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abstract: |
We determine the maximum size $A_2(8,6;4)$ of a binary subspace code of packet length $v=8$, minimum subspace distance $d=6$, and constant dimension $k=4$ to be $257$. There are two isomorphism types of optimal codes. Both of them are extended LMRD codes. In finite geometry terms, the maximum number of solids in $\operatorname{PG}(7,2)$ mutually intersecting in at most a point is $257$. The result was obtained by combining the classification of substructures with integer linear programming techniques. This result implies that the maximum size $A_2(8,6)$ of a binary mixed-dimension subspace code of packet length $8$ and minimum subspace distance $6$ is $257$ as well.
**Keywords:** network coding, constant-dimension codes, subspace distance, classification, integer linear programming.
**MSC:** 51E20 94B65 05B25 51E23
author:
- Daniel Heinlein
- Thomas Honold
- Michael Kiermaier
- Sascha Kurz
- 'Alfred Wassermann [^1]'
title: 'Classifying optimal binary subspace codes of length 8, constant dimension 4 and minimum distance 6'
---
Introduction
============
Let $q>1$ be a prime power, [[${\ensuremath{\mathbb{F}}}_q$]{}]{} the field with $q$ elements, and $V\cong{{\ensuremath{\mathbb{F}}}}_q^v$ a $v$-dimensional vector space over [[${\ensuremath{\mathbb{F}}}_q$]{}]{}. By ${{\ensuremath{\operatorname{L}({\ensuremath{V}})}}}$ we denote the set of all subspaces of $V$. The set ${{\ensuremath{\operatorname{L}({\ensuremath{V}})}}}$ forms a lattice with respect to the inclusion order $U \le W \Leftrightarrow U \subseteq W$, the lattice of flats of the projective geometry $\operatorname{PG}(V)\cong
\operatorname{PG}({\ensuremath{{\ensuremath{\mathbb{F}}}_q}}^v)=\operatorname{PG}(v-1,q)$, and a metric space with respect to the subspace distance $${\mathrm{d}_{\mathrm{s}}}(U,W)= \dim(U+W)-\dim(U \cap W)=\dim(U)+\dim(W)-2\dim(U \cap W),$$ which may be viewed as a $q$-analogue of the Hamming space $({\ensuremath{\mathbb{F}}}_2^v,{\mathrm{d}_{\mathrm{Ham}}})$.
The metric space $({\ensuremath{\operatorname{L}({\ensuremath{V}})}},{\mathrm{d}_{\mathrm{s}}})$ plays an important role in network coding. It was introduced as part of the subspace channel model in [@MR2451015] to describe error-resilient data transmission in packet networks employing random linear network coding.
For $k\in\{0,\ldots,v\}$, [[${\left[\begin{smallmatrix}{{\ensuremath{V}}}\\{k}\end{smallmatrix}\right]}$]{}]{} denotes the set of all $k$-dimensional subspaces in [[$V$]{}]{}. We have $$\#{\ensuremath{{\left[\begin{smallmatrix}{{\ensuremath{V}}}\\{k}\end{smallmatrix}\right]}}}={\left[\begin{smallmatrix}{v}\\{k}\end{smallmatrix}\right]_{q}}= \prod_{i=1}^{k} \frac{q^{v-k+i}-1}{q^i-1}\text{.}$$
A subset $\mathcal{C}$ of ${\left[\begin{smallmatrix}{V}\\{k}\end{smallmatrix}\right]}$ is called a $k$-dimensional *constant-dimension code* (*CDC*). As usual, the elements of $\mathcal{C}$ are called *codewords*. For $\#\mathcal{C} \geq 2$, the minimum distance of $\mathcal{C}$ is defined as ${\mathrm{d}_{\mathrm{s}}}(\mathcal{C}) = \min\{{\mathrm{d}_{\mathrm{s}}}(U,W) \mid U,W\in\mathcal{C}, U\neq W\}$. The most important parameters of a CDC $\mathcal{C}$ are the order $q$ of the base field, the dimension $v$ of the ambient space $V$, the minimum (subspace) distance $d =
{\mathrm{d}_{\mathrm{s}}}(\mathcal{C})$ of $\mathcal{C}$, the cardinality $N=\#\mathcal{C}$, and the constant dimension $k$ of each element in $\mathcal{C}$. We denote them by $(v,N,d;k)_q$. In a $(v,N,d;k)_q$ CDC the minimum distance $d$ is always an even number satisfying $2\le d\le 2\min\{k,v-k\}$.
The determination of the corresponding maximal size $A_q(v,d;k)$ and the classification of the optimal codes is known as the *main problem of subspace coding*, since it forms a $q$-analogue of the *main problem of classical coding theory* (cf. [@MR0465509 page 23]).
Without the restriction of all codewords having the same dimension, i.e., $\mathcal{C}\subseteq{\ensuremath{\operatorname{L}({\ensuremath{V}})}}$, the code $\mathcal{C}$ is called a *subspace code* (per se) or a *mixed-dimension code* (*MDC*). The maximal cardinality of an MDC in $V$ having subspace distance $d$ is denoted as $A_q(v,d)$. Clearly, $A_q(v,d;k) \leq A_q(v,d)$ for all $k$.
In the following, let $\beta$ be a fixed non-degenerate symmetric bilinear form on $V$ and $\pi : L(V) \to L(V), U \mapsto U^\perp$ the corresponding polarity. The *orthogonal code* or *dual code* of a subspace code $\mathcal{C}$ is defined as $$\mathcal{C}^\perp = \pi(\mathcal{C}) = \{U^\perp \mid U \in \mathcal{C}\}\text{.}$$ Up to isomorphism of subspace codes as defined further below, the code $\mathcal{C}^\perp$ does not depend on the particular choice of $\beta$.
Considering orthogonal codes allows us to almost halve the parameter space: If $\mathcal{C}$ is a $(v,N,d;k)_q$ CDC then $\mathcal{C}^\perp$ has the parameters $(v,N,d;v-k)_q$, i.e., $A_q(v,d;k)=A_q(v,d;v-k)$, so that we can assume $k\le \frac{v}{2}$ in the following. The iterative application of the so-called Johnson type bound II ([@xia2009johnson Theorem 3], [@MR2810308 Theorem 4,5]), which is a $q$-generalization of [@johnson1962new Inequality (5)], gives the upper bound $$\label{ie_r_johnson}
A_q(v,d;k) \le
\left\lfloor \frac{q^{v}-1}{q^{k}-1} \left\lfloor \frac{q^{v-1}-1}{q^{k-1}-1} \left\lfloor \ldots
\left\lfloor \frac{q^{v'+1}-1}{q^{\frac{d}{2}+1}-1} A_q(v',d;\frac{d}{2}) \right\rfloor
\ldots \right\rfloor \right\rfloor \right\rfloor$$ where $v' = v - k +
\frac{d}{2}$. It is attained with equality at $v=ak$ and $d=2k$, i.e., for spreads, and also at $v=13$, $k=3$, $d=4$ with $A_2(13,4;3)=1597245$, see [@MR3542513]. Using $q^r$-divisible linear codes over ${\ensuremath{\mathbb{F}}}_q$ with respect to the Hamming metric, this bound was sharpened very recently, see [@kiermaier2017improvement], to $$\label{ie_best_upper_bound}
A_q(v,d;k) \!\le
\!\left\{\! \frac{q^{v}-1}{q^{k}-1} \!\left\{\! \frac{q^{v-1}-1}{q^{k-1}-1} \!\left\{\! \!\ldots\!
\!\left\{\! \frac{q^{v'+1}-1}{q^{\frac{d}{2}+1}-1} A_q(v',d;\frac{d}{2}) \!\right\}_{\!\!\frac{d}{2}+1}
\!\!\!\!\!\!\!\ldots \!\right\}_{\!\!k-2} \!\right\}_{\!\!k-1}\! \right\}_{\!\!k}\!\!,$$ where $\left\{a / {\left[\begin{smallmatrix}{k}\\{1}\end{smallmatrix}\right]_{q}}\right\}_k:=b$ with maximal $b\in\mathbb{N}$ permitting a representation of $a-b\cdot {\left[\begin{smallmatrix}{k}\\{1}\end{smallmatrix}\right]_{q}}$ as non-negative integer combination of the summands $q^{k-1-i}\cdot\frac{q^{i+1}-1}{q-1}$ for $0\le i\le k-1$.[^2] Of course, Inequality (\[ie\_r\_johnson\]) is implied by Inequality (\[ie\_best\_upper\_bound\]). Both bounds refer back to bounds for so-called partial spreads, i.e., $(v,N,2k;k)_q$ codes, for which the minimum distance has the maximal value $d=2k$. For upper bounds in this special subclass of CDCs, there is a recent series of improvements [@kurz2017improved; @kurz2017packing; @nastase2016maximum]. The underlying techniques can be explained using the language of projective $q^{k-1}$-divisible codes and the linear programming method, see [@honold2016partial]. While a lot of upper bounds for the maximum sizes of CDCs have been proposed in the literature, most of them are dominated by Inequality (\[ie\_r\_johnson\]), see [@heinlein2017asymptotic]. Indeed, besides Inequality (\[ie\_best\_upper\_bound\]), the only known improvements were $A_2(6,4;3)=77<81$ [@MR3329980] and $A_2(8,6;4)\le 272<289$ [@new_bounds_subspaces_codes]. The latter result is improved in this paper. For numerical values of the known lower and upper bounds on the sizes of subspace codes we refer the reader to the online tables at <http://subspacecodes.uni-bayreuth.de> associated with [@HKKW2016Tables]. The tables in particular contain representatives for the two isomorphism types of $(8,257,6;4)_2$ CDCs. A survey on Galois geometries and coding theory can be found in [@Etzion2016].
This article investigates binary CDCs with $v=8$, $d=6$ and $k=4$. The so-called Echelon–Ferrers construction, see e.g. [@MR2589964], gives $A_2(8,6;4)\ge 257$. More precisely, a corresponding code is given by a lifted maximum rank distance code (LMRD code), extended by a single codeword. In Corollary \[cor\_isomorphism\_types\] we will show that up to isomorphism there are two such codes. By [@MR3015712 Theorem 10], this construction is optimal for subspace codes containing an LMRD code. Our main theorem states that this construction is optimal even without the restriction of containing an LMRD code and, moreover, that all subspace codes of maximum possible size $257$ are extended LMRD codes.
\[main\_thm\] $A_2(8,6;4) = 257$, and up to isomorphism there are two maximum codes, both are extended LMRD codes.
Theorem \[main\_thm\] is the main theorem of this paper.
\[fact\] If $v=2k \ge 8$ then $A_q(v,v-2)=A_q(v,v-2;k)$.
Theorem \[main\_thm\] and Fact \[fact\] together give the maximum cardinality in the corresponding mixed-dimension case:
$A_2(8,6)=257$.
Given Theorem \[main\_thm\], one may ask whether there exists an integer $k\ge 4$ with $A_2(2k,2k-2;k)>2^{2k}+1$.
The remaining part of the paper is structured as follows. In Section \[sec\_preliminaries\] we provide the necessary preliminaries on lifted maximum rank distance codes, acting symmetry groups, and upper bounds for code sizes based on the number of incidences of codewords with a fixed subspace. As in [@MR3329980], we want to apply integer linear programming methods in order to determine the exact maximum size of CDCs with the specified parameters. Since this algorithmic approach suffers from the presence of a large symmetry group[^3], we use the inherent symmetry to prescribe some carefully chosen substructures up to isomorphism. A general outline of the proof of Theorem \[main\_thm\] is presented in Section \[sec\_general\]. The substructures involved are described in Section \[sec\_substructures\] and the integer linear programming formulations are described in Section \[sec\_ILP\]. All these parts are put together in the proof of our main theorem in Section \[sec\_main\_thm\].
Preliminaries {#sec_preliminaries}
=============
Let $m,n$ be positive integers. The *rank distance* of $m\times n$ matrices $A$ and $B$ over ${\ensuremath{\mathbb{F}}}_q$ is defined as ${\mathrm{d}_{\mathrm{r}}}(A,B)=\operatorname{\operatorname{rk}}(A-B)$. The rank distance provides a metric on ${\ensuremath{\mathbb{F}}}_q^{m\times n}$. Any subset $C$ of the metric space $({\ensuremath{\mathbb{F}}}_q^{m\times n},{\mathrm{d}_{\mathrm{r}}})$ is called *rank metric code*. Its minimum distance $d$ is the minimum of the rank distances between pairs of distinct codewords (defined for $\#C \ge 2$). If $C$ is a subspace of the ${\ensuremath{\mathbb{F}}}_q$-vector space ${\ensuremath{\mathbb{F}}}_q^{m\times n}$, then $C$ is called *linear*. If $m\le n$ (otherwise transpose), then $\# C\le q^{(m-d+1)n}$ by [@delsarte1978bilinear Theorem 5.4]. Codes achieving this bound are called *maximum rank distance* (MRD) codes. In fact, MRD codes do always exist. A suitable construction has independently been found in [@delsarte1978bilinear; @gabidulin1985theory; @roth1991maximum]. Today these codes are known as *Gabidulin codes*. In the square case $m=n$, after the choice of an ${\ensuremath{\mathbb{F}}}_q$-basis of ${\ensuremath{\mathbb{F}}}_{q^n}$ the Gabidulin code is given by the matrices representing the ${\ensuremath{\mathbb{F}}}_q$-linear maps given by the $q$-polynomials $a_0 x^{q^0}+ a_1 x^{q^1} +\dots+ a_{n-d} x^{q^{n-d}} \in
{\ensuremath{\mathbb{F}}}_{q^n}[x]$. The lifting map $\Lambda\colon {\ensuremath{\mathbb{F}}}_q^{m\times n}\to {\left[\begin{smallmatrix}{{\ensuremath{\mathbb{F}}}_q^{m+n}}\\{m}\end{smallmatrix}\right]}$ maps an $(m\times n)$-matrix $A$ to the row space $\langle (I_m | A)\rangle$, where $I_m$ denotes the $m\times m$ identity matrix. The mapping $\Lambda$ is injective and its image is given by all $m$-dimensional subspaces of ${\ensuremath{\mathbb{F}}}_q^{m\times n}$ having trivial intersection with the special subspace $S=\langle e_{m+1},\dots,e_{m+n}\rangle$ of ${\ensuremath{\mathbb{F}}}_q^{m+n}$ ($e_i$ denoting the $i$th unit vector). In fact, the lifting map defines an isometry from $({\ensuremath{\mathbb{F}}}_q^{m\times n},2{\mathrm{d}_{\mathrm{r}}})$ into $(\operatorname{L}({\ensuremath{\mathbb{F}}}_q^{m+n}),{\mathrm{d}_{\mathrm{s}}})$. Of particular interest are the LMRD codes, i.e., CDCs obtained by lifting MRD codes, which are CDCs of fairly large, though not of maximal size.
Although we use the algebraic dimension $v$ instead of the geometric dimension $v-1$ in this paper, we adopt the use of geometric language: Abbreviating $k$-dimensional subspaces as $k$-spaces, we call $1$-spaces points, $2$-spaces lines, $3$-spaces planes, $4$-spaces solids, and $(v-1)$-spaces hyperplanes.
For dimensions $v\ge 3$ the automorphism group of the metric space $({{\ensuremath{\operatorname{L}({\ensuremath{V}})}}},{\mathrm{d}_{\mathrm{s}}})$ is generated by $\operatorname{P\Gamma{}L}({\ensuremath{V}})$ and the polarity $\pi$. It carries the structure of a semidirect product $\operatorname{P\Gamma{}L}({\ensuremath{V}})\rtimes \langle \pi\rangle \cong \operatorname{P\Gamma{}L}(v,q)\rtimes
\mathbb{Z}/2\mathbb{Z}$. Hence, for classifications of CDCs in ${\left[\begin{smallmatrix}{V}\\{k}\end{smallmatrix}\right]}$ up to isomorphism, the relevant acting group is $\operatorname{P\Gamma{}L}({\ensuremath{V}})$, except for the case $v = 2k$ in which it is the larger group $\operatorname{P\Gamma{}L}({\ensuremath{V}})\rtimes \langle \pi\rangle$.
In order to describe suitable substructures of $(8,N,6;4)_2$ codes with $\textit{large}$ cardinality $N$, we will consider incidences with fixed subspaces. To this end, let ${\ensuremath{\mathcal{I}\left(\mathcal{S},X\right)}}$ be the set of subspaces in $\mathcal{S} \subseteq {\ensuremath{\operatorname{L}({\ensuremath{V}})}}$ that are incident with $X \le {\ensuremath{V}}$, i.e., ${\ensuremath{\mathcal{I}\left(\mathcal{S},X\right)}} = \{U \in \mathcal{S} \mid U \le X \,\lor\, X \le U\}$. As special subspaces we explicitly label a point $\widetilde{P}=\langle (0,0,0,0,0,0,0,1) \rangle$ and a hyperplane $\widetilde{H}=\{x \in V \mid x_8 = 0\}$. Note that $\widetilde{P}$ and $\widetilde{H}$ are not incident. By $\iota: \mathbb{F}_2^7 \rightarrow \widetilde{H}$ we denote the canonical embedding, which we will apply to subspaces and sets of subspaces.
To keep the paper self-contained, we restate upper bounds for $\# {\ensuremath{\mathcal{I}\left(\mathcal{S},X\right)}}$ and $N$ from the earlier conference paper [@new_bounds_subspaces_codes] with their complete but short proofs.
\[lem:deg\_upper\_bound\] Let $\mathcal{C}$ be a $(v,\#\mathcal{C},d;k)_q$ CDC and $X \le {\ensuremath{V}}$. Then we have $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},X\right)}}\le A_q(\dim(X),d;k)$ if $\dim(X)\ge k$ and $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},X\right)}}\le A_q(v-\dim(X),d;k-\dim(X))$ otherwise.
Note that ${\ensuremath{\mathcal{I}\left(\mathcal{C},X\right)}}$ is a $(\dim(X),\#{\ensuremath{\mathcal{I}\left(\mathcal{C},X\right)}},d;k)_q$ CDC. For the second part we write $V=X\oplus V'$ and $U_i=X\oplus U_i'$ for all $U_i\in {\ensuremath{\mathcal{I}\left(\mathcal{C},X\right)}}$. With this we have ${\mathrm{d}_{\mathrm{s}}}(U_i,U_j)=2k-2\dim(U_i\cap U_j)\le 2\left(k-\dim(X)\right)-2\dim(U_i'\cap U_j')={\mathrm{d}_{\mathrm{s}}}(U_i',U_j')$.
\[cor\_one\_incidence\] Let $\mathcal{C}$ be a $(2k,\#\mathcal{C},2k-2;k)_q$ CDC for $k\ge 1$. Then $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},H\right)}} \le q^k+1$ and $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},P\right)}} \le q^k+1$ for all hyperplanes $H$ and points $P$.
We have $A_q(v,2k;k) = \frac{q^v-q}{q^k-1}-q+1$ for $v \equiv 1 \pmod{k}$ and $2 \le k \le v$, see [@MR0404010], so that Lemma \[lem:deg\_upper\_bound\] gives $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},P\right)}} \le A_q(2k-1,2k-2;k-1)=q^k+1$ and $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},H\right)}} \le A_q(2k-1,2k-2;k)=A_q(2k-1,2k-2;k-1)=q^k+1$.
In particular, Corollary \[cor\_one\_incidence\] shows that each point and hyperplane is incident with at most $17$ codewords of an $(8,N,6;4)_2$ CDC. The next lemma refines this counting by including points which are not incident with a fixed hyperplane.
\[lem\_minilemma\] Let $\mathcal{C}$ be an $(8,\#\mathcal{C},6;4)_2$ CDC of size $\#\mathcal{C} \ge 255$. For each hyperplane $H$, there is a point $P' \not \le H$ with $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},P'\right)}} \ge 14$. Moreover, if $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},P\right)}} \le 16$ for all points $P$, then for each hyperplane $H$ there is a point $P''\not \le H$ with $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},P''\right)}} \ge 15$.
Let $H$ be a hyperplane and $\mathcal{P} = {\left[\begin{smallmatrix}{{\ensuremath{\mathbb{F}}}_2^8}\\{1}\end{smallmatrix}\right]}$ be the set of points. Double counting of the set $\{(P,U) \in \mathcal{P} \times \mathcal{C} \mid P \leq U\}$ gives $$\!\!\!\!\!\sum_{P \in {\ensuremath{\mathcal{I}\left(\mathcal{P},H\right)}}}\!\!\!\!\! \#{\ensuremath{\mathcal{I}\left(\mathcal{C},P\right)}} + \!\!\!\!\!\sum_{P \not\in {\ensuremath{\mathcal{I}\left(\mathcal{P},H\right)}}} \!\!\!\!\!\#{\ensuremath{\mathcal{I}\left(\mathcal{C},P\right)}} = \#\mathcal{C} \cdot {\left[\begin{smallmatrix}{4}\\{1}\end{smallmatrix}\right]_{2}} \geq 255\cdot 15\text{.}$$ By Corollary \[cor\_one\_incidence\], $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},P\right)}} \le 17$ for all points $P$. Assuming $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},P\right)}} \le 13$ for all points $P\not\le H$, the left hand side is $\le 127 \cdot 17 + 128 \cdot 13 = 255\cdot 15 - 2$, which is a contradiction. If $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},P\right)}} \leq 16$ for all points $P$, the assumption $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},P\right)}} \leq 14$ for all $P\not\le H$ leads to a left hand side $\le 127 \cdot 16 + 128 \cdot 14 = 255 \cdot 15 - 1$, which is again a contradiction.
Furthermore we need the following lemma to split a difficult problem into multiple small problems.
\[lem:split\] Let $X$ be a finite set and $f\colon 2^X\to\{0,1\}$ be a function. A bijection $\pi\colon X\to X$ is called an automorphism (with respect to $f$) if $f(S)=f(\pi(S))$ for all $S\subseteq X$. Let $\Gamma$ be a group of automorphisms, $T=\{t_1, \ldots, t_m\}$ be a transversal of $\Gamma$ acting on $X$, where the corresponding orbit sizes are decreasing, and $\tau\colon X\to \{1,\dots, m\}$ such that $x\in X$ is in the same orbit as $t_{\tau(x)}$. If $\tilde{S}\subseteq X$ and $i=\min\{\tau(x) \mid x\in \tilde{S}\}$, then there exists an automorphism $\gamma\in\Gamma$ with $\{t_i\}\subseteq \gamma(\tilde{S})$, $f(\tilde{S})=f(\gamma(\tilde{S}))$, and $\min\{\tau(x) \mid x\in \gamma(\tilde{S})\}=i$.
Choose $x\in X$ with $\tau(x)=i$ and $\gamma\in\Gamma$ with $\gamma(x)=t_i$. Note that $\tau(\gamma'(x'))=\tau(x')$ for all $\gamma'\in\Gamma$ and all $x'\in X$.
This lemma will be applied in Section \[sec\_main\_thm\] to exploit the symmetry for the computation of representatives of cliques of maximal size as well as for the solving of a binary linear program, cf. Section \[sec\_ILP\]. If $G=(V,E)$ is a graph with nontrivial automorphism group $\operatorname{Aut}(G)$, we use Lemma \[lem:split\] with $X=V$, $f$ defined by $f(S)=1$ iff $S$ is a clique, and $\Gamma \le \operatorname{Aut}(G)$. Let $T$ be a transversal for the action of $\Gamma$ on $V$. Then any nonempty clique of size $c$ is in the same orbit as the clique $\{t\} \dot\cup S'$ with $t \in T$ and $\#S' = c-1$. This argument can also be applied recursively.
General outline of the proof of Theorem \[main\_thm\] {#sec_general}
=====================================================
In the first phase we try to extend the $715+14445$ hyperplane configurations from Theorems \[theo:7\_17\_6\_3\_2\] and \[theo:7\_16\_6\_3\_2\] to $(8,N,6;4)_2$ CDCs with $N \ge 257$. This is accomplished by using the linear programming relaxation of the integer linear programming model from Lemma \[lem:phase\_1\]. It turns out that such an extension is not possible for all but $38$ of those hyperplane configurations.
For the remaining $38$ hyperplane configurations the integer linear programming model for the extension to an $(8,N,6;4)_2$ CDC with $N \ge 257$ is used. This test fails for all but seven of the $38$ cases.
In the second phase we try to enlarge the remaining hyperplane configurations to larger substructures. Overall, we get $73\,234$ possible $31$-point-hyperplane configurations.
In the third phase it is again tested if these configurations can be extended to $(8,N,6;4)_2$ CDCs with $N \ge 257$. For this, the linear programming relaxation of the integer linear programming model from Lemma \[lem:phase\_2\] is used. All but three hyperplane configurations with $195 + 98 + 240$ $31$-point-hyperplane configurations fail this test.
Finally, the integer linear programming model shows that from the remaining $195 + 98 + 240$ cases exactly two give $(8,257,6;4)_2$ CDCs. All other configurations lead to smaller codes.
Substructures of $(8,N,6;4)_2$ CDCs for $N\ge 257$ {#sec_substructures}
==================================================
Let $\mathcal{C}$ be an $(8,N,6;4)_2$ CDC with $N\ge 257$. From Corollary \[cor\_one\_incidence\] we conclude $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},H\right)}}\le 17$ for any hyperplane $H$. If $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},H\right)}}\le 15$ for each hyperplane $H$, then $\# \mathcal{C}\le {\left[\begin{smallmatrix}{8}\\{1}\end{smallmatrix}\right]_{2}}\cdot 15 / {\left[\begin{smallmatrix}{4}\\{1}\end{smallmatrix}\right]_{2}}=255<257$, since every solid is contained in ${\left[\begin{smallmatrix}{8-3}\\{7-4}\end{smallmatrix}\right]_{2}}={\left[\begin{smallmatrix}{4}\\{1}\end{smallmatrix}\right]_{2}}$ hyperplanes. So, there exists at least one hyperplane $H$ with $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},H\right)}}\in\{16,17\}$. Since $\operatorname{P\Gamma{}L}({\ensuremath{\mathbb{F}}}^{8}_{2}) = \operatorname{GL}({\ensuremath{\mathbb{F}}}^{8}_{2})$ acts transitively on the set of hyperplanes, we can assume $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},\widetilde{H}\right)}}\in\{16,17\}$. Then $\left(\iota^{-1}\left({\ensuremath{\mathcal{I}\left(\mathcal{C},\widetilde{H}\right)}}\right)\right)^\perp$, i.e., the corresponding dual in $\widetilde{H}$, is a set of pairwise disjoint planes in $\widetilde{H}$, i.e., a $(7,N',6;3)_2$ CDC with $N'\in\{16,17\}$, which have already been classified:
([@honold2016classification Theorem 1])\[theo:7\_17\_6\_3\_2\] \[thm\_class\_1\] $A_2(7,6;3)=17$ and there are $715$ isomorphism types of $(7,17,6;3)_2$ CDCs. Their automorphism groups have orders: $1^{551}\allowbreak{}2^{70}\allowbreak{}3^{27}\allowbreak{}4^{19}\allowbreak{}6^{6}\allowbreak{}7^{1}\allowbreak{}8^{8}\allowbreak{}12^{2}\allowbreak{}16^{7}\allowbreak{}24^{6}\allowbreak{}32^{5}\allowbreak{}42^{1}\allowbreak{}48^{5}\allowbreak{}64^{2}\allowbreak{}96^{1}\allowbreak{}112^{1}\allowbreak{}128^{1}\allowbreak{}192^{1}\allowbreak{}2688^{1}$.
([@honold2016classification Theorem 2])\[theo:7\_16\_6\_3\_2\] \[thm\_class\_2\] There are $14445$ isomorphism types of $(7,16,6;3)_2$ CDCs. Their automorphism groups have orders: $1^{13587}\allowbreak2^{511}\allowbreak3^{143}\allowbreak4^{107}\allowbreak6^{20}\allowbreak7^{4}\allowbreak8^{19}\allowbreak9^{3}\allowbreak12^{24}\allowbreak16^{1}\allowbreak18^{1}\allowbreak20^{1}\allowbreak21^{1}\allowbreak24^{9}\allowbreak36^{1}\allowbreak42^{1}\allowbreak48^{3}\allowbreak64^{1}\allowbreak96^{1}\allowbreak112^{1}\allowbreak168^{2}\allowbreak288^{1}\allowbreak384^{1}\allowbreak960^{1}\allowbreak2688^{1}$.
We call those configurations *hyperplane configurations* and denote a transversal of the isomorphism classes of sets of planes in Theorems \[theo:7\_17\_6\_3\_2\] and \[theo:7\_16\_6\_3\_2\] by $\mathcal{A}_{17}$ and $\mathcal{A}_{16}$, respectively. So, $\left(\iota^{-1}\left({\ensuremath{\mathcal{I}\left(\mathcal{C},\widetilde{H}\right)}}\right)\right)^\perp$ is isomorphic to exactly one set in $\mathcal{A}_{16} \cup \mathcal{A}_{17}$. Computing the LP relaxation of a suitable integer linear programming formulation, see the next section, one can check easily that all but $38$ of the $715+14445$ hyperplane configurations can not be extended to $(8,257,6;4)_2$ CDCs. These $38$ remaining elements are listed in Table \[tab:ausgeschrieben\] and their LP values are stated Table \[tab:details\]. By $F_i$ we denote the corresponding sets of solids in ${\ensuremath{\mathbb{F}}}_2^8$ for $1 \le i \le 38$.
Next we want to enlarge some of the possible hyperplane configurations to larger substructures, more precisely those with indices $1\le i\le 7$ in Table \[tab:ausgeschrieben\]. Therefore we distinguish both possibilities for $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},\widetilde{H}\right)}}$. If it is $17$, then Lemma \[lem\_minilemma\] guarantees a point $P \not \le \widetilde{H}$ such that $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},\widetilde{H}\right)}}+\#{\ensuremath{\mathcal{I}\left(\mathcal{C},P\right)}} \ge 17+14=31$. If $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},\widetilde{H}\right)}}=16$ then we can assume w.l.o.g. that $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},P\right)}}\le 16$ for all points $P$, since otherwise we can apply the orthogonality and have the first case. Then Lemma \[lem\_minilemma\] guarantees a point $P \not \le \widetilde{H}$ such that $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},\widetilde{H}\right)}}+\#{\ensuremath{\mathcal{I}\left(\mathcal{C},P\right)}} \ge 16+15=31$. Since the stabilizer of $\widetilde{H}$ in $\operatorname{GL}({\ensuremath{\mathbb{F}}}^{8}_{2})$ acts transitively[^4] on the set of points not incident with $\widetilde{H}$, we can assume $\#{\ensuremath{\mathcal{I}\left(\mathcal{C},\widetilde{P}\right)}}+\#{\ensuremath{\mathcal{I}\left(\mathcal{C},\widetilde{H}\right)}}\ge 31$. We call sets of $a$ solids in $\widetilde{H}$ and $b$ solids containing $\widetilde{P}$ with $16\le a\le 17$, $a+b=31$, and minimum subspace distance $d=6$ briefly *$31$-point-hyperplane configurations*.
Anticipating the results from Section \[sec\_main\_thm\], we mention that altogether just $242$ non-isomorphic $31$-point-hyperplane configurations can be extended to CDCs with cardinality $257$. Moreover, we will verify indirectly that in all those extensions there exists a codeword $U$ such that $\mathcal{C}\backslash \{U\}$ is isomorphic to an LMRD code.
([@class_mrd]) \[thm\_mrd\_4\_4\_3\] The Gabidulin construction gives the unique isomorphism type of (not necessarily linear) $4\times 4$ MRD codes over ${\ensuremath{\mathbb{F}}}_2$ with minimum rank distance $3$.
This result has been achieved computationally in the context of the work [@class_mrd]. However, to make this article as self-contained as possible, we decided to include the idea of the proof.
Let $C$ be a $4\times 4$ MRD code over ${\ensuremath{\mathbb{F}}}_2$ of minimum rank distance $3$. Then $\#C = 256$. For each vector $v\in{\ensuremath{\mathbb{F}}}_2^4$, there are exactly $16$ matrices in $C$ having $v$ as their last row. After removing this common row, these $16$ matrices form a binary $3\times 4$ MRD code of minimum rank distance $3$. These MRD codes have been classified in [@honold2016classification] into $37$ isomorphism classes.
Let $C'$ be one of these codes, extended to size $4\times 4$ by appending the zero vector as the last row to all the matrices in $C'$. Up to isomorphism, $C$ is the extension of one of these $37$ codes $C'$ by $256-16 = 240$ matrices. In particular, for each $v\in{\ensuremath{\mathbb{F}}}_2^4\backslash\{\mathbf{0}\}$, it must be possible to add $16$ matrices of size $4\times 4$ with last row $v$ without violating the rank distance condition. For fixed $v$, this question can be formulated as a clique problem: We define a graph $G_v$ whose vertex set is given by all $4\times 4$ matrices with last row $v$ having rank distance $\geq 3$ to all matrices in $C'$. Two vertices are connected by an edge if the corresponding matrices have rank distance $\geq 3$. Now the question is whether all graphs $G_v$, $v\in{\ensuremath{\mathbb{F}}}_2^4\backslash\{\mathbf{0}\}$, admit a clique of size $16$. Using the software [@cliquer], we found that out of the $37$ types of codes $C'$, this is possible only for a single type.
For this remaining type, the full extension problem to a $4\times 4$ MRD code is again formulated as a clique problem. The graph is defined in a similar way, but without the restriction on the last row of the matrices in the vertex set. This yields a graph with $1920$ vertices. The maximum clique problem is solved within seconds for this graph [^5] The result are $8$ cliques of maximum possible size $240$. In other words, there are $8$ extensions to a rank distance $3$ code of size $16 + 240 = 256$, i.e., an MRD code. All $8$ codes turned out to be isomorphic to the Gabidulin code.
By the last theorem, in our setting there is only a single type of LMRD code, which is the lifted Gabidulin code. It is iso-dual (isomorphic to its orthogonal code).
\[cor\_isomorphism\_types\] Let $\mathcal{C}$ be an $(8,257,6;4)_2$ CDC that contains an LMRD code $\mathcal{C}'$. Then $\mathcal{C}$ is isomorphic to either $\{ \langle (I_4 \mid B) \rangle \mid B \in M \} \cup \{ \langle
(0_{4\times 4} \mid I_4) \rangle \}$ or $\{\langle (I_4 \mid B) \rangle \mid B \in M \} \cup \{ \langle
(0_{4\times 3} \mid I_4 \mid 0_{4\times 1}) \rangle \}$, where $M$ is the $4\times 4$ Gabidulin code with minimum rank distance $3$, $I_4$ is the $4\times 4$ identity matrix, and $0_{m\times n}$ is the $m\times n$ all-zero matrix.
From Theorem \[thm\_mrd\_4\_4\_3\] we conclude that $\mathcal{C}'$ is the lifted Gabidulin code $M$. The automorphism group $A$ of $\mathcal{C}'$ has order $4\cdot 15^2\cdot
2^8=230\,400$. Identifying $V$ with ${\ensuremath{\mathbb{F}}}_{16}\times{\ensuremath{\mathbb{F}}}_{16}$ and denoting by $\alpha$ a generator of ${\ensuremath{\mathbb{F}}}_{16}^\times$, $A$ is generated by $(x,y)\mapsto(x^2,y^2)$, $(x,y)\mapsto(\alpha x,y)$, $(x,y)\mapsto(x,\alpha y)$, and the “translations” $(x,y)\mapsto (x,a_0x+a_1x^2+y)$ with $a_0x+a_1x^2\in M$. From this it is readily seen that $A$ partitions the $451$ solids intersecting each codeword of $\mathcal{C}'$ in at most a point (these are precisely the solids intersecting the special solid $S$ of $\mathcal{C}'$ in at least a plane) into two orbits: An orbit of size $1$ containing $S$, which is fixed by $A$, and an orbit of size $450$ containing the solids that meet $S$ in a plane. This accounts for the two indicated isomorphism classes of $\mathcal{C}$.
Integer linear programming models {#sec_ILP}
=================================
It is well known that the determination of $A_q(v,d;k)$ can be formulated as an integer linear programming problem with binary variables (BLP). If all constraints of the form $x\in\{0,1\}$ are replaced by $x\in\mathbb{R}_{\ge 0}$, we speak of the corresponding linear programming relaxation (LP). Suppose that we already know that a CDC $\mathcal{C}$ contains the solids from $F \subseteq {\left[\begin{smallmatrix}{\mathbb{F}_2^8}\\{4}\end{smallmatrix}\right]}$ and that each point and hyperplane is incident with at most $f$ codewords. Then we can state the following upper bounds on $\# \mathcal{C}$:
\[lem:phase\_1\] Let $F \subseteq {\left[\begin{smallmatrix}{\mathbb{F}_2^8}\\{4}\end{smallmatrix}\right]}$ and $f\in\mathbb{N}$. Then any $(8,\#\mathcal{C},6;4)_2$ CDC $\mathcal{C}$ with $F \subseteq \mathcal{C}$ and such that each point and hyperplane is incident with at most $f$ codewords has $\#\mathcal{C} \le z_8^{\operatorname{BLP}}(F,f) \le z_8^{\operatorname{LP}}(F,f)$, where $\operatorname{Var}_8 = {\left[\begin{smallmatrix}{\mathbb{F}_2^8}\\{4}\end{smallmatrix}\right]}$, $z_8^{\operatorname{LP}}$ is the LP relaxation of $z_8^{\operatorname{BLP}}$, and $$\begin{aligned}
z_8^{\operatorname{BLP}}(F,f) := \max
\sum_{U \in \operatorname{Var}_8} &x_U \\
\text{subject to}
\sum_{U \in {\ensuremath{\mathcal{I}\left(\operatorname{Var}_8,W\right)}}} &x_U \le f &&\forall W \in {\left[\begin{smallmatrix}{\mathbb{F}_2^8}\\{w}\end{smallmatrix}\right]} &&\forall w \in \{1,7\} \\
\sum_{U \in {\ensuremath{\mathcal{I}\left(\operatorname{Var}_8,W\right)}}} &x_U \le 1 &&\forall W \in {\left[\begin{smallmatrix}{\mathbb{F}_2^8}\\{w}\end{smallmatrix}\right]} &&\forall w \in \{2,6\} \\
&x_U = 1 &&\forall U \in F \\
&x_U \in \{0,1\} &&\forall U \in \operatorname{Var}_8.\end{aligned}$$
Interpreting $(x_U)_{U \in \operatorname{Var}_8}$ as characteristic vector of $\mathcal{C}$, the objective function equals $\#\mathcal{C}$. The first two sets of constraints are feasible by Lemma \[lem:deg\_upper\_bound\] and the choice of $f$. The third set of constraints is feasible since $F \subseteq \mathcal{C}$.
If $\# F$ is rather small, then the computation of $z_8^{\operatorname{BLP}}(F,f)$ takes too much time, so that we also consider a linear programming formulation for $\# \{U\cap \widetilde{H} \mid U\in \mathcal{C}\}$, i.e., we consider the image of $\mathcal{C}$ in $\widetilde{H}$.
\[lem:phase\_2\] For $F \subseteq {\left[\begin{smallmatrix}{\mathbb{F}_2^7}\\{4}\end{smallmatrix}\right]}$ let $\operatorname{Var}_7(F):=\left.\left\{ U \in {\left[\begin{smallmatrix}{\mathbb{F}_2^7}\\{3}\end{smallmatrix}\right]} \,\right|\, \dim(U \cap S) \le 1 \,\forall S \in F \right\}$ and $\omega(F,W) = \max\{ \#\Omega \mid \Omega \subseteq {\ensuremath{\mathcal{I}\left(\operatorname{Var}_7(F),W\right)}} \land \dim(U_1 \cap U_2) \le 1 \,\forall U_1 \ne U_2 \in \Omega \}$. If $\#F \in \{16,17\}$, then any $(8,\#\mathcal{C},6;4)_2$ CDC $\mathcal{C}$ with $\#\mathcal{C} \ge 255$ and $\iota(F) \subseteq \mathcal{C}$ and such that each point and hyperplane is incident with at most $\#F$ codewords satisfies $\#\mathcal{C} \le z_7^{\operatorname{BLP}}(F)$, where $$\begin{aligned}
z_7^{\operatorname{BLP}}(F) := \max \!\!\!\!\!\!\!
\sum_{U \in \operatorname{Var}_7(F)} &x_U + \#F \\
\text{subject to}
\sum_{U \in {\ensuremath{\mathcal{I}\left(\operatorname{Var}_7(F),W\right)}}} &x_U \le \#F-\#{\ensuremath{\mathcal{I}\left(F,W\right)}} &&\forall W \in {\left[\begin{smallmatrix}{\mathbb{F}_2^7}\\{1}\end{smallmatrix}\right]} \\
\sum_{U \in {\ensuremath{\mathcal{I}\left(\operatorname{Var}_7(F),W\right)}}} &x_U \le 1 &&\forall W \in {\left[\begin{smallmatrix}{\mathbb{F}_2^7}\\{2}\end{smallmatrix}\right]} \setminus (\cup_{S \in F} {\left[\begin{smallmatrix}{S}\\{2}\end{smallmatrix}\right]}) \\
\sum_{U \in {\ensuremath{\mathcal{I}\left(\operatorname{Var}_7(F),W\right)}}} &x_U \le 1 &&\forall W \in {\left[\begin{smallmatrix}{\mathbb{F}_2^7}\\{4}\end{smallmatrix}\right]} \setminus F \\ \sum_{U \in {\ensuremath{\mathcal{I}\left(\operatorname{Var}_7(F),W\right)}}} &x_U \le \min\{\omega(F,W),7\} &&\forall W \in {\left[\begin{smallmatrix}{\mathbb{F}_2^7}\\{5}\end{smallmatrix}\right]} : S \not \le W \,\forall S \in F \\
\sum_{U \in {\ensuremath{\mathcal{I}\left(\operatorname{Var}_7(F),W\right)}}} &x_U \le 2(\#F-\#{\ensuremath{\mathcal{I}\left(F,W\right)}}) &&\forall W \in {\left[\begin{smallmatrix}{\mathbb{F}_2^7}\\{6}\end{smallmatrix}\right]} \\
\sum_{U \in \operatorname{Var}_7(F)} &x_U + \#F \ge 255 \\
&x_U \in \{0,1\} &&\forall U \in \operatorname{Var}_7(F) \\\end{aligned}$$
Interpreting $(x_U)_{U \in \operatorname{Var}_7(F)}$ as characteristic vector of $\{ U \cap \widetilde{H} \mid U \in \mathcal{C} \land U \not \le \widetilde{H} \}$, one can check the correctness of the objective function and the last two lines. Since two solids in $\mathcal{C}$ intersect in at most a point, any two elements in $\{ U \cap \widetilde{H} \mid U \in \mathcal{C} \}$ also intersect in at most a point, which gives the constraints with $\dim(W) \in \{2,4\}$.
Any $5$-space $W$ contains at most $\omega(F,W)$ planes by choice of $\omega$, also $\iota(W)$ is incident with ${\left[\begin{smallmatrix}{8-5}\\{6-5}\end{smallmatrix}\right]_{2}}=7$ $6$-spaces, which in turn contain at most one codeword of $\mathcal{C}$. If $W$ contains a solid of $F$, then any plane in $W$ meets this solid in at least a line. This gives the constraints with $\dim(W) = 5$.
For any point $W$ its embedding $\iota(W)$ is incident with at most $\#F$ codewords of $\mathcal{C}$ giving the constraints with $\dim(W) = 1$.
For any $6$-subspace $W$ its embedded $\iota(W)$ is contained in ${\left[\begin{smallmatrix}{8-6}\\{7-6}\end{smallmatrix}\right]_{2}}=3$ hyperplanes in $\mathbb{F}_2^8$ of which one of them is $\widetilde{H}$. Since each hyperplane is incident with at most $\#F$ codewords and $\bar{H}$ is incident with exactly $\#F$ codewords, i.e., $\iota(F)$, the other two hyperplanes are each incident with either $\#F$ codewords if $W$ contains no element of $F$ or $\#F-1$ codewords if $W$ contains one element of $F$. Obviously two solids in a $6$-space intersect in at least a line and hence $W$ contains at most one element of $F$. This gives the constraints with $\dim(W) = 6$.
The last inequality allows the BLP solver to cut the branch & bound tree early since we are only interested in solutions of cardinality at least $255$.
Proof of the main theorem {#sec_main_thm}
=========================
The algorithmic proof of Theorem \[main\_thm\] is split into several phases that are described in detail in the following subsections; Subsection 6.i corresponds to Phase i. The (integer) linear programming problems are solved with `CPLEX` [@citeulike:8436868].
Let $\mathcal{C}$ be an $(8,\#\mathcal{C},6;4)_2$ CDC with $\#\mathcal{C}\ge 257$. As argued at the beginning of Section \[sec\_substructures\], $\mathcal{C}$ has to contain one of the $715+14445$ hyperplane configurations from $\mathcal{A}_{17}\cup\mathcal{A}_{16}$. This list is reduced in Phase 1, see Section \[subsec\_phase\_1\], and then extended to $31$-point-hyperplane configurations in Phase 2, see Section \[subsec\_phase\_2\]. The resulting list is reduced in Phase 3, see Section \[subsec\_phase\_3\]. Then we deduce that $\mathcal{C}$ must be an LMRD code extended by a single codeword, see Section \[subsec\_phase\_4\]. The classification of such structures at the end of Section \[sec\_substructures\] concludes the proof. Let us mention that the termination of Phase 1 proves $A_2(8,6;4)\le 271$ and the termination of Phase 3 proves $A_2(8,6;4)=257$. The required computation times for the four phases are $42\,087$, $2\,214$, $1\,804$, and $2\,168$ hours, respectively, i.e., $48\,273$ hours in total.
Besides the internal parallelization performed by the ILP solvers, we employed parallelization only by setting up independent subproblems. We used the cluster of the University of Bayreuth[^6] for solving the subproblems and other computers for the management and generation of the subproblems.
Excluding hyperplane configurations {#subsec_phase_1}
-----------------------------------
For all $A \in \mathcal{A}_{16} \cup \mathcal{A}_{17}$ we computed $z_8^{\operatorname{LP}}(\iota(A^\perp),\#A)$ and found that all but $33$ elements in $\mathcal{A}_{16}$ ($37\,251$ hours) and $5$ elements in $\mathcal{A}_{17}$ ($1021$ hours) have an optimal value smaller than $256.9$, i.e., we have implemented a safety threshold of $\varepsilon=0.1$. These $38$ elements are listed in Table \[tab:ausgeschrieben\] and their LP values are stated in Table \[tab:details\].
For indices $1\le i\le 38$ we computed $z_7^{\operatorname{BLP}}(\iota(F_i))$ and obtained $6$ elements in $\mathcal{A}_{16}$ and $2$ elements in $\mathcal{A}_{17}$ that may allow $z_7^{\operatorname{BLP}}(\iota(F_i))\ge 256.9$, cf. Table \[tab:details\] for details. This computation was aborted after $100$ hours of wall time for each of these $38$ subproblems.
$\operatorname{Var}_7(\iota(F_8))$ has exactly $948$ planes which form $56$ orbits ($4^3 8^{13} 16^{28} 32^{12} $) under the action of the automorphism group of order $32$. We apply Lemma \[lem:split\] to obtain $56$ subproblems. Less than $15$ hours were needed to verify $z_7^{\operatorname{BLP}}\le 256$ in all cases.
Extending hyperplane configurations to $31$-point-hyperplane configurations {#subsec_phase_2}
---------------------------------------------------------------------------
The seven hyperplane configurations, with indices $1\le i\le 7$ remaining after Phase 1, are extended to $31$-point-hyperplane configurations.
We define a graph $G_i=(V_i,E_i)$, whose vertex set $V_i$ consists of all solids in ${\left[\begin{smallmatrix}{\mathbb{F}_2^8}\\{4}\end{smallmatrix}\right]}$ that contain $\widetilde{P}$ and intersect the elements from $F_i$ in at most a point. For $U,W\in V_i$ we set $\{U,W\}\in E_i$ iff $U\cap W=\widetilde{P}$. Using `Cliquer` [@cliquer], we enumerated all cliques of size $31-\#F_i$ of $G_i$ and computed a transversal $T(F_i)$ of the action of the stabilizer of $F_i$. The clique computations for $1\le i\le 7$, $i\neq 5$ took between $27$ and $589$ hours (see Table \[tab:detailsphase2\] for details about the running times and $\#V_i$; the computation time for the transversal was negligible). The transversal is denoted by $T(F_i)$; see Column 6 of Table \[tab:details\] for the corresponding orbit lengths.
The clique computation for $G_5$ was aborted after $600$ hours and then performed in parallel by applying Lemma \[lem:split\] with $X$ as the vertex set of $G_5$, $\Gamma$ the automorphism group of $F_5$, and the function $f$ defined by $f(S)$ equals $1$ iff $S$ is a clique in $G_5$. In general, we label the elements of $T$ in decreasing order of the corresponding orbit lengths, since large orbits admit small stabilizers and forbid many elements from $X$ in the subsequent subproblems, resulting in few rather asymmetric large subproblems and many small subproblems. The $1258$ vertices of $G_5$ are partitioned into $24$ orbits of size $1$ and $617$ orbits of size $2$ by $\Gamma$, leaving $641$ graphs where we have to enumerate all cliques of size $31-\# F_5-1=14$. Since some of these graphs still consist of *many* vertices, we iteratively apply Lemma \[lem:split\] with the identity group as $\Gamma$ for at most two further times: After the first round we split the $68$ subproblems, which lead to graphs with at least $700$ vertices. Then, we split the $81$ subproblems, which lead to graphs with at least $600$ vertices. We are left with $104\,029$ graphs, for which we have to enumerate all cliques of size $14$, $13$ or $12$. All of these instances were solved in parallel with `Cliquer` to get a superset of the transversal of all cliques of size $15$ of $G_5$. Applying the action of the automorphism group of order $2$ then allowed us to obtain a transversal as well as all cliques, simply as union of the orbits. This took about $750$ hours of CPU time, the smaller problems being preprocessed on a single computer and the remaining $55\,420$ larger subproblems being processed in parallel with $16$ cores.
The extension of the configuration with index $5$ took $750$ hours, and the extension of the other indices took $1464$ hours; see Table \[tab:detailsphase2\] for details.
Excluding $31$-point-hyperplane configurations {#subsec_phase_3}
----------------------------------------------
For the $73\,234$ $31$-point-hyperplane configurations resulting from Section \[subsec\_phase\_2\], we computed $z_8^{\operatorname{LP}}(.)$ in $953$ hours. The maximum value aggregated by the contained hyperplane configuration with index $i$ is stated in Column 7 of Table \[tab:details\], see also Table \[tab:detailsphase2\]. For the configuration with index $1$ there are $195$, for the configuration with index $3$ there are $98$, and for the configuration with index $7$ there are $240$ $31$-point-hyperplane configurations with $z_8^{\operatorname{LP}}\ge 256.9$.
Next we computed $z_8^{\operatorname{BLP}}$ (see Column 8 of Table \[tab:details\]) for these remaining $195+98+240$ cases in $851$ hours (see Table \[tab:detailsphase2\]).
The counts for value exactly $257$ are $2+0+240$.
Structural results for $(8,N,6;4)_2$ CDCs with $N\ge 257$ {#subsec_phase_4}
---------------------------------------------------------
So far we know that the hyperplane configuration of $\mathcal{C}$ in $\tilde{H}$ is either $F_1 \in \mathcal{A}_{16}$ or $F_7 \in \mathcal{A}_{17}$ with $2$ and $240$ possible $31$-point-hyperplane configurations, respectively.
For $F_1$ there exists a unique solid $S$ in ${\ensuremath{\mathbb{F}}}_2^8$ which is disjoint from the $31$ prescribed solids in both cases. Adding the constraint $x_S=0$ to the BLP of Lemma \[lem:phase\_1\] gives an upper bound of $256$, i.e., $S$ has to be a codeword in $\mathcal{C}$, after about $2$ hours of computation time in each of the two cases. The codeword $S$ covers $15$ contained points. Via $x_S=1$ and $
\sum_{P \in {\ensuremath{\mathcal{I}\left({\left[\begin{smallmatrix}{V}\\{1}\end{smallmatrix}\right]},S\right)}}}
\sum_{U \in {\ensuremath{\mathcal{I}\left(\operatorname{Var}_8,P\right)}}}
x_U
\ge 16
$ we can ensure that another codeword of $\mathcal{C}$ meets $S$ in a point. This modification of the BLP of Lemma \[lem:phase\_1\] gives again an upper bound of $256$ after about two hours of computation time in both cases. Thus $\mathcal{C}\backslash\{S\}$ has to be an LMRD code.
For $F_7$ there exists a unique solid $S$ in ${\ensuremath{\mathbb{F}}}_2^8$ which is disjoint from $30$ of the prescribed solids and meets the other prescribed solid $S'$ in a plane, in all $240$ cases. By adding $
\sum_{P \in {\ensuremath{\mathcal{I}\left({\left[\begin{smallmatrix}{V}\\{1}\end{smallmatrix}\right]},S\right)}}}
\sum_{U \in {\ensuremath{\mathcal{I}\left(\operatorname{Var}_8,P\right)}}}
x_U
\ge 8
$ we can ensure that $S$ meets another codeword from $\mathcal{C}$ in a point. The augmented BLP of Lemma \[lem:phase\_1\] needs $9$ hours computation time and ends with $z_8^{\operatorname{BLP}}\le 256$ for each of the $240$ cases. Thus $\mathcal{C}\backslash\{S'\}$ has to be an LMRD code.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank the *High Performance Computing group* of the University of Bayreuth for providing the excellent computing cluster and especially Bernhard Winkler for his support.
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Appendix {#appendix .unnumbered}
========
It is well known that any plane in $\mathbb{F}_2^7$ has a unique binary $3 \times 7$ generator matrix in reduced row echelon form and vice versa. In Table \[tab:ausgeschrieben\], we list the $38$ $(7,16,6;3)_2$ and $(7,17,6;3)_2$ CDCs with $z_8^{\operatorname{LP}}(.)\ge 256.9$. Each plane is denoted by an integer with at most seven digits, one for each column of the generator matrix in such a way that the three entries in each column are coefficients of a $2$-adic number, i.e., $(c_1, c_2, c_3)^T \leftrightarrow c_1 \cdot 2^0 + c_2 \cdot 2^1 + c_3 \cdot 2^2$. Leading zeroes are omitted. For example the number $1024062$ denotes the subspace $
\left(
\begin{smallmatrix}
1&0&0&0&0&0&0\\
0&0&1&0&0&1&1\\
0&0&0&1&0&1&0\\
\end{smallmatrix}
\right)
$. Note that since we are encoding matrices in reduced row echelon form, the three pivot columns are the first numbers $1$, $2$, and $4$ appearing in this order and no digit is larger than $7$. Table \[tab:details\] lists for these CDCs whether it is in $\mathcal{A}_{16}$ or $\mathcal{A}_{17}$, the size of their automorphism group, the relaxations $z_8^{\operatorname{LP}}(.)$ and $z_7^{\operatorname{BLP}}(.)$, which are applied to the hyperplane configurations, then the orbits of the extension to point-hyperplane configurations of each hyperplane configuration and finally the maximum of $z_8^{\operatorname{LP}}(.)$ with prescribed point-hyperplane configuration grouped by the contained hyperplane configuration and, if needed, the maximum $z_8^{\operatorname{LP}}(.)$, again for prescribed point-hyperplane configuration grouped by the contained hyperplane configuration. Details for the extension of one of the first seven hyperplane configurations to the corresponding point-hyperplane configurations are shown in Table \[tab:detailsphase2\].
----- --------- --------- --------------- ----------------
$i$ $\#V_i$ Phase 2 LP in Phase 3 BLP in Phase 3
1 1231 144 51 328
2 1303 589 78
3 1194 217 21 519
4 1243 278 22
5 1258 750 419
6 1251 209 13
7 864 27 349 4
----- --------- --------- --------------- ----------------
: Details for the computation of all $31$-point-hyperplane configurations in Phase 2 and Phase 3.[]{data-label="tab:detailsphase2"}
Index Type Aut $z_8^{\operatorname{LP}}(.)$ $z_7^{\operatorname{BLP}}(.)$ Orbits of Phase 2 $\max z_8^{\operatorname{LP}}(\text{\lq\lq 31\rq\rq})$ $\max z_8^{\operatorname{BLP}}(\text{\lq\lq 31\rq\rq})$
------- ------ ----- ------------------------------ ------------------------------- --------------------------------------- -------------------------------------------------------- ---------------------------------------------------------
1 16 960 272 271.1856 $ 16^{2},240^{6},480^{47},960^{242} $ 263.0287799 257
2 16 384 266.26086957 267.4646 $ 96^{6},192^{91},384^{711} $ 206.04279728
3 16 4 270.83786676 265.3281 $ 1^{13},2^{29},4^{2638} $ 257.20717665 254
4 16 48 271.43451032 262.082 $ 4^{3},12^{11},24^{59},48^{1104} $ 200.5850228
5 16 2 263.8132689 259.8044 $ 1^{5},2^{59966} $ 206.39304042
6 16 20 267.53272206 259.394 $ 5,10^{9},20^{1843} $ 199.98690666
7 17 64 282.96047431 259.1063 $ 16^{10},32^{145},64^{6293} $ 259.45364626 257
8 17 32 268.0388109 257.2408
9 16 1 263.82742528 256.392
10 16 1 263.36961743 255.8305
11 16 1 264.25957151 $\le$ 254
12 16 1 263.85869815 $\le$ 254
13 16 2 263.07052878 $\le$ 254
14 16 12 261.91860556 $\le$ 254
15 16 4 261.62648174 $\le$ 254
16 16 12 261.31512837 $\le$ 254
17 17 4 261.11518721 $\le$ 254
18 16 1 260.96388752 $\le$ 254
19 16 1 260.82432878 $\le$ 254
20 16 2 260.65762276 $\le$ 254
21 16 4 260.43036283 $\le$ 254
22 16 2 260.19475349 $\le$ 254
23 16 1 260.08583792 $\le$ 254
24 16 1 260.04857193 $\le$ 254
25 16 1 259.75041996 $\le$ 254
26 16 2 259.55230081 $\le$ 254
27 16 2 259.46335297 $\le$ 254
28 16 12 259.11945025 $\le$ 254
29 16 1 258.89395938 $\le$ 254
30 17 24 258.75142045 $\le$ 254
31 16 8 258.35689437 $\le$ 254
32 16 1 257.81420526 $\le$ 254
33 16 2 257.75126819 $\le$ 254
34 16 4 257.63965018 $\le$ 254
35 16 1 257.57663803 $\le$ 254
36 16 1 257.2820438 $\le$ 254
37 16 4 257.01931801 $\le$ 254
38 17 128 257 $\le$ 254
Index $16$ or $17$ planes in ${\ensuremath{\mathbb{F}}}_2^7$
------- -----------------------------------------------------------------------------------------------------------------------------------
1 1240000,1240124,1241062,1241146,1242463,1242547,1243401,1243525,1244635,1244711,1245657,1245773,1246256,1246372,1247234,1247310
2 1240000,1240124,1241062,1241146,1242647,1242763,1243625,1243701,1244234,1244310,1245256,1245372,1246473,1246557,1247411,1247535
3 124,1240000,1240124,1241447,1241563,1242631,1242715,1243276,1243352,1244230,1244314,1245753,1246401,1246525,1247046,1247162
4 1240000,1240524,1241042,1241566,1242237,1242403,1243165,1243751,1244270,1244354,1245632,1245716,1246127,1246313,1247441,1247675
5 124,1240124,1241046,1241162,1242637,1242713,1243671,1243755,1244230,1244314,1245276,1245352,1246407,1246523,1247441,1247565
6 1240000,1240124,1241370,1241757,1242605,1242721,1243276,1243451,1244017,1244133,1245263,1245345,1246534,1246612,1247446,1247562
7 124,124000,124124,1024062,1024146,1214452,1214746,1224403,1224727,1241572,1241633,1242557,1242615,1245461,1245724,1246476,1246730
8 124,124000,124124,1024062,1024146,1214546,1214652,1224503,1224627,1241471,1241730,1242416,1242754,1245527,1245662,1246575,1246633
9 124,1240000,1240124,1241157,1242634,1242756,1243673,1243710,1244211,1244335,1245262,1245347,1246463,1246501,1247425,1247546
10 124,1240000,1240124,1241072,1241157,1242634,1242756,1243673,1243710,1244211,1244335,1245347,1246463,1246501,1247425,1247546
11 124,1240000,1241072,1241157,1242634,1242756,1243673,1243710,1244211,1244335,1245262,1245347,1246463,1246501,1247425,1247546
12 124,1240000,1240124,1241072,1241157,1242634,1242756,1243673,1243710,1244211,1245262,1245347,1246463,1246501,1247425,1247546
13 124,1240000,1240124,1241241,1241630,1242415,1242561,1243166,1244023,1244452,1245613,1245737,1246354,1246775,1247206,1247372
14 124,1240000,1240124,1241241,1241630,1242415,1242561,1243166,1243547,1244023,1244452,1245737,1246354,1246775,1247206,1247372
15 124,1240000,1241437,1241513,1242661,1242745,1243252,1243376,1244230,1244314,1245647,1245763,1246051,1246175,1247422,1247506
16 124,1240000,1241241,1241630,1242415,1242561,1243166,1243547,1244023,1244452,1245613,1245737,1246354,1246775,1247206,1247372
17 124,124000,124124,1024466,1024553,1204267,1204342,1234506,1234713,1240570,1240721,1243437,1243565,1245042,1245126,1246453,1246634
18 124,1240000,1241664,1241740,1242427,1242503,1243165,1243243,1244076,1244757,1245516,1245632,1246372,1246451,1247235,1247311
19 124,1240000,1240124,1241367,1241446,1242521,1243243,1243562,1244076,1244757,1245311,1245734,1246150,1246673,1247235,1247412
20 124,1240000,1240124,1241367,1241446,1242521,1242605,1243243,1243562,1244757,1245311,1245734,1246150,1246673,1247235,1247412
21 124,1240000,1240124,1241367,1241446,1242521,1242605,1243243,1243562,1244076,1244757,1245311,1245734,1246150,1247235,1247412
22 124,1240000,1240124,1241664,1241740,1242427,1242503,1243165,1244076,1244757,1245516,1245632,1246372,1246451,1247235,1247311
23 124,1240000,1240124,1241664,1241740,1242427,1242503,1243165,1243243,1244076,1244757,1245516,1245632,1246372,1246451,1247311
24 124,1240000,1240124,1241367,1241446,1242521,1242605,1243243,1244076,1244757,1245311,1245734,1246150,1246673,1247235,1247412
25 124,1240000,1240124,1241664,1241740,1242427,1242503,1243165,1243243,1244076,1244757,1245516,1245632,1246372,1247235,1247311
26 124,1240000,1240124,1241664,1242427,1242503,1243165,1243243,1244076,1244757,1245516,1245632,1246372,1246451,1247235,1247311
27 124,1240000,1240124,1241740,1242427,1242503,1243165,1243243,1244076,1244757,1245516,1245632,1246372,1246451,1247235,1247311
28 124,1240000,1240124,1241437,1241513,1242661,1242745,1243376,1244230,1244314,1245647,1245763,1246051,1246175,1247422,1247506
29 124,1240124,1241664,1241740,1242427,1242503,1243165,1243243,1244076,1244757,1245516,1245632,1246372,1246451,1247235,1247311
30 124,124000,124124,1024341,1024630,1204526,1204653,1234367,1234644,1240046,1240135,1243474,1243726,1245237,1245664,1246512,1246605
31 124,1240000,1240124,1241057,1241173,1242655,1242771,1243602,1243726,1244230,1244314,1245267,1245343,1246465,1246541,1247516
32 124,1240000,1240124,1241664,1241740,1242427,1242503,1243165,1243243,1244076,1245516,1245632,1246372,1246451,1247235,1247311
33 124,1240000,1240124,1241664,1241740,1242427,1242503,1243243,1244076,1244757,1245516,1245632,1246372,1246451,1247235,1247311
34 124,1240000,1240124,1241367,1241446,1242521,1242605,1243243,1243562,1244076,1244757,1245311,1245734,1246673,1247235,1247412
35 124,1240000,1240124,1241367,1241446,1242521,1242605,1243243,1243562,1244076,1244757,1245311,1245734,1246150,1246673,1247235
36 124,1240000,1240124,1241664,1241740,1242427,1242503,1243165,1243243,1244076,1244757,1245632,1246372,1246451,1247235,1247311
37 10024,1202436,1211471,1221433,1232464,1240776,1243450,1243712,1244143,1244522,1245307,1245660,1246021,1246615,1247267,1247546
38 124,124000,124124,1024062,1024146,1214466,1214772,1224437,1224713,1241561,1241620,1242574,1242636,1245407,1245742,1246423,1246765
[^1]: Supported by the grants KU 2430/3-1, WA 1666/9-1 – “Integer Linear Programming Models for Subspace Codes and Finite Geometry” – from the German Research Foundation and by Grant no. 61571006 – “Research on Subspace Codes and Related Combinatorial Structures” – from the National Natural Science Foundation of China. D. Heinlein, M. Kiermaier, S. Kurz, A. Wassermann: Department of Mathematics, University of Bayreuth, Bayreuth, Germany; firstname.lastname@uni-bayreuth.de T. Honold: ZJU-UIUC Institute, Zhejiang University, Haining, China; <honold@zju.edu.cn> The final publication is available at <https://link.springer.com/article/10.1007%2Fs10623-018-0544-8>.
[^2]: As an example we consider $A_2(9,6;4)\le
\left\{{\left[\begin{smallmatrix}{9}\\{1}\end{smallmatrix}\right]_{2}}A_2(8,6;3)/{\left[\begin{smallmatrix}{4}\\{1}\end{smallmatrix}\right]_{2}}\right\}_4=
\left\{\frac{17374}{15}\right\}_4$, using $A_2(8,6;3)=34$. We have $\left\lfloor \frac{17374}{15} \right\rfloor=1158$, $17374-1158\cdot 15=4$, $17374-1157\cdot 15=19$, and $17374-1156\cdot 15=34$. Since $4$ and $19$ cannot be written as a non-negative linear combination of $8$, $12$, $14$, and $15$, but $34=14+12+8$, we have $A_2(9;6;4)\le 1156$, which improves upon the iterative Johnson bound by two. Let us remark that [@kiermaier2017improvement] contains an easy and fast algorithm to check the representability as non-negative integer linear combination as specified above.
[^3]: Algorithmic methods taking into account known symmetries of integer linear programming formulations automatically are presented in the literature. However, we are not aware of any paper, where those approaches have been successfully applied to compute tightened upper bounds for CDCs.
[^4]: Since $\operatorname{Stab}_{\operatorname{GL}\left({\ensuremath{\mathbb{F}}}^{8}_{2}\right)}\left(\widetilde{H}\right)
=
\left\{
\left(\begin{smallmatrix}A & 0 \\ b & 1\end{smallmatrix}\right)
\in \operatorname{GL}\left({\ensuremath{\mathbb{F}}}^{8}_{2}\right)
\,\left|\,
A \in \operatorname{GL}\left({\ensuremath{\mathbb{F}}}^{7}_{2}\right) \text{ and } b\in {\ensuremath{\mathbb{F}}}_2^{7}
\right.\right\}
$, any point that is not incident with $\widetilde{H}$, i.e., $\langle(p\mid 1)\rangle$ with $p \in {\ensuremath{\mathbb{F}}}_2^{7}$, can be mapped via $\left(\begin{smallmatrix}I_7 & 0 \\ p & 1\end{smallmatrix}\right)^{-1}$ to $\widetilde{P}$.
[^5]: We noticed that the order of the vertices makes a huge difference for the running time. For fast results, matrices with the same last row should be numbered consecutively.
[^6]: <http://www.hpc.uni-bayreuth.de>
|
---
abstract: 'We measure and analyze the energy, momentum, and mass feedback efficiencies due to radiation from active galactic nuclei (AGN) . Our measurements are based on the two-dimensional (axisymmetric) and time-dependent radiation-hydrodynamical simulations recently presented in Kurosawa & Proga. In that paper, we studied outflows from a slowly rotating (sub-Keplerian) infalling gas driven by the energy and pressure of the radiation emitted by the AGN. These simulations follow dynamics of gas under the influence of the gravity of the central $10^8~\MSUN$ black hole on scales from $\sim0.01$ to $\sim 10$ pc. They self-consistently couple the accretion-luminosity with the mass inflow rate at the smallest radius (our proxy for the mass-accretion rate, $\dot{M}_{\mathrm{a}}$). Over thirty simulations have been performed to investigate how the results depend on the gas density at the outer radius, $\rho_{\mathrm{o}}$. A key feature of these simulations is that the radiation field and consequently the gas dynamics are axisymmetric, but not spherically symmetric. Therefore, the gas inflow and outflow can occur at the same time. We compare our $\dot{M}_{\mathrm{a}}$-$\rho_{\mathrm{o}}$ relation with that predicted by the Bondi accretion model. For high luminosities comparable to the Eddington limit, the power-law fit ($\dot{M}_{\mathrm{a}} \propto \rho_{\mathrm{o}}^{q}$) to our models yields $q\approx 0.5$ instead of $q=1.0$ which is predicted by the Bondi model. This difference is caused by the outflows which are important for the overall mass budget at high luminosities. The maximum momentum and mass feedback efficiencies found in our models are $\sim 10^{-2}$ and $\sim 10^{-1}$, respectively. However, the outflows are much less important energetically: the thermal and kinetic powers in units of the radiative luminosity are $\sim 10^{-5}$ and $\sim 10^{-4}$, respectively. In addition, the efficiencies do not increase monotonically with the accretion luminosity but rather peak around the Eddington limit beyond which a steady state disk-wind-like solution exists. Our energy feedback efficiencies are significantly lower than 0.05, which is required in some cosmological and galaxy merger simulations. The low feedback efficiencies found here could have significant implications on the mass growth of super massive black holes in the early universe.'
author:
- 'Ryuichi Kurosawa, Daniel Proga, and Kentaro Nagamine'
title: On the Feedback Efficiency of Active Galactic Nuclei
---
= ==1=1=0pt =2=2=0pt
Introduction {#sec:Introduction}
============
The central location of AGN in their host galaxies and the fact that they can produce a large amount of energy imply that AGN can play a very important role in setting the physical conditions in their vicinity as well as on larger, galactic and even intergalactic scales (e.g., @Igumenshchev:1993; @ciotti:1997, [-@ciotti:2001], [-@ciotti:2007]; @king:2003; @Murray:2005; @Sazonov:2005; @Springel:2005b; @Begelman:2005; @Hopkins:2005; @Wang:2006; @Thacker:2006; @Fabian:2006b, [-@Fabian:2008]; @Pelupessy:2007; @Krolik:2007; @Merloni:2008; @Booth:2009, and references therein). There are many indications that support this idea. For example, the presence of broad and narrow emission lines, broad and narrow absorption lines, in AGN spectra suggests that AGN continuum radiation affects the immediate environment of AGN (see @Krolik:1999 for an overview). In addition, the tight correlation between the mass ($M_\mathrm{BH}$) of the central black hole (BH) in a galactic nucleus and the velocity dispersion $\sigma$ of the galaxy’s bulge or spheroid, the so-called “$M_{\mathrm{BH}}-\sigma$” relation (e.g., @Ferrarese:2000; @Gebhardt:2000; @Tremaine:2002) can be explained by the feedback between AGN and the infalling material from large distances. This feedback can quench both BH accretion and star formation in the galaxy when BH reaches a certain mass. AGNs could provide such feedback because they are very powerful sources of energy and momentum (e.g., @Silk:1998; @Blandford:1999; @Sazonov:2005; @Fabian:1999; @Fabian:2002; @king:2003; @Scannapieco:2004; @Murray:2005; @Springel:2005b; @DiMatteo:2005; @Booth:2009).
AGN are powered by mass accretion onto a super massive black hole (SMBH). To illustrate how the growth of SMBH can be self-regulated and how AGN feedback can be characterized, let us first express the radiation luminosity due to accretion as $$L_\mathrm{a}= \epsilon_{\mathrm{r}} c^2 \MDOT_{\mathrm{a}},
\label{eq:L-acc}$$ where we invoke the simplest assumption such that the luminosity is proportional to the mass accretion rate ($\MDOT_{\mathrm{a}}$) and a radiative (or the rest-mass conversion) efficiency ($\epsilon_{\mathrm{r}}$). Both $\epsilon_{\mathrm{r}}$ and $\MDOT_\mathrm{a}$ are uncertain. For example, $\epsilon_{\mathrm{r}}$ ranges from $\sim 10^{-1}$ in a standard, radiatively efficient thin disk to $\sim 10^{-11}$ for spherically symmetric accretion from a low density medium (e.g., @shakura:1973; @Shapiro:1973; @Meszaros:1975; @Soltan:1982; @Yu:2002) while the mass accretion rate depends on poorly constrained physical conditions and geometry at large distances from the black hole.
A common method to estimate $\MDOT_\mathrm{a}$ is to adopt the analytic formula by @Bondi:1952 who considered spherically symmetric accretion from a non-rotating polytropic gas with uniform density $\rho_\infty$ and sound speed $c_\infty$ at infinity. Under these assumptions, a steady state solution to the equations of mass and momentum conservation exists with a mass accretion rate of $$\MDOT_\mathrm{B}= \lambda \, 4 \pi r^2_\mathrm{B} \rho_\infty c_\infty,
\label{eq:mdot_bondi}$$ where $\lambda$ is a dimensionless parameter that, for the Newtonian potential, depends only on the adiabatic index (cf. @Bondi:1952; @Shu:1992; @Frank:1992). The Bondi radius, $r_\mathrm{B}$, is defined as $$r_\mathrm{B}=\frac{G M}{c^2_\infty}
\label{eq:r_bondi}$$ where $G$ is the gravitational constant and $M$ is the mass of the accretor.
One can quantify AGN feedback by measuring its efficiency in affecting the flow of energy, momentum, and mass. In this work, we consider only the energy and momentum carried out by matter. The total energy feedback efficiency $\epsilon_{\mathrm{t}}$ is defined as the ratio between the accretion luminosity of the system $L_{\mathrm{a}}$ (Eq. [\[]{}\[eq:L-acc\]\]) and the sum of the kinetic power (kinetic energy flux) $P_{\mathrm{k}}$ and thermal energy power (thermal energy flux) $P_{\mathrm{th}}$, i.e., $$\epsilon_{\mathrm{t}}=\left(P_{\mathrm{k}}+P_{\mathrm{th}}\right)/L_{\mathrm{a}}\,.
\label{eq:eff-total-energy}$$
Similarly, the kinetic energy and thermal energy feedback efficiencies ($\epsilon_{\mathrm{k}}$ and $\epsilon_{\mathrm{th}}$) are defined as $$\epsilon_{\mathrm{k}}=P_{\mathrm{k}}/L_{\mathrm{a}}
\label{eq:eff-kinetic-energy}$$ and $$\epsilon_{\mathrm{th}}=P_{\mathrm{th}}/L_{\mathrm{a}}\,,
\label{eq:eff-thermal-energy}$$ respectively. From these definitions, it obviously follows that $\epsilon_{\mathrm{t}}=\epsilon_{\mathrm{k}}+\epsilon_{\mathrm{th}}$.
The momentum feedback efficiency ($\epsilon_{\mathrm{p}}$) is defined as the ratio of the total wind momentum $p_{\mathrm{w}}$ to the total radiation momentum ($L_{\mathrm{a}}/c$), i.e., $$\epsilon_{\mathrm{p}}=p_{\mathrm{w}}/\left(L_{\mathrm{a}}/c\right)\,.
\label{eq:eff-momentum}$$ Lastly, the mass feedback efficiency $\epsilon_{\mathrm{m}}$ is defined as the ratio of the mass-outflow rate at the outer boundary $\dot{M}_{\mathrm{out}}$ to the mass-inflow rate at the inner boundary $\dot{M}_{\mathrm{in}}$, i.e., $$\epsilon_{\mathrm{m}}=
\dot{M}_{\mathrm{out}}/\dot{M}_{\mathrm{in}}\,.
\label{eq:eff-mass}$$
The most advanced studies of feedback effects were carried out by @Springel:2005b (SDH05 hereafter), @DiMatteo:2005 and @Booth:2009 (BS09 hereafter). These studies used computer simulations of merging galaxies in which they linked local and global processes. This was possible because they adopted relatively crude phenomenological realizations of star formation, radiative cooling in a complex multi-phase medium, BH accretion and feedback, and because their spatial resolution is larger than $r_\mathrm{B}$. In particular, they assumed values of the above introduced efficiencies instead of directly computing them. A main result of these simulations is that the $M_{\mathrm{BH}}-\sigma$ relation can be reproduced remarkably well and that the relation is insensitive to the gas fraction in the model galaxies. In addition, the BH mass appears to be little affected by the details of star formation and supernova feedback.
@Begelman:2005 argued that the results from simulations of merging galaxies suggest that the feedback regulating BH accretion operates on local scales, comparable to $r_\mathrm{B}$ or closer in, rather than solely on the global scales usually considered (see also @Murray:2005). They also presumed that the insensitivity to gas fraction occurs in the galaxy merger simulations because the gas mass is somehow “maximized” on the scales where the accretion rate is determined. Thus on these scales, BH feedback can be much more important than that due to stars. If this is correct then for state-of-the-art models, the key feedback processes represent “subgrid” physics. This is a limitation of current models of AGN feedback in large scale cosmological and galaxy merger simulations because they cannot be directly related to AGN physics.
On the other hand, simulations that aim to provide insights to AGN physics do not include galaxy but rather focus on $r_\mathrm{B}$ or even smaller scales (e.g., @Proga:2007b; @Proga:2008; @Kurosawa:2008; @Kurosawa:2009b [@Kurosawa:2009]). Thus they cannot be directly related to AGN feedback on large scales. However, these smaller scale simulations can be used directly to measure the feedback efficiencies listed above. Consequently, they can be used to quantify the effects that are assumed or parametrized in large scale simulations.
The goal of this paper is to present measurements of the mass accretion rate, and various feedback efficiencies based on direct simulations of inflows and outflows in AGN, on sub-parsec and parsec scales performed by @Kurosawa:2009b (KP09 hereafter). In other words, we wish to determine if AGN can supply energy in the form and amount required by the cosmological and galaxy merger simulations.
AGN Models in Current Cosmological Simulations {#sec:AGN-SPH}
==============================================
Mass Accretion Rates {#sub:Mdot-SPH}
--------------------
In recent cosmological and galaxy merger simulations (e.g., SDH05; @Robertson:2006; @Sijacki:2007; @Khalatyan:2008; @DiMatteo:2008; @Johansson:2009; BS09), the actual physical process of the mass-accretion onto the BH is not explicitly modeled because of a relatively poor resolution. Often, these simulations rely on a separate analytical model to describe the small scale physical processes. The unresolved accretion process is usually described by a Bondi-Hoyle-Little formulation (@Hoyle:1939; @Bondi:1944; @Bondi:1952). Here, we consider the case in which the accreting BH does not move with respect to the surrounding gas. This reduces the process to a simpler Bondi spherical accretion problem. Then, the mass-accretion rate can be written as Equation (\[eq:mdot\_bondi\]). The dimensionless constant $\lambda$ (in Eq. \[\[eq:mdot\_bondi\]\]) depends on the adiabatic index $\gamma$. For the gas with $\gamma=5/3,$ $\lambda=1/4$ (see e.g., @Frank:1992). The Bondi accretion formula relates the mass-accretion rate of a BH located at the center to the gas density and the sound speed (or equivalently the temperature) of the gas at a large scale.
