamma,N})=0$ in this case. In what follows, we thus assume that $\mathcal{A}_{\gamma,N}$ is not entanglement breaking.
First, the SDP dual to the one in is $$\label{eq:R-max-dual-SDP}
\hat{\Delta}(\mathcal{N})=\left\{\begin{array}{l l} \text{max}. & {\operatorname{Tr}}[\Gamma_{AB}^{\gamma,N}R_{AB}]\\
\text{subject to} & -\rho_A\otimes\mathbbm{1}_B\leq R_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B}\leq \rho_A\otimes\mathbbm{1}_B,\\
& \rho_A\geq 0,
{\operatorname{Tr}}[\rho_A]\leq 1. \end{array}\right.$$ By strong duality, it holds that $\hat{\Delta}(\mathcal{N})=\Delta(\mathcal{N})$.
Let $a\in[0,1]$, which we will specify in more detail later as a function of $\gamma$ and $N$. We pick $$\begin{aligned}
\rho_A&=\begin{pmatrix} a & 0 \\ 0 & 1-a \end{pmatrix},\\
R_{AB}&=\begin{pmatrix} a & 0 & 0 & 2a(1-a) \\ 0 & a(1-2a) & 0 & 0 \\ 0 & 0 & -(1-a)(1-2a) & 0 \\ 2a(1-a) & 0 & 0 & 1-a \end{pmatrix}.
\end{aligned}$$ Note that $\rho_A\geq 0$ and ${\operatorname{Tr}}[\rho_A]=1$. Also, consider that $$\begin{aligned}
R_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B}&=\begin{pmatrix} a & 0 & 0 & 0 \\ 0 & a(1-2a) & 2a(1-a) & 0 \\ 0 & 2a(1-a) & -(1-a)(1-2a) & 0 \\ 0 & 0 & 0 & 1-a \end{pmatrix},\\
\rho_A\otimes\mathbbm{1}_B&=\begin{pmatrix} a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & 1-a & 0 \\ 0 & 0 & 0 & 1-a \end{pmatrix},
\end{aligned}$$ implying that $$R_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B}+\rho_A\otimes\mathbbm{1}_B=2\begin{pmatrix} a & 0 & 0 & 0 \\ 0 & a(1-a) & a(1-a) & 0 \\ 0 & a(1-a) & a(1-a) & 0 \\ 0 & 0 & 0 & 1-a \end{pmatrix},$$ which is positive semi-definite since $a\in[0,1]$. Also, we have that $$\rho_A\otimes\mathbbm{1}_B-R_{AB}^{{{\scriptscriptstyle\mathsf{T}}}_B}=2\begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & a^2 & -a(1-a) & 0 \\ 0 & -a(1-a) & (1-a)^2 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix},$$ which has eigenvalues equal to zero and $2(1-2a(1-a))$, the latter being nonnegative for all $a\in[0,1]$. Thus, our choice of $\rho_A$ and $R_{AB}$ satisfies the constraints in . Now, computing ${\operatorname{Tr}}[\Gamma_{AB}^{\gamma,N}R_{AB}]$, we find that $$\begin{gathered}
{\operatorname{Tr}}[\Gamma_{AB}^{\gamma,N}R_{AB}]= g(a,\gamma,N)\\
\equiv 1-2(1-N)\gamma -2a^2\left(2\sqrt{1-\gamma}+\gamma\right)\\
+4a(\sqrt{1-\gamma}+\gamma(1-N)).
\end{gathered}$$ We now choose $a$ such that the equation $$1-\frac{\gamma}{2}+\frac{1}{2}\sqrt{(\gamma(2N-1))^2+4(1-\gamma)}=g(a,\gamma,N)$$ is satisfied. It has solutions $$\label{eq-R_max_pf1}
a=\frac{c_1\pm \sqrt{c_1^2+c_2((4N-3)\gamma- c_3)}}{c_2},$$ where $$\begin{aligned}
c_1&\equiv 4\left(\sqrt{1-\gamma}+\gamma(1-N)\right),\\
c_2&\equiv 4\left(2\sqrt{1-\gamma}+\gamma\right),\\
c_3&\equiv \sqrt{(\gamma(2N-1))^2+4(1-\gamma)}
\end{aligned}$$ Note that the solutions for $a$ in satisfy $a\in[0,1]$ for all $\gamma,N$ such that the GADC is not entanglement breaking. Thus, for this choice of $a$, we conclude that $$\hat{\Delta}(\mathcal{N})\geq 1-\frac{\gamma}{2}+\frac{1}{2}\sqrt{(\gamma(2N-1))^2+4(1-\gamma)}.$$ We thus have that $$\begin{gathered}
R_{\max}(\mathcal{A}_{\gamma,N})=E_{\max}(\mathcal{A}_{\gamma,N})\\
=1-\frac{\gamma}{2}+\frac{1}{2}\sqrt{(\gamma(2N-1))^2+4(1-\gamma)},
\end{gathered}$$ as required.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Alexander Shekhovtsov, Paul Swoboda, and Bogdan Savchynskyy [^1]'
bibliography:
- 'bib/strings.bib'
- 'bib/persistence-nips2014.bib'
- 'bib/books.bib'
- 'bib/optim.bib'
- 'bib/max-plus-en.bib'
- 'bib/kiev-en.bib'
---
[14]{}(1,0.5) Copyright notice.
[^1]:
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study the observability of the $\tautau$ decay mode of a Higgs boson produced in the $s$-channel at a muon collider. We find that the spin correlations of the $\tautau$ in $\tau\to \pi\nu_{\tau},\ \rho\nu_{\tau}$ decays are discriminative between the Higgs boson signal and the Standard Model background. Observation of the predicted distinctive distribution can confirm the spin-0 nature of the Higgs resonance. The relative coupling strength of the Higgs boson to $b$ and $\tau$ can also be experimentally determined.'
address: |
Department of Physics, University of Wisconsin\
1150 University Avenue, Madison, WI 53706, USA
author:
- 'V. Barger, T. Han and C.-G. Zhou[^1]'
date: 'March, 2000'
title: |
Higgs Boson Decays to $\tau$-pairs\
in the $s$-channel at a Muon Collider
---
\[1\][0= 0=0 1= 1=1 0>1 \#1 / ]{}
\#1\#2\#3\#4\]\#5
Introduction
============
The Higgs boson is a crucial ingredient for electroweak symmetry breaking in the Standard Model (SM) and in supersymmetric (SUSY) theories. In the minimal supersymmetric standard model (MSSM), the mass of the lightest Higgs boson must be less than about 135 GeV [@hmass], and in a typical weakly coupled SUSY theory $m_h$ should be lighter than about 150 GeV [@mass2]. On the experimental side, the non-observation of Higgs signal at the LEP-II experiments has established a lower bound on the SM Higgs boson mass of 106.2 GeV at a 95% Confidence Level (CL) [@95GeV] and future searches at LEP-II may eventually be able to explore a SM Higgs boson with a mass up to 110 GeV. If the Fermilab Tevatron can reach an integrated luminosity of $10-30$ fb$^{-1}$, then it should be possible to observe a Higgs boson with 5$\sigma$ signal for $m_h<130$ GeV and even possibly to 190 GeV with a weaker signal [@run2]. The CERN Large Hadron Collider (LHC) is believed to be able to cover up to the full $m_h$ range of theoretical interest, to about 1000 GeV [@LHC], although it may be challenging to discover a Higgs boson in the “intermediate” mass region 110 GeV $<m_h<$ 150 GeV due to the huge SM background to $h\to b\bar b$ and the requirement of excellent di-photon mass resolution for the $h\to \gamma\gamma$ signal.
Once a Higgs boson is discovered, it will be of major importance to determine its properties to high precision. It has been pointed out that precision measurements of the Higgs mass, width and the primary decay rates such as $h\to b\bar b,\ WW^*$ and $ZZ^*$, can be obtained via the $s$-channel resonant production of a neutral Higgs boson at the first muon collider (FMC) [@FMC]. To determine the Higgs boson couplings and other properties, it is necessary to observe as many decay channels as possible.
A particularly important channel is the $\tautau$ final state $$\mumu \to \tautau.$$ In the SM at tree level, this $s$-channel process proceeds in two ways, via $\gamma/Z$ exchange and Higgs boson exchange. The former involves the SM gauge couplings and presents a characteristic $FB$ (forward-backward in the scattering angle) asymmetry and a $LR$ (left-right in beam polarization) asymmetry; the latter is governed by the Higgs boson couplings to $\mumu,\tautau$ proportional to the fermion masses and is isotropic in phase space due to spin-0 exchange. With the possibility for beam polarizations of a muon collider, the asymmetries were studied in Ref. [@Marciano] to improve the Higgs boson signal to background ratio. The unambiguous establishment of the $\tautau$ signal would allow a determination of the relative coupling strength of the Higgs boson to $b$ and $\tau$ and thus test the usual assumption of $\tau-b$ unification in SUSY GUT theories. The angular distribution would probe the spin property of the Higgs resonance.
In this paper, we propose to make additional use of spin correlations in the final state $\tautau$ events. We will demonstrate the significant difference of spin correlations between the background events from the spin-1 $\gamma/Z$ exchange and the signal events from the spin-0 Higgs exchange. The correlation is particularly strong for the two-body decay modes for $\tau\to \pi\nu_{\tau},\ \rho\nu_{\tau}$. In Sec. II, we analyze the $\tautau$ production and decay and present our results. In sec. III, we provide further discussions on the results and draw our conclusion.
analysis
========
The $s$-channel Higgs boson (spin-0) exchange populates the $\mumu$ helicity combinations of left-left ($LL$) and right-right ($RR$). This results in the correlation of $\tautau$ polarization of $LL$ and $RR$ by angular momentum conservation. In contrast, the SM background channel yields $\tautau$ polarization combination of left-right ($LR$) and right-left ($RL$). By studying the decay products from the correlated and polarized $\tau^\pm$, we can effectively distinguish these two channels and gain information about the spin of the resonance.
Production cross section for $\mu^- \mu^+ \to \tau^- \tau^+$
------------------------------------------------------------
The differential cross section for $\mu^- \mu^+ \to \tau^- \tau^+$ via $s$-channel Higgs ($h$) exchange can be expressed as $${{d\sigma_h(\mumu \to h\to\tautau)}\over{d\cos\theta}}
={1\over 2}{\overline \sigma_h}\ (1+P_-P_+)
\label{higgs}$$ where $\theta$ is the scattering angle between $\mu^-$ and $\tau^-$, $P_\mp$ the percentage longitudinal polarizations of the initial $\mu^\mp$ beams, with $P=-1$ purely left-handed, $P=+1$ purely right-handed and $P=0$ unpolarized. ${\overline\sigma_h}$ is the integrated unpolarized cross section convoluted with the collider energy distribution [@FMC], $${\overline\sigma_h} \approx {4\pi\over m^2_h}\
{B(h\to \mumu)B(h\to\tautau)\over
\left[1+ {8\over \pi}\left({\sigma_{\sqrt s}
\over \Gamma_h}\right)^2\right]^{1/2}}
\label{reson}$$ where $B(h\to\ell^-\ell^+)$ is the Higgs decay branching fraction and $\Gamma_h$ is the total width. The Gaussian rms spread in the beam energy $\sqrt s$ is given by $$\sigma_{\sqrt s}^{} = {R\over \sqrt 2}{\sqrt s},
\label{sigmas}$$ with $R$ the energy resolution of each beam, anticipated in the range $R\sim 0.05\% - 0.005\%$. Note that for a very narrow Higgs boson, like that of the SM for $m_h< 140$ GeV, the cross section in Eq. (\[reson\]) is proportional to ${\Gamma_h/\sigma_{\sqrt s}}$.
The unpolarized cross section for the SM background can be written as $${d\sigma_{SM}^{}(\mumu \to \gamma^*/Z^*\to\tautau)
\over {d\cos\theta}}={3\over 8}\sigma_{QED}^{}\
[A(1+\cos^2\theta)+B\cos\theta] ,
\label{bkgrnd}$$ where $\sigma_{QED}^{}$ is the QED cross section for $\mumu \to \gamma^*\to\tautau$ and the coefficients $A$ and $B$ are functions of the c. m. energy and gauge couplings [@SM]. The interference between the vector current and the axial-vector current leads to a forward-backward asymmetry characterized by $${A_{FB}} =
{{\int_0^1d\cos\theta(d\sigma/d\cos\theta)-
\int_{-1}^0d\cos\theta(d\sigma/d\cos\theta)}
\over {\int_{-1}^1d\cos\theta(d\sigma/d\cos\theta)}}
={3\over 8}{B \over A}\ .
\label{FB-asym}$$ Furthermore, the chiral neutral current couplings lead to a left-right asymmetry which can be characterized by $A_{LR}$ defined as $$A_{LR}^{}={{\sigma_{LR\to LR+RL}^{} - \sigma^{}_{RL\to LR+RL}}
\over{\sigma^{}_{LR\to LR+RL}+\sigma^{}_{RL\to LR+RL}}} \ .
\label{LR-asym}$$ Again with longitudinal polarizations $P_\mp$ for the $\mu^\mp$ beams, the differential cross section for the SM background is $${d\sigma_{SM}^{} \over
d\cos\theta}={3\over8}\sigma_{QED}^{} A
[1-P_+P_-+(P_+-P_-)A_{LR}](1+\cos^2\theta+
{8\over 3} \cos\theta A^{eff}_{FB}).
\label{eff-bkgrnd}$$ Here the effective $FB$ asymmetry factor is $$A^{eff}_{FB}={{A_{FB}+P_{eff}A_{LR}^{FB}} \over {1+P_{eff}A_{LR}}},
\label{eff-FB}$$ the effective polarization is $$P_{eff}= {{P_+-P_-}\over{1-P_+P_-}},
\label{effP}$$ and $$A_{LR}^{FB}= {{\sigma_{LR + RL \to LR}-\sigma_{LR + RL
\to RL}}\over{\sigma_{LR + RL \to LR}+\sigma_{LR + RL \to RL}}}\ .
\label{factors}$$ For the case of interest where initial and final state particles are leptons, $A_{LR}=A_{LR}^{FB}$.
From the cross section formulas of Eqs. (\[higgs\]) and (\[eff-bkgrnd\]), the enhancement factor of the signal-to-background ratio ($S/B$) due to the beam polarization effects is $${S\over B} \sim {1+P_-P_+\over 1-P_-P_+ +(P_+-P_-)A_{LR}}\ .
\label{pmu}$$
The normalized differential cross section for $\mumu \to \tautau$ at $\sqrt s =m_h= 120$ GeV is shown in Fig. [\[one\]]{} for both the SM $\gamma/Z$ exchange (solid curve) and a scalar $h$ exchange (dashed line). We see that the SM distribution exhibits a clear forward-backward asymmetry; while the scalar exchange is flat, as expected. Calculation shows that at this c. m. energy, the SM process yields $A_{FB}\sim 0.7$ while $A_{LR}\sim 0.15$. Using the initial polarized beam and the forward-backward asymmetry to improve the precision measurement has been discussed in [@Marciano].
$\tau$ decay and final state spin correlation
---------------------------------------------
$\tau$ decay modes $\mu\bar\nu_\mu\nu_\tau$ $e\bar\nu_e \nu_{\tau}$ $ \pi\nu_\tau$ $\rho\nu_{\tau}$ $a_1 \nu_\tau $
-------------------------------- -------------------------- ------------------------- ---------------- ------------------ -----------------
branching fraction $B_i\ (\%)$ 17.37 17.81 11.08 25.02 18.38
: $\tau$ decay modes and branching fractions from Ref. [@databook]. []{data-label="taudecay"}
As noted previously, the final state polarization configurations of $\tautau$ from the Higgs signal and the SM background are very different. The $\tau$ decay modes and their branching fractions ($B_i$) are listed in Table I. The vector and axial vector resonances $\rho$ and $a_1$ subsequently decay into $2\pi$ and $3\pi$ respectively and the vector meson masses can be reconstructed from the final state poins. There is always a charged track to define a kinematical distribution for the decay. In the $\tau$-rest frame, the normalized differential decay rate can be written as $${1\over \Gamma}{d\Gamma_i \over {d\cos\theta}}=
{B_i\over 2} (a_i+b_iP_\tau \cos\theta)
\label{tau-decay}$$ where $\theta$ is the angle between the momentum direction of the charged decay product in the $\tau$-rest frame [@TauDecay] and the $\tau$-momentum direction, $B_i$ is the branching fraction listed in Table I, and $P_\tau=\pm 1$ is the $\tau$ helicity. For the two-body decay modes, $a_i$ and $b_i$ are constant and given by $$\begin{aligned}
&& a_\pi=b_\pi=1,\\
&& a_i=1\quad {\rm and}\quad b_i=-{m_\tau^2-2m_i^2\over m_\tau^2+2m_i^2}\quad
{\rm for}\quad i=\rho,\ a_1^{}.\end{aligned}$$ For the three-body leptonic decays, the $a_{e,\mu}^{}$ and $b_{e,\mu}^{}$ are not constant for a given three-body kinematical configuration and are obtained by the integration over the energy fraction carried by the invisible neutrinos. One can quantify the event distribution shape by defining a “sensitivity” ratio parameter $$r_i= {b_i\over a_i}.
\label{sens}$$ For the two-body decay modes, the sensitivities are $r_\pi=1,\ r_\rho=0.45$ and $r_{a_1}=0.007$. The $\tau\to a_1\nu_\tau$ mode is consequently less useful in connection with the $\tau$ polarization study. As to the three-body leptonic modes, although experimentally readily identifiable, the energy smearing from the decay makes it hard to reconstruct the $\tautau$ final state spin correlation.
The differential distribution for the two charged particles ($i,j$) in the final state from $\tautau$ decays respectively can be expressed as $$\begin{aligned}
{d\sigma \over {d\cos\theta_id\cos\theta_j}}
\sim \sum_{P_\tau=\pm 1} {B_{i}B_{j}\over 4}\ (a_i+b_iP_{\tau^-} \cos\theta_i)
(a_j+b_jP_{\tau^+}\cos\theta_j),\end{aligned}$$ where $\cos\theta_i\ (\cos\theta_j)$ is defined in $\tau^-\ (\tau^+)$ rest frame as in Eq. (\[tau-decay\]). For the Higgs signal channel, $\tautau$ helicities are correlated as $LL\ (P_{\tau^-}=P_{\tau^+}=-1)$ and $RR\ (P_{\tau^-}=P_{\tau^+}=+1)$. This yields the spin-correlated differential cross section $$\begin{aligned}
{d\sigma_h \over {d\cos\theta_id\cos\theta_j}}
= (1+P_-P_+)\sigma_h\ {B_{i}B_{j}\over 4}\
[a_ia_j+b_ib_j\cos\theta_i\cos\theta_j],
\label{LL&RR}\end{aligned}$$ where the factor $(1+P_-P_+)$ comes from the initial $\mu^\mp$ beam polarization; $\sigma_h$ is the unpolarized total cross section. We expect that the distribution reaches maximum near $\cos\theta_i=\cos\theta_j=\pm 1$ and minimum near $\cos\theta_i=-\cos\theta_j=\pm 1$. How significant the peaks are depends on the sensitivity parameter in Eq. (\[sens\]). Here we simulate the double differential distribution of Eq. (\[LL&RR\]) for $\mumu\to h \to \tautau \to \rho^-\nu_\tau\rho^+\bar \nu_\tau$ and the result is shown in Fig. \[two\]. Here we take $\sqrt s=m_h=120$ GeV for illustration. The Higgs production cross section is convoluted with Gaussian energy distribution [@FMC] for a resolution $R=0.05\%$. We see distinctive peaks in the distribution near $\cos\theta_{\rho^-}=\cos\theta_{\rho^+}=\pm 1$, as anticipated. In this demonstration, we have taken $\mu^\mp$ beam polarizations to be $P_-=P_+=25\%$, which is considered to be natural with little cost to beam luminosity [@collider].
In contrast, the SM background via $\gamma^*/Z^*$ produces $\tautau$ with helicity correlation of $LR\ (P_{\tau^-}=-P_{\tau^+}=-1)$ and $RL\ (P_{\tau^-}=-P_{\tau^+}=+1)$. Furthermore, the numbers of the left-handed and right-handed $\tau^-$ at a given scattering angle are different because of the left-right asymmetry, so the initial muon beam polarization affects the $\tautau$ spin correlation non-trivially. Summing over the two polarization combinations in $\tautau$ decay to particles $i$ and $j$, we have $$\begin{aligned}
{d\sigma_{SM}^{} \over {d\cos\theta_id\cos\theta_j}} =
&& (1-P_-P_+)\sigma_{SM}^{}\ (1+P_{eff}A_{LR})\times
\nonumber\\
&&{B_{i}B_{j}\over 4}[(a_ia_j-b_ib_j\cos\theta_i\cos\theta_j) +
A^{eff}_{LR}(a_ib_j\cos\theta_j-a_jb_i\cos\theta_i)].
\label{LR&RL}\end{aligned}$$ The effective $LR$-asymmetry factor is given by $$A^{eff}_{LR} \equiv
{{\sigma^{eff}_{LR+RL\to LR}-\sigma^{eff}_{LR + RL \to RL}} \over
{\sigma^{eff}_{ LR+RL\to LR}+\sigma^{eff}_{LR + RL \to RL}}}
={{A_{LR}^{FB}+P_{eff}A_{FB}}\over {1+P_{eff}A_{LR}}},
\label{eff-LR}$$ with $\sigma^{eff}$ the cross section including the percentage beam polarization $P_\pm$. The final state spin correlation for $\mumu\to \gamma^*/Z^*\to \tautau$ decaying into $\rho^-\rho^+$ pairs is shown in Fig. \[three\]. The maximum regions near $\cos\theta_{\rho^-}=-\cos\theta_{\rho^+}=\pm 1$ are clearly visible. Most importantly, the peak regions in Figs. \[two\] and \[three\] occur exactly in the opposite positions from the Higgs signal. We also note that the spin correlation from the Higgs signal is symmetric, while that from the background is not. The reason is that the effective $LR$-asymmetry in the background channel changes the relative weight of the two maxima, which becomes transparent from the last term in Eq. (\[LR&RL\]).
$\sqrt{s}=m_h$ (GeV) $100$ $110$ $120$ $130$
---------------------------- ------- ------- ------- -------
$\sigma^{}_B$ 55000 19300 12000 8900
$\sigma^{}_S\ (R=0.05\%)$ 478 380 286 189
$S/B\ (\%)$ 0.87 2.0 2.4 2.1
$S/\sqrt B\ (1~\fbi)$ 2.0 2.7 2.6 2.0
$\sigma^{}_S\ (R=0.01\%)$ 2140 1690 1250 806
$S/B\ (\%)$ 3.9 8.8 10 9.0
$S/\sqrt B\ (1~\fbi)$ 9.1 12 11 8.5
$\sigma^{}_S\ (R=0.005\%)$ 3750 2970 2170 1350
$S/B\ (\%)$ 6.8 15 18 15
$S/\sqrt B\ (1~\fbi)$ 16 21 20 14
: Total cross sections (in units of fb) of $\mu^- \mu^+ \to \tau^- \tau^+$ for the $s$-channel Higgs signal for $\sqrt{s}=m_h=100-130$ GeV and the SM background. Also shown are the signal-to-background ratio ($S/B$) and the signal statistical significance ($S/\sqrt B$) for an integrated luminosity of 1 $\fbi$. The Higgs channel cross sections are evaluated for three different beam resolutions ($R$). The polarization of the initial $\mu$ beams is taken to be zero. []{data-label="mu-tau"}
Results
-------
The total cross sections of the $s$-channel Higgs signal ($\sigma^{}_S$) and the SM background ($\sigma^{}_B$) for $\mumu\to \tautau$ are listed in Table \[mu-tau\] with $\sqrt{s}=m_h=100-130$ GeV. We show the signal results for three different beam energy resolutions $R=0.05\%,\ 0.01\%$ and $0.005\%$. A better beam energy resolution significantly improves the signal rate for the very narrow Higgs resonance with a width of order of a few MeV, while it has a negligible effect on the background rate. Also shown in the Table are the signal-to-background ratios ($S/B$) and the signal statistical significances ($S/\sqrt B$) for an integrated luminosity of 1 $\fbi$. We expect signals that are quite statistically significant. Even if we consider a luminosity of only 0.1 $\fbi$ and include only the clean channels listed in Table \[taudecay\] that count for $90\%$ of the branching fraction, the $R=0.005\%$ case still gives a significance better than 3$\sigma$.
As a further refinement in the analyses, we demand the final state $\tautau$ to be away from the beam hole by $15^\circ$, or equivalently $$|\cos\theta|<0.97.$$ This reduces the signal rate by about $3\%$ and the background rate by about $5\%$. One could expect to improve the signal observability by imposing more stringent cuts on $\cos\theta$ [@Marciano].
$\sqrt{s}=m_h$ (GeV) $100$ $110$ $120$ $130$
------------------------------------------ ------- ------- ------- -------
$P_+=P_-=0,\ {\rm no\ cut}$
$\sigma^{}_B$ 3450 1210 754 559
$\sigma^{}_S\ (R=0.05\%)$ 29.9 23.8 17.8 11.9
$S/B\ (\%)$ 0.87 2.0 2.4 2.1
$\sigma^{}_S\ (R=0.01\%)$ 134 106 78.5 52.9
$S/B\ (\%)$ 3.8 8.8 10 9.4
$\sigma^{}_S\ (R=0.005\%)$ 235 186 136 84.3
$S/B\ (\%)$ 6.8 15 18 15
$P_+=P_-=0,\ {\rm cut}\ (\ref{cuts})$
$\sigma^{}_B$ 1630 573 357 265
$\sigma^{}_S\ (R=0.05\%)$ 15.7 12.5 9.38 6.23
$S/B\ (\%)$ 0.96 2.2 2.6 2.4
$\sigma^{}_S\ (R=0.01\%)$ 70.3 55.8 41.3 26.6
$S/B\ (\%)$ 4.3 9.7 12 10
$\sigma^{}_S\ (R=0.005\%)$ 123 97.6 71.2 44.3
$S/B\ (\%)$ 7.6 17 20 17
$P_+=P_-=0.25,\ {\rm cut}\ (\ref{cuts})$
$\sigma^{}_B$ 1530 537 335 249
$\sigma^{}_S\ (R=0.05\%)$ 16.7 13.3 9.97 6.62
$S/B\ (\%)$ 1.1 2.5 3.0 2.7
$\sigma^{}_S\ (R=0.01\%)$ 74.7 59.3 43.9 28.2
$S/B\ (\%)$ 4.9 11 13 11
$\sigma^{}_S\ (R=0.005\%)$ 131 104 75.7 47.1
$S/B\ (\%)$ 8.6 19 23 19
: Total cross sections (in units of fb) of $\mu^- \mu^+ \to \tau^- \tau^+ \to\rho^-\nu_\tau\rho^+\bar \nu_\tau $ for the $s$-channel Higgs signal ($S$) at $\sqrt{s}=m_h=100-130$ GeV and the SM background ($B$). The polarization of the initial $\mu$ beams is taken to be 0 and $25\%$ for comparison. The Higgs channel cross sections are evaluated for three different beam resolutions ($R$). The signal-to-background ratios ($S/B$) are also given.[]{data-label="mu-tau-rho"}
We explore another approach instead to exploit the $\tautau$ spin correlation. From Figs. \[two\] and \[three\], we see that if we focus on the kinematical region $\cos\theta_i=\cos\theta_j\approx \pm 1$, we can substantially improve the ratio $S/B$. We need to preserve a sufficient signal rate by not taking too tight angular cuts. For illustration, we apply the acceptance cuts on the decay angles in $\tau$ rest frame with $$\begin{aligned}
&\cos&\theta_1 \geq 0,\quad \cos\theta_2 \geq 0, \nonumber\\
{\rm or}\quad &\cos&\theta_1 \leq 0,\quad \cos\theta_2 \leq 0.
\label{cuts}\end{aligned}$$ In Table \[mu-tau-rho\], we give the results for the $\rho^-\nu_\tau\rho^+\bar \nu_\tau$ final state from $\tautau$ decay. Because this mode has the largest branching fraction of about $25\%$, the cross section is larger than for other modes. However, the sensitivity parameter in Eq. (\[sens\]) for this mode is not maximal, being about 0.45. The surviving signal (background) is $53\%$ ($48\%$) after the acceptance cuts of Eq. (\[cuts\]). The $S/B$ after the cuts does not improve much. The $\pi^-\nu_\tau\pi^+\bar \nu_\tau$ mode has the maximal sensitivity factor of one, but a rather small cross section due to the low branching fraction. The surviving signal (background) is $63\%$ ($38\%$) after the acceptance cuts, and the $S/B$ is appreciably improved. The results are shown in Table \[mu-tau-pi\].
$\sqrt{s}=m_h$(GeV) $100$ $110$ $120$ $130$
------------------------------------------- ------- ------- ------- -------
$P_+=P_-=0,\ {\rm no\ cut}$
$\sigma^{}_B$ 677 237 148 110
$\sigma^{}_S\ (R=0.05\%)$ 5.86 4.67 3.50 2.32
$S/B\ (\%)$ 0.87 2.0 2.4 2.1
$\sigma^{}_S\ (R=0.01\%)$ 26.2 20.8 15.4 9.90
$S/B\ (\%)$ 3.9 8.8 10 9.0
$\sigma^{}_S\ (R=0.005\%)$ 46.0 36.4 26.6 16.5
$S/B\ (\%)$ 6.8 19 18 15
$P_+=P_-=0,\ {\rm cut}\ (\ref{cuts}) $
$\sigma^{}_B$ 254 88.9 55.5 41.1
$\sigma^{}_S\ (R=0.05\%)$ 3.66 2.92 2.19 1.45
$S/B\ (\%)$ 1.4 3.3 3.9 3.5
$\sigma^{}_S\ (R=0.01\%)$ 16.4 13.0 9.62 6.19
$S/B\ (\%)$ 6.5 15 17 15
$\sigma^{}_S\ (R=0.005\%)$ 28.8 22.8 16.6 10.3
$S/B\ (\%)$ 11 26 30 25
$P_+=P_-=0.25,\ {\rm cut}\ (\ref{cuts}) $
$\sigma^{}_B$ 238 83.4 52.0 38.6
$\sigma^{}_S\ (R=0.05\%)$ 3.89 3.10 2.32 1.54
$S/B\ (\%)$ 1.6 3.7 4.5 4.0
$\sigma^{}_S\ (R=0.01\%)$ 17.4 13.8 10.2 6.58
$S/B\ (\%)$ 7.3 17 20 17
$\sigma^{}_S\ (R=0.005\%)$ 30.6 24.2 17.7 11.0
$S/B\ (\%)$ 13 29 34 29
: Total cross sections (in units of fb) of $\mu^- \mu^+ \to \tau^- \tau^+ \to\pi^-\nu_\tau\pi^+\bar \nu_\tau $ for the $s$-channel Higgs signal at $\sqrt{s}=m_h=100-130$ GeV and the SM background. The polarization of the initial $\mu$ beams is taken to be 0 and $25\%$ for comparison. The Higgs channel cross sections are evaluated for three different beam resolutions ($R$). The signal-to-background ratios ($S/B$) are also given.[]{data-label="mu-tau-pi"}
A $25\%$ polarization of both beams only slightly improves the signals and decreases the backgrounds, as implied by Eq. (\[pmu\]). Higher beam polarization could help improve $S/B$, but perhaps at a significant cost to the luminosity [@collider].
We next estimate the luminosity needed for signal observation of a given statistical significance. The results are shown in Fig. \[lum\]. The integrated luminosity ($L$ in $\fbi$) needed for observing the characteristic two-body decay channels $\tau \to \rho\nu_\tau$ and $\tau \to \pi\nu_\tau$ at $3\sigma$ (solid) and $5\sigma$ (dashed) significance is calculated for both signal and SM background with $\sqrt s=m_h$. Beam energy resolution $R=0.005\%$ and a $25\%$ $\mu^\pm$ beam polarization are assumed.
Based on Tables \[mu-tau-rho\] and \[mu-tau-pi\], we estimate the statistical error on the cross section measurement. If we take the statistical error to be given by $$\epsilon = {\sqrt{S+B}\over S} =
{1\over \sqrt L}\ {\sqrt{\sigma_S+\sigma_B} \over \sigma_S},$$ summing over both $\rho\nu_\tau$ and $\pi\nu_\tau$ channels for $R=0.005\%$, a $25\%$ beam polarization with 1 $\fbi$ luminosity, we obtain $$\begin{array}{lcccc}
\sqrt s = m_h\ (\gev)\quad& 100 & 110 & 120 & 130 \\
\epsilon\ (\%)\quad & 27 & 21 & 23 & 32
\end{array}
\label{eps}$$ The uncertainties on the cross section measurements determine the extent to which the $h\tautau$ coupling can be measured.
Discussion and conclusion
=========================
If we only consider the $\pi\nu_\tau$ and $\rho\nu_\tau$ channels that best preserve the $\tautau$ spin correlation, the effective branching fraction is only $36\%$ and we may be limited by statistics. However, the distinctively different double differential distributions of the signal and the SM backgrounds may provide definitive information for determining the spin of the resonant Higgs particle. When analyzing the data sample, one may consider a sophisticated fitting to the superposition of the signal and background distributions.
The characteristic angular distributions of polarized $\tau$ decays are only simply manifest in the $\tau$ rest frame. It is thus desirable to infer the $\tau^\pm$ momenta in order to boost the final state particles ($\rho,\ \pi,\ \ell^\pm$ etc.) to the parent $\tau$ rest frame. Because of the excellent energy calibration of a muon collider, it is a good approximation to assume each $\tau$ to have an energy of $\sqrt s/2$. However, it may be experimentally challenging to determine the $\tau$ momentum direction. One of the possible methods is to locate the secondary vertices for $\tau$ decays. The impact parameter for $\tau$ decays is $\ell/\gamma \approx \beta c\tau_\tau\approx 87\ \mu$m. This should be sufficiently large to be resolved by vertex detectors.
It is important to note that it is not necessary to fully reconstruct the $\tau$ momenta for the clean two-body channels. This is because the polar angles ($\theta_1,\theta_2$) in the $\tau$ rest frame can be uniquely determined by the charged particle energy [@TauDecay]. If the lab-frame energy for $\rho,\pi$ is $E_i$, then the relation to the polar angle is $$\cos\theta_i = {2z_i -1 -a^2\over \beta (1-a^2)},$$ where the energy fraction $z_i=2E_i/\sqrt s$, $a=m_i/m_\tau$ and $\beta$ is the velocity of the decay product. Due to the unique linear relation between $\cos\theta_i$ and $z_i$, two dimensional correlation plots for $z_i-z_j$ can be obtained in a similar fashion as Figs. \[two\] and \[three\].
The $s$-channel Higgs signal at the FMC could provide a precision measurement for the Higgs total width [@FMC], and thus lead to the determination of the coupling strength parameter $\tan\beta$ in SUSY theories. The observation of the $h\to \tautau$ channel in addition to the channel $h\to b\bar b$ is very important: The relative strength of the Higgs couplings to $b$ and to $\tau$ could be an indicator to the underlying physics, such as the possibly large non-universal radiative effects in MSSM [@ttbb] from the chargino and gluino loops, and radiatively generated Yukawa couplings [@yukawa]. We expect that the measurement of the coupling ratio is robust, and only statistically limited in the $\tautau$ mode. If a high degree of transverse polarization of the beams is achievable, one could consider the possibility to determine the CP properties of the Higgs boson coupling [@FMC; @cp] by making use of the $\tautau$ mode.
In summary, we have demonstrated the feasibility of observing the resonant channel $h\to \tautau$ at a muon collider. For a narrow resonance like the SM Higgs boson, a good beam energy resolution is crucial for a clear signal. On the other hand, a moderate beam polarization would not help much for the signal identification. The integrated luminosity needed for a signal observation is presented in Fig. \[lum\]. Estimated statistical errors for the $\mumu \to h \to \tautau$ cross section measurement are given in Eq. (\[eps\]). We emphasized the importance of final state spin correlation to purify the signal of a scalar resonance and to confirm the nature of its spin. It is also important to carefully study the $\tautau$ channel of a supersymmetric Higgs boson which would allow a determination of the relative coupling strength of the Higgs to $b$ and $\tau$.
0.5cm
[*Acknowledgments*]{}: We thank Dieter Zeppenfeld for discussions on the $\tau$ polarization. This work was supported in part by a DOE grant No. DE-FG02-95ER40896 and in part by the Wisconsin Alumni Research Foundation.
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[^1]: Current address, Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854-0849
|
{
"pile_set_name": "ArXiv"
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|
---
abstract: 'In this article, we derive two spectral gap bounds for the reduced Laplacian of a general simplicial complex. Our two bounds are proven by comparing a simplicial complex in two different ways with a larger complex and with the corresponding clique complex respectively. Both of these bounds lead to generalizations of the result of Aharoni et al. (2005) [@ABM] which is valid only for clique complexes. As an application, we decrease by a logarithmic factor, the upper bound for the threshold for vanishing of cohomology of the neighborhood complex of the Erdös-Rényi random graph derived by Kahle (2007) [@kahle]. We also increase the lower bound for the above threshold by a polynomial factor.'
author:
- 'Samir Shukla[^1], D. Yogeshwaran[^2]'
title: Spectral bounds for vanishing of cohomology and the neighborhood complex of a random graph
---
[**Keywords:**]{} spectral bounds, Laplacian, cohomology, neighborhood complex, Erdös-Rényi random graphs.
[**AMS MSC 2010:**]{} 05C80. 05C50. 05E45. 55U10.
Introduction
============
Let $G$ be a graph with vertex set $V(G)$ (often to be abbreviated as $V$) and let $L(G)$ denotes the (unnormalized) [*Laplacian*]{} of $G$. Let $\lambda_1(G) \leq \lambda_2(G) \leq \ldots \leq\lambda_{|V|}(G)$ denote the eigenvalues of $L(G)$. Here, the second smallest eigenvalue $\lambda_2(G)$ is called the [*spectral gap*]{}. The [*clique complex*]{} of a graph $G$ is the simplicial complex whose simplices are all subsets $\sigma \subset V$ which spans a complete subgraph of $G$. We shall denote the $k$th reduced cohomology of a simplicial complex $X$ by $\widetilde{H}^k(X)$. In this article, we always consider the reduced cohomology with real coefficients. For more detailed definitions, see section \[sec:prel\].
In [@ABM], Aharoni et al. proved the following result which guarantees the vanishing of cohomology of a clique complex, provided the spectral gap of its $1$-skeleton is large enough.
[@ABM Theorem 1.2] \[t:abm\] Let $X$ be the clique complex of a graph $G$. If $\lambda_2(G) > \frac{k|V|}{k+1}$, then $\widetilde{H}^k(X) = 0$.
Aharoni et al. ([@ABM]) used Theorem \[t:abm\] to find a lower bound of the homological connectivi-ty of the independence complex of a graph $G$ (a simplicial complex whose simplices are the independent sets of $G$), which implies Hall type theorems for systems of disjoint representati-ves in hypergraphs.
Theorem \[t:abm\] can be viewed as a global counterpart for clique complexes of spectral gap results of Garland ([@G Theorem 5.9]) and Ballman-[' S]{}wiatkowski ([@BS Theorem 2.5]) for vanishing of cohomology of a simplicial complex. In their simplest form, these results say that for a pure $k$-dimensional finite simplicial complex $\Delta$, if the spectral gap of link $lk_{\Delta}(\tau)$ is sufficiently large for every ($k-2$)-dimensional simplex $\tau$, then $\widetilde{H}^{k-1}(\Delta) = 0$. A very powerful application of the afore-mentioned result is by Kahle ([@MK1]) to derive sharp vanishing thresholds for cohomology of random clique complexes. See [@Hoffman2017] for more applications of this spectral gap result in random topology. Recently, Hino and Kanazawa ([@HK Theorem 2.5]) generalized this result of Garland and Ballman-[' S]{}wiatkowski, and upper bounded the $(d-1)^{th}$ Betti number of a pure $d$-dimensional simplicial complex $\Delta$ by the sum (taken over all $(d-2)$-dimensional simplices $\tau$) of the number of ‘suitably small’ eigenvalues in the spectrum of the laplacian of the link $lk_{\Delta}(\tau)$. They used this quantitative version of the spectral gap result to prove weak laws for (persistent) lifetime sums of randomly weighted clique and $d$-dimensional complexes.
Motivated by applications of spectral gap bounds to random complexes, we seek to generalize Theorem \[t:abm\] to more general simplicial complexes. We achieve two different generalizations (see Corollaries \[cor:subcomplex\] and \[cor:general\]) by comparing an arbitrary simplicial complex with a larger complex and the corresponding clique complex in two different ways. Our aim in exploring this generalization was to obtain vanishing threshold for cohomology in other random complex models. We use one of our generalizations to improve the vanishing threshold for cohomology of a random neighbourhood complex (see Theorem \[t:main\]) by a logarithmic factor. After the result, we also discuss why it is difficult to apply Garland’s method and hence a different spectral gap result is needed. By computing the probabilities involved more precisely than [@kahle], we also improve the lower bound by a polynomial factor.
The paper is organised as follows. In section \[subsec:notation\], we introduce some notation, which we shall use in rest of the paper. In section \[subsec:spectral\], we state our results, which relates the cohomology and spectral gap. In section \[subsec:randomneighborhood\], we state the results about the neighborhood complexes of a random graph. We also discuss our improvements in relation to the results of [@kahle]. We give the necessary preliminaries from graph theory and topology in section \[sec:prel\] . section \[sec:proof\] is dedicated to the proofs of the results stated in section \[subsec:spectral\] and \[subsec:randomneighborhood\].
Notations {#subsec:notation}
---------
We shall use the following notations throughout this paper. Let $X$ be a (simplicial) complex on $n$ vertices. We denote $G_X$ as the $1$-skeleton of $X$, i.e., $G_X$ is the graph whose vertices are the $0$-dimensional simplices and edges are the $1$-dimensional simplices of $X$. Let $X(k)$ denotes the set of all $k$-dimensional oriented simplices of $X$. $X$ is said to be a [*clique complex*]{} if for all $k \geq 0$, $X(k)$ is the set of $(k+1)$-cliques in the graph $G_X$. For $k \geq -1$, let $C^k(X; {\mathbb{R}})$ denote the space of real valued $k$-cochains of $X$. Let $\delta_k(X) : C^k(X;{\mathbb{R}}) \to C^{k+1}(X;{\mathbb{R}})$ denote the coboundary operator.
For $k \geq 0$, let $\delta_k^{\ast}(X)$ denote the adjoint of $\delta_k(X)$ and let $\Delta_k(X) = \delta_{k-1}(X) \delta_{k-1}^{\ast}(X) + \delta_{k}^{\ast}(X)\delta_k(X)$ (see section \[sec:prel\] for details). Let $\mu_k(X)$ denote the minimal eigenvalue of $\Delta_k(X)$. Observe that $\lambda_2(G_X) = \mu_0(X)$. We again emphasize that we consider reduced cohomology with real coefficients.
We shall now define two ways to measure the difference between two complexes. The first compares a complex to its subcomplex whereas the second compares a complex $X$ to the corresponding clique complex of $G_X$. Denoting the indicator function by $ \1[ \ \cdot \ ]$, define for $k \geq 1$ $$\label{e:Sk}
S_k(X,X') := \max \limits_{\sigma \in X'(k)} \bigg\{ \sum\limits_{\tau \in X(k+1) \setminus X'(k+1)} \1[ \sigma \subset \tau ] \bigg\},$$ where $X'$ is a subcomplex of $X$. Throughout the article, we shall use $X$ to denote a complex and we shall denote a subcomplex of $X$ by $X'$.
For a simplex $\eta \in X$, the [*link*]{} of $\eta$ is the complex defined as $$lk_X(\eta) = \{\sigma \in X \ | \ \sigma \cup \eta \in X \ \text{and} \ \sigma \cap \eta = \emptyset\}.$$ For $k \geq 1$ and $1 \leq j \leq k+1$, define $$\begin{aligned}
\label{e:Dkj}
D_k(X,j) := & \max\limits_{\sigma \in X(k)} \bigg\{\sum\limits_{u} \1[u \notin lk_X(\sigma) \ \text{and} \ \exists \ \mbox{exactly} \ j \ \mbox{vertices} \ v_1, \ldots, v_j \in \sigma \ \mbox{such that} \nonumber\\
& \, \, u \in lk_X(\sigma \setminus \{v_i\}) \ \forall \ 1\leq i \leq j ] \bigg\}.\end{aligned}$$
\[r:Dk\] If the $k$-skeleton of $X$ is a clique complex of $G_X$, i.e., for $i \leq k$, any $(i+1)$-tuple of vertices of $X$ form a simplex in $X$ if and only if they induce a complete subgraph of $G_X$, then $u \in lk_X(\sigma \setminus \{v\}) \cap lk_X(\sigma \setminus \{w\})$ for some $\{v, w\} \subseteq \sigma$ implies that $u \in lk_X(\sigma \setminus \{v\}) \ \forall \ v \in \sigma$, [*i.e.*]{}, any $(k+1)$-subset of $\sigma \cup \{u\}$ will be a $k$-simplex. Therefore, in this case $D_k(X, j) = 0$ for all $2 \leq j \leq k$ and $$\begin{aligned}
\label{def:Dk}
D_k(X,k+1)= &\max\limits_{\sigma \in X(k)} \bigg\{ \sum\limits_{w} \1[ w \notin lk_X(\sigma) \ \text{and any} \ (k+1)\text{-subset of} \ \sigma \cup \{w\} \ \text{is a} \ k\text{-simplex} ] \bigg\}. \end{aligned}$$ Thus, if $X$ is a clique complex then $D_k(X, j) = 0$ for all $2 \leq j \leq k+1$.
Spectral gap and cohomology {#subsec:spectral}
---------------------------
We shall now present our two spectral gap results and corollaries that generalize Theorem \[t:abm\]. We first recall the following result from [@ABM], which formed the crux of the proof of Theorem \[t:abm\].
[@ABM Theorem $1.1$] \[t:abm1\] Let $X$ be the clique complex of $G_X$. For $k \geq 1$, $$k \mu_k(X) \geq (k+1) \mu_{k-1}(X) - |V(G_X)|.$$
We prove our first main spectral gap result by directly comparing the operators $\delta_k(X)$, $\delta_k(X')$. Following the theorem, we state a simple corollary which generalizes Theorem \[t:abm\]..
\[t:subcomplex\] For $k \geq 1$, $$\label{e:subcomplexname}
\mu_k(X') \geq \mu_k(X) -(k+2)S_k(X,X').$$
\[cor:subcomplex\] Let $X$ be the clique complex of $G_X$ and the $1$-skeleton of $X'$ is $G_X$. If $\lambda_2(G_X)> \frac{kn}{k+1} + \frac{k+2}{k+1} S_k(X,X')$, then $\widetilde{H}^k(X') = 0$.
If $X'= X$, then $S_k(X,X') = 0$ and so Corollary \[cor:subcomplex\] implies Theorem \[t:abm\]. Now, we present our generalization of Theorem \[t:abm1\] using $D_k(X,j)$’s and as before a simple corollary for later use.
\[t:general\] For $k \geq 1$, $$\begin{aligned}
k \mu_k(X) \geq (k+1) \mu_{k-1}(X) - n - (k(k+1)+1) \sum\limits_{j=2}^{k+1} D_k(X,j).\end{aligned}$$
\[cor:general\] Let $k$-skeleton of $X$ is the clique complex of $G_X$. If $\lambda_2(G_X) > \frac{kn}{k+1} + (k+ \frac{1}{k+1}) D_k(X,k+1)$, then $\widetilde{H}^k(X) = 0$.
Since for a clique complex $X$, $D_k(X, j) = 0 \ \forall \ 2 \leq j \leq k+1 $, we see that in this case Theorem \[t:general\] implies Theorem \[t:abm1\] and Corollary \[cor:general\] implies Theorem \[t:abm\]. The proof of Theorem \[t:general\] follows the ideas of [@ABM] but some of the terms that cancel out in the case of clique complexes do not cancel out for a general simplicial complex. Hence, it requires more careful bounding to derive suitable bounds.
Neighborhood complex of a random graph {#subsec:randomneighborhood}
--------------------------------------
We shall now introduce neighborhood complex of random graphs, recall results from [@kahle] and state our results about the cohomology of the same. For more on random graphs, we refer the reader to [@Janson11; @Frieze16] and refer to [@Kahle14a] for a survey on random simplicial complexes.
The [*neighborhood complex*]{}, ${\mathcal{N}}(G)$ of a graph $G$ is the simplicial complex whose simplices are those subsets $\sigma$ of $V$ which have a common neighbor. The concept of neighborhood complex was introduced by Lov[' a]{}sz ([@lovasz]) in his proof of the famous Kneser conjecture. We now introduce the Erdös-Rényi random graph $G(n,p)$ on $n$ vertices and with edge-probability $p$. $G(n,p)$ is constructed by deleting edges of the complete graph on $n$ vertices independently of each other with probability $1-p$ or equivalently the edges are retained independently of each other with probability $p$. In this article, we consider $p$ as a function of $n$. A [*graph property $\mathcal{P}$*]{} is a class of graphs such that for any two isomorphic graphs either both belong to the class or both do not belong to the class. For any graph property $\mathcal{P}$, we say that $G(n, p) \in \mathcal{P}$ [*with high probability*]{} ([*w.h.p.*]{}) if $\mathbb{P} (G(n, p) \in \mathcal{P}) \to 1$ as $n \to \infty$. We shall also say $\mathcal{P}$ holds for $G(n,p)$ instead of $G(n, p) \in \mathcal{P}$. In [@kahle], M. Kahle considered the neighborhood complex of the Erdös-Rényi random graph. He showed that (see [@kahle Theorem 2.1]), if ${n \choose k+2} (1-p^{k+2})^{n-(k+2)} = o(1)$, then w.h.p. $\widetilde{H}^i({\mathcal{N}}(G(n,p))) = 0$, for $i \leq k$. In particular, if $p = \Big(\frac{(k+2)\log n +c_n}{n}\Big)^{\frac{1}{k+2}}$ for $c_n \to \infty$, then w.h.p. $\widetilde{H}^i(\mathcal{N}(G(n, p))) = 0$ for $i \leq k$. Using Corollary \[cor:general\], we achieve the following improvement on Kahle’s result.
\[t:main\] Let $k \geq 1$. If $p = \Big(\frac{(k+1)\log n +c_n}{n}\Big)^{\frac{1}{k+2}}$ with $c_n \to \infty$, then $\widetilde{H}^i(\mathcal{N}(G(n, p))) = 0$ w.h.p. for $i \leq k$.
Note that in Theorem \[t:main\], ${n \choose k+2} (1-p^{k+2})^{n-(k+2)} \to \infty$ and therefore we cannot apply Kahle’s result in this case. His proof involves showing that for $p$ satisfying ${n \choose k+2} (1-p^{k+2})^{n-(k+2)} = o(1)$, ${\mathcal{N}}(G(n, p))$ has the full $(k+1)$-skeleton, i.e., any $t$-tuple of vertices form a $(t-1)$-simplex in ${\mathcal{N}}(G(n,p))$ for $t \leq k+2$. This trivially yields that $\widetilde{H}^i(\mathcal{N}(G(n, p))) = 0$ for $i \leq k$. But one would expect that this is a very strong condition for vanishing of cohomology and our theorems shows that this can be reduced a little. We expect that our bound for the threshold for vanishing of cohomology to be reduced even further.
The above theorem is one of our motivations to prove spectral bounds for vanishing of cohomology for general complexes. This was inspired by the proof of a sharp threshold result for vanishing of cohomology of clique complexes of Erdös-Rényi random graphs in [@MK1] which was proven using the spectral gap result of Garland and Ballman-[' S]{}wiatkowski. This required to show that the spectral gap of the normalized Laplacian of the $1$-skeleton of all the links are sufficiently large. For a clique complex of an Erdös-Rényi random graph, it is easy to see that the $1$-skeleton of a link is also an Erdös-Rényi random graph and hence by proving suitable spectral bounds for the normalized Laplacian of Erdös-Rényi random graphs, the result of Garland and Ballman-[' S]{}wiatkowski was used. But it is not easy to use the same argument to prove Theorem \[t:main\], as the $1$-skeleton of the link of a simplex in neighborhood complex of an Erdös-Rényi random graph is not an Erdös-Rényi random graph. It has a complicated dependency structure making it harder to analyse the spectral gap of the corresponding random graph.
Kahle (see [@kahle Corollary 2.9]) also showed that for $p= n^{\alpha}$, if $\frac{-2}{k+1} < \alpha < \frac{-4}{3(k+1)}$, then w.h.p. $\widetilde{H}^k(\mathcal{N}(G(n, p)))$ $ \neq 0$. We derive more exact bounds for the probabilities involved but still use the same argument as that of [@kahle] to extend this result as well.
\[p:extension\] Let $p = n^{\alpha}$. If $\frac{-2}{k+1} < \alpha < \frac{-1}{k+1}$, then w.h.p. $\widetilde{H}^k(\mathcal{N}(G(n, p))) \neq 0$.
Despite the improvement of the bounds presented here, it is still an open problem to determine sharp bounds for vanishing of cohomology of neighborhood complexes. From Theorem \[t:main\], Proposition \[p:extension\] and [@kahle Corollary 2.5], we summarize the known bounds as follows : For $k \geq 1$, $$\begin{aligned}
\widetilde{H}^k(\mathcal{N}(G(n, p))) & = 0 \ \ \mbox{w.h.p. if $p = n^{\alpha}$ with $\alpha < \frac{-4}{k+2}$ for $k$ even and $\alpha < \frac{-4(k+2)}{(k+1)(k+3)}$ for $k$ odd},\\
\widetilde{H}^k(\mathcal{N}(G(n, p))) & \neq 0 \ \ \mbox{w.h.p. if $p = n^{\alpha}$ with $\frac{-2}{k+1} < \alpha < \frac{-1}{k+1}$}, \\
\widetilde{H}^k(\mathcal{N}(G(n, p))) & = 0 \ \ \mbox{w.h.p. if $p = \Big(\frac{(k+1)\log n +c_n}{n}\Big)^{\frac{1}{k+2}}$ with $c_n \to \infty$}. \end{aligned}$$
In [@kahle Corollary 2.5] homology groups are given with integer coefficients, but the proof was by showing that ${\mathcal{N}}(G(n,p))$ deformation retracts onto a subcomplex of dimension $k-1$. Hence, the same proof is valid irrespective of the coefficients of homology. Also, we have used that the homology and cohomology groups with real coefficients are isomorphic to each other.
Preliminaries {#sec:prel}
=============
A [*graph*]{} $G$ is a pair $(V, E)$, where $V$ is the set of vertices of $G$ and $E \subset V \times V$ called the set of edges. If $(u, v) \in E$, it is also denoted by $u \sim v$ and we say that $u$ is adjacent to $v$. For any $A \subset V$, the neighborhood of $A$, $N(A):= \{u \in V \ | \ u \sim a
\,\,\forall\,\, a \in A \}$. The [*degree*]{} of a vertex $v$ is denoted by $deg(v)$. For a subset $X \subset V$, the induced subgraph $G[X]$ is the subgraph whose set of vertices $V(G[X]) = X$ and the set of edges $E(G[X]) = \{(u, v) \in E \ | \ u, v \in X\}$. The [*complete graph*]{} or a [*clique*]{} of order $n$ is a graph on $n$ vertices, where any two distinct vertices are adjacent and it is denoted by $K_n$. All the graphs in this article are assumed to be simple i.e., $(x, y) \in E$ implies $x \neq y$.
The (unnormalized) [*Laplacian*]{} of a graph $G$ is the $|V| \times |V|$ matrix $L(G)$ given by $$L(G)(x, y) := \begin{cases}
\ deg(x)& x=y ,\\
\ -1 & (x,y) \in E,\\
\ 0 & \text{otherwise}.\\
\end{cases}$$ For details about Laplacian we refer the reader to [@bapat]. We next introduce the concept of simplicial complexes, which are higher dimensional counterparts of graphs.
A finite (abstract) [*simplicial complex X*]{} is a family of subsets of a finite set, which is closed under the deletion of elements, i.e., if $\alpha \in X$ and $\beta \subset \alpha$, then $\beta \in X$. For $\sigma \in X$, the dimension of $\sigma$ is defined to be $|\sigma| - 1$ and denoted by $dim(\sigma)$. If dim($\sigma$) = $k$, then it is said to be a $k$-dimensional simplex or $k$-simplex. The $0$-dimensional simplices are called vertices of $X$. We denote the set of vertices of $X$ by $V(X)$. The [*boundary*]{} of a $k$-dimensional simplex $\sigma $ is the simplicial complex, consisting of all simplices $\tau \subset \sigma$ of dimension $\leq k-1$. We refer the reader to book by Kozlov ([@dk]) for more details about simplicial complexes. Let $X$ be a simplicial complex. Two ordering of vertices of a simplex $\sigma = \{v_0, v_1, \ldots, v_k\}$ called equivalent if they differ from one another by an even permutation. Thus the ordering of these vertices of simplex divided into two equivalences classes. Each of these classes is called an orientation of $\sigma$. An oriented simplex is a simplex $\sigma$ together with an orientation and we denote it by $[ v_0, \ldots, v_k ]$.
Let each simplex of $X$ having arbitrary but fixed orientation. Let $X(k)$ denote the set of oriented $k$-simplices of $X$. For $k \geq 0$, let $C_k(X)$ denote the free abelian group with basis $X(k)$, with the relation $[ v_0,v_1, \ldots, v_k ] = - [ v_1, v_0, \ldots, v_k ]$ for each $k$-simplex $\sigma = \{v_0, \ldots, v_k\}$.
For $k \geq 0$, let $C^k(X; {\mathbb{R}})$ be the dual group Hom$(C_k(X); {\mathbb{R}})$. The elements of $C^k(X;{\mathbb{R}})$ are called $k$-cochains of $X$. For an ordered $(i+1)$-simplex $\sigma = [v_0, \ldots, v_{i+1}]$ the $j$-face of $\sigma$ is an ordered $i$-simplex $\sigma_j=[v_0, \ldots, \hat{v_j}, \ldots, v_{i+1}]$. The coboundary operator $\delta_k (X) : C^k(X;{\mathbb{R}}) \to C^{k+1}(X;{\mathbb{R}})$ is given by $$\delta_k(X) \phi (\sigma) := \sum\limits_{j=0}^{k+1} (-1)^j \phi(\sigma_j).$$ By letting $C^{-1}(X;{\mathbb{R}}) = {\mathbb{R}}$, define $\delta_{-1}(X) :C^{-1}(X;{\mathbb{R}}) \to C^{0}(X;{\mathbb{R}}) $ by $\delta_{-1}(X)(x)(v) = x$ for all $x \in {\mathbb{R}}$ and $v \in X(0)$. It is well known that $\delta_{k} \delta_{k-1} = 0 $ for all $k \geq 1$. For $k \geq 0$, the quotient Ker $\delta_k(X)$ / Im $\delta_{k-1}(X)$ is called the $k$-th reduced cohomology group of $X$ with real coefficients and it is denoted by $\widetilde{H}^k(X)$. For more details about cohomology we refer the reader to [@Munkres].
For each $k \geq -1$ we can defined the standard inner product on $C^k(X;{\mathbb{R}})$ by $ \langle \phi, \psi \rangle := \sum_{\sigma \in X(k)} \phi(\sigma)\psi(\sigma)$ and the corresponding $L^2$ norm $|| \phi || := (\sum_{\sigma \in X(k)} \phi(\sigma)^2)^{\frac{1}{2}}$.
Let $\delta_k^{\ast}(X): C^{k+1}(X;{\mathbb{R}}) \to C^{k}(X;{\mathbb{R}})$ denote the adjoint of $\delta_k(X)$ with respect to these standard inner product, i.e., the unique operator satisfying $ \langle \delta_k(X)\phi, \psi \rangle = \langle \phi, \delta_k^{\ast}(X) \psi \rangle$ for all $\phi \in C^k(X; {\mathbb{R}})$ and $\psi \in C^{k+1}(X;{\mathbb{R}})$. The reduced $k$-Laplacian of $X$ is the mapping $$\Delta_k (X) := \delta_{k-1}(X) \delta_{k-1}^{\ast}(X) + \delta_k^{\ast}(X)\delta_k(X) : C^{k}(X;{\mathbb{R}}) \to C^{k}(X;{\mathbb{R}}).$$ It can be easily verify that if $\mathbb{I}$ denotes the $|V(G_X)| \times $ $|V(G_X)|$ matrix with all entries $1$, then $\mathbb{I}+ L(G_X)$ represents $\Delta_0(X)$ with respect to the standard basis. In particular the minimal eigenvalue of $\Delta_0(X)$ ([*i.e.*]{}, $\mu_0(X)$) is $\lambda_2(G_X)$. More details about the operator $\Delta_k(X)$ can be found in [@BS] and [@G].
We now recall the following well known simplicial Hodge theorem.
\[p:hodge\] For $k \geq 0$, Ker $\Delta_k(X) \cong \widetilde{H}^k(X)$.
Proofs {#sec:proof}
======
Proofs of the results of section \[subsec:spectral\] {#sec:spectralgap}
----------------------------------------------------
Throughout this article, for any positive integer $m$, we denote the set $\{1, \ldots, m\}$ by $[m]$. Recall that, $X$ is complex and $X'$ is a subcomplex of $X$. For two oriented simplices $\eta \in X$ and $\tau \in lk_X(\eta)$, $\eta\tau$ denotes their oriented union, i.e., if $\eta = [v_0, \ldots, v_k]$ and $\tau = [u_0, \ldots, u_l]$, then $\eta\tau = [v_0, \ldots, v_k, u_0, \ldots, u_l]$.
Throughout this article, for any $k$-cochain $\phi$ of $X'$, we also consider $\phi$ as a cochain of $X$ by simply taking $\phi(\sigma) = 0$ whenever $\sigma \in X(k) \setminus X'(k)$ and a cochain $\psi$ of $X$ considered as a cochain of $X'$ by taking restriction of $\psi$ on $X'$.
In the rest of the section, we shall abbreviate as follows : $\delta_k = \delta_k(X), \delta'_k =\delta_k(X'), \delta_k^{\ast} = \delta_k^{\ast}(X),\delta_k^{'\ast} = \delta_k^{\ast}(X'), \Delta_k = \Delta_k(X)$ and $ \Delta_k' = \Delta_k'(X)$.
\[l:lemma1\] For $\phi \in C^{k}(X';{\mathbb{R}})$ $$\label{e:equal1}
||\delta_{k-1}^{\ast}\phi||^2 = ||\delta_{k-1}^{'\ast}\phi||^2.$$
For any $\tau \in X(k-1)$ and $\sigma \in X'(k-1)$, by the definition of $\phi$ on $X$, we have that $$\begin{aligned}
\delta_{k-1}^{\ast} \phi(\tau) = \sum_{v \in lk_{X}(\tau)} \phi(v\tau) = \sum_{v \in lk_{X'}(\sigma)} \phi(v\sigma) = \delta_{k-1}^{'\ast} \phi(\sigma).\end{aligned}$$
We shall require the following simple inequality : For any real numbers $x_1, x_2, \ldots, x_n$, it holds that $$\begin{aligned}
\sum\limits_{\{i, j\} , i \neq j} x_i x_j \leq \frac{(n-1)}{2} \sum\limits_{i=1}^{n} x_i^2. \label{e:inequality}\end{aligned}$$
\[l:lemma2\] For $ \phi \in C^{k}(X';{\mathbb{R}})$, recalling $S_k(X,X')$ as defined in , we have that $$\label{e:lessequal1}
||\delta_k \phi||^2 - ||\delta'_k \phi||^2 \leq (k+2) S_k(X, X') ||\phi||^2.$$
$$\begin{aligned}
||\delta_k \phi||^2 - ||\delta'_k \phi||^2 & = \sum_{\tau \in X(k+1) \setminus X'(k+1)} \Big( \delta_k \phi(\tau) \Big)^2 \nonumber\\
& = \sum_{\tau \in X(k+1) \setminus X'(k+1)} \sum_{i=0}^{k+2} (-1)^i \phi(\tau_i) \sum_{j=0}^{k+2} (-1)^j \phi(\tau_j) \nonumber\\
& = \sum_{\tau \in X(k+1) \setminus X'(k+1)} \Big(\sum_{i=0}^{k+2} (-1)^{2i} \phi(\tau_i)^2 + \sum_{ i \neq j} (-1)^{i+j} \phi(\tau_i) \phi(\tau_j) \Big) \nonumber\\
& = \sum_{\tau \in X(k+1) \setminus X'(k+1)} \Big(\sum_{i=0}^{k+2} \phi(\tau_i)^2 + 2 \sum_{ \{i, j \}, i \neq j } (-1)^{i+j} \phi(\tau_i)\phi(\tau_j)\Big) \nonumber \\
& \leq \sum_{\tau \in X(k+1) \setminus X'(k+1)} \Big( \sum_{i=0}^{k+2} \phi(\tau_i)^2 + (k+1) \sum_{i=0}^{k+2} \phi(\tau_i)^2 \Big) \notag \label{middle}
\end{aligned}$$
where last inequality follows from . Hence, we derive that $$\begin{aligned}
||\delta_k \phi||^2 - ||\delta'_k \phi||^2 & \leq (k+2) \sum_{\tau \in X(k+1) \setminus X'(k+1)} \sum_{i=0}^{k+2} \phi(\tau_i)^2 \nonumber\\
& = (k+2)\sum_{\sigma \in X(k)} \phi(\sigma)^2 \sum_{\tau \in X(k+1) \setminus X'(k+1)} \1[\sigma \subset \tau] \nonumber\\
& = (k+2)\sum_{\sigma \in X'(k)} \phi(\sigma)^2 \sum_{\tau \in X(k+1) \setminus X'(k+1)} \1[\sigma \subset \tau] \nonumber\\
& \leq (k+2) S_k(X, X') ||\phi||^2 \nonumber.\end{aligned}$$
We now recall the following well known Minmax principle.
(Minmax principle ; [@RB Corollary III.1.2 & Exercise III.1.3]) \[p:maxmin\] Let $A$ be the self-adjoint operator on inner product space $(V, \langle \ \rangle)$. Let $\lambda_{min}$ be the minimum eigenvalue of $A$. For $0 \neq x \in V$, $$\lambda_{min} \leq \frac{\langle Ax, x \rangle}{\langle x, x \rangle}.$$
Let $0 \neq \phi \in C^{k}(X';{\mathbb{R}})$ be an eigenvector of $\Delta'_k = \Delta_k(X')$ with eigenvalue $\mu'_k = \mu_k(X')$. Using and for the first inequality below along with definition of Laplacian and min-max principle (Proposition \[p:maxmin\]), we derive that $$\begin{split}
\mu'_k ||\phi||^2 & = \langle \Delta'_k \phi, \phi \rangle = ||\delta'_k \phi||^2 + ||\delta_{k-1}^{' \ast} \phi||^2\\
& \geq ||\delta_k \phi||^2 + ||\delta_{k-1}^{\ast} \phi||^2 - (k+2)S_k(X,X') ||\phi||^2\\
& = \langle \Delta_k \phi, \phi \rangle - (k+2)S_k(X,X') ||\phi||^2\\
& \geq \mu_k ||\phi||^2 - (k+2)S_k(X,X') ||\phi||^2.
\end{split}$$
By applying induction on $k$ in Theorem \[t:abm1\], we derive that $\mu_k(X) \geq (k+1)\mu_0(X) - kn$. Now, substituting the above bound in Theorem \[t:subcomplex\] and using the fact that $\mu_0(X) = \lambda_2(G_X)$, we obtain $$\mu_k(X') \geq (k+1)\lambda_2(G_X) -kn -(k+2)S_k(X,X').$$
Hence, if $\lambda_2(G_X) > \frac{kn}{k+1} + \frac{k+2}{k+1} S_k(X,X')$, then we have that $\mu_k(X') > 0$ and Proposition \[p:hodge\] implies that $\widetilde{H}^k(X') = 0$.
For an $i$-simplex $\eta \in X$ let deg($\eta$) denote the number of $(i+1)$-simplices in $X$ which contain $\eta$. For $\phi \in C^k(X)$ and a vertex $u \in V(X)$ define $\phi_u \in C^{k-1}(X;{\mathbb{R}})$ by $$\phi_u(\tau)= \begin{cases}
\phi(u\tau) & \text{if $u \in lk_X(\tau) $},\\
\ 0 & \text{otherwise}.\\
\end{cases}$$
We now recall some results from [@ABM], which was stated and proved for a clique complex but the same proof is also valid for any general simplicial complex.
\[claim3.1\] [@ABM Claim $3.1$] For $\phi \in C^k(X;{\mathbb{R}})$ $$\label{e:ABM1}
||\delta_k \phi||^2 \ = \sum_{\sigma \in X(k)} \mbox{deg}(\sigma) \phi(\sigma)^2 - 2 \sum_{\eta \in X(k-1)} \sum_{vw\in lk_X(\eta)} \phi(v\eta)\phi(w\eta).$$
\[claim3.3\] [@ABM Claim $3.3$] For $\phi \in C^k(X;{\mathbb{R}})$ $$\label{e:ABM3}
\sum_{u \in V(X)} ||\delta_{k-2}^{\ast} \phi_u||^2 = k ||\delta_{k-1}^{\ast} \phi||^2.$$
\[claim3.2\] [@ABM page $7$, upto second equality in the proof of Claim $3.2$] $$\label{e:ABM2}
\begin{split}
\sum_{u \in V(X)} ||\delta_{k-1} \phi_u||^2 & = \sum_{\sigma \in X(k)} \Big(\sum_{\tau \in \sigma(k-1)} \mbox{deg}(\tau)\Big) \phi(\sigma)^2 \\
& \ \ \ \ - 2 \sum_{\eta \in X(k-2)} \sum_{vw \in lk_X(\eta)} \sum_{u\in lk_X(v\eta) \cap lk_X(w\eta)} \phi(vu\eta)\phi(wu\eta).
\end{split}$$
Let $0 \neq \psi \in C^k(X;{\mathbb{R}})$ be an eigenvector of $\Delta_k$ with eigenvalue $\mu_k(X)$. By double counting $$\begin{aligned}
\label{e:doublecounting}
\sum\limits_{v \in V(X)} ||\psi_v||^2 = (k+1)||\psi||^2.
\end{aligned}$$
We first derive the expression for $\sum\limits_{u \in V(X)} \langle \Delta_{k-1} \psi_u, \psi_u \rangle$. We shall use in the second equality below. $$\label{e:ABMequation1}
\begin{split}
\sum\limits_{u \in V(X)} \langle \Delta_{k-1} \psi_u, \psi_u \rangle & = \sum\limits_{u \in V(X)}(||\delta_{k-1} \psi_u||^2 + ||\delta_{k-2}^{\ast}\psi_u||^2)\\
& = \sum_{u \in V(X)}||\delta_{k-2}^{\ast}\psi_u||^2 + \sum_{\sigma \in X(k)} \Big(\sum_{\tau \in \sigma(k-1)} \mbox{deg}(\tau)\Big)\psi(\sigma)^2\\
& \ \ - 2 \sum_{\eta \in X(k-2)} \sum_{vw \in lk_X(\eta)} \sum_{u \in lk_X(v \eta) \cap lk_X(w \eta)} \psi(vu \eta)\psi(wu \eta).
\end{split}$$ Now, we relate $\sum\limits_{u \in V(X)} \langle \Delta_{k-1} \psi_u, \psi_u \rangle$ to $k\langle \Delta_k \psi, \psi \rangle$. In the following derivation, we shall use and for the second equality and the third equality will follow from . $$\begin{split}
k\langle \Delta_k \psi, \psi \rangle & = k(||\delta_{k} \psi||^2 + ||\delta_{k-1}^{\ast}\psi||^2) \\
& = k\Big(\sum_{\sigma \in X(k)}\mbox{deg}(\sigma)\psi(\sigma)^2-2\sum_{\eta\in X(k-1)} \sum_{vw \in lk_X(\eta)} \psi(v\eta)\psi(w\eta)\Big)\\
& ~~~~ + \sum_{u \in V(X)} ||\delta_{k-2}^{\ast}\psi_u||^2\\
& = k\sum_{\sigma \in X(k)}\mbox{deg}(\sigma)\psi(\sigma)^2 - 2k\sum_{\eta\in X(k-1)} \sum_{vw \in lk_X(\eta)} \psi(v\eta)\psi(w\eta)\\
& ~~~~ + \sum_{u \in V(X)} \langle \Delta_{k-1}\psi_u, \psi_u\rangle - \sum_{\sigma \in X(k)}\Big( \sum_{\tau \in \sigma(k-1)} \mbox{deg}(\tau)\Big) \psi(\sigma)^2 \\
& ~~~~ + 2 \sum_{\eta \in X(k-2)} \sum_{vw \in lk_X(\eta)} \sum_{u \in lk_X(v\eta) \cap lk_X(w\eta)} \psi(vu\eta)\psi(wu\eta).
\end{split}$$ Thus, from the previous two derivations, we obtain that $$k \langle \Delta_k \psi, \psi \rangle = \sum\limits_{u \in V(X)} \langle \Delta_{k-1}\psi_u, \psi_u \rangle + I_1-I_2-T,$$ where $$\begin{aligned}
T & := \sum_{\sigma \in X(k)}\Big( \sum_{\tau \in \sigma(k-1)} \mbox{deg}(\tau) - k \ \mbox{deg}(\sigma)\Big) \psi(\sigma)^2, \label{e:T}\\
I_1 & := 2 \sum_{\eta \in X(k-2)} \sum_{vw \in lk_X(\eta)} \sum_{u \in lk_X(v\eta) \cap lk_X(w\eta)} \psi(vu\eta)\psi(wu\eta) \label{e:I1}
\end{aligned}$$ and $$\begin{aligned}
I_2 & := 2k\sum_{\eta\in X(k-1)} \sum_{vw \in lk_X(\eta)} \psi(v\eta)\psi(w\eta). \label{e:I2}
\end{aligned}$$ We now use the bounds for $|I_1 -I_2|$ and $T$ given in Claims \[i1i2\] and \[T1\] which we prove later at the end of this section. Combining Claims \[i1i2\] and \[T1\], we have the following. $$\begin{aligned}
k\langle\Delta_k \psi, \psi \rangle & \geq \sum_{v \in V(X)} \langle\Delta_{k-1}\psi_v, \psi_v\rangle -(|V(X)|+(k(k+1)+1) \sum\limits_{j=2}^{k+1} D_k(X,j)) ||\psi||^2. \label{e:Deltak}
\end{aligned}$$
From and we have $$\begin{aligned}
k \mu_k(X) ||\psi||^2 & = k\langle\Delta_k \psi, \psi\rangle \geq \sum_{v \in V(X)} \langle \Delta_{k-1}\psi_v, \psi_v \rangle-(n+(k(k+1)+1) \sum\limits_{j=2}^{k+1} D_k(X,j)) ||\psi||^2 \\
& \geq \mu_{k-1}(X) \sum_{v \in V(X)} ||\psi_v||^2 - (n+(k(k+1)+1) \sum\limits_{j=2}^{k+1} D_k(X,j)) ||\psi||^2 \\
& = ((k+1) \mu_{k-1}(X) - n - (k(k+1)+1) \sum\limits_{j=2}^{k+1} D_k(X,j)) ||\psi||^2.\end{aligned}$$
Since the $k$-skeleton of $X$ is a clique complex, $D_i(X,i) = 0$ for all $2 \leq i \leq k$. Hence, Theorem \[t:general\] implies that $\mu_k(X) \geq (k+1) \mu_0(X) - kn - (k(k+1)+1) D_k(X,k+1)$. Therefore, if $\mu_0(X) = \lambda_2(G) > \frac{kn}{k+1} + (k+ \frac{1}{k+1}) D_k(X,k+1)$, then $\mu_k(X) > 0$ and result follows from Proposition \[p:hodge\].
Now, we shall give proofs of Claims \[i1i2\] and \[T1\]. For a $k$-simplex $\sigma$ and $v_1, \ldots, v_l \in \sigma$, $\hat{\sigma}_{v_1 \ldots v_l} := \sigma \setminus \{v_1, \ldots, v_l\}$ is a ($k-l$)-simplex. Recalling the definition of $D_i(X, j), i \geq 1$ and $ 1 \leq j \leq i+1$ from and the definitions of $T, I_1$ and $I_2$ from , and respectively.
\[i1i2\] $$\label{I2I1}
|I_1-I_2| \leq k(k+1) \sum\limits_{j=2}^{k+1} D_k(X,j) ||\psi||^2.$$
In this proof, we use the convention that $\psi(\tau) = 0$, whenever $\tau \notin X(k)$. Observe that the expression for $I_2$ given in (\[e:I2\]) can be rewritten as, $$\begin{aligned}
I_2 = 2 \sum_{\eta \in X(k-2)} \sum_{\substack{\{v,w\} \\ vw \in lk_X(\eta)}} \sum_{\substack{u \in lk_X(v\eta) \cap lk_X(w\eta) \\ u \in lk_X(vw\eta)}} \psi(vu\eta)\psi(wu\eta).
\end{aligned}$$ By recalling the definition of $I_1$ from (\[e:I1\]), we obtain, $$\begin{aligned}
I_1-I_2 & = 2 \sum_{\eta \in X(k-2)} \sum_{\substack{\{v,w\} \\ vw \in lk_X(\eta)}} \sum_{\substack{u \in lk_X(v\eta) \cap lk_X(w\eta) \\ u \notin lk_X(vw\eta)}} \psi(vu\eta)\psi(wu\eta)\\
&= 2 \sum_{\sigma \in X(k)} \sum_{\{v, w\} \subseteq \sigma} \sum_{\substack{u \in lk_X(\hat{\sigma}_w) \cap lk_X(\hat{\sigma}_v) \\ u \notin lk_X(\sigma)}} \psi(uv\hat{\sigma}_{vw})\psi(uw\hat{\sigma}_{vw})\\
& = 2 \sum_{\sigma \in X(k)} \sum_{u \notin lk_X(\sigma)} \sum_{\{v, w\} \subseteq \sigma} \1[u \in lk_X(\hat{\sigma}_v) \cap lk_X(\hat{\sigma}_w)]\psi(uv\hat{\sigma}_{vw})\psi(uw\hat{\sigma}_{vw}) \\
&= 2 \sum_{\sigma \in X(k)} \sum\limits_{j = 2}^{k+1} \sum_{u \notin lk_X(\sigma)} \1[u \in \bigcap\limits_{i=1}^{j} lk_X(\hat{\sigma}_{v_i})\ \mbox{for exactly} \ j \ \mbox{vertices} \ v_1, \ldots, v_j \in \sigma]\\
& ~~~~~\sum_{\{v, w\} \subseteq \sigma} \1[u \in lk_X(\hat{\sigma}_v) \cap lk_X(\hat{\sigma}_w)]\psi(uv\hat{\sigma}_{vw})\psi(uw\hat{\sigma}_{vw})\\
&= 2 \sum_{\{v_1, \ldots, v_{k+2}\} \notin X(k+1)}\sum_{i \in [k+2]} \1[\gamma^i = \{v_1, \ldots, \hat{v_i}, \ldots, v_{k+2}\} \in X(k)]\\
& ~~~~~\sum\limits_{j=2}^{k+1} \1[v_i \in \bigcap\limits_{l=1}^{j} lk_X(\hat{\gamma^i}_{v_{i_l}}) \ \mbox{for exactly} \ j \ \mbox{vertices} \ v_{i_1}, \ldots, v_{i_j} \in \gamma^i] \\
& ~~~~~ \sum_{i \notin \{p, q\} \subseteq[k+2]} \1[v_i \in lk_X(\hat{\gamma^i}_{v_p}) \cap lk_X(\hat{\gamma^i}_{v_q}) ]\psi(v_iv_p \hat{\gamma^i}_{v_pv_q})\psi(v_iv_q\hat{\gamma^i}_{v_pv_q}) \\
& = 2 \sum_{\{v_1, \ldots, v_{k+2}\} \notin X(k+1)} \1[\gamma^i = \{v_1, \ldots, \hat{v_i}, \ldots, v_{k+2}\} \in X(k)] \\
& ~~~~~ \sum\limits_{j=2}^{k+1} \1[v_i \in \bigcap\limits_{l=1}^{j} lk_X(\hat{\gamma^i}_{v_{i_l}}) \ \mbox{for exactly} \ j \ \mbox{vertices} \ v_{i_1}, \ldots, v_{i_j} \in \gamma^i] \\ &~~~~~ \sum_{i \in [k+2]} \sum_{ \{p, q\} \in [k+2] \setminus\{i\}} \psi(v_iv_p \hat{\gamma^i}_{v_pv_q})\psi(v_iv_q\hat{\gamma^i}_{v_pv_q}) \notag.
\end{aligned}$$ Hence, using we obtain $$\begin{aligned}
|I_1-I_2| & \leq 2\cdot\frac{k}{2}\sum_{\{v_1, \ldots, v_{k+2}\} \notin X(k+1)} \1[\gamma^i = \{v_1, \ldots, \hat{v_i}, \ldots, v_{k+2}\} \in X(k)]\\
& ~~~~~ \sum\limits_{j=2}^{k+1} \1[v_i \in \bigcap\limits_{l=1}^{j} lk_X(\hat{\gamma^i}_{v_{i_l}}) \ \mbox{for exactly} \ j \ \mbox{vertices} \ v_{i_1}, \ldots, v_{i_j} \in \gamma^i] \\
&~~~~~~ \sum_{i \in [k+2]} \sum_{p \in [k+2]\setminus\{i\}} \psi(\gamma^p)^2\\
& = k(k+1)\sum_{\sigma \in X(k)} \psi(\sigma)^2 \sum\limits_{j=2}^{k+1} \\
&~~~~~ \sum\limits_{w \notin lk_X(\sigma)} \1[w \in \bigcap\limits_{l=1}^{j} lk_X(\hat\sigma_{v_{i_l}}) \ \mbox{for exactly} \ j \ \mbox{vertices} \ v_{i_1}, \ldots, v_{i_j} \in \sigma] \\
& \leq k(k+1)\sum_{\sigma \in X(k)} \psi(\sigma)^2 \sum\limits_{j=2}^{k+1} D_k(X,j) \\
& = k(k+1)\sum\limits_{j=2}^{k+1} D_k(X,j)||\psi||^2 .
\end{aligned}$$
\[T1\] $$\begin{aligned}
\label{T}
T & \leq (|V(X)|+ \sum\limits_{j=2}^{k+1} D_k(X,j)) ||\psi||^2.
\end{aligned}$$
$$\begin{aligned}
T & = \sum_{\sigma \in X(k)} \Big(\sum_{\tau \in \sigma(k-1)} \mbox{deg}(\tau) - k \ \mbox{deg}(\sigma) \Big) \psi(\sigma)^2 \\
& = \sum_{\sigma \in X(k)} \Big(\sum_{v \in \sigma} \sum_{u \in lk_X(\hat{\sigma}_v)} \psi(\sigma)^2 - k \sum_{v \in lk_X(\sigma)} \psi(\sigma)^2 \Big)\\
& = \sum_{\sigma \in X(k)} \sum_{u \in lk_X(\sigma)} \psi(\sigma)^2 + \sum_{\sigma \in X(k)} \sum_{v \in \sigma}\sum_{\substack{u \in lk_X(\hat{\sigma}_v) \\ u \notin lk_X(\sigma)}} \psi(\sigma)^2 \\
& = \sum_{\sigma \in X(k)} \sum_{u \in lk_{X}(\sigma)} \psi(\sigma)^2 + \sum_{\sigma \in X(k)} \Big(\sum_{u \notin lk_{X}(\sigma)} \sum_{v\in \sigma} \1[u \in lk_X(\hat{\sigma}_v)] \Big) \psi(\sigma)^2\\
& = T_1 + T_2
\end{aligned}$$
where, $$\begin{aligned}
T_1 & := \sum_{\sigma \in X(k)} \Big(\sum_{u \in lk_{X}(\sigma)} \psi(\sigma)^2 + \sum_{u \notin lk_X(\sigma)}\1[u \in lk_X(\hat{\sigma}_v) \ \mbox{for exactly one} \ v \in \sigma] \psi(\sigma)^2\Big) \nonumber\\
& \leq |V(X)| \sum_{\sigma \in X(k)} \psi(\sigma)^2 = |V(X)| ||\psi||^2 \nonumber
\end{aligned}$$ and $$\begin{aligned}
T_2 & := \sum_{\sigma \in X(k)} \sum\limits_{j=2}^{k+1} \sum_{u \notin lk_X(\sigma)} \1[u \in \bigcap\limits_{i=1}^{j} lk_X(\hat{\sigma}_{v_i})\ \mbox{for exactly} \ j \ \mbox{vertices} \ v_1, \ldots, v_j \in \sigma]\psi(\sigma)^2 \nonumber\\
& \leq \sum_{\sigma \in X(k)} \sum\limits_{j=2}^{k+1} D_k(X,j)\psi(\sigma)^2 \nonumber\\
& \leq \sum\limits_{j = 2}^{k+1}D_k(X,j) ||\psi||^2 \nonumber.\end{aligned}$$
Proofs of the results of section \[subsec:randomneighborhood\] {#sec:random}
--------------------------------------------------------------
Let $p = \Big(\frac{(k+1)\log n +c_n}{n}\Big)^{\frac{1}{k+2}}$, where $c_n \to \infty$. The two main proof steps are (i) $\mathcal{N}(G(n, p))$ has full $k$-skeleton and (ii) $n^{-1}\mathbb{E}(D_k({\mathcal{N}}(G(n,p)),k+1)) \to 0$.
From (ii) and Markov’s inequality, we have that for all $\epsilon > 0$, ${\mathbb{P}}(D_k({\mathcal{N}}(G(n,p)),k+1) \geq \epsilon \frac{ n}{k(k+1)+1}) \to 0$ and hence w.h.p. $D_k({\mathcal{N}}(G(n,p)),k+1) < \frac{ n}{k(k+1)+1}$. Let $G$ be the $1$-skeleton of $\mathcal{N}(G(n, p))$. Since $G$ is a complete graph w.h.p. (because of (i)), the spectral gap $\lambda_2(G) = n$ and therefore Corollary \[cor:general\] and (i) imply that $\widetilde{H}^k(\mathcal{N}(G(n, p))) = 0$. This completes the proof provided we establish (i) and (ii).
First, we show (i). The expected number of $(k+1)$-tuples of vertices in $G(n, p)$ with no neighbor is $$\begin{aligned}
{n \choose k+1}(1-p^{k+1})^{n-k-1} & \leq {n \choose k+1} e^{-(n-k-1)p^{k+1}} \\
& = {n \choose k+1} e^{-n\big(\frac{(k+1)\log n + c_n}{n}\big)^{\frac{k+1}{k+2}}} e^{(k+1)\big(\frac{(k+1)\log n + c_n}{n}\big)^{\frac{k+1}{k+2}}}\\
&={n \choose k+1}e^{-n^{\frac{1}{k+2}} ((k+1)\log n + c_n)^{\frac{k+1}{k+2}}}e^{(k+1)\big(\frac{(k+1)\log n + c_n}{n}\big)^{\frac{k+1}{k+2}}}\\
& = o(1).
\end{aligned}$$ Hence, w.h.p. $\mathcal{N}(G(n, p))$ has full $k$-skeleton. In particular, w.h.p. $k$-skeleton of ${\mathcal{N}}(G(n, p))$ is a clique complex. It is well known that, if a complex has full $k$-skeleton then it has trivial cohomology in all dimensions less than $k$. Therefore $\widetilde{H}^i(\mathcal{N}(G(n, p))) = 0$ for $i < k$.
Now, we shall establish (ii). Let $B_k$ be the number of subcomplexes of ${\mathcal{N}}(G(n, p))$, which are isomorphic to the simplicial boundary of a $(k+1)$-simplex. Since, $k$-skeleton of ${\mathcal{N}}(G(n,p))$ is a clique complex, from we observe that $D_k({\mathcal{N}}(G(n, p)),k+1) \leq B_k$. Thus it suffices to show that $n^{-1}{\mathbb{E}}(B_k) \to 0$ to prove (ii).
The rest of the proof is to compute ${\mathbb{E}}(B_k)$ and show the above. For $0 \leq i \leq { k+2 \choose 2}$, let $C_i$ denotes the number of graphs on $k+2$ vertices with $i$ edges. For any $\{v_1, \ldots, v_{k+2}\} \subset [n]$, the probability that the induced subcomplex of ${\mathcal{N}}(G(n,p))$ on $\{v_1, \ldots, v_{k+2}\}$ is isomorphic to the simplicial boundary of a $(k+1)$-simplex, is bounded above by $(1-p^{k+2})^{n-k-2}$. Therefore $$\begin{aligned}
\frac{\mathbb{E} B_k}{n} & \leq \frac{1}{n} {n \choose k+2} \sum\limits_{i=0}^{{k+2 \choose 2}}C_i p^i(1-p)^{{k+2 \choose 2} - i} (1-p^{k+2})^{n-k-2} \\
& \leq \frac{1}{n}{n \choose k+2} \sum\limits_{i=0}^{{k+2 \choose 2}} C_i p^i(1-p)^{{k+2 \choose 2} - i} e^{-(n-k-2)p^{k+2}}\\
& = \frac{1}{n}{n \choose k+2} \sum\limits_{i=0}^{{k+2 \choose 2}} C_i p^i(1-p)^{{k+2 \choose 2} - i} e^{-(n-k-2)\frac{(k+1)\log n + c_n}{n}}\\
& = {n \choose k+2} n^{-(k+2)} e^{-c_n} e^{(k+2)\frac{(k+1)\log n + c_n}{n}} \sum\limits_{i=0}^{{k+2 \choose 2}} C_i (1-p)^{{k+2 \choose 2} - i} p^i.
\end{aligned}$$ Now using that $c_n \to \infty$ and $p \to 0$, we derive that $n^{-1}{\mathbb{E}}(B_k) \to 0$ completing the proof of (ii) as well as that of the theorem.
Let $U = \{u_1, \ldots, u_r\}, V = \{v_1, \ldots, v_r\}$ be subsets of $[n]$ such that $U \cap V = \emptyset.$ The graph $X_{U,V}$ is defined as the graph with vertex set $U \cup V$ and edges $u_i \sim u_j$ and $u_i \sim v_j$ for all $ 1 \leq i \neq j \leq r$. To prove Proposition \[p:extension\], we need the following result relating $X_{U,V}$ graphs to cohomology. For two sequences of real numbers $a_n, b_n, n \geq 1$, we shall use the notation $a_n = o(b_n)$ to denote that $a_n / b_n \to 0$ as $n \to \infty$.
\[kahleprop\][@kahle Theorem $2.7$] If $H$ is any graph containing a maximal clique of order $r$ that cannot be extended to an $X_{U,V}$ subgraph for some $U,V$, then ${\mathcal{N}}(H)$ retract onto a sphere $\mathbb{S}^{r-2}$.
Let $p = n^{\alpha}$, where $ \frac{-2}{r-1} < \alpha < \frac{-1}{r-1}, r \geq 2$. We shall set $G_n = G(n,p)$. Let $$\begin{aligned}
\Lambda_r := & |\{ A \subset [n] \ | \ |A| = r, G_n[A] \ \text{is a maximal clique and} \ G_n[A] \not\subseteq X_{A,A'} \ \text{for all} \ A' \ \text{disjoint} \\
& \text{and} \ |A'| = r \}|.
\end{aligned}$$ For a $A \subset [n]$ with $|A| = r$, let $$\begin{aligned}
I_A & := \1[\mbox{$G_n[A]$ is a $r$-clique}], \, \, J_A := \prod_{A' : A' \supsetneq A} \1[\mbox{$G_n[A']$ is not a clique}], \\
K_A &:= \prod_{ \substack{ A' : |A'| = r \\ A' \cap A = \emptyset}}\1[[G_n[A] \not\subseteq X_{A,A'} ].
\end{aligned}$$
Then, $\Lambda_r = \sum\limits_{A \subset [n], |A| = r} I_A J_A K_A$. By Proposition \[kahleprop\], the proof is complete provided we show that $\Lambda_r \geq 1$ w.h.p.. To do so, we shall use the second moment bound, i.e., $$\mathbb{P}(\Lambda_r \geq 1) \geq \frac{(\mathbb{E} \Lambda_r)^2} {\mathbb{E}\Lambda_r^2}.$$ To use the second moment bound, we first derive a lower bound for ${\mathbb{E}}(\Lambda_r)$. Fix $A = [r]$ in the below derivation. $$\begin{aligned}
{\mathbb{E}}(\Lambda_r) &= {n \choose r} {\mathbb{E}}(I_AJ_AK_A) = {n \choose r} {\mathbb{E}}\left(I_A (J_A - J_A\1[\cup_{\substack{A' : |A'| = r \\ A' \cap A =\emptyset}} \{G_n[A] \subset X_{A,A'} \} ]) \right) {\nonumber}\\
& \geq {n \choose r} {\mathbb{E}}\Bigg( I_A \Bigg( J_A - {\nonumber}\\
& ~~~~~~~~~~ \sum_{\substack{A' = \{u_1, \ldots, u_r\} \\ A' \cap A =\emptyset}} \1[G_n[A] \subset X_{A,A'} ] \1[i \nsim u_i \ \forall i] \prod_{v \notin A' \cup A}\1[ i \nsim v \ \mbox{for some} \ i \in A] \Bigg) \Bigg) {\nonumber}\\
& \geq {n \choose r} {\mathbb{E}}\left( \1[i \sim j \ \forall \ 1 \leq i \neq j \leq r] {\mathbb{E}}\left( \prod_{v \notin A}\1[ i \nsim v \ \mbox{for some $i \in A$}] - \right. \right. {\nonumber}\\
& \left. \left. \sum_{\substack{A' = \{u_1, \ldots, u_r\} \\ A' \cap A =\emptyset}} \1[u_i \sim j \ \forall \ 1 \leq i \neq j \leq r]\1[i \nsim u_i \ \forall i] \prod_{v \notin A' \cup A}\1[ i \nsim v \ \mbox{for some $i \in A$}] \right) \right) {\nonumber}\\
& = {n \choose r} p^{{r \choose 2}} ((1-p^r)^{n-r} - {n-r \choose r} p^{r(r-1)}(1-p)^r(1-p^r)^{n-2r} ) {\nonumber}\\
& \geq {n \choose r} p^{{r \choose 2}}(1-p^r)^{n-r}(1 - n^{r} p^{r(r-1)}(1-p)^r(1-p^r)^{-r}), {\nonumber}\end{aligned}$$ where in the equality in the penultimate line we have used the independence between the corresponding indicator random variables as they depend on disjoint sets of edges. Since $p = n^{\alpha}$ for $\alpha < \frac{-1}{r-1}$, we have that $n^{r} p^{r(r-1)}(1-p)^r(1-p^r)^{-r} \to 0$ and hence for large enough $n$, we derive that $$\label{eq:ELambda}
{\mathbb{E}}(\Lambda_r) \geq {n \choose r} p^{{r \choose 2}}(1-p^r)^{n-r} (1 - o(1)).$$ Now, we proceed to derive upper bounds for the second moment. $$\Lambda_r^2 =\sum\limits_{i = 0}^{r} \sum\limits_{\substack{ |A_1| = |A_2| = r \\ |A_1 \cap A_2| = i } } I_{A_1}J_{A_1} K_{A_1} I_{A_2} J_{A_2}K_{A_2} \leq \sum\limits_{i=0}^r Y_i,$$ where $Y_i := \sum\limits_{\substack{|A_1| = |A_2| = r \\ |A_1 \cap A_2| = i} } I_{A_1} I_{A_2}$ for $0 \leq i \leq r$. Hence, using , we derive that for large enough $n$ $$\begin{aligned}
\frac{\mathbb{E}\Lambda_r^2} {(\mathbb{E} \Lambda_r)^2} &\leq \frac{1} {(\mathbb{E} \Lambda_r)^2} \sum\limits_{i=0}^r \mathbb{E} Y_i \nonumber \\
& \leq \frac{1}{({n \choose r} p^{{r \choose 2}}(1-p^r)^{n-r} (1 - o(1)))^2} \sum\limits_{i=0}^r{n \choose 2r-i} p^{r(r-1) - \frac{i(i-1)}{2}} \nonumber \\
& = \frac{{n \choose 2r} }{{n \choose r}^2(1-p^r)^{2(n-r)}(1 - o(1))^2} + \sum\limits_{i=1}^r{n \choose 2r-i} \frac{ p^{r(r-1) - \frac{i(i-1)}{2}} } {{n \choose r}^2 p^{r(r-1)}(1-p^r)^{2(n-r)}(1 - o(1))^2} \nonumber \\
& = \frac{{n \choose 2r} }{{n \choose r}^2(1-p^r)^{2(n-r)}(1 - o(1))^2} + \sum\limits_{i=1}^r \frac{ {n \choose 2r-i} } {{n \choose r}^2 n^{\alpha \frac{i(i-1)}{2}}(1-p^r)^{2(n-r)}(1 - o(1))^2} \nonumber \\
& \leq \frac{{n \choose 2r}}{{n \choose r}^2(1-p^r)^{2(n-r)}(1 -o(1))^2} + C \sum\limits_{i=1}^r \frac{n^{2r}} {{n \choose r}^2 n^{ i + \alpha \frac{i(i-1)}{2}}(1-p^r)^{2(n-r)}(1 -o(1))^2} \nonumber \\
& = 1 + o(1) \label{lambda0},
\end{aligned}$$ where $C$ is a constant and follows as $1+\alpha(r-1) < 0$, $i + \alpha \frac{i(i-1)}{2} > 0$ (because $\alpha > \frac{-2}{r-1}$) and further $(1-p^r)^{2(n-r)} = e^{-2np^r}(1 + o(1)) = 1 + o(1)$ for large $n$. From , we conclude that $\liminf_{n \to \infty}\frac{(\mathbb{E} \Lambda_r)^2} {\mathbb{E}\Lambda_r^2} = 1$. Therefore by the second moment bound, we derive that $\mathbb{P}(\Lambda_r \geq 1) \to1$ as $n \to \infty$ as required.
Acknowledgements {#acknowledgements .unnumbered}
================
The first author was financially supported partially by the Indian Statistical Institute Bangalore, India, where this work was done and partially by the DST INSPIRE Faculty award of Dr. D. Y.. D.Y. was supported by DST INSPIRE Faculty award and CPDA from the Indian Statistical Institute.
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[^1]: Department of Mathematics, Indian Institute of Technology Bombay, India. [email protected]
[^2]: Statistics and Mathematics Unit, Indian Statistical Institute Bangaluru, India. [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Splitting and rephrasing a complex sentence into several shorter sentences that convey the same meaning is a challenging problem in NLP. We show that while vanilla seq2seq models can reach high scores on the proposed benchmark [@shashi2017], they suffer from memorization of the training set which contains more than 89% of the unique simple sentences from the validation and test sets. To aid this, we present a new train-development-test data split and neural models augmented with a copy-mechanism, outperforming the best reported baseline by 8.68 BLEU and fostering further progress on the task.'
author:
- |
Roee Aharoni & Yoav Goldberg\
Computer Science Department\
Bar-Ilan University\
Ramat-Gan, Israel\
`{roee.aharoni,yoav.goldberg}@gmail.com`\
bibliography:
- 'sprp.bib'
title: 'Split and Rephrase: Better Evaluation and a Stronger Baseline'
---
@topnum0 @botnum0
Introduction {#sec:intro}
============
Processing long, complex sentences is challenging. This is true either for humans in various circumstances [@inui2003text; @watanabe2009facilita; @de2010text] or in NLP tasks like parsing [@tomita1986efficient; @mcdonald2011analyzing; @jelinek2014improvements] and machine translation [@chandrasekar1996motivations; @pougetabadie-EtAl:2014:SSST-8; @koehn2017six]. An automatic system capable of breaking a complex sentence into several simple sentences that convey the same meaning is very appealing.
A recent work by introduced a dataset, evaluation method and baseline systems for the task, naming it “[[S]{}plit-and-[R]{}ephrase]{}”. The dataset includes 1,066,115 instances mapping a single complex sentence to a sequence of sentences that express the same meaning, together with RDF triples that describe their semantics. They considered two system setups: a text-to-text setup that does not use the accompanying RDF information, and a semantics-augmented setup that does. They report a BLEU score of 48.9 for their best text-to-text system, and of 78.7 for the best RDF-aware one. We focus on the text-to-text setup, which we find to be more challenging and more natural.
We begin with vanilla [<span style="font-variant:small-caps;">Seq2Seq</span>]{} models with attention [@Bahdanau2014NeuralMT] and reach an accuracy of 77.5 BLEU, substantially outperforming the text-to-text baseline of and approaching their best RDF-aware method. However, manual inspection reveal many cases of unwanted behaviors in the resulting outputs: (1) many resulting sentences are *unsupported* by the input: they contain correct facts about relevant entities, but these facts were not mentioned in the input sentence; (2) some facts are *repeated*—the same fact is mentioned in multiple output sentences; and (3) some facts are *missing*—mentioned in the input but omitted in the output.
The model learned to *memorize entity-fact pairs* instead of learning to split and rephrase. Indeed, feeding the model with examples containing entities alone without any facts about them causes it to output perfectly phrased but unsupported facts (Table \[tab:issues\]). Digging further, we find that 99% of the simple sentences (more than 89% of the unique ones) in the validation and test sets also appear in the training set, which—coupled with the good memorization capabilities of [<span style="font-variant:small-caps;">Seq2Seq</span>]{} models and the relatively small number of distinct simple sentences—helps to explain the high [<span style="font-variant:small-caps;">BLEU</span>]{} score.
To aid further research on the task, we propose a more challenging split of the data. We also establish a stronger baseline by extending the [<span style="font-variant:small-caps;">Seq2Seq</span>]{} approach with a copy mechanism, which was shown to be helpful in similar tasks [@gu2016incorporating; @merity2016pointer; @abc2017]. On the original split, our models outperform the best baseline of by up to 8.68 BLEU, without using the RDF triples. On the new split, the vanilla <span style="font-variant:small-caps;">Seq2Seq</span> models break completely, while the copy-augmented models perform better. In parallel to our work, an updated version of the dataset was released (v1.0), which is larger and features a train/test split protocol which is similar to our proposal. We report results on this dataset as well. The code and data to reproduce our results are available on Github.[^1] We encourage future work on the split-and-rephrase task to use our new data split or the v1.0 split instead of the original one.
Preliminary Experiments
=======================
#### Task Definition
In the split-and-rephrase task we are given a complex sentence $C$, and need to produce a sequence of simple sentences $T_1,...,T_n$, $n\geq2$, such that the output sentences convey all and only the information in $C$. As additional supervision, the split-and-rephrase dataset associates each sentence with a set of RDF triples that describe the information in the sentence. Note that the number of simple sentences to generate is not given as part of the input.
Model BLEU \#S/C \#T/S
------------------------------------------------------------------------ ------- ------- -------
<span style="font-variant:small-caps;">Source</span> 55.67 1.0 21.11
<span style="font-variant:small-caps;">Reference</span> – 2.52 10.93
[<span style="font-variant:small-caps;">HybridSimpl</span>]{} 39.97 1.26 17.55
[<span style="font-variant:small-caps;">Seq2Seq</span>]{} 48.92 2.51 10.32
[<span style="font-variant:small-caps;">MultiSeq2Seq</span>]{}\* 42.18 2.53 10.69
[<span style="font-variant:small-caps;">Split-MultiSeq2Seq</span>]{}\* 77.27 2.84 11.63
[<span style="font-variant:small-caps;">Split-Seq2Seq</span>]{}\* 78.77 2.84 9.28
This work
<span style="font-variant:small-caps;">Seq2Seq128</span> 76.56 2.53 10.53
<span style="font-variant:small-caps;">Seq2Seq256</span> 77.48 2.57 10.56
<span style="font-variant:small-caps;">Seq2Seq512</span> 75.92 2.59 10.59
: BLEU scores, simple sentences per complex sentence (\#S/C) and tokens per simple sentence (\#T/S), as computed over the test set. <span style="font-variant:small-caps;">Source</span> are the complex sentences and <span style="font-variant:small-caps;">Reference</span> are the reference rephrasings from the test set. Models marked with \* use the semantic RDF triples.[]{data-label="tab:results"}
#### Experimental Details {#sec:experimental_details}
We focus on the task of splitting a complex sentence into several simple ones *without* access to the corresponding RDF triples in either train or test time. For evaluation we follow and compute the averaged individual multi-reference BLEU score for each prediction.[^2] We split each prediction to sentences[^3] and report the average number of simple sentences in each prediction, and the average number of tokens for each simple sentence. We train vanilla sequence-to-sequence models with attention [@Bahdanau2014NeuralMT] as implemented in the <span style="font-variant:small-caps;">Opennmt-py</span> toolkit [@2017opennmt].[^4] Our models only differ in the LSTM cell size (128, 256 and 512, respectively). See the supplementary material for training details and hyperparameters. We compare our models to the baselines proposed in . [<span style="font-variant:small-caps;">HybridSimpl</span>]{} and [<span style="font-variant:small-caps;">Seq2Seq</span>]{} are text-to-text models, while the other reported baselines additionally use the RDF information.
#### Results
As shown in Table \[tab:results\], our 3 models obtain higher BLEU scores then the [<span style="font-variant:small-caps;">Seq2Seq</span>]{} baseline, with up to 28.35 BLEU improvement, despite being single-layer models vs. the 3-layer models used in . A possible explanation for this discrepancy is the [<span style="font-variant:small-caps;">Seq2Seq</span>]{} baseline using a dropout rate of 0.8, while we use 0.3 and only apply it on the LSTM outputs. Our results are also better than the [<span style="font-variant:small-caps;">MultiSeq2Seq</span>]{} and [<span style="font-variant:small-caps;">Split-MultiSeq2Seq</span>]{} models, which use explicit RDF information. We also present the macro-average[^5] number of simple sentences per complex sentence in the reference rephrasings (<span style="font-variant:small-caps;">Reference</span>) showing that the [<span style="font-variant:small-caps;">Split-MultiSeq2Seq</span>]{} and [<span style="font-variant:small-caps;">Split-Seq2Seq</span>]{} baselines may suffer from over-splitting since the reference splits include 2.52 simple sentences on average, while the mentioned models produced 2.84 sentences. [ ]{}
#### Analysis {#sec:analysis}
We begin analyzing the results by manually inspecting the model’s predictions on the validation set. This reveals three common kinds of mistakes as demonstrated in Table \[tab:issues\]: unsupported facts, repetitions, and missing facts. All the unsupported facts seem to be related to entities mentioned in the source sentence. Inspecting the attention weights (Figure \[fig:vanilla\_attn\]) reveals a worrying trend: throughout the prediction, the model focuses heavily on the first word in of the first entity (“A wizard of Mars”) while paying little attention to other cues like “hardcover”, “Diane” and “the ISBN number”. This explains the abundance of “hallucinated” unsupported facts: rather than learning to split and rephrase, the model learned to identify entities, and spit out a list of facts it had memorized about them. To validate this assumption, we count the number of predicted sentences which appeared as-is in the training data. We find that 1645 out of the 1693 (97.16%) predicted sentences appear verbatim in the training set. Table \[tab:stats\] gives more detailed statistics on the [<span style="font-variant:small-caps;">WebSplit</span>]{} dataset.
To further illustrate the model’s recognize-and-spit strategy, we compose inputs containing an entity string which is duplicated three times, as shown in the bottom two rows of Table \[tab:issues\]. As expected, the model predicted perfectly phrased and correct facts about the given entities, although these facts are clearly not supported by the input.
New Data-split {#sec:split}
==============
The original data-split is not suitable for measuring generalization, as it is susceptible to “cheating” by fact memorization. We construct a new train-development-test split to better reflect our expected behavior from a split-and-rephrase model. We split the data into train, development and test sets by randomly dividing the 5,554 distinct complex sentences across the sets, while using the provided RDF information to ensure that:
1. Every possible RDF relation (e.g., <span style="font-variant:small-caps;">BornIn</span>, <span style="font-variant:small-caps;">LocatedIn</span>) is represented in the training set (and may appear also in the other sets).
2. Every RDF triplet (a complete fact) is represented only in one of the splits.
While the set of complex sentences is still divided roughly to 80%/10%/10% as in the original split, now there are nearly no simple sentences in the development and test sets that appear verbatim in the train-set. Yet, every relation appearing in the development and test sets is supported by examples in the train set. We believe this split strikes a good balance between challenge and feasibility: to succeed, a model needs to learn to identify relations in the complex sentence, link them to their arguments, and produce a rephrasing of them. However, it is not required to generalize to unseen relations. [^6]
The data split and scripts for creating it are available on Github.[^7] Statistics describing the data split are detailed in Table \[tab:RDF\_split\_stats\].
Copy-augmented Model {#sec:copy}
====================
To better suit the split-and-rephrase task, we augment the [<span style="font-variant:small-caps;">Seq2Seq</span>]{} models with a copy mechanism. Such mechanisms have proven to be beneficial in similar tasks like abstractive summarization [@gu2016incorporating; @abc2017] and language modeling [@merity2016pointer]. We hypothesize that biasing the model towards copying will improve performance, as many of the words in the simple sentences (mostly corresponding to entities) appear in the complex sentence, as evident by the relatively high BLEU scores for the <span style="font-variant:small-caps;">Source</span> baseline in Table \[tab:results\].
Copying is modeled using a “copy switch” probability $p(z)$ computed by a sigmoid over a learned composition of the decoder state, the context vector and the last output embedding. It interpolates the $p_{softmax}$ distribution over the target vocabulary and a copy distribution $p_{copy}$ over the source sentence tokens. $p_{copy}$ is simply the computed attention weights. Once the above distributions are computed, the final probability for an output word $w$ is: $$\begin{aligned}
p(w) = p(z=1) p_{copy}(w) + p(z=0) p_{softmax}(w)
\label{eq:pw}\end{aligned}$$ In case $w$ is not present in the output vocabulary, we set $p_{softmax}(w)=0$. We refer the reader to for a detailed discussion regarding the copy mechanism.
Experiments and Results
=======================
Models with larger capacities may have greater representation power, but also a stronger tendency to memorize the training data. We therefore perform experiments with copy-enhanced models of varying LSTM widths (128, 256 and 512). We train the models using the negative log likelihood of $p(w)$ as the objective. Other than the copy mechanism, we keep the settings identical to those in Section \[sec:experimental\_details\]. We train models on the original split, our proposed data split and the v1.0 split.
-- ----------------------------------------------------------------- -------------------------------------------------------- ------- -------
[<span style="font-variant:small-caps;">BLEU</span>]{} \#S/C \#T/S
<span style="font-variant:small-caps;">Source</span> 55.67 1.0 21.11
<span style="font-variant:small-caps;">Reference</span> – 2.52 10.93
[<span style="font-variant:small-caps;">Split-Seq2Seq</span>]{} 78.77 2.84 9.28
<span style="font-variant:small-caps;">Seq2Seq128</span> 76.56 2.53 10.53
<span style="font-variant:small-caps;">Seq2Seq256</span> 77.48 2.57 10.56
<span style="font-variant:small-caps;">Seq2Seq512</span> 75.92 2.59 10.59
<span style="font-variant:small-caps;">Copy128</span> 78.55 2.51 10.29
<span style="font-variant:small-caps;">Copy256</span> 83.73 2.49 10.66
<span style="font-variant:small-caps;">Copy512</span> 87.45 2.56 10.50
<span style="font-variant:small-caps;">Source</span> 55.66 1.0 20.37
<span style="font-variant:small-caps;">Reference</span> – 2.40 10.83
<span style="font-variant:small-caps;">Seq2Seq128</span> 5.55 2.27 11.68
<span style="font-variant:small-caps;">Seq2Seq256</span> 5.28 2.27 10.54
<span style="font-variant:small-caps;">Seq2Seq512</span> 6.68 2.44 10.23
<span style="font-variant:small-caps;">Copy128</span> 16.71 2.0 10.53
<span style="font-variant:small-caps;">Copy256</span> 23.78 2.38 10.55
<span style="font-variant:small-caps;">Copy512</span> 24.97 2.87 10.04
<span style="font-variant:small-caps;">Source</span> 56.1 1.0 20.4
<span style="font-variant:small-caps;">Reference</span> – 2.48 10.69
<span style="font-variant:small-caps;">Copy512</span> 25.47 2.29 11.74
-- ----------------------------------------------------------------- -------------------------------------------------------- ------- -------
: Results over the test sets of the original, our proposed split and the v1.0 split[]{data-label="tab:results_shuffled"}
#### Results
Table \[tab:results\_shuffled\] presents the results. On the original data-split, the <span style="font-variant:small-caps;">Copy512</span> model outperforms all baselines, improving over the previous best by 8.68 BLEU points. On the new data-split, as expected, the performance degrades for all models, as they are required to generalize to sentences not seen during training. The copy-augmented models perform better than the baselines in this case as well, with a larger relative gap which can be explained by the lower lexical overlap between the train and the test sets in the new split. On the v1.0 split the results are similar to those on our split, in spite of it being larger (1,331,515 vs. 886,857 examples), indicating that merely adding data will not solve the task.
#### Analysis {#analysis}
We inspect the models’ predictions for the first 20 complex sentences of the original and new validation sets in Table \[tab:analysis\_copy\]. We mark each simple sentence as being “correct” if it contains all and only relevant information, “unsupported” if it contains facts not present in the source, and “repeated” if it repeats information from a previous sentence. We also count missing facts. Figure \[fig:copy\_mech\_attn\] shows the attention weights of the <span style="font-variant:small-caps;">Copy512</span> model for the same sentence in Figure \[fig:vanilla\_attn\]. Reassuringly, the attention is now distributed more evenly over the input symbols.
On the new splits, all models perform catastrophically. Table \[tab:issues\_new\_split\] shows outputs from the <span style="font-variant:small-caps;">Copy512</span> model when trained on the new split. On the original split, while <span style="font-variant:small-caps;">Seq2Seq128</span> mainly suffers from missing information, perhaps due to insufficient memorization capacity, <span style="font-variant:small-caps;">Seq2Seq512</span> generated the most unsupported sentences, due to overfitting or memorization. The overall number of issues is clearly reduced in the copy-augmented models.
Conclusions
===========
We demonstrated that a [<span style="font-variant:small-caps;">Seq2Seq</span>]{} model can obtain high scores on the original split-and-rephrase task while not actually learning to split-and-rephrase. We propose a new and more challenging data-split to remedy this, and demonstrate that the cheating [<span style="font-variant:small-caps;">Seq2Seq</span>]{} models fail miserably on the new split. Augmenting the [<span style="font-variant:small-caps;">Seq2Seq</span>]{} models with a copy-mechanism improves performance on both data splits, establishing a new competitive baseline for the task. Yet, the split-and-rephrase task (on the new split) is still far from being solved. We strongly encourage future research to evaluate on our proposed split or on the recently released version 1.0 of the dataset, which is larger and also addresses the overlap issues mentioned here.
### Acknowledgments {#acknowledgments .unnumbered}
We thank Shashi Narayan and Jan Botha for their useful comments. The work was supported by the Intel Collaborative Research Institute for Computational Intelligence (ICRI-CI), the Israeli Science Foundation (grant number 1555/15), and the German Research Foundation via the German-Israeli Project Cooperation (DIP, grant DA 1600/1-1).
Appendix A {#appendix-a .unnumbered}
==========
Training details {#training-details .unnumbered}
================
Outr models are trained with early stopping by running the proposed evaluation method on the development set after every epoch. We use a single layer LSTM for the encoder and decoder. We tie the embeddings of the encoder and the decoder, and preliminary experiments showed similar results without tying. In all models The size of the embedding vectors is similar to the size of the LSTM units (128/256/512). We decode using beam search with a beam size of 12. All model parameters, including the embeddings are randomly initialized and learned during training. For optimization we use SGD with an initial learning rate of 1.0 and decay the learning rate by 0.5 when there is no improvement on the validation set.
[^1]: <https://github.com/biu-nlp/sprp-acl2018>
[^2]: Note that this differs from “normal” multi-reference BLEU (as implemented in ) since the number of references differs among the instances in the test-set.
[^3]: Using <span style="font-variant:small-caps;">NLTK</span> v3.2.5 <https://www.nltk.org/>
[^4]: <https://github.com/OpenNMT/OpenNMT-py> commit d4ab35a
[^5]: Since the number of references varies greatly from one complex sentence to another, (min: 1, max: 76,283, median: 16) we avoid bias towards the complex sentences with many references by performing macro average, i.e. we first average the number of simple sentences in each reference among the references of a specific complex sentence, and then average these numbers.
[^6]: The updated dataset (v1.0, published by Narayan et al. after this work was accepted) follows (2) above, but not (1).
[^7]: <https://github.com/biu-nlp/sprp-acl2018>
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The symplectic Stiefel manifold, denoted by ${\mathrm{Sp}(2p,2n)}$, is the set of linear symplectic maps between the standard symplectic spaces $\mathbb{R}^{2p}$ and $\mathbb{R}^{2n}$. When $p=n$, it reduces to the well-known set of $2n\times 2n$ symplectic matrices. Optimization problems on $\mathrm{Sp}(2p,2n)$ find applications in various areas, such as optics, quantum physics, numerical linear algebra and model order reduction of dynamical systems. The purpose of this paper is to propose and analyze gradient-descent methods on ${\mathrm{Sp}(2p,2n)}$, where the notion of gradient stems from a Riemannian metric. We consider a novel Riemannian metric on ${\mathrm{Sp}(2p,2n)}$ akin to the canonical metric of the (standard) Stiefel manifold. In order to perform a feasible step along the antigradient, we develop two types of search strategies: one is based on quasi-geodesic curves, and the other one on the symplectic Cayley transform. The resulting optimization algorithms are proved to converge globally to critical points of the objective function. Numerical experiments illustrate the efficiency of the proposed methods.'
author:
- 'Bin Gao[^1]'
- 'Nguyen Thanh Son[^2]'
- 'P.-A. Absil'
- 'Tatjana Stykel[^3]'
bibliography:
- 'bibfile.bib'
title: 'Riemannian Optimization on the Symplectic Stiefel Manifold[^4]'
---
Riemannian optimization, symplectic Stiefel manifold, quasi-geodesic, Cayley transform
65K05, 70G45, 90C48
Introduction {#sec:intro}
============
We consider the following optimization problem with symplectic constraints: $$\begin{aligned}
\label{prob:original}
\begin{array}{cl}
\min\limits_{X\in{\mathbb{R}^{2n\times 2p}}}&f(X)\\
{\mathrm{s.\,t.}\,\,}& X{^{\top}}J_{2n} X = J_{2p},
\end{array}\end{aligned}$$ where $p\le n$, $J_{2m}={\left[\begin{smallmatrix}
0&I_m\\ -I_m & 0
\end{smallmatrix}\right]}$, and $I_m$ is the $m\times m$ identity matrix for any positive integer $m$. When there is no confusion, we omit the subscript of $J_{2m}$ and $I_m$ for simplicity. We assume that the objective function $f: {\mathbb{R}^{2n\times 2p}}\rightarrow{\mathbb{R}}$ is continuously differentiable. The feasible set of problem is known as the [*symplectic Stiefel manifold*]{} [@sigrist1973cross; @ajayi2013explicit] $${\mathrm{Sp}(2p,2n)}:={\left\{X\in{\mathbb{R}^{2n\times 2p}}: X{^{\top}}J_{2n} X = J_{2p}\right\}}.$$ Whereas the term usually refers to the $p=n$ case, we call a matrix $X$ *symplectic* whenever $X\in{\mathrm{Sp}(2p,2n)}$, as [@PengM16] does. When $p=n$, the symplectic Stiefel manifold ${\mathrm{Sp}(2p,2n)}$ becomes a matrix Lie group, termed the *symplectic group* and denoted by ${\mathrm{Sp}(2n)}$. Note that the name “symplectic Stiefel manifold” is also found in the literature [@hsiang1968classification] to denote the set of orthogonal frames in the quaternion [$n$-space]{}. This definition differs fundamentally from ${\mathrm{Sp}(2p,2n)}$ since the latter is noncompact, as we will see in \[sec:geometry\].
Optimization with symplectic constraints appears in various fields. In the case $p=n$, i.e., the symplectic group, most applications are found in physics. In the study of optical systems, such as human eyeballs [@harris2004averageeye; @fiori2016riemannian], the problem of averaging optical transference matrices can be formulated as an optimization problem with symplectic constraints. Another application can be found in accelerator design and charged-particle beam transport [@draft1988lie], where symplectic constraints are required for characterizing beam dynamics. A recent application is also found in the optimal control of quantum symplectic gates [@wu2008optimal], where one has to find a (square) symplectic matrix such that the distance from this sought matrix to a given symplectic matrix is minimal. It is indeed an optimization problem on ${\mathrm{Sp}(2n)}$, where $n$ corresponds to the number of quantum observables. Furthermore, several problems in scientific computing require solving for $p<n$. For instance, [@williamson1936algebraic] states that there exists a symplectic matrix that diagonalizes a given symmetric positive definite matrix, which is popularly termed the symplectic eigenvalue problem. In many cases, one is interested in computing only a few extreme eigenvalues. In [@bhatia2015symplectic], it is established that the sum of the $p$ smallest symplectic eigenvalues is equal to the minimal value of an optimization problem on $\mathrm{Sp}(2p,2n)$. We also mention the projection-based symplectic model order reduction problem in systems and control theory [@PengM16; @AfkhH17; @BuchBH19]. This problem requires to reduce the order ($p\ll n$) of a Hamiltonian system, and at the same time, preserve the Hamiltonian structure. This can only be done via finding so-called symplectic projection matrices, and it is formulated as the problem where the objective function describes the projection error.
The symplectic constraints make the problem out of the reach of several optimization techniques: the feasible set is nonconvex and, in contrast with the (standard) Stiefel manifold [@bhattacharya_bhattacharya_2012 Theorem 10.2], the projection onto the symplectic Stiefel manifold does not admit a known closed-form expression. It appears that all the existing methods that explicitly address restrict either to a specific objective function or to the case $p=n$ (symplectic group). Optimality conditions of the Brockett function [@brockett1989least] over quadratic matrix Lie groups, which include the symplectic group, were studied in [@machado2002optimization]. In [@wu2010critical], the critical landscape topology for optimization on the symplectic group [was]{} investigated, where the corresponding cost function is the Frobenius distance from a target symplectic transformation. In [@fiori2011solving], Fiori studied the geodesics of the symplectic group under a pseudo-Riemannian metric and [proposed]{} a geodesic-based method for computing the empirical mean of a set of symplectic matrices. Follow-up work can be found in [@fiori2016riemannian; @wang2018riemannian]. More recently, Birtea et al. [@birtea2018optimization] studied the first and second order optimality conditions for optimization problems on the symplectic group. Their proposed method computes the steepest-descent direction with respect to a left-invariant metric, and adopts the symplectic Cayley transform [@machado2002optimization; @de2014metaplectic] as a retraction to preserve the symplectic group constraint. In this paper, we propose Riemannian gradient methods for optimization problems on ${\mathrm{Sp}(2p,2n)}$. To this end, we first prove that ${\mathrm{Sp}(2p,2n)}$ is a closed unbounded embedded submanifold of the Euclidean space ${\mathbb{R}^{2n\times 2p}}$. Then, leveraging two explicit characterizations of its tangent space, we develop a class of novel canonical-like Riemannian metrics, with respect to which we obtain expressions for the normal space, the tangent and normal projections, and the Riemannian gradient. We propose two strategies to select a search curve on ${\mathrm{Sp}(2p,2n)}$ along a given tangent direction. One is based on a quasi-geodesic curve, and it needs to compute two matrix exponentials. The other is based on the symplectic Cayley transform, which requires to solve a [$2n\times 2n$]{} linear system. By exploiting the low-rank structure of the tangent vectors, we construct a numerically efficient update for the symplectic Cayley transform. In addition, we find that the Cayley transform can be interpreted as an instance of the trapezoidal rule for solving ODEs on quadratic Lie groups. We develop and analyze Riemannian gradient algorithms that combine each of the two curve selection strategies with a non-monotone line search scheme. We prove that the accumulation points of the sequences of iterates produced by the proposed algorithms are critical points of .
Note that all these results subsume the case of the symplectic group ($p=n$). Along the way, we extend the convergence analysis of general non-monotone Riemannian gradient methods to the case of retractions that are not globally defined; see \[theorem:convergence\]. Numerical experiments investigate the impact of various algorithmic choices—curve selection strategy, Riemannian metric, line search scheme—on the convergence and feasibility of the iterates. Moreover, tests on various instances of —nearest symplectic matrix problem, minimization [of]{} the Brockett cost function, and symplectic eigenvalue problem—illustrate the efficiency of the proposed algorithms.
The paper is organized as follows. In \[sec:geometry\], we study the geometric structure of ${\mathrm{Sp}(2p,2n)}$. The Riemannian geometry of ${\mathrm{Sp}(2p,2n)}$ endowed with the canonical-like metric is investigated in \[section:canonical\]. We construct two different curve selection strategies and propose a Riemannian gradient framework with a non-monotone line search scheme in \[sec:optimization\]. Numerical results on several problems are reported in \[sec:Numerical Experiment\]. Conclusions are drawn in \[sec:conclusion\].
Notation
========
The Euclidean inner product of two matrices $X, Y\in {\mathbb{R}^{2n\times 2p}}$ is denoted by ${\left\langleX,Y\right\rangle}=\operatorname*{tr}(X{^{\top}}Y)$, where $\operatorname*{tr}(\cdot)$ denotes the trace of the matrix argument. The Frobenius norm of $X$ is denoted by ${\left\|X\right\|}{_{\mathrm{F}}}:=\sqrt{{\left\langleX,X\right\rangle}}$. Given $A\in {\mathbb{R}}^{m\times m}$, $e^{A}$ or $\exp(A)$ represents the matrix exponential of $A$. Moreover, $$\operatorname*{sym}(A):=\frac{1}{2}(A+A{^{\top}})\quad\mbox{and}\quad\operatorname*{skew}(A):=\frac{1}{2}(A-A{^{\top}})$$ stand for the symmetric part and the skew-symmetric part of $A$, respectively, and $\det(A)$ denotes the determinant of $A$. We let $\mathrm{diag}(v)\in{\mathbb{R}}^{m\times m}$ denote the diagonal matrix with the components of $v\in{\mathbb{R}}^m$ on the diagonal. We use $\operatorname*{span}(A)$ to express the subspace spanned by the columns of $A$. Furthermore, ${{\cal S}_{\mathrm{sym}}}(n)$ and ${{\cal S}_{\mathrm{skew}}}(n)$ denote the sets of all symmetric and skew-symmetric $n\times n$ matrices, respectively. Let ${\cal E}_1$ and ${\cal E}_2$ be two finite-dimensional vector spaces over ${\mathbb{R}}$. The Fréchet derivative of [a map]{} $F: {\cal E}_1 \rightarrow {\cal E}_2$ at $X\in{\cal E}_1$ is the linear operator $$\mathrm{D}F(X): {\cal E}_1 \rightarrow {\cal E}_2 : Z \mapsto \mathrm{D}F(X)[Z]$$ satisfying $F(X+Z)=F(X)+\mathrm{D}F(X)[Z]+o({\left\|Z\right\|})$. The rank of $F$ at $X\in{\cal E}_1$, denoted by ${\operatorname*{rank}(F)(X)}$, is the dimension of the range of $\mathrm{D}F(X)$. The domain of $F$ is denoted by $\operatorname*{dom}(F)$.
The symplectic Stiefel manifold {#sec:geometry}
===============================
In this section, we investigate the submanifold structure of ${\mathrm{Sp}(2p,2n)}$. The matrix $J$ defined in \[sec:intro\] satisfies the following properties $$J{^{\top}}=-J,\quad J{^{\top}}J=I,\quad J^2=-I,\quad J{^{-1}}= J{^{\top}},$$ [which imply]{} that $J$ is [skew-symmetric and]{} orthogonal. \[tab:notation\] collects the notation and definition of several matrix manifolds that appear in this paper.
------------------------------------- ------------------------ -------------------------------------------------------------
[Space]{} [Symbol]{} [Element]{}
\[2mm\] Orthogonal group $\mathcal{O}(n)$ $Q\in{\mathbb{R}}^{n\times n}: Q{^{\top}}Q=I_n$
\[1mm\] Stiefel manifold ${\mathrm{St}(p,n)}$ $V\in{\mathbb{R}^{n\times p}}: V{^{\top}}V=I_p$
\[1mm\] Symplectic group ${\mathrm{Sp}(2n)}$ $U\in{\mathbb{R}}^{2n\times 2n}: U{^{\top}}J_{2n} U=J_{2n}$
\[1mm\] Symplectic Stiefel manifold ${\mathrm{Sp}(2p,2n)}$ $X\in{\mathbb{R}^{2n\times 2p}}: X{^{\top}}J_{2n} X=J_{2p}$
------------------------------------- ------------------------ -------------------------------------------------------------
: Notation for [matrix]{} manifolds\[tab:notation\]
First we show that ${\mathrm{Sp}(2p,2n)}$ is an embedded submanifold of the Euclidean space ${\mathbb{R}^{2n\times 2p}}$.
\[proposition:dim\] The symplectic Stiefel manifold ${\mathrm{Sp}(2p,2n)}$ is a closed embedded submanifold of [the]{} Euclidean space ${\mathbb{R}^{2n\times 2p}}$. Moreover, it has dimension $4np-p(2p-1)$.
Consider the map $$\label{eq:mapF}
F:{\mathbb{R}^{2n\times 2p}}\rightarrow {{\cal S}_{\mathrm{skew}}}(2p):
X \mapsto X{^{\top}}JX-J.$$ We have $$\label{eq:F-inv0}
{\mathrm{Sp}(2p,2n)}=F{^{-1}}(0),$$ which implies that ${\mathrm{Sp}(2p,2n)}$ is closed since it is the inverse image of the closed set ${\left\{0\right\}}$ under the continuous map $F$.
Next, we prove that the rank of $F$ is $p(2p-1)$ at every point of ${\mathrm{Sp}(2p,2n)}$. Let $ X\in{\mathrm{Sp}(2p,2n)}$. It is sufficient to show that $\mathrm{D}F(X)$ is a surjection, i.e., for all $\bar{Z}\in{{\cal S}_{\mathrm{skew}}}(2p)$, there exists $Z\in{\mathbb{R}^{2n\times 2p}}$ such that $\mathrm{D}F(X)[Z]=\bar{Z}$. We have $$\label{eq:DF(X)Z}
\mathrm{D}F(X)[Z]=X{^{\top}}JZ+Z{^{\top}}JX$$ for all $Z\in{\mathbb{R}}^{2n\times 2p}$. [Let $\bar{Z}\in{{{\cal S}_{\mathrm{skew}}}(2p)}$. Then substituting $Z={\frac{1}{2}}XJ{^{\top}}\bar{Z}$ into the above equation]{}, we obtain $\mathrm{D}F(X)[{\frac{1}{2}}XJ{^{\top}}\bar{Z}]= {\frac{1}{2}}X{^{\top}}JXJ{^{\top}}\bar{Z}+{\frac{1}{2}}\bar{Z}{^{\top}}J X{^{\top}}JX={\frac{1}{2}}\bar{Z}-{\frac{1}{2}}\bar{Z}{^{\top}}=\bar{Z}.$ Hence $F$ has full rank, namely, ${\operatorname*{rank}(F)(X)}=\operatorname*{dim}({{\cal S}_{\mathrm{skew}}}(2p))=p(2p-1)$. Using and [the]{} submersion theorem [@absil2009optimization Proposition 3.3.3], it follows that ${\mathrm{Sp}(2p,2n)}$ is a closed embedded submanifold of ${\mathbb{R}^{2n\times 2p}}$. Its dimension is $\operatorname*{dim}({\mathrm{Sp}(2p,2n)})=\operatorname*{dim}(F{^{-1}}(0))=\operatorname*{dim}({\mathbb{R}^{2n\times 2p}})-\operatorname*{dim}({{\cal S}_{\mathrm{skew}}}(2p))=4np-p(2p-1).$
Observe that the dimension of the symplectic Stiefel manifold ${\mathrm{Sp}(2p,2n)}$ is larger than the dimension of Stiefel manifold $\mathrm{St}(2p,2n)$, which is equal to $4np-p(2p+1)$. Another essential difference between ${\mathrm{Sp}(2p,2n)}$ and $\mathrm{St}(2p,2n)$ is that the symplectic Stiefel manifold is unbounded, hence noncompact. We show it for the simplest case, $p=n=1$. For $X={\left[\begin{smallmatrix}
a & b\\
c & d
\end{smallmatrix}\right]} \in {\mathbb{R}}^{2\times 2}$, we readily obtain that $X\in\mathrm{Sp}(2)$ if and only if $ad-bc=1$. Hence, $$\mathrm{Sp}(2)={\left\{{\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}} \in {\mathbb{R}}^{2\times 2} : ad-bc=1\right\}}$$ and it has dimension $3$. In particular, the matrix ${\left[\begin{smallmatrix}
a & 0\\
0 & 1/a
\end{smallmatrix}\right]}$ is symplectic for all [$a\in{\mathbb{R}}\setminus\{0\}$]{}, which implies that $\mathrm{Sp}(2)$ is unbounded. On the other hand, the orthogonal group $$\mathcal{O}(2)={\left\{{\begin{bmatrix}
\sin\theta & \cos\theta\\
\cos\theta & -\sin\theta
\end{bmatrix}}: \theta\in[0,2\pi]\right\}}$$ has dimension $1$ and is compact. We now set the scene for the descriptions of the tangent space that will come in \[prop:tangent\]. Given $X\in{\mathrm{Sp}(2p,2n)}$, we let $X_\perp\in{\mathbb{R}}^{2n\times (2n-2p)}$ be a full rank matrix such that $\operatorname*{span}(X_{\perp})$ is the orthogonal complement of $\operatorname*{span}(X)$, and we let $$\label{eq:E}
E := \left[XJ\,\,\, JX_{\perp}\right] \in{\mathbb{R}}^{2n\times 2n}.$$ Note that $X_\perp$ is not assumed to be an orthonormal matrix. The next lemma gathers basic linear algebra results that will be useful later on.
\[lemma:E\] The matrix $E = \left[XJ\,\,\, JX_{\perp}\right]$ defined in has the following properties.
1. $E$ [is invertible;]{}
2. $E{^{\top}}JE={\left[\begin{smallmatrix}
J & 0\\
0 & X{^{\top}}_\perp J X_\perp
\end{smallmatrix}\right]}$ and $X_\perp{^{\top}}JX_\perp$ is invertible;
3. $E{^{-1}}={\left[\begin{smallmatrix}
X{^{\top}}J{^{\top}}\\
{\left(X{^{\top}}_\perp J X_\perp\right)}{^{-1}}X_\perp{^{\top}}\end{smallmatrix}\right]}$;
4. [Every matrix]{} $Z\in{\mathbb{R}^{2n\times 2p}}$ [can]{} be represented as $Z = E {\left[\begin{smallmatrix} W \\ K \end{smallmatrix}\right]}$, i.e., $$\begin{aligned}
\label{eq:tangent-decomp}
Z=XJW+JX_\perp K,
\end{aligned}$$ where $W\in{\mathbb{R}}^{2p\times 2p}$ and $K\in{\mathbb{R}}^{(2n-2p)\times 2p}$. Moreover, we have $$\begin{aligned}
\label{eq:tangent-cood}
W=X{^{\top}}J{^{\top}}Z, \quad K={\left(X{^{\top}}_\perp J X_\perp\right)}{^{-1}}X{^{\top}}_\perp Z.
\end{aligned}$$
\(i) Suppose $E {\left[\begin{smallmatrix} y_1 \\ y_2 \end{smallmatrix}\right]} = 0$. Multiplying this equation from the left by $X{^{\top}}J$ yields $y_1=0$. The equation thus reduces to $J X_\perp y_2 = 0$, which yields $y_2=0$ since $J X_\perp$ has full [column]{} rank. This shows that $E$ has full rank.
\(ii) By using $X{^{\top}}J X=J$ and $X{^{\top}}X_\perp=0$, we have $E{^{\top}}JE={\left[\begin{smallmatrix}
J & 0\\
0 & X{^{\top}}_\perp J X_\perp
\end{smallmatrix}\right]}$. From (i), we know that $E$ is invertible, hence $E{^{\top}}JE$ is invertible, and so is $X{^{\top}}_\perp J X_\perp$.
\(iii) From (ii), we have $E{^{-1}}={\left[\begin{smallmatrix}
J & 0\\
0 & X{^{\top}}_\perp J X_\perp
\end{smallmatrix}\right]}{^{-1}}E{^{\top}}J$, and the result follows.
\(iv) The first claim follows from the invertibility of $E$. Using (iii), we have $$\begin{aligned}
\begin{bmatrix}
W\\
K
\end{bmatrix}=E{^{-1}}Z=\begin{bmatrix}
X{^{\top}}J{^{\top}}Z\\
{\left(X{^{\top}}_\perp J X_\perp\right)}{^{-1}}X{^{\top}}_\perp Z
\end{bmatrix}.
\end{aligned}$$
Given $X\in{\mathrm{Sp}(2p,2n)}$, there are infinitely many possible choices of $X_\perp$. The choice of $X_\perp$ affects $E$ in and $K$ in the decomposition of $Z$. However, it does not affect $JX_\perp K$. In fact, it follows from (iii) in \[lemma:E\] that $$\label{eq:JXpK-1}
I=EE{^{-1}}=XJX{^{\top}}J{^{\top}}+ J X_\perp {\left(X{^{\top}}_\perp J X_\perp\right)}{^{-1}}X{^{\top}}_\perp,$$ which further implies that $JX_\perp K = J X_\perp {\left(X{^{\top}}_\perp J X_\perp\right)}{^{-1}}X{^{\top}}_\perp Z = (I-XJX{^{\top}}J{^{\top}}) Z$, where we used the expression of $K$ in .
The tangent space of the symplectic Stiefel manifold at $X\in{\mathrm{Sp}(2p,2n)}$, denoted by ${{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}$, is defined by $${{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}:={\left\{\gamma'(0): \gamma(t) \mbox{~is a smooth curve in~} {\mathrm{Sp}(2p,2n)}\mbox{~with~} \gamma(0)=X\right\}}.$$ The next result gives an implicit form and two explicit forms of ${{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}$.
\[prop:tangent\] Given $X\in{\mathrm{Sp}(2p,2n)}$, the tangent space of ${\mathrm{Sp}(2p,2n)}$ at $X$ admits the following expressions
$$\begin{aligned}
{{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}&=\{Z\in {\mathbb{R}^{2n\times 2p}}: Z{^{\top}}J X + X{^{\top}}J Z=0\} \label{eq:tangent-1}\\
&= \{XJW+JX_\perp K: W\in{{\cal S}_{\mathrm{sym}}}(2p), K\in{\mathbb{R}}^{(2n-2p)\times 2p}\} \label{eq:tangent-2}\\
&= \{SJX: S\in{{\cal S}_{\mathrm{sym}}}(2n)\}. \label{eq:tangent-3}
\end{aligned}$$
Let $F$ be as in . According to , we observe that the right-hand side of is the null space of $\mathrm{D}F(X)$, namely, $\{Z\in\mathbb{R}^{2n\times 2p}: \mathrm{D}F(X)[Z]=0\}$. It follows from [@absil2009optimization (3.19)] that it coincides with ${{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}$.
Using , relation is equivalent to $W=W{^{\top}}$, which yields .
Finally, we prove that the form is equivalent to . We readily obtain $$\label{eq:tangent-relation}
\{SJX: S\in{{\cal S}_{\mathrm{sym}}}(2n)\} \subset \{Z\in {\mathbb{R}^{2n\times 2p}}: Z{^{\top}}J X + X{^{\top}}J Z=0\}.$$ Since both sets are linear subspaces of ${\mathbb{R}^{2n\times 2p}}$, it remains to show that they have the same dimension in order to conclude that they are equal. The dimension of the right-hand set is the dimension of ${{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}$, which is the dimension of ${\mathrm{Sp}(2p,2n)}$, i.e., $4np- p(2p-1)$; see \[proposition:dim\]. As for the dimension of the left-hand set, choose $X_\perp$ as above and observe that $P := JE{\left[\begin{smallmatrix} J{^{\top}}& 0 \\ 0 & -I \end{smallmatrix}\right]} =[JX~X_\perp]\in{\mathbb{R}}^{2n\times 2n}$ is invertible. Let $B:= P{^{\top}}S P=\left[
\begin{smallmatrix}
B_{11} &B_{21}{^{\top}}\\
B_{21} &B_{22}\\
\end{smallmatrix}
\right]$. Then we have $$\begin{aligned}
& \operatorname*{dim}\{SJX: S\in{{\cal S}_{\mathrm{sym}}}(2n)\} = \operatorname*{dim}\{P{^{\top}}S JX: S\in{{\cal S}_{\mathrm{sym}}}(2n)\} \\
& \qquad = \operatorname*{dim}\{P{^{\top}}SP \left[
\begin{smallmatrix}
I_{2p}\\
0
\end{smallmatrix}
\right] : S\in{{\cal S}_{\mathrm{sym}}}(2n)\} = \operatorname*{dim}\{B\left[
\begin{smallmatrix}
I_{2p}\\
0\\
\end{smallmatrix}
\right]: B\in{{\cal S}_{\mathrm{sym}}}(2n)\} \\
& \qquad = \operatorname*{dim}\{\left[
\begin{smallmatrix}
B_{11}\\
{B_{21}}\\
\end{smallmatrix}
\right]: B_{11}\in{{\cal S}_{\mathrm{sym}}}(2p), B_{21}\in{\mathbb{R}}^{(2n-2p)\times2p}\} \\
& \qquad = {p(2p+1)} +(2n-2p)2p = 4np-p(2p-1),
\end{aligned}$$ yielding the equality of dimensions that concludes the proof. The first equality follows from $\operatorname*{dim}({P\mathcal{S}})=\operatorname*{dim}(\mathcal{S})$ for any subspace $\mathcal{S}\subset {\mathbb{R}}^{2n\times 2p}$ and invertible matrix $P$. The second equality follows from $JX=P\left[
\begin{smallmatrix}
I_{2p}\\
0\\
\end{smallmatrix}
\right]$. The third equality is due to $P{^{\top}}{{\cal S}_{\mathrm{sym}}}(2n) P = \mathcal{S}_{sym}(2n)$, and the last equality comes from the fact $\operatorname*{dim}({{\cal S}_{\mathrm{sym}}}(2p))=p(2p+1)$.
In fact, [it follows]{} from the proof above [that]{} the derivation of also works for $\{JSX: S\in{{\cal S}_{\mathrm{sym}}}(2n)\}$, namely, ${{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}=\{JSX: S\in{{\cal S}_{\mathrm{sym}}}(2n)\},$ but we will restrict to in the following. Unlike , when $p<n$, is an over-parameterization, i.e., $\operatorname*{dim}({{\cal S}_{\mathrm{sym}}}(2n)) > \operatorname*{dim}({{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}})$. However, $S$ can be chosen with a low-rank structure. Indeed, letting $P := JE{\left[\begin{smallmatrix} J{^{\top}}& 0 \\ 0 & -I \end{smallmatrix}\right]} =[JX~X_\perp]$ and $B := P{^{\top}}SP$, we have that $SJX = P^{-\top}BP{^{-1}}JX = P^{-\top}B{\left[\begin{smallmatrix}
I_{2p}\\
0
\end{smallmatrix}\right]}$, which shows that only the first $2p$ columns of the symmetric matrix $B$ have an impact on $SJX$. Hence there is no restriction on $SJX$, $S\in{{\cal S}_{\mathrm{sym}}}(2n)$, if we restrict $B$ to the form $B = M {\left[\begin{smallmatrix} I_{2p} & 0\end{smallmatrix}\right]} + {\left[\begin{smallmatrix}I_{2p} \\ 0 \end{smallmatrix}\right]} M{^{\top}}$, with $M\in{\mathbb{R}}^{2n\times 2p}$, or equivalently, if we restrict $S$ to the form $$\label{eq:S-rank2p}
S=P^{-\top}BP{^{-1}}=L(XJ){^{\top}}+XJL{^{\top}}=\begin{bmatrix}
L & XJ
\end{bmatrix}
\begin{bmatrix}
(XJ){^{\top}}\\ L{^{\top}}\end{bmatrix},$$ where $L=P^{-\top}M \in{\mathbb{R}}^{2n\times 2p}$ and the second equality follows from \[lemma:E\]. Note that such $S$ has rank at most $4p$. This low-rank structure will have a crucial impact on the computational cost of the Cayley retraction introduced in \[subsec:cayley\].
[The canonical-like metric]{} {#section:canonical}
=============================
Given an objective function $f:{\mathrm{Sp}(2p,2n)}\to{\mathbb{R}}$, finding the steepest-descent direction at a point $X\in{\mathrm{Sp}(2p,2n)}$ amounts to finding $Z\in{{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}$ subject to ${\left\|Z\right\|}_X=1$ that minimizes $\mathrm{D}f(X)[Z]$. In order to define ${\left\|\cdot\right\|}_X$, it is customary to endow ${{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}$ with an inner product ${{\left\langle\cdot,\cdot\right\rangle}_{X}}$ that depends smootly on $X$; ${\left\|\cdot\right\|}_X$ is then the norm induced by the inner product. Such an inner product is termed a Riemanian metric—*metric* for short—and turns ${\mathrm{Sp}(2p,2n)}$ into a Riemannian manifold. In this section, we propose a class of metrics on ${\mathrm{Sp}(2p,2n)}$ that are inspired from the canonical metric on the Stiefel manifold. Then we work out formulas for the normal space and gradient, and we propose a class of curves that we term quasi-geodesics. For the Stiefel manifold ${\mathrm{St}(p,n)}$, any tangent vector at $V\in{\mathrm{St}(p,n)}$ has a unique expression $\Delta=VA+V_\perp B$, where $V_\perp$ satisfies $V_\perp^\top V=0$ and $V_\perp^\top V_\perp^{}=I$, $A\in{{\cal S}_{\mathrm{skew}}}(p)$ and $B\in{\mathbb{R}}^{(n-p)\times p}$, see [@edelman1998geometry]. The canonical metric [@edelman1998geometry (2.39)] on ${\mathrm{St}(p,n)}$ is defined as $$g_c(\Delta_1,\Delta_2)=\operatorname*{tr}{\left(\Delta_1{^{\top}}(I-{\frac{1}{2}}VV{^{\top}})\Delta_2\right)}= {\frac{1}{2}}\operatorname*{tr}(A_1{^{\top}}A_2)+\operatorname*{tr}(B_1{^{\top}}B_2)$$ for $\Delta_i= VA_i+V_\perp B_i$, $i=1,2$.
By using the tangent vector representation in , we develop a similar metric for the symplectic Stiefel manifold ${\mathrm{Sp}(2p,2n)}$. We choose the inner product on ${{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}$ to be $$\begin{aligned}
\label{eq:rmetric}
g_{\rho,X_\perp}(Z_1,Z_2) &\equiv {{\left\langleZ_1, Z_2\right\rangle}_{X}} :={\frac{1}{\rho}\, \operatorname*{tr}}(W_1{^{\top}}W_2)+\operatorname*{tr}(K_1{^{\top}}K_2),\end{aligned}$$ where $\rho>0$ is a parameter and $W_i\in{{\cal S}_{\mathrm{sym}}}(2p)$ and $K_i\in{\mathbb{R}}^{(2n-2p)\times 2p}$ are obtained from $Z_i$ as in , i.e., ${\left[\begin{smallmatrix}
W_i\\
K_i
\end{smallmatrix}\right]}=E{^{-1}}Z_i $ for $i=1,2$. Hence, we have $$\begin{aligned}
g_{\rho,X_\perp}(Z_1,Z_2) &= \operatorname*{tr}{\left(\begin{bmatrix}
W_1\\
K_1
\end{bmatrix}{^{\top}}\begin{bmatrix}
\frac{1}{\rho} I & 0\\
0 & I
\end{bmatrix} \begin{bmatrix}
W_2\\
K_2
\end{bmatrix} \right)}
= \operatorname*{tr}{\left((E{^{-1}}Z_1){^{\top}}\begin{bmatrix}
\frac{1}{\rho} I & 0\\
0 & I
\end{bmatrix} E{^{-1}}Z_2\right)}\\
&= \operatorname*{tr}{\left(Z_1{^{\top}}E^{-\top} \begin{bmatrix}
\frac{1}{\rho} I & 0\\
0 & I
\end{bmatrix} E{^{-1}}Z_2 \right)} = \operatorname*{tr}(Z_1{^{\top}}B_X Z_2),\end{aligned}$$ where $$\begin{aligned}
\label{eq:BX}
B_X := E^{-\top} {\begin{bmatrix}
\frac{1}{\rho} I & 0\\
0 & I
\end{bmatrix}} E{^{-1}}=\frac{1}{\rho} JXX{^{\top}}J{^{\top}}-X_\perp (X{^{\top}}_\perp J X_\perp)^{-2}X{^{\top}}_\perp.\end{aligned}$$ The last expression of $B_X$ follows from (iii) in \[lemma:E\]. In view of its definition, $B_X$ is positive definite; this confirms that is a bona-fide inner product. The expressions of $B_X$ also confirm that $g_{\rho,X_\perp}$ depend on $\rho$ and $X_\perp$. There is however an invariance: $g_{\rho,X_\perp Q} = g_{\rho,X_\perp}$ for all $Q\in\mathcal{O}(2n-2p)$.
In order to make $g_{\rho,X_\perp}$ a bona-fide Riemannian metric, it remains to choose the “$X_\perp$ map” ${\mathrm{Sp}(2p,2n)}\ni X\mapsto X_\perp$ in such a way that $B_X$ smoothly depends on $X$. Choosing a smooth $X\mapsto X_\perp$ map would be sufficient, but it is unknown whether such a smooth map globally exists. However, if orthonormalization conditions are imposed on $X_\perp$ such that the $X_\perp$-term of $B_X$ can be rewritten as a smooth expression of $X\in{\mathrm{Sp}(2p,2n)}$, then $g_{\rho,X_\perp}$ becomes a bona-fide Riemannian metric, which we term *canonical-like metric*. We will consider the following two such orthonormalization conditions on $X_\perp$, which are readily seen to be achievable since the set of all admissible $X_\perp$ matrices has the form $\{X_\perp M: M\in{\mathbb{R}}^{(2n-2p)\times (2n-2p)} \text{~{is invertible}}\}$:
- $X_\perp$ is orthonormal.
- $X_\perp (X_\perp{^{\top}}JX_\perp)^{-1}$ is orthonormal.
The announced smooth expressions of $B_X$ are given next.
Under the orthonormalization condition (I), the $X_\perp$-term of $B_X$ in , namely, $-X_\perp (X_\perp{^{\top}}J X_\perp)^{-2} X_\perp{^{\top}}$, is equal to $-(JXJX{^{\top}}J{^{\top}}-J)^2$. Under the orthonormalization condition (II), it is equal to $I - X(X{^{\top}}X)^{-1} X{^{\top}}$.
For the first claim, since $X_\perp$ is restricted to be orthonormal, it follows that $$X_\perp (X_\perp{^{\top}}J X_\perp)^{-2} X_\perp{^{\top}}= X_\perp (X_\perp{^{\top}}J X_\perp)^{-1} X_\perp{^{\top}}X_\perp (X_\perp{^{\top}}J X_\perp)^{-1} X_\perp{^{\top}}= (JXJX{^{\top}}J{^{\top}}-J)^2.$$ The last equality is due to the fact $X_\perp (X_\perp{^{\top}}J X_\perp)^{-1} X_\perp{^{\top}}= JXJX{^{\top}}J{^{\top}}-J$ which is readily derived from . For the second claim, since $X_\perp (X_\perp{^{\top}}J X_\perp)^{-1}$ is now restricted to be orthonormal, we have that $(X_\perp{^{\top}}J X_\perp)^{-\top} X_\perp{^{\top}}X_\perp (X_\perp{^{\top}}J X_\perp)^{-1}=I$, which yields $(X_\perp{^{\top}}J X_\perp)^{2}=-X_\perp{^{\top}}X_\perp$. Consequently, we obtain that $$X_\perp (X_\perp{^{\top}}J X_\perp)^{-2} X_\perp{^{\top}}=-X_\perp (X_\perp{^{\top}}X_\perp)^{-1} X_\perp{^{\top}}= X(X{^{\top}}X)^{-1} X{^{\top}}- I,$$ where the last equality can be checked by observing that multiplying each side by the invertible matrix $[X~X_\perp]$ yields $-[0~X_\perp]$.
We will use $g_\rho$ to denote $g_{\rho,X_\perp}$ if there is no confusion. In the particular case $p=n$, where ${\mathrm{Sp}(2p,2n)}$ reduces to the symplectic group ${\mathrm{Sp}(2n)}$, the $K$-term in disappears. Further choose $\rho=1$. Then, for all $U\in{\mathrm{Sp}(2n)}$, we have $B_U=JUU{^{\top}}J{^{\top}}$. Since $U{^{-1}}=J U{^{\top}}J{^{\top}}$, we also have $B_U=U^{-\top} U{^{-1}}$, and the canonical-like metric reduces to ${\left\langleZ_1, Z_2\right\rangle}_U :=\operatorname*{tr}{\left(Z_1{^{\top}}B_U Z_2\right)}={\left\langleU{^{-1}}Z_1, U{^{-1}}Z_2\right\rangle},$ which is the left-invariant metric used in [@wang2018riemannian; @birtea2018optimization].
[Normal space and projections]{}
--------------------------------
The symmetric matrix $B_X$ in is positive definite if and only if $X\in{\mathbb{R}^{2n\times 2p}}_\star := \{X\in{\mathbb{R}^{2n\times 2p}}: \det(X{^{\top}}J X)\neq0\}$. Hence the Riemannian metric $g_{\rho}$ defined in , extended to ${\mathbb{R}^{2n\times 2p}}_\star$, remains a Riemannian metric, which we also denote by $g_{\rho}$. In this section, we give an expression for the normal space of ${\mathrm{Sp}(2p,2n)}$ viewed as a submanifold of $({\mathbb{R}^{2n\times 2p}}_\star,g_\rho)$, and we obtain expression for the projections onto the tangent and normal spaces. This will be instrumental in the expression of the gradient in \[subsubsec:rgrad\]. The normal space at $X\in{\mathrm{Sp}(2p,2n)}$ with respect to $g_\rho$ is defined as $${{\left({{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}\right)}^\perp}:={\left\{N\in{\mathbb{R}^{2n\times 2p}}: g_{\rho}(N,Z)=0 \mbox{~for all~} Z\in{{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}\right\}},$$ where we have used the fact that $\mathrm{T}_X {\mathbb{R}^{2n\times 2p}}_\star \simeq \mathrm{T}_X {\mathbb{R}^{2n\times 2p}}\simeq {\mathbb{R}^{2n\times 2p}}$.
\[proposition:normal\] Given $X\in{\mathrm{Sp}(2p,2n)}$, we have $$\label{eq:normal}
{{\left({{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}\right)}^\perp}= {\left\{XJ\varOmega: \varOmega\in{{\cal S}_{\mathrm{skew}}}(2p)\right\}}.$$
Using \[lemma:E\] and , we have that $N\in{\mathbb{R}^{2n\times 2p}}$ and $Z\in{{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}$ if and only if $$\begin{aligned}
N&=XJ\varOmega+JX_\perp K_N, \quad \varOmega\in{\mathbb{R}}^{2p\times 2p}, \quad K_N\in{\mathbb{R}}^{(2n-2p)\times 2p},\\
Z&=XJW+JX_\perp K_Z, \quad W\in{{\cal S}_{\mathrm{sym}}}(2p), \quad K_Z\in{\mathbb{R}}^{(2n-2p)\times 2p}.
\end{aligned}$$ The normal space condition, $g_{\rho}(N,Z)=0$ for all $Z\in{{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}$, is thus equivalent to $$\begin{aligned}
\frac{1}{\rho} \operatorname*{tr}(\varOmega{^{\top}}W)+\operatorname*{tr}(K_N{^{\top}}K_Z) = 0 \quad \text{for all $W\in{{\cal S}_{\mathrm{sym}}}(2p)$ and $K_Z\in{\mathbb{R}}^{(2n-2p)\times 2p}$}.
\end{aligned}$$ In turn, this equation is equivalent to $\varOmega\in{{\cal S}_{\mathrm{skew}}}(2p)$ (since ${{\cal S}_{\mathrm{skew}}}(2p)$ is the orthogonal complement of ${{\cal S}_{\mathrm{sym}}}(2p)$ with respect to the Euclidean inner product) and $K_N=0$.
Every $Y\in{\mathbb{R}^{2n\times 2p}}$ admits a decomposition $Y={\mathcal{P}_X}(Y)+{{\mathcal{P}_X}^\perp}(Y)$, where ${\mathcal{P}_X}$ and ${{\mathcal{P}_X}^\perp}$ denote the orthogonal (in the sense of $g_\rho$) projections onto the tangent and normal space, respectively. Next, we derive explicit expressions of ${\mathcal{P}_X}$—in the forms and —and ${{\mathcal{P}_X}^\perp}$.
\[proposition:projection\] Given $X\in{\mathrm{Sp}(2p,2n)}$ and $Y\in{\mathbb{R}^{2n\times 2p}}$, the orthogonal projection onto ${{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}$ and ${{\left({{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}\right)}^\perp}$ of $Y$ with respect to the metric $g_{\rho}$ has the following expressions $$\begin{aligned}
\label{eq:project-tangent}
{\mathcal{P}_X}(Y) &= XJ\operatorname*{sym}(X{^{\top}}J{^{\top}}Y)+(I- XJX{^{\top}}J{^{\top}})Y \stepcounter{equation}\tag{\theequation a}\\
&= S_{X,Y}JX, \label{eq:pj-tg-S} \tag{\theequation b}\\
{{\mathcal{P}_X}^\perp}(Y) &= XJ\operatorname*{skew}(X{^{\top}}J{^{\top}}Y), \label{eq:project-normal}
\end{aligned}$$ where $S_{X,Y} = G_XY(XJ){^{\top}}+ XJ (G_XY){^{\top}}$ and $G_X = I-\frac12 XJX{^{\top}}J{^{\top}}$.
First we prove and . For all $Y\in{\mathbb{R}^{2n\times 2p}}$, according to and \[proposition:normal\], we [have]{} $$\begin{aligned}
{\mathcal{P}_X}(Y) &= XJW_Y+JX_\perp K_Y,\\
{{\mathcal{P}_X}^\perp}(Y)&= XJ\varOmega_Y,
\end{aligned}$$ where $W_Y\in{{\cal S}_{\mathrm{sym}}}(2p)$, $K_Y\in{\mathbb{R}}^{(2n-2p)\times 2p}$ and $\varOmega_Y\in{{\cal S}_{\mathrm{skew}}}(2p)$. We thus have $$\begin{aligned}
\label{eq:Y=PY+PPY}
Y = {\mathcal{P}_X}(Y)+{{\mathcal{P}_X}^\perp}(Y)= XJW_Y+JX_\perp K_Y +XJ\varOmega_Y.
\end{aligned}$$ Multiplying from the left by $X{^{\top}}J{^{\top}}$ yields $X{^{\top}}J{^{\top}}Y = W_Y+\varOmega_Y$, hence $W_Y = \operatorname*{sym}(X{^{\top}}J{^{\top}}Y)$ and $\varOmega_Y = \operatorname*{skew}(X{^{\top}}J{^{\top}}Y)$. Replacing these expressions in yields $JX_\perp K_Y=Y-XJW_Y-XJ\varOmega_Y=(I- XJX{^{\top}}J{^{\top}})Y$. We have thus obtained and .
It remains to prove . A fairly straightforward development confirms that is equal to $Y-XJ\operatorname*{skew}(X{^{\top}}J{^{\top}}Y) = Y - {{\mathcal{P}_X}^\perp}(Y) = {\mathcal{P}_X}(Y)$. Instead we give a constructive proof which explains how we obtained the expression of $S_{X,Y}$. We thus seek $S_{X,Y}$ symmetric such that $S_{X,Y}JX = {\mathcal{P}_X}(Y)$. In view of and \[lemma:E\], we can restrict our search to $S_{X,Y} = L(XJ){^{\top}}+ (XJ)L{^{\top}}$ with $L = X J W_L + JX_\perp K_L$, and the task thus reduces to finding $W_L\in{\mathbb{R}}^{2p\times 2p}$ and $K_L\in{\mathbb{R}}^{(2n-2p)\times 2p}$ such that $$\begin{aligned}
\label{eq:pj-tg-S-3}
2XJ\operatorname*{sym}(W_L)+JX_\perp K_L&={\mathcal{P}_X}(Y) = Y - {{\mathcal{P}_X}^\perp}(Y)=Y - XJ\operatorname*{skew}(X{^{\top}}J{^{\top}}Y),
\end{aligned}$$ where the last equality comes from . Multiplying from the left by $X{^{\top}}J{^{\top}}$ yields $2\operatorname*{sym}(W_L)=\operatorname*{sym}(X{^{\top}}J{^{\top}}Y)$, a solution of which is $W_L={\frac{1}{2}}X{^{\top}}J{^{\top}}Y$. It further follows from that $JX_\perp K_L=(I- XJX{^{\top}}J{^{\top}})Y$. Replacing the obtained expressions in the above-mentioned decomposition of $L$ yields $L=X J W_L + JX_\perp K_L={\frac{1}{2}}XJ X{^{\top}}J{^{\top}}Y+(I- XJX{^{\top}}J{^{\top}})Y=(I-\frac12 XJX{^{\top}}J{^{\top}})Y$. Replacing this expression in yields the sought expression of $S_{X,Y}$.
Note that the projections do not depend on $\rho$ nor $X_\perp$, though the metric $g_\rho$ does. Nevertheless, by using , we obtain $$\begin{aligned}
\label{eq:project-tangent-1}
{\mathcal{P}_X}(Y) = XJ\operatorname*{sym}(X{^{\top}}J{^{\top}}Y) + JX_\perp{\left(X{^{\top}}_\perp J X_\perp\right)}{^{-1}}X{^{\top}}_\perp Y.\end{aligned}$$ Therefore, the projection ${\mathcal{P}_X}(Y)$ in can [alternatively be]{} computed by involving $X_\perp$.
For later reference, we record the following result which follows from and the fact that ${\mathcal{P}_X}(Z)=Z$ when $Z\in{{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}$. It shows how to write a tangent vector in the form .
\[cor:ZtoS\] If $X\in{\mathrm{Sp}(2p,2n)}$ and $Z\in{{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}$, then $Z = S_{X,Z}JX$ with $S_{X,Z}$ as in \[proposition:projection\].
Riemannian gradient {#subsubsec:rgrad}
-------------------
All is now in place to specify how, given the Euclidean gradient $\nabla \bar{f}(X)$ where $\bar{f}$ is any smooth extension of $f:{\mathrm{Sp}(2p,2n)}\to{\mathbb{R}}$ around $X$ in ${\mathbb{R}^{2n\times 2p}}$, one can compute the steepest-descent direction of $f$ at $X$ in the sense of the metric $g_\rho$ given in , as announced at the beginning of \[section:canonical\].
It is well known (see, e.g., [@absil2009optimization §3.6]) that the steepest-descent direction is along the opposite of the Riemannian gradient. Denoted by $\mathrm{grad}f(X)$, the Riemannian gradient of $f$ with respect to a metric $g$ is defined to be the unique element of ${{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}$ satisfying the condition $$\label{eq:Riemannian_gradient}
g(\mathrm{grad}f(X), Z)=\mathrm{D}\bar{f}(X)[Z]={\left\langle\nabla \bar{f}(X), Z\right\rangle} \quad\mbox{for all}\; Z\in{{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}},$$ where the last expression is the standard Euclidean inner product. (The second equality follows from the definition of Fréchet derivative in ${\mathbb{R}^{2n\times 2p}}$.)
\[proposition:rgrad\] The forms and of the Riemannian gradient of [a]{} function $f:{\mathrm{Sp}(2p,2n)}\to{\mathbb{R}}$ with respect to the metric $g_\rho$ defined in are
\[eq:rgrad-parent\] $$\begin{aligned}
\label{eq:rgrad-1}
{\mathrm{grad}_\rho f(X)} &= \rho XJ \operatorname*{sym}(J{^{\top}}X{^{\top}}\nabla\bar{f}(X)) + JX_\perp X{^{\top}}_\perp J{^{\top}}\nabla\bar{f}(X)\\
&= S_X JX, \label{eq:rgrad-2}
\end{aligned}$$
where $\nabla\bar{f}(X)$ is the Euclidean gradient of any smooth extension $\bar{f}$ of $f$ around $X$ in ${\mathbb{R}^{2n\times 2p}}$, and $S_X={\left(H_X\nabla\bar{f}(X)\right)} (XJ){^{\top}}+XJ {\left(H_X\nabla\bar{f}(X)\right)}{^{\top}}$ with ${H_X}=\frac{\rho}{2}XX{^{\top}}+JX_\perp X{^{\top}}_\perp J{^{\top}}$.
According to the definitions of [the]{} Riemannian gradient and $g_\rho$, for all $Z\in{\mathbb{R}^{2n\times 2p}}$, it follows that ${\left\langleB_X {\mathrm{grad}_\rho \bar{f}(X)},Z\right\rangle}=g_{\rho}({\mathrm{grad}_\rho \bar{f}(X)}, Z)=\mathrm{D}\bar{f}(X)[Z]={\left\langle\nabla\bar{f}(X), Z\right\rangle}$. Hence, ${\mathrm{grad}_\rho \bar{f}(X)} = B_X{^{-1}}\nabla \bar{f}(X)$. The expression of $B_X$ in yields ${\mathrm{grad}_\rho \bar{f}(X)} = B_X{^{-1}}\nabla\bar{f}(X)=E{\left[\begin{smallmatrix}
{\rho} I & 0\\
0 & I
\end{smallmatrix}\right]} E{^{\top}}\nabla\bar{f}(X)=(\rho XX{^{\top}}+JX_\perp X_\perp{^{\top}}J{^{\top}})\nabla\bar{f}(X)$. Owing to [@absil2009optimization (3.37)], it holds that ${\mathrm{grad}_\rho f(X)}={\mathcal{P}_X}({\mathrm{grad}_\rho \bar{f}(X)}).$ By fairly straightforward developments, the expression of ${\mathcal{P}_X}$ yields while the expression of ${\mathcal{P}_X}$ yields .
Just as the Riemannian metric $g_{\rho,X_\perp}$ in , the gradient ${\mathrm{grad}_\rho f(X)}$ depends on $\rho$ and $X_\perp$. The dependence on $X_\perp$ is through the factor $JX_\perp X_\perp{^{\top}}J{^{\top}}$. However, we have seen that when we impose the orthonormalization condition (I) or (II), the metric $g_{\rho,X_\perp}$ no longer depends on the remaining freedom in the choice of $X_\perp$. Hence the same can be said of ${\mathrm{grad}_\rho f(X)}$. Specifically:
- $X_\perp$ is orthonormal. Then $X_\perp X_\perp{^{\top}}$ is the symmetric projection matrix onto the orthogonal complement of $\operatorname*{span}(X)$, i.e., $$X_\perp X_\perp{^{\top}}= I - X(X{^{\top}}X)^{-1}X{^{\top}};$$
- $X_\perp (X_\perp{^{\top}}JX_\perp)^{-1}$ is orthonormal. Then the factor $$\begin{aligned}
JX_\perp X_\perp{^{\top}}J{^{\top}}&= JX_\perp (X_\perp{^{\top}}JX_\perp)^{-\top}X_\perp{^{\top}}X_\perp (X_\perp{^{\top}}JX_\perp)^{-1} X_\perp{^{\top}}J{^{\top}}\\
&= (I-XJX{^{\top}}J{^{\top}})(I-XJX{^{\top}}J{^{\top}}){^{\top}},
\end{aligned}$$ where the last equality follows from .
In practice, we compute the gradient by means of above two formulations. Note that we do not need to compute $X_\perp$. Even when a restriction on $X_\perp$ makes ${\mathrm{grad}_\rho f(X)}$ independent of the choice of $X_\perp$, it is still impacted by the choice of $\rho$. The influence of $\rho$ on the algorithmic performance will be investigated in \[subsection:default setting\]. As we have seen, in the particular case $p=n$ and $\rho=1$, the matrix $X_\perp$ disappears and the canonical-like metric $g_\rho$ reduces to the left-invariant metric used in [@wang2018riemannian; @birtea2018optimization]. In that particular case, the Riemannian gradient is thus also the same [as that]{} developed in [@wang2018riemannian; @birtea2018optimization].
Quasi-geodesics {#subsec:geo}
---------------
Searching along geodesics can be viewed as the most “geometrically natural” way to minimize a function on a manifold. In this section, we observe that it is difficult, perhaps impossible, to obtain a closed-form expression of the geodesics of ${\mathrm{Sp}(2p,2n)}$ with respect to $g_\rho$ in . Instead, we develop an approximation, termed quasi-geodesics, defined by imposing that its *Euclidean* second derivative (in lieu of the Riemannian second derivative in ${\mathbb{R}^{2n\times 2p}}$ endowed with $g_\rho$) belongs to the normal space .
The geodesics on $({\mathrm{Sp}(2p,2n)},g_\rho)$ are the smooth curves on ${\mathrm{Sp}(2p,2n)}$ with zero Riemannian acceleration. Since $({\mathrm{Sp}(2p,2n)},g_\rho)$ is a Riemannian submanifold of $({\mathbb{R}^{2n\times 2p}},g_\rho)$, it follows that the Riemannian acceleration of a curve on $({\mathrm{Sp}(2p,2n)},g_\rho)$ is the orthogonal projection (in the sense of $g_\rho$) onto ${{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}$ of its Riemannian acceleration as a curve in $({\mathbb{R}^{2n\times 2p}},g_\rho)$; see, e.g., [@absil2009optimization Prop. 5.3.2 and (5.23)]. In other words, the geodesics on $({\mathrm{Sp}(2p,2n)},g_\rho)$ are the curves $Y(t)$ in ${\mathrm{Sp}(2p,2n)}$ that satisfy $$\label{eq:geo-normal-true}
\frac{\mathcal{D}}{\mathrm{d}t}\dot{Y}(t)+Y(t)J\varOmega=0 \quad\mbox{with~} \varOmega\in{{\cal S}_{\mathrm{skew}}}(2p),$$ where we have used the characterization of the normal space, and where $\frac{\mathcal{D}}{\mathrm{d}t}$ denotes the Riemannian covariant derivative [@absil2009optimization (5.23)] on ${\left({\mathbb{R}^{2n\times 2p}},g_\rho\right)}$.
Finding a closed-form solution of is elusive. Instead, we replace $\frac{\mathcal{D}}{\mathrm{d}t}\dot{Y}$ in by the classical (Euclidean) second derivative $\ddot{Y}$ in ${\mathbb{R}^{2n\times 2p}}$, yielding $$\label{eq:geo-normal}
\ddot{Y}(t)+Y(t)J\varOmega=0 \quad\mbox{with~} \varOmega\in{{\cal S}_{\mathrm{skew}}}(2p),$$ which we term *quasi-geodesic* equation.
The purpose of the rest of this section is to obtain a closed-form solution of .
For ease of notation, we [write $Y$ instead of $Y(t)$ if there is no confusion]{}. Since the curve $Y(t)$ remains on ${\mathrm{Sp}(2p,2n)}$, we have $Y(t){^{\top}}J Y(t)=J$ for all $t$. Differentiating this equation twice with respect to $t$, we have $$\label{eq:geo-differential}
Y{^{\top}}J \dot{Y} + \dot{Y}{^{\top}}J Y=0,\quad Y{^{\top}}J \ddot{Y}+2 \dot{Y}{^{\top}}J \dot{Y} +\ddot{Y}{^{\top}}J Y=0.$$ Substituting $\ddot{Y}$ of into the second equation of , we arrive at $\varOmega=-\dot{Y}{^{\top}}J \dot{Y}$. Therefore, can equivalently be written as $$\label{eq:geo-ODE}
\ddot{Y}-YJ(\dot{Y}{^{\top}}J \dot{Y})=0.$$ Following an approach similar to the one used in [@edelman1998geometry] to obtain the geodesics on the Stiefel manifold, we rewrite the differential equation by involving three types of [first integrals]{}. Let $$C=Y{^{\top}}J Y,\quad W=Y{^{\top}}J \dot{Y},\quad S=\dot{Y}{^{\top}}J \dot{Y}.$$ According to , it is straightforward to verify that $$\dot{C} = W-W{^{\top}},\quad \dot{W} = S+CJS, \quad \dot{S} = SJW-W{^{\top}}JS.$$ Since $Y\in{\mathrm{Sp}(2p,2n)}$, we have that $C = J$ and therefore $\dot{C}=0$. This in turn implies that $W=W{^{\top}}$ and $\dot{W}=0$. [As a consequence, we get $W=W(0)$.]{} Thus, integrals of the above motion are $$\begin{aligned}
\label{eq:geo-integral}
{C(t)} = J,\quad {W(t)} = W(0), \quad {S(t)} = e^{(JW){^{\top}}t} S(0) e^{JWt}.\end{aligned}$$
Since $W$ is symmetric, the matrix $JW$ is a Hamiltonian matrix. We will make use of the following classical result stating that the exponential of every Hamiltonian matrix is symplectic.
\[proposition:exponential\] For every symmetric matrix $W\in{\mathbb{R}}^{2p\times 2p}$, the matrix exponential $e^{JW}$ is symplectic.
A proof can be found, e.g., in [@meyer1992linear Corollary 3.1] and [@deGosson2006 Proposition 2.15]. Briefly, letting $U(t) = (e^{tJW}){^{\top}}J e^{tJW}$, it follows from $\frac{\mathrm{d}}{\mathrm{d}t} e^{At} = Ae^{At} = e^{At}A$ that $\dot{U}(t) = (e^{tJW}){^{\top}}((JW){^{\top}}J+ J(JW)) e^{tJW} = (e^{tJW}){^{\top}}(W{^{\top}}-W) e^{tJW} = 0$ for all $t$. Hence $U(t)$ is constant, and since $U(0)=J$, the claim follows.
Next, we reorganize as an integrable equation to derive the closed-form expression of quasi-geodesics.
\[proposition:geo\] Let $Y(t)$ on ${\mathrm{Sp}(2p,2n)}$ satisfy the quasi-geodesic equation with the initial conditions $Y(0)=X\in{\mathrm{Sp}(2p,2n)}$ and $\dot{Y}(0)=Z\in{{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}$. Then $$\label{eq:geo}
Y(t) = Y_X^{\operatorname*{qgeo}}(t;Z):={\left[X, Z\right]} \exp{\left(t\begin{bmatrix}
-JW & JZ{^{\top}}J Z \\ I_{2p} & -JW
\end{bmatrix}\right)}
\begin{bmatrix}
I_{2p} \\ 0
\end{bmatrix}
e^{JWt},$$ where $W=X{^{\top}}J Z$.
Using the notation and results introduced above, we have $$\begin{aligned}
\arraycolsep=2pt
\begin{array}{rcl}
\frac{\mathrm{d}}{\mathrm{d}t}{\left(Y(t)e^{-JWt}\right)} & = &\dot{Y}{(t)} e^{-JWt} - Y{(t)}e^{-JWt} JW, \\
\frac{\mathrm{d}}{\mathrm{d}t}{\left(\dot{Y}(t)e^{-JWt}\right)} & = & \ddot{Y}{(t)} e^{-JWt} - \dot{Y}{(t)}e^{-JWt} JW\\
&=& Y{(t)}JS{(t)}e^{-JWt} - \dot{Y}(t)e^{-JWt} JW\\
&=& Y{(t)}J e^{(JW){^{\top}}t} S(0) e^{JW t} e^{-JWt} - \dot{Y}(t)e^{-JWt} JW \\
&=& Y{(t)} e^{-JW t} JS(0)- \dot{Y}(t)e^{-JWt} JW,
\end{array}
\end{aligned}$$ where the last equality uses the identity $J e^{(JW){^{\top}}t}=e^{-JW t} J$ follows from the fact (\[proposition:exponential\]) that $e^{JWt}$ is symplectic. Stacking the above equations together yields $$\frac{\mathrm{d}}{\mathrm{d}t} {\left[Y(t)e^{-JWt}, \dot{Y}(t)e^{-JWt}\right]} = {\left[Y(t)e^{-JWt}, \dot{Y}(t)e^{-JWt}\right]} \begin{bmatrix}
-JW & JS(0) \\ I_{2p} & -JW
\end{bmatrix},$$ which implies that $$Y(t)={\left[Y(0), \dot{Y}(0)\right]}\exp{\left(t\begin{bmatrix}
-JW & JS(0) \\ I_{2p} & -JW
\end{bmatrix}\right)}
\begin{bmatrix}
I_{2p} \\ 0
\end{bmatrix}
e^{JWt}.$$ Since $Y(0)=X$, $\dot{Y}(0)=Z$, $W=W(0)=X{^{\top}}J Z$, $S(0)=Z{^{\top}}J Z$, we arrive at .
Note that if a tangent vector is [given]{} by $Z=XJW_Z+JX_\perp K_Z$ [with]{} $W_Z\in{{\cal S}_{\mathrm{sym}}}(2p)$ and $K_Z\in{\mathbb{R}}^{(2n-2p)\times 2p}$, then $W=-W_Z$ in .
In the case of [the]{} symplectic group ($p=n$), quasi-geodesics can be computed directly by solving $W=Y{^{\top}}J \dot{Y}=W(0)$, which leads to $Y_U^{\operatorname*{qgeo}}(t;Z)=Ue^{-JWt},$ where $Y(0)=U\in{\mathrm{Sp}(2n)}$ and $W=U{^{\top}}J Z$. It turns out that the quasi-geodesic curves for $p=n$ coincide with the geodesic curves corresponding to the indefinite Khvedelidze–Mladenov metric ${\left\langleZ_1, Z_2\right\rangle}_U :=\operatorname*{tr}{\left(U{^{-1}}Z_1 U{^{-1}}Z_2\right)}$; see [@fiori2011solving Theorem 2.4].
Optimization on the symplectic Stiefel manifold {#sec:optimization}
===============================================
With the geometric tools of \[sec:geometry\] and \[section:canonical\] at hand, almost everything is in place to apply to problem first-order Riemannian optimization methods such as those proposed and analyzed in [@absil2009optimization; @RinWir2012; @iannazzo2018riemannian; @HuaAbsGal2017; @BouAbsCar2018; @hu2019brief]. The remaining task is to choose a mechanism which, given a search direction in the tangent space ${{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}$, returns a curve on ${\mathrm{Sp}(2p,2n)}$ along that direction. In Riemannian optimization, it is customary to provide such a mechanism by choosing a retraction. In \[subsec:quasi\] and \[subsec:cayley\], we propose two retractions on ${\mathrm{Sp}(2p,2n)}$. Then, in \[subsection:non-monotone\], we work out a non-monotone gradient method for problem .
First we briefly recall the concept of (first-order) retraction on a manifold ${{\cal M}}$; see [@ADM2002] or [@absil2009optimization §4.1] for details. Let $\mathrm{T}{{\cal M}}:=\bigcup_{X\in{{\cal M}}}\mathrm{T}_X {{\cal M}}$ be the tangent bundle to ${{\cal M}}$. A smooth mapping ${\cal R}: \mathrm{T}{{\cal M}}\rightarrow {{\cal M}}$ is called a *retraction* if the following properties hold for all $X\in{{\cal M}}$:
- ${\cal R}_X(0_X)=X$, where $0_X$ is the origin of $\mathrm{T}_X {{\cal M}}$;
- $\frac{\text{d}}{\text{dt}}{\cal R}_X(tZ)|_{t=0}=Z$ for all $Z\in \mathrm{T}_X {{\cal M}}$,
where ${\cal R}_X$ denotes the restriction of ${\cal R}$ to $\mathrm{T}_X {{\cal M}}$. Then, given a search direction $Z$ at a point $X$, the above-mentioned mechanism simply returns the curve $t\mapsto {\cal R}_X(tZ)$.
In most analyses available in the literature, the retraction ${\cal R}$ is assumed to be globally defined, i.e., defined everywhere on the tangent bundle $\mathrm{T}{{\cal M}}$. In [@BouAbsCar2018], however, ${\cal R}_X$ is only required to be defined locally, in a closed ball of radius $\varrho(X)>0$ centered at $0_X\in \mathrm{T}_X{{\cal M}}$, provided that $\inf_k \varrho(X^k)>0$ where the $X^k$’s denote the iterates of the considered method. Henceforth, unless otherwise stated, we only assume that, for all $X\in{{\cal M}}$, ${\cal R}_X$ is defined in a neighborhood of $0_X$ in $\mathrm{T}_X{{\cal M}}$.
Quasi-geodesic curve {#subsec:quasi}
--------------------
In view of the results of \[subsec:geo\], we define ${\cal R}^{\operatorname*{qgeo}}$ by $$\begin{aligned}
\label{eq:geo-exp}
{\cal R}^{\operatorname*{qgeo}}_X(Z) := Y_X^{\operatorname*{qgeo}}(1;Z)={\left[X, Z\right]} \exp{\left(\begin{bmatrix}
-JW & JZ{^{\top}}J Z \\ I_{2p} & -JW
\end{bmatrix}\right)}
\begin{bmatrix}
I_{2p} \\ 0
\end{bmatrix}
e^{JW}, \end{aligned}$$ where $Z\in{{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}$ and $W=X{^{\top}}J Z$. The concept is illustrated in \[fig:geodesic\]. We prove next that ${\cal R}^{\operatorname*{qgeo}}$ is a well-defined retraction on ${\mathrm{Sp}(2p,2n)}$.
(1.5,6) – (-1,4) – (9,3) – (11,5.3) – (1.5,6); (1.5,6) – (-1,4) – (9,3) – (11,5.3) – (1.5,6);
\(X) at (3,4.8); (Y) at (8,2); (Z) at (8,4.3); (M) at (5.3,1); (T) at (0,5); (C) at ($ (X) ! .5 ! (Z) $); (A) at ($ (X) ! .5 ! (Y) $); (C) – (Z); (X) – (Z); at ($(Y) + (-1.7,0.1)$) [$\mathcal{R}_X^{\operatorname*{qgeo}}(tZ)$]{}; at (X) ;
(0,2) .. controls (0.5,5) and (3,5) .. (6,0.5); (0,2) .. controls (1,2.5) and (2,3) .. (3.92,3); (6,0.5) .. controls (7.5,1.5) and (9.5,2) .. (12,2); (0.5,3.4) .. controls (1.7,5.5) and (11,8) .. (12,2);
\(X) .. controls ($(A)+(-0.5,1)$) and ($(A)+(0.5,0.7)$) .. (Y);
\[lemma:retraction-geo\] The map ${\cal R}^{\operatorname*{qgeo}}: \mathrm{T}{\mathrm{Sp}(2p,2n)}\to {\mathrm{Sp}(2p,2n)}$ defined in is a globally defined retraction.
In view of , ${\cal R}^{\operatorname*{qgeo}}$ is well defined and smooth on $\mathrm{T}{\mathrm{Sp}(2p,2n)}$. From the power series definition of the matrix exponential, we obtain that $Y_X^{\operatorname*{qgeo}}(1;0_X) = X$. It also follows from this definition (or from the Baker–Campbell–Hausdorff formula) that $\frac{\mathrm{d}}{\mathrm{d}t} \exp(A(t))|_{t=0} = \exp(A(0)) \dot{A}(0)$ when $A(0)$ and $\dot{A}(0)$ commute. This property can be exploited along with the product rule to deduce that $\frac{\mathrm{d}}{\mathrm{d}t} Y_X^{\operatorname*{qgeo}}(1;tZ)|_{t=0} = Z$.
Numerically, computing the exponential will dominate the complexity when $p$ is relatively large; see \[fig:geoVSsymplecticCayley\] for timing experiments.
Symplectic Cayley transform {#subsec:cayley}
---------------------------
In this section, we present another retraction, based on the Cayley transform. It follows naturally from the Cayley transform on quadratic Lie groups [@hairer2006geometric Lemma 8.7] and the Cayley retraction on the Stiefel manifold [@WenYin2013 (7)], with the crucial help of the tangent vector representation given in \[cor:ZtoS\].
Given $Q\in{\mathbb{R}^{n\times n}}$, and considering the quadratic Lie group $\mathcal{G}_Q:=\{X\in{\mathbb{R}^{n\times n}}: X{^{\top}}QX=Q\}$ and its Lie algebra $\mathfrak{g}_Q:=\{A\in{\mathbb{R}^{n\times n}}: QA+A{^{\top}}Q=0\}$, the Cayley transform is given by [@hairer2006geometric Lemma 8.7] $$\label{eq:cay}
\operatorname*{cay}: \mathfrak{g}_Q\to\mathcal{G}_Q: A \mapsto \operatorname*{cay}(A) := (I-A){^{-1}}(I+A),$$ which is well defined whenever $I-A$ is invertible. As for the Cayley retraction on the Stiefel manifold, it is defined, in view of by [@WenYin2013 (7)], by $\mathcal{R}_X(Z) = \operatorname*{cay}(A_{X,Z}) X$ where $A_{X,Z} = (I-\frac12 XX{^{\top}})ZX{^{\top}}- X Z{^{\top}}(I-\frac12 XX{^{\top}})$. This inspires the following definition.
\[def:R-cay\] The *Cayley retraction* on the symplectic Stiefel manifold ${\mathrm{Sp}(2p,2n)}$ is defined, for $X\in{\mathrm{Sp}(2p,2n)}$ and $Z\in{{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}$, by $$\label{eq:R-cay}
{\cal R}^{\operatorname*{cay}}_X(Z)
:= {\left(I-\frac12 S_{X,Z}J\right)}{^{-1}}{\left(I+\frac12 S_{X,Z}J\right)} X,$$ where $S_{X,Z}$ is as in \[proposition:projection\], i.e., $S_{X,Z} = G_XZ(XJ){^{\top}}+ XJ (G_XZ){^{\top}}$ and $G_X = I-\frac12 XJX{^{\top}}J{^{\top}}$. It is well defined whenever $I-\frac12 S_{X,Z}J$ is invertible.
In other words, the selected curve along $Z\in{{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}$ is $$\label{eq:cay-Lie}
Y_X^{\operatorname*{cay}}(t;S) := {\left(I -\frac{t}{2}SJ\right)}{^{-1}}{\left(I +\frac{t}{2}SJ\right)}X = \operatorname*{cay}{\left(\frac{t}{2}SJ\right)}X,$$ where $S = S_{X,Z}$ as defined above.
We confirm right away that ${\cal R}^{\operatorname*{cay}}$ is indeed a retraction.
The map ${\cal R}^{\operatorname*{cay}}: \mathrm{T}{\mathrm{Sp}(2p,2n)}\to {\mathrm{Sp}(2p,2n)}$ in is a retraction.
When $Z=0$, we have $S_{X,Z}=0$ and we obtain ${\cal R}^{\operatorname*{cay}}_X(Z)=X$, which is the first defining property of retractions. For the second property, we have $\frac{\mathrm{d}}{\mathrm{d}t} {\cal R}^{\operatorname*{cay}}_X(tZ)|_{t=0} = \frac{\mathrm{d}}{\mathrm{d}t} \operatorname*{cay}(\frac{t}{2} S_{X,Z}J) X|_{t=0} = \mathrm{D} \operatorname*{cay}(\frac{t}{2} S_{X,Z}J)[\frac12 S_{X,Z}J] X|_{t=0} = S_{X,Z}JX = Z$, where the last two equalities come from $\mathrm{D}\operatorname*{cay}(A)[\dot{A}] = 2(I-A)^{-1} \dot{A} (I+A)^{-1}$ (see [@hairer2006geometric Lemma 8.8]) and \[cor:ZtoS\].
Incidentally, we point out that the Cayley transform can be interpreted as the trapezoidal rule for solving ODEs on quadratic Lie groups. According to [@hairer2006geometric IV. (6.3)], notice that $\mathrm{T}_{X}\mathcal{G}_Q=\{AX: A\in\mathfrak{g}_Q\}$. Hence, the following defines a differential equation on $\mathcal{G}_Q$: $$\label{eq:ODE-Lie}
\dot{Y}(t) = A Y(t),\quad Y(0)=X\in\mathcal{G}_Q,$$ where $A\in\mathfrak{g}_Q$. To solve it numerically, we adopt one step of the trapezoidal rule over $[0,t]$, $$\label{eq:trapezoidal}
Y(t) = X + \frac{t}{2} {\left(AX+AY(t)\right)}.$$ Its solution is given by $Y(t)=\operatorname*{cay}{\left(\frac{t}{2}A\right)}X$ whenever $I-\frac{t}{2} A$ is invertible. Since $\operatorname*{cay}(\frac{t}{2}A)$, $X\in \mathcal{G}_Q$, it follows that $Y(t)\in\mathcal{G}_Q$. This means that the trapezoidal rule for is indeed achieved by the Cayley transform and remains on $\mathcal{G}_Q$. In particular, letting $A=S_{X,Z}J\in\mathfrak{g}_{{\mathrm{Sp}(2n)}}$, $Z=AX\in{{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}$ and $Y={\cal R}^{\operatorname*{cay}}_X(Z)$, the Cayley retraction can be exactly recovered by the same trapezoidal rule ; see \[fig:trapezoidal\] for an illustration.
(1.5,6) – (-1,4) – (9,3) – (11,5.3) – (1.5,6); (1.5,6) – (-1,4) – (9,3) – (11,5.3) – (1.5,6); (X) at (3,4.8); (Y) at (8,2); (Z) at (9,4.1); (M) at (5.3,1); (T) at (0,5); (C) at ($ (X) ! .5 ! (Z) $); (A) at ($ (X) ! .5 ! (Y) $);
\(X) – (C); (C) – (Z); (C) – (Y); at ($(C) + (-0.9,0.6)$) [$\frac{1}{2}AX$]{}; at ($(Y) + (-0.3,1.5)$) [$\frac{1}{2}AY$]{}; at ($(Z) + (0.2,0.5)$) [$Z=AX$]{}; at ($(Y) + (-1.3,-0.3)$) [${\cal R}^{\operatorname*{cay}}_X(Z)$]{}; at (X) ; at (Y) ;
(0,2) .. controls (0.5,5) and (3,5) .. (6,0.5); (0,2) .. controls (1,2.5) and (2,3) .. (3.92,3); (6,0.5) .. controls (7.5,1.5) and (9.5,2) .. (12,2); (0.5,3.4) .. controls (1.7,5.5) and (11,8) .. (12,2);
\(X) .. controls ($(A)+(-0.5,1)$) and ($(A)+(0.5,0.7)$) .. (Y);
In the case of the symplectic group ($p=n$), it can be shown that ${\cal R}^{\operatorname*{cay}}$ reduces to the retraction proposed in [@birtea2018optimization Proposition 2].
The retraction ${\cal R}^{\operatorname*{cay}}_X(tZ)$ in is not globally defined. It is defined if and only if $I-\frac{t}{2}S_{X,Z}J$ is invertible, i.e., $\frac2t$ is not an eigenvalue of the Hamiltonian matrix $S_{X,Z}J$. Since the eigenvalues of Hamiltonian matrices come in opposite pairs, we have that $Y_X^{\operatorname*{cay}}(t;S)$ exists for all $t\geq0$ if and only if $SJ$ has no real eigenvalue. This situation constrasts with the Cayley retraction on the (standard) Stiefel manifold, which is everywhere defined due to the fact that $I-A$ is invertible for all skew-symmetric $A$.
The fact that ${\cal R}^{\operatorname*{cay}}$ is not globally defined does not preclude us from applying the convergence and complexity results of [@BouAbsCar2018]. However, we have to ensure that ${\cal R}_X$ is defined in a closed ball of radius $\varrho(X)>0$ centered at $0_X\in \mathrm{T}_X{{\cal M}}$, with $\inf_k \varrho(X^k)>0$ where the $X^k$’s denote the iterates of the considered method. An assumption that guarantees this condition is that (i) the objective function $f$ in has compact sublevel sets and (ii) the considered optimization scheme guarantees that $f(X^k)\leq f(X^0)$ for all $k$. Indeed, in that case, since $\{X^k\}_{k=0,1,...}$ remains in a compact subset of ${\mathrm{Sp}(2p,2n)}$, it is possible to find $\rho$ such that, for all $k$, if $\|Z\|_{X^k}\leq\rho$, then the spectral radius of $\frac12 S_{X^k,Z}J$ is stricly smaller than one, making $I-\frac12 S_{X^k,Z}J$ invertible. The main computational cost of ${\cal R}^{\operatorname*{cay}}_X(Z)$ (\[def:R-cay\]) is the symplectic Cayley transform, which requires solving linear systems with the $2n\times 2n$ system matrix $(I-\frac12 S_{X,Z}J)$. In general, [this]{} has complexity $O(n^3)$. However, as we now show, the low-rank structure of $S_{X,Z}$ can be exploited to reduce the linear system matrix to the size $4p\times 4p$, thereby inducing a considerable reduction of computational cost when $p\ll n$. The development parallels the one given in [@WenYin2013 Lemma 4(1)] for the (standard) Stiefel manifold.
Let $S=LR{^{\top}}+ RL{^{\top}}=UV{^{\top}}$, where $L,R\in{\mathbb{R}^{2n\times 2p}}$ and $U=[L~R]\in{\mathbb{R}}^{2n\times 4p}$, $V=[R~L]\in{\mathbb{R}}^{2n\times 4p}$. If $I+\frac{t}{2} V{^{\top}}J{^{\top}}U$ is invertible, then admits the expression $$\label{eq:cayley-simple}
Y_X^{\operatorname*{cay}}(t;S)=X+tU{\left(I+\frac{t}{2} V{^{\top}}J{^{\top}}U\right)}{^{-1}}V{^{\top}}JX.$$ In particular, if we choose $L=-H_X \nabla \bar{f}(X)$ and $R=XJ$, then we get $S=-S_X$ with $S_X$ given in the gradient formula (\[proposition:rgrad\]), and we have that ${\cal R}^{\operatorname*{cay}}_X(-t{\mathrm{grad}_\rho f(X)})$ is given by $$\label{eq:cayley-grad-simple}
Y_X^{\operatorname*{cay}}(t;-S_X)=X+t[-P_{f}~XJ] {\left(I+\frac{t}{2}
\begin{bmatrix}
E_\rho & J{^{\top}}\\ P_f{^{\top}}J{^{\top}}P_f & -E{^{\top}}_\rho
\end{bmatrix}
\right)}{^{-1}}\begin{bmatrix}
I \\ -E{^{\top}}_\rho J
\end{bmatrix},$$ where $H_X$ is defined in \[proposition:rgrad\], $P_f:=H_X \nabla \bar{f}(X)$, and $E_\rho:=\frac{\rho}{2} X{^{\top}}\nabla \bar{f}(X)$.
By using Sherman–Morrison–Woodbury (SMW) formula [@golub2013matrix (2.1.4)]: $$\label{eq:SMW}
(A+\bar{U}\bar{V}{^{\top}}){^{-1}}=A{^{-1}}-A{^{-1}}\bar{U} {\left(I+\bar{V}{^{\top}}A{^{-1}}\bar{U}\right)} {^{-1}}\bar{V}{^{\top}}A{^{-1}}$$ with $A=I$, $\bar{U}=-\frac{t}{2}U$ and $\bar{V}=J{^{\top}}V$, it follows that ${\left(I -\frac{t}{2}SJ\right)}{^{-1}}={\left(I -\frac{t}{2}UV{^{\top}}J\right)}{^{-1}}=(A+\bar{U}\bar{V}{^{\top}}){^{-1}}=I-\frac{t}{2} U(I+\frac{t}{2} V{^{\top}}J{^{\top}}U){^{-1}}V{^{\top}}J{^{\top}}.$ In view of , it turns out that $$\begin{aligned}
\arraycolsep=2pt
\begin{array}{rcl}
Y_X^{\operatorname*{cay}}(t;S) &=& \operatorname*{cay}{\left(\frac{t}{2}SJ\right)}X = {\left(I -\frac{t}{2}SJ\right)}{^{-1}}{\left(I +\frac{t}{2}SJ\right)}X \\
&=& X+\frac{t}{2}U{\left(I+(I+\frac{t}{2} V{^{\top}}J{^{\top}}U){^{-1}}{(I-\frac{t}{2}V{^{\top}}J{^{\top}}U)}\right)}V{^{\top}}JX\\
&=& X+tU{\left(I+\frac{t}{2} V{^{\top}}J{^{\top}}U\right)}{^{-1}}V{^{\top}}JX,
\end{array}
\end{aligned}$$ which completes the proof of . Substituting $L=-H_X \nabla \bar{f}(X)$ and $R=XJ$ into $Y_X^{\operatorname*{cay}}(t;S)$, it is straightforward to arrive at .
A non-monotone line search scheme on manifolds {#subsection:non-monotone}
----------------------------------------------
Now, since we can compute a gradient ${\mathrm{grad} f(X)}$ and a retraction ${\cal R}_X(Z)$, various first-order Riemannian optimization methods can be applied to problem . Here, we adopt an approach called non-monotone line search [@zhang2004nonmonotone]. It was extended to the Stiefel manifold in [@WenYin2013] and to general Riemannian manifolds in [@iannazzo2018riemannian] and [@hu2019brief §3.3]. The non-monotone approach has been observed to work well on various Riemannian optimization problems. In this section, we first present and analyze the non-monotone line search algorithm on general Riemannian manifolds endowed with a retraction that need not be globally defined. Then we apply the algorithm and its analysis to the case of the symplectic Stiefel manifold ${\mathrm{Sp}(2p,2n)}$.
Given $\beta\in(0,1)$, a backtracking parameter $\delta\in(0,1)$, a search direction $Z^k$ and a trial step size $\gamma_k>0$, the non-monotone line search procedure proposed in [@hu2019brief] uses the step size $t_k = \gamma_k \delta^h$, where $h$ is the smallest integer such that $$\label{eq:non-monotone}
f{\left({\cal R}_{X^k}(t_k Z^k)\right)} \le c_k + \beta t_k {{\left\langle{\mathrm{grad} f(X^k)},Z^k\right\rangle}_{X^k}},$$ and the next iterate is given by $X^{k+1}={\cal R}_{X^k}(t_k Z^k)$. The scalar $c_{k}$ in is a convex combination of $c_{k-1}$ and $f(X^k)$. Specifically, we set $q_0=1 , c_0= f(X^0)$, $$\begin{aligned}
\label{eq:non-monotone-CQ}
\begin{array}{l}
q_k = \alpha q_{k-1} + 1,\\
c_k = \frac{\alpha q_{k-1}}{q_k} c_{k-1} + \frac{1}{q_k} f(X^k)
\end{array}\end{aligned}$$ with a parameter $\alpha\in[0,1]$. When $\alpha=0$, it follows that $q_k=1$ and $c_k=f(X^k)$, and the non-monotone condition reduces to the standard Armijo backtracking line search $$\label{eq:monotone}
f{\left({\cal R}_{X^k}(t_k Z^k)\right)} \le f(X^k) + \beta t_k {{\left\langle{\mathrm{grad} f(X^k)},Z^k\right\rangle}_{X^k}}.$$
The corresponding Riemannian gradient method is presented in . The search direction is chosen as the Riemannian antigradient in line \[alg:nmg:grad\]. On the other hand, there is no restriction on the choice of the trial step size $\gamma_k$ in line \[alg:nmg:gamma\]. One possible strategy is the Barzilai–Borwein (BB) method [@BB], which often accelerates the convergence of gradient methods when the search space is a Euclidean space. This method was extended to general Riemannian manifolds [in]{} [@iannazzo2018riemannian]. We implement and compare several choices of $\gamma_k$; see \[subsection:default setting\] for details.
**Input:** $X^0\in{{\cal M}}$.\
**Require:** Continuously differentiable function $f:{{\cal M}}\to{\mathbb{R}}$; retraction $\mathcal{R}$ on ${{\cal M}}$ defined on $\operatorname*{dom}(\mathcal{R})$; $\beta, \delta\in(0,1)$, $\alpha \in [0,1]$, $0<\gamma_\mathrm{min}<\gamma_\mathrm{max}$, $c_0 = f(X^0)$, $q_0=1$, $\gamma_0 = f(X^0)$.\
**Output:** Sequence of iterates $\{X^k\}$.
Next we prove the convergence of on a general Riemannian manifold ${{\cal M}}$. Note that, in terms of convergence analysis, the only relevant difference between and [@hu2019brief Algorithm 1] is that the latter assumes that the retraction $\mathcal{R}$ is globally defined, whereas we only assume that, for all $X\in{{\cal M}}$, ${\cal R}_X$ is defined in a neighborhood of $0_X$ in $\mathrm{T}_X{{\cal M}}$. In other words, we only assume that, for every $X\in{{\cal M}}$, there exists a ball $\mathcal{B}^r_{0_{X}}:=\{Z\in\mathrm{T}_X{{\cal M}}: {\left\|Z\right\|}_{X}< r\}\subseteq\operatorname*{dom}(\mathcal{R})$. Hence, the convergence result in [@hu2019brief Theorem 3.3] does not directly apply to our case.
First, we show that does not abort.
\[proposition:infinite-sequence\] generates an infinite sequence of iterates.
Let $X^k$ be the current iterate. In view of $Z^k=-{\mathrm{grad} f(X^k)}$, by applying [@zhang2004nonmonotone Lemma 1.1] to the Riemannian case, it yields $f(X^k)\leq c_k$. Since $\mathcal{R}$ is locally defined and $\delta,\beta\in(0,1)$, it follows that $$\begin{aligned}
\lim\limits_{h\to +\infty}\frac{f{\left({\cal R}_{X^k}(\gamma_k \delta^h Z^k)\right)}-c_k}{\gamma_k \delta^h}&\leq\lim\limits_{h\to +\infty}\frac{f{\left({\cal R}_{X^k}(\gamma_k \delta^h Z^k)\right)}-f(X^k)}{\gamma_k \delta^h}\\
&={{\left\langle{\mathrm{grad} f(X^k)},Z^k\right\rangle}_{X^k}}
<\beta{{\left\langle{\mathrm{grad} f(X^k)},Z^k\right\rangle}_{X^k}}.
\end{aligned}$$ It implies that there exists $\bar{h}\in\mathbb{N}$ such that $t_k=\gamma_k\delta^{\bar{h}}\in(0,\frac{r}{\|Z^k\|_{X^k}})$ and the non-monotone condition hold. It means that $t_k Z^k\in\mathcal{B}^r_{0_{X^k}}\subseteq\operatorname*{dom}(\mathcal{R})$, and hence it is accepted.
Next, we give the proof of the convergence for .
\[theorem:convergence\] Let $\{X^k\}$ be an infinite sequence of iterates generated by . Then every accumulation point $X^*$ of $\{X^k\}$ such that $0_{X^*}\in\mathrm{T}_{X^*}{{\cal M}}$ in the interior of $\operatorname*{dom}(\mathcal{R})$, is a critical point of $f$, i.e., ${\mathrm{grad} f(X^*)}=0$.
We adapt the proof strategy of [@hu2019brief Theorem 3.3] (that invokes [@absil2009optimization Theorem 4.3.1], which itself is a generalization to manifolds of the proof of [@Ber95 Proposition 1.2.1]) in order to handle the locally defined retraction. The adaptation consists in defining and making use of the neighborhood $\mathcal{N}_{0_{X^*}}$.
Since $X^*$ is an accumulation point of $\{X^k\}$, there exists a subsequence $\{X^k\}_{k\in\mathcal{K}}$ that converges to it. In view of and , it holds that $q_{k+1}=1+\alpha q_k=1+\alpha+\alpha^2q_{k-1}=\dots=\sum_{i=0}^{k+1}\alpha^i
\le k+2
$. Moreover, it follows that $$\begin{aligned}
c_{k+1}-c_k&=\frac{\alpha q_{k}c_{k}+f(X^{k+1})}{q_{k+1}} -c_k=\frac{\alpha q_{k}c_{k}+f(X^{k+1})}{q_{k+1}} -\frac{\alpha q_{k}+1}{q_{k+1}}c_k\\
&=\frac{f(X^{k+1})-c_k}{q_{k+1}} \leq -\frac{\beta t_k{\left\|{\mathrm{grad} f(X^k)}\right\|}^2_{X^k}}{q_{k+1}}<0.
\end{aligned}$$ Hence, $\{c_k\}$ is monotonically decreasing. Since $f(X^k)\leq c_k$ (in the proof of \[proposition:infinite-sequence\]) and $\{f(X^k)\}_{k\in\mathcal{K}}\to f(X^*)$, it readily follows that $ c_\infty:=\lim_{k\to+\infty}c_k>-\infty$. Summing above inequalities and using $q_{k+1}\le k+2$, it turns out $$\sum_{k=0}^{\infty}\frac{\beta t_k{\left\|{\mathrm{grad} f(X^k)}\right\|}^2_{X^k}}{k+2}
\leq \sum_{k=0}^{\infty}\frac{\beta t_k{\left\|{\mathrm{grad} f(X^k)}\right\|}^2_{X^k}}{q_{k+1}}
\leq
\sum_{k=0}^{\infty}(c_k-c_{k+1})
=c_0-c_\infty<\infty.$$ This inequality implies that $$\label{eq:convergence-1}
\lim\limits_{k\to+\infty} t_k{\left\|{\mathrm{grad} f(X^k)}\right\|}^2_{X^k}=0, \quad k\in\mathcal{K}.$$
For the sake of contradiction, suppose that $X^*$ is not a critical point of $f$, i.e., that ${\mathrm{grad} f(X^*)}\neq0$. It readily follows from that $\{t_k\}_{k\in\mathcal{K}}\to0$. By the construction of , the step size $t_k$ has the form of $t_k=\gamma_k \delta^h$ with $\gamma_k\ge\gamma_{\min}>0$. Since $0_{X^*}$ is in the interior of $\operatorname*{dom}(\mathcal{R})$, there exists a neighborhood $\mathcal{N}_{0_{X^*}}:=\{(X,Z)\in\mathrm{T}{{\cal M}}: \mathrm{dist}^2(X,X^*)+ {\left\|Z\right\|}_{X}^2 < r\}\subseteq\operatorname*{dom}(\mathcal{R})$. Due to $\{t_k\}_{k\in\mathcal{K}}\to0$, it follows that there exists $\bar{k}\in\mathcal{K}$ and $h_k\in\mathbb{N}$ with $h_k\geq1$ such that for all $k>\bar{k}$, $t_k=\gamma_k \delta^{h_k}$ and $(X^k,\frac{t_k}{\delta} Z^k)\in\mathcal{N}_{0_{X^*}}$. Since $h_k$ is the smallest integer such that the non-monotone condition holds, it follows that $\frac{t_k}{\delta} Z^k$ does not satisfy , i.e., $$\begin{aligned}
\label{eq:non-monotone-contradiction}
f(X^k) - f{\left({\cal R}_{X^k}(\frac{t_k}{\delta} Z^k)\right)}
\leq
c_k - f{\left({\cal R}_{X^k}(\frac{t_k}{\delta} Z^k)\right)}
< \beta \frac{t_k}{\delta} {\left\|{\mathrm{grad} f(X^k)}\right\|}^2_{X^k}.
\end{aligned}$$ The mean value theorem for ensures that there exists $\bar{t}_k\in{\left[0,\frac{t_k}{\delta}\right]}$ such that $$\mathrm{D}(f\circ {\cal R}_{X^k})(\bar{t}_k Z^k)[Z^k]
<\beta {\left\|{\mathrm{grad} f(X^k)}\right\|}^2_{X^k},\quad \mbox{for all~} {k}\in\mathcal{K}, k>\bar{k}.$$ We now take the limit in the above inequality as $k\to\infty$ over $\mathcal{K}$. Using the fact that $\mathrm{D}(f\circ {\cal R}_{X^*})(0_{X^*}) = \mathrm{D}f(X^*)$ in view of the defining properties of a retraction, we obtain ${\left\|{\mathrm{grad} f(X^*)}\right\|}^2_{X^*}\leq\beta {\left\|{\mathrm{grad} f(X^*)}\right\|}^2_{X^*}$. Since $\beta<1$, this is a contradiction with the supposition that ${\mathrm{grad} f(X^*)}\neq0$.
applies to ${{\cal M}}= {\mathrm{Sp}(2p,2n)}$ as follows. Pick a constant $\rho>0$ and one of the orthonormalization conditions (I) or (II) for $X_\perp$ in order to make a bona-fide Riemannian metric. In line \[alg:nmg:grad\], the gradient is then as stated in \[proposition:rgrad\]. Finally, choose the retraction $\mathcal{R}$ as either the quasi-geodesic curve or the symplectic Cayley transform . The initialization techniques for $X^0\in{\mathrm{Sp}(2p,2n)}$ and the stopping criterion will be discussed in \[subsection:Stopping criteria\].
\[theorem:convergence-symplectic\] Apply to ${\mathrm{Sp}(2p,2n)}$ as specified in the previous paragraph. Let $\{X^k\}$ be an infinite sequence of iterates generated by . Then every accumulation point $X^*$ of $\{X^k\}$ is a critical point of $f$, i.e., ${\mathrm{grad} f(X^*)}=0$.
First, the quasi-geodesic retraction is globally defined. Hence, \[theorem:convergence\] can be directly applied.
Next, we consider the Cayley-based algorithm. In order to conclude by invoking \[theorem:convergence\], it is sufficient to show that, for all $X\in{\mathrm{Sp}(2p,2n)}$, $0_X\in{{\mathrm{T}_{X}}{\mathrm{Sp}(2p,2n)}}$ is in the interior of $\operatorname*{dom}({\cal R}^{\operatorname*{cay}})$. In view of the properties of retractions, we have that $0_X\in\operatorname*{dom}(\mathcal{R}^{\operatorname*{cay}})$, and we finish the proof by showing that all points of $\operatorname*{dom}(\mathcal{R}^{\operatorname*{cay}})$ are in its interior, namely, $\operatorname*{dom}(\mathcal{R}^{\operatorname*{cay}})$ is open. To this end, let $F:\mathrm{T}{\mathrm{Sp}(2p,2n)}\to{\mathbb{R}}:(X,Z)\mapsto\det(I-\frac12 S_{X,Z})$, where $S_{X,Z}$ is as in \[def:R-cay\]. Since $F$ is continuous and $\{0\}$ is closed, it follows that $F{^{-1}}(0)=\{(X,Z)\in\mathrm{T}{\mathrm{Sp}(2p,2n)}: F(X,Z)=0\}$ is a closed set. Its complement in the tangent bundle is thus open, and it is also the domain of $\mathcal{R}^{\operatorname*{cay}}$.
Numerical experiments {#sec:Numerical Experiment}
=====================
In this section, we report the numerical performance of . Both methods based on the [quasi-geodesics]{} and the symplectic Cayley transform are evaluated. We first introduce implementation details in \[subsection:Stopping criteria\]. To determine default settings, we investigate the parameters of our proposed algorithms in \[subsection:default setting\]. Finally, the efficiency of is assessed by solving several different problems. The experiments are performed on a workstation with two Intel(R) Xeon(R) Processors Silver 4110 (at 2.10GHz$\times 8$, 12M Cache) and 384GB of RAM running MATLAB R2018a under Ubuntu 18.10. The code that produced the results is available from <https://github.com/opt-gaobin/spopt>.
Implementation details {#subsection:Stopping criteria}
----------------------
As we mentioned in \[subsubsec:rgrad\], we propose two strategies to compute the Riemannian gradient ${\mathrm{grad}_\rho f(X)}$. Both algorithms with type (I) and type (II) perform well in our preliminary experiments. In this section, with the symplectic Cayley transform is denoted by “Sp-Cayley", and its instances using type (I) and (II) for the gradient are represented as “Sp-Cayley-I" and “Sp-Cayley-II", respectively. Similarly, quasi-geodesic algorithms are denoted by “Quasi-geodesics", “Quasi-geodesics-I" and “Quasi-geodesics-II".
We adopt formula , i.e., $H_X\nabla\bar{f}(X) (XJ){^{\top}}JX +XJ (H_X\nabla\bar{f}(X)){^{\top}}JX$, to assemble ${\mathrm{grad}_\rho f(X)}$ for all the algorithms. In order to save flops and obtain a good feasibility (see \[fig:geoVSsymplecticCayley\]), we choose to compute the Cayley retraction. Note that we keep the calculation of $(XJ){^{\top}}JX$ in the first term of ${\mathrm{grad}_\rho f(X)}$, although it is trivial that $(XJ){^{\top}}JX=I$ for $X\in{\mathrm{Sp}(2p,2n)}$. This is due to our observation that the feasibility of Quasi-geodesics gradually degrades when we omit this calculation. At the beginning of , we need a feasible point $X^0\in{\mathrm{Sp}(2p,2n)}$ to start the iteration. The easiest way to generate a symplectic matrix is to choose the “identity" matrix in ${\mathrm{Sp}(2p,2n)}$, namely, $I^0={\left[\begin{smallmatrix}
I_p & 0 & 0 & 0 \\ 0& 0 & I_p & 0
\end{smallmatrix}\right]}{^{\top}}$. Moreover, by using \[proposition:exponential\], we suggest the following strategies to generate an initial point:
- $X^0=I^0$;
- $X^0=I^0 e^{J(W+W{^{\top}})}$, where $W$ is randomly generated by `W=randn(2*p,2*p)`;
- $X^0$ is assembled by the first $p$ columns and $(n+1)$-th to $(n+p)$-th columns of $e^{J(W+W{^{\top}})}$, where $W$ is randomly generated by `W=randn(2*n,2*n)`.
The matrix exponential is computed by the function $\texttt{expm}$ in MATLAB. Unless otherwise specified, we choose strategy 2) as our initialization.
For a stopping criterion, we check the following two conditions: $$\begin{aligned}
\label{eq:stop}
&{\left\|{\mathrm{grad}_\rho f(X^k)}\right\|}{_{\mathrm{F}}}\le\epsilon,&\\
\label{eq:stop1}
&\frac{{\left\|X^k-X^{k+1}\right\|}{_{\mathrm{F}}}}{\sqrt{2n}} < \epsilon_x \quad \mbox{and}\quad
\frac{{\left|f(X^k)-f(X^{k+1})\right|}}{{\left|f(X^k)\right|}+1} < \epsilon_f &
$$ with given tolerances $\epsilon,\epsilon_x, \epsilon_f>0$. [We terminate the algorithms [once]{} one of the criteria -]{} or a maximum iteration number $\mathrm{MaxIter}$ is reached. The default tolerance parameters are chosen as $\epsilon = 10^{-5}$, $\epsilon_x = 10^{-5}$, $\epsilon_f = 10^{-8}$ and $\mathrm{MaxIter}=1000$. For parameters to control the non-monotone line search, we follow the choices in the code OptM[^5] [@WenYin2013], specifically, $\beta=10^{-4}, \delta=0.1, \alpha=0.85$, as our default settings. In addition, we choose $\gamma_\mathrm{min}=10^{-15}$, $\gamma_\mathrm{max}=10^{15}$, and a trial step size $\gamma_k$ as in .
Default settings of the algorithms {#subsection:default setting}
----------------------------------
In this section, we study the performance and robustness of our algorithms by choosing different parameters. All the comparisons and results are based on a test problem, called the nearest symplectic matrix problem, which aims to calculate the nearest symplectic matrix to a target matrix $A\in{\mathbb{R}^{2n\times 2p}}$ with respect to the Frobenius norm, i.e., $$\label{eq:nearest}
\min\limits_{X\in{\mathrm{Sp}(2p,2n)}} {\left\|X-A\right\|}{^2_{\mathrm{F}}}.$$ [The special case of this problem on the symplectic group (i.e., $p=n$) was studied in [@wu2010critical]]{}. In our experiments, $A$ is randomly generated by `A=randn(2*n,2*p)`, then it is scaled using `A=A/norm(A)`.
### A comparison between quasi-geodesics and symplectic Cayley transform {#subsection:geoVScayley}
In , we can choose between two different retractions: quasi-geodesic and Cayley. In order to investigate the numerical performance of the two alternatives, we solve the nearest symplectic matrix problem of size $2000 \times 400$ and choose the parameter for the metric $g_{\rho}$ to be $\rho=1$. We stop our algorithms only when $\mathrm{MaxIter}=120$ is reached. The evolution of the norm of the gradient and the feasibility violation ${\left\|X{^{\top}}JX-J\right\|}{_{\mathrm{F}}}$ for both algorithms is shown in \[fig:geoVSsymplecticCayley\]. It illustrates that Sp-Cayley performs better than Quasi-geodesics in terms of efficiency and feasibility. Therefore, we choose Sp-Cayley as our default algorithm and the following experiments will focus on Sp-Cayley.
### A comparison of different metrics {#sec:exper-metrics}
In this section, we compare the performance of Sp-Cayley with different metrics. Namely, we compare Sp-Cayley-I and Sp-Cayley-II for a set of parameters $\rho=2^l$ with $l$ chosen from $\{-3,-2,-1,0,1,2,3\}$. We run [both]{} algorithms 100 times on randomly generated nearest symplectic matrix problems of size $2000\times 40$. Note that for each instance, Sp-Cayley-I and Sp-Cayley-II use the same initial guess. In order to get an average performance, we stop algorithms only when ${\left\|{\mathrm{grad}_\rho f(X^k)}\right\|}{_{\mathrm{F}}}\le10^{-4}$. A summary of numerical results is reported in \[fig:metric\]. It displays average iteration numbers and the feasibility violation for the two algorithms with different $\rho$. We can learn from [the]{} figures that:
- The value $\rho^*$ at which [Sp]{}-Cayley-I and Sp-Cayley-II achieve the best performance is approximately equal to $1/2$ and 1, respectively. The difference of $\rho^*$ is due to the different choice of $X_\perp$, which also has an effect on the metric $g_\rho$. We have observed that $\rho^*$ may vary [for]{} different objective functions. This indicates that tuning $\rho$ for the problem class of interest may significantly improve the performance of .
- Sp-Cayley-I has a lower average iteration number than Sp-Cayley-II when $\rho\le 1$. Over all the variants considered in \[fig:metric\], Sp-Cayley-I with $\rho=1/2$ has the best performance.
- Both Sp-Cayley-I and Sp-Cayley-II show a loss of feasibility when $\rho$ becomes large. A possible reason is that the non-normalized second term of $H_X=JX_\perp X{^{\top}}_\perp J{^{\top}}+\frac{\rho}{2}XX{^{\top}}$ in the Riemannian gradient becomes dominant.
According to the above observations, we choose $\rho=1/2$ for Sp-Cayley-I and $\rho=1$ for Sp-Cayley-II as our default settings.
### A comparison of different line search schemes {#subsec:line-search}
The non-monotone line search strategy (\[subsection:non-monotone\]) depends on several parameters. The purpose of this section is to investigate which among those parameters have a significant impact on the performance of . First, we consider the choice of [the]{} trial step size $\gamma_k$ in . In our case, the ambient space is Euclidean. Thus, we can use the BB method proposed in [@WenYin2013] [and define]{} $$\begin{aligned}
\label{eq:BB}
{\gamma_k^{\mathrm{BB}1}} := \frac{{\left\langleS^{k-1},S^{k-1}\right\rangle}}{{\left|{\left\langleS^{k-1},{Y^{k-1}}\right\rangle}\right|}}, \quad
{\gamma_k^{\mathrm{BB}2}} :=\frac{{\left|{\left\langleS^{k-1},Y^{k-1}\right\rangle}\right|}}{{{\left\langleY^{k-1},{Y^{k-1}}\right\rangle}}},\end{aligned}$$ where $S^{k-1} = X^k - X^{k-1}$ and $Y^{k-1} ={\mathrm{grad}_\rho f(X^k)} - {\mathrm{grad}_\rho f(X^{k-1})}$. Note that this differs from the Riemannian BB method in [@iannazzo2018riemannian] since it adopts the Euclidean inner product ${\left\langle\cdot,\cdot\right\rangle}$ rather than $g_\rho$. This choice is cheaper in flops and we have observed that it speeds up the algorithm. Owing to its efficiency, we further adopt the *alternating* BB strategy [@Dai_Fletcher_2005] [and choose the trial step size as]{} $$\begin{aligned}
\label{eq:ABB}
\gamma_k^{\mathrm{ABB}}:= \left\{
\begin{array}{cl}
\gamma_k^{\mathrm{BB}1},& \mbox{for odd}~~k,\\
\gamma_k^{\mathrm{BB}2},& \mbox{for even}~~k.
\end{array}
\right.\end{aligned}$$ We next compare $\gamma_k^{\mathrm{BB1}}$, $\gamma_k^{\mathrm{BB2}}$, $\gamma_k^{\mathrm{ABB}}$, and the step size $$\gamma_k^{\mathrm{M}}:=2{\left|\frac{f(X^k)-f(X^{k-1})}{\mathrm{D}f(X)[Z]}\right|}$$ proposed in [@nocedal2006numerical (3.60)], where $\gamma_k^{\mathrm{M}}$ is also used in the line search function in Manopt[^6] [@manopt]. In this test, we opt for the monotone line search ($\alpha=0$) adopted with tolerances $\epsilon = 10^{-10}$, $\epsilon_x = 10^{-10}$, $\epsilon_f = 10^{-14}$. \[fig:gamma\] reveals that the BB strategies greatly improve the performance of the Riemannian gradient method, and outperform the classical initial step size $\gamma^{\mathrm{M}}$ in iteration number and function value decreasing. We have obtained similar results, omitted here, for Sp-Cayley-II. [Since]{} $\gamma_k^{\mathrm{ABB}}$ is the most efficient choice in this experiment, we employ it as our default setting henceforth.
Next we investigate the impact of the parameter $\alpha$, which controls the degree of non-monotonicity. If $\alpha=0$, then condition reduces to the usual monotone condition . Here we scale the problem as `A=2*A/norm(A)` because we found that it reveals better the advantage that the non-monotone approach can have. We test Sp-Cayley-II with $\alpha=0,0.85$, i.e., the monotone and non-monotone schemes. The results are shown in \[fig:monotone\]. The purpose of the non-monotone strategy is to make the line search condition more prone than the monotone strategy to accept the trial step size $\gamma_k$, here $\gamma_k^{\mathrm{ABB}}$. We see that this results in a faster convergence in this experiment. Since the non-monotone condition works well in our problem, thus we choose it as a default setting henceforth.
Nearest symplectic matrix problem
---------------------------------
In this section, we still focus on the nearest symplectic matrix problem . We first compare the algorithms Sp-Cayley-I and Sp-Cayley-II on an open matrix dataset: SuiteSparse Matrix Collection[^7]. Due to the different size and scale of data matrices, we choose the column number $p$ from the set $\{5,10,20,40,80\}$, and the target matrix $A\in{\mathbb{R}^{2n\times 2p}}$ is generated by the first $2p$ columns of an original data matrix. In order to obtain a comparable error, we [normalize all matrices as $A/\|A\|_{\max}$]{}, where ${\|A\|_{\max}}:=\max_{i,j}{\left|A_{ij}\right|}$. Numerical results are presented in \[tab:nearest sympectic matrix\] for representative problem instances. Here, “fval" represents the function value, “gradf", “feasi", “iter", and “time" stand for ${\left\|\mathrm{grad}_{{\rho}}f\right\|}{_{\mathrm{F}}}$, ${\left\|X{^{\top}}JX-J\right\|}{_{\mathrm{F}}}$, the number of iteration[s]{}, and the wall-clock time in second[s]{}, respectively. From the table, we find that [both algorithms]{} perform well on most of the instances, and Sp-Cayley-I performs better than Sp-Cayley-II with fewer iteration number and fewer running time. In the largest problem “2cubes\_sphere", [both]{} methods converge and obtain comparable results for the function value and gradient error. In addition, Sp-Cayley-II diverges on the problem “msc23052" with $p=10$, while Sp-Cayley-I converges. Therefore, we conclude that for [the nearest symplectic matrix problem,]{} Sp-Cayley-I is more robust and efficient than [Sp-Cayley-II]{}.
[rrrrrrrrrrr]{} & Sp-Cayley-I &&&&& Sp-Cayley-II &&&&\
(r)[2-6]{}(r)[7-11]{} & [fval]{} & [gradf]{} & feasi & [iter]{} & [time]{} & [fval]{} & [gradf]{} & feasi & [iter]{} & [time]{}\
\
5 & 3.995e+00 & 8.38e-05 & 1.39e-14 & 24 & 2.11 & 3.995e+00 & 1.40e-04 & 4.53e-15 & 23 & 2.18\
10 & 7.331e+00 & 2.23e-04 & 1.95e-14 & 27 & 4.14 & 7.331e+00 & 5.28e-04 & 9.13e-15 & 28 & 4.60\
20 & 1.602e+01 & 2.19e-04 & 4.76e-14 & 30 & 8.81 & 1.602e+01 & 6.72e-04 & 5.26e-14 & 36 & 12.53\
40 & 3.423e+01 & 2.51e-03 & 7.08e-14 & 32 & 23.60 & 3.423e+01 & 7.20e-04 & 4.58e-14 & 40 & 31.79\
80 & 1.056e+02 & 7.98e-04 & 4.05e-13 & 39 & 55.97 & 1.056e+02 & 1.50e-03 & 2.81e-10 & 41 & 71.11\
\
5 & 2.649e+00 & 2.03e-04 & 1.00e-14 & 22 & 0.78 & 2.649e+00 & 1.49e-03 & 3.15e-15 & 27 & 1.01\
10 & 6.230e+00 & 2.16e-04 & 1.65e-14 & 26 & 1.54 & 6.230e+00 & 6.35e-04 & 6.35e-15 & 36 & 2.30\
20 & 1.289e+01 & 2.86e-04 & 3.00e-14 & 26 & 2.79 & 1.289e+01 & 6.85e-04 & 1.08e-14 & 38 & 4.31\
40 & 2.542e+01 & 1.33e-03 & 5.74e-14 & 28 & 8.19 & 2.542e+01 & 6.75e-04 & 3.72e-14 & 36 & 11.78\
80 & 5.156e+01 & 6.38e-04 & 1.08e-13 & 26 & 19.81 & 5.156e+01 & 1.47e-03 & 7.33e-14 & 36 & 30.05\
\
5 & 6.204e+00 & 1.23e-04 & 2.47e-04 & 92 & 1.30 & 6.204e+00 & 7.61e-04 & 5.26e-06 & 66 & 1.04\
10 & 1.361e+01 & 5.87e-04 & 7.06e-08 & 67 & 1.66 & 4.556e-01 & 2.72e-02 & 4.24e+00 & 637 & 17.69\
20 & 2.732e+01 & 9.82e-04 & 1.48e-04 & 80 & 3.70 & 2.730e+01 & 1.14e-02 & 8.69e-03 & 112 & 5.91\
40 & 5.721e+01 & 2.18e-03 & 1.74e-06 & 70 & 6.78 & 5.721e+01 & 4.36e-03 & 1.98e-03 & 90 & 9.90\
80 & 1.302e+02 & 9.96e-03 & 8.89e-03 & 98 & 27.87 & 1.302e+02 & 1.60e-02 & 1.00e-02 & 88 & 28.35\
\
5 & 2.170e+00 & 3.34e-04 & 1.29e-14 & 34 & 0.16 & 2.170e+00 & 1.05e-03 & 3.79e-15 & 35 & 0.17\
10 & 4.378e+00 & 1.53e-04 & 3.62e-14 & 35 & 0.25 & 4.378e+00 & 4.34e-03 & 9.72e-15 & 41 & 0.29\
20 & 9.033e+00 & 3.54e-04 & 6.24e-14 & 34 & 0.44 & 9.033e+00 & 1.46e-03 & 1.41e-14 & 37 & 0.53\
40 & 1.828e+01 & 7.82e-04 & 8.58e-14 & 40 & 1.09 & 1.828e+01 & 1.75e-03 & 2.81e-14 & 44 & 1.24\
80 & 3.731e+01 & 3.42e-04 & 3.32e-13 & 48 & 3.00 & 3.731e+01 & 6.74e-04 & 5.79e-14 & 56 & 3.92\
We next compare our algorithms on randomly generated datasets. Given a set of samples $\{X_1,X_2,\dots,X_N\}$ with $X_i\in{\mathrm{Sp}(2p,2n)}$ for $i=1,\dots,N$, the extrinsic mean problem [@bhattacharya2003large Section 3] on ${\mathrm{Sp}(2p,2n)}$ is defined as $$\label{eq:minimal-distance}
\min\limits_{X\in{\mathrm{Sp}(2p,2n)}} \frac{1}{N}\sum_{i=1}^{N}{\left\|X-X_i \right\|}{^2_{\mathrm{F}}}.$$ In view of [@bhattacharya2003large Section 3], the solutions of are those of with $A=\frac{1}{N}\sum_{i=1}^{N}X_i$. This allows us to reuse the code that addressed . We test Sp-Cayley-I and Sp-Cayley-II for solving the problem with three different random sample sets: (i) $N=100, n=p=2$; (ii) $N=100, n=p=10$; (iii) $N=1000, n=1000,p=20$. In each dataset, samples are randomly generated around a selected center $Y^0\in{\mathrm{Sp}(2p,2n)}$. Specifically, we choose $X_i=Y^0 e^{J(W_i+W_i{^{\top}})}$, where `W_i=0.1*randn(2*p,2*p)`. The initial point $X^0$ and the center $Y^0$ are calculated by the strategy 3) (\[subsection:Stopping criteria\]) for datasets (i)-(ii), and 2) for dataset (iii). In the first two sets (the symplectic group), Sp-Cayley-I and Sp-Cayley-II reduce to the same algorithm due to the same Riemannian gradient. Therefore, we omit the results of Sp-Cayley-II. We run our algorithms twice with different stopping tolerances: default and $\{\epsilon = 10^{-10}, \epsilon_x = 10^{-10}, \epsilon_f = 10^{-14}\}$. The detailed results for these two setting are presented in \[tab:LS\]. It reveals that our algorithms converge for three different sample sets with different stopping tolerances. Moreover, we show the initial and final errors of each sample in \[fig:LS\], where $X^*$ denotes the solution obtained by Sp-Cayley. From the figure, we observe that for both dataset (i) and (ii), the sample error greatly decreases with Sp-Cayley.
[crrrrrrrrrr]{} & Sp-Cayley-I &&&&& Sp-Cayley-II &&&&\
(r)[2-6]{}(r)[7-11]{} & [fval]{} & [gradf]{} & feasi & [iter]{} & [time]{} & [fval]{} & [gradf]{} & feasi & [iter]{} & [time]{}\
\
1e-05 & 1.627e+00 & 3.74e-05 & 1.07e-14 & 82 & 0.01 & - & - & - & - & -\
1e-10 & 1.627e+00 & 9.14e-10 & 8.84e-15 & 158 & 0.01 & - & - & - & - & -\
\
1e-05 & 3.068e+01 & 1.13e-04 & 8.97e-14 & 158 & 0.02 & - & - & - & - & -\
1e-10 & 3.068e+01 & 6.91e-09 & 1.12e-13 & 316 & 0.03 & - & - & - & - & -\
\
1e-05 & 1.333e+02 & 1.54e-04 & 3.56e-13 & 154 & 0.96 & 1.333e+02 & 2.54e-04 & 3.38e-13 & 178 & 1.05\
1e-10 & 1.333e+02 & 3.67e-08 & 5.32e-13 & 256 & 1.33 & 1.333e+02 & 2.83e-07 & 4.63e-13 & 262 & 1.50\
Minimization of the Brockett cost function {#ssec:Brokett}
------------------------------------------
In [@brockett1989least], Brockett investigated least squares matching problems on matrix Lie groups [with an objective function]{} $$\label{eq:brockett}
f(X):=\operatorname*{tr}(X{^{\top}}AXN-2BX{^{\top}}),$$ where $A,N,B\in{\mathbb{R}^{n\times n}}$ are given matrices. This function is widely known as the Brockett cost function. Recently, in [@machado2002optimization], the results of [@brockett1989least] were extended to $P$-orthogonal matrices satisfying $X{^{\top}}PX=P$ with a given orthogonal matrix $P\in{\mathbb{R}^{n\times n}}$. For $P=J$, the problem reduces to an optimization problem on the symplectic group ${\mathrm{Sp}(2n)}$. Such a problem was more recently considered in [@birtea2018optimization], where several optimization algorithms were proposed.
[In]{} this section, we study a well-defined (bounded from below) minimization problem based on the Brockett cost function . Specifically, we consider the following optimization problem $$\label{eq:brockett-symplectic}
\min\limits_{X\in{\mathrm{Sp}(2p,2n)}} \operatorname*{tr}(X{^{\top}}AX),$$ where $A\in{\mathbb{R}}^{2n\times 2n}$ is a symmetric positive definite matrix. Such a problem arises, for example, in the symplectic eigenvalue problem [@williamson1936algebraic; @bhatia2015symplectic] which will be investigated in more detail in \[ssec:sev\]. In this test, the matrix $A\in {\mathbb{R}}^{2n\times 2n}$ is randomly generated [as ]{}, where $Q\in {\mathbb{R}}^{2n\times 2n}$ is an orthogonal matrix computed from a QR factorization `Q=qr(randn(2*n,2*n)`), and $\Lambda\in {\mathbb{R}}^{2n\times 2n}$ is a diagonal matrix with diagonal elements $\Lambda_{ii}=\lambda^{1-i}$ for $i=1,2,\dots,2n$. The parameter $\lambda \ge 1$ determines the decay of eigenvalues of $A$. Th experiments are divided into two parts. At first, we solve the problem on the symplectic group, i.e., $n=p$, for matrices of different size $20\times20$, $80\times80$ and $160\times160$ and different parameters $\lambda\in\{1.01,1.04,1.07,1.1\}$. The numerical results are [presented]{} in \[tab:brockett\]. From the table, we observe that Sp-Cayley (since Sp-Cayley-I and Sp-Cayley-II reduce to the same method when $n=p$) works well on different problems. In the second part, we move to the symplectic Stiefel manifold and test [our algorithms on different problems with parameters]{} $n=1000,2000,3000$, $p=5,10,20,40,80$, and $\lambda$ [as above]{}. The corresponding results are also presented in \[tab:brockett\]. It illustrates that for relatively large problems, Sp-Cayley-I and Sp-Cayley-II still perform well, and have the comparable function values and feasibility violations.