On the other hand, in the Bondi accretion prescription used by e.g., SDH05 and BS09, the mass-accretion rate is written as $$\dot{M_{\mathrm{S}}}=\alpha\,\frac{4\pi
G^{2}M_{\mathrm{BH}}^{2}\rho}{c_{\mathrm{s}}^{3}}
\label{eq:mdot_bondi_sph}$$ where $\rho$ and $c_{\mathrm{s}}$ are the density and the sound speed estimated near the BH using the surrounding smoothed particle hydrodynamics (SPH: @Gingold:1977; @Lucy:1977) gas particles. Note that the expression contains “the dimensionless parameter” $\alpha$ which is different from $\lambda$ in Equation (\[eq:mdot\_bondi\]). SDH05 introduced $\alpha$ parameter to overcome the gap in the scale sizes between the numerical resolution and the Bondi accretion regime. In a typical cosmological or galaxy merger SPH simulation, the smoothing length ($\sim10^{3}$ pc) is much larger than the gravitational radius of influence or the Bondi radius, $r_{\mathrm{B}}$, which is $\sim2$ pc. If we assume the gas located at a large distance, heated by the AGN radiation, is “Comptonized” ($T\approx2\times10^{7}$ K), the corresponding speed of sound (assuming $\gamma=5/3$) is relatively high ($\sim500\,\mathrm{km\, s^{-1}}$).
SDH05, BS09 and others (e.g., @Robertson:2006; @Sijacki:2007; @Khalatyan:2008; @DiMatteo:2008; @Johansson:2009) find that a very large factor of $\alpha$ is required for low-mass BHs to grow their masses; hence, the problem is not strictly a Bondi accretion problem. Most of the AGN feedback model in the cosmological simulations mentioned above assume a constant value of $\alpha=100$ (see also Table 2 in BS09) except for BS09 who allow $\alpha$ to depend on the value of local gas density. The assumption of a very large value of $\alpha$ becomes inadequate when the local gas density is higher than that required by the formation of the a cold interstellar gas phase, and when the cosmological simulation does resolve the Jean length and the Bondi radius (BS09).
By noting these, BS09 abandoned the assumption of constant $\alpha$ in Equation (\[eq:mdot\_bondi\_sph\]), and introduced the following parametrization of $\alpha$. $$\alpha=
\begin{cases}
1 & \mathrm{for\, n_{\mathrm{H}}<n_{\mathrm{H}}^{*}}\\
\left(n_{\mathrm{H}}/\mathrm{n_{\mathrm{H}}^{*}}\right)^{\beta} &
\mathrm{for}\, n_{\mathrm{H}}\geq n_{\mathrm{H}}^{*}
\end{cases}
\label{eq:alpha-booth}$$ where $n_{\mathrm{H}}$ and $n_{\mathrm{H}}^{*}$ are the number density of hydrogen and the critical hydrogen number density above which the gas is expected to become multi-phase, and star formation is expected to begin via contraction of gas due to thermo-gravitational instability (cf. @Schaye:2004; @Schaye:2008). The critical density is chosen as $n_{\mathrm{H}}^{*}=0.1\,\mathrm{cm^{-3}}$, i.e., the corresponding critical hydrogen density is $\rho_{\mathrm{H}}^{*}=1.7\times10^{-25}\,\mathrm{g\, cm^{-3}}$. The best fit models of BS09 to some observations (e.g., the $M_{\mathrm{BH}}$–$\sigma$ relation) gives $\beta=2.0$. Note that the new parametrization of $\alpha$ in Equation (\[eq:alpha-booth\]) provides an additional density dependency of the mass-accretion rate in Equation (\[eq:mdot\_bondi\_sph\]). In the formulation of BS09, the mass-accretion rate steeply depends on the density of the surrounding gas i.e., $\dot{M}\propto\rho^{3}$, for $\rho>\rho_{\mathrm{H}}^{*}$ while the formulation of SDH05 and the original Bondi accretion model (Eq. [\[]{}\[eq:mdot\_bondi\]\]) always give a linear dependency i.e., $\dot{M}\propto\rho$. In most of the AGN accretion models in the cosmological simulations (e.g., SDH05; BS09), the highest mass-accretion rate is limited to the Eddington rate, i.e., $$\dot{M}_{\mathrm{Edd}}=\frac{4\pi
GM_{\mathrm{BH}}m_{\mathrm{p}}}{\epsilon_{\mathrm{r}}\sigma_{\mathrm{T}}c}
\label{eq:mdot-Eddington}$$ where $m_{\mathrm{p}}$, $\epsilon_{\mathrm{r}}$, $\sigma_{\mathrm{T}}$ and $c$ are the proton mass, the radiative efficiency (the rest mass to radiation conversion efficiency), the Thomson cross-section and the speed of light, respectively. The Eddington ratio ($\Gamma$) is defined as the ratio of a system mass-accretion rate to the Eddington rate, i.e., $\Gamma = \MDOT_{\mathrm{a}} / \dot{M}_{\mathrm{Edd}}$.
Figure \[fig:mdot-models\] illustrates how the mass-accretion rate depends on the density ($\rho$) in the models by @Bondi:1952, SDH05 and BS09. The mass of the BH is assumed as $M_{\mathrm{BH}}=10^{8}\,{\mathrm{M_{\sun}}}$. In all three models, the speed of sound $c_{\mathrm{s}}$ is set to $520\,\mathrm{km\, s^{-2}}$, which corresponds to that of Comptonized gas temperature $T\approx2\times10^{7}$ K with the adiabatic index of gas $\gamma=5/3$. In the modified Bondi accretion models of SDH05 and BS09, the mass-accretion rates are limited to the Eddington rate (Eq. [\[]{}\[eq:mdot-Eddington\]\]), and the radiative efficiency $\epsilon_{\mathrm{f}}$ is set to $0.1$. The dimensionless parameter $\alpha=100$ is adopted for the model of SDH05, and $\beta=2.0$ is adopted in the model of BS09. The figure shows that the mass-accretion rate of SDH05 is larger than the Bondi accretion rate for $\rho\simless10^{-20}\,\mathrm{g\, cm^{-3}}$. On the other hand, the mass-accretion rate by BS09 is similar to that of the Bondi accretion rate (off by a factor of $\lambda=1/4$ in Eq. [\[]{}\[eq:mdot\_bondi\]\]) in the low-density regime ( $\rho\simless10^{-25}\,\mathrm{g\, cm^{-3}}$), but it is significantly larger than the Bondi accretion rate for the density range of $10^{-25}\simless\rho\simless10^{-20}\,\mathrm{g\, cm^{-3}}$. The densities above which the accretion proceeds at the Eddington rate are $\sim9.5\times10^{-23}\,\mathrm{g\, cm^{-3}}$ and $\sim6.3\times10^{-24}\,\mathrm{g\, cm^{-3}}$ for the models of SDH05 and BS09, respectively. However, these values change depending on the adopted value of $c_{\mathrm{s}}$. In the Bondi accretion model, the mass-accretion rates reaches the Eddington rate at much higher density ($\rho\sim10^{-20}\,\mathrm{g\, cm^{-3}}$). We note that adding a rotation to the gas can reduce the mass-accretion rate. As shown by @Proga:2003c and @Ryu:1995, the reduction of $\MDOT_{\mathrm{a}}$ can be significant, i.e., by one order of magnitude or more, compared to $\MDOT_{\mathrm{B}}$
![Comparison of the mass-accretion rates from the Bondi accretion model (@Bondi:1952) (*solid*), the modified Bondi accretion model by SDH05 (*dashed*) and that by BS09 (*dotted*), as described in Eqs. (\[eq:mdot\_bondi\]), (\[eq:mdot\_bondi\_sph\]) and (\[eq:alpha-booth\]), respectively. The mass-accretion rates are computed as a function of density $\rho$ at a radius $r_{\mathrm{o}}$ which is much larger than the Bondi radius ($r_{\mathrm{B}}$), i.e., $r_{\mathrm{o}}\gg r_{\mathrm{B}}$. In all models, the speed of sound $c_{\mathrm{s}}$ at $r_{\mathrm{o}}$ is set to $520\,\mathrm{km\, s^{-2}}$, which corresponds to that of Comptonized gas temperature $T\approx2\times10^{7}$ K with the adiabatic index of gas $\gamma=5/3$. The mass of the BH is set to $M_{\mathrm{BH}}=10^{8}\,{\mathrm{M_{\sun}}}$. While $\alpha=100$ is adopted in the model of SDH05, $\beta=2.0$ is adopted in the model of BS09. In the modified Bondi accretion models of SDH05 and BS09, the mass-accretion rates are limited to the Eddington rate (Eq. [\[]{}\[eq:mdot-Eddington\]\]). []{data-label="fig:mdot-models"}](f01.eps){width="45.00000%"}
AGN Feedback Models {#sub:AGN-FB-SPH}
-------------------
In SDH05 and BS09 (also in many others, cf. Table 2 in BS09), $\epsilon_{\mathrm{r}}=0.1$ (e.g., @shakura:1973; see also @Soltan:1982) is assumed, and is fixed. A higher value of $\epsilon_{\mathrm{r}}$ ($\sim0.2$) can be achieved in an accretion model with a thin disk and a rapidly rotating BH (e.g., @Thorne:1974). Recent observations suggest a wide range of $\epsilon_{\mathrm{r}}$: $0.07$ (@Martinez-Sansigre:2009), $0.30$–$0.35$ (@Wang:2006b), $0.16$ (@Yu:2008), $0.15$ (@Elvis:2002) and $\sim 0.1$ or $\sim 0.2$ (@Yu:2002). On the other hand, @Cao:2008 find $\epsilon_{\mathrm{r}}$ is relatively low ($\sim0.08$) for $M_{\mathrm{BH}}<10^{8}\,{\mathrm{M_{\sun}}}$ and relatively high ($\simgreat0.18$) for $M_{\mathrm{BH}}\simgreat10^{9}\,{\mathrm{M_{\sun}}}$. Since the exact mechanism of how the accretion luminosity of a BH couples to the surrounding gas is not well known, SDH05 and BS09 simply assume that $L_{\mathrm{a}}$ couples only thermally (and isotropically) to the surrounding. Using Equation (\[eq:L-acc\]), the rate of energy deposition to the surrounding (the AGN feedback rate) in SDH05 is written as $$\dot{E}_{\mathrm{f}}=\epsilon_{\mathrm{f}}\,
L_{\mathrm{a}}=\epsilon_{\mathrm{f}}\epsilon_{\mathrm{r}}\dot{M}_{\mathrm{BH}}c^{2}
\label{eq:Edot-Feedback}$$ where $\epsilon_{\mathrm{f}}$ is the efficiency of the AGN energy deposition to the surrounding gas, and is a free parameter which is to be constrained by observations. BS09 find the models with $\epsilon_{\mathrm{f}}=0.15$ matches observations (e.g., the Magorrian relation and the $M_{\mathrm{BH}}-\sigma$ relations) very well, and similarly SDH05 find $\epsilon_{\mathrm{f}}=0.05$ matches observations well (see also @DiMatteo:2005). In the study of BS09, they find that the global BH number density at the current era (zero redshift) and the BH scaling relations are very sensitive to a choice of $\epsilon_{\mathrm{f}}$, and the former is nearly inversely proportional to the value of $\epsilon_{\mathrm{f}}$.
Our Model {#sec:Model}
=========
Our approach is to use physical two-dimensional (axisymmetric) and time-dependent hydrodynamical (HD) simulations of AGN flows to investigate the dependency of the BH mass-accretion rate on the surrounding gas density, and to find the AGN feedback efficiencies in converting the accretion luminosity into the outward fluxes of energy, momentum and mass. Here, we simply analyze the simulations results previously published in KP09 for this purpose. KP09 used a modified version of the [ZEUS-MP]{} code [cf. @Hayes:2006] for their numerical simulations. In the following, we briefly summarize their main model assumptions and results.
In KP09, we studied axisymmetric hydrodynamical simulations of a slowly rotating gas that is under the influence of the gravity of a $10^8~\MSUN$ black hole and is irradiated by a geometrically thin UV accretion disk and a spherical X-ray corona. We ran a set of simulations for various values of the gas density ($\rho_{\mathrm{o}}$) at the outer radius of the computational domain, $r_{\mathrm{o}}\approx$ 7 pc. After the initial transient stage, this density determines the key characteristics of our solutions such as the accretion luminosity and the outflow properties. We compute the accretion luminosity of a system based on the accretion-rate which is assumed to be equal to the mass-supply rate at the inner radius of the computational domain $r_{\mathrm{i}}\approx 10^{-2}$ pc, (i.e., we used Eq. [\[]{}\[eq:L-acc\]\] where we assumed $\epsilon_{\mathrm{r}}=1/12$ and $\MDOT_{\mathrm{a}}=\MDOT_{\mathrm{in}}[r_{\mathrm{i}}]$).
For the models with high temperature gas at large radii and with high luminosities, we found a strong correlation between $\MDOT_{\mathrm{out}}$ and $L_{\mathrm{a}}$ (see Fig. 1 in KP09). The power law index describing the correlation is very similar to that for radiation-driven stellar and disk wind models (e.g., @Castor:1975; @Proga:1998; @Proga:1999). More surprisingly, for the models with high density at large radii, we found that steady state solutions with $L_{\mathrm a}$ exceeding the Eddington limit. The super-Eddington accretion proceeds in the equatorial region and is possible because the radiation flux from the disk is significantly reduced in the equatorial direction due to the geometrical foreshortening effect.
In all simulations performed by KP09, an outflow is driven from an inflow with sub-Keplerian rotation. For the models with high temperatures at large radii, the inflow occurs over a wide range of the polar angles, whereas the outflow occurs in a relatively narrow polar angle range (see the left panel in Fig. \[fig:samples\]). However, for the super-Eddington cases with low temperature at large radii, the inflow persists only very close to the equatorial plane, resembling a thin accretion disk, while the outflow arises in a wide range of radii and polar angles (see the right panel in Fig. \[fig:samples\]). The geometry of this extreme inflow-outflow solution is very similar to a radiation-driven wind from a luminous Keplerian accretion disk (e.g., @Woods:1996; @Proga:1998; @Proga:2002b). For the cold super-Eddington solutions, $\MDOT_{\mathrm{out}}$ is only very weakly correlated. The weaker correlation is mainly caused by a mismatch between with the direction of escaping photons and the inflowing gas. In other words, the radiation is emitted mostly in the polar directions whereas the inflowing gas occurs mainly in the equatorial region.
As it has been discussed and shown in the past, we find that self-consistently determined preheating/cooling from the quasar radiation can significantly reduce the rate at which the central BH is fed with matter. However, our results also emphasize a little-appreciated feature. Namely, quasar radiation does drive a non-spherical, multi-temperature and very dynamic flow.
In the following, we present the mass-accretion rates and various (energy, momentum and mass) AGN feedback efficiencies computed from the simulations. For this purpose, we use a subset of the models in KP09. Here, we concentrate on the models in which *the outer boundary temperature is not fixed at a constant value, but it is self-consistently determined from the radiative and adiabatic heating* (Models 28–34 in KP09). Note that the flow solutions for these models are, in general, very similar to those with the fixed outer boundary temperature at $2\times10^6\,{\mathrm{K}}$ (the low temperature models, i.e., Models 1-9 in KP09).
{width="98.00000%"}
------------- ----------------------------------------------- ----------------------------------- ---------------------------------------------------- -------------------------- ----------------------------------------------------- ----------------------------------------------------
$\rho_{\mathrm{o}}$ $T_{\mathrm{o}}^{**}$ $\dot{M}_{\mathrm{in}}\left(r_{\mathrm{i}}\right)$ $\Gamma$ $\dot{M}_{\mathrm{out}}\left(r_{\mathrm{o}}\right)$ $\dot{M}_{\mathrm{in}}\left(r_{\mathrm{o}}\right)$
Model$^{*}$ $\left(10^{-21}\,\mathrm{g\, cm^{-3}}\right)$ $\left(10^{7}\,\mathrm{K}\right)$ $\left(10^{25}\mathrm{\, g\, s^{-1}}\right)$ $\cdots$ $\left(10^{25}\,\mathrm{g\, s^{-1}}\right)$ $\left(10^{25}\,\mathrm{g\, s^{-1}}\right)$
35 2 0.02 $3.4\left(0.1\right)^{\dagger}$ $0.20\left(0.01\right)$ $0.0\left(0.0\right)$ $3.4\left(0.1\right)$
36 4 0.07 $5.5\left(0.6\right)$ $0.32\left(0.04\right)$ $0.11\left(0.02\right)$ $5.4\left(0.2\right)$
28 10 0.14 $8.9\left(0.68\right)$ $0.52\left(0.032\right)$ $1.0\left(0.032\right)$ $9.4\left(0.1\right)$
29 20 0.14 $12\left(0.58\right)$ $0.71\left(0.034\right)$ $3.5\left(0.52\right)$ $15\left(0.3\right)$
30 40 0.36 $18\left(1.1\right)$ $1.1\left(0.1\right)$ $7.1\left(1.4\right)$ $25\left(0.2\right)$
31 80 0.80 $25\left(2.6\right)$ $1.4\left(0.2\right)$ $10\left(2.3\right)$ $35\left(2.1\right)$
32 160 0.98 $36\left(4.9\right)$ $2.1\left(0.3\right)$ $9.9\left(2.6\right)$ $49\left(5.3\right)$
33 320 0.85 $52\left(2.6\right)$ $3.1\left(0.2\right)$ $11\left(0.48\right)$ $63\left(1.1\right)$
34 640 1.30 $72\left(0.56\right)$ $4.3\left(0.03\right)$ $9.5\left(0.19\right)$ $82\left(0.9\right)$
------------- ----------------------------------------------- ----------------------------------- ---------------------------------------------------- -------------------------- ----------------------------------------------------- ----------------------------------------------------
([\*]{}) The model numbers are identical to those in @Kurosawa:2009b except for Models 35 and 36 which are additional models presented here for the first time.
([\*]{}[\*]{}) Self-consistently determined temperature at the outer boundary.
($\dagger$) Values in brackets are the standard deviations of the time series values.
Results {#sec:Results}
=======
We analyze the dependency of the mass-accretion rate on the gas density at a large distance from a BH and AGN feedback efficiencies based on the axisymmetric hydrodynamical simulations presented in KP09. The main results of the models along with the input outer boundary density $\rho_{\mathrm{o}}$ are summarized in Table \[tab:Model-Summary\].
Mass-Accretion Rates {#sub:result-mdot}
--------------------
The mass-inflow rates at the inner boundary $\dot{M}_{\mathrm{in}}\left(r_{\mathrm{i}}\right)$ and those at outer boundary $\dot{M}_{\mathrm{in}}\left(r_{\mathrm{o}}\right)$ from the HD simulations are plotted as a function of the outer boundary density $\rho_{\mathrm{o}}$ in Figure \[fig:mdot\] (see Tab. \[tab:Model-Summary\] for the numerical values). For a given value of $\rho_{\mathrm{o}}$, $\dot{M}_{\mathrm{in}}\left(r_{\mathrm{i}}\right)$ and $\dot{M}_{\mathrm{in}}\left(r_{\mathrm{o}}\right)$ are not equal to each other, but rather $\dot{M}_{\mathrm{in}}\left(r_{\mathrm{i}}\right)<\dot{M}_{\mathrm{in}}\left(r_{\mathrm{o}}\right)$ because of an outflow. The lowest density model (Model 35) is an exception since no outflow is formed in this model. For the higher density models, an outflow forms, and not all the material entering from the outer boundary reaches the inner boundary. A fraction of gas experiences a strong radiation pressure and radiative heating, and the direction of flow changes, forming an outflow.
The figure also shows the the mass-accretion rates predicted by the Bondi accretion model (Eq. [\[]{}\[eq:mdot\_bondi\]\]) and those computed from the formulations of SDH05 and BS09 (Eqs. [\[]{}\[eq:mdot\_bondi\_sph\]\] and [\[]{}\[eq:alpha-booth\]\]). The outer radius is much smaller than that of a typical smoothing scale on a SPH cosmological simulation ($\sim10^{3}$ pc), and the outer density values used in our simulations are much larger than a typical local density at BH in the SPH simulations. In our simulations, the higher density at a 10 pc scale is required for a system to produce an outflow. For example, as we can see in Table \[tab:Model-Summary\], $\rho_{\mathrm{o}}$ must be greater than $2\times10^{-21}\,\mathrm{g\, cm^{-3}}$, which corresponds to ($n_{\mathrm{H}}\simgreat1.2\times10^{3}\,\mathrm{cm^{-3}}$), to form an outflow with our system setup. In the density range of the models considered here, the mass-accretion rates adopted by SDH05 and BS09 are limited by the Eddington rate (Eq. [\[]{}\[eq:mdot-Eddington\]\]); hence, the line is flat (cf. Fig. \[fig:mdot-models\]). Note that the radiative efficiency $\epsilon_{\mathrm{r}}=1/12$ instead of 0.1 is adopted for the Eddington rate (Eq. [\[]{}\[eq:mdot-Eddington\]\]) in the models of SDH05 and BS05 to be consistent with our simulations. This moves the Eddington rate only slightly upward.
Our models include the effects of radiative heating and radiation force. Therefore, we do not, in general, expect our solution to reproduce an exactly same density dependency of the mass-inflow rate as that of the Bondi model. However, the figure shows the mass-inflow rates from our models are very similar to those of the Bondi rates, i.e., the rates are of the same order of magnitude. Interestingly our $\dot{M}_{\mathrm{in}}\left(r_{\mathrm{i}}\right)$ and the Bondi mass-accretion rate matches around $\rho_{\mathrm{o}}=4\times10^{-20}\,\mathrm{g\, cm^{-3}}$ which coincidentally corresponds to $\Gamma\approx1$. Since the accretion rates from SDH05 and BS09 are the Eddington rates (the rates corresponding to $\Gamma=1$) in this density range, their lines also cross at the same point.
The figure clearly shows that our models have a weaker dependency of the mass-inflow rates on the density than that of the Bondi accretion. The power-law fits of data points give the slope $q=0.52\left(\pm0.01\right)$ for $\dot{M}_{\mathrm{in}}\left(r_{\mathrm{i}}\right)$ and $q=0.56\left(\pm0.02\right)$ for $\dot{M}_{\mathrm{in}}\left(r_{\mathrm{o}}\right)$, which are indeed much smaller than that of the Bondi accretion model, i.e., $q=1$ (cf. Eq. [\[]{}\[eq:mdot\_bondi\]\]). Although not shown here, if we turn off the rotation, radiation force and radiative heating in our models, we obtain $q\approx1$, as this is equivalent to the Bondi accretion problem (see also @Proga:2003c; @Janiuk:2008).
![Comparison of the mass-inflow rates found in the HD simulations (Tab. \[tab:Model-Summary\]) with those predicted by the Bondi accretion model (@Bondi:1952) (*solid line*) and with those adopted by SDH05 and BS09 (*dashed line*). In the density range of the models considered here, the mass-accretion rates adopted by SDH05 and BS09 are limited by the Eddington rate (Eq. [\[]{}\[eq:mdot-Eddington\]\]); hence, the line is flat (cf. Fig. \[fig:mdot-models\]). The mass-inflow rates at the inner boundary (*circles*) and those at the outer boundary (*squares*) of the computational domain are shown as a function of the outer boundary density $\rho_{\mathrm{o}}$. The mass-accretion from the HD simulations are very similar to the Bondi accretion rates, but the HD models has a less steeper dependency on the density. The Bondi mass-accretions rates and those of SDH05 and BS09 are computed for the gas with the Comptonized temperature $T=2\times10^{7}$ K and with the adiabatic index $\gamma=5/3$. []{data-label="fig:mdot"}](f03.eps){width="45.00000%"}
Feedback Efficiencies {#sub:efficiency-HD}
---------------------
Next, we compute AGN feedback efficiencies in energy, momentum and mass using the simulation results, as defined in Equations (\[eq:eff-total-energy\])–(\[eq:eff-mass\]). Since the models used here show some degree of variability (typically $\sim10\%$ level, cf. Tab \[tab:Model-Summary\]), the physical quantities used to compute the feedback efficiencies are based on the time averaged values.
------- ---------- ------------------------- -------------------------- ------------------------- ------------------------- -------------------------
Model $\Gamma$ $\epsilon_{\mathrm{k}}$ $\epsilon_{\mathrm{th}}$ $\epsilon_{\mathrm{t}}$ $\epsilon_{\mathrm{p}}$ $\epsilon_{\mathrm{m}}$
35 $0.20$ $0$ $0$ $0$ $0$ $0$
36 $0.32$ $1.4\times10^{-8}$ $9.1\times10^{-8}$ $1.0\times10^{-7}$ $7.8\times10^{-5}$ 0.02
28 $0.52$ $8.1\times10^{-7}$ $4.6\times10^{-7}$ $1.3\times10^{-6}$ $1.3\times10^{-3}$ 0.11
29 $0.71$ $1.4\times10^{-5}$ $1.0\times10^{-6}$ $1.5\times10^{-5}$ $8.3\times10^{-3}$ 0.29
30 $1.1$ $9.0\times10^{-5}$ $1.8\times10^{-6}$ $9.2\times10^{-5}$ $2.9\times10^{-2}$ 0.39
31 $1.4$ $1.3\times10^{-4}$ $9.0\times10^{-6}$ $1.4\times10^{-4}$ $3.0\times10^{-2}$ 0.40
32 $2.1$ $1.0\times10^{-4}$ $2.0\times10^{-6}$ $1.0\times10^{-4}$ $1.8\times10^{-2}$ 0.28
33 $3.1$ $5.8\times10^{-5}$ $1.7\times10^{-6}$ $6.0\times10^{-5}$ $9.8\times10^{-3}$ 0.21
34 $4.3$ $8.2\times10^{-5}$ $1.5\times10^{-6}$ $8.4\times10^{-5}$ $7.7\times10^{-3}$ 0.13
------- ---------- ------------------------- -------------------------- ------------------------- ------------------------- -------------------------
### Energy Feedback Efficiency {#subsub:eff_energy}
Figure \[fig:eff\_energy\] shows the energy feedback efficiencies, $\epsilon_{\mathrm{t}}$, $\epsilon_{\mathrm{k}}$ and $\epsilon_{\mathrm{th}}$ computed based on our models (Table \[tab:Model-Summary\]), as a function of the Eddington ratio ($\Gamma$). The numerical values of the efficiencies are listed in Table \[tab:efficiency\]. For systems with relatively low Eddington ratio ($\Gamma$$\simless0.4$), the thermal feedback efficiency is higher than the kinetic feedback efficiency ($\epsilon_{\mathrm{th}}>\epsilon_{\mathrm{k}}$). On the other hand, for systems with relatively high Eddington ratio ($\Gamma$$\simgreat0.6$), the kinetic feedback dominates the thermal feedback by a factor of $\sim10$ to $\sim100$. The model with $\Gamma=0.2$ does not form an outflow, indicating an approximate $\Gamma$ value below which no outflow forms (with our system setup).
The energy feedback efficiencies increase as $\Gamma$ increases, but the efficiencies saturate for $\Gamma\simgreat1$. The total energy feedback efficiency peaks at $\Gamma\approx1$ with $\epsilon_{\mathrm{t}}\sim10^{-4}$. The flattening of the efficiencies for $\Gamma\simgreat1$ is caused by the transition of the inflow-outflow morphology to a “disk wind” like configuration (cf. Fig. \[fig:samples\]) for the higher $\Gamma$ models (KP09). As briefly mentioned in § \[sec:Model\], because of the mismatch between the direction in which most of the radiation escapes (in polar direction) and the direction in which the most of the accretion occurs in the system (the equatorial direction), the radiatively driven outflows in the disk-wind-like configuration cannot increase the outflow efficiency by increasing the accretion luminosity or equivalently $\Gamma$. A similar behavior is found in the $\dot{M}_{\mathrm{out}}\left(r_{\mathrm{o}}\right)$– $\Gamma$ relation of KP09 (see their Fig. 7).
![The efficiencies of converting the BH accretion luminosity $L_{\mathrm{a}}$ to the rate of energy deposition to the surrounding gas are plotted as a function of the Eddington ratio ($\Gamma$). The panel shows the kinetic energy feedback efficiency $\epsilon_{\mathrm{k}}$ (*circles*), the thermal energy feedback efficiency $\epsilon_{\mathrm{th}}$ (*squares*) and the total energy feedback efficiency $\epsilon_{\mathrm{t}}=\epsilon_{\mathrm{k}}+\epsilon_{\mathrm{th}}$ (*triangles*), separately (see Eqs. [\[]{}\[eq:eff-total-energy\]\], [\[]{}\[eq:eff-kinetic-energy\]\], and [\[]{}\[eq:eff-thermal-energy\]\]). The maximum total energy feedback efficiency is $\sim10^{-4}$. For the models with relatively low Eddington ratio ($\Gamma$$\simless0.4$), the thermal feedback is more efficient than the kinetic feedback ($\epsilon_{\mathrm{th}}>\epsilon_{\mathrm{k}}$). For the models with relatively high Eddington ratio ($\Gamma \simgreat0.6$), the kinetic feedback is more efficient than the thermal feedback by a factor of $\sim10$ to $\sim100$. The model with $\Gamma=0.2$ does not form an outflow, and the vertical line (*dashed*) at $\Gamma=0.2$ indicates an approximate $\Gamma$ value below which no outflow forms. The flattening of the efficiencies at $\Gamma\approx1$ is caused by the transition of the inflow-outflow morphology to a “disk wind” like configuration for the larger $\Gamma$ models (cf. Fig. \[fig:samples\]).[]{data-label="fig:eff_energy"}](f04.eps){width="45.00000%"}
### Momentum Feedback Efficiency {#subsub:eff_momentum}
Figure \[fig:eff\_momentum\] shows the momentum feedback efficiency $\epsilon_{\mathrm{p}}$ as a function $\Gamma$. The numerical values of $\epsilon_{\mathrm{p}}$ for each model are listed in Table \[tab:efficiency\]. The dependency of $\epsilon_{\mathrm{p}}$ on $\Gamma$ is similar to that of the energy feedback efficiency. For $\Gamma\simless1$, $\epsilon_{\mathrm{p}}$ increases as $\Gamma$ increases, but it decreases as $\Gamma$ increases for $\Gamma\simgreat1$. No outflow is formed for the models with $\Gamma=0.2$ and below. The cause of the peaking of $\epsilon_{\mathrm{p}}$ at $\Gamma\approx1$ (and the declining for $\Gamma\simgreat1$) is again due to the change in the inflow-outflow morphology to a disk-wind-like (KP09; see also Fig. \[fig:samples\]). The momentum deposition of the photons becomes less efficient once the flow has the disk-wind-like configuration since a major fraction of the radiation escapes in the polar direction, but gas is not present in that direction since it has been already blown away by the strong radiation. The maximum momentum feedback efficiency is $\epsilon_{\mathrm{p}}\approx0.03$ which is about 2 orders of magnitude larger than the total *energy* feedback efficiency found in § \[subsub:eff\_energy\].
![The momentum feedback efficiency ($\epsilon_{\mathrm{p}}$), which is defined as the ratios of the total wind momentum (at the outer boundary) to the total radiation momentum ($L_{\mathrm{a}}/c$), is plotted as a function of the Eddington ratio ($\Gamma$). The efficiency peaks at $\Gamma\approx1$ with $\epsilon_{\mathrm{p}}\approx0.03$, and it decreases for larger $\Gamma$ values. The decline of the curve beyond $\Gamma\approx1$ is caused by the change in the inflow-outflow morphology to a “disk wind” like configuration for the larger $\Gamma$ models (cf. Fig. \[fig:samples\]). The model with $\Gamma=0.2$ does not form an outflow, and the vertical line (*dashed*) at $\Gamma=0.2$ indicates an approximate $\Gamma$ value below which no outflow forms.[]{data-label="fig:eff_momentum"}](f05.eps){width="45.00000%"}
### Mass Feedback Efficiency {#subsub:eff_mass}
Figure \[fig:eff\_mass\] shows the mass feedback efficiency ($\epsilon_{\mathrm{m}}$) plotted as a function $\Gamma$. The dependency of $\epsilon_{\mathrm{m}}$ on $\Gamma$ is very similar to that of the momentum feedback efficiency (Fig. \[fig:eff\_momentum\]). The numerical values of $\epsilon_{\mathrm{m}}$ are listed in Table \[tab:efficiency\]. For $\Gamma\simless1$, the mass feedback efficiency $\epsilon_{\mathrm{m}}$ increases as $\Gamma$ increases, but it starts to decreases slightly beyond $\Gamma\approx1$. The efficiency peaks around $\Gamma=1$ with the maximum efficiency value $\sim0.4$. In other words, about $40\%$ of the total inflowing mass is redirected to an outflow. The turn-around of $\epsilon_{\mathrm{m}}$ values is caused by the transition of the outflow morphology to a disk-wind-like configuration (cf. Fig. \[fig:samples\]) for the larger $\Gamma$ models, as in the cases for the energy and momentum feedback efficiencies (§§ \[subsub:eff\_energy\] and \[subsub:eff\_momentum\]).
The accretion process in our model is fundamentally different from that of the Bondi accretion model and those adopted in the cosmological simulations (e.g., SDH05; BS09) because in our model an outflow and an inflow can simultaneously be formed while the Bondi accretion model can only form an inflow. The model of SDH05 and others can form either an inflow or an accretion (but not both simultaneously).
![The mass feedback efficiency ($\epsilon_{\mathrm{m}}$) plotted as a function the Eddington ratio ($\Gamma$). The efficiency $\epsilon_{\mathrm{m}}$ is defined as the ratio of the mass-outflow rate at the outer boundary $\dot{M}_{\mathrm{out}}\left(r_{\mathrm{o}}\right)$ to the mass-inflow rate at the inner boundary $\dot{M}_{\mathrm{in}}\left(r_{\mathrm{i}}\right)$, i.e., $\epsilon_{\mathrm{m}}=\dot{M}_{\mathrm{out}}\left(r_{\mathrm{o}}\right)/\dot{M}_{\mathrm{in}}\left(r_{\mathrm{i}}\right)$. The efficiency peaks around $\Gamma=1$ with the maximum efficiency value $\sim0.4$, i.e., $40\%$ of the total inflowing mass is converted to the outflows. The turn-around of $\epsilon_{\mathrm{m}}$ values is caused by the transition of the outflow morphology to a “disk wind” like configuration for the larger $\Gamma$ models (cf. Fig. \[fig:samples\]). The model with $\Gamma=0.2$ does not form an outflow, and the vertical line (*dashed*) at $\Gamma=0.2$ indicates an approximate $\Gamma$ value below which no outflow forms.[]{data-label="fig:eff_mass"}](f06.eps){width="45.00000%"}
Discussion {#sec:Discussion}
==========
{#subsec:compare-others}
A main purpose of this study is to measure the efficiency of the AGN feedback in our radiation hydrodynamical simulations of accretion flows irradiated by AGN. Especially, we are interested in the energy feedback efficiency ($\epsilon_{\mathrm{f}}$) as this is an important parameter to determine the BH growth rate and the BH number density at the current epoch (redshift zero) in the cosmological simulations (e.g., SDH05; BS09). BS09 find that the BH growth rate is nearly inversely proportional to the value of $\epsilon_{\mathrm{f}}$, which is important for a self-regulation of a BH growth (e.g., @Silk:1998; @Fabian:1999).
The approximate ratios of the maximum feedback efficiencies in energy, momentum and mass using the results from § \[sub:efficiency-HD\] are: $$\epsilon_{\mathrm{m}}:\epsilon_{\mathrm{p}}:\epsilon_{\mathrm{t}}:\epsilon_{\mathrm{k}}:\epsilon_{\mathrm{th}}=1000:100:1:1:0.1
\label{eq:eff-ratios}$$ where the total energy feedback efficiency $\epsilon_{\mathrm{t}}\sim10^{-4}$. Compared to the mass feedback efficiency, the thermal energy feedback efficiency is $10^{4}$ times smaller. Thus the coupling between the radiation and the thermal energy of the gas, on scales between $10^{-2}$ and a few parsecs, is relatively inefficient.
The efficiency of the AGN energy deposition to the surrounding gas $\epsilon_{\mathrm{f}}$ is usually treated as a free parameter and is somewhat constrained by observations, e.g., by fits to the Magorrian relation and the $M_{\mathrm{BH}}-\sigma$ relations. SDH05, @Robertson:2006, @Sijacki:2007, @DiMatteo:2008 and @Johansson:2009 find that their numerical simulations with $\epsilon_{\mathrm{f}}=0.05$ match observations. BS09, who adopted a slightly different BH accretion model (§ \[sub:Mdot-SPH\]), find a larger value $\epsilon_{\mathrm{f}}=0.15$ produces a good match with observations. In their models, the energy is assumed to be released only in the form of thermal energy; hence, their $\epsilon_{\mathrm{f}}$ is equivalent to our $\epsilon_{\mathrm{th}}$. Our thermal energy feedback efficiency $\epsilon_{\mathrm{th}}$ is about $5\times10^{3}$ times smaller than their $\epsilon_{\mathrm{f}}$. Even when we compare it with our total energy feedback efficiency $\epsilon_{\mathrm{t}}$, their $\epsilon_{\mathrm{f}}$ is still about $5\times10^{2}$ times larger.
The relatively small value of the kinetic energy feedback efficiency $\epsilon_{\mathrm{k}}$ found here ($\sim10^{-4}$) is consistent with those found in @Chelouche:2008 who analyzed the X-ray spectra (*Chandra/HETG*) of five AGNs (type-I) with $M_{\mathrm{BH}}\sim10^{7}\,{\mathrm{M_{\sun}}}$. He found $\epsilon_{\mathrm{k}}$ ranges between $10^{-6}$ and $10^{-3}$. Our models also agree with the relatively low energy and mass feedback efficiencies found by @Krongold:2007 and @Stoll:2009 who studied of X-ray and UV absorbing outflows in Seyfert galaxies. On the other hand, the AGN evolution and the BH growth synthesis model of @Merloni:2008, combined with the observed star formation rate history of the universe, suggests that $\epsilon_{\mathrm{k}}=\left(3-5\right)\times10^{-3}$, which is about an order of magnitude larger than the value found in our models.
While SDH05 and BS09 assumed that the energy feedback efficiency is constant (independent of accretion luminosity), recent studies by @Ciotti:2009 and @Shin:2009, who performed one-dimensional hydrodynamical simulations of co-evolution of SMBH and elliptical galaxy, include the dependency of the AGN feedback efficiency (in a form of kinetic energy) on the accretion luminosity or more precisely on the Eddington ratio ($\Gamma$) of the system. Their parametrization of $\epsilon_{\mathrm{k}}$ with $\Gamma$ somewhat resembles our result (Fig. \[fig:eff\_energy\]), in a sense that $\epsilon_{\mathrm{k}}$ monotonously increases for $\Gamma \simless
1$. @Ciotti:2009 find that models with varying $\epsilon_{\mathrm{k}}$ are in general more consistent with observations than those with a constant $\epsilon_{\mathrm{k}}$. Interestingly, $\epsilon_{\mathrm{k}}$ values found by @Shin:2009 are only about 5 to 10 times larger[^1] than our values. This is a much better agreement than that with the models of SDH05 and BS09, as mentioned above. In summary, @Ciotti:2009 and @Shin:2009 concluded that the models need both radiation and mechanical feedback mechanisms included at the same time to account for observations such as the ratio of the central BH mass to the stellar mass ratio ($M_{\mathrm{BH}}/M_{*}$), and the X-ray luminosity of hot diffused gas.
{#subsec:cause-low-eff}
{#subsec:Mdot_a_explain}
Conclusions {#sec:Conclusions}
===========
We have presented and analyzed the AGN feedback efficiencies of energy, momentum and mass based on our axisymmetric and time-dependent hydrodynamical simulations (see Tab. \[tab:Model-Summary\]) presented in @Kurosawa:2009b. The simulations capture the radiation-driven outflows formed from a slowly rotating (sub-Keplerian) infalling gas under the influence of the gravity of the central SMBH. The accretion-luminosity and the outer boundary temperature are self-consistently determined in these models. The radial range of the simulations spans from $\sim
10^{-2}$ to $\sim 10$ pc.
The dependency of the mass-accretion rate on the density of the surrounding gas (or the outer boundary density $\rho_{o}$) has been examined (Fig. \[fig:mdot\]). The result is compared with the Bondi mass-accretion rate (Eq. \[\[eq:mdot\_bondi\]\]), and with those adopted in the cosmological simulations of SDH05 and BS09 (see also Fig. \[fig:mdot-models\]). The density dependency of the mass-accretion rate in our models is somewhat similar to that of the Bondi accretion model. For the density range of $10^{-21} \simless
\rho_{o} \simless 10^{-18}\,\mathrm{g cm^{-3}}$ (or correspondingly for $0.2 \simless
\Gamma \simless 5$), the differences between the two models are within a factor of 10. At $\Gamma \approx
1$ ($\rho_{o} \approx 4 \times 10^{-20}\,\mathrm{g cm^{-3}}$), the accretion rate of our model and that of the Bondi accretion model agree with each other. An important difference between the two models is the steepness of the dependency on the density. The power-law fit ($\dot{M}_{\mathrm{a}} \propto \rho_{\mathrm{o}}^{q}$) of our models results yields $q\approx 0.5$ instead of $q=1.0$ which is predicted by the Bondi accretion model. This difference is due to outflows in our model. We note that in this density range, the mass-accretion rates adopted by SDH05 and BS09 have no density dependency because their accretion rates are limited by the Eddington rate. The accretion rates of SDH05 are artificially boosted up ($\alpha$ factor in Eq. \[\[eq:mdot\_bondi\_sph\]\]) by a factor of 100 in comparison with the Bondi mass-accretion rates. Consequently, their rates reach the Eddington limit at much smaller $\rho_{o}$ than in our simulations (see Fig. \[fig:mdot-models\]).
We find the energy feedback efficiency of our models depends on the accretion luminosity of the system (Fig. \[fig:eff\_energy\]). For $\Gamma \simless 1$, the dependency is similar to the parametrization of the feedback efficiency adopted by @Ciotti:2009. Both kinetic and thermal energy efficiencies ($\epsilon_{\mathrm{k}}$ and $\epsilon_{\mathrm{th}}$) peak at around $\Gamma=1$. The maximum efficiency values are $\epsilon_{\mathrm{k}}\approx 10^{-4}$ and $\epsilon_{\mathrm{th}}\approx 10^{-5}$ respectively (see Table \[tab:efficiency\]). For systems with relatively low Eddington ratio ($\Gamma$$\simless0.4$), the thermal feedback efficiency is higher than the kinetic feedback efficiency ($\epsilon_{\mathrm{th}}>\epsilon_{\mathrm{k}}$). On the other hand, for systems with relatively high Eddington ratio ($\Gamma$$\simgreat0.6$), the kinetic feedback dominates the thermal feedback by a factor of $\sim10$ to $\sim100$.
The dependency of the momentum feedback efficiency $\epsilon_{\mathrm{p}}$ on $\Gamma$ is similar to that of the energy feedback efficiency (Fig. \[fig:eff\_momentum\]). The maximum efficiency is $\epsilon_{\mathrm{p}}\approx 10^{-2}$ which is about $\sim 100$ times larger than that of the total energy feedback efficiency. The dependency of the mass feedback efficiency $\epsilon_{\mathrm{m}}$ on $\Gamma$ is also similar to that of $\epsilon_{\mathrm{p}}$ (Fig. \[fig:eff\_mass\]). The maximum value of $\epsilon_{\mathrm{m}}$ found is $\sim0.4$ at around $\Gamma=1$, i.e., about $40\%$ of the total mass that moves inward at large radii, does not reach the inner boundary of our computational domain, but rather is turned into outflows.
Compared to the energy (thermal only) feedback efficiencies ($\epsilon_{\mathrm{f}}=0.05$) required in the recent cosmological and galaxy mergers simulations (e.g., SDH05; @Robertson:2006, @Sijacki:2007, @DiMatteo:2008 and @Johansson:2009), our thermal energy feedback efficiency $\epsilon_{\mathrm{th}}$ at the peak value is about $5\times10^{3}$ times smaller than their $\epsilon_{\mathrm{f}}$. Our total and kinetic energy efficiencies are about $5\times10^{2}$ times smaller than their values. These large discrepancies would suggest a few things. For example, our models are missing important elements. In particular, we do not include effects of dust which could make the outflows stronger. In addition, we focus here on axisymmetric models which could differ from fully three-dimensional (3-D) models. Our preliminary 3-D simulations show that in 3-D, the wind kinetic energy is smaller than in 2-D while the opposite is true for the thermal energy (@Kurosawa:2009). Although these changes are small (less than a factor of 2) in one of the cases studied by @Kurosawa:2009, they could be more significant in other cases (i.e., for the luminosity higher and lower than $\Gamma=0.6$ assumed by @Kurosawa:2009).
It is also possible that the AGN feedback may not be as effective as one might have had expected. Instead, other forms of feedback may be more significant than the AGN feedback via radiation on scales between $10^{-2}$ and a few parsecs. They include, the supernova feedback, the radiative feedback from star formation, the strong stellar wind from massive stars, and strong accretion disk winds or jets from AGN. The last two forms will introduce magnetic fields which may carry outward fluxes of energy and momentum. It is also possible that the AGN feedback efficiencies are indeed low and the AGNs take a long time to influence their environment. We note that in our models the AGNs do not shut off the mass supply completely even at very high luminosities. This indicates that the AGNs could operate on a very long time scale over which their impact on the environment can accumulate, and eventually become significant.
Finally, we conclude by noting that AGN feedback has two distinct modes: (1) radiation-driven (quasar) mode and (2) magnetic driven (radio jet) mode. In the current cosmological simulations, it is particularly difficult to deal with the latter, as it requires a full magnetohydrodynamical (MHD) treatment, and we do not have an adequate resolution and more importantly the full understanding of the jet mechanism. Nevertheless, @Sijacki:2007 extended the original implementation of the quasar mode feedback in SDH05 by injecting energy at random positions within a sphere centered around a BH to emulate the hot gas bubbles created by AGN jets. The radio mode could be very important in certain situations, such as the cooling flows in clusters of galaxies where channeling the energy within a very narrow jet helps to transport the energy outside a galaxy. However, this occurs in the low accretion rate regime, and it is subdominant in terms of the total BH mass growth. On the other hand, our simulations do not include the MHD treatment of the collimated jet; hence, the focus of this paper is on the physical mechanism of the quasar mode feedback. Since the quasar mode is the dominant process for the total BH mass growth, our results of low feedback efficiencies would have significant implications on the mass growth of SMBHs in the early universe. In the near future, we plan to further investigate the implications of our results by taking the boundary conditions of our simulations from full cosmological hydrodynamic simulations, and thereby making more direct connections with the physical conditions in a cosmological context.
Authors thank the anonymous referee for constructive comments and suggestions for improving the clarity of the manuscript. We thank J. M. Stone for suggesting to carry out this study. We also thank J. Ostriker, J.-H. Choi, and L. Ciotti for useful discussions. This work was supported by NASA through grant HST-AR-11276 from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. We acknowledge support from NSF (grant AST-0807491) and the National Aeronautics and Space Administration under grant/Cooperative Agreement No. NNX08AE57A issued by the Nevada NASA EPSCoR program.
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[^1]: The feedback efficiency defined by @Ciotti:2009 and @Shin:2009 is equivalent to our $\epsilon_{\mathrm{r}} \times \epsilon_{\mathrm{k}}$ where $\epsilon_{\mathrm{r}}\approx 0.1$ used in our model.
|
---
author:
- |
Jared Duker Lichtman\
\
title: On the Multidimensional Stable Marriage Problem
---
Introduction
============
The stable marriage problem (SM) is a famous problem in mathematics in which there exists a community of $n$ men and $n$ women, all of whom are to be paired with each other heterogeneously in marriage. Each individual provides a complete preference list ranking the members of the opposite sex according to his or her preference for marriage. The final matching sought is one in which no two people would rather be married to each other over their current spouses, and is thus called **stable**. It is of note that stability is a heuristic approach to optimizing matching.
In 1962, David Gale and Lloyd Shapley [@gs] presented an algorithm to solve SM. In the Gale-Shapley (GS) algorithm, each man proposes to his favorite woman, and each woman is temporarily matched to the man that proposes to her. If a woman is proposed to by multiple men, she is temporarily matched to the proposing man she prefers the most. If there remain any unmatched people after the first round of matching, the single men propose again, this time to their second choices. If a woman who has already been matched is proposed to by a man whom she prefers over her current partner, she will leave her current partner and become paired with the new man. This process is repeated, with each unmatched man proposing to the women in order of his preference list and each woman choosing her best possible mate who has proposed to her, until everyone has been matched.
In [@gs], the following properties of the GS algorithm were shown:
- It terminates
- It is $O(n^2)$; maximal number of rounds is $n^2 - 2n + 2$
- Resultant matching is stable
- matching is optimal (of stable) for proposing party
- matching is pessimal (of stable) for responding party
For a thorough introduction to the stable marriage problem, including the Gale-Shapley algorithm, we refer the reader to [@gusfieldirving].
In 1976, Donald Knuth proposed twelve open questions [@knuth] on SM, one of which asked to generalize SM from two to three parties—the 3-dimensional SM (3DSM). Given the open-ended wording, the question has been addressed under various interpretations with respect to structure of preferences and definitions of stability. With the addition of indifference in preferences [@irving], there have been many different constructions of 3DSM, most of which have been shown to be $\mathcal{NP}$-complete [@nghirsch][@huang].
However, 3DSM has been shown to work under a simple scheme [@danilov] in which each individual provides two simple preference lists for the other two parties. Along this vein, this paper will consider the $p$-dimensional SM ($p$DSM), where $p \geq 2$, and propose two types of algorithms to deal with it. Before presenting the algorithms in their entirety, we motivate them with the simplest nontrivial consideration, when $p=3$.
In an instance of 3DSM, we have a community of men, women, and dogs. One possible way to match the men, women, and dogs together is to choose two parties, say men and women, and create a matching between them using the Gale-Shapley algorithm.
Repeating this with another pair of parties, say women and dogs, we arrive at sets of man-woman and woman-dog pairs, from which we can deduce a matching for the entire community. This is the idea motivating the **elemental algorithm**.
Alternatively, we can take the man-woman pairs and view each as members of a compound party, the humans. Each human’s preferences are constructed by combining each (associated) man’s and woman’s preferences for dogs. Similarly, we can modify each dog’s preferences by combining its preference of men and women into preferences for humans. This is the idea motivating the **compound algorithm**.
Here the names elemental and compound are chosen in allusion to chemistry; elements are the fundamental building blocks, and compounds are created by bonding elements together. In the elemental algorithm, each party is treated as an individual element in “pure" form, whereas in the compound algorithm, parties are “bonded" together via stable matchings.
Problem Definition
==================
We proceed with a formal construction of the multidimensional stable marriage problem. An instance of the $p$-dimensional stable marriage problem ($p$DSM) is an ordered pair $(\mathcal{P}, L)$ with a set $\mathcal{P}$ of $p$ disjoint parties with $n$ elements each, and a set $L$ of preference lists for every element (to be defined below). An element of a given party is said to be a **member** of that party. Let $U_\mathcal{P} =\bigcup_{P \in \mathcal{P} } P$ be the **community** of $\mathcal{P}$. For all $x \in U_\mathcal{P}$, let $prt(x)$ return the party of which $x$ is a member.
Associated with each $x \in U_\mathcal{P}$ is a $p-1 \times n$ **strictly ordered preference array** $L_x$ of $x$’s preferences defined as follows. For all $y \in P \in \mathcal{P}$ where $x \notin P$, let $L_x(y)=j$ denote that $y$ is $x$’s $j^{th}$ preferred member of $P$. Let $$L_x(P)=\{ y \in P\: |\: \text{strictly ordered by } L_x(y) \}$$ $$L_x=\{ L_x(P) \: | \: P \in \mathcal{P} \}$$ $$L=\{ L_x \: | \: x \in U_\mathcal{P} \}$$ Furthermore, given $a,b \in P$, let $a \succ_x b$ denote that $L_x(a) < L_x(b)$ and let $a \succeq_x b$ denote that $L_x(a) \leq L_x(b)$.
A **family** $F \subseteq U_\mathcal{P}$ is a set of $p$ elements, one member from each party. A **matching** $\mathcal{F}$ over $\mathcal{P}$ is a partition of $U_\mathcal{P}$ into $n$ families. Elements of a single family are said to be **relatives** in $\mathcal{F}$. Let $rel_\mathcal{F}(x, P)$ return $x$’s relative in $\mathcal{F}$ from $P$.
For a given $\mathcal{F}$, a family $F \notin \mathcal{F}$ is **blocking** if and only if
- $x \succeq_y rel_\mathcal{F}\big( y, prt(x)\big) \; , \; y \succeq_x rel_\mathcal{F}\big( x, prt(y)\big) \quad \text{ for all } x,y \in F$
- for each $x$, there exists $z \in F$ such that $z \succ_x rel_\mathcal{F}\big( x, prt(z)\big)$
A matching $\mathcal{F}$ is **unstable** if there exists a blocking family in $U_\mathcal{P}$. $\mathcal{F}$ is otherwise **stable**.
There are several possible ways to define the problem, with regards to both manner of describing preferences and definition of stability. The manner of preferences was chosen, in part, because of its simplicity. In 3DSM, simple preference lists leads to an efficient algorithm, while more permutation-oriented (Cartesian product) setups have generated complex problems proven to be $\mathcal{NP}$-complete. The definition of stability chosen resembles that of the traditional problem most closely. This is because if there is a blocking family of elements who all prefer each other to their corresponding partners in $\mathcal{F}$, they would all “elope" and desert their established families in real life. Whereas setups that permit indifference among individuals have led to $\mathcal{NP}$-complete problems as well [@irving].
Two types of algorithms will be presented for $p$DSM, elemental and compound, both of which are novel extensions of the original GS algorithm. Given any algorithm $A$, let $\langle A\rangle$ denote the matching generated by $A$. Given a set of algorithms $\mathcal{A}$, let $\langle\mathcal{A}\rangle = \{ \langle A \rangle \: | \: A \in \mathcal{A} \}$.
Elemental Algorithms
====================
A tentative definition of an elemental algorithm will initially be provided, which will prompt a deeper understanding and a more rigorous definition. Let $GS(P, Q)$ denote that $P$ proposes to $Q$ according to the GS algorithm. An important way of viewing $GS(P, Q)$ is that it establishes a bijection between $P$ and $Q$.
An **elemental algorithm** is a set $\upvarepsilon$ of bijections $GS(R, S)$ such that for all $P, Q \in \mathcal{P}$ there exists a unique bijection (either directly, or indirectly by composition) between $P$ and $Q$. To execute $\upvarepsilon$, each element in $\upvarepsilon$ is executed, generating a unique 1-1 correspondence between each pair of parties, and thus a matching for the problem.
In SM, the GS algorithm is executed on the pair of men and women. This may be viewed as a directed graph with a vertex for each party and an edge directed from the proposing into the responding vertex. Unless otherwise specified, all graphs are assumed to be simple and labeled.
In $p$DSM, these graphs may be similarly constructed. Given an elemental algorithm $\upvarepsilon$, an **elemental graph** $G=\{ V,E\}$ is generated according to the following bijections. $$\mathcal{P} \rightarrow V : P \mapsto v_p \qquad \quad \upvarepsilon \rightarrow E : GS(P,Q) \mapsto e_{pq}$$
The following theorem will provide a basis for the formal definition of an elemental algorithm.
$G$ is an elemental graph $\Longleftrightarrow G$ is a tree.
This bidirectional proof will be split into two parts. The first will be proven by contradiction, and the second by direct construction.
**part 1:** Take an elemental graph $G$. Since $e_{pq}$ represents a bijection between $P$ and $Q$, an (undirected) walk $W$ in $G$ establishes a composition of bijections, relating all parties with corresponding vertices in $W$ to each other. Since a bijection is established between all pairs of parties, $G$ must be connected. Further, since each bijection is unique, there can be no cycles in $G$. So $G$ is a tree.
**part 2:** Take a tree $T=\{ V,E\}$. Let $T$ generate a set $\updelta$ of bijections $GS(P, Q)$ according to the following bijections: $$V \rightarrow \mathcal{P} : v_p \mapsto P \qquad \quad E \rightarrow \updelta : e_{pq} \mapsto GS(P,Q)$$ Since $T$ is a tree, there exists a unique walk between $v_p$ and $v_q$, corresponding to a unique bijection between $P$ and $Q$. Thus, by definition, $\updelta$ is an elemental algorithm. Since the maps used to generate $\updelta$ from $T$ were bijective, $\updelta$ may be thought to generate $T$ using the inverse maps. Then, since $\updelta$ is an elemental algorithm, $T$ is an elemental graph.
The formal definition follows from the reasoning in part 2 of Theorem 1. Given a directed tree $T=\{ V,E\}$ with $p$ vertices, an **elemental algorithm** $\upvarepsilon_T$ over $\mathcal{P}$ is a set of bijections $GS(P, Q)$ generated according to the following bijections: $$V \rightarrow \mathcal{P} : v_p \mapsto P \qquad \quad E \rightarrow \upvarepsilon_T : e_{pq} \mapsto GS(P,Q)$$ The execution of $\upvarepsilon_T$ remains unchanged; each element of $\upvarepsilon_T(\mathcal{P})$ is executed, generating a matching $\langle\upvarepsilon_T(\mathcal{P})\rangle$.
It is important to note here that a *directed* tree is specified. The direction of the edge indicates which party proposes, an important fact when executing the algorithm. However, such specification was unnecessary in the proof of Theorem 1 because both are viable elemental algorithms.
Having provided a definition, we now show that the elemental algorithm shares many of the nice properties the GS algorithm has. Let $\mathcal{T}_p$ be the set of all undirected trees with $p$ vertices. Let $\mathcal{E}(P)=\{\upvarepsilon_T(\mathcal{P}) \: |\: T \in \mathcal{T}_p \}$. For ease (given appropriate context), let $\upvarepsilon$, $\mathcal{T}$ and $\mathcal{E}$ be short for $\upvarepsilon_T$, $\mathcal{T}_p$, and $\mathcal{E}(\mathcal{P})$, respectively.
All elemental algorithms terminate.
Take $\upvarepsilon \in\mathcal{E}$. By construction, $\upvarepsilon$ contains $p-1$ instances of the GS algorithm. Since the GS algorithm terminates [@gs] and $p-1$ is finite, $\upvarepsilon$ must terminate.
All elemental algorithms yield stable matchings.
Consider an arbitrary $\upvarepsilon \in \mathcal{E}$. Suppose $\langle\upvarepsilon\rangle$ is unstable; there exists a blocking family $F \notin \langle\upvarepsilon\rangle$. Take $x,y \in F$ where $x \in P, y \in Q$.
**case 1**: $GS(P, Q) \in \upvarepsilon$. Since $GS(P, Q)$ is stable [@gs], we have $x = rel_{\langle\upvarepsilon\rangle}(y, P)$.
**case 2**: $GS(P, Q) \notin \upvarepsilon$. Take the tree $T=\{ V,E \}$ that generated $\upvarepsilon$. There is a unique walk $W$ in $T$ from $v_p$ to $v_q$. By relabeling vertices, without loss of generality, let
$W=u_1 u_2 u_3 \cdots u_w$ for some $1 < w < p$, where $u_1=v_p$ and $u_w=v_q$
Relabeling parties accordingly, we have that
$GS(P_k, P_{k+1}) \in \upvarepsilon$ for $1 \leq k < w$, where $P_1=P$ and $P_w=Q$
We now apply case 1 $w-1$ times on $x_k,x_{k+1}$, where $x_k \in P_k, x_1=x$ and $x_w=y$. This gives $x_k=rel_{\langle\upvarepsilon\rangle}(x_{k+1}, P_k)$. Then by transitivity, $x_1=rel_{\langle\upvarepsilon\rangle}(x_w, P_1)$, or $x=rel_{\langle\upvarepsilon\rangle}(y, P)$.
Combining both cases, we have that $F \in \langle\upvarepsilon\rangle$, which is a contradiction. Therefore $\langle\upvarepsilon\rangle$ is stable.
For 3DSM, this setup was criticized for its simplicity: “It is not hard to see that we can apply the Gale-Shapley algorithm twice to get a weak stable matching: letting the men propose to women and then propose to dogs. Women and dogs make the decision of acceptance or rejection based on their simple lists of men." [@huang] However, previous literature has overlooked the multitude of combinations of matchings available, allowing one to customize the algorithm to fit the given task at hand. The following result computes the exact number of such options.
$|\mathcal{E}|=2^{p-1}p^{p-2}$
Since each $T \in \mathcal{T}$ generates a distinct $\upvarepsilon (T) \in \mathcal{E}$, we have that $|\mathcal{T}|=|\mathcal{E}|$. By Cayley’s Formula, there are $p^{p-2}$ undirected trees with $p$ vertices. Since there are $p-1$ edges to a tree, each undirected tree corresponds to $2^{p-1}$ distinct directed trees. Thus $|\mathcal{E}|=|\mathcal{T}|=2^{p-1}p^{p-2}$.
An elemental algorithm is $O(pn^2)$.
Recall that the GS algorithm is $O(n^2)$ and the maximal number of rounds is $n^2 - 2n + 2$. Since an elemental algorithm applies the GS algorithm $p-1$ times, the maximal number of rounds in an elemental algorithm is $(p-1)(n^2 - 2n + 2)$. Therefore, it is $O(pn^2)$.
Problem Structure
=================
Before defining compound algorithms formally, we need to develop a vocabulary for some of the natural structure of a multidimensional stable marriage instance.
Given a $p$DSM $\mathcal{P}$, a $q$DSM $\mathcal{Q}$ is said to be a **subproblem** of $\mathcal{P}$ if and only if $\mathcal{Q} \subseteq \mathcal{P}$. Given $\mathcal{P}$, a set partition $\pi(\mathcal{P})\neq \mathcal{P}, \{\mathcal{P}\}$ is said to be a **problem partition** containing subproblems of $\mathcal{P}$. Let $\Pi(\mathcal{P})$ be the set of all problem partitions. For ease, let $\pi$, $\Pi$ be short for $\pi(\mathcal{P})$, $\Pi(\mathcal{P})$, respectively.
Given a problem partition $\pi(\mathcal{P})$ containing $1\leq p' < p$ subproblems with a matching $\mathcal{F}$ over each $\mathcal{Q} \in \pi$, then a **reduced problem** $(\mathcal{P'}, L')$ is a $p'$DSM where $\mathcal{P'}=\pi$ and $L'$ is defined as follows. Given matchings $\mathcal{F}, \mathcal{G}$ for subproblems $\mathcal{Q}, \mathcal{R} \in \mathcal{P'}$, respectively, for all families $F \in \mathcal{F}, G \in \mathcal{G}$, let $$L'_F(G)=\sum\limits_{x \in F, \: y \in G} L_x(y)$$ $$L'_F(\mathcal{R})=\{G \in \mathcal{G} \: | \: \text{strictly ordered according to } L'_F(G) \}$$ $$L'_F = \{ L'_F(\mathcal{R}) \: | \: \mathcal{R} \in \mathcal{P'}\}$$ $$L' = \{ L'_F \: | \: F \in U_\mathcal{P'} \}$$ The definition of the reduced problem effectively collapses each subproblem $\mathcal{Q} \in \mathcal{P'}$ into a single party and each family $F \in \mathcal{F}$ into a single individual.
Given a matching $\mathcal{F'}$ over reduced problem $(\mathcal{P'}, L')$, $\mathcal{F'}$ may be **expanded** to $\mathcal{F}$ according to $$\mathcal{F}=\bigcup_{F'\in\mathcal{F'}} F'$$ to give the equivalent matching $\mathcal{F}$ over the original problem $(\mathcal{P}, L)$.
Compound Algorithms
===================
Given a $p$DSM $(\mathcal{P}, L)$, a **compound algorithm** $C$ is executed over $\mathcal{P}$ according to the following recursive procedure. Two counters $i, c$ are both intially set to zero.
1. Take an elemental algorithm $\upvarepsilon \in \mathcal{E}(\mathcal{P})$. If $\Pi(\mathcal{P})\neq\emptyset$, take a problem partition $\pi \in \Pi(\mathcal{P})$. Else, let $\langle C\rangle=\langle\upvarepsilon\rangle$ and go to step 6.
2. Create reduced problem $(\mathcal{P'}, L')$: set $\mathcal{P'}=\pi$; for each subproblem $\mathcal{Q} \in \pi$, take an elemental algorithm $\upvarepsilon\in\mathcal{E}(\mathcal{Q})$ and generate $L'$ using matchings $\langle\upvarepsilon(\mathcal{Q})\rangle$.
3. Index $i$ by one.
4. If $|\pi|>1$, repeat from step 1 letting $(\mathcal{P}, L)=(\mathcal{P'}, L')$. Else, let $c=i$.
5. Given $\langle\upvarepsilon(\mathcal{P'}) \rangle$ used in the $c^{th}$ execution of step 2, construct $\langle C\rangle$ as the result of expanding $\langle\upvarepsilon(\mathcal{P'}) \rangle \: c$ times.
6. Return $\langle C\rangle$.
Having provided a definition, we now show that compound algorithms share many of the nice properties the elemental algorithms have. Let $\mathcal{C}(\mathcal{P})$ be the set of all compound algorithms over $\mathcal{P}$.
All compound algorithms terminate.
The statement will be proven by induction on $|\mathcal{P}|$. Take $C \in \mathcal{C}(\mathcal{P})$.
**Base case:** $|\mathcal{P}|=2$
Let $\mathcal{P}=\{P,Q\}$. Without loss of generality, $\upvarepsilon=GS(P,Q)$. The only possible set partitions are $\mathcal{P}, \: \{\mathcal{P}\}$, neither of which are problem partitions. Thus $\Pi(\mathcal{P})=\emptyset$; for the execution of $C$, we go from step 1 to step 6, return $\langle GS(P,Q)\rangle$, and terminate.
**Inductive step:** Given $|\mathcal{P}|=k+1$ and $C(\mathcal{Q})$ terminates when $|\mathcal{Q}|=i$ for all $2 \leq i \leq k$.
Recall that a problem partition omits the trivial partition $\pi=\{\mathcal{P}\}$. This ensures, for all $\pi \in \Pi(\mathcal{P})$, that $|\mathcal{P}| > |\mathcal{Q}|$ for all $\mathcal{Q} \in \pi$. Thus $k+1 > |\mathcal{Q}|$, meaning that $|\mathcal{Q}|=i$ for some $2 \leq i \leq k$. By the inductive hypothesis, $C(\mathcal{Q})$ terminates. Therefore, so does $C(\mathcal{P})$.
Before showing stability, we need a lemma.
Given $p$DSM $(\mathcal{P}, L)$ reduced to $(\mathcal{P'}, L')$ using stable matchings $\mathcal{F}_\mathcal{Q}$ for all $\mathcal{Q} \in \mathcal{P'}$. If matching $\mathcal{F'}$ over $\mathcal{P'}$ is stable, then the expanded matching $\mathcal{F}$ over $\mathcal{P}$ is stable.
The proof will be given by taking any $\mathcal{Q}, \mathcal{R} \in \mathcal{P'}$ and showing that the expansion of $\mathcal{F'} \cap (\mathcal{Q} \cup \mathcal{R})$, which is $\mathcal{F} \cap (\mathcal{Q} \cup \mathcal{R})$, is stable. For ease of notation within the proof itself, assume that $\mathcal{F'}$ stands for $\mathcal{F'} \cap (\mathcal{Q} \cup \mathcal{R})$, and $\mathcal{F}$ for $\mathcal{F} \cap (\mathcal{Q} \cup \mathcal{R})$.
Take $\mathcal{Q},\mathcal{R} \in \mathcal{P'}$. Let $\mathcal{F}_\mathcal{Q},\mathcal{F}_\mathcal{R}$ be the stable matchings over $\mathcal{Q},\mathcal{R}$, respectively.
Suppose $\mathcal{F}$ is unstable; there exists a blocking family $F \notin \mathcal{F}$. Since $F$ is blocking and $\mathcal{F'}$ is stable, $F$ cannot be an expansion of some $F'\in\mathcal{F'}$. Therefore, since $\mathcal{F}_\mathcal{Q},\mathcal{F}_\mathcal{R}$ are stable, $F=(G \cap \mathcal{Q}) \cup (H \cap \mathcal{R})$, for some $G, H \in \mathcal{F'}$.
For ease of notation, let $$F_1=G \cap \mathcal{Q},\; F_2=G \cap \mathcal{R},\; F_3=H \cap \mathcal{Q},\; F_4=H \cap \mathcal{R}$$ Thus, without loss of generality, $F=F_1 \cup F_4$. By definition of $\mathcal{P'}$, we have that $F_1, F_2, F_3, F_4 \in U_\mathcal{P'}$.
Consider $x \in F_1, y \in F_4$. Since $x,y \in F$ and $F$ is blocking, we have that $x \succeq_y rel_{\mathcal{F}}\big( y, prt(x)\big)$ and $y \succeq_x rel_{\mathcal{F}}\big( x, prt(y)\big)$. This means that $$L_x(y) \leq L_x\Big( rel_{\mathcal{F}}\big( x, prt(y)\big) \Big)$$ $$L_y(x) \leq L_y\Big( rel_{\mathcal{F}}\big( y, prt(x)\big) \Big)$$ Summing over all such $x,y$ pairs, $$\sum_{x \in F_1 ,\: y \in F_4} L_x(y) \leq \sum_{x \in F_1 ,\: y \in F_4} L_x\Big( rel_{\mathcal{F}}\big( x, prt(y)\big) \Big)$$ $$\sum_{x \in F_1 ,\: y \in F_4} L_y(x) \leq \sum_{x \in F_1 ,\: y \in F_4} L_y\Big( rel_{\mathcal{F}}\big( y, prt(x)\big) \Big)$$ Additionally, for each $x \in F_1, y \in F_4$ there exist $z \in F_4, w \in F_1$ such that $z \succ_x rel_{\mathcal{F}}(x, prt(z))$ and $w \succ_y rel_{\mathcal{F}}(y, prt(w))$. This implies that $$\sum_{x \in F_1 ,\: y \in F_4} L_x(y) < \sum_{x \in F_1 ,\: y \in F_4} L_x\Big( rel_{\mathcal{F}}\big( x, prt(y)\big) \Big)$$ $$\sum_{x \in F_1 ,\: y \in F_4} L_y(x) < \sum_{x \in F_1 ,\: y \in F_4} L_y\Big( rel_{\mathcal{F}}\big( y, prt(x)\big) \Big)$$ Then, by definition of $L'$, we have $$L'_{F_1}(F_4) < L'_{F_1}(F_2)$$ $$L'_{F_4}(F_1) < L'_{F_4}(F_3)$$ But this contradicts the fact that $\mathcal{F'}$ is stable. Therefore there can be no blocking family in $\mathcal{F}$, and thus $\mathcal{F}$ is stable.
All compound algorithms are stable.
This statement will be proven by tracing the steps of the compound algorithm and proving that stability is preserved throughout. Take $C \in \mathcal{C}(\mathcal{P})$. Upon executing the algorithm, let $c$ be the final number of times that steps $1-4$ were repeated.
Nothing occurs with regards to stability in steps 1, 3, 4, and 6. In step 2, by Theorem 3 each matching $\langle\upvarepsilon(\mathcal{Q})\rangle$ is stable for all $\mathcal{Q} \in \pi$. In step 5, $\langle\upvarepsilon(\mathcal{P'})\rangle$ is stable by Theorem 3. The initial algorithm conditions and steps 2 and 5 satisfy the givens for Lemma 1, so by applying Lemma 1 $c$ times on $\langle\upvarepsilon(\mathcal{P'})\rangle$, we have that the $c^{th}$ expanded matching of $\langle\upvarepsilon(\mathcal{P'})\rangle$, $\langle C\rangle$, is stable over the original $\mathcal{P}$.
Open Problems
=============
There are several interesting questions we can now ask.
- determine optimal and pessimal matchings for a given party
- determine egalitarian matchings
- determine efficiency of the compound algorithm
- determine additional structure of matchings under elemental and compound algorithms (dependent on chosen directed tree, etc.)
- find applications
[9]{} V. I. Danilov. “Existence of stable matchings in some three-sided systems". *Mathematical Social Science* 46.2 (2003), pp. 145–148.
D. Gale and L. S. Shapley. “College admission and the stability of marriage". *The American Mathematical Monthly* 69.1 (1962), pp.9–15.
D. Gusfield and R. W. Irving. *The stable marriage problem: structure and algorithms.* Cambridge, Massachusetts: MIT Press, 1989.
C. C. Huang. “Two’s company, three’s a crowd: stable family and threesome roomates problems". Technical Report TR2007-598, Computer Science Department, Dartmouth College (2007).
R. W. Irving. “Stable marriage and indifference". *Discrete Applied Mathematics* 48 (1994), pp. 261–272.
D. E. Knuth. *Mariages stables et leurs relations avec d’autre problèmes combinatoires.* Les Presses de l’université de Montréal, 1976.
C. Ng and D. S. Hirschberg. “Three-dimensional Stable Matching Problems". *SIAM Journal on Discrete Mathematics* 4.2 (1991), pp. 245–252.
|
---
abstract: 'We show that the recent results of \[Int. J. Mod. Phys. D 25 (2016) 1650051\] on the application of Lie/Noether symmetries in scalar field cosmology are well-known in the literature while the problem could have been solved easily under a coordinate transformation. That follows from the property, that the admitted group of invariant transformations of dynamical system is independent on the coordinate system.'
address:
- |
Instituto de Ciencias Físicas y Matemáticas,\
Universidad Austral de Chile, Valdivia, Chile\
anpaliath@phys.uoa.gr
- |
Academy of Athens, Research Center for Astronomy and Applied Mathematics,\
Soranou Efesiou 4, 11527, Athens, Greece\
svasil@academyofathens.gr
- |
Faculty of Physics, Department of Astrophysics - Astronomy - Mechanics,\
University of Athens,\
Panepistemiopolis, Athens 157 83, Greece\
mtsampa@phys.uoa.gr
author:
- ANDRONIKOS PALIATHANASIS
- SPYROS BASILAKOS
- MICHAEL TSAMPARLIS
title: 'Comment on A study of phantom scalar field cosmology using Lie and Noether symmetries \[Int. J. Mod. Phys. D 25 (2016) 1650051\]'
---
In [@dutta] the authors consider the action$$S=\int d^{4}x\sqrt{-g}\left[ R-\frac{1}{2}\lambda\left( \phi\right)
g^{\mu\nu}\phi_{;\mu}\phi_{;\nu}-V\left( \phi\right) \right] +S_{m}%
\label{eq.1}%$$ where $S_{m}$, corresponds to the matter source, and $\lambda\left(
\phi\right) $ is an unknown function in a spatially flat Friedmann–Robertson–Walker (FRW) spacetime with signature $\left(
-,+,+,+\right) $, scale factor $a\left( t\right) $, and a perfect fluid with constant equation of state parameter, $p=\left( \gamma-1\right) \rho$. From this action for the lapse time $N\left( t\right) =1$, and for the comoving observers, $u^{a}=\delta_{t}^{a}$, it follows that the Lagrangian of the field equations is$$L\left( a,\dot{a},\phi,\dot{\phi}\right) =-3a\dot{a}^{2}+\frac{1}{2}%
\lambda\left( \phi\right) a^{3}\dot{\phi}^{2}-a^{3}V\left( \phi\right)
-\rho_{m0}a^{3\left( 1-\gamma\right) },\label{eq.2}%$$ while the corresponding field equations are the Hamiltonian function, which is the first Friedmann’s equation $$3a\dot{a}^{2}-\frac{1}{2}\lambda\left( \phi\right) a^{3}\dot{\phi}^{2}%
-a^{3}V\left( \phi\right) -\rho_{m0}a^{3\left( 1-\gamma\right)
}=0,\label{eq.3}%$$ and the Euler-Lagrange equations of Lagrangian (\[eq.2\]) for the variables $\left\{ a,\phi\right\} $. In [@dutta], the authors claim that the existence of Lie point symmetries for the field equations or Noether point symmetries for the Lagrangian (\[eq.2\]) result in constraints which provide the unknown parameters of the model, that is $\gamma$, $V\left( \phi\right)
$ and the function $\lambda\left( \phi\right) $. The purpose of this short note is to show that this is not true and the correct answer is that the only parameters which can be constrained are $\gamma$, and $V\left( \phi\right)
$, while the last will be $V\left( \lambda\left( \phi\right) \right) $, or more precisely, $V\left( \phi\right) =V\left( \int\sqrt{\lambda\left(
\phi\right) }d\phi\right) $.
First we note that the Lie algebra of the Lie/Noether symmetries of a dynamical system are independent on the coordinate system. Therefore if in the Lagrangian (\[eq.2\]) we define the new field $\psi,~$such as $d\psi
=\sqrt{\lambda\left( \phi\right) }d\phi$ we obtain$$L\left( a,\dot{a},\psi,\dot{\psi}\right) =-3a\dot{a}^{2}+\frac{1}{2}%
a^{3}\dot{\psi}^{2}-a^{3}V\left( \psi\right) -\rho_{m0}a^{3\left(
1-\gamma\right) }\label{eq.10}%$$ where $V\left( \psi\right) =V\left( \int\sqrt{\lambda\left( \phi\right)
}d\phi\right) .$ This is the classical Lagrangian of a minimally coupled scalar field cosmological model in a spatially flat FRW spacetime. Furthermore in the case in which $\psi\rightarrow i\psi$, or $\lambda\left( \phi\right)
<0,$ we have the Lagrangian of a phantom field.
Hence the symmetry analysis of (\[eq.2\]) or (\[eq.10\]) will give the same results, however of a different form of the potentials $V\left(
\psi\right) $, $V\left( \phi\right) $, but which they will be related under the transformation $\phi\rightarrow\psi$. Furthermore, if the latter transformation is not complex, the solution for the scale factor will be exactly the same.
Concerning the application of Noether symmetries of the gravitational Lagrangian (\[eq.2\]), the potential $V(\phi)$ in (\[eq.1\]) the authors of [@dutta] find is the Unified Dark Matter potential (UDM) (for instance see [@berta; @gorini; @basill]). For instance, for $\lambda\left(
\phi\right) =\frac{\lambda_{0}}{\phi^{2}},~$in (\[eq.2\]), they have found the potentials $$\begin{aligned}
V\left( \phi\right) & =V_{0}\sinh^{2}\left( \frac{3}{8}\left\vert
\lambda_{0}\right\vert \ln\phi+p_{1}\right) ,~\lambda_{0}>0\\
V\left( \phi\right) & =V_{0}\sin^{2}\left( \frac{3}{8}\left\vert
\lambda_{0}\right\vert \ln\phi+p_{1}\right) ,~\lambda_{0}<0\end{aligned}$$ which under the coordinate transformation $\psi=\ln\phi$, which is the same result when $\lambda\left( \phi\right) =\lambda_{0}$, i.e. a constant, becomes $$\begin{aligned}
V\left( \psi\right) & =V_{0}\sinh^{2}\left( \frac{3}{8}\psi+p_{1}\right)
\\
V\left( \psi\right) & =V_{0}\sin^{2}\left( \frac{3}{8}\psi+p_{1}\right)
.\end{aligned}$$
We would like to remark that the application of the complete Noether’s theorem in scalar field cosmology can be found in [@paper1] and the application of Lie point symmetries in [@paper2], hence the results of [@dutta] are not new in the literature.
Furthermore we wish to draw attention to the paper by Capozziello et al. [@cap96] on Scalar-tensor theories in which an extended discussion on the application of Noether symmetries in cosmology is presented; which includes also the results on Noether symmetries of [@dutta].
Finally we refer the authors of [@dutta] to the original work on symmetries of differential equations of S. Lie [@lie] and its application to the Action Integral which has been done by E. Noether [@noe]. [**[Acknowledgments:]{}**]{} AP acknowledges financial support of FONDECYT postdoctoral grant no. 3160121.
[9]{}
S. Dutta and S. Chakraborty, A study of phantom scalar field cosmology using Lie and Noether symmetries, Int. J. Mod. Phys. D. **25** 1650051 (2016) (DOI: 10.1142/S0218271816500516)
S. Capozziello, E. Piedipalumbo, C. Rubano and P. Scudellaro, Phys. Rev. D. **80** 104030 (2009)
D. Bertacca, S. Matarese and M. Pietroni, Mod. Phys. Lett. A **22** 2893 (2007)
V. Gorini, A. Kamenshchik, U. Moschella, V. Pasquier and A. Starobinsky Phys. Rev. D. **72** 103518 (2005)
S. Basilakos and G. Lukes-Gerakopoulos, Phys. Rev. D. **78** 083509 (2008)
S. Basilakos, M. Tsamparlis and A. Paliathanasis, Phys. Rev. D. **83** 103512 (2011)
M. Tsamparlis and A. Paliathanasis, Class. Quant. Gravit. **29** 015006 (2012)
S. Capozziello, R. De Ritis, C. Rubano and P. Scudellaro, Riv. Nuovo Cim. **19** 1 (1996)
S.M. Lie *Differentialgleichungen* (Chelsea, New York, 1967)
E. Noether, Nachr. v.d. Ges. d. Wiss. zu Gottingen **235**, (1918)
|
---
address: 'ARTEMIS, Université Côte d’Azur, CNRS and Observatoire de la Côte d’Azur, Boulevard de l’Observatoire F-06304 Nice, France'
author:
- 'M. TURCONI, T. HARDER, R. SOULARD, W. CHAIBI'
title: MITIGATION OF PARAMETRIC INSTABILITY
---
Introduction
============
Parametric Instability in Gravitational Wave Detectors
------------------------------------------------------
Gravitational wave detectors based on laser interferometry have reached a sensitivity h = $\Delta$L/L $\approx$ 10$^{-23}$/$\sqrt{\rm{Hz}}$ at about 100Hz which makes detection of sources in 10Hz-10kHz bandwidth such as black holes and neutron stars binaries, possible, opening a new era for astronomy [@Abbott2016; @Abbott2017b; @Abbott2017a; @Abbott2018]. Advanced detectors aLIGO and AdVirgo still haven’t reached their nominal performances and improvements of sensitivity are foreseen in order to push the limits of the observable universe and to increase the detection rate. Moreover, since about ten years, researchers work at the design of a third generation GW ground-based detector: the Cosmic Explorer [@Abbott2017c] and the Einstein Telescope (ET) [@Hild2010]. The ET, a proposed European project, aims at achieving a sensitivity ten times better than the advanced detectors on a broader spectral bandwidth (1-10kHz). A key action for improving the signal to noise ratio is to increase the laser power. The power inside the Fabry-Pérot cavities composing the interferometer arms of the advance detectors is foreseen to approach 1 MW and 3 MW for ET. This high circulating power is needed to decrease the shot noise level which limits the detector sensitivity at high frequency (above 200 Hz). But increasing the optical power also means to deal with parametric instability. PI consists in the amplification of the mirror mechanical modes, initially thermally excited, by radiation pressure. This pressure is exerted by the intracavity laser field composed by the main optical mode and one or several high order modes. The latter are created by scattering of the main mode by the mirror vibrations themselves. PI is analogous to Stimulated Brillouin Scattering (SBS[@Bai; @Valley]), the parametric interaction involved is the conversion of the main laser mode $\omega_0$ into one phonon (acoustic excitation) $\omega_m$ and one photon at lower energy $\omega_s$ = $\omega_0 – \omega_m$. Two conditions need to be met for the instability to occur: all the three modes involved must be resonant in the cavity (for the optical modes) and in the mirror (for the mechanical mode) and there must be a significant spatial overlap between the beat note and the mechanical mode profiles.\
The first one-dimensional analysis of PI in the context of advanced GW laser interferometers detectors was described by Braginsky [@Braginsky2001]. Since then, several theoretical and experimental studies have been carried-out in order to understand the PI impact on the gravitational wave detectors $^{16-24}$. This impact was eventually observed in 2015 at aLIGO and reported by [@Evans2015]: an acoustic mode around 15 kHz becomes unstable preventing the detector functioning for an intracavity power of 50kW, which is much lower than the design power (800 kW). On the other hand, PI hasn’t been observed on AdVirgo detector yet. They are foreseen to appear in the next detection run O4, in two years, when the laser power will be increased and the signal recycling cavity will be installed. Therefore, in order to operate the advanced detectors at their design sensitivity and to realize the next generation of GW detectors, it is necessary to mitigate the PI effects by damping the unstable mirror modes.
PI Mitigation Strategies
------------------------
A wide variety of techniques have been suggested to overcome PI in GW detectors; one can divide them in two categories: passive and active techniques. The passive techniques aim at preventing the instabilities by changing the cavity parameters of the detector so that the PI no longer occurs. One kind of these techniques acts by increasing the loss of the mirror mechanical modes with passive dampers. Ring dampers [@Gras2008] and piezoelectric acoustic mode dampers [@Gras2015] have been proposed. These methods are able to decrease the parametric gain, but their effect depends on the frequency of the mechanical mode and care has to be taken on their design to limit the cost in terms of thermal noise. Among them the acoustic mode dampers seem promising [@Biscans2018] and have been applied on LIGO mirrors. Another passive technique is the thermal tuning. It consists in changing the radius of curvature of one or more optics by means of thermal actuators [@Degallaix2007]. This method has been used during the first observation run of LIGO [@Evans2015]. By tuning the resonant frequency of the high order mode involved in the parametric interaction, the amplification of the mechanical mode has been stopped, the intracavity power could be increased up to 100 kW allowing the first detection of GWs. This technique, though, is not enough to avoid all the unstable modes at the nominal power $\approx$ 1MW; because of the high density of the mechanical unstable modes, a detuning which is effective for one mode will bring other modes into resonance. The active techniques consist in monitoring the onset of the instability and suppress it by a feedback that damps the mirror unstable mode. Miller et al. [@Miller2011] proposed the active damping by means of the electrostatic actuators of LIGO mirrors. Recently Blair et al. [@Blair2017] proved the effectiveness of this technique. The unstable mirror vibration mode at 15kHz was successfully damped by using the electrostatic control system, normally used to provide longitudinal actuation on the mirrors for the interferometer lock. But this technique is limited by the fixed position of the four actuators. Not all the mechanical modes can be efficiently damped.\
Within the Virgo collaboration different groups are doing research on PI: Università di Roma Sapienza is involved in simulations and development of passive dampers; Institut Foton in Rennes is studying the issue of PI early detection. A group in LKB is also involved in the PI simulations. Our team in Artemis laboratory is working on the active mitigation solutions. The main goal of our research program is to build a flexible attenuation system for the PI, based on the radiation pressure applied by an auxiliary laser.
PI Damping System Based on Radiation Pressure
=============================================
Concept
-------
Such attenuator device consists of two cascaded acousto-optic modulators (AOM) used as deflectors (for a 2D scan in X and Y direction). An auxiliary laser beam is placed by the AOMs in phase quadrature on the mirror’s areas where the surface deformation, due to the mechanical mode, is maximal. The auxiliary beam amplitude is modulated at the mechanical mode frequency f$_m$ by the AOM itself or by an intensity modulator, it is injected through the mirror’s back surface and reflected on the high reflectivity surface, with a 90$^o$ phase shift with respect to the mechanical mode oscillations. In such a way, it applies a viscous damping force by radiation pressure, meaning that dissipation increases for that particular mode which eventually is damped. Moreover, the auxiliary laser can be used for detection of the unstable modes. For this, the auxiliary beam is split before the contact with the mirror substrate and it is recombined after the reflection on the high reflectivity coating. The output of the Mach-Zehnder interferometer will allow to locally monitor the mirror deformation amplitude, phase and frequency.\
In the Figure \[fig:princi\], a simplified scheme of the PI-damping device is represented (a). Here the mechanical mode induces a longitudinal displacement dz at mirror center (blue straight line in (b)) which is damped by the force applied by the auxiliary laser (red dashed line). If the mechanical mode shows a more complicated spatial profile like in (c), the laser is continuously moved from one lobe to the other.\
![PI damping system principle. a) An auxiliary laser is reflected on the suspended end-cavity mirror exerting a radiation pressure force where the mechanical mode induces a deformation of the mirror surface, dz, along the cavity axis. The Mach-Zehnder interferometer allows local detection of the mirror displacements. b) The laser amplitude (red dashed line) is modulated at the same frequency, in phase quadrature to damp the mechanical mode (blue line). This plot is for illustrative purposes only. c) Spatial profile of a higher order mechanical mode, the red dots represent the areas where the radiation pressure of the auxiliary beam must be applied.[]{data-label="fig:princi"}](scheme0.png){width="0.8\linewidth"}
Compared to the active electrostatic damping system, this attenuation device can be more effective since the overlap between the applied force and the spatial shape of the unstable modes can be finely adjusted by the X-Y deflectors. It is also more flexible: the optical power, frequency and phase used for the actuation can be controlled by the AOMs themselves or by an additional intensity modulator.
First Results
-------------
Since November 2018, our team has started the experimental investigation of AOM deflectors in order to define the best characteristics needed for the PI damping purpose. The auxiliary laser should be moved fast in order to hit the mirror on different areas during one period of the mirror mechanical mode. This means that the RF driving frequency of the AOM must change value rapidly. We estimated a scan frequency at 10 MHz is enough for damping 1kHz-100kHz range modes. Moreover the full scan angle should be wide so that a 35 cm wide mirror like the GW detector’s ones can be covered.\
We have begun by studying the performances of an AOM already available in our laboratory: the model MT110-A1.5-1064 from AA Opto Electronic. Its central driving frequency is 110 MHz, this determines the central position around which the diffracted beam will be moved. It is not trivial to generate radio frequency (RF) signals (for this AOM at 110 MHz) whose frequency is modulated at a rate of 10 MHz or more. In order to do this we exploited the beating of two optical signals as explained below.\
The experimental setup used for the AOM characterization is depicted in the left panel of Figure \[fig:setup\].The laser source is a fiber laser with a wavelength of 1064nm amplified to 100mW. The laser is split in three beams, one that goes to the AOM under test, the beam spot size is 300 $\mu$m at the AOM input, this is the smaller accepted beam-size according to the AOM datasheet. The smaller the spot size, the larger is the AOM response bandwidth. The other two beams are used for a Mach-Zhender interferometer. In one arm the laser beam is frequency shifted by 110 MHz by an AOM and in the other arm the phase of the beam is modulated at the frequency f$_m$, which is set from 1 to 10 MHz. The amplitude of the phase modulation $\phi_m$ is set by adjusting the amplitude of the RF generator. The two arms overlap on a Photodiode which detects the beating signal whose frequency varies sinusoidally from 110 MHz - f$_m$ to 110 MHz + f$_m$.
![Left: schematic drawing of the experimental setup. The yellow area is the Mach-Zhender interferometer used for the generation of the RF signal which drives the AOM deflector MT110-A1.5-1064. A Photodetector (PD) on a translation stage is used to monitor the beam displacement. BS: beam Splitter; PM: Phase Modulator. RF Gen: Radio Frequency Generator at frequency f$_m$; RF Amp: Radio Frequency Amplifier; +1: AOM diffraction order.\
Right: Beam displacement as function of time for three values of f$_m$: 1, 3, 6 MHz.[]{data-label="fig:setup"}](exp_setup_0.png){width="\linewidth"}
![Left: schematic drawing of the experimental setup. The yellow area is the Mach-Zhender interferometer used for the generation of the RF signal which drives the AOM deflector MT110-A1.5-1064. A Photodetector (PD) on a translation stage is used to monitor the beam displacement. BS: beam Splitter; PM: Phase Modulator. RF Gen: Radio Frequency Generator at frequency f$_m$; RF Amp: Radio Frequency Amplifier; +1: AOM diffraction order.\
Right: Beam displacement as function of time for three values of f$_m$: 1, 3, 6 MHz.[]{data-label="fig:setup"}](fig.png){width="\linewidth"}
The photodiode signal is used to drive the tested AOM. We detect the deflected beam with a 1mm wide photodiode on a translation stage, recording the signal every 500 $\mu$m on the transverse plane we are able to track the beam displacement (see right panel of Figure \[fig:setup\]). The total scan angle is of 15 mrad, beyond this range, the diffraction efficiency of our AOM drops significantly (not shown). This kind of AOM is thus not suitable for our application.
Perspectives
------------
The reported results have been used to define the best AOM specifications for the PI attenuation device. Our choice is a trade-off between a big scan-angle and rapidity of the AOM response.The chosen AOM, model 4225-2 from Gooch&Housego has a scan-angle of 35 mrad for a rather flat efficiency curve and a response time of 33ns. Since the optical power needed for PI attenuation is unknown, the chosen AOM has a high damage threshold allowing up to few tens of watts of injected power. The other key element of the PI damping system based on radiation pressure is the RF generator for driving the AOMs. The Mach-Zhender interferometer shown in this report will be replaced by a software defined radio reconfigurable device from National Instruments which is able to generate a signal in the range 1MHz-6GHz and change frequency within 5 ns. The AOMs will be controlled by a computer program interfaced with the RF device.\
In the short-term, the auxiliary laser position control will be assembled, including a power stabilization loop in order to overcome the angle dependency of the AOMs efficiency.
Conclusions
===========
PI is a harmful phenomenon for GW detectors. We have reported on the research program of Artemis laboratory on PI mitigation. The goal is to develop a flexible device for PI mitigation based on radiation pressure. The device is under development, it will be characterized and then tested. Exploitable results are waited before starting of O4.
Acknowledgments {#acknowledgments .unnumbered}
===============
The research program on PI mitigation based on radiation pressure has received the financial support of Labex First-TF, Fédération Doblin, Observatoire de la Côte d’Azur, Univérsité de Nice-Sophia Antipolis. We acknowledge European Gravitational Observatory for a three year PhD grant on this topic.
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|
---
abstract: 'In this paper, we consider a multi-agent consensus problem with an active leader and variable interconnection topology. The state of the considered leader not only keeps changing but also may not be measured. To track such a leader, a neighbor-based local controller together with a neighbor-based state-estimation rule is given for each autonomous agent. Then we prove that, with the proposed control scheme, each agent can follow the leader if the (acceleration) input of the active leader is known, and the tracking error is estimated if the input of the leader is unknown.'
address:
- |
Key Laboratory of Systems and Control, Institute of Systems Science\
Chinese Academy of Sciences, Beijing 100080, China
- 'Institute of Systems Science, Wenzhou University, Zhejiang, China'
author:
- Yiguang Hong
- Jiangping Hu
- Linxin Gao
title: 'Tracking Control for Multi-Agent Consensus with an Active Leader and Variable Topology'
---
,
,
Multi-agent systems, consensus, state estimation, active leader.
Introduction
============
In recent years, there has been an increasing research interest in the control design of multi-agent systems. Many results have been obtained with local rules applied to each agent in a considered multi-agent system. These neighbor rules for each agent are based on the average of its own information and that of its neighbors or its leader (Fax & Murray, 2004; Jadbabaie, Lin, & Morse, 2003; Lin, Broucke, & Francis, 2004; Olfati-Saber & Murray, 2004; Savkin, 2004). For example, Jadbabaie [*et al.*]{} (2003) demonstrated that a simple neighbor rule makes all agents eventually move in the same direction despite the absence of centralized coordination and each agent’s set of neighbors changing with time as the system evolves under a joint connection condition. Also, with a similar technique, Lin [*et al.*]{} (2004) studied three formation strategies for groups of mobile autonomous agents. The stability analysis of multi-vehicle formations was given with a Nyquist-type criterion in (Fax & Murray, 2004). Moreover, by a Lyapunov-based approach, Olfati-Saber [*et al.*]{} (2004) solved the average-consensus problem with directed interconnection graphs or time-delays.
In reality, some variables of the agents and/or the leader in a multi-agent system may not be able to be measured. Fax [*et al.*]{} (2004) raised this important issue regarding observer design for multi-agent systems, and first tackled this problem. However, many works remain to be done for the distributed observer design of networks of multiple agents.
With this background, we consider a consensus problem with an active leader with an underlying dynamics. Here, some variables (that is, the velocity and maybe the acceleration) of an active leader cannot be measured, and each agent only gets the measured information (that is, the position) of the leader once there is a connection between them. In this paper, we propose an “observer" by inserting an integrator into the loop for each agent to estimate the leader’s velocity. To analyze the problem, a Lyapunov-based approach is developed. With the proposed estimation rule and a selected Lyapunov function, the leader-following problem can be solved if the leader’s input is known, while the tracking error can also be analyzed if the input is unknown.
Problem Formulation
===================
To solve coordination problems, graph theory is helpful. An undirected graph $\mathcal{G}$ on vertex set $\mathcal{V}=\{1,2,\cdots,n\}$ contains $\mathcal{V}$ and a set of unordered pairs $\mathcal{E}=\{(i,j):i,j\in \mathcal{V}\}$, which are called $\mathcal{G}$’s edges. If there is an edge between two vertices, the two vertices are called adjacent. A graph is simple if it has no self-loops or repeated edges. If there is a path between any two vertices of a graph $\mathcal{G}$, then $\mathcal{G}$ is connected, otherwise disconnected. A subgraph ${\mathcal X}$ of $\mathcal{G}$ is an induced subgraph if two vertices of ${\mathcal V(X)}$ are adjacent in ${\mathcal X}$ if and only if they are adjacent in $\mathcal{G}$. An induced subgraph ${\mathcal X}$ of $\mathcal{G}$ that is maximal, subject to being connected, is called a component of $\mathcal{G}$.
Here we consider a system consisting of $n$ agents and a leader. In the sequel, the state of agent $i$ is denoted by $x_i$ for $i=1,...,n$. With regarding the $n$ agents as the vertices in $\mathcal{V}$, the relationships between $n$ agents can be conveniently described by a simple and undirected graph ${\mathcal
G}$, which is defined so that $(i,j)$ defines one of the graph’s edges in case agents $i$ and $j$ are neighbors. $N_{i}(t)$ denotes the set of labels of those agents which are neighbors of agent $i\; (i=1,...,n)$ at time $t$. The weighted adjacency matrix of ${\mathcal G}$ is denoted by $A=[a_{ij}]\in R^{n\times
n}$, where $a_{ii}=0$ and $a_{ij}=a_{ji}\geq 0$ ($a_{ij}>0$ if there is an edge between agent $i$ and agent $j$). Its degree matrix $D=diag\{ d_1,...,d_n\}\in R^{n\times n}$ is a diagonal matrix, where diagonal elements $d_i=\sum_{j=1}^n a_{ij}$ for $i=1,...,n$. Then the Laplacian of the weighted graph is defined as $$L=D-A,$$ which is symmetric. In what follows, we mainly concern a graph $\bar{\mathcal G}$ associated with the system consisting of $n$ agents and one leader. In fact, $\bar{\mathcal G}$ contains $n$ agents (related to graph ${\mathcal G}$) and the leader with directed edges from some agents to the leader. By “the graph, $\bar{\mathcal G}$, of this system is connected", we mean that at least one agent in each component of ${\mathcal G}$ is connected to the leader.
For the multi-agent system under consideration, the relationships between neighbors (and the interconnection topology) change over time. Suppose that there is an infinite sequence of bounded, non-overlapping, contiguous time-intervals $[t_{i},t_{i+1}), \;
i=0,1,\cdots$, starting at $t_{0}=0$.
Denote ${\mathcal{S}}=\{\bar {\mathcal G}_1, \bar {\mathcal
G}_2,\cdots,\bar {\mathcal G}_N\}$ as a set of the graphs with all possible topologies, which includes all possible interconnection graphs (involving $n$ agents and a leader), and denote $\mathcal{P}=\{1,2,\cdots, N\}$ as its index set. To describe the variable interconnection topology, we define a switching signal $\sigma: [0,\infty)\rightarrow \mathcal{P}$, which is piecewise-constant. Therefore, $N_i$ and the connection weight $a_{ij}\; (i=1,...,n, j=1,...,n)$ are time-varying, and moreover, Laplacian $L_{p}\; (p\in {\mathcal P})$ associated with the switching interconnection graph is also time-varying (switched at $t_i,\; i=0,1,\cdots$), though it is a time-invariant matrix in any interval $[t_{i},t_{i+1})$. In our problem, we assume that there are fixed positive constants $\alpha_{ij}\; (i=1,...,n;
j=1,...,n)$ such that $$\label{alpha} a_{ij}(t)=\begin{cases} \alpha_{ij}=\alpha_{ji},&\mbox{if agents $i$ and $j$}\\
&\mbox{\qquad are connected at $t$} \\
0,& \mbox{otherwise}\end{cases}$$ Meanwhile, the connection weight between agent $i$ and the leader, denoted by $b_i$, is time-varying, too. We assume that there are fixed positive constants $\beta_i\; (i=1,...,n)$ such that $$\label{beta} b_i(t)=\begin{cases} \beta_i &\mbox{if agent $i$ is
connected to the leader at $t$} \\ 0 & \mbox{otherwise}\end{cases}$$
The next lemma was given in Horn and Johnson (1985), to check the positive definiteness of a matrix.
\[lem1\] Suppose that a symmetric matrix is partitioned as $$E=\begin{pmatrix}E_1&E_2 \\ E_2^{T}&E_3 \end{pmatrix}$$ where $E_1$ and $E_3$ are square. $E$ is positive definite if and only if both $E_1$ and $E_3- E_2^{T}E_1^{-1}E_2$ are positive definite.
The following result is well-known in algebraic graph theory (Godsil & Royle, 2001) and establishes a direct relationship between the graph connectivity and its Lapalcian.
\[lem2\] Let $\mathcal{G}$ be a graph on $n$ vertices with Laplacian $L$. Denote the eigenvalues of $L$ by $\lambda_{1}(L),\cdots, \lambda_{n}(L)$ satisfying $\lambda_{1}(L)\leq\cdots\leq\lambda_{n}(L)$. Then $\lambda_{1}(L)=0$ and $\textbf{1}=[1,1,\cdots,1]^{T}\in R^n$ is its eigenvector. Moreover, if ${\mathcal G}$ is connected, $\lambda_2>0$.
In this paper, all the considered agents move in a plane: $$\label{modela} \dot x_i=u_i\in R^2,\quad i=1,...,n,$$ where $u_i$ is the control input. The leader of this considered multi-agent system is active; that is, its state variables keep changing. Its underlying dynamics can be expressed as follows: $$\label{model0} \begin{cases}\dot{x}_0=v_0\\ \dot v_0=a(t)=
a_0(t)+\delta(t)\\ y=x_0\end{cases}\quad x_0,\, v_0,\, \delta\in
R^2$$ where $y(t)=x_0(t)$ is the measured output and $a(t)$ is the (acceleration) input. Note that (\[model0\]) is completely different from the agent dynamics (\[modela\]). In other words, the agents will track a leader with a different dynamics.
In our problem formulation, the input $a(t)$ may not be completely known. We assume that $a_0(t)$ is known and $\delta(t)$ is unknown but bounded with a given upper bound $\bar\delta$ (that is, $||\delta(t)||\leq \bar \delta$). The input $a(t)$ is known if and only if $\bar\delta=0$. On the other hand, $y=x_0$ is the only variable that can be obtained directly by the agents when they are connected to the leader. Our aim here is to propose a decentralized control scheme for each agent to follow the leader (i.e., $x_i \to x_0$).
Since $v_0(t)$ cannot be measured even when the agents are connected to the leader, its value cannot be used in the control design. Instead, we have to estimate $v_0$ during the evolution. Note that, each agent has to estimate $v_0$ only by the information obtained from its neighbors in a decentralized way. The estimate of $v_0(t)$ by agent $i$ is denoted by $v_i(t)$ ($i=1,...,n$). Therefore, for each agent, the local control scheme consists of two parts:
- a neighbor-based feedback law: $$\label{con} \begin{split} u_i=&-k[\sum_{j\in N_i(t)}a_{ij}(t)(x_i-
x_j)+b_i(t)(x_i-x_0)]\\
&+v_i,\quad k>0, \; i=1,\cdots,n,
\end{split}$$ where $N_i$ is the set consisting of agent $i$’s neighbor agents;
- a dynamic neighbor-based system to estimate $v_0$ $$\label{update1}
\begin{split}
\dot v_i=&a_0-\gamma k[\sum_{j\in
N_i(t)}a_{ij}(t)(x_i- x_j)+b_i(t)\cdot\\
&(x_i-x_0)],\quad i=1,\cdots,n,
\end{split}$$ for some positive constant $\gamma<1$. In fact, (\[update1\]), can be viewed as an “observer" in some sense.
Note that $u_i$ in (\[con\]) is a local controller of agent $i$, which only depends on the information from its neighbors, and, in fact, when $v_0=0, a=0$, the proposed control law (\[con\]) is consistent with the one given in Olfati-Saber and Murray (2004). In addition, with the neighbor-based estimation rule in a form of observer (\[update1\]) to estimate the leader’s velocity, each agent relies only on the locally available information at every moment. In other words, each agent cannot “observe" or “estimate" the leader directly based on the measured information of the leader if it is not connected to the leader. In fact, it has to collect the information of the leader in a distributed way from its neighbor agents.
Take $$x=\begin{pmatrix} x_1\\ \vdots \\ x_n\end{pmatrix}, \quad v=\begin{pmatrix} v_1\\
\vdots \\ v_n\end{pmatrix},\quad u=\begin{pmatrix} u_1\\
\vdots \\ u_n\end{pmatrix}.$$ Regarding the switching interconnection graphs, the closed-loop system can be expressed as: $$\label{model1}
\begin{cases}
\dot{x}=u=-k(L_{\sigma}+B_{\sigma})\otimes I_2 x+k
B_{\sigma}\textbf{1}\otimes x_{0}+v
\\
\dot v =\textbf{1}\otimes a_{0}-\gamma k(L_\sigma+B_\sigma)\otimes
I_2 x+\gamma k (B_{\sigma}\textbf{1})\otimes x_{0}
\end{cases}$$ where $I_l\in R^{l\times l}$ (for any positive integer $l$) is the identity matrix and $\otimes$ denotes the Kronecker product, $\sigma: [0,\infty)\to \mathcal{P}=\{1,2,\cdots,N\}$ is a piecewise constant switching signal with successive switching times, $B_{\sigma}$ is an $n\times n$ diagonal matrix whose $i$th diagonal element is $b_i(t)$ at time $t$, $L_{\sigma}$ is the Laplacian for the $n$ agents. Note that, even in the case when the interconnection graph is connected, $b_i(t)$ may be always 0 for some $i$, and therefore, $B_{\sigma}$ may not be of full rank.
Denote $\bar x=x-\textbf{1}\otimes x_{0}$ and $\bar v=v-
\textbf{1}\otimes v_{0}$. Because $-k (L_\sigma+B_\sigma)\otimes
I_2 x+k B_{\sigma}\textbf{1}\otimes x_{0} =-k
(L_\sigma+B_\sigma)\otimes I_2\bar x$ (invoking Lemma \[lem2\]), we can obtain an error dynamics of (\[model1\]) as follows: $$\label{model2} \dot{\epsilon}=F_{\sigma}\epsilon+g,\quad
g=\begin{pmatrix} 0\\ -{\bf 1}\otimes \delta
\end{pmatrix}$$ where $$\epsilon=\left(\begin{array}{c} \bar x\\
\bar v\end{array}\right),\quad F_{\sigma}
=\begin{pmatrix} -k(L_\sigma+B_\sigma)&I_{n}\\
-\gamma k(L_\sigma+B_\sigma)&0
\end{pmatrix}\otimes I_2.$$
Main Results
============
In this section, we investigate the consensus problem of multi-agent system (\[model1\]), or the convergence analysis of system (\[model2\]). If the information of the input $a(t)$ can be used in local control design, we can prove that all the agents can follow the leader, though the leader keeps changing. If not, we can also get some estimation of the tracking error. We first assume that the interconnection graph $\bar{\mathcal G}$ is always connected, though the interconnection topology keeps changing; and then we consider an extended case.
As mentioned above, $\bar{\mathcal G}$ is connected if at least one agent in each of its component is connected with the leader. To be specific, if there are $m\geq 1$ components, then the Laplacian $L_{p}$ (for any $p\in {\mathcal P}$) of the graph associated with $n$ agents have $m$ zero eigenvalues. For simplicity, we can rearrange the indices of $n$ agents such that $L_{p}$ can be rewritten as a block diagonal matrix: $$L_{p}=\left(
\begin{array}{cccc}
L_p^{1}&&&\\
&L_p^{2}&&\\
&&\ddots&\\
&&&L_p^{m}
\end{array}
\right)$$ where each block matrix $L_p^i$ is also a Laplacian of the corresponding component. For convenience, denote $M_{p}=L_{p}+B_{p}$, where $L_p$ is the weighted Laplacian and $B_p\; (p\in {\mathcal P})$ is the diagonal matrix as defined in Section 2. The next lemma is given for $M_p$.
\[lem3\] If graph $\bar {\mathcal G}_p$ is connected, then the symmetric matrix $M_p$ associated with $\bar {\mathcal G}_p$ is positive definite.
Proof: We only need to prove the case when $m=1$. Let $\lambda_1,\cdots,\lambda_{n}$ be the eigenvalues of Laplacian $L_p$ in the increasing order. From Lemma \[lem2\], $\lambda_{1}=0$ and $\lambda_{i}>0,i\geq 2$. Denote $n$ eigenvectors of $L_p$ by $\zeta_{i},\; i=1,...,n$, with $\zeta_1=\textbf{1}$, an eigenvector of $L_p$ corresponding to $\lambda_1=0$. Then any nonzero vector $z\in R^{n}$ can be expressed by $z=\sum_{i=1}^{n}c_i\zeta_{i}$ for some constants $c_i,i=1,2,\cdots,n$. Moreover, $B_p\neq 0$ since there is at least one agent connected to the leader. Without loss of generality, we assume $b_j> 0$ for some $j$, and it is obvious $\zeta_1^TB_p\zeta_1\geq b_j$. Therefore, in either the case when $c_2=...=c_n=0$ (so $c_1\neq 0$) or the case when $c_i\neq 0$ for some $i\geq 2$, we always have $$\label{esti1} z^{T}M_pz=z^{T}L_pz+z^{T}B_pz \geq
\sum_{i=2}^{n}\lambda_{i}c_{i}^{2}\zeta_i^T\zeta_i +z^{T}B_pz>0$$ for $z\neq 0,$ which implies the conclusion.
------------------------------------------------------------------------
Based on Lemma \[lem3\] and the fact that the set ${\mathcal P}$ is finite, $$\label{barlamb} \begin{split} \bar
\lambda=&\min\{\mbox{eigenvalues of}\; M_p\in R^{n\times n},\;
\forall \mbox{$\bar {\mathcal G}_p$
is} \\
& \mbox{connected}\} >0,
\end{split}$$ is fixed and depends directly on the constants $\alpha_{ij}$ and $\beta_i$ for $i=1,...,n,\; j=1,...,n$ given in (\[alpha\]) and (\[beta\]). Its estimation is also related to the minimum nonzero eigenvalue of Laplacian $L_p$, which has been widely studied in different situations (Merris, 1994).
In some existing works, including Jadbabaie, Lin, & Morse (2003) and Lin, Broucke, & Francis (2004), the convergence analysis depends on theory of nonnegative matrices or stochastic matrices. However, $F_{p}$ of system (\[model2\]) fails to be transformed easily to a matrix with some properties related to stochastic matrices, and therefore, the effective methods used in Jadbabaie, Lin, & Morse (2003) or Lin, Broucke, & Francis (2004) may not work. Here, we propose a Lyapunov-based approach to deal with the problem.
\[thm1\] For any fixed $0<\gamma<1$ and $\bar\lambda$ defined in (\[barlamb\]), we take a constant $$\label{constk} k> \frac{1}{4\gamma(1-\gamma^{2})\bar{\lambda}}.$$ If the switching interconnection graph keeps connected, then $$\label{result12} \lim_{t\to \infty}||\epsilon(t)||\leq C,$$ for some constant $C$ depending on $\bar\delta$. Moreover, if $a(t)$ is known (i.e., $a(t)=a_0(t)$ or $\bar\delta= 0$), $$\label{result11} \lim_{t\rightarrow \infty}\epsilon(t)=0.$$
Proof: Take a Lyapunov function $V(\epsilon)=\epsilon^{T}(t)P\epsilon(t)$ with symmetric positive definite matrix $$\label{matrixp} P= \left(\begin{array}{cc}I_n&-\gamma I_n\\-\gamma
I_n&I_n\end{array}\right)\otimes I_2.$$
The interconnection graph is time-varying, but the interconnection graph associated with $F_p$ for some $p\in \mathcal{P}$ is connected on an interval $[t_i, t_{i+1})$ with its topology unchanged. Consider the derivative of $V(\epsilon)$: $$\label{dotv}
\begin{split}
\dot{V}(\epsilon)|_{(\ref{model2})}&=\epsilon^{T}(F_{p}^{T}P+PF_{p})\epsilon+
2\epsilon^{T}F_pg \\
&\leq -\epsilon^{T}Q_{p}\epsilon+2(1+\gamma)\bar\delta
||\epsilon||
\end{split}$$ where $$\label{matrixq} Q_{p}=-(F_{p}^{T}P+PF_{p})=\begin{pmatrix}
2k(1-\gamma^{2}) M_{p}&-I_n\\
-I_n&2\gamma I_n
\end{pmatrix}\otimes I_2$$ is a positive definite matrix because $2\gamma
I-\frac{1}{2k(1-\gamma^2)}M_{p}^{-1}$ and $M_p$ are positive definite (by virtue of (\[constk\]), Lemma \[lem1\] and Lemma \[lem3\]).
Let $\mu_{i,j},i=1,\cdots,n, j=1,2$ denote the (at most) $2n$ different eigenvalues of $Q_p$ though $Q_p\in R^{4n\times 4n}$ defined in (\[matrixq\]). Based on $\lambda_i(M_p)$, the eigenvalues of $M_{p}$, we have the $2n$ eigenvalues in the following forms: $$\begin{split}
&\mu_{i,1}=(1-\gamma^{2})k\lambda_i(M_p)+\gamma\\
&+\sqrt{[(1-\gamma^{2})k\lambda_i(M_p)+\gamma]^{2}-4\gamma(1-\gamma^{2})k\lambda_i(M_p)+1},
\end{split}$$ $$\begin{split}
&\mu_{i,2}=(1-\gamma^{2})k\lambda_i(M_p)+\gamma\\
&-\sqrt{[(1-\gamma^{2})k\lambda_i(M_p)+\gamma]^{2}-4\gamma(1-\gamma^{2})k\lambda_i(M_p)+1},
\end{split}$$ for $i=1,...,n$. Clearly, the smallest eigenvalue of $Q_p$ will be found in the form of $\mu_{i,2}$ for some $i$.
Note that (\[constk\]) implies $k\lambda_i(M_p)>
\frac{1}{4\gamma(1-\gamma^{2})}$. In this case, $\mu_{i,2}$ increases as $k\lambda_i(M_p)$ increases. Therefore, the minimum eigenvalue of $Q_p$ will be no less than $$\label{barmu} \bar \mu=(1-\gamma^{2})k\bar \lambda+\gamma-
\sqrt{[(1-\gamma^{2})k\bar\lambda-\gamma]^{2}+1}>0,$$ which is obtained by taking $\lambda_i(M_p)=\bar \lambda$ with a given $k$ satisfying (\[constk\]). In addition, since the eigenvalues of $P$ are either $\mu_{min}=1-\gamma$ or $\mu_{max}=1+\gamma$, we have $$\label{relation} (1-\gamma)\|\e\|^2\leq V(\e) \leq
(1+\gamma)||\e||^2.$$ Therefore, $$\min\frac{\epsilon^TQ_p\epsilon}{\epsilon^TP\epsilon}\geq
\frac{\bar\mu}{\mu_{max}} =2\beta,$$ where $\beta=\frac{\bar \mu}{2(1+\gamma)}>0$ with $\bar\mu$ defined in (\[barmu\]).
Due to (\[relation\]), $$||\epsilon|| \leq \frac{1}{\sqrt{1-\gamma}}\sqrt{V(\epsilon)}.$$ Therefore, from (\[dotv\]), $$\begin{split} \dot V(\epsilon)|_{(\ref{model2})}&\leq -2\beta
V(\epsilon)+2\sqrt{\frac{(1+\gamma)^2V(\e)}{1-\gamma}}\bar\delta\\
&\leq -\beta
V(\epsilon)+\frac{(1+\gamma)^2\bar{\delta}^2}{(1-\gamma)\beta}
\end{split}$$ or equivalently, $$\begin{split}
V(\epsilon(t))\leq
V(\epsilon(t_i))e^{-\beta(t-t_i)}+\frac{(1+\gamma)^2\bar\delta^2}{(1-\gamma)\beta^2}
&(1-e^{-\beta(t-t_i)}),\\
&t\in [t_{i}, t_{i+1}).
\end{split}$$ Thus, with $t_0=0$, $$\label{expf} V(\epsilon(t))\leq V(\epsilon(0))e^{-\beta
t}+\frac{(1+\gamma)^2\bar\delta^2}{(1-\gamma)\beta^2}(1-e^{-\beta
t}),$$ which implies (\[result12\]) with taking $C=\frac{1+\gamma}{(1-\gamma)\beta}\bar\delta$.
Furthermore, if $\bar\delta=0$, then (\[result11\]) is obtained.
------------------------------------------------------------------------
Next, we consider an extended case: the interconnection graph is not always connected. Let $T>0$ be a (sufficient large) constant, and then we have a sequence of interval $[T_j,T_{j+1}),\;
j=0,1,\cdots$ with $T_0=t_0,T_{j+1}=T_j+T$. Each interval $[T_j,T_{j+1})$ consists of a number of intervals (still expressed in the form of $[t_i,t_{i+1})$, during which the interconnection graph is time-invariant), including the intervals during which the graphs are connected and those during which the graphs are not. We assume that there is a constant $\tau>0$, often called dwell time, with $t_{i+1}-t_i\geq \tau, \;\forall i$.
Denote the total length of the intervals associated with the connected graphs as $T_j^c$ in $[T_j,T_{j+1})$ and the total length of the intervals with the unconnected graphs as $T_j^d$ in $[T_j,T_{j+1})$. In what follows, we denote an upper bound of $T_{j}^{d}\; (j=0,1,\cdots)$ as $T^d(<T)$, and a lower bound of $T_j^c\; (j=0,1,\cdots)$ as $T^c(=T-T^d)$.
\[thm2\] During each time interval $[T_{j},T_{j+1})$, if the total period that the interconnection graph is connected (i.e., $T^c$) is sufficient large, then (\[result12\]) still holds with $k$ given in (\[constk\]). Moreover, (\[result11\]) holds if $\bar\delta= 0$ (or equivalently $a(t)=a_0(t)$).
Proof: Still take a Lyapunov function $V(\epsilon)=\epsilon^{T}P\epsilon$ with $P$ defined in (\[matrixp\]), and then we have (\[dotv\]). If the graph associated with $F_p$ for some $p\in \mathcal{P}$ is connected during $[t_i,t_{i+1})$, then, according to Theorem \[thm1\], we have $$V(\epsilon(t_{i+1}))\leq e^{-\beta(t_{i+1}-t_i)}
V(\epsilon(t_{i}))+ \frac{(1+\gamma)^{2}}{(1-\gamma)\beta^{2}}
\bar\delta^{2}.$$
If the graph associated with $F_q$ for some $q\in \mathcal{P}$ is not connected during $[t_l,t_{l+1})$. The minimum eigenvalue of $Q_q$ is $\gamma-\sqrt{1+\gamma^{2}}(<0)$ and, by (\[relation\]), we have $$-\epsilon^{T} Q_q \epsilon \leq
(\sqrt{1+\gamma^2}-\gamma)\epsilon^{T} \epsilon \leq
\frac{\alpha}{2} V(\epsilon)$$ where $\alpha=\frac{2\sqrt{1+\gamma^2}-2\gamma}{1-\gamma}$.
Similarly, with (\[relation\]), $$\dot V(\epsilon(t))|_{(\ref{model2})}\leq \alpha V(\epsilon(t))+
\frac{2(1+\gamma)^{2}}{\alpha(1-\gamma)}\bar\delta^{2},\quad t\in
[t_l,t_{l+1}),$$ and therefore, $$V(\epsilon(t_{l+1}))\leq e^{\alpha (t_{l+1}-t_l)}
V(\epsilon(t_{l}))+
\frac{2(1+\gamma)^{2}}{\alpha^{2}(1-\gamma)}(e^{\alpha
T^d}-1)\bar\delta^{2}.$$
Denote $\eta=\max\{\frac{(1+\gamma)^{2}}{\beta^{2}(1-\gamma)},
\frac{2(1+\gamma)^{2}}{\alpha^{2}(1-\gamma)}(e^{\alpha T^d}-1)\}$. It is not hard to see that there are at most $m_d=[\frac{T^d}{\tau}]+1$ intervals (in $[T_j,T_{j+1})$) associated with unconnected graphs. Therefore, we have $$\begin{split} V(\epsilon(T_{j+1})) \leq& e^{-\beta
T_{j}^{c}+\alpha T_{j}^{d}
}V(\epsilon(T_{j}))+(1+e^{T^d}+e^{2T^d}\\
&+\cdots+e^{m_dT^d})\eta \bar\delta^2\\
\leq &e^{-\beta T^{c}+\alpha
(T-T^{c}) }V(\epsilon(T_{j}))+\bar\eta\bar\delta^2
\end{split}$$ with $\bar\eta=\frac{e^{(m_d+1)T^d}-1}{e^{T^d}-1}\eta>0$.
If $\beta T_{c}>\alpha (T-T^{c})$ or $T_c>\frac{\alpha
T}{\alpha+\beta}$, then $\nu=e^{-\beta T^{c}+\alpha (T-T^{c})}<1$. Thus, $$\begin{aligned}
V(\epsilon(T_{j+1})) &\leq
\nu^{j+1}V(\epsilon(T_{0}))+(\nu^{j}+\cdots+1)\bar\eta\bar\delta^2\\
&\leq
\nu^{j+1}V(\epsilon(T_{0}))+\frac{1-\nu^{j+1}}{1-\nu}\bar\eta\bar\delta^2.\end{aligned}$$ For any $t>0$, there is $j$ such that $T_j<t<T_{j+1}$ with $$V(\epsilon(t)) \leq e^{\alpha T^{d}}
V(\epsilon(T_{j}))+\bar\eta\bar\delta^2$$ Thus, (\[result12\]) is obtained with taking $C=\sqrt{\frac{(e^{\alpha T^{d}}+1-\nu)
\bar\eta}{(1-\nu)(1-\gamma)}}\bar\delta$.
Furthermore, if $\bar\delta=0$, then $C=0$, which implies (\[result11\]), or $\epsilon \to 0$ as $t\to \infty$.
------------------------------------------------------------------------
In fact, the proposed estimation idea can be extended to the case of an active leader with the following dynamics: $$\label{general}
\begin{cases}
\dot{x}_0^1=x_0^2 \\
\dot{x}_0^2=x_0^3\\
\vdots\\
\dot{x}_0^{\kappa}=a(t)=a_0(t)+\delta(t)\\
y=x_0=x_0^1\in R^2
\end{cases}$$ where $y(t)$ is the measured output variable of the leader and $a(t)$ is its input variable. The dynamics of each agent is still taken in the form of (\[modela\]). Then we will construct an observer as we did for system (\[model0\]). Here, for the space limitations, we only give the corresponding error system, which can be expressed as: [$$\begin{pmatrix} \dot{\bar x}^1\\
\dot{\bar x}^2\\\vdots\\ \dot{\bar x}^{\kappa}\end{pmatrix}=
\begin{pmatrix}-kM_{p}&I_n&&\\
-\gamma_1 kM_{p}&0&I_n&\\
\vdots & &&\\
-\gamma_{\kappa-2}k M_{p}&&0&I_n\\
-\gamma_{\kappa-1}k M_{p}&&&0
\end{pmatrix}\otimes I_2\begin{pmatrix} {\bar x^1}\\
{\bar x}^2\\\vdots\\ {\bar x}^{\kappa}\end{pmatrix}+\begin{pmatrix} 0\\
0\\\vdots\\ - {\bf 1}\otimes \delta\end{pmatrix}$$ ]{} or equivalently in a compact form: $$\dot\epsilon=F_p\epsilon+g\in R^{2n\kappa}$$ where $k>0$ and $0<\gamma_j<1, \; (j=1,...,\kappa-1)$ are suitable real numbers and ${\bar x}^i=x^i-\textbf{1}\otimes x_{0}^i\in
R^{2n}$ with $x^1=x$ and $x^i\; (2\leq i\leq \kappa)$ as the vector whose components are the respective estimated values of $x_0^i$ by $n$ agents.
To obtain the results similar to Theorems \[thm1\] or \[thm2\], we need to find a suitable quadratic Lyapunov function; that is, to construct an appropriate positive definite matrix $P$ such that $F_{p}^{T}P+PF_{p}$ is negative definite then the corresponding graph $\bar {\mathcal G}_p$ is connected. For example, when $\kappa=3$, we can choose [$$F_p=\left(\begin{array}{ccc}
-kM_{p}&I_n&0\\
-\frac{8k}{9}M_{p}&0&I_n\\
-\frac{4k}{9}M_{p}&0&0
\end{array}\right)\otimes I_2,\; P=\left(\begin{array}{ccc}I_n&-\frac{2}{3}I&0
\\-\frac{2}{3}I_n&I_n&-\frac{1}{2}I_n\\0&-\frac{1}{2}I_n&I_n\\\end{array}\right)\otimes
I_2.$$]{}
Conclusions
===========
This paper studied the consensus problem of a group of autonomous agents with an active leader, whose velocity cannot be measured. To solve the problem, a distributed feedback (i.e., (\[con\])) along with a distributed state-estimation rule (i.e., (\[update1\])) was proposed for each continuous-time dynamical agent in a varying interconnection topology. In fact, generalized cases, including those when the interconnection graphs are directed and when the dynamics of each agent is more complex than system (\[modela\]), will be considered in future.
The authors wish to thank the reviewers for their constructive suggestions, especially the reviewer who introduced us a framework of distributed observer design for multi-agent systems. This work was supported by the NNSF of China under Grants 60425307, 50595411 and 60221301.
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|
---
abstract: 'It has been suggested that materials which break spatial inversion symmetry, but not time reversal symmetry, will be optically gyrotropic and display a nonlocal Hall effect. The associated optical rotary power and the suggested possibility of inducing a Kerr effect in such materials, in turn are central to recent discussions about the nature of the pseudogap phases of various cuprate high-temperature superconductors. In this letter, we show that optical gyrotropy and the nonlocal Hall effect provide a sensitive probe of broken inversion symmetry in $1T$-TiSe$_2$. This material was recently found to possess a chiral charge ordered phase at low temperatures, in which inversion symmetry is spontaneously broken, while time reversal symmetry remains unbroken throughout its phase diagram. We estimate the magnitude of the resulting gyrotropic constant and optical rotary power and suggest that $1T$-TiSe$_2$ may be employed as a model material in the interpretation of recent Kerr effect measurements in cuprate superconductors.'
author:
- Martin Gradhand$^1$
- 'Jasper van Wezel$^{2}$'
title: 'Optical Gyrotropy and the Nonlocal Hall Effect in Chiral Charge Ordered TiSe$_2$'
---
*Introduction* — The measurement of a Kerr effect in the pseudogap phase of several high-temperature superconductors constrains the symmetries that this state may exhibit [@Xia_2008; @He_2011; @Karapetyan_2012; @Hosur_2013]. Although the particular experimental setup used in these studies allows for a non-zero linear response to arise under equilibrium conditions only in the presence of broken time reversal symmetry [@Halperin_1992; @Armitage_2014; @Fried_2014; @Hosur_2015], it has been argued that the observed optical activity may nonetheless be fundamentally linked to a breakdown of spatial inversion symmetry, related to the presence of charge order [@Hosur_2015]. That it is possible for a charge ordered state to spontaneously break inversion symmetry even in the absence of magnetism or electrostatic polarisation, has only recently become clear, with the discovery of chiral charge order in the low temperature phase of $1T$-TiSe$_2$ [@Ishioka_2010; @vanWezel_2011; @Ishioka_2011; @vanWezel_2012; @Iavarone_2012; @Castellan_2013]. This unexpected emergence of a spiral configuration among the scalar charge density was explained theoretically by the simultaneous presence of orbital order, yielding a vectorial combined order parameter [@vanWezel_2011; @Fukutome_1984]. The charge and orbital ordered state in this scenario must arise through a sequence of phase transitions, as indicated schematically in Fig. \[Fig.Phase\], and confirmed experimentally by specific heat, transport and diffraction experiments [@Castellan_2013].
The chiral charge and orbital order in $1T$-TiSe$_2$ forms an ideal test case for studying the types of phases that have been argued to dominate the optical activity of high-temperature superconductors. It provides an experimentally accessible setting in which charge order breaks spatial inversion symmetry, without the complication of nearby phases with broken time reversal symmetry. In this paper, we show that the chiral order in $1T$-TiSe$_2$ causes the material to be optically gyrotropic. The gyrotropy is evidenced by a non-local Hall effect, and gives rise to non-zero optical rotary power (*i.e.* a Faraday effect at zero magnetic field). Although there cannot be a related non-zero Kerr effect in the presence of time reversal symmetry [@Halperin_1992; @Armitage_2014; @Fried_2014], we argue that the presented results indicate that $1T$-TiSe$_2$ can be used as a model system to investigate the relation between the presence of chiral charge order and the observed optical activity of high-temperature superconductors. It allows the effects of non-equilibrium conditions or explicitly broken time reversal symmetry to be studied in a well-understood material within the specific experimental setup used to measure the Kerr effect in cuprate high-temperature superconductors, and can thus be employed to shed light on the interpretation of these measurements.
*Symmetry Considerations* — The normal state of $1T$-TiSe$_2$ consists of quasi two-dimensional layers with hexagonal planes of Ti atoms sandwiched between hexagonal planes of Se atoms. Individual sandwich layers are separated from each other by a Van der Waals gap [@Pehlke_1987; @Fang_1997; @Goli_2012]. The local coordination of the Ti atoms is octahedral (see inset of Fig. \[Fig.DOS\]), resulting in an overall $P\overline{3̄}m1$ space group, characterized by three two-fold rotation axes in the Ti plane, and a three-fold rotation axis perpendicular to that plane. Importantly, the structure preserves both spatial and temporal inversion symmetry. It falls into the point group D$_{3d}$, and accordingly the Laue group $321'$ [@Kleiner_1966]. For this Laue group, general group theoretical arguments show that the conductivity tensor must have vanishing off-diagonal elements [@Kleiner_1966; @Orenstein_2013], which rules out the occurrence of a polar Kerr effect.
This general statement still holds for the non-chiral, triple-Q charge density wave (CDW) forming below T$_{\text{CDW}}$ and is even valid for the chiral state observed below T$_{\text{Chiral}}$ (see left of Fig. \[Fig.Phase\]). Although the latter state breaks spatial inversion symmetry, reducing the point group to C$_2$ and the associated Laue group to $21'$, the symmetry class only allows for symmetric contributions to the conductivity tensor, and hence still forbids any polar Kerr effect [@Kleiner_1966; @Orenstein_2013]. It was recently pointed out however, that materials without inversion symmetry may be optically gyrotropic [@Orenstein_2013]. They then display a nonlocal Hall effect which is closely related to the optical Kerr effect, and is defined by the linear response: $$\begin{aligned}
j_x=\lambda^G_{xyz}\frac{dE_y}{dz}.\end{aligned}$$ Here $j_x$ is an electric current in the $x$ direction, flowing in response to the gradient in the $z$ direction of the $y$-component $E_y$ of an applied electric field. The gyrotropic coupling constant $\lambda_{xyz}^G$ can be expressed in terms of the Berry curvature of filled bands as: $$\begin{aligned}
\label{Eq_lambda}
\lambda_{xyz}^G &= \frac{e^2}{\hbar}\frac{1}{(1-i \omega\tau_z)^2}\int\limits_{-k_{Fz}}^{+k_{Fz}}dk_z\ \phi(k_z,\omega)v_z(k_z)\tau_z \notag \\
\phi(k_z,\omega) &= \int\limits_0^{k_F(k_z)}d^2k\ \Omega( {\bf k} ,k_z,\omega).\end{aligned}$$ Here, $\Omega( {\bf k},k_z,\omega)$ is the frequency dependent Berry curvature at ${\bf k}=(k_x,k_y)$ within a plane of constant $k_z$. For systems preserving both temporal and spatial inversion symmetry, $\lambda_{xyz}^G$ vanishes, since the Berry curvature vanishes at every point. For a system preserving time reversal symmetry, but not spatial inversions, $\Omega({\bf k},k_z,\omega)$ will be odd under inversion of both ${\bf k}$ and $k_z$. The local Hall conductivity, which is proportional to the average Berry curvature over all filled states, is therefore strictly zero. The nonlocal Hall conductivity arising from the gyrotropic $\lambda_{xyz}^G$ on the other hand, involves the product of two odd functions, $\phi(k_z)$ and $v_z(k_z)$, and is in general nonzero. In the following, we will evaluate the integrals in Eq. for a phenomenological tight-binding model of the chiral CDW wave state in $1T$-TiSe$_2$.
*Tight binding model* — We solve Eq. in the clean limit, approximating $\tau_z \approx 50$ fs to be constant in momentum and the band velocity $v_f\approx 1 \cdot 10^{-6}$ m/s to equal its value at the Fermi energy. In addition, we focus only on the odd part of $\phi(k_z)$, which can contribute to the nonlocal Hall effect. We thus evaluate the expression: $$\begin{aligned}
\lambda_{xyz}^G = \frac{e^2}{\hbar}\frac{2\tau_z v_F}{(1-i \omega\tau_z)^2}\int\limits_{0}^{+k_{Fz}}dk_z\ \left[\phi(k_z,\omega)-\phi(-k_z,\omega)\right].\end{aligned}$$
We employ a three-dimensional, nine band, tight-binding model to describe the electronic structure of the normal state above T$_{\text{CDW}}$. This includes three Ti-$d$ orbitals and three Se-$p$ orbitals each for the upper and lower Se atoms (see the unit cell in Fig. \[Fig.DOS\]). This model was previously introduced and described in detail for the case of a single quasi two-dimensional sandwich layer, neglecting the weak couplings between TiSe$_2$ planes [@vanWezel_2010]. In the present case, the dispersion in the $k_z$ direction is crucial for the emergence of the nonlocal Hall effect. We therefore extend the previous model by introducing weak interplane overlap integrals between the upper and lower Se atoms in consecutive layers. For concreteness, we take $c=6$ $\AA$ as the inter-layer distance. The tight-binding parameters are given in Table \[Tab:para\] and result in the orbital-resolved density of states (DOS) presented in the left of Fig. \[Fig.DOS\]. The parameters are chosen in order to create a small semiconducting gap ($\sim 0.04$ eV) to avoid spurious effects arising around the Fermi energy at low frequencies.
------------------------ --------------------- --- ------------------------- ---------
$\varepsilon_{\rm Ti}$ $-0.5625$ $t_{\rm Ti}$ $0.20$
$\varepsilon_{\rm Se}$ $-0.10\phantom{25}$ $t_{\rm Se}$ $0.30$
$E_{\text{F}}$ $0.97\phantom{25}$ $t_{\rm Ti-Se}$ $1.05$
$\Delta$ $2.3\phantom{625}$ $t_{\rm Se-Se}$ $-0.35$
$U$ $0.02\phantom{25}$ $t^{\perp}_{\rm Se-Se}$ $0.25$
------------------------ --------------------- --- ------------------------- ---------
: The numerical parameters for the tight-binding model of TiSe$_2$ in units of $\text{eV}$. Here, $\varepsilon_{\rm Ti}$ and $\varepsilon_{\rm Se}$ are the on-site energies for Ti and Se atoms. The nearest-neighbour hoppings within the layer are given by $t_{\rm Ti}$, $t_{\rm Se}$, $t_{\rm Ti-Se}$, and $t_{\rm Se-Se}$, while the inter-layer coupling is given by $t^{\perp}_{\rm Se-Se}$. The Fermi energy lies at $E_{\text{F}}$, and the difference in chemical potential between Ti and Se is $\Delta$. The CDW transition is driven by an imposed (mean field) coupling $U$. []{data-label="Tab:para"}
The CDW consists of three components, with propagation vectors connecting the Se states at $\Gamma$ to Ti states at the three inequivalent $L$ points in the first Brillouin zone (see right of Fig. \[Fig.DOS\]). The relative phase differences between the three CDW components and their corresponding frozen phonons in real space, are not fixed by the electronic structure alone. Based on free energy arguments, it was shown that two successive phase transitions at $T_{\text{CDW}}$ and $T_{\text{Chiral}}$ separate the high-temperature homogeneous state from a non-chiral CDW phase in which all phase differences vanish, and a chiral CDW at low temperatures [@vanWezel_2011]. In the chiral state, non-zero phase differences imply a breakdown of spatial inversion symmetry, which in turn allows for the presence of optical gyrotropy.
The three CDW components depicted in the right of Fig. \[Fig.DOS\] do not form a closed set by themselves. We therefore also consider all four higher harmonics, given by Q$_4=$Q$_1$+Q$_2$, Q$_5=$Q$_1$+Q$_3$, Q$_6=$Q$_2$+Q$_3$, and Q$_7=$Q$_1$+Q$_2$+Q$_3$. All together, this makes the tight-binding Hamiltonian a 72$\times$72 matrix, based on eight sectors for the normal state and all Q vectors, with nine orbital entries each. The coupling between different Q sectors is induced by an imposed mean CDW field along the three vectors Q$_1$, Q$_2$, and Q$_3$. To describe the chiral phase, we allow for a nonzero phase difference $\varphi$ between the Q$_1$ and Q$_2$ CDW components, and $-\varphi$ between the Q$_1$ and Q$_3$ components. For the transport calculations presented below, this eigenvalue problem has to be solved on a very dense mesh of roughly $8 \cdot 10^6$ [**k**]{}-points in $1/8$ of the normal state Brillouin Zone.
*Results* — In the right of Fig. \[Fig.Phase\] we present the gyrotropic coupling constant resulting from the tight-binding model, as a function of the relative phase $\varphi$ between the three Q vectors. As expected, the non-chiral phase at $\varphi=0$, which does not break spatial inversion symmetry, does not show any gyrotropic response. For nonzero relative phase difference a nonzero response does develop, which may be roughly described by a skewed sinusoidal form with a maximum around $\varphi \approx 0.34 \pi$. In the following we will fix the phase to this value.
In the left of Fig. \[Fig.Faraday\] we present the real and imaginary parts of the gyrotropic response as a function of chemical potential, for fixed optical frequency $\omega=0.8$ eV. While the imaginary part varies weakly over the considered energy range, the real part shows strong features around the Fermi energy. This is due to the small normal state semiconducting gap, which allows small shifts in chemical potential to result in remarkable changes of the response function. Evidently, this makes any quantitative prediction difficult.
Fixing the chemical potential and instead varying the frequency of the probing light results in the nonlocal Hall conductivity shown in the middle of Fig. \[Fig.Faraday\]. Both the real and imaginary parts are smooth functions of the optical frequency up to $1$ eV. The response drastically increases to lower frequency and the imaginary part changes sign around $\omega=0.56$ eV.
Although the nonlocal Hall conductivity is a clean and direct indication of the presence of broken spatial inversion symmetry and hence chiral charge and orbital order in this system, it may be a difficult quantity to access experimentally. We therefore also present the optical rotary power upon transmission with normal incidence. The transmission geometry limits this application to thin films. The weak Van der Waals bonding between sandwich layers in $1T$-TiSe$_2$ however, allows for relatively straightforward production of thin films with thicknesses ranging from micrometers to nanometers [@Goli_2012; @Peng_2014]. Even in single atomic layers, the chiral charge ordered phase is expected to survive [@vanWezel_2012], and the optical rotary power can therefore provide a direct means of establishing the broken inversion symmetry in this phase. It is calculated using the expression [@Bungay_1993]: $$\begin{aligned}
\theta_F / d &= -\frac{\mu_0}{2} ~ \textrm{Im}\left[ \gamma^G_{xyz} \omega \right].
\label{Eq.:Faraday}\end{aligned}$$ Here, we changed the notation with respect to Ref. , in order to keep SI units and to avoid the need to introduce a nonlocal dielectric tensor. In the right of Fig. \[Fig.Faraday\] we present the optical rotary power in $1T$-TiSe$_2$ resulting from Eq. . It is of the order of $10^{-2}$ rad/m at low frequencies and tracks the imaginary part of the nonlocal conductivity. It diminishes sharply with increasing frequency and changes sign around $0.6$eV, while remaining of the order of $10^{-3}$ rad/m for higher energies. For films of several micrometers thickness, the rotation of polarization at low frequencies lies at the limit of what is detectable with current state of the art techniques [@Li_2014].
The closely related optical Kerr effect, or rotation of polarization of reflected light with normal incidence, has previously been suggested as an alternative probe of optical gyrotropy [@Orenstein_2013], which would not require thin films. It was recently pointed out however, that the standard expression used to calculate the Kerr response [@Bungay_1993], is incomplete, as it ignores the variation of gyrotropy at the sample interface [@Halperin_1992; @Armitage_2014; @Fried_2014; @Hosur_2015]. Taking into account these boundary effects, the linear Kerr effect is always identically zero in time reversal symmetric materials under equilibrium conditions. Nevertheless, it has been argued recently that in the Kerr effect measurements of cuprate high-temperature superconductors, the condition of thermal equilibrium may have been weakly violated, allowing for a Kerr response even in the absence of time reversal symmetry breaking [@Hosur_2015].
In the case of $1T$-TiSe$_2$, a second way to observe a Kerr effect would be to independently satisfy the two symmetry conditions for obtaining a non-zero response [@Hosur_2015], broken time reversal symmetry and the absence of mirror symmetries. While an applied magnetic field suffices to break time reversal symmetry, it will not affect the mirror symmetries present in the high-temperature atomic lattice. Upon cooling, the lattice symmetries will be spontaneously broken as the chiral transition temperature is traversed, and the onset of a Kerr effect within a magnetic field thus provides a probe for the onset of the chiral charge and orbital ordered state.
In both the scenario of violated equilibrium conditions, and that of explicitly broken time reversal symmetry, the precise size and shape of the response will depend sensitively on the details of the experimental configuration, and are beyond the scope of the present investigation. The strength and frequency dependence of the cancelled contribution to the standard expression for the Kerr effect on the other hand [@Bungay_1993], are accessible within the current tight binding model, and are presented in the Supplemental Material. The obtained values in $1T$-TiSe$_2$ are comparable to those in cuprate superconductors, making it plausible that the non-equilibrium and in-field responses of the two classes are likewise of a similar order of magnitude.
*Conclusions* — In summary, we have shown that the breakdown of spatial inversion symmetry and emergence of chirality in the low-temperature charge and orbital ordered phase of $1T$-TiSe$_2$ renders that state optically gyrotropic. The susceptibility to a nonlocal Hall effect serves as a probe for the optical gyrotropy, being zero in both the normal and non-chiral charge ordered phases. In the chiral phase, the nonlocal susceptibility increases with growing relative phase difference between charge density wave components, which can be used as an order parameter for the emerging chirality [@vanWezel_2011; @vanWezel_2012]. The gyrotropic response was also found to depend sensitively on the value of the chemical potential, which can be tuned experimentally using various types of intercalants [@Taguchi_1981; @Morosan_2006; @Morosan_2010; @Iavarone_2012]. For non-zero probing frequency, the maximum response is found if the chemical potential lies within a small semiconducting gap, which is close to the condition in the pristine material [@Kidd_2002; @Li_2007; @May_2011].
Direct detection of the nonlocal Hall effect is challenging experimentally. The imaginary parts of the nonlocal susceptibility however, is closely tracked by the rotated polarization of normally incident light upon transmission. This optical rotary power is shown to be strongest at low frequencies, and to be of the order of $10^{-2}$ rad/m. For thin films, the optical rotary power may be experimentally accessible, and can serve as a probe of the broken inversion symmetry in $1T$-TiSe$_2$ associated with the onset of chiral order.
The closely related rotation of linear polarization upon reflection of normally incident light, the Kerr effect, was recently shown to be identically zero under equilibrium conditions unless time reversal symmetry is broken [@Halperin_1992; @Armitage_2014; @Fried_2014; @Hosur_2015]. In spite of this result, it has been argued that the observation of a Kerr effect in cuprate high-temperature superconductors may be due to the presence of chiral charge order, without broken time reversal symmetry [@Hosur_2015]. This could be possible if the measurement induced slight non-equilibrium conditions, or if time reversal symmetry was broken explicitly by contaminating magnetic fields or defects. To experimentally test these hypotheses, and thus determine with certainty the symmetries of the pseudogap phase in high-temperature superconductors, a well-understood reference system is required. The chiral charge ordered phase of $1T$-TiSe$_2$ provides such a reference, as it spontaneously breaks inversion symmetry and becomes chiral without breaking time reversal symmetry. At the same time, it can be easily and controllably intercalated with magnetic atoms, subjected to external fields, or driven out of equilibrium. An investigation of the Kerr response in $1T$-TiSe$_2$ thus provides an ideal test bed that can be used to interpret and understand the observed Kerr effect in high-temperature cuprates.
[**Acknowledgements**]{} — MG acknowledges financial support from the Leverhulme Trust via an Early Career Research Fellowship (ECF-2013-538). JvW acknowledges support from a VIDI grant financed by the Netherlands Organisation for Scientific Research (NWO).
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abstract: 'These lectures provide an informal introduction into the notions and tools used to analyze statistical properties of eigenvalues of large random Hermitian matrices. After developing the general machinery of orthogonal polynomial method, we study in most detail Gaussian Unitary Ensemble (GUE) as a paradigmatic example. In particular, we discuss Plancherel-Rotach asymptotics of Hermite polynomials in various regimes and employ it in spectral analysis of the GUE. In the last part of the course we discuss general relations between orthogonal polynomials and characteristic polynomials of random matrices which is an active area of current research.'
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0.5cm
**Introduction to the Random Matrix Theory:**
**Gaussian Unitary Ensemble and Beyond**
0.4cm
**Yan V. Fyodorov**
0.3cm
Department of Mathematical Sciences, Brunel University,
Uxbridge, UB8 3PH, United Kingdom.
0.3cm
Preface
=======
Gaussian Ensembles of random Hermitian or real symmetric matrices always played a prominent role in the development and applications of Random Matrix Theory. Gaussian Ensembles are uniquely singled out by the fact that they belong both to the family of invariant ensembles, and to the family of ensembles with independent, identically distributed (i.i.d) entries. In general, mathematical methods used to treat those two families are very different.
In fact, all random matrix techniques and ideas can be most clearly and consistently introduced using Gaussian case as a paradigmatic example. In the present set of lectures we mainly concentrate on consequences of the invariance of the corresponding probability density function, leaving aside methods of exploiting statistical independence of matrix entries. Under these circumstances the method of orthogonal polynomials is the most adequate one, and for the Gaussian case the relevant polynomials are Hermite polynomials. Being mostly interested in the limit of large matrix sizes we will spend a considerable amount of time investigating various asymptotic regimes of Hermite polynomials, since the latter are main building blocks of various correlation functions of interest. In the last part of our lecture course we will discuss why statistics of characteristic polynomials of random Hermitian matrices turns out to be interesting and informative to investigate, and will make a contact with recent results in the domain.
The presentation is quite informal in the sense that I will not try to prove various statements in full rigor or generality. I rather attempt outlining the main concepts, ideas and techniques preferring a good illuminating example to a general proof. A much more rigorous and detailed exposition can be found in the cited literature. I will also frequently employ the symbol $\propto$. In the present set of lectures it always means that the expression following $\propto$ contains a multiplicative constant which is of secondary importance for our goals and can be restored when necessary.
Introduction
============
In these lectures we use the symbol $^T$ to denote matrix or vector transposition and the asterisk $^*$ to denote Hermitian conjugation. In the present section the bar $\overline{z}$ denotes complex conjugation.
Let us start with a square complex matrix $\hat{Z}$ of dimensions $N\times N$, with complex entries $z_{ij}=x_{ij}+iy_{ij},\, 1\le
i,j\le N$. Every such matrix can be conveniently looked at as a point in a $2N^2$-dimensional Euclidean space with real Cartesian coordinates $x_{ij},\,y_{ij}$, and the length element in this space is defined in a standard way as: $$\label{1}
(ds)^2=\mbox{Tr}\left(d\hat{Z}d\hat{Z}^*\right)=\sum_{ij}dz_{ij}
\overline{dz_{ij}}=\sum_{ij}\left[(dx)^2_{ij}+(dy)^2_{ij}\right].$$
As is well-known (see e.g.[@Fom]) any surface embedded in an Euclidean space inherits a natural Riemannian metric from the underlying Euclidean structure. Namely, let the coordinates in a $n-$dimensional Euclidean space be $(x_1,\ldots,x_n)$, and let a $k-$dimensional surface embedded in this space be parameterized in terms of coordinates $(q_1,\ldots,q_k),\, k\le n$ as $x_i=x_i(q_1,\ldots,q_k),\, i=1,\ldots n$. Then the Riemannian metric $g_{ml}=g_{lm}$ [*on the surface*]{} is defined from the Euclidean length element according to $$\label{2}
(ds)^2=\sum_{i=1}^n(dx_i)^2=\sum_{i=1}^n\left(\sum_{m=1}^k
\frac{\partial x_i}{\partial q_m}dq_m\right)^2 =\sum_{m,l=1}^k
g_{mn}dq_{m}dq_{l}.$$ Moreover, such a Riemannian metric induces the corresponding integration measure on the surface, with the volume element given by $$\label{3}
d\mu=\sqrt{|g|}dq_1\ldots dq_k,\quad
g=\det{\left(g_{ml}\right)_{l,m=1}^k}.$$
For $k=n$ these are just the familiar formulae for the lengths and volume associated with change of coordinates in an Euclidean space. For example, for $n=2$ we can pass from Cartesian coordinates $-\infty<x,\, y<\infty $ to polar coordinates $r>0$, $0\le \theta<2\pi$ by $x=r\cos{\theta}$, $y=r\sin{\theta}$, so that $dx=dr\cos{\theta}-r\sin{\theta}d\theta$, $dy=
dr\sin{\theta}+r\cos{\theta}d\theta$, and the Riemannian metric is defined by $(ds)^2=(dx)^2+(dy)^2=(dr)^2+r^2(d\theta)^2$. We find that $g_{11}=1, \, g_{12}=g_{21}=0, \, g_{22}=r^2$, and the volume element of the integration measure in the new coordinates is $d\mu=rdrd\theta$; as it should be. As the simplest example of a “surface" with $k<n=2$ embedded in such a two-dimensional space we consider a circle $r=R=const$. We immediately see that the length element $(ds)^2$ restricted to this “surface" is $(ds)^2=R^2(d\theta)^2$, so that $g_{11}=R^2$, and the integration measure induced on the surface is correspondingly $d\mu=Rd\theta$. The “surface" integration then gives the total “volume" of the embedded surface (i.e. circle length $2\pi R$).
Similarly, we can consider a two-dimensional ($k=2$) sphere $R^2=x^2+y^2+z^2$ embedded in a three-dimensional Euclidean space $(n=3)$ with coordinates $x,y,z$ and length element $(ds)^2=(dx)^2+(dy)^2+(dz)^2$. A natural parameterization of the points on the sphere is possible in terms of the spherical coordinates $\phi,\theta$ (see Fig. \[fig1\]) $$x=R\sin{\theta}\cos{\phi},\,\, y=R\sin{\theta}\sin{\phi},\,\,
z=R\cos{\theta};\quad 0\le \theta\le\pi,\,\, 0\le \phi < 2\pi,$$ which results in $(ds)^2=R^2(d\theta)^2+R^2\sin^2{\theta}(d\phi)^2$. Hence the matrix elements of the metric are $g_{11}=R^2,\,g_{12}=g_{21}=0,
g_{22}=R^2\sin^2{\theta}$, and the corresponding “volume element" on the sphere is the familiar elementary area $d\mu=R^2\sin{\theta}d\theta d\phi$.
As a less trivial example to be used later on consider a $2-$dimensional manifold formed by $2\times 2$ unitary matrices $\hat{U}$ embedded in the $8$ dimensional Euclidean space of $Gl(2;C)$ matrices. Every such matrix can be represented as the product of a matrix $\hat{U}_c$ from the coset space $U(2)/U(1)\times U(1)$ parameterized by $k=2$ real coordinates $0\le \phi<2\pi,\, 0\le \theta\le \pi/2$, and a diagonal unitary matrix $U_d$, that is $\hat{U}=\hat{U}_d\hat{U}_c$, where $$\label{4}
\hat{U_c}=\left(\begin{array}{cc}
\cos{\theta}&-\sin{\theta}e^{-i\phi}\\
\sin{\theta}e^{i\phi}&\cos{\theta}\end{array}\right),\quad
\hat{U_d}=\left(\begin{array}{cc} e^{-i\phi_1}&0\\
0&e^{i\phi_2}\end{array}\right).$$ Then the differential $d\hat{U}$ of the matrix $\hat{U}=\hat{U}_d\hat{U}_c$ has the following form: $$\label{5}
\hat{dU}=\left(\begin{array}{cc}
-[d\theta \sin{\theta}+i\cos{\theta}d\phi_1]e^{-i\phi_1}
& e^{-i(\phi_1+\phi)}[-d\theta\cos{\theta}+i(d\phi_1+d\phi)\sin{\theta}]
\\ e^{i(\phi+\phi_2)}
[d\theta\cos{\theta}+i(d\phi+d\phi_2)\sin{\theta}]
& [-d\theta \sin{\theta}+id\phi_2\cos{\theta}] e^{i\phi_2}
\end{array}\right),$$ which yields the length element and the induced Riemannian metric: $$\label{6}
(ds)^2=\mbox{Tr}\left(d\hat{U}d\hat{U}^*\right)=
2(d\theta)^2+(d\phi_1)^2+(d\phi_2)^2+
2\sin^2{\theta}(d\phi)^2+2\sin^2{\theta}(d\phi\, d\phi_1 +d\phi\,
d\phi_2).$$ We see that the nonzero entries of the Riemannian metric tensor $g_{mn}$ in this case are $g_{11}=2,\, g_{22}=g_{33}=1, g_{44}=2\sin^2{\theta}, \,
g_{24}=g_{42}=g_{34}=g_{43}=\sin^2{\theta}$, so that the determinant $\det{[g_{mn}]}=4\sin^2{\theta}\cos^2{\theta}$. Finally, the induced integration measure on the group $U(2)$ is given by $$\label{7}
d\mu(\hat{U})=2\sin{\theta}\cos{\theta}\,d\theta\, d\phi\,
d\phi_1\, d\phi_2.$$ It is immediately clear that the above expression is invariant, by construction, with respect to multiplications $\hat{U}\to
\hat{V}\hat{U}$, for any fixed unitary matrix $V$ from the same group. Therefore, Eq.(\[7\]) is just the Haar measure on the group.
We will make use of these ideas several times in our lectures. Let us now concentrate on the $N^2-$dimensional subspace of Hermitian matrices in the $2N^2-$ dimensional space of all complex matrices of a given size $N$. The Hermiticity condition $\hat{H}=\hat{H}^*\equiv\overline{\hat{H}^T}$ amounts to imposing the following restrictions on the coordinates: $x_{ij}=x_{ji},\,\,y_{ij}=-y_{ji}$. Such a restriction from the space of general complex matrices results in the length and volume element on the subspace of Hermitian matrices: $$\label{8}
(ds)^2=\mbox{Tr}\left(d\hat{H}d\hat{H}^*\right)=
\sum_{i}(dx_{ii})^2+2\sum_{i<j}\left[(dx_{ij})^2+(dy_{ij})^2\right]$$ $$\label{8a}
d\mu(\hat{H})=2^{\frac{N(N-1)}{2}}\prod_{i}dx_{ii}\prod_{i<j}dx_{ij}dy_{ij}.$$ Obviously, the length element $(ds)^2=\mbox{Tr}d\hat{H}d\hat{H}^*$ is invariant with respect to an automorphism (a mapping of the space of Hermitian matrices to itself) by a similarity transformation $\hat{H}\to U^{-1}\hat{H}\hat{U}$, where $\hat{U}\in U(N)$ is any given unitary $N\times N$ matrix: $\hat{U}^*=\hat{U}^{-1}$. Therefore the corresponding integration measure $d\mu(\hat{H})$ is also invariant with respect to all such “rotations of the basis".