----------------------- ------------- ----------- ---------- ---------- ---------- -------------- ----------- ---------- ---------- ----------
Sp-Cayley-I Sp-Cayley-II
(r)[2-6]{}(r)[7-11]{} [fval]{} [gradf]{} feasi [iter]{} [time]{} [fval]{} [gradf]{} feasi [iter]{} [time]{}
${\lambda}$
1.01 3.642e+01 5.73e-06 3.98e-14 12 0.01 - - - - -
1.04 2.793e+01 5.75e-06 5.19e-14 14 0.01 - - - - -
1.07 2.168e+01 1.10e-05 2.20e-14 14 0.01 - - - - -
1.10 1.737e+01 6.26e-06 4.85e-14 19 0.01 - - - - -
1.01 1.094e+02 4.25e-06 4.90e-13 14 0.01 - - - - -
1.04 4.197e+01 1.61e-04 6.56e-13 25 0.02 - - - - -
1.07 2.032e+01 4.69e-05 6.15e-13 46 0.04 - - - - -
1.10 1.222e+01 1.00e-04 5.28e-13 72 0.07 - - - - -
1.01 1.531e+02 2.98e-05 3.30e-12 19 0.07 - - - - -
1.04 3.256e+01 1.46e-04 3.95e-12 59 0.21 - - - - -
1.07 1.421e+01 2.06e-04 2.89e-12 154 0.58 - - - - -
1.10 8.442e+00 1.90e-04 2.39e-12 252 0.94 - - - - -
${p}$
5 3.150e-04 2.10e-04 1.20e-14 182 0.90 2.015e-04 1.62e-04 1.50e-14 208 1.05
10 3.631e-04 1.61e-04 3.07e-14 293 2.05 4.028e-04 1.95e-04 4.40e-14 260 1.93
20 5.902e-04 1.85e-04 9.02e-14 362 3.18 5.350e-04 1.81e-04 1.37e-13 280 2.97
40 7.764e-04 1.61e-04 3.82e-13 484 7.06 6.669e-04 1.52e-04 7.34e-13 548 9.77
80 1.037e-03 5.72e-04 1.65e-12 649 24.33 1.094e-03 1.72e-04 3.02e-12 619 21.14
10 1.241e-04 1.77e-04 2.95e-14 146 5.15 1.105e-04 1.85e-04 3.08e-14 174 6.22
20 1.748e-04 1.84e-04 2.12e-13 164 7.06 1.671e-04 2.06e-04 1.99e-13 234 10.61
40 2.216e-04 1.77e-04 3.37e-13 220 11.93 1.499e-04 1.36e-04 3.13e-13 270 17.54
80 2.948e-04 1.69e-04 1.91e-12 303 27.72 2.031e-04 1.47e-04 1.75e-12 273 24.48
5 1.794e-05 7.72e-04 9.63e-15 87 5.65 3.433e-05 1.70e-04 7.48e-15 74 4.98
10 5.133e-05 1.55e-04 5.01e-14 110 8.19 3.337e-05 2.34e-04 4.91e-14 125 8.65
20 7.352e-05 1.49e-04 1.58e-13 141 12.33 4.253e-05 1.32e-04 1.53e-13 142 12.41
40 1.226e-04 1.80e-04 4.43e-13 176 20.75 1.085e-04 1.91e-04 4.14e-13 134 16.43
80 1.661e-04 1.70e-04 3.96e-12 180 31.62 1.637e-04 1.91e-04 3.48e-12 188 35.02
----------------------- ------------- ----------- ---------- ---------- ---------- -------------- ----------- ---------- ---------- ----------
: [Numerical]{} results in the Brockett cost function minimization\[tab:brockett\]
The [symplectic]{} eigenvalue [problem]{} {#ssec:sev}
-----------------------------------------
It was shown in [@williamson1936algebraic] that for every symmetric positive definite matrix $M\in\mathbb{R}^{2n\times 2n}$, there exists $X\in{\mathrm{Sp}(2n)}$ such that $$X{^{\top}}MX = \begin{bmatrix} D & \\
& D\end{bmatrix},\label{eq:sev}$$ where $D = {\mathrm{diag}}(d_1,\dots,d_n)$ and $0<d_1\le\dots\le d_n$. These entries are called [*symplectic eigenvalues*]{}, and is referred to as the [*symplectic eigenvalue problem*]{}. Note that the symplectic eigenvalues are uniquely defined, whereas the symplectic transformation $X$ is not unique. It can be shown (see [@hiroshima2006 (12)]) that the symplectic eigenvalues of $M$ coincide with the positive (standard) eigenvalues of the matrix $G = \mathrm{i} J{^{\top}}M$, where $\mathrm{i}=\sqrt{-1}$ is the imaginary unit. In practice, it can be of interest to compute only a few extreme symplectic eigenvalues. They can be determined by exploiting the following relationship between the $p\leq n$ smallest symplectic eigenvalues of $M$ and a symplectic optimization problem $$\label{eq:symplectic-eig}
2\sum_{j=1}^{p}d_j = \min_{X\in{\mathrm{Sp}(2p,2n)}} \operatorname*{tr}(X{^{\top}}MX)$$ which was first established in [@hiroshima2006] and further investigated in [@bhatia2015symplectic]. Based on this relation, we aim to compute the smallest symplectic eigenvalue $d_1$ using with $p=1$. We test this algorithm on different data matrices from the MATLAB matrix gallery: (i) the Lehmer matrix; (ii) the Wilkinson matrix; (iii) the companion matrix of the polynomial whose coefficients are $1,\ldots,2n+1$; (iv) the central finite difference matrix. Whenever $M$ is not positive definite, which happens to the second and third case, we use $M{^{\top}}M$ instead of $M$ to generate the appropriate problem. The parameters in are default settings. For a comparison, we [also]{} compute the smallest positive eigenvalue of $G$ by using the MATLAB function `eig`. The obtained results are shown in \[Tab:smallestSymplEig\]. We observe that the symplectic eigenvalues computed by Sp-Cayley-I are comparable with that provided by `eig`.
Model matrix $n$ `eig` Sp-Cayley-I
--------------------- ----- ------------------- -------------------
Lehmer 50 7.67480301454e-03 7.67480302204e-03
Wilkinson 75 1.53471652403e+01 1.53471650305e+01
Companion 500 5.47240371331e-02 5.47244189951e-02
Centr. Finite Diff. 500 2.23005375485e-05 2.23005375834e-05
: The smallest symplectic eigenvalues\[Tab:smallestSymplEig\]
Conclusion and perspectives {#sec:conclusion}
===========================
We have developed the ingredients—retraction and Riemannian gradient—that turn general first-order Riemannian optimization methods into concrete numerical algorithms for the optimization problem on the symplectic Stiefel manifold ${\mathrm{Sp}(2p,2n)}$. The algorithms only need to be provided with functions that evaluate the objective function $f$ and the Euclidean gradient $\nabla \bar{f}$. In order to cover the case of the Cayley retraction, we have extended the convergence analysis of a Riemannian non-monotone gradient descent method to encompass the situation where the retraction is not globally defined. This extended analysis leads to the conclusion that, for the sequences generated by the proposed algorithms, every accumulation point is a stationary point. Numerical experiments demonstrate the efficiency of the proposed algorithms. All the results in this paper apply to the symplectic group as the special case $p=n$.
This paper opens several perspectives for further research. In particular, it is tempting to further exploit the leeway in the choice of the metric and the retraction. Extensions to quotients of other quadratic Lie groups are also worth considering, as well as other applications.
[^1]: ICTEAM Institute, UCLouvain, Louvain-la-Neuve, Belgium ([email protected], [email protected]).
[^2]: ICTEAM Institute, UCLouvain, Louvain-la-Neuve, Belgium; Department of Mathematics and Informatics, Thai Nguyen University of Sciences, Thai Nguyen, Vietnam ([email protected]).
[^3]: Institute of Mathematics, University of Augsburg, Augsburg, Germany ([email protected]).
[^4]: Submitted to the editors June 26, 2020.
[^5]: Available from <https://github.com/wenstone/OptM>.
[^6]: A MATLAB toolbox for optimization on manifolds (available from <https://www.manopt.org/>).
[^7]: Available from <https://sparse.tamu.edu/>.
|
{
"pile_set_name": "ArXiv"
}
|
epsf
Introduction
============
Anderson’s suggestion[@Anderson_1987_1196] that the copper-oxygen planes of the high-temperature superconductors[@Bednorz_1986_189] are strongly correlated systems has sparked renewed interest in the two-dimensional Hubbard model. Much of our understanding of the strong-coupling limit of the model, and the related $t$-$J$ model, has been obtained by numerical work (reviewed in Ref. ). Although the single-hole properties have been studied extensively, exact-diagonalization studies of small systems are hindered by large finite-size effects in the parameter region of interest, and Monte-Carlo studies of larger systems are hindered by the minus-sign problem; other methods have also been used, but there is still no general agreement on these properties, particularly for $t/J$ values in the physical region. For this reason, we have studied the single-hole properties using the hopping basis of Trugman[@Trugman_1988_1597; @Trugman_1990_892] and compared them with results obtained by other methods.
In the limit $U\gg t$, the Hubbard Hamiltonian can be approximated by the strong-coupling Hamiltonian[@Hirsch_1985_1317] $H_{sc} =
H_{t\mbox{\small -}J} + H_3$; this differs from the $t$-$J$ Hamiltonian $H_{t\mbox{\small -}J}$ (which has its own justification) by the three-site terms in $H_3$: $$\begin{aligned}
\label{t-J_Hamiltonian_eq}
H_{t\mbox{\small -}J} &=&
-t \sum_{i,\delta,\sigma}
c_{i+\delta,\sigma}^\dagger c_{i,\sigma}^{\phantom\dagger}
+ J \sum_{\langle ij\rangle}({\bf S}_i \cdot {\bf S}_j
- \textstyle {1\over 4} n_i n_j)\ , \\
\label{three_site_term_eq}
H_3 &=& -{{J}\over{4}}\sum_{i,\sigma}\sum_{\delta,\delta^\prime}
(c_{i+\delta,\sigma}^\dagger c_{i,-\sigma}^\dagger
c_{i,-\sigma}^{\phantom\dagger}c_{i+\delta^\prime,\sigma}^{\phantom\dagger}
\nonumber \\
& & \qquad\qquad\ \ -c_{i+\delta,-\sigma}^\dagger c_{i,\sigma}^\dagger
c_{i,-\sigma}^{\phantom\dagger}c_{i+\delta^\prime,\sigma}^{\phantom\dagger})\ ;\end{aligned}$$ here sites $i+\delta$ and $i+\delta^\prime$ are distinct nearest neighbors of site $i$, $\langle ij \rangle$ are nearest-neighbor pairs, and $J = 4t^2/U$. The $t$-$J$ and strong-coupling Hamiltonians operate in the reduced Hilbert space with no doubly occupied sites; this restriction is implicit in the above. Validity of the strong-coupling approximation requires $U\gg t$; the parameter range believed appropriate to the high-temperature superconductors is $2<t/J<10$, or $8<U/t<40$. We present results for the $t$-$J$ model in the region $0.1 \leq t/J \leq 10$ and for the strong-coupling model in the region $1 \leq t/J \leq 10$.
The single-hole properties in the $t$-$J$ and strong-coupling models have been studied previously, the first having received more attention. Methods include exact-diagonalization studies of small lattices, studies of infinite lattices using a restricted basis set, Monte-Carlo methods, and other methods. Properties discussed include the ground-state energy, the bandwidth, the dispersion, the band masses, the nearest-neighbor spin-spin correlations and the spectral function. As well, there is an extensive literature on the Hubbard model itself, including recent finite-temperature Monte-Carlo results[@Bulut_1994_705; @Bulut_1994_748; @Bulut_1994_7215].
This paper studies the one-hole properties on an infinite lattice, using a restricted basis set (in effect a variational method). Section \[hopping\_basis\] describes the basis, and Sections \[dispersion\]-\[correlations\] give results for the bandwidth, the dispersion, the band masses, and the nearest-neighbor spin-spin correlations respectively. For both models, the band minimum is at ${\bf k}=(\pi/2,\pi/2)$ and the maximum at ${\bf k}=(0,0)$ for the $t/J$ values investigated. The bandwidth is approximately $2J$ at large $t/J$, in agreement with loop-expansion results[@Marsiglio_1991_10882; @Martinez_1991_317; @Liu_1995_3156] and in disagreement with variational Monte-Carlo results[@Boninsegni_1992_4877]. At large $t/J$, the effects of three-site terms on the bandwidth are well described by first-order perturbation theory using the $t$-$J$ ground-state wavefunction; that is, the three-site terms appear to have little effect on the ground-state wavefunction at large $t/J$. The band mass parallel to the zone face is much larger than the perpendicular mass. The spin-spin correlations near the hole are reduced relative to the starting state, but remain antiferromagnetic.
Hopping Basis {#hopping_basis}
=============
We study a system of $N-1$ electrons on a square lattice of $N$ sites with periodic boundary conditions; the Hilbert space is restricted to the $S_z=1/2$ sector with no doubly occupied sites. We use the same basis for both models, namely the hopping basis[@Trugman_1988_1597; @Trugman_1990_892] which has been used previously. This method allows the study of infinite systems (eliminating finite-size effects), but only certain properties, like the bandwidth and the band masses, can be studied.
In zeroth order, the basis (denoted $B_0$) consists of a single state (denoted $|cN\rangle$), the Néel state with a missing down-spin electron. Higher-order bases are generated by repeatedly applying the $t$ term in the Hamiltonian (which hops the hole to a nearest-neighbor site). The first-order basis $B_1$ contains the $|cN\rangle$ state plus the four states generated by hopping the hole. The $n$-th order basis $B_n$ consists of the states in the basis $B_{n-1}$ plus those generated by applying the hopping operator to the states in the difference $B_{n-1}-B_{n-2}$. The basis size (values are given in Table \[basissize\_table\]) grows exponentially with order. The hopping basis, which emphasizes states differing from the $|cN\rangle$ state only near the hole, cannot give a good value of the ground-state energy (because, for example, it does not generate spin interchanges far from the hole in reasonable order); the expectation is that it describes well properties like the dispersion and the nearest-neighbor spin-spin correlations near the hole.
We have used the bases from $B_6$ to $B_{13}$ for most quantities, going to such large bases because some properties were still changing significantly; even with basis $B_{13}$ ($\sim2\times10^6$ states), however, some properties are incompletely converged. Various extrapolation schemes were considered but judged unreliable, and so we usually present values for the three largest bases to provide an estimate of the error due to the truncation of the basis. The system size ($16\times16$; the lattice constant $a$ is unity) is effectively infinite since there are no paths which wrap around the system even in 13-th order. Since the hole moves in an antiferromagnetic background, the Brillouin zone is reduced to the square formed by the points $(\pm\pi,0)$ and $(0,\pm\pi)$. The symmetries of the lattice reduce the independent part of the Brillouin zone to the triangle with corners at $(0,0)$, $(\pi,0)$, and $(\pi/2,\pi/2)$, denoted $\bf\Gamma$, $\bf M$, and $\bf S$ respectively. Each state $|n\rangle$ in the basis is a Bloch state, an eigenstate of the translation operator corresponding to an allowed value of the momentum. For each basis, and each value of the momentum [**k**]{}, the lowest eigenvalue and eigenvector were found using a conjugate-gradient method to minimize the function $\langle\Psi|H|\Psi\rangle/\langle\Psi|\Psi\rangle$ with respect to the expansion coefficients in $|\Psi\rangle=a_n|n\rangle$; this method is reported to converge more rapidly than others commonly used[@Nightingale_1993_7696], but gives the eigenvector to only single precision. Where necessary, the eigenvector was improved by a Lanczos method.
The dispersion (in the energy as a function of [**k**]{}) results from several processes. The Trugman paths[@Trugman_1988_1597; @Trugman_1990_892] translate the hole to a next-nearest-neighbor site or a third-nearest-neighbor site on the same sub-lattice, restoring the original configuration. In the lowest-order path, the hole hops six times around the smallest square to a next-nearest-neighbor site; as a result, matrix elements like $\langle B_2 | c_{i+\delta,\sigma}^\dagger
c_{i,\sigma}^{\phantom\dagger} | B_3\rangle$ are momentum-dependent. Momentum dependence can also arise from the $J$ term in $H$; for example, the basis $B_2$ contains states with the hole translated by $2a$ and a pair of flipped spins, and so matrix elements like $\langle
B_0|{\bf S}_i\cdot{\bf S}_j|B_2\rangle$ depend on [**k**]{}. The results show odd-even effects in the order of the basis; as the basis size increases, Trugman paths of higher order, and also states differing from the starting state by nearest-neighbor spin interchanges, are generated.
Related bases were also studied, in an effort to determine which states are important for the hole properties. The hopping basis can be described symbolically as $B_n=\sum_{k=0}^n h^k |cN\rangle$ where $h$ is the hole hopping operator. We define also operators ${\cal
S}_8$, ${\cal S}_{12}$ and ${\cal S}_{20}$; the first scrambles the 8 spins at distances $a$ and $\sqrt2 a$ from the hole (giving 70 states when operating on the $|cN\rangle$ state), the second these spins plus the four at distance $2a$, and the third the 20 spins inside a $5\times 5$ square minus the four corner sites. If hole properties like the bandwidth are determined primarily by configurations which differ from the $|cN\rangle$ state only near the hole, then the bases $\sum_k h^k{\cal S}_m|cN\rangle$, or (likely better) ${\cal
S}_m\sum_kh^k|cN\rangle$, should converge more rapidly than the hopping basis; we find the opposite: when the bandwidth is plotted against the inverse of the log of the basis size, these modified bases behave like the hopping basis, except that properties are shifted toward larger basis sizes. We considered also two other bases, both of which reduce the importance of string states (in which the hole wanders without looping): (i) the basis $\sum_kM_m h^k|cN\rangle$ where the operator $M_m$ removes states in which the Manhattan displacement ($|x|+|y|$) of the hole relative to its initial position is greater than $ma$, and (ii) the basis $\sum_{k=0}^\infty (I_n h)^k
|cN\rangle$, where the operator $I_n$ removes states with more than $n$ “bad bonds” (that is, it filters states according to their Ising energy relative to the $|cN\rangle$ state; the limit $\infty$ means that the hop-filter combination is applied until the basis no longer grows, for given $n$). Neither the Manhattan nor the Ising filters improved the convergence. We conclude from these numerical experiments that the single-hole properties are determined not so much by the spin configurations near the hole as by loop and string paths. It appears that the hopping basis, whether in its original form or in the modified forms we have investigated, is capable of only limited accuracy even if carried to very high order.
Bandwidth
=========
Because the lattice is effectively infinite, the lowest energy can be found for any $\bf k$. For both models, we found $E({\bf k})$ at 81 independent $\bf k$ values of the form $(2\pi n/L, 2\pi m/L)$ with $n$ and $m$ integers and $L = 32$, for $t/J$ values in the range $0.1\leq
t/J\leq10$ for the $t$-$J$ model and in the range $1\leq t/J\leq10$ for the strong-coupling model (for which the lower values of $t/J$ are of little interest).
For the $t$-$J$ model, the energy is a minimum at ${\bf k}={\bf S}$ (and a maximum at ${\bf\Gamma}$) for $0.1\leq t/J\leq10$, for all bases used ($B_6$ to $B_{13}$), in agreement with all previous work.
For the strong-coupling model, the energy is also a minimum at ${\bf
k}={\bf S}$ (and a maximum at ${\bf\Gamma}$) for all $t/J$ in the range $1.0\leq t/J\leq10$, but only for the largest bases at small $t/J$; this result disagrees with predictions (based on fits to exact-diagonalization results for small systems[@Fehske_1991_8473]) that the minimum is at ${\bf M}$ for $t/J \leq 5$. For the smaller bases, particularly for the smaller values of $t/J$, the minimum can be at ${\bf M}$ or elsewhere along the zone face; for example, the minimum is at ${\bf S}$ only in 11-th order and higher for $t/J=1$.
Figure \[bw\_ours\_figure\] plots the bandwidth $W=E({\bf\Gamma})-E({\bf S})$ for both models as found using the bases $B_{11}$, $B_{12}$, and $B_{13}$. The convergence is good for the $t$-$J$ model at all $t/J$ investigated; it is moderately good for the strong-coupling model at larger $t/J$, but worsens at smaller $t/J$. The $t$-$J$ bandwidth is approximately $t$ for $t/J<2$ and $2J$ for $t/J>2$, but decreases weakly at large $t/J$. The strong-coupling bandwidth is also about $2J$ (though about 20% larger) and also decreases as $t/J$ increases. The hopping-basis results are incompletely converged, however; the bandwidth is still increasing with basis size, and the trend is greater at larger $t/J$. It is possible then that the slight decrease which we find is due to the finite size of the hopping basis.
Figure \[bw\_comp\_figure\] compares our values for the $t$-$J$ bandwidth with those obtained by other methods; major differences occur in the physical region $t/J>2$. The hopping-basis results agree best with loop-expansion results[@Marsiglio_1991_10882; @Martinez_1991_317; @Liu_1995_3156] and poorly with variational Monte-Carlo results[@Boninsegni_1992_4877] (for unknown reasons); the $4\times4$ exact-diagonalization results[@Elrick_1993_6004] at large $t/J$ are unreliable due to finite-size effects. Our results at large $t/J$ are qualitatively consistent with the mean-field result[@Schrieffer_1989_11663] $W\approx4J$ for strong coupling.
From Figure 1, the normalized bandwidth difference $(W_{sc}-W_{t\mbox{\small -}J})/J$ is almost independent of $t/J$ for $t/J\agt4$. Since $(H_{sc} - H_{t\mbox{\small -}J})/J=H_3/J$ has no explicit dependence on $t$ or $J$, this suggests treating the three-site terms as a perturbation to the $t$-$J$ model. The error in the first-order result for the bandwidth difference $\Delta W_1= \Delta E_1({\bf\Gamma}) - \Delta E_1({\bf M})$, where $\Delta E_1({\bf k}) = \langle\Psi_{t-J}|H_3|\Psi_{t-J}\rangle({\bf k})$, is less than 2% at $t/J=10$ and $t/J = 8$, but is much larger at smaller $t/J$ (52% at $t/J=$ 4). Of course the estimate for the strong-coupling bandwidth itself is much better (errors are 0.3%, 0.3%, and 11% at $t/J=$ 10, 8, and 4). It appears then that the three-site terms can be treated in first order for $t/J\agt6$.
Further investigation revealed that the first-order estimates of the energy at $\bf S$ are excellent; $(\langle H_{sc}
\rangle_{t\mbox{\small -}J} - E_{sc})/W_{sc}$ is $0.1\%$, $0.09\%$, $0.06\%$ and $0.04\%$ at $t/J=10$, 8, 4, and 1 respectively; the corresponding values at $\bf\Gamma$ are $0.4\%$, $0.4\%$, $11\%$ and $41\%$. For unknown reasons, at intermediate $t/J$ values the three-site terms appear to affect the $\bf\Gamma$ ground state strongly and the ${\bf M}$ ground state very weakly.
Dispersion
==========
The Fourier coefficients $a_{lm}$ defined by $$\label{dispersion_equation}
E({\bf k}) = \sum_{l,m=0}^{L/2} a_{lm} \cos{l k_x} \cos{m k_y}$$ are easily obtained by inversion from the energy as a function of ${\bf k}$. The symmetries of the lattice give $a_{lm} = a_{ml}$, and $a_{lm} = 0$ for $l+m$ odd. The independent coefficients are then the 81 $a_{lm}$ with $0\leq l\leq16$, $0\leq m\leq l$, and $l+m$ even. The coefficient $a_{00}$ depends strongly on the order of the basis, as more states important for the ground-state energy are generated; it affects none of our results since we look only at quantities (like the dispersion) which depend on energy differences.
Of the other coefficients, $a_{11}$ and $a_{20}$ (both positive) are the largest, with the ratio $a_{20}/a_{11}$ less than about 0.6 for both models for the range of $t/J$ values studied. The remaining coefficients are less than about $0.1a_{11}$ in magnitude for both models at the $t/J$ values studied. Figures \[a11\_figure\] and \[a20\_figure\] plot the two leading coefficients as functions of $t/J$ for the two models. The convergence is of course qualitatively the same as for the bandwidth, good for the $t$-$J$ model at all $t/J$ and for the strong-coupling model for $t/J\agt4$, but increasingly poor for the latter with decreasing $t/J$.
At large $t/J$, the values $a_{20}/J$ are almost independent of $t/J$, whereas the coefficients $a_{11}/J$ decrease with increasing $t/J$. The strong-coupling coefficients are larger than the $t$-$J$ coefficients, reflecting the enhanced mobility due to the three-site terms. Also, at larger $t/J$, the difference $(a_{20}/J)_{sc}-(a_{20}/J)_{t\mbox{\small -}J}$ for the two models is almost independent of $t/J$, as is the difference in the values of $a_{11}/J$, for the reason discussed in Section \[bandwidth\]. Figures \[a11\_figure\] and \[a20\_figure\] also plot other results[@Marsiglio_1991_10882; @Martinez_1991_317] for the $t$-$J$ Fourier coefficients; the agreement is as expected from Section \[bandwidth\]. Recent Monte-Carlo results[@Dagotto_1994_728; @Giamarchi_1993_2775], available only at $t/J = 2.5$, are about 25% higher than ours.
Band Masses {#bandmass}
===========
The band masses at the band minimum, which is at $\bf S$ for both models in the region $1\leq t/J\leq10$, are defined in terms of the second derivatives of $E({\bf k})$ with respect to ${\bf k}$: $$m_{\mu\nu} = \hbar^2 \left(\frac{\partial^2 E({\bf k})}
{\partial k_\mu\partial k_\nu}\right)^{-1}.$$ The masses were obtained by calculating $E({\bf k})$ at additional points near $\bf S$ and using finite-difference approximations for the derivatives. Figures \[mper\_figure\] and \[mpar\_figure\] give results for the masses perpendicular and parallel to the zone face respectively, in units of the bare mass $m_0=\hbar^2/2t$. The parallel mass is much larger than the perpendicular mass, as found previously.
The perpendicular mass is well converged for both models. For the $t$-$J$ model, $m_\bot/m_0$ is almost linear in $t/J$ at large $t/J$, but flattens out at small $t/J$. For the strong-coupling model, $m_\bot/m_0$ is almost proportional to $t/J$; the smaller effective mass reflects again the increased hole mobility relative to that in the $t$-$J$ model.
The parallel mass is much more poorly converged, especially at smaller $t/J$; even at $t/J=10$ (the most favorable value), the masses change by over $5\%$ between the bases $B_{12}$ and $B_{13}$. The poor convergence results because the energies are nearly independent of ${\bf k}$ (the mass is large). For large $t/J$, though, it appears that $m_\parallel/m_0$ increases only weakly with $t/J$ for both models and that the two models have the same parallel mass.
Figures \[mper\_figure\] and \[mpar\_figure\] also give the results from Ref. , derived from their dispersion results (Table II of Ref. ) using the free mass $m_0=\hbar^2/2t$, rather than the effective masses of their Table III. The difference is due in part to a genuinely different dispersion, but part of it arises because they used only two components in the Fourier expansion (the parallel mass, being large, is particularly sensitive to small changes in the energy).
Spin-spin Correlations {#correlations}
======================
Figures \[corr\_tj\_figure\] and \[corr\_sc\_figure\] show the nearest-neighbor spin-spin correlation $\langle {\bf S}_i \cdot {\bf
S}_j \rangle$ for pairs of sites $i$ and $j$ near the hole, for the $t$-$J$ model and strong-coupling model respectively. The momentum is ${\bf k} = {\bf S}$ (the band minimum), $t/J = 2.5$, and the basis is $B_{13}$. In the units of $\hbar^2 = 1$ used, the spin-spin correlation is -0.75 for a singlet pair of spins, $0.25$ for a triplet pair, and -0.25 for a Néel pair. The correlations are antiferromagnetic, and moderately less than in the starting state. The “cigar” polaron in Figures 7 and 8 is well known from other studies.
This research was supported by the Natural Sciences and Engineering Research Council of Canada. Computations were done on a Kendall Square Research 1-32 computer provided by University of Toronto Instructional and Research Computing; we are grateful to UTIRC staff for aid in making efficient use of the parallel architecture.
e-mail addresses: [email protected] and [email protected]
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---------------- ----------------- ---------------- -----------------
[*Order*]{} [*Number*]{} [*Order*]{} [*Number*]{}
[*of Basis*]{} [*of states*]{} [*of Basis*]{} [*of states*]{}
0 1 8 9786
1 5 9 27990
2 17 10 80196
3 49 11 228196
4 141 12 650022
5 405 13 1842326
6 1177 14 5225938
7 3389
---------------- ----------------- ---------------- -----------------
: Number of states in the hopping basis versus order of the basis. \[basissize\_table\]
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a large-scale dataset for the task of rewriting an ill-formed natural language question to a well-formed one. Our multi-domain question rewriting ([<span style="font-variant:small-caps;">MQR</span>]{}) dataset is constructed from human contributed Stack Exchange question edit histories. The dataset contains 427,719 question pairs which come from 303 domains. We provide human annotations for a subset of the dataset as a quality estimate. When moving from ill-formed to well-formed questions, the question quality improves by an average of 45 points across three aspects. We train sequence-to-sequence neural models on the constructed dataset and obtain an improvement of 13.2% in BLEU-4 over baseline methods built from other data resources. We release the MQR dataset to encourage research on the problem of question rewriting.[^1]'
author:
- |
Zewei Chu,[^2] **Mingda Chen**,$^*$[^3] **Jing Chen**,$^\dagger$ **Miaosen Wang**,$^\dagger$\
**Kevin Gimpel**,**Manaal Faruqui**,**Xiance Si**\
The University of Chicago, 5730 S Ellis Ave, Chicago, IL 60637, USA\
Toyota Technological Institute at Chicago, 6045 S Kenwood Ave, Chicago, IL 60637, USA\
Google Assistant, 1600 Amphitheatre Pkwy, Mountain View, CA 94043, USA\
[email protected], {mchen, kgimpel}@ttic.edu\
{chenjin, miaosen, mfaruqui, sxc}@google.com
bibliography:
- 'aaai.bib'
title: |
How to Ask Better Questions?\
A Large-Scale Multi-Domain Dataset for Rewriting Ill-Formed Questions
---
Introduction {#sec:intro}
============
Ill-formed Well-formed Category
------------------------------------------------- --------------------------------------------------------------- -------------
Spaghetti carbonara, mixing How to mix a spaghetti carbonara? cooking
Ethical Investing... where to begin? How to begin ethical investing? money
charging canon sx 700 battery through powerbank Can I charge a Canon SX 700 battery using a mobile powerbank? photo
H1B Visa consulate interview timeline What is the timeline for an H1B visa consulate interview? expatriates
Hanging weight from drywall ceiling How much weight can I hang from a drywall ceiling? diy
Understanding text and voice questions from users is a difficult task as it involves dealing with “word salad” and ill-formed text. Ill-formed questions may arise from imperfect speech recognition systems, search engines, dialogue histories, inputs from low bandwidth devices such as mobile phones, or second language learners, among other sources. However, most downstream applications involving questions, such as question answering and semantic parsing, are trained on well-formed natural language. In this work, we focus on rewriting textual ill-formed questions, which could improve the performance of such downstream applications.
@faruqui2018identifying introduced the task of identifying well-formed natural language questions. In this paper, we take a step further to investigate methods to rewrite ill-formed questions into well-formed ones without changing their semantics. We create a multi-domain question rewriting dataset ([<span style="font-variant:small-caps;">MQR</span>]{}) from human contributed Stack Exchange question edit histories.[^4] This dataset provides pairs of questions: the original ill-formed question and a well-formed question rewritten by the author or community contributors. The dataset contains 427,719 question pairs which come from 303 domains. The [<span style="font-variant:small-caps;">MQR</span>]{}dataset is further split into TRAIN and DEV/TEST, where question pairs in DEV/TEST have less $n$-gram overlap but better semantic preservation after rewriting. Table \[tab:example\_mqr\] shows some example question pairs from the [<span style="font-variant:small-caps;">MQR</span>]{}DEV split. Our dataset enables us to train models directly for the task of question rewriting. We train neural generation models on our dataset, including Long-Short Term Memory networks (LSTM; [@Hochreiter:1997:LSM:1246443.1246450]) with attention [@luong-etal-2015-effective] and transformers [@vaswani2017attention]. We show that these models consistently improve the well-formedness of questions although sometimes at the expense of semantic drift. We compare to approaches that do not use our training dataset, including general-purpose sentence paraphrasing, grammatical error correction (GEC) systems, and round trip neural machine translation. Methods trained on our dataset greatly outperform those developed from other resources. Augmenting our training set with additional question pairs such as Quora or Paralex question pairs [@fader2013paraphrase] has mixed impact on this task. Our findings from the benchmarked methods suggest potential research directions to improve question quality.
To summarize our contributions:
- We propose the task of question rewriting: converting textual ill-formed questions to well-formed ones while preserving their semantics.
- We construct a large-scale multi-domain question rewriting dataset [<span style="font-variant:small-caps;">MQR</span>]{}from human generated Stack Exchange question edit histories. The development and test sets are of high quality according to human annotation. The training set is of large-scale. We release the [<span style="font-variant:small-caps;">MQR</span>]{}dataset to encourage research on the question rewriting task.
- We benchmark a variety of neural models trained on the [<span style="font-variant:small-caps;">MQR</span>]{}dataset, neural models trained with other question rewriting datasets, and other paraphrasing techniques. We find that models trained on the [<span style="font-variant:small-caps;">MQR</span>]{}and Quora datasets combined followed by grammatical error correction perform the best in the [<span style="font-variant:small-caps;">MQR</span>]{}question rewriting task.
Related Work
============
Query and Question Rewriting
----------------------------
Methods have been developed to reformulate or expand search queries [@jones2006generating]. Sometimes query rewriting is performed for sponsored search [@zhang2007query; @zhangcomparing]. This work differs from our goal as we rewrite ill-formed questions to be well-formed.
Some work rewrites queries by searching through a database of query logs to find a semantically similar query to replace the original query. @de2010learning compute query similarities for query ranking based on user click information. @dong-etal-2017-learning learn paraphrases of questions to improve question answering systems. @kumar2018translating translate queries from search engines into natural language questions. They used Bing’s search logs and their corresponding clicked question page as a query-to-question dataset. We work on question rewriting without any database of question logs.
Actively rewriting questions with reinforcement learning has been shown to improve QA systems [@buck2017ask]. This work proposes to rewrite questions to fulfill more general quality criteria.
Paraphrase Generation
---------------------
A variety of paraphrase generation techniques have been proposed and studied [@barzilay2003learning; @bannard2005paraphrasing; @androutsopoulos2010survey; @madnani-dorr-2010-generating; @malakasiotis2011generate; @li2019decomposable]. Recently, @gupta2018deep use a variational autoencoder to generate paraphrases from sentences and @li2018paraphrase use deep reinforcement learning to generate paraphrases. Several have generated paraphrases by separately modeling syntax and semantics [@iyyer2018adversarial; @chen-etal-2019-controllable].
Paraphrase generation has been used in several applications. @cho-etal-2019-paraphrase use paraphrase generation as a data augmentation technique for natural language understanding. @iyyer2018adversarial and @ribeiro2018semantically generate adversarial paraphrases with surface form variations to measure and improve model robustness. @wieting-gimpel-2018-paranmt generate paraphrases using machine translation on parallel text and use the resulting sentential paraphrase pairs to learn sentence embeddings for semantic textual similarity.
Our work focuses on question rewriting to improve question qualities, which is different from general sentence paraphrasing.
Text Normalization
------------------
Text normalization [@sproat2001normalization] is the task of converting non-canonical language to “standard” writing. Non-canonical language frequently appears in informal domains such as social media postings or other conversational text, user-generated content, such as search queries or product reviews, speech transcriptions, and low-bandwidth input settings such as those found with mobile devices. Text normalization is difficult to define precisely and therefore difficult to provide gold standard annotations and evaluate systems for [@eisenstein-2013-bad]. In our setting, rewriting questions is defined implicitly through the choices made by the Stack Exchange community with the goals of helpfulness, clarity, and utility.
Task Definition: Question Rewriting {#sec:questionrewrite}
===================================
Given a question $q_i$, potentially ill-formed, the question rewriting task is to convert it to a well-formed natural language question $q_w$ while preserving its semantics and intention. Following @faruqui2018identifying, we define a well-formed question as one satisfying the following constraints:
- The question is grammatically correct. Common grammatical errors include misuse of third person singular or verb tense.
- The question does not contain spelling errors. Spelling errors refer specifically to typos and other misspellings, but not to grammatical errors such as third person singular or tense misuse in verbs.
- The question is explicit. A well-formed question must be explicit and end with a question mark. A command or search query-like fragment is not well-formed.
Question Spelling Grammar Explicit Remark
---------------------------------------------------------------- ---------- --------- ---------- ---------------------
How to remove water-based paint? 1 1 1
how can I make quark the music player to work in unity, natty? 0 0 1
What is the value to checking in broken unit tests? 1 0 1 to $\rightarrow$ of
No room for RO drain saddle? 1 1 0
Question 1 Question 2 Equivalent
---------------------------------------------------- --------------------------------------------------------------- ------------
How to add a lightbox? How to add a lightbox to class mix? 0
how to get md5sum of a string directly in terminal How to get the MD5 hash of a string directly in the terminal? 1
-------------- -------------- -------------- -------------- -------------- -------------- --------------
Spelling Grammar Explicit Spelling Grammar Explicit
139/310=0.45 166/310=0.54 175/310=0.56 282/290=0.97 271/290=0.93 286/290=0.99 183/200=0.92
-------------- -------------- -------------- -------------- -------------- -------------- --------------
TRAIN DEVTEST
-------------------------------- ------- ---------
\# Categories 303 166
Mean \# instances per category 320.4 25.5
Std \# instances per category 754.8 47.1
Min \# instances per category 1 1
Max \# instances per category 6237 295
: \[tab:dataset\_categories\]Statistics of question pairs (“instances”) from Stack Exchange categories in the [<span style="font-variant:small-caps;">MQR</span>]{}dataset.
[<span style="font-variant:small-caps;">MQR</span>]{}Dataset Construction and Analysis {#sec:stack_exchange}
======================================================================================
We construct our Multi-Domain Question Rewriting ([<span style="font-variant:small-caps;">MQR</span>]{}) dataset from human contributed Stack Exchange question edit histories. Stack Exchange is a question answering platform where users post and answer questions as a community. Stack Exchange has its own standard of good questions,[^5] and their standard aligns well with our definition of well-formed questions. If questions on Stack Exchange do not meet their quality standards, members of the community often volunteer to edit the questions. Such edits typically correct spelling and grammatical errors while making the question more explicit and easier to understand.
We use 303 sub areas from Stack Exchange data dumps.[^6] The full list of area names is in the appendix. We do not include Stack Overflow because it is too specific to programming related questions. We also exclude all questions under the following language sub areas: *Chinese*, *German*, *Spanish*, *Russian*, *Japanese*, *Korean*, *Latin*, *Ukrainian*. This ensures that the questions in [<span style="font-variant:small-caps;">MQR</span>]{}are mostly English sentences. Having questions from 303 Stack Exchange sites makes the [<span style="font-variant:small-caps;">MQR</span>]{}dataset cover a broad range of domains.
We use “PostHistory.xml” and “Posts.xml” tables of each Stack Exchange site data dump. If a question appears in both “PostHistory.xml” and “Posts.xml”, it means the question was modified. We treat the most up-to-date Stack Exchange questions as a well formed-question and treat its version from “PostHistory.xml” as ill-formed. “PostHistory.xml” only keeps one edit for each question, so the [<span style="font-variant:small-caps;">MQR</span>]{}dataset does not contain duplicated questions.
The questions in the Stack Exchange raw data dumps do not always fulfill our data quality requirements. For example, some questions after rewriting are still not explicit. Sometimes rewriting introduces or deletes new information and cannot be done correctly without more context or the question description. We thus perform the following steps to filter the question pairs:
1. All well-formed questions in the pairs must start with “how”, “why”, “when”, “what”, “which”, “who”, “whose”, “do”, “where”, “does”, “is”, “are”, “must”, “may”, “need”, “did”, “was”, “were”, “can”, “has”, “have”, “are”. This step is performed to make sure the questions are explicit questions but not statements or commands.
2. To ensure there are no sentences written in non-English languages, we keep questions that contain 80% or more of valid English characters, including punctuation.[^7]
This yields the [<span style="font-variant:small-caps;">MQR</span>]{}dataset. We use the following heuristic criteria to split [<span style="font-variant:small-caps;">MQR</span>]{}into TRAIN, DEV, and TEST sets:
1. The BLEU scores between well-formed and ill-formed questions (excluding punctuation) are lower than 0.3 in DEV and TEST to ensure large variations after rewriting.
2. The lists of verbs and nouns between well-formed and ill-formed questions have a Jaccard similarity greater than 0.8 in DEV and TEST. We split DEV and TEST randomly and equally. This yields 2,112 instances in DEV and 2,113 instances in TEST.
3. The rest of the question edit pairs (423,495 instances) are placed in the TRAIN set.
Examples are shown in Table \[tab:example\_mqr\]. We release our TRAIN/DEV/TEST splits of the [<span style="font-variant:small-caps;">MQR</span>]{}dataset to encourage research in question rewriting.
Dataset Quality {#subsec:dataset_quality}
---------------
To understand the quality of the question rewriting examples in the [<span style="font-variant:small-caps;">MQR</span>]{}dataset, we ask human annotators to judge the quality of the questions in the DEV and TEST splits (abbreviated as DEVTEST onward). Specifically, we take both ill-formed and well-formed questions in DEVTEST and ask human annotators to annotate the following three aspects regarding each question [@faruqui2018identifying]:
1. Is the question grammatically correct?
2. Is the spelling correct? Misuse of third person singular or past tense in verbs are considered grammatical errors instead of spelling errors. Missing question mark in the end of a question is also considered as spelling errors.
3. Is the question an explicit question, rather than a search query, a command, or a statement?
The annotators were asked to annotate each aspect with a binary (0/1) answer. Examples of questions provided to the annotators are in Table \[tab:annotation\_quality\]. We consider all “How to” questions (“How to unlock GT90 in Gran Turismo 2?”) as grammatical. Although it is not a complete sentence, this kind of question is quite common in our dataset and therefore we choose to treat it as grammatically correct.
The ill-formed and well-formed questions are shuffled so the annotators do not have any prior knowledge or bias regarding these questions during annotation. We randomly sample 300 questions from the shuffled DEVTEST questions, among which 145 examples are well-formed and 155 are ill-formed. Two annotators produce a judgment for each of the three aspects for all 300 questions.
The above annotation task considers a single question at a time. We also consider an annotation task related to the quality of a question *pair*, specifically whether the two questions in the pair are semantically equivalent. If rewriting introduces additional information, then the question rewriting task may require additional context to be performed, even for a human writer. This may happen when a user changes the question content or the question title is modified based on the additional description about the question. In the [<span style="font-variant:small-caps;">MQR</span>]{}dataset, we focus on question rewriting tasks that can be performed without extra information.
We randomly sample 100 question pairs from DEVTEST for annotation of semantic equivalence. Two annotators produced binary judgments for all 100 pairs. Example pairs are shown in Table \[tab:annotation\_semantics\].
Table \[tab:dataset\_quality\] summarizes the human annotations of the quality of the DEVTEST portion of the [<span style="font-variant:small-caps;">MQR</span>]{}dataset. We summed up the binary scores from two annotators. There are clear differences between ill-formed and well-formed questions. Ill-formed question are indeed ill-formed and well-formed questions are generally of high quality. The average score over three aspects improves by 45 points from ill-formed to well-formed questions. Over 90% of the question pairs possess semantic equivalence, i.e., they do not introduce or delete information. Therefore, the vast majority of rewrites can be performed without extra information.
The Cohen’s Kappa inter-rater reliability scores [@mchugh2012interrater] are 0.83, 0.77, and 0.89 respectively for the question quality annotations, and 0.86 for question semantic equivalence. These values show good inter-rater agreement on the annotations of the qualities and semantic equivalences of the [<span style="font-variant:small-caps;">MQR</span>]{}question pairs.
Dataset Domains
---------------
As the [<span style="font-variant:small-caps;">MQR</span>]{}dataset is constructed from 303 sub areas of the Stack Exchange networks, it covers a wide range of question domains. Table \[tab:dataset\_categories\] summarizes the number of categories in the TRAIN and DEVTEST portions of the [<span style="font-variant:small-caps;">MQR</span>]{}dataset, as well as the mean, standard deviation, minimum, and maximum number of instances per categories.
The number of questions from each sub area is not evenly distributed due to the fact that some sub areas are more popular and have more questions than the others, but the DEV/TEST splits still cover a reasonably large range of domains.
The most common categories in DEV and TEST are “diy”(295), “askubuntu”(288), “math”(250), “gaming”(189), and “physics”(140). The least common categories are mostly “Meta Stack Exchange” websites where people ask questions regarding the policies of posting questions on Stack Exchange sites. The most common categories in TRAIN are “askubuntu”(6237), “math”(5933), “gaming”(3938), “diy”(2791), and “2604”(scifi).
-- --------------------------------------------------------------------------- -- -- -- --
**BLEU-4 & **ROUGE-1 & **ROUGE-2 & **ROUGE-L & **METEOR\
Ill-formed & 5.9 & 50.9 & 19.4 & 45.5 & 33.4\
\
LSTM seq-to-seq with attention & 19.2 & 55.8 & 28.3 & 52.8 & 32.7\
Transformer & **22.1 & **59.8 & **32.2 & **56.6 & **36.4\
\
Grammatical error correction & 13.1 & 52.4 & 24.4 & 47.5 & 34.4\
Round trip NMT (Pivot: De) & 9.9 & 41.6 & 16.8 & 38.2 & 28.4\
Round trip NMT (Pivot: Fr) & 9.3 & 40.4 & 15.7 & 36.9 & 27.5\
Paraphrase generator trained on ParaNMT & 4.9 & 24.8 & 7.5 & 21.8 & 18.8\
********************
-- --------------------------------------------------------------------------- -- -- -- --
--------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- -- --
**Training Dataset & **BLEU-4 & **ROUGE-1 & **ROUGE-2 & **ROUGE-L & **METEOR\
[<span style="font-variant:small-caps;">MQR</span>]{}TRAIN & 22.1 & 59.8 & 32.2 & 56.6 & 36.4\
[<span style="font-variant:small-caps;">MQR</span>]{}TRAIN + $\langle$well-formed, well-formed$\rangle$ pairs & 21.1 & **61.4 & 32.1 & **58.0 & **36.8\
[<span style="font-variant:small-caps;">MQR</span>]{}TRAIN + Quora & **23.6 & 60.5 & **33.4 & 57.5 & **36.8\
[<span style="font-variant:small-caps;">MQR</span>]{}TRAIN + Paralex & 21.7 & 58.3 & 31.3 & 55.3 & 35.7\
[<span style="font-variant:small-caps;">MQR</span>]{}TRAIN + Quora + Paralex & 23.1 & 60.3 & 33.0 & 57.2 & 36.7\
************************
--------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- -- --
-- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
**BLEU-4 & **ROUGE-1 & **ROUGE-2 & **ROUGE-L & **METEOR\
Transformer ([<span style="font-variant:small-caps;">MQR</span>]{}+ Quora) & 23.6 & 60.5 & 33.4 & 57.5 & 36.8\
GEC & 13.1 & 52.4 & 24.4 & 47.5 & 34.4\
GEC $\rightarrow$ Transformer ([<span style="font-variant:small-caps;">MQR</span>]{}+ Quora) & 24.8 & 60.2 & 33.9 & 57.3 & 36.8\
Transformer ([<span style="font-variant:small-caps;">MQR</span>]{}+ Quora) $\rightarrow$ GEC & **26.3 & **61.0 & **35.4 & **58.1 & **37.3\
Transformer ([<span style="font-variant:small-caps;">MQR</span>]{}) $\rightarrow$ Transformer ([<span style="font-variant:small-caps;">MQR</span>]{}) & 20.4 & 55.8 & 29.2 & 52.5 & 35.1\
********************
-- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
Spelling Grammar Explicit Semantics wrt. ill-formed Semantics wrt. well-formed
---------------------------------------------------------------------------------------------- ---------- --------- ---------- --------------------------- ----------------------------
Ill formed 0.31 0.41 0.61 - -
GEC 0.39 0.56 0.59 1.00 0.84
Transformer ([<span style="font-variant:small-caps;">MQR</span>]{}+ Quora) 0.96 0.75 1.00 0.67 0.56
Transformer ([<span style="font-variant:small-caps;">MQR</span>]{}+ Quora) $\rightarrow$ GEC 0.96 0.91 1.00 0.71 0.63
model question S G E Semantics
----------------------------------------------------------------------------------------- ----------------------------------------------------------- --- ----- --- -----------
Ill-formed best way of widening butcherblock countertop? 0 0 0 1
Well-formed What’s the best way to widen a butcherblock countertop? 1 1 1 -
Trans. How can I widen a butcherblock countertop? 1 1 1 0
LSTM What is the best way of widening butcherblock countertop? 1 0 1 1
GEC best way of widening butcherblock countertop? 0 1 0 1
Round trip NMT (Pivot: De) best way to extend the racquet counter pole? 0 1 0 0
Round trip NMT (Pivot: Fr) What is the best way to expand the butcherblock? 1 1 1 0？
ParaNMT the best way to expand the countertop of the butcher ? 1 1？ 0 1？
Trans. ([<span style="font-variant:small-caps;">MQR</span>]{}+ Quora) What is the best way of widebutcherblock countertop? 0 0 0 0
Trans. ([<span style="font-variant:small-caps;">MQR</span>]{}+ Quora) $\rightarrow$ GEC What is the best way of widebitcherblock countertop? 0 0 0 0
Ill-formed drawing polygons from python console 0 1 0 1
Well-formed How to draw polygons from the python console? 1 1 1 -
Trans. How to draw polygons from a python console? 1 1 1 1
LSTM How can I draw polygons from a Python console? 1 1 1 1
GEC drawing polygons from python console 0 0 0 1
Round trip NMT (Pivot: De) Drawing polygons from the Python console 0 1 0 1
Round trip NMT (Pivot: Fr) polygons of the python console 0 1 0 0
ParaNMT drawing polygons from python console 0 0 0 1
Trans. ([<span style="font-variant:small-caps;">MQR</span>]{}+ Quora) How to draw polygons from python console? 1 0 1 1
Trans. ([<span style="font-variant:small-caps;">MQR</span>]{}+ Quora) $\rightarrow$ GEC How to draw polygons from a python console? 1 1 1 1
Models and Experiments
======================
In this section, we describe the models and methods we benchmarked to perform the task of question rewriting.
To evaluate model performance, we apply our trained models to rewrite the ill-formed questions in TEST and treat the well-formed question in each pair as the reference sentence. We then compute BLEU-4 [@papineni2002bleu], ROUGE-1, ROUGE-2, ROUGE-L [@linrouge], and METEOR [@banerjee2005meteor] scores.[^8] As a baseline, we also evaluate the original ill-formed question using the automatic metrics.
Models Trained on [<span style="font-variant:small-caps;">MQR</span>]{}
-----------------------------------------------------------------------
#### Transformer.
We use the Tensor2Tensor [@tensor2tensor] implementation of the transformer model [@vaswani2017attention]. We use their “transformer\_base” hyperparameter setting. The details are as follows: batch size 4096, hidden size 512, 8 attention heads, 6 transformer encoder and decoder layers, learning rate 0.1 and 4000 warm-up steps. We train the model for 250,000 steps and perform early stopping using the loss values on the DEV set.
In following sections, when a transformer model is used, we follow the same setting as described above.
#### LSTM Sequence to Sequence Model with Attention.
We use the attention mechanism proposed by [@luong-etal-2015-effective]. We use the Tensor2Tensor implementation [@tensor2tensor] with their provided Luong Attention hyperparameter settings. We set batch size to 4096. The hidden size is 1000 and we use 4 LSTM hidden layers following [@luong-etal-2015-effective].
Methods Built from Other Resources
----------------------------------
We also benchmark other methods involving different training datasets and models. All the methods in this subsection use transformer models.
#### Round Trip Neural Machine Translation.
Round trip neural machine translation is an effective approach for question or sentence paraphrasing [@mallinson-etal-2017-paraphrasing; @dong-etal-2017-learning; @iyyer2018adversarial]. It first translates a sentence to another pivot language, then translates it back to the original language. We consider the use of both German (De) and French (Fr) as the pivot language, so we require translation systems for En$\leftrightarrow$De and En$\leftrightarrow$Fr.
The English-German translation models are trained on WMT datasets, including News Commentary 13, Europarl v7, and Common Crawl, and evaluated on newstest2013 for early stopping. On the newstest2013 dev set, the En$\rightarrow$De model reaches a BLEU-4 score of 19.6, and the De$\rightarrow$En model reaches a BLEU-4 score of 24.6.
The English-French models are trained on Common Crawl 13, Europarl v7, News Commentary v9, Giga release 2, and UN doc 2000. On the newstest2013 dev set, the En$\rightarrow$Fr model reaches a BLEU-4 score of 25.6, and the Fr$\rightarrow$En model reaches a BLEU-4 score of 26.1.
#### Grammatical Error Correction (GEC).
As some ill-formed questions are not grammatical, we benchmark a state-of-the-art grammatical error correction system on this task. We use the system of [@lichtarge2019corpora], a GEC ensemble model trained from Wikipedia edit histories and round trip translations.
#### Paraphrase Generator Trained on ParaNMT.
We also train a paraphrase generation model on a subset of the ParaNMT dataset [@wieting-gimpel-2018-paranmt], which was created automatically by using neural machine translation to translate the Czech side of a large Czech-English parallel corpus. We use the filtered subset of 5M pairs provided by the authors. For each pair of paraphrases (S1 and S2) in the dataset, we train the model to rewrite from S1 to S2 and also rewrite from S2 to S1. We use the [<span style="font-variant:small-caps;">MQR</span>]{}DEV set for early stopping during training.
Results
-------
Table \[tab:experiment\_models\] shows the performance of the models and methods described above. Among these methods models trained on [<span style="font-variant:small-caps;">MQR</span>]{}work best. GEC corrects grammatical errors and spelling errors, so it also improves the question quality in rewriting. Round trip neural machine translation is a faithful rewrite of the questions, and it naturally corrects some spelling and grammatical errors during both rounds of translation due to the strong language models present in the NMT models. However, it fails in converting commands and statements into questions.
The paraphrase generator trained on ParaNMT does not perform well, likely because of domain difference (there are not many questions in ParaNMT). It also is unlikely to convert non-question sentences into explicit questions.
Additional Training Data
------------------------
We consider two additional data resources to improve question rewriting models.
The first resource is the Quora Question Pairs dataset.[^9] This dataset contains question pairs from Quora, an online question answering community. Some question pairs are marked as duplicate by human annotators and other are not. We consider all Quora Question Pairs (Q1 and Q2) marked as duplicate as additional training data. We train the model to rewrite from Q1 to Q2 and also from Q2 to Q1. This gives us 298,364 more question pairs for training.
The second resource is the Paralex dataset [@fader2013paraphrase]. The questions in Paralex are scraped from WikiAnswers,[^10] where questions with similar content are clustered. As questions in the Paralex dataset may be noisy, we use the annotation from [@faruqui2018identifying]. Following their standard, we treat all questions with scores higher than 0.8 as well-formed questions. For each well-formed question, we take all questions in the same Paralex question cluster and construct pairs to rewrite from other questions in the cluster to the single well-formed question. This gives us 169,682 extra question pairs for training.
We also tried adding “identity” training examples in which the well-formed questions from the [<span style="font-variant:small-caps;">MQR</span>]{}TRAIN set are repeated to form a question pair.
The results of adding training data are summarized in Table \[tab:experiment\_datasets\]. Adding the identity pairs improves the ROUGE and METEOR scores, which are focused more on recall, while harming BLEU, which is focused on precision. We hypothesize that adding auto-encoding data improves semantic preservation, which is expected to help the recall-oriented metrics. Adding Quora Question Pairs improves performance on TEST but adding Paralex pairs does not. The reason may stem from domain differences: WikiAnswers (used in Paralex) is focused on factoid questions answered by encyclopedic knowledge while Quora and Stack Exchange questions are mainly answered by community contributors. Semantic drift occurs more often in Paralex question pairs as Paralex is constructed from question clusters, and a cluster often contains more than 5 questions with significant variation.
Combining Methods
-----------------
In addition to the aforementioned methods, we also try combining multiple approaches. Table \[tab:experiment\_ensemble\] shows results when combining GEC and the Quora-augmented transformer model. We find that combining GEC and a transformer question rewriting model achieves better results than each alone. In particular, it is best to first rewrite the question using the transformer trained on [<span style="font-variant:small-caps;">MQR</span>]{}+ Quora, then run GEC on the output.
We also tried applying the transformer (trained on [<span style="font-variant:small-caps;">MQR</span>]{}) twice, but it hurts the performance compared to applying it only once (see Table \[tab:experiment\_ensemble\]).
Human Evaluation
----------------
To better evaluate model performance, we conduct a human evaluation on the model rewritten questions following the same guidelines from the “Dataset Quality” subsection. Among the 300 questions annotated earlier, we chose the ill-formed questions from the TEST split, which yields 75 questions. We evaluate questions rewritten by three methods (Transformer ([<span style="font-variant:small-caps;">MQR</span>]{}+ Quora), GEC, and Transformer ([<span style="font-variant:small-caps;">MQR</span>]{}+ Quora) $\rightarrow$ GEC), and ask annotators to determine the qualities of the rewritten questions. To understand if question meanings change after rewriting, we also annotate whether a model rewritten question is semantically equivalent to the ill-formed question or equivalent to the well-formed one.
Table \[tab:human\_evaluation\] shows the annotations from two annotators. When the two annotators disagree, a judge makes a final decision. Note that the examples annotated here are a subset of those annotated in Table \[tab:dataset\_quality\], so the first row is different from the ill-formed questions in Table \[tab:dataset\_quality\]. According to the annotations, the GEC method slightly improves the question quality scores. Although Table \[tab:experiment\_models\] shows that GEC improves the question quality by some automatic metrics, it simply corrects a few grammatical errors and the rewritten questions still do not meet the standards of human annotators. However, the GEC model is good at preserving question semantics.
The Transformer ([<span style="font-variant:small-caps;">MQR</span>]{}+ Quora) model and Transformer ([<span style="font-variant:small-caps;">MQR</span>]{}+ Quora) $\rightarrow$ GEC excel at improving question quality in all three aspects, but they suffer from semantic drift. This suggests that future work should focus on solving the problem of semantic drift when building question rewriting models. Table \[tab:example\_model\_rewrite\] shows two example questions rewritten by different methods. The questions rewritten by GEC remain unchanged but are still of low quality, whereas ParaNMT and round trip NMT make a variety of changes, resulting in large variations in question quality and semantics. Methods trained on [<span style="font-variant:small-caps;">MQR</span>]{}excel at converting ill-formed questions into explicit ones (e.g., adding “What is” in the first example and “How to” in the second example), but sometimes make grammatical errors (e.g., Trans. ([<span style="font-variant:small-caps;">MQR</span>]{}+ Quora) misses “a” in the second example). According to Table \[tab:experiment\_ensemble\], combining neural models trained on [<span style="font-variant:small-caps;">MQR</span>]{}and GEC achieves the best results in automatic metrics. However, they still suffer from semantic drift. In the first example of Table \[tab:example\_model\_rewrite\], the last two rewrites show significant semantic mistakes, generating non-existent words “widebutcherblock” and “widebitcherblock”.
Conclusion and Future Work
==========================
We proposed the task of question rewriting and produced a novel dataset [<span style="font-variant:small-caps;">MQR</span>]{}to target it. Our evaluation shows consistent gains in metric scores when using our dataset compared to systems derived from previous resources. A key challenge for future work is to design better models to rewrite ill-formed questions without changing their semantics. Alternatively, we could attempt to model the process whereby question content changes. Sometimes community members do change the content of questions in online forums. Such rewrites typically require extra context information, such as the question description. Additional work will be needed to address this context-sensitive question rewriting task.
Acknowledgments
================
We thank Shankar Kumar, Zi Yang, Yiran Zhang, Rahul Gupta, Dekang Lin, Yuchen Lin, Guan-lin Chao, Llion Jones, and Amarnag Subramanya for their helpful discussions and suggestions.
[^1]: <https://github.com/ZeweiChu/MQR>
[^2]: Work performed during internship at Google.
[^3]: Contributed equally.
[^4]: <https://archive.org/download/stackexchange>
[^5]: <https://meta.stackexchange.com/questions/92074/what-can-i-do-when-getting-this-question-body-does-not-meet-our-quality-standar>
[^6]: <https://archive.org/download/stackexchange>
[^7]: The list of valid characters after lowercasing is: 0123456789abcdefghijklmnopqrstuvwxyz . , / ? : ; ’ \_ + - = ! @ \# \$ % \^ & \* ( ) $\vert$ { } $< >$ ‘ " ’ ” and space
[^8]: The BLEU-4 and METEOR scores are calculated using <https://github.com/Maluuba/nlg-eval>. ROUGE-1, ROUGE-2, and ROUGE-L are calculated using <https://github.com/pltrdy/rouge>.
[^9]: <https://www.quora.com/q/quoradata/First-Quora-Dataset-Release-Question-Pairs>
[^10]: <http://wiki.answers.com/>
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In his last article [*Against ‘Measurement’*]{} [J. S. Bell]{} sums up his well known critique of the problem of explaining the measurement process within the framework of quantum theory. In this article I will discuss the measurement process by analysing the concept of measurement from the epistemological point of view and I will argue against [Bell]{} that it belongs to the preconditions of experience to necessarily end up with a ”reduction of the wavefunction”. I will consider the ”chain of reduction” in detail – from pure states of $\cal S \otimes A$ (system $\cal S$ and measuring apparatus $\cal A$) via different kinds of mixtures to pure states of $\cal A(S)$. It turns out that decoherence is not sufficient to explain reduction, but that this can be done in terms of the concept of information within a transcendental approach.'
---
\#1[ | \#1 ]{} \#1[ \#1 | ]{} \#1\#2[ \#1 | \#2 ]{} \#1[ | \#1 ]{}
[****]{}[^1]
[Holger Lyre]{}[^2]
[September 1997]{}
> [Contents]{}\
> 1 Introduction 2
>
> 2 The Measurement Problem 2
>
> 3 The Chain of Reduction 3
>
> 4 Against Measurement? – Some Interpretations 5
>
> 5 Information as a Transcendental Concept 7
>
> 6 The Measurement Problem – Revisited 8
Introduction
============
Quantum theory is the ‘hard core’ of physics today. Its experimental success is overwhelming. But although the mathematical framework and its pragmatical application is non-controversial, it suffers from an obstinate interpretation problem: what does the formalism really mean philosophically? One part of this open question is the [measurement problem]{}. In this article I first like to discuss the measurement problem in terms of measurement theory and in comparison with some actual interpretation approaches. Since [John S. Bell]{} in his famous article [*Against ‘Measurement’*]{} [@bell90] gives a very sophisticated (and also humorous) analysis of the problems of quantum theory according to measurement theory, I like to deal with some of his clever arguments – including the usage of his abbreviation
[*FAPP = for all practical purposes*]{}.
The final aim is to stress the epistemological point of view in discussing the measurement problem. This will be done using a transcendental argument in the spirit of [Kant]{}. It turns out that in opposition to [Bell]{} the concept of information plays a key role in quantum theory.
The Measurement Problem {#problem}
=======================
The measurement problem is caused by the [*universality*]{} of quantum theory, which means that there exists no inherent reason not to apply it to any physical system or object. In the theory of measurement the world is divided into distinct parts: a system $\cal S$, on which the measurement is performed, a measuring apparatus $\cal A$, and the ‘rest of the world’ $\cal R$. Because of its universality all parts must in principle be describable by quantum theory.
Let us briefly review the steps of the measurement process. I will denote the quantum states of $\cal S$ by $\ket{\psi}$ and the states of $\cal A$ by $\ket{\chi}$. The observable $\hat A$ satisfies the eigenvalue equation A = a\_i . To perform a measurement the [Hilbert]{} space of $\cal S$ must be enlarged to the [Hilbert]{} space of the compound system $\cal S \otimes A$ by forming the tensor product which is spanned by the states \[S+A\] = = \_i c\_i with \[Phi\_i\] = . The tensor product also contains interference terms, which represent the typical quantum correlations between $\cal S$ and $\cal A$. The state (\[S+A\]) is a pure state with a projection operator \[Phi\_rein\] P\_ = . After the measurement interaction $\hat H_{int}$ between $\cal S$ and $\cal A$ we obtain \[S+A’\] = e\^[ i H\_[int]{} t ]{} with states = instead of (\[Phi\_i\]). In case of an ideal measurement we can replace $\ket{ \psi' } \stackrel{i.m.}{=} \ket{ \psi }$. Note that now the states $\Bket{ \chi'( \psi_i) }$ of the measuring apparatus are not independent from those of the system $\ket{ \psi }$.
After the measurement interaction the compound system (\[S+A’\]) must be separated into the subsystems $\cal S$ and $\cal A$ in order to read $\cal A$. This is done by a cut. Firstly, $\hat P_{\Phi'}$ transforms into the mixed states of $\cal S$ and $\cal A$ with the density operators $\hat\rho_{\psi}$ and $\hat\rho_{\chi}$ P\_[’]{} \_ = \_[i,k]{} w\_[ik]{} . Due to decoherence the interference terms will become immensely small (but do not vanish exactly).
In principle the mixed state $\hat\rho_{\psi}$ allows an infinite number of possible decompositions into states $\ket{ \psi_i }$ of $\cal S$ (resp. into states $\Bket{ \chi( \psi_i) }$ of $\cal A$). By picking out one special decomposition the mixed state transforms into a mixture of states[^3] \[mixture\] \_ \_ = { ( w\_i, P\_[\_i]{} ) }. As a last step the system is in a new state $\hat P_{\psi_i}$ \_ P\_[\_i]{} = . Likewise the measuring apparatus will show one result and is therefore in a definite pointer state $\hat P_{ \chi( \psi_i) }$. Finally, at the end of the whole measuring act, the systems $\cal A$ and $\cal S$ are in new pure states.
The Chain of Reduction {#chain}
======================
As a result of the preceding section we see that strictly speaking the so called ‘reduction of the wavefunction’ has to be considered as a [*chain of reduction*]{} of at least three steps: $$\ba{ccccccl}
& & \hat\rho_{\psi} & \longrightarrow & \hat\gamma_{\psi} & \longrightarrow & \ \hat P_{\psi_i} \\
\hat P_{\Phi'} & {\displaystyle \nearrow \atop \displaystyle \searrow} \\
& & \hat\rho_{\chi} & \longrightarrow & \hat\gamma_{\chi} & \longrightarrow & \ \hat P_{\chi(\psi_i)} \\
& \mbox{\footnotesize (step 1)} & & \mbox{\footnotesize (step 2a)} & & \mbox{\footnotesize (step 2b)}
\ea$$
Let us consider this chain step by step.
#### Step 1.
Due to the measurement interaction and due to the coupling between systems $\cal S$, $\cal A$, and $\cal R$ this step leads to the cancellation FAPP of the interference terms between $\cal S$ and $\cal A$, i.e. the typical quantum correlations disappear (FAPP, but not exactly). Thus, step 1 can be looked upon as a unitary temporal development according to some wave equation.
#### Step 2.
Due to the objectification the mixed state must be replaced by a mixture $\hat\gamma$ according to (\[mixture\]). It is first of all important to note that the quantum theoretical framework gives no possibility to describe this step as a pure quantum process (i.e. as a time evolution by a unitary operator). [Bell]{} points out that, logically speaking, a system described by $\hat\rho$ is in a state
[*$\Psi_1 \Psi_1^*$ and $\Psi_2 \Psi_2^*$ and ... ,*]{}
whereas $\hat\gamma$ refers to a state
[*$\Psi_1 \Psi_1^*$ or $\Psi_2 \Psi_2^*$ or ... .*]{}
Although step 2 is certainly not involved in step 1 it is indeed astonishing that this problem is very often not discussed in the common measurement theory – or as [Bell]{} puts it: [*”The idea of elimination of coherence, in one way or another, implies the replacement of ‘and’ by ‘or’, is a very common one among the solvers of the ‘measurement problem’. It has always puzzled me.”*]{} [@bell90 p. 36].
Step 2 can be split into two logical steps 2a and 2b ([Bell]{} makes no explicit distinction between them):
#### Step 2a.
A mixed state $\hat\rho$ represents a class of equivalent mixtures $\hat\gamma$ – such that from the mathematical point of view no particular mixture is distinguished. Step 2a stresses the point that one particular has to be choosen.
#### Step 2b.
At the end of the whole measuring act the apparatus must show exactly one result (on a display for instance) – what else could be the meaning of the term ‘measurement’? Step 2b stresses the point that any satisfying theory of measurement must explain the final occurrence of exacly [*one*]{} measuring outcome out of the set of the many possible ones.
To make a distinction between steps 2a and 2b could be confusing. The mixture $\hat\gamma$ can be considered as a statistical description of different wavefunctions $\Psi_i$ – such as the density matrix in classical statistical thermodynamics. This means, ontologically speaking, that the system already exists in an actual state $\Psi_i$, but it is not known (only with probabilities) to the observer. Thus, the so called [*ignorance interpretation*]{} is valid for $\hat\gamma$ but not for $\hat\rho$. From this viewpoint step 2b would just describe the reading of the measuring device by an observer and would be of no philosophical interest. But it should be emphasized that one cannot distinguish between $\hat\rho$ and $\hat\gamma$ by any observation. With respect to this the distinction between step 2a and 2b indicates the logical difference between ‘picking out a certain mixture $\hat\gamma$’ and ‘actually being in one certain state’ (the final pointer state of $\cal A$ for instance).
Against Measurement? – Some Interpretations {#against}
===========================================
What do the most prominent interpretation programs of quantum theory today answer to the above steps in the chain of reduction? I will briefly discuss some of them.
#### The Decoherence Program.
Nowadays, the concept of decoherence to explain step 1 is very popular with physicists. It seems indeed very useful to scrutinize the conditions and orders of magnitude under which quantum systems decohere, but one has to keep in mind that decoherence is essentially FAPP. The question arises wether a FAPP description will be a satisfactory explanation. [Bell]{} was obviously not satisfied – although he and nobody, I presume, would deny the [*”... absence FAPP of interference between macroscopically different states”*]{} [@bell90 p. 36]. All in all decoherence is sufficient to explain step 1, whereas step 2 is by no means explained.
#### The [Bohm-de Broglie]{} Program or the Hidden Variable Program.
It is the first of [Bell]{}’s favourites in his article. Of course a hidden variable argument is an objection which can always be raised: there could be something we do not know today! But, as a consequence of [Bell]{}’s own invention – his famous inequalities [@bell65] – [*local*]{} hidden variables are nowadays experimentally excluded. But does a theory with non-local hidden variables really show a conceptual difference to common quantum theory? New arguments support the idea that this is indeed not the case [@englert_etal92] – would [Bell]{} have believed in them?
#### The [Ghirardi-Rimini-Weber]{} Model.
This is the second of [Bell]{}’s favourites in his article. Its starting point is the [*ad-hoc-assumption*]{} that the wavefunction will ’collapse’ after a given small and stochastic time interval by a spontaneous localization process. The parameters of the model are choosen for it to be in good correspondence with the ordinary quantum predictions. The model ‘explains’ step 1 in the bandwidth of its parameters – i.e. very well, but FAPP. Insofar the spontanous processes are [*stochastic*]{}, the central questions behind step 2a and, most of all, 2b remain unanswered – thus the situation resembles the decoherence program. The question persists how a mere ad-hoc-model could be a satisfactory explanation of the deep problem of quantum measurement.
#### The Many Worlds Interpretation.
This interpretation leads essentially to [*quantum cosmology*]{}, i.e. the reduction problem is considered for the universe as a whole. It does ‘explain’ – in a very broad-minded meaning of this word – step 2, but it does not explain step 1. Consider for instance the following open questions: At what time steps does the ‘branching’ of the universes occur and how many universes do occur at each time step? Today many authors combine the idea of decoherence with the concept of many worlds. But nevertheless the ‘ontological costs’ of assuming many universes are very high! Should this really be the right answer? In any case, even [Bell]{}, also in combination with the [Bohm-de Broglie]{} theory, [*”... did not like it”*]{} [@bell76].
#### The Copenhagen Interpretation (CI).
This is the orthodox interpretation of quantum theory, as far as it refers to its founders [Werner Heisenberg]{} and, most of all, [Niels Bohr]{}. Since there does not exist a kind of ‘codification’ of CI, there is still a certain confusion about its basic concepts. I like to propose the following as the central assumption of CI: [*The outcomes of measurements must be described in classical terms, i.e. the measuring apparatus must be described classically.*]{}[^4] What is the meaning of this assumption? It certainly means not that apparatuses and measuring devices are non-quantum systems. But it means that the apparatuses must [*necessarily*]{} be described classically in order to give an appropriate description of the outcomes of a measurement. ‘Appropriate’ here means that the outcomes have to be communicable and understandable to each observer. E.g., the idea that the pointer of a device should be in a superposition of different pointer positions is obviously senseless and non-communicable. In this sense experimental data must be described classically.
Moreover, CI can be read as just offering the [*minimal semantics*]{} to quantum theory, i.e. semantics which is necessary in order to apply the theory to reality [@goernitz+cfw91b]. Generally speaking, a physical theory contains two parts: the mathematical structure of the formalism and the related physical concepts. The minimal semantics of CI is:
Mathematical structure Physical concepts
-------------------------------------------------------------- ----------------------------------------------------
[Hilbert]{} space $\cal H$ object (system)
(topological) tensor product ${\cal H}_1 \otimes {\cal H}_2$ composition of the objects 1 and 2
(self-adjoint) operator observable
unitary U(1) transformation temporal development
vector $\ket{\Psi} \in \cal H$ state of an object
scalar $p=\Big| \braket{\Psi_i}{\Psi_f} \Big|^2$ probability for the transition of state $i$ to $f$
The last row refers to the central concept of CI: [*probability*]{}. The scalar product of two states gives the amplitude of the transition probability between them. Thus, the [Hilbert]{} space is provided with a probability metric.
A further remark should be made: probability can be seen as the mathematical quantification of ‘possibility’. Interestingly, the many worlds interpretation is in some sense not richer than CI, since there exists a one-to-one terminological mapping between both interpretations: the many world interpreters have simply replaced the term ‘possibility’ by the term ‘world’. It seems that ‘many worlds’ is just a fancy way of saying something very trivial, namely ‘many possibilities’ (i.e. probability).
But a crucial question remains: how can a probability theory be understood without the concept of measurement? How can physics be [*”against measurement”*]{}?
Information and the Transcendental Approach
===========================================
As shown in section \[chain\], the ‘reduction of the wavefunction’ is logically more than one single step. CI gives a necessary minimal semantics to quantum theory, but the alternative interpretive attempts in section \[against\] do not go beyond it to provide a satisfactory understanding of the entire chain of reduction. Thus, maybe a philosophical invention would be helpful: the transcendental point of view. As I like to argue in the following, this will not contradict CI, but it will extend the [*epistemological aspect*]{} of quantum theory.
At the end of section \[against\] probability was identified as the central concept of quantum theory – at least according to CI, its minimal semantics. ”Against [Bell]{}” I like to stress the point that in any probability theory one can neither renounce the concept of measurement, nor the concepts microscopic, macroscopic, reversible, irreversible, observable, or observer – some of the [*”... worst terms”*]{} in [Bell]{}’s understanding. Moreover, I like to propose [*information*]{} – decidedly against [Bell]{} a really [*good word*]{} – as the central concept behind it all [@lyre98]. In view of the main question, what the quantum wavefunction $\Psi$ really stands for, the following basic assumption should be made
$\Psi$ [*is*]{} information.
What are the reasons for believing this? Probability means – due to a logarithm - the same as the syntactic aspect of information $I_{syn} \sim - \ln p$. There also exist a semantic and a pragmatic aspect. Thus, probability can be seen as a sub-concept of information. Information, moreover, involves the terms subject and object. Let us try to answer [Bell]{} (1990, p. 34): [*”Information? Whose information?”*]{} Information for any subject with conceptual and empirical competence! [*”Information about what?”*]{} Information about empirically knowable objects, which are constituted just by the information which can be gained from them! In short: Objects are constituted by information which is available to subjects.
In order to make the above answers plausible the direction of arguments must be changed. This can be done by using a transcendental argument in the manner of [Immanuel Kant]{}: the foundations of empirical science are based on the preconditions of experience [@kant_KrV] – and, certainly nowadays, the foundations of empirical science are the foundations of quantum theory [@drieschner79]. The key idea is that since experience, and moreover empirical science, is obviously possible and successful, certain preconditions of experience must hold, which, in the last analysis, make experience possible. These preconditions will of course never fail in experience – per definition they can never become empirically falsified. The crucial question then is: What are good candidates for preconditions of experience? I like to propose just these two: [*distinguishability*]{} and [*temporality*]{}. Why? Whithout the possibility of making distinctions we could never be able to have speech, concepts, and communicable thoughts. If science is possible distinguishability is one of its most rudimentary methodological preconditions. Further on, without implicitly using the already known difference between past and future we could never give any meaning to the word experience – or as [Carl Friedrich von Weizsäcker]{} puts it: [*”... experience means to learn from the past for the future, then any empirical science presupposes an understanding of past and future”*]{} [@goernitz+cfw91b]. This understanding may be called temporality.
Translated into an information theoretic language the difference between past and future can be expressed as the difference between [*actual*]{} and [*potential information*]{}. More detailed than the above statement we can say that the wavefunction or, in general, density operators represent potential information. Still more detailed, it is [*quantum information*]{}, since quantum bits are composed by the tensor product of two dimensional [Hilbert]{} spaces, i.e. quantum bits are indistinguishable in contrast to bits of classical potential information. In an earlier paper [@lyre97] I have argued that the so called [*complete concept of information*]{}, i.e. the syntactic, semantic, pragmatic, and temporal aspect of information, can conceptually be deduced from distinguishability and temporality alone. In this sense information can be based transcendentally, i.e. concerning the preconditions of experience.
The Measurement Problem – Revisited
===================================
What are the advantages in expanding CI strictly towards an information theoretic interpretation of quantum theory? In this last section I like to revisit the measurement problem – and especially the chain of reduction. I will do this in a list of eight theses:
1. [*Thesis. The chain of reduction is not a chain of physical interaction steps but of methodological ones.*]{} The usual aim of measurement theory is to describe the measurement process as a quantum physical process by itself, i.e. to describe it as an interaction between $\cal S$ and $\cal A$. Surely, first of all the physical interaction $\hat H_{int}$ – as expressed in (\[S+A’\]) – establishes the measuring act. But this has nothing to do with the problem of reduction! Otherwise there should exist a unitary transformation for each step of the chain of reduction. But as analysed in section \[chain\], step 1 can only be considered as a time development FAPP, whereas step 2 certainly cannot. For the whole procedure described in section \[problem\] we should better speak of measuring ‘act’ instead of ‘process’ (to speak with [Bell]{}: ‘process’ is a bad word at this stage).
2. [*Thesis.*]{} According to step 1: [ *Quantum theory does not predetermine the cut between $\cal S$ and $\cal A$, nevertheless the cut is necessary in order to apply quantum theory to reality*]{}. Because of the universality of quantum theory, any system can in principle be described by quantum theory and, consequently, each measuring apparatus as well. As it was explained in section \[against\] this is thoroughly compatible with CI. But in CI it is clearly seen that the universality of quantum theory is in a certain contradiction to its meaning as a theory of empirical science. Therefore from the CI viewpoint the idea of a wavefunction of the universe is a physically senseless extrapolation of the mathematical formalism.[^5] Is is, in principle, not forbidden to apply quantum theory to the world as a whole, but this would be no information for anybody since no subject is left. Thus, for any measurement the cut is a necessary precondition.
3. [*Thesis.*]{} According to step 2a: [ *The measuring apparatus must be suitable as such.*]{} E.g., the pointer states must be orthogonal. Once a system is chosen to act as a measuring apparatus the decomposition of the density operator into the spectrum of the pointer variable is fixed. This is involved in the CI’s central assumption of describing the measuring apparatus classically.
4. [*Thesis.*]{} According to step 2b: [ *Measurements must lead to irreversible facts.*]{} Irreversibility in this context characterizes any documents of the past, e.g. pointer devices, printer outputs, computer memories, human brain states... . Facticity means that a measurement can lead to one and only one outcome. Thus, facts are classical (at least FAPP).
5. [*Thesis. ‘Measurement’ is a necessary term in any empirical science. It is related to information as the key concept of quantum theory.*]{} Experience means to learn from the facts of the past for the possibilities of the future. In empirical science this will be done by measurements. Thus, in empirical science the term ‘measurement’ is methodologically irreducible. Quantum physics, the hard core of empirical science, describes possibilities of the future in terms of potential information (”$\Psi$ is information”). Potential information is information which can be gained, i.e. can become actual, if a measurement is performed.
6. [*Thesis. A measurement represents the transition from potential to actual information.*]{} This thesis can mathematically be quantified. Quantum information is measured in terms of the entropy of the density operator $S \left[\hat\rho \right] = - k_B \ Sp \, ( \hat\rho \ \ln \hat\rho )$. A pure state contains no potential information, i.e. the initial as well as the final state of the chain of reduction represents $S\left[\hat P\right]=0$ bit. During step 2 the potential information amounts $S\left[ \hat\gamma \right] > 0$ bits. Since a measuring act represents the transition from potential to actual information, the reduction is a presupposition of the measurement.
7. [*Thesis. The subject is an irreducible element in any empirical science.*]{} According to the logic of the transcendental argument subjects in empirical science must be equipped at least with conceptual and empirical competence – a [*”PhD”*]{} [@bell90 p. 34] is of course not a necessary precondition of experience! It should be noted that being a subject in this sense and being conscious is not necessarily the same. Thus, the assertion is not the reduction taking place in the consciousness of the observer as in the [London-Bauer]{} or [Wigner]{} approach, since in these approaches human consciousness is excluded from (quantum) physical description. But this contradicts the universality of quantum theory. The idea of the proposed transcendental approach is not to exclude anything from quantum theory, i.e. any subject can in principle be described by quantum theory – but then, of course, as an object for and from another subject. This is the key point of the argument: subjects can be described as objects, but not all of them at the same time! Physics without any subjects would be meaningless. In that sense the subject is irreducible.
8. [*Thesis. Physics is essentially FAPP, but ‘FAPPness’ is no sufficient explanation to the measurement problem.*]{} Objects are constituted by information which exists for subjects. Any (object) information presupposes a certain semantics under which the information can be understood. But for the same reason the information invested in the semantics needs other semantics before and so on. In a finite world, constituted by a huge but finite amount of information this leads to an inherent circularity [@lyre97]. Therefore, empirical science is by no means as exact as its mathematical framework suggests. Strictly speaking there exist no isolated quantum objects. Quantum theory is a holistic theory which, in the empirical application, must necessarily be FAPP.[^6] But ‘FAPPness’ is no sufficient explanation to the measurement problem. It explains step 1, whereas my proposal is that step 2 should be seen under the transcendental approach.
All in all the measurement ‘problem’ could be soluble or even vanishes, if it is not seen as a problem on the intrinsic-physical level, i.e. described as a physical interaction process, but on the meta-level, i.e. seen on the basis of the methodological and epistemological presuppositions of physics. Thus, [Bell]{}’s list of bad words in fact appears to be a list of [*necessary*]{} terms of any empirical science – most of all the terms ‘measurement’ and ‘information’.
(1965). On the [E]{}instein-[P]{}odolsky-[R]{}osen [P]{}aradox. , 1:195–200. Reprinted in [@bell87].
(1976). The [M]{}easurement [T]{}heory of [E]{}verett and de [B]{}roglies’s [P]{}ilot [W]{}ave. In [@bell87].
(1987). . Cambridge University Press, Cambridge.
(1990). Against ’[M]{}easurement’. , 8:33–40.
(1949). . In [Schilpp, P. A.]{} (ed.): [*Albert Einstein: Philosopher-Scientist*]{}, The Library of Living Philosophers, Vol. VII, p. 201–241, Evanston, Illinois.
(1997). Spektrum Akademischer Verlag, Heidelberg.
(1979). Lecture Notes in Physics 99. Springer, Berlin.
, [Scully, M. O.]{}, [S[ü]{}ssmann, G.]{}, and [ Walther, H.]{} (1992). Surrealistic [B]{}ohm [T]{}rajectories. , 47a:1175–1186. Comment: [Dürr]{} et al., [*Z. Naturf.*]{} 48a, 1261 (1993). Reply to Comment: [*Z. Naturf.*]{} 48a, 1263 (1993).
and [Weizs[ä]{}cker, C. F. v.]{} (1991). Steps in the [P]{}hilosophy of [Q]{}uantum [T]{}heory. In [Hennig, J.]{}, [L[ü]{}cke, W.]{}, and [Tolar, J.]{} (eds.), [*Differential Geometry, Group Representations, and Quantization*]{}, Lecture Notes in Physics 379. Springer, Berlin.
(1930). . Hirzel, Leipzig.
(1781). . Riga.
(1997). Time and [I]{}nformation. In [Atmanspacher, H.]{} and [Ruhnau, E.]{} (eds.). . Springer, Berlin.
(1998). . Springer, Wien.
(1998). . Cambridge University Press, Cambridge.
(1958). Über den [M]{}eßvorgang. , Heft 88.
[^1]: Talk at the X-th Max Born Symposium ”Quantum Future”, Wroc[ł]{}aw, September 24 - 27, 1997\
(http://xxx.lanl.gov/abs/quant-ph/9709059)
[^2]: Institut für Philosophie, Ruhr-Universität Bochum, D-44780 Bochum, Germany,\
e-mail: [email protected], http://www.ruhr-uni-bochum.de/philosophy/staff/lyre.htm
[^3]: In German: ”Gemisch” (mixed state) vs. ”Gemenge” (mixture of states) [@heisenberg30 p. 43], [@suessmann58].
[^4]: Compare [Bohr]{} who emphasized that [*”... however far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms”*]{} [@bohr49 p. 209].
[^5]: Compare [Heisenberg]{}: [*”... wenn man das ganze Universum in das System einbezöge – dann ist ... die Physik verschwunden und nur noch ein mathematisches Schema geblieben”*]{} [@heisenberg30 p. 44], [*”... if the whole universe were to be included into the system then physics would vanish and just a mathematical scheme remains”*]{} (translation by the author).
[^6]: Recent research even shows the universality of quantum theory leading to problems of self-referentiality for inner observers. E.g., an inner observer $\cal A$ cannot distinguish between a pure state and a mixture of $\cal S \otimes A$, i.e. they are indistinguishable FAPP. Moreover, self-referentiality seems to be connected to incompleteness of quantum physics [@breuer97], [@mittelstaedt98].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In the Lagrangian theory of guiding center motion, an effective magnetic field $\bm{B}^* = \bm{B}+(m/e)v_\parallel\nabla \times {\bm{b}}$ appears prominently in the equations of motion. Because the parallel component of this field can vanish, there is a range of parallel velocities where the Lagrangian guiding center equations of motion are either ill-defined or very badly behaved. Moreover, the velocity dependence of $\bm{B}^*$ greatly complicates the identification of canonical variables, and therefore the formulation of symplectic integrators for guiding center dynamics. This Letter introduces a simple coordinate transformation that alleviates both of these problems simultaneously. In the new coordinates, the Liouville volume element is equal to the toroidal cotravariant component of the magnetic field. Consequently, the large-velocity singularity is completely eliminated. Moreover, passing from the new coordinate system to canonical coordinates is extremely simple, even if the magnetic field is devoid of flux surfaces. We demonstrate the utility of this approach to regularizing the guiding center Lagrangian by presenting a new and stable one-step variational integrator for guiding centers moving in arbitrary time-dependent electromagnetic fields.'
author:
- 'J. W. Burby'
- 'C. L. Ellison'
bibliography:
- 'SI\_refs.bib'
title: Toroidal regularization of the guiding center Lagrangian
---
Without loss of generality, the Lagrangian for both guiding centers and gyrocenters in time-dependent electromagnetic fields $(\bm{A},\varphi)$ may be written $$\begin{aligned}
\ell&=(e\bm{A}(\bm{X},t)+mv_\parallel \bm{b}(\bm{X},t))\cdot\dot{\bm{X}}-H(\bm{X},v_\parallel,t),
$$ where $H$ is either the guiding center or gyrocenter Hamiltonian, as appropriate. [@Littlejohn_1983; @Cary_2009; @Hahm_Lee_Brizard_1988; @Brizard_2007] The Hamiltonian can always be written as $H=e\varphi+K(\bm{X},v_\parallel;\bm{E},\bm{B})$, where the gyrocenter kinetic energy $K$ depends parametrically on the potentials only through the gauge-invariant $\bm{E}=-\partial_t\bm{A}-\nabla\varphi$ and $\bm{B}=\nabla\times\bm{A}$. (The parametric dependence is in general nonlinear and nonlocal; see Ref. for the leading-order nonlocal terms.) This Letter is concerned with addressing a pair of computational and theoretical challenges associated with $\ell$.
The first challenge is theoretical in nature, and it concerns unphysical infinities that appear in the Euler-Lagrange equations, which read
\[eq:standard\_gc\] $$\begin{aligned}
\dot{v}_\parallel&=\frac{e}{m}\frac{\bm{E}^*\cdot\bm{B^*}}{B_\parallel^*}\label{normal_a_par}\\
\dot{\bm{X}}&=\frac{\bm{B}^*}{B_\parallel^*}\frac{\partial H/\partial v_\parallel}{m}+\frac{\bm{E}^*\times\bm{b}}{B_\parallel^*}.\label{normal_drift}\end{aligned}$$
Here the effective magnetic field is given by $$\begin{aligned}
\bm{B}^*&=\bm{B}+\frac{m}{e}v_\parallel\nabla\times\bm{b}\\
B^*_\parallel&=\bm{b}\cdot\bm{B}^*=B+\frac{m}{e}v_\parallel\bm{b}\cdot\nabla\times\bm{b},\end{aligned}$$ and the effective electric field is given by $$\begin{aligned}
\bm{E}^*=\bm{E}-\frac{m}{e}v_\parallel \partial_t\bm{b}-\frac{1}{e}\nabla K.\end{aligned}$$ Because the Hamiltonian $H$ is a smooth function of $\bm{X}$ and $v_\parallel$, these equations of motion become infinite whenever $B_\parallel^*=0$. While such infinities occur only at values of $v_\parallel$ that technically violate the guiding center and gyrocenter ordering assumptions, they nevertheless lead to vexing inconsistencies and complications in kinetic theories built on top of $\ell$. For instance, because the support of a Maxwellian distribution function contains all of phase space, gyrokinetic theory necessarily allows for a small number of particles to sample the problematic range of parallel velocities.
The second challenge is related to the problem of developing symplectic integrators for guiding center and gyrocenter motion. While the dynamical equations for guiding centers and gyrocenters, -, possess a Hamiltonian structure, they are not written in canonical Hamiltonian form. Thus, the Hamiltonian structure is non-canonical, and standard symplectic integration techniques cannot be applied.[@Karasozen_2004]
Previous authors have addressed each of these challenges with either computational or theoretical applications in mind. On the theoretical side, Correa-Restrepo and Wimmel[@Restrepo_1985] proposed a method for regularizing the infinities associated with $B_\parallel^*$ based on multiplying the $v_\parallel \bm{b}$-term in $\ell$ by a specially-designed form factor. White and Zakharov[@White_2003] proposed a system of canonical coordinates for guiding centers that may be used if the magnetic field admits nested toroidal flux surfaces. Zhang *et al.*[@Zhang_R_2014] offer an alternative approach to guiding center canonical coordinates that is able to handle arbitrary magnetic fields, but leads to a cumbersome relationship between $(\bm{X},v_\parallel)$ and the canonical coordinates. On the computational side, the pioneering work on the development of structure-preserving integrators for guiding center motion was done by Qin and Guan,[@Qin_2008] who developed a variational integrator that performs well in axisymmetric magnetic field configurations. Subsequently, Ellison[@Ellison_thesis] investigated the prospect of extending the Qin-Guan technique to allow for arbitrary magnetic fields, but found the existing integrators to be numerically unstable. The latter work went on to uncover an unknown and serious gap in variational integration theory, and was able to identify stable variational integrators for a broad (yet not fully general) class of non-axisymmetric magnetic fields. The most flexible currently-available structure-preserving integrators for guiding center dynamics are the canonical-coordinate-based integrator of Zhang *et al.*,[@Zhang_R_2014], and a special subset of the projected variational integrators developed recently by Kraus,[@Kraus_2017] all of which are capable of handling arbitrary magnetic fields. While each of these integrators is symplectic, none of them are derived directly from a discrete variational principle. Therefore, stable variational integrators that preserve a symplectic form on the guiding center phase space have yet to be identified.
In this Letter, we will argue that it is possible to simplify, unify, and generalize much of the previous work on regularization and canonization of guiding center theory. We will show that by applying a simple coordinate change to $\ell$, all $B_\parallel^*$ singularities, and all difficulties associated with finding canonical coordinates, are eliminated simultaneously; the theory is *toroidally-regularized*. In order to illustrate the power of this result, we will then construct a novel and simple one-step variational integrator for guiding centers moving in arbitrary time-dependent electromagnetic fields without flux surfaces.
Derivation of the toroidally-regularized guiding center Lagrangian comprises two coordinate transformations. The first transformation is near-identity. The second transformation is non-perturbative, but extremely simple. The only required assumptions are (a) that the standard guiding center ordering parameter satisfies $\epsilon=\rho/L\ll 1$ and (b) that the guiding centers of interest move in a toroidal region with a toroidal angle $\phi$ that satisfies $$\begin{aligned}
\label{eq:nonzero_bphi}
|\bm{B}\cdot\nabla\phi|\equiv |B^\phi|>0.\end{aligned}$$ Neither flux surfaces nor time-independence of the fields need to be assumed. The second assumption (b) is generally valid in the interior of devices envisioned for achieving magnetic confinement fusion, in particular in tokamaks and stellarators.[^1]
The near-identity transformation maps the coordinates $(\bm{X},v_\parallel)$ to the new coordinates $(\bm{\mathsf{X}},\mathsf{v}_\parallel)$ using the Lie transform $$\begin{aligned}
(\bm{\mathsf{X}},\mathsf{v}_\parallel)=\exp(G)(\bm{X},v_\parallel),\end{aligned}$$ where $G$ is a time-dependent undetermined $O(\epsilon)$ vector field on $(\bm{X},v_\parallel)$-space. The guiding center Lagrangian is transformed into $$\begin{gathered}
\bar{\ell}(\bm{\mathsf{X}},\mathsf{v}_\parallel,\dot{\bm{\mathsf{X}}},\dot{\mathsf{v}}_\parallel)=(e\bm{A}(\bm{\mathsf{X}})+m \mathsf{v}_\parallel \bm{b}(\bm{\mathsf{X}}))\cdot\dot{\bm{\mathsf{X}}}\nonumber\\
+e G^{\bm{X}}\times\bm{B}(\bm{\mathsf{X}})\cdot\dot{\bm{\mathsf{X}}}-(H(\bm{\mathsf{X}},\mathsf{v}_\parallel)+e G^{\bm{X}}\cdot\bm{E})+O(\epsilon).\end{gathered}$$ The compontents of $G$ are chosen according to $$\begin{aligned}
G^{v_\parallel}&=0\\
G^{\bm{X}}&=-\frac{m}{e B^\phi} \mathsf{v}_\parallel\nabla\phi\times\bm{b}.\end{aligned}$$ Because the unit vector along the magnetic field may be expressed (without approximation) as $$\begin{aligned}
\bm{b}&=\frac{B}{B^\phi} \nabla\phi-\frac{B}{B^\phi}\bm{b}\times(\nabla\phi\times\bm{b}),
$$ the transformed guiding center Lagrangian becomes $$\begin{gathered}
\bar{\ell}(\bm{\mathsf{X}},\mathsf{v}_\parallel,\dot{\bm{\mathsf{X}}},\dot{\mathsf{v}}_\parallel)=\left(e\bm{A}(\bm{\mathsf{X}})+m \mathsf{v}_\parallel \frac{B}{ B^\phi} \nabla\phi\right)\cdot\dot{\bm{\mathsf{X}}}\nonumber\\
-\left(H(\bm{\mathsf{X}},\mathsf{v}_\parallel)-m\mathsf{v}_\parallel\bm{b}\cdot \frac{\bm{E}\times\nabla\phi}{B^\phi}\right)+O(\epsilon).\label{lie_tran_lag}\end{gathered}$$ Note that the explicit form of the near-identity transformation is $$\begin{aligned}
\bm{\mathsf{X}}&=\bm{X}-\frac{m}{e B^\phi(\bm{X})}v_\parallel(\nabla\phi\times\bm{b})(\bm{X})+O(\epsilon^2)\\
\mathsf{v}_\parallel&=v_\parallel.\end{aligned}$$ Apparently this transformation amounts to a $v_\parallel$-dependent modification of the usual gyroradius vector.
The non-perturbative transformation maps the coordinates $(\bm{\mathsf{X}},\mathsf{v}_\parallel)$ to the coordinates $(\bm{\mathsf{X}},\mathsf{v}_\parallel^*)$ according to $$\begin{aligned}
\label{non_pert_trans}
\mathsf{v}_\parallel^*=\mathsf{v}_\parallel \frac{B}{R_o B^\phi},\end{aligned}$$ where $R_0$ is an arbitrary constant with the dimensions of length. The Lagrangian finally becomes $$\begin{aligned}
\ell^*(\bm{\mathsf{X}},\mathsf{v}_\parallel^*,\dot{\bm{\mathsf{X}}},\dot{\mathsf{v}}_\parallel^*)&=\left(e\bm{A}(\bm{\mathsf{X}})+m \mathsf{v}_\parallel^* R_o\nabla\phi\right)\cdot\dot{\bm{\mathsf{X}}}\nonumber\\
&\hspace{6em}-H^*(\bm{\mathsf{X}},\mathsf{v}_\parallel^*)+O(\epsilon),\end{aligned}$$ where the new Hamiltonian is given by $$\begin{aligned}
H^*(\bm{\mathsf{X}},\mathsf{v}_\parallel^*)&=e\varphi +K^*(\bm{\mathsf{X}},\mathsf{v}_\parallel^*)\\
K^*(\bm{\mathsf{X}},\mathsf{v}_\parallel^*)&=K(\bm{\mathsf{X}},\mathsf{v}_\parallel^*(R_o B^\phi/B))\nonumber\\
&\quad- m v_\parallel^* \bm{b}\cdot\frac{\bm{E}\times R_o\nabla\phi}{B}+O(\epsilon).\end{aligned}$$ For guiding centers with $E\times B$ speed that is comparable to the thermal speed, the leading-order toroidally-regularized guiding center kinetic energy is given by $$\begin{aligned}
K^*_{\text{gc}}(\bm{\mathsf{X}},\mathsf{v}_\parallel^*)=&\frac{1}{2}m\frac{(R_o B^\phi)^2}{B^2}\mathsf{v}_\parallel^{*2}+\mu B\nonumber\\
&-\frac{1}{2}m \frac{|\bm{E}_\perp|^2}{B^2}- m v_\parallel^* \bm{b}\cdot\frac{\bm{E}\times R_o\nabla\phi}{B}.\end{aligned}$$
The Euler-Lagrange equations associated with the toroidally-regularized Lagrangian $\ell^*$ are given by
\[eq:regularized\_gc\] $$\begin{aligned}
\dot{\mathsf{v}}_\parallel^*&=\frac{e}{m}\frac{\bm{B}\cdot \bm{E}^*}{R_o B^\phi}\\
\dot{\bm{\mathsf{X}}}&=\frac{\bm{B}}{R_o B^\phi}\frac{\partial H^*/\partial \mathsf{v}_\parallel^*}{m}+\frac{\bm{E}^*\times R_o\nabla\phi}{R_oB^\phi},\end{aligned}$$
where the effective electric field is given by $\bm{E}^*=\bm{E}-e^{-1}\nabla K^*$. By the assumption , these equtions of motion are free of singularities. We have therefore succeeded in eliminating the infinities present in the standard variational guiding center equations of motion. Moreover, the $v_\parallel$-dependent $B_\parallel^*$ denominators have been replaced with $\mathsf{v}_\parallel^*$-*independent* denominators $R_o B^\phi$. This significantly simplifies the process of computing the current density generated by a distribution of gyrocenters in variational gyrokinetics and drift kinetics. In contrast, the regularization proposed in Ref. retains $v_\parallel$ dependence in the denominators. It is also instructive to compare what we have done here with Ref., specifically the discussion surrounding Eqs.(42) and (94) therein. There it is explained that in a low beta and large aspect ratio tokamak it is justifiable to replace $\bm{b}$ with $R\nabla\phi$. This enables one to introduce a transformation akin to that eliminates $v_\parallel$-dependent denominators from the gyrocenter equations of motion. Therefore toroidal regularization may be viewed as a generalization of the ideas in Ref. that allows for high-beta, arbitrary aspect ratio, fully-three-dimensional field configurations.
To verify the proposed transformation, Fig. \[fig:banana\_comparison\] demonstrates that the regularized Lagrangian recovers familiar guiding center dynamics. We solve both the standard guiding center equations (Eq.) and the regularized guiding center equations (Eq. ) for a trapped particle in an axisymmetric tokamak magnetic field. We use a system of toroidal coordinates $(r, \theta, \phi)$ and a magnetic field defined by the vector potential: [@Qin_2009] $$\begin{aligned}
\label{eq:axisymmetric_vector_potential}
\bm{A}(r, \theta, \phi) =& \frac{B_0 R_0}{\cos^2 \theta} \left( r \cos \theta - R_0 \log\left(1 + \frac{r \cos \theta}{R_0}{}\right) \right) \nabla \theta\nonumber\\
&\hspace{11em}- \frac{B_0 r^2}{2 q_0} \nabla \phi,\end{aligned}$$ where $B_0$ is a magnetic field amplitude and $R_0$ is the major radius. In this demonstration, both of the guiding center theories accurately represent the gyro-averaged particle motion.
![The toroidally-regularized guiding center equations accurately recover the familiar trapped particle “banana orbit". Conditions: 2 keV proton, $B_0 = 1 $T, $R_0 = 100$cm, $q_0 = \sqrt{2}$, $(\bm{\mathsf{X}}, v_\parallel)$ = (5 cm, 0, 0, -12.9 cm/$\mu$s), $h= 0.3 \mu$s (100x smaller for full orbit).[]{data-label="fig:banana_comparison"}](banana_comparison.pdf){width="45.00000%"}
The successful elimination of the singular behavior is highlighted in Fig. \[fig:singular\_behavior\], where large-$v_\parallel$ trajectories are initialized near the $B_\parallel^*=0$ singularity. The toroidally-regularized Lagrangian produces a smooth trajectory that remains in good agreement with the full orbit calculation despite violating the guiding center ordering assumptions. Meanwhile, the trajectory generated by the conventional equations discontinuously leaps onto a different — and more energetic — trajectory upon encountering the singularity.
![The large-$v_\parallel$ singularity is manifest in a passing-particle trajectory generated by the standard guiding center equations. Same conditions as Fig. \[fig:banana\_comparison\] except $q_0 = 0.1$, $v_\parallel=-600$cm/$\mu$s and timesteps reduced by a factor of ten.[]{data-label="fig:singular_behavior"}](singular_trajectory.pdf){width="45.00000%"}
Aside from the elimination of infinities, a significant benefit of toroidal regularization is that it greatly simplifies the identitification of canonical coordinates. Continuing with the $(r, \theta, \phi)$ toroidal coordinates (not necessarily field aligned) choose a gauge where $\bm{A}=A_r\nabla r+A_\theta\nabla\theta+A_\phi\nabla\phi$ satisfies $A_r=0$. In this gauge the Lagrangian $\ell^*$ becomes $$\begin{aligned}
\label{eq:l_gauge}
\ell^*(\bm{\mathsf{X}},\mathsf{v}_\parallel^*,\dot{\bm{\mathsf{X}}},\dot{\mathsf{v}}_\parallel^*)=& e\bar{A}_\theta\dot{\theta}+(e\bar{A}_\phi+m\mathsf{v}_\parallel^* R_o)\dot{\phi}\nonumber\\
&\hspace{2em}-H^*(r,\phi,\theta,\mathsf{v}_\parallel^*)+O(\epsilon).\end{aligned}$$ A viable set of canonical coordinates for guiding center motion, even in time-dependent fields, is therefore $(\theta,\phi,p_\theta,p_\phi)$, where $$\begin{aligned}
p_\theta&=e \bar{A}_\theta\\
p_\phi&=e\bar{A}_\phi+m \mathsf{v}_\parallel^* R_o.\label{p_phi_eqn}\end{aligned}$$ We have included the overbars here to emphasize the requirement of choosing the gauge $A_r=0$. Toroidal regularization also does not eliminate the usual benefits associated with Lagrangian guiding center theory. If the electromagnetic field is axisymmetric, and if the toroidal angle $\phi$ is assumed to be the symmetry angle, then the quantity $p_\phi$ in is conserved exactly. If the electromagnetic field is time-independent, the Hamiltonian $H=e\varphi+K$ is conserved exactly. Finally, phase space volume computed using the Liouville volume element $$\begin{aligned}
\Omega_L=e m R_0 B^\phi d^3\bm{\mathsf{X}}\,d\mathsf{v}_\parallel^*\end{aligned}$$ is conserved. Note that $\Omega_L$ is free of $\mathsf{v}_\parallel^*$-depedendence.
Although symplectic integration of the regularized guiding center equations is facilitated by the preceding identification of canonical coordinates, it is simpler and computationally more efficient to directly advance the non-canonical coordinates. Toward that end, we turn now to the construction of a non-canonical symplectic integrator using the recently developed technique of “degenerate variational integration.”[@Ellison_thesis; @Ellison_2017] Degenerate variational integrators, or DVIs, were developed to remedy the numerical instabilities discovered in the initial (non-degenerate) variational guiding center integrators.[@Qin_2009; @Li_2011; @Ellison_2015_PPCF] Whereas it is only known how to construct a DVI for conventional guiding center dynamics under a restricted set of magnetic coordinates/magnetic field configurations, [@Ellison_thesis; @Ellison_2017] the toroidally regularized Lagrangian is amenable to the method with no restrictions beyond the $A_r = 0$ gauge transformation.
To construct a DVI for the regularized system, begin with the Lagrangian in Eq. . Further, for notational compactness, let $u = \mathsf{v}_\parallel^*$ and $\bm{A}^*({\bm{\mathsf{X}}}, u) = \bar{\bm{A}} + m u R_0 \nabla \phi$. Next, choose a discrete Lagrangian according to $$\begin{aligned}
& \ell_{\text{d}}({\bm{\mathsf{X}}}_{k}, u_k, {\bm{\mathsf{X}}}_{k+1}, u_{k+1}) = \nonumber\\
& \hspace{0.5em} e \bm{A}^*({\bm{\mathsf{X}}}_{k+1}, u_{k+1}) \cdot \frac{{\bm{\mathsf{X}}}_{k+1} - {\bm{\mathsf{X}}}_{k}}{h} - H({\bm{\mathsf{X}}}_{k+1}, u_{k+1}), \end{aligned}$$ where $h$ is the numerical step size. The discrete action corresponding to this choice is $$\begin{aligned}
&S_d({\bm{\mathsf{X}}}_0, u_0, {\bm{\mathsf{X}}}_1, u_1, ..., {\bm{\mathsf{X}}}_N, u_N) = \nonumber\\
&\hspace{7em}\sum_{k=0}^{N-1} h \ell_d({\bm{\mathsf{X}}}_k, u_k, {\bm{\mathsf{X}}}_{k+1}, u_{k+1}).\end{aligned}$$ A variational integrator is obtained by requiring the variation of the discrete action with respect to each of the four coordinate functions $({\bm{\mathsf{X}}}, u)$ to be zero for all $k=1, ..., N-1$. The resulting *discrete Euler-Lagrange equations* are given by:
\[eq:del\] $$\begin{aligned}
&\nabla \bm{A}^*({\bm{\mathsf{X}}}_k, u_k) \cdot({\bm{\mathsf{X}}}_k - {\bm{\mathsf{X}}}_{k-1}) - \bm{A}^*({\bm{\mathsf{X}}}_{k+1}, u_{k+1}) \nonumber\\
&\hspace{6em}+ \bm{A}^*({\bm{\mathsf{X}}}_k, u_k) - h \nabla H({\bm{\mathsf{X}}}_k, u_k) = 0 \label{eq:del_x} \\
&\nabla_u \bm{A}^*({\bm{\mathsf{X}}}_k, u_k) \cdot({\bm{\mathsf{X}}}_k - {\bm{\mathsf{X}}}_{k-1})\nonumber\\
&\hspace{11.5em} - h \nabla_u H({\bm{\mathsf{X}}}_k, u_k) = 0. \label{eq:del_u}\end{aligned}$$
At first glance, this algorithm appears to be a multistep method, requiring specification of ${\bm{\mathsf{X}}}, u$ at two instances in time before the time advance may be iterated. The crux of the DVI method is, however, that it avoids this multistep character. It is in fact possible to rearrange these equations into a one-step method as follows. Because $\bm{A}^*$ has only two non-zero components (namely, the $\theta$ and $\phi$ components), variables at time $t_{k+1}$ only appear in two components of Eq. . The procedure for constructing a one-step method involves eliminating the $t_{k-1}$ dependence in these two equations. Let $$\Delta = {\left(\begin{array}{c}\Delta^\theta \\ \Delta^\phi \end{array} \right)} = {\left(\begin{array}{c}\theta_k - \theta_{k-1} \\ \phi_k - \phi_{k-1} \end{array} \right)}.$$ Then by Eq. , $\Delta$ satisfies $${\left(\begin{array}{cc} e A_{\theta,r}({\bm{\mathsf{X}}}_k) & e A_{\phi,r}({\bm{\mathsf{X}}}_k) \\ 0 & m R_0 \end{array} \right)} {\left(\begin{array}{c}\Delta^\theta \\ \Delta^\phi \end{array} \right)} = {\left(\begin{array}{c}h H^*_{,r}({\bm{\mathsf{X}}}_k, u_k) \\ h H^*_{,u}({\bm{\mathsf{X}}}_k, u_k) \end{array} \right)}.$$ Pertinently, we can eliminate the $t_{k-1}$ dependence in Eq. by expressing $\Delta$ as a function of the variables at time $t_k$. The one-step DVI, advancing $({\bm{\mathsf{X}}}_k, u_k)$ to $({\bm{\mathsf{X}}}_{k+1}, u_{k+1})$, is then given by:
$$\begin{aligned}
& e A_{\theta, r}({\bm{\mathsf{X}}}_{k+1})\left(\theta_{k+1} - \theta_{k} \right) + e A_{\phi, r}({\bm{\mathsf{X}}}_{k+1})\left(\phi_{k+1} - \phi_{k} \right) \nonumber\\
&\hspace{10em}- h H^*_{,r}({\bm{\mathsf{X}}}_{k+1}, u_{k+1}) = 0 \label{eq:onestepdel_r}\\
&e A_{\theta, \theta}({\bm{\mathsf{X}}}_{k})\Delta^{\theta} + e A_{\phi, \theta}({\bm{\mathsf{X}}}_{k})\Delta^\phi + eA_\theta({\bm{\mathsf{X}}}_k)\nonumber\\
&\hspace{6em} - e A_\theta({\bm{\mathsf{X}}}_{k+1}) - h H^*_{,\theta}({\bm{\mathsf{X}}}_k, u_k) = 0 \label{eq:onestepdel_theta} \\
&e A_{\theta, \phi}({\bm{\mathsf{X}}}_{k})\Delta^\theta + e A_{\phi, \phi}({\bm{\mathsf{X}}}_{k})\Delta^\phi +
eA^*_\phi({\bm{\mathsf{X}}}_k, u_k)\nonumber\\
&\hspace{4em} - e A^*_\phi({\bm{\mathsf{X}}}_{k+1}, u_{k+1}) - h H^*_{,\phi}({\bm{\mathsf{X}}}_k, u_k) = 0 \label{eq:onestepdel_phi}\\
&mR_0 \left(\phi_{k+1} - \phi_{k} \right) - h H^*_{,u}({\bm{\mathsf{X}}}_{k+1}, u_{k+1}) = 0. \label{eq:onestepdel_u}\end{aligned}$$
For time-dependent fields, the algorithm is unchanged except the field evaluations become, e.g., $A({\bm{\mathsf{X}}}_k) \mapsto A({\bm{\mathsf{X}}}_k, t_k)$.
The DVI possesses, by construction, desirable conservation properties. For one, it can be observed in Eq. that the scheme exactly preserves the regularized version of the canonical toroidal momentum whenever toroidal symmetry is present. Additionally, the variational formulation of the algorithm implies that it preserves a symplectic two-form [@Marsden_2001] — a fundamental property of Hamiltonian systems. Variational integrators constructed in this way preserve a two-form that is nearby to the one preserved by the continuous system, approaching it as the numerical step size tends to zero. [@Ellison_thesis; @Ellison_2017] By preserving a symplectic two-form, the DVI retains the Hamiltonian character of the dynamics.
To illustrate the benefits of non-canonical symplectic integration of guiding center trajectories, the final numerical study evolves passing particle trajectories in a resonantly perturbed tokamak. The resonantly perturbed field is described by the magnetic vector potential: $$\label{eq:rmp_vector_potential}
\bm{A}(r, \theta, \phi) = \bm{A}_0 - \frac{B_0 r^2}{2 q_0} \sum_i \delta_i \sin(m_i \theta - n_i \phi) \nabla \phi,$$ where $A_0$ is the axisymmetric vector potential given in Eq. and $\delta_i$ is the size of the $i$’th resonant perturbation. In this example we consider two perturbative harmonics: an $m=3, n=2$ harmonic and an $m=7, n=5$ harmonic, both of amplitude $\delta = 4 \times 10^{-4}$. Figure \[fig:poincare\] depicts a contant-energy Poincaré section formed by intersecting the particle trajectories with a plane of constant toroidal angle $\phi$. In the unperturbed, axisymmetric limit, the particle trajectories reside on circular KAM tori analogous to magnetic flux surfaces. Hamiltonian theory — specifically, the KAM theorem — dictates that these KAM tori should persist throughout a majority of the phase space when small perturbations are introduced. Because the DVI retains the Hamiltonian character, its Poincaré section can be seen to manifest this behavior for indefinitely long times. The same cannot be expected of non-symplectic algorithms, which eventually lose the Hamiltonian character of the dynamics to dissipative truncation error.
![By retaining the Hamiltonian character of the dynamics, the symplectic DVI generates integrable and stochastic guiding center trajectories in the resonantly perturbed tokamak fields. Conditions: varying initial radii and poloidal angles; $q_0=1.35$; $\mathsf{v}_\parallel = 12.9$ cm/$\mu$s; zero magnetic moment; $h=3.5 \mu$s; $10^6$ steps taken.[]{data-label="fig:poincare"}](poinc_portrait_dvi.pdf){width="45.00000%"}
To summarize, we have removed the unphysical infinities from variational guiding center theory in a simple and physically-appealing manner. As a result, we were able to find a very simple structure-preserving integrator for guiding centers. This integrator is capable of handling guiding center motion in arbitrary electric and magnetic fields, even those with time dependence and without nested magnetic flux surfaces. While we have not provided the explicit expressions here, analogous results apply in the context of variational gyrocenter motion. Thus, our results should enable the development of structure-preserving integrators for (at least) electrostatic drift kinetics and gyrokinetics. In addition, we have presented empirical evidence that our regularized guiding center theory performs surprisingly well at large parallel velocities, accurately capturing the shape (although not the phase) of the true orbit. Because such large parallel velocities violate the guiding center ordering assumption, it would be interesting to understand the reason for this good behavior in the future.
This research was supported by the U. S. Department of Energy, Office of Science, Fusion Energy Sciences under Award No. DE-FG02-86ER53223 and the U.S. Department of Energy Fusion Energy Sciences Postdoctoral Research Program administered by the Oak Ridge Institute for Science and Education (ORISE) for the DOE. ORISE is managed by Oak Ridge Associated Universities (ORAU) under DOE contract number DE-AC05-06OR23100. All opinions expressed in this paper are the author’s and do not necessarily reflect the policies and views of DOE, ORAU, or ORISE. This work was also performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. LLNL-JRNL-737871-DRAFT
[^1]: Poloidal regularization is possible in reversed field pinch configurations, but is theoretically more cumbersome because the poloidal angle is not well-defined at the magnetic axis.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Let $A_n$ be the alternating group of even permutations of $X:=\{1,2,\cdots,n\}$ and ${\mathcal E}_n$ the set of even derangements on $X.$ Denote by $A\T_n^q$ the tensor product of $q$ copies of $A\T_n,$ where the Cayley graph $A\T_n:=\T(A_n,{\mathcal E}_n)$ is called the even derangement graph. In this paper, we intensively investigate the properties of $A\T_n^q$ including connectedness, diameter, independence number, clique number, chromatic number and the maximum-size independent sets of $A\T_n^q.$ By using the result on the maximum-size independent sets $A\T_n^q$, we completely determine the full automorphism groups of $A\T_n^q.$'
author:
- |
Yun-Ping Deng$^{1}$, Fu-Ji Xie$^2$ and Xiao-Dong Zhang$^{1,\dagger}$\
[$^1$Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, P.R.China]{}\
[$^2$ Antai College of Economics & Management Shanghai Jiao Tong University, Shanghai 200052, P.R.China]{}\
[Emails: [email protected], [email protected], [email protected]]{}\
title: ' Maximum-Size Independent Sets and Automorphism Groups of Tensor Powers of the Even Derangement Graphs [^1]'
---
\[section\] \[theorem\][Corollary]{} \[theorem\][Definition]{} \[theorem\][Conjecture]{} \[theorem\][Question]{} \[theorem\][Lemma]{} \[theorem\][Proposition]{} \[theorem\][Example]{} \[theorem\][Problem]{}
c Ø ø $H$ ¶[$P$ ]{} ł
[[**Key words:**]{} Automorphism group; Cayley graph; tensor product; maximum-size independent sets; alternating group. ]{}\
[[**AMS Classifications:**]{} 05C25, 05C69]{} 0.5cm
Introduction
============
For a simple graph $\T,$ we use $V(\T), E(\T)$ and $\Aut(\T)$ to denote its vertex set, edge set and full automorphism group, respectively. We denote by $N_{\T}(v)$ the neighbourhood of a vertex $v$ in $\T.$ Let be a finite group and $S$ a subset of not containing the identity element $1$ with $S=S^{-1}.$ The [*Cayley graph*]{} $\T:=\T(G,S)$ on with respect to $S$ is defined by $$V(\T){=}G,~E(\T){=}\{(g,sg): g{\in} G,\ s{\in} S\}.$$ Clearly, $\T$ is a $|S|$-regular and vertex-transitive graph, since $\Aut(\T)$ contains the right regular representation $R(G)$ of . Moreover, $\T$ is connected if and only if is generated by $S.$
Let $S_n$ be the [*symmetric group*]{} and $A_n$ the [*alternating group*]{} on $X=\{1,2,\cdots,n\}.$ Let ${\mathcal
D}_n:=\{\o\in S_n:x^{\,\sigma}\neq x,\forall x\in X\}$ and ${\mathcal E}_n:={\mathcal D}_n\cap A_n$ denote the degrangements and the even derangements on respectively. Then the graph $\T_n:=\T(S_n,{\mathcal D}_n)$ and $A\T_n:=\T(A_n,{\mathcal E}_n)$ are called the [*derangement graph*]{} [@Paul] and the [*even derangement graph*]{} on respectively.
The [*tensor product*]{} $\T_1\otimes\T_2$ of two graphs $\T_1$ and $\T_2$ is the graph with vertex set $V(\T_1)\times V(\T_2)$ and edge set consisting of those pairs of vertices $(u_1,u_2),\,(v_1,v_2)$ where $u_1$ is adjacent to $v_1$ in $\T_1$ and $u_2$ is adjacent to $v_2$ in $\T_2$. A [*projection*]{} is a homomorphism $pr_{i,n}:\T^q\rightarrow\T$ given by $pr_{i,n}(x_1,x_2,\cdots,x_q)=x_i$ for some $i,$ where $\T^q$ is the tensor product of $q$ copies of a graph $\T.$ By the definition of tensor product, it is easy to see that $A\T_n^q$ is the Cayley graph $\T(A_n^q,{\mathcal E}_n^q),$ where $A_n^q$ is the direct product of $q$ copies of $A_n$ and ${\mathcal
E}_n^q:=\{(\sigma_1,\sigma_2,\cdots,\sigma_q):\sigma_i\in{\mathcal
E}_n,i=1,2,\cdots,q\}.$
A family $I\subseteq S_n$ is [*intersecting*]{} if any two elements have at least one common entry. It is easy to see that an intersecting family of maximal size in $S_n$ corresponds to a maximum-size independent set in $\T_n.$ In [@Cameron], Cameron and Ku showed that the only intersecting families of maximal size in $S_n$ are the cosets of point stabilizers. In [@Ku], Ku and Wong proved that analogous results hold for the alternating group and the direct product of symmetric groups, which equivalently shows that the structure of maximum-size independent sets of $A\T_n$ is as follows:
(Theorem 1.2 in ${{\fs\cite{Ku}}}$)\[pr-1.1\] All the maximum-size independent sets of $A\T_n~(n\geq5)$ are $B_{i,j}=\{\sigma\in A_n:\,i^{\sigma}=j\},\,i,j=1,2,\cdots,n.$ In particular, each $|B_{i,j}|=\frac{(n-1)!}{2}.$
In this paper, we prove that the result analogous to [@Cameron] holds for the direct product of the alternating groups, which can be equivalently stated as follows:
\[dl-1.2\] All the maximum-size independent sets of $A\T_n^q~(q\geq1,n\geq5)$ are $$B_{i,j}^{(k)}=\{(\sigma_1,\sigma_2,\cdots,\sigma_q)\in A_n^q:
\,i^{\sigma_k}=j\},\,i,j=1,2,\cdots,n;\,k=1,2\cdots,q.$$ In particular, the independence number of $A\T_n^q$ is $$|B_{i,j}^{(k)}|=\frac{(n-1)!n!^{q-1}}{2^q}.$$
[**Remark.**]{} Generally speaking, for a graph $\T,$ all maximum-size independent sets of $\T^q$ are not necessarily preimages of maximum-size independent sets of $\T$ under projections (see [@Ku2; @Larose2]). Theorem \[dl-1.2\] shows that all maximum-size independent sets of $A\T_n^q$ are preimages of maximum-size independent sets of $A\T_n$ under projections.
Many researchers (see [@Cameron; @deza1977; @Eggleton1985; @Larose; @Paul; @Rasmussen]) have studied the properties of $\T_n,$ such as the clique number, the chromatic number, the independence number, maximum-size independent sets and so on. Motivated by the nice structures of $\T_n,$ here we show that $A\T_n^q$ have the similar nice structures. For example, we obtain that the diameter $D(A\T_n^q)=2,$ the clique number $\omega(A\T_n^q)=n$ and the chromatic number $\chi(A\T_n^q)=n.$
Cayley graphs are of general interest in the field of Algebraic Graph Theory due to their good properties, especially their high symmetry. One difficult problem in Algebraic Graph Theory is to determine the automorphism groups of Cayley graphs. Although there are some nice results on the automorphism groups of Cayley graphs (see [@Fang; @Feng2; @Feng1; @Godsil2; @Huan; @Xu; @Z; @Zhang]), we still lack enough understanding on them. In this paper, we completely determine the automorphism groups of $A\T_n^q,$ which in fact gives a kind of method on the computation of automorphism group of Cayley graph by using the characterization of the maximum-size independent sets. Another main result of this paper is as follows:
\[dl-1.3\] Define the mapping $\varphi_k:\,A_n^q\rightarrow A_n^q$ as $(\sigma_1,\cdots,\sigma_{k-1},\sigma_k,\sigma_{k+1},\cdots,\sigma_q)^{\varphi_k}=(\sigma_1,\cdots,\sigma_{k-1},\sigma_k^{-1},\sigma_{k+1},\cdots,\sigma_q)$ for $k=1,2,\cdots,q.$ For $q\geq1$ and $n\geq 5,$ $$\Aut(A\T_n^q)=(R(A_n^q)\rtimes (\Inn(S_n)\wr S_q))\rtimes Z_2^q,$$ where $\Inn(S_n)\,(\,\cong S_n)$ is the inner automorphism group of $S_n,$ $Z_2^q=\langle\varphi_1\rangle\times\langle\varphi_2\rangle\times\cdots\langle\varphi_q\rangle$ and $\,\Inn(S_n)\wr S_q$ denotes the wreath product of $\,\Inn(S_n)$ and $S_q.$
[**Remark.**]{} Sanders and George [@Sanders] showed that for a graph $\T,$ $\Aut(\T^2)\geq\Aut(\T)\wr S_2,$ where $\wr$ denotes the wreath product, however, the equality cannot hold in most situations. Theorem \[dl-1.3\] implies that $\Aut(A\T_n^q)=\Aut(A\T_n)\wr S_q.$
The rest part of this paper is organized as follows. In Section 2, we give the connectedness and diameter of $A\T_n^q.$ In Section 3, we determine the independence number and the structure of maximum-size independent sets of $A\T_n^q,$ as its corollary, we obtain the clique number and chromatic number of $A\T_n^q.$ In section 4, we completely determine the full automorphism groups of $A\T_n^q.$
The connectedness and diameter
==============================
In this section, we give the connectedness and diameter of $A\T_n^q.$
For a group $G,$ we denote the automorphism group and the inner automorphism group of $G$ by $\Aut(G)$ and $\Inn(G),$ respectively. Next we need the following known result:
${{\fs\cite{Suzuki}\,[III,\,(2.17)-(2.20)]}}$ \[pr-2.1\] If $n\geq2$ and $n\neq6,$ then $\Aut(A_n)=\Inn(S_n).$ If $n=6,$ then $|\Aut(A_6):\Inn(S_6)|=2,$ and for each $\alpha\in
\Aut(A_6){\setminus}\Inn(S_6),$ $\alpha$ maps a $3$-cycle to a product of two disjoint $3$-cycles.
\[yl-2.2\] If $n\geq 5,$ then the even derangement graph $A\T_n$ is connected.
By Theorem 2.8 of page 293 in [@Suzuki], the alternating group $A_n~(n\geq 5)$ is generated by the totality of $3$-cycles. Clearly $(1\,2\,3)=(1\,2\,\cdots\,n)^2\cdot(n\,n-1\,\cdots\,1)^2(1\,2\,3)$ and $(1\,2\,\cdots\,n)^2,\,(n\,n-1\,\cdots\,1)^2(1\,2\,3)\in
{\mathcal E}_n$ by $n\geq 5.$
For any $3$-cycle $(i\,j\,k),$ there exists a $\phi\in \Inn(S_n)$ such that $(1\,2\,3)^{\phi}=(i\,j\,k).$ By Proposition \[pr-2.1\], we have $\Aut(A_n,{\mathcal E}_n)=\{\phi\in
\Aut(A_n):\,{\mathcal E}_n^{\,\phi}={\mathcal E}_n\}=\Inn(S_n).$ Thus
$$(i\,j\,k)=(1\,2\,3)^{\phi}=[(1\,2\,\cdots\,n)^2]^{\phi}\cdot[(n\,n-1\,\cdots\,1)^2(1\,2\,3)]^{\phi}$$ and $$[(1\,2\,\cdots\,n)^2]^{\phi},\,[(n\,n-1\,\cdots\,1)^2(1\,2\,3)]^{\phi}
\in {\mathcal E}_n.$$ So the alternating group $A_n~(n\geq 5)$ is generated by ${\mathcal E}_n,$ which implies that $A\T_n$ is connected.
[**Remark.**]{} If $n=3,$ clearly $A_3=\lg {\mathcal E}_3\rg,$ so $A\T_3$ is connected. If $n=4,$ then $A_4\neq\lg {\mathcal
E}_4\rg=\{1,\,(1\,2)(3\,4),\,(1\,3)(2\,4),\,(1\,4)(2\,3)\},$ so $A\T_4$ is not connected.
${}^{{\fs\cite{Graham}}}$\[yl-2.3\] (i) The tensor product of two connected graphs is bipartite if and only if at least one of them is bipartite.
\(ii) The tensor product of two connected graphs is disconnected if and only if both factors are bipartite.
\[dl-2.4\] $A\T_n^q$ is connected and non-bipartite for any $q\geq 1$ and $n\geq 5.$
Since $A\T_n^q=\underbrace{A\T_n\otimes\cdots\otimes A\T_n}_q$ and $A\T_n$ is connected and non-bipartite for $n\geq 5$ by Lemma \[yl-2.2\], the assertion holds by Lemma \[yl-2.3\].
\[yl-2.5\] For any $g_1,g_2\in A_n~(n\geq 5),$ there exists a $g\in A_n$ such that $g\in N_{A\T_n}(g_1)\cap N_{A\T_n}(g_2).$
If $n=5,$ we have $$(a_1,a_2,a_3,a_4,a_5)=(a_1,a_4,a_2,a_5,a_3)^2,$$ $$(a_1,a_2)(a_3,a_4)=(a_5,a_1,a_3,a_2,a_4)(a_1,a_5,a_3,a_2,a_4),$$ $$(a_1,a_2,a_3)=(a_1,a_5,a_3,a_4,a_2)(a_1,a_3,a_5,a_2,a_4),$$ $$1=(a_1,a_2,a_3,a_4,a_5)(a_5,a_4,a_3,a_2,a_1),$$ that is, for any $x\in A_5,$ there exist $s_1,s_2\in {\mathcal E}_5$ such that $x=s_1s_2.$ Now for $x=g_1g_2^{-1},$ we have $g_1g_2^{-1}=s_1s_2,\,s_1,s_2\in {\mathcal E}_5,$ i.e. $g_1=s_1s_2g_2.$ Set $g:=s_2g_2.$ Clearly $g\in N_{A\T_5}(g_1)\cap
N_{A\T_5}(g_2).$
If $n\geq 6,$ by proposition 6 in [@Cameron], for any $g_1,g_2\in A_n,$ there exists a $g\in S_n$ such that $g\in
N_{\T_n}(g_1)\cap N_{\T_n}(g_2).$ That is, there exist $s_1,\,s_2\in
{\mathcal D}_n$ such that $g=s_1g_1=s_2g_2.$ If $g\in A_n,$ then $s_1,\,s_2\in {\mathcal E}_n,$ so $g\in N_{A\T_n}(g_1)\cap
N_{A\T_n}(g_2)$ and the assertion holds. If $g\in S_n\setminus A_n,$ then $s_1,\,s_2\in {\mathcal D}_n\setminus {\mathcal E}_n.$ For any $i\in X=\{1,2,\cdots,n\},$ select a $j\in\{i,i^{s_1},i^{s_2},i^{s_1^{-1}},i^{s_2^{-1}}\}\neq
\emptyset~(n\geq 6).$ Set $$g':=(i\,j)g,\,s_1':=(i\,j)s_1,\,s_2':=(i\,j)s_2.$$ Thus $g'=s_1'g_1=s_2'g_2$ and $s_1',s_2'\in {\mathcal E}_n$ by $j\in
X\setminus\{i,i^{s_1},i^{s_2},i^{s_1^{-1}},i^{s_2^{-1}}\}.$ Hence $g'\in N_{A\T_n}(g_1)\cap N_{A\T_n}(g_2)$ and the assertion holds.
\[dl-2.6\] If $n\geq5,$ then $diam(A\T_n^q)=2,$ where $diam(A\T_n^q)$ is the diameter of $A\T_n^q.$
For any $(\sigma_1,\sigma_2,\cdots,\sigma_q),(\tau_1,\tau_2,\cdots,\tau_q)\in
A_n^q,$ by Lemma \[yl-2.5\], there exist $\varsigma_i\in
A_n~(i=1,2,\cdots,q)$ such that $\varsigma_i\in
N_{A\T_n}(\sigma_i)\cap N_{A\T_n}(\tau_i).$ So there exists a $(\varsigma_1,\varsigma_2,\cdots,\varsigma_q)\in A_n^q$ such that $$(\varsigma_1,\varsigma_2,\cdots,\varsigma_q)\in
N_{A\T_n^q}((\sigma_1,\sigma_2,\cdots,\sigma_q))\cap
N_{A\T_n^q}((\tau_1,\tau_2,\cdots,\tau_q)),$$ which implies that any two vertices in $A\T_n^q$ have at least a common neighbourhood. Hence $diam(A\T_n^q)=2.$
The stucture of maximum-size independent sets
=============================================
In this section we characterize the structure of maximum-size independent sets of $A\T_n^q$ for $q\geq 1,$ which is a generalization of Theorem 1.2 in [@Ku]. First we give the independence number of $A\T_n^q$ as follows:
\[yl-3.1\] For any $q\geq 1,\,n\geq 5,$ the independence number of $A\T_n^q$ is given by $$\alpha(A\T_n^q)=\frac{(n-1)!n!^{q-1}}{2^q}.$$
By Proposition 1.3 in [@Alon], we have
$$\frac{\alpha(A\T_n^q)}{|A_n^q|}=\frac{\alpha(A\T_n)}{|A_n|}
\Rightarrow\alpha(A\T_n^q)=\frac{\alpha(A\T_n)\cdot|A_n^q|}{|A_n|}.$$ Then by Proposition \[pr-1.1\], we obtain $$\alpha(A\T_n^q)=\frac{\frac{(n-1)!}{2}\cdot(\frac{n!}{2})^q}{\frac{n!}{2}}
=\frac{(n-1)!n!^{q-1}}{2^q}.$$ Thus the assertion holds.
For any two graphs $H_1$ and $H_2,$ a map $\phi$ from $V(H_1)$ to $V(H_2)$ is [*homomorphism*]{} if $\{u^{\phi},v^{\phi}\}\in E(H_2)$ whenever $\{u,v\}\in E(H_1),$ i.e. $\phi$ is a edge-preserving map. Next we need the following fundamental result of Albertson and Collins [@Albertson] which is also called ’No-Homomorphism Lemma’.
${{\fs\cite{Albertson}}}$\[yl-3.2\] Let $H_1$ and $H_2$ be graphs such that $H_2$ is vertex transitive and there exists a homomorphism $\phi:~V(H_1)\rightarrow V(H_2).$ Then $$\label{2}
\frac{\alpha(H_1)}{|V(H_1)|}\geq\frac{\alpha(H_2)}{|V(H_2)|}$$ Furthermore, if equality holds in (\[2\]), then for any independent set $I$ of cardinality $\alpha(H_2)$ in $H_2$, $I^{{\phi}^{-1}}$ is an independent set of cardinality $\alpha(H_1)$ in $H_1.$
\[yl-3.3\] Let $H=(V_1,V_2,E)$ be a $d$-regular bipartite graph whose partition has the parts $V_1$ and $V_2$ with $|V_1|=|V_2|.$ If $H$ is connected, then $|S|<|N_H(S)|$ for any $S\subsetneq V_1,$ where $N_H(S)$ is the neighborhood of $S$ in $H.$
Let $T=N_H(S)$ and $E(S,T)=\{(s,t)\in E:s\in S,t\in T\}.$ Then $$d|S|=|E(S,T)|\leq |E(V_1,T)|=d|T|.$$ If $|S|=|T|,$ then $E(S,T)|=|E(V_1,T)|,$ i.e. any vertex $u\in S\cup T$ is not adjacent to any vertex $v\not\in S\cup T,$ which contradicts the connectedness of $H.$ Thus $|S|<|T|=|N_H(S)|.$
\[yl-3.4\] All the maximum-size independent sets of $A\T_n^2~(n\geq7)$ are $$B_{i,j}^{(k)}=\{(g_1,g_2)\in A_n^2:
\,i^{g_k}=j\},\,i,j=1,2,\cdots,n;\,k=1,2.$$
Set ${\mathcal B}=\{B_{i,j}^{(k)}:i,j=1,2,\cdots,n;\,k=1,2\}.$ Clearly $|B_{i,j}^{(k)}|=\frac{(n-1)!n!}{4},$ which is equal to $\alpha(A\T_n^2)$ by Lemma \[yl-3.1\]. That is, $B_{i,j}^{(k)}$ is a maximum-size independent set of $A\T_n^2.$ Next for any maximum-size independent set $I$ of $A\T_n^2,$ it suffices to show that $I\in{\mathcal B}.$
Define a homomorphism $\phi$ from $A\T_n$ to $A\T_n^2$ as $g^{\phi}=(g,g).$ Without loss of generality, we may assume that the identity $Id=(id,id)\in I.$ By Proposition \[pr-1.1\], Lemmas \[yl-3.1\] and \[yl-3.2\], $I^{{\phi}^{-1}}$ is a maximum independent set of $A\T_n.$ So $I^{{\phi}^{-1}}=\{g\in
A_n:~i_0^g=j_0\}$ for some $i_0,\,j_0$ by Proposition \[pr-1.1\]. Since $id=(Id)^{{\phi}^{-1}}\in I^{\phi^{-1}},$ $I^{\phi^{-1}}=\{g\in A_n:~i_0^g=i_0\}$ for some $i_0.$ Therefore $I\supseteq I_0:=(I^{{\phi}^{-1}})^{\phi}=\{(g,g)\in
A_n^2:~i_0^g=i_0\}.$ Next we shall show that $I\in{\mathcal B}$ by the following four Claims:\
[**Claim 1.**]{} For any $(g_1,g_2)\in I,$ either $i_0^{g_1}=i_0$ or $i_0^{g_2}=i_0.$
Suppose on the contrary that $i_0^{g_1}\neq i_0$ and $i_0^{g_2}\neq
i_0.$ By Lemma \[yl-2.5\], there exists a $g\in A_n$ such that $g\in N_{A\T_n}(g_1)\cap N_{A\T_n}(g_2).$ That is, there exist $s_1,s_2\in {\mathcal E}_n$ such that $g=s_1g_1=s_2g_2.$
If $i_0^g=i_0,$ then $(g,g)\in I_0\subseteq I$ and $\{(g,g),(g_1,g_2)\}\in E(A\T_n^2),$ which contradicts the fact that $(g_1,g_2)\in I$ and $I$ is an independent set.
If $i_0^g\neq i_0,$ select a $j\in X\setminus
\{i_0,i_0^{g^{-1}},i_0^{s_1},i_0^{s_2},
i_0^{g^{-1}s_1^{-1}},i_0^{g^{-1}s_2^{-1}}\}\neq\emptyset~(n\geq 7).$ Set $$g'=(i_0,i_0^{g^{-1}},j)g,\,s_1'=(i_0,i_0^{g^{-1}},j)s_1,\,s_2'=(i_0,i_0^{g^{-1}},j)s_2.$$ Thus $i_0^{g'}= i_0,\,g'=s_1'g_1=s_2'g_2$ and $s_1',s_2'\in
{\mathcal E}_n$ by $j\in X\setminus
\{i_0,i_0^{g^{-1}},i_0^{s_1},i_0^{s_2},
i_0^{g^{-1}s_1^{-1}},i_0^{g^{-1}s_2^{-1}}\}.$ So $(g',g')\in
I_0\subseteq I$ and$\{(g',g'),(g_1,g_2)\}\in E(A\T_n^2),$ which as above yields a contradiction.
Hence Claim 1 holds.\
Set $$J_0=\{(g_1,g_2)\in A_n^2:i_0^{g_1}=i_0~and~i_0^{g_2}=i_0\},$$ $$J_1=\{(g_1,g_2)\in A_n^2:i_0^{g_1}=i_0~and~i_0^{g_2}\neq i_0\},$$ $$J_2=\{(g_1,g_2)\in A_n^2:i_0^{g_1}\neq i_0~and~i_0^{g_2}=i_0\}.$$
Clearly $|J_0|=\frac{(n-1)!^2}{4},|J_1|=|J_2|=\frac{(n-1)(n-1)!^2}{4}.$\
[**Claim 2.**]{} $A\T_n^2[J_1\cup J_2]$ is connected, where $A\T_n^2[J_1\cup J_2]$ denote the induced subgraph of $A\T_n^2$ by $J_1\cup J_2.$
For any $(\sigma_1,\sigma_2),(\tau_1,\tau_2)\in J_1,$ clearly they are not adjacent in $A\T_n^2.$ By Theorem \[dl-2.6\], there exists a $(\varsigma_1,\varsigma_2)\in A_n^2$ such that $\{(\sigma_1,\sigma_2),(\varsigma_1,\varsigma_2)\}$ and $\{(\tau_1,\tau_2),(\varsigma_1,\varsigma_2)\}\in E(A\T_n^2).$ That is, there exist $s_1,s_2,t_1,t_2\in {\mathcal E}_n$ such that $\varsigma_1=s_1\sigma_1=t_1\tau_1,\,\varsigma_2=s_2\sigma_2=t_2\tau_2.$ Clearly $i_0^{\varsigma_1}=i_0^{s_1\sigma_1}\neq i_0.$
If $i_0^{\varsigma_2}=i_0,$ then $(\varsigma_1,\varsigma_2)\in J_2.$
If $i_0^{\varsigma_2}\neq i_0,$ then select a $j\in X\setminus
\{i_0,i_0^{\varsigma_2^{-1}},i_0^{s_2},i_0^{t_2},
i_0^{\varsigma_2^{-1}s_2^{-1}},i_0^{\varsigma_2^{-1}t_2^{-1}}\}\neq\emptyset~(n\geq
7).$ Set $$\varsigma_2'=(i_0,i_0^{\varsigma_2^{-1}},j)\varsigma_2,\,s_2'=(i_0,i_0^{\varsigma_2^{-1}},j)s_2,\,t_2'=(i_0,i_0^{\varsigma_2^{-1}},j)t_2.$$ Thus and $i_0^{\varsigma_2'}=
i_0,\,\varsigma_2'=s_2'\sigma_2=t_2'\tau_2$ and $s_2',t_2'\in
{\mathcal E}_n$ by $j\in X\setminus
\{i_0,i_0^{\varsigma_2^{-1}},i_0^{s_2},i_0^{t_2},
i_0^{\varsigma_2^{-1}s_2^{-1}},i_0^{\varsigma_2^{-1}t_2^{-1}}\}.$ So $(\varsigma_1,\varsigma_2')\in J_2$ and $\{(\sigma_1,\sigma_2),(\varsigma_1,\varsigma_2')\},\{(\tau_1,\tau_2),(\varsigma_1,\varsigma_2')\}\in
E(A\T_n^2[J_1\cup J_2]).$
Similarly, for any $(\sigma_1,\sigma_2),(\tau_1,\tau_2)\in J_2,$ their exists $(\varsigma_1,\varsigma_2)\in J_1$ such that $\{(\sigma_1,\sigma_2),(\varsigma_1,\varsigma_2)\},\\
\{(\tau_1,\tau_2),(\varsigma_1,\varsigma_2)\}\in
E(A\T_n^2[J_1\cup J_2]).$
Hence Claim 2 holds.\
[**Claim 3.**]{} Either $I\cap J_1=\emptyset$ or $I\cap
J_2=\emptyset.$
Suppose on the contrary that $I\cap J_1\neq\emptyset$ and $I\cap
J_2\neq\emptyset,$ consider the following two possible cases:
[**Case 1.**]{} $I\cap J_1=J_1$ or $I\cap J_2=J_2.$
Since $I\cap(J_1\cup J_2)$ is an independent set, this case cannot happen.
[**Case 2.**]{} $I\cap J_1\subsetneq J_1$ and $I\cap J_2\subsetneq
J_2.$
It is easy to see that $A\T_n^2[J_1\cup J_2]$ is a regular bipartite graph whose partition has the parts $J_1$ and $J_2$ with $|J_1|=|J_2|.$ By Claim 2 and Lemma \[yl-3.3\], we have $$|I\cap J_1|<|N_{A\T_n^2[J_1\cup J_2]}(I\cap J_1)|.$$ Since $I\cap(J_1\cup J_2)$ is an independent set, we have $$\begin{aligned}
&&I\cap J_2\subseteq J_2\setminus N_{A\T_n^2[J_1\cup J_2]}(I\cap J_1)\\
&\Rightarrow&|N_{A\T_n^2[J_1\cup J_2]}(I\cap J_1)|+|I\cap J_2|\leq
|J_2|\\
&\Rightarrow&|I\cap J_1|+|I\cap J_2|<|J_2|\end{aligned}$$ By Claim 1, $I=\bigcup_{i=0}^2(I\cap J_i).$ Since $J_i~(i=0,1,2)$ are pairwise disjoint, we have $$\begin{aligned}
|I|&=&|I\cap J_0|+|I\cap J_1|+|I\cap J_2|\\
&<&|J_0|+|J_2|\\
&=&\frac{(n-1)!^2}{4}+\frac{(n-1)(n-1)!^2}{4}\\
&=&\frac{(n-1)!n!}{4}\end{aligned}$$ which is a contradiction, since $|I|=\frac{(n-1)!n!}{4}$ by Lemma \[yl-3.1\]. Hence Claim 3 holds.\
[**Claim 4.**]{} Either $I=J_0\cup J_1$ or $I=J_0\cup J_2.$
By Claim 3, either $I\cap J_1=\emptyset$ or $I\cap J_2=\emptyset.$ If $I\cap J_1=\emptyset,$ then we have $$\begin{aligned}
&&\frac{(n-1)!n!}{4}=|I|=|I\cap J_0|+|I\cap J_2|\leq |J_0|+|J_2|=\frac{(n-1)!n!}{4}\\
&\Rightarrow&I\cap J_0=J_0,\,I\cap J_2=J_2.\end{aligned}$$
So $I=\bigcup_{i=0}^2(I\cap J_i)=J_0\cup J_2.$
Similarly, if $I\cap J_2=\emptyset,$ then $I=J_0\cup J_1.$ Hence Claim 4 holds.\
By Claim 4, we have $I\in B,$ which conclude the proof.
${{\fs\cite{Alon}}}$\[yl-3.5\] Let $\T$ be a connected d-regular graph on n vertices and let $d=\mu_1\geq\mu_2\geq\cdots\mu_n$ be the eigenvalues of the adjacency matrix of $\T.$ If $$\frac{\alpha(\T)}{n}=\frac{-\mu_n}{d-\mu_n},$$ then for every integer $q\geq1,$ $$\frac{\alpha(\T^q)}{n^q}=\frac{-\mu_n}{d-\mu_n}.$$ Moreover, if $\T$ is also non-bipartite, and if $I$ is an independent set of size $\frac{-\mu_n}{d-\mu_n}n^q$ in $\T^q,$ then there exists a coordinate $i\in\{1,2,\cdots,q\}$ and a maximum-size independent set $J$ in $\T,$ such that $$I=\{(v_1,\cdots,v_q\in V(\T^q):v\in J\}.$$
${{\fs\cite{Ku2}}}$\[yl-3.6\] Let $\T$ be a connected, non-bipartite vertex-transitive graph. Suppose that the only independent sets of maximal cardinality in $H^2$ are the preimages of the independent sets of maximal cardinality in $\T$ under projections. Then the same holds for all powers of $\T.$
(of Theorem \[dl-1.2\]). For $n=5,6,$ it is easy to see that $A\T_n$ is connected, non-bipartite and $e(n)$-regular graph with $e(5)=24$ and $e(6)=130.$ Moreover, a [**Matlab**]{} computation shows that the least eigenvalue of the adjacency matrix of $A\T_5$ and $A\T_6$ are $-6$ and $-26,$ respectively. Thus the assertion holds by Lemmas \[yl-3.1\] and \[yl-3.5\].
For $n\geq 7,$ combining Proposition \[pr-1.1\], Lemma \[yl-3.4\] and \[yl-3.6\], the assertion holds.
\[co-3.7\] Let $\omega(A\T_n^q)$ and $\chi(A\T_n^q)$ denote the clique number and chromatic number of $A\T_n^q~(n\geq 5).$ Then we have $$\omega(A\T_n^q)=\chi(A\T_n^q)=n.$$
By [@Ku], we have $\omega(A\T_n)=n.$ Let $\{\sigma_1,\sigma_2,\cdots,\sigma_n\}$ be a clique of $A\T_n.$ Then clearly $\{(\sigma_1,\sigma_1,\cdots,\sigma_1),(\sigma_2,\sigma_2,\cdots,\sigma_2),\cdots,(\sigma_n,\sigma_n,\cdots,\sigma_n)\}$ is a clique of $A\T_n^q.$ So we have $\omega(A\T_n^q)\geq n.$ On the other hand, by Theorem \[dl-1.2\], we know that the independence number $\alpha(A\T_n^q)=\frac{(n-1)!n!^{q-1}}{2^q}.$ By Corollary 4 in [@Cameron], we have $\omega(A\T_n^q)\alpha(A\T_n^q)\leq
|V(A\T_n^q)|,$ that is $\omega(A\T_n^q)\cdot\frac{(n-1)!n!^{q-1}}{2^q}\leq
\frac{n!^q}{2^q},$ so $\omega(A\T_n^q)\leq n.$ Thus $\omega(A\T_n^q)=n.$
In addition, by Corollary 6.1.3 in [@Godsil], for any Cayley graph $\T:=\T(G,S),$ if $S$ is closed under conjugation and $\alpha(\T)\omega(\T)=|V(\T)|,$ then $\chi(\T)=\omega(\T).$ Note that for $A\T_n^q=\T(A_n^q,{\mathcal E}_n^q),\,{\mathcal E}_n^q$ is closed under conjugation and $\alpha(A\T_n^q)\omega(A\T_n^q)=|V(A\T_n^q)|.$ Hence $\chi(A\T_n^q)=\omega(A\T_n^q)=n.$
The automorphism group of $A\T_n^q$
===================================
In this section, we completely determine the full automorphism group of $A\T_n^q~(n\geq 5).$ First we introduce some definitions. Let $\Sym(\Omega)$ denote the set of all permutations of a set $\Omega.$ A [*permutation representation*]{} of a group is a homomorphism from into $\Sym(\Omega)$ for some set $\Omega.$ A permutation representation is also referred to as an action of on the set $\Omega,$ in which case we say that acts on $\Omega.$ Furthermore, if $\{g\in G:x^g=x,\,\forall x\in \Omega\}=1,$ we say the action of $G$ on $\Omega$ is [*faithful*]{}, or $G$ acts [*faithfully*]{} on $\Omega.$
Next we need the following known results:
${{\fs\cite{Johnson}}}$ \[pr-4.1\] Let $G^q=G\times G\times\cdots\times G$ be the external direct product of $q$ copies of the nontrivial group $G.$ If $G$ has the following properties:
\(i) the center $Z(G)$ of $G$ is trivial;
\(ii) G cannot be decomposed as a nontrivial direct product.\
Then $\Aut(G^q)=\Aut(G)\wr S_q.$
${{\fs\cite{Godsil3}}}$ \[pr-4.2\] Let $N_{Aut(\T(G,S)}(R(G))$ be the normalizer of $R(G)$ in $\Aut(\T(G,S)).$ Then $$N_{Aut(\T(G,S)}(R(G))=R(G)\rtimes \Aut(G,S)\leq \Aut(\T(G,S)),$$ where $\Aut(G,S)=\{\phi\in \Aut(G):\,S^{\phi}=S\}.$
\[yl-4.3\] Define the mapping $\varphi_k:\,A_n^q\rightarrow A_n^q$ as $(\sigma_1,\cdots,\sigma_{k-1},\sigma_k,\sigma_{k+1},\cdots,\sigma_q)^{\varphi_k}=(\sigma_1,\cdots,\sigma_{k-1},\sigma_k^{-1},
\sigma_{k+1},\cdots,\sigma_q)$ for $k=1,2,\cdots,q.$ For $n\geq 5,$ $$(R(A_n^q)\rtimes
(\Inn(S_n)\wr S_q))\rtimes Z_2^q\leq \Aut(A\T_n^q),$$ where $\Inn(S_n)~\cong S_n$ and $Z_2^q=\langle\varphi_1\rangle\times\langle\varphi_2\rangle\times\cdots\langle\varphi_q\rangle.$ In particular, $|\Aut(A\T_n^q)|\geq |(R(A_n^q)\rtimes (\Inn(S_n)\wr
S_q))\rtimes Z_2^q|=q!n!^{2q}.$
By Proposition \[pr-2.1\] and \[pr-4.1\], we have $$\begin{aligned}
\Aut(A_n^q,{\mathcal E}_n^q)&=&\{\phi\in \Aut(A_n^q):\,({\mathcal E}_n^q)^{\,\phi}={\mathcal E}_n^q\}\\
&=&\{\phi\in \Aut(A_n)\wr Sq:\,({\mathcal E}_n^q)^{\,\phi}={\mathcal E}_n^q\}\\
&=&\Inn(S_n)\wr Sq.\end{aligned}$$
Using Proposition \[pr-4.2\], we obtain $R(A_n^q)\rtimes
(\Inn(S_n)\wr S_q)\leq \Aut(A\T_n^q).$
Next we show that $\varphi_k$ is an automorphism of $A\T_n^q.$ $$\begin{aligned}
&&\{(\sigma_1,\cdots,\sigma_k,\cdots,\sigma_q),(\tau_1,\cdots,\tau_k,\cdots,\tau_q)\}\in E(A\T_n^q)\\
&\Leftrightarrow&\forall~i\in\{1,2,\cdots,n\},\forall~k\in\{1,2,\cdots,q\},\,i^{\sigma_k}\neq i^{\tau_k}\\
&\Leftrightarrow&\forall~i\in\{1,2,\cdots,n\},\forall~k\in\{1,2,\cdots,q\},\,(i^{{\sigma_k}^{-1}})^{\sigma_k}\neq (i^{{\sigma_k}^{-1}})^{\tau_k}\\
&\Leftrightarrow&\forall~i\in\{1,2,\cdots,n\},\forall~k\in\{1,2,\cdots,q\},\,i\neq i^{{\sigma_k}^{-1}\tau_k}\\
&\Leftrightarrow&\forall~i\in\{1,2,\cdots,n\},\forall~k\in\{1,2,\cdots,q\},\,i^{{\tau_k}^{-1}}\neq i^{{\sigma_k}^{-1}}\\
&\Leftrightarrow&\{(\sigma_1,\cdots,\sigma_k^{-1},\cdots,\sigma_q),(\tau_1,\cdots,\tau_k^{-1},\cdots,\tau_q)\}\in E(A\T_n^q)\\
&\Leftrightarrow&\{(\sigma_1,\cdots,\sigma_k,\cdots,\sigma_q)^{\varphi_k},(\tau_1,\cdots,\tau_k,\cdots,\tau_q)^{\varphi_k}\}\in
E(A\T_n^q).\end{aligned}$$ It is easy to see that $\varphi_k\not\in R(A_n^q)$ and $\varphi_k\not\in \Inn(S_n)\wr S_q.$ Hence $$(R(A_n^q)\rtimes
(\Inn(S_n)\wr S_q))\rtimes Z_2^q\leq \Aut(A\T_n^q),$$ where $Z_2^q=\langle\varphi_1\rangle\times\langle\varphi_2\rangle\times\cdots\langle\varphi_q\rangle.$ The assertion holds.
\[yl-4.4\] Let ${\mathcal
B}=\{B_{i,j}^{(k)},\,i,j=1,2,\cdots,n;\,k=1,2\cdots,q\}$, where $B_{i,j}^{(k)}=\{(\sigma_1,\sigma_2,\cdots,\sigma_q)\in
A_n^q:\,i^{\sigma_k}=j\}.$ Then the action of $\Aut(A\T_n^q)$ on ${\mathcal B}$ can be induced by the natural action of $\Aut(A\T_n^q)$ on $A_n^q$, and is faithful. Furthermore, any $\phi\in \Aut(A\T_n^q)$ is a permutation of ${\mathcal B}.$
Obviously, any $\phi\in\Aut(A\T_n^q)$ maps a maximum-size independent set of $A\T_n^q$ to a maximum-size independent set of $A\T_n^q.$ So by Theorem \[dl-1.2\], for any $B_{i,j}^{(k)}\in
{\mathcal B}$ and $\phi\in\Aut(\T_n^q),$ we have ${B_{i,j}^{(k)}}^{\,\phi}\in {\mathcal B}.$
Next we show that if $\phi\in\Aut(A\T_n^q)$ satisfies ${B_{i,j}^{(k)}}^{\,\phi}=B_{i,j}^{(k)}$ for each $B_{i,j}^{(k)}\in
{\mathcal B},$ then $\phi$ is the identity map. In fact, clearly, $$\forall\,(\sigma_1,\sigma_2,\cdots,\sigma_q)\in A_n^q,\,\{(\sigma_1,\sigma_2,\cdots,\sigma_q)\}=\bigcap_{k=1}^q\bigcap_{i=1}^n
B_{i,i^{\,\sigma_k}}^{(k)}.$$ So $$\begin{aligned}
\{(\sigma_1,\sigma_2,\cdots,\sigma_q)^{\,\phi}\}&=&(\bigcap_{k=1}^q\bigcap_{i=1}^n B_{i,i^{\,\sigma_k}}^{(k)})^{\,\phi}\\
&\subseteq& \bigcap_{k=1}^q\bigcap_{i=1}^n {B_{i,i^{\,\sigma_k}}^{(k)}}^{\phi}\\
&=&\bigcap_{k=1}^q\bigcap_{i=1}^nB_{i,i^{\,\sigma_k}}^{(k)}\\
&=&\{(\sigma_1,\sigma_2,\cdots,\sigma_q)\}.\end{aligned}$$ Thus $\phi$ is the identity map.
For any $B_{i,j}^{(k)},\,B_{i^{'},j^{'}}^{(k^{'})}\in {\mathcal B}$ and $\phi\in \Aut(A\T_n^q),$ we have $$\begin{aligned}
B_{i,j}^{(k)}\neq B_{i^{'},j^{'}}^{(k^{'})}&\Leftrightarrow&|B_{i,j}^{(k)}\cup B_{i^{'},j^{'}}^{(k^{'})}|>\frac{(n-1)!n!^{q-1}}{2^q}\\
&\Leftrightarrow&|(B_{i,j}^{(k)}\cup B_{i^{'},j^{'}}^{(k^{'})})^{\phi}|>\frac{(n-1)!n!^{q-1}}{2^q}\\
&\Leftrightarrow&|{B_{i,j}^{(k)}}^{\,\phi}\cup {B_{i^{'},j^{'}}^{(k^{'})}}^{\phi}|>\frac{(n-1)!n!^{q-1}}{2^q}\\
&\Leftrightarrow&{B_{i,j}^{(k)}}^{\,\phi}\neq
{B_{i^{'},j^{'}}^{(k^{'})}}^{\phi}.\end{aligned}$$ Thus $\phi$ is a permutation of ${\mathcal B}.$
\[yl-4.5\] $B_{i,j}^{(k)}\cap B_{i^{'},j^{'}}^{(k^{'})}=\emptyset$ if and only if $k=k^{'}$ and exactly one of $i=i^{'}$ and $j=j^{'}$ holds.
If $k=k^{'}$ and exactly one of $i=i^{'}$ and $j=j^{'}$ holds, then $B_{i,j}^{(k)}\cap B_{i^{'},j^{'}}^{(k^{'})}=\emptyset.$
If $k\neq k^{'},$ then $|B_{i,j}^{(k)}\cap
B_{i^{'},j^{'}}^{(k^{'})}|=|\{(\sigma_1,\sigma_2,\cdots,\sigma_q)\in
A_n^q:\,i^{\sigma_k}=j,\,{i^{'}}^{\sigma_{k^{'}}}=j^{'}\}|=\frac{(n-1)!^2n!^{q-2}}{2^q}.$
If $k=k^{'},\,i= i^{'}$ and $j= j^{'},$ then $B_{i,j}^{(k)}=
B_{i^{'},j^{'}}^{(k^{'})},$ so $B_{i,j}^{(k)}\cap
B_{i^{'},j^{'}}^{(k^{'})}\neq\emptyset.$
If $k=k^{'},\,i\neq i^{'}$ and $j\neq j^{'},$ then $$|B_{i,j}^{(k)}\cap
B_{i^{'},j^{'}}^{(k^{'})}|=\{(\sigma_1,\sigma_2,\cdots,\sigma_q)\in
A_n^q:\,i^{\sigma_k}=j,\,{i^{'}}^{\sigma_k}=j^{'}\}|=\frac{(n-1)!n!^{q-1}}{2^q}.$$
Thus the assertion holds.
\[yl-4.6\] Let ${\mathcal
B}^{(k)}=\{B_{i,j}^{(k)},\,i,j=1,2,\cdots,n\},\,k=1,2,\cdots,q.$ For any $\phi\in\Aut(A\T_n^q),$ There exists a $B_{i,j}^{(k)}\in
{\mathcal B}^{(k)}$ such that ${B_{i,j}^{(k)}}^{\,\phi}\in {\mathcal
B}^{(k^{'})}$ if and only if ${B_{i,j}^{(k)}}^{\,\phi}\in {\mathcal
B}^{(k^{'})}$ for any $B_{i,j}^{(k)}\in {\mathcal B}^{(k)}.$
Suppose on the contrary that there exist two distinct $B_{i,j}^{(k)},\,B_{i^{'},j^{'}}^{(k)}\in {\mathcal B}^{(k)}$ such that ${B_{i,j}^{(k)}}^{\,\phi}\in {\mathcal
B}^{(k^{'})},\,{B_{i^{'},j^{'}}^{(k)}}^{\phi}\in {\mathcal
B}^{(k^{''})}$ with $k^{'}\neq k^{''}.$
Since $B_{i,j}^{(k)}\neq B_{i^{'},j^{'}}^{(k)},$ we have $|B_{i,j}^{(k)}\cap
B_{i^{'},j^{'}}^{(k)}|=0~or~\frac{(n-2)!n!^{q-1}}{2^q}$ by using Lemma \[yl-4.5\] and its proof. So $$\begin{aligned}
|{B_{i,j}^{(k)}}^{\,\phi}\cup
{B_{i^{'},j^{'}}^{(k)}}^{\phi}|&=&|(B_{i,j}^{(k)}\cup
B_{i^{'},j^{'}}^{(k)})^{\,\phi}|=|B_{i,j}^{(k)}\cup
B_{i^{'},j^{'}}^{(k)}|\\
&=&\frac{2(n-1)!n!^{q-1}}{2^q}~or~\frac{2(n-1)!n!^{q-1}-(n-2)!n!^{q-1}}{2^q}.\end{aligned}$$ On the other hand, $$\begin{aligned}
{B_{i,j}^{(k)}}^{\,\phi}\in
B^{(k^{'})},{B_{i^{'},j^{'}}^{(k)}}^{\phi}\in B^{(k^{''})}\,
(k^{'}\neq k^{''})&\Rightarrow& |{B_{i,j}^{(k)}}^{\,\phi}\cap
{B_{i^{'},j^{'}}^{(k)}}^{\phi}|=\frac{(n-1)!^2n!^{q-2}}{2^q}\\
&\Rightarrow& |{B_{i,j}^{(k)}}^{\,\phi}\cup
{B_{i^{'},j^{'}}^{(k)}}^{\phi}|=\frac{2(n-1)!n!^{q-1}-(n-1)!^2n!^{q-2}}{2^q},\end{aligned}$$ which is a contradiction. Thus the assertion holds.
\[yl-4.7\] Let ${\mathcal
R}_i^{(k)}=\{B_{i,1}^{(k)},B_{i,2}^{(k)},\cdots,B_{i,n}^{(k)}\}$ and ${\mathcal
C}_j^{(k)}=\{B_{1,j}^{(k)},B_{2,j}^{(k)},\cdots,B_{n,j}^{(k)}\},\,k=1,2,\cdots,q.$ Then for any $x_1,x_2,\cdots,x_n\in {\mathcal B},$ we have $$x_1\cup x_2\cup\cdots\cup x_n=A_n^q$$ if and only if there exist some $k\in\{1,2,\cdots,q\}$ and some $i$ or $j\in \{1,2,\cdots,n\}$ such that $\{x_1,x_2,\cdots,x_n\}={\mathcal R}_i^{(k)}$ or ${\mathcal
C}_j^{(k)}.$
Clearly if $\{x_1,x_2,\cdots,x_n\}={\mathcal R}_i^{(k)}$ or ${\mathcal C}_j^{(k)}$ for some $k\in\{1,2,\cdots,q\}$ and some $i$ or $j\in \{1,2,\cdots,n\},$ then $x_1\cup x_2\cup\cdots\cup
x_n=A_n^q.$
Assume that $x_1\cup x_2\cup\cdots\cup x_n=A_n^q.$ Since $\forall\,i,\,|x_i|=\frac{(n-1)!n!^{q-1}}{2^q}$ and $|A_n^q|=\frac{n!^q}{2^q},$ we have $x_i\cap
x_j=\emptyset,\,\forall i,j,\,i\neq j.$ Applying Lemma \[yl-4.5\], we obtain $\{x_1,x_2,\cdots,x_n\}={\mathcal R}_i^{(k)}$ or ${\mathcal C}_j^{(k)}.$
\[yl-4.8\] Let $\Omega=\{{\mathcal C}_i^{(k)},{\mathcal
R}_j^{(k)},i,j=1,2,\cdots,n;\,k=1,2,\cdots,q\}.$ Then the action of $\Aut(A\T_n^q)$ on $\Omega$ can be induced by the action of $\Aut(A\T_n^q)$ on ${\mathcal B}$ in Lemma \[yl-4.4\], and it is faithful. Furthermore, any $\phi\in \Aut(A\T_n^q)$ is a permutation of $\Omega.$
First for any ${\mathcal R}_i^{(k)}\in \Omega$ and $\phi\in\Aut(A\T_n^q),$ we have $${B_{i,1}^{(k)}}^{\,\phi}\cup{B_{i,2}^{(k)}}^{\,\phi}\cup\cdots\cup{B_{i,n}^{(k)}}^{\,\phi}=
(B_{i,1}^{(k)}\cup B_{i,2}^{(k)}\cup\cdots\cup
B_{i,n}^{(k)})^{\,\phi}=(A_n^q)^{\,\phi}=A_n^q.$$ So by Lemma \[yl-4.7\], we have ${{\mathcal
R}_i^{(k)}}^{\,\phi}=\{{B_{i,1}^{(k)}}^{\,\phi},{B_{i,2}^{(k)}}^{\,\phi},\cdots,{B_{i,n}^{(k)}}^{\,\phi}\}\in\Omega.$
Similarly, for any ${\mathcal C}_j^{(k)}\in \Omega$ and $\phi\in\Aut(A\T_n^q),$ we have ${{\mathcal C}_j^{(k)}}^{\,\phi}\in
\Omega.$
Assume that $\phi\in \Aut(A\T_n^q)$ satisfies ${{\mathcal
R}_i^{(k)}}^{\,\phi}={\mathcal R}_i^{(k)}$ and ${{\mathcal
C}_j^{(k)}}^{\,\phi}={\mathcal C}_j^{(k)}$ for any $i,j\in
\{1,2,\cdots,n\}$ and $k\in \{1,2,\cdots,q\}.$ Then it suffices to show that $\phi$ is the identity map.
Since for any $B_{i,j}^{(k)}\in {\mathcal B},$ we have $\{{B_{i,j}^{(k)}}^{\,\phi}\}=({\mathcal R}_i^{(k)}\cap {\mathcal
C}_j^{(k)})^{\,\phi} \subseteq {{\mathcal R}_i^{(k)}}^{\,\phi}\cap
{{\mathcal C}_j^{(k)}}^{\,\phi} ={\mathcal R}_i^{(k)}\cap {\mathcal
C}_j^{(k)}=\{B_{i,j}^{(k)}\}.$ By Lemma \[yl-4.4\], the action of $\Aut(A\T_n^q)$ on ${\mathcal B}$ is faithful. Thus $\phi$ is the identity map.
For any $\omega_1,\omega_2\in \Omega$ and $\phi\in \Aut(A\T_n^q),$ $$\begin{aligned}
\omega_1\neq\omega_2&\Leftrightarrow&|\omega_1\cup\omega_2|>n\\
&\Leftrightarrow&|(\omega_1\cup\omega_2)^{\phi}|>n\\
&\Leftrightarrow&|\omega_1^{\phi}\cup\omega_2^{\phi}|>n\\
&\Leftrightarrow&\omega_1^{\phi}\neq\omega_2^{\phi}.\end{aligned}$$ Thus $\phi$ is a permutation of $\Omega.$
\[yl-4.9\] Let ${\mathcal R}^{(k)}=\{{\mathcal R}_1^{(k)},{\mathcal
R}_2^{(k)},\cdots,{\mathcal R}_n^{(k)}\},\,{\mathcal
C}^{(k)}=\{{\mathcal C}_1^{(k)},{\mathcal
C}_2^{(k)},\cdots,{\mathcal C}_n^{(k)}\}$ and $\Omega^{(k)}={\mathcal R}^{(k)}\cup {\mathcal
C}^{(k)},\,k=1,2,\cdots,q.$ For any $\phi\in \Aut(A\T_n^q),$ the following (i)-(iii) hold:
\(i) There exists a $\sigma\in S_q$ such that ${\Omega^{(k)}}^{\,\phi}=\Omega^{(k^{\sigma})},\,k=1,2,\cdots,q.$
\(ii) There exists some ${\mathcal R}_i^{(k)}\in {\mathcal R}^{(k)}$ such that ${{\mathcal R}_i^{(k)}}^{\,\phi}\in {\mathcal
R}^{(k^{'})}$ if and only if ${{\mathcal R}_i^{(k)}}^{\,\phi}\in
{\mathcal R}^{(k^{'})}$ for any ${\mathcal R}_i^{(k)}\in {\mathcal
R}^{(k)};$
\(iii) There exists some ${\mathcal R}_j^{(k)}\in {\mathcal R}^{(k)}$ such that ${{\mathcal R}_j^{(k)}}^{\,\phi}\in {\mathcal
C}^{(k^{'})}$ if and only if ${{\mathcal R}_j^{(k)}}^{\,\phi}\in
{\mathcal C}^{(k^{'})}$ for any ${\mathcal R}_j^{(k)}\in {\mathcal
R}^{(k)}.$
\(i) By Lemma \[yl-4.6\], for any $k\in \{1,2,\cdots,q\}$ there exists a $l\in \{1,2,\cdots,q\}$ such that ${{\mathcal
B}^{(k)}}^{\,\phi}={\mathcal B}^{(l)}.$ Moreover, if $k\neq k^{'},$ then by Lemma \[yl-4.4\], we have ${{\mathcal
B}^{(k)}}^{\,\phi}\neq{{\mathcal B}^{(k^{'})}}^{\,\phi}.$ Thus there exists a $\sigma\in S_q$ such that ${{\mathcal
B}^{(k)}}^{\,\phi}={\mathcal B}^{(k^{\sigma})},\,k=1,2,\cdots,q.$ By Lemma \[yl-4.8\], the assertion holds.
\(ii) First by (i), there exists some ${\mathcal R}_i^{(k)}\in
{\mathcal R}^{(k)}$ such that ${{\mathcal R}_i^{(k)}}^{\,\phi}\in
\Omega^{(k^{'})}$ if and only if ${{\mathcal R}_i^{(k)}}^{\,\phi}\in
\Omega^{(k^{'})}$ for any ${\mathcal R}_i^{(k)}\in {\mathcal
R}^{(k)}.$
Suppose on the contrary that there exist $i,j\,(\neq i)\in
\{1,2,\cdots,n\}$ such that ${{\mathcal R}_i^{(k)}}^{\,\phi}\in
{\mathcal R}^{(k^{'})}$ and ${{\mathcal R}_j^{(k)}}^{\,\phi}\in
{\mathcal C}^{(k^{'})}.$
Note that $${\mathcal
R}_i^{(k)}\cap {\mathcal R}_j^{(k)}=\emptyset~for~i\neq j,$$ $${\mathcal
R}_i^{(k)}\cap {\mathcal C}_j^{(k)}=\{B_{i,j}^{(k)}\}~for~any~i,j.$$ Then $$i\neq j\Rightarrow {\mathcal
R}_i^{(k)}\cap {\mathcal R}_j^{(k)}=\emptyset \Rightarrow |{\mathcal
R}_i^{(k)}\cup {\mathcal R}_j^{(k)}|=2n\Rightarrow |{{\mathcal
R}_i^{(k)}}^{\,\phi}\cup {{\mathcal
R}_j^{(k)}}^{\,\phi}|=|({\mathcal R}_i^{(k)}\cup {\mathcal
R}_j^{(k)})^{\,\phi}|=2n.$$ On the other hand, $${{\mathcal
R}_i^{(k)}}^{\,\phi}\in {\mathcal R}^{(k^{'})},\,{{\mathcal
R}_j^{(k)}}^{\,\phi}\in {\mathcal C}^{(k^{'})}\Rightarrow
|{{\mathcal R}_i^{(k)}}^{\,\phi}\cap {{\mathcal
R}_j^{(k)}}^{\,\phi}|=1\Rightarrow |{{\mathcal
R}_i^{(k)}}^{\,\phi}\cup {{\mathcal R}_j^{(k)}}^{\,\phi}|=2n-1,$$ which is a contradiction. Thus the assertion holds.
\(iii) The proof of (iii) is similar to that of (ii).
\[yl-4.10\] For $n\geq 5,$ we have $$|\Aut(A\T_n^q)|\leq q!n!^{2q}.$$
By (i) of Lemma \[yl-4.9\], for any $\phi\in \Aut(A\T_n^q),$ there exists a $\sigma\in S_q$ such that ${\Omega^{(k)}}^{\,\phi}=\Omega^{(k^{\sigma})}~(k=1,2,\cdots,q).$ Using (ii) and (iii) of Lemma \[yl-4.9\] we obtain the following disjoint alternatives:
\(i) ${{\mathcal R}^{(k)}}^{\,\phi}={\mathcal R}^{(k^{\sigma})}$ and ${{\mathcal C}^{(k)}}^{\,\phi}={\mathcal C}^{(k^{\sigma})};$
\(ii) ${{\mathcal R}^{(k)}}^{\,\phi}={\mathcal C}^{(k^{\sigma})}$ and ${{\mathcal C}^{(k)}}^{\,\phi}={\mathcal R}^{(k^{\sigma})}.$
So $\Aut(A\T_n^q)=\bigcup_{\sigma\in S_q}\{\phi\in
\Aut(A\T_n^q):{\Omega^{(k)}}^{\,\phi}=\Omega^{(k^{\sigma})},k=1,2,\cdots,q\}.$ Hence, if we can prove the last two inequalities, then we have $$\begin{aligned}
|\Aut(A\T_n^q)|&\leq&\Sigma_{\sigma\in S_q}|\{\phi\in
\Aut(A\T_n^q):{\Omega^{(k)}}^{\,\phi}=\Omega^{(k^{\sigma})},k=1,2,\cdots,q\}|\\
&\leq&\Sigma_{\sigma\in S_q}\Pi_{k=1}^q(|\{\phi\in
\Aut(A\T_n^q):{{\mathcal R}^{(k)}}^{\,\phi}={\mathcal
R}^{(k^{\sigma})},\,{{\mathcal C}^{(k)}}^{\,\phi}={\mathcal
C}^{(k^{\sigma})}\}|+\\
&&~~~~~~~~~~~~~~~~~|\{\phi\in \Aut(A\T_n^q):{{\mathcal
R}^{(k)}}^{\,\phi}={\mathcal C}^{(k^{\sigma})},\,{{\mathcal
C}^{(k)}}^{\,\phi}={\mathcal
R}^{(k^{\sigma})}\}|)\\
&\leq&\Sigma_{\sigma\in S_q}\Pi_{k=1}^q(\frac{n!^2}{2}+\frac{n!^2}{2})\\
&=&\Sigma_{\sigma\in S_q}n!^{2q}\\
&=&q!n!^{2q}.\end{aligned}$$
Now we show that $$|\{\phi\in \Aut(A\T_n^q):{{\mathcal
R}^{(k)}}^{\,\phi}={\mathcal R}^{(k^{\sigma})},\,{{\mathcal
C}^{(k)}}^{\,\phi}={\mathcal C}^{(k^{\sigma})}\}|\leq
\frac{n!^2}{2},$$ $$|\{\phi\in \Aut(A\T_n^q):{{\mathcal
R}^{(k)}}^{\,\phi}=C^{(k^{\sigma})},\,{C^{(k)}}^{\,\phi}={\mathcal
R}^{(k^{\sigma})}\}|\leq \frac{n!^2}{2}.$$
Indeed, for any $\phi\in \Aut(A\T_n^q)$ such that ${{\mathcal
R}^{(k)}}^{\,\phi}={\mathcal R}^{(k^{\sigma})},\,{{\mathcal
C}^{(k)}}^{\,\phi}={\mathcal C}^{(k^{\sigma})},$ define $\phi_1,\phi_2\in S_n$ as ${{\mathcal R}_i^{(k)}}^{\phi}={\mathcal
R}_{i^{\,\phi_1}}^{(k^{\sigma})},\,{{\mathcal
C}_j^{(k)}}^{\phi}={\mathcal C}_{j^{\,\phi_2}}^{(k^{\sigma})}.$
Since $\{{B_{ij}^{(k)}}^{\phi}\}=({\mathcal R}_i^{(k)}\cap {\mathcal
C}_j^{(k)})^{\phi}\subseteq {{\mathcal
R}_i^{(k)}}^{\phi}\cap{{\mathcal C}_j^{(k)}}^{\phi}={\mathcal
R}_{i^{\,\phi_1}}^{(k^{\sigma})}\cap {\mathcal
C}_{j^{\,\phi_2}}^{(k^{\sigma})}=\{B_{i^{\,\phi_1}j^{\,\phi_2}}^{(k^{\sigma})}\},$ we have $$\{(1,1,\cdots,1)^{\phi}\}\in (\bigcap_{i=1}^n B_{ii}^{(k)})^{\phi}
\subseteq\bigcap_{i=1}^n {B_{ii}^{(k)}}^{\phi} =\bigcap_{i=1}^n
B_{i^{\,\phi_1}i^{\,\phi_2}}^{(k^{\sigma})}
=\{(\tau_1,\tau_2,\cdots,\tau_q)\in
A_n^q:\tau_{k^{\sigma}}=\phi_1^{-1}\phi_2\}.$$ So $(1,1,\cdots,1)^{\phi}=(\tau_1,\cdots,\tau_{k^{\sigma}-1},\phi_1^{-1}\phi_2,\tau_{k^{\sigma}+1}\cdots,\tau_q)\in
A_n^q,$ which implies that $\phi_1^{-1}\phi_2\in A_n.$
Thus $$\begin{aligned}
&&|\{\phi\in \Aut(A\T_n^q):{{\mathcal
R}^{(k)}}^{\,\phi}={\mathcal R}^{(k^{\sigma})},\,{{\mathcal
C}^{(k)}}^{\,\phi}={\mathcal
C}^{(k^{\sigma})}\}|\\
&=&|\{\phi\in \Aut(A\T_n^q):{{\mathcal R}^{(k)}}^{\,\phi}={\mathcal
R}^{(k^{\sigma})},\,{{\mathcal C}^{(k)}}^{\,\phi}={\mathcal
C}^{(k^{\sigma})},\,\phi_1^{-1}\phi_2\in
A_n\}|\\
&\leq& \frac{n!^2}{2}.\end{aligned}$$
Similarly, $$|\{\phi\in \Aut(A\T_n^q):{{\mathcal
C}^{(k)}}^{\,\phi}={\mathcal R}^{(k^{\sigma})},\,{{\mathcal
C}^{(k)}}^{\,\phi}={\mathcal R}^{(k^{\sigma})}\}|\leq
\frac{n!^2}{2}.$$
Thus the assertion holds.
(of Theorem \[dl-1.3\]). By Lemma \[yl-4.3\], we have $|\Aut(A\T_n^q)|\geq q!n!^{2q}.$ On the other hand, by Lemma \[yl-4.10\], we obtain $|\Aut(A\T_n^q|\leq q!n!^{2q}.$ Hence $|\Aut(A\T_n^q)|=q!n!^{2q},$ and by Lemma \[yl-4.3\] again, the assertion holds.
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[^1]: This work is supported by National Natural Science Foundation of China (No:10971137), the National Basic Research Program (973) of China (No.2006CB805900), and a grant of Science and Technology Commission of Shanghai Municipality (STCSM, No: 09XD1402500) . $^{\dagger}$Correspondent author: Xiao-Dong Zhang (Email: [email protected])
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We derive the capacity region of arbitrarily varying multiple-access channels with conferencing encoders for both deterministic and random coding. For a complete description it is sufficient that one conferencing capacity is positive. We obtain a dichotomy: either the channel’s deterministic capacity region is zero or it equals the two-dimensional random coding region. We determine exactly when either case holds. We also discuss the benefits of conferencing. We give the example of an AV-MAC which does not achieve any non-zero rate pair without encoder cooperation, but the two-dimensional random coding capacity region if conferencing is possible. Unlike compound multiple-access channels, arbitrarily varying multiple-access channels may exhibit a discontinuous increase of the capacity region when conferencing in at least one direction is enabled.'
author:
- 'Moritz Wiese, and Holger Boche, [^1] [^2] [^3] [^4]'
title: 'The Arbitrarily Varying Multiple-Access Channel with Conferencing Encoders'
---
Base station cooperation, Channel uncertainty, Compound channels, Arbitrarily Varying Channels, Conferencing Encoders.
Introduction
============
Multiple-Access Channels (MACs) and similar multi-sender channels with conferencing encoders have attracted attention recently due to the inclusion of base-station cooperation methods in standards for future wireless systems [@BLW; @MYK; @SGPGS; @Wig]. The original conferencing protocol for the discrete memoryless MAC is due to Willems [@Wi1; @Wi2]. The conferencing MAC with imperfect channel state information was modeled as a compound MAC with conferencing encoders and considered in [@WBBJ11], a different model for channel state uncertainty is given in [@PSSB].
This paper covers a very high degree of channel uncertainty in MACs: the channel states may vary arbitrarily over time. The task is to use coding to enable reliable communication for every possible state sequence. The corresponding information-theoretic channel model is the Arbitrarily Varying MAC (AV-MAC). The random coding capacity region of the AV-MAC without encoder cooperation was determined in [@J]. Building on this result, the deterministic coding capacity region of some AV-MACs without cooperation was determined in [@AC]. In general, it is still open. We will use the “robustification” and “elimination of correlation” techniques developed by Ahlswede in [@A1; @A2], and partly already used in [@J] in a multi-user setting, in order to characterize both the deterministic and random coding capacity regions of any AV-MAC with conferencing encoders, i.e. of any AV-MAC where encoding is done using a Willems conference as in [@Wi1; @Wi2] with at least one positive conferencing capacity. Thus none of the techniques we apply in this paper is completely new, but in contrast to the non-conferencing situation, they allow for the complete solution of the problems considered here. The rather general “robustification” technique establishes the random coding capacity region of the AV-MAC with conferencing encoders. Both single- and multi-user arbitrarily varying channels are special in that random coding as commonly used in information theory does not yield the same results as deterministic coding. This shows that common randomness shared at the senders and the receiver is an important additional resource. There is a dichotomy: either reliable communication at any non-zero rate pair is impossible with the application of deterministic codes, or the deterministic capacity region coincides with the random coding capacity region, which then is two-dimensional. In the latter case, one needs the non-standard “elimination of correlation” [@A1] for derandomization. It is a two-step protocol which achieves the random coding capacity region if this is possible.
The combination of the elimination technique with conferencing proves to be very fruitful. Here lies the main difference between the AV-MAC with and without conferencing. One can show that there exist channels which only achieve the zero rate pair without transmitter cooperation, but where derandomization using the elimination technique is possible if the transmitters may have a conference. The reason for this is symmetrizability. This can be interpreted in terms of an adversary knowing the channel input symbols and randomizing over the channel states. There are three kinds of symmetrizability for multiple-access channels. The capacity region of the AV-MAC without conferencing equals $\{(0,0)\}$ if all three symmetrizability conditions are satisfied. In contrast, the elimination of correlation technique works if the AV-MAC with Willems conferencing encoders does not satisfy the conditions for the first of the three kinds of symmetrizabilities. The two others do not matter. By conferencing with at least one positive conferencing capacity, the AV-MAC gets closer to a single-sender arbitrarily varying channel where only one symmetrizability condition exists [@CN]. This induced change of the channel structure is also reflected in the counter-intuitive fact that conferencing with rates tending to zero in blocklength can enlarge the capacity region. The adversary interpretation of symmetrizability highlights the importance of the AV-MAC for the theory of information-theoretic secrecy: if a channel is symmetrizable, an adversary can completely prevent communication.
The paper is organized as follows: the next section is devoted to the formalization of the channel model and the coding problems. We present the main theorems and several auxiliary coding results. The direct parts of the random and deterministic coding theorems are solved in Section \[sect:ach\]. Section \[sect:conv\] gives the converses of the random and deterministic AV-MAC coding theorems. Section \[sect:concl\] concludes the paper with a discussion. In particular, the gains of conferencing are analyzed there.
*Notation:* In the information-theoretic setting, we also use the terms “encoders” for the senders and “decoder” for the receiver. For any positive integer $m$, we write $[1,m]$ for the set $\{1,\ldots,m\}$. For a set $A\subset{\mathcal{X}}$, we denote its complement by $A^c:={\mathcal{X}}\setminus A$. For real numbers $x$ and $y$, we set $x\wedge y:=\min(x,y)$ and $x\vee y:=\max\{x,y\}$. $\mathcal{P}({\mathcal{X}})$ denotes the set of probability measures on the discrete set ${\mathcal{X}}$.
Problem Setting
===============
The Main Coding Problems
------------------------
Let ${\mathcal{X}},{\mathcal{Y}},{\mathcal{Z}}$ be finite alphabets, let ${\mathcal{S}}$ be another finite set. For every $s\in{\mathcal{S}}$, let a stochastic matrix $$W(z\vert x,y\vert s):\quad(x,y,z)\in{\mathcal{X}}\times{\mathcal{Y}}\times{\mathcal{Z}}$$ be given with inputs from ${\mathcal{X}}\times{\mathcal{Y}}$ and outputs from ${\mathcal{Z}}$. ${\mathcal{S}}$ is to be interpreted as the set of channel states. We set $${\mathcal{W}}:=\{W(\,\cdot\,\vert\,\cdot\,,\,\cdot\,\vert s):s\in{\mathcal{S}}\}.$$ We assume that the channel state varies arbitrarily from channel use to channel use. Given words ${\mathbf{x}}=(x_1,\ldots,x_n)\in{\mathcal{X}}^n$, ${\mathbf{y}}=(y_1,\ldots,y_n)\in{\mathcal{Y}}^n$, and ${\mathbf{z}}=(z_1,\ldots,z_n)\in{\mathcal{Z}}^n$, the probability that ${\mathbf{z}}$ is received upon transmission of ${\mathbf{x}}$ and ${\mathbf{y}}$ depends on the sequence ${\mathbf{s}}\in{\mathcal{S}}^n$ of channel states attained during the transmission. It equals $$W^n({\mathbf{z}}\vert{\mathbf{x}},{\mathbf{y}}\vert{\mathbf{s}}):=\prod_{m=1}^nW(z_m\vert x_m,y_m\vert s_m).$$
The set of stochastic matrices $$\{W^n(\,\cdot\,\vert\,\cdot\,,\,\cdot\,\vert{\mathbf{s}}):{\mathbf{s}}\in{\mathcal{S}}^n,n=1,2,\ldots\}$$ is called the *Arbitrarily Varying Multiple Access Channel (AV-MAC)* determined by ${\mathcal{W}}$.
In the traditional non-cooperative encoding schemes used for multiple-access channels, none of the senders has any information about the other sender’s message. The goal here is to characterize the capacity region of the AV-MAC achievable when limited information can be exchanged between the encoders. We use Willems conferencing for this exchange [@Wi1; @Wi2]. If the encoders’ message sets are $[1,M_1]$ and $[1,M_2]$, respectively, then this can be described as follows. Let positive integers $V_1$ and $V_2$ be given which can be written as products $$V_\nu=V_{\nu,1}\cdots V_{\nu,I}$$ for some positive integer $I$ which does not depend on $\nu$. A pair $(c_1,c_2)$ of Willems conferencing functions is determined in an iterative manner via sequences of functions $c_{1,1},\ldots,c_{1,I}$ and $c_{2,1},\ldots,c_{2,I}$. The function $c_{1,i}$ describes what encoder 1 tells the other encoder in the $i$-th conferencing iteration given the knowledge accumulated so far at encoder 1. Thus in general, using the notation $$\bar\nu:=\begin{cases}
1&\text{if }\nu=2,\\
2&\text{if }\nu=1,
\end{cases}$$ these functions satisfy for $\nu=1,2$ and $i=2,\ldots,I$ $$\begin{aligned}
c_{\nu,1}&:[1,M_\nu]\rightarrow[1,V_{\nu,1}],\\
c_{\nu,i}&:[1,M_\nu]\times[1,V_{\bar\nu,1}]\times\ldots\times[1,V_{\bar\nu,i-1}]\rightarrow[1,V_{\nu,i}].\end{aligned}$$ These functions recursively define other functions $$\begin{aligned}
c_{\nu,1}^*&:[1,M_\nu]\rightarrow[1,V_{\nu,1}],\\
c_{\nu,i}^*&:[1,M_1]\times[1,M_2]\rightarrow[1,V_{\nu,i}]\end{aligned}$$ by $$\begin{aligned}
c_{1,1}^*(j)&=c_{1,1}(j),\\
c_{2,1}^*(k)&=c_{2,1}(k),\\
c_{1,i}^*(j,k)&=c_{1,i}\bigl(j,c_{2,1}^*(k),\ldots,c_{2,i-1}^*(j,k)\bigr),\\
c_{2,i}^*(j,k)&=c_{2,i}\bigl(k,c_{1,1}^*(j),\ldots,c_{1,i-1}^*(j,k)\bigr).\end{aligned}$$ Then we set $$c_\nu(j,k):=\bigl((c_{\nu,1}^*(j,k),\ldots,c_{\nu,I}^*(j,k)\bigr).$$ Observe that given a message pair $(j,k)$, the conferencing outcome $(c_1(j,k),c_2(j,k))$ is known at both transmitters. If all conferencing protocols were allowed, the encoders could inform each other precisely about their messages, so this would turn the MAC into a single-sender channel. Thus for nonnegative numbers $C_1,C_2$, if conferencing is used in a blocklength-$n$ code, Willems introduces the restrictions $$\label{confcap}
\frac{1}{n}\log V_\nu\leq C_\nu.$$ $C_1,C_2$ are called the conferencing capacities. Having introduced Willems conferencing, we can now define the codes we are going to consider.
\[defn\_code\]
1. Let $n,M_1,M_2$ be positive integers and $C_1,C_2\geq0$. A *deterministic code$_{\textnormal{CONF}}(n,M_1,M_2,C_1,C_2)$ with blocklength $n$, codelength pair $(M_1,M_2)$, and conferencing capacities $C_1,C_2$* is given by functions $c_1,c_2,f_1,f_2,\Phi$. Here, $(c_1,c_2)$ is a Willems conferencing protocol satisfying . $f_1,f_2$ are the encoding functions $$\begin{aligned}
f_1&:[1,M_1]\times[1,V_2]\rightarrow{\mathcal{X}}^n,\\
f_2&:[1,M_2]\times[1,V_1]\rightarrow{\mathcal{Y}}^n.\end{aligned}$$ The *decoding function* $\Phi$ is a function $$\Phi:{\mathcal{Z}}^n\rightarrow[1,M_1]\times[1,M_2].$$
2. A *random code$_{\textnormal{CONF}}(n,M_1,M_2,C_1,C_2)$ with blocklength $n$, codelength pair $(M_1,M_2)$, and conferencing capacities $C_1,C_2$* is a pair $(C,G)$, where $C=\{C(\gamma):\gamma\in\Gamma\}$ is a finite family of deterministic codes$_{\textnormal{CONF}}(n,M_1,M_2,C_1,C_2)$, and where $G$ is a random variable taking values in $\Gamma$.
![The AV-MAC ${\mathcal{W}}$ with conferencing encoders.[]{data-label="fig:confbild"}](CONF-AV-MAC_Bild){width="\columnwidth"}
Note that a code$_{\textnormal{CONF}}(n,M_1,M_2,0,0)$ is a traditional MAC code without conferencing. An AV-MAC together with the above coding procedure is called an *AV-MAC with conferencing encoders*, see Fig. \[fig:confbild\]. A code$_{\textnormal{CONF}}$ $(n,M_1,M_2,C_1,C_2)$ defined by $(c_1,c_2,f_1,f_2,\Phi)$ gives rise to a family $$\label{codeform}
\{({\mathbf{x}}_{jk},{\mathbf{y}}_{jk},F_{jk}):(j,k)\in[1,M_1]\times[1,M_2]\},$$ where $$\begin{aligned}
&{\mathbf{x}}_{jk}:=f_1(j,c_2(j,k))\in{\mathcal{X}}^n,\\&{\mathbf{y}}_{jk}:=f_2(k,c_1(j,k))\in{\mathcal{Y}}^n,\\&F_{jk}:=\Phi^{-1}\{(j,k)\}\subset{\mathcal{Z}}^n.\end{aligned}$$ If the message pair $(j,k)$ is present at the senders, the *codewords* ${\mathbf{x}}_{jk}$ and ${\mathbf{y}}_{jk}$ are sent. The decoding sets $\{F_{jk}:(j,k)\in[1,M_1]\times[1,M_2]\}$ give a partition of ${\mathcal{Z}}^n$ which, just like $\Phi$, assigns to every channel output ${\mathbf{z}}\in{\mathcal{Z}}^n$ a message pair which the receiver will decide for upon reception of ${\mathbf{z}}$.
Note that every family , where the $F_{jk}$ are disjoint, together with a Willems conferencing protocol $(c_1,c_2)$ satisfying defines a code$_{\textnormal{CONF}}(n,M_1,M_2,C_1,C_2)$ if $$\begin{aligned}
{\mathbf{x}}_{jk}&={\mathbf{x}}_{jk'}\qquad\text{if }c_2(j,k)=c_2(j,k'),\label{conf1}\\
{\mathbf{y}}_{jk}&={\mathbf{y}}_{j'k}\qquad\text{if }c_1(j,k)=c_1(j',k).\label{conf2}\end{aligned}$$ Thus a code$_{\textnormal{CONF}}$ can equivalently be defined by a family together with a conferencing protocol $(c_1,c_2)$ such that and are satisfied. We will often refer to a code$_{\textnormal{CONF}}$ using the description , and usually without specifying the corresponding conferencing protocol by just assuming that there is one.
The first example of this convention is encountered in our definition of the average error, where the explicit form of the conferencing protocol is irrelevant.
1. A code$_{\textnormal{CONF}}(n,M_1,M_2,C_1,C_2)$ defining a family has an average error probability less than $\lambda\in(0,1)$ if $$\frac{1}{M_1M_2}\sum_{j,k}W^n(F_{jk}^c\vert{\mathbf{x}}_{jk},{\mathbf{y}}_{jk}\vert {\mathbf{s}})\leq\lambda\quad\text{for all }{\mathbf{s}}\in{\mathcal{S}}^n.$$
2. Let a random code$_{\textnormal{CONF}}(n,M_1,M_2,C_1,C_2)$ with the form $(C,G)$ be given. Assume that the deterministic code$_{\textnormal{CONF}}$ $C(\gamma)$ has the form $$\{({\mathbf{x}}_{jk}^\gamma,{\mathbf{y}}_{jk}^\gamma,F_{jk}^\gamma):(j,k)\in[1,M_1]\times[1,M_2]\}.$$ Then for any ${\mathbf{s}}\in{\mathcal{S}}^n$, define $$\label{P_edefn}
P_e(C(\gamma)\vert{\mathbf{s}}):=\frac{1}{M_1M_2}\sum_{j,k}W^n((F_{jk}^\gamma)^c\vert{\mathbf{x}}_{jk}^\gamma,{\mathbf{y}}_{jk}^\gamma\vert{\mathbf{s}})$$ to be the average error incurred by $C(\gamma)$ under channel conditions ${\mathbf{s}}$. Assume that $G$ has distribution $p_G$. We say that the random code$_{\textnormal{CONF}}$ defined by $(C,G)$ has an average error smaller than $\lambda\in(0,1)$ if $$\sum_{\gamma\in\Gamma}P_e(C(\gamma)\vert{\mathbf{s}})p_G(\gamma)\leq\lambda\qquad\text{for every }{\mathbf{s}}\in{\mathcal{S}}^n.$$
This means that uniformly for every interfering sequence, transmission using this code is reliable up to the average error level $\lambda$. The possible state sequences are not weighted by any probability measure. One can interpret this in a communication setting with an adversary who knows which words ${\mathbf{x}},{\mathbf{y}}$ are input into the channel by the senders and then can choose any state sequence ${\mathbf{s}}\in{\mathcal{S}}^n$ in order to obstruct the transmission of ${\mathbf{x}}$ and ${\mathbf{y}}$. The goal of the encoders then is to enable reliable communication no matter what sequence ${\mathbf{s}}$ the bad guy might use.
The concept of achievability of a rate pair is the usual one except that conferencing codes$_{\textnormal{CONF}}$ are allowed for code construction.
A rate pair $(R_1,R_2)$ is *achievable by the AV-MAC with conferencing encoders and conferencing capacities $C_1,C_2$ under deterministic/random coding* if for every $\lambda\in(0,1)$ and for every ${\varepsilon}>0$, for $n$ sufficiently large, there is a deterministic/random code$_{\textnormal{CONF}}(n,M_1,M_2,C_1,C_2)$ with $$\frac{1}{n}\log M_\nu\geq R_\nu-{\varepsilon}\quad(\nu=1,2),$$ and with an average error smaller than $\lambda$. The set of achievable rates under deterministic/random coding is called the *deterministic/random capacity region of the AV-MAC with conferencing encoders and conferencing capacities $C_1,C_2$* and denoted by ${\mathcal{C}}_d({\mathcal{S}},C_1,C_2)$ (for deterministic coding) and ${\mathcal{C}}_r({\mathcal{S}},C_1,C_2)$ (for random coding).
We can now formulate the coding problems which are at the center of this work:
*Characterize the deterministic/random capacity regions ${\mathcal{C}}_d({\mathcal{S}},C_1,C_2)$ and ${\mathcal{C}}_r({\mathcal{S}},C_1,C_2)$ of the AV-MAC with conferencing capacities $C_1,C_2$.*
Of course, the main focus is on the deterministic capacity region ${\mathcal{C}}_d({\mathcal{S}},C_1,C_2)$ as the random capacity region ${\mathcal{C}}_r({\mathcal{S}},C_1,C_2)$ requires common randomness shared at the encoders and the receiver. For both ${\mathcal{C}}_d({\mathcal{S}},C_1,C_2)$ and ${\mathcal{C}}_r({\mathcal{S}},C_1,C_2)$, we need to consider the convex hull $\WQ$ of ${\mathcal{W}}$. It is parametrized by the set of probability distributions ${\mathcal{P}}({\mathcal{S}})$ on ${\mathcal{S}}$, so one can regard ${\mathcal{P}}({\mathcal{S}})$ as its “state space”. The stochastic matrix from $\WQ$ assigned to the “state” $q\in{\mathcal{P}}({\mathcal{S}})$ is the matrix with inputs from ${\mathcal{X}}\times{\mathcal{Y}}$ and outputs from ${\mathcal{Z}}$ having the form $$\begin{gathered}
W(z\vert x,y\vert q):=\sum_{s\in{\mathcal{S}}}W(z\vert x,y\vert s)q(s),\\(x,y,z)\in{\mathcal{X}}\times{\mathcal{Y}}\times{\mathcal{Z}}.\end{gathered}$$ We have $\W\subset\WQ$ by identifying $s\in{\mathcal{S}}$ with the Dirac measure $\delta_s\in{{\mathcal{P}}({\mathcal{S}})}$, so that $W(\,\cdot\,\vert\,\cdot\,,\,\cdot\,\vert s)=W(\,\cdot\,\vert\,\cdot\,,\,\cdot\,\vert\delta_s)$.
Next we define a set of rates ${\mathcal{C}}^*({\mathcal{S}},C_1,C_2)$. Let $\Pi$ be the set consisting of probability distributions $p\in{\mathcal{P}}({\mathcal{U}}\times{\mathcal{X}}\times{\mathcal{Y}})$, where ${\mathcal{U}}$ ranges over the finite subsets of the integers and where $p$ has the form $$p(u,x,y)=p_0(u)p_1(x\vert u)p_2(y\vert u).$$ To each $p\in\Pi$ and $q\in{{\mathcal{P}}({\mathcal{S}})}$ one can associate a generic random vector $(U,X,Y,Z_{ q})$ with distribution $$\label{pbars}
p_{ q}(u,x,y,z)=p(u,x,y)W(z\vert x,y\vert q).$$ In this way every $p\in\Pi$ and $ q\in{{\mathcal{P}}({\mathcal{S}})}$ define a set ${\mathcal{R}}(p,q,C_1,C_2)$ consisting of those pairs $(R_1,R_2)$ of nonnegative real numbers which satisfy $$\begin{aligned}
R_1&\leq I(Z_{ q};X\vert Y,U)+C_1,\\
R_2&\leq I(Z_{ q};Y\vert X,U)+C_2,\\
R_1+R_2&\leq (I(Z_{ q};X,Y\vert U)+C_1+C_2)
\wedge I(Z_{ q};X,Y).\end{aligned}$$ Then set $${\mathcal{C}}^*({\mathcal{S}},C_1,C_2):=\bigcup_{p\in\Pi}\;\bigcap_{ q\in{{\mathcal{P}}({\mathcal{S}})}}{\mathcal{R}}(p, q,C_1,C_2).$$
\[thmconfrand\] For the AV-MAC determined by $\W$ with conferencing capacities $C_1,C_2\geq0$, we have $${\mathcal{C}}_r({\mathcal{S}},C_1,C_2)={\mathcal{C}}^*({\mathcal{S}},C_1,C_2).$$ More precisely, for every $(R_1,R_2)\in{\mathcal{C}}^*({\mathcal{S}},C_1,C_2)$ and every ${\varepsilon}>0$ there is a $\zeta>0$ and a sequence $(C_n,G_n)$ of random codes$_{\textnormal{CONF}}(n,M_1^{(n)},M_2^{(n)},C_1,C_2)$ with an average error at most $2^{-n\zeta}$ such that $$\frac{1}{n}\log M_\nu^{(n)}\geq R_\nu-{\varepsilon}\quad(\nu=1,2).$$ Additionally the $(C_n,G_n)$ can be chosen such that for every $n$, the constituent deterministic codes$_{\textnormal{CONF}}$ share the same non-iterative Willems conferencing protocol $(c_1^{(n)},c_2^{(n)})$ given by $$\label{conffunct}
c_\nu^{(n)}:[1,M_\nu^{(n)}]\rightarrow[1,V_\nu^{(n)}]\qquad(\nu=1,2).$$
The simple form of conferencing means that no complicated conferencing protocol needs to be designed.
${\mathcal{C}}^*({\mathcal{S}},C_1,C_2)$ was analyzed in [@WBBJ11]. It is convex and the auxiliary sets ${\mathcal{U}}$ can be restricted to have cardinality at most $(\lvert{\mathcal{X}}\rvert\lvert{\mathcal{Y}}\rvert+2)\wedge(\lvert{\mathcal{Z}}\rvert+3)$. Moreover, one can determine finite $C_1,C_2$ such that
1. the full-cooperation sum rate, or
2. the full-cooperation capacity region
are achievable. The first statement can be phrased as $$\begin{aligned}
\max_{p\in\Pi}&\min_{q\in{\mathcal{P}}({\mathcal{S}})}\bigl\{(I(Z_q;X,Y\vert U)+C_1+C_2)\wedge I(Z_q;X,Y)\bigr\}\\&=\max_{p\in\Pi}\min_{q\in{\mathcal{P}}({\mathcal{S}})}I(Z_q;X,Y)=:C_\infty.\end{aligned}$$ If for $p\in\Pi$ we set ${\mathcal{Q}}_p:=\{q\in{\mathcal{P}}({\mathcal{S}}):I(Z_q\wedge X,Y)=C_\infty\}$, then a simple calculation shows that the above condition is satisfied if $$C_1+C_2\geq C_\infty-\min_{p\in\Pi}\max_{q\in{\mathcal{Q}}_p}I(Z\wedge X,Y\vert U).$$ Statement 2) is valid if both $R_1=C_\infty$ and $R_2=C_\infty$ are possible. For this one needs $$\begin{aligned}
C_1&\geq C_\infty-\max_{p\in\Pi}\min_{q\in{\mathcal{Q}}_p}I(Z_q\wedge X\vert Y,U),\\
C_2&\geq C_\infty-\max_{p\in\Pi}\min_{q\in{\mathcal{Q}}_p}I(Z_q\wedge Y\vert X,U).\end{aligned}$$
\[randconv\] Theorem \[thmconfrand\] has a weak converse.
Determining the general deterministic capacity region ${\mathcal{C}}_d({\mathcal{S}},C_1,C_2)$ is more complex. We give the solution in Theorem \[thmconf\] below for $C_1\vee C_2>0$. For $C_1=C_2=0$ a partial solution is given in [@J],[@G],[@G2],[@AC]. The relation between the cases $C_1=C_2=0$ and $C_1\vee C_2>0$ is discussed in detail in Section \[sect:concl\].
\[defsymm\]
1. $\W$ is called *$({\mathcal{X}},{\mathcal{Y}})$-symmetrizable* if there is a stochastic matrix $$\sigma(s\vert x,y):\quad(x,y,s)\in{\mathcal{X}}\times{\mathcal{Y}}\times{\mathcal{S}}$$ such that for every $z\in{\mathcal{Z}}$ and $x,x'\in{\mathcal{X}}$ and $y,y'\in{\mathcal{Y}}$, $$\sum_s\!W(z\vert x,y\vert s)\sigma(s\vert x',y')\!=\!\sum_s\!W(z\vert x',y'\vert s)\sigma(s\vert x,y).$$
2. ${\mathcal{W}}$ is called *${\mathcal{X}}$-symmetrizable* if there is a stochastic matrix $$\sigma_1(s\vert x):\quad(x,s)\in{\mathcal{X}}\times{\mathcal{S}}$$ such that for every $z\in{\mathcal{Z}}$ and $x,x'\in{\mathcal{X}}$ and $y\in{\mathcal{Y}}$, $$\sum_sW(z\vert x,y\vert s)\sigma_1(s\vert x')=\sum_sW(z\vert x',y\vert s)\sigma_1(s\vert x).$$
3. ${\mathcal{W}}$ is called *${\mathcal{Y}}$-symmetrizable* if there is a stochastic matrix $$\sigma_2(s\vert y):\quad(y,s)\in{\mathcal{Y}}\times{\mathcal{S}}$$ such that for every $z\in{\mathcal{Z}}$ and $x\in{\mathcal{X}}$ and $y,y'\in{\mathcal{Y}}$, $$\sum_sW(z\vert x,y\vert s)\sigma_2(s\vert y')=\sum_sW(z\vert x,y'\vert s)\sigma_2(s\vert y).$$
\[thmconf\] For the deterministic capacity region of the AV-MAC determined by ${\mathcal{W}}$ with conferencing capacities $C_1\vee C_2>0$, we have $$\begin{aligned}
{\mathcal{C}}_d({\mathcal{S}},C_1,C_2)&={\mathcal{C}}^*({\mathcal{S}},C_1,C_2)
\intertext{if $\W$ is not $({\mathcal{X}},{\mathcal{Y}})$-symmetrizable and}
{\mathcal{C}}_d({\mathcal{S}},C_1,C_2)&=\{(0,0)\}\end{aligned}$$ if $\W$ is $({\mathcal{X}},{\mathcal{Y}})$-symmetrizable. As for ${\mathcal{C}}_r({\mathcal{S}},C_1,C_2)$, the Willems conferencing protocols can be assumed to have the simple non-iterative form .
\[zweidim\] If ${\mathcal{W}}$ is not $({\mathcal{X}},{\mathcal{Y}})$-symmetrizable, then ${\mathcal{C}}_d({\mathcal{S}},C_1,C_2)={\mathcal{C}}^*({\mathcal{S}},C_1,C_2)$ is at least one-dimensional. As $C_1\vee C_2>0$, in order to show this it clearly suffices to check that $$\label{verletzt}
\max_{p\in\Pi}\;\min_{q\in{{\mathcal{P}}({\mathcal{S}})}}I(Z_q;X,Y)>0$$ if ${\mathcal{W}}$ is not symmetrizable. If were violated, then by [@CK Lemma 1.3.2] there would be a $q\in{{\mathcal{P}}({\mathcal{S}})}$ such that $$W(z\vert x,y\vert q)=W(z\vert x',y'\vert q)$$ for all $x,x'\in{\mathcal{X}},y,y'\in{\mathcal{Y}},z\in{\mathcal{Z}}$. Thus ${\mathcal{W}}$ would be $({\mathcal{X}},{\mathcal{Y}})$-symmetrizable using the stochastic matrix $$\sigma(s\vert x,y)=q(s),\quad(x,y,s)\in{\mathcal{X}}\times{\mathcal{Y}}\times{\mathcal{S}}.$$ But this would contradict our assumption, so must hold.
One can regard symmetrizability as the single-letterization of the adversary interpretation of the AV-MAC given above. There, a complete input word pair has to be known to the adversary who can then choose the state sequence. In the definition of $({\mathcal{X}},{\mathcal{Y}})$-symmetrizability, the stochastic matrix $\sigma:{\mathcal{X}}\rightarrow{\mathcal{S}}$ means that given a *letter* $x\in{\mathcal{X}}$, the adversary chooses a *random* state $s\in{\mathcal{S}}$. If $\W$ is $({\mathcal{X}},{\mathcal{Y}})$-symmetrizable, the adversary can thus produce a useless single-state MAC $\tilde W:({\mathcal{X}}\times{\mathcal{Y}})^2\rightarrow{\mathcal{Z}}$ defined by $$\tilde W(z\vert x,y,x',y')=\sum_{s\in{\mathcal{S}}}W(z\vert x,y\vert s)\sigma(s\vert x',y').$$ $\tilde W$ is useless because it is symmetric in $(x,y)$ and $(x',y')$. Thus for word pairs $({\mathbf{x}},{\mathbf{y}})$ and $({\mathbf{x}}',{\mathbf{y}}')$, the receiver cannot decide which of the pairs was input into the channel by the senders and which was induced by the adversary’s random state choice.
The above adversary interpretation of symmetrizability makes AV-MACs relevant for information-theoretic secrecy. Clearly, we do not say anything about the decodability of communication taking place in an AV-MAC for non-legitimate listeners. However, reliable communication can be completely prevented in the case the AV-MAC is symmetrizable. A discussion of the single-sender arbitrarily varying wiretap channel can be found in [@BBSW].
\[subexpconf\] By the definition of Willems conferencing, setting $C_1=C_2=0$ yields the traditional MAC coding, i.e. no conferencing at all is allowed. An inspection of the elimination technique applied in \[subsect:randdet\] shows that actually it suffices to have conferencing with $V_1=n^2$, so $C_1=(2\log n)/n$ (or, by symmetry, $V_2=n^2$). Using conferencing with this non-constant rate tending to zero in non-$({\mathcal{X}},{\mathcal{Y}})$-symmetrizable AV-MACs yields the capacity region ${\mathcal{C}}^*({\mathcal{S}},0,0)$.
\[detconv\] If $\W$ is $({\mathcal{X}},{\mathcal{Y}})$-symmetrizable, then Theorem \[thmconf\] almost has a strong converse: it is possible to show that every code that encodes more than one message incurs an average error at least $1/4$. If $\W$ is not $({\mathcal{X}},{\mathcal{Y}})$-symmetrizable, then we have a weak converse.
Theorem \[thmconf\] does not carry over to the case $C_1=C_2=0$, which is the traditional AV-MAC with non-cooperative coding. To our knowledge, the full characterization of the deterministic capacity region ${\mathcal{C}}_d({\mathcal{S}},0,0)$ of the AV-MAC without cooperation is still open. We summarize here what has been found out in [@AC], [@G2], [@G], and [@J]. For notation, observe that $$\begin{aligned}
&\mathrel{\hphantom{=}}\max_{p\in\Pi}\inf_{q\in{{\mathcal{P}}({\mathcal{S}})}}I(Z_q;X\vert Y,U)\\
&=\max_{p\in\Pi}\inf_{q\in{{\mathcal{P}}({\mathcal{S}})}}I(Z_q;X\vert Y)\\
&=\max_{y\in{\mathcal{Y}}}\max_{r\in{\mathcal{P}}({\mathcal{X}})}\inf_{q\in{{\mathcal{P}}({\mathcal{S}})}}I(Z_q;X\vert Y=y),\end{aligned}$$ where in the last term, the random vector $(X,Z_q)$ has the distribution $r(x)W(z\vert x,y\vert q)$.
\[AC\_Satz\]
1. If ${\mathcal{W}}$ is neither $({\mathcal{X}},{\mathcal{Y}})$- nor ${\mathcal{X}}$- nor ${\mathcal{Y}}$-symmetrizable, then ${\mathcal{C}}_d({\mathcal{S}},0,0)={\mathcal{C}}^*({\mathcal{S}},0,0)$ and ${\mathcal{C}}^*({\mathcal{S}},0,0)$ has nonempty interior.
2. If ${\mathcal{W}}$ is neither $({\mathcal{X}},{\mathcal{Y}})$- nor ${\mathcal{X}}$-symmetrizable, but ${\mathcal{Y}}$-symmetrizable, then $$\begin{aligned}
&{\mathcal{C}}_d({\mathcal{S}},0,0)\\&\quad\subset[0,\max_{y\in{\mathcal{Y}}}\max_{r\in{\mathcal{P}}({\mathcal{X}})}\inf_{q\in{{\mathcal{P}}({\mathcal{S}})}}I(Z_q;X\vert Y=y)]\times\{0\}.\end{aligned}$$
3. If ${\mathcal{W}}$ is neither $({\mathcal{X}},{\mathcal{Y}})$- nor ${\mathcal{Y}}$-symmetrizable, but ${\mathcal{X}}$-symmetrizable, then $$\begin{aligned}
&{\mathcal{C}}_d({\mathcal{S}},0,0)\\&\quad\subset\{0\}\times[0,\max_{x\in{\mathcal{X}}}\max_{r\in{\mathcal{P}}({\mathcal{Y}})}\inf_{q\in{{\mathcal{P}}({\mathcal{S}})}}I(Z_q;Y\vert X=x)].\end{aligned}$$
4. If $\W$ is $({\mathcal{X}},{\mathcal{Y}})$-symmetrizable, then ${\mathcal{C}}_d({\mathcal{S}},0,0)=\{(0,0)\}$.
In particular if $\W$ is both ${\mathcal{X}}$- and ${\mathcal{Y}}$-symmetrizable, then ${\mathcal{C}}_d({\mathcal{S}},0,0)=\{(0,0)\}$.
1\) from Theorem \[AC\_Satz\] is due to [@AC] and [@J]. The other points are due to [@G2; @G]. The precise characterization of ${\mathcal{C}}_d({\mathcal{S}},0,0)$ in points 2) and 3) is still open.
The relation between the three kinds of symmetrizability from Definition \[defsymm\] is treated in Section \[sect:concl\]. There we provide the example of an AV-MAC which is both ${\mathcal{X}}$- and ${\mathcal{Y}}$-symmetrizable but not $({\mathcal{X}},{\mathcal{Y}})$-symmetrizable.
Related Coding Results
----------------------
The set $\WQ$ also determines a compound MAC. This channel differs from the AV-MAC in that it does not change its state during the transmission of a codeword, only constant state sequences are possible. Thus the probability that ${\mathbf{z}}\in{\mathcal{Z}}^n$ is received given the transmission of words ${\mathbf{x}}\in{\mathcal{X}}^n$, ${\mathbf{y}}\in{\mathcal{Y}}^n$ only depends on the given state $ q\in{{\mathcal{P}}({\mathcal{S}})}$. It equals $$\label{Wprodmass}
W^n({\mathbf{z}}\vert{\mathbf{x}},{\mathbf{y}}\vert{\mathbf{q}}_q):=\prod_{m=1}^nW(z_m\vert x_m,y_m\vert q),$$ where we denote elements of ${\mathcal{P}}({\mathcal{S}})^n$ by ${\mathbf{q}}$ and set ${\mathbf{q}}_q:=(q,\ldots,q)\in{\mathcal{P}}({\mathcal{S}})^n$.
The set of stochastic matrices $$\{W^n(\,\cdot\,\vert\,\cdot\,,\,\cdot\,\vert{\mathbf{q}}_q):q\in{{\mathcal{P}}({\mathcal{S}})},n=1,2,\ldots\}$$ is called the *compound Multiple Access Channel (compound MAC)* determined by $\WQ$.
One uses the same deterministic codes$_{\textnormal{CONF}}$ as for the AV-MAC. Let $$\{({\mathbf{x}}_{jk},{\mathbf{y}}_{jk},F_{jk}):(j,k)\in[1,M_1]\times[1,M_2]\}$$ be such a code$_{\textnormal{CONF}}$. It has an average error less than $\lambda$ for the compound MAC determined by $\WQ$ if $$\frac{1}{M_1M_2}\sum_{j,k}W^n(F_{jk}^c\vert{\mathbf{x}}_{jk},{\mathbf{y}}_{jk}\vert{\mathbf{q}}_q)\leq\lambda$$ for every $q\in{{\mathcal{P}}({\mathcal{S}})}$. Using this average error criterion, the concept of achievability and the definition of the capacity region is analogous to that for deterministic coding for AV-MACs.
Comparing the error criteria for the AV-MAC and the compound MAC from the adversary perspective, one observes that the AV-MAC yields a significantly more robust performance. Theorem \[thmconf\] describes the region achievable if transmission is reliable for every possible sequence the adversary might choose, whereas Theorem \[thmcomp\] describes the region which is achievable if the adversary is restricted to constant state sequences.
In [@WBBJ11] the following theorem was proved.
\[thmcomp\] The capacity region of the compound MAC determined by $\WQ$ with conferencing capacities $C_1,C_2\geq 0$ equals ${\mathcal{C}}^*({\mathcal{S}},C_1,C_2)$. More precisely, for every achievable rate pair $(R_1,R_2)\in{\mathcal{C}}^*({\mathcal{S}},C_1,C_2)$ and every ${\varepsilon}>0$, there is a $\zeta>0$ and a sequence of codes$_{\textnormal{CONF}}$ $(n,M_1^{(n)},M_2^{(n)},C_1,C_2)$ with an average error at most $2^{-n\zeta}$ and $$\frac{1}{n}\log M_\nu^{(n)}\geq R_\nu-{\varepsilon}\quad(\nu=1,2).$$ These codes$_{\textnormal{CONF}}$ can be chosen such that their conferencing protocols have the form .
Finally we have to recall the definition and a corollary of the deterministic coding result for single-user Arbitrarily Varying Channels (AVCs). Let ${\mathcal{A}}$ be a finite input alphabet, ${\mathcal{B}}$ a finite output alphabet, and ${\mathcal{S}}$ a finite state set. Let a family $${\mathcal{H}}:=\{H(\,\cdot\,\vert\,\cdot\,\vert s):s\in{\mathcal{S}})\}$$ of stochastic matrices $$H(b\vert a\vert s):\quad(a,b)\in{\mathcal{A}}\times{\mathcal{B}}$$ be given. As for AV-MACs, every state sequence ${\mathbf{s}}=(s_1,\ldots,s_n)\in{\mathcal{S}}^n$ determines a new stochastic matrix $$ H^n({\mathbf{b}}\vert{\mathbf{a}}\vert{\mathbf{s}}):=\prod_{m=1}^nH(b_m\vert a_m\vert s_m):\quad({\mathbf{a}},{\mathbf{b}})\in{\mathcal{A}}^n\times{\mathcal{B}}^n.$$
The set of stochastic matrices $$\{H^n(\,\cdot\,\vert\,\cdot\,\vert{\mathbf{s}}):{\mathbf{s}}\in{\mathcal{S}}^n,n=1,2,\ldots\}$$ is called the *Arbitrarily Varying Channel (AVC)* determined by ${\mathcal{H}}$.
The admissible codes are classical single-user codes as used for discrete memoryless channels. If such a code with blocklength $n$ and codelength $M$ has the form $$\{({\mathbf{a}}_\ell,F_\ell):\ell\in[1,M]\},$$ then the average error incurred by this code is smaller than $\lambda\in(0,1)$ if $$\frac{1}{M}\sum_\ell H^n(F_\ell^c\vert{\mathbf{a}}_\ell\vert{\mathbf{s}})\leq\lambda\qquad\text{for all }{\mathbf{s}}\in{\mathcal{S}}^n.$$ Then it is obvious what is meant by “achievable rates” and “capacity” for ${\mathcal{H}}$. The capacity of single-user AVCs, which was determined in [@CN], exhibits a dichotomy similar to the one claimed in Theorem \[thmconf\]. It is described by the original symmetrizability concept from [@E].
\[susymm\] ${\mathcal{H}}$ is called *symmetrizable* if there is a stochastic matrix $$\sigma(s\vert a):\quad(a,s)\in{\mathcal{A}}\times{\mathcal{S}}$$ such that for every $b\in{\mathcal{B}}$ and $a,a'\in{\mathcal{A}}$ $$\sum_sH(b\vert a\vert s)\sigma(s\vert a')=\sum_sH(b\vert a'\vert s)\sigma(s\vert a).$$
\[rem:equi\] Clearly, the $({\mathcal{X}},{\mathcal{Y}})$-symmetrizability of the MAC ${\mathcal{W}}$ means nothing but symmetrizability of ${\mathcal{W}}$ when considered as a set of stochastic matrices with inputs from the alphabet ${\mathcal{A}}={\mathcal{X}}\times{\mathcal{Y}}$.
\[CN\_Satz\] The deterministic capacity of the single-user AVC determined by ${\mathcal{H}}$ is positive if and only if ${\mathcal{H}}$ is not symmetrizable. If ${\mathcal{H}}$ is symmetrizable, then every code with at least two codewords incurs an average error at least $1/4$.
The Direct Parts {#sect:ach}
================
We derive the direct part of Theorem \[thmconfrand\] from Theorem \[thmcomp\] in Subsections \[comparb\] and \[subsect:amount\]. Then, if ${\mathcal{W}}$ is not $({\mathcal{X}},{\mathcal{Y}})$-symmetrizable, we derandomize in Subsections \[subsect:posrat\] and \[subsect:randdet\] to obtain the direct part of Theorem \[thmconf\].
From Compound to Arbitrarily Varying {#comparb}
------------------------------------
Here we prove the direct part of Theorem \[thmconfrand\]. We use Ahlswede’s “robustification lemma”. Let $S_n$ be the symmetric group (the group of permutations) on the set $[1,n]$. $S_n$ operates on ${\mathcal{S}}^n$ by $\pi({\mathbf{s}}):=(s_{\pi(1)},\ldots,s_{\pi(n)})$ for any $\pi\in S_n$ and ${\mathbf{s}}=(s_1,\ldots,s_n)\in{\mathcal{S}}^n$. Further recall the notation ${\mathbf{q}}_q$ defined in .
\[robusti\] If $h:{\mathcal{S}}^n\rightarrow[0,1]$ satisfies for a $\lambda\in(0,1)$ and for all $q\in{\mathcal{P}}({\mathcal{S}})$ the inequality $$\label{h-bed}
\sum_{{\mathbf{s}}\in{\mathcal{S}}^n}h({\mathbf{s}}){\mathbf{q}}_q({\mathbf{s}})\geq1-\lambda,$$ then it also satisfies the inequality $$\frac{1}{n!}\sum_{\pi\in S_n}h(\pi({\mathbf{s}}))\geq 1-(n+1)^{\lvert{\mathcal{S}}\rvert}\lambda\qquad\text{for all }{\mathbf{s}}\in{\mathcal{S}}^n.$$
Now let $(R_1,R_2)\in{\mathcal{C}}^*({\mathcal{S}},C_1,C_2)$. Theorem \[thmcomp\] states that for any ${\varepsilon}>0$ there is a $\zeta>0$ such that for sufficiently large $n$ there is a code$_{\textnormal{CONF}}(n,M_1,M_2,C_1,C_2)$ with an average error at most $2^{-n\zeta}$ and satisfying $$\frac{1}{n}\log M_\nu\geq R_\nu-{\varepsilon}\quad(\nu=1,2).$$ Writing this code$_{\textnormal{CONF}}$ in the form , this means for every $ q\in{{\mathcal{P}}({\mathcal{S}})}$ that $$\label{comperr}
\frac{1}{M_1M_2}\sum_{j,k}W^n(F_{jk}\vert{\mathbf{x}}_{jk},{\mathbf{y}}_{jk}\vert{\mathbf{q}}_q)\geq 1-2^{-n\zeta}.$$ We would like to apply Lemma \[robusti\] with $\lambda=2^{-n\zeta}$ to the function $h:{\mathcal{S}}^n\rightarrow[0,1]$ defined by $$h({\mathbf{s}}):=\frac{1}{M_1M_2}\sum_{j,k}W^n(F_{jk}\vert{\mathbf{x}}_{jk},{\mathbf{y}}_{jk}\vert{\mathbf{s}}).$$ Thus we need to show that $h$ satisfies . Let $q\in{\mathcal{P}}({\mathcal{S}})$. By , one obtains $$\begin{aligned}
&\mathrel{\hphantom{=}}\sum_{{\mathbf{s}}\in{\mathcal{S}}^n}h({\mathbf{s}})q^n({\mathbf{s}})\\
&=\frac{1}{M_1M_2}\sum_{j,k}\sum_{{\mathbf{z}}\in F_{jk}}\sum_{{\mathbf{s}}\in{\mathcal{S}}^n}W^n({\mathbf{z}}\vert{\mathbf{x}}_{jk},{\mathbf{y}}_{jk}\vert{\mathbf{s}}){\mathbf{q}}_q({\mathbf{s}})\\
&=\frac{1}{M_1M_2}\sum_{j,k}\sum_{{\mathbf{z}}\in F_{jk}}W^n({\mathbf{z}}\vert{\mathbf{x}}_{jk},{\mathbf{y}}_{jk}\vert{\mathbf{q}}_q)\\
&\geq 1-2^{-n\zeta},\end{aligned}$$ and is satisfied. Applying Lemma \[robusti\], one obtains $$\label{randerr}
\frac{1}{n!}\sum_{\pi\in S_n}h(\pi({\mathbf{s}}))\geq 1-(n+1)^{\lvert{\mathcal{S}}\rvert}2^{-n\zeta}\qquad\text{for all }{\mathbf{s}}\in{\mathcal{S}}^n.$$ Recall that $\pi^{-1}$ also is an element of $S_n$. Writing $\pi^{-1}(F_{jk})=\{\pi^{-1}({\mathbf{z}}):{\mathbf{z}}\in F_{jk}\}$, the left side of equals $$\begin{aligned}
&\hphantom{\mathrel{=}}\frac{1}{n!}\sum_{\pi\in S_n}\biggl(\frac{1}{M_1M_2}\sum_{j,k}W^n(F_{jk}\vert{\mathbf{x}}_{jk},{\mathbf{y}}_{jk}\vert\pi({\mathbf{s}}))\biggr)\notag\\
&=\frac{1}{n!}\sum_{\pi\in S_n}\biggl(\frac{1}{M_1M_2}\times\notag\\
&\mathrel{\hphantom{=}}\times\sum_{j,k}W^n\left(\pi^{-1}(F_{jk})\vert\pi^{-1}({\mathbf{x}}_{jk}),\pi^{-1}({\mathbf{y}}_{jk})\vert{\mathbf{s}}\right)\biggr).\label{expr}\end{aligned}$$ Because of the bijectivity of $\pi^{-1}$, the family of sets $\{\pi^{-1}(F_{jk}):(j,k)\in[1,M_1]\times[1,M_2]\}$ is disjoint. Thus is the average error expression of the random code $(C,G)$ when applied in the AV-MAC determined by $\W$, where $G$ is uniformly distributed on $\Gamma:=S_n$ and where for every $\pi\in S_n$, $$\begin{aligned}
C(\pi):=\{\bigl(\pi^{-1}({\mathbf{x}}_{jk}),&\pi^{-1}({\mathbf{y}}_{jk}),\pi^{-1}(F_{jk})\bigr):\\&(j,k)\in[1,M_1]\times[1,M_2]\}.\end{aligned}$$ The conferencing protocol remains the same for every $C(\pi)$, as the conference only concerns the messages and not the codewords. By the average error of this random code is less than $(n+1)^{\lvert{\mathcal{S}}\rvert}2^{-n\zeta}$, hence it tends to zero exponentially. Thus ${\mathcal{C}}^*({\mathcal{S}},C_1,C_2)\subset{\mathcal{C}}_r({\mathcal{S}},C_1,C_2)$, which proves the direct part of Theorem \[thmconfrand\].
Bounding the amount of correlation {#subsect:amount}
----------------------------------
As a first derandomization step to proving the direct part of Theorem \[thmconf\], we have to show the following lemma.
\[n\^2\] To every random code$_{\textnormal{CONF}}(n,M_1,M_2,C_1,C_2)$ with average error at most $\lambda$ which is given as a pair $(C,G)$ there exists a random code$_{\textnormal{CONF}}(n,M_1,M_2,C_1,C_2)$ with an average error smaller than $3\lambda$ given as a pair $(C',G')$ where $C'\subset C$ and $\lvert C'\rvert= n^2$ and where $G'$ is uniformly distributed on $[1,n^2]$.
For the proof of Lemma \[n\^2\], we need a simple result from [@J Section IV].
\[Jahnslemma\] Let $N$ i.i.d. random variables $T_1,\ldots,T_N$ with values in $[0,1]$ and underlying probability measure ${\mathbb{P}}$ be given. Let $\bar\lambda>0$. Denote by ${\mathbb{E}}$ the expectation corresponding to ${\mathbb{P}}$. Then $${\mathbb{P}}\left[\frac{1}{N}\sum_{m=1}^NT_m>\bar\lambda\right]\leq\exp\left(-(\bar\lambda-e{\mathbb{E}}[T_1])N\right).$$
Let a random code$_{\textnormal{CONF}}$ $(C,G)$ with blocklength $n$ and average error smaller than $\lambda$. Recalling our notation , the fact that $(C,G)$ has an average error less than $\lambda$ can be stated as $${\mathbb{E}}[P_e(C(G)\vert {\mathbf{s}})]\leq\lambda\quad\text{for every }{\mathbf{s}}\in{\mathcal{S}}^n.$$ Let $G_1,\ldots,G_{n^2}$ be independent copies of $G$. This induces a family of $n^2$ independent copies of $(C,G)$. The goal is to show $$\label{goal}
{\mathbb{P}}\biggl[\;\frac{1}{n^2}\sum_{m=1}^{n^2} P_e(C(G_m)\vert {\mathbf{s}})\leq3\lambda\text{ for all }{\mathbf{s}}\in{\mathcal{S}}^n\biggr]>0.$$ Given , there is a realization $(\gamma_1,\ldots,\gamma_{n^2})$ of $(G_1,\ldots,G_{n^2})$ such that $$\label{reform}
\frac{1}{n^2}\sum_{m=1}^{n^2} P_e(C(\gamma_m)\vert {\mathbf{s}})\leq3\lambda$$ for every ${\mathbf{s}}\in{\mathcal{S}}^n$. Then one defines a random code $(C',G')$ by setting $$C':=\{C(\gamma_m):m \in[1,n^2]\}$$ and by taking $G'$ to be uniformly distributed on $[1,n^2]$. The expression then is nothing but the statement that the average error of the random code $(C',G')$ is smaller than $3\lambda$, and we are done.
It remains to prove . ${\mathcal{S}}$ is finite by assumption, so $\lvert{\mathcal{S}}^n\rvert$ grows exponentially with blocklength. Hence it suffices to show that $$\label{suffices}
{\mathbb{P}}\biggl[\;\frac{1}{n^2}\sum_m P_e(C(G_m)\vert {\mathbf{s}})>3\lambda\biggr]$$ is superexponentially small uniformly in ${\mathbf{s}}\in{\mathcal{S}}^n$. Let us fix an ${\mathbf{s}}\in{\mathcal{S}}^n$. The $G_m$ are i.i.d. copies of $G$, so by Lemma \[Jahnslemma\], the term is smaller than $$\label{bound}
\exp\bigl(-\left(3\lambda-e\,{\mathbb{E}}[P_e(C(G)\vert {\mathbf{s}})]\right)n^2\bigr).$$ By assumption $${\mathbb{E}}[P_e(C(G)\vert {\mathbf{s}})]\leq\lambda,$$ so the exponent in is negative. This gives the desired superexponential bound on .
Note that we cannot require the codes$_{\textnormal{CONF}}$ with at most $n^2$ values of $G$ to have an exponentially small probability of error. This is due to the fact that the exponent in must not decrease exponentially in order for the proof to work. Thus there is a trade-off between the error probability and the number of deterministic component codes$_{\textnormal{CONF}}$ of the random codes$_{\textnormal{CONF}}$ used to achieve the random capacity region of the AV-MAC with conferencing encoders.
A Positive Rate {#subsect:posrat}
---------------
In the second derandomization step, we show that if ${\mathcal{W}}$ is not $({\mathcal{X}},{\mathcal{Y}})$-symmetrizable and $C_1>0$ or $C_2>0$ then the encoder with the positive conferencing capacity achieves a positive rate by deterministic coding. Without loss of generality we may assume that $C_1>0$.
\[thmposrat\] Let $C_1>0$. If ${\mathcal{W}}$ is not $({\mathcal{X}},{\mathcal{Y}})$-symmetrizable, then there exists an $R$ with $0<R<C_1$ such that the rate pair $(R,0)\in{\mathcal{C}}^*({\mathcal{S}},C_1,C_2)$ is deterministically achievable using codes$_{\textnormal{CONF}}$ with conferencing capacity pair $(C_1,0)$ such that the conferencing function $c_1$ is the identity on the message set.
By Remark \[rem:equi\] and Theorem \[CN\_Satz\], ${\mathcal{W}}$ considered as a single-user AVC with vector inputs from the alphabet ${\mathcal{X}}\times{\mathcal{Y}}$ has positive capacity. The idea of the proof is to construct from a code for this single-user AVC a code$_{\textnormal{CONF}}$ such that the first transmitter achieves a positive rate. There is a positive rate $R<C_1$ which is deterministically achievable by the single-user AVC determined by ${\mathcal{W}}$. This means that for every $\lambda\in(0,1)$ and every ${\varepsilon}>0$, for $n$ large enough, there is a single-user code $$\{({\mathbf{x}}_\ell,{\mathbf{y}}_\ell,F_\ell):\ell\in[1,M_1]\}$$ for ${\mathcal{W}}$ with $$2^{n(R-{\varepsilon}/2)}\leq M_1\leq2^{nR}$$ and with $$\frac{1}{M_1}\sum_{\ell=1}^{M_1}W^n(F_\ell^c\vert{\mathbf{x}}_\ell,{\mathbf{y}}_\ell\vert{\mathbf{s}})\leq\lambda\qquad\text{for all }{\mathbf{s}}\in{\mathcal{S}}^n.$$
By setting $c_1$ to be the identity on $[1,M_1]$, this code becomes a code$_{\textnormal{CONF}}(n,M_1,1,C_1,0)$. This is allowed because $\log M_1\leq nR\leq nC_1$. The encoding and decoding functions are defined in the obvious way. Thus the positive rate pair $(R,0)$ is achievable.
From Random to Deterministic {#subsect:randdet}
----------------------------
Finally we can show that if ${\mathcal{W}}$ is not $({\mathcal{X}},{\mathcal{Y}})$-symmetrizable, then ${\mathcal{C}}^*({\mathcal{S}},C_1,C_2)\subset{\mathcal{C}}_d({\mathcal{S}},C_1,C_2)$. To do so we follow Ahlswede’s “Elimination Technique” [@A1], whose idea is to use random codes and to replace the randomness needed there by a prefix code with small blocklength which encodes the set of constituent deterministic codes. We again assume that $C_1>0$.
Theorem \[thmposrat\] implies that there is a $0<R<C_1$ such that for any ${\varepsilon}\in(0,R)$ and any $\lambda\in(0,1)$, if $n$ is large, there is a code$_{\textnormal{CONF}}$ $$\{({\mathbf{x}}_{\gamma}^*,{\mathbf{y}}_{\gamma}^*,F_{\gamma}^*):\gamma\in[1,n^2]\}$$ with blocklength $m$, $$\label{m-bed}
\frac{2}{R}\log n\leq m\leq\frac{2}{R-{\varepsilon}}\log n,$$ with codelength pair $(n^2,0)$ and with average error smaller than $\lambda$. Further, the conferencing function $c_1^*$ is the identity on the set $[1,n^2]$.