The above-given measure $d\mu(\hat{H})$ written in the coordinates $x_{ii},\, x_{i<j},\, y_{i<j}$ is frequently referred to as the “flat measure". Let us discuss now another, very important coordinate system in the space of Hermitian matrices which will be in the heart of all subsequent discussions. As is well-known, every Hermitian matrix $\hat{H}$ can be represented as $$\label{81}
\hat{H}=\hat{U}\hat{\Lambda}\hat{U}^{-1},\quad
\hat{\Lambda}=\mbox{diag} (\lambda_1,\ldots,\lambda_N),\,\,
\hat{U}^*\hat{U}=\hat{I},$$ where real $-\infty<\lambda_k<\infty,\,k=1,\ldots,N$ are eigenvalues of the Hermitian matrix, and rows of the unitary matrix $\hat{U}$ are corresponding eigenvectors. Generically, we can consider all eigenvalues to be simple (non-degenerate). More precisely, the set of matrices $\hat{H}$ with non-degenerate eigenvalues is dense and open in the $N^2$-dimensional space of all Hermitian matrices, and has full measure (see [@Deift], p.94 for a formal proof). The correspondence $\hat{H}\to
\left(\hat{U}\in U(N),\hat{\Lambda}\right)$ is, however, not one-to-one, namely $\hat{U}_1\hat{\Lambda}\hat{U}_1^{-1}=\hat{U}_2\hat{\Lambda}\hat{U}_2^{-1}$ if $\hat{U}^{-1}_1\hat{U}_2=\mbox{diag}\left(e^{i\phi_1},\ldots,e^{i\phi_N}\right)$ for any choice of the phases $\phi_1,\ldots,\phi_N$. To make the correspondence one-to-one we therefore have to restrict the unitary matrices to the coset space $U(N)/U(1)
\otimes\ldots\otimes U(1)$, and also to order the eigenvalues, e.g. requiring $\lambda_1<\lambda_2<\ldots<\lambda_N$. Our next task is to write the integration measure $d\mu(\hat{H})$ in terms of eigenvalues $\hat{\Lambda}$ and matrices $\hat{U}$. To this end, we differentiate the spectral decomposition $\hat{H}=\hat{U}\hat{\Lambda}\hat{U}^{*}$, and further exploit: $d\left(\hat{U}^*\hat{U}\right)=d\hat{U}^*\hat{U}+\hat{U}^*d\hat{U}=0$. This leads to $$\label{82}
d\hat{H}=\hat{U}\left[d\hat{\Lambda}+\hat{U}^{*}d\hat{U}\hat{\Lambda}
-\hat{\Lambda}\hat{U}^{*}d\hat{U}\right]\hat{U}^{*}.$$ Substituting this expression into the length element $(ds)^2$, see Eq.(\[8\]), and using the short-hand notation $\delta\hat{U}$ for the matrix $\hat{U}^*d\hat{U}$ satisfying anti-Hermiticity condition $\delta\hat{U}^*=-\delta\hat{U}$, we arrive at: $$\label{83}
(ds)^2=\mbox{Tr}\left[\left(d\hat{\Lambda}\right)^2+2d\hat{\Lambda}
\left(\delta\hat{U}\hat{\Lambda}
-\hat{\Lambda}\delta\hat{U}\right)+\left(\delta\hat{U}\hat{\Lambda}\right)^2
+\left(\hat{\Lambda}\delta\hat{U}\right)^2-2\delta\hat{U}
\hat{\Lambda}^2\delta\hat{U}\right].$$ Taking into account that $\hat{\Lambda}$ is purely diagonal, and therefore the diagonal entries of the commutator $\left(\delta\hat{U}\hat{\Lambda}
-\hat{\Lambda}\delta\hat{U}\right)$ are zero, we see that the second term in Eq.(\[83\]) vanishes. On the other hand, the third and subsequent terms when added up are equal to $$2\mbox{Tr}\left[\delta\hat{U}\hat{\Lambda}\delta\hat{U}\hat{\Lambda}-
\delta\hat{U}^2\hat{\Lambda}^2\right]=2\sum_{ij}\left[
\delta U_{ij}\lambda_j\delta U_{ji}\lambda_i-\lambda_i^2
\delta U_{ij}\delta U_{ji}\right]=-\sum_{ij}\left(\lambda_i-
\lambda_j\right)^2\delta U_{ji}\delta U_{ij}\,,$$ which together with the first term yields the final expression for the length element in the “spectral" coordinates $$\label{84}
(ds)^2=\sum_{i}\left(d\lambda_i\right)^2+\sum_{i<j}\left(\lambda_i-
\lambda_j\right)^2\overline{\delta U_{ij}}\delta U_{ij}\,$$ where we exploited the anti-Hermiticity condition $-\delta U_{ji}=
\overline{\delta U_{ij}}$. Introducing the real and imaginary parts $\delta U_{ij}=\delta p_{ij}+i\delta q_{ij}$ as independent coordinates we can calculate the corresponding integration measure $d\mu(\hat{H})$ according to the rule in Eq.(\[3\]), to see that it is given by $$\label{85}
d\mu(\hat{H})=\prod_{i<j}\left(\lambda_i-
\lambda_j\right)^2\prod_{i}d\lambda_i \times d{\cal M}(\hat{U})\,.$$ The last factor $d{\cal M}(U)$ stands for the part of the measure depending only on the $U-$variables. A more detailed consideration shows that, in fact, $d{\cal M}(\hat{U})\equiv d\mu(\hat{U})$, which means that it is given (up to a constant factor) by the invariant Haar measure on the unitary group $U(N)$. This fact is however of secondary importance for the goals of the present lecture.
Having an integration measure at our disposal, we can introduce a probability density function (p.d.f.) ${\cal P}(\hat{H})$ on the space of Hermitian matrices, so that ${\cal P}(\hat{H})
d\mu(\hat{H})$ is the probability that a matrix $\hat{H}$ belongs to the volume element $d\mu(\hat{H})$. Then it seems natural to require for such a probability to be invariant with respect to all the above automorphisms, i.e. ${\cal P}(\hat{H})={\cal
P}\left(\hat{U}^*\hat{H}\hat{U}\right)$. It is easy to understand that this “postulate of invariance" results in ${\cal P}$ being a function of $N$ first traces $\mbox{Tr}\hat{H}^n,\, n=1,\ldots,N$ (the knowledge of first $N$ traces fixes the coefficients of the characteristic polynomial of $\hat{H}$ uniquely, and hence the eigenvalues. Therefore traces of higher order can always be expressed in terms of the lower ones). Of particular interest is the case $$\label{9}
{\cal P}(\hat{H})=C\exp{-Tr\,Q(\hat{H})},\quad
Q(x)=a_{2j}x^{2j}+\ldots +a_0,$$ where $2j\le N$, the parameters $a_{2l}$ and $C$ are real constants, and $a_{2j}>0$. Observe that if we take $$\label{10}
Q(x)=ax^{2}+bx+c,$$ then $e^{-Tr\,Q(\hat{H})}$ takes the form of the product $$\label{11}
e^{-a\left[\sum_ix^2_{ii}+2
\sum_{i<j}(x^2_{ij}+y^2_{ij})\right]}e^{-b\sum_ix_{ii}}e^{-cN}
=e^{-cN}\prod_{i=1}^N\left( e^{-ax_{ii}^2-bx_{ii}}\right)
\prod_{i<j}e^{-2ax_{ij}^2}\prod_{i<j}e^{-2ay_{ij}^2}.$$ We therefore see that the probability distribution of the matrix $\hat{H}$ can be represented as a product of factors, each factor being a suitable Gaussian distribution depending only on one variable in the set of all coordinates $x_{ii},\, x_{i<j},\,
y_{i<j}$. Since the same factorization is valid also for the integration measure $d\mu(\hat{H})$, see Eq.(\[8a\]), we conclude that all these $N^2$ variables are statistically independent and Gaussian-distributed.
A much less obvious statement is that if we impose [*simultaneously*]{} two requirements:
- The probability density function ${\cal P}(\hat{H})$ is invariant with respect to all conjugations $\hat{H}\to
\hat{H}'=U^{-1}\hat{H}\hat{U}$ by unitary matrices $\hat{U}$, that is ${\cal P}(\hat{H}')={\cal P}(\hat{H})$; and
- the $N^2$ variables $x_{ii},\, x_{i<j},\, y_{i<j}$ are statistically independent, i.e. $$\label{11a}
{\cal P}(\hat{H})=\prod_{i=1}^Nf_i(x_{ii})
\prod_{i<j}^Nf^{(1)}_{ij}(x_{ij})f^{(2)}_{ij}(y_{ij}),$$
then the function ${\cal P}(\hat{H})$ is [*necessarily*]{} of the form ${\cal P}(\hat{H})=Ce^{-\left(a\mbox{Tr}\hat{H}^2+b\mbox{Tr}
\hat{H}+cN\right)}$, for some constants $a>0,\, b,\, c$. The proof for any $N$ can be found in [@Mehta], and here we just illustrate its main ideas for the simplest, yet nontrivial case $N=2$. We require invariance of the distribution with respect to the conjugation of $\hat{H}$ by $\hat{U}\in U(2)$, and first consider a particular choice of the unitary matrix $\hat{U}=
\left(\begin{array}{cc} 1&-\theta\\ \theta&1\end{array}\right)$ corresponding to $\phi=\phi_1=\phi_2=0$, and small values $\theta
\ll 1$ in Eq.(\[4\]). In this approximation the condition $\hat{H}'=U^{-1}\hat{H}\hat{U}$ amounts to $$\label{11b}
\left(\begin{array}{cc}
x'_{11}& x'_{12}+iy'_{12} \\ x'_{12}-iy'_{12}&
x'_{22}\end{array}\right)=\left(\begin{array}{cc}
x_{11}+2\theta x_{12}&
x_{12}+iy_{12}+\theta\left(x_{22}-x_{11}\right)
\\ x_{12}-iy_{12}+\theta\left(x_{22}-x_{11}\right)&
x_{22}-2\theta x_{12}\end{array}\right),$$ where we kept only terms linear in $\theta$. With the same precision we expand the factors in Eq.(\[11a\]): $$f_1(x'_1)=f_1(x_1)\left[1+2\theta x_{12}\frac{1}{f_1}
\frac{df_1}{dx_{11}}\right],\,\,\,
f_2(x'_{22})=f_2(x_{22})\left[1-2\theta x_{12}\frac{1}{f_2}
\frac{df_2}{dx_{22}}\right]$$ $$f^{(1)}_{12}(x'_{12})=f^{(1)}_{12}(x_{12})
\left[1+\theta (x_{22}-x_{11})\frac{1}{f^{(1)}_{12}}
\frac{df^{(1)}_{12}}{dx_{12}}\right],\,\,\,
f^{(2)}_{12}(y'_{21})=f^{(2)}_{12}(y_{12}).$$ The requirements of statistical independence and invariance amount to the product of the left-hand sides of the above expressions to be equal to the product of the right-hand sides, for any $\theta$. This is possible only if: $$\label{11c}
2x_{12}\left[\frac{d\ln{f_1}}{dx_{11}}-\frac{d\ln{f_2}}{dx_{22}}\right]
+(x_{22}-x_{11})\frac{d\ln{f^{(1)}_{12}}}{dx_{12}}=0,$$ which can be further rewritten as $$\label{11d}
\frac{1}{(x_{22}-x_{11})}
\left[\frac{d\ln{f_1}}{dx_{11}}-\frac{d\ln{f_2}}{dx_{22}}\right]=const=
\frac{1}{2x_{12}}\frac{d\ln{f^{(1)}_{12}}}{dx_{12}},$$ where we used that the two sides in the equation above depend on essentially different sets of variables. Denoting $const_1=-2a$, we see immediately that $$f^{(1)}_{12}(x_{12})\propto e^{-2ax^2_{12}},$$ and further notice that $$\frac{d\ln{f_1}}{dx_{11}}+2ax_{11}=const_2=\frac{d\ln{f_2}}{dx_{22}}+2ax_{22}$$ by the same reasoning. Denoting $const_2=-b$, we find: $$\label{11e}
f_{1}(x_{11})\propto e^{-a x^2_{11}-bx_{11}},\quad
f_{2}(x_{22})\propto e^{-a x^2_{22}-bx_{22}},$$ and thus we are able to reproduce the first two factors in Eq.(\[11\]). To reproduce the remaining factors we consider the conjugation by the unitary matrix $
\hat{U_d}=\left(\begin{array}{cc} 1-i\alpha&0\\
0&1+i\alpha\end{array}\right)$, which corresponds to the choice $\theta=0,\,\phi_1=\phi_2=-\alpha=$ in Eq.(\[4\]), and again we keep only terms linear in the small parameter $\alpha\ll 1$. Within such a precision the transformation leaves the diagonal entries $x_{11},\, x_{22}$ unchanged, whereas the real and imaginary parts of the off-diagonal entries are transformed as $$x'_{12}=x_{12}-2\alpha y_{12},\quad y'_{12}=y_{12}+2\alpha x_{12}.$$ In this case the invariance of the p.d.f. ${\cal P}(\hat{H})$ together with the statistical independence of the entries amount, after straightforward manipulations, to the condition $$\frac{1}{x_{12}}\frac{d\ln{f^{(1)}_{12}}}{dx_{12}}
=\frac{1}{y_{12}}\frac{d\ln{f^{(2)}_{12}}}{dy_{12}}$$ which together with the previously found $f^{(1)}_{12}(x_{12})$ yields $$f^{(2)}_{12}(y_{12})\propto e^{-2ay^2_{12}},$$ completing the proof of Eq.(\[11\]).
The Gaussian form of the probability density function, Eq.(\[11\]), can also be found as a result of rather different lines of thought. For example, one may invoke an information theory approach [*a la*]{} Shanon-Khinchin and define the amount of information ${\cal I} [{\cal P}(\hat{H})]$ associated with any probability density function ${\cal P}(\hat{H})$ by $$\label{12}
{\cal I} [{\cal P}(\hat{H})]=-\int d\mu(\hat{H})\,{\cal P}(\hat{H})
\ln{{\cal P}(\hat{H})}$$ This is a natural extension of the corresponding definition ${\cal I} [p_1,\ldots,p_m]=-\sum_{l=1}^mp_m\ln{p_m}$ for discrete events $1,...,m$.
Now one can argue that in order to have matrices $\hat{H}$ as random as possible one has to find the p.d.f. minimizing the information associated with it for a certain class of ${\cal
P}(H)$ satisfying some conditions. The conditions usually have a form of constraints ensuring that the probability density function has desirable properties. Let us, for example, impose the only requirement that the ensemble average for the two lowest traces $\mbox{Tr}\hat{H},\mbox{Tr}\hat{H}^2$ must be equal to certain prescribed values, say $E\left[\mbox{Tr}\hat{H}\right]=b$ and $E\left[\mbox{Tr}\hat{H}^2\right]=a>0$, where the $E\left[\ldots
\right]$ stand for the expectation value with respect to the p.d.f. ${\cal P}(H)$. Incorporating these constraints into the minimization procedure in a form of Lagrange multipliers $\nu_1,\nu_2$, we seek to minimize the functional $$\label{13}
{\cal I} [{\cal P}(\hat{H})]=-\int d\mu(\hat{H})\,{\cal
P}(\hat{H}) \left\{\ln{{\cal P}(\hat{H})}-\nu_1
\mbox{Tr}\hat{H}-\nu_2\mbox{Tr}\hat{H}^2\right\}.$$ The variation of such a functional with respect to $\delta{\cal
P}(\cal{H})$ results in $$\label{14}
\delta{\cal I} [{\cal P}(\hat{H})]=-\int
d\mu(\hat{H})\,\delta{\cal P}(\hat{H}) \left\{1+\ln{{\cal
P}(\hat{H})}-\nu_1
\mbox{Tr}\hat{H}-\nu_2\mbox{Tr}\hat{H}^2\right\}=0$$ possible only if $${\cal P}(\hat{H})\propto\exp\{\nu_1
\mbox{Tr}\hat{H}+\nu_2\mbox{Tr}\hat{H}^2\}$$ again giving the Gaussian form of the p.d.f. The values of the Lagrange multipliers are then uniquely fixed by constants $a,b$, and the normalization condition on the probability density function. For more detailed discussion, and for further reference see [@Mehta], p.68.
Finally, let us discuss yet another construction allowing one to arrive at the Gaussian Ensembles exploiting the idea of Brownian motion. To start with, consider a system whose state at time $t$ is described by one real variable $x$, evolving in time according to the simplest linear differential equation $\frac{d}{dt}x=-x$ describing a simple exponential relaxation $x(t)=x_0e^{-t}$ towards the stable equilibrium $x=0$. Suppose now that the system is subject to a random additive Gaussian white noise $\xi(t)$ function of intensity $D$ [^1], so that the corresponding equation acquires the form $$\label{Br1}
\frac{d}{dt}x=-x+\xi(t),\quad E_{\xi}\left[
\xi(t_1)\xi(t_1)\right]=D\delta(t_1-t_2),$$ where $E_{\xi}[\ldots]$ stands for the expectation value with respect to the random noise. The main characteristic property of a Gaussian white noise process is the following identity: $$\label{Br2}
E_{\xi}\left[\exp\left\{\int_a^b\xi(t)v(t)dt\right\}\right]=
\exp\left\{\frac{D}{2}\int_a^bv^2(t)dt\right\}$$ valid for any (smooth enough) test function $v(t)$. This is just a direct generalization of the standard Gaussian integral identity: $$\label{Gauint0}
\int_{-\infty}^{\infty}
\,\frac{dq}{\sqrt{2\pi a}}\, e^{-\frac{1}{2a}q^2+qb}=e^{\frac{ab^2}{2}}.$$ valid for $\mbox{Re}\,a>0$, and any (also complex) parameter $b$.
For any given realization of the Gaussian random process $\xi(t)$ the solution of the stochastic differential equation Eq.(\[Br1\]) is obviously given by $$\label{Br3}
x(t)=e^{-t}\left[x_0+\int_{0}^te^{\tau}\xi(\tau)d\tau\right].$$ This is a random function, and our main goal is to find the probability density function ${\cal P}(t,x)$ for the variable $x(t)$ to take value $x$ at any given moment in time $t$, if we know surely that $x(0)=x_0$. This p.d.f. can be easily found from the characteristic function $$\label{Br4}
{\cal F}(t,q)= E_{\xi}\left[e^{-iqx(t)}\right]
=\exp\left\{-iqx_0e^{-t}-\frac{Dq^2}{4}(1-e^{-2t})\right\}$$ obtained by using Eqs. (\[Br2\]) and (\[Br3\]). The p.d.f. is immediately recovered by employing the inverse Fourier transform: $$\label{Br5}
{\cal P}(t,x)=\int_{-\infty}^{\infty} \frac{dq}{2\pi} e^{iqx}
E_{\xi}\left[e^{-iqx(t)}\right] =\frac{1}{\sqrt{\pi D(1-e^{-2t})}}
\exp\left\{-\frac{\left(x-x_0e^{-t}\right)^2}{D(1-e^{-2t})}\right\}.$$
The formula Eq.(\[Br5\]) is called the Ornstein-Uhlenbeck (OU) probability density function, and the function $x(t)$ satisfying the equation Eq.(\[Br1\]) is known as the O-U process. In fact, such a process describes an interplay between the random “kicks" forcing the system to perform a kind of Brownian motion and the relaxation towards $x=0$. It is easy to see that when time grows the OU p.d.f. “forgets" about the initial state and tends to a [*stationary*]{} (i.e. time-independent) universal Gaussian distribution: $$\label{Br6}
{\cal P}(t\to\infty,x)=\frac{1}{\sqrt{\pi D}}
\exp\left\{-\frac{x^2}{D}\right\}.$$
Coming back to our main topic, let us consider $N^2$ independent OU processes: $N$ of them denoted as $$\label{Br7}
\frac{d}{dt}x_{i}=-x_{i}+\xi_i(t),\quad 1\le i\le N$$ and the rest $N(N-1)$ given by $$\label{Br8}
\frac{d}{dt}x_{ij}=-x_{ij}+\xi^{(1)}_{ij}(t), \quad
\frac{d}{dt}y_{ij}=-y_{ij}+\xi^{(2)}_{ij}(t),$$ where the indices satisfy $\quad 1\le i<j\le N$. Stochastic processes $\xi(t)$ in the above equations are taken to be all mutually independent Gaussian white noises characterized by the correlation functions: $$\label{Br9}
E_{\xi}\left[\xi_{i_1}(t_1)\xi_{i_2}(t_2)\right]=2D\delta_{i_1,i_2}
\delta(t_1-t_2),\,\,E_{\xi}\left[\xi^{\sigma_1}_{ij}(t_1)
\xi^{\sigma_2}_{kl}(t_2)\right]=
D\delta_{\sigma_1,\sigma_2}\delta_{i,k}\delta_{j,l}\delta(t_1-t_2).$$ As initial values $x_i(0),x_{ij}(0),y_{ij}(0)$ for each OU process we choose diagonal and off-diagonal entries $H_{ii}^{(0)},\,\,
i=1,\ldots,N$ and $\mbox{Re}H_{i<j}^{(0)},\,
\mbox{Im}H_{i<j}^{(0)}$ of a fixed $N\times N$ Hermitian matrix $\hat{H}^{(0)}$. Let us now consider the Hermitian matrix $\hat{H}(t)$ whose entries are $H_{ii}(t)=x_i(t),\,
H_{i<j}(t)=x_{i<j}(t)+iy_{i<j}(t)$ for any $t\ge 0$. It is immediately clear that the joint p.d.f. ${\cal
P}\left(t,\hat{H}\right)$ of the entries of such a matrix $\hat{H}(t)$ will be given for any $t\ge 0 $ by the OU-type formula: $$\label{Br10}
{\cal P}(t,\hat{H}) \propto Const\times
\frac{1}{\sqrt{(1-e^{-2t})^{N^2}}}
\exp\left\{-\frac{1}{D(1-e^{-2t})}\mbox{Tr}\left(\hat{H}-\hat{H}_0
e^{-t}\right)^2\right\}.$$ In the limit $t\to\infty$ this p.d.f. converges to a stationary, $t-$independent expression $$\label{Br11}
{\cal P}(t,\hat{H})
\propto C\, e^{-\frac{1}{D}\mbox{Tr}\hat{H}^2}$$ [*independent*]{} of the initial matrix $\hat{H}_0$. We see therefore that the familiar Gaussian ensemble in the space of Hermitian matrices arises as the result of the stochastic relaxation from any initial condition, in particular, from any diagonal matrix with uncorrelated entries. In the next step one may try to deduce the stochastic dynamics of the [*eigenvalues*]{} of the corresponding matrices. Those eigenvalues obviously evolve from completely uncorrelated to highly correlated patterns. This very interesting set of question goes beyond our present goals and we refer to [@Mehta] for an introduction into the problem.
After specifying the probability density function ${\cal P}(H)$ the main question of interest is to characterize the statistical properties of the sequence of eigenvalues $\lambda_1,\ldots,\lambda_N$ of $\hat{H}$. A convenient way of doing this is to start with the joint p.d.f. of all these eigenvalues. Because of the “rotational invariance" assumption the function ${\cal P}(\hat{H})$ depends in fact only on the eigenvalues, for example for the “symmetric" Gaussian case $b=0$ we have ${\cal P}(\hat{H})\propto e^{-a\sum_{i=1}^N\lambda_i^2}$. Moreover, we have seen that the integration measure $d\mu(\hat{H})$ when expressed in terms of eigenvalues and eigenvectors effectively factorizes, see Eq.(\[85\]). Collecting all these facts we arrive at the conclusion, that the relevant joint p.d.f of all eigenvalues can be always written, up to a normalization constant, as $$\label{JPDG}
{\cal P}(\lambda_1,\ldots,\lambda_N)\,d\lambda_1 \ldots d\lambda_N
\propto e^{-\sum_{i=1}^NQ(\lambda_i)}
\prod_{i<j}\left(\lambda_i-\lambda_j\right)^2
\,\,d\lambda_1 \ldots d\lambda_N$$ for a general, non-gaussian weight $e^{-\mbox{Tr}Q(\hat{H})}$. We immediately see that the presence of the “Jacobian factor" $\prod_{i<j}\left(\lambda_i-\lambda_j\right)^2$ is responsible of the fact that the eigenvalues are correlated in a non-trivial way. In what follows we are going to disregard the fact that eigenvalues $\lambda_i$ were initially put in increasing order. More precisely, for any [*symmetric*]{} function $f$ of $N$ real variables $\lambda_1,\ldots,\lambda_N$ the expected value will be calculated as $$\int_{{\bf R}^N} f(\lambda_1,\ldots,\lambda_N) {\cal
P}(\lambda_1,\ldots,\lambda_N)d\lambda_1\,\ldots\, d\lambda_N.$$ Indeed, with p.d.f. being symmetric with respect to permutations of the eigenvalue set, disregarding the ordering amounts to a simple multiplicative combinatorial factor $n!$ in the normalization constant.
Our main goal is to extract the information about these eigenvalue correlations in the limit of large size $N$. From this point of view it is pertinent to discuss a few quantitative measures frequently used to characterize correlations in sequences of real numbers.
Characterization of Spectral Sequences
======================================
Let $-\infty<\lambda_1,\lambda_2,\ldots,\lambda_N<\infty$ be the positions of $N$ points on the real axis, characterized by the joint probability density function (JPDF) $${\cal P}(\lambda_1,\lambda_2,\ldots,\lambda_N)
\,\,d\lambda_1\,\ldots\,d\lambda_N$$ of having, [*regardless of labelling*]{}, one point in the interval $[\lambda_1,\lambda_1+d\lambda_1]$, another in the interval $[\lambda_2,\lambda_2+d\lambda_2]$,..., another in $[\lambda_N,\lambda_N+d\lambda_N]$. Since in this section we deal exclusively with real variables, the bar will stand for the expectation value with respect to such a JPDF.
The statistical properties of the sequence $\{\lambda_i\}$ are conveniently characterized by the set of $n-$point [*correlation functions*]{}, defined as $$\label{21}
{\cal R}_{n}(\lambda_1,\lambda_2,\ldots,\lambda_n)
=\frac{N!}{(N-n)!}\int{\cal
P}(\lambda_1,\lambda_2,\ldots,\lambda_N) \,\,
d\lambda_{n+1}\,\ldots\,d\lambda_N.$$ It is obvious from this definition that the lower correlation functions can be obtained from the higher-order ones: $$\label{22}
{\cal R}_{n}(\lambda_1,\lambda_2,\ldots,\lambda_n)
=\frac{1}{(N-n)}\int{\cal R}_{n+1}(\lambda_1,\lambda_2,\ldots,\lambda_{n+1})
\,\, d\lambda_{n+1}.$$
To provide a more clear interpretation of these correlation functions we relate them to the statistics of the number $N_B$ of points of the sequence $\{\lambda_i\}$ within any set $B$ of the real axis (e.g an interval $[a,b]$). Let $\chi_B(x)$ be the characteristic function of the set $B$, equal to unity if $x\in B$ and zero otherwise. Introduce the exact density function $\rho_N(\lambda)$ of the points $\{\lambda_i\}$ around the point $\lambda$ on the real axis. It can be conveniently written using the Dirac’s $\delta-$function as $\rho_N(\lambda)=\sum_{i=1}^N\delta(\lambda-\lambda_i)$. Then $N_B=\int\chi_B(\lambda)\rho_N(\lambda)d\lambda$.
On the other hand, consider $$\begin{aligned}
\label{23}
\nonumber &&\int_B{\cal R}_{1}(\lambda_1)d\lambda_1=
\int\chi_B(\lambda_1){\cal R}_{1}(\lambda_1)d\lambda_1
=N\int \chi_B(\lambda_1){\cal P}(\lambda_1,\lambda_2,\ldots,\lambda_N)
\,\, d\lambda_{1}\,\ldots\,d\lambda_N\\
&&
=\int\, \sum_{i=1}^N\chi_B(\lambda_i){\cal P}
(\lambda_1,\lambda_2,\ldots,\lambda_N) \,\,
d\lambda_{1}\,\ldots\,d\lambda_N\end{aligned}$$ and therefore $$\begin{aligned}
\label{24}
\int_B{\cal R}_{1}(\lambda_1)d\lambda_1=\overline{ N_B}=
\mbox{expectation of the number of points in B}.\end{aligned}$$
Similarly, consider $$\begin{aligned}
\label{25}
\nonumber &&\int\chi_B(\lambda_1)\chi_B(\lambda_2) {\cal
R}_{2}(\lambda_1,\lambda_2)d\lambda_1d\lambda_2 =N(N-1)\int
\chi_B(\lambda_1)\chi_B(\lambda_2){\cal P}
(\lambda_1,\lambda_2,\ldots,\lambda_N)
\,\, d\lambda_{1}\,\ldots\,d\lambda_N\\
&& = \int\, \sum_{i\ne j}^N\chi_B(\lambda_i)\chi_B(\lambda_j)
{\cal P} (\lambda_1,\lambda_2,\ldots,\lambda_N) \,\,
d\lambda_{1}\,\ldots\,d\lambda_N,\end{aligned}$$ which can be interpreted as $$\begin{aligned}
\label{26}
\int_{B\times B}{\cal R}_{2}(\lambda_1,\lambda_2)d\lambda_1d\lambda_2=
\mbox{expectation of the number of pairs of points in B}\end{aligned}$$ where if, say, $\lambda_1$ and $\lambda_2$ are in $B$, then the pair $\{1,2\}$ and $\{2,1\}$ are both counted.
To relate the two-point correlation function to the variance of $N_B$ we notice that in view of Eq.(\[24\]) the one-point correlation function ${\cal R}_1(\lambda)$ coincides with the mean density $\overline{\rho}_N(\lambda)$ of the points $\{\lambda_i\}$ around the point $\lambda$ on the real axis. Similarly, write the mean square $\overline{N^2_B}$ $$\label{2p}
\overline{N^2_B}=\int\chi_B(\lambda)\chi_B(\lambda')
\overline{\rho_N(\lambda)\rho_N(\lambda')}\,\,d\lambda d\lambda'$$ and notice that $$\begin{aligned}
\label{27}
\nonumber &&\overline{\rho_N(\lambda)\rho_N(\lambda')}=\overline{\sum_{ij}
\delta(\lambda-\lambda_i)\delta(\lambda'-\lambda_j)}
=\delta(\lambda-\lambda')\overline{\sum_{i}
\delta(\lambda-\lambda_i)}+\overline{\sum_{i\ne j}
\delta(\lambda-\lambda_i)\delta(\lambda'-\lambda_j)}\\
&& =\delta(\lambda-\lambda'){\cal R}_1(\lambda)+
{\cal R}_2(\lambda,\lambda').\end{aligned}$$ In this way we arrive at the relation: $$\overline{N^2_B}=\overline{N_B}+\int_{B\times B}
{\cal R}_{2}(\lambda,\lambda')\,\,d\lambda d\lambda'.$$ In fact, it is natural to introduce the so-called “number variance" statistics $\Sigma_2(B)=\overline{N^2_B}-\left[\overline{N_B}\right]^2$ describing the variance of the number of points of the sequence inside the set $B$. Obviously, $$\label{sigma2}
\Sigma_2(B)=\overline{N_B}+\int_{B\times B}\left[ {\cal
R}_{2}(\lambda,\lambda')-{\cal R}_{1}(\lambda){\cal
R}_{1}(\lambda') \right]\,\,d\lambda d\lambda'\equiv
\overline{N_B}-\int_{B\times B}
Y_{2}(\lambda,\lambda')\,\,d\lambda d\lambda'$$ where we introduced the so-called [*cluster function*]{} $Y_{2}(\lambda,\lambda')= {\cal R}_{1}(\lambda){\cal
R}_{1}(\lambda')-{\cal R}_{2}(\lambda,\lambda')$ frequently used in applications.
Finally, in principle the knowledge of all $n-$point correlation functions provides one with the possibility of calculating an important characteristic of the spectrum known as the “hole probability" $A(L)$. This quantity is defined as the probability for a random matrix to have [*no*]{} eigenvalues in the interval $(-L/2,L/2)$ [^2]. Define $\chi_L(\lambda)$ to be the characteristic function of this interval. Obviously, $$\begin{aligned}
\label{hole}
&& A(L)=\int\ldots\int\, {\cal
P}(\lambda_1,\ldots,\lambda_N)\prod_{k=1}^N
\left(1-\chi_L(\lambda_k)\right)\,\, d\lambda_1 \ldots d\lambda_N\\
&& =\sum_{j=0}^N(-1)^j\, \int\ldots\int\, {\cal
P}(\lambda_1,\ldots,\lambda_N)h_j\left\{
\chi_L(\lambda_1),\ldots,\chi_L(\lambda_N)\right\}\,\, d\lambda_1
\ldots d\lambda_N,\end{aligned}$$ where $h_j\{x_1,\ldots,x_N\}$ is the $j-th$ symmetric function: $$h_0\{x_1,\ldots,x_N\}=1,\,\,h_1\{x_1,\ldots,x_N\}=\sum_{i=1}^Nx_i\, ,$$ $$h_2\{x_1,\ldots,x_N\}=\sum_{i<j}^Nx_ix_j, \quad \ldots ,\quad
h_N\{x_1,\ldots,x_N\}=x_1x_2\ldots x_N.$$ Now, for $1\le j\le N$ $$\begin{aligned}
\label{hole1}
\nonumber && \int\ldots\int\,\prod_{k=1}^j \chi_L(\lambda_k)\,\,
{\cal
P}(\lambda_1,\ldots,\lambda_N) d\lambda_1 \ldots d\lambda_N\\
&&=\frac{(N-j)!}{N!} \int\ldots\int \prod_{k=1}^j
\chi_L(\lambda_k) {\cal R}_j(\lambda_1,\ldots,\lambda_j)\,\,
d\lambda_1 \ldots d\lambda_j\\&& =
\frac{(N-j)!}{N!}\int_{|x_1|<L/2}\ldots \int_{|x_j|<L/2}\, {\cal
R}_j(\lambda_1,\ldots,\lambda_j)\,\, d\lambda_1 \ldots d\lambda_j.\end{aligned}$$ As $h_j$ contains $\left(\begin{array}{c}N \\ j\end{array}\right)$ terms and as ${\cal R}_j(\lambda_1,\ldots,\lambda_j)$ is invariant under permutations of the arguments, it follows that $$\begin{aligned}
\label{hole2}
\nonumber && \int\ldots\int {\cal
P}(\lambda_1,\ldots,\lambda_N)h_j\left\{
\chi_L(\lambda_1)\ldots\chi_L(\lambda_N)\right\}\,\, d\lambda_1
\ldots d\lambda_j\\ && =\frac{(N-j)!}{N!}\left(\begin{array}{c}N
\\ j\end{array}\right)
\int_{|x_1|<L/2}\ldots \int_{|x_j|<L/2}\,
{\cal R}_j(\lambda_1,\ldots,\lambda_j)\,\,
d\lambda_1 \ldots d\lambda_j=\\ && \frac{1}{j!}
\int_{|x_1|<L/2}\ldots \int_{|x_j|<L/2}\,
{\cal R}_j(\lambda_1,\ldots,\lambda_j)\,\,
d\lambda_1 \ldots d\lambda_j.\end{aligned}$$ Thus, we arrive at the following relation between the hole probability and the $n-$point correlation functions: $$\begin{aligned}
\label{hole3}
&& A(L)=\sum_{j=0}^N\frac{(-1)^j}{j!}
\int_{-L/2}^{L/2}\ldots \int_{-L/2}^{L/2}\,
{\cal R}_j(\lambda_1,\ldots,\lambda_j)\,\,d\lambda_1 \ldots d\lambda_j.\end{aligned}$$
One of the main goals of this set of lectures is to develop a method allowing to evaluate all the $n-$point correlation functions of the eigenvalues for any JPDF corresponding to unitary invariant ensembles of the form Eq.(\[9\]). After that we will concentrate on a particular case of Gaussian weight and will investigate the limiting behaviour of the kernel function $K_n(\lambda,\lambda')$ as $N\to \infty$. But even before doing this it is useful to keep in mind for reference purposes the results corresponding to completely uncorrelated (a.k.a. Poissonian) spectra. Those are described by a sequence of real points $\lambda_1,\ldots,\lambda_N$, characterized by the fully factorized JPDF: $$\label{uncor}
{\cal P}(\lambda_1,\lambda_2,\ldots,\lambda_N)
\,\,=p(\lambda_1)\,\ldots\,p(\lambda_N).$$ The normalization condition requires $\int_{-\infty}^{\infty}
p(\lambda)\,d\lambda=1$, and we further assume $p(\lambda)$ to be a smooth enough integrable function. Obviously, for this case $$\label{uncor1}
{\cal R}_{n}(\lambda_1,\lambda_2,\ldots,\lambda_n)
=\frac{N!}{(N-n)!}p(\lambda_1)\,\ldots\,p(\lambda_n).$$ In particular, ${\cal R}_{n}(\lambda)=Np(\lambda)$ which is just the mean density $\overline{\rho(\lambda)}$ of points $\{\lambda_i\}$ around the point $\lambda$ on the real axis, and ${\cal
R}_{n}(\lambda_1,\lambda_2)=N(N-1)p(\lambda_1) p(\lambda_2)$, etc.. From this we easily find for the number of levels in the domain $B$ and for its mean square: $$\label{uncor2}
\overline{N_B}=N\int_{B}p(\lambda)\,d\lambda,\quad
\overline{N^2_B}=\overline{N_B}(N-1)/N+\left[\overline{N_B}\right]^2$$ and for the hole probability $$\begin{aligned}
\label{uncorhole}
&& A(L)=\sum_{j=0}^N\frac{(-1)^j}{j!}\frac{N!}{(N-j)!}
\left[\int_{-L/2}^{L/2} p(\lambda)\,d\lambda\right]^j=\left[1-
\int_{-L/2}^{L/2} p(\lambda)\,d\lambda\right]^N.\end{aligned}$$ Finally, let us specify $B$ to be the interval $[-L/2,L/2]$ around the origin, and being interested mainly in large $N\gg 1$ consider the length $L$ comparable with the mean spacing between neighbouring points in the sequence $\{\lambda_i\}$ close to the origin, given by $\Delta\equiv
\left[\overline{\rho(0)}\right]^{-1}= 1/[Np(0)]$. In other words $s=L/\Delta=LNp(0)$ stays finite when $N\to\infty$. On the other hand, for large enough $N$ the function $p(\lambda)$ can be considered practically constant through the interval of the length $L=O(1/N)$, and therefore the mean number of points of the sequence $\{\lambda_i\}$ inside the interval $[-L/2,L/2]$ will be asymptotically given by $\overline{N(s)}\approx N\,L\,p(0)=s$. Similarly, using Eq.(\[uncor2\]) one can easily calculate the “number variance" $\Sigma_2(s)=\overline{N^2_{[-\frac{L}{2},\frac{L}{2}]}} -
\left[\overline{N_{[-\frac{L}{2},\frac{L}{2}]}}\right]^2=
(N-1)\int_{-L/2}^{L/2} p(\lambda)\,d\lambda\approx s$. In the same approximation the hole probability, Eq.(\[uncorhole\]), tends asymptotically to $A(s)\approx e^{-s}$. Later on we shall compare these results with the corresponding behaviour emerging from the random matrix calculations.
The method of orthogonal polynomials
====================================
In the heart of the method developed mainly by Dyson, Mehta and Gaudin lies an “integrating-out" Lemma [@Mehta]. In presenting this material I follow very closely [@Deift], pp.103-105.
- [*Let $J_n=J_n({\bf x})=(J_{ij})_{1\le i,j\le n}$ be an $n\times n$ matrix whose entries depend on a real vector ${\bf
x}=(x_1,x_2,\ldots,x_n)$ and have the form $J_{ij}=f(x_i,x_j)$, where $f$ is some (in general, complex-valued) function satisfying for some measure $d\mu(x)$ the “reproducing kernel" property: $$\label{31}
\int f(x,y)f(y,z)\,d\mu(y)=f(x,z).$$ Then $$\label{32}
\int \mbox{det}\,J_n({\bf x})\,d\mu(x_n)=[q-(n-1)]\mbox{det}\,J_{n-1}$$ where $q=\int f(x,x)\,d\mu(x)$, and the matrix $J_{n-1}=(J_{ij})_{1\le i,j\le n-1}$ have the same functional form as $J_n$ with ${\bf x}$ replaced by $(x_1,x_2,\ldots,x_{n-1}).$* ]{}
Before giving the idea of the proof for an arbitrary $n$ it is instructive to have a look on the simplest case $n=2$, when $$J_2=\left(\begin{array}{cc}f(x_1,x_1)&f(x_1,x_2)\\
f(x_2,x_1)&f(x_2,x_2)\end{array} \right),\quad \mbox{hence}\quad
\det{J_n}=f(x_1,x_1)f(x_2,x_2)-f(x_1,x_2)f(x_2,x_1).$$ Integrating the latter expression over $x_2$, and using the “reproducing kernel" property, we immediately see that the result is indeed just $(q-1)f(x,x)=(q-1)\det{J_1}$ in full agreement with the statement of the Lemma.
For general $n$ one should follow essentially the same strategy and expand the determinant as a sum over $n!$ permutations $P_{n}(\sigma)=(\sigma_1,\ldots\sigma_n)$ of the index set $1,\ldots,n$ as $$\label{33}
\int \mbox{det}\,J_n({\bf x})\,d\mu(x_n)=\sum_{P_{n}}
(-1)^{P_n}\int f(x_1,x_{\sigma_1})\ldots f(x_n,x_{\sigma_n})
\,d\mu(x_n),$$ where $(-1)^{P_n}$ stands for the sign of permutations. Now, we classify the terms in the sum according to the actual value of the index $\sigma_n=k,\,k=1,2,\ldots,n$. Consider first the case $\sigma_n=n$, when effectively only the last factor $f(x_n,x_n)$ in the product is integrated yielding $d$ upon the integration. Summing up over the remaining $(n-1)!$ permutations $P_{n-1}(\sigma)$ of the index set $(1,2,...,n-1)$ we see that: $$\sum_{P_{n-1}}(-1)^{P_{n}}\int \, f(x_1,x_{\sigma_1})\ldots
f(x_n,x_{n}) \,d\mu(x_n) =q\sum_{P_{n-1}}(-1)^{P_{n-1}}\,
f(x_1,x_{\sigma_1})\ldots f(x_{n-1},x_{\sigma_{n-1}}),$$ which is evidently equal to $q\det{J_{n-1}}$. Now consider $(n-1)!$ terms with $\sigma_n=k<n$, when we have $\sigma_j=n$ for some $j<n$. For every such term we have by the “reproducing property" $$\int \,f(x_1,x_{\sigma_1}) \ldots f(x_j,x_{n})\ldots
f(x_n,x_{k}) \,d\mu(x_n)
=f(x_1,x_{\sigma_1})\ldots f(x_j,x_{k})
\ldots f(x_{n-1},x_{\sigma_{n-1}}).$$ Therefore $$\label{35}
\int \mbox{det}\,J_n({\bf x})\,d\mu(x_n)=
q\det{J_{n-1}}+\sum_{k=1}^{n-1}
\sum_{P_{n}:\sigma_n=k)}
(-1)^{P_n}f(x_1,x_{\sigma_1})\ldots f(x_j,x_k)
\ldots f(x_{n-1},x_{\sigma_{n-1}}).$$ It is evident that the structure and the number of terms is as required, and the remaining task is to show that the summation over all possible $(n-1)!$ permutation of the index set for fixed $k$ yields always $-\det{J_{n-1}}$, see [@Deift]. Then the whole expression is indeed equal to $[q-(n-1)]\det{J_{n-1}}$ as required.
Our next step is to apply this Lemma to calculating the $n-$point correlation functions of the eigenvalues $\lambda_1,\ldots,\lambda_n$ starting from the JPDF, Eq.(\[JPDG\]).
For this we notice that $$\label{vdm}
\prod_{i<j}^N(\lambda_i-\lambda_j)=(-1)^{\frac{N(N-1)}{2}}
\det{\left(\begin{array}{ccc}1&\ldots & 1\\ \lambda_1&\ldots & \lambda_N\\
.&.&.\\ .&.&.\\.&.&.\\\lambda^{N-1}_1&\ldots & \lambda^{N-1}_N
\end{array}\right)}\equiv \Delta_N(\lambda_1,\ldots,\lambda_N),$$ where the determinant in the right-hand side is the famous van der Monde determinant. Since the determinant cannot change upon linearly combining its rows, the entries $\lambda_i^k$ in $(k+1)-th$ row of the van der Monde determinant can be replaced, up to a constant factor $a_0a_1...a_{N-1}$, by a polynomial of degree $k$ of the form: $\pi_k(\lambda_i)=a_k\lambda_i^k+\mbox{any
polynomial in }\, \lambda_i\,
\mbox{of degree less than
k}$, with any choice of the coefficients $a_l,\,l=0,\ldots,k$. Therefore: $$\label{vdm1}
\prod_{i<j}^N(\lambda_i-\lambda_j)=\frac{(-1)^{\frac{N(N-1)}{2}}}
{a_0a_1...a_{N-1}}\det{
\left(\begin{array}{ccc}\pi_0(\lambda_1)&\ldots& \pi_0(\lambda_N)
\\ \pi_1(\lambda_1)&\ldots &\pi_1(\lambda_N)\\
.&.&.\\ .&.&.\\.&.&.\\ \pi_{N-1}(\lambda_1)&\ldots& \pi_{N-1}(\lambda_1)
\end{array}\right)}\equiv \frac{(-1)^{\frac{N(N-1)}{2}}}
{a_0a_1...a_{N-1}}\det{\left(
\pi_{i-1}(\lambda_j)\right)_{1\le i,j\le N}}.$$ Multiplying every entry in $j_{th}$ column in the above determinant with the factor $e^{-\frac{1}{2}Q(\lambda_j)}$ we see that the JPDF can be conveniently written, up to a multiplicative constant, as $$\label{JPD1}
{\cal P}(\lambda_1,\ldots,\lambda_N)\propto
\left[\det{\left(e^{-\frac{1}{2}Q(\lambda_j)}\pi_{i-1}(\lambda_j)
\right)_{1\le i,j\le N}}\right]^2.$$ If we let $\hat{A}$ be the matrix with the entries $A_{ij}=
\left(\phi_{i-1}(x_j)\right)_{1\le i,j\le N}$, then $$\begin{aligned}
\label{orp}
[\det{A}]^2=\det{\hat{A}^T\hat{A}}=
\det{\left(\sum_{j=1}^n A_{ji}A_{jk}\right)}.\end{aligned}$$ This implies the following form of the JPDF: $$\label{JPD2}
{\cal P}(\lambda_1,\ldots,\lambda_N)\propto
\det{\left(\sum_{j=1}^N \phi_{j-1}(\lambda_i)\phi_{j-1}(\lambda_k)
\right)_{1\le i,k\le N}}\equiv
\det{\left(K_N(\lambda_i,\lambda_k)\right)_{1\le i,k\le N}}$$ where we introduced the notation: $$\label{kern}
K_N(\lambda,\lambda')=\sum_{j=0}^{N-1}\phi_j(\lambda)\phi_j(\lambda')$$ usually called “kernel" in the literature. In our particular case $$\label{orp1}
\phi_{i-1}(\lambda)=
e^{-\frac{1}{2}Q(\lambda)}\pi_{i-1}(\lambda)$$ so that the kernel is given explicitly by $$\label{kern1}
K_N(\lambda,\lambda')=e^{-\frac{1}{2}\left(Q(\lambda)+Q(\lambda')\right)}
\sum_{j=0}^{N-1}\pi_{j}(\lambda)\pi_{j}(\lambda').$$ Now it is easy to see that if we take the polynomials $\pi_{i}(x)$ such that they form an [*orthonormal system*]{} with respect to the weight $e^{-Q(x)}$, the corresponding kernel will be a “reproducing" one with respect to the measure $d\mu(x)\equiv dx$, in the sense of the “integrating-out" Lemma. Indeed, suppose that $\pi_i(x)$ satisfy the orthonormality conditions: $$\label{orp2}
\int e^{-Q(x)}\pi_i(x)\pi_j(x)\,dx=\delta_{ij},$$ for any indices $i\ge 1,\,j\ge 1$. Then we obviously have $$\begin{aligned}
\label{kern2}
\nonumber &&
\int K_N(x,y)K_N(y,z)dy=\sum_{j=0}^{N-1}\sum_{k=0}^{N-1}
e^{-\frac{1}{2}\left(Q(x)+Q(z)\right)}\pi_{j}(x)\pi_{k}(z)
\int \pi_{j}(y)\pi_{k}(y)e^{-Q(y)}dy\\
&&=\sum_{j=0}^{N-1}e^{-\frac{1}{2}\left(Q(x)+Q(z)\right)}
\pi_{j}(x)\pi_{j}(z)=K_N(x,z)\end{aligned}$$ exactly as required by the reproducing property. Moreover, in this case obviously $$q_N=\int K_N(x,x)dx=\sum_{j=0}^{N-1}\int\,
e^{-Q(x)}\pi_j(x)\pi_j(x)\,dx=N,$$ and therefore the relation (\[32\]) amounts to $$\label{kern3}
\int \mbox{det}\left(K_N(x_i,x_j)\right)_{1\le i,j\le N}
\,dx_N=\mbox{det}\left(K_N(x_i,x_j)\right)_{1\le i,j\le N-1}.$$ Continuing this process one step further we see $$\begin{aligned}
\label{kern4}
&&\int\ \int \mbox{det}\left(K_N(x_i,x_j)\right)_{1\le i,\,j\le N}
\,dx_{N-1}dx_{N}=
\int \mbox{det}\left(K_N(x_i,x_j)\right)_{1\le i,\,j\le N-1}dx_{N-1}
\\
&=& [N-(N-2)]\mbox{det}\left(K_N(x_i,x_j)\right)_{1\le i,\,j\le N-2}\end{aligned}$$ and continuing by induction $$\label{kern5}
\int\ldots \int\mbox{det}
\left(K_N(x_i,x_j)\right)_{1\le i,\,j\le N}
\,dx_{k+1}\ldots dx_{N}=
(N-k)!\,\mbox{det}\left(K_N(x_i,x_j)\right)_{1\le i,\,j\le k}$$ for $k=1,2,\ldots$, and the result is $N!$ for $k=0$. Remembering the expression of the JPDF, Eq.(\[JPD2\]), in terms of the kernel $K_N(x_i,x_j)$ we see that, in fact, the theory developed provided simultaneously the explicit formulae for all $n-$point correlation functions of the eigenvalues ${\cal R}_n(\lambda_1,
\ldots, \lambda_n)$, introduced by us earlier, Eq.(\[23\]): $$\label{kern6}
{\cal R}_n(\lambda_1, \ldots, \lambda_n)=
\mbox{det}\left(K_N(\lambda_i,\lambda_j)\right)_{1\le i,j\le n}$$ expressed, in view of the relations Eq.(\[kern1\]) effectively in terms of the orthogonal polynomials $\pi_k(\lambda)$. In particular, remembering the relation between the mean eigenvalue density and the one-point function derived by us earlier, we have: $$\label{kern7}
\overline{\rho_N(\lambda)}=K_N(\lambda,\lambda)=
\sum_{j=1}^{N-1}
e^{-Q(\lambda)}\pi_{j-1}(\lambda)\pi_{j-1}(\lambda).$$ The latter result allows to represent the “connected" (or “cluster") part of the two-point correlation function introduced by us in Eq.(\[sigma2\]) in the form: $$\label{kern8}
Y_2(\lambda_1,\lambda_2)=\overline{\rho_N(\lambda_1)}\,\,\,
\overline{\rho_N(\lambda_2)} -{\cal R}_2(\lambda_1, \lambda_2)
=\left[K_N(\lambda_1,\lambda_2)\right]^2.$$
Finally, combining the relation Eq.(\[hole3\]) between the hole probability $A(L)$ and the n-point correlation functions, and on the other hand the expression of the latter in terms of the kernel $K_N(\lambda,\lambda')$, see Eq.(\[kern6\]), we arrive at $$\begin{aligned}
\label{holekern}
A(L)=\sum_{j=0}^N\frac{(-1)^j}{j!} \int_{-L/2}^{L/2}\ldots
\int_{-L/2}^{L/2}\, \det{\left(\begin{array}{ccc}
K_N(\lambda_1,\lambda_1)&\ldots&K_N(\lambda_1,\lambda_j)\\
.&.&.\\.&.&.\\.&.&.\\
K_N(\lambda_j,\lambda_1)&\ldots&K_N(\lambda_j,\lambda_j)
\end{array}\right)}
\,\, d\lambda_1 \ldots d\lambda_j.\end{aligned}$$ In fact, the last expression can be written in a very compact form by noticing that it is just a Fredholm determinant $\det{\left({\cal I}-{\cal K}_N\right)}$, where ${\cal K}_N$ is a (finite rank) integral operator with the kernel $K_N(\lambda,\lambda')=\sum_{i=0}^{N-1}\phi_i(\lambda)\phi_i(\lambda')$ acting on square-integrable functions on the interval $\lambda\in (-L/2,L/2)$.
Properties of Hermite polynomials
=================================
Orthogonality, Recurrent Relations and Integral Representation
--------------------------------------------------------------
Consider the set of polynomials $h_k(x)$ defined as [^3] $$\label{Her1}
h_k(x)=(-1)^ke^{N\frac{x^2}{2}}
\frac{d^k}{dx^k}\left(e^{-N\frac{x^2}{2}}\right)=N^kx^k+\cdots,$$ and consider, for $k\ge l$ $$\begin{aligned}
\label{Her2}
&&\int_{-\infty}^{\infty}
e^{-N\frac{x^2}{2}}h_l(x)h_k(x)dx=
(-1)^k\int_{-\infty}^{\infty}\,dx\, h_l(x)
\frac{d^k}{dx^k}\left(e^{-N\frac{x^2}{2}}\right)\\ \nonumber
&&=(-1)^{k+1}\int_{-\infty}^{\infty}\,dx\, h'_l(x)
\frac{d^{k-1}}{dx^{k-1}}\left(e^{-N\frac{x^2}{2}}\right)
=\ldots=(-1)^{2k}\int_{-\infty}^{\infty}\,dx\, \,e^{-N\frac{x^2}{2}}
\frac{d^k}{dx^{k}} h_l(x).\end{aligned}$$ Obviously, for $k>l$ we have $\frac{d^{k}}{dx^{k}} h_l(x)=0$, whereas for $k=l$ we have $\frac{d^{k}}{dx^{k}} h_k(x)=k!N^k$. In this way we verified the orthogonality relations and the normalization conditions $$\begin{aligned}
\label{Her3}
\int_{-\infty}^{\infty}
e^{-N\frac{x^2}{2}}\tilde{h}_l(x)\tilde{h}_k(x)dx= \delta_{kl}\end{aligned}$$ for normalized polynomials $$\label{normher}
\tilde{h}_k(x)=:\frac{1}{\left[k!N^k\sqrt{\frac{2\pi}{N}}\right]^{1/2}}
h_k(x).$$ In the theory of orthogonal polynomials an important role is played by recurrence relations: $$\begin{aligned}
\label{Her4}& h_{k+1}(x)=(-1)^{k+1} e^{N\frac{x^2}{2}}
\frac{d^k}{dx^k} \left(\frac{d}{dx}e^{-N\frac{x^2}{2}}\right)=
(-1)^{k+2}N e^{N\frac{x^2}{2}}\frac{d^k}{dx^k}
\left(xe^{-N\frac{x^2}{2}}\right)\\ &
\nonumber =(-1)^{k+2}N e^{N\frac{x^2}{2}}
\left[\left(\begin{array}{c}0\\ k\end{array}
\right)x\frac{d^k}{dx^k}\left(e^{-N\frac{x^2}{2}}\right)+
\left(\begin{array}{c}1\\ k\end{array}
\right)\frac{d^{k-1}}{dx^{k-1}}\left(e^{-N\frac{x^2}{2}}\right)\right]
=N\left[x\,h_k(x)-k\,h_{k-1}(x)\right],\end{aligned}$$ where we exploited the Leibniz formula for the $k-$th derivative of a product. After normalization we therefore have $$\label{Her5}
\left[\frac{k+1}{N}\right]^{1/2}\tilde{h}_{k+1}(x)=x\,
\tilde{h}_{k}(x)-\left[\frac{k}{N}\right]^{1/2}\tilde{h}_{k-1}(x).$$ Let us multiply this relation with $\tilde{h}_k(y)$, and then replace $x$ by $y$. In this way we arrive at two relations: $$\begin{aligned}
\label{Her6}
&& \left[\frac{k+1}{N}\right]^{1/2}\tilde{h}_{k+1}(x)
\tilde{h}_{k}(y)=x\,
\tilde{h}_{k}(x)\tilde{h}_{k}(y)-
\left[\frac{k}{N}\right]^{1/2}\tilde{h}_{k-1}(x)\tilde{h}_{k}(y),\\
&& \left[\frac{k+1}{N}\right]^{1/2}\tilde{h}_{k+1}(y)
\tilde{h}_{k}(x)=y\,
\tilde{h}_{k}(x)\tilde{h}_{k}(y)-
\left[\frac{k}{N}\right]^{1/2}\tilde{h}_{k-1}(y)\tilde{h}_{k}(x).\end{aligned}$$ The difference between the upper and the lower line can be written for any $k=1,2,\ldots $ as $$(x-y)\tilde{h}_{k}(x)\tilde{h}_{k}(y)=A_{k+1}-A_k\,,\quad A_k=
\left[\frac{k}{N}\right]^{1/2}\{\tilde{h}_{k-1}(y)\tilde{h}_{k}(x)-
\tilde{h}_{k-1}(x)\tilde{h}_{k}(y)\}.$$ Summing up these expressions over $k$: $$(x-y)\sum_{k=1}^{n-1}\tilde{h}_{k}(x)\tilde{h}_{k}(y)=(A_2+\ldots+A_{n})-
(A_1+\ldots+A_{n-1})=A_{n}-A_1$$ and remembering that $A_1=\sqrt{\frac{N}{2\pi}}(x-y)=(x-y)\tilde{h}_0(x)\tilde{h}_0(y)$ we arrive at a very important relation: $$\label{darboux}
\sum_{k=0}^{n-1}\tilde{h}_{k}(x)\tilde{h}_{k}(y)=
\sqrt{\frac{n}{N}}\frac{\tilde{h}_{n-1}(y)\tilde{h}_{n}(x)-
\tilde{h}_{n-1}(x)\tilde{h}_{n}(y)}{x-y},$$ or, for the original (not-normalized) polynomials: $$\label{darboux1}
\sum_{k=0}^{n-1}\frac{1}{k!N^k}h_{k}(x)h_{k}(y)=\frac{1}{(n-1)!N^{n}}
\frac{h_{n-1}(y)h_{n}(x)-h_{n-1}(x)h_{n}(y)}{x-y},$$ which are known as the [*Christoffel-Darboux formulae*]{}. Finally, taking the limit $x\to y$ in the above expression we see that $$\label{darboux2}
\sum_{k=0}^{n-1}\frac{1}{k!N^k}h^2_{k}(x)=\frac{1}{(n-1)!N^{n}}
\left[h_{n-1}(x)h'_{n}(x)-h'_{n-1}(x)h_{n}(x)\right].$$
Most of the properties and relations discussed above for Hermite polynomials have their analogues for general class of orthogonal polynomials. Now we are going to discuss another very useful property which is however shared only by few families of [*classical*]{} orthogonal polynomials: Hermite, Laguerre, Legendre, Gegenbauer and Jacoby. All these polynomials have one of few [*integral representations*]{} which are frequently exploited when analyzing their properties. For the case of Hermite polynomials we can most easily arrive to the corresponding representation by using the familiar Gaussian integral identity, cf. Eq.(\[Gauint0\]): $$\label{Gauint}
e^{-N\frac{x^2}{2}}=\sqrt{\frac{N}{2\pi}}\int_{-\infty}^{\infty}
\, dq\, e^{-\frac{N}{2}q^2+ixqN}.$$ Substituting such an identity to the original definition, Eq.(\[Her1\]), we immediately see that $$\label{intrep}
h_k(x)=(-iN)^k\sqrt{\frac{N}{2\pi}}\,
e^{N\frac{x^2}{2}}\int_{-\infty}^{\infty}
\, dq\, q^k\,e^{-\frac{N}{2}q^2+ixqN},$$ which is the required integral representation, to be mainly used later on when addressing the large-$N$ asymptotics of the Hermite polynomials. Meanwhile, let us note that differentiating the above formula with respect to $x$ one arrives at the useful relation $\frac{d}{dx}h_k(x)=Nxh_k(x)-h_{k+1}(x)=Nk\,h_{k-1}(x)$. This can be further used to simplify the formula Eq.(\[darboux2\]) bringing it to the form $$\label{darboux3}
\sum_{k=0}^{n-2}\frac{1}{k!N^k}h^2_{k}(x)=\frac{1}{(n-2)!N^{n-1}}
\left[h^2_{n}(x)-h_{n-1}(x)h_{n+1}(x)\right].$$
Saddle-point method and Plancherel-Rotach asymptotics of Hermite polynomials
----------------------------------------------------------------------------
In our definition, the Hermite polynomials $h_k(x)$ depend on two parameters: explicitly on the order index $k=0,1,\ldots$ and implicitly on the parameter $N$ due to the fact that the weight function $e^{-N\frac{x^2}{2}}$ contains this parameter. Invoking the random matrix background for the use of orthogonal polynomials, we associate the parameter $N$ with the size of the underlying random matrix. From this point of view, the limit $N\gg 1$ arises naturally as we are interested in investigating the spectral characteristics of large matrices. A more detailed consideration reveals that, from the random matrix point of view, the most interesting task is to extract the asymptotic behaviour of the Hermite polynomials with index $k$ large and comparable with $N$, i.e. $k=N+n$, where the parameter $n$ is considered to be of the order of unity. Such behaviour is known as Plancherel-Rotach asymptotics.
To understand this fact it is enough to invoke the relation (\[kern7\]) expressing the mean eigenvalue density in terms of the set of orthogonal polynomials: $$\begin{aligned}
\label{kern77}
&&\overline{\rho_N(\lambda)}=K_N(\lambda,\lambda)=
e^{-\frac{N}{2}\lambda^2}\,\sum_{j=0}^{N-1}
\tilde{h}^2_{j}(\lambda),
\\
&& =e^{-\frac{N}{2}\lambda^2}\frac{\sqrt{N/2\pi}}{(N-1)!N^{N}}
\left[h^2_{N}(\lambda)-h_{N-1}(\lambda)h_{N+1}(\lambda)\right],
\label{kern777}\end{aligned}$$ where we used the expressions pertinent to the Gaussian weight: $Q(\lambda)\equiv\frac{N}{2}\lambda^2\,,\, \pi_k(\lambda)\equiv
\tilde{h}_k(\lambda)$, and further exploited the variant of the Christoffel-Darboux formula, Eq.(\[darboux3\]). It is therefore evident that the limiting shape of the mean eigenvalue density for large random matrices taken from the Gaussian Unitary Ensemble is indeed controlled by the Plancherel-Rotach asymptotics of the Hermite polynomials. In fact, similar considerations exploiting the original Christoffel-Darboux formula, Eq.(\[darboux\]), show that our main object of interest -the kernel $K_N(\lambda,\lambda')$ - can be expressed as $$\label{kernher}
K_N(\lambda,\lambda')=e^{-\frac{N}{4}(\lambda^2+\lambda'^2)}
\frac{\tilde{h}_{N-1}(\lambda)\tilde{h}_{N}(\lambda')-
\tilde{h}_{N-1}(\lambda)\tilde{h}_{N}(\lambda')}{\lambda-\lambda'}$$ and therefore all the higher correlation functions are controlled by the Plancherel-Rotach asymptotics as well.
For extracting the required asymptotics we are going to use the integral representation for the Hermite polynomials. We start with rewriting the expression Eq.(\[intrep\]) as $$\begin{aligned}
\label{intrep1}
&& h_{N+n}(x)=(-iN)^{N+n}\sqrt{\frac{N}{2\pi}}\,
\int_{-\infty}^{\infty}
\, dq\, q^{N+n}\, e^{-\frac{N}{2}\left(q-ix\right)^2}
\\&=&(-iN)^{N+n}\sqrt{\frac{N}{2\pi}}\,\left[I_{N+n}(x)+
(-1)^{N+n}I_{N+n}(-x) \right],\end{aligned}$$ where $$\label{integral}
I_{N+n}(x)=\int_{0}^{\infty}
\, dq\, q^{n}\, e^{Nf(q)},\quad
f(q)=\ln{q}-\frac{1}{2}\left(q-ix\right)^2.$$ The latter form is suggestive of exploiting the so-called [*saddle-point*]{} method (also known as the method of [*steepest descent*]{} or method of [*stationary phase*]{}) of asymptotic evaluation of integrals of the form $$\label{sp1}
\int_{\Gamma}\phi(z)e^{NF(z)}dz,$$ where the integration goes along a contour $\Gamma$ in the complex plane, $F(z)$ is an analytic function of $z$ in some domain containing the contour of integration, and $N$ is a large parameter. The main idea of the method can be informally outlined as follows. Suppose that the contour $\Gamma$ is such that: (i) the value of $\mbox{Re}F$ has its [*maximum*]{} at a point $z_0\in
\Gamma$, and decreases fast enough when we go along $\Gamma$ away from $z_0$, and (ii) the value of $\mbox{Im}F$ stays constant along $\Gamma$ (to avoid fast oscillations of the integrand). Then we can expect the main contribution for $N\gg 1$ to come from a small vicinity of $z_0=x_0+iy_0$.
Since the function $\mbox{Re}F$ is a harmonic function of $x=\mbox{Re}z,\, y=\mbox{Im}z$, it can have only [*saddle points*]{} (see Fig. \[fig2\]) found from the condition of stationarity $F'(z_0)=0$. Let us suppose that there exists only one such saddle point $z=z_0$, close to which we can expand $F(z)\approx F(z_0)+C(z-z_0)^2$, where $C=\frac{1}{2}F''(z_0)$. Consider the level curves $[\mbox{Re}F](x,y)
=[\mbox{Re}F](x_0,y_0)$, which are known either to go to infinity, or end up at a boundary of the domain of analyticity. In the vicinity of the chosen saddle-point the equation for the level curves is $\mbox{Re}[F(z)-F(z_0)]=0$, hence $$\mbox{Re}[|C|e^{i\theta}(z-z_0)^2]=\left[(x-x_0)^2
(y-y_0)^2\right]\cos{\theta}-2\,(x-x_0)(y-y_0)\sin{(\theta)}=0,$$ which describes two orthogonal straight lines passing through the saddle-point $$y=y_0+\tan{\left(\frac{\pi}{4}-\frac{\theta}{2}\right)}(x-x_0),\quad
y=y_0-\tan{\left(\frac{\pi}{4}+\frac{\theta}{2}\right)}(x-x_0)$$
partitioning the $x,y$ plane into four sectors: two “positive" ones: $\mbox{Re}F(z)>\mbox{Re}F(z_0)$, and two “negative" ones $\mbox{Re}F(z)<\mbox{Re}F(z_0)$, see Fig. \[fig3\]. If the “edge points" of the integration contour $\Gamma$ (denoted $z_1$ and $z_2$) both belong to the [*same*]{} sector, and $\mbox{Re}F(z_1)\ne \mbox{Re}F(z_2)$, one always can deform the contour in such a way that $\mbox{Re}F(z)$ is monotonically increasing along the contour. Then obviously the main contribution to the integral comes from the vicinity of the endpoint (of the largest value of $\mbox{Re}F(z)$). Essentially the same situation happens when $z_1$ belongs to a negative (positive) sector, and $z_2$ is in a positive (resp., negative) sector. And only if the two endpoints belong to two [*different negative*]{} sectors, we can deform the contour in such a way, that $\mbox{Re}F(z)$ has its maximum along the contour at $z=z_0$, and decays away from this point. Moreover, it is easy to understand that the fastest decay away from $z_0$ will occur along the [*bi-sector*]{} of the negative sectors, i.e. along the line $y-y_0=\tan{\frac{\pi-\theta}{2}}(x-x_0)$. Approximating the integration contour in the vicinity of $z_0$ as this bi-sector, i.e. by $z=z_0+(x-x_0)\frac{e^{-i\frac{\pi-\theta}{2}}}{\sin{(\theta/2)}}$, we get the leading term of the large-$N$ asymptotics for the original integral by extending the limits of integration in the variable $\tilde{x}=x-x_0$ from $-\infty$ to $\infty$: $$\begin{aligned}
\label{sp2}
&&\nonumber \int_{\Gamma}\phi(z)e^{NF(z)}dz\approx \phi(z_0)e^{NF(z_0)}
\frac{e^{-i\frac{\pi-\theta}{2}}}{\sin{(\theta/2)}}
\int_{-\infty}^{\infty}d\tilde{x}e^{-N|C|\tilde{x}^2}{\sin^2{\theta/2}}\\
&&=\phi(z_0)\sqrt{\frac{2\pi}{N|F''(z_0)|}}
\exp{\{NF(z_0)+\frac{i}{2}(\pi-Arg[F''(z_0)/2)])\}}.\end{aligned}$$ It is not difficult to make our informal consideration rigorous, and to calculate systematic corrections to the leading-order result, as well as to consider the case of several isolated saddle-points, the case of a saddle-point coinciding with an end of the contour, etc., see [@saddlepoint] for more detail.
After this long exposition of the method we proceed by applying it to our integral, Eq.(\[integral\]). The saddle-point equation and its solution in that case amount to: $$F'(q)=\frac{1}{q}-q+ix=0,\quad q=q_{\pm}=\frac{1}{2}\left(
ix\pm \sqrt{4-x^2}\right).$$ It is immediately clear that we have essentially three different cases: a) $|x|<2$ (b) $|x|>2$ and (c) $|x|=2$.
1. [**$|x|<2$**]{}. In this case we can introduce $x=2\cos{\phi},\,\, 0<\phi<\pi$, so that $q_{\pm}=i\cos{\phi}\pm
\sin{\phi}$, or $q_{+}=e^{-i(\phi-\pi/2)},
\,q_{-}=e^{i(\phi+\pi/2)}$. It is easy to understand that we are interested only in $q_{+}$ (see Fig.4) and to calculate that $\mbox{Re}f(q_{+})=\frac{1}{2}\cos{(2\phi)}$. On the other hand $\mbox{Re}f(q)\to-\infty$ when either $q\to \infty$ or $q\to 0$, so that both endpoints belong to negative sectors. To understand whether they belong to the same or different sectors, we consider the values of $\mbox{Re}f(q)=\ln{R}-\frac{1}{2}(R^2-x^2)$ along the real axis, $q=R$-real. As a function of the variable $R$ this expression has its maximal value $\mbox{Re}f(q=1)=-\frac{1}{2}+2\cos^2{\phi}$ at $q=R=1$.
Noting that $\mbox{Re}f(q=1)-\mbox{Re}f(q_{+})=\cos^2{\phi}>0$, we conclude that the point $q=1$ belongs to a positive sector, and therefore the existence of this positive sector makes the endpoints $q=0$ and $q=\infty$ belonging to two [*different*]{} negative sectors, as required by the saddle-point method. Calculating $$f''(q_{+})=-\left(1+\frac{1}{q^2_{+}}\right)=2i\sin{\phi}e^{i\phi}$$ we see that $|C|=\sin{\phi},\, \theta=\phi+\pi/2$, and further $$f''(q_{+})=\frac{1}{2}\cos{(2\phi)}+i\left[\frac{1}{2}\sin{(2\phi)}-
\phi+\pi/2\right].$$ Now we have all the ingredients to enter in Eq.(\[sp2\]), and can find the leading order contribution to $I_{N+n}(x)$. Further using $I_{N+n}(-x)=\overline{I_{N+n}(x)}$, valid for real $x$, we obtain the required Plancherel-Rotach asymptotics of the Hermite polynomial: $$\begin{aligned}
\label{planrot1}
h_{N+n}(x)\approx N^{N+n}\sqrt{\frac{2}{\sin{\phi}}}
e^{\frac{N}{2}\cos{2\phi}}\cos{\left\{(n+1/2)\phi-\pi/4+N
\left(\phi-\frac{1}{2}\sin{2\phi}\right)\right\}},\end{aligned}$$ where $x=2\cos{\phi},\quad 0<\phi<\pi,\quad n\ll N$.
Now we consider the opposite case:
2. $|x|>2$. It is enough to consider explicitly the case $x>2$ and parameterize $x=2\cosh{\phi},\quad
0<\phi<\infty$. The saddle points in this case are purely imaginary: $$\label{out1}
q_{\pm}=\frac{i}{2}(2\cosh{\phi}\pm 2\sinh{\phi})=ie^{\pm \phi}.$$ One possible contour of the constant phase passing through both points is just the imaginary axis $q=iy$, where $\mbox{Im}f(q)=\pi/2$ and $\mbox{Re}f(q)=\ln{y}+\frac{1}{2}(y-x)^2.$ Simple consideration gives that $y_{-}=e^{-\phi}$ corresponds to the maximum, and $y_{+}=e^{\phi}$ to the minimum of $\mbox{Re}f(q)$ along such a contour. It is also clear that for $q=iy_{+}$ the expression $\mbox{Re}f(q)$ has a local maximum along the path going through this point in the direction [*transverse*]{} to the imaginary axis. The “topography" of $\mbox{Re}f(q)$ in the vicinity of the two saddle-points is sketched in Fig. \[fig5\]
This discussion suggests a possibility to deform the path of integration $\Gamma$ to be a contour of constant phase $\mbox{Im}f(q)$ consisting of two pieces - $\Gamma_1=\{q=iy,\,0\le
y\le y_{+}\}$ and $\Gamma_2$ starting from $q=iy_{+}$ perpendicular to the imaginary axis and then going towards $q=\infty$. Correspondingly, $$\label{offspec}
I_{N+n}(x>2)=\int_{0}^{\infty}\, dq\, q^{n}\, e^{Nf(q)}
=\int_{0}^{iy_{+}}
\, dq\, q^{n}\, e^{Nf(q)}+\int_{\Gamma_2}
\, dq\, q^{n}\, e^{Nf(q)}.$$ The second integral is dominated by the vicinity of the saddle-point $q=iy_{+}$, and its evaluation by the saddle-point technique gives: $$\int_{\Gamma_2} \, dq\, q^{n}\, e^{Nf(q)}\approx
\frac{1}{2}\sqrt{\frac{\pi e^{\phi}}{N\sinh{\phi}}} i^{n+N}
e^{n\phi+N\left(\phi+\frac{1}{2}e^{-2\phi}\right)},$$ where the factor $\frac{1}{2}$ arises due to the saddle-point being simultaneously the end-point of the contour. As to the first integral, it is dominated by the vicinity of $iy_{-}$, and can also be evaluated by the saddle-point method. However, it is easy to verify that when calculating $h_{N+n}(x)\propto
\left[I_{N+n}(x)+ (-1)^{N+n}I_{N+n}(-x) \right] $ the corresponding contribution is cancelled out. As a result, we recover the asymptotic behaviour of Hermite polynomials for $x>2$ to be given by: $$\label{offspec1}
h_{N+n}(x=2\cosh{(2\phi)}>2)=\frac{N^{n+N}
e^{-\frac{N}{2}}}{\sqrt{\sinh{\phi}}}
e^{\left(n+\frac{1}{2}\right)\phi-
\frac{N}{2}\left(\sinh{(2\phi)}-2\phi\right)}.$$
Now we come to the only remaining possibility,
3. $|x|=2$. It is again enough to consider only the case $x=2$ explicitly. In fact, this is quite a special case, since for $x\to 2$ two saddle-points $q_{\pm}$ degenerate into one: $q_{+}\to q_{-}\to
i$. Under such exceptional circumstances the standard saddle-point method obviously fails. Indeed, the method assumed that different saddle-points do not interfere, which means the distance $|q_{+}-q_{-}|=\sqrt{|4-x^2|}$ is much larger than the typical widths $W\sim \frac{1}{\sqrt{N|f''(q_{\pm})|}}$ of the regions around individual saddle-points which yield the main contribution to the integrand. Simple calculation gives $|f''(q_{\pm})|=|1+q^{-2}_{\pm}|=\sqrt{|4-x^2|}$, and the criterion of two separate saddle-points amounts to $|x-2|\gg
N^{-2/3}$. We therefore see that in the vicinity of $x=2$ such that $|x-2|\sim N^{-2/3}$ additional care must be taken when extracting the leading order behaviour of the corresponding integral $I_{N+n}(x=2)$ as $N\to\infty$.
To perform the corresponding calculation, we introduce a new scaling variable $\xi=N^{2/3}(2-x)$, and consider $\xi$ to be fixed and finite when $N\to\infty$. We also envisage from the discussion above that the main contribution to the integral comes from the domain around the saddle-point $q_{sp}=i$ of the widths $|q-i|\sim \sqrt{|2-x|}\sim N^{-1/3}$. The integral we are interested in is given by $$\begin{aligned}
\label{ai1}
J_N(\xi)&=&\int_{-\infty}^{\infty}
\, dq\, q^{N+n}\, e^{-\frac{N}{2}\left(q-ix\right)^2}\\
&=&N^{-1/3} \int_{-\infty}^{\infty}
\, dt\, \left(i+\frac{t}{N^{1/3}}\right)^n\,
e^{N\left[\ln{(i+\frac{t}{N^{1/3}})}-
\frac{1}{2}\left\{i+\frac{t}{N^{1/3}}-i\left(2-\frac{\xi}{N^{2/3}}
\right)\right\}^2\right]}\end{aligned}$$ where we shifted the contour of integration from the real axis to the line $q=i+\frac{t}{N^{1/3}}\,, -\infty<t<\infty$ to ensure that it passes through the expected saddle-point $q_{sp}=i$, and also scaled the integration variable appropriately. Now we can consider $\xi,t$-finite when $N\gg 1$, and expand the integrand accordingly. A simple computation yields: $$\begin{aligned}
\label{ai2}
J_{N\gg 1}(\xi)\approx
N^{-1/3}i^{N+n}\,e^{N/2-N^{1/3}\xi} \int_{-\infty}^{\infty}
\, dt\, e^{-i\xi\,t+i\frac{t^3}{3}}.\end{aligned}$$ Up to a constant factor the integral appearing in this expression is, in fact, a representation of a special function known as Airy function $Ai(\xi)$: $$\label{Airydef}
Ai(\xi)=\frac{1}{\pi}\int_{0}^{\infty}
\, dt\,\, \cos{\left(\xi\,t+\frac{t^3}{3}\right)}$$ which is a solution of the second-order linear differential equation $\frac{d^2}{d\xi^2}F(\xi)-\xi\,F(\xi)=0$. A typical behaviour of such a solution is shown in Fig. \[fig6\].
All this results in the asymptotic behaviour of the Hermitian polynomials in the so-called “scaling vicinity" of the point $x=2$: $$\begin{aligned}
\label{Air3}
h_{N+n}\left(x=2-\frac{\xi}{N^{2/3}}\right)\approx
\frac{N^{1/6}}{\sqrt{2\pi}}N^{N+n}\,e^{N/2-N^{1/3}\xi}\,Ai(-\xi).\end{aligned}$$ Such scaling vicinity of $x=2$ is what gives room for a transitional regime between the oscillating asymptotics of the Hermite polynomials for $|x|<2$, see Eq.(\[planrot1\]), and the exponential decay typical for $|x|>2$ as described in Eq.(\[offspec1\]). Formula (\[Air3\]) indeed matches Eq.(\[planrot1\]) as $\xi\to\infty$ and Eq.(\[offspec1\]) as $\xi\to-\infty$. This statement is most easily verified by invoking the known asymptotics of the Airy function: $$\label{Airyas}
Ai(-\xi)\approx \left\{\begin{array}{c}\xi^{-1/4}\pi^{-1/2}
\cos{\left(-\frac{2}{3}\xi^{3/2}+\frac{\pi}{4}\right)},\quad \xi\to \infty\\
\frac{1}{\pi^{1/2}|\xi|^{1/4}}\,e^{-\frac{2}{3}|\xi|^{3/2}},\quad
\xi\to-\infty\end{array}\right.$$ and identifying $\phi=|\xi|^{1/2}N^{-1/3}\ll 1$ in the corresponding expressions.
Now we are going to apply the derived formulae for extracting the large-N behaviour of the mean eigenvalue density and the kernel as described in Eqs.(\[kern77\]) and(\[kernher\]), respectively. In fact, it is more conventional in the random matrix literature to use the mean density to be normalized to unity, rather than to $N$. Such a density will have a well-defined large-N limit which we will denote as $\rho_{\infty}(\lambda)$.
Scaling regimes for GUE
=======================
Bulk scaling: Wigner semicircle and Dyson kernel.
-------------------------------------------------
The first case to be considered is the spectral parameter $|\lambda|<2$ when we can parameterize $\lambda=2\cos{\phi}$, and exploit the Plancherel-Rotach expression (\[planrot1\]) for the Hermite polynomials. Furthermore, denoting $\alpha=\frac{1}{2}\phi-\frac{\pi}{4}+N
\left(\phi-\frac{1}{2}\sin{2\phi}\right)$, and using the identity $\cos^2{\alpha}-\cos{(\alpha+\phi)}\cos{(\alpha-\phi)}=\sin^2{\phi}$ we find that $h_N^2(\lambda)-h_{N-1}(\lambda)h_{N+1}(\lambda)\approx
2N^{2N}\sin{\phi} e^{N\cos{(2\phi)}}$. Furthermore, using for large $N$ the Stirling formula: $(N-1)!\approx
\sqrt{\frac{2\pi}{N}}N^Ne^{-N}$ and remembering that $\sin{\phi}=\frac{1}{2}\sqrt{4-\lambda^2}$ we arrive, after collecting all factors, to the famous Wigner semicircular law for the mean (normalized) spectral density: $$\label{semi}
\lim_{N\to\infty}\left[
\frac{1}{N}\overline{\rho(\lambda)}\right]=\rho_{\infty}(\lambda)=
\frac{1}{2\pi}\sqrt{4-\lambda^2},
\quad |\lambda|<2.$$ We see that in the limit of large $N$ all $N$ eigenvalues of GUE matrices are concentrated in the interval $[-2,2]$, and the typical separation of two neighbouring eigenvalues close to an “internal“ point $\lambda \in (-2,2)$ is $\Delta=\frac{1}{N\rho_{\infty}(\lambda)}=O(N^{-1})$, see Fig. \[fig7\]. That is why the case $\lambda \in (-2,2)$ is frequently referred to as the ”bulk of the spectrum" regime.
Let us now follow the same strategy for obtaining, under the same conditions, the limiting expression for the kernel $K(\lambda,\lambda')$ using for this goal formula (\[kernher\]). We have: $$\begin{aligned}
\label{kernher11}
&& h_N(\lambda)h_{N-1}(\lambda')-h_N(\lambda')h_{N-1}(\lambda)
\\ &&\approx 2N^{2N}\frac{1}{\sqrt{\sin{\phi}\sin{\phi'}}}
e^{\frac{N}{2}\left(\cos{(2\phi)}+\cos{(2\phi')}\right)}
\left[\cos{\alpha_1^{+}}\cos{\alpha_2^{-}}
-\cos{\alpha_1^{-}}\cos{\alpha_2^{+}}\right]\end{aligned}$$ where $\alpha^{\pm}_{1}=\pm\frac{1}{2}\phi-\frac{\pi}{4}+N
\left(\phi-\frac{1}{2}\sin{2\phi}\right),\,\,
\alpha^{\pm}_{2}=\pm\frac{1}{2}\phi'-\frac{\pi}{4}+N
\left(\phi'-\frac{1}{2}\sin{2\phi'}\right)$. The next step is to introduce $\psi=(\phi+\phi')/2$ and $\Omega= (\phi-\phi')/2$, and to consider the parameter $\Omega$ to be of the order of $O(N^{-1})$ when taking the limit. This choice ensures that the distance $\lambda-\lambda'= 2[\cos{\phi}-\cos{\phi'}]\approx
4\Omega \sin{\psi}\approx 4\Omega \pi\rho_{\infty}(\lambda)$ is of the order of the mean eigenvalue separation $\Delta$ - the typical scale for the correlations between the eigenvalues in the [*bulk*]{} of the spectrum- and thus must be reflected in the structure of the kernel. To this end we denote $\Omega=\omega/N$, and keep in the expressions for $\alpha^{\pm}_{1,2}$ terms up to the order $O(1)$, i.e. writing $\alpha^{\pm}_{1,2}=N\beta+\left[
\pm\frac{1}{2}\psi-\frac{\pi}{4}\pm 2\omega \sin^2{\psi}\right]$, where $\beta=\left(\psi-\frac{1}{2}\sin{2\psi}\right)$. With the same precision: $$\cos{\alpha_1^{+}}\cos{\alpha_2^{-}}
-\cos{\alpha_1^{-}}\cos{\alpha_2^{+}}\approx \sin{\psi}
\sin{\left(4\omega\sin^2{\psi}\right)}\approx \sin{\psi}
\sin{\left[\pi \rho_{\infty}(\lambda)N(\lambda_1-\lambda_2)\right]}$$ and $\cos{2\phi_1}+\cos{2\phi_2}\approx
2\left(\frac{\lambda^2}{2}-1\right)$ substituting all these factors back into Eq.(\[kernher11\]), we get $$\begin{aligned}
\label{kernher12}
h_N(\lambda)h_{N-1}(\lambda')- h_N(\lambda')h_{N-1}(\lambda)
\approx 2N^{2N} e^{2N\left(\frac{\lambda^2}{2}-1\right)}
\sin{\left[\pi \rho_{\infty}(\lambda)N(\lambda-\lambda')\right]}.\end{aligned}$$ Now, taking into account the normalization factors in $\tilde{h}_{N}(\lambda)$ and $\tilde{h}_{N-1}(\lambda),$ see Eq.(\[normher\]), using again the Stirling formula and invoking Eq.(\[kern7\]) we arrive at the following asymptotic expression for the kernel, Eq.(\[kernher\]): $$\label{Dyson}
\lim_{N\to\infty} \left[\frac{K_N(\lambda,\lambda')}
{K_N(\lambda,\lambda)}\right]=K_{\infty}\left[
N\rho_{\infty}(\lambda)(\lambda-\lambda')\right],\quad
K_{\infty}(r)=\frac{\sin{\pi r}}{\pi r}$$ where $K_{\infty}(r)$ is the famous [*Dyson scaling*]{} form for the kernel. The formula is valid as long as both $\lambda$ and $\lambda'$ are within the range $(-2,2)$, and $\lambda-\lambda'=O(N^{-1})$. Such choice of the parameters is frequently referred to as the “bulk scaling" limit.
Having at our disposal the limiting form of both mean eigenvalue density and the two-point kernel we can analyse such important statistical characteristics of the spectra as e.g. the “number variance", see Eq.(\[sigma2\]), for an interval of the length $L$ comparable with the mean spacing close to the origin $\Delta=[N\rho_{\infty}(0)]^{-1}$. Under such a condition we can legitimately employ the scaling form Eq.(\[Dyson\]) of the kernel when substituting it into formula (\[kern8\]) for the cluster function $Y_2(\lambda,\lambda')$. In this way we arrive at $$\begin{aligned}
\label{sigmasc}
\nonumber && \Sigma_2(L)=N\int_{-L/2}^{L/2} \rho_{\infty}(\lambda)d\lambda
-N^2 \int_{-L/2}^{L/2}d\lambda \, \int_{-L/2}^{L/2} d\lambda'
\rho_{\infty}(\lambda)\rho_{\infty}(\lambda')
K^2_{\infty}\left[
N\rho_{\infty}(\lambda)(\lambda-\lambda')\right]\\
&& =s-\int_{-s/2}^{s/2} du \, \int_{-s/2}^{s/2} \, du'
K^2_{\infty}\left[(u-u')\right].\end{aligned}$$ Here we used the fact that with the same precision we can put $\rho_{\infty}(\lambda)\approx \rho_{\infty}(\lambda')\approx
\rho_{\infty}(0)$ in the above expression, and introduced the natural scaling variables: $u=\lambda/\Delta,\, u=\lambda'/\Delta$ as well as the scaled length of the interval $s=L/\Delta$ (cf. a similar procedure for Poissonian sequences after Eq.(\[hole3\])). To simplify this expression further we introduce $u_{+}=(u+u')/2,\, r=u-u'$ as integration variables, and use that, in fact $K_{\infty}(r)\equiv K_{\infty}(|r|)$. The number variance takes the final form: $$\Sigma_2(s)=s-\int_{-s}^{s}\,dr\,
\int_{-\frac{s}{2}+\frac{|r|}{2}}^{\frac{s}{2}-\frac{|r|}{2}}\,
du_{+} K^2_{\infty}(|r|)=s-2 \int_{0}^{s}\,dr\, (s-r)
\left[\frac{\sin{\pi r}}{\pi r}\right]^2.$$ In fact, we are mainly interested in the large-$s$ behaviour of this expression. To extract it, we use the identity: $2
\int_{0}^{\infty}\,dr\, \left[\frac{\sin{\pi r}}{\pi r}\right]^2=
\frac{2}{\pi} \int_{0}^{\infty}\,dx\,
\left[\frac{\sin{x}}{x}\right]^2=1$, and rewrite the above expression as $$\Sigma_2(s)=\frac{2s}{\pi} \int_{\pi s}^{\infty}\,dx\,
\frac{\sin^2{x}}{x^2}+\frac{1}{\pi^2}\int_{0}^{2\pi s}\frac{1-\cos{x}}{x}\,dx.$$ The second integral obviously grows logarithmically with $s$ and dominates at large $s$. A more accurate evaluation gives the asymptotic formula: $$\Sigma_2(s\gg 1)=\frac{1}{\pi^2}\left[\ln{2\pi s}+\gamma+1\right]+O(1/s).$$ where $\gamma=0.5772...$ is Euler’s constant. This is much slower than the linear growth $\Sigma_2(s\gg 1)=s$ typical for uncorrelated (Poissonian) sequence, see Fig. \[fig8\]. The explanation of the slow growth is that the sequence of eigenvalues is, in fact, quite ordered, with quite regular spacings of the order of $\Delta$, and therefore the number of points in the interval does not fluctuate as much as it does for uncorrelated sequence.
As to another important and frequently used statistical characteristic of spectral sequences - the “hole probability"- its calculation amounts to investigating the asymptotics of the Fredholm determinant of the kernel $K_{N\to\infty}$, see Eq.(\[holekern\]). This is a very difficult mathematical problem, and the most elegant solution uses an advanced mathematical technique known as the Riemann-Hilbert method[@Deift]. Let us just quote the result: $$A(s\gg 1)\propto \frac{1}{s^{1/4}}e^{-\frac{\pi^2}{8}s^2}.$$ This Gaussian decay should be again contrasted with a much slower exponential decay typical for uncorrelated sequences as indeed in full correspondence with a “quasiregular" structure of the random matrix spectrum.
Edge scaling regime and Airy kernel
------------------------------------
As we already know, in the vicinity of the “spectral edge" $x=2$ (and its counterpart $x=-2$) the Plancherel-Rotach asymptotics of the Hermitian polynomials changes, and is basically given by the Airy function, see Eq.(\[Air3\]). This certainly results in essential modifications of the large$-N$ behaviour of the mean eigenvalue density and of the two-point kernel as long as $|\lambda-2|\sim N^{-2/3}$. To extract the explicit formulae for this so-called “edge scaling" limit one may try the same strategy as in the bulk. However, one immediately discovers that simple substitution of Eq.(\[Air3\]) into formula (\[kern777\]) for the mean density yields zero. A possible way out may be to calculate the next-to-leading order corrections to the asymptotics of $h_N(x)$, but we will rather follow a slightly different (and more direct) route and consider the integral representation for the main combination of interest: $$\begin{aligned}
\label{intrep2}
{\cal D}_N(\lambda)=h^2_{N}(\lambda)-h_{N-1}(\lambda)h_{N+1}(\lambda)=
\frac{(-1)^{N}}{2\pi}N^{2N}\,
\int_{-\infty}^{\infty}\, dq_1
\int_{-\infty}^{\infty}\,dq_2\frac{q_1-q_2}{q_1}
e^{N\left[f(q_1)+f(q_2)\right]}\end{aligned}$$ where we exploited Eq.(\[intrep1\],\[integral\]), and defined, as before, $f(q)=\ln{q}-\frac{1}{2}\left(q-i\lambda\right)^2$. To evaluate this integral in the edge scaling limit, we follow a familiar procedure: introduce the scaling variable $\xi=N^{2/3}(\lambda-2)$, shift the contours of integration from the real axis to the lines $q_{1,2}=i+\frac{t_{1,2}}{N^{1/3}}\,,
-\infty<t_{1,2}<\infty$, consider $\xi,\,t_{1,2}$ to be fixed and finite when $N\to\infty$, and expand the integrand accordingly around the saddle-points $t_{1,2}=0$. Simple calculation yields, in complete analogy with Eq.(\[ai2\]), the expression: $$\begin{aligned}
\label{ai22}
\nonumber
&& {\cal D}_N(\xi)\approx \frac{N^{2N}}{2\pi}
N^{-1/3}\,e^{N+2N^{1/3}\xi} \\
&& \times \left\{\int_{\Gamma}
\, dt_1\, e^{i\xi\,t_1+i\frac{t_1^3}{3}}\int_{\Gamma}
\, dt_2\, t_2^2\,e^{i\xi\,t_2+i\frac{t_2^3}{3}}-
\int_{\Gamma}\, dt_1\, t_1\, e^{i\xi\,t_1+i\frac{t_1^3}{3}}\int_{\Gamma}
\, dt_2\, t_2\,e^{i\xi\,t_2+i\frac{t_2^3}{3}}\right\}.\end{aligned}$$
The only essential difference from Eq.(\[ai2\]) which deserves mentioning is the choice of the integration contour $\Gamma$ which ensures the existence of all the integrals involved. Obviously, one can not simply take $\Gamma=(-\infty,\infty)$, but a more detailed investigation shows that the correct contour must be chosen in such a way as to be asymptotically tangent to the line $\mbox{Arg}{(t)}=5\pi/6$ for $\mbox{Re}\,t\to -\infty$, and asymptotically tangent to $\mbox{Arg}{(t)}=\pi/6$ for $\mbox{Re}\,t\to \infty$, see Fig.9. It is then evident, that $$Ai(\xi)=\frac{1}{\pi}\int_{\Gamma}
\, dt\, e^{i\xi\,t+i\frac{t^3}{3}},\,\,
-iAi'(\xi)=\frac{1}{\pi}\int_{\Gamma}
\, dt\,t\, e^{i\xi\,t+i\frac{t^3}{3}},\,\,
-Ai''(\xi)=\frac{1}{\pi}\int_{\Gamma}
\, dt\,t^2\, e^{i\xi\,t+i\frac{t^3}{3}}$$ and collecting all factors we find the expression of the mean eigenvalue density close to the “spectral edge": $$\label{denedge}
\overline{\rho}(\lambda=2+\xi\,N^{-2/3})\propto \rho_{e}(\xi)=Ai'(\xi)^2-
Ai''(\xi)Ai(\xi).$$
For $\xi<0$ the function shows noticeable oscillations , see Fig.10, with neighbouring maxima separated by distance of the order of $\lambda_i-\lambda_{i-1}\propto \Delta_{edge}=\sim
N^{-1/3}$ and reflecting typical positions of individual eigenvalues close to the “spectral edge". In contrast, for $\xi>0$ the mean density decays extremely fast, reflecting the typical absence of the eigenvalues beyond the spectral edge.
A very similar calculation shows that under the same conditions the kernel $K_{N}(\lambda,\lambda')$ assumes the form: $$\label{keredge}
K(\xi_1,\xi_2)= \frac{Ai(\xi_1)Ai'(\xi_2)-
Ai(\xi_2)Ai'(\xi_1)}{\xi_1-\xi_2}$$ known as the [*Airy kernel*]{}, see [@Tracy].
Orthogonal polynomials versus characteristic polynomials
========================================================
Our efforts in studying Hermite polynomials in detail were amply rewarded by the provided possibility to arrive at the bulk and edge scaling forms for the matrix kernel in the corresponding large-N limits. It is those forms which turn out to be [*universal*]{}, which means independent of the particular detail of the random matrix probability distribution, provided size of the corresponding matrices is large enough. This is why one can hope that the Dyson kernel would be relevant to many applications, including properties of the Riemann $\zeta$-function. An important issue for many years was to prove the universality for unitary-invariant ensembles which was finally achieved, first in [@PS].
In fact, quite a few basic properties of the Hermite polynomials are shared also by any other set of orthogonal polynomials $\pi_{k}(x)$. Among those worth of particular mentioning is the Christoffel-Darboux formula for the combination entering the two-point kernel Eq.(\[kern1\]), (cf. Eq.(\[darboux1\])): $$\label{darboux11}
\sum_{k=0}^{n-1}\pi_{k}(x)\pi_{k}(y)=b_n
\frac{\pi_{n-1}(y)\pi_{n}(x)-\pi_{n-1}(x)\pi_{n}(y)}{x-y},$$ where $b_n$ are some constants. So the problem of the universality of the kernel (and hence, of the n-point correlation functions) amounts to finding the appropriate large-$N$ scaling limit for the right-hand side of Eq.(\[darboux11\]) (in the “bulk" of the spectrum, or close to the spectral “edge").
The main dissatisfaction is that explicit formulas for orthogonal polynomials (most important, an integral representation similar to Eq.(\[intrep\])) are not available for general weight functions $dw(\lambda)=e^{-Q(\lambda)}d\lambda$. For this reason we have to devise alternative tools of constructing the orthogonal polynomials and extracting their asymptotics. Any detailed discussion of the relevant technique goes far beyond the modest goals of the present set of lectures. Nevertheless, some hints towards the essence of the powerful methods employed for that goal will be given after a digression.
Namely, I find it instructive to discuss first a question which seems to be quite unrelated,- the statistical properties of the characteristic polynomials $$\label{Z}
Z_N(\mu)=\det\left(\mu {\bf 1}_N-\hat{H}\right)=\prod_{i=1}^N(\mu-\lambda_i)$$ for any Hermitian matrix ensemble with invariant JPDF ${\cal
P}(\hat{H})\propto \exp\{-N\mbox{Tr}Q(\hat{H})\}$. Such objects are very interesting on their own for many reasons. Moments of characteristic polynomials for various types of random matrices were much studied recently, in particular due to an attractive possibility to use them, in a very natural way, for characterizing “universal" features of the Riemann $\zeta$-function along the critical line, see the pioneering paper[@KSn] and the lectures by Jon Keating in this volume. The same moments also have various interesting combinatorial interpretations, see e.g. [@Strahov; @DG], and are important in applications to physics, as I will elucidate later on.
On the other hand, addressing those moments will allow us to arrive at the most natural way of constructing polynomials orthogonal with respect to an arbitrary weight $dw(\lambda)=e^{-Q(\lambda)}d\lambda$. To understand this, we start with considering the lowest moment, which is just the expectation value of the characteristic polynomial: $$\label{Z1}
E\left[Z_N(\mu)\right]=\int_{-\infty}^{\infty}dw(\lambda_1)\ldots
\int_{-\infty}^{\infty}dw(\lambda_N)
\prod_{i<j}^N(\lambda_i-\lambda_j)^2\,\prod_{i=1}^N(\mu-\lambda_i)$$ We first notice that $$\label{vdm11}
\prod_{i<j}^N(\lambda_i-\lambda_j)\,\prod_{i=1}^N(\mu-\lambda_i)
\propto \det{\left(\begin{array}{cccc}1&\ldots & 1&1\\
\lambda_1&\ldots & \lambda_N& \mu\\
.&.&.&.\\ .&.&.&.\\.&.&.&.\\ \lambda^{N-1}_1&\ldots & \lambda^{N-1}_N
& \mu^{N-1}\\ \lambda^{N}_1&\ldots & \lambda^{N}_N
& \mu^{N}
\end{array}\right)}$$ Indeed, the right-hand side is obviously a polynomial of degree $N$ in the variable $\mu$, with roots at $\mu=\mu_1,\mu_2,\ldots,\mu_N$. Therefore it must be of the form $C \times \prod_{i=1}^N(\mu-\lambda_i)$, with prefactor $C$ being a function of $\lambda_1,...,\lambda_N$. The value of such a prefactor can be easily established by comparing both sides as $\mu\to \infty$: the left-hand side behaves as $C\,\mu^N$, whereas expanding the determinant with respect to the last column and using the expression for the van der Monde determinant, Eq.(\[vdm\]), we see that the right-hand side grows as $\mu^N
\prod_{i<j}^N(\lambda_i-\lambda_j)$.
Exploiting Eq.(\[vdm11\]) allows us to rewrite the expectation value for the characteristic polynomial as $$\label{Z2}
E\left[Z_N(\mu)\right]\propto\int_{-\infty}^{\infty}\prod_{i=1}^N
dw(\lambda_i)
\det{\left(\begin{array}{ccc}1&\ldots & 1\\
\lambda_1&\ldots & \lambda_N\\
.&.&.\\ .&.&.\\.&.&.\\ \lambda^{N-1}_1&\ldots & \lambda^{N-1}_N
\end{array}\right)}
\det{\left(\begin{array}{cccc}1&\ldots & 1&1\\
\lambda_1&\ldots & \lambda_N& \mu\\
.&.&.&.\\ .&.&.&.\\.&.&.&.\\ \lambda^{N}_1&\ldots & \lambda^{N}_N
& \mu^{N}
\end{array}\right)},$$ which can be further written down as the standard sum over all permutations $P_{\sigma}=(\sigma_1,\,\ldots,\, \sigma_N)$ of the index set $(1,2,...,N)$: $$\label{Z22}
E\left[Z_N(\mu)\right]\propto\sum_{P_{\sigma}}(-1)^{|P_{\sigma}|}
\int_{-\infty}^{\infty}\prod_{i=1}^N
dw(\lambda_i)
\lambda^{0}_{\sigma_1}\ldots \lambda^{N-1}_{\sigma_N}
\det{\left(\begin{array}{cccc}1&\ldots & 1&1\\
\lambda_1&\ldots & \lambda_N& \mu\\
.&.&.&.\\ .&.&.&.\\.&.&.&.\\ \lambda^{N}_1&\ldots & \lambda^{N}
& \mu^{N}
\end{array}\right)},$$ where $|P_{\sigma}|=0(1)$ for even(odd) permutations. The symmetry of the remaining determinant with respect to permutation of its columns ensures that every term in the sum above yields exactly the same contribution, and it is enough to consider only the first term with $P_{\sigma}=(1,2,...,N)$, and multiply the result with $N!$. For such a choice, the product of factors $\lambda^{0}_{1}\ldots \lambda^{N-1}_{N}$ can be “absorbed" in the determinant by multiplying the $j-$th column of the latter with the factor $\lambda^{j-1}_{j}$, for all $j=1,\ldots,N$. This gives $$\label{Z4}
E\left[Z_N(\mu)\right]\propto
\int_{-\infty}^{\infty}\prod_{i=1}^N
dw(\lambda_i)
\det{\left(\begin{array}{ccccc}1&\lambda_2&\ldots & \lambda_N^{N-1}&1\\
\lambda_1&\lambda_2^2&\ldots & \lambda^N_N& \mu\\
.&.&.&.&\\ .&.&.&.&\\.&.&.&.&\\ \lambda^{N-1}_1&
\lambda^{N}_2&\ldots & \lambda^{2N-2}_N
& \mu^{N-1}\\
\lambda^{N}_1&
\lambda^{N+1}_2&\ldots & \lambda^{2N-1}_N
& \mu^{N}
\end{array}\right)}.$$ The integral in the right-hand side is obviously a polynomial of degree $N$ in $\mu$, which we denote $D_N(\mu)$ and write in the final form as $$\label{Z5}
D_N(\mu)=
\det{\left(\begin{array}{ccccc}\int_{-\infty}^{\infty}
dw(\lambda)
&\int_{-\infty}^{\infty}
dw(\lambda)\lambda &\ldots & \int_{-\infty}^{\infty}
dw(\lambda)\lambda^{N-1}&1\\
\int_{-\infty}^{\infty}
dw(\lambda)\lambda&\int_{-\infty}^{\infty}
dw(\lambda)\lambda^2&\ldots &\int_{-\infty}^{\infty}
dw(\lambda) \lambda^N& \mu\\
.&.&.&.&\\ .&.&.&.&\\.&.&.&.&\\ \int_{-\infty}^{\infty}
dw(\lambda)\lambda^{N-1}&
\int_{-\infty}^{\infty}
dw(\lambda)\lambda^{N}&\ldots &\int_{-\infty}^{\infty}
dw(\lambda) \lambda^{2N-2}
& \mu^{N-1}\\
\int_{-\infty}^{\infty}
dw(\lambda)\lambda^{N}&
\int_{-\infty}^{\infty}
dw(\lambda)\lambda^{N+1}&\ldots & \int_{-\infty}^{\infty}
dw(\lambda)\lambda^{2N-1}
& \mu^{N}
\end{array}\right)}.$$
The last form makes evident the following property. Multiply the right-hand side with $dw(\mu)\mu^p$ and integrate over $\mu$. By linearity, the factor and the integration can be “absorbed" in the last column of the determinant. For $p=0,1,\ldots,N-1$ this last column will be identical to one of preceding columns, making the whole determinant vanishing, so that $$\label{z6}
\int_{-\infty}^{\infty} dw(\mu)\mu^p D_N(\mu)=0,\quad
p=0,1,\ldots,N-1.$$
Moreover, it is easy to satisfy oneself that the polynomial $D_N(\mu)$ can be written as $D_N(\mu)=D_{N-1}\mu^N+\ldots $, where the leading coefficient $D_{N-1}=\det{\left(\int_{-\infty}^{\infty}
dw(\lambda)\lambda^{i+j}\right)_{i,j=0}^N}$ is necessarily positive: $D_{n-1}>0$. The last fact immediately follows from the positivity of the quadratic form: $$G(x_1,\ldots,x_N)=\int_{-\infty}^{\infty}
dw(\lambda)\left(\sum_{i=1}^N\,x_i\lambda^{i}\right)^2=
\sum_{i,j}^Nx_ix_j\int_{-\infty}^{\infty}
dw(\lambda)\lambda^{i+j}$$ Finally, notice that $$\int_{-\infty}^{\infty}
dw(\mu)D_N^2(\mu)=\int_{-\infty}^{\infty}
dw(\mu)D_N(\mu)\left[D_{N-1}\mu^N+lower\,\, powers\right]$$ $$=D_{N-1}\int_{-\infty}^{\infty}
dw(\mu)D_N(\mu)\mu^N=D_{N-1}D_N$$ where we first exploited Eq.(\[z6\]) and at the last stage Eq.(\[Z5\]). Combining all these facts together we thus proved that the polynomials $\pi_N(\lambda)=\frac{1}{\sqrt{D_{N-1}D_N}}D(\lambda)$ form the orthogonal (and normalized to unity) set with respect to the given measure $dw(\lambda)$. Moreover, our discussion makes it immediately clear that the expectation value of the [*characteristic*]{} polynomial $Z_N(\mu)$ for any given random matrix ensemble is nothing else, but just the corresponding [*monic*]{} orthogonal polynomial: $$\label{z8}
E\left[Z_N(\mu)\right]=\pi^{(m)}_{N}(\mu),$$ whose leading coefficient is unity. Leaving aside the modern random matrix interpretation the combination of the right hand sides of the formulas Eq.(\[z8\]) and Eq.(\[Z1\]) goes back, according to [@Szego], to Heine-Borel work of 1878, and as such is completely classical.
The random matrix interpretation is however quite instructive, since it suggests to consider also higher moments of the characteristic polynomials, and even more general objects like the correlation functions $$\label{corrf}
{\cal C}_k(\mu_1,\mu_2,\ldots,\mu_k)=E\left[Z_N(\mu_1)Z_N(\mu_2)
\ldots Z_N(\mu_k)\right].$$ Let us start with considering $$\label{corrf1}
{\cal C}_2(\mu_1,\mu_2)=
\int_{-\infty}^{\infty}dw(\lambda_1)\ldots
\int_{-\infty}^{\infty}dw(\lambda_N)
\prod_{i<j}^N(\lambda_i-\lambda_j)^2\,\prod_{i=1}^N(\mu_1-\lambda_i)
\prod_{i=1}^N(\mu_2-\lambda_i).$$ Using the notation $\Delta_N(\lambda_1,\ldots,\lambda_N)$ for the van der Monde determinant, see Eq.(\[vdm\]), we further notice that $$\Delta_{N+2}(\lambda_1,\ldots,\lambda_N,\mu_1,\mu_2)=
\Delta_N(\lambda_1,\ldots,\lambda_N)\times (\mu_1-\mu_2)
\prod_{i=1}^N(\mu_1-\lambda_i) \prod_{i=1}^N(\mu_2-\lambda_i),$$ which allows us to rewrite the correlation function as $${\cal C}_2(\mu_1,\mu_2)=
\frac{1}{(\mu_1-\mu_2)}\int_{-\infty}^{\infty}\prod^N_{i=1}
dw(\lambda_i)\,\Delta_{N}(\lambda_1,\ldots,\lambda_N)
\Delta_{N+2}(\lambda_1,\ldots,\lambda_N,\mu_1,\mu_2).$$ Now we replace each entry $\lambda_i^j$ in both van der Monde determinant factors with the orthogonal polynomial $\pi_j(\lambda_i)$ (cf. eq.(\[vdm1\])), and further expand the first factor as a sum over permutations: $\Delta_{N}(\lambda_1,\ldots,\lambda_N)\propto \sum_{P}(-1)^{|P|}
\pi_{0}(\lambda_{\sigma_1})\ldots \pi_{N-1}(\lambda_{\sigma_N})$. Further using permutational symmetry of the second determinant, we again see that every term yields after integration the same contribution. Up to a proportionality factor we can therefore rewrite the correlation function as $$\begin{aligned}
{\cal C}_2(\mu_1,\mu_2)&=&
\frac{1}{(\mu_1-\mu_2)}\int_{-\infty}^{\infty}\prod^N_{i=1}
dw(\lambda_i)\,\pi_{0}(\lambda_{1})\ldots \pi_{N-1}(\lambda_{N})\\
\nonumber
&\times&\det{\left(\begin{array}{ccccc}\pi_{0}(\lambda_{1})&
\pi_{0}(\lambda_2)&\ldots &\pi_{0}(\mu_1)&\pi_{0}(\mu_2)\\
\pi_{1}(\lambda_1)&\pi_{1}(\lambda_2)&\ldots &
\pi_{1}(\mu_1)& \pi_{1}(\mu_2) \\
.&.&.&.&\\ .&.&.&.&\\.&.&.&.&\\ \pi_{N}(\lambda_1)&
\pi_{N}(\lambda_2)&\ldots & \pi_{N}(\mu_1)
& \pi_{N}(\mu_2)\\
\pi_{N+1}(\lambda_1)&
\pi_{N+1}(\lambda_2)&\ldots &\pi_{N+1}(\mu_1)
& \pi_{N+1}(\mu_2)
\end{array}\right)}.\end{aligned}$$ At the next step we absorb the factors $
\pi_{0}(\lambda_{1}),\dots, \pi_{N-1}(\lambda_N)$ inside the determinant by multiplying the first column with $\pi_{0}(\lambda_{1})$,..., the $N-th$ column with $\pi_{N-1}(\lambda_N)$, and leaving the last two columns intact. By linearity, we can also absorb the product of the integrals inside the determinant by integrating the first column over $\lambda_1$,..., and $N-th$ column over $\lambda_N$. Due to the orthogonality, the first $N$ columns of the resulting determinant after integration contain zero components off-diagonal, whereas the entries on the main diagonal are equal to the normalization constants $c_k=\int_{-\infty}^{\infty}
dw(\lambda)\,\pi^2_{k}(\lambda),\,\, k=0,\dots,N$. Therefore, the resulting determinant is easy to calculate and, up to a multiplicative constant we arrive to the following simple formula: $$\label{corrf2}
{\cal C}_2(\mu_1,\mu_2)\propto \frac{1}{(\mu_1-\mu_2)}
\det{\left(\begin{array}{cc} \pi_{N}(\mu_1) & \pi_{N}(\mu_2)\\
\pi_{N+1}(\mu_1) & \pi_{N+1}(\mu_2)\end{array}\right)}.$$ In particular, for the second moment of the characteristic polynomial we have the expression $$\label{mom2}
E[Z^2(\mu)]=\lim_{\mu_1\to\mu_2=\mu} {\cal
C}_2(\mu_1,\mu_2)\propto \det{\left(\begin{array}{cc} \pi_{N}(\mu)
& \pi'_{N}(\mu)\\ \pi_{N+1}(\mu) &
\pi'_{N+1}(\mu)\end{array}\right)}.$$
This procedure can be very straightforwardly extended to higher order correlation functions[@BH; @MN], and higher order moments[@FW] of the characteristic polynomials. The general structure is always the same, and is given in the form of a determinant whose entries are orthogonal polynomials of increasing order.
One more observation deserving mentioning here is that the structure of the two-point correlation function of characteristic polynomials is identical to that of the Christoffel-Darboux, which is the main building block of the kernel function, Eq.(\[kern1\]). Moreover, comparing the above formula (\[mom2\]) for the gaussian case with expressions (\[kern77\],\[darboux3\]), one notices a great degree in similarity between the structure of mean eigenvalue density and that for the second moment of the characteristic polynomial. All these similarities are not accidental, and there exists a general relation between the two types of quantities as I proceed to demonstrate on the simplest example. For this we recall that the mean eigenvalue density $\overline{\rho_N(\lambda)}$ is just the one-point correlation function, see Eq.(\[24\]), and according to Eq.(\[21\]) and Eq.(\[JPDG\]) can be written as $$\begin{aligned}
\nonumber {\cal R}_{1}(\lambda) &=&N \int \,{\cal
P}(\lambda,\lambda_2,\ldots,\lambda_N) \,\,
d\lambda_{2}\,\ldots\,\lambda_N\\ &\propto& e^{-Q(\lambda)} \int
d\lambda_{2}\,\ldots\,\lambda_N e^{-\sum_{i=2}^NQ(\lambda_i)}
\prod_{i=2}^N(\lambda-\lambda_i)^2 \prod_{2\le i<j\le
N}\left(\lambda_i-\lambda_j\right)^2. \label{211}\end{aligned}$$ It is immediately evident after simple renumbering $(\lambda_2,\ldots,\lambda_N)\to (\lambda_1,\ldots,\lambda_{N-1})$ that the integral in the second line allows a clear interpretation as the second moment of the characteristic polynomial $E[Z^2_{N-1}(\lambda)]$ of a random matrix $H_{N-1}$ distributed according to the same joint probability density function ${\cal
P}\left(H_{N-1}\right) d\hat{H}_{N-1},$ but of reduced size $N-1$, see Eq.(\[Z1\]) for comparison. We therefore have a general relation between the mean eigenvalue density and the second moment of the characteristic polynomial of the reduced-size matrix: $$\overline{\rho_N(\lambda)}\propto e^{-Q(\lambda)}
\overline{\left[\det{\left(\lambda {\bf 1}_N-\hat{H}_{N-1}\right)}
\right]^2}$$ which explains the observed similarity. This type of relations, and their natural generalizations to higher-order correlation functions hold for general invariant ensembles and were found helpful in several applications; e.g. for the so-called “chiral" ensembles (notion of such ensembles is shortly discussed in the very end of these notes) in [@AK], for non-Hermitian matrices with complex eigenvalues see examples and further references in [@FS]); for real symmetric matrices see the recent paper[@fluc].
Now let us discuss another important class of correlation functions involving characteristic polynomials, - namely one combining both positive and negative moments, the simplest example being the expectation value of the ratio: $$\label{ratio}
{\cal K}_{N}(\mu,\nu)= E\left[ \frac{ Z_N(\mu)}{Z_N(\nu)}\right].$$ For such an object to be well-defined it is necessary to regularize the characteristic polynomial in the denominator $Z_N(\nu)=\det\left(\nu {\bf 1}_N-\hat{H}\right)$ by considering the complex-valued spectral parameter $\nu$ such that $\mbox{Im}\nu\ne 0$. Further generalizations include more than one polynomial in numerator and/or denominator.
Such objects turned out to be indispensable tools in applications of random matrices to physical problems. In fact, in all applications a very fundamental role is played by the resolvent matrix $(\mu{\bf 1}_N- \hat{H})^{-1}$, and statistics of its entries is of great interest. In particular, the familiar eigenvalue density $\rho(\nu)$ can be extracted from the trace of the resolvent as $$\label{rho}
\rho(\nu)=\frac{1}{\pi}\lim_{\mbox{Im}\mu\to 0^{-}}
\mbox{Im}\mbox{Tr}\frac{1}{\mu{\bf 1}_N- \hat{H}}.$$
It is easy to understand that one can get access to such an object, and more general correlation functions of the traces of the resolvent by using the identity: $$\label{rho1}
\mbox{Tr}\frac{1}{\mu{\bf 1}_N- \hat{H}}=-\frac{\partial}{\partial
\nu} \frac{Z_N(\mu)}{Z_N(\nu)}|_{\mu=\nu}.$$ We conclude that the products of ratios of characteristic polynomials can be used to extract the multipoint correlation function of spectral densities (see an example below). Moreover, distributions of some other interesting quantities as, e.g. individual entries of the resolvent, or statistics of eigenvalues as functions of some parameter can be characterized in terms of general correlation functions of ratios, see [@AS] for more details and examples. Thus, that type of the correlation function is even more informative than one containing products of only positive moments of the characteristic polynomials.
In fact, it turns out that there exists a general relation between the two types of the correlation functions, which is discussed in full generality in recent papers [@FS1; @SF2; @BDS; @BS]. Here we would like to illustrate such a relation on the simplest example, Eq.(\[ratio\]). To this end let us use the following identity: $$\label{ident}
\left[Z_N(\nu)\right]^{-1}=\frac{1}{\prod_{i=1}^N{(\nu-\lambda_i)}} =
\sum_{k=1}^N\frac{1}{\nu-\lambda_k}\prod_{i\ne k}^N
\frac{1}{\lambda_i-\lambda_k}$$ and integrate the ratio of the two characteristic polynomials over the joint probability density of all the eigenvalues. When performing integrations, each of $N$ terms in the sum in Eq.(\[ident\]) produces identical contributions, so that we can take one term with $k=1$ and multiply the result by $N$. Representing $\Delta^2(\lambda_1,\ldots, \lambda_N)=\prod_{2\le
i}(\lambda_1-\lambda_i)^2 \prod_{2\le i<j}
(\lambda_i-\lambda_j)^2$, and observing some cancellations, we have $$\begin{aligned}
\nonumber && {\cal K}_{N}(\mu,\nu)\propto \int
dw(\lambda_1) \frac{\mu-\lambda_1}{\nu-\lambda_1} \int
dw(\lambda_2)\, ...\, dw(\lambda_N)
\prod_{2\le i<j}^N(\lambda_i-\lambda_j)^2
\prod_{i=2}^N{(\lambda_1-\lambda_i)}{(\mu-\lambda_i)}
\\ &&\propto \int
dw(\lambda_1) \frac{\mu-\lambda_1}{\nu-\lambda_1} \times
\overline{\mbox{det}\left(\lambda_1-\hat{H}_{N-1}\right)\mbox{det}
\left(\mu-\hat{H}_{N-1}\right)}. \label{Herspe}\end{aligned}$$ The average value of the products of two characteristic polynomials found by us in Eq.(\[corrf2\]) can now be inserted into the integral entering Eq.(\[Herspe\]), and the resulting expression can be again written in the form of a $2\times 2$ determinant: $$\label{Her44}
K_{N}(\mu,\nu)\propto
\mbox{det}\left(\begin{array}{cc}
\pi_{N-1}\left(\mu\right)&
f_{N-1}(\nu)
\\ \pi_{N}\left(\mu \right)&
f_N(\nu)
\end{array}\right)$$ where $f_N(\nu)$ stands for the so-called Cauchy transform of the orthogonal polynomial $$\label{fdef}
f_N(\nu)=\frac{1}{2\pi i}
\int^{\infty}_{-\infty}\frac{dw(\lambda)}{{\nu-\lambda}}
\,\pi_{N}\left(\lambda\right)$$
The emerged functions $f_{N}(\nu)$ is a rather new feature in Random Matrix Theory. It is instructive to have a closer look at their properties for the simplest case of the Gaussian Ensemble, $Q(\lambda)=N\lambda^2/2$. It turns out that for such a case the functions $f_N(\nu)$ are, in fact, related to the so-called generalized Hermite functions ${\cal H}_{N}$ which are second -non-polynomial- solutions of the same differential equation which is satisfied by Hermite polynomials themselves. The functions also have a convenient integral representations, which can be obtained in the most straightforward way by substituting the identity $$\frac{1}{\nu-\lambda}\propto\int_0^{\infty} dt
e^{it\mbox{\small sgn}{\left[\mbox{\small Im}\nu\right]}(\nu-\lambda)}$$ into the definition (\[fdef\]), replacing the Hermite polynomial with its integral representation, Eq.(\[intrep\]), exchanging the order of integrations and performing the $\lambda-$integral explicitly. Such a procedure results in $$f_{N+n}(\nu)\propto \int_0^{\infty} dt t^{N+n} e^{-N
\left(\frac{t^2}{2}-it\mbox{\small sgn}{\left[ \mbox{\small
Im}\nu\right]}\nu\right)}.$$ Note, that this is precisely the integral (\[integral\]) whose large-N asymptotics for real $\nu$ we studied in the course of our saddle-point analysis. The results can be immediately extended to complex $\nu$, and in the “bulk scaling" limit we arrive to the following asymptotics of the correlation function (\[Her44\]) close to the origin $$\label{FS}
\lim_{N\to\infty} {\cal K}_N(\mu,\nu)={\cal K}_{\infty}\left[
N\rho_{\infty}(0)(\mu-\nu)\right],\quad
K_{\infty}(r)\propto\left\{\begin{array}{cc}
e^{-i\pi r}& \mbox{if Im}(\nu)>0 \\
e^{i\pi r} & \mbox{if Im}(\nu)>0 \end{array}\right..$$ In a similar, although more elaborate way one can calculate an arbitrary correlation function containing ratios and products of characteristic polynomials [@FS1; @BDS; @BS]. The detailed analysis shows that the kernel $ S(\mu,\nu)={\cal
K}(\mu,\nu)/(\mu-\nu)$ and its scaling form $S_{\infty}(r)\propto\frac{K_{\infty}(r)}{r}$ play the role of a building block for more general correlation functions involving ratios, in the same way as the Dyson kernel (\[Dyson\]) plays similar role for the n-point correlation functions of eigenvalue densities. This is a new type of “kernel function" with structure different from the standard random matrix kernel Eq.(\[kern\]). The third type of such kernels - made from functions $f_N(\nu)$ alone - arises when considering only negative moments of the characteristic polynomials.
To give an instructive example of the form emerging consider $$\label{ratio2}
{\cal K}_N(\mu_1,\mu_2,\nu_1,\nu_2)=E\left[
\frac{ Z_N(\mu_1)}{Z_N(\nu_1)}\frac{ Z_N(\mu_2)}{Z_N(\nu_2)}\right]=
\frac{(\mu_1-\nu_1)(\mu_1-\nu_2)
(\mu_2-\nu_1)(\mu_2-\nu_2)}{(\mu_1-\mu_2)(\nu_1-\nu_2)}$$ $$\times\mbox{det}\left(\begin{array}{cc}
S\left(\mu_1,\nu_1\right)&
S\left(\mu_1,\nu_2\right)
\\ S\left(\mu_2,\nu_1\right)&
S\left(\mu_2,\nu_2\right)
\end{array}\right).$$
Assuming Im$\,\nu_1>0$, Im$\,\nu_2<0$, both infinitesimal, we find in the bulk scaling limit such that both $N\rho(0)\mu_{1,2}=\zeta_{1,2}$ and $N\rho(0)\nu_{1,2}=\kappa_{1,2}$ are finite the following expression (see e.g. [@SF2], or [@AS]) $$\label{ratio3}
\lim_{N\to\infty}{\cal
K}_N(\mu_1,\mu_2,\nu_1,\nu_2)=
{\cal K}_{\infty}(\zeta_1,\zeta_2,\kappa_1,\kappa_2)$$ $$=\frac{e^{i\pi(\zeta_1-\zeta_2)}}{\zeta_1-\zeta_2}
\left[e^{i\pi(\kappa_1-\kappa_2)}\frac{(\kappa_1-\zeta_1)
(\kappa_2-\zeta_2)}
{\kappa_1-\kappa_2}-e^{-i\pi(\kappa_1-\kappa_2)}
\frac{(\kappa_1-\zeta_2)(\kappa_2-\zeta_1
)}{\kappa_1-\kappa_2}\right]. \nonumber$$
This formula can be further utilized for many goals. For example, it is a useful exercise to understand how the scaling limit of the two-point cluster function (\[kern8\]) can be extracted from such an expression (hint: the cluster function is related to the correlation function of eigenvalue densities by Eq.(\[2p\]); exploit the relations (\[rho\]),(\[rho1\])).
All these developments, - important and interesting on their own, indirectly prepared the ground for discussing the mathematical framework for a proof of universality in the large-$N$ limit. As was already mentioned, the main obstacle was the absence of any sensible integral representation for general orthogonal polynomials and their Cauchy transforms. The method which circumvents this obstacle in the most elegant fashion is based on the possibility to define [*both*]{} orthogonal polynomials [*and*]{} their Cauchy transforms in a way proposed by Fokas, Its and Kitaev, see references in [@Deift], as elements of a (matrix valued) solution of the following (Riemann-Hilbert) problem. The latter can be introduced as follows. Let the contour $\Sigma$ be the real axis orientated from the left to the right. The upper half of the complex plane with respect to the contour will be called the positive one and the lower half - the negative one. Fix an integer $n\geq 0$ and the measure $w(z)=e^{-Q(z)}$ and define the Riemann-Hilbert problem as that of finding a $2\times 2$ matrix valued function $Y=Y^{(n)}(z)$ satisfying the following conditions:
- $Y^{(n)}(z)- \mbox{analytic}\;\mbox{in}\;
\textsc{C}\setminus\Sigma $
- $Y^{(n)}_{+}(z)=Y^{(n)}_{-}(z)\left(
\begin{array}{cc}
1 & w(z) \\
0 & 1 \
\end{array}
\right),\;z\in \Sigma$
- $Y^{(n)}(z)\mapsto\left(I+{\mathcal{O}}(z^{-1})\right)
\left(\begin{array}{cc}
z^n & 0 \\
0 & z^{-n} \
\end{array}
\right)\;\; \mbox{as}\;\; z\mapsto \infty $
Here $Y^{(n)}_{\pm}(z)$ denotes the limit of $Y^{(n)}(z')$ as $z'\mapsto z\in \Sigma$ from the positive/negative side of the complex plane. It may be proved (see [@Deift]) that the solution of such a problem is unique and is given by $$\label{R-H Solution}
Y^{(n)}(z)=\left(
\begin{array}{cc}
\pi_n(z) & f_n(z) \\
\gamma_{n-1}\pi_{n-1}(z) & \gamma_{n-1}f_{n-1}(z)
\end{array}
\right),\;\;\;\mbox{Im}\;z\neq 0$$ where the constants $\gamma_n$ are simply related to the normalization of the corresponding polynomials: $\gamma_n=-2\pi i
[\int_{-\infty}^{\infty} dw \pi^2_n]^{-1}$.
On comparing formulae (\[Her44\]) and (\[R-H Solution\]) we observe that the structure of the correlation function ${\mathcal{K}}_{N}(\mu,\nu)$ is very intimately related to the above Riemann-Hilbert problem. In fact, for $\mu=\nu=z$ the matrices involved are identical (even the constant $\gamma_{n-1}$ in Eq.(\[Her44\]) emerges when we replace $\propto$ with exact equality sign). Actually, all three types of kernels can be expressed in terms of the solution of the Riemann-Hilbert problem. The original works [@Deift; @BI] dealt only with the standard kernel built from polynomials alone. From that point of view the presence of Cauchy transforms in the Riemann-Hilbert problem might seem to be quite mysterious, and even superfluous. Now, after revealing the role and the meaning of more general kernels the picture can be considered complete, and the presence of the Cauchy transforms has its logical justification.
The relation to the Riemann-Hilbert problem is the starting point for a very efficient method of extracting the large$-N$ asymptotics for essentially any potential function $Q(x)$ entering the probability distribution measure. The corresponding machinery is known as the variant of the steepest descent/stationary phase method introduced for Riemann-Hilbert problems by Deift and Zhou. It is discussed at length in the book by Deift[@Deift] which can be recommended to the interested reader for further details. In this way the universality was verified for all three types of kernels pertinent to the random matrix theory not only for bulk of the spectrum[@SF2], but also for the spectral edges . In our considerations of the Gaussian Unitary Ensemble we already encountered the edge scaling regime where the spectral properties were parameterized by the Airy functions $Ai(x)$. Dealing with ratios of characteristic polynomials in such a regime requires second solution of the Airy equation denoted by $Bi(x)$, see [@AF].
We finish our exposition by claiming that there exist other interesting classes of matrix ensembles which attracted a considerable attention recently, see the paper[@sycl] for more detail on the classification of random matrices by underlying symmetries. In the present framework we only mention one of them - the so-called [*chiral*]{} GUE. The corresponding $2N\times 2N$ matrices are of the form $\hat{H}_{ch}=\left(\begin{array}{cc}{\bf 0}_N&\hat{J}\\
\hat{J}^{\dagger}& {\bf 0}_N \end{array}\right)$, where $\hat{J}$ of a general complex matrix. They were introduced to provide a background for calculating the universal part of the microscopic level density for the Euclidian QCD Dirac operator, see [@Ver] and references therein, and also have relevance for applications to condensed matter physics. The eigenvalues of such matrices appear in pairs $\pm\lambda _k\,\,,\,\, k=1,...,N$. It is easy to understand that the origin $\lambda=0$ plays a specific role in such matrices, and close to this point eigenvalue correlations are rather different from those of the GUE, and described by the so-called Bessel kernels[@T2]. An alternative way of looking essentially at the same problem is to consider the random matrices of Wishart type $\hat{W}= \hat{J}^{\dagger}\hat{J}$, where the role of the special point is again played by the origin (in such context the origin is frequently referred to as the “hard spectral edge“, since no eigenvalues are possible beyond that point. This should be contrasted with the Airy regime close to the semicircle edge, the latter being sometimes referred to as the ”soft edge" of the spectrum. ). The corresponding problems for products and ratios of characteristic polynomials were treated in full rigor by Riemann-Hilbert technique by Vanlessen[@vanlessen], and in a less formal way in [@AF].
Acknowledgement
---------------
My own understanding in the topics covered was shaped in the course of numerous discussions and valuable contacts with many colleagues. I am particularly grateful to Gernot Akemann, Jon Keating, Boris Khoruzhenko and Eugene Strahov for fruitful collaborations on various facets of the Random Matrix theory which I enjoyed in recent years. I am indebted to Nina Snaith and Francesco Mezzadri for their invitation to give this lecture course, and for careful editing of the notes. It allowed me to spend a few wonderful weeks in the most pleasant and stimulating atmosphere of the Newton Institute, Cambridge whose hospitality and financial support I acknowledge with thanks. Finally, it is my pleasure to thank Guler Ergun for her assistance with preparing this manuscript for publication.
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[^1]: The following informal but instructive definition of the white noise process may be helpful for those not very familiar with theory of stochastic processes. For any positive $t>0$ and integer $k\ge 1$ define the random function $\xi_k(t)=\sqrt{2/\pi}\sum_{n=0}^k a_n\cos{nt}$, where real coefficients $a_n$ are all independent, Gaussian distributed with zero mean $E[a_n]=0$ and variances $E[a^2_0]=D/2$ and $E[a^2_n]=D$ for $1\le n\le k$. Then one can, in a certain sense, consider white noise as the limit of $\xi_k(t)$ for $k\to \infty$. In particular, the Dirac $\delta(t-t')$ is approximated by the limiting value of $\frac{\sin{[(k+1/2)(t-t')]}}{2\pi
\sin{(t-t')/2}}$
[^2]: Sometimes one uses instead the interval $[-L,L]$ to define $A(L)$, see e.g. [@Deift].
[^3]: The standard reference to the Hermite polynomials uses the definition $$H_k(x)=(-1)^ke^{x^2}
\frac{d^k}{dx^k}\left(e^{-x^2}\right)=2^kx^k+\cdots,$$ Such a choice ensures $H_k(x)$ to be orthogonal with respect to the weight $e^{-x^2}$. Our choice is motivated by random matrix applications, and is related to the standard one as $h_k(x)=H_k\left(\sqrt{\frac{N}{2}x}\right)$.
|
---
abstract: 'We give a proof that every space of weighted square-integrable holomorphic functions admits an equivalent weight whose Bergman kernel has zeroes. Here the weights are equivalent in the sense that they determine the same space of holomorphic functions. Additionally, a family of radial weights in $L^1(\mathbb{C})$ whose associated Bergman kernels have infinitely many zeroes is exhibited.'
address: 'Department of Mathematics, Texas A&M University, College Station TX 77843-3368'
author:
- 'Blake J. Boudreaux'
bibliography:
- 'VBK.bib'
title: Equivalent Bergman Spaces With Inequivalent Weights
---
Introduction
============
Since the paper of L. Qi-Keng [@Qi-Keng1966], there has been interest in constructing domains in $\mathbb{C}^N$ whose Bergman kernel has zeroes. After the observation that weighted Bergman kernels correspond to unweighted Bergman kernels in higher dimension, it is natural to consider the same questions involving weighted Bergman kernels.
A positive measurable function $\mu$ defined on a domain $D\subset\mathbb{C}^N$ is called a *weight*. (We merely require a weight to be measurable; some authors require weights to additionally be integrable.) To every weight $\mu$ on $D$ corresponds a Hilbert space $L^2(D,\mu)$ of measurable functions determined by the inner product $$\left\langle f,g\right\rangle_{\mu} :=\int_{D}f(\zeta)\overline{g(\zeta)}\mu(\zeta)dA(\zeta).$$ Let $L^2_H(D,\mu)$ denote the subspace of $L^2(D,\mu)$ consisting of those functions that are also holomorphic.
We are interested in weights that determine a space on which a weighted Bergman kernel can be defined. We call a weight $\mu$ *admissible* if for each $z\in D$ the evaluation functional $E_z:f\mapsto f(z)$ is continuous on $L^2_H(D,\mu)$, and if $L^2_H(D,\mu)$ is a closed subspace of $L^2(D,\mu)$. In the case that $\mu$ is admissible, for each $z\in D$ the Riesz representation theorem provides a unique $B_z^{D,\mu}(\,\cdot\,)\in L^2_H(D,\mu)$ such that $$f(z)=\left\langle f,B_z^{D,\mu}\right\rangle_{\mu}.$$ It is common to write $K_{D,\mu}(z,\zeta)=B^{D,\mu}_z(\zeta)$ and view it as a function on $D\times D$. We call $K_{D,\mu}$ the weighted Bergman kernel of $D$ (with respect to the weight $\mu$). We typically write $K_{\mu}$ in place of $K_{D,\mu}$ if the domain is clear from context. As in the unweighted case [@Krantz2001; @Range1986], the kernel $K_{\mu}(z,\zeta)$ possesses the following properties:
1. $K_{\mu}(z,\zeta)$ is holomorphic in $z$ and conjugate-holomorphic in $\zeta$,
2. $K_{\mu}(z,\zeta)=\sum_{j}\varphi_{j}(z)\overline{\varphi_{j}(\zeta)}$ for any complete orthonormal system $\{\varphi_j\}$ of $L^2_H(D,\mu)$, with convergence uniform on compact sets of $D\times D$,
3. and $\overline{K_{\mu}(z,\zeta)}=K_{\mu}(\zeta,z)$.
A comprehensive reference on the theory of admissible weights is the paper, “On the Dependence of the Reproducing Kernel on the Weight of Integration”[@PW1990].
Following the convention of A. Perälä [@Perälä2017], we say that two admissible weights $\mu_1$ and $\mu_2$ are *equivalent*, or $\mu_1\sim\mu_2$, if $L^2_H(D,\mu_1)=L^2_H(D,\mu_2)$ as sets. For example, if $g$ is a positive measurable function on $D$ with the property that $\text{ess inf}_{z\in D}g(z)>0$ and $\text{ess sup}_{z\in D}g(z)<\infty$, then $g\cdot \mu\sim\mu$ for any weight $\mu$ on $D$.
The purpose of this note is to answer two questions [@Perälä2017]. The firsts asks if every space $L^2_H(D,\mu)$ can be equipped with an equivalent weight $\mu^*$ so that the Bergman kernel $K_{D,\mu^*}$ has zeroes. The second asks if there exists a radial weight $W$ on $\mathbb{C}$ such that the kernel $K_{\mathbb{C},W}(\,\cdot\, ,z)$ has infinitely many zeros for a fixed $z\in D$. We answer both questions in the affirmative.
The plan of attack to answering the first question above is best illustrated by considering the case when $D\subset\mathbb{C}^N$ is a bounded set containing zero and $\mu$ is continuous. In this situation, the weight $\nu(z):=\mu(z)/\|z\|^{2N}$ is not locally integrable at zero. Indeed, by continuity $\mu$ is uniformly bounded away from zero in a sufficiently small neighborhood of $z=0$. It follows that every member of $L^2_H(D,\nu)$in particular $K_{\nu}(\,\cdot\,,\zeta)$ for each $\zeta\in D$vanishes at zero. Moreover, for each $n\in\mathbb{N}$ the weight $\nu_n(z):=\min(n,\|z\|^{-2N})\cdot\mu(z)$ is equivalent to $\mu(z)$: each $\nu_n$ is simply the product of $\mu$ and a bounded function that is uniformly bounded away from zero (recall we are assuming $D$ is bounded). Since $\nu_n$ increases to $\nu$ as $n\to\infty$, we may apply a weighted generalization of the Ramadanov theorem [@PWW2016] to see that $K_{\nu_n}\to K_{\nu}$ uniformly on compact subsets of $D\times D$. Regarding $\zeta\in D$ as fixed, the function $K_{\nu}(\,\cdot\,,\zeta)$ vanishes at zero, so a variant of Hurwitz’s theorem shows that for large $n$ we have $K_{\nu_n}(0,\zeta)=0$, completing the proof. The general case requires a more delicate approach, as a general measurable function may have extremely pathlogical behavior near every point in its domain, but the main idea remains the same.
The approach to answering the second question involves carefully choosing the weight $W$ so that for each fixed nonzero $w$ in the plane the kernel $K_{\mathbb{C},W}(\,\cdot\,,w)$ is an entire function of finite but non-integer order, which by a consequence of the Hadamard factorization theorem has infinitely many zeros. As a special case of our construction, we exhibit a weight with kernel $\cos(i\sqrt{z\bar{w}})$; this is a function satisfying the necessary criteria by elementary means.
I would like to thank my advisor, Dr. Harold Boas, for bringing these questions to my attention, as well as providing direction on how one might approach them. I would also like to thank the referee, whose insightful comments transformed the third section of this note from a single example to a large family.
The Bergman Kernels of Equivalent Weights
=========================================
Let us first show that our definition of an admissible weight is consistent with another standard definition [@PW1990].
Let $\mu$ be a weight on a domain $D\subset\mathbb{C}^N$. Then $\mu$ is admissible if and only if the norm of the point evaluation functional $E_z:f\mapsto f(z)$ is locally bounded (if thought of as a function on $D$).
Suppose that $\mu$ is admissible. Fix $z\in D$ and let $V_z$ be an open set containing $z$ with $\overline{V_z}\subset D$. Since $$\sup_{w\in V_z}|E_w(f)|=\sup_{w\in V_z}|f(w)|<\infty$$ for every $f\in L^2_H(D,\mu)$, an application of the uniform boundedness principle to the family $\{E_w\,:\,w\in V_z\}$ of continuous linear functionals shows that $$\sup_{w\in V_z}\|E_w\|<\infty.$$ The converse follows from known results ([@PW1990] Proposition 2.1).
Before proceeding we require a lemma. It allows us in many cases to assume the given weight is bounded below by a more well behaved weight.
Let $\mu_1$ be an admissible weight on $D$, and suppose that $\mu_1$ is integrable on a bounded open neighborhood $U$ of some point $z_0\in D$ with $\overline{U}\subset D$. Let $\mu_2$ be a weight on $U$. Then the weight $\tilde{\mu}_1$ defined by $$\tilde{\mu}_1(z):= \begin{cases}
\max(\mu_1(z),\mu_2(z))\,& \text{ if }z\in U\\
\mu_1(z) & \text{ if } z\in D\setminus U \end{cases}$$ is an admissible weight with $L^2_H(D,\tilde{\mu}_1)\subset L^2_H(D,\mu_1)$ and continuous inclusion, such that $\mu_2\leq \tilde{\mu}_1$ on $U$. Furthermore, if $\mu_2$ is integrable over $U$ as well, then $L^2_H(D,\mu_1)=L^2_H(D,\tilde{\mu}_1)$ and $\tilde{\mu}_1$ determines an equivalent norm to $\mu_1$.
Observe that $\|f\|_{\mu_1}\leq\|f\|_{\tilde{\mu}_1}$ for every measurable function $f$. Therefore $L^2(D,\tilde{\mu}_1)\subset L^2(D,\mu_1)$ with continuous inclusion. It follows that $\|E_{z}\|_{\tilde{\mu}_1}\leq\|E_{z}\|_{\mu_1}$ for every $z\in D$, and hence $\tilde{\mu}_1$ is an admissible weight.
Now suppose that $\mu_2$ is integrable over $U$ as well. By applying the uniform boundedness principle to the family of continuous functionals given by evaluation at each point of $U$, we may find a $C>0$ such that $$\begin{aligned}
\|f\|_{\tilde{\mu}_1}^2 &= \int_{U}|f(\zeta)|^2\max(\mu_1(\zeta),\mu_2(\zeta))dA(\zeta) +\int_{D\setminus U}|f(\zeta)|^2\mu_1(\zeta) dA(\zeta)\\
&\leq\sup_{z\in U}|f(z)|^2\cdot\int_{U}\max(\mu_1(\zeta),\mu_2(\zeta))dA(\zeta) +\|f\|_{\mu_1}^2\\
&\leq C\cdot\left(\int_{U}\max(\mu_1(\zeta),\mu_2(\zeta))dA(\zeta)\right)\cdot \|f\|_{\mu_1}^2+\|f\|^2_{\mu_1}\\
&\leq \left[C\left(\int_{U}\max(\mu_1(\zeta),\mu_2(\zeta))dA(\zeta)\right)+1\right]\|f\|^2_{\mu_1}\end{aligned}$$ holds for each $f\in L^2_H(D,\mu_1)$. Observe that $\max(\mu_1,\mu_2)$ is integrable over $U$ since both $\mu_1$ and $\mu_2$ are.
By setting $\mu_2\equiv 1$ above, we have the immediate corollary that $\tilde{\mu}_1$ is an equivalent weight to $\mu_1$ with equivalent norm having the property that $1\leq \tilde{\mu}_1$ on $U$.
We also require a weighted generalization of the Ramadanov theorem [@PWW2016], whose statement is included for the convenience of the reader.
\[Ramadanov\] Let $\{D_i\}_{i=1}^{\infty}$ be a sequence of domains in $\mathbb{C}^N$ and set $D:=\bigcup_{j}D_j$. Let $\mu$ be an admissible weight on $D$, and $\mu_k$ be an admissible weight on $D_k$ for each $k$. Extend $\mu_k$ by $\mu$ on $D$. Assume moreover that
- For any $n\in\mathbb{N}$ there is $N=N(n)$, such that $D_n\subset D_m$ and $\mu_n(z)\leq\mu_m(z)\leq\mu(z)$ for $m\geq N(n)$, $z\in D_n$.
- $\mu_k\xrightarrow[k\to\infty]{}\mu$ pointwise almost everywhere on $D$.
Then $$\lim_{k\to\infty}K_{D_k,\mu_k}=K_{D,\mu}$$ locally uniformly on $D\times D$.
We now have all the necessary tools to prove the main result of this section.
\[Main\] Let $\mu$ be an admissible weight on a domain $D\subset\mathbb{C}^N$. Then there exists an admissible weight $\mu^*$, with $\mu\sim\mu^*$, such that $K_{D,\mu^*}$ has zeroes.
We assume that $L_H^2(D,\mu)\neq \{0\}$, otherwise the Bergman kernel vanishes identically. By translating if necessary, we may assume that $0\in D$. If $\mu$ is not integrable in any neighborhood of $z=0$, then every $f\in L^2_H(D,\mu)$ must satisfy $f(0)=0$; in particular this implies that $K_{\mu}(0,\zeta)=0$ for each $\zeta\in D$. Therefore we may assume that $\mu$ is integrable on some neighborhood $U$ of $z=0$. By shrinking $U$ if necessary, we may assume that $U$ is bounded with $\overline{U}\subset D$. The Lemma now allows us to assume that $1\leq \mu$ on $U$.
Set $g(z)=\max \left(1,1/\|z\|^{2N}\right)$ and consider $$\nu(z)=g(z)\mu(z).$$ Since $1\leq g(z)$ everywhere, we have $\|f\|^2_{\mu}\leq\|f\|^2_{\nu}$. This shows that the inclusion $L^2(D,\nu)\subset L^2(D,\mu)$ is continuous, implying that $\nu$ is an admissible weight (as in the proof of the Lemma). Since we assume that $1\leq\mu(z)$ on $U$, $\nu$ is not integrable in any neighborhood of $z=0$ and hence $K_{D,\nu}(\,\cdot\,,w)=0$ for each $w\in D$.
The function $\min(n,g(z))$ is bounded above and uniformly bounded away from zero on $D$, so $$\nu_n(z)=\min\left(n,g(z)\right)\cdot \mu(z)$$ is equivalent to $\mu$ for each $n\in\mathbb{N}$.
Next, we apply Theorem \[Ramadanov\]. Since $L^2_H(D,\nu)\neq \{0\}$ (e.g. $z^{\alpha}f\in L^2_H(D,\nu)$ whenever $f\in L^2_H(D,\mu)$ and $\alpha$ is a multindex with $|\alpha|=N$), we may find a $w\in D$ so that $K_{\nu}(z,w)$ is a nontrivial holomorphic function of $z$. We claim that $K_{\nu_{m}}(z,w)$ has zeros for some $m\in\mathbb{N}$. Seeking a contradiction, suppose that $K_{\nu_{n}}(z,w)$ has no zeros for every $n\in\mathbb{N}$. We have chosen $w\in D$ so that $K_{\nu}(z,w)$ is not identically zero, so fix $z_0\in D$ with $K_{\nu}(z_0,w)\neq 0$. Applying Hurwitz’s theorem of one complex variable to connected component of $\{\lambda z_0\in D\,:\,\lambda\in\mathbb{C}\}$ containing the origin, we see that $K_{\nu}(\lambda z_0,w)$ has no zeros. However this implies that $K_{\nu}(0,w)\neq 0$, a contradiction to what was shown above: every $f\in L^2_H(D,\nu)$ vanishes at zero. This shows the claim, and setting $\mu^*=\nu_{m}$ completes the proof.
**Remark.** Observe that something slightly stronger than the conclusion of the theorem has been shown: that we mayup to any positive erroractually prescribe the point at which the zero occurs. Furthermore, by carrying out this construction at finitely many points simultaneously, one can show that an equivalent weight exists whose Bergman kernel has zeroes at finitely many predetermined points up to any positive error.
Radial Weights with Kernel Having Infinitely Many Zeroes in The Plane
=====================================================================
It is known [@Perälä2017] that the kernel $B_{\mathbb{D},\mu}(\,\cdot\,,w)$ of an integrable radial weight $\mu$ on the unit disk $\mathbb{D}\subset\mathbb{C}$ cannot have infinitely many zeroes for a fixed $w\in\mathbb{D}$. In this section we show that the analogous result fails when $\mathbb{D}$ is replaced with the complex plane. In fact, we exhibit a family of radial weights $\mathcal{W}\subset L^1(\mathbb{C})$ such that for every $W\in\mathcal{W}$, the associated weighted Bergman kernel $B_{\mathbb{C},W}(\,\cdot\,,w)$ has infinitely many zeroes for each fixed nonzero $w$ in the plane. This is achieved by following a similar construction to that of H. Bommier-Hato, M. Engliš, and E.-H. Youssfi [@BHEEH].
Given two real and positive parameters $\beta,\gamma$, we may define a holomorphic function $E_{\beta,\gamma}(z)$ by the power series $$E_{\beta,\gamma}:=\sum_{k=0}^{\infty}\frac{z^k}{\Gamma(\beta k+\gamma)}.$$ This is known as the Mittag-Leffler function associated to $\beta$ and $\gamma$. $E_{\beta,\gamma}$ is an entire function with order $1/\beta$ and type 1. A comprehensive treatise on the theory of Mittag-Leffler functions is *Mittag-Leffler functions, Related Topics and Applications* [@GAM2014].
Let $\mathcal{W}\subset L^1(\mathbb{C})$ be the family of weights of the form $$W(z)=\frac{1}{2\pi}|z|^n\exp(-\alpha|z|^{2m}),$$ where $n\in (-2,\infty)$, $\alpha,m>0$, and $m\not\in\mathbb{Z}$. Every member $W$ of $\mathcal{W}$ is admissible and induces a kernel $K_{\mathbb{C},W}(\,\cdot\,,w)$ having infinitely many zeroes for each nonzero $w$ in the plane.
Fix $W\in\mathcal{W}$. We first show that $W$ is admissible. By a result of Z. Pasternak-Winiarski ([@PW1990(2)] Corollary 3.1), it suffices to show that there exists a $c>0$ such that $W^{-c}$ is locally integrable; setting $c=1/n$ if $n>0$ and $c=1$ otherwise will work. Since $W$ is radial, the monomials are an orthonormal basis for $L_H^2(D,W)$, and the representation $$B_{W}(z,w)=\sum_{k=0}^{\infty}\frac{1}{W_k}(z\bar{w})^k,$$ with $W_k=2\pi\int_{0}^{\infty} r^{2k+1}W(r)dr$, holds. Now $$W_k=\int_{0}^{\infty}r^{2k+1+n}\exp(-\alpha r^{2m})=\frac{1}{2m}\alpha^{\tfrac{2m-2k-n-3}{2m}}\cdot\Gamma\left(\frac{2k+2+n}{2m}\right).$$ Note that $W_1=\|W\|_{L^1}<\infty$, so $W\in L^1(\mathbb{C})$ and hence $\mathcal{W}$ is well defined. Comparing (2) with (3) yields $$\begin{aligned}
B_{W}(z,w)&=2m\alpha^{\tfrac{3-2m+n}{2m}}\sum_{k=0}^{\infty}\alpha^{k/m}\frac{(z\bar{w})^k}{ \Gamma\left(\frac{2k+2+n}{2m}\right)}\\\nonumber &=2m\alpha^{\tfrac{3-2m+n}{2m}}\sum_{k=0}^{\infty}\frac{\big(\alpha^{1/m}(z\bar{w})\big)^k}{ \Gamma\left(\tfrac{k}{m}+\tfrac{2+n}{2m}\right)}.\end{aligned}$$ We may write this in terms of the Mittag-Leffler function (1) as $$B_{W}(z,w)=2m\alpha^{\tfrac{3-2m+n}{2m}}E_{\tfrac{1}{m},\tfrac{2+n}{2m}}\big(\alpha^{1/m}(z\bar{w})\big).$$ Fix a nonzero $w$ in the plane. It follows from (4) that $B_{W}(z,w)$ is an entire function of order $m$. Since $m\not\in\mathbb{Z}$ by construction, it is a consequence of the Hadamard factorization theorem that the kernel $B_{W}(\,\cdot\,, w)$ has infinitely many zeroes ([@C1978] Theorem XI.3.7).
Observe that setting $m=1/2$, $\alpha=1$, and $n=-1$ in (4) shows $$B_{W}(z,w)=\sum_{k=0}^{\infty}\frac{(z\bar{w})^k}{\Gamma(2k+1)}=\sum_{k=0}^{\infty}\frac{(z\bar{w})^k}{(2n)!}=\cos(i\sqrt{z\bar{w}}),$$ which provides a concrete member of $\mathcal{W}$ that can be shown to satisfy the conclusion of Theorem 3 without having to invoke Hadamard’s theorem.
**Remark.** The construction of this family of examples required solving a Stieltjes moment problem whose solution is absolutely continuous with respect to Lebesgue measure. There has been much work done on solving the Stieltjes moment problem [@D1989; @ST1943], so it would be interesting to see if one could characterize the entire functions $f$ for which there corresponds an admissible weight $\mu_f$ on $\mathbb{C}$ with $B_{\mathbb{C},\mu_f}(z,w)=f(z\bar{w})$. For instance, it is clear that a necessary condition on such a function $f$ is that its Maclaurin series coefficients be all real and positive.
|
---
abstract: |
We study minimizers of the energy functional $$\int_{D}{|x_n|^a |\nabla u|^2} + \int_{D \cap ({{\mathbb R}}^{n-1} \times \{0\} )}{\lambda^+ \chi_{ \{u > 0\} } + \lambda^- \chi_{ \{u<0\} }} \ d{\mathcal{H}}^{n-1}$$ without any sign restriction on the function $u$. The main result states that the two free boundaries $$\Gamma^+ = \partial \{u(\ \cdot \ , 0) > 0\} \text{ and }
\Gamma^- = \partial \{u(\ \cdot \ , 0) < 0\}$$ cannot touch. i.e. $\Gamma^+ \cap \Gamma^- = \emptyset$
address: 'Department of Mathematics, Purdue University, West Lafayette, IN 47907'
author:
- Mark Allen
bibliography:
- 'Bibliography.bib'
title: Separation of a lower dimensional free boundary in a two phase problem
---
Introduction {#S: introduction}
============
This paper aims to study the local properties of a two phase free boundary problem for the fractional Laplacian. Recently, in [@mA11] the following free boundary problem for the half laplacian has been studied. For a function $u \in C({{\mathbb R}}^N)$ and domain $D$ consider the problem $$\label{E: fractional1}
\begin{aligned}
(-\Delta)^{1/2} u(x)=0 &\quad\text{in }D \cap \{u > 0\}\\
\lim_{y \to x} \frac{u(y)}{((y-x) \cdot \nu(x))^{1/2}} = A &\quad\text{ if } x \in D \cap \partial\{u=0\}
\end{aligned}$$ The study of presents certain difficulties since the fractional laplacian is a global operator. Many of the common techniques for studying free boundaries are unavailable. However, one may work in ${{\mathbb R}}^n = {{\mathbb R}}^{N+1}$ and a common reformulation of the half laplacian is the following $$(- \Delta)^{1/2}u(x') = \lim_{x_n \to 0} \tilde{u}_{x_n}(x',x_n) \text{ for } x' \in {{\mathbb R}}^N$$ where $$\begin{aligned}
\Delta \tilde{u} =0 \text { in } {{\mathbb R}}_{+}^{N+1} \\
\tilde{u}(x',0) = u(x')
\end{aligned}$$ By adding the extra dimension one may then study a localized version of the free boundary problem by studying minimizers of the functional $$\int_{D}{|\nabla u|^2} + \int_{D \cap ({{\mathbb R}}^{n-1} \times \{0\} )}{ \chi_{ \{u > 0\} } } \ d{\mathcal{H}}^{n-1}$$ Since the above functional gives study to a one phase problem, it is natural to study the corresponding two phase problem $$\int_{D}{|\nabla u|^2} + \int_{D \cap ({{\mathbb R}}^{n-1} \times \{0\} )}{\lambda^+ \chi_{ \{u > 0\} } + \lambda^- \chi_{ \{u<0\} }} \ d{\mathcal{H}}^{n-1}$$ which has been done in [@mA11]. One may generalize the study of the free boundary problem by considering other powers $0<s<1$ of the fractional laplacian. In [@CS] the appropriate extension theorem was proven enabling one to give a reformulation of the fractional laplacian by adding an extra dimension. By adding an extra dimension, the study of with $s$ replacing $1/2$ is reduced to the study of the minimizers of the functional $$\int_{D}{|x_n|^a|\nabla u|^2} + \int_{D \cap ({{\mathbb R}}^{n-1} \times \{0\} )}{\chi_{ \{u > 0\} }} \ d{\mathcal{H}}^{n-1}$$ where $a=1-2s$ and $n=N+1$. This problem has been recently studied in [@lC10]. This paper will study the corresponding two-phase problem and extend one of the main results in [@mA11] to the general fractional case when $0<s<1$. This paper will then focus on the localized two phase problem which is to consider minimizers of the functional $$\label{E: functional}
\int_{D}{|x_n|^a|\nabla u|^2} + \int_{D \cap ({{\mathbb R}}^{n-1} \times \{0\} )}{\lambda^+ \chi_{ \{u > 0\} } + \lambda^- \chi_{ \{u<0\} }} \ d{\mathcal{H}}^{n-1}$$ over the class $$H^1(a,D) \overset{\text{def}}{=} \{v \in L^2(D) \mid |y|^{a/2} \nabla v \in L^2(D)\}$$ and such that $u - \phi \in H_0^1(a,D)$ for a prescribed $\phi$. From the relation $0<s<1$ and $s=(1-a)/2$ it follows that $a$ will vary in the range $-1<a<1$. Throughout the paper we assume that $\lambda^+$ and $\lambda^-$ are positive constants. By use of the extension theorem given in [@CS] it is natural to make the assumptions that $D$ and $\phi$ are symmetric about the hyperplane ${{\mathbb R}}^{n-1} \times \{0\}$; however, we will not make these assumptions in this paper since the proofs presented will not rely on even symmetry.
The main study of this paper concerns the local properties of the two free boundaries $$\Gamma^+ = \partial \{u( \ \cdot \ , 0) > 0 \} \text{ and } \Gamma^- = \partial \{u( \ \cdot \ , 0) < 0 \}$$ with the boundary being defined by the topology of ${{\mathbb R}}^{n-1} \times \{0\}$.
Motivation and Applications {#motivation-and-applications .unnumbered}
---------------------------
The motivation for studying the problem comes from recognizing the similarity to the problem of studying minimizers of the functional $$\label{E: classical}
J(u) = \int_{D}{|\nabla u|^2 + \lambda^+ \chi_{ \{u>0\} } + \lambda^- \chi_{ \{u<0\} }}$$ which has been done in [@MR732100]. The minimizers of are generalized solutions of a classical two-phase free boundary problem $$\label{E: FBP-classic}
\begin{aligned}
\Delta u=0&\quad\text{in }\{u>0\}\cup\{u<0\}\\
|\nabla u^+|^2-|\nabla u^-|^2=M&\quad\text{on }\partial\{u>0\}\cup\partial\{u<0\},
\end{aligned}$$ with $M=(\lambda^+)^2-(\lambda^-)^2$. The study of problem has applications in two dimensional flow problems as well as in heat flow. In one specific application, the problem arises in a simplified model for premixed equidiffusional flames, in the stationary case, in the limit as $\epsilon \to 0+$ of a singular perturbation problem, see e.g. [@CLW]. By measuring the positivity and negativity on the boundary ${{\mathbb R}}^{n-1} \times \{0\}$, minimizers of can be seen as the limit of solutions to a boundary reaction problem, see e.g. [@mA11].
In modeling, when long range interactions are present, it is relevant to replace the Laplacian by nonlocal operators, such as the fractional Laplacian. See survey papers [@MR1937584] and [@MR1081295].
Main Results {#main-results .unnumbered}
------------
As previously mentioned the two phase case of has been recently studied in [@mA11] under the additional assumption that $a=0$ (or $s=1/2$). By restricting the two phase problem to the case in which $a=0$, the authors in [@mA11] were able to use more technical tools such as the Alt-Caffarelli-Friedman monotonicity formula. One of the main results in [@mA11] is that if $a=0$, then $\Gamma^+ \cap \Gamma^- = \emptyset$. i.e. the free boundaries $\Gamma^+$ and $\Gamma^-$ cannot touch. This result is in complete contrast to many two phase free boundary problems. Often the interphase $\Gamma^+ \cap \Gamma^-$ is difficult to study. In the classical two-phase free boundary problem in , the two-phase points create a major complication even in the proof of the optimal (Lipschitz in that case) regularity of solutions, see [@MR732100]. The separation of $\Gamma^+$ and $\Gamma^-$ is useful in that it reduces the two-phase free boundary problem to the one-phase free boundary problem. That is, locally minimizers have a sign in ${{\mathbb R}}^{n-1} \times \{0\}$, and so we may assume either $\lambda^+=0$ or $\lambda^- =0$. In the case $a=0$, after one establishes optimal regularity and nondegeneracy, the separation of $\Gamma^+$ and $\Gamma^-$ is an immediate consequence of the Alt-Caffarelli-Friedman (ACF) monotonicity formula which was introduced and proven in [@MR732100]. The main result of this paper is the separation of the free boundaries for the more general case in which $a \neq 0$, namely
\[T: acase\] Let $-1<a<1$ and let $u$ be a minimizer to the functional in . Then $\Gamma^+ \cap \Gamma^- = \emptyset$. Furthermore, if $x_0 \in \Gamma^+ \ (x_0 \in \Gamma^-)$ then there exists $r>0$ such that $u \geq 0 \ (u \leq 0)$ in $B_r(x_0)$.