For any $0<\delta<C_1$, let $(R_1,R_2)\in{\mathcal{R}}(p, q,C_1-\delta,C_2)$ for some $p\in\Pi$ and all $q\in{{\mathcal{P}}({\mathcal{S}})}$. By Theorem \[thmconfrand\], this rate pair is achievable with conferencing capacities $C_1-\delta$ and $C_2$ under random coding. For every ${\varepsilon}>0$, $\lambda\in(0,1)$, and large $n$ this implies the existence of a random code$_{\textnormal{CONF}}(n,M_1,M_2,C_1-\delta,C_2)$ defined by a pair $(C,G)$ such that $$\frac{1}{n}\log M_\nu\geq R_\nu-{\varepsilon}$$ and with an average error smaller than $\lambda$. Again by Theorem \[thmconfrand\], we may further assume that the deterministic component codes$_{\textnormal{CONF}}$ $C(\gamma)$ of $C$ share the same conferencing protocol $(c_1,c_2)$. As we do not need an exponential decrease of the average error, we may by Lemma \[n\^2\] assume that $G$ is uniformly distributed on $\Gamma=[1,n^2]$. Let every deterministic component code$_{\textnormal{CONF}}$ $C(\gamma)$ be given as $$\{({\mathbf{x}}_{jk}^{\gamma},{\mathbf{y}}_{jk}^{\gamma},F_{jk}^{\gamma}):(j,k)\in[1,M_1]\times[1,M_2]\}.$$
We now construct a deterministic code$_{\textnormal{CONF}}$ $(\tilde c_1,\tilde c_2,\tilde f_1,\tilde f_2,\tilde\Phi)$ with blocklength $m+n$, message sets $[1,n^2]\times[1,M_1]$ and $[1,M_2]$ (yielding the codelength pair $(n^2M_1,M_2)$), conferencing capacities $C_1,C_2$, and average error smaller than 2$\lambda$. It is defined via concatenation. We define the conferencing functions to be $$\begin{aligned}
\tilde c_1(\gamma,j)&:=(\gamma,c_1(j))\in[1,n^2]\times[1,V_1],\\\tilde c_2(k)&:=c_2(k)\in[1,V_2].\end{aligned}$$ Note that $(\tilde c_1,\tilde c_2)$ has the form . It is a permissible conferencing protocol if $$\frac{1}{2\,R^{-1}\log n+n}\log n\leq\delta,$$ because then $$\frac{1}{m+n}\log(n^2V_1)\leq\frac{1}{2\,R^{-1}\log n+n}\log n+\frac{1}{n}\log V_1\leq C_1.$$
If the encoders have the messages $(\gamma,j)$ and $k$, respectively, they use the codewords $$\bigl({\mathbf{x}}^*_{\gamma},{\mathbf{x}}^{\gamma}_{jk}\bigr)\in{\mathcal{X}}^{m+n}\quad\text{and}\quad\bigl({\mathbf{y}}^*_{\gamma},{\mathbf{y}}^{\gamma}_{jk}\bigr)\in{\mathcal{Y}}^{m+n}.$$ Together with the conferencing protocol $(\tilde c_1,\tilde c_2)$ defined above, this fixes encoding functions $f_1$ and $f_2$, as and are satisfied. The decoding set of the code$_{\textnormal{CONF}}$ deciding for the pair $\bigl((\gamma,j),k\bigr)$ is defined to be $F^*_{\gamma}\times F_{jk}^{\gamma}\subset{\mathcal{Z}}^{m+n}$. Thus the deterministic code$_{\textnormal{CONF}}$ achieving the rate pair $(R,0)$ is used as a prefix code which distinguishes the deterministic component codes$_{\textnormal{CONF}}$ of the random code$_{\textnormal{CONF}}$. In this way, derandomization can be seen as a two-step protocol. Setting $a:=2/(R-{\varepsilon})$, the rates of the new code are $$\frac{1}{m+n}\log(nM_\nu)
\geq\frac{1}{a\frac{\log n}{n}+1}\cdot\frac{1}{n}\log M_\nu\geq R_\nu-2{\varepsilon},$$ where the second inequality holds for all $n$ large enough such that $$\frac{1}{a\frac{\log n}{n}+1}\geq\frac{R_\nu-2{\varepsilon}}{R_\nu-{\varepsilon}}.$$
The randomness of the random code is needed in the estimation of the average error incurred by this coding procedure. Recall Ahlswede’s Innerproduct Lemma [@A1]:
\[innerprod\] Let $(\alpha_1,\ldots,\alpha_N)$ and $(\beta_1,\ldots,\beta_N)$ be two vectors with $0\leq\alpha_m,\beta_m\leq 1$ for $m=1,\ldots,N$ which for some $\lambda\in(0,1)$ satisfy $$\label{geschbed}
\frac{1}{N}\sum_{m=1}^N\beta_m\geq 1-\lambda,\qquad\frac{1}{N}\sum_{m=1}^N\alpha_m\geq 1-\lambda,$$ then $$\frac{1}{N}\sum_{m=1}^N\alpha_m\beta_m\geq1-2\lambda.$$
We use this lemma with $N=n^2$ and replace the index $m$ by $\gamma\in[1,n^2]$. Fix an ${\mathbf{s}}\in{\mathcal{S}}^n$ and set $$\begin{aligned}
\alpha_{\gamma}&=W^m(F^*_{\gamma}\vert{\mathbf{x}}^*_{\gamma},{\mathbf{y}}^*_{\gamma}\vert{\mathbf{s}}),\\
\beta_{\gamma}&=\frac{1}{M_1M_2}\sum_{j,k}W^n\bigl(F_{jk}^{\gamma}\vert{\mathbf{x}}^{\gamma}_{jk},{\mathbf{y}}^{\gamma}_{jk}\vert{\mathbf{s}}\bigr).\end{aligned}$$ Then the conditions in are satisfied because both the deterministic prefix code $(c_1^*,c_2^*,f_1^*,f_2^*,\Phi^*)$ and the random code $(C,G)$ with constituent codes $(c_1,c_2,f_1^{\gamma},f_2^{\gamma},\Phi^{\gamma})$ have an average error smaller than $\lambda$. Lemma \[innerprod\] now implies that the code$_{\textnormal{CONF}}$ $(\tilde c_1,\tilde c_2,\tilde f_1,\tilde f_2,\tilde\Phi)$ constructed above has an average error probability smaller than $2\lambda$.
This shows that the rate pair $(R_1,R_2)$ is achievable for ${\mathcal{W}}$ with conferencing capacities $C_1,C_2$. Consequently one obtains $$\bigcup_{\delta>0}{\mathcal{C}}^*({\mathcal{S}},C_1-\delta,C_2)\subset{\mathcal{C}}_d({\mathcal{S}},C_1,C_2).$$ As the capacity region is closed, ${\mathcal{C}}^*({\mathcal{S}},C_1,C_2)$, which is the closure of the set on the left-hand side, is contained in ${\mathcal{C}}_d({\mathcal{S}},C_1,C_2)$ as well. This proves the direct part of Theorem \[thmconf\].
Converses for the AV-MAC with Conferencing Encoders {#sect:conv}
===================================================
Here we prove the converses claimed in Remark \[randconv\] and \[detconv\]. Recall that a weak converse means that, depending on the situation, any deterministic or random code$_{\textnormal{CONF}}(n,M_1,M_2,C_1,C_2)$ such that the real two-dimensional vector $((1/n)\log M_1,(1/n)\log M_2)$ is at least distance ${\varepsilon}>0$ from the achievable rate region incurs an average error at least $\lambda({\varepsilon})>0$ if $n$ is large.
Random Coding
-------------
Here we prove the weak converse for Theorem \[thmconfrand\] (see Remark \[randconv\]). The idea of the proof is to reduce it to the weak converse for the compound MAC with conferencing encoders defined by $\WQ$ where the use of random codes is allowed. This is proved in the Appendix.
Every ${\mathbf{q}}\in{\mathcal{P}}({\mathcal{S}})^n$ induces a product measure via ${\mathbf{q}}({\mathbf{s}})=q_1(s_1)\cdots q_n(s_n)$. The notation carries over to these general ${\mathbf{q}}$. Further, recall the notation introduced in . We generalize this notation by setting $$P_e(C(\gamma)\vert{\mathbf{q}}):=\frac{1}{M_1M_2}\sum_{j,k}W^n((F_{jk}^\gamma)^c\vert{\mathbf{x}}_{jk}^\gamma,{\mathbf{y}}_{jk}^\gamma\vert{\mathbf{q}}).$$ The following lemma is a generalized version of Lemma 2.6.3 in [@CK].
\[auxlem\] For any random code$_{\textnormal{CONF}}$ which is defined by $(C,G)$ and whose components have the form $$\{({\mathbf{x}}_{jk}^\gamma,{\mathbf{y}}_{jk}^\gamma,F_{jk}^\gamma):(j,k)\in[1,M_1]\times[1,M_2]\},$$ one has $$\begin{aligned}
\sup_{{\mathbf{s}}\in{\mathcal{S}}^n}\sum_{\gamma\in\Gamma}P_e(C(\gamma)\vert{\mathbf{s}})&p_G(\gamma)\\&=\sup_{{\mathbf{q}}\in{\mathcal{P}}({\mathcal{S}})^n}\sum_{\gamma\in\Gamma}P_e(C(\gamma)\vert{\mathbf{q}})p_G(\gamma).\end{aligned}$$
The direction “$\leq$” is clear. In order to prove “$\geq$”, let ${\mathbf{q}}\in{\mathcal{P}}({\mathcal{S}})^n$. Clearly $$W^n({\mathbf{z}}\vert{\mathbf{x}},{\mathbf{y}}\vert{\mathbf{q}})
=\sum_{{\mathbf{s}}\in{\mathcal{S}}^n}{\mathbf{q}}({\mathbf{s}})W^n({\mathbf{z}}\vert{\mathbf{x}},{\mathbf{y}}\vert{\mathbf{s}}).$$ Thus $$\begin{aligned}
\sum_{\gamma\in\Gamma}P_e(C(\gamma)\vert{\mathbf{q}})p_G(\gamma)&=\sum_{{\mathbf{s}}\in{\mathcal{S}}^n}{\mathbf{q}}({\mathbf{s}})\sum_{\gamma\in\Gamma}P_e(C(\gamma)\vert{\mathbf{s}})p_G(\gamma)\\
&\leq\sup_{{\mathbf{s}}\in{\mathcal{S}}^n}\sum_{\gamma\in\Gamma}P_e(C(\gamma)\vert{\mathbf{s}})p_G(\gamma).\end{aligned}$$ Upon taking the supremum over ${\mathbf{q}}\in{\mathcal{P}}({\mathcal{S}})^n$ on the left-hand side, the lemma is proved.
Now let a random code$_{\textnormal{CONF}}$ $(n,M_1,M_2,C_1,C_2)$ be given defined by $(C,G)$ and with average error at most $\lambda$. Assume that the pair $((1/n)\log M_1,(1/n)\log M_2)$ is at distance at least ${\varepsilon}$ from ${\mathcal{C}}^*({\mathcal{S}},C_1,C_2)$. Because of Lemma \[auxlem\], $$\label{canonly}
\lambda_0:=\sup_{q\in{\mathcal{P}}({\mathcal{S}})}\sum_{\gamma\in\Gamma}P_e(C(\gamma)\vert{\mathbf{q}}_q)p_G(\gamma)\leq\lambda.$$ Thus the random code$_{\textnormal{CONF}}$ $(C,G)$ has an average error at most $\lambda$ for the compound MAC with conferencing encoders defined by $\WQ$. But the weak converse for the compound MAC with conferencing encoders and random coding, which is proved in the Appendix, implies that can only hold if $\lambda\geq\lambda_0\geq\lambda({\varepsilon})>0$. This concludes the weak converse for the AV-MAC with conferencing encoders using random codes$_{\textnormal{CONF}}$, and Theorem \[thmconfrand\] is proved.
Deterministic Coding
--------------------
### If ${\mathcal{W}}$ is $({\mathcal{X}},{\mathcal{Y}})$-symmetrizable
If ${\mathcal{W}}$ is $({\mathcal{X}},{\mathcal{Y}})$-symmetrizable, then by Remark \[rem:equi\] it is also symmetrizable if considered as a single-user AVC with input alphabet ${\mathcal{X}}\times{\mathcal{Y}}$. Thus Theorem \[CN\_Satz\] implies any single-user code with at least two codewords incurs an average error greater than $1/4$. Finally, note that every code$_{\textnormal{CONF}}$ for the AV-MAC with conferencing encoders determined by ${\mathcal{W}}$ also is a code for the single-user AVC determined by $\W$, so this carries over to the multi-user situation. This proves Theorem \[thmconf\] if ${\mathcal{W}}$ is $({\mathcal{X}},{\mathcal{Y}})$-symmetrizable.
### If ${\mathcal{W}}$ is not $({\mathcal{X}},{\mathcal{Y}})$-symmetrizable
We show that the weak converse for the compound MAC determined by $\WQ$ implies the weak converse for the AV-MAC determined by ${\mathcal{W}}$. Let a code$_{\textnormal{CONF}}$ $(n,M_1,M_2,C_1,C_2)$ be given. If the rate pair $((1/n)M_1,(1/n)M_2)$ is at least distance ${\varepsilon}$ away from ${\mathcal{C}}^*({\mathcal{S}},C_1,C_2)$ and if $n$ is sufficiently large, then there is a $q\in{{\mathcal{P}}({\mathcal{S}})}$ such that $$\frac{1}{M_1M_2}\sum_{j,k}W^n(F_{jk}^c\vert{\mathbf{x}}_{jk},{\mathbf{y}}_{jk}\vert{\mathbf{q}}_q)\geq\lambda({\varepsilon})$$ for some $\lambda({\varepsilon})>0$ because of the weak converse for the compound MAC. Lemma \[auxlem\] now implies that $$\sup_{{\mathbf{s}}\in{\mathcal{S}}^n}\frac{1}{M_1M_2}\sum_{j,k}W^n(F_{jk}^c\vert{\mathbf{x}}_{jk},{\mathbf{y}}_{jk}\vert{\mathbf{s}})\geq\lambda({\varepsilon})$$ must hold. Thus the proof of Theorem \[thmconf\] is complete.
Discussion and Conclusion {#sect:concl}
=========================
The goal of this paper was to characterize the capacity region of an AV-MAC whose encoders may exchange limited information about their messages. This topic is motivated by the increasing interest of cooperative networks which are subject to exterior interference. For example, spectrum sharing has been discussed for inclusion into future wireless system standards. We saw above that the AV-MAC can be interpreted as a channel suffering from attacks by an adversary who may choose the state sequence given the channel inputs. The reliability requirements for AV-MACs are very strict – coding is done such that the average error is small for every possible state sequence. The resulting capacity region is the same as that for the conferencing compound MAC determined by the convex hull of the set of channel matrices of the original AV-MAC *if* the latter is not $({\mathcal{X}},{\mathcal{Y}})$-symmetrizable. Otherwise the AV-MAC is useless. In contrast, for AV-MACs without conferencing, the complete characterization of the deterministic capacity region is still open.
The dichotomy in the form of the deterministic capacity regions of arbitrarily varying multiple-access channels does not occur if random coding is used. However, using a random code requires both the senders and the receiver to have access to a common source of randomness. Thus random coding is usually only used as a mathematical tool for finding good deterministic codes. It is well known that derandomization is no problem for compound channels (including discrete memoryless channels as a special case) – this is nothing but the well-known “random coding method”. It builds on the fact that the finite number of channel states does not increase with blocklength. This is not so in the case of arbitrarily varying channels. The number of states *per channel use* remains constant, but the number of states per transmission of a codeword increases *exponentially* in blocklength. This is the reason why the deterministic capacity region of arbitrarily varying multiple-access channels may be strictly contained in the random coding capacity region. In fact, if derandomization is not possible, then no positive rates are achievable at all.
In contrast to the derandomization technique used for simpler channels, Ahlswede’s elimination technique gives rise to a two-step protocol. In order to approximate a given achievable rate pair $(R_1,R_2)$, one only needs the constituent deterministic codes of a random code whose rate pair approximates $(R_1,R_2)$. The randomness of the random code is used in the average error estimate. (On the other hand, this shows how much weaker the average error criterion is compared with the maximal error requirement – the randomized part can be “hidden” in the average error.)
It is noteworthy that for the arbitrarily varying multiple-access channel, the conferencing protocols needed to achieve any rate pair within the capacity region remain as simple as for the compound multiple-access channel with conferencing encoders. There are no iterative steps, so the implementation of such a conference is straightforward.
Finally we would like to analyze the benefits of Willems conferencing. We compare the gains obtained in AV-MACs to the gains obtained in compound MACs. For both compound and AV-MACs, conferencing may help to achieve positive rates where only the rate pair $(0,0)$ is achievable without transmitter cooperation. This effect is similar to the “superactivation” of quantum channels as observed in [@SYQuantum], where it was shown that there are pairs of quantum channels with zero quantum capacity each which achieve positive rates when used together.
Every compound MAC with conferencing capacities $C_1\vee C_2>0$ and $$\label{comp>0}
\max_{p\in\Pi}\min_{q\in{\mathcal{P}}({\mathcal{S}})}I(Z_q;X,Y)>0$$ has an at least one-dimensional capacity region. If is not satisfied, then the capacity region equals $\{(0,0)\}$. No matter what dimension the corresponding ${\mathcal{C}}^*({\mathcal{S}},0,0)$ has, the gains of conferencing are continuous in $C_1,C_2$, in particular in $(C_1,C_2)=(0,0)$. This is in contrast to the AV-MAC. The changes in in the deterministic capacity region ${\mathcal{C}}_d({\mathcal{S}},C_1,C_2)$ are continuous in all $(C_1,C_2)$ with $C_1\vee C_2>0$ because either ${\mathcal{C}}_d({\mathcal{S}},C_1,C_2)={\mathcal{C}}^*({\mathcal{S}},C_1,C_2)$ for all $(C_1,C_2)$ with $C_1\vee C_2>0$ or ${\mathcal{C}}_d({\mathcal{S}},C_1,C_2)=\{(0,0)\}$ for all $C_1,C_2$. However, there may be a discontinuity in $(C_1,C_2)=(0,0)$.
This corresponds to the two roles conferencing plays in AV-MACs. The “traditional” role is to generate a common message and to use the coding result for the (compound) MAC with common message to enlarge the capacity region. For AV-MACs, it does even more – it changes the channel structure. Recall Remark \[subexpconf\]. For a conferencing rate pair with $C_1\vee C_2=(2\log n)/n$, the capacity region of the compound MAC stays as it is. Under the conditions that $\W$ is not $({\mathcal{X}},{\mathcal{Y}})$-symmetrizable and ${\mathcal{C}}_d({\mathcal{S}},0,0)\neq{\mathcal{C}}^*({\mathcal{S}},0,0)$, though, we can strictly enlarge the capacity region of the AV-MAC with this kind of conferencing.
General conditions for ${\mathcal{C}}_d({\mathcal{S}},0,0)\neq{\mathcal{C}}^*({\mathcal{S}},0,0)$ to hold cannot be given because an exact characterization of ${\mathcal{C}}_d({\mathcal{S}},0,0)$ is generally unavailable. We certainly know by Theorem \[AC\_Satz\] that if ${\mathcal{C}}_d({\mathcal{S}},0,0)$ is two-dimensional, then ${\mathcal{C}}_d({\mathcal{S}},0,0)={\mathcal{C}}^*({\mathcal{S}},0,0)$. We can further say that if in addition to not being $({\mathcal{X}},{\mathcal{Y}})$-symmetrizable, $\W$ is both ${\mathcal{X}}$- and ${\mathcal{Y}}$-symmetrizable, then ${\mathcal{C}}_d({\mathcal{S}},0,0)=\{(0,0)\}$, again by Theorem \[AC\_Satz\]. This is a situation where already the conferencing from Remark \[subexpconf\] helps. With the same argumentation as in Remark \[zweidim\] it can easily be seen that $$\max_{p\in\Pi}\min_{q\in{\mathcal{P}}({\mathcal{S}})}I(Z_q;X,Y\vert U)>0,$$ so ${\mathcal{C}}^*({\mathcal{S}},0,0)$ is at least one-dimensional. Thus there is a discontinuity in $(C_1,C_2)=(0,0)$ in this case. Gubner [@G] has found the example of a ${\mathcal{W}}$ which is both ${\mathcal{X}}$- and ${\mathcal{Y}}$-symmetrizable, but not $({\mathcal{X}},{\mathcal{Y}})$-symmetrizable.
Let ${\mathcal{X}}={\mathcal{Y}}={\mathcal{S}}=\{0,1\}$ and ${\mathcal{Z}}=\{0,1,2,3\}$. For $s\in{\mathcal{S}}$ set $$W(z\vert x,y\vert s)=\delta(z-x-y-s),$$ where $\delta(t)=1$ if $t=0$ and $\delta(t)=0$ else. An equivalent description of this is $$z=x+y+s.$$ Gubner shows that ${\mathcal{W}}$ is not $({\mathcal{X}},{\mathcal{Y}})$-symmetrizable, but that it is both ${\mathcal{X}}$- and ${\mathcal{Y}}$-symmetrizable. Thus this channel is useless if coding is done without conferencing, even though the interfering signal is only added to the sum of the transmitters’ signals – the reliable transmission of messages through the channel is completely prevented. This shows that even the structure of rather simple AV-MACs can be changed by conferencing so as to produce discontinuous jumps at $(C_1,C_2)=(0,0)$.
\[appconv\]
Here we prove the weak converse for the compound MAC with conferencing encoders defined by $\WQ$ and with random coding. Let a random code$_{\textnormal{CONF}}(n,M_1,M_2,C_1,C_2)$ be given which is defined by the pair $(C,G)$. Let this code have average error at most $\lambda$. Denote the conferencing function of the deterministic component code$_{\textnormal{CONF}}$ with index $\gamma$ by $c_\gamma$. The set $c_\gamma$ maps into is denoted by $[1,V_\gamma]$.
We assume that the pair $((1/n)\log M_1,(1/n)\log M_2)$ is at least distance ${\varepsilon}$ away from ${\mathcal{C}}^*({\mathcal{S}},C_1,C_2)$. As all norms are equivalent on the plane, we can without loss of generality work with the $\ell^1$-norm. That means that we assume that $$\begin{aligned}
&\sup_{(R_1,R_2)\in{\mathcal{C}}^*({\mathcal{S}},C_1,C_2)}\!\left\{\left\lvert\frac{1}{n}\log M_1-R_1\right\rvert+\left\lvert\frac{1}{n}\log M_2-R_2\right\rvert\right\}\\&\qquad\quad\geq{\varepsilon}.\end{aligned}$$ This statement is equivalent to the fact that for every $p\in\Pi$ there is some $q\in{{\mathcal{P}}({\mathcal{S}})}$ such that one of the following inequalities holds: $$\begin{aligned}
\frac{1}{n}\log M_1&\geq C_1+I(Z_q; X\vert Y,U)+{\varepsilon},\label{erstegute}\\
\frac{1}{n}\log M_2&\geq C_2+I(Z_q; Y\vert X,U)+{\varepsilon},\\
\frac{1}{n}\log M_1M_2\\\geq\{C_1+&C_2+I(Z_q;X,Y\vert U)\wedge I(Z_q;X,Y\vert U)\}+{\varepsilon}.\label{letztegute}\end{aligned}$$
Our goal is to mainly use arguments already known from the weak converse for deterministic coding, so that we can refer to [@WBBJ11]. From the random code$_{\textnormal{CONF}}$, we define several random variables in addition to $G$:
- the pair $(T_1,T_2)$, which is uniformly distributed on $[1,M_1]\times[1,M_2]$ and independent of $G$,
- the pair $(\tilde U_1,\tilde U_2):=(c_1^G(T_1,T_2),c_2^G(T_1,T_2))$ taking values in $[1,V_1]\times[1,V_2]$, where for $\nu=1,2$ we define $V_\nu=\max_{\gamma\in\Gamma}V_\nu^\gamma$,
- $\tilde X:={\mathbf{x}}_{T_1T_2}^G$, $\tilde Y:={\mathbf{y}}_{T_1T_2}^G$,
- a random variable $\tilde Z\in{\mathcal{Z}}^n$ which satisfies $$\begin{aligned}
&\mathrel{\hphantom{=}}{\mathbb{P}}[\tilde Z={\mathbf{z}}\vert\tilde X={\mathbf{x}},\tilde Y={\mathbf{y}},\tilde U_1=v_1,\tilde U_2=v_2,\\
&\qquad\qquad\qquad T_1=j,T_2=k,G=\gamma]\\
&=W^n({\mathbf{z}}\vert{\mathbf{x}},{\mathbf{y}}).\end{aligned}$$
Every $\gamma\in\Gamma$ corresponds to a deterministic code $C(\gamma)$ with average error at most $\lambda_\gamma$. For each of these codes, we can proceed as in [@WBBJ11]. That means that we first apply Fano’s inequality and then obtain single-letter bounds on the code rates. More precisely, writing ${\mathcal{U}}=[1,n]\times[1,V_1]\times[1,V_2]$, we can construct for each $\gamma$ a probability distribution $p(u,x,y\vert \gamma)$ on ${\mathcal{U}}\times{\mathcal{X}}\times{\mathcal{Y}}$ which is contained in $\Pi$. This is due to the fact proved in [@Wi1] that conditional on $\gamma$ and $(\tilde U_1,\tilde U_2)$, the random variables $\tilde X$ and $\tilde Y$ are independent. Thus we have $$p(u,x,y\vert\gamma)=p_0(u\vert\gamma)p_1(x\vert u,\gamma)p_2(y\vert u,\gamma).$$ Further, for each $q\in{{\mathcal{P}}({\mathcal{S}})}$, we construct the random vector $(U,X,Y,Z_q)$ which together with $G$ has the distribution $$\begin{aligned}
\label{Zgut}
&\mathrel{\hphantom{=}}{\mathbb{P}}[Z_q=z,Y=y,X=x,U=u,G=\gamma]\notag\\
&=W(z\vert x,y\vert q)p(u,x,y\vert\gamma)p_G(\gamma).\end{aligned}$$ By construction, this random vector satisfies for every $q\in{{\mathcal{P}}({\mathcal{S}})}$ $$\begin{aligned}
\frac{1}{n}\log M_1&\leq C_1+I(Z_q;X\vert Y,U,G=\gamma)+\frac{1}{n}\Delta_\gamma,\\
\frac{1}{n}\log M_2&\leq C_2+I(Z_q;Y\vert X,U,G=\gamma)+\frac{1}{n}\Delta_\gamma,\\
\frac{1}{n}\log M_1M_2&\leq\{(C_1+C_2+I(Z_q;X,Y\vert U,G=\gamma))\\&\qquad\qquad\wedge I(Z_q;X,Y\vert G=\gamma)\}+\frac{1}{n}\Delta_\gamma,\end{aligned}$$ where $$\Delta_\gamma:=2h(2\lambda_\gamma)+4\lambda_\gamma\log M_1M_2.$$ Next we take the expectation over $G$. Using the concavity of $h$ and , this can be transformed into $$\begin{aligned}
\frac{1}{n}\log M_1&\leq C_1+I(Z_q;X\vert Y, U,G)+\frac{1}{n}\Delta,\\
\frac{1}{n}\log M_2&\leq C_2+I(Z_q;Y\vert X, U,G)+\frac{1}{n}\Delta,\\
\frac{1}{n}\log M_1M_2&\leq\{(C_1+C_2+I(Z_q;X,Y\vert U,G))\notag\\&\qquad\qquad\wedge I(Z_q;X,Y\vert G)\}+\frac{1}{n}\Delta,\label{wrgstf}\end{aligned}$$ with $$\Delta:=2h(2\lambda)+4\lambda\log M_1M_2.$$ As $I(Z_q;X,Y\vert G)=H(Z_q\vert G)-H(Z_q\vert X,Y)$, the concavity of entropy implies that the bound in is relaxed if one replaces $I(Z_q;X,Y\vert G)$ by $I(Z_q;X,Y)$. We now set $\hat U:=(U,G)$ and observe that the distribution of $(\hat U,X,Y)$ is contained in $\Pi$. Comparing the resulting set of inequalities with a valid one among - and using the same simple arguments as in [@WBBJ11], we can now show that $\lambda\geq\lambda({\varepsilon})>0$. This finishes the proof of the weak converse.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors would like to thank Frans Willems for the fruitful discussions about this work at the Banff workshop “Interactive Information Theory” in January 2012. They would also like to thank the associate editor Yossef Steinberg for his valuable comments given during the review process of the paper.
[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}
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[Moritz Wiese]{} (S’09) received the Dipl.-Math. degree in mathematics from the university of Bonn, Germany, in 2007. He has been pursuing the PhD degree since then. From 2007 to 2010, he was a research assistant at the Heinrich-Hertz-Lehrstuhl für Mobilkommunikation, Technische Universität Berlin, Germany. Since 2010, he is a research and teaching assistant at the Lehrstuhl für Theoretische Informationstechnik, Technische Universität München, Munich, Germany.
[Holger Boche]{} (M’04-SM’07-F’11) received the Dipl.-Ing. and Dr.-Ing. degrees in electrical engineering from the Technische Universität Dresden, Dresden, Germany, in 1990 and 1994, respectively. He graduated in mathematics from the Technische Universität Dresden in 1992. From 1994 to 1997, he did postgraduate studies in mathematics at the Friedrich-Schiller Universität Jena, Jena, Germany. He received his Dr. rer. nat. degree in pure mathematics from the Technische Universität Berlin, Berlin, Germany, in 1998. In 1997, he joined the Heinrich-Hertz-Institut (HHI) für Nachrichtentechnik Berlin, Berlin, Germany. Starting in 2002, he was a Full Professor for mobile communication networks with the Institute for Communications Systems, Technische Universität Berlin. In 2003, he became Director of the Fraunhofer German-Sino Lab for Mobile Communications, Berlin, Germany, and in 2004 he became the Director of the Fraunhofer Institute for Telecommunications (HHI), Berlin, Germany. Since October 2010 he has been with the Institute of Theoretical Information Technology and Full Professor at the Technische Universität München, Munich, Germany. He was a Visiting Professor with the ETH Zurich, Zurich, Switzerland, during the 2004 and 2006 Winter terms, and with KTH Stockholm, Stockholm, Sweden, during the 2005 Summer term. Prof. Boche is a Member of IEEE Signal Processing Society SPCOM and SPTM Technical Committee. He was elected a Member of the German Academy of Sciences (Leopoldina) in 2008 and of the Berlin Brandenburg Academy of Sciences and Humanities in 2009. He received the Research Award “Technische Kommunikation” from the Alcatel SEL Foundation in October 2003, the “Innovation Award” from the Vodafone Foundation in June 2006, and the Gottfried Wilhelm Leibniz Prize from the Deutsche Forschungsgemeinschaft (German Research Foundation) in 2008. He was co-recipient of the 2006 IEEE Signal Processing Society Best Paper Award and recipient of the 2007 IEEE Signal Processing Society Best Paper Award.
[^1]: M. Wiese and H. Boche are with the Lehrstuhl für Theoretische Informationstechnik, Technische Universität München, Munich, Germany (e-mail: {wiese,boche}@tum.de)
[^2]: The results of this paper were presented at the 2011 IEEE International Symposium on Information Theory (ISIT ’11), St. Petersburg, Russia.
[^3]: This work was supported by the German Ministry of Education and Research (BMBF) under Grant 01BQ1050 and the DFG COIN Project.
[^4]: Copyright (c) 2012 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected].
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Detailed fluctuation theorem, a microscopic version of the steady state fluctuation theorem, has been proposed by Jarzynski and demonstrated in the case of Hamiltonian systems weakly coupled with reservoirs. We show that an identical theorem for phase space compressibility rate can be derived for systems evolving under non-Hamiltonian extended system dynamics, without certain limiting assumptions made in the original work. Our derivation is based on the non-Hamiltonian phase space formulation of statistical mechanics and does not rely on any assumptions of thermodynamic nature. This version of the detailed fluctuation theorem is shown to be generic enough to be applicable to several thermostatting schemes. It is shown that in equilibrium, this detailed fluctuation theorem boils down to the detailed balance equation and it is further shown to reproduce the Jarzynski’s work theorem for driven systems.'
author:
- 'K. Gururaj, G. Raghavan and M.C. Valsakumar'
title: 'Generalization of the detailed fluctuation theorem for Non-Hamiltonian Dynamics'
---
Introduction
============
A family of remarkable results in non-equilibrium statistical mechanics, collectively called *Fluctuation theorems,* has been obtained in recent years. These theorems address the issue of how macroscopic irreversibility arises from microscopic time reversible dynamics, and in a sense quantify the probability of observing Second law violating events. These fluctuation theorems are valid for systems arbitrarily far from thermodynamic equilibrium and have been demonstrated for both deterministic and stochastic evolution. It is this sweeping generality that make these results quite extraordinary. One version of the fluctuation theorem was first discovered by Evans and Searles in the context of molecular dynamics simulation [@key-11] of a steady state system subject to a Gaussian isokinetic thermostat. The system under consideration was a steady state system subjected to external forcing. Under such circumstances, stationarity can be attained only under constant dissipation of energy. In a dynamical system, this dissipation manifests itself as a contraction of the phase space volume. Considering a set of phase points centered around $\Gamma_{0}(p_{0},q_{0})$ at time t=0, under the dynamics, we have the mapping d$\Gamma_{0}$$\rightarrow$d$\Gamma_{t}$ with:
$$\left|\frac{d\Gamma_{0}}{d\Gamma_{t}}\right|=exp(-\intop_{0}^{t}\Lambda(\Gamma_{s})ds)\label{eq:1}$$
where $$\Lambda(\Gamma_{s})=\mathit{-\mathit{\mathsf{\frac{\partial}{\partial\Gamma_{s}}\dot{\Gamma_{t}}}}}\label{eq:2}$$ then the heat exchange rate with the thermostat, at temperature T, is $$\dot{Q}(\Gamma,t)=-T\Lambda(\Gamma,t)\label{eq:3}$$ which provides the link between phase space compression and entropy production rate. Evans and Searles then define a dissipation function :
$$\Omega_{t}(\Gamma_{0})=ln\left[\frac{f(\Gamma_{0,}0)}{f(\Gamma_{t}^{*},0)}\right]-\overset{t}{\underset{0}{\int}}\Lambda(\Gamma_{s},s)ds\label{eq:4}$$
where $f(\Gamma_{s},s)$ is the phase space probability density.
For a dissipative system, $$\left\langle \Omega_{t}\right\rangle \geqslant0\label{eq:5}$$ When the initial probability density $f(\Gamma_{0},0)$ is drawn from an equilibrium distribution and the system is driven away from it by external forcing, the statement of the Evans-Searles (transient) fluctuation theorem takes the form:
$$\frac{P(\Omega_{t}=A)}{P(\Omega_{t}=-A)}=exp(At)\label{eq:6}$$
In the case where the initial phase space points are drawn from a steady state distribution, Evans and Searles derived a rearranged form of the above equation called the steady state fluctuation theorem valid for the long time limit, given by the expression:
$$\underset{t\rightarrow\infty}{lim}\frac{1}{t}ln\frac{P(\overline{\Omega}=A)}{P(\overline{\Omega}=-A)}=A\label{eq:7}$$
Independent of this work, Galavotti and Cohen derived the steady state fluctuation theorem using average compression as:
$$\underset{t\rightarrow\infty}{lim}\frac{1}{t}ln\frac{P(-\bar{\Lambda}=A)}{P(-\bar{\Lambda}=-A)}=A\label{eq:8}$$
These fluctuation theorem are valid in the non-linear, nonequilibrium regime where very general results are available. Fluctuation relation have been derived under a variety of conditions [\[]{}[@key-11][\]]{} and have been demonstrated experimentally in small systems.
Jarzynski’s detailed fluctuation theorem
========================================
In an interesting development, Jarzynski**[@key-7; @key-6]**derived a hybrid fluctuation theorem*,* which is also a statement of detailed balance. This *detailed fluctuation theorem* shows that under a nonequilibrium process, the ratio of the probability of observing specific trajectory - anti-trajectory pairs goes as an exponential of the entropy produced. For a trajectory that starts in a microstate $z_{A}$ and evolves into $z_{B}$ in a duration $\tau$, resulting in the production of entropy $\triangle S$, its *anti-trajectory* is the one that starts from $z_{B}^{*}$ and evolves into $z_{A}^{*}$ causing an entropy consumption of $\triangle S$ . Here $(\mathbf{q},\mathbf{p})^{*}$ stands for reversal of momenta $(\mathbf{q},-\mathbf{p})$ . The detailed fluctuation theorem, like all fluctuation theorems, shows that it is more likely that entropy is generated rather than consumed in a non-equilibrium process. This theorem is however distinct from the other forms of fluctuation theorems in that it makes specific reference to the initial and final microstates.
Jarzynski has derived this result for a Hamiltonian system weakly coupled to a set of Hamiltonian reservoirs. The system is manipulated by an external protocol which involves making or breaking contact with external forces and heat reservoirs in a specified sequence. In this scheme, the application of the external protocol in the reversed order amounts to the realization of the time-reversed trajectories. A process $\Pi^{+}$ is defined to be the execution of a given protocol for a time interval $\tau$ and correspondingly, the process $\Pi^{-}$ is the same protocol executed in the reverse order. The detailed fluctuation theorem then assumes the form$$\frac{P_{+}(\mathbf{z}_{B},\triangle S|\mathbf{z}_{A})}{P_{-}(\mathbf{z}_{A}^{*},-\Delta S\mid\mathbf{z_{\mathbf{B}}^{*}})}=e^{\triangle S/k_{B}}\label{eq:9}$$ where the numerator (denominator) in LHS is the joint conditional probability that, under the process $\Pi^{+}$$(\Pi^{-})$, that the system starts from $z_{A}$ $(z_{B}^{*})$ and ends at $z_{B}$ $(z_{A}^{*})$, producing (consuming) an entropy $\triangle S.$ The derivation of the theorem follows directly from the assumptions that the reservoir temperatures do not change and their degrees of freedom at both initial and final times are Maxwell-Boltzmann distributed.The entropy generated by the dynamical evolution is then assumed to be:
$$\triangle S=-\overset{N}{\underset{n=1}{\sum}}\frac{\Delta Q_{n}}{T_{n}}\label{eq:10}$$
where $\Delta Q_{n}$ is the the change in the internal energy of the n$^{th}$ heat reservoir. These assumptions, coupled with the fact that Hamiltonian evolution is time reversible and phase-space conserving, lead directly to the detailed fluctuation theorem. Regarding the particular choice made for $\triangle S$, eq, Jarzynski explicitly states that it is valid only when the system is weakly coupled with the heat reservoirs possessing infinite number of degrees of freedom. Towards the end of his paper, Jarzynski suggests that it would be desirable to perform experiments to test the detailed fluctuation theorem. This requires experimental control and manipulation of the full microstates of the system which is very difficult to realize experimentally. This might explain only very few results are available on the experimental verification of the detailed fluctuation theorem in contrast to work fluctuation theorem of Jarzynski. In contrast to the experimental scenario, in molecular dynamics one has full control and information about the specific microstate and complete dynamics of the system and hence one can attempt to verify the detailed fluctuation theorem in such a setting.
Detailed Fluctuation theorem in Molecular Dynamics
==================================================
In the context of molecular dynamics simulations, thermostatting is realized by appending a few extra degrees of freedom to the system, the time evolution of which is of a non-Hamiltonian character and hence cannot be understood as a simple perturbation of the original uncoupled dynamics. Hence, the question of whether these fluctuation theorems can be captured in molecular dynamics simulations has been attracting a lot of attention in recent years. For example, the *work fluctuation theorem* has been rederived for Nose-Hoover dynamics without the weak coupling assumption [@key-10]. Similarly, the validity of these fluctuation theorems for different molecular dynamics ensembles has appeared in literature [@key-1; @key-2; @key-3; @key-15]. Within the context of detailed fluctuation theorem, Jarzynski [@key-6; @key-7], has suggested that it would be interesting to see whether the DFT manifests itself in the Nose-Hoover thermostatted dynamics. Nose-Hoover dynamics belong to a class of thermostatting schemes called Extended Phase Space Methods [@key-8] and we realize that in the case of such systems, there are no coupling terms in the total Hamiltonian and hence one cannot talk of a weak coupling or even delineate the system energy from the heat bath energy, as was done in the original derivation. Therefore one cannot use the canonical definition of entropy (eq 10) . Further, in the extended system dynamics, temperature enters only as an external parameter and hence taking recourse to thermodynamic definition of entropy or internal energy is not viable. Given a set of autonomous equations of motion, perhaps the only quantity that can be related to entropy is the phase space compressibility. Although the absolute entropy for such systems may not be definable, one can identify, under certain assumptions, phase space compression rate with the entropy production rate[@key-14; @key-13; @key-12]. Independent of whether such an identification is to be made or not, we demonstrate that a detailed fluctuation theorem can be formulated for the phase space compressibility that takes exactly the same form as that of the original.
Derivation of the detailed fluctuation theorem for the phase compression rate
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The present derivation follows essentially the same methodology as that of the original, but differs significantly in that no use is made of the Liouville theorem since, in general, phase space volume is not conserved under a non-Hamiltonian evolution. The notations and symbols used are identical to the ones used in the original, so as to make the similarities and distinctions between the two derivations easily observable.
We start by considering a system $\psi$ coupled to a thermostat such that, the extended system undergoes a time-reversible, non-Hamiltonian dynamics for which a constant *energy functional* $\mathcal{H}$, hence-forth referred to as the psuedo-Hamiltonian, can be identified. It is possible to cast many of the present day thermostats and ergostats into this form, see [@key-9]. Let $\mathbf{z=(}\mathbf{r,p)}$ denote the phase space of the system $\psi$ and let the degrees of freedom of the thermostat be represented by $\mathbf{Y}$. Let $\mathbf{\Gamma}=(\mathbf{z},\mathbf{Y})$ denote a point in the phase space of the extended system, evolving under the deterministic non-Hamiltonian dynamics: [@key-9]
$$\dot{\mathbf{\mbox{\ensuremath{\Gamma}}}_{i}}=\overset{2Nd}{\underset{j=1}{\sum}}\mathbf{B_{\mathbf{ij}}\frac{\partial\mathcal{H}}{\partial\Gamma_{\mathbf{\mathbf{j}}}}}\label{eq:11}$$
for $i=1,...,2Nd$, where $\mathcal{H}$($\Gamma$**)** is the generalized energy and **$B$** is a $2Nd\times2Nd$ anti-symmetric matrix. The equations are known to generate energy preserving dynamics. For instance, when $\mathbf{B=\left[\begin{array}{cc}
\mathbf{0} & \mathbf{1}\\
\mathbf{-1} & \mathbf{0}\end{array}\right]}$ where $\mathbf{0}$ and $\mathbf{1}$ indicate the zero and identity matrices of appropriate dimensions, these equations reduce to the usual Hamilton’s equations of motion. Any other choice of matrix **$\mathbf{B}$,** so long as it is anti-symmetric, still generates a non-Hamiltonian dynamics that conserves **$\mathcal{H}$:**
$$\frac{d\mathcal{H}}{dt}=\overset{2Nd}{\underset{i=1}{\sum}}\overset{2Nd}{\;\underset{j=1}{\sum}}\frac{\partial\mathcal{H}}{\partial\Gamma_{i}}\mathbf{B}_{ij}\frac{\partial\mathcal{H}}{\partial\Gamma_{j}}=0\label{eq:12}$$
We further note that time reversibility of the underlying dynamics enforces the condition
$$\mathcal{H}(\Gamma)=\mathcal{H}(\Gamma^{*})\label{eq:13}$$
where $\Gamma^{*}$ is obtained from $\Gamma$ by reversing all the momentum-like variables. The phase space contraction rate $\Lambda(\Gamma^{s})$ at time $t=s$, defined earlier, may be written as
$$\Lambda(\Gamma)_{t=s}=-\underset{i=1}{\overset{2Nd}{\sum}}\dfrac{\partial\dot{\Gamma}_{i}(t)}{\partial\Gamma_{i}(t)}\mid_{t=s}=-\overset{2Nd}{\underset{i,j=1}{\sum}}\frac{\partial\mathbf{B}_{ij}}{\partial\Gamma_{i}}\frac{\partial\mathcal{H}}{\partial\Gamma_{j}}\mid_{t=s}\label{eq:14}$$
We now define entropy production in a time interval $\tau$ under a non-Hamiltonian evolution be$$\Delta S=k_{B}\underset{0}{\overset{\tau}{\int}}\Lambda(\Gamma^{t})dt\label{eq:15}$$ where $k_{B}$ denotes the Boltzmann constant. For a given state $\mathbf{\Gamma}=(\mathbf{z},\mathbf{Y})$, we define a time reversed state $\mathbf{\Gamma^{*}}$ to be $(\mathbf{z}^{*},\mathbf{Y}^{*})$ . Let $\Gamma_{+}(t)$ denote the state of the extended system at time $t$ starting from an initial extended microstate $\mathbf{\Gamma}^{0}$ and let $\Gamma_{-}(t)$ denote state of the system at time $t$ but evolving from the momentum reversed final state $\mathbf{\Gamma}^{\tau*}$. With this as the background, we are interested in computing the probability that the system $\psi$, at $t=0$, starts in a particular microstate $\mathbf{\mathbf{z}}(0)=\mathbf{z}_{A}$ and after evolving for a time $t=\tau$, reaches a state $\mathbf{\mathbf{z}}(\tau)=\mathbf{z}_{B}$, generating an entropy $\triangle S$. Note that the evolution of $\mathbf{\Gamma}=(\mathbf{z},\mathbf{Y})$ is deterministic, eq. We resort to probabilistic description when we are focusing only on the evolution of $\mathbf{z}$. For a particular choice of the initial condition $\mathbf{z}=\mathbf{z}_{A}$, the probability distribution is over the initial conditions of heatbath variables **Y**$$P_{+}(\mathbf{z}_{B},\triangle S|\mathbf{z}_{A})=\int d\mathbf{Y}p(\mathbf{Y})\delta[\mathbf{z}_{B}-\mathbf{z_{+}^{\tau}(}\mathbf{z_{A},}\mathbf{Y)].}\delta[\triangle S-\hat{\triangle S}(\mathbf{z_{A},}\mathbf{Y})]\label{eq:16}$$ where $\mathbf{z_{+}^{\tau}(}\mathbf{z_{A},}\mathbf{Y)}$ and $\hat{\triangle S}(\mathbf{z_{A},}\mathbf{Y})$ denote the final state of the system $\psi$ and the net entropy produced respectively, after $\psi$ has evolved for a time $t=\tau$ starting at $\mathbf{z_{A}}$ with the reservoir starting at $\mathbf{Y}.$ Using the identity $\mathbf{z_{A}}=\mathbf{\hat{z}_{+}^{0}(}\mathbf{z,}\mathbf{Y)}$ we can now integrate over the full phase space vector $\mathbf{\Gamma}=(\mathbf{z},\mathbf{Y})$:$$P_{+}(\mathbf{z}_{B},\triangle S|\mathbf{z}_{A})=\int d\mathbf{\Gamma}p(\mathbf{\Gamma})\delta[\mathbf{z_{A}}-\mathbf{\hat{z}_{+}^{0}(}\mathbf{\Gamma)}].\delta[\mathbf{z}_{B}-\mathbf{z_{+}^{\tau}(}\mathbf{\Gamma})].\delta[\triangle S-\hat{\triangle S}(\mathbf{\Gamma})]\label{eq:17}$$ if the dynamical system eq possess $n_{c}$ independent conservation laws:
$$\Lambda_{k}(\Gamma)=\lambda_{k},\quad k=1,..,n_{c}\label{eq:18}$$
then the probability distribution of the mirostates can be written as: [@key-8]
$$p(\Gamma)=\overset{n_{c}}{\underset{k=1}{\prod}}\delta(\Lambda_{k}(\Gamma)-\lambda_{k})\label{eq:19}$$
In instances where $\mathcal{H}$ is the only conserved quantity of the dynamics, the states $\mathbf{\Gamma}$ are those drawn from the constant $\mathcal{H}$ surface
$$p(\Gamma^{0})=\delta(H(\mathbf{\Gamma^{0}})-E)\label{eq:20}$$
hence, eq may be rewritten as
$$P_{+}(\mathbf{z}_{B},\triangle S|\mathbf{z}_{A})=\int d\mathbf{\Gamma^{0}}\delta(\mathcal{H}(\Gamma^{0})-E)\delta[\mathbf{z_{A}}-\mathbf{\hat{z}^{0}(}\mathbf{\Gamma^{0})}].\delta[\mathbf{z}_{B}-\mathbf{z^{\tau}(}\Gamma^{0})].\delta[\triangle S-\hat{\triangle S}(\Gamma^{0})]\label{eq:21}$$
As the dynamics is time reversal invariant, and $\mathcal{H}$ is conserved and we have:
$$\triangle\hat{S}(\Gamma^{\tau*})=-\hat{\triangle S}(\Gamma^{0})\label{eq:22}$$
$$\mathcal{H}(\mathbf{\Gamma^{0}})=\mathcal{H}(\mathbf{\Gamma}^{\tau*})\label{eq:23}$$
$$\mathbf{\hat{z}^{0}(}\mathbf{\Gamma^{0})}=\left[\mathbf{z^{\tau}(}\Gamma^{\tau*})\right]^{*}\label{eq:24}$$
With these expressions at hand, we may now recast the above integral as:
$$P_{+}(\mathbf{z}_{B},\triangle S|\mathbf{z}_{A})=\int d\mathbf{\Gamma^{0}}\delta(\mathcal{H}(\mathbf{\Gamma}^{\tau*})-E)\delta[\mathbf{z_{A}}^{*}-\mathbf{\hat{z}^{\tau}(}\mathbf{\mathbf{\Gamma}^{\tau*})}].\delta[\mathbf{z}_{B}^{*}-\mathbf{\hat{z}^{\mathbf{0}}(}\mathbf{\mathbf{\Gamma}^{\tau*}})].\delta[\triangle S+\hat{\triangle S}(\mathbf{\Gamma}^{\tau*})]\label{eq:25}$$
where we have cast all the variables in the r.h.s of the above integral with the time reversed counterparts of the final states, except for the integration volume element $d\Gamma^{0}$. In general, as the phase space volume is not conserved in a non-Hamiltonian evolution, $d\Gamma^{0}$ is not an invariant volume under eq and hence $d\Gamma^{0}$cannot be directly replaced by $d\Gamma^{\tau*}$. The Jacobian of transformation from the initial to the final time reversed phase space coordinates $J(\mathbf{\Gamma}^{\tau*};\mathbf{\Gamma}^{0})$ is not unity and we have
$$d\mathbf{\Gamma}^{\tau*}=J(\mathbf{\Gamma}^{\tau*};\mathbf{\Gamma}^{0})d\mathbf{\Gamma}^{0}=J(\mathbf{\Gamma}^{\tau*};\mathbf{\Gamma}^{\tau})J(\mathbf{\Gamma}^{\tau};\mathbf{\Gamma}^{0})d\Gamma^{0}\label{eq:26}$$
As the dynamical system eq is assumed to be time reversal invariant, $$J(\mathbf{\Gamma}^{\tau*};\mathbf{\Gamma}^{\tau})=1\label{eq:27}$$ therefore, $$J(\mathbf{\Gamma}^{\tau*};\mathbf{\Gamma}^{0})=J(\mathbf{\Gamma}^{\tau};\mathbf{\Gamma}^{0})\label{eq:28}$$ The time evolution of Jacobian of transformation is: [@key-8]
$$\dfrac{dJ}{dt}=J\underset{l=1}{\overset{2Nd}{\sum}}\dfrac{\partial\dot{\Gamma_{l}}(t)}{\partial\Gamma_{l}(t)}\label{eq:29}$$
The sum on the right is nothing but the negative of the phase space compressibility rate $\Lambda(\Gamma^{t})$ , hence:
$$J(\Gamma^{\tau};\Gamma^{0})=e^{\underset{0}{\overset{\tau}{-\int}}\Lambda(\Gamma^{t})dt}=e^{-\frac{\hat{\triangle S(}\mathbf{\Gamma}^{0})}{k_{B}}}\label{eq:30}$$
From eq, we know that $\triangle\hat{S}(\Gamma^{\tau*})=-\hat{\triangle S}(\Gamma^{0})$ and we have,
$$d\mathbf{\Gamma}^{\tau*}=e^{\frac{\hat{\triangle S}(\mathbf{\Gamma}^{\tau^{*}})}{k_{B}}}d\mathbf{\Gamma}^{0}\label{eq:31}$$
We may now replace $d\mathbf{\Gamma}^{0}$ in eq with $e^{-\frac{\hat{\triangle S}(\mathbf{\Gamma}^{\tau^{*}})}{k_{B}}}d\mathbf{\Gamma}^{\tau*}$ and we have:
$$P_{+}(\mathbf{z}_{B},\triangle S|\mathbf{z}_{A})=\int e^{-\frac{\hat{\triangle S}(\mathbf{\Gamma}^{\tau^{*}})}{k_{B}}}d\mathbf{\mathbf{\Gamma}^{\tau*}}\delta(\mathcal{H}(\mathbf{\Gamma}^{\tau*})-E)\delta[\mathbf{z_{A}}-\mathbf{\hat{z}^{0}(}\mathbf{\mathbf{\Gamma}^{\tau*})}].\delta[\mathbf{z}_{B}-\mathbf{z^{\tau}(}\mathbf{\mathbf{\Gamma}^{\tau*}})].\delta[\triangle S+\hat{\triangle S}(\mathbf{\Gamma}^{\tau*})],\label{eq:32}$$
We can pull out the factor $e^{\frac{\triangle S}{k_{B}}}$ out of the integration due to the presence of the delta function $\delta[\triangle S+\hat{\triangle S}(\mathbf{\Gamma}^{\tau*})]$.
$$P_{+}(\mathbf{z}_{B},\triangle S|\mathbf{z}_{A})=e^{\frac{\triangle S}{k_{B}}}\int d\mathbf{\mathbf{\Gamma}^{\tau*}}\delta(\mathcal{H}(\mathbf{\Gamma}^{\tau*})-E)\delta[\mathbf{z_{A}}-\mathbf{\hat{z}^{0}(}\mathbf{\mathbf{\Gamma}^{\tau*})}].\delta[\mathbf{z}_{B}-\mathbf{z^{\tau}(}\mathbf{\mathbf{\Gamma}^{\tau*}})].\delta[\triangle S+\hat{\triangle S}(\mathbf{\Gamma}^{\tau*})],\label{eq:33}$$
The right hand side of the above equation is nothing but $e^{\frac{\Delta S}{k_{B}}}$ times the probability distribution $P(\mathbf{z}_{B}^{*},-\triangle S|\mathbf{z}_{A}^{*})$ . Thus we have the final result of the Detailed Fluctuation theorem:
$$P(\mathbf{z}_{B},\triangle S|z_{A})=e^{\frac{\Delta S}{k_{B}}}P(\mathbf{z}_{B}^{*},-\triangle S|\mathbf{z}_{A}^{*}),\label{eq:34}$$
In summary, from the ingredients of the above derivation we list all the requirements that a dynamical system has to satisfy for the present derivation to go through:
1\. The phase space of the extended system is of even number of dimensions.
2\. The evolution is deterministic and equations of motion are time reversal invariant.
3\. The phase space compressibility of the system is non-zero.
4\. A constant of motion, usually a “psuedo-Hamiltonian” can be identified for the system that is time reversal invariant.
From the above points, it can be see that no assumptions have been made about the specific type of interactions present in the system and there are no restrictions on system size and time scale $\tau$ over which these fluctuations are realized. It should also be appreciated that this detailed fluctuation theorem is valid arbitrarily far from equilibrium, as no assumptions of thermodynamic nature has been made in the derivation. Further, unlike in the original derivation, there is no requirement for the system to be driven out of equilibrium through a given protocol. As there are fluctuations in the phase space compressibility even at equilibrium, one can capture the detailed fluctuation theorem even in an equilibrium simulation. This result is in contrast to many other fluctuation theorems which are valid far from equilibrium and at equilibrium they boil down to trivial identities as there is no average entropy production or consumption at equilibrium.
Another issue is the choice of the microcanonical ensemble for the extended system. This is in contrast with Jarzynski’s suggestion, which is to sample the reservoir degree of freedom from a Gaussian distribution. Instead, the present derivation treats all the degrees of freedom on equal footing and hence is more appealing, where the only assumptions made are the usual ones of ergodicity and equal apriori probabilities[@key-8; @key-10].
Illustration of detailed fluctuation theorem in Different ensembles
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For a better appreciation of our results, we shall study the detailed fluctuation theorem in the context of a few popular extended phase space methods. All the examples mentioned below have the non-Hamiltonian evolution such that the system of interest evolves consistent with the appropriate ensemble the equations are supposed to mimic. The statistical mechanical properties of these systems are very extensively studied in the earlier papers [\[]{}[@key-16; @key-8; @key-18][\]]{} and hence our interest here will be limited to examining them in the light of the applicability of the detailed fluctuation theorem.
NOSE HOOVER DYNAMICS
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In the Nose-Hoover thermostatting scheme [@key-8], an N particle system in $d$ spatial dimensions with Cartesian positions $\mathbf{r}\equiv\{\mathbf{r_{1}},...,\mathbf{r_{N}}\mathbf{\}}$ and momenta $\mathbf{p\equiv\{p}_{1},...,\mathbf{p_{N}\}}$ interacting through N particle potential $\Phi(\mathbf{r}_{1},...,\mathbf{r}_{N})$ is augmented with 2 heat bath variables $\mathbf{Y=}(\eta,p_{\eta})$ such that the $2dN+2$ dimensional extended phase space vector $\mathbf{\Gamma}=(\mathbf{r},\mathbf{p},\eta,p_{\eta})$ evolves as follows:
$$\mathbf{\dot{r}_{i}=}\dfrac{\mathbf{p_{i}}}{m}\label{eq:35}$$
$$\mathbf{\dot{p}_{i}}=-\dfrac{\partial\Phi(\{\mathbf{r_{i}\})}}{\partial\mathbf{r_{i}}}-\dfrac{p_{\eta}}{Q}\mathbf{p_{i}}\label{eq:36}$$
$$\dot{\eta}=\dfrac{p_{n}}{Q}\label{eq:37}$$
$$\dot{p_{\eta}}=\left[\overset{N}{\underset{i=1}{\sum}}\dfrac{\mathbf{p}_{i}^{2}}{m}-dNk_{B}T\right]\label{eq:38}$$
The parameter $Q$ represents the strength of the Nose-Hoover coupling which controls the time scale over which the equilibration takes place and $T$ is the temperature at which we wish to maintain the system of interest. A curious point that can be noted in these equations is that the variable $\eta$ does not explicitly get connected to other degrees of freedom. Still, it is profitable to retain this variable in the dynamical equations of motion as it facilitates the casting of these equations in the desired form
$$\left(\begin{array}{c}
\dot{\mathbf{r}_{i}}\\
\dot{\eta}\\
\mathbf{\dot{p_{i}}}\\
\dot{p_{\eta}}\end{array}\right)=\left[\begin{array}{cccc}
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
-1 & 0 & 0 & -\mathbf{p_{i}}\\
0 & -1 & \mathbf{p_{i}} & 0\end{array}\right]\left(\begin{array}{c}
\frac{\partial\Phi(\{\mathbf{r_{i}}\})}{\partial\mathbf{r_{i}}}\\
dNk_{B}T\\
\frac{\mathbf{p}_{i}}{m}\\
\frac{p_{\eta}}{Q}\end{array}\right)\label{eq:39}$$
for$$\mathcal{H}(\Gamma)=\overset{N}{\underset{i=1}{\sum}}\dfrac{\mathbf{p}_{i}^{2}}{2m_{i}}+\Phi(\{\mathbf{r\}})+\dfrac{p_{\eta}^{2}}{2Q}+dNk_{B}T\eta\label{eq:40}$$ Identifying the first two terms of the above as the system Hamiltonian,
$$H^{s}(\mathbf{\{r}\},\{\mathbf{p\}})=\overset{N}{\underset{i=1}{\sum}}\dfrac{\mathbf{p}_{i}^{2}}{2m_{i}}+\Phi(\{\mathbf{r\}})\label{eq:41}$$
we have $$\mathcal{H}(\Gamma)=H^{s}(\{\mathbf{r\}},\{\mathbf{p\}})+\dfrac{p_{\eta}^{2}}{2Q}+dNk_{B}T\eta\label{eq:42}$$ Nose-Hoover dynamics is time reversal invariant and possess a “psuedo-Hamiltonian” as a constant of motion and hence, satisfies the essential requirements needed for the present analysis. The phase space compressibility of this system is given by eq
$$\Lambda(\Gamma^{s})=-\overset{2n+2}{\underset{i=1}{\sum}}\frac{\partial\dot{\Gamma^{i}}}{\partial\Gamma^{i}}=-\underset{i}{\sum}\frac{\partial\mathbf{p_{i}}}{\partial\mathbf{p}_{i}}=\dfrac{dNp_{\eta}}{Q}=dN\dot{\eta}\label{eq:43}$$
so the total phase space compression during an evolution from t=0 to t=$\tau$ is from eq
$$\Delta S=k_{B}\underset{0}{\overset{\tau}{\int}}\Lambda(\Gamma^{s})ds=k_{B}dN(\eta(\tau)-\eta(0))\label{eq:44}$$
There are several points in order. Given a time evolution of $\Gamma^{0}=(\mathbf{r_{i}^{0},p_{i}^{0}},\eta^{0},p_{\eta}^{0})$ to $\Gamma^{t}=(\mathbf{r_{i}^{t},p_{i}^{t}},\eta^{t},p_{\eta}^{t})$, there is a phase space compression of $k_{B}dN(\eta^{t}-\eta^{0})$. Note that this quantity is path independent and depends only on the initial and final state of the position variable of the heat bath, and does not explicitly depend upon the functional form of the potential.
With this definition, one can gather all the trajectories that start at $(\mathbf{z_{A},}\eta,p_{\eta})$ and evolve to $(\mathbf{z_{B},}\eta+\triangle S,\acute{p}_{\eta})$ where $\mathbf{z_{A}}$and $\mathbf{z_{B}}$ are fixed initial and final states (positions and momenta) of the system of interest and $\eta$,$p_{\eta}$and $\acute{p_{\eta}}$ are arbitrary (subject to the requirement that the extended state vector $\Gamma$ stays on a constant energy hypersurface). The probability of obtaining such a trajectory is $P_{+}(\mathbf{z_{B}},+\triangle S|\mathbf{z_{A})}$. To obtain the anti-trajectories, from the time reversed final states one gathers all those states that start from $(\mathbf{z_{B}^{*},}\eta,\acute{p}_{\eta})$ and evolve into $(\mathbf{z_{A}^{*},}\eta-\triangle S,p_{\eta})$. The probability of obtaining such a trajectory is $P_{-}(\mathbf{z_{A}^{*}},-\triangle S|\mathbf{z_{B}^{*})}$. The detailed fluctuation theorem requires the ratio of these two probabilities go like $e^{\frac{\triangle S}{k_{B}}}$.
Further, in the limit of $Q\rightarrow\infty$, the equations of motion eqand eq are reduced to that of a Hamiltonian evolution and we have, from eq, $\eta(\tau)=\eta(0)$ and hence, from eq , $\Delta S=0$. This means that the probabilities of forward and backward trajectories are identical:
$$P(\mathbf{z}_{A}\rightarrowtail\mathbf{z_{B}})=P(\mathbf{z}_{B}^{*}\mathbf{\rightarrowtail z_{A}^{*}})\label{eq:45}$$
as can be expected for an isolated system.
Now, consider the probability that the system $\psi$ evolves from state $\mathbf{z_{A}}$to state $\mathbf{z_{B}}$ in a time $\tau$. This probability, let it be denoted by $P(\mathbf{z}_{A}\rightarrowtail\mathbf{z_{B}})$ , is
$$P(\mathbf{z}_{A}\rightarrowtail\mathbf{z_{B}})=\int P(\mathbf{z}_{B},\triangle S|\mathbf{z}_{A})d(\Delta S)\label{eq:46}$$
Now, applying the detailed fluctuation theorem on the right hand side, we have
$$P(\mathbf{z}_{A}\rightarrowtail\mathbf{z_{B}})=\int e^{\frac{\Delta S}{k_{B}}}P(\mathbf{z}_{A}^{*},-\triangle S|\mathbf{z}_{B}^{*})d(\Delta S)\label{eq:47}$$
For the Nose-Hoover thermostat, we know that $\triangle S=k_{B}dN(\eta(\tau)-\eta(0))$ and from eq we have
$$\eta(0)=\frac{\left[E-H^{s}(\mathbf{z_{A}})-\dfrac{p_{\eta}^{2}(0)}{2Q}\right]}{dNk_{B}T},\qquad\qquad\eta(\tau)=\frac{\left[E-H^{s}(\mathbf{z_{B}})-\dfrac{p_{\eta}^{2}(\tau)}{2Q}\right]}{dNk_{B}T}\label{eq:48}$$
where E is the constant energy over which the microcanonical ensemble of the full system is defined. Hence we have,
$$\frac{\Delta S}{k_{B}}=dN(\eta(\tau)-\eta(0))=\frac{H^{s}(\mathbf{z_{A}})-\dfrac{p_{\eta}^{2}(0)}{2Q}-H^{s}(\mathbf{z_{B}})+\dfrac{p_{\eta}^{2}(\tau)}{2Q}}{k_{B}T}\label{eq:49}$$
Inserting this into the right hand side of eq,
$$P(\mathbf{z}_{A}\rightarrowtail\mathbf{z_{B}})=\int e^{\frac{H^{s}(\mathbf{z_{A}})-\dfrac{p_{\eta}^{2}(0)}{2Q}-H^{s}(\mathbf{z_{B}})+\dfrac{p_{\eta}^{2}(\tau)}{2Q}}{k_{B}T}}P(\mathbf{z}_{A}^{*},-\triangle S|\mathbf{z}_{B}^{*})d(\Delta S)\label{eq:50}$$
Since the $\mathbf{z}_{A}$ and $\mathbf{z}_{B}$ are independent of the integration variable, they can be pulled out of the integration and we have
$$P(\mathbf{z}_{A}\rightarrowtail\mathbf{z_{B}})=e^{\frac{H^{s}(\mathbf{z_{A}})-H^{s}(\mathbf{z_{B}})}{k_{B}T}}\int e^{\frac{p_{\eta}^{2}(\tau)-p_{\eta}^{2}(0)}{2Qk_{B}T}}P(\mathbf{z}_{A}^{*},-\triangle S|\mathbf{z}_{B}^{*})d(\Delta S)\label{eq:51}$$
Re-arranging the above,
$$e^{\frac{-H^{s}(\mathbf{z_{A}})}{k_{B}T}}P(\mathbf{z}_{A}\rightarrowtail\mathbf{z_{B}})=e^{\frac{-H^{s}(\mathbf{z_{B}})}{k_{B}T}}\int e^{\frac{p_{\eta}^{2}(\tau)-p_{\eta}^{2}(0)}{2Qk_{B}T}}P(\mathbf{z}_{A}^{*},-\triangle S|\mathbf{z}_{B}^{*})d(\Delta S)\label{eq:52}$$
Assuming that the times $t=0$ and $t=\tau$ are chosen sufficiently long time after the system has equilibrated, we can assume that the kinetic energy distribution is consistent with the equipartition theorem, and hence the total kinetic energy is equal to the $\frac{1}{2}dNk_{B}T$.
$$\overset{N}{\underset{i=1}{\sum}}\dfrac{\mathbf{p}_{i}^{2}}{2m}=\frac{1}{2}dNk_{B}T\label{eq:53}$$
With this assumption, and from the last of the Nose-Hoover equations eq, we see that $\dot{p_{\eta}}=0$, and hence $e^{\frac{p_{\eta}^{2}(\tau)-p_{\eta}^{2}(0)}{2Qk_{B}T}}=1.$ Substituting this in eq , we get
$$e^{\frac{-H^{s}(\mathbf{z_{A}})}{k_{B}T}}P(\mathbf{z}_{A}\rightarrowtail\mathbf{z_{B}})=e^{\frac{-H^{s}(\mathbf{z_{B}})}{k_{B}T}}\int P(\mathbf{z}_{A}^{*},-\triangle S|\mathbf{z}_{B}^{*})d(\triangle S)\label{eq:54}$$
The integral in right hand side of the above equation can readily be identified as the probability $P(\mathbf{z}_{B}^{*}\rightarrowtail\mathbf{z_{A}^{*}})$ and since the system Hamiltonian also the time reversal invariance, this probability is also equal to $P(\mathbf{z}_{B}\rightarrowtail\mathbf{z_{A}}).$ So, we have:
$$e^{\frac{-H^{s}(\mathbf{z_{A}})}{k_{B}T}}P(\mathbf{z}_{A}\rightarrowtail\mathbf{z_{B}})=e^{\frac{-H^{s}(\mathbf{z_{B}})}{k_{B}T}}P(\mathbf{z}_{B}\rightarrowtail\mathbf{z_{A}})\label{eq:55}$$
Which is nothing but the statement of detailed balance, which is valid for any system at equilibrium.
NOSE HOOVER CHAIN DYNAMICS
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If more than one conservation laws are obeyed by the dynamical system, it is well known that the Nose-Hoover thermostatting scheme fails to generate the canonical ensemble. This happens because the accessible phase space gets restricted by the conservation and hence the system fails to access all the regions of the phase space in the course of its dynamics. This problem can be overcome by extending the number of heat bath variables. One such method is the Nose-Hoover chain thermostat. Its equations of motion are given by:
$$\mathbf{\dot{r}_{i}=}\dfrac{\mathbf{p_{i}}}{m}\label{eq:56}$$
$$\mathbf{\dot{p}_{i}}=-\dfrac{\partial\Phi(\{\mathbf{r_{i}\}})}{\partial\mathbf{r_{i}}}-\dfrac{p_{\eta_{1}}}{Q_{1}}\mathbf{p_{i}}\label{eq:57}$$
$$\dot{\eta_{j}}=\frac{p_{\eta_{j}}}{Q_{j}},\quad j=1,...,M\label{eq:58}$$
$$\dot{p_{\eta_{1}}}=\left[\overset{N}{\underset{i=1}{\sum}}\dfrac{\mathbf{p}_{i}^{2}}{m_{i}}-dNk_{B}T\right]-\frac{p_{\eta_{2}}}{Q_{2}}p_{\eta_{1}}\label{eq:59}$$
$$\dot{p_{\eta_{j}}}=\left[\dfrac{p_{\eta_{j-1}}^{2}}{Q_{j-1}}-k_{B}T\right]-\frac{p_{\eta_{j+1}}}{Q_{j+1}}p_{\eta_{j}}\quad j=2,...,M-1\label{eq:60}$$
$$\dot{p_{\eta_{M}}}=\left[\dfrac{p_{\eta_{M-1}}^{2}}{Q_{M-1}}-k_{B}T\right]\label{eq:61}$$
It can be shown that these equations of motion, preserve the psuedo-Hamiltonian:
$$\mathcal{H}(\Gamma)=\overset{N}{\underset{i=1}{\sum}}\dfrac{\mathbf{p}_{i}^{2}}{2m_{i}}+\Phi(\{\mathbf{r\}})+\overset{N}{\underset{j=1}{\sum}}\dfrac{p_{\eta_{j}}^{2}}{2Q_{j}}+dNkT\eta_{1}+k_{B}T(\overset{M}{\underset{k=2}{\sum}}\eta_{k})\label{eq:62}$$
The phase space compressibility for this system of equations is given by
$$\Lambda(\Gamma^{s})=(dN\dot{\eta}_{1}+\dot{\eta}_{2}+\dot{\eta}_{3}+...+\dot{\eta}_{M})\label{eq:63}$$
Again, the phase space compressibility rate for this system is also a total time derivative of the heat bath variables and the psuedo-Hamiltonian is invariant under time reversal. Hence we can derive the detailed fluctuation theorem in the context of Nose-Hoover chain thermostat.
MTK ISOBARIC ENSEMBLE
---------------------
The Martyna, Tobias and Klein ensemble is also based on the Non-Hamiltonian phase space formulation and is known to generate the correct isobaric ensemble. The equations of motion read ([@key-18])
$$\mathbf{\dot{r}_{i}=}\dfrac{\mathbf{p_{i}}}{m}+\frac{\mathbf{p}_{g}}{W_{g}}\mathbf{r_{i}}\label{eq:64}$$
$$\mathbf{\dot{p}_{i}}=-\dfrac{\partial\Phi(\{\mathbf{r_{i}\})}}{\partial\mathbf{r_{i}}}-\frac{\mathbf{p}_{g}}{W_{g}}\mathbf{p_{i}}-\frac{1}{N_{f}}\frac{Tr(\mathbf{p}_{g})}{W_{g}}\mathbf{p_{i}}-\frac{p_{\eta_{1}}}{Q_{1}}\mathbf{p_{i}}\label{eq:65}$$
$$\mathbf{\dot{h}=}\frac{\mathbf{p_{g}h}}{W_{g}}\label{eq:66}$$
$$\mathbf{\dot{p}_{g}=}det[\mathbf{h}](\mathbf{P^{(int)}-I}P)+\frac{1}{N_{f}}\sum\mathbf{\frac{p_{i}^{2}}{\mathbf{m}}I}-\frac{p_{\xi_{1}}}{\acute{Q}_{1}}\mathbf{p}_{g}\label{eq:67}$$
$$\dot{\eta_{j}}=\frac{p_{\eta_{j}}}{Q_{j}}\qquad\dot{\xi}_{j}=\frac{p_{\xi_{j}}}{\acute{Q_{j}}}\label{eq:68}$$
$$\dot{p_{\eta_{M}}}=G_{M}\label{eq:69}$$
$$\dot{p_{\xi_{j}}}=\acute{G}_{j}-\frac{p_{\xi_{j+1}}}{\acute{Q_{j+1}}}p_{\xi_{j}}\qquad\dot{p}_{\xi_{M}}=\acute{G}_{M}\label{eq:70}$$
where **$\mathbf{P}^{(int)}$**is the internal pressure, **$I$** is the 3x3 identity matrix and $G_{j}$are the thermostat forces, given by
$$G_{1}=\overset{N}{\underset{i=1}{\sum}}\dfrac{\mathbf{p}_{i}^{2}}{m_{i}}-dk_{B}T\qquad G_{j}=\frac{p_{\eta_{j-1}}^{2}}{Q_{j-1}}-k_{B}T\label{eq:71}$$
and
$$\acute{G}_{1}=\frac{Tr[\mathbf{p_{g}^{T}p_{g}]}}{W_{g}}-d^{2}kT\qquad\acute{G}_{j}=\frac{p_{\xi_{j-1}}^{2}}{Q_{j-1}}-k_{B}T\label{eq:72}$$
These equations have the psuedo-Hamiltonian as
$$H=\underset{i}{\sum}\dfrac{\mathbf{p}_{i}^{2}}{2m_{i}}+\Phi(\{\mathbf{r\}})+\frac{Tr(\mathbf{p_{g}^{T}p_{\mathbf{g}}})}{2W_{g}}+Pdet(\mathbf{h})+\underset{j=2}{\overset{M}{\sum}}\left[\dfrac{p_{\eta_{j}}^{2}}{2Q_{j}}+\dfrac{p_{\xi_{j}}^{2}}{2\acute{Q_{j}}}\right]+N_{f}k_{B}T\eta_{1}+d^{2}k_{B}T\xi_{1}+k_{B}T\underset{k=2}{\overset{M}{\sum}}(\eta_{k}+\xi_{k})\label{eq:73}$$
The phase space compression rate for this system of equations comes out to be
$$\Lambda=(d-1)\frac{d}{dt}ln[det(\mathbf{h})]+dN\dot{\eta}_{1}+d^{2}\dot{\xi_{1}}+\underset{k=2}{\overset{M}{\sum}}\left[\dot{\eta_{k}}+\dot{\xi_{k}}\right]\label{eq:74}$$
It is readily evident that, as with Nose-Hoover, Nose-Hoover Chain and massive thermostatting schemes, the phase space compression rate is again a total time derivative of the heat bath variables alone and the psuedo-Hamiltonian is invariant under time reversal operation. Further, the phase space compressibility is a function of position-like variable of the heat bath and hence satisfies the assumption, eq. This implies that the MTK isobaric ensemble is capable of capturing the detailed fluctuation theorem.
Also, from eq , eq and eq, we see that the phase space compression for all the extended system dynamics is dependent only on position-like variables of the heat bath and not on the system variables per se. This is evident from the extended phase space formulation itself where there are no coupling terms between the heat bath variables and our Hamiltonian system variables. The phase space compression for the Hamiltonian systems is always zero, hence the contribution towards the phase space compression, eq \[eq:14\] is from the heat bath variables alone.
As is evident from the examples above, it is important to identify all the conservation laws satisfied by a given set of dynamical equations. For instance, in many thermostatting schemes, the psuedo-Hamiltonian, $\mathcal{H}$ is usually not the only conserved quantity. In such cases, the probability distribution has to be sampled from the hypersurface defined in eq [@key-8]. We note here that all conserved quantities have to be time reversal invariant for the present proof to go through.
GAUSSIAN ISOKINETIC ENSEMBLE
----------------------------
Another example where the non-Hamiltonian phase space formalism can be readily applied is the case of Gaussian isokinetic ensemble, which keeps the kinetic energy of the system constrained to a particular value but generates a canonical distribution in the coordinate space. The equations of motion read
$$\mathbf{\dot{r}_{i}=}\dfrac{\mathbf{p_{i}}}{m_{i}}\quad i=1,..,N\label{eq:75}$$
$$\mathbf{\dot{p_{i}}=F_{i}-\left[\frac{\overset{N}{\underset{j=1}{\sum}}F_{j}.p_{j}/\mathbf{m_{j}}}{\overset{N}{\underset{j=1}{\sum}}p_{j}^{2}/\mathbf{m_{j}}}\right]}\mathbf{p_{i}}\quad i=1,..,N\label{eq:76}$$
This isokinetic ensemble method different from other non-Hamiltonian phase space methods in that there are no extra degrees of freedom appended to the system, and also the total energy of the system is also not conserved. But, by construction, one has the conservation of the total kinetic energy and the unnormalized microcanonical probability density can still be defined as:
$$p(\mathbf{p})=\delta(\overset{N}{\underset{i=1}{\sum}}\frac{\mathbf{p_{i}^{2}}}{m}-K)\label{eq:77}$$
where $K$ is an arbitrary constant. The phase space compressibility rate of this system, can be obtained as
$$\Lambda=\overset{N}{-\underset{i=1}{\sum}}\nabla_{\mathbf{r_{i}}}.\mathbf{\dot{r_{i}}}+\nabla_{\mathbf{p_{i}}}.\mathbf{\dot{p_{i}}}\label{eq:78}$$
from eq and eq this becomes gives
$$\Lambda=-\frac{3N-1}{K}\frac{d\phi(\{\mathbf{r}\})}{dt}\label{eq:79}$$
where the assumption is that there is no explicit time dependence of the potential on time, so that the partial derivative of the potential is zero.
The Compressibility for this system is given as
$$\triangle S=-\frac{3N-1}{K}\overset{\tau}{\underset{0}{\int}}\frac{d\phi(\{\mathbf{r}\})}{dt}=-k_{B}\frac{3N-1}{K}\left[\phi(\mathbf{r}_{B})-\phi(\mathbf{r}_{A})\right]\label{eq:80}$$
The above equation implies, as the system starts from $\mathbf{z_{A}\equiv(}\mathbf{r}_{A},\mathbf{p_{A})}$ and evolves to $\mathbf{z_{B}\equiv(}\mathbf{r_{B}},\mathbf{p_{B})}$, the entropy generated is proportional to the change in the potential energy at the end points, $\phi(\mathbf{r}_{A})-\phi(\mathbf{r}_{B})$. As the constraint, total kinetic energy, is invariant under time reversal and the phase space compressibility of the system eq satisfies eq, the present derivation of the detailed fluctuation theorem applies for the system evolving under the Gaussian isokinetic ensemble.
The absence of external degrees of freedom in this example means that the phase space vector evolves deterministically and hence the probabilities $P(\mathbf{z}_{B},\triangle S|\mathbf{z}_{A})$ and $P(\mathbf{z}_{B}^{*},-\triangle S|\mathbf{z}_{A}^{*})$ are reduced to just product of delta functions as there are no heat bath variables to integrate over. The joint probability $P(\mathbf{z}_{B},\triangle S|\mathbf{z}_{A})$ is actually just a function of $\mathbf{z_{B}}$ and $\mathbf{z_{A}}$alone as $\triangle S$ itself is a function of $\mathbf{z_{B}}$ and $\mathbf{z_{A}}$. In that case, consider the probability $P(\mathbf{r}_{A}\rightarrowtail\mathbf{r}_{B})\equiv P(\mathbf{r}_{B}|r_{A})$ that the system evolves from a position$\mathbf{r}_{A}$ to a position $\mathbf{r}_{B}$ in a time $\tau$. This involves integrating over all momentum variables and all possible phase space compression values.
$$P(\mathbf{r}_{A}\rightarrowtail\mathbf{r}_{B})=\int d\mathbf{p_{A}}d\mathbf{p_{B}}\int P(\mathbf{\triangle}S)d(\mathbf{\triangle}S)P(\mathbf{z}_{B}\equiv(\mathbf{r}_{B},\mathbf{p_{B}}),\triangle S|\mathbf{z}_{A}\equiv(\mathbf{r}_{A},\mathbf{p_{A}}))\label{eq:81}$$
where $P(\mathbf{\triangle}S)$ is the probability distribution of $\mathbf{\triangle}S$. Since the phase space compression depends only on the initial and final coordinates, eq , we have $$P(\mathbf{\triangle}S)=\delta\left[\Delta S-\hat{\Delta S}(\mathbf{r}_{A},\mathbf{r}_{B})\right]\label{eq:82}$$ where
$$\hat{\Delta S}(\mathbf{r}_{A},\mathbf{r}_{B})=-k_{B}\frac{3N-1}{K}\overset{\tau}{\underset{0}{\int}}\left(\frac{d\phi(\mathbf{r})}{dt}\right)dt\label{eq:83}$$
Inserting this into the above equation
$$P(\mathbf{r}_{A}\rightarrowtail\mathbf{r}_{B})=\int d\mathbf{p_{A}}d\mathbf{p_{B}}d(\mathbf{\triangle}S)\delta\left[\Delta S-\hat{\Delta S}(\mathbf{r}_{A},\mathbf{r}_{B})\right]e^{\frac{\triangle S}{k_{B}}}P(\mathbf{z}_{A}^{*},-\triangle S|\mathbf{z}_{B}^{*})\label{eq:84}$$
Now, $\hat{\Delta S}(\mathbf{r}_{A},\mathbf{r}_{B})=-\hat{\Delta S}(\mathbf{r}_{B},\mathbf{r}_{A})$ hence
$$P(\mathbf{r}_{A}\rightarrowtail\mathbf{r}_{B})=e^{\frac{\hat{\Delta S}(\mathbf{r}_{A},\mathbf{r}_{B})}{k_{B}}}\int d\mathbf{p_{A}}d\mathbf{p_{B}}\int d(\mathbf{\triangle}S)\delta\left[\Delta S+\hat{\Delta S}(\mathbf{r}_{B},\mathbf{r}_{A})\right]P(\mathbf{z}_{A}^{*},-\triangle S|\mathbf{z}_{B}^{*})\label{eq:85}$$
The RHS can be rearranged, remembering that the delta function is even in its arguments,
$$P(\mathbf{r}_{A}\rightarrowtail\mathbf{r}_{B})=e^{\frac{\hat{\Delta S}(\mathbf{r}_{A},\mathbf{r}_{B})}{k_{B}}}\int d\mathbf{(-p_{A}})d\mathbf{(-p_{B})}\int d(\mathbf{-\triangle}S)\delta\left[\Delta S-\hat{\Delta S}(\mathbf{r}_{B},\mathbf{r}_{A})\right]P(\mathbf{z}_{A}^{*},\triangle S|\mathbf{z}_{B}^{*})\label{eq:86}$$
The integral on the right hand side can be readily identified as the probability of system evolving from $r_{B}$ to $r_{A}$and hence
$$P(\mathbf{r}_{A}\rightarrowtail\mathbf{r}_{B})=e^{\frac{\hat{\Delta S}(\mathbf{r}_{A},\mathbf{r}_{B})}{k_{B}}}P(\mathbf{r}_{B}\rightarrowtail\mathbf{r}_{A})\label{eq:87}$$
Now, from eq $$\hat{\Delta S}(\mathbf{r}_{A},\mathbf{r}_{B})=-k_{B}\frac{3N-1}{K}\left[\phi(\mathbf{r_{B}})-\phi(\mathbf{r_{A}})\right]\label{eq:88}$$ Choosing the arbitrary constant $K=(3N-1)k_{B}T$, where T is the desired temperature, we have
$$P(\mathbf{r}_{A}\rightarrowtail\mathbf{r}_{B})=e^{\frac{\phi(\mathbf{r_{A}})-\phi(\mathbf{r_{B}})}{k_{B}T}}P(\mathbf{r}_{B}\rightarrowtail\mathbf{r}_{A})\label{eq:89}$$
Rearranging the above equation, we see that the detailed fluctuation theorem just boils down to the detailed balance equation in the configuration space:$$e^{-\frac{\phi(\mathbf{r_{A})}}{k_{B}T}}P(\mathbf{r}_{A}\rightarrowtail\mathbf{r}_{B})=e^{-\frac{\phi(\mathbf{r_{B})}}{k_{B}T}}P(\mathbf{r}_{B}\rightarrowtail\mathbf{r}_{A})\label{eq:90}$$
The choice $K=(3N-1)k_{B}T$ is natural from the equipartitioning theorem which says that every independent momentum degree of freedom has an average kinetic energy of $\frac{1}{2}k_{B}T$. The system has $(3N-1)$ independent momentum degrees of freedom ($3N$ momentum variables and one constraint on the total kinetic energy), hence the total kinetic energy is $\frac{1}{2}(3N-1)k_{B}T$. The Gaussian Isokinetic ensemble is a simple yet powerful example to appreciate that the Non-Hamiltonian characteristic alone is sufficient to bring out all the seemingly counter-intuitive features of the statistical mechanics like the entropy production and consumption anisotropy, direction of time emerging from the time reversible dynamics, anisotropy in the transition probabilities, emergence of Boltzmann distribution in the configuration space etc.
DETAILED FLUCTUATION THEOREM AND CONSERVATION LAWS
==================================================
Consider the Nose Hoover thermostatting scheme. If the forces acting on the system are derivable from a two body potential,
$$\Phi(\mathbf{r}_{1},...,\mathbf{r}_{N})=\overset{N}{\underset{\overset{i,j=1}{i\neq j}}{\frac{1}{2}\sum}}\phi_{2}(|\mathbf{r}_{i}-\mathbf{r}_{j}|)\label{eq:91}$$
such that the net force acting on the system is zero, then there are $d$ additional conserved quantities that emerge in the Nose-Hoover dynamics:
$$\mathbf{P}e^{\eta}=\mathbf{K}\label{eq:92}$$
where $\mathbf{P}=\overset{n}{\underset{i=1}{\sum}}\mathbf{p}_{i}$ is the total momentum of the system and $\mathbf{K}$ is an arbitrary vector in $d$ dimensions. In our context, this would mean that we can no longer use eqs but instead use $p(\mathbf{\Gamma})=\delta(H^{'}(\mathbf{\Gamma})-E)\delta(\mathbf{P}e^{\eta}-\mathbf{K})$ as the correct distribution function. The problems associated with the presence of additional conservation laws in the context of generating the dynamics appropriate to a desired ensemble is well studied [@key-8], where it is shown that the presence of hidden conservation laws in the dynamics will lead to an ensemble different from the required canonical ensemble. It should be noted that the suggested solution of appending an extended chain of thermostats, instead of one, though known to generate the correct distribution in the system subspace, will not invalidate the presence of additional conservation laws. So, the augmentation of the Nose-Hoover to Nose-Hoover chain will not help in demonstration of the detailed fluctuation theorem. The problem arises from the fact $\mathbf{P}e^{\eta}$ is not invariant under time reversal: $(\mathbf{P}e^{\eta})^{*}\neq\mathbf{P}e^{\eta}$ and it is because of this problem, the present method of derivation hits a roadblock. In fact, for all systems which have conserved quantities which do not have a definite parity or are of odd parity under time reversal would fail to capture fluctuation theorems of the usual type, as the time reversed states are not accessible to the system. But for systems which have conserved quantities which are of odd parity under time reversal, say$\mathbf{K}$, it is easy to see that the Detailed Fluctuation Theorem takes the form:
$$P_{+}(\mathbf{z}_{B},\triangle S|z_{A},\mathbf{K})=e^{\frac{\Delta S}{k_{B}}}P_{-}(\mathbf{z}_{B}^{*},-\triangle S|\mathbf{z}_{A}^{*},\mathbf{K^{*})}\label{eq:93}$$
Free Energy Relations from the Detailed Fluctuation theorem
===========================================================
It would be worthwhile to investigate whether the free energy equality, Jarzynski’s identity, is derivable from the detailed fluctuation theorem in the non-Hamiltonian framework. For simplicity, we shall attempt to derive the Jarzynski’s identity from the detailed fluctuation theorem result for a system coupled to Nose-Hoover thermostat. The Jarzynski’s identity **[@key-21]** reads,
$$<exp(-\beta W)>=exp(-\beta\triangle F_{AB}),\qquad\beta=\frac{1}{kT}\label{eq:94}$$
where $\triangle F_{AB}$is the equilibrium free energy difference between A and B:
$$\triangle F_{AB}=-k_{B}Tln\left(\frac{Z_{A}}{Z_{B}}\right)\label{eq:95}$$
where $Z_{A}$and $Z_{B}$ are canonical partition functions of the systems A and B.
$$Z_{A}=\int e^{-\beta H_{A}^{s}(\mathbf{r_{\mathbf{A}},}\mathbf{p_{\mathbf{A}})}}d\mathbf{r_{A}}d\mathbf{p_{A}}\label{eq:96}$$
and similarly
$$Z_{B}=\int e^{-\beta H_{B}^{s}(\mathbf{r_{B},}\mathbf{p_{B})}}d\mathbf{r_{B}}d\mathbf{p_{B}}\label{eq:97}$$
where $H_{A}^{s}$and $H_{B}^{s}$ are the Hamiltonian of the two systems whose free energy difference is to be computed. The Hamiltonians are superscripted to make the distinction between the Hamiltonian of the system of interest and the psuedo-Hamiltonian, which contains reservoir degrees of freedom also. We assume that there is a single time-dependent Hamiltonian, which at the time $t=0$ is the Hamiltonian corresponding to the state A, $H_{A}$ and at a time $t=\tau$ transforms to the Hamiltonian corresponding to the state B, $H_{B}$. This variation can be brought about by, for example, a time dependent potential $\Phi(\mathbf{r}_{1},...,\mathbf{r}_{N},t)$, such at $\Phi(\mathbf{r}_{1},...,\mathbf{r}_{N},0)=\Phi_{A}(\mathbf{r}_{1},...,\mathbf{r}_{N})$ and $\Phi(\mathbf{r}_{1},...,\mathbf{r}_{N},\tau)=\Phi_{B}(\mathbf{r}_{1},...,\mathbf{r}_{N})$ where $\Phi_{A}$and $\Phi_{B}$ are the potentials of states A and B respectively.
The change in the energy of the “system of interest” due to this time variation of the potential is given by [@key-2; @key-10]:
$$H_{B}-H_{A}=H^{S}(\mathbf{z}_{B},\tau)-H^{S}(\mathbf{z}_{A},0)=\overset{\tau}{\underset{0}{\int}}dt\frac{\partial H^{s}(\mathbf{z}_{t},t)}{\partial\mathbf{z}_{t}}\dot{\mathbf{z}_{t}}+\overset{\tau}{\underset{0}{\int}}dt\frac{\partial H^{s}(\mathbf{z}_{t},t)}{\partial t}=Q+W\label{eq:98}$$
The above equation can be called the mathematical formulation of the First law of Thermodynamics, where the term on the left hand side is identified with the change in the internal energy of the system, the first term of the right is identified as the heat Q transferred from the bath to the system and the second term is the work performed on the the system. Note that both, work and heat are defined in terms of the system Hamiltonian alone. The effect of thermostatting is felt only through the coupling of evolution of the system variables and heat bath variables.
From the explicit functional form for the system Hamiltonian and the Nose-Hoover equations of motion, we can calculate the first term on the right hand side of eq as
$$Q=\overset{\tau}{\underset{0}{\int}}dt\left(-\dfrac{p_{\eta}(t)}{Q_{1}}\right)\overset{N}{\underset{i=1}{\sum}}\frac{\mathbf{p_{i}}^{2}(t)}{m}\label{eq:99}$$
Identifying the term $\dfrac{p_{\eta}(t)}{Q_{1}}$ as $\dot{\eta}$ and $\overset{N}{\underset{i=1}{\sum}}\frac{\mathbf{p_{i}}^{2}(t)}{m}$ as $\dot{p_{\eta}}+dNk_{B}T$ from the Nose-Hoover equations of motion eq (38) and eq (39) we get
$$Q=-\overset{\tau}{\underset{0}{\int}\dot{\eta}}(\dot{p_{\eta}}+dNk_{B}T)dt\label{eq:100}$$
As discussed earlier, If the times $t=0$ and $t=\tau$ are such long after the system has reached steady state such that the average kinetic energy is determined by the temperature, we have, from eq and eq
$$Q(\tau)=-dNk_{B}T\overset{\tau}{\underset{0}{\int}}dt\dot{\eta}(t)=-dNk_{B}T(\eta(\tau)-\eta(0))\label{eq:101}$$
Identifying $dN\dot{\eta}(t)$ as the phase space compression rate for this system, eq, we have $$Q(\tau)=k_{B}T\overset{\tau}{\underset{0}{\int}}\Lambda(t)dt=-T\triangle S\label{eq:102}$$ Thus we see that $k_{B}T$ times the total phase space compression can be identified with the heat lost by the system to the thermostat. This is consistent with the assumption of eq . If the term $\triangle S$ can be identified by the change in the entropy, then the above equation boils to the Second Law of Thermodynamics, [@key-19; @key-20]
With this identification, we are ready to take-on the Jarzynski’s work theorem in the context of Nose-Hoover thermostatted system.
$$<e^{(-\beta W)}>=\int d\mathbf{z}_{A}p(\mathbf{z}_{A})P_{+}(\mathbf{z}_{B},\triangle S|\mathbf{z}_{A})e^{(-\beta W(\mathbf{z_{A},z_{B},\triangle S)}))}d(\mathbf{z_{B}})d(\triangle S)\label{eq:103}$$
It is easy to understand how the above integral is constructed. the quantity $P_{+}(\mathbf{z}_{B},\triangle S|\mathbf{z}_{A})$ gives the probability of system making a transition from $\mathbf{z}_{A}$ at time $t=0$ to $\mathbf{z}_{B}$ at time $t=\tau$ and in the process generates a phase space compression of $\triangle S$. $p(\mathbf{z}_{A})$ denotes the probability that the system is found in the state $\mathbf{z}_{A}$at time $t=0.$ If the times are so chosen that the system is fully equilibrated at time t=0, this probability is actually the probability of the canonical ensemble:
$$p(\mathbf{z}_{A})=\frac{e^{-\beta H_{A}^{s}(\mathbf{Z_{A})}}}{\mathbf{Z}_{A}}\label{eq:104}$$
where $\mathbf{Z}_{A}$ is the partition function for the system in the state A, eq . Substituting this and also the detailed fluctuation theorem, we have
$$<e^{-\beta W}>=\frac{1}{\mathbf{Z}_{A}}\int d\mathbf{z}_{A}e^{-\beta H_{A}^{s}(\mathbf{Z_{A})}}e^{\frac{\triangle S}{k_{B}}}P(\mathbf{z}_{A}^{*},-\triangle S|\mathbf{z}_{B}^{*})e^{(-\beta W(\mathbf{z_{A},z_{B},\triangle S)}))}d(\mathbf{z_{B}})d(\triangle S)\label{eq:105}$$
Consider the terms in the exponentials, from eq and eq we have $$-\beta\left[H_{A}^{s}(\mathbf{z}_{A})-T\triangle S+W(\mathbf{z_{A},z_{B},\triangle}S)\right]=-\beta(H_{A}^{s}(\mathbf{z}_{B})\label{eq:106}$$ and thus
$$<e^{-\beta W}>=\frac{1}{\mathbf{Z}_{A}}\int d\mathbf{z}_{A}e^{-\beta H_{B}^{s}(\mathbf{z_{B})}}P(\mathbf{z}_{A}^{*},-\triangle S|\mathbf{z}_{B}^{*})d(\mathbf{z_{B}})d(\triangle S)\label{eq:107}$$
But $$\int P(\mathbf{z}_{A}^{*},-\triangle S|\mathbf{z}_{B}^{*})d\mathbf{z}_{A}d(\triangle S)=1\label{eq:108}$$ And hence we have
$$<e^{-\beta W}>=\frac{1}{\mathbf{Z}_{A}}\int d\mathbf{z}_{B}e^{-\beta H_{B}^{s}(\mathbf{z_{B})}}=\frac{\mathbf{Z}_{B}}{\mathbf{Z}_{A}}=exp(-\beta\triangle F_{AB})\label{eq:109}$$
This is the Jarzynski’s Work theorem, eq we set out to prove.
Although the connection between the detailed Fluctuation theorem and the Jarzynski’s identity has been established here only for the case of Nose-Hoover thermostatting scheme, it should be evident that the Jarzynski’s work theorem can be derived in all contexts where the Detailed Fluctuation theorem is applicable.
For illustration, consider the case of Gaussian Isokinetic ensemble. As mentioned above, these equations of motion fail to generate the proper canonical sampling in the momentum space but generates a canonical distribution in the coordinate space. But from eq, we see that the Free energy differences depends on the logarithm of the ratio of the two partition functions and hence the momentum partition function cancels out in the ratio and we are left with the ratio of the configuration partition functions at the states A and B. So we can see that one can realize the Jarzynski’s identity in the Gaussian Isokinetic ensemble.
As with the Nose-Hoover thermostat example above, Consider a system given by the Hamiltonian, $H^{s}(\mathbf{\{r}\},\{\mathbf{p\},}t)=\overset{N}{\underset{i=1}{\sum}}\dfrac{\mathbf{p}_{i}^{2}}{2m_{i}}+\Phi(\{\mathbf{r\}},t)$ we have the Gaussian Isokinetic equations of motion of the form,
$$\mathbf{\dot{r}_{i}=}\dfrac{\mathbf{p_{i}}}{m_{i}}\quad i=1,..,N\label{eq:110}$$
$$\mathbf{\dot{p_{i}}=-\nabla_{\mathbf{r}_{j}}\phi(\{\mathbf{r}\},t)}+\mathbf{\alpha(}\{\mathbf{r\}},\mathbf{\{p}\},t)\mathbf{p_{i}}\quad i=1,..,N\label{eq:111}$$
where $$\mathbf{\alpha(}\{\mathbf{r\}},\mathbf{\{p}\},t)=\left[\frac{\overset{N}{\underset{j=1}{\sum}}\nabla_{\mathbf{r}_{j}}\phi(\{\mathbf{r}\},t).p_{j}/\mathbf{m_{j}}}{\overset{N}{\underset{j=1}{\sum}}p_{j}^{2}/\mathbf{m_{j}}}\right]\label{eq:112}$$
and the potential $\phi(\{\mathbf{r}\},t)$ is such that $\Phi(\mathbf{r}_{1},...,\mathbf{r}_{N},0)=\Phi_{A}(\mathbf{r}_{1},...,\mathbf{r}_{N})$ and $\Phi(\mathbf{r}_{1},...,\mathbf{r}_{N},\tau)=\Phi_{B}(\mathbf{r}_{1},...,\mathbf{r}_{N})$ where $\Phi_{A}$and $\Phi_{B}$ are the potentials of states A and B respectively. The time variation of this potential indicates that the work is done on the system. Consider again, eq, we have the heat lost to the thermostat given by
$$Q=\overset{\tau}{\underset{0}{\int}}dt\frac{\partial H^{s}(\mathbf{z}_{t},t)}{\partial\mathbf{z}_{t}}\dot{\mathbf{z}_{t}}=\overset{\tau}{\underset{0}{\int}}dt\mathbf{\alpha(}\{\mathbf{r\}},\mathbf{\{p}\},t)\overset{N}{\underset{i=1}{\sum}}p_{i}^{2}/\mathbf{m_{i}}\label{eq:113}$$
substituting for $\mathbf{\alpha(}\{\mathbf{r\}},\mathbf{\{p}\},t)$ and simplifying we have $$Q=\overset{\tau}{\underset{0}{\int}}dt\left[\overset{N}{\underset{j=1}{\sum}}\nabla_{\mathbf{r}_{j}}\phi(\{\mathbf{r}\},t).p_{j}/\mathbf{m_{j}}\right]\label{eq:114}$$ Identifying $\overset{N}{\underset{j=1}{\sum}}\nabla_{\mathbf{r}_{j}}\phi(\{\mathbf{r}\},t).p_{j}/\mathbf{m_{j}}=\frac{K}{3N-1}\Lambda(t)$ we have $Q=\frac{K}{3N-1}\overset{\tau}{\underset{0}{\int}}dt\Lambda(t)$
Together with eq this gives $$Q=\frac{-K}{3N-1}\frac{\Delta S}{k_{B}}\label{eq:115}$$ Choosing the arbitrary constant $K=(3N-1)k_{B}T$ (for reasons already discussed ) we have
$$Q=-T\Delta S\label{eq:116}$$
Again, we see that the heat lost by the system is proportional to the total phase space compression, as with the Nose-Hoover thermostat. With this identification, the procedure to calculate $<e^{-\beta W}>$ is essentially unchanged from the Nose-Hoover case, and we have the Jarzynski’s identity in the case of a system coupled to a Gaussian Isokinetic ensemble.
Conclusion
==========
Detailed Fluctuation Theorem has been extended to a class of thermostatted systems, evolving under the extended system dynamics. It is demonstrated that this theorem retains the same form as for the original DFT for entropy production when one replaces the thermodynamic entropy with phase space compressibility. This theorem is of a wider applicability than its original counterpart and can be applied even to the systems at equilibrium. It is shown that this detailed fluctuation theorem is formally equivalent to the detailed balance equation for systems at equilibrium. Rederivation of the Jarzynski’s identity through the Detailed Fluctuation theorem has been demonstrated for both Nose-Hoover thermostat and the Gaussian Isokinetic ensembles.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Twelve single chambered, air-cathode Tubular Microbial Fuel Cells (TMFCs) have been filled up with fruit and vegetable residues. The anodes were realized by means of a carbon fiber brush, while the cathodes were realized through a graphite-based porous ceramic disk with Nafion membranes (117 Dupont). The performances in terms of polarization curves and power production were assessed according to different operating conditions: percentage of solid substrate water dilution, adoption of freshwater and a 35mg/L NaCl water solution and, finally, the effect of an initial potentiostatic growth.
All TMFCs operated at low pH (pH$=3.0 \pm 0.5$), as no pH amendment was carried out. Despite the harsh environmental conditions, our TMFCs showed a Power Density (PD) ranging from 20 to 55 mW/m$^2 \cdot$kg$_{\text{waste}}$ and a maximum CD of 20 mA/m$^2 \cdot$kg$_{\text{waste}}$, referred to the cathodic surface.\
COD removal after a $28-$day period was about $45 \%$.
The remarkably low pH values as well as the fouling of Nafion membrane very likely limited TMFC performances. However, a scale-up estimation of our reactors provides interesting values in terms of power production, compared to actual anaerobic digestion plants. These results encourage further studies to characterize the graphite-based porous ceramic cathodes and to optimize the global TMFC performances, as they may provide a valid and sustainable alternative to anaerobic digestion technologies.
author:
- Nicole Jannelli
- Rosa Anna Nastro
- Viviana Cigolotti
- Mariagiovanna Minutillo
- Giacomo Falcucci
date: 'Sept., 1st, 2016'
---
**Keywords**:\
microbial fuel cell; waste to energy; power generation; direct conversion of solid organic waste\
Introduction
============
The quest for sustainable and efficient energy sources has been the object of great research efforts in the last decades, [@An_2011; @Srirangan_2012]. Along with the increasing need of a more sustainable waste management, new approaches able to combine waste treatment with energy recovery can contribute to the establishment of a green economy [@Karagiannidis_2009; @Koufodimos_2002; @Falcucci_CCHP_2014; @Nastro_et_al_2016]. Generally, waste-to-energy technologies based on incineration, gasification or anaerobic digestion are recommended for large-scale plants, to produce thermal energy or to convert waste in biogas, syngas and other secondary fuels, [@Karagiannidis_2009; @Koufodimos_2002; @Pandey_2016]. Nevertheless, the large part of these processes do not require the selection of incoming waste, with a consequent, unavoidable production of pollutants, toxic leachate, ash and greenhouse gases, [@Pavlas_2009]. The treatment of such pollutants makes the management and control of these processes complex and expensive, [@Cormos_2014; @Dong_2014].\
In this context, bio-reactors based on natural processes like fermentation and/or methanogenesis, can play an important role towards the establishment of a more sustainable waste management. Microbial Fuel Cells (MFCs) are based on the ability of microorganisms to use inorganic compounds as electronic acceptor to obtain electric power directly from microbe metabolism, without any combustion. Recent researches confirm the potential of such systems in turning “waste” into a “resource”, by treating different types of substrates and even recovering by-products with a non-negligible economic potential, [@Nastro_et_al_2016].\
As biomass-based systems, MFCs (like other Bioelectrochemical Systems) are considered carbon neutral: the bio-transformation of organic matter into chemicals through microbial metabolism, in fact, prevents the primary production of CO$_2$ emissions. Moreover, MFCs do not involve CH$_4$ production and combustion, as opposed to traditional anaerobic digestion plants. Thus, this newborn technology is characterized by remarkable features: the direct electrical power production, the conversion of the chemical energy contained in any form of biomass [@Mohan_2014], low environmental impact, low operating temperatures and simple architectures [@Du_2007]. All these characteristics, together with the environmental advantages ingrained with this technology are supposed to largely overcome the costs for MFC development and implementation, [@Nastro_et_al_2016].
However, even though a wide number of papers deals with the set-up and the study of MFCs fed with wastewaters, few attempts to apply this innovative technology to solid organic waste treatment have been carried out. In 2011, Mohan & Chandrasekhar studied the operational factors affecting the performances of MFCs fed with canteen food waste, focusing on electrodes distance and feedstock pH, [@Mohan_2011]. After this seminal work, other researchers started working on the application of MFCs to the Organic Fraction of Municipal Solid Waste (OFMSW) using different approaches, but all confirming the effectiveness of MFC technology as tool for energy recovery and organic load removal from OFMSW, [@Nastro_et_al_2016]. MFCs, in fact, represent a valid alternative to achieve small-scale, distributed and efficient conversion of organic waste into electricity [@Logan_2006], even in developing Countries, where solid waste is often dumped rather than landfilled.\
Nevertheless, the industrial implementation of this technology is still challenging, [@Di_Maria_2015; @Di_Palma_2015]. The use of solid or liquid food waste in MFCs generally leads to anodic environmental conditions highly unfavorable for bioelectricity production, due to low pH and high salinity and, as a consequence, high ionic strength, [@Mohan_2014; @Li_2016; @Karthikeyan_2016; @Choi_2015].\
Thus, the scale-up of reactors, necessary for industrial power applications, is limited by several issues, so that further research is required to improve the reactor geometry, the economic feasibility and the response of the system to unfavorable conditions. In our previous works, we assessed the possible adoption of MFC technology for the treatment of OFMSW, highlighting the importance of appropriate reactor layout and electrode design, in order to minimize the internal resistance, [@Frattini_2016; @Falcucci_2013_MFC; @Nastro_EFC_2015; @Nastro_Cleaner_2015].\
In this work, we explored the efficiency of tubular single-chambered, air-cathode MFCs (TMFCs) fed by a feedstock composed of vegetable and fruit slurry. All MFCs were provided with Nafion membranes adhering to the cathode, which was realized by means of a graphite-based porous ceramic disk: this innovative material was ad-hoc designed and realized in order to be used in scaled-up MFCs, as well. Power production was studied as a function of different parameters, such as solid substrate inoculum, liquid-to-solid ratio, salinity of amending solution and the adoption of a preliminary potentiostatic growth phase. An ad-hoc measurement set-up has been realized to test the MFC reactors in order to prove evidence of their feasibility and reliability in standard and also harsh anodic environment.
Materials & Methods
===================
TMFC Assembly
-------------
Twelve tubular MFC bioreactors, adapted from [@Logan_2007; @Kim_2009; @Frattini_2016], were realized by using standard 50 mL polypropylene Falcon test tubes, supplied by BD Corning Inc. (Tewksbury, USA), sterile and suitable for biological cultures. Two 20 mL Falcon tubes were used to sample the organic feedstock and to monitor pH (see Fig. \[Fig\_1a\]).
![\[Fig\_1a\] Layout of the Tubular MFC, realized by means of one 50 mL and two 20 mL Falcon test tubes.](Fig_1.pdf "fig:"){height="5.5cm"}\
The electrodes were made by carbon-fiber anode brush, realized with a high strength carbon fiber from FIDIA s.r.l. (Perugia, ITALY) and unpolished stainless steel wire (ASTM A313) with 0.5 mm section, while for the cathode a porous ceramic disk was developed starting from graphite powder type GK 2 Ultra-fine, by AMG Mining AG (Hauzenberg, Germany). The brush anode had an estimated surface area of 0.22 m$^2$/g while the cathode disk had a surface area of 60.75 m$^2$/g, [@Logan_2007], (Figure \[Fig:1b\](b)). The electrodes were placed at a distance of $\sim 3$ cm. A standard Nafion 117 membrane by DuPont Inc. (Richmond, USA, [@Nafion]) was used to seal the porous cathode.
\
Data Acquisition System
-----------------------
The data collection hardware was based on the Arduino board MEGA 2560, [@arduino], composed by a load array (for polarization curve acquisition) with 6 resistors, ranging from $\sim 10^6$ $\Omega$ to $\sim 10 \ \Omega$. The software for data acquisition was developed with LabVIEW Interface For Arduino, (LIFA) package. Figure \[Fig\_arduino\] sketches the acquisition system and the measurement chain.
![\[Fig\_arduino\] Electrical Scheme of the acquisition and measurement system.](Fig_3.pdf){width="73.00000%"}
Experimental Setup {#setup}
------------------
Apples, pumpkins, chickpeas and zucchini in a 1:1:1:1 ratio were used to prepare a slurry by mechanically mixing the vegetable residues with water (see Tab \[Tab\_2\] for the details of water composition).\
In order to verify the effect of salt concentration on TMFC performances, six reactors were filled with a slurry prepared using fresh water; in the other six cells, a NaCl solution (35 mg/L NaCl) was added to improve the slurry conductivity and test TMFC performances in presence of high Na$^+$ concentration, as salts concentration can increase, in leachate, along with the organic matter degradation, [@Mohan_2014; @Li_2016; @Karthikeyan_2016; @Choi_2015].\
The Liquid-to-Solid ratio, and the specific conditions used in each cell are reported in Table \[Tab\_3\]. Cells 4, 5 and 10, 11 have been added with spent substrate (10$\%$ of the overall feedstock) from TMFC reactors filled with the same organic substrate and previously operated for a 28-day period.\
Cells 1, 2, 3, 6, 7, 8 and 9 were filled with 30 g of solid waste and 30 g of water, while cells 4, 5, 10, 11 and 12 were filled by 15 g of solid waste and 45 g of water, to obtain the desired Liquid-to-Solid ratio. Finally, cells 3 and 9 were subject to a potentiostatic growth phase for the first 7 days of operation, by imposing an external potential difference between the anode and the cathode of $\Delta V_{ext}= 0.8$ V: this procedure was aimed at verifying the possible effects of narrowing the bacteria consortium to a mixed culture of exoelectrogenic micro-organisms.
---------------------------------------------- ---------------
pH 6.65 $[-]$
Electrical Conductibility (at 20$^{\circ}$C) 420 $\mu$S/cm
Fixed residue (at 180$^{\circ}$C) 341 mg/L
Free CO$_2$ at source 125 mg/L
Bicarbonates 254 mg/L
Potassium 26.9 mg/L
Calcium 31.5 mg/L
Magnesium 9.4 mg/L
Fluorides 1.1 mg/L
Nitrates 7 mg/L
---------------------------------------------- ---------------
----------- ------- ------- ---------------- ----------------- -----------------
Cell $\#$ Fresh Salt Potentiostatic Spent substrate Liquid-to-Solid
water water conditioning amendment ratio
1 X 1:1
2 X 1:1
3 X X 1:1
4 X X 3:1
5 X X 3:1
6 X 3:1
7 X 1:1
8 X 1:1
9 X X 1:1
10 X X 3:1
11 X X 3:1
12 X 3:1
----------- ------- ------- ---------------- ----------------- -----------------
Chemical Analyses
-----------------
Slurry organic content in terms of COD was measured according to Standard Methods (2012), [@Standard_Methods_2012], at beginning of the experiment and after 28 days in order to assess MFCs efficiency in organic load removal. NO$_3^-$ and NO$_2^-$ concentrations were evaluated as well, in order to provide further information regarding the micro-organism metabolism inside the TMFC reactors.
COD NO$_3^-$ NO$_2^-$
----------------- ------- ---------- ----------
Fresh Substrate 48960 40 9.6
Spent Substrate 27520 180 12.2
Mixture 48320 140 11.4
Results & Discussion
====================
Polarization Behavior
---------------------
Figure \[Fig\_3\] reports the polarization curves of the different cells, according to the feedsotck concentration and to its salinity; in the Figure, the performance of the cells subject to the initial potentiostatic growth process are reported, as well.\
According to Fig. \[Fig\_3\], the cells highlight similar polarization trends, despite the differences in the substrate. This can be ascribed to the layout of the cells and gives evidence of the reproducibility of TMFCs performances.\
Salt-water TMFCs were characterized by a different behavior, compared to fresh-water cells: salty reactors, in fact, showed a lower value of internal resistance (R$_{\text{int}}$, given by the slope of polarization curves) in the first 7 days of the experiment but, during the last week of operation, the internal resistance of these cells became higher than the R$_{\text{int}}$ value of freshwater TMFCs. It is known that an increase in the feedstock conductivity results in a decrease in the cell resistance: in fact, cations concentration increases and, thus, their flow towards the cathode, where they take part to the electrochemical reactions.\
However, in presence of high cation concentration, Nafion sulfonated groups are rapidly saturated, preventing the protons linkage to the membrane itself, [@Chae_et_al_2008]. This mechanism, known as Nafion *fouling*, together with the biofilm growth, leads to the saturation of the membrane, which eventually prevents cations and protons from taking part to the cathodic reactions: this causes a decrease of cathode functionality, an increase in the cell internal resistance (compared to freshwater reactors) and, as a consequence, a degradation of TMFC performance.\
![\[Fig\_int\_res\] Evolution of the internal resistance of the cells 1$\&$2, 4$\&$5 and 7$\&$8: the values of internal resistance and their evolution in time are very similar, confirming the assessed reproducibility of our tests.](resistenze_interne.pdf){width="75.00000%"}
Effect of salt water amendment
------------------------------
Figure \[Fig\_pH\] reports the trends of internal pH value for fresh-water cells (left panel) and for salt-water reactors (right panel). The salt-water TMFCs were characterized by a higher internal acidity, due to the membrane fouling. Nafion, in fact, has negatively-charged chemical groups which attract the positive ions in the organic substrate. Since only H$^+$ can pass through the membrane, the other positive ions accumulate at the cathode, preventing the further passage of protons: this causes the lower pH values shown in Fig. \[Fig\_pH\](b).
The effect of water composition on TMFC performance is reported in Figure \[fresh-salt\](a) for cells 1 and 2 vs cells 7 and 8 at day 14 and in Fig. \[fresh-salt\](b) for cells 4 and 5 vs 11 and 12. In both cases, the cells with NaCl addition provide lower performance in terms of power production.
Effect of Potentiostatic Growth
-------------------------------
In Figure \[pot\_growth\], the performance of TMFCs that were subject to a potentiostatic growth phase in the first week of operation are compared to the behavior of similar reactors (according to Tab. \[Tab\_2\]).\
From this Figure, it is evident that carrying out the potentiostatic growth at the beginning of the experiment does not bring sensible enhancements in the power production of the reactors along the experiment, both for fresh and salt water systems.
According to the results in the literature, the potentiostatic growth is known to have a positive effect on the performance of MFCs operating with complex substrates, such as wastewaters with metal pollutants, [@Huang_2011_MFC]. In our case, on the contrary, it has a detrimental effect on the cell performance: this can be ascribed to the different characteristics of our substrate, which is rich in carbohydrates, sugar and proteins (typical with fruit and vegetable slurries), and to the presence of the Nafion membrane at the cathode.
Effect of spent feedstock inoculum
----------------------------------
Cells 4, 5 and 10, 11 have been prepared with the same water as 1, 2, 3 and 7, 8, 9 respectively, according to the details reported in the previous Sections.\
The aim of inoculating pre-digested feedstock was to investigate whether an enriched microflora could increase the power production of TMFCs.\
Figure \[inoc\] shows the comparison of cell performance in terms of power production at different days of cells 4 and 5 vs 6 and 11 vs 12. Cell 10 has not been included, as its performances were too different from all other TMFCs. According to Fig. \[inoc\], the addition of pre-digested organic matter provides a performance decrease of $\sim 10 \%$ for freshwater reactors. In the case of salt-water TMFCs, the cell with the inculum has provided almost half the power production as the corresponding reactor with fresh solid waste. Nevertheless, a certain reduction of the cell activation energy was detected, as highlighted by the slope of the polarization curves at low current densities in Fig. \[inoc\].
Such a complex behavior in presence of a spent feedstock amendment can be ascribed to the very low pH values of the pre-digested substrate (pH $\sim 2.5$) which contributes to further increasing the overall acidity inside the reactors, making the environmental conditions even harsher.\
In the salty reactors, the increase in salt concentration leads to the further decrease in performance showed in Fig. \[inoc\]: further studies are in progress to shed light on these outcomes.
\
Effect of water dilution
------------------------
Figure \[water\_dil\] shows the comparison between power curves at the 14th day of operation for cells characterized by 1:1 and 3:1 Liquid-to-Solid ratio, for fresh water (Fig. \[water\_dil\](a)) and salt water (Fig. \[water\_dil\](b)) reactors.\
In the case of fresh water, the two trends at the 14th day of operation are very close, while in the case of salt-water reactors, the power trend in more diluted MFCs is considerably above that of 1:1 reactors. It is worth noting that in both the fresh- and salt-water reactors, the higher the feedstock dilution, the higher the power density.
Industrial outlook
==================
It is interesting to foresee an industrial application for energy generation from MFCs based on solid organic waste. Very likely, low pH values negatively affected tubular MFCs performances, nevertheless a power production at $\sim 40$ mW/m$\cdot$kg$_\text{waste}$ can be estimated: in all the cases, in fact, such a power density value is easily reached both with fresh and salt water, with the solid organic content ranging between 15 g and 30 g . We can, thus, focus on this power value to study the feasibility of the scale-up of such reactors for large power production applications. As a benchmark, we consider the energy produced at a modern anaerobic digestion plant, which is accredited of processing 33000 tons per year of solid waste, producing $\sim 1$ MW of power, for a global energy production of $\sim 8000$ MWh/year, [@Caivano]. If we consider the same amount of waste, our TMFCs would produce: $$40 \times 10^{-3} \ \text{W/m}^2 \cdot \text{kg} \; \times \; 33 \cdot 10^6 \ \text{kg} \sim \; 1000 \ \text{kW/m}^2 \; , \nonumber$$ which is comparable to a modern anaerobic digestion plant of the same capability.\
It is well known that scaling up MFCs causes issues related to a considerable increase in internal resistance, but the realization of proper anodic and cathodic surfaces can limit this problem. Our research on an innovative cathodic material is aimed at achieving a sustainable and economically cheap solution to this issue.
Conclusions
===========
In this work we have assessed the performance of air cathode, single-chambered, Tubular Microbial Fuel Cells (TMFCs) provided with Nafion membrane, according to different operating conditions.\
All the reactors have worked for 28 days at ambient temperature ($T = 20 \pm 2^{\circ}C$) with feedstocks characterized by low values of pH (pH$=3.0 \pm 0.5$) and high NaCl content (35 mg/L in 6 of the 12 reactors).\
Our TMFCs proved to be capable of producing power even with such harsh internal conditions, showing a higher power yield in the cells with fresh water: the salty feedstock, in fact, is characterized by considerably low values of pH, probably due to the progressive deterioration and fouling of Nafion membrane. For the industrial scaling-up of our reactors, then, the Nafion membrane will not be adopted both for its deterioration and for its prohibitive costs.\
The potentiostatic growth at the beginning has provided some slight increase in power production only in presence of salt water and only for the first half of the experiment: thus, such a procedure is not considered as a valid tool to enhance MFC power productivity.\
The inoculum of 10$\%$ of a solid organic waste previously digested for a 28-day period in an analogous experiment has not shown a positive effect on MFC performance, due to the low pH value reached at the end of the digestion period (pH $\sim 2.5$). Such a procedure, which is standard in anaerobic digestion plants for the development of methanogenic bacteria, is not recommended to improve MFC performance working in low-pH environments.
Finally, we have developed and employed a novel porous material at the cathode, which has shown promising performances and could be adopted in scaled-up MFC plants; nevertheless, further studies are needed to better characterize the graphite-ceramic material and to improve its chemical and mechanical properties.
The obtained results provide a further evidence of the versatility of MFC technology.
Acknowledgments
===============
This work was supported by the Italian Government Research Project PON03PE\_00109\_1 “FCLab - Sistemi innovativi e tecnologie ad alta efficienza per la poligenerazione”, with Prof. Elio Jannelli as the Scientific Responsible.\
The precious contributions of Dr. Giovanni Erme and Dr. Enzo De Santis for reactor realization are kindly acknowledged.
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---
abstract: 'Microbunching instability usually exists in the linear accelerator (linac) of a free electron laser (FEL) facility. If it is not controlled effectively, the beam quality will be damaged seriously and the machine will not operate properly. In the electron linac of a soft X-Ray FEL device, because the electron energy is not very high, the problem can become even more serious. As a typical example, the microbunching instability in the linac of the proposed Shanghai Soft X-ray Free Electron Laser facility (SXFEL) is investigated in detail by means of both analytical formulae and simulation tools. In the study, a new mechanism of introducing random noise into the beam current profile as the beam passes through a chicane-type bunch compressor is proposed. The higher-order modes that appear in the simulations suggest that further improvement of the current theoretical model of the instability is needed.'
author:
- Dazhang Huang
- Qiang Gu
- Zhen Wang
- Meng Zhang
- King Yuen Ng
title: 'Analysis of the microbunching instability in a mid-energy electron linac'
---
\[sec:level1\]Introduction
==========================
The microbunching instability that occurs in the linac of a free-electron Laser (FEL) facility has always been a problem that degrades the quality of electron beams. The instability is driven by various effects, such as the longitudinal space charge (LSC) [@Saldin], coherent synchrotron radiation (CSR) [@Heifets; @ZHuang1], and linac wakefields. As the beam passes through a bunch compressor, for example, a magnetic chicane, the energy modulation introduced by those effects is transformed into density modulation and thus the instability develops. In a FEL facility, there is usually more than one bunch compressor and the overall growth of the instability is the product of all the gains in each compressor. As a result, the final gain of the instability can become significant. On the other hand, the FEL process has a high demand for electron beam quality in terms of peak current, emittance, energy spread, etc. Therefore without effective control, the microbunching instability can damage the beam quality so seriously that the whole FEL facility fails.
The Shanghai Soft X-Ray FEL Facility (SXFEL) project has been proposed and the construction will start soon. It is a cascading high-gain harmonic generation (HGHG) FEL facility operating at the wavelength of 9 nm in the soft-X-ray region with 840 MeV electron beam energy at the linac exit. In this article, as a typical example, the study of the microbunching instability based on the design parameters of the SXFEL linac is carried out in order to gain better understanding of the instability process and to analyze the limitations of analytic formulae and numerical simulations. The study shows that the initial noise level of the beam current has a big impact on the final behavior of the instability, which makes the analysis of the initial current noise imperative. On the other hand, we realize that higher-order harmonics of the beam current can be excited in a bunch compressor, which may modify the linear-theory modeling of the gain. Moreover, the noise in one transverse plane can also be transferred into the longitudinal direction due to the transverse-longitudinal coupling in each dipole magnets within a magnetic bunch compressor. This transfer reduces the smoothness of the current and its effect will be considered. Also because of this, extra noise can be introduced into the beam current when the beam passes through a chicane-type laser heater. The structure of this paper is as follows: In Sec. II, we introduce the basic structure of the SXFEL linac and the fundamental theory of microbunching instability. Comparisons of the gain curves with and without the initial current noise are provided based on the present design parameters and the results are analyzed. In Sec. III, the higher-order modes of the instability exhibited in the study are presented, In Sec. IV, a new mechanism of introducing random noise into the beam current is proposed and discussed. Summaries and concluding remarks are given in Sec. V.
Microbunching instability in SXFEL
==================================
The basic mechanism of the microbunching instability driven by various effects has been studied and their corresponding theories have been derived in Refs. . The development of the instability is similar to the amplification process in a klystron amplifier. The initial density modulation and/or white noise are transformed into energy modulation by various kinds of impedance (e.g., LSC, CSR, linac wakefields) during beam transportation. The energy modulation accumulates and is fed-back into the beam after passing through a dispersive section such as a bunch compressor. This accumulation results in much stronger amplified density modulation. Additionally, the CSR effect in the dispersive section forms a positive feedback which in turn enhances the instability. More dispersive sections in a linac result in stronger instability growth. In the linac of the SXFEL, both S-band and C-band rf structures are used to accelerate the electron beam. One X-band structure is implemented to suppress the second-order nonlinear components in the longitudinal phase space to avoid undesired growth of transverse emittance and slice energy spread. Two magnetic chicane-type bunch compressors (BC1 and BC2) are used to compress the beam to arrive at the required peak current. The layout of the SXFEL linac is shown in Figure \[linac\], and the magnet layout and some of the Twiss parameters of the SXFEL linac are shown in Figure \[magnet\]. Note that S-band accelerating structures are used in sections L1 and L2, whereas a C-band accelerating structure is used in section L3. The following discussions are based on the design parameters shown in Table \[parameters\], which is taken from the SXFEL feasibility study report. Moreover, since the length scale in which the structural impedance is effective is much longer than that of microbunching wavelength [@feasibility; @Venturini1], we may neglect the effects from the linac wakefields in the following discussions without compromising accuracy.
![(Color) Layout of the SXFEL linac.[]{data-label="linac"}](linac.eps "fig:"){width="80mm"} -0.05in
![(Color) Magnet layout and Twiss parameters of the SXFEL linac.[]{data-label="magnet"}](sxfelmag.eps "fig:"){width="70mm"} -0.05in
Parameter Value
-------------------------------------------------------- ------------- -- -- --
bunch charge (nC) 0.5
beam energy out of injector (MeV) 130
bunch length (FWHM) at the exit of injector (ps) 8
peak current before BC1 (A) 60
beam radius (rms) before BC1 (mm) 0.40
beam energy before BC1 (MeV) 208.4
local (slice) rms energy spread right before BC1 (keV) 6.5
linac length up to BC1 (m) 17.3
$R_{56}$ of BC1 (mm) 48
beam energy before BC2 (MeV) 422.0
local (slice) rms energy spread right before BC2 (keV) 43.4
linac length up to BC2 (m) 60.3
$R_{56}$ of BC2 (mm) 15
linac length after BC2 (m) 63.7
compression ratio (BC1$\times {\rm BC2}$) $5\times 2$
beam radius (rms) before BC2 (mm) 0.26
: \[parameters\]Main beam parameters used in the microbunching-instability study for the SXFEL (obtained by [ELEGANT]{} [@elegant], with laser heater turned off in simulation).
The microbunching instability for the case of linear compression has been discussed by Saldin et al. [@Saldin] phenomenologically by comparing the energy distributions before and after the compression. Consider a density modulation at wavenumber $k$. Without higher harmonics of beam current taken into account, the gain of the instability driven by the wake fields upstream of the compressor reads [@Saldin] G=Ck|R\_[56]{}|(-C\^2k\^2R\^2\_[56]{}). \[saldin\] Here, $\gamma$ is the nominal relativistic factor of the electron beam with rms local energy spread $\sigma_\gamma$ in front of the bunch compressor, $C=1/(1+hR_{56})$ is the compression ratio, $h$ is the linear energy chirp, $R_{56}$ is the 5-6 element of the transport matrix, $I_0$ is the initial peak current of the beam, $Z_0=377 \ \Omega$ is the free-space impedance, $Z_{\rm tot}$ is the overall impedance upstream of compressor including those of the LSC, linac wake, etc., and $I_A=17$ kA is the Alfven current. The longitudinal space-charge impedance per unit length in free space takes the form [@ZHuang; @Venturini]: $$\begin{aligned}
Z_{\rm LSC}(k)&=\frac{iZ_0}{\pi kr^2_b}\Bigg[1-\frac{kr_b}{\gamma}K_1\Bigg(\frac{kr_b}{\gamma}\Bigg)\Bigg]
\nonumber \\
&\approx\left\{\begin{array}{lll}
\frac{iZ_0}{\pi kr^2_b} & \frac{kr_b}{\gamma}\gg1,\\
~&~ \\
\frac{iZ_0k}{4\pi\gamma^2}(1+2{\rm ln}\frac{\gamma}{r_bk}) & \frac{kr_b}{\gamma}\ll1,
\end{array}\right.
\label{LSCimp}\end{aligned}$$ where $r_b$ is the radius of the beam, $K_1$ is the first-order modified Bessel function of the second kind. Based on Eqs. (\[saldin\]) and (\[LSCimp\]), using the parameters output by the [ELEGANT]{} simulation starting from a beam with $\sim\pm1\%$ noise fluctuation in current and 1 – 2 keV uncorrelated (slice) energy spread (Figure \[inputbeam\]), the gains of microbunching instability in the region around the peak current induced by LSC impedance at the exits of BC1 and BC2 (first and second bunch compressors) are computed by the analytic formula. The beam parameters at the exit of the SXFEL injector are prepared by [PARMELA]{} [@feasibility; @parmela] simulation employing one million macro-particles. The LSC impedances are computed separately in drift space and the accelerating section.
The reason why we purposely choose to use a noisy input instead of a smooth one is because the real beam is usually not ideally smooth, it always includes the ripples introduced by the random noise fluctuation, the quantum process of field emission, the laser power jitter, etc. On the other hand, the linear theory is applied in the computation because: I. in general, the initial density modulation is very small, and II. the gain in the first compressor BC1 is small as well.
Besides the LSC-induced microbunching instability, when a charged particle beam passes through a bunch compressor, coherent synchrotron radiation (CSR) can be emitted at wavelengths much shorter than the bunch itself. The density of bunch particles is modulated at those wavelengths. As we have already discussed, since the CSR effect in a bunch compressor introduces positive-feedback to the microbunching instability, the gain of the instability therefore rises rapidly.
The gain of CSR-driven microbunching instability in a bunch compressor has been derived analytically [@ZHuang1] in terms of beam energy, current, emittance, energy spread and chirp, as well as initial lattice and chicane parameters. Assuming a beam uniform in the $z$-direction and Gaussian in transverse and in energy distributions, the CSR-driven microbunching instability growth follows the expression [@ZHuang1]: $$\begin{aligned}
G_f&\approx\bigg |\exp\bigg [-\frac{{\bar{\sigma}}^2_\delta}{2(1+hR_{56})^2}\bigg ]+A{\bar{I}}_f\bigg [\bigg (F_0({\bar{\sigma}}_x) \nonumber \\
&+\frac{1-e^{-{\bar{\sigma}}^2_x}}{2{\bar{\sigma}}^2_x}\bigg )\exp{\bigg (-\frac{{\bar{\sigma}^2_\delta}}{2(1+hR_{56})^2}\bigg )} \nonumber \\
&+F_1(hR_{56},{\bar{\sigma}}_x,\alpha_0,\phi,{\bar{\sigma}}_\delta)\bigg ] \nonumber \\
&+A^2{\bar{I}}^2_fF_0({\bar{\sigma}_x})F_2(hR_{56},{\bar{\sigma}}_x,\alpha_0,\phi,{\bar{\sigma}}_\delta)\bigg |.
\label{CSRgain}\end{aligned}$$
As described in Ref. , the first term on right side of Eq. (\[CSRgain\]) represents the loss of microbunching in the limit of vanishing current, the second term (linear in current) provides a one-stage amplification at low current (low gain), and the last term (quadratic in current) corresponds to the two-stage amplification at high current (high gain). The functions of $F_0$, $F_1$, and $F_2$ are defined in Ref. , and $\bar{\sigma}_\delta$ and $\bar{\sigma}_x$ are related to, respectively, the local (slice) energy spread and rms transverse size of the beam.
In our calculation for the SXFEL linac, since the high-gain term in Eq. (\[CSRgain\]) is much smaller than the low-gain term (possibly due to the rapid increase of slice energy spread in the bunch compressor), we just ignore the contribution of the two-stage amplification.
As we mentioned in the previous section, noise is always introduced inevitably when electrons are emitted from the cathode. Therefore, in order to have a clearer picture about the effect of the initial noise on the final gain, the gain with an initially smoothed electron beam is computed as well. Figure \[inputbeam\] shows the current profile of the input beam with $\sim1\%$ noise fluctuation in rms, and figure \[inputbeam\] is the longitudinal phase space of the beam. Our study shows that the uncorrelated (slice) energy spread before the second bunch compressor (BC2) becomes larger for an initially noisy (non-smoothed) beam input. As a result, its final gain is smaller than the one computed with an initially smoothed beam. Figures \[L1dgamma\] and \[L2dgamma\] show, respectively, the uncorrelated (slice) energy spread of the smoothed and the non-smoothed beam before BC1 and BC2 by [ELEGANT]{}. All the parameters in the analytical calculation are obtained from [ELEGANT]{}. The term “smoothed beam” means the whole six-dimensional beam distribution is smoothed by removing the noise fluctuation, which is done by the sdds command smoothDist6s [@sddstoolkit]. With this command, the noise fluctuation of the beam current profile in both the longitudinal and the transverse directions, and that of the uncorrelated (slice) energy spread are all suppressed to almost zero. As we have already known, due to the smaller slice energy spread, the peak gain of the smoothed beam is much higher than that of the noisy beam, and the wavelength where the peak resides is also much shorter. However, although the total gain is small for the noisy beam, the microbunching instability can still be a problem because of the significant initial noise. As a demonstration, Figure \[finalcurrentFFT\] shows that the final current fluctuation of the non-smoothed beam is comparable to that of the smoothed one. Thus it is also consistent with the analytical results (Figure \[gaincurve\]) in terms of the wavelength where the peak resides. In summary, careful studies are needed to decide how much noise should be included in the computation.
![(Color) The slice energy spreads of the smoothed (red dot) and the non-smoothed beam (black) before BC1.[]{data-label="L1dgamma"}](L1dgamma.eps "fig:"){width="60mm"} -0.10in
![(Color) The slice energy spreads of the smoothed (red dots) and the non-smoothed beam (black) before BC2.[]{data-label="L2dgamma"}](L2dgamma.eps "fig:"){width="60mm"} -0.10in
![(Color) The current spectra of the smoothed (red dots) and non-smoothed beam (black) at the exit of the SXFEL linac. The wavelength is reduced by the factor of 10 from the initial value because of compression.[]{data-label="finalcurrentFFT"}](finalcurrentFFT.eps "fig:"){width="60mm"} -0.10in
![The layout of the SXFEL laser heater at 130 MeV.[]{data-label="laserheater"}](SXFELHeater.eps "fig:"){width="80mm"} -0.10in
Currently, a laser heater has been a common device to suppress microbunching instability [@ZHuang] by increasing the uncorrelated energy spread of the electron beam in the linac of a FEL facility. Figure \[laserheater\] shows the structure of the SXFEL laser heater including an infrared injection laser of wavelength around 1000 nm, an undulator, and a chicane-type bunch compressor. To study the behavior of the laser heater, the gain curves as functions of laser power are computed separately for the smoothed and noisy beams. Figure \[gaincurve\] shows the gain curves of a noisy beam as functions of laser power, and Figure \[gaincurve\] provides those of a smoothed beam as a function of laser power. In comparison of the two plots in Figure \[gaincurve\], we find that as the laser power increases, the peak of the gain curve of the smoothed beam decreases. This is consistent with the instability theory since the laser increases the initial uncorrelated energy spread of the beam, whereas the gain curve of the noisy beam reaches a maximum at around 0.06 MW before it starts to drop off (Figure \[gainpeak\]).
Figure \[gainpeak\] can be explained as follows: The cause of the inconsistency comes from the initial noise of the beam current. During beam transportation, the initial current noise is turned into the uncorrelated energy noise of the beam by various impedances such as the LSC impedance, CSR impedance, etc. On the other hand, on top of this, the laser heater introduces extra energy noise, which reduces the gain of the instability after BC1. The result is that the uncorrelated energy spread observed before BC2 goes down simultaneously because of the decrease of the instability gain through BC1. This explains why the peak of the gain increases as the laser power rises. When the laser heater power becomes higher, the energy noise introduced by the heater dominates and becomes larger and larger. Therefore the slice energy spread before BC2 starts to rise and the magnitude of peak drops with the appearance of a maximum.
![The behavior of the peak of the gain curve of the noisy beam.[]{data-label="gainpeak"}](gain_peak.eps "fig:"){width="60mm"} -0.10in
Higher order harmonics of beam current
======================================
Equation (\[saldin\]) only takes into account the gain of the fundamental mode of beam current. In other words, the gain curve obtained from it merely describes the behavior of the fundamental mode. However, in our simulations, higher-order harmonics of beam current are also observed along with the initial modulation. We therefore believe that, in certain cases, higher harmonics should be included in the computation of the gain. A rough analysis tell us that the gain of the higher harmonics depends not only on the beam current, but also on the modulation depth. Thus the inclusion of the higher-harmonics becomes a bit more complicated than the fundamental mode. As an example, Figure \[harmonics\] shows the excitation of higher harmonics simulated by [ELEGANT]{} after the first bunch compressor (BC1) with initial modulation wavelength of 100 $\mu$m (corresponding to 20 $\mu$m in Figure \[harmonics\] after compression) which is in the vicinity of the peak (Figure \[gaincurve\]). The initial modulation depth is 5% in rms and the fast Fourier transform (FFT) spectrum of the initial beam current is illustrated in Figure \[initcurFFT\]. We see in Figure \[harmonics\] that, besides the peak gain of the fundamental residing around 20 $\mu$m, there are also the first, second, and third harmonics excited at, respectively, 10, 6.7, and 5 $\mu$m. Similar excitations are observed as well at other initial modulation wavelengths such as $80~\mu$m, $60~\mu$m, and $35~\mu$um. Thus we need to keep in mind that higher-order of the initial modulation can also be excited along with the initial modulation. The linear-growth theory should therefore be extended.
![The fast Fourier transform (FFT) spectrum of the beam current with initial modulation wavelength of 100 $\mu$m at the exit of BC1, where 20 $\mu$m in this figure corresponds to the initial 100 $\mu $m because of the compression ratio of 5.[]{data-label="harmonics"}](100um_BPM06.eps "fig:"){width="60mm"} -0.10in
![The fast Fourier transform (FFT) spectrum of the beam current with initial modulation wavelength of 100 $\mu$m at the exit of the injector.[]{data-label="initcurFFT"}](100um_BPM01.eps "fig:"){width="60mm"} -0.10in
Transverse-longitudinal coupling
================================
One important topic in microbunching-instability study is noise. Like the self-amplified spontaneous emission (SASE) process, micro-bunches can be generated from random noise. If the beam is ideally smoothed in all directions, i.e., without any noise, fluctuation, modulation, etc., microbunching instability can hardly be excited. To reduce microbunching, there are ways to smooth the longitudinal current profile from the cathode by improving the temporal stability of driving laser. However, our investigation indicates that the density noise in the transverse direction can also introduce roughness to the longitudinal beam density distribution as a result of transverse-longitudinal coupling. The reason can be the following: Although it is well-known that the $R_{51}$ and $R_{52}$ elements in the transfer matrix of an ideal chicane-type bunch compressor are both zero, however, these two numbers are not zero for each dipole inside the chicane. As the result, the noise in one transverse plane is transferred into the longitudinal. Although the matrix elements themselves cancel out each other along the chicane, the noise transfer will not be reversed due to its random nature.
![(Color) The current (black) and energy (red dots) spectra of a beam smoothed in all directions at the exit of BC1 with the laser heater turned off.[]{data-label="smoothall"}](noLH_smth_BC1_spec.eps "fig:"){width="65mm"} -0.10in
Assuming a beam smoothed in all directions, and another beam with $\thicksim5\%$ noise level in the transverse planes but smoothed longitudinally, Figures \[smoothall\] and \[smoothlong\] show the current and energy spectra of the beams at the exit of the first chicane (BC1) without laser heating. In the figures, one can see that the levels of density and energy fluctuation (modulation) of the 3-D smoothed beam are undoubtedly smaller than those when the beam is smoothed only longitudinally.
![(Color) The current (black) and energy (red dots) spectra of a beam smoothed only in the longitudinal direction at the exit of BC1 with the laser heater turned off.[]{data-label="smoothlong"}](noLH_zsmth_BC1_spec.eps "fig:"){width="65mm"} -0.10in
Because of the transverse-longitudinal coupling, some extra noise is introduced into the longitudinal direction and the amplitudes of the modulation are also amplified when beam passing through the laser heater. Figures \[LH\] and \[noLH\] show the current and energy spectra of a beam longitudinally smoothed but transversely noisy at the location right after the designed location of the laser heater with and without the laser heater included in the simulations. We can clearly see that the laser heater introduces extra peaks at various wavelengths and also increases the amplitudes of the current and energy modulation, which will compromise the beam quality thereafter. For this reason, the $R_{51}$ and $R_{52}$ element of the dipoles in the magnetic chicane should be revisited with more care. The parameters of the SXFEL laser heater used in the simulation are: length 0.55 m, periods 10, undulator magnetic peak field 0.31 T, laser peak power 0.80 MW, laser spot size at waist 0.30 mm.
![(Color) The current (black) and energy (red dots) spectra of a beam longitudinally smoothed but transversely noisy at the first BPM right after the laser heater in the SXFEL linac, with the laser heater turned on in the simulation.[]{data-label="LH"}](0um_BPM01_zsmth_LH.eps "fig:"){width="65mm"} -0.10in
![(Color) The current (black) and energy (red dots) spectra of a beam longitudinally smoothed but transversely noisy at the first BPM right after the location of the laser heater in the SXFEL linac, with the laser heater substituted by a drift in the simulation.[]{data-label="noLH"}](0um_BPM01_zsmth_noLH.eps "fig:"){width="65mm"} -0.10in
conclusions
===========
The study of microbunching instability in the SXFEL linac is performed in detail as a classic example. The potential problems and some new effects are uncovered including the initial beam distribution, the higher order modes in the noise amplification, the transverse-longitudinal coupling and the plasma effect in the computation of LSC. More investigations are needed to obtain deeper insights of those problems.
Computations using both analytic expressions and numerical simulations show that the gain of the microbunching instability indicates large discrepancy between the noisy- and the smoothed-beam input. Since noise in the beam can always be introduced by the random noise, the laser power jitters, etc. in the real case, one should consider how much noise to be included in the initial input. On the other hand, our work shows that in terms of the final density/energy fluctuation (or the bunching factor), the difference between the noisy and the smoothed input is not large.
Higher-order harmonics of the current modulation are also observed in the simulations. This suggests that the linear theory may not be adequate in making estimate of the gain, especially in the vicinity of the peak. The transverse-longitudinal coupling in a magnetic chicane will transport random noise from the transverse dimension to the longitudinal, and this effect needs to be considered in the design of a laser heater. In this sense, we believe that microbunching instability in an electron linac cannot be completely avoided unless the beam is ideally smooth in the whole 6-dimensional phase space.
Because both the analytic and numerical methods exhibit limitations in the estimate of the microbunching instability, systematic experimental measurements are desired to provide solid and full understanding of the physical essence. Work is ongoing at the Shanghai Institute of Applied Physics (SINAP) to prepare these experiments.
We wish to acknowledge the help of many colleagues in SINAP for the discussions on the analytical and simulation results, and many useful suggestions from the experts in the other institutes. The work is partially supported by National Natural Science Foundation of China (NSFC), grant No. 11275253, and Natural Science Foundation of Shanghai City, grant No. 12ZR1436600.
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|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
Robert Ziegler\
CERN, Theoretical Physics Department, 1 Esplanade des Particules, Geneva 23, Switzerland\
E-mail:
bibliography:
- 'Mybib.bib'
title: Flavored Axions
---
Introduction
============
One of the best motivated particle beyond the SM is arguably the QCD axion. Originally proposed as a solution to the strong CP Problem [@PQ1; @PQ2; @WW1; @WW2], it was soon realized that the axion can also serve as a viable Dark Matter candidate in vast parts of the parameter space [@AxionDM1; @AxionDM2; @AxionDM3]. In the past years axion searches have received renewed interest, and many new ideas have been pushed forward to look for the axion with small-scale experiments (see Ref. [@ExpReviewGraham] for a review). Besides the imperative axion couplings to gluons, most searches employ the axion couplings to photons, which are expected to be sizable in the majority of axion models [@Luca1; @Luca2]. Although more sensitive to the underlying UV axion model, also *flavor-violating axion couplings* allow to look for the QCD axion in rare decays with high-precision flavor experiments. In the following we discuss various aspects of such flavored axions; we begin with a brief review of axion phenomenology and the present experimental constraints, before discussing the theoretical origin of flavor-violating axion couplings. We finally review a particularly predictive flavored axion model where the Peccei-Quinn (PQ) symmetry is identified with the simplest flavor symmetry addressing Yukawa hierarchies.
Axion Couplings
===============
At energies much below the PQ breaking scale, the relevant effective axion Lagrangian reads $${\cal L} = \frac{a}{f_a} \frac{\alpha_s}{8 \pi} G \tilde{G} + \frac{E}{N} \frac{a}{f_a} \frac{\alpha_{\rm em}}{8 \pi} F \tilde{F} + \frac{\partial_\mu a}{2 f_a} \overline{f}_i \gamma^\mu \left( C^V_{ij} + C^A_{ij} \gamma_5 \right) f_j \, ,
\label{La}$$ where $f_i$ runs over SM quark and lepton generations and $E/N$ is the ratio between the electromagnetic and color anomaly coefficients. In complete generality, the axion couplings to fermions are parameterized by two hermitian $3\times3$ matrices $C^V$ and $C^A$ in each fermion sector. Note that flavor-diagonal vector couplings are unphysical, because they can be absorbed by non-anomalous field redefinitions.
The first term in Eq. (\[La\]) defines the axion decay constant $f_a$, and gives the only contribution to the axion potential, which can be conveniently calculated using chiral perturbation theory. This potential has a trivial minimum, thus dynamically setting the QCD $\theta$-term to zero, which explains the absence of CP violation in strong interactions. The same potential generates an axion mass whose parametric size is set by $m_a \propto m_\pi f_\pi/f_a$, and including higher-order corrections [@Villadoro] given by $$m_a = 5.7 \, {\rm \mu eV} \left( \frac{10^{12} {\, \rm{GeV}}}{f_a} \right) \, .$$ The second term in Eq. (\[La\]) gives rise to axion couplings to photons, which at energies much below the QCD scale are given by [@Villadoro] $${\cal L} \supset C_\gamma \frac{a}{f_a} \frac{\alpha_{\rm em}}{8 \pi} F_{\mu \nu} \tilde{F}^{\mu \nu} \, , \qquad \qquad C_\gamma = \left| E/N - 1.92(4) \right| \, .
\label{Lagamma}$$ These couplings are constrained by astrophysics [@RaffeltAstro], in particular from the evolution of HB stars in globular clusters [@HBbounds]. This bound, which is of the same order as the constraint from the CAST experiment [@CAST], translates into a lower bound on $C_\gamma/f_a$ or equivalently an upper bound on $C_\gamma m_a$ $$m_a < \frac{0.3}{C_\gamma} {\, \rm{eV}}\, .$$ The next generation of axion helioscopes such as IAXO [@IAXO1; @IAXO2; @IAXO3] will be able to improve this bound by about an order of magnitude.
Turning to the third term in Eq. (\[La\]), it is again astrophysics that provides the strongest constraints on (flavor-diagonal) axion couplings to ordinary matter, i.e. nucleons and electrons. From the shape of the White Dwarf luminosity function [@WDbound] one obtains a bound on the axion coupling to electrons $C_e \equiv |C_{ee}^A|$ $$m_a < \frac{3 \cdot 10^{-3}}{C_e} {\, \rm{eV}}\, ,$$ while the burst duration of the SN 1987A neutrino signal provides a constraint on the axion coupling to nucleons [@SNbound], defined as $${\cal L} = \frac{\partial_\mu a}{2 f_a} \left[ C_p \overline{p} \gamma^\mu \gamma_5 p + C_n \overline{n} \gamma^\mu \gamma_5 n \right] \, .$$ The proton and neutron couplings $C_{p,n}$ in turn are given by the (flavor-diagonal) axial vector couplings to quarks $C^A_{qq}$, apart from a model-independent contribution due to axion-gluon couplings [@Villadoro]. Using the bound from Ref. [@CoolingAnom3], the average axion couplings to nucleons defined as $C_N \equiv \sqrt{C_p^2 + C_n^2}$ is constrained at a level comparable to the bound on electron couplings $$m_a < \frac{4 \cdot 10^{-3}}{C_N} {\, \rm{eV}}\, .$$ Finally we discuss the constraints on flavor off-diagonal axion couplings, which arise from flavor-violating decays with invisible and practically massless final state axions. This signature is very similar to very rare meson decays in the SM like $K \to \pi \nu \overline{\nu}$, which are strongly constrained by experiments. Using the latest bound from E787/E949 on $K \to \pi a$ [@Kpia], one obtains for $s-d$ transitions $$m_a < \frac{2 \cdot 10^{-5}}{|C^V_{sd}|} {\, \rm{eV}}\, .$$ Interestingly this (already stringent) bound could be further improved in the near future with the NA62 experiment by almost an order of magnitude [@NA621; @NA622], which allows to test axion decay constants up to $10^{12} {\, \rm{GeV}}$. Similarly, $b-s$ transitions are constrained from $B \to K a$ searches at CLEO [@BKa], giving $$m_a < \frac{9 \cdot 10^{-2}}{|C^V_{bs}|} {\, \rm{eV}}\, .$$ Also this bound can be further improved by the BELLE II experiment in the near future, presumably by at least an order of magnitude. Turning to the charged lepton sector, the experimental situation becomes more challenging, as the main decay channels in the SM are similar to the signal. Nevertheless constraints on e.g. $\mu-e$ transitions have been obtained in the late 80’s with the Crystal Box detector [@mueaga1; @mueaga2], which allows to put bounds on the averaged couplings $C_{\mu e} \equiv \sqrt{|C_{\mu e}^V|^2 + |C_{\mu e}^A|^2 }$ $$m_a < \frac{3 \cdot 10^{-3}}{C_{\mu e}} {\, \rm{eV}}\, .$$ These bounds are likely to be further improved by the MEG II [@MEG2] and/or Mu3e experiments [@Mu3e].
A summary plot of the most relevant constraints[^1] discussed so far is shown in Fig. \[fig1\], which shows the upper bound on the axion mass from various processes by setting the respective dimensionless couplings $C_i = \{ C_\gamma, C_e, C_N, C^V_{sd}, C^V_{bs} \}$ to 1. Also shown is the region where the axion can naturally account for the present Dark Matter abundance through the misalignment mechanism [@AxionDM1; @AxionDM2; @AxionDM3].
![Sketch of present and future constraints on axion couplings for $C_i = 1$, see text for details. []{data-label="fig1"}](plot.pdf){width=".6\textwidth"}
From Fig. \[fig1\] it is clear that rare flavor-violating decays are very important to constrain flavor-violating axion couplings to matter, and can compete with the stringent constraints on flavor-diagonal couplings from astrophysics. In the case of $s-d$ transitions, the present bounds on $f_a$ from $K \to \pi + a$ decays are actually about two orders of magnitude stronger, for equal sizes of the respective dimensionless couplings. Most interestingly, these bounds are expected to be improved in the very near future by various experiments such as NA62 and Belle II. Therefore precision flavor experiments provide the exciting possibility to look for the QCD axion in a way that is complementary to the usual searches with helio- and haloscopes.
Flavored Axions
===============
In this section we investigate the expected size of flavour-violating axion couplings in UV axion models. In general these couplings arise from the PQ current, which has to be rotated to the fermion mass basis. In this basis, given by unitary rotations $V_{f}$ defined by $V_{f_L}^\dagger M_f V_{f_R}= M_f^{\rm diag}$, one obtains $$C_{f_i f_j}^{V,A} = \frac{1}{2N} \left( V_{f_R}^\dagger X_{f_R} V_{f_R} \pm V_{f_L}^\dagger X_{f_L} V_{f_L} \right)_{ij} \, ,
\label{Cdef}$$ where $X_{f_L, f_R}$ denote the (flavor-diagonal) PQ charge matrices of left- and right-handed fermions. Therefore flavor-violating couplings are present whenever SM fermions carry PQ charges that represent a new source of flavor violation beyond SM Yukawas, i.e. *whenever PQ charges do not commute with Yukawa matrices*. In this case the off-diagonal couplings depend on the unitary rotations that connect interaction and mass basis, and thus can be quantitatively predicted only in a theory of flavor.
The minimal scenario is to completely disentangle the solution to the strong CP problem from the origin of SM flavor puzzle, i.e. axion and flavor physics, as in common axion benchmark models. Indeed in KSVZ models [@KSVZ1; @KSVZ2] SM fermions do not carry PQ charges, and thus do not couple to the axion at all[^2]. In the simplest DFSZ models [@DFSZ1; @DFSZ2] SM fermions are given flavor-universal PQ charges, which also implies that flavor-violating couplings vanish at tree-level.
The simplest example of scenarios with flavor-violating axion couplings are DFSZ models with non-universal PQ charges (which of course provide an equally good solution to the strong CP problem as long as PQ is anomalous under QCD). In such cases the flavor-violating axion couplings depend on the unitary rotations, but can be effectively parametrized by a couple of free parameters under certain assumptions, e.g. if PQ allows all Yukawa couplings and only two Higgs doublets are present. Such scenarios can be motivated by other features, for example the possibility to suppress the axion couplings to nucleons and/or electrons [@Astrophobic].
More predictive are scenarios where PQ is also (partially) responsible for explaining the peculiar pattern of SM Yukawas, which has been proposed already long time ago by F. Wilczek [@Wilczek]. A particular simple realization[^3] is based on the identification of the PQ symmetry with the smallest flavor symmetry able to address Yukawa hierarchies, i.e. a horizontal $U(1)$ symmetry à la Froggatt-Nielsen (FN) [@Japs; @Axiflavon]. In the next section we briefly summarize the scenario in Ref. [@Axiflavon].
PQ=FN: The Axiflavon
====================
As in usual FN models [@FN; @SeibergNir1] we assume that the hierarchies of the Yukawa couplings are due to a global horizontal symmetry $U(1)_H$, under which SM Weyl fermions carry positive, flavor-dependent charges $[q]_i, [u]_i, [d]_i, [l]_i, [e]_i$, respectively, while the Higgs is neutral. The $U(1)_H$ symmetry is spontaneously broken at high scales by the vev $V_\Phi$ of a complex scalar field $\Phi$ with $U(1)_H$ charge of $-1$. SM Yukawas arise then from higher-dimensional operators involving appropriate powers of $\Phi$ to be invariant under $U(1)_H$, suppressed by the UV cutoff scale $\Lambda$. After plugging in the vev of $\Phi = V_\Phi/\sqrt{2}$, this gives rise to SM Yukawa couplings $$y^{u,d,e}_{ij} = a^{u,d,e}_{ij} {\epsilon}^{[L]_i + [R]_j} \,,
\label{yuks}$$ where $[L]_i = [q]_i, [R]_i = [u]_i, [d]_i$ in the quark sectors, $[L]_i = [l]_i, [R]_i = [e]_i$ in the charged lepton sector and we have defined the small parameter ${\epsilon}\equiv V_\Phi/(\sqrt 2 \Lambda)$. Here $a^{u,d,e}_{ij}$ are unknown Wilson coefficients, of the effective operators, assumed to be ${\cal O}(1)$. While to some extent they can be determined together with the fermion charges by a numerical fit to fermion masses and mixings, here we focus on analytical results that aim to incorporate the uncertainties present in such a fit.
In this scenario, upon the identification of PQ with $U(1)_H$, the Goldstone boson contained in $\Phi$ plays the role of the QCD axion, and its couplings to gluons, photons and fermion are determined by the horizontal fermion charges. Although the precise value of these charges depend on the fermion mass fit, one can find a pretty narrow range for the ration of electromagnetic and color anomaly coefficient, given by $$\frac{E}{N} \in \left[ 2.4 , 3.0 \right] \, .
\label{ENpred}$$ This surprisingly narrow window can be obtained by calculating the determinants of the fermion mass matrices, and expressing them in terms of the anomaly coefficients. This leads to $$\frac{E}{N} = \frac{8}{3} - 2 \frac{\log \frac{{\rm det} \, m_d}{{\rm det} \, m_e} - \log \alpha_{de}}{\log \frac{{\rm det} \, m_u {\rm det} \, m_d}{v^6} - \log \alpha_{ud}} \, ,
\label{ENprediction}$$ where $\alpha_{ud} = {\rm det} \, a_u {\rm det} \, a_d $ and $\alpha_{de} = {\rm det} \, a_d/{\rm det} \, a_e$ contain the ${\cal O}(1)$ uncertainties from the coefficients in Eq. (\[yuks\]). Independently of the precise values of these parameters, it is clear that the second term on the right-hand side of Eq. (\[ENprediction\]) is strongly suppressed by the large denominator ($\log {\rm det} \, m_u {\rm det} \, m_d/v^6 \approx -44 $), so that $E/N$ is expected to be close to $8/3$.
Axion couplings to fermions arise from Eq. (\[Cdef\]), with unitary rotations that are themselves determined (at least parametrically) by $U(1)_H$ charges according to $$(V_{f_L})_{ij} \sim {\epsilon}^{|[L]_i - [L]_j|} \, , \qquad \qquad (V_{f_R})_{ij} \sim {\epsilon}^{|[R]_i - [R]_j|} \, .$$ Therefore all flavor-diagonal couplings are expected to be ${\cal O}(1)$, while off-diagonal couplings are suppressed by small rotation angles. Using typical values of $U(1)_H$ charges needed to give a good fit to quark masses and mixings, it is clear that flavor-violating $s-d$ couplings are just suppressed by a Cabibbo angle and therefore sizable $$C^V_{sd} \sim \lambda \approx 0.2.$$ This implies an upper bound on the axion mass of about $m_a \lesssim 10^{-4} {\, \rm{eV}}$, which is just at the edge of the natural axion DM window (see Fig. (\[fig1\])). Thus the “axiflavon" in this scenario will be tested in the near future complementarily by precision flavor physics at NA62 and axion haloscopes with the ADMX upgrade [@ADMXfuture].
Conclusions
===========
To summarize, we have argued that precision flavor experiments allow to look for the QCD axion complementarily to the usual searches with axion helio- and haloscopes. The strongest sensitivity is obtained for flavor-violating $s-d$ transitions, where NA62 is expected to test PQ breaking scales as high as $10^{12} {\, \rm{GeV}}$ by looking for $K \to \pi a$ decays. Such decays are sensitive to flavor-violating axion couplings, which are generic and potentially sizable whenever SM fermions carry flavor non-universal PQ charges. In this case the axion couplings depend on the misalignment of PQ charges and Yukawa couplings, i.e. a theory of flavor is need in order to make quantitative predictions for off-diagonal couplings. A particularly predictive scenario is obtained when PQ is identified with the simplest FN flavor symmetry, in which case all flavor-violating axion couplings are directly related to Yukawa hierarchies, up to ${\cal O}(1)$ coefficients.
[^1]: Constraints on the remaining flavor-violating axion couplings to quarks and leptons can be found in Refs. [@Murayama; @King].
[^2]: Still axion couplings to nucleons are induced as a result of the axion couplings to gluons.
[^3]: For other possibilities in the context of larger flavor symmetry groups see e.g. Refs. [@Celis1; @Matthias; @Carone],
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Raw data on the cumulative number of deaths at a country level generally indicate a spatially variable distribution of the incidence of COVID-19 disease. An important issue is to determine whether this spatial pattern is a consequence of environmental heterogeneities, such as the climatic conditions, during the course of the outbreak. Another fundamental issue is to understand the spatial spreading of COVID-19. To address these questions, we consider four candidate epidemiological models with varying complexity in terms of initial conditions, contact rates and non-local transmissions, and we fit them to French mortality data with a mixed probabilistic-ODE approach. Using standard statistical criteria, we select the model with non-local transmission corresponding to a diffusion on the graph of counties that depends on the geographic proximity, with time-dependent contact rate and spatially constant parameters. This original spatially parsimonious model suggests that in a geographically middle size centralized country such as France, once the epidemic is established, the effect of global processes such as restriction policies, sanitary measures and social distancing overwhelms the effect of local factors. Additionally, this modeling approach reveals the latent epidemiological dynamics including the local level of immunity, and allows us to evaluate the role of non-local interactions on the future spread of the disease. In view of its theoretical and numerical simplicity and its ability to accurately track the COVID-19 epidemic curves, the framework we develop here, in particular the non-local model and the associated estimation procedure, is of general interest in studying spatial dynamics of epidemics.'
author:
- |
Lionel Roques$^{\hbox{\small{ a,*}}}$, Olivier Bonnefon$^{\hbox{\small{ a}}}$, Virgile Baudrot$^{\hbox{\small{ a}}}$,\
Samuel Soubeyrand$^{\hbox{\small{ a}}}$ and Henri Berestycki$^{\hbox{\small{ b}}}$\
\
\
\
\
title: 'A parsimonious model for spatial transmission and heterogeneity in the COVID-19 propagation'
---
*Keywords.* COVID-19 $|$ Spatial diffusion $|$ Mechanistic-statistical model $|$ Non-local transmission $|$ Immunity rate
Introduction
============
In France, the first cases of COVID-19 epidemic have been reported on 24 January 2020, although it appeared latter that some cases were already present in December 2019 [@DesBer20]. Since then, the first important clusters were observed in February in the Grand Est region and the Paris region. A few months later, at the beginning of June, the spatial pattern of the disease spread seems to have kept track of these first introductions. As this spatial pattern may also be correlated with covariates such as climate [@DemFle20] (see also SI 1), a fundamental question is to assess whether this pattern is the consequence of a heterogeneous distribution of some covariates or if it can be explained by the heterogeneity of the initial introduction points. In the latter case, we want to know if the epidemic dynamics simply reflects this initial heterogeneity and can be modeled without taking into account any spatial heterogeneity in the local conditions.
SIR epidemiological models and their extensions have been proposed to study the spread of the COVID-19 epidemic at the country or state scale (e.g., [@RoqKle20] in France and [@BerFra20] in three US states), and at the regional scale ([@MaiBro20; @PreLiu20] in China, [@SalTra20] in France and [@GatBer20] in Italy). In all cases, a different set of parameters has been estimated for each considered region/province. One of the main goals of our study is to check whether the local mortality data at a thin spatial scale can still be well explained by a single set of parameters, at the country scale, but spatially varying initial conditions. In this study we consider SIR models with time-dependent contact rate, to track changes over time in the dynamic reproductive number as in the branching process considered in [@BerFra20]. Using standard statistical criteria, we compare four types of models, whose parameters are either global at the country scale or spatially heterogeneous, and with non-local transmission or not. We work with mortality data only, as these data appear to be more reliable and less dependent on local testing strategies than confirmed cases, in settings where cause of death is accurately determined [@GarCha20]. Additionally, it was already shown that for this type of data, SIR models outperform other classes of models (an SEIR models and a branching process) in the three US states considered in [@BerFra20]. The approach we develop here is in line with the general principle of using parsimonious models eloquently emphasized in [@BerFra20].
Another important issue that such models may help to address is the quantification of the relative effects of restrictions on inter-regional travels, [*vs*]{} reductions in the probability of infection per contact at the local scale. France went into lockdown on March 17, 2020, which was found to be very effective in reducing the spread of the disease. It divided the effective reproduction number (number $\Rt_t$ of secondary cases generated by an infectious individual) by a factor 5 to 7 at the country scale by May 11 [@RoqKle20b; @SalTra20]. This is to be compared with the estimate of the basic reproduction number $\Rt_0$ carried out in France at the early onset of the epidemic, before the country went into lockdown (with values $\Rt_0=2.96$ in [@SalTra20] and $\Rt_0=3.2$ in [@RoqKle20]). Contact-surveys data [@ZhaLit20] for Wuhan and Shanghai, China, have found comparable estimates of the reduction factor. The national lockdown in France induced important restrictions on movement, with e.g. a mandatory home confinement except for essential journeys, leading to a reduction of the number of contacts. In parallel, generalized mask wearing and use of hydroalcoholic gels reduced the probability of infection per contact.
After the lockdown, restrictions policies were generally based on raw data rather than modeling. An efficient regulation at the local scale would need to know the current number of infectious and the level of immunity at the scale of counties. The French territory is divided into administrative units called ‘départements’, analogous to counties. We use ’counties’ in the sequel for départements. These quantities cannot be observed directly in the absence of large scale testing campaigns or spatial random sampling. In particular, there is a large number of unreported cases. Previous studies developed a mixed mechanistic-probabilistic framework to estimate these quantities at the country scale. These involved estimating the relative probability of getting tested for an infected individual [*vs*]{} a healthy individual, leading to a factor x8 between the number of confirmed cases and the actual number of cases before the lockdown [@RoqKle20], and x20 at the end of the lockdown period [@RoqKle20b]. This type of framework – often referred to as ‘mechanistic-statistical modeling’ – aims at connecting the solution of continuous state models such as differential equations with complex data, such as noisy discrete data, and identifying latent processes such as the epidemiological process under consideration here. Initially introduced for physical models and data [@Ber03], it is becoming standard in ecology [@SouLai09].
Using this framework the objectives of our study are (i) to assess whether the spatial pattern observed in France is due do some local covariates or is simply the consequence of the heterogeneity in the initial conditions together with global processes at the country scale; (ii) to evaluate the role of non-local interactions on the spread of the disease; (iii) to propose a tool for real-time monitoring the main components of the disease in France, with a particular focus on the local level of immunity.
Materials and Methods {#materials-and-methods .unnumbered}
=====================
Data {#data .unnumbered}
----
Mainland France (excluding Corsica island) is made of 94 counties called ’départements’. The daily number of hospital deaths – excluding nursing homes – at the county scale are available from Santé Publique France since 18 March 2020 (and available as Supplementary Material). The daily number of observed deaths (still excluding nursing homes) in county $k$ during day $t$ is denoted by $\hD_{k,t}$.
We denote by $[t_i,t_f]$ the observation period and by $n_d$ the number of considered counties. To avoid a too large number of counties with 0 deaths at initial time, the observation period ranges from $t_i=$ March 30 to $t_f=$ June 11, corresponding to $n_t=74$ days of observation. All the counties of mainland France (excluding Corsica) are taken into account, but the Ile-de-France region, which is made of 8 counties with a small area is considered as a single geographic entity. This leads to $n_d=87$.
Mechanistic-statistical framework\[sec:framework\] {#mechanistic-statistical-frameworksecframework .unnumbered}
--------------------------------------------------
The mechanistic-statistical framework is a combination of a mechanistic model that describes the epidemiological process, a probabilistic observation model and a statistical inference procedure. We begin with the description of four mechanistic models, whose main characteristics are given in Table \[table:defmodels\].
### Mechanistic models {#mechanistic-models .unnumbered}
#### Model $\M_{0}$: SIR model for the whole country.
The first model is the standard mean field SIRD model that was used in [@RoqKle20; @RoqKle20b]: $$\baco{l} \label{eq:modela}
{\displaystyle}S'(t)=- \frac{\alpha(t)}{N} \, S \, I, \vspace{1mm}\\
{\displaystyle}I'(t)= \frac{\alpha(t)}{N} \, S \, I - (\beta+\gamma) \, I, \vspace{1mm}\\
{\displaystyle}R'(t)=\beta \, I,\vspace{1mm}\\
{\displaystyle}D'(t)=\gamma \, I,
{\end{array} \right.}$$ with $S$ the susceptible population, $I$ the infectious population, $R$ the recovered population, $D$ the number of deaths due to the epidemic and $N$ the total population, in the whole country. For simplicity, we assume that $N$ is constant, equal to the current population in France, thereby neglecting the effect of the small variations of the population on the coefficient $\alpha(t)/N$. The parameter $\alpha(t)$ is the contact rate (to be estimated) and $1/\beta$ is the mean time until an infectious becomes recovered. The results in [@ZhoYu20] show that the median period of viral shedding is 20 days, but the infectiousness tends to decay before the end of this period: the results in [@HeLau20] indicate that infectiousness starts from 2.5 days before symptom onset and declines within 7 days of illness onset. Based on these observations we assume here that $1/\beta=10$ days. The parameter $\gamma$ corresponds to the death rate of the infectious. It was estimated independently in [@RoqKle20; @RoqKle20b] and [@SalTra20], leading to a value $\gamma=5 \cdot 10^{-4}$. This value only takes into account the deaths at hospital, and is therefore consistent with the data that we used here.
#### Model $\M_{1}$: SIR model at the county scale with globally constant contact rate and no spatial transmission.
The model $\M_{0}$ is applied at the scale of each county $k$, leading to compartments $S_k$, $I_k$, $R_k$, $D_k$ that satisfy an equation of the form , with $N$ replaced by $N_k$, the total population in the county $k$. In this approach, the contact rate $\alpha(t)$ is assumed to be the same in all of the counties.
---------- -------------- -------------- -------------- ------------------ --
Heterog. Heterog. Intercounty Nb. parameters
initial data contact rate transmission
$\M_{0}$ no no no $ n_t$
$\M_{1}$ yes no no $ n_t$
$\M_{2}$ yes yes no $n_d \times n_t$
$\M_{3}$ yes no yes $n_t+2$
---------- -------------- -------------- -------------- ------------------ --
: Main characteristics of the four models. The quantity $n_t=74$ corresponds to the number of days of the observation period and $n_d=87$ corresponds to the number of administrative units.[]{data-label="table:defmodels"}
#### Model $\M_{2}$: SIR model at the county scale with spatially heterogeneous contact rate and no spatial transmission.
With this approach, the model $\M_{1}$ is extended by assuming that the contact rate $\alpha_k(t)$ depends on the considered county.
#### Model $\M_{3}$: County scale model with globally constant contact rate and spatial transmission.
The model $\M_{1}$ is extended to take into account disease transmission events between the counties: $$\baco{l} \label{eq:model3}
{\displaystyle}S_k'(t)=- \frac{\bx(t)}{N_k} \, S_k \, \sum_{j=1}^{n_d} w_{j,k}\, I_j, \vspace{1mm}\\
{\displaystyle}I_k'(t)=\frac{\bx(t)}{N_k} \, S_k \, \sum_{j=1}^{n_d} w_{j,k}\, I_j - (\beta+\gamma) \, I_k , \vspace{1mm}\\
{\displaystyle}R_k'(t)=\beta \, I_k,\vspace{1mm}\\
{\displaystyle}D_k'(t)=\gamma \, I_k.
{\end{array} \right.}$$ The weights $w_{j,k}$ describe the dependence of the contagion rates with respect to the distance between counties. We assume a power law decay with the distance: $$w_{j,k}=\frac{1}{1+(\hbox{dist}(j,k)/d_0)^{\delta}},$$ with $\hbox{dist}(j,k)$ the geographic distance (in km) between the centroids of counties $j$ and $k$, $d_0>0$ a proximity scale, and $\delta>0$. Thus, the model involves two new global parameters, $d_0$ and $\delta$, compared to model $\M_1$. This model extends the Kermack-McKendrick SIR model to take into account non-local spatial interactions. It was introduced by Kendall [@Ken57] in continuous variables. The model we adopt here is inspired from the study of [@BonBer18] in a different context where the same types of weights have been used. We thus take into account diffusion on the weighted graph of counties in France. This amounts to considering that individuals in a given county are infected by individuals form other counties with a probability that decreases with distance as a power law, in addition to contagious individuals from their own county. This dependence of social spatial interactions with respect to the distance is supported, notably, by [@BroHuf06] that analyzed the short-time dispersal of bank notes in the US. We also refer to [@MeyHel14] for a thorough discussion on the various applications of power law dispersal kernels since they were introduced by Pareto [@Par1896].
With this non-local contagion model, in contradistinction to epidemiological models with dispersion such as reaction-diffusion epidemiological models [@GauGha10], the movements of the individuals are not modeled explicitly. The model implicitly assumes that infectious individuals may transmit the disease to susceptible individuals in other counties, but eventually return to their county of origin. This has the advantage of avoiding unrealistic changes in the global population density.
### Observation model {#observation-model .unnumbered}
We denote by $\Db_k(t)$ the expected cumulative number of deaths given by the model, in county $k$. With the mean-field model $\M_{0}$, we assume that it is proportional to the population size: $\Db_k(t)=D(t) \, N_k/N,$ with $N_k$ the population in county $k$ and $N$ the total French population. With models $\M_{1},$ $\M_{2}$ and $\M_{3}$, we simply have $\Db_k(t)=D_k(t)$. The expected daily increment in the number of deaths given by the models in a county $k$ is $\Db_k(t)-\Db_k(t-1)$.
The observation model assumes that the daily number of new observed deaths $\hD_{k,t}$ in county $k$ follows a Poisson distribution with mean value $\Db_k(t)-\Db_k(t-1)$: $$\label{eq:model_poisson}
\hD_{k,t}\sim \text{Poisson}(\Db_k(t)-\Db_k(t-1)).$$ Note that the time $t$ in the mechanistic models is a continuous variable, while the observations $\hD_{k,t}$ are reported at discrete times. For the sake of simplicity, we used the same notation $t$ for the days in both the discrete and continuous cases. In the formula $\Db_k(t)$ (resp. $\Db_k(t-1)$) is computed at the end of day $t$ (resp. $t-1$).
Initial conditions {#initial-conditions .unnumbered}
------------------
In models $\M_{1}$, $\M_{2}$, $\M_{3}$, at initial time $t_i$, we assume that the number of susceptible cases is equal to the number of inhabitants in county $k$: $S_k(t_i)=N_k$, the number of recovered is $R(t_i)=0$ and the number of deaths is given by the data: $D_k(t_i)=\hD_{k,t_i}$. To initialise the number of infectious, we use the equation $D'(t)=\gamma \, I(t)$, and we define $I(t_i)$ as $1/\gamma$ $\times$ (mean number of deaths over the period ranging from $t_i$ to $29$ days after $t_i$): $$\label{eq:IO}
I_k(t_i)=\frac{1}{\gamma}\frac{1}{30} \sum\limits_{s=t_i,\ldots,t_i+29}\hD_{k,s}.$$ The 30-days window was chosen such that there was at least one infectious case in each county. In model $\M_{0}$, the initial conditions are obtained by adding the initial conditions of model $\M_{1}$ (or equivalently, $\M_{2}$) over all the counties.
Statistical inference {#statistical-inference .unnumbered}
---------------------
#### Real-time monitoring of the parameters and data assimilation procedure.
To smooth out the effect of small variations in the data, and to avoid identifiability issues due to the large number of parameters, while keeping the temporal dependence of the parameters, the parameters $\alpha(t)$ and $\alpha_k(t)$ of the ODE models $\M_{0}$, $\M_{1}$, $\M_{2}$ are estimated by fitting auxiliary problems with time-constant parameters over moving windows $(t-\tau/2,t+\tau/2)$ of fixed duration equal to $\tau$ days. These auxiliary problems are denoted respectively by $\tilde{\M}_{0,t}$, $\tilde{\M}_{1,t}$, and $\tilde{\M}_{2,t}$ (see SI 2 for a precise formulation of these problems). The initial conditions associated with this system, at the date $t-\tau/2$ are computed iteratively from the solution of $\M_{0}$, $\M_{1}$ and $\M_{2}$, respectively.
#### Inference procedure.
For simplicity, in all cases, we denote by $$f_{\Db_k,\hD_k}(s):= \frac{(\Db_k(s)-\Db_k (s-1))^{\hD_{k,s }}}{\hD_{k,s}!}\, e^{-(\Db_k(s)-\Db_k(s-1))}$$the probability mass function associated with the observation process at date $s$ in county $k$, given the expected cumulative number of deaths $\Db_k$ given by the considered model in county $k$.
In models $\tilde{\M}_{0,t}$ and $\tilde{\M}_{1,t}$, the estimated parameter is $\tA$. The likelihood $\mathcal{L}$ is defined as the probability of the observations (here, the increments $\{\hD_{k,s}\}$) conditionally on the parameter. Using the assumption that the increments $\hD_{k,s}$ are independent conditionally on the underlying SIRD process $\tilde{\M}_{0,t}$ (resp. $\tilde{\M}_{1,t}$), we get: $$\begin{aligned}
\mathcal{L}(\tA){\displaystyle}& :=P(\{\hD_{k,s}, \, k=1,\ldots,n_d, \, s=t-\tau/2,\dots,t+\tau/2\} |\tA) \\ &
=\prod_{k=1}^{n_d}\prod_{s=t-\tau/2}^{t+\tau/2} f_{\Db_k,\hD_k}(s).\end{aligned}$$ We denote by $\tA_{t}^*$ the corresponding maximum likelihood estimator, and we set $\alpha(t)=\tA_{t}^* \hbox{ in model }\M_{0} \hbox{ (resp. }\M_{1}).$
For model $\tilde{\M}_{2,t}$, the inference of the parameters $\tA_k$ is carried out independently in each county, leading to the likelihoods: $$\begin{aligned}
\mathcal{L}_k(\tA_k){\displaystyle}&:=P(\{\hD_{k,s}, \ s=t-\tau/2,\dots,t+\tau/2\} |\tA_k)\\
&=\prod_{s=t-\tau/2}^{t+\tau/2} f_{\Db_k,\hD_k}(s).\end{aligned}$$ We denote by $\tA_{k,t}^*$ the corresponding maximum likelihood estimator, and we set $\alpha_k(t)=\tA_{k,t}^*$ in model $\M_{2}.$
For model $\M_{3}$, we apply a two-stage estimation approach. We first use the estimate obtained with model $\M_1$ by setting $\bx(t)= C \, \alpha(t),$ where $\alpha(t)$ is the estimated contact rate of model $\M_1$ and $C$ is a constant (to be estimated; note that estimating $\rho(t)$ means estimating $n_t$ parameters). Thus, given $\alpha(t)$, the only parameters to be estimated are the constant $C$, the proximity scale $d_0$ and the exponent $\delta$. They are estimated by maximizing: $$\begin{aligned}
\label{eq:likelihoodEDP}
\mathcal{L}(C,d_0,\delta){\displaystyle}& :=P(\{\hD_{k,s}, \ k=1,\ldots,n_d, \, s=t_i,t_f\} |C,d_0,\delta) \\
& =\prod_{k=1}^{n_d}\prod_{s=t_i}^{t_f}f_{\Db_k,\hD_k}(s).\end{aligned}$$
#### Model selection.
We use the Akaike information criterion (AIC) [@Aka74] and the Bayesian information criterion (BIC) [@Sch78] to compare the models. For both criteria, we need to compute the likelihood function associated with the model, with the parameters determined with the above inference procedure: $$\L(\M_m)=\prod_{k=1}^{n_d}\prod_{s=t_i}^{t_f}f_{\Db_k,\hD_k}(s),$$ with $m=0\,,1,\,2,\,3$ and $\Db_k$ the expected (cumulative) number of deaths in county $k$ given by model $(\M_m)$. Given the number of parameters $\sharp \M_m$ estimated in model $\M_m$, the AIC score is defined as follows: $$AIC(\M_m)=2\, \sharp\M_m -2 \, \ln(\L(\M_m)),$$ and the BIC score: $$BIC(\M_m)=\sharp\M_m\, \ln(K)-2 \, \ln(\L(\M_m)),$$with $K=n_d \times n_t=6\,438$ the number of data points.
*Numerical methods*. To find the maximum likelihood estimator, we used a BFGS constrained minimization algorithm, applied to $ - \ln (\L)$ via the Matlab ^^ function *fmincon*. The ODEs were solved thanks to a standard numerical algorithm, using Matlab^^ *ode45* solver. The Matlab ^^ codes are available as Supplementary Material.
Results {#results .unnumbered}
=======
*Model fit and model selection.* To assess model fit, we compared the daily increments in the number of deaths given by each of the four models with the data. For each model, we used the parameters corresponding to the maximum likelihood estimators. The comparisons are carried out at the regional scale: mainland France is divided into 12 administrative regions, each of which is made of several counties. The results are presented in Fig. \[fig:model\_fit\_region\]. In all the regions, the models $\M_1,$ $\M_2$ and $\M_3$ lead to a satisfactory visual fit of the data, whereas the mean field model $\M_0$ does not manage to reproduce the variability of the dynamics among the regions.
The log-likelihood, AIC and BIC values are given in Table \[table:AIC\]. Models $\M_1$, $\M_2$, $\M_3$ lead to significantly higher likelihood values than model $\M_0$. This reflects the better fit obtained with these three models, compared to model $\M_0$ and shows the importance of taking into account the spatial heterogeneities in the initial densities of infectious cases. On the other hand, the log-likelihood, though higher with model $\M_2$ is close to that obtained with $\M_1$, and the model selection criteria are both strongly in favor of model $\M_1$. This shows that the spatial heterogeneity in the contact rate does not have a significant effect on the epidemic dynamics within mainland France.
Model $\M_3$ with spatial transmission leads to an intermediate likelihood value, between those of models $\M_1$ and $\M_2$, with only $2$ additional parameters with respect to model $\M_1$. As a consequence, the model selection criteria exhibit a strong evidence in favor of the selection of model $\M_3$. This means a large part of the difference between models $\M_1$ and $\M_2$ can be captured by taking into account the spatial transmission, which therefore seems to have a significant effect on the epidemic dynamics.
As a byproduct of the estimation of the parameter $\alpha(t)$ (resp. $\alpha_k(t)$) of model $\M_1$ (resp. $\M_2$), we get an estimate of the effective reproduction number in each county, which is given by the formula [@NisGer09]:$$\Rt_{t}^k=\frac{\alpha(t)}{\beta+\gamma}\frac{S_k(t)}{N_k},$$so that $I_k'(t)<0$ (the epidemic tends to vanish) whenever $\Rt_{t}^k<1$, whereas $I_k'(t)>0$ whenever $\Rt_{t}^k>1$ (the number of infectious cases in the population follows an increasing trend). The dynamics of $\Rt_{t}^k$ obtained with model $\M_1$ are depicted in Fig. \[fig:Rt\], clearly showing a decline in $\Rt_{t}^k$, as already observed in [@RoqKle20b; @SalTra20], but there the computation was at a fixed date.
For model $\M_3$, the maximum likelihood estimation gives $C=0.87$, $d_0=2.16~$km and $\delta=1.85$, which yields a nearly quadratic decay of the weights with the distance. The value of $d_0$, indicates that non-local contagion plays a secondary role compared to within-county contagion: the minimum distance between two counties is 36 km, leading to a weight of $5.5/1000$, to be compared with the weight 1 for within-county contagion. However, the fact that the parameter $C$ is significantly smaller than $1$ (recall that the contact rate in model $\M_3$ is $\rho(t)=C\, \alpha(t)$ with $\alpha(t)$ the contact rate in model $\M_2$) shows that the non-local contagion term plays an important role on the spreading of the epidemic.
*Immunity rate.* Using model $\M_3$, which leads to the best fit, we derive the number of recovered individuals (considered here as immune) at a date $t$, in each county, and the immunity rate $R_k/N_k$. It is presented at time $t_f$ in Fig. \[fig:immunity\]. The full timeline of the dynamics of immunity obtained with model $\M_3$ since the beginning of April is available as Supplementary Material (see SI 1).
{width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"}
Model AIC BIC Log-likelihood $\Delta$AIC
---------- ------------------- -------------------- -------------------- --------------------
$\M_{0}$ $2.68 \cdot 10^4$ $2.73 \cdot 10^4$ $-13.4 \cdot 10^3$ $-9.57 \cdot 10^3$
$\M_{1}$ $ 1.74\cdot 10^4$ $ 1.79\cdot 10^4$ $-8.62 \cdot 10^3$ $-220$
$\M_{2}$ $2.97\cdot 10^4$ $ 7.36 \cdot 10^4$ $-8.41 \cdot 10^3$ $-1.25 \cdot 10^4$
$\M_{3}$ $ 1.72\cdot 10^4$ $ 1.77\cdot 10^4$ $-8.52 \cdot 10^3$ 0
: Log-likelihood, AIC and BIC values for the four models. The last column $\Delta$AIC corresponds to the difference with the AIC value of the best model (here $\M_3$).[]{data-label="table:AIC"}
![Dynamics of the effective reproduction rate $\Rt_{t}^k$ given by model $\M_1$ over the 87 considered counties.[]{data-label="fig:Rt"}](Images/Rt_M1.png){width="50.00000%"}
![Estimated immunity rate, at the county scale, by June 11, 2020, using model $\M_3$. []{data-label="fig:immunity"}](Images/immun_M3.png){width=".8\linewidth"}
*Limiting movement vs limiting the probability of transmission per contact.* Before the lockdown, the basic reproduction number $\Rt_0$ in France was about 3 [@SalTra20; @RoqKle20], and was then reduced by a factor 5 to 7, leading to values around 0.5 (see [@RoqKle20b; @SalTra20] and Fig. \[fig:Rt\]). This corresponds to a contact rate $\alpha(t)\approx \beta \, \Rt_t\approx 0.3$ before the lockdown and $\alpha(t)\approx 0.05$ after the lockdown in model $\M_1$. Let us now consider a hypothetical scenario of a new outbreak with an effective reproduction number that rises again to reach values above 1. Due to the higher awareness of the population with respect to epidemic diseases and the new sanitary behaviors induced by the first COVID-19 wave, the reproduction number will probably not reach again values as high as 3.
The new outbreak scenario is described as follows: we start from the state of the epidemic at June 11, and we assume a ’local’ contact rate $\rho(t)$ jumping to the value $0.11$ in model $\M_3$ (corresponding to a 2-fold increase compared to the previous 30 days). In parallel, to describe the lifting of restrictions on individual movements, we set $d_0=20$ km for the proximity scale in model $\M_3$. This new outbreak runs during 10 days, and then, we test four strategies:
- Strategy 1: no restriction. The parameters remain unchanged: $\rho(t)=0.11$ and $d_0=20$ km;
- Strategy 2: restriction on intercounty movement. The parameter $\rho(t)=0.11$ is unchanged, but $d_0=2.16$ km, corresponding to its estimated value during the period $(t_i,t_f)$;
- Strategy 3: reduction of the contact rate within each county (e.g., by wearing masks), but no restriction on intercounty movement: $\rho(t)=0.05$ and $d_0=20$ km;
- Strategy 4: reduction of the contact rate within each county and restriction on intercounty movement: $\rho(t)=0.05$ and $d_0=2.16$ km.
The daily number of deaths corresponding to each scenario is presented in Fig. \[fig:scenarios\]. We estimate in each case a value of the effective reproduction number $\Rt_t$ over the whole country by fitting the global number of infectious cases with an exponential function over the last 30 days. As expected, the more restrictive the strategy, the less the number of deaths. After 30 days, the cumulative number of deaths with the first strategy is $17\, 271$, and $\Rt_t\approx 2$. Restriction on intercounty movement (strategy 2) leads to a 81% decrease in the cumulative number of deaths ($3\, 281$ deaths) and $\Rt_t\approx 1.2$; reducing the contact rate within each county leads to a 88% decrease (strategy 3, $2\, 139$ deaths) and $\Rt_t\approx 0.8$; finally, control strategy 4, which combines both types of restrictions leads to a 91% decrease ($1\, 503$ deaths) and $\Rt_t\approx 0.4$.
![Daily number of deaths due to a new outbreak in logarithmic scale; comparison between four management strategies. The number of deaths is computed over the whole country.[]{data-label="fig:scenarios"}](Images/scenarios.png){width=".8\linewidth"}
Discussion {#discussion .unnumbered}
==========
We show here that a parsimonious model can reproduce the local dynamics of the COVID-19 epidemic in France with a relatively high goodness of fit. This is achieved despite the spatial heterogeneity across French counties of some environmental factors potentially influencing the disease propagation. Indeed, our model only involves the initial spatial distribution of the infectious cases and spatially-homogeneous (i.e. countrywide) parameters. For instance, the mean temperature during the considered period ranges from $12.0\degree$C to $18.4\degree$C depending on the region. We do observe a negative correlation between the mean temperature and the immunity rate (see Fig. S1), however it does not reflect a causality. Actually, our study shows that if there is an effect of local covariates such as the mean temperature on the spread of the disease once it emerged, its effect is of lower importance compared to the global processes at work at the country scale. Such covariates might play a major role in the emergence of the disease, but our work focuses on the disease dynamics after the emergence.
Hence, we find that initial conditions and spatial diffusion are the main drivers of the spatial pattern of the COVID-19 epidemic. This result may rely on specific circumstances: e.g., mainland France covers a relatively middle-sized area, with mixed urban and countryside populations across the territory, a relatively homogeneous population age distribution, and a high level of centralism for public decision (in particular regarding the disease-control strategy). Of course, these features are not universal. In other countries with more socio-environmental diversity within which environmental drivers and state decentralization could significantly induce spatial variations in disease spread and, consequently, in which countrywide parameters would not be appropriate. Moreover, at the global scale, the COVID-19 dynamics in different countries seem highly contrasted as illustrated by the data of Johns Hopkins University Center for Systems Science and Engineering [@DonDu20] —see also <http://covid19-forecast.biosp.org/> [@SouRib20]— and could probably not be explained with a unique time-varying contact rate parameter. Nonetheless, this model could be adapted to many other situations at an appropriate geographical level, with a single well defined political decision process.
Herd immunity requires that a fraction $1-1/\Rt_0\approx 70\%$ of the population has been infected. It is far from being reached at the country scale in France, but we observe that the fraction of immune individuals strongly varies across the territory, with possible local immunity effects. For instance, in the most impacted county the immunity rate is 16%, whereas it is less than $1$% in less affected counties. At a thinner grain scale, even higher rates may be observed, for instance, by April 4 the proportion of people with confirmed SARS-CoV-2 infection based on antibody detection was 41% in a high-school located in Northern France [@FonTon20].
Real-time monitoring of the immunity level will be crucial to define efficient management policies, if a new outbreak occurs. We propose such a tool which is based on the modeling approach $\M_3$ of this paper (see Immunity tab in <http://covid19-forecast.biosp.org/>). Remarkably, the estimated levels of immunity are comparable to those observed in Spain by population-based serosurveys [@PolPer20], with values ranging from $0.5\%$ to $13.0\%$ at the beginning of May at the provincial resolution. Such a large-scale serological testing campaign has not yet been carried out in France. However, if such data become available in France, our predictions could be evaluated and our model updated accordingly by including this new dataset in our estimation procedure. The mechanistic-statistical approach that we followed here can indeed be easily adapted to multiple observation processes.
Our results indicate that —at the country scale— travel restriction alone, although they may have a significant effect on the cumulative number of deaths and the reproduction number over a definite period, are less efficient than social distancing and other sanitary measures. Obviously, these results may strongly depend on the parameter values, which have been chosen here on the basis of values estimated during the lockdown period. This is consistent with results for China [@ChiDav20], where travel restrictions to and from Wuhan have been shown to have a modest effect unless paired with other public health interventions and behavioral changes.
The model selection criteria led to a strong evidence in favor of the selection of model $\M_3$ with non-local transmission and spatially-constant contact rate. It is much more parsimonious than the fully heterogeneous model $\M_2$ and is therefore better suited to isolating key features of the epidemiological dynamics [@BerFra20]. Despite important restrictions on movement during the considered period (mandatory home confinement except for essential journeys until 11 May and a 100 km travel restriction until 2 June), the model $\M_3$ was also selected against the model $\M_1$ which does not take into account non-local transmission. This shows that intercounty transmission is one of these key-features that the non-local model manages to take into account.
More generally, in France just as in Italy [@GatBer20], the spatial pattern of COVID-19 incidence indicates that spatial processes play a key role. At this stage, only a few models can address this aspect. Some have adopted a detailed spatio-temporal modeling approach and use mobility data (see [@GatBer20] for an SEIR-like model with 9 compartments). The framework we develop here, including the non-local model and the associated estimation procedure, should be of broad interest in studying the spatial dynamics of epidemics, due to its theoretical and numerical simplicity and its ability to accurately track the epidemics. This approach applies when geographic distance matters, which may not be the case at the scale of countries like the US. However, it does at more regional scales. Furthermore, we can envision natural extensions of the approach that would take into account long range dispersal events in the interaction term.
Data availability {#data-availability .unnumbered}
=================
All data used in this manuscript are publicly available. French mortality data at the county scale are available at https://www.gouvernement.fr/info-coronavirus/carte-et-donnees and are also available as Supplementary Material.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was funded by INRAE (MEDIA network) and EHESS. We thank Jean-François Rey for assistance in developing the web app.
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Supplementary Material {#supplementary-material .unnumbered}
======================
SI 1. Supplementary figures {#si-1.-supplementary-figures .unnumbered}
===========================
- The mean temperatures during the observation period are presented in Fig. \[fig:temp\], together with a scatter plot of the mean temperature vs immunity rate.
(source: https://www.data.gouv.fr/fr/datasets/temperature-quotidienne-departementale-depuis-janvier-2018/).
- In Fig. \[fig:timeline\], we describe the timeline of the spatio-temporal dynamics of the immunity rate during the observation period.
![(a) Mean temperature in each county over the period ranging from 30 March 2020 to 11 June 2020. (b) Immunity rate vs mean temperature: we observe a negative correlation between the mean temperature and the immunity rate (Pearson correlation coefficient: $-0.24$).[]{data-label="fig:temp"}](Images/temperature.png "fig:"){width="48.00000%"} ![(a) Mean temperature in each county over the period ranging from 30 March 2020 to 11 June 2020. (b) Immunity rate vs mean temperature: we observe a negative correlation between the mean temperature and the immunity rate (Pearson correlation coefficient: $-0.24$).[]{data-label="fig:temp"}](Images/immun_temperature.png "fig:"){width="48.00000%"}
![Dynamics of the immunity rate given by model $\M_3$, from 6 April 6 to 8 June.[]{data-label="fig:timeline"}](Images/immun_M3_k1.png "fig:"){width="32.00000%"} ![Dynamics of the immunity rate given by model $\M_3$, from 6 April 6 to 8 June.[]{data-label="fig:timeline"}](Images/immun_M3_k2.png "fig:"){width="32.00000%"} ![Dynamics of the immunity rate given by model $\M_3$, from 6 April 6 to 8 June.[]{data-label="fig:timeline"}](Images/immun_M3_k3.png "fig:"){width="32.00000%"} ![Dynamics of the immunity rate given by model $\M_3$, from 6 April 6 to 8 June.[]{data-label="fig:timeline"}](Images/immun_M3_k4.png "fig:"){width="32.00000%"} ![Dynamics of the immunity rate given by model $\M_3$, from 6 April 6 to 8 June.[]{data-label="fig:timeline"}](Images/immun_M3_k5.png "fig:"){width="32.00000%"} ![Dynamics of the immunity rate given by model $\M_3$, from 6 April 6 to 8 June.[]{data-label="fig:timeline"}](Images/immun_M3_k6.png "fig:"){width="32.00000%"} ![Dynamics of the immunity rate given by model $\M_3$, from 6 April 6 to 8 June.[]{data-label="fig:timeline"}](Images/immun_M3_k7.png "fig:"){width="32.00000%"} ![Dynamics of the immunity rate given by model $\M_3$, from 6 April 6 to 8 June.[]{data-label="fig:timeline"}](Images/immun_M3_k8.png "fig:"){width="32.00000%"} ![Dynamics of the immunity rate given by model $\M_3$, from 6 April 6 to 8 June.[]{data-label="fig:timeline"}](Images/immun_M3_k9.png "fig:"){width="32.00000%"} ![Dynamics of the immunity rate given by model $\M_3$, from 6 April 6 to 8 June.[]{data-label="fig:timeline"}](Images/immun_M3_k10.png "fig:"){width="32.00000%"} ![Dynamics of the immunity rate given by model $\M_3$, from 6 April 6 to 8 June.[]{data-label="fig:timeline"}](Images/colorbar2.png "fig:"){width="5.00000%"}
SI 2. Auxiliary models {#si-2.-auxiliary-models .unnumbered}
======================
At each date $t$, the time-dependent parameters $\alpha(t)$ in $\M_{0}$ and $\M_{1}$ and $\alpha_k(t)$ in $\M_{2}$ maximum likelihood estimators (MLEs) in the window $(t-\tau/2,t+\tau/2)$ of the following auxiliary models: $$\label{eq:Modab} \tag{$\tilde{\M}_{0,t}$}
\baco{l}
{\displaystyle}\tS'(s)=- \frac{\tA}{N} \, \tS \, \tI, \vspace{1mm}\\
{\displaystyle}\tI'(s)= \frac{\tA}{N} \, \tS \, \tI - (\beta+\gamma) \, \tI, \vspace{1mm}\\
{\displaystyle}\tR'(s)=\beta \, \tI,\vspace{1mm}\\
{\displaystyle}\tD'(s)=\gamma \, \tI,
{\end{array} \right.}\hbox{ for }s\in (t-\tau/2,t+\tau/2),$$ and $$\label{eq:Mod0b} \tag{$\tilde{\M}_{1,t}$}
\baco{l}
{\displaystyle}\tS_k'(s)=- \frac{\tA}{N_k} \, \tS_k \, \tI_k, \vspace{1mm}\\
{\displaystyle}\tI_k'(s)= \frac{\tA}{N_k} \, \tS_k \, \tI_k - (\beta+\gamma) \, \tI_k, \vspace{1mm}\\
{\displaystyle}\tR_k'(s)=\beta \, \tI_k,\vspace{1mm}\\
{\displaystyle}\tD_k'(s)=\gamma \, \tI_k,
{\end{array} \right.}\hbox{ for }s\in (t-\tau/2,t+\tau/2),$$ and $$\label{eq:Mod1b} \tag{$\tilde{\M}_{2,t}$}
\baco{l}
{\displaystyle}\tS_k'(s)=- \frac{\tA_k}{N_k} \, \tS_k \, \tI_k, \vspace{1mm}\\
{\displaystyle}\tI_k'(s)= \frac{\tA_k}{N_k} \, \tS_k \, \tI_k - (\beta+\gamma) \, \tI_k, \vspace{1mm}\\
{\displaystyle}\tR_k'(s)=\beta \, \tI_k,\vspace{1mm}\\
{\displaystyle}\tD_k'(s)=\gamma \, \tI_k,
{\end{array} \right.}\hbox{ for }s\in (t-\tau/2,t+\tau/2).$$ The initial condition in these models is computed iteratively from the solutions of $\M_{0}$, $\M_{1}$ and $\M_{2}$, respectively, over the period $[t_i,t-\tau/2]$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider a one-dimensional optical lattice of three-dimensional Harmonic Oscillators which are loaded with neutral fermionic atoms trapped into two hyperfine states. By means of a standard variational coherent-state procedure, we derive an effective Hamiltonian for this quantum model and the hamiltonian equations describing its evolution. To this end, we identify the algebra $\mathcal L$ of two-fermion operators –describing the relevant microscopic quantum processes of our model– whereby the natural choice for the trial state appears to be a so(2r) coherent state. The coherent-state parameters, playing the role of dynamical variables for the effective Hamiltonian, are shown to identify with the $\mathcal L$-operator expectation values thus providing a clear physical interpretation of this algebraic mean-field picture.'
address: 'Dipartimento di Fisica, Torino Politecnico, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy'
author:
- 'F. P. Massel, V. Penna'
title: 'Mean-field algebraic approach to the dynamics of fermions in a 1D optical lattice'
---
Introduction
============
Recently, a hierarchy of Hubbard-like Hamiltonians has been proposed to describe the behavior of ultracold fermions in one-dimensional optical lattices [@Massel].
These lattices can be realized with a pair of lasers propagating at a given angle $\theta$ ($\theta=\pi$ represents the familiar counterpropagating case), with global confinement ensured by a magnetic trap (see Fig. \[fig:Setup\], for a detailed description of this setup see [@Peil]). The pair of lasers give rise to a interference pattern needed to obtain a periodic potential by AC Stark effect. The lattice constant can be adjusted tuning the angle $\theta$ according to the relation $d=\lambda/2\sin
(\theta/2)$ where $\lambda$ is the laser wavelength. In addition, it is possible to control both the barrier height of the periodic potential (as a function of the laser intensity) and the interaction between fermions via an external magnetic field (Feshbach resonance, see, e.g. [@Kokkelmans]).
These simple considerations allow one to understand how ultracold-atoms physics offers the possibility to explore experimentally a wide range of parameters set that would be unattainable in other contexts, such as the Hubbard model in condensed matter physics.
As a first step towards the description of the cited models, we propose here a mean-field algebraic approach based on coherent-states procedure [@Gilmore] for a fermionic one-dimensional array of harmonic wells. Although the analytical approach followed here may be regarded as completely general, future numerical analysis will concentrate on a dimer with a six-level structure per well as depicted in Fig. \[fig:levels\]). The approach followed here allows one a straightforward reformulation of the the usual mean-field approach for quantum system (based on the ‘linearization’ of the Hamiltonian and the subsequent solving appropriate self-consistency equations) in terms of a corresponding classical dynamical system. While, for fermions, the interpretation of the aforementioned classical dynamical system as a semiclassical approximation seems not beyond need of justification, it is possible to give a precise physical interpretation to the dynamical variables of the classical problem in terms of expectation values of quantum operators.
In general, it is possible to consider a mean-field approach to a given problem as the constrained minimization of the Hamiltonian $\hat{H}$ over a algebra $\mathcal{L}$. A different choice of $\mathcal{L}$ will lead to different mean-field solutions ([@Rasetti; @Gilmore; @MontoPe]). In particular, we will focus on the so(2r) coherent states that, as it will be shown, will lead to the Hartree-Fock-Bogoliubov [@Lieb] mean-field approximation, whose effectiveness has been proven for a single spherical harmonic trap in [@Grasso].
The paper is organized as follows. In section \[sec:ModDesc\] a brief discussion of the general model considered will be given, along with some possible approximations in different physical situations. As we already mentioned, the fully-analytical control over the physical parameters allows to conceive various Hamiltonians that may have direct experimental relevance. In section \[sec:CohSt\] so(2r) coherent states and the relevant algebra will be defined. The end of this rather technical section will be devoted to the physical interpretation of the choice of so(2r) as the algebra for the mean-field procedure. In section \[sec:ClH\] the classical Hamiltonian $\mathcal{H}_{cl}$ will be deduced and the functional dependence in terms of quantum operators expectation values will be investigated. Finally, in section \[sec:EvZeta\] the analysis of the classical dynamical system whose Hamiltonian is $\mathcal{H}_{cl}$ is performed: Lie-Poisson brackets (namely the ‘classical’ commutators) and, consequently, the evolution equations for the dynamical variables are given.
Model Description {#sec:ModDesc}
=================
In [@Massel], along the lines introduced in [@Albus], a generalized Hubbard Hamiltonian has been introduced to describe the behavior of alkali-metal fermionic atoms in a one-dimensional optical lattice of oblate three dimensional (2+1D) Harmonic Oscillators (pancakes) $$\label{eq:RHH}
\hat{\mathsf{H}}= \sum_{\alpha}
\lambda_{\alpha}
\hat{n}_\alpha+
\sum_{\alpha,\beta}
T_{\alpha,\beta}
\hat{c}^\dagger_\alpha
\hat{c}_\beta
+\sum_{\alpha,\beta,\gamma,\delta}
U_{\alpha,\beta,\gamma,\delta}
\hat{c}^\dagger_\alpha
\hat{c}^\dagger_\beta
\hat{c}_\delta
\hat{c}_\gamma \, .$$
In Eq. (\[eq:RHH\]), $\alpha$ must be considered as a multiple index $\alpha=\left\{i_\alpha,n_\alpha=0,J_ \alpha,m_\alpha,\sigma_\alpha \right\}$ whose origin can be traced back to the space(+local)-modes approximation. In this picture $n_\alpha,J_\alpha$ and $m_\alpha$ are the local 2+1 D Harmonic Oscillator quantum numbers, $i_\alpha$ is the site quantum number and $\sigma_\alpha$ is the spin quantum number. In the following we will confine our analysis to situation where radial modes only are involved in the system dynamics (i.e. we will “freeze” the axial quantum number $n_\alpha$ to zero). The validity of this assumption is guaranteed as long as the radial trapping frequency $\Omega_\perp$ is much smaller than the axial trapping frequency $\omega_x$, i.e. $\Omega_\perp/\omega_x \ll 1$. In this case the tunneling coefficient assumes the following form: $$\label{eq:Tdelta}
T_{\alpha,\beta}=\delta_{J_\alpha,J_\beta}
\delta_{m_\alpha,m_\beta}
\delta_{\sigma_\alpha,\sigma_\beta}
T_{i_\alpha,i_\beta}$$ if we allow nearest-neighbor hopping only $$\label{eq:Tdelta2}
T_{i_\alpha,i_\beta}=T\delta_{i_\alpha+1,i_\beta} \, ,$$ where $T$ is a known function of the external parameters. Another assumption concerns the two-body interaction term $U_{\alpha,\beta,\gamma,\delta}$, which is treated within the pseudopotential approximation, leading to a delta-like spatial dependence, thus excluding nearest-neighbor two-body interaction terms. If the fermionic nature of the interacting particles is taken into account we have $$\label{eq:U}
U_{\alpha,\beta,\gamma,\delta}=
\delta_{i_\alpha,i_\beta,i_\gamma,i_\delta}
\delta_{\sigma_\alpha,\sigma_\gamma}\delta_{\sigma_\beta,\sigma_\delta}
\delta_{\sigma_\alpha,-\sigma_\beta}
U_{\{J_\alpha,m_\alpha\},\{J_\beta,m_\beta\},\{J_\gamma,m_\gamma\},\{J_\delta,m_\delta\}}\,.$$ Finally, we give the expression for the one-particle energy term which is essentially given by the single-particle energy of the 3D Harmonic Oscillator $$\label{eq:lambda2}
\lambda_{i_\alpha,n_\alpha,J_\alpha,m_\alpha,\sigma_\alpha}=
\left[
\hbar\omega_x
\left(
n_\alpha+\frac{1}{2}
\right)+
\hbar\Omega_\perp\left(2J_\alpha+1\right) - T_{i_\alpha,i_\alpha}
\right] \, ,$$ where $T_{i_\alpha,i_\alpha}$ represents a “hopping correction” to the single particle energy term. For the case considered in Fig. \[fig:levels\] the selection rules imposed on the two-body interaction term select three possible values on $U_{\alpha,\beta,\gamma,\delta}$ that can be classified as: i) lowest-level/ lowest-level interaction terms, ii) lowest-level/highest-level interaction terms, iii) highest-level/highest-level interaction terms.
Coherent States {#sec:CohSt}
===============
Since in our mean-field analysis we would like to keep trace of the (possible) atom pairing, the most appropriate coherent-states algebra seems to be, according to [@Gilmore], the algebra spanned by the $r(2r-1)$ operators $\{\hat{c}_\alpha^\dagger\hat{c}_\beta (1\leq i,j\leq r),
\hat{c}_\alpha\hat{c}_\beta,\hat{c}^\dagger_\alpha\hat{c}^\dagger_\beta \}$, i.e. so(2r). Its commutation relations can be written as $$\begin{aligned}
\label{eq:CommY}
&&\left[ Y^1_{\alpha\beta},Y^1_{\gamma\delta}\right]=\left[
Y^2_{\alpha\beta},Y^2_{\gamma\delta}\right]=0 \nonumber\\
&&\left[ Y^1_{\alpha\beta},Y^2_{\gamma\delta}\right]=
Y^3_{\gamma\beta}\delta_{\alpha\delta}+Y^3_{\delta\alpha}\delta_{\beta\gamma}
-Y^3_{\gamma\alpha}\delta_{\beta\delta}-Y^3_{\delta\beta}\delta_{\alpha\gamma}\nonumber\\
&&\left[Y^1_{\alpha\beta},Y^3_{\gamma\delta}\right]=
Y^1_{\alpha\delta}\delta_{\beta\gamma}-Y^1_{\beta\delta}\delta_{\alpha\gamma}\nonumber\\
&&\left[Y^2_{\alpha\beta},Y^3_{\gamma\delta}\right]=Y^2_{\beta\gamma}\delta_{\alpha\delta}-Y^2_{\alpha\gamma}\delta_{\beta\delta}\, ,
\end{aligned}$$ having defined $$\begin{aligned}
\label{eq:AlgComm}
Y^1_{\alpha\beta}=\hat{c}_\alpha\hat{c}_\beta\, ,\\
Y^2_{\alpha\beta}=\hat{c}^\dagger_\beta\hat{c}^\dagger_\alpha\, , \\
Y^3_{\alpha\beta}=\hat{c}^\dagger_\alpha\hat{c}_\beta \,.
\end{aligned}$$ With the above definitions the coherent states can be expressed as $$\label{eq:CohSt}
|\phi\rangle=\exp
\left [ \, - \sum_{1\leq \alpha \neq \beta \leq r}
\left( \eta_{\alpha,\beta}\hat{c}^\dagger_\alpha
\hat{c}^\dagger_\beta-H.c.\right) \right ] |0>\,.$$
To evaluate the expectation value of the Hamiltonian $\hat{H}$ defined by equation (\[eq:RHH\]) over the coherent state of the form (\[eq:CohSt\]), it is necessary to evaluate the action of the operator $$\label{eq:Omega}
\hat{\Omega} =
\exp
\left [ \, -\sum_{1\leq \alpha \neq \beta \leq r}
\left( \eta_{\alpha,\beta}\hat{c}^\dagger_\alpha
\hat{c}^\dagger_\beta-H.c.\right) \right ]$$ over the fermionic raising and lowering operators. Namely $$\begin{aligned}
\label{eq:ExplOmOp}
\hat{\Omega}^\dagger \hat{c}^\dagger_\alpha \hat{\Omega} =
\exp &&\left[
\sum_{1\leq i \neq j \leq r}
\left(
\eta_{i,j}
\hat{c}^\dagger_i \hat{c}^\dagger_j -
\eta^*_{i,j}
\hat{c}_j\hat{c}_i
\right)
\right]
\hat{c}^\dagger_\alpha \nonumber \\
&&\exp \left[
-\sum_{1\leq i \neq j \leq r}
\left(
\eta_{i,j}
\hat{c}^\dagger_i \hat{c}^\dagger_j -
\eta^*_{i,j}
\hat{c}_j\hat{c}_i
\right)
\right]
\end{aligned}$$ which, exploiting the BCH formula [@Gilmore] can be written as $$\label{eq:OpBCH}
\hat{\Omega}^\dagger \hat{c}^\dagger_\alpha \hat{\Omega} =
\sum_m \frac{1}{m} \left[ \sum_{ij} \eta_{i,j}
\hat{c}^\dagger_i \hat{c}^\dagger_j -
\eta^*_{i,j}
\hat{c}_j\hat{c}_i \right]_m \,.$$ It can be shown that in the last summation the two first terms only survive, leading to to the following expression for $\hat{\Omega}
\hat{c}^\dagger_\alpha \hat{\Omega}^\dagger$ and $\hat{\Omega} \hat{c}_\gamma
\hat{\Omega}^\dagger$ respectively $$\begin{aligned}
\label{eq:OmExpl4}
&&\hat{\Omega}^\dagger \hat{c}^\dagger_\alpha \hat{\Omega}=
\hat{c}^\dagger_\alpha +\sum_i \zeta^*_{i\alpha}\hat{c}_i \, ,\\
&&\hat{\Omega}^\dagger \hat{c}_\gamma \hat{\Omega}=
\hat{c}_\gamma + \sum_i\zeta_{m\gamma} \hat{c}^\dagger_m
\end{aligned}$$ with $\zeta_{ij}=2\eta_{ij}$.
We are now in the position to evaluate $$\label{eq:ExpVal}
\mathcal{H}_{cl}=\langle\phi|\hat{\mathsf{H}}-\mu \hat{N}|\phi\rangle\, ,$$ where the term $\mu \hat{N}$ has been added to take into account the particle number constraint.
With Eq. (\[eq:CohSt\]), Eq. (\[eq:ExpVal\]) becomes $$\label{eq:ExpVal2}
\mathcal{H}_{cl}=\langle0|\Omega\left[\left(\hat{\mathsf{H}}_0+\hat{\mathsf{H}}_I\right)\right]\Omega^\dagger|0\rangle
\, ,$$ where $$\begin{aligned}
\hat{\mathsf{H}}_0= \sum_{\alpha,\beta}
\Gamma_{\alpha,\beta}
\hat{c}^\dagger_\alpha
\hat{c}_\beta \, , \\
\hat{H}_I=\sum_{\alpha,\beta,\gamma,\delta}
U_{\alpha,\beta,\gamma,\delta}
\hat{c}^\dagger_\alpha
\hat{c}^\dagger_\beta
\hat{c}_\delta
\hat{c}_\gamma \, ,
\end{aligned}$$ with $\Gamma_{\alpha,\beta}=\lambda_\alpha\delta_{\alpha,\beta}-T_{\alpha,\beta}-\mu \delta_{\alpha,\beta}.$ Since $\Omega$ is a unitary operator, we can write $$\begin{aligned}
\label{eq:OmH}
\hat{\Omega}^\dagger \hat{H}_0 \hat{\Omega} =
\sum_{\alpha,\beta}
\Gamma_{\alpha,\beta}
\hat{\Omega}^\dagger
\hat{c}^\dagger_\alpha
\hat{\Omega}
\hat{\Omega}^\dagger
\hat{c}_\beta
\hat{\Omega}\, , \nonumber \\
\hat{\Omega}^\dagger \hat{H}_I \hat{\Omega} =
\sum_{\alpha,\beta,\gamma,\delta}
U_{\alpha,\beta,\gamma,\delta}
\hat{\Omega}^\dagger \hat{c}^\dagger_\alpha \hat{\Omega}
\hat{\Omega}^\dagger \hat{c}^\dagger_\beta \hat{\Omega}
\hat{\Omega}^\dagger \hat{c}_\delta \hat{\Omega}
\hat{\Omega}^\dagger \hat{c}_\gamma \hat{\Omega} \, .
\end{aligned}$$
The following expectation values must then be evaluated. For the one-body term $$\label{cs2}
\hat{\Omega}^\dagger \hat{H}_0 \hat{\Omega} =
\sum_{\alpha,\beta}
\Gamma_{\alpha,\beta}
\left[
\hat{c}^\dagger_\alpha +\sum_i \zeta^*_{i\alpha} \hat{c}_i
\right] \cdot
\left[
\hat{c}_\beta + \sum_k \zeta_{k\beta} \hat{c}^\dagger_k
\right]$$ and for the interaction term $$\begin{aligned}
\label{cs2.1}
\hat{\Omega}^\dagger \hat{H}_I \hat{\Omega} =&&\sum_{\alpha,\beta,\gamma,\delta}
U_{\alpha,\beta,\gamma,\delta}
\left[
\hat{c}^\dagger_\alpha +\sum_i \zeta^*_{i\alpha} \hat{c}_i
\right] \cdot
\left[
\hat{c}^\dagger_\beta +\sum_j \zeta^*_{j\beta} \hat{c}_j
\right] \cdot \nonumber\\
&&\hspace{-1cm} \left[
\hat{c}_\delta + \sum_k \zeta_{k\delta} \hat{c}^\dagger_k
\right] \cdot
\left[
\hat{c}_\gamma + \sum_l \zeta_{l\gamma} \hat{c}^\dagger_l
\right] \,.
\end{aligned}$$
As it can be directly verified, in the calculation of the expectation values over the vacuum state $|0\rangle$ only the following terms survive $$\label{eq:cs3}
\langle0|\hat{\Omega}^\dagger \hat{H}_0
\hat{\Omega} |0\rangle=\sum_{\alpha,\beta} \sum_{ij}
\Gamma_{\alpha,\beta}\zeta^*_{i\alpha}\zeta_{j\beta}
\langle0|\hat{c}_i\hat{c}^\dagger_j|0\rangle \, ,$$ $$\begin{aligned}
\label{eq:cs3.1}
&&\langle0|\hat{\Omega}^\dagger \hat{H}_I \hat{\Omega} |0\rangle=
\sum_{\alpha,\beta,\gamma,\delta}
U_{\alpha,\beta,\gamma,\delta}
\left(
\sum_{i,j,k,l}
\zeta^*_{i\alpha}\zeta^*_{j\beta}\zeta_{k\delta}\zeta_{l\gamma}
\langle0| \hat{c}_i \hat{c}_j\hat{c}^\dagger_k
\hat{c}^\dagger_l|0\rangle \right. \nonumber \\
&&\hspace{5.5cm}+\left.
\sum_{i,j}\zeta^*_{i\alpha}\zeta_{j\gamma}\langle0|\hat{c}_i\hat{c}^\dagger_\beta\hat{c}_\delta\hat{c}^\dagger_j|0 \rangle
\right)\,.
\end{aligned}$$ The two expectation values over the vacuum state give $$\begin{aligned}
\label{eq:ExpVal3}
&& \langle0| \hat{c}_i \hat{c}^\dagger_j|0\rangle=\delta_{ij} \\
&&\langle0| \hat{c}_i \hat{c}_j\hat{c}^\dagger_l \hat{c}^\dagger_k|0\rangle =
\delta_{il}\delta_{jk}-\delta_{jk}\delta_{il} \\
&&\langle0|\hat{c}_i\hat{c}^\dagger_\beta\hat{c}_\delta\hat{c}^\dagger_j|0
\rangle=
\delta_{i\beta}\delta_{j\delta}
\end{aligned}$$ hence $$\label{eq:Hcl}
\hspace{-2cm}
\mathcal{H}^{cl}=\sum_{\alpha\beta} \Gamma_{\alpha\beta}\sum_{i}
\zeta^*_{i\alpha}\zeta_{i\beta} +
\sum_{\alpha,\beta,\gamma,\delta}
U_{\alpha,\beta,\gamma,\delta}
\left[
\sum_{i,j} \left( \zeta^*_{i\alpha}\zeta^*_{j\beta}\zeta_{j\delta}\zeta_{i\gamma}-\zeta^*_{i\alpha}\zeta^*_{j\beta}\zeta_{i\delta}\zeta_{j\gamma} \right) +
\zeta^*_{\beta\alpha}\zeta_{\delta\gamma}
\right]$$
Effective Hamiltonian {#sec:ClH}
======================
Hamiltonian (\[eq:Hcl\]) can be shown to represent the effective Hamiltonian associated with $\hat H$ within the time-dependent variational principle procedure [@Gilmore]. The latter is based on approximating the quantum states of the system by a trial state $|\Psi \rangle$ satisfying the weak form of the Schrödinger equation $ \langle \Psi |i \hbar \partial_t -{\hat H}| \Psi \rangle = 0 $. Here, we assume that $| \Psi \rangle$, up to an irrelevant phase factor, is the coherent state defined in equation (\[eq:CohSt\]). The variational procedure allows one to derive the effective Lagrangian ${\dot S} = \langle\Psi |i \hbar \partial_t -{\hat H}| \Psi\rangle $, depending on dynamical variables $\zeta_{\alpha \beta}$, which in turn supplies the effective Hamiltonian (\[eq:Hcl\]). Such a procedure provides as well the dynamical equations pertaining to Hamiltonian (\[eq:Hcl\]) and the relevant Lie-Poisson brackets. The latter exhibit the same algebraic structure of commutators (\[eq:CommY\]) and will be defined below.
A quite direct physical insight about coherent-state parameters $\zeta_{\alpha \beta}$ is achieved when considering the expectation values for the elements of the Lie algebra so(2r) over the coherent states $|\phi\rangle$. We have $$\begin{aligned}
\label{eq:Exp1}
&&\langle \phi|\hat{c}^\dagger_\alpha\hat{c}^\dagger_\beta |\phi
\rangle=\zeta^*_{\beta\alpha} \, ,\\
&&\langle \phi|\hat{c}^\dagger_\alpha\hat{c}_\beta |\phi \rangle=\sum_i
\zeta^*_{i\alpha}\zeta_{i\beta}=\xi_{\alpha\beta} \, ,\\
&&\langle \phi|\hat{c}_\alpha\hat{c}_\beta |\phi \rangle=\zeta_{\alpha\beta}
\, ,
\end{aligned}$$ showing how parameters $\zeta_{\alpha \beta}$ are related to microscopic physical processes of creation/destruction of lattice fermions. Moreover Eq. (\[eq:Hcl\]) can be written as $$\label{eq:HClSpin}
\mathcal{H}_{cl}=\sum_{\alpha\beta} \Gamma_{\alpha\beta} \xi_{\alpha\beta} +
\sum_{\alpha,\beta,\gamma,\delta}
U_{\alpha,\beta,\gamma,\delta}
\left[
\left( \xi_{\alpha\gamma}\xi_{\beta\delta}-
\xi_{\alpha\delta}\xi_{\beta\gamma}
\right) +
\zeta^*_{\beta\alpha}\zeta_{\delta\gamma}
\right]\, .$$ The three terms represent the direct, the exchange and the pairing term in the HFB mean-field approximation.
Evolution Equations for the canonical variables {#sec:EvZeta}
-----------------------------------------------
According to [@Gilmore] the variables $\zeta_{\alpha\beta}$, $\zeta^*_{\alpha\beta}$ and $\xi_{\alpha \beta}$ represent the canonical variable for the classical Hamiltonian $\mathcal{H}_{cl}$. With a well-known procedure [@Perelomov] it is possible to describe the time evolution of those canonical variables in terms of their Poisson brackets with $\mathcal{H}_{cl}$. To write the Lie-Poisson brackets for the given dynamical system we can make explicit the structure constants for the so(2r) algebra $$\begin{aligned}
\label{eq:StrConst}
&&c_{1,\alpha,\beta;1,\gamma,\delta}^{\phantom{1,\alpha,\beta;1,\gamma,\delta}q,\mu,\nu}=
c_{2,\alpha,\beta;2,\gamma,\delta}^{\phantom{2,\alpha,\beta;2,\gamma,\delta}q,\mu,\nu}=0
\nonumber \, ,\\
&&c_{1,\alpha,\beta;2,\gamma,\delta}^{\phantom{1,\alpha,\beta;2,\gamma,\delta}q,\mu,\nu}=\delta_{q,3}\left(
\delta_{\mu\gamma}\delta_{\nu\beta}\delta_{\alpha\delta}+\delta_{\mu\delta}\delta_{\nu\alpha}
\delta_{\beta\gamma}-
\delta_{\mu\gamma}\delta_{\nu\alpha}\delta_{\beta\delta}-\delta_{\mu\delta}\delta_{\nu\beta}
\delta_{\alpha\gamma}
\right) \nonumber \, ,\\
&&c_{1,\alpha,\beta;3,\gamma,\delta}^{\phantom{1,\alpha,\beta;3,\gamma,\delta}q,m,n}=\delta_{q,1}\left(
\delta_{\mu\alpha}\delta_{\nu\delta}\delta_{\beta\gamma}-
\delta_{\mu\beta}\delta_{\nu\delta}\delta_{\alpha\gamma}
\right) \nonumber \, ,\\
&&c_{2,\alpha,\beta;3,\gamma,\delta}^{\phantom{2,\alpha,\beta;3,\gamma,\delta}q,m,n}=\delta_{q,2}\left(
\delta_{\mu\beta}\delta_{\nu\gamma}\delta_{\alpha\delta}-
\delta_{\mu\alpha}\delta_{\nu\gamma}\delta_{\beta\delta}
\right) \, .
\end{aligned}$$
Thus the Poisson brackets have the following form $$\begin{aligned}
\label{eq:PoissBrack}
\hspace{-2cm}
\left\{f,g \right\} = && \sum_{\alpha\beta\gamma\delta}
%\nonumber \\&&
\left(\xi_{\gamma\beta}\delta_{\alpha\gamma}-
\xi_{\gamma\alpha}\delta_{\beta\delta}+
\xi_{\delta\alpha}\delta_{\beta\gamma}-
\xi_{\delta\beta}\delta_{\alpha\gamma}
\right)
\left(
\frac{\partial f}{\partial \zeta_{\alpha\beta}}
\frac{\partial g}{\partial \zeta^*_{\gamma\delta}}-
\frac{\partial f}{\partial \zeta^*_{\gamma\delta}}
\frac{\partial g}{\partial \zeta_{\alpha\beta}}
\right) + \nonumber \\
&& +\left(
\zeta_{\alpha\delta}\delta_{\gamma\beta}-\zeta_{\beta\delta}\delta_{\gamma\alpha}
\right)
\left(
\frac{\partial f }{\partial \zeta_{\alpha\beta}}
\frac{\partial g }{\partial \xi_{\gamma\delta}}-
\frac{\partial f }{\partial \xi_{\gamma\delta}}
\frac{\partial g }{\partial \zeta_{\alpha\beta}}
\right)+ \nonumber \\
&&+ \left(
\zeta_{\gamma\beta}\delta_{\alpha\delta}-\zeta_{\alpha\gamma}\delta_{\beta\delta}
\right)
\left(
\frac{\partial f }{\partial \zeta^*_{\alpha\beta}}
\frac{\partial g }{\partial \xi_{\gamma\delta}}-
\frac{\partial f }{\partial \xi_{\gamma\delta}}
\frac{\partial g }{\partial \zeta^*_{\alpha\beta}}
\right)+ \nonumber \\
&& +\left(
\xi_{\beta\delta}\delta_{\gamma\alpha}-\xi_{\alpha\gamma}\delta_{\beta\delta}
\right)
\left(
\frac{\partial f }{\partial \xi_{\alpha\beta}}
\frac{\partial g }{\partial \xi_{\gamma\delta}}-
\frac{\partial f }{\partial \xi_{\gamma\delta}}
\frac{\partial g }{\partial \xi_{\alpha\beta}}
\right) \, .\end{aligned}$$ Remembering that $$\label{eq:EvZeta}
\dot{\zeta}_{\rho,\theta}=\left\{\zeta_{\rho,\theta},H\right\} \, ,$$ it is possible to write $$\begin{aligned}
\label{eq:EvZeta2}
\dot{\zeta}_{\rho,\theta}=&&
\sum_\alpha
\left(\Gamma_{\rho\alpha}\zeta_{\theta\alpha}-\Gamma_{\theta\alpha}\zeta_{\rho\alpha}\right)+ \nonumber
\\ &&
\sum_{\alpha\gamma\eta}\left[U_{\alpha\rho\gamma\eta}\left(\zeta_{\theta\gamma}\xi_{\alpha\eta}-\zeta_{\theta\eta}\xi_{\alpha\gamma}\right)
+U_{\alpha\theta\gamma\eta}\left(\zeta_{\rho\eta}\xi_{\alpha\gamma}-\zeta_{\theta\gamma}\xi_{\alpha\eta}\right)\right]
\nonumber \\ &&
\sum_{\beta\gamma\eta}\left[U_{\rho\beta\gamma\eta}\left(\zeta_{\theta\eta}\xi_{\beta\gamma}-\zeta_{\theta\gamma}\xi_{\beta\eta}\right)
+U_{\theta\beta\gamma\eta}\left(\zeta_{\rho\eta}\xi_{\beta\gamma}-\zeta_{\rho\gamma}\xi_{\beta\eta}\right)\right]+
\nonumber \\&&
\sum_{\alpha\gamma\eta}
\zeta_{\eta\gamma}\left(U_{\alpha\theta\gamma\eta}\xi_{\alpha\rho}-U_{\alpha\rho\gamma\eta}\xi_{\alpha\theta}\right)+
\nonumber \\&&
\sum_{\beta\gamma\eta}
\zeta_{\eta\gamma}\left(U_{\rho\beta\gamma\eta}\xi_{\beta\theta}-U_{\theta\rho\gamma\eta}\xi_{\beta\rho}\right)\end{aligned}$$ that provide the set of dynamical equations governing the evolution of the coherent state that approximates the system quantum state. In particular, they allow to find the mean-field ground the state for the system and to perform a weakly-excited state analysis.
Conclusions {#sec:Concl}
===========
In this paper we have formulated an HFB mean-field approximation for a one-dimensional array of oblate Harmonic Oscillators loaded with neutral fermionic atoms. As already pointed out by Grasso et al. [@Grasso], the numerical solution to Eq. (\[eq:EvZeta2\]) appears to be rather demanding from a computational point of view. It seems then appropriate for future work to concentrate on the simplest situation beyond known models like, as already mentioned, a dimer with a six-level local structure.
In this case the evaluation of ground-state properties in this mean-field picture as a function of the relevant parameters(i.e $T_{\alpha,\beta}$, $U_{\alpha\beta\gamma\delta}$, $\mu$) reduces to the fixed-point analysis of Eq.(\[eq:EvZeta2\]). Moreover, an extension to finite-temperature properties does not seem beyond the possibilities of the analytical techniques here outlined and may represent one of the future lines of research.
Expectation values calulation
=============================
In the present section we will explicitly calculate the terms obtained form Eqs. (\[cs2\],\[cs2.1\]) leading to Eqs. (\[eq:cs3\]). To evaluate (\[cs2\]) we need to perform the following product calculation $$\label{eq:app1}
\hat{\Omega}^\dagger \hat{H}_0
\hat{\Omega} = \sum_{\alpha\beta ij}
\Gamma_{\alpha,\beta} \left[
\hat{c}^\dagger_\alpha +\sum_i \zeta^*_{i\alpha}\hat{c}_i
\right] \cdot
\left[
\hat{c}_\beta +\sum_j \zeta_{j\beta}\hat{c}^\dagger_j
\right]$$ which is equal to
$$\label{eq:app2}
\hat{\Omega}^\dagger \hat{H}_0
\hat{\Omega} = \sum_{\alpha\beta}\left[\hat{c}^\dagger_\alpha \hat{c}_\beta + \sum_i \zeta^*_{i\alpha}\hat{c}_i
\hat{c}_\beta+
\sum_j \zeta_{j\beta} \hat{c}^\dagger_\alpha\hat{c}^\dagger_j +\sum_{ij}
\zeta^*_{i\alpha}\zeta_{j\beta}\hat{c}_i \hat{c}^\dagger_j\right].$$
The evaluation of Eq. (\[eq:app2\]) over the vacuum state $|0\rangle$ leads to vanishing contributions for all non number-conseving terms and for all the terms with a lowering operator on the right-hand side (or a raising operator on the left-hand side). Namely (see Eq. (\[eq:cs3\]) $$\label{eq:app4}
\langle0|\hat{\Omega}^\dagger \hat{H}_0
\hat{\Omega} |0\rangle=\sum_{\alpha\beta} \sum_{ij}
\Gamma_{\alpha,\beta} \zeta^*_{i\alpha}\zeta_{j\beta} \langle0|\hat{c}_i\hat{c}^\dagger_j|0\rangle.$$
With an analogous procedure it is possible to evaluate the expression given by Eq. (\[cs2.1\]) $$\begin{aligned}
\label{eq:app5}
\hat{\Omega}^\dagger \hat{H}_I \hat{\Omega} =\sum_{\alpha,\beta,\gamma,\delta}
U_{\alpha,\beta,\gamma,\delta}
\left[
\hat{c}^\dagger_\alpha +\sum_i \zeta^*_{i\alpha} \hat{c}_i
\right]&& \cdot
\left[
\hat{c}^\dagger_\beta +\sum_j \zeta^*_{j\beta} \hat{c}_j
\right] \cdot \nonumber\\
&&\hspace{-1cm} \left[
\hat{c}_\delta + \sum_k \zeta_{k\delta} \hat{c}^\dagger_k
\right] \cdot
\left[
\hat{c}_\gamma + \sum_l \zeta_{l\gamma} \hat{c}^\dagger_l
\right]
\end{aligned}$$ leading to $$\begin{aligned}
\label{eq:app6}
&& \hat{\Omega}^\dagger \hat{H}_I \hat{\Omega} =
\sum_{\alpha,\beta,\gamma,\delta}
U_{\alpha,\beta,\gamma,\delta} \nonumber \\
&&\left[\hat{c}^\dagger_\alpha \hat{c}^\dagger_\beta
\hat{c}_\delta \hat{c}_\gamma +
\sum_k \zeta_{k\delta}
\hat{c}^\dagger_\alpha \hat{c}^\dagger_\beta
\hat{c}^\dagger_k \hat{c}_\gamma+
\sum_{l} \zeta_{l\gamma}
\hat{c}^\dagger_\alpha \hat{c}^\dagger_\beta
\hat{c}_\delta \hat{c}^\dagger_l+
\sum_{kl} \zeta_{l\gamma} \zeta_{k\delta}
\hat{c}^\dagger_\alpha \hat{c}^\dagger_\beta
\hat{c}^\dagger_k \hat{c}^\dagger_l+ \right. \nonumber\\
&& \sum_j \zeta^*_{j\beta}
\hat{c}^\dagger_\alpha \hat{c}_j
\hat{c}_\delta \hat{c}_\gamma +
\sum_{jk} \zeta^*_{j\beta} \zeta_{k\delta}
\hat{c}^\dagger_\alpha \hat{c}_j
\hat{c}^\dagger_k \hat{c}_\gamma +
\sum_{jl} \zeta^*_{j\beta}\zeta_{l\gamma}
\hat{c}^\dagger_\alpha \hat{c}_j
\hat{c}_\delta \hat{c}^\dagger_l +
\sum_{jkl} \zeta^*_{j\beta}\zeta_{l\gamma}\zeta_{k\delta}
\hat{c}^\dagger_\alpha \hat{c}_j
\hat{c}^\dagger_k \hat{c}^\dagger_l +\nonumber\\
&&\sum_{i}\zeta^*_{i\alpha}
\hat{c}_i \hat{c}^\dagger_\beta
\hat{c}_\delta \hat{c}_\gamma+
\sum_{ik} \zeta^*_{i\alpha} \zeta_{k\delta}
\hat{c}_i \hat{c}^\dagger_\beta
\hat{c}^\dagger_k \hat{c}_\gamma +
\sum_{il} \zeta^*_{i\alpha}\zeta_{l\gamma}
\hat{c}_i \hat{c}^\dagger_\beta
\hat{c}_\delta \hat{c}^\dagger_l +
\sum_{ikl} \zeta^*_{i\alpha}\zeta_{l\gamma}\zeta_{k\delta}
\hat{c}_i \hat{c}^\dagger_\beta
\hat{c}^\dagger_k \hat{c}^\dagger_l +\nonumber\\
&&\sum_{ij} \zeta^*_{i\alpha} \zeta^*_{j\beta}
\hat{c}_i \hat{c}_j
\hat{c}_\delta \hat{c}_\gamma +
\sum_{ijk} \zeta^*_{i\alpha} \zeta^*_{j\beta} \zeta_{k\delta}
\hat{c}_i \hat{c}_j
\hat{c}_\gamma \hat{c}^\dagger_k +
\sum_{ijl} \zeta^*_{i\alpha} \zeta^*_{j\beta} \zeta_{l\gamma}
\hat{c}_i \hat{c}_j
\hat{c}_\delta \hat{c}^\dagger_l + \nonumber\\
&&\left.\sum_{ijkl} \zeta^*_{i\alpha} \zeta^*_{j\beta}
\zeta_{k\delta} \zeta_{l\gamma}
\hat{c}_i \hat{c}_j
\hat{c}^\dagger_k \hat{c}^\dagger_l \right] \,.\end{aligned}$$ With the same argument needed to obtain Eq. (\[eq:app4\]) we can write the expectation value of the operator defined by Eq. (\[eq:app6\]) over the vacuum state $|0\rangle$ as $$\begin{aligned}
\label{eq:app7}
&&\langle0|\hat{\Omega}^\dagger \hat{H}_I \hat{\Omega} |0\rangle=
\sum_{\alpha,\beta,\gamma,\delta}
U_{\alpha,\beta,\gamma,\delta}
\left(
\sum_{i,j,k,l}
\zeta^*_{i\alpha}\zeta^*_{j\beta}\zeta_{k\delta}\zeta_{l\gamma}
\langle0| \hat{c}_i \hat{c}_j\hat{c}^\dagger_k
\hat{c}^\dagger_l|0\rangle + \right. \nonumber \\
&&\hspace{5.5cm}+\left.
\sum_{i,j}\zeta^*_{i\alpha}\zeta_{j\gamma}\langle0|\hat{c}_i
\hat{c}^\dagger_\beta\hat{c}_\delta\hat{c}^\dagger_j|0 \rangle
\right)\end{aligned}$$ which is the expression given by Eq. (\[eq:cs3.1\]).
[10]{}
F. Massel and V. Penna. Hubbard-like [H]{}amiltonian for ultracold atoms in a one-dimensional optical lattice. , 72:053619, 2005.
S. Peil, J. V. Porto, B. Laburthe Tolra, J. M. Obrecht, B. E. King, 2 S. L. Rolston M. Subbotin, and W. D. Phillips. Patterned loading of a bose-einstein condensate into an optical lattice. , 67:051603(R), 2003.
S. J. J. M. F. Kokkelmans, J. N. Milstein, M. L. Chiofalo, R. Walser, and M. J. Holland. Resonance superfluidity: Renormalization of resonance scattering theory. , 65:053617, 2002.
WM Zhang, DH Feng, and R. Gilmore. Coherent states: [T]{}heory and some applications. , 62:867–927, 1990.
, [Rasetti M.]{}, and [Solomon A. I.]{} Dynamical [S]{}uperalgebra and [S]{}upersymmetry for a [M]{}any-[F]{}ermion [S]{}ystem. , 59(20):2244, 1987.
A. Montorsi and V. Penna. Spin picture of the one-dimensional [H]{}ubbard model: [T]{}wo-fluid structure and phase dynamics. , 60:12069, 1999.
V. Bach, EH Lieb, and JP Solovej. Generalized [H]{}artree-[F]{}ock [T]{}heory and the [H]{}ubbard [M]{}odel. , 76:3–90, 1994.
and [Urban M.]{} Hartree-[F]{}ock-[B]{}ogoliubov theory versus local-density approximation for superfluid trapped fermionic atoms. , 68:033610, 2003.
Alexander Albus, Fabrizio Illuminati, and Jens Eisert. Mixtures of bosonic and fermionic atoms in optical lattices. , 68:023606, 2003.