The ACF monotonicity formula provides a simple proof to Theorem I when $a=0$. Therefore, it would be natural in seeking to prove Theorem I to try to prove a generalization of the ACF monotonicity formula that applies to solutions of div$(|x_n|^a \nabla u) \geq 0$. Unfortunately, the proof of such a formula would require much more than mere adaptations to the proof of the classical ACF monotonicity formula. In this paper we provide an alternate method that gives a relatively simple proof of Theorem I and does not utilize a generalization of the ACF formula. In its place we utilize a Weiss-type monotonicity formula (defined in Section \[S: nondegeneracy\]) that is an adaptation of the Weiss-type monotonicity formula given in [@mA11]. The proof that the functional in is monotone requires only slight modifications of the proof provided in [@mA11].
Outline of Paper {#outline-of-paper .unnumbered}
----------------
The outline of this paper is as follows.
- In Section \[S: weights\] we state known results for the weight $|x_n|^a dx$ and solutions of div$(|x_n|^a \nabla u) =0$ that we will need.
- In section \[S: optimalregularity\] we prove the optimal regularity of minimizers and its corollaries.
- In section \[S: nondegeneracy\] we use nondegeneracy and the Weiss monotonicity formula to prove that “blow-ups” (see ) of minimizers are homogeneous of degree $s=(1-a)/2$
- In section \[S: courant\] we use the Courant-Fischer maximum-minimum principle to establish a lower bound for the degree of homogeneity for homogeneous solutions of div$(|x_n|^a \nabla u)=0$.
- In section \[S: separation\] we use the results from the previous sections to provide a simple proof of Theorem I.
Notation and Terminology {#notation-and-terminology .unnumbered}
------------------------
For the remainder of the paper it will be useful to use the following notation. $B_r(x_0) := \{x \in {{\mathbb R}}^n \mid \ |x-x_0| \leq r \}$ and $B_r = B_r(0)$ the ball centered at the origin with radius r. We denote a point $x \in {{\mathbb R}}^n$ by $(x',x_n)$ where $x' = (x_1, \ldots, x_{n-1})$.
For any set $\Omega \subset {{\mathbb R}}^n$, we define $$\Omega' \overset{ \text{def} }{=} \Omega \cap ({{\mathbb R}}^{n-1} \times \{0\} )$$ Throughout the paper we will refer to the plane ${{\mathbb R}}^{n-1} \times \{0\}$ as the thin space. Likeweise, we will call $B_{r}'$ the thin ball where as $B_r$ will be the solid ball. For minimizers of we will call the set $$\Lambda(u) = ({{\mathbb R}}^{n-1} \times \{0\}) \cap \{u=0\}$$ the coincidence set.
We define the following two spaces $$\begin{aligned}
H^1(a,D) & \overset{\text{def}}{=} \{v \in L^2(D) \mid |x_n|^{a/2} \nabla v \in L^2(D)\} \\
L^2(a,D) & \overset{\text{def}}{=} \{|x_n|^{a/2} v \in L^2(D)\}
\end{aligned}$$ We will also use $\mathcal{L}_a u$ to denote the operator div$(|x_n|^a \nabla u)$. Throughout the paper $s=(1-a)/2$.
p-admissible weights and a-harmonic functions {#S: weights}
=============================================
We begin this section by noting that the measure $|x_n|^a dx$ is a Muckenhoupt $A_2$ weight. In [@MR1207810] it is shown that Muckenhoupt $A_p$ weights are $p$-admissible weights; therefore, we have the following Sobolev inequality from [@MR1207810] $$\label{E: asobolev}
\left( \frac{1}{| B |_{a}} \int_{B}{|\phi|^{2 \varkappa} |x_n|^a \ dx} \right)^{\frac{1}{2 \varkappa}} \leq cr
\left( \frac{1}{| B |_{a}} \int_{B}{|\nabla \phi|^{2} |x_n|^a \ dx } \right)^{\frac{1}{2}}$$ whenever $B=B(x_0,r)$ is a ball and $\phi \in H_0^1(a,B)$. $\varkappa >1$ and $c$ are two constants depending on $n$ and $a$. Here, $| B |_a = \int_{B}{|x_n|^a dx}$.
The following proposition is a consequence of the compactness theorem for admissable p-weights proven in [@MR1455468].
\[P: compact\] Let $u_k$ be a bounded sequence in $H_0^1(a, D)$ for $D \Subset {{\mathbb R}}^n$. Then there exists a convergent subsequence such that $u_k \to u$ pointwise $a.e.$ and in norm in $L^q(a,D)$ for all $q<2 \varkappa$ for $\varkappa$ as in .
We call a function $u$ $a$-harmonic if $\mathcal{L}_a u =0$. These functions share many properties with classical harmonic functions. In [@MR643158] it is shown that $a$-harmonic functions are Hölder continuous. It was also shown that $a$-harmonic functions have the maximum principle, Harnack inequality, and Boundary Harnack inequality. We also have the following Almgren’s type monotonicity formula which was proven in [@CS].
\[L: almgren\] Let $\mathcal{L}_a u =0$ in $B_1$. Then $$N(r,u)=r\frac{\int_{B_r}{|x_n|^a |\nabla u|^2}}{\int_{\partial B_r}{|x_n|^a u^2}} = r \frac{D(r)}{H(r)}$$ is monotone increasing in $r$. $N(r,u)$ is constant if and only if $u$ is homogeneous of degree $k$.
Our assumptions are slightly different from those given in [@CS]; namely we do not assume even symmetry in the $x_n$ variable. The modified proof is therefore placed in the appendix.
\[L: lowerbound\] Let $\mathcal{L}_a u =0$ in $B_1$ with $u$ not identically zero. Assume also that $u(0)=0$. Then $$\lim_{r \to 0} N(r,u) = N(0+,u) \geq \min \{1, 1-a\}$$
It is easy to verify that $N(\rho, u_r)=N(r\rho, u)$ for the rescalings $$u_r(x) := \frac{u(rx)}{\frac{1}{r^{n-1+a}}\int_{\partial B_r}{|x_n|^a u^2}}$$ and $\| u_r \|_{L^2(a,\partial B_1)} =1$. Now $\mathcal{L}_a u_r =0$ in $B_{1/r}$ and from the uniform Hölder continuity provided in [@MR643158] and the Sobolev inequality , we may extract a subsequence such that $u_r \to u_0$ in $C^{\beta}(B_{\rho})$ and weakly in $H^1(a, B_{\rho})$ for $\rho <1$. The strong convergence in $H^1(a, B_{\rho})$ follows by using the Caccioppoli inequality for $a$-harmonic functions $$\int_{B_{\rho}}{|\nabla(u_r - u_0)|^2 |x_n|^a} \leq \frac{C}{(r-\rho)^2}\int_{B_r}{|u_r-u_0|^2 |x_n|^a}$$ Now $$N(\rho, u_0) = \lim_{r \to 0} N(\rho, u_r) = \lim_{r \to 0} N(r \rho, u) = N(0+,u)$$ So $\mathcal{L}_a u_0 =0$ in $B_1$ and is homogenous of degree $k = N(0+,u)$. $u_0$ is not identically zero since $\| u_0 \|_{L^2(a,\partial B_1)} =1$. We now only need to conclude that $k \geq \min \{1, 1-a\}$. Since $u_0(0)=0$, this is a direct consequence of Theorem \[T: originregularity\]. Theorem \[T: originregularity\] has been placed in Section \[S: courant\] for purposes of readibility of the paper.
From Almgren’s monotonicity formula we may prove the following Lemma.
\[L: menergy\] If $\mathcal{L}_a u =0$ in $B_R(y',0)$ then $$\frac{1}{r^{n-|a|}} \int_{B_r(y',0)}{|x_n|^a |\nabla u|^2}$$ is monotone increasing in $r$.
A few remarks need to be said. First, if we add the additional assumption for even symmetry, namely that that $u(x',x_n) = u(x', -x_n)$, then $$\label{E: na}
\frac{1}{r^{n+a}}\int_{B_r(y',0)}{|x_n|^a |\nabla u|^2} \quad \text{ is monotone increasing in } r$$ The solution $v=\frac{x_n}{|x_n|^a}$ for $a>0$ shows that if there is not even symmetry, then is not true. Likewise, the hypothesis that that the ball be centered on the ${{\mathbb R}}^{n-1} \times \{0\}$ plane is essential. $v$ as given above with center $(y',y_n)$ with suitably chosen $y_n \neq 0$ will be a counterexample. is also not true if the ball is not centered on the thin space, and a counterexample is much easier to provide: off the thin space solutions are $C^1$, so if $a>0$ and $y_n \neq 0$ then $$\lim_{r \to 0} \frac{1}{r^{n+a}} \int_{B_r(y',y_n)}{|x_n|^a |\nabla u|^2} \to \infty$$ and so it is clear that can only be true if $y_n=0$.
Following the notation in Lemma \[L: almgren\] we have $$H'(r)= \frac{(n-1+a)}{r} H(r) + 2D(r)$$ This equality comes from and . This implies that $rH'(r)/H(r) = n-1+a+2N(r)$ is also monotone increasing. Hence $rH'(r)/H(r) \geq n-1+a+2k$ where $k=N(0+)$. Then $r^{-(n-1+a+2k)}H(r)$ is monotone increasing and therefore also $$\frac{1}{r^{n-2+a+2k}}D(r) = \frac{1}{r^{n-1+a+2k}}H(r)N(r)$$ is monotone increasing in $r$. By subtracting the constant $u(0)$ which is a solution of $\mathcal{L}_a$ we may use Lemma \[L: lowerbound\] to conclude that $k \geq \min \{1,1-a\}$ and the Lemma is proven.
Optimal Regularity {#S: optimalregularity}
==================
In studying free boundary problems it becomes useful to utilize the so called “blow-up” process. If $u$ is a minimizer of the functional in $B_1(x_{0}',0)$, then the rescaled function $$\label{E: rescale}
u_r(x) \overset{\text{def}}{=} \frac{u((x_{0}',0)+rx)}{r^{s}}$$ is a minimizer in $B_{1/r}$. Here $s=(1-a)/2$. By taking a sequence $r_k \to 0$ we may hope to find a subsequence $u_{r_k} \to u_0$ where $u_0$ is a minimizer in all compact subsets of ${{\mathbb R}}^n$. By considering properties of the free boundary of $u_0$ one may gather information on the free boundary of $u$ close to the point $x_0$. Theorem \[T: regularity\] will guarantee that $u_0$ does exist.
\[T: regularity\] Let $u$ be a minimizer in $B_1$. Then $u \in C^{0, s }(U)$ for all $U \Subset B_1$.
For minimizers of we follow the method provided in [@lC10] for the one phase case.
Throughout the beginning of the proof $C$ will be any constant depending on dimension $n$ and $a$. Let $u$ be a minimizer in $B_2$. For every $0<r<1$ we consider the harmonic replacement $v$ of $u$ in $B_r=B_r(x',0)$. That is $\mathcal{L}_a v=0$ and $v=u$ on $\partial B_r$. Since $u$ is a minimizer, $J(u) \leq J(v)$ in $B_r$, so $$\int_{B_r}{|x_n|^a |\nabla u|^2} \leq \int_{B_r}{|x_n|^a |\nabla v|^2} + C r^{n-1}$$ We now use that $\mathcal{L}_a v =0$, so $$\int_{B_r}{|x_n|^a \langle \nabla v, \nabla (v-u)\rangle} = 0$$ and this allows us to conclude $$\int_{B_r}{|x_n|^a |\nabla (u-v)|^2} \leq C r^{n-1}$$ If we now choose $\rho < r < 1$ $$\begin{aligned}
{2}
\int_{B_{\rho}}{|x_n|^a |\nabla u|^2} &= \int_{B_{\rho}}{|x_n|^a |\nabla (u-v+v)|^2} \\
&\leq 2 \left(\int_{B_r}{|x_n|^a |\nabla (u-v)|^2} + \int_{B_{\rho}}{|x_n|^a |\nabla v|^2} \right) \\
&\leq C r^{n-1} + 2 \left(\frac{\rho}{r}\right)^{n-|a|} \int_{B_r}{|x_n|^a |\nabla v|^2} \text{ by Lemma \ref{L: menergy}} \\
&\leq C r^{n-1} + C \left(\frac{\rho}{r}\right)^{n-|a|} \int_{B_r}{|x_n|^a |\nabla u|^2}\end{aligned}$$ We now choose $\delta < 1/2$ with $$r = \delta^k , \qquad \rho = \delta^{k+1} , \qquad \mu \equiv \delta^{n-1}$$ to obtain $$\label{E: iteratebound}
\int_{B_{\delta^{k+1}}}{|x_n|^a |\nabla u|^2} \leq C \mu^{k} + C \mu \delta^{1-|a|}
\int_{B_{\delta^k}}{|x_n|^a |\nabla u|^2}$$ We now may choose $\delta$ such that $C \delta^{1-|a|} < 1$. Using a simple induction argument we conclude $$\int_{B_{\delta^k}}{|x_n|^a |\nabla u|^2} \leq \frac{C^2}{1-C\delta^{1-|a|}} \mu^{k-1}$$ Then for all $r<1/2$ and a different constant which will also depend on the $L^2(a,B_2)$ norm of $\nabla u$ $$\label{E: ebound1}
\int_{B_r(x',0)}{|x_n|^a |\nabla u|^2} \leq C r^{n-1}$$ and so we may conclude as in [@lC10] that $$\label{E: ebound2}
\int_{B_r(x',0)}{|\nabla u|} \leq Cr^{n-1 + s }$$ Since the estimate is only true for balls centered on the thin space we cannot use Morrey’s theorem to immediately conclude $C^{0,s}$ regularity for $u$ inside the solid ball $B_{1/2}$. However, one may use the proof of Morrey’s theorem (as outlined in [@mZ97]) with the estimate to conclude $$\label{E: campanato}
|u(x',0)-\overline{u}_B| \leq Cr^{s}$$ so that $u$ is $C^{0, s}$ on the thin space ${{\mathbb R}}^{n-1} \times \{0\}$. Equation and hence also will hold for $|u|$. We now aim to conclude that we have the same Hölder growth off the thin space. By optimal Hölder regularity along the thin space, we only need to show Hölder growth in the pure $|x_n|$ direction. For a fixed point $(y',0)$, we consider the rescaled functions $$u_r(x) \equiv \frac{u(y',0) + xr)-u(y',0)}{r^{s}}$$ which have a universal (unweighted) $L^2$ gradient bound in $B^* = B_{1/2}(0, \ldots ,0,1)$ by . Using estimate for $|u_r|$, we may deduce that the average value of $|u_r|$ over $B_{3/2}(0)$ is universally bounded; consequently, the average value of $|u_r|$ over $B^*$ will also be universally bounded. By using the (unweighted) Poincare inequality in $B^*$ we obtain $$\|u_r\|_{W^{1,2}(B_{1/2}(0, \ldots ,0,1))} \leq C$$ By first variation $\mathcal{L}_a u_r =0$ if $|x_n>0|$. By staying away from the thin space we may use regularity theory for uniformly elliptic equations and conclude that each $u_r$ is continuous in $B^*$ and we have the weak Harnack inequality $$\| u_r \|_{L^{\infty}(B_{1/4}(0,\ldots ,0,1))} \leq C$$ This proves the Hölder growth off the thin space. That is, $$\label{E: hgrowth}
\frac{|u(y',0)-u(x)|}{|(y',0)-x|^{s}} \leq C$$ Let now $x,y \in B_1$. If $|y_n| \leq |x-y|$ we may use to bound $$\frac{|u(x)-u(y)|}{|x-y|^{s}}$$ If $|y_n|>|x-y|$ then we may rescale with $$u_r = \frac{u((x',0) + rx)-u(x',0)}{r^{s}}$$ and use interior gradient bounds (in $B^*$ as defined before) on uniformly elliptic equations to conclude $$\frac{|u(x)-u(y)|}{|x-y|^{s}} \leq C$$
The Hölder regularity of minimizers allows us to conclude the following about the convergence of sequences of minimizers.
\[C: holderconv\] \[C: conv\] Let $\{u_k\}$ be a sequence of minimizers of the functional in the domain $D$ with $\left\| u_k \right\|_{L^{\infty}(\partial D)}
\leq M$. Then there exists a subsequence and a function $u_0$ such that for every open $U \Subset D$ $$\begin{aligned}
{2}
&(1) &\quad & u_0 \in H^1(a,U) \cap C^{s}(\overline U) \\
&(2) && u_k \to u_0 \text{ in } C^{\beta}(\overline U) \text{ for } \beta < s\\
&(3) && u_k \rightharpoonup u_0 \text{ in } H^{1}(a,U)\\
\end{aligned}$$
Properties (1) and (2) follow immediately from the Hölder-regularity proven in Theorem \[T: regularity\]. Property (3) follows from the inequalities and .
Since minimizers are continuous, we may use the first-variation to conclude
\[P: solution\] Let $u$ be a minimizer of in $\Omega$. Then $$\mathcal{L}_a u=0 \quad \text{in } \Omega \setminus \Lambda(u)$$
From Proposition \[P: solution\] one expects the following
\[P: byparts\] Let $u$ be a minimizer in $\Omega$. For any ball $B \Subset \Omega$ $$\int_{B}{|x_n|^a |\nabla u|^2} = \int_{\partial B}{|x_n|^a uu_{\nu}}$$
Proposition \[P: byparts\] holds for more general domains than a ball; however, the assumption that the domain is a ball will suffice for our purposes.
We define the following sequence of cutoff functions $$\eta_k(x) =
\begin{cases}
0 , &\text{ if } d_x \leq 1/k \\
kd_x - 1, &\text{ if } 1/k \leq d_x \leq 2/k \\
1 , &\text{ otherwise }
\end{cases}$$ Where $d_x = \text{dist}(x,\Lambda(u))$. Then $|\nabla \eta_k|=k$ when $1/k \leq d_x \leq 2/k$ and zero otherwise. We now use optimal regularity of $u$ to establish that the sequence $\eta_k u$ is bounded in $H^{1}(a,B)$. $$\begin{aligned}
{2}
\int_{B}{|x_n|^a |\nabla (\eta_k u)|^2}
&\leq \int_{B}{2|x_n|^a \left( \eta_k^2|\nabla u|^2 + u^2 |\nabla \eta_k|^2 \right)} \\
&\leq \int_{B}{2|x_n|^a |\nabla u|^2}
+ \int_{B \cap \{d_x \leq 2/k \}}{8|x_n|^a Ck^{a-1} k^2} \\
&\leq \int_{B}{2|x_n|^a |\nabla u|^2}
+ \int_{B \cap \{|x_n| \leq 2/k \}}{8|x_n|^a Ck^{1+a} } \\
&\leq \int_{B}{2|x_n|^a |\nabla u|^2} + C \quad \text{ for some new constant } C\end{aligned}$$ Then there exists $v$ such that $\eta_k u \rightharpoonup v$ in $H^{1}(a,B)$ and $\eta_k u \to v$ pointwise by Proposition \[P: compact\]. Since $\eta_k u \to u$ pointwise, then $v = u$. Now using the divergence theorem and that $\mathcal{L}_{a}u =0$ away from the coincidence set $\Lambda(u)$ we obtain $$\int_{B}{|x_n|^a \langle \nabla (\eta_k u) , \nabla u \rangle} = \int_{\partial B}{|x_n|^a \eta_k u u_\nu}$$ Then let $k \to \infty$ to obtain the result.
Nondegeneracy and Weiss Monotonicity {#S: nondegeneracy}
====================================
When we have a blow-up sequence $u_r \to u_0$, it is not immediately obvious if $u_0$ could be degenerate, that is $u_0 \equiv 0$. If $u_0 \equiv 0$, then we would be unable to gather any information on the free boundary of $u$ near $x_0$. Theorem \[T: nondegeneracy\] will guarantee that $u_0$ will not be degenerate.
\[T: nondegeneracy\] Fix $t > 0$, and let $u$ be a minimizer of $J$. There exists $\epsilon > 0$ with $\epsilon$ depending only on $\{\lambda^+, \lambda^-, t\}$ such that if $u|_{\partial B_r} \leq \epsilon r^{s} $ $(u|_{\partial B_r} \geq -\epsilon r^{s})$ then $$u(x) \leq 0 \quad (u(x) \geq 0) \qquad \text{for } x \in B_{tr}'$$
The proof of Theorem \[T: nondegeneracy\] is included in the appendix and only requires slight modifications from the proof presented in [@mA11].
\[C: 1/2 growth\] If $u$ is a minimizer and $0 \in \Gamma^+$ $(0 \in \Gamma^-)$, then $$\label{E: 1/2 growth}
\sup_{\partial B_r}u \geq C r^{s} \qquad \left( \inf_{\partial B_r}u \leq -C r^{s} \right)$$ Where $C$ depends only on $\lambda^+, \lambda^-$ and $n$.
In [@gW98] G. Weiss introduced a monotonicity formula for a free boundary problem that allowed one to conclude that blow-ups were homogeneous. Theorem \[T: weiss\] gives a modified Weiss-type monotonicity formula that allows us to conclude Corollary \[C: monotone\] namely, that all blow-ups are homogeneous of degree $s=(1-a)/2$. Corollary \[C: monotone\] is crucial in proving Theorem I.
\[T: weiss\] Let $B_r = B_r(x_0,0)$. Define $W(r,u,x_0) = $ $$\label{E: weiss}
\frac{1}{r^{n-1}} \left( \int_{B_r}{|x_n|^a|\nabla u|^2}
+ \int_{B_r'}{\lambda^+ \chi_{ \{u>0\} } + \lambda^- \chi_{ \{u<0\} } } \right)
-\frac{s}{r^n} \int_{\partial B_r}{|x_n|^a u^2}$$ $W(r,u,x_0)$ is finite and monotone increasing in $r$. Furthermore, if $r_1 < r_2$, then $W(r_1, u) = W(r_2, u)$ if and only if $u$ is homogeneous of degree $s=(1-a)/2$ on the ring $r_1 < |x| < r_2$.
If $u_r(x) = \frac{u(rx)}{r^{s}}$, then $W(r,u) = W(1,u_r)$.
The proof of Theorem \[T: weiss\] requires only slight modifications from the proof presented in [@mA11] for the case in which $a=0$ and therefore the proof is contained in the appendix.
\[C: monotone\] Let $u_r \to u_0$ a blow-up at $(x_0,0)$. Then $u_0$ is homogeneous of degree $s$
By and optimal regularity it is easy to verify that $W(0+,u,x_0)$ is bounded from below, so that $$W(2r/3,u,x_0)-W(r/3,u,x_0) \to 0 \text{ as } r \to 0$$ From the explicit representation of $W'$ provided in the proof of Theorem \[T: weiss\] we may write $$\begin{aligned}
{2}
W(2r/3)-W(r/3)
& = \int_{r/3}^{2r/3}{\frac{1}{\rho^{n-1}}
\int_{\partial B_{\rho}}{|x_n|^a \left(\frac{(1-a)u}{\sqrt{2}\rho} - \sqrt{2} u_{\nu} \right)^2} \ d\rho} \\
& = \int_{1/3}^{2/3}{\frac{1}{(rt)^{n-1}}
\int_{\partial B_{rt}}{|x_n|^a \left(\frac{(1-a)u}{\sqrt{2}rt} - \sqrt{2} u_{\nu} \right)^2} r \ dt} \\
& = \int_{1/3}^{2/3}{\frac{1}{t^{n-1}}
\int_{\partial B_{t}}{(rx_n)^a \left(\frac{(1-a)u(rx)}{\sqrt{2}rt} - \sqrt{2} \nabla u(rx)\cdot \nu \right)^2} r \ dt} \\
& = \int_{1/3}^{2/3}{\frac{1}{t^{n-1}}
\int_{\partial B_{t}}{|x_n|^a \left(\frac{(1-a)u_r}{\sqrt{2}t} - \sqrt{2} \nabla u_r\cdot \nu \right)^2} \ dt} \\
& \geq \int_{1/3}^{2/3}{
\int_{\partial B_{t}}{|x_n|^a \left(\frac{(1-a)u_r}{\sqrt{2}t} - \sqrt{2} \nabla u_r\cdot \nu \right)^2} \ dt} \\
& = \int_{B_{2/3} \setminus B_{1/3}}{ |x_n|^a\left(\frac{u_r}{\sqrt{2}t} - \sqrt{2} \nabla u_r\cdot \nu \right)^2} dx\end{aligned}$$ Now we use that $u_r \rightharpoonup u_0$ in $H^1(a, B_1)$ and $u_r \to u_0$ in $L^2(a,B_1)$ by Corollary \[C: holderconv\], so $u_0$ is homogeneous of degree $s$.
Courant-Fischer Maximum-Minimum Principle {#S: courant}
=========================================
We may decompose the operator $\mathcal{L}_a$ into its radial and spherical parts similar to the case for the Laplacian. If $u \in H^1(a,S^{n-1})$, then $\mathcal{L}_a^{\theta} u = f$ is to be interpreted as $$-\int_{S^{n-1}}{|x_n|^a \langle \nabla_{\theta} u, \nabla_{\theta} v \rangle} =
\int_{S^{n-1}}{|x_n|^a fv}
\quad \text{for all } v \in H^1(a, S^{n-1})$$ Corollary \[C: monotone\] shows that all blow-ups are homogeneous. Homogeneous solutions of $\mathcal{L}_a u =0$ correspond to eigenfunctions on the sphere. Specifically, if $\mathcal{L}_a u=0$ and $u=r^{\alpha}f(\theta)$, then $$-\mathcal{L}_a^{\theta} f = \lambda f$$ where $\lambda = \alpha(\alpha +n-2+a)$. We also have the converse.
\[L: spheresolution\] Suppose $-\mathcal{L}_{a}^{\theta} f = \lambda f$. If $u=r^{\alpha}f$ with $\lambda=\alpha(\alpha+n-2+a)>0$, $\alpha>0$, then $u$ is a weak solution to $\mathcal{L}_a u =0$ in ${{\mathbb R}}^n$.
Let $v \in H_{0}^1(a,B_R)$. Then $$\begin{aligned}
{2}
\int_{B_R}{|x_n|^a \langle \nabla u , \nabla v \rangle}
&=\int_{0}^{R} \int_{\partial B_{r}}{|x_n|^a \left( \frac{\langle \nabla_{\theta} u , \nabla_{\theta} v \rangle}{r^2} + u_{\nu} v_{\nu} \right) } \\
&=\int_{0}^{R} \int_{\partial B_{r}}{|x_n|^a \left( r^{\alpha -2}\lambda fv + \alpha r^{\alpha -1} f v_{\nu} \right)} \\
&=\int_{\partial B_1}{\cos^a(\theta_{n-1}) \alpha f} \left(\int_{0}^{R}{\frac{d}{dr}\left(r^{\alpha + n-2+a}v(r \theta) \right)dr} \right) d\sigma\end{aligned}$$ Now $$\int_{0}^{R}{\frac{d}{dr}\left(r^{\alpha + n-2+a}v(r \theta) \right)dr} =0$$ for a.e. $\theta$ since $v \in H_{0}^{1}(a, B_R)$, and thus the lemma is proven.
To utilize the Courant-Fischer maximum-minimum principle we will need the following lemma.
Let $\Omega \subset S^{n-1}$ be open. The spectrum of $$-\mathcal{L}_{a}^{\theta} u: H^1(a, \Omega) \subset L^2(a,\Omega) \hookrightarrow L^2(a,\Omega)$$ is a nonnegative sequence that is either finite or increases to infinity.
From the Reisz representation theorem, for every $f \in L^2(a,\Omega)$ there exists a unique $u \in H^1(a, \Omega)$ such that for all $v \in H^1(a, \Omega)$ the following identity holds $$\int_{\Omega}{|x_n|^a \left(\langle \nabla u, \nabla v \rangle + uv \right)} = \int_{\Omega}{|x_n|^a fv }$$ We now aim to conclude that the operator $K: L^2(a,\Omega) \hookrightarrow L^2(a,\Omega)$ given by $K(f)=u$ is compact. To obtain a compactness theorem on $S^{n-1}$, for any $u \in H^1(a, \Omega)$ we extend $u$ radially by defining $\tilde{u} = \eta(r) u$ for $\eta$ a bump function on ${{\mathbb R}}$. Then for a sequence $u_k \in H^1(a, \Omega)$ we obtain a bounded sequence $\tilde{u}_{k} \in H_0^1(a, B_2 \setminus B_{1/2})$ and by Proposition \[P: compact\] we obtain that for a subsequence $\tilde{u}_{k} \to \tilde{u} \in L^2(a,B_2 \setminus B_{1/2})$ and pointwise almost everywhere. Since $\tilde{u} = \eta(r) u$ for some $u \in H^1(a, \Omega)$ we conclude that $u_k \to u$ in $L^2(a,\Omega)$. We may therefore conclude that $K$ is compact.
Now $-\mathcal{L}_a^{\theta} u = \lambda u$ in $\Omega$ if and only if $K(u)= \frac{1}{\lambda + 1} u$. From the theory of self-adjoint nonnegative compact operators we know that the spectrum of $K$ is either finite or a nonnegative sequence decreasing to zero. Then we obtain that the spectrum of $-\mathcal{L}_a^{\theta}$ is either finite or a nonnegative sequence increasing to infinity.
If $\Omega = S^{n-1}$ then the first eigenvalue $\lambda_1 =0$. If $\Omega$ is a proper subset of $S^{n-1}$ such that $\Omega^c$ has positive capacity, then the first eigenvalue $\lambda_1 > 0$ and corresponds to the principle eigenfunction that is nonnegative. Let $\Omega \subset S^{n-1}$ be open and define $W=H_0^1(a,\Omega)$. To compare the eigenvalues of $V=H^1(a,S^{n-1})$ to those of the subspace $W=H_0^1(a,\Omega)$ we employ the Courant-Fischer maximum-minimum principle.
Let $\Omega \subset S^{n-1}$ be open. The k-th eigenvalue of $-\mathcal{L}_a^{\theta} u$ associated to the domain $\Omega$ is determined by $$\lambda_k = \max_{S \in \Sigma_{k-1}}
\min_{\overset{ v \in S^{\perp}}{\left\| v \right\|_{\mathcal{L}_a^2}=1}}
\int_{S^{n-1}}{|x_n|^a|\nabla v|^2}$$
where $\Sigma_{k-1}$ is the collection of all $k-1$ dimesnional subspaces of $H_0^1(a, \Omega)$.
This principle is proven in [@MR0065391].
From this principle we conclude
\[P: eigencomparison\] If $0 = \lambda_1 < \lambda_2 \leq \ldots$ are the eigenvalues of $V$ and $\gamma_1 < \gamma_2 \leq \ldots$ are the eigenvalues of $W$, then $$\lambda_k \leq \gamma_k \text{ for all } k$$
The proof is along the same lines of the proof of the maximum-minimum principle provided in [@MR0065391]. The only difference is we take a linear combination of the first $k$ eigenvectors in $W$ rather than in $V$. Specifically, let $S$ be any $k-1$ dimensional subspace of $V$. Let $w_1, \ldots , w_k$ be the normalized eigenfunctions corresponding to the first $k$ eigenvalues of the subspace $W$. We may then construct $$w = \sum_{i=1}^{k}{c_i w_i} \quad \text{ with } \sum_{i=1}^{k}{c_{i}^2} = 1$$ and such that $w \in S^{\perp}$. Since the $w_i$ are orthogonal to each other we obtain that $$\int_{S^{n-1}}{|x_n|^a |\nabla w|^2} = \sum_{i=1}^{k}{\gamma_i c_i^2 \leq \gamma_k}$$ Thus we have shown that for $S$ any $k-1$ dimensional subspace of $V$ $$\min_{\overset{ v \in S^{\perp}}{\left\| v \right\|_{\mathcal{L}_a^2}=1}}
\int_{S^{n-1}}{|x_n|^a|\nabla v|^2}
\leq \gamma_k$$ Then by the Courant-Fischer maximum-minimum principle, $\lambda_k \leq \gamma_k$, and the proposition is proven.
\[T: originregularity\] Let $\mathcal{L}_a u = 0$ in all of ${{\mathbb R}}^n$ and let $u$ be homogeneous of degree $\alpha$ with $u(0)=0$. If $\alpha < \min \{1, 1-a\}$, then $u \equiv 0$.
Solutions of $\mathcal{L}_a u =0$ are $C^1$ in any $(x',0)$ direction [@MR2367025]. Since $u$ is homogeneous of degree $\alpha < 1$, we may conclude that $u(x',0) \equiv 0$. (We must be differentiable in any $(x',0)$ direction at the origin.) We note that $x_n^{1-a}$ is the principle eigenfunction on $H_0^1(a,S_+^{n-1})$ since it is positive. Here $S_+^{n-1} = S^{n-1} \cap \{x_n > 0\}$. Now $u \in H_0^1(a,S_+^{n-1})$ and $\alpha < 1-a$, so the eigenvalue associated to $u$ is strictly less than that of the eigenvalue associated to that of $x_n^{1-a}$. Then $u \equiv 0$.
\[C: homogeneity\] Let $u$ be homogeneous of degree $\alpha$, having nontrivial positive and negative parts, continuous, and such that $$\mathcal{L}_a u(x) = 0$$ whenever $x \notin \Lambda(u)$. Then $\alpha \geq \min \{1, 1-a\}$
Since $u$ is homogeneous, then $u$ is an eigenfunction of $\mathcal{L}_a^{\theta}$ on $\Omega = S^{n-1} \setminus \Lambda(u) $. If $\Omega$ is not connected, then $\Lambda(u)= B_{1}'$. Then by comparison with the principle eigenfunction $x_n^{1-a}$ (as in the proof of Theorem \[T: originregularity\]) $\alpha \geq 1-a$. If $\Omega$ is connected, then since $u$ has nontrivial positive and negative parts, $u$ cannot be the principle eigenfunction. By Proposition \[P: eigencomparison\] the eigenvalue $\gamma_2$ of $u$ is such that $$\lambda_2 \leq \gamma_2$$ where $\lambda_2$ is the eigenvalue corresponding to the first free eigenfunction $g$ on $S^{n-1}$. That is $\mathcal{L}_a^{\theta} g = \lambda_2 g$ on $S^{n-1}$. We may then define $v = r^{\beta}g$ where $\lambda_2 = \beta(\beta +n-2+a)$. Then $\mathcal{L}_a v = 0$ in ${{\mathbb R}}^n$ by Lemma \[L: spheresolution\], and so by Theorem \[T: originregularity\] we know $\beta \geq \min\{1, 1-a \}$. Since $\gamma_2 = \alpha (\alpha + n-2+a)$ and $\lambda_2 = \beta(\beta +n-2+a)$, we see then that $\alpha \geq \beta \geq \min \{1,1-a\}$.
Separation of the Free Boundaries {#S: separation}
=================================
We may now prove the main theorem of the paper. We first show the separation of the phases.
\[T: separation\] Let $u$ be a minimizer. Then $\Gamma^+ \cap \Gamma^- = \emptyset$.
Suppose by way of contradiction that $x_0 \in \Gamma^+ \cap \Gamma^-$. Let $u_r \to u_0$ be a blow-up. By nondegeneracy (Corollary \[C: 1/2 growth\]) and $C^{\beta}$ convergence (Corollary \[C: conv\]) $u_0$ has nontrivial positive and negative parts. Also it follows from Corollary \[C: conv\] that $\mathcal{L}_a u_0(x) =0$ if $x \notin \Lambda(u_0)$. By Corollary \[C: monotone\] we know that $u_0$ is homogeneous of degree $s=(1-a)/2$. Since $(1-a)/2 < \min\{1, 1-a\}$, we obtain a contradiction to Corollary \[C: homogeneity\].
We may now prove the second half of Theorem I. Namely, in a small neighborhood of each free boundary point a minimizer has a sign in the solid ball.
\[T: solidsep\] Let $x_0 \in \Gamma^+ \ (x_0 \in \Gamma^-)$ then there exists $r>0$ depending on $x_0$ such that $u \geq 0 \ (u \leq 0)$ in the solid ball $B_r(x_0)$.
Without loss of generality we may assume $x_0 =0$. Let $u_{r_k} \to u_0$ be a blow-up of $u$ at the origin. Since $u_0$ is homogeneous of degree $s$, Corollary \[C: homogeneity\] allows us to conclude $u_0 \geq 0$ in all of ${{\mathbb R}}^n$. Since each $u_{r_k}(x',x_n)$ is $a$-harmonic in the open set $\{x \in B_{1/ r_k} \mid x_n \neq 0\}$, then $u_0$ will be $a$-harmonic in the open set $\{x \in {{\mathbb R}}^n \mid x_n
\neq 0\}$. We define $$\delta = \inf u_0 \text{ over the set } B_1 \cap \{|x_n| \geq 1/2\}.$$ We claim that $\delta>0$. Indeed, otherwise by the strong minimum principle (or Harnack inequality) $u_0\equiv 0$ in ${{\mathbb R}}^n_+$ or ${{\mathbb R}}^n_-$, and therefore $u_0\equiv 0$ on ${{\mathbb R}}^{n-1}\times\{0\}$. By nondegeneracy we know that on either ${{\mathbb R}}^n_+$ or ${{\mathbb R}}^n_-$ we have $u_0 >0$. Then by odd reflection we obtain a homogeneous (of degree $s=(1-a)/2$) function $\tilde{u}_0$ that is $a$-harmonic in all of ${{\mathbb R}}^n$. This is a contradiction to Theorem \[T: originregularity\]. So $\delta >0$.
Then, by $C^{\alpha}$ convergence, for large enough $k$, $u_{r_k}(x',x_n) \geq \delta /2$ for $|x_n| \geq 1/2$ in $B_1$. Also by $C^{\alpha}$ convergence, $\inf_{B_1} u_{r_k} \to
0$. Now by thin separation, for large enough $k$, $$u_{r_k}(x',0) \geq 0 \text{ in } B_1'$$ Without loss of generality it suffices to show that $u_{r_k} \geq 0$ in $B_{1/2}^+$. Let $v_k$ be the $a$-harmonic function such that $$v_k|_{B_1'}=0 \text{, and } v_k |_{\partial B_1^+} = u_{r_k}$$ Then $ v_k \leq u_{r_k} $ in all of $B_1^+$. We show for $k$ large enough that $v_k \geq 0$ in $B_{1/2}^+$. To this end, consider two subsets $E_1$ and $E_2$ of $\partial({B_1^+})$: $$E_1=\partial(B_1^+)\cap \{x_n\geq 1/2\},\quad E_2=\partial(B_1^+)\cap
\{0<x_n<1/2\},\quad$$ and there $a$-harmonic measures $\omega_1$ and $\omega_2$ with respect to the domain $B_1^+$. The latter means that $\omega_i$ are $a$-harmonic functions in $B_1^+$ satisfying $$\omega_i|_{\partial (B_1^+) }=\chi_{E_i},\quad i=1,2.$$ By using the boundary Harnack inequality, one then has that $$c |x_n|^{1-a} \leq \omega_i(x)\leq C |x_n|^{1-a} \quad\text{in }B_{1/2}^+.$$ for some positive constants $c$ and $C$ depending on $n$ and $a$. Now, by using the maximum principle we then can write that in $B_{1/2}^+$ $$\begin{aligned}
v_k(x)&\geq (\delta/2) \omega_1(x)+\omega_2(x)\inf_{B_1^+}v_k\\
& \geq |x_n|^{1-a} [(\delta/2)c- C\sup_{(\partial B_1)^+} u_{r_k}^-].\end{aligned}$$ Since $u_{r_k}^- \to 0$ uniformly on compact subsets of ${{\mathbb R}}^n$, we obtain that $v_k(x)\geq 0$ in $B_{1/2}^+$ for large $k$. This completes the proof.
Proof of Almgren’s Formula
==========================
The proof of Lemma \[L: almgren\] relies on the following equality $$\label{E: almgrenlemma}
D'(r)= \frac{n-2+a}{r}D(r) + \int_{\partial B_r}{|x_n|^a 2u_{\nu}^2}$$ is in the case that $\lambda^+ = \lambda^- =0$ ($a$-harmonic functions are minimizers of when $\lambda^+ = \lambda^- =0$). We also have $$\label{E: almgrenlemma2}
\int_{B_r}{|x_n|^a |\nabla u|^2} = \int_{\partial B_r}{|x_n|^a u u_{\nu}}$$ We obtain by recalling that $\mathcal{L}_a u =0$ in $B_1$ and using $\eta_k u$ as a test function where $\eta_k$ is defined as in , then $$\int_{B_r}{|x_n|^a \langle \nabla u, \nabla (u \eta_k) \rangle} = 0$$ By letting $k \to 0$ we obtain . The monotonicity of $N(r)$ as well as case of equality then follow from and exactly as shown in [@CS].
Proof of Nondegeneracy
======================
We begin this section with the so called Lattice principle. Since minimizers are not necessarily unique, we may not necessarily conclude that if $u$ and $v$ are two minimizers with $u \leq v$ on $\partial D$, then $u \leq v$ in $D$. Instead we have the following theorem.
\[T: Maximumprinciple\] Let $u,v$ be two minimizers of the functional $J$ with $u|_{\partial D} \leq v$. If we define $w_1 \equiv \max \{u,v\}$ and $w_2 \equiv \min \{u,v\}$, then $w_1$ and $w_2$ are minimizers of the functional $J$.
It is fairly straightforward to check that $$J(w_1) + J(w_2) = J(u) + J(v)$$ Since $w_1 |_{\partial D} = v$ and $w_2 |_{\partial D} = u$, we conclude that $w_1$ and $w_2$ are minimizers of the functional $J$.
\[C: nondegeneracy\] If the boundary data are symmetric about the line $(0,\dots ,0,x_n)$, then there is a maximal (minimal) minimizer, i.e. there exists a minimizer $u^*$ such that $v \leq u^*$ $(v \geq u^*)$ in $B$ for all other minimizers such that $v|_{\partial B} = u^*$. Furthermore, $u^*$ will be symmetric about the line $(0,\dots, 0 ,x_n)$
By Theorem \[T: Maximumprinciple\] the maximum (minimum) of rotations will be a minimizer. $u^*$ may be obtained by a limiting procedure.
To prove Theorem \[T: nondegeneracy\] we will need the following two lemmas.
\[L: collapse\] There exists a modulus of continuity $\sigma $ with $\sigma (0)=0$ such that if $u_{\epsilon}$ is any minimizer such that $u |_{\partial B_1} \equiv \epsilon$, then $$\int_{B_{1}'}{\lambda^+ \chi_{ \{u_{\epsilon} > 0 \} }} \leq \sigma (\epsilon)$$
Define $$v_{\epsilon} =
\begin{cases}
0 & \text{for } |x| \leq 1 - \sqrt{\epsilon} \\
\sqrt{\epsilon}(|x|-1) + \epsilon & \text{ otherwise}
\end{cases}$$ It is easy to see that $J(v_\epsilon) \to 0$ as $\epsilon \to 0$. Now since $$\int_{B_{1}'}{\lambda^+ \chi_{ \{u_{\epsilon} > 0 \} }} \leq J(u_{\epsilon}) \leq J(v_\epsilon)$$ the lemma is proven.
This next lemma will strengthen Corollary \[C: nondegeneracy\] in the case when our boundary values are identically constant.
\[L: steiner\] Let $u$ be a minimizer such that the values of $u|_{\partial B} = M$. Then $u$ is symmetric about the line $(0, \dots, 0, x_n)$, and the coincidence set $\Lambda(u) = \overline{B}_{\rho}'$ for some $\rho \geq 0$.
Extend $u$ to be a function on the cube $Q$ with side length 2, by defining $u(x) = M$ for $x \notin B$. We now apply Steiner symmetrization (as defined in [@bK85 page 82]) to the function $w = M - u$ on lines parallel to ${{\mathbb R}}^{n-1} \times \{0\} $.
If we only consider $\{x \mid \ |x_n| > \epsilon\}$, then $w$ is Lipschitz. Then by [@bK85 page 82], if we Steiner symmetrize $w$ to obtain $v$ we get: $$\int_{B \cap \{|x_n|> \epsilon \} }{|x_n|^a|\nabla u|^2} = \int_{B \cap \{|x_n|> \epsilon \} }{|x_n|^a |\nabla w|^2}
\geq \ \int_{B \cap \{|x_n|> \epsilon \} }{|x_n|^a|\nabla v|^2}$$ Equality is only achieved if $w$ (and hence $u$) is already Steiner symmetric along the lines we symmetrize. Furthermore, $v$ will have the same boundary values as $w$ on $\partial B$. Then by letting $\epsilon \to 0$ we obtain $$\int_{B}{|x_n|^a|\nabla u|^2} = \int_{B}{|x_n|^a|\nabla w|^2} \geq \ \int_{B}{|x_n|^a|\nabla v|^2}$$ Finally, we note that ${\mathcal{H}}^{n-1}(\{u=0\})$ is invariant under Steiner symmetrization. Then by a limiting process, we see that $u$ is a minimizer if and only if $u$ is symmetric about the line $(0, \dots ,0, x_n)$ and $\{u=0\}$ is a connected thin ball and centered at the origin.
We are now able to prove the nondegeneracy result.
First we note that by rescaling we only need to prove Theorem \[T: nondegeneracy\] on the unit ball $B$. Also, Theorem \[T: Maximumprinciple\] and Corollary \[C: nondegeneracy\] reduce Theorem \[T: nondegeneracy\] to proving the theorem for the maximal minimizer $u_{\epsilon}^*$ where $u_{\epsilon}^*|_{\partial B} = \epsilon$. Lemma \[L: steiner\] proves that $$\Lambda(u_{\epsilon}^*)= \overline{B}_{\rho}'$$ for some $\rho < 1$. Lemma \[L: collapse\] shows $$\int_{B_1'} {\lambda^+ \chi_{ \{u_{\epsilon}^*>0\} }} \ \to 0 \text{ as } \epsilon \to 0$$ Then there exists $\epsilon$ depending only on $\{t,\lambda^+ \}$ such that if $u |_{\partial B} = \epsilon$ then $$u |_{ B_{t}'} \equiv 0$$ The case for which $u \geq -\epsilon$ is proven similarly.
Proof of Weiss monotonicity formula
===================================
The proof is a slight modification of the proof for the case $a=0$ given in [@mA11]. Since $u$ is not necessarily differentiable we follow the ideas of using domain variation given by G. Weiss in [@gW98]. Since our formula is defined for a ball centered on the ${{\mathbb R}}^{n-1} \times \{0\}$ plane, we may assume without loss of generality that $x_0 =0$. Let $\tau_{\epsilon}(x) = x + \epsilon \eta_{k} x$ where $$\label{E: etak}
\eta_{k}(x) = \max \left(0 , \min ( 1, \frac{r- |x|}{k} ) \right)$$ Then $\eta_{k}(x) = 0$ outside of $B_r (0)$, and $$\eta_{k}(x) \to \chi_{ \{B_r (0)\} } \text{ as } k \to 0$$ Notice that $\tau_{\epsilon}(x) = x(1+ \epsilon \eta_k (x))$ leaves ${{\mathbb R}}^{n-1} \times \{0\}$ invariant. Now $$\nabla \eta_k (x) = \frac{-x}{|x|k} \chi_{ \{B_r \setminus B_{r-k} \} }$$ and $$D \tau_{\epsilon} (x) = I + \epsilon \left(\eta_k (x) I + x \nabla \eta_k (x) \right) + o(\epsilon)$$ Now let $u_{\epsilon} \left( \tau_{ \epsilon} (x) \right) = u(x)$ and $y = \tau_{\epsilon}(x)$. Then $$\frac{1}{\epsilon} \left(J(u_{\epsilon}) - J(u) \right) \geq 0$$ and $$\begin{aligned}
{2}
J(u_{\epsilon}) - J(u)
& = \int_{D}{|y_n|^a|\nabla u_\epsilon (y)|^2} + \int_{D'}{\lambda^+ \chi_{\{u_\epsilon > 0 \} }
+ \lambda^- \chi_{\{u_\epsilon < 0 \}} } \\
& \quad - \int_{D}{|x_n|^a|\nabla u (x)|^2} - \int_{D'}{\lambda^+ \chi_{\{u > 0 \} }
+ \lambda^- \chi_{\{u < 0 \}} }\end{aligned}$$ Now $$\begin{aligned}
{2}
\text{det } D \tau_{\epsilon}(x) &= 1 + \epsilon \ \text{ trace } D(\eta_k(x)x) + o(\epsilon) \\
\text{trace } D(\eta_k(x)x) &= \text{ div } (\eta_k(x)x) \\
D \tau_{\epsilon}^{-1} &= I - \epsilon D(\eta_k(x)x) + o(\epsilon) \end{aligned}$$ Then substituting these into the equality above we obtain that $J(u_\epsilon) -J(u)$ $$\begin{aligned}
{2}
& = \int_{D}{|x_n + \epsilon \eta_k (x)x_n|^a|\nabla u(x)(D \tau_\epsilon (x))^{-1}|^2 \text{ det } D \tau_\epsilon} +o(\epsilon) \\
& \quad + \int_{D'}{\left(\lambda^+ \chi_{ \{u>0\} } + \lambda^- \chi_{ \{u<0\} } \right) \text{ det } D \tau_{\epsilon}' (x)} + o(\epsilon) \\
& \quad - \int_{D}{|x_n|^a | \nabla u|^2} - \int_{D'}{\lambda^+ \chi_{ \{u>0\} } + \lambda^- \chi_{ \{u<0\} }} \\
& = \int_{D}{|x_n + \epsilon \eta_k (x)x_n|^a\left(|\nabla u|^2 - 2 \epsilon \nabla u D(\eta_k (x)x) \nabla u \right)
\left( 1+ \epsilon \text{ div }\eta_k (x)x \right)} \\
& \quad + \int_{D'}{\left(\lambda^+ \chi_{ \{u>0\} } + \lambda^- \chi_{ \{u<0\} } \right)
\left(1 + \epsilon \text{ div } \eta_{k}' (x',0)x' \right)} + o(\epsilon) \\
& \quad - \int_{D}{|x_n|^a| \nabla u|^2} - \int_{D'}{\lambda^+ \chi_{ \{u>0\} } + \lambda^- \chi_{ \{u<0\} }} \\
& = \int_{D}{|x_n + \epsilon \eta_k (x)x_n|^a|\nabla u|^2 - |x_n|^a|\nabla u|^2} \\
& \quad + \epsilon \int_{D}{ |x_n + \epsilon \eta_k (x)x_n|^a
\left( |\nabla u|^2 \text{ div } \eta_k (x)x -2 \nabla u D(\eta_k (x)x) \nabla u \right)} + o(\epsilon) \\
& \quad + \epsilon \int_{D'}{ \left(\lambda^+ \chi_{ \{u>0\} } + \lambda^- \chi_{ \{u<0\} } \right)
\left( \text{ div } \eta_{k}' (x',0)x' \right)} + o(\epsilon)\end{aligned}$$
Now we may let $\epsilon$ be both positive and negative and the limit is the same, so $$\lim_{\epsilon \to 0} \frac{1}{\epsilon} \left[J(u_\epsilon) - J(u) \right] = 0$$ Then we obtain the following equality: $$\begin{aligned}
{2}
0 & = \int_{D}{a|x_n|^a |\nabla u|^2 \eta_k(x)} +
\int_{D}{|x_n|^a \left(|\nabla u|^2 \text{ div } \eta_k (x)x -2 \nabla u D(\eta_k (x)x) \nabla u \right)} \\
& \quad + \int_{D'}{\left(\lambda^+ \chi_{ \{u>0\} } + \lambda^- \chi_{ \{u<0\} } \right)
\left( \text{ div } \eta_{k}' (x',0)x' \right)}\end{aligned}$$ We have $$\begin{aligned}
{2}
\text{div } \eta_k (x)x = n \eta_k (x) - \frac{|x|}{k}\chi_{B_r \setminus B_{r-k}} \\
\text{div }(\eta_k (x',0)x') =
(n-1) \eta_{k}' - \frac{|x'|}{k} \chi_{B_{r}' \setminus B_{r-k}'}\end{aligned}$$ Then $$\begin{aligned}
{2}
0 &= (n-2+a)\int_{B_r}{|x_n|^a |\nabla u|^2 \eta_{k}} - \frac{1}{k}\int_{B_r \setminus B_{r-k}}{|x| |x_n|^a \left( |\nabla u|^2 - 2|\langle \nabla u, \frac{x}{|x|} \rangle|^2 \right)} \\
& + \quad (n-1)\int_{B_{r}'}{(\lambda^+ \chi_{ \{u>0\} } + \lambda^- \chi_{ \{u<0\} } )\eta_{k}'} \\
& \quad - \frac{1}{k}\int_{B_{r}' \setminus B_{r-k}'}{|x'|(\lambda^+ \chi_{ \{u>0\} } + \lambda^- \chi_{ \{u<0\} } )}\end{aligned}$$ so as $k \to 0$ $$\label{E: almgrenmin}
\begin{aligned}
0 & = (n-2+a)\int_{B_r}{|x_n|^a |\nabla u|^2} - r \int_{\partial B_r}{|x_n|^a \left( |\nabla u|^2 -2u_{\nu}^2 \right)} \\
& \quad + (n-1)\int_{B_{r}'}{\lambda^+ \chi_{ \{u>0\} } + \lambda^- \chi_{ \{u<0\} }}
-r \int_{\partial B_{r}'}{\lambda^+ \chi_{ \{u>0\} } + \lambda^- \chi_{ \{u<0\} }} \\
& = (n-1) \int_{B_r}{|x_n|^a|\nabla u|^2} - r \int_{\partial B_r}{|x_n|^a|\nabla u|^2} \\
& \quad + (n-1) \int_{B_{r}'}{\lambda^+ \chi_{ \{u>0\} } + \lambda^- \chi_{ \{u<0\} }}
-r \int_{\partial B_{r}'}{\lambda^+ \chi_{ \{u>0\} } + \lambda^- \chi_{ \{u<0\} }} \\
& \quad -(1-a) \int_{B_r}{|x_n|^a|\nabla u|^2} + 2r \int_{\partial B_r}{|x_n|^a u_{\nu}^2}
\end{aligned}$$ By Proposition \[P: byparts\] $$\int_{B_r}{|x_n|^a |\nabla u|^2} = \int_{\partial B_r}{|x_n|^a u u_{\nu}}$$ so $$\begin{aligned}
{2}0 & = (n-1) \int_{B_r}{|x_n|^a |\nabla u|^2} - r \int_{\partial B_r}{|x_n|^a|\nabla u|^2} \\
& \quad + (n-1) \int_{B_{r}'}{\lambda^+ \chi_{ \{u>0\} } + \lambda^- \chi_{ \{u<0\} }}
-r \int_{\partial B_{r}'}{\lambda^+ \chi_{ \{u>0\} } + \lambda^- \chi_{ \{u<0\} }} \\
& \quad - (1-a)\int_{\partial B_r}{|x_n|^a u \cdot u_{\nu}} + 2r \int_{\partial B_r}{|x_n|^a u_{\nu}^2} \end{aligned}$$ Now multiply both sides of the equation by $-r^{-n}$ to obtain that for almost every $r$ $$\begin{aligned}
{2}
0 & = \left[\frac{1}{r^{n-1}} \int_{B_r}{|x_n|^a|\nabla u|^2} \right]'
+ \left[\frac{1}{r^{n-1}} \int_{B_r '}{\lambda^+ \chi_{ \{u>0\} } + \lambda^- \chi_{ \{u<0\} }} \right]' \\
& \quad
- \frac{1}{r^{n-1}} \int_{\partial B_r}{|x_n|^a\left(\frac{(1-a)u u_\nu}{r} - 2 u_{\nu}^2 \right)}\end{aligned}$$ For $\epsilon < r$ we may integrate and use Fubini’s theorem to obtain $$\begin{aligned}
{2}
\int_{\epsilon}^{r} \frac{1}{\rho^{n-1+a}} \int_{\partial B_{\rho}}{|x_n|^a 2uu_{\nu}} d\sigma \ d\rho
&=\int_{\partial B_1} \int_{\epsilon}^{r}{|x_n|^a 2u(\rho x)u_{\nu}(\rho x)} d\rho d\sigma \\
&=\int_{\partial B_1} |x_n|^a \int_{\epsilon}^{r}{ 2u(\rho x)u_{\nu}(\rho x)} d\rho d\sigma \\
&=\int_{\partial B_1}{ |x_n|^a \left(u^2(rx) - u^2(\epsilon x) \right) d \sigma } \\
&=- c + \frac{1}{r^{n-1+a}} \int_{\partial B_r}{|x_n|^a u^2} d \sigma \end{aligned}$$ So for almost every $r$ $$\label{E: der}
\frac{\text{d}}{\text{dr}}\left[\frac{1-a}{2r^n} \int_{\partial B_r}{|x_n|^a u^2} \right] =
\frac{1}{r^{n-1}} \int_{\partial B_r}{|x_n|^a \left( \frac{(1-a)u u_{\nu}}{r} - \frac{(1-a)^2u^2}{2r^2} \right)}$$ We then add and subtract the piece from to obtain for almost every $r$ $$\begin{aligned}
{2}
0 & = \left[\frac{1}{r^{n-1}} \int_{B_r}{|x_n|^a|\nabla u|^2} \right]'
+ \left[\frac{1}{r^{n-1}} \int_{B_r '}{\lambda^+ \chi_{ \{u>0\} } + \lambda^- \chi_{ \{u<0\} }} \right]' \\
& \quad - \left[\frac{1-a}{2r^n} \int_{\partial B_r}{|x_n|^a u^2} \right]'
- \frac{1}{r^{n-1}} \int_{\partial B_r}{|x_n|^a \left(\frac{(1-a)u}{\sqrt{2}r} - \sqrt{2} u_{\nu} \right)^2}\end{aligned}$$
Thus, $W' \geq 0$, and $W' = 0$ on the interval $r_1 <r< r_2$ if and only if $u$ is homogeneous of degree $s=(1-a)/2$ on the ring $r_1 < |x| < r_2$.
|
---
abstract: 'The doubts concerning validity of gas approximation for strong interaction (for example, hard spheres) are expressed. A contradictory example - a Bose system in a lattice model - is considered. Namely, the $X-Y$ model for spin $1/2$ is taken. A state with spins directed downwards is considered to be vacuum with respect to the particles. An inverse spin (+1/2) corresponds to a particle. There is an usual band spectrum (one band) for a single particle. A trial function is written for a multiple particle system. This function was shown to be a good one within a macroscopic limit, when the particle and lattice site numbers tend to infinity (the ratio of energy standard deviation to average energy tends to zero). A result within the gas limit (particle number is small compared with the lattice site number) is compared with that obtained via generally accepted approach.'
author:
- 'E.G. Batyev'
title: |
Properties of Bose gas in a lattice model\
(strong interaction)
---
Introduction
============
As is known, a Bose gas model with a weak interaction (repulsion), considered by Bogoliubov [@1], is generalized for a case of an arbitrary interaction value provided that a so-called gas approximation is correct (for example, see [@2]). The main point of this generalization is a transition from true interaction to scattering length (which is small in comparison with inter-particle distance within the gas approximation). However, the gas approximation, being good for a classical case, is not always suitable for a quantum one. The reason is the absence of a conception about trajectories (no quasi-classics) in ultra-quantum limit (which takes place for Bose system). Therefore one can hardly speak about binary collisions. In fact, interaction of every particle with all the particles at once is rather probable: when particle number is small, most particles are in Bose condensate, i.e. these particles are characterized by an infinite wave length. The aim of the present work is to show this phenomenon within the framework of a simple model. Namely, a Bose gas in the lattice model with infinite interaction is considered, so that no more than one particle can be at each cite (description of interaction like for hard spheres). This model is equivalent to X-Y model for spin $1/2$ , where particle vacuum is a state with all spins directed downwards and spin directed upwards $(+1/2)$ corresponds to a particle. An average energy value is calculated using a trial function at given particle number written for the X-Y model and a ratio of a mean-square energy deviation to an average energy value is shown to tend to zero within a macroscopic limit. The result of calculation of the basic state energy via the traditional approach (according to the accepted rules) is shown to be different.
Bose gas in the hard spheres model was considered, for example, in the work \[3\], where pseudo-potential method was used. The pseudo-potential value is selected so that the scattering of two particles at each other was the same as in the case of hard spheres. It is the pseudo-potential, for which the corresponding multi-particle Hamiltonian is written, then the interaction is presented in a more simple form and, finally, the basic state energy value (calculated from this simple form) is given (\[3\], section 1). This model and mathematical treatment is given in \[4\] also. Nevertheless, no accuracy estimate of the traditional approach was made as nobody used the trial function for initial interaction (hard spheres). This raises the question of whether this approach is correct in the case of Bose condensate, when no conception about particle trajectory and accordingly about binary collisions exists.
The model
=========
The Hamiltonian of the $X-Y$ model is: $$\begin{aligned}
\label{1}H =
-t\sum_{<nn'>}S^+(n)S^-({n'})\ \ \ \ \ (t>0)\ .\end{aligned}$$ Here $<nn'>$ denotes nearest neighbors (the sum takes place over nearest neighbors), and operators $S^{\pm}(n)$ relate to spin $1/2$ at the site with number $n$: $$S^{\pm}(n) = S_x(n)\pm iS_y(n)$$ (corresponding radius - vector of the site is ${\bf R}_n$). These operators commute at different sites and at one site we have: $$\begin{aligned}
\label{2}S^-(n)S^+(n)-S^+(n)S^-(n)=-2S_z(n)\ ,\\
\nonumber S_z(n)S^+(n)-S^+(n)S_z(n)=S^+(n)\ . \end{aligned}$$
The following Hamiltonian can be written for the Bose particles in the lattice model instead of (\[1\]): $$\begin{aligned}
\label{3}
H\rightarrow -t\sum_{<nn'>}A_n^+A_{n'} +\\ \nonumber
+U\sum_nA_n^+A_n^+A_nA_n \ \ \ \ \ \ (U\rightarrow\infty)\
.\end{aligned}$$
Operators $A_n^+,\ A_n$ are the operators of creation and destruction of a Bose particle at a site with number n. It is obvious that these statements of the problem are equivalent. Further the spin approach (\[1\]) will be used mainly.
The state $\Phi_0$ with all spins directed down (-1/2) is taken as initial one, i.e.: $$\Phi_0\equiv |\downarrow>\ \ \ \ \ (H\Phi_0=0)\
.$$ It is vacuum with respect to the particles. A wave function $S^+(n)|\downarrow>$ corresponding to one inverse spin is equivalent to a particle (a particle at a site with number n). As the state $\Bigl(S^+(n)\Bigr)^2|\downarrow> =0$, there can be only one particle at a site (two or more particles can not exist at one site), i.e. it can be considered to be the model with interaction of the hard spheres (equivalent of the condition $U\rightarrow\infty$ in expression (\[3\])).
The spectrum $\epsilon$ of a single particle is found by a conventional method: the wave function $\Phi_1$ has the form: $$\Phi_1=\sum_nC_nS^+(n)|\downarrow>\ ,\ \ \ \ \ \ H\Phi_1=\epsilon\Phi_1\ .$$ Hence we have the following: $$-t\sum_{<nm>}C_mS^+(n)|\downarrow> =
\epsilon\sum_nC_nS^+(n)|\downarrow>\ ,$$$$\epsilon C_n
=-t\sum_{\nu} C_{n+\nu} \ .$$ The sum with respect to $\nu$ in the last expression is the sum with respect to the nearest neighbors to the site with number $n$. The solution in the form $C_n\sim\exp\Bigl(i{\bf kR}_n\Bigr)$ is sought. Hence we have the following: $$\begin{aligned}
\label{4}\epsilon({\bf
k})=-t\sum_\nu\exp\Bigl(i{\bf k R}_n-i{\bf k
R}_{n+\nu}\Bigr)\rightarrow \\ \nonumber
-2t\Bigl\{\cos(k_xa)+\cos(k_ya)+\cos(k_za) \Bigr\}\ . \end{aligned}$$ Here the expression for a simple cubic lattice is presented ($a$ is the lattice period). In the vicinity to the bottom of the band we have: $$\epsilon({\bf k})\approx -t\nu_0 +\frac{k^2}{2m}\ \ \ \ \ \
\ \Bigl(1/m=2ta^2\Bigr)\ .$$
The following values correspond to the ground state of one particle: $$\Phi_1\sim \sum_{n=1}^{N_0}S^+(n) |\downarrow>\ ,\ \ \ \
\epsilon(0)=-\nu_0 t$$ ($\nu_0$ is the number of nearest neighbors, $N_0$ is the number of the lattice sites).
The value $\epsilon(0)$ is the beginning of the particle energy counting (insignificant value). If the particle number $N$ is small ($N<<N_0$), one can talk about the Bose gas. The particle number operator is $$\begin{aligned}
\label{5}\hat{ N}=\frac{1}{2}\
\sum_{n=1}^{N_0} \Bigl(1+2S_z(n)\Bigr) = \frac{N_0}{2} + S_z \\
\nonumber \Bigl(S_z\equiv \sum_{n=1}^{N_0}S_z(n)\Bigr)\
.\end{aligned}$$ It is integral of motion (commutates with the Hamiltonian). This is obvious from the Hamiltonian form and can be directly confirmed.
Trial function
==============
Above mentioned taken into account, the trial function has the following form: $$\begin{aligned}
\label{6} \Phi_N= \Bigl(
S^+\Bigr)^N |\downarrow> \ \ \ \ \ \ \ \ \ \ \biggl( S^+\equiv
\sum_{n=1}^{N_0}S^+(n)\biggr) \ . \end{aligned}$$ This approximation can be expected to be good (at least for small particle number $N<<N_0$), as all the particles are in Bose condensate.
First let us consider normalization, i.e. the value $(\Phi_N,\Phi_N)$. The initial state $|\downarrow>$ (vacuum to particles) corresponds to maximum system spin ($N_0/2$) with maximum negative projection ($-N_0/2$). This state is normalized. $S^+$ operator raises the projection by a unity without changing of the full spin value. Matrix elements of the operator (calculated by the normalized functions) are known from the general courses of quantum mechanics: $$\Bigl( S^+\Bigr)_{M,M-1} = \sqrt{(S+M)(S-M+1)}\ .$$ Here $M$ is $S_z$ spin projection value. For example, the effect of operation on the particle’s vacuum ($M-1=-N_0/2,\ S=N_0/2$) is: $$\Bigl( S^+\Bigr)_{M,M-1} = \sqrt{N_0}\ \ \rightarrow\ \ S^+\Phi_0
= \sqrt{N_0}\Phi_1$$$$(M=-N_0/2+1)\ .$$ Consequently, having denoted the corresponding to $\Phi_N$ normalized function by $\widetilde{\Phi}_N\ (\widetilde{\Phi}_N\equiv D_N\Phi_N)$, one can derive: $$\frac{1}{D_N} = \prod_{M=1-N_0/2}^{N-N_0/2}
\sqrt{(S+M)(S-M+1)} =$$$$=\prod_{n=1}^N\sqrt{n(N_0-n+1)}\ .$$ And, finally: $$\begin{aligned}
\label{7} \widetilde{\Phi}_N=D_N\Phi_N\
;\\ \nonumber D_N = \sqrt{\frac{(N_0-N)!}{N!N_0!}}\ .\end{aligned}$$
Auxiliary relations
-------------------
Some relations are necessary in what follows. The most simple is the calculation of, for example, $\Bigl(\Phi_N,\Phi_N(n) \Bigr)$, where $\Phi_N(n)\equiv
S^+(n)\Phi_{N-1}$. This value does not depend on the cite number $n$, as all the cites are equivalent. Therefore one can write: $$\Phi_N(n)\equiv S^+(n)\Phi_{N-1}\ ;$$ $$\begin{aligned}
\label{8} \Bigl(\Phi_N,\Phi_N(n) \Bigr) =
\frac{1}{N_0}\sum_n \Bigl(\Phi_N,\Phi_N(n) \Bigr) =\frac{1}{N_0} (\Phi_N,\Phi_N) ,\\
\nonumber \Bigl(\Phi_N,\Phi_N(n,n') \Bigr) = \frac{1}{N_0(N_0-1)}
(\Phi_N,\Phi_N)\ .\end{aligned}$$ Analogous relation can be written for the case of both functions containing the cite number, for example: $$\Bigl(\Phi_N(n'),\Phi_N(n) \Bigr)_{n\neq n'} =
\frac{1}{N_0-1}\biggl\{\sum_{n'}\Bigl(\Phi_N(n'),\Phi_N(n)
\Bigr)-$$$$- \Bigl(\Phi_N(n),\Phi_N(n) \Bigr)\biggr\} =$$ $$=\
\frac{1}{N_0-1}\biggl\{\Bigl(\Phi_N,\Phi_N(n) \Bigr)
-\Bigl(\Phi_N(n),\Phi_N(n) \Bigr) \biggr\}\ .$$ As for the value with coinciding numbers ($n'=n$), one can notice that one of these states is occupied a fortiori, so a norm of the state is got using the relation (\[7\]) and substituting $N_0\rightarrow (N_0-1),\ N
\rightarrow (N-1)$. The result is the following: $$\begin{aligned}
\label{9} \Bigl(\Phi_N(n),\Phi_N(n) \Bigr) =
\frac{1}{NN_0}(\Phi_N,\Phi_N)\ ;\\ \nonumber
\Bigl(\Phi_N(n'),\Phi_N(n) \Bigr)_{n\neq n'} =
\frac{N-1}{NN_0(N_0-1)}\Bigl(\Phi_N,\Phi_N \Bigr)\ .\end{aligned}$$ The correctness of the relation is tested by substitution of $N=1$.
Energy
=======
Now an average energy value can be calculated: $$E =
(\widetilde{\Phi}_N,H\widetilde{\Phi}_N) = D_N^2(\Phi_N,H\Phi_N) =$$$$=-tD_N^2N_0\nu_0\ \Bigl(\Phi_{N+1}(n'),\Phi_{N+1}(n) \Bigr)_{n'\neq
n}\ .$$ Hence taking into account (\[7\]), (\[9\]) the following relation is derived: $$\begin{aligned}
\label{10} E = -tD_N^2\
\frac{N \nu_0}{(N+1)(N_0-1)}\ \Bigl(\Phi_{N+1},\Phi_{N+1} \Bigr) =
\\ \nonumber =(-t\nu_0)\ N\Biggl\{1-\frac{N-1}{N_0-1} \Biggr\}\ .\end{aligned}$$ Note natural symmetry at the substitution $N\rightarrow(N_0-N)$. The contribution linear in relation to the particle number $N$ is just particle energy at the band bottom, quadratic contribution is a result of particle interaction being taken into account (the test of correctness: true result after substitution of $N$ by $1$).
It is easy to see, that similar energy value is obtained by simplified approach, namely, when using the trial function in the form: $$\begin{aligned}
\label{11}\Phi^{(0)} = \prod_{n=1}^{N_0}
(u+vS^+(n))|\downarrow>\\ \nonumber (u^2+v^2 = 1\ , \ \ v^2=N/N_0)\
;\end{aligned}$$ $$E^{(0)}=(\Phi^{(0)},H\Phi^{(0)})=
(-tN_0\nu_0)(uv)^2 =$$$$=(-t\nu_0)\ N\Biggl\{1-\frac{N}{N_0}
\Biggr\}\ .$$ The function (\[11\]) is a self-consistent field approximation.
It is interesting to note, that foregoing is true in a two-dimensional case (square lattice and three-dimensional spin).
Distribution function
---------------------
A distribution function can be found for the state (\[11\]). The trial function (\[11\]) can be rewritten using Bose particles and presented in the form: $$\Phi^{(0)} \rightarrow \prod_{n=1}^{N_0}
(u+vA^+_n)|0>\ .$$ The number of particles with given quasi-momentum is: $$<A^+({\bf k})A({\bf k})> =
\frac{1}{N_0}\sum_{n,n'}<A^+_nA_{n'}>\times$$$$\times \exp\biggl\{
i{\bf k}\Bigl[{\bf R}(n')- {\bf R}(n)\Bigr] \biggr\} ;$$$$\sum_{n,n'} =\sum_{n\neq n'}+\sum_{n=n'}\ .$$
The forbidding of two(many)fold occupation of the cites should be taken into account. The result of the calculation is: $$\begin{aligned}
\label{12} n(0)=<A^+(0)A(0)> = N\biggl(1-
\frac{N}{N_0}\biggr)+ \biggl( \frac{N}{N_0} \biggr)^2\ ,\\
\nonumber n({\bf k})= <A^+({\bf k})A({\bf k)}>\Bigl|_{{\bf k}\neq
0}=\biggl( \frac{N}{N_0} \biggr)^2\ .\end{aligned}$$ The summation gives the required result: $$<A^+(0)A(0)>+\sum_{{\bf k}\neq
0}<A^+({\bf k})A({\bf k})> = N$$ (state number $N_0-1$ should be taken into account in the sum over ${\bf k}\neq 0$). Then the energy value is the same: $$E^{(0)} \rightarrow -t\nu_0 <A^+(0)A(0)> +
\sum_{{\bf k}\neq 0}\epsilon({\bf k})<A^+({\bf k})A({\bf k})> .$$
It should be emphasized, that though most particles are in the condensate (see (\[12\])), the approximate wave function of the system cannot be written in the form $(A^+(0))^N|0>$ (in contrast to weak interaction [@1]). Otherwise, the forbidden case can take place, i.e. the particles can meet at one cite.
The traditional approach
------------------------
It is interesting to compare obtained energy value with the one obtained using the traditional approach (see [@2]) within the gas approximation (in our case it takes place at $N<<N_0$). The system energy can be estimated within the gas approximation using the scattering amplitude. This means to find a vertex function in stair approximation and then to write interaction energy in main approximation, provided that all the particles are in the condensate.
For this purpose Bose particles and their interaction according to Hubbard is used (see (\[3\])). The relation for interaction energy is: $$\begin{aligned}
\label{13} H_{int} = U\sum_n A^+_n A^+_nA_nA_n
=\ \ \ \ \ \ \ \ \ \\ \nonumber =\frac{U}{N_0}\sum_{{\bf p}_1+{\bf
p}_2={\bf p}_3+{\bf p}_4} A^+({\bf p}_1)A^+({\bf p}_2)A({\bf
p}_3)A({\bf p}_4)\Bigl|_{U\rightarrow\infty} .\end{aligned}$$
According to [@2] a full vertex function $\Gamma$, describing mutual scattering of two particles, should be found in gas approximation. It is $\Gamma$, that should be used for estimation of the interaction role within the gas limit instead of the initial interaction $U$. For this purpose diagram technique is used and calculations in stair approximation are made (sum frequency is equal to double particle energy in the band bottom, total momentum is zero): $$\Gamma = U + 2i\ \frac{U^2}{N_0}\ <GG>+...=
\frac{U}{1-2i(U/N_0)<GG>}\ ;$$$$\Gamma\bigl|_{U\rightarrow\infty}\rightarrow\ \frac{iN_0}{2<GG>}\
,$$ $$<GG>=\sum_{\bf p}\int \frac{d \omega}{2\pi}G(\Omega+\omega,{\bf
p})G(-\omega,-{\bf p})\ ,$$$$G(\omega,{\bf
p})=\frac{1}{\omega-\epsilon({\bf p})+i\delta} \ .$$
Here sum frequency is $\Omega=2\epsilon(0)$. The result of the calculation is: $$\begin{aligned}
\label{14}\Gamma^{-1}=\frac{-2i}{N_0}<GG> =\frac{1}{N_0} \sum_{\bf
p}\frac{1}{\epsilon({\bf p})-\epsilon(0)}\ ;\\ \nonumber
\Gamma^{-1}\rightarrow \frac{0.505}{2t}\ .\ \ \ \ \ \\end{aligned}$$ The last value is given for simple cubic lattice. The expression for energy (all the particles are in the condensate $A^+(0)=A(0)\rightarrow\sqrt{N}$) is: $$E\rightarrow
\epsilon(0)<A^+(0)A(0)> + \frac{\Gamma}{N_0}<A^+(0)A^+(0)A(0)A(0)>$$$$= -t\nu_0 N +\frac{\Gamma}{N_0}\ N^2=\ -t\nu_0 N\biggl(1
-\frac{2}{0.505\nu_0}\ \frac{N}{N_0}\biggr)\ .$$
One can see, that contribution of interaction for cubic lattice ($\nu_0 = 6$) is one and a half times less than for used trial function. It should be emphasized, that it is the consequence of binary collision approximation.
Accuracy evaluation
-------------------
Corrections to energy can be estimated using the functions resulting from Hamiltonian action on the function $\Phi_N$. Thus: $$S^-(n)\Phi_N\equiv S^-(n)S^+\Phi_{N-1} =\Bigl[S^+S^-(n)-2S_z(n)
\Bigr] \Phi_{N-1}\ .$$ First, the value $S_z(n)\Phi_N$ is found. It is easy to see, that: $$S_z(n)\Phi_N = \Phi_N(n) +
S^+\Bigl\{S_z(n)\Phi_{N-1} \Bigr\}\ .$$ From this recurrent relation follows: $$\begin{aligned}
\label{15}S_z(n)\Phi_N = N\Phi_N(n)
-\frac{1}{2} \Phi_N\ .\end{aligned}$$ It is verified directly or by summation by $n$. Thus: $$S^-(n)\Phi_N = \Phi_{N-1} -
2(N-1)\Phi_{N-1}(n) + S^+ \Bigl\{ S^-(n)\Phi_{N-1} \Bigr\}\ .$$ From this recurrent relation follows: $$\begin{aligned}
\label{16}
S^-(n)\Phi_N = N \Phi_{N-1} - N(N-1) \Phi_{N-1}(n)\ .\end{aligned}$$ It is verified by a direct substitution as well as at $N=1,\ N=2$.
The result is: $$\begin{aligned}
\label{17} H \Phi_N =
-t\biggl\{N\nu_0 \Phi_N - N(N-1) \sum_{<nn'>}\Phi_N(n,n')\biggr\}
.\end{aligned}$$ The first term arises from the particles at the band bottom, the second - from interaction of these particles and orthogonal to $\Phi_N$ states ($\Phi_{N\perp}$). The last should be determined for corrections to the energy of an initial state to be found. Noteworthily, that the same energy value (\[10\]) is obtained.
We may write: $$\begin{aligned}
\label{18} H\widetilde{\Phi}_N =
E\widetilde{\Phi}_N + w\widetilde{\Phi}_{N\perp}\ .\end{aligned}$$ Here $\widetilde{\Phi}_{N\perp}$ is a normalized function, $w$ is a transition matrix element between the states $\widetilde{\Phi}_N$ and $\widetilde{\Phi}_{N\perp}$. This element should be found for the corresponding two-level problem to be considered.