A. Perelomov. . Springer, Berlin, 1986.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Background
: The $^{14}$N(p,$\gamma$)$^{15}$O reaction plays a vital role in various astrophysical scenarios. Its reaction rate must be accurately known in the present era of high precision astrophysics. The cross section of the reaction is often measured relative to a low energy resonance, the strength of which must therefore be determined precisely.
Purpose
: The activation method, based on the measurement of $^{15}$O decay, has not been used in modern measurements of the $^{14}$N(p,$\gamma$)$^{15}$O reaction. The aim of the present work is to provide strength data for two resonances in the $^{14}$N(p,$\gamma$)$^{15}$O reaction using the activation method. The obtained values are largely independent from previous data measured by in-beam gamma-spectroscopy and are free from some of their systematic uncertainties.
Method
: Solid state TiN targets were irradiated with a proton beam provided by the Tandetron accelerator of Atomki using a cyclic activation. The decay of the produced $^{15}$O isotopes was measured by detecting the 511keV positron annihilation $\gamma$-rays.
Results
: The strength of the E$_p$=278keV resonance was measured to be $\omega\gamma_{278}$=(13.4$\pm$0.8)meV while for the E$_p$=1058keV resonance $\omega\gamma_{1058}$=(442$\pm$27)meV.
Conclusions
: The obtained E$_p$=278keV resonance strength is in fair agreement with the values recommended by two recent works. On the other hand, the E$_p$=1058keV resonance strength is about 20% higher than the previous value. The discrepancy may be caused in part by a previously neglected finite target thickness correction. As only the low energy resonance is used as a normalization point for cross section measurements, the calculated astrophysical reaction rate of the $^{14}$N(p,$\gamma$)$^{15}$O reaction and therefore the astrophysical consequences are not changed by the present results.
author:
- 'Gy. Gyürky'
- 'Z. Halász'
- 'G.G. Kiss'
- 'T. Szücs'
- 'A. Csík'
- 'Zs. Török'
- 'R. Huszánk'
- 'M.G. Kohan'
- 'L. Wagner'
- 'Zs. Fülöp'
title: ' Resonance strengths in the $^{14}$N(p,$\gamma$)$^{15}$O astrophysical key reaction measured with activation '
---
Introduction {#sec:intro}
============
Catalytic cycles of hydrogen burning represent an alternative way to the pp-chains for converting four protons into one alpha particle in stellar interiors and for providing thus the energy source of stars. The simplest cycle is the first CNO or Bethe-Weizsäcker cycle [@Wiescher2018] where carbon, nitrogen and oxygen isotopes are involved. The CNO cycle is the dominant energy source of main sequence stars more massive than about 1.3 solar masses but it also plays an important role in various astrophysical scenarios including quiescent and explosive burning processes [@DEGLINNOCENTI200413; @Fields_2018 e.g.].
In the 21$^{\rm th}$ century, astronomical observations as well as astrophysical models are becoming more and more precise. The insufficient knowledge of nuclear reaction rates often represent the largest uncertainty of stellar models. Increasing the precision of experimental nuclear cross sections is thus needed in order to provide accurate reaction rates for the models. For solar models, for example, a precision well below 5% is required for the $^{14}$N(p,$\gamma$)$^{15}$O reaction discussed in the present work [@Haxton_2008].
The slowest reaction of the CNO cycle is the radiative proton capture of $^{14}$N and therefore the rate of this $^{14}$N(p,$\gamma$)$^{15}$O reaction determines the rate of the cycle and hence its efficiency and its contribution to the stellar energy generation. Realizing its importance, many experiments have been devoted to the measurement of its cross section (the full list of references can be found in [@RevModPhys.83.195] and the three latest sets of results are published in [@PhysRevC.94.025803; @PhysRevC.93.055806; @PhysRevC.97.015801]).
Depending on the astrophysical site, the relevant temperature where the CNO cycle is active and important is between about 15 and 200MK. This translates into astrophysically relevant center-of-mass energy ranges (the Gamow-window) of the $^{14}$N(p,$\gamma$)$^{15}$O reaction between 20 and 200 keV. Measured cross sections are available only down to 70keV. Consequently, for lower temperature environments (like for example our Sun with its 15.7MK core temperature) theoretical cross sections or extrapolation of the available data are necessary.
At low energies the $^{14}$N(p,$\gamma$)$^{15}$O reaction proceeds mostly through the direct capture mechanism with contribution from wide resonances. The total cross section is dominated by the capture to the $E_x$=6.79MeV excited states in $^{15}$O but the capture to the ground state and to the $E_x$=6.17MeV excited state also contributes significantly. At higher energies, on the other hand, where cross section data are available, transitions to other states as well as narrow and wide resonances play important roles.
The importance of the $E_p$=278keV resonance {#subsec:278kev_res}
--------------------------------------------
The extrapolation of the cross section to astrophysical energies is typically carried out using the R-matrix approach. For a reliable R-matrix extrapolation, high precision experimental data in a wide energy range is needed for all the relevant transitions and for the resonances as well as for the direct capture.
At $E$=259keV center-of-mass energy the $^{14}$N(p,$\gamma$)$^{15}$O reaction exhibits a strong narrow resonance. Its energy is too high for any direct astrophysical relevance, however, it plays an important role in the experiments targeting the $^{14}$N(p,$\gamma$)$^{15}$O cross section measurement. In direct kinematics this resonance is observed at $E_p$=278keV proton energy and often serves as a normalization point for the measured non-resonant cross section data, i.e. the cross section is measured relative to the strength of this resonance. Therefore, the precision of this resonance strength directly influences the precision of the cross section data at low energies.
The available measured $E_p$=278keV resonance strength values are summarized in Table\[tab:278strength\_literature\]. Based on several measurements, the recommended value of the strength has an uncertainty of 4.6%, as given by the latest compilation of solar fusion reactions [@RevModPhys.83.195]. In the paper of Daigle *et al.* [@PhysRevC.94.025803] published after the above cited compilation, a new result was presented and the literature data were also critically re-analyzed. Their recommended value is in agreement with that of [@RevModPhys.83.195] but its uncertainty is reduced to 2.4%. This uncertainty seems surprisingly low considering on one hand the stopping power uncertainty which is common to almost all the experiments and on the other hand the difficulty in characterizing the implanted targets used by Daigle *et al.* [@PhysRevC.94.025803].
In the present work the $E_p$=278keV resonance strength is measured using an independent method, the activation technique [@Gyurky2019]. As an additional result, the strength of the $E_p$=1058keV resonance in $^{14}$N(p,$\gamma$)$^{15}$O reaction is also measured. This resonance plays no role in astrophysics, but it can also be used as a reference point for the $^{14}$N(p,$\gamma$)$^{15}$O non-resonant cross section measurements and an R-matrix extrapolation of experimental data also requires the knowledge of the parameters of this resonance. Although this resonance is stronger than the $E_p$=278keV one, its strength has been measured in fewer experiments (see Table\[tab:1058strength\_literature\]) and the precision of the strength recommended by Marta *et al.* is not better than about 5%.
------------------------------------------------------------------ ------ -------------- ------ ------- -----
Reference Year Method
E.J. Woodbury [*et al.*]{} [@woodbury1949] 1949 activation
D.B. Duncan, J.E. Perry [@PhysRev.82.809] 1951 activation
S. Bashkin [*et al.*]{} [@PhysRev.99.107] 1955 prompt gamma 13 $\pm$ 3
D.F. Hebbard [*et al.*]{} [@HEBBARD1963666] 1963 prompt gamma 14 $\pm$ 2
H.W. Becker [*et al.*]{} [@Becker1982] 1982 prompt gamma 14 $\pm$ 1
R. C. Runkle [*et al.*]{} [@PhysRevLett.94.082503] 2005 prompt gamma 13.5 $\pm$ 1.2
G. Imbriani [*et al.*]{} [@Imbriani2005] 2005 prompt gamma 12.9 $\pm$ 0.9
D. Bemmerer [*et al.*]{} [@BEMMERER2006297] 2006 prompt gamma 12.8 $\pm$ 0.6
S. Daigle [*et al.*]{} [@PhysRevC.94.025803] 2016 prompt gamma 12.6 $\pm$ 0.6
recommended by E.G. Adelberger [*et al.*]{} [@RevModPhys.83.195] 2011 13.1 $\pm$ 0.6
recommended by S. Daigle [*et al.*]{} [@PhysRevC.94.025803] 2016 12.6 $\pm$ 0.3
------------------------------------------------------------------ ------ -------------- ------ ------- -----
------------------------------------------------------------ ------ -------------- ----- ------- ----
Reference Year Method
D.B. Duncan, J.E. Perry [@PhysRev.82.809] 1951 activation
D.F. Hebbard [*et al.*]{} [@HEBBARD1963666] 1963 prompt gamma
U. Schröder [*et al.*]{} [@SCHRODER1987240] 1987 prompt gamma 310 $\pm$ 40
M. Marta [*et al.*]{} [@PhysRevC.81.055807] 2010 prompt gamma 364 $\pm$ 21
recommended by M. Marta [*et al.*]{} [@PhysRevC.81.055807] 2010 353 $\pm$ 18
------------------------------------------------------------ ------ -------------- ----- ------- ----
The activation method for the study of the $^{14}$N(p,$\gamma$)$^{15}$O reaction {#subsec:actmethod}
--------------------------------------------------------------------------------
The proton capture of $^{14}$N at the $E_p$=278keV and $E_p$=1058keV resonances leads to the formation of $^{15}$O in excited states of $E_x$=7556keV and $E_x$=8284keV, respectively. These excited states decay to the $^{15}$O ground state by the emission of prompt $\gamma$-radiation through several possible cascades. The detection of this $\gamma$-radiation has been used in almost all the experiments to determine the resonance strengths (see the entries in tables \[tab:278strength\_literature\] and \[tab:1058strength\_literature\] labeled as ’prompt gamma’). For a precise resonance strength measurement the $\gamma$-detection efficiency (up to 8MeV $\gamma$-energy), the angular distribution of the various transitions as well as the branching ratio of these transitions (including the weak ones) must be known precisely. All these factors introduce systematic uncertainties in the resonance strength determination.
Since the reaction product of the $^{14}$N(p,$\gamma$)$^{15}$O reaction is radioactive, the resonance strength can also be measured by activation. $^{15}$O decays by positron emission to $^{15}$N with a half-life of 122.24$\pm$0.16s [@AJZENBERGSELOVE19911]. The decay is not followed by the emission of $\gamma$-radiation, however, the 511keV $\gamma$-ray following the positron annihilation provides a possibility for the reaction strength measurement with activation. This method is free from some uncertainties encumbering the prompt gamma experiments. The decay occurs isotropically, thus no angular distribution needs to be measured. The $\gamma$-detection efficiency must be known only at a single, low energy point (511keV) where it is measured more easily than at several MeV’s. Since by the activation method the number of produced isotopes is measured, this technique provides directly the total reaction cross section or resonance strength (independent from the decay scheme of the exited levels) and therefore no uncertainty arises from weak transitions.
The activation method was used only by some very first studies of the $^{14}$N(p,$\gamma$)$^{15}$O reaction about 70 years ago (see tables \[tab:278strength\_literature\] and \[tab:1058strength\_literature\]) and these measurements did not lead to precise resonance strengths. In the present work this method is used again to provide precise resonance strength values which are largely independent from the ones measured with prompt gamma detection. The next section provides details of the experimental procedure while the data analysis is presented in Sec.\[sec:analysis\]. The final results, their comparison with available data and conclusions are given in Sec.\[sec:results\].
Experimental procedure {#sec:experiment}
======================
Target preparation and characterization {#subsec:target}
---------------------------------------
In many of the past experiments solid state titanium-nitride (TiN) targets proved to be an excellent choice to carry out ion-beam induced reaction studies on nitrogen isotopes [@PhysRevC.97.015801; @PhysRevC.93.055806; @PhysRevC.81.055807; @2004PhLB..591...61F e.g.]. TiN can be produced at the required thickness and purity and these targets can withstand intense beam bombardment. The Ti:N atomic ratios are typically found to be very close to 1:1 in such targets.
Considering the advantages, solid state TiN targets were used also in the present work. They were prepared by reactive sputtering of TiN onto 0.5mm thick Ta backings at the Helmholtz-Zentrum Dresden-Rossendorf, Germany. The nominal thicknesses of the TiN layers were between 100 and 300nm, but for the purpose of the resonance strength measurements presented here, only 300nm thick targets were used. This corresponds to roughly 1.5$\times$10$^{18}$ N atoms/cm$^2$.
For the resonance strength determination, the total thickness of the targets does not play a role (as long as the thick target assumption holds, see Sec.\[sec:analysis\]). The stoichiometry, i.e. the Ti:N ratio, on the other hand, is a crucial parameter as discussed in Sec.\[sec:analysis\]. The Ti:N ratio and the amount of impurities were therefore measured with three independent methods: SNMS, RBS and PIXE, as described below. For these measurements, TiN layers sputtered onto Si wafers were used. These samples had been prepared together with the actual targets on Ta backings in the same sputtering geometry and therefore the layer compositions are the same.
Secondary Neutral Mass Spectrometry (SNMS) technique was used to measure the target composition as a function of the depth of the layer. The measurement was done with the INA-X type (SPECS GmbH, Berlin) SNMS facility of Atomki [@OECHSNER1993250; @vad2009]. Figure\[fig:SNMS\] shows a typical SNMS profile of a 300nm target. Besides the small amount of oxygen contamination on the surface, no elements other than nitrogen and titanium were observed. The Ti:N ratio was found to be uniform along the thickness of the target within the statistical fluctuation of the data and the average ratio is 1.015$\pm$0.051. The uncertainty includes the statistical component (less than 1%) and a 5% systematic uncertainty.
![\[fig:SNMS\] Concentration of the various elements in the target as a function of depth measured with the SNMS technique.](fig1_SNMS_spectrum.eps "fig:"){width="\columnwidth"}\
Rutherford Backscattering Spectrometry (RBS) was also used to measure the target composition. A 1.6MeV $\alpha$-beam provided by the 5MV Van de Graaff accelerator of Atomki was focused onto the targets in an Oxford type microbeam setup [@Huszank2016]. The scattered $\alpha$-particles were detected by two ion-implanted Si detectors positioned at 135 and 165 degrees with respect to the incoming beam direction. Figure\[fig:RBS\] shows a typical RBS spectrum. Based on the evaluation of the RBS spectra with the SIMNRA code [@SIMNRA], a Ti:N ratio of 0.976$\pm$0.048 was determined. The uncertainty includes the fit uncertainty and a 3% systematic one characterizing the general accuracy of the used RBS system which was assessed based on the measured thickness reproducibility of several RBS standards.
![\[fig:RBS\] A typical measured and simulated RBS spectrum. ](fig2_RBS_spectrum.eps "fig:"){width="\columnwidth"}\
Using the same microbeam setup as for the RBS measurement, the targets were also studied with Proton Induced X-ray Emission (PIXE) [@Huszank2016]. The targets were bombarded by a 2.0MeV proton beam and the induced X-rays were detected by a silicon drift X-ray detector. A typical X-ray spectrum can be seen in Fig.\[fig:PIXE\]. Owing to the thin window of the detector, characteristic X-rays of nitrogen could be detected with good accuracy and a Ti:N ratio of 0.981$\pm$0.064 was obtained. Here the uncertainty include a 3% systematic component.
![\[fig:PIXE\] PIXE spectrum of a TiN target. Major elements included in the fit are labeled.](fig3_PIXE_spectrum.eps "fig:"){width="\columnwidth"}\
Table\[tab:targets\] summarizes the results of the target stoichiometry measurements. All three results are in excellent agreement with the expected 1:1 ratio. Based on the weighted average, 0.991$\pm$0.031 is adopted as the Ti:N ratio for the resonance strength calculations.
Method
--------- ------- ------- -------
SNMS 1.015 $\pm$ 0.051
RBS 0.976 $\pm$ 0.048
PIXE 0.981 $\pm$ 0.064
adopted 0.991 $\pm$ 0.031
: \[tab:targets\] Measured Ti:N ratios of the used targets. The adopted ratio is the weighted average of the results of the three methods.
Activations {#subsec:activations}
-----------
The proton beams for the excitation of the studied resonances in the $^{14}$N(p,$\gamma$)$^{15}$O reaction were provided by the Tandetron accelerator of Atomki. The energy calibration of the accelerator has been carried out recently [@RAJTA2018125] and that was used for setting the energies for the resonance studies. As the resonances are relatively strong, no high beam intensity was necessary which was useful to avoid target deterioration. The typical beam intensity was 5$\mu$A on target.
![\[fig:chamber\] Drawing of the target chamber used for the activations](fig4_chamber.eps "fig:"){width="\columnwidth"}\
The applied beam energies for the study of the 278keV and 1058keV resonances were $E_p$=300keV and $E_p$=1070keV, respectively. These values correspond to the middle of the yield curve plateau, where the maximum yield can be reached. See the discussion in Sec.\[sec:analysis\].
The schematic drawing of the target chamber can be seen in Fig.\[fig:chamber\]. The beam enters the chamber through a water cooled collimator of 5mm in diameter. Behind the collimator, an electrode biased at -300V is placed to suppress secondary electrons emitted from the target or from the collimator. After the collimator the whole chamber serves as a Faraday-cup to measure the charge carried by the beam to the target. The measured charge was used to determine the number of protons impinging on the target.
Detection of the annihilation radiation {#subsec:detection}
---------------------------------------
As the half-life of the $^{15}$O reaction product is rather short (about two minutes), the induced activity was measured without removing the target from the activation chamber. A 100% relative efficiency HPGe $\gamma$-detector was therefore placed close behind the target. The distance between the target and the detector end-cap was about 1cm.
In order to increase the number of detected decay, the cyclic activation method was applied. The target was irradiated for 5 minutes and then the beam was stopped in the low energy Faraday cup of the accelerator and the decay was measured for 10 or 20 minutes. This cycle was repeated many times (up to 30 cycles in a singe irradiation campaign).
In order to follow the decay of $^{15}$O, the number of events in the region of 511 keV peak (selected by gating with a single channel analyzer) was recorded in five second time intervals using an ADC in multichannel scaling mode. Figure\[fig:decay\] shows typical examples of the recorded number of counts as a function of time. The upper panel shows a case measured on the $E_p$=278keV resonance with 10 minute counting intervals, while the lower panel represents a measurement on the $E_p$=1058keV resonance with 20 minute counting intervals. During the 5 minutes irradiation intervals the events in the detector were disregarded as in these periods the counts were dominated by beam induced background.
![\[fig:decay\] Number of events detected by the HPGe detector at the 511keV peak region as a function of time using 5 second time bins. The fit to the data including a time-independent laboratory background component and using the known half-life of $^{15}$O is also shown. The upper and lower panels show the $E_p$=278keV and $E_p$=1058keV measurements, respectively. For the latter case the fit residuals are also plotted in order to indicate that the decay of $^{15}$O alone fits well the measured data, no other radioactivity was present in significant amount. This is also confirmed by the reduced $\chi^2$ value of the fit being very close to unity.](fig5a_278keV.eps "fig:"){width="\columnwidth"}\
![\[fig:decay\] Number of events detected by the HPGe detector at the 511keV peak region as a function of time using 5 second time bins. The fit to the data including a time-independent laboratory background component and using the known half-life of $^{15}$O is also shown. The upper and lower panels show the $E_p$=278keV and $E_p$=1058keV measurements, respectively. For the latter case the fit residuals are also plotted in order to indicate that the decay of $^{15}$O alone fits well the measured data, no other radioactivity was present in significant amount. This is also confirmed by the reduced $\chi^2$ value of the fit being very close to unity.](fig5b_1058keV.eps "fig:"){width="\columnwidth"}
Determination of the detector efficiency {#subsec:efficiency}
----------------------------------------
For the absolute measurement of the resonance strengths the absolute detection efficiency of the HPGe detector must be known in the counting geometry used. In the present case of a positron decaying isotope, the positron annihilation does not occur in a point-like geometry and hence the precise determination of the detection efficiency is not trivial.
The positrons leave the decaying $^{15}$O nucleus with typically several hundreds of keV energy (the positron end-point energy is 1732keV [@AJZENBERGSELOVE19911]). The positrons which travel towards the target backing stop within a few 100$\mu$m (thus well inside the backing) and annihilate therefore in a quasi point-like geometry. On the other hand, those positrons which leave towards the other direction, will move into the vacuum chamber and travel freely until they hit the walls of the chamber. Therefore, their annihilation takes place in an extended and not well defined geometry.
In such a situation the direct efficiency measurement with calibrated radioactive sources is not possible. Instead, an indirect method using the following procedure was applied. As a first step, longer-lived positron emitters were produced in the activation chamber. For this purpose $^{18}$F (t$_{1/2}$=109.77$\pm$0.05min, produced by the $^{18}$O(p,n)$^{18}$F reaction) and $^{13}$N (t$_{1/2}$=9.965$\pm$0.004min, produced by the $^{12}$C(p,$\gamma$)$^{13}$N reaction) were chosen. The decay of these sources was measured with the HPGe detector standing next to the chamber (the one which was used for the $^{14}$N(p,$\gamma$)$^{15}$O reaction) for typically 1-2 half-lives. This measurement gives information about the efficiency in the non-trivial extended geometry. Then the sources were removed from the chamber, transferred to another HPGe detector (used in many recent experiments and characterized precisely, see e.g. [@PhysRevC.95.035805]) where the decay was followed for several half-lives. At this detector the sources were placed in a position which guaranteed the point-like geometry, i.e. the sources were placed between 0.5mm thick Ta sheets which stopped the positrons completely. The absolute efficiency of this HPGe detector in the used geometry was measured with calibrated radioactive sources to a precision of 3%. The measurement with the second detector provided the absolute activity of the sources and – knowing precisely the half-lives and the elapsed time between the two countings – the absolute efficiency of the first detector could be obtained. The efficiency obtained with $^{18}$F and $^{13}$N sources were in agreement within the statistical uncertainties of 0.7% and 1.5%, respectively.
In addition to the measurements with $^{18}$F and $^{13}$N, the same procedure was followed also with the actual $^{15}$O isotope produced by the $^{14}$N(p,$\gamma$)$^{15}$O reaction. Here the short half-life resulted in a higher statistical uncertainty of 2.5%, but the obtained efficiency was in agreement with the results from $^{18}$F and $^{13}$N. Based on these measurements the final efficiency is determined with a precision of 4%.
Data analysis {#sec:analysis}
=============
As it can be seen by the red line in Fig.\[fig:decay\], the 511keV count rate can be well fitted with a constant background plus an exponential decay with the $^{15}$O half-life. With the known detection efficiency and the decay parameters of $^{15}$O the only free parameter of the fit is the yield of the reaction (i.e. the number of reactions per incident proton) which can be related to the resonance strength.
With the thick target assumption (i.e. when the energetic thickness of the target $\Delta E$ is much larger than the natural width $\Gamma$ of the resonance), the resonance strength $\omega\gamma$ can be related to the yield $Y$ measured on the top of the resonance by the following formula: $$\omega\gamma=\frac{2 \epsilon_{\rm eff} Y}{\lambda^2}$$ where $\lambda$ is the de Broglie wavelength at the resonance energy in the center of mass system and $\epsilon_{\rm eff}$ is the effective stopping power. If the thick target condition is not met, the maximum yield $Y_{max}$ measured in the middle of the resonance curve plateau can be related to the ideal thick target yield leading to the following correction factor [@RevModPhys.20.236]: $$f \equiv \frac{Y_{max}}{Y} = \frac{2}{\pi} \tan^{-1} \frac{\Delta E}{\Gamma}.$$ Based on the target characterizations presented in Sec.\[subsec:target\], the energetic thickness of the targets used for the present experiments at the two studied resonances was $\Delta E_{278}$=51.9keV and $\Delta E_{1058}$=24.4keV, respectively with about 5% uncertainty. The natural widths of the two resonances (taken from the literature) are $\Gamma_{278}$=1.12$\pm$0.03keV [@doi:10.1063/1.3087064] and $\Gamma_{1058}$=3.8$\pm$0.5keV [@PhysRevC.81.055807]. These lead to correction factors of $f_{278}$=0.986$\pm$0.001 and $f_{1058}$=0.902$\pm$0.015. Here the uncertainties take into account the resonance widths, the target thickness and the beam energy uncertainties. The latter one measures how precisely the maximum of the resonance curve is found. It is evident that in the case of the astrophysically more important low energy resonance this correction is very close to unity and its uncertainty is negligible. This is owing to the larger target thickness at this low energy and the small natural width of the resonance.
The effective stopping power $\epsilon_{\rm eff}$ in the case of a target composed of Ti and N can be obtained as $$\epsilon_{\rm eff}=\epsilon_{\rm N} + \frac{N_{\rm Ti}}{N_{\rm N}}\epsilon_{\rm Ti}$$ where $\epsilon_{\rm N}$ and $\epsilon_{\rm Ti}$ are the stopping powers of N and Ti, respectively, taken at the resonance energy and ${N_{\rm Ti}}/{N_{\rm N}}$ is the Ti:N atomic ratio as discussed in Sec.\[subsec:target\].
The stopping power was taken from the 2013 version of the SRIM code [@SRIM]. The following values were used:
- $\epsilon_{\rm N}$(278 keV)=10.72eV/(10$^{15}$ atoms/cm$^2$),
- $\epsilon_{\rm N}$(1058 keV)=4.733eV/(10$^{15}$ atoms/cm$^2$),
- $\epsilon_{\rm Ti}$(278 keV)=22.81eV/(10$^{15}$ atoms/cm$^2$),
- $\epsilon_{\rm Ti}$(1058 keV)=10.80eV/(10$^{15}$ atoms/cm$^2$).
As for Ti, the stopping power was measured by N. Sakamoto *et al.* [@SAKAMOTO2000250] with very good precision of better than 1%. The measured values are in excellent agreement with the SRIM data, never deviating more than 2%, Therefore, an uncertainty of 2% is assigned to $\epsilon_{\rm Ti}$ in the present work. The situation is somewhat worse in the case of nitrogen. The stopping power measured with gaseous N can be different from the solid form (see e.g. [@REITER1987287] for the stopping power dependence on the chemical form). Therefore, 4% uncertainty is assigned to $\epsilon_{\rm N}$ as recommended by SRIM. The uncertainty of the effective stopping power was calculated taking into account the uncertainty of the measured Ti:N ratio and considering the $\epsilon_{\rm Ti}$ and $\epsilon_{\rm N}$ values uncorrelated. The isotopic abundance of $^{14}$N in natural nitrogen (99.6337%) was taken into account in the calculation of $\epsilon_{\rm eff}$.
For the determination of the resonance strength, the non-resonant component of the reaction yield must be subtracted from the resonant yield. In the case of the $E_p$=1058keV resonance the yields below and above the resonance – at $E_p$=1000keV and $E_p$=1150keV – were measured. Based on these measurements a 2.7% nonresonant contribution to the resonant yield was determined. This non-resonant yield was subtracted from the resonant yield and a conservative relative uncertainty of 20% was assigned to it, leading to a 0.6% uncertainty of the determined resonance strength.
In the case of the $E_p$=278keV resonance the off-resonant reaction yield was below the detection limit. Based on some recent experiments, a cross section of about 1.5$\times10^{-8}$ barns can be expected at this energy [@PhysRevC.93.055806; @PhysRevC.97.015801]. Such a cross section leads to a calculated non-resonant yield which is 0.3% of the resonant yield. This tiny contribution is subtracted from the yield and - as this value is not based on our own measurement - a 100% relative uncertainty is assigned to it.
Table\[tab:uncert\] lists the uncertainties of the final resonance strength values. As the studied resonances are relatively strong and the cyclic activations were carried out many times, the statistical uncertainty of the $\gamma$-counting is very low compared to the other sources of uncertainty. The quoted total uncertainty is the quadratic sum of the components. Other uncertainties (like for example the uncertainties of the $^{15}$O decay parameters) are well below 1% and are therefore neglected.
------------------------------------ -------- ---------
Source 278keV 1058keV
counting statistics 1.0% 0.7%
effective stopping power
HPGe detector efficiency
current integration
finite target thickness correction 0.1% 1.7%
non-resonant yield subtraction 0.3% 0.6%
total uncertainty 5.8% 6.0%
------------------------------------ -------- ---------
: \[tab:uncert\] Components of the resonance strength uncertainties
Results and conclusions {#sec:results}
=======================
The obtained strengths of the two studied resonances are the following:
- $\omega\gamma_{278}$=(13.4$\pm$0.8)meV,
- $\omega\gamma_{1058}$=(442$\pm$27)meV.
If we take into account the total uncertainties, the new result for the $E_p$=278keV resonance strength is in good agreement with the adopted values recommended by the Solar Fusion II compilation [@RevModPhys.83.195] as well as by the more recent work of S. Daigle *et al.* [@PhysRevC.94.025803] (see Table\[tab:278strength\_literature\]). We do not quote here a new recommended value, we just note that considering our new value determined with an independent technique, the strength recommended by the Solar Fusion II compilation [@RevModPhys.83.195] and especially its somewhat higher uncertainty seems more appropriate than the value of S. Daigle *et al.* [@PhysRevC.94.025803] with its very small error bar. The results of those experiments where the $E_p$=278keV resonance is used as a normalization point, do not change by the present result. Therefore, the astrophysical consequences are also unchanged.
The strength of the $E_p$=1058keV resonance, on the other hand, was measured to be significantly higher than the ones determined in the two most recent works (see Table\[tab:1058strength\_literature\]). One reason can be that the finite target thickness correction might not have been done in those experiments. In the case of U. Schröder [*et al.*]{} [@SCHRODER1987240] there is no information about this in the paper. In the case of M. Marta [*et al.*]{} [@PhysRevC.81.055807] it is confirmed that no such a correction has been applied [@bemmerer_private]. Based on the information available in [@PhysRevC.81.055807] and [@marta_thesis], the correction should be about 7%. This would lead to a resonance strength of $\omega\gamma_{1058}$=(389$\pm$22)meV. This value still differs from the present one by about two standard deviations [^1]. Consequently, as opposed to the $E_p$=278keV resonance, this strength value is rather uncertain and further measurements would be required.
As a summary, the activation technique was successfully used in the present work for the $^{14}$N(p,$\gamma$)$^{15}$O reaction and precise resonance strength values were derived. This technique can also be applied for the measurement of the non-resonant $^{14}$N(p,$\gamma$)$^{15}$O cross section. Such an experiment is in progress using the setup introduced here. The results will be presented in a forthcoming publication. As the activation method provides data which are complementary to the prompt gamma data, the combination of the results can lead to more precise cross section of the $^{14}$N(p,$\gamma$)$^{15}$O astrophysical key reaction.
This work was supported by NKFIH grants K120666 and NN128072, by the ÚNKP-18-4-DE-449 New National Excellence Program of the Human Capacities of Hungary and by the COST Association (ChETEC, CA16117). G.G. Kiss acknowledges support form the János Bolyai research fellowship of the Hungarian Academy of Sciences. The authors thank I. Rajta, I. Vajda, G. Soltész and Zs. Szűcs for providing excellent beams and working conditions at the Tandetron accelerator.
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, .
, Ph.D. thesis, ().
[^1]: The subtraction of the non-resonant component in [@PhysRevC.81.055807] may also be overestimated. According to [@marta_thesis], the off-resonance measurement below the resonance was done at 1049keV, where the contribution from the low energy tail of the resonance can still be about 5%
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{
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, , , and
We investigate the capability of the [*UBVRIJHK*]{} photometric system to quantify star clusters in terms of age, metallicity and color excess by their integrated photometry in the framework of PÉGASE single stellar population (SSP) models. The age-metallicity-extinction degeneracy was analyzed for various parameter combinations, assuming different levels of photometric accuracy. We conclude, that most of the parameter degeneracies, typical to the [*UBVRI*]{} photometric system, are broken in the case when the photometry data are supplemented with at least one infrared magnitude of the [*JHK*]{} passbands, with an accuracy better than $\sim$0.05 mag. The presented analysis with no preassumptions on the distribution of photometric errors of star cluster models, provides estimate of the intrinsic capability of any photometric system to determine star cluster parameters from integrated photometry.
techniques: photometric – methods: data analysis – galaxies: star clusters
Broad-band photometric data of extragalactic star clusters are applicable to derive their evolutionary parameters via comparison with single stellar population (SSP) models. However, this procedure is restricted by the fact that in some parameter domains strong age-metallicity-extinction degeneracies (see, e.g., Worthey 1994) remain unsolved. For a study of broad-band color indices, most sensitive to various star and cluster parameters, see, e.g., Jordi et al. (2006) and Li et al. (2007).
It is known (Anders et al. 2004; Kaviraj et al. 2007) that infrared and ultraviolet passbands are helpful for breaking the age-metallicity-extinction degeneracies. In this respect infrared observations have the priority: in some atmospheric windows they are accessible for ground-based telescopes, while ultraviolet photometry shorter than 300 nm is possible only from space. In some extragalactic studies of star clusters published to date various combinations of optical and/or near-infrared passbands have been used (e.g., Kodaira et al. 2004; Hempel & Kissler-Patig 2004; Fan et al. 2006; Hempel et al. 2007; Narbutis et al. 2008; Pessev et al. 2008).
However, in comparison to photometry in the optical range, ground-based infrared photometry usually has larger errors due to variable humidity, and this requires a careful calibration (e.g., Kodaira et al. 1999; Kidger et al. 2006). The reliability of secondary standards used for photometric calibration of wide field images, should be verified by several independent sources, if they are available (e.g., Narbutis, Stonkutė & Vansevičius 2006).
In the present paper we continue to investigate a possibility of determining star cluster parameters (age, metallicity and color excess) by comparison of their integrated color indices with the SSP models computed with the PÉGASE (v. 2.0; Fioc & Rocca-Volmerange 1997) code package. In our previous study (Narbutis et al. 2007b, hereafter Paper I) it was found that the [*UBVRI*]{} system enables us to estimate cluster parameters over a wide range of their values, when the overall accuracy of color indices is better than $\sim$0.03 mag. In the following we discuss how the adding of the [*JHK*]{} passbands affects the accuracy of cluster parameter (age, metallicity and color excess) determination. We analyze degeneracies at various accuracy levels of photometry for the same values of cluster parameters as in Paper I.
In the similar study Anders et al. (2004) have used the so-called AnalySED method for determining cluster parameters with a different approach to the degeneracy problem. In Section 2 we discuss the main differences between the Anders et al. and our methods and note, that the parameter degeneracy analysis presented in this study is based on minimum assumptions.
The analysis method used in this study is similar to that of Paper I. The SSP models were computed with the PÉGASE program package, applying its default options and the universal initial mass function (UIMF; Kroupa 2001). The integrated color indices in all pass-bands in respect to the $V$-band of SSP models were reddened by taking into account the dependence of color-excess ratio (e.g., $E_{U-B}/E_{B-V}$) on color index $B$–$V$ of SSP model and assumed color excess, $E_{B-V}$, applying the standard extinction law (Cardelli et al. 1989). The three-parameter (3-D) SSP model grid of $\sim$$5\times10^{5}$ models was constructed at the following nodes: (i) 76 age, $t$, values from 1Myr to 20Gyr with a constant step of $\log\kern1pt(t/{\rm Myr})
= 0.05$;[^1] (ii) 31 metallicity[^2], \[M/H\], values from $-2.3$ to $+0.7$, with a step of 0.1dex; (iii) 201 color excess, $E_{B-V}$, values from 0.0 to 2.0, with a step of 0.01.
The procedure of determination of cluster parameters ($t$, \[M/H\], $E_{B-V}$) was implemented as a C++ code in the data analysis and the graphing software package ‘Origin’ (OriginLab Corporation). It is based on a similar technique developed for star quantification by Vansevičius & Bridžius (1994), i.e., the comparison of the observed color indices of a star cluster with color indices of the SSP from the model grid. For this purpose we use the quantification quality criterion, $\delta$, calculated by the formula: $${\delta=\sqrt{\sum_{}^{}w_{i}(CI_{i}^{\rm obs}-CI_{i}^{\rm mod})^{2}\over\sum_{}^{}w_{i}}}\,,$$ where $CI_{i}^{\rm obs}$ stands for the “observed” color indices $U$–$V$, $B$–$V$, $V$–$R$, $V$–$I$, $V$–$J$, $V$–$H$ and $V$–$K$; $CI_{i}^{\rm mod}$ – the corresponding color indices of the SSP models from the grid; $w _{i}$ – the weights for the observed color indices.
We investigate the possibility to determine cluster parameters using the [*UBVRIJHK*]{} photometric system for 54 models taken from the SSP model grid as “observed” objects with the following parameters: $t=0.02$, 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10 Gyr, \[M/H\] = 0.0, $-0.7$, $-1.7$ and $E_{B-V}=0.1$, 1.0. Their color indices are denoted by $CI_{i}^{\rm obs}$, see Eq. 1. We have assigned the weights of $w_{i}
= 1$ for $U-V$, $B-V$, $V-R$ and $V-I$ color indices, and $w_{i} =
1/9$ for $V$–$J$, $V$–$H$ and $V$–$K$ of the “observed” star clusters. Such weighting scheme assumes that the accuracy of the infrared [*JHK*]{} observations is $\sim$3 times lower than that of the optical [*UBVRI*]{} – a typical condition in photometry of extragalactic star clusters.
Anders et al. (2004) have extensively analyzed the possibilities of their AnalySED method, which has a different approach to the degeneracy problem, comparing to the method presented in this paper. First, AnalySED is based on the comparison of observed magnitudes with the model magnitudes, while for the same aim we are using color indices. This allows us to avoid fitting of an additional free parameter – the cluster mass. Second, Anders et al. (2004) define the “observed” magnitudes of model clusters by assigning to them random errors. However, we did not apply “observation errors” to color indices, since it is reasonable to assume that the quantification quality criterion, $\delta$, reflects the effect of photometric accuracy sufficiently well. In general, $\delta$ represents the lowest possible average photometric error of all color indices, which in the case of real clusters can be larger, mostly due to the influence of crowding by contaminating field stars (e.g., Narbutis et al. 2007a). Therefore, the parameter degeneracy analysis presented in this study is based on minimum assumptions.
We have used several $\delta_{\rm max}$ values as the upper threshold levels of photometric errors. The cluster parameters were determined independently at each accuracy level by averaging the corresponding parameter values of SSP models at the grid nodes, which have $\delta
\leq \delta_{\rm max}$. The weights for the calculation of the parameter averages and the standard deviations were assigned 1 and $10^{-4}/\delta^{2}$ for the nodes with $\delta\leq0.01$ and $>$0.01 mag, respectively.
Furthermore, for the parameter averaging we used only the nodes, which reside in a single continuous ‘island’ around the true cluster position in the 3-D parameter space of the SSP model grid. Boundaries of ‘islands’ at each $\delta_{\rm max}$ level were determined automatically by the [*clustering procedure*]{}, which finds discontinuities in the parameter space, starting from the true position of the cluster. Such a procedure excludes nodes, which are located in the secondary $\delta$ minima, arising due to the age-metallicity-extinction degeneracies in the 3-D parameter space.
The results of parameter determinations of clusters are provided in Figures 1 and 2 for color excess values $E_{B-V}=0.1$ and 1.0, respectively. In each panel the differences of parameters (determined minus true) are shown in groups of five filled circles, the middle circle is positioned at the true age of the cluster, i.e., the circles indicate parameters determined at $\delta_{\rm max}$ threshold values from 0.01 to 0.05 mag, with a step of 0.01 mag, plotted from left to right. Open and black circles correspond to the [*UBVRI*]{} and [*UBVRIJHK*]{} passband set cases, respectively. Error-bars indicate standard deviations of the determined parameters and characterize the integrated ‘size’ of the $\delta\leq\delta_{\rm max}$ ‘island’ in the corresponding parameter space.
It is clearly visible in both Figures 1 and 2, that additional usage of [*JHK*]{} passbands significantly improves the precision of cluster parameter determination with respect to the [*UBVRI*]{} passbands alone. Especially good results are in the case of the small color excess, $E_{B-V}=0.1$, value. Note, that strong age-metallicity degeneracies at ages $t=0.1$, 0.2 and 10 Gyr, which are present in the [*UBVRI*]{} case, completely disappear in the [*UBVRIJHK*]{} case. A similar, but a slightly weaker effect is also visible for models of ages $t=2$ and 5 Gyr. However, if objects are highly reddened, $E_{B-V}=1.0$, there are still notable parameter degeneracies for some cases. The strongest age-extinction degeneracy is for clusters with: $t=1$ Gyr and \[M/H\] = 0.0; $t=2$ Gyr and \[M/H\] = $-0.7$; $t=5$ Gyr and \[M/H\] = $-1.7$. Actually, the quantification results of the last case are much more accurate, when SSP models older than 15 Gyr are not included in the calculation of the average.
The impact of adding the [*JHK*]{} passbands to the [*UBVRI*]{} system on the accuracy of the quantification of cluster parameters is illustrated in Figures 3–10 for the clusters of ages $t=0.2$, 1, 2, 5, 10 Gyr for solar metallicity, and $t=0.5$ Gyr for \[M/H\] = $-1.7$. The figures display cluster parameter quantification maps at the threshold levels of $\delta_{\rm max}=0.01$, 0.03 and 0.05 mag.
In Figures 3 and 5 for the [*UBVRI*]{} set, the left and the central panels look very similar, although the left panel displays quantification map of a cluster with $t=200$ Myr and \[M/H\] = 0.0, while the central panel – for a cluster with $t=500$ Myr and \[M/H\] = $-1.7$. This means that due to age-metallicity degeneracy for both clusters we obtain similar and wrong parameters. However, the situation changes, when the [*JHK*]{} magnitudes are added. Figures 4 and 6 display quantification maps of the same clusters, but for the [*UBVRIJHK*]{} passband set. Note, that degeneracies disappear, even at $\delta_{\rm
max}=0.05$ mag level.
An interesting example is the cluster with the parameters $t=2$ Gyr and \[M/H\] = 0.0. Figures 1 and 2 show that the addition of the [*JHK*]{} magnitudes improves the age determination, but $E_{B-V}$ determination becomes of lower accuracy, comparing to the [*UBVRI*]{} set alone. Rightmost panels in Figures 3–6 clarify this situation. In the [*UBVRI*]{} case, strong age-metallicity degeneracy takes place: note the additional ‘tail’ of suitable SSP models in the $E_{B-V}$ vs. $t$ plane towards older ages and lower $E_{B-V}$ values. This helps to compensate the influence of SSP models at higher $E_{B-V}$ values, and the quantification produces a ‘correct’ reddening. In the case of [*UBVRIJHK*]{}, no such ‘tail’ is seen, therefore the determined $E_{B-V}$ value is shifted from its true position.
Figures 7–10 display three similar cases of parameter degeneracies for clusters of solar metallicity and ages $t=1$, 5 and 10 Gyr. Only the 10 Gyr age sample is sensitive enough to the addition of the [*JHK*]{} passbands, when parameter degeneracies become broken even at $\delta_{\rm max}=0.05$ mag level. On the contrary, for the ages of 1 Gyr (except for the case of $E_{B-V}=0.1$; see Figures 7 and 8) and 5 Gyr, parameter degeneracies exist even for $\delta_{\rm
max}=0.01$ and 0.03 mag, respectively. This implies, that the overall accuracy of cluster color indices must be better than 0.03 mag, which is rather difficult to achieve in infrared photometry.
We also investigated separately the influence of each infrared passband to the parameter determination precision with the purpose to find a minimum passband set, which would be sufficient for the quantification of clusters. The result is that all three color indices ($V$–$J$, $V$–$H$, $V$–$K$), added to the [*UBVRI*]{} photometric system separately, reduce the quantification errors, but $V$–$K$ is most important. Figure 11 displays the quantification results for clusters using the [*UBVRIJHK*]{} (filled circles) and the [*UBVRIK*]{} (open circles) passband sets, both pictures are quite similar. Thus the [*K*]{} passband helps to solve majority of the parameter degeneracy problems. The error-bars in the [*UBVRIK*]{} case show the sensitivity of the parameters to the change of $V$–$K$ by $\pm$0.03 mag.
Recently an updated version of stellar isochrones, compared to those used in PÉGASE, has been released by the Padova group, which includes a more accurate treatment of thermally pulsating AGB stars (Marigo et al. 2008). It has been shown that the interpretation of integrated colors of unresolved galaxies, by applying galaxy populations synthesis method and the new isochrone set, can be significantly altered due to improved models of AGB stars (e.g., Tonini et al. 2008). In the case of single stellar population analysis of star clusters one can expect similar changes. Our method of the parameter degeneracy analysis, which is much simpler than the methods described in the literature (see, e.g., de Grijs et al. 2005), can be used to study the effects AGB stars on cluster colors by intercomparing parameter degeneracies in various SSP model frameworks.
We conclude that additional photometric information from the [*JHK*]{} passbands can significantly improve the accuracy of the determination of cluster parameters based on the PÉGASE SSP models, in comparison with the results when photometry only the [*UBVRI*]{} passbands is available. Even one additional [*K*]{} passband can improve significantly the capability of the [*UBVRI*]{} photometric system to eliminate age-metallicity and age-extinction degeneracies in the majority of the investigated cluster models, when the overall accuracy of color indices is better than $\sim$0.05 mag.
Note, however, that this condition of photometric accuracy is broken for young low-mass star clusters, where the stochastic effects, arising due to a few bright stars, dominate (e.g., Cerviño & Luridiana 2004; Deveikis et al. 2008). This implies, that even in case of ideal photometric calibrations, the uncertainty of cluster colors due to stochastic effects limits the applicability of SSP model fitting to derive evolutionary parameters of low-mass star clusters.
[^3]
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[^1]: For the ages $<$12 Myr the step was equal to 1Myr. The SSP models for the ages larger than the oldest known globular clusters are used to avoid marginal effects.
[^2]: \[M/H\] was computed applying the approximation ${\rm [M/H]}=\log\kern1pt(Z/Z_{\sun})$.
[^3]: We thank the anonymous referee for critical comments and numerous suggestions, which helped to improve the paper. This work was financially supported in part by a Grant of the Lithuanian State Science and Studies Foundation.
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---
abstract: 'We study synchronization in the context of network traffic on a $2-d$ communication network with local clustering and geographic separations. The network consists of nodes and randomly distributed hubs where the top five hubs ranked according to their coefficient of betweenness centrality (CBC) are connected by random assortative and gradient mechanisms. For multiple message traffic, messages can trap at the high CBC hubs, and congestion can build up on the network with long queues at the congested hubs. The queue lengths are seen to synchronize in the congested phase. Both complete and phase synchronization is seen, between pairs of hubs. In the decongested phase, the pairs start clearing, and synchronization is lost. A cascading master-slave relation is seen between the hubs, with the slower hubs (which are slow to decongest) driving the faster ones. These are usually the hubs of high CBC. Similar results are seen for traffic of constant density. Total synchronization between the hubs of high CBC is also seen in the congested regime. Similar behavior is seen for traffic on a network constructed using the Waxman random topology generator. We also demonstrate the existence of phase synchronization in real Internet traffic data.'
author:
- Satyam Mukherjee
- Neelima Gupte
title: Queue Length Synchronization in a Communication Network
---
Introduction
============
The phenomenon of synchronization has been studied in contexts ranging from the synchronization of clocks and the flashing of fire-flies [@synch] to synchronization in oscillator networks [@carol] and in complex networks [@kurths]. Synchronized states have been seen in the context of traffic flows as well [@kerner], and investigations of traffic flow on substrates of various geometries have been the focus of recent research interest [@tadic; @wang; @Jiang; @moreno]. The synchronization of processes at the nodes, or hubs, of complex networks can have serious consequences for the performance of the network [@pipa]. In the case of communication networks, the performance of the networks is assessed in terms of their efficiency at packet delivery. Such networks can show a congestion-decongestion transition [@congest]. We note that an intimate connection between congestion and synchronization effects has been seen in the case of real networks [@TCP; @huang].
The aim of this paper is to study the interplay of congestion and synchronization effects on each other, and examine their effect on the efficiency of the network for packet delivery in the context of two model networks based on two dimensional grids. The first network consists of nodes and hubs, with the hubs being connected by random assortative or gradient connections[@BrajNeel]. In the case of the second network, in addition to nearest neighbour connections between nodes, the nodes are connected probabilistically to other nodes, with the probability of a connection between nodes being dependent on the Euclidean distance between them[@waxmangraph]. Such networks are called Waxman networks and are popular models of internet topology[@lakhina]. Synchronisation effects are observed in the congested phase of both these model networks. In addition to these two networks, we also discuss synchronisation effects seen in actual internet data.
We first study synchronization behavior in a two dimensional communication network of nodes and hubs. Such networks have been considered earlier in the context of search algorithms [@kleinberg] and of network traffic with routers and hosts [@Ohira; @Sole2; @fuks]. Despite the regular $2-d$ geometry such models have shown log-normal distribution in latency times as seen in Internet dynamics [@sole1]. The lattice consists of two types of nodes, the regular or ordinary nodes, which are connected to each of their nearest neighbors, and the hubs, which are connected to all the nodes in a given area of influence, and are randomly distributed in the lattice. Thus, the network represents a model with local clustering and geographical separations [@warren; @cohen]. Congestion effects are seen on this network when a large number of messages travel between multiple sources and targets due to various factors like capacity, band-width and network topology [@Huang]. Decongestion strategies, which involve the manipulation of factors like capacity and connectivity have been set up for these networks. Effective connectivity strategies have focused on setting up random assortative[@braj1], or gradient connections[@sat] between hubs of high betweenness centrality.
We introduce the ideas of phase synchronization and complete synchronization in the context of the queue lengths at the hubs. The queue at a given hub is defined to be the number of messages which have the hub as a temporary target. During multiple message transfer, when many messages run simultaneously on the lattice, the network tends to congest when the number of messages exceed a certain critical number, and the queue lengths tend to build up at hubs which see heavy traffic. The hubs which see heavy traffic are ranked by the co-efficient of betweenness centrality ($CBC$), which is the fraction of messages which pass through a given hub. We focus on the top five hubs ranked by CBC. Phase synchronization is seen between pairs of hubs of comparable betweenness centrality. The hub which is slowest to decongest (generally the hub of highest CBC) drives the slower hubs with a cascading master-slave effect in the hub hierarchy. When the network starts decongesting, the queue lengths decrease, and synchronization is lost. These results are reflected in the global synchronization parameter. When decongestion strategies which set up random assortative, or gradient, connections between hubs are implemented, complete synchronization is seen between some pairs of these hubs in the congested phase, and phase synchronization is seen between others. We demonstrate our results in the context of the gradient decongestion strategy, but the results remain unaltered for decongestion strategies based on random assortative connections. Similar results are seen for constant density traffic where a fixed number of messages are fed on the system at regular intervals. Total synchronization is also seen in the queue lengths of the hubs of high CBC.
All the results obtained for the first model are observed for message transport on Waxman topology network, where again synchronisation of hubs of high CBC is observed in the congested state. We demonstrate these results. Finally we study internet traffic data and demonstrate that phase synchronisation is seen in this data as well. Intermittent phase synchronization is also seen in this data.
A communication network with local clustering and geographic separation
=========================================================================
We first study traffic congestion for a model network with local clustering developed in Ref.[@BrajNeel]. This network consists of a two[-]{}dimensional lattice with ordinary nodes and hubs (See Fig. \[fig:asgr\]). Each ordinary node is connected to its nearest-neighbors, whereas the hubs are connected to all nodes within a given area of influence defined as a square of side $2k$ centered around the hub[@BrajNeel]. The hubs are randomly distributed on the lattice such that no two hubs are separated by less than a minimum distance, $d_{min}$. Constituent nodes in the overlap areas of hubs acquire connections to all the hubs whose influence areas overlap. The source S$(is,js)$ and target T$(it,jt)$ are chosen from the lattice and separated by a fixed distance $D_{st}$ which is defined by the Manhattan distance $D_{st}$ = $|is-it|$ + $|js-jt|$ . It is useful to identify and rank hubs which see the maximum traffic. This is done by defining the co-efficient of betweenness centrality (CBC) where the CBC of a given hub $k$ is defined as $CBC=\frac{N_{k}}{N}$, i.e. the ratio of the number of messages that go through a hub $k$ to the total number of messages running on the lattice. These are listed in Table \[tab:table1\].
Efficient decongestion strategies have been set up by connecting hubs of high CBC amongst themselves, or to randomly chosen other hubs via assortative connections [@braj1]. Gradient mechanisms [@grad] can also be used to decongest traffic [@danila; @sat](See Fig. \[fig:asgr\](b)).
Hub label CBC value Rank
----------- ----------- ------
x 0.827 1
y 0.734 2
z 0.726 3
u 0.707 4
v 0.705 5
In all the simulations here, we consider a lattice of size ${100\times
100}$ with 4$\%$ hub density and $D_{st}$ = 142, $d_{min}=1$. The critical message density which congests this lattice is $N_c=1530$. The studies carried out here correspond to the congested phase, where $2000$ or $4000$ messages run on the lattice. We first consider the baseline lattice as in Fig. \[fig:asgr\](a) where there are no short-cuts between the hubs. The message holding capacity of ordinary nodes and hubs is unity for the baseline lattice.
A given number $N$ of source and target pairs separated by a fixed distance $D_{st}$ are randomly selected on the lattice. Here, all source nodes start sending messages to the selected recipient nodes simultaneously, however, each node can act as a source for only one message during a given run. The routing takes place by a distance based algorithm in which each node holding a message directed towards a target tries to identify the hub nearest to itself, and in the direction of the target as the temporary target, and tries to send the message to the temporary target through the connections available to it. During peak traffic, when many messages run, some of the hubs, which are located such that many paths pass through them, have to handle more messages than they are capable of holding simultaneously. Messages tend to jam in the vicinity of such hubs (usually the hubs of high CBC) leading to formation of transport traps which leads to congestion in the network. Other factors like the opposing movement of messages from sources and targets situated on different sides of the lattice, as well as edge effects ultimately result in the formation of transport traps. We have studied trapping configurations for the same $2-d$ network in Ref.[@sat]. Fig. \[fig:trap1\](a) shows a situation in which messages are trapped in the vicinity of high CBC hubs. Fig. \[fig:trap1\](b) shows the number of messages running on the lattice as a function of time. It is clear that the messages are trapped for the baseline case. We study the network for situations which show this congested phase.
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Queue lengths and synchronization
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As mentioned in the introduction, the queue at a given hub is defined to be the number of messages which have the hub as a temporary target. As traffic increases in the network, hubs which see heavy traffic start getting choked due to capacity limitations, and are unable to transfer messages aimed towards them to the next temporary target. Thus queue lengths start to build up at these hubs. If these hubs are not decongested quickly, so that the queue lengths start falling, the congestion starts spilling over to other hubs. If the number of messages increases beyond a certain critical number, messages get trapped irretrievably and the entire lattice congests. A plot of the queue lengths as a function of time can be seen in Fig. \[fig:psmodel1\](a). Here, the queues at the $1_{st}$, $2_{nd}$ and $5_{th}$ hubs ranked by the CBC are plotted for the base-line network with no decongestion strategies implemented. Thus the network congests very easily. Since the queue length is defined as the number of messages with the given hub as the temporary target, the queue starts dropping as soon as the hub starts clearing messages and reaches a minimum. Meanwhile, other hubs which were temporary targets have cleared their messages, and some new messages pick up the hub of interest as their temporary target. The queue thus starts building up here, and reaches a maximum. After this the messages start clearing, and the queues drop sharply. However, since the number of messages is sufficiently large for the network to congest, some messages get trapped in the vicinity of the hub, and the queues saturate to a constant value. Similar phenomena can be seen at the hubs of lower CBC (see Fig. \[fig:psmodel1\](a)). Here again three distinct scales can be seen with values of the same order as those for the highest ranked hub. An important difference can be seen in the queues of the fifth ranked hub (Fig. \[fig:psmodel1\](a)) as well as the fourth ranked hub (not shown). Since these hubs have lower CBC values, and thus fewer messages take them as the temporary targets, the queues at these hubs clear completely. Thus, the saturation value at these hubs, is zero. It should also be noted that the time at which the last two hubs clear completely, i.e. the queue length drops to zero, is substantially earlier than the saturation time of the top two hubs. The above results are observed for a typical configuration and is valid for different configurations as well.
Synchronization
---------------
We now study the synchronization between the queues at different hubs. We see phase synchronization between queues at pairs of high CBC hubs for the baseline, and complete synchronization between some pairs once decongestion strategies are implemented. The usual definitions of complete synchronization and phase synchronization in the literature are as follows.
Complete synchronization (CS) in coupled identical systems appears as the equality of the state variables while evolving in time. Other names were given in the literature, such as [*conventional synchronization*]{} or [*identical synchronization*]{}[@pyragas]. It has been observed that for chaotic oscillators starting from uncoupled non-synchronized oscillatory systems, with the increase of coupling strength, a weak degree of synchronization, the [*phase synchronization*]{}(PS) where the phases become locked is seen [@pikovsky; @rosa] . Classically, the phase synchronization of coupled periodic oscillators is defined as the locking of phases ${\phi_{1,2}}$ with a ratio $n:m$ ($n$ and $m$ are integers), i.e. $|n\phi_{1}-m\phi_{2}|< $Const.
These two concepts of synchronization are applied to the queue lengths of the top five hubs. The plot of $q_{i}(t)$ as a function of average queue length $<q(t)>$ shows a loop in the congested phase, similar to that observed in coupled chaotic oscillators [@maria]. We define a phase as in [@maria] ${\Phi}_{i}(t)=tan^{-1}(\frac{q_{i}(t)}{<q(t)>})$, where $q_{i}(t)$ is the queue length of $i_{th}$ hub at time $t$, and $<q(t)>=\frac{1}{N_{h}}\displaystyle\sum_{i}q_{i}(t)$ where the average is calculated over the top five hubs ($N_{h}=5$). The queue lengths are phase synchronized if $$|\Phi_{i}(t)-\Phi_{j}(t)| < Const$$ where $\Phi_{i}$(t) and $\Phi_{j}$(t) are the phase at time $t$ of the $i_{th}$ and $j_{th}$ hub respectively.
Two queue lengths $q_{i}(t)$ and $q_{j}(t)$ are said to be completely synchronized if $$q_{i}(t)=q_{j}(t)$$
Fig. \[fig:psmodel1\](b) shows that the queue lengths of the first and fifth ranked hubs are not completely synchronized. Fig. \[fig:psmodel1\](c) shows the phase difference between the top pair of hubs as a function of time for the base-line case. It is clear that the two hubs are phase synchronized in the regimes where the queues congest. There are three distinct time scales in the problem. The two hubs are phase synchronized up to the first time scale $t_1$, where the queues cross each other first, they lose synchronization after this. The point at which the phase difference is maximum is $t_2$. This is the point at which the first hub saturates, but the second hub is still capable of clearing its queue. At $t_3$ both the hubs get trapped and the phases lock again.
Fig. \[fig:psmodel1\](d) shows a similar plot for the hubs of the two remaining ranks. It’s clear that the hubs phase synchronize. The synchronization behavior of the remaining hubs for a typical configuration is listed in Table \[tab:table2\]. It is clear that the hubs synchronize pair wise, and that the slower hubs drive the hubs which clear faster. Since the queues at the fourth and fifth hub clear faster than the first hub saturates, there is no peak in the $\Delta \Phi$[^1] plot for the $(1,5)$ pair and hence no scale $t_2$. The phase synchronization between hubs three and four shows similar behavior. This is valid for different configurations as well.
PS pairs $t_{1}$ $t_{2}$ $t_{3}$
---------- --------- --------- ---------
$(1,2)$ 440 589 675
$(1,3)$ 225 595 727
$(1,4)$ 360 595 720
$(1,5)$ 472 - 590
$(2,3)$ 295 495 727
$(2,4)$ 405 620 727
$(2,5)$ 450 590 675
$(3,4)$ 285 - 727
$(3,5)$ 270 585 727
$(4,5)$ 360 585 727
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It is also interesting to compare the synchronization effects between these hubs of high CBC, and randomly selected hubs on the lattice. Fig. \[fig:randhub\] shows the phase difference between the hub of highest CBC (hub ‘x’, ranked 1) and a randomly chosen hub (with CBC value $0.56$ ). It is clear that there is no synchronization between these two hubs. However, this randomly chosen hub shows excellent phase synchronization with another randomly chosen hub (with comparable CBC value $0.67$). Similar results are seen for larger number of messages.
Decongestion strategies and the role of connections
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As discussed earlier, the addition of extra connections between the hubs of high betweenness centrality can ease congestion. This leads to two effects. The time scales of the problem, the rate at which the queues build up and clear, and the way in which correlations occur between different hubs are altered due to the addition of extra connections. We see the effects of this in the synchronization between the queues at the hubs. We illustrate the effects seen for gradient connections between the hubs (Fig. \[fig:asgr\](b)). To set up the gradient mechanism, we enhance the capacities of top five hubs ranked by their CBC values, proportional to their CBC values by a factor of 10. A gradient flow is assigned from each hub to all the hubs with the maximum capacity ($C_{max}$). Thus, the hubs with lower capacities are connected to the hubs with highest capacity $C_{max}$ by the gradient mechanism. Hence the hub with highest CBC value is maximally connected. Fig. \[fig:trap1\](b) and Fig. \[fig:bslgrad\] shows that connecting the top five hubs by the gradient mechanism relieves the system of congestion rapidly when 2000 messages are traveling in the lattice for 4% hub density and run time of $10D_{st}$.
The most striking observation is that now complete synchronization is seen between at least one pair of hubs, and phase synchronization is seen between the remaining pairs. In Fig. \[fig:csps1\](a) we plot a pair of queue length $q_{i}$ vs $q_{j}$. If these two quantities lie along the $y=x$ line with a standard deviation less than one, we call them completely synchronized. It is clear from the Fig. \[fig:csps1\](a) and the value of the standard deviation that the queue lengths of the $2_{nd}$ and the $3_{rd}$ ranked hubs are completely synchronized. These two hubs are of comparable CBC values (See Table \[tab:table1\]) and are indirectly connected via the top most ranked hub, to which each of the lower ranked hubs is connected via a gradient. If the standard deviation is greater than one, the queue lengths are not completely synchronized. In Fig. \[fig:csps1\](b) we observe phase synchronization when the top five hubs are connected by the gradient mechanism. Phase synchronization is observed when the queues congest. As soon as the queues decongest they are no longer phase synchronized. This observation is true for the complete synchronization as well. In the gradient scheme we see a star-like geometry where the central hub is connected to the hubs of low capacity. This central hub gets congested leading to the congestion of the rest of the hubs. Once this hub gets decongested the rest of the hubs of high CBC get cleared. Thus, the central hub, which is the hub of highest CBC, drives the rest.
The finite time Lyapunov exponent
---------------------------------
The queue lengths increase in the congested phase and the difference between two queue lengths (i.e. queue lengths at distinct hubs) is small in this phase, as compared to the decongested phase, where the difference between queue lengths is large. This is analogous to the behaviour of trajectories in the chaotic regime where the separation between two co-evolving trajectories with neighbouring initial conditions increases rapidly, as compared to the separation in the periodic regime where it rapidly decreases. Hence the stability of the completely synchronized state seen in the gradient case can studied by calculating the finite time Lyapunov exponent of the separation of queue lengths for the top five pairs of hubs. The finite time Lyapunov exponent is given by
$$\lambda(t)=\frac{1}{t}\ln(\frac{\delta(t)}{\delta(0)})$$
where $\delta(t)$ = $|q_{i}(t)-q_{j}(t)|$ and $\delta(0)$ is the initial difference in queue lengths [^2]. If $\lambda(t) < 0$ then queue lengths are completely synchronized (Fig. \[fig:lyap1\](a)) and if $\lambda(t) > 0$ then queue lengths are not completely synchronized (Fig. \[fig:lyap1\](b)). The time is calculated from the time ($t=15$) at which the queue starts building up in the lattice. It is clear from the Fig. \[fig:lyap1\](a) that complete synchronization exists till $t_{c}=720$. This is the time at which queues are cleared. In Fig. \[fig:lyap1\](b) complete synchronization exists till $t_{cs}=150$, when queues are building up in the lattice. No complete synchronization is observed after this, but queues are cleared at $t_{c}=740$ .
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Global synchronization
----------------------
It is useful to define an over-all characterizer of emerging collective behavior. The usual characterizer of global synchronization is the order parameter [@Arenas] defined by
$$r\exp{i\psi}=\frac{1}{N_{h}}\sum_{j=1}^{N} \exp{i \Phi_{j}}$$
$N_{h}$=5, where we consider the top five hubs. Here $\psi$ represents the average phase of the system, and the $\Phi_j$-s are the phases defined in Eq. 1. Here the parameter $0 \leq r \leq 1$ represents the order parameter of the system with the value $r=1$ being the indicator of total synchronization.
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We plot the order parameter $r$ and average phase $\psi$ as a function of time for the baseline mechanism in Fig. \[fig:globalbsl\](a) and the gradient connections in Fig. \[fig:globalbsl\](b). It is clear that the order parameter goes to one up to the time $t_{synch}\simeq 300 $ indicating that the queues at all the top $5$ hubs synchronize up to this point. As discussed earlier, this point is also the time at which the network congests. Thus the intimate connection between congestion and synchronization is clearly demonstrated by the order parameter[^3]. It is to be noted that the order parameter $r$ and average phase $\psi$ are calculated for hubs of comparable CBC values (in this case the top five hubs).
Other decongestion schemes
--------------------------
Decongestion schemes based on random assortative connections between the top five hubs ($CBC_a$(one way) and $CBC_c$ (two way)) and the top $5$ hubs and randomly chosen other hubs ($CBC_b$(one way) and $CBC_d$ (two way)) have also proved to be effective. The phenomena of complete synchronization and phase synchronization can be seen for these schemes as well. (See Table \[tab:table3\]). Apart from the gradient mechanism complete synchronization is seen for the $CBC_{c}$ mechanism as well, where the $4_{th}$ and $5_{th}$ ranked hubs (ranked by $CBC$) are completely synchronized. Unlike the gradient mechanism, the $4_{th}$ and $5_{th}$ ranked hubs have a direct two way connection for this realization of the $CBC_{c}$ mechanism. Both these hubs have comparable CBC values (See Table \[tab:table1\]) and therefore, we see that the queue lengths are completely synchronized. The error to the fit to the $y=x$ line is 0.926. The FTLE of the queue lengths of these hubs is less than zero indicating complete synchronization. No complete synchronization is observed for the other assortative mechanisms. Its clear from Table \[tab:table3\] that the top most hub (labeled x) drives the rest of the top five hubs. Global synchronization emerges for these cases as well. Thus it is seen that synchronization in queue lengths is a robust phenomena. Irrespective of the nature of connections between high CBC hubs, synchronization in queue lengths of highly congested hubs exists during the congested phase.
Mechanism Complete Synchronization Phase Synchronization
----------- -------------------------- -------------------------------------------------------------
$CBC_{a}$ - (x,y),(x,z),(x,u),(x,v),(y,z),(y,u),(y,v),(z,u),(z,v),(u,v)
$CBC_{b}$ - (x,y),(x,z),(x,u),(x,v),(y,z),(y,u),(y,v),(z,u),(z,v),(u,v)
$CBC_{c}$ (u,v)\[0.926\] (x,y),(x,z),(x,u),(x,v),(y,z),(y,u),(y,v),(z,u),(z,v)
$CBC_{d}$ - (x,y),(x,z),(x,u),(x,v),(y,z),(y,u),(y,v),(z,u),(z,v),(u,v)
Gradient (y,z)\[0.992\] (x,y),(x,z),(x,u),(x,v),(y,u),(y,v),(z,u),(z,v),(u,v)
A network with random Waxman topology
=====================================
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The random network topology generator introduced by Waxman [@waxmangraph] is a geographic model for the growth of a computer network. In this model the nodes of the network are uniformly distributed in the plane and edges are added according to probabilities that depend on the distances between the nodes. Such networks are useful for Internet modeling due to the distance dependence in link formation which is characteristic of real world networks [@lakhina] and have been widely used to model the topology of intra-domain networks [@verma; @neve; @shin; @guo]. We study queue length synchronization on this network, and compare it with the synchronization seen for the geographically clustered network of Section II. We consider the case where the Waxman graphs are generated on a rectangular coordinate grid of side $L$ with a probability $P(a,b)$ of an edge from node $a$ to node $b$ given by
$$P(a,b)=\beta\exp(-\frac{d}{\alpha M})$$
where the parameters $0< \alpha,\beta < 1$, $d$ is the Euclidean distance from $a$ to $b$ and $M=\sqrt2\times L$ is the maximum distance between any two nodes [@waxmangraph; @naldi; @waxsquare]. Larger values of $\beta$ results in graphs with larger link densities and smaller values of $\alpha$ increase the density of short links as compared to the longer ones.
Here we select a $100 \times 100$ lattice. A topology similar to Waxman graphs is generated by selecting randomly a pre-determined number $N_{w}$ of nodes in the lattice. The nodes are then connected by the Waxman algorithm, resulting in a topology which is similar to Waxman graphs (Fig. \[fig:waxfig\]). Additionally, each node has a connection to its nearest neighbors. We study message transfer by the same routing algorithm as used in Section II. We evaluate the coefficient of betweenness centrality of the nodes and select the five top most nodes ranked by their CBC values. We compare the synchronization in queue lengths of these nodes for different values of $\beta$ and $\alpha$ for simultaneous message transfer.
The phenomenon of phase synchronization in queue lengths is again studied for simultaneous message transfer where $N=2000$ messages flow simultaneously on the lattice with $N_{w}=100$ points chosen randomly in the lattice and connected by Waxman algorithm. The source target separation is $D_{st}=142$ as before, and is again the Manhattan distance between source and target. If $\alpha=0.05$ and $\beta=0.05$, the number of links between the randomly distributed nodes are very few as in the geographically clustered network. In such a situation, messages are cleared slowly and we observe strong phase synchronization (Fig. \[fig:waxphasesynch\](a)(i)). An increment in the values of $\alpha$ and $\beta$ increases the density of links. Messages are cleared faster due to the presence of a large number of short cuts which leads to larger fluctuations in phase, and weaker phase synchronization is seen (Fig. \[fig:waxphasesynch\](a)(ii)). For both the situations messages get trapped and after some time and the phase gets locked. We see the saturation in the plot of $\Delta\Phi$. Global synchronization is also seen in this system (Fig. \[fig:waxphasesynch\](b)).
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Constant Density Traffic
========================
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In the previous sections we discussed synchronization in queue lengths for simultaneous message transfer where $N$ messages are deposited simultaneously on the lattice and no further messages are fed on to the system. In this section we study synchronization in queue lengths for the constant density traffic. For the model of section II we consider $100$ messages fed at every $120$ time steps with 100 hubs and $D_{st}$ = 142 for a total run time of $12000$. Again, two phases, viz. the decongested phase and the congested phase are seen. In the decongested phase all messages are delivered to their respective targets, despite the fact that new messages are coming in at regular intervals. The queue lengths are not phase synchronized during this phase. In the congested phase, messages tend to get trapped in the vicinity of the hubs of high CBC, due to the reasons discussed in Section II. As more messages come in, the number of undelivered messages increase and the queue lengths start increasing until total trapping occurs in the system. During this phase, the creation of messages is stopped and the system attains maximal congestion. The queue lengths show phase synchronization during this phase (Fig. \[fig:cdt1\](a)). Initially the fluctuations in $\Delta\Phi$ are large. After a time $t\simeq 2000$ the queue lengths start increasing and the fluctuations are reduced indicating stronger phase synchronization. As soon as maximal congestion takes place ($\simeq 7800$), the phase difference attains a constant value. Global synchronization is also seen in this system as can be seen from Fig. \[fig:cdt1\](b). Note that the scales on which the phase difference and the global synchronization parameter fluctuate is very small indicating a much stronger version of synchronization than in the earlier case.
The results are compared with constant density traffic for the Waxman topology network as discussed earlier. We consider 100 messages fed continuously at every $120$ time steps for a total run time of $50000$ with $N_{w}=500$ points and $D_{st}$ = 142. If $\alpha=0.05$ and $\beta=0.05$ messages get trapped in the system very fast ($t_{c}=8000$). Phase synchronization in queue lengths is observed in such cases (Fig. \[fig:cdt1\](c)(i)). If the values of $\alpha$ and $\beta$ increase the number of links increase. Phase synchronization in queue lengths take place at much higher time for $(i)$ $\alpha=0.05$ and $\beta=0.4$ and $(ii)$ $\alpha=0.4$ and $\beta=0.05 $. If $\alpha=0.4$ and $\beta=0.4$ the density of links is very large and all the top five nodes have approximately equal queue lengths. Hence we observe a stronger phase synchronization where the fluctuation of $\Delta\Phi$ is well below the predetermined constant $C=0.05$ (Fig. \[fig:cdt1\](c)(ii)). Global synchronization is also observed in this system (Fig. \[fig:cdt1\](d)).
Thus, the model networks studied here show phase synchronisation as well as global synchronisation in the congested phase. The two traffic patterns studied here are those of a single time deposition, and that of constant density traffic. Real life networks can have traffic patterns which wax and wane several times in a single day. However, synchronisation phenomena can be seen in real networks as well. We demonstrate this phenomenon in terms of the number of views at different web-sites, in the next section.
Synchronization for real traffic data
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We discuss phase synchronization in the Internet traffic data in the Indian Institute of Technology Madras, India. The data is collected for the number of views to different websites. The websites are $www.google.com$, $www.gmail.com$, $www.yahoo.com$, $www.youtube.com$ and $www.rediffmail.com$. The data is counted specifically for the given sites and not for subdomains [^4]. Fig. \[fig:nviews\](a) shows the total number of views for the five websites per day for a period of $92$ days ,from $01/10/2008$ to $31/12/2008$ ($t$ in days on the x-axis). In Fig. \[fig:nviews\](b) we plot the number of views per minute for $10^{th}$ November $2008$ ($t$ in minutes on the x-axis).
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As is evident from the plots the number of views for the website google is very large as compared to the rest. In Fig. \[fig:nviews\] it is seen that $N_{views}$ for google show abrupt high peaks at $t_{1}=13$ ($13^{th}$ October), $t_{2}=24$ ($24^{th}$ October), $t_{3}=28$ ($28^{th}$ October), $t_{4}=51$ ($20^{th}$ November) and $t_{5}=53$ ($22^{nd}$ November) (See [^5]).
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In Fig. \[fig:synchreal\] we plot the phase synchronization in number of views for different websites. The phase is defined as in Eq.1. It is observed that the phase locking condition holds for the pairs (yahoo, gmail) and (youtube, rediffmail) (See Fig. \[fig:synchreal\](a)). No such phase locking condition exists between the pairs (google, yahoo) and (google, rediffmail) (See Fig. \[fig:synchreal\](c)). This is due to two facts. First the number of views for $www.google.com$ are much higher than those of the other websites. Secondly, the presence of abrupt peaks for google leads to larger fluctuations. The plot of global synchronization parameters $r$ and $\Psi$ shows that the websites are synchronized in terms of number of views (See Fig. \[fig:synchreal\](b, d)). Larger fluctuations in $r$ and $\Psi$ are seen when all the five websites are taken into account (See Fig. \[fig:synchreal\](b)). The fluctuations are reduced when the website $www.google.com$ is not taken into account (See Fig. \[fig:synchreal\](d)).
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We observe similar behavior when the data is studied for $10^{th}$ Nov $2008$. It is observed that the two websites (youtube, rediffmail) peak together during the time interval $t=400 - 600$ (Fig. \[fig:phasesynch1\](a)). During this time interval the two websites are synchronized as shown in Fig. \[fig:phasesynch1\](a(ii)). As soon as the values of $N_{views}$ start decreasing for both the sites, phase synchronization is lost. When we compared the sites google and yahoo it was observed that for yahoo, the number of views increases intermittently. Hence intermittent behavior of phase synchronization is observed for yahoo and google (See Fig. \[fig:phasesynch1\](b)). No phase synchronization is observed between google and rediffmail. This is similar to the absence of phase synchronization in queue lengths between higher CBC hubs and hubs of low CBC values as in Fig. \[fig:randhub\] for the $2-d$ communication network. Also we observed that hubs of comparable CBC phase synchronize. Similarly in the Internet data, websites of comparable volume of traffic phase synchronize.
We also study the traffic data of the global top five websites ranked by the percentage of global Internet users who visit the respective website. Fig. \[fig:phasesynch2\](a) shows the plot of percentage of global Internet users,$f_{views}$ for the top ranked websites for a period of $16$ days from 08/02/2009 to 24/02/2009. The data has been collected from the website $www.alexa.com$. The websites are $www.google.com$, $www.yahoo.com$, $www.youtube.com$, $www.live.com$ and $www.msn.com$. Here also phase locking is observed for the pairs of websites (Fig. \[fig:phasesynch2\](b)).
Conclusions
===========
To summarize, we have established a connection between synchronization and congestion in the case of two communication networks based on $2-d$ geometries, a locally clustered network, and a network based on random graphs, viz. the Waxman topology network.
We first considered the case where many messages are deposited simultaneously on the lattice. We observed that the queue lengths of the top five hubs get phase synchronized when the system is in the congested phase i.e. the queue lengths at the hubs start piling up. Phase synchronization is lost when messages start getting delivered to their destinations and queue lengths start decreasing. Complete synchronization in queue lengths between certain pairs of top five hubs is observed when decongestion strategies are implemented by connecting the top five hubs by gradient connections or by assortative linkages. Phase synchronization is also seen between nodes of comparable CBC in the Waxman topology despite the fact that the degree distribution of the Waxman graphs is entirely different from that of the locally clustered network[^6]. The phenomenon of synchronization in queue length in the congested phase is thus a robust one. We also observed that the phase synchronization does not exist between hubs of widely separated CBC-s, as can be seen from the lack of synchronization between the high CBC hubs and any randomly chosen hub (which turn out to have lower CBC values).
It is has been seen, for typical configurations, that the central most region of the lattice is the most congested due to flow of messages from both sides of the lattice[@sat]. The top hubs ranked by their CBC values were also seen to be located at the central region of the lattice and were close to each other. In such cases, the hub with the highest CBC value gets congested first and the remaining hubs follow, resulting in increase in queue lengths at the remaining hubs after a time lag resulting in a phase difference between the queues at different hubs. This phase difference tends to a constant in the congested phase. Thus, phase synchronization in queue lengths exists pairwise for hubs of comparable CBC values. Global synchronization is also seen between hubs of comparable CBC-s. This is demonstrated by the behavior of the global synchronization parameters for the top five hubs.
Similar phase synchronized behavior is observed when messages are fed on the lattice at a constant rate. The parameters of synchronization, such as the phase difference and the global synchronization parameters, show small fluctuations in this case, indicating stronger synchronization as compared to simultaneous deposition.
Thus, synchronization is associated with the inefficient phase of the system. Similar phenomena can be found in the context of neurophysiological systems [@zemanova] and in computer networks [@flyod]. We observed that the most congested hub drives the rest. As soon as this hub decongests, the synchronization is lost. In the case of our communication network, this is usually the hub of highest CBC. Due to the master-slave relation between the most congested hub and the rest, there is cascading effect by which successive pairs lose synchronization. Cascading effects have been seen in other systems such as power grids [@lynch] and the Internet [@vespi]. It will be interesting to see if synchronization effects can be observed in these contexts.
Finally we studied the Internet traffic data obtained in Indian Institute of Technology, Madras, India as well as the global Internet data obtained from the Alexa website. Despite the irregular pattern of traffic in this data, phase synchronization in the number of views is observed between two websites of comparable volume of traffic. Phase synchronization breaks down if the volume of traffic changes abruptly. Global synchronization is also observed for the Internet data. This data also shows the existence of intermittent synchronization.
These observations can be of great utility in the practical situations. For example, e.g. synchronization can be considered as a predictor of congestion. Synchronization can also be used to detect changes in the pattern of traffic or to detect abnormal traffic from a given hub. The hub from where the attack originates can be easily identified via a synchronization effect. Synchronization in transport may also provide information about the way in which the network is connected. Thus our study may prove to be useful in a number of application contexts.
We wish to acknowledge the support of DST, India under the project SP/S2/HEP/10/2003. We thank A. Prabhakar and S. R. Singh for sharing the Internet data of the campus.
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[^1]: $\Delta \Phi$ = $|\Phi_{i}(t)-\Phi_{j}(t)|$
[^2]: The queue lengths $q(t)$ are of the order $10^2$ in the congested phase, whereas the $\delta(t)-s$ are of order $1$. Thus $\delta(t) << q(t)$.
[^3]: The value of $r$ is seen to increase at the end of the decongestion phase. This is due to the fact that for the gradient mechanisms all queues are cleared and thus take the value zero at the end of the run. For the baseline mechanism the queues of $4_{th}$ and $5_{th}$ hubs are cleared while rest are trapped leading to a constant value of $r$ which is less than one at the end of the run.
[^4]: The data is obtained from the log files (generated by the SQUID software) of the proxy server in the IIT Madras campus
[^5]: $13^{th}$ October, $24^{th}$ October, $20^{th}$ November and $22^{nd}$ November were all dates for semester examinations in the Indian Institute of Technology Madras, India and students tend to access google more for tutorials and solutions available in the web, at these times. Again $28^{th}$ October is the festival of $Diwali$ which is a national holiday in India. The Internet users on campus appear to have spent most of this holiday browsing. The value of $N_{views}$ for all the websites reach their peak during the day time but decreases during night, contrary to the notion that web browsing increases during the night. This is due to the fact that Internet is unavailable between 20:00 hours to 04:00 hours in the student hostels.
[^6]: the geographically clustered network has a bimodal degree distribution[@BrajNeel],whereas the Waxman graph has degree distributions which depend on the values of $\alpha$ and $\beta$[@skim].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We investigate a laser model for a resonant system of photons and ion cluster-solvated rotating water molecules in which ions in the cluster are identical and have very low, non-relativistic velocities and direction of motion parallel to a static electric field induced in a single direction. This model combines Dicke superradiation with wave-particle interaction. As the result, we find that the equations of motion of the system are expressed in terms of a conventional free electron laser system. This result leads to a mechanism for dynamical coherence, induced by collective instability in the wave-particle interaction.'
author:
- 'Eiji Konishi[^1]'
title: 'Wave-particle Interactions in a Resonant System of Photons and Ion-solvated Water'
---
The mechanisms behind [*coherence*]{} in physical systems are mainly classified into two types. In the first type, coherence is due to long-range order; in the second type, it is due to the collective instability in a many-body system with a long-range interaction. In laser physics, Dicke superradiation belongs to the first type and the free electron laser (FEL) belongs to the second type[@Dicke; @Super; @FEL; @Bonifacio; @BookFEL]. In this Letter, we present a new laser model that incorporates both types of dynamical coherence mechanism from the aspect of wave-particle interactions[@Book; @FELPR; @FEL1; @FEL2; @FEL3]. The ingredients of this system are [*photons*]{}, [*solvated ions*]{} and [*water molecules*]{}.
In Ref.[@GPV] the many-body system of electric dipoles of water molecules that interact with the electromagnetic field radiated from rotating water molecules was studied using a quantum field theoretical approach based on an analogy to an FEL[@FEL; @Bonifacio; @FELPR; @FEL1; @FEL2; @FEL3; @Nature]. In simplified settings, it was shown that, around an impurity that carries a sizeable electric dipole and induces a static electric field oriented in the $z$-direction, a permanent electric polarization of water molecules in the $z$-direction emerges in the limit cycle of the system, due to the coherent and collective interaction of the water molecules with the selected modes of the radiation field.
Using this result, in this Letter we consider an FEL-like model for a cluster of solvated identical ions with very low, non-relativistic velocities and a uniform direction of motion. The electric dipoles of the water molecules which solvate each moving ion behave as $XY$ spins which interact directly only with the transverse electromagnetic field radiated from the rotating water molecules. On the particle side of the wave-particle (i.e., radiation field-particle) interactions, there are two kinds of elements: water molecules and ions around which the water molecules are trapped by a Lennard-Jones potential.
In the FEL model[@FEL; @Bonifacio; @BookFEL; @FELPR], each relativistic unbound electron has the phase degree of freedom of its $XY$ spin-like direction on the $x$-$y$ plane orthogonal to its direction of motion (called the $z$-direction). Due to the presence of an undulator, that is, a transverse magnetic field created by a periodic arrangement of magnets with alternating poles, the unbound electrons are accelerated, producing synchrotron radiation.
In our case, the variable corresponding to the phase coordinate of an unbound electron is the phase $\theta$ of the direction of the electric dipole vector $\vec{d}$ on the $x$-$y$ plane for each water molecule. Classically, we have $$d_{x}+id_y=d_0|\sin \varphi| e^{i\theta}\;,$$ where $\varphi$ is the zenith angle and $d_0=2ed_e$ with $d_e\approx 0.2$ \[Å\][@Franks]; however, we consider it quantum mechanically. Here, we assume that ions move along the $z$-axis with velocity $v\ll c$[^2] and a static electric field $\vec{E}_0$ is induced in the $z$-direction. Then, we can invoke the result of Ref.[@GPV], that is, the emergence of a permanent electric polarization of the solvent and non-screening water[^3] in the $z$-direction.
This Letter is based on the assertion that, [*[to a good approximation in our system, the radiation field exchanges energy with water molecules only through the excitation and de-excitation of water molecules between the two lowest levels of the internal rotation of the hydrogen atoms of each water molecule around its electric dipole axis. The energy difference between these two levels is ${\cal E}$, such that ${\cal E}/(\hbar c) \approx 160$ \[cm$^{-1}$\]]{}*]{}[@GPV; @Franks; @JPY]. To describe this [*resonant*]{} interaction, in the quantum mechanical regime, we introduce the [*energy*]{} spin variables of the $j$-th water molecule as [^4] $$\begin{aligned}
\widehat{s}^1_j&=&\frac{1}{2}[|e{\rangle}{\langle}g|+|g{\rangle}{\langle}e|]_j\;,\\
\widehat{s}^2_j&=&\frac{1}{2i}[|e{\rangle}{\langle}g|-|g{\rangle}{\langle}e|]_j\;,\\
\widehat{s}^3_j&=&\frac{1}{2}[|e{\rangle}{\langle}e|-|g{\rangle}{\langle}g|]_j\;,\label{eq:Spin}\end{aligned}$$ where $|g{\rangle}_j$ and $|e{\rangle}_j$ are, respectively, the low-lying energy ground state and the excitation energy state of the $j$-th water molecule in the two-level approximation, and the superscripts of the energy spins represent fictitious dimensions[@PRSR; @JPY]. We introduce an electric dipole moment operator $\vec{\widehat{d}}_j$ for the $j$-th water molecule such that its third axis is the quantization axis of the rotating water molecule. One of the off-diagonal matrix elements $({\langle}e|\vec{\widehat{d}}|g{\rangle})_j=\overline{({\langle}g|\vec{\widehat{d}}|e{\rangle})_j}$ of the electric dipole moment operator $\vec{\widehat{d}}_j$ of the $j$-th water molecule is given by[^5]
$$\begin{aligned}
d_0\int_{-1}^1d(\cos \varphi)\int_0^{2\pi}d\theta (\bar{Y}_{1,1}(\varphi,\theta)(\vec{e}_1\sin\varphi \cos \theta+\vec{e}_2\sin \varphi \sin \theta+\vec{e}_3\cos \varphi)_jY_{0,0}(\varphi,\theta))={d_0}\sqrt{\frac{1}{6}}(-\vec{e}_1+i\vec{e}_2)_j\;.\end{aligned}$$
Then, the electric dipole moment operator can be represented as an off-diagonal matrix in the two-dimensional energy state space:[@Dover] $$\vec{\widehat{d}}_j=\frac{\tilde{d}_0}{2}(-\vec{e}_1(2\widehat{s}^1)-\vec{e}_2(2\widehat{s}^2))_j\;,$$ where we define $$\tilde{d}_0={d_0}\sqrt{\frac{2}{3}} \approx 0.82\cdot d_0\;.$$
The Hamiltonian for the interaction between ion-solvated water and the transversal radiation vector field $\vec{\widehat{A}}$ (i.e., the electromagnetic field in the radiation gauge) can be written using the total relevant electric charge current $\vec{\widehat{j}}$ of the solvated ions and water molecules as $$\widehat{H}_{{\rm int}}^{(p-i-w)}=-\sum_{I=1}^{N}\sum_{i=0}^{n_I}\vec{\widehat{A}}\cdot {\vec{\widehat{j}}}_{{I,i}}\;,\label{eq:Hint}$$ where the natural number $n_I$ is the number of water molecules solvating the $I$-th ion and is observed to be $20$ to $40$[@Number0; @Number1; @Number2; @Number3]. The total relevant electric polarization current $\vec{\widehat{j}}$ is the sum of the ions’ electric charge currents $\vec{\widehat{j}}_I$ and the $1$,$2$-components of the water molecules’ electric polarization currents $\dot{\vec{\widehat{d}}}_{I,i}$: $$\begin{aligned}
\vec{\widehat{j}}_{I,0}&=&\vec{\widehat{j}}_I\;,\\
\vec{\widehat{j}}_{I,i}&=&(\vec{e}_1 \dot{\widehat{d}}_1+\vec{e}_2 \dot{\widehat{d}}_2)_{I,i}\ \ (i\neq 0)\;,\label{eq:dd}\end{aligned}$$ where the time derivative $\dot{\vec{\widehat{d}}}_{I,i}$ is taken in the interaction picture. In the interaction Hamiltonian Eq.(\[eq:Hint\]), the odd-parity operator $\vec{\widehat{j}}_{I,i}$ ($i\neq 0$) has only off-diagonal matrix elements in the representation where the water molecule’s free Hamiltonian in the two-level approximation ${\cal E}\widehat{s}_{I,i}^3$ is diagonal[@JPY].
It is a crucial point that the photon-water molecule part of Eq.(\[eq:Hint\]) is the Dicke interaction Hamiltonian for superradiance that works via the long-range order of electric dipoles in the system[@Dicke].
The characteristic length for the long-range order of water molecules, that is, the [*coherence length*]{} (i.e., the wavelength of a resonant photon), denoted by $l_c$, is estimated to be the inverse of the wavenumber of a resonant photon ${\cal E}/(\hbar c)$. As ${\cal E}/(\hbar c)\approx 160$ \[cm$^{-1}$\], $l_c\approx 63$ \[$\mu$m\][@Franks]. In our model, we assume that the $x$-$y$ dimensions of the system fall within this length $l_c$.
In the semi-classical treatment, the generic form of the Hamiltonian of the water molecule system relevant to our mechanism consists of the rotational kinetic energy of water molecules with average moment of inertia $I^{(w)}=2m_pd_g^2$ (where $m_p$ is the proton mass and $d_g\approx 0.82$ \[Å\])[@Franks], the water solvation potential of ions and the interaction Hamiltonian of water molecules with the radiation field and the static electric field:
$$\widehat{H}^{(w)}=\sum_{I=1}^{N}\sum_{i=1}^{n_I}\frac{1}{2I^{(w)}}\widehat{L}_{I,i}^2+\sum_{I=1}^{N}\sum_{i=1}^{n_I}{v}_{I,i}^{(i-w)}+\sum_{I=1}^{N}\sum_{i=1}^{n_I}{v}_{I,i}^{(LJ)}-\sum_{I=1}^{N}\sum_{i=1}^{n_I}\Bigl\{\vec{{A}}\cdot {\vec{\widehat{j}}}_{{I,i}}+\vec{E}_0\cdot \vec{d}_{I,i}\Bigr\}\;,\label{eq:Hw}$$
where $\vec{d}_{I,i}=(\vec{e}_3)_{I,i}d_0$ and $v_{I,i}^{(i-w)}$ is the screened Coulomb potential between the $I$-th ion and the charges on a water molecule. The Lennard-Jones potential $v_{I,i}^{(LJ)}=4\epsilon_{LJ}[(\sigma_{LJ}/r_{I,i})^{12}-(\sigma_{LJ}/r_{I,i})^6]$ causes the $I$-th ion be solvated by $n_I$ water molecules. It is a function of the distance $r_{I,i}$ between the $I$-th ion and the $i$-th water molecule and is attractive for $r_{I,i}>\sigma_{LJ}$ and repulsive for $\sigma_{LJ}>r_{I,i}$, where $\sigma_{LJ}$ is a very short distance of the order of $1$ \[Å\][@Water]. When there is no external electromagnetic field, ${v}_{I,i}^{(i-w)}$ determines the configuration of electric dipoles of water molecules that are in contact with an ion. The solvation potentials are translationally invariant with respect to phase coordinates $\theta_{I,i}$. It is a significant point that, in our system, the bulk water molecules that screen the charges of ions do not form part of the laser mechanism, due to thermal noise that prevents ordered motion.
Now, due to the FEL-like mechanism, the collective and coherent behavior of the $x$-$y$ phase coordinates of water molecules follows. This mechanism consists of two interlocked parts in a positive feedback cycle. The first part is the long-range collective ordering of dipole vectors of water molecules that solvate each ion moving along the $z$-axis. As a consequence, the radiation from the clusters of ordered rotating water molecules is almost monochromatic and the time-dependent process of the radiation field and order of water molecules approximates a coherent wave amplified along the $z$-axis. These approximations are improved by positive feedback in the second part of the mechanism: that is, the FEL-like wave-particle interaction process where an exponential instability of the fluctuation around the dynamic equilibrium state (i.e., our ready state) accompanies both the magnification of the radiation intensity and the water molecule’s $XY$-phase bunching that produces long-range ordering[@Bonifacio; @Kim]. Finally, we will find that this positive feedback cycle leads to laser radiation.
Under the assumption of monochromaticity of the radiation, the details of the first part of the mechanism are as follows. A part of the classical radiation field is a transverse wave in the $x$-$y$ plane, which can be written as $$A_x+iA_y=A_0e^{-i\phi_0}\;,$$ where $A_0$ is positive and real. The photon-water molecule part of $\widehat{H}_{{\rm int}}^{(p-i-w)}$ can be written as $$\begin{aligned}
\widehat{H}_{{\rm int}}^{(p-w)}=-\sum_{I=1}^{N}\sum_{i=1}^{n}\vec{{A}}\cdot(\vec{e}_1 \dot{\widehat{d}}_1+\vec{e}_2 \dot{\widehat{d}}_2)_{I,i}\label{eq:Hint0}\end{aligned}$$ in the representation in which the water molecule’s free Hamiltonian is diagonalized by ${\cal E}\widehat{s}_{I,i}^3$. To approximate this equation, we drop the $I$-dependence of the number $n_I$ ($I=1,2,\ldots,N$) and assume that the permanent polarization of the solvent water-molecule’s electric dipoles in the $z$-direction, $d_0^{({{\rm ave}})}$, is uniform over all solvated-ions. Due to this approximation, the expectation value of Eq.(\[eq:Hint0\]) can be written using the density matrix $\widehat{\varrho}^{(w)}$ of the system of water molecules as $$\begin{aligned}
{\rm tr}\Bigl[\widehat{H}_{{\rm int}}^{(p-w)}\widehat{\varrho}^{(w)}\Bigr]\approx \sum_{I=1}^{N}A_0\omega_c{\Delta n}\tilde{d}_0^{({\rm ave})}\frac{1}{2}\sin(\theta_I+\phi)\;,\label{eq:ave}\end{aligned}$$ where we have introduced new phase variables $\theta_I$ ($I=1,2,\ldots,N$), the shifted phase $\phi=\phi_0+\delta$ with $\delta=\omega_ct$ for resonance angular frequency $\omega_c=c/l_c$, the rescaled permanent polarization $\tilde{d}_0^{({\rm ave})}=d_0^{({\rm ave})} \tilde{d}_0/d_0$, and $\Delta n=n_--n_+$ with $n_+$ and $n_-$ referring to the numbers of excited and ground state water molecules, respectively, arithmetically averaged over all ions. Here, we assume thermal equilibrium (i.e., the Boltzmann distribution). Then, we obtain $$\begin{aligned}
\Delta n=n\tanh\biggl(\frac{{\cal E}}{k_BT}\biggr)\;,\end{aligned}$$ where ${\cal E}/k_BT\approx 0.12$ at room temperature ($T=300$ \[K\]) and $\Delta n\approx 3.6$ for $n=30$.
In the following, we will derive Eq.(\[eq:ave\]). The basis of the restricted Hilbert subspace of the water molecule’s quantum pure states is the set of the symmetrized collective energy spin states[@PRSR] with respect to each solvated ion: $$\begin{aligned}
|L,M{\rangle}&=&\sqrt{\frac{(L+M)!}{n!(L-M)!}}\nonumber\\
&&\times\Biggl\{\sum_{j=1}^n\widehat{s}_j^{(-)}\Biggr\}^{(L-M)}|(e,e,e\ldots e)_n{\rangle}\end{aligned}$$ with $\widehat{s}_j^{(-)}=\widehat{s}_j^1-i\widehat{s}_j^2$. In the collective energy spin state $|L,M{\rangle}$, $L$ is an integer or half-integer $n/2$ and $M$ is an integer or half-integer $\Delta n/2$ which runs over $-L\le M\le L$. Here, to symmetrize the quantum pure state basis of the system of water molecules, which defines the coupling between water molecules and the radiation field, we have used the fact that a photon wavelength of the order of $l_c$ is much longer than the dimensions of the system of one ion and its solvent water molecules, that is, of the order of $1$ \[Å\][@Dim].[@PRSR] With respect to the $j$-th water molecule, the most general forms of the internal parts of the wave functions $\psi^{(g)}_j$ and $\psi^{(e)}_j$ excited and de-excited, respectively, from thermal equilibrium into a set of superradiant states with $M$ having been reduced to $0$ by classical radiation at the $I$-th ion are $$\begin{aligned}
\psi^{(g)}_j&=&\frac{1}{\sqrt{2}}\Bigl\{|e{\rangle}_je^{i\vartheta_1}-|g{\rangle}_je^{i\vartheta_2}\Bigr\}\;,\\
\psi^{(e)}_j&=&\frac{1}{\sqrt{2}}\Bigl\{|g{\rangle}_je^{-i\vartheta_1}+|e{\rangle}_je^{-i\vartheta_2}\Bigr\}\;,\end{aligned}$$ where $$\vartheta_2-\vartheta_1=\theta_I+\delta\;.$$ Then we have $$\begin{aligned}
\sum_{i=1}^{n}\langle (\vec{e}_{1}\dot{\widehat{d}}_{1}+\vec{e}_{2}
\dot{\widehat{d}}_{2})_{I,i}\rangle
&\approx&\omega_{c}{\Delta n}\tilde{d}
_{0}\frac{1}{2}\Biggl[-\frac{1}{n}\Biggl\{\sum_{i=1}^n\vec{e}_{1,i}\Biggr\}\sin (\theta +\delta )
\nonumber
\\
&&{}+\frac{1}{n}\Biggl\{\sum_{i=1}^n\vec{e}_{2,i}\Biggr\}\cos (\theta +\delta )\Biggr]_{I}\;,
\label{eq:dipole}\end{aligned}$$ where we have used $$\begin{aligned}
&&\frac{1}{n}\sum_{i=1}^n\vec{e}_{k,I,i}\approx\frac{1}{n_+}\sum_{\bigl\{\psi^{(e)}\bigr\}_I}\vec{e}_{k,I,i}\approx\frac{1}{n_-}\sum_{\bigl\{\psi^{(g)}\bigr\}_I}\vec{e}_{k,I,i}\;,\nonumber\\
&&k=1,2\;.\end{aligned}$$ Eq.[(\[eq:dipole\])]{} leads to Eq.[(\[eq:ave\])]{}. As a significant point here, Eq.(\[eq:ave\]) is due to purely quantum mechanical processes and arises from the off-diagonal elements of the density matrix (i.e., quantum coherence).
Now, after straightforward calculations, we find the Schr${\ddot{{\rm o}}}$dinger equations for superradiant $\psi^{(g)}_{j}$ ($\vec{e}_{k,I,j}$, for $k=1,2$, is reduced to its arithmetic average over $j$ due to our restriction of the Hilbert space) in the interaction picture: $$\begin{aligned}
\frac{i}{2}\omega_c+i\dot{\vartheta}_{1,I}&=&-\frac{1}{2\hbar}A_0\omega_c\tilde{d}_0^{({\rm ave})}e^{i(\theta_I+\phi)}\;,\\
\frac{i}{2}\omega_c-i\dot{\vartheta}_{2,I}&=&-\frac{1}{2\hbar}A_0\omega_c\tilde{d}_0^{({\rm ave})}e^{-i(\theta_I+\phi)}\end{aligned}$$ under the two-level approximation with a fixed rotational energy spectrum. These equations can be combined into $$i\hbar\dot{\theta}_I=A_0\omega_c\tilde{d}_0^{({\rm ave})}\cos (\theta_I+\phi)\;.$$ This equation allows the following pulse form radiation solutions only. $$\begin{aligned}
\theta_I&=&\theta_0\;,\\
-\phi_0&=&\omega_ct+\theta_0+\frac{\pi}{2}+n\pi\;,\end{aligned}$$ where $\theta_0$ is time-independent and $n$ is an integer.
As can be seen from this result, to treat the general radiation solutions, we need to relax the rotational energy spectrum of the water molecules in the two-level approximation. To do this, by a classical mechanical procedure, we change the rotational energy gap ${\cal E}$ as $$\frac{I^{(w)}\omega_c^2}{2}\to\frac{I^{(w)}(\omega_c+\dot{\theta}_I)^2}{2}$$ in ${\langle}\widehat{H}^{(w)}{\rangle}$ such that $\dot{\theta}_I\ll \omega_c$. According to this change, we can write down the equations of motion of the energy spin system as the canonical equations of ${\langle}\widehat{H}^{(w)}{\rangle}$ with respect to the variables $\theta_I$ and their canonical conjugates $L_I$ and start to describe the second part of the FEL-like mechanism. (Note that the water molecule’s ground state energy is $-{\cal E}/2$.) Here, we take the origin of the coordinate system to be near the center of the system of radiators. Then, under the suppression of the second time derivatives of the complex amplitude of the ansatz[^6] $$(A_x+iA_y)(r,t)={A}(t)\frac{e^{i(ct-r)/l_c}}{r}\;,$$ and the suppressions according to the assumptions $$\dot{\theta}_I\;,\ \ \frac{\dot{A}}{A}\ll \omega_c\;,\label{eq:ll}$$ the canonical equations of the phase coordinates and the angular momenta of water molecules and the equation of motion of the radiation field are $$\begin{aligned}
\frac{n}{2}\bigl(\omega_c+\dot{\theta}_{I}\bigr)&=&\frac{L_{I}}{I^{(w)}}\;,\label{eq:th}\\
\dot{{L}}_{I}&=&-A_0{\omega_c}{\Delta n}\tilde{d}_0^{({\rm ave})}\frac{1}{2}\cos(\theta_I+\phi)\;,\\
-\frac{i\omega_c}{r}\dot{\tilde{{A}}}e^{i\delta}&=&-\sum_{I=1}^{N}\sum_{i=1}^n\mu c^2\tilde{j}_{{I,i}}\label{eq:EEq}\end{aligned}$$ with $\tilde{{A}}=rA_0e^{-i\phi}$, a complexification of the electric charge current density $\tilde{j}=j_x+ij_y$, and $\mu\approx \mu_0$ being the magnetic permeability in water. The equation of motion of $A_{x,y}$, Eq.(\[eq:EEq\]), is equivalent to $$\begin{aligned}
\dot{A}_0&=&\frac{\mu c^2 N{\Delta n}\tilde{d}_0^{({\rm ave})}}{V}\frac{1}{2}{\langle}\cos (\theta_{I}+\phi){\rangle}_{I}\;,\label{eq:CE3}\\
\dot{\phi}&=&-\frac{\mu c^2 N\Delta n \tilde{d}_0^{({\rm ave})}}{A_0V}\frac{1}{2}{\langle}\sin (\theta_{I}+\phi){\rangle}_{I}\;,\label{eq:CE4}\end{aligned}$$ where $V$ is the volume of the system. In order to facilitate analysis, we introduce dimensionless variables: $${\cal A}_0=A_0\biggl(\frac{\alpha}{2\beta^2}\biggr)^{1/3}\;,\ \ \tau=t\biggl(\frac{\alpha\beta}{2}\biggr)^{1/3}\;,$$ where $$\begin{aligned}
\alpha=\frac{\Delta n\omega_c\tilde{d}_0^{({\rm ave})}}{nI^{(w)}}\;,\ \ \beta=\frac{\mu c^2N\Delta n \tilde{d}_0^{({{\rm ave}})}}{2V}\;.\end{aligned}$$ We denote the scaled time derivative by a prime. Then, the set of equations of motion becomes $$\begin{aligned}
\theta_I^{\prime\prime}&=&-2{\cal A}_0\cos (\theta_I+\phi)\;,\label{eq:EOM1}\\
{\cal A}_0^\prime&=&{\langle}\cos (\theta_I+\phi){\rangle}_I\;,\label{eq:EOM2}\\
\phi^\prime&=&-\frac{1}{{\cal A}_0}{\langle}\sin(\theta_I+\phi){\rangle}_I\;,\label{eq:EOM3}\end{aligned}$$ which is exactly the same as that of the conventional FEL model as described in Ref.[@Nature].
In this paragraph, we qualitatively explain the FEL-like mechanism, following the discussion in Ref.[@Nature]. When the radiation intensity is initially zero, since the phase coordinates $\theta_{I}$ with respect to $I$ are distributed randomly and uniformly within $0\le \theta_I<2\pi$, the system does not change over time. However, when the initial values of ${\cal A}_0$ and $\phi$ satisfy the conditions $0<{\cal A}_0\ll 1$ and $\phi=0$, respectively, the ponderomotive potential $\Phi=\Phi(\theta)$ for water molecules is subtly perturbed around $\theta=3\pi/2$ due to a small bunching induced by the forces. At this point, ${\phi}^\prime$ is positive and though $-{\langle}\sin(\theta_{I}+\phi){\rangle}_{I}\ll 1$ holds, since ${\cal A}_0\ll 1$ holds too, their ratio ${\phi}^\prime$ can be significant. Then, the phase of the radiation field $\phi$ starts to change and the total phase ${\langle}\theta_{I}{\rangle}_{I}+\phi$ starts to grow from $3\pi/2$. Then, ${\cal A}_0^\prime$ becomes positive. Due to this instability, the periodic ponderomotive potential wells $\Phi$ deepen and the bunched water molecules for solvated ions fall to the bottom of $\Phi$ on the space of phase coordinate $0\le \theta<2\pi$. Then, the rotational kinetic energy of water molecules is transferred to $\Phi$ and this gives a positive feedback cycle: this is the FEL-like mechanism.
Due to this mechanism, when the system reaches ${\cal A}_0\approx 1$, the positive feedback loop is expected to close; the system then enters the non-linear saturated regime. At the same time, the coherent dynamics of the radiation field and maximally bunched water molecules arises: the quantum coherence of water molecules (i.e., the sum of Eq.(\[eq:dipole\]) over the system of $N$ ions) is coupled over the system of ion-solvated water, and the intensity of the radiation field is magnified by a multiplicative factor of the order of $N^{4/3}$. As the conclusive formulae for this regime, we obtain $$\begin{aligned}
\biggl(\frac{\alpha}{2\beta^2}\biggr)^{-1/3}&=&c_A\cdot \rho^{2/3}\cdot P^{1/3}_z\;,\label{eq:FA}\\
c_A&\approx&2.6\cdot 10^{-22}\ [{\rm m}^3\cdot {\rm kg}\cdot{\rm s}^{-2}\cdot {\rm A}^{-1}]\;,\\
\biggl(\frac{\alpha\beta}{2}\biggr)^{-1/3}&=&c_t\cdot \rho^{-1/3}\cdot P^{-2/3}_z\;,\label{eq:FT}\\
c_t&\approx&2.4\cdot 10^{-4}\ [{\rm m}^{-1}\cdot{\rm s}]\;,\end{aligned}$$ where $\rho=N/V$ is the ion number concentration in the system and $P_z=d_0^{({\rm ave})}/d_0$ is the permanent electric polarization of water molecules under the static electric field $E_{0,z}$. The first formula Eq.(\[eq:FA\]) refers to the value of $A_0$ at the time when ${\cal A}_0\approx 1$, and the second formula Eq.(\[eq:FT\]) refers to its gain time. Notably, both formulae do not contain the ion velocity $v$. Here, according to the formulae in Ref.[@GPV], we obtain the dependence of $P_z$ on $E_{0,z}$ for a realistic value of $E_{0,z}$[^7] $$\begin{aligned}
P_z&\approx&c_P\cdot E_{0,z}\;,\\
c_P&\approx&9.1\cdot 10^{-9}\ [{\rm m}\cdot{\rm V}^{-1}]\;.\end{aligned}$$
Finally, we have to consider saturation effects. Our system satisfies the condition $$l_b\ll l_s={(c-v)}\frac{l_g}{v}\;,$$ where $l_b$ and $l_g$ are the bunch and gain lengths, respectively. In this case, the radiation emitted by a sufficiently small bunch of water molecules could quickly escape from it due to slippage; so saturation effects would be reduced[@FELSR1; @FELSR2]. In a dissipative system whose original system is governed by the same equations of motion (i.e., Eqs.(\[eq:EOM1\]) to (\[eq:EOM3\])) as our system, it has been argued that superradiation, avoiding saturation effects, can be realized[@FELSR1; @FELSR2]. However, in our system its growth rate is too slow due to the largeness of the ratio of the slippage distance $l_s$ to the bunch length $l_b$; so this superradiance scenario [for the system of all ions]{} has to be abandoned.
Now, we summarize the overall results.
In this Letter, we studied the model of a resonant system of photons and ion cluster-solvated rotating water molecules for the case where ions in the cluster have very low, non-relativistic velocities and direction of motion parallel to a static electric field induced in a single direction. The system of water molecules solvating an ion is reduced to an $XY$ energy spin system under the two-level approximation in the rotational spectrum. By incorporating the mechanism of Dicke superradiation [for each ion]{}, we found that the equations of motion of the $XY$ energy spin systems, over all ions, coupled to the radiation field are expressed in terms of a conventional free electron laser. This result leads to a dynamical coherence mechanism, induced by collective instability in the wave-particle interaction.
In the application of our result, a key earlier result is Ref.[@GPV], which shows the existence of a permanent electric polarization $P_z$ of water molecules under a strong static electric field induced in one direction. This may occur, for example, around an impurity that carries a sizeable electric dipole. We use this earlier result for the $XY$ spin picture of water molecules. In conclusion, our coherence mechanism has desirable properties when it is applied to models of the cluster current of a large number of ions solvated in water. As described earlier, this is assumed to be under a strong static electric field induced parallel to the ion current, with the dimensions of the ion cluster falling within the coherence length (i.e., the wavelength of a resonant photon) $l_c$, and the condition Eq.(\[eq:ll\]) must also be satisfied.
[99]{} R. H. Dicke, Phys. Rev. [**93**]{}, 99 (1954). N. E. Rehler and J. H. Eberly, Phys. Rev. A [**3**]{}, 1735 (1971). W. B. Colson, Phys. Lett. A [**59**]{}, 187 (1976). R. Bonifacio, C. Pellegrini and L. M. Narducci, Opt. Commun. [**50**]{}, 373 (1984). H. Wiedemann, [*Particle Accelerator Physics*]{}, 4th edn. (Springer-Verlag, Berlin Heidelberg, 2015). A. Campa [*et al.*]{}, [*Physics of Long-range Interacting Systems*]{} (Oxford University Press, Oxford, 2014). E. L. Saldin, E. A. Schneidmiller and M. V. Yurkov, Phys. Rep. [**260**]{}, 187 (1995). Z. Huang and K. J. Kim, Phys. Rev. ST Accel. Beams [**10**]{}, 034801 (2007). P. de Buyl [*et al.*]{}, Phys. Rev. ST Accel. Beams [**12**]{}, 060704 (2009). C. Pellegrini, A. Marinelli and S. Reiche, Rev. Mod. Phys. [**88**]{}, 015006 (2016). E. D. Giudice, G. Preparata and G. Vitiello, Phys. Rev. Lett. [**61**]{}, 1085 (1988). B. W. J. McNeil and N. R. Thompson, Nature. Photonics. [**4**]{}, 814 (2010). F. Franks, [*Water: A Comprehensive Treatise*]{} (Plenum, New York, 1972). B. Hribar [*et al.*]{}, J. Am. Chem. Soc. [**124**]{}, 12302 (2002). M. Jibu, K. H. Pribram and K. Yasue, Int. J. Mod. Phys. B [**10**]{}, 1735 (1996). M. Gross and S. Haroche, Phys. Rep. [**93**]{}, 301, (1982). L. Allen and J. H. Eberly, [*Optical Resonance and Two-Level Atoms*]{} (Wiley, New York, 1975). R. J. Cooper, T. M. Chang and E. R. Williams, J. Phys. Chem. A [**117**]{}, 6571 (2013). G. H. Peslherbe. B. M. Ladanyi and J. T. Hynes, Chem. Phys. [**258**]{}, 201 (2000). P. R. Smirnov, O. V. Grechin and I. L. Kritskii, Russ. J. Phys. Chem. A [**89**]{}, 630 (2015). O. N. Pestova, Yu. P. Kostikov and M. K. Khripun, Russian J. Appl. Chem. [**77**]{}, 1066 (2004). K. J. Kim, Phys. Rev. Lett. [**57**]{}, 1871 (1986). Y. Shi and T. L. Beck, J. Chem. Phys. [**139**]{}, 044504 (2013). For the conditions for FEL superradiance, see R. Bonifacio and F. Casagrande, Nucl. Instrum. Methods A [**239**]{}, 36 (1985). R. Bonifacio, B. W. J. McNeil and P. Pierini, Phys. Rev. A [**40**]{}, 4467 (1989).
[^1]: [email protected]
[^2]: In this Letter, quantities of the order of $(v/c)^n$ ($n\ge 1$) are ignored.
[^3]: The solvent water obeys an electrostatic ordering mechanism (see the explanation below Eq.(\[eq:Hw\]))[@Water].
[^4]: In this Letter, a hat indicates that a variable is a quantum mechanical operator.
[^5]: In its definition, the frame $(\vec{e}_1,\vec{e}_2,\vec{e}_3)$ takes paths in $O(3)$ so that its distribution contains $(\vec{e}_x,\vec{e}_y,\vec{e}_z)$, $(\vec{e}_z,\vec{e}_x,\vec{e}_y)$, $(\vec{e}_y,\vec{e}_z,\vec{e}_x)$, $(-\vec{e}_x,-\vec{e}_y,-\vec{e}_z)$, $(-\vec{e}_z,-\vec{e}_x,-\vec{e}_y)$, and $(-\vec{e}_y,-\vec{e}_z,-\vec{e}_x)$ and is statistically symmetric with respect to $1$, $2$, and $3$ in the cyclic order.
[^6]: This suppression is attributed to the assumption that the characteristic time for the change of the complex amplitude of $A_{x,y}$ is much longer than the radiation wave period which is of the order of $2\pi/\omega_c\approx 1.3\cdot 10^{-12}$ \[s\].
[^7]: This formula holds for $E_{0,z}\lesssim 10^7$ $[{\rm m}^{-1}\cdot{\rm V}]$ in itself under the setting of Ref.[@GPV].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Angular distribution of photoelectrons released by the ionization of randomly-oriented molecules with two laser fields of carrier frequencies $\omega$ and $2\omega$, which are linearly polarized in two mutually-orthogonal directions, is analyzed in the perturbation limit for the case of one- vs two-photon ionization process. In particular, we focus on the recently predicted \[Ph.V. Demekhin [*et al.*]{}, Phys. Rev. Lett. **121**, 253201 (2018)\] forward-backward asymmetry in the photoelectron emission, which is induced by the interference of two fields and depends on the external relative phase between them. The present theoretical analysis proves a chiral origin of the effect and suggests that a molecule introduces an additional internal relative phase between two ionizing fields.'
author:
- 'Philipp V. Demekhin'
title: 'Photoelectron circular dichroism with Lissajous-type bichromatic fields: One- vs two-photon ionization of chiral molecules'
---
Introduction
============
The photoelectron circular dichroism (PECD, [@Ritchie]) is a very promising tool for the chiral recognition of molecules in the gas phase [@CDbook; @REV1; @REV2; @REV3]. The effect consists in the forward-backward asymmetry in the emission of photoelectrons and can be triggered via one-photon [@Expt1; @Expt2] or multiphoton [@Lux12; @Lehmann13] ionization of chiral molecules. Traditionally, investigation of PECD utilizes circularly (or elliptically [@Lux15CPC; @Comby16]) polarized light. The observed effect emerges in the electric-dipole approximation [@Ritchie] as an incomplete compensation of the amplitudes for the emission of partial photoelectron waves with positive and negative projections $m$ of the carried angular momentum $\ell$. Here, a typical forward-backward asymmetry can reach about 10% of the total photoionization signal [@CDbook; @REV1; @REV2; @REV3].
Recently [@PRLw2w], it was suggested that PECD can also be observed by a Lissajous-type electric field configuration consisting of carrier frequencies $\omega$ and $2\omega$ linearly polarized in two mutually-orthogonal directions: $$\mathcal{\vec{E}}(t)= \hat{e}_x \mathcal{E}_x \cos(2\omega t) + \hat{e}_y \mathcal{E}_y \cos(\omega t +\phi).
\label{eq:field1}$$ Depending on the relative phase $\phi$ between two fields, the resulting electric field vector mimics rotational motions in different directions in the dipole $xy$-plane, which is perpendicular to the propagation of the light (along $z$-axis). For instance, for $\phi=\pm \frac{\pi}{4}$, the electric field (\[eq:field1\]) forms a ‘butterfly’ that is oriented along the $y$-axis \[[see Fig. \[fig\](b)]{}\]. Such a field possesses in the upper and lower hemispheres two different rotational directions and induces thereby two opposite forward-backward asymmetries. This subcycle chiral asymmetry [emerges due]{} to the interference between two fields.
Numerical calculations, performed in Ref. [@PRLw2w] with the time-dependent single center method [@TDSC1; @TDSC2] for the one- vs two-photon ionization of a model methane-like chiral system, demonstrate a sizable PECD which depends on the handedness of a chiral molecule, rotational direction of the field, and the relative phase $\phi$ between two fields. In the present work, we investigate the chiral asymmetry predicted in Ref. [@PRLw2w] for the one- vs two-photon ionization process in details. In particular, we derive and analyze the emerging PECD signal in the weak-field limit in terms of the partial photoionization amplitudes. [Energy scheme of the considered process is depicted in Fig. \[fig\](a).]{}
![[Panel (a): Energy scheme of the one- vs two-photon ionization of a chiral molecule from its ground electronic state (set to the energy origin) in the electronic continuum state of energy $\varepsilon=2\omega-IP$, where $IP$ stays for the ionization potential. Panel (b): Different configurations of the total electric field (\[eq:field1\]) for different values of the relative phase $\phi$. Directions of rotation along the field trajectory are indicated by arrows. Panel (c): The system of coordinates and notations utilized in the present work. The field propagates along $z$-axis, while $x$- and $y$-axes are defined according to Eq. (\[eq:field1\]).]{}[]{data-label="fig"}](fig.png)
Theoretical analysis
====================
Below, we utilize the system of coordinates and notations introduced in [Figs. \[fig\](b,c)]{}. To assist tracing the present derivation, the most important used relations, [adjusted from the textbook Ref. [@Varshalovich] to the present notations,]{} are collected in the appendix.
Basic equations
---------------
In the weak-field limit, the electric field (\[eq:field1\]) can be split into two parts, which in the rotating wave approximation are responsible for the absorption and emission of photons: $$\mathcal{\vec{E}}= \frac{1}{2} \left(\hat{e}_x \mathcal{E}_x e^{-2i\omega t} + \hat{e}_y \mathcal{E}_y e^{-i\omega t} e^{-i\phi} \right) +cc(\mathrm{emission}).$$ In order to unambiguously introduce a unique direction of the propagation of both fields, we make use of [spherical basis]{} (\[eq:cyclicA\]): $$\label{eq:cyclic}
\hat{e}_x=\frac{1}{\sqrt 2}\left(-\hat{e}_+ + \hat{e}_- \right), ~~~~\hat{e}_y=\frac{i}{\sqrt 2}\left(\hat{e}_+ + \hat{e}_- \right).$$ Here, ‘$\pm$’ correspond to the circularly polarized light with positive or negative helicity, both propagating along the laboratory $z$-axis. In the dipole length gauge, the respective light-matter interaction reads $$\label{eq:operqtor}
\vec{r}\cdot\mathcal{\vec{E}}= \frac{\mathcal{E}_x }{2\sqrt 2}\left(- \mathbf{d}_{+1} + \mathbf{d}_{-1} \right) + \frac{i\mathcal{E}_y }{2\sqrt 2}\left( \mathbf{d}_{+1} + \mathbf{d}_{-1} \right)e^{-i\phi} ,$$ with the electric dipole operator $ \mathbf{d}_{k}=r \sqrt{\frac{4\pi}{3}}Y_{1k}$. Here, $k=0$ and $k=\pm1$ stand for the linear and circular polarizations, respectively.
The wave function of a photoelectron in the continuum spectrum of energy $\varepsilon={p^2}/{2}$ is described by the incoming-wave-normalized [@Starace] superposition of spherical waves with given angular momentum quantum numbers $\ell$ and $m^\prime$ (known as the partial electron waves [@Cherepkov81]): $$\label{eq:photelectron}
\Psi_\mathbf{p}^-(\mathbf{r^\prime}) = \sum_{\ell m^\prime} (i)^{\ell} R_{\varepsilon \ell m^\prime}(r^\prime) \, Y_{\ell m^\prime}(\theta^\prime,\varphi^\prime) \,Y^\ast_{\ell m ^\prime}(\theta_p^\prime,\varphi_p^\prime).$$ Here, prime refers to the quantum numbers and coordinates as defined in the frame of the molecule. Note also that the normalization coefficient and the phases of the partial waves are incorporated in the respective radial parts $R_{\varepsilon \ell m^\prime}$, for brevity.
In the frame of molecule, the electric dipole transition matrix elements for the emission of the partial wave $\varepsilon \ell m^\prime$ by the absorption of one photon of energy $2\omega$ and polarization $k^\prime$ reads: $$\label{eq:onephotonME}
d_{\ell m ^\prime k^\prime}=\langle R_{\varepsilon \ell m^\prime}Y_{\ell m^\prime} \vert \mathbf{d}_{k^\prime} \vert \Phi_0 \rangle _{_{\mathbf{r}^\prime}}.$$ [Here and below, the subscript $\mathbf{r}^\prime$ indicates an integration over spatial coordinates in the frame of molecule.]{} Since absorption of one photon of energy $\omega$ is not sufficient to ionize the initial molecular orbital $\Phi_0$, the respective two-photon transition matrix element reads: $$\label{eq:twophotonME}
{t}_{\ell m ^\prime k_1^\prime k_2^\prime}=\sum_I \frac{\langle R_{\varepsilon \ell m^\prime} Y_{\ell m^\prime} \vert \mathbf{d}_{k_2^\prime} \vert \Phi_I \rangle_{_{\mathbf{r}^\prime}} \langle \Phi_I \vert \mathbf{d}_{k_1^\prime} \vert \Phi_0 \rangle_{_{\mathbf{r}^\prime}}}{E_I-E_0-\omega}.$$ Here, summation must be performed over all virtual intermediate electronic states $\Phi_I$ in the discrete and continuum electron spectrum.
In order to proceed, we transform all electric dipole operators entering Eqs. (\[eq:onephotonME\]) and (\[eq:twophotonME\]) from the laboratory to the molecular frame and the emitted partial waves from Eq. (\[eq:photelectron\]) in the opposite way. The transformations (\[eq:photonA\]) and (\[eq:electronA\]) are performed with the help of the Wigner’s rotation matrices $\mathcal{D}^\ell_{i,j}$ for given Euler orientation angles $\left\{\alpha,\beta,\gamma\right\}$. Making use of the introduced basic equations, the total amplitude of the considered process can be written as a coherent superposition of the amplitudes for the one-photon (\[eq:onephotonME\]) and two-photon (\[eq:twophotonME\]) ionizations: $$\begin{gathered}
\label{eq:TOTAMPL}
\mathbb{D}_{T}=\sum_{\ell_1 m_1^\prime} (-i)^{\ell_1} \sum_{m_1} \mathcal{D}^{\ell_1\ast}_{m_1^\prime,m_1} Y_{\ell_1 m_1 }(\hat{p})\left[ \frac{\mathcal{E}_x }{2\sqrt 2} \sum_{k^\prime}\left( - \mathcal{D}^1_{k^\prime,+1} + \mathcal{D}^1_{k^\prime,-1} \right) d_{\ell_1 m_1 ^\prime k^\prime}\right. \\ \left.-e^{-2i\phi} \frac{\mathcal{E}^2_y }{ 8} \sum_{k_1^\prime k_2^\prime}\left( \mathcal{D}^1_{k_1^\prime,+1} \mathcal{D}^1_{k_2^\prime,+1} + \mathcal{D}^1_{k_1^\prime,+1} \mathcal{D}^1_{k_2^\prime,-1} + \mathcal{D}^1_{k_1^\prime,-1} \mathcal{D}^1_{k_2^\prime,+1}+\mathcal{D}^1_{k_1^\prime,-1} \mathcal{D}^1_{k_2^\prime,-1} \right) {t}_{\ell_1 m_1 ^\prime k_1^\prime k_2^\prime}\right].\end{gathered}$$ As one can see from Eq. (\[eq:TOTAMPL\]), the interference between the one- and two-photon routes (the sum of respective terms in the square brackets) can be controlled by the relative phase $\phi$ between two fields (i.e., by the factor $e^{-2i\phi}$).
Analysis of the asymmetry
-------------------------
We first notice that the product $\mathbb{D}_{T}\mathbb{D}^\ast_{T}$ contains the product of two spherical functions which define the direction of the emission of the photoelectron $\hat{p}=(\theta_p,\varphi_p)$ in the laboratory frame. The latter product can be reduced to the sum over spherical functions via Eq. (\[eq:directionA\]). As a consequence, the differential cross section for the emission of photoelectrons can be expanded as $$\label{eq:dcs}
\frac{d\sigma}{d\Omega_p} =2\pi \vert \mathbb{D}_{T}\vert^2=\sum_{LM} B_{LM}Y^\ast_{LM }(\hat{p}).$$ Expansion (\[eq:dcs\]) has three different contributions. The product of the one-photon ionization amplitudes (\[eq:onephotonME\]) includes terms with $L\leq2$, and the product of the two-photon ionization amplitudes (\[eq:twophotonME\]) is restricted by $L\leq 4$ [[@AD1; @AD2]]{}. The cross-products of the one- and two-photon ionization amplitudes (i.e., the interference terms between two fields which are responsible for the effect predicted in Ref. [@PRLw2w]) are restricted by $L\leq3$. In addition, according to symmetry considerations, a combined [*forward-backward*]{} (with respect to the $\pm z$ directions) and [*up-down*]{} (with respect to the $\pm y$ directions) asymmetry is described by the expansion terms $B_{LM}$ with even values of $L$ and odd values of $M$. Below, this combined asymmetry is referred to as $FBUD$ asymmetry.
As justified above, in the case of one- vs two-photon ionization process, the $FBUD$ asymmetry dictated by the interference between two fields is given by: $$\label{eq:FBUD1}
FBUD(\hat{p})=B_{2,+1}Y^\ast_{2,+1 }(\hat{p})+B_{2,-1}Y^\ast_{2,-1 }(\hat{p}),$$ [which are the only terms with even $L$ and odd $M$ values from the expansion limited by $L\leq3$]{}. In order to keep this observable real, the condition $B_{2,-1}=-B^\ast_{2,+1}$ should be fulfilled. Using now the explicit expressions (\[eq:Y2mp1A\]) for the spherical functions $Y^\ast_{2,\pm1 }$, we obtain $$\label{eq:FBUD2}
FBUD(\theta_p,\varphi_p)= -\sqrt\frac{15}{2\pi}\cos\theta_p\sin\theta_p \left[\mathrm{Re}(B_{2,+1})\cos\varphi_p + \mathrm{Im}(B_{2,+1})\sin\varphi_p\right].$$ Further symmetry considerations suggest that a combined [*forward-backward*]{} and [*left-right*]{} (with respect to the $\pm x$ directions) asymmetry is absent for all relative phases $\phi$. [This is because electric field (\[eq:field1\]) always possesses two opposite rotational directions for equal periods of time in each of the left and right hemispheres. Thus, the term with $\cos\varphi_p$ in Eq. (\[eq:FBUD2\]) must vanish.]{} This holds only if $B_{2,+1}$ is purely imaginary, i.e., $\mathrm{Re}(B_{2,+1})=0$. Taking into account that there are two complex conjugate cross-terms in the $\mathbb{D}_{T}\mathbb{D}^\ast_{T}$ product, this expansion coefficient can be represented as: $$\label{eq:b21}
B_{2,+1}=\mathcal{A}e^{-2i\phi}-\mathcal{A}^\ast e^{2i\phi}= -2i{A}\sin(2\phi-\delta) ,$$ with the complex coefficient $\mathcal{A}={A}e^{i\delta}$. We finally arrive at the following equation for the $FBUD$ asymmetry: $$\label{eq:FBUD3}
FBUD(\theta_p,\varphi_p)= A \cos\theta_p\sin\theta_p \sin\varphi_p \sin(2\phi-\delta).$$ Note, that the remaining prefactor $\sqrt{30/\pi}$ is now explicitly included in the complex coefficient $\mathcal{A}$ from the expression (\[eq:b21\]) for $B_{2,+1}$.
Magnitude of the asymmetry
---------------------------
In order to compute the coefficient $\mathcal{A}$, we set $L=2$ and $M=+1$ in the expansion (\[eq:dcs\]) and collect all terms in front of the phase $e^{-2i\phi}$:
$$\begin{gathered}
\label{eq:AAA1}
\mathcal{A }=-2\pi\sqrt\frac{30}{\pi} \sum_{\ell_1 m_1^\prime m_1}(-i)^{\ell_1} \mathcal{D}^{\ell_1\ast}_{m_1^\prime,m_1} \frac{\mathcal{E}^2_y }{ 8} \sum_{k_1^\prime k_2^\prime}\left( \mathcal{D}^1_{k_1^\prime,+1} \mathcal{D}^1_{k_2^\prime,+1} + \mathcal{D}^1_{k_1^\prime,+1} \mathcal{D}^1_{k_2^\prime,-1} + \mathcal{D}^1_{k_1^\prime,-1} \mathcal{D}^1_{k_2^\prime,+1}\right. \\ \left. + \mathcal{D}^1_{k_1^\prime,-1} \mathcal{D}^1_{k_2^\prime,-1} \right) {t}_{\ell_1 m_1 ^\prime k_1^\prime k_2^\prime} \sum_{\ell_2 m_2^\prime m_2} (i)^{\ell_2} \mathcal{D}^{\ell_2}_{m_2^\prime,m_2} \frac{\mathcal{E}_x }{2\sqrt 2} \sum_{k^{\prime\prime}}\left( - \mathcal{D}^{1\ast}_{k^{\prime\prime},+1} + \mathcal{D}^{1\ast}_{k^{\prime\prime},-1} \right) d^\ast_{\ell_2 m_2 ^\prime k^{\prime\prime}} \\ \times (-1)^{m_2}\sqrt\frac{(\ell_1)(\ell_2)5}{4\pi}\left( \begin{array}{ccc} \ell_1 & \ell_2 &2 \\0 & 0& 0\end{array} \right)\left( \begin{array}{ccc} \ell_1 & \ell_2 &2 \\m_1 & -m_2& +1\end{array} \right).\end{gathered}$$
By reducing the products of the rotation matrices via Eqs. (\[eq:DDD1A\]) and (\[eq:DDD2A\]) and performing a summation over the indices $m_1$ and $m_2$ via Eq. (\[eq:DDD3A\]), we simplify Eq. (\[eq:AAA1\]) to the following form (note that, according to Eq. (\[eq:DDD3A\]), $J=2$ and $M_J=1$): $$\begin{gathered}
\label{eq:AAA2}
\mathcal{A }=\frac{5\sqrt{3}}{16} \mathcal{E}_x\mathcal{E}^2_y \sum_{\ell_1 m_1^\prime}\sum_{\ell_2 m_2^\prime } \sum_{k_1^\prime k_2^\prime k^{\prime\prime}} \sum_{M_J^\prime} \sqrt{(\ell_1)(\ell_2)} \, (i)^{\ell_1+\ell_2} \left( \begin{array}{ccc} \ell_1 & \ell_2 &2 \\0 & 0& 0\end{array} \right) \left( \begin{array}{ccc} \ell_1 & \ell_2 &2 \\-m_1^\prime & m_2^\prime& -M_J^\prime\end{array} \right) \mathcal{D}^{2}_{M_J^\prime,+1} \\ \times (-1)^{k^{\prime\prime}}\left[ -\mathcal{D}^{1}_{-k^{\prime\prime},-1} + \mathcal{D}^{1}_{-k^{\prime\prime},+1} \right] d^\ast_{\ell_2 m_2 ^\prime k^{\prime\prime}} \sum_{TM_TM^\prime_T} (-1)^{\ell_2+m_1^\prime-M_J^\prime+M_T-M^\prime_T} (T) \left( \begin{array}{ccc} 1 & 1 &T \\ k_1^\prime& k_2^\prime& -M^\prime_T\end{array} \right) \mathcal{D}^{T}_{M^\prime_T,M_T} \\ \times \left[ \left( \begin{array}{ccc} 1 & 1 &T \\ +1 & +1 & -M_T\end{array} \right) + \left( \begin{array}{ccc} 1 & 1 &T \\ +1 & -1 & -M_T\end{array} \right) +\left( \begin{array}{ccc} 1 & 1 &T \\ -1 & +1 & -M_T\end{array} \right) +\left( \begin{array}{ccc} 1 & 1 &T \\ -1 & -1 & -M_T\end{array} \right) \right] {t}_{\ell_1 m_1 ^\prime k_1^\prime k_2^\prime}.\end{gathered}$$
Finally, we average the product of the three remaining rotation matrices over all molecular orientations via Eq. (\[eq:3rotmatA\]). Equations (\[eq:AAA2\]) and (\[eq:3rotmatA\]) suggest that only the values $T=2$ and $M_T=0,-2$ are allowed, which yields: $$\begin{gathered}
\label{eq:AAA3}
\mathcal{A }=\frac{25\sqrt{3}}{16} \mathcal{E}_x\mathcal{E}^2_y \sum_{\ell_1 m_1^\prime}\sum_{\ell_2 m_2^\prime } \sum_{M_J^\prime M^\prime_T} \sqrt{(\ell_1)(\ell_2)}\,(i)^{\ell_1+\ell_2} (-1)^{\ell_2+m_1^\prime}\left( \begin{array}{ccc} \ell_1 & \ell_2 &2 \\0 & 0& 0\end{array} \right) \left( \begin{array}{ccc} \ell_1 & \ell_2 &2 \\-m_1^\prime & m_2^\prime& -M_J^\prime\end{array} \right) \\ \times
\left[ -\left( \begin{array}{ccc} 2& 1&2\\ +1 & -1&0\end{array} \right) \left\{ \left( \begin{array}{ccc} 1 & 1 &2 \\ +1 & -1 & 0\end{array} \right) +\left( \begin{array}{ccc} 1 & 1 &2 \\ -1 & +1 & 0\end{array} \right)\right\}
+\left( \begin{array}{ccc} 2& 1&2\\ +1 & +1&-2\end{array} \right) \left( \begin{array}{ccc} 1 & 1 &2 \\ -1 & -1 & 2\end{array} \right) \right] \\ \times\sum_{k_1^\prime k_2^\prime k^{\prime\prime}} \left( \begin{array}{ccc} 1 & 1 &2 \\ k_1^\prime& k_2^\prime& -M^\prime_T\end{array} \right)\left( \begin{array}{ccc} 2& 1&2\\ M_J^\prime & -k^{\prime\prime} & M^\prime_T \end{array} \right)d^\ast_{\ell_2 m_2 ^\prime k^{\prime\prime}} {t}_{\ell_1 m_1 ^\prime k_1^\prime k_2^\prime}.\end{gathered}$$ By making use of the following relations $M^\prime_T=k_1^\prime+ k_2^\prime $, $M_J^\prime=k^{\prime\prime} - M^\prime_T =k^{\prime\prime} -k_1^\prime- k_2^\prime$, and $ M_J^\prime= m_2^\prime -m_1^\prime$, the summations over indices $M_J^\prime$ and $M^\prime_T$ in Eq. (\[eq:AAA3\]) can be omitted. We now replace the second line of this equation by its explicit value of $\frac{2}{5\sqrt3}$ and arrive at the following final expression for the complex coefficient $\mathcal{A }$: $$\begin{gathered}
\label{eq:AAA4}
\mathcal{A }=\frac{5}{8} \mathcal{E}_x\mathcal{E}^2_y \sum_{\ell_1\ell_2 } \sqrt{(\ell_1)(\ell_2)}\, (i)^{\ell_1+\ell_2} (-1)^{\ell_2^\prime} \left( \begin{array}{ccc} \ell_1 & \ell_2 &2 \\0 & 0& 0\end{array} \right)\sum_{ m_1^\prime m_2^\prime } \sum_{k_1^\prime k_2^\prime k^{\prime\prime}} (-1)^{m_1} d^\ast_{\ell_2 m_2 ^\prime k^{\prime\prime}} {t}_{\ell_1 m_1 ^\prime k_1^\prime k_2^\prime} \\ \times\left( \begin{array}{ccc} \ell_1 & \ell_2 &2 \\-m_1^\prime & m_2^\prime& ( k_1^\prime+k_2^\prime- k^{\prime\prime})\end{array} \right) \left( \begin{array}{ccc} 2& 1&2\\ (m_2^\prime -m_1^\prime ) & -k^{\prime\prime} & ( k_1^\prime+k_2^\prime) \end{array} \right)\left( \begin{array}{ccc} 1 & 1 &2 \\ k_1^\prime& k_2^\prime& -(k_1^\prime+k_2^\prime)\end{array} \right).\end{gathered}$$
Discussion and outlook
======================
We first prove the chiral origin of the $FBUD$ asymmetry. For this purpose, we notice that only the partial waves of the same parity can contribute to the asymmetry, which is different from the case of a traditional PECD [@Ritchie]. This fact follows from the very fist 3j-symbol in Eq. (\[eq:AAA4\]), which suggests that $\ell_1+\ell_2$ must be even. We further notice that the second 3j-symbol in the second line of Eq. (\[eq:AAA4\]) changes its sign according to $$\label{eq:asym3j}
\left( \begin{array}{ccc} 2& 1&2\\ (m_2^\prime -m_1^\prime ) & -k^{\prime\prime} & ( k_1^\prime+k_2^\prime) \end{array} \right) = -\left( \begin{array}{ccc} 2& 1&2\\ -(m_2^\prime -m_1^\prime ) & k^{\prime\prime} & - (k_1^\prime+k_2^\prime) \end{array} \right).$$ Simultaneously, the remaining two 3j-symbols in the second line of Eq. (\[eq:AAA4\]) are symmetric with respect to the interchange of signs in the lower rows (since sum of upper indices is always even). Therefore, when performing summations over the complete range of indices $\left\{m_1^\prime,m_2^\prime,k^{\prime\prime}, k_1^\prime,k_2^\prime\right\}$, the respective contributions from the positive and negative angular momentum projections and light helicities will cancel out. Such a complete cancellation would not occur if the respective electric dipole transition amplitudes were inequivalent:
$$\label{eq:chiral1}
d_{\ell_2 m_2 ^\prime k^{\prime\prime}} \ne d_{\ell_2, -m_2 ^\prime, -k^{\prime\prime}},$$
$$\label{eq:chiral2}
{t}_{\ell_1 m_1 ^\prime k_1^\prime k_2^\prime} \ne {t}_{\ell_1, -m_1 ^\prime, -k_1^\prime, -k_2^\prime},$$
which is the case for chiral molecules [@Ritchie].
Numerical calculations, performed in Ref. [@PRLw2w] for the one- vs two-photon ionization of a model chiral system, showed that the $FBUD$ asymmetry is maximal for the relative phases $\phi=\pm \frac{\pi}{4}$ \[‘butterfly’ form [in Fig. \[fig\](b)]{}\], and it vanishes for $\phi=0$ and $\phi=\pm\frac{\pi}{2}$ \[‘horseshoe’ form [in Fig. \[fig\](b)]{}\]. This result is in agreement with Eq. (\[eq:FBUD3\]), if the argument $\delta$ of the complex coefficient $\mathcal{A}$ vanishes. In general, the chiral asymmetry (\[eq:FBUD3\]) is proportional to $\sin(2\phi-\delta)$, and for particular values of $\delta$ it maximizes and minimizes at different relative phases $\phi$. The argument $\delta$ of the complex coefficient $\mathcal{A}$ \[given by Eq. (\[eq:AAA4\])\] depends on the transition amplitudes for the one- and two-photon ionization, which are inherent electronic properties of a molecule at a given frequency $\omega$. It, therefore, can be considered as an internal phase, which is introduced to the fields by a molecule in additional to the external phase $\phi$. In the other words, for $\delta\ne 0$ a molecule ‘sees’ a field configuration which is different from that prepared by the experimental setup. Such an internal molecular phase [[@MolPhase]]{} is commonly utilized in different coherent control schemes [@CoCo1; @CoCo2; @Goetz19]. It is especially relevant for resonance-enhanced multiphoton ionization (REMPI) schemes, where different intermediate electronic states introduce additional phases to the interference between a manifold of multiphoton ionization pathways.
As a final point, we discuss the optical-regime photoionization with bichromatic fields (\[eq:field1\]), where higher-order multiphoton processes need to be involved. In this case, a manifold of interference terms between different multiphoton ionization pathways will contribute to the chiral asymmetry by several expansion terms $B_{LM}$ with even values of $L$ and odd values of $M$. The respective combined forward-backward and up-down asymmetry will be angularly-structured by the higher order $(\cos\theta_p)^{2n+1}$ and $\sin([2n+1]\varphi_p)$ terms. Moreover, the contributions from different multiphoton pathways will possess different dependencies on the external phase. As photoionization in the optical regime likely involves intermediate resonances, a molecule will introduce different internal phases to those pathways. The respective contributions, which scale as $\sin([2n]\phi-\delta_n)$, will minimize and maximize at different external phases.
The author acknowledges T. Baumert and O. Smirnova for many valuable discussions. This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Projektnummer 328961117 – SFB 1319 ELCH (subproject C1).
Useful relations {#sec:app}
================
Definition of [spherical basis]{}: $$\label{eq:cyclicA}
\hat{e}_\pm = \mp\frac{1}{\sqrt 2}\left(\hat{e}_x \pm i \hat{e}_y \right).$$
Transformation of the electric dipole operator from the laboratory to molecular frame: $$\label{eq:photonA}
\mathbf{d}_k = \sum_{k^\prime} \mathcal{D}^1_{k^\prime,k}(\alpha,\beta,\gamma)\, \mathbf{d}_{k^\prime} .$$
Transformation of the partial emission waves from the molecular to laboratory frame: $$\label{eq:electronA}
Y_{\ell m ^\prime}(\hat{p}^\prime) = \sum_{m} \mathcal{D}^{\ell\ast}_{m^\prime,m}(\alpha,\beta,\gamma)\, Y_{\ell m }(\hat{p}) .$$
Reduction of the product of two spherical functions: $$\begin{gathered}
\label{eq:directionA}
Y_{\ell_1 m_1 }(\hat{p})Y^\ast_{\ell_2 m_2 }(\hat{p})=\\(-1)^{m_2}\sum_{LM}\sqrt\frac{(\ell_1)(\ell_2)(L)}{4\pi}\left( \begin{array}{ccc} \ell_1 & \ell_2 &L \\0 & 0& 0\end{array} \right) \left( \begin{array}{ccc} \ell_1 & \ell_2 &L \\m_1 & -m_2& M\end{array} \right)Y^\ast_{LM }(\hat{p}),\end{gathered}$$ where $(\ell)=2\ell+1$ for brevity.
Explicit expressions for spherical functions: $$\label{eq:Y2mp1A}
Y^\ast_{2,\pm1 }(\hat{p})=\mp\frac{1}{2}\sqrt\frac{15}{2\pi}\cos\theta_p\sin\theta_p e^{\mp i\varphi_p}.$$
Reduction of the product of two rotation matrices: $$\begin{gathered}
\label{eq:DDD1A}
\mathcal{D}^{\ell_1\ast}_{m_1^\prime,m_1} \mathcal{D}^{\ell_2}_{m_2^\prime,m_2}=\\ (-1)^{m_1^\prime-m_1}\sum_{JM_JM^\prime_J}(-1)^{M_J-M^\prime_J} (J) \left( \begin{array}{ccc} \ell_1 & \ell_2 &J \\-m_1^\prime & m_2^\prime& -M^\prime_J\end{array} \right) \left( \begin{array}{ccc} \ell_1 & \ell_2 &J \\-m_1& m_2& -M_J\end{array} \right)\mathcal{D}^{J}_{M^\prime_J,M_J}\end{gathered}$$ $$\begin{gathered}
\label{eq:DDD2A}
\mathcal{D}^1_{k_1^\prime,\pm 1} \mathcal{D}^1_{k_2^\prime,\pm1}=\sum_{TM_TM^\prime_T}(-1)^{M_T-M^\prime_T} (T) \left( \begin{array}{ccc} 1 & 1 &T \\ k_1^\prime& k_2^\prime& -M^\prime_T\end{array} \right) \left( \begin{array}{ccc} 1 & 1 &T \\ \pm1 & \pm1 & -M_T\end{array} \right)\mathcal{D}^{T}_{M^\prime_T,M_T}\end{gathered}$$
Summation over indices $m_1$ and $m_2$: $$\begin{gathered}
\label{eq:DDD3A}
\sum_{m_1 m_2}(-1)^{m_2 -m_1} \left( \begin{array}{ccc} \ell_1 & \ell_2 &2 \\m_1 & -m_2& +1\end{array} \right) \left( \begin{array}{ccc} \ell_1 & \ell_2 &J \\-m_1& m_2& -M_J\end{array} \right) = (-1)^{M_J}(-1)^{\ell_1+ \ell_2 +J }\frac{\delta_{2,J} \delta_{1,M_J}}{(J)}.\end{gathered}$$
Averaging of the product of three rotation matrices over all molecular orientation angles: $$\label{eq:3rotmatA}
\frac{1}{8\pi^2} \int d^3(\alpha\beta\gamma)\, \mathcal{D}^{2}_{M_J^\prime,+1} \mathcal{D}^{1}_{-k^{\prime\prime},\mp1} \mathcal{D}^{T}_{M^\prime_T,M_T} = \left( \begin{array}{ccc} 2& 1&T\\ M_J^\prime & -k^{\prime\prime} & M^\prime_T \end{array} \right)\left( \begin{array}{ccc} 2& 1&T\\ +1 & \mp1& M_T\end{array} \right).$$
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This paper introduces a number of new intrinsically 3-linked graphs through five new constructions. We then prove that intrinsic 3-linkedness is not preserved by ${{\text{Y}}}\nabla$ moves. We will see that the graph $M$, which is obtained through a ${{\text{Y}}}\nabla$ move on $(PG)^*_*(PG)$, is not intrinsically 3-linked.'
address: 'Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA'
author:
- 'D. O’Donnol'
title: 'Intrinsic 3-linkedness is Not Preserved by ${{\text{Y}}}\nabla$ moves'
---
[^1]
Introduction
============
A graph, $G$, is *intrinsically knotted* if every embedding of $G$ in ${{\mathbb{R}}}^3$ contains a nontrivial knot. A link $L$ is *splittable* if there is an embedding of a 2-sphere $F$ in ${{\mathbb{R}}}^3\smallsetminus L$ such that each component of ${{\mathbb{R}}}^3\smallsetminus F$ contains at least one component of $L$. If $L$ is not splittable it is called *non-split*. A graph, $G$, is *intrinsically linked* if every embedding of $G$ in ${{\mathbb{R}}}^3$ contains a non-split link. A graph, $G$, is *minor minimal with respect to being intrinsically linked* (or simply *minor minimal intrinsically linked*) if $G$ is intrinsically linked and no minor of $G$ is intrinsically linked. The combined work of Conway and Gordon [@CG], Sachs [@Sa], and Robertson, Seymour, and Thomas [@RST] fully characterizes intrinsically linked graphs. The graphs in the Petersen family, shown in Figure \[PF\], are the complete set of minor minimal intrinsically linked graphs. So no minor of one of the graphs of the Petersen family is intrinsically linked, and every graph that is intrinsically linked contains one of these graphs as a minor. Let the set of the seven graphs of the Petersen family be denoted by $\mathcal{PF}$.
The concept of a graph being intrinsically linked can be generalized to a graph that intrinsically contains a link of more than two components. A graph $G$ is *intrinsically $n$-linked* if every embedding of $G$ in ${{\mathbb{R}}}^3$ contains a non-split $n$-component link. From here forward we will use *$n$-link* to mean a non-split $n$-component link. In this paper we focus on intrinsically 3-linked graphs. In Section \[back\], we discuss the set of known intrinsically 3-linked graphs and introduce our five new constructions. In Section \[i3-l\], we prove that each of the new constructions results in an intrinsically 3-linked graph.
(260, 330) (0,0)[![This figure shows the graphs in the Petersen family, and the arrows indicate $\nabla{{\text{Y}}}$ moves. []{data-label="PF"}](Pfamily "fig:")]{} (65,305)[$K_6$]{} (190,310)[$K_{3,3,1}$]{} (125,250)[$G_7$]{} (245,220)[$G_8$]{} (180,100)[$G_9$]{} (65,130)[$K_{4,4}^-$]{} (130,15)[$G_{10}=PG$]{}
(200, 70) (0,-10)[![The ${{\text{Y}}}\nabla$ move and $\nabla\text{Y}$ move.[]{data-label="Ytri"}](Ytri "fig:")]{} (72, 41)[**${{\text{Y}}}\nabla$ move**]{} (72, 20)[**$\nabla {{\text{Y}}}$ move**]{}
A *$\text{Y}\nabla$ move* on an abstract graph is where a valance 3 vertex, $v$, together with its adjacent edges are deleted, and three edges are added, one between each pair of vertices that had been adjacent to $v$. The reverse move is called a *$\nabla\text{Y}$ move*. See Figure \[Ytri\]. In [@Sa], Sachs showed that each graph in the Petersen family, i.e. all those graphs obtained from $K_6$ by ${{\text{Y}}}\nabla$ and $\nabla {{\text{Y}}}$ moves, is also minor minimal intrinsically linked. Motwani, Raghunathan, and Saran [@MRS] showed that both intrinsic linkedness and intrinsic knottedness are preserved by $\nabla{{\text{Y}}}$ moves. Their proof that intrinsic linkedness is preserved by $\nabla{{\text{Y}}}$ moves immediately generalizes to show that intrinsic $n$-linkedness is also preserved by $\nabla{{\text{Y}}}$ moves. Robertson, Seymour, and Thomas [@RST] showed that ${{\text{Y}}}\nabla$ moves also preserve intrinsic linkedness. On the other hand, Flapan and Naimi [@FN] showed that ${{\text{Y}}}\nabla$ moves do not preserve intrinsic knottedness. It is not known if intrinsic $n$-linkedness is preserved by ${{\text{Y}}}\nabla$ moves in general. The work in [@RST] showed that intrinsic $2$-linkedness is preserved by ${{\text{Y}}}\nabla$ moves. While the family of minor minimal intrinsically linked graphs (also minor minimal intrinsically 2-linked graphs) is connected by ${{\text{Y}}}\nabla$ and $\nabla {{\text{Y}}}$ moves, the family of minor minimal intrinsically 3-linked graphs is not [@FFNP]. It is also known that if the graph resulting from a ${{\text{Y}}}\nabla$ move on a minor minimal intrinsically $n$-linked graph is intrinsically $n$-linked, then it is minor minimal intrinsically $n$-linked [@BDLST]. In Section \[Ytri\], we prove that intrinsic 3-linkedness is not preserved by ${{\text{Y}}}\nabla$ moves.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The author would like to thank Dorothy Buck, Erica Flapan, Kouki Taniyama and R. Sean Bowman for helpful conversations and their continued support.
Intrinsically 3-linked graphs {#back}
=============================
There are a number of graphs already known to be intrinsically 3-linked. Figure \[I3L\] shows all those graphs that have been shown to be intrinsically 3-linked, where no minor is known to be intrinsically 3-linked (only $G(2)$ is known to be minor minimal intrinsically 3-linked). In [@FNP], Flapan, Naimi, and Pommersheim investigate intrinsically 3-linked graphs (or *intrinsically triple linked* graphs). They proved that the complete graph on ten vertices, $K_{10}$ is the smallest complete graph to be intrinsically 3-linked. Bowlin and Foisy [@BF] also looked at intrinsically 3-linked graphs. They exhibited two different subgraphs of $K_{10}$ that are also intrinsically 3-linked, the graph resulting from removing two disjoint edges from $K_{10}$, call it $K_{10}-\{2\text{ edges}\}$, and the graph obtained by removing four edges incident to a common vertex from $K_{10}$, call it $K_{10}^*$. So $K_{10}$ is not minor minimal intrinsically 3-linked. They also described two constructions that give intrinsically 3-linked graphs, the first being the graph that results from identifying an edge of $G_1$ with an edge of $G_2$, when $G_1$ and $G_2$ are either $K_7$ or $K_{4,4}$; we will call this graph $G_1|G_2$. Since all of the edges of $K_7$ are equivalent as are those of $K_{4,4}$ this gives rise to three graphs. One of these graphs, $K_{4,4}|K_{4,4}$ (also called $J$), was previously shown to be intrinsically 3-linked in [@FFNP]. The second is the graph obtained by connecting two graphs $G_1,G_2\in \mathcal{PF}$ by a 6-cycle where the vertices of the 6-cycle alternate between $G_1$ and $G_2$. We will call such a graph $(G_1CG_2)_i$. The subscript $i$ is given because there can be multiple ways to combine the same two graphs in this way. Not all of the vertices of each of the graphs of the Petersen family are equivalent, so there are many different ways to form the 6-cycle. Thus, there will be many more than $7+{7\choose 2}=28$ such graphs that one might first expect from combining the seven different graphs of $\mathcal{PF}$ in this construction. Due to the large number of graphs of this form they are not drawn in Figure \[I3L\] but instead a pictorial representation of any such graph is shown. Flapan, Foisy, Naimi, and Pommersheim addressed the question of minor minimal intrinsically $n$-linked graphs in [@FFNP], where they constructed a family of minor minimal intrinsically $n$-linked graphs. The minor minimal intrinsically $3$-linked graph they constructed was called $G(2)$, shown in Figure \[I3L\].
(260, 330) (0,0)[![The set of of previously known intrinsically 3-linked graphs, for which no known minor is intrinsically 3-linked. (The graph $G(2)$ is known to be minor minimal.)[]{data-label="I3L"}](I3Ln "fig:")]{} (33,215)[$K_{10}-\{2\text{ edges}\}$]{} (180,207)[$K_{10}^*$]{} (50,144)[$G(2)$]{} (175,148)[$G_1CG_2$]{} (25,7)[**$K_7|K_7$**]{} (109,11)[**$K_7|K_{4,4}$**]{} (178,23)[**$J=K_{4,4}|K_{4,4}$**]{}
In Section \[i3-l\], we prove that the following five constructions give rise to intrinsically 3-linked graphs: Let $G_1, G_2 \in \mathcal{PF}$ and let $v_i$ be a vertex in the graph $G_i$. Let the set of adjacent vertices to the vertex $v_i$ be $A_i$, for $i=1, 2$.
: Let the graph obtained by adding the edges between $v_1$ and all but one of the vertices of $A_2$ and the edges between $v_2$ and all but one of the vertices of $A_1$ to the graphs $G_1$ and $G_2$ be called $(G_1,v_1)^*_*(G_2, v_2)$. We call this the *double star construction*.
: Let the graph obtained by identifying the vertices $v_1$ and $v_2$, adding a vertex $x$ and the edges from $x$ to all but one of the vertices in the set $A_1$ and all but one of the vertices in the set $A_2$ be called $(G_1,v_1)^*_x(G_2, v_2)$.
: The graph $K_{4,4}^-$ has two vertices of valence three, label them $x$ and $y$. Let $G$ be one of the graphs of the Petersen family, let $v$ be one of the vertices of $G$, and let $A$ be the set of vertices adjacent to $v$ in $G$. Let the graph obtained by identifying the two vertices $x$ and $v$, and adding edges between $y$ and all but one of the vertices in $A$ be called $K_{4,4}(G,v)$.
: Let the *vertex identification construction* be the construction where a graph $H$ is formed by adding edges between the sets of vertices $A_1$ and $A_2$, such that between every pair of vertices of $A_1$ and pair of vertices of $A_2$ there is at least one edge joining a vertex from $A_1$ to a vertex from $A_2$ and then identifying the vertices $v_1$ and $v_2$ to get a single vertex $x$. Let the set of added edges between $A_1$ and $A_2$ be $E_{n,m}$, where $|A_1|=n$ and $|A_2|=m$.
: Let $V_i$ be the full vertex set of $G_i$. Let $(G_1)^\equiv_=(G_2)$ be the graph obtained by adding five disjoint edges between $V_1$ and $V_2$ to the graphs $G_1$ and $G_2$.
The vertex is dropped from the notation for Constructions 1, 2 and 3, if the vertices of the graph $G_1$ or $G_2$ are equivalent. These constructions introduce numerous new intrinsically 3-linked graphs. For example, the graphs $(K_6)^*_*(K_6)$, $(K_6)^*_x(K_6)$ are both subgraphs of $K_7|K_7$. See Figure \[mm\]. So these two graphs together with all of the graphs that can be obtained from them by $\nabla{{\text{Y}}}$ moves were previously unknown to be intrinsically 3-linked. Similarly, $K_{4,4}(K_6)$ is a subgraph of $K_7|K_{4,4}$. So the graph $K_{4,4}(K_6)$ introduces a set of new intrinsically 3-linked graphs.\
New intrinsically 3-linked graphs {#i3-l}
=================================
In this section we prove that the five new constructions explained in Section \[back\] give intrinsically 3-linked graphs. These constructions exploit some nice properties of the graphs in the Petersen family.
For each $G\in\mathcal{PF}$ every pair of disjoint cycles contains all of the vertices of $G$.
Thus every embedding of a graph from the Petersen family in ${{\mathbb{R}}}^3$ not only contains a 2-link but contains a 2-link which contains all of the vertices of the graph. It is known that, for any $G\in\mathcal{PF}$, every embedding of $G$ in ${{\mathbb{R}}}^3$ contains a two component link with odd linking number [@CG; @Sa]. So we will work with linking mod (2) and denote the mod (2) linking number of two simple closed curves $L$ and $J$ by $\omega (L,J)$.
![The graphs (a) $(K_6)^*_*(K_6)$, (b) $(K_6)^*_x(K_6)$, and (c) $K_{4,4}(K_6)$.[]{data-label="mm"}](newgraphs)
We will use the following lemma, proved in [@BF], to prove that Construction 1 of the previous section gives rise to an intrinsically 3-linked graph.
\[2path\] In an embedded graph with mutually disjoint simple closed curves, $C_1$, $C_2$, $C_3,$ and $C_4$, and two disjoint paths $x_1$ and $x_2$, such that $x_1$ and $x_2$ begin in $C_2$ and end in $C_3$, if $\omega(C_1,C_2)=\omega(C_3,C_4)=1$ then the embedded graph contains a non-splittable 3-component link.
\[starstar\] Let $(G_1,v_1)^*_*(G_2, v_2)$ be a graph obtained via Construction 1. Then $(G_1,v_1)^*_*(G_2, v_2)$ is intrinsically 3-linked.
Fix an arbitrary embedding of $(G_1,v_1)^*_*(G_2, v_2)$. Since $G_1$ is a graph in the Petersen family we know it must contain a 2-link and that $v_1$ must be in one of the components of the 2-link. Let the component that contains $v_1$ be called $C_2$ and the other component be called $C_1$. Note that $\omega(C_1,C_2)=1$. Similarly, $G_2$ must contain a 2-link and that $v_2$ must be in one of the components of the 2-link. Let the component that contains $v_2$ be called $C_3$ and the other component be called $C_4$, note that $\omega(C_3,C_4)=1$. Since $v_1$ is in $C_2$ two of the vertices adjacent to $v_1$ are also in $C_2$. At least one of these vertices must be adjacent to $v_2$, call it $a$. Note, the edge $\overline{v_1 a}$ goes between $C_2$ and $C_3$. Next, since $v_2$ is in $C_3$ two of the vertices adjacent to $v_2$ are also in $C_3$ and at least one of them is adjacent to $v_1$, call it $b$. The edge $\overline{b v_2}$ also goes between $C_2$ and $C_3$. Notice that $\overline{v_1 a}$ and $\overline{b v_2}$ cannot be the same edges by construction. Thus by Lemma \[2path\] we see that the chosen embedding contains a 3-link, and so $(G_1,v_1)^*_*(G_2, v_2)$ is intrinsically 3-linked.
To prove Proposition \[con2\], we will use the following lemma which appears in [@FFNP]:
\[3Lpath\] Suppose that $G$ is a graph embedded in ${{\mathbb{R}}}^3$ and contains the simple closed curves $C_1, C_2, C_3,$ and $C_4$. Suppose that $C_1$ and $C_4$ are disjoint from each other and both are disjoint from $C_2$ and $C_3$, and that $C_2$ and $C_3$ intersect in precisely one vertex $x$. Also, suppose there are vertices $u\neq x$ in $C_2$ and $v\neq x$ in $C_3$ and a path P in $G$ with endpoints $u$ and $v$ whose interior is disjoint from each $C_i$. If $\omega(C_1,C_2)=\omega(C_3,C_4)=1$, then there is a non-splittable 3-component link in $G$.
\[con2\] Let $(G_1,v_1)^*_x(G_2, v_2)$ be a graph obtained via Construction 2, then $(G_1,v_1)^*_x(G_2, v_2)$ is intrinsically 3-linked.
Let $A_i$ be the sets vertices and $x$ be the vertex as described in Construction 2, in the previous section. Fix an arbitrary embedding of $(G_1,v_1)^*_x(G_2, v_2)$. Since $G_1$ is in the Petersen family we know it must contain a 2-link and that $v_1$ must be in one of the components of the 2-link. Let the component that contains $v_1$ be called $C_2$ and the other component be called $C_1$; note that $\omega(C_1,C_2)=1$. Similarly, $G_2$ must contain a 2-link and that $v_2$ must be in one of the components of the 2-link. Let the component that contains $v_2$ be called $C_3$ and the other component be called $C_4$; note that $\omega(C_3,C_4)=1$. Since $v_1$ is in $C_2$ there are two vertices in $A_1$ that are also in $C_2$ and at least one of them is adjacent to $x$. Call it $a_i$. Similarly, since $v_2$ is in $C_3$ there are two vertices of $A_2$ that are also in $C_3$ and at least one of them is adjacent to $x$. Call it $b_i$. Let the path consisting of the two edges $\overline{a_ix}$ and $\overline{xb_i}$ be called $P$. The path $P$ goes from $C_2$ to $C_3$. So by Lemma \[3Lpath\] we see that the embedding contains a 3-link. Thus $(G_1,v_1)^*_x(G_2, v_2)$ is intrinsically 3-linked.
The graph $K_{4,4}(K_6)$ is shown in Figure \[mm\]. We will use the following lemmas in the proof of the next proposition about Construction 3. See Section \[back\] for the constructions.
\[k44\] [@Sa] Let $K_{4,4}$ be embedded in ${{\mathbb{R}}}^3$, then every edge of $K_{4,4}$ is in a component of a 2-link.
\[3l\] [@FNP] Suppose that $G$ is a graph embedded in ${{\mathbb{R}}}^3$ that contains the simple closed curves $C_1, C_2, C_3,$ and $C_4$. Suppose that $C_1$ and $C_4$ are disjoint from each other and both are disjoint from $C_2$ and $C_3$, and that $C_2\cap C_3$ is an arc. If $\omega(C_1,C_2)=1$ and $\omega(C_3,C_4)=1$, then there is a non-split 3-component link in $G$.
Let $K_{4,4}(G,v)$ be a graph obtained via Construction 3, then $K_{4,4}(G,v)$ is intrinsically 3-linked.
Let $x$, $y$, and $A$ be as described in Construction 3. Fix an arbitrary embedding of $K_{4,4}(G,v)$. Since $G$ is one of the graphs from the Petersen family it contains a 2-link $C_1\cup C_2$ which contains the vertex $v=x$, without loss of generality let $x$ be in $C_2$. Since $x$ is in $C_2$ two of the vertices adjacent to $x$, are also in $C_2$. So at least one of these vertices in $C_2$ is adjacent to $y$, call the vertex $a$. Label the path P, that is comprised of the two edges $\overline{xa}$ and $\overline{ay}$. Notice $\overline{xa}\in C_2$ and $\overline{ay}\notin C_1\cup C_2$. Now the subgraph $K_{4,4}^-$ together with the path $P$ form a subdivision of $K_{4,4}$, i.e. this can be viewed as $K_{4,4}$ where $P$ is one if the edges. By Lemma \[k44\] for every embedding of $K_{4,4}$ each edge is contained in a 2-link, so $P$ is contained in a 2-link $C_3\cup C_4$. Without loss of generality, let $P$ be an edge of $C_3$. So $C_2\cap C_3=\overline{xa}$ is an arc, $\omega(C_1,C_2)=1$ and $\omega(C_3,C_4)=1$. Thus by Lemma \[3l\] the embedding of $K_{4,4}(G,v)$ contains a 3-link.
![Let $|A_1|=n$ and $|A_2|=m$. This figure shows possible sets of added edges $E_{n,m}$ for the vertex identification constructions for all different possible sizes of the vertex set $A_1$ and $A_2$. For each of them the vertex sets $A_1$ and $A_2$ are shown vertically and additional edges $E_{n,m}$ are shown. []{data-label="addedge"}](add_edges)
Recall Construction 4, the vertex identification construction, where a set of edges $E_{n,m}$ is added between the two sets of vertices $A_1$ and $A_2$. For the full definition refer to Section \[back\]. The graphs in the Petersen family have vertices of valence 3, 4, 5, and 6. Let $|A_1|=n$ and $|A_2|=m$. We want to construct $E_{n,m}$, a set of edges between $A_1$ and $A_2$ such that given any pair of vertices from $A_1$ and any pair of vertices from $A_2$ there is an edge between two of the vertices from the chosen pair. So each pair of vertices from $A_1$ must be connected to $m-1$ vertices from $A_2$. To reduce the total number of edges needed we divide the edges evenly between the two vertices, so each vertex of $A_1$ is connected to $\frac{m-1}{2}$ vertices of $A_2$. Suppose $m\geq n$, this gives a lower bound of $|E_{n,m}|=(\frac{m-1}{2})n$, if $m$ is odd, and $|E_{n,m}|=(\frac{m}{2})(n-1)+\frac{m-2}{2}$ if $m$ is even. Figure \[addedge\] shows possible sets of edges $E_{n,m}$ to be added between $A_1$ and $A_2$ for all possible combinations of valence. It can be checked that there sets of vertices satisfy the criterion. However these are not the only possible $E_{n,m}$ sets, and it is not known if they are optimal. In the case of $n=m=3$ then $|E_{3,3}|=3$ is the lower bound but in all other examples given $|E_{n,m}|$ is greater than the lower bound obtained.
\[VIdI3L\]Any graph $H$ constructed through the vertex identification construction of graphs $G_1$ and $G_2$ in the Petersen family is intrinsically 3-linked.
Consider an arbitrary embedding of $H$. Let the notation be as in construction 4: the identified vertex is labelled $x$, the sets of vertices adjacent to $x$ in the subgraphs $G_1$ and $G_2$, respectively, are labelled $A_1$ and $A_2$, and the set of edges between $A_1$ and $A_2$ is $E_{n,m}$. Since every vertex of a Petersen graph is contained in every link in the embedding, $x$ is in one of the components of the link in each $G_i$. Let the components of the link in $G_1$ be labelled $C_1$ and $C_2$, with the vertex $x$ in the component $C_2$, and let the components of the link in $G_2$ be labelled $C_3$ and $C_4$, with the vertex $x$ in the component $C_3$. A pair of vertices from the set $V_1$ is also part of $C_2$ and, similarly, a pair of vertices from the set $V_2$ is also part of $C_3$. By construction, there is an edge, $e$, of the set $E_{n,m}$ between $C_2$ and $C_3$. Since none of the edges of $E_{n,m}$ are contained in $G_1$ or $G_2$, the interior of the edge $e$ is disjoint from the links $C_1\cup C_2$ and $C_3\cup C_4$. Thus, by Lemma \[3Lpath\], H contains a 3-link.
The graph $(G_1)^\equiv_=(G_2)$ obtained by Construction 5, is intrinsically 3-linked.
Let $V_i$ be the vertex set of $G_i$, let the set of five added edges be $E$. Fix an embedding of $(G_1)^\equiv_=(G_2)$. Since $G_1,G_2\in \mathcal{PF}$, $G_1$ contains a 2-link that contains all of the vertices of $V_1$, call the link $C_1\cup C_2$, and $G_2$ contains a 2-link that contains all of the vertices of $V_2$, call the link $C_3\cup C_4$. Because all of the vertices are in one of the 2-links each edge of $E$ will go between components of the different 2-links. There are four different pairs of components that can be connected by the said edges, so by the pigeonhole principle two of the edges must go between the same pair of components. By Lemma \[2path\] the embedding contains a 3-link. Thus $(G_1)^\equiv_=(G_2)$ is intrinsically 3-linked.
Intrinsic 3-linkedness is not preserved by ${{\text{Y}}}\nabla$ moves {#Ytri}
=====================================================================
In this section, we show that intrinsic 3-linkedness is not preserved by ${{\text{Y}}}\nabla$ moves.
(260, 220) (10,0)[![The graph $M$ is obtained from $(PG)^*_*(PG)$ by a ${{\text{Y}}}\nabla$ move on the indicated bold edges. []{data-label="Ytri"}](PGYtri "fig:")]{} (200,140)[$(PG)^*_*(PG)$]{} (200,15)[**$M$**]{}
(260, 130) (0,10)[![The spatial graph $f(M)$. An embedding of $M$ that does not contain a 3-link. []{data-label="f(M)"}](Memb "fig:")]{} (25,88)[**1**]{} (45,94)[**2**]{} (63,104)[**3**]{} (107,90)[**4**]{} (107,72)[**5**]{} (107,50)[**6**]{} (65,7)[**7**]{} (45,20)[**8**]{} (25,35)[**9**]{} (150,108)[**$a$**]{} (200,118)[**$b$**]{} (133,72)[**$c$**]{} (180,93)[**$d$**]{} (230,72)[**$e$**]{} (157,55)[**$f$**]{} (187,75)[**$g$**]{} (203,57)[**$h$**]{} (150,23)[**$j$**]{} (207,23)[**$k$**]{}
Intrinsic 3-linkedness is not preserved by ${{\text{Y}}}\nabla$ moves.
We begin with $(PG)^*_*(PG)$, since all of the vertices of $PG$ are equivalent there is a single graph that can be obtained throughout the double star construction with two Petersen graphs. By Proposition \[starstar\], $(PG)^*_*(PG)$ is intrinsically 3-linked. Let the graph obtained by a ${{\text{Y}}}\nabla$ move on $(PG)^*_*(PG)$ as indicated in Figure \[Ytri\] be called $M$.
We claim that, the graph $M$ is not intrinsically 3-linked. Consider the embedding $f(M)$ shown in Figure \[f(M)\]. Let the vertices be labelled as indicated. Let $K_1$ be the embedded subgraph defined by the vertices 1, 2, 3, 4, 5, 6, 7, 8, 9 and the edges between them in $f(M)$, and let $K_2$ be the embedded subgraph defined by the vertices $a, b, c, d, e, f, g, h, j, k$ and the edges between them. For $f(M)$ to contain a 3-link, the 3-link must be in both the embedded subgraphs $K_1$ and $K_2$, since neither contains three disjoint simple closed curves on their own. Since $K_1$ and $K_2$ are disjoint and there is no linking between them, two of the edges joining the subgraphs must also be in the 3-link. The subgraph $K_1$ contains a single linked pair of cycles, indicated with thickened edges. Similarly, in $K_2$ there is a single linked pair of cycles, indicated with thickened edges. All other cycles in the subgraphs $K_1$ and $K_2$ bound disks that do not intersect the graph in their interiors. No pair of edges between the two subgraphs $K_1$ and $K_2$ connects two of the linked cycles. So there is no 3-link in $f(M)$.
Notice that there are many graphs that can be constructed with the double star construction that are a ${{\text{Y}}}\nabla$ move away from a graph that is not intrinsically 3-linked. Consider $(PG)^*_*G$ for any $G\in\mathcal{PF}$, a similar ${{\text{Y}}}\nabla$ move to that is the proof above will produce a graph that is not intrinsically 3-linked. More generally, this can be done with any double star construction where the vertex of $A_i$ that is not connected to the vertex $v_j$ is trivalent.
[mmI3L]{}
G. Bowlin and J. Foisy, [*Some new intrinsically 3-linked graphs*]{}, J. of Knot Theory Ramifications [**13**]{}(8) (2004), 1021–1027.
A. Brouwer, R. Davis, A. Larkin, D. Studenmund, C. Tucker, [*Intrinsically $S^1$ 3-linked graphs and other aspects of $S^1$ embeddings*]{}, Rose-Hulman Undergrad. Math. J. [**8**]{} (2007).
J. Conway and C. Gordan, [*Knots and links in spatial graphs*]{}, J. of Graph Theory [**7**]{} (1983), 445–453.
E. Flapan, J. Foisy, R. Naimi, and J. Pommersheim, [*Intrinsically n-linked graphs*]{}, J. of Knot Theory Ramifications [**10**]{}(8) (2001), 1143–1154.
E. Flapan, and R. Naimi, [*The Y-triangle move does not preserve intrinsic knottedness*]{}, Osaka J. Math. [**45**]{} (2008) 107-111.
E. Flapan, R. Naimi, and J. Pommersheim, [*Intrinsically triple linked complete graphs*]{}, Topol. Appl. [**115**]{} (2001), 239–246.
R. Motwani, A. Raghunathan and H. Saran, [*Constructive results for graph minors: Linkless embeddings*]{}, 29th Annual Symposium on Foundations of Computer Science, IEEE (1988), 398–409.
N. Robertson, P. Seymour, and R. Thomas, [*Sachs’ linkless embedding conjecture*]{}, J. of Combinatorial Theory, Series B [**64**]{} (1995), 185–227.
H. Sachs, [*On a spatial analogue of Kuratowski’s Theorem on planar graphs – an open problem*]{}, Graph Theory, Lag$\acute{o}$w, 1981, Lecture Notes in Mathematics, Vol. 1018 (Springer-Verlag, Berlin, Heidelberg, 1983), 649–662.
H. Sachs, [*On spatial representations of finite graphs*]{}, Colloq. Math. Soc. János Bolyai, Vol. 37 (North-Holland, Budapest, 1984), 649–662.
[^1]: Supported in part by a NSF-AWM Mathematics Mentoring Travel Grant
|
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"pile_set_name": "ArXiv"
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---
abstract: 'The scaling properties of direct photon production in $pp, \bar pp$ and $pA$ collisions at high energies is reviewed. The experimental data on the cross sections obtained at ISR, SpS and Tevatron are used in the analysis. The properties of data $z$-presentation, the energy and angular independencies, the power law, and $A$-dependence are discussed. The use of $z$-scaling to search for new physics phenomena in hadron-hadron, hadron-nucleus and nucleus-nucleus collisions is suggested. The violation of $z$-scaling characterized by the change of the fractal dimension is considered as a new and complimentary signature of a nuclear phase transition. The cross sections of direct photon, $\pi^0$- and $\eta^0$-meson production in $pp$ and $pPb$ collisions at RHIC and LHC energies are predicted.'
author:
- 'M.Tokarev, G.Ef[i]{}mov'
title: ' $Z$-SCALING, HIGH-$pT$ DIRECT PHOTON AND $ \pi^0$-MESON PRODUCTION AT RHIC AND LHC '
---
INTRODUCTION
============
Direct photon production due to small cross section of electromagnetic interactions is one among very few signals which can provide direct information on the partonic, early phase of interaction. ’Penetrating probes’, direct photons and dilepton pairs are traditionally considered to be one of the best probes for Quark-Gluon Plasma (QGP) [@Feinberg; @Shuryak; @McLerran].
Direct photons are produced through different mechanisms (Compton scattering, quark annihilation, bremsstrahlung of quarks and gluons). Decay of hadrons such as $\pi^0$- and $\eta^0$-mesons is a source of background for direct photon production. Therefore it is important to establish general features of photon production in hadron-hadron ($pp, \bar pp, \pi p$ etc.) collisions and then to search for their violation at extremal conditions (high multiplicity particle density, high $p_T$ etc.) taking place in hadron-nucleus and nucleus-nucleus collisions.
The description of direct photon production in the framework of QCD [@Auren; @Jalil; @Jeon] and comparison with available experimental data reveal numerous ambiguities . Some of them are theoretical and connected with the choice of the factorization scheme, renormalization, factorization and fragmentation scales and with considerations of higher order QCD corrections. The other ones are relevant to consistency among different experimental data sets. The third ones are phenomenological and are introduced to correct theory by the model dependent manner (for example “$k_T$”-smearing effect).
Nuclear effects such as multiple parton interactions, nuclear shadowing , energy loss are not small [@Jalil] and should be taken into account for calculations of direct photon cross section in heavy ion collisions. These effects modify nuclear structure functions, photon fragmentation functions and add uncertainties in the theoretical calculations of cross sections.
We use the concept of $z$-scaling to analyze numerous experimental data on direct photon cross sections for $pp$, $\bar pp$ and $pA$ collisions at high $ p_T$ and to make some predictions for $\gamma, \pi^0$ and $\eta^0$ produced in $pp$ and $pA$ collisions at the RHIC and LHC energies. The method of data analysis is complementary to a method of direct calculations developed in the framework of QCD and methods based on Monte Carlo generators. The use of the method allow us to obtain additional constraints in order to reduce the theoretical uncertainties and to estimate more reliably the photon cross section and background.
$Z$-SCALING
===========
The idea of $z$-scaling [@Z96]-[@Z00] is based on the assumptions that inclusive particle distribution of the process (\[eq:r1\]) at high energies and high $p_T$ can be described in terms of the corresponding kinematic characteristics $$P_{1}+P_{2} \rightarrow q + X \label{eq:r1}$$ of the exclusive elementary sub-process [@Stavinsky] and that the scaling function depending on a single variable $z$ exists and can be expressed via the dynamic quantities, invariant inclusive cross section $Ed^3\sigma/{dq^{3}}$ of the process (\[eq:r1\]) and particle multiplicity density $\rho(s,\eta)$.
The elementary parton-parton collision is considered as a binary sub-process which satisfies the condition
$$(x_{1}P_{1} + x_{2}P_{2} - q)^{2} = (x_{1}M_{1} + x_{2}M_{2} +
m_{2})^{2}.
\label{eq:r5}$$
The equation reflects minimum recoil mass hypothesis in the elementary sub-process. To connect kinematic and structural characteristics of the interaction, the coefficient $\Omega$ is introduced. It is chosen in the form $$\Omega(x_1,x_2) = m(1-x_{1})^{\delta_1}(1-x_{2})^{\delta_2},
\label{eq:r8}$$ where $m$ is a mass constant and $\delta_1$ and $\delta_2$ are factors relating to the fractal structure of the colliding objects [@Z99]. The fractions $x_{1}$ and $x_{2}$ are determined to maximize the value of $\Omega(x_1,x_2)$, simultaneously fulfilling the condition (\[eq:r5\]) $${d\Omega(x_1,x_2)/ dx_1}|_{x_2=x_2(x_1)} = 0.
\label{eq:r9}$$ The variables $x_{1,2}$ are equal to unity along the phase space limit and cover the full phase space accessible at any energy.
Scaling function $\psi(z)$ and variable $z$
-------------------------------------------
The scaling function $\psi(z)$ is written in the form [@Z99] $$\psi(z) = - \frac{\pi s_A}{\rho_A(s,\eta) \sigma_{inel}}J^{-1}
E\frac{d\sigma}{dq^{3}}.
\label{eq:r20}$$ Here $\sigma_{inel}$ is the inelastic cross section, $s_A \simeq
s \cdot A$ and $s$ are the center-of-mass energy squared of the corresponding $ h-A $ and $ h-N $ systems, $A$ is the atomic weight and $\rho_A(s,\eta)$ is the average particle multiplicity density. The factor $J$ is the known function of the kinematic variables, the momenta and masses of the colliding and produced particles [@Z99].
The function is normalized as $$\int_{z_{min}}^{\infty} \psi(z) dz = 1. \label{eq:b6}$$ The equation allows us to give the physical meaning of the function $\psi$ as a probability density to form a particle with the corresponding value of the variable $z$.
In accordance with the approach suggested in [@Z99], the variable $z$ is taken in the form (\[eq:r28\]) as a simple physically meaningful variable reflecting self-similarity and fractality as a general pattern of hadron production at high energies $$z = \frac{ \sqrt{ {\hat s}_{\bot} }} {\Omega \cdot \rho_A(s) }.
\label{eq:r28}$$ Here $\sqrt{ {\hat s}_{\bot} } $ is the minimal transverse energy of colliding constituents necessary to produce a real hadron in the reaction (\[eq:r1\]). The factor $\Omega$ is given by (\[eq:r8\]) and $\rho_A(s) =\rho_A(s, \eta)|_{\eta=0}$. The form of $z$ determines its variation range $(0,\infty) $. These values are scale independent and kinematically accessible at any energy.
One of the features of the procedure to construct $\psi(z)$ and $z$ described above is the joint use of the experimental quantities characterizing hard ($Ed^3\sigma/{dq^{3}}$) and soft ($\rho_A(s,
\eta)$) processes of particle interactions. Let us clarify the physical meaning of the variable $z=\sqrt{\hat s_{\bot}}/(\Omega\rho_A)$. The value $ \sqrt {\hat s_{\bot}}$ is the minimal transverse energy of colliding constituents necessary to produce a real hadron in the reaction (\[eq:r1\]). It is assumed that two point-like and massless elementary constituents interact with each other in the initial state and convert into real hadrons in the f[i]{}nal state. The conversion is not instant process and is called hadronization or particle formation. The microscopic space-time picture of the hadronization is not understood enough at present time. We assume that number of hadrons produced in the hard interaction of constituents is proportional to $\rho_A$. Therefore the value $ \sqrt {\hat s_{\bot}}/\rho_A$ corresponds to the energy density per one hadron produced in the sub-process. The factor $\Omega $ is relative number of all initial configurations containing the constituents which carry the momentum fractions $x_1$ and $x_2$. This factor thus represents a tension in the considered sub-system with respect to the whole system. Taking into account the qualitative scenario of hadron formation as a conversion of a point-like constituent into a real hadron we interpreted the variable $z$ as particle formation length.
PROPERTIES of $z$-SCALING
=========================
In this section we discuss properties of the $z$-scaling for direct photons produced in $pp$, $\bar pp$ and $pA$ collisions. They are the energy and angular independencies of data $z$-presentation, the power law of the scaling function at very high-$p_T$ and $A$- dependence of $z$-scaling. All properties are asymptotic ones because they reveal themselves at extreme conditions (high $\sqrt s$ and $p_T$). Numerous experimental data obtained at ISR [@WA70]-[@R807], SpS [@UA6p]-[@UA2] and Tevatron [@E704]-[@E706g] were used in the analysis.
Energy independence
-------------------
The energy independence of data $z$-presentation means that the scaling function $\psi(z)$ has the same shape for different $\sqrt s$ over a wide $p_T$ range.
Figures 1(a,c) show the dependence of the cross section of direct photon production in $pp$ and $\bar pp$ interactions on transverse momentum $p_T$ at different $\sqrt s $ over a central rapidity range. We would like to note that the data cover a wide transverse momentum range, $p_T = 19-63~GeV/c$ and $p_T=24-1800~GeV/c$ for $pp$ and $\bar pp$, respectively.
![ Dependence of inclusive cross sections of direct photon production on transverse momentum in $pp$ (a) and $\bar pp$ (c) collisions at $\sqrt s = 19-1800~GeV$. Experimental data are taken from [@WA70]-[@E706g]. The corresponding scaling functions $\psi(z)$ for $pp$ and $\bar pp$ collisions are shown in (b) and (d).](fig1a.eps "fig:"){width="6.5cm"} -6.cm ![ Dependence of inclusive cross sections of direct photon production on transverse momentum in $pp$ (a) and $\bar pp$ (c) collisions at $\sqrt s = 19-1800~GeV$. Experimental data are taken from [@WA70]-[@E706g]. The corresponding scaling functions $\psi(z)$ for $pp$ and $\bar pp$ collisions are shown in (b) and (d).](fig1b.eps "fig:"){width="6.5cm"}
0.5cm
a\) b)
![ Dependence of inclusive cross sections of direct photon production on transverse momentum in $pp$ (a) and $\bar pp$ (c) collisions at $\sqrt s = 19-1800~GeV$. Experimental data are taken from [@WA70]-[@E706g]. The corresponding scaling functions $\psi(z)$ for $pp$ and $\bar pp$ collisions are shown in (b) and (d).](fig1c.eps "fig:"){width="6.5cm"} -6.5cm ![ Dependence of inclusive cross sections of direct photon production on transverse momentum in $pp$ (a) and $\bar pp$ (c) collisions at $\sqrt s = 19-1800~GeV$. Experimental data are taken from [@WA70]-[@E706g]. The corresponding scaling functions $\psi(z)$ for $pp$ and $\bar pp$ collisions are shown in (b) and (d).](fig1d.eps "fig:"){width="6.5cm"}
c\) d)
Let us note some features of the photon spectra. The first one is the strong dependence of the cross section on energy $\sqrt
s$. The second feature is a tendency that the difference between photon yields increases with the transverse momentum $p_T$ and the energy $\sqrt s$. The third one is a non-exponential behavior of the spectra at $q_{T}>4~GeV/c$.
Figures 1(b,d) show $z$-presentation of the same data sets. Taking into account the experimental errors we can conclude that the scaling function $\psi(z)$ demonstrates energy independence over a wide energy and transverse momentum range at $\theta_{cms} \simeq 90^0$.
Angular independence
--------------------
The angular independence of data $z$-presentation means that the scaling function $\psi(z)$ has the same shape for different values of the angle $\theta_{cms} $ of produced photon over a wide $p_T$ range and $\sqrt s$. Taking into account the energy independence of $\psi(z)$ it will be enough to verify the property at some $\sqrt s$.
To analyze the angular dependence of the scaling function $\psi(z)$ we use some data sets. The first one obtained at Tevatron [@D0pho] includes the results of measurements of the invariant cross section $Ed^3\sigma/dq^3$ at $\sqrt s =
1800~GeV$ over a momentum and angular ranges of $p_T=10-115~GeV/c$ and $0.0<|\eta|<2.5$. The second one is the E706 data set [@E706g] for direct photons produced in $pp$ collisions at $\sqrt s = 31.6$ and $38.8~GeV$ and in the rapidity range $(-1.0,0.75)$.
![ (a) Dependence of inclusive cross sections of direct photon production on transverse momentum in $pp$ collisions at $\sqrt s
= 24.3,31.6, 38.8$ and $63.~GeV$. Experimental data $+, \star $ and $\circ, \triangle $ are taken from [@R806; @UA6p] and [@E706g], respectively. (b) The corresponding scaling function $\psi(z)$.](fig2a.eps "fig:"){width="6.5cm"} -6.5cm ![ (a) Dependence of inclusive cross sections of direct photon production on transverse momentum in $pp$ collisions at $\sqrt s
= 24.3,31.6, 38.8$ and $63.~GeV$. Experimental data $+, \star $ and $\circ, \triangle $ are taken from [@R806; @UA6p] and [@E706g], respectively. (b) The corresponding scaling function $\psi(z)$.](fig2b.eps "fig:"){width="6.5cm"} a) b)
![ (a) Dependence of inclusive cross sections of direct photon production on transverse momentum in $pp$ collisions at $\sqrt s
= 24.3,31.6, 38.8$ and $63.~GeV$. Experimental data $+, \star $ and $\circ, \triangle $ are taken from [@R806; @UA6p] and [@E706g], respectively. (b) The corresponding scaling function $\psi(z)$.](fig3a.eps "fig:"){width="6.5cm"} -6.cm ![ (a) Dependence of inclusive cross sections of direct photon production on transverse momentum in $pp$ collisions at $\sqrt s
= 24.3,31.6, 38.8$ and $63.~GeV$. Experimental data $+, \star $ and $\circ, \triangle $ are taken from [@R806; @UA6p] and [@E706g], respectively. (b) The corresponding scaling function $\psi(z)$.](fig3b.eps "fig:"){width="6.5cm"}
c\) d)
Figures 2(a) and 3(a) show the dependence of the cross section of direct-$\gamma$ production in $\bar pp$ and $pp$ collisions on transverse momentum at the fixed $\sqrt s$ and for different rapidity intervals. A strong angular dependence of the cross section was experimentally observed for D0 data and was found to be much smaller for E706 data. The last data have been averaged over the central rapidity range and therefore the angular dependence of the data is weak.
Figures 2(b) and 3(b) demonstrate $z$-presentation of the same data sets. The obtained results show that the function $\psi(z)$ is independent of the angle $\theta_{cms}$ over a wide range of transverse momentum $p_T$ and energy $\sqrt s$. This is the experimental confirmation of the angular scaling of data $z$-presentation.
We would like to note that absolute normalization factors for [@WA70; @R806] and [@E706g] data sets are found to be different. The ratio is about factor 0.5.
Power law
---------
Here, we discuss a new feature of data $z$-presentation for direct-$\gamma$ production. This is the power law of the scaling function, $\psi(z) \sim z^{-\beta}$.
As seen from Figures 2(b) and 3(b) the data sets demonstrate a linear $z$-dependence of $\psi(z)$ on the log-log scale at high $z$. The quantity $\beta $ is a slope parameter.
Taking into account the accuracy of the available experimental data, we can conclude that the behavior of $\psi(z)$ for direct photons produced in $\bar pp$ and $pp$ collisions reveals a power dependence and the value of the slope parameter is independent of the energy $\sqrt s$ over a wide range of high transverse momentum. It was also found that $\beta_{pp}^{\gamma} > \beta_{\bar pp}^{\gamma}$.
Direct photons are mainly produced in $p-p$ and $\bar p-p$ collisions through the Compton and annihilation processes, respectively. This fact causes different values of the slope parameters $\beta_{pp}^{\gamma}$ and $\beta_{\bar pp}^{\gamma}$.
The existence of the power law, $\psi(z) \sim z^{-\beta }$, means, from our point of view, that the mechanism of particle formation reveals fractal behavior.
A-dependence
------------
A study of $A$-dependence of particle production in $hA$ and $AA$ collisions is traditionally connected with nuclear matter influence on particle formation. The difference between the cross sections of particle production on free and bound nucleons is normally considered as an indication of unusual physics phenomena like EMC-effect $J/\psi$-suppression and Cronin effect [@Cronin].
A-dependence of $z$-scaling for particle production in $pA$ collisions was studied in [@Z01]. It was established $z$-scaling for different nuclei ($A=D-Pb$) and type of produced particles ($\pi^{\pm,0}, K^{\pm}, \bar p$). The symmetry transformation of the scaling function $\psi(z)$ and variable $z$ under the scale transformation $z\rightarrow \alpha_A z$, $\psi
\rightarrow \alpha_A^{-1} \psi $ was suggested to compare the scaling functions for different nuclei. It was found that $\alpha$ depends on the atomic number only and can be parameterized by the formula $\alpha(A)=0.9A^{0.15}$ [@Z01].
We use the parameterization $\alpha (A)$ to study $A$-dependence of direct photon production in $pA$ collisions. New data [@E706g] obtained by E706 Collaboration are used in the analysis. The experimental cross sections have been measured for $pBe$ and $pCu$ collisions at $\sqrt s = 31.6$ and $38.8~GeV$ and cover the $p_T$-range $(3-11)~GeV/c$.
Figure 4(a) demonstrates the spectra of photons produced in proton-nucleus collisions. As seen from Figure 4(a) the $p_T$-spectra shows the strong energy dependence. The difference between spectra at $\sqrt s = 31.6$ and $38.8~GeV$ increases with $p_T$. The $z$-presentation of the same data is shown in Figure 4(b). The scaling functions for both targets, $Be$ and $Cu$, coincide each other. This is the direct confirmation that a nuclear effect for direct photon production can be described by the same function $\alpha (A)$ as for hadrons produced in proton-nucleus collisions [@Z01]. The shape of the scaling functions is found to be a linear one on the log-log scale for both cases. The fit of the data is shown by the solid line in Figure 4(b).
![ Dependence of inclusive cross sections of direct photon (a) and $\pi^0$-meson (c) production on transverse momentum in $pBe$ and $pCu$ collisions at $\sqrt s = 31.6$ and $38.8~GeV$. Experimental data are taken from [@E706g]. The corresponding scaling functions $\psi(z)$ for $\gamma$ and $\pi^0$ are shown in (b) and (d). ](fig4a.eps "fig:"){width="6.5cm"} -6.cm ![ Dependence of inclusive cross sections of direct photon (a) and $\pi^0$-meson (c) production on transverse momentum in $pBe$ and $pCu$ collisions at $\sqrt s = 31.6$ and $38.8~GeV$. Experimental data are taken from [@E706g]. The corresponding scaling functions $\psi(z)$ for $\gamma$ and $\pi^0$ are shown in (b) and (d). ](fig4b.eps "fig:"){width="6.5cm"}
0.5cm
a\) b)
![ Dependence of inclusive cross sections of direct photon (a) and $\pi^0$-meson (c) production on transverse momentum in $pBe$ and $pCu$ collisions at $\sqrt s = 31.6$ and $38.8~GeV$. Experimental data are taken from [@E706g]. The corresponding scaling functions $\psi(z)$ for $\gamma$ and $\pi^0$ are shown in (b) and (d). ](fig4c.eps "fig:"){width="6.5cm"} -6.5cm ![ Dependence of inclusive cross sections of direct photon (a) and $\pi^0$-meson (c) production on transverse momentum in $pBe$ and $pCu$ collisions at $\sqrt s = 31.6$ and $38.8~GeV$. Experimental data are taken from [@E706g]. The corresponding scaling functions $\psi(z)$ for $\gamma$ and $\pi^0$ are shown in (b) and (d). ](fig4d.eps "fig:"){width="6.5cm"}
c\) d)
The value of the slope parameter $\beta_{pBe}^{\gamma}$ is constant over a wide $p_T$ range and equal to 7.07. The fact means that the nuclear matter changes the probability of photon formation with different formation length $z$ and does not change the fractal dimension of the mechanism of photon formation (photon “dressing”).
Taking into account an experimental accuracy of data used in the analysis, the obtained results show that the fractal dimension $\delta$ and the slope parameter $\beta$ is independent of A. Therefore the experimental investigations of $A$-dependence of $z$-scaling for direct photons produced in hadron-nucleus collisions at RHIC and LHC energies are very important to obtain any indications on nuclear phase transition and formation of QGP.
The main source of the background for direct photon production are $\pi^0$ and $\eta^0$-mesons decay. Therefore it is important to obtain a reliable estimation of the background. This can be done by using the scaling function of $\pi^0$-mesons for calculation of cross sections at the corresponding energy. We use experimental data [@E706g] on $\pi^0$-meson cross sections to construct the scaling function $\psi(z)$.
Figure 4 shows the dependence of inclusive cross section of $\pi^0$-meson produced in proton-nucleus collisions on transverse momentum and the results of $z$-presentation of the same data sets. The values of the fractal dimension $\delta$ were found to be different, 0.5 and 0.8, for $\pi^0$ and $\gamma$ production, respectively. The values of the slope parameter $\beta$ of $\psi(z) \sim z^{-\beta}$ were found to be different as well.
DIRECT $\gamma$, $\pi^0$ and $\eta^0$ PRODUCTION
================================================
The properties of $z$-scaling found for direct-$\gamma$, $\pi^0$- and $\eta^0$-meson production allow us to calculate spectra of photons produced in $pp$ and $pA$ collisions at RHIC and LHC energies.
$pp$ and $\bar p$ collisions
-----------------------------
Results of our analysis of numerous experimental data on direct photon production in $pp$ and $\bar pp$ collisions in the framework of $z$-scaling scheme show that the fractal dimension $\delta$ is independent of energy $\sqrt s$ over a wide range of transverse photon momentum. Therefore the violation of $z$-scaling could give indications on modification of the mechanism of direct photon formation by a new type of interaction beyond Standard Model. A change of the fractal dimension $\delta$ is suggested to be the quantitative measure of the $z$-scaling violation.
Figure 5(a) shows that the scaling functions for direct photon, $\pi^0$ and $\eta^0$ reveal the power law $\psi(z) \sim z^{-\beta}$ in high-$z$ range. It was found that $\beta_{pp}^{\pi^0} \simeq \beta_{pp}^{\eta^0}$ and $\beta_{pp}^{\gamma} > \beta_{pp}^{\pi^0}$. Figure 5(b) demonstrates the power law for direct photon production in $\bar pp$ collisions as well. The slope parameter $\beta_{\bar pp}^{\gamma}$ is found to be 4.58 and $\beta_{pp}^{\gamma} > \beta_{\bar pp}^{\gamma}$. As seen from Figure 5(c) the asymptotic shapes of $\psi(z)$ for $\pi^0$-meson production in $pp$ and $\bar pp$ are different. Both ones have a power dependence and $\beta_{pp}^{\pi^0} > \beta_{\bar pp}^{\pi^0}$. The properties of the scaling functions for direct $\gamma$ and $\pi^0$ were used to estimate the dependence of the $\gamma / \pi^0$ ratio of inclusive cross sections on transverse momentum at $\sqrt s = 5.5$ and $14.0~TeV$. Figure 5(d) shows that the ratio increases with $p_T$ and it is different for $pp$ and $\bar pp$ collisions. The ratio has the crossover point at $p_T\simeq 60-70~GeV/c$ and $p_T \simeq
110-130~GeV/c$ for $pp$ and $\bar pp$ collisions, respectively.
![ (a) The scaling function of direct photon, $\pi^0$-meson and $\eta^0$-meson production in $pp$ collisions. (b) The scaling function of direct photon production in $pp$ and $\bar pp$ collisions. (c) The scaling function of $\pi^0$-mesons produced in $pp$ and $\bar pp$ collisions. (d) The $\gamma / \pi^0 $ ratio versus transverse momentum in $pp$ and $\bar pp$ collisions at $\sqrt s = 5.5$ and $14.0~TeV$. Experimental data are taken from [@WA70]-[@Banner] . ](fig5a.eps "fig:"){width="6.5cm"} -6.cm ![ (a) The scaling function of direct photon, $\pi^0$-meson and $\eta^0$-meson production in $pp$ collisions. (b) The scaling function of direct photon production in $pp$ and $\bar pp$ collisions. (c) The scaling function of $\pi^0$-mesons produced in $pp$ and $\bar pp$ collisions. (d) The $\gamma / \pi^0 $ ratio versus transverse momentum in $pp$ and $\bar pp$ collisions at $\sqrt s = 5.5$ and $14.0~TeV$. Experimental data are taken from [@WA70]-[@Banner] . ](fig5b.eps "fig:"){width="6.5cm"}
0.5cm
a\) b)
![ (a) The scaling function of direct photon, $\pi^0$-meson and $\eta^0$-meson production in $pp$ collisions. (b) The scaling function of direct photon production in $pp$ and $\bar pp$ collisions. (c) The scaling function of $\pi^0$-mesons produced in $pp$ and $\bar pp$ collisions. (d) The $\gamma / \pi^0 $ ratio versus transverse momentum in $pp$ and $\bar pp$ collisions at $\sqrt s = 5.5$ and $14.0~TeV$. Experimental data are taken from [@WA70]-[@Banner] . ](fig5c.eps "fig:"){width="6.5cm"} -6.cm ![ (a) The scaling function of direct photon, $\pi^0$-meson and $\eta^0$-meson production in $pp$ collisions. (b) The scaling function of direct photon production in $pp$ and $\bar pp$ collisions. (c) The scaling function of $\pi^0$-mesons produced in $pp$ and $\bar pp$ collisions. (d) The $\gamma / \pi^0 $ ratio versus transverse momentum in $pp$ and $\bar pp$ collisions at $\sqrt s = 5.5$ and $14.0~TeV$. Experimental data are taken from [@WA70]-[@Banner] . ](fig5d.eps "fig:"){width="6.5cm"}
c\) d)
Figure 6 shows our predictions of the dependence of the inclusive cross section $Ed^3\sigma /dq^3$ on transverse momentum $p_T$ for direct photon (a), $\pi^0$ (b) and $\eta^0$ (c) in $pp$ collisions at RHIC and LHC energies and at the angle of $\theta_{cms}=90^0$. The results for the cross sections at ISR energy of $\sqrt s = 24-63~GeV$ are also shown for comparison. The verification of the predictions is very important because it allows us to confirm or disconfirm the new scaling of photon production and to determine the region of the scaling validity.
![ Dependence of inclusive cross sections of direct photon (a), $\pi^0$-meson (b) and $\eta^0$-meson (c) production on transverse momentum in $pp$ collisions at $\sqrt s =
24-14000~GeV$. Experimental data are taken from [@R806; @UA6p; @Angel; @Kourk]. Solid lines and points $\star,
\triangle, *, \times$ are the calculated results. (d) Dependence of the variable $z$ of direct photons produced in $pp$ collisions on transverse momentum $p_{T}$ at energy $\sqrt s = 24-14000~GeV$ and $\theta_{cms} \simeq 90^0$. ](fig6a.eps "fig:"){width="6.5cm"} -6.cm ![ Dependence of inclusive cross sections of direct photon (a), $\pi^0$-meson (b) and $\eta^0$-meson (c) production on transverse momentum in $pp$ collisions at $\sqrt s =
24-14000~GeV$. Experimental data are taken from [@R806; @UA6p; @Angel; @Kourk]. Solid lines and points $\star,
\triangle, *, \times$ are the calculated results. (d) Dependence of the variable $z$ of direct photons produced in $pp$ collisions on transverse momentum $p_{T}$ at energy $\sqrt s = 24-14000~GeV$ and $\theta_{cms} \simeq 90^0$. ](fig6b.eps "fig:"){width="6.5cm"}
0.5cm
a\) b)
![ Dependence of inclusive cross sections of direct photon (a), $\pi^0$-meson (b) and $\eta^0$-meson (c) production on transverse momentum in $pp$ collisions at $\sqrt s =
24-14000~GeV$. Experimental data are taken from [@R806; @UA6p; @Angel; @Kourk]. Solid lines and points $\star,
\triangle, *, \times$ are the calculated results. (d) Dependence of the variable $z$ of direct photons produced in $pp$ collisions on transverse momentum $p_{T}$ at energy $\sqrt s = 24-14000~GeV$ and $\theta_{cms} \simeq 90^0$. ](fig6c.eps "fig:"){width="6.5cm"} -6.5cm ![ Dependence of inclusive cross sections of direct photon (a), $\pi^0$-meson (b) and $\eta^0$-meson (c) production on transverse momentum in $pp$ collisions at $\sqrt s =
24-14000~GeV$. Experimental data are taken from [@R806; @UA6p; @Angel; @Kourk]. Solid lines and points $\star,
\triangle, *, \times$ are the calculated results. (d) Dependence of the variable $z$ of direct photons produced in $pp$ collisions on transverse momentum $p_{T}$ at energy $\sqrt s = 24-14000~GeV$ and $\theta_{cms} \simeq 90^0$. ](fig6d.eps "fig:"){width="6.5cm"}
c\) d)
$pA$ collisions
----------------
It is assumed that direct photons produced in heavy ion collisions at RHIC and LHC could give a direct indication of phase transition to the new state of nuclear matter, QGP. Taking into account an experimental accuracy of data used in the analysis, the obtained results show that the fractal dimension $\delta$ and the slope parameter $\beta$ are independent of A.
Figure 7 demonstrates our predictions of the dependence of the inclusive cross section $Ed^3\sigma /dq^3$ on transverse momentum $p_T$ for direct photon (a), $\pi^0$ (b) and $\eta^0$ (c) in $pPb$ collisions at RHIC and LHC energies and at the angle of $\theta_{cms}=90^0$.
A change of the shape of photon $p_T$ spectra means a modification of the mechanism of photon formation in transition region.
{width="6.5cm"} -6.cm {width="6.5cm"}
0.5cm
a\) b)
{width="6.5cm"} c)
$z-p_T$ PLOT
============
The $z-p_T$ plot allows us to determine the high transverse momentum range interesting for searching for the kinematic region where the $z$-scaling can be violated. Figure 6(d) shows the $z-p_{T}$ plot for the $pp\rightarrow \gamma X$ process at $\sqrt s = 24-14000~GeV$. As seen from Figure 3(b) the scaling function $\psi(z)$ is measured up to $z\simeq 20$. The function $\psi(z)$ demonstrates the power behavior in the range. Therefore the kinematic range $ z > 20$ is of more preferable for experimental investigations of $z$-scaling violation. The condition determines the low boundaries for $p_T$ ranges, $p_T > 5, 10, 16, 22, 35, 45$ and $52~GeV/c$ at different energy $\sqrt s = 24, 63, 200, 500, 2000, 7000$ and $14000~GeV$, respectively.
CONCLUSION
==========
Analysis of the numerous experimental data on high-$p_T$ direct photon and $\pi^0$-meson production in $pp, \bar pp$ and $pA$ collisions obtained at ISR, SpS and Tevatron in the framework of $z$-scaling concept was presented. It was shown that the general concept of $z$-scaling is valid for photon production in hadron-hadron and hadron-nucleus collisions.
The scaling function $\psi(z)$ and scaling variable $z$ are expressed via the experimental quantities, momenta and masses of colliding and produced particles and the invariant inclusive cross section $Ed^3\sigma/dq^3$ and the multiplicity density of charged particles $\rho_A(s,\eta)$. The physics interpretation of the scaling function $\psi$ as a probability density to produce a particle with the formation length $z$ is argued. The quantity $z$ has the property of the fractal measure and $\delta $ is the anomalous fractal dimension describing the intrinsic structure of the interaction constituents revealed at high energies. The fractal dimensions of nuclei satisfy the relation $\delta_{A} = A\cdot \delta_N$.
It was shown that the properties of $z$-scaling, the energy and angular independence, the power law $\psi(z)\sim z^{-\beta}$ and $A$-dependence are confirmed by the numerous experimental data obtained at ISR, SpS and Tevatron.
A comparison of the scaling function of direct photon and $\pi^0$-meson production in $pp$ and $\bar pp$ was performed and different asymptotic behavior of $\psi(z)$ was found. It was shown that $p_T$-dependence of the ratio of direct photon and $\pi^0$-meson inclusive cross sections for $pp$ and $\bar pp$ collisions has the different crossover points. Based on the universality of the scaling function, the predictions of direct photon, $\pi^0$- and $\eta^0$-meson cross sections in $pp$ and $pPb$ collisions at RHIC and LHC energies were made. The $z-p_T$ plot was used to establish the kinematic range that is of more preferable for experimental investigations of $z$-scaling violation.
The violation of $z$-scaling due to the change of the value of the fractal dimension $\delta$ is suggested to search for a new physics phenomena such as quark compositeness, new type of interactions, nuclear phase transition in $pp, pA$ and $AA$ collisions at RHIC and LHC.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
One of the authors (M.T.) would like to thank the organizers of the workshop “Hard Probes in Heavy Ion Collisions at LHC” U. Wiedemann, H. Satz, M. Mangano and convenors of the working groups P. Aurenche, P. Levai and K.J. Eskola for excellent atmosphere during the workshop and for many interesting and useful discussions.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The systematics of strangeness enhancement is calculated using the HIJING and VENUS models and compared to recent data on $\,pp\,$, $\,pA\,$ and $\,AA\,$ collisions at CERN/SPS energies ($200A\,\, GeV\,$). The HIJING model is used to perform a [*linear*]{} extrapolation from $pp$ to $AA$. VENUS is used to estimate the effects of final state cascading and possible non-conventional production mechanisms. This comparison shows that the large enhancement of strangeness observed in $S+Au$ collisions, interpreted previously as possible evidence for quark-gluon plasma formation, has its origins in non-equilibrium dynamics of few nucleon systems. A factor of two enhancement of $\Lambda^{0}$ at mid-rapidity is indicated by recent $pS$ data, where on the average [*one*]{} projectile nucleon interacts with only [*two*]{} target nucleons. There appears to be another factor of two enhancement in the light ion reaction $SS$ relative to $pS$, when on the average only two projectile nucleons interact with two target ones.'
address:
- |
Department of Physics,Columbia University,[**New York**]{}, N.Y.10027\
and\
Dipartamento di Fisica “G.Galilei”,Via Marzolo 8-35131, Padova [**Italy**]{}
- 'Department of Physics,Columbia University,[**New York**]{}, N.Y.10027'
- |
Nuclear Science Division,Lawrence Berkeley Laboratory,\
University of California,[**Berkeley**]{},CA 94720
- |
Dipartamento di Fisica “G.Galilei”,Via Marzolo 8 -35131, Padova [**Italy**]{}\
and\
INFN Sezione di Padova, [**Italy**]{}
author:
- '[**V.Topor Pop**]{}[^1]'
- '**M.Gyulassy**'
- '**X.N.Wang**'
- '[**A.Andrighetto,M.Morando,F.Pellegrini,R.A.Ricci,G.Segato**]{}'
date: 17 February 1995
title: |
[**Strangeness Enhancement in $p+A$ and $S+A$\
Interactions at SPS Energies**]{}$^*$[^2]
---
= 16truecm = 24truecm
CU-TP-676\
DFPD 95/NP/20
[**Strangeness Enhancement in $p+A$ and $S+A$\
Interactions at SPS Energies**]{}\
[**V.Topor Pop**]{}$^1$\
[Department of Physics,Columbia University,[**New York**]{}, N.Y.10027]{}\
[and]{}\
[Dipartamento di Fisica “G.Galilei”,Via Marzolo 8-35131, Padova [**Italy**]{} ]{}\
[**M.Gyulassy**]{}\
[Department of Physics,Columbia University,[**New York**]{}, N.Y.10027]{}\
[**X.N.Wang**]{}\
[Nuclear Science Division,Lawrence Berkeley Laboratory,\
University of California,[**Berkeley**]{},CA 94720]{}\
[**A.Andrighetto,M.Morando,F.Pellegrini,R.A.Ricci,G.Segato**]{}\
[Dipartamento di Fisica “G.Galilei”,Via Marzolo 8 -35131, Padova [**Italy**]{}]{}\
[and ]{}\
[INFN Sezione di Padova,[**Italy**]{}]{}\
[*Work to be submitted to Phys.Rev.[**C**]{}*]{}
Introduction
============
The search for new states of dense nuclear matter is one of the most active areas of research in nuclear physics [@pro6],[@pro5] . Enhanced strangeness production in ultra-relativistic heavy ion collisions was suggested long ago [@raf1] as a signal for quark-gluon plasma formation[@quer1],[@quer2], and has been observed at both the AGS and SPS. There is extensive data from both the SPS at CERN [@1na35]-[@mur94] and the AGS at BNL [@3e802]-[@bare94] on strangeness yields from reactions ranging from elementary $p+p$ to $p+A_T$ and $A_B+A_T$ for targets ranging up to $A_T\approx 200$ and beams up to $A_B=30$. Detailed rapidity and transverse momentum spectra of ($K^{+}$,$K^{-}$, $K^{0}_{s}$,$\Lambda$ ,$\bar{\Lambda}$) are available and spectra of $\,\Xi^{-}\,$ and even $\,\Omega^{-}\,$ are becoming available. In all cases their yield relative to pions or negative hadrons are larger in nucleus-nucleus than expected from geometrically scaled proton-proton collisions. New experiments with truly heavy ion projectiles are in progress with Au beams at BNL[@dieb93],[@nam94] and with Pb beams at CERN ($Pb(170A\, GeV)+Pb$) [@quer2] and will soon extend considerably the data base.
These and other data on nuclear reactions have stimulated the development of many hadronic transport models to address the problem of multi-particle production in nuclear collisions. These include Dual Partons Models(DPM) [@dpm94]-[@ranft4], Quark Gluon String Models(QGSM)[@kai1]-[@ame2], VENUS [@wer7], FRITIOF [@ander]-[@nils], ATTILA [@Gyu1], HIJING[@wang0]-[@wang4] RQMD models [@aich1]-[@sor94], Parton String Model (PSM)[@ame94],HIJET[@fol94], Parton Cascade Model(PCM) [@gei1]-[@gei3]. An excellent review and detailed comparison of the models is given by Werner in Ref.[@wer7].
At present no conventional explanation of the large enhancement of hyperons or antihyperons has been found. The Pomeron exchange picture has motivated the development of many of the above models with the Pomeron modeled in terms of colored strings. However, the string picture itself suggests the possibility of new dynamical mechanisms ranging from string fusion to color rope formation. Some of the above transport models like RQMD[@sor94] and VENUS[@wer7] include such non-conventional mechanisms as default options. These proposed novel [*non-equilibrium*]{} dynamical mechanisms were shown to be able to reproduce many features of the observed strangeness enhancement [@braun4]-[@merino2], [@sor93a],[@sor94a], [@wer7]. On the other hand, there have been many attempts (see, e.g., review by Heinz in [@pro6] p. 205c and references therein) to attribute the strangeness enhancement to the formation of an equilibrated fireball containing a quark-gluon plasma state [@pro6],[@rafel].
Therefore it appears that either non-conventional multi-particle mechanisms or the existence of a new form of matter seems to be indicated by the observed strangeness enhancement. Either case is of basic interest. The goal of the present study is to clarify which of these alternatives is more compelling. We use the HIJING model[@wang0]-[@wang4] to perform a [*linear*]{} extrapolation of strangeness production dynamics from $pp$ to $AA$ taking into account essential nuclear geometry and kinematical constraints. At higher collider energies it includes pQCD semi-hard processes, but in the SPS range it reduces essentially to a hybrid version of the FRITIOF and DPM models. We use the VENUS model[@wer7] to estimate possible effects of final state cascading and new mechanisms of strangeness production in few nucleon processes. The non-conventional mechanism in VENUS4.13 is the occurrence of “double strings” which may form when one projectile nucleon interacts with two or more target nucleons. A double string is defined as a color singlet baryon configuration consisting of one projectile quark connected to two different valence quarks in the target via a three gluon vertex. In earlier versions of the model the parameterization of the vertex kinematics led to anomalously large baryon stopping power. In the present version, the double string phenomenology is constrained to reproduce the $pA\rightarrow pX$ data. However, the new feature, see eq. 15.52 in ref. [@wer7], is the assumption that the probability for hyperon production in the fragmentation regions is enhanced by a factor of two relative to the single string rates. This enhanced strangeness production mechanism due to double strings is similar to that postulated in the color rope model[@biro] and incorporated into the RQMD model. The hyperon enhancement in VENUS is however more confined to the fragmentation regions.
Both HIJING and VENUS have been compared to a wide variety of data in $\,pp\,$,$\,pA\,$ and $\,AA\,$ collisions [@wang2],[@wang3], [@wer7]. However, no systematic study of strangeness production at SPS-CERN energies were performed up to now. In addition, there have been substantial changes in the final published data [@na3594] relative to earlier comparisons to preliminary data [@1na35], [@2na35]. In this paper, we calculate the rapidity and transverse momentum spectra of strange particles for $\,pp\,$, minimum bias collisions of $\,pS,pAg,pAu\,$ and central collisions of $S+S,Ag,Au,W\,\,$ at the energy of $\,200\,\,AGeV$ and $Pb+Pb$ at the energy of $\,170\,\,AGeV$. We focus special emphasis on the comparison with the data on $pp$, $pS$, and $SS$ from Alber et al. [@na3594]. That comparison reveals that much of the enhancement of strangeness in heavy ion collisions can be traced back to the enhancement of strangeness in the lightest nontrivial ion collisions, $p+S$. Our main conclusion based on these data is that the enhancement of strangeness observed in $S+Au$ is therefore most likely due to new non-equilibrium multi-particle production mechanisms in processes involving [*few*]{} nucleon systems.
This paper is organized as follows: A brief description of the HIJING Monte Carlo model and theoretical background are given in Section II. For a detailed discussion of the VENUS model, we refer to the review in [@wer7]. In Section III, detailed numerical results with HIJING and VENUS for $\,pp\,$,$\,pA\,$ and $\,AA\,$ reactions at CERN-SPS energies ($\,\sqrt{s}\simeq 20A \,\,GeV\,$) for strangeness production are compared to experimental data and other model predictions . Section IV concludes with a summary and discussion of results.
Outline of the HIJING Model
===========================
A detailed discussion of the HIJING Monte Carlo model was reported in references[@wang0]-[@wang4]. The formulation of HIJING was guided by the LUND-FRITIOF and Dual Parton Model(DPM) phenomenology for soft nucleus-nucleus reactions at intermediate energies ($\sqrt{s}<20\,\, GeV$) and implementation of perturbative QCD(PQCD) processes in the PHYTHIA model[@sjos94] for hadronic interactions. We give in this section a brief review of the aspect of the model relevant to hadronic interaction:
1. Exact diffuse nuclear geometry is used to calculate the impact parameter dependence of the number of inelastic processes [@Gyu1].
2. Soft beam jets are modeled by quark-diquark strings with gluon kinks along the lines of the DPM and FRITIOF models. Multiple low $\,p_{T}\,$ exchanges among the end point constituents are included.
3. The model includes multiple mini-jet production with initial and final state radiation along the lines of the PYTHIA model and with cross sections calculated within the eikonal formalism.
4. Hadronization is performed via the JETSET7.2 algorithm [@sjos94] that summarizes data on $e^+e^-$.
5. HIJING does not incorporate any mechanism for final state interactions among low $\,p_{T}\,$ produced particles nor does it have color rope formation.
The rate of multiple mini-jet production in HIJING is constrained by the cross sections in nucleon-nucleon collision. Within an eikonal formalism [@hwa1] the total elastic cross sections $\sigma_{el}$,total inelastic cross sections $\sigma_{in}$ and total cross sections $\sigma_{tot}$ can be expressed as: $$\sigma_{el}=\pi\int_{0}^{\infty}\:db^{2}(1-\exp(-\chi(b,s)))^{2}
\label{e1}$$ $$\sigma_{in}=\pi\int_{0}^{\infty}\:db^{2}(1-\exp(-2\,\chi(b,s)))
\label{e2}$$ $$\sigma_{tot}=2\,\pi\int_{0}^{\infty}\:db^{2}(1-\exp(-\chi(b,s)))
\label{e3}$$ Strong interactions involved in hadronic collisions can be generally divided into two categories depending on the scale of momentum transfer $q^{2}$ of the processes. If $q^{2}< \Lambda_{QCD}^{2}$ the collisions are nonperturbative and considered [*soft*]{} and modeled by beam jet fragmentation via the string model. If $q^{2}\gg \Lambda_{QCD}^{2}$ the subprocesses on the parton level are considered [*hard*]{} and calculated via pQCD [@wang3].
In the limit that the real part of the scattering amplitude is small and the eikonal function $\chi(b,s)$ is real, the factor $$g(b,s)=1-\exp(-2\,\chi(b,s))
\label{e4}$$ can be interpreted in terms of semi-classical probabilistic model as [*the probability for an inelastic event of nucleon–nucleon collisions at impact parameter $b$*]{} which may be caused by hard, semi-hard or soft parton interactions.
To calculate the probability of multiple mini-jet, the main dynamical assumption is that they are independent. This holds as long as their average number is not too large as is the case below LHC energies [@wang3]. When shadowing can be neglected,the probability of no jets and $j$ independent jet production in an inelastic event at impact parameter $b$,can be written as : $$g_{0}(b,s)=(1-\exp(-2\,\chi_{s}(b,s)))\exp(-2\,\chi_{h}(b,s))
\label{e5}$$ $$g_{j}(b,s)=\frac{\left[2\,\chi_{h}(b,s)\right]^{j}}{j!}\cdot
exp(-2\,\chi_{h}(b,s))\,\,\,\,\,\, j \geq 1
\label{e6}$$ where $\chi_{s}(b,s)$ –is the eikonal function for soft interaction, $2\,\chi_{h}(b,s)$ –is the average number of hard parton interactions at a given impact parameter, $\exp(-2\,\chi_{s}(b,s))$ –is the probability for no soft interaction. Summing eqs.(5) and (6) over all values of $j$ leads to : $$\sum_{j=0}^{\infty}\,\,g_{j}(b,s)=1-\exp(-2\,\chi_{s}(b,s)-2\,
\chi_{h}(b,s))
\label{e7}$$ Comparing with eq.(\[e4\]) one has : $$\chi(b,s)=\chi_{s}(b,s)+\chi_{h}(b,s)
\label{e8}$$
Assuming that the parton distribution function is factorizable in longitudinal and transverse directions and that the shadowing can be neglected the average number of hard interaction $2\chi_{h}(b,s)$ at the impact parameter $b$ is given by : $$\chi_{h}(b,s)=\frac{1}{2}\,\,\sigma_{jet}(s)\,T_{N}(b,s)
\label{e9}$$ where $T_{N}(b,s)$ is the effective partonic overlap function of the nucleons at impact parameter $b$. $$T_{N}(b,s)=\int d^{2}b'\rho(b')\rho(\left|b-b'\right|)
\label{10}$$ with normalization $\int\,\,d^{2}b\,T_{N}(b,s)=1$ and $ \sigma_{jet}\,\,$ is the pQCD cross section of parton interaction or jet production [@wang2],[@wang3]. Note that $\,\,\xi=b/b_{0}(s)$, where $b_{0}(s)$ provides a measure of the geometrical size of the nucleon $\pi b_{0}^{2}(s)=\sigma_{s}(s)/2$ assuming the same geometrical distribution for both soft and hard overlap functions $$\chi_{s}(\xi,s)\equiv \frac{\sigma_{s}}{2\sigma_{0}}\chi_{0}(\xi)
\label{e11}$$ $$\chi_{h}(\xi,s)\equiv \frac{\sigma_{jet}}{2\sigma_{0}(s)}\chi_{0}(\xi)
\label{e12}$$ $$\chi(\xi,s)\equiv \frac{1}{2\sigma_{0}}\left [{\sigma_{s}(s)
+\sigma_{jet}(s)}\right ]\chi_{0}(\xi)
\label{e13}$$ We note that $\,\chi(\xi,s)\,$ is a function not only of $\,\xi\,$ but also of $\,\sqrt{s} \,$ because of the $\,\sqrt{s}\,$ dependence on the jet cross section $\,\sigma_{jet}(s)\,$.Geometrical scaling implies on the other hand that $\,\chi_{s}(\xi,s)=\chi_{0}(\xi)\,$ is only a function of $\,\xi\,$. Therefore, geometrical scaling is broken at high energies by the introduction of $\,\sigma_{jet}(s)\,$ of jet production.
The cross sections of nucleon - nucleon collisions can in this case be expressed as: $$\sigma_{el}=\sigma_{0}(s)\int_{0}^{\infty}d\,\xi^{2}\left ( 1-exp(-\chi
(\xi,s)\right )^{2}
\label{e14}$$ $$\sigma_{in}=\sigma_{0}(s)\int_{0}^{\infty}d\,\xi^{2}\left ( 1-exp(-2\,
\chi(\xi,s)\right))
\label{e15}$$ $$\sigma_{tot}=2\,\sigma_{0}(s)\int_{0}^{\infty}d\,\xi^{2}\left (1-exp
(-\chi(\xi,s)\right))
\label{e16}$$ The calculation of these cross sections requires specifying $\sigma_{s}(s)$ with a corresponding value of cut - off momenta $p_{0}\approx 2$ GeV/c [@wang4].
In the energy range $10\,\, GeV<\sqrt{s}<70\,\, GeV$, where only soft parton interactions are important, the soft cross section $\sigma_{s}(s)$ is fixed by the data on total cross sections $\sigma_{tot}(s)$ directly. In and above the $Sp\bar{p}S$ energy range $\sqrt{s}\geq 200\,\, GeV$, a fixed $\sigma_{s}(s)= 57$ mb and a mini-jet cutoff scale $p_{0}=2 \,\,GeV/c$, leads to observed energy dependence of the cross sections and inclusive distributions. Between the two regions $70\,\, GeV < \sqrt{s}<200\,\, GeV$, a smooth extrapolation for $\,\sigma_{s}(s)\,$ is used.
In HIJING, a nucleus-nucleus collisions is decomposed into a sequence of binary collisions involving in general excited or wounded nucleons. Wounded nucleon are assumed to be $\,q-qq\,$ string like configurations that decay on a slow time scale compared to the collision time of the nuclei. In the FRITIOF scheme wounded nucleon interactions follow the same excitation law as the original hadrons. In the DPM scheme subsequent collisions essentially differ from the first since they are assumed to involve sea partons instead of valence ones. The HIJING model adopts a hybrid scheme, iterating string-string collisions as in FRITIOF but utilizing DPM like distributions. In the SPS range the HIJING results for nuclear collisions are very similar to those of FRITIOF. However, HIJING provides an interpolation model between the nonperturbative beam jet fragmentation physics at intermediate CERN-SPS energies and perturbative QCD mini-jet physics at the highest collider energies ($\,RHIC\,,\,LHC\,$).
NUMERICAL RESULTS
=================
STRANGENESS IN PROTON - PROTON INTERACTION
------------------------------------------
We used the program HIJING with default parameters: IHPR2(11)=1 gives the baryon production model with diquark-antidiquark pair production allowed, initial diquark treated as unit; IHPR2(12)=1, decay of particle such as $\,\pi^{0}\,$,$\,K_{s}^{0}\,$,$\,\Lambda\,$,$\,\Sigma\,$, $\,\Xi\,$, $\,\Omega\,$ are allowed;IHPR2(17)=1 - Gaussian distribution of transverse momentum of the sea quarks ;IHPR2(8)=0 - jet production turned off for theoretical predictions denoted by HIJING model, and IHPR2(8)=10-the maximum number of jet production per nucleon-nucleon interaction for for theoretical predictions denoted by $\,HIJING^{(j)}\,$ for comparison.
In Table I the calculated average multiplicities of particle at $E_{lab}=200\,\, GeV\,$ in proton-proton($pp$) interaction are compared to data. The theoretical values $\,HIJING\,$ and $\,HIJING^{(j)}\,$ are obtained for $\,\,10^{5}\,\,$ generated events and in a full phase space. The values $\,\,HIJING^{(j)}\,\,$ include the very small possibility of mini jet production at these low SPS energies. The experimental data are taken from Gazdzicki and Hansen [@11na35].
The small kaon to pion ratio is due to the suppressed strangeness production basic to string fragmentation. Positive pions and kaons are more abundant than the negative ones due to charge conservation. We note that the [*integrated*]{} multiplicities for neutral strange particle $\,\,<\Lambda>,<\bar{\Lambda}>,<K_{s}^{0}>\,\,$ are reproduced at the level of three standard deviations for $\,pp\,$ interactions at $\,200\, GeV\,$. However the values for $\,<\bar{p}>\,$ and $\,<\bar{\Lambda}>\,$ are significantly over predicted by the model. This is important since as we shall see the $\bar{\Lambda}$ in $S+S$ is significantly underestimated by HIJING.
For completeness we include a comparison of hadron yields at collider energies$\,\,\sqrt{s}=546\,\, GeV$($\,Sp\bar{p}S\,$-energies), for $\,\,\bar{p}p\,\,$ interactions, where mini-jet production plays a much more important role. From different collider experiments Alner et al.(UA5 Collaboration))[@1ua5] attempted to piece together a picture of the composition of a typical soft event at the $\,Sp\bar{p}S\,$ [@ward]. The measurements were made in various different kinematic regions and have been extrapolated in the full transverse momenta($\,p_{T}\,$) and rapidity range for comparison as described in reference [@1ua5]. The experimental data are compared to theoretical values obtained with $HIJING^{(j)}$ in Table II. It was stressed by Ward [@ward] that the data show a substantial excess of photons compared to the mean $\,\,\pi^{+}+\pi^{-}\,\,$. It was suggested as a possible explanation of such enhancement a gluon Cerenkov radiation emission in hadronic collision [@drem]. Our calculations rules out such hypothesis. Taking into account decay from resonances and direct gamma production, good agreement is found within the experimental errors. The experimental ratio $\,\frac{K^{+}}{\pi^{+}} =0.095 \pm 0.009\,$ is also reproduced by$\,\, HIJING^{(j)}\,\,$ model (0.099). We note that a detailed study of the ratios of invariant cross sections of kaons to that of pions as a function of transverse momenta in the central region was presented in [@wang3].
In the following plots the kinematic variable used to describe single particle properties are the transverse momentum $\,p_{T}\,$ and the rapidity $\,y\,$ defined as usual as: 0.3cm $$y=\frac{1}{2}ln \frac{E+p_{3}}{E-p_{3}}=ln\frac{E+p_{3}}{m_{T}}
\label{e17}$$ 0.3cm with $\,E,p_{3}\,$,and $\,m_{T}\,$ being energy,longitudinal momentum and transverse mass $ m_{T}=\sqrt{m_{0}^{2}+p_{T}^{2}}$ with $\,m_{0}\,$ being the particle rest mass.
In Fig.1a, 3a, 4a, and 6a, we show rapidity and transverse momentum distributions for $\,\,\Lambda\,\,$’s (Fig.1a,4a) and $K_{s}^{0}$’s (Fig.3a,6a) produced in $ pp$ scattering at$\,\, 200\,\,$ GeV. The theoretical histograms obtained with HIJING (solid) and VENUS-4.13 (dashed) are compared with experimental data taken from Jaeger et al.[@jagl]. The HIJING spectra for $\,\Lambda\,$,$\,K_{s}^{0}\,$ are close to the data at mid rapidity [@jagl], although the dip in the $\,\,K_{s}^{0}\,\,$ yield at mid-rapidity and the $\,\,\Lambda\,\,$ peak in the fragmentation regions are not well reproduced (see also ref. [@wer7]). Unfortunately, more precise data are not available in $pp$ interactions and those features could reflect experimental acceptance cuts. Similarly no detailed $\bar{\Lambda}$ spectra are as yet available in $pp$.
In comparison with VENUS (taking $10^4$ events) we note that this version seems to over-predict the $pp\rightarrow \Lambda^0$ rapidity density at mid-rapidity by $50-100\%$ in Fig. 1a, even though the rapidity integrated transverse momentum distribution in Fig 4a seems closer to the data. The $K_s^0$ yields in Figs. 3a and 6a are similar to those of HIJING with the dip structure in the data absent.
The very sparse data base on $pp$ strangeness production at SPS energies should be expanded in the future to improve the test of dynamical models before they are applied to the more complex nuclear collision case. Without $\,\bar{\Lambda}\,$ spectra in $pp$, for example, the need for the new dynamical mechanisms in that channel cannot be confirmed.
Multiplicities in $pA$ and $AA$ collisions
------------------------------------------
In this section, we compare strange particle production in the HIJING and VENUS models to $pA$ and $AA$ data. Again we limit the study to $\,\,\Lambda,\bar{\Lambda},K_{s}^{0}\,\,$ to compare with recent data from Alber et al.[@na3594]. First we consider the average integrated multiplicities for negative hadrons $\,<h^{-}>\,$, negative pions $\,<\pi^{-}>\,$ and neutral strange particles $\,<K_{s}^{0}>\,$, $\,<\Lambda>\,$, $\,<\bar{\Lambda}>\,$ in $\,pp,pS,pAg,pAu\,$ (minimum bias collisions) and $\,SS,SAg,SAu\,$ (central collisions) at $\,200$ AGeV. The default parameters of HIJING were used without mini-jet production (IHPR2(8)=0). The number of Monte Carlo generated events was $\,10^{5}\,$ for HIJING and $\,10^4\,$ for VENUS for $pp$,$pA$ interactions and $\, 5\cdot 10^{3}$ for $\,SS,$ and $\,10^{3}\,$ for $\,S+Ag,W,Au\,$ and $\,PbPb\,$ collisions.
The mean multiplicities are compared in Table III (for $pp$ and $pA$ interactions) and in Table IV (for $AA$ interactions) with experimental data from Alber et al.[@na3594]. Note that while the HIJING model describes well the integrated neutral strange particle multiplicities (except for $<\bar{\Lambda}>$) in $pp$ and $pA$ interactions, there is a large discrepancy already for the light ion $S+S$ reaction.
It is worthwhile to mention that theoretical calculations have been done for $\,\,pA\,\,$ ’minimum bias’ collisions and the experimental data are for the events with charged particle multiplicity greater than five,which contain a significant fraction (about 90 %) of the ’minimum bias’ events [@na3594].
In Table III and IV the data are compared also with other theoretical values obtained in some models : VENUS (as computed here), RQMD [@na3594], QGSM [@ame2],[@na3594] and DPM models. The theoretical values$\,DPM^{1}\,$ are from the Mohring et al.[@mohr1], version of DPM which include additionally $\,\,(qq)-(\bar{q}\bar{q})\,\,$ production from the sea into the chain formation process and the values $\,DPM^{2}\,$ are from the Mohring et al. [@mohr2], version of DPM which include chain fusion , as a mechanism to explain the anomalous antihyperon production.
Alber et al. [@na3594] have considered that the total production of strangeness should be treated in a model independent way using all available experimental information for ratio $\,\,E_{S}\,\,$ expressed as : 0.3cm $$E_{S}=\frac{<\Lambda>+4<K_{s}^{0}>}{3<\pi^{-}>}
\label{e18}$$
0.3cm We have calculated this ratio in HIJING approach for the above interactions and the corresponding numerical predictions are shown in Table V. We note that there is much less discrepancy between HIJING and the data for this particular ratio. We conclude from this that such ratio is insensitive to the underlying physics and therefore should NOT be used for any further tests of models! This ratio hides very effectively the gross deficiencies of the HIJING model in $SS$ reactions pointed out later in the comparison to the rapidity and transverse momentum distributions. We include Table V only to prove the futility of studying the systematics of such ratios in the search for novel dynamics in nuclear collisions!
Single inclusive distributions for neutral strange particles in $\,pA\,$ and $\,AA\,$
--------------------------------------------------------------------------------------
The main results of the present study. are contained in Figs. 1-6. Figure 1 is our most important result revealing the systematics of $\Lambda$ enhancement from (a) $pp$ to (d) $SAu$. In part (a) the pp data at mid rapidity are seen to be well reproduced by HIJING. However, the new minimum bias $pS$ data[@na3594] in Fig. 1b clearly shows a factor of $2-3$ discrepancy with respect to the linear extrapolation from $pp$ as performed by HIJING. The effect of double string fragmentation and final state cascading, as modeled with VENUS is seen on the other hand to account for the observed $\Lambda$ enhancement. We note, however, that in $pp$, VENUS over-predicts the $\Lambda$ yield at mid rapidity. Some fraction of the aggreement in $pS$ with VENUS may be due to this effect. The overprediction of midrapidity $\Lambda$’s in $pp$ by VENUS was shown in Fig. 10.20b of ref.[@wer7], but was not emphasized there. If both the $pp$ and $pS$ data on $\,\Lambda\,$ production are correct,then the most striking increase of hyperon production therefore occurs between $pp$ to $pS$ reactions.
The strangeness enhancement in minimum bias $p+S$ is striking because the number of target nucleons struck by the incident proton is on the average only two! The step from single $p+p$ to triple $p+p+p$ reactions therefore apparently leads a substantial enhancement of midrapidity $\Lambda$’s which obviously cannot have anything to do with equilibrium physics.
In central $S+S$ reactions shown in Fig. 1c, the discrepancy relative to HIJING grows by another factor of two. We note that the new data[@na3594]shown here have increased substatially relative to earlier data [@1na35],[@2na35] due to inclusion of lower transverse momentum regions and $\Lambda$’s originating from the decay of $\Sigma,\Xi$ in the analysis. Including these decay channels, VENUS is seen to reproduce the new data as well. We note that with RQMD the excess $\Lambda$’s is also reproduced with the introduction of rope formation (see Table IV). For heavier targets, $\,Ag,Au\,$, in Fig. 1d, the discrepancy relative to HIJING is in fact less dramatic than in $\,S+S\,$.
In central $\,S+S\,$, on the average each projectile nucleon interacts with only two target one, but each target nucleon also interacts with two projectile ones. In effect, then $\,S+S\,$ reactions probe strangeness production in four nucleon interactions $p+p+p+p\rightarrow \Lambda + X$. Such reactions appear to be approximately four times as efficient in producing midrapidity $\Lambda$’s as two nucleon interactions in part (a). Our main conclusion therefore is that strangeness enhancement is a nonequilibrium dynamical effect clearly revealed in the lightest ion interactions.
Further support for this conclusion is shown in Figs. 4 a,b,c, where the transverse momentum distributions are compared. We see that there is an enhancement of the $\Lambda$ transverse momentum relative to $pp$ in $pS$. Comparing to VENUS we can interpret Fig. 4b as evidence that the enhanced transverse momentum of $\Lambda$ in $pS$ is due cascading. The discrepency in Fig. 4c between VENUS and the data in $SS$ may be due to the rapidity cuts in the data, which we have not included in the calculated spectra. In all cases the deficiency of the linear extrapolation via the HIJING model is clearly evident. For heavier targets, $S+Ag,Au$, the transverse momentum distribution predicted by VENUS is close to the data.
The same general conclusion emerges from the systematics of $\bar{\Lambda}$ and $K_{s}^{0}$ production s in Figs 2,3 and in Figs 5,6 respectively. In Fig.2a, the agreement between HIJING and VENUS and the data on the $p+S\rightarrow
\bar{\Lambda}$ must be viewed with caution since as shown in Table III, both models overpredict the integrated $\bar{\Lambda}$ multiplicity by a factor $2-3$. Given the absence of more detailed rapidity and transverse momentum distributions for $\bar{\Lambda}$, it is not possible to determine whether the $pp$ and $pA$ data are compatible. However, at least the step form $pS$ to $SS$ in Fig 2b indicates a possible factor of two enhancement of $\bar{\Lambda}$ similar to the comparison of Fig. 1b,1c for ${\Lambda}$. As in the case of ${\Lambda}$ production, there appears to be no further $\bar{\Lambda}$ enhancement from $SS$ to $SAu$. As regards to the transverse momentum distributions in Fig. 5, we note that as in Fig. 4 the $\bar{\Lambda}$ emerge with higher $p_\perp$ in $pS$ than in $pp$ in accord with the VENUS model. We note that in Figs 5 c,d, the norm theoretical curves is obtained intergrating over the full rapidity interval, while the norm data are limited to a smaller domain as shown in Fig. 2 b,c.
In the case of $K_{s}^{0}$ production in Figs 3, 6, the same general trends are seen but in a less dramatic form.
We conclude that the new data indicate that the origin of strangeness enhancement in heavy ion collisions may be traced back to non-conventional and necessarily non-equilibrium dynamical effects that arise in collisions of three or more nucleons. However, this conclusion is forced upon us by the sytematics of the new light ion data on $p+S$ and $S+S$ reported in [@na3594]. As shown in Figs 1b, 2a, and 3b, those systematics, especially in $pA$, differ considerably from the trends of earlier NA5 data[@10na35] and preliminary NA36 data[@1ander94]-[@greiner95]. Those data for [*heavier*]{} target nuclei incidate substantially less enhancement of midrapidity $\Lambda,\bar{\Lambda},K^0_s$ than do the NA35 data on $p+S$. Part of the difference between these data sets may be due to different acceptance cuts and the inclusion or rejection of fragments from decay of higher mass hyperons. Obviously, the difference between these data sets must be resolved. Until then, the NA36 data must be regarded as an important caveat on our conclusions.
For completeness we show also in Figs 7 the linear extrapolations of HIJING to $Pb+Pb$ at 170 AGeV for all positives (Fig 7a) and all negatives charges (Fig 7b), for $\Lambda$ (Fig 7c) and for $\bar{\Lambda}$ (Fig 7d). It will be interesting to compare these extrapolations with upcoming data to test if the strangeness enhancement increases from $SS$ to $PbPb$.
We include the two dimensional distributions in Fig. 8 to emphasize that strangeness enhancement analyses restricted to narrow rapidity and transverse momentum cuts, especially with simplistic fireball models, may completely miss the global non-equilibrium character of the data.
Conclusions
===========
In this paper we performed a systematic analysis of strange particle production in $\,pp\,$,$\,pA\,$ and $\,AA\,$ collisions at SPS CERN-energies using the HIJING and VENUS models. The most surprising result is that the breakdown of the linear extrapolation from $pp$ data to nucleus-nucleus in the strangeness channel already occurs in minimum bias $pS$ ! The apparent enhancement of $\Lambda$, $\bar{\Lambda}$ and $K_{s}^0$ at midrapidities in $pS$ reactions by a factor of 2 indicates that the mechanism for strangeness enhancement in heavier ion collisions must be associated with non-equilibrium dynamics involving multiparticle production and not with equilibrium quark-gluon fireball. In minimum bias $pS$ one projectile nucleon interacts on the average with only two target ones. The data[@na3594] on $pS$ therefore indicate the existence of new dynamical mechanisms for strangeness production that becomes operative in $p+p+p$ collisions. The new data[@na3594] on central $S+S$ show another factor of 2 enhancement of strangeness production relative to $pS$. This light ion reaction basically probes multiparticle production in $p+p+p+p$. The strangeness enhancement in heavier target systems apparently saturates at the S+S level. We also showed that traditional analysis of strangeness enhancement in terms of ratios of integrated multiplicities is very ineffective since those ratios hide well defects of the detailed rapidity and transverse momentum distributions predicted by models.
The agreement with VENUS and RQMD results suggests color rope formation as a possible mechanism. However, to clarify the new physics much better quality data on elementary $p+p$ as well as on other light ion $p+\alpha,C,S$ and $\alpha+\alpha,C,S$ reactions will be needed. Especially, the discrepancy between NA35 and NA36 must be resolved. Only then can strangeness enhancement systematics used meaningfully in the search for signatures of quark-gluon plasma formation in future experiments with $Au+Au$ and $Pb+Pb$.
Acknowledgements
================
We are grateful to Klaus Werner for providing the source code of VENUS. One of the authors (VTP) would like to expresses his gratitude to Professor C.Voci for kind invitation and acknowledge financial support from INFN-Sezione di Padova, Italy where this work was initiated. VTP is also indebted to Professor E.Quercigh for his kind hospitality in CERN (May 1994), and for very useful discussions. Finally, VTP greatfully acknowledges partial financial support from the Romanian Soros Foundation for Open Society Bucharest,Romania.
[199]{} Edited by E. Stenlund,H. A. Gustafsson,A. Oskarsson and I. Otterlund\
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[**pp**]{} [**Exp.data**]{} [**HIJING**]{} ${\bf HIJING^{(j)}}$
------------------------------------ ------------------- ---------------- ----------------------
$<\pi^{-}>$ $2.62\pm 0.06$ $2.61$ $2.65$
$<\pi^{+}>$ $3.22\pm 0.12$ $3.18$ $3.23$
$<\pi^{0}>$ $3.34 \pm 0.24$ $3.27$ $3.27$
$<h^{-}>$ $2.86 \pm 0.05$ $2.99$ $ 3.03$
$<K^{+}>$ $0.28 \pm 0.06$ $0.32$ $0.32$
$<K^{-}>$ $0.18 \pm 0.05$ $0.24$ $0.25$
$<\Lambda + \Sigma^{0}>$ $0.096 \pm 0.015$ $0.16$ $0.165$
$<\bar{\Lambda}+\bar{\Sigma^{0}}>$ $0.013 \pm 0.01$ $0.03$ $0.037$
$<K_{s}^{0}>$ $0.17 \pm 0.01$ $ 0.26$ $0.27$
$<p>$ $ 1.34\pm 0.15 $ $1.43$ $1.45$
$<\bar{p}>$ $0.05 \pm 0.02$ $0.11$ $0.12$
: Particle multiplicities for $\,pp\,$ interaction at $\,\,200\,$ GeV are compared with data from Gazdzicki and Hansen[]{data-label="Tab1"}
[**Particle type**]{} ${\bf <n>}$ $ {\bf Exp.data}$ $ {\bf HIJING^{(j)}}$
----------------------------------------------------------- ------------------- ------------------- -----------------------
[**All charged**]{} $29.4 \pm 0.3$ [@1ua5] $28.2$
$K^{0}+\bar{K^{0}}$ $2.24 \pm 0.16$ [@1ua5] $1.98$
$K^{+}+K^{-}$ $ 2.24 \pm 0.16 $ [@1ua5] $2.06$
$p+\bar{p}$ $1.45 \pm 0.15$ [@ward] $1.55$
$\Lambda+\bar{\Lambda}$ $0.53 \pm 0.11$ [@1ua5] $0.50$
$\Sigma^{+}+\Sigma^{-}+\bar{\Sigma^{+}}+\bar{\Sigma^{-}}$ $0.27 \pm 0.06$ [@ward] $0.23$
$\Xi^{-}$ $0.04 \pm 0.01$ [@1ua5] $0.037$
$\gamma$ $33 \pm 3$ [@1ua5] $29.02$
$\pi^{+}+\pi^{-}$ $23.9 \pm 0.4$ [@1ua5] $23.29$
$K_{s}^{0}$ $1.1 \pm 0.1$ [@1ua5] $0.99$
$\pi^{0}$ $11.0\pm 0.4$ [@ward] $13.36$
: Particle composition of $p+\bar{p}$ interactions at 540 GeV in cm.[]{data-label="Tab2"}
-------------------- ------------------ ------------------- ---------------------- -------------------------- ------------------------
${ \bf Reaction}$ ${ \bf <h^{-}>}$ $ { \bf <\Lambda>}$ ${ \bf <\bar{\Lambda}>}$ ${ \bf <K_{s}^{0}> }$
[**DATA**]{} $ 2.85 \pm 0.03$ $ 0.096 \pm 0.015$ $ 0.013 \pm 0.005$ $0.17 \pm 0.01$
[**HIJING**]{} $ 2.99$ $ 0.16$ $ 0.030$ $ 0.26 $
[**VENUS**]{} $2.79$ $0.181$ $0.033$ $0.27$
[**RQMD**]{} $2.59$ $0.11$ $ 0.21$
${\bf p+p}$ [**QGSM**]{} $2.85$ $0.15$ $0.015$ $0.21$
$ {\bf DPM^{1}}$ $3.52$ $0.155$ $0.024$ $0.18$
${\bf DPM^{2}}$ $3.52$ $0.155$ $0.024$ $0.18$
${\bf p+S}$ [**DATA**]{} $ 5.7 \pm 0.2$ $ 0.28 \pm 0.03$ $ 0.049 \pm 0.006$ $0.38\pm0.05$
$'min. bias' $ ${\bf HIJING}$ $4.83$ $0.255$ $0.046$ $0.400 $
[**VENUS**]{} $5.40$ $0.340$ $0.065$ $0.510$
[**QGSM**]{} $5.87$ $0.240$ $0.023$ $0.340$
$ {\bf DPM^{1}}$ $5.53$ $0.300$ $0.043$ $ 0.360$
${\bf DPM^{2}}$ $5.54$ $0.32$ $0.060$ $ 0.360$
${\bf p+Ag}$ [**DATA**]{} $6.2\pm0.2$ $0.37\pm0.06$ $ 0.05 \pm 0.02 $ $0.525 \pm 0.07 $
$'min. bias' $ ${\bf HIJING}$ $6.28$ $0.34$ $0.054$ $ 0.505$
${\bf p+Au}$ [**DATA**]{} $9.6 \pm 0.2$
$'central'$ ${\bf HIJING}$ $11.25$ $0.67$ $ 0.090$ $0.88$
-------------------- ------------------ ------------------- ---------------------- -------------------------- ------------------------
: Average multiplicities for negative charged hadrons and neutral strange hadrons in $\,pp\,$ and $\,pA\,$ interactions. HIJING and VENUS model results are compared with others recent estimates using RQMD, QGSM, DPM and with data from Alber et al..[]{data-label="Tab3"}
-------------------- ----------------- ------------------- ---------------------- -------------------------- ------------------------
${ \bf Reaction}$ ${ \bf <h^{-}>}$ $ { \bf <\Lambda>}$ ${ \bf <\bar{\Lambda}>}$ ${ \bf <K_{s}^{0}> }$
[**DATA**]{} $95 \pm 5 $ $9.4\pm1.0$ $ 2.2\pm0.4$ $10.5\pm1.7 $
${\bf S+S}$ ${\bf HIJING}$ $88.8$ $4.58$ $0.86$ $ 7.23$
$'central'$ ${\bf VENUS}$ $ 94.06$ $8.20$ $2.26$ $11.94$
${\bf RQMD}$ $110.2$ $7.76$ $10.0$
${\bf QGSM}$ $120.0$ $4.70$ $0.35$ $7.0$
${\bf DPM^{1}}$ $109.8$ $6.83$ $0.80$ $10.6$
${\bf DPM^{2}}$ $107.0$ $7.18$ $1.57$ $10.24$
[**DATA**]{} $160\pm8$ $15.2 \pm 1.2 $ $ 2.6 \pm 0.3$ $ 15.5 \pm 1.5 $
${\bf S+Ag}$ ${\bf HIJING}$ $ 164.35$ $ 8.61$ $1.48$ $13.20$
$'central'$ ${\bf RQMD}$ $192.3$ $13.4$ $18.30$
${\bf DPM^{1}}$ $195.0$ $13.3$ $1.45$ $19.40$
${\bf DPM^{2}}$ $186.90$ $14.06$ $3.65$ $15.73$
${\bf S+Au}$ ${\bf HIJING}$ $213.2$ $11.3$ $1.81$ $16.55$
$'central'$ ${\bf VENUS}$ $201.6$ $14.0$ $3.01$ $21.52$
${\bf S+W}$ ${\bf HIJING}$ $210.0$ $10.64$ $ 1.71$ $16.05$
$'central'$
${\bf Pb+Pb}$ ${\bf HIJING}$ $725.15$ $36.44$ $5.93$ $54.86$
$'central'$
-------------------- ----------------- ------------------- ---------------------- -------------------------- ------------------------
: Average multiplicities for negative charged hadrons and neutral strange hadrons in $\,AA\,$ interactions. HIJING and VENUS model results are compared with others recent estimates using RQMD, QGSM, DPM and with data from Alber et al..[]{data-label="Tab4"}
${\bf Reaction}$ $<\pi^{-}>$ $<E_{S}>$
------------------ ----------------- ------------------ --------------------
${\bf p+p}$ [**DATA**]{} $ 2.62 \pm 0.06$
[**HIJING**]{} $ 2.61$ $0.153$
${\bf N+N}$ [**DATA**]{} $3.06 \pm 0.08 $ $0.100\pm 0.01$
$ {\bf HIJING}$ $2.89$ $0.140$
${\bf p+S}$ [**DATA**]{} $5.26\pm0.13$ $ 0.086 \pm 0.008$
$'min. bias' $ ${\bf HIJING}$ $4.3$ $0.144$
${\bf p+Ag}$ [**DATA**]{} $6.4\pm0.11$ $0.108\pm0.009$
$'min. bias' $ ${\bf HIJING}$ $5.59$ $0.141$
${\bf p+Au}$ [**DATA**]{} $9.3 \pm 0.2$ $0.073\pm0.015$
$'central'$ ${\bf HIJING}$ $10.22$ $0.136$
${\bf S+S}$ [**DATA**]{} $88\pm5 $ $0.183\pm0.012$
$'central'$ ${\bf HIJING}$ $79.6$ $0.140$
${\bf S+Ag}$ [**DATA**]{} $149\pm8$ $0.173\pm0.017$
$'central'$ ${\bf HIJING}$ $147.8$ $0.138$
: The mean multiplicities of negative pions and $\,\,E_{S}\,\,$ ratios(see the text for definition) for nuclear collisions at $\,\, 200\,$ AGeV. The data are from Alber et al.and the $\,\,NN\,\,$ data are from Gazdzicki and Hansen .[]{data-label="Tab5"}
[^1]: On leave from absence from Institute for Space Sciences,P.O.Box MG - 6,Bucharest, [**Romania**]{} E-MAIL [email protected](EARN );[email protected]
[^2]: This work was supported in part by the Director, Office of Energy Research,Division of Nuclear Physics of the Office of High Energy and Nuclear Physics of the U.S Department of Energy under Contract No.DE-FG02-93ER40764.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'On large angular scales, the polarization of the CMB contains information about the evolution of the average ionization during the epoch of reionization. Interpretation of the polarization spectrum usually requires the assumption of a fixed functional form for the evolution, e.g. instantaneous reionization. We develop a model-independent method where a small set of principal components completely encapsulate the effects of reionization on the large-angle $E$-mode polarization for any reionization history within an adjustable range in redshift. Using Markov Chain Monte Carlo methods, we apply this approach to both the 3-year [*WMAP*]{} data and simulated future data. [*WMAP*]{} data constrain two principal components of the reionization history, approximately corresponding to the total optical depth and the difference between the contributions to the optical depth at high and low redshifts. The optical depth is consistent with the constraint found in previous analyses of [*WMAP*]{} data that assume instantaneous reionization, with only slightly larger uncertainty due to the expanded set of models. Using the principal component approach, [*WMAP*]{} data also place a 95% CL upper limit of 0.08 on the contribution to the optical depth from redshifts $z>20$. With improvements in polarization sensitivity and foreground modeling, approximately five of the principal components can ultimately be measured. Constraints on the principal components, which probe the entire reionization history, can test models of reionization, provide model-independent constraints on the optical depth, and detect signatures of high-redshift reionization.'
author:
- 'Michael J. Mortonson$^{1,2}$ and Wayne Hu$^{1,3}$'
title: 'Model-independent constraints on reionization from large-scale CMB polarization'
---
Introduction {#sec:intro}
============
The amplitude of the $E$-mode component of the cosmic microwave background (CMB) polarization on large scales provides the current best constraint on the Thomson scattering optical depth to reionization, $\tau$. Using the first three years of data from the *Wilkinson Microwave Anisotropy Probe* ([*WMAP*]{}) and making the simple assumption that the universe was reionized instantaneously, [@Speetal06] find $\tau = 0.09 \pm 0.03$. Theoretical studies suggest that the process of reionization was too complex to be well described as a sudden transition [e.g., @BarLoe01]. Previous studies have examined how the constraint on $\tau$ depends on the evolution of the globally-averaged ionized fraction during reionization, $x_e(z)$, for a variety of specific theoretical scenarios. If the assumed form of $x_e(z)$ is incorrect, the estimated value of $\tau$ can be biased; this bias can be lessened by considering a wider variety of reionization histories at the expense of increasing the uncertainty in $\tau$ [@Kapetal03; @Holetal03; @Coletal05].
The angular scales on which CMB polarization from reionization is correlated depend on the horizon size at the redshift of the free electrons: the higher the redshift, the higher the multipole, $\ell$ [e.g., @Zal97; @HuWhi97a]. Varying $x_e(z)$ changes the relative contributions to the polarization coming from different redshifts, and therefore changes the shape of the large-scale $E$-mode angular power spectrum, ${C_{\ell}^{EE}}$. Because of this dependence, measurements of the low-$\ell$ $E$-mode spectrum should place at least weak constraints on the global reionization history in addition to the constraint on the total optical depth. Recent studies suggest that [*WMAP*]{} data provide little information about $x_e(z)$ beyond $\tau$ [e.g., @LewWelBat06], but it is worth asking what we can ultimately expect to learn about reionization from CMB polarization.
@HuHol03 proposed using a principal component decomposition of the ionization history to quantify the information contained in the large-scale $E$-mode polarization. The effect of any ionization history on the $E$-modes can be completely described by a small number of eigenmode parameters, unlike a direct discretization of $x_{e}(z)$ in redshift bins. Here we extend the methods of [@HuHol03] using Markov Chain Monte Carlo techniques to find constraints on the principal components of $x_e(z)$ using both the 3-year [*WMAP*]{} observations and simulated future data.
Analytic studies and simulations indicate that reionization is an inhomogeneous process, and this inhomogeneity is expected to contribute to the small-scale CMB temperature and polarization anisotropies [e.g., @Hu00; @Ilietal06a; @MorHu07; @Doretal07]. Here we focus on the large-scale $E$-modes only ($\ell \lesssim 100$) where such effects can be neglected, so we only consider the evolution of the globally-averaged ionized fraction as a function of redshift.
In the following section, we describe the principal component method for parameterizing the ionization history and show that the effects of $x_e(z)$ on large-scale $E$-mode polarization can be encapsulated in a small set of parameters. The method allows $x_e(z)$ to be a free function of redshift that is not bounded by physical considerations, so in § \[sec:phys\] we derive limits that can be placed on the model parameters to eliminate most of the unphysical models where $x_e>1$ or $x_e<0$. We outline several ways to apply the principal component approach to constrain the reionization history with large-scale $E$-mode data in § \[sec:appl\]. Using Markov Chain Monte Carlo methods, we examine some of these applications in more detail in § \[sec:mcmc\] using both 3-year [*WMAP*]{} data and simulated future CMB polarization data. We summarize our findings and conclude in § \[sec:discuss\].
Ionization History Eigenmodes {#sec:eigenmodes}
=============================
Models with similar total optical depth but different ionization histories can produce markedly different predictions for the $E$-mode power spectrum. In an instantaneous reionization scenario, the contribution to the optical depth from $x_e(z)$ is concentrated at the lowest redshifts possible for a given $\tau$, and the $E$-mode power spectrum for such a model is sharply peaked on large scales, at $\ell \lesssim 10$. The main effect of shifting some portion of the reionization history to higher redshifts while keeping $\tau$ fixed is to reduce the $E$-mode power on the largest scales and increase it on smaller scales.
This redistribution of power is illustrated in Figure \[fig:cl\] by the ionization histories and ${C_{\ell}^{EE}}$ for two models, one with nearly instantaneous reionization and the other with comparable optical depth but with $x_e(z)$ concentrated at higher $z$. In general, a flatter large-scale $E$-mode spectrum with power extending out to $\ell \sim 10$-20 is a sign of a large ionized fraction at high redshift. However, there is not a one-to-one correspondence between $\ell$ and $z$; the ionized fraction within any particular narrow redshift bin affects ${C_{\ell}^{EE}}$ over a wide range of angular scales. Ionized fractions in adjacent redshift bins have highly correlated effects on ${C_{\ell}^{EE}}$, which makes it difficult to extract from $E$-mode polarization data a constraint on $x_e$ at a specific redshift.
As suggested by [@HuHol03], one can use principal components of the reionization history as the model parameters instead of $x_e$ in redshift bins. These components are defined to have uncorrelated contributions to the $E$-mode power; since each has a unique effect on ${C_{\ell}^{EE}}$, the amplitudes of the components can be inferred from measurements of the large-scale power spectrum. Principal component analysis of $x_e(z)$ also indicates which components can be determined best from the data. In this section, we describe the principal component method and introduce the notation that we will use throughout the paper.
Consider a binned ionization history $x_e(z_i)$, $i\in \{1,2,\ldots,N_z\}$, with redshift bins of width ${\Delta z}$ spanning ${z_{\rm min}}\leq z \leq {z_{\rm max}}$, where $z_1=z_{\rm min}+{\Delta z}$ and $z_{N_z}=z_{\rm max}-{\Delta z}$ so that $N_z+1=(z_{\rm max}-z_{\rm min})/\Delta z$. Throughout this paper we assume that the ionized fraction is $x_e\approx 1$ for redshifts $z\leq z_{\rm min}$ and $x_e\approx 0$ at $z\geq z_{\rm max}$. We take $z_{\rm min}=6$, consistent with observations of quasar spectra [@FanCarKea06].
The principal components of $x_e(z_i)$ are eigenfunctions of the Fisher matrix, computed by taking derivatives of ${C_{\ell}^{EE}}$ with respect to $x_e(z_i)$: $$F_{ij}=\sum_{\ell=2}^{\ell_{\rm max}}\left(\ell+\frac{1}{2}\right)
\frac{\partial \ln {C_{\ell}^{EE}}}{\partial x_e(z_i)}
\frac{\partial \ln {C_{\ell}^{EE}}}{\partial x_e(z_j)} \,,
\label{eq:fisher1}$$ assuming full sky coverage and neglecting noise. Since $x_e(z_i)$ only significantly contributes to the $E$-mode spectrum at small $\ell$, we typically truncate the sum in equation (\[eq:fisher1\]) at $\ell_{\rm max}=100$ where ${C_{\ell}^{EE}}$ is dominated by the first acoustic peak. The derivatives are evaluated at a fiducial reionization history, ${x_e^{\rm fid}}(z_i)$. Following [@HuHol03] we typically choose ${x_e^{\rm fid}}(z_i)$ to be constant during reionization, although other functional forms may be used.
Since the effects on ${C_{\ell}^{EE}}$ of $x_e(z_i)$ in adjacent redshift bins are highly correlated, the Fisher matrix contains large off-diagonal elements. The principal components $S_{\mu}(z_i)$ are the eigenfunctions of $F_{ij}$, $$F_{ij}=(N_z+1)^{-2}\sum_{\mu=1}^{N_z}
S_{\mu}(z_i)\sigma^{-2}_{\mu}S_{\mu}(z_j),
\label{eq:fisher2}$$ where the factor $(N_z+1)^{-2}$ is included so that the eigenfunctions and their amplitudes have certain convenient properties. The inverse eigenvalues, $\sigma^2_{\mu}$, give the estimated variance of each principal component eigenmode from the measurement of low-$\ell$ $E$-modes. We order the modes so that the best-constrained principal components (smallest $\sigma^2_{\mu}$) have the lowest values of $\mu$, starting at $\mu=1$. The noise level and other characteristics of an experiment can be included in the construction of the eigenfunctions, but since the effect on $S_{\mu}(z)$ is small we always use the noise-free eigenfunctions here.
The eigenfunctions satisfy the orthogonality and completeness relations $$\begin{aligned}
\int _{{z_{\rm min}}}^{{z_{\rm max}}} dz
~S_{\mu}(z)S_{\nu}(z)&=&(z_{\rm max}-z_{\rm min})\delta_{\mu\nu}\,,
\label{eq:orthog1}\\
\sum_{\mu=1}^{N_z} S_{\mu}(z_i)S_{\mu}(z_j)&=& (N_z+1) \delta_{ij}\,.
\label{eq:orthog2}\end{aligned}$$ The normalization of $S_{\mu}(z)$ is chosen so that the eigenfunctions are independent of bin width as ${\Delta z}\rightarrow 0$. In equation (\[eq:orthog1\]) and elsewhere in this paper where there are sums over redshift, we assume the continuous limit, replacing $\sum_i \Delta z$ by $\int dz$. As long as the bin width is chosen to be sufficiently small, the final results we obtain are independent of the redshift binning. We adopt $\Delta z=0.25$ as the default bin width.
The three lowest-variance eigenfunctions for two different fiducial models are shown in Figure \[fig:eigfn\]. The lowest eigenmode ($\mu=1$) is an average of the ionized fraction over the entire redshift range, weighted at high $z$. The $\mu=2$ mode can be thought of as a difference between the amount of ionization at high $z$ and at low $z$, and higher modes follow this pattern with weighted averages of $x_e(z)$ that oscillate with higher and higher frequency in redshift. Eigenfunctions of fiducial models with different values of ${z_{\rm max}}$ have similar shapes with the redshift axis rescaled according to the width of $({z_{\rm max}}-{z_{\rm min}})$. The eigenfunctions are mostly insensitive to the choice of ionized fraction in the constant-$x_e$ fiducial histories.
An arbitrary reionization history can be represented in terms of the eigenfunctions as $$x_e(z)={x_e^{\rm fid}}(z)+\sum_{\mu}m_{\mu}S_{\mu}(z),
\label{eq:mmutoxe}$$ where the amplitude of eigenmode $\mu$ for a perturbation $\delta x_e(z)\equiv x_e(z)-{x_e^{\rm fid}}(z)$ is $$m_{\mu}=\frac{1}{{z_{\rm max}}-{z_{\rm min}}}
\int _{{z_{\rm min}}}^{{z_{\rm max}}} dz~S_{\mu}(z)\delta x_e(z).
\label{eq:xetommu}$$ \[Note that our conventions for the normalization of $m_{\mu}$ and $S_{\mu}(z)$ differ from those of [@HuHol03] by factors of $(N_z+1)^{1/2}$.\] Any global ionization history $x_e(z)$ over the range $z_{\rm min}<z<{z_{\rm max}}$ is completely specified by a set of mode amplitudes $m_{\mu}$.
If perturbations to the fiducial history are small, $\delta x_e(z) \ll 1$, then the mode amplitudes are uncorrelated, with covariance matrix $\langle m_{\mu}m_{\nu}\rangle=\sigma_{\mu}^2 \delta_{\mu\nu}$. For a fixed fiducial model, however, arbitrary reionization histories generally have $\delta x_e\sim 1$, in which case the amplitudes of different modes can become correlated as we discuss in § \[sec:fisher\].
The main advantage of using the principal component eigenmodes of $x_e(z)$ instead of some other parameterization is that most of the information relevant for large-scale $E$-modes is contained in the first few modes. This means that if one constructs $x_e(z)$ from equation (\[eq:mmutoxe\]) keeping only the first few terms in the sum over $\mu$, then the $E$-mode spectrum of the resulting ionization history will closely match that of $x_e(z)$ with all modes included in the sum. The effect of each eigenmode on ${C_{\ell}^{EE}}$ becomes smaller as $\mu$ increases, as shown in Figure \[fig:zmaxa\] for the first two modes. [@HuHol03] demonstrated that for a specific fiducial ionization history ${x_e^{\rm fid}}(z)$ and assumed true history, only the first three modes of $x_e(z)$ are needed to produce ${C_{\ell}^{EE}}$ indistinguishable from the true $E$-mode spectrum.
The ionization histories and corresponding $E$-mode spectra in Figure \[fig:clcomp\] demonstrate this completeness for a fairly extreme model in which the first ten eigenmodes all have significant amplitudes. Even in the simplest case where a single eigenmode is used in place of the original $x_e(z)$, the error in ${C_{\ell}^{EE}}$ is only $\sim 10\%$. With 3-5 modes, the error is a few percent or less at all multipoles and safely smaller than the cosmic variance of $${\Delta {C_{\ell}^{EE}}\over {C_{\ell}^{EE}}} = \sqrt{2 \over 2 \ell +1} \,.$$ The top panel of Figure \[fig:clcomp\] shows that this completeness does not extend to the ionization history itself: $x_e(z)$ constructed from as many as five eigenmodes is a poor approximation to the full reionization history.
From this and other similar tests on the completeness in ${C_{\ell}^{EE}}$ of the lowest-variance eigenmodes we conclude that the first 3-5 modes contain essentially all of the information about the reionization history that is relevant for large-scale $E$-mode polarization. This fact is particularly useful for constraining the global reionization history with CMB polarization data using Markov Chain Monte Carlo techniques. Since the number of parameters that must be added to a Monte Carlo chain to describe an arbitrary $x_e(z)$ is relatively small, we can obtain constraints from the data that are independent of assumptions about the reionization history with minimal added computational expense [@HuHol03]. The exact number of modes required varies depending on the true ionization history and the fiducial history, so when analyzing data it is a good idea to check that the results do not change significantly when the next modes are included in the sum in equation (\[eq:mmutoxe\]).
The main caveat to this model independence is that in practice we must set some maximum redshift for histories in any particular chain of Monte Carlo samples, ignoring any contribution to the observed low-$\ell$ ${C_{\ell}^{EE}}$ from ionization at $z>{z_{\rm max}}$. Since the eigenfunctions of the ionization history are stretched in redshift as ${z_{\rm max}}$ increases (Fig. \[fig:eigfn\]), at higher ${z_{\rm max}}$ more modes are needed to accurately represent any particular feature in $x_e(z)$. For example, take the true ionization history to be instantaneous reionization at $z=11.5$. Figure \[fig:zmaxb\] shows the error in ${C_{\ell}^{EE}}$ if we truncate the eigenmode sum of equation (\[eq:mmutoxe\]) at $N$ modes using fiducial histories with ${z_{\rm max}}=30$ and ${z_{\rm max}}=20$. For each fiducial model, the error decreases as the number of modes in the sum increases, but the error at fixed $N$ is larger for ${z_{\rm max}}=30$ than ${z_{\rm max}}=20$. The requirement of retaining a larger set of parameters as ${z_{\rm max}}$ increases makes it less practical to study models with significant reionization at extremely high redshifts \[${z_{\rm max}}\gtrsim 100$; e.g., [@NasChi04; @KasKawSug04]\], but even for ${z_{\rm max}}$ as high as $\sim 40$ the number of eigenmodes needed is reasonably small ($N\lesssim 5$).
While theories of reionization provide useful priors on ${z_{\rm max}}$ in the context of specific models, it would be better to be able to constrain ${z_{\rm max}}$ empirically by measuring the $E$-mode power accurately up to $\ell \sim 100$. The current 3-year [*WMAP*]{} polarization data have high enough signal-to-noise to be useful for parameter constraints only at $\ell < 24$ [@Pagetal06], so their sensitivity to high-$z$ reionization is limited. Given that ionization at some redshift generates polarization out to a maximum $\ell$, [*WMAP*]{} data can still place weak bounds on the total optical depth contribution above a certain redshift, even if it cannot distinguish whether these contributions arise from redshifts above a chosen ${z_{\rm max}}$ as we discuss in § \[sec:wmap\]. The values we choose here for ${z_{\rm max}}$ are partly influenced by the fact that few theoretical reionization scenarios predict an ionized fraction at $z\gtrsim 30$ that would significantly affect large-scale ${C_{\ell}^{EE}}$. Future data should better constrain ${C_{\ell}^{EE}}$ at higher multipoles, allowing useful limits to be placed on ${z_{\rm max}}$ from the data alone.
The need to consider a limited range in redshift is shared by other methods, for example those that constrain binned $x_e(z_i)$ instead of the eigenmode amplitudes [@LewWelBat06]. As already mentioned, the principal component approach has the unique advantages that results can be made independent of bin width and that relatively few extra parameters are required. However, this approach also has a unique difficulty in that the physicality of the reionization history \[$0\leq x_e(z)\leq 1$\] is not built in to the method. Hence constraints derived from measurements of the eigenmodes can be weaker than those from a method that enforces physicality. We can, however, place some prior constraints on the set of eigenmode amplitudes that must be satisfied by any physical model, as we discuss in the next section.
Priors from Physicality {#sec:phys}
=======================
Although the actual ionized fraction must be between 0 and 1 (neglecting helium reionization and the small residual ionized fraction after recombination), there is nothing in the construction of $x_e(z)$ from the eigenmodes in equation (\[eq:mmutoxe\]) that ensures that the ionized fraction will obey these limits. Whether or not $x_e$ has a physical value at a particular redshift depends on the amplitudes of all of the principal components. Even for a physical ionization history, the truncated sum up to mode $N$, $$x_e^{(N)}(z)\equiv {x_e^{\rm fid}}(z)+\sum_{\mu=1}^N m_{\mu} S_{\mu}(z),
\label{eq:trunc}$$ is not necessarily bounded by 0 and 1 at all redshifts. While formally it is possible to evaluate ${C_{\ell}^{EE}}$ for reionization histories with unphysical values of $x_e$, we would like to eliminate as much as possible those models for which the full sum of the eigenmodes, $x_e(z)=\lim_{N\to\infty}x_e^{(N)}(z)$, is unphysical.
We find the largest and smallest values of $m_{\mu}$ that are consistent with $x_e(z)\in [0,1]$ for all $z$ using the definition of $m_{\mu}$ in equation (\[eq:xetommu\]). We are free to choose ${x_e^{\rm fid}}(z)\in [0,1]$ so that $-{x_e^{\rm fid}}(z)\leq\delta x_e(z)\leq 1-{x_e^{\rm fid}}(z)$, where the lower limit is strictly negative or zero and the upper limit is positive or zero. For a particular mode $\mu$, the choice of $\delta x_e(z)$ that maximizes $m_{\mu}$ is $$\delta x_e^{\rm max}(z)=\left\{ \begin{array}{rl}
-{x_e^{\rm fid}}(z), & S_{\mu}(z)\leq 0 \\
1-{x_e^{\rm fid}}(z), & S_{\mu}(z)>0.
\end{array}\right.
\label{eq:dxemax}$$ Using this in equation (\[eq:xetommu\]) gives an upper limit on $m_{\mu}$. Similarly, a lower limit can be obtained by reversing the signs of the inequalities in equation (\[eq:dxemax\]). The resulting physicality bounds are $m_{\mu}^{(-)}\leq m_{\mu}\leq m_{\mu}^{(+)}$, where $$m_{\mu}^{(\pm)} = \int _{{z_{\rm min}}}^{{z_{\rm max}}} dz
\frac{S_{\mu}(z)[1-2{x_e^{\rm fid}}(z)]\pm |S_{\mu}(z)|}{2({z_{\rm max}}-{z_{\rm min}})}.
\label{eq:physbounds}$$ If $m_{\mu}$ violates these bounds for any $\mu$, the reionization history is guaranteed to be unphysical for some range in redshift. The opposite is not true, however: even if all $m_{\mu}$ satisfy equation (\[eq:physbounds\]), $x_e(z)$ may still be unphysical for some $z$.
The parameter space that physical models may occupy is restricted further by an inequality that must be satisfied by all eigenmodes simultaneously. Assume for simplicity that the fiducial model has a constant ionized fraction, ${x_e^{\rm fid}}\in [0,1]$, for ${z_{\rm min}}<z<{z_{\rm max}}$. Any physical reionization history $x_e(z)$ must satisfy $$\int _{{z_{\rm min}}}^{{z_{\rm max}}} dz [x_e(z)-{x_e^{\rm fid}}]^2 \leq ({z_{\rm max}}-{z_{\rm min}}) f,
\label{eq:phys2}$$ where $f\equiv \max[({x_e^{\rm fid}})^2,(1-{x_e^{\rm fid}})^2]$. Using equation (\[eq:mmutoxe\]), the left side of the inequality can also be written $$\begin{aligned}
\int _{{z_{\rm min}}}^{{z_{\rm max}}} dz [x_e(z)-{x_e^{\rm fid}}]^2&=&\int _{{z_{\rm min}}}^{{z_{\rm max}}} dz
\left[\sum_{\mu} m_{\mu} S_{\mu}(z) \right]^2 \nonumber\\
&=& ({z_{\rm max}}-{z_{\rm min}}) \sum_{\mu} m_{\mu}^2 \,,
\label{eq:phys3}\end{aligned}$$ where the second line follows from the orthogonality of the eigenfunctions (eq. \[\[eq:orthog1\]\]). Comparing equations (\[eq:phys2\]) and (\[eq:phys3\]) we obtain a constraint on the sum of the squares of the mode amplitudes, $$\sum_{\mu} m_{\mu}^2 \leq f,
\label{eq:physbounds2}$$ where $0.25\leq f\leq 1$, depending on the value of ${x_e^{\rm fid}}$. As with the physicality bounds of equation (\[eq:physbounds\]), this upper limit is a necessary but not sufficient condition for physicality.
Since in practice we can only constrain a limited set of eigenmodes, the uncertainty in modes higher than the first few prevents us from simply excluding all models where $x_e<0$ or $x_e>1$ at any redshift because the higher modes can have a significant effect on $x_e(z)$. However, as shown in § \[sec:eigenmodes\], the higher modes do not affect the polarization power spectrum since the high-frequency oscillations in redshift of higher modes are averaged out. Similarly, such eigenmodes have a small effect on the optical depth from a sufficiently large range in redshift. For example, the optical depth from $15<z<30$ due to modes $\mu>5$ subject to the physicality constraints of this section can be no larger than $\sim 0.01$, and is likely to be smaller for realistic reionization scenarios. Because of this, we assume that we can place priors on the *optical depth* that correspond to $0 \le x_e \le 1$ over the relevant range in redshift. Reionization histories with an unphysical optical depth over a large range in redshift are considered to be unphysical models since the addition of higher modes can not perturb the optical depth enough to give it a physical value.
Applications of the Principal Component Method {#sec:appl}
==============================================
Once constraints on the principal components of the reionization history have been obtained from CMB polarization data, there are several ways to use those constraints to place limits on observables such as $\tau$ or to test theories of reionization [@HuHol03]. We describe some possible applications in this section, and in § \[sec:mcmc\] we put these ideas into practice using the 3-year [*WMAP*]{} data and simulated future data.
As mentioned in § \[sec:intro\], the constraint on the total optical depth to reionization depends on the assumed model for $x_e(z)$. The principal component method allows us to explore all globally-averaged ionization histories within a chosen redshift range, ${z_{\rm min}}<z<{z_{\rm max}}$. For a given set of eigenmode amplitudes, $\{m_{\mu}\}$, equation (\[eq:mmutoxe\]) yields the corresponding ionization history which can then be integrated to find the optical depth between any two redshifts $z_{1}$ and $z_{2}$, $$\tau(z_1,z_2) = 0.0691(1-Y_p)\Omega_b h
\int_{z_1}^{z_2} dz \frac{(1+z)^2}{H(z)/H_0}x_e(z).
\label{eq:tauz1z2}$$ In particular, the total optical depth to reionization is $$\tau = \tau(0,{z_{\rm min}})+\tau({z_{\rm min}},{z_{\rm max}}),
\label{eq:tau}$$ where $\tau(0,{z_{\rm min}})\approx 0.04$ for ${z_{\rm min}}=6$. The principal component approach provides a model-independent way to constrain $\tau$, so we expect the results to be unbiased and the uncertainty in $\tau$ to accurately reflect the present uncertainty about $x_e(z)$. Moreover, the information about $\tau({z_{\rm min}},{z_{\rm max}})$ is encapsulated in the first few eigenmodes, i.e. the truncated ionization history of equation (\[eq:trunc\]), $x_e^{(N)}(z)$, with $N=3$-5 for typical fiducial models.
The total optical depth and the first principal component amplitude, $m_1$, are similar quantities in that they are both averages of $x_e(z)$ weighted at high $z$. As we show in the next section, CMB $E$-mode polarization data can constrain higher eigenmodes as well ($\mu \geq 2$). In particular, the second mode should be the next best constrained quantity since it is constructed to have the smallest variance after $m_1$. Since $m_2$ is related to the difference in $x_e$ between high redshift and low redshift (see Fig. \[fig:eigfn\]), we might guess that besides total $\tau$ the data are mainly constraining the fraction of optical depth coming from high redshift versus low redshift. Given any set of mode amplitudes defining an ionization history, we can compute a low-$z$ optical depth, $\tau({z_{\rm min}},{z_{\rm mid}})$, and a high-$z$ optical depth, $\tau({z_{\rm mid}},{z_{\rm max}})$, for some choice of intermediate redshift, ${z_{\rm mid}}$.
Note that if either of the redshift ranges $[{z_{\rm min}},{z_{\rm mid}}]$ or $[{z_{\rm mid}},{z_{\rm max}}]$ is too narrow, constraints on the partial optical depths will be influenced by the physicality priors (since $0\leq x_e\leq 1$ sets limits on $\tau$ in redshift intervals as discussed in § \[sec:phys\]) and by the uncertainty in modes higher than those included in the chain, which have greater effect on the optical depth in narrower redshift intervals. For these reasons, ${z_{\rm mid}}$ should be chosen to be not too close to either ${z_{\rm min}}$ or ${z_{\rm max}}$.
Given an appropriately chosen value of ${z_{\rm mid}}$, constraints on the principal components can be converted into constraints on the optical depths at high and low redshift. These partial optical depths are observables in the sense that high-$z$ and low-$z$ optical depth affect the large-scale $E$-modes over different ranges of multipoles. For example, compare the two reionization models in Figure \[fig:cl\]. Both have similar total optical depth, but in one case $\tau$ only comes from $z<15$, resulting in more power at $\ell < 8$ and less power at $\ell > 8$ than the other model in which the optical depth primarily comes from $z>15$.
Besides learning about the relative amount of ionization at $z<{z_{\rm mid}}$ and $z>{z_{\rm mid}}$, the partial optical depth constraints also provide a way to empirically set the maximum redshift, ${z_{\rm max}}$. If some set of data are found to place a tight upper bound on $\tau({z_{\rm mid}},{z_{\rm max}})$ for fairly conservative (i.e. large) values of ${z_{\rm mid}}$ and ${z_{\rm max}}$, then for analyses of future data this value of ${z_{\rm mid}}$ can be used as a new, lower value for ${z_{\rm max}}$ since optical depth from higher redshifts is small. This approach assumes that there is essentially no reionization earlier than the original ${z_{\rm max}}$, but as long as this initial maximum redshift is taken to be large the presence or absence of high-$z$ ionization can be tested in this way over a wide range of redshifts. We explore this idea further in § \[sec:mcmc\].
While we do not examine constraints on specific reionization models in this paper, the principal component approach is well suited to model testing. Consider a model of the global reionization history, $x_e(z;\bm{\theta})$, parameterized by $\bm{\theta}$. This could be a simple toy model (for example, instantaneous reionization where $\bm{\theta}$ is a single parameter, the redshift of reionization) or a more physical model where $\bm{\theta}$ might include parameters that govern properties of the ionizing sources. For a particular choice of ${x_e^{\rm fid}}(z)$, the principal component amplitudes of the reionization model follow from equation (\[eq:xetommu\]), giving a set of mode amplitudes that depend on the model parameters, $\{m_{\mu}(\bm{\theta})\}$. Then constraints on the mode amplitudes from CMB polarization data (defined using the same fiducial history) can be mapped to constraints on the parameters of the reionization model.
Applying this method to a model that has as one of its parameters a maximum redshift allows one to obtain constraints on ${z_{\rm max}}$ within a class of theoretical models. Although obtaining good constraints on model parameters does not necessarily imply validation of the model class, this procedure provides a straightforward method by which different model classes can tested and falsified within a single analysis.
Generating Monte Carlo chains to find constraints on $\{m_{\mu}\}$ as described in the next section can be a somewhat time-consuming process, but it only needs to be done once per redshift range, after which any model of the global reionization history within this range ${z_{\rm min}}<z<{z_{\rm max}}$ can be tested using the same parameter chains. In cases where constraints from the data turn out to be close to Gaussian, the covariance matrix of the principal components can be used in place of the full Monte Carlo chains, reducing the amount of information needed for model testing from $\sim 10^5$ numbers in chains of Monte Carlo samples to only $N(N+3)/2$ numbers when $N$ eigenmodes are included in the chains \[$N$ mean values plus $N(N+1)/2$ entries in the covariance matrix, $\langle m_{\mu}m_{\nu}\rangle$\]. However, near-Gaussian constraints on $\{m_{\mu}\}$ are likely to be possible only for certain realizations of future data, as discussed in the next section.
Markov Chain Monte Carlo Constraints on Eigenmodes {#sec:mcmc}
==================================================
We find reionization constraints from CMB polarization data using Markov Chain Monte Carlo (MCMC) techniques to explore the principal component parameter space [see e.g. @Chretal01; @KosMilJim02; @Dunetal04]. Chains of Monte Carlo samples are generated using the publicly available code CosmoMC[^1] [@LewBri02], which includes the code CAMB [@Lewetal00] for computing the theoretical angular power spectrum at each point in the $\{m_1,\ldots,m_N\}$ parameter space. We have modified both codes to allow specification of an arbitrary reionization history calculated from a set of mode amplitudes using equation (\[eq:mmutoxe\]).
The principal component amplitudes are the only parameters allowed to vary in the chains. Since nearly all the information in ${C_{\ell}^{EE}}$ from reionization is contained within the first few eigenmodes of $x_e(z)$, we include the modes $\{m_1,\ldots,m_N\}$ in each chain with $3\leq N\leq 5$. We use only the $E$-mode polarization data for parameter constraints, and assume that the values of the standard [$\Lambda$CDM]{} parameters (besides $\tau$) are fixed by measurements of the CMB temperature anisotropies. This leads to a slight underestimate of the error on $\tau$ as we discuss later, but to a good approximation the effect of $x_e(z)$ on the large-scale $E$-modes is independent of the other parameters.
Specifically, we take $\Omega_bh^2=0.0222$, $\Omega_ch^2=0.106$, $100\theta=1.04$ (corresponding to $h=0.73$), $A_s e^{-2\tau}=1.7 \times 10^{-9}$, and $n_s=0.96$, consistent with results from the most recent version of the [*WMAP*]{} 3-year likelihood code. When computing the optical depth to reionization we take the primordial helium fraction to be $Y_p=0.24$. We also assume that ${z_{\rm min}}=6$, so that the optical depth contributed at lower redshifts is fixed at $\tau({z_{\rm min}})\approx 0.04$. The remaining total optical depth from reionization, $\tau({z_{\rm min}},{z_{\rm max}})$, is determined by the values of $\{m_{\mu}\}$ for each sample in the chains. The default bin width for our fiducial models ${x_e^{\rm fid}}(z)$ is ${\Delta z}=0.25$, which is small enough that numerical effects related to binning should be negligible.
To get accurate results from MCMC analysis, it is important to make sure that the parameter chains contain enough independent samples covering a sufficient volume of parameter space so that the density of the samples converges to the actual posterior probability distribution. For each scenario that we study, we run 4 separate chains until the Gelman and Rubin convergence statistic $R$, corresponding to the ratio of the variance of parameters between chains to the variance within each chain, satisfies $R-1<0.01$ [@GelRub92; @BroGel98]. The convergence diagnostic of [@RafLew92] is used to determine how much each chain must be thinned to obtain independent samples. Both of these statistics and other diagnostic measures are computed automatically by CosmoMC.
Since $x_e(z)$ during reionization must match onto $x_e\approx 1$ at $z=z_{\rm min}$ and $x_e\approx 0$ at $z={z_{\rm max}}$, there are often sharp transitions at these redshifts since nothing in equation (\[eq:mmutoxe\]) forces $x_e(z)$ to satisfy these boundary conditions. To avoid problems with the time integration in CAMB, we smooth the reionization history by convolving $x_e(z)$ with a Gaussian of width $\sigma_z=0.5$.
As described in previous sections, we assume that $0\leq x_e\leq 1$ between ${z_{\rm min}}$ and ${z_{\rm max}}$. This assumption neglects helium reionization, which can make $x_e$ slightly larger than unity, and the small residual ionized fraction remaining after recombination that prevents $x_e$ from ever being exactly zero. These are relatively small effects, especially since we are not placing constraints on $x_e(z)$ directly but rather on weighted averages of $x_e$ over redshift. The residual ionized fraction at $z>{z_{\rm max}}$ is accounted for in the Monte Carlo exploration of reionization histories.
In § \[sec:wmap\], we examine the current constraints from the 3-year [*WMAP*]{} data. We then provide forecasts for principal component constraints with idealized, cosmic variance-limited, simulated data in § \[sec:cvdata\]. In each case the likelihood computation includes only the $E$-mode polarization data, up to $\ell=100$ for simulated data and $\ell=23$ for [*WMAP*]{}; the likelihood code for [*WMAP*]{} does not use ${C_{\ell}^{EE}}$ at smaller scales due to low signal-to-noise. At multipoles $\ell \gtrsim 100$ the global reionization history only affects the amplitude of the angular power spectra, which we fix by setting $A_s e^{-2\tau}$ constant in the Monte Carlo chains. A comparison of the MCMC results with the Fisher matrix approximation follows in § \[sec:fisher\].
Constraints from [*WMAP*]{} {#sec:wmap}
---------------------------
In our analysis of [*WMAP*]{} data we use the 3-year [*WMAP*]{} likelihood code, with settings chosen so that likelihoods include only contributions from the low-$\ell$ $E$-mode polarization. In this regime, the code computes model likelihoods with a pixel-based method instead of using the angular power spectrum [@Pagetal06]. Since the maximum redshift at which there is still a significant ionized fraction is uncertain, we generate Monte Carlo chains using different fiducial models with $10\leq {z_{\rm max}}\leq 40$. To avoid possible bias due to neglecting the possibility of high-redshift reionization, we focus here on the results obtained using the more conservative values of ${z_{\rm max}}=30$ and 40.
The MCMC constraints on the first three principal components from [*WMAP*]{} are shown in Figure \[fig:mcmcwmap\] using fiducial models with ${z_{\rm max}}=30$ and ${z_{\rm max}}=40$. The marginalized constraints are plotted within the physicality bounds of equation (\[eq:physbounds\]), $m_{\mu}^{(-)}\leq m_{\mu}\leq m_{\mu}^{(+)}$, so the size of the contours inside each box gives an idea of the constraining power of the data within the space of potentially physical models. For both fiducial histories, the data place a strong upper limit on $m_1$ and weakly constrain $m_2$. It is important to note that although the parameters $\{m_1,m_2,m_3\}$ have the same names in both the ${z_{\rm max}}=30$ and ${z_{\rm max}}=40$ plots, they are defined with respect to different fiducial models and so we do not expect the contours in the left and right plots in Figure \[fig:mcmcwmap\] to agree exactly. The qualitative similarity between the contours simply reflects the fact that the eigenfunctions for different ${z_{\rm max}}$ have similar shapes (Figure \[fig:eigfn\]).
Since $m_1$ and $\tau$ are both averages of $x_e$ from ${z_{\rm min}}$ to ${z_{\rm max}}$ with more weight at high $z$ than low $z$, the strong constraint on $m_1$ mostly reflects the ability of the data to constrain the total optical depth to reionization. Indeed, $m_1$ and $\tau$ are strongly correlated in the Monte Carlo chains, while the correlations between $\tau$ and higher principal components of $x_e(z)$ are weaker.
Relative to the physicality bounds, the constraints in Figure \[fig:mcmcwmap\] are stronger for the ${z_{\rm max}}=40$ chains than for ${z_{\rm max}}=30$; this is to be expected since the physicality bounds on $\{m_{\mu}\}$ permit a greater variety of ionization histories, and in particular a wider range in $\tau$, when ${z_{\rm max}}$ is larger. The range of eigenmode amplitudes allowed by the limits of equation (\[eq:physbounds\]) is nearly independent of ${z_{\rm max}}$, but the effect of a unit-amplitude principal component on ${C_{\ell}^{EE}}$ increases with ${z_{\rm max}}$ as illustrated in Figure \[fig:zmaxa\].
As described in § \[sec:appl\], principal component constraints from MCMC yield model-independent constraints on the total optical depth. For each Monte Carlo sample of the principal components we compute $\tau$ using equation (\[eq:tau\]). The constraints on $\tau$ are listed in Table \[tab:tau\] for various values of ${z_{\rm max}}$ and $N$, the number of modes in the Monte Carlo chains. The uncertainty in $\tau$ from the Monte Carlo chains is slightly underestimated because we fix all cosmological parameters that are not directly related to reionization. To estimate the effect of this assumption, we compare constraints on $\tau$ in two cases where instantaneous reionization is assumed instead of using the model-independent principal component approach: one in which the non-reionization parameters are fixed as in the rest of our Monte Carlo chains, and the other, from the 3-year [*WMAP*]{} analysis of [@Speetal06], in which these parameters are allowed to vary. These constraints, listed in rows 4 and 5 of Table \[tab:tau\], suggest that fixing [$\Lambda$CDM]{} parameters besides $\tau$ reduces $\sigma_{\tau}$ by $\sim 10\%$.
[llcclc]{} & Use PCs & & & Fix other &\
Data & of $x_e(z)$? & ${z_{\rm max}}$ & $N$ & parameters? & $\tau$\
WMAP3 & Yes & 20 & 3 & Yes & $0.098\pm0.025$\
WMAP3 & Yes & 30 & 5 & Yes & $0.106\pm0.027$\
WMAP3 & Yes & 40 & 5 & Yes & $0.107\pm0.029$\
WMAP3 & No & – & – & Yes & $0.096\pm0.027$\
WMAP3 & No & – & – & No & $0.089\pm0.030$\
CV & Yes & 20 & 3 & Yes & $0.103\pm0.004$\
CV & Yes & 30 & 5 & Yes & $0.108\pm0.005$\
CV & No & – & – & Yes & $0.108\pm0.003$\
\[tab:tau\]
In all cases where we fix the other cosmological parameters, the uncertainty in $\tau$ is similar regardless of whether we use principal components to explore a variety of reionization histories or restrict the analysis to instantaneous $x_e(z)$. This is somewhat surprising, since in general one would expect that expanding the model space for $x_e(z)$ would increase the estimated error on $\tau$. The physicality priors may be responsible for reducing $\sigma_{\tau}$ slightly — top-hat priors on $\{m_{\mu}\}$ induce a prior on $\tau$ that is flat over a certain range but falls off approximately linearly at the edges — but even after accounting for priors, the optical depth constraint is robust to replacing the instantaneous reionization assumption with a model-independent analysis.
The insensitivity of the constraint on $\tau$ to the set of models considered is partly due to the fact that the degeneracy in the eigenmode constraints is aligned in the direction of constant $\tau$, as shown in Figure \[fig:tauinst\] in the $m_1-m_2$ plane. The set of instantaneous reionization models, plotted as a curve in Figure \[fig:tauinst\], cuts across the [*WMAP*]{} constraints on more general models in a region of high posterior probability where the distribution of samples varies slowly along lines of constant $\tau$. This large overlap between the general reionization histories favored by the data and the line of instantaneous models is the main reason why the probability distributions $P(\tau)$, plotted in the right panel of Figure \[fig:tauinst\], are similar for the two classes of models. \[The sharp cutoff at low $\tau$ in the instantaneous reionization $P(\tau)$ comes from our ${z_{\rm min}}=6$ prior.\] Note that the fact that the instantaneous reionization curve passes through the middle of the 68% confidence region also indicates that models with rapid reionization are not at all disfavored by the 3-year [*WMAP*]{} data. We use two methods to find the posterior distribution for $\tau$ in the instantaneous reionization case: one is the usual approach of varying the optical depth (or reionization redshift) in a Monte Carlo chain, and the other involves computing $P(\tau)$ for a subset of samples from chains in which principal components of $x_e(z)$ are varied, selecting only those samples with $\{m_{\mu}\}$ values close to the 1D instantaneous reionization curve. Both approaches produce consistent probability distributions; the former method is used for $P(\tau)$ plotted in Figure \[fig:tauinst\].
The weak $m_2$ constraint visible in Figure \[fig:mcmcwmap\] suggests that in addition to determining the total optical depth, [*WMAP*]{} data may provide useful limits on high-$z$ and low-$z$ optical depth as defined in the previous section. The 68 and 95% posterior probability contours for the partial optical depths from [*WMAP*]{} are shown in Figure \[fig:tauwmap\] for ${z_{\rm mid}}= 20$ and ${z_{\rm max}}= \{30,40\}$. The $x_e=0$ physicality prior cuts off the contours at $\tau(z_1,z_2)=0$; in both panels, the upper limits set by $x_e=1$ are outside the plotted area, so the contours are not strongly influenced by those priors. The error “ellipses” are narrowest in the direction along which the total optical depth is constrained, as shown by dashed lines of constant total optical depth at $\tau=0.06$ and $\tau=0.12$ \[approximately the upper and lower 1 $\sigma$ limits from the 3-year [*WMAP*]{} analysis of @Speetal06\].
Comparison of the two panels in Figure \[fig:tauwmap\] reveals that the choice of ${z_{\rm max}}$ does not have a large effect on constraints on the optical depth above and below $z=20$. This suggests that $\tau(20,{z_{\rm max}})$ should be interpreted as $\tau(z>20)$. We provide further justification for this interpretation in § \[sec:cvdata\].
The 95% upper limits on the high-redshift optical depth, $\tau(z>20)$, are $0.076$ and $0.078$ for ${z_{\rm max}}=30$ and 40, respectively, after marginalizing over all other parameters. This is not a particularly strong constraint, since this result is only marginally inconsistent with all of the optical depth from reionization coming from $z>20$, but with future data it should be possible to either reduce the upper limit on high-$z$ optical depth or to detect the presence of a substantial ionized fraction at high redshift (see § \[sec:cvdata\]).
Models of reionization can be tested by computing the principal components of proposed ionization histories and comparing with constraints on $\{m_{\mu}\}$ from Monte Carlo chains. The [*WMAP*]{} constraints in Figure \[fig:mcmcwmap\] are non-Gaussian, in part because the physicality priors on $\{m_{\mu}\}$ intersect the posterior probability contours where the likelihood is large. Because the constraints do not have a simple Gaussian form, it appears that the full parameter chains are necessary for accurate model testing, although the viability of various models can be estimated by comparing their eigenmode amplitudes with marginalized constraints such as those in Figure \[fig:mcmcwmap\], keeping in mind that models favored by marginalized constraints could be disfavored in the full $N$-D parameter space.
Cosmic variance-limited data {#sec:cvdata}
----------------------------
To forecast how well principal components of the reionization history could be measured by future CMB polarization experiments, we repeat the analysis of § \[sec:wmap\] using simulated realizations of the full-sky, noiseless $E$-mode angular power spectrum instead of [*WMAP*]{} data. Any real experiment will of course involve sky cuts, noise, foregrounds, and other complications, so the results presented here represent an optimistic limit on what we can learn about the global reionization history from low-$\ell$ $E$-mode polarization. At the end of this section, we estimate how much the constraints might be degraded from this idealized case for an experiment with characteristics similar to those proposed for the *Planck* satellite.
We generate simulated realizations of ${C_{\ell}^{EE}}$ drawn from $\chi^2$ distributions with cosmic variance determined by the theoretical angular power spectra that we compute using CAMB. For the $j$th sample in a chain, the likelihood including only $E$-mode polarization is $$-\ln L_{(j)}=\sum_{\ell}\left(\ell+\frac{1}{2}\right)
\left(\frac{\hat{C}_{\ell}^{EE}}
{C_{\ell(j)}^{EE}}+\ln\frac{C_{\ell(j)}^{EE}}
{\hat{C}_{\ell}^{EE}}-1\right),
\label{eq:like}$$ where $\hat{C}_{\ell}^{EE}$ is the spectrum of the simulated data and $C_{\ell(j)}^{EE}$ is the theoretical spectrum calculated with the parameter values at step $j$ in the chain.
We have run Monte Carlo chains for multiple realizations of ${C_{\ell}^{EE}}$ drawn from spectra computed assuming a variety of “true” reionization histories, ${x_e^{\rm true}}(z)$. We start by taking the true history to be a model with nearly instantaneous reionization and $\tau=0.105$. As a contrasting model for comparison we also use an extended, double reionization model with $\tau=0.090$. Figure \[fig:cl\] shows ${C_{\ell}^{EE}}$ and $x_e(z)$ for each of these models.
Since parameter constraints derived using a single draw of ${C_{\ell}^{EE}}$ from the underlying power spectrum may contain features unique to that realization, we generated Monte Carlo chains for 10 random realizations of the instantaneous reionization power spectrum. Two of these realizations are plotted as points in Figure \[fig:clcv\], along with the theoretical spectrum with cosmic variance bands. The thick curves in Figure \[fig:clcv\] are spectra of the best-fit models from the Monte Carlo chains for each realization. The overall best-fit models are similar for the two realizations and both agree closely with the theoretical ${C_{\ell}^{EE}}$. The dashed curve in the right panel of Figure \[fig:clcv\] shows an “alternative” best-fit model that we discuss later.
The 2D marginalized MCMC constraints on principal components for these two realizations are shown in Figure \[fig:mcmccv\]. The plotted regions are bounded by the physical top-hat priors on $\{m_{\mu}\}$ as in Figure \[fig:mcmcwmap\], but the eigenmodes are different here with ${z_{\rm max}}=20$ instead of 30 or 40 for the fiducial history. Since we have constructed ${x_e^{\rm true}}(z)$ to have no ionization at $z\gtrsim 15$, we know that ${z_{\rm max}}=20$ should be a large enough maximum redshift; this choice of ${z_{\rm max}}$ also reflects the fact that improved empirical determination of ${z_{\rm max}}$ will allow it to be set at lower redshift for future data analysis, assuming that high-$z$ reionization is not detected.
The contours in Figure \[fig:mcmccv\] show that cosmic variance-limited data can constrain models within the physically allowed parameter space much better than current data can. The chains used for the results shown here vary the first 3 eigenmodes; by running chains with $N>3$ we find that cosmic variance-limited data can provide $2~\sigma$ constraints that are tighter than the physicality bounds for the first 4 eigenmodes of a ${z_{\rm max}}=20$ fiducial model and for at least 5 eigenmodes when ${z_{\rm max}}=30$.
Although the only difference in the MCMC setup for the left and right panels of Figure \[fig:mcmccv\] is the realization of ${C_{\ell}^{EE}}$ (Fig. \[fig:clcv\]), there are some significant differences in the resulting eigenmode constraints, particularly in the $m_2-m_3$ plane. This kind of variation is typical among the realizations of simulated data, and is related to whether features in the theoretical ${C_{\ell}^{EE}}$ remain intact with the inclusion of cosmic variance. The relevant features in the spectrum are the small oscillations in the low-power “valley” in ${C_{\ell}^{EE}}$ on scales $10\lesssim \ell \lesssim 30$. In the example shown here, the feature that matters most is the bump in the theoretical spectrum at $\ell \approx 13$. In the left panel of Figure \[fig:clcv\], the ${C_{\ell}^{EE}}$ realization retains this bump, while in the right panel the bump is washed out by cosmic variance. The result is that the left realization is better able to pick out those reionization models in Monte Carlo chains that reproduce the true ${C_{\ell}^{EE}}$, leading to the tight constraints on the left side of Figure \[fig:mcmccv\]. The realization in the right panel of Figure \[fig:clcv\] lacks this important “fingerprint” for identifying models that match the theoretical ${C_{\ell}^{EE}}$, so the data allow a wider variety of models as reflected in the constraints on the right side of Figure \[fig:mcmccv\]. In this case, there are two best-fit models corresponding to the two $68\%$ contours separated in the $m_2$ direction. The spectrum of the overall best-fit model (with smaller $m_2$), which is close to the theoretical spectrum, is plotted as a thick solid curve in the right panel of Figure \[fig:clcv\], while the other best-fit model (at larger $m_2$) is plotted as a dashed curve. These two models have significantly different spectra, but because of the scatter in the random draw of ${C_{\ell}^{EE}}$ they are each able to fit the data better at some multipoles and worse at others in such a way that the high-$m_2$ best-fit model is only a slightly worse fit than the low-$m_2$ best-fit model.
Among the realizations of simulated data that we explored with MCMC methods, constraints like those in the right panel of Figure \[fig:mcmccv\] are a fairly extreme case, occurring about 10-20% of the time. In general the realizations form a sort of continuum between the two presented here, with about half showing some displacement of the $95\%$ contour towards larger $m_2$ but not as much as in the second realization that we show as an example.
The extent to which differences between realizations show up in principal component constraints depends on ${x_e^{\rm true}}(z)$. For example, the ${C_{\ell}^{EE}}$ for the double reionization model (Fig. \[fig:cl\]) have more pronounced oscillations at $\ell\sim 10$-$30$ than the instantaneous reionization spectrum, so they are not as easily erased by cosmic variance or noise and the resulting principal component constraints tend to be more consistent from one realization to the next and more like the left panel of Figure \[fig:mcmccv\].
The differences between constraints from various realizations of simulated data suggest the interesting possibility that our ability to learn about the reionization history from large scale $E$-mode polarization may ultimately depend on the luck of the draw of ${C_{\ell}^{EE}}$ at our particular vantage point. In an unlucky draw, cosmic variance can distort subtle features in the power spectrum in a way that would limit constraints on reionization eigenmodes to be worse than we might expect from the Fisher approximation or a more typical draw. The good news is that even a realization like the one in the right panels of Figures \[fig:clcv\] and \[fig:mcmccv\] would permit constraints that are far better than what is currently possible. It is also important to note that although such constraints are weaker than in the best-case scenario, they are still consistent with the true parameter values and in general would not lead us to rule out the true reionization history based on a principal component analysis of the data.
As with the principal components of $x_e(z)$, MCMC analysis indicates that constraints on the total optical depth to reionization would be greatly improved with cosmic variance-limited data: $\sigma_{\tau}\approx 0.005$, down a factor of six from the 3-year [*WMAP*]{} value, $\sigma_{\tau}\approx 0.03$. The constraints on $\tau$ using the realization of instantaneous reionization ${C_{\ell}^{EE}}$ in the left panel of Figure \[fig:clcv\] are listed in Table \[tab:tau\] for two fiducial models with ${z_{\rm max}}=20$ and 30. Optical depth constraints from the other draw of ${C_{\ell}^{EE}}$ are similar, so estimates of $\tau$ and $\sigma_{\tau}$ appear not to be biased for atypical realizations. It makes sense that differences in realizations do not significantly affect $\tau$ since the optical depth constraint comes mainly from the peak in ${C_{\ell}^{EE}}$ on the largest scales whereas differences in principal component constraints between realizations arise from the low-power part of the spectrum at higher $\ell$.
The model-independent constraints on $\tau$ are neither biased nor significantly weaker relative to the constraint obtained assuming instantaneous reionization (bottom row in Table \[tab:tau\]), as is true for the [*WMAP*]{} data. This should not be surprising since we have assumed that ${x_e^{\rm true}}(z)$ is an instantaneous reionization model. However, if ${x_e^{\rm true}}(z)$ is in fact very different from the instantaneous model, the estimate of $\tau$ obtained under the assumption of instantaneous reionization will be incorrect. For example, if we take ${x_e^{\rm true}}(z)$ to be the double reionization model of Figure \[fig:cl\], the model-independent approach yields an optical depth constraint consistent with the true value of $\tau=0.090$. On the other hand, MCMC analysis restricted to instantaneous reionization histories gives a significantly biased estimate, $\tau=0.135\pm 0.005$. In this case, the 1D curve of instantaneous reionization models in the space of eigenmodes never lies near the principal component values favored by the data, so any analysis that only considers such models would never find a good fit to the data. This illustrates the importance of a model-independent approach: although results from current data may not be significantly affected by the assumption of a specific form of $x_e(z)$, future data will be sensitive enough to the reionization history that the choice of a specific model can greatly impact the results [@Holetal03; @Coletal05].
Though constraints on the contributions to $\tau$ from high and low $z$ are currently limited to weak upper bounds, future polarization data will likely enable more definitive detections of high-$z$ reionization if it is present, or tighter upper limits if absent. The left and right panels of Figure \[fig:taucv1\] show these constraints from Monte Carlo chains using the ${C_{\ell}^{EE}}$ plotted in the left and right panels of Figure \[fig:clcv\], respectively. The true ionization history has zero ionized fraction at $z>15$, and the MCMC analysis places a 95% CL upper limit on $\tau(z>15)$ of $\sim 0.03$. Figure \[fig:taucv2\] shows MCMC constraints on the same partial optical depths using simulated data where the true model is instead taken to be the extended, double reionization history plotted as a thin curve in the inset of Figure \[fig:cl\]. In this case, most of $\tau({z_{\rm min}},{z_{\rm max}})$ comes from $z>15$, and using the principal components of $x_e(z)$ the high-$z$ optical depth can be detected at a high level of confidence and measured to an accuracy of $\sim 0.01$. Note that the constraints in both Figures \[fig:taucv1\] and \[fig:taucv2\] are consistent with the [*WMAP*]{} constraints in Figure \[fig:tauwmap\].
As with [*WMAP*]{} data, most of the constraints from simulated data in Figures \[fig:taucv1\] and \[fig:taucv2\] do not appear to depend strongly on the choice of ${z_{\rm max}}$. For the instantaneous reionization model used for Figure \[fig:taucv1\], this is to be expected since we know that the simulated ${C_{\ell}^{EE}}$ have no contribution from $z \gtrsim 20$. Each set of contours in Figure \[fig:taucv1\] is consistent with the values of $\tau(6,15)$ and $\tau(15,{z_{\rm max}})$ for ${x_e^{\rm true}}(z)$. The constraints in the second panel in Figure \[fig:taucv1\] extend to larger high-$z$ optical depth than those in the first panel because the effect of the physicality priors is not as strong at larger ${z_{\rm max}}$, as noted in § \[sec:wmap\]; the leftmost ${z_{\rm max}}=20$ contours would stretch to larger $\tau(15,20)$ and smaller $\tau(6,15)$ if we did not apply the physicality bounds. The two panels on the right side of Figure \[fig:taucv1\] show weaker constraints on the partial optical depths because the principal component constraints for that realization of ${C_{\ell}^{EE}}$ are weaker.
In the extended, double reionization model used for Figure \[fig:taucv2\], reionization actually starts at $z=23$, so the choice of ${z_{\rm max}}=20$ is not appropriate for this model. The result of such an error is that the constraint on $\tau(15,20)$ is inconsistent with the true value, marked by a cross in the left panel of Figure \[fig:taucv2\]. The ${z_{\rm max}}=30$ constraints in the right panel are consistent with the expected value, although not in perfect agreement due to cosmic variance. It is interesting that although the value chosen for ${z_{\rm max}}$ in the left panel is too low for this model, the constraints on $\tau(z>15)$ for ${z_{\rm max}}=20$ and ${z_{\rm max}}=30$ are still consistent with each other. This supports the interpretation of constraints on $\tau({z_{\rm mid}},{z_{\rm max}})$ as $\tau(z>{z_{\rm mid}})$ in § \[sec:wmap\]
While the cosmic variance-limited simulated data are useful for determining how well the reionization history could possibly be constrained by large-scale $E$-mode polarization measurements, it is also interesting to ask how well we can do with future experiments that fall somewhat short of the idealized case that we have considered so far. In particular, the upcoming *Planck* satellite is expected to improve our knowledge of the large-scale $E$-mode spectrum substantially [@Planck]; what does this imply for constraints on $x_e(z)$? To estimate what might be possible with *Planck* data, we assume that after subtracting foregrounds a single foreground-free frequency channel remains for constraining the low-$\ell$ $E$-mode polarization. We take this to be the 143 GHz channel with a white noise power level of $w_P^{-1/2}=81~\mu$K$'$ and beam size $\theta_{\rm FWHM}=7.1'$, and we assume that the sky coverage is $f_{\rm sky}=0.8$ after cutting out the Galactic plane [@Planck; @Albetal06]. We compute the likelihood of Monte Carlo samples using the routines provided in CosmoMC and analyze parameter chains with the principal component method as described for [*WMAP*]{} and cosmic variance-limited data.
For various choices of ${x_e^{\rm fid}}(z)$ and ${x_e^{\rm true}}(z)$ we find that going from full-sky cosmic variance-limited data to *Planck*-like data increases the uncertainty in $x_e(z)$ principal components, $\sigma_{\mu}$, and in the optical depth, $\sigma_{\tau}$, by roughly a factor of two. Based on these results, it should be possible to constrain about three eigenmodes at $\sim 2~\sigma$ within the space of physical models, and we expect $\tau$ to be determined to an accuracy of about $\pm 0.01$. This is consistent with previous estimates of the *Planck* optical depth uncertainty when considering more general reionization models than instantaneous reionization [e.g., @Holetal03; @HuHol03].
Fisher matrix predictions versus MCMC results {#sec:fisher}
---------------------------------------------
The Fisher matrix analysis of § \[sec:eigenmodes\] predicts that the principal components of $x_e(z)$ should be uncorrelated, with errors given by $\sigma_{\mu}$ from equation (\[eq:fisher2\]) (in the idealized limit of full sky, noiseless observations). These characteristics rely on the assumption that the difference between the true and fiducial reionization histories is small: $\delta {x_e^{\rm true}}(z)={x_e^{\rm true}}(z)-{x_e^{\rm fid}}(z) \ll 1$. Clearly this assumption will not hold for any single fiducial history if one wants to consider a variety of possible true histories, and the consequence is that $\langle m_{\mu}m_{\nu} \rangle$ differs from $\sigma_{\mu}^2 \delta_{\mu\nu}$.
As an example of the difference between the Fisher approximation and the full Monte Carlo analysis, in Figure \[fig:fisher2d\] the marginalized constraints in the $m_1-m_2$ plane from the left panel of Figure \[fig:mcmccv\] are compared with the Fisher matrix error ellipses, centered on the true $(m_1,m_2)$ for the same choices of ${x_e^{\rm true}}(z)$ (instantaneous reionization with $\tau=0.105$) and ${x_e^{\rm fid}}(z)$ (constant $x_e=0.15$ out to ${z_{\rm max}}=20$). The MCMC constraints have correlated values of $m_1$ and $m_2$ with somewhat larger uncertainties than the Fisher matrix $\{\sigma_{\mu}\}$. The eigenfunctions, $S_{\mu}(z)$, are constructed to have orthogonal effects on ${C_{\ell}^{EE}}$ in the vicinity of the fiducial model, but for large $\delta x_e(z)$ the orthogonality breaks down. Fortunately the errors on $\tau$ remain largely unaffected because the $m_{1}-m_{2}$ correlation is oriented such that the increased errors are along the direction of constant total $\tau$ (see Figure \[fig:tauinst\]).
One can compute new eigenfunctions by evaluating ${\partial \ln {C_{\ell}^{EE}}}/{\partial x_e(z)}$ at ${x_e^{\rm true}}(z)$ and see that their dependence on redshift differs from the original $S_{\mu}(z)$. It is possible to decorrelate the principal components by diagonalizing the covariance matrix from the Monte Carlo chains and rotating the eigenmodes into the new basis. However, this procedure is unnecessary for most applications; the most important property of the principal components for constraining $x_e(z)$ is that the first few modes form a complete basis for ${C_{\ell}^{EE}}$ on large scales, accurate within cosmic variance limits. Completeness of the eigenmodes holds true even if they are not exactly orthogonal.
Finally, note that if the true ionization history and the draw of ${C_{\ell}^{EE}}$ permit constraints on principal components similar to those in the left panel of Figure \[fig:mcmccv\], then these constraints may be close enough to Gaussian so that the covariance matrix would be sufficient to describe the information in the ${C_{\ell}^{EE}}$ for finding constraints on reionization models. However, if what we see looks more like the right panel of Figure \[fig:mcmccv\] then the full chains of Monte Carlo samples may be necessary for further applications even if we can measure the $E$-mode polarization perfectly.
Discussion {#sec:discuss}
==========
Observations of the large-scale $E$-mode polarization of the CMB in the near future are expected to yield new information about the spatially-averaged reionization history of the universe. The principal components of the reionization history are a promising tool for extracting as much of that information as possible from the data. We have shown that the principal component method can be usefully applied to real, currently available data, and forecasts from simulated data suggest that there is room to substantially improve constraints on the reionization history using this method as measurements of the large-scale $E$-modes improve.
We find that the key features of the principal component analysis put forward by [@HuHol03] continue to apply when we go from the Fisher matrix approximation to an exploration of the full likelihood surface using Markov Chain Monte Carlo methods. For fairly conservative choices of the maximum redshift of reionization (${z_{\rm max}}\sim 30$-40), only the first five principal components at most are needed for a complete representation of the $E$-mode angular power spectrum to within cosmic variance. To account for arbitrary reionization histories in the analysis of CMB data, only a few additional parameters must be included in chains of Monte Carlo samples if those parameters are taken to be the lowest-variance principal components of $x_e(z)$.
Specific models of reionization can be tested easily by computing their eigenmode amplitudes and comparing with constraints on the eigenmodes from the data. Constraints on derived parameters, such as total optical depth or the optical depth from a certain range in redshift, represent other applications of the MCMC constraints on principal components of $x_e(z)$.
Often, estimates of the optical depth to reionization are computed assuming instantaneous reionization or some other simple form for $x_e(z)$. Here we extend the analysis of the 3-year [*WMAP*]{} data to allow a more general set of models of the global reionization history. We find that expanding the model space does not significantly widen the uncertainty in $\tau$ beyond the instantaneous reionization value of $\sigma_{\tau} = 0.03$. Robust $\tau$ constraints are important for tests of the dark energy based on the growth of structure since they control the uncertainty on the amplitude of the initial spectrum.
Moreover, even with current data the principal component constraints are beginning to show the possibility of determining properties of reionization in addition to $\tau$. By comparing the optical depth from high $z$ with that from low $z$, we obtain an upper limit on the contribution to the optical depth from high redshift: $\tau(z>20) < 0.08$ at 95% confidence, assuming that there is no significant episode of reionization at $z\gg 40$.
Due to the limitations of noise and foreground contamination, only the first two eigenmodes of the reionization history, $m_1$ and $m_2$, can be determined with present polarization data to any reasonable degree of accuracy. Constraints on these two modes come primarily from the main, broad peak in $E$-mode power at low $\ell$ from reionization.
As measurements of the $E$-mode polarization improve, for example from additional [*WMAP*]{} data or through planned future experiments such as *Planck*, better knowledge of the low-power “trough” in ${C_{\ell}^{EE}}$ between the main reionization peak and the first acoustic peak should enable constraints on the third and higher principal components, up to about $m_5$ for near cosmic variance-limited data. Since constraints on these higher modes rely on the ability to identify subtle features in the trough of ${C_{\ell}^{EE}}$, the ultimate accuracy to which the eigenmodes can be determined may depend on whether or not the necessary features are well reproduced in the particular random draw of ${C_{\ell}^{EE}}$ that is available to us. However, even if we are unlucky enough to have a realization in which some of the important features of the spectrum are washed out by randomness, it should still be possible to measure several of the principal components to better accuracy than is currently possible. Knowledge of the $\mu\geq 3$ eigenmodes of $x_e(z)$, along with improved constraints on $m_1$ and $m_2$, will allow more stringent tests of reionization models and a better understanding of the global reionization history.
: We thank Cora Dvorkin, Gilbert Holder, Dragan Huterer, Hiranya Peiris, and Jochen Weller for useful discussions. MJM was supported by a National Science Foundation Graduate Research Fellowship. WH was supported by the KICP through the grant NSF PHY-0114422, the DOE through contract DE-FG02-90ER-40560 and the David and Lucile Packard Foundation.
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[^1]: http://cosmologist.info/cosmomc/
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[*A. Introduction*]{}. Monopole and domain wall problems are some of the central issues in the modern astroparticle physics. The problem of monopoles is especially serious since it is generic to the idea of grand unification [@p79]. The popular solution based on the idea of inflation cannot be implemented in the minimal Grand Unified Theories (GUTs), and even if it does work it would imply a sad prediction of essentially no monopoles in the Universe, and thus eliminate a prospect of observing this exciting aspect of charge quantization. Of course, it is hard to imagine a Universe without ever having passed through an era of inflation; we simply take here the point of view that this may have happened before the time of grand unification in the thermal history of the Universe. Similarly, the problem of domain walls [@zko74] in theories with a spontaneous breaking of discrete symmetries requires inflation to take place after the phase transitions that cause the production of these defects, which is difficult to achieve in general. Recently, a possible solution of the monopole problem was suggested [@gv97], based on the possibility that unstable domain walls sweep away the monopoles.
There is another possible way out of these problems and it is based on an unusual picture of non restoration of symmetries at high temperatures. It has been known for a long time that in theories with more than one Higgs multiplet, which seems to be a necessary feature of all theories beyond the Standard Model (SM), broken symmetries may remain broken at high temperature $T$ in some regions of the parameter space, and even the unbroken ones may get broken as the system in question is heated up [@w74; @ms79].
The idea of symmetry nonrestoration provides a simple way out of the domain wall problem [@ds95; @dms96], but unfortunately in the case of the monopole problem the situation is far from clear [@dms95], since next to the leading order effects tend to invalidate this picture for local symmetries [@bl95b]. While in the case of discrete symmetries the original lattice calculations spoke against nonrestoration at high $T$ [@bitu97], the latest results give full support of this idea [@jl98], as do the other nonperturbative results [@roos96]. However, it can be shown that this scenario does not work in supersymmetry. More precisely, there is a rigorous proof at the renormalizable level [@h82; @m84], and the simple counter examples at the non-renormalizable level [@dt96] have been shown not to work [@bms96].
A manifestation of nonrestoration is an old idea [@lp80] of $U(1)_{em}$ breaking at temperatures above $M_W$. Unfortunately, this suffers from the same next-to-leading order effects mentioned above [@dms96]. There is a simple variation of this scenario where $U(1)_{em}$ is broken only in a very narrow range of temperatures around the electroweak scale [@ds92], [@fkwy92]. In this case the monopoles get produced with the hope of being annihilated fast enough through the strings attached to monopole-antimonopole pairs. However, there is a serious question whether the annihilation does the job [@gkt92], [@hkr92].
The situation becomes much more promising if one accepts the possibility of having a large background charge in the Universe, large in a sense of being comparable to the entropy [@l76]. The presence of some sizeable charge asymmetry may postpone symmetry restoration in nonsupersymmetric theories [@hw82] or, even more remarkably, it can lead to symmetry breaking of internal symmetries at high temperature [@bbd91]. Furthermore, the phenomenon of symmetry nonrestoration at high $T$ in presence of large charge asymmetries has been recently shown to work in supersymmetry too [@rs97]. The principal candidate for a large charge is the lepton number which today could reside in the form of neutrinos. This has inspired Linde in his original work to point out that large enough lepton number of the Universe would imply the nonrestoration of symmetry even in the SM [@l79]. While one could naively think that the large lepton number would be washed out by the sphaleron effects at the temperature above the weak scale, it turns out that the nonrestoration of symmetry prevents this from happening [@ls94], and remarkably enough up to this day this still remains a consistent possibility. Indeed, the successful predictions of primordial nucleosynthesis are not jeopardized as long as the lepton number is smaller than $\sim 7\: T^3$ at temperatures of the order of 1 MeV [@ks92]. It is therefore natural to ask ourselves whether a large lepton asymmetry in the Universe may play any significant role in solving the monopole problem. The main point we wish to make in this Letter is that the answer may be positive if these two basic requirements are satisfied: the large lepton asymmetry leads to the symmetry nonrestoration of the SM gauge symmetry and some charge field condensation takes place. While it is not clear whether this happens in SM, it is certainly true for its minimal extensions (such as an additional charged scalar). Thus, if the lepton number of the Universe were to turn out large, there would be no monopole problem whatsoever.
Now, if Nature has chosen the option that the lepton number is large enough so that SM symmetry is not restored at high $T$, but without any charge field condensation, even in this case the cosmological consequence would be remarkable, for this would suffice to nonrestore the symmetry in the minimal model of spontaneous CP violation with two Higgs doublets [@l73]. Namely, without the external charge, in this particular model CP is necessarily restored at high temperature [@dms96] leading to the domain wall problem.
As we mentioned before, it was shown recently that the phenomenon of nonrestoration at high $T$ in the presence of a large charge works in supersymmetry too. We have exemplified our findings on simple U(1) models [@rs97]. It can be shown that this is true in general, and all that we say above works also in Minimal Supersymmetric Standard Model (MSSM). In this Letter we wish to avoid any model building, but rather concentrate on SM showing that its cosmology may be something entirely different from what one normally imagines.
Let us now discuss in some detail what happens at high temperature if the lepton number is large. Notice first that, since we can assume that the lepton number $L$ is conserved (the sphaleron effects are suppressed [@ls94]), then the ratio of the lepton density $n_L$ to entropy density $s$ is constant, too. Now, we are interested in the temperatures above the weak scale when the number of light degrees of freedom grows by another order of magnitude with respect to $T\simeq 1$ MeV. Thus, the above cited limit $n_L/T^3< 7$ at the time of nucleosynthesis becomes for us an order of magnitude bigger: $n_L/T^3<70$.
In order to study symmetry breaking, we need to compute the effective potential at high $T$ and high chemical potential. We employ the approximation $\mu_L < T$ (where $\mu_L$ is the chemical potential associated with the lepton number), since in this case one can obtain the solutions in a closed form. With increased $\mu_L$ the physical effect of symmetry breaking gets only stronger [@l79].
The baryon number of the Universe is negligible, $n_B/s \simeq
10^{-11}$, thus we work in the approximation $B=0$. Since the gauge potentials act essentially as the chemical potentials at high $T$, we include them in our $V_{eff}(T,\mu)$. A word of caution is in order. Although $B=0$, since the quarks carry non trivial baryon number, one must include the associated chemical potential $\mu_B$ and we will see below that it does not vanish. We will see that quarks carry a nonvanishing electric charge at high $T$, similarly to the $W$ bosons and the charged Higgs scalar.
Using the techniques of [@hw82; @bbd91] the effective potential at high $T$ ($T >\mu \gg M_W$) and large $n_L$ for the Higgs doublet $H$ in the direction of its neutral vacuum expectation value $H=(0, v/\sqrt{2})^T$ reads
$$\begin{aligned}
\label{veffall}
V_{eff}&=&\lambda{v^4\over 4}+
{g^2\over 2}\left[(A_0^aA_0^b)(A_i^aA_i^b)-(A_0^aA_0^a)
(A_i^b A_i^b)\right]\nonumber\\
&+&{g^2\over 4}\left[(A_i^aA_i^a)(A_j^bA_j^b)-(A_i^aA_i^b)
(A_j^aA_j^b)\right]\nonumber\\
&-&{v^2\over 8}\left[g^2(A_0^aA_0^a)+g'^2
(B_0B_0)\right]+{gg'\over 2}B_0A_0^3{v^2\over 2}\nonumber\\
&+&{g^2\over 4}\left[A_i^1A_i^1+A_i^2A_i^2\right]{v^2\over 2}
+\mu_Ln_L\nonumber\\
&-&T^2\left({\mu_L^2\over 4}+{\mu_B^2\over 9}\right)+
{T^2\over 3}(g'B_0)\left(\mu_L-{\mu_B\over 3}\right)\nonumber\\
&-&{13\over 36}(g'B_0)^2T^2
-{7\over 12}g^2A_0^aA_0^aT^2+\lambda'T^2{v^2\over 2}.\end{aligned}$$
where $A_\mu^a$ and $B_\mu$ are the $SU(2)_L$ and $U(1)_Y$ gauge potentials. In the above we have used $g' B_i = g A_i^3$ which follows from the equation of motion for $B_i$ and
$$\lambda'={1\over 12} \left[6\lambda+
y_e^2+3y_u^2+3y_d^2+
{3\over 4}(g'^2+3g^2)\right],$$
where $y_f$ are the fermionic Yukawa couplings. For simplicity, we take only the third generation of fermions since its couplings are dominant. The inclusion of the first two generations is straightforward and does not change our conclusions.
Notice the point we made before. Although we take $B=0$ , the associated chemical potential plays an important role in the above expression. The equations of motion for the gauge fields $A_\mu^a$ show that the solution discussed in [@l79] – all gauge potentials zero except for $A_0^3$ and $A_1^1$ – is consistent with the above constraints.
Using the constraints $\partial V_{eff}/\partial x=0$ for $x=\mu_L$, $\mu_B$, $g'B_0$ and $gA_0^3$ we can rewrite the effective potential as a function of $v$ and $C=\langle A_1^1\rangle$ only:
$$\begin{aligned}
V_{eff}&=&{\lambda\over 4}v^4 +{\lambda'\over 2} T^2 v^2 +
{g^2\over 8}v^2 C^2+{n_L^2\over T^2}\nonumber\\
&+&{4 n_L^2(3 v^2 + 12 C^2 + 14 T^2)
\over 54 v^2 C^2 + (87 v^2 + 96 C^2) T^2 + 112 T^4}\;.\end{aligned}$$
The effective potential is manifestly bounded from below and it is a simple exercise to minimize it. We work with a small $(H^\dagger H)^2$ coupling – only for the sake of presenting simple analytic expressions. We find the results presented below. In discussing them, it will turn out useful to have the individual distribution of the various charges. Namely, here lies an important point that was overlooked before [@l79] and that plays a significant role for our considerations about the monopole problem, [*i.e.*]{} the fact that quarks, the charged Higgs and $W$ carry electromagnetic charge in spite of having lepton number zero. Since the $\mu$-dependent part of the effective potential can be written [@hw82; @bbd91] $$\label{veffmu}
V_{eff}^{(\mu)}=-{T^2\over 12}\sum_{f}\mu_i^2-{T^2\over 6}
\sum_{b}\mu_i^2-\sum_{b}\mu_i^2|\phi_i|^2+\mu_Ln_L,$$ one can find for the distribution of fermionic and bosonic charges
$$\begin{aligned}
\label{qf}
({\cal Q}_F^a)_i&=&q_i^a\mu_i\left({T^2\over 6}\right)\;,\\
\label{qb}
({\cal Q}_B^a)_i&=&q_i^a\mu_i\left({T^2\over 3}+2|\phi_i|^2\right),\end{aligned}$$
where $q_i^a$ denote the transformation property $
\varphi_i \rightarrow e^{i q_i^a T^a } \varphi_i$, $\varphi_i$ stands for any field, $a$ goes over all the relevant charges ($L,B, Y/2, T_{3W}$) and $\mu_i=\sum_aq_i^a\mu_a$.
Let us first shortly discuss the case of small lepton asymmetry or, more precisely, $n_L < (n_L)_1 \equiv (4/3) \sqrt{\lambda'} T^3$. In such a case only the trivial solution is possible: $v = C = 0$. This is the usual scenario of small charge densities.
It is an easy exercise to compute the distribution of charges. One finds $
L (\nu_L) = L (e_L) = {3 \over 8} L$ and $ L (e_R) = {2 \over 8} L$ for the $L$ number distribution (notice that $L({\nu_L}) = L({e_L})$ since $SU(2)$ is not broken). For the electromagnetic charge (we list only the nonvanishing ones)
$$\begin{aligned}
Q(e_L) = -{3 \over 8} L & \;,\; Q(e_R) = -{2 \over 8} L & \;,\nonumber \\
Q(u_R) = {2 \over 8} L & \;,\; Q(d_R) = {1 \over 8} L & \;,\;
Q(h^+) = {2 \over 8} L \;,\end{aligned}$$
so that $Q_{tot} =0$ as it should be, but the charge is distributed among both fermions and charged Higgs bosons (we find later $W^{\pm}$ participating too).
Let us now focus on the following intermediate range $(n_L)_1 < n_L < (n_L)_2 \equiv
(n_L)_1( 1 + {203 \over 192}g^2 /\lambda')$. It is easy to show that now $v \neq 0$, but $C$ still vanishes
$$\begin{aligned}
\label{vb}
V_{eff}(v\neq 0)&=&V_{eff}(v=0)-{21\over 58T^2}
[n_L - (n_L)_1]^2\;,\nonumber\\
v^2&=&{112\over 87}{n_L-(n_L)_1\over
(n_L)_1}T^2,\;\;\; C=0.
\label{cb}\end{aligned}$$
Clearly, $SU(2)_L\times U(1)_Y$ is spontaneously broken down to $U(1)_{em}$, but there is [*no*]{} condensation of $W$ bosons. This means that for such values of the lepton number it is energetically preferable for the system to cancel the electric charge by means of the asymmetries in the quarks and charged Higgs boson, but no spontaneous breaking of electromagnetism takes place.
Finally, let us consider the case of large lepton asymmetry, $n_L >
(n_L)_2$. Now, on top of the Higgs mechanism, we have also the $W$ condensation [@l79]: $C\neq 0$. Notice that $(n_L)_2$ depends very mildly on the Higgs mass in the physically interesting range between $80$ GeV and $500$ GeV: $(n_L)_2\approx
(2.0-2.5)\: T^3$. This is clearly much below the upper limit $70\: T^3$. Strictly speaking, for $n_L > (n_L)_1$ we have $\mu_L > T$ so that our analytic formulae are not exact. Thus, we have also performed numerical computations for the case of large chemical potentials and finite $\lambda$, which prevents exact analytic results. This amounts to including the terms in the effective potential of the order of $\mu^4$. Our findings from this standard procedure are shown in the table below, where we give the corrections to the critical densities calculated analytically.
$m_H (\hbox{\rm GeV})$ $(n_L)_1^{num}/(n_L)_1$ $(n_L)_2^{num}/(n_L)_2$ $v_2^{num}/v_2$
------------------------ ------------------------- ------------------------- -----------------
100 1.05 1.28 1.01
200 1.07 1.34 0.92
400 1.13 1.52 0.96
600 1.21 1.66 1.06
Table 1: The ratios between the exact numerical solutions (numerators) and the approximate analytic solutions (denominators) described in the text as functions of the Higgs mass. $v_2^{(num)}$ is the Higgs vev for $n_L=(n_L)_2^{(num)}$. 0.5cm
Clearly, the numerical study confirms our analytical findings of symmetry breaking for large densities. Although the precise value of the second critical density (the first critical density is almost unchanged) is increased about $30\%$ for a reasonable values of the Higgs mass, this does not affect the possibility of symmetry restoration. Namely, the critical density remains still an order of magnitude below the allowed value of $70T^3$.
We have seen that a large enough lepton density implies symmetry breaking at high temperature, which opens the door for the solution of the monopole problem. The simplest possibility is to follow the scenario [@lp80] for the high T breaking of $U(1)_{em}$. The essential point here is that if the $U(1)_{em}$ symmetry is broken due to a large external charge, it would be broken for the whole parameter space of the theory and for all temperatures above $M_W$ all the way to the GUT scale. Thus, monopoles may never be created and there would obviously be no problem at all. Even if they did get produced they would surely have time to annihilate. In this sense it is only our scenario that guarantees the solution to the monopole problem. Of course one must make sure that $U(1)_{em}$ is really broken. If we restrict ourselves to the SM and work in the regime of $W$-condensation, it is not clear to us what the precise situation is. First of all, the fact that the $W$ has condensed implies the breakdown of the rotational invariance and the description of the formation, if any, of the monopoles at the GUT scale might be completely different from the usual one. Secondly, if monopoles do get formed, they might not annihilate rapidly enough or might not annihilate at all due to the anti-screening effects of the $W$-background. These issues are extremely important and certainly deserve a separate investigation. We would like to point out, however, that the situation is more transparent if we consider a simple extension of the SM where an electrically charged field $S$ is present (a similar extension would be to add another doublet). In a grand unified theory this singlet would be embedded in a larger representation, such as an $SU(5)$ (a doublet would belong to another [5]{}). The idea of one singlet in addition to the SM Higgs was already pursued in [@ds92], [@fkwy92]. However, as we axplained in the introduction, without the external charge this mechanism may not work [@gkt92], [@hkr92].
We have explicitly checked that, for large enough lepton number, the SM gauge group is broken at high $T$. Moreover, since the field $S$ gets a VEV, $U(1)_{em}$ is spontaneously broken. More important, similarly to what we have described before for the SM, there exists a range of values of the lepton number for which the $W$-condensation does [*not*]{} take place. Under these conditions, the monopole problem is solved. Namely, at $T\simeq M_X$ when the GUT symmetry (say $SU(5)$) breaks down, $U(1)_{em}$ is broken and there will be no creation of monopoles. It is intriguing that a realistic realization of this idea may take place within the Minimal Supersymmetric Standard Model (MSSM) where charged Higgs fields are present. The only price to pay is to accept the idea that the lepton number may be large enough. Once this step is made, the monopole problem is no longer with us. This is a remarkable result.
What about the domain wall problem? Clearly, the presence of large lepton number asymmetry through the nonrestoration at high temperature solves the domain wall problem in an analogous manner. For example, this would solve the well-known domain wall problems associated with the spontaneous violation of CP [@l73] or the $Z_2$ natural flavour conservation symmetry [@gw77].
We have argued that a large lepton asymmetry in the Universe may mean an automatic solution of the monopole and domain wall problems through symmetry nonrestoration at high T. As far as the monopole problem is concerned, this idea works for simple extensions of the SM and in particular in the MSSM.
For all we know the lepton number of the Universe may be comparable, if not bigger than, the entropy of the Universe. The fact that the large lepton number can be consistent with the small baryon number in the context of grand unification has been pointed out a long time ago [@hk81] and recently a model for producing large $L$ and small $B$ has been presented [@ccg97]. We stress, however, that our findings should remain valid if, instead of the lepton number we consider any other conserved charge in the system under consideration.
0.3cm
We would like to thank G. Dvali, M. Shaposhnikov and G.G. Ross for interesting discussions. This work was partially supported by the British Royal Society and by the Ministry of Science and Technology of Slovenia (B.B.) and by EEC under the TMR contract ERBFMRX-CT960090 (G.S.). This work was completed during the Extended Workshop on the Highlights in Astroparticle Physics, held at ICTP from October 15 to December 15, 1997. B.B. and A.R. thank ICTP for hospitality during the course of this work.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'The exchange interaction between magnetic ions and charge carriers in semiconductors is considered as prime tool for spin control. Here, we solve a long-standing problem by uniquely determining the magnitude of the long-range $p-d$ exchange interaction in a ferromagnet-semiconductor (FM-SC) hybrid structure where a 10 nm thick CdTe quantum well is separated from the FM Co layer by a CdMgTe barrier with a thickness on the order of 10 nm. The exchange interaction is manifested by the spin splitting of acceptor bound holes in the effective magnetic field induced by the FM. The exchange splitting is directly evaluated using spin-flip Raman scattering by analyzing the dependence of the Stokes shift $\Delta_S$ on the external magnetic field $B$. We show that in strong magnetic field $\Delta_S$ is a linear function of $B$ with an offset of $\Delta_{pd} = 50-100~\mu$eV at zero field from the FM induced effective exchange field. On the other hand, the $s-d$ exchange interaction between conduction band electrons and FM, as well as the $p-d$ contribution for free valence band holes, are negligible. The results are well described by the model of indirect exchange interaction between acceptor bound holes in the CdTe quantum well and the FM layer mediated by elliptically polarized phonons in the hybrid structure.'
author:
- 'I. A. Akimov'
- 'M. Salewski'
- 'I. V. Kalitukha'
- 'S. V. Poltavtsev'
- 'J. Debus'
- 'D. Kudlacik'
- 'V. F. Sapega'
- 'N. E. Kopteva'
- 'E. Kirstein'
- 'E. A. Zhukov'
- 'D. R. Yakovlev'
- 'G. Karczewski'
- 'M. Wiater'
- 'T. Wojtowicz'
- 'V. L. Korenev'
- 'Yu. G. Kusrayev'
- 'M. Bayer'
title: 'Long-range p-d exchange interaction in a ferromagnet-semiconductor Co/CdMgTe/CdTe quantum well hybrid structure'
---
Introduction
============
The integration of magnetism into semiconductor electronics would initiate a new generation of computers based on advanced functional elements where the magnetic memory and electronic data processor are located on a single chip [@Dietl-2010; @Zutic-2004; @Korenev-UFN-2005; @Johnson-Spinbook]. One approach in this direction is based on hybrid systems where a thin ferromagnetic film is placed on top of a semiconductor. In such a system one expects to detect emergent functional properties which appear and benefit from bringing the primary constituents together, i.e. the magnetism as in ferromagnets (FM) with the optical and electrical tunability as in semiconductors (SC) [@Dzhioev-FTT-1995; @Kawakami-2001; @Petrou-2003; @Crooker-2005; @Crowell-2007; @Jonker-2007; @Ciorga-2009; @Song-2011]. For that purpose it is mandatory to establish a strong exchange interaction between the charge carriers in the SC and the magnetic ions in the FM. Control of the concentration of the charge carriers and the penetration of their wavefunction into the FM layer should consequently change the magnitude of the exchange coupling between FM and SC [@Korenev-2003]. As a result of the coupling, the following interdependencies are established: spin polarization of charge carriers in the SC by the magnetized FM layer and inverse action of the spin polarized carriers to control the FM magnetization. Previously, it was demonstrated that the stray fields of a FM layer influence the spin polarization of conduction band electrons in bulk GaAs [@Dzhioev-FTT-1995; @Jansen-2011] and diluted magnetic semiconductors [@Crowell-1997; @Henne-2007]. In turn, illumination of a GaAs SC changed the coercive force of a nickel-based interfacial FM layer (photocoercivity), which was attributed to optical control of the exchange coupling at the interface between FM and SC [@Dzhioev-FTT-1995].
A novel type of hybrid structure with a thickness of a few tens of nanometers only is obtained by combining a FM layer and a SC quantum well (QW) that are located in close proximity of each other, separated by a SC barrier with a few nanometer thickness [@Crowell-1997; @Henne-2007; @Myers-2004; @Aronzon-2009; @Zaitsev-2010; @NC-Korenev-2012; @NP-Korenev-2016]. Such structures with a well-defined profile along the growth axis can be fabricated with monolayer precision. The stray fields from the FM layer are weak so that they contribute significantly to the carrier spin polarization only in combination with a magnetic SC QW [@Crowell-1997; @Henne-2007]. For non-magnetic SCs, the contributing mechanisms are a direct exchange interaction generating an equilibrium spin polarization [@Korenev-2003; @Myers-2004; @Zaitsev-2010; @Aronzon-2009] and a spin dependent tunneling into the FM layer [@NC-Korenev-2012; @Rozhansky-2015]. In Ref. it was demonstrated that in hybrid structures based on combining a GaMnAs FM with a InGaAs QW the conduction band electrons in the QW are spin polarized due to spin-dependent capture into the FM layer. Another mechanism leading to an equilibrium spin polarization of a two-dimensional hole gas in an InGaAs QW due to the $p-d$ exchange interaction was reported in Ref. . Also, the exchange fields in graphene layers coupled to yttrium iron garnet were used to achieve a strong modulation of spin currents [@Kawakami-2017].
{width="17cm"}
Recently, a new type of proximity effect was observed in a hybrid structure composed of a few nanometer thick Co layer which is deposited on top of a CdTe/CdMgTe semiconductor QW structure. The proximity effect was manifested in a FM induced spin polarization of holes bound to shallow acceptors in the QW [@NP-Korenev-2016]. The polarization of the holes takes place due to an [*effective $p-d$ exchange interaction*]{} between the FM ($d$-system) and the QW holes ($p$-system). In this case, the FM produces an effective magnetic field which acts on the acceptor-hole spins and consequently leads to an equilibrium spin polarization of the holes. The main feature of this indirect exchange interaction is its long-range character, i.e. the proximity effect is almost constant with increasing thickness of the CdMgTe spacer between the FM and the QW layers up to 30 nm. This length scale is significantly larger than the 1-2 nm distance required for a significant overlap of wavefunctions in the direct exchange interaction between the QW holes and the magnetic ions in the FM. In Ref. it was conjectured that the long-range indirect exchange originates from [*exchange*]{} of elliptically polarized acoustic phonons which exist in the FM layer close to the magnon-phonon resonance [@Kittel-1958] and can penetrate into the SC layer. This mechanism was used later to explain the influence of elliptically polarized phonons on the magnetic properties of materials [@Cavaleri-2017]. However, the spin polarization of the acceptors and the resulting circular polarization of the photoluminescence (PL) depend not only on the exchange splitting between the spin levels of the holes $\Delta_{pd}$ but also on other factors such as the temperature, the ratio of lifetime and spin relaxation time of the holes etc. Therefore, the polarization of the PL evaluated in Ref. can be considered only as rough estimate for the splitting $\Delta_{pd} \approx 50~\mu$eV, and it is necessary to perform a direct measurement of the spin splitting of the acceptor holes using complementary techniques.
In this paper, we report on the investigation of the FM induced spin splitting of the acceptor bound holes in a CdTe QW located in close proximity of a Co layer. While previous optical and electrical measurements were indirect requiring additional model assumptions for analysis, here we perform a direct measurement using spin-flip Raman scattering giving the dependence of the Stokes shift $\Delta_S$ on external magnetic field $B$. In strong magnetic fields, $\Delta_S(B)$ scales linearly with $B$. Extrapolation of these data to zero magnetic field reveals a finite offset of the Stokes shift due to the FM induced effective exchange field with a magnitude of $\Delta_{pd} = 50-100~\mu$eV. This offset varies only weakly on the CdMgTe spacer thickness also in ranges where wavefunction overlap is negligible so that it has to be attributed to a long-range $p-d$ interaction. In addition, we show that the $s-d$ exchange interaction between conduction band electrons and the FM as well as the corresponding $p-d$ contribution for free valence band holes are negligible. These results are surprising from the viewpoint of standard theory of exchange interaction which is proportional to the overlap of the wavefunctions of the interacting particles. However, they are in line with the conjecture of an indirect exchange mediated by elliptically polarized phonons in FM-SC hybrid structures [@NP-Korenev-2016] and therefore corroborate this model.
The paper is organized as follows. First, in Sec. II we describe the proximity effect based on PL data recorded in a wide range of magnetic fields up to 3 T. Next, we present the results on spin-flip Raman scattering in Sec. III. In Sec. IV time-resolved data on pump-probe Kerr rotation are given where we evaluate the influence of the FM on the Larmor precession of the optically oriented holes and electrons. Finally, the results are discussed in Sec. V.
Ferromagnetic proximity effect
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The studied CdTe/Cd$_{0.8}$Mg$_{0.2}$Te QW structures were grown by molecular-beam epitaxy (MBE) on top of (100)-oriented GaAs substrates. The subsequent deposition of Co at room temperature was done without any intermediate contact to ambient atmosphere. Details on growth and characterization are given in Ref. . A schematic presentation of the structure and of the geometry for PL measurements is shown in Fig. \[fig:PL-plus\](a). The used gradient growth technique allowed variation of the thickness of both the Co layer and the Cd$_{0.8}$Mg$_{0.2}$Te spacer up to 10 and 30 nm, respectively. The 10 nm thick CdTe QW is sandwiched between Cd$_{0.8}$Mg$_{0.2}$Te barriers. The thickness of the Cd$_{0.8}$Mg$_{0.2}$Te buffer is about 3 $\mu$m. Most of studies are performed on samples with a Co layer thickness of about 4 nm and a spacer thickness of $d_S=5-10$ nm. The samples are mounted in a split-coil helium bath cryostat with a variable temperature insert. The magnetic field is applied in the Faraday geometry parallel to the structure growth axis ($\mathbf{B}\|z$). In the PL measurements, excitation of electron-hole pairs in the QW layer is accomplished by picosecond optical pulses emitted by a tunable Ti:Sapphire laser at a repetition frequency of 75.75 MHz. The photon energy $\hbar\omega_{\rm exc}$ is kept below the band gap energy of the Cd$_{0.8}$Mg$_{0.2}$Te barriers ($\sim 1.9$ eV) in order to generate carriers in the QW layer only. The emission is analyzed and detected by a spectrometer equipped with a charge-coupled-device camera and a streak camera for time-integrated and time-resolved measurements, respectively.
Figure \[fig:PL-plus\] summarizes the time-integrated data on the ferromagnetic proximity effect. These PL data are measured in the Faraday geometry on the sample with $d_S=10$ nm at a bath temperature of $T_{\rm bath}=2$ K. The total PL intensity $I_0=I^\pi_++I^\pi_-$ and degree of circular polarization $\rho^\pi_c$ spectra are shown in Fig. \[fig:PL-plus\](b). The degree of circular polarization is defined as $\rho_c^\pi = (I^\pi_+-I^\pi_-)/(I^\pi_++I^\pi_-)$, where $I^\pi_+$ and $I^\pi_-$ are the $\sigma^+$- and $\sigma^-$-polarized emission intensities of the PL under linear polarized excitation, as indicated with the $\pi$ in the superscript. Already in weak magnetic fields $B_F=\pm40$ mT a circular polarization of several percent appears in the spectral range of the low energy PL band from $1.57-1.62$ eV, which corresponds to recombination of conduction band electrons with holes bound to acceptors (the $e-A^0$ band). This effect was studied in detail in our previous work where we demonstrated that:[@NP-Korenev-2016] (i) the circular polarization appears due to a FM induced spin polarization of the acceptor bound holes; (ii) the effect is induced by an [*interfacial*]{} FM which is formed at the Co/CdMgTe interface with a magnetization $\mathbf{M} || z$ and an out-of-plane (perpendicular) anisotropy (see Fig. \[fig:PL-plus\]). In weak magnetic fields the magnetization of the Co layer $\mathbf{M_{\rm Co}}$ is located in the plane of the structure ($\mathbf{M_{\rm Co}} \perp z$) and does not contribute to the circular polarization of the PL.
![(Color online) (a) Transients of circular polarization $\bar{\rho}_c^\pi(t)$ for different magnetic fields $|B_F|=$ 0.5, 1.7, 2.5 T. Solid lines are fits to the data with Eq. . (b) Magnetic field dependence of $\rho_{\rm MCD}$ and $A$. Dashed line is a fit to the data for $A$ using Eq. with $\Delta_{pd}=50\pm10~\mu$eV, $|g_A|=0.4\pm0.1$ and $T=5$ K.[]{data-label="Fig:PL-TRPL"}](fig-2.eps){width="0.9\columnwidth"}
Here, we extend the measurements of $\rho_c^\pi(B_F)$ to a larger magnetic field range up to 3 T, where an out-of-plane magnetization of the Co FM layer is present. Figure \[fig:PL-plus\](c) shows the FM induced dependence $\rho_c^\pi(B_F)$ averaged across the spectral range from $1.57 - 1.62$ eV as function of the magnetic field $B_F$ in the Faraday configuration. In strong fields $B_F>0.25$ T the polarization increases with $B_F$ and changes its slope to a weaker dependence around 2 T, which is close to the saturation field of Co $4\pi M_{\rm Co}=1.7$ T (see Ref. ). At first glance this behaviour could be attributed to a spin polarization of the holes due to exchange interaction with the Co where the exchange constant $J_{pd}$ has the opposite sign as that of the interfacial FM. However, care should be exercised here because there is a significant contribution of magnetic circular dichroism (MCD) to the data as follows from time-resolved photoluminescence (TRPL) measurements.
Figure \[Fig:PL-TRPL\](a) shows transients of the antisymmetric term of the polarization degree $\bar{\rho}_c^\pi(|B_F|) = [\rho_c^\pi(+B_F) - \rho_c^\pi(-B_F)]/2$. The data can be well described with the following expression $$\label{eq:MCD}
\bar{\rho}_c^\pi(t) = \rho_{\rm MCD} + A(1-e^{-t/\tau_S}),$$ where the instantaneous polarization degree $\rho_{\rm MCD}$ results from the difference in absorption of $\sigma^+$ and $\sigma^-$ polarized light in the Co layer, the amplitude $A$ corresponds to the equilibrium polarization of the acceptor holes induced by the external magnetic field $B_F$ and the FM induced effective exchange field. $\tau_S$ is the spin relaxation time of polarized carriers, during which equilibrium populations of the spin levels are reached.
The magnetic field dependencies of $\rho_{\rm MCD}$ and $A$ evaluated from fits to the $\bar{\rho}_c^\pi(t)$ transients are shown in Fig. \[Fig:PL-TRPL\](b). Obviously the MCD saturates at $B_F \approx 1.7$ T, while the amplitude $A$ continuously grows with $B_F$. The increase of $A$ with magnetic field is related to an additional equilibrium polarization of the holes due to thermalization between the spin levels. For small splittings ($A \ll 1$) the field dispersion of $A$ can be approximated by $$\label{eq:PL-Delta}
A = \frac{\mu_B |g_A| B - \Delta_{pd}}{2 k_B T}$$ where $\mu_B>0$ is the Bohr magneton, $k_B$ is the Boltzmann constant, and $g_A$ is the Landé factor of the acceptor which determines the splitting of the heavy hole states with angular momentum projections $J_z=\pm3/2$ onto the quantization axis of the QW. Since the amplitude $A$ does not saturate in magnetic fields $B_F>1.7$ T (see Fig. \[Fig:PL-TRPL\](b)), we conclude that the contribution of the Co film to the proximity effect is negligible. Using Eq. we obtain $\Delta_{pd}=50\pm10~\mu$eV and $|g_A|=0.4\pm0.1$ (see the dashed line in Fig. \[Fig:PL-TRPL\](b)). This evaluation depends, however, sensitively on the actual temperature of the crystal lattice $T$ in the illumination area which we assumed to be $T=5$ K, i.e. about 3 K higher than the bath temperature of $T_{\rm bath}=2$ K. Laser heating of the crystal lattice due to optical excitation is in agreement with our previous studies on optical orientation of Mn ions in GaAs [@Akimov-2011].
Thus, TRPL polarization measurements as applied up to now can be used to estimate the exchange energy splitting $\Delta_{pd}$ but this requires an accurate knowledge of $T$. Such a precise assessment is, however, hardly possible, but every determination of the crystal temperature is subject of considerable inaccuracies. In the following, we therefore present other methods that can be used for a direct measurement of the exchange energy which does not require any estimates.
Evaluation of ${\bf p-d}$ exchange interaction via spin-flip Raman scattering
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Resonant spin-flip Raman scattering (SFRS) allows measuring the magnetic field induced splitting of the spin levels of charge carriers in semiconductor QW structures [@SapegaPRB94; @Akimov-2011; @Sirenko-1997; @Debus2013]; moreover, it can be also exploited to evaluate exchange energies by which different spin configurations are separated [@Debus2014]. As we will demonstrate in the following, in contrast to polarization-resolved PL measurements, SFRS grants access to the effective $p-d$ exchange constant in the hybrid structures studied here. The physics of SFRS for a hole bound to an acceptor is shown in Fig. \[Fig:SFRS-schema\].
![(Color online) Schematic presentation of (a) SFRS for acceptor bound heavy hole; (b) geometry of SFRS experiment with $\Theta=20^\circ$. Black bold arrows $\Uparrow$ and $\Downarrow$ correspond to $z$-projections of angular momentum of acceptor hole, $J_z$, equal to $+3/2$ and $-3/2$, respectively. Red bold arrow $\Uparrow$ corresponds to $z$-projection of angular momentum of heavy hole in exciton which is equal to +3/2. Thin arrows $\uparrow$ and $\downarrow$ correspond to electron spin projection on $z$ axis, $+1/2$ and $-1/2$, respectively. Coefficients $\alpha$ and $\beta$ determine the mixing of electron spin states in external magnetic field and depend on angle $\Theta$.[]{data-label="Fig:SFRS-schema"}](fig-3.eps){width="\columnwidth"}
Initially, the exciting photon in state $|\omega_1, \mathbf{k_1},\sigma^+ \rangle$ with optical frequency $\omega_1$ and circular polarization $\sigma^+$ propagates along the magnetic field direction $\mathbf{k_1}\|\mathbf{B}$. The $|\pm3/2\rangle$ ground states of the heavy hole bound to an acceptor $A^0$ in the QW are the eigenstates of the angular momentum projection onto the direction $z$ perpendicular to the QW plane, $J_z=\pm3/2$ (black bold arrows $\Uparrow$ and $\Downarrow$ in Fig. \[Fig:SFRS-schema\](a)). In the absence of $p-d$ exchange interaction, the Zeeman splitting of the spin levels is given by $E_{\pm3/2}=\pm\frac{1}{2}\mu_Bg_AB$. In the intermediate SFRS state the $A^0X$ complex given by an exciton bound to a neutral acceptor is created. For $\sigma^+$ excitation, the angular momentum projection of the heavy hole in the exciton is equal to +3/2 (red bold arrow $\Uparrow$ in Fig. \[Fig:SFRS-schema\](a)), while the spin of the acceptor bound hole is equal to $J_z=-3/2$ (see Fig. \[Fig:SFRS-schema\](a))[@SapegaPRB94]. The exchange interaction between the exciton heavy hole and the acceptor bound hole can lead to a mutual flip of their spins with conservation of total angular momentum. In the next step, the exciton is annihilated and a photon is emitted with optical frequency $\omega_2$ and opposite circular polarization $\sigma^-$. Here, energy conservation is fulfilled only for the initial and final states (photon and acceptor), but not in the intermediate state (exciton bound to neutral acceptor). In the final state we obtain the emitted photon $|\omega_2, \mathbf{k_2},\sigma^-\rangle$ and the acceptor with $J_z=+3/2$. Thus, the energy of the emitted photon is $\hbar\omega_2 = \hbar\omega_1 - \mu_B |g_A| B$, which is shifted into the Stokes region.
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In Faraday geometry ($\mathbf{B}\|z$) the transition described above is forbidden because the angular momentum of the hole in the $A^0X$ complex should change by three quanta, $\Delta J_z =3$, while the angular momentum of the photon $\Delta l$ in the back-scattering geometry ($\mathbf{k_1}=-\mathbf{k_{2}}$) changes by 0 or $\pm2$ only. For the observation of an SFRS line corresponding to the transition of the hole between its Zeeman levels we use therefore an oblique field geometry, namely an angle $\Theta$ between the $z$-axis and magnetic field $B$ of $20^\circ$ is chosen (see Fig. \[Fig:SFRS-schema\](b)). In this geometry, the magnetic field induces a mixing of the electron states with spin projections $+1/2$ and $-1/2$ along $z$ (thin arrows $\uparrow$ and $\downarrow$ in Fig. \[Fig:SFRS-schema\](a)) which allows for observing SFRS in crossed circular polarizations [@SapegaPRB94]. For an efficient SFRS process, it is necessary to tune the laser photon energy into resonance with the $A^0X$ transition (1.610 eV). In case of a noticeable $p-d$ exchange interaction between the magnetic ions in the FM layer ($d$-system) and the holes bound to acceptors in the QW ($p$-system) the splitting $\Delta_S(B)$ of the $A^0$ states is determined not only by the external magnetic field $B$, but also by the additional contribution due to the effective exchange field from the FM. Therefore, the resulting splitting is given by $$\label{eq:Exchange}
\Delta_S(B)=\mu_B |g_A| B - \Delta_{pd}m_z,$$ where $m_z$ is the $z$-projection of the unit vector $\mathbf m$ along the magnetization $\mathbf M$. Eq. is valid for large magnetic fields, when the first term on the right hand side is larger than the second one, i.e. $\Delta_S(B)>0$. Here, we use the fact that the $p-d$ exchange interaction between the magnetic ions and the heavy holes in a QW structure is strongly anisotropic, i.e., it is described by the Ising Hamiltonian $\frac{1}{3}\Delta_{pd}m_zJ_z$ [@Merkulov-1995]. In strong magnetic fields, the FM is fully magnetized along the $B$-direction and the dependence $\Delta_S(B)$ is a straight line with an offset given by the exchange constant $\Delta_{pd}$. For small $\Theta$, the projection $m_z=\cos\Theta\approx1$ and $|g_A|$ corresponds to the longitudinal acceptor $g$ factor, which determines the Zeeman splitting for $B$ applied along the $z$-direction. In our case $\Theta=20^\circ$ which allows one to use $\cos\Theta = 1$ in the evaluation of the exchange energy $\Delta_{pd}$ with an accuracy of 7%.
Figure \[Fig:SFRS-shift\] summarizes the data on the SFRS corresponding to the spin flip of the electron ($e$) and the hole bound to an acceptor ($A^0$). For $B=10$ T under resonant excitation of the $A^0X$ transition with photon energy $\hbar\omega_{\rm exc}= 1.610$ eV, the spin flip of the acceptor bound hole is observed for crossed orientations of polarizer and analyzer ($\sigma^+,\sigma^-$) as shown in Fig. \[Fig:SFRS-shift\](a). The signal is given by the broad line with a Raman shift of $\Delta_S =160~\mu$eV close to the laser line. Figure \[Fig:SFRS-shift\](b) shows the magnetic field dependences of the Raman shift of the acceptor bound hole $\Delta_S$ for various temperatures $T_{\rm bath}$. The data are well described by Eq. with the hole g factor $|g_A|=0.4$ which determines the slope of the line. The offset $\Delta_{pd}\approx 50~\mu$eV for $T_{\rm bath}=5$ K and depends weakly on temperature. A weak dependence of $\Delta_{pd}$ on $T_{\rm bath}$ follows also from Fig. \[Fig:SFRS-shift\](d), where the temperature dependence of the Raman shift $\Delta_{S}$ for a fixed magnetic field $B=10$ T is shown. Such behavior cannot be attributed to an exchange interaction with paramagnetic ions or FM Co clusters diffused into the QW during the growth process. The magnetization of ions should decrease strongly with increasing temperature from 2 to 25 K, which is in contrast to our observations (see Fig. \[Fig:SFRS-shift\](b) and (d)). Thus, we conclude that the SFRS demonstrates the splitting of the acceptor bound hole in the FM induced exchange field. The striking feature of this interaction is its long range nature. Figure \[Fig:SFRS-shift\](e) shows the splitting $\Delta_{pd}$ ${\sl vs}$ the spacer thickness evaluated from magnetic field dependences of $\Delta_S(B)$ measured on corresponding samples. We observe a splitting of about 100 $\mu$eV even for spacers as large as 10 nm. This distance is significantly larger than the penetration depth of electron and hole wavefunctions of maximum 1-2 nm into a FM layer that would be required to obtain a considerable direct exchange interaction [@NP-Korenev-2016].
![(Color online) Scheme of optical transitions involved in SFRS of acceptor bound hole (red arrows). Lower energy doublet with angular momentum projections $\pm 3/2$ corresponds to heavy hole states where splitting in magnetic field is depicted for the case of $g_A>0$. Upper energy doublet with angular momentum projections $\pm1/2$ corresponds to light hole states. $\Delta_{lh}$ is energy splitting between heavy and light holes bound to acceptor.[]{data-label="Fig:SFRS-acceptor"}](fig-5.eps){width="0.7\columnwidth"}
The offset in the magnetic field dependence of the acceptor bound hole Raman shift $\Delta_S(B)$ has to be considered with considerable care. Apart from the FM induced exchange field, the offset may result from the energy splitting between the heavy and light holes bound to an acceptor. The magnitude of this splitting is about $\Delta_{lh}\approx 1$ meV [@SapegaPRB94]. For the magnetic fields $B \leq 10$ T used in our experiments, the Zeeman splitting of the hole states $\mu_B |g_A| B$ is clearly less than $\Delta_{lh}$, which results in the transition scheme shown in Fig. \[Fig:SFRS-acceptor\]. At low temperatures, the lowest energy heavy hole state with angular momentum projection $J_z=-3/2$ is populated. From this state, there are three possible transitions which are shown with the red arrows in Fig. \[Fig:SFRS-acceptor\]. It follows that a decrease of the magnetic field leads to vanishing of the $|-3/2\rangle \rightarrow |+3/2\rangle$ spin flip transition energy. However, the transitions $|-3/2\rangle \rightarrow |-1/2\rangle$ and $|-3/2\rangle \rightarrow |+1/2\rangle$ have a positive offset corresponding to $\Delta_{lh}$. We emphasize that our results cannot be attributed to such behaviour because: (i) the offset in Fig. \[Fig:SFRS-shift\](b) is negative and (ii) the magnitude of exchange energy $\Delta_{pd} < 100~\mu$eV is significantly smaller than $\Delta_{lh}$. Moreover, the magnetic field dependence of $\Delta_S(B)$ in CdTe QW structures without Co layer shows linear behavior which approaches zero when extrapolated to zero field, i.e. no offset is detected in this case.
Therefore, the observation of SFRS on the acceptor bound hole corresponds to the spin-flip transition $|-3/2\rangle \rightarrow |+3/2\rangle$ and the offset is related to the heavy hole splitting in the effective exchange field from the FM. Transitions to the light hole states with $J_z=\pm1/2$ were not detected in the investigated samples which may be attributed to spectral broadening of the Raman line due to fluctuations of $\Delta_{lh}$.
The SFRS signal related to the heavy hole spin flip disappears when the exciting laser photon energy is increased and approaches the exciton resonance $X$ (see the PL spectrum in Fig. \[fig:PL-plus\](b)). In this case the spin flip of the electron dominates the SFRS spectrum, which is shown in Fig. \[Fig:SFRS-shift\](c) for $\hbar\omega_{\rm exc}=1.615$ eV. Figure \[Fig:SFRS-shift\](f) presents the magnetic field dependence of the Stokes shift for the electron spin-flip $\Delta_S^e(B)$. The shift follows a linear dependence with the electron $g$ factor $|g_e|=1.58$ and does not show any measurable offset [@Sirenko-1997]. This indicates that the effective $s-d$ interaction between the conduction band electrons in the QW and the FM layer is negligibly small as compared with the $p-d$ interaction of the QW heavy holes.
We also note that we do not observe a SFRS signal related to the free heavy hole which is not bound to the acceptor. Its absence may be due to strong fluctuations of the free hole $g$ factor leading to a significant broadening of the SFRS line. For detecting the spin splitting of the unbound heavy hole $\hbar\Omega_h$, we use a transient pump-probe technique as described below.
Larmor spin precession of valence band holes
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Transient pump-probe Kerr rotation in the vicinity of the exciton resonance allows us to measure the frequency of the Larmor precession of electrons $\Omega_e$ and holes $\Omega_h$ in CdTe/(Cd,Mg)Te QWs [@Zhukov-2007]. Thereby circularly polarized pump pulses photoexcite carriers with optically oriented spin polarization parallel to the growth direction ($z$-axis). In a transverse magnetic field $\mathbf{B}\|x$ the subsequent spin precession leads to transient oscillations of the $z$-component, $S_z$, of the spin polarization which is detected by the Kerr rotation of the linearly polarized probe beam when the delay between the pump and probe pulses $t$ is varied. The electron and hole spins precess with different Larmor precession frequencies due to the difference in their $g$ factors. The electron $g$ factor in CdTe QW is close to isotropic, while the heavy hole one has a strong anisotropy.
Our experiment requires an oblique magnetic field since the $z$-component of magnetic field has to induce a magnetization of the FM layer, while the $x$-component is required to observe the oscillatory precession signal. We stress that the pump-probe signal is observed in the studied FM-SC hybrid structures only when the excitation photon energy is tuned to the QW exciton resonance. This indicates that the experimental data monitor the spin dynamics of photoexcited carriers in the QW and not in the FM. Moreover, we get exclusively access to the Larmor precession of the conduction band electrons and valence band holes because an efficient optical orientation of the photoexcited carriers occurs only for resonant excitation of the excitons, whose oscillator strength is significantly larger than that of the excitons bound to acceptors.
![(Color online) Transient pump-probe Kerr rotation signal measured as function of pump-probe delay for various magnetic fields. Photon energy of pump and probe $\hbar\omega_p=1.627$ eV is tuned in resonance with exciton transition. $T_{\rm bath}=2$ K, $d_S=10$ nm and $\Theta=70^\circ$. Inset shows schematically geometry of experiment.[]{data-label="Fig:PP-transients"}](fig-6.eps){width="\columnwidth"}
Figure \[Fig:PP-transients\] shows corresponding transient Kerr rotation signals in different magnetic fields. The inset shows schematically the geometry of the experiment where the magnetic field is tilted by an angle $\Theta=70^\circ$ with respect to the $z$-axis. The transient signals comprise two contributions. The first one corresponds to a signal with high oscillation frequency and is attributed to the electron spin precession. The second contribution oscillates quite slowly and corresponds to the heavy hole spin dynamics with a small $g$ factor. Each of these oscillatory signals is well described with $A_i\cos(\Omega_it+\phi_i)$ which allows us to determine the magnetic field dependence of the Larmor precession frequencies $\Omega_i$ for the electrons ($i=e$) and holes ($i=h$). The data are summarized in Fig. \[Fig:PP-Freq\].
![(Color online) Magnetic field dependence of Zeeman splitting for (a) holes $\hbar\Omega_h$ for two temperatures of 2 K and 12 K and (b) electrons $\hbar\Omega_e$ evaluated from pump-probe transients. Dashed lines are linear fits to the data with $|g_h|=0.17$ in (a) and $|g_e|=1.31$ in (b). $d_S=10$ nm and $\Theta=70^\circ$.[]{data-label="Fig:PP-Freq"}](fig-7.eps){width="0.8\columnwidth"}
For the holes, $\Omega_h(B)$ dependencies are shown for $\Theta=70^\circ$ at two different temperatures, 2 K and 12 K (Fig. \[Fig:PP-Freq\](a)). At first glance the dependences appear to be linear across the whole range of magnetic fields with the corresponding $g$ factor $|g_h|=0.17$ which weakly depends on temperature. However, a closer look shows that $\Omega_h(B)$ shows small wiggles above about $B\approx1.5$ T. One possible explanation for the non-linear behavior of $\Omega_h(B)$ is the exchange interaction of the heavy holes with magnetic ions in the FM layer whose magnetization slowly varies with magnetic field. However, even if this effect is present its magnitude is expected to be rather small. Therefore, we conclude that the valence band holes are weakly coupled to the FM layer which is in contrast to the strongly interacting holes bound to acceptors as demonstrated in Section III. The value of the heavy hole $g$ factor is determined from the relation $|g_h|= \sqrt{g_z^2\cos^2\Theta + g_x^2\sin^2\Theta}$. Taking $g_x\approx0$ we obtain $|g_z|\approx 0.5$ thereby. This value is slightly larger than the $g$ factor of the acceptor bound hole $|g_A|=0.4$ extracted from the SFRS data, which indicates that indeed the pump-probe signal addresses the spin dynamics of unbound, free valence band holes.
The Larmor precession frequency of the electrons $\Omega_e(B)$ depends linearly on magnetic field (Fig. \[Fig:PP-Freq\](b)), from which we evaluate the electron $g$ factor to be $|g_e|=1.31$. The slight difference between the values obtained from pump-probe and SFRS is related to the anisotropy of the electron $g$ factor [@Sirenko-1997]. Note that the magnetic field dependence of $\Omega_e$ also does not show any offset. Thus, the electrons do not experience a $s-d$ exchange interaction which is in accord with the SFRS data.
Discussion
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The main result of our study is the direct measurement of the exchange energy $\Delta_{pd} = 50-100~\mu$eV for the effective $p-d$ interaction between the magnetic ions in the FM layer and the holes bound to acceptors in the semiconductor QW, without involving any model. This energy splitting of the hole spin levels is in agreement with our previous estimates in Ref. , where $\Delta_{pd} \approx 50~\mu$eV was evaluated from polarization- and time-resolved PL measurements in weak longitudinal magnetic fields in which the interfacial FM layer resulted in a magnetization of the acceptor holes. Here, SFRS measurements have been performed in strong magnetic fields and, therefore, it is expected that an additional contribution from the Co layer to the exchange interaction is expected. This is because magnetic fields larger than 2 T are sufficient to saturate the out of plane magnetization of the Co film. However, in contrast to MCD the amplitude of the proximity effect $A(B)$ in Fig. \[Fig:PL-TRPL\](b) increases linearly with magnetic field. Therefore, we conclude that the main contribution to the $p-d$ exchange interaction comes from the interfacial FM. The origin of the interfacial magnetic layer requires further studies. Currently, it is reasonable to assume that its formation is caused by chemical reaction of Co atoms with the Cd$_{0.8}$Mg$_{0.2}$Te material.
We observe no FM induced splitting of the spin levels of the valence band holes which are not bound to acceptors as well as of the conduction band electrons. The splitting of valence band holes has been evaluated from degenerate pump-probe Kerr rotation measurements under resonant excitation of excitons in the QW structure. This experiment differs significantly from SFRS which probes the acceptor bound holes under resonant excitation of excitons bound to neutral acceptors $A^0X$. Figure \[Fig:PP-Freq\](a) demonstrates that the magnetic field dependence $\Omega_h(B)$ does not show a detectable offset and a deviation from a linear behavior. Thus, the pump-probe measurements clearly demonstrate that the exchange interaction between the valence band holes in the QW and the FM layer is small. The same result is obtained for the conduction band electrons where as well no offset in the magnetic field dependence of their Zeeman splitting is detected, both in SFRS and pump-probe.
A further result obtained from SFRS is that the exchange energy $\Delta_{pd}$ does not decrease with increasing spacer thickness for $d_S \le 10$ nm (see Fig. \[Fig:SFRS-shift\](e)). This is in accord with our previous studies in Ref. , where the suppression of PL intensity with decreasing $d_S$ gives a characteristic length of $1-2$ nm for the wavefunction penetration into the Co-layer. This distance is much smaller than the spacer range of $d_S=5-10$ nm addressed in the present study. Also, the FM induced polarization of the PL depends only weakly on $d_S =5-30$ nm [@NP-Korenev-2016]. Therefore, we conclude that the effective $p-d$ exchange interaction between the Co ions in the FM and the holes bound to acceptors in the QW is not determined by their wavefunction overlap. These results are surprising from the viewpoint of the standard theory of exchange interaction whose strength is proportional to this overlap [@Miotkowska-2001; @Kossacki-Co-CdTe-2013]. Note, however, that this does not represent a contradiction because the exchange reported here is observed for holes bound to acceptors but is absent for conduction band electrons and valence band holes.
![(Color online) Optical (a) and phonon (b) [*ac*]{} Stark effect in semiconductors with $\sigma^+$ photons and phonons (blue arrows), respectively. In (a) the energy of photons is tuned below the band gap energy $E_g$ of semiconductor and results in a shift of electronic states in the conduction band (CB) with spin projection $S_z=-1/2$ and the valence band (VB) with angular momentum projection $J_z=-3/2$. In (b) phonons couple to transitions between between heavy (hh) and light (lh) hole states which are split by $\Delta_{lh}$. For $\sigma^+$ polarized phonons the energy shift occurs for the heavy hole state with $J_z=-3/2$ and the light hole state with $J_z=-1/2$. The $z$-axis is dictated by the photon or phonon propagation direction. The repulsion of energy levels is indicated with dashed lines. Both [*ac*]{} Stark effects result in a splitting of spin states and can be considered as generation of an effective [*dc*]{} magnetic field.[]{data-label="Fig:ac-Stark"}](fig-8.eps){width="\columnwidth"}
In Ref. we proposed that this kind of long-range interaction can be mediated by elliptically polarized acoustic phonons. The latter are strongly polarized in the vicinity of the magnon-phonon resonance in the FM [@Kittel-1958]. In addition, phonons do not experience the electronic barrier between the QW and the FM layer. The characteristic frequencies of these elliptically polarized phonons (about 1 meV) are close to the energy splitting between the acceptor bound heavy $|\pm3/2\rangle$ and light $|\pm1/2\rangle$ holes (quasi-resonant case) and significantly smaller than the corresponding splitting between the confined valence band states in the QW with 10 nm width. For example, if the phonons are mainly $\sigma^+$ polarized (with positive $z$-projection of angular momentum) the interaction with the holes couples the ground state $|-3/2\rangle$ with the excited state $|-1/2\rangle$ which consequently leads to an energy shift of the levels. This results in lifting of the Kramers degeneracy of the $|\pm3/2\rangle$ doublet in zero external magnetic field which is the phonon analog of the optical [*ac*]{} Stark effect and the inverse Faraday effect which occurs in case of illumination with elliptically polarized light in transparency region.
The optical Stark effect is a well established phenomenon in semiconductors [@Combescot-1989]. It takes place when an electromagnetic wave with $\sigma^+$ polarization couples the electronic states with angular momentum projection $-3/2$ in the valence band and $-1/2$ state in the conduction band as shown in Fig. \[Fig:ac-Stark\](a). Due to the interaction with light these states experience an energy shift $\Delta \propto P^2/\delta$, where $\delta = E_g - \hbar\omega$. Here, $\hbar\omega$ is the photon energy and $E_g$ is the energy gap of the semiconductor, $P$ is the dipole matrix element of the optical transitions between valence and conduction bands. For photons with $\hbar\omega < E_g$ repulsion between the electronic states takes place, i.e. $\Delta>0$. Similarly, in the case of the phonon Stark effect [@NP-Korenev-2016], a circularly polarized phonon couples the heavy (hh) and light (lh) hole acceptor states with angular momentum projections $-3/2$ and $-1/2$, respectively (see Fig. \[Fig:ac-Stark\](b)). The spin-phonon interaction for holes occurs due to the spin-orbit coupling of hole states in the valence band. In this case, the level shift is proportional to the square of the matrix element of the spin-phonon interaction divided by the detuning of the phonon frequency at the magnon-phonon resonance in the FM relative to the energy separation between the heavy and light hole acceptor levels in the QW.
In conclusion, our results are in agreement with the proposed model of an [*effective $p-d$ exchange interaction*]{} mediated by elliptically polarized phonons. Here the energy splitting of the acceptor bound holes has been measured directly and amounts to $\Delta_{pd} = 50-100~\mu$eV. This model explains the absence of a long-range $s-d$ exchange interaction because the spin-orbit interaction in the conduction band is much smaller than the one in the valence band.
Acknowledgements
================
We acknowledge the financial support by the Deutsche Forschungsgemeinschaft through the International Collaborative Research Centre 160. The partial financial support from the Russian Foundation for Basic Research Grant No. 15-52-12017 NNIOa and the Russian Ministry of Education and Science (Contract No. 14.Z50.31.0021) is acknowledged as well. N.E.K. acknowledges support from RFBR (grant No. 15-52-12019). The research in Poland was partially supported by the National Science Centre (Poland) through Grants No. DEC- 2012/06/A/ST3/00247 and No. DEC-2014/14/M/ST3/00484, as well as by the Foundation for Polish Science through the IRA Programme co-financed by EU within SG OP.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper we investigate expanding Bianchi type I models with two tilted fluids with linear equations of state. Individually the fluids have non-zero energy fluxes w.r.t. the symmetry surfaces, but these cancel each other because of the Codazzi constraint. Asymptotically toward the past the solutions approach Kasner states if the speeds of sound are less than that of light. If one of the fluids has a speed of sound that is less or equal to $1/3$ of the speed of light (radiation) then the models isotropize toward the future, but if both fluids are stiffer than radiation then the final state is anisotropic with non-zero Hubble-normalized shear. The significance of these results is discussed in a broader context.'
author:
- |
Patrik Sandin $^{1}$[^1], and Claes Uggla$^{1}$[^2]\
$^{1}$[*Department of Physics, University of Karlstad,*]{}\
[*S-651 88 Karlstad, Sweden*]{}
title: Bianchi type I models with two tilted fluids
---
PACS numbers: 04.20.-q, 04.20.Dw, 04.20.Ha, 98.80.-k, 98.80.Bp, 98.80.Jk
Introduction {#Sec:intro}
============
The construction of a relativistic model of gravity contains the following ingredients: (i) a 4-dimensional manifold ${\cal
M}$ endowed with a Lorentzian metric, (ii) a matter source description, (iii) dynamical laws—Einstein’s field equations and, if needed, matter equations. A general relativistic [*cosmological*]{} model requires that one in addition attempts to describe the universe at a particular scale.
At the largest spatial scales present observational data suggest that the ‘standard’ $\Lambda{\rm CDM}$ model of cosmology provides the most simple consistent description of the universe today. This model is spatially homogeneous (SH) and isotropic with flat spatial geometry; the matter content consists of a dark energy component, modeled by a positive cosmological constant, supplemented with dark matter and atoms, described by pressureless fluids with $n^a$, the unit normal to the symmetry surfaces, as the common 4-velocity. Density fluctuations, described by linear scalar perturbations, are seeded by an almost Gaussian, adiabatic, nearly scale invariant process, see e.g. [@hinetal08],[@tegetal06], and references therein.
However, this description does not hold on all scales, neither spatial nor temporal. On smaller spatial scales matter has to be described by many components, with energy fluxes in different directions, e.g., our galaxy is moving w.r.t. to the CMB. In the very early universe, and perhaps also in the distant future, the $\Lambda{\rm CDM}$ model does not give a correct matter description, indeed, although radiation can be treated as a gravitational test field[^3] today it has been an important gravitational source in the past, and inflationary proponents suggest that matter could be extremely different in the very early universe. Clearly the matter description in the ‘standard’ scenario is not exact, and even with a few matter components the associated energy fluxes cannot all be [*exactly*]{} aligned, not even on the largest spatial scales. What then happens with the matter and its associated energy fluxes in the far future and what was the situation in the distant past?
We believe that we understand how radiation and matter interacts, at least after the very early universe when we think our empirical experience holds. Presumably this interaction explains why the 4-velocities associated with radiation and matter today are fairly well aligned with each other on large spatial scales, or maybe this alignment was produced in the very early universe by some unknown process, perhaps inflation. But is it obvious that this alignment should have persisted to the extent present observations indicate after recombination, and is it going to persist in the far future? Do linear vector perturbations of SH and isotropic models suffice to determine this? In the early universe interactions presumably played an important role in aligning energy fluxes of different matter components, but it is unclear what those interactions were; is it possible to shed any light on this issue without knowing the details of these interactions?
Here we are going to consider two non-interacting perfect fluids that in general move w.r.t. each other and a non-negative cosmological constant, which includes the $\Lambda{\rm CDM}$ matter content as a special case. Although this may not be a good matter description at all times, it is still a useful step since it allows a comparative study of the effects of various types of interactions in possible future projects, an issue we return to in the concluding remarks.
Two matter source components with energy fluxes in different directions produce an anisotropic source which excludes the isotropic standard model and forces one to consider anisotropic geometries. There are several reasons that suggest that the natural anisotropic models to start with are the SH Bianchi type I models. One reason is their geometric simplicity since this sheds light on more general models—if type I turns out to be complicated, then more general models will be even worse. But more importantly is that they are the foundation in a hierarchy of ever more geometrically complex models.
The SH Bianchi models (models that admit simply transitive 3-dimensional symmetry groups) form a crucial level in the geometric complexity hierarchy, and within this level the Bianchi type I models is the common ingredient since they can be obtained from all other Bianchi models by Lie algebra contractions. The consequences of this property are revealed when one casts Einstein’s equations into a dynamical system where the type I models appear as part of a state space boundary that describes asymptotic features of all other Bianchi models, see e.g. [@waiell97] and [@col03], and references therein.
Moreover, the Bianchi models themselves serve as building blocks for understanding asymptotic dynamics of more general inhomogeneous models, the primary reason being the following: In the very early universe near a generic or isotropic singularity, or in the very late universe in an inflationary epoch, horizons form and asymptotically shrink, asymptotically prohibiting communication—a phenomenon naturally referred to as asymptotic silence—generically pushing inhomogeneities outside the horizons leading to that the asymptotic evolution can locally be described by SH models—asymptotic locality, see [@uggetal03],[@andetal05],[@heietal07]. In the dynamical systems approach these features are formally captured by recasting the field equations into an infinite dimensional dynamical system that at each spatial point has a boundary—the silent boundary, which asymptotically attracts generic asymptotically local dynamics. The dynamics on the silent boundary, which determine the generic asymptotically local dynamics, is described by a finite dimensional dynamical system that is identical to that of the Bianchi models and hence type I is a common key ingredient in a very general context [@uggetal03],[@andetal05],[@heietal07],[@rohugg05]. Furthermore, there are hints, analytical and numerical, that Bianchi models are important for describing future asymptotic states, even in the absence of inflation and asymptotic silence and locality, that sheds light on spatial structure formation.
This is not the first study using dynamical systems techniques for studying multi-fluid models. Bianchi models with two fluids with both 4-velocities being orthogonal to the symmetry surfaces were studied in [@colwai92]. However, it is not difficult to predict what is going to happen when the non-interacting fluids are aligned with each other, even in the general inhomogeneous case. Let us introduce a length scale $\ell$ defined by $\dot{\ell}/\ell=H$, where $H$ is the Hubble scalar and where the dot refers to the time derivative w.r.t. the proper clock time along the common fluid congruence. For simplicity we consider fluids with linear equations of state such that $p=w\rho$, where $p$, $\rho$ is the pressure and energy density, respectively, and where the constant $w$ describes the speed of sound $c_s$ according to $w=c_s^2$ when $w\geq 0$. Interesting examples of equations of state are: $w=-1$, which corresponds to a positive cosmological constant; dust, $w=0$; radiation, $w=\frac{1}{3}$; and a stiff fluid, $w=1$, for which the speed of sound is equal to that of light. Local energy-momentum conservation then yields that $\rho\propto \ell^{-3(1+w)}$, see e.g. [@waiell97], [@heietal05]. Thus if $\ell\rightarrow 0$ the energy densities of fluids with smaller $w$ become asymptotically negligible compared to those with larger $w$, while if $\ell\rightarrow \infty$ the opposite holds.
However, the situation is much more complicated when the non-interacting fluids are not aligned. In [@golnil00] Bianchi type V models with two fluids and a positive cosmological constant were studied; one of the fluids had a flow orthogonal to the symmetry surfaces while the other had a ‘tilted’ flow, i.e., its 4-velocity was not aligned with the normal to the SH surfaces. In [@colher04], where brane-world cosmology was invoked to motivate the study of multiple fluids, Bianchi type VI$_0$ with two non-interacting tilted fluids was investigated. Perhaps because the focus in these papers was on the quite interesting late time behavior, there was no mentioning about the possibility of having two (or more) tilted fluids in Bianchi type I.
The present paper is organized as follows: In the next section we derive a reduced dynamical system that describes Bianchi type I with two non-interacting fluids with linear equation of state and a cosmological constant. In the subsequent section we describe the associated state space and the influence of a positive cosmological constant; in addition we list the invariant subsets and fix points that are essential for understanding the present models. In section \[Sec:mon\] we give several monotonic functions that are useful for determining the asymptotic dynamics; we also briefly discuss some reasons why they exist since this allows one to produce monotonic functions for other models as well. Section \[Sec:attractor\] takes the results in the previous sections as the starting point for a dynamical systems analysis which yield our main results about asymptotic dynamics toward the past and toward the future, in the absence of a cosmological constant. We conclude with a discussion in section \[Sec:concl\] about the significance of our results in a more general context. Appendix \[Sec:locstab\] contains detailed information about the fix points and their stability properties. Throughout we use units such that $c=1=8 \pi G$.
Derivation of the dynamical system {#Sec:dynsysder}
==================================
In the orthonormal frame approach one uses a tetrad of four orthogonal unit basis vector fields $\{\,{\bf e}_a\,\}$ and the associated dual one-forms $\{\,{\bom}^a\,\}$ ($a=0,1,2,3$), which, when expressed in a local coordinate basis, take the form ${\bf e}_a = e_a{}^\mu\ptl/\ptl x^\mu\ =
e_a{}^\mu\ptl_\mu$, $\bom^a = e^a{}_\mu\,{\bf d}x^\mu$ ($\mu=0,1,2,3$), where the tetrad components $e_a{}^\mu
(x^\nu)$ and their inverse components $e^a{}_\mu (x^\nu)$ satisfy the duality relations $e_a{}^\mu\,e^a{}_\nu =
\delta^\mu{}_\nu \,\Leftrightarrow \, e_a{}^\mu\,e^b{}_\mu =
\delta^b{}_a$, and where the orthogonality conditions are given by $g_{ab} = {\bf e}_a\cdot {\bf e}_b =
g_{\mu\nu}\,e_a{}^\mu\,e_b{}^\nu = \eta_{ab}$; $g_{\mu\nu} =
\eta_{ab}\,e^a{}_\mu\,e^b{}_\nu$, $\eta_{ab} = {\rm
diag}\,(\,-1,1,1,1\,)$. The commutator functions $c^a{}_{bc}(x^\mu)$, defined as $[{\bf e}_a,{\bf e}_b] =
c^c{}_{ab}\,{\bf e}_c$, are typically ‘elevated’ to dependent variables satisfying the Jacobi identities, ${\bf
e}_{[a}\,c^d{}_{bc]} - c^d{}_{e[a}\,c^e{}_{bc]} = 0$.
Let us now consider SH Bianchi models, i.e., models with a foliation of SH hypersurfaces invariant under a simply transitive group action $G_3$, and let us also introduce an orthonormal basis of vector fields $\{\mathbf{e}_a\}$ that is invariant under the group action such that the timelines are orthogonal to the SH hypersurfaces with ${\bf e}_0={\bf n} =
\ptl/\ptl t$, where $t$ is the proper time along the geodesic timelines (the geodesic property follows from the symmetries). This yields the line-element: $ds^2 = - dt^2 +
\delta_{\alpha\beta}\,\bom^\alpha\otimes \bom^\beta$ ($\alpha,\beta =1,2,3$), where $\bom^\alpha$ (with components $e^\alpha{}_i$) are the one-forms dual to the triad $\vece_\alpha$ (with components $e_\alpha{}^i$), tangential to the symmetry surfaces, i.e., $e_\alpha{}^i e^\beta{}_i =
\delta^\alpha{}_\beta$ ($i=1,2,3$).
A 3+1 split of the commutator equations w.r.t. ${\bf e}_0={\bf
n}$ yields:
$$\begin{aligned}
\lb{dcomts0} [\,\vece_{0}, \vece_{\alpha}\,] &= -
[\,H\,\d_{\alpha}{}^{\beta} + \sig_{\alpha}{}^{\beta} +
\eps_{\alpha}{}^{\beta}{}_{\gam}\,\Omega^{\gam}\,]\,
\vece_{\beta}\:, \\
\lb{dcomtsa} [\,\vece_{\alpha}, \vece_{\beta}\,] &=
c^\gam{}_{\alpha\beta}\,\vece_{\gam} =
2a_{[\alpha}\,\d_{\beta]}{}^{\gam} +
\eps_{\alpha\beta\delta}\,n^{\delta\gam}\:.\end{aligned}$$
Here $H$ is the Hubble variable, which is related to the expansion $\theta$ of ${\bf n}$ according to $H=\frac{1}{3}\theta$; $\sigma_{\alpha\beta}$ is the shear associated with ${\bf n}$; $\Omega^\alpha$ is the Fermi rotation which describes how the spatial triad rotates with respect to a gyroscopically fixed so-called Fermi frame;[^4] $n^{\alpha\beta}$ and $a_\alpha$ describe the Lie algebra of the 3-dimensional simply transitive Lie group and determine the spatial three-curvature, see e.g. [@waiell97].
Due to the symmetries $e_\alpha{}^i$ can be written as $e_\alpha{}^i=\tilde{e}_\alpha{}^\beta(t)\hat{e}_\beta{}^i$, where $\hat{e}_\alpha{}^i$ are functions of the spatial coordinates $x^i$ alone such that $[\,\hat{\vece}_{\alpha},
\hat{\vece}_{\beta}\,] =
\hat{c}^\gam{}_{\alpha\beta}\,\hat{\vece}_{\gam} =
2\hat{a}_{[\alpha}\,\d_{\beta]}{}^{\gam} +
\eps_{\alpha\beta\delta}\,\hat{n}^{\delta\gam}$, where $\hat{\vece}_\alpha = \hat{e}_\alpha{}^i\partial/\partial x^i$, and where $\hat{c}^\gam{}_{\alpha\beta}$, parameterized by $\hat{a}_\alpha , \hat{n}^{\alpha\beta}$, are the structure constants of the symmetry group. The symmetries lead to that the equations for the variables $\tilde{e}_\alpha{}^\beta(t)$ ($d{\tilde{e}}_\alpha{}^\beta(t)/dt =
-[\,H\,\d_{\alpha}{}^{\gam} + \sig_{\alpha}{}^{\gam} +
\eps_{\alpha}{}^{\gam}{}_{\delta}\,\Omega^{\delta}\,]\tilde{e}_\gam{}^\beta$, as follows from ) decouple from the remaining field equations, and because of this they are not usually considered when discussing Bianchi models from an orthonormal frame perspective.
A 3+1 split of the total stress-energy tensor $T_{ab}$ w.r.t. $n^a$ yields:
\[Tirreduc\] $$\begin{aligned}
T_{ab} &= \rho\,n_a\,n_b + 2q_{(a}\,n_{b)} + p\,h_{ab} + \pi_{ab}\:,\\
\rho &= n^a\,n^b\,T_{ab}\:,\qquad q_a = -h_a{}^b\,n^c\,T_{bc}\:,\qquad
p = {{\textstyle{1\over3}}}\,h^{ab}\,T_{ab}\:,\qquad \pi_{ab} = h_{\la
a}{}^c\,h_{b\ra}{}^d\,T_{cd}\:,\end{aligned}$$
where $h_{ab} = n_a n_b + g_{ab}$; $\rho, p$ is the total energy density and total effective pressure, respectively, measured in the rest space of $n^a$; $\la..\ra$ stands for the trace-free part of a symmetric spatial tensor, i.e. $A_{\la
\alpha\beta \ra}=A_{\alpha\beta} -
\frac{1}{3}\delta_{\alpha\beta}\,A^\gamma{}_\gamma$. In general $T_{ab}$ consists of several components $T^{(i)}_{ab}$, such that $T_{ab} = \sum_i T^{(i)}_{ab}$. If the components are non-interacting, then $\bna_a T_{(i)}^{ab}=0$. A cosmological constant $\Lambda$ can be formally regarded as a component of $T_{ab}$ such that $\rho_\Lambda=\Lambda,\, p_\Lambda =
-\Lambda$, while $q_{(\Lambda)}^\alpha=0,\,
\pi_{(\Lambda)}^{\alpha\beta}=0$.
In the Hubble normalized approach one factors out the Hubble variable $H$ by means of a conformal transformation which yields dimensionless quantities [@rohugg05]. In the present SH case this amounts to the following: (\_, R\^, N\^,A\_) = (\_, \^, n\^,a\_),(, P, Q\_, \_) = (,p,q\_,\_),where we have chosen to normalize the stress-energy quantities with $3H^2$ rather than $H^2$ in order to conform with the usual definition of $\Omega$. In addition to this we choose a new dimensionless time variable $\tau$ according to = H. Since $H$ is the only variable with dimension, its evolution equation decouples from the remaining equations for dimensional reasons: H\^= -(1+q)H;q = 2\^[2]{} + (+3P),\^2=\_\^, where a prime henceforth denotes $d/d\tau$ and where $q$ is the deceleration parameter, obtained by means of one of Einstein’s equations—the Raychaudhuri equation; note that $\Omega$ and $P$ in the expression for $q$ refers to the total Hubble-normalized stress-energy content. A 3+1 split of the remaining Einstein’s field equations ($G_{ab}=T_{ab}$, where $G_{ab}$ is the Einstein tensor and $T_{ab}$ the total stress-energy tensor) and the Jacobi identities, yields the following reduced system of coupled equations for the Hubble-normalized variables:
$$\begin{aligned}
\Sigma_{\alpha\beta}^\prime &= -(2-q)\Sigma_{\alpha\beta} +
2\epsilon^{\gamma\delta}{}_{\la \alpha}\,\Sigma_{\beta\ra
\delta}\,R_\gamma - \,^3{\cal R}_{\la\alpha\beta\ra} + 3\Pi_{\alpha\beta}\:,\lb{HspatE}\\
A_{\alpha}^\prime &= [q\,\d_{\alpha}{}^{\beta} -
\Sig_{\alpha}{}^{\beta} - \eps_{\alpha}{}^{\beta}{}_{\gam}\,R^{\gam}] A_\beta\:,\lb{Hajac}\\
(N^{\alpha\beta})^\prime &= [q\,\d_{\gamma}{}^{(\alpha} +
2\Sig_{\gam}{}^{(\alpha} + 2
\eps_{\gam}{}^{(\alpha}{}_{\delta}\,R^{\delta}] N^{\beta )\gamma}\lb{Hnjac}\:,\\
0 &= 1 - \Sigma^2 + {{\textstyle{1\over6}}}\,^3{\cal R} - \Omega\:,\label{dGauss}\\
0 &= (3\delta_\alpha{}^\gamma\,A_\beta +
\epsilon_\alpha{}^{\delta\gamma}
\,N_{\delta\beta})\,\Sigma^\beta{}_\gamma - 3Q_\alpha\:,\label{dCodazzi}\\
0 &= A_\beta\, N^\beta{}_\alpha\:,\lb{HJacobi}\end{aligned}$$
where $^3{\cal R}_{\la\alpha\beta\ra}$ and $^3{\cal R}$ describe the trace-free and scalar parts of the Hubble-normalized three-curvature, respectively, according to: \^3[R]{}\_ = B\_ + 2\^\_N\_A\_,\^3[R]{} = -B\^\_- 6A\^2;B\_ = 2 N\_N\^\_- N\^\_N\_; are the Hubble-normalized spatial and trace-free Einstein equations; and are evolution equations obtained from the Jacobi identities; and are the Hubble-normalized Gauss and Codazzi constraints, respectively, while the constraint stems from the Jacobi identities. The conservation law $\bna_a T^{ab}=0$ for the total stress-energy tensor yields:
$$\begin{aligned}
\lb{dlomdot} \Om^\prime &= (2q-1)\,\Om - 3P +
2A_{\alpha}\,Q^{\alpha} - \Sig_{\alpha\beta}\Pi^{\alpha\beta}\:,\\
\lb{dlqmalpha} Q_{\alpha}^\prime &= -[2(1-q)\,\d_{\alpha}{}^{\beta}
+ \Sig_{\alpha}{}^{\beta} +
\eps_{\alpha}{}^{\beta}{}_{\gam}\,R^{\gam}]\,Q_{\beta} +
(3A_\beta\,\delta_\alpha{}^\delta -
\epsilon_{\alpha\beta}{}^\gamma\,N_\gamma{}^\delta)\,\Pi_\delta{}^\beta\:.\end{aligned}$$
Let us now restrict ourselves to the [*Bianchi type I*]{} case with expansion ($H>0$), characterized by A\_=0,N\^=0. In type I reduces to $Q_{\alpha}^\prime =
-[2(1-q)\,\d_{\alpha}{}^{\beta} + \Sig_{\alpha}{}^{\beta} +
\eps_{\alpha}{}^{\beta}{}_{\gam}\,R^{\gam}]\,Q_{\beta}$, and hence the reduction of the Codazzi constraint to $Q_\alpha=0$ is consistent since it is preserved during evolution. Thus there is no total energy flux in Bianchi type I, and hence the type I SH frame is an energy frame, in the nomenclature of Landau and Lifshitz [@lanlif63]. Even so, a matter source can consist of several components that individually have non-zero energy fluxes, as long as they add up to zero.
Let us now specialize to a source that consists of a non-negative [*cosmological constant*]{} $\Lambda\geq 0$ [*and two non-interacting perfect fluids*]{}, i.e., $T^{ab}_{(i)}
= (\tilde{\rho}_{(i)} + \tilde{p}_{(i)})
\tilde{u}^a_{(i)}\tilde{u}^b_{(i)} + \tilde{p}_{(i)} g^{ab}$; $\bna_a T^{ab}_{(i)}=0$ ($i=1,2$), where $\tilde{\rho}_{(i)},
\tilde{p}_{(i)}$, is the energy density and pressure, respectively, in the rest frame of the $i$:th fluid, while $\tilde{u}^a_{(i)}$ is its 4-velocity. We assume that $\tilde{\rho}_{(i)}\geq 0$, and for simplicity also a [*linear equations of state*]{}, $\tilde{p}_{(i)} = w_{(i)}
\tilde{\rho}_{(i)}$, where $w_{(i)}=const$. The most interesting equations of state are dust, $w=0$, and radiation, $w=\frac{1}{3}$, but it is useful to not restrict oneself to these values in order to study structural stability, however, we do restrict ourselves to 0 w\_[(2)]{} < w\_[(1)]{} <1; since $w_{(1)}=1$, $w_{(1)}=w_{(2)}$ are associated with bifurcations that needs special treatment, to be dealt with elsewhere.[^5]
Making a 3+1 split with respect to ${\bf n}= \vece_0$ yields \^a\_[(i)]{} = \_[(i)]{}(n\^a + v\^a\_[(i)]{}); n\_a v\^a\_[(i)]{}=0,\_[(i)]{} = 1/ (v\^2\_[(i)]{}= \_v\_[(i)]{}\^v\_[(i)]{}\^), which gives \[pfrel\] Q\_[(i)]{}\^= (1 + w\_[(i)]{}) (G\^[(i)]{}\_+)\^[-1]{} v\_[(i)]{}\^\_[(i)]{}, P\_[(i)]{} = w\_[(i)]{}\_[(i)]{} + (1 - 3w\_[(i)]{})Q\^[(i)]{}\_v\_[(i)]{}\^, \^[(i)]{}\_ = Q\^[(i)]{}\_v\^[(i)]{}\_, where $G^{(i)}_\pm = 1 \pm w_{(i)} \, v_{(i)}^2$; $\Omega_{(i)}=\rho_{(i)}/(3H^2)$. The cosmological constant contributes $\Omega_\Lambda = \Lambda/(3H^2)=-P_\Lambda$ to the total $\Omega$ and $P$, while $Q^\alpha_\Lambda=0=\Pi^{\alpha\beta}_\Lambda$. Due to its definition and equation , $\Omega_\Lambda$ satisfies \_\^= 2(1+q)\_. The Codazzi constraint, $Q_\alpha= Q_\alpha^{(1)} +
Q_\alpha^{(2)}= 0$, taken in combination with forces the two fluids 3-velocities to be anti-parallel. Kinematically the situation is similar to that of Bianchi type I with a general magnetic field studied in [@leb97], and it is therefore natural to exploit the same mathematical structures in the present problem. We therefore choose the spatial triad so that one of the frame vectors is aligned with the fluid velocities, which we choose to be $\vece_3$, in agreement with what is usually done in physics, i.e. $v_{(i)}^{\alpha}= (0,0,v_{(i)})$. Demanding that these conditions on $v_{(i)}^{\alpha}$ hold for all times lead to the following conditions[^6] R\_1 = -\_[23]{},R\_2 = \_[31]{}. This leaves $R_3$ undetermined, however, we still have the freedom of arbitrary rotations in the 1-2-plane, which we use to set R\_3=0.Following [@leb97], we introduce the variables $\Sigma_+,\Sigma_A,\Sigma_B,\Sigma_C$ according to \_+ = (\_[11]{}+\_[22]{}),\_[31]{} + \_[23]{} = \_A e\^ ,\_- + \_[12]{}= (\_B + \_C) e\^[2]{}, where $\Sigma_-=(\Sigma_{11}-\Sigma_{22})/(2\sqrt{3})$, which leads to \^2= \_+\^2 + \_A\^2 + \_B\^2 + \_C\^2. The above decomposition of $\Sigma_{\alpha\beta}$ has the advantage that the equation for $\phi$, $d\phi/d\tau =
-\Sigma_C$, decouples from the other equations, leaving the following reduced constrained dynamical system of coupled equations for the Hubble-normalized shear variables $\Sigma_+,\Sigma_A,\Sigma_B,\Sigma_C$, the fluid 3-velocities $v_{(1)}, v_{(2)}$, the Hubble-normalized energy densities $\Omega_{(1)},\Omega_{(2)}$, and the Hubble-normalized cosmological constant $\Omega_\Lambda$:
[*Evolution equations*]{}:
\[evolBI\] $$\begin{aligned}
\Sigma_+^\prime &= -(2-q)\Sigma_+ + 3\Sigma_A^2 - Q_{(1)}v_{(1)} - Q_{(2)}v_{(2)}\:, \label{Sigp}\\
\Sigma_A^\prime &= -(2 - q + 3\Sigma_+ + \sqrt{3}\Sigma_B) \Sigma_A \:, \label{SigA}\\
\Sigma_B^\prime &= -(2-q)\Sigma_B + \sqrt{3}\Sigma_A^2 -
2\sqrt{3}\Sigma_C^2\:, \\
\Sigma_C^\prime &= -(2 - q - 2\sqrt{3}\Sigma_B) \Sigma_C\:, \label{SigC}\\
v_{(i)}^\prime &= (G^{(i)}_-)^{-1} (1-v_{(i)}^2)(3w_{(i)} - 1 + 2\Sigma_+)v_{(i)}\:, \label{vieq}\\
\Omega_{(i)}^\prime &= (2q - 1 - 3w_{(i)})\Omega_{(i)} + (3w_{(i)} - 1 + 2\Sigma_+)Q_{(i)}v_{(i)}\:,
\label{Omieq}\\
\Omega_{\Lambda}^\prime &= 2(1 + q)\Omega_{\Lambda}\:.\label{lambdaeq}\end{aligned}$$
[*Constraint equations*]{}:
\[constrBI\] $$\begin{aligned}
0 &= 1 -\Sigma^2 - \Omega_{(1)} - \Omega_{(2)} -\Omega_{\Lambda}
\:,\label{gausssys}\\
0 &= Q_{(1)} + Q_{(2)}\label{codazzisys}\:,\end{aligned}$$
where q = 2\^2 + (\_[m]{} + 3P\_[m]{}) -\_ = 2 - (\_[m]{} - P\_[m]{}) -3\_; \_[m]{} = \_[(1)]{} + \_[(2)]{}, P\_[m]{} = P\_[(1)]{} + P\_[(2)]{}. Equations and were obtained by using that takes the same form for non-interacting individual matter components, where, however, the total matter content enters into $q$, together with the type I conditions and the relations for the individual perfect fluids. The assumption of non-negative energy densities and a non-negative cosmological constant, $\Omega_{(i)},\Omega_\Lambda \geq 0$, together with and , yields that $-1\leq q \leq 2$, and hence that $2-q\geq 0$, where $q=-1$ only when $\Omega_\Lambda=1,
\Omega_m=0,\Sigma^2=0$. It follows that $\tau \in
(-\infty,\infty)$ and $H \rightarrow \infty$ when $\tau\rightarrow -\infty$ (if $\Omega_\Lambda\neq 1$ initially).
The auxiliary equation, \[rhoidot\] \_[(i)]{}\^= -(1+w\_[(i)]{})(G\_+\^[(i)]{})\^[-1]{}\[3 + v\_[(i)]{}\^2 - 2\_+ v\_[(i)]{}\^2\]\_[(i)]{},implies that $\rho_{(i)}$ is a monotonically decreasing function such that $\rho_{(i)}\rightarrow \infty$ ($\rho_{(i)}\rightarrow 0$) when $\tau\rightarrow -\infty$ ($\tau\rightarrow \infty$); hence the models begin with an initial curvature singularity, where $\Lambda$ becomes negligible when compared to $\rho_{(i)}$ when $\tau\rightarrow -\infty$, and then expand forever to a state where the ordinary matter is infinitely diluted, leading to that $\Omega_{\rm m}$ becomes negligible compared to $\Omega_\Lambda$.
State space properties {#Sec:statprop}
======================
The state space
---------------
The reduced state space consists of ${\bf S} = \{
\Sigma_+,\Sigma_A,\Sigma_B,\Sigma_C,v_{(1)},v_{(2)},
\Omega_{(1)},\Omega_{(2)}, \Omega_{\Lambda}\}$, subject to the two constraints , i.e., the state space is seven-dimensional. From the definitions and the constraints it follows that the state space is bounded. Our primary concern in this paper is the ‘interior’ state space for the case of two [*tilted*]{} fluids for which $\Omega_{(1)}\Omega_{(2)}>0,\,0<|v_{(1)}v_{(2)}|<1$, however, the solutions belonging to the interior state space often asymptotically approach its boundary. To understand the interior dynamics we therefore consider the closure of the interior state space, $\bar{{\bf S}}$, thus obtaining a compact state space, which is possible because of the regularity of the evolution equations. Hence $\Sigma^2 \leq 1, 0\leq
v_{(i)}^2\leq 1$; $0\leq\Omega_{(1)}\leq 1,
0\leq\Omega_{(2)}\leq 1, 0\leq\Omega_\Lambda\leq 1$, in such a way so that the constraints are satisfied; note that the Codazzi constraint leads to that $v_{(1)}v_{(2)}\leq 0$ when $\Omega_{(1)}\Omega_{(2)}>0$, a condition on $v_{(i)}$ that we extend to the boundary. The dynamical system , is invariant under the following discrete symmetries: \[discrete\] \_A -\_A,\_C -\_C;(v\_[(1)]{},v\_[(2)]{}) -(v\_[(1)]{},v\_[(2)]{}). We therefore assume without loss of generality that $\Sigma_A
\in [0,1], \, \Sigma_C \in [0,1], \, v_{(1)} \in [0,1]$, and $v_{(2)} \in [-1,0]$; the solutions in the other sectors of the state space are easily obtained by means of the discrete symmetries.
The influence of a cosmological constant
----------------------------------------
Equation implies that $\Omega_\Lambda$ is monotonically increasing from zero to one. Hence $$\label{Wald}
\Omega_{\Lambda} \rightarrow 1, \quad \Sigma^2 \rightarrow 0, \quad
\Omega_{(i)} \rightarrow 0 \quad \text{when} \quad \tau\rightarrow
\infty\:,$$ as follows from combining $\Omega_{\Lambda} \rightarrow 1$ with the Gauss constraint , i.e., the solutions approach a de Sitter state when $\tau\rightarrow \infty$. This result is a special case of the proof by Wald [@wal83], which holds for Bianchi types I-VIII. In the present case the fluids behave as test fields on a de Sitter background at late times, obeying the equations: $v_{(i)}^\prime =
(G^{(i)}_-)^{-1}(1-v_{(i)}^2)(3w_{(i)} - 1) \,v_{(i)}$. It follows that $v_{(i)}=const$ if $w_{(i)}=\frac{1}{3}$; $v_{(1)}$ is monotonically increasing (decreasing) from 0 to 1 (1 to 0) if $w_{(1)}>\frac{1}{3}$ ($w_{(1)}<\frac{1}{3}$); $v_{(2)}$ is monotonically decreasing (increasing) from 0 to $-1$ ($-1$ to 0) if $w_{(2)}>\frac{1}{3}$ ($w_{(2)}<\frac{1}{3}$). Thus if one of the fluids has a soft equation of state, $w_{(2)}<\frac{1}{3}$, and the other has a sufficiently stiff equation of state, $w_{(1)}\geq
\frac{1}{3}$, then the fluids will obtain a relative velocity w.r.t each other (in general when $w_{(1)}=\frac{1}{3}$ and always if $w_{(1)}>\frac{1}{3}$); this is an invariant statement, and it is not possible to eliminate this effect with any choice of reference congruence—if one has two fluids, one with a sufficiently soft and one with a sufficiently stiff equation of state, then it follows that the fluids will asymptotically form anisotropies on a de Sitter background irrespectively of the choice of reference congruence. We note that this result is compatible with the analysis of Bianchi type V in [@golnil00], and that it reflects a bifurcation that takes place at $w=\frac{1}{3}$ for a fluid in any, homogeneous or inhomogeneous, forever expanding model with a cosmological constant, see [@limetal04].
At early times $\Lambda$ has a negligible effect compared to normal matter and hence it suffices to study the $\Omega_\Lambda=0$ subset (it follows from and the application of the monotonicity principle, see e.g. [@waiell97; @heiugg06] and references therein,[^7] that the $\alpha$-limit for all orbits (solutions) must reside on this subset (assuming that $\Omega_\Lambda\neq 1$ initially); cf. also the discussion after equation ). Since $\Lambda$ therefore has no effect on the past asymptotic dynamics and since it is of interest to also study late time behavior when one does not have a cosmological constant, we will from now on assume $\Lambda=0$. The state space we henceforth therefore consider is given by |[**S**]{} = { \_+,\_A,\_B,\_C,v\_[(1)]{},v\_[(2)]{}, \_[(1)]{},\_[(2)]{}}, subject to the constraints , i.e., the state space when one does not have a cosmological constant is six-dimensional; since the discrete symmetries still hold we continue to assume that $\Sigma_A \in [0,1], \,
\Sigma_C \in [0,1], \, v_{(1)} \in [0,1]$, and $v_{(2)} \in
[-1,0]$. When $\Lambda=0$ the deceleration parameter $q$ is given by q = 2\^2 + (\_[m]{} + 3P\_[m]{}) = 2 - (\_[m]{} - P\_[m]{}) q 2.
Invariant subsets
-----------------
The dynamical system , , with $\Omega_\Lambda=0$, admits a number of invariant subsets, conveniently divided into three classes: (i) ‘geometric subsets’, i.e., sets associated with conditions on the shear and hence the metric since the type I models are intrinsically flat; (ii) invariant sets on the boundary of the physical state space for two tilted fluids that do not belong to (i); (iii) subsets that can be obtained by intersections of the subsets belonging to (i) and (ii). We will introduce a notation where the kernel suggests the type of subset and where a subscript, when existent, suggests the values of $v_{(1)}$ and $v_{(2)}$.
[*Geometric subsets*]{}
- ${\cal TW}$: The ‘twisting’ subset, characterized by $\Sigma_C=0,\, \Sigma_{A}\neq 0$, which leads to that the decoupled $\phi$-variable satisfies $\phi=const$ and hence $\Sigma_{12} \propto\Sigma_{11}-\Sigma_{22}$.
- ${\cal RD}$: The constantly rotated diagonal subset, given by $\Sigma_A=0,\,\Sigma_C\neq 0$ ($R_\alpha=0$). This subset is the diagonal subset, discussed next, rotated with a constant angle around $\vece_3$.
- ${\cal D}$: The diagonal subset, defined by $\Sigma_A=\Sigma_C=0;\, \Sigma_B = \Sigma_-$, and hence $R_\alpha=0$.
- ${\cal LRS}$: The locally rotationally symmetric subset. This plane symmetric subset of the diagonal subset is characterized by the additional condition $\Sigma_B=\Sigma_-=0$. This is the simplest subset compatible with two tilted fluids.
- ${\cal FLO}$, ${\cal FLT}_{0v_{(2)}}$and ${\cal
FLT}_{v_{(1)}0}$: The demand that $\Sigma^2=0$, and hence $\Omega_{\rm m}=1$, holds for all times enforces either that $v_{(1)}=v_{(2)}=0$, which defines the orthogonal Friedmann-Lemaître subset ${\cal FLO}$, or $v_{(1)}=\Omega_{(2)}=0,\, \Omega_{(1)}=1$ ($v_{(2)}=\Omega_{(1)}=0,\, \Omega_{(2)}=1$), which gives the ${\cal FLT}_{0v_{(2)}}$ (${\cal
FLT}_{v_{(1)}0}$) Friedmann-Lemaître subset with one orthogonal fluid and a test vector field $v_{(2)}$ ($v_{(1)}$); these subsets belong to the boundary of the two tilted fluid case and thus there exists no Friedmann-Lemaître subset with two tilted fluids.
[*Boundary subsets*]{}
- ${\cal O}$: The orthogonal subset for which $v_{(1)}=v_{(2)}=0$. In general this subset is expressed in a non-Fermi frame for which $\Sigma_A,\Sigma_C\neq 0$, however, usually when dealing with this case one makes a rotation to a Fermi frame in which the shear and the metric are diagonal so that ${\cal O}$ belongs to ${\cal D}$.
- ${\cal OT}_{v_{(1)}0}$ and ${\cal OT}_{0v_{(2)}}$: The ${\cal OT}_{v_{(1)}0}$ subset describes a single orthogonal fluid, $\Omega_{(2)} \geq 0,\ v_{(2)}=0$, and a test vector field $v_{(1)}$ ($\Omega_{(1)}=0$), and similarly for ${\cal OT}_{0v_{(2)}}$.
- ${\cal ET}_{1v_{(2)}}$ and ${\cal ET}_{v_{(1)}1}$: The subset ${\cal ET}_{1v_{(2)}}$ describes a fluid that consists of particles with zero rest mass moving with the speed of light $v_{(1)}=1 \,\Rightarrow \,
Q_{(1)}=\Omega_{(1)} = 3P_{(1)}$, as follows from the Codazzi constraint ; similar statements hold for ${\cal ET}_{v_{(1)}1}$.
- ${\cal K}$: The vacuum subset is called the Kasner subset and is defined by $\Omega_{\rm m}=0;
\,\Sigma^2=1$; it describes the Kasner solutions, but in general in a non-Fermi propagated frame, and with $v_{(i)}$ as test fields.
The lists above are far from complete; intersections of subsets are possible in many cases, which then form invariant subsets of lower dimension; an important example is:
- ${\cal ET}_{11}={\cal ET}_{1v_{(2)}} \cap {\cal
ET}_{v_{(1)}1}$: The double extreme tilt subset where both fluids propagate with the speed of light, $v_{(1)}=1=-v_{(2)}\,\Rightarrow\, \Omega_{(1)} =
\Omega_{(2)}=3P_{(1)}=3P_{(2)}$, and hence $\rho_{(1)}=\rho_{(2)}=3p_{(1)}=3p_{(2)}$.
There are also a number of fix points which we denote by a kernel that is related to a subset to which the fix point belong together with a subscript that indicates the fix point values of $v_{(1)}$ and $v_{(2)}$; sometimes we also use a superscript. The fix points and their stability properties are given in Appendix \[Sec:locstab\]; here we give a brief summary:
- There are a number of Kasner points, all satisfying $\Omega_m=0,\, \Sigma^2=1,\, q=2$. The four Kasner circles: ${\rm K}^{\ocircle}_{00},\, {\rm
K}^{\ocircle}_{10},\, {\rm K}^{\ocircle}_{01},\, {\rm
K}^{\ocircle}_{11}$, and the eight Kasner lines: ${\rm
KL}_{v_{(1)}0}^\pm,\, {\rm KL}_{v_{(1)}1}^\pm,\, {\rm
KL}_{0v_{(2)}}^\pm,\, {\rm KL}_{1v_{(2)}}^\pm$, where the superscript denotes the sign of $\Sigma_B$.
- There also are a number of Friedmann points with $\Sigma^2=0$. The four Friedmann points: ${\rm
F}^{10}_{00},\, {\rm F}^{10}_{01}$, for which $q=\frac{1}{2}(1+3w_{(1)})$, and ${\rm
F}^{01}_{00},\,{\rm F}^{01}_{10}$, for which $q=\frac{1}{2}(1+3w_{(2)})$. When $w_{(2)}=\frac{1}{3}$ there is a line of fix points, ${\rm
FL}^{10}_{0v_{(2)}}$, that connects ${\rm F}^{10}_{00}$ with ${\rm F}^{10}_{01}$, with $q=\frac{1}{2}(1+3w_{(1)})$, and similarly when $w_{(1)}=\frac{1}{3}$ then ${\rm FL}^{01}_{v_{(1)}0}$ connects ${\rm F}^{01}_{00}$ with ${\rm F}^{01}_{10}$, with $q=\frac{1}{2}(1+3w_{(2)})$. The superscript denotes the values of $\Omega_{(1)}$ and $\Omega_{(2)}$.
- When $\frac{1}{3}<w_{(2)}<w_{(1)}$ there exists two fix points: ${\rm LRS}_{v_{(1)}^*1}$ with $\Sigma^2=\frac{1}{4}(3w_{(1)}-1)^2,\,q=\frac{1}{2}(1+3w_{(1)})$, and ${\rm LRS}_{1v_{(2)}^*}$ with $\Sigma^2=\frac{1}{4}(3w_{(2)}-1)^2,\,q=\frac{1}{2}(1+3w_{(2)})$.
- On the twisting subset there exists the extremely tilted fix point ${\rm TW}_{11}$, for which $\Sigma^2=\frac{2}{5},\,q=\frac{7}{5}$; ${\rm
TW}_{v_{(1)}^*1}$, which exists when $\frac{1}{2}<w_{(1)}<\frac{3}{5}$ with $\Sigma^2=\frac{1}{4}(3w_{(1)}-1)(15w_{(1)}-7),\,
q=\frac{1}{2}(1+3w_{(1)})$; ${\rm TW}_{1v_{(2)}^*}$, which exists when $\frac{1}{2}<w_{(2)}<\frac{3}{5}$ with $\Sigma^2=\frac{1}{4}(3w_{(2)}-1)(15w_{(2)}-7),\,
q=\frac{1}{2}(1+3w_{(2)})$.
- Finally there exists the extremely tilted fix point ${\rm G}_{11}$ with $\Sigma^2=\frac{1}{3},\,q=\frac{4}{3}$; the line of fix points ${\rm GL}_{v_{(1)}^*1}$ exists when $w_{(1)}=\frac{5}{9}$ with $\Sigma^2=\frac{7}{3}v_{(1)}/(3+4v_{(1)}),\,q=\frac{4}{3}$, while the line of fix points ${\rm GL}_{1v_{(2)}^*}$ exists when $w_{(2)}=\frac{5}{9}$ with $\Sigma^2=\frac{7}{3}|v_{(2)}|/(3+4|v_{(2)}|),\,q=\frac{4}{3}$.
Monotone functions and their consequences {#Sec:mon}
=========================================
In the analysis of Bianchi type VI$_0$ in [@colher04] a monotone function is defined:
$$\begin{aligned}
\chi &= \frac{\beta_{(2)}\Omega_{(2)} -
\beta_{(1)}\Omega_{(1)}}{\beta_{(2)}\Omega_{(2)} +
\beta_{(1)}\Omega_{(1)}} \qquad \text{where} \qquad \beta_{(i)}
= (G_+^{(i)})^{-1} (1-v_{(i)}^2)^{\frac{1}{2}(1-w_{(i)})}\:,\\
\chi^\prime &= {{\textstyle{3\over2}}}(w_{(1)} - w_{(2)})(1-\chi^2)\:,
\qquad\qquad\qquad -1\leq \chi \leq 1\:.\lb{chi}\end{aligned}$$
The above holds whether or not we include a cosmological constant. If $w_{(1)} = w_{(2)}$, then $\chi$ is a constant of the motion, however, here our concern is with the case $w_{(1)}
> w_{(2)}$, and then $\chi$ is a monotonic function that increases from $-1$ to $1$, which leads to: $$\begin{aligned}
\lim_{\tau\rightarrow -\infty}\, \chi &= -1\quad \Rightarrow\quad
\lim_{\tau\rightarrow -\infty}\,
(\beta_{(2)}\Omega_{(2)}/\beta_{(1)}\Omega_{(1)})=0\quad
\Rightarrow\quad \text{at early times\/}\quad \beta_{(2)}\Omega_{(2)}
\rightarrow 0\:,\label{chicondpast}\\
\lim_{\tau\rightarrow \infty}\, \chi &= 1\quad\,\,\,\, \Rightarrow\quad
\lim_{\tau\rightarrow \infty}\,\,\,\,\:
(\beta_{(1)}\Omega_{(1)}/\beta_{(2)}\Omega_{(2)})=0\quad
\Rightarrow \quad \text{at late times\/}\quad\,\,\, \beta_{(1)}\Omega_{(1)}
\rightarrow 0\:.\label{chicondfuture}\end{aligned}$$ Combined with the Codazzi constraint this leads to the following possibilities if $\tau\rightarrow
-\infty$:
- $\lim_{\tau\rightarrow
-\infty}(\Omega_{(1)},\Omega_{(2)})=(0,0)$, i.e., the solutions $\alpha$-limits reside on ${\cal K}$,
- $\lim_{\tau\rightarrow
-\infty}(\Omega_{(2)},v_1)=(0,0)$, i.e., the solutions $\alpha$-limits reside on ${\cal OT}_{0v_{(2)}}$,
- $\lim_{\tau\rightarrow -\infty}v_{(2)}= -1$, $\lim_{\tau\rightarrow -\infty}Q_{(1)}=\Omega_{(2)}$, i.e., the solutions $\alpha$-limits reside on ${\cal
ET}_{v_{(1)}1}$,
or combinations/intersections thereof. If $\tau\rightarrow
\infty$ then:
- $\lim_{\tau\rightarrow
\infty}(\Omega_{(1)},\Omega_{(2)})=(0,0)$, i.e., the solutions $\omega$-limits reside on ${\cal K}$,
- $\lim_{\tau\rightarrow
\infty}(\Omega_{(1)},v_2)=(0,0)$, i.e., the solutions $\omega$-limits reside on ${\cal OT}_{v_{(1)}0}$,
- $\lim_{\tau\rightarrow \infty}v_{(1)}= 1$, $\lim_{\tau\rightarrow \infty}Q_{(1)}=\Omega_{(1)}$, i.e., the solutions $\omega$-limits reside on ${\cal
ET}_{1v_{(2)}}$,
or combinations/intersections thereof.
Another monotonic function is given by V = v\_[(1)]{}\^2(1-v\_[(2)]{}\^2)\^[1-w\_[(2)]{}]{}v\_[(2)]{}\^[-2]{}(1-v\_[(1)]{}\^2)\^[-(1-w\_[(1)]{})]{};V\^= 6(w\_[(1)]{} - w\_[(2)]{})V, where $V$ asymptotically increases from zero to infinity. Combining $V$ with $\chi$ to obtain a constant of the motion leads to $Q_{(1)}/Q_{(2)}=const=-1$, where the latter equality is imposed by the Codazzi constraint, so unfortunately we obtain nothing new. However, it follows that when $\tau\rightarrow -\infty$ ($\tau\rightarrow\infty$) then $v_{(1)}\rightarrow 0$ or/and $v_{(2)}\rightarrow -1$ ($v_{(1)}\rightarrow 1$ or/and $v_{(2)}\rightarrow 0$), i.e., these limits also hold in the above ${\cal K}$ cases.
Before giving the next monotonic functions it is useful to give the following auxiliary equations:
\[auxm\] $$\begin{aligned}
Q_{(i)}^\prime &= 2(q-1+\Sigma_+)Q_{(i)}\:,\qquad
T_{(i)}^\prime = 2(2q-1-3w_{(i)})T_{(i)}\:, \qquad
\text{where}\lb{QTi}\\
T_{(i)} &=
Q_{(i)}^2(1-v_{(i)}^2)^{1-w_{(i)}}v_{(i)}^{-2}=
(1+w_{(i)})^2(G_+^{(i)})^{-2}\Omega_{(i)}^2(1-v_{(i)}^2)^{1-w_{(i)}}\:.\lb{Ti}\end{aligned}$$
If $\Sigma_A,\Sigma_C\neq 0$ there exist two more monotonic functions: M\_[AC]{}\^[(i)]{} = Q\_[(i)]{}\^[-12]{}T\_[(i)]{}\^9\_A\^[-8]{}\_C\^[-4]{} = Q\_[(i)]{}\^6(1-v\_[(i)]{}\^2)\^[9(1-w\_[(i)]{})]{} v\_[(i)]{}\^[-18]{}\_A\^[-8]{}\_C\^[-4]{};(M\_[AC]{}\^[(i)]{})\^= 6(5-9w\_[(i)]{})M\_[AC]{}\^[(i)]{}, where $M_{AC}^{(i)}$ asymptotically increases from zero to infinity if $w_{i}<\frac{5}{9}$ while it decreases from infinity to zero if $w_{i}>\frac{5}{9}$; at $w_{i}=\frac{5}{9}$ $M_{AC}^{(i)}$ is a constant of the motion, reflecting that we have bifurcations when $w_{i}=\frac{5}{9}$, see Appendix \[Sec:locstab\]. The above four monotonic functions, $\chi,\,V=T_{(2)}/T_{(1)},\,M_{AC}^{(1)},\,M_{AC}^{(2)}$, can be combined to yield three constants of the motion, but one of these is just the Codazzi constraint, so there only are two independent ‘non-trivial’ constants of the motion; here are two possible representations of these constants of the motion: C\_[AB]{}= (M\_[AC]{}\^[(1)]{})\^[5-9w\_[(2)]{}]{}(M\_[AC]{}\^[(2)]{})\^[9w\_[(1)]{}-5]{}= const,D\_[AB]{}= V\^[9w\_[(1)]{}-5]{}(M\_[AC]{}\^[(1)]{})\^[w\_[(1)]{} - w\_[(2)]{}]{}=const.In addition to these monotonic functions there also exist several monotonic functions on the various subsets.
The existence of monotone functions is not coincidental, a fact that will be discussed elsewhere, but let us here comment on $\chi$, which is a monotonic function for all Class A models (i.e., Bianchi models for which $A_\alpha=0$, see section \[Sec:dynsysder\] and e.g. [@waiell97]). Its existence is a consequence of that $\chi$ is expressible as a dimensionless ratio of the spatial volume density and the dimensional constants $\ell_{(i)}$ in class A, where $\ell_{(i)}$ is related to particle conservation of the $i$:th fluid, see e.g. [@jan01]. Interestingly there exists one more constant of the motion for each fluid in Class A, however, these constants of the motion, together with the constants $\ell_{(i)}$, only lead to an integral related to the Codazzi constraint , which therefore, unfortunately, is of no use. Incidentally, other constants of the motion exist in class B and hence, based on the above insight, there should exist a monotonic function also in this case, again related to particle conservation, but in a more complicated way.
The ${\cal K}$ subset
---------------------
Before continuing it is useful to discuss the Kasner subset ${\cal K}$. The state space of ${\cal K}$ is given by ${\bf K}
= \{\Sigma_+,\Sigma_A,\Sigma_B,\Sigma_C,v_{(1)},v_{(2)}\}$ subjected to the Gauss constraint $\Sigma^2=\Sigma_+^2+\Sigma_A^2+\Sigma_B^2+\Sigma_C^2=1$. The equations for the test fields $v_{(1)}\in [0,1]$, $v_{(2)}\in
[-1,0]$ decouple from those of the shear and from each other. The state space therefore can be written as the following Cartesian product: = [**KP**]{}{v\_[(1)]{}}{v\_[(2)]{}} , = {\_+,\_A,\_B,\_C}, where ${\bf KP}$ is the projected Kasner state space, which of course is subjected to $\Sigma^2=1$. By determining the $\alpha$- and $\omega$-limits for solutions on ${\bf KP}$ one can then determine the asymptotic states of $v_{(1)}$ and $v_{(2)}$ separately, and thus the $\alpha$- and $\omega$-limits for solutions on ${\cal K}$. Let us therefore first turn to the equations on ${\bf KP}$: \[KP\] \_+\^= 3\_A\^2;\_A\^= -(3\_+ + \_B) \_A ;\_B\^= \_A\^2 - 2\_C\^2;\_C\^= 2\_B\_C . This system admits a circle of fix points, the projected Kasner circle: ${\rm KP}^\ocircle$, see Figure \[Ksectors\]. It is described by $\Sigma_A=\Sigma_C=0$, $\Sigma_+=\hat{\Sigma}_+,\Sigma_B=\hat{\Sigma}_-$, where the constants $\hat{\Sigma}_\pm$ satisfy $\hat{\Sigma}_+^2 +
\hat{\Sigma}_-^2=1$.
The subset $\Sigma_A=0$ yields that $\Sigma_+=\hat{\Sigma}_+,\,\Sigma_B^2+\Sigma_C^2=\hat{\Sigma}_-^2$, where $\Sigma_B$ is monotonically decreasing. The subset $\Sigma_C=0$ leads to $\Sigma_+ -
\sqrt{3}\Sigma_B=\hat{\Sigma}_+ - \sqrt{3}\hat{\Sigma}_-$ while $\Sigma_+$ and $\Sigma_B$ are monotonically increasing and $\Sigma_A^2=1-\Sigma_+^2-\Sigma_B^2$. Projected onto the $\Sigma_+-\Sigma_B$-plane this yields the straight lines—single frame transitions, using the nomenclature of [@heietal07], given in Figures \[KtransC\] and \[KtransA\] (a frame transition preserves a Kasner state while permuting the spatial axes). As discussed in [@heietal07], the general case can be regarded as multiple frame transitions that yield the same result as combinations of single transitions which therefore determine the general asymptotic solution structure on ${\cal KP}$, for details see [@heietal07]. From this we conclude that the the $\alpha$-limits for all solutions with $\Sigma_A\Sigma_C\neq 0$ on ${\cal KP}$ resides on the segment ${\rm KP}^\ocircle$, yielding a segment on ${\rm KP}^\ocircle$ characterized by $-1\leq\Sigma_+=\hat{\Sigma}_+\leq
-\frac{1}{2},\, 0\leq \hat{\Sigma}_-\leq\frac{\sqrt{3}}{2}$, i.e., the segment consists of sector $(213)$ together with the fix points ${\rm Q}_2$ and ${\rm T}_3$ on ${\rm KP}^\ocircle$, see Figure \[Kasnerattractor\].
\[cc\]\[cc\][$\Sigma_+$]{} \[cc\]\[cc\][$\Sigma_3$]{} \[cc\]\[cc\][$\Sigma_B$]{} \[cc\]\[cc\][$(321)$]{} \[cc\]\[cc\][$(231)$]{} \[cc\]\[cc\][$(213)$]{} \[cc\]\[cc\][$(123)$]{} \[cc\]\[cc\][$(132)$]{} \[cc\]\[cc\][$(312)$]{} \[cc\]\[cc\][0]{} \[cc\]\[cc\][${\rm T}_3$]{}\[cc\]\[cc\][${\rm T}_2$]{} \[cc\]\[cc\][${\rm T}_1$]{} \[cc\]\[cc\][${\rm Q}_3$]{} \[cc\]\[cc\][${\rm Q}_2$]{}\[cc\]\[cc\][${\rm Q}_1$]{}
The $\alpha$-limits for solutions on ${\cal K}$ are determined by the $\alpha$-limits on ${\cal KP}$ which determine the asymptotic limits for $v_{(i)}$. The equation for $|v_{(i)}|\in[0,1]$ on ${\rm KP}^\ocircle$ is given by: $|v_{(i)}|^\prime = (G^{(i)}_-)^{-1} (1-v_{(i)}^2)(3w_{(i)} - 1
+ 2\hat{\Sigma}_+) \,|v_{(i)}|$. It follows that the $\alpha$-limits for all orbits on ${\cal K}$ on the general geometric set with $\Sigma_A\Sigma_C\neq 0$ resides on the global past attractor ${\cal A}_{\{**\}}$, where the subscript denotes the range of values of $w_{(1)}$ and $w_{(2)}$, given by
\[Kattr\] $$\begin{aligned}
{\cal A}_{\{w_{(2)}<w_{(1)}<{{\textstyle{2\over3}}}\}} &=
\{{\rm K}^\ocircle_{11}: \hat{\Sigma}_+\in [-1,-{{\textstyle{1\over2}}}]\}\:,\\
{\cal A}_{\{w_{(2)}<w_{(1)}={{\textstyle{2\over3}}}\}} &=
\{{\rm K}^\ocircle_{11}: \hat{\Sigma}_+\in [-1,-{{\textstyle{1\over2}}})\}\cup
\{{\rm KL}^+_{v_{(1)}1}: \hat{\Sigma}_+=-{{\textstyle{1\over2}}})\}\:,\\
{\cal A}_{\{w_{(2)}<{{\textstyle{2\over3}}}<w_{(1)}\}} &=
\{{\rm K}^\ocircle_{11}: \hat{\Sigma}_+\in [-1,-{{\textstyle{1\over2}}}(3w_{(1)}-1)\}\cup
\{{\rm KL}^+_{v_{(1)}1}: \hat{\Sigma}_+=-{{\textstyle{1\over2}}}(3w_{(1)}-1)\}\cup\nonumber\\
& \quad\,\, \{{\rm K}^\ocircle_{01}:(-{{\textstyle{1\over2}}}(3w_{(1)}-1),-{{\textstyle{1\over2}}}]\}\:,\\
{\cal A}_{\{{{\textstyle{2\over3}}}=w_{(2)}<w_{(1)}\}} &=
\{{\rm K}^\ocircle_{11}: \hat{\Sigma}_+\in [-1,-{{\textstyle{1\over2}}}(3w_{(1)}-1)\}\cup
\{{\rm KL}^+_{v_{(1)}1}: \hat{\Sigma}_+=-{{\textstyle{1\over2}}}(3w_{(1)}-1)\}\cup\nonumber\\
& \quad\,\, \{{\rm K}^\ocircle_{01}:(-{{\textstyle{1\over2}}}(3w_{(1)}-1), -{{\textstyle{1\over2}}})\}
\cup \{{\rm KL}^+_{0v_{(2)}}: \hat{\Sigma}_+= -{{\textstyle{1\over2}}}\}\:,\\
{\cal A}_{\{{{\textstyle{2\over3}}}<w_{(2)}<w_{(1)}\}} &=
\{{\rm K}^\ocircle_{11}: \hat{\Sigma}_+\in [-1,-{{\textstyle{1\over2}}}(3w_{(1)}-1)\}\cup
\{{\rm KL}^+_{v_{(1)}1}: \hat{\Sigma}_+=-{{\textstyle{1\over2}}}(3w_{(1)}-1)\}\cup\nonumber\\
& \quad\,\, \{{\rm K}^\ocircle_{01}:(-{{\textstyle{1\over2}}}(3w_{(1)}-1),-{{\textstyle{1\over2}}}(3w_{(2)}-1))\}
\cup \nonumber\\
& \quad\,\,
\{{\rm KL}^+_{0v_{(2)}}: \hat{\Sigma}_+= -{{\textstyle{1\over2}}}(3w_{(2)}-1)\}\cup
\{{\rm K}^\ocircle_{00}: \hat{\Sigma}_+\in(-{{\textstyle{1\over2}}}(3w_{(2)}-1),-{{\textstyle{1\over2}}}]\}\:.\end{aligned}$$
As toward the past, the results in [@heietal07] implies that all orbits on ${\cal KP}$, on the generic geometric set as well as all the Kasner compatible geometric subsets, asymptotically also approach ${\rm KP}^\ocircle$ toward the future. From this it easily follows from the decoupled $v_{(i)}$ equations that the $\omega$-limit for any orbit on ${\cal K}$ is one of the Kasner fix points on ${\rm
K}^\ocircle_{**},\, {\rm KL}^\pm_{**}$. But according to the local stability analysis in Appendix \[Sec:locstab\] all fix points on ${\cal K}$ are destabilized toward the future by the matter degrees of freedom in the full state space, leading to that the $\omega$-limit points on ${\cal K}$ become saddles in the full state space such that no matter solutions with $Q_{(1)}> 0,\, v_{(1)}^2<1,\,v_{(2)}^2<1$ initially are attracted to any part of ${\cal K}$ when $\tau\rightarrow
\infty$, and thus the $\omega$-limits for all ‘interior’ matter solutions either resides on ${\cal OT}_{v_{(1)}0}$ or ${\cal
ET}_{1v_{(2)}}$ such that $q<2$ when $\tau\rightarrow\infty$, since $q=2$ only on ${\cal K}$.
Future and past dynamics {#Sec:attractor}
========================
Future dynamics {#futattr}
---------------
The following theorem is easy to prove, but is nevertheless of interest.
\[noniso\] If $\frac{1}{3}<w_{(2)}<w_{(1)}<1$, and if $Q_{(1)}> 0,\,
v_{(1)}^2<1,\,v_{(2)}^2<1$ initially, then no models isotropize when $\tau\rightarrow\infty$, i.e., $\Sigma^2\neq 0$ when $\tau\rightarrow\infty$.
Assume that all solutions of the above type isotropize, i.e. that the $\omega$-limit set for each solution resides on a Friedmann-Lemaître subset. The equations for $v_{(1)}$ and $v_{(2)}$ then yield $(v_{(1)},v_{(2)})\rightarrow (1,-1)$ when $\tau\rightarrow\infty$, which is a contradiction since no Friedmann-Lemaître subset has $v_{(1)}v_{(2)}\neq 0$. Hence none of the solutions described in theorem \[noniso\] isotropize when $\tau\rightarrow\infty$. $\Box$
The above theorem does not tell us where the solutions end up when $\frac{1}{3}<w_{(2)}<w_{(1)}<1$. This turns out to depend on what geometric set they belong to, leading to a division of the models into three classes: (i) The ${\cal RD}$, ${\cal D}$, ${\cal LRS}$ subsets (ii) the ${\cal TW}$ subset, and, (iii) the general case. Unfortunately we have not been able to prove what the global attractors are, but our local analysis in Appendix \[Sec:locstab\] together with numerical simulations lead to the following conjectures:
\[nonisoid\] If $Q_{(1)}> 0,\, v_{(1)}^2<1,\,v_{(2)}^2<1$ initially, then the $\omega$-limit for all orbits that belong to the geometric subsets ${\cal RD}$, ${\cal D}$, and ${\cal LRS}$ is the fix point ${\rm LRS}_{1v_{(2)}^*}$ if $\frac{1}{3}<w_{(2)}<w_{(1)}<1$.
\[nonisoit\] If $Q_{(1)}> 0,\, v_{(1)}^2<1,\,v_{(2)}^2<1$ initially, then the $\omega$-limit for all orbits that belong to the geometric subset ${\cal TW}$ ($\Sigma_A\neq 0$) is the fix point ${\rm
LRS}_{1v_{(2)}^*}$ if $\frac{1}{3}<w_{(2)}\leq \frac{1}{2}$ and $w_{(2)}<w_{(1)}<1$; the fix point ${\rm TW}_{1v_{(2)}^*}$ if $\frac{1}{2}<w_{(2)}<\frac{3}{5}$ and $w_{(2)}<w_{(1)}<1$; the fix point ${\rm TW}_{11}$ if $\frac{3}{5}\leq
w_{(2)}<w_{(1)}<1$.
\[nonisoig\] If $Q_{(1)}> 0,\, v_{(1)}^2<1,\,v_{(2)}^2<1$ initially, then the $\omega$-limit for all orbits that belong to the general geometric set ($\Sigma_A,\Sigma_C\neq 0$) is the fix point ${\rm LRS}_{1v_{(2)}^*}$ if $\frac{1}{3}<w_{(2)}\leq
\frac{1}{2}$ and $w_{(2)}<w_{(1)}<1$; the fix point ${\rm
TW}_{1v_{(2)}^*}$ if $\frac{1}{2}<w_{2}<\frac{5}{9}$ and $w_{(2)}<w_{1}<1$; the line of fix points ${\rm GL}_{1v_{(2)}}$ if $\frac{5}{9}=w_{(2)}<w_{(1)}<1$; the fix point ${\rm
G}_{11}$ if $\frac{5}{9}<w_{(2)}<w_{(1)}<1$.
However, models for which $Q_{(1)}> 0,\,
v_{(1)}^2<1,\,v_{(2)}^2<1$ initially and with $0\leq
w_{(2)}\leq \frac{1}{3}$ [*do*]{} isotropize (this is also true if $Q_{(1)}=0$, even if the equations of state are stiffer than radiation), as shown in the following lemma:
If $Q_{(1)}> 0,\, v_{(1)}^2<1,\,v_{(2)}^2<1$ initially, and if $0\leq w_{(2)}\leq \frac{1}{3}$, then all models isotropize when $\tau\rightarrow\infty$, i.e., $\Sigma^2\rightarrow 0$ when $\tau\rightarrow\infty$.
In section \[Sec:mon\] we showed that the future $\omega$-limit of a ‘matter’ orbit has to reside on either ${\cal OT}_{v_{(1)}0}$ or ${\cal ET}_{1v_{(2)}}$ with $q<2$ and $\Omega_{\rm m}>0$. Assume that the $\omega$-limit of an orbit resides on ${\cal OT}_{v_{(1)}0}$. The equations for the $\Sigma,\Omega_{(2)}$-variables on this subsets are just those for a single orthogonal fluid, but in general in a non-Fermi propagated frame. However, in a Fermi frame the single orthogonal fluid case is easily solved and one finds that $\Omega_{(2)}\rightarrow 1$ and $\Sigma^2\rightarrow 0$ when $\tau\rightarrow\infty$. This statement is frame invariant and therefore holds for any frame, and hence it follows that the $\omega$-limit resides on the Friedmann-Lemaître subset ${\cal FLT}_{v_{(1)}0}$ and that $\Sigma^2\rightarrow 0$. Let us now assume that the $\omega$-limit for a matter orbit resides on ${\cal ET}_{1v_{(2)}}$. Then, since $v_{(1)}=1$, $q=2 -\Omega_{(1)} -
\frac{3}{2}(\Omega_{(2)}-P_{(2)})=2-\Omega_{\rm m} -
\frac{1}{2}(1-3w_{(2)})(\Omega_{(2)} - Q_{(2)}v_{(2)})$, and hence $2q-1-3w_{(2)}= 2(1-\Omega_{\rm m})+
(1-3w_{(2)})(1-\Omega_{(2)} + Q_{(2)}v_{(2)})\geq 0$, since $w_{(2)}\leq\frac{1}{3}$, where the inequality is strict if $\Omega_{\rm m}<1$, which we now assume. Then $T_{(2)}$ in is strictly monotonically increasing and grows without bounds, but this is impossible since $T_{(2)}$ is finite, and hence $\Omega_{\rm m}\rightarrow 1$, and thus $\Sigma^2\rightarrow 0$ when $\tau\rightarrow \infty$.$\Box$
\[iso\] If $Q_{(1)}> 0,\, v_{(1)}^2<1,\,v_{(2)}^2<1$ initially, then the $\omega$-limit for all orbits is the fix point ${\rm
F}_{00}^{01}$ if $0\leq w_{(2)}<w_{(1)}<\frac{1}{3}$; one of the fix points on the line ${\rm FL}_{v_{(1)}0}^{01}$ if $0\leq
w_{(2)}<w_{(1)}=\frac{1}{3}$; the fix point ${\rm F}_{10}^{01}$ if $0\leq w_{(2)}\leq \frac{1}{3}<w_{(1)}<1$.
According to Lemma \[liso\] $\Sigma^2=0$ asymptotically toward the future. Imposing this condition on the ${\cal ET}_{1v_{(2)}}$ subset yields the ${\cal
FLT}_{v_{(1)}0}$ subset with $v_{(1)}=1$, which is a special case of the other possibility that the $\omega$-limit of an arbitrary orbit with $Q_{(1)}\neq 0$ initially resides on the ${\cal OT}_{v_{(1)}0}$ subset, and hence that the $\omega$-limit resides on ${\cal FLT}_{v_{(1)}0}$ with $v_{(1)}$ so far undetermined ($\Omega_{(2)}=1$). To find the desired $\omega$-limit we only need to find the asymptotic limit of $v_{(1)}$, which, according to , is determined by the signature of $3w_{(1)}-1$ when $\Sigma^2=0$, immediately leading to the theorem.
The above theorems and conjectures are summarized in the global attractor bifurcation diagrams in figure \[bifur\].
\[cc\]\[cc\][$0$]{} \[cc\]\[cc\][$\frac{1}{3}$]{} \[cc\]\[cc\][$\frac{1}{2}$]{} \[cc\]\[cc\][$\frac{5}{9}$]{} \[cc\]\[cc\][$1$]{} \[cc\]\[cc\][$w_{(1)}$]{} \[cc\]\[cc\][$w_{(2)}$]{} \[cc\]\[cc\][${\rm F}^{01}_{00}$]{} \[cc\]\[cc\][${\rm FL}^{01}_{v_{(1)}0}$]{} \[cc\]\[cc\][${\rm F}^{01}_{10}$]{} \[cc\]\[cc\][${\rm LRS}_{1v^{\ast}_{(2)}}$]{} \[cc\]\[cc\][${\rm TW}_{1v^{\ast}_{(2)}}$]{} \[cc\]\[cc\][${\rm GL}_{1v_{(2)}}$]{} \[cc\]\[cc\][${\rm G}_{11}$]{} \[cc\]\[cc\][${\rm TW}_{11}$]{} \[cc\]\[cc\][$\frac{3}{5}$]{}
Past dynamics {#pastattr}
-------------
Based on the local analysis in Appendix \[Sec:locstab\], the previous analysis of ${\cal K}$, and a numerical analysis, we make the following conjecture:
The $\alpha$-limit for each orbit with $Q_{(1)}> 0,\,
v_{(1)}^2<1,\,v_{(2)}^2<1$ initially on the general geometric set with $\Sigma_A\Sigma_C\neq 0$ is one of the fix points on the global past attractor ${\cal A}_{\{**\}}$ for the Kasner subset ${\cal K}$ given in equation .
For the various geometric subsets other parts of the projected Kasner circle are the relevant building blocks for producing the global attractor for each subset, in a similar way as in the generic case (e.g., in the ${\cal RD}$ case, with $\Sigma_A=0,\Sigma_C\neq 0$, $0\leq\hat{\Sigma}_-\leq 1$ is the restriction on $\hat{\Sigma}_-$, in contrast to the generic case where $0\leq \hat{\Sigma}_-\leq\frac{\sqrt{3}}{2}$).
Concluding remarks {#Sec:concl}
==================
In this paper we have shown that the type I models with two tilted fluids exhibit a rich bifurcation structure, hinting at the complexity one can expect from models with more realistic sources and more general geometries. Some of our results reflect features that hold under more general circumstances, while others are particular for the Bianchi type I models with two-non-interacting fluids, but in this latter instance the present models yield a natural reference with which to compare results from more general settings.
The asymptotically silent regimes of generic spacelike singularities and of an inflationary future share some properties: in the inflationary case all other matter fields than the inflationary field become test fields and do not influence the spacetime geometry—hence matter that is not inflationary matter does not matter for the spacetime geometry; in the case of a generic singularity fluids with speeds of sound less than that of light also become test fields, in this case gravity alone creates gravity to a larger extent than matter, and hence ‘matter does not matter’ in this case either [@bkl70], [@bkl82], [@uggetal03]. However, that matter fields asymptotically become test fields does not mean that they do not matter observationally, on the contrary, today to a good approximation the CMB can be regarded as a test field although it is the prime observational source for cosmology!
In the present case a cosmological constant has yielded a final de Sitter state—this is a typical feature in a forever expanding model, as is the bifurcation at the radiation value $w=\frac{1}{3}$. Hence if one has several test fields, some less stiff and some as stiff or stiffer than radiation, one obtains anisotropies on a de Sitter background. However, one would perhaps not expect fields that are stiffer than radiation after an inflationary period in the early universe or in the far future, but does the bifurcation at the radiation value hint at that e.g. atomic matter and/or cold dark matter and radiation develop observationally significant relative velocities, perhaps non-linearly? As regards generic singularities, the Bianchi type I Kasner singularity is transformed into a singularity of ‘Mixmaster’ type when one considers geometrically more general models that admit Bianchi type II models on the silent boundary in such a combination with possible frame transitions so that the whole projected Kasner circle becomes unstable toward the past. But it is by no means uninteresting to examine the past behavior of type I, since matter sometimes lead to bifurcations such that matter sometimes does matter, as illustrated by e.g., a magnetic field [@leb97], or by a kinematic description of matter [@heiugg06] where matter mattered non-generically in a very subtle way, illustrating that it was not quite obvious that there would not be any non-generic subtle effects in the present case; the lack of such effects suggest that the past dynamics in general is structurally stable under a change from one to several fluids as long as $0\leq w_{(i)}<1$.
Asymptotic scenarios where non-interacting matter components do not matter may have interesting consequences when one introduces more realistic interacting sources. If the interactions only contribute source terms that are proportional to the non-interacting parts of the source, then the interactions presumably also become negligible for the determination of the geometry; it is only when interactions contribute more to the total stress-energy than the sources themselves that the matter does not matter property would be broken. Hence the approximation of non-interacting fields may be asymptotically less restrictive than one may initially think.
There are no (quasi-) isotropic singularities, see e.g. [@khaetal03],[@limetal04] and references therein, in the present case when $\Sigma^2\neq 0$ initially. The reason for this is that the shear completely destabilizes such singularities in Bianchi type I, which therefore is extremely misleading in a (quasi-) isotropic singularity context.
The result that models with fluids stiffer than radiation asymptotically produce anisotropies toward the future is mathematically interesting, and shows that the isotropization results for a single fluid are structurally unstable within the Bianchi type I context, although from a physical point of view one would not expect such equations of states at late times. The result suggests that tilted fluids may become as anisotropically significant as spatial curvature at late times (in the absence of inflation) when one considers more general models than Bianchi type I, leading to considerable complexity, further illustrated by the type VI$_0$ investigation in [@colher04]. Our results about isotropization for soft equations of state may be regarded as a non-linear Bianchi type I generalization of perturbations of flat FRW models with two fluids, a reasonable approximation before dark energy has becomes significant, and it is of interest then to point out that one again has radiation bifurcations.
Fix points and local stability analysis {#Sec:locstab}
=======================================
In this section we use the Gauss constraint to eliminate $\Omega_{(2)}$ globally, however, the Codazzi constraint cannot, unfortunately, be analytically solved globally, but we can follow ch. 7 in [@waiell97] and use it to locally eliminate one variable, usually $\Omega_{(1)}$, at each fix point when is non-singular. There are several features that are similar for many of the fix points. All fix points, except one, have $\Sigma_C=0$; several fix points have $\Sigma_A=0$. Linearization of when $\Sigma_C=0$, and when $\Sigma_A=0$, yield the eigenvalues \_[\_C]{} = -\[2 - q\_0 - 2(\_B)\_0\],\_[\_A]{} = -\[2 - q\_0 + 3(\_+)\_0 + (\_B)\_0\], where $q_0,\, (\Sigma_B)_0,\, (\Sigma_+)_0$ are the fix point values of $q,\, \Sigma_B,\,\Sigma_+$, respectively. For many fix points $v_{(i)}=0$ or $|v_{(i)}|=1$. In these cases linearization of yields \_[v\_[(i)]{}]{}\^0 = 3w\_[(i)]{} - 1 + 2(\_+)\_0,\_[v\_[(i)]{}]{}\^1 = -2(3w\_[(i)]{} - 1 + 2(\_+)\_0)/(1-w\_[(i)]{}),where the subscript refers to the $v_{(i)}$ variable the eigenvalue is connected with while the superscript denotes its absolute fix point value. Let us now turn to the various individual fix points; throughout kernel subscripts give an indication of the absolute fix point values for $v_{(1)}$ and $v_{(2)}$.
[**Kasner fix points**]{}: There are four circles of Kasner points and eight lines of fix points when $0\leq
w_{(2)}< w_{(1)}$. The Kasner circles are characterized by $\Sigma_+ = \hat{\Sigma}_+,\, \Sigma_B = \hat{\Sigma}_-,\,
\Sigma_A = \Sigma_C = 0,\, \Omega_{(1)} = \Omega_{(2)} = 0$, where $\hat{\Sigma}_\pm$ are constants that satisfy $\hat{\Sigma}_+^2 + \hat{\Sigma}_-^2=1$, and the following values of $v_{(i)}$:
$$\begin{aligned}
{\rm K}^{\ocircle}_{00}:\quad v_{(1)} &= v_{(2)} =
0\:,\qquad\qquad\, {\rm K}^{\ocircle}_{10}:\,\, v_{(1)} = 1\:,\, v_{(2)}
= 0\:,\\
{\rm K}^{\ocircle}_{01}:\quad v_{(1)} &= 0\:,\,
v_{(2)} = -1\:,\qquad {\rm K}^{\ocircle}_{11}:\,\, v_{(1)} =
-v_{(2)} = 1\:.\end{aligned}$$
The eigenvalues for the four cases are:
\[Kasenrcirc\] $$\begin{aligned}
& {\rm K}^{\ocircle}_{00}:\quad 0\:;\quad \lambda_{\Sigma_A}\:;\quad
\lambda_{\Sigma_C}\:;\quad \lambda_{v_{(1)}}^0\:;\quad
\lambda_{v_{(2)}}^0\:;\quad 3(1-w_{(1)})\:; \quad 3(1-w_{(2)})\:,\\
& {\rm K}^{\ocircle}_{10}:\quad 0\:;\quad \lambda_{\Sigma_A}\:;\quad
\lambda_{\Sigma_C}\:;\quad \lambda_{v_{(1)}}^1\:;\quad
\lambda_{v_{(2)}}^0\:;\quad 3(1-w_{(2)})\:,\\
& {\rm K}^{\ocircle}_{01}:\quad 0\:;\quad \lambda_{\Sigma_A}\:;\quad
\lambda_{\Sigma_C}\:;\quad \lambda_{v_{(1)}}^0\:;\quad
\lambda_{v_{(2)}}^1\:;\quad 3(1-w_{(1)})\:,\\
& {\rm K}^{\ocircle}_{11}:\quad 0\:;\quad \lambda_{\Sigma_A}\:;\quad
\lambda_{\Sigma_C}\:;\quad \lambda_{v_{(1)}}^1\:;\quad
\lambda_{v_{(2)}}^1\:;\quad 2(1+\hat{\Sigma}_+)\:,\end{aligned}$$
where
$$\begin{aligned}
\lambda_{\Sigma_A} &=
-(3\hat{\Sigma}_+ + \sqrt{3}\hat{\Sigma}_-)\:, \qquad
\lambda_{\Sigma_C} = 2\sqrt{3}\hat{\Sigma}_-\:,\\
\lambda_{v_{(i)}}^0 &= 3w_{(i)} - 1 + 2\hat{\Sigma}_+\:,\qquad\,\,
\lambda_{v_{(i)}}^1 = -2(3w_{(i)} - 1
+ 2\hat{\Sigma}_+)/(1-w_{(i)})\:.\end{aligned}$$
In the ${\rm K}^{\ocircle}_{00}$ case the Codazzi constraint is singular and hence it cannot be locally solved; in all other cases has been used to eliminate $\Omega_{(1)}$. The zero eigenvalue corresponds to that one has a one-parameter set of fixed points. The eight lines of Kasner fix points are characterized by $\Sigma_A = \Sigma_C = 0,\, \Omega_{(1)} = \Omega_{(2)} =
0,\, \Sigma^2=1$, and
\[Kasnerlines\] $$\begin{aligned}
{\rm KL}_{v_{(1)}0}^\pm:\quad \Sigma_+ &= {{\textstyle{1\over2}}}(1-3w_{(1)})\:,\quad
\Sigma_B = \pm\sqrt{1-\Sigma_+^2}\:,\quad 0 \leq v_{(1)} \leq 1\:, \quad v_{(2)} = 0\:,\\
{\rm KL}_{v_{(1)}1}^\pm:\quad \Sigma_+ &= {{\textstyle{1\over2}}}(1-3w_{(1)})\:,\quad
\Sigma_B = \pm\sqrt{1-\Sigma_+^2}\:, \quad 0 \leq v_{(1)} \leq 1\:,\quad v_{(2)} = -1\:,\\
{\rm KL}_{0v_{(2)}}^\pm:\quad \Sigma_+ &= {{\textstyle{1\over2}}}(1-3w_{(2)})\:,\quad
\Sigma_B = \pm\sqrt{1-\Sigma_+^2}\:,\quad v_{(1)} = 0\:, \quad -1 \leq v_{(2)} \leq 0\:,\\
{\rm KL}_{1v_{(2)}}^\pm:\quad \Sigma_+ &= {{\textstyle{1\over2}}}(1-3w_{(2)})\:,\quad
\Sigma_B = \pm\sqrt{1-\Sigma_+^2}\:,\quad v_{(1)} = 1\:, \quad -1 \leq v_{(2)} \leq 0\:,\end{aligned}$$
where the superscript denotes the sign of $\Sigma_B$. After eliminating $\Omega_{(1)}$ locally by means of the Codazzi constraint , the eigenvalues for the eight Kasner lines are:
\[KL\] $$\begin{aligned}
&{\rm KL}_{v_{(1)}0}^\pm:\quad 0\:;\quad 0\:;\quad
\lambda_{\Sigma_A}\:;\quad \lambda_{\Sigma_C}\:;\quad
3(1-w_{(2)})\:;\quad -3(w_{(1)} - w_{(2)})\:,\\
&{\rm KL}_{v_{(1)}1}^\pm:\quad 0\:;\quad 0\:;\quad
\lambda_{\Sigma_A}\:;\quad \lambda_{\Sigma_C}\:;
\quad 3(1-w_{(1)})\:;\quad
6\frac{w_{(1)} - w_{(2)}}{1-w_{(2)}}\:,\\
&{\rm KL}_{0v_{(2)}}^\pm:\quad 0\:;\quad 0\:;\quad
\lambda_{\Sigma_A}\:;\quad \lambda_{\Sigma_C}\:;\quad
3(1-w_{(1)})\:; \quad 3(w_{(1)} - w_{(2)})\:,\\
&{\rm KL}_{1v_{(2)}}^\pm:\quad 0\:;\quad 0\:;\quad
\lambda_{\Sigma_A}\:;\quad \lambda_{\Sigma_C}\:;\quad
3(1-w_{(2)})\:; \quad -6\frac{w_{(1)} - w_{(2)}}{1-w_{(1)}}\:,\end{aligned}$$
where again $\lambda_{\Sigma_A} = -(3\Sigma_+ +
\sqrt{3}\Sigma_B)\:,\, \lambda_{\Sigma_C} = 2\sqrt{3}\Sigma_B$, where $\Sigma_+,\, \Sigma_B$ take the fix point values for the relevant line of fix points. Here one zero eigenvalue corresponds to that one has a line of fix points while the second is associated with the existence of a one parameter set of solutions that are anti-parallel w.r.t. each other on each side of the line of fix points.
[**Friedmann fix points**]{}: All four Friedmann fix points satisfy $\Sigma_+=\Sigma_A=\Sigma_B=\Sigma_C=0,\,
\Omega_{(1)}\Omega_{(2)}=0,\, \Omega_{\rm m} = 1,\,
v_{(1)}v_{(2)}=0$. They are distinguished by their $\Omega_{(i)}$ and $v_{(i)}$ values according to:
$$\begin{aligned}
& {\rm F}_{00}^{10}:\quad v_{(1)}=0\:,\quad v_{(2)}=0\:,\qquad
\Omega_{(1)}=1\:,\quad \Omega_{(2)}=0\:,\\
& {\rm F}_{01}^{10}:\quad v_{(1)}=0\:,\quad
v_{(2)}=-1\:,\quad\,
\Omega_{(1)}=1\:,\quad \Omega_{(2)}=0\:,\\
& {\rm F}_{00}^{01}:\quad v_{(1)}=0\:,\quad v_{(2)}=0\:,\qquad
\Omega_{(1)}=0\:,\quad \Omega_{(2)}=1\:,\\
& {\rm F}_{10}^{01}:\quad v_{(1)}=1\:,\quad v_{(2)}=0\:,\qquad
\Omega_{(1)}=0\:,\quad \Omega_{(2)}=1\:,\end{aligned}$$
where the superscript refers to the values of $\Omega_{(1)}$ and $\Omega_{(2)}$. The associated eigenvalues are:
$$\begin{aligned}
& {\rm F}_{00}^{10}:\quad \lambda_{1,2,3,4} = -{{\textstyle{3\over2}}}(1-w_{(1)})\:;\quad
3w_{(2)} - 1\:; \quad 3(w_{(1)}-w_{(2)})\:;\qquad v_{(1)}\quad \text{eliminated}\:,\\
& {\rm F}_{01}^{10}:\quad \lambda_{1,2,3,4} =
-{{\textstyle{3\over2}}}(1-w_{(1)})\:;\quad
3w_{(1)}-1\:;\quad \frac{2(1-3w_{(2)})}{1-w_{(2)}}\:;\qquad\,\, \Omega_{(1)}\quad \text{eliminated}\:,\\
& {\rm F}_{00}^{01}:\quad \lambda_{1,2,3,4} =
-{{\textstyle{3\over2}}}(1-w_{(2)})\:;
\quad 3w_{(1)} - 1\:; \quad -3(w_{(1)}-w_{(2)})\:;\quad\, v_{(2)}\quad \text{eliminated}\:,\\
& {\rm F}_{10}^{01}:\quad \lambda_{1,2,3,4} = -{{\textstyle{3\over2}}}(1-w_{(2)})\:; \quad
3w_{(2)} - 1\:;\quad \frac{2(1-3w_{(1)})}{1-w_{(1)}}\:;\qquad\,\, \Omega_{(1)}\quad \text{eliminated}\:.\end{aligned}$$
Here the last entry for each line of fix points refers to the variable that has been eliminated by means of the Codazzi constraint . Two of the eigenvalues of $\lambda_{1,2,3,4}$ refer to $\lambda_{\Sigma_A}$ and $\lambda_{\Sigma_C}$. If $w_{(2)} = \frac{1}{3}$ there exists a line of Friedmann points, parameterized by $v_{(2)}$, ${\rm
FL}_{0v_{(2)}}^{10}$, that connects ${\rm F}_{00}^{10}$ and ${\rm F}_{01}^{10}$. Similarly if $w_{(1)} = \frac{1}{3}$ there exists a line of fix points, ${\rm FL}_{v_{(1)}0}^{01}$, that connects ${\rm F}_{00}^{01}$ and ${\rm F}_{10}^{01}$. They are given by
$$\begin{aligned}
& {\rm FL}_{0v_{(2)}}^{10}:\quad && v_{(1)}=0\:,\quad v_{(2)}=const\:,\qquad
\Omega_{(1)}=1\:,\quad \Omega_{(2)}=0\:, \quad w_{(2)}=1/3\:,\\
& {\rm FL}_{v_{(1)}0}^{01}:\quad && v_{(1)}=const\:,\quad
v_{(2)}=0\:,\qquad
\Omega_{(1)}=0\:,\quad \Omega_{(2)}=1\:, \quad w_{(1)}=1/3\:.\end{aligned}$$
The eigenvalues associated with the two lines are:
$$\begin{aligned}
& {\rm FL}_{0v_{(2)}}^{10}:\quad \lambda_{1,2,3,4} = -{{\textstyle{3\over2}}}(1-w_{(1)})\:;\quad
3w_{(1)}-1\:;\quad 0\:;\qquad\,\, \Omega_{(1)}\quad \text{eliminated}\:,\\
& {\rm FL}_{v_{(1)}0}^{01}:\quad \lambda_{1,2,3,4} =
-{{\textstyle{3\over2}}}(1-w_{(2)})\:;
\quad 3w_{(2)} - 1\:; \quad 0\:;\qquad\, \Omega_{(1)}\quad \text{eliminated}\:.\end{aligned}$$
We now turn to fix points for which $0<\Sigma^2<1$.
[**Fix points on ${\cal LRS}\cap {\cal
ET}_{v_{(1)}1}$ and ${\cal LRS}\cap {\cal ET}_{1v_{(2)}}$**]{}: When $\frac{1}{3} < w_{(2)} < w_{(1)}$ there are two additional fix points, ${\rm LRS}_{v^{\ast}_{(1)}1}$ and ${\rm
LRS}_{1v^{\ast}_{(2)}}$, which enter the physical state space via ${\rm F}_{01}^{10}$ and ${\rm F}_{10}^{01}$ when $w_{(1)}=\frac{1}{3}$, $w_{(2)}=\frac{1}{3}$, respectively, and move into the ${\cal LRS}$-subset with increasing values of $w_{(i)}$. In the stiff perfect fluid limit ($w_{(1)}=1$, $w_{(2)}=1$) the lines merge with the coalesced Kasner lines ${\rm KL}_{v_{(1)}1}^+={\rm KL}_{v_{(1)}1}^-$, ${\rm
KL}_{1v_{(2)}}^+={\rm KL}_{1v_{(2)}}^-$, respectively. The two fix points are characterized by $\Sigma_A = \Sigma_B = \Sigma_C
= 0$, and:
$$\begin{aligned}
& {\rm LRS}_{v^{\ast}_{(1)}1}: \quad \Sigma_+ =-{{\textstyle{1\over2}}}(3w_{(1)}-1)\:, \qquad
v_{(1)} =\frac{3w_{(1)}-1}{5w_{(1)}+1}\:,\quad v_{(2)} = -1\:,\nonumber \\
& \Omega_{(1)} = \frac{3(1- w_{(1)})(9w_{(1)}+1)(1+w_{(1)})}{32\,w_{(1)}}\:, \qquad
\Omega_{(2)} = \frac{3(1-w_{(1)})(5w_{(1)}+1)(3w_{(1)}-1)}{32\,w_{(1)}}\:,\\
& {\rm LRS}_{1v^{\ast}_{(2)}}: \quad \Sigma_+ = -{{\textstyle{1\over2}}}(3w_{(2)}
- 1)\:, \qquad v_{(1)} =1\:, \qquad
v_{(2)} = -\frac{3 w_{(2)} - 1}{5w_{(2)}+1}\:, \nonumber \\
& \Omega_{(1)} =
\frac{3(1-w_{(2)})(5w_{(2)}+1)(3w_{(2)}-1)}{32\,w_{(2)}}\:,
\qquad \Omega_{(2)} = \frac{3(1- w_{(2)})( 9w_{(2)}+1)(1+w_{(2)})}{32\,w_{(2)}}\:.\end{aligned}$$
After eliminating $\Omega_{(1)}$ locally the eigenvalues for the two LRS-points are:
$$\begin{aligned}
{\rm LRS}_{v^{\ast}_{(1)}1}: \quad & \lambda_{\Sigma_A} = 3(2w_{(1)}-1)\:; \quad
\lambda_{\Sigma_B} = \lambda_{\Sigma_C} = -{{\textstyle{3\over2}}}(1-w_{(1)})\:;
\quad 6 \frac{w_{(1)}-w_{(2)}}{1-w_{(2)}}\:; \nonumber \\
&-{{\textstyle{3\over4}}}(1-w_{(1)}) \left(1 \pm
\sqrt{A(w_{(1)})} \right)\:,\nonumber \\
{\rm LRS}_{1v^{\ast}_{(2)}}: \quad & \lambda_{\Sigma_A} =
3(2w_{(2)}-1)\:; \quad \lambda_{\Sigma_B} = \lambda_{\Sigma_C}
= -{{\textstyle{3\over2}}}(1-w_{(2)})\:;
\quad -6 \frac{w_{(1)}-w_{(2)}}{1-w_{(1)}}\:; \nonumber \\
& -{{\textstyle{3\over4}}}(1-w_{(2)})\left(1 \pm
\sqrt{A(w_{(2)})} \right)\:,\end{aligned}$$
where ${\rm Re}\, A(w_{(i)})<1$; since the expression for $A(w_{(i)})$ is rather messy we will refrain from giving it.
[**Fix point on ${\cal TW}\cap {\cal ET}_{11}$**]{}:
\_[11]{}: \_+ = -, \_C = 0, \_A = \_B = , v\_[(1)]{}=1, v\_[(2)]{} = -1, \_[(1)]{}=\_[(2)]{}=. Local elimination of $\Omega_{(1)}$ by means of the Codazzi constraint yields the eigenvalues: \_[\_C]{}=; -; -(1 i );; .
[**Fix points on ${\cal TW}\cap {\cal
ET}_{v_{(1)}1}$ and ${\cal TW}\cap {\cal ET}_{1v_{(2)}}$**]{}: When $\frac{1}{2} < w_{(1)} < \frac{3}{5}$ there exists one more fix point on ${\cal TW}$: ${\rm TW}_{v_{(1)}1}$. This fix point comes into existence when the point LRS$_{v^{\ast}_{(1)}1}$ bifurcate into two points at $w_{(1)}=\frac{1}{2}$; it then wanders away from ${\cal D}$ when $w_{(1)}$ increases and eventually leaves the physical state space through ${\rm TW}_{11}$ when $w_{(1)}=\frac{3}{5}$. Yet another similar fix point exists on ${\cal TW}$ if $\frac{1}{2} < w_{(2)} < \frac{3}{5}$: ${\rm
TW}_{1v^{\ast}_{(2)}}$. The fix points are characterized by $\Sigma_C = 0$ and
$$\begin{aligned}
{\rm TW}_{v^{\ast}_{(1)}1}: \quad \Sigma_+ &=
-{{\textstyle{1\over2}}}(3w_{(1)}-1)\:,\qquad \Sigma_A =
\sqrt{{{\textstyle{3\over2}}}(1-w_{(1)})(2w_{(1)}-1)}\:,\qquad
\Sigma_B = \sqrt{3}(2w_{(1)} -1)\:,\nonumber \\
v_{(1)} &= v_{(1)}^*=
\frac{(1-w_{(1)})(15w_{(1)}-7)}{-25w_{(1)}^2+18w_{(1)}-1}\:,
\qquad v_{(2)} = -1\:,\nonumber\\
\Omega_{(1)} &= 1 - {{\textstyle{1\over4}}}(3w_{(1)}-1)(15w_{(1)}-7) - B(w_{(1)})\:,\qquad
\Omega_{(2)} = B(w_{(1)})\:,\\
{\rm TW}_{1v^{\ast}_{(2)}}: \quad \Sigma_+ &=
-{{\textstyle{1\over2}}}(3w_{(2)}-1)\:,\qquad \Sigma_A =
\sqrt{{{\textstyle{3\over2}}}(1-w_{(2)})(2w_{(2)}-1)}\:,\qquad
\Sigma_B = \sqrt{3}(2w_{(2)} -1)\:,\nonumber \\
v_{(1)} &= 1\:,\qquad
v_{(2)} = v_{(2)}^* = -\frac{(1-w_{(2)})(15w_{(2)}-7)}{-25w_{(2)}^2+18w_{(2)}-1}\:,\nonumber \\
\Omega_{(1)} &= B(w_{(2)})\:,\qquad
\Omega_{(2)} = 1 - {{\textstyle{1\over4}}}(3w_{(2)}-1)(15w_{(2)}-7) - B(w_{(2)})\:,\end{aligned}$$
where B(w\_[(i)]{})= -.
Local elimination of $\Omega_{(1)}$ yields the following eigenvalues:
$$\begin{aligned}
& {\rm TW}_{v^{\ast}_{(1)}1}:\, \lambda_{\Sigma_C}=-{{\textstyle{3\over2}}}(5-9w_{(1)})\:; \quad
6\frac{w_{(1)}-w_{(2)}}{1-w_{(2)}}\:; \,\, \lambda_{3,4,5,6} =
-{{\textstyle{3\over4}}}(1-w_{(1)})\left(1 \pm \sqrt{C_{(1)}
\pm D_{(1)}}\right)\:,\\
& {\rm TW}_{1v^{\ast}_{(2)}}:\, \lambda_{\Sigma_C}= -{{\textstyle{3\over2}}}(5-9w_{(2)})\:; \,\,
-6\frac{w_{(1)}-w_{(2)}}{1-w_{(1)}}\:; \,\, \lambda_{3,4,5,6} =
-{{\textstyle{3\over4}}}(1-w_{(2)})\left(1 \pm \sqrt{C_{(2)}
\pm D_{(2)}}\right)\:,\end{aligned}$$
where $C_{(i)}=C_{(i)}(w_{(i)}), D_{(i)}=D_{(i)}(w_{(i)})$ exhibit quite messy expressions, which we therefore refrain from giving, such that real parts of the associated eigenvalues always are negative.