Thus, according to (\[17\]): $$H\widetilde{\Phi}_N = -t
N\nu_0\biggl\{1-\frac{N-1}{N_0-1} \biggr\}\widetilde{\Phi}_N +$$$$+
t D_N N(N-1)\biggl\{ \sum_{<nn'>}
\Phi_N(n,n')-\frac{\nu_0}{N_0-1}\Phi_N\biggr\}\ .$$ The second term is a sought quantity: $$\begin{aligned}
\label{19}
w\widetilde{\Phi}_{N\perp}=t D_N N(N-1)\Phi'_N \ ;\\
\nonumber \Phi'_N \equiv\biggl\{ \sum_{<nn'>} \Phi_N(n,n')
-\frac{\nu_0}{N_0-1}\Phi_N\biggr\}\ .\end{aligned}$$
First let us find a norm of function $\Phi'_N$: $$\Bigl(\Phi'_N,\Phi'_N \Bigr) = \sum_{<m' n'>}\sum_{<m
n>}\Bigl(\Phi_N(m',n'),\Phi_N(m,n) \Bigr)-$$$$-\frac{\nu_0^2}{(N_0-1)^2}\Bigl(\Phi_N,\Phi_N \Bigr)\ .$$ Here the orthogonality of functions $\Phi_N$ and $\Phi'_N$ is used.
Various cases should be taken into account in the calculations: all the numbers are different ($W_N^{(0)}$), two numbers are equal ($W_N^{(1)}$), two pairs of the coinciding numbers ($W_N^{(2)}$). The result is the following: $$\sum_{<m' n'>}\sum_{<m
n>}\Bigl(\Phi_N(m',n'),\Phi_N(m,n) \Bigr) =$$$$=W_N^{(2)}\
2N_0\nu_0\ +\ W_N^{(1)}\ 4N_0\nu_0(\nu_0-1) +$$$$+ W_N^{(0)}\
\Bigl\{N_0^2\nu_0^2 - 4N_0\nu_0(\nu_0-1) - 2N_0\nu_0\Bigr\}\ .$$
Calculation of coefficients can be illustrated by example of $W_N^{(0)}$. We have: $$W_N^{(0)}= \Bigl( \Phi_{N}(m'n'), \Phi_{N}(n
m ) \Bigr)\Bigl|_{\neq} =$$$$= \frac{1}{N_0-3}\sum_{l\neq m n}\Bigl(
\Phi_{N}(ln'), \Phi_{N}(n m ) \Bigr)\ =$$$$=\
\frac{1}{N_0-3}\biggl\{\sum_{l}\Bigl( \Phi_{N}(ln'), \Phi_{N}(n m )
\Bigr)-$$$$-\Bigl( \Phi_{N}(mn'), \Phi_{N}(n m ) \Bigr) - \Bigl(
\Phi_{N}(nn'), \Phi_{N}(n m ) \Bigr)\biggr\}\rightarrow$$ $$\rightarrow\ \frac{1}{N_0-3}\biggl\{ \Bigl( \Phi_{N}(n'),
\Phi_{N}(n m ) \Bigr) -2 W_N^{(1)}\biggr\} =$$$$=
\frac{1}{N_0-3}\Biggl\{ \frac{(N-2)\Bigl( \Phi_{N}, \Phi_{N}
\Bigr)}{NN_0(N_0-1)(N_0-2)} -2 W_N^{(1)}\Biggr\}\ .$$
Similar operations are made in the other cases: $$\begin{aligned}
\nonumber W_N^{(0)} = \frac{1}{N_0-3}\Biggl\{
\frac{(N-2)\Bigl( \Phi_{N}, \Phi_{N} \Bigr)}{NN_0(N_0-1)(N_0-2)} -2
W_N^{(1)}\Biggr\}; \\ \label{20} W_N^{(1)}=
\frac{1}{N_0-2}\biggl\{\frac{\Bigl( \Phi_{N}, \Phi_{N}
\Bigr)}{NN_0(N_0-1)} -W_N^{(2)}\biggr\}; \\ \nonumber W_N^{(2)}
=\frac{\Bigl( \Phi_{N}, \Phi_{N} \Bigr)}{N(N-1)N_0(N_0-1)}\
.\end{aligned}$$
The following relations for the sought quantity and for $w$, defined in (\[19\]), result from combining of all above mentioned relations: $$\begin{aligned}
\label{21} \Bigl(\Phi'_N,\Phi'_N \Bigr)
\approx 2\nu_0\ \frac{(N_0-N)^2}{N^2N_0^3}\ \Bigl(\Phi_N,\Phi_N \Bigr)\ ;\\
\nonumber w\approx t \sqrt{2\nu_0}\ \frac{N(N_0-N)}{N_0^{3/2}}\ \
\ \ \ \ \ \ (N,N_0\rightarrow\infty)\ .\end{aligned}$$
Now the correction to the system energy can be estimated. The value $w$ tends to infinity within the macroscopic limit, therefore it is a fortiori much higher than the difference of $\widetilde{\Phi}'_N$ state energy and the initial one. Consequently, the correction to energy is approximately equal to $- w$. And this value is proportional to the square root of volume, i.e. is much lower than the energy of interest (proportional to volume). Therefore, one can draw a conclusion, that initial test function is a good approximation of the problem.
For reliability, the energy root-mean-square value (i.e. $<H^2>$) is found and compared with the value $<H>^2=E^2$. There are all the data for the following relations: $$\Bigl(\widetilde{\Phi}_N,H^2\widetilde{\Phi}_N \Bigr) =
\Bigl(E\widetilde{\Phi}_N+w\widetilde{\Phi}_{
N\bot},E\widetilde{\Phi}_N+w\widetilde{\Phi}_{ N\bot} \Bigr)=
E^2+w^2 ;$$ $$\begin{aligned}
\label{22}\frac{<(H-E)^2>}{E^2} =
2\nu_0 t^2 \frac{N^2(N_0-N)^2}{E^2N_0^3}\biggl|_{N,N_0\rightarrow\infty} \rightarrow 0
.\end{aligned}$$ Here a finite concentration is meant (the ratio $N/N_0$ is constant).
In (\[22\]) the system ground state is shown to be well described by the trial function used. It is true for an $X-Y$ model in general.
Conclusion
==========
Within the framework of the model used, the conventional Bose gas theory for the particles with strong interaction was shown to be inconsistent. If most particles are in Bose condensate, description of their interaction based on binary collisions (using binary scattering amplitude) does not suit, as there is no quasi-classics for the particles with an infinite wave length. It turns out, as if every particle interacts with all the particles at once. This fact leads to the increase of the interaction energy, as shown in the model used here.
It should be emphasized, that in present work the conclusion about the accuracy of the approach is made after writing of a trial function. In [@3] first the problem is simplified by a transition to a pseudo-potential, then solved by the perturbation theory. However, there are no successful attempts to write a multi-particle function for initial interaction (hard spheres).
Note, that the results obtained suit for two-dimensional case (three-dimensional spin).
So far it is not clear, how the spectrum of elementary excitations can be got. Perhaps, a diagram Belyaev - type technique (see, for example, [@5]) or its modified form (?) should be used.
Acknowledgement. The author gratefully acknowledges the discussion of A.V. Chaplik, M.V. Entin and V.M. Kovalev and also the financial support of RFBR (Grant 11-02-00060) and Russian Academy of Sciences (Programs).
[99]{} N.N. Bogoliubov, J. Phys. U.S.S.R. **11**, 23 (1947). E.M. Lifshitz, L.P. Pitaevskii, *Statistical physics, part 2* (Pergamon, 1980). T.D. Lee, K. Huang and C.N. Yang, Phys. Rev. **106**, 1135 (1957). Kerson Huang, *Statistical mechanics* (John Wiley & Sons, Inc., New York - London, 1963). A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinski, *Methods of Quantum Field Theory in Statistical Physics.* (Englewood Cliffs: Prentice Hall, 1963).
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abstract: |
We use Kashiwara-Nakashima’s combinatorics of crystal graphs associated to the roots sytems $B_{n}$ and $D_{n}$ to extend the results of [@lec3] and [@Mor] by showing that Morris type recurrence formulas also exist for the orthogonal root systems. We derive from these formulas a statistic on Kashiwara-Nakashima’s tableaux of types $B_{n},C_{n}$ and $D_{n}$ generalizing Lascoux-Schützenberger’s charge and from which it is possible to compute the Kostka-Foulkes polynomials $K_{\lambda,\mu}(q)$ with restrictive conditions on $(\lambda,\mu)$ . This statistic is different from that obtained in [@lec3] from the cyclage graph structure on tableaux of type $C_{n}$. We show that such a structure also exists for the tableaux of types $B_{n}$ and $D_{n}$ but can not be simply related to the Kostka-Foulkes polynomials. Finally we give explicit formulas for $K_{\lambda,\mu}(q)$ when $\left|
\lambda\right| \leq3,$ or $n=2$ and $\mu=0$.
author:
- |
Cédric Lecouvey\
lecouvey@math.unicaen.fr
title: ' Combinatorics of crystal graphs and Kostka-Foulkes polynomials for the root systems $B_{n},C_{n}$ and $D_{n}.$'
---
Introduction
============
The multiplicity $K_{\lambda,\mu}$ of the weight $\mu$ in the irreducible finite dimensional representation $V(\lambda)$ of the simple Lie algebra $g$ can be written in terms of the ordinary Kostant’s partition function $\mathcal{P}$ defined from the equality:$$\prod_{\alpha\text{ positive root}}\dfrac{1}{(1-x^{\alpha})}=\sum_{\beta
}\mathcal{P}(\beta)x^{\beta}$$ where $\beta$ runs on the set of nonnegative integral combinations of positive roots of $g$. Thus $\mathcal{P}(\beta)$ is the number of ways the weight $\beta$ can be expressed as a sum of positive roots. Then we have$$K_{\lambda,\mu}=\sum_{\sigma\in W}(-1)^{l(\sigma)}\mathcal{P}(\sigma
(\lambda+\rho)-(\mu+\rho))$$ where $W$ is the Weyl group of $g.$
There exists a $q$-analogue $K_{\lambda,\mu}(q)$ of $K_{\lambda,\mu
}$ obtained by substituting the ordinary Kostant’s partition function $\mathcal{P}$ by its $q$-analogue $\mathcal{P}_{q}$ satisfying$$\prod_{\alpha\text{ positive root}}\dfrac{1}{(1-qx^{\alpha})}=\sum_{\beta
}\mathcal{P}_{q}(\beta)x^{\beta}.$$ So we have$$K_{\lambda,\mu}(q)=\sum_{\sigma\in W}(-1)^{l(\sigma)}\mathcal{P}_{q}(\sigma(\lambda+\rho)-(\mu+\rho)).$$ As shown by Lusztig [@Lut] $K_{\lambda,\mu}(q)$ is a polynomial in $q$ with non negative integer coefficients.
For type $A_{n-1}$ the positivity of the Kostka-Foulkes Polynomials can also be proved by a purely combinatorial method. Recall that for any partitions $\lambda$ and $\mu$ with $n$ parts the number of semi-standard tableaux of shape $\lambda$ and weight $\mu$ is equal to the multiplicity of the weight $\mu$ in $V(\lambda).$
In [@LSc1], Lascoux and Schützenberger have introduced a beautiful statistic $\mathrm{ch}_{A}$ on dominant evaluation words $w$ that is on words $w=x_{1}\cdot\cdot\cdot x_{l}$ whose letters $x_{i}$ are positive integers such that for any $i\geq1$ with $i$ a letter of $w,$ $w$ contains more letters $i$ than letters $i+1.$ Recall that the plactic monoid is the quotient set of the free monoid on the positive integers by Knuth’s relations$$abx\equiv\left\{
\begin{tabular}
[c]{l}$bax$ if $a<x\leq b$\\
$axb$ if $x\leq a<b$\end{tabular}
\right. .$$ The statistic $\mathrm{ch}_{A}$ is the unique function from dominant evaluation words to non-negative integers such that $$\left\{
\begin{tabular}
[c]{l}$\mathrm{ch}_{A}(\emptyset)=0$\\
$\mathrm{ch}_{A}(xu)=\mathrm{ch}_{A}(ux)+1$ if $x$ is not the lowest letter of
$w$\\
$\mathrm{ch}_{A}(xu)=\mathrm{ch}_{A}(u)$ if $x$ is the lowest letter of $w$\\
$\mathrm{ch}_{A}(\sigma w)=\mathrm{ch}_{A}(w)$ for any $\sigma\in
\mathcal{S}_{n}$\\
$\mathrm{ch}_{A}(w_{1})=\mathrm{ch}_{A}(w_{2})$ if $w_{1}\equiv w_{2}$\end{tabular}
\right. \label{def_chA_ini}$$ [@kill]. Then the charge of the semi-standard tableau $T$ of dominant weight verifies $\mathrm{ch}_{A}(T)=\mathrm{ch}_{A}(\mathrm{w}(T))$ where $\mathrm{w}(T)$ is the word obtained by column reading the letters of $T$ from top to bottom and right to left. Lascoux and Schützenberger have proved the equality$$K_{\lambda,\mu}(q)=\sum_{T}q^{\mathrm{ch}_{A}(T)} \label{th_ls}$$ where $T$ runs on the set of semi-standard tableaux of shape $\lambda$ and weight $\mu.\;$The proof of (\[th\_ls\]) is based on Morris recurrence formula which permits to express each Kostka-Foulkes polynomials related to the root system $A_{n}$ in terms of Kostka-Foulkes polynomials related to the root system $A_{n-1}.$
The compatibility of the charge with plactic relations provides alternative ways to compute $\mathrm{ch}_{A}(T)$. By applying the reverse bumping algorithm on the boxes contained in the longest row of $T$ we obtain a pair $(R,T^{\prime})$ with $R$ a row tableau whose length is equal to the longest row of $T$ and $T^{\prime}$ a semi-standard tableau which does not contain the lowest letter $t$ of $T$ such that $\mathrm{w}(T)\equiv
\mathrm{w}(R)\otimes\mathrm{w}(T^{\prime})$. Let $R^{\prime}$ be the row tableau obtained by erasing all the letters $t$ in $R.$ Then the catabolism of $T$ is the unique semi-standard tableau $\mathrm{cat}(T)$ such that $\mathrm{w}(\mathrm{cat}(T))\equiv\mathrm{w}(T^{\prime})\otimes\mathrm{w}(R^{\prime})$ computed via the bumping algorithm. We have $$\mathrm{ch}_{A}(\mathrm{cat}(T))=\mathrm{ch}_{A}(T)+r^{\prime}$$ where $r^{\prime}$ is the length of $R^{\prime}.$ Since the number of boxes of $\mathrm{cat}(T)$ is strictly less than that of $T,$ $\mathrm{ch}_{A}(T)$ can be obtained from $T$ by computing successive catabolism operations$.$ In fact this is this characterization of the charge which is needed to prove (\[th\_ls\]).
The charge may also be obtained by endowing $ST(\mu)$ the set of semi-standard tableaux of weight $\mu$ with a structure of graph. We draw an arrow $T\rightarrow T^{\prime}$ between the two tableaux $T$ and $T^{\prime}$ of $ST(\mu),$ if and only if there exists a word $u$ and a letter $y$ which is not the lowest letter of $T$ such that $\mathrm{w}(T)\equiv xu$ and $\mathrm{w}(T^{\prime})\equiv ux$. Then we say that $T^{\prime}$ is a cocyclage of $T.$ The essential tool to define this graph structure is yet the bumping algorithm for the semi-standard tableaux. The cyclage graph $ST(\mu)$ contains a unique row tableau $L_{\mu}$ which can not be obtained as the cocyclage of another tableau of $ST(\mu)$. Let $T_{\mu}$ be the unique semi-standard tableau of shape $\mu$ belonging to $ST(\mu)$. Then there is no cocyclage of $T_{\mu}.$ For any $T\in ST(\mu)$ all the paths joining $L_{\mu}$ to $T$ have the same length. This length is called the cocharge of $T$ and denoted $\mathrm{coch}_{A}(T).$ Similarly, all the paths joining $T$ to $T_{\mu}$ have the same length which is equal to the charge of $T.$ The maximal value of $\mathrm{ch}_{A}$ is $\left\| \mu\right\| =\mathrm{ch}_{A}(L_{\lambda})=\sum_{i}(i-1)\mu_{i}.\;$Moreover the charge and the cocharge are related by the equality $\mathrm{ch}_{A}(T)=\left\| \mu\right\|
-\mathrm{coch}_{A}(T)$ for any $T\in ST(\mu).$
Analogues of semi-standard tableaux also exit for the other classical root systems. They have been introduced by Kashiwara and Nakashima [@KN] via crystal bases theory. For each classical root system these tableaux naturally label the vertices of the crystal graph $B(\lambda)$ associated to the dominant weight $\lambda.$
In [@lec3] we have proved that an analogue of Morris recurrence formula exists for the root system $C_{n}.$ Moreover it is also possible to endow the corresponding set of tableaux with a structure of cyclage graph. From these graphs we have introduced a natural statistic on Kashiwara-Nakashima’s tableaux of type $C_{n}$ and have conjectured that this statistic yields an analogue of Lascoux-Schützenberger’s theorem.
This article is an attempt to look at possible generalizations and extensions of these results to the orthogonal roots systems. We establish Morris type recurrence formula for the root systems $B_{n}$ and $D_{n}.$ Moreover we show that is possible to endow the set of tableaux of types $B_{n}$ and $D_{n}$ with a structure of cyclage graph. Nevertheless the situation is more complicated than for the root system $C_{n}$ and we are not able to deduce from these graphs a natural statistic relevant for computing the Kostka-Foulkes polynomials. To overcome this problem we change our strategy and define a new statistic $\chi_{n}$ on tableaux of types $B_{n},C_{n}$ and $D_{n}$ based on the catabolism operation. Then we prove that this statistic can be used to compute the Kostka-Foulkes polynomials $K_{\lambda,\mu}(q)$ with restrictive conditions on $(\lambda,\mu)$. Note that the analogue of (\[th\_ls\]) with $\chi_{n}$ is false in general. In particular $\chi_{n}$ is not equal to the statistic defined in [@lec3] for the tableaux of type $C_{n}$ even if the two statistics can be regarded as generalizations of $\mathrm{ch}_{A}$ since they coincide on semi-standard tableaux.
In Section $1$ we recall the Background on Kostka-Foulkes polynomials and combinatorics of crystal graphs that we need in the sequel. We also summarize the basic properties of the insertion algorithms and plactic monoids for the root systems $B_{n},C_{n}$ and $D_{n}$ introduced in [@Lec] and [@lec2]. Section $2$ is devoted to Morris type recurrence formulas for types $B_{n}$ and $C_{n}$. In Section $3$ we define the catabolism operation for the tableaux of type $B_{n},C_{n}$ and $D_{n}.$ Then we introduce the statistics $\chi_{n}^{B},\chi_{n}^{C}$ and $\chi_{n}^{D}$ and prove that analogues of (\[th\_ls\]) hold for these statistics if $\lambda$ and $\mu$ satisfy restrictive conditions. We also introduce the cyclage graph structure on tableaux of types $B_{n}$ and $D_{n}$ and show that a charge statistic related to Kostka-Foulkes polynomials can not be obtained in a similar way that in [@lec3]. Finally we give in Section $4$ explicit simple formulas for the Kostka-Foulkes polynomials $K_{\lambda,\mu}(q)$ when $\left|
\lambda\right| \leq3,$ or $n=2$ and $\mu=0$ deduced from the results of Sections $2$ and $3.$
**Notation:** In the sequel we frequently define similar objects for the root systems $B_{n}$ $C_{n}$ and $D_{n}$. When they are related to type $B_{n}$ (resp. $C_{n},D_{n}$), we implicitly attach to them the label $B$ (resp. the labels $C,D$). To avoid cumbersome repetitions, we sometimes omit the labels $B,C$ and $D$ when our definitions or statements are identical for the three root systems.
Background
==========
Kostka-Foulkes polynomials associated to a root system
------------------------------------------------------
Let $g$ be a simple Lie algebra and $\alpha_{i},$ $i\in I$ its simple roots. Write $Q^{+}$ and $R^{+}$ for the set of nonnegative integral combinations of positive roots and for the set of positive roots of $g$. Denote respectively by $P$ and $P^{+}$ its weight lattice and its cone of dominant weights. Let $\{s_{i},i\in I\}$ be a set of generators of the Weyl group $W$ and $l$ the corresponding length function.
The $q$-analogue $\mathcal{P}_{q}$ of the Kostant function partition is such that$$\prod_{\alpha\in R^{+}}\dfrac{1}{1-qx^{\alpha}}=\sum_{\beta\in Q^{+}}\mathcal{P}_{q}(\beta)x^{\beta}\text{ and }\mathcal{P}_{q}(\beta)=0\text{ if
}\beta\notin Q^{+}.$$
\[def\_kost\]Let $\lambda,\mu\in P^{+}.$ The Kostka-Foulkes polynomial $K_{\lambda,\mu}(q)$ is defined by $$K_{\lambda,\mu}(q)=\sum_{\sigma\in W}(-1)^{l(\sigma)}\mathcal{P}_{q}(\sigma(\lambda+\rho)-(\mu+\rho)).$$ where $\rho$ is the half sum of positive roots.
Let $\beta\in P.\;$We set $$a_{\beta}=\sum_{\sigma\in W}(-1)^{l(\sigma)}(\sigma\cdot x^{\beta})$$ where $\sigma\cdot x^{\mu}=x^{\sigma(\mu)}.$ The Schur function $s_{\beta}$ is defined by $$s_{\beta}=\dfrac{a_{\beta+\rho}}{a_{\rho}}.$$ When $\lambda\in P^{+},$ $s_{\lambda}$ is the Weyl character of $V(\lambda)$ the finite dimensional irreducible $g$-module with highest weight $\lambda.$ For any $\sigma\in W,$ the dot action of $\sigma$ on $\beta\in P$ is defined by $\sigma\circ\beta=\sigma\cdot(\beta+\rho)-\rho.$ We have the following straightening law for the Schur functions. For any $\beta\in P$, $s_{\beta}=0$ or there exists a unique $\lambda\in P^{+}$ such that $s_{\beta}=(-1)^{l(\sigma)}s_{\lambda}$ with $\sigma\in W$ and $\lambda=\sigma\circ
\beta.$ Set $\mathbb{K}=\mathbb{Z}[q,q^{-1}]$ and write $\mathbb{K}[P]$ for the $\mathbb{K}$-module generated by the $x^{\beta}$, $\beta\in P.$ Set $\mathbb{K}[P]^{W}=\{f\in\mathbb{K}[P],$ $\sigma\cdot f=f$ for any $\sigma\in
W\}.$ Then $\{s_{\lambda}\}$ is a basis of $\mathbb{K}[P]^{W}.$
To each positive root $\alpha,$ we associate the raising operator $R_{\alpha}:P\rightarrow P$ defined by$$R_{\alpha}(\beta)=\alpha+\beta.$$ Given $\alpha_{1},...,\alpha_{p}$ positive roots and $\beta\in P,$ we set $(R_{\alpha_{1}}\cdot\cdot\cdot R_{\alpha_{p}})s_{\beta}=s_{R_{\alpha_{1}}\cdot\cdot\cdot R_{\alpha_{p}}(\beta)}.$ For all $\beta\in P,$ we define the Hall-Littelwood polynomial $Q_{\beta}$ by$$Q_{\beta}=\left( \prod_{\alpha\in R^{+}}\dfrac{1}{1-qR_{\alpha}}\right)
s_{\beta}$$ where $\dfrac{1}{1-qR_{\alpha}}=\sum_{k=0}^{+\infty}q^{k}R_{\alpha}^{k}.$
\[th\_hall\_kostka\][@mac]For any $\lambda,\mu\in P^{+},$ $K_{\lambda
,\mu}(q)$ is the coefficient of $s_{\lambda}$ in $Q_{\mu}$ that is, $$Q_{\mu}=\sum_{\lambda\in P^{+}}K_{\lambda,\mu}(q)s_{\lambda}.$$
Kostka-Foulkes polynomials for the root systems $B_{n},C_{n}$ and $D_{n}\label{subsec_KF}$
------------------------------------------------------------------------------------------
We choose to label respectively the Dynkin diagrams of $so_{2n+1}$, $sp_{2n}$ and $so_{2n}$ by $$\overset{0}{\circ}\Longleftarrow\overset{1}{\circ}-\overset{2}{\circ}-\overset{3}{\circ}-\overset{4}{\circ}-\cdot\cdot\cdot\overset{n-1}{\circ
}\text{, }\overset{0}{\circ}\Longrightarrow\overset{1}{\circ}-\overset
{2}{\circ}-\overset{3}{\circ}-\overset{4}{\circ}-\cdot\cdot\cdot\overset
{n-1}{\circ}\text{ and }\begin{tabular}
[c]{l}$\overset{1}{\circ}$\\
$\ \ \backslash$\\
$\ \ \ \ \overset{2}{\circ}$\\
$\ \ /$\\
$\underset{0}{\circ}$\end{tabular}
-\overset{3}{\circ}-\overset{4}{\circ}-\cdot\cdot\cdot\overset{n-2}{\circ
}-\overset{n-1}{\circ}. \label{DD}$$ The weight lattices for the root systems $B_{n},C_{n}$ and $D_{n}$ can be identified with $P_{n}=\mathbb{Z}^{n}$ equipped with the orthonormal basis $\varepsilon_{\overline{i}},$ $i=1,...,n$. We take for the simple roots$$\left\{
\begin{tabular}
[c]{l}$\alpha_{0}^{B_{n}}=\varepsilon_{\overline{1}}\text{ and }\alpha_{i}^{B_{n}}=\varepsilon_{\overline{i+1}}-\varepsilon_{\overline{i}}\text{,
}i=1,...,n-1\text{ for the root system }B_{n}$\\
$\alpha_{0}^{C_{n}}=2\varepsilon_{\overline{1}}\text{ and }\alpha_{i}^{C_{n}}=\varepsilon_{\overline{i+1}}-\varepsilon_{\overline{i}}\text{,
}i=1,...,n-1\text{ for the root system }C_{n}$\\
$\alpha_{0}^{D_{n}}=\varepsilon_{\overline{1}}+\varepsilon_{\overline{2}}\text{ and }\alpha_{i}^{D_{n}}=\varepsilon_{\overline{i+1}}-\varepsilon
_{\overline{i}}\text{, }i=1,...,n-1\text{ for the root system }D_{n}$\end{tabular}
\right. . \label{simple_roots}$$ Then the set of positive roots are$$\left\{
\begin{tabular}
[c]{l}$R_{B_{n}}^{+}=\{\varepsilon_{\overline{i}}-\varepsilon_{\overline{j}},\varepsilon_{\overline{i}}+\varepsilon_{\overline{j}}\text{ with }1\leq
j<i\leq n\}\cup\{\varepsilon\overline{_{i}}\text{ with }1\leq i\leq n\}\text{
for the root system }B_{n}$\\
$R_{B_{n}}^{+}=\{\varepsilon_{\overline{i}}-\varepsilon_{\overline{j}},\varepsilon_{\overline{i}}+\varepsilon_{\overline{j}}\text{ with }1\leq
j<i\leq n\}\cup\{2\varepsilon\overline{_{i}}\text{ with }1\leq i\leq n\}\text{
for the root system }C_{n}$\\
$R_{D_{n}}^{+}=\{\varepsilon_{\overline{i}}-\varepsilon_{\overline{j}},\varepsilon_{\overline{i}}+\varepsilon_{\overline{j}}\text{ with }1\leq
j<i\leq n\}\text{ for the root system \ }D_{n}$\end{tabular}
\right. .$$ Denote respectively by $P_{B_{n}}^{+},P_{C_{n}}^{+}$ and $P_{D_{n}}^{+}$the sets of dominant weights of $so_{2n+1},sp_{2n}$ and $so_{2n}.$ Write $\Lambda_{0}^{B_{n}},...,\Lambda_{n-1}^{B_{n}}$ for the fundamentals weights of $so_{2n+1},$ $\Lambda_{0}^{C_{n}},...,\Lambda_{n-1}^{C_{n}}$ for the fundamentals weights of $sp_{2n}$ and $\Lambda_{0}^{D_{n}},...,\Lambda
_{n-1}^{D_{n}}$ for the fundamentals weights of $so_{2n+1}$.
We have $\Lambda_{i}^{B_{n}}=\Lambda_{i}^{C_{n}}=\Lambda_{i}^{D_{n}}=\varepsilon_{\overline{n}}+\cdot\cdot\cdot+\varepsilon_{\overline{i+1}}$ for $2\leq i\leq n-1$, $\Lambda_{0}^{B_{n}}=\Lambda_{0}^{D_{n}}=\dfrac{1}{2}(\varepsilon_{\overline{n}}+\cdot\cdot\cdot+\varepsilon_{\overline{2}}+\varepsilon_{\overline{1}})$, $\Lambda_{0}^{C_{n}}=\varepsilon_{\overline
{n}}+\cdot\cdot\cdot+\varepsilon_{\overline{2}}+\varepsilon_{\overline{1}},$ $\Lambda_{1}^{B_{n}}=\Lambda_{1}^{C_{n}}=\varepsilon_{\overline{n}}+\cdot
\cdot\cdot+\varepsilon_{\overline{2}}$ and $\Lambda_{1}^{D_{n}}=\dfrac{1}{2}(\varepsilon_{\overline{n}}+\cdot\cdot\cdot+\varepsilon_{\overline{2}}-\varepsilon_{\overline{1}}).$
Consider $\lambda\in P_{B_{n}}^{+}$ and write $\lambda=\sum
_{i=0}^{n-1}\widehat{\lambda}_{i}\Lambda_{i}^{B}$ with $\widehat{\lambda}_{i}\in\mathbb{N}.\;$Set $\lambda_{\overline{1}}=\dfrac{\widehat{\lambda}_{0}}{2}$ and $\lambda_{\overline{i}}=\dfrac{\widehat{\lambda}_{0}}{2}+\widehat{\lambda}_{1}+\cdot\cdot\cdot+\widehat{\lambda}_{i-1},$ $i=2,...,n.$ The dominant weight $\lambda$ is characterized by the generalized partition $(\lambda_{\overline{n}},...,\lambda_{\overline{1}})$ such that $\lambda
_{\overline{n}}\geq\cdot\cdot\cdot\geq\lambda_{\overline{1}}$ and $\lambda_{\overline{i}}\in\dfrac{\mathbb{N}}{2},$ $i=1,...,n.$ In the sequel we will identify $\lambda$ and $(\lambda_{\overline{n}},...,\lambda
_{\overline{1}})$ by setting $\lambda=(\lambda_{\overline{n}},...,\lambda
_{\overline{1}}).$ Then $\lambda=\lambda_{\overline{1}}\varepsilon
_{\overline{1}}+\cdot\cdot\cdot+\lambda_{\overline{n}}\varepsilon
_{\overline{n}}$ that is, the $\lambda_{i}$’s are the coordinates of $\lambda$ on the basis $(\varepsilon_{\overline{n}},...,\varepsilon_{\overline{1}})$. The half sum of positive roots verifies $\rho_{B_{n}}=(n-\dfrac{1}{2},n-\dfrac{3}{2},...,\dfrac{1}{2}).$
Consider $\lambda\in P_{C_{n}}^{+}$ and write $\lambda=\sum
_{i=0}^{n-1}\widehat{\lambda}_{i}\Lambda_{i}^{C}$ with $\widehat{\lambda}_{i}\in\mathbb{N}.\;$The dominant weight $\lambda$ is characterized by the partition $(\lambda_{\overline{n}},...,\lambda_{\overline{1}})$ where $\lambda_{\overline{1}}=\widehat{\lambda}_{0}$ and $\lambda_{\overline{i}}=\widehat{\lambda}_{0}+\widehat{\lambda}_{1}+\cdot\cdot\cdot+\widehat
{\lambda}_{i-1},$ $i=2,...,n.$ We set $\lambda=(\lambda_{\overline{n}},...,\lambda_{\overline{1}}).$ Then $\lambda=\lambda_{\overline{1}}\varepsilon_{\overline{1}}+\cdot\cdot\cdot+\lambda_{\overline{n}}\varepsilon_{\overline{n}}$ and the half sum of positive roots verifies $\rho_{C_{n}}=(n,n-1,...,1).$
Now consider $\lambda\in P_{D_{n}}^{+}$ and write $\lambda
=\sum_{i=0}^{n-1}\widehat{\lambda}_{i}\Lambda_{i}^{D}$ with $\widehat{\lambda
}_{i}\in\mathbb{N}.\;$Set $\lambda_{\overline{1}}=\dfrac{\widehat{\lambda}_{0}-\widehat{\lambda}_{1}}{2}$, $\lambda_{\overline{2}}=\dfrac{\widehat
{\lambda}_{0}+\widehat{\lambda}_{1}}{2}$ and $\lambda_{\overline{i}}=\dfrac{\widehat{\lambda}_{0}+\widehat{\lambda}_{1}}{2}+\widehat{\lambda}_{2}+\cdot\cdot\cdot+\widehat{\lambda}_{i-1},$ $i=3,...,n.$ The dominant weight $\lambda$ is characterized by the generalized partition $(\lambda
_{\overline{n}},...,\lambda_{\overline{1}})$ such that $\lambda_{\overline{n}}\geq\cdot\cdot\cdot\geq\lambda_{\overline{1}}$, $\lambda_{\overline{i}}\in\dfrac{\mathbb{N}}{2}$ $i=2,...,n$ and $\lambda_{\overline{1}}\in
\dfrac{\mathbb{Z}}{2}.$ Note that we can have $\lambda_{\overline{1}}<0.$ We set $\lambda=(\lambda_{\overline{n}},...,\lambda_{\overline{1}}).$ Then $\lambda=\lambda_{\overline{1}}\varepsilon_{\overline{1}}+\cdot\cdot
\cdot+\lambda_{\overline{n}}\varepsilon_{\overline{n}}$ and the half sum of positive roots verifies $\rho_{D_{n}}=(n-1,n-2,...,0).$
For any generalized partition $\lambda=(\lambda_{\overline{n}},...,\lambda
_{\overline{1}})\in P_{n}^{+},$ we write $\lambda^{\prime}\in P_{n-1}^{+}$ for the generalized partition obtained by deleting $\lambda_{\overline{n}}$ in $\lambda.$ Moreover we set $\left| \lambda\right| =\lambda_{\overline{1}}+\lambda_{\overline{2}}+\cdot\cdot\cdot+\lambda_{\overline{n}}$ if $\lambda_{\overline{1}}\geq0,$ $\left| \lambda\right| =-\lambda
_{\overline{1}}+\lambda_{\overline{2}}+\cdot\cdot\cdot+\lambda_{\overline{n}}$ otherwise.
The Weyl group $W_{B_{n}}=W_{C_{n}}$ of $so_{2n+1}$ can be regarded as the sub group of the permutation group of $\{\overline{n},...,\overline
{2},\overline{1},1,2,...,n\}$ generated by $s_{i}=(i,i+1)(\overline
{i},\overline{i+1}),$ $i=1,...,n-1$ and $s_{0}=(1,\overline{1})$ where for $a\neq b$ $(a,b)$ is the simple transposition which switches $a$ and $b.$ We denote by $l_{B}$ the length function corresponding to the set of generators $s_{i},$ $i=0,...n-1.$
The Weyl group $W_{D_{n}}$ of $so_{2n}$ can be regarded as the sub group of the permutation group of $\{\overline{n},...,\overline{2},\overline{1},1,2,...,n\}$ generated by $s_{i}=(i,i+1)(\overline{i},\overline{i+1}),$ $i=1,...,n-1$ and $s_{0}^{\prime}=(1,\overline
{2})(2,\overline{1})$. We denote by $l_{D}$ the length function corresponding to the set of generators $s_{0}^{\prime}$ and $s_{i},$ $i=1,...n-1.$
Note that $W_{D_{n}}\subset W_{B_{n}}$ and any $\sigma\in W_{B_{n}}$ verifies $\sigma(\overline{i})=\overline{\sigma(i)}$ for $i\in\{1,...,n\}.$ The action of $\sigma$ on $\beta=(\beta_{\overline{n}},...,\beta_{\overline
{1}})\in P_{n}$ is given by$$\sigma\cdot(\beta_{\overline{n}},...,\beta_{\overline{1}})=(\beta
_{\overline{n}}^{\sigma},...,\beta_{\overline{1}}^{\sigma})$$ where $\beta_{\overline{i}}^{\sigma}=\beta_{\sigma(\overline{i})}$ if $\sigma(\overline{i})\in\{\overline{1},...,\overline{n}\}$ and $\beta
_{\overline{i}}^{\sigma}=-\beta_{\sigma(i)}$ otherwise.
For any $\beta=(\beta_{\overline{n}},...,\beta_{\overline{1}})\in
P_{n}$ we set $x^{\beta}=x_{n}^{\beta_{\overline{n}}}\cdot\cdot\cdot
x_{1}^{\beta_{\overline{1}}}$ where $x_{1},...,x_{n}$ are fixed indeterminates.
The following lemma is a consequence of Definition \[def\_kost\].
\[prop\_degreeK\]The Kostka-Foulkes polynomial $K_{\lambda},_{\mu}(q)$ is monic of degree
- $\sum_{i=1}^{n}i(\lambda\overline{_{i}}-\mu\overline{_{i}})$ for the root system $B_{n}$
- $\sum_{i=1}^{n}i(\lambda\overline{_{i}}-\mu\overline{_{i}})-\dfrac{1}{2}(\left| \lambda\right| -\left| \mu\right| )$ for the root system $C_{n}$
- $\sum_{i=2}^{n}(i-1)(\lambda\overline{_{i}}-\mu\overline{_{i}})$ for the root system $D_{n}$
It is similar to that given in Example 4 page 243 of [@mac] for the degree of Kostka-Foulkes polynomials associated to the root system $A_{n}.$
**Remarks:**
$\mathrm{(i):}$ The above proposition suffices to determinate $K_{\lambda,\mu}(q)$ when $\dim V(\lambda)_{\mu}=1.$ In particular we have $K_{\lambda,\mu}(q)=1$ for each minuscule representation $V(\lambda).$
$\mathrm{(ii):}$ If $\left| \lambda\right| =\left| \mu\right| $ then $K_{\lambda,\mu}^{B_{n}}(q)=K_{\lambda,\mu}^{C_{n}}(q)=K_{\lambda,\mu
}^{D_{n}}(q)=K_{\lambda,\mu}^{A_{n-1}}(q).$
$\mathrm{(iii):}$ Suppose $\lambda,\mu\in P_{D_{n}}^{+}$. Set $\lambda^{\ast}=(\lambda_{\overline{n}},...,\lambda_{\overline{2}},-\lambda_{\overline{1}})$ and $\mu^{\ast}=(\mu_{\overline{n}},...,\mu
_{\overline{2}},-\mu_{\overline{1}})$ then $$K_{\lambda,\mu}^{D_{n}}(q)=K_{\lambda^{\ast},\mu^{\ast}}^{D_{n}}(q)
\label{K=K*}$$ This is due to the symmetric role played by the simple roots $\alpha_{0}$ and $\alpha\overline{_{1}}$ in the root system $D_{n}.$ Moreover when $\lambda_{\overline{1}}=0,$ $\lambda=\lambda^{\ast}$ thus $K_{\lambda,\mu
}^{D_{n}}(q)=K_{\lambda^{\ast},\mu^{\ast}}^{D_{n}}(q)=K_{\lambda^{\ast},\mu
}^{D_{n}}(q)=K_{\lambda,\mu^{\ast}}^{D_{n}}(q).$
$\mathrm{(iv):}$ Consider $\lambda,\mu\in P_{n}^{+}$ such that $\lambda_{\overline{n}}=\mu_{\overline{n}}.$ Then $K_{\lambda,\mu
}(q)=K_{\lambda^{\prime},\mu^{\prime}}(q).$
Convention for crystal graphs
-----------------------------
In the sequel $g$ is any of the Lie algebras $so_{2n+1},sp_{2n}$ or $so_{2n}$. The crystal graphs for the $U_{q}(g)$-modules are oriented colored graphs with colors $i\in\{0,...,n-1\}$. An arrow $a\overset{i}{\rightarrow}b$ means that $\widetilde{f}_{i}(a)=b$ and $\widetilde{e}_{i}(b)=a$ where $\widetilde{e}_{i}$ and $\widetilde{f}_{i}$ are the crystal graph operators (for a review of crystal bases and crystal graphs see [@Ka2]). A vertex $v^{0}\in B$ satisfying $\widetilde{e}_{i}(v^{0})=0$ for any $i\in\{0,...,n-1\}$ is called a highest weight vertex. The decomposition of $V$ into its irreducible components is reflected into the decomposition of $B$ into its connected components. Each connected component of $B$ contains a unique highest weight vertex. The crystals graphs of two isomorphic irreducible components are isomorphic as oriented colored graphs. The action of $\widetilde{e}_{i}$ and $\widetilde{f}_{i}$ on $B\otimes B^{\prime}=\{b\otimes b^{\prime};$ $b\in
B,b^{\prime}\in B^{\prime}\}$ is given by: $$\begin{aligned}
\widetilde{f_{i}}(u\otimes v) & =\left\{
\begin{tabular}
[c]{c}$\widetilde{f}_{i}(u)\otimes v$ if $\varphi_{i}(u)>\varepsilon_{i}(v)$\\
$u\otimes\widetilde{f}_{i}(v)$ if $\varphi_{i}(u)\leq\varepsilon_{i}(v)$\end{tabular}
\right. \label{TENS1}\\
& \text{and}\nonumber\\
\widetilde{e_{i}}(u\otimes v) & =\left\{
\begin{tabular}
[c]{c}$u\otimes\widetilde{e_{i}}(v)$ if $\varphi_{i}(u)<\varepsilon_{i}(v)$\\
$\widetilde{e_{i}}(u)\otimes v$ if $\varphi_{i}(u)\geq\varepsilon_{i}(v)$\end{tabular}
\right. \label{TENS2}$$ where $\varepsilon_{i}(u)=\max\{k;\widetilde{e}_{i}^{k}(u)\neq0\}$ and $\varphi_{i}(u)=\max\{k;\widetilde{f}_{i}^{k}(u)\neq0\}$. The weight of the vertex $u$ is defined by $\mathrm{wt}(u)=\underset{i=0}{\overset{n-1}{\sum}}(\varphi_{i}(u)-\varepsilon_{i}(u))\Lambda_{i}$.
The following lemma is a straightforward consequence of (\[TENS1\]) and (\[TENS2\]).
\[lem\_plu\_hp\]Let $u\otimes v$ $\in$ $B\otimes B^{\prime}$ $u\otimes v$ is a highest weight vertex of $B\otimes B^{\prime}$ if and only if for any $i\in\{0,...,n-1\}$ $\widetilde{e}_{i}(u)=0$ (i.e. $u$ is of highest weight) and $\varepsilon_{i}(v)\leq\varphi_{i}(u).$
The Weyl group $W$ acts on $B$ by: $$\begin{aligned}
s_{i}(u) & =(\widetilde{f_{i}})^{\varphi_{i}(u)-\varepsilon_{i}(u)}(u)\text{
if }\varphi_{i}(u)-\varepsilon_{i}(u)\geq0,\label{actionW}\\
s_{i}(u) & =(\widetilde{e_{i}})^{\varepsilon_{i}(u)-\varphi_{i}(u)}(u)\text{
if }\varphi_{i}(u)-\varepsilon_{i}(u)<0.\nonumber\end{aligned}$$ We have the equality $\mathrm{wt}(\sigma(u))=\sigma(\mathrm{wt}(u))$ for any $\sigma\in W$ and $u\in B.$ For any $\lambda\in P^{+},$ we denote by $B(\lambda)$ the crystal graph of $V(\lambda).$
Kashiwara-Nakashima’s tableaux
------------------------------
Accordingly to (\[DD\]) the crystal graphs of the vector representations are:$$\begin{gathered}
B(\Lambda_{n-1}^{B}):\overline{n}\overset{n-1}{\rightarrow}\overline
{n-1}\overset{n-2}{\rightarrow}\cdot\cdot\cdot\cdot\rightarrow\overline
{2}\overset{1}{\rightarrow}\overline{1}\overset{0}{\rightarrow}0\overset
{0}{\rightarrow}1\overset{1}{\rightarrow}2\cdot\cdot\cdot\cdot\overset
{n-2}{\rightarrow}n-1\overset{n-1}{\rightarrow}n\\
B(\Lambda_{n-1}^{C}):\overline{n}\overset{n-1}{\rightarrow}\overline
{n-1}\overset{n-2}{\rightarrow}\cdot\cdot\cdot\cdot\rightarrow\overline
{2}\overset{1}{\rightarrow}\overline{1}\overset{0}{\rightarrow}1\overset
{1}{\rightarrow}2\cdot\cdot\cdot\cdot\overset{n-2}{\rightarrow}n-1\overset
{n-1}{\rightarrow}n\\
B(\Lambda_{n-1}^{D}):\overline{n}\overset{n-1}{\rightarrow}\overline
{n-1}\overset{n-2}{\rightarrow}\cdot\cdot\cdot\overset{3}{\rightarrow
}\overline{3}\overset{2}{\rightarrow}\begin{tabular}
[c]{c}$1$ \ \ \\
\ \ $\overset{0}{\nearrow}$ $\ \ \ \overset{\text{ \ \ \ }1}{\text{
\ }\searrow}$ \ \ \ \\
$\overline{2}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2$\\
\ $\underset{1\text{ \ \ \ }}{\searrow}$ \ \ \ $\underset{0}{\nearrow}$
\ \ \ \\
$\overline{1}$ \
\end{tabular}
\overset{2}{\rightarrow}3\overset{3}{\rightarrow}\cdot\cdot\cdot\overset
{n-2}{\rightarrow}n-1\overset{n-1}{\rightarrow}n.\end{gathered}$$ Kashiwara-Nakashima’s combinatorial description of the crystal graphs $B(\lambda)$ is based on a notion of tableaux analogous for each root system $B_{n},C_{n}$ or $D_{n}$ to semi-standard tableaux.
We define an order on the vertices of the above crystal graphs by setting$$\begin{gathered}
\mathcal{A}_{n}^{B}=\{\overline{n}<\cdot\cdot\cdot<\overline{1}<0<1<\cdot
\cdot\cdot<n\}\\
\mathcal{A}_{n}^{C}=\{\overline{n}<\cdot\cdot\cdot<\overline{1}<1<\cdot
\cdot\cdot<n\}\text{ and }\\
\mathcal{A}_{n}^{D}=\{\overline{n}<\cdot\cdot\cdot<\overline{2}<\begin{tabular}
[c]{l}$1$\\
$\overline{1}$\end{tabular}
<2<\cdot\cdot\cdot<n\}.\end{gathered}$$ Note that $\mathcal{A}_{n}^{D}$ is only partially ordered. For any letter $x$ we set $\overline{\overline{x}}=x.$ Our convention for labelling the crystal graph of the vector representations are not those used by Kashiwara and Nakashima. To obtain the original description of $B(\lambda)$ from that used in the sequel it suffices to change each letter $k\in\{1,...,n\}$ into $\overline{n-k+1}$ and each letter $\overline{k}\in\{\overline{1},...,\overline{n}\}$ into $n-k+1.\;$The interest of this change of convention is to yield a natural extension of the above alphabets$.$
For types $B_{n},C_{n}$ and $D_{n},$ we identify the vertices of the crystal graph $G_{n}^{B}=\underset{l}{{\textstyle\bigoplus}
}B(\Lambda_{n-1}^{B})^{\bigotimes l},G_{n}^{C}=\underset{l}{{\textstyle\bigoplus}
}B(\Lambda_{n-1}^{C})^{\bigotimes l}$ and $G_{n}^{D}=\underset{l}{{\textstyle\bigoplus}
}B(\Lambda_{n-1}^{D})^{\bigotimes l}$ respectively with the words on $\mathcal{A}_{n}^{B},\mathcal{A}_{n}^{C}$ and $\mathcal{A}_{n}^{D}$.$\;$For any $w\in G_{n}$ we have $\mathrm{wt}(w)=d_{\overline{n}}\varepsilon
_{\overline{n}}+d_{\overline{n-1}}\varepsilon_{\overline{n-1}}\cdot\cdot
\cdot+d_{\overline{1}}\varepsilon_{\overline{1}}$ where for all $i=1,...,n$ $d_{\overline{i}}$ is the number of letters $\overline{i}$ of $w$ minus its number of letters $i.$
Consider $\lambda$ a generalized partition with nonnegative parts. Suppose first that $\lambda$ is a partition. Write $T_{\lambda}$ for the filling of the Young diagram of shape $\lambda$ whose $k$-th row contains only letters $\overline{n-k+1}.$ Let $b_{\lambda}$ be the vertex of $B(\Lambda
_{n-1})^{\bigotimes\left| \lambda\right| }$ obtained by column reading $T_{\lambda}$ from right to left and top to bottom$.$ Kashiwara and Nakashima realize $B(\lambda)$ as the connected component of the tensor power $B(\Lambda_{n-1})^{\bigotimes\left| \lambda\right| }$ of highest weight vertex $b_{\lambda}$. For each roots system $B_{n},C_{n}$ and $D_{n},$ the Kashiwara-Nakashima tableaux of type $B_{n},C_{n}$, $D_{n}$ and shape $\lambda$ are defined as the tableaux whose column readings are the vertices of $B(\lambda)$. We will denote by $\mathrm{w}(T)$ the column reading of the tableau $T.$
Now suppose that $\lambda$ belongs to $P_{+}^{B_{n}}$ or $P_{+}^{D_{n}}$ and its parts are half nonnegative integers. In this case we can write $\lambda=\lambda{{}^\circ}+(1/2,...,1/2)$ with $\lambda{{}^\circ}$ a partition and $B(\lambda)$ can be realized as the connected component of the crystal graph $\frak{G}_{n}^{0}=B(\Lambda_{n-1})^{\bigotimes\left|
\lambda{{}^\circ}\right| }\otimes B(\Lambda_{0})$ of highest weight vertex $b_{\lambda
}=b_{\lambda{{}^\circ}}\otimes b_{\Lambda_{0}}$ where $b_{\Lambda_{0}}$ is the highest weight vertex of $B(\Lambda_{0})$ the crystal graph of the spin representation $V(\Lambda_{0})$ of the corresponding Lie algebra$.$ The vertices of $B(\Lambda_{0})$ are labelled by spin columns which are special column shaped diagrams of width $1/2$ and height $n.$ Then the vertices of $B(\lambda)$ can be identified with the column readings of Kashiwara-Nakashima’s spin tableaux of types $B_{n},D_{n}$ and shape $\lambda$ obtained by adding a column shape diagram of width $1/2$ to the Young diagram associated to $\lambda{{}^\circ}.$
Finally suppose that $\lambda$ belongs to $P_{+}^{D_{n}}$ and verifies $\lambda_{\overline{1}}<0.$ The above description of $B(\lambda)$ remain valuable up to the following minor modifications. If the parts of $\lambda$ are integers the letters $\overline{1}$ must be changed into letters $1$ in the above definition of $T_{\lambda}.$ Otherwise we set $\lambda
=\lambda{{}^\circ}+(1/2,...,1/2,-1/2)$ where $\lambda{{}^\circ}$ is generalized partition with integer parts. Then $B(\lambda)$ is realized as the connected component of the crystal graph $\frak{G}_{n}^{1}=B(\Lambda_{n-1})^{\bigotimes\left| \lambda{{}^\circ}\right| }\otimes B(\Lambda_{1}^{D_{n}})$ of highest weight vertex $b_{\lambda}=b_{\lambda^{{{}^\circ}}}\otimes b_{\Lambda_{1}^{D_{n}}}$ where $b_{\Lambda_{1}^{D_{n}}}$ is the highest weight vertex of $B(\Lambda_{1}^{D_{n}})$ the crystal graph of the spin representation $V(\Lambda_{1}^{D_{n}})$.
For any generalized partition $\lambda$ of length $n,$ write $\mathbf{T}^{B_{n}}(\lambda),$ $\mathbf{T}^{C_{n}}(\lambda)$ and $\mathbf{T}^{D_{n}}(\lambda)$ respectively for the sets of Kashiwara-Nakashima’s tableaux of shape $\lambda$. Set $\mathbf{T}^{B_{n}}=\underset{\lambda\in P_{B_{n}}^{+}}{\cup}\mathbf{T}^{B_{n}}(\lambda),$ $\mathbf{T}^{C_{n}}=\underset{\lambda\in
P_{C_{n}}^{+}}{\cup}\mathbf{T}^{C_{n}}(\lambda)$ and $\mathbf{T}^{D_{n}}=\underset{\lambda\in P_{D_{n}}^{+}}{\cup}\mathbf{T}^{D_{n}}(\lambda).$ In the sequel we only summarize the combinatorial description of the partition shaped tableaux that is, tableaux of $\mathbf{T}^{n}(\lambda)$ where the parts of $\lambda$ are integers (with eventually $\lambda_{\overline{1}}<0$ for the root system $D_{n}).$ We refer the reader to [@Ba], [@KN], [@Lec] and [@lec2] for the complete description of $\mathbf{T}^{n}(\lambda)$ which necessitates a large amount of combinatorial definitions especially when the parts of $\lambda$ are half integers.
So consider $\lambda$ a generalized partition with integer parts. Suppose first that $\lambda_{\overline{n}}=1.$ Then the tableaux of $\mathbf{T}^{n}(\lambda)$ are called the $n$-admissible columns. The $n$-admissible columns of types $B_{n},C_{n}$ and $D_{n}$ are in particular columns of types $B_{n},C_{n}$ and $D_{n}$ that is have the form$$C=\begin{tabular}
[c]{|l|}\hline
$C_{-}$\\\hline
$C_{0}$\\\hline
$C_{+}$\\\hline
\end{tabular}
,C=\begin{tabular}
[c]{|l|}\hline
$C_{-}$\\\hline
$C_{+}$\\\hline
\end{tabular}
\text{ and }C=\begin{tabular}
[c]{|c|}\hline
$D_{-}$\\\hline
$D$\\\hline
$D_{+}$\\\hline
\end{tabular}
\label{col}$$ where $C_{-},C_{+},C_{0},D_{-},D_{+}$ and $D$ are column shaped Young diagrams such that$$\left\{
\begin{tabular}
[c]{l}$C_{-}$ is filled by strictly increasing barred letters from top to bottom\\
$C_{+}$ is filled by strictly increasing unbarred letters from top to bottom\\
$C_{0}$ is filled by letters $0$\\
$D_{-}$ is filled by strictly increasing letters $\leq\overline{2}$ from top
to bottom\\
$D_{+}$ is filled by strictly increasing letters $\geq2$ from top to bottom\\
$D$ is filled by letters $\overline{1}$ or $1$ with differents letters in two
adjacent boxes
\end{tabular}
\right. .$$ Note that all the columns are not $n$-admissible even if their letters $a$ satisfy $\overline{n}\leq a\leq n.$ More precisely a column $C$ of (\[col\]) is $n$-admissible if and only if it can be duplicated following a simple algorithm described in [@lec2] into a pair $(lC,rC)$ of columns without pair of opposite letters $(x,\overline{x})$ (the letter $0$ is counted as the pair $(0,\overline{0})$) and containing only letters $a$ such that $\overline{n}\leq a\leq n.$
For the column $C=\begin{tabular}
[c]{|l|}\hline
$$\\\hline
$$\\\hline
$$\\\hline
$$\\\hline
$$\\\hline
\end{tabular}
$ of type $B$ we have $lC=$
\[c\][|l|]{}$\mathtt{\bar{5}}$\
$\mathtt{\bar{4}}$\
$\mathtt{\bar{3}}$\
$\mathtt{1}$\
$\mathtt{2}$\
and $rC=$
\[c\][|l|]{}$\mathtt{\bar{3}}$\
$\mathtt{\bar{1}}$\
$\mathtt{3}$\
$\mathtt{4}$\
$\mathtt{5}$\
. Hence $C$ is $5$-admissible but not $n$-admissible for $n\leq4.$
Now for a general $\lambda$ with integer parts, a tableau $T\in\mathbf{T}^{n}(\lambda)$ can be regarded as a filling of the Young diagram of shape $\lambda$ if $\lambda_{\overline{1}}\geq0$ (of shape $\lambda^{\ast}$ otherwise) such that
- $T=C_{1}\cdot\cdot\cdot C_{r}$ where the columns $C_{i}$ of $T$ are $n$-admissible,
- for any $i\in\{1,...r-1\}$ the columns of the tableau $r(C_{i})l(C_{i+1})$ weakly increase from left to right and do not contain special configurations (detailed in [@KN] and [@lec2]) when $T$ is of type $D_{n}.$
**Remark:** We have $\mathbf{T}^{n}(\lambda)\subset
\mathbf{T}^{n+1}(\lambda^{\#})$ where $\lambda^{\#}=(\lambda_{\overline{n}},...,\lambda_{\overline{1}},0)$ since the $n$-admissible columns are also $(n+1)$-admissible and the duplication process of a column does not depend on $n.$ To simplify the notation we will write in the sequel $\mathbf{T}^{n+1}(\lambda)$ instead of $\mathbf{T}^{n+1}(\lambda^{\#})$ for any $\lambda\in P_{n}^{+}.$
Insertion schemes and plactic monoids
-------------------------------------
There exist insertion schemes related to each classical root system [@Ba], [@Lec] and [@lec2] analogous for Kashiwara-Nakashima’s tableaux to the well known bumping algorithm on semi-standard tableaux.
Denote by $\sim_{n}^{B},\sim_{n}^{C}$ and $\sim_{n}^{D}$ the equivalence relations defining on the vertices of $G_{n}^{B},G_{n}^{C}$ and $G_{n}^{D}$ by $w_{1}\sim_{n}w_{2}$ if and only if $w_{1}$ and $w_{2}$ belong to the same connected component of $G_{n}.\;$For any word $w,$ the insertions schemes permit to compute the unique tableau $P_{n}(w)$ such that $w\sim
_{n}\mathrm{w}(P_{n}(w))$. In fact $\sim_{n}^{B},\sim_{n}^{C}$ and $\sim
_{n}^{D}$ are congruencies $\equiv_{n}^{B},\equiv_{n}^{C}$ and $\equiv_{n}^{D}$ [@Ba] [@Lec] [@lec2] [@lit] obtained respectively as the quotient of the free monoids of words on $\mathcal{A}_{n}^{C},\mathcal{A}_{n}^{B}$ and $\mathcal{A}_{n}^{D}$ by two kinds of relations.
The first is constituted by relations of length $3$ analogous to Knuth relations defining Lascoux-Schützenberger’s plactic monoid. In fact these relations are precisely those which are needed to describe the insertion $x\rightarrow C$ of a letter $x$ in a $n$-admissible column $C=\begin{tabular}
[c]{|l|}\hline
$a$\\\hline
$b$\\\hline
\end{tabular}
$ such that
\[c\][|l|]{}$a$\
$b$\
$x$\
is not a column. This can be written$$x\rightarrow\begin{tabular}
[c]{|l|}\hline
$a$\\\hline
$b$\\\hline
\end{tabular}
=\begin{tabular}
[c]{c|c|}\cline{2-2}& $a$\\\hline
\multicolumn{1}{|c|}{$x$} & $b$\\\hline
\end{tabular}
=\begin{tabular}
[c]{|l|l}\hline
$a^{\prime}$ & \multicolumn{1}{|l|}{$x^{\prime}$}\\\hline
$b^{\prime}$ & \\\cline{1-1}\end{tabular}
\label{trans_ele}$$ and contrary to the insertion scheme for the semi-standard tableaux the sets $\{a^{\prime},b^{\prime},x^{\prime}\}$ and $\{a,b,c\}$ are not necessarily equal (i.e. the relations are not homogeneous in general).
Next we have the contraction relations which do not preserve the length of the words. These relations are precisely those which are needed to describe the insertion $x\rightarrow C$ of a letter $x$ such that $\overline{n}\leq x\leq n$ in a $n$-admissible column $C$ such that
\[c\][|l|]{}$C$\
$x$\
(obtained by adding the letter $x$ on bottom of $C)$ is a column which is not $n$-admissible. In this case
\[c\][|l|]{}$C$\
$x$\
is necessarily $(n+1)$-admissible and have to be contracted to give a $n$-admissible column. We obtain $x\rightarrow C=\widetilde{C}$ with $\widetilde{C}$ a $n$-admissible column of height $h(C)$ or $h(C)-1.$
The insertion of the letter $x$ in a $n$-admissible column $C$ of arbitrary height such that
\[c\][|l|]{}$C$\
$x$\
is not a column can then be pictured by$$x\rightarrow\begin{tabular}
[c]{|c|}\hline
$a_{1}$\\\hline
$\cdot$\\\hline
$a_{k-2}$\\\hline
$a_{k-1}$\\\hline
$a_{k}$\\\hline
\end{tabular}
=\begin{tabular}
[c]{c|c|}\cline{2-2}\ \ \ \ \ & $a_{1}$\\\cline{2-2}& $\cdot$\\\cline{2-2}& $a_{k-2}$\\\cline{2-2}& $a_{k-1}$\\\hline
\multicolumn{1}{|c|}{$x$} & $a_{k}$\\\hline
\end{tabular}
=\begin{tabular}
[c]{c|c}\cline{2-2}& \multicolumn{1}{|c|}{$a_{1}$}\\\cline{2-2}& \multicolumn{1}{|c|}{$\cdot$}\\\cline{2-2}& \multicolumn{1}{|c|}{$a_{k-2}$}\\\hline
\multicolumn{1}{|c|}{$\delta_{k-1}$} & \multicolumn{1}{|c|}{$y$}\\\hline
\multicolumn{1}{|c|}{$d_{k}$} & \\\cline{1-1}\end{tabular}
=\cdot\cdot\cdot=\begin{tabular}
[c]{|c|c}\hline
$d_{1}$ & \multicolumn{1}{|c|}{$\ \ z$ \ \ }\\\hline
$\cdot$ & \\\cline{1-1}$\cdot$ & \\\cline{1-1}$d_{k-1}$ & \\\cline{1-1}$d_{k}$ & \\\cline{1-1}\end{tabular}$$ that is, one elementary transformation (\[trans\_ele\]) is applied to each step. One proves that $x\rightarrow C$ is then a tableau of $\mathbf{T}^{(n)}$ with two columns respectively of height $h(C)$ and $1$.
Now we can define the insertion $x\rightarrow T$ of the letter $x$ such that $\overline{n}\leq x\leq n$ in the tableau $T\in\mathbf{T}^{n}(\lambda)$. Set $T=C_{1}\cdot\cdot\cdot C_{r}$ where $C_{i},$ $i=1,...,r$ are the $n$-admissible columns of $T.\;$
1. When
\[c\][|l|]{}$C_{1}$\
$x$\
is not a column, write $x\rightarrow C=\begin{tabular}
[c]{|l|l|}\hline
$C\_[1]{}\^$ & $y$\\\hline
\end{tabular}
$ where $C_{1}^{\prime}$ is an admissible column of height $h(C_{1})$ and $y$ a letter. Then $x\rightarrow T=C_{1}^{\prime}(y\rightarrow C_{2}\cdot
\cdot\cdot C_{r})$ that is, $x\rightarrow T$ is the juxtaposition of $C_{1}^{\prime}$ with the tableau $\widehat{T}$ obtained by inserting $y$ in the tableau $C_{2}\cdot\cdot\cdot C_{r}.$
2. When
\[c\][|l|]{}$C_{1}$\
$x$\
is a $n$-admissible column, $x\rightarrow T$ is the tableau obtained by adding a box containing $x$ on bottom of $C_{1}$.
3. When
\[c\][|l|]{}$C_{1}$\
$x$\
is a column which is not $n$-admissible, write $x\rightarrow C=\widetilde{C}$ and set $\mathrm{w}(\widetilde{C})=y_{1}\cdot\cdot\cdot y_{s}$ where the $y_{i}$’s are letters. Then $x\rightarrow T=y_{s}\rightarrow(y_{s-1}\rightarrow(\cdot\cdot\cdot y_{1}\rightarrow\widehat{T}))$ that is $x\rightarrow T$ is obtained by inserting successively the letters of $\widetilde{C}$ into the tableau $\widehat{T}=C_{2}\cdot\cdot\cdot C_{r}$. Note that there is no new contraction during this $s$ insertions.
**Remarks:**
$\mathrm{(i)}\mathbf{:}$ The $P_{n}$-symbol defined above can be computed recursively by setting $P_{n}(w)=\begin{tabular}
[c]{|l|}\hline
$w$\\\hline
\end{tabular}
$ if $w$ is a letter and $P_{n}(w)=x\rightarrow P_{n}(u)$ where $w=ux$ with $u$ a word and $x$ a letter otherwise.
$\mathrm{(ii)}\mathbf{:}$ Consider $T\in\mathbf{T}^{n}(\lambda
)\subset\mathbf{T}^{n+1}(\lambda)$ and a letter $x$ such that $\overline
{n}\leq x\leq n.$ The tableau obtained by inserting $x$ in $T$ may depend wether $T$ is regarded as a tableau of $\mathbf{T}^{n}(\lambda)$ or as a tableau of $\mathbf{T}^{n+1}(\lambda)$. Indeed if
\[c\][|l|]{}$C_{1}$\
$x$\
is not $n$-admissible then it is necessarily $(n+1)$-admissible since $C_{1}$ is $n$-admissible. Hence there is no contraction during the insertion $x\rightarrow T$ when it is regarded as a tableau of $\mathbf{T}^{n+1}(\lambda).$
$\mathrm{(iii)}\mathbf{:}$ Consider $w\in\mathcal{A}_{n}$, from $\mathrm{(ii)}\mathbf{\ }$we deduce that there exists an integer $m\geq n$ minimal such that $P_{m}(w)$ can be computed without using contraction relation. Then for any $k\geq m,$ $P_{k}(w)=P_{m}(w).$
$\mathrm{(iv)}\mathbf{:}$ Similarly to the bumping algorithm for semi-standard tableaux, the insertion algorithms described above are reversible.
$\mathrm{(v)}\mathbf{:}$ There also exit insertion algorithms for the spin tableaux of types $B_{n}$ and $D_{n}$ [@lec2]. To make the paper more readable we only establish the combinatorial results contained in the sequel for the partition shaped tableaux. Nevertheless note that they can be extended to take also into account the spin tableaux associated to the root systems $B_{n}$ and $D_{n}.$
\[lem\_fact\_row\]Consider $\lambda,\mu\in P_{n}^{+}.$ Let $T\in
\mathbf{T}^{n}(\lambda).$ If $\lambda$ and $\mu$ have integer parts, then there exists a unique pair $(R,T^{\prime})$ such that$$\mathrm{w}(T)\equiv_{n}\mathrm{w}(R)\otimes\mathrm{w}(T^{\prime})$$ where $R\in\mathbf{T}^{n}(\lambda_{\overline{n}}\Lambda_{n-1})$ is a row tableau of length $\lambda_{\overline{n}}$ and $T^{\prime}\in\mathbf{T}^{n-1}(\lambda^{\prime})$ with $\lambda^{\prime}=(\lambda_{\overline{n-1}},...,\lambda_{\overline{1}}).$
When $\lambda_{\overline{1}}\geq0$ we have $$b_{\lambda}\equiv_{n}(\overline{n})^{\otimes\lambda_{\overline{n}}}\otimes\left( (\overline{1})^{\otimes\widehat{\lambda}_{\overline{1}}^{\prime}}\otimes(\overline{1}\ \overline{2})^{\otimes\widehat{\lambda
}_{\overline{2}}^{\prime}}\otimes\cdot\cdot\cdot\otimes(\overline
{1}\ \overline{2}\cdot\cdot\cdot\overline{n-1})^{\otimes\widehat{\lambda
}_{\overline{n-1}}^{\prime}}\right) \equiv_{n}b_{\lambda_{\overline{n}}\Lambda_{n-1}}\otimes b_{\lambda^{\prime}}$$ with the notation used in \[subsec\_KF\]. Indeed the plactic relations on words containing only barred letters coincide with Knuth relations. This implies the existence of the required pair $(R,T^{\prime})$. Now if $\mathrm{w}(T)\equiv_{n}\mathrm{w}(R)\otimes\mathrm{w}(T^{\prime}),$ we deduce from \[lem\_plu\_hp\] that the highest weight vertex of the connected component of $G_{n}$ containing $\mathrm{w}(R)\otimes\mathrm{w}(T^{\prime})$ is necessarily $b_{\lambda_{\overline{n}}\Lambda_{n-1}}\otimes b_{\lambda
^{\prime}}$. Thus the pair $(R,T^{\prime})$ is unique.
**Remark:** The pair $(R,T^{\prime})$ can be explicitly computed by using the reverse insertion schemes.
\[sec\_morris\]Morris type recurrence formulas for the orthogonal root systems
==============================================================================
In this section we introduce recurrence formulas for computing Kostka-Foulkes polynomials analogous for types $B_{n}$ and $D_{n}$ to Morris recurrence formula. They allow to explain the Kostka-Foulkes polynomials for types $B_{n}$ and $D_{n}$ respectively as combinations of Kostka-Foulkes polynomials for types $B_{n-1}$ and $D_{n-1}.$ We essentially proceed as we have done in [@lec3] for the root system $C_{n}$. So we only sketch the arguments except for Theorems \[Th\_rec\_morrB\] and \[Th\_rec\_morrD\] for which the proofs necessitate refinements of the proof of Theorem 3.2.1 of [@lec3].
We classically realize $so_{2n-1},sp_{2n-2}$ and $so_{2n-2}$ respectively as the sub-algebras of $so_{2n+1},sp_{2n}$ and $so_{2n}$ generated by the Chevalley operators $e_{i},f_{i}$ and $t_{i},$ $i=0,...n-2.$ The weight lattice $P_{n-1}$ of these algebras of rank $n-1$ is the $\mathbb{Z}$-lattice generated by the $\varepsilon_{\overline{i}},$ $i=1,...,n-1$ and $P_{n-1}^{+}=P_{n}^{+}\cap P_{n-1}$ is the set of dominant weights. The Weyl group $W_{n-1}$ is the sub-group of $W_{n}$ generated by the $s_{i},$ $i=0,...n-2$ and we have $R_{n-1}^{+}=R_{n}\cap P_{n-1}^{+}.$
Given any positive integer $r,$ set $B^{B_{n}}(r)=B(r\Lambda
_{n-1}^{B_{n}}),B^{C_{n}}(r)=B(r\Lambda_{n-1}^{C_{n}}),$ and $B^{D_{n}}(r)=B(r\Lambda_{n-1}^{D_{n}})$. To obtain our recurrence formulas we need to describe the decomposition $B(\gamma)\otimes B(r)$ with $\gamma\in P_{n}^{+}$ and $r\geq0$ an integer into its irreducible components. This is analogous for types $B_{n}$ and $D_{n}$ to Pieri rule.
Pieri rule for types $B_{n}$ and $D_{n}$
----------------------------------------
It follows from [@KN] that the vertices of $B^{B_{n}}(r),$ $B^{C_{n}}(r)$ and $B^{D_{n}}(r)$ can be respectively identified to the words $$L=(n)^{k_{n}}\cdot\cdot\cdot(2)^{k_{2}}(1)^{k_{1}}(\overline{1})^{k_{\bar{1}}}(\overline{2})^{k_{\bar{2}}}\cdot\cdot\cdot(\overline{n})^{k_{\bar{n}}}\text{, }L=(n)^{k_{n}}\cdot\cdot\cdot(2)^{k_{2}}(1)^{k_{1}}(0)(\overline
{1})^{k_{\bar{1}}}(\overline{2})^{k_{\bar{2}}}\cdot\cdot\cdot(\overline
{n})^{k_{\bar{n}}} \label{LB}$$$$L=(n)^{k_{n}}\cdot\cdot\cdot(2)^{k_{2}}(1)^{k_{1}}(\overline{1})^{k_{\bar{1}}}(\overline{2})^{k_{\bar{2}}}\cdot\cdot\cdot(\overline{n})^{k_{\bar{n}}}
\label{LC}$$ and $$L=(n)^{k_{n}}\cdot\cdot\cdot(2)^{k_{2}}(\overline{1})^{k_{\overline{1}}}(\overline{2})^{k_{\bar{2}}}\cdot\cdot\cdot(\overline{n})^{k_{\bar{n}}}\text{, }L=(n)^{k_{n}}\cdot\cdot\cdot(2)^{k_{2}}(1)^{k_{1}}(\overline
{2})^{k_{\bar{2}}}\cdot\cdot\cdot(\overline{n})^{k_{\bar{n}}} \label{LD}$$ of length $r$ where $k_{\overline{i}},k_{i}$ are positive integers, $(x)^{k}$ means that the letter $x$ is repeated $k$ times in $L.$ Note that there can be only one letter $0$ in the vertices of $B^{B_{n}}(r)$ and the letters $\overline{1}$ and $1$ can not appear simultaneously in the vertices of $B^{D_{n}}(r).$
Let $\gamma=(\gamma_{\overline{n}},...,\gamma_{\overline{1}})\in P_{n}^{+}.$ When $\gamma\in P_{B_{n}}^{+}$ set $B(\gamma)\otimes B^{B_{n}}(r)=\underset
{\lambda\in P_{B_{n}}^{+}}{\cup}B(\lambda)^{b_{\gamma,r}^{\lambda}}$ that is $b_{\gamma,r}^{\lambda}$ is the multiplicity of $V(\lambda)$ in $V(\gamma
)\otimes V(r\Lambda_{n-1}^{B}).$ Similarly set $B(\gamma)\otimes B^{C_{n}}(r)=\underset{\lambda\in P_{C_{n}}^{+}}{\cup}B(\lambda)^{c_{\gamma
,r}^{\lambda}}$ and $B(\gamma)\otimes B^{D_{n}}(r)=\underset{\lambda\in
P_{D_{n}}^{+}}{\cup}B(\lambda)^{d_{\gamma,r}^{\lambda}}$ when $\gamma$ belongs respectively to $P_{C_{n}}^{+}$ and $P_{D_{n}}^{+}.$
Write $b_{\gamma}$ for the highest weight vertex of $B(\gamma).$ The two following lemmas and their corollaries are consequences of Lemma \[lem\_plu\_hp\].
\[lem\_b\_gamm\_tens\_Lb\]$b_{\gamma}\otimes L$ is a highest weight vertex of $B(\gamma)\otimes B^{B_{n}}(r)$ if and only if the following conditions holds:
$\mathrm{(i):}$ $\gamma_{\overline{1}}-k_{1}\geq0$ if $k_{0}=0$, $\gamma\overline{_{1}}-k_{1}>0$ otherwise
$\mathrm{(ii):}$ $\gamma_{\overline{i+1}}-k_{i+1}\geq\gamma
_{\overline{i}}$ for $i=1,...,n-1$
$\mathrm{(iii):}$ $\gamma_{\overline{i}}-k_{i}+k_{\overline{i}}\leq\gamma_{\overline{i+1}}-k_{i+1}$ for $i=1,...,n-1$
\[cor\_pieriB\]The multiplicity $b_{\gamma,r}^{\lambda}$ is the number of vertices $L\in B^{B_{n}}(r)$ such that $k_{\overline{i}}-k_{i}=\lambda
_{\overline{i}}-\gamma_{\overline{i}}$ for $i=1,....,n$ and
$\mathrm{(i)}:$ $\lambda_{\overline{i}}\leq\lambda_{\overline{i+1}}-k_{\overline{i+1}}$ for $i=1,...,n-1$,
$\mathrm{(ii)}:$ $\lambda_{\overline{i+1}}-k_{\overline{i+1}}\geq\lambda_{\overline{i}}+k_{i}-k_{\overline{i}}$ for $i=1,...,n-1$,
$\mathrm{(iii)}:$ $\lambda_{\overline{1}}-k_{\overline{1}}\geq0$ if $k_{0}=0$ (i.e. $k_{\overline{1}}+\cdot\cdot\cdot+k_{\overline{n}}+k_{1}+\cdot\cdot\cdot+k_{n}=r)$ and $\lambda_{\overline{1}}-k_{\overline{1}}>0$ otherwise (i.e. $k_{\overline{1}}+\cdot\cdot\cdot+k_{\overline{n}}+k_{1}+\cdot\cdot\cdot+k_{n}=r-1$).
\[lem\_b\_gamm\_tens\_Ld\]$b_{\gamma}\otimes L$ is a highest weight vertex of $B(\gamma)\otimes B^{D_{n}}(r)$ if and only if the following conditions holds:
$\mathrm{(i):}$ $\gamma_{\overline{2}}-k_{2}\geq\gamma_{\overline{1}}$ if $\gamma_{\overline{1}}\geq0$ and $\gamma_{\overline{2}}-k_{2}\geq
-\gamma_{\overline{1}}$ otherwise,
$\mathrm{(ii):}$ $\gamma_{\overline{i+1}}-k_{i+1}\geq\gamma
_{\overline{i}}$ for $i=2,...,n-1$,
$\mathrm{(iii):}\gamma_{\overline{1}}+k_{\overline{1}}\leq
\gamma_{\overline{2}}-k_{2}$ if $k_{1}=0$ and $-\gamma_{\overline{1}}+k_{1}\leq\gamma_{\overline{2}}-k_{2}$ otherwise,
$\mathrm{(iii):}$ $\gamma_{\overline{i}}-k_{i}+k_{\overline{i}}\leq\gamma_{\overline{i+1}}-k_{i+1}$ for $i=2,...,n-1$
\[cor\_pieriD\]The multiplicity $d_{\gamma,r}^{\lambda}$ is the number of vertices $L\in\otimes B^{D_{n}}(r)$ such that $k_{\overline{i}}-k_{i}=\lambda_{\overline{i}}-\gamma_{\overline{i}}$ for $i=1,....,n$, and
$\mathrm{(i)}:$ $\lambda_{\overline{1}}\leq\lambda_{\overline{2}}-k_{\overline{2}}$ if $k_{1}=0$ and $-\lambda_{\overline{1}}\leq
\lambda_{\overline{2}}-k_{\overline{2}}$ otherwise
$\mathrm{(ii)}:\lambda_{\overline{i}}\leq\lambda_{\overline{i+1}}-k_{\overline{i+1}}$ for $i=2,...,n-1$,
$\mathrm{(iii)}:$ $\lambda_{\overline{i+1}}-k_{\overline{i+1}}\geq\lambda_{\overline{i}}+k_{i}-k_{\overline{i}}$ for $i=2,...,n-1$,
$\mathrm{(iv)}:$ $\left\{
\begin{tabular}
[c]{l}$(a):\_-k\_\_-k\_$ if $k\_[1]{}=0$ and $\_0$\\
$(b):\_-k\_\_+k\_[1]{}$
if $k\_=0$ and $\_0$\end{tabular}
\right\} $, $\left\{
\begin{tabular}
[c]{l}$(c):\_-k\_-\_-k\_[1]{}$
if $k\_=0$ and $\_<0$\\
$(d):\_-k\_-\_+k\_$ if $k\_[1]{}=0$ and $\_<0$\end{tabular}
\right\} .$
**Remarks:**
$\mathrm{(i):}$ **** In the above corollaries, $b_{\gamma
,r}^{\lambda}$ and $d_{\gamma,r}^{\lambda}$ are the number of ways of starting with $\gamma,$ removing a horizontal strip to obtain a partition $\nu$ (corresponding to the unbarred letters of $L)$ and then adding a horizontal strip (corresponding to the barred letters of $L$) to obtain $\lambda.$
$\mathrm{(ii):}$ $B(\gamma)\otimes B((r)_{n})$ is not multiplicity free in general.