[**Fix point in the generic geometric manifold**]{}: There exists one fix point ${\rm G}_{11}$ for which all the off-diagonal components of the shear are non-zero. It thus exists on the generic ‘geometric’ manifold, but on the ‘matter boundary’ ${\cal ET}_{11}$ where both fluids are extremely tilted. It is characterized by: \_[11]{}: \_+ = -, \_A = , \_B = \_C =,v\_[(1)]{}= 1, v\_[(2)]{} = -1, \_[(1)]{} = \_[(2)]{} = . Local elimination of $\Omega_{(1)}$ yields the eigenvalues: \_[1,2,3,4]{} = -(1 i );-;-. At $w_{(2)}<w_{(1)}=\frac{5}{9}$ ($w_{(2)}=\frac{5}{9}<w_{(1)}$) there exists a line of fix points, ${\rm GL}_{v_{(1)}1}$ (${\rm GL}_{1v_{(2)}}$), connecting ${\rm TW}_{v_{(1)}^*1}$ (${\rm TW}_{1v_{(2)}^*}$) with $G_{11}$; ${\rm GL}_{v_{(1)}1}$ and ${\rm GL}_{1v_{(2)}}$ are given by:
$$\begin{aligned}
{\rm GL}_{v_{(1)}1}: \quad \Sigma_+ & = -{{\textstyle{1\over3}}}\:,\qquad
\Sigma_A = \frac{1}{3\sqrt{3}}\sqrt{\frac{34v_{(1)}-6}{3+4v_{(1)}}}\:,\qquad
\Sigma_B = {{\textstyle{1\over3\sqrt{3}}}}\:, \quad \Sigma_C =
\frac{1}{3\sqrt{3}}\sqrt{\frac{13v_{(1)}-6}{3+4v_{(1)}}}\nonumber \\
{{\textstyle{6\over13}}} & \leq v_{(1)} = const \leq 1\:, \qquad v_{(2)} = -1\:, \nonumber \\
\Omega_{(1)} &= \frac{9+5v_{(1)}^2}{3(1+v_{(1)})(3+4v_{(1)})}
\:, \quad
\Omega_{(2)} = \frac{14v_{(1)}}{3(1+v_{(1)})(3+4v_{(1)})}\:,\\
{\rm GL}_{1v_{(2)}}: \quad \Sigma_+ & = -{{\textstyle{1\over3}}}\:,\qquad
\Sigma_A = \frac{1}{3\sqrt{3}}\sqrt{\frac{-34v_{(2)}-6}{3-4v_{(2)}}}\:,\qquad
\Sigma_B = {{\textstyle{1\over3\sqrt{3}}}}\:, \quad \Sigma_C =
\frac{1}{3\sqrt{3}}\sqrt{\frac{-13v_{(2)}-6}{3-4v_{(2)}}} \nonumber \\
v_{(1)} &= 1\:,\qquad -1\leq v_{(2)} = const\leq -{{\textstyle{6\over13}}}\:,\nonumber \\
\Omega_{(1)} &= \frac{-14v_{(2)}}{3(1-v_{(2)})(3-4v_{(2)})}\:,\qquad
\Omega_{(2)} = \frac{9+5v_{(2)}^2}{3(1-v_{(2)})(3-4v_{(2)})}\:.\end{aligned}$$
Local elimination of $\Omega_{(1)}$ yields the eigenvalues:
$$\begin{aligned}
& {\rm GL}_{v_{(1)}1}: \,\, 0\:; \qquad\,\,\,\,
\frac{2}{3}\frac{5-9w_{(2)}}{1-w_{(2)}}\:; \quad \lambda_{3,4,5,6} =
-{{\textstyle{1\over3}}}\left(1 \pm \sqrt{F_{(1)} \pm G_{(1)}}\right)\:,\\
& {\rm GL}_{1v_{(2)}}: \,\, 0\:; \quad\,\,\,\,\,
-\frac{2}{3}\frac{9w_{(1)}-5}{1-w_{(1)}}\:; \quad \lambda_{3,4,5,6} =
-{{\textstyle{1\over3}}}\left(1 \pm \sqrt{F_{(2)} \pm G_{(2)}}\right)\:,\end{aligned}$$
where $F_{(i)}=F_{(i)}(v_{(i)}), G_{(i)}=G_{(i)}(v_{(i)})$ exhibit quite messy expressions, which we therefore refrain from giving, such that real parts of the associated eigenvalues always are negative.
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[^1]: Electronic address: [[email protected]]{}
[^2]: Electronic address: [[email protected]]{}
[^3]: A gravitational test field is a field that does not affect the metric, i.e., it is a field for which we can neglect: (i) its source contribution to Einstein’s equations, (ii) its influence on gravitational sources.
[^4]: The sign in the definition of $\Omega^\alpha$ is the same as in [@ellels99], but opposite of that in [@waiell97].
[^5]: We could have extended the range of the equation of state to include $-1/3<w_{(i)}<0$, but the well-posedness of the Einstein equations for this range has been questioned, see [@friren00].
[^6]: In the case of a magnetic field, aligned along $\vece_3$, one obtains $R_1 = \Sigma_{23},\,R_2 =
-\Sigma_{31}$, i.e., the signs are opposite of those of the two tilted fluid case! This dynamical result in turn leads to sign differences in the $\Sigma$-equations. Note that the kinematic results in [@leb97] still hold, hence e.g. fix points correspond to transitively self-similar models.
[^7]: The monotonicity principle gives information about the global asymptotic behavior of the dynamical system. If $M: X\rightarrow \mathbb{R}$ is a ${\mathcal C}^1$ function which is strictly decreasing along orbits (solutions) in $X$, then $$\omega(x) \subseteq \{\xi \in \bar{X}\backslash X\:|\:
\lim\limits_{\zeta\rightarrow \xi} M(\zeta) \neq
\sup\limits_{X} M\}\:, \quad \alpha(x) \subseteq \{\xi \in
\bar{X}\backslash X\:|\:\lim\limits_{\zeta\rightarrow \xi}
M(\zeta) \neq \inf\limits_{X} M\}$$ for all $x\in X$, where $\omega(x)$ \[$\alpha(x)$\] is the $\omega$-limit \[$\alpha$-limit\] set of a point $x\in X$, defined as the set of all accumulation points of the future \[past\] orbit of $x$; and analogously for strictly increasing monotonic functions.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We investigate dynamic magneto-optic effects in a ferromagnetic nematic liquid crystal experimentally and theoretically. Experimentally we measure the magnetization and the phase difference of the transmitted light when an external magnetic field is applied. As a model we study the coupled dynamics of the magnetization, $\bf M$, and the director field, $\bf n$, associated with the liquid crystalline orientational order. We demonstrate that the experimentally studied macroscopic dynamic behavior reveals the importance of a dynamic cross-coupling between $\bf M$ and $\bf n$. The experimental data are used to extract the value of the dissipative cross-coupling coefficient. We also make concrete predictions about how reversible cross-coupling terms between the magnetization and the director could be detected experimentally by measurements of the transmitted light intensity as well as by analyzing the azimuthal angle of the magnetization and the director out of the plane spanned by the anchoring axis and the external magnetic field. We derive the eigenmodes of the coupled system and study their relaxation rates. We show that in the usual experimental set-up used for measuring the relaxation rates of the splay-bend or twist-bend eigenmodes of a nematic liquid crystal one expects for a ferromagnetic nematic liquid crystal a mixture of at least two eigenmodes.'
author:
- |
Tilen Potisk$^{1,2,*}$, Alenka Mertelj$^3$, Nerea Sebastián$^3$, Natan Osterman$^{1,3}$,\
Darja Lisjak$^3$, Helmut R. Brand$^2$, Harald Pleiner$^4$, and Daniel Svenšek$^1$
title: ' Magneto-optic dynamics in a ferromagnetic nematic liquid crystal'
---
Introduction
============
In Ref. [@bropgdg] Brochard and de Gennes suggested and discussed a ferromagnetic nematic phase combining the long range nematic orientational order with long range ferromagnetic order in a fluid system. The synthesis and experimental characterization of ferronematics and ferrocholesterics, a combination of low molecular weight nematic liquid crystals (NLCs) with magnetic liquids leading to a superparamagnetic phase, started immediately [@rault] and continued thereafter [@amer; @antonio87; @antonio2004; @kopcansky; @ouskova; @buluy; @podoliak]. These studies made use of ferrofluids or magnetorheological fluids (colloidal suspensions of magnetic particles) [@rosensweig]; their experimental properties [@rosensweig; @odenbach2004] have been studied extensively in modeling [@mario2001; @luecke2004; @hess2006; @sluckin2006; @mario2008; @klapp2015] using predominantly macroscopic descriptions [@mario2001; @luecke2004; @hess2006; @mario2008].
On the modeling side, the macroscopic dynamics of ferronematics was given first for a relaxed magnetization [@jarkova2] followed by taking into account the magnetization as a dynamic degree of freedom [@jarkova] as well as incorporating chirality effects leading to ferrocholesterics [@fink2015]. In parallel a Landau description including nematic as well as ferromagnetic order has been presented [@hpmhd].
Truly ferromagnetic NLCs have been generated [@alenkanature] in 2013 followed by reports of further ferromagnetic NLCs in [@shuaiNC; @smalyukhPNAS], and their macroscopic static properties were characterized in detail [@alenkasoftmatter]. Quite recently ferromagnetic cholesteric liquid crystals have been synthesized and investigated [@smalyukhPRL; @smalyukhNatMat; @alenkascience]. For a review on ferromagnetic NLCs see Ref. [@alenkarev].
In the present paper we describe in detail experimentally and theoretically the static and dynamic properties of ferromagnetic NLCs [@tilenshort]. We analyze the coupled dynamics of the magnetization and the director, initiated and controlled by an external magnetic field. We show experimentally and theoretically that dissipative dynamic coupling terms influence qualitatively the dynamics. Experimentally this is done by measuring the temporal evolution of the normalized phase difference associated with the dynamics of the director. Quantitative agreement between the experimental results and the model is reached and a dissipative cross-coupling coefficient between the magnetization and the director is accurately evaluated. It is demonstrated that this cross-coupling is crucial to account for the experimental results thus underscoring the importance of such off-diagonal effects in this first multiferroic fluid system. We also make concrete theoretical predictions of how the reversible dynamic cross-coupling terms between magnetization and director influence the macroscopic dynamics and how these effects can be detected experimentally. The experimental and theoretical dynamic results discussed in some detail in this paper for low magnetic fields in ferromagnetic NLCs demonstrate the potential for applications of these materials in displays and magneto-optic devices as well as in the field of smart fluids.
The paper is organized as follows. In Section \[sec:experiments\] we describe the experimental set-up followed in Section \[sec:model\] by the macroscopic model. The connection between the measurements and the model is established in Section \[sec:connection\]. In Section \[sec:statics\] we analyze the statics and in Sections \[sec:ON\] and \[sec:OFF\] we analyze in detail the coupled macroscopic dynamics of the magnetization and the director field when switching the external magnetic field on and off, respectively. Section \[sec:flucts\] is dedicated to a theoretical analysis of fluctuations and light scattering and in the conclusions we give a summary of the main results and a perspective.
Experiments {#sec:experiments}
===========
The experimental samples have been prepared along the lines described in detail in Refs. [@alenkanature; @alenkasoftmatter]. In brief, the BaSc$_x$Fe$_{12-x}$O$_{19}$ nanoplatelets were suspended in the liquid crystal mixture E7 (Merck, nematic – isotropic transition temperature $T_{\mathrm{NI}}=58\,^\circ$C). The suspension was filled in liquid crystal cells with rubbed surfaces (thickness $d=20\,\mu$m, Instec Inc.), which induced homogeneous in-plane orientation of the NLC. The volume concentration of the magnetic platelets in the nematic low-molecular-weight liquid crystal E7 (Merck) has been estimated to be $\sim 1.3 \times 10^{-3}$ from the measurements of the magnetization magnitude [@alenkasoftmatter] which was $M_0 \sim 200$A/m. E7 suspensions show long term stability, with a homogeneous response to magnetic fields and no aggregates for a period of several months. A surfactant (dodecylbenzene sulfonic acid) was used for the treatment of the nanoplatelets which favors a perpendicular orientation of the NLC molecules with respect to the nanoplatelets. Quantitative values for the Frank coefficients for E7 are available in the literature [@km].
![ (Color online) Top: sketch of the experimental set-up and definition of coordinate axes [@tilenshort]. The thick yellow arrows indicate the direction of the light passing through the polarizer and the analyzer. In the absence of an applied magnetic field (${\bf H}$, $z$ direction), the equilibrium director (${\bf n}$) and magnetization (${\bf M}$) fields are only slightly pretilted from the $x$ direction. Inset: distortion of the NLC director (ellipsoids, schematic) prevents flocculation of the suspended nanoplatelets carrying a magnetic moment ${\bf p}_m$ parallel to $\bf n$ in equilibrium. Bottom: ferromagnetic E7 nematic 20$\mu$m sample placed between crossed polarizers, with the director at an angle of 45 degrees. Polarizing optical microscopy image width corresponds to 700$\mu$m. The spheres are cell spacers. []{data-label="Bild0"}](fig1 "fig:"){width="3.5in"} ![ (Color online) Top: sketch of the experimental set-up and definition of coordinate axes [@tilenshort]. The thick yellow arrows indicate the direction of the light passing through the polarizer and the analyzer. In the absence of an applied magnetic field (${\bf H}$, $z$ direction), the equilibrium director (${\bf n}$) and magnetization (${\bf M}$) fields are only slightly pretilted from the $x$ direction. Inset: distortion of the NLC director (ellipsoids, schematic) prevents flocculation of the suspended nanoplatelets carrying a magnetic moment ${\bf p}_m$ parallel to $\bf n$ in equilibrium. Bottom: ferromagnetic E7 nematic 20$\mu$m sample placed between crossed polarizers, with the director at an angle of 45 degrees. Polarizing optical microscopy image width corresponds to 700$\mu$m. The spheres are cell spacers. []{data-label="Bild0"}](fig1b "fig:"){width="3.35in"}
Dynamics of the director was measured by inducing director reorientation in planarly treated 20$\mu$m cells (pretilt in the range 1-3 degrees) when applying a magnetic field perpendicular to the cell plates, Fig. \[Bild0\] (top). Experiments were performed on monodomain samples (see Ref. [@alenkarev] for a description of monodomain sample preparation) so that the director is initially at 45 degrees with respect to the crossed polarizers, Fig. \[Bild0\] (bottom). Using polarizing microscopy, the monochromatic light intensity transmitted through the sample was recorded with a CMOS camera (IDS Imaging UI-3370CP, 997 fps) as a function of time on switching the magnetic field on and off. An interference filter (623.8nm) was used to filter the light from the halogen lamp used in the microscope. The transmitted light intensity is related to the phase difference between the ordinary and the extraordinary light as will be explained below. The advantage of using polarization microscopy is that the measurements are performed in the homogeneous region of the sample without spacers or other impurities. Recording the image of the sample during the measurements also allows us to simultaneously monitor the homogeneity of the response. With the use of a vibrating sample magnetometer [@alenkasoftmatter] (LakeShore 7400 Series VSM) also the equilibrium $z$ component of the magnetic moment of the sample is measured. We note that this technique is not suitable for measuring the magnetization dynamically, as several seconds per measurement are required for ambient magnetic noise averaging.
Macroscopic model {#sec:model}
=================
Throughout the present paper we take into account the magnetization ${\bf M}$ and the director field $ {\bf n}$ as macroscopic variables; in the following we focus on the essential ingredients of their dynamics necessary to capture the experimental results we will discuss. That is we assume isothermal conditions and discard flow effects. For a complete set of macroscopic dynamic equations for ferronematics we refer to Ref. [@jarkova].
The static behavior is described by the free energy density $f({\bf M}, {\bf n}, \nabla{\bf n})$, $$f = - \mu_0{\bf M}\cdot{\bf H} - {\textstyle{1\over 2}}A_1 ({\bf M}\cdot{\bf n})^2
+ {\textstyle{1\over 2}}A_2 \left(|{\bf M}|-M_0\right)^2
+ f^F, \label{f}
$$ where $\mu_0$ is the magnetic constant, ${\bf H} = H \hat{\bf e}_z$ is the applied magnetic field, and $A_{1,2}>0$ will be assumed constant. The first term represents the coupling of the magnetization and the external magnetic field. Since $H\gg M_0$, the local magnetic field is equal to $\bf H$, which is fixed externally, and is thus independent of the ${\bf M}({\bf r})$ configuration. The second term describes the static coupling between the director field and the magnetization (originating from the magnetic particles). The third term describes the energy connected with the deviation of the modulus of the magnetization from $M_0$. The last term is the Frank elastic energy associated with director distortions [@degennesbook] $$\begin{aligned}
f^F &=& {\textstyle{1\over 2}}K_1(\nabla\cdot{\bf n})^2
+ {\textstyle{1\over 2}}K_2\left[{\bf n}\cdot(\nabla\times{\bf n})\right]^2\nonumber \\
&+& {\textstyle{1\over 2}}K_3\left[{\bf n}\times(\nabla\times{\bf n})\right]^2, \label{f^F}\end{aligned}$$ with positive elastic constants for splay ($K_1$), twist ($K_2$), and bend ($K_3$). The saddle-splay elastic energy [@degennesbook] is zero in the considered geometry.
While it is a good approximation to assume that $|{\bf M}|=M_0$, we will take into account small variations of $|{\bf M}|$ (corresponding to large values of $A_2$).
The anchoring of the director at the plates is taken into account using a finite surface anchoring energy [@rapini] $$f^S = - {\textstyle{1\over 2}} W ({\bf n}_S\cdot{\bf n})^2,
\label{f^S}$$ where $W$ is the anchoring strength and ${\bf n}_{S}=\hat{{\bf e}}_{z}\sin\varphi_{s}+\hat{{\bf e}}_{x}\cos\varphi_{s}$ is the preferred direction specified by the director pretilt angle $\varphi_{s}$.
For the total free energy we have $F = \int\!\!f\,{\rm d}V+\int\!\!f^S\,{\rm d}S$ and the equilibrium condition requires $\delta F = 0$.
The macroscopic dynamic equations for the magnetization and the director read [@pleinerbrandchapter; @jarkova] $$\begin{aligned}
\dot M_i + X_i^R + X_i^D &=& 0, \label{Mdot}\\
\dot n_i + Y_i^R + Y_i^D &=& 0, \label{ndot}\end{aligned}$$ where the quasi-currents have been split into reversible ($X_i^R$, $Y_i^R$) and irreversible, dissipative ($X_i^D$, $Y_i^D$) parts. The reversible (dissipative) parts have the same (opposite) behavior under time reversal as the time derivatives of the corresponding variables, i.e, Eqs. (\[Mdot\])-(\[ndot\]) are invariant under time reversal only if the dissipative quasi-currents vanish.
The quasi-currents are expressed as linear combinations of conjugate quantities (thermodynamic forces); they take the form $$\begin{aligned}
h_i^M &\equiv& {\delta f\over\delta M_i} = {\partial f\over\partial M_i}, \label{h^M}\\
h_i^n &\equiv& \delta_{ik}^\perp {\delta f\over\delta n_k} =
\delta_{ik}^\perp\left({\partial f\over\partial n_k}-\partial_j\Phi_{kj}\right), \label{h^n}\end{aligned}$$ with $\Phi_{kj} = {\partial f/\partial (\partial_j n_k)}$ and where the transverse Kronecker delta $\delta_{ik}^\perp = \delta_{ik} - n_i n_k$ projects onto the plane perpendicular to the director owing to the constraint ${\bf n}^2=1$.
In Ref. [@tilenshort] we focused on the dissipative quasi-currents as they had a direct relevance for the explanation of the experimental results discussed there. In the present paper we also include the reversible quasi-currents, which give rise to transient excursions of $\bf M$ and $\bf n$ out of the switching plane.
The dissipative quasi-currents take the form [@jarkova] $$\begin{aligned}
\label{dyndis}
X_i^D &=& b_{ij}^D h_j^M + \chi_{ji}^D h_j^n, \label{X}\\
Y_i^D &=& {1\over\gamma_1} h_i^n + \chi_{ij}^D h_j^M, \label{Y}\end{aligned}$$ with $$\begin{aligned}
\chi_{ij}^D &=& \chi_1^D\delta_{ik}^\perp M_k n_j + \chi_2^D\delta_{ij}^\perp M_k n_k,\label{chiD}\\
b_{ij}^D &=& b_\parallel^D n_i n_j + b_\perp^D \delta_{ij}^\perp \label{bijD}\end{aligned}$$ Throughout the present paper we will discard the biaxiality of the material which arises for ${\bf n}\nparallel{\bf M}$.
The reversible quasi-currents are obtained by requiring that the entropy production $Y_ih_i^n+X_ih_i^M$ is zero [@jarkova]: $$\begin{aligned}
X_i^R &=& b_{ij}^R h_j^M + \chi^R \epsilon_{ijk} n_j h_k^n, \label{XR}\\
Y_i^R &=& (\gamma_1^{-1})_{ij}^R h_j^n + \chi^R \epsilon_{ijk} n_j h_k^M, \label{YR}\end{aligned}$$ where [@jarkova2] $$\begin{aligned}
b_{ij}^R &=& b_1^R \epsilon_{ijk} M_k + b_2^R \epsilon_{ijk} n_k n_p M_p \label{b_ij^R} \\
&&+ b_3^R(\epsilon_{ipq}M_p n_q n_j-\epsilon_{jpq}M_p n_q n_i),\nonumber\\
(\gamma_1^{-1})_{ij}^R &=&
(\gamma_1^{-1})_1^R \epsilon_{ijk} n_k n_p M_p\label{gamma_inv} \\
&& +
(\gamma_1^{-1})_2^R (\epsilon_{ijp} + \epsilon_{ipk} n_k n_j
- \epsilon_{jpk} n_k n_i ) M_p \nonumber.\end{aligned}$$
For solving the system Eqs. (\[Mdot\])-(\[ndot\]) a simple numerical method was used. We first discretized space into slices of width $\Delta z=d/(N-1)$, where $N$ is the number of discretization points. Empirically it was found that using $N=50$ is already sufficient. After discretizing space one obtains $N$ ordinary differential equations. Due to its simplicity, we use the Euler method. One step of the Euler method for the i-th component of the director field at $z$ is $$n_i(t+\delta t,z)=n_i(t,z)-\delta tY_i(t,z)+\mathcal{O}(\delta t^2),$$ where $\delta t$ is the time step. An analogous equation holds for the magnetization field and the equations are solved simultaneously. Since the numerical scheme for the director field is not norm preserving, we normalize the director field after each time step: $n_i \rightarrow n_i/\sqrt{n_jn_j}$.
In the discrete version, the two surface points are best treated by satisfying the same dynamic equations Eqs. (\[Mdot\])-(\[ndot\]) as the internal points, with the addition of the surface anchoring energy Eq. (\[f\^S\]) expressed as a volume density. The divergence part of the force Eq. (\[h\^n\]) is then replaced by its surface flux (the volume density thereof, again): $$h_i^{n\, {\rm surf.}} = \delta_{ik}^\perp\left[{\partial f\over\partial n_k}+{1\over\Delta z}
\left(\nu_j\Phi_{kj}+{\partial f^S\over\partial n_k}\right)\right],$$ where $\bnu$ is the surface normal pointing down (up) at the bottom (top) plate.
Connection between measurements and the model {#sec:connection}
=============================================
In equilibrium the magnetic-field-distorted director and magnetization fields are lying in the $xz$ plane, $\mathbf{n}=(\sin \theta, 0 , \cos \theta)$ and $\mathbf{M}=M(\sin \psi, 0, \cos \psi)$. In the absence of the magnetic field, the director is tilted from the $x$ axis by the pretilt $\varphi_s$, Eq. (\[f\^S\]). The coordinate system used here is shown in Fig. \[Bild0\]. As explained earlier, the average $z$ component of the magnetization, $M_z$, is measured by the vibrating sample magnetometer. In modeling, it is obtained by averaging the $z$ component of the magnetization field, $$\label{eq0}
M_z=\frac{1}{d}\int_{0}^d M\cos \psi(z)\, \mathrm{d}z.$$
To derive the expression for the phase difference we start with an electric field $\mathbf{E}$, which is linearly polarized after the light passes through the polarizer, $$\mathbf{E}=E_0{\mathbf{j}}\, {\rm e}^{\mathrm{i}(\mathbf{k}_i\cdot\mathbf{r}-\omega t)},$$ where $E_0$ is the electric field amplitude, ${\mathbf{j}}$ the initial polarization, $\mathbf{k}_i$ the wave vector and $\omega$ the frequency of the incident light. In our case the wave vector points in the $z$ direction,$\mathbf{k}_i=k_0 \hat{\mathbf{e}}_z$, with $k_0=\frac{2\pi}{\lambda}$ being the wave number. The polarization of the light therefore lies in the $xy$ plane and is described by the two-component complex vector ${\bf j}=j_x(z)\hat{\bf e}_x+j_y(z)\hat{\bf e}_y$. As the light passes through the sample also the components of this (Jones) polarization vector change and we analyze these changes using the Jones matrix formalism (assuming perfectly polarized light) [@hechtbook]. The incident light first goes through the polarizer oriented at $45^\circ$ with respect to the $x$ axis, Fig. \[Bild0\], and is linearly polarized with the initial Jones vector being $\mathbf{j}=\frac{1}{\sqrt{2}}(1,1)^T$. The optical axis is parallel to the director and generally varies through the cell. For any ray direction we can decompose the polarization into a polarization perpendicular to the optical axis (ordinary ray) and a polarization which is partly in the direction of the optical axis (extraordinary ray). The ordinary ray experiences an ordinary refractive index $n_o$ and the extraordinary ray experiences a refractive index $n_e$, $$\label{refract}
n_e^{-2}(z)=n_{e0}^{-2}\sin^2\theta(z)+n_o^{-2}\cos^2\theta(z),$$ where $n_{e0}$ is the extraordinary refractive index.
To calculate the intensity of the transmitted light, one first divides the liquid crystal cell into $N$ thin slices of width $h=d/N$ and describes the effect of each slice on the polarization by the phase matrix $$\textsf{W}(z) =
\begin{pmatrix}
{\rm e}^{\mathrm{i}k_0[n_e(z)-n_o]h/2} &0 \\
0 & {\rm e}^{-\mathrm{i}k_0[n_e(z)-n_o]h/2}
\end{pmatrix}.$$ In the limit $N\to \infty$ we can express the transmission matrix of the liquid crystal cell as $$\label{transm}
\textsf{T} =
\begin{pmatrix}
{\rm e}^{\mathrm{i}\phi/2} &0 \\
0 & {\rm e}^{-\mathrm{i}\phi/2}
\end{pmatrix},$$ where we have introduced the phase difference $$\label{eqfaza}
\phi=k_0 \int_0^d [n_e(z)-n_o]\mathrm{d}z.$$
In general, as we will see, the director can have also a nonzero component in the $y$ direction. In this case the simple expression for the transmission matrix Eq. (\[transm\]) does not hold anymore and must be generalized.
We start the derivation of the general transmission matrix by assuming a general orientation of the director, $$\mathbf{n}=(\sin\theta \cos \varphi, \sin\theta \sin \varphi, \cos\theta).$$ The azimuthal angle of the director $\varphi$ can vary through the cell and the transformation matrix at point $z$ is $$\textsf{T}(z) =
\textsf{R}[-\varphi(z)]\textsf{W}(z)\textsf{R}[\varphi(z)],$$ where $\textsf{R}$ is the rotation matrix $$\textsf{R}(\varphi) =
\begin{pmatrix}
\cos(\varphi) & \sin(\varphi) \\
-\sin(\varphi) & \cos(\varphi)
\end{pmatrix}.$$ Our goal is to find the transfer matrix for the whole cell, $$\label{eeq1}
\textsf{T}=\prod_{z \in [0,d]}^{\longleftarrow}\textsf{T}(z),$$ where the arrow denotes the ordered product starting from $\textsf{T}(0)$ at the right side. We first notice that $$\textsf{T}(z)\approx I+\mathrm{i} \frac{k_0[n_e(z)-n_o]h}{2} \begin{pmatrix}
\cos[2\varphi(z)]& \sin[2\varphi(z)] \\
\sin[2\varphi(z)] & -\cos[2\varphi(z)]
\end{pmatrix},$$ where $I$ is the identity matrix. Consequently we can write $\textsf{T}(z)$ as an exponential, $$\textsf{T}(z) = \lim_{h\to 0}\exp [\mathrm{i}\textsf{A}(z)h],$$ where $\textsf{A}$ is defined by $$\textsf{A}(z)=\frac{k_0[n_e(z)-n_o]}{2} \begin{pmatrix}
\cos[2\varphi(z)] & \sin[2\varphi(z)] \\
\sin[2\varphi(z)] & -\cos[2\varphi(z)]
\end{pmatrix}.$$ We can now rewrite Eq. (\[eeq1\]) as $$\label{eeq2}
\textsf{T}= \lim_{h\to 0}\exp \left[\mathrm{i}\sum_{z\in [0,d]}\textsf{A}(z)h\right] = \exp [\mathrm{i}\int_{0}^d \textsf{A}(z)\mathrm{d}z],$$ where we used $${\rm e}^{\textsf{A}h}{\rm e}^{\textsf{B}h}={\rm e}^{(\textsf{A}+\textsf{B})h}+\frac{1}{2}[\textsf{A},\textsf{B}]h^2 + \mathcal{O}(h^3).$$ The exponential of the $2\times2$ matrix from Eq. (\[eeq2\]) reads $$\textsf{T}= \begin{pmatrix}
\cos(c)+\mathrm{i}\frac{a}{c}\sin(c) &\mathrm{i}\frac{b}{c}\sin(c) \\
\mathrm{i}\frac{b}{c}\sin(c) & \cos(c)-\mathrm{i}\frac{a}{c}\sin(c)
\end{pmatrix},$$ where $c=\sqrt{a^2+b^2}$ with $$\label{ab}
\begin{split}
a&=\frac{k_0}{2}\int_{0}^d [n_e(z)-n_o]\cos[2\varphi(z)]\mathrm{d} z, \\
b&=\frac{k_0}{2}\int_{0}^d [n_e(z)-n_o]\sin[2\varphi(z)]\mathrm{d} z.
\end{split}$$ We then let the light pass through an analyzer $\textsf{P}_\alpha$ at an angle $\alpha$, $$\textsf{P}_\alpha=\begin{pmatrix}
\cos^2\alpha & \sin \alpha \cos \alpha \\
\sin \alpha \cos \alpha & \sin^2\alpha
\end{pmatrix},$$ which gives for the final Jones vector ($\alpha=-45^\circ$) $$\mathbf{j}'=\frac{\mathrm{i}a\sin (c)}{\sqrt{2}c}\begin{pmatrix}
1 \\
-1
\end{pmatrix}.$$ This yields the measured normalized intensity $$\label{normint}
\frac{I}{I_0}=\mathbf{j'^*}^T\mathbf{j'}=\frac{a^2}{c^2}\sin^2(c).$$
Next we evaluate the relation between the phase difference and the measured intensity. Let $\mathbf{j}$ be the Jones vector after the liquid crystal cell, $$\mathbf{j}=\begin{pmatrix}
z_1{\rm e}^{\mathrm{i}\phi} \\
z_2
\end{pmatrix},$$ where $z_1$ and $z_2$ are real and $z_1^2+z_2^2=1$. Generally $|z_1|\neq |z_2|$. After an analyzer with $\alpha = -45^\circ$ we have a Jones vector $$\mathbf{j}'=\frac{1}{2}(z_1{\rm e}^{\mathrm{i}\phi}-z_2)\begin{pmatrix}
1 \\
-1
\end{pmatrix}$$ and the intensity is related to the phase difference as $$\frac{I}{I_0}=\frac{1}{2}\left[1-2z_1z_2\cos(\phi)\right].$$ Only if the director is restricted to the $xz$ plane, $z_1=z_2$ and we have $$\frac{I}{I_0}=\frac{1}{2}\left[1-\cos(\phi)\right]=\sin^2\left(\frac{\phi}{2}\right),$$ such that the relation between the intensity and the phase difference is $$\label{eqff}
\phi=m\pi \pm 2\arcsin\left[\sqrt{\frac{I}{I_0}}\right],$$ where $m\in \mathbb{Z}$ and the sign $\pm$ is determined by demanding that $\phi$ is sufficiently smooth. Generally however, the quantity obtained from the measured intensity by Eq. (\[eqff\]) is not the phase difference. It is the phase difference only when the director field is in the $xz$ plane. For the analysis of the dynamics not confined to the $xz$ plane, Sec. \[sec:reversible\], we will therefore use the normalized intensity Eq. (\[normint\]).
In the case when the dynamics is in the $xz$ plane, to compare the numerical results with the experiments and also to compare the dynamics of the director with the dynamics of the magnetization, it is convenient to introduce the normalized phase difference $$r(H) = 1 - \frac{\phi(H)}{\phi_0},$$ where $\phi_0$ is the phase difference at zero magnetic field. The normalized phase difference is zero at $t=0$ and is always smaller or equal to 1. It can also assume negative values as we will see.
Statics {#sec:statics}
=======
In this Section we present experimental and numerical results of statics and derive analytic formulae for the equilibrium configurations in the low and large external magnetic field limits.
![ (Color online) Comparison of experimental and theoretical static results: (top) normalized phase difference $r(H)$ and (bottom) magnetization component $M_z$ as functions of the magnetic field $\mu_0 H$. []{data-label="Bild1"}](fig2){width="3.3in"}
In Fig. \[Bild1\] we compare the numerical results of the equilibrium normalized phase difference to the experimental data. Below we will show in Eqs. (\[up1\])-(\[up11\]) that the equilibrium normalized phase difference is quadratically dependent on the applied magnetic field at small magnetic fields. The normalized phase difference saturates quickly above $\mu_0H$ = 10 mT at a value which is less than 1, which means there is a limit to how much the director field deforms. We also observe that the dependence of the equilibrium normalized phase difference is not symmetric with respect to the $\mu_0H=0$ axis, which is seen in experiments as well. The reason for this is the nonzero pretilt at both glass plates.
>From the fits to the model we extract values for the anchoring strength $W$, the pretilt angle $\varphi_s$, the Frank elastic constant $K\equiv K_1=K_3$ in the one constant approximation, and the static coupling coefficient $A_1$: $$\begin{aligned}
W &\sim& 4 \times 10^{-5}\,{\rm J/m}^2, \label{Wextracted}\\
\varphi_s &\sim& -0.05,\\
K &\sim& 17\,{\rm pN},\\
A_1 &\sim& 140 \mu_0.\label{A1extracted}\end{aligned}$$ The extracted parameters Eqs. (\[Wextracted\])-(\[A1extracted\]) correspond to the (local) minimum of the sum of squares of residuals between the numerical and experimental values of the normalized phase difference. This minimum was sought in sensible parameter ranges (for example the Frank elastic constant was sought in the range between 5pN and 25pN). There are several indications that this minimum is at least very close to the global one. First, the extracted value of the Frank elastic constant is close to the value of $K_3$ in the pure E7 NLC. Secondly, the extracted pretilt is within the range specified by the cell provider. Moreover the value of the static coupling is similar to that estimated for the ferromagnetic NLC based on 5CB [@alenkasoftmatter].
The limiting behaviors of the normalized phase difference and the normalized $z$ component of the magnetization as the magnetic field goes to zero or infinity can be calculated analytically. In all cases the boundary condition is $$K \frac{\partial \theta}{\partial z}\nu_z+\frac{\partial f^S}{\partial \theta}=0,$$ where $\nu_z$ is the $z$ component of the surface normal pointing upwards at $z=d$ and downwards at $z=0$.
Low magnetic fields
-------------------
The free energy density in lowest order in deviations of magnetization and director field from the equilibrium is $$\label{free1}
f =\frac{1}{2}K \left(\frac{\partial \theta}{\partial z}\right)^2+\frac{1}{2}A_1M_0^2(\theta-\psi)^2+\mu_0HM_0\psi.$$ The equilibrium solutions for the angles are $$\begin{aligned}
\label{angles}
&\theta(z)=\frac{1}{2}\frac{\mu_0HM_0}{K}z(z-d) -\frac{\mu_0HM_0d}{2W}+ \frac{\pi}{2}-\varphi_s,\\
&\psi(z)=\theta(z)-\frac{\mu_0HM_0}{A_1M_0^2}.\end{aligned}$$ After inserting the solutions Eqs. (\[angles\]) in equations for the normalized phase difference and magnetization, one gets $$\begin{aligned}
\label{up1}
&r(H)=r_0 \frac{\mu_0HM_0d^2}{6K} \nonumber \\
&\times\left[\frac{\mu_0HM_0d^2}{20K}\left(1+10 \frac{\xi}{d}+30 \frac{\xi^2}{d^2}\right)+\left(1+6 \frac{\xi}{d}\right)\varphi_s\right],\end{aligned}$$ where $\xi=K/W$ is the so called anchoring extrapolation length and $r_0=n_{e0}(n_{e0}+n_o)/(2n_o^2)$. In the limit of infinite anchoring the normalized phase difference reads $$r(H)=r_0 \frac{\mu_0HM_0d^2}{6K}\left(\frac{\mu_0HM_0d^2}{20K}+\varphi_s \right).
\label{up11}$$ One can also observe that the location of the minimum of the normalized phase difference is shifted to a value $\mu_0H_{\mathrm{min}}$ determined by the pretilt: $$\label{up2}
-\frac{10K\varphi_s\left(1+6\frac{\xi}{d}\right)}{M_0d^2 \left(1+10 \frac{\xi}{d}+30\frac{\xi^2}{d^2}\right)}\xrightarrow{W\to \infty}- \frac{10K\varphi_s}{M_0d^2}.$$ Eqs. (\[up1\]) and (\[up2\]) are useful for determining the anchoring strength $W$ and the pretilt $\varphi_s$.
>From the behavior of the normalized phase difference at low fields Eqs. (\[up1\])-(\[up11\]) one cannot determine the value of the static coupling $A_1$. It can on the other hand be determined from the low-field behavior of the magnetization. In Fig. \[Bild1\] we see that the behavior is linear for low magnetic fields as can be shown analytically: $$\label{maglow}
\frac{M_z}{M_0}=\varphi_s+\left(\frac{1}{A_1M_0^2}+\frac{1}{12}\frac{d^2}{K}+\frac{d}{2W}\right)\mu_0H M_0.$$
Large magnetic fields
---------------------
In the large magnetic field limit we assume that both the polar angle of the director and the magnetization are either close to 0 if the applied magnetic field is positive ($+$) or close to $\pi$ if the applied magnetic field is negative ($-$). The corresponding solutions will be denoted as $\theta^+(z),\theta^-(z),\psi^+(z)$, $\psi^-(z)$, $M_z^+$, $M_z^-$, $r^+$, and $r^-$.
The free energy in the case of a positive magnetic field is $$\label{free2}
f \approx\frac{1}{2}K \left(\frac{\partial \theta}{\partial z}\right)^2+\frac{1}{2}A_1M_0^2(\theta-\psi)^2+\frac{1}{2}\mu_0HM_0 \psi^2.$$ The equilibrium solutions for the angles $\theta^+(z)$ and $\psi^+(z)$ are $$\begin{aligned}
\label{angles2}
&\theta^+(z)=\frac{\frac{\pi}{2}-\varphi_s}{1+q\xi\tanh\left(\frac{qd}{2}\right)}\frac{\cosh\left[q(z-\frac{d}{2})\right]}{\cosh\left(\frac{qd}{2}\right)},\\
&\psi^+(z)=\frac{\theta^+(z)}{1+\frac{\mu_0|H|M_0}{A_1M_0^2}},\end{aligned}$$ where $$\label{qnum}
q^2=q_0^2\frac{\mu_0|H|M_0}{\mu_0|H|M_0+A_1M_0^2}$$ with $q_0=\sqrt{A_1M_0^2/K}$ (which is proportional to the inverse “magnetization coherence length” of the director). The normalized $z$ component of the magnetization for large fields is $$\begin{aligned}
\label{maghigh}
&&\frac{M_z^+}{M_0}=1-\\
&&\frac{[\frac{\pi}{2}-\varphi_s]^2(qd+\sinh(qd))A_1^2M_0^4}{4qd\left[1+q\xi\tanh\left(\frac{qd}{2}\right)\right]^2\cosh^2\left(\frac{qd}{2}\right)(A_1M_0^2+\mu_0|H|M_0)^2}\nonumber\end{aligned}$$ and the normalized phase difference is $$\begin{aligned}
\label{phasehigh}
&&r^+(H) = 1-\frac{n_or_\infty k_0d}{2\phi_0}\frac{\left[\frac{\pi}{2}-\varphi_s \right]^2}{\left[1+q\xi\tanh(\frac{qd}{2})\right]^2}\frac{qd+\sinh(qd)}{2qd\cosh(\frac{qd}{2})^2}\nonumber\\
&&-\frac{n_or_\infty k_0d}{4\phi_0}(3r_\infty/4-1/3)\frac{\left[\frac{\pi}{2}-\varphi_s \right]^4}{\left[1+q\xi\tanh(\frac{qd}{2})\right]^4}\nonumber\\ &&\times \frac{6qd+8\sinh(qd)+\sinh(2qd)}{8qd\cosh\left(\frac{qd}{2}\right)^4},\end{aligned}$$ where $r_\infty=(n_{e0}^2-n_o^2)/n_{e0}^2$.
It follows from symmetry that $\theta^-(\varphi_s) = \pi-\theta^+(-\varphi_s)$, $\psi^-(\varphi_s) = \pi-\psi^+(-\varphi_s)$, $M_z^-(\varphi_s) = -M_z^+(-\varphi_s)$, and $r^-(\varphi_s) = r^+(-\varphi_s)$
Since the magnetization is not anchored at the boundary, in Eq. (\[maghigh\]) it was sufficient to consider terms not higher than $(\psi^{+})^2$. On the other hand, due to the anchoring of the director field, in Eq. (\[phasehigh\]) we expanded the phase difference to the order $(\theta^+)^4$. It should be noted, that the approximation for the phase difference is better if the anchoring $W$ is low, i.e., $q\xi \gg 1$ or $W \ll \sqrt{A_1M_0^2K}$.
In the large magnetic field limit, where $qd \gg 1$, and if $q\xi \gg 1$ in addition, one can study asymptotic behavior of Eqs. (\[maghigh\]) and (\[phasehigh\]): $$\begin{aligned}
r^+(H) &\asymp& r^+(\infty)-\frac{f^+(q_0)}{\mu_0|H|},\label{asympph}\\
\frac{M_z^+}{M_0} &\asymp& 1-\frac{h^+(q_0)}{(\mu_0H)^2},\label{asympm}\end{aligned}$$ where $f^+$ and $h^+$ are functions of static parameters for positive magnetic fields and $r^+(\infty) = \lim_{\mu_0H\to \infty}r^+(H)$. The behavior of the magnetization $M_z$, Fig. \[Bild1\], may at a first glance look like the Langevin function, often observed in magnetic systems. Eq. (\[asympm\]) tells us that this is not the case, since the Langevin function saturates with the first power in magnetic field, whereas here the saturation Eq. (\[asympm\]) is of second order in $H$.
![(Color online) For low magnetic fields, the numerically calculated polar angle of the director is in agreement with Eq. (\[angles\]).[]{data-label="Bild3"}](fig3){width="3.3in"}
![(Color online) For large magnetic fields, the numerically calculated polar angle of the director is in agreement with Eq. (\[angles2\]).[]{data-label="Bild4"}](fig4){width="3.3in"}
Comparison of analytic approximations with numerics
---------------------------------------------------
A comparison of analytic and numeric results for the director polar angle $\theta(z)$ is made in Figs. \[Bild3\] and \[Bild4\] for small and large magnetic fields, respectively. We find a good agreement for small magnetic fields up to 0.7mT and for large magnetic fields above 4mT. It should be emphasized that the values of the magnetic fields at which the approximations become valid depend on the values of the static parameters. We use the values Eqs. (\[Wextracted\])-(\[A1extracted\]) extracted from the fits to the macroscopic model.
In Fig. \[Bild5\] we compare analytic and numeric results for the $z$ component of the magnetization and the normalized phase difference. Again we find a good agreement between the results at similar ranges of the magnetic field. From the insets of Fig. \[Bild5\] one can conclude that for our system a magnetic field as small as 1mT can be considered as large already. The notable discrepancy of the numeric and analytic normalized phase difference at large magnetic fields is due to the fact that one has expanded the expression for the phase difference, Eq. (\[eqfaza\]), up to the order $\theta^4$. Since $\theta$ does not saturate to zero, this means that the constant term of Eq. (\[phasehigh\]) is slightly different from the actual value determined numerically.
![(Color online) Comparison of numeric and analytic results at low and high values of the applied magnetic field: (top) magnetic field dependence of the normalized phase difference for small magnetic fields is in agreement with Eq. (\[up1\]) below 0.5mT, whereas the approximation for large magnetic fields, Eq. (\[phasehigh\]), is within one percent of the numerical value already when above 0.8mT. (bottom) Magnetic field dependence of the $z$ component of the magnetization for small magnetic fields is in agreement with Eq. (\[maglow\]) below 0.5mT, whereas the approximation for large magnetic fields, Eq. (\[maghigh\]), is within one percent of the numerical value already when above 0.8mT.[]{data-label="Bild5"}](fig5){width="3.3in"}
The agreement between experimental data and the model for two key static properties underscores that we have solid ground for the analysis of the dynamic results which now follows.
Switch-on dynamics {#sec:ON}
==================
In this Section we present the experimental and theoretical results of the dynamics that takes place when the magnetic field is switched on.
In Fig. \[Bild19\] we plot the comparison of experimental and theoretical data for the dynamics of the normalized phase difference (top) as well as the theoretical results for the normalized $z$ component of the magnetization (bottom) for two values of the applied magnetic field. As an inset we show that for small times the magnetization grows linearly, which is also obtained analytically in Sec. \[sec-initial\]. As expected the rise time for the magnetization is reduced as the applied magnetic field is increased. The inset for the top graph shows that the initial phase difference is quadratic in time, which is again obtained also analytically, Sec. \[sec-initial\].
The fits for the comparison of the experimental and theoretical normalized phase difference are performed by varying the dynamic parameters taking into account the fundamental restrictions [@tilenshort] on their values, at fixed values of the static parameters Eqs. (\[Wextracted\])-(\[A1extracted\]). The model captures the dynamics very well for all times from the onset to the saturation. The extracted values of the dynamic parameters are $$\begin{aligned}
\gamma_1 &\sim& 0.13\,{\rm Pa\,s},\label{labgam}\\
b_\perp^D &\sim& 1.5\times 10^5\,{\rm Am/Vs^2},\label{labbperp}\\
\chi_2^D &\sim& 4\,({\rm Pa\,s})^{-1}.\end{aligned}$$ The dissipative cross-coupling coefficient $\chi_2^D$ is within the allowed interval determined by the restriction [@tilenshort] $$|\chi_2^D|<\sqrt{\frac{b_\perp^D}{\gamma_1M_0^2}}\approx 5.4\,({\rm Pa\,s})^{-1}.$$ The remaining two dynamic parameters do not affect the dynamics significantly and are set to $b_\parallel=b_\perp$ and $\chi_1^D=0$.
![ (Color online) Top: time evolution of the measured normalized phase difference, $r(H)$, fitted by the dynamic model Eqs. (\[f\])-(\[bijD\]). The linear-quadratic onset of $r(H)$ is in accord with the analytic result given in Eq. (\[short\]). Bottom: the corresponding theoretical time evolution of $M_{z}/M_{0}$, initially growing linearly as given in Eq. (\[linear\]). []{data-label="Bild19"}](fig6){width="3.3in"}
To extract from the time evolution of the normalized phase difference, Fig. \[Bild19\] (top), a switching time $\tau$ as a measure of an overall relaxation rate of the dynamics, we use a squared sigmoidal model function $$f(t) = {C'}\left[1-{1+C\over 1+C\exp(-2t/\tau)}\right]^2.
\label{sigmoidal}$$ Remarkably, the relaxation rate, $1/\tau(H)$, shows a linear dependence on $H$, Fig. \[Bild22\]. We were first interested in the effect of the dissipative cross coupling on $1/\tau(H)$. We find that a reasonably strong dynamic cross coupling $\chi_2^D$ is needed in order to obtain the observed linear magnetic field dependence of the relaxation rate. In the absence of this dynamic cross coupling, Fig. \[Bild22\], the relaxation rate levels off already at low fields as expected since the transient angle between $\bf M$ and $\bf n$ gets larger, and starts to decrease for even higher magnetic fields.
![(Color online) The overall relaxation rate, $1/\tau(H)$, as a function of the magnetic field $\mu_0H$, extracted from the experimental data and the theoretical results using the fitting function Eq. (\[sigmoidal\]). Inset: without the dynamic cross-coupling, the relaxation rate levels off already at low fields (dashed).[]{data-label="Bild22"}](fig7){width="3.4in"}
The best match of the relaxation rates $1/\tau(H)$ extracted from the experimental data and the model, Fig. \[Bild22\], allows for a robust evaluation of the dissipative cross-coupling between the magnetization and the director: $$\chi_2^D = (4.0\pm 0.7)\,({\rm Pa\,s})^{-1}.$$
Initial dynamics {#sec-initial}
----------------
We investigate the initial dynamics of the normalized phase difference and magnetization upon application of the magnetic field. Up to linear order we also take into account the pretilt. Initially, $\mathbf{n}$ and $\mathbf{M}$ are parallel to $\mathbf{n}_S$. Keeping the modulus of the magnetization exactly fixed, the initial thermodynamic forces Eqs. (\[h\^M\]) and (\[h\^n\]) are $$\begin{split}
\label{inforco}
&\mathbf{h}^n=0,\\
&\mathbf{h}^{\perp M}=\mu_0H(\varphi_s,0,-1).\end{split}$$ where $\mathbf{h}^{\perp M}$ is the projection of $\mathbf{h}^M$ perpendicular to $\bf M$. With that, the initial quasi-currents are $$\begin{aligned}
Y_i&=&\chi_{ij}^Dh_j^{\perp M}+\chi^R\epsilon_{ijk}n_jh^{\perp M}_k\quad \Rightarrow\nonumber \\
{\bf Y}&=&\mu_0H(\chi_2^DM_0 \varphi_s,\chi^R,-\chi_2^DM_0),\label{intokoY}\\
X_i&=&b_{ij}^Dh_j^{\perp M}+b_{ij}^Rh_j^{\perp M} \quad\Rightarrow\nonumber \\
{\bf X}&=&\mu_0H(b_\perp^D\varphi_s,-(b_1^R+b_2^R)M_0,-b_\perp^D).\label{intokoX}\end{aligned}$$ At finite $\chi_{2}^D$ and zero $\chi^R$ it follows from Eq. (\[intokoY\]) that the $z$ component of the director field responds linearly in time as well as linearly in the magnetic field for small times: $$n_z(t)\approx \varphi_s + \chi_{2}^{D}M_0\mu_0 H\, t.
\label{nzlinear}$$
As a contrast, if $\chi_{2}^D$ is zero, the director responds through the nonzero molecular field $h_z^n$ due to the static coupling $A_1$, $$\label{nzquadratic'}
h_z^n=-A_1M_0 M_z(t) = -A_1M_0b_\perp^D\mu_0H\, t,$$ where $M_z(t) = b_\perp^D\mu_0H t$ is the initial response of the $z$ component of the magnetization, Eq. (\[intokoX\]). The $z$ component of the director field thus responds quadratically in time rather than linearly, $$n_z(t)\approx \varphi_s + \frac{A_1M_0b_\perp^D\mu_0H}{2\gamma_1}t^2.
\label{nzquadratic}
$$
For small times $t$ we can express the refractive index Eq. (\[refract\]) as $$n_e(t)\approx n_{e0}\left[1-\frac{n_{e0}^2-n_o^2}{2n_o^2}\left(\varphi_s + \chi_2^DM_0\mu_0Ht\right)^2\right].$$ The coefficients $a$ and $b$ from Eq. (\[ab\]) are then $$\begin{split}
\label{abiniti}
&a\approx \frac{k_0d}{2}\left[n_e(t)-n_o\right]\left[1-2\left(\chi^R\mu_0H\right)^2t^2\right],\\
&b\approx \frac{k_0d}{2}\left[n_e(t)-n_o\right]\left(-2\chi^R\mu_0H\right)t\\
\end{split}$$ and the normalized intensity of the transmitted light for small times is $$\begin{split}
\label{intrev}
&\frac{I}{I_0}\approx \sin^2\left(\frac{\phi_0}{2}\right)-r_0\varphi_s\chi_2^D\mu_0HM_0\phi_0\sin(\phi_0)t\\
&-\left[\frac{r_0}{2}(\chi_2^D\mu_0HM_0)^2\phi_0\sin(\phi_0)+4(\chi^R\mu_0H)^2\sin^2\left(\frac{\phi_0}{2}\right)\right]t^2.
\end{split}$$ In the lowest order of $t$, for the phase difference one gets a linear term that is also linear in pretilt and a quadratic term which does not vanish if the pretilt is zero: $$\begin{aligned}
\label{short}
r(H)&\approx &r_0\left[\left(\chi_2^DM_0\mu_0H\right)^2t^2+2\varphi_s \chi_2^DM_0\mu_0H t\right]\nonumber \\
&\equiv& k^2t^2+pt.\end{aligned}$$ Eq. (\[short\]) will be used to extract the dissipative cross coupling coefficient $\chi_2^D$ and the pretilt $\varphi_s$ from the experimental data. Furthermore, from Eq. (\[short\]) one can see that in the case of positive (negative) pretilt the normalized phase difference has a minimum at negative (positive) magnetic fields. By measuring the time of this minimum, Fig. \[Bild17\], $$\label{tmin}
t_{\mathrm{min}}=-\frac{\varphi_s}{\chi_2^D\mu_0HM_0},$$ one can calculate the ratio of the pretilt and the dissipative cross coupling.
![ (Color online) Inverse of the time of the minimum determined from the measured normalized phase difference as a function of the magnetic field. The linear behavior in magnetic field is in agreement with Eq. (\[tmin\]).[]{data-label="Bild17"}](fig8){width="3.3in"}
If $\chi_2^D=0$, the time of the minimum decreases more slowly with increasing magnetic field: $$t_{\mathrm{min}}=\sqrt{-\frac{2\gamma_1\varphi_s}{A_1 b_\perp^D\mu_0HM_0}}.$$
The normalized phase difference evaluated at $t_{\mathrm{min}}$ is of second order in the pretilt: $$\label{eqpret}
r(H)_{\mathrm{min}}=-r_0\varphi_s^2.$$ The minimum value Eq. (\[eqpret\]) is independent of the applied magnetic field. This can be explained by the fact that the director field goes through an intermediate state which is approximately aligned with the glass plates of the cell.
![ (Color online) Pretilt, determined from experimental data using Eq. (\[eqpret\]).[]{data-label="Bild18"}](fig9){width="3.3in"}
We note that if both the dissipative cross coupling coefficient $\chi_2^D$ and the pretilt $\varphi_s$ are zero, the normalized phase difference initially grows as $t^4$.
Assuming a negative pretilt, Eq. (\[short\]) predicts a minimum for positive magnetic fields, which is also seen in experiments, Fig. \[Bild19\] (top). In Fig. \[Bild17\] we show experimental inverse times of the minima. The large error at high magnetic fields is due to the time resolution limitations (1ms). >From the linear behavior predicted by Eq. (\[tmin\]) we can extract the ratio between the dissipative cross-coupling and the pretilt. Independently we can extract the pretilt by measuring the values of the minima, Fig \[Bild18\]. Fitting Eq. (\[short\]) to the initial time evolution of measured normalized phase differences (like those presented in Fig. \[Bild19\]) for several values of the magnetic field $\mu_0H$, we determine the parameters $k$ and $p$ shown in Fig. \[Bild20\] and Fig. \[Bild21\], respectively. Therefrom we extract the value of the dissipative cross-coupling parameter $\chi_2^D$ between director and magnetization, $$\chi_2^D \sim (4.0\pm 0.5)\,({\rm Pa\,s})^{-1},$$ and from the parameter $p$ of Eq. (\[short\]) we extract the pretilt, $$\varphi_s \sim -0.065 \pm 0.01.$$
![ (Color online) The coefficient $k$ of Eq. (\[short\]) as a function of the magnetic field $\mu_0H$. The straight line fits are used to extract $\chi_2^D$.[]{data-label="Bild20"}](fig10){width="3.3in"}
![ (Color online) The coefficient $p$ of Eq. (\[short\]) as a function of the magnetic field $\mu_0H$. The straight line fit is used to extract $\varphi_s$.[]{data-label="Bild21"}](fig11){width="3.3in"}
The normalized $z$ component of the magnetization Eq. (\[eq0\]) is linear in $t$: $$\label{linear}
\frac{M_z}{M_0}=\varphi_s + \frac{b_{\perp}^D}{M_0}\mu_0Ht, $$ which is in accord with Fig. \[Bild19\] (bottom). >From the initial behavior one can therefore directly determine the dissipative coefficient $b_\perp^D$.
Let us define the initial rate of the director reorientation as the time derivative of the director $z$ component at $t=0$, $$\frac{1}{\tau_{d}}=\left .\frac{\partial n_z}{\partial t}\right |_{t=0}.$$ For a nonzero dissipative cross-coupling coefficient $\chi_2^D$ the initial rate, Eq. (\[nzlinear\]), is $$\label{dinamre}
\frac{1}{\tau_{d}}=\chi_{2}^{D}M_0\mu_0|H|.$$ However if $\chi_2^D=0$, the initial rate of the director reorientation is proportional to the $z$ component of the magnetization, Eq. (\[nzquadratic’\]), $$\label{staticre}
\frac{1}{\tau_{s}}=\frac{A_1M_0}{\gamma_1}|M_z(t)|.$$ The relaxation rates Eqs. (\[dinamre\]) and (\[staticre\]) describe two different mechanisms of the director reorientation. The former is associated with the dynamic coupling of the director and the magnetization, whereas the latter is governed by the static coupling $A_1$ of the director and the magnetization. Here a deviation of the magnetization from the director is needed to exert a torque on the director.
Dissipative cross-coupling
--------------------------
We have demonstrated that the dissipative cross-coupling of the director and the magnetization, i.e., the $\chi_{ij}^D$ terms of Eqs. (\[X\]) and (\[Y\]), affects the dynamics decisively and is crucial to explain the experimental results. It is described by the parameters $\chi_1^D$ and $\chi_2^D$ of Eq. (\[chiD\]). Here we check the sensitivity of the dynamics to the values of these two parameters. Varying $\chi_1^D$ while keeping $\chi_2^D=0$, Fig. \[Bild9\], we see that the influence of $\chi_1^D$ is rather small and is not substantial. Moreover, the initial dynamics is not affected, Fig. \[Bild9\] (inset).
On the other hand, increasing $\chi_2^D$ strongly reduces the rise time of the normalized phase difference, Fig. \[Bild10\], and also strongly affects the initial behavior (inset). For large values of $\chi_2^D$ one also observes an overshoot in the normalized phase difference.
By inspecting Eq. (\[chiD\]) one sees that the influence of $\chi_1^D$ is largest when ${\bf M}\perp{\bf n}$, ${\bf h}^n\parallel{\bf M}$ and ${\bf h}^M\parallel{\bf n}$. On the other hand, the influence of $\chi_2^D$ is largest when ${\bf M}\parallel{\bf n}$. Since $\bf M$ and $\bf n$ are initially parallel and moreover the transient angle between them never gets large due to the strong static coupling compared to the magnetic fields applied, it is understandable that $\chi_2^D$ affects the dynamics more than $\chi_1^D$.
![Normalized phase difference at different values of the dissipative cross-coupling parameter $\chi_{1}^D$, $\chi_2^D=0$, $\mu_0 H=50$mT. Inset: the initial behavior is not affected.[]{data-label="Bild9"}](fig12){width="3.3in"}
![Normalized phase difference at different values of the dissipative cross-coupling parameter $\chi_{2}^D$, $\chi_1^D=0$, $\mu_0 H=50$mT. Inset: the initial behavior is strongly affected as well.[]{data-label="Bild10"}](fig13){width="3.3in"}
Reversible cross-coupling {#sec:reversible}
-------------------------
The reversible cross-coupling of the director and the magnetization, described by the $\chi^R$ terms of Eqs. (\[XR\]) and (\[YR\]), has not been considered up to this point. We focus on the reversible cross coupling coefficient $\chi^R$ and put both reversible tensors $b_{ij}^R$ and $(\gamma^{-1})_{ij}^R$ of Eqs. (\[b\_ij\^R\]) and (\[gamma\_inv\]) to zero.
If the reversible currents are included, both variables wander out of the $xz$ plane dynamically, which will be described by the azimuthal angles $\delta$ and $\varphi$ of the magnetization and the director, respectively, defined by $\mathbf{M}=M_0(\cos\delta\sin\psi,\sin\delta\sin\psi,\cos\psi)$, $\mathbf{n}=(\cos\varphi\sin\theta,\sin\varphi\sin\theta,\cos\theta)$. The dynamic behavior of both azimuthal angles is shown in Fig. \[Bild15\].
![ (Color online) The time dependences of the azimuthal angles (degrees) of the director ($\varphi$) and the magnetization ($\delta$) at $z=d/2$ for different values of $\chi^R$, $\chi_1^D=\chi_2^D=0$, $\mu_0H=10$mT.[]{data-label="Bild15"}](fig14){width="3.3in"}
Contrary to the polar angles we find that the response of the azimuthal angle of the director is faster than that of the magnetization. >From Fig. \[Bild15\] we read off that the maximum azimuthal angles increase with $\chi^R$, being higher for the magnetization than for the director. We note again that here we only included the reversible cross-coupling $\chi^R$. >From the initial quasi-currents Eqs. (\[intokoY\]) and (\[intokoX\]) one can see that the initial azimuthal response of the magnetization can be faster than that of the director if the coefficients of the tensor $b_{ij}^R$ are sufficiently large, $$|b_1^R+b_2^R|>|\chi^R|/M_0.$$
There exists a direct way of detecting the possible dynamics in the $xy$ plane. The intensity of the transmitted light in the experiments with crossed polarizers at $45^\circ$ and $-45^\circ$ is given by Eq. (\[normint\]), $$\frac{I}{I_0}=\frac{a^2}{c^2}\sin^2(c).$$ It is this quantity that is typically measured. On the other hand, crossed polarizers at $0^\circ$ and $90^\circ$ give us the intensity $$\label{intxy}
\frac{I}{I_0}=\frac{b^2}{c^2}\sin^2(c),$$ with a and b given by Eq. (\[ab\]). This method is better suited for detecting the $xy$ dynamics, since $b$ is more sensitive to the deviation of the director field from the $xz$ plane.
Our numerical calculations have revealed that, owing to the reversible dynamics, the magnetization and the director are not confined to the $xz$ plane. As a consequence, the maxima of the time-dependent intensity of transmitted light are lower than unity, Fig. \[Bild16\], in contrast to the case of a purely in-plane (dissipative) dynamics. Observation of the lower maxima could thus be an indication of the azimuthal dynamics. This effect is more prominent at higher magnetic fields and at higher values of the reversible cross coupling coefficients.
In recent experiments no clear-cut consequences of the azimuthal dynamics have been found using crossed polarizers at $0^\circ$ and $90^\circ$. In the following we will therefore discard the reversible dynamics.
![ (Color online) Time dependence of the normalized intensity of transmitted light for zero and nonzero values of the reversible cross coupling coefficient $\chi^R$; $\mu_0H=5$mT.[]{data-label="Bild16"}](fig15){width="3.3in"}
Switch-off dynamics {#sec:OFF}
===================
Dynamics of the normalized phase difference after switching off the magnetic field has been also measured. In experiments, the initial state is obtained by switching on the desired magnetic field and waiting for a couple of seconds. Contrary to the previous experiments, here the initial state is not homogeneous.
In Fig. \[Bild25\] we compare the experimental and numerical normalized phase difference at two different fields. We observe, similarly to the switch-on case, that the normalized phase difference goes through a minimum. This is again explained by the fact that the director field goes through a state, which is approximately aligned with the surfaces of the glass plates.
![ (Color online) Experimental and numerical normalized phase difference as a function of time at different values of the applied magnetic field.[]{data-label="Bild25"}](fig16){width="3.3in"}
![ (Color online) Normalized phase difference as a function of time at 5 mT, calculated with $\chi_2^D=0$ and $\chi_2^D=4.0$(Pas)$^{-1}$.[]{data-label="Bild23"}](fig17){width="3.3in"}
Numerical calculations reveal that a strong dissipative cross-coupling causes the initial behavior of the normalized phase difference to be a linear function in time, Fig. \[Bild23\], as found experimentally, Fig. \[Bild25\].
To extract a relaxation time $\tau$ of the normalized phase difference we use an exponential function $$f(t)=f(0){\rm e}^{-t/\tau}.$$ The relaxation rate $1/\tau$ for the experimental data is shown in Fig. \[Bild26\]. It saturates at a finite value as one increases the magnetic field. This is expected since the initial director and magnetization fields do not change much with magnetic field any more when the field is large.
![ (Color online) Experimental switch-off relaxation rate of the normalized phase difference as a function of the applied magnetic field.[]{data-label="Bild26"}](fig18){width="3.3in"}
In Fig. \[Bild28\] the relaxation rate of both the computed phase difference and the magnetization is shown. One can see that the relaxation rate of the magnetization is smaller than that of the normalized phase difference, owing to the fact that it is the director that is driven by the nonzero elastic force, while the magnetization only follows. This is true for all allowed values of the dynamic cross coupling parameters.
![ (Color online) Relaxation rate of the normalized phase difference and $z$ component of the magnetization after switching off the magnetic field of strength $\mu_0H$ at $\chi_2^D=4$(Pas)$^{-1}$ and $\varphi_s=0$.[]{data-label="Bild28"}](fig19){width="3.3in"}
One can derive analytic formulas for the relaxation rate in the limit of low magnetic fields. With the assumption that the relaxation follows a simple exponential function, it is possible to extract the relaxation rate $1/\tau^{off}$ from the initial time derivative of the normalized phase difference, $$\label{eqfitx}
r(H,t)\approx r(H,t=0)\left(1-\frac{t}{\tau^{off}}\right).$$ Note that Eq. (\[eqfitx\]) is defined only when $r(H,t=0)\neq 0$.
One starts with the director quasi-current $\bf Y$, Eq. (\[Y\]). The response of the $z$ component of the director field is $n_z\approx n_z(z,t=0)-Y_z(z,t=0)\,t$, which one uses in the equation Eq. (\[eqfaza\]) for the phase difference, $$\label{tauoffdef}
\frac{1}{\tau^{off}}=\frac{2k_0r_0(n_{e0}-n_o)}{\phi_0 r(H,t=0)}\int_{0}^d \mathrm{d}z\frac{n_z(z)Y_z(z) }{\left(1+\frac{n_{e0}^2-n_o^2}{n_o^2}n_z^2(z)\right)^{3/2}},$$ where all $z$-dependent quantities are evaluated at $t=0$. In the last step the integrand is expanded up to linear order in time and the relaxation rate in the low-magnetic-field limit is finally expressed as $$\label{tauoff}
\frac{1}{\tau^{off}}=\frac{\left(1+r_0\varphi_s^2\right)\left[\mu_0HM_0\left(1+6\frac{\xi}{d}\right)+12\frac{K_1\varphi_s}{d^2}\right]\chi_2^D}{\frac{\mu_0HM_0d^2}{20K_1}\left(1+10\frac{\xi}{d}+30 \frac{\xi^2}{d^2}\right)+\left(1+6\frac{\xi}{d}\right)\varphi_s},$$ which is linear in the dissipative cross-coupling coefficient $\chi_2^D$.
Not only does the dissipative cross-coupling make the switching process faster when switching on the field, this can be also true for switching off the field, Figs. \[Bild23\] and \[Bild29\]. Fig. \[Bild29\] shows the relaxation rate of the normalized phase difference at a high magnetic field as a function of the dissipative cross coupling coefficient $\chi_2^D$. As expected, the relaxation rate decreases with increasing rotational viscosity $\gamma_1$. The relaxation rate at first increases with increasing values of $\chi_{2}^D$, which seems also to be the case for small magnetic fields described by Eq. (\[tauoff\]). For values above approximately $\chi_2^D=3.5$(Pas)$^{-1}$, the relaxation rate starts to decrease rather rapidly. This is in contrast with the field switch-on case, where the response is faster for increasing values of $\chi_2^D$.
The increasing part of the dependence $\tau^{-1}(\chi_2^D)$ in Fig. \[Bild29\] is due to the director elastic forces, which drive the switch-off dynamics and also enter Eq. (\[Mdot\]) through the dissipative cross-coupling governed by $\chi_2^D$. At higher values of $\chi_2^D$ one must however also consider the part of the thermodynamic forces corresponding to the static ($A_1$) coupling between the director and the magnetization. Focusing only on the director equation Eq. (\[ndot\]), one sees that the director relaxes towards the magnetization with a characteristic time set by the rotational viscosity and the static coupling ($A_1$). On the other hand, the positive value of $\chi_2^D$ has the opposite effect. While both fields eventually relax to the ground state parallel to $x$, the angle between them is decreasing slower and slower as the dynamic cross-coupling ($\chi_2^D$) gets larger. For small magnetic fields one can study the relaxation rate of the dynamic eigenmodes, Eq. (\[eqflu\]) of the next section. The value of $\chi_2^D$ above which the relaxation rate starts to decrease then reads $$\chi_2^D=
\begin{cases}
\frac{A_1b_\perp^D}{A_1M_0^2+K(\pi/d)^2}& \text{if $\frac{1}{\gamma_1}>\frac{b_\perp^D}{M_0^2}$,} \\
\frac{1}{\gamma_1} & \text{if $\frac{1}{\gamma_1}<\frac{b_\perp^D}{M_0^2}$.}
\end{cases}$$ In our case $\frac{1}{\gamma_1}>\frac{b_\perp^D}{M_0^2}$ holds and the maximum is at $\chi_2^D \approx 3.5$(Pas)$^{-1}$.
The switch-on case is different in that the dynamics is driven by the external magnetic field. If the external field is sufficiently high (large compared to $A_1M_0$), the static cross-coupling effects, which decrease the relaxation rate in the switch-off case through the increasing dynamic cross-coupling $\chi_2^D$, can be neglected and hence the relaxation rate is monotonically increasing with $\chi_2^D$.
![ (Color online) Relaxation rate at $\mu_0H=50$mT as a function of the dissipative coefficient $\chi_2^D$ for different values of the director rotational viscosity $\gamma_1$.[]{data-label="Bild29"}](fig20){width="3.3in"}
Fluctuations and light scattering {#sec:flucts}
=================================
Nematic liquid crystals appear turbid in sufficiently thick layers [@degennesbook]. The scattering of light is caused by strong director fluctuations which cause fluctuations in the dielectric tensor $$\label{vareps}
\varepsilon_{ij}=\varepsilon_\perp\delta_{ij}^\perp+\varepsilon_\parallel n_in_j,$$ where $\varepsilon_\perp$ and $\varepsilon_\parallel$ are dielectric susceptibilities for the electric field perpendicular and parallel to the director, respectively. Fluctuations are easy to observe experimentally and are used to determine the viscoelastic properties of liquid crystals [@alenkaliqcryst].
In this paper we derive the relaxation rates of the fluctuations without taking into account the effects of flow. Since the director is coupled to the magnetization, we now have two fluctuation modes for each director fluctuation mode of the usual nematic [@alenkanature].
The fluctuating director and magnetization fields are linearized as $$\begin{split}
\label{flukss}
\mathbf{n}&=\mathbf{n}_0+\delta \mathbf{n},\\
\mathbf{M}&=\mathbf{M}_0+M_0\delta \mathbf{m},
\end{split}$$ where the equilibrium director $\mathbf{n}_0$ and magnetization $\mathbf{M}_0$ fields point in $x$ direction in which a magnetic field is applied, whereas fluctuations $\delta \mathbf{n}$ and $\delta \mathbf{m}$ are perpendicular, $\mathbf{n}_0\cdot\delta \mathbf{n}=\mathbf{M}_0\cdot\delta \mathbf{m}=0$. The ansatz for the director fluctuations is $$\delta \mathbf{n}(\mathbf{r})=\frac{1}{V}\sum_{\mathbf{q}}\delta \mathbf{n}(\mathbf{q})e^{\mathrm{i}\mathbf{q}\cdot\mathbf{r}},$$where $\mathbf{q}=q_x \hat{\mathbf{e}}_x+q_y \hat{\mathbf{e}}_y+q_z \hat{\mathbf{e}}_z$ is the wave vector of the fluctuation. A similar ansatz is used for the fluctuations of the magnetization. In a confined system, the fluctuation spectrum generally depends on the interaction of the nematic with the surface [@alenkaliqcryst]. For simplicity we will use the infinite anchoring limit, so that $q_z=n\pi/d, n\in \mathbb{N}$, while $q_x$ and $q_y$ are in principle arbitrary. For details regarding the anchoring effect we refer to Ref. [@alenkaliqcryst].
To understand the static light scattering experiments one must determine thermal averages of the fluctuations. This is done by finding linear combinations of the variables in terms of which the free energy functional Eq. (\[f\]) is expressed as a sum of quadratic terms, and making use of equipartition. Such linear combinations are uncorrelated (statistically independent). A systematic way to perform this decomposition is to write the free energy of a fluctuation $\bf q$-mode as a quadratic form and find the corresponding eigenvalues and eigenvectors, $$\label{eq1}
F({\bf q})=\frac{1}{2}\mathbf{\delta x}({\bf q})^H \textsf{E}(\mathbf{q}) \mathbf{\delta x}({\bf q}),$$ where $\mathbf{\delta x}({\bf q})=\{\delta n_z({\bf q}),\delta m_z({\bf q}),\delta n_y({\bf q}),\delta m_y({\bf q})\}$, in short $\mathbf{\delta x}({\bf q})\equiv\{n_z,m_z,n_y,m_y\}$, is the vector of the fluctuation amplitudes, $\textsf{E}({\bf q})$ is a self-adjoint matrix and superscript $^H$ is the conjugate transpose.
In lowest order of fluctuations, the contributions Eq. (\[eq1\]) of the free energy Eq. (\[f\]) are [@degennesbook] $$\begin{aligned}
F({\bf q})=\frac{1}{2V}\Bigg[&(K_1 q_y^2+K_2q_z^2+K_3q_x^2+A_1M_0^2)|n_y|^2\nonumber\\
&+(K_1q_z^2+K_2 q_y^2+K_3q_x^2+A_1M_0^2)|n_z|^2\nonumber\\
&+(K_1-K_2)q_zq_y(n_yn_z^*+n_y^*n_z)\nonumber\\
&+(\mu_0HM_0+A_1M_0^2)(|m_y|^2+|m_z|^2)\nonumber\\
&- A_1M_0^2(n_ym_y^*+n_y^*m_y+n_zm_z^*+n_z^*m_z)\Bigg].\end{aligned}$$ For completeness (not needed here), the volume-integrated free energy is $F = \sum_{\bf q}F({\bf q})$.
Before giving the eigenvectors of the quadratic form $\textsf{E}$, we perform a rotation in the $yz$ plane, $(n_y,n_z)\to (n_1,n_2)$ and $(m_y,m_z)\to (m_1,m_2)$, where the new bases in this plane are $\{\hat{\bf e}^{n}_1,\hat{\bf e}^{n}_2\}$ and $\{\hat{\bf e}^{M}_1,\hat{\bf e}^{M}_2\}$. Vectors $\hat{\bf e}^{n}_2$ and $\hat{\bf e}^{M}_2$ are normal to ($\mathbf{q},\mathbf{n}_0$) and ($\mathbf{q},\mathbf{m}_0$) plane, respectively and vectors $\hat{\bf e}^{n}_1$ and $\hat{\bf e}^{M}_1$ are normal to $\hat{\bf e}^{n}_2$ and $\hat{\bf e}^{M}_2$, respectively. It should be emphasized that we are studying the case $\mathbf{n}_0||\mathbf{m}_0$, so the planes ($\mathbf{q},\mathbf{n}_0$) and ($\mathbf{q},\mathbf{m}_0$) are identical. In the confined system, this would not be the case if the external magnetic field were applied in any direction other than parallel to the initial homogeneous state.
A general fluctuation $\delta \mathbf{x}({\bf q})$ can be written as $$\label{amplit}
\delta \mathbf{x} = t_1 \mathbf{t}_1+p_1 \mathbf{p}_1+t_2 \mathbf{t}_2+p_2 \mathbf{p}_2,$$ where $\mathbf{t}_1,\mathbf{t}_2,\mathbf{p}_1$, $\mathbf{p}_2$ are the eigenvectors of the quadratic form $\textsf{E}$ and $t_1,t_2,p_1$, $p_2$ are the amplitudes of these uncorrelated excitations. The eigenvectors are $$\begin{aligned}
\mathbf{t}_\alpha &=& a^t_\alpha \hat{\mathbf{e}}_\alpha^n + b^t_\alpha \hat{\mathbf{e}}_\alpha^M\label{eigenmod1} \\
&=&\frac{Z_\alpha^-}{\sqrt{1+(Z_\alpha^-)^2}} \hat{\mathbf{e}}_\alpha^n-\frac{1}{\sqrt{1+(Z_\alpha^-)^2}}\hat{\mathbf{e}}_\alpha^M,\nonumber\\
\mathbf{p}_\alpha &=& a^p_\alpha \hat{\mathbf{e}}_\alpha^n + b^p_\alpha \hat{\mathbf{e}}_\alpha^M \label{eigenmod2}\\
&=&\frac{Z_\alpha^+}{\sqrt{1+(Z_\alpha^+)^2}} \hat{\mathbf{e}}_\alpha^n-\frac{1}{\sqrt{1+(Z_\alpha^+)^2}}\hat{\mathbf{e}}_\alpha^M,\nonumber\end{aligned}$$ where $$Z_\alpha^\pm=\frac{-\mu_0HM_0+K_\alpha q_ \perp^2+K_3q_x^2\pm s_\alpha}{2A_1M_0^2},
\label{Z_alpha}$$ with $q_\perp^2=q_y^2+q_z^2$ and $$s_\alpha^2=4A_1^2M_0^4+\left(K_\alpha q_\perp^2+K_3q_x^2-\mu_0HM_0\right)^2.
\label{s_alpha}$$
The excitation modes $\mathbf{t}_1$ and $\mathbf{p}_1$ are the analogues of the splay-bend mode in the usual NLCs, whereas $\mathbf{t}_2$ and $\mathbf{p}_2$ are the analogues of the twist-bend mode.
It is found that in the limit of large magnetic fields these excitations become decoupled, i.e., one eigenvector only contains the fluctuation of the director field and the other the fluctuation of the magnetization field, Figs. \[Bild29a\] and \[Bild29b\].
![ (Color online) The normalized coefficients Eq. (\[eigenmod1\]) of the eigenvectors $\mathbf{t}_1$ and $\mathbf{t}_2$ as a function of the applied magnetic field with $q_x=0$ and $q_\perp =\pi/2$; $K_1=K_2$. []{data-label="Bild29a"}](fig21){width="3.3in"}
![ (Color online) The normalized coefficients Eq. (\[eigenmod2\]) of the eigenvectors $\mathbf{p}_1$ and $\mathbf{p}_2$ as a function of the applied magnetic field with $q_x=0$ and $q_\perp =\pi/2$; $K_1=K_2$. []{data-label="Bild29b"}](fig22){width="3.3in"}
The thermal averages of the squared amplitudes of the independent excitations read $$\begin{aligned}
\label{thermal}
\langle |t_\alpha(\mathbf{q})|^2\rangle &= \frac{k_BTV}{\frac{1}{2}\left(2A_1M_0^2+\mu_0HM_0+K_\alpha q_\perp^2+K_3 q_x^2-s_\alpha\right)},\\
\langle |p_\alpha(\mathbf{q})|^2\rangle &= \frac{k_BTV}{\frac{1}{2}\left(2A_1M_0^2+\mu_0HM_0+K_\alpha q_\perp^2+K_3 q_x^2+s_\alpha\right)},\label{pms}\end{aligned}$$ with $k_B$ the Boltzmann constant and $T$ the temperature, whereas their thermal cross-correlations are zero.
If $K_1=K_2$, the splay-bend $(\alpha=1)$ and the twist-bend $(\alpha=2)$ excitation modes have the same structure Eqs. (\[Z\_alpha\])-(\[s\_alpha\]), Figs. \[Bild29a\] and \[Bild29b\], as well as the same energy and thermal amplitude Eqs. (\[thermal\])-(\[pms\]). The same is true in the degenerate case when $\mathbf{q}=q\,\hat{\mathbf{e}}_x$, i.e., for a pure bend excitation (in an unconfined system), where there is no difference between the modes $\alpha=1,2$ and the bases $\{\hat{\bf e}^{n}_1,\hat{\bf e}^{n}_2\}$ and $\{\hat{\bf e}^{M}_1,\hat{\bf e}^{M}_2\}$ are chosen arbitrarily in the $yz$ plane.
The space correlations are expressed as $$\begin{aligned}
\langle t_\alpha(\mathbf{r})t_{\alpha'}(\mathbf{r}')\rangle & = \frac{1}{V^2}\sum_{\mathbf{q},\mathbf{q}'}\langle t_\alpha(\mathbf{q})t_{\alpha'}(\mathbf{q}')\rangle e^{-\mathrm{i}(\mathbf{q}\cdot\mathbf{r}+\mathbf{q}'\cdot\mathbf{r}')}\nonumber \\
&=\frac{\delta_{\alpha,\alpha'}}{V^2}\sum_{\mathbf{q}}\langle t_\alpha(\mathbf{q})t_{\alpha}(-\mathbf{q})\rangle e^{-\mathrm{i}\mathbf{q}\cdot(\mathbf{r}-\mathbf{r}')},
\label{corr}\end{aligned}$$ and similarly for $\langle p_\alpha(\mathbf{r})p_{\alpha'}(\mathbf{r}')\rangle$, whereas $\langle t_\alpha(\mathbf{r})p_{\alpha'}(\mathbf{r}')\rangle=0$. In the large magnetic field limit these correlations are $$\begin{aligned}
\label{realcorel2}
\langle t_\alpha(\mathbf{r})t_\alpha(\mathbf{r}')\rangle &\approx \frac{k_BT}{4\pi K}\frac{1}{r}e^{-q_0r},\\
\langle p_\alpha(\mathbf{r})p_\alpha(\mathbf{r}')\rangle &\approx \frac{k_BT}{(2\pi)^3\mu_0HM_0} \delta(r),\end{aligned}$$ where $r=|\mathbf{r}-\mathbf{r}'|$ and $q_0=\sqrt{A_1M_0^2/K}$.
In experiments one measures the intensity of the scattered light. To calculate this intensity we need an expression for the amplitude of the outgoing electric field. We start with an incident electric field $\mathbf{E}_i$, described by a plane wave: $\mathbf{E}=E_0\hat{\mathbf{i}}\,{\rm e}^{\mathrm{i}(\mathbf{k}_i\cdot \mathbf{r}-\omega t)}$, where $\mathbf{k}_i$ is the wave vector, $E_0$ the amplitude and $\omega$ the frequency of the incident light. We then proceed with a summation of the electric field contributions of the scattered light through the whole cell, treating every point $\bf r$ as a radiating dipole. Last, we project the electric field on the axis $\hat{\mathbf{f}}$ of the analyzer. The electric field amplitude of the scattered light is [@degennesbook]: $$\begin{aligned}
\label{eqfield}
&E_f(\mathbf{q},t)\nonumber\\ &=\frac{E_0\omega^2}{c^2R}{\rm e}^{\mathrm{i}(\mathbf{k}_f\cdot\mathbf{r}'-\omega t)}\int_V\!\! \mathrm{d}^3r\,{\rm e}^{-\mathrm{i}\mathbf{q}\cdot \mathbf{r}}\hat{f}_i\, [\varepsilon_{ij}(\mathbf{r},t)-\delta_{ij}]\,\hat{i}_j \nonumber\\
&=\frac{E_0\omega^2}{c^2R}{\rm e}^{\mathrm{i}(\mathbf{k}_f\cdot\mathbf{r}'-\omega t)}\,\hat{f}_i \varepsilon_{ij}(\mathbf{q},t)\,\hat{i}_j,\end{aligned}$$ where $\mathbf{k}_f$ is the wave vector of the scattered light, $R$ is the distance from the sample to the detector at ${\bf r}'$ and $\mathbf{q}=\mathbf{k}_f-\mathbf{k}_i$ is the fluctuation wave vector. In the last line of Eq. (\[eqfield\]) we discarded the Fourier contribution of $\delta_{ij}$, since it is nonzero only if $\mathbf{q}=0$. We have assumed that $R$ is large compared to the size of the scattering region which in turn is much larger than the wave length of the light, and that we are in the limit of small dielectric anisotropy.
In our calculations below, we will be using details of an experimental set-up usually used for measuring splay-bend fluctuations in a NLC, which in our geometry have $\delta{\bf n} = \delta n_z\hat{\bf e}_z$, $q_y = 0$, $\hat{\mathbf{e}}_2^{n,M}=\hat{\mathbf{e}}_y$ and $\hat{\mathbf{e}}_1^{n,M}=\hat{\mathbf{e}}_z$. In this case we have a polarizer and an analyzer that are both in the $xz$ plane. The polarizer $\hat{\mathbf{i}}$ is parallel to the $x$ axis, whereas the analyzer $\hat{\mathbf{f}}$ is at an angle $\zeta$ from the $x$ axis. In Eq. (\[eqfield\]), the projection of the fluctuating part of the dielectric tensor Eq. (\[vareps\]) reads $$\hat{f}_i\, \varepsilon_{ij}(\mathbf{q},t)\,\hat{i}_j=\varepsilon_a f_z\delta n_z,
\label{projection}$$ where $f_z=\hat{\mathbf{f}}\cdot \hat{\mathbf{e}}_z$. Using the expansion $$\delta n_z=(t_1\mathbf{t}_1 +p_1\mathbf{p}_1) \cdot \hat{\mathbf{e}}_1^n,$$ the scattering cross section $\sigma=\langle E_f^*(\mathbf{q},t)E_f(\mathbf{q},t)\rangle$ with ${\bf q}\cdot\hat{\bf e}_y=0$ is $$\begin{aligned}
\sigma&=\frac{\varepsilon_a^2\omega^4}{c^4}\langle |\delta n_z(\mathbf{q})|^2\rangle f_z^2\nonumber \\&=\frac{\varepsilon_a^2\omega^4}{c^4}\left(C_1^+\langle |t_1(\mathbf{q})|^2\rangle+C_1^-\langle |p_1(\mathbf{q})|^2\rangle\right) f_z^2,\label{scatt}\end{aligned}$$ with the coefficient $$\begin{aligned}
C_1^\pm&=\frac{(Z_1^\mp)^2}{1+(Z_1^\mp)^2}.\end{aligned}$$ In the usual experimental set-up one observes two splay-bend modes, ${\bf t}_1$ and ${\bf p}_1$, as opposed to the usual NLC, where one observes only one splay-bend mode.
Asymptotic behaviors of the coefficients $C_1^+$ and $C_1^-$ at large magnetic fields, $$\begin{aligned}
C_1^+&\asymp 1-\frac{A_1^2M_0^4}{(\mu_0HM_0)^2},\label{eqasc1}\\
C_1^-&\asymp \frac{2(K_\alpha q_\perp^2+K_3 q_x^2)^2-3A_1^2M_0^4}{(\mu_0HM_0)^2},\label{eqasc2}\end{aligned}$$ reveal that in the large magnetic field limit only the eigenmode $\mathbf{t}_1$ contributes to the scattering cross section Eq. (\[scatt\]). The dynamics of the fluctuations is probed by dynamic light scattering, where one measures the time correlation of the light intensity $I(t)$, $$g^{(2)}(t)=\frac{\langle I(0)I(t)\rangle}{\langle I(0) \rangle^2}.
\label{g2}$$ Assuming Gaussian fluctuations it follows $$g^{(2)}(t)=1+\left|g^{(1)}(t)\right|^2,
\label{g21}$$ where $$g^{(1)}(t)=\frac{\langle E_f^*(\mathbf{q},0)E_f(\mathbf{q},t)\rangle}{\langle |E_f(\mathbf{q},0)|^2 \rangle}\label{g1c}$$ is the time correlation of the scattered light electric field.
To calculate the time dependence of the fluctuations, we first linearize the system of dynamic equations and determine the dynamic eigenmodes. Considering only the dissipative dynamics, Eqs. (\[X\])-(\[Y\]), and using $\delta \mathbf{n}=\delta n_1 \hat{\mathbf{e}}_1^n+ \delta n_2 \hat{\mathbf{e}}_2^n$, $\delta \mathbf{m}=\delta m_1 \hat{\mathbf{e}}_1^M+ \delta m_2 \hat{\mathbf{e}}_2^M$, we find a 2$\times$2 homogeneous system for each $\alpha=1,2$, $$\begin{split}
\label{eqflu}
\frac{1}{\tau}\delta n_\alpha&=\left[\frac{1}{\gamma_1}\left(K_\alpha q_\perp^2+K_3q_x^2+A_1M_0^2\right)-\chi_2^DA_1M_0^2\right]\delta n_\alpha \\ &+\left[A_1M_0^2\left(\chi_2^D-\frac{1}{\gamma_1}\right)+\chi_2^D\mu_0HM_0\right]\delta m_\alpha, \\
\frac{1}{\tau}\delta m_\alpha&=\left[-b_\perp^DA_1+\chi_2^D\left(K_\alpha q_\perp^2+K_3q_x^2+A_1M_0^2\right)\right]\delta n_\alpha\\ &+\left[b_\perp^DA_1 \left(1+\frac{\mu_0HM_0}{A_1M_0^2}\right)-\chi_2^DA_1M_0^2\right]\delta m_\alpha,
\end{split}$$ which can be rewritten as $$\label{flu2}
\left(\textsf{A}-\frac{1}{\tau}\textsf{I}\right)\begin{pmatrix}
\delta n_\alpha \\
\delta m_\alpha
\end{pmatrix}=\mathbf{0}$$ and has nontrivial solutions if $\mathrm{det}(\textsf{A}-\frac{1}{\tau}\textsf{I})=0$. The dynamic eigenmodes are the eigenvectors of the matrix $\textsf{A}$, $$\begin{aligned}
\mathbf{t}_\alpha^h &=& c^{t}_\alpha \hat{\mathbf{e}}_\alpha^n + d^{t}_\alpha \hat{\mathbf{e}}_\alpha^M,\label{dyneigenmod1} \\
\mathbf{p}_\alpha^h &=& c^{p}_\alpha \hat{\mathbf{e}}_\alpha^n + d^{p}_\alpha \hat{\mathbf{e}}_\alpha^M, \label{dyneigenmod2} \end{aligned}$$ where the components $c^{t}_\alpha, c^{p}_\alpha, d^{t}_\alpha, d^{p}_\alpha$ are functions of the static and dynamic material parameters and will not be given explicitly. It is important to realize that the dynamic fluctuation modes Eqs. (\[dyneigenmod1\])-(\[dyneigenmod2\]) in general differ from the statistically independent excitation modes Eqs. (\[eigenmod1\])-(\[eigenmod2\]). If the reversible dynamics Eqs. (\[XR\])-(\[YR\]) is included, a 4$\times$4 eigensystem is obtained coupling both $\alpha$’s. In that case splay-bend and twist-bend dynamic modes are no longer decoupled and each eigenmode spans all directions $\{\hat{\bf e}_1^ {n,M}, \hat{\bf e}_2^{n,M}\}$.
The time dependence of a fluctuation is first expressed in terms of the dynamic eigenmodes Eqs. (\[dyneigenmod1\])-(\[dyneigenmod2\]), which are then further expressed by the uncorrelated excitation modes Eqs. (\[eigenmod1\])-(\[eigenmod2\]). Using Eqs. (\[eqfield\])-(\[projection\]) and expressing $\delta n_z(t)$ of the splay-bend fluctuation as just explained, the electric field time correlation Eq. (\[g1c\]) becomes $$|g^{(1)}(t)|=\frac{D_1^+(t)\langle |t_1(\mathbf{q},0)|^2\rangle +D_1^-(t)\langle |p_1(\mathbf{q},0)|^2\rangle }{C_1^+\langle |t_1(\mathbf{q},0)|^2\rangle+C_1^-\langle |p_1(\mathbf{q},0)|^2\rangle},$$ where $$\begin{aligned}
D_1^+(t)&=& (\mathbf{t}_1 \cdot \hat{\mathbf{e}}_1^n)^2f_{\rm I}(t) +(\mathbf{t}_1 \cdot \hat{\mathbf{e}}_1^n)(\mathbf{t}_1 \cdot \hat{\mathbf{e}}_1^M)f_{\rm II}(t),\nonumber\\
D_1^-(t)&=& (\mathbf{p}_1 \cdot \hat{\mathbf{e}}_1^n)^2f_{\rm I}(t) +(\mathbf{p}_1\cdot \hat{\mathbf{e}}_1^n)(\mathbf{p}_1 \cdot \hat{\mathbf{e}}_1^M)f_{\rm II}(t).\nonumber\\\end{aligned}$$ The functions $f_{\rm I}(t)$ and $f_{\rm II}(t)$ are expressed using the components $c^{t}_1, c^{p}_1, d^{t}_1, d^{p}_1$ and the relaxation times of the dynamic eigenmodes denoted by $\tau_1^t$ and $\tau_1^p$: $$\begin{aligned}
f_{\rm I}(t)&=\frac{c^{t}_1 d^{p}_1 {\rm e}^{-t/\tau_1^t}-c^{p}_1 d^{t}_1 {\rm e}^{-t/\tau_1^p}}{c^{t}_1 d^{p}_1-c^{p}_1 d^{t}_1},\\
f_{\rm II}(t)&=\frac{d^{t}_1d^{p}_1({\rm e}^{-t/\tau_1^t}- {\rm e}^{-t/\tau_1^p})}{c^{t}_1 d^{p}_1-c^{p}_1 d^{t}_1}.\end{aligned}$$
In the limit of large magnetic fields one gets $D_1^\pm \to C_1^\pm {\rm e}^{-t/\tau_1^t}$. Taking into account also the large magnetic field dependence of the coefficients $C_1^\pm$, Eqs. (\[eqasc1\])-(\[eqasc2\]), the intensity correlation function Eq. (\[g21\]) is a single exponential $$g^{(2)}(t)=1+{\rm e}^{-2t/\tau_1^t}.$$
![Relaxation rates of almost pure bend fluctuations ($q_x\gg q_\perp$) and the corresponding dynamic eigenmodes as a function of the applied magnetic field. The dashed lines represent the limiting behavior of the relaxation rates, described by Eqs. (\[lim1\]) and (\[lim2\]). For clarity, a smaller value of the rotational viscosity was used to make the asymptotic behavior set in sooner.[]{data-label="Bild30"}](fig23){width="3.3in"}
It is found that the dynamics of the eigenmodes $\mathbf{t}_\alpha^h$ slows down ($\tau_\alpha^t \to \infty$) at a negative critical magnetic field, here given for ${\bf q}=q_z \hat{\bf e}_z$: $$\label{bcrit}
\mu_0H_c^{(\alpha)}=-\frac{A_1M_0K_\alpha q_z^2}{K_\alpha q_z^2+A_1M_0^2}.$$ Negative value of the critical magnetic field means that it is pointing in the direction opposite to the magnetization. If the applied magnetic field is more negative than the critical field, the magnetization starts to reverse. In NLCs, $K_2<K_1$ usually holds and it is the twist mode $\mathbf{t}_2^h$ that slows down at a less negative magnetic field. With the smallest wave number $q_z = \pi/d$ we get $\mu_0H_c^{(2)}=-2.5$mT.
In Fig. \[Bild30\] we present the magnetic field dependence of the relaxation rate of almost pure bend ($q_x\gg q_\perp$) fluctuations. We also depict the corresponding eigenmodes at a small positive field and at large magnetic fields.
For a general fluctuation, in the limit of large magnetic fields the relaxation rate of the faster (magnetization-like) ${\bf p}_\alpha^h$ mode is proportional to the applied magnetic field (Fig. \[Bild30\] presents the bend fluctuation as an example), $$\begin{aligned}
\label{lim1}
\frac{1}{\tau_\alpha^p}&=&\frac{A_1(b_\perp^D-\chi_2^DM_0)^2+(\chi_2^DM_0)^2(K_\alpha q_\perp^2+ K_3q_x^2)}{b_\perp^D}\nonumber \\ &+&\frac{b_\perp^D}{M_0}\mu_0H.\end{aligned}$$ The relaxation rate of the slower (director-like) ${\bf t}_\alpha^h$ mode saturates at a finite value (Fig. \[Bild30\] presents the bend fluctuation as an example), $$\label{lim2}
\frac{1}{\tau_\alpha^t}=\frac{A_1M_0^2+(K_\alpha q_\perp^2+K_3q_x^2)}{\gamma_1}\left(1-\frac{(\chi_2^DM_0)^2\gamma_1}{b_\perp^D}\right).$$
It is also illuminating to study the relaxation rates of general fluctuations at zero magnetic field, $H=0$. Expanding the relaxation rates to second order in $q_x$ and $q_\perp$ one gets $$\begin{aligned}
\frac{1}{\tau_\alpha^p}&=&\frac{A_1M_0^2}{\gamma_1}\left(1-2\chi_2^D\gamma_1+\frac{b_\perp^D\gamma_1}{M_0^2}\right)\nonumber\\
&+&\frac{(K_\alpha q_\perp^2+K_3q_x^2)\Xi_p}{\gamma_1},\label{zero1}\\
\frac{1}{\tau_\alpha^t}&=&\frac{(K_\alpha q_\perp^2+K_3q_x^2)\Xi_t}{\gamma_1},\label{zero2}\end{aligned}$$ where $$\begin{aligned}
\Xi_p&= \frac{(\chi_2^D\gamma_1-1)^2M_0^2}{b_\perp^D\gamma_1+(1-2\chi_2^D\gamma_1)M_0^2},\\
\Xi_t&=\frac{\gamma_1(b_\perp^D-(\chi_2^DM_0)^2\gamma_1)}{b_\perp^D\gamma_1+(1-2\chi_2^D\gamma_1)M_0^2}.\end{aligned}$$ >From Eqs. (\[zero1\]) and (\[zero2\]) one can see that the relaxation rate $1/\tau_\alpha^p$ of the faster (optic) mode ${\bf p}_\alpha^h$ stays finite in the limit $\mathbf{q}\to 0$. The slower mode ${\bf t}_\alpha^h$ is on the other hand acoustic, i.e., $1/\tau_\alpha^t \to 0$ as $\mathbf{q}\to 0$.
Summary and Perspective {#sec:summary}
=======================
In the present extensive study we have presented detailed experimental and theoretical investigations of the dynamics of the magnetization and the director in a ferromagnetic liquid crystal in the absence of flow. We have shown that a dissipative cross-coupling between these two macroscopic variables, which has been determined quantitatively, is essential to account for the experimental results also for the compound E7 as a nematic solvent for the ferromagnetic nematic phase. Before, this was demonstrated for 5CB as a nematic solvent [@tilenshort]. We also find that all the experimental results presented here for E7 complement well and are consistent with the previous ones using 5CB as the nematic component. Remarkably, the dissipative cross-coupling ($\chi_2^D$) found for the E7-based ferromagnetic nematic liquid crystal is about a factor of 5 smaller than that of the 5CB-based, while the dissipative coefficient of the magnetization ($b_\perp^D$) is (only) twice as large. This leads to an interesting suggestion for future experimental work, namely to address the question of which molecular features determine the strength of this dissipative cross-coupling. The nematic phases of 5CB and E7, respectively, show one qualitatively different feature: the nematic phase of 5CB is well-known to favour the formation of transient pair-like aggregates [@cladis1981] because of its nitrile group, while such tendencies are reduced in E7 since it is mixture of four different compounds and also contains a terphenyl. A natural experiment to study these features in more detail would be to investigate the dependence of the dissipative cross-coupling on the magnetic particle concentration on one hand and to investigate mixtures of the nematic solvents 5CB and E7 on the other to learn more about the coupling mechanisms between the nematic order and the magnetic order.
We have also analyzed the consequences of an out-of-plane dynamics, i.e., out of the plane spanned by the magnetic field and the spontaneous magnetization. We give predictions for both, the azimuthal angles of director and magnetization as well as for the intensity change related to the reversible dynamic cross-coupling terms between the two order parameters, the magnetization and the director. We find that from both measurements a value for the reversible cross-coupling terms can be extracted.
>From the present analysis the next steps in this field appear to be quite well-defined. First of all, the incorporation of flow effects appears to be highly desirable, both from a theoretical as well as from an experimental point of view. Early experimental results in this direction have been described in Ref. [@alenkaapl], where it has been shown that viscous effects can be tuned by an external magnetic field of about $10^{-2}$T by more than a factor of two. >From a theoretical perspective questions like the analogues of the Miesowicz viscosities and flow alignment are high on the priority list [@tilenflow].
Moreover, it will be important to realize, although perhaps experimentally challenging, a nematic or cholesteric liquid crystalline version of uniaxial magnetic gels and rubbers [@collin2003; @bohlius2004]. Cross-linking a ferromagnetic nematic would give rise to the possibility to obtain a soft ferromagnetic gel, opening the door to a new class of magnetic complex fluids. This way one could combine the macroscopic degrees of freedom of the first liquid multiferroic, namely the ferromagnetic nematic liquid crystal, with the strain field as well as with relative rotations. In a step towards this goal, we will derive macroscopic dynamic equations generalizing those for uniaxial magnetic gels and ferronematics to obtain the macroscopic dynamics for ferromagnetic nematic and cholesteric gels [@tilengel].
Partial support of this work by H.R.B., H.P., T.P. and D.S. through the Schwerpunktprogramm SPP 1681 ’Feldgesteuerte Partikel-Matrix-Wechselwirkungen: Erzeugung, skalen�bergreifende Modellierung und Anwendung magnetischer Hybridmaterialien’ of the Deutsche Forschungsgemeinschaft is gratefully acknowledged, as well as the support of the Slovenian Research Agency, Grants N1-0019, J1-7435 (D.S.), P1-0192 (A.M. and N.O.) and P2-0089 (D.L.). N.S. thanks the “EU Horizon 2020 Framework Programme for Research and Innovation” for its support through the Marie Curie Individual fellowship No. 701558 (MagNem). We thank the CENN Nanocenter for use of the LakeShore 7400 Series VSM vibrating-sample magnetometer.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Using the stabilized jellium model with self-compression, we have calculated the dissociation energies and the barrier heights for the binary fragmentation of charged silver clusters. At each step of calculations, we have used the relaxed-state sizes and energies of the clusters. The results for the doubly charged Ag clusters predict a critical size, at which evaporation dominates the fission, in good agreement with the experiment. Comparing the dissociation energies and the fission barrier heights with the experimental ones, we conclude that in the experiments the fragmentation occurs before the full structural relaxation expected after the ionization of the cluster. In the decays of Ag$_N^{4+}$ clusters, the results predict that the charge-symmetric fission processes are dominant for smaller clusters, and the charge-asymmetric fission processes become dominant for sufficiently larger clusters.'
address: |
Center for Theoretical Physics and Mathematics, Atomic Energy Organization of Iran,\
P. O. Box 11365-8486, Tehran, Iran
author:
- 'M. Payami'
title: ' Fragmentation of positively charged metal clusters in stabilized jellium model with self-compression [^1]'
---
Introduction {#sec1}
============
The fragmentation of ionic metal clusters[@Naher; @Uzi99] as well as other properties of metallic clusters have been extensively studied using the jellium model (JM).[@ekardt; @knight; @brack93] In this model, the discrete ions are replaced by a uniform positive charge background of density $n=3/4\pi r_s^3$ in which $r_s$ is the bulk value of the Wigner-Seitz (WS) radius of the valence electrons of the metal. The simplest geometry for the positive background is spherical which is appropriate for closed-shell clusters or large enough clusters in which the Jahn-Teller [@Jahn] deformation has negligible contribution. However, using this simple spherical JM, a lot of information on the properties of metal clusters has been obtained. A refined version of the JM, the stabilized[@pertran; @Shore] jellium model (SJM), which was introduced by Perdew [*et. al.*]{} in 1990, has improved some drawbacks[@lang70; @ashlang67] of the JM (For a recent review on SJM, see Ref.\[\[Kiejna99\]\]). In recent years, the SJM has been used to predict the properties of bulk metals [@pertran; @Perdew2001], metal surfaces[@Fiolhais92; @Kiejna99; @Sarria], metal clusters[@Perdew93; @Seidl; @PayamiJPC] and metallic voids.[@Ziesche] The fragmentation of charged metal clusters has been also studied[@Vieira95; @Vieira98] by Vieira [*et. al.*]{} using the SJM. However, since the surface effects have a large contribution in the energetics and sizes of small clusters, and also since in a fission process the competition between the surface tension and coulomb repulsion leads to the existence of a barrier, a more sophesticated use[@Perdew93; @PayamiJPC] of the SJM is needed to predict the correct energetics of the clusters and the barrier heights (BH) in the study of the fragmentation processes. This method, which is called SJM with self-compression (SJM-SC), has been used to predict the equilibrium sizes and energies of charged[@Braj96] or spin-polarized[@PayamiJPC] metal clusters as well as the calculation of chemical potentials of metallic clusters.[@Kiejna96] The SJM-SC has been also used by Sarria [*et. al.*]{}[@Sarria] to calculate the surface energies and the work functions of metals. In contrast to the JM and the SJM in which the $r_s$ value is borrowed from the bulk system, in the SJM-SC, the density parameter $r_s$ of the jellium sphere is assumed to be a free parameter which can be adjusted in such a way that a cluster with a given number of electrons and specific electronic configuration achieves its equilibrium state. The SJM-SC calculations on neutral metal clusters[@Perdew93; @PayamiJPC] has shown that the equilibrium $r_s$ value of the jellium sphere is less than the bulk value and tends to its bulk value for infinitely large cluster. This phenomenon is called self-compression which is due to the dominant effect of surface tension in small metal clusters. However, it has been shown that[@Braj96] charging a small metal cluster can result in an equilibrium $r_s$ value which is larger than the bulk value. This effect is called self-expansion. The self-expansion has been also predicted for highly polarized metal clusters[@PayamiJPC; @Payami99]. These two effects have different origins. In the former, the repulsive coulomb force dominates the surface tension whereas, in the latter, the Pauli force is responsible for the self-expansion.
In this work, using SJM-SC, we have studied the binary decay processes of positively charged Ag clusters containing up to 100 atoms in all possible channels. We have considered the following possible decay processes for singly ionized Ag clusters $${\rm Ag}_N^{1+}\to {\rm Ag}_{N-p}^{1+} + {\rm Ag}_p^0,
\;\;\;\;\;\;\;\;p=1,2,\cdots,N-2.
\label{eq1}$$ For doubly charged clusters, the decays can proceed via two different processes. The first one is the evaporation process $${\rm Ag}_N^{2+}\to {\rm Ag}_{N-p}^{2+} + {\rm Ag}_p^0,
\;\;\;\;\;\;\;\;p=1,2,\cdots,N-3
\label{eq2}$$ and the second one is fission into two charged products $${\rm Ag}_N^{2+}\to {\rm Ag}_{N-p}^{1+} + {\rm Ag}_p^{1+},
\;\;\;\;\;\;\;\;p=2,3,\cdots,[N/2]
\label{eq3}$$
In general, for the binary decay of $Z$-ply charged ($Z$ is a positive integer) cluster, we have $${\rm Ag}_N^{Z}\to {\rm Ag}_{N-p}^{Z-z_1} + {\rm Ag}_p^{z_1},
\;\;\;\;\;\;\;\;z_1=0,1,\cdots,[Z/2];\;\; p=z_1+1,\cdots,N-Z+z_1-1.
\label{decaygen}$$ For an even value of $Z$ with $z_1=Z/2$, the range of $p$ reduces to $p=z_1+1,\cdots,[N/2]$. The processes for which $z_1=0$ ( i.e., one of the fragments is neutral), are called evaporation processes and others ( both fragments are charged) are fission processes. In evaporation processes, the negativity of the difference between total energies before and after fragmentation is sufficient to have a spontaneous decay. However, in fission processes a negative value for the difference energy is not sufficient for the fission of the parent cluster. This is because, the competition between the short-range surface tension and the long-range repulsive coulomb force may give rise to a fission barrier ( i.e., one should supply energy to overcome the barrier).
The organization of this paper is as follows. In section \[sec2\] we explain the method of calculating the total energies and fission barriers. To obtain the total energy of a given cluster, we solve the self-consistent Kohn-Sham (KS) equations[@KohnSham] in the density functional theory[@Kohn64] (DFT) with local spin density approximation (LSDA) for the exchange-correlation (XC) functional. To calculate the fission barrier, we use the two-touching-spheres model for the saddle configuration.[@Naher] In section \[sec3\], we discuss the results, and finally, we conclude this work in section \[sec4\].
Calculational Scheme {#sec2}
====================
Total energy of a cluster
-------------------------
In the context of the SJM, the average energy per valence electron in the bulk with density parameter $r_s$ and polarization $\zeta$ is given by[@Payami98]
$$\varepsilon(r_s,\zeta,r_c)=t_s(r_s,\zeta)+\varepsilon_{xc}(r_s,\zeta)+\bar
w_R(r_s,r_c)+\varepsilon_{\rm M}(r_s),
\label{eq7}$$
where
$$t_s(r_s,\zeta)=\frac{c_k}{r_s^2} \left[(1+\zeta)^{5/3}+(1-\zeta)^{5/3}\right]
\label{eq8}$$
$$\varepsilon_{xc}(r_s,\zeta)=\frac{c_x}{r_s}
\left[(1+\zeta)^{4/3}+(1-\zeta)^{4/3}\right]+\varepsilon_c(r_s,\zeta)
\label{eq9}$$
$$c_k=\frac{3}{10} \left(\frac{9\pi}{4}\right)^{2/3}
;\;\;\;\;c_x=\frac{3}{4}\left(\frac{9}{4\pi^2}\right)^{1/3}.
\label{eq10}$$
All equations throughout this paper are expressed in Rydberg atomic units. Here $t_s$ and $\varepsilon_{xc}$ are the mean noninteracting kinetic energy and the exchange-correlation energy per particle, respectively. For $\varepsilon_c$ we use the Perdew-Wang parametrization.[@perwan] For a $z$-valent metal the average Madelung energy, $\varepsilon_{\rm M}$, is defined as $\varepsilon_{\rm M}=-9z/5r_0$, in which $r_0$ is the radius of the WS sphere, $r_0=z^{1/3}r_s$. In Eq.(\[eq7\]), $\zeta=(n_\uparrow-n_\downarrow)/(n_\uparrow+n_\downarrow)$ in which $n_\uparrow$ and $n_\downarrow$ are the spin densities of the homogeneous system with total density $n=n_\uparrow+n_\downarrow$. The quantity $\bar w_R$ is the average value (over the WS cell) of the repulsive part of the Ashcroft empty core[@ash66] pseudopotential,
$$w(r)=-\frac{2z}{r}+w_R,\;\;\;\;\;w_R=+\frac{2z}{r}\theta(r_c-r),
\label{eq11}$$
and is given by $\bar
w_R=3r_c^2/r_s^3$ where, $z$ is the valence of the atom, $\theta(x)$ is the ordinary step function which assumes the value of unity for positive arguments, and zero for negative values.
The core radius is fixed to the bulk value, $r_c^B$, by setting the pressure of the unpolarized bulk system equal to zero at the observed equilibrium density $\bar{n}=3/4\pi[\bar{r}_s^B(0)]^3$:
$$\left.\frac{\partial}{\partial r_s}\varepsilon(r_s,0,r_c)\right|_{
r_s=\bar r_s^B(0),r_c=r_c^B}=0.
\label{eq12}$$
Here, $\bar{r}_s^B(0)\equiv\bar{r}_s^B(\zeta=0)$ is the observed equilibrium density parameter for the unpolarized bulk system, and takes the value of 3.02 for Ag. The derivative is taken at fixed $r_c$, and the solution of the above equation gives $r_c^B$ as a function of $\bar{r}_s^B(0)$
$$r_c^B[\bar r_s^B(0)]=\frac{1}{3}[\bar r_s^B(0)]^{3/2}\left\{\left[
-2t_s(r_s,0)-\varepsilon_x(r_s,0)+ r_s
\frac{\partial}{\partial
r_s}\varepsilon_c(r_s,0)-\varepsilon_M(
r_s)\right]_{r_s=\bar r_s^B(0)}\right\}^{1/2}.
\label{eq13}$$
The SJM energy for a spin-polarized system with boundary surface is given by [@pertran]
$$\begin{aligned}
E_{\rm SJM}\left[\nup,\ndo,n_+\right]&=&
E_{\rm JM}\left[\nup,\ndo,n_+\right]+\left(\varepsilon_M(r_s)+\bar
w_R(r_s,r_c^B)\right)\int d\rr\;n_+(\rr) \nonumber \\
&&+\langle\delta v\rangle_{\rm WS}(r_s,r_c^B)\int
d\rr\;\Theta(\rr)\left[n(\rr)-n_+( \rr)\right],
\label{eq14}\end{aligned}$$
where $$\begin{aligned}
E_{\rm
JM}\left[\nup,\ndo,n_+\right]&=&T_s\left[\nup,\ndo\right]+E_{xc}\left[\nup,\ndo
\right] \nonumber\\ &&+\frac{1}{2}\int
d\rr\;\phi\left([n,n_+];\rr\right)\left[n(\rr)-n_+(\rr)\right]
\label{eq15}\end{aligned}$$ and $$\phi\left([n,n_+];\rr\right)=2\int
d\rrp\;\frac{\left[n(\rrp)-n_+(\rrp)\right]}{\left|\rr-\rrp\right|}.
\label{eq16}$$ Here, $n=n_\uparrow+n_\downarrow$ and $n_+$ is the jellium density. $\Theta(\rr)$ takes the value of unity inside the jellium background and zero, outside. The first and second terms in the right hand side of Eq.(\[eq15\]) are the non-interacting kinetic energy and the exchange-correlation energy, and the last term is the Coulomb interaction energy of the system. The quantity $\langle\delta v\rangle_{\rm WS}$ is the average of the difference potential over the Wigner-Seitz cell and the difference potential, $\delta v$, is defined as the difference between the pseudopotential of a lattice of ions and the electrostatic potential of the jellium positive background. The effective potential, used in the self-consistent KS equations, is obtained by taking the variational derivative of the SJM energy functional with respect to the spin densities as
$$\begin{aligned}
v_{eff}^\sigma\left(\left[n_\uparrow,n_\downarrow,n_+\right];\rr\right)&=&
\frac{\delta} {\delta n_\sigma(\rr)}(E_{\rm SJM} -T_s)\nonumber\\
&=&\phi\left(\left[n,n_+\right];\rr\right)+
v_{xc}^\sigma\left(\left[n_\uparrow,n_\downarrow\right];\rr\right)
+\Theta(\rr)\langle\delta v\rangle_{\rm WS} (r_s,r_c^B),
\label{eq17}\end{aligned}$$
where $\sigma=\uparrow,\downarrow$. By solving the KS equations $$\left(\nabla^2+v_{eff}^\sigma(\rr)\right)\phi_i^\sigma(
\rr)=\varepsilon_i^\sigma
\phi_i^\sigma(\rr),\;\;\;\;\;\;\;\sigma=\uparrow,\downarrow,
\label{eq18}$$
$$n(\rr)=\sum_{\sigma=\uparrow,\downarrow}n_\sigma(\rr),
\label{eq19}$$
$$n_\sigma(\rr)=\sum_{i(occ)}\left|\phi_i^\sigma(\rr)\right|^2,
\label{eq20}$$
and finding the self-consistent values for $\varepsilon_i^\sigma$ and $\phi_i^\sigma$, one obtains the total energy.
In our spherical JM, we have
$$n_+(\rr)=\frac{3}{4\pi r_s^3}\theta(R-r)
\label{eq21}$$
in which $R=(zN)^{1/3}r_s$ is the radius of the jellium sphere, and $n(\rr)$ denotes the electron density at point $\rr$ in space. Using the Eq. (21) of Ref. \[\[pertran\]\], this average value is given by
$$\langle\delta v\rangle_{\rm WS}(r_s,r_c^B)=\frac{3(r_c^B)^2}{r_s^3}-
\frac{3}{5r_s}.
\label{eq22}$$
Applying Eq. (\[eq14\]) to a metal cluster which contains $N_\uparrow$ spin-up, $N_\downarrow$ spin-down and $N$ $(=N_\uparrow+N_\downarrow)$ total electrons in the ground state, the SJM energy becomes a function of $N$, $r_s$, and $r_c^B$. The equilibrium density parameter, $\bar r_s(N)$, for a cluster in the ground state electronic configuration, is the solution of the equation
$$\left.\frac{\partial}{\partial r_s}E(N,r_s,r_c^B)\right|_{r_s=\bar
r_s(N)}=0.
\label{eq23}$$
Here, the derivative is taken at fixed values of $N$ and $r_c^B$. For an $N$-electron cluster in its ground state electronic configuration, we have solved the KS equations[@KohnSham] self-consistently for various $r_s$ values and obtained the equilibrium density parameter, $\bar r_s(N)$, and its corresponding energy, $\bar E(N)\equiv E(N,\bar r_s(N),r_c^B)$.
Dissociation energy and fission barrier
---------------------------------------
The dissociation energy (DE) for the general binary decay process (\[decaygen\]), defined as the difference in the sum of total energies of the products and the total energy of the parent cluster, is given by $$D^Z_{z_1}(N,p)=(E_{N-p}^{Z-z_1} + E_p^{z_1}) - E_N^{Z},
\label{deltagen}$$ In evaporation processes ($z_1=0$), a negative value for the DE implies that the parent cluster is unstable against that particular decay channel and therefore, the fragmentation is spontaneous. On the other hand, a positive DE in a particular decay channel means that the parent cluster is stable against the decay in that particular channel. That is, one should somehow supply energy to the system to induce the fragmentation. For processes (\[eq1\]) and (\[eq2\]) the DE becomes $$D^{1+}_0(N,p)=(E_{N-p}^{1+} + E_p^0) - E_N^{1+}
\label{eq4}$$ and $$D^{2+}_0(N,p)=(E_{N-p}^{2+} + E_p^0) - E_N^{2+},
\label{eq5}$$ respectively. However, in the fission processes ($z_1>0$) as in Eq.(\[eq3\]), a negative DE does not mean that the cluster would decay. It is because of the existence of a fission barrier which originates from the short-range attractive (due to binding energy) and long-range repulsive (due to coulomb repulsion) forces between the charged products. The situation is shown in Fig. \[fig1\]. Any excitation above the barrier, which may be induced by collisions or radiation, will eventually make the expected decay possible. One of the main deficiencies of the JM is that it gives negative values[@lang70] of surface energies for $r_s\le 2$. Our using of SJM-SC is expected, therefore, to give more realistic values of surface energies and barrier heights. The fission barrier $B_{z_1}^Z(N,p)$ is approximated by the Coulomb interaction $E_c$ of two touching spheres (i.e., the fission products) and the DE as
$$B_{z_1}^Z(N,p)=D^Z_{z_1}(N,p)+E_c
\label{eq24}$$
For the Coulomb interaction between the two fission products we take into account their polarizabilities. The interaction energy of two charged conducting spheres can be calculated numerically using image charge method.[@Naher] An equally good but much simpler approach is the use of the analytical expression[@Bott; @Kruck] for the interaction between charges $z_1$ and $z_2$ with polarizabilities $\alpha_1$ and $\alpha_2$ at a distance $s$
$$E_{1,2}^{\rm
pol}(s)=\frac{A_2}{A_4}\frac{2z_1z_2}{s}
-\frac{\alpha_1}{s^3}\frac{1}{A_4}\frac{z_2^2}{s}
-\frac{\alpha_2}{s^3}\frac{1}{A_4}\frac{z_1^2}{s}.
\label{eq25}$$
Here $A_j$ is given by
$$A_j=1-\frac{j\alpha_1\alpha_2}{s^6},
\label{eq26}$$
and the polarizability of a conducting sphere (i.e., the metal cluster) with radius $R$ is [@Uwe] $\alpha=R^3$. An other formula which was used [@Naher; @Nakamura] for the Coulomb interaction of two touching conducting spheres is given by
$$E_{1,2}=\frac{2z_1z_2}{R_1+R_2+2\delta R}
\label{eq27}$$
where, for silver, the value $\delta R=0.94 $ takes the polarizability into account. The BH’s for small clusters, obtained from this formula, are somewhat smaller than those we obtained using Eqs. (\[eq25\]), (\[eq26\]). Koizumi [*et. al.*]{}[@Koizumi1; @Koizumi2] have calculated the barrier heights for the fission of doubly charged silver clusters using a shape function in the LDM with shell correction.
Results and discussion {#sec3}
======================
After an extensive self-consistent SJM-SC calculations, we have calculated the equilibrium $r_s$ values and the energies of Ag$_N^Z$ ($Z$=0,1,2,3,4) for different cluster sizes $(1\le N\le 100)$. To show the main differences in the equilibrium $r_s$ values of these clusters, which are appreciable for relatively small clusters, we have plotted, in Fig. \[fig2\], the corresponding $\bar r_s(N)$ values only up to $N=34$. As is obviously seen in the figure, the neutral and singly ionized clusters are self-compressed for all values of $N$. This is because of the dominant effect of the surface tension. However, for multiply charged clusters, the $\bar r_s(N)$ values cross the bulk border (i.e., $r_s=3.02$) at some $N$ which we show it by $N_0$. Our results show that, in general, for larger values of charging, the self-expansion persists up to larger values of $N_0$. That is, $N_0^{2+}<N_0^{3+}<N_0^{4+}<\cdots$. For example, here we have obtained the values of 7, 17, 23 for $N_0^{2+},\;N_0^{3+},\;N_0^{4+}$, respectively. This means that, for larger charging values, the coulomb repulsion between excess charges dominates the surface tension up to larger values of $N_0$. It is also clearly seen in Fig. \[fig2\] that for clusters with the same numbers of electrons but different numbers of atoms, $N$, the following inequality holds
$$\bar r_s^0(N)<\bar r_s^{1+}(N+1)<\bar r_s^{2+}(N+2)<\bar
r_s^{3+}(N+3)<\cdots.
\label{eq28}$$
For Ag$_N^{4+}$ clusters, we could not find any solution of the Eq. (\[eq23\]) for $N=5$. That is, the single remaining electron is not able to bind the 5 constituent ions to each other in the Ag$_5^{4+}$ system. However, the solutions of $\bar r_s^{4+}(6)=14.76$ and $\bar r_s^{3+}(4)=29.0$ have been obtained for Eq. (\[eq23\]) which are so large that one can not realize the corresponding bound states experimentally and we ignore these bound states.
Figure \[fig3\] shows the equilibrium energies per atom in electron-volts for Ag$_N^0$, Ag$_N^{1+}$, Ag$_N^{2+}$, Ag$_N^{3+}$, and Ag$_N^{4+}$ with different cluster sizes ($1\le N\le 34$). For comparison, we have also plotted the bulk value ($\varepsilon=-7.89eV$) by a dashed line. As is seen, by increasing the charge of a given $N$-atom cluster, the coulomb repulsion between the excess charge induces an inflation in the cluster (see Fig. \[fig2\]) and therefore, the density of the material in the cluster decreases which, in turn, leads to a smaller binding energies.
In Fig. \[fig4\](a) we have plotted the DE’s of the most favored decay channels for the process ${\rm Ag}_N^{1+}\to {\rm Ag}_{N-p}^{1+} + {\rm Ag}_p^0$. By definition, the DE is minimum in the most favored channel. We have shown the most favored value of $p$ by $p^*$. The solid small square symbols show the most favored values $p^*$ on the right vertical axis whereas, the corresponding DE’s, $D^{1+}_0(N,p^*)$, are shown on the left vertical axis by large open squares. The dashed line is the result of a fitting to the quantal DE’s. As is seen, the magority of the clusters have positive DE’s and therefore, they are stable against the spontaneous decay. However, the remaining clusters have negative DE’s and accorgingly, they decay into smaller fragments. Clusters close to the closed-shell ones, decay by emitting a monomer or dimer. On the other hand, clusters that are far from being a closed-shell, can break into two fragments each of which are close or identical to closed-shell ones. For example, [Ag]{}$_3^{1+}$ emits a neutral monomer and the remaining is a singly charged dimer, [Ag]{}$_{44}^{1+}$ emits an [Ag]{}$_8^0$ which is a closed-shell and the remaining is [Ag]{}$_{36}^{1+}$ which is close to a closed-shell, and finally, [Ag]{}$_{80}^{1+}$ emits [Ag]{}$_{20}^0$ and the situation is similar to the latter one. Except for the closed-shell singly ionized cluster [Ag]{}$_{69}^{1+}$, all other closed-shell singly ionized clusters, [Ag]{}$_N^{1+}$ ($N$=3, 9, 19, 21, 35, 41, 59, 91, 93) are stable against the spontaneous decay. The dashed fitted line which resembles the result of liquid-drop model (LDM) calculations ( see Fig. 4 of Ref. \[\[Vieira95\]\]), predicts that all singly ionized clusters are stable and the asymptotic value of DE is constant and equal to $0.20eV$.
Figure \[fig4\](b) compares the experimental[@Kruck99] dissociation energies with the monomer DE’s $D^{1+}_0(N,1)$, dimer DE’s $D^{1+}_0(N,2)$, and the most favored DE’s $D^{1+}_0(N,p^*)$. As is seen, the most favored fragments are somewhere monomers, somewhere dimers and somewhere none of them. The general trend of the calculated monomer dissociation energies is similar to the experimental one and has a better agreement with the experiment than the other two DE’s. That is, from $N=3$ to $N=4$ the energy decreases; from $N=4$ to $N=9$ the energy increases in the mean; a decrease on going from 9 to 10; an increase from 10 to 21; and finally, a decrease from $N=21$ to $N=22$ and again increasing from $N=22$. However, our results lack the odd-even staggering because, it originates from the nonspherical shapes for the jellium. Resorting to non-spherical shapes also decreases the pronounced shell effects.[@Nakamura] The relative smallness of our calculated DE’s can be explained in terms of the details of the experimental setup. If the experiment starts using neutral Ag$_N$ clusters, then the equilibrium $r_s$ values would be smaller than the bulk value (see the $Z=0$ plot in Fig. \[fig2\]), and therefore, the total energies would be more negative (see the $Z=0$ plot in Fig. \[fig3\]). Now, irradiating the parent neutral cluster with a high power laser beam would lead to the ionization of the neutral cluster. If the photons also interact with the ionized cluster before the ionized cluster achieves its relaxed state, then the equilibrium $r_s$ value of the ionized cluster would be less than the relaxed value (In our calculations we have used the relaxed values at all steps.), and therefore, the magnitude of the total energy of the ionized parent cluster would be larger. This fact would lead to larger values of the DE’s. Comparing the experimental data with our results we conclude that the photo-dissociation occurs before the relaxation of the parent ionized cluster is completed. The other extreme is that we consider the ‘[*sudden*]{}’ approximation in which we assume that the relaxation time for the ionized cluster is infinite and the cluster undergoes the dissociation without changing the volume (i.e., the saturation approximation which is used in nuclear fragmentations and ordinary jellium calculations for clusters). In reality, neither of these extreme ‘[*relaxed*]{}’ or ‘[*sudden*]{}’ approximations are at work but something in between.
In Fig. \[fig5\](a) we have shown the most favored products Ag$_{p^*}^0$ and the dissociation energies $D^{2+}_0(N,p^*)$ for the decay of [Ag]{}$_N^{2+}$ via evaporation channel. For this process, as in singly ionized case, the dashed fitted line predicts no spontaneous decays and shows a higher constant asymptotic DE as $0.40 eV$. The most favored products are mainly monomers, dimers and octamers.
Figure \[fig5\](b) shows the barrier heights $B^{2+}_{1+}(N,p^*)$ for the most favored channels of the process ${\rm Ag}_N^{2+}\to {\rm Ag}_{N-p}^{1+} + {\rm Ag}_p^{1+}$. By definition, the most favored fission channel has a minimum value for the BH. As is seen, some of the BH’s are negative. The negativity of a BH means that we need no energy to supply the system to initiate the fission. The dashed line which shows the mean behavior of BH intersects the zero line at $N_a^{\rm mean}\approx 29$ (the mean appearance size). This means that on the average, all Ag$_N^{2+}$ clusters with $N<29$ are unstable against spontaneous fission. The values of $p^*$ show that most of the emitted fragments are closed-shell [Ag]{}$_N^{1+}$ clusters with $N=3,9,21$.
In Fig. \[fig5\](c) we have compared the most favored decays of Figs. \[fig5\](a) and \[fig5\](b). It is clearly seen that in a certain size range, the fission and evaporation definitly start their competition. Our quantal results in Fig. \[fig5\](c) show that in the size range $21\le N\le 26$ the evaporation dominates the fission which is in good agreement with the Katakuse [*et. al.*]{} experimental results[@Katakuse] that reveal fission for $N\le 22$. However, our result is slightly larger than the Krückeberg [*et. al.*]{}[@Kruck] experimental data which show that the fission occurs for $N\le 16$. This difference in the experimental results depends on the details of the experiment. For $N>26$, our results show that fission dominates again. To estimate the size range at which evaporation completely dominates the fission, we simply find the intersection point of the two mean behaviors (dashed lines) in Figs. \[fig5\](a) and \[fig5\](b). A simple calculation gives this mean critical value as $N_c^{\rm mean}\approx 50$. That is, in an induced fragmentation experiment of Ag$_N^{2+}$ clusters, the evaporation dominates the fission for $N>50$.
In Fig. \[fig5\](d) we have compared the most favored values $D^{2+}_0(N,p^*)$ and $B^{2+}_{1+}(N,p^*)$ with the experimental threshold energies[@Kruck]. Here, also we have smaller DE’s and BH’s compared to the experiment. One reason for this behavior is that the equilibrium volume of the parent cluster is not equal to the sum of the equilibrium volumes of the product clusters (i.e., the ‘[*relaxed*]{}’ approximation) but is larger. The larger value of the equilibrium $r_s$ leads to a smaller magnitude of the initial energy and therefore, by Eq. (\[eq24\]) to a smaller barrier heights. In other words, the energy needed to deform the parent cluster toward the fission ( In deformation the surface area increases.) is partly paied as a result of self-expansion of the parent cluster. Our calculations show that in almost all decay channels, the sum of volumes of the decay products is smaller than that of the parent cluster which can be explained by the fact that in smaller clusters the surface effect is higher than that in larger clusters. The calculated results for the most favored values of DE’s at $N=9$ and $N=10$ are very close to the experimental values. The most favored products at $N=9$ and $N=10$ are neutral dimer and monomer, respectively (see Fig. \[fig5\](a)).
In Fig. \[fig5\](e) we have compared the calculated monomer DE’s $D^{2+}_0(N,1)$ and the singly charged trimer BH’s $B^{2+}_{1+}(N,3)$ with the experiment. Here also the difference is appreciable.
In Fig. \[fig6\](a) we have plotted the $p^*$ and the $D^{3+}_0(N,p^*)$ for different cluster sizes. The situation is similar to other previous evaporation processes. Here also the asymptotic behavior of the fitted line predicts no decay and has a constant value of about $0.50 eV$.
In Fig. \[fig6\](b), we have plotted the BH’s $B^{3+}_{1+}(N,p^*)$ for the most favored channel of the binary fission of the process ${\rm Ag}_N^{3+}\to{\rm Ag}_{N-p}^{2+}+{\rm Ag}_p^{1+}$. Here, the fission products with smaller charge are more or less the same as those in the fission of Ag$_N^{2+}$ clusters. The mean behavior dashed line intersects the zero axis at $N_a^{\rm mean}\approx 53$. The slope of this line is larger than that of Fig. \[fig5\](b).
Figure \[fig6\](c) compares the most favored decays of Figs. \[fig6\](a) and \[fig6\](b). To our knowledge, there is no experimental results in the literature on the decay of Ag$_N^Z$ with $Z\ge 3$. It is seen that at $N=38$, evaporation dominates and from $N=39$ to $N=42$ evaporation and fission are equally probable. From $N=43$ to $N=58$, except for $N=54$, fission dominates again. From $N=59$ to $N=67$ the evaporation process overcomes and so on. To obtain the mean critical value, we find the intersection point of the two mean behaviors (dashed lines) in Figs. \[fig6\](a) and \[fig6\](b) which results in the value $N_c^{\rm mean}\approx 80$.
Fig. \[fig7\](a) plots the $p^*$ and the $D^{4+}_0(N,p^*)$ for the evaporation processes of Ag$_N^{4+}$. In this figure, one notes the unstability of two smallest sized Ag$_7^{4+}$, Ag$_8^{4+}$ and the stability of almost all others against the evaporation. The fitted dashed line predicts no evaporation and has the asymptotic value of $0.75 eV$. The evaporation products are seen to be mostly neutral dimers and a few monomers and octamers.
In Fig. \[fig7\](b) we have plotted the $p^*$ and the $B^{4+}_{1+}(N,p^*)$ as functions of $N$. We see that at $N=21$ the BH becomes positive for the first time. The dashed line shows the mean behavior of the fission barries. This fitted line has crossed the zero axis at $N_1^{\rm mean}\approx 74$. That is, on the average, the Ag$_N^{4+}$ clusters are stable against the charge-asymmetric fission channel for $N>74$. The most favored charged products are mostly magic clusters, Ag$_N^{1+}$ with $N=3,9,21$.
The results of charge-symmetric binary fission of Ag$_N^{4+}$ for most favored decays are shown in Fig. \[fig7\](c). The smallest positive BH occurs at $N=22$. The mean behavior (the dashed line) intersects the zero line at $N_2^{\rm mean}\approx 51$.
In Fig. \[fig7\](d) we have compared the results for the three different decay processes of Ag$_N^{4+}$. It is seen that these clusters smaller than $N\approx 50$ are highly unstable. As is seen from the figure, it is difficult to specify the competitions for the quantal values. However, to give quantitative values, we have compared the fitted lines in Fig. \[fig7\](e). It is seen that for $N<20$ the charge-symmetric fission is the dominant spontaneous decay process but, for $20<N<51$ the dominant spontaneous fission process changes to the charge-asymmetric one. For $51<N<73$ the only spontaneous fission decay process is the charge-asymmetric one. Clusters larger than $N_a^{\rm mean}\approx 73$ are stable against any spontaneous decay process. In an induced fragmentation experiment of Ag$_N^{4+}$ clusters, the dominant process for $73<N<107$ is charge-asymmetric fission, and for $N$ larger than $N_c^{\rm mean}=107$ the evaporation process dominates. To summarize, for smaller clusters the charge-symmetric fission is dominant, and larger clusters prefer to decay via a charge-asymmetric fission process.
Besides the most favored quantities which are strongly related to the stability of the charged cluster and were explained in the above lines, it is also interesting to calculate the DE’s and BH’s for a process in which the fragment products are specified. Consider the process ${\rm Ag}_N^{Z}\to {\rm Ag}_{N-1}^{Z} + {\rm Ag}^{0}$ in which one of the products is a neutral monomer. We have calculated the dissociation energies $D^Z_0(N,1)$ for all values of $Z$=1, 2, 3, 4 and $N\le 100$. The calculated values show pronounced shell effects as in previous figures of the most favored channels. However, the mean behaviors have asymptotic constant values. For $Z=1,2,3,4$, these asymptotic values in electron-volts are 0.95, 1.00, 1.08, 1.15, respectively. This means that the monomer evaporation from a singly charged cluster needs a smaller energy than from a doubly charged and so on. The same analysis for the dimer evaporation in the process ${\rm Ag}_N^{Z}\to {\rm Ag}_{N-2}^{Z} + {\rm Ag}_2^{0}$ shows also constant asymptotic mean behaviors for the $D^Z_0(N,2)$. The obtained values in electron-volts are 0.49, 0.59, 0.72, 0.87 for $Z$=1, 2, 3, 4, respectively. This means that, as in the monomer evaporation, the detachment of a dimer from singly ionized cluster is easier than from a doubly ionized cluster and so on. However, comparing the dissociation energies for monomer and dimer evaporation (keeping the charge constant) shows that atomic evaporation needs more energy than dimer evaporation.
Now, we consider the fission processes ${\rm Ag}_N^{Z}\to {\rm Ag}_{N-2}^{Z-1} + {\rm Ag}_2^{1+}$ and ${\rm Ag}_N^{Z}\to {\rm Ag}_{N-3}^{Z-1} + {\rm Ag}_3^{1+}$ in which one of the fission products is a singly ionized dimer or a singly ionized trimer. The mean behaviors of the BH’s for these processes are plotted in Fig. \[fig8\]. As is seen, the energy needed to detach a singly ionized dimer decreases by increasing the charge of the parent cluster. This behavior should be contrasted to the behavior in the monomer or dimer evaporations. We recall that in the monomer or dimer evaporation, the dissociation energy increases by increasing the charge of the parent cluster.
It is now easy to find the mean sizes at which atomic evaporation process dominates the fission into singly ionized dimer or trimer for each charging value of the parent cluster. In doubly charged silver clusters, the monomer evaporation dominates the singly charged dimer and trimer detachments at $N=11$ and $N=31$, respectively. The corresponding numbers for triply charged clusters are 21 and 66. For parent clusters Ag$_N^{4+}$, the numbers $N=38$ and $N=120$ have been obtained. To summarize, by increasing the charge of the parent cluster the competition occurs at larger values of $N$.
Conclusion {#sec4}
==========
In this work, we have studied the fragmentations of multiply charged silver clusters taking into account the structural relaxations of the neutral and charged parent as well as daughter clusters. To calculate the relaxed-state sizes and energies of the clusters we have employed the stabilized jellium model with self-compression using a spherical geometry for the jellium background. Using these relaxed-state radius and energy for the clusters, we have calculated the dissociation energies and barrier heights for evaporation and fission processes in all possible channels. For the barrier heights, we have used the two-touching-spheres model with taking into account the polarizabilities of the two charged products. Comparison of our most favored results with the experimental data shows that our results lie under the experimetal results but, the critical size for the competition of the evaporation and the fission of doubly charged silver clusters is predicted in good agreement with the experiment. This comparison also reveals that the fragmentation processes mostly occur before the complete relaxation of the charged parent clusters. That is, in the above-mentioned experiments the structural relaxation time is larger than the average time elapsed for the fragmentation of the ionized parent cluster. Having the initial (just after ionization) and the relaxed sizes $r_{s,\rm ini}^Z$, $r_{s,\rm rela}^Z$ of a $Z$-ply ionized cluster, one may choose an $r_{s,\rm frag}^Z$ value ($r_{s,\rm ini}^Z\le r_{s,\rm frag}^Z\le r_{s,\rm rela}^Z$) for the ionized cluster (just before the fragmentation) such that the calculated values coincides the experimental ones. Then using a linear interpolation it is possible to calculate the relative fragmentation time for an ionized cluster. In ordinary jellium model calculations, the assumption
$$r_s^0(N)=r_s^Z(N)=
r_{s,\rm ini}^Z(N)=r_{s,\rm frag}^Z(N)=r_{s,\rm rela}^Z(N)=r_{s,\rm bulk}
\label{eq29}$$
is used. It should be mentioned that for exact matching of the calculated and experimental values one should use non-spherical shapes.
We have obtained the asymptotic DE’s for the most favored channels in evaporation processes by fitting a simple curve on the quantal results. The result shows that the asymptotic values increase by increasing the charge of the parent cluster. In the case of Ag$_N^{4+}$, we have shown that for relatively small clusters the charge-symmetric fission process is dominant and then, before dominating the evaporation process the charge-asymmetric fission process overcomes. In general, the critical size (at which the evaporation dominates the fission) increases by increasing the charge of the parent cluster. The results show that the neutral $p$-mer dissociation energy increases by increasing the charge of the parent cluster; and for a given charged parent, the atomic evaporation needs more energy than a dimer evaporation. Finally, it has been shown that the energy needed for the detachment of a singly charged dimer or singly charged trimer decreases by increasing the charge of the parent cluster. However, for a given parent cluster, the detachment of a singly charged trimer is easier than that of a singly charged dimer. [**Acknowledgement**]{}
[The author would like to thank John P. Perdew for the useful discussions and comments during this work. He also thanks Adam Kiejna for providing me with his recent review article on the stabilized jellium model.]{}
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[^1]: This work is dedicated to the memory of my mother, Gohar and the 68$^{th}$ birthday of my father, Bahram.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We show that that the jackknife variance estimator $v_{jack}$ and the the infinitesimal jackknife variance estimator are asymptotically equivalent if the functional of interest is a smooth function of the mean or a smooth trimmed L-statistic. We calculate the asymptotic variance of $v_{jack}$ for these functionals.'
author:
- 'Alex D. Gottlieb'
title: Asymptotic Accuracy of the Jackknife Variance Estimator for Certain Smooth Statistics
---
Introduction
============
Let $p$ be a probability measure on a sample space $\X$. Given $n$ samples from $\X$, sampled independently under the probability law $p$, one desires to estimate the value $T(p)$ of some real functional $T$ on the space $\P(\X)$ of all probability measures on $\X$. Denote by $\e_n$ the map that converts $n$ data points $x_1,x_2,\ldots,x_n$ into the empirical measure $$\label{e}
\e_n(x_1,x_2,\ldots,x_n) \ = \ \oon \sum_{i=1}^n \delta(x_i)$$ where $\delta(x_i)$ denotes a point-mass at $x_i$. The [*plug-in estimate*]{} of $T(p)$ given the data $\bx =
(x_1,\ldots,x_n)$ is $$\label{plug-in}
T_n \ = \ T(\e_n(\bx)).$$ Suppose $T_n$ is an asymptotically normal estimator of $T(p)$, so that the distribution of $n^{1/2}(T_n - T(p))$ tends to $\N(0,\sigma^2)$. The jackknife is a computational technique for estimating $\sigma^2$: one transforms the $n$ original data points into $n$ pseudovalues and computes the sample variance of those pseudovalues.
Given the data $\bx = x_1,x_2,\ldots,x_n$, the [*jackknife pseudovalues*]{} are $$Q_{ni} \ = \ n T_n(\e_n) \ - \
(n-1)T(\e_{ni}) \qquad \qquad i = 1,2,\ldots,n$$ with $\e_n$ as in (\[e\]) and $$\label{e-ni}
\e_{ni} \ = \ \frac{1}{n-1} \sum_{j\ne i} \delta(x_j).$$ The [*jackknife variance estimator*]{} is $$\label{JackVar}
v_{jack}(x_1,x_2,\ldots,x_n) \ = \ \frac{1}{n-1}\sum_{i=1}^n
\left( Q_{ni} - \overline{Q_n} \ \right)^2$$ where $\overline{Q_n} = \oon \sum Q_{nj}$. The variance estimator $v_{jack}$ is said to be [*consistent*]{} if $v_{jack}\longrightarrow \sigma^2$ almost surely as $n \rightarrow
\infty$. Sufficient conditions for the consistency of $v_{jack}$ are given in terms of the functional differentiability of $T$. An early result of this kind states that $v_{jack}$ is consistent if $T$ is strongly Fréchet differentiable \[Parr85\], and it is now known that $v_{jack}$ is consistent even if $T$ is only continuously Gâteaux differentiable as in Definition \[contGat\] below \[ST95\].
A functional derivative of $T$ at $p$, denoted $\partial T_p$, is a linear functional that best approximates the behavior of $T$ near $p$ in some sense. For instance, a functional $T$ on the space of bounded signed measures $\M(\X)$ is [*Gâteaux differentiable*]{} at $p$ if there exists a continuous linear functional $\partial T_p$ on $\M(X)$ such that $$\lim_{t \rightarrow 0} \big| t^{-1}\left(T(p + t m) - T(p)\right) \ - \ \partial T_p(m)
\big| \ = \ 0$$ for all $m \in \M(\X)$. More relevant to mathematical statistics is the concept of Hadamard differentiability, for the fluctuations of $T(\e_n)$ about $T(p)$ are asymptotically normal if $T$ is Hadamard differentiable at $p$. A functional $T:\P(\RR)
\longrightarrow \RR$ is [*Hadamard differentiable*]{} at $p$ if there exists a continuous linear functional $\partial T_p$ on $\M(\RR)$ such that $$\lim_{t\rightarrow 0} \big|t^{-1} \left(T(p + t m_t) - T(p)\right) \ - \ \partial T_p(m)
\big| \ = \ 0$$ whenever $\{m_t\}_{t \in \RR}$ is such that $\lim\limits_{t\rightarrow 0} m_t = m$ and $m_t(\RR)=0$ for all $t$, the topology on $\M(\RR)$ being the one induced by the norm $\|m\| = \sup\limits_{t \in
\RR}\left\{\big|m((-\infty,t])\big|\right\}$. If $T$ is Hadamard differentiable at $p$, the variance of $n^{1/2}T(\e_n)$ tends to $$\label{sigmasquared}
\sigma^2 \ = \ \EE_p\phi_p^2$$ as $n \longrightarrow \infty$, where $\phi_p(x)$ is the [*influence function*]{} $$\label{influence}
\phi_p(x) \ = \ \partial T_p (\delta(x)\ - \ p)$$ (this can be shown via the Delta method \[vdW98\] using Donsker’s theorem).
If $T$ is smooth enough then $
n^{1/2}\big( v_{jack} \ - \ \sigma^2 \big)
$ is also asymptotically normal. In this note we calculate the asymptotic variance of $v_{jack}$ (i.e., the limit as $n
\longrightarrow \infty$ of the variance of $n^{1/2} v_{jack} $) for two very well behaved functionals $T$: smooth functions of the mean $ T(p) \ = \ g \left(\overline{p}\right)$ and smooth trimmed L-functionals. In these cases, the asymptotic variance of $v_{jack}$ equals that of $ \EE_{\e_n} \phi_{\e_n}^2 $, the estimator of $\sigma^2$ obtained from (\[sigmasquared\]) by substituting the empirical measure for $p$. This is known as the [*infinitesimal jackknife*]{} estimator \[ST95, p 48\]. We are tempted to conjecture that $v_{jack}$ and the infinitesimal jackknife variance estimator are asymptotically equivalent for sufficiently regular functionals $T$, but we have no general results in this direction.
The literature does not address the accuracy of $v_{jack}$ adequately. In fact, \[ST95, Section 2.2.3\] gets it wrong, conjecturing that the asymptotic variance of $ v_{jack} $ should equal $\hbox{Var}\ \phi_p^2$ for sufficiently regular functionals! However, Theorem 2 of \[Ber84\] does contain a general formula for the variance of $v_{jack}$ which is valid when the functional $T$ has a kind of second-order functional derivative. The theorem there applies to the trimmed L-functionals we discuss in Section \[L\], and to many other functionals besides, but it is hampered by the hypothesis that $p$ have bounded support. We recommend Theorem 2 of \[Ber84\] for its generality and its revelation of the role of second-order differentiability, but our particular results cannot be derived from it directly.
The text \[ST95, p 43\] purports to prove that the asymptotic variance of $n^{1/2} \left( v_{jack} - \sigma^2 \right)$ equals $\hbox{Var}\ \phi_p^2$ when $T$ is of the form (\[LFunctional\]), but there is a mistake there. We paraphrase the following definition from \[ST95, p 43\]: [*For probability measures $p$ and $q$ on the line, let $\rho(p,q)$ denote the $L^{\infty}$ distance between the cdf’s of $p$ and $q$. A functional $T:\P(\RR) \longrightarrow \RR$ is*]{} $\rho$-[**Lipschitz differentiable at**]{} $q$ [*if $$\label{LipschitzDiff}
T(p_k) - T(q_k) \ - \ \partial T_q(p_k-q_k) \ = \
O\left(\rho(p_k,q_k)^2 \right)$$ for all sequences $\{p_k\}$ and $\{q_k\}$ such that $\rho(p_k,q)$ and $\rho(q_k,q)$ converge to $0$*]{}. Assuming that $\hbox{Var}\
\phi_p^2 < \infty$ and $T$ is $\rho$-Lipschitz differentiable, the authors prove (correctly) that $n^{1/2}\big( v_{jack} - \sigma^2
\big)$ is asymptotically normal with variance $\hbox{Var}\
\phi_p^2$. They go on to assert that smooth trimmed L-functionals are $\rho$-Lipschitz differentiable, but this is false (it is not difficult to construct counterexamples).
A close look at the definition of $\rho$-Lipschitz differentiability leads one to wonder whether there are any functionals (besides trivial, linear ones) that satisfy the definition. The problem is that $q$ appears on the left hand side of (\[LipschitzDiff\]) but not on the right; it is easy to imagine $p_k$ and $q_k$ that are close to one another in the $\rho$ metric, yet far enough from $q$ that $\partial
T_q(p_k-q_k)$ badly approximates $T(p_k) - T(q_k)$. Replacing $\partial T_q(p_k-q_k)$ by $\partial T_{q_k}(p_k-q_k)$ in the left-hand-side of (\[LipschitzDiff\]) might result in a more useful characteristic of smoothness for a functional $T$. Indeed, it was this observation that guided our calculations in Sections \[meanie\] and \[L\].
In this note we work with modified pseudovalues $$\label{pseudovalue}
Q'_{ni}(x_1,x_2,\ldots,x_n) \ = \ (n -1)\left[ T(\e_n) -
T(\e_{ni})\right].$$ Substituting $Q'_{ni}$ for $Q_{ni}$ and $\overline{Q_n'} = \oon
\sum Q'_{nj}$ for $\overline{Q_n} = \oon \sum Q_{nj}$ in (\[JackVar\]) does not change the value of $v_{jack}$, so one may compute $v_{jack}$ by the same formula using the $Q'_{ni}$. Using the modified pseudovalues $Q'_{ni}$ makes it easier to take advantage of the magic formula $
(n-1)\left( \e_n - \e_{ni}\right) \ = \ \delta_{x_i} - \e_n
$.
Using pseudovalues to estimate the variance of $\phi_p^2$
==========================================================
One aim of this letter is to emphasize that $\hbox{Var}\ \phi_p^2$ is typically [*not*]{} the asymptotic variance of $n^{1/2}\big(
v_{jack} \ - \ \sigma^2 \big)$, contrary to the assertion of \[ST95, p 42\]. However, should one desire an estimate of $\hbox{Var}\ \phi_p^2$ for some reason, the pseudovalues can be used to this end. Once one has already computed $v_{jack}$, the variance of $\phi_p^2$ is easy to estimate with very little additional labor: just compute the sample variance of the squares of the pseudovalues. We prove this, assuming that the functional $T$ is [*continuously Gâteaux differentiable*]{} and $\phi_p$ is bounded (trimmed L-functionals satisfy these requirements, for instance). This section is an interlude whose results will not be invoked in Sections \[meanie\] and \[L\], the main part of this note.
Continuous Gâteaux differentiability is introduced in \[ST95\] as a sufficient condition for the strong consistency of the jackknife variance estimator.
\[contGat\] A functional $T$ is [**continuously Gâteaux differentiable**]{} at $p$ if it has Gâteaux derivative $\partial
T_p$ at $p$ and if $$\label{contGatEq}
\lim_{k \rightarrow \infty} \sup_{x \in \RR}\left\{ \Big|
\frac{T(p_k + t_k (\delta(x)-p_k)) - T(p_k)}{t_k} \ - \
\partial T_p(\delta(x)-p_k) \Big|\right\} \
\ = \ \ 0$$ for any sequence of probability measures $p_k$ whose cdf’s converge uniformly to that of $p$ and for any sequence of real numbers $t_k$ that converges to $0$.
The proof in \[ST95\] that continuous Gâteaux differentiability implies strong consistency of the jackknife \[ST95, Theorem 2.3\] also serves to prove the following proposition.
\[P1\] Suppose that $T:\P(\RR) \longrightarrow \RR$ is continuously Gâteaux differentiable at $p$, with influence function $
\phi_p(x) \ = \ \partial T_p(\delta(x) - p)
$ satisfying $$\int |\phi_p(x) | p(dx) < \infty \qquad \int \phi_p(x)p(dx) =
0.$$ If the data $X_1, X_2, X_3,\ldots$ are iid $p$ then the empirical measures of the jackknife pseudovalues obtained from the data converge almost surely to $p\circ \phi_p^{-1}$: $$\e_n(Q'_{n1},Q'_{n2},\ldots,Q'_{nn}) \ \longrightarrow \ p\circ \phi_p^{-1}
\qquad
\mathrm{a.s.}$$
[**Proof**]{}: Omitted, but cf. the proof of Theorem 2.3 in \[ST95\]. $\square$
Now, suppose that $T:\P(\RR) \longrightarrow \RR$ has a bounded influence function and satisfies the conditions of Proposition \[P1\]. Given iid $p$ data $
X_1,\ X_2,\ldots,\ X_n
$ compute the jackknife pseudovalues $$Q'_{n,1},\ Q'_{n,2},\ldots,\ Q'_{n,n}$$ and the jackknife estimate $v_{jack}$ based on these pseudovalues. Set $$\S(x) =
\min\{x^2,\|\phi_p\|^2_{\infty}\} ,$$ and $$\tau^2 \ = \ \oon \sum_{j=1}^n \left(
\S(Q'_{n,j}) -
\oon \sum \S(Q'_{n,j})\right)^2.$$ By Proposition \[P1\], the empirical measure of the jackknife pseudovalues converges almost surely in $\P(\RR)$ to $p\circ
\phi_p^{-1} $. It follows that $ \tau^2
\longrightarrow \hbox{Var}\ \phi_p^2 $ almost surely.
One may also estimate $\hbox{Var}\ \phi_p^2$ by applying the bootstrap to the pseudovalues themselves, just as if the pseudovalues were actually iid. To bootstrap, sample $n$ times with replacement from the empirical measure of the pseudovalues $Q'_{n,1},\ldots,Q'_{n,n}$, to produce a bootstrap sample $$Q^*_{n,1},\ Q^*_{n,2},\ldots,Q^*_{n,n}$$ and compute $$\label{boot}
\frac{1}{n^{1/2}}\sum_{i=1}^n \left(\S(Q^*_{n,i})
-\S(Q'_{n,i}) \right).$$ Given a triangular array of pseudovalues $Q'_{n,j}$ having the property that $
\e_n(Q'_{n,1},\ldots,Q'_{n,n})\ \longrightarrow \ p\circ \phi_p^{-1}
$ as $n \longrightarrow \infty$, one may define $Y_{n,i} =
\S(Q^*_{n,i}) - \oon\sum_j \S(Q'_{n,j})$ and apply the Lindeberg-Feller Central Limit Theorem to the array $\{Y_{n,i}\}_{n,i}$ to show that (\[boot\]) converges in distribution to $\N(0,\hbox{Var}\ \phi_p^2)$. But $\e_n(Q'_{n,1},\ldots,Q'_{n,n})$ almost surely converges to $p\circ \phi_p^{-1}$ by Proposition \[P1\]. It follows that, almost surely, (\[boot\]) converges in distribution to $\N(0,\hbox{Var}\ \phi_p^2)$.
Functions of the mean {#meanie}
=====================
When $q$ is a measure, we denote $\int x q(dx)$ by $\overline{q}$ if the integral is defined. Let $g \in C^1(\RR)$ and let $$T(m) \ = \ g \left(\overline{m}\right)$$ be defined for all finite signed measures $m$ with finite first moment. The functional derivative at $m$ of $T$, evaluated at $q$, is $
\partial T_m(q) = g'\left(\overline{m}
\right)\overline{q}
$; the influence function (\[influence\]) is $\phi_m(x) =
g'\left(\overline{m}
\right) \left(x - \overline{m}\right)$. Suppose that $x_1,x_2,\ldots$ are iid $p$, and $p$ has a finite second moment. Let $T_n$ denote the plug-in estimator defined in (\[plug-in\]). Then the asymptotic variance of $n^{1/2}\left(T_n-T(p)\right)$ is $$\label{sigsquared}
\sigma^2 \ = \ g'(\olp)^2\Big\{ \int x^2 p(dx) - \olp^2 \Big\}.$$ Let $v_{jack}$ denote the jackknife variance estimator for $\sigma^2$.
\[mean1\] If $g'$ is (globally) Hölder continuous of order $h > 1/2$ and $p$ has a finite moment of order $2(1+h)$ then $n^{1/2} ( v_{jack} - \sigma^2 )$ and $n^{1/2} \left(
\EE_{\e_n}\phi_{\e_n}^2 - \sigma^2 \right)$ have the same limit in distribution, if any.
[**Proof**]{}: Set $\Delta_{ni} = \left(Q'_{ni}
-\overline{Q_n'}\right) - \phi_{\e_n}(x_i)$ and note that $$v_{jack} \ = \ \frac{1}{n-1} \sum_{i=1}^n
\big(Q_{ni}' - \overline{Q_n'}\big)^2 \ = \ \frac{n}{n-1} \left\{ \EE_{\e_n}\phi_{\e_n}^2 \ + \ \oon
\sum_{i=1}^n \phi_{\e_n}(x_i)\Delta_{ni} \ + \
\oon \sum_{i=1}^n \Delta_{ni}^2
\right\},$$ whence $$n^{1/2} \left( v_{jack} - \sigma^2 \right)
\ = \
n^{1/2} \left( \EE_{\e_n}\phi_{\e_n}^2 - \sigma^2 \right) \ + \ \frac{n^{1/2}}{n-1}\ \EE_{\e_n}\phi_{\e_n}^2
\nonumber \ + \
\frac{n^{3/2}}{n-1} \left\{ \oon \sum_{i=1}^n \phi_{\e_n}(x_i)\Delta_{ni} \ + \
\oon \sum_{i=1}^n \Delta_{ni}^2 \right\}.$$ To prove that $n^{1/2} ( v_{jack} - \sigma^2 )$ and $n^{1/2}
\left( \EE_{\e_n}\phi_{\e_n}^2 - \sigma^2 \right)$ have the same limit in distribution (if any) it suffices to show that $$\label{tendstozero}
\frac{n^{1/2}}{n-1}\ \EE_{\e_n}\phi_{\e_n}^2 \ + \
\frac{n^{3/2}}{n-1} \left( \oon \sum_{i=1}^n \phi_{\e_n}(x_i)\Delta_{ni} \ + \
\oon \sum_{i=1}^n \Delta_{ni}^2 \right)$$ converges almost surely to $0$.
Recall the notation $\e_n$ and $\e_{ni}$ of (\[e\]) and (\[e-ni\]). The first term in (\[tendstozero\]) converges almost surely to $0$ since $$\EE_{\e_n}\phi_{\e_n}^2 \ = \
\oon \sum_{i=1}^n \phi_{\e_n}^2(x_i) \ = \ \oon \sum_{i=1}^n
g'\left(\olen \right)^2(x_i - \olen)^2$$ converges almost surely to $\sigma^2$.
To show that the other terms tend to zero we need a bound on $\Delta_{ni}$. Since $g$ is differentiable, $ g \left( \olenj \right) - g \left(\oleni \right)
\ = \ g'\left(\eta_{ji}\right)\left(\olenj - \oleni \right) $ for some $\eta_{ji}$ between $\oleni$ and $\olenj$, so that $$Q'_{ni} - \overline{Q_n'} \ = \
\frac{n-1}{n} \sum_{j=1}^n
\big( g \left( \olenj \right) - g \left(\oleni \right)\big)
\ = \
\frac{n-1}{n} \sum_{j=1}^n
g'\left(\eta_{ji}\right)\left(\olenj - \oleni \right).$$ Therefore, since $\phi_{\e_n}(x_i) = g'\left(\olen \right)(x_i -
\olen) = \oon\sum_j g'\left(\olen \right)(x_i - x_j)$, $$\begin{aligned}
\Delta_{ni}
\ = \
\left(Q'_{ni} - \overline{Q_n'}\right) - \phi_{\e_n}(x_i)
& = &
\frac{n-1}{n} \sum_{j=1}^n
g'\left(\eta_{ji}\right)\left(\olenj - \oleni \right)
\ - \ \oon\sum_{j=1}^n g'\left(\olen \right)(x_i - x_j) \\
& = &
\oon \sum_{j=1}^n
\left( g'\left(\eta_{ji}\right) - g'\left(\olen \right)\right)(x_i - x_j
).\end{aligned}$$ But $g'$ is Hölder continuous of order $h$ and $|\eta_{ji}-\olen|< \max\{|\olenj - \olen|,|\oleni - \olen|\}$, so $$\left| g'\left(\eta_{ji}\right) - g'\left(\olen
\right)\right| \ \le \ C \big( |\olenj - \olen|^h +
|\oleni - \olen|^h \big)
\ \le \ C (n-1)^{-h} \big( |\olen - x_j|^h +
|\olen - x_i|^h \big),$$ where $C$ is a global Hölder constant for $g'$. It follows that $$\left| \Delta_{ni} \right|
\ = \
C (n-1)^{-h} \oon \sum_{j=1}^n \big( |\olen - x_j|^h +
|\olen - x_i|^h \big) \big( |\olen - x_j| +
|\olen - x_i| \big).$$ With this bound on $\Delta_{ni}$, and assuming that $p$ has a finite moment of order $2(1+h)$, it may be shown that $$\oon \sum_{i=1}^n \Delta_{ni}^2 \ = \ O_s\big(n^{-2h}),$$ and then, by the Cauchy-Schwartz inequality, that $$\Big\vert \oon \sum_{i=1}^n \phi_{\e_n}(x_i)\Delta_{ni}
\Big\vert \ = \ O_s\big( n^{-h} \big).$$ The preceding estimates and the assumption that $h > 1/2$ imply that the last two terms in (\[tendstozero\]) converge to almost surely to $0$. Thus, $n^{1/2} \left( v_{jack} - \sigma^2 \right)$ and $n^{1/2} \left(
\EE_{\e_n}\phi_{\e_n}^2 - \sigma^2 \right)$ have the same limit in distribution, if any. $\square$
If we strengthen the smoothness assumption on $g$ and the moment assumption on $p$ then we can calculate the limit in distribution of $n^{1/2}\left( \EE_{\e_n}\phi_{\e_n}^2 - \sigma^2 \right)$. Suppose that $g''$ is bounded (so that $g'$ is globally Lipschitz) and Hölder continuous of order $r > 0$, and suppose that $p$ has a finite fourth moment. Then $$\phi_{\e_n}(x_i)
\ = \
g'\left(\olen \right) \left( x_i - \olen \right)
\ = \
\left[ g'\left( \olp \right) + g''\left(\olp\right)\left( \olen - \olp \right)
+ O_s \big( n^{-(r+1)/2} \big) \right] \left( x_i - \olen
\right),$$ so that $$\begin{aligned}
\EE_{\e_n}\phi_{\e_n}^2
& = &
\oon \sum_{i=1}^n \phi_{\e_n}^2(x_i)
\ = \
\left[ g'\left( \olp \right) + g''\left(\olp\right) \left( \olen - \olp
\right)\right]^2
\oon \sum_{i=1}^n \left( x_i - \olen \right)^2
\ + \ O_s \big( n^{-(r+1)/2} \big)
\\
& = &
\left[
g'\left( \olp \right)^2 + 2 g'\left( \olp \right)
g''\left(\olp\right)\left( \olen - \olp \right)
\right] \oon \sum_{i=1}^n\left( x_i - \olen \right)^2
\ + \ O_s \big( n^{-(r+1)/2} \big).
\\\end{aligned}$$ From formula (\[sigsquared\]) for $\sigma^2$ we see that $$\begin{aligned}
n^{1/2}\left( \EE_{\e_n}\phi_{\e_n}^2 - \sigma^2 \right)
& = &
g'\left( \olp \right)^2 n^{1/2}\left( \oon \sum_{i=1}^n \left( x_i - \olen
\right)^2 \ - \ \Big\{ \int x^2 p(dx) - \olp^2 \Big\}\right)
\nonumber \\
& & \ + \
2 g'\left( \olp \right) g''\left(\olp\right)
n^{1/2}\left( \olen - \olp \right) \oon \sum_{i=1}^n\left( x_i - \olen
\right)^2
\ + \ O_s \left( n^{-r/2}\right) . \label{fluctuations}\end{aligned}$$ Set $Z_n = n^{1/2}\left( \olen - \olp \right)$ and $$Y_n \ = \ n^{1/2}\left( \oon \sum_{i=1}^n \left( x_i - \olen
\right)^2 \ - \ \Big\{ \int x^2 p(dx) - \olp^2
\Big\}\right).$$ Since $p$ has a finite fourth moment, the random vector $(Y_n,Z_n)$ has a Gaussian limit by the Central Limit Theorem. Equation (\[fluctuations\]) shows that $n^{1/2}\left(
\EE_{\e_n}\phi_{\e_n}^2 - \sigma^2 \right)$ is asymptotically normal with variance $ (a,b) \Gamma (a,b)^{tr} $, where $(a,b) =
\left( g'( \olp )^2,\ 2 g'( \olp ) g''(\olp) \right)$ and $\Gamma$ denotes the asymptotic covariance matrix for $(Y_n,Z_n)$.
In view of Proposition \[mean1\], we find that if $g''$ is bounded and Hölder continuous of order $r > 0$, and if $p$ has a finite fourth moment, then the asymptotic variance of $n^{1/2}
\left( v_{jack} - \sigma^2 \right)$ equals $ (a,b) \Gamma
(a,b)^{tr} $. In contrast, under the same conditions on $p$ and $g$ it may be shown that $\hbox{Var}\ \phi_p^2 = a^2
\Gamma_{1,1}$.
Trimmed L-statistics {#L}
====================
Suppose that $\ell:(0,1) \longrightarrow \RR$ is supported on $[\alpha,1-\alpha]$ for some $0 < \alpha < 1/2$, and let $$\label{LFunctional}
L(p) \ = \ \int_0^1 P^{-1}(s)\ell(s)ds.$$ Here $P^{-1}$ denotes the quantile function for $p$, i.e., $
P^{-1}(s) = \min\{x: P(x) \ge s\}
$ for $0<s<1$ where $P$ denotes the cdf of $p$. A plug-in estimate for $L$ is called a [*trimmed L-statistic*]{}, or a trimmed [*linear combination of quantiles*]{}. (It is called [*trimmed*]{} because the restricted support of $\ell$ discards outliers.) L-statistics are good for robust estimation of a location parameter.
Now assume that $\ell$ is continuous. Then $L$ is Hadamard differentiable (and the L-statistics are asymptotically normal) at all $p \in \P(\RR)$ \[vdW98, Lemma 22.10\]. The functional derivative at $p$ of $L$, evaluated at a bounded signed measure $m$, is $$\partial L_p(m) \ = \ -\int \ell(P(x))M(x)dx$$ where $M(x) = m((-\infty,x])$. The asymptotic variance of the L-statistics is $$\sigma^2 \ = \ \int\int \ell(P(y))\Gamma(y,z)\ell(P(z))dydz,$$ where $$\label{Gamma}
\Gamma(y,z) \ = \ P(y)\wedge P(z) \ - \ P(y)P(z).$$ This formula is obtained via Donsker’s Theorem: [*Let $P_n$ denote the cdf of $\e_n$, a random bounded function. Then $
n^{1/2}(P_n(t)-P(t))$ converges in law to a Gaussian process $\{\bB(t)\}_{t \in \RR}$ with covariance*]{} $$\Gamma(s,t) \ = \ \EE_p\left[\bB(s)\bB(t) \right] \ = \ P(s)\wedge P(t) -
P(s)P(t).
\label{BB}$$ Finally, the influence function is $$\label{LFunInfluence}
\phi_p(x) \ = \ \partial L_p(\delta(x) - p) \ = \
-\int \ell(P(y))(H_x - P)(y)dy,$$ where $H_{x}$ denotes the cdf of $\delta(x)$. Note that $
\sigma^2 = \EE_p \phi_p^2 $ and $$\EE_{\e_n}\phi_{\e_n}^2 \ = \ \int\int
\ell(P_n(y)) \left[ P_n(y)\wedge P_n(z) - P_n(y)P_n(z)
\right]
\ell(P_n(z))dydz.$$
Let $v_{jack}$ denote the jackknife variance estimator for $\sigma^2$. We find that the $v_{jack}$ is asymptotically equivalent to $\EE_{\e_n}\phi_{\e_n}^2$ and asymptotically normal:
\[LFunProp\] Suppose $p$ has no point masses and $\ell'$ is Hölder continuous of order $h > 1/2$. Then $$\label{equivalent}
n^{1/2} \left( v_{jack} - \sigma^2 \right) \ = \
n^{1/2}\left( \EE_{\e_n}\phi_{\e_n}^2 - \sigma^2 \right) \ + \
O_s\big(n^{1/2-h}\big)$$ and converges in law to the Gaussian random variable $ Y + Z$, where $$\begin{aligned}
Y & = &
\int \int \ell(P(y))
\left\{ \bB(y\wedge z) - P(y)\bB(z) - \bB(y)P(z) \right\} \ell(P(z))
dydz \nonumber \\
Z & = &
2 \int \int
\ell'(P(y))\bB(y)\Gamma(y,z)
\ell(P(z))dydz \label{YnZ}\end{aligned}$$ and $\bB$ denotes the Brownian Bridge (\[BB\]).
[**Proof**]{}: We prove first that $n^{1/2}\left(
\EE_{\e_n}\phi_{\e_n}^2 - \sigma^2 \right)$ converges in law to $Y+Z$, and afterwards we establish (\[equivalent\]).
Define $$\begin{aligned}
Y_n & = & n^{1/2}\Big( \sum_{i=1}^n \phi_p(x_i)^2 - \sigma^2 \Big) \nonumber \\
Z_n & = & -2n^{-1/2} \sum_{i=1}^n \phi_p(x_i)
\int \ell'(P(y))(P_n-P)(y)\left( H_{x_i} - P_n \right)(y)
dy. \label{YnandZn}\end{aligned}$$ We claim that $Y_n$ converges in law to $Y$ and $Z_n$ converges in law to $Z$. To see this, substitute (\[LFunInfluence\]) for $\phi_p$ in the definitions of $Y_n$ and $Z_n$, and apply Donkser’s Theorem. Substituting (\[LFunInfluence\]) for $\phi_p$ yields $$\begin{aligned}
Y_n
& = & \int \int \ell(P(y))
n^{1/2}\Big(\oon \sum_{i=1}^n H_{x_i}(y)H_{x_i}(z) \ - \
P(y)\wedge P(z)\Big) \ell(P(z)) dydz \\
& & \ - \ \int \int \ell(P(y)) P(y)n^{1/2}(P_n-P)(z) \ell(P(z)) dydz \\
& & \qquad \ - \ \int \int \ell(P(y)) n^{1/2}(P_n-P)(y)P(z) \ell(P(z))
dydz \\
Z_n & = & 2 n^{-1/2}
\sum_{i=1}^n \int \int \ell'(P(y))(P_n-P)(y)\left( H_{x_i} - P_n \right)(y)
\ell(P(z))(H_{x_i} - P)(z) dydz \\
& = & 2 \int \int
\ell'(P(y))n^{1/2}(P_n-P)(y)\Big( \oon \sum_{i=1}^n
H_{x_i}(y)H_{x_i}(z) - P_n
(y)P_n(z) \Big) \ell(P(z))dydz.\end{aligned}$$ Note that $ \oon \sum H_{x_i}(y)H_{x_i}(z) - P_n(y)P_n(z)$ in the expression for $Z_n$ converges almost surely to $\Gamma(y,z)$ of (\[Gamma\]). Also, in the expression for $Y_n$, $$n^{1/2}\Big(\oon \sum_{i=1}^n H_{x_i}(y)H_{x_i}(z) -
P(y)\wedge P(z)\Big) \ = \ n^{1/2}\left( P_n(y)\wedge P_n(z) -
P(y)\wedge P(z) \right)$$ converges in law to the Gaussian process $ \bB(y\wedge z)$. Writing $M_{ni} =
H_{x_i} - P_n$, we find that $$\begin{aligned}
\phi_{\e_n}(x_i)
& = & - \int \left\{ \ell(P(y) + \ell'(P(y))(P_n - P)(y) +
O_s(n^{-h})\right\} M_{ni}(y) dy \nonumber \\
& = & \phi_p(x_i) \ - \ \int \ell'(P(y))(P_n - P)(y)M_{ni}(y)
dy \ + \ O_s\big(n^{-h}\big). \label{parTy}\end{aligned}$$ Equations (\[parTy\]) and (\[YnandZn\]) imply that $$\begin{aligned}
n^{1/2} \left(\EE_{\e_n}\phi_{\e_n}^2 -
\sigma^2 \right) \ = \ Y_n \ + \ Z_n
& + &
n^{-1/2}
\sum_{i=1}^n \left( \int \ell'(P(y))(P_n(y) - P(y))M_{ni}(y) dy \right)^2
\\
& + & O_s\big(n^{1/2-h}\big).\end{aligned}$$ But the third term on the right hand side of the last equation is $O_s\big(n^{-1/2}\big)$, since $$\begin{aligned}
&& \oon \sum_{i=1}^n \left( \int \ell'(P(y))(P_n(y) - P(y))M_{ni}(y) dy
\right)^2 \\
&& \ = \ \oon \sum_{i=1}^n \int\int \ell'(P(y))(P_n - P)(y)
\ell'(P(z))(P_n - P)(z)M_{ni}(y) M_{ni}(z) dydz \\
&& \ = \ \int\int \ell'(P(y))(P_n - P)(y)
\ell'(P(z))(P_n - P)(z)\oon \sum_{i=1}^n M_{ni}(y) M_{ni}(z) dydz \\\end{aligned}$$ and $$\oon \sum_{i=1}^n M_{ni}(y) M_{ni}(z) \ = \ \oon
\sum_{i=1}^n H_{x_i}(y)H_{x_i}(z) \ - \ P_n(y)P_n(z),$$ converges almost surely to $\Gamma(y,z)$. Thus, $$n^{1/2} \left( \EE_{\e_n}\phi_{\e_n}^2 - \sigma^2 \right) \ = \ Y_n \ + \ Z_n
\ + \ O_s\big(n^{-h}\big),$$ so that $n^{1/2} \left( \EE_{\e_n}\phi_{\e_n}^2 - \sigma^2
\right)$ converges in law to $ Y + Z$, a Gaussian random variable.
It remains to establish (\[equivalent\]). To this end it suffices to show that $$\label{suffices}
\max_{1\le i \le n} \left\{ \left| Q_{ni}' -
\overline{Q_n'} - \phi_{\e_{n}}(x_i) \right| \right\} = O_s\big(
n^{-h}\big),$$ for then, since $v_{jack} = (n-1)^{-1} \sum
(Q_{ni}' - \overline{Q_n'})^2$, it would follow that $$\begin{aligned}
n^{1/2} \left( v_{jack} -
\sigma^2 \right) & = &
n^{1/2}\Big( \frac{1}{n-1} \sum_{i=1}^n \phi_{\e_n}^2(x_i)
\ - \ \sigma^2\Big) \ + \ O_s\big(n^{1/2-h}\big) \nonumber \\
& = & n^{1/2}\left( \EE_{\e_n}\phi_{\e_n}^2 - \sigma^2 \right) \ + \
O_s\big(n^{1/2-h}\big).\end{aligned}$$ Let $P_{ni}$ denote the cdf of $\e_{ni}$. Integration by parts of (\[LFunInfluence\]) shows that $$\label{Influence2}
\phi_{\e_n}(x_i) \ = \ \int x d \left[ \ell(P_n)(
H_{x_i} - P_n)(y)\right]$$ (the boundary term vanishes because (\[LFunctional\]) is trimmed). Suppose $x_1,x_2,x_3,\ldots$ are distinct (we are assuming that $p$ has no point masses, so this is the case almost surely). Then (\[Influence2\]) becomes $$\begin{aligned}
\phi_{\e_n}(x_i)
& = &
x_i \ell (P_n(x_i)) \ + \ \sum_{j:\ x_j > x_i} x_j \left\{ \ell(P_n(x_j))
\ - \ \ell\big(P_n(x_j)- 1/n \big) \right\} \\
& &
- \ \sum_{j=1}^n x_j \left\{ \ell(P_n(x_j)) P_n(x_j)
\ - \ \ell\big(P_n(x_j)- 1/n \big) \big(P_n(x_j)- 1/n\big) \right\},\end{aligned}$$ which we rewrite as $\phi_{\e_n}(x_i) = A + B_i + C_i + D_i \ $ with $$\begin{aligned}
A & = &
- \ \oon \sum_{j=1}^n x_j \ell\big(P_n(x_j)- 1/n \big) \nonumber \\
B_i & = &
x_i \ell (P_n(x_i)) \nonumber \\
C_i & = &
- \ \sum_{j:x_j \le x_i} x_j \left\{ \ell(P_n(x_j)) - \ell(P_n(x_j)- 1/n )\right\} P_n(x_j) \nonumber \\
D_i & = &
\sum_{j:x_j > x_i} x_j \left\{ \ell(P_n(x_j)) - \ell(P_n(x_j)- 1/n )\right\} \left( 1 -P_n(x_j)\right) .
\label{ABCD}\end{aligned}$$ For $1 \le i \le n$, let $$\zeta_{ni}(x) \ = \ (n-1) \int_{P_{ni}(x) - \frac{1}{n-1}}^{P_{ni}(x)} \ell(s)
ds.$$ Observe that $\ell\big(\zeta_{ni}(x)\big) =
\ell\big(\zeta_{nk}(x)\big)$ if $x < \min\{x_i, x_k\}$ or if $x
> \max\{x_i, x_k\}$, and $$\begin{aligned}
\zeta_{nk}(x) - \zeta_{ni}(x) \ = \
(n-1)\int_{P_{ni}(x)}^{P_{ni}(x) + \frac{1}{n-1}}
\ell(s)-\ell\big(s - 1/(n-1)\big)ds
&\quad \hbox{if} &
x_i < x < x_k \nonumber \\
\zeta_{nk}(x) - \zeta_{ni}(x) \ = \
-(n-1)\int_{P_{ni}(x) - \frac{1}{n-1}}^{P_{ni}(x)}
\ell(s)-\ell\big(s - 1/(n-1)\big)ds
& \quad \hbox{if} &
x_k < x < x_i.
\label{observation}\end{aligned}$$ Thus $L(\e_{ni}) = \frac{1}{n-1}\sum\limits_{j:j\ne i} x_j
\zeta_{ni}(x_j)$ and $$\begin{aligned}
Q'_{ni} - \overline{Q_n'} & = &
-\sum_{j:j \ne i} x_j \zeta_{ni}(x_j)
\ + \ \oon \sum_{k=1}^n \sum_{j:j \ne k}
x_j \zeta_{nk}(x_j) \\
& = &
-\ \oon \sum_{k=1}^n x_k \zeta_{ni}(x_k)
\ + \
\oon \sum_{k=1}^n x_i \zeta_{nk}(x_i)
\ + \ \oon \sum_{k=1}^n
\sum_{j:j \ne k,i} x_j \left\{ \zeta_{nk}(x_j)-
\zeta_{ni}(x_j)\right\} \\
& = &
-\ \oon \sum_{k=1}^n x_k \zeta_{ni}(x_k)
\ + \
\oon \sum_{k=1}^n x_i \zeta_{nk}(x_i)
\ + \ \oon \sum_{j:x_j < x_i} \ \sum_{k:x_k < x_j}
x_j \left( \zeta_{nk}(x_j)-
\zeta_{ni}(x_j) \right) \\
& &
\ + \ \oon \sum_{j:x_j > x_i} \ \sum_{k:x_k > x_j}
x_j \left( \zeta_{nk}(x_j)-
\zeta_{ni}(x_j) \right).\end{aligned}$$ Using (\[observation\]) we find that $Q'_{ni} - \overline{Q_n'}
= A_i'+B_i'+C_i'+D_i'\ $ with $$\begin{aligned}
A_i' & = &
-\ \oon \sum_{j=1}^n x_j \zeta_{ni}(x_j) \nonumber \\
B_i' & = &
\oon \sum_{j=1}^n x_i \zeta_{nj}(x_i)
\nonumber \\
C_i' & = &
(n-1) \sum_{j:x_j < x_i}
x_j \left(P_n(x_j) - 1/n\right) \int_{P_{ni}(x_j) - \frac{1}{n-1}}^{P_{ni}(x_j)}
\ell(s)-\ell\big(s - 1/(n-1)\big)ds \nonumber \\
D_i' & = &
(n-1) \sum_{j:x_j > x_i} x_j \left(1 - P_n(x_j)\right)
\int_{P_{ni}(x_j)}^{P_{ni}(x_j) + \frac{1}{n-1}}
\ell(s)-\ell\big(s - 1/(n-1)\big)ds.
\label{A'B'C'D'}\end{aligned}$$ The sequence $\{P_n\}$ converges almost surely to $P$ and hence it is almost surely tight. Thus there exists a (random) bound $M>0$ such that $P_n(x) < \alpha/2$ if $x<M$ and $P_n(x) > 1 - \alpha/2$ if $x > M$. Since $\ell$ vanishes off of $[\alpha,1-\alpha]$, it follows that $B_i = 0$ if $|x_i| > M$, and $B'_i = 0$ if $|x_i| >
M$ and $1/(n-1)<\alpha/4$. Similarly, if $n$ is sufficiently large, the sums defining $A',C',D',A_i',C_i'$ and $D_i'$ in (\[ABCD\]) and (\[A’B’C’D’\]) may be replaced with sums over $j$ such that $|x_i| > M$. Thus $$\begin{aligned}
|A'_i - A|
& \le &
M \frac{n-1}{n} \sum_{j=1}^n \int_{P_{ni}(x_j) - \frac{1}{n-1}}^{P_{ni}(x_j)}
\left| \ell(s) - \ell\big(P_n(x_j)- 1/n \big) \right| ds
\\
|B'_i - B_i|
& \le &
M \frac{n-1}{n} \sum_{j=1}^n
\int_{P_{nj}(x_i) - \frac{1}{n-1}}^{P_{nj}(x_i)} \left| \ell(s) - \ell (P_n(x_i))
\right|ds\end{aligned}$$ are both $O_s(1/n)$ since $\ell$ is differentiable. For $n > 1
\NN$ and $s \in [1/n,1]$, let $t_n(s)$ be a number between $s
-1/n$ and $s$ such that $
\ell'(t_n(s)) = n \left( \ell(s)-\ell\big(s -
1/n \big)\right) $. (The functions $t_n$ may be chosen to be continuous, since $\ell'$ is continuous.) We now have $$\begin{aligned}
|C'_i - C_i|
& \le &
M \big| \ell'(t_n(P_n(x_i)))\big| P_n(x_i)
\ + \
\frac{M}{n} \sum_{j:x_j < x_i}
\int_{P_{ni}(x_j) - \frac{1}{n-1}}^{P_{ni}(x_j)}
\big| \ell'(t_{n-1}(s)) \big| ds
\\
& &
+ \
M \sum_{j:x_j < x_i}
P_n(x_j) \int_{P_{ni}(x_j) - \frac{1}{n-1}}^{P_{ni}(x_j)}
\Big| \ell'(t_{n-1}(s)) - \ell'(t_n(P_n(x_j))) \Big|
ds
\\
|D'_i - D_i|
& \le &
M \sum_{j:x_j > x_i} \left(1 - P_n(x_j)\right)
\int_{P_{ni}(x_j)}^{P_{ni}(x_j) + \frac{1}{n-1}}
\Big| \ell'(t_{n-1}(s)) - \ell'(t_n(P_n(x_j))) \Big|ds.\end{aligned}$$ But $ \ell'(t_{n-1}(s)) - \ell'(t_n(P_n(x_j))) = O\big(
n^{-h}\big)$ throughout the interval of integration because of the Hölder continuity of $\ell'$, and so $|C'_i - C_i|$ and $|D'_i -
D_i|$ are both $O_s\big( n^{-h}\big)$ uniformly in $i$. The preceding estimates show that $$\left| Q_{ni}' -
\overline{Q_n'} - \phi_{\e_{n}}(x_i) \right|
\ \le \
|A'_i - A| + |B'_i - B| + |C'_i - C_i| + |D'_i - D_i|
\ = \
O_s\big( n^{-h}\big)$$ uniformly in $i$, establishing (\[suffices\]). $\square$
Proposition \[LFunProp\] is also true as stated for $ L(p)= \int
x \ell(P(x))p(dx) $, which is not exactly the same as the L-functional (\[LFunctional\]) but has the same functional derivative. An argument similar to the one above shows that the asymptotic variance of $n^{1/2} \left( v_{jack} - \sigma^2
\right)$ equals $\hbox{Var}\ (Y+Z)$ with $Y$ and $Z$ as in (\[YnZ\]). On the other hand, one can show that $\hbox{Var}\
\phi_p^2 = \hbox{Var}\ Y$. This is contrary to \[ST95, p 43\], where it is asserted that $\hbox{Var}\ Y$ is the asymptotic variance of $n^{1/2} \left( v_{jack} - \sigma^2 \right)$.
Acknowledgments
===============
Thanks to Steve Evans for his advice and encouragement. Thanks to Rudolf Beran. The author is supported by the Austrian START project [*Nonlinear Schrödinger and quantum Boltzmann equations*]{}.
References
==========
\[Ber84\] R. Beran. Jackknife approximations to bootstrap estimates.
12 (1): 101 - 118, 1984.
\[Parr85\] W. Parr. Jackknifing differentiable statistical functions.
47 (1): 56 - 66, 1985.
\[ST95\] J. Shao and D. Tu. [*The Jackknife and Bootstrap*]{}. Springer-Verlag, New York, 1995.
\[vdW98\] A.W. van der Waart. [*Asymptotic Statistics*]{}. Cambridge University Press, 1998.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
The many-anyons wavefunction is constructed via the superposition of all the permutations on the direct product of single anyon states and its interchange properties are examined. The phase of permutation is not a representation but the *word metric* of the permutation group . Amazingly the interchange phase yields a finite *capacity* of one quantum state interpolating between Fermion and Boson and the mutual exchange phase has no explicit effect on statistics. Finite capacity of quantum state is defined as *Gentile statistics* and it is different from the fractional exclusion statistics. Some discussion on the general model is also given.
PACS numbers
: 05.30.Pr, 05.30.-d, 05.70.Ce, 03.65.Vf
author:
- Qiang Zhang
- Bin Yan
bibliography:
- 'anyon.bib'
title: 'Many-Anyons Wavefunction, State Capacity and Gentile Statistics'
---
In the past three decades, the topic of anyon[@wilczek90] and fractional statistics is continually of great interest for physics community[@laughlin88; @nayak08]. Anyon is defined through the interchange phase[@wilczek82; @wu84; @halperin84; @arovas84; @leinaas77]: quasi-particles with arbitrary interchange phase $\theta$ could exist and lead to an intermediate statistics[@wilczek82] interpolating between Boson($\theta=0$) and Fermion($\theta=\pi$). The widely analyzed model is the charge-flux model[@wilczek82; @wu84m] and spin lattice[@hatsugai96; @kitaev06; @Han07], although the interchange property for many-body wavefunction is never carefully examined. Alternatively, in condensed matter system the one-particle Hilbert space dimension $d_\alpha$ in general depends on the occupation of state, $\Delta d_\alpha=-\sum_\beta g_{\alpha\beta}\Delta N_\beta$, yielding fractional exclusion statistics proposed by Haldane-Wu[@haldane91; @wu94]. The statistical parameter $g_{\alpha\beta}$ might relate to the interchange phase, e.g. $g_{\alpha\beta}=\theta_{\alpha\beta}/(2\pi)$ in FQHE[@haldane91] and $g_{\alpha\beta}=\delta_{\alpha\beta}\theta/\pi $ in spin chain[@wu94; @hatsugai96] with possible verification[@cooper15; @trovato13]. Yet, an universal relation between the interchange phase and statistics effect is still lacking. The aim of this paper is to construct the many-anyons wavefunction and explore its statistics.\
*Many-anyons wavefunction*-The physical interchange of anyons may give any(rational) phase[@wilczek82], $\phi^\dagger_a\phi^\dagger_b=e^{i\theta}\phi^\dagger_b\phi^\dagger_a$, specifically in two dimension as a representation of the braid group[@wu84; @leinaas77]. Analogy to the Slater determinant form of many-Fermions wavefunction, we could build up our many-anyons wavefunction(unnormalized) by summing over all the permutations on the direct product of single particle states: $$\begin{aligned}
\label{psi}
\Psi\equiv\sum_P(e^{-i\theta})^{l(P)}P\Phi_0\end{aligned}$$ here $\Phi_0 $ is the direct product of the orthonormal single anyon states $\phi_a$ with a preset order, let’s say $\Phi_0=\phi_a\bigotimes\phi_b\cdots$(omit $\bigotimes $ for later convenience). $P$ belongs to the permutation group $S_N$. $P\Phi_0$ is a permutation of the ordered anyons and its coefficient is a phase $e^{-i\theta l(P)}$ to guarantee their equal probability. Once we select the phase of $\Phi_0$ as zero, then we will expect the phase of $P\Phi_0$ with once adjacent interchange to be $\theta$. Thus, it is reasonable to define $l(P)$ as the number of *least adjacent interchange* from order $\Phi_0 $ to $P\Phi_0 $. This is consistent with the field operator language where we are barely allowed to interchange neighbor operators one by one. $l(P)$ is called *length* of $P$ in *word metric*[@billey07] and it equals to the number of *inversion pairs* (nm) in ordered $P\Phi_0$(anyon-$m$ appears on the left of anyon-$n$ in $P\Phi_0$ for $m>n$)[@BH01]. Typically, $l(PP^\prime)\leq l(P)+l(P^\prime)$ and surely $e^{-i\theta l(P)}$ is *not* a representation of the permutation group, unless $e^{-i\theta}=\pm1$. The negative sign in the phase is to ensure the positive interchange phase of the total wavefunction.\
To gain more reliability, we label the anyons with upper indices and let $P$ permute on anyons, equivalent to on the states, to check the interchange properties of the wavefunction. $l(P)$ now is the sum over the inversion pairs parameter $g_{nm}$, since those anyons must be interchanged to obtain $P\Phi_0$. For instance, two-anyons $$\label{2}
\Psi_{ab}^{12}\equiv\sum_P e^{-i\theta l(P)}P\phi_a^i\phi_b^j= \phi_a^1\phi_b^2+e^{-i\theta g_{12}}\phi_a^2\phi_b^1$$ and three-anyons $$\begin{aligned}
\label{3}
\Psi_{abc}^{123}
=& \phi_a^1\phi_b^2\phi_c^3+e^{-i\theta g_{12}}\phi_a^2\phi_b^1\phi_c^3+e^{-i\theta g_{12}}e^{-i\theta g_{13}}\phi_a^2\phi_b^3\phi_c^1\nonumber\\
& +e^{-i\theta g_{12}}e^{-i\theta g_{13}}e^{-i\theta g_{23}}\phi_a^3\phi_b^2\phi_c^1+e^{-i\theta g_{23}} \phi_a^1\phi_b^3\phi_c^2\nonumber\\
&+e^{-i\theta g_{23}}e^{-i\theta g_{13}} \phi_a^3\phi_b^1\phi_c^2\end{aligned}$$ For two-anyons wavefunction, interchange anyon 1 and 2 denoting as $\hat{T}^{12}$, $$\begin{aligned}
\hat{T}^{12}\Psi_{ab}^{12}=&\phi_a^2\phi_b^1+e^{-i\theta g_{21}}\phi_b^2\phi_a^1\nonumber\\
=&e^{-i\theta g_{21}}(\phi_a^1\phi_b^2+e^{i\theta g_{21}}\phi_b^1\phi_a^2)\end{aligned}$$ If $g_{21}=-g_{12}$, then the interchange yields a phase shift $\hat{T}^{12}\Psi^{12}_{ab}=e^{-i\theta g_{21}}\Psi^{12}_{ab}$. Certainly interchange twice $$\begin{aligned}
\hat{T}^{12}(\hat{T}^{12}\Psi^{12}_{ab})=e^{-i\theta g_{12}}e^{-i\theta g_{21}}\Psi^{12}_{ab}=\Psi^{12}_{ab}\end{aligned}$$ gives no effect. We shall emphasize here that $\hat{T}^{ij}$ also interchange the indices in the phase term. Let $g_{nm}=-g_{mn}=1$ for $n\leq m$ in general, corresponding to the clockwise/count-clockwise winding. Then the permutation of anyons leads to a total phase shift(see the proof in Appendix.I\[app:int\]), $$\begin{aligned}
\hat{T}^{ij}\Psi=&e^{i\theta(2|i-j|-1)}\Psi\label{interchange}\\
P\Psi=&e^{i\theta l(P)}\Psi\end{aligned}$$ For different permutation, the phase shift for the wavefunction is not identical, otherwise it must be a trivial representation of permutation group as Boson. The interchange phase of the wavefunction(\[interchange\]) is anyons label dependent and this is due to the selection of anyons order in $\Phi_0$. Different selection of $\Phi_0$ gives different wavefunction $\Psi^\prime$ and yields different interchange phases. Yet, this does *not* hurt the indistinguishability and statistics since $\Psi^\ast\Psi=\Psi^{\prime\ast}\Psi^\prime $.\
In the *totally ordered* coordinate space $X$, the anyon state $\phi_a^i$ is projected as wavefunction $\phi_a(x_i)$. Now the coordinate $x_i$ works as label and $g_{ij}$ is replaced with $g(x_i,x_j)$. $g(x_i,x_j)=-g(x_j,x_i)=1$ for $x_i\preceq x_j$. Here $\preceq$ is the order relation symbol. In the region $x_1\preceq x_2\preceq\cdots \preceq x_N$, all the involved inversion pair parameter $g(x_i,x_j)=1$ and the wavefunction is equation(\[psi\]). In other regions we could permute the coordinates to increasing order and the wavefunction equals to the permutation on wavefunction(\[psi\]). Compare with the charge-flux model[@wu84m; @wilczek90], our many-anyons wavefunction can ensure the permutation properties for many anyons and work for any Hamiltonian system manifesting the anyonic interchange phase.\
*Capacity of state*-The many-anyons wavefunction(\[psi\]) is neither symmetric as Boson nor anti-symmetric as Fermion. Consider two and three anyons at the same state($a=b=c$), equations (\[2\]) and (\[3\])becomes $$\begin{aligned}
\label{23same}
\Psi_{aa}=&(1+e^{-i\theta})\phi_a\phi_a\\
\Psi_{aaa}=&(1+e^{-i\theta})(1+e^{-i\theta}+e^{-2i\theta})\phi_a\phi_a\phi_a\end{aligned}$$ By induction(or see the proof through generating function in Appendix.II\[app:same\]) the wavefunction of $N_a$ anyons at the same state-$a$, $$\begin{aligned}
\label{manysame}
\Psi_{N_a}=&\prod_{n=0}^{N_a-1}(\sum_{k=0}^ne^{-i*k\theta})\times\phi_a...\phi_a\nonumber\\
=& \prod_{n=1}^{N_a}(\frac{1-e^{-i*n\theta}}{1-e^{-i\theta}})\times\phi_a...\phi_a\end{aligned}$$ Indeed, this is the *Q-factorial* form for the number operator in quantum group[@macfarlane89] with $Q=e^{-i\theta}$. Analogy to the exclusion of Fermion, where more than two Fermions at the same state vanish the total wavefunction(\[23same\]), i.e.$1+e^{-i\pi}=0$, the many-anyons wavefunction(\[manysame\]) vanishes when $N_a\geqslant q+1$, providing $(e^{-i\theta})^{1+q}=1$. As a consequence, the maximal occupation or *capacity* of state-$a$ is $q$. This is the *general exclusion principle* we obtained from the interchange phase!\
In order that a thermodynamic limit can be achieved via a sequence of systems with different particle numbers[@haldane91; @wilczek82], the phase $\theta/\pi$ shall be a rational number $r/p$. The capacity $q$ of quantum state is always an integer, $$\begin{aligned}
q=\left\{
\begin{array}{ll}
2p-1,\ \ r\ is\ odd\\
p-1,\ \ \ r\ is \ even\\
\end{array}
\right.
\end{aligned}$$ Irrational phase factor $\theta/\pi$ can not vanish the total wavefunction, thus yielding no constraint on the occupation. In the (real)Q-analogy quantum group approach[@macfarlane89; @fivel90; @greenberg91]($aa^\dagger +Qa^\dagger a=Q^N$ with $-1<Q<1$), it is noticed that the number operator is positive definite unless $Q$ is the root of unity. If we could extend their $Q$ to the anyonic interchange phase $e^{-i\theta}$, the capacity of quantum state should be found.\
Many-anyons wavefunction with mutual exchange phases $\theta_{\alpha\beta} $ among species $\alpha, \ \beta$ could be similarly achieved. Basically we construct the wavefunction via equation(\[psi\]) with $\theta l(P)$ replaced by $\Theta(P)=\sum \theta_{\sigma_i\sigma_j} $. $(ji)$ is inversion pair in $P\Phi_0$ likewise and $\sigma_i=\alpha,\beta\cdots$ is the specie anyon-$i$ belonging to. For $N_a$ anyons at state-$a$ of specie $\alpha$ and $N_b$ anyons at state-$b$ of specie $\beta$, $$\begin{aligned}
\label{mut}
\Psi_{N_a^\alpha N_b^\beta}=&\prod_{n=0}^{N_a-1}(\sum_{k=0}^ne^{-i*k\theta_{\alpha\alpha}})\prod_{n=0}^{N_b-1}(\sum_{k=0}^ne^{-i*k\theta_{\beta\beta}})\nonumber\\&
\times\sum e^{-i\Theta(P)}P\phi_\alpha^{N_a}\phi_\beta^{N_b}\end{aligned}$$ In the last sum of permutation terms, there are $(N_a+N_b)!/(N_a!N_b!)$ distinct permutations and all the mutual exchange phases $\theta_{\alpha\beta} $ appear in these terms. They are mutual orthogonal and the superposition will never cancel each other. Thus the wavefunction can only vanish due to the first and second Q-factorial terms of the same state interchange phase. Consequently, the occupation of the same species different states($\alpha=\beta,\ a\neq b$) and of the different species($\alpha\neq \beta$) do not mutually affect. Only the interchange phase of the same species $\theta_{\alpha\alpha}$ constrain the capacity of each single state, quite different from the exclusion statistics parameter $g_{\alpha\beta}$ from fractional exclusion statistics[@haldane91; @wu94].\
As to the generalized ideal gas of fractional exclusion statistics without mutual statistics[@wu94] $g_{\alpha\beta}=\alpha\delta_{\alpha\beta} $, specifically, a Fermi-like step distribution is found at $T=0$. Below the Fermi surface, the occupation number is $1/\alpha$. The statistics parameter is mapped to the phase by $\alpha=\theta/\pi=r/p$, so the maximal occupation $\bar{n}_{\epsilon<\varepsilon_F}$ is $p/r$, not necessary an integer. This is not the same to the capacity we find here either. For instance, when $\theta=2\pi/3$, the state capacity is $q=2$ while from fractional exclusion statistics, $\bar{n}_{\epsilon<\varepsilon_F}=3/2$ and for $\theta=\pi/2$, we get $q=3$ while $\bar{n}_{\epsilon<\varepsilon_F}=2$.\
*Statistics*- From our construction of many-anyons wavefunction, the explicit statistical effect for anyon is that the interchange phase $\theta$ gives a finite capacity of each quantum state. Finite capacity $q$ of quantum state is defined as Gentile statistics[@gentile40; @dai04; @khare05]. The grand canonical partition function is $$\begin{aligned}
Z=&\sum_{\{n_i\}}\sum_iz_i^{n_i}=\prod_i\sum_{n_i=0}^q z_i^{n_i}\\
=&\sum_{\{N_j\}}W(\{N_j\})z_j^{N_j}=\prod_j\sum_{N_j=0}^{qG_j}W_jz_j^{N_j}\label{partition}\end{aligned}$$ here $z_i\equiv e^{-\beta(\epsilon_i-\mu)}$, $\{n_i\}$ denote all the possible configuration of $n_i $ anyons at quantum state-$i$ and $\{N_j\}$ denote all the possible configuration of $N_j$ anyons at energy level $\epsilon_j$ with degeneracy $G_j$. The statistical weight $W(\{N_j\})=\prod_jW_j$ and $W_j$ is thus(*combination with limited repetition*[@lavenda91]) $$\label{we}
W_j=\sum_{k=0}^{k=\lfloor N_j/(q+1)\rfloor}(-1)^kC^k_{G_j}*C^{G_j-1}_{G_j-1+N_j-k(q+1)}$$ here Binomial constant $C^k_G\equiv G!/(k!(G-k)!)$. Then the mean occupation is $$\begin{aligned}
\label{mean}
\langle n_i\rangle\equiv &\sum_{\{n_i\}}n_i\sum_iz_i^{n_i}=\frac{z_i}{1-z_i}+\frac{(1+q)z^{1+q}}{z_i^{1+q}-1}=\langle N_i\rangle/G_i\end{aligned}$$ The thermodynamics behavior such as the heat capacity, equation of state and condensation temperature[@lavagno00] are well studied[@khare05; @dai04]. From equation(\[partition\]), the most probable occupation $\bar{N}_j$ is determinant from the identity $$\begin{aligned}
\label{mp}
\delta_{N_j} \log (W_jz_j^{N_j})=\delta_{N_j}\log W_j|_{\bar{N}_j}+\log z_j=0\end{aligned}$$ Compare equations (\[mean\]) and (\[mp\]), we can estimate the variance between the mean occupation and the most probable occupation $\Delta\equiv \langle N_j\rangle/\bar{N}_j-1$. It can be numerically verified that the variance is almost zero(filled square in FIG.\[fig:comp\]) coinciding with intuition.
![\[fig:comp\] Numerical comparison of the variance: $\Delta_1\equiv \langle N\rangle/\bar{N}-1$ is the variance between mean occupation and most probable occupation for Gentile statistics. $\Delta_2\equiv \langle N\rangle /\bar{N^\prime}-1$ compares the mean occupation of Gentile statistics and the most probable occupation of fractional exclusion statistics. Here we set $G=10^4$ and $q=3$ as an example. The inset is the relative difference of particle number differentiation $\delta\equiv \delta_N \log W/\delta_N\log W^\prime-1$, which represents the difference of most probable occupation between Gentile and fractional exclusion statistics.](fig1){width="\linewidth"}
Certainly, at $T=0$, $\langle n_i\rangle=q$ for those states $\epsilon_i<\mu$, similar to the fractional exclusion statistics[@haldane91; @wu94]. Plausibly, one may consider that $N_j$ particles occupy at least $\lceil N_j/q\rceil$(ceiling function) states among $G_j$ states, then the effective Fock space dimension $d_B=G_j+N_j-\lceil N_j/q\rceil\approx G_j+(N_j-1)(1-1/q)$ and the number of ways $W^\prime_j=C^{N_j}_{d_B}$ is the same to Wu’s counting[@wu94], providing $q=1/\alpha$.\
Yet numerically $W_j\gg W^\prime_j$ in general. From identity(\[mp\]), the most probable occupation is controled by the differentiation on particle number $\delta_N \log W(N)$. It is close for the two different kinds of counting only in the limit of $G_j\gg N_j$ as shown in the inset of FIG.\[fig:comp\]. This corresponds to the most probable occupation number of high energy state and both of them reduce to the classical Boltzmann distribution. For the low energy state, $N_j\approx G_j $, Gentile statistics and fractional exclusion statistics yield completely different occupation number(see also empty circle data in FIG.\[fig:comp\]).\
*Discussion*- It is confusing to talk about many anyons at the same state due to the interchange phase, since we would have $\Psi_{aa}=e^{i\theta}\Psi_{aa}$. For $\theta\neq 0$, it seems that the wavefunction vanishes. In some proposed model, such as the magnon excitation in Heisenberg spin chain via Bethe ansatz[@hatsugai96; @karbach00], the charge-flux model[@wu84m] and the spin-anyon mapping from Jordan-Wigner transformation[@fradkin89; @batista01], people always preinstall the hard-core condition, resulting in the conventional Fermi statistics[@hatsugai96]. Indeed in the coordinate space $X^N/S_N$[@leinaas77] of $N$ indistinguishable particles, the permutation of coordinates at point$(x_1,x_2..x_N)$ gives the identical point while the wavefunction feels a phase shift[@leinaas77]. The wavefunction is multivalued[@wu84m; @leinaas77] and the interchange property($\Psi=e^{i\theta}\Psi$) does not directly indicate a vanishing wavefunction. The hard-core condition is a too strong constraint for identical particles and it makes sense to discuss many-anyons at the same state.\
We also noticed that the capacity of quantum state is super sensitive to the interchange phase. Two very close phases $\theta$ and $\theta^\prime$ might yield completely different capacities. Thus to obtain a stable capacity and statistics, the interchange phase shall be exact. If the phase factor $r/p$ could be looked as the ratio of electron and flux, thus the resistance, then the resistance must be exactly on the plateau, well known result in FQHE[@tsui82; @stern08].\
In our construction of many-anyons wavefunction(\[psi\]) from single anyon state, the permutation is decomposed into least adjacent interchange. We applied the winding interpretation $g_{nm}=-g_{mn}=1$ for $n<m$ and (\[length\]) reduces to the length of $P$ in the word metric. Then anyons shall obey Gentile statistics. Yet, the permutation property(\[permute\]) is valid for any $g_{nm}=-g_{mn}$, e.g. $g_{nm}=-g_{mn}=(-1)^{n-m-1}$ for $n<m$. The statistics effect may be discussed in future work.\
More generally, the many-anyons wavefunction is $$\begin{aligned}
\label{anyon}
\Psi\equiv \sum_Pe^{-i\Theta(P)}P\Phi_0\end{aligned}$$ here $\Phi_0$ could be any function(not necessary to be the product of single anyons state) and $P$ permutes on any indistinguishable indices. The permutation $P^\prime$ on the wavefunction $$\begin{aligned}
P^\prime\Psi=&\sum_P e^{-iP^\prime\Theta(P)}P^\prime P\Phi_0\nonumber\\
=&\sum_P e^{-iP^\prime\Theta(P)+i\Theta(P^\prime P)}e^{-i\Theta(P^\prime P)}P^\prime P\Phi_0\nonumber\\
=&e^{i\Theta^\prime(P^\prime)}\sum_{P}e^{-i\Theta(P^\prime P)}P^\prime P\Phi_0\end{aligned}$$ if $$\begin{aligned}
\label{gel}
\Theta(P^\prime P)-P^\prime\Theta(P)=\Theta^\prime(P^\prime)\end{aligned}$$ is independent of $P$, then the permutation $P^\prime$ yield a total phase shift $\Theta^\prime(P^\prime)$ for $\Psi$. Equations (\[anyon\]) and (\[gel\]) are the general definition for anyons we proposed. There might be other form for $\Theta(P)$ satisfying equation(\[gel\]) and thus controlling other anyonic statistics.\
The many-anyons wavefunction equations (\[psi\]) and (\[anyon\]) may be used in the consensed matter system and the thermodynamics of Gentile statistics might be compared with experiments.\
I.Interchange properties {#app:int}
========================
For convenience, we write the ordered anyons sequence $\phi^i_a\phi^j_b\phi^k_c\phi^l_d..$ as $P^{ijkl..}\Phi_0$. Sequence $ijkl..$ could represent the permutation $P$. Any permutation could be decomposed into least adjacent interchange called *reduced word* in word metric[@billey07]. Although the decomposition is not unique, the interchanged anyons pairs are identical. They are the *inversion* pairs $(mn)$ in sequence $ijkl..$, if $m$ is on the left of $n$ for $m>n$[@BH01]. In terms of inversion pair parameter $g_{nm}$ $$\begin{aligned}
\label{length}
l(P^{ijkl..})=\sum_{(mn)} g_{nm}\end{aligned}$$ Now we will prove the interchange property of the many-anyons wavefunction $\Psi^{1234\cdots}_{abcd\cdots} $. Consider the following two sequences($i>j$): $$\begin{aligned}
\circledS1=\underbrace{a_1a_2...a_{k-1}}_{s1}\ i\ \underbrace{a_{k+1}a_{k+2}..a_{m-1}}_{s2}\ j\ \underbrace{a_{m+1}a_{m+2}..a_n}_{s3}\\
\circledS2= \underbrace{a_1a_2...a_{k-1}}_{s1}\ j\ \underbrace{a_{k+1}a_{k+2}..a_{m-1}}_{s2}\ i\ \underbrace{a_{m+1}a_{m+2}..a_n}_{s3}\end{aligned}$$
$$\begin{aligned}
l(P^{\circledS1})=&\sum_{m_1>j}g_{jm_1}+\sum_{m_1>i}g_{im_1}+\sum_{m_2<j}g_{m_2j}+\sum_{m_2>i}g_{im_2}\nonumber\\
&+\sum_{m_3<j}g_{m_3j}+\sum_{m_3<i}g_{m_3i}+\textit{ij-indept terms.}\\
l(P^{\circledS2})=&\sum_{m_1>i}g_{im_1}+\sum_{m_1>j}g_{jm_1}+\sum_{m_2<i}g_{m_2i}+\sum_{m_2>j}g_{jm_2}\nonumber\\
&+\sum_{m_3<i}g_{m_3i}+\sum_{m_3<j}g_{m_3j}+g_{ji}+\textit{ji-indept}.\end{aligned}$$
$m_\alpha$($\alpha=1,2,3$) are the anyons in segment $s_\alpha$ of the sequences. Interchange anyon-$i$ and $j$ by $\hat{T}^{ij}$, sequences $\circledS2$ and $\circledS1$ are mutually mapped. The phase shift shall be equal, $$\begin{aligned}
l_{ij}\equiv\hat{T}^{ij}l(P^{\circledS2})-l(P^{\circledS1})=\hat{T}^{ij}l(P^{\circledS1})-l(P^{\circledS2})\end{aligned}$$ The above identity shall be independent of the position of $i,j$ and the order of other anyons. The only solution for the above equation is $g_{mn}=-g_{nm}$ and the overall phase shift is $e^{-i\theta l_{ij}}$, $$\begin{aligned}
l_{ij}=\sum_{m>j}^{m<i}(g_{mj}+g_{im})+g_{ij}\end{aligned}$$ $\hat{T}^{ij}$ is a 2-*circle* permutation as an element of the permutation group and it is easy to check $l(\hat{T}^{ij})=-l_{ij}$. Thus $\hat{T}^{ij}\Psi= e^{i\theta l(\hat{T}^{ij})}\Psi$. Any permutation could be decomposed as some independent cyclic permutation $P=T^{ijk\cdots}T^{mn\cdots}$. Use the same method, we can prove that for any $P$ $$\begin{aligned}
\label{permute}
P\Psi=e^{i\theta l(P)}\Psi\end{aligned}$$ Let $g_{nm}=-g_{mn}=1$ for $n<m$ as the clockwise/cout-clockwise interpretation, then $l(P)$ is the length of $P$ and the wavefunction reduces to equation(\[psi\]). Replace $\theta l(P)$ with $\Theta(P)=\sum \theta_{\sigma_i\sigma_j} $, the many-anyons wavefunction with mutual interchange phase $\theta_{\alpha\beta}$ can be obtained in the same spirit.
II.Occupying the same states {#app:same}
----------------------------
For $N$ anyons sequences, denote $I_N(k)$ the number of permutation sequences with $k\leq C^2_N$ inverse pairs. $C_N^m\equiv N!/((N-m)!m!)$ is the binomial constant. Clearly, $I_N(0)=1$ and the generating function $\Psi_N(x)$ of $I_N(k)$ satisfying the recurrence relation[@BH01] $$\begin{aligned}
\Psi_N(x)&\equiv\sum_{k=0}^{C^2_N}I_N(k)x^k\\
&=(1+x+x^2+\cdots+x^{N-1})\Psi_{N-1}(x)\label{rec}\end{aligned}$$ For $N_a$-anyons at the same state, equation(\[psi\]) $$\begin{aligned}
\Psi_N&=\sum_Pe^{-i\theta l(P)}P\phi_a...\phi_a \\
&=\sum_{l=0}^{C^l_N}I_N(l)(e^{-i\theta })^l\phi_a...\phi_a\end{aligned}$$ is indeed the generating function of $I_N(k)$ with argument $e^{-i\theta}$. $\Psi_1=1$, use equation(\[rec\]), $$\begin{aligned}
\Psi_{N_a}=&\prod_{n=0}^{N_a-1}(\sum_{k=0}^ne^{-i*k\theta})\times\phi_a...\phi_a\nonumber\\
=& \prod_{n=1}^{N_a}(\frac{1-e^{-i*n\theta}}{1-e^{-i\theta}})\times\phi_a...\phi_a\end{aligned}$$ For the wavefunction with different states or mutual interchange phases, the identity equation(\[mut\]) can be checked out in the same way.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper, we investigate Gaussian process regression models where inputs are subject to measurement error. In spatial statistics, input measurement errors occur when the geographical locations of observed data are not known exactly. Such sources of error are not special cases of “nugget” or microscale variation, and require alternative methods for both interpolation and parameter estimation. Gaussian process models do not straightforwardly extend to incorporate input measurement error, and simply ignoring noise in the input space can lead to poor performance for both prediction and parameter inference. We review and extend existing theory on prediction and estimation in the presence of location errors, and show that ignoring location errors may lead to Kriging that is not “self-efficient”. We also introduce a Markov Chain Monte Carlo (MCMC) approach using the Hybrid Monte Carlo algorithm that obtains optimal (minimum MSE) predictions, and discuss situations that lead to multimodality of the target distribution and/or poor chain mixing. Through simulation study and analysis of global air temperature data, we show that appropriate methods for incorporating location measurement error are essential to valid inference in this regime.'
address: 'Department of Statistics, Harvard University, Cambridge, MA, USA .'
author:
-
-
bibliography:
- 'gpreg\_bib.bib'
title: Gaussian Process Regression with Location Errors
---
Introduction
============
Gaussian process models assume an output variable of interest varies smoothly over an input space (*e.g.*, percipitation totals across geographical coordinates, crop yield across factor levels of an experimental design). Such models appear frequently in areas as diverse as climate science \[@mardia1993spatial\], epidemiology \[@lawson1994using\], and black-box problems such as computer experiments, and Bayesian optimization \[@sacks1989design [@srinivas2009gaussian]\]. See [@stein1999interpolation; @cressie1993statistics; @baner14] and [@gelman2014bayesian] for more detailed treatments.
Noisy spatial input data are common in many applications; for example, geostatistical data is often imprecisely spatially referenced, “binned” to the nearest latitude/longitude grid point, or referenced to maps with distorted coordinates \[@veregin1999data [@barber2006modelling]\]. Accounting for measurement error on covariates in the context of regression models is a well studied theme \[@carroll2006measurement\]; however, despite their importance in applications, surprisingly little work has been done on interpolation or Gaussian process regression problems in the presence of (spatial) location measurement error. As we show in this paper, Gaussian process models do not straightforwardly extend to incorporate input measurement error, and simply ignoring noise in the input space can lead to poor performance.
Previous research on such error sources has mostly focused on demonstrating their existence and quantifying their magnitude \[@bonner2003positional [@ward2005positional]\]. For regression problems, @gabrosek2002effect (and later @Cressie2003) adjust Kriging equations for the presence of location errors, and @fanshawe2011spatial further develop research for this regime to include problems where the locations of future observations or predictions are subject to error. Location errors have also been studied in the context of point process data \[@zimmerman2006estimating [@zimmerman2010spatial]\].
Properly accounting for location errors is essential for optimal interpolation and uncertainty quantification, as well precise and efficient parameter estimation when parameters of the covariance function are unknown. Using theoretical results and extensive simulations, our paper provides guidelines on situations when location errors are most impactful for data analysis, and suggestions for incorporating this source of error into inference and prediction. We expand the research in @Cressie2003 on best linear unbiased prediction (Kriging) to include procedures for obtaining interval forecasts and for quantifying the cost of ignoring location errors. We also discuss Markov Chain Monte Carlo (MCMC) methods for optimal (minimum mean squared error (MSE)) predictions, which average over the conditional distribution of (latent) location errors given the observed data. Section \[sec:notation\] establishes notation and describes the basic model with location errors used throughout the paper. In Section \[sec:kriging\], we discuss Kriging using the covariance structure of the location-error induced process. Section \[sec:hmc\] considers MCMC methods for obtaining minimum MSE predictions, and thus improving upon Kriging. We compare these methods through simulation study in Section \[sec:simul\], and explore an application to interpolating northern hemisphere temperature anomolies in Section \[sec:cru\]. The proofs of all of the theoretical results are given in an Appendix.
The Model {#sec:notation}
=========
We will write ${\mathbf{s}}_n = (s_1 \: \: s_2 \: \ldots \: s_n)'$ to denote a $n$-vector of locations in the input space ${\mathbb{S}}\subset {\mathbb{R}}^p$, and ${\mathbf{x}}_n = (x(s_1) \: \: x(s_2) \: \ldots \: x(s_n))'$ as the associated vector of observations at ${\mathbf{s}}_n$. Similarly, we will denote ${\mathbf{x}}^*_k = (x(s^*_1) \: \ldots \: x(s^*_k))'$, or simply $x^* = x(s^*)$ where $\{s^*_i, i=1, \ldots, k\}$ are unobserved locations. The process $x: {\mathbb{S}}\rightarrow {\mathbb{R}}$ is called a *Gaussian process* if, for any $s_1, \ldots s_n \in {\mathbb{S}}$, ${\mathbf{x}}_n = (x(s_1) \: \: x(s_2) \: \ldots \: x(s_n))'$ is jointly Normally distributed. Typically, the form of this joint distribution is specified by a deterministic or parametric mean function (for now, taken without loss of generality to be 0) and a covariance function $c: {\mathbb{S}}^2 \rightarrow {\mathbb{R}}$, so that $$\label{GPdef}
\begin{pmatrix} x(s_1) \\ \vdots \\ x(s_n) \end{pmatrix}
\sim {\mathcal{N}}\left( {\mathbf{0}},
\begin{pmatrix} c(s_1, s_1) & \cdots & c(s_1, s_n) \\
\vdots & \ddots & \\
c(s_n, s_1) & & c(s_n, s_n) \end{pmatrix}
\right).$$ For $c$ to be a valid covariance function, the covariance matrix in Equation must be positive semi-definite for all input vectors ${\mathbf{s}}_n = (s_1 \: \: s_2 \: \ldots \: s_n)'$. Gaussian process regression is primarily used as a method for interpolating (predicting) values ${\mathbf{x}}^*_k$ at unobserved points ${\mathbf{s}}^*_k = (s^*_1 \: \ldots \: s^*_k)'$ in the input space, given all available observations. Such conditional distributions are easily obtained by exploiting the joint normality of the response $x$ at observed and unobserved locations: $$\begin{aligned}
{\mathbf{x}}^*_k | {\mathbf{x}}_n & \sim {\mathcal{N}}\big( {\mathbf{C}}({\mathbf{s}}^*_k, {\mathbf{s}}_n){\mathbf{C}}({\mathbf{s}}_n, {\mathbf{s}}_n)^{-1} {\mathbf{x}}_n, \nonumber \\
& \hspace{1.75cm} {\mathbf{C}}({\mathbf{s}}^*_k, {\mathbf{s}}^*_k) - {\mathbf{C}}({\mathbf{s}}^*_k, {\mathbf{s}}_n){\mathbf{C}}({\mathbf{s}}_n, {\mathbf{s}}_n)^{-1}{\mathbf{C}}({\mathbf{s}}_n, {\mathbf{s}}^*_k)\big).
\label{GPconditional}\end{aligned}$$ In Equation , ${\mathbf{C}}({\mathbf{s}}_n, {\mathbf{s}}_n)$ denotes the covariance matrix of ${\mathbf{x}}_n$, ${\mathbf{C}}({\mathbf{s}}^*_k, {\mathbf{s}}_n)$ denotes the $k \times n$ covariance matrix between ${\mathbf{x}}^*_k$ and ${\mathbf{x}}_n$.
When the locations in the input space ${\mathbb{S}}$ are affected by error, we observe a surrogate process $y: {\mathbb{S}}\rightarrow {\mathbb{R}}$, $$y(s_i) = x(s_i + u_i),$$ where $s_i$ is a known location in ${\mathbb{S}}$ and $u_i \in {\mathbb{S}}$ is unobserved location error. The problem of Gaussian process regression with location errors addressed in this paper is to predict $x$ at unobserved (exact) locations $x(s^*)$ given observations from the noise-corrupted process $y$.
When $x$ is assumed to be a Gaussian process, there is no nontrivial structure for $u$ that results in $y$ being a Gaussian process. Additionally, it is not possible to write $y$ as a convolution of $x$ and a white noise process as differences between the surfaces $y$ and $x$ will generally be correlated across space, *i.e.,* $\mathrm{Cov} [y(s_1) - x(s_1), y(s_2) - x(s_2)] \neq 0$. Gaussian process regression with location errors therefore cannot be thought of as a classical or Berkson errors-in-variables problem \[[@carroll2006measurement]\]. Interestingly, in some cases, the process $y$ may be more informative for prediction at a new location $x(s^*)$ than the process $x$ is. Thus, appropriate methods can deliver lower MSE interpolations in a location-error regime than the MSE of the usual methods in an error-free regime.
Kriging the Location Error Induced Process $y$ {#sec:kriging}
==============================================
As shown in @Cressie2003, the second moment properties of $y$ can be used to perform Kriging (they named this “Kriging adjusting for location error” or KALE), noting that measurement errors $u$ induce a new covariance function $$\begin{aligned}
{2}
k(s_1, s_2) &= {\mathrm{Cov}}[y(s_1), y(s_2)] & &= {\mathbb{E}}[c(s_1 + u_1, s_2 + u_2)] \text{ for $s_1 \neq s_2$ } \nonumber \\
k(s, s) &= {\mathbb{V}}[y(s)] & &= {\mathbb{E}}[c(s + u, s + u)] \nonumber \\
k^*(s, s^*) &= {\mathrm{Cov}}[y(s), x(s^*)] & &= {\mathbb{E}}[c(s + u, s^*)].
\label{cov_y}\end{aligned}$$ The expectation here is taken over the input errors $u$, which are assumed to have some joint distribution $g_{{\mathbf{s}}_n}$. The following result shows that if $c$ is a valid covariance function, then so is $k$, regardless of the error distribution $g(\cdot)$.
\[validcovfn\] Assume for all $n$ and ${\mathbf{s}}_n \in {\mathbb{S}}$, ${(u_1, u_2, \ldots, u_n) \sim g_{{\mathbf{s}}_n}\in \mathcal{G}}$, where $\mathcal{G}$ is any family of probability measures on ${\mathbb{S}}$. Then $k$ is a valid covariance function if $c$ is.
Regardless of the form of $c$, $k$ always exhibits the “nugget” effect, or discontinuities in the covariance function \[@matheron1962traite\] ${\lim_{s_2 \to s_1} k(s_2, s_1) \neq k(s_1, s_1).}$ In fact, several authors cite location/positional error as a justification for including a nugget term in an arbitrary covariance function $c$ \[[@cressie1993statistics; @stein1999interpolation]\], alongside independent measurement error in observing the response, $x(s) + \epsilon$. Location errors, however, cause $k$ to differ from $c$ throughout the spatial domain ${\mathbb{S}}^2$ (this is shown in Figure \[fig:c\_vs\_k\]), meaning that while they induce a nugget, a nugget term alone cannot capture the effect of location errors.
![Comparison of $c$ and $k$ for ${\mathbb{S}}= {\mathbb{R}}^2$ and $c(s_1, s_2) = \exp(-\beta\|s_1 - s_2\|^2)$, with $u_i \stackrel{iid}{\sim} {\mathcal{N}}(0, \sigma^2_u\mathbf{I}_2)$. Location errors $\sigma^2_u > 0$ cause $c$ and $k$ to differ as a function of distance, and induce a nugget discontinuity at 0.\[fig:c\_vs\_k\]](figure/c_vs_k){width="\maxwidth"}
Using $k$, we get the Kriging estimator adjusting for location error for $x(s^*)$ at an unobserved location of $x$: $$\label{ykriging}
{\hat{x}_{\text{\tiny\textsc{KALE}}}}(s^*) = {\mathbf{K}}^*(s^*, {\mathbf{s}}_n) {\mathbf{K}}({\mathbf{s}}_n, {\mathbf{s}}_n)^{-1} {\mathbf{y}}_n.$$ In Equation , ${\mathbf{K}}$ and ${\mathbf{K}}^*$ respectively denote the covariance matrices corresponding to the kernels $k$ and $k^*$. The quantity ${\hat{x}_{\text{\tiny\textsc{KALE}}}}(s^*)$ is the best linear unbiased predictor for $x(s^*)$ (in terms of MSE) and has all the usual Kriging properties. When there are no location errors, the Kriging estimator is equivalent to the conditional expectation of $x(s^*)$ given ${\mathbf{x}}_n$ (see Equation ).
In general, the covariance functions $k$ and $k^*$ can be evaluated using Monte Carlo integration by repeatedly sampling ${\mathbf{u}}_n$ from $g$. For several common combinations of covariance function and location error models, however, it is possible to arrive at the expressions in Equation in closed form. In particular, if $$c(s_1, s_2) = \tau^2 \exp(-\beta d(s_1, s_2)),$$ then we can define a random variable $Z = d(s_1 + u_1, s_2 + u_2)$ and find its moment generating function $M_Z(t)$. If we can evaluate $M_Z(t)$ at $t = -\beta$, then this yields $k(s_1, s_2)$. For instance, for the squared exponential covariance function $d(s_1, s_2) = \|s_1 - s_2\|^2$ and Normal location errors $u \sim {\mathcal{N}}(0, \sigma^2_u \mathbf{I}_p)$, $Z$ has a scaled noncentral $\chi^2_p$ distribution and $$\begin{aligned}
k(s_1, s_2) &= \frac{\tau^2}{(1 + 4\beta\sigma^2_u)^{p/2}}\exp \left( - \frac{\beta}{1 + 4 \beta \sigma^2_u} \|s_1 - s_2\|^2 \right) \text{ for } s_1 \neq s_2 \nonumber \\
k(s, s) &= \tau^2
\label{cov_y_sqexp}\end{aligned}$$ with a similar expression for $k^*(s, s^*)$. Thus the covariance function for $y$ is also squared exponential (it is not generally true that $c$ and $k$ will share the same functional form). Note, however, that not all parameters are identifiable—we must know at least one of $(\tau^2, \beta, \sigma_u^2)$ in order to estimate the others. Interestingly, it is possible for the KALE to yield lower MSE predictions than those given from an error-free regime, where ${\mathbf{u}}_n \equiv 0$ and $x = y$. In other words, ${\mathbf{y}}_n$ can be more informative than ${\mathbf{x}}_n$ for predicting $x(s^*)$. Heuristically, this happens when ${\mathbf{y}}_n$ is more strongly correlated with $x(s^*)$ than is ${\mathbf{x}}_n$. Below we characterize the conditions for observing this phenomenon in a simple model with one observed data point (Figure \[fig:c\_vs\_k\] provides an illustration); it seems difficult to generalize this to larger observed location samples and covariance/error structures.
Assume $n=1$, $\|s - s^*\|^2 = \Delta^2$, $c(s, s^*) = \tau^2 \exp(-\beta \Delta^2)$ for all $s, s^* \in {\mathbb{S}}$, and $u \sim {\mathcal{N}}(0, \sigma^2_u \mathbf{I}_p)$. Without location error ($\sigma^2_u = 0$), the *MSE* in predicting $x(s^*)$ from $x(s)$ is $c_0 = \tau^2(1 - \exp(-2\beta\Delta^2))$. There exists $\sigma_u>0$ such that ${\mathbb{E}}[({\hat{x}_{\text{\tiny\textsc{KALE}}}}(s^*) - x(s^*))^2] < c_0$ if and only if $\beta\Delta^2 > p/2$. \[prop1\]
Interval predictions
--------------------
For many applications of Gaussian process regression, particularly in geostastics and environmental modeling, both point and interval predictions are of interest. However, Kriging, being strictly a moment-based procedure, does not provide uncertainty quantification for predictions other than variance. In a location-error Gaussian process regime, KALE predictions will always be non-Gaussian, thus variance alone is not sufficient to provide distributional or interval predictions.
However, it is relatively straightforward to derive confidence intervals for predictions at unobserved locations $x(s^*)$ given measurements ${\mathbf{y}}_n$ at locations ${\mathbf{s}}_n$. The following proposition provides the exact distribution function (CDF) for prediction errors $x(s^*) - {\hat{x}_{\text{\tiny\textsc{KALE}}}}(s^*)$, which can be inverted to obtain a confidence interval for $x(s^*)$ based on ${\hat{x}_{\text{\tiny\textsc{KALE}}}}(s^*)$.
Let $$\begin{aligned}
V({\mathbf{u}}_n) & = \sigma^2 + \gamma'{\mathbf{C}}({\mathbf{s}}_n + {\mathbf{u}}_n, {\mathbf{s}}_n + {\mathbf{u}}_n)\gamma - 2\gamma'{\mathbf{C}}({\mathbf{s}}_n + {\mathbf{u}}_n, s^*) \\
\text{where } \: \gamma &= {\mathbf{K}}({\mathbf{s}}_n, {\mathbf{s}}_n)^{-1}{\mathbf{K}}^*({\mathbf{s}}_n, s^*), \sigma^2 = {\mathbb{V}}[x(s^*)].\end{aligned}$$ Then $$\label{Wcdf}
{\mathbb{P}}(x(s^*) - {\hat{x}_{\text{\tiny\textsc{KALE}}}}(s^*) < z) = {\mathbb{E}}\left[ \Phi \left( \frac{z}{\sqrt{V({\mathbf{u}}_n)}} \right) \right],$$ where $\Phi$ is the CDF of the standard Normal distribution. \[coverprop\]
It may be necessary to evaluate Equation using Monte Carlo; if so, it is practical to use the same draws of ${\mathbf{u}}_n$ when evaluating different quantiles $z$, as this guarantees a Monte Carlo estimate of the distribution function be non-decreasing.
While intervals based on provide exact coverage (modulo Monte Carlo error), such coverage is achieved by averaging over all data, both observed (${\mathbf{y}}_n$) and unobserved ($x(s^*)$) as well as the location errors, ${\mathbf{u}}_n$. This is in contrast to usual interval estimates from Gaussian process regression without location error, which are exact probability statements conditional on the observed data ${\mathbf{x}}_n$. The reason this is an important distinction is because the usual Gaussian process conditional probability intervals yield the proper coverage rate across multiple prediction intervals (when predicting $x$ at a collection of unobserved locations ${\mathbf{s}}^*_k$), whereas the confidence intervals corresponding to KALE may not.
Advantages over Kriging while Ignoring Location Errors
------------------------------------------------------
Failing to adjust for location errors when Kriging (@Cressie2003 called this “Kriging ignoring location errors” or KILE) can lead to poor performance. A data analyst ignoring the location errors will use (see Equation ) $$\label{xkriging}
{\hat{x}_{\text{\tiny\textsc{KILE}}}}(s^*) = {\mathbf{C}}(s^*, {\mathbf{s}}_n){\mathbf{C}}({\mathbf{s}}_n, {\mathbf{s}}_n)^{-1} {\mathbf{y}}_n.$$ Since ${\hat{x}_{\text{\tiny\textsc{KALE}}}}(s^*)$ is the best linear unbiased estimator for $x(s^*)$ and ${\hat{x}_{\text{\tiny\textsc{KILE}}}}$ is also an unbiased linear estimator, KALE dominates KILE and always yields a reduced MSE. Figure \[fig:ignore\_mspe\_plot\] illustrates the disparity in MSE for a simple model; intuitively, the relative cost of ignoring location errors increases as the magnitude of the location errors increases. We also see in panel (B), illustrating Proposition \[prop1\], that for small values of $\sigma^2_u$, the MSE for both KALE and KILE decreases in $\sigma^2_u$.
Besides yielding suboptimal predictions relative to KALE, ignoring location errors also leads to an estimator for $x(s^*)$ that is not self-efficient \[*p*. 549, @meng1994multiple\]. Following @meng1994multiple, an estimator $T$ for parameter $\theta$ is self-efficient if for any $\lambda \in [0,1]$ and subset of the observed data $X_c \subset X$, we have $${\mathbb{E}}[(\lambda T(X) + (1 - \lambda)T(X_c) - \theta)^2] \geq {\mathbb{E}}[(T(X) - \theta)^2].$$ Thus, roughly speaking, self-efficient estimators are those that cannot be improved by using only a subset of the original data \[@meng2014got\].
The following theorem states that the KILE MSE is unbounded as a function of any single spatial location $s_i$ for $i=1, \ldots, n$. This is a stronger result than just the lack of self-efficiency. A consequence of Theorem \[theorem1\] is that, assuming only simple continuity conditions on the covariance function and location error model, the KILE MSE can always increase when observing more data, regardless of the locations of the existing observations or the locations at which we want to make predictions.
Suppose that the following conditions hold:
- $c$ is continuous and bounded in ${\mathbb{S}}^2$,
- the location error model $g$ satisfies $(u_1^m, u_2^m) \stackrel{D}{\rightarrow} (u_1, u_2)$ for all $s_1, s_2 \in {\mathbb{S}}$ and sequences $(s_1^m, s_2^m)$ such that $\lim_{m \to \infty} (s_1^m, s_2^m) = (s_1, s_2)$,
- and ${\mathbb{P}}(u_1 \neq u_2) < 1$ for all $s_1, s_2 \in {\mathbb{S}}$.
Let ${\hat{x}_{\text{\tiny\textsc{KILE}}}}(s^*)$ be the *KILE* estimator for $x(s^*)$ given ${\mathbf{y}}_n$. Then for any $M > 0$, $n \geq 2$, and $s_2, \ldots, s_n \in {\mathbb{S}}$, there exists $s_1$ such that ${\mathbb{E}}[(x(s^*) - {\hat{x}_{\text{\tiny\textsc{KILE}}}}(s^*))^2] > M$. \[theorem1\]
Note that the condition that $c$ is continuous excludes a nugget term from the distribution of $x$. We prove Theorem \[theorem1\] (in the Appendix) by showing that when observed locations are very close together, the corresponding covariance matrix is nearly singular, and this increases MSE. Without location errors, the usual Kriging estimator does not exhibit this behavior since the difference between values of $x(s)$ for points that are close together also converges in probability to 0. This is not the case for the noise-corrupted process, as $y(s_2) - y(s_1)$ does not converge to 0.
![Here we assume $x(s)$ is a Gaussian process with mean 0 and covariance function $c(s, s^*) = \exp(-(s - s^*)^2) + \sigma^2_x \mathbf{1}_{s = s^*}$. Location errors have the form $u_i \sim {\mathcal{N}}(0, 0.04)$. We use *KILE* to predict $x(5)$ based on ${\mathbf{y}}_{\text{obs}} = \{y(0), \ldots, y(4), y(6), \ldots, y(10)\}$, as well as an additional observation $y_(s)$. The MSE in predicting $x(5)$ given ${\mathbf{y}}_{\text{obs}}$ and $y(s)$ is plotted as a function of $s$, while the red line denotes the *MSE* based only on ${\mathbf{y}}_{\text{obs}}$. Despite the magnitude of the location errors being relatively small, observing another measurement of $y$ at some locations can increase (possibly dramatically) the *MSE*.\[fig:no\_self\_efficiency\]](figure/no_self_efficiency){width="\maxwidth"}
Simulation results suggest that even when $c$ contains a nugget term $\sigma^2_x$, KILE is still not self-efficient, and additional observations can increase MSE. Figure \[fig:no\_self\_efficiency\] illustrates the change in MSE as a function of the location of an additional observation of $y$. Following Theorem \[theorem1\], we see the MSE is unbounded when $\sigma^2_x = 0$. But even when $\sigma^2_x = 1$, it is possible for an additional observation to (slightly) increase MSE.
Parameter Estimation for Kriging
--------------------------------
In typical applied settings, some or all parameters of the covariance function are unknown and must be estimated by the analyst in order to obtain Kriging equations. For Gaussian process models without a location error component, parameter estimation can be accomplished using likelihood methods. This can be computationally challenging for large data sets, as each likelihood evaluation requires a Cholesky factorization of the covariance matrix (or equivalent operations), which is $\mathcal{O}(n^3)$ except in special cases. An alternative is to choose parameters by maximizing goodness of fit between the empirical variogram and the theoretical (parametric) variogram, though this is less efficient for parametric Gaussian models.
Location errors present challenges for both such procedures as the covariance function for the observed provess $y$ may not be available in closed form, meaning neither the likelihood function or variogram can be evaluated exactly. While Monte Carlo methods surely offer effective approaches in theory \[@fanshawe2011spatial\], they muliply the computational expense of the problem, as each evaluation of the likelihood requires $M$ matrix factorizations, where $M$ is the number of Monte Carlo samples used to approximate the likelihood. @Cressie2003 advocate a pseudo-likelihood procedure \[[@carroll2006measurement]\] that uses a Gaussian likelihood approximation based on the first two moments of $y$, $$\label{pseudo}
\tilde{L}(\theta; {\mathbf{y}}_n) \propto |{\mathbf{K}}_{\theta}({\mathbf{s}}_n, {\mathbf{s}}_n)|^{-1/2}\exp \left( - \frac{1}{2} {\mathbf{y}}_n ' {\mathbf{K}}_{\theta}({\mathbf{s}}_n, {\mathbf{s}}_n)^{-1} {\mathbf{y}}_n \right),$$ where we write ${\mathbf{K}}_{\theta}$ to explicitly mark the dependence of the covariance function $k$ on unknown parameters $\theta$. This pseudo-likelihood requires inverting ${\mathbf{K}}$ only once per pseudo-likelihood evaluation, even when ${\mathbf{K}}_{\theta}$ is computed by Monte Carlo.
We can work out inferential properties of the maximum pseudo-likelihood estimator $\hat{\tilde{\theta}} = \text{argmax}_{\theta} \tilde{L}(\theta; {\mathbf{y}}_n)$. First, it is straightforward to check that the pseudo-score pertains to an unbiased estimating equation: $$\label{pseudoscore}
{\mathbb{E}}[\tilde{S}(\theta; {\mathbf{y}}_n)] = {\mathbb{E}}[\nabla \log( \tilde{L}(\theta; {\mathbf{y}}_n))] = \mathbf{0}.$$ Moreover, one can show the covariance matrix of the pseudo-score is given by $$\begin{aligned}
\label{varscore}
\tilde{G}(\theta) =& {\mathbb{E}}[\tilde{S}(\theta; {\mathbf{y}}_n) \tilde{S}(\theta; {\mathbf{y}}_n)'] \nonumber \\
\tilde{G}(\theta)_{ij} =& {\mathbb{E}}\left[\frac{1}{2}\text{Tr}\{\Omega_i {\mathbf{C}}_{\theta}({\mathbf{u}}_n) \Omega_j {\mathbf{C}}_{\theta}({\mathbf{u}}_n)\}\right]
\nonumber \\
& + \frac{1}{4}\Big( {\mathbb{E}}[ \text{Tr}\{\Omega_i {\mathbf{C}}_{\theta}({\mathbf{u}}_n)\}\text{Tr}\{\Omega_j {\mathbf{C}}_{\theta}({\mathbf{u}}_n)\} ] - \text{Tr}\{\Omega_i {\mathbf{K}}_{\theta}\}\text{Tr}\{\Omega_j {\mathbf{K}}_{\theta}\} \Big),\end{aligned}$$ using the notational abbreviations ${\mathbf{C}}_{\theta}({\mathbf{u}}_n) = {\mathbf{C}}_{\theta}({\mathbf{s}}_n + {\mathbf{u}}_n, {\mathbf{s}}_n + {\mathbf{u}}_n)$, ${\mathbf{K}}_{\theta} = {\mathbf{K}}_{\theta}({\mathbf{s}}_n, {\mathbf{s}}_n) = {\mathbb{E}}[{\mathbf{C}}_{\theta}({\mathbf{u}}_n)]$, and $\Omega_i = {\mathbf{K}}_{\theta}^{-1} \left(\frac{\partial}{\partial \theta_i} {\mathbf{K}}_{\theta} \right) {\mathbf{K}}_{\theta}^{-1}$. Lastly, the expected negative Hessian of the log pseudo-likelihood is $$\begin{aligned}
\label{hessian}
\tilde{H}(\theta)_{ij} &= {\mathbb{E}}\left[- \frac{\partial^2}{\partial \theta_i \partial \theta_j} \log( \tilde{L}(\theta; {\mathbf{y}}_n)) \right] \nonumber \\
&= \frac{1}{2}\text{Tr}\{\Omega_i {\mathbf{K}}_{\theta} \Omega_j {\mathbf{K}}_{\theta}\}.
\end{aligned}$$
If there are no location errors (${\mathbf{u}}_n \equiv \mathbf{0}$), $\tilde{L}$ is an exact likelihood and the second term in the right hand side of Equation vanishes so that $\tilde{G}(\theta) = \tilde{H}(\theta)$, confirming the second Bartlett identity \[@ferguson1996course\]. For non-zero location errors, however, we construct the Godambe information matrix as an analog to the Fisher information matrix \[@varin2011overview\], $$\tilde{I}(\theta) = \tilde{H}(\theta) [\tilde{G}(\theta)]^{-1} \tilde{H}(\theta).$$ Evaluating $\tilde{I}(\theta)$ for different location error models illustrates the information loss in estimating covariance function parameters $\theta$ relative to the error-free case, where $\tilde{I}(\theta) = \tilde{G}(\theta) = \tilde{H}(\theta)$ is equivalent to the Fisher information matrix.
General theory of unbiased estimation equations \[@heyde1997quasi\] suggests the asymptotic behavior of the pseudo-likelihood procedure satisfies $$\tilde{I}(\theta)^{1/2}(\hat{\tilde{\theta}} - \theta) \stackrel{D}{\rightarrow} {\mathcal{N}}(\mathbf{0}, \mathbf{I}).
\label{asym}$$ However, Expression does not hold in general even in an error-free regime ${\mathbf{u}}_n \equiv 0$, as asymptotic results for Gaussian process covariance parameters depend on the spatial sampling scheme used and the specific form of the covariance function \[[@stein1999interpolation]\]. We nevertheless expect to hold for suitably well-behaved processes under increasing-domain asymptotics. [@guyon1982parameter] gives an applicable result when locations ${\mathbf{s}}_n$ are on a lattice. We are not aware of other theoretical results in this context.
Markov Chain Monte Carlo Methods {#sec:hmc}
================================
Markov Chain Monte Carlo methods offer an alternative to Kriging for prediction in a regime with noisy inputs. They allow us to compute the MSE-optimal prediction $$\begin{aligned}
\hat{x}(s^*) &= {\mathbb{E}}[x(s^*) | {\mathbf{y}}_n ] \nonumber \\
&=\int \left( {\mathbf{C}}(s^*, {\mathbf{s}}_n + {\mathbf{u}}_n)[{\mathbf{C}}({\mathbf{s}}_n + {\mathbf{u}}_n, {\mathbf{s}}_n + {\mathbf{u}}_n)]^{-1}{\mathbf{y}}_n \right) \pi({\mathbf{u}}_n | {\mathbf{y}}_n)d{\mathbf{u}}_n,
\label{bayesrule}\end{aligned}$$ which will dominate the KALE estimator in terms of MSE for any model and set of observed and predicted locations. The optimality of $\hat{x}(s^*)$ in is due to the fact that the conditional mean ${\mathbb{E}}[x(s^*) | {\mathbf{y}}_n]$ obtains the minimum MSE for any estimator of $x(s^*)$ that is a function of ${\mathbf{y}}_n$. This estimator is not linear, and MCMC methods are necessary for evaluating as the density for the conditional distribution $\pi({\mathbf{u}}_n | {\mathbf{y}}_n)$ will not be available in closed form (no possible “conjugate“ form for the distribution of ${\mathbf{u}}_n$ is known to us). When model parameters, such as in the covariance function $c$ or the distribution of $u$ are unknown, the distribution $\pi({\mathbf{u}}_n | {\mathbf{y}}_n)$ implicitly averages over the posterior distributions of such parameters.
MCMC methods also allow us to compute prediction intervals $(z_{\text{low}}, z_{\text{high}})$ such that ${\mathbb{P}}(z_{\text{low}} < x(s^*) < z_{\text{high}} | {\mathbf{y}}_n) = 1 - \alpha$. When the covariance function $c$ and location error model $g$ are known, these intervals are exact probability conditional probability statements, providing a stronger coverage guarantee than that achieved with the KALE procedure in Proposition \[coverprop\], where coverage is achieved only by averaging over ${\mathbf{y}}_n$.
Distributional Assumptions
--------------------------
MCMC inference for requires the assumption that ${\mathbf{x}}_n$ is Gaussian. While this is a common assumption in practice and has been assumed throughout this paper, it is not necessary to derive the KALE equations and their MSE (but it is necessary to produce coverage intervals as in Proposition \[coverprop\]). Thus, Kriging approaches, including KALE, are attractive when there is information about the joint distribution of $x$ beyond its first two moments.
In this scenario, however, we can still advocate—from a decision-theoretic perspective—a Gaussian assumption when the goal of the analysis is minimum MSE prediction. Specifically, let $\pi \in \Pi_{\mathbf{0}, {\mathbf{C}}}$ be a choice of joint distribution for ${\mathbf{x}}_n$ with the appropriate first two moments $\mathbf{0}$ and ${\mathbf{C}}$. The minimum MSE prediction of $x(s^*)$ assuming ${\mathbf{x}}_n \sim \pi$ is the conditional mean ${\mathbb{E}}_{\pi}[x(s^*) | {\mathbf{x}}_n]$. Let $\mathrm{R}_{\pi_0}(\pi)$ be the risk (MSE) of this minimum MSE predictor under the assumption that ${\mathbf{x}}_n \sim \pi$ when in fact ${\mathbf{x}}_n \sim \pi_0$; that is, $$\mathrm{R}_{\pi_0}(\pi) = {\mathbb{E}}_{\pi_0}[({\mathbb{E}}_{\pi}[x(s^*) | {\mathbf{x}}_n] - x(s^*))^2].$$ Thus $R_{\pi_0}(\pi)$ represents the risk (MSE) in a misspecified joint distribution for ${\mathbf{x}}_n$, where the analyst assumes ${\mathbf{x}}_n \sim \pi$, but in fact ${\mathbf{x}}_n \sim \pi_0$. We then have the following proposition, based on Theorem 5.5 of [@morris1983natural]:
Let $\pi_0 \in \Pi_{\mathbf{0}, {\mathbf{C}}}$ be Gaussian. Then for all $\pi \in \Pi_{\mathbf{0}, {\mathbf{C}}}$ we have $$\mathrm{R}_{\pi}(\pi) \leq \mathrm{R}_{\pi}(\pi_0) = \mathrm{R}_{\pi_0}(\pi_0) \leq \mathrm{R}_{\pi_0}(\pi).$$ \[morris\_prop\]
Unlike traditional decision theory problems, here we are fixing the estimator (Kriging), and considering the costs of different distributional assumptions ($\pi$). Given that the analyst has decided to use Kriging for predicting $x(s^*)$, then the risk in making an incorrect distributional assumption is $\mathrm{R}_{\pi_0}(\pi_0) - \mathrm{R}_{\pi}(\pi_0) = 0$. This reflects the fact that the Kriging MSE depends only on the first two moments of $\pi$. However, there is an “opportunity cost” in making any non-Gaussian assumption $\mathrm{R}_{\pi}(\pi_0) - \mathrm{R}_{\pi}(\pi) > 0$ for $\pi \neq \pi_0$, which represents the reduction in MSE under $\pi$ that could be achieved by using a different estimator other than Kriging.
Obviously, if there is a strong reason to believe a non-Gaussian $\pi$ is true, then analysis should proceed with this assumption, ideally leveraging an estimator that is optimal under these assumptions (instead of Kriging). However, without strong distributional knowledge, the analyst can assume Gaussianity without risking increased MSE or paying an opportunity cost for using an inefficient method.
Hybrid Monte Carlo
------------------
Hybrid Monte Carlo \[[@neal2005hamiltonian]\] is well-suited for the problem of sampling $\pi({\mathbf{u}}_n | {\mathbf{y}}_n) \propto \pi({\mathbf{y}}_n | {\mathbf{u}}_n) \pi({\mathbf{u}}_n)$ in order to evaluate . This is because while $\pi({\mathbf{u}}_n | {\mathbf{y}}_n)$ is computationally expensive (requiring inversion of the covariance matrix ${{\mathbf{C}}_{\theta}({\mathbf{u}}_n)}= {\mathbf{C}}_{\theta}({\mathbf{s}}_n + {\mathbf{u}}_n, {\mathbf{s}}_n + {\mathbf{u}}_n)$), the gradient $\nabla \log (\pi({\mathbf{u}}_n | {\mathbf{y}}_n))$ is a relatively cheap byproduct of this calculation. Often the conditional distribution ${\mathbf{u}}_n | {\mathbf{y}}_n$ is correlated across components, making gradient-based MCMC methods more efficient for generating samples. Other gradient-based MCMC sampling methods, such as the Metropolis-adjusted Langevin algorithm \[@roberts2001optimal\] and variants, may also be well-suited to this problem.
Bayes rule provides $\pi(\theta, {\mathbf{u}}_n | {\mathbf{y}}_n) \propto \pi({\mathbf{y}}_n | \theta, {\mathbf{u}}_n)\pi(\theta, {\mathbf{u}}_n)$, where $\theta$ here represents any unknown parameter(s) of the covariance function $c$. In most situations it will be reasonable to assume ${\mathbf{u}}_n$ and $\theta$ are independent a priori—this is trivially true in the case that $\theta$ is assumed known. Assuming this, and recognizing that $\pi({\mathbf{y}}_n|\theta, {\mathbf{u}}_n)$ is Gaussian, we can write the log posterior and its gradient: $$\begin{aligned}
\log(\pi(\theta, {\mathbf{u}}_n | {\mathbf{y}}_n)) &= - \frac{1}{2} \log(|{{\mathbf{C}}_{\theta}({\mathbf{u}}_n)}|) - \frac{1}{2} {\mathbf{y}}_n' {{\mathbf{C}}_{\theta}({\mathbf{u}}_n)}^{-1} {\mathbf{y}}_n + \text{const.}\\
{\frac{\partial}{\partial u_i}}\log(\pi(\theta, {\mathbf{u}}_n | {\mathbf{y}}_n)) &= \\
& \hspace{-3cm} \frac{1}{2}\text{Tr}\left({{\mathbf{C}}_{\theta}({\mathbf{u}}_n)}^{-1} \left[ {\frac{\partial}{\partial u_i}}{{\mathbf{C}}_{\theta}({\mathbf{u}}_n)}\right] \left({{\mathbf{C}}_{\theta}({\mathbf{u}}_n)}^{-1} {\mathbf{y}}_n {\mathbf{y}}_n' - \mathbf{I}_n \right) \right) + {\frac{\partial}{\partial u_i}}\log(\pi({\mathbf{u}}_n)) \\
{\frac{\partial}{\partial \theta_i}}\log(\pi(\theta, {\mathbf{u}}_n | {\mathbf{y}}_n)) &= \\
& \hspace{-3cm} \frac{1}{2}\text{Tr}\left({{\mathbf{C}}_{\theta}({\mathbf{u}}_n)}^{-1} \left[ {\frac{\partial}{\partial \theta_i}}{{\mathbf{C}}_{\theta}({\mathbf{u}}_n)}\right] \left({{\mathbf{C}}_{\theta}({\mathbf{u}}_n)}^{-1} {\mathbf{y}}_n {\mathbf{y}}_n' - \mathbf{I}_n \right) \right) + {\frac{\partial}{\partial \theta_i}}\log(\pi(\theta)).\end{aligned}$$
The computational cost of both the likelihood and gradient are dominated by solving ${{\mathbf{C}}_{\theta}({\mathbf{u}}_n)}$ (*e.g.*, Cholesky factorization), which is $\mathcal{O}(n^3)$. Every likelihood evaluation computes this term, which can then be re-used in the gradient equations. Thus, the computational cost of computing both the likelihood and gradient remains $\mathcal{O}(n^3)$.
Multimodality
-------------
The posterior distribution $\pi(\theta, {\mathbf{u}}_n | {\mathbf{y}}_n)$ is often multimodal, more so if the distrubution $\pi({\mathbf{u}}_n)$ is diffuse. This is because if there is a local mode at $(\hat{\theta}, \hat{{\mathbf{u}}}_n)$, there may be a local mode at any $(\theta, {\mathbf{u}}_n)$ such that ${{\mathbf{C}}_{\theta}({\mathbf{u}}_n)}= C_{\hat{\theta}}(\hat{{\mathbf{u}}}_n)$, as the likelihood is constant for such $(\theta, {\mathbf{u}}_n)$. In particular, for isotropic covariance models, the likelihood is constant for additive shifts in ${\mathbf{u}}_n$ or rotations of ${\mathbf{s}}_n + {\mathbf{u}}_n$, as these operations preserve pairwise distances. Additionally, multimodality can be induced by the many-to-one mapping of the set of true locations $\{s_i + u_i, i=1, \ldots, n\}$ to the set of observed locations $\{s_i, i=1, \ldots, n\}$. For instance, with $n=2$ and an isotropic covariance function, for any choice of $u_1, u_2$ we get the same likelihood with $\tilde{u}_1 = s_2 + u_2 - s_1$ and $\tilde{u}_2 = s_1 + u_1 - s_2$. Moreover, for fixed ${\mathbf{u}}_n$, for many common covariance functions it is possible for the posterior of $\theta$ to be multimodal \[@warnes1987problems\].
HMC (and other gradient MCMC methods) can efficiently sample from multiple modes, however this becomes difficult when the modes are isolated by regions of extremely low likelihood \[@neal2011mcmc\]. Isolated modes can occur in the location-error GP regime. For example, assume one-dimensional locations ($p=1$) and an isotroptic covariance model with known parameters $\theta$ and nugget $\sigma^2_x$. Marginally, as $ \| s_1 + u_1 - (s_2 + u_2)\| \rightarrow 0$, $y_1 - y_2 \stackrel{D}{\rightarrow} {\mathcal{N}}(0, 2\sigma^2_x)$; that is, the scaled difference $|y_1 - y_2|/(\sigma_x)$ must be reasonably small. When this is not the case (e.g. $\sigma^2_x = 0$), then the log-likelihood asymptotes at $s_1 + u_1 = s_2 + u_2$ almost surely. Thus, the Markov chain can only sample ${\mathbf{u}}_n$ such that the ordering of $\{s_i + u_i, i=1, \ldots, n\}$ is preserved. Note that when $p > 1$, while the log-likelihood may still asymptote at $s_1 + u_1 = s_2 + u_2$, this no longer constrains the space of ${\mathbf{u}}_n$ (except on sets with measure 0).
Figure \[fig:multi\_modes\_plot\] demonstrates the modal behavior for this simple example with $p=1$ and $n=2$. When location errors are large in magnitude and the nugget tern $\sigma^2_x$ is small, the modes of $(u_1, u_2)$ are separated by a contour of near 0 density (panel A). A higher nugget $\sigma^2_x$ increases the density between the modes, making it easier for the same MCMC chain to travel between them (panel B). Decreasing the magnitude of the (Gaussian) location errors, $\sigma^2_u$, puts more mass on a single mode, as the unimodal distribution $\pi({\mathbf{u}}_n)$ has a greater influence on $\pi({\mathbf{u}}_n | {\mathbf{y}}_n)$ (panel C).
Thus, as with any MCMC application, for the location-error GP problem it is advisible to run separate chains in parallel, with different, diffuse starting points, and monitor mixing diagnostics \[@gelman2011inference\]. Multiple chains failing to mix is likely a symptom of multiple isolated modes, in which case we should modify the HMC algorithm to include tempering \[@salazar1997simulated\] or non-local proposals that allow for mode switching \[@qin2001multipoint [@lan2013wormhole]\]. Another strategy to overcome multiple isolated modes is importance sampling: as Figure \[fig:multi\_modes\_plot\] shows, increasing the nugget variance $\sigma_x^2$ increases the density between modes. If we generate samples according to $\tilde{\pi}(\theta, {\mathbf{u}}_n | {\mathbf{y}}_n) \propto \tilde{\pi}({\mathbf{y}}_n | \theta, {\mathbf{u}}_n) \pi(\theta) \pi({\mathbf{u}}_n)$ where $\tilde{\pi}({\mathbf{y}}_n | \theta, {\mathbf{u}}_n)$ is the density corresponding to ${\mathcal{N}}({\mathbf{0}}, {{\mathbf{C}}_{\theta}({\mathbf{u}}_n)}+ \kappa \mathbf{I}_n)$ for some fixed $\kappa$, then it is straightforward to compute importance weights $\pi(\theta, {\mathbf{u}}_n | {\mathbf{y}}_n) / \tilde{\pi}(\theta, {\mathbf{u}}_n | {\mathbf{y}}_n)$. This is because ${{\mathbf{C}}_{\theta}({\mathbf{u}}_n)}^{-1}$ is easy to compute from $({{\mathbf{C}}_{\theta}({\mathbf{u}}_n)}+ \kappa\mathbf{I}_n)^{-1}$ (and vice versa) using the Woodbury formula. Either standard importance sampling, or Hamiltonian importance sampling \[@neal2005hamiltonian\], could be used to generate parameter estimates, point/interval predictions, and any other posterior estimates of interest.
Simulation study {#sec:simul}
================
We compare Kriging (both KALE and KILE) and HMC methods for point/interval forecasts for Gaussian process regression in a simulation study. For various combinations of parameter values for the covariance function $c(s_1, s_2)$ and location error model $g(u)$ we simulate observations ${\mathbf{y}}_n$ where $y_i = x(s_i + u_i)$ and make predictions for values of $x$ at unobserved locations: ${\mathbf{x}}^*_k = (x(s^*_1) \: \: \ldots \: \: x(s^*_k))'$.
We simulate data using the squared exponential covariance function $c(s_1, s_2) = \tau^2 \exp(-\beta \|s_1 - s_2\|^2) + \sigma^2_x \mathbf{1}_{s_1 = s_2}$ and an i.i.d. Gaussian location error model $u_i \stackrel{iid}{\sim} {\mathcal{N}}(0, \sigma^2_u\mathbf{I}_p)$. The squared exponential covariance function and Gaussian location error model combine to form a convenient regime, as we can evaluate $k$ in closed form . Without loss of generality, we can use $\tau^2 = 1$ for all simulations as it is simply a scale parameter. We consider a $p=2$ dimensional location space, $s_i \in {\mathbb{R}}^2$. On a $8 \times 8$ grid, we randomly select 54 locations at which we observe $y$, and target the remaining 10 locations for interpolating $x$. Figure \[fig:sim\_plot\] illustrates a range of data samples for processes used in our simulations on this space, while Table \[param\_table\] provides a full summary of all the parameter value combinations we consider. Data from each parameter combination is simulated 100 times.
Parameter Values simulated Prior support
-------------- ----------------------------- ---------------
$\tau^2$ 1 $(0, 10)$
$\beta$ 0.001, 0.01, 0.1, 0.5, 1, 2 $(0.0005, 3)$
$\sigma^2_x$ 0.0001, 0.01, 0.1, 0.5, 1 $(0, 10)$
$\sigma^2_u$ 0.0001, 0.01, 0.1, 0.5, 1 $(0, 10)$
: Parameter values used in simulation study. The range $(0.0005, 3)$ for $\beta$ guarantees that at least one pair of points among our observed data has a correlation in the range $(0.05, 0.95)$. This eliminates modes corresponding to white noise processes from the likelihood surface.[]{data-label="param_table"}
![Samples of $x(s)$ for different values of the length-scale parameter $\beta$ with the squared exponential covariance function, $c(s_1, s_2) = \exp(-\beta\|s_1 - s_2\|^2) + \sigma^2_x \mathbf{1}_{s_1 = s_2}$. Black points are where we have observed $y(s)$ and white points are where we wish to predict $x(s)$. Observed/predicted locations were randomly sampled from an $8 \times 8$ grid.\[fig:sim\_plot\]](figure/sim_plot){width="\maxwidth"}
We evaluate the three prediction methods—KALE, KILE, and HMC—using both adjusted root mean squared error (RMSE) and the coverage probability of a $95\%$ interval. “Adjusted” RMSE is based on the MSE with $\sigma^2_x$ subtracted out, as this term appears in the MSE for any prediction method. For every parameter combination of interest used, these statistics are calculated first by averaging over each of the $k=10$ prediction targets in each simulated draw of new data, and then over the $J=100$ independent data draws.
Both evaluation statistics can be evaluated more precisely during simulation by utilizing a simple Rao-Blackwellization. For iteration $j$, instead of drawing ${\mathbf{x}}^*_k$ in addition to ${\mathbf{y}}_n$ and calculating $\text{rmse}_j = \|{\mathbf{x}}^*_k - \hat{{\mathbf{x}}}^*_k\|/k$, we simply condition on the simulated location errors ${\mathbf{u}}_n$ to get $\text{rmse}_j = {\mathbb{E}}[\|{\mathbf{x}}^*_k - \hat{{\mathbf{x}}}^*_k\|/k \mid {\mathbf{y}}_n, {\mathbf{u}}_n]$. Similarly, to calculate coverage of an interval $(L_{s^*}({\mathbf{y}}_n), U_{s^*}({\mathbf{y}}_n))$ for $x(s^*)$, for iteration $j$ we use $$\text{cov}_j = \frac{1}{k} \sum_{i=1}^k {\mathbb{E}}[\mathbf{1}[x(s_i^*) \in (L_{s_i^*}({\mathbf{y}}_n), U_{s_i^*}({\mathbf{y}}_n))] \mid {\mathbf{y}}_n, {\mathbf{u}}_n].$$
HMC is implemented using the software `RStan` \[@rstan-software:2014\], which implements the “no-U-turn” HMC sampler \[@homan2014no\]. 10000 samples were drawn during each simulation iteration, which (for most parameter values) takes a few minutes on a single 2.50Ghz processor.
Known covariance parameters
---------------------------
We first simulate point and interval prediction for KALE, KILE, and HMC using the same parameter values that generated the data. By doing so, we leave aside the issue of parameter inference and simply compare the extent to which different methods leverage the information in the location-error corrupted data ${\mathbf{y}}_n$ to infer $x(s^*)$. Figure \[fig:mspe\_known\_0-0001\] compares RMSE for the three methods when there is a very small nugget, $\sigma_x^2 = 0.0001$.
We can see that there is little difference among the three methods when $\sigma_u^2$ is sufficiently small ($0.0001$), or when $\beta$ is sufficiently large ($2$). This makes sense, as in the former case, with small location errors the potential improvement over KILE (which is exact for $\sigma^2_u = 0$) is negligible, and in the latter case, observations are too weakly correlated for nearby points to be informative. Larger values of $\sigma_u^2$ give KALE a significant reduction in RMSE versus KILE, with the reduction as large as $79\%$ for the case of large magnitude location errors ($\sigma^2_u = 1$) and a moderately smooth signal ($\beta = 0.1$).
The idea of a moderately smooth signal requires further elaboration: for a given $\sigma^2_u$, when $x$ is very smooth ($\beta$ very small), the process is roughly constant within small neighborhoods, meaning $y(s) \approx x(s)$ and location errors are less of a concern for accurate inference and prediction. On the other hand, when $\beta$ is very large and the process is highly variable in small regions of the input space, location errors are less of a concern because there is very little information in the data to begin with. Location errors are most influential when the process $x$ has more moderate variation across neighborhoods corresponding to the plausible range of the location errors. HMC offers further reductions in RMSE over KALE in roughly the same regions of the parameter space in which KALE improves over KILE, although the additional improvement is less dramatic. The maximum RMSE reduction we observe is about $28\%$, once again for a moderately smooth signal with larger magnitude location errors.
When the nugget variance $\sigma^2_x$ is increased (Figure \[fig:mspe\_known\_0-1\] shows results for $\sigma^2_x = 0.1$), differences in RMSE among the three methods become smaller (the differences are wiped out entirely at $\sigma^2_x = 1$, which is not pictured). This is not due to a shared $\sigma^2_x$ term in the RMSE value for all methods, as this is subtracted out. Rather, the similarity of all three methods reflects the fact that a larger nugget leaves less information in the data that can be effectively used for prediction. However, the differences that we do observe (both comparing KALE to KILE and HMC to KALE) occur primarily when the magnitude of location errors $\sigma^2_u$ is large.
In the case where all parameters are fixed and known, both KALE and HMC produce intervals with exact coverage (subject to Monte Carlo or numerical approximation errors) in all simulations. KILE, however, can severely undercover in the presence of location errors. Figure \[fig:cov\_known\_none\] shows coverage as low as $4\%$ when the magnitude of the location errors is high ($\sigma^2_u = 1$), $\beta = 0.1$, and the nugget variation is minimal ($\sigma^2_x = 0.0001$). Undercoverage still persists in this region of the parameter space for $\sigma^2_x = 1$, the largest nugget variance used in our simulations.
Unknown covariance parameters
-----------------------------
In typical applied settings, the analyst will not know model parameters such as those of the covariance function ($\tau^2, \beta$), the nugget variance $\sigma^2_x$, or even the variance of the location errors $\sigma^2_u$. Due to identifiability issues with our choice of covariance function in this simulation , we assume $\sigma^2_u$ is known but estimate all other parameters before making predictions at unobserved locations.
For KILE and KALE, parameter estimation is accomplished through maximum (pseudo-) likelihood, as in . Parameter estimates are then plugged into Kriging equations – to obtain corresponding point and interval estimates. Because $c$ and $k$ are both squared exponential , the pseudolikelihood estimation procedure estimates the same covariance function for $y$, however the estimated parameters (and therefore Kriging equations, based on $k^*$) will differ. The plug-in approach ignores uncertainty in parameter estimates, so plug-in MSE estimates will be too optimistic. Various techniques exist for adjusting MSE from estimated parameters \[@smith2004asymptotic [@zhu2006spatial]\], though there is no need to incorporate such techniques into our analysis since exact (up to Monte Carlo error) MSEs are provided by simulation.
For HMC, we supply unknown parameters with prior distributions and sample parameters and predictions jointly from the posterior distribution $\pi(\theta, {\mathbf{x}}^*_k | {\mathbf{y}}_n)$. The priors we use are flat over a reasonable range (see Table \[param\_table\]), which guarantees both a proper posterior and a posterior mode that agrees with the maximum likelihood estimate of $\theta$. This second condition supports fair comparisons between predictions derived from HMC parameter estimates versus those based on the maximum (psueolikelihood) parameter values.
Figure \[fig:mspe\_unknown\_1\] provides the relative RMSE of KALE *vs.* KILE, and HMC *vs.* KALE, for predictions when parameters must first be estimated (using $\sigma_x^2 = 0.0001$). We notice that there does not appear to be a great advantage in KALE over KILE when parameters are first estimated. This is because, as mentioned earlier, the marginal process $y$ still has a squared exponential covariance function \[cov\_y\_sqexp\], so Kriging equations for KALE and KILE will be very similar. On the other hand, we notice a modest improvement when using HMC over Kriging, except in a small region of the parameter space ($\sigma^2_u \leq .01$ and $\beta \in [0.5, 1]$).
When the nugget variance is increased to $\sigma^2_x = 0.1$, we see the results in Figure \[fig:mspe\_unknown\_2\]. We still see relatively similar performances from KALE and KILE. HMC offers a small improvement over KALE when $\beta \geq 0.01$, though for $\beta = 0.001$ we actually see significantly higher MSEs with HMC. At $\beta = 0.001$ the process is extremely smooth, as the most distant pairs of observations still have a correlation of $0.88$. We are thus more concerned with overestimating $\beta$ than underestimating it; as the former shrinks predictions towards 0 while the latter shrinks towards (approximately) the mean of all observations. As we use a flat prior for $\beta$, where almost all mass is located $\beta > .001$, the posterior tends to overestimate $\beta$, leading to draws with relatively high MSE.
Neither Kriging or HMC guarantees prediction intervals with the correct coverage in the regime where parameters must first be estimated (though HMC would give proper “Bayes coverage” when simulating $\theta$ according to the prior used). We nevertheless present coverage results in Figures \[fig:cov\_false\_none\]–\[fig:cov\_false\_hmc\]. While we do not expect any method used to provide exact coverage, Kriging (both KALE and KILE) suffer from significant undercoverage for some regions of the parameter space, while HMC is consistent in offering at least $85\%$ coverage throughout our simulations. In a regime without location errors, @zimmerman1992mean advocate Bayesian procedures under non-informative priors over frequentist procedures in order to obtain interval estimates with good coverage; our simulation results, albeit in the context of location errors, agree with this finding.
Summary
-------
Our simulation results confirm the theoretical guarantee of KALE dominating KILE in prediction MSE when the covariance function is known, and furthermore HMC dominating KALE. The magnitude of differences in MSE between these methods is greatest when the process is moderately smooth relative to the spatial sampling (*e.g.*, $0.01 \leq \beta \leq 0.5$), when the magnitude of location errors $\sigma^2_u$ is largest, and when nugget variation $\sigma^2_x$ is smallest. For such regions of the parameter space, KILE fails to deliver prediction intervals with proper coverage, whereas KALE and HMC can give valid prediction intervals for any parameter values.
An important consequence in adjusting for location errors with a known covariance function is the corresponding adjustment to the nugget. The discussion in (Sections 3.6 and 3.7 of) @stein1999interpolation emphasizes the importance of correctly specifying the high-frequency behavior of the process when interpolating (correctly specifying the low-frequency behavior is less crucial), including the nugget term. Estimating parameters, including the nugget term $\sigma^2_x$, implicitly corrects for model misspecification when ignoring location errors. Thus we see little difference in predictive performance between KALE and KILE when parameters are first estimated. Depending on the choice of prior, KALE/KILE may give lower MSE predictions than HMC, which averages over posterior parameter uncertainty; however, interval coverage is better for HMC (using weak prior information) than for KALE/KILE.
Interpolating Northern Hemisphere Temperature Anomolies {#sec:cru}
=======================================================
To illustrate the methods discussed in this paper, we consider interpolating northern hemisphere temperature anomolies during the summer of 2011 using the publicly available CRUTEM3v data set[^1] \[@brohan2006uncertainty\]. Figure \[fig:cru\_data\_plot\] shows our data. These data are used in geostatistical reconstructions of the Earth’s temperature field, which interpolate temperatures at unobserved points in space-time in order to better understand the historical behavior of climate change (see, *e.g.*, @tingley2010bayesian and @richard2012new). Each observation is a spatiotemporal average: temperature readings are averaged over the April–September period and each $5^{\circ} \times 5^{\circ}$ longitude-latitude grid cell. These values are then expressed as anomolies relative to the global average during the period 1850–2009, which is calculated using an ANOVA model \[@tingley2012bayesian\]. Apart from this spatiotemporal averaging, numerous other preprocessing steps adjust this data for differences in altitude, timing, equipment, and measurement practices between sites, along with other potential sources of error; please see @morice2012quantifying and @jones2012hemispheric for more details.
Our analysis, restricted to interpolating a single year of data, and without using external data such as temperature proxies \[@mann2008proxy\], is intended as a proof of concept rather than as a refinement or improvement to existing analyses of these data. We wish to illustrate the potential impact of location errors on conclusions drawn from these data.
![CRUTEM3v data for summer 2011, with 2011 mean subtracted so that measurements represent spatial anomolies. Generally speaking, we see lower (cooler) anomolies in North America and positive (warmer) anomolies in Europe. Higher latitudes also tend to have positive anomolies.\[fig:cru\_data\_plot\]](figure/cru_data_plot){width="\maxwidth"}
The “gridding”, or spatial averaging across $5^{\circ} \times 5^{\circ}$ cells, complicates analyses using Gaussian process models \[@director2015connecting\]. However, assuming a smooth temperature field, we know that the recorded spatial average must be realized exactly at some location in each grid box (closer to the center if a lot of points have been averaged together). This frames the spatial averaging problem as a location measurement error problem: instead of observing the temperature $x(s)$ at each grid center $s$, we observe the temperature at an unknown location displaced from the grid center: $y(s) = x(s + u)$.
Following @tingley2010bayesian, we assume an exponential covariance function for $x(s)$, where distance is calculated along the Earth’s surface. As $s$ is given in terms of longitude/latitude ($s = (\psi, \phi)$), this has the form $$\begin{aligned}
c(s_1, s_2) &= \tau^2 \exp(-\beta \Delta) + \sigma^2_x \mathbf{1}_{s_1 = s_2}\nonumber \\
\Delta & = 2r\arcsin \sqrt{\sin^2\left(\frac{\phi_2 - \phi_1}{2}\right) + \cos(\phi_1) \cos(\phi_2)\sin^2\left(\frac{\psi_2 - \psi_1}{2}\right)}, \label{haversine}\end{aligned}$$ where $r=6371$ is the radius of the earth (in km). At higher latitudes ($\phi$), the centers of each grid cell are closer together, so nearby observations are more strongly correlated. The nugget term $\sigma^2_x$ represents some combination of measurement error in temperature readings and high-frequency spatial variation that is inestimable using the gridded observation samples.
We assume the following model for location errors $u_i$, which are additive displacements of longitude/latitude coordinates $s_i = (\psi_i, \phi_i)$: $$\label{u_geo}
u_i \sim {\mathcal{N}}\left( \mathbf{0}, \sigma^2_u \left(\frac{180}{\pi r}\right)^2 \begin{pmatrix}\frac{1}{\cos^2(\phi_i)} & 0 \\ 0 & 1 \end{pmatrix} \right).$$ This prior is equivalent to assuming that distance along the Earth’s surface (great-circle distance) between each grid center and the corresponding observation location has a scaled chi distribution, $d(s_i, s_i + u_i) \sim \sigma_u \chi$. Combining and , we use Monte Carlo to compute $k$.
We treat parameters $\tau^2, \beta, \sigma^2_x$ as unknown, but fix $\sigma^2_u = 7500$. At this value, the median magnitude of the location errors in great-circle distance is 102km, which is consistent with analyzing the coordinates of the temperature recording sites used to compile the CRUTEM3v data[^2].
Kriging
-------
We first apply Kriging approaches to interpolate the CRUTEM3v data, both adjusting for and ignoring location errors . Because parameters $\tau^2, \beta, \sigma_x^2$ are unknown, we first need to estimate them using maximum likelihood (when ignoring location errors) or maximum pseudo-likelihood (when adjusting for location errors). These can then be plugged in to covariance functions $c$ and $k$ to obtain “empirical” Kriging equations we can use for interpolation \[@zimmerman1992mean\].
We find small differences in parameter estimates when ignoring location errors (assuming $\sigma_u^2 = 0$) and adjusting for them (assuming $\sigma_u^2 = 7500$):
$\sigma_u^2$ $\hat{\tau}^2$ $\hat{\beta}$ $\hat{\sigma}_x^2$
-------------- ---------------- ------------------------ --------------------
0 1.1671 $1.4275\times 10^{-4}$ 0.0747
7500 1.1649 $1.4677\times 10^{-4}$ 0.0699
: Covariance function parameter estimates when ignoring location errors (assuming $\sigma_u^2 = 0$) and adjusting for location errors (assuming $\sigma_u^2 = 7500$).[]{data-label="paramtable"}
Consequently, when we interpolate data at the centers of grid cells for which no data was observed, we see differences between the KALE and KILE approaches. Figure \[fig:cru\_krig\] shows the differences between KALE and KILE interpolations (both point and interval estimates). Relative to the range of the data (most anomolies are in the interval $(-1, 1)$), the discrepency between KALE and KILE does not seem very significant.
![Kriging results for interpolating temperature anomolies from summer 2011. The top plot shows interpolations at unobserved grid centers given by *KALE*. The bottom left plot shows the difference in estimates between *KALE* and *KILE* (*KALE* – *KILE*), and the bottom right plot shows difference in 95% interval widths between *KALE* and *KILE*.[]{data-label="fig:cru_krig"}](figure/cru_krig.pdf){width="100.00000%"}
HMC
---
Using HMC, parameter inference and interpolations are made simultaneously. The resulting point and interval predictions differ substantially from the Kriging results. However, because HMC incorporates parameter uncertainty in predictions, this comparison is not sufficient to illustrate the impact of location errors on conclusions from this data. A more appropriate comparison is between HMC with a location error model ($\sigma_u^2 = 7500$), and HMC assuming with no location errors ($\sigma_u^2 = 0$). These results are plotted in Figure \[fig:cru\_hmc\].
![Results for interpolating temperature anomolies from summer 2011 using HMC. The top plot shows interpolations at unobserved grid centers, assuming location errors $\sigma^2_u = 7500$. The bottom left plot shows the difference in estimates between the location error model and the model with $\sigma^2_u = 0$. The bottom right plot shows difference in 95% interval widths.[]{data-label="fig:cru_hmc"}](figure/cru_hmc.pdf){width="100.00000%"}
Using HMC, accounting for location errors produces more significant differences in inference/prediction than was observed for Kriging. This is particularly true for interval predictions, where adjusting for location errors yields intervals as much as 0.1 wider, which is a significant discrepency when most observations lie in $(-1, 1)$. Figure \[fig:cru\_params\] shows posterior densities for unknown parameters of the covariance function based on HMC draws from the $\sigma^2_u = 7500$ and $\sigma^2_u = 0$ models (the Kriging estimates of these parameters are vertical lines). HMC under location error model ($\sigma^2_u = 7500$) gives slightly larger $\beta$ estimates than when using $\sigma^2_u = 0$, meaning observations are inferred to be less strongly correlated. This yields prediction intervals that tend to be wider (see Figure \[fig:cru\_hmc\]). The most extreme descrepencies occur in the arctic, where distances between grid points are closest. The fact that modeling location errors adds additional uncertainty to arctic predictions is of particular interest to climate scientists, as accurate climate reconstruction for the arctic region is essential for understanding recent climate change patterns \[@cowtan2014coverage\].
![Density of posterior draws from HMC using $\sigma^2_u = 7500$ (blue) and $\sigma^2_u = 0$ (red). Point estimates of these parameters from Kriging (Table \[paramtable\]) are shown as vertical lines.[]{data-label="fig:cru_params"}](figure/cru_params.pdf){width="100.00000%"}
The difference between predictions obtained under the $\sigma^2 = 0$ and $\sigma^2_u = 7500$ models using HMC suggests that modeling location errors, even when they are small in magnitude, meaningfully impacts parameter estimates and predictions at unobserved locations. The fact that results for HMC (assuming $\sigma^2_u = 7500$) also differ from the results using KALE, while the KILE results do so less, demonstrates that moment procedures such as Kriging may be ineffective in adjusting for these errors.
Conclusion {#sec:conclusion}
==========
In this paper, we have explored the issue of Gaussian process regression when locations in the input space ${\mathbb{S}}$ are subject to error. Even when location errors are quite small in magnitude, it is essential to adjust Kriging equations in order to obtain good point and interval estimates; further improvements can be made by using MCMC to sample directly from the distribution of the measurement of interest given the sampled data.
Both MCMC and Kriging will be infeasible for large data sets, due to the cost of the covariance matrix inversion. A useful future study would be to adapt the procedures discussed in this paper to methods for inference and prediction for large spatial data sets, such as the predictive process approach \[[@banerjee2008gaussian]\], low rank representations \[@cressie2008fixed\], likelihood approximations \[@stein2004approximating\], and Markov random field approximations \[@lindgren2011explicit\]. It will also be useful to extend the analysis of this paper to regimes where location errors may be correlated with the process of interest $x$. For example, in climate data, regions with extreme climates will be harder to sample, thus there may be greater error in the spatial refencing of such sampling than for regions that are easier to sample.
Proofs of results
=================
Proof of Proposition \[validcovfn\]
-----------------------------------
$k$ is a valid covariance function if and only if for all $n$, ${\mathbf{s}}_n$, and $\{a_i \in {\mathbb{R}}, i=1, \ldots, n\}$, we have $$\sum_{i=1}^n \sum_{j=1}^n a_i a_j k(s_i, s_j) \geq 0.$$ From , this condition can be rewritten: $$\begin{aligned}
\sum_{i=1}^n \sum_{j=1}^n a_i a_j k(s_i, s_j) &= \sum_{i=1}^n \sum_{j=1}^n a_i a_j \int_{{\mathbb{S}}} c(s_i + u_i, s_j + u_j) dg_{{\mathbf{s}}_n}({\mathbf{u}}_n) \\
&= \int_{{\mathbb{S}}} \sum_{i=1}^n \sum_{j=1}^n a_i a_j c(s_i + u_i, s_j + u_j) dg_{{\mathbf{s}}_n}({\mathbf{u}}_n)\end{aligned}$$ As $c$ is a valid covariance function, the integrand in this expression is always non-negative, so the integral is also non-negative. Thus $k$ is a valid covariance function.
Note that for the common scenario where location errors are independent, so that $g_{{\mathbf{s}}_n}$ is a product measure ${g_{s_1} \times \ldots \times g_{s_n}}$, then Proposition \[validcovfn\] is a special case of kernel convolution \[@rasmussen2006gaussian\].
Proof of Proposition \[prop1\]
------------------------------
Without loss of generality, we can assume $\tau^2 = 1$ and fix $\beta, \Delta >0$. Using the fact that $k^*(s, s^*) = {\mathbb{E}}[\exp(-\beta \|s + u - s^*\|^2)]$, evaluating the moment generating function of a non-central $\chi_p^2$ random variable $\|s + u - s^*\|^2$ yields $$c(\sigma_u^2) \equiv {\mathbb{E}}[({\hat{x}_{\text{\tiny\textsc{KALE}}}}(s^*) - x(s^*))^2] = 1 - \left(\frac{1}{1 + 2\beta\sigma^2_u}\right)^p \exp \left( \frac{-2\beta\Delta^2}{1 + 2\beta\sigma_u^2} \right).$$ Differentiating, we get $$c'(\sigma_u^2) = \frac{2\beta [2\beta(p\sigma^2_u - \Delta^2) + p]}{(1 + 2\beta\sigma_u^2)^{p + 2}} \exp \left( \frac{-2\beta \Delta^2}{1 + 2\beta\sigma_u^2}\right).$$ If $\beta \Delta^2 \leq p/2$, then $c'(\sigma_u^2) > 0$ for all $\sigma_u^2 > 0$. Since $c(\sigma_u^2)$ is left continuous at 0, continuous on ${\mathbb{R}}_+$, and $c(0) = c_0$, this means $\beta \Delta^2 \leq p/2$ implies $c(\sigma_u^2) \geq c_0$ for all $\sigma_u^2$.
Otherwise, if $\beta \Delta^2 > p/2$, then for all $0 < \sigma^2_u < \frac{\Delta^2}{k} - \frac{1}{2\beta}$, we have $c'(\sigma^2_u) < 0$. Once again, because $c(\sigma_u^2)$ is left continuous at 0, continuous on ${\mathbb{R}}_+$, and $c(0) = c_0$, this means $c(\sigma_u^2) < c_0$ for $\sigma_u^2$ in this interval.
Proof of Proposition \[coverprop\]
----------------------------------
Let $W = x(s^*) - {\hat{x}_{\text{\tiny\textsc{KALE}}}}(s^*)$. We can explicitly write the dependence of $W$ on ${\mathbf{u}}_n$: $$W | {\mathbf{u}}_n \sim {\mathcal{N}}( 0, V({\mathbf{u}}_n))$$ where $$\begin{aligned}
V({\mathbf{u}}_n) & = \sigma^2 + \gamma'{\mathbf{C}}({\mathbf{s}}_n + {\mathbf{u}}_n, {\mathbf{s}}_n + {\mathbf{u}}_n)\gamma - 2\gamma'{\mathbf{C}}({\mathbf{s}}_n + {\mathbf{u}}_n, s^*), \\
\gamma &= {\mathbf{K}}({\mathbf{s}}_n, {\mathbf{s}}_n)^{-1}{\mathbf{K}}^*({\mathbf{s}}_n, s^*),\end{aligned}$$ and $\sigma^2 = {\mathbb{V}}[x(s^*)]$. Thus $$\begin{aligned}
{\mathbb{P}}(W < z) &= {\mathbb{E}}[{\mathbb{P}}(W < z | {\mathbf{u}}_n)] \\
&= {\mathbb{E}}\left[ \Phi \left( \frac{z}{\sqrt{V({\mathbf{u}}_n)}} \right) \right].\end{aligned}$$
Proof of Theorem \[theorem1\]
-----------------------------
First, our assumptions in the hypothesis imply that $k$ is continuous everywhere in ${\mathbb{S}}^2$ except where $s_1 = s_2$. To see this, take any distinct $s_1, s_2 \in {\mathbb{S}}$ and sequence $s_1^m, s_2^m$ converging to $(s_1, s_2)$. The sequence $c(s_1^m + u_1^m, s_2^m + u_2^m)$ is bounded and converges in distribution to $c(s_1 + u_1, s_2 + u_2)$. Thus, by the Dominated Convergence theorem, $k(s_1^m, s_2^m) \rightarrow k(s_1, s_2)$.
Now, for any $n \in \mathbb{N}$, the KILE MSE in predicting $x(s^*)$ given ${\mathbf{y}}_n$ is $$\begin{aligned}
{\mathbb{E}}[(x(s^*) - {\hat{x}_{\text{\tiny\textsc{KILE}}}}(x^*))^2] &= {\mathbb{V}}[x(s^*)] - 2{\mathbf{C}}(s^*, {\mathbf{s}}_n) {\mathbf{C}}({\mathbf{s}}_n, {\mathbf{s}}_n)^{-1} {\mathbf{K}}^*({\mathbf{s}}_n, s^*) \nonumber \\
& \hspace{-1cm} + {\mathbf{C}}(s^*, {\mathbf{s}}_n) {\mathbf{C}}({\mathbf{s}}_n, {\mathbf{s}}_n)^{-1}{\mathbf{K}}({\mathbf{s}}_n, {\mathbf{s}}_n) {\mathbf{C}}({\mathbf{s}}_n, {\mathbf{s}}_n)^{-1} {\mathbf{C}}({\mathbf{s}}_n, s^*).
\label{kile_mse}
\end{aligned}$$ The matrix ${\mathbf{C}}({\mathbf{s}}_n, {\mathbf{s}}_n)$ is symmetric and positive definite, and thus it can be written as ${\mathbf{C}}({\mathbf{s}}_n, {\mathbf{s}}_n) = {\mathbf{Q}}{\boldsymbol{\Lambda}}{\mathbf{Q}}'$, where ${\mathbf{Q}}$ is an orthogonal matrix and ${\boldsymbol{\Lambda}}$ is diagonal. Assume without loss of generality the entries are ${\boldsymbol{\Lambda}}$ are ordered $0 < \lambda_1 < \ldots < \lambda_n$. Similarly, write ${\mathbf{K}}({\mathbf{s}}_n, {\mathbf{s}}_n) = {\mathbf{R}}{\boldsymbol{\Omega}}{\mathbf{R}}'$. Further letting ${\mathbf{a}}= {\mathbf{C}}(s^*, {\mathbf{s}}_n) {\mathbf{Q}}$, ${\mathbf{b}}= {\mathbf{Q}}' {\mathbf{K}}({\mathbf{s}}_n, s^*)$, and ${\mathbf{D}}= {\mathbf{Q}}'{\mathbf{R}}{\boldsymbol{\Omega}}^{\frac{1}{2}}$, we can write as $$\begin{aligned}
{\mathbb{E}}[(x(s^*) - {\hat{x}_{\text{\tiny\textsc{KILE}}}}(x^*))^2] &= {\mathbb{V}}[x(s^*)] - 2{\mathbf{a}}' {\boldsymbol{\Lambda}}^{-1} {\mathbf{b}}+ {\mathbf{a}}' {\boldsymbol{\Lambda}}^{-1} {\mathbf{D}}{\mathbf{D}}' {\boldsymbol{\Lambda}}^{-1} {\mathbf{a}}\nonumber \\
&= {\mathbb{V}}[x(s^*)] - 2\sum_{i=1}^n \frac{a_ib_i}{\lambda_i} + \sum_{j=1}^n \left( \sum_{i=1}^n \frac{a_i D_{ij}}{\lambda_i} \right)^2.
\label{kile_mse_2}\end{aligned}$$ Let $\xi = \lambda_1^{-1}$; then Equation $\eqref{kile_mse_2}$ can be expressed as $${\mathbb{E}}[(x(s^*) - {\hat{x}_{\text{\tiny\textsc{KILE}}}}(x^*))^2] = \xi^2 \left(\sum_{j=1}^n a_1^2 D_{1 j}^2\right) + h(\xi),
\label{quadratic}$$ where $h$ is linear in $\xi$. Without loss of generality, assume $s_1 \rightarrow s_2$. Thus ${\mathbf{C}}({\mathbf{s}}_n, {\mathbf{s}}_n)$ becomes rank $n-1$, with $\lambda_{1} \rightarrow 0$ and for all $i > 1$, $\lambda_{i} \rightarrow \lambda^*_{i} > 0$. Thus $\xi \rightarrow \infty$. However, ${\mathbf{K}}({\mathbf{s}}_n, {\mathbf{s}}_n)$ does not become singular, since ${\mathbb{P}}(u_1 \neq u_2) < 1$ implies $$\lim_{s_1 \to s_2} k(s_1, s_2) \neq {\mathbb{V}}[y(s_2)].$$ Since $k$ is continuous and ${\mathbf{K}}({\mathbf{s}}_n, {\mathbf{s}}_n)$ nonsingular in the limit, all of the terms besides $\xi$ in converge as $s_1 \rightarrow s_2$; that is ${\mathbf{a}}\rightarrow {\mathbf{a}}^*$, ${\mathbf{b}}\rightarrow {\mathbf{b}}^*$, and ${\mathbf{D}}\rightarrow {\mathbf{D}}^*$ as $s_1 \rightarrow s_2$. Moreover, we cannot have $D^*_{1 j} = 0$ for all $j$, as this contradicts ${\mathbf{K}}({\mathbf{s}}_n, {\mathbf{s}}_n)$ remaining full-rank. Lastly, since ${\mathbf{C}}(s^*, {\mathbf{s}}_n) \neq \mathbf{0}$ and ${\mathbf{Q}}$ is orthogonal, $a_i \neq 0$ and $a^*_i \neq 0$ for all $i=1, \ldots, n$.
Thus the quadratic coefficient in , $\sum_{j=1}^n a_1^2 D_{1 j}^2$ is strictly positive, and $h(\xi) = \mathcal{O}(\xi)$. Because $\xi \rightarrow \infty$, we get $$\lim_{s_1 \to s_2} {\mathbb{E}}[(x(s^*) - {\hat{x}_{\text{\tiny\textsc{KILE}}}}(x^*))^2] = \infty.$$
For pathological choices of $g_{{\mathbf{s}}_n}$ where $k$ is not continuous everywhere and limits for ${\mathbf{b}}$ and ${\mathbf{D}}$ may not exist, all components of these terms can be still be bounded, which is sufficient for Theorem \[theorem1\] to hold.
Proof of Proposition \[morris\_prop\]
-------------------------------------
Bayes rule predictors by definition satisfy $\mathrm{R}_{\pi}(\pi) \leq \mathrm{R}_{\pi}(\tilde{\pi})$, which confirms the two inequalities in the statement of Proposition \[morris\_prop\]. The equality $\mathrm{R}_{\pi}(\pi_0) = \mathrm{R}_{\pi_0}(\pi_0)$ holds since the risk of the Bayes estimator under $\pi_0$ is a quadratic form, and therefore constant for all $\pi \in \Pi_{\mathbf{0}, {\mathbf{C}}}$: $$\begin{aligned}
\mathrm{R}_{\pi_0}(\pi_0) &= {\mathbb{E}}_{\pi}[({\mathbb{E}}_{\pi_0}[x(s^*)|{\mathbf{x}}_n ] - x(s^*))^2] \\
&= {\mathbb{E}}_{\pi}[({\mathbf{C}}(s^*, {\mathbf{s}}_n){\mathbf{C}}({\mathbf{s}}_n, {\mathbf{s}}_n)^{-1}{\mathbf{x}}_n - x(s^*))^2] \\
&= c(s^*, s^*) - {\mathbf{C}}(s^*, {\mathbf{s}}_n){\mathbf{C}}({\mathbf{s}}_n, {\mathbf{s}}_n)^{-1}{\mathbf{C}}({\mathbf{s}}_n, s^*) \\
&= \mathrm{R}_{\pi}(\pi_0).
\end{aligned}$$
Acknowledgements {#acknowledgements .unnumbered}
================
[We thank Luke Bornn and Peter Huybers for helpful comments and encouragement. NSP was partially supported by an ONR grant. DC was partially supported by a research grant from the Harvard University Center for the Environment.]{}
[^1]: http://www.cru.uea.ac.uk/cru/data/temperature/
[^2]: Station locations are vieweable at https://www.ncdc.noaa.gov/oa/climate/ghcn-daily/
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We investigate a measurement-feedback process of repeated operations with time delay. During a finite-time interval, measurement on the system is performed and the feedback protocol derived from the measurement outcome is applied with time delay. This protocol is maintained into the next interval until a new protocol from the next measurement is applied. Unlike a feedback process without delay, both memories associated with previous and present measurement outcomes are involved in the system dynamics, which naturally brings forth a joint system described by a system state and two memory states. The thermodynamic second law provides a lower bound for heat flow into a thermal reservoir by the (3-state) Shannon entropy change of the joint system. However, as the feedback protocol depends on memory states sequentially, we can deduce a tighter bound for heat flow by integrating out irrelevant memory states during dynamics. As a simple example, we consider the so-called cold damping feedback process where the velocity of a particle is measured and a dissipative feedback protocol is applied to decelerate the particle. We confirm that the heat flow is well above the tightest bound. We also examine the long-time limit of this feedback process, which turns out to exhibit an interesting instability transition as well as heating by controlling parameters such as measurement errors, time interval, protocol strength, and time delay length. We discuss the underlying mechanism for instability and heating, which might be unavoidable in reality.'
author:
- 'Chulan Kwon$^{1,2}$'
- 'Jaegon Um$^{2}$'
- 'Hyunggyu Park$^{2,3}$'
title: 'Information thermodynamics for a multi-feedback process with time delay'
---
The recent information thermodynamics has been proven to resolve the paradox of Maxwell’s demon [@maxwell] which was a long-lived problem in spite of enormous research works [@maxwell; @szilard; @brillouin; @landauer; @leff_rex]. Replacing Maxwell’s demon by a physical memory device, that was refined by Landauer [@landauer], one is able to describe measurement inside a memory device and feedback after measurement acting on the system (engine) as thermodynamic processes. In the measurement process, information acquisition is realized as mutual information gain in the entropy of the joint system (system and memory device). In the subsequent feedback process, mutual information is expended through relaxation out of initial state producing work outside. The work production is balanced energetically by heat dissipation into the reservoir, which may be negative like in the Szilard engine [@szilard], resulting in entropy loss in the reservoir. It was shown that such entropy loss in the reservoir, if any, be compensated sufficiently by the entropy gain of the joint system through mutual information decrease so as to satisfy the second law of thermodynamics. Hence the paradox of Maxwell’s demon is resolved. It is the main feature of the information thermodynamics developed by Sagawa and Ueda [@sagawa; @sagawa_new; @sagawa_network; @sagawa_bipartite]. The increase of the total entropy of the joint system and reservoir was proven with the aid of the fluctuation theorem (FT), which was discovered about two decades ago and has been regarded as a principle of nonequilibrium statistical mechanics [@evans; @jarzynski; @crooks; @kurchan; @lebowitz]. The role of mutual information in feedback processes has also been confirmed in experiments [@toyabe; @koski].
Memory is usually assumed to reach local equilibrium so fast that system state does not change during measurement. In a feedback process, system state changes in time subject to a fixed protocol given from memory state picked out of its local equilibrium. In this sense, measurement and feedback can reasonably be regarded as processes with separated time periods [@sagawa_bipartite; @UHKP] and the fluctuation theorem for the total entropy production was shown to hold separately for the two bipartite periods [@shiraishi].
In real situations, however, measurement process takes a finite time and the feedback protocol ought to be applied afterwards. This naturally generates time gap between the start of measurement and feedback. In the present work, we consider a realistic feedback process composed of multiple steps repeated in a finite-time interval, in each of which a feedback protocol is applied with time delay. As an example, we consider a simple cold-damping problem where the velocity of a particle is measured and a dissipative protocol is applied. In repeated feedback steps, the temperature of the system is expected to be cooled down below the reservoir temperature.
![(Color online) A schematic picture for repeated measurement-feedback processes. $\mathbf{m}_i$ is a measurement outcome for an initial state $\mathbf{s}_i$ of step $i$, which is applied as a protocol with time delay $\delta$. This protocol is maintained into the next step until the next protocol is applied. $\mathbf{m}_0=\mathbf{0}$ when there is no previous measurement.[]{data-label="fig1"}](fig1){width="\columnwidth"}
Consider that both system state $\mathbf{s}(t)$ and memory state $\mathbf{m}(t)$ in $d$ dimensions coevolve in time $t$ by their own dynamics. At measurement time $t=t_i$, memory starts to measure or copy $\mathbf{s}(t_i)=\mathbf{s}_i $ that acts as a protocol to drive memory into a copied state. One may think of the Langevin dynamics for such a process: $\dot{\mathbf{m}}=-\tau_{\textrm{m}}^{-1}(\mathbf{m}-\mathbf{s}_i)+\boldsymbol{\xi}(t)$ where $\langle {\xi}_a(t){\xi}_b(t')\rangle=2\tau_{\textrm{m}}^{-1} T_{\textrm{m}}\delta_{ab}\delta(t-t')$ with component indices $a,b=1,\cdots, d$ and temperature $T_{\textrm{m}}$ of the reservoir surrounding memory. The Boltzmann constant is set to unity here and also in the following. Waiting for a long enough time $\delta$ compared to relaxation time $\tau_{\textrm{m}}$, the memory reaches a local equilibrium with the conditional probability density function (PDF) given as $p_{\textrm{m}}(\mathbf{m}_i|\mathbf{s}_i)=(2\pi \sigma)^{-d/2} e^{-(\mathbf{m}_i-\mathbf{s}_i)^2/(2\sigma)}$ for $\sigma=\tau_{\textrm{m}} T_{\textrm{m}}$, which can be interpreted as measurement probability. For this period, the system undergoes a transition to state $\mathbf{s}_i'$ at $t=t_i+\delta$ under a previous protocol $\mathbf{m}_{i-1}$. A new protocol $\mathbf{m}_i$ chosen from the distribution $p_{\textrm{m}}(\mathbf{m}_i|\mathbf{s}_i)$ is applied in turn to the dynamics of the system for $t_i+\delta <t< t_{i+1}=t_i+\Delta$. Since the measurement process at step $i$ depends only on $\mathbf{s}_i$, as seen in the above Langevin equation, intermediate memory states between $\mathbf{m}_{i-1}$ and $\mathbf{m}_i$ can be averaged out for $t_i-(\Delta-\delta) \le t <t_i+\delta$ without influencing the dynamics of $\mathbf{s}(t)$. In Fig. \[fig1\], the corresponding path of $\mathbf{s}(t)$ is shown with $\mathbf{m}_{i-1}$ and $\mathbf{m}_i$ coexisting in step $i$.
We introduce an adjoint dynamics with time-reverse protocols in which the probability of the system tracing the time-reverse path conjugate to a given (forward) path will be considered. The time-reversed path is defined as $\bar{\mathbf{s}}(t)=\varepsilon\mathbf{s}(t_N+t_1-t)$ conjugate to a (forward) path $\mathbf{s}(t)$, where $\varepsilon$ is the parity operator giving $+1$ ($-1$) if it is applied to a even (odd) parity state in time reversal such as position (momentum). The time-reverse protocols are defined as $\bar{\mathbf{m}}_i=\varepsilon\mathbf{m}_{N-i+1}$. For each of time-reverse protocols, not only the order in time is reversed, but also the parity is multiplied, copying a time-reverse state.
Let $\Pi_{\mathbf{s}_i,\mathbf{s}_i'}^{\mathbf{m}_{i-1}}[\mathbf{s}(t)]$ ($\Pi_{\mathbf{s}_i',\mathbf{s}_{i+1}}^{\mathbf{m}_{i}}[\mathbf{s}(t)]$) be the conditional probability for a partial path from $\mathbf{s}_i$ ($\mathbf{s}_i'$) to $\mathbf{s}_i'$ ($\mathbf{s}_{i+1}$) under a protocol $\mathbf{m}_{i-1}$ ($\mathbf{m}_{i}$) for $t_i\le t < t_{i}+\delta$ ($t_i+\delta \le t < t_{i+1}$) in step $i$. Similarly, we define the conditional path probabilities for time-reverse paths and protocols as $\Pi_{\varepsilon\mathbf{s}_{i+1},\varepsilon\mathbf{s}_i'}^{\varepsilon\mathbf{m}_{i}}[\bar{\mathbf{s}}(t)]$ and $\Pi_{\varepsilon\mathbf{s}_i',\varepsilon\mathbf{s}_{i}}^{\varepsilon\mathbf{m}_{i-1}}[\bar{\mathbf{s}}(t)]$. For usual thermodynamic process without feedback, the change in the total entropy of system and reservoir is known as the log-ratio of the path probabilities of the forward and time-reverse path. Extending to the joint system of system and memory, the corresponding [*total entropy*]{} change may be written as $$\begin{aligned}
\sum_{i=1}^N\Delta S_{\textrm{tot},i}&=&\prod_{i=1}^N\ln \left[\frac{\rho_i(\mathbf{s}_i)\rho_i(\mathbf{m}_{i-1}|\mathbf{s}_i)p_{\textrm{m}}(\mathbf{m}_{i}|\mathbf{s}_i)}
{\rho_{i+1}(\mathbf{s}_{i+1})\bar{\rho}(\varepsilon\mathbf{m}_{i},\varepsilon\mathbf{m}_{i-1}|\bar{\mathbf{s}}(t))}\right.
\nonumber\\
&&\left.~~~~~~~~\times \frac{\Pi_{\mathbf{s}_i,\mathbf{s}_i'}^{\mathbf{m}_{i-1}}[\mathbf{s}(t)]\Pi_{\mathbf{s}_i',\mathbf{s}_{i+1}}^{\mathbf{m}_{i}}[\mathbf{s}(t)]}
{\Pi_{\varepsilon\mathbf{s}_{i+1},\varepsilon\mathbf{s}_i'}^{\varepsilon\mathbf{m}_{i}}[\bar{\mathbf{s}}(t)]\Pi_{\varepsilon\mathbf{s}_i',\varepsilon\mathbf{s}_{i}}^{\varepsilon\mathbf{m}_{i-1}}[\bar{\mathbf{s}}(t)]}
\right]
\nonumber\\
&=&\sum_{i=1}^{N}\left[ \Delta S_{\textrm{sm},i}+\Delta S_{\textrm{env},i}\right]\end{aligned}$$ where $\Delta S_{\textrm{tot},i}$ denotes the contribution from step $i$ and $\rho_i$ is the PDF at $t=t_i$. A conditional probability $\bar{\rho}$ for time-reverse protocols in the adjoint dynamics can be chosen in various ways, which will be discussed later.
The environmental entropy production for step $i$ is defined as $$\Delta S_{\textrm{env},i}=\ln \left[\frac{\Pi_{\mathbf{s}_i,\mathbf{s}_i'}^{\mathbf{m}_{i-1}}[\mathbf{s}(t)]\Pi_{\mathbf{s}_i',\mathbf{s}_{i+1}}^{\mathbf{m}_{i}}[\mathbf{s}(t)]}
{\Pi_{\varepsilon\mathbf{s}_{i+1},\varepsilon\mathbf{s}_i'}^{\varepsilon\mathbf{m}_{i}}[\bar{\mathbf{s}}(t)]\Pi_{\varepsilon\mathbf{s}_i',\varepsilon\mathbf{s}_{i}}^{\varepsilon\mathbf{m}_{i-1}}[\bar{\mathbf{s}}(t)]}
\right]~.
\label{env_EP}$$ In the absence of odd-parity states, $\Delta S_{\textrm{env},i}$ is equal to $Q_i/T$ for heat production $Q_i$ into the reservoir at temperature $T$. However, it may contain an unconventional contribution due to an odd-parity force induced by an odd-parity protocol [@KYKP]. We will encounter this situation for a cold-damping problem where the velocity of a particle is measured.
$\Delta S_{\textrm{sm},i}$ is the entropy change of the joint system for step $i$, which reads $\Delta S_{\textrm{sys},i}-\Delta I_i$. Here $\Delta I_i$ is the mutual information change between system and memory. Note that the memory state does not change during each step. We find the first term to be the Shannon entropy change of the system, given as $$\Delta S_{\textrm{sys},i}=-[\ln \rho_{i+1}(\mathbf{s}_{i+1})-\ln \rho_i(\mathbf{s}_i)],
\label{shannon}$$ resulting from choosing the initial PDF of the time-reverse dynamics to be the final PDF $\rho_{i+1}(\mathbf{s}_{i+1})$ of the given dynamics. $\Delta I$ depends on how $\bar{\rho}{(\varepsilon\mathbf{m}_i,\varepsilon\mathbf{m}_{i-1}|\bar{\mathbf{s}}(t))}$ is chosen in the time-reverse dynamics.
We consider two choices in setting the distribution of protocols in the time-reverse dynamics, each of which yields mutual information as a part of $\Delta S_{\textrm{sm},i}$. The first one is given by $$\bar{\rho}(\varepsilon\mathbf{m}_i,\varepsilon\mathbf{m}_{i-1}|\bar{\mathbf{s}}(t))=\rho_{i+1}(\mathbf{m}_{i-1},\mathbf{m}_i|\mathbf{s}_{i+1})~,$$ which is the conditional PDF of the joint system at time $t_{i+1}$ for the given dynamics found as $\rho_{i+1}(\mathbf{s}_{i+1},\mathbf{m}_{i-1},\mathbf{m}_i)\rho_{i+1}(\mathbf{s}_{i+1})^{-1}$. Then, we have $$\Delta I_i^{(1)}=\ln\frac{\rho_{i+1}(\mathbf{m}_{i-1},\mathbf{m}_i|\mathbf{s}_{i+1})}{\rho_i(\mathbf{m}_{i-1}|\mathbf{s}_i)p_{\textrm{m}}(\mathbf{m}_i|\mathbf{s}_{i})}~,
\label{MI_1}$$ which is the change in mutual information between system and two-state memory. The second choice is $$\bar{\rho}(\varepsilon\mathbf{m}_i,\varepsilon\mathbf{m}_{i-1}|\bar{\mathbf{s}}(t))
=\rho_{i+1}(\mathbf{m}_i|\mathbf{s}_{i+1})
\rho_{i'}(\mathbf{m}_{i-1}|\mathbf{s}_{i}',\mathbf{m}_{i})$$ where the first (second) factor determines the distribution of $\epsilon\mathbf{m}_i$ ($\epsilon\mathbf{m}_{i-1}$) for the period $\Delta -\delta$ ($\delta$) of step $N-i$ in the time-reverse dynamics. Then, we have $$\Delta I^{(2)}_i= \ln \frac{\rho_{i'}(\mathbf{m}_{i-1},\mathbf{m}_i|\mathbf{s}_i')}{\rho_i(\mathbf{m}_{i-1}|\mathbf{s}_i)p_{\textrm{m}}(\mathbf{m}_i|\mathbf{s}_{i})}+ \ln \frac{\rho_{i+1}(\mathbf{m}_i|\mathbf{s}_{i+1})}{\rho_{i'}(\mathbf{m}_i|\mathbf{s}_{i}')}~,
\label{MI_2}$$ where $\rho_{i'}$ is the PDF at $t=t_i+\delta$ and $\rho_{i'}(\mathbf{m}_{i-1},\mathbf{m}_i|\mathbf{s}_i')/\rho_{i'}(\mathbf{m}_i|\mathbf{s}_{i}')=\rho_{i'}(\mathbf{m}_{i-1}|\mathbf{s}_{i}',\mathbf{m}_{i})$ is used. The first term is the change in mutual information between system and two-state memory coexisting in the delay period, and the second is that between system and new memory in the remaining period. Writing $\Delta S_{\textrm{tot},i}^{(1,2)}=\Delta S_{\textrm{sm},i}^{(1,2)}+\Delta S_{\textrm{env},i}$ with $\Delta S_{\textrm{sm},i}^{(1,2)}=\Delta S_{\textrm{sys},i}-\Delta I_i^{(1,2)}$. We can show both satisfy the FT such that $\langle e^{-\sum_i\Delta S_{\textrm{tot},i}^{(1,2)}}\rangle=1$ and also $\langle e^{-\Delta S_{\textrm{tot},i}^{(1,2)}}\rangle=1$, leading to the inequality $\langle\Delta S_{\textrm{tot},i}^{(1,2)}\rangle \ge 0$, the generalized thermodynamic second law.
Another choice is given from $$\begin{aligned}
\Delta S_{\textrm{tot},i}^{\delta(3)}&=&\ln\left[\frac{\rho_i(\mathbf{s}_i)\rho_i(\mathbf{m}_{i-1}|\mathbf{s}_i)
\Pi_{\mathbf{s}_i,\mathbf{s}_i'}^{\mathbf{m}_{i-1}}[\mathbf{s}(t)]}
{\rho_{i'}(\mathbf{s}_i')\rho_{i'}(\mathbf{m}_{i-1}|\mathbf{s}_i')\Pi_{\varepsilon\mathbf{s}_i',\varepsilon\mathbf{s}_i}^{\varepsilon\mathbf{m}_{i-1}}[\bar{\mathbf{s}}(t)]}
\right]~,\\
S_{\textrm{tot},i}^{\Delta-\delta(3)}&=&\ln\left[\frac{\rho_{i'}(\mathbf{s}_i')\rho_{i'}(\mathbf{m}_{i}|\mathbf{s}_i')
\Pi_{\mathbf{s}_i',\mathbf{s}_{i+1}}^{\mathbf{m}_{i}}[\mathbf{s}(t)]}{\rho_{i+1}(\mathbf{s}_{i+1})
\rho_{i+1}(\mathbf{m}_i|\mathbf{s}_{i+1})\Pi_{\varepsilon\mathbf{s}_{i+1},\varepsilon\mathbf{s}_i'}^{\varepsilon\mathbf{m}_{i}}
[\bar{\mathbf{s}}(t)]}\right]~,\end{aligned}$$ which are defined for $t_i\le t\le t_i+\delta$ and $t_i+\delta\le t\le t_{i+1}$, respectively. The FT can be shown to hold separately for the two as $\langle e^{\Delta S_{\textrm{tot},i}^{\delta(3)}}\rangle=1$ and $\langle e^{\Delta S_{\textrm{tot},i}^{\Delta-\delta(3)}}\rangle=1$, but not for the sum of them, $\langle e^{-\Delta S_{\textrm{tot},i}^{(3)}}\rangle\neq 1$, for $\Delta S_{\textrm{tot},i}^{(3)}=\Delta S_{\textrm{sm},i}^{(3)}+\Delta S_{\textrm{env},i}$. However, the inequality holds for the sum, $\langle\Delta S_{\textrm{tot},i}^{(3)}\rangle\ge 0$. We can similarly write $\Delta S_{\textrm{sm},i}^{(3)}=\Delta S_{\textrm{sys},i}-\Delta I_i^{(3)}$ where $$\Delta I_i^{(3)}= \ln \frac{\rho_{i'}(\mathbf{m}_{i-1}|\mathbf{s}_i')}{\rho_i(\mathbf{m}_{i-1}|\mathbf{s}_i)}+ \ln \frac{\rho_{i+1}(\mathbf{m}_i|\mathbf{s}_{i+1})}{\rho_{i'}(\mathbf{m}_i|\mathbf{s}_{i}')}.
\label{MI_3}$$ As presented in Fig. \[fig2\], $-\Delta I_i^{(3)}$ is found to have the lowest bound to the change in total entropy among the three representations. One can say that the total entropy change is overestimated as considered is mutual information between system and protocol having no influence on the dynamics. Overestimated are mutual information due to new protocol in time delay and that due to past protocol in new feedback period, labeled by 0 and 5 in the figure, respectively.
![(Color online) Venn diagrams for Shannon entropies (discs) and mutual informations (intersections). $I_i^{(1)}$ is presented by blue areas in (a) at $t=t_i$ and in (b) at $t=t_{i+1}$. The figure (b) presents that initial state $\mathbf{s}_i$ evolves to $\mathbf{s}_i'$ and subsequently to $\mathbf{s}_{i+1}$. $-\Delta I_i^{(1)}$ is represented by the whole red area, $-\Delta I_i^{(2)}$ by the areas labeled by 1, 2, 3, 4, 5, and $-\Delta I_i^{(3)}$ by those labeled by 1, 2, 3, 4. []{data-label="fig2"}](fig2){width="\columnwidth"}
We apply our theory to a cold-damping problem where a feedback force is applied in the opposite direction to the measured velocity [@khkim; @jourdan; @ito]. From now on, we investigate the problem within a single step, say for $t_1\le t \le t_2$. We consider the one-dimensional motion of a particle described by the Langevin equation for the velocity $v$, $$\dot{v}=-\gamma v-\tilde{\gamma}y_i +\xi(t)~,
\label{langevin}$$ where mass is set to unity. Then, $\mathbf{s}=v$ and $\mathbf{m}_i=y_i$ where $i=0$ ($i=1$) denotes past (new) protocol. $y_0$ is applied for $t_1\le t \le t_1+\delta$ and $y_1$ for the remaining period. This feedback process can be realized in experiment for a colloidal particle with charge $q$ where $\tilde{\gamma}$ is a control parameter for an electric field $E=\tilde{\gamma}y/q$. $\xi$ is a usual stochastic force with mean zero and variance $\langle \xi(t)\xi(t')\rangle=2T\gamma\delta(t-t')$. $\tilde{\gamma}>0$ is used for the purpose of cold damping.
One can find various PDF’s and moments recursively given the initial PDF $\rho(v_1, y_0)$ with initial moments, $$T_1 =\langle v^2_1 \rangle ~,~ P_1 =\langle y_0^2 \rangle~,~
R_1 = \langle v_1 y_0 \rangle~.
\label{initial_mom}$$ It is convenient to consider composite states at $t=t_1$ and $t=t_2$, given as $\mathbf{c}_1=(v_1,y_0,y_1)$ and $\mathbf{c}_2=(v_2,y_0,y_1)$. Then, $\rho(\mathbf{c}_{1})$ is equal to the product of $\rho(v_1, y_0)$ and $p_{\textrm{m}}(y_1|v_1)= (2\pi\sigma)^{-1/2}e^{-(y_1-v_1)^2/(2\sigma)}$. The Onsager-Machlup theory [@onsager] gives the conditional probability for path $v(t)$ from $v(\tau)=v$ to $v(\tau')=v'$ as $\Pi_{v,v'}^{y_i}[v(t)]\propto
\exp[-(4\gamma T)^{-1}\int_{\tau}^{\tau'}dt\left(\dot{u}+\gamma u\right)^2 ]$, where $u(t)=v(t)+(\tilde{\gamma}/\gamma)y_i$. Then, the path integral of $\Pi_{v_1,v_1'}^{y_0}[v(t)]\Pi_{v_1',v_2}^{y_1}[v(t)]$ over all paths gives rise to the transition probability of $v(t_2)=v_2$ given a composite state $\mathbf{c}_1$. We find $$\label{eq:propagator}
\rho(v_2|\mathbf{c}_{1})=\frac{1}{\sqrt{2\pi w_\Delta}}e^{-\left(
v_2- e^{-\gamma \Delta} v_{1} + (\tilde{\gamma}/\gamma) f \right)^2/(2w_\Delta) },$$ where $w_{\Delta}=T(1-e^{-2\gamma\Delta})$ and $$f = \left( e^{-\gamma (\Delta-\delta)} - e^{-\gamma \Delta} \right)y_0
+ \left( 1 - e^{-\gamma (\Delta-\delta) } \right) y_1.$$ Using this, the PDF of $\mathbf{c}_2$ is given as $$\rho(\mathbf{c}_2) = \int dv_1 \, \rho(\mathbf{c}_{1}) \rho(v_2|\mathbf{c}_1)
= \sqrt{ \frac{{\rm det} {\mathsf D}_2}{(2\pi)^3}}e^{-\mathbf{c}_2
\mathsf{D}_2 \mathbf{c}_2^{\textrm{t}}/2},
\label{rho_c2}$$ where the superscript t denotes the transpose. Using the property of multi-variate Gaussian integral, the inversion of the matrix $\mathsf{D}_2$ yields six moments such that $$\label{eq:matrix}
\mathsf{D}_2^{-1}=\left(
\begin{array}{ccc}
\langle v_2^2 \rangle & \langle v_2 y_0 \rangle
& \langle v_2 y_1 \rangle \\
\langle v_2 y_0 \rangle& \langle y_0^2 \rangle & \langle y_0y_1\rangle\\
\langle v_2y_1 \rangle & \langle y_0y_1 \rangle& \langle y_1^2 \rangle
\end{array}\right)~,$$ which can be found in terms of $T_1$, $P_1$, and $R_1$ given in Eq. (\[initial\_mom\]).
![(Color online) The diagram is drawn for $\delta / \Delta =0.25$, $\sigma=0.1$. C denotes the region for $T^{\textrm{av}}_{\infty}<T$, W for $T<T^{\textrm{av}}_{\infty}<\infty$, and I for $T^{\textrm{av}}_{\infty}=\infty$. The three points are picked from the three regions, for which $\langle v(t)^2\rangle$ versus $\gamma t$ are shown.[]{data-label="fig3"}](fig3){width="\columnwidth"}
In particular, $T_2=\langle v_2^2\rangle$. $P_2=\langle y_1^2\rangle$, and $R_2=\langle v_2 y_1\rangle$ are found to satisfy the linear recursion relation: $$\begin{aligned}
\label{recursion}
T_2 &=& w_\Delta + \sigma h^2 +K^2 T_1 +L^2 P_1 -2K L R_1, \nonumber\\
P_2 &=& \sigma+T_1, \\
R_2 &=& -\sigma h +K T_1 -L R_1, \nonumber\end{aligned}$$ where $K = e^{-\gamma \Delta} -H$ with $H=(\tilde{\gamma}/\gamma) \left(1-e^{-\gamma(\Delta-\delta)} \right) $ and $L= (\tilde{\gamma}/\gamma) e^{-\gamma \Delta }\left( e^{\gamma \delta} -1\right) $. The recursion relation can be rewritten as $\mathbf{Z}_{2}=\mathsf{G}\cdot\mathbf{Z}_1+\mathbf{A}$ for $\mathbf{Z}_i=(T_i,P_i,R_i)^{\textrm{t}}$ where the matrix $\mathsf{G}$ and the vector $\mathbf{A}$ are given from Eq. (\[recursion\]). $T_i=\langle v_i^2\rangle$ is defined as the effective temperature at $t=t_i$ and is updated through feedback steps as $T_1\to T_2\to T_3\to\cdots$. The recursion relation will leads to a fixed value $T_{\infty}$ only if $|\lambda_a|<1$ for eigenvalues $\lambda_a$ of $\mathsf{G}$ for $ a=1,2, 3$. The average effective temperature at step $i$ can be found as $T^{\textrm{av}}_i=\Delta ^{-1}\int_{t_i}^{t_{i+1}}dt \langle v(t)^2\rangle$. Cold damping will be successful if $T^{\textrm{av}}_{\infty}<T$. In Fig. [\[fig3\]]{}, C (cold) stands for the region for $T^{\textrm{av}}_{\infty}<T$, W (warm) for $T^{\textrm{av}}_{\infty}>T$, and I (instability) for the instability region with $|\lambda_a|\ge1$.
We can compute the parts of the total entropy change. The Shannon entropy for $\rho(\mathbf{c}_{2})$ in Eq. (\[rho\_c2\]) can be written as $-\langle \ln \rho(\mathbf{c}_2) \rangle = (1/2) \left( -\ln {\rm det}\mathsf{D}_2+ 3 \ln 2\pi +3 \right)$, and similarly for $\rho(\mathbf{c}_{1})$. Then, we obtain $\langle \Delta S_{\textrm{sm}}^{(1)}\rangle =\langle\ln [\rho(\mathbf{c}_1)/\rho(\mathbf{c}_2)]\rangle$. By integrating $\rho(v(t),y_0,y_1)$ over $y_0$ or $y_1$, one can find $\rho(v(t),y_i)$. Then, we find $$\label{eq:vl}
\left< \ln \frac{\rho(v_1, y_0)}{\rho(v'_1, y_0) }\right> = \frac{1}{2}\ln \left[ e^{-2\gamma \delta }+ \frac{w_\delta P_1}{T_1 P_1 -R_1^2}\right],$$ and $$\begin{aligned}
\label{eq:vn}
\left< \ln \frac{\rho(v_1',y_1)}{\rho(v_2, y_1)}\right> &=& \frac{1}{2}\ln \left[ e^{-2\gamma (\Delta-\delta) } \right. \\
&&\left. +\frac{w_{\Delta-\delta}(T_1 + \sigma) }{w_\delta T_1
+ \sigma \langle v_{1}'^2 \rangle+ H_{\delta}^2(T_1 P_1 -R^2_1)}\right], \nonumber\end{aligned}$$ where $H_{\delta} = (\tilde{\gamma}/\gamma) (1-e^{-\gamma \delta} )$. Adding Eqs. (\[eq:vl\]) and (\[eq:vn\]) leads to $\langle \Delta S_{\textrm{sm}}^{(3)}\rangle $. $\langle \Delta S_{\textrm{sm}}^{(2)}\rangle$ in Eq.(\[MI\_2\]) can be found by adding the one in Eq. (\[eq:vn\]) and $\langle\ln \rho(\mathbf{c}_1)- \ln \rho(\mathbf{c}'_1) \rangle$. The three representations of the Shannon entropy change for the joint system are shown in Fig. \[fig4\].
The average environmental entropy production in Eq. (\[env\_EP\]) is given as $$\begin{aligned}
\label{eq:se_cold}
\lefteqn{\langle\Delta S_{\textrm{env}}\rangle=
\left\langle \!\! \ln \! \left[\frac{\Pi_{v_1,v_1'}^{y_0}[v(t)]\Pi_{v_1',v_2}^{y_1}[v(t)]}
{\Pi_{-v_2,-v_1'}^{-y_1}[-v(t)]\Pi_{-v_1',-v_1}^{-y_0}[-v(t)]} \!
\right] \! \right\rangle}\\
&=&
\frac{T_{1} -T_{2} }{2T}
-\frac{\tilde{\gamma}}{\gamma T} \bigl [ \left < v_2 y_1 \right> -\left< v'_1 y_1 \right> \bigr]
-\frac{\tilde{\gamma}}{\gamma T} \bigl[\left< v'_1 y_0 \right> -\left< v_1 y_0 \right> \bigr ]. \nonumber\end{aligned}$$ $\langle v'_1 y_1\rangle$, and $\langle v'_1 y_0 \rangle$ can be obtained from $\langle v_2 y_1\rangle$, and $\langle v_2 y_0 \rangle$ in Eq. (\[eq:matrix\]) by putting $\Delta=\delta$.
The average heat production is found from $\int_{t_1}^{t_2}dt\langle [\gamma v(t)-\xi(t)]\circ v(t)\rangle$ with $\circ$ denoting the Stratonovich calculus [@Strat]. We find $\langle Q\rangle=\gamma\Delta (T^{\textrm{av}}-T)$. When the average effective temperature is lower than the reservoir temperature, meeting the need of cold damping, the average heat becomes negative, which is the situation in which the paradox of Maxwell’s demon is raised.
$\langle\Delta S_{\textrm{uc}}\rangle=\langle\Delta S_{\textrm{env}}\rangle-\langle Q/T\rangle$ is an unconventional entropy production which is known to appear in the presence of an odd-parity force; $-\tilde{\gamma}y_i$ in our case [@KYKP]. Without feedback control, $\langle \Delta S_{\textrm{tot}}\rangle$ maintains positivity even for a negative $\langle Q\rangle/T$ thanks to $\langle \Delta S_{\textrm{uc}}\rangle$. For feedback process, $-\langle \Delta I\rangle$ plays an additive role in compensating entropy loss in reservoir together with $\langle \Delta S_{\textrm{uc}}\rangle$.
![(Color online) The components of $\Delta S_{\textrm{tot}}$ as functions of $\tilde{\gamma}/\gamma$ at the fixed point in the recursion procedure where $T_i=T_{i+1}$. Here, the plot is drawn for $\sigma =0.1$, $\gamma \Delta = 0.2$, $\gamma \delta =0.05$, and $T=1$. For simplicity, we use $S_i=\langle\Delta S^{(\alpha)}_{\textrm{sm}}\rangle$ for $\alpha=1,2,3$, $S_{\textrm{odd}}=\langle\Delta S_{\textrm{uc}}\rangle$, and $S_{\textrm{h}}=\langle Q/T\rangle$. []{data-label="fig4"}](fig4){width="\columnwidth"}
In Fig. \[fig4\], we display the components comprising the total entropy change at the fixed point of the recursive feedback process for $T_1=T_2$ in the above equations. In the figure, $\langle \Delta S^{(\alpha)}_{\textrm{sm}}+\Delta S_{\textrm{uc}}\rangle$ is shown to be greater than $-\langle Q/T\rangle$ for all $\alpha$, which confirms the generalized second law of thermodynamics. As expected from Fig. \[fig2\], $\langle \Delta S^{(3)}_{\textrm{tot}}\rangle$ is shown to yield the tightest bound.
We examine the generalized thermodynamic second law in the presence of coexisting past and present memories. We show the total entropy change to have the tightest bound as only mutual informations influencing the dynamics are considered, which is confirmed in the cold-damping problem. For the cold-damping using a multi-step feedback, the effective temperature can be reduced below reservoir temperature for a certain range of parameters, while it can reach a higher value or even diverge unlimitedly due to overshooting caused by large $\tilde{\gamma}$ and $\Delta$, as shown in Fig. \[fig3\]. We derive the stability condition for the convergence of feedback. An intriguing role of $\delta$ to enhance the stability for large $\Delta$ will be further investigated in a future study [@cold_damping]. We expect overshooting and instability to take place in general feedback processes for finite $\delta$ and $\Delta$, which are unavoidable in reality.
This research was supported by the NRF Grant No. 2013R1A1A2011079 (C.K.) and 2013R1A1A2A10009722 (H.P.).
[99]{} J. C. Maxwell, [*Theory of Heat*]{} (London:Appleton) (1871). L. Szilard, Z. Phys. [**53**]{}, 840 (1929); Behavioral Science [**9**]{}, 301 (1964), translated in English. L. Brillouin, J. Appl. Phys. [**22**]{}, 334 (1951). R. Landauer, IBM J. Res. Dev. [**5**]{}, 183 (1961); Phys. Today 44 , 23 (1991); Science 272, 1914 (1996) , edited by H. S. Leff and A. F. Rex (IOP Publishing, 2003). T. Sagawa and M. Ueda, Phys. Rev. Lett. [**100**]{}, 080403 (2008); [*ibid*]{}. [**102**]{}, 250602 (2009); [*ibid*]{}. [**104**]{}, 090602 (2010); Phys. Rev. E [**85**]{}, 021104 (2012); [*Nonequilibrium Statistical Physics of Small Systems: Fluctuation Relations and Beyond*]{}, edited by R. Klages, W. Just, C. Jarzynski (Wiley-VCH, Weinheim, 2012). T. Sagawa and M. Ueda, Phys. Rev. Lett. [**109**]{}, 180602 (2012). S. Ito and T. Sagawa, Phys. Rev. Lett. [**110**]{}, 180603 (2013). T. Sagawa and M. Ueda, New J. Phys. [**15**]{}, 125012 (2013). D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Phys. Rev. Lett. [**71**]{}, 2401 (1993). C. Jarzynski, Phys. Rev. Lett. [**78**]{}, 2690 (1997). G. E. Crooks, J. Stat. Phys. [**90**]{}, 1481 (1998). J. Kurchan, J. Phys. A [**31**]{}, 3719 (1998). J. L. Lebowitz and H. Spohn, J. Stat. Phys. [**95**]{}, 333 (1999). S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano, Nat. Phys. [**6**]{}, 988 (2010). J. V. Koski, V. F. Maisi, T. Sagawa, and J. P. Pekola, Phys. Rev. Lett. [**113**]{}, 030601 (2014). D. Mandal and C. Jarzynski, Proc. Natl. Acad. Sci. [**109**]{}, 11641 (2012). J. Um, H. Hinrichsen, C. Kwon, and H. Park, New J. Phys. [**17**]{}, 085001 (2015). N. Shiraishi and T. Sagawa, Phys. Rev. E [**91**]{}, 012130 (2015). C. Kwon, J. H. Yeo, H. Lee, and H. Park, J. Kor. Phys. Soc. [**68**]{}, 633 (2016). K. H. Kim and H. Qian, Phys. Rev. E [**75**]{}, 022102 (2007). G. Jourdan, G. Torricelli, J. Chevrier, and F. Comin, Nanotechnology [**18**]{}, 475502 (2007). S. Ito and M. Sano, Phys. Rev. E [**84**]{}, 021123 (2011). L. Onsager and S. Machlup, Phys. Rev. [**91**]{}, 1505 (1953); [*ibid*]{}. [bf 91]{}, 1512 (1953). For the Wiener process $dW$ defined by $\int_t^{t+\epsilon}ds \xi(s)$ in $\epsilon\to 0$ limit, $\langle dW\circ v(t)\rangle =\langle dW[v(t)+v(t+\epsilon)]/2\rangle=\langle dW\cdot dv/2\rangle$, where $\langle dW\cdot v(t)\rangle=0$ is used. Using $dv\simeq dW$ from Eq. (\[langevin\]), $\langle dW\circ v(t)\rangle\simeq \langle (dW)^2\rangle /2=\epsilon\gamma T$. J. Um, J. D. Noh, C. Kwon, H. Park, unpublished.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Using the [*Spitzer*]{} telescope, we have conducted a high-resolution spectroscopic study of 18 bulgeless (Hubble type of Sd or Sdm) galaxies that show no definitive signatures of nuclear activity in their optical spectra. This is the first systematic mid-infrared search for weak or hidden active galactic nuclei (AGNs) in a statistically significant sample of bulgeless (Sd/Sdm) disk galaxies. Based on the detection of the high-ionization \[NeV\] 14.3 $\mu$m line, we report the discovery of an AGN in one out of the 18 galaxies in the sample. This galaxy, NGC 4178, is a nearby edge-on Sd galaxy, which likely hosts a prominent nuclear star cluster (NSC). The bolometric luminosity of the AGN inferred from the \[NeV\] line luminosity is $\sim$ 8$\times$10$^{41}$ ergs s$^{-1}$. This is almost two orders of magnitude greater than the luminosity of the AGN in NGC 4395, the best studied AGN in a bulgeless disk galaxy. Assuming that the AGN in NGC 4178 is radiating below the Eddington limit, the lower mass limit for the black hole is $\sim$ 6$\times$10$^3$M$_{\odot}$. The fact that none of the other galaxies in the sample shows any evidence for an AGN demonstrates that while the AGN detection rate based on mid-infrared diagnostics is high (30-40%) in optically quiescent galaxies with pseudobulges or weak classical bulges (Hubble type Sbc and Sc), it drops drastically in Sd/Sdm galaxies. Our observations therefore confirm that AGNs in completely bulgeless disk galaxies are [*not*]{} hidden in the optical but truly are rare. Of the three Sd galaxies with AGNs known so far, all have prominent NSCs, suggesting that in the absence of a well-defined bulge, the galaxy must possess a NSC in order to host an AGN. On the other hand, while the presence of a NSC appears to be a requirement for hosting an AGN in bulgeless galaxies, neither the properties of the NSC nor those of the host galaxy appear exceptional in late-type AGN host galaxies. The recipe for forming and growing a central black hole in a bulgeless galaxy therefore remains unknown.'
author:
- 'S. Satyapal, T. Böker, W. Mcalpine, M. Gliozzi , N. P. Abel, & T. Heckman'
title: ' The Incidence of Active Galactic Nuclei in Pure Disk Galaxies: The [*Spitzer*]{} View'
---
Introduction
============
We now know that supermassive black holes lurk in the centers of most bulge-dominated galaxies in the local Universe and that their black hole masses, M$_{\rm BH}$, and the stellar velocity dispersions, $\sigma$, of their host galaxies are strongly correlated (Gebhardt et al. 2000, Ferrarese & Merritt 2000). This discovery has launched numerous speculations that the formation and evolution of galaxies and supermassive black holes are fundamentally linked, and that perhaps the presence of a bulge is a necessary ingredient for a black hole to form and grow. Indeed, M33, the most nearby example of a truly bulgeless disk galaxy shows no evidence of a supermassive black hole, and the upper limit on the black hole mass determined by stellar dynamical studies is significantly below that predicted by the M$_{\rm BH}$-$\sigma$ relation established in early-type galaxies (e.g., Gebhardt et al. 2001). In contrast, the disk galaxy NGC 4395 shows no evidence for a bulge and yet does contain an active nucleus (e.g., Filippenko & Ho 2003). However, this galaxy has remained until recently the only case of a truly bulgeless disk galaxy with an accreting black hole, leaving open the possibility that it is an anomaly. Indeed, prior to the launch of [*Spitzer*]{}, the vast majority of known accreting black holes - i.e., active galactic nuclei (AGN) - in the local Universe were found in galaxies with prominent bulges (e.g. Heckman 1980; Ho, Filippenko, & Sargent 1997; Kauffmann et al. 2003).
However, these studies were based on spectroscopic observations at optical wavelengths, which can be severely limited in the study of bulgeless galaxies, where a putative AGN is likely to be both energetically weak and deeply embedded in the center of a dusty late-type spiral. In such systems, the traditional optical emission lines used to identify AGN can be dominated by emission from star formation regions, in addition to being significantly attenuated by dust in the host galaxy. As a result, it is by no means clear what fraction of late-type galaxies host AGN. Therefore, some key fundamental questions on the connection between black holes and galaxy formation and evolution have yet to be answered, such as: What fraction of late-type galaxies host AGNs? Do black holes form and grow in galaxies without a bulge? How are the incidence and properties of the black hole related to the host galaxy in cases where there is no bulge?
Motivated by these questions, and by the possibility that optical studies may fail at finding AGNs in the latest Hubble types, we have previously conducted an exhaustive archival mid-infrared (MIR) spectroscopic investigation of 34 late-type (Sbc or later) galaxies observed by [*Spitzer*]{} to search for AGNs (Satyapal et al. 2007; Satyapal et al. 2008 - henceforth S07; S08, respectively). Remarkably, these observations revealed the presence of the high ionization \[NeV\] 14 and/or 24 lines - which are not generally produced in ionized gas surrounding hot stars - in a significant number of galaxies that have no clear signatures of an AGN in their optical spectra. Using detailed photoionization models with both an input AGN and an extreme EUV-bright radiation field from a young starburst, we demonstrated that the MIR spectrum of these galaxies cannot be replicated unless an AGN contribution, in some cases as weak as 10% of the total galaxy luminosity, is included (S08, Abel & Satyapal 2007). [*This implies that the AGN detection rate in late-type galaxies is possibly more than 4 times larger than what optical spectroscopic studies alone indicate*]{}. We have obtained follow-up X-ray observations of a subset of these galaxies, which in all cases confirm the presence of an AGN (Gliozzi et al. 2009; Satyapal et al. 2009). A more recent [*Spitzer*]{} study has also uncovered a significant population of AGNs in optically quiescent galaxies of earlier Hubble type (Goulding & Alexander 2009), demonstrating the power of mid-infrared spectroscopy in AGN searches. Other recent multiwavelength studies have also shown that AGNs are significantly more common in late-type galaxies than once thought (e.g., Greene, Ho, & Barth 2009; Shields et al. 2008; Ghosh et al. 2008; Barth et al. 2009; Dewangan et al. 2008; Desroches & Ho 2009). It is therefore clear that classical bulges are [*not*]{} required for black holes to form and grow.
While it is evident that AGNs do reside in a significant number of late-type galaxies, most of the galaxy hosts appear to have a pseudobulge component, i.e. a central light excess characterized by an exponential surface brightness profile. These pseudobulges are thought to form via secular processes, in contrast to the violent merger-driven formation history of classical bulges (Kormendy& Kennicutt 2004). Amongst our previous archival Spitzer sample, there were only 4 very late-type spirals (Hubble type Sd/Sdm) without any obvious sign of a pseudo-bulge. We discovered prominent \[NeV\] emission from only one of these sources - the nearby Sd galaxy NGC 3621 (S07). Follow-up X-ray (Gliozzi et al. 2009) and high spatial resolution optical (Barth et al. 2009) observations confirm the presence of an AGN in this source. NGC 3621 is similar to NGC 4395 in that both galaxies are essentially bulgeless and contain a massive nuclear star cluster (Barth et al. 2009). However, with only two examples and only a total of 4 truly bulgeless disk galaxies observed, it is not possible to determine robustly the fraction of AGNs in pure disk galaxies and to understand how the incidence and properties of black holes relate to the host galaxy in the absence of a bulge.
In this paper, we present results from a recent [*Spitzer*]{} MIR spectroscopic investigation of 18 optically quiescent, truly bulgeless disk galaxies in order to search for previously undetected low luminosity and/or embedded AGN. This is the first systematic MIR search for weak or hidden AGN in a statistically significant sample of ”pure“ disk galaxies. The primary goal of this paper is to refine the incidence of AGNs in this type of galaxy. As our previous [*Spitzer*]{} work has demonstrated, optical studies miss a significant fraction of AGNs in late-type galaxies, leaving open the possibility that there are a significant number of active black holes in the centers of completely bulgeless galaxies that are as yet undiscovered.
This paper is structured as follows. In Section 2, we summarize the properties of the [*Spitzer*]{} sample presented in this paper. In Section 3, we summarize the observational details and data analysis procedure, followed by a description of our results in Section 4. In Section 5, we discuss the origin of the \[NeV\] emission and the evidence for an AGN in our sample, followed by a discussion of the AGN detection rate in pure disk galaxies in Section 6. In Section 7, we investigate the demographics of late-type galaxies with AGNs, followed by an exploration in Section 8 of the host characteristics of the few AGNs that reside in definitively bulgeless galaxies. A summary of our major conclusions is given in Section 9.
The Sample
==========
Our goals in selecting a sample were to 1) obtain a statistically significant sample of pure (i.e. bulgeless) disk galaxies to provide meaningful estimates of the fraction that host AGNs, 2) select close-by objects to enable detailed follow-up of potential AGN discoveries, 3) select isolated disk galaxies to avoid the effects of interactions on triggering black hole formation and growth and to test whether black holes routinely form through purely secular processes, and 4) select well-studied sources with extensive optical spectroscopic and multiwavelength data available in the literature. Our target sources were selected from the Palomar survey of nearby bright galaxies (Ho et al. 1997; henceforth H97). Of the 486 galaxies in the Palomar survey, a little more than two-dozen are of Hubble type of Sd/Sdm. Excluding galaxies with irregular morphologies or other signs of interactions, and excluding the well-studied Seyfert NGC 4395, our final sample contained 18 galaxies.
Table 1 summarizes the basic properties of the galaxies in our sample. All targets are nearby, ranging in distance from 2.4 to 21.6 Mpc, with an average distance of $\sim$ 11 Mpc. The aperture of the optical measurements from H97 was 2”$\times$4”. The extinction-corrected absolute B-band magnitude ranges from $\sim$ -17 to -20. We estimate and list in Table 1 galaxy stellar masses using the extinction-corrected B-V color and absolute B magnitude using the mass-to-light ratios from Bell et al. (2003). The inferred galaxy masses for our sample range from $\sim$ 8$\times$10$^8$M$_{\odot}$ to $\sim$ 1.6$\times$10$^{10}$M$_{\odot}$. The distribution of derived galaxy masses of the sample is shown in Figure 1. We also list in Table 1 the nuclear star formation rate (SFR), estimated using the extinction-corrected H$\alpha$ luminosity from H97, assuming all of the H$\alpha$ luminosity arises from star forming regions, and using the prescription given in Kennicutt (1998). The SFR ranges from $\sim$ 6$\times$10$^{-5}$M$_{\odot}$/yr to $\sim$ 2$\times$10$^{-3}$M$_{\odot}$/yr. The (inclination-corrected) line width of the HI profile, also from H97, ranges from 21 to 314 km/s, suggesting that the sample spans a large range of dark matter mass. To determine whether the presence or properties of potential AGNs are related in any way to gas mass, we list in Table 1 the HI mass, estimated from the HI fluxes compiled in H97.
[{width="9cm"}]{}
[lclcccccccccc]{}\
Galaxy & Distance & \[OIII\]/H$_{\beta}$ & \[OI\]/H$_\alpha$ & \[NII\]/H$_\alpha$ & \[SII\]/H$_\alpha$ & Optical & M$_{B_T}^0$ & M$_{Gal}$ & M$_{HI}$ & $\Delta{\rm V}_{rot}$ & NC & SFR\
Name & (Mpc) & & & & & Class & & (M$_{\odot}$) & M$_{\odot}$ & (km/s) & & (10$^{-3}{\rm M}_{\odot}$/yr)\
IC 2574& 3.4& 0.23& 0.025& 0.07& 0.25& H& -17.33& 8.91& 9.05& 131& Unlikely & 0.16\
NGC 2500& 10.1& 2.42& 0.068& 0.33& 0.45& H& -18.03& 9.40& 8.91& 228& Yes & 0.14\
NGC 2537& 9& 1.83& 0.008& 0.15& 0.18& H& -17.75& 9.29& 8.59& 205& $\cdots$ & 27.81\
NGC 3027& 19.5& 1.16& 0.037& 0.19& 0.53& H& -19.77& 10.10& 9.93& 247& $\cdots$ & 3.99\
NGC 3432& 7.8& 1.71& 0.012& 0.14& 0.22& H& -18.43& 9.56& 9.26& 260& Maybe & 5.47\
NGC 3495& 12.8& 0.38& 0.05& 0.42& 0.43& H& -19.8& 10.11& 9.17& 314& $\cdots$ & 0.67\
NGC 4145& 20.7& 1.09& 0.13& 0.61& 0.8& T2& -20.04& 10.21& 9.82& 21& $\cdots$ & 1.68\
NGC 4178& 16.8& 0.35& 0.022& 0.32& 0.5& H& -19.78& 10.10& 9.66& 299& Yes & 5.16\
NGC 4242& 7.5& 1& 0.27& 0.27& 0.91& H& -18.08& 9.42& 8.78& 206& Yes & 0.06\
NGC 4618& 7.3& 1.82& 0.012& 0.16& 0.23& H& -18.19& 9.47& 9.04& 220& Yes & 4.75\
NGC 4656& 7.2& 4.02& 0.022& 0.05& 0.18& H& -19.19& 9.87& 9.62& 179& Unlikely & 2.05\
NGC 4713& 17.9& 1.01& 0.087& 0.44& 0.67& T2& -19.41& 9.96& 9.64& 242& Yes & $<$ 3.14\
NGC 5147& 21.6& 0.44& 0.035& 0.37& 0.52& H& -19.38& 9.94& 9.30& 259& $\cdots$ & 4.14\
NGC 5204& 4.8& 0.96& 0.099& 0.18& 0.56& H& -16.93& 8.96& 8.76& 154& Maybe & $<$ 0.38\
NGC 5585& 7& 1.7& 0.017& 0.15& 0.32& H& -18.18& 9.46& 9.19& 200& Yes & 2.63\
NGC 6689& 12.2& 1.87& 0.067& 0.4& 0.62& H& -18.48& 9.58& 9.13& 220& Unlikely & 0.79\
NGC 784& 4.7& 4.9& 0.006& 0.03& 0.09& H& -17.2& 9.07& 8.54& 116& Unlikely & $<$ 0.96\
NGC 959& 10.1& 1.16& 0.032& 0.34& 0.37& H& -17.66& 9.26& 8.45& 196& Yes & 2.09\
[ Col(1): Common Source Names; Col(2): Distance to the source in units of Mpc are all taken directly from H97 where distances are adopted from Tully & Shaya (1984); Col(3): \[OIII\] to H$_\beta$ ratio taken from H97; Col(4): \[OI\] to H$_\alpha$ ratio taken from H97; Col (5): \[NII\] to H$_\alpha$ ratio taken from H97); Col(6): \[SII\] to H$_\alpha$ ratio taken from H97); Col(7): Optical classification of the source; “H” signifies HII region ratios, “T” represents transitional spectra between LINERs and HII regions, and “2” indicates that broad permitted lines were not found in the optical spectrum. Col(8): Total absolute B magnitude corrected for extinction, adopted from H97; Col(9): Galactic Mass obtained from B-V color and B-magnitude from H97 using mass-to-light ratios from Bell et al. (2003); Col(10): HI mass taken from H97; Col(11): Inclination-corrected Hi rotational amplitude taken directly from Table 10, col(7) in H97; Col(12) Presence of Nuclear Cluster based on archival [*HST*]{} observations. Col (13) Nuclear star formation rate estimated using the extinction-corrected H$\alpha$ luminosity from H97, assuming all of the H$\alpha$ luminosity arises from star forming regions, and using the prescription given in Kennicutt (1998)]{}
The majority of galaxies in our sample are classified in H97 as “HII” stellar-powered galaxies; only two are “T2” transition galaxies. T2 galaxies have optical line ratios intermediate between HII galaxies and low-ionization nuclear emission-line regions (LINERs) and have no broad permitted lines (e.g. H$\alpha$) in their optical spectrum. There is therefore no firm optical spectroscopic evidence for AGNs in any of the galaxies in our sample. This is illustrated in Figure 2 which shows the standard optical line ratio diagnostic diagrams (Veilleux & Osterbrock 1987) widely used to classify AGNs for the entire H97 sample. Our Spitzer sample is highlighted, together with the theoretical starburst limit line from Kewley et al. 2001, i.e. the maximum line ratios allowed by starburst photoionization models using the hardest possible radiation field. Note that the majority of galaxies in our sample have optical line ratios well to the left of this line, indicating that the optical line ratios do not require the presence of [*any*]{} AGN contribution.
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As can be seen from the DSS images in Figure 3, the morphologies of the sample galaxies are varied, with some objects having prominent disks and clear photocenters, and others displaying a more diffuse structure with no obvious photocenter. Archival [*HST*]{} images are available for 14 out of the 18 galaxies in the sample, taken mostly with the [*Advanced Camera for Surveys*]{} (ACS). We inspected these images for the presence of a well-defined nuclear star cluster (NSC). An unambiguous detection of such a source was possible in 7 of the 14 sample galaxies. These galaxies are identified in Table 1. We also performed elliptical isophote fitting to check for a notable bulge component in the one-dimensional surface brightness profiles, and confirmed that all of the sample galaxies are pure disk galaxies.
[ccc]{} {width="30.00000%"} & {width="30.00000%"} & {width="30.00000%"}\
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{width="30.00000%"} & {width="30.00000%"} & {width="30.00000%"}\
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\
Observations and Data Reduction
===============================
Data for all but one of the galaxies in our sample were obtained from the [*Spitzer*]{} Cycle 5 GO program ID 50339 (PI: Satyapal). These observations were all executed between 2008 June and 2008 November. One of the galaxies in our sample (NGC 4618) was previously observed with the IRS in 2005 December (PID 20140). We used the archival data for this galaxy. All observations were carried out in staring mode using both the short-wavelength (SH, 4.7”$\times$11.3”, $\lambda$ = 9.9-19.6 $\mu$m) and long-wavelength (LH, 11.1”$\times$22.3”, $\lambda$ = 18.7-37.2 $\mu$m) high-resolution modules of the Infrared Spectrograph (IRS; Houck et al. 2004) which have a spectral resolution of R $\sim$ 600. Exposure times were chosen to provide a signal-to-noise ratio of at least 5 for the \[NeV\] 14.3 $\mu$m line, assuming the lowest \[NeV\] luminosity detected to date in any galaxy (S08). Each observation was followed by a background sky observations located 2 $\arcmin$ from the source in order to enable backround-subtraction.
All on-source observations were centered on the galaxy nuclei, i.e. the photocenter coordinates from H97, which agree well with the 2MASS coordinates. Figure 3 overlays the SH and LH slit apertures onto the DSS images for all our targets, demonstrating that the nucleus of the galaxy always falls well within the slit. The slit size for the median distance of 10 Mpc corresponds to a projected extraction aperture of 0.2 kpc$\times$0.5 kpc and 0.5 kpc$\times$1.1 kpc for the SH and LH modules, respectively. The projected slit sizes, as well as a number of other observational details, are listed in Table 2.
We note that the SH and LH staring observations are dithered, i.e. the integration is split into two slit positions overlapping by one third of a slit. Unless the emission originates from a compact source that falls entirely within the slit for both pointings, the two spectra cannot be averaged. The procedure for flux extraction for staring observations was the following: 1) If the fluxes measured from the two slits differed by no more than the calibration error of the instrument, then the fluxes were averaged; otherwise, the slit with the highest measured line flux was chosen. 2) If an emission line was detected in one slit, but not in the other, then the detection was selected. The overall calibration uncertainty for the fluxes we report in this paper is 15%.
The raw data were preprocessed by the IRS pipeline (version 17.2) at the [*Spitzer*]{} Science Center (SSC) prior to download. Preprocessing includes ramp fitting, dark-sky subtraction, droop correction, linearity correction, flat-fielding, and flux calibration. The [*Spitzer*]{} spectra were further processed using the SMART v. 6.3.0 analysis package (Higdon et al. 2004) and the corresponding version of the calibration files (v.1.5.0), which were used to obtain final line fluxes. Each spectra was individually inspected and any bad pixels remaining after pipeline processing were removed. The fine-structure line fluxes presented in this work were obtained from Gaussian fits to the spectral line and linear fits to the baseline continuum.
Many of the galaxies in our sample display prominent PAH features and molecular hydrogen emission lines. In this work, we limit the discussion to the fine-structure emission lines relevant for identification of potential AGN. We defer discussion of all other spectral diagnostics and the star-formation properties of bulgeless galaxies to a future paper.
[lccrlcc]{}\
\
Galaxy & Exposure Time & Exposure Time & & Extraction Aperture & [Extraction Aperture]{}\
Name & SH (seconds) & LH (seconds) & & & Size SH (pc) & Size LH (pc)\
\
IC2574 & 6$\times$3 & 6$\times$3 & 10 28 23.50 & +68 24 44.00 & 77$\times$186 & 183$\times$368\
NGC2500 & 30$\times$3 & 14$\times$3 & 08 01 53.21 & +50 44 13.60 & 230$\times$553 & 544$\times$1092\
NGC2537 & 30$\times$2 & 14$\times$3 & 08 13 14.60 & +45 59 23.30 & 205$\times$493 & 484$\times$973\
NGC3027 & 120$\times$3 & 240$\times$3 & 09 55 40.60 & +72 12 12.80 & 444$\times$1068 & 1049$\times$2108\
NGC3432 & 30$\times$2 & 14$\times$2 & 10 52 31.13 & +36 37 07.60 & 178$\times$427 & 420$\times$843\
NGC3495 & 30$\times$4 & 60$\times$3 & 11 01 16.23 & +03 37 40.50 & 292$\times$701 & 689$\times$1384\
NGC4145 & 120$\times$4 & 240$\times$4 & 12 10 01.52 & +39 53 01.90 & 472$\times$1134 & 1114$\times$2238\
NGC4178 & 120 $\times$2 & 240$\times$2 & 12 12 46.40 & +10 51 57.50 & 383$\times$920 & 904$\times$1816\
NGC4242 & 30$\times$2 & 14$\times$2 & 12 17 30.18 & +45 37 09.50 & 171$\times$411 & 404$\times$811\
NGC4618 & 30$\times$2 & 60$\times$2 & 12 11 32.85 & +41 09 02.80 & 166$\times$400 & 393$\times$789\
NGC4656 & 30$\times$2 & 14$\times$2 & 12 43 57.73 & +32 10 05.30 & 164$\times$394 & 387$\times$778\
NGC4713 & 120$\times$2 & 240$\times$2 & 12 49 57.87 & +05 18 41.10 & 408$\times$981 & 963$\times$1935\
NGC5147 & 120$\times$4 & 240$\times$5 & 13 26 19.71 & +02 06 02.60 & 492$\times$1183 & 1162$\times$2335\
NGC5204 & 6$\times$3 & 6$\times$3 & 13 29 36.51 & +58 25 07.40 & 109$\times$263 & 258$\times$519\
NGC5585 & 30$\times$2 & 14$\times$2 & 14 19 48.20 & +56 43 44.60 & 160$\times$383 & 377$\times$757\
NGC6689 & 30$\times$3 & 60$\times$2 & 18 34 50.25 & +70 31 26.10 & 278$\times$668 & 657$\times$1319\
NGC784 & 6$\times$3 & 6$\times$3 & 02 01 16.93 & +28 50 14.10 & 107$\times$257 & 253$\times$508\
NGC959 & 30$\times$2 & 14$\times$3 & 02 32 23.94 & +35 29 40.70 & 230$\times$553 & 544$\times$1092\
\
[ Col(1): Common Source Names; Col(2) & (3): On-source exposure time per pointing in seconds given for the SH and LH modules, respectively; Col(4) & (5): Coordinates used for each observations; Col(6) & (7): Extraction apertures for the SH and LH modules, which correspond to the full aperture of the slit. Values given are in parsecs, using galaxy distances listed in Table 1.]{}
Results
=======
Fine-Structure Line Fluxes
----------------------------
In Table 3 we list the line fluxes, statistical errors and 3$\sigma$ upper limits for the strongest MIR fine-structure lines. The apertures from which these fluxes were extracted are listed in Table 2. In all cases, detections were defined when the line flux was at least 3$\sigma$. The strongest emission features in the spectra were the \[NeII\] 12.8 $\mu$m, \[NeIII\] 15.5 $\mu$m, \[SIII\] 18.7 $\mu$m, \[SIII\] 33.5 $\mu$m, and \[SiIII\] 34.8 $\mu$m lines, detected in $\sim$ 60%-80% of the galaxies in the sample. The \[FeII\] 26 $\mu$m, \[NeV\] 24 $\mu$m, and \[OIV\] 25.9 $\mu$m emission lines were not detected in any galaxy in the sample. We note that the spectral resolution of the SH and LH modules of the IRS is insufficient to resolve the velocity structure for most of the lines. We detected the \[NeV\] 14.3$\mu$m line in only 1 out of the 18 galaxies, providing strong evidence for the presence of an AGN in this one galaxy. We discuss the IR spectral line fluxes and flux ratios for this galaxy, NGC 4178, separately in Section 5 below.
[lrrrrrrrrrr]{}\
\
Name & \[SIV\] 10.51& \[NeII\] 12.81& \[NeV\] 14.32& \[NeIII\] 15.56& \[SIII\] 18.71& \[NeV\] 24.32& \[OIV\] 25.89& \[FeII\] 25.99& \[SIII\] 33.48& \[SiII\] 34.82\
\
IC2574 & $<$19.95 & $<$10.25 & $<$19.31 & $<$13.18 & $<$30.29 & $<$22.96 & $<$24.47 & $<$24.56 & $<$89.63 & $<$119.89\
NGC2500 & $<$6.46 & $<$3.87 & $<$5 & $<$3.69 & $<$5.34 & $<$9.69 & $<$10.32 & $<$10.36 & 90$\pm$19 & $<$27.51\
NGC2537 & $<$5.88 & 9.6$\pm$2.76 & $<$3.4 & $<$3.95 & $<$4.15 & $<$8.16 & $<$8.69 & $<$8.72 & 43$\pm$13 & $<$43.89\
NGC3027 & 7.6$\pm$1.02 & 6.13$\pm$0.62 & $<$0.48 & 3.1$\pm$0.6 & 4.81$\pm$0.55 & $<$2.62 & $<$2.8 & $<$2.81 & 21.5$\pm$2.88 & 21.2$\pm$2.46\
NGC3432 & 8.1$\pm$1.75 & 28.1$\pm$2.5 & $<$3.71 & 30.2$\pm$3.4 & 28.9$\pm$2.16 & $<$7.75 & $<$8.25 & $<$8.29 & 198.5$\pm$15.16 & 122.5$\pm$8.59\
NGC3495 & $<$2.95 & 13.3$\pm$1.3 & $<$1.88 & 12.2$\pm$1.6 & 14$\pm$3.8 & $<$5.45 & $<$5.81 & $<$5.83 & 32.9$\pm$9 & 39.7$\pm$7.5\
NGC4145 & 5.13$\pm$1.42 & 7.55$\pm$0.87 & $<$0.64 & 2.77$\pm$0.36 & 3.65$\pm$0.55 & $<$1.89 & $<$2.02 & $<$2.03 & 17$\pm$7.19 & 22.8$\pm$0.65\
NGC4178 & $<$1.23 & 20.9$\pm$1.95 & 2.43$\pm$0.39 & 5.95$\pm$0.45 & 12.82$\pm$0.89 & $<$2.79 & $<$2.11 & $<$2.14 & 80.7$\pm$4.6 & 83.7$\pm$2.53\
NGC4242 & $<$2.83 & $<$3.78 & $<$3.47 & $<$4.58 & $<$5.26 & $<$9.44 & $<$10.05 & $<$10.09 & $<$26.44 & $<$31.15\
NGC4496 & $<$5.49 & $<$6.53 & $<$3.58 & $<$4.5 & $<$6.6 & $<$5.44 & $<$5.8 & $<$5.82 & $<$9.19 & $<$15.38\
NGC4618 & $<$8.24 & 36.59$\pm$6.62 & $<$4.78 & 24.91$\pm$3.88 & 24.51$\pm$1.94 & $<$8.84 & $<$9.6 & $<$9.64 & 63.23$\pm$6.82 & 49.41$\pm$6.25\
NGC4656 & 47.3$\pm$5.25 & 40$\pm$1.7 & $<$3.5 & 41.8$\pm$2.65 & 23.8$\pm$3.7 & $<$6.82 & $<$7.26 & $<$7.29 & 128$\pm$15.49 & 97.8$\pm$14\
NGC4713 & $<$1.97 & 15.85$\pm$1.58 & $<$1.85 & 5.47$\pm$0.92 & 7.96$\pm$1.07 & $<$2.21 & $<$2.36 & $<$2.37 & 92.2$\pm$4.18 & 94.9$\pm$2.63\
NGC5147 & $<$1.46 & 9.06$\pm$1.09 & $<$1 & 3.6$\pm$0.42 & 4.96$\pm$0.7 & $<$2.3 & $<$2.45 & $<$2.46 & 28.8$\pm$3.47 & 32.8$\pm$2.13\
NGC5204 & $<$16.99 & $<$8.06 & $<$9.01 & $<$11.16 & 51.4$\pm$7.25 & $<$10.57 & $<$11.26 & $<$11.31 & 165$\pm$54.3 & $<$78.34\
NGC5585 & $<$2.52 & 11.6$\pm$2.9 & $<$2.04 & 11.7$\pm$2.15 & 17.1$\pm$2.74 & $<$7.48 & $<$7.97 & $<$8 & 53.5$\pm$14 & 67.8$\pm$16.3\
NGC6689 & $<$3.68 & $<$3.32 & $<$3.14 & $<$2.68 & $<$4.96 & $<$5.94 & $<$6.33 & $<$6.36 & 36.35$\pm$9.59 & 32$\pm$3.8\
NGC784 & $<$20.97 & $<$10.25 & $<$17.93 & 45$\pm$18.3 & $<$26.21 & $<$16.4 & $<$17.48 & $<$17.54 & $<$34.07 & $<$82.68\
NGC959 & $<$5.15 & $<$3.42 & $<$3.69 & 9.25$\pm$1.9 & $<$5.92 & $<$7.7 & $<$8.21 & $<$8.24 & 60.3$\pm$12 & 47.7$\pm$14.2\
\
[ Col(1): Common Source Names; Col(2)-Col(11)): Fluxes are in units of 10$^{-22}$ W cm$^{-2}$. 3 $\sigma$ upper limits are reported for nondetections.]{}
Incidence of AGN
------------------
The absence of \[NeV\] (ionization potential 97 eV) emission in our sample strongly suggests that with the exception of NGC 4178, none of the galaxies in our sample harbor AGNs. In Table 4, we list the \[NeV\] 14.3 $\mu$m luminosities corresponding to the 3$\sigma$ upper limits on the fluxes for all galaxies in the sample with the exception of NGC 4178. The luminosities were obtained using the galaxy distances listed in Table 1. The upper limits to the line luminosity are well below 10$^{38}$ ergs s$^{-1}$. Using the compilations of MIR line fluxes of standard AGN from Sturm et al. (2002), Haas et al. (2005), Weedman et al. (2005), Ogle et al. (2006), Cleary et al. (2007), Armus et al. (2007), Gorjian et al. (2007), Deo et al. (2007), Tommasin et al. (2008), and Dale et al. (2009) , there are 82 standard AGNs (optically classified as type 1 or type 2 AGN) with measured \[NeV\] 14$\mu$m line fluxes. The \[NeV\] 14 $\mu$m line luminosities for these AGNs range from $\sim$ 2$\times$10$^{38}$ ergs s$^{-1}$ to $\sim$ 8$\times$10$^{42}$ ergs s$^{-1}$ with a median value of $\sim$ 5$\times$10$^{40}$ ergs s$^{-1}$, more than two orders of magnitude above the \[NeV\] limiting sensitivities listed in Table 4. The \[NeV\] luminosity of NGC 3621, our one and only previously discovered Sd galaxy with a weak AGN is $\sim$ 5$\times$10$^{37}$ ergs s$^{-1}$ (S07), consistent with or above the limiting sensitivities of $\sim$ 90% of our sample. Our non-detections thus firmly imply that these galaxies do not host AGNs with luminosities comparable to the weakest known in any galaxy.
[lcccc]{}\
\
Name & $L_ {\rm [NeV]}$ & ${\rm [OIV]_{25.89}/[NeII]_{12.81}}$ & ${\rm [NeV]_{14.32}/[NeII]_{12.81}}$ & ${\rm [SIII]_{18.71}/[SIII]_{33.48}}$\
\
IC2574 & $<$2.52 &&&\
NGC2500 & $<$5.77 &&&\
NGC2537 & $<$3.12 & $<$0.91 & $<$0.35 &\
NGC3027 & $<$2.06 & $<$0.46 & $<$0.08 & 1.67\
NGC3432 & $<$2.55 & $<$0.29 & $<$0.13 & 1.91\
NGC3495 & $<$3.48 & $<$0.44 & $<$0.14 & 1.56\
NGC4145 & $<$3.09 & $<$0.27 & $<$0.08 & 0.51\
NGC4242 & $<$2.21 &&&\
NGC4496 & $<$6.94 &&&\
NGC4618 & $<$2.88 & $<$0.26 & $<$0.13 & 3.59\
NGC4656 & $<$2.05 & $<$0.18 & $<$0.09 & 1.54\
NGC4713 & $<$6.72 & $<$0.15 & $<$0.12 & 1.90\
NGC5147 & $<$5.28 & $<$0.27 & $<$0.11 & 1.43\
NGC5204 & $<$2.35 &&& 0.95\
NGC5585 & $<$1.13 & $<$0.69 & $<$0.18 & 1.22\
NGC6689 & $<$5.28 &&&\
NGC784 & $<$4.48 &&&\
NGC959 & $<$4.25 &&&\
\
[ Col(1): Common Source Names; Col(2): \[NeV\] 14.32 luminosity 3$\sigma$ upper limits in units of 10$^{37}~{\rm ergs ~s^{-1}}$; Col(3) & (4) & (5): Line flux ratios using fluxes from full apertures listed in Table 3.]{}
There are a number of MIR diagnostics used to characterize the dominant ionizing radiation field in galaxies. Since the flux ratio of emission lines from high-ionization to low-ionization ions depends on the nature of the ionizing source, the \[NeV\] 14.3 $\mu$m/\[NeII\] 12.8 $\mu$m and the \[OIV\]25.9$\mu$m/\[NeII\]12.8$\mu$m line flux ratios have been widely used to characterize the nature of the dominant ionizing source in galaxies (Genzel et al. 1996; Sturm et al. 2002; Satyapal et al. 2004; Dale et al. 2006,2009). We can compare our line flux ratio upper limits to the ratios in standard AGNs. Again, using the recent compilations of MIR line fluxes of standard AGNs observed by [*Spitzer*]{} from Deo et al. (2007), Tommasin et al. (2008), and Dale et al. (2009), there are 56 AGNs with measured \[NeII\] 12.8$\mu$m and \[NeV\] 14$\mu$m line fluxes. The \[NeV\]/\[NeII\] line flux ratio in these galaxies ranges from 0.02 to 2.97, with a median value of 0.73. As can be seen from Table 4, all of the galaxies with \[NeV\] 14.3 $\mu$m upper limits have \[NeV\]/\[NeII\] upper limits well below the median value in standard AGNs, supporting the hypothesis that these galaxies lack an AGN. Similarly, using the fluxes compiled in Verma et al. (2003), Deo et al. (2007), Tommasin et al. (2008), Dale et al. (2009), and Melendez et al. (2008), there are over 100 AGNs with measured \[NeII\] 12.8$\mu$m and \[OIV\] 25.9 $\mu$m line fluxes. The \[OIV\]/\[NeII\] line flux ratio in these galaxies ranges from 0.02 to 11.1, with a median value of 1.33. As can be seen from Table 4, the upper limits for the \[OIV\]/\[NeII\] flux ratio in our sample are also all well below the median value in standard AGNs, again strongly suggesting that these galaxies lack AGN.
An alternative diagnostic proposed by Dale et al. (2006) to classify the ionizing source in galaxies involves the \[NeIII\] 15.5$\mu$m/\[NeII\] 12.8 $\mu$m flux ratio and the \[SIII\] 33.48$\mu$m/ \[SiII\]34.82$\mu$m flux ratio. They find that AGNs display lower \[SIII\] 33.48$\mu$m/\[SiII\] 34.82$\mu$m line flux ratios than “pure star-forming” nuclei, presumably due to enhanced \[SiII\] emission in X-ray dominated regions around AGNs (Maloney et al. 1996). In Figure 4 we plot the \[NeIII\] 15.5$\mu$m/\[NeII\] 12.8 $\mu$m flux ratio versus the \[SIII\] 33.48$\mu$m/\[SiII\]34.82$\mu$m flux ratio for standard AGN, starburst and HII galaxies (based on the compilations listed above), and for those galaxies in our sample for which the lines were detected. We delineate the four regions defined by Dale et al. (2006), where regions I and II are exclusively occupied by LINERs and Seyferts, and regions III and IV are exclusively occupied by HII nuclei and extranuclear HII regions. As can be seen, all of the galaxies in our sample (including NGC 4178) fall entirely within region III, exhibiting similar ratios to HII nuclei.
To summarize, with the exception of NGC 4178, the MIR spectra of all our sample galaxies strongly suggest that they do not contain AGN.
[{width="9cm"}]{}
Density of the Ionized Gas
---------------------------
Abundance-independent density estimates have been obtained using infrared fine-structure transitions from like ions in the same ionization state with different critical densities. The density diagnostic available in the IRS spectra of our objects are the \[SIII\]18.71$\mu$m and 33.48 $\mu$m lines (where n$_{crit}$ $\sim 1.5 \times10^4 {\rm~cm^{-3}}$, and 4.1 $\times
10^3 {\rm~cm^{-3}}$, respectively, where n$_{crit}$ = A$_{ul}$/$\gamma$$_{ul}$, with A$_{ul}$ the Einstein A coefficient and $\gamma$$_{ul}$ the rate coefficient for collisional de-excitation from the upper to the lower level). The results are largely unaffected by the shape of the ionizing continuum. However, in Dudik et al. (2007), we showed that since the \[SIII\] emission is generally extended and the LH slit is larger than the SH slit, any analysis derived using line fluxes obtained from apertures of different sizes is ambiguous for most nearby galaxies. Furthermore, differential extinction towards the emitting gas in very obscured sources will also affect the line flux ratio, resulting in further ambiguities in the interpretation of the ratio. Nonetheless, for comparative purposes, we list in Table 4 the \[SIII\]18.71 $\mu$m/ \[SIII\] 33.48 $\mu$m line flux ratios for our sample of galaxies. . From Table 4, we see that for the galaxies in which both lines are detected, the line flux ratio ranges from 0.5 to 3.6, with an average value of 1.6. As shown in Dudik et al. (2007), at the distances for most of our sources, the \[SIII\] emission will likely extend beyond the SH aperture, resulting possibly in an artificially [*lower*]{} \[SIII\]18.71 $\mu$m/ \[SIII\]33.48 $\mu$m line flux ratio. As a comparison, the [*aperture-matched*]{} \[SIII\]18.71 $\mu$m/ \[SIII\] 33.48 $\mu$m line flux ratio in the SINGS sample of 75 galaxies from Dale et al. (2009) ranges from 0.3 to 2, with an average value of 0.8, significantly [*lower*]{} than the values listed in Table 4. Indeed, 97% of the SINGS sample has line flux ratios below the average value found in our sample. Since aperture affects should in principle lower the line flux ratios in our sample compared to the aperture-matched values in Dale et al. (2009), the higher \[SIII\]18.71 $\mu$m/ \[SIII\] 33.48 $\mu$m line flux ratios found in our sample of bulgeless galaxies possibly implies higher gas densities toward the \[SIII\]-emitting gas. The gas densities derived using the models from Dudik et al. (2007) for a gas temperature of $T = 10^4~K$ (obtained using the collision strengths from Tayal & Gupta (1999) and radiative transition probabilities from Mendoza & Zeippen 1982) range from $\sim 100~{\rm cm^{-3}}$ - 3 $\times10^3{\rm~ cm^{-3}}$ for our sample. We emphasize that the ambiguities inherent in the use of the \[SIII\] ratio, particularly significant in nearby galaxies, preclude us from making definitive statements about the actual gas densities in the ionized gas in our sample. Nevertheless, the \[SIII\] ratios appear to imply higher ionized gas densities compared with standard AGN and normal/starburst galaxies of earlier Hubble type. We note that Dale et al. (2009) find that the average \[SIII\] ratios is independent of whether the region probed is a star-forming or AGN environment.
The AGN in NGC 4178
===================
Line Fluxes and Spectral Line Fits
-----------------------------------
In Figure 5, we show the spectra extracted from the full SH aperture near 13 , 14 , and 15 , showing three emission lines from different ionization states of Neon. As can be seen, there is a clear detection ($\sim$ 6$\sigma$) of the \[NeV\] 14.32 line, providing strong evidence for an AGN in this galaxy. The \[NeV\] line is not resolved with the R=600 [*Spitzer*]{} SH spectral resolution. The \[NeV\]/\[NeII\] line flux ratio for NGC 4178 is 0.12, within the range but at the low end of the values observed in standard AGNs as discussed above. The \[NeV\]/\[NeII\] line flux ratio in NGC 4178 is a factor of $\sim$ 2 higher than the same ratio in NGC 3621, our previously discovered Sd galaxy with \[NeV\] emission. The infrared spectra (S07, Abel & Satyapal 2008) together with newly acquired additional multiwavelenth observations (Barth et al. 2009; Gliozzi et al. 2009) have made the case for an AGN in NGC 3621 secure. It is thus very likely that NGC 4178 also harbors an AGN, and that it is possibly slightly more energetically significant than the one in NGC 3621. The upper limit to the \[OIV\]/\[NeII\] line flux ratio in NGC 4178 is 0.1, still within range of the observed values in standard AGNs as discussed above. As in the case of NGC 3621, the low \[NeV\]/\[NeII\] and \[OIV\]/\[NeII\] line flux ratios in NGC 4178 suggests that the [*Spitzer*]{} spectrum is dominated by regions of star formation, and that there is significant contamination of the lower ionization emission lines from star formation within the [*Spitzer*]{} aperture.
The \[NeV\] 14.32 luminosity observed from NGC 4178 is 8.23$\times$10$^{37}$ ergs s$^{-1}$, slightly higher than the value observed in NGC 3621 (5$\times$10$^{37}$ ergs s$^{-1}$; S07) and almost 3 orders of magnitude lower than the median value observed in standard AGNs (see Section 3.1.1). It is also on the low end of the luminosities observed in other recently discovered late-type galaxies showing \[NeV\] emission presented in S08. Like NGC 3621, NGC 4178 likely harbors a very weak AGN.
[cc]{} {width="45.00000%"} & {width="45.00000%"}\
\
Other Evidence
--------------
There does not appear to be any previously published evidence for an AGN in NGC 4178. This source was not observed by [*Chandra*]{} or [*XMM-Newton*]{}. The galaxy was not detected by Einstein and the upper limit to the X-ray luminosity, ( L$_{\rm X}$ $\sim$ 2.5$\times$10$^{40}$ ergs s$^{-1}$ ; Fabbiano, Kim, & Trinchieri 1992) is consistent with a low luminosity AGN (e.g. Ho et al. 2001, Dudik et al. 2005). There is no evidence of a central source at radio wavelengths. The radio emission at 2.8, 6.3 and 20 cm peaks 55 " away from the optical center, and is associated with optically bright knots (Niklas et al. 1995a). There is no evidence of an excess in the radio-FIR correlation as is seen in AGN (Niklas et al. 1995b) and the radio spectral index is typical of star forming galaxies (Vollmer et al. 2004). The \[NeV\] detection reported in this work is the first observation suggesting the presence of an AGN.
In Abel & Satyapal (2008), we used the spectral synthesis code CLOUDY to model the emission line spectrum from gas ionized by both an input AGN radiation field and a young starburst. In the case of NGC 3621, we showed that the MIR spectrum cannot be replicated unless 30-50% of the bolometric luminosity within the [*Spitzer*]{} IRS aperture is due to an AGN. In Figure 6, we show the predicted \[NeV\]14.3$\mu$m/\[NeII\]12.8$\mu$m flux ratio versus the \[OI\]/ H$\alpha$ and \[SII\]/ H$\alpha$ optical line flux ratios based on the models from Abel & Satyapal (2008) for varying values of the ionization parameter (U; the dimensionless ratio of ionizing flux to gas density) and AGN luminosity contribution. We display only a narrow range of ionization parameters that generate line flux ratios within the range observed in NGC 4178 and NGC 3621. A more extensive grid of theoretical calculations, with all standard optical line flux ratios plotted, is presented in Abel & Satyapal (2007). As can be seen from Figure 6, the MIR and optical emission line spectra of NGC 4178 cannot be replicated with a pure starburst ionizing radiation field. An AGN contribution of $\sim$ 30-90% is required.
![The \[NeV\]14.3$\mu$m/\[NeII\]12.8$\mu$m line flux ratio versus the optical (a) \[NII\]/ H$\alpha$ flux ratio and (b) the \[SII\]/ H$\alpha$ flux ratio from the models from Abel & Satyapal (2008). The solid, dashed, and dotted line display model results for ionization parameters of - 1.5, -2.5, and -3.5, respectively. The dashed-dotted lines show the fraction of the total luminosity due to the AGN. The line attached to the circles represent 3% AGN, inverted triangles 10% AGN, squares 30% AGN, diamond 50% AGN, and triangles 100% AGN, as indicated in the Figure. The line flux ratios for NGC 4178 and NGC 3621 are shown. We note that the optical line flux ratios for NGC 4178 are taken directly from Table 4 in H97. The \[NII\]/H$\alpha$ flux ratio for NGC 3621 is taken from the large aperture (20“$\times$20”) measurements from Dale et al. 2006. The \[SII\]/H$\alpha$ measurement for NGC 3621 is from the higher spatial resolution (1 ) observations from Barth et al. (2009).](f6a.ps "fig:"){width="45.00000%"} ![The \[NeV\]14.3$\mu$m/\[NeII\]12.8$\mu$m line flux ratio versus the optical (a) \[NII\]/ H$\alpha$ flux ratio and (b) the \[SII\]/ H$\alpha$ flux ratio from the models from Abel & Satyapal (2008). The solid, dashed, and dotted line display model results for ionization parameters of - 1.5, -2.5, and -3.5, respectively. The dashed-dotted lines show the fraction of the total luminosity due to the AGN. The line attached to the circles represent 3% AGN, inverted triangles 10% AGN, squares 30% AGN, diamond 50% AGN, and triangles 100% AGN, as indicated in the Figure. The line flux ratios for NGC 4178 and NGC 3621 are shown. We note that the optical line flux ratios for NGC 4178 are taken directly from Table 4 in H97. The \[NII\]/H$\alpha$ flux ratio for NGC 3621 is taken from the large aperture (20“$\times$20”) measurements from Dale et al. 2006. The \[SII\]/H$\alpha$ measurement for NGC 3621 is from the higher spatial resolution (1 ) observations from Barth et al. (2009).](f6b.ps "fig:"){width="45.00000%"}
Although it is clear that gas photoionized solely by even the youngest starburst ionizing radiation field cannot simultaneously reproduce the optical and mid-infrared spectrum of NGC 4178, the possibility that the \[NeV\] emission originates from shocked gas associated with a starburst-driven superwind (e.g Veilleux, Cecil, & Bland-Hawthorn 2005) needs to be explored. Since the large aperture of the Spitzer IRS modules precludes us from using morphological arguments to rule out shocks, we consider whether the combined optical and mid-infrared fine structure line ratios are consistent with radiative shock models. Using the extensive grid of models from the most recent MAPPINGS III shock and photoionization code (Allen et al. 2008), we find that the optical spectrum of NGC 4178 cannot be reproduced for most of the parameter space they studied. In fact, the only shock models which can simultaneously reproduce the observed \[O III\]/H$\beta$, \[N II\]/H$\alpha$, and \[S II\]/H$\alpha$ line ratios for NGC 4178 are for high-density (n(H) = 1000 cm-3) shocks (Allen et al. 2008; Figure 22a). However, at these densities, the mid-infrared spectrum, in particular the \[Ne III\]/\[Ne II\] is 1-2 orders of magnitude higher than the observed value of 0.28 for NGC 4178. Thus, the mid-infrared spectrum and the optically “normal” spectrum of NGC 4178 cannot be simultaneously replicated by any shock model. Although high density radiative shocks may play a role in the emission line spectrum of NGC 4178 (and other AGN), it appears that the combined optical and mid-infrared spectrum cannot be produced without an AGN radiation field. Follow-up [*Chandra*]{} observations are crucial to confirm the presence of the AGN and constrain its location.
Bolometric Luminosity and Black Hole mass limit
------------------------------------------------
We can obtain an order of magnitude estimate of the bolometric luminosity of the AGN in NGC 4178 using the \[NeV\] line luminosity. Assuming that the line emission arises exclusively from the AGN, we follow the procedure adopted by S07 and S08 to estimate the nuclear bolometric luminosity of the AGN. Using the tight correlation between the \[NeV\] 14 $\mu$m line luminosity and the AGN bolometric luminosity found in a large sample of standard AGN (Equation 1 in S07), the AGN bolometric luminosity of NGC 4178 is $\sim$ 8$\times$10$^{41}$ ergs s$^{-1}$, slightly greater than the estimate for the AGN bolometric luminosity of NGC 3621 (S07). This estimate assumes that the relationship between the \[NeV\] 14 $\mu$m line luminosity and the bolometric luminosity established in more luminous AGN (see S07) extends to the lower \[NeV\] luminosity range characteristic of NGC 4178 and other late-type galaxies. The nuclear bolometric luminosities of the AGNs discovered in the late-type galaxies from S08 range from $\sim 3\times10^{41}{\rm~ ergs s^{-1}}$ to $\sim 2\times10^{43}{\rm~
ergs s^{-1}}$, with a median value of $\sim 9\times10^{41}{\rm~ ergs s^{-1}}$. As can be seen, the luminosity of the AGN in NGC 4178 is typical of other recently discovered AGN in low-bulge galaxies.
If we assume that the AGN is radiating below the Eddington limit, we can estimate the lower limit to the mass of the black hole based on the AGN bolometric luminosity estimate. The Eddington mass estimate in NGC 4178 is $\sim 6\times10^3{\rm~M_{\odot}}$ , well within the range of lower mass limits found in other late-type galaxies with AGN (S07, S08). There appears to be a nuclear star cluster in NGC 4178 (see section 7.1). However, there are no measurements of the central velocity dispersion. We therefore cannot determine if the lower mass limits derived for the black hole mass are incompatible with the M$_{BH}$-$\sigma$ relation, assuming a linear extrapolation to the low mass range.
Comparison to other AGNs in Bulgeless Galaxies
----------------------------------------------
NGC 4178 is one of less than a handful of completely bulgeless disk galaxies showing evidence for an AGN. The best-studied definitively bulgeless disk galaxy with an AGN is the galaxy NGC 4395, which shows the hallmark signatures of a type 1 AGN (e.g. Filippenko & Ho 2003; Lira et al. 1999; Moran et al. 1999). The bolometric luminosity of the AGN is $\sim 10^{40}{\rm~ergs s^{-1}}$ (Filippenko & Ho 2003), almost two orders of magnitude lower than the estimated bolometric luminosity of the AGN in NGC 4178. The estimated bolometric luminosity of the AGN in NGC 3621 is a factor of 1.6 less (S07) than that of the AGN in NGC 4178, making NGC 4178 the most luminous AGN in an Sd galaxy currently known. The black hole mass of NGC 4395, determined by reverberation mapping, is M$_{BH}=(3.6\pm 1.1)\times10^5 {\rm~M_{\odot}}$ (Peterson et al. 2005). This value for the black hole mass would be consistent with the lower limit of the black hole mass derived for NGC 4178 of 6$\times$10$^3$M$_{\odot}$ if the AGN is accreting at a high rate. Our recent X-ray observations, when combined with our [*Spitzer*]{} observations, suggest that the black hole mass in NGC 3621 is $\sim$ 2$\times$10$^4$M$_{\odot}$ and that it is accreting at a high rate (L$_{bol}$/L$_{Edd}$ $>$ 0.2) (Gliozzi et al. 2009). If NGC 4178 is similar to NGC 3621, then its black hole mass is comparable to that in NGC 4395 and is inline with those inferred for nuclear black holes in other pseudobulge galaxies (Greene & Ho 2007).
Mid-Infrared AGN Detection Rate in Sd Galaxies
==============================================
In S08, we showed that optical studies significantly miss AGN in late-type galaxies. From the H97 sample, out of the full sample of 486 galaxies, 207 are of Hubble type Sbc or later, and only 16 (8%) are optically classified as AGN. Using MIR diagnostics, we demonstrated that the AGN detection rate in optically normal disk galaxies of Hubble Type Sbc or later, is $\sim$ 30%, implying that the overall fraction of late-type (Sbc or later) galaxies hosting AGNs is possibly more than 4 times larger than what optical spectroscopic studies indicate. Although it is now clear that AGNs do reside in a significant number of late-type galaxies, virtually all of the newly discovered AGNs are in galaxies with Hubble type of Scd or earlier. Prior to the current work, there were only a handful of Sd galaxies observed by the high resolution modules of [*Spitzer’s*]{} IRS, precluding us from determining based on MIR diagnostics the true AGN fraction in galaxies with essentially [*no*]{} bulge.
In Figure 7, combining our current sample with that from S08, we show the AGN detection fraction in optically normal galaxies as a function of Hubble type. Since the sensitivity of the observations varied across the sample, we also indicate with a downward arrow in Figure 6 the number of galaxies with \[NeV\] 14 $\mu$m 3$\sigma$ line sensitivity of 10$^{38}$ ergs s$^{-1}$ or better. As can be seen, all of the Sd/Sdm galaxies were observed with the highest sensitivity. There are a total of 22 Sd/Sdm galaxies observed by [*Spitzer*]{} IRS and only 1 with a \[NeV\] detection (NGC 4178). Figure 7 shows that the AGN detection rate in optically normal galaxies drops dramatically for pure disk galaxies, with a detection rate of only 4.5%. From the H97 sample, out of the full sample of 486 galaxies, excluding interacting and irregular galaxies, 26 are of Hubble type Sd/Sdm and only one is optically identified as an AGN (NGC 4395). With the discovery of only one additional AGN in an Sd galaxy from the H97 sample based on MIR diagnostics, the overall detection rate of AGN in pure disk galaxies is only $\sim$ 8%, significantly lower than the detection rate in late-type galaxies with some bulge component. Our study thus shows that AGNs in pure disk galaxies do not appear to be hidden but are indeed truly rare.
[![The distribution of Hubble types for the current sample combined with that from S08. The galaxies with \[NeV\] detections are indicated by the filled histogram. Since the sensitivity of the observations varied across the sample, we also indicate with a downward arrow in Figure 6 the number of galaxies with \[NeV\] 14 $\mu$m line sensitivity of 10$^{38}$ ergs s$^{-1}$or better. As can be seen the AGN detection rate in optically normal galaxies drops dramatically for galaxies with no bulge component.](f7.ps "fig:"){width="9cm"}]{}
Demographics of Late-type Galaxies with AGNs
============================================
A Nuclear Star Cluster in NGC 4178?
-----------------------------------
With the discovery of an AGN in NGC 4178, there are now only 3 known AGN in Sd galaxies. As mentioned earlier, the two other Sd galaxies with AGNs, NGC 4395 and NGC 3621, both have prominent NSCs. In Figure 8, we show the [*HST*]{} NICMOS image of NGC 4178 (Böker et al. 1999). The image reveals a prominent source close to the apparent photocenter, indicated by an arrow. As will be discussed in Section 8, the luminosity of this source is consistent with that of a NSC. Unfortunately, the NICMOS image is not centered well, and thus does not allow an unambiguous determination of the photocenter.
However, visual inspection and a cursory isophote analysis over the limited NICMOS field of view suggest that the location of the NSC is indeed consistent with that of the photocenter of NGC 4178. Because the spatial resolution of the [*Spitzer*]{} data precludes us from determining the spatial location of the \[NeV\] peak to determine whether it coincides with the NSC, follow-up [*Chandra*]{} observations are crucial to confirm that the AGN indeed resides within the putative NSC. If we assume for the moment that the prominent central source in the NICMOS image is a NSC, then one can infer that all known AGNs in Sd galaxies reside in NSCs, possibly suggesting that in the absence of any bulge, an NSC is required for an AGN to be present.
We can estimate a rough mass for the NSC in NGC 4178 if we assume the average I-band M/L ratio for NSCs in late-type galaxies with measured dynamical masses (Walcher et al. 2005). Based on the rough I-band magnitude estimate (see Section 8), the nuclear cluster mass in NGC 4178 is $\sim 0.5\times10^6{\rm~M_{\odot}}$, comparable to the nuclear cluster mass in NGC 4395 (Seth et al. 2008) and an order of magnitude less than the one in NGC 3621 (Barth et al. 2009). Seth et al. (2008) find that in cases with known nuclear cluster and black hole masses, the ratio of the black hole mass to nuclear cluster mass, M$_{BH}$/M$_{NC}$, ranges from 0.1-1. This ratio is $\sim$ 1/3 in NGC 4395. Using the lower limit to the black hole mass for NGC 4178, we find M$_{BH}$/M$_{NC} > 10^{-2}$, which is possibly consistent with the ratio found in galaxies with measured nuclear cluster and black hole masses (Seth et al. 2008).
[![[*HST*]{} NICMOS H-band image of NGC 4178. The location of the NSC is indicated by the arrow and is consistent with the apparent photocenter. The field of view of the image is 51“$\times$51”.](f8.ps "fig:"){width="9cm"}]{}
AGNs in Late-type Bulges
------------------------
It is well established that AGN are common in early-type galaxies and that there is a trend of increasing AGN activity with bulge mass (e.g. H97; Kauffman et al. 2003). Based on our study and other recent studies, it is also now clear that AGN do exist in late-type galaxies that lack a classical bulge and that they are significantly more common than previously thought (S07;S08; Greene, Ho, & Barth (2009); Shields et al. 2008; Ghosh et al. 2008; Barth et al. 2008; Dewangan et al. 2008; Desroches & Ho 2009). Late-type galaxies are often characterized by so-called pseudobulges, with exponential surface brightness profiles similar to disks rather than classical bulges. These pseudobulges are thought to have formed from quiescent secular processes within the host galaxy, in contrast to the violent merger-driven events thought to have formed classical bulges (see review in Kormendy & Kennicutt 2004). It is therefore relevant to ask whether the incidence and properties of BHs in late-type galaxies are related to the presence and properties of pseudobulges.
Combining the samples from S08, and including NGC 3621 (S07), and this work, there are a total of 52 nearby optically normal late-type galaxies in which the presence or absence of AGN has been determined by MIR diagnostics. With a total of 9 AGN in this combined sample, we can attempt to investigate the relationship between AGN activity and the host galaxy properties for galaxies of Hubble type Sbc or later. We emphasize, that the AGNs in this sample were previously unknown based on optical spectroscopic studies. The study of the demographics of these newly discovered AGNs allows us to investigate whether there are any trends in AGN activity with host galaxy properties in late-type galaxies that were previously unseen in studies based on optical observations.
For those 13 galaxies in our current sample with existing [*HST*]{} imagery, we have used the archival [*HST*]{} data to establish the presence or absence of a nuclear star cluster (see Table 1). None of these [*HST*]{} images show any evidence for a bulge component, confirming our selection criterium, and we therefore assume the remaining 5 galaxies in our sample are bulgeless as well. For the remaining galaxies in our combined [*Spitzer*]{} sample, we searched the literature for all information on the structural properties of the host galaxies. Of the 52 late-type galaxies, 32 had published surface brightness profile fits to characterize the bulge properties (Knapen et al. 2003; Laurikainen et al. 2004; Scarlata et al. 2004; Dong & De Robertis 2006; Drory & Fisher 2007). We point out that several of the published bulge properties are based on ground-based imagery with low spatial resolution, which can significantly compromise the inferred bulge parameters. In addition, the presence or absence of NSCs has not been investigated in all galaxies. We therefore do not carry out a quantitative analysis of the relationship between bulge parameters and AGN presence and properties. Instead, we investigate whether there are any evident [*trends*]{} between the host galaxy properties and the incidence of AGN activity. Most of the galaxies in our previous sample (S08) have pseudobulges or weak classical bulges and four have identified NSCs. We point out that the NSC presence has not been investigated in the majority of galaxies in the sample and it is likely that most of these late-type galaxies do have NSCs. Of the 8 AGNs in our S08 sample, 6 have published surface brightness profile fits. Amongst these sources, 2 are reported to have classical bulges (NGC 3367, NGC 4414; Dong & De Robertis 2006, Laurikainen et al. 2004, respectively), and the remaining 4 are reported to have surface brightness profiles consistent with pseudobulges (NGC 3938, NGC 4321, NGC 4536, and NGC 5055; Dong & De Robertis 2006, Scarlata et al. 2004, Drory & Fisher 2007). One of the AGN galaxies is reported in the literature to host a NSC (NGC 4321; Knapen et al. 1995), but the presence of absence of a NSC in the remaining AGN galaxies has not been established. Although the small sample size and the ambiguity in the published bulge properties precludes us from conducting a quantitative investigation of the relationship between bulge parameters and nuclear cluster presence and AGN activity, it appears that most AGNs in late-type galaxies reside in galaxies with pseudobulges or weak classical bulges. It is also clear that NSCs and AGNs coexist, consistent with the findings from Seth et al. (2008) in more massive galaxies spanning a wide range of Hubble types.
Most importantly, our current study robustly shows that AGNs are extremely rare in bulgeless galaxies and that for the few cases where one does exist (NGC 3621, NGC 4178, and NGC 4395), there is a prominent NSC. These findings possibly suggest that if there is no bulge of any kind in a galaxy, the galaxy [*must have a NSC*]{} in order to host an AGN.
Is NGC 4178 Special?
====================
Our findings demonstrate that AGN are truly rare in bulgeless galaxies. An important question to then ask is what distinguishes bulgeless disk galaxies [*with*]{} AGN from those [*without*]{} AGN? Is the presence and properties of the black hole in any way related to the properties of the host galaxies in cases where there truly is [*no*]{} bulge? With 18 bulgeless disk galaxies in our sample and only one AGN (NGC 4178), the question arises whether NGC 4178 is in some way special in our sample. Of course the absence of an AGN does not imply the absence of a quiescent massive black hole in any of the galaxies in our sample. The only dynamical study that rules out the presence of a massive black hole (M$_{BH} < 1500 {\rm~M_{\odot}}$) in a bulgeless disk galaxy was carried out for the nearby galaxy M33 (Gebhardt et al. 2001).
We investigated whether NGC 4178 is unique in any way in its basic host galaxy properties listed in Table 1. The total estimated galaxy mass for NGC 4178 is $\sim 10^8{\rm~M_{\odot}}$, on the high end but not the highest of the galaxies in our sample (see Figure 1). From Table 1, we can see that the estimated HI mass is high but again comparable to or lower than several other galaxies in the sample, implying that the disk mass does not appear to be related to the presence or absence of an AGN in bulgeless disk galaxies. Similarly, the inclination-corrected HI rotational amplitude of NGC 4178 is high but not the highest in the sample (see Table 1), suggesting that the total dark matter mass is not the determining factor in whether or not a bulgeless disk galaxy hosts an AGN. Finally the nuclear SFR in NGC 4178 listed in Table 1, estimated using the exintction-corrected H$\alpha$ luminosity, is only slightly larger than the median value for the entire sample, but well below the highest value in the sample. There is thus no clear indication that the basic host galaxy properties in NGC 4178 are exceptional in any way compared to the rest of our sample.
If the basic host galaxy properties in NGC 4178 do not distinguish themselves from the rest of sample, we can ask whether the nuclear cluster properties do. Using the [*HST*]{} NICMOS image of NGC 4178, we estimate a magnitude of m$_{H}$=18.24 using a circular aperture of 3 pixels centered on the putative NSC. We point out that in this analysis, we are assuming that the AGN in NGC 4178 is coincident with the putative NSC - an assumption which is not possible to confirm with the poor spatial resolution of the [*Spitzer*]{} data. Follow-up high spatial resolution [*Chandra*]{} observations are crucial to confirm this hypothesis. Using the latest stellar population synthesis models from Bruzual & Charlot (2009), the I-H color for single age population older than $10^8$ years, assuming solar metalicity and a Chabrier (2003) initial mass function does not exceed 1.7. This means that the nuclear cluster in NGC 4718 has an absolute I-band magnitude of M$_{I}$ $\sim$ -11, typical of the I-band magnitudes of nuclear clusters in the extensive sample of nuclear clusters in late-type galaxies from Böker et al. (2002) (see their Figure 5). We note that we have not made any extinction correction, which could be significant in an edge-on galaxy such as NGC 4178. We also assumed an old stellar population. Since NGC 4178 shows prominent low ionization emission lines indicating the presence of young stars, the nuclear cluster color might be bluer than assumed above, resulting in an underestimate of the I-band luminosity. Finally, since the AGN is weak and hidden at optical wavelengths, we have assumed that the AGN contribution to the central luminosity in the H-band is negligible. Our estimate of the nuclear cluster luminosity should therefore be considered approximate. However, it does appear based on this rough estimate, that the nuclear cluster luminosity in NGC 4178 is typical of a nuclear cluster in late-type galaxies. There is thus no clear indication that NGC 4178 distinguishes itself from the rest of our sample of disk galaxies both in terms of the overall galaxy properties or its nuclear cluster luminosity. The recipe for forming and growing a massive black hole in a truly bulgeless disk galaxy is still unknown.
Summary and Conclusion
======================
We conducted a MIR spectroscopic investigation of 18 completely bulgeless disk galaxies showing no signatures of AGN in their optical spectra in order to search for low luminosity and/or embedded AGN. This is the first systematic search for weak or hidden AGN in a statistically significant sample of esssentially bulgeless disk galaxies. The primary goal of our study was to determine the incidence of AGNs in galaxies in the absence of a significant bulge. Our high resolution [*Spitzer*]{} spectroscopic observations reveal that while AGNs in galaxies with pseudobulges or weak classical bulges are significantly more common than once thought, AGNs in truly bulgeless disk galaxies are exceedingly rare. Our main results are summarized below:
1. We detected the high ionization \[NeV\] 14.3 $\mu$m emission line in only one out of the 18 galaxies in the sample, providing strong evidence for an AGN in this one source. This galaxy, NGC 4178, is a nearby (d=16.8 Mpc) edge-on disk galaxy with optical emission line ratios in the normal star formation regime, indicating that there is absolutely no hint of an AGN based on its optical spectrum.
2. With the exception of NGC 4178, none of the galaxies in the sample shows any evidence in the MIR for a weak or embedded AGN, suggesting that they lack AGN. Instead, most galaxies show signs of active star formation and possibly higher ionized gas densities than galaxies of earlier Hubble type.
3. Our work suggests that the AGN detection rate based on MIR diagnostics in late-type optically normal galaxies is high (30-40%) in galaxies of Hubble type Sbc and Sc but drops drastically in Sd/Sdm galaxies (4.5%). Our observations confirm that AGNs in completely bulgeless disk galaxies are not hidden in the optical but truly are rare.
4. The AGN bolometric luminosity of NGC 4178 inferred using our \[NeV\] line luminosity is $\sim$ 8$\times$10$^{41}$ ergs s$^{-1}$, a factor of 1.6 times greater than the estimated bolometric luminosity in the Sd galaxy NGC 3621, and almost two orders of magnitude greater than the AGN bolometric luminosity of NGC 4395, the best-known AGN in an Sd galaxy. This makes the AGN in NGC 4178 the most luminous known in a bulgeless disk galaxy. Assuming that the AGN is radiating below the Eddington limit, this c orresponds to a lower mass limit for the black hole of $\sim 6\times 10^3{\rm~M_{\odot}}$. There are no published measurements of the central stellar velocity dispersion. It is therefore unknown if the lower mass limit for the black hole in NGC 4178 violates the M$_{\rm BH}$-$\sigma$ relation established in early-type galaxies.
5. NGC 4178 is now one of only 3 known Sd galaxies showing evidence for an AGN. [*HST*]{} images of this galaxy suggests that it has a prominent NSC, similar to the ones seen in the other two known Sd galaxies with AGNs (NGC 3621, NGC 4395). If follow-up [*Chandra*]{} observations confirm that the AGN is coincident with the putative NSC in this source, this finding suggests that if there is [*no*]{} bulge of any kind in a galaxy, the galaxy must have a NSC in order to host an AGN.
6. We find that NGC 4178 is not exceptional in our sample of 18 bulgeless galaxies based both on its basic host galaxy properties (galaxy mass, disk mass, dark matter halo mass, nuclear SFR) and nuclear cluster properties. The recipe for forming and growing a black hole in a truly bulgeless disk galaxy still remains a mystery.
It is a pleasure to thank Rachel Dudik for her invaluable help in consulting with us on data analysis issues, for troubleshooting various software installation roadblocks, for her technical assistance in planning the observations, and for stimulating science discussions. This work would not have been possible without her expertise in IRS data analysis. We are also very grateful to the [*Spitzer*]{} helpdesk for numerous emails in support of our data analysis questions. Brian O’Halloran and Dan Watson were very helpful in consulting with us on data analysis procedures. This work is based on observations taken with the [*Spitzer*]{} Space Telescope, which is operated by JPL/Caltech under a contract with NASA. The thoughtful suggestions of the anonymous referee helped improve this paper. We are also very grateful for a fruitful discussion with Dave Alexander and Andy Goulding, which led us to outline more explicitly our data analysis procedure in Section 3. 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. SS gratefully acknowledges financial support from NASA grant RSA 1345391.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We investigate the non-universal part of the orbital entanglement spectrum (OES) of the $\nu = 1/3$ fractional quantum Hall effect (FQH) ground-state with Coulomb interactions. The non-universal part of the spectrum is the part that is missing in the Laughlin model state OES whose level counting is completely determined by its topological order. We find that the OES levels of the Coulomb interaction ground-state are organized in a hierarchical structure that mimic the excitation-energy structure of the model pseudopotential Hamiltonian which has a Laughlin ground state. These structures can be accurately modeled using Jain’s “composite fermion” quasihole-quasiparticle excitation wavefunctions. To emphasize the connection between the entanglement spectrum and the energy spectrum, we also consider the thermodynamical OES of the model pseudopotential Hamiltonian at finite temperature. The observed good match between the thermodynamical OES and the Coulomb OES suggests a relation between the entanglement gap and the true energy gap.'
address:
- '$^1$ Laboratoire Pierre Aigrain, ENS and CNRS, 24 rue Lhomond, 75005 Paris, France'
- '$^2$ Department of Physics, Princeton University, Princeton, NJ 08544'
author:
- 'A. Sterdyniak$^{1}$, B.A. Bernevig$^{2}$, N. Regnault$^{1}$ and F. D. M. Haldane$^2$'
bibliography:
- 'CoulombES.bib'
title: Hierarchical structure in the orbital entanglement spectrum in Fractional Quantum Hall systems
---
10000
Introduction
============
Most condensed matter phases (states of matter) can be characterized using local order parameters. However, in some cases, such as systems with topological order, it is not possible to obtain a full characterization of a state using only local operators. It has been proposed that non-local measurements borrowed from quantum information theory[@RevModPhys.80.517], such as entanglement, can provide new insights into topological phases phases. The most commonly-used quantifier of entanglement, the Von Neumann entanglement entropy, measures the entanglement between two blocks of the system. In systems exhibiting the Fractional Quantum Hall (FQH) effect, the first topological phases to be experimentally realized, such an entanglement measure providing a single number does not provide a unique characterization of the many possible states that can occur. A few years ago, it was suggested that generalizing the entanglement entropy to the entanglement spectrum [@li-08prl010504], could reveal much more of the physical properties of a FQH state.
The notion of entanglement spectrum has now been applied to many different systems: Quantum Hall mono-layers [@li-08prl010504; @regnault-09prl016801; @thomale-10pr180502; @lauchli-10prl156404; @lauchli-NJP-1367-2630; @bergholtz-arXiv1006.3658B; @PhysRevLett.106.100405; @rodriguez-arXiv1007.5356R; @papic-arXiv1008.5087P; @kargarian-82prb085106; @hermanns-1009arXiv4199H; @PhysRevB.80.201303; @2011arXiv1103.5437Q; @2011arXiv1103.0772Z], Quantum Hall bilayers[@PhysRevB.83.115322; @thomale-2010arXiv1010.4837T], quantum spin systems [@thomale-105prl116805; @poilblanc-105prl077202; @turner-2010arXiv1008.4346T; @fidkowski-81prb134509; @yao-105prl080501; @pollmann-njp-1367-2630; @2010PhRvB..81f4439P; @2008PhRvA..78c2329C; @2010arXiv1010.4508H; @PhysRevB.83.045110; @2011arXiv1104.2544S; @2010arXiv1011.2147P; @2010arXiv1002.2931F; @2011arXiv1104.1139H; @2011arXiv1103.3427C], superconductor[@bray-ali-80prb180504; @2011arXiv1105.4808D], topological insulators [@prodan-105prl115501; @fidkowski-104prl130502; @2010PhRvB..82x1102T] and ultra-cold gases [@PhysRevA.83.013620; @2011arXiv1104.5157D]. For a system described by a density matrix $\rho$, the orbital entanglement spectrum (OES) is defined using an angular-momentum orbital decomposition that cuts the system into two regions $A$ and $B$. Such an orbital cut mimics a geometrical cut only when the orbitals are localized in space. The reduced density matrix $\rho_A$ is obtained by tracing out the $B$ subsystem degrees of freedom, which yields $\rho_A=\Tr_B \rho$. As the eigenvalues of $\rho_A$ are non-negative, one can write $\rho_A=\exp(- \mathcal{H})$, thus introducing a fictitious Hamiltonian $\mathcal{H}$, whose spectrum is the OES.
For several FQH states, the count of the low-lying states in the entanglement spectrum was numerically shown to match that of the state’s edge-theory. Moreover, for a realistic Hamiltonian ground state that has a large overlap with a FQH model wavefunction, the OES exhibits a low-lying branch with the same entanglement state-count as the model wavefunction, but now accompanied by higher entanglement energy-levels which were previously not believed to provide useful information about the system. In this work, we show that the higher energy levels in the Coulomb OES are organized into branches whose structure can be related to virtual particle-hole excitations that dress the simpler entanglement spectrum of the model ground-state that just characterizes (in its purest form) the topological order of the FQH state.
This paper is organized as follows. In Section \[section\_ESinFQHE\], we introduce the sphere geometry, present the concept of the entanglement spectrum for the fractional quantum Hall effect, and summarize the main results that have already been obtained in the literature. In Section \[section\_JainCF\], we introduce Jain’s composite fermion wavefunctions and their neutral excitations. In Section \[section\_laughlinCoulombES\], we use the composite fermion construction to interpolate between Laughlin state and Coulomb interaction ground state entanglement spectrum. This interpolation explains the hierarchical structure observed in the OES. In Section \[section\_thermalES\], we investigate the behavior of the entanglement spectrum at finite temperature and, based on it, conjecture a relation between the energy gap and the entanglement gap. In the final section \[section\_discussion\], we present a discussion of these results.
Entanglement spectrum of fractional quantum Hall States on the sphere {#section_ESinFQHE}
=====================================================================
We will consider a system of $N$ particles moving on the surface of a sphere $(\theta, \phi)$ through which $N_{\Phi}$ magnetic flux quanta pass. The radius of the sphere is equal to $R=\sqrt{N_{\Phi}/2}$. In the lowest Landau level (LLL), the one-particle orbitals can be expressed as $$\psi_l(u,v)= \left[\frac{N_{\Phi}+1}{4\pi}\binom{N_{\Phi}}{l}\right]^{1/2}(-1)^{N_{\Phi}-l}v^{N_{\Phi}-l}u^{l}$$ where $\binom{n}{k}$ is the binomial coefficient, $u=\cos(\theta/2)\rme^{\rmi \phi/2}$ and $v=\sin(\theta/2)\rme^{-\rmi \phi/2}$ are the spinor variables. These orbitals are eigenstates of $L_Z$, the $z$-component of the angular momentum, with eigenvalues given by $l- N_{\Phi}/2$, where $l$ ranges from $0$ to $N_{\Phi}$. The orbitals $\psi_l$ form an approximate real-space partition of the sphere into rings: the north pole (resp. south) corresponds to $l=N_{\Phi}$ (resp. $l=0$).
Generic fermionic (bosonic) many-body wave functions of $N$ particles and total azimuthal angular momentum $L_z^{tot}$ can be expressed as linear combinations of Fock states in the occupancy basis of the single particle orbitals. Each Fock state can be labeled either by $\lambda$, a partition of $L_z^{tot}$ into $N$ components, or the occupation number configuration $n(\lambda)=\{n_l(\lambda), l = N_{\Phi},...,0\}$, where $n_l(\lambda)$ is the number of times $l$ appears in $\lambda$. We define “*squeezing*” as a two-particle operation on partitions that moves a particle from orbital $l_1$ (resp. $l_2$) to orbital $l'_1$ (resp. $l'_2$) such that $l_1+l_2=l'_1+l'_2$ and $l_1 < l'_1 \leq l'_2<l_2$ for bosons or $l_1 < l'_1 < l'_2<l_2$ for fermions. Squeezing defines a partial ordering on partitions: if a partition $\mu$ can be obtained from a partition $\lambda$ using successive squeezing operations, the partition $\lambda$ is said to *dominate* the partition $\mu$ ($\lambda > \mu$). Certain model wave functions have a [*root*]{} partition $\lambda_0$: in their expansion on the Fock states, only partitions dominated by $\lambda_0$ can have a non-zero weight. For instance, this is the case in the $1/m$ Laughlin state, in which the occupation number configuration of the root partition is given by $n(\lambda_0)=\{10^{m-1}10^{m-1}1\dots \}$, where $0^{m-1}$ denotes $m-1$ consecutive empty orbitals. This root partition is “$(1,m)$-*admissible*” - it obeys a generalized Pauli principle which does not allow more than one particle to occupy $m$ consecutive orbitals.
The orbital entanglement spectrum (OES) is obtained by cutting the sphere into two parts $A$ and $B$. Part $A$ contains the $l_A$ first orbitals from the north pole while part $B$ contains the $l_B=N_{\Phi}+1 -l_A$ remaining orbitals. Due to the localized nature of the Landau level (LL) orbitals, this cut is a reasonable approximation to a spatial one [@haque-07prl060401]. Tracing over the $l_B$ orbitals gives a reduced density matrix $\rho_A$ with a block-diagonal structure. Each block is characterized by two quantum numbers, $(L_{z,A},N_A)$: the $z$-component of the angular momentum for the part-$A$ orbitals and the number of particles in part $A$ are the only symmetry-generators present after the partial trace (the full $\vec{L}$ symmetry of the original state is lost). For the sphere geometry, the OES is the plot of the negative logarithm $\xi_i$ of $\rho_A$ eigenvalues as a function of $L_{z,A}$ for a fixed $N_A$. The $\xi_i$’s are called “entanglement (pseudo)energies”.
For numerical efficiency we use an algorithm (described in Table \[algo\]) that computes each block of the reduced density matrix independently, as for a given pair of quantum numbers of the $A$ region, $(L_{z,A},N_A)$, the only $B$-region subspace to consider is that defined by $(L_{z,B}=L_z-L_{z,A},N_B=N-N_A)$. When we consider a system that is not in a pure state, this procedure is repeated for all states present in the density matrix. The reduced density matrix thus obtained is then diagonalized using standard full diagonalization techniques.
For each basis states ${\left|\Psi^B_i\right\rangle} \in B$ ($i$ denotes the position in the basis)\
for each basis states ${\left|\Psi^A_j\right\rangle} \in A$\
create the basis state ${\left|\Psi\right\rangle}={\left|\Psi^A_j\right\rangle}\otimes{\left|\Psi^B_i\right\rangle}$\
find $k$, the index of ${\left|\Psi\right\rangle}$ in the basis describing $A\otimes B$ and the coefficient $c_k$ of\
in the full state ${\left|\Phi\right\rangle}$, including sign from reordering if needed (fermionic case)\
for each index $k$ in the basis of $A$\
for each index $p$ in the basis of $A$ with $p \geq k$\
add to the matrix element $\rho(k,p)$: $c_k*c_p$
![\[figure1\]Orbital entanglement spectrum for the Laughlin state, (a), and for the ground state of the Coulomb interaction, (b) for $N=8$ fermions, $N_{\Phi} = 21$, $N_A = 4$ and $l_A=11$. A small system-size has been selected for pedagogical purposes. The low-lying part of these spectra are almost identical, and exhibit the same structure and state-count. In addition to the Laughlin-like branch starting at $L_{z,A}=24$, the Coulomb spectrum contains at least two other clearly-defined branches starting at $L_{z,A}=28$ and $L_{z,A}=30$.](fermions_haldane_laughlin_n_8_2s_21_lz_0_0_la_11_na_4_entspec.eps "fig:"){width="7.5"} ![\[figure1\]Orbital entanglement spectrum for the Laughlin state, (a), and for the ground state of the Coulomb interaction, (b) for $N=8$ fermions, $N_{\Phi} = 21$, $N_A = 4$ and $l_A=11$. A small system-size has been selected for pedagogical purposes. The low-lying part of these spectra are almost identical, and exhibit the same structure and state-count. In addition to the Laughlin-like branch starting at $L_{z,A}=24$, the Coulomb spectrum contains at least two other clearly-defined branches starting at $L_{z,A}=28$ and $L_{z,A}=30$.](fermions_coulomb_n_8_2s_21_lz_0_0_la_11_na_4_entspec.eps "fig:"){width="7.5"}
Figure \[figure1\] shows a typical OES at $\nu=1/3$ for both the Laughlin state (fig. \[figure1\].a) and the Coulomb interaction ground state (fig. \[figure1\].b). In the Laughlin state, the entanglement stops at a maximum value of $L_{z,A}=L^{max}_{z,A}$, the $z$ component the angular momentum of the complementary region B. At this value, a single state is found for any values of the total particle number, despite the fact that the corresponding Hilbert space dimension of states in $A$ grows exponentially with $N_A$, the number of particles in the sector investigated. When applied to the Laughlin state root configuration, the orbital partitioning results in the root configuration $10010010010$ for the region $A$, which corresponds to the single state found at the highest possible value of $L_{z,A}$. While the existence of a root partition makes it obvious that no state can be found with a higher $L_{z,A}$, the fact that there is a unique state for this value is nontrivial and is related to a strong constraint on the state decomposition on the Fock basis called the “product rule”[@bernevig-09prl206801; @thomale-2010arXiv1010.4837T]. The coefficient of a configuration whose two parts after a cut can be independently-obtained by squeezing from the state root partition is equal to the product of the coefficients of the two disconnected pieces. When decreasing $L_{z,A}$ from this value, the counting of the eigenvalues (the number of eigenvalues at a given $L_{z,A}$) of the model state is the same as the number of levels of the corresponding edge conformal field theory (CFT) in the thermodynamic limit; empirically, it consists of counting the number of $(1,3)$-admissible partitions with the correct angular momentum in $l_A + \Delta L_z$ orbitals, where $\Delta L_z = L^{max}_{z,A}-L_{z,A}$. This conjecture can be proved using bulk-edge correspondence[@2011arXiv1102.2218C]. In the finite size systems, the equivalence between thermodynamic CFT edge counting and orbital entanglement spectrum counting does not hold for all $L_{z,A}$ values. The counting of the spectrum develops finite-size effects that have been recently [@hermanns-1009arXiv4199H] related to the encoding of the Haldane exclusion principle within the model state.
The entanglement spectrum of the Coulomb interaction ground state at filling $\nu=1/3$ exhibits a branch of low-lying levels displaying the same CFT counting as the Laughlin state, separated from higher energy states by an entanglement gap which was conjectured to remain finite in the thermodynamic limit[@li-08prl010504]. However, this entanglement gap closes as $L_{z,A}$ is reduced, making its definition ambiguous. Using a different Fock state normalization in which each LL orbital is normalized by the same factor, it has been shown that, in the Coulomb ground state OES, a full gap emerges between low-lying states whose state-count is the same as that of the Laughlin state, and higher entanglement energy states[@thomale-10pr180502] which had been previously deemed non-universal. In this paper (in section \[section\_laughlinCoulombES\]) we show that even the higher entanglement energy states in fact present a universal structure related to particle-hole excitations of the Haldane pseudopotential Hamiltonian for the Laughlin state. For now, we notice in the Coulomb OES the presence of well separated structures (“christmas-tree”-like branches) with states starting at a larger $L^{max}_{z,A}$ than that of the Laughlin-like branch (for example, the branches starting at $L^{max}_{z,A}=28$ and $L^{max}_{z,A}=30$ in Figure \[figure1\].b). Understanding these branches is the purpose of this paper.
Composite fermions wavefunctions and their excitations {#section_JainCF}
======================================================
Jain’s “composite fermion” (CF) picture [@jain89prl199] provides a nice heuristic explanation of many features of quantum Hall effect, including the observed incompressible states at $\nu=p/(2p+1)$ as well as the existence of a compressible the state at $\nu=1/2$ (see reference [@Jain_CF] for an extensive review on CF). The CF ansatz replaces the strongly interacting electrons or bosons by “composite fermions” formed by binding them to $n$ flux quanta, where $n$ is even or odd, depending on whether the original “bare” particles are bosons ($n$ odd) or fermions ($n$ even). The generic Jain states are given by: $$\Psi_{CF}={\cal P}_{{\rm LLL}} \left[\prod_{i<j}\left(u_iv_j - u_jv_i\right)^n \Phi^{CF}_p\right]
\label{jaincfwf}$$ where $\prod_{i<j}(u_iv_j - u_jv_i)^n$ binds $n$ flux quanta to each original particle. ${\cal P}_{{\rm LLL}}$ is the projection operator onto the LLL. $\Phi^{CF}_p$ is the wave function of the free CF in $p$ effective Landau levels, which Jain has called “Lambda levels” ($\Lambda$L). When $p$ such levels are fully occupied, equation \[jaincfwf\] gives rise to a model state that remarkably accurately approximates the ground state at filling $\nu=p/(np+1)$. The observed incompressible states at $\nu=p/(2p+1)$ can then be thought of as incompressible integer QH CF states at $\nu^*=p$ where $\nu^*$ is the filling factor of the CFs. Moreover, the compressibility of the $\nu=1/2$ state can be attributed[@halperin-PhysRevB.47.7312] to the formation of a CF Fermi sea state as the effective magnetic field felt by the CFs vanishes. In this picture, the $\nu=1/m$ Laughlin wave function is interpreted as the limiting case $n=m-1$ with $p=1$ filled CF $\Lambda$ levels. A schematic view of this picture of the Laughlin state is shown in figure \[fig\_cf\]a. Except for the Laughlin state, the Jain wavefunctions are not known to be unique zero-energy states of a model Hamiltonian. However, it has been recently been shown that they do have some special “squeezing” properties: the bosonic Jain states at $\nu=p/(p+1)$ and the bosonic counterpart $\Psi_B=\Psi_F/(\prod_{i<j}(u_iv_j - u_jv_i))$ of the fermionic one at $\nu=p/(2p+1)$ have a single root partition and vanish with a power 2 when $p+1$ particles are brought to the same point [@regnault-09prl016801].
The natural way to build quasihole-quasiparticle excitations within the CF construction is to assume that the $\Lambda$L’s are separated by an effective cyclotron energy $\hbar\omega^*_c$ and to then sort the different excited states with respect to their energy. Thus, the lowest energy excited states above the Laughlin ground-state, with energy is $\hbar\omega^*_c$, are obtained by exciting one CF in the second $\Lambda$L as shown in figure \[fig\_cf\]b. For $2\hbar\omega^*_c$ energy states, there are two possibilities: two CFs can be put in the second $\Lambda$L (fig. \[fig\_cf\]c) or one CF in the third $\Lambda$L - all other CFs remains in the lowest one (fig. \[fig\_cf\]c). In general, the $n^{th}$ excited states branch involves the $n+1$ lowest $\Lambda$ levels. Although this method is just a simple phenomenological sketch (as interactions between “composite fermions” should surely also be taken into account, and there is no phenomenological reason for equal spacing of the levels), the resulting wavefunctions remarkably-well reproduce the low energy structure of the Coulomb-interaction Hamiltonian when their variational energies are evaluated using the exact Hamiltonian and CF diagonalization[@Jain_CF].
 Schematic representation of the Laughlin state as a CF filled lowest $\Lambda$ level for $N=5$ CFs. (b) Creation of a quasiparticle-quasihole excitation above the Laughlin state in the $L_z=0$ sector. (c) and (d) Two different possibilities to create an excited state with energy $2\hbar\omega^*_c$.](cfexcitations.eps){width="9"}
Monte Carlo methods have been used extensively to quantitatively explore the predictions of the CF picture[@jain-97ijmpb156404], proving useful in computing quantities such as predicted energies of CF states and their overlaps with states obtained by exact diagonalization. However, calculations of entanglement spectra requires a high accuracy in the decomposition of the wave function on the Fock basis. With the exception of several special cases [@regnault-09prl016801], Monte Carlo techniques fail to reach the accuracy needed and we needed to implement a new exact projection method. The first step of this method is to use the Jack recursion formulas of [@bernevig-09prl206801; @stanley89advm76] to expand $\prod_{j<k}(u_jv_k-v_ju_k)^n$ (which is a Jack polynomial) in terms of symmetric monomials or slater determinants (for $n$ even or odd respectively). The product with $\Phi^{CF}_p$ is computed explicitly, using only the resulting wave function symmetry properties to reduce the computation time. Finally, the projection is applied and only terms fully in the LLL are kept. We found that the clustering properties characterizing how “composite particle” quasihole-quasiparticle excitation states vanish as particles come together are determined by the number of the $\Lambda$ level involved: the bosonic states and the bosonic counterpart of the fermionic states (fermionic states divided by a vandermonde determinant) constructed from the $k$’th $\Lambda$L vanish when $k+1$ particles are brought to the same point as the second power of the difference in the particle coordinates. Therefore the excited states also exhibit a non-trivial root configuration. In all the cases we have studied, we noticed the following properties of states of energies less or equal to $2\hbar\omega^*_c$: such excited states can be realized in a number of different ways: all CFs in the lowest $\Lambda$L, one or two CFs in the second $\Lambda$L or one CF in the third $\Lambda$L. The last possibility does not introduce any new states compared to the ones that are produced using the first two ways. To obtain this result, the LLL projection is crucial as it creates linear dependencies between the projected states and greatly reduces the number of independent states. For instance, for $N=6$ fermions and $N_{\Phi}=15$ in the $L_z^{tot}=0$ sector, there are $51$ states with $2$ CFs or less in the second $\Lambda$L and none in the third LL whereas there are $57$ states whose energy is less than or equal to $2\hbar\omega^*_c$; after projection and re-orthonormalization these two numbers are reduced to $36$ and the states spaces spanned by these two sets of states are identical.
From Laughlin to Coulomb entanglement spectrum {#section_laughlinCoulombES}
==============================================
In this section, we use the admixture of virtual quasiparticle-quasihole excited states into the ground state to interpolate between the Laughlin state and the Coulomb interaction ground state at the same filling factor, and examine the effect of this on the orbital entanglement spectrum. As both states have $L=0$ we only consider the zero angular momentum sector . The $\nu=1/3$ Laughlin state is known to be the densest zero-energy eigenstate of the Haldane pseudopotentials[@haldane83prl605] by $V_1=1$ and $V_{n>1}=0$. Using this interaction, we could obtain the low energy $L=0$ states and use them to reconstruct the Coulomb interaction ground state. However, these states do not have an exact model wavefunction construction that would lead to a simple understanding of the additional structure in the orbital entanglement spectrum of the Coulomb state. In particular the $L=0$ excited states of this model interaction do not feature a non trivial root configuration, contrary to the CF approach. Therefore, we first use the CF quasihole-quasiparticle excitations states whose construction was described in the previous section to iteratively construct a basis of $L=0$ states (the iteration being the number of $\Lambda$L levels occupied) whose effective cyclotron energy is less than a given effective cyclotron energy. We approximate the Coulomb ground state using: $${\left|\Psi_{n}\right\rangle}= \sum_{{\left|\Psi^{n\hbar\omega^*_c}_{CF}\right\rangle}} {\left\langle \Psi^{n\hbar\omega^*_c}_{CF} | \Psi_{coulomb} \right\rangle}{\left|\Psi^{n\hbar\omega^*_c}_{CF}\right\rangle}
\label{Coulombfit}$$ where the sum runs over all the $L=0$ states of the iterative basis which spans the space of states whose effective cyclotron energy is less than $n\hbar\omega^*_c$. This state is then normalized, which translates into a global shift of the entanglement spectrum. As the effective cyclotron energy is increased, the partition that dominates all the basis states changes and successively dominates the previous basis (see Table \[table\_cf\]). The $L^{max}_{z,A}$’s of the successive root partitions correspond to those of the additional structures observed on Coulomb ground state entanglement spectrum - thereby allowing us to predict the starting point of the higher entanglement energy branches. In the Table \[table\_cf\], the number of $L=0$ states, the topmost partitions and the corresponding $L^{max}_{z,A}$ values for the accessible effective cyclotron energy are given for $N=8$ and $N_{\Phi}=21$.
$n$ number of $L=0$ states Topmost partition $L^{max}_{z,A}$
----- ------------------------ ------------------------ -----------------
1 1 1001001001001001001001 24
2 4 1100010010000100100011 28
3 8 1100010010001000010011 28
4 14 1100100100000010010011 30
: \[table\_cf\] Characteristics of the iterative basis which spans the space of states whose effective cyclotron energy is less than $n\hbar\omega^*_c$ for $N=8$ and $N_{\Phi}=21$. The last column indicates the $L^{max}_{z,A}$ value corresponding to the topmost partition with respect to the cut defined by $L_A=11$ and $N_A=4$ - shown by a vertical line in the partitions. The total Hilbert space has $31$ $L=0$ states - we are using at most $14$ of them.
![\[oes\_cf\] Orbital entanglement spectrum of different ${\left|\Psi_{n}\right\rangle}$: $n=2$ (a), $n=3$ (b) and $n=4$ (c) for $N=8$ fermions, $N_{\Phi} = 21$, $N_A = 4$ and $l_A=11$. The structures observed on the Coulomb state OES progressively appear as $n$ is increased and we obtained completely seperated branches when $n$ is even. The last observed branch in the Coulomb spectrum, for which $L^{max}_{z,A}=32$, is expected to be described by an higher effective cyclotron energy CF wavefunctions.](fermions_super2_n_8_2s_21_lz_0_0_la_11_na_4_entspec.eps "fig:"){width="5.3"} ![\[oes\_cf\] Orbital entanglement spectrum of different ${\left|\Psi_{n}\right\rangle}$: $n=2$ (a), $n=3$ (b) and $n=4$ (c) for $N=8$ fermions, $N_{\Phi} = 21$, $N_A = 4$ and $l_A=11$. The structures observed on the Coulomb state OES progressively appear as $n$ is increased and we obtained completely seperated branches when $n$ is even. The last observed branch in the Coulomb spectrum, for which $L^{max}_{z,A}=32$, is expected to be described by an higher effective cyclotron energy CF wavefunctions.](fermions_super3cf_n_8_2s_21_lz_0_0_la_11_na_4_entspec.eps "fig:"){width="5.3"} ![\[oes\_cf\] Orbital entanglement spectrum of different ${\left|\Psi_{n}\right\rangle}$: $n=2$ (a), $n=3$ (b) and $n=4$ (c) for $N=8$ fermions, $N_{\Phi} = 21$, $N_A = 4$ and $l_A=11$. The structures observed on the Coulomb state OES progressively appear as $n$ is increased and we obtained completely seperated branches when $n$ is even. The last observed branch in the Coulomb spectrum, for which $L^{max}_{z,A}=32$, is expected to be described by an higher effective cyclotron energy CF wavefunctions.](fermions_super4cf_n_8_2s_21_lz_0_0_la_11_na_4_entspec.eps "fig:"){width="5.3"}
The comparison of the orbital entanglement spectrum for the first three interesting ${\left|\Psi_{n}\right\rangle}$ displayed in Figure \[oes\_cf\], and the Coulomb state spectrum (Fig. \[figure1\]) shows that the CF quasihole-quasiparticle excitation basis allows a step-by-step reconstruction of the different Coulomb entanglement spectrum structures. Given the method used, it is obvious that the spectra of ${\left|\Psi_{n}\right\rangle}$ and $ {\left|\Psi_{coulomb}\right\rangle}$ would get closer as the effective cyclotron energy is increased; in that case, we include more and more $\Lambda$L’s and, in a finite-size calculation, we start diagonalizing in larger and larger parts of the full Hilbert space. However, nontrivially, the structures of the Coulomb ground state entanglement spectrum are sorted by the effective cyclotron energy, in the sense that the first, second, *etc.* branches of the Coulomb spectrum can be obtained by only using CF states with $n=1,2$, *etc*. We checked these properties for the fermionic $\nu=1/3$ Laughlin state for up to $N = 8$ particles and for the bosonic $\nu = 1/2$ one for up to $N = 10$ particles (Fig. \[oes\_cf\_bosons\]). The different scales in the eigenvalues of the Coulomb interaction ground state density matrix seem to be linked to the different effective cyclotron energy scales involved.
![\[oes\_cf\_bosons\] Orbital entanglement spectrum of for the Coulomb interaction ground state (a) and different ${\left|\Psi_{n}\right\rangle}$: $n=2$ (b) and $n=4$ (c) for $N=10$ bosons, $N_{\Phi} = 18$, $N_A = 5$ and $l_A=9$.](bosons_coulomb_n_10_2s_18_lz_0_0_la_9_na_5_entspec.eps "fig:"){width="5.3"} ![\[oes\_cf\_bosons\] Orbital entanglement spectrum of for the Coulomb interaction ground state (a) and different ${\left|\Psi_{n}\right\rangle}$: $n=2$ (b) and $n=4$ (c) for $N=10$ bosons, $N_{\Phi} = 18$, $N_A = 5$ and $l_A=9$.](bosons_super_energylessthan_2_n_10_2s_18_lz_0_0_la_9_na_5_entspec.eps "fig:"){width="5.3"} ![\[oes\_cf\_bosons\] Orbital entanglement spectrum of for the Coulomb interaction ground state (a) and different ${\left|\Psi_{n}\right\rangle}$: $n=2$ (b) and $n=4$ (c) for $N=10$ bosons, $N_{\Phi} = 18$, $N_A = 5$ and $l_A=9$.](bosons_super_energylessthan_4_n_10_2s_18_lz_0_0_la_9_na_5_entspec.eps "fig:"){width="5.3"}
Orbital entanglement spectrum at finite temperature {#section_thermalES}
===================================================
We now try to reproduce the Coulomb ground state entanglement spectrum using a thermal density matrix. For a system at finite temperature $T$, the density matrix is given by: $$\rho = \frac{1}{Z}\exp(-\beta H)\label{thermaldens}$$ where $\beta=1/T$, $Z=\Tr\left[\exp(-\beta H)\right]$ and $H$ denotes the Hamiltonian of the system. In our case, we want to compare the entanglement spectrum of the Coulomb-interaction ground state to the entanglement spectrum of the density matrix describing finite-temperature corrections to the Laughlin state and hence the simplest choice is to take $H$ to be the pseudopotential interaction Hamiltonian. At $T=0$ we recover the pure Laughlin-state entanglement results. In contrast to the previous section, we now have to take into account not only the $L=0$ states, but states with all possible values of $L$, and all the $L_z$ sectors of the pseudopotential Hamiltonian. This involves a very large number of states and Equation \[thermaldens\] can be realized exactly only for small systems. By choosing an *ad-hoc* “entanglement temperature” temperature $T$, the entanglement spectrum of $\rho$ and the one of the Coulomb ground state can be made very similar as shown on figure \[fullspectrum\]. Although the full shape of the spectrum is very similar, we notice that the Coulomb spectrum exhibits degeneracies which are not present in the thermal density matrix approach. For example, the rightmost levels are degenerate for the Coulomb-interaction ground state and are no longer degenerate when the thermal density matrix is used, despite its rotational-invariance. It can be analytically shown that these degeneracies are linked to the $L = 0$ states that enter the density matrix. When the sum of Equation \[thermaldens\] is restricted to just the $L=0$ states, these degeneracies are recovered (fig. \[fullspectrum\]c).
To analyze bigger systems, we consider a simpler model. In the thermal density matrix, we only kept the two lowest energy branches of the incompressible state: the ground state wave function and the magneto-roton mode. This is the single-mode approximation of the entanglement spectrum. The dispersion relation of the magneto-roton mode is that obtained through by the $V_1$ interaction. The magneto-roton states ${\left|\Psi^{mag}_{L,lz}\right\rangle}$, obtained by putting one CF in the second $\Lambda$L, consist of $N-1$ multiplets whose total angular momentum ranges from $L=2$ to $L=N$. Even though a $L=1$ state is naively expected, this multiplet is systematically suppressed by the $LLL$ projection, as pointed out in [@PhysRevLett.69.2843]. Thus we use the following approximation for $\rho$: $$\label{eqmagndens}\rho=\frac{1}{Z} \left({\left|\Psi_{Laugh}\right\rangle}{\left\langle\Psi_{Laugh}\right|} + \sum_{L,l_z} e^{-\beta E_L} {\left|\Psi^{mag}_{L,lz}\right\rangle} {\left\langle\Psi^{mag}_{L,lz}\right|}\right)$$ where $$Z=1 +\sum_{L,l_z}e^{-\beta E_L}.$$ $L$ is the total angular momentum of the multiplet states and $E_L$ is their energy with respect to the pseudopotential interaction Hamiltonian. We then compute the orbital entanglement spectrum associated with this density matrix for various $\beta$ values and for up to $N=10$ fermions (see Fig. \[ESMagn\]).
\
In every case we have studied, when the temperature is infinite, the universal Laughlin-like part of the entanglement spectrum cannot be distinguished. As temperature is decreased, two branches of levels split: a lower one whose state-count is the same as that of the Laughlin state, and an upper one whose average “entanglement energy” goes to infinity as $T$ reaches zero. In the sector $L_{z,A}=L^{max}_{z,A}$ (in Fig\[\[ESMagn\]\] $L^{max}_{z,A}=24$), and the neighboring ones, it is possible even at infinite temperature (see figure \[ESMagn\]), to define an entanglement gap [@li-08prl010504], given by $\delta_{L_{z,A}}=\xi_{n+1,L_{z,A}}-\xi_{n,L_{z,A}}$ where $\xi_{n,L_{z,A}}$ is highest entanglement energy at $L_{z,A}$ that belongs to the Laughlin-state structure. For example, in the infinite temperature limit in figure \[ESMagn\], this can be done for $L_{z,A}=24,23,22$, at which the Laughlin-like levels of counting $1,1,2$ respectively are separated from the higher energy states by a small but visible gap. Moreover, the upper-branch state-count, and the $L_{z,A}$ value at which it starts, are the same as those of the first branch after the Laughlin one in the Coulomb ground state entanglement spectrum. As the magneto-roton mode can be obtained from excitations to the first $\Lambda$L level, only the first entanglement spectrum above the Laughlin one can be fitted in this single-mode approximation.
To more precisely characterize the behavior of the two branches as a function of the temperature, we calculated the entanglement gap $\delta_{L^{max}_{z,A}}$ as a function of $\beta=1/T$. The entanglement gap, shown in Figure \[gapbeta\], decreases as the temperature is increased from $T=0$ but, starting from a certain temperature value of the order of the energy gap, it reaches a plateau. This behavior is the same for all the cases we studied, independent of the number of fermions. Moreover, to see how our crude single-mode approximation affects the entanglement spectrum, we considered the next energy-level feature, taking all CF excitations states whose effective cyclotron energies are less than or equal to $2\hbar\omega^*_c$. These spectra are presented in Figure \[2cf\], where we observe the emergence of a third branch in the thermal density matrix, matching the third branch of the Coulomb entanglement spectrum. The main behavior as the temperature is varied is not substantially affected by these additional states. However, it should be noticed that an additional structure appears and that the entanglement gap at high temperature no longer exhibits a plateau but just a change of slope (shown in the Figure \[gapbeta\]).
Discussion {#section_discussion}
==========
The entanglement spectrum can be used to determine the universality class of realistic Hamiltonians in a topologically ordered phase: by identifying the state-count of the lowest-lying entanglement branch of the spectrum with the state-count (characters) of the CFT of an edge theory, we can in principle predict that the ground-state of the system lies in a certain topological phase. In this paper we showed that, for the case of the $\nu=1/3$ Coulomb interaction, not only the low-lying branch (with the Laughlin-state state-count), but also the higher’ “entanglement-energy” branches exhibit a nontrivial characteristic structure, related to dressing of the simple model ground state that gives rise to the lowest branch of the entanglement spectrum by zero-point fluctuations of particle-hole collective excitations, when the difference between the model Hamiltonian and the Coulomb Hamiltonian is added back as a “perturbation” . We have explicitly showed that the higher-energy branches correspond to the wavefunctions for adding particle-hole excitations on top of the Laughlin ground-state. The correspondence exhibits quantum number matching ($L_{z,A}$ and counting of the levels in a specific branch) with the Coulomb spectrum if the model wavefunctions we use for the excitations are Jain’s CF wavefunctions. We then performed a single-mode approximation of the Coulomb spectrum by calculating the reduced thermal density matrix of the Laughlin state augmented by the magnetoroton mode. We found that by this method we could obtain the first branch above the Laughlin-state branch in the entanglement spectrum of the Coulomb state. This exercise also appears to supports the idea that the entanglement spectrum of the ground-state of a realistic Hamiltonian contains information not only about the universality class of the ground-state but also about its excitations, which in a generic Hamiltonian (unlike in free fermion Hamiltonians FQH model Hamiltonians) will be represented in the ground state properties though zero-point fluctuations of collective modes.
Acknowledgment {#acknowledgment .unnumbered}
==============
BAB was supported in part by the Alfred P. Sloan Foundation, NSF CAREER DMR-095242, and by NSF-MRSEC DMR-0819860 at the Princeton Center for Complex Materials. FDMH was supported by DOE grant DE-SC0002140. BAB also wishes to thank Microsoft Station Q for generous hosting during the last stages of preparation of this work.
References {#references .unnumbered}
==========
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present an approach for imputation of missing items in multivariate categorical data nested within households. The approach relies on a latent class model that (i) allows for household-level and individual-level variables, (ii) ensures that impossible household configurations have zero probability in the model, and (iii) can preserve multivariate distributions both within households and across households. We present a Gibbs sampler for estimating the model and generating imputations. We also describe strategies for improving the computational efficiency of the model estimation. We illustrate the performance of the approach with data that mimic the variables collected in typical population censuses.'
author:
- 'Olanrewaju Akande, Jerome Reiter and Andrés F. Barrientos [^1]'
bibliography:
- 'mybib.bib'
title: Multiple Imputation of Missing Values in Household Data with Structural Zeros
---
Key words: categorical, census, edit, latent, mixture, nonresponse.
Introduction
============
In many population censuses and demographic surveys, statistical agencies collect data on individuals grouped within houses. In the U. S. decennial census, for example, the Census Bureau collects the age, race, sex, and relationship to the household head for every individual in the household, as well as whether or not the residents own the house. After collection, agencies share these datasets for secondary analysis, either as tabular summaries, public use microdata samples, or restricted access files.
When creating these data products, agencies typically have to deal with item nonresponse both for individual-level variables and household-level variables. They typically do so using some type of imputation procedure. Ideally, these procedures satisfy three desiderata. First, the imputations preserve the joint distribution of the variables as best as possible. As part of this, the procedure should preserve relationships within households. For example, the missing race of a spouse likely, but certainly not definitely, matches the race of the household head; the imputation procedure should reflect that. Second, the imputations respect structural zeros. For example, a daughter’s age cannot exceed her biological mother’s age. The imputations should not create impossible combinations of individuals in the same household. Third, the imputation procedure allows for appropriate uncertainty to be propagated in subsequent analyses of the data.
Typical approaches to imputation of missing household items use some variant of hot deck imputation [@KaltonKasprzyk1986; @AndridgeLittle2010]. However, depending on how the hot deck is implemented, it may not satisfy one or more of the desiderata. Indeed, we are not aware of any hot deck imputation procedure for household data that satisfies all three explicitly. An alternative is to estimate a model that describes the joint distribution of all the variables, and impute missing values from the implied predictive distributions in the model. For household data, one such model is the nested data Dirichlet process mixture of products of multinomial distributions (NDPMPM) model of @HuEtAl2018, which assumes that (i) each household is a member of a household-level latent class, and (ii) each individual is a member of an individual-level latent class nested within its household-level latent class. The model assigns zero probability to combinations corresponding to structural zeros, and also handles both household-level and individual-level variables simultaneously. The NDPMPM is appealing as an imputation engine, as it can preserve multivariate associations while avoiding imputations that result in impossible households. The NDPMPM is related to models proposed by @Vermunt2003 [@Vermunt2008] and @BenninkEtAl2016, although these are used for regression rather than multivariate imputation and do not deal with structural zeros.
@HuEtAl2018 use the NDPMPM to generate synthetic datasets [@Rubin1993; @RaghunathanRubin2001; @ReiterRaghunathan2007] for statistical disclosure limitation, but they do not describe how to use it for imputation of missing data. We do so in this article. With structural zeros in the NDPMPM, the conditional distributions of the missing values given the observed values are not available in closed form. We therefore add a rejection sampling step to the Gibbs sampler used by @HuEtAl2018, which generates completed datasets as byproducts of the Markov chain Monte Carlo (MCMC) algorithms used to estimate the model. These completed datasets can be analyzed using multiple imputation inferences [@Rubin1987]. We also present two new strategies for speeding up the computations with NDPMPMs, namely (i) turning data for the household head into household-level variables rather than individual-level variables, and (ii) using an approximation to the likelihood function. These scalable innovations are necessary, as the NDPMPM is computationally quite intensive even without missing data. The speed-up strategies also can be employed when using the NDPMPM to generate synthetic data.
The remainder of this article is organized as follows. In Section \[NDPMPM\], we review the NDPMPM model in the presence of structural zeros and the MCMC sampler for fitting the model without missing data. In Section \[MissingData\], we extend the MCMC sampler for the NDPMPM model to allow for missing data. In Section \[SpeedUp\], we present the two strategies for speeding up the MCMC sampler. In Section \[simulations\], we present results of simulation studies used to examine the performance of the NDPMPM as a multiple imputation engine, using the two strategies for speeding up the run time. In Section \[Discussion\], we discuss findings, caveats and future work.
Review of the NDPMPM Model {#NDPMPM}
==========================
@HuEtAl2018 present the NDPMPM model including motivation for how it can preserve associations across variables and account for structural zeros. Here, we summarize the model without detailed motivations, referring the reader to @HuEtAl2018 for more information. We begin with notation needed to understand the model and the Gibbs sampler, assuming complete data. The presentation closely follows that in @HuEtAl2018.
Notation and model specification
--------------------------------
Suppose the data contain $n$ households. Each household $i=1, \dots, n$ contains $n_i$ individuals, so that there are $\sum_{i=1}^n n_i = N$ individuals in the data. Let $X_{ik} \in \{1, \ldots, d_k\}$ be the value of categorical variable $k$ for household $i$, which is assumed to be identical for all $n_i$ individuals in household $i$, where $k = p+1, \ldots, p+q$. Let $X_{ijk} \in \{1, \ldots, d_k\}$ be the value of categorical variable $k$ for person $j$ in household $i$, where $j = 1, \ldots, n_i$ and $k = 1, \ldots, p$. Let $\textbf{X}_i = (X_{i(p+1)}, \dots, X_{i(p+q)}, X_{i11}, \dots, X_{in_ip})$ include all household-level and individual-level variables for the $n_i$ individuals in household $i$.
Let $\mathcal{H}$ be the set of all household sizes that are possible in the population. For all $h \in \mathcal{H}$, let $\mathcal{C}_h$ represent the set of all combinations of individual-level and household-level variables for households of size $h$, including impossible combinations; that is, $\mathcal{C}_h = \prod_{k=p+1}^{p+q} \{1, \ldots, d_k\} \prod_{j=1}^{h} \prod_{k=1}^{p} \{1, \ldots, d_k\}$. Let $\mathcal{S}_h \subset \mathcal{C}_h$ represent the set of impossible combinations, i.e., those that are structural zeros, for households of size $h$. These include combinations of variables within any individual, e.g., a three year old person cannot be a spouse, or across individuals in the same household, e.g., a person cannot be older than his biological parents. Let $\mathcal{C} = \bigcup_{h \in \mathcal{H}} \mathcal{C}_h$ and $\mathcal{S} = \bigcup_{h \in \mathcal{H}} \mathcal{S}_h$.
Although the NDPMPM model we use restricts the support of $\textbf{X}_i$ to $\mathcal{C} - \mathcal{S}$, it is helpful for understanding the model to begin with no restrictions on the support of $\textbf{X}_i$. Each household $i$ belongs to one of $F$ classes representing latent household types. For $i=1, \dots, n$, let $G_i \in \{1, \dots, F\}$ indicate the household class for household $i$. Let $\pi_g = \Pr(G_i = g)$ be the probability that household $i$ belongs to class $g$. Within any class, all household-level variables follow independent, multinomial distributions. For any $k \in \{p+1,\ldots,p+q\}$ and any $c \in \{1,\ldots,d_k\}$, let $\lambda_{gc}^{(k)} = \Pr(X_{ik} = c | G_i = g)$ for any class $g$, where $\lambda_{gc}^{(k)}$ is the same value for every household in class $g$. Let $\pi = \{\pi_1, \ldots \pi_F\}$, and $\lambda = \{\lambda^{(k)}_{gc}: c = 1, \ldots, d_k; k = p+1, \ldots, p+q; g = 1, \ldots, F\}$.
Within each household class, each individual belongs to one of $S$ individual-level latent classes. For $i=1, \dots, n$ and $j=1, \dots, n_i$, let $M_{ij}$ represent the individual-level latent class of individual $j$ in household $i$. Let $\omega_{gm} = \Pr(M_{ij} = m | G_i = g)$ be the probability that individual $j$ in household $i$ belongs to individual-level class $m$ nested within household-level class $g$. Within any individual-level class, all individual-level variables follow independent, multinomial distributions. For any $k \in \{1,\ldots,p\}$ and any $c \in \{1,\ldots,d_k\}$, let $\phi_{gmc}^{(k)} = \Pr(X_{ijk} = c | (G_i, M_{ij}) = (g,m))$ for the class pair $(g,m)$, where $\phi_{gmc}^{(k)}$ is the same value for every individual in the class pair $(g,m)$. Let $\omega = \{\omega_{gm}: g = 1, \ldots, F; m=1, \ldots, S \}$, and $\phi = \{\phi^{(k)}_{gmc}: c = 1, \ldots, d_k; k = 1, \ldots, p; m=1, \ldots, S ; g = 1, \ldots, F\}$.
For purposes of the Gibbs sampler in Section \[NDPMPMSampler\], it is useful to distinguish values of $\mathbf{X}_i$ that satisfy all the structural zero constraints from those that do not. Let the superscript “$1$” indicate that a random variable has support only on $\mathcal{C} - \mathcal{S}$. For example, $\textbf{X}_i^1$ represents data for a household with values restricted only on $\mathcal{C} - \mathcal{S}$, i.e., not an impossible household, whereas $\textbf{X}_{i}$ represents data for a household with any values in $\mathcal{C}$. Let $\mathcal{X}^1$ be the observed data comprising $n$ households, that is, a realization of $(\textbf{X}^1_{1}, \ldots, \textbf{X}^1_{n})$. The kernel of the NDPMPM, $\Pr(\mathcal{X}^1 | \theta)$, is $$\label{StrucZeroLikelihood}
\textrm{L}(\mathcal{X}^1 | \theta) = \prod_{i=1}^n \sum_{h \in \mathcal{H}} \mathds{1}\{n_i = h \} \mathds{1}\{\textbf{X}_i^1 \notin \mathcal{S}_h \} \left[ \sum_{g=1}^F \pi_g \prod^{p+q}_{k = p+1} \lambda^{(k)}_{gX^1_{ik}} \prod^{h}_{j=1} \sum_{m=1}^S \omega_{gm}\prod^p_{k = 1} \phi^{(k)}_{gmX^1_{ijk}} \right],$$ where $\theta$ includes all the parameters, and $\mathds{1}\{.\}$ equals one when the condition inside the $\{\}$ is true and equals zero otherwise.
For all $h \in \mathcal{H}$, let $n_{1h} = \sum_{i=1}^n \mathds{1}\{n_i = h\}$ be the number of households of size $h$ in $\mathcal{X}^1$ and $\pi_{0h}(\theta) = \Pr(\textbf{X}_i \in \mathcal{S}_h | \theta) $ As stated in @HuEtAl2018, the normalizing constant in the likelihood in (\[StrucZeroLikelihood\]) is $\prod_{h \in \mathcal{H}}(1 - \pi_{0h}(\theta))^{n_{1h}} $. Therefore, the posterior distribution is $$\label{NDPMPMTruncatedPosterior}
\Pr(\theta | \mathcal{X}^1, T(\mathcal{S})) \propto \Pr(\mathcal{X}^1 | \theta) \Pr(\theta) = \dfrac{1}{\prod_{h \in \mathcal{H}}(1 - \pi_{0h}(\theta))^{n_{1h}} } \textrm{L}(\mathcal{X}^1 | \theta) \Pr(\theta)$$ where $T(\mathcal{S})$ emphasizes that the density is for the NDPMPM with support restricted to $\mathcal{C} - \mathcal{S}$.
The likelihood in can be written as a generative model of the form $$\begin{aligned}
\phantom{X_{ijk} | G_i, M_{ij}, \phi, n_i}
&\begin{aligned} \label{ModelSpecification1}
\mathllap{X_{ik} | G_i, \lambda} & \sim \textrm{Discrete}(\lambda^{(k)}_{G_i1}, \ldots, \lambda^{(k)}_{G_id_k}) \\
&\qquad \forall i = 1, \ldots, n \ \textrm{and} \ k = p + 1, \ldots, p + q
\end{aligned}\\
&\begin{aligned} \label{ModelSpecification2}
\mathllap{X_{ijk} | G_i, M_{ij}, \phi, n_i} & \sim \textrm{Discrete}(\phi^{(k)}_{G_iM_{ij}1}, \ldots, \phi^{(k)}_{G_iM_{ij}d_k}) \\
&\qquad \forall i = 1, \ldots, n \ , \ j = 1, \ldots, n_i \ \textrm{and} \ k = 1, \ldots, p
\end{aligned}\\
&\begin{aligned} \label{ModelSpecification3}
\mathllap{G_i | \pi} & \sim \textrm{Discrete}(\pi_1, \ldots, \pi_F) \\
&\qquad \forall i = 1, \ldots, n
\end{aligned}\\
&\begin{aligned} \label{ModelSpecification4}
\mathllap{M_{ij} | G_i, \omega, n_i} & \sim \textrm{Discrete}(\omega_{G_i1}, \ldots, \omega_{G_iS}) \\
&\qquad \forall i = 1, \ldots, n \ \textrm{and} \ j = 1, \ldots, n_i
\end{aligned}\end{aligned}$$ where the Discrete distribution refers to the multinomial distribution with sample size equal to one. We restrict the support of each $\mathbf{X}_i$ to ensure the model assigns zero probability to all combinations in $\mathcal{S}$ as desired. The model in (\[ModelSpecification1\]) to (\[ModelSpecification4\]) can be used without restricting the support to $\mathcal{C} - \mathcal{S}$. This ignores all structural zeros. While not appropriate for the joint distribution of household data, this model turns out to useful for the Gibbs sampler. We refer to the generative model in (\[ModelSpecification1\]) to (\[ModelSpecification4\]) with support on all of $\mathcal{C}$ as the untruncated NDPMPM. For contrast, we call the model in the truncated NDPMPM.
For prior distributions, we follow the recommendations of @HuEtAl2018. We use independent uniform Dirichlet distributions as priors for $\lambda$ and $\phi$, and the truncated stick-breaking representation of the Dirichlet process as priors for $\pi$ and $\omega$ [@Sethuraman1994; @DunsonXing2009; @SiReiter2013; @Manrique-VallierReiter2014b], $$\begin{aligned}
\phantom{\omega_{gm} }
&\begin{aligned} \label{DirichletPriorI}
\mathllap{\lambda_g^{(k)} } & = (\lambda^{(k)}_{g1}, \ldots, \lambda^{(k)}_{gd_k}) \sim \textrm{Dirichlet}(1,\ldots, 1) \\
\end{aligned}\\
&\begin{aligned} \label{DirichletPriorII}
\mathllap{\phi_{gm}^{(k)} } & = (\phi^{(k)}_{gm1}, \ldots, \phi^{(k)}_{gmd_k}) \sim \textrm{Dirichlet}(1,\ldots, 1) \\
\end{aligned}\\
&\begin{aligned}
\mathllap{\pi_g } & = u_g \prod_{f < g} (1 - u_f) \ \textrm{for} \ g = 1, \ldots F \\
\end{aligned}\\
&\begin{aligned}
\mathllap{u_g } & \sim \textrm{Beta}(1,\alpha) \ \textrm{for} \ g = 1, \ldots, F-1, \ u_F = 1 \\
\end{aligned}\\
&\begin{aligned} \label{GammaPriorI}
\mathllap{\alpha } & \sim \textrm{Gamma}(0.25, 0.25) \\
\end{aligned}\\
&\begin{aligned}
\mathllap{\omega_{gm} } & = v_{gm} \prod_{s < m} (1 - v_{gs}) \ \textrm{for} \ m = 1, \ldots S \\
\end{aligned}\\
&\begin{aligned}
\mathllap{v_{gm} } & \sim \textrm{Beta}(1,\beta_g) \ \textrm{for} \ m = 1, \ldots, S-1, \ v_{gS} = 1 \\
\end{aligned}\\
&\begin{aligned} \label{GammaPriorII}
\mathllap{\beta_g } & \sim \textrm{Gamma}(0.25, 0.25). \\
\end{aligned}\end{aligned}$$
We set the parameters for the Dirichlet distributions in (\[DirichletPriorI\]) and (\[DirichletPriorII\]) to $\mathbf{1}_{d_k}$ (a $d_k$-dimensional vector of ones) and the parameters for the Gamma distributions in (\[GammaPriorI\]) and (\[GammaPriorII\]) to $0.25$ to represent vague prior specifications. We also set $\beta_g = \beta$ for computational expedience. For further discussion on prior specifications, see @HuEtAl2018.
Conceptually, the latent household-level classes can be interpreted as clusters of households with similar compositions, e.g., households with children or households in which no one is related. Similarly, the latent individual-level classes can be interpreted as clusters of individuals with similar characteristics, e.g., older male spouses or young female children. However, for purposes of imputation, we do not care much about interpreting the classes, as they serve mainly to induce dependence across variables and individuals in the joint distribution.
It is important to select $F$ and $S$ to be large enough to ensure accurate estimation of the joint distribution. However, we also do not want to make $F$ and $S$ so large as to produce many empty classes in the model estimation. Allowing many empty classes increases computational running time without any corresponding increase in estimation accuracy. This can be especially problematic in the Gibbs sampler for the truncated NDPMPM, as these empty classes can introduce mass in regions of the space where impossible combinations are likely to be generated. This slows down the convergence of the Gibbs sampler.
We therefore recommend following the strategy in @HuEtAl2018 when setting $(F, S)$. Analysts can start with moderate values for both, say between 10 and 15, in initial tuning runs. After convergence, analysts examine posterior samples of the latent classes to check how many individual-level and household-level latent classes are occupied. Such posterior predictive checks can provide evidence for the case that larger values for $F$ and $S$ are needed. If the numbers of occupied household-level classes hits $F$, we suggest increasing $F$. If the number of occupied individual-level classes hits $S$, we suggest increasing $F$ first but then increasing $S$, possibly in addition to $F$, if increasing $F$ alone does not suffice. When posterior predictive checks do not provide evidence that larger values of $F$ and $S$ are needed, analysts need not increase the number of classes, as doing so is not expected to improve the accuracy of the estimation. We note that similar logic is used in other mixture model contexts [@Walker2007; @SiReiter2013; @Manrique-VallierReiter2014b; @MurrayReiter2016].
MCMC sampler for the NDPMPM {#NDPMPMSampler}
---------------------------
@HuEtAl2018 use a data augmentation strategy [@Manrique-VallierReiter2014b] to estimate the posterior distribution in (\[NDPMPMTruncatedPosterior\]). They assume that the observed data $\mathcal{X}^1$, which includes only feasible households, is a subset from a hypothetical sample $\mathcal{X}$ of $(n + n_0)$ households directly generated from the untruncated NDPMPM. That is, $\mathcal{X}$ is generated on the support $\mathcal{C}$ where all combinations are possible and structural zeros rules are not enforced, but we only observe the sample of $n$ households $\mathcal{X}^1$ that satisfy the structural zero rules and do not observe the sample of $n_0$ households $\mathcal{X}^0 = \mathcal{X} - \mathcal{X}^1$ that fail the rules.
We use the strategy of @HuEtAl2018 and augment the data as follows. For each $h \in \mathcal{H}$, we simulate $\mathcal{X}$ from the untruncated NDPMPM, stopping when the number of simulated feasible households in $\mathcal{X}$ directly matches $n_{1h}$ for all $h \in \mathcal{H}$. We replace the simulated feasible households in $\mathcal{X}$ with $\mathcal{X}^1$, thus, assuming that $\mathcal{X}$ already contains $\mathcal{X}^1$ and we only need to generate the part $\mathcal{X}^0$ that fall in $\mathcal{S}$. Given a draw of $\mathcal{X}$, we draw $\theta$ from posterior distribution defined by the untruncated NDPMPM, treating $\mathcal{X}$ as the observed data. This posterior distribution can be estimated using a blocked Gibbs sampler [@IshwaranJames2001; @SiReiter2013].
We now present the full MCMC sampler for fitting the truncated NDPMPM. Let $\textbf{G}^0$ and $\textbf{M}^0$ be vectors of the latent class membership indicators for the households in $\mathcal{X}^0$ and $n_{0h}$ be the number of households of size $h$ in $\mathcal{X}^0$, with $n_0 = \sum_h n_{0h}$. In each full conditional, let “–” represent conditioning on all other variables and parameters in the model. At each MCMC iteration, we do the following steps.
1. Set $\mathcal{X}^0 = \textbf{G}^0 = \textbf{M}^0 = \emptyset$. For each $h \in \mathcal{H}$, repeat the following:
1. Set $t_0 = 0$ and $t_1 = 0$.
2. Sample $G_i^0 \in \{1, \ldots, F \} \sim \textrm{Discrete}(\pi_1^{\star\star}, \ldots, \pi_F^{\star\star})$ where $\pi_g^{\star\star} \ \propto \ \lambda^{(k)}_{gh} \pi_g $ and $k$ is the index for the household-level variable “household size”.
3. For $j = 1, \ldots, h$, sample $M^0_{ij} \in \{1, \ldots, S\} \sim \textrm{Discrete}(\omega_{G^0_i1}, \ldots, \omega_{G^0_iS})$.
4. Set $X^0_{ik} = h$, where $X^0_{ik}$ corresponds to the variable for household size. Sample the remaining household-level and individual-level values using the likelihoods in (\[ModelSpecification1\]) and (\[ModelSpecification2\]). Set the household’s simulated value to $\textbf{X}^0_i$.
5. If $\textbf{X}^0_i \in \mathcal{S}_h$, let $t_0 = t_0 + 1$, $\mathcal{X}^0 = \mathcal{X}^0 \cup \textbf{X}^0_i$, $\textbf{G}^0 = \textbf{G}^0 \cup G^0_i$ and $\textbf{M}^0 = \textbf{M}^0 \cup \{M_{i1}^0, \ldots, M_{ih}^0 \}$. Otherwise set $t_1 = t_1 + 1$. \[StepF\]
6. If $t_1 < n_{1h}$, return to step (b). Otherwise, set $n_{0h} = t_0$. \[StepG\]
2. For observations in $\mathcal{X}^1$,
1. Sample $G_i \in \{1, \ldots, F \} \sim \textrm{Discrete}(\pi_1^\star, \ldots, \pi_F^\star)$ for $i = 1, \ldots, n$, where $$\pi_g^\star = \Pr(G_i = g | - ) = \dfrac{\pi_g \left[\prod\limits^q_{k=p+1} \lambda^{(k)}_{gX^1_{ik}} \left(\prod\limits^{n_i}_{j=1} \sum\limits^S_{m=1}\omega_{gm}\prod\limits^p_{k=1} \phi^{(k)}_{gmX^1_{ijk}} \right) \right] }{\sum\limits^F_{f=1} \pi_f \left[\prod\limits^q_{k=p+1} \lambda^{(k)}_{fX^1_{ik}} \left(\prod\limits^{n_i}_{j=1} \sum\limits^S_{m=1}\omega_{gm}\prod\limits^p_{k=1} \phi^{(k)}_{fmX^1_{ijk}} \right) \right] }$$ for $g = 1, \ldots, F$. Set $G_i^1 = G_i$.
2. Sample $M_{ij} \in \{1, \ldots, S\} \sim \textrm{Discrete}(\omega_{G_i^11}^\star, \ldots, \omega_{G_i^1S}^\star)$ for $i = 1, \ldots, n$ and $j = 1, \ldots, n_i$, where $$\omega_{G_i^1m}^\star = \Pr(M_{ij} = m | - ) = \dfrac{\omega_{G_i^1m}\prod\limits^p_{k=1} \phi^{(k)}_{G_i^1mX^1_{ijk}} }{ \sum\limits^S_{s=1}\omega_{G_i^1s}\prod\limits^p_{k=1} \phi^{(k)}_{G^1_isX^1_{ijk}}}$$ for $m = 1, \ldots, S$. Set $M^1_{ij} = M_{ij}$
3. Set $u_F = 1$. Sample $$\begin{split}
u_g | - \ & \sim \textrm{Beta} \left(1 + U_g, \alpha + \sum^F_{f=g+1} U_f \right), \ \ \pi_g = u_g \prod_{f<g} (1 - u_f) \\
\textrm{where} \ \ U_g & = \sum^{n}_{i=1} \mathds{1}(G^1_i = g) + \sum\limits^{n_{0}}_{i=1} \mathds{1}(G_i^0 = g)
\end{split}$$ for $g = 1, \ldots, F-1$.
4. Set $v_{gM} = 1$ for $g = 1, \ldots, F$. Sample $$\begin{split}
v_{gm} | - \ & \sim \textrm{Beta} \left(1 + V_{gm}, \beta + \sum^S_{s=m+1} V_{gs} \right), \ \ \omega_{gm} = v_{gm} \prod_{s<m} (1 - v_{gs}) \\
\textrm{where} \ \ V_{gm} & = \sum^{n}_{i=1} \mathds{1}(M^1_{ij} = m, G^1_i = g) + \sum\limits^{n_{0}}_{i=1} \mathds{1}(M_{ij}^0 = m, G_i^0 = g)
\end{split}$$ for $m = 1, \ldots, S-1$ and $g = 1, \ldots, F$.
5. Sample $$\begin{split}
\lambda_g^{(k)} | - & \sim \textrm{Dirichlet}\left(1 + \eta^{(k)}_{g1}, \ldots, 1 + \eta^{(k)}_{gd_k} \right) \\
\textrm{where} \ \ \eta^{(k)}_{gc} & = \sum^{n}_{i|G_i^1 = g} \mathds{1}(X^1_{ik} = c) + \sum\limits^{n_{0}}_{i|G_i^0 = g} \mathds{1}(X_{ik}^0 = c)
\end{split}$$ for $g = 1, \ldots, F$ and $k = p+1, \ldots, q$.
6. Sample $$\begin{split}
\phi_{gm}^{(k)} | - & \sim \textrm{Dirichlet}\left(1 + \nu^{(k)}_{gm1}, \ldots, 1 + \nu^{(k)}_{gmd_k} \right) \\
\textrm{where} \ \ \nu^{(k)}_{gmc} & = \sum^{n}_{i,j |\substack{G_i^1 = g, \\ M_{ij}^1 = m}} \mathds{1}(X^1_{ijk} = c) + \sum\limits^{n_{0}}_{i,j |\substack{G_i^0 = g, \\ M_{ij}^0 = m}} \mathds{1}(X_{ijk}^0 = c)
\end{split}$$ for $g = 1, \ldots, F$, $m = 1, \ldots, S$ and $k = 1, \ldots, p$.
7. Sample $$\alpha | - \sim \textrm{Gamma}\left(a_\alpha + F - 1, b_\alpha - \sum^{F-1}_{g=1} \textrm{log}(1-u_g) \right).$$
8. Sample $$\beta | - \sim \textrm{Gamma}\left(a_\beta + F \times (S - 1), b_\beta - \sum^{S-1}_{m=1} \sum^{F}_{g=1} \textrm{log}(1-v_{gm}) \right).$$
This Gibbs sampler is implemented in the R software package “NestedCategBayesImpute” [@WangEtAl2016]. The software can be used to generate synthetic versions of the original data, but it requires all data to be complete.
Handling Missing Data Using the NDPMPM {#MissingData}
======================================
We modify the Gibbs sampler for the truncated NDPMPM to incorporate missing data. For $i = 1, \ldots, n$, let $\textbf{a}_i = (a_{i(p+1)}, \ldots, a_{i(p+q)})$ be a vector with $a_{ik} = 1$ when household-level variable $k \in \{p+1, \ldots, p+q\}$ in $\textbf{X}_i^1$ is missing, and $a_{ik} = 0$ otherwise. For $i = 1, \dots, n$ and $j = 1, \dots, n_i$, let $\textbf{b}_{ij} = (b_{ij1}, \ldots, b_{ijp})$ be a vector with $b_{ijk} = 1$ when individual-level variable $k \in \{1, \ldots, p\}$ for individual $j \in \{1, \ldots, n_i\}$ in $\textbf{X}_i^1$ is missing, and $b_{ijk} = 0$ otherwise. For each household $i$, let $\textbf{X}_i^1 = (\textbf{X}_i^{\textrm{obs}},\textbf{X}_i^{\textrm{mis}})$, where $\textbf{X}_i^{\textrm{obs}}$ comprise all data values corresponding to $a_{ik} = 0$ and $b_{ijk} = 0$, and $\textbf{X}_i^{\textrm{mis}}$ comprises all data values corresponding to $a_{ik} = 1$ and $b_{ijk} = 1$. We assume that the data are missing at random [@Rubin1976].
To incorporate missing values in the Gibbs sampler, we need to sample from the full conditional of each variable in $\textbf{X}_i^{\textrm{mis}}$, conditioned on the variables for which $a_{ik} = 0$ and $b_{ijk} = 0$, at every iteration. Thus, we add the ninth step,
1. For $i = 1, \ldots, n$, sample $\textbf{X}_i^{\textrm{mis}}$ from its full conditional distribution $$\Pr(\textbf{X}_i^{\textrm{mis}} | - ) \ \propto \ \mathds{1}\{\textbf{X}_i^1 \notin \mathcal{S}_h \} \ \left( \pi_{G^1_i} \prod\limits^{p+q}_{k | a_{ik} = 1} \lambda^{(k)}_{G^1_iX^1_{ik}} \prod\limits^{n_i}_{j=1} \omega_{G^1_iM^1_{ij}}\prod\limits^p_{k | b_{ijk} = 1} \phi^{(k)}_{G^1_iM^1_{ij}X^1_{ijk}} \right)$$
Sampling from this conditional distribution is nontrivial because of the dependence among variables induced by the structural zero rules in each $\mathcal{S}_h$. Because of the dependence, we cannot simply sample each variable independently using the likelihoods in (\[ModelSpecification1\]) and (\[ModelSpecification2\]). If we could generate the set of all possible completions for all households with missing entries, conditional on the observed values, then calculating the probability of each one and sampling from the set would be straightforward. Unfortunately, this approach is not practical when the size of each $\mathcal{S}_h$ is large. Even when the size of each $\mathcal{S}_h$ is modest, each household could have different sets of completions, necessitating significant computing, storage, and memory requirements.
However, the full conditional in S9 takes a similar form as the kernel of the truncated NDPMPM in (\[StrucZeroLikelihood\]), so that we can generate the desired samples through a second rejection sampling scheme. Essentially, we sample from an untruncated version of the full conditional $P^\star_{\textbf{X}_i^{\textrm{mis}}} = \pi_{G^1_i}\prod^{p+q}_{k | a_{ik} = 1} \lambda^{(k)}_{G^1_iX^1_{ik}} (\prod^{n_i}_{j=1} \omega_{G^1_iM^1_{ij}}\prod^p_{k | b_{ijk} = 1} \phi^{(k)}_{G^1_iM^1_{ij}X^1_{ijk}} ) $, until we obtain a valid sample that satisfies $\textbf{X}_i^1 \notin \mathcal{S}_h$; see the supplementary materials for a proof that this rejection sampling scheme results in a valid Gibbs sampler. Notice that since $P^\star_{\textbf{X}_i^{\textrm{mis}}}$ itself is untruncated, we can generate samples from it by sampling each variable independently using (\[ModelSpecification1\]) and (\[ModelSpecification2\]). We therefore replace step S9 with S9$^\prime$.
1. For $i = 1, \ldots, n$, sample $\textbf{X}_i^{\textrm{mis}}$ as follows.
1. For each missing household-level variable, that is, each variable where $k \in \{p+1, \ldots, p+q \}$ with $a_{ik} = 1$, sample $X_{ik}^{1}$ using (\[ModelSpecification1\]).
2. For each missing individual-level variable, that is, each variable where $j = 1, \ldots, n_i$ and $k \in \{1, \ldots, p\}$ with $b_{ijk} = 1$, sample $X_{ijk}^{1}$ using (\[ModelSpecification2\]).
3. Set the sampled household-level and individual-level values to $\textbf{X}_i^{\textrm{mis}\star}$.
4. Combine $\textbf{X}_i^{\textrm{mis}\star}$ with the observed $\textbf{X}_i^{\textrm{obs}}$, that is, set $\textbf{X}_i^{1\star} = (\textbf{X}_i^{\textrm{obs}},\textbf{X}_i^{\textrm{mis}\star})$. If $\textbf{X}_i^{1\star} \notin \mathcal{S}_h$, set $\textbf{X}_i^{\textrm{mis}} = \textbf{X}_i^{\textrm{mis}\star}$, otherwise, return to step (9$^\prime$a).
To initialize each $\textbf{X}_i^{\textrm{mis}}$, we suggest sampling from the empirical marginal distribution of each variable $k$ using the available cases for each variable, and requiring that the household satisfies $\textbf{X}_i^1 \notin \mathcal{S}_h$.
Strategies for Speeding Up the MCMC Sampler {#SpeedUp}
===========================================
The rejection sampling step in the Gibbs sampler in Section \[NDPMPMSampler\] can be inefficient when $\mathcal{S}$ is large [@Manrique-VallierReiter2014b; @HuEtAl2018], as the sampler tends to generate many impossible households before getting enough feasible ones. In addition, it takes computing time to check whether or not each sampled household satisfies all the structural zero rules. These computational costs are compounded when the sampler also incorporates missing values. In this section, we present two strategies that can reduce the number of impossible households that the algorithm generates, thereby speeding up the sampler. The supplementary material includes simulation studies showing that both strategies can speed up the MCMC significantly.
Moving the household head to the household level {#SpeedUpMoveHH}
------------------------------------------------
Many datasets include a variable recording the relationship of each individual to the household head. There can be only one household head in any household. This restriction can account for a large proportion of the combinations in $\mathcal{S}$. As a simple working example, consider a dataset that contains $n=1000$ households of size two, resulting in a total of $N=2000$ individuals. Suppose the data contain no household-level variables and two individual-level variables, age and relationship to household head. Also, suppose age has 100 levels while relationship to household head has 13 levels, which include household head, spouse of the household head, etc. Then, $\mathcal{C}$ contains $13^2 \times 100^2 = 1.69 \times 10^6$ combinations. Suppose the rule, “each household must contain exactly one head,” is the only structural zero rule defined on the dataset. Then, $\mathcal{S}$ contains $1.45 \times 10^6$ impossible combinations, approximately $86\%$ the size of $\mathcal{C}$. If, for example, the model assigns uniform probability to all combinations in $\mathcal{C}$, we would expect to sample about $(.86/.14) * 1000 \approx 6,143$ impossible households at every iteration to augment the $n$ feasible households.
Instead, we treat the variables for the household head as a household-level characteristic. This eliminates structural zero rules defined on the household head alone. Using the working example, moving the household head to the household level results in one new household-level variable, age of household head, which has 100 levels. The relationship to household head variable can be ignored for household heads. For others in the household, the relationship to household head variable now has 12 levels, with the level corresponding to “household head” removed. Thus, $\mathcal{C}$ contains $12 \times 100^2 = 1.20 \times 10^5$ combinations, and $\mathcal{S}$ contains zero impossible combinations. We wouldn’t even need to sample impossible households in the Gibbs sampler in Section \[NDPMPMSampler\].
In general, this strategy can reduce the size of $\mathcal{S}$ significantly, albeit usually not to zero as in the simple example here since $\mathcal{S}$ usually contains combinations resulting from other types of structural zero rules. This strategy is not a replacement for the rejection sampler in Section \[NDPMPMSampler\]; rather, it is a data reformatting technique that can be combined with the sampler.
Setting an upper bound on the number of impossible households to sample {#SpeedUpCapping}
-----------------------------------------------------------------------
To reduce computation time, we can put an upper bound on the number of sampled cases in $\mathcal{X}^0$. One way to achieve this is to replace $n_{1h}$ in step S1(f) of Section \[NDPMPMSampler\] with $\lceil n_{1h} \times \psi_h \rceil$, for some $\psi_h$ such that $1/\psi_h$ is a positive integer, so that we sample only approximately $\lceil n_{0h} \times \psi_h \rceil$ impossible households for each $h \in \mathcal{H}$. However, doing so underestimates the actual probability mass assigned to $\mathcal{S}$ by the model. We can illustrate this using the simple example of Section \[SpeedUpMoveHH\]. Suppose the model assigns uniform probability to all combinations in $\mathcal{C}$ as before. We set $\psi_2 = 0.5$, so that we sample approximately $3,072 = \lceil 6143 \times 0.5 \rceil$ impossible households in every iteration of the MCMC sampler. The probability of generating one impossible household is $3072 / (1000 + 3072) = 0.75$, a decrease from the actual value of 0.86. Therefore, we would underestimate the true contribution of $\{\mathcal{X}^0, \textbf{G}^0, \textbf{M}^0 \}$ to the likelihood.
To use the cap-and-weight approach, we need to apply a correction that re-weights the contribution of $\{\mathcal{X}^0, \textbf{G}^0, \textbf{M}^0 \}$ to the full joint likelihood. We do so using ideas akin to those used by @ChambersSkinner2003 [@Savitskytoth2016], approximating the likelihood of the full unobserved data with a “pseudo” likelihood using weights (the $1/\psi_h$’s). The impossible households only contribute to the full joint likelihood through the discrete distributions in (\[ModelSpecification1\]) to (\[ModelSpecification4\]). The sufficient statistics for estimating the parameters of the discrete distributions in (\[ModelSpecification1\]) to (\[ModelSpecification4\]) are the observed counts for the corresponding variables in the set $\{\mathcal{X}^1, \textbf{G}^1, \textbf{M}^1, \mathcal{X}^0, \textbf{G}^0, \textbf{M}^0 \}$, within each latent class for the household-level variables and within each latent class pair for the individual-level variables. Thus, for each $h \in \mathcal{H}$, we can re-weight the contribution of impossible households by multiplying the observed counts for households of size $h$ in $\{\mathcal{X}^0, \textbf{G}^0, \textbf{M}^0 \}$ by $1/\psi_h$ for the corresponding variable and latent classes. This raises the likelihood contribution of impossible households of size $h$ to the power of $1/\psi_h$. Clearly, $1/\psi_h$ need not be a positive integer. We require that only to make its multiplication with the observed counts free of decimals. We modify the Gibbs sampler to incorporate the cap-and-weight approach by replacing steps S1, S3, S4, S5 and S6; see the supplementary materials for the modified steps.
Setting each $\psi_h = 1$ corresponds to the original rejection sampler, so that the two approaches should provide very similar results when $\psi_h$ near $1$. Based on our experience, results of the cap-and-weight approach become significantly less accurate than the regular rejection sampler when $\psi_h < 1/4$. The time gained using this speedup approach in comparison to the regular sampler depends on the features of the data and the specified values for the weights $\{\psi_h: h \in \mathcal{H} \}$. To select the $\psi_h$’s, we suggest trying out different values—starting with values close to one—in initial runs of the MCMC sampler on a small random sample of the data. Analysts should examine the convergence and mixing behavior of the chains in comparison to the chain with all the $\psi_h$’s set to one, and select values that offer reasonable speedup while preserving convergence and mixing. This can be done quickly by comparing trace plots of a random set of parameters from the model that are not subject to label switching, such as $\alpha$ and $\beta$, or by examining marginal, bivariate and trivariate probabilities estimated from synthetic data generated from the MCMC.
Empirical Study {#simulations}
===============
To evaluate the performance of the NDPMPM as an imputation method, as well as the speed up strategies, we use data from the public use microdata files from the 2012 ACS, available for download from the United States Census Bureau (<http://www2.census.gov/acs2012_1yr/pums/>). We construct a population of 764,580 households of sizes $\mathcal{H} = \{2, 3, 4\}$, from which we sample $n=5,000$ households comprising $N=13,181$ individuals. We work with the variables described in Table \[MissingDataVariableDef\], which mimic those in the U. S. decennial census. The structural zeros involve ages and relationships of individuals in the same house; see the supplementary material for a full list of rules that we used. We move the household head to the household level as in Section \[SpeedUpMoveHH\] to take advantage of the computational gains.
[p[0.4]{}p[0.5]{}]{} Description of variable & Categories\
&\
\
Ownership of dwelling & 1 = owned or being bought, 2 = rented\
Household size & 2 = 2 people, 3 = 3 people, 4 = 4 people\
Gender of HH & 1 = male, 2 = female\
Race of HH & 1 = white, 2 = black,\
& 3 = American Indian or Alaska native,\
& 4 = Chinese, 5 = Japanese,\
& 6 = other Asian/Pacific islander, 7 = other race,\
& 8 = two major races,\
& 9 = three or more major races\
Hispanic origin of HH & 1 = not Hispanic, 2 = Mexican,\
& 3 = Puerto Rican, 4 = Cuban, 5 = other\
Age of HH & 1 = less than one year old, 2 = 1 year old,\
& 3 = 2 years old, …, 96 = 95 years old\
&\
\
Gender & same as “Gender of HH”\
Race & same as “Race of HH”\
Hispanic origin & same as “Hispanic origin of HH”\
Age & same as “Age of HH”\
Relationship to head of household & 1 = spouse, 2 = biological child,\
& 3 = adopted child, 4 = stepchild, 5 = sibling,\
& 6 = parent, 7 = grandchild, 8 = parent-in-law,\
& 9 = child-in-law, 10 = other relative,\
& 11 = boarder, roommate or partner,\
& 12 = other non-relative or foster child\
We introduce missing values using the following scenario. We let household size and age of household heads be fully observed. We randomly and independently blank 30% of each variable for the remaining household-level variables. For individuals other than the household head, we randomly and independently blank 30% of the values for gender, race and Hispanic origin. We make age missing with rates 50%, 20%, 40% and 30% for values of the relationship variable in the sets {2}, {3,4,5,10}, {7,9} and {6,8,11,12,13}, respectively. We make the relationship variable missing with rates 40%, 25%, 10%, and 55% for values of age in the sets {$x : x \leq 20$}, {$x : 20 < x \leq 50$}, {$x : 50 < x \leq 70$}, and {$x : x > 70$}, respectively. This results in approximately 30% missing values for both variables. About 8% of the individuals in the sample are missing both the age and relationship variable, and 2% are missing gender, age, and relationship jointly. This mechanism results in data that technically are not missing at random, but we use the NDPMPM approach regardless to examine its potential in a complicated missingness mechanism. Actual rates of item nonresponse in census data tend to be smaller than what we use here, but we use high rates to put the NDPMPM through a challenging stress test. We also introduce missing values using a missing completely at random scenario with rates in the 10% range across all the variables. In short, the results are similar to those here, though more accurate due to the lower rates of missingness. See the supplementary material for the results.
We estimate the NDPMPM using two approaches, both using the rejection step S9$^\prime$ in Section \[MissingData\]. The first approach considers $\psi_2 = \psi_3 = \psi_4 = 1$, i.e., without using the cap-and-weight approach, while the second approach considers $\psi_2 = \psi_3 = 1/2$ and $\psi_4 = 1/3$. For each approach, we run the MCMC sampler for 10,000 iterations, discarding the first 5,000 as burn-in and thinning the remaining samples every five iterations, resulting in 1,000 MCMC post burn-in iterates. We set $F = 30$ and $S = 15$ for each approach based on initial tuning runs. Across the approaches, the effective number of occupied household-level clusters usually ranges from 13 to 16 with a maximum of 25, while the effective number of occupied individual-level clusters across all household-level clusters ranges from 3 to 5 with a maximum of 10. For convergence, we examined trace plots of $\alpha$, $\beta$, and weighted averages of a random sample of the multinomial probabilities in (\[ModelSpecification1\]) and (\[ModelSpecification2\]) (since the multinomial probabilities themselves are prone to label switching).
For both methods, we generate $L = 50$ completed datasets, $\textbf{Z} = (\textbf{Z}^{(1)}, \ldots, \textbf{Z}^{(50)})$, using the posterior predictive distribution of the NDPMPM, from which we estimate all marginal distributions, bivariate distributions of all possible pairs of variables, and trivariate distributions of all possible triplets of variables. We also estimate several probabilities that depend on within household relationships and the household head to investigate the performance of the NDPMPM in estimating complex relationships. We obtain confidence intervals using multiple imputation inferences [@Rubin1987]. As a brief review, let $q$ be the completed-data point estimator of some estimand $Q$, and let $u$ be the estimator of variance associated with $q$. For $l=1, \dots, L$, let $q^{(l)}$ and $u^{(l)}$ be the values of $q$ and $u$ in completed dataset $\textbf{Z}^{(l)}$. We use $\bar{q}_L = \sum_{l=1}^L q^{(l)}/L$ as the point estimate of $Q$. We use $T_L = (1 + 1/L)b_L + \bar{u}_L$ as the estimated variance of $\bar{q}$, where $b_L = \sum_{l=1}^L (q^{(l)} - \bar{q}_L)^2/(L-1)$ and $\bar{u}_L = \sum_{l=1}^L u^{(l)}/L$. We make inference about $Q$ using $(\bar{q}_L - Q) \sim t_{v}(0, T_L)$, where $t_{v}$ is a $t$-distribution with $v = (L-1)(1 + \bar{u}_L/ [(1+1/L) b_L])^2$ degrees of freedom.
Figures \[AllProbs\] and \[AllProbs\_Weighted\] display the value of $\bar{q}_{50}$ for each estimated marginal, bivariate and trivariate probability plotted against its corresponding estimate from the original data, without missing values. Figure \[AllProbs\] shows the results for the NDPMPM with the rejection sampler, and Figure \[AllProbs\_Weighted\] shows the results for the NDPMPM using the cap-and-weight approach. For both approaches, the point estimates are close to those from the data before introducing missing values, suggesting that the NDPMPM does a good job of capturing important features of the joint distribution of the variables. Figure \[AllProbs\_Weighted\] in particular also shows that the cap-and-weight approach did not degrade the estimates.
[L[0.49]{}|R[0.03]{}R[0.115]{}R[0.115]{}R[0.115]{}]{} & $Q$ & No Missing & NDPMPM & NDPMPM Capped\
All same race household: & & & &\
$n_i = 2$ & .942 & (.932, .949) & (.891, .917) & (.884, .911)\
$n_i = 3$ & .908 & (.907, .937) & (.843, .890) & (.821, .870)\
$n_i = 4$ & .901 & (.879, .917) & (.793, .851) & (.766, .828)\
SP present & .696 & (.682, .707) & (.695, .722) & (.695, .722)\
Same race CP & .656 & (.641, .668) & (.640, .669) & (.634, .664)\
SP present, HH is White & .600 & (.589, .616) & (.603, .632) & (.604, .634)\
White CP & .580 & (.569, .596) & (.577, .606) & (.574, .604)\
CP with age difference less than five & .488 & (.465, .492) & (.341, .371) & (.324, .355)\
Male HH, home owner & .476 & (.456, .484) & (.450, .479) & (.451, .480)\
HH over 35, no CH present & .462 & (.441, .468) & (.442, .470) & (.443, .471)\
At least one biological CH present & .437 & (.431, .458) & (.430, .459) & (.428, .456)\
HH older than SP, White HH & .322 & (.309, .335) & (.307, .339) & (.311, .343)\
Adult female w/ at least one CH under 5 & .078 & (.070, .085) & (.062, .078) & (.061, .077)\
White HH with Hisp origin & .066 & (.064, .078) & (.062, .079) & (.062, .078)\
Non-White CP, home owner & .058 & (.050, .063) & (.038, .052) & (.037, .051)\
Two generations present, Black HH & .057 & (.053, .066) & (.052, .066) & (.052, .067)\
Black HH, home owner & .052 & (.046, .058) & (.044, .058) & (.044, .059)\
SP present, HH is Black & .039 & (.032, .042) & (.032, .044) & (.031, .043)\
White-nonwhite CP & .034 & (.029, .039) & (.038, .053) & (.043, .059)\
Hisp HH over 50, home owner & .029 & (.025, .034) & (.023, .034) & (.024, .034)\
One grandchild present & .028 & (.023, .033) & (.024, .035) & (.023, .035)\
Adult Black female w/ at least one CH under 18 & .027 & (.028, .038) & (.025, .036) & (.025, .036)\
At least two generations present, Hisp CP & .027 & (.022, .031) & (.022, .032) & (.023, .033)\
Hisp CP with at least one biological CH & .025 & (.020, .028) & (.019, .029) & (.020, .030)\
At least three generations present & .023 & (.020, .028) & (.017, .026) & (.017, .026)\
Only one parent & .020 & (.016, .024) & (.013, .021) & (.013, .021)\
At least one stepchild & .019 & (.018, .026) & (.019, .030) & (.019, .030)\
Adult Hisp male w/ at least one CH under 10 & .018 & (.017, .025) & (.014, .022) & (.014, .022)\
At least one adopted CH, White CP & .008 & (.005, .010) & (.004, .010) & (.004, .011)\
Black CP with at least two biological children & .006 & (.003, .007) & (.003, .007) & (.003, .007)\
Black HH under 40, home owner & .005 & (.005, .009) & (.006, .013) & (.007, .013)\
Three generations present, White CP & .005 & (.004, .008) & (.004, .010) & (.004, .009)\
White HH under 25, home owner & .003 & (.002, .005) & (.003, .007) & (.003, .007)\
Table \[CI:MissingData\] displays $95\%$ confidence intervals for several probabilities involving within-household relationships, as well as the value in the full population of 764,580 households. The intervals include the two based on the NDPMPM imputation engines and the interval from the data before introducing missingness. For the latter, we use the usual Wald interval, $\hat{p} \pm 1.96 \sqrt{\hat{p}(1-\hat{p})/n}$, where $\hat{p}$ is the corresponding sample percentage. For the most part, the intervals from the NDPMPM with the full rejection sampling are close to those based on the data without any missingness. They tend to include the true population quantity. The NDPMPM imputation engine results in noticeable downward bias for the percentages of households where everyone is the same race, with bias increasing as the household size gets bigger. This is a challenging estimand to estimate accurately via imputation, particularly for larger households. @HuEtAl2018 identified biases in the same direction when using the NDPMPM (with household head data treated as individual-level variables) to generate fully synthetic data, noting that the bias gets smaller as the sample size increases. The NDPMPM fits the joint distribution of the data better and better as the sample size grows. Hence, we expect the NDPMPM imputation engine to be more accurate with larger sample sizes, as well as with smaller fractions of missing values.
The interval estimates from the cap-and-weight method are generally similar to those for the full rejection sampler, with some degradation particularly for the percentages of same race households by household size. This degradation comes with a benefit, however. Based on MCMC runs on a standard laptop, the NDPMPM using the cap-and-weight approach and moving household heads’ data values to the household level is about $42\%$ faster than the NDPMPM with household heads’ data values moved to the household level.
Discussion {#Discussion}
==========
The empirical study suggests that the NDPMPM can provide high quality imputations for categorical data nested within households. To our knowledge, this is the first parametric imputation engine for nested multivariate categorical data. The study also illustrates that, with modest sample sizes, agencies should not expect the NDPMPM to preserve all features of the joint distribution. Of course, this is the case with any imputation engine. For the NDPMPM, agencies may be able to improve accuracy for targeted quantities by recoding the data used to fit the model. For example, one can create a new household-level variable that equals one when everyone has the same race and equals zero otherwise, and replace the individual race variable with a new variable that has levels “1 = race is the same as race of household head,” “2 = race is white and differs from race of household head,” “3 = race is black and differs from race of household head,” and so on. The NDPMPM would be estimated with the household-level same race variable and the new individual-level race variable. This would encourage the NDPMPM to estimate the percentages with the same race very accurately, as it would be just another household-level variable like home ownership. It also would add structural zeros involving race to the computation. Evaluating the trade offs in accuracy and computational costs of such recodings is a topic for future research.
The NDPMPM can be computationally expensive, even with the speed-ups presented in this article. The expensive parts of the algorithm are the rejection sampling steps. Fortunately, these can be done easily by parallel processing. For example, we can require each processor to generate a fraction of the impossible cases in Section \[NDPMPMSampler\]. We also can spread the rejection steps for the imputations over many processors. These steps should cut run time by a factor roughly equal to the number of processors available.
The empirical study used households up to size four. We have run the model on data with households up to size seven in reasonable time (a few hours on a standard laptop). Accuracy results are similar qualitatively. As the household sizes get large, the model can generate hundreds or even thousands times as many impossible households as there are feasible ones, slowing the algorithm. In such cases, the cap-and-weight approach is essential for practical applications.
Acknowledgments
===============
This research was supported by grants from the National Science Foundation (NSF SES 1131897) and the Alfred P. Sloan Foundation (G-2-15-20166003).
Supplementary Materials
=======================
This is a supplementary material to the paper. It contains proof that the rejection sampling step S9$^\prime$ in Section \[MissingData\] generates samples from the correct posterior distribution. It also contains the modified Gibbs sampler for the cap-and-weight approach and a list of the structural zero rules used in fitting the NDPMPM model. Finally, we include empirical results for the speedup approaches mentioned in the paper, using synthetic data, and additional results for handling missing data using the NDPMPM under a missing completely at random scenario.
Proof that the rejection sampling step S9$^\prime$ in Section \[MissingData\] generates samples from the correct posterior distribution
---------------------------------------------------------------------------------------------------------------------------------------
The $X^1_{ik}$ and $X^1_{ijk}$ values generated using the rejection sampler in Step S9$^\prime$ are generated from the full conditionals, resulting in a valid Gibbs sampler. The proof follows from the properties of rejection sampling (or simple accept reject). The target distribution is the full conditional for $\textbf{X}_i^{\textrm{mis}}$. It can be re-expressed as $$p(\textbf{X}_i^{\textrm{mis}}) \ = \dfrac{\mathds{1}\{\textbf{X}_i^1 \notin \mathcal{S}_h \}}{\Pr(\textbf{X}_i \notin \mathcal{S}_h | \theta)} g(\textbf{X}_i^{\textrm{mis}})$$ where $$g(\textbf{X}_i^{\textrm{mis}}) = \pi_{G^1_i}\prod^{p+q}_{k | a_{ik} = 1} \lambda^{(k)}_{G^1_iX^1_{ik}} \left(\prod^{n_i}_{j=1} \omega_{G^1_iM^1_{ij}}\prod^p_{k | b_{ijk} = 1} \phi^{(k)}_{G^1_iM^1_{ij}X^1_{ijk}} \right).$$ Our rejection scheme uses $g(\textbf{X}_i^{\textrm{mis}})$ as a proposal for $p(\textbf{X}_i^{\textrm{mis}})$. To show that the draws are indeed from $p(\textbf{X}_i^{\textrm{mis}})$, we need to verify that $w(\textbf{X}_i^{\textrm{mis}}) = p(\textbf{X}_i^{\textrm{mis}})/g(\textbf{X}_i^{\textrm{mis}}) < M$, where $1 < M < \infty$, and that we are accepting each sample with probability $w(\textbf{X}_i^{\textrm{mis}})/M$. In our case,
1. $w(\textbf{X}_i^{\textrm{mis}}) = p(\textbf{X}_i^{\textrm{mis}})/g(\textbf{X}_i^{\textrm{mis}}) = \mathds{1}\{\textbf{X}_i^1 \notin \mathcal{S}_h \}/\Pr(\textbf{X}_i \notin \mathcal{S}_h | \theta) \leq 1/\Pr(\textbf{X}_i \notin \mathcal{S}_h | \theta)$, and $0 < \Pr(\textbf{X}_i \notin \mathcal{S}_h | \theta) < 1 \ \Rightarrow \ 1 < 1/\Pr(\textbf{X}_i \notin \mathcal{S}_h | \theta) < \infty$ necessarily.
2. By sampling until we obtain a valid sample that satisfies $\textbf{X}_i^1 \notin \mathcal{S}_h$, we are indeed sampling with probability $w(\textbf{X}_i^{\textrm{mis}})/M = \mathds{1}\{\textbf{X}_i^1 \notin \mathcal{S}_h \}$.
Modified Gibbs sampler for the cap-and-weight approach
------------------------------------------------------
The modified Gibbs sampler for the cap-and-weight approach replaces steps S1, S3, S4, S5 and S6 of the Gibbs sampler in the main text as follows.
1. For each $h \in \mathcal{H}$, repeat steps S1(a) to S1(e) as before but modify step S1(f) to: if $t_1 < \lceil n_{1h} \times \psi_h \rceil$, return to step (b). Otherwise, set $n_{0h} = t_0$.
2. Set $u_F = 1$. Sample $$\begin{split}
u_g | - \ & \sim \textrm{Beta} \left(1 + U_g, \alpha + \sum^F_{f=g+1} U_f \right), \ \ \pi_g = u_g \prod_{f<g} (1 - u_f) \\
\textrm{where} \ \ U_g & = \sum^{n}_{i=1} \mathds{1}(G^1_i = g) + \sum_{h \in \mathcal{H}} \dfrac{1}{\psi_h} \sum\limits_{i | n^0_i = h} \mathds{1}(G_i^0 = g)
\end{split}$$ for $g = 1, \ldots, F-1$.
3. Set $v_{gM} = 1$ for for $g = 1, \ldots, F$. Sample $$\begin{split}
v_{gm} | - \ & \sim \textrm{Beta} \left(1 + V_{gm}, \beta + \sum^S_{s=m+1} V_{gs} \right), \ \ \omega_{gm} = v_{gm} \prod_{s<m} (1 - v_{gs}) \\
\textrm{where} \ \ V_{gm} & = \sum^{n}_{i=1} \mathds{1}(M^1_{ij} = m, G^1_i = g) + \sum_{h \in \mathcal{H}} \dfrac{1}{\psi_h} \sum\limits_{i | n_i^0 = h} \mathds{1}(M_{ij}^0 = m, G_i^0 = g)
\end{split}$$ for $m = 1, \ldots, S-1$ and $g = 1, \ldots, F$.
4. Sample $$\begin{split}
\lambda_g^{(k)} | - & \sim \textrm{Dirichlet}\left(1 + \eta^{(k)}_{g1}, \ldots, 1 + \eta^{(k)}_{gd_k} \right) \\
\textrm{where} \ \ \eta^{(k)}_{gc} & = \sum^{n}_{i|G^1_i = g} \mathds{1}(X^1_{ik} = c) + \sum_{h \in \mathcal{H}} \dfrac{1}{\psi_h} \sum\limits_{i \big| \substack{n_i^0 = h, \\ G_i^0 = g}} \mathds{1}(X_{ik}^0 = c)
\end{split}$$ for $g = 1, \ldots, F$ and $k = p+1, \ldots, q$.
5. Sample $$\begin{split}
\phi_{gm}^{(k)} | - & \sim \textrm{Dirichlet}\left(1 + \nu^{(k)}_{gm1}, \ldots, 1 + \nu^{(k)}_{gmd_k} \right) \\
\textrm{where} \ \ \nu^{(k)}_{gmc} & = \sum^{n}_{i \big| \substack{G^1_i = g, \\ M^1_{ij} = m}} \mathds{1}(X_{ijk}^1 = c) + \sum_{h \in \mathcal{H}} \dfrac{1}{\psi_h} \sum\limits_{i \big| \substack{n_i^0 = h, \\ G_i^0 = g, \\ M^0_{ij} = m}} \mathds{1}(X_{ijk}^0 = c)
\end{split}$$ for $g = 1, \ldots, F$, $m = 1, \ldots, S$ and $k = 1, \ldots, p$.
List of structural zeros
------------------------
We fit the NDPMPM model using structural zeros which involve ages and relationships of individuals in the same house. The full list of the rules used is presented in Table \[SynStructuralZeros\]. These rules were derived from the 2012 ACS by identifying combinations involving the relationship variable that do not appear in the constructed population. This list should not be interpreted as a “true” list of impossible combinations in census data.
Description
----- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1. Each household must contain exactly one head and he/she must be at least 16 years old.
2. Each household cannot contain more than one spouse and he/she must be at least 16 years old.
3. Married couples are of opposite sex, and age difference between individuals in the couples cannot exceed 49.
4. The youngest parent must be older than the household head by at least 4.
5. The youngest parent-in-law must be older than the household head by at least 4.
6. The age difference between the household head and siblings cannot exceed 37.
7. The household head must be at least 31 years old to be a grandparent and his/her spouse must be at least 17. Also, He/she must be older than the oldest grandchild by at least 26.
8. The household head must be older than the oldest child by at least 7.
9. The household head must be older than the oldest biological child by at least 7.
10. The household head must be older than the oldest adopted child by at least 11.
11. The household head must be older than the oldest stepchild by at least 9.
: List of structural zeros.[]{data-label="SynStructuralZeros"}
Empirical study of the speedup approaches {#SpeedUpResults}
-----------------------------------------
We evaluate the performance of the two speedup approaches mentioned in the main text using synthetic data. We use data from the public use microdata files from the 2012 ACS, available for download from the United States Census Bureau (<http://www2.census.gov/acs2012_1yr/pums/>) to construct a population of $857,018$ households of sizes $\mathcal{H} = \{2, 3, 4, 5, 6\}$, from which we sample $n=10,000$ households comprising $N=29,117$ individuals. We work with the variables described in Table \[NDPMPMVariableDef\]. We evaluate the approaches using probabilities that depend on within household relationships and the household head.
[p[0.4]{}p[0.5]{}]{} Description of variable & Categories\
&\
\
Ownership of dwelling & 1 = owned or being bought, 2 = rented\
Household size & 2 = 2 people, 3 = 3 people, 4 = 4 people,\
& 5 = 5 people, 6 = 6 people\
&\
\
Gender & 1 = male, 2 = female\
Race & 1 = white, 2 = black,\
& 3 = American Indian or Alaska native,\
& 4 = Chinese, 5 = Japanese,\
& 6 = other Asian/Pacific islander, 7 = other race,\
& 8 = two major races,\
& 9 = three or more major races\
Hispanic origin & 1 = not Hispanic, 2 = Mexican,\
& 3 = Puerto Rican, 4 = Cuban, 5 = other\
Age & 1 = less than one year old, 2 = 1 year old,\
& 3 = 2 years old, …, 96 = 95 years old\
Relationship to head of household & 1 = household head, 2 = spouse, 3 = child,\
& 4 = child-in-law, 5 = parent, 6 = parent-in-law,\
& 7 = sibling, 8 = sibling-in-law, 9 = grandchild,\
& 10 = other relative, 11 = partner/friend/visitor,\
& 12 = other non-relative\
We consider the NDPMPM using two approaches, both moving the values of the household head to the household level as in Section \[SpeedUpMoveHH\] of the main text and also using the cap-and-weight approach in Section \[SpeedUpCapping\] of the main text. The first approach considers $\psi_2 = \psi_3 = \psi_4 = \psi_5 = \psi_6 = 1$ while the second approach considers $\psi_2 = \psi_3 = 1/2$ and $\psi_4 = \psi_5 = \psi_6 = 1/3$. We compare these approaches to the NDPMPM as presented in [@HuEtAl2018]. For each approach, we create $L = 50$ synthetic datasets, $\textbf{Z} = (\textbf{Z}^{(1)}, \ldots, \textbf{Z}^{(50)})$. We generate the synthetic datasets so that the number of households of size $h \in \mathcal{H}$ in each $\textbf{Z}^{(l)}$ exactly matches $n_h$ from the observed data. Thus, $\textbf{Z}$ comprises partially synthetic data [@Little1993; @Reiter2003], even though every released $Z_{ijk}$ is a simulated value. We combine the estimates using using the approach in @Reiter2003. As a brief review, let $q$ be the point estimator of some estimand $Q$, and let $u$ be the estimator of variance associated with $q$. For $l=1, \dots, L$, let $q_l$ and $u_l$ be the values of $q$ and $u$ in synthetic dataset $\mathbf{Z}^{(l)}$. We use $\bar{q} = \sum_{l=1}^L q_l/L$ as the point estimate of $Q$ and $T = \bar{u} + b/L$ as the estimated variance of $\bar{q}$, where $b = \sum_{l=1}^L (q_l - \bar{q})^2/(L-1)$ and $\bar{u} = \sum_{l=1}^L u_l/L$. We make inference about $Q$ using $(\bar{q} - Q) \sim t_{v}(0, T)$, where $t_{v}$ is a $t$-distribution with $v = (L-1)(1 + L\bar{u}/b])^2$ degrees of freedom.
[L[0.3]{}|R[0.14]{}R[0.14]{}R[0.14]{}R[0.14]{}]{} & Original & NDPMPM & NDPMPM w/ HH moved & NDPMPM capped w/ HH moved\
All same race & & & &\
$n_i = 2$ & (.939, .951) & (.918, .932) & (.912, .928) & (.910, .925)\
$n_i = 3$ & (.896, .920) & (.859, .888) & (.845, .875) & (.844, .874)\
$n_i = 4$ & (.885, .912) & (.826, .860) & (.813, .848) & (.817, .852)\
$n_i = 5$ & (.879, .922) & (.786, .841) & (.786, .841) & (.777, .834)\
$n_i = 6$ & (.831, .910) & (.701, .803) & (.718, .819) & (.660, .768)\
SP present & (.693, .711) & (.678, .697) & (.676, .695) & (.677, .695)\
SP with white HH & (.589, .608) & (.577, .597) & (.576, .595) & (.575, .595)\
SP with black HH & (.036, .043) & (.035, .043) & (.034, .042) & (.034, .042)\
White couple & (.570, .589) & (.560, .580) & (.553, .573) & (.552, .572)\
White couple, own & (.495, .514) & (.468, .488) & (.461, .481) & (.463, .483)\
Same race couple & (.655, .673) & (.636, .655) & (.626, .645) & (.625, .644)\
White-nonwhite couple & (.028, .035) & (.028, .035) & (.034, .041) & (.036, .044)\
Nonwhite couple, own & (.057, .067) & (.047, .056) & (.045, .053) & (.045, .054)\
Only mother present & (.017, .022) & (.014, .019) & (.014, .019) & (.013, .018)\
Only one parent present & (.021, .026) & (.026, .032) & (.026, .033) & (.027, .033)\
Children present & (.507, .527) & (.493, .512) & (.517, .537) & (.511, .531)\
Siblings present & (.022, .028) & (.027, .034) & (.027, .033) & (.027, .033)\
Grandchild present & (.041, .049) & (.051, .060) & (.049, .058) & (.050, .059)\
Three generations present & (.036, .044) & (.037, .045) & (.042, .050) & (.040, .048)\
White HH, older than SP & (.309, .327) & (.283, .301) & (.294, .313) & (.302, .321)\
Nonhisp HH & (.882, .894) & (.875, .888) & (.879, .891) & (.876, .889)\
White, Hisp HH & (.071, .082) & (.074, .085) & (.072, .082) & (.073, .084)\
Same age couple & (.087, .098) & (.027, .034) & (.023, .029) & (.024, .031)\
For each approach, we run the MCMC sampler for 20,000 iterations, discarding the first 10,000 as burn-in and thinning the remaining samples every five iterations, resulting in 2,000 MCMC post burn-in iterates. We create the $L = 50$ synthetic datasets by randomly sampling from the 2,000 iterates. We set $F = 40$ and $S = 15$ for each approach based on initial tuning runs. For convergence, we examined trace plots of $\alpha$, $\beta$ and weighted averages of a random sample of the multinomial probabilities in the NDPMPM likelihood. Across the approaches, the effective number of occupied household-level clusters usually ranges from 20 to 33 with a maximum of 38, while the effective number of occupied individual-level clusters across all household-level clusters ranges from 5 to 9 with a maximum of 12.
Based on MCMC runs on a standard laptop, moving household heads’ data values to the household level alone results in a speedup of about $63\%$ on the default rejection sampler while the cap-and-weight approach alone results in a speedup of about $40\%$.
Table \[CI:SyntheticData\] shows the $95\%$ confidence intervals for each approach. Essentially, all three approaches result in similar confidence intervals, suggesting not much loss in accuracy from the speedups. Most intervals also are reasonably similar to confidence intervals based on the original data, except for the percentage of same age couples. The last row is a rigorous test of how well each method can estimate a probability that can be fairly difficult to estimate accurately. In this case, the probability that a household head and spouse are the same age can be difficult to estimate since each individual’s age can take 96 different values. All three approaches are thus off from the estimate from the original data in this case. These results suggest that we can significantly speedup the sampler with minimal loss in accuracy of estimates and confidence intervals of population estimands.
Empirical study of missing data imputation under MCAR {#MissingDataResults}
-----------------------------------------------------
We also evaluate the performance of the NDPMPM as an imputation method under a missing completely at random (MCAR) scenario. We use the same data as in Section 5 of the main text. As a reminder, the data contains $n=5,000$ households of sizes $\mathcal{H} = \{2, 3, 4\}$, comprising $N=13,181$ individuals. We introduce missing values using a MCAR scenario. We randomly select 80% households to be complete cases for all variables. For the remaining 20%, we let the variable “household size” be fully observed and randomly – and independently – blank 50% of each variable for the remaining household-level and individual-level variables. We use these low rates to mimic the actual rates of item nonresponse in census data.
[L[0.49]{}|R[0.03]{}R[0.115]{}R[0.115]{}R[0.115]{}]{} & $Q$ & No Missing & NDPMPM & NDPMPM Capped\
All same race household: & & & &\
$n_i = 2$ & .942 & (.932, .949) & (.924, .944) & (.925, .946)\
$n_i = 3$ & .908 & (.907, .937) & (.887, .924) & (.890, .925)\
$n_i = 4$ & .901 & (.879, .917) & (.854, .900) & (.855, .900)\
SP present & .696 & (.682, .707) & (.683, .709) & (.683, .709)\
Same race CP & .656 & (.641, .668) & (.637, .664) & (.638, .665)\
SP present, HH is White & .600 & (.589, .616) & (.590, .618) & (.590, .618)\
White CP & .580 & (.569, .596) & (.568, .596) & (.568, .597)\
CP with age difference less than five & .488 & (.465, .492) & (.422, .451) & (.422, .450)\
Male HH, home owner & .476 & (.456, .484) & (.455, .483) & (.456, .485)\
HH over 35, no CH present & .462 & (.441, .468) & (.438, .466) & (.438, .466)\
At least one biological CH present & .437 & (.431, .458) & (.432, .460) & (.432, .460)\
HH older than SP, White HH & .322 & (.309, .335) & (.308, .335) & (.306, .333)\
Adult female w/ at least one CH under 5 & .078 & (.070, .085) & (.068, .084) & (.067, .083)\
White HH with Hisp origin & .066 & (.064, .078) & (.064, .079) & (.064, .079)\
Non-White CP, home owner & .058 & (.050, .063) & (.048, .061) & (.048, .061)\
Two generations present, Black HH & .057 & (.053, .066) & (.053, .066) & (.053, .067)\
Black HH, home owner & .052 & (.046, .058) & (.046, .059) & (.046, .059)\
SP present, HH is Black & .039 & (.032, .042) & (.032, .043) & (.032, .042)\
White-nonwhite CP & .034 & (.029, .039) & (.032, .044) & (.032, .044)\
Hisp HH over 50, home owner & .029 & (.025, .034) & (.025, .035) & (.025, .035)\
One grandchild present & .028 & (.023, .033) & (.024, .034) & (.024, .034)\
Adult Black female w/ at least one CH under 18 & .027 & (.028, .038) & (.027, .037) & (.027, .037)\
At least two generations present, Hisp CP & .027 & (.022, .031) & (.022, .031) & (.022, .031)\
Hisp CP with at least one biological CH & .025 & (.020, .028) & (.019, .028) & (.019, .028)\
At least three generations present & .023 & (.020, .028) & (.019, .028) & (.019, .028)\
Only one parent & .020 & (.016, .024) & (.016, .024) & (.016, .024)\
At least one stepchild & .019 & (.018, .026) & (.018, .027) & (.018, .027)\
Adult Hisp male w/ at least one CH under 10 & .018 & (.017, .025) & (.016, .025) & (.016, .025)\
At least one adopted CH, White CP & .008 & (.005, .010) & (.005, .010) & (.005, .010)\
Black CP with at least two biological children & .006 & (.003, .007) & (.003, .007) & (.003, .007)\
Black HH under 40, home owner & .005 & (.005, .009) & (.005, .010) & (.005, .011)\
Three generations present, White CP & .005 & (.004, .008) & (.004, .010) & (.004, .009)\
White HH under 25, home owner & .003 & (.002, .005) & (.004, .009) & (.004, .009)\
Similar to the main text, we estimate the NDPMPM using two approaches, both combining the rejection step in Section 4.1 of the main text with the cap-and-weight approach in Section 4.2 of the main text. The first approach considers $\psi_2 = \psi_3 = \psi_4 = 1$ while the second approach considers $\psi_2 = \psi_3 = 1/2$ and $\psi_4 = 1/3$. For each approach, we run the MCMC sampler for 10,000 iterations, discarding the first 5,000 as burn-in and thinning the remaining samples every five iterations, resulting in 1,000 MCMC post burn-in iterates. We set $F = 30$ and $S = 15$ for each approach based on initial tuning runs. We monitor convergence as in the main text. For both methods, we generate $L = 50$ completed datasets, $\textbf{Z} = (\textbf{Z}^{(1)}, \ldots, \textbf{Z}^{(50)})$, using the posterior predictive distribution of the NDPMPM, from which we estimate the same probabilities as in the main text.
Figures \[AllProbsII\] and \[AllProbs\_WeightedII\] display each estimated marginal, bivariate and trivariate probability $\bar{q}_{50}$ plotted against its corresponding estimate from the original data, without missing values. Figure \[AllProbsII\] shows the results for the NDPMPM with the rejection sampler, and Figure \[AllProbs\_WeightedII\] shows the results for the NDPMPM using the cap-and-weight approach. For both approaches, the NDPMPM does a good job of capturing important features of the joint distribution of the variables as the point estimates are very close to those from the data before introducing missing values. In short, the results are very similar to those in the main text, though more accurate.
Table \[CI:MissingDataII\] displays $95\%$ confidence intervals for selected probabilities involving within-household relationships, as well as the value in the full population of 764,580 households. The intervals include the two based on the NDPMPM imputation engines and the interval from the data before introducing missingness. The intervals are generally more accurate than those presented in the main text. This is expected since we use lower rates of missingness in the MCAR scenario. For the most part, the intervals from the NDPMPM with the two approaches tend to include the true population quantity. Again, the NDPMPM imputation engine results in downward bias for the percentages of households where everyone is the same race. As mentioned in the main text, this is a challenging estimand to estimate accurately via imputation, particularly for larger households.
[^1]: Olanrewaju M. Akande is PhD Candidate, Department of Statistical Science, Duke University, Durham, NC 27708 (E-mail: <[email protected]>); Jerome P. Reiter is Professor of Statistical Science, Duke University, Durham, NC 27708 (E-mail: <[email protected]>); and Andrés F. Barrientos is Postdoctoral Associate, Department of Statistical Science, Duke University, Durham, NC 27708 (E-mail: <[email protected]>).
|
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"pile_set_name": "ArXiv"
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Introduction
============
This part is a continuation of the Part I [@anso11] where resolutions of identity for certain non-Hermitian Hamiltonians were constructed of biorthogonal sets of their eigen- and associated functions. The spectral problem was defined on entire axis. Non-Hermitian Hamiltonians were taken with continuous spectrum and they were endowed with an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum or an exceptional point situated inside of continuous spectrum. In the present work (Part II) the detailed rigorous proofs are given for the resolutions of identity in both cases. Moreover the reductions of the derived resolutions of identity under narrowing of the classes of employed test functions in the Gel’fand triple [@gelfand] are built. In Section \[section2\] the definitions of the employed spaces of test functions and distributions are given. In Section \[section3\] the proofs of the initial resolution of identity and of its reduced forms for restricted spaces of test functions are elaborated for an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum. In Section \[section4\] the analogous proofs of resolutions of identity are presented for an exceptional point situated inside of continuous spectrum.
Definition of spaces of test functions and distributions {#section2}
========================================================
In this paper we shall use the following spaces of test functions and distributions.
Let $CL_\gamma=C^\infty_{\Bbb R}\cap L^2(\Bbb R;(1+|x|)^\gamma)$, $\gamma\in\Bbb R$, be the space of test functions. The sequence $\varphi_n(x)\in CL_\gamma$, $n=1, 2, 3, \dots$ is called convergent in $CL_\gamma$ to $\varphi(x)\in CL_\gamma$, $$\mathop{{\lim}_\gamma}_{n\to+\infty}\varphi_n(x)=\varphi(x)$$ if $$\lim_{n\to+\infty}\int_{-\infty}^{+\infty}|\varphi_n(x)-\varphi(x)|^2
(1+|x|)^\gamma dx=0,$$ and for any $x_1, x_2\in \Bbb R$, $x_1<x_2$ and any $l=0, 1, 2, \dots$, $$\lim_{n\to+\infty}\max_{[x_1,x_2]}\big|\varphi^{(l)}_n(x)-\varphi^{(l)}(x)\big|=0.$$
We shall denote the value of a functional $f$ on $\varphi\in
CL_\gamma$ conventionally as $(f,\varphi)$. A linear functional $f$ is called continuous if for any sequence $\varphi_n\in CL_\gamma$, $n=1, 2, 3, \dots$ convergent in $CL_\gamma$ to zero the equality $$\lim_{n\to+\infty}(f,\varphi_n)=0$$ is valid. The space of distributions over $CL_\gamma$, i.e. of linear continuous functionals over $CL_\gamma$, is denoted $CL'_\gamma$. The sequence $f_n\in CL'_\gamma$, $n=1, 2, 3, \dots$ is called convergent in $CL'_\gamma$ to $f\in CL'_\gamma$, $$\mathop{\lim\nolimits'_\gamma}_{n\to+\infty}f_n=f,$$ if for any $\varphi \in CL_\gamma$ the relation takes place, $$\lim_{n\to+\infty}(f_n,\varphi)=(f,\varphi).$$
A functional $f\in CL'_\gamma$ is called regular if there is $f(x)\in L^2(\Bbb R;(1+|x|)^{-\gamma})$ such that for any $\varphi\in CL_\gamma$ the equality $$(f,\varphi)=\int_{-\infty}^{+\infty}f(x)\varphi(x)\,dx$$ holds. In this case we shall identify the distribution $f\in
CL'_\gamma$ with the function $f(x)\in L^2(\Bbb
R;(1+|x|)^{-\gamma})$. In virtue of the Bunyakovskii inequality, $$\left|\int_{-\infty}^{+\infty}f(x)\varphi(x)\,dx\right|^2\leqslant
\int_{-\infty}^{+\infty}{{|f^2(x)|\,dx}\over{(1+|x|)^\gamma}}
\int_{-\infty}^{+\infty}\big|\varphi^2(x)\big|(1+|x|)^\gamma\,dx,$$ it is evident that $L_2(\Bbb R;(1+|x|)^{-\gamma})\subset CL'_\gamma$ and this inclusion is continuous.
For any $\gamma_1<\gamma_2$ there is a continuous inclusion $CL_{\gamma_2}\subset CL_{\gamma_1}$. Let us also notice that the Dirac delta function $\delta(x-x')$ belongs to $CL'_\gamma$ for any $\gamma\in\Bbb R$.
Proofs of resolutions of identity for the model Hamiltonians\
with exceptional point of arbitrary multiplicity\
at the bottom of continuous spectrum {#section3}
=============================================================
Proof of the biorthogonality relations between eigenfunctions\
for continuous spectrum {#section3.2}
--------------------------------------------------------------
Let us start proofs by proving of the biorthogonality relation $$\begin{gathered}
\int_{-\infty}^{+\infty}\big[k^n\psi_n(x;k)\big]
\big[(k')^n\psi_n(x;-k')\big]\,dx= (k')^{2n}\delta(k-k')\label{ort4}\end{gathered}$$ between eigenfunctions $\psi_n(x;k)$ for the continuous spectrum of the Hamiltonian $h_n$, $n=0, 1,$ $2,\dots$ (see (2.17) of Part I). Proof of this biorthogonality relation (\[ort4\]) is based on the following Lemmas \[lemma3.1\]–\[lemma3.3\].
\[lemma3.1\] Suppose that the functions $\psi_n(x;k)$, $n=0, 1,2, \dots$ are defined by the formula $(2.6)$ of Part [I]{} for any $x\in\Bbb R$, $k\in\Bbb C$, $k\ne0$ and fixed $z\in\Bbb C$, ${\rm{Im}}\,z\ne0$. Then for any $n=1, 2, 3,\dots$, $R>0$, $k\in\Bbb C$ and $k'\in\Bbb
C$ the following relation holds, $$\begin{gathered}
\int_{-R}^R\big[k^n\psi_n(x;k)\big]\big[(k')^n\psi_n(x;-k')\big]\,dx=
(k')^{2n} {{\sin R(k-k')}\over{\pi(k-k')}}\nonumber\\
\qquad{}-i\sum_{l=0}^{n-1}(k')^{2l}
\big[k^{n-1-l}\psi_{n-1-l}(x;k)\big]\big[(k')^{n-l}\psi_{n-l}(x;-k')\big]
\Big|_{-R}^R.\label{chas'}
\end{gathered}$$
Let us check first that $$\begin{gathered}
\int_{-R}^R\big[k^n\psi_n(x;k)][(k')^n\psi_n(x;-k')\big]\,dx=
-i\big[k^{n-1}\psi_{n-1}(x;k)\big]\big[(k')^n\psi_n(x;-k')\big]
\Big|_{-R}^R\nonumber\\
\qquad{} +(k')^2\int_{-R}^R\big[k^{n-1}\psi_{n-1}(x;k)\big]
\big[(k')^{n-1}\psi_{n-1}(x;-k')\big]\,dx.\label{chas1'}\end{gathered}$$ This equality can be derived with the help of (2.3), (2.6) and (2.9) of Part I and integration by parts, $$\begin{gathered}
\int_{-R}^R\big[k^n\psi_n(x;k)\big]\big[(k')^n\psi_n(x;-k')\big]\,dx\\
\qquad{} =i\int_{-R}^R\left\{\left(-\partial+{n\over{x-z}}\right)
[k^{n-1}\psi_{n-1}(x;k)]\right\}\big[(k')^n\psi_n(x;-k')\big]\,dx\\
\qquad{}
=-i\big[k^{n-1}\psi_{n-1}(x;k)\big]\big[(k')^n\psi_n(x;-k')\big] \Big|_{-R}^R\\
\qquad\quad{} +i\int_{-R}^R\big[k^{n-1}\psi_{n-1}(x;k)\big]\left\{\left(\partial
+{n\over{x-z}}\right)\big[(k')^n\psi_n(x;-k')\big]\right\} dx\\
\qquad{} =-i\big[k^{n-1}\psi_{n-1}(x;k)\big]\big[(k')^n\psi_n(x;-k')\big] \Big|_{-R}^R\\
\qquad\quad{} +\int_{-R}^R\big[k^{n-1}\psi_{n-1}(x;k)\big]\big\{q_n^-q_n^+
\big[(k')^{n-1}\psi_{n-1}(x;-k')\big]\big\}\,dx\\
\qquad{} =-i\big[k^{n-1}\psi_{n-1}(x;k)\big]\big[(k')^n\psi_n(x;-k')\big] \Big|_{-R}^R\\
\qquad\quad{} +\int_{-R}^R\big[k^{n-1}\psi_{n-1}(x;k)\big]\big\{h_{n-1}
\big[(k')^{n-1}\psi_{n-1}(x;-k')\big]\big\}\,dx\\
\qquad{} =-i\big[k^{n-1}\psi_{n-1}(x;k)\big]\big[(k')^n\psi_n(x;-k')\big] \Big|_{-R}^R\\
\qquad\quad{} +(k')^2\int_{-R}^R\big[k^{n-1}\psi_{n-1}(x;k)\big]
\big[(k')^{n-1}\psi_{n-1}(x;-k')\big]\,dx.\end{gathered}$$ The equality (\[chas’\]) follows from (\[chas1’\]) by induction, in view of the relation $$\int_{-R}^R\psi_0(x;k)\psi_0(x;-k')\,dx=
{1\over{2\pi}}\int_{-R}^Re^{ix(k-k')}\,dx={{\sin
R(k-k')}\over{\pi(k-k')}}.$$ Lemma 3.1 is proved.
\[lemma3.2\] For any $k'\in\Bbb R$, $z\in\Bbb C$, ${\rm{Im}}\,z\ne0$, $j=0, 1, 2, \dots$, $l=0, 1, 2, \dots$, $m=0, 1, 2,\dots$ and $\gamma>1+2l-2m$ the following relation holds, $$\mathop{{\lim}'_\gamma}_{x\to\pm\infty}{k^le^{ix(k-k')}\over
{(1+k^2)^{m/2}(x-z)^j}}=0.$$
It is true that $${k^le^{ix(k-k')}\over
{(1+k^2)^{m/2}(x-z)^j}}\in L^2(\Bbb R;(1+|k|)^{-\gamma})\subset
CL'_\gamma,\qquad\gamma>1+2l-2m.$$ Thus, to prove Lemma \[lemma3.2\], it is sufficient to prove that for any $\varphi(k)\in CL_\gamma$ the function $k^l\varphi(k)/(1+k^2)^{m/2}$ belongs to $L^1_{\Bbb R}$ in view of the Riemann theorem. The latter is valid by virtue of the Bunyakovskii inequality: $$\left(\int_{-\infty}^{+\infty}{{|k^l\varphi(k)|}\over
{(1+k^2)^{m/2}}}\,dk\right)^2
\leqslant\int_{-\infty}^{+\infty}|\varphi^2(k)|(1+|k|)^\gamma\,dk
\int_{-\infty}^{+\infty}{{k^{2l}\,dk}\over{(1+k^2)^m
(1+|k|)^\gamma}}<+\infty,$$ where the condition $\gamma>1+2l-2m$ is taken into account. Lemma \[lemma3.2\] is proved.
\[corollary3.1\] In the conditions of Lemma [\[lemma3.1\]]{}, in view of $(2.6)$ of Part [I]{} by virtue of Lemma [\[lemma3.2\]]{} for any $m=0, 1, 2, \dots$, $n=1, 2, 3, \dots$, $l=0, \dots,n-1$ and $\gamma>-2l-2m+2n-1$, the following relation holds, $$\mathop{{\lim}'_\gamma}_{x\to\pm\infty}(k')^{2l}
\left[{{k^{n-1-l}\psi_{n-1-l}(x;k)}\over{(1+k^2)^{m/2}}}\right]
\left[{{(k')^{n-l}\psi_{n-l}(x;-k')}\over{(1+(k')^2)^{m/2}}}\right]=0.$$
\[lemma3.3\] For any $k'\in\Bbb R$, $m=0, 1,
2, \dots$, $n=0, 1, 2, \dots$ and $\gamma>-2m-1$ the following relation is valid, $$\mathop{{\lim}'_\gamma}_{R\to+\infty}{{(k')^{2n}}\over{(1+k^2)^{m/2}
(1+(k')^2)^{m/2}}} {{\sin R(k-k')}
\over{\pi(k-k')}}={{(k')^{2n}}\over{(1+(k')^2)^{m}}} \delta(k-k').$$
The proof of Lemma \[lemma3.3\] is quite analogous to the one for Lemma \[lemma3.6\] from Section \[section3.5\].
Validity of the biorthogonality relation (\[ort4\]) is a corollary of the following theorem.
\[theorem3.1\] Suppose that the functions $\psi_n(x;k)$, $n=0,1, 2, \dots$ are defined by the formula $(2.6)$ of Part [I]{} for any $x\in\Bbb R$, $k\in\Bbb C$, $k\ne0$ and fixed $z\in\Bbb C$, ${\rm{Im}}\,z\ne0$. Then for any $k'\in\Bbb R$, $m=0,1, 2, \dots$, $n=0, 1, 2, \dots$ and $\gamma>-2m+2n-1$ the following relation takes place, $$\begin{gathered}
\mathop{{\lim}'_\gamma}_{R\to+\infty}\int_{-R}^R
\left[{{k^n\psi_n(x;k)}\over{(1+k^2)^{m/2}}}\right]
\left[{{(k')^n\psi_n(x;-k')}\over{(1+(k')^2)^{m/2}}}\right] dx=
{{(k')^{2n}}\over{(1+(k')^2)^m}}\,\delta(k-k').\label{ort5}\end{gathered}$$
The statement of Theorem \[theorem3.1\] follows from Lemmas \[lemma3.1\] and \[lemma3.3\] and from Corollary \[corollary3.1\].
\[remark3.1\] The parameter $m$ in Theorem \[theorem3.1\] regulates the class of test functions for which the biorthogonality relation (\[ort5\]) takes place. One can prove as well this relation for any fixed $m$ for test functions from a wider class than in Theorem \[theorem3.1\] with the help of the technique of Theorem \[theorem3.3\] and Remark \[remark3.2\] from Section \[section3.5\].
Proofs of the resolutions of identity {#section3.5}
-------------------------------------
Proof of the initial resolution of identity (2.18) of Part I is based on the following Lemmas .
\[lemma3.4\] Suppose that
1. the functions $\psi_n(x;k)$, $n=0, 1, 2, \dots$ are defined by the formula $(2.6)$ of Part [I]{} for any $x\in\Bbb R$, $k\in\Bbb C$, $k\ne0$ and fixed $z\in\Bbb C$, ${\rm{Im}}\,z\ne0$;
2. ${\cal L}(A)$ is a path in complex $k$ plane, made of the segment $[-A,A]$ by its deformation near the point $k=0$ upwards or downwards and the direction of ${\cal L}(A)$ is specified from $-A$ to $A$.
Then for any $n=1, 2, 3, \dots$, $x\in\Bbb R$ and $x'\in\Bbb R$ the following relation holds, $$\begin{gathered}
\int_{{\cal
L}(A)}\psi_n(x;k)\psi_n(x';-k)\,dk\nonumber\\
\qquad{} =\sum_{l=0}^{n-1}\left({{x'-z}\over{x-z}}\right)^{\!l}
{{\psi_{n-1-l}(x;k)\psi_{n-l}(x';-k)}
\over{i(x-z)}} \Big|_{-A}^A+\left({{x'-z}\over{x-z}}\right)^{\!n} {{\sin
A(x-x')}
\over{\pi(x-x')}}.\label{chas}\end{gathered}$$
Let us check first that $$\begin{gathered}
\int_{{\cal L}(A)}\psi_n(x;k)\psi_n(x';-k)\,dk\nonumber\\
\qquad{} ={{\psi_{n-1}(x;k)\psi_n(x';-k)}
\over{i(x-z)}} \Big|_{-A}^A+{{x'-z}\over{x-z}}\int_{{\cal
L}(A)}\psi_{n-1}(x;k)\psi_{n-1}(x';-k)\,dk.\label{chas1}\end{gathered}$$ This equality can be derived with the help of (2.3), (2.6) and (2.9) of Part I and of integration by parts: $$\begin{gathered}
\int_{{\cal
L}(A)}\! \psi_n(x;k)\psi_n(x';-k)\,dk =\int_{{\cal
L}(A)}\!{e^{ikz}\over{i(x-z)}}\!\left[\left({\partial\over{\partial
k}}-{n\over
k}\right)\big(e^{-ikz}\psi_{n-1}(x;k)\big)\right]\psi_n(x';-k)\,dk\\
={{\psi_{n-1}(x;k)\psi_n(x';-k)}\over{i(x-z)}} \Big|_{-A}^A
-\int_{{\cal
L}(A)}{{e^{-ikz}\psi_{n-1}(x;k)}\over{i(x-z)}}\left[\left({\partial\over{\partial
k}}+{n\over
k}\right)\big(e^{ikz}\psi_n(x';-k)\big)\right] dk\\
={{\psi_{n-1}(x;k)\psi_n(x';-k)}\over{i(x-z)}} \Big|_{-A}^A \\
\quad{}
+i{{x'-z}\over{x-z}}\int_{{\cal
L}(A)} {{e^{-ikz}\psi_{n-1}(x;k)}\over{x'-z}} \left[\left({\partial\over{\partial
k}}+{n\over
k}\right)\big(e^{ikz}\psi_n(x';-k)\big)\right] dk\\
={{\psi_{n-1}(x;k)\psi_n(x';-k)}\over{i(x-z)}} \Big|_{-A}^A\\
\quad{} +i{{x'-z}\over{x-z}}\int_{{\cal
L}(A)}e^{-ikz}\psi_{n-1}(x;k)\left[\left({1\over
k}{\partial\over{\partial
x'}}+{n\over{k(x'-z)}}\right)\big(e^{ikz}\psi_n(x';-k)\big)\right] dk\\
={{\psi_{n-1}(x;k)\psi_n(x';-k)}\over{i(x-z)}} \Big|_{-A}^A\\
\quad{} +{{x'-z}\over{x-z}}\int_{{\cal
L}(A)}{1\over k^2}\,\psi_{n-1}(x;k)\left[\left({\partial\over{\partial
x'}}+{n\over{x'-z}}\right)\left(-{\partial\over{\partial
x'}}+{n\over{x'-z}}\right)\psi_{n-1}(x';-k)\right] dk\\
={{\psi_{n-1}(x;k)\psi_n(x';-k)}\over{i(x-z)}} \Big|_{-A}^A
+{{x'-z}\over{x-z}}\int_{{\cal
L}(A)}{1\over
k^2} \psi_{n-1}(x;k)\big[h_{n-1}\psi_{n-1}(x';-k)\big]\,dk\\
={{\psi_{n-1}(x;k)\psi_n(x';-k)}\over{i(x-z)}} \Big|_{-A}^A
+{{x'-z}\over{x-z}}\int_{{\cal
L}(A)}\psi_{n-1}(x;k)\psi_{n-1}(x';-k)\,dk.\end{gathered}$$ The equality (\[chas\]) follows from (\[chas1\]) by induction in view of the relation $$\begin{gathered}
\int_{{\cal
L}(A)}\psi_0(x;k)\psi_0(x';-k)\,dk=
{1\over{2\pi}}\int_{-A}^Ae^{ik(x-x')}\,dk={{\sin
A(x-x')}\over{\pi(x-x')}}.\label{int00}\end{gathered}$$ Lemma \[lemma3.4\] is proved.
\[lemma3.5\] For any $x'\in\Bbb R$, $z\in\Bbb C$, ${\rm{Im}}\,z\ne0$, $l=0, 1, 2, \dots$, $m=1, 2, 3, \dots$ and $\gamma>1-2m$ the following relation takes place, $$\mathop{{\lim}'_\gamma}_{k\to\pm\infty}{e^{ik(x-x')}\over{k^l(x-z)^m}}=0.$$
It is true that $${e^{ik(x-x')}\over{k^l(x-z)^m}}\in L^2(\Bbb R;(1+|x|)^{-\gamma})
\subset CL'_\gamma, \qquad\gamma>1-2m.$$ Thus, in view of the Riemann theorem, in order to prove the lemma, it is sufficient to prove that for any $\varphi(x)\in CL_\gamma$ the fraction $\varphi(x)/(x-z)^m$ belongs to $L^1_{\Bbb R}$. The latter is valid by virtue of the Bunyakovskii inequality: $$\left(\int_{-\infty}^{+\infty}{{|\varphi(x)|}\over{|x-z|^m}}\,dx\right)^2
\leqslant\int_{-\infty}^{+\infty}|\varphi^2(x)|(1+|x|)^\gamma\,dx
\int_{-\infty}^{+\infty}{{dx}\over{|x-z|^{2m}(1+|x|)^\gamma}}<+\infty,$$ where the condition $\gamma>1-2m$ is taken into account. Lemma \[lemma3.5\] is proved.
\[corollary3.2\] In the conditions of Lemma [\[lemma3.4\]]{}, in view of $(2.6)$ from Part [I]{} by virtue of Lemma [\[lemma3.5\]]{} for any $n=1, 2,
3, \dots$, $l=0, \dots,n-1$ and $\gamma>-2l-1$, the following relation holds, $$\mathop{{\lim}'_\gamma}_{k\to\pm\infty}\left({{x'-z}\over{x-z}}\right)^{\!l}
{{\psi_{n-1-l}(x;k)\psi_{n-l}(x';-k)} \over{i(x-z)}}=0.$$
\[lemma3.6\] For any $x'\in\Bbb R$, $z\in\Bbb C$, ${\rm{Im}}\,z\ne0$, $n=0, 1, 2, \dots$ and $\gamma>-2n-1$ the following relation is valid, $$\mathop{{\lim}'_\gamma}_{A\to+\infty}\left({{x'-z}\over{x-z}}\right)^{\!n} {{\sin
A(x-x')}
\over{\pi(x-x')}}=\delta(x-x').$$
It is true that $$\left({{x'-z}\over{x-z}}\right)^{\!n} {{\sin A(x-x')}
\over{\pi(x-x')}}\in L^2(\Bbb R;(1+|x|)^{-\gamma})\subset
CL'_\gamma,\qquad\gamma>-2n-1.$$ Thus, to prove the lemma, it is sufficient to prove that for any $\varphi(x)\in CL_{\gamma}$, $\gamma>-2n-1$ the equality $$\lim_{A\to+\infty}\int_{-\infty}^{+\infty}{{\sin A(x-x')}
\over{\pi(x-x')}} \left({{x'-z}
\over{x-z}}\right)^{n}\varphi(x)\,dx=\varphi(x')$$ takes place. For this purpose let us consider the function $$\psi(x)=\left({{x'-z}
\over{x-z}}\right)^{n}\varphi(x).$$ By virtue of the Bunyakovskii inequality for arbitrary $\delta>0$, $$\begin{gathered}
\left[\left(\int_{-\infty}^{x'-\delta}+\int_{x'+\delta}^{+\infty}\right)
{{|\psi(x)|}\over{|x-x'|}}\,dx\right]^2
\leqslant\left(\int_{-\infty}^{x'-\delta}+\int_{x'+\delta}^{+\infty}\right)
|\varphi^2(x)|(1+|x|)^\gamma\,dx\\
\qquad{}\times
\left(\int_{-\infty}^{x'-\delta}+\int_{x'+\delta}^{+\infty}\right)
{{|x'-z|^{2n}\,dx}\over{|x-z|^{2n}|x-x'|^{2}(1+|x|)^\gamma}}<+\infty\end{gathered}$$ the following inclusion is valid, $${{\psi(x)}\over{x-x'}}\in
L^1({\Bbb R}\setminus(x'-\delta,x'+\delta)),\qquad \delta>0,$$ and, moreover, it is evident that $${{\psi(x)-\psi(x')}\over{x-x'}}\in L^1([x'-\delta,x'+\delta]),\qquad \delta>0.$$ Hence, by virtue of the Riemann theorem, $$\begin{gathered}
\lim_{A\to+\infty}\int_{-\infty}^{+\infty}{{\sin A(x-x')}
\over{\pi(x-x')}}\left({{x'-z}
\over{x-z}}\right)^{n}\varphi(x)\,dx\\
\qquad{} =\lim_{A\to+\infty}\left[\psi(x')
\int_{x'-\delta}^{x'+\delta} {{\sin A(x-x')}
\over{\pi(x-x')}}\,dx\right]
={2\over\pi} \varphi(x')\int_0^{+\infty}{{\sin t}\over t}\,dt=\varphi(x').
\end{gathered}$$ Thus, Lemma \[lemma3.6\] is proved.
Validity of the resolution of identity (2.18) of Part I in $CL'_\gamma$ for any $\gamma>-1$ is a corollary of the following theorem.
\[theorem3.2\] Suppose that
1. the functions $\psi_n(x;k)$, $n=0, 1, 2, \dots$ are defined by the formula $(2.6)$ of Part [I]{} for any $x\in\Bbb R$, $k\in\Bbb C$, $k\ne0$ and fixed $z\in\Bbb C$, ${\rm{Im}}\,z\ne0$;
2. ${\cal L}(A)$ is a path in complex $k$ plane, made of the segment $[-A,A]$, $A>0$ by its deformation near the point $k=0$ upwards or downwards and the direction of ${\cal L}(A)$ is specified from $-A$ to $A$.
Then for any $\gamma>-1$, $x'\in\Bbb R$ and $n=0, 1,
2, \dots$ the following relation holds, $$\begin{gathered}
\mathop{{\lim}'_\gamma}_{A\to+\infty}\int_{{\cal
L}(A)}\psi_n(x;k)\psi_n(x';-k)\,dk=\delta(x-x').$$
The statement of Theorem \[theorem3.2\] follows from Lemmas \[lemma3.4\] and \[lemma3.6\] and from Corollary \[corollary3.2\].
The applicability of the resolution of identity (2.18) of Part I for some bounded and slowly increasing test functions is based on the next theorem.
\[theorem3.3\] Suppose that
1. the functions $\psi_n(x;k)$, $n=0, 1, 2, \dots$ are defined by the formula $(2.6)$ of Part [I]{} for any $x\in\Bbb R$, $k\in\Bbb C$, $k\ne0$ and fixed $z\in\Bbb C$, ${\rm{Im}}\,z\ne0$;
2. ${\cal L}(A)$ is a path in complex $k$ plane, made of the segment $[-A,A]$, $A>0$ by its deformation near the point $k=0$ upwards or downwards and the direction of ${\cal L}(A)$ is specified from $-A$ to $A$;
3. the function $\eta(x)\in C^\infty_{\Bbb R}$, $\eta(x)\equiv0$ for any $x\leqslant1$, $\eta(x)\in[0,1]$ for any $x\in[1,2]$ and $\eta(x)\equiv1$ for any $x\geqslant2$.
Then for any $\varkappa\in[0,1)$, $k_0\in\Bbb R$, $x'\in\Bbb R$ and $n=0, 1, 2, \dots$ the following relation is valid, $$\begin{gathered}
\lim_{A\to+\infty}\int_{-\infty}^{+\infty}
\left[\int_{{\cal
L}(A)}\psi_n(x;k)\psi_n(x';-k)\,dk\right]\big[\eta(\pm
x)e^{ik_0x}|x|^\varkappa\big]\,dx=
\eta(\pm x')e^{ik_0x'}|x'|^\varkappa.\label{int52}\end{gathered}$$
In the case $n=0$ in view of (\[int00\]) the proof can be easily realized in the same way as for Theorem 2 from Appendix B of [@andcansok10]. Thus, we present the proof for the case $n=1, 2, 3,\dots$ with upper signs in (\[int52\]) only, taking into account that the proof for the case with lower signs is quite similar. In order to prove Theorem \[theorem3.3\] in this case we employ Lemmas \[lemma3.4\] and \[lemma3.6\], Corollary \[corollary3.2\] and the fact that $$\begin{gathered}
\eta(x)e^{ik_0x}|x|^\varkappa\in CL_\gamma,\qquad
-3< \gamma<-1-2\varkappa.\label{inkl}\end{gathered}$$ Then it is sufficient to prove that $$\lim_{A\to+\infty}\int_{-\infty}^{+\infty}\left[
{{\psi_{n-1}(x;k)\psi_n(x';-k)}
\over{i(x-z)}}\Big|_{-A}^A\right]\big[\eta(
x)e^{ik_0x}|x|^\varkappa\big]\,dx=0.$$ In turn, to prove the latter, in view of (2.6) from Part I, (\[inkl\]) and Lemma \[lemma3.5\], it is sufficient to prove that $$\begin{gathered}
\lim_{A\to+\infty}\int_{-\infty}^{+\infty}\left[{e^{\pm
iA(x-x')} \over{x-z}}\right]\big[\eta(
x)e^{ik_0x}|x|^\varkappa\big]\,dx=0.\label{lim34}\end{gathered}$$ The equality (\[lim34\]) follows from the Riemann theorem and the chain of transformations, $$\begin{gathered}
\int_{-\infty}^{+\infty}\left[{e^{\pm
iA(x-x')} \over{x-z}}\right]\big[\eta(
x)e^{ik_0x}|x|^\varkappa\big]\,dx=e^{ik_0x'}\int_{-\infty}^{+\infty}
{{|t+x'|^\varkappa \eta(t+ x') }\over{t+x'-z}} e^{i(k_0\pm
A)t}\,dt\\
\qquad{} =e^{ik_0x'}\int_{-\infty}^{+\infty}
{{|t+x'|^\varkappa \eta(t+ x') }\over{t+x'-z}} d{e^{i(k_0\pm
A)t}\over{i(k_0\pm A)}}\\
\qquad{} =-{e^{ik_0x'}\over{i(k_0\pm
A)}}\int_{-\infty}^{+\infty} e^{i(k_0\pm
A)t} d{{|t+x'|^\varkappa \eta(t+ x') }\over{t+x'-z}},\qquad
A>|k_0|,\end{gathered}$$ derived with help of integration by parts. Thus, Theorem \[theorem3.3\] is proved.
\[remark3.2\] Theorems \[theorem3.2\] and \[theorem3.3\] provide the validity of the resolution of identity (2.18) from Part I for test functions which are linear combinations of functions $\eta(\pm
x)e^{ik_0x}|x|^\varkappa$, in general, with different $\varkappa\in[0,1)$ and $k_0\in\Bbb R$ and functions from $CL_\gamma$, in general, with different $\gamma>-1$. In particular, these theorems guarantee applicability of (2.18) from Part I for the eigenfunctions (2.6) from Part I and for the associated function (2.2) from Part I (in the case of even $n$) of the Hamiltonian $h_n$, $n=0, 1, 2, \dots$.
\[remark3.3\] The first of resolutions of identity (2.19) of Part I follows from (2.18) of Part I and Lemma \[lemma3.4\]. The second of resolutions of identity (2.19) of Part I and (2.35) of Part I can be derived from the first of resolutions of identity (2.19) of Part I with the help of calculation of the substitution $|_{-\varepsilon}^\varepsilon$ and of identical transformations. The resolution of identity (2.20) of Part I follows trivially from (2.19) of Part I.
The resolutions of identity (2.21) and (2.36) of Part I are corollaries of the resolutions of identity (2.19) and (2.35) of Part I respectively and of the following Lemma \[lemma3.7\].
\[lemma3.7\] For any $\gamma>-1$ and $x'\in\Bbb R$ the relation holds, $$\mathop{{\lim}'_\gamma}\limits_{\varepsilon\downarrow0}
{{\sin\varepsilon(x-x')}\over{x-x'}}=0 .$$
Proof of Lemma \[lemma3.7\] is analogous to the proof of Lemma 2 from Appendix B of [@andcansok10].
The resolution of identity (2.37) of Part I is a corollary of the resolution of identity (2.36) of Part I and of the following Lemma \[lemma3.8\].
\[lemma3.8\] For any $\gamma>1$, $x'\in\Bbb R$ and $z\in\Bbb C$, ${\rm{Im}}\,z\ne0$ the relation takes place, $$\mathop{{\lim}'_\gamma}\limits_{\varepsilon\downarrow0}
{{\sin^2{\varepsilon\over2}(x-x')}\over{\varepsilon(x-z)(x'-z)}}=0 .$$
Proof of Lemma \[lemma3.8\] is analogous to the proof of Lemma 3 from Appendix of [@ancansok06].
The resolution of identity (2.38) of Part I is a corollary of the resolution of identity (2.37) of Part I and of the following Lemmas \[lemma3.9\] and \[lemma3.10\].
\[lemma3.9\] For any $\gamma>3$, $x'\in\Bbb R$ and $z\in\Bbb C$, ${\rm{Im}}\,z\ne0$ the relation is valid, $$\mathop{{\lim}'_\gamma}\limits_{\varepsilon\downarrow0}
{{(x-x')\sin^2
{\varepsilon\over4}(x-x')\sin{\varepsilon\over2}(x-x')}\over
{\varepsilon^2(x-z)^2(x'-z)^2}}=0 .$$
It is true that $${{(x-x')\sin^2
{\varepsilon\over4}(x-x')\sin{\varepsilon\over2}(x-x')}\over
{\varepsilon^2(x-z)^2(x'-z)^2}}\in L^2({\Bbb
R};(1+|x|)^{-\gamma})\subset CL_\gamma,\qquad\gamma>3.$$ Thus, to prove the lemma it is sufficient to establish that for any $\varphi(x)\in CL_\gamma$, $\gamma>3$, the relation $$\lim_{\varepsilon\downarrow0}\int_{-\infty}^{+\infty}{{(x-x')\sin^2
{\varepsilon\over4}(x-x')\sin{\varepsilon\over2}(x-x')}\over
{\varepsilon^2(x-z)^2(x'-z)^2}} \varphi(x)\,dx=0$$ is valid. But its validity follows from the chain of inequalities, $$\begin{gathered}
\left|\int_{-\infty}^{+\infty}{{(x-x')\sin^2
{\varepsilon\over4}(x-x')\sin{\varepsilon\over2}(x-x')}\over
{\varepsilon^2(x-z)^2(x'-z)^2}}\,\varphi(x)\,dx\right|^2\\
{} \leqslant \int_{-\infty}^{+\infty}{{\sin^4
{\varepsilon\over4}(x-x')\sin^2{\varepsilon\over2}(x-x')}\over
{\varepsilon^4|x-x'|^\alpha}}\,dx\int_{-\infty}^{+\infty}
{{|x-x'|^{\alpha+2}|\varphi^2(x)|}\over {|x-z|^4|x'-z|^4}}\,dx\\
{} =\varepsilon^{\alpha-5}\int_{-\infty}^{+\infty}{{\sin^4
{t\over4}\,\sin^2{t\over2}}\over
{|t|^\alpha}}\,dt\int_{-\infty}^{+\infty}
{{|x-x'|^{\alpha+2}|\varphi^2(x)|}\over {|x-z|^4|x'-z|^4}}\,dx\\
{} \leqslant\varepsilon^{\alpha-5}\sup_{x\in\Bbb R}\left[{{|x-x'|^{\alpha+2}}\over
{|x-z|^4|x'-z|^4(1+|x|)^\gamma}}\right]
\int_{-\infty}^{+\infty}{{\sin^4
{t\over4}\,\sin^2{t\over2}}\over
{|t|^\alpha}}\,dt\int_{-\infty}^{+\infty}
|\varphi^2(x)|(1+|x|)^\gamma\,dx,\\
5<\alpha<\min\{7,\gamma+2\},
\end{gathered}$$ derived with the help of the Bunyakovskii inequality. Lemma \[lemma3.9\] is proved.
\[lemma3.10\] For any $\gamma>3$, $x'\in\Bbb R$ and $z\in\Bbb C$, ${\rm{Im}}\,z\ne0$ the relation takes place, $$\mathop{{\lim}'_\gamma}\limits_{\varepsilon\downarrow0}
{{[\varepsilon(x-x')-2\sin{\varepsilon\over2}(x-x')]^2}\over
{\varepsilon^3(x-z)^2(x'-z)^2}}=0 .$$
It is true that $${{[\varepsilon(x-x')-2\sin{\varepsilon\over2}(x-x')]^2}\over
{\varepsilon^3(x-z)^2(x'-z)^2}}\in L^2({\Bbb
R};(1+|x|)^{-\gamma})\subset CL_\gamma,\qquad\gamma>3.$$ Thus, to prove the lemma it is sufficient to establish that for any $\varphi(x)\in CL_\gamma$, $\gamma>3$, the relation $$\lim_{\varepsilon\downarrow0}\int_{-\infty}^{+\infty}
{{[\varepsilon(x-x')-2\sin{\varepsilon\over2}(x-x')]^2}\over
{\varepsilon^3(x-z)^2(x'-z)^2}}\,\varphi(x)\,dx=0$$ is valid. But its validity follows from the chain of inequalities, $$\begin{gathered}
\left|\int_{-\infty}^{+\infty}{{[\varepsilon(x-x')-2
\sin{\varepsilon\over2}(x-x')]^2}\over
{\varepsilon^3(x-z)^2(x'-z)^2}} \varphi(x)\,dx\right|^2\\
{} \leqslant \int_{-\infty}^{+\infty}{{[\varepsilon(x-x')-2
\sin{\varepsilon\over2}(x-x')]^4}\over
{\varepsilon^6|x-x'|^\alpha}}\,dx\int_{-\infty}^{+\infty}
{{|x-x'|^\alpha|\varphi^2(x)|}\over {|x-z|^4|x'-z|^4}}\,dx\\
{} =\varepsilon^{\alpha-7}\int_{-\infty}^{+\infty}{{[t-2\sin{t\over2}]^4}\over
{|t|^\alpha}}\,dt\int_{-\infty}^{+\infty}
{{|x-x'|^\alpha|\varphi^2(x)|}\over {|x-z|^4|x'-z|^4}}\,dx\\
{} \leqslant\varepsilon^{\alpha-7}\sup_{x\in\Bbb R}\left[{{|x-x'|^\alpha}\over
{|x-z|^4|x'-z|^4(1+|x|)^\gamma}}\right]
\int_{-\infty}^{+\infty}{{[t-2\sin{t\over2}]^4}\over
{|t|^\alpha}}\,dt\int_{-\infty}^{+\infty}
|\varphi^2(x)|(1+|x|)^\gamma\,dx,\\
7<\alpha<\min\{13,\gamma+4\},
\end{gathered}$$ derived with the help of the Bunyakovskii inequality. Lemma \[lemma3.10\] is proved.
\[remark3.4\] Let us consider the functionals $$\begin{gathered}
\mathop{{\lim}''_\gamma}_{\varepsilon\downarrow0} {{(x-x')\sin^2
{\varepsilon\over4}(x-x')\sin{\varepsilon\over2}(x-x')}\over
{\varepsilon^2(x-z)^2(x'-z)^2}}, \qquad
\mathop{{\lim}''_\gamma}_{\varepsilon\downarrow0}
{{[\varepsilon(x-x')-2\sin{\varepsilon\over2}(x-x')]^2}\over
{\varepsilon^3(x-z)^2(x'-z)^2}}, \label{fun96}\end{gathered}$$ each of which is defined by a related expression in the set $$\begin{gathered}
\lim_{\varepsilon\downarrow0} \int_{-\infty}^{+\infty}{{(x-x')\sin^2
{\varepsilon\over4}(x-x')\sin{\varepsilon\over2}(x-x')}\over
{\varepsilon^2(x-z)^2(x'-z)^2}} \varphi(x)\,dx,\nonumber\\
\lim_{\varepsilon\downarrow0}\int_{-\infty}^{+\infty}
{{[\varepsilon(x-x')-2\sin{\varepsilon\over2}(x-x')]^2}\over
{\varepsilon^3(x-z)^2(x'-z)^2}} \varphi(x)\,dx,\label{lim97}\end{gathered}$$ for all test functions $\varphi(x)\in CL_\gamma$, $\gamma\in\Bbb R$, for which the limit from (\[lim97\]) corresponding to (\[fun96\]) exists. It follows from Lemmas \[lemma3.9\] and \[lemma3.10\] that these functionals are trivial (equal to zero) for any $\gamma>3$, but at the same time in view of the formulae (2.39) and (2.40) from [@anso11] these functionals are nontrivial (different from zero) for any $\gamma<3$. By virtue of Lemmas \[lemma3.9\] and \[lemma3.10\] the restrictions of the functionals (\[fun96\]) on the standard space ${\cal{D}}(\Bbb
R)\subset CL_\gamma$, $\gamma\in\Bbb R$ are equal to zero. Hence, the supports of these functionals for any $\gamma\in\Bbb R$ do not contain any finite real number. On the other hand, one can represent a test function $\varphi(x)\in CL_\gamma$, $\gamma\in\Bbb
R$ for any $R>0$ as a sum of two functions from $CL_\gamma$ in the form $$\begin{gathered}
\varphi(x)=\eta(|x|-R)\varphi(x)+[1-\eta(|x|-R)]\varphi(x),\qquad
R>0,\label{repr98}\end{gathered}$$ where $\eta(x)\in C^\infty_{\Bbb
R}$, $\eta(x)\equiv1$ for any $x<0$, $\eta(x)\in[0,1]$ for any $x\in[0,1]$ and $\eta(x)\equiv0$ for any $x>1$. In view of Lemmas \[lemma3.9\] and \[lemma3.10\] the values of the functionals (\[fun96\]) for $\varphi(x)$ are equal to their values for the second term of (\[repr98\]) for any arbitrarily large $R>0$. Hence, the values of the functionals (\[fun96\]) for a test function depend only on the behavior of this function in any arbitrarily close (in the conformal sense) vicinity of the infinity and are independent of values of the function in any finite interval of real axis. In this sense the supports of the functionals (\[fun96\]) for any $\gamma<3$ consist of the unique element which is the infinity. At last, since (i) for any $\varphi(x)\in CL_\gamma$ and $\gamma\in\Bbb
R$ the relation $$\mathop{{\lim}_\gamma}\limits_{R\to+\infty}\eta(|x|-R)\varphi(x)=
\varphi(x)$$ holds; (ii) the restrictions of the functionals (\[fun96\]) on ${\cal D}(\Bbb R)$ are zero for any $\gamma\in\Bbb
R$ and (iii) the functionals (\[fun96\]) are nontrivial for any $\gamma<3$, so the latter functionals are discontinuous for any $\gamma<3$.
Proofs of resolutions of identity for the model Hamiltonian\
with exceptional point inside of continuous spectrum {#section4}
============================================================
Proof of the initial resolution of identity (3.7) of Part I is based on the following Lemmas \[lemma4.1\]–\[lemma4.3\].
\[lemma4.1\] Suppose that
1. the functions $\psi(x;k)$, $\psi_0(x)$ and $\psi_1(x)$ are defined by the formulas $(3.1)$ and $(3.2)$ of Part [I]{} for fixed $\alpha>0$, $z\in\Bbb C$, ${\rm{Im}}\,z\ne0$ and any $x\in\Bbb R$, $k\in\Bbb C$, $k\ne\pm\alpha$;
2. ${\cal L}(A)$ is an integration path in complex $k$ plane, obtained from the segment $[-A,A]$, $A>\alpha$ by its simultaneous deformation near the points $k=-\alpha$ and $k=\alpha$ upwards or downwards and the direction of ${\cal L}(A)$ is specified from $-A$ to $A$.
Then for any $x,x'\in\Bbb R$ and $A>\alpha$ the following relation is valid, $$\begin{gathered}
\int_{{\cal L}(A)}\psi(x;k)\psi(x';-k)\,dk\nonumber\\
\qquad{} ={{\sin A(x-x')}
\over{\pi(x-x')}}-{1\over{2\pi\alpha}}
\left\{{{\cos[(A+\alpha)(x-x')]}\over{A+\alpha}}+
{{\cos[(A-\alpha)(x-x')]}\over{A-\alpha}}\right\}\psi_0(x)\psi_0(x')\nonumber\\
\qquad{} -{1\over{\pi}}
\left[\int_{A-\alpha}^{A+\alpha}\cos t(x-x') {{dt}\over
t}\right][\psi_0(x)\psi_1(x')+\psi_1(x)\psi_0(x')].
\label{chasA}\end{gathered}$$
With the help of (3.1) and (3.2) of Part I and certain identical transformations one can rearrange the left-hand part of (\[chasA\]) to the form, $$\begin{gathered}
\int_{{\cal
L}(A)}\psi(x;k)\psi(x';-k)\,dk ={1\over{2\pi}}\int_{{\cal
L}(A)}e^{ik(x-x')}\,dk\\
{} -{1\over{4\pi\alpha}} \psi_0(x)\psi_0(x')\left[\int_{{\cal
L}(A)}{\partial\over{\partial
k}}\left({e^{i(k-\alpha)(x-x')}\over{k-\alpha}}\right) dk+\int_{{\cal
L}(A)}{\partial\over{\partial
k}}\left({e^{i(k+\alpha)(x-x')}\over{k+\alpha}}\right) dk\right]\\
{} +{1\over{2\pi}} [\psi_0(x)\psi_1(x')+\psi_1(x)\psi_0(x')]
\left[\int_{{\cal
L}(A)}{e^{i(k-\alpha)(x-x')}\over{k-\alpha}} dk-\int_{{\cal
L}(A)}{e^{i(k+\alpha)(x-x')}\over{k+\alpha}} dk\right],\end{gathered}$$ where from the equality (\[chasA\]) follows trivially. Lemma \[lemma4.1\] is proved.
\[lemma4.2\] In the conditions of Lemma [\[lemma4.1\]]{} for any $x'\in\Bbb R$ and $\gamma>-1$ the following relation holds, $$\mathop{{\lim}'_\gamma}_{k\to\pm\infty}\left[{e^{ik(x-x')}\over k}
\psi_0(x)\right]
=0.$$
Proof of Lemma \[lemma4.2\] in view of (3.2) of Part I is quite similar to the proof of a more complicated Lemma \[lemma3.2\] from Section \[section3.2\].
\[lemma4.3\] In the conditions of Lemma [\[lemma4.1\]]{} for any $x'\in\Bbb R$ and $\gamma>-1$ the following relation takes place, $$\mathop{{\lim}'_\gamma}_{A\to+\infty}\left\{\left[
\int_{A-\alpha}^{A+\alpha}\cos t(x-x')\,{{dt}\over
t}\right][\psi_0(x)\psi_1(x')+\psi_1(x)\psi_0(x')]\right\}
=0.$$
Let us use the estimate (B12) from [@andcansok10], $$\begin{gathered}
\left|\int_{A-\alpha}^{A+\alpha}\cos
t(x-x')\,{{dt}\over
t}\right|\leqslant{{AC}\over{[1+(A-\alpha)|x-x'|](A-\alpha)}},
\\
x,x'\in{\Bbb R},\qquad A>\alpha,\qquad C=2\sup_{\xi>0}\left|(1+\xi)
{{\sin\xi}\over\xi}\right|.\end{gathered}$$ Therefrom it follows that $$\begin{gathered}
\left|\left[ \int_{A-\alpha}^{A+\alpha}\cos
t(x-x')\,{{dt}\over
t}\right][\psi_0(x)\psi_1(x')+\psi_1(x)\psi_0(x')]\right|\leqslant
{{AD}\over{[1+(A-\alpha)|x-x'|](A-\alpha)}},\nonumber\\
x,x'\in{\Bbb R},\qquad A>\alpha,\qquad
D=2C\sup_{x,x'\in\Bbb
R}|\psi_0(x)\psi_1(x')|,\label{est}\end{gathered}$$ where $D$ is a finite constant by virtue of (3.2) of Part I. The statement of Lemma \[lemma4.3\] is valid in view of the following chain of inequalities obtained with the help of (\[est\]) and the Bunyakovskii inequality, $$\begin{gathered}
\left|\int_{-\infty}^{+\infty}\left\{\left[
\int_{A-\alpha}^{A+\alpha}\cos t(x-x')\,{{dt}\over
t}\right][\psi_0(x)\psi_1(x')+\psi_1(x)\psi_0(x')]\right\}
\varphi(x)\,dx\right|^2\\
\qquad{} \leqslant {{A^2D^2}\over{(A-\alpha)^2}}
\int_{-\infty}^{+\infty}|\varphi^2(x)|(1+|x-x'|)^\gamma\,dx
\int_{-\infty}^{+\infty}{{dx}\over{(1+|x-x'|)^\gamma
[1+(A-\alpha)|x-x'|]^2}}\\
\qquad{} = {{2A^2D^2}\over{(A-\alpha)^3}}
\int_{-\infty}^{+\infty}|\varphi^2(x)|(1+|x-x'|)^\gamma\,dx
\int_{0}^{+\infty}{{dt}\over{[1+t/(A-\alpha)]^\gamma
(1+t)^2}}\\
\qquad{} \leqslant {{2A^2D^2}\over{(A-\alpha)^3}}
\int_{-\infty}^{+\infty}|\varphi^2(x)|(1+|x-x'|)^\gamma\,dx\\
\qquad\quad{}\times
\begin{cases}\displaystyle \int_{0}^{+\infty}{{dt}\over{
(1+t)^{2+\gamma}}},&-1<\gamma<0,
A\geqslant\alpha+1, \\
\displaystyle
\int_{0}^{+\infty}
{{dt}\over{(1+t)^2}},&\gamma\geqslant0
\end{cases} \to0,\qquad A\to+\infty,\end{gathered}$$ where $\varphi(x)$ is any function from $CL_\gamma$, $\gamma>-1$. Lemma \[lemma4.3\] is proved.
Validity of the resolution of identity (3.7) of Part I in $CL'_\gamma$ for any $\gamma>-1$ is a corollary of the following theorem.
\[theorem4.1\] Suppose that
1. the function $\psi(x;k)$ is defined by the formula $(3.1)$ of Part [I]{} for fixed $\alpha>0$, $z\in\Bbb C$, ${\rm{Im}}\,z\ne0$ and any $x\in\Bbb R$, $k\in\Bbb C$, $k\ne\pm\alpha$;
2. ${\cal L}(A)$ is an integration path in complex $k$ plane, obtained from the segment $[-A,A]$, $A>\alpha$ by its simultaneous deformation near the points $k=-\alpha$ and $k=\alpha$ upwards or downwards and the direction of ${\cal L}(A)$ is specified from $-A$ to $A$.
Then for any $\gamma>-1$ and $x'\in\Bbb R$ the following relation holds, $$\begin{gathered}
\mathop{{\lim}'_\gamma}_{A\to+\infty}\int_{{\cal
L}(A)}\psi(x;k)\psi(x';-k)\,dk=\delta(x-x').$$
Theorem 4.1 follows from Lemmas \[lemma4.1\]–\[lemma4.3\] and from the case $n=0$ of Lemma \[lemma3.6\]. Proof of the resolution of identity (3.7) of Part I for some bounded and slowly increasing test functions is based on the following lemma.
\[lemma4.4\] In the conditions of Lemma [\[lemma4.1\]]{} for any $x\in\Bbb R$, $x'\in\Bbb R$ and $A>\alpha$ the inequalities take place, $$\begin{gathered}
\left|\int_{A-\alpha}^{A+\alpha}\cos t(x-x') {{dt}\over t} -
\left\{{{\sin[(A+\alpha)(x-x')]}\over{(A+\alpha)(x-x')}} -
{{\sin[(A-\alpha)(x-x')]}\over{(A-\alpha)(x-x')}}\right\}\right|\nonumber\\
\qquad{} \leqslant{{6}\over{(A-\alpha)^2(x-x')^2}},\label{ner1}\\
|\psi_0(x)|\leqslant{{(2\alpha)^{3/2}}\over{|\sin2\alpha x+2\alpha(x-z)|}},
\label{ner2}\\
\left|\psi_0(x)-\sqrt{2\alpha} {{\cos\alpha x}\over{x-z}}\right|\leqslant
{{\sqrt{2\alpha}}\over{|x-z| |\sin2\alpha
x+2\alpha(x-z)|}}
\label{ner3}\end{gathered}$$ and $$\begin{gathered}
\left|\psi_1(x)-{1\over\sqrt{2\alpha}} \sin\alpha x\right|\leqslant
{1\over{\sqrt{2\alpha} |\sin2\alpha
x+2\alpha(x-z)|}}.\label{ner4}\end{gathered}$$
The inequality (\[ner1\]) can be derived with the help of integration by parts, $$\begin{gathered}
\left|\int_{A-\alpha}^{A+\alpha}\cos t(x-x') {{dt}\over t} -
\left\{{{\sin[(A+\alpha)(x-x')]}\over{(A+\alpha)(x-x')}} -
{{\sin[(A-\alpha)(x-x')]}\over{(A-\alpha)(x-x')}}\right\}\right|\\
=\left|\int_{A-\alpha}^{A+\alpha}{{\sin t(x-x')}\over{x-x'}} {{dt}\over
{t^2}}\right|= \left| 2\int_{A-\alpha}^{A+\alpha}{1\over
{t^2}} d{{\sin^2[t(x-x')/2]}\over{(x-x')^2}}\right|\\
=\left| 2\left\{{{\sin^2[(A+\alpha)(x-x')/2]}\over{(A+\alpha)^2(x-x')^2}}-
{{\sin^2[(A-\alpha)(x-x')/2]}\over{(A-\alpha)^2(x-x')^2}}\right\}+
4\! \int_{A-\alpha}^{A+\alpha}\! {{\sin^2[t(x-x')/2]}\over{(x-x')^2}}
{{dt}\over{t^3}}\right|\\
\leqslant {4\over{(A-\alpha)^2(x-x')^2}}+{4\over{(x-x')^2}}
\int_{A-\alpha}^{A+\alpha}{{dt}\over
t^3}\leqslant{6\over{(A-\alpha)^2(x-x')^2}}.\end{gathered}$$ The inequality (\[ner2\]) follows trivially from (3.2) of Part I. The inequalities (\[ner3\]) and (\[ner4\]) can be obtained with the help of (3.2) of Part I, $$\begin{gathered}
\left|\psi_0(x)-\sqrt{2\alpha} {{\cos\alpha x}
\over{x-z}}\right|={{\sqrt{2\alpha} |\sin2\alpha x \cos\alpha x|}
\over{|(x-z)[\sin2\alpha x+2\alpha(x-z)]|}} \leqslant
{{\sqrt{2\alpha}} \over{|x-z|\,|\sin2\alpha x+2\alpha(x-z)|}},\\
\left|\psi_1(x)-{1\over\sqrt{2\alpha}} \sin\alpha x\right|=
{{|\cos2\alpha x \cos\alpha x|}\over{\sqrt{2\alpha} |\sin2\alpha
x+2\alpha(x-z)|}}\leqslant {1\over{\sqrt{2\alpha} |\sin2\alpha
x+2\alpha(x-z)|}}.\end{gathered}$$ Lemma \[lemma4.4\] is proved.
The applicability of the resolution of identity (3.7) of Part I for some bounded and slowly increasing test functions is based on the next theorem.
\[theorem4.2\] Suppose that
1. the function $\psi(x;k)$ is defined by the formula $(3.1)$ of Part [I]{} for fixed $\alpha>0$, $z\in\Bbb C$, ${\rm{Im}}\,z\ne0$ and any $x\in\Bbb R$, $k\in\Bbb C$, $k\ne\pm\alpha$;
2. ${\cal L}(A)$ is an integration path in complex $k$ plane, obtained from the segment $[-A,A]$, $A>\alpha$ by its simultaneous deformation near the points $k=-\alpha$ and $k=\alpha$ upwards or downwards and the direction of ${\cal L}(A)$ is specified from $-A$ to $A$;
3. the function $\eta(x)\in C^\infty_{\Bbb R}$, $\eta(x)\equiv0$ for any $x\leqslant1$, $\eta(x)\in[0,1]$ for any $x\in[1,2]$ and $\eta(x)\equiv1$ for any $x\geqslant2$.
Then for any $\varkappa\in[0,1)$, $k_0\in\Bbb R$ and $x'\in\Bbb R$ the following relation holds, $$\begin{gathered}
\lim_{A\to+\infty}\int_{-\infty}^{+\infty}
\left[\int_{{\cal L}(A)}\psi(x;k)\psi(x';-k)\,dk\right]\big[\eta(\pm
x)e^{ik_0x}|x|^\varkappa\big]\,dx=
\eta(\pm x')e^{ik_0x'}|x'|^\varkappa.$$
Proof of Theorem \[theorem4.2\] is quite analogous to the proof of Theorem 2 from Appendix B of [@andcansok10] and it is based on the inequalities from Lemma \[lemma4.4\].
\[remark4.1\] Theorems \[theorem4.1\] and \[theorem4.2\] provide the validity of the resolution of identity (3.7) of Part I for test functions which are linear combinations of functions $\eta(\pm x)e^{ik_0x}|x|^\varkappa$, in general, with different $\varkappa\in[0,1)$ and $k_0\in\Bbb R$ and functions from $CL_\gamma$, in general, with different $\gamma>-1$. In particular, these theorems guarantee applicability of (3.7) of Part I to the eigenfunctions $\psi(x;k)$ and to the associated function $\psi_1(x)$ of the Hamiltonian $h$ (see Part I).
The resolutions of identity (3.8) and (3.9) of Part I are corollaries of the resolution of identity (3.7) of Part I and of the following Lemma \[lemma4.5\].
\[lemma4.5\] Suppose that
1. the functions $\psi(x;k)$, $\psi_0(x)$ and $\psi_1(x)$ are defined by the formulas $(3.1)$ and $(3.2)$ of Part [I]{} for fixed $\alpha>0$, $z\in\Bbb C$, ${\rm{Im}}\,z\ne0$ and any $x\in\Bbb R$, $k\in\Bbb C$, $k\ne\pm\alpha$;
2. ${\cal L}_\pm(k_0;\varepsilon)$ with fixed $k_0\in\Bbb
R$ and $\varepsilon>0$ is an integration path in complex $k$ plane defined by $$k=k_0+\varepsilon[\cos(\pi-\vartheta)\pm i\sin(\pi-\vartheta)],
\qquad 0\leqslant\vartheta\leqslant\pi,$$ where the upper $($lower$)$ sign corresponds to the upper $($lower$)$ index of ${\cal
L}_\pm(k_0;\varepsilon)$, and the direction of ${\cal
L}_\pm(k_0;\varepsilon)$ is specified from $\vartheta=0$ to $\vartheta=\pi$.
Then for any $x,x'\in\Bbb R$ and $\varepsilon\in(0,\alpha)$ the following relation is valid, $$\begin{gathered}
\left(\int_{{\cal L}_\pm(-\alpha;\varepsilon)}+\int_{{\cal L}_\pm
(\alpha;\varepsilon)}\right) \psi(x;k)\psi(x';-k)\,dk\nonumber\\
\qquad{} ={2\over\pi} \cos\alpha(x-x'){{\sin\varepsilon(x-x')}
\over{x-x'}}-{1\over{\pi\alpha}} \psi_0(x)\psi_0(x')
\bigg\{{1\over\varepsilon}\left[1-2\sin^2{\varepsilon\over2} (x-x')\right]
\nonumber\\
\qquad\quad{} -{\varepsilon\over{4\alpha^2-\varepsilon^2}} \cos2\alpha(x-x')
\cos\varepsilon(x-x')-{{2\alpha}\over{4\alpha^2-\varepsilon^2}} \sin2\alpha(x-x')
\sin\varepsilon(x-x')\bigg\}\nonumber\\
\qquad\quad{} -{1\over{\pi}} [\psi_0(x)\psi_1(x')+\psi_1(x)\psi_0(x')]
\int_{2\alpha-\varepsilon}^{2\alpha+\varepsilon}\cos
t(x-x') {{dt}\over t}.\label{chasA1}\end{gathered}$$
(\[chasA1\]) follows trivially from the same representation of the integrand $\psi(x;k)\psi(x';-k)$ as in the proof of Lemma \[lemma4.1\]. Proof of the resolution of identity (3.10) of Part I is based on the following Lemmas \[lemma4.6\] and \[lemma4.7\].
\[lemma4.6\] In the conditions of Lemma [\[lemma4.5\]]{} for any $x'\in\Bbb R$ and $\gamma>-1$ the following relation takes place, $$\mathop{{\lim}'_\gamma}_{\varepsilon\downarrow0}\big\{\psi_0(x)
\big[\varepsilon\cos2\alpha(x-x')
\cos\varepsilon(x-x')+2\alpha\sin2\alpha(x-x')
\sin\varepsilon(x-x')\big]\big\}=0.$$
The fact that for any $\gamma>-1$ the relation $$\mathop{{\lim}'_\gamma}_{\varepsilon\downarrow0}\big\{\psi_0(x)
\big[2\alpha\sin2\alpha(x-x') \sin\varepsilon(x-x')\big]\big\}=0$$ holds follows from Lemma \[lemma3.7\] in view of (3.2) of Part I. Hence, to prove Lemma \[lemma4.6\], it is sufficient to show that for any $\gamma>-1$ the relation $$\begin{gathered}
\mathop{{\lim}'_\gamma}_{\varepsilon\downarrow0}
\big\{\psi_0(x)\big[\varepsilon\cos2\alpha(x-x')
\cos\varepsilon(x-x')\big]\big\}=0\label{lim67}\end{gathered}$$ is valid. It is true that $$\psi_0(x)
\big[\varepsilon\cos2\alpha(x-x') \cos\varepsilon(x-x')\big]\in
L^2({\Bbb R};(1+|x|)^{-\gamma})\subset CL'_\gamma,\qquad\gamma>-1.$$ Thus, to prove (\[lim67\]) it is sufficient to establish that for any $\varphi(x)\in CL_\gamma$, $\gamma>-1$, the relation $$\lim_{\varepsilon\downarrow0}\int_{-\infty}^{+\infty}\psi_0(x)
\big[\varepsilon\cos2\alpha(x-x')
\cos\varepsilon(x-x')\big] \varphi(x)\,dx=0$$ holds. But in view of (3.2) of Part I its validity follows from the chain of inequalities, $$\begin{gathered}
\left|\int_{-\infty}^{+\infty}\psi_0(x)
\left[\varepsilon\cos2\alpha(x-x')
\cos\varepsilon(x-x')\right] \varphi(x)\,dx\right|^2\leqslant
\varepsilon^2\left(\int_{-\infty}^{+\infty}|\psi_0(x)
\varphi(x)|\,dx\right)^2\\
\qquad{} \leqslant
\varepsilon^2\int_{-\infty}^{+\infty}{{|\psi^2_0(x)|}\over
{(1+|x|)^\gamma}}\,dx\int_{-\infty}^{+\infty}
|\varphi^2(x)|(1+|x|)^\gamma\,dx\to0,\qquad\varepsilon\downarrow0,\end{gathered}$$ derived with the help of the Bunyakovskii inequality. Thus, Lemma \[lemma4.6\] is proven.
\[lemma4.7\] In the conditions of Lemma [\[lemma4.5\]]{} for any $x'\in\Bbb R$ and $\gamma>-1$ the following relation holds, $$\mathop{{\lim}'_\gamma}_{\varepsilon\downarrow0}\left\{
[\psi_0(x)\psi_1(x')+\psi_1(x)\psi_0(x')]
\int_{2\alpha-\varepsilon}^{2\alpha+\varepsilon}\cos
t(x-x') {{dt}\over t}\right\}=0.$$
Proof of Lemma \[lemma4.7\] with the help of the estimate from Lemma 3 from Appendix B of [@andcansok10] and the Bunyakovskii inequality is quite analogous to the proof of Lemma 4 from Appendix B of [@andcansok10].
\[corollary4.1\] The resolution of identity $(3.10)$ of Part [I]{} follows from the resolution of identity $(3.8)$ of Part [I]{} and from Lemmas [\[lemma4.6\]]{}, [\[lemma4.7\]]{} and [\[lemma3.7\]]{}.
The resolution of identity (3.11) of Part I is a corollary of the resolution of identity (3.10) of Part I and of the following Lemma \[lemma4.8\].
\[lemma4.8\] In the conditions of Lemma [\[lemma4.5\]]{} for any $x'\in\Bbb R$ and $\gamma>1$ the following relation takes place, $$\mathop{{\lim}'_\gamma}_{\varepsilon\downarrow0}\left\{\psi_0(x)
\left[ {1\over\varepsilon}\sin^2{\varepsilon\over2} (x-x')\right]\right\}=0.$$
Proof of Lemma \[lemma4.8\] is analogous to the proof of Lemma 3 from Appendix of [@ancansok06].
\[remark4.2\] Let us consider the functional $$\begin{gathered}
\mathop{{\lim}''_\gamma}_{\varepsilon\downarrow0}
\left[{2\over{\pi\varepsilon\alpha}}
\sin^2{\varepsilon\over2} (x-x') \psi_{0}(x)\psi_{0}(x')\right],
\label{funA96}\end{gathered}$$ where $\psi_0(x)$ is the eigenfunction (3.2) of Part I, which is defined by the expression $$\begin{gathered}
\lim_{\varepsilon\downarrow0}
\int_{-\infty}^{+\infty}\left[{2\over{\pi\varepsilon\alpha}}
\sin^2{\varepsilon\over2} (x-x')\,\psi_{0}(x)\psi_{0}(x')\right]
\varphi(x)\,dx\label{limA97}\end{gathered}$$ for all test functions $\varphi(x)\in CL_\gamma$, $\gamma\in\Bbb R$, for which the limit (\[limA97\]) exists. It follows from Lemma \[lemma4.8\] that the functional (\[funA96\]) is trivial (equal to zero) for any $\gamma>1$, but at the same time, in view of the formula (3.12) from [@anso11], this functional is nontrivial (different from zero) for any $\gamma<1$. By virtue of Lemma \[lemma4.8\] the restriction of the functional (\[funA96\]) on the standard space ${\cal{D}}(\Bbb R)\subset CL_\gamma$, $\gamma\in\Bbb R$ is equal to zero. Hence, the support of this functional for any $\gamma\in\Bbb
R$ does not contain any finite real number. On the other hand, one can represent any test function $\varphi(x)\in CL_\gamma$, $\gamma\in\Bbb R$ for any $R>0$ as a sum of two functions from $CL_\gamma$ in the form $$\begin{gathered}
\varphi(x)=\eta(|x|-R)\varphi(x)+[1-\eta(|x|-R)]\varphi(x),\qquad
R>0,\label{reprA98}\end{gathered}$$ where $\eta(x)\in C^\infty_{\Bbb
R}$, $\eta(x)\equiv1$ for any $x<0$, $\eta(x)\in[0,1]$ for any $x\in[0,1]$ and $\eta(x)\equiv0$ for any $x>1$. In view of Lemma \[lemma4.8\] the value of the functional (\[funA96\]) for $\varphi(x)$ is equal to its value for the second term of (\[reprA98\]) for any arbitrarily large $R>0$. Hence, the value of the functional (\[funA96\]) for a test function depends only on the behavior of this function in any arbitrarily close (in the conformal sense) vicinity of the infinity and is independent of values of the function in any finite interval of real axis. In this sense the support of the functional (\[funA96\]) for any $\gamma<1$ consists of the unique element which is the infinity. At last, since (i) for any $\varphi(x)\in CL_\gamma$ and $\gamma\in\Bbb R$ the relation $$\mathop{{\lim}_\gamma}\limits_{R\to+\infty}\eta(|x|-R)\varphi(x)=
\varphi(x)$$ holds; (ii) the restriction of the functional (\[funA96\]) on ${\cal D}(\Bbb R)$ is zero for any $\gamma\in\Bbb
R$ and (iii) the functional (\[funA96\]) is nontrivial for any $\gamma<1$, so the functional (\[funA96\]) for any $\gamma<1$ is discontinuous.
Acknowledgments {#acknowledgments .unnumbered}
---------------
This work was supported by Grant RFBR 09-01-00145-a and by the SPbSU project 11.0.64.2010.
[99]{}
Andrianov A.A., Sokolov A.V., Resolutions of identity for some non-Hermitian Hamiltonians. I. Exceptional point in continuous spectrum, [[*SIGMA*]{}](http://dx.doi.org/10.3842/SIGMA.2011.111) [**7**]{} (2011), 111, 19 pages, [arXiv:1107.5911](http://arxiv.org/abs/1107.5911).
Gel’fand I.M., Vilenkin N.J., Generalized functions, Vol. 4, Some applications of harmonic analysis, Academic Press, New York, 1964.
Sokolov A.V., Andrianov A.A., Cannata F., Non-Hermitian quantum mechanics of non-diagonalizable Hamiltonians: puzzles with self-orthogonal states, [[*J. Phys. A: Math. Gen.*]{}](http://dx.doi.org/10.1088/0305-4470/39/32/S20) [**39**]{} (2006), 10207–10227, .
Andrianov A.A., Cannata F., Sokolov A.V., Spectral singularities for non-Hermitian one-dimensional Hamiltonians: puzzles with resolution of identity, [[*J. Math. Phys.*]{}](http://dx.doi.org/10.1063/1.3422523) [**51**]{} (2010), 052104, 22 pages, [arXiv:1002.0742](http://arxiv.org/abs/1002.0742).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Thermodynamic bulk measurements of binding reactions rely on the validity of the law of mass action and the assumption of a dilute solution. Yet important biological systems such as allosteric ligand-receptor binding, macromolecular crowding, or misfolded molecules may not follow these assumptions and require a particular reaction model. Here we introduce a fluctuation theorem for ligand binding and an experimental approach using single-molecule force-spectroscopy to determine binding energies, selectivity and allostery of nucleic acids and peptides in a model-independent fashion. A similar approach could be used for proteins. This work extends the use of fluctuation theorems beyond unimolecular folding reactions, bridging the thermodynamics of small systems and the basic laws of chemical equilibrium.'
author:
- 'J. Camunas-Soler'
- 'A. Alemany'
- 'F. Ritort'
title: 'Experimental measurement of binding energy, selectivity and allostery using fluctuation theorems'
---
Binding energies are key quantities determining the fate of intermolecular reactions [@stormo2010determining]. Bulk experimental approaches such as surface plasmon resonance, isothermal titration calorimetry and fluorescent ligand binding assays, allow the extraction of binding energies ($\Delta G^0_{{\rm bind}}$) from measurements of the dissociation constant ($K_d$) with accuracy $\sim$1 kcal/mol through the expression: $$\Delta G^0_{{\rm bind}}=-k_{\rm B}T\log\left[K_d\right]~,
\label{eq:masact}$$ where $k_{\rm B}$ is the Boltzmann constant, and $T$ the temperature [@leavitt2001direct; @mcdonnell2001surface]. However, many ligands such as DNA-binding proteins display different binding modes with varying affinities, or require the concerted action of several subunits, making quantitative measurements challenging [@kalodimos2004structure; @kim2013probing].
Force techniques such as optical tweezers can be used to pull on individual ligand-DNA complexes allowing detection of binding events one-at-a-time (Figure 1[**a**]{}, inset) [@bustamante2003ten; @junker2005influence; @cao2007functional; @koirala2011single; @ainavarapu2005ligand; @hann2007effect]. However, force-induced ligand unbinding usually takes place in non-equilibrium conditions, and binding energies cannot be directly inferred from the measured work values. The Crooks fluctuation theorem and the Jarzynski equality [@jarzynski1997nonequilibrium; @crooks2000path] are tools to extract equilibrium free energy differences from work distributions obtained far from equilibrium, allowing the measurement of folding free energies of nucleic acids and proteins, both from fully equilibrated [@liphardt2002equilibrium; @collin2005verification; @shank2010folding] and kinetic states [@maragakis2008differential; @junier2009recovery; @alemany2012experimental]. However, to date the use of fluctuation theorems remains restricted to unimolecular reactions (e.g. folding).
Here we introduce a fluctuation theorem for ligand binding (FTLB) that allows us to directly extract binding energies of bimolecular or higher-order reactions from irreversible work measurements in pulling experiments (see S1.1 in [@suppmaterials]). We first show how cyclic protocols allow an unambiguous classification of experimental pathways in relation to the initial and final state, which is an essential step in the application of these theorems. We then apply the FTLB to directly verify the validity of the law of mass action for dilute ligand solutions. Next we use the FTLB to accurately measure specific and nonspecific binding energies, as well as allosteric effects due to the cooperative binding of ligand pairs. Finally, we show how the FTLB is also applicable to extract binding energies to non-native structures (e.g. misfolded states, prions, chaperones), a measurement inaccesible to most bulk techniques [@orte2008direct; @fierz2012stability; @yu2012direct].
As a proof of principle we investigated the binding of the restriction endonuclease EcoRI to a 30-bp DNA hairpin that contains its recognition site (GAATTC) (see S1.2,S1.3 in [@suppmaterials]). Restriction endonucleases, which bind their cognate sequences with high affinity, are a paradigm of protein-DNA interactions [@lesser1990energetic; @jen1997protein]. In a typical experiment, the hairpin is unfolded (refolded) by increasing (decreasing) the distance ($\lambda$) between the optical trap and the micropipette (Figure 1[**a**]{}). In the absence of ligand, the hairpin folds and unfolds in the force range $F_c\sim12-15$ pN. The binding of EcoRI increases the stability of the hairpin leading to higher unfolding forces ($\sim$23 pN). During a pulling experiment, EcoRI binds DNA when the hairpin is folded. However, since there is no net change in molecular extension upon binding/unbinding, the native ($N$) and bound ($B$) states cannot be distinguished at low forces. In contrast, at forces above $F_c\sim12-15$ pN the bound state ($B$) can be unambiguously distinguished from the unfolded state ($U$), as the hairpin remains folded when the protein is bound but unfolds when it is unbound (Figure 1[**a**]{}, empty, blue and cyan dots respectively).
The FTLB is based on the extended fluctuation relation [@maragakis2008differential; @junier2009recovery; @alemany2012experimental], and relates the work ($W$) performed along a pulling protocol connecting the different EcoRI-hairpin binding states ($N$, $B$, $U$) to their thermodynamic free-energy differences. We performed cyclic protocols that start and end at a force $\sim$21 pN (Figure 1[**b**]{} inset and Fig. S1). This force is well above $F_c\sim12-15$ pN, resulting in paths that connect states $U$ and $B$, that are then classified into four different sets according to their initial and final states ($U\rightarrow U$, $U\rightarrow B$, $B\rightarrow U$, $B\rightarrow B$, Figure 1[**c**]{}). We repeatedly pulled the hairpin and measured the partial work distributions and fraction of paths connecting states $U$ and $B$ ($P^{B\rightarrow U}(W)$, $P^{U\rightarrow B}(W)$, $\phi^{B\rightarrow U}$, $\phi^{U\rightarrow B}$ respectively) and extracted the free-energy difference between states $B$ and $U$, $\Delta
G_{BU}$ ($=G_U-G_B$) using the FTLB (see S1.4 in [@suppmaterials]): $$\frac{\phi^{B\rightarrow U}}{\phi^{U \rightarrow
B}}\frac{P^{B\rightarrow U}(W)}{P^{U\rightarrow
B}(-W)}=\exp\left[\frac{W-\Delta G_{BU}}{k_{\rm
B}T}\right]~~~~.\label{eq:FT}$$
We performed experiments at different EcoRI concentrations, and determined $\Delta G_{BU}$ from the work value ($\widetilde{W}$) at which the partial work distributions cross ($P^{B\rightarrow
U}(\widetilde{W})=P^{U\rightarrow B}(-\widetilde{W})$) by taking $\Delta G_{BU}=\widetilde{W}+k_{\rm B}T\log\left(\phi^{U\rightarrow
B}/\phi^{B \rightarrow U}\right)$ (Figure 1[**d**]{}, S1.5 in [@suppmaterials]). The term $\Delta G_{BU}$ includes all the energetic contributions involved in going from $B$ to $U$ (e.g. binding energy, conformational changes, elastic terms, see S1.6 in [@suppmaterials]). By subtracting the elastic contributions and the energy of formation of the hairpin from the measured $\Delta G_{BU}$ value, we extract the binding energy at zero force ($\Delta G_{{\rm bind}}$) at different EcoRI concentrations (Figure 1[**e**]{} and Tables S1, S2). As shown in Figure 1[**e**]{}, $\Delta G_{{\rm bind}}$ follows the law of mass action (Eq. \[eq:masact\]), $\Delta G_{{\rm
bind}}=\Delta G_{{\rm bind}}^0+k_{\rm B}T\log(C/C_0)$ with $\Delta
G^0_{{\rm bind}}=26\pm0.5$ $k_{\rm B}T$, providing a direct test of its validity. This value is independent on the start/end force of the cyclic protocol and relies on a correct classification of paths (Fig. S2-S3). We also performed titration experiments with varying NaCl concentration showing that EcoRI binding energy has a pronounced salt-dependency with slope $m_{\rm [NaCl]}=-11\pm 2$ $k_{\rm B}T$ (Figure 1[ **f**]{} and Tables S3, S4), in agreement with previous bulk experiments [@terry1983thermodynamic; @koch2002probing]. Finally, we repeated experiments with hairpins containing non-cognate DNA sequences which did not show binding in the same range of EcoRI concentrations, proving the specificity of the interaction [@bustamante2003ten; @koirala2011single].
To further test the validity of Eq. \[eq:FT\], we investigated a model system consisting of a short oligonucleotide of 10 bases that binds a DNA hairpin. The oligonucleotide can bind the substrate by base-pairing complementarity when the hairpin is in the unfolded ($U$) state, thereby inhibiting the refolding of the hairpin at low forces. At forces below the critical force range of the hairpin ($F_c\sim8-10$ pN), the oligo-bound state ($B$) competes with the formation of the native hairpin ($N$), and states $B$ and $N$ can be distinguished due to their different molecular extension (Figure 2[**a**]{}). To apply Eq. \[eq:FT\], we considered cyclic protocols that start and end at a force lower than the range $F_c$ (Figure 2[**b**]{}). From the measured partial work distributions and fractions of paths connecting $N$ and $B$ (Figure 2[**c**]{}) we extracted the binding energies at zero force ($\Delta G_{{\rm bind}}$) (Figure 2[**d**]{} and Tables S5, S6). Measured binding energies again follow the law of mass action with $\Delta G^0_{{\rm bind}}=22\pm1$ $k_{\rm B}T$ (Figure 2[**d**]{}). This agrees with theoretical predictions using the nearest-neighbour model ($\Delta G^0_{\rm th}=22$ $k_{\rm
B}T$) [@santalucia1998unified; @huguet2010single] and equilibrium experiments performed at the coexistence force of the hairpin, where hopping due to binding/unbinding is observed (Figs. S4-S6 and Table S7). The inclusion of the ratio $\phi^{N\rightarrow B}/\phi^{B \rightarrow N}$ (Figure 2[**d**]{}, inset) is essential to recover the correct binding energies. Previous attempts to derive binding energies using unidirectional work measurements and the Jarzynski equality did not account for concentration-dependent effects in the chemical potential that are essential in Equation \[eq:FT\] [@koirala2011single] .
To prove the general power of the method, we studied echinomycin, a small DNA bis-intercalator with selectivity for CG steps [@van1984echinomycin] that binds contiguous ACGT sites cooperatively [@bailly1996cooperativity]. We performed experiments with a 12-bp DNA hairpin containing a single CG-step (SP hairpin) that shows rupture forces in the range $F_c\sim6-8$ pN (Figure 3[**a**]{} and Fig. S7). In the presence of echinomycin the histogram of rupture forces is shifted to higher values and shows a bimodal distribution, indicating two binding modes: a high-affinity binding to the specific CG-site (high-force peak, $\sim18$ pN), and a low-affinity binding to other non-specific sites (low-force peak, $\sim12$ pN). To confirm this, we pulled a hairpin in which we removed the specific binding site by inverting the CG-motif (NSP hairpin). In the presence of ligand only the low affinity peak is observed (Figure 3[**a**]{}).
To extract the binding energy of each mode, we performed cyclic protocols that start at a force high enough to discriminate both binding modes: we used $\sim18$ pN ($\sim13$ pN) for the SP (NSP) hairpin in order to extract both the specific and nonspecific binding energy of the ligand. In this way, we obtained paths connecting states $B$ and $U$, and extracted the binding energy of the specific and nonspecific modes (Tables S8-S11). For both binding modes, $\Delta G_{{\rm
bind}}$ follows the law of mass action with $\Delta G^0_{{\rm bind,
SP}}=20.0\pm0.8$ $k_{\rm B}T$ and $\Delta G^0_{{\rm bind,
NSP}}=13.2\pm0.5$ $k_{\rm B}T$ (Figure 3[**b**]{}), which give affinities of 2 nM and 1.8 $\mu$M respectively (Eq. \[eq:masact\]). This measurement of an affinity in the nM range for the specific binding is compatible with quasi-equilibrium experiments (Fig. S8) and improves previous studies where accurate measurements could not be obtained due to the concurrent action of both modes [@leng2003energetics].
The FTLB allows us to go beyond free-energy measurements of single ligands, and measure allosteric effects between ligands binding at nearby positions [@kim2013probing] . For this, we designed hairpin $NC$ which contains two ACGT sites separated by 2 bp (Figure 3[**c**]{}). The simultaneous binding of two ligands can be distinguished from the binding of a single ligand from the force rips observed in the force-distance curve (Fig. S9). By applying the FTLB we extracted the binding energy per ligand in the single and double bound states, and found that binding is favoured by the presence of a neighbouring ligand. The FTLB allows us to quantitatively test the distance-dependence of this allosteric effect by performing a differential measurement of binding energies with hairpin $C$, which contains two contiguous sites (Figure 3[**c**]{}, Tables S12-S14). The binding energy per ligand we obtain in the double bound state in hairpin $C$ is $2.4\pm0.5$ $k_{\rm B}T$ higher than in hairpin $NC$, providing a direct experimental measurement of cooperativity effects in ligand pairs as a function of their distance.
Single-molecule manipulation is particularly suited to observe the formation of misfolded structures (e.g. prions, amyloids) [@yu2012direct; @heidarsson2014direct], but methods to characterize binding to these species are currently lacking. By applying the FTLB it is possible to extract the binding energy to these kinetically stabilized non-native structures. By using a DNA hairpin with two binding sites separated by 4bp, we observe the formation of a misfolded structure consisting of two short (4bp) hairpins in series (Figure 3[**d**]{}, hairpin M). Such an off-pathway kinetic state is unobservable in the absence of ligand due to its low energy of formation, however it is kinetically stabilized by the binding of the ligand. We applied Eq. \[eq:FT\] by choosing a starting point of the cyclic protocol where the native-bound and misfolded-bound conformations are distinguishable ($F\sim10$ pN), and found that the energy of binding to both configurations are equal ($\Delta G_{\rm bind,M-N} = 2\pm1$ $k_{\rm B}T$, Fig. S10 and Tab. S15, S16).
In this work, we have introduced a fluctuation theorem for ligand binding (FTLB) to directly determine binding energies as a function of ligand concentration in single-molecule experiments. Using different biomolecular systems of increasing complexity we provide a single-molecule verification of the law of mass action, and show how the FTLB can account for mass exchange between a molecular system and the environment. We can resolve binding energies to specific and non-specific sites with affinities spanning six orders of magnitude. The FTLB provides a direct experimental measurement of binding energies without assuming any model or reaction scheme, which is particularly useful in cases where the law of mass action does not hold. To show this, we applied the FTLB in two situations where this may happen: the cooperative binding of multiple ligands to the same substrate and the stabilization of kinetic structures through ligand binding - both measurements inaccessible to bulk methods and relevant to many interactions between proteins and ligands.
The use of an inherently non-equilibrium method to obtain equilibrium binding energies also grants access to molecular interactions that equilibrate over very long timescales (e.g. nucleosome assembly) and that can only be currently measured by indirect techniques such as competition assays [@leavitt2001direct; @fierz2012stability; @thaastrom2004histone]. The FTLB relates work measurements to binding energies without making any assumption on reaction kinetics or the ideal solution limit. Therefore it might be also used to test the explicit breakdown of the law of mass action in conditions where it is not applicable, for instance in crowded environments, where ligands exhibit compartmentalized dynamics due to steric hindrance interactions [@schnell2004reaction]. Lastly, the applicabilty of the FTLB is not restricted to biomolecular reactions, and might be directly applied to other interacting systems that can only be explored through non-equilibrium methods.
Acknowledgements {#sec:Acknow}
================
All authors acknowledge funding from grants ERC MagReps 267 862, FP7 grant Infernos 308850, Icrea Academia 2013 and FIS2013-47796- P.
All data used in this study are included in the main text and in the supplementary materials.
Author Contributions
====================
J.C.-S. and A.A. equally contributed to this work. \[sec:Author\]
Figure Captions
===============
Figure 1
--------
[**EcoRI binding to DNA.**]{} [**(a)**]{} Unfolding/refolding force-distance curves of a DNA hairpin in the absence (magenta/black) and presence (blue/cyan) of EcoRI protein. The bound ($B$) and unfolded ($U$) states are discriminated at high force by the presence of two distinct force branches. [**(b)**]{} Cyclic pulling curves classified according to their initial (blue dot) and final state (cyan dot) that start and end at a high force ($\sim$21 pN). Work equals the enclosed area between the two curves and is shown in dark/light gray for positive/negative values. [**(c)**]{} Paths of a non-equilibrium cyclic protocol connecting different initial and final states. [**(d)**]{} Partial work distributions of $U\rightarrow B$ (green) and $B\rightarrow U$ (magenta) transitions at different EcoRI concentrations. [**(e)**]{} Binding energy of EcoRI (blue) and fit to the law of mass action (red line) at (130 mM $\rm{Na}^{+}$, $25^{\circ}$C, $C_0=1$ M). [**(f)**]{} Binding energy of EcoRI at varying \[NaCl\] (1 nM EcoRI). Error bars were obtained from bootstrap using 1000 re-samplings of size N (N is total number of pulls for each condition shown in Tables S1 and S3).
Figure 2
--------
[**Oligo binding to DNA.**]{} [**(a)**]{} Scheme of native ($N$), unfolded ($U$) and oligo bound ($B$) states. [**(b)**]{} Cyclic pulling curves that start and end at low forces ($\sim$6 pN) classified according to their initial (blue dot) and final state (cyan dot). [ **(c)**]{} Partial work distributions of $B\rightarrow N$ (green) and $N\rightarrow B$ (magenta) transitions. [ **(d)**]{} Binding energy of the 10-base oligo (blue) and fit to the law of mass action (red line). The value obtained from hopping equilibrium experiments at \[oligo\]=400 nM (see S1.7 in [@suppmaterials]) is shown in cyan. (Inset) Contribution of the ratio $\phi^{N\to B}/\phi^{B\to N}$ to the binding energy. Error bars were obtained from bootstrap as described in Fig. 1.
Figure 3
--------
[**Binding specificity, allostery and kinetic stabilization of misfolded states for the peptide Echinomycin.**]{} [**(a)**]{} First rupture force distribution of hairpins SP (red) and NSP (blue) in the absence (light) and presence (dark) of Echinomycin. [**(b)**]{} Binding energy of Echinomycin to a specific (red) and nonspecific (blue) site, and fit to the law of mass action. [**(c)**]{} Hairpins C and NC contain two specific binding sites (red boxes) placed contiguously or separated by 2-bp respectively. Binding energy per ligand when one (blue) or two ligands (green) are bound to hairpins NC or C (magenta) (\[Echninomycin\]=3 $\mu$M). Gaussian distributions are reconstructed from mean and variance of measurements. The wider distribution for the single bound state in hairpin NC (blue) is due to the lower number of paths reaching this state at high ligand concentration, increasing measurement error. [**(d)**]{} Pulling cycle of hairpin M in the presence of Echinomycin. The unfolding curve (blue) shows two force rips at $F\sim20$ pN corresponding to the unbinding of two ligands bound to specific sites. In the refolding curve (cyan), the hairpin does not fold back to the native state, and misfolds into a kinetically stabilized configuration of longer molecular extension ($\sim40\%$ of refolding curves at \[Echninomycin\]=10 $\mu$M). Error bars were obtained from bootstrap as described in Fig. 1.
Materials and Methods
=====================
Mathematical proof of the Fluctuation Theorem for Ligand Binding (FTLB)
-----------------------------------------------------------------------
The FTLB is derived following the same steps as in [@junier2009recovery]. Consider a system with a fluctuating number of particles $N$, which correspond to the ligand molecules. The system evolves under an experimental protocol $\lambda(t)$, where $\lambda$ denotes the control parameter and in our case corresponds to the position of the optical trap relative to the pipette. We discretize in time the protocol as $\lambda(t)=\{\lambda_0, \lambda_1,\dots,\lambda_{t_f}\}$, where $\lambda_i$ ($i=0,1,\dots,t_f$) denotes the value of $\lambda$ at the time of the protocol $t=i\Delta t$ (being $\Delta t$ the time discretization unit), and $t_f$ denotes the duration of the protocol. Along the protocol $\lambda(t)$ the system follows a given trajectory $\Gamma$, where a sequence of configurations ${{\mathcal{C}}}$ are sampled. The trajectory can be discretized as $\Gamma = \{ {{\mathcal{C}}}_0, {{\mathcal{C}}}_1, \dots, {{\mathcal{C}}}_{t_f}\}$. Each configuration ${{\mathcal{C}}}_i$ ($i\in 0, 1, \dots, t_f$) is characterized by the number of particles, $N_i$, and the degrees of freedom of each particle.
The equilibrium probability to be in a given configuration ${{\mathcal{C}}}_i$ at $\lambda$ can be written, according to the grand-canonical ensemble, as: $$\begin{aligned}
\label{eq: total equilibrium}
P^{\rm eq}({{\mathcal{C}}}_i)=\frac{z^{N_i}e^{-\beta E_\lambda({{\mathcal{C}}}_i)}}{Z^{GC}},& &Z^{GC} = \sum_{N_i}\sum_{\mathcal{C}_i(N_i)}z^{N_i} e^{-\beta E_\lambda(\mathcal{C}_i)}\end{aligned}$$ where $\beta=({k_{\rm B}T})^{-1}$ (being ${k_{\rm B}}$ the Boltzmann constant and $T$ the absolute temperature), $z$ is the fugacity of the system (equal to $e^{\beta\mu}$, being $\mu$ the chemical potential of the ligand molecules), $Z^{GC}$ is the grand canonical partition function, and $E_\lambda({{\mathcal{C}}}_i)$ is the energy of the configuration ${{\mathcal{C}}}_i$ at $\lambda$.
We suppose that the dynamics of the system satisfy the following detailed balance condition: $$\frac{P({{\mathcal{C}}}_t\to {{\mathcal{C}}}_{t+1})}{P({{\mathcal{C}}}_{t+1}\to {{\mathcal{C}}}_t)}=z^{N_{t+1}-N_t}e^{-\beta\left(E_{\lambda(t+1)}({{\mathcal{C}}}_{t+1})-E_{\lambda(t+1)}({{\mathcal{C}}}_t)\right)}.$$ Therefore, the probability of the system to follow a given trajectory $\Gamma$ (without imposing any initial and final configuration), and the probability to follow its time reversed $\hat{\Gamma}$ is: \[eq: 1\] $$P(\Gamma)= \prod_{t=0}^{t_f-1}P({{\mathcal{C}}}_t\to {{\mathcal{C}}}_{t+1}),
\qquad \text{ } \qquad
\hat{P}(\hat\Gamma)= \prod_{t=0}^{t_f-1}P(\hat {{\mathcal{C}}}_t\to \hat {{\mathcal{C}}}_{t+1}),
\tag{\theequation a,b}\label{eq: 2}$$ where $\hat {{\mathcal{C}}}_t={{\mathcal{C}}}_{t_f-t}$.
We assume that in the forward protocol $\lambda(t)$ the system starts in partial equilibrium at ${{\mathcal{C}}}_0$, while in the reversed protocol $\hat\lambda(t)$ it starts in partial equilibrium at $\hat {{\mathcal{C}}}_0={{\mathcal{C}}}_{t_f}$ (and $\hat\lambda(0)=\lambda(t_f)$). The partial equilibrium probability density function of a given configuration ${{\mathcal{C}}}_i\in S$, where $S$ is a subset of configurations accessible by the system, can be written as [@junier2009recovery]:
$$\begin{aligned}
\label{eq: partial}
P^{\rm eq}_{S}({{\mathcal{C}}}_i)&=\chi_{S}({{\mathcal{C}}}_i)\frac{z^{N_i}e^{-\beta E({{\mathcal{C}}}_i)}}{\sum_{N_i}\sum_{{{\mathcal{C}}}_i\in S}z^{N_i}e^{-\beta E({{\mathcal{C}}}_i)}}=\chi_{S}({{\mathcal{C}}}_i)P^{\rm eq}({{\mathcal{C}}}_i)\frac{Z^{GC}}{Z^{GC}_{S}},\end{aligned}$$
where $\chi_{S}({{\mathcal{C}}}_i)$ is equal to one if ${{\mathcal{C}}}_i\in S$ and zero otherwise, and $Z^{GC}_S$ is the grand canonical partition function restricted to the subset $S$.
Suppose that the system starts in non-equilibrium conditions. Particularly, the system starts in partial equilibrium at the kinetic state $S_0$ in the forward trajectory and at at the kinetic state $S_{t_f}$ in the reversed one. A kinetic state is a partially equilibrated region of configurational space, meaning that during a finite amount of time the system is confined and thermalized within that region. This is mathematically described by a Boltzmann-Gibbs distribution restricted to configurations contained in that region. Then:
$$\begin{aligned}
\frac{P^{\rm eq}_{\lambda(0),S_0}({{\mathcal{C}}}_0)P(\Gamma)}{P^{\rm eq}_{\lambda(t_f),S_{t_f}}({{\mathcal{C}}}_{t_f})\hat P(\hat\Gamma)} &=
\frac{\chi_{S_0}({{\mathcal{C}}}_0)P^{\rm eq}_{\lambda(0)}({{\mathcal{C}}}_0)Z^{GC}_{\lambda(0)}}{Z^{GC}_{\lambda(0),S_0}}
\frac{Z^{GC}_{\lambda(t_f),S_{t_f}}}{\chi_{S_{t_f}}({{\mathcal{C}}}_{t_f})P^{\rm eq}_{\lambda(t_f)}({{\mathcal{C}}}_{t_f})Z^{GC}_{\lambda(t_f)}}
\frac{P(\Gamma)}{\hat P(\hat\Gamma)}\\
&=\frac{\chi_{S_0}({{\mathcal{C}}}_0)Z^{GC}_{\lambda(t_f),S_{t_f}}}{\chi_{S_{t_f}}({{\mathcal{C}}}_{t_f})Z^{GC}_{\lambda(0),S_0}}e^{\beta W(\Gamma)},\end{aligned}$$
where: $$\label{eq: work}
W(\Gamma)=\sum_{t=0}^{t_f-1}\left(E_{\lambda(t+1)}({{\mathcal{C}}}_{t})-E_{\lambda(t)}({{\mathcal{C}}}_t)\right),$$ is as the work exerted upon the system along the forward process.
Now we compute the average of the quantity $\mathcal{O}e^{-\beta W}$ over the forward trajectories that start in partial equilibrium in a configuration ${{\mathcal{C}}}_0\in S_0$ and end in ${{\mathcal{C}}}_{t_f}\in S_{t_f}$. Therefore: $$\begin{aligned}
\langle \mathcal{O}e^{-\beta W} \rangle^{S_0\to S_{t_f}} &= \frac{\sum_\Gamma P^{\rm eq}_{\lambda(0), S_0}({{\mathcal{C}}}_0)P(\Gamma)\chi_{S_{t_f}}({{\mathcal{C}}}_{t_f}) \mathcal{O}(\Gamma)e^{-\beta W(\Gamma)}}{\sum_\Gamma P^{\rm eq}_{\lambda(0), S_0}({{\mathcal{C}}}_0)P(\Gamma)\chi_{S_{t_f}}({{\mathcal{C}}}_{t_f})}\\
&=\frac{\phi^{S_{t_f}\to S_0}}{\phi^{S_0\to S_{t_f}}} \frac{Z^{GC}_{\lambda(t_f),S_{t_f}}}{Z^{GC}_{\lambda(0),S_0}}
\frac{\sum_{\Gamma}P^{\rm eq}_{\hat\lambda(t_0),\hat S_{0}}(\hat {{\mathcal{C}}}_{0})\hat P(\hat\Gamma)\chi_{\hat S_{t_f}}(\hat {{\mathcal{C}}}_{t_f}) \hat{\mathcal{O}}(\hat\Gamma)}{\sum_{\Gamma}P^{\rm eq}_{\hat\lambda(t_0),\hat S_{0}}(\hat {{\mathcal{C}}}_{0})\hat P(\hat\Gamma)\chi_{\hat S_{t_f}}(\hat {{\mathcal{C}}}_{t_f})}\\
&=\frac{\phi^{S_{t_f}\to S_0}}{\phi^{S_0\to S_{t_f}}} \frac{Z^{GC}_{\lambda(t_f),S_{t_f}}}{Z^{GC}_{\lambda(0),S_0}} \langle \hat{\mathcal{O}}\rangle^{S_{t_f}\to S_0}.\end{aligned}$$
By defining $\mathcal{O}(\Gamma)=\delta(W-W(\Gamma))$ we obtain the extended Crooks relation in the grand-canonical ensemble, or equivalently the Fluctuation Theorem for Ligand Binding (FTLB):
\[eq: crooks\] $$\begin{aligned}
\frac{\phi^{S_0\to S_{t_f}}}{\phi^{S_{t_f}\to S_0}}\frac{P^{S_0\to S_{t_f}}(W)}{P^{S_{t_f}\to S_0}(-W)}&=e^{\beta W}\frac{Z^{GC}_{\lambda(t_f),S_{t_f}}}{Z^{GC}_{\lambda(0),S_0}}\\
&=\exp\left[\beta \left(W-\Delta G_{S_0S_{t_f}}\right)\right],\end{aligned}$$
where $\Delta G_{S_0S_{t_f}} = G(\lambda(t_f),S_{t_f})- G(\lambda(0),S_0)$.
Force spectroscopy experiments with optical tweezers
----------------------------------------------------
Experiments are performed with a highly stable miniaturized dual-beam optical tweezers described in previous studies [@huguet2010single]. The DNA hairpins are tethered between two beads by using short dsDNA handles (29-bp) that are differentially end-labelled with biotin and digoxigenin to attach each handle to a different bead (see Supp. Section \[sec: Summary of DNA hairpins used in this work\] for hairpin sequences and synthesis details) [@forns2011improving]. Pulling speed is set at 190 nm/s in all the experiments. For the EcoRI experiments, longer dsDNA handles ($\sim$600-bp) are used to reduce nonspecific interactions mediated by the protein and beads. For these experiments, the microfluidics chamber was also coated with Poly(ethylene glycol) (PEG) to avoid protein loss due to nonspecific absorption on the glass surface [@cheng2011single]. Experimental conditions for each interaction were chosen to be comparable to previous ensemble and single-molecule studies: for EcoRI (Hepes 10 mM pH7.5, EDTA 1 mM, NaCl 130 mM, BSA 0.1 mg/ml, 100 $\mu$M, DTT, 0.01% NaN$_3$) and salt titrated in the range 60-180 mM NaCl; for the oligonucleotide (Tris 10 mM pH7.5, EDTA 1 mM, 100 mM NaCl, 0.01% NaN$_3$); and for echinomycin (Tris 10 mM pH7.5, EDTA 1 mM, 100 mM NaCl, 2% DMSO, 0.01% NaN$_3$). All ligands were obtained from commercial sources and used without further purification: EcoRI (New England Biolabs, 100 U/$\mu$l, $\sim$800 nM dimer), oligonucleotide (Eurofins MWG Operon, HyPur grade), Echinomycin (Merck Millipore). Concentrations were confirmed using a spectrophotometric analysis for the oligonucleotide (extinction coefficient 97400 M$^{-1}$cm$^{-1}$ at $\lambda$=260 nm) and echinomycin (extinction coefficient 11500 M$^{-1}$cm$^{-1}$ at $\lambda$=325 nm), whereas for EcoRI we performed an electrophoretic mobility shift assay using previously described protocols [@sidorova1996differences]. All experiments were performed at 25. In all the cases, the number of cycles obtained per molecule range between 20 and 1300. A minimum of 2 and a maximum of 10 molecules were pulled in each case (see tables \[tab: EcoRI N\], \[tab: ecori N sal\], \[tab: oligo N\], \[tab: echi N\], \[tab: echins N\], \[tab: echi c nc N\] \[tab: echi m N\]).
DNA hairpins used in this work {#sec: Summary of DNA hairpins used in this work}
------------------------------
DNA hairpin sequences, secondary structure, binding sites, thermodynamic information and synthesis details:
{width="7.7cm"}
Haipin ${\Delta G}_{NU}^0$ (${k_{\rm B}T}$) Handle length
------------------- -------------------------------------- ---------------
EcoRI 65 $\pm$ 3 long
Rev 67 $\pm$ 2 long
Rnd 68 $\pm$ 2 long
Oligo 28 $\pm$ 1 short
SP 19.7 $\pm$ 0.5 short
NSP 19.4 $\pm$ 0.3 short
C 18 $\pm$ 1 short
NC 18 $\pm$ 1 short
M ([*Native*]{}) 33 $\pm$ 1 short
([*Misfolded*]{}) 10 $\pm$ 1
[**Blue-Top.**]{} Hairpins used to study the binding of EcoRI to its recognition sequence (5’-GAATTC-3’). Hairpin EcoRI contains the specific binding site, indicated in red; Hairpin Rev contains its reversed sequence (red); and Hairpin Rnd contains a random sequence (red). We did not observe binding of EcoRI to hairpins Rnd and Rev in pulling experiments using these hairpins.\
[**Red-Middle.**]{} Hairpin (left) used to study the specific binding of an oligonucleotide (right) to its complementary ssDNA sequence. The binding site of the oligo is indicated in red.\
[**Green-Bottom.**]{} Hairpins used to study the binding of echinomycin to different binding motifs. Hairpin SP contains a specific binding site 5’-ACGT-3’ (red). In hairpin NSP the specific binding site is removed by doing a sequence permutation (red). Hairpin C contains two contiguous binding sites (red), while hairpin NC contains two binding sites separated by two basepairs (red). Hairpin M contains two binding sites that are separated by four base pairs (hairpin M, native). In the presence of echinomycin, a misfolded structure containing two serially connected hairpins (hairpin M, misfolded) becomes kinetically stabilized by the binding of the ligand to two 4-bp hairpins containing the 5’-ACGT-3’ motif (red).
Mean values for the free energy of formation at zero force at 25 and 130 mM NaCl (which are the experimental conditions unless stated otherwise) have been obtained by averaging over the results provided by the nearest-neighbor model and the unified oligonucleotide set of basepair free energies measured in bulk [@mfold; @santalucia1998unified] and unzipping [@huguet2010single] experiments. Error bars are standard errors obtained between the two different estimations. To pull on the hairpins, handles of two different length are used in the experiments: short (29 bp) and long (500 bp) dsDNA handles.
For the experiments with echinomycin and the oligonucleotide we used a short handle construct (total handle length: 58-bp). This short handles construct is better suited to study small ligands that might non-specifically bind to the dsDNA handles. Due to the short length of this construct, it can be synthesized by direct annealing and ligation of three partially complementary oligonucleotides that create the hairpin structure and dsDNA handles, as described in previous studies [@forns2011improving].
For the experiments performed with EcoRI we used a long handle construct to maintain a larger separation between the two beads at low forces (total handle length: 1322-bp). The synthesis is similar to the protocol described in [@forns2011improving]. Briefly, the two handles are performed by PCR amplification of plasmid pBR322 to obtain DNA fragments that contain a restriction site for TspRI or Tsp45I respectively, but do not contain potential binding sites for the ligand (i.e. EcoRI). A biotin tag is introduced in one of the handles using a 5’-biotinylated primer on the PCR reaction. The other handle is tailed with digoxigenin-dUTP using the 3’-5’ exonuclease activity of T4 DNA polymerase. After digestion of the labelled products with each enzyme, the TspRI/Tsp45I cohesive ends are used to anneal and ligate the handles to the the hairpin structure, that is assembled using oligonucleotides.
Application of the FTLB: how to {#sec: Application of the Fluctuation relation to our experiments}
-------------------------------
To apply the FTLB to cyclic pulling protocols we follow the next steps:
1. Identification of initial and final states in the non-equilibrium protocol.
In the particular cases studied in our work, initial and final states in the cyclic pulling experiments are:
- Specific binding of EcoRI to dsDNA: $B$ and $U$ (high forces).
- Specific binding of an oligo to ssDNA: $B$ and $N$ (low forces).
- Specific binding of echinomycin to dsDNA: $B$ and $U$ (high forces).
- Non-specific binding of echinomycin to dsDNA: $B$ and $U$ (high forces).
- Cooperative binding of echinomycin to dsDNA: $B_2$, $B$ and $U$ (high forces).
- Kinetically-stabilized non-native structures due to the simultaneous binding of two echinomycin ligands to dsDNA, misfolded: $N$, $M$.
2. Classification of trajectories.
Each trajectory can be classified as a function of the initial and final state. For instance, in EcoRI experiments we have four types of trajectories: ($i$) start at $B$ and end at $U$; ($ii$) start at $B$ and end at $B$ ; ($iii$) start at $U$ and end at $B$ ; and ($iv$) start at $U$ and end at $U$.
3. Obtain partial work distributions.
In each cyclic trajectory, the work is calculated as the area enclosed in the trajectory. The partial work distribution is the histogram of work values restricted to each type of trajectory.
4. Obtain the grand-canonical partial partition function for the intial and the final state.
The partial partition function for a state $S$ is computed in the restricted subset of configurations ${{\mathcal{C}}}_j$ that characterize state $S$: $$\label{eq: partial ZGC}
Z^{GC}_S = \sum_{N_i\in S}\sum_{{{\mathcal{C}}}_j(N_i)\in S} z^{N_i}e^{-\beta E({{\mathcal{C}}}_j)},$$ where $N_i$ and $E(C_i)$ are the number of bound particles and the energy in configuration $C_i$, $z=e^{\beta\mu}$ is the fugacity, and $\mu$ is the chemical potential of the binding agent.
In the particular cases studied in this work these are:
- Specific binding of EcoRI and echinomycin to dsDNA. \[eq: ZGC EcoRI\] $$Z^{GC}_U = e^{-\beta G_U(\lambda_0)},
\qquad \text{ } \qquad
Z^{GC}_B = ze^{-\beta\left({\varepsilon}+ G_{n}(\lambda_0)\right)}=e^{-\beta\left({\varepsilon}-\mu+ G_{n}(\lambda_0)\right)}.
\tag{\theequation a,b}\label{eq: ZGC EcoRIab}$$ The term $G_U(\lambda_0)$ is the free energy of the hairpin in state $U$ at $\lambda_0$. The term $G_{n}(\lambda_0)$ is the free energy of the hairpin at $\lambda_0$ with $n$ unfolded basepairs in the hairpin stem prior to the binding site, ${\varepsilon}$ is the binding energy of the ligand and $\mu$ is its chemical potential. We refer to $\mu-{\varepsilon}$ as the binding free energy ${\Delta G}_{\rm bind}$.
- Specific binding of a short oligo to its complementary ssDNA sequence. \[eq: ZGC oligo\] $$Z^{GC}_N = e^{-\beta G_{N}(\lambda_0)},
\qquad \text{ } \qquad
Z^{GC}_B = ze^{-\beta\left({\varepsilon}+G_U(\lambda_0)\right)}=e^{-\beta\left({\varepsilon}-\mu+ G_U(\lambda_0)\right)},
\tag{\theequation a,b}\label{eq: 3}$$ where $G_N(\lambda_0)$ and $G_U(\lambda_0)$ are the free energies of the hairpin at $\lambda_0$ in the folded ($N$) or unfolded ($U$) state respectively, and ${\Delta G}_{\rm bind}=\mu-{\varepsilon}$ is the binding free energy of the oligonucleotide.
- Non-specific binding of echinomycin to dsDNA. \[eq: ZGC EcoRI nons\] $$Z^{GC}_U = e^{-\beta G_U(\lambda_0)},
\qquad \text{ } \qquad
Z^{GC}_{B_i} = ze^{-\beta\left({\varepsilon}_i+ G_{n_i}(\lambda_0)\right)}=e^{-\beta\left({\varepsilon}_i-\mu+ G_{n_i}(\lambda_0)\right)},
\tag{\theequation a,b}\label{eq: ZGC EcoRIab2}$$ where $B_i$ ($i=1,\dots,\mathcal{N}$) is a possible binding state (echinomycin binding to any site in dsDNA), ${\varepsilon}_i$ is the binding free energy at position $i$ and $G_{n_i}(\lambda_0)$ is the free energy of the hairpin at $\lambda_0$ where the $n_i$ basepairs prior to the position binding site of echinomycin are unfolded.
The FTLB applies to each pair os states $B_i$ and $U$, therefore: $$\label{eq: FT sum}
\sum_{i=1}^{{\mathcal{N}}}\frac{\phi^{B_i\to U}}{\phi^{U\to B_i}}\frac{P^{B_i\to U}(W)}{P^{U\to B_i}(-W)} = \sum_{i=1}^{{\mathcal{N}}}e^{\beta W}e^{-\beta\left(\mu-{\varepsilon}_i +{\Delta G}_{n_iU}(\lambda_0)\right)}.$$ By assuming that $\phi^{B_i\to U} = \phi^{B\to U}$ and $\phi^{U\to B_i} = \frac{1}{{{\mathcal{N}}}}\phi^{U\to B}$, ${\varepsilon}_i={\varepsilon}_n$ ($i=1\dots\mathcal{N}$) and: $$\begin{aligned}
\label{eq: PBUW}
P^{B\to U}(W) &= \frac{1}{{{\mathcal{N}}}}\sum_{i=1}^{{\mathcal{N}}}P^{B_i\to U}(W),\end{aligned}$$ $$P^{U\to B}(-W)\simeq P^{U\to B_i}(-W),~~\forall i,$$ it can be shown that:
$$\begin{aligned}
\label{eq: FT sum2}
\frac{\phi^{B\to U}}{\phi^{U\to B}}\frac{P^{B\to U}(W)}{P^{U\to B}(-W)} &= {e^{\beta W}}\frac {e^{-\beta(\mu-{\varepsilon}_n)}}{{{\mathcal{N}}}^2}\sum_{i=1}^{{\mathcal{N}}}e^{-\beta{\Delta G}_{n_iU}(\lambda_0)}\\
&=e^{\beta \left(W-{\Delta G}_{BU}\right)}\end{aligned}$$
where ${\Delta G}_{BU} = {\Delta G_{\rm bind}}-{k_{\rm B}T}\log\sum_{i=1}^{{\mathcal{N}}}e^{-\beta{\Delta G}_{n_iU}(\lambda_0)} + 2{k_{\rm B}T}\log {{\mathcal{N}}}$.
- Simultaneous specific binding of two echinomycin molecules to sequential dsDNA sites (cooperativity). \[eq: ZGC Coop t\] $$Z^{GC}_U = e^{-\beta G_U(\lambda_0)},
\qquad \text{ } \qquad
Z^{GC}_{B^2} = z^2e^{-\beta(2{\varepsilon}+G_{n}(\lambda_0))},
\tag{\theequation a,b}\label{eq: ZGC Coop}$$
- Stabilization of misfolded states through simultaneous binding of two echinomycin molecules to sequential dsDNA sites. \[eq: ZGC Misfolded\] $$Z^{GC}_B = z^2e^{-\beta\left(2{\varepsilon}+G_N(\lambda_0)\right)},
\qquad \text{ } \qquad
Z^{GC}_M = z'^2e^{-\beta\left(2{\varepsilon}'+G_M(\lambda_0)\right)}.
\tag{\theequation a,b}\label{eq: ZGC misfab}$$
5. Plug everything into the FTLB.
According to Eq. , we write the FTLB for trajectories $A\to B$ ($B\to A$) as: $$\label{eq: FR EcoRI}
\frac{\phi^{A\to B}}{\phi^{B\to A}}\frac{P^{A\to B}(W)}{P^{B\to A}(-W)} = e^{\beta W}\frac{Z^{GC}_B}{Z^{GC}_A}=e^{\beta (W-{\Delta G}_{AB})},$$ Use the Bennett acceptance ratio method for a better estimation of ${\Delta G}_{AB}$ (section \[sec: Bennett\]).
6. Extract the elastic contributions to ${\Delta G}_{AB}$ in order to get the binding free energy ${\Delta G_{\rm bind}}$ (Section \[sec: energy contributions\]).
Bennett acceptance ratio method {#sec: Bennett}
-------------------------------
The Bennett acceptance ratio method is used to estimate the free-energy difference $\Delta G_{AB}$ between two states that satisfies Eq. from non-equilibrium work measurements. Given a set of $n_F$ ($n_R$) forward (reversed) work measurements $W_i$, it is shown in [@bennett; @shirts] that the solution $u$ of the following transcendental equation: $$\frac{u}{{k_{\rm B}T}} = -\log\left(\frac{\phi^{A\to B}}{\phi^{B\to A}}\right) + z_R(u) - z_F(u),$$ where:
$$z_R(u)=\log\frac{1}{n_R}\sum_{i=1}^{n_R}\left( \frac{e^{-\beta W_i}}{1+\frac{n_F}{n_R}e^{-\beta(W_i+u)}}\right)$$
$$z_F(u)=\log\frac{1}{n_F}\sum_{i=1}^{n_F}\left( \frac{1}{1+\frac{n_F}{n_R}e^{\beta(W_i-u)}} \right)$$
minimizes the statistical variance of the free energy estimation for $u=\Delta G_{AB}$.
Energetic contributions to the binding free energy {#sec: energy contributions}
--------------------------------------------------
The free energy difference $\Delta G_{AB}$ obtained by using the FTLB contains the binding free energy of the ligand to the given substrate ${\Delta G_{\rm bind}}$ plus elastic and thermodynamic energetic contributions of the experimental setup ${\Delta G}_{AB}(\lambda_0)$, which can be described as follows:
$$\begin{aligned}
\label{eq: contributions}
{\Delta G}_{AB}(\lambda_0) &= G_B(\lambda_0)-G_A(\lambda_0) \\
&= {\Delta G}_{AB}^0 + {\Delta W}_{AB}^{\rm handles} + {\Delta W}_{AB}^{\rm bead} + {\Delta W}_{AB}^{\rm ssDNA} + {\Delta W}_{AB}^{\rm d}.\end{aligned}$$
Here, $A$ ($B$) stands for the configuration of the hairpin at the beginning (ending) of the cyclic protocol at $\lambda_0$. In what follows, $f_A$ ($f_B$) is the force acting on the molecular setup when the hairpin is in state $A$ ($B$) at $\lambda_0$.
The term ${\Delta G}_{AB}^0=G_B^0-G_A^0$ is the difference between the free energy of formation of the conformations of the DNA hairpin in states $A$ and $B$. This term depends on the sequence of the hairpin and is usually calculated using the nearest-neighbor model and the unified oligonucleotide set of basepair free energies [@huguet2010single; @mfold] or can be recovered from pulling experiments performed in the absence of binding agents using fluctuation relations [@collin2005verification].
The two terms ${\Delta W}_{AB}^{\rm handles}$ and ${\Delta W}_{AB}^{\rm bead}$ correspond to the reversible work needed to stretch the handles and move the bead captured in the optical trap from state $A$ to state $B$. For short handles: $${\Delta W}_{AB}^{\rm handles} + {\Delta W}_{AB}^{\rm bead} = \frac{f^2_B-f^2_A}{2k_{\rm eff}},$$ where $k_{\rm eff}$ is the effective stiffness of the experimental setup, equal to the slope of the force-distance curve measured in the force-branch corresponding to the native state of the hairpin. For long handles: $$\begin{aligned}
\label{eq: DWhOT}
{\Delta W}_{AB}^{\rm handles} + {\Delta W}_{AB}^{\rm bead} &= \int_{x_{\rm h}(f_A)}^{x_{\rm h}(f_B)} f(x')dx' + \frac{f^2_B-f^2_A}{2k_b},\end{aligned}$$ where $x_{\rm h}(f_A)$ ($x_{\rm h}(f_B)$) is the equilibrium end-to-end distance of the handles at force $f_A$ ($f_B$), which is calculated according to the worm-like chain model using a persistence length equal to 43.7 nm, a contour length equal to 446.08 nm and a Young modulus of 1280 pN [@joanthio]; and $k_b=0.068$ pN/nm is the stiffness of the optical trap in our setup [@forns2011improving].
The term ${\Delta W}_{AB}^{\rm ssDNA}=W_{B}^{\rm ssDNA}-W_{A}^{\rm ssDNA}$ corresponds to the difference between the reversible work needed to stretch the released single stranded DNA in configurations $B$ and $A$ from zero force to $f_A$ and $f_B$, respectively. This is calculated according to: $$\label{eq: DWss}
{\Delta W}_{AB}^{\rm ssDNA} = \int_0^{x_{\rm ssDNA}(f_B)}f(x')dx' - \int_0^{x_{\rm ssDNA}(f_A)}f(x')dx',$$ where equilibrium relation between the force and the end-to-end distance $f(x)$ and its inverse $x_{\rm ssDNA}(f)$ are modeled according to the worm-like chain ideal elastic model with a persistence length equal to 1.35 nm and an inter-phosphate distance of 0.59 nm/base, and the number of bases released as single-stranded DNA depends on state $B$ or $A$ [@alemany14].
The term ${\Delta W}_{AB}^{\rm d}$ is the difference between reversible work needed to orient the double-helix diameter between states $A$ and $B$: $$\label{eq: DWd}
{\Delta W}_{AB}^{\rm d} = \int_0^{x_{\rm d}(f_B)}f(x')dx' - \int_0^{x_{\rm d}(f_A)}f(x')dx'.$$ The helix diameter is modeled as a single bond of length $d=2$ nm that is oriented due to the action of an external force $f$ [@alemany14; @forns2011improving].
Equilibrium experiments for the hairpin-oligonucleotide system
--------------------------------------------------------------
In equilibrium experiments in passive-mode the position of the optical trap is held constant. Hairpin “Oligo” (section \[sec: Summary of DNA hairpins used in this work\]) hops rapidly between the unfolded (low forces) and the folded (high forces) states (Fig. \[fig: equilibrium expts\]a) [@forns2011improving]. The binding and unbinding of the 10-bp oligonucleotide to its complementary sequence in the hairpin occur at a slower timescale (several seconds), and consequently binding/unbinding events can be readily identified in experiments in which a concentration of binding oligonucleotide is present (Fig. \[fig: equilibrium expts\]b).
To extract the binding free energy of the oligo to the unfolded DNA hairpin we consider the reaction pathway $N \leftrightarrows U \leftrightarrows B$, where $N$ corresponds to the state where the hairpin is in its native state, $U$ corresponds to the state where the hairpin is unfolded (and the oligo is not bound), and $B$ corresponds to the state where the hairpin is unfolded and an oligo bound. Since in equilibrium experiments we cannot distinguish between states $U$ and $B$ due to the very similar extension of dsDNA and ssDNA at the relevant range of forces for these experiments (Fig. \[fig: equilibrium expts\]b) [@comstock2011ultrahigh], we define the joint probability $\rho_{UB}=\rho_U+\rho_B$. By considering that detailed balance is verified, it can be shown that: $$\beta{\Delta G_{\rm bind}}= \log \left(\frac{\rho_{UB}}{\rho_N}e^{\beta{\Delta G}_{NU}}-1\right) + \beta{\Delta G}_{UB}.$$ In Fig. \[fig: pf eq\] we show force-time trajectories with the corresponding experimental probability density functions obtained at different values of $\lambda$ at 400 nM oligo. From the fit to a double Gaussian (blue dashed line) we can extract the weights $\rho_N$ and $\rho_{UB}$, and determine the two forces levels ($f_N$ and $f_U=f_B$) as the average force of each Gaussian peak. In Table \[tab: equilibrium expts\] we summarize the different contributions to extract ${\Delta G_{\rm bind}}$ for three different experimental traces. Since $f_U=f_B$, the terms $\Delta W_{UB}^{\rm handles}$, $\Delta W_{UB}^{\rm bead}$ and $\Delta W_{UB}^{\rm d}$ equal zero. In average, we find that at a concentration of 400 nM oligo $\langle\beta\left(\mu-{\varepsilon}\right)\rangle_{\rm 400 nM [oligo]} = 7\pm 1$, in good agreement with the theoretical predictions and with the results obtained by applying the FTLB in non-equilibrium pulling experiments.
Supplementary Figures
=====================
![[**Force-distance curves of EcoRI binding to DNA.**]{} Example of cyclic pulling curves classified according to their initial (blue dot) and final state (cyan dot) that start and end at a high force ($\sim$21 pN). Work is calculated by integrating the area between the two curves and is shown in dark/light gray for positive/negative work values.[]{data-label="ecori_1"}](SuppFigEcoRI)
![[**Effect of the upper unfolding force on the derivation of the binding free energy.**]{} The intitial/final value of $\lambda$ can be set to any position where we can unambiguously distinguish states $B$ and $U$. For each value of $\lambda$, a corresponding force $f_U$ is observed in the unfolded branch (gray area in top-left panel). We then apply the FTLB to extract the binding energy of EcoRI for different positions of $\lambda$ at high forces. The rest of the panels show the dependence on force $f_U$ of the prefactor $\log \left(\phi^{U\to B}/\phi^{B\to U}\right)$; the forward and reversed mean work values $\langle W_F\rangle$ and $\langle W_R\rangle$; the free energy difference recovered with the direct application of the FTLB, $\Delta G$; the contribution to $\Delta G$ due to the released ssDNA, the hairpin diameter and the handles of the system $W_{\rm{ssDNA}}$, $W_{\rm{d}}$ and $W_{\rm {h}}$ respectively; and finally, the free energy of binding ${\Delta G_{\rm bind}}$. It can be seen that the resulting value of ${\Delta G_{\rm bind}}$ does not depend on $f_U$, therefore does not depend on the initial/final position of $\lambda$ in the cyclic pulling protocol. Data shown corresponds to experiments performed at 4.8 nM for one molecule where a total of 413 cycles where recorded. []{data-label="fig: ecoRI force effect"}](forceEffect)
![[**Effect of random classification of B, U states in transitions close to the initial/final $\lambda$.**]{} **(a)** Example of folding (left) and unfolding (right) force-distance curves where a transition $B\to U$ occurs close to the initial/final value of $\lambda$ (indicated with a vertical black line respectively). Probabily density function of values for ${k_{\rm B}T}\log\left(\phi^{U\to B}/\phi^{B\to U}\right)$ **(b)** and ${\Delta G_{\rm bind}}$ **(c)** obtained by of randomly assigning 500 independent times states B and U to trajectories where an unbinding transition is observed at $\lambda\pm10$ nm (such as the ones depicted in panel a). Vertical arrows indicate the value recovered with the correct classification of the initial/final states along the cyclic pulling protocol. We observed that a random classification of states in trajectories were transitions are observed close the the inifial/final value of $\lambda$ persistently leads to lower values for the binding free energy. []{data-label="fig: ecoRI random"}](FigRrandom)
![[**Experimental traces of equilibrium experiments of oligonucleotide binding.**]{} [**(a)**]{} Equilibrium experiments performed without oligo with hairpin “Oligo”. [**(b)**]{} Equilibrium experiments performed at 400 nM \[oligo\]. Two time-scales are revealed when the hairpin is in the unfolded state (low forces).[]{data-label="fig: equilibrium expts"}](eqexpts2)
![[**Equilibrium experiments at different forces.**]{} In gray we show an averaged experimental trace; in red we highlight experimental data points (acquisition rate: 1 kHz) where the hairpin is in the folded state (and therefore no oligo is bound), whereas in blue we highlight data points where the hairpin is in the unfolded state (either with the oligo bound or not bound). Two time-scales are observed in the experiments showing that the oligo binds and unbinds from the hairpin in an stochastic manner and with a timescale much longer than the folding/unfolding rate of the hairpin. The panels on the right show an histogram of the probability density (red) and a double gaussian fit to the data (blue).[]{data-label="fig: pf eq"}](p123456)
![[**Free energy of branches $B$, $N$ and $U$.**]{} Free energy branches are shown as a function of force and computed relative to the total free energy of the system (also called potential of mean force, equal to $-\log\left[\exp(-\beta\Delta G_B) + \exp(-\beta\Delta G_N) + \exp(-\beta\Delta G_U)\right]$). The free energy of state $B$ is computed by taking into account the free energy of binding (equal to 20 ${k_{\rm B}T}$) plus the elastic response of a DNA made of two ligated ssDNA and dsDNA segments. The first segment is a 24 bases-long ssDNA chain (modeled with the worm-like chain (WLC) model with persistence length equal to 1.5 nm and inter-phosphate distance equal to 0.59 nm/base). The second segment is the elastic response of a 10 basepair-long dsDNA chain (modeled with the WLC model with persistence length equal to 50 nm and inter-phosphate distance equal to 0.34 nm/base). The free energy of state $U$ is computed by taking only into account the elastic response of a 34 bases-long ssDNA chain. Finally, the free energy of state N contains the folding free energy of the hairpin (28 ${k_{\rm B}T}$) plus the elastic response of the hairpin diameter, modeled as a bond of length 2.0 nm (equal to the hairpin diameter) that is oriented in the presence of a force. It can be seen that at low forces state $N$ is the most stable. However, at $\sim$5 pN state $B$ becomes the most stable until $\sim$62 pN, where state $U$ becomes more stable. Therefore, with this simple model we predict that the threshold force above which the oligo will not bind is $\sim$62 pN. However, at those high forces potential perturbations of the force into the ssDNA structure neglected in the model might change this value. ](FigRbranches)
![[**Force-distance curves of hairpins SP and NSP.**]{} [**(a)**]{} FDCs of hairpin SP in the absence (left) and presence of ligand (right). [**(b)**]{} FDCs of hairpin NSP in the absence (left) and presence of ligand (right). In each FDC blue/green is unfolding and cyan/magenta refolding. Pulling speed is 70 nm/s in (a) and 250 nm/s in (b).[]{data-label="echi"}](SuppFig_Echi1)
![[**Specific ligand binding of echinomycin to DNA.**]{} [**(a)**]{} Fast mechanical unfolding is performed after incubating the hairpin at $\sim$5 pN during a fixed time interval $\Delta t$. The unfolding force of the hairpin, indicated with an arrow, relates to the bound state of echinomycin: unfolding forces above (below) 15 pN (red(blue) curve) indicate that an echinomycin molecules has (has not) bound to the hairpin (Fig. 3a, main document). [**(b)**]{} Fraction of bound states as a function of the time $\Delta t$ and concentration of echinomycin and fit to to a first order reaction kinetics model (DNA + I$\rightleftarrows$DNA$\cdot$I) where $\phi(t) = \frac{k_\to [I]}{k_\leftarrow [I] + k_\leftarrow}\left(1-\exp[(k_\to[I] + k_\leftarrow)t]\right)$. From the fit, we obtain $k_\to=(4.9\pm0.4)\times 10^{-4}$ nM$^{-1}$s$^{-1}$ and $k_\leftarrow=(2.0\pm 0.5)\times 10^{-2}$ s$^{-1}$, which implies $K_d=k_\leftarrow/k_\to= 41 \pm 10$ nM and ${\Delta G_{\rm bind}}=17\pm1$ ${k_{\rm B}T}$. This result is in good agreement with the value obtained using the FTLB (${\Delta G_{\rm bind}}=20\pm1$ ${k_{\rm B}T}$). [**(c)**]{} Binding isotherm of echinomycin determined from optical trapping. The red curve has been obtained from the fit in panel b. Blue points are the fraction of bound population measured as described in panel a at the largest measured time $\Delta t=30$s. The disagreement between theory and experiments observed at low concentrations shows that binding kinetics is still out of equilibrium at the largest measured time of 30s. Error bars are standard errors computed by averaging over different molecules. []{data-label="fig: echi titration"}](Fig_echititration)
![[**Force-distance curves of hairpins C and NC.**]{} [**(a)**]{} FDCs of hairpin C in the absence (left) and presence of ligand (right) [**(b)**]{} FDCs of hairpin NC in the absence (left) and presence of ligand. In the presence of ligand, FDCs of hairpin C show higher unfolding forces than those of hairpin NC, in agreement with the proposed cooperative effect between ligand pairs. Similarly a partially unfolded intermediate with just one ligand bound is observed in hairpin NC, whereas hairpin C cooperatively unfolds in a single step. In each FDC blue/green is unfolding and cyan/magenta refolding. Pulling speed is 70 nm/s in (a) and 250 nm/s in (b).[]{data-label="coop"}](SuppFig_Echicoop)
![[**Force-distance curves of hairpin M at \[Echinomycin\]=10 $\mu$M.**]{} The ligand can kinetically trap a misfolded state consisting of two 4-bp DNA hairpins serially connected. Characteristic pulling curves connecting the native (N) and misfolded (M) state are shown. The different molecular configurations observed during the pulling curve are indicated in the scheme. Pulling speed is 70 nm/s.[]{data-label="misfolding"}](SuppFig_misfolded)
Supplementary Tables
====================
(nM) pulled molecules (number of cycles per molecule) total cycles $N$
------ ------------------------------------------------------ ------------------
0.25 5 (141, 87, 95, 360, 101) 784
0.50 8 (297, 93, 43, 101, 40, 18, 137, 27) 756
1.00 8 (852, 555, 486, 48, 153, 76, 159, 404) 2733
2.40 8 (170, 356, 350, 433, 212, 93, 245, 126) 1985
4.80 3 (413, 339, 349) 1101
10.0 10 (470, 317, 290, 242, 87, 200, 248, 401, 102, 396) 2753
20.0 3 (257, 242, 668) 1167
: [**Number of experiments performed at 130 mM NaCl for different concentrations of EcoRI.**]{} Number of molecules measured at each concentration of EcoRI, corresponding cycles per molecule shown in parenthesis, and total number of cycles used for computing the binding energy.[]{data-label="tab: EcoRI N"}
$f_B$ (pN) $f_U$ (pN) ${\Delta G}_{nU}^0$ ${\Delta W}_{nU}^{\rm handles}+{\Delta W}_{nU}^{\rm bead}$ ${\Delta W}_{nU}^{\rm ssDNA}$ ${\Delta W}_{nU}^d$
---------------- ---------------- --------------------- ------------------------------------------------------------ ------------------------------- ---------------------
19.77$\pm$0.01 18.54$\pm$0.01 42$\pm$2 -92$\pm$1 21.51$\pm$0.01 -1.955$\pm$0.001
: [**Contributions to the binding free energy of EcoRI to dsDNA as a function of \[EcoRI\] at 130 mM NaCl.**]{} For all the pulling experiments performed for different molecules at different concentrations of EcoRI, the initial/final value of the control parameter $\lambda$ was chosen so that forces $f_B$ and $f_U$ are on average the same. Hence, numerical values for ${\Delta G}_{nU}^0$, ${\Delta W}_{nU}^{\rm handles}+{\Delta W}_{nU}^{\rm bead}$ (Eq. \[eq: DWhOT\]), ${\Delta W}_{nU}^{\rm ssDNA}$ (Eq. \[eq: DWss\]), and ${\Delta W}_{nU}^{\rm d}$ (Eq. \[eq: DWd\]) are also on average the same for different molecules pulled at different concentrations of EcoRI. In contrast, $\phi^{B\to U}$, $\phi^{U\to B}$, $\log \left(\phi^{B\to U}/\phi^{U\to B}\right)$, ${\Delta G}_{BU}$ and ${\Delta G_{\rm bind}}$ depend on the concentration of ligand. Here $n=7$. []{data-label="tab: EcoRI 1"}
(nM) $\phi^{B\to U}$ $\phi^{U\to B}$ $\log \left(\phi^{B\to U}/\phi^{U\to B}\right)$ ${\Delta G}_{BU}$ ${\Delta G_{\rm bind}}$
------ ----------------- ----------------- ------------------------------------------------- ------------------- -------------------------
0.25 0.4$\pm$0.1 0.12$\pm$0.03 1.2$\pm$0.3 -29$\pm$2 2$\pm$4
0.50 0.24$\pm$0.05 0.22$\pm$0.02 0.1$\pm$0.2 -25$\pm$2 5$\pm$4
1.00 0.15$\pm$0.03 0.31$\pm$0.02 -0.8$\pm$0.1 -23$\pm$1 5$\pm$3
2.40 0.24$\pm$0.04 0.26$\pm$0.03 -0.2$\pm$0.2 -24$\pm$2 8$\pm$2
4.80 0.24$\pm$0.04 0.27$\pm$0.03 -0.1$\pm$0.2 -27$\pm$2 5$\pm$3
10.0 0.17$\pm$0.02 0.71$\pm$0.05 -1.5$\pm$0.1 -22$\pm$1 8$\pm$1
20.0 0.13$\pm$0.02 0.83$\pm$0.03 -1.9$\pm$0.1 -23$\pm$1 8$\pm$1
: [**Contributions to the binding free energy of EcoRI to dsDNA as a function of \[EcoRI\] at 130 mM NaCl.**]{} For all the pulling experiments performed for different molecules at different concentrations of EcoRI, the initial/final value of the control parameter $\lambda$ was chosen so that forces $f_B$ and $f_U$ are on average the same. Hence, numerical values for ${\Delta G}_{nU}^0$, ${\Delta W}_{nU}^{\rm handles}+{\Delta W}_{nU}^{\rm bead}$ (Eq. \[eq: DWhOT\]), ${\Delta W}_{nU}^{\rm ssDNA}$ (Eq. \[eq: DWss\]), and ${\Delta W}_{nU}^{\rm d}$ (Eq. \[eq: DWd\]) are also on average the same for different molecules pulled at different concentrations of EcoRI. In contrast, $\phi^{B\to U}$, $\phi^{U\to B}$, $\log \left(\phi^{B\to U}/\phi^{U\to B}\right)$, ${\Delta G}_{BU}$ and ${\Delta G_{\rm bind}}$ depend on the concentration of ligand. Here $n=7$. []{data-label="tab: EcoRI 1"}
(mM) pulled molecules (number of cycles per molecule) total cycles $N$
------ ----------------------------------------------------------------------- ------------------
60 15 (156, 69, 525, 60, 61, 35, 21, 43, 713, 258, 433, 58, 78, 162, 83) 2755
75 6 (263, 312, 519, 506, 156, 302) 2058
100 7 (276, 273, 516, 577, 128, 494, 235) 2499
130 8 (852, 555, 486, 48, 153, 76, 159, 404) 2733
180 8 (654, 323, 88, 298, 61, 423, 211, 80) 2138
: [**Number of experiments performed at 1 nM EcoRI for different concentrations of NaCl.**]{}[]{data-label="tab: ecori N sal"}
$f_B$ (pN) $f_U$ (pN) ${\Delta W}_{nU}^{\rm handles}+{\Delta W}_{nU}^{\rm bead}$ ${\Delta W}_{nU}^{\rm ssDNA}$ ${\Delta W}_{nU}^d$
---------------- ---------------- ------------------------------------------------------------ ------------------------------- ---------------------
19.76$\pm$0.01 18.56$\pm$0.01 -91$\pm$1 21.54$\pm$0.01 -1.955$\pm$0.001
: [**Contributions to the binding free energy of EcoRI to dsDNA as a function of \[NaCl\] at 1 nM EcoRI.**]{} Caption as in Table \[tab: EcoRI 1\].[]{data-label="eq: ecori g nacl"}
(nM) ${\Delta G}_{nU}^0$ $\phi^{B\to U}$ $\phi^{U\to B}$ $\log \left(\phi^{B\to U}/\phi^{U\to B}\right)$ ${\Delta G}_{BU}$ ${\Delta G_{\rm bind}}$
------ --------------------- ----------------- ----------------- ------------------------------------------------- ------------------- -------------------------
60 39$\pm$2 0.056$\pm$0.008 0.8$\pm$0.1 -2.5$\pm$0.4 -19$\pm$3 13$\pm$3
75 40$\pm$2 0.06$\pm$0.01 0.7$\pm$0.1 -2.5$\pm$0.4 -20$\pm$1 11$\pm$2
100 41$\pm$2 0.14$\pm$0.02 0.29$\pm$0.05 -0.9$\pm$0.2 -25$\pm$2 5$\pm$2
130 42$\pm$2 0.15$\pm$0.03 0.31$\pm$0.02 -0.8$\pm$0.1 -23$\pm$1 6$\pm$1
180 44$\pm$2 0.44$\pm$0.07 0.06$\pm$0.02 1.6$\pm$0.4 -27$\pm$2 0$\pm$2
: [**Contributions to the binding free energy of EcoRI to dsDNA as a function of \[NaCl\] at 1 nM EcoRI.**]{} Caption as in Table \[tab: EcoRI 1\].[]{data-label="eq: ecori g nacl"}
(nM) pulled molecules (number of cycles per molecule) total cycles $N$
------ -------------------------------------------------- ------------------
25 2 (25, 244) 269
50 7 (57, 131, 72, 242, 311, 98 46) 957
100 6 (182, 198, 22, 435, 368, 386) 1591
200 5 (28, 17, 59, 67, 87) 258
400 4 (264, 123, 213, 109) 709
1000 10 (59, 56, 288, 293, 325, 224, 83, 432, 317) 2077
2000 5 (185, 603, 74, 273, 147) 1282
: [**Number of experiments performed at different concentrations of oligo.**]{}[]{data-label="tab: oligo N"}
$f_B$ (pN) $f_N$ (pN) ${\Delta G}_{UN}^0$ ${\Delta W}_{UN}^{\rm handles}+{\Delta W}_{UN}^{\rm bead}$ ${\Delta W}_{UN}^{\rm ssDNA}$ ${\Delta W}_{UN}^d$
--------------- --------------- --------------------- ------------------------------------------------------------ ------------------------------- ---------------------
6.20$\pm$0.06 6.80$\pm$0.06 -27.5$\pm$0.8 16.5$\pm$0.5 -4.81$\pm$0.05 0.896$\pm$0.008
: [**Contributions to the binding free energy of oligo to complementary ssDNA as a function of the concentration of the oligo.**]{} For all the pulling experiments performed for different molecules at different concentrations of oligo, the initial/final value of the control parameter $\lambda$ was chosen so that forces $f_B$ and $f_N$ are on average the same same. Numerical values for ${\Delta G}_{UN}^0$, ${\Delta W}_{UN}^{\rm handles}+{\Delta W}_{UN}^{\rm bead}$ (Eq. \[eq: DWhOT\]), ${\Delta W}_{UN}^{\rm ssDNA}$ (Eq. \[eq: DWss\]), and ${\Delta W}_{UN}^{\rm d}$ (Eq. \[eq: DWd\]) are also on average the same for different molecules pulled at different concentrations of oligo. In contrast, $\phi^{B\to N}$, $\phi^{N\to B}$, $\log \left(\phi^{B\to N}/\phi^{N\to B}\right)$, ${\Delta G}_{BN}$ and ${\Delta G_{\rm bind}}$ de depend on the concentration of ligand.[]{data-label="tab: ecori g"}
(nM) $\phi^{B\to N}$ $\phi^{N\to B}$ $\log \left(\phi^{B\to N}/\phi^{N\to B}\right)$ ${\Delta G}_{BN}$ ${\Delta G_{\rm bind}}$
------ ----------------- ----------------- ------------------------------------------------- ------------------- -------------------------
25 1.0$^*$ 0.030$\pm$0.004 3.6$\pm$0.1 -11.3$\pm$0.7 4$\pm$1
50 0.96$\pm$0.04 0.037$\pm$0.006 3.4$\pm$0.2 -11.8$\pm$0.6 3$\pm$1
100 0.94$\pm$0.03 0.062$\pm$0.006 2.79$\pm$0.08 -11.0$\pm$0.6 4$\pm$1
200 0.87$\pm$0.08 0.09$\pm$0.03 2.3$\pm$0.3 -9.2$\pm$0.4 6$\pm$1
400 0.86$\pm$0.06 0.18$\pm$0.03 1.6$\pm$0.3 -8.4$\pm$0.3 7$\pm$1
1000 0.49$\pm$0.05 0.45$\pm$0.04 0.37$\pm$0.07 -7.7$\pm$0.5 7$\pm$1
2000 0.37$\pm$0.02 0.60$\pm$0.04 -0.5$\pm$0.1 -6.5$\pm$0.8 8$\pm$1
: [**Contributions to the binding free energy of oligo to complementary ssDNA as a function of the concentration of the oligo.**]{} For all the pulling experiments performed for different molecules at different concentrations of oligo, the initial/final value of the control parameter $\lambda$ was chosen so that forces $f_B$ and $f_N$ are on average the same same. Numerical values for ${\Delta G}_{UN}^0$, ${\Delta W}_{UN}^{\rm handles}+{\Delta W}_{UN}^{\rm bead}$ (Eq. \[eq: DWhOT\]), ${\Delta W}_{UN}^{\rm ssDNA}$ (Eq. \[eq: DWss\]), and ${\Delta W}_{UN}^{\rm d}$ (Eq. \[eq: DWd\]) are also on average the same for different molecules pulled at different concentrations of oligo. In contrast, $\phi^{B\to N}$, $\phi^{N\to B}$, $\log \left(\phi^{B\to N}/\phi^{N\to B}\right)$, ${\Delta G}_{BN}$ and ${\Delta G_{\rm bind}}$ de depend on the concentration of ligand.[]{data-label="tab: ecori g"}
trace
------- ------- ------ ------- ------ ------- -------
a 11.22 0.05 10.62 0.01 0.038 0.006
b 9.92 0.01 9.28 0.01 0.364 0.004
c 9.18 0.01 8.53 0.01 0.901 0.005
: [**Determination of $\beta(\mu-{\varepsilon})$ in equilibrium experiments.**]{} Results obtained at 400 nM \[oligo\].[]{data-label="tab: equilibrium expts"}
----- --- ------ ----- ------ ------ ------ ----- ----- ---
-29 2 10.4 0.5 1.40 0.02 -2.4 0.5 7.5 1
-28 2 9.3 0.5 1.27 0.02 -2.1 0.5 6 1
-25 3 8.7 0.5 1.19 0.02 -1.9 0.5 6 1
----- --- ------ ----- ------ ------ ------ ----- ----- ---
: [**Determination of $\beta(\mu-{\varepsilon})$ in equilibrium experiments.**]{} Results obtained at 400 nM \[oligo\].[]{data-label="tab: equilibrium expts"}
(nM) pulled molecules (number of cycles per molecule) total cycles $N$
------ --------------------------------------------------- ------------------
100 7 (121, 408, 369, 277, 98, 484, 261) 2018
300 6 (496, 271, 201, 341, 600, 511) 2420
1000 9 (1302, 1272, 486, 978, 420, 121, 327, 470, 196) 5572
3000 6 (670, 175, 1010, 399, 1268, 475) 3997
: [**Number of experiments performed at different concentrations of echinomycin with hairpin SP.**]{}[]{data-label="tab: echi N"}
$f_B$ (pN) $f_U$ (pN) ${\Delta G}_{nU}^0$ ${\Delta W}_{nU}^{\rm handles}+{\Delta W}_{nU}^{\rm bead}$ ${\Delta W}_{nU}^{\rm ssDNA}$ ${\Delta W}_{nU}^d$
---------------- ---------------- --------------------- ------------------------------------------------------------ ------------------------------- ---------------------
16.77$\pm$0.02 16.25$\pm$0.02 10.5$\pm$0.3 -33.0$\pm$0.8 8.25$\pm$0.03 -1.790$\pm$0.006
: [**Contributions to the binding free energy of echinomycin to dsDNA as a function of ligand concentration \[Echi\].**]{} Caption as in Table \[tab: EcoRI 1\]. Here $n=1$.[]{data-label="tab: echi g"}
(nM) $\phi^{B\to U}$ $\phi^{U\to B}$ $\log \left(\phi^{B\to U}/\phi^{U\to B}\right)$ ${\Delta G}_{BU}$ ${\Delta G_{\rm bind}}$
------ ----------------- ----------------- ------------------------------------------------- ------------------- -------------------------
100 0.5$\pm$0.1 0.046$\pm$0.005 -2.2$\pm$0.4 -10.8$\pm$0.4 3$\pm$2
300 0.46$\pm$0.07 0.112$\pm$0.008 -1.3$\pm$0.2 -10.6$\pm$0.6 5$\pm$2
1000 0.27$\pm$0.04 0.18$\pm$0.05 0.5$\pm$0.4 -10.3$\pm$0.8 5$\pm$2
3000 0.158$\pm$0.008 0.59$\pm$0.04 1.3$\pm$0.1 -8.6 $\pm$0.6 10$\pm$1
: [**Contributions to the binding free energy of echinomycin to dsDNA as a function of ligand concentration \[Echi\].**]{} Caption as in Table \[tab: EcoRI 1\]. Here $n=1$.[]{data-label="tab: echi g"}
(nM) pulled molecules (number of cycles per molecule) total cycles $N$
------ -------------------------------------------------- ------------------
100 6 (131, 145, 761, 492, 230, 567) 2326
300 8 (504, 506, 528, 455, 46, 320, 219, 275) 2853
1000 8 (779, 560, 547, 814, 666, 342, 377, 519) 4604
3000 7 (310, 150, 268, 872, 241, 584, 533) 2958
: [**Number of experiments performed at different concentrations of echinomycin with hairpin NSP.**]{}[]{data-label="tab: echins N"}
$i$ $f_{B_i}$ (pN) $f_U$ (pN) ${\Delta G}_{n_iU}^0$ ${\Delta W}_{n_iU}^{\rm handles}+{\Delta W}_{n_iU}^{\rm bead}$ ${\Delta W}_{n_iU}^{\rm ssDNA}$ ${\Delta W}_{n_iU}^d$
----- ---------------- ---------------- ----------------------- ---------------------------------------------------------------- --------------------------------- -----------------------
0 10.73$\pm$0.06 10.17$\pm$0.05 19.9$\pm$0.2 -26.5$\pm$0.4 8.20$\pm$0.04 -1.345$\pm$0.006
1 10.69$\pm$0.06 10.17$\pm$0.05 16.8$\pm$0.2 -24.3$\pm$0.3 7.59$\pm$0.04 -1.340$\pm$0.006
2 10.64$\pm$0.06 10.17$\pm$0.05 13.7$\pm$0.2 -22.1$\pm$0.3 6.90$\pm$0.03 -1.336$\pm$0.006
3 10.60$\pm$0.06 10.17$\pm$0.05 11.64$\pm$0.07 -20.0$\pm$0.3 6.39$\pm$0.03 -1.332$\pm$0.006
4 10.55$\pm$0.06 10.17$\pm$0.05 10.5$\pm$0.1 -17.8$\pm$0.3 5.79$\pm$0.03 -1.327$\pm$0.006
5 10.50$\pm$0.06 10.17$\pm$0.05 8.43$\pm$0.02 -15.6$\pm$0.4 5.19$\pm$0.02 -1.323$\pm$0.006
6 10.48$\pm$0.06 10.17$\pm$0.05 4.63$\pm$0.06 -13.8$\pm$0.5 4.61$\pm$0.02 -1.320$\pm$0.006
7 10.43$\pm$0.06 10.17$\pm$0.05 2.5$\pm$0.1 -11.6$\pm$0.5 4.02$\pm$0.02 -1.316$\pm$0.006
8 10.38$\pm$0.06 10.17$\pm$0.05 1.4$\pm$0.1 -9.4$\pm$0.5 3.44$\pm$0.02 -1.311$\pm$0.006
9 10.33$\pm$0.06 10.17$\pm$0.05 -0.11$\pm$0.06 -7.2$\pm$0.6 2.87$\pm$0.01 -1.307$\pm$0.006
10 10.29$\pm$0.06 10.17$\pm$0.05 -1.18$\pm$0.03 -5.1$\pm$0.6 2.29$\pm$0.01 -1.302$\pm$0.006
11 10.24$\pm$0.06 10.17$\pm$0.05 -2.71$\pm$0.03 -2.9$\pm$0.7 1.73$\pm$0.01 -1.296$\pm$0.006
: [**Contributions to the binding free energy of echinomycin to dsDNA as a function of ligand concentration \[Echi\].**]{} The term ${{\mathcal{N}}}$ is taken equal to ${{\mathcal{N}}}=12$, which is equal to the number of basepairs of H.NSP. The term $\log\sum_i\exp\left(-\beta\Delta G_{n_iU}\right)$ is equal to 7.6$\pm$0.5 ${k_{\rm B}T}$.[]{data-label="tab: echins g"}
(nM) $\phi^{B\to U}$ $\phi^{U\to B}$ $\log \left(\phi^{B\to U}/\phi^{U\to B}\right)$ ${\Delta G}_{BU}$ ${\Delta G_{\rm bind}}$
------ ----------------- ----------------- ------------------------------------------------- ------------------- -------------------------
100 0.70$\pm$0.07 0.12$\pm$0.03 1.9$\pm$0.2 -4.5$\pm$0.2 -2$\pm$1
300 0.53$\pm$0.07 0.30$\pm$0.01 0.5$\pm$0.1 -3.3$\pm$0.2 -1$\pm$1
1000 0.37$\pm$0.02 0.46$\pm$0.03 -0.2$\pm$0.1 -2.6$\pm$0.2 0$\pm$1
3000 0.21$\pm$0.04 0.70$\pm$0.05 -1.3$\pm$0.3 -1.2$\pm$0.4 1$\pm$1
: [**Contributions to the binding free energy of echinomycin to dsDNA as a function of ligand concentration \[Echi\].**]{} The term ${{\mathcal{N}}}$ is taken equal to ${{\mathcal{N}}}=12$, which is equal to the number of basepairs of H.NSP. The term $\log\sum_i\exp\left(-\beta\Delta G_{n_iU}\right)$ is equal to 7.6$\pm$0.5 ${k_{\rm B}T}$.[]{data-label="tab: echins g"}
Hairpin pulled molecules (number of cycles per molecule) total cycles $N$
--------- -------------------------------------------------- ------------------
HC 7 (40, 348, 312, 324, 603, 352, 374) 2353
HNC 6 (997, 300, 193, 325, 845, 125) 2785
: [**Number of experiments performed at 3000 nM \[Echi\] with hairpins C and NC.**]{}[]{data-label="tab: echi c nc N"}
Hairpin $f_{B^2}$ (pN) $f_U$ (pN) ${\Delta G}_{{n_2}U}^0$ ${\Delta W}_{{n_2}U}^{\rm handles}+{\Delta W}_{n_2U}^{\rm bead}$ ${\Delta W}_{n_2U}^{\rm ssDNA}$ ${\Delta W}_{B^2U}^d$
--------- ---------------- ---------------- ------------------------- ------------------------------------------------------------------ --------------------------------- -----------------------
C 20.0$\pm$0.2 19.4$\pm$0.2 18.3$\pm$0.8 -51.1$\pm$0.4 12.10$\pm$0.07 -1.967$\pm$0.008
NC 16.67$\pm$0.05 15.94$\pm$0.05 18.3$\pm$0.8 -41$\pm$1 10.56$\pm$0.03 -1.785$\pm$0.003
: **Contributions to the binding free energy for double binding to dsDNA at 3000 nM echinomycin.** At 3000 nM \[Echi\] double binding events of echinomycin to hairpins C and NC are always observed. $n_2=4$ is the number of open basepairs when two echinomycin molecules bind to specific sites in H.C or H.NC.[]{data-label="tab: echi c nc g"}
Hairpin $\phi^{B^2\to U}$ $\phi^{U\to B^2}$ $\log \left(\phi^{B^2\to U}/\phi^{U\to B^2}\right)$ ${\Delta G}_{B^2U}$ ${\Delta G_{\rm bind}}$
--------- ------------------- ------------------- ----------------------------------------------------- --------------------- -------------------------
C 0.26$\pm$0.05 0.72$\pm$0.03 -1.1$\pm$0.3 -14$\pm$1 4.5$\pm$0.5
NC 0.4$\pm$0.1 0.15$\pm$0.02 0.9$\pm$0.3 -8$\pm$1 2.6$\pm$0.4
: **Contributions to the binding free energy for double binding to dsDNA at 3000 nM echinomycin.** At 3000 nM \[Echi\] double binding events of echinomycin to hairpins C and NC are always observed. $n_2=4$ is the number of open basepairs when two echinomycin molecules bind to specific sites in H.C or H.NC.[]{data-label="tab: echi c nc g"}
Hairpin $f_B$ (pN) $f_U$ (pN) ${\Delta G}_{n_1U}^0$ ${\Delta W}_{n_1U}^{\rm handles}+{\Delta W}_{n_1U}^{\rm bead}$ ${\Delta W}_{n_1U}^{\rm ssDNA}$ ${\Delta W}_{n_1U}^d$
--------- -------------- ---------------- ----------------------- ---------------------------------------------------------------- --------------------------------- -----------------------
NC 16.1$\pm$0.2 15.95$\pm$0.06 6.6$\pm$0.4 -15$\pm$2 5.65$\pm$0.01 -1.757$\pm$0.006
: **Contributions to the binding free energy for single binding to dsDNA at 3000 nM echinomycin.** Even though when pulling hairpin NC at 3000 nM \[Echi\] all binding events correspond to double binding, single binding events are also transiently observed with hairpin NC at large forces when the echimoycin molecule bound at the start of the hairpin stem spontaneously unbinds. Forward and reverse trajectories connecting such transient single binding state and the unfolded state also provide a measurement of ${\Delta G_{\rm bind}}$ for hairpin H.NC. []{data-label="tab: EchiS-SNC"}
$\phi^{B\to U}$ $\phi^{U\to B}$ $\log \left(\phi^{B\to U}/\phi^{U\to B}\right)$ ${\Delta G}_{BU}$ ${\Delta G_{\rm bind}}$
----------------- ----------------- ------------------------------------------------- ------------------- -------------------------
0.6$\pm$0.2 0.031$\pm$0.006 3.1$\pm$0.3 -9$\pm$1 -5$\pm$2
: **Contributions to the binding free energy for single binding to dsDNA at 3000 nM echinomycin.** Even though when pulling hairpin NC at 3000 nM \[Echi\] all binding events correspond to double binding, single binding events are also transiently observed with hairpin NC at large forces when the echimoycin molecule bound at the start of the hairpin stem spontaneously unbinds. Forward and reverse trajectories connecting such transient single binding state and the unfolded state also provide a measurement of ${\Delta G_{\rm bind}}$ for hairpin H.NC. []{data-label="tab: EchiS-SNC"}
pulled molecules (number of cycles per molecule) total cycles $N$
-------------------------------------------------- ------------------
3 (101, 300, 314) 715
: [**Number of experiments performed at 10 $\mu$M \[Echi\] with hairpins M.**]{}[]{data-label="tab: echi m N"}
$f_N$ (pN) $f_M$ (pN) ${\Delta G}_{NM}^0$ ${\Delta W}_{NM}^{\rm handles}+{\Delta W}_{NM}^{\rm bead}$ ${\Delta W}_{NM}^{\rm ssDNA}$ ${\Delta W}_{NM}^d$
--------------- --------------- --------------------- ------------------------------------------------------------ ------------------------------- ---------------------
7.32$\pm$0.03 6.75$\pm$0.05 23$\pm$1 -16$\pm$1 3.48$\pm$0.02 0.606$\pm$0.006
: **Double binding of two echinomycin molecules to hairpin M stabilizes a misfolded state.** In the simultaneous binding of two echinomycin molecules to the folded hairpin M two unrelated structures are observed: one corresponds to the native state whereas the other correspond to a misfolded structure (Fig. S1 and \[misfolding\]). The difference in binding energy of echinomycin to each of the two structures is ${\Delta G}_{\text{bind,}NM}=2\pm1~{k_{\rm B}T}$ according to our results. []{data-label="tab: echi m g"}
$\phi^{N\to M}$ $\phi^{N\to M}$ $\log \left(\phi^{N\to M}/\phi^{M\to N}\right)$ ${\Delta G}_{NM}$ ${\Delta G}_{\text{bind,}NM}$
----------------- ----------------- ------------------------------------------------- ------------------- ------------------------------- --
0.46$\pm$0.06 0.56$\pm$0.03 -0.2$\pm$0.2 17$\pm$1 2$\pm$1
: **Double binding of two echinomycin molecules to hairpin M stabilizes a misfolded state.** In the simultaneous binding of two echinomycin molecules to the folded hairpin M two unrelated structures are observed: one corresponds to the native state whereas the other correspond to a misfolded structure (Fig. S1 and \[misfolding\]). The difference in binding energy of echinomycin to each of the two structures is ${\Delta G}_{\text{bind,}NM}=2\pm1~{k_{\rm B}T}$ according to our results. []{data-label="tab: echi m g"}
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2.1em
[c]{} [**Chuan-Hung Chen$^a$ and C. Q. Geng$^b$**]{}\
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[*${}^a$Department of Physics, National Cheng Kung University*]{}\
[*$\ $Tainan, Taiwan, Republic of China* ]{}\
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[*${}^b$Department of Physics, National Tsing Hua University*]{}\
[*$\ $ Hsinchu, Taiwan, Republic of China* ]{}\
5.3em
[**Abstract**]{}
We investigate the rare baryonic exclusive decays of $\Lambda_b\to
\Lambda l^+ l^-\ (l=e,\mu,\tau )$ with polarized $\Lambda $. Under the approximation of the heavy quark effective theory (HQET), in the standard model we derive the differential decay rates and various polarization asymmetries by including lepton mass effects. We find that with the long-distance effects the decay branching ratios are $5.3\times 10^{-5}$ for $\Lambda_b\to \Lambda l^+ l^-\
(l=e,\mu)$ and $1.1\times 10^{-5}$ for $\Lambda_b\to \Lambda
\tau^+\tau^-$. The effects of new physics in the decay rates are also discussed. The integrated longitudinal $\Lambda$ polarizations are $-0.31$ and $-0.12$, while that of the normal ones $0.02$ and $0.01$, for di-muon and tau modes, respectively. The CP-odd transverse polarization of $\Lambda$ is zero in the standard model but it is expected to be sizable in some theories with new physics.
Introduction
=============
It is known that the recent interest in flavor physics has been focused in the rare decays related to $b\to s l^+l^-$ induced by the flavor changing neutral current (FCNC) due to the CLEO measurement of the radiative $b \to
s\gamma$ decay [@cleo]. In the standard model, these rare decays occur at loop level and provide us information on the parameters of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements [@ckm] as well as various hadronic form factors. In the literature, most of studies have been concentrated on the corresponding exclusive rare B meson decays such as $B\to K^{(*)}l^+l^-$ [@Bmeson].
In this paper, we investigate the baryon decays of $\Lambda_b\to\Lambda
l^+l^-$ with $\Lambda $ being polarized. Unlike mesonic decays, the baryonic decays could maintain the helicity structure of interactions in transition matrix elements. Through this property, we will show that the polarization asymmetries of $\Lambda$ are sensitive to right-handed couplings which are suppressed in the standard model. Thus, these baryonic decays could be used to search for physics beyond the standard model.
To study the exclusive bayonic decays, one of the most difficulties is to evaluate the hadronic matrix elements. It is known that there are many form factors for the matrix elements of $\Lambda_b$ to $\Lambda$, which are hard to be calculated since they are related to the non-perturbative effect of QCD. However, in heavy particle decays, HQET could reduce the number of form factors and supply the information with respect to their relative size. In our numerical calculations, we shall use the results in HQET. It is also know that a large theoretical uncertainty in our calculation to the decays arises from the long-distance (LD) effect. To reduce the uncertainty, we shall study various kinematic regions to distinguish the LD contributions. In our calculations, as a completeness, we will include the lepton mass, which is important for the tau lepton mode.
The paper is organized as follows. In Sec. 2, we study the effective Hamiltonian for the di-lepton decays of $\Lambda_b \to \Lambda l \bar{l} $ and form factors in the $\Lambda_b \to \Lambda$ transition. In Sec. 3, we derive the general forms of the differential decay rates and the $\Lambda$ polarizations. In Sec. 4, we give the numerical analysis. We present our conclusions in Sec. 5.
Effective Hamiltonian and Form factors
======================================
The effective Hamiltonian for the inclusive decay of $b\rightarrow
sl^{+}l^{-}$ is given by
$${\cal H}=-4\frac{G_{F}}{\sqrt{2}}V_{tb}V_{ts}^{*}\sum_{i=1}^{10}C_{i}\left(
\mu \right) O_{i}\left( \mu \right) \,, \label{Ham}$$
where the expressions for the renormalized Wilson coefficients $C_{i}\left(
\mu \right) $ and operators $O_{i}\left( \mu \right) $ can be found in Ref. [@Buras]. In terms of the Hamiltonian in Eq. (\[Ham\]), the decay amplitude is written as $$\begin{aligned}
{\cal M} &=&\frac{G_{F}\alpha _{em}}{\sqrt{2}\pi }V_{tb}V_{ts}^{*}\left[
\bar{s}\left( C_{9}^{eff}\left( \mu \right) \gamma _{\mu }P_{L}-\frac{2m_{b}}{q^{2}}C_{7}\left( \mu \right) i\sigma _{\mu \nu }q^{\nu }P_{R}\right) b\;\bar{l}\gamma ^{\mu }l\right. \nonumber \\
&&\left. +\bar{s}C_{10}\gamma _{\mu }P_{L}b\;\bar{l}\gamma ^{\mu }\gamma
_{5}l\right] \label{Am}\end{aligned}$$ with $P_{L(R)}=(1\mp \gamma _{5})/2$. We note that in Eq. (\[Am\]), only the term associated with Wilson coefficient $C_{10}$ is independent of the $\mu $ scale. We also note that the dominant contribution to the decay rate is from the long-distance (LD) such as the cc resonant states of $\Psi
,\Psi ^{\prime }...etc.$ To find out the LD effects for the B-meson decays, in the literature [@DTP; @LMS; @AMM; @OT; @KS; @Geng1], both the factorization assumption (FA) and the vector meson dominance (VMD) approximation have been used. For the LD contributions in baryonic decays, we assume that the parametrization is the same as that in the B meson decays. Hence, we include the resonant effects (RE) by absorbing it to the corresponding Wilson coefficients. The effective Wilson coefficient of $C_{9}^{eff}$ has the standard form $$C_{9}^{eff}=C_{9}\left( \mu \right) +\left( 3C_{1}\left( \mu \right)
+C_{2}\left( \mu \right) \right) \left( h\left( x,s\right) +\frac{3}{\alpha
_{em}^{2}}\sum_{j=\Psi ,\Psi ^{\prime }}k_{j}\frac{\pi \Gamma \left(
j\rightarrow l^{+}l^{-}\right) M_{j}}{q^{2}-M_{j}^{2}+iM_{j}\Gamma _{j}}\right) \,, \label{effc9}$$ where $h(x,s)$ describes the one-loop matrix elements of operators $O_{1}=\bar{s}_{\alpha }\gamma ^{\mu }P_{L}b_{\beta }\ \bar{c}_{\beta
}\gamma _{\mu }P_{L}c_{\alpha }$ and $O_{2}=\bar{s}\gamma ^{\mu
}P_{L}b\ \bar{c}\gamma _{\mu }P_{L}c$ as shown in Ref. [@Buras], $M_{j}$ and $\Gamma _{j}$ are the masses and widths of intermediate states, and the factors $k_{j}$ are phenomenological parameters for compensating the approximations of FA and VMD and reproducing the correct branching ratios of $B(\Lambda _{b}\to \Lambda J/\Psi \to \Lambda
l^{+}l^{-})=B(\Lambda_b\to \Lambda J/\Psi )\times B(J/\Psi \to
l^{+}l^{-})$ when we study the $\Lambda_b$ decays. We note that by taking $k_{\Psi }\simeq -1/(3C_{1}+C_{2})$ and $B(\Lambda_b\to
\Lambda J/\Psi )=\left( 4.7\pm 2.8\right) \times 10^{-4}$, the $k_{j}$ factors in the $\Lambda_b$ case are almost the same as that in the B meson one. In this paper we take the Wilson coefficients at the scale of $\mu \sim m_{b}\sim
5.0 $ GeV and their values are $C_{1}\left( m_{b}\right) =-0.226,$ $C_{2}\left( m_{b}\right) =1.096,$ $C_{7}\left( m_{b}\right) =-0.305,$ $C_{9}\left( m_{b}\right) =4.186,$ and $C_{10}\left( m_{b}\right) =-4.599$, respectively.
It is clear that one of the main theoretical uncertainties in studying exclusive decays arises from the calculation of form factors. In general there are many form factors in exclusive baryon decays. However, the number of the form factors can be reduced by the heavy quark effective theory (HQET). With HQET, the hadronic matrix elements for the heavy baryon decays could be parametrized as follows [@MR] $$<\Lambda (p,s)\ |\ \bar{s}\ \Gamma \ b\ |\ \Lambda _{b}(\upsilon ,s^{\prime
})>=\bar{u}_{\Lambda }(p,s)\ \left\{ F_{1}(p\cdot v)+\not{v}\ F_{2}(p\cdot
v)\right\} \ \Gamma \ u_{\Lambda _{b}}(v,s^{\prime }) \label{hq}$$ with $R=F_{2}\left( p.v\right) /F_{1}\left( p.v\right) $ where $v$ is the four-velocity of heavy baryon and $\Gamma $ denotes the possible Dirac matrix. Note that in terms of HQET there are only two independent form factors in Eq. (\[hq\]) for each $\Gamma $. In the following, we shall adopt the HQET approximation to analyze the behavior of $\Lambda _{b}\to \Lambda l^{+}l^{-}$.
Differential Decay Rate and Polarizations
==========================================
In this section we present the formulas for the differential decay rates and the longitudinal and normal $\Lambda$ polarizations of $\Lambda_b(p_{\Lambda_b})\to \Lambda (p_{\Lambda},s) l^{+}(p_{l^+})l^{-}(p_{l^-})$. In our calculations, we have included the lepton masses. To study the $\Lambda $ spin polarization, we write the $\Lambda $ four-spin vector in terms of a unit vector, $\hat{\xi}$, along the $\Lambda $ spin in its rest frame, as $$\begin{aligned}
s_{0}\,=\,\frac{\vec{p}_{\Lambda }\cdot \hat{\xi}}{M_{\Lambda }},\qquad \vec{s}\,=\,\hat{\xi}+\frac{s_{0}}{E_{\Lambda }+M_{\Lambda }}\vec{p}_{\Lambda },\end{aligned}$$ and choose the unit vectors along the longitudinal, normal, transverse components of the $\Lambda $ polarization, to be $$\begin{aligned}
\hat{e}_{L} &=&\frac{\vec{p}_{\Lambda }}{\left| \vec{p}_{\Lambda }\right| },
\nonumber \\
\hat{e}_{N} &=&\frac{\vec{p}_{\Lambda }\times \left( \vec{p}_{l^{-}}\times
\vec{p}_{\Lambda }\right) }{\left| \vec{p}_{\Lambda }\times \left( \vec{p}_{l^{-}}\times \vec{p}_{\Lambda }\right) \right| }, \nonumber \\
\hat{e}_{T} &=&\frac{\vec{p}_{l^{-}}\times \vec{p}_{\Lambda }}{\left| \vec{p}_{l^{-}}\times \vec{p}_{\Lambda }\right| }\,, \label{uv}\end{aligned}$$ respectively. The partial decay width for $\Lambda_b\rightarrow \Lambda \
l^{+}\ l^{-}\ ( l=e$ or $\mu$ or $\tau)$ is given by $$\begin{aligned}
d\Gamma &=&\frac{1}{4M_{\Lambda _{b}}}\left| {\cal M}\right| ^{2}\left( 2\pi
\right) ^{4}\delta \left( p_{\Lambda _{b}}-p_{\Lambda
}-p_{l^+}-p_{l^-}\right) \nonumber \\
&&\times \frac{d\vec{p}_{\Lambda }}{\left( 2\pi \right) ^{3}2E_{\Lambda }}
\frac{d\vec{p}_{l^+}}{\left( 2\pi \right) ^{3}2E_{1}}\frac{d\vec{p}_{l^-}}{\left( 2\pi \right) ^{3}2E_{2}} \label{Dr0}\end{aligned}$$ with $$\begin{aligned}
|{\cal M}| ^{2}&=&\frac{1}{2}\left| {\cal M}^{0}\right| ^{2}\left[ 1+\left(
P_L\hat{e}_L+P_N\hat{e}_N+ P_T\hat{e}_T\right) \cdot \hat{\xi}\right] ,
\label{M1}\end{aligned}$$ where $|{\cal M}^0| ^{2}$ is related to the decay rate for the unpolarized $\Lambda $ and $P_i\ (i=L,N,T)$ denote the longitudinal, normal and transverse polarizations of $\Lambda$, respectively. Introducing dimensionless variables of $\lambda _{t}=V_{tb}V_{ts}^{*}$, $\hat{t}=E_{\Lambda}/M_{\Lambda_{b}}$, $r=M_{\Lambda }^{2}/M_{\Lambda_{b}}^{2}$, $\hat{m}_l=m_{l}/M_{\Lambda _{b}}$, $\hat{m}_{b}=m_{b}/M_{\Lambda _{b}}$ and $\hat{s}=1+r-2 \hat{t}$, and integrating the angle dependence of the lepton, the differential decay width in Eq. (\[Dr0\]) can be rewritten as $$\begin{aligned}
d\Gamma &=&\frac{1}{2}d\Gamma^0\left[ 1+\vec{P}\cdot \hat{\xi}\right]
\nonumber \\
d\Gamma^0 &=&\frac{G_F^2\alpha_{em}^2|\lambda_t|^2}{192\pi^5}
M_{\Lambda_b}^5 \sqrt{(\hat{t}^2-r)\left(1-\frac{4\hat{m}_l^2}{\hat{s}}\right)} \rho_0( \hat{t}) d\cos \theta _{\Lambda }d\hat{t}, \label{diffrate}\end{aligned}$$ with $$\begin{aligned}
\vec{P}&=& P_L\hat{e}_L+P_N\hat{e}_N+ P_T\hat{e}_T\end{aligned}$$ and $$\begin{aligned}
\rho_{0}\left( \hat{t}\right) &=& (\Gamma _{1}+\Gamma
_{2}+\Gamma_{3}+\Gamma_{4})\end{aligned}$$ where $$\begin{aligned}
\Gamma_{1} &=&4\frac{\hat{m}_{b}^{2}}{\widehat{s}}|C_{7}|^{2} \left\{
-\left( F_1^2-F_2^2\right) \left( \hat{s}\ \hat{t}-4(1-\hat{t})(\hat{t}-r)\right) \right. \nonumber \\
&& -2F_2(F_1\sqrt{r}+F_2\hat{t})\ \left( \hat{s}\ -4(1-\hat{t})^{2}\right) +8\frac{\hat{m}_l^2}{\hat{s}} \left( (F_{1}^{2}-F_{2}^{2})(1-\hat{t})(\hat{t}-r)\right. \nonumber \\
&& \left. \left. +2F_2(F_1\sqrt{r}+F_2\hat{t})(1-\hat{t})^2 \right) -2\hat{m}_l^2 \left( (F_1^2+F_2^2)\ \hat{t}+2F_{1}F_{2}\sqrt{r}\right) \right\} ,
\nonumber \\
\Gamma_2 &=&12\hat{m}_{b}\mathop{\rm Re}C_{9}^{eff}C_{7}^{*}\left( 1+2\frac{\hat{m}_{l}^{2}}{\hat{s}}\right) \left[ (F_{1}^{2}-F_{2}^{2})(\hat{t}-r)+2F_{2}(F_{1}\sqrt{r}+F_{2}\hat{t})(1-\hat{t})\right] , \nonumber \\
\Gamma _3 &=& \left( |C_{9}^{eff}|^{2}+|C_{10}|^{2}\right) \left\{ \left(1-4\frac{\hat{m}_{l}^{2}}{\hat{s}}\right) \hat{s} \left[(F_{1}^{2}+F_{2}^{2})\
\hat{t}+2F_{1}F_{2}\sqrt{r}\right] \right. \nonumber \\
&&+2(1+2\frac{\hat{m}_l^2}{\hat{s}}) \left( 1-\hat{t}\right) \left.\left[
\left( \hat{t}-r\right) (F_{1}^{2}-F_{2}^{2})+2F_{2}(F_{1}\sqrt{r}+F_{2}\hat{t}) \left( 1-\hat{t}\right) \right] \right\}, \nonumber \\
\Gamma _{4} &=&6\hat{m}_{l}^{2}\left( |C_{9}^{eff}|^{2}-|C_{10}|^{2}\right)
\left[ (F_{1}^{2}+F_{2}^{2})\ \hat{t}+2F_{1}F_{2}\sqrt{r}\right] \,.
\label{rate}\end{aligned}$$ Here the form factors and Wilson coefficients in Eq. (\[rate\]) depend on the $\Lambda$ energy ($E_{\Lambda }$) and the scale of $\mu$. The ranges of $\hat{t}$ and $\hat{s}$ are as follows: $$\begin{aligned}
\sqrt{r} &\leq &\hat{t}\leq \frac{1}{2}\left( 1+r-4\hat{m}_{l}^{2}\right) ,
\nonumber \\
4\hat{m}_{l}^{2} &\leq &\hat{s}\leq \left( 1-\sqrt{r}\right) ^{2}\,.\end{aligned}$$ We note that our result for the differential decay rate in Eq. (\[diffrate\]) is consistent with that given in Refs. [@Lb1; @Lb2] when one takes the limit of massless lepton.
The longitudinal, normal and transverse $\Lambda $ polarization asymmetries in Eq. (\[M1\]) can be defined by $$\begin{aligned}
P_{i}\left( \hat{t}\right)& =&\frac{d\Gamma \left( \hat{e}_{i}\cdot \hat{\xi}=1\right) -d\Gamma \left( \hat{e}_{i}\cdot \hat{\xi}=-1\right) }{d\Gamma
\left( \hat{e}_{i}\cdot \hat{\xi}=1\right) +d\Gamma \left( \hat{e}_{i}\cdot
\hat{\xi}=-1\right) }\,. \label{asy}\end{aligned}$$ From Eqs. (\[diffrate\]) and (\[asy\]), we obtain the polarizations of $P_L$ and $P_N$ to be $$\begin{aligned}
P_L( \hat{t}) &=&\frac{\sqrt{t^{2}-r}}{\sqrt{r}\rho _{0}\left( \hat{t}\right) }D_{L}\end{aligned}$$ and $$\begin{aligned}
P_{N}\left( \hat{t}\right) &=&\frac{-3}{2\rho _{0}\left( \hat{t}\right) }\pi
\sqrt{1-4\frac{\hat{m}_{l}^{2}}{\hat{s}}}\sqrt{\hat{s}\ }\left[ \left(
F_{1}^{2}+F_{2}^{2}\right) \sqrt{r}+2F_{1}F_{2}\ \hat{t}\right] \nonumber \\
&&\times \left[ \mathop{\rm Re}C_{9}^{eff}C_{10}^{*}\left( 1-\hat{t}\right)
+2\hat{m}_{b}\mathop{\rm Re}C_{10}C_{7}^{*}\right] \,, \label{PN}\end{aligned}$$ respectively, where $D_{L}=L_{1}+L_{2}+L_{3}+L_{4}$ with $$\begin{aligned}
L_{1} &=&-4{\frac{\hat{m}_b^2}{\widehat{s}}} \left(1-2\frac{\hat{m}_l^2}{\hat{s}}\right) |C_{7}|^{2}\sqrt{r} \left\{ - \left( 1-4\frac{\hat{m}_l^2}{\hat{s}} \right) \left( F_1^2-F_2^2\right) \ \hat{s} \right. \nonumber \\
&& \left. +4\left( 1-\frac{\hat{m}_l^2}{\hat{s}}\right)
\left(F_1^2-F_2^2+2F_2^{2}\ \hat{t}+2F_{1}F_{2}\sqrt{r}\right) \left(
1-t\right) \right\} \nonumber \\
&& +8\frac{\hat{m}_l^2\hat{m}_b^2}{\hat{s}}|C_{7}|^{2}\sqrt{r} \left\{
\left(F_1^2+F_2^2\right) \left( 1-10\frac{1-\hat{t}}{\hat{s}}\right)
+3\left( 1-2\frac{\hat{m}_l^2}{\hat{s}}\right)
\left(F_{1}^{2}-F_{2}^{2}\right) \right. \nonumber \\
&& \left. -2\left( 1-4\frac{\hat{m}_l^2}{\hat{s}}\right) F_{2}^{2}+4\left(
5-2\frac{\hat{m}_l^2}{\hat{s}}\right) \left(\frac{1-\hat{t}}{\hat{s}}\right)
\left( F_2^2\left( 1-t\right) -F_1F_2\sqrt{r}\right) \right\} , \nonumber \\
L_{2} &=&-12\ \hat{m}_{b}\mathop{\rm Re}C_{9}^{eff}C_{7}^{*}\left( 1+2\frac{\hat{m}_{l}^{2}}{\hat{s}}\right) \sqrt{r}\left[ \left(
F_{1}^{2}-F_{2}^{2}\right) +2\ \hat{t}\ F_{2}^{2}+2\ \sqrt{r}\
F_{1}F_{2}\right] , \nonumber \\
L_{3} &=&-\left( |C_{9}^{eff}|^{2}+|C_{10}|^{2}\right) \sqrt{r}\left\{
\left( 1-4\frac{\hat{m}_{l}^{2}}{\hat{s}}\right) (F_{1}^{2}-F_{2}^{2})\ \
\hat{s}+2\ \left( 1+2\frac{\hat{m}_{l}^{2}}{\hat{s}}\right) \left( 1-\hat{t}\right) \times \right. \nonumber \\
&&\left. \left[ (F_{1}^{2}-F_{2}^{2}+2\ \hat{t}\ F_{2}^{2}+2F_{1}F_{2}\
\sqrt{r}\right] \right\} , \nonumber \\
L_{4} &=&-6\hat{m}_{l}^{2}\left( |C_{9}^{eff}|^{2}-|C_{10}|^{2}\right)
(F_{1}^{2}-F_{2}^{2})\ \sqrt{r}\,. \label{L}\end{aligned}$$ For the T-odd transverse $\Lambda$ polarization, we have that $$\begin{aligned}
P_{T} &\sim &m_{s} \mathop{\rm Im} C_{10}C_{7}^{*}\,. \label{Pt}\end{aligned}$$ It is clear that $P_T$ is zero in the standard model since there is no phase in $C_{10}C_7^*$. We remark that even there is a phase in a theory of the standard model like, due to the suppression of $m_s$, $P_T$ is expected to be small. However, a possible CP violating right-handed interaction could induce a sizable $P_T$ [@chen1]. Therefore, observing $P_T$ could indicate new physics beyond the standard model.
It is interesting to point out that we can also discuss $\Lambda_b\rightarrow \Lambda \ \bar{\nu}\ \nu $ by taking the limits of $$\begin{aligned}
m_l\rightarrow 0,\ C_7\rightarrow 0,\ C_9^{eff}\rightarrow \frac{X(x_t)}{\sin ^2\theta _W},\ C_{10}\rightarrow -\frac{X(x_t)}{\sin ^2\theta _W}\end{aligned}$$ in Eqs. (\[Dr0\])-(\[Pt\]), where $X(x_t) =0.65x_{t}^{0.575}$ [@Buras] and $x_t=m_t^2/M_W^2$. Explicitly, we have $$\begin{aligned}
d\Gamma \left( \Lambda _{b}\rightarrow \Lambda \ \bar{\nu}\ \nu \right) &=&\frac{1}{2}d\Gamma ^{0}\left( \Lambda _{b}\rightarrow \Lambda \ \bar{\nu}\
\nu \right) \left[ 1+\vec{P}^{\nu \nu }\cdot \hat{\xi}\right] , \nonumber \\
d\Gamma ^{0}\left( \Lambda _{b}\rightarrow \Lambda \ \bar{\nu}\ \nu \right)
&=&3\frac{G_{F}^{2}\alpha _{em}^{2}\lambda _{t}^{2}}{192\pi ^{5}}M_{\Lambda
_{b}}^{5}\sqrt{\hat{t}^{2}-r}\ \rho ^{\nu \nu }\left( \hat{t}\right) d\cos
\theta _{\Lambda }d\hat{t} \label{nunu}\end{aligned}$$ where $$\begin{aligned}
\rho ^{\nu \nu }\left( \hat{t}\right) &=&2\left( \frac{X\left( x_{t}\right)
}{\sin ^{2}\theta _{W}}\right) ^{2}\left\{ \left[ (F_{1}^{2}+F_{2}^{2})\
\hat{t}+2F_{1}F_{2}\sqrt{r}\right] \hat{s}+2\left( 1-\hat{t}\right) \right.
\nonumber \\
&&\left. \times \left[ \left( \hat{t}-r\right)
(F_{1}^{2}-F_{2}^{2})+2F_{2}(F_{1}\sqrt{r}+F_{2}\hat{t})\left( 1-\hat{t}\right) \right] \right\} \label{rnunu}\end{aligned}$$ and $$\begin{aligned}
P_{L}^{\nu \nu }\left( \hat{t}\right) &=&-2\frac{\sqrt{t^{2}-r}}{\rho ^{\nu
\nu }\left( \hat{t}\right) }\left( \frac{X\left( x_{t}\right) }{\sin
^{2}\theta _{W}}\right) ^{2}\left\{ (F_{1}^{2}-F_{2}^{2})\ \hat{s}+2\ \left(
1-t\right) \times \right. \nonumber \\
&&\left. \left[ (F_{1}^{2}-F_{2}^{2}+2\ \hat{t}\ F_{2}^{2}+2F_{1}F_{2}\
\sqrt{r}\right] \right\}\,. \label{nnunu}\end{aligned}$$ Here we have only listed the longitudinal polarization of $\Lambda$ because the momentum of the neutrino cannot be measured experimentally.
Numerical Analysis
==================
In order to analyze the decay rate and polarization asymmetries, we use the Wilson coefficients at the scale $\mu \approx m_{b}$ as stated in Sec. 2. The other parameters used in our numerical calculations are listed in Table 1.
------------------------ ------------------------ ------------
$M_{\Lambda _{b}}$ $5.64$ GeV
$M_{\Lambda }$ $1.116$ GeV
$m_{t}$ $165$ GeV
$m_{b}$ $4.8$ GeV
$m_{\tau }$ $1.777$ GeV
$m_{\mu }$ $1.05$ GeV
$m_{c}$ $1.4$ GeV
$\alpha _{em}$ $1/129$
$\tau _{\Lambda _{b}}$ $1.8848\times 10^{12}$ GeV$^{-1}$
$V_{tb}V_{ts}^{*}$ $0.04$
------------------------ ------------------------ ------------
: Input parameters used in our numerical calculations.
As to the $\Lambda _{b}\rightarrow \Lambda $ transition form factors, we adopt the results and input parameters given in Ref. [@Lb1], in which the QCD sum rule approach based on the framework of HQET was used. However, there is a undetermined parameter, Borel parameter (M), in the approach, which is introduced to suppress the contribution from the higher excited and continuum states. According to the analysis of Ref. [@Lb1], it could be $1.5\ GeV\leq M\leq 1.9\ GeV$. For simplicity, we will take $M=1.7\ GeV$ in our numerical analysis. As a comparison, we will also present the results with the dipole form assumption [@MR].
Decay Rates and Polarizations of $\Lambda $
-------------------------------------------
From Eqs. (\[diffrate\]) and ([\[rate\]), by integrating the whole range of $\Lambda $ energy and setting phenomenological factor $\kappa =-1/\left( 3C_{1}+C_{2}\right) $, the branching ratios of the dilepton decays are summarized in Table 2 and the distributions of the differential decay rates are shown in Figures 1 and 2 for $\Lambda _{b}\to \Lambda \mu ^{+}\mu ^{-}$ and $\Lambda _{b}\to \Lambda \tau ^{+}\tau ^{-}$, respectively. Here we have also illustrated the results from the pole model [@MR]. The form factors with the dipole forms in the model are given by $$F_{1,2}\left( p_{\Lambda }\cdot v\right) =N_{1,2}\left( \frac{\Lambda _{QCD}}{\Lambda _{QCD}+p_{\Lambda }\cdot v}\right) ^{2} \label{dipole}$$ where $p_{\Lambda }\cdot v=E_{\Lambda }$ and $\Lambda _{QCD}$ is chosen to be around $200$ $MeV$. From Eq. (\[dipole\]), one obtains that $R=F_{2}/F_{1}=N_{2}/N_{1}\sim -0.25$ [@MR; @CLEO2]. In terms of HQET the form factors of $\Lambda _{b}\rightarrow \Lambda $ should be the same as that of $\Lambda _{c}\rightarrow \Lambda $ at the maximal momentum transfer. Therefore, by using the measured branching ratio of $\Lambda _{c}\rightarrow \Lambda l\nu $, we extract that $|N_{1}|\sim 52.32$ with the same dipole forms. ]{}
----------------------------------------------------------------------------------------------------------------------------------------------------------
Model Decay Br $\Lambda_b\to\Lambda\nu\bar{\nu}$ $\Lambda_b\to \Lambda $\Lambda_{b}\to \Lambda \mu^{+}\mu^{-}$ $\Lambda _{b}\to \Lambda
e^+e^-$ \tau^+\tau^-$
------------ ------------ ----------------------------------- ----------------------- ----------------------------------------- --------------------------
QCD without LD
Sum Rule with LD
Pole Model without LD
with LD
----------------------------------------------------------------------------------------------------------------------------------------------------------
: Decay branching ratios (Br) based on the form factors from the QCD sum rule approach and the dipole model, respectively
From Table 2, we find that the branching ratios with including LD contributions are about $1-2$ orders of magnitude larger than that without LD ones and the results from the pole model are close to those from the QCD sum rule.
If it is not mentioned, we shall use the form factors from the QCD sum rule approach in the rest of our numerical analysis.
To estimate the contributions to the decay branching ratios by excluding the resonances of $J/\psi $ and $\psi^{\prime}$, we choose five separate regions in terms of the masses of $J/\psi $ and $\psi ^{\prime }$, and they are given as follows $$\begin{aligned}
I &:&\ M_{\Lambda}\leq E_{\Lambda }\leq E|_{\max }-\delta _{\psi ^{\prime
}}^{1}\,, \nonumber \\
II &:&\ E|_{\max }-\delta _{\psi ^{\prime }}^{1}\leq E_{\Lambda }\leq
E|_{\max }-\delta _{\psi ^{\prime }}^{2}\,, \nonumber \\
III&:&\ E|_{\max }-\delta _{\psi ^{\prime }}^{2}\leq E_{\Lambda}\leq
E|_{\max }-\delta _{J/\psi }^{1}\,, \nonumber \\
IV&:&\ E|_{\max}-\delta _{J/\psi}^1\leq E_{\Lambda}\leq E|_{\max} -\delta
_{J/\psi}^2, \nonumber \\
V&:& E|_{\max}-\delta_{J/\psi }^{2}\leq E_{\Lambda }\leq E|_{\max }\,,\end{aligned}$$ where $$\begin{array}{ll}
E|_{\max }=M_{\Lambda _{b}}\left( 1+r-4\hat{m}_{l}^{2}\right) /2, & \\
\delta _{\psi ^{\prime }}^{1}=\left( M_{\psi ^{\prime }}+\sqrt{\sqrt{2}M_{\psi ^{\prime }}\Gamma _{\psi ^{\prime }}}\right) ^{2}/2M_{\Lambda _{b}},
& \delta _{\psi ^{\prime }}^{2}=\left( M_{\psi ^{\prime }}-\sqrt{\sqrt{2}M_{\psi ^{\prime }}\Gamma _{\psi ^{\prime }}}\right) ^{2}/2M_{\Lambda _{b}},
\\
\delta _{J/\psi }^{1}=\left( M_{J/\psi }+\sqrt{\sqrt{2}M_{J/\psi }\Gamma
_{J/\psi }}\right) ^{2}/2M_{\Lambda _{b}}, & \delta _{J/\psi }^{2}=\left(
M_{J/\psi }-\sqrt{\sqrt{2}M_{J/\psi }\Gamma _{J/\psi }}\right)
^{2}/2M_{\Lambda _{b}}.
\end{array}$$ The factor of $\sqrt{2}$ in $\delta _{V}^{i}$ is a typical value and one may take a larger value to reduce the LD contributions in the regions of $I$ and $V$. The estimations of the decay branching ratios in the different regions are listed in Table 3. From the table, We find that the RE in region I is about $20\%$ for the $e^{+}e^{-}$ and $\mu ^{+}\mu ^{-}$ modes and $25\%$ for $\tau ^{+}\tau^{-}$. The larger RE for the $\tau $ pair arises from $\Gamma _{4}$ in Eq. (\[rate\]), which is proportional to the lepton mass. Moreover, this term also yields different distributions between the electron (or muon) and tau modes in region I when a large deviation from $(|C_9(m_b)|-|C_{10}|)$ appears. Therefore, studying the region with lower RE could distinguish the SD Wilson coefficients from the standard model.
------ ------- -- -- ---- -- -------------------- -- -- -- --
Br ($\times 10^{-7}$)
Mode I
SR
$2.7$
$3.4$
$2.7$
$3.4$
$1.2$
$1.6$
------ ------- -- -- ---- -- -------------------- -- -- -- --
: Decay branching ratios for QCD sume rule (SR) and Pole model (PM) with and without LD in different regions of $\Lambda $ energy with $\kappa =-1/\left( 3C_{1}+C_{2}\right) $.
As we can see from Eq. (\[effc9\]), the LD effects have been absorbed into the Wilson coefficient of $C_{9}^{eff}$ and they are parametrized in the form of the phenomenological Breit-Wigner Ansatz. To compensate FA and VMD approximation, one phenomenological factor $\kappa $ is also introduced. In Table 4, we show the decay branching ratios by taking $\kappa=-3.5$ and $-1.9 $. It is easily seen that in the regions of $I$ and $V$ the differences for the branching ratios with lower and higher $\kappa$ are between $5\% -
16\%$. This tells us that, as expected, the uncertainty from the LD effect is small outside the resonance region.
------------ ------- ---- -------------------- -- --
Br ($\times 10^{-7}$)
Decay Mode I
$2.6$
$2.8$
$2.6$
$2.8$
$1.1$
$1.2$
------------ ------- ---- -------------------- -- --
: Decay branching ratios in the whole range of $\Lambda $ energy including LD with two different values of $\kappa $.
In order to study how the effects arising from new physics beyond the standard model will affect the baryonic dilepton decays, we consider cases where the Wilson coefficients are different from those in the standard model. The results for the distributions of the differential branching rates are shown in Figures $3-6$.
We now discuss our results as follows:
$\bullet$ According to the results in Table 3 and Figures 1 and 2, we clearly see that outside the resonant regions the uncertainties arising from the QCD models are larger than that from the LD effects. $\bullet$ We first compare our results in baryon decays with those in the B meson dilepton ones of $B\rightarrow
K^{*}l^{+}l^{-}$ [@DTP; @LMS; @AMM; @OT; @KS; @Geng1]. In the meson decays, the pole of $\hat{s}$ is related to $\left| \hat{m}_{b}C_{7}/\hat{s}\right| ^{2}$ and $\hat{m}_{b}C_{7}/\hat{s}$, respectively, and thus with the requirement $\hat{s}\geq
4\hat{m}_{l}$ from the phase space, the processes of $B\rightarrow K^{*}\mu
^{+}\mu ^{-}$ and $B\rightarrow K^{*}e^{+}e^{-}$ have very different decay rates. However, for the decays of $\Lambda _{b}\rightarrow\Lambda l^{+}l^{-}$, the associated terms are proportional to $\left| \hat{m}_{l}\hat{m}_{b}C_{7}\right| ^{2}/\hat{s}^{2}$ and $\left| \hat{m}_{b}C_{7}\right| ^{2}/\hat{s}$. Clearly, due to the mass suppression for the light lepton, the main pole dependence is $\sim \left| \hat{m}_{b}C_{7}\right| ^{2}/\hat{s}$ so that the rate difference between $\Lambda_b\to \Lambda \mu^{+}\mu^{-}$ and $\Lambda_b\to \Lambda e^{+}e^{-}$ is small.
$\bullet$ The differential decay rates of $\Lambda _{b}\rightarrow \Lambda
l^{+}l^{-}$ are sensitive to the signs of $C_9$ and $C_7$. Although $C_{7}\ll C_{9}$ and $C_{10}$, there exists an enhanced factor of $12\mathop{\rm Re}C_{9}C_{7}^{*}\sim 15.3$ in $\Gamma _{2}$ of Eq. (\[rate\]). When the sign of $C_{7}$ is opposite to that in the standard model, there is a deviation of 50% in branching ratio for neglecting RE. Thus, the contribution from electromagnetic part cannot be neglected. As changing the sign of $C_{9}$ to be opposite to that in the standard model, only a deviation of 27% on the branching ratios occurs. However, from Figures 3 and 5, we see that the distributions are different from each others.
$\bullet$ From Eq. (\[rate\]), we find that the differential decay rates cannot have the information in the sign of $C_{10}$ since they are always related to $\left| C_{10}\right| ^{2}$.
$\bullet$ From Figures 4 and 6, we find that with $C_{10}=2C_{9}|_{SM}$ the distribution for the differential decay rate of the $\tau ^{+}\tau ^{-}$ mode is higher than that with $C_{9}=2C_{10}|_{SM}$ in region I but it is reversed in that of the $\mu^{+}\mu ^{-}$ distribution. The origin of this difference is from the $\Gamma _{4}$ in Eq. ( \[rate\]) which is proportional to $6\hat{m}_{l}^{2}\left( \left| C_{9}\right| ^{2}-\left|
C_{10}\right| ^{2}\right) $. This effect can be neglected in the standard model since $|C_{9}| \sim |C_{10}|$ and the light lepton modes as well. Although $\hat{m}_{\tau }^{2}\sim 10\%$, this factor will become important. if there is a large deviation between $C_{9}$ and $C_{10}$. $\bullet$ The decay width distributions for the longitudinal polarized $\Lambda$ with and without LD effects as the function of $\Lambda $ energy are shown in Figures 7 and 8. Comparing the figures with the differential decay branching rates in Figures 1 and 2, respectively, we find that the distributions are very similar to each other except the opposite sign.
$\bullet$ Finally, as usual, from Eq. (\[diffrate\]) we may also write the partial decay rate as $$\begin{aligned}
d\Gamma _{\Lambda _{b}}&=&\frac{1}{2}\Gamma _{0}\left( 1-\alpha _{\Lambda }\
{\rm \hat{p}\cdot \hat{s}} d\cos \theta _{\Lambda }\right)\,,\end{aligned}$$ where $\Gamma _{0}$ is related to the decay width of $\Lambda
_{b}\rightarrow \Lambda \ l^{+}\ l^{-}$, p is the unit direction of $\Lambda $ momentum in the $\Lambda _{b}$ rest frame, and $\alpha _{\Lambda }$, called $\Lambda $ polarization, is defined by $$\begin{aligned}
\alpha _{\Lambda }=\frac{\int_{r}^{t_{\max }}D_{L}(1-4 \hat{m}_l^2/\hat{s})\left( \hat{t}^{2}-r\right) /\sqrt{r}d\hat{t}}{\int_{r}^{t_{\max }}\sqrt{1-4\hat{m}_l^2/\hat{s}}\sqrt{\hat{t}^{2}-r}\rho _{0}\left( \hat{t}\right) d\hat{t}},\end{aligned}$$ where $t_{\max }=\left( 1+r-4\hat{m}_{l}^{2}\right) /2$. Numerically, we find that the polarizations of $\Lambda $ in $\Lambda _{b}\rightarrow
\Lambda \ l^{+}\ l^{-}$ $(l=e,\ \mu ,\ \tau)$ decays are all unity, $\alpha
_{\Lambda }\approx 1$.
Polarization asymmetries
------------------------
In this subsection we will discuss longitudinal and normal polarization asymmetries and their implications and we will study the transverse polarization elsewhere [@chen1] since it is zero in the standard model as mentioned in Sec. 3. From Eq. (\[asy\]), we show the the distributions of $P_L$ and $P_N$ with respect to the dimensionless kinematic variable $\hat{t}$ in Figures $9-12$, respectively. From the figures, we find the following interesting results:
$\bullet$ The polarization asymmetries are insensitive to the LD effects.
$\bullet$ The values of $P_{L}$ are near unity except a narrow region with a small $\Lambda $ momentum.
$\bullet$ $P_N$ approaches zero as the $\Lambda $ energy increases. This is because the polarization is proportional to $\sqrt{\hat{s}}$ as shown in Eq. (\[PN\]).
$\bullet$ The values of $P_{L,N}$ from the QCD sum rule and the pole models shown in the figures are close to each other. The results imply that both $P_L$ and $P_N$ are not very sensitive to the form factors. Therefore, one would like to use $P_{L,N}$ to probe the short-distance (SD) physics due to the smallness of the uncertainties from the strong interaction.
We now discuss the sensitivity for the longitudinal polarization of $P_{L}$ to new physics. We first notice that by using different values of the Wilson coefficients from the standard model, the polarizations do not change. The reason is that the coefficients get canceled out between the denominator and numerator in Eq. (\[asy\]). However, in our derivation for the differential decay rate, we have assumed the $V-A$ hadronic current and neglected the contribution of left-handed electromagnetic moment since it is proportional to the strange quark mass. If we include the interaction with the right-handed current, the polarization will behave quit different from that in the standard model, which can be understand easily by Eq. (26) of Ref. [@Lb1] as $h_A\neq h_V$. Finally, we define the integrated longitudinal and normal polarization asymmetries as $$\begin{aligned}
\bar{P}_{L}&=&\int d\hat{t} P_L\,, \nonumber \\
\bar{P}_{N}&=&\int d\hat{t} P_N\,. \label{plpn}\end{aligned}$$ In the standard model, we obtain that $\bar{P}_{L(N)}=-0.31\ ( 0.02) $ and $\bar{P}_{L(N) }=-0.12\ (0.01)$ for $\mu \mu $ and $\tau \tau $ modes, respectively. If deviations from the standard model predictions for the integrated polarization asymmetries are measured, it is clear that there exit some kinds of new physics.
Conclusions
===========
We have studied the rare baryonic exclusive decays of $\Lambda_b\to \Lambda
l^+ l^-\ (l=e,\mu,\tau )$ with polarized $\Lambda $. Under the approximation of HQET, in the standard model we have derived the differential decay rates and the polarization asymmetries of $\Lambda $ by including lepton mass effects.
We have found that with the LD effects the decay branching ratios of $\Lambda_b\to \Lambda l^+ l^-\ (l=e,\mu,\tau )$ are $5.3\times 10^{-5}$, $5.3\times 10^{-5}$, and $1.1\times 10^{-5}$ from the QCD sum rule approach and $1.2\times 10^{-5}$, $1.2\times 10^{-5}$, and $3.2\times 10^{-6}$ from the the pole model, respectively. We have also estimated the decay branching ratio of $\Lambda_b\to \Lambda \nu \bar{\nu}$ to be $1.6\times 10^{-5}$ and $3.3\times 10^{-6}$ in the two models, respectively. In physics beyond the standard model, we have studied various cases of different Wilson coefficients. We have shown that the decay rates as well as the distributions can be very different from those in the standard model.
The integrated longitudinal $\Lambda$ polarizations are $-0.31$ and $-0.12$, while that of the normal ones $0.02$ and $0.01$, for di-muon and tau modes, respectively. The CP-odd transverse polarization of $\Lambda$ is zero in the standard model but it is expected to be sizable in new physics such as the CP violating theories with right-handed interactions. We have demonstrated that the polarization asymmetries are insensitive to LD contributions but sensitive to the right-handed couplings. It is clear that one could probe new physics through measurements of the $\Lambda$ polarizations in the decays of $\Lambda_b\to \Lambda l^+ l^-$.
[**Acknowledgments**]{}
This work was supported in part by the National Science Council of the Republic of China under contract numbers NSC-89-2112-M-007-054 and NSC-89-2112-M-006-033.
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The differential decay branching ratio of $\Lambda_b\to\Lambda
\mu^+\mu^-$ as a function of $\Lambda$ energy. The solid and dashed curves stand for the QCD sum rule and pole models, respectively.
Same as Figure 1 but for $\Lambda_b\to\Lambda \tau^+\tau^-$.
The differential decay branching ratio of $\Lambda_b\to\Lambda
\mu^+\mu^-$ as a function of $\Lambda$ energy with the Wilson coefficients being different from those in the standard model. The solid, dashed, dotted, long-dashed, and dash-dotted curves represent the results of the standard model (SM), $C_{10}=0$, $C_9=-C_9|_{SM}$, $C_7=-C_7|_{SM}$, and $C_{7}=0$, respectively.
Same as Figure 3 but the dashed, dotted, long-dashed, and dash-dotted curves are for $C_9=-2C_9|_{SM}$, $C_9=-2C_9|_{SM}$ and $C_7=0$, $C_9=2C_9|_{SM}$, and $C_{10}=2C_{10}|_{SM}$, respectively.
Same as Figure 3 but for $\Lambda_b\to\Lambda \tau^+\tau^-$.
Same as Figure 4 but for $\Lambda_b\to\Lambda \tau^+\tau^-$.
The decay width distribution of $\Lambda_b\to\Lambda \mu^+\mu^-$ for the longitudinal polarized $\Lambda$ as a function of $\Lambda$ energy with and without RE.
Same as Figure 7 but for $\Lambda_b\to\Lambda \tau^+\tau^-$.
The longitudinal polarization asymmetry of $\Lambda_b\to\Lambda
\mu^+\mu^-$ as a function of $E_{\Lambda}/M_{\Lambda}$. Legend is the same as Figure 1.
Same as Figure 9 but for $\Lambda_b\to\Lambda \tau^+\tau^-$.
The normal polarization asymmetry of $\Lambda_b\to\Lambda
\mu^+\mu^-$ as a function of $E_{\Lambda}/M_{\Lambda}$. Legend is the same as Figure 1.
Same as Figure 11 but for $\Lambda_b\to\Lambda \tau^+\tau^-$.
15.5cm
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