$\mathrm{(iii):}$ Consider $\gamma=(\gamma_{\overline{n}},...,\gamma_{\overline{1}})\in P_{B_{n}}$ (resp. $P_{D_{n}})$ such that $\lambda=(\lambda_{\overline{n}},...,\lambda_{\overline{1}})\in P_{B_{n}}$ (resp. $P_{D_{n}})$ defined by $\lambda_{\overline{i}}=\gamma_{\overline{i}}+k_{\overline{i}}-k_{\overline{i}},$ $i=1,...,n$ verifies conditions $\mathrm{(i),(ii)}$ and $\mathrm{(iii)}$ of Corollary \[cor\_pieriB\] (resp. \[cor\_pieriD\]). Then $\gamma\in P_{B_{n}}^{+}$ (resp. $P_{D_{n}}^{+})$ that is $\gamma$ is a generalized partition.
Recurrence formulas
-------------------
Consider $\gamma\in P_{n}^{+}$ and $r$ a positive integer. We set$$\begin{gathered}
\left( \gamma\otimes r\right) _{B_{n}}=\{\lambda\in P_{B_{n}}^{+},\text{
}b_{\gamma,r}^{\lambda}\neq0\},\left( \gamma\otimes r\right) _{C_{n}}=\{\lambda\in P_{C_{n}}^{+},\text{ }c_{\gamma,r}^{\lambda}\neq0\}\\
\text{and }\left( \gamma\otimes r\right) _{D_{n}}=\{\lambda\in P_{D_{n}}^{+},\text{ }d_{\gamma,r}^{\lambda}\neq0\}.\end{gathered}$$
For the root system $C_{n}$ and $\mu=(\mu_{\overline{n}},...,\mu_{\overline{1}})$, we have established in [@lec3] the following analogue of Morris recurrence formula:$$Q_{\mu}^{C_{n}}=\sum_{\gamma\in P_{C_{n-1}}^{+}}\sum_{R=0}^{+\infty}\sum_{r+2m=R}q^{m+r}\sum_{\lambda\in\left( \gamma\otimes r\right) _{C_{n-1}}}c_{\gamma,r}^{\lambda}K_{\lambda,\mu^{\prime}}^{C_{n-1}}(q)s_{(\mu
_{\overline{n}}+R,\gamma)}$$ where $\mu^{\prime}=(\mu_{\overline{n-1}},...,\mu_{\overline{1}})\in
P_{C_{n-1}}^{+}.$
\[Th\_rec\_morrB\]Let $\mu\in P_{B_{n}}^{+}.$ Then$$Q_{\mu}^{B_{n}}=\sum_{\gamma\in P_{B_{n-1}}^{+}}\sum_{R=0}^{+\infty}\sum_{r+2m=R}q^{R}\sum_{\lambda\in\left( \gamma\otimes r\right) _{B_{n-1}}}b_{\gamma,r}^{\lambda}K_{\lambda,\mu^{\prime}}^{B_{n-1}}(q)s_{(\mu
_{\overline{n}}+R,\gamma)}. \label{rec_mor_b}$$
From $Q_{\mu}=\left( \prod_{\alpha\in R_{B_{n}}^{+}}\dfrac{1}{1-qR_{\alpha}}\right) s_{\mu}$ and Proposition 3.5 of [@NR] we can write $$Q_{\mu}=\left( \underset{\alpha\notin R_{B_{n-1}}^{+}}{\prod_{\alpha\in
R_{B_{n}}^{+}}}\dfrac{1}{1-qR_{\alpha}}\right) \left[ \left( \underset
{\alpha\in R_{B_{n-1}}^{+}}{\prod}\dfrac{1}{1-qR_{\alpha}}\right) s_{\mu
}\right] .$$ Then by applying Theorem \[th\_hall\_kostka\], we obtain$$Q_{\mu}=\left( \underset{\alpha\notin R_{B_{n-1}}^{+}}{\prod_{\alpha\in
R_{B_{n}}^{+}}}\dfrac{1}{1-qR_{\alpha}}\right) \left( \sum_{\lambda\in
P_{B_{n-1}}^{+}}K_{\lambda,\mu^{\prime}}^{B_{n-1}}(q)s_{(\mu_{\overline{n}},\lambda)}\right) . \label{for_pro_q_mu}$$ Set $R_{\overline{i}}=R_{\varepsilon_{\overline{n}}-\varepsilon_{\overline{i}}}$ for $i=1,...,n-1$ $R_{n}=R_{\varepsilon\overline{_{n}}}$ and $R_{i}=R_{\varepsilon_{\overline{n}}+\varepsilon_{\overline{i}}}$ for $i=1,...,n.$ Recall that for any $\beta\in P_{B_{n-1}},$ $R_{\overline{i}}(\beta)=\beta+\varepsilon_{\overline{n}}-\varepsilon_{\overline{i}}$ and $R_{i}(\beta)=\beta+\varepsilon_{\overline{n}}+\varepsilon_{\overline{i}}.$ Then (\[for\_pro\_q\_mu\]) implies$$\begin{gathered}
Q_{\mu}=\sum_{\lambda\in P_{B_{n-1}}^{+}}K_{\lambda,\mu^{\prime}}^{B_{n-1}}(q)\times\\
\left( \sum_{r=0}^{+\infty}\sum_{b=0}^{+\infty}\sum_{k_{\overline{1}}+\cdot\cdot\cdot+k_{\overline{n-1}}+k_{1}+\cdot\cdot\cdot+k_{n-1}=r}q^{r+b}(R_{n})^{b}(R_{1})^{k_{1}}(R_{\overline{1}})^{k_{\overline{1}}}\cdot\cdot\cdot(R_{n-1})^{k_{n-1}}(R_{\overline{n-1}})^{k_{\overline{n-1}}}s_{(\mu_{\overline{n}},\lambda)}\right) .\end{gathered}$$$$\begin{gathered}
Q_{\mu}=\sum_{r=0}^{+\infty}\sum_{b=0}^{+\infty}q^{r+b}\sum_{\lambda\in
P_{B_{n-1}}^{+}}K_{\lambda,\mu^{\prime}}^{B_{n-1}}(q)\sum_{k_{\overline{1}}+\cdot\cdot\cdot+k_{\overline{n-1}}+k_{1}+\cdot\cdot\cdot+k_{n-1}=r}s_{(\mu_{\overline{n}}+r+b,\lambda_{\overline{n-1}}+k_{n-1}-k_{\overline{n-1}},\cdot\cdot\cdot,\lambda_{\overline{1}}+k_{1}-k_{\overline{1}})}=\\
\sum_{R=0}^{+\infty}\sum_{r=0}^{R}q^{R}\sum_{\lambda\in P_{B_{n-1}}^{+}}K_{\lambda,\mu^{\prime}}^{B_{n-1}}(q)\sum_{k_{\overline{1}}+\cdot\cdot
\cdot+k_{\overline{n-1}}+k_{1}+\cdot\cdot\cdot+k_{n-1}=r}s_{(\mu_{\overline
{n}}+R,\lambda_{\overline{n-1}}+k_{n-1}-k_{\overline{n-1}},\cdot\cdot
\cdot,\lambda_{\overline{1}}+k_{1}-k_{\overline{1}})}$$ by setting $R=r+b.$ Now fix $\lambda,R>0$ and $0<r\leq R$ and write $$\begin{aligned}
S_{1} & =\sum_{k_{\overline{1}}+\cdot\cdot\cdot+k_{\overline{n-1}}+k_{1}+\cdot\cdot\cdot+k_{n-1}=r}s_{(\mu_{\overline{n}}+R,\lambda
_{\overline{n-1}}+k_{n-1}-k_{\overline{n-1}},\cdot\cdot\cdot,\lambda
_{\overline{1}}+k_{1}-k_{\overline{1}})}\text{,}\\
S_{2} & =\sum_{k_{\overline{1}}+\cdot\cdot\cdot+k_{\overline{n-1}}+k_{1}+\cdot\cdot\cdot+k_{n-1}=r-1}s_{(\mu_{\overline{n}}+R,\lambda
_{\overline{n-1}}+k_{n-1}-k_{\overline{n-1}},\cdot\cdot\cdot,\lambda
_{\overline{1}}+k_{1}-k_{\overline{1}})},\end{aligned}$$ $S_{R,r}=S_{1}+S_{2}$ and $\gamma=(\lambda_{\overline{n-1}}+k_{n-1}-k_{\overline{n-1}},...,\lambda_{\overline{1}}+k_{1}-k_{\overline{1}}).$
$\mathrm{(a):}$Consider$\gamma$ appearing in $S_{1}$ or $S_{2}$ and suppose that there exists $i\in\{1,...,n-2\}$ such that $\lambda_{\overline{i}}>\lambda_{\overline{i+1}}-k_{\overline{i+1}}.$ Set $\widetilde{\gamma}=s_{i}\circ\gamma$ that is $$\widetilde{\gamma}=s_{i}(\gamma_{\overline{n-1}}+n-3/2,...,\gamma
_{\overline{i+1}}+n-i+1/2,\gamma_{\overline{i}}+n-i,...,\gamma_{\overline{1}}+1/2)-(n-3/2,...,1/2).$$ Then $\gamma_{\overline{s}}=\widetilde{\gamma}_{\overline{s}}$ for $s\neq
i+1,i$, $\widetilde{\gamma}_{\overline{i+1}}=\gamma_{\overline{i}}-1$ and $\widetilde{\gamma}_{\overline{i}}=,\gamma_{\overline{i+1}}+1$ that is $$\left\{
\begin{tabular}
[c]{l}$\widetilde{\gamma}_{\overline{i+1}}=\lambda_{\overline{i}}+k_{i}-k_{\overline{i}}-1$\\
$\widetilde{\gamma}_{\overline{i}}=\lambda_{\overline{i+1}}+k_{i+1}-k_{\overline{i+1}}+1$\end{tabular}
\right. .$$ Write $\widetilde{k}_{i+1}=k_{i}$, $\widetilde{k}_{i}=k_{i+1}$, $\widetilde
{k}_{\overline{i+1}}=\lambda_{\overline{i+1}}-\lambda_{\overline{i}}+k_{\overline{i}}+1$ and $\widetilde{k}_{\overline{i}}=\lambda_{\overline{i}}-\lambda_{\overline{i+1}}+k_{\overline{i+1}}-1.$ To make our notation homogeneous set $\widetilde{k}_{t}=k_{t}$ for any $t\neq i,i+1,\overline
{i},\overline{i+1}.$ Then $\lambda_{\overline{i}}>\lambda_{\overline{i+1}}-\widetilde{k}_{\overline{i+1}}.$ We have $\widetilde{k}_{\overline{i+1}}\geq0$ and $\widetilde{k}_{\overline{i}}=\lambda_{\overline{i}}-\lambda
_{\overline{i+1}}+k_{\overline{i+1}}-1\geq0$ since $\lambda_{\overline{i}}>\lambda_{\overline{i+1}}-k_{\overline{i+1}}$.$\;$Moreover $\widetilde
{k}_{\overline{1}}+\cdot\cdot\cdot+\widetilde{k}_{\overline{n-1}}+\widetilde{k}_{1}+\cdot\cdot\cdot+\widetilde{k}_{n-1}=k_{\overline{1}}+\cdot\cdot\cdot+k_{\overline{n-1}}+k_{1}+\cdot\cdot\cdot+k_{n-1}$ and for any $s\in\{1,...,n-2\}$$$\widetilde{\gamma}_{\overline{s}}=\lambda_{\overline{s}}+\widetilde{k}_{s}-\widetilde{k}_{\overline{s}}.$$ $\mathrm{(b):}$ Consider$\gamma$ appearing in $S_{1}$ and suppose that $\lambda_{\overline{i}}\leq\lambda_{\overline{i+1}}-k_{\overline{i+1}}$ for all $i=1,...,n-2$ and $\lambda_{\overline{1}}-k_{\overline{1}}<0$. Set $\widetilde{\gamma}=s_{0}\circ\gamma.$ Then $\gamma_{\overline{s}}=\widetilde{\gamma}_{\overline{s}}$ for $s\neq1$ and $\widetilde{\gamma
}_{\overline{1}}=-\lambda_{\overline{1}}-k_{1}+k_{\overline{1}}-1$. Write $\widetilde{k}_{i}=k_{i},$ $\widetilde{k}_{\overline{i}}=k_{\overline{i}}$ for all $i=2,...,n-1$ and set $\widetilde{k}_{1}=k_{\overline{1}}-\lambda
_{\overline{1}}-1,$ $\widetilde{k}_{\overline{1}}=k_{1}+\lambda_{\overline{1}}.$ We have $\widetilde{k}_{1}\geq0$, $\lambda_{\overline{i}}\leq
\lambda_{\overline{i+1}}-\widetilde{k}_{\overline{i+1}}$ for all $i=1,...,n-2$ and $\lambda_{\overline{1}}-\widetilde{k}_{\overline{1}}\leq0.\;$Moreover $\widetilde{k}_{\overline{1}}+\cdot\cdot\cdot+\widetilde{k}_{\overline{n-1}}+\widetilde{k}_{1}+\cdot\cdot\cdot+\widetilde{k}_{n-1}=r-1$ (thus $\gamma$ appears in $S_{2})$ and $\widetilde{\gamma}_{\overline{1}}=\lambda
_{\overline{1}}+\widetilde{k}_{1}-\widetilde{k}_{\overline{1}}.$
$\mathrm{(c):}$ Consider$\gamma$ appearing in $S_{2}$ and suppose that $\lambda_{\overline{i}}\leq\lambda_{\overline{i+1}}-k_{\overline{i+1}}$ for all $i=1,...,n-2$ and $\lambda_{\overline{1}}-k_{\overline{1}}\leq0$. Set $\widetilde{\gamma}=s_{0}\circ\gamma.$ Then $\gamma_{\overline{s}}=\widetilde{\gamma}_{\overline{s}}$ for $s\neq1$ and $\widetilde{\gamma}_{\overline{1}}=-\lambda_{\overline{1}}-k_{1}+k_{\overline{1}}-1$. Write $\widetilde{k}_{i}=k_{i},$ $\widetilde
{k}_{\overline{i}}=k_{\overline{i}}$ for all $i=2,...,n-1$ and set $\widetilde{k}_{1}=k_{\overline{1}}-\lambda_{\overline{1}},$ $\widetilde
{k}_{\overline{1}}=k_{1}+\lambda_{\overline{1}}+1.$ We have $\widetilde{k}_{1}\geq0$, $\lambda_{\overline{i}}\leq\lambda_{\overline{i+1}}-\widetilde
{k}_{\overline{i+1}}$ for all $i=1,...,n-2$ and $\lambda_{\overline{1}}-\widetilde{k}_{\overline{1}}<0.\;$Moreover $\widetilde{k}_{\overline{1}}+\cdot\cdot\cdot+\widetilde{k}_{\overline{n-1}}+\widetilde{k}_{1}+\cdot
\cdot\cdot+\widetilde{k}_{n-1}=r$ (thus $\gamma$ appears in $S_{1})$ and $\widetilde{\gamma}_{\overline{1}}=\lambda_{\overline{1}}+\widetilde{k}_{1}-\widetilde{k}_{\overline{1}}.$
$\mathrm{(d):}$Now consider$\gamma$ appearing in $S_{1}$ or $S_{2}$ and suppose that $\lambda_{\overline{s}}\leq\lambda
_{\overline{s+1}}-k_{\overline{s+1}}$ for any $s\in\{1,...,n-2\}$, $\lambda_{\overline{1}}-k_{\overline{1}}\geq0$ (resp. $\lambda_{\overline{1}}-k_{\overline{1}}>0)$ if $\gamma$ appears in $S_{1}$ (resp. in $S_{2})$ and there exists $i\in\{1,...,n-2\}$ such that $\lambda_{\overline{i+1}}-k_{\overline{i+1}}<\lambda_{\overline{i}}+k_{i}-k_{\overline{i}}.$ Define $\widetilde{\gamma}=s_{i}\circ\gamma$ as above. Set $\widetilde{k}_{\overline{i+1}}=k_{\overline{i+1}}$, $\widetilde{k}_{\overline{i}}=k_{\overline{i}}$, $\widetilde{k}_{i+1}=\lambda_{\overline{i}}-\lambda_{\overline{i+1}}-k_{\overline{i}}+k_{i}+k_{\overline{i+1}}-1$ and $\widetilde{k}_{i}=(\lambda_{\overline{i+1}}-\lambda_{\overline{i}}-k_{\overline{i+1}})+k_{i+1}+k_{\overline{i}}+1.$ Write $\widetilde{k}_{t}=k_{t}$ for any $t\neq i,i+1,\overline{i},\overline{i+1}.$ We obtain $\widetilde{k}_{i}\geq0$ and $\widetilde{k}_{i+1}\geq0$ since $\lambda
_{\overline{i}}\leq\lambda_{\overline{i+1}}-k_{\overline{i+1}}$ and $\lambda_{\overline{i+1}}-k_{\overline{i+1}}<\lambda_{\overline{i}}+k_{i}-k_{\overline{i}}.$ Since $\widetilde{k}_{\overline{s}}=k_{\overline{s}}$ for all $s=1,...,n-1,$ we have $\lambda_{\overline{s}}\leq\lambda
_{\overline{s+1}}-\widetilde{k}_{\overline{s+1}}$ for any $s\in\{1,...,n-2\}$ and $\lambda_{\overline{1}}-\widetilde{k}_{\overline{1}}\geq0.\;$Moreover the assertion $\lambda_{\overline{i+1}}-\widetilde{k}_{\overline{i+1}}<\lambda_{\overline{i}}+\widetilde{k}_{i}-\widetilde{k}_{\overline{i}}$ holds since it is equivalent to $0<k_{i+1}+1.$ Finally $\widetilde{k}_{\overline{1}}+\cdot\cdot\cdot+\widetilde{k}_{\overline{n-1}}+\widetilde{k}_{1}+\cdot
\cdot\cdot+\widetilde{k}_{n-1}=k_{\overline{1}}+\cdot\cdot\cdot+k_{\overline
{n-1}}+k_{1}+\cdot\cdot\cdot+k_{n-1}$ and for any $s\in\{1,...,n-2\}$$$\widetilde{\gamma}_{\overline{s}}=\lambda_{\overline{s}}+\widetilde{k}_{s}-\widetilde{k}_{\overline{s}}.$$
Denote by $E_{a},E_{d}$ the sets of multi-indices $(k_{\overline{1}},...,k_{\overline{n-1}},k_{1},...,k_{n-1})$ such that $k_{\overline{1}}+\cdot\cdot\cdot+k_{\overline{n-1}}+k_{1}+\cdot\cdot\cdot+k_{n-1}=r$ and satisfying respectively the assertions $\mathrm{(a)}$ $\mathrm{(d)}$Let$f$ be the map defined on $E_{a}\cup E_{d}$ by $$f(\gamma)=\widetilde{\gamma}.$$ Then by the above arguments $f$ is a bijection which verifies $f(E_{a})=E_{a}$ and $f(E_{d})=E_{d}$. Now the pairing $\gamma\longleftrightarrow
\widetilde{\gamma}$ provides the cancellation of all the $s_{\gamma}$ with $\gamma=(\lambda_{\overline{n-1}}+k_{n-1}-k_{\overline{n-1}},...,\lambda
_{\overline{1}}+k_{1}-k_{\overline{1}})$ such that $(k_{\overline{1}},...,k_{\overline{n}},k_{1},...,k_{n})\in E_{a}\cup E_{d}$ appearing in $S_{1}.$ Indeed $s_{(\mu_{\overline{n}}+R,\gamma)}=-s_{(\mu_{\overline{n}}+R,\widetilde{\gamma})}.$ We obtain similarly the cancellation of all the $s_{\gamma}$ such that $\gamma$ verifies the assertions $\mathrm{(a)}$ or $\mathrm{(d)}$ appearing in $S_{2}.$
Now write $E_{b}$ (resp. $E_{c})$ for the set of multi-indices $(k_{\overline{1}},...,k_{\overline{n-1}},k_{1},...,k_{n-1})$ such that $k_{\overline{1}}+\cdot\cdot\cdot+k_{\overline{n-1}}+k_{1}+\cdot\cdot
\cdot+k_{n-1}=r$ (resp. $r-1)$ and satisfying assertion $\mathrm{(b)}$(resp. $\mathrm{(c))}$Let$e$ be the map defined on $E_{b}\cup E_{c}$ by $$e(\gamma)=\widetilde{\gamma}.$$ Then $e$ is a bijection which verifies $e(E_{b})=E_{c}$ and $e(E_{c})=E_{b}$ and the $s_{\gamma}$ such that $\gamma$ verifies the assertions $\mathrm{(b)}$ or $\mathrm{(c)}$ cancel in $S_{R,r}.$ Finally by Corollary \[cor\_pieriB\] and Remark $\mathrm{(iii)}$ following Corollary \[cor\_pieriD\] we obtain $$S_{R,r}=\sum_{\gamma\in P_{B_{n-1}}^{+},\lambda\in\left( \gamma\otimes
r\right) _{B_{n-1}}}b_{\gamma,r}^{\lambda}s_{(\mu_{\overline{n}}+R,\gamma).}$$ Note that this equality is also true when $R=r=0$ if we set $S_{0,0}=s_{(\mu_{\overline{n}},\lambda)}.$ Thus we have$$Q_{\mu}=\sum_{R=0}^{+\infty}\sum_{\underset{r\equiv R\operatorname{mod}2}{0\leq r\leq R}}q^{R}\sum_{\gamma\in P_{B_{n-1}}^{+},\lambda\in\left(
\gamma\otimes r\right) _{B_{n-1}}}b_{\gamma,r}^{\lambda}K_{\lambda
,\mu^{\prime}}^{B_{n-1}}(q)s_{(\mu_{\overline{n}}+R,\gamma)}$$ which is equivalent to (\[rec\_mor\_b\]). So the theorem is proved.
\[Th\_rec\_morrD\]Let $\mu\in P_{D_{n}}^{+}.$ Then$$Q_{\mu}^{D_{n}}=\sum_{\gamma\in P_{D_{n-1}}^{+}}\sum_{R=0}^{+\infty}\sum_{r+2m=R}q^{R}\sum_{\lambda\in\left( \gamma\otimes r\right) _{n-1}}d_{\gamma,r}^{\lambda}K_{\lambda,\mu^{\prime}}^{D_{n-1}}(q)s_{(\mu
_{\overline{n}}+R,\gamma)}. \label{rec_mor_D}$$
Set $R_{\overline{i}}=R_{\varepsilon_{\overline{n}}-\varepsilon_{\overline{i}}}$ for $i=1,...,n-1$ and $R_{i}=R_{\varepsilon_{\overline{n}}+\varepsilon
_{\overline{i}}}$ for $i=1,...,n.$ We obtain as in proof of Theorem \[Th\_rec\_morrB\]$$\begin{gathered}
Q_{\mu}=\sum_{\lambda\in P_{D_{n-1}}^{+}}K_{\lambda,\mu^{\prime}}(q)\times\\
\left( \sum_{R=0}^{+\infty}\sum_{\kappa_{\overline{1}}+k_{\overline{2}}+\cdot\cdot\cdot+k_{\overline{n-1}}+\kappa_{1}+k_{2}+\cdot\cdot\cdot
+k_{n-1}=R}q^{R}(R_{1})^{\kappa_{1}}(R_{\overline{1}})^{\kappa_{\overline{1}}}(R_{2})^{k_{2}}(R_{\overline{2}})^{k_{\overline{2}}}\cdot\cdot\cdot
(R_{n-1})^{k_{n-1}}(R_{\overline{n-1}})^{k_{\overline{n-1}}}s_{(\mu
_{\overline{n}},\lambda)}\right) =\\
\sum_{R=0}^{+\infty}\sum_{\lambda\in P_{D_{n-1}}^{+}}q^{R}K_{\lambda
,\mu^{\prime}}(q)\sum_{\kappa_{\overline{1}}+k_{\overline{2}}+\cdot\cdot
\cdot+k_{\overline{n-1}}+\kappa_{1}+k_{2}+\cdot\cdot\cdot+k_{n-1}=R}s_{(\mu_{\overline{n}}+R,\lambda_{\overline{n-1}}+k_{n-1}-k_{\overline{n-1}},\cdot\cdot\cdot,\lambda_{\overline{2}}+k_{2}-k_{\overline{2}},\lambda
_{\overline{1}}+\kappa_{1}-\kappa_{\overline{1}})}.\end{gathered}$$ Fix $\lambda,R$ and consider$$S_{R}=\sum_{\kappa_{\overline{1}}+k_{\overline{2}}+\cdot\cdot\cdot
+k_{\overline{n-1}}+\kappa_{1}+k_{2}+\cdot\cdot\cdot+k_{n-1}=R}s_{(\mu
_{\overline{n}}+R,\lambda_{\overline{n-1}}+k_{n-1}-k_{\overline{n-1}},\cdot\cdot\cdot,\lambda_{\overline{2}}+k_{2}-k_{\overline{2}},\lambda
_{\overline{1}}+\kappa_{1}-\kappa_{\overline{1}})}.$$ Set $\gamma=(\lambda_{\overline{n-1}}+k_{n-1}-k_{\overline{n-1}},\cdot
\cdot\cdot,\lambda_{\overline{2}}+k_{2}-k_{\overline{2}},\lambda_{\overline
{1}}+\kappa_{1}-\kappa_{\overline{1}}).$
$\mathrm{(a):}$Consider$\gamma$ appearing in $S_{R}$ and suppose that there exists $i\in\{2,...,n-2\}$ such that $\lambda_{\overline{i}}>\lambda_{\overline{i+1}}-k_{\overline{i+1}}.$ Then we associate a $\widetilde{\gamma}$ verifying $\lambda_{\overline{i}}>\lambda_{\overline{i+1}}-\widetilde{k}_{\overline{i+1}}$ to $\gamma$ as we have done in case $\mathrm{(a)}$ of the above proof. This is possible since $s_{i}=(\overline{i+1},\overline{i})(i,i+1)\in W_{D_{n}}.$
$\mathrm{(b):}$Consider$\gamma$ appearing in $S_{R}$ such that $\lambda_{\overline{1}}>\lambda_{\overline{2}}-k_{\overline{2}}.$ We set $\widetilde{\gamma}=s_{1}\circ\gamma,$ $\widetilde{k}_{2}=\kappa_{1},$ $\widetilde{\kappa}_{1}=k_{2},$ $\widetilde
{k}_{\overline{2}}=\lambda_{\overline{2}}-\lambda_{\overline{1}}+\kappa_{\overline{1}}+1$ and $\widetilde{\kappa}_{\overline{1}}=\lambda_{\overline{1}}-\lambda_{\overline{2}}+k_{\overline{2}}-1.$ Then $\gamma$ appears in $S_{R}$ and verifies $\lambda_{\overline{1}}>\lambda_{\overline{2}}-\widetilde{k}_{\overline{2}}$ whatever the sign of $\lambda_{\overline{1}}.$
$\mathrm{(c):}$Consider$\gamma$ appearing in $S_{R}$ such that $-\lambda_{\overline{1}}>\lambda_{\overline{2}}-k_{\overline{2}}.$ We set $\widetilde{\gamma}=s_{0}\circ\gamma,$ $\widetilde{k}_{2}=\kappa_{\overline{1}},$ $\widetilde{\kappa}_{\overline{1}}=k_{2},$ $\widetilde{k}_{\overline{2}}=\lambda_{\overline{2}}+\lambda
_{\overline{1}}+\kappa_{1}+1$ and $\widetilde{\kappa}_{1}=-\lambda
_{\overline{1}}-\lambda_{\overline{2}}+k_{\overline{2}}-1.$ Then $\gamma$ appears in $S_{R}$ and verifies $-\lambda_{\overline{1}}>\lambda_{\overline
{2}}-\widetilde{k}_{\overline{2}}$ whatever the sign of $\lambda_{\overline
{1}}.$
$\mathrm{(d):}$ Consider$\gamma$ appearing in $S_{R}$ and suppose that $\lambda_{\overline{s}}\leq\lambda_{\overline{s+1}}-k_{\overline{s+1}}$ for any $s\in\{2,...,n-2\}$, $\pm\lambda_{\overline{1}}>\lambda_{\overline{2}}-k_{\overline{2}},$ and there exists $i\in
\{1,...,n-2\}$ such that $\lambda_{\overline{i+1}}-k_{\overline{i+1}}<\lambda_{\overline{i}}+k_{i}-k_{\overline{i}}.$ We set $\widetilde{\gamma
}=s_{i}\circ\gamma$ and proceed as in case $\mathrm{(d)}$ of the above proof.
$\mathrm{(e):}$ Consider$\gamma$ appearing in $S_{R}$ and suppose that $\lambda_{\overline{s}}\leq\lambda_{\overline{s+1}}-k_{\overline{s+1}}$ for any $s\in\{2,...,n-2\}$, $\pm\lambda_{\overline{1}}>\lambda_{\overline{2}}-k_{\overline{2}},$ and $\lambda_{\overline{2}}-k_{\overline{2}}<\lambda_{\overline{1}}+\kappa_{1}-\kappa_{\overline{1}}.$ We set $\widetilde{\gamma}=s_{1}\circ\gamma$, $\widetilde{k}_{\overline{2}}=k_{\overline{2}},$ $\widetilde{\kappa}_{\overline{1}}=\kappa_{\overline{1}},$ $\widetilde{k}_{2}=\lambda_{\overline{1}}-\lambda_{\overline{2}}-\kappa_{\overline{1}}+\kappa_{1}-1$ and $\widetilde{\kappa}_{1}=\lambda_{\overline{2}}-\lambda_{\overline{1}}-k_{\overline{2}}+k_{2}+\kappa_{\overline{1}}+1.$ Then $\gamma$ appears in $S_{R}$ and verifies $\lambda_{\overline{s}}\leq\lambda_{\overline{s+1}}-\widetilde{k}_{\overline{s+1}}$ for any $s\in\{2,...,n-2\}$, $\pm\lambda_{\overline{1}}>\lambda_{\overline{2}}-\widetilde{k}_{\overline{2}},$ and $\lambda
_{\overline{2}}-\widetilde{k}_{\overline{2}}<\lambda_{\overline{1}}+\widetilde{\kappa}_{1}-\widetilde{\kappa}_{\overline{1}}$ whatever the sign of $\lambda_{\overline{1}}.$
$\mathrm{(f):}$ Consider$\gamma$ appearing in $S_{R}$ and suppose that $\lambda_{\overline{s}}\leq\lambda_{\overline{s+1}}-k_{\overline{s+1}}$ for any $s\in\{2,...,n-2\}$, $\pm\lambda_{\overline{1}}>\lambda_{\overline{2}}-k_{\overline{2}},$ and $\lambda_{\overline{2}}-k_{\overline{2}}<-\lambda_{\overline{1}}-\kappa_{1}+\kappa_{\overline{1}}.$ We set $\widetilde{\gamma}=s_{0}\circ\gamma$, $\widetilde{k}_{\overline{2}}=k_{\overline{2}},$ $\widetilde{\kappa}_{1}=\kappa_{1},$ $\widetilde{k}_{2}=-\lambda_{\overline{1}}-\lambda_{\overline{2}}+\kappa_{\overline{1}}-\kappa_{1}-1$ and $\widetilde{\kappa}_{1}=\lambda_{\overline{2}}+\lambda_{\overline{1}}-k_{\overline{2}}+k_{2}+\kappa_{1}+1.$ Then $\gamma$ appears in $S_{R}$ and verifies $\lambda_{\overline{s}}\leq\lambda
_{\overline{s+1}}-\widetilde{k}_{\overline{s+1}}$ for any $s\in\{2,...,n-2\}$, $\pm\lambda_{\overline{1}}>\lambda_{\overline{2}}-\widetilde{k}_{\overline{2}},$ and $\lambda_{\overline{2}}-\widetilde{k}_{\overline{2}}<-\lambda
_{\overline{1}}+\widetilde{\kappa}_{1}-\widetilde{\kappa}_{\overline{1}}$ whatever the sign of $\lambda_{\overline{1}}.$
By considering the pairing $\gamma\longleftrightarrow\widetilde
{\gamma},$ the $s_{\gamma}$ appearing in $S_{R}$ cancel if they do not verify simultaneously all the following conditions$$\left\{
\begin{tabular}
[c]{l}$1:\lambda_{\overline{1}}\leq\lambda_{\overline{2}}-\widetilde{k}_{\overline{2}}\text{ and }-\lambda_{\overline{1}}\leq\lambda_{\overline{2}}-\widetilde{k}_{\overline{2}}$\\
$2:\lambda_{\overline{i}}\leq\lambda_{\overline{i+1}}-k_{\overline{i+1}}$ for
$i=2,...,n-1$\\
$3:\lambda_{\overline{i+1}}-k_{\overline{i+1}}\geq\lambda_{\overline{i}}+k_{i}-k_{\overline{i}}$ for $i=2,...,n-2$\\
$4:\lambda_{\overline{2}}-k_{\overline{2}}\leq\lambda_{\overline{1}}+\kappa_{1}-\kappa_{\overline{1}}$ and $\lambda_{\overline{2}}-k_{\overline
{2}}\leq-\lambda_{\overline{1}}+\kappa_{1}-\kappa_{\overline{1}}$\end{tabular}
\right. . \label{cond}$$ Note that conditions $1,2$ and $3$ are precisely conditions $\mathrm{(i)},\mathrm{(ii)}$ and $\mathrm{(iii)}$ of Corollary \[cor\_pieriD\]. Let $E_{R}$ be the set multi-indices $M=(\kappa_{\overline{1}},...,k_{\overline
{n}},\kappa_{1},...,k_{n})$ such that $\kappa_{\overline{1}}+\cdot\cdot
\cdot+k_{\overline{n-1}}+\kappa_{1}+\cdot\cdot\cdot+k_{n-1}=R$ and satisfying (\[cond\]). We can write $S=\sum_{M\in E_{R}}s_{(\mu_{\overline{n}}+R,\gamma_{M})}.$Set $E_{R}^{-}=\{M\in E_{R},\kappa_{1}-\kappa_{\overline{1}}\leq0\}$ and $E_{R}^{+}=\{M\in E_{R},\kappa_{1}-\kappa_{\overline{1}}>0\}.$
Let $m$ be an integer such that $0\leq m\leq R/2.$ Set $r=R-2m.$ Consider the multi-indices $M\in E_{R}^{-}$ such that $\kappa_{1}=m$. Set $k_{\overline{1}}=\kappa_{\overline{1}}-m=\kappa_{\overline{1}}-\kappa_{1}.$ If $\gamma_{\overline{1}}=\lambda_{\overline{1}}-k_{\overline{1}}\geq0$ (resp. $\gamma_{\overline{1}}<0)$ then condition $4$ of (\[cond\]) is equivalent to condition $\mathrm{(iv,(a))}$ of Corollary \[cor\_pieriD\] (resp. to condition $\mathrm{(iv,(d)).}$ Moreover $k_{\overline{1}}+\sum_{2\leq i\leq
n}(k_{\overline{i}}+k_{i})=r.$ Write $B^{-}(r)$ for the sub-graph of $B^{D_{n-1}}(r)$ defined by the vertices which does not contain any letter $1.$ Set $B(\gamma)\otimes B^{-}(r)=\underset{\lambda\in P_{D_{n-1}}^{+}}{\cup}B(\lambda)^{d_{\gamma,r}^{\lambda,-}}$ and $\left( \gamma\otimes
r\right) _{D_{n-1}}^{-}=\{\lambda\in P_{D_{n}}^{+},$ $d_{\gamma,r}^{\lambda,-}\neq0\}.$ By Remark $\mathrm{(iii)}$ following Corollary \[cor\_pieriD\] we know that $\gamma\in P\in P_{D_{n-1}},$ so we obtain $$\sum_{M\in E_{R}^{-},\kappa_{1}=m}s_{(\mu_{\overline{n}}+R,\gamma_{M})}=\sum_{\gamma\in P_{D_{n-1},}\lambda\in\left( \gamma\otimes r\right)
_{D_{n-1}}^{-}}d_{\gamma,r}^{\lambda,-}s_{(\mu_{\overline{n}}+R,\gamma)}.$$ Now consider the multi-indices $M\in E_{R}^{+}$ such that $\kappa
_{\overline{1}}=m$. Set $r=R-2m$ and $B(\gamma)\otimes B^{+}(r)=\underset
{\lambda\in P_{D_{n-1}}^{+}}{\cup}B(\lambda)^{d_{\gamma,r}^{\lambda,+}}$ where $B^{+}(r)$ is the sub-graph of $B^{D_{n-1}}(r)$ defined by the vertices which does not contain any letter $\overline{1}.$ Write $\left( \gamma\otimes
r\right) _{D_{n-1}}^{+}=\{\lambda\in P_{D_{n}}^{+},$ $d_{\gamma,r}^{\lambda,+}\neq0\}.$ We obtain similarly$$\sum_{M\in E_{R}^{+},\kappa_{\overline{1}}=m}s_{(\mu_{\overline{n}}+R,\gamma_{M})}=\sum_{\gamma\in P_{D_{n-1},}\lambda\in\left( \gamma\otimes
r\right) _{D_{n-1}}^{+}}d_{\gamma,r}^{\lambda,+}s_{(\mu_{\overline{n}}+R,\gamma)}.$$ Finally$$S=\sum_{r+2m=R}\ \sum_{\gamma\in P_{D_{n-1},},\lambda\in\left( \gamma\otimes
r\right) _{D_{n-1}}^{-}\cup\left( \gamma\otimes r\right) _{D_{n-1}}^{-}}\ (d_{\gamma,r}^{\lambda,-}+d_{\gamma,r}^{\lambda,+})s_{(\mu_{\overline{n}}+R,\gamma)}=\sum_{r+2m=R}\ \sum_{\gamma\in P_{D_{n-1},},\lambda\in\left(
\gamma\otimes r\right) _{D_{n-1}}}d_{\gamma,r}^{\lambda}s_{(\mu_{\overline
{n}}+R,\gamma)}$$ since $\left( \gamma\otimes r\right) _{D_{n-1}}$ is the disjoint union of $\left( \gamma\otimes r\right) _{D_{n-1}}^{-}$ and $\left( \gamma\otimes
r\right) _{D_{n-1}}^{+}.$ So the theorem is proved.
Consider $\nu,\mu$ two generalized partitions of length $n.$ Write $p$ for the lowest integer in $\{1,...,n\}$ such that $\nu_{\overline{p}}+p-\mu
_{\overline{n}}-n\geq0.$ For any $k\in\{p,p+1,...,n\}$ let $\sigma_{k}$ be the signed permutation defined by$$\sigma_{k}(i)=\left\{
\begin{tabular}
[c]{l}$i+1$ if $k\leq i\leq n-1$\\
$i$ if $1\leq i\leq k-1$\\
$k$ if $i=n$\end{tabular}
\right. .$$ Note that $(-1)^{l_{B}(\sigma_{k})}=(-1)^{l_{D}(\sigma_{k})}=(-1)^{n-k}.$ Let $\gamma_{k}$ be the generalized partition of length $n-1$$$\gamma_{k}=(\nu_{\overline{n}}+1,\nu_{\overline{n-1}}+1,...,\nu_{\overline
{k+1}}+1,\nu_{\overline{k-1}},...,\nu_{\overline{1}}).$$ Finally set $R_{k}=\nu_{\overline{k}}+k-\mu_{\overline{n}}-n.$
From the above recurrence formulas it is possible to express any Kostka-Foulkes polynomial $K_{\nu,\mu}(q)$ associated to a classical root system of rank $n$ in terms of Kostka-Foulkes polynomials associated to the corresponding root system of rank $n-1.$
\[Th\_mor\_expli\]With the above notation we have $$\begin{gathered}
\mathrm{(i)}:K_{\nu,\mu}^{B_{n}}(q)=\sum_{k=p}^{n}(-1)^{n-k}\times q^{R_{k}}\times\sum_{r+2m=R_{k}}\sum_{\lambda\in\left( \gamma_{r}\otimes r\right)
_{B_{n-1}}}b_{\gamma_{r},r}^{\lambda}K_{\lambda,\mu^{\prime}}^{B_{n-1}}(q),\\
\mathrm{(ii)}:K_{\nu,\mu}^{C_{n}}(q)=\sum_{k=p}^{n}(-1)^{n-k}\times
\sum_{r+2m=R_{k}}\sum_{\lambda\in\left( \gamma_{r}\otimes r\right)
_{C_{n-1}}}q^{R_{k}-m}\times c_{\gamma_{r},r}^{\lambda}K_{\lambda,\mu^{\prime
}}^{C_{n-1}}(q)\\
\mathrm{(iii)}:K_{\nu,\mu}^{D_{n}}(q)=\sum_{k=p}^{n}(-1)^{n-k}\times q^{R_{k}}\times\sum_{r+2m=R_{k}}\sum_{\lambda\in\left( \gamma_{r}\otimes r\right)
_{D_{n-1}}}d_{\gamma_{r},r}^{\lambda}K_{\lambda,\mu^{\prime}}^{D_{n-1}}(q).\end{gathered}$$
In case $\mathrm{(i),}$ write $E_{\nu}$ for the set of pairs $(\gamma,R)$ such that there exists $\sigma_{(\gamma,R)}\in W_{B_{n}}$ verifying $\sigma
_{(\gamma,R)}\circ(\mu_{\overline{n}}+R,\gamma)=\nu.$ We obtain from Theorems \[th\_hall\_kostka\] and \[Th\_rec\_morrB\]$$K_{\nu,\mu}(q)=\sum_{(\gamma,R)\in E_{\nu}}\sum_{r+2m=R}q^{R}\sum_{\lambda
\in\left( \gamma\otimes r\right) _{B_{n-1}}}b_{\gamma,r}^{\lambda
}(-1)^{l(\sigma_{(\gamma,R)})}K_{\lambda,\mu^{\prime}}(q).
\label{rec_mor_expli}$$ Consider $(\gamma,R)\in E_{\nu}.$ We must have$$\sigma\left( \mu_{\overline{n}}+R+n-\dfrac{1}{2},\gamma_{\overline{n-1}}+n-\dfrac{3}{2},...,\gamma_{\overline{1}}+\dfrac{1}{2}\right) =\left(
\nu_{\overline{n}}+n-\dfrac{1}{2},\nu_{\overline{n-1}}+n-\dfrac{3}{2},...,\nu_{\overline{1}}+\dfrac{1}{2}\right) .$$ The strictly decreasing subsequence $(\gamma_{\overline{n-1}}+n-\dfrac{3}{2},...,\gamma_{\overline{1}}+\dfrac{1}{2})$ must be sent under the action of $\sigma$ on a strictly decreasing subsequence $I_{\gamma}$ of $(\nu
_{\overline{n}}+n-\dfrac{1}{2},\nu_{\overline{n-1}}+n-\dfrac{3}{2},...,\nu_{\overline{1}}+\dfrac{1}{2}).$ These subsequences correspond to the choice of a $\nu_{\overline{k}}+\dfrac{2k-1}{2}$ (for the image of $\mu_{\overline{n}}+R+n-\dfrac{1}{2}$ under the action of $\sigma)$ which does not belong to $I_{\gamma}.$ For such a subsequence we must have $\mu
_{\overline{n}}+R+n-\dfrac{1}{2}=\nu_{\overline{k}}+\dfrac{2k-1}{2}.$ Since $R=\nu_{\overline{k}}+k-\mu_{\overline{n}}-n\geq0$ this implies that $k\in\{p,...n\}$, $R=R_{k},$ $\sigma=\sigma_{k}$ and $\gamma=\gamma_{k}.$
We prove $\mathrm{(ii)}$ and $\mathrm{(iii)}$ similarly.
The statistics $\chi_{n}^{B},\chi_{n}^{C}$ and $\chi_{n}^{D}$
=============================================================
In this section we introduce a statistic on partition shaped Kashiwara-Nakashima’s tableaux verifying$$K_{\nu,\mu}(q)=\sum_{T\in\mathbf{T}(\lambda)_{\mu}}q^{\chi_{n}(T)}$$ when $(\nu,\mu)$ satisfies restrictive conditions. Although the statistic $\chi_{n}$ can be regarded as a generalization of Lascoux-Schützenberger’s charge for semi-standard tableaux, it does not permit to recover the Kostka-Foulkes polynomial $K_{\nu,\mu}(q)$ for any $(\nu,\mu).$
Catabolism
----------
From Theorem \[Th\_mor\_expli\] we derive the following lemma:
\[lem\_K(q)\_perf\]Let $\nu,\mu\in P_{n}^{+}$ be such that $\mu
_{\overline{n}}\geq\nu_{\overline{n-1}}.$ Set $l=\nu_{\overline{n}}-\mu_{\overline{n}}\geq0$ (otherwise $K_{\nu,\mu}(q)=0$).$\;$Then:$$\begin{aligned}
\mathrm{(i)}:K_{\nu,\mu}^{B_{n}}(q)=q^{l}\sum_{r+2m=l}\text{\ }\sum
_{\lambda\in\left( \nu^{\prime}\otimes r\right) _{B_{n-1}}}b_{\nu^{\prime
},r}^{\lambda}K_{\lambda,\mu^{\prime}}^{B_{n-1}}(q),\\
\mathrm{(ii)}:K_{\nu,\mu}^{C_{n}}(q)=\sum_{r+2m=l}q^{r+m}\text{\ }\sum_{\lambda\in\left( \nu^{\prime}\otimes r\right) _{C_{n-1}}}c_{\nu^{\prime},r}^{\lambda}K_{\lambda,\mu^{\prime}}^{C_{n-1}}(q),\\
\mathrm{(iii)}:K_{\nu,\mu}^{D_{n}}(q)=q^{l}\sum_{r+2m=l}\text{\ }\sum
_{\lambda\in\left( \nu^{\prime}\otimes r\right) _{D_{n-1}}}d_{\nu^{\prime
},r}^{\lambda}K_{\lambda,\mu^{\prime}}^{D_{n-1}}(q).\end{aligned}$$
Assertions $\mathrm{(i),(ii)}$ and $\mathrm{(iii)}$ follow by applying Theorem \[Th\_mor\_expli\] with $p=n.$
From now $\nu$ and $\mu$ are generalized partitions with integers parts. Consider $T\in\mathbf{T}^{n}(\nu)_{\mu}.$ Accordingly to Lemma \[lem\_fact\_row\], we can write $$\mathrm{w}(T)\equiv_{n}\mathrm{w}(R)\otimes\mathrm{w}(T^{\prime}).
\label{fact_tab}$$ Let $R^{\prime}$ be the row tableau obtained by erasing all the letters $\overline{n}$ and $n$ in $R.$ The catabolism of the tableau $T$ is defined by $$\mathrm{cat}(T)=P_{n-1}(\mathrm{w}(T^{\prime})\otimes\mathrm{w}(R^{\prime})).$$ The tableau $\mathrm{cat}(T)$ is well defined and belongs to $\mathbf{T}^{n-1}(\lambda)_{\mu^{\prime}}$ where $\lambda$ is the shape of $\mathrm{cat}(T)$ since $T^{\prime}$ and $R^{\prime}$ do not contain any letter $\overline{n}$ or $n.$
In the sequel we denote by $\mathrm{ch}_{A}$ the Lascoux-Schützenberger’s charge statistic on semi-standard tableaux. Note that $\mathrm{ch}_{A}$ may be used to compute Kostka-Foulkes polynomials for the root systems $B_{1}=C_{1}=A_{1}$ and $D_{3}=A_{3}.$
Consider $T\in\mathbf{T}^{n}(\nu)_{\mu}.$ The statistics $\chi
_{n}^{B},\chi_{n}^{C}$ and $\chi_{n}^{D}$ are defined recursively by:$$\begin{gathered}
\chi_{n}^{B}(T)=\left\{
\begin{tabular}
[c]{l}$\mathrm{ch}_{A}(T)$ if $n=1$\\
$\chi_{n-1}^{B}(\mathrm{cat}(T))+\nu_{\overline{n}}-\mu_{\overline{n}}$
otherwise
\end{tabular}
\right. ,\text{ }\chi_{n}^{D}(T)=\left\{
\begin{tabular}
[c]{l}$\mathrm{ch}_{A}(T)$ if $n=3$\\
$\chi_{n-1}^{D}(\mathrm{cat}(T))+\nu_{\overline{n}}-\mu_{\overline{n}}$
otherwise
\end{tabular}
\right. \text{ and}\\
\chi_{n}^{C}(T)=\left\{
\begin{tabular}
[c]{l}$\mathrm{ch}_{A}(T)$ if $n=1$\\
$\chi_{n-1}^{C}(\mathrm{cat}(T))+\nu_{\overline{n}}-\mu_{\overline{n}}-m$
otherwise
\end{tabular}
\right. \text{ where }m\text{ is the number of letters }n\text{ in }R.\end{gathered}$$
**Remark:**
$\mathrm{(i):}$ The statistics $\chi_{n}^{B},\chi_{n}^{C}$ and $\chi_{n}^{D}$ can be regarded as extensions of $\mathrm{ch}_{A}$. More precisely we have $\chi_{n}^{B}(T)=\chi_{n}^{C}(T)=\chi_{n}^{D}(T)=\mathrm{ch}_{A}(T)$ for the tableaux $T$ which contain only barred letters.
$\mathrm{(ii)}:$ To obtain $\chi_{1}^{B},\chi_{1}^{C}$ and $\chi
_{3}^{D}$ we need to compute $\mathrm{ch}_{A}$ on tableaux which are not semi-standard. This can be done from the characterization of $\mathrm{ch}_{A}$ in terms of crystal graphs given in [@LLT] or more directly by using the crystal graphs isomorphisms:$$B(\Lambda_{0}^{B_{1}})\simeq B(2\Lambda_{1}^{A_{1}})\text{, }B(\Lambda
_{0}^{C_{1}})\simeq B(\Lambda_{1}^{A_{1}}),\text{ }B(\Lambda_{0}^{D_{3}})\simeq B(\Lambda_{3}^{A_{3}})\text{, }B(\Lambda_{1}^{D_{3}})\simeq
B(\Lambda_{1}^{A_{3}})\text{ and }B(\Lambda_{2}^{D_{3}})\simeq B(\Lambda
_{2}^{A_{3}}) \label{isom}$$ which permit to turn each tableau $T$ related to types $B_{1},C_{1}$ and $D_{3}$ into its corresponding tableau $\tau_{T}$ of type $A_{1}$ or $A_{3}$ via bumping algorithm on semi-standard tableaux.
Consider the tableau of type $D_{3}$ and shape $(3,2,1),$ $T=\begin{tabular}
[c]{|l|ll}\hline
$$ & $$ & \multicolumn{1}{|l|}{$\mathtt{\bar
{1}}$}\\\hline
$$ & $$ & \multicolumn{1}{|l}{}\\\cline{1-1}\cline{1-2}$$ & & \\\cline{1-1}\end{tabular}
.$ Then $\mathrm{w}(T)=\overline{1}(\overline{2}2)(\overline{3}1\overline
{1}).$ We have $\overline{1}\in B(\Lambda_{2}^{D_{3}}),$ $(\overline{2}2)\in
B(\Lambda_{1}^{D_{3}}+\Lambda_{0}^{D_{3}})\simeq B(\Lambda_{1}^{A_{3}}+\Lambda_{3}^{A_{3}})$ and $(\overline{3}1\overline{1})\in B(2\Lambda
_{0}^{D_{3}})\simeq B(2\Lambda_{3}^{A_{3}}).$ Thus the semi-standard tableau $\tau$ corresponding to $T$ is obtained by applying the bumping algorithm to the word $w=(23)(3124)(124123).$ Finally $\tau_{T}=\begin{tabular}
[c]{|l|l|l|ll}\hline
$$ & $$ & $$ & $$ &
\multicolumn{1}{|l|}{$\mathtt{3}$}\\\hline
$$ & $$ & $$ & $$ &
\multicolumn{1}{|l}{}\\\cline{1-3}\cline{1-4}$$ & $$ & $$ & & \\\cline{1-3}\end{tabular}
.$
Catabolism and Kostka-Foulkes polynomials
-----------------------------------------
Consider $T\in\mathbf{T}^{n}(\nu)_{\mu}$ and suppose $n\geq2.$ For any integer $p\leq n$ consider the sequence of tableaux defined by $T_{n}=T$ and $T_{k}=\mathrm{cat}(T_{k+1})$ for $k=n-1,...,p$. Denote by $v^{(k)}\in
P_{k}^{+}$ the shape of $T_{k}.$ Then $T_{k}\in\mathbf{T}^{k}(\nu^{(k)})_{\mu^{(k)}}$ with $\mu^{(k)}=(\mu_{\overline{k}},...,\mu_{\overline{1}}).$
\[lem\_tech\]If $\mu_{\overline{p}}\geq v_{\overline{n-1}}$ then for every $k=n,...,p$ we have $\mu_{\overline{p}}\geq v_{\overline{k-1}}^{(k)}.$
We proceed by induction on $k.$ The lemma is true for $k=n$. Consider $k\in\{p+1,...,n\}$ such that $\mu_{\overline{p}}\geq v_{\overline{k-1}}^{(k)}.$ Then we must have $\nu_{\overline{k-2}}^{(k-1)}\leq\nu_{\overline
{k-1}}^{(k)}$ by Lemmas \[lem\_b\_gamm\_tens\_Lb\] and \[lem\_b\_gamm\_tens\_Ld\] since the shape $\nu^{(k-1)}$ is obtained by adding or deleting boxes on distinct columns of the shape obtained by deleting the longest row of $\nu^{(k)}$. Hence $\nu_{\overline{k-2}}^{(k-1)}\leq\nu_{\overline{k-1}}^{(k)}\leq\mu_{\overline{p}}$.
\[prop\_xhi\]Consider $\nu,\mu$ verifying one of the following conditions
$\mathrm{(i):}\nu,\mu\in P_{B_{n}}^{+}$ $n=1$ or, $n\geq2$ and $\mu_{\overline{2}}\geq\nu_{\overline{n-1}}$
$\mathrm{(ii):}\nu,\mu\in P_{C_{n}}^{+}$ $n=1$ or, $n\geq2$ and $\mu_{\overline{2}}\geq\nu_{\overline{n-1}}$
$\mathrm{(iii):}\nu,\mu\in P_{D_{n}}^{+}$ $n=3$ or, $n\geq4$ and $\mu_{\overline{4}}\geq\nu_{\overline{n-1}}$
then$$K_{\nu,\mu}(q)=\sum_{T\in\mathbf{T}^{n}(\nu)_{\mu}}q^{\chi_{n}(T)}.
\label{K(q)_xhi}$$
The assertion is proved by induction on $n$.
Case $\mathrm{(ii).}$ The proposition is true for the root system $C_{1}=A_{1}$. Now suppose that (\[K(q)\_xhi\]) is true for the root system $C_{n-1}$ with $n\geq2$ and consider $\nu,\mu$ two partitions of length $n$ such that $\mu_{\overline{2}}\geq\nu_{\overline{n-1}}$. Set $l=\nu
_{\overline{n}}-\mu_{\overline{n}}.$ From Lemma \[lem\_K(q)\_perf\] $\mathrm{(i)}$ we obtain$$K_{\nu,\mu}(q)=\sum_{r+2m=l}q^{r+m}\sum_{\lambda\in\left( \nu^{\prime}\otimes
r\right) _{C_{n-1}}}c_{\nu^{\prime},r}^{\lambda}K_{\lambda,\mu^{\prime}}(q)$$ since $\mu_{\overline{n}}\geq\mu_{\overline{2}}\geq\nu_{\overline{n-1}}.$ Set $$K(q)=\sum_{T\in\mathbf{T}^{n}(\nu)_{\mu}}q^{\chi_{n}(T)}.$$ Accordingly to Lemma \[lem\_fact\_row\], the reading of any $T\in
\mathbf{T}^{n}(\nu)_{\mu}$ can be factorized as $$\mathrm{w}(T)\equiv_{n}\mathrm{w}(R)\otimes\mathrm{w}(T^{\prime}).$$ Set $\mathcal{T}_{m}=\{T\in\mathbf{T}^{n}(\nu)_{\mu},\mathrm{w}(R)$ contains $m$ letters $n\}.$ We must have $0\leq m\leq l/2$ since all the letters $\overline{n}$ or $n$ of $T$ belong to $R$ and the number of letters $\overline{n}$ minus that of letters $n$ in $R$ must be equal to $\mu_{\overline{n}}$. For any $T\in\mathcal{T}_{m}$ we can write $\mathrm{cat}(T)=P_{n-1}(\mathrm{w}(T^{\prime})\otimes\mathrm{w}(R^{\prime}))$ where $R^{\prime}$ is a row tableau of length $r=l-2m$. The first row of $T\ $contains at least $\mu_{\overline{n}}$ letters $\overline{n}$. Moreover we have $\mu_{\overline{n}}\geq\mu_{\overline{2}}\geq\nu_{\overline{n-1}}$. This means that $\{\mathrm{w}(R^{\prime})\otimes\mathrm{w}(T^{\prime}),T\in\mathcal{T}_{m}\}=\left( B((r)_{n-1})\otimes B((\nu^{\prime})\right)
_{\mu^{\prime}}.$ Thus we have $\{\mathrm{w}(T^{\prime})\otimes\mathrm{w}(R^{\prime}),T\in\mathcal{T}_{m}\}=\left( B(\nu^{\prime})\otimes
B((r)_{n-1})\right) _{\mu^{\prime}}$ and $\{(\mathrm{cat}(T),T\in
\mathcal{T}_{m}\}$ is exactly the set of tableaux of shape $\lambda\in\left(
\nu^{\prime}\otimes r\right) _{C_{n-1}}$ and weight $\mu^{\prime}.$ By lemma \[lem\_tech\] we have $\mu_{\overline{2}}\geq\lambda_{\overline{n-2}}$ for any $\lambda\in B(\nu^{\prime})\otimes B((r)_{n-1})$ when $n-1\geq2$. So we can use the induction hypothesis and obtain $$\begin{gathered}
K(q)=\sum_{m=0}^{l/2}\sum_{T\in\mathcal{T}_{m}}q^{\chi_{n}(T)}=\sum
_{r+2m=l}\sum_{T\in\mathcal{T}_{m}}q^{\chi_{n-1}(\mathrm{cat}(T))+l-m}=\sum_{r+2m=l}q^{r+m}\sum_{T\in\mathcal{T}_{m}}q^{\chi_{n-1}(\mathrm{cat}(T))}=\\
\sum_{r+2m=l}q^{r+m}\sum_{\lambda\in\left( \nu^{\prime}\otimes r\right)
_{C_{n-1}}}c_{\nu^{\prime},r}^{\lambda}K_{\lambda,\mu^{\prime}}(q)=K_{\nu,\mu
}(q).\end{gathered}$$
Assertions $\mathrm{(i)}$ and $\mathrm{(iii)}$ are proved similarly by induction on $n$ starting respectively from $n=1$ and $n=3.$
Set $\nu=(4,1)$ and $\mu=(1,0)$ for type $B_{2}.$ For the $5$ corresponding tableaux of shape $\lambda$ and weight $\mu$ we obtain:
$\chi_{2}^{B}\left(
\begin{tabular}
[c]{|l|lll}\hline
$$ & $$ & \multicolumn{1}{|l}{$\mathtt{\bar
{1}}$} & \multicolumn{1}{|l|}{$\mathtt{1}$}\\\hline
$$ & & & \\\cline{1-1}\end{tabular}
\right) =\chi_{2}^{B}\left( 1\bar{1}\bar{1}\bar{2}\otimes1\right)
=\mathrm{ch}_{A}(1\otimes1\bar{1}\bar{1})+3=4+3=7,\vspace{0.1cm}$
$\chi_{2}^{B}\left(
\begin{tabular}
[c]{|l|lll}\hline
$$ & $$ & \multicolumn{1}{|l}{$\mathtt{0}$} &
\multicolumn{1}{|l|}{$\mathtt{2}$}\\\hline
$$ & & & \\\cline{1-1}\end{tabular}
\right) =\chi_{2}^{B}\left( 20\bar{2}\bar{2}\otimes0\right) =\mathrm{ch}_{A}(0\otimes0)+3=1+3=4,\vspace{0.1cm}$
$\chi_{2}^{B}\left(
\begin{tabular}
[c]{|l|lll}\hline
$$ & $$ & \multicolumn{1}{|l}{$\mathtt{0}$} &
\multicolumn{1}{|l|}{$\mathtt{1}$}\\\hline
$$ & & & \\\cline{1-1}\end{tabular}
\right) =\chi_{2}^{B}\left( 10\bar{1}\bar{2}\otimes0\right) =\mathrm{ch}_{A}(0\otimes10\bar{1})+3=3+3=6,\vspace{0.1cm}$
$\chi_{2}^{B}\left(
\begin{tabular}
[c]{|l|lll}\hline
$$ & $$ & \multicolumn{1}{|l}{$\mathtt{\bar
{1}}$} & \multicolumn{1}{|l|}{$\mathtt{2}$}\\\hline
$$ & & & \\\cline{1-1}\end{tabular}
\right) =\chi_{2}^{B}\left( 2\bar{1}\bar{2}\bar{2}\otimes1\right)
=\mathrm{ch}_{A}(1\otimes\bar{1})+3=2+3=5$,$\vspace{0.1cm}$
$\chi_{2}^{B}\left(
\begin{tabular}
[c]{|l|lll}\hline
$$ & $$ & \multicolumn{1}{|l}{$\mathtt{1}$} &
\multicolumn{1}{|l|}{$\mathtt{1}$}\\\hline
$$ & & & \\\cline{1-1}\end{tabular}
\right) =\chi_{2}^{B}\left( 11\bar{1}\bar{2}\otimes\bar{1}\right)
=\mathrm{ch}_{A}(\bar{1}\otimes11\bar{1})+3=2+3=5\vspace{0.1cm}$
$\chi_{2}^{B}\left(
\begin{tabular}
[c]{|l|lll}\hline
$$ & $$ & \multicolumn{1}{|l}{$\mathtt{1}$} &
\multicolumn{1}{|l|}{$\mathtt{2}$}\\\hline
$$ & & & \\\cline{1-1}\end{tabular}
\right) =\chi_{2}^{B}\left( 21\bar{2}\bar{2}\otimes\bar{1}\right)
=\mathrm{ch}_{A}(\bar{1}\otimes1)+3=0+3=3.$
Finally $K_{\nu,\mu}^{B_{2}}(q)=q^{7}+q^{6}+2q^{5}+q^{4}+q^{3}.$
The following corollary makes clear $K_{\nu,\mu}(q)$ when $\nu$ is a row partition.
\[cor\_xhi\_L\]Let $\nu,\mu$ be two partitions such that $\nu$ is a row partition and $\mu_{\overline{1}}\geq0$. Set $h_{n}(\mu)=\underset
{i=1}{\overset{n}{\sum}}(n-i)\mu\overline{_{i}}.$ Then for any $R\in
\mathbf{T}^{n}(\nu)_{\mu}$ we have
$\mathrm{(i):}$ $\chi_{n}^{B}(R)=h_{n}(\mu)+2\underset{i=1}{\overset{n}{\sum}}(n-i+1)k_{i}$ if $0\notin R$ and $\chi_{n}^{B}(R)=h_{n}(\mu)+2\underset
{i=1}{\overset{n}{\sum}}(n-i+1)k_{i}+n$ otherwise,
$\mathrm{(ii):}$ $\chi_{n}^{C}(R)=h_{n}(\mu)+\underset{i=1}{\overset{n}{\sum}}(2(n-i)+1)k_{i}$
$\mathrm{(iii):}$ $\chi_{n}^{D}(R)=h_{n}(\mu)+2\underset{i=2}{\overset{n}{\sum}}(n-i+1)k_{i}$
where $k_{i}$ is the number of letters $i$ which belong to $R.$
We proceed by recurrence on $n.$
Suppose first $n=1$ for cases $\mathrm{(i)}$ and $\mathrm{(ii)}$. We deduce from proposition \[prop\_degreeK\] that $K_{\nu,\mu}^{C_{1}}(q)=q^{\tfrac{\nu-\mu}{2}}$ and $K_{\nu,\mu}^{B_{1}}(q)=q^{\nu-\mu}.$ Thus $\chi_{1}^{C}(R)=\tfrac{\nu-\mu}{2}=k_{1}$, $$\chi_{1}^{B}(R)=\nu-\mu=\left\{
\begin{tabular}
[c]{l}$2k_{1}$ if $0\notin R$\\
$2k_{1}+1$ otherwise
\end{tabular}
\right.$$ and the Corollary holds for $n=1.$ The rest of the proof is similar to that of proposition 3.2.3 in [@lec3].
Now suppose $n=3$ for case $\mathrm{(iii)}.$ We can write $$R=\begin{tabular}
[c]{|l|l|l|l|l|}\hline
$\overline{3}^{k_{\overline{3}}}$ & $\overline{2}^{k_{\overline{2}}}$ &
$\overline{1}^{k_{\overline{1}}}$ & $2^{k_{2}}$ & $3^{k_{3}}$\\\hline
\end{tabular}$$ where
\[c\][|l|]{}$a^{k}$\
means that there are $k$ boxes containing the letter $a$ in $R.$ Then the semi-standard tableau associated to $R$ by (\[isom\]) is $$R_{A}=\begin{tabular}
[c]{|l|l|l|l|l|}\hline
$1^{k_{\overline{3}}}$ & $1^{k_{\overline{2}}}$ & $2^{k_{\overline{1}}}$ &
$2^{k_{2}}$ & $3^{k_{3}}$\\\hline
$2^{k_{\overline{3}}}$ & $3^{k_{\overline{2}}}$ & $3^{k_{\overline{1}}}$ &
$4^{k_{2}}$ & $4^{k_{3}}$\\\hline
\end{tabular}
.$$ By using the definition of the charge for semi-standard tableaux one verifies that $\mathrm{ch}(R_{A})=\mu_{\overline{2}}+2\mu_{\overline{1}}+2k_{3}+4k_{2}=\chi_{3}^{D}(R).$ Thus the corollary holds for $n=3$ and we terminate as in proof of proposition 3.2.3 in [@lec3].
**Remarks:**
$\mathrm{(i):}$ Write $(r)$ for the row partition whose non zero part is equal to $r.$ From Proposition \[prop\_xhi\] and Corollary \[cor\_xhi\_L\], we deduce that for any partition $\mu\in P_{+}$ we have $K_{(r),\mu
}(q)=q^{h_{n}(\mu)}\times K_{(l),0}(q)$ with $l=r-\left| \mu\right| .$ If $l$ is even we obtain $K_{(l),0}^{B_{n}}(q)=q^{l/2}K_{(l),0}^{C_{n}}(q)$ since the row tableaux of types $B_{n}$ and $C_{n}$ are then identical. Moreover the map $t$ defined from $\mathbf{T}^{B_{n-1}}((l))$ to $\mathbf{T}^{D_{n}}((l))$ by changing each barred letter $\overline{x}$ (resp. unbarred letter $x$) of $R$ into $\overline{x+1}$ (resp. $x+1)$ is a bijection. Hence we have $$K_{(l),0}^{D_{n}}(q)=\sum_{R\in\mathbf{T}^{D_{n}}((l))_{0}}q^{2\underset
{i=2}{\overset{n}{\sum}}(n-i+1)k_{i}}=\sum_{t^{-1}(R)\in\mathbf{T}^{B_{n-1}}((l))_{0}}q^{l+\underset{j=1}{\overset{n-1}{\sum}}2(n-1-j)k_{j}}=K_{(l),0}^{B_{n-1}}(q)=q^{l/2}K_{(l),0}^{C_{n-1}}(q).$$
$\mathrm{(ii):}$ The statistic $\chi_{n}$ can not be used to compute any Kostka-Foulkes polynomial. For type $C_{2},$ $\lambda=(3,1)$ and $\mu=(0,0)$ we have $K_{\lambda,\mu}(q)=q^{5}+q^{4}+q^{3}.$ By considering the $3$ tableaux of type $C_{2},$ shape $\lambda$ and weight $\mu$ we obtain$$\chi_{2}^{C}\left(
\begin{tabular}
[c]{|l|ll}\hline
$\mathtt{\bar{1}}$ & $\mathtt{\bar{1}}$ & \multicolumn{1}{|l|}{1}\\\hline
$\mathtt{1}$ & & \\\cline{1-1}\end{tabular}
\right) =5,\text{ }\chi_{2}^{C}\left(
\begin{tabular}
[c]{|l|ll}\hline
$\mathtt{\bar{2}}$ & $\mathtt{\bar{1}}$ & \multicolumn{1}{|l|}{2}\\\hline
$\mathtt{1}$ & & \\\cline{1-1}\end{tabular}
\right) =3\text{ and }\chi_{2}^{C}\left(
\begin{tabular}
[c]{|l|ll}\hline
$\mathtt{\bar{2}}$ & $\mathtt{1}$ & \multicolumn{1}{|l|}{2}\\\hline
$\mathtt{\bar{1}}$ & & \\\cline{1-1}\end{tabular}
\right) =2$$ and $K_{\lambda,\mu}(q)\neq q^{5}+q^{3}+q^{2}.$
Cyclage graphs for the orthogonal root systems
----------------------------------------------
In [@lec3] we have introduced a (co)-cyclage graph structure on tableaux of type $C.$ We are going to see that such a structure also exists for the partition shaped tableaux of types $B$ and $D$. For any $n\geq1$ we embed the finite alphabets $\mathcal{A}_{n}^{B},\mathcal{A}_{n}^{C}$ and $\mathcal{A}_{n}^{D}$ respectively into the infinite alphabets$$\begin{gathered}
\mathcal{A}_{\infty}^{B}=\{\cdot\cdot\cdot<\overline{n}<\cdot\cdot
\cdot<\overline{1}<0<1<\cdot\cdot\cdot<n<\cdot\cdot\cdot\}\\
\mathcal{A}_{\infty}^{C}=\{\cdot\cdot\cdot<\overline{n}<\cdot\cdot
\cdot<\overline{1}<1<\cdot\cdot\cdot<n<\cdot\cdot\cdot\}\\
\mathcal{A}_{\infty}^{D}=\{\cdot\cdot\cdot<\overline{n}<\cdot\cdot
\cdot<\overline{2}<\begin{tabular}
[c]{l}$\overline{1}$\\
$1$\end{tabular}
<2<\cdot\cdot\cdot<n<\cdot\cdot\cdot\}.\end{gathered}$$ The vertices of the crystal $G_{\infty}^{B}=\underset{n\geq0}{{\textstyle\bigoplus}
}G_{n}^{B},G_{\infty}^{C}=\underset{n\geq0}{{\textstyle\bigoplus}
}G_{n}^{C}$ and $G_{\infty}^{D}=\underset{n\geq0}{{\textstyle\bigoplus}
}G_{n}^{D}$ can be regarded as the words respectively on $\mathcal{A}_{\infty
}^{B},\mathcal{A}_{\infty}^{C}$ and $\mathcal{A}_{\infty}^{D}$.$\;$The congruences obtained by identifying the vertices of $G_{\infty}^{B},G_{\infty
}^{C}$ and $G_{\infty}^{D}$ equal up to the plactic relations of length $3$ are respectively denoted by $\equiv_{B},\equiv_{C}$ and $\equiv_{D}.$ Set $\mathbf{T}^{B}=\underset{n\geq0}{\cup}\mathbf{T}_{n}^{B},$ $\mathbf{T}^{C}=\underset{n\geq0}{\cup}\mathbf{T}_{n}^{C}$ and $\mathbf{T}^{D}=\underset{n\geq0}{\cup}\mathbf{T}_{n}^{D}.$
By Remark $\mathrm{(iii)}$ before Lemma \[lem\_fact\_row\], there exits a unique tableau $P(w)$ such that $w\equiv\mathrm{w}(P(w))$ computed from $w$ without using contraction relation.
In the sequel $\mu$ is a partition with $n$ integers parts. A tableau $T\in\mathbf{T}$ is of weight $\mathrm{wt}(T)=\mu$ if $T\in
\mathbf{T}_{m}$ with $m\geq n$, $d_{\overline{i}}=\mu_{\overline{i}}$ for $1\leq i\leq n$ and $d_{\overline{i}}=0$ for $i>m.$ Set $\mathbf{T}^{B}[\mu]=\{T\in\mathbf{T}^{B}$ of weight $\mu\},$ $\mathbf{T}^{C}[\mu
]=\{T\in\mathbf{T}^{C}$ of weight $\mu\}$ and $\mathbf{T}^{D}[\mu
]=\{T\in\mathbf{T}^{D}$ of weight $\mu\}.$
Consider $T=C_{1}\cdot\cdot\cdot C_{r}\in\mathbf{T}_{\mu}$ with $r>1$ columns. The cocyclage operation is authorized for $T$ if $T$ contains at least a column with a letter $n$ or without letter $\overline{n}.$ In this case, let $x$ be the rightmost letter of the longest row of $T.$ We can write $\mathrm{w}(T)=x\mathrm{w}(T_{\ast})$ where $T_{\ast}\in\mathbf{T}$. Then we set$$U(T)=P(\mathrm{w}(T_{\ast})x).$$ This means that $U(T)$ is obtained by column inserting $x$ in $T_{\ast}$ without using contraction relation.
**Remarks:**
$\mathrm{(i)}\mathbf{:}$ If $\mathrm{wt}(T)=0$ then the cocyclage operation is always authorized.
$\mathrm{(ii)}\mathbf{:}$ By convention there is no cocyclage operation on the columns.
We endow the set $\mathbf{T}[\mu]$ with a structure of graph by drawing an array $T\rightarrow T^{\prime}$ if and only if the cocyclage operation is authorized on $T$ and $U(T)=T^{\prime}.$ Write $\Gamma(T)$ for the connected component containing $T.$
\[cont\_ex\_ch\_B\]For $\mu=(0,0,0)$ the following graphs are connected components of $\mathbf{T}^{B}[\mu]:$$$\begin{gathered}
\text{\begin{tabular}
[c]{|l|l|l|}\hline
$\mathtt{\bar{1}}$ & $\mathtt{0}$ & $\mathtt{1}$\\\hline
\end{tabular}
}\rightarrow\text{\begin{tabular}
[c]{|l|l}\hline
$\mathtt{\bar{1}}$ & \multicolumn{1}{|l|}{$\mathtt{0}$}\\\hline
$\mathtt{1}$ & \\\cline{1-1}\end{tabular}
}\rightarrow\text{\begin{tabular}
[c]{|l|l}\hline
$\mathtt{\bar{2}}$ & \multicolumn{1}{|l|}{$\mathtt{2}$}\\\hline
$\mathtt{0}$ & \\\cline{1-1}\end{tabular}
}\rightarrow\text{\begin{tabular}
[c]{|l|}\hline
$\mathtt{\bar{2}}$\\\hline
$\mathtt{0}$\\\hline
$\mathtt{2}$\\\hline
\end{tabular}
,
\begin{tabular}
[c]{|l|l|l|}\hline
$\mathtt{\bar{2}}$ & $\mathtt{0}$ & $\mathtt{2}$\\\hline
\end{tabular}
}\rightarrow\text{\begin{tabular}
[c]{|l|l}\hline
$\mathtt{\bar{2}}$ & \multicolumn{1}{|l|}{$\mathtt{0}$}\\\hline
$\mathtt{2}$ & \\\cline{1-1}\end{tabular}
}\rightarrow\text{\begin{tabular}
[c]{|l|l}\hline
$\mathtt{\bar{3}}$ & \multicolumn{1}{|l|}{$\mathtt{3}$}\\\hline
$\mathtt{0}$ & \\\cline{1-1}\end{tabular}
}\rightarrow\text{\begin{tabular}
[c]{|l|}\hline
$\mathtt{\bar{3}}$\\\hline
$\mathtt{0}$\\\hline
$\mathtt{3}$\\\hline
\end{tabular}
}\\\begin{tabular}
[c]{|l|l|l|}\hline
$\mathtt{\bar{3}}$ & $\mathtt{0}$ & $\mathtt{3}$\\\hline
\end{tabular}
\rightarrow\begin{tabular}
[c]{|l|l}\hline
$\mathtt{\bar{3}}$ & \multicolumn{1}{|l|}{$\mathtt{0}$}\\\hline
$\mathtt{3}$ & \\\cline{1-1}\end{tabular}
\rightarrow\begin{tabular}
[c]{|l|l}\hline
$\mathtt{\bar{4}}$ & \multicolumn{1}{|l|}{$\mathtt{4}$}\\\hline
$\mathtt{0}$ & \\\cline{1-1}\end{tabular}
\rightarrow\begin{tabular}
[c]{|l|}\hline
$\mathtt{\bar{4}}$\\\hline
$\mathtt{0}$\\\hline
$\mathtt{4}$\\\hline
\end{tabular}
,\text{ }\begin{tabular}
[c]{|l|l}\hline
$\mathtt{\bar{1}}$ & \multicolumn{1}{|l|}{$\mathtt{1}$}\\\hline
$\mathtt{0}$ & \\\cline{1-1}\end{tabular}
\rightarrow\begin{tabular}
[c]{|l|}\hline
$\mathtt{\bar{1}}$\\\hline
$\mathtt{0}$\\\hline
$\mathtt{1}$\\\hline
\end{tabular}
,\text{ }\begin{tabular}
[c]{|l|}\hline
$\mathtt{0}$\\\hline
$\mathtt{0}$\\\hline
$\mathtt{0}$\\\hline
\end{tabular}
.\end{gathered}$$ All these tableaux belong to $T_{3}^{B}$ except
\[c\][|l|]{}$\mathtt{\bar{3}}$\
$\mathtt{0}$\
$\mathtt{3}$\
,
\[c\][|l|l]{}$\mathtt{\bar{3}}$ &\
$\mathtt{3}$ &\
,
\[c\][|l|l]{}$\mathtt{\bar{4}}$ &\
$\mathtt{0}$ &\
which belong to $T_{4}^{B}$ and
\[c\][|l|]{}$\mathtt{\bar{4}}$\
$\mathtt{0}$\
$\mathtt{4}$\
which belongs to $T_{5}^{B}.$
The following proposition is proved in the same way than Proposition 4.2.2 of [@lec3].
\[prop\_cyc\]Let $T_{0}\in\mathbf{T}[0]$ and let $T_{k+1}=U(T_{k})$. Then the sequence $(T_{n})$ is finite without repetition and there exists an integer $e$ such that $T_{e}$ is a column of weight $0.$
In [@lec3] we introduce another statistic $\mathrm{ch}_{C_{n}}$ on Kashiwara-Nakashima’s tableaux of type $C_{n}$ based on cocyclage operation$.$ From $T\in\mathbf{T}^{C}[\mu]$ we define a finite sequence of tableaux $(T_{k})_{0\leq k\leq p}$ whose last tableau $T_{p}$ is a column of weight $0$. When $\mu=0$ this sequence $(T_{k})_{0\leq k\leq p}$ is precisely that given in Proposition \[prop\_cyc\]. Then the statistic $\mathrm{ch}_{C_{n}}$ is first defined on the columns of weight $0$ next on the tableaux by setting $$\mathrm{ch}_{C_{n}}(T)=\mathrm{ch}_{C_{n}}(C_{T})+p.$$ We conjecture that (\[K(q)\_xhi\]) holds if we replace $\chi_{n}^{C}$ by $\mathrm{ch}_{C_{n}}$ whatever the partitions $\lambda$ and $\mu.$ In particular $\mathrm{ch}_{C_{n}}(T)\neq\chi_{n}(T)$ in general.
Unfortunately such a statistic defined in the same way for computing Kostka-Foulkes polynomials can not exist for the orthogonal root systems. This can be verified by considering the case $\left| \lambda\right| =3,$ $\mu=0$ for type $B_{3}$. Set $\lambda_{1}=(3,0,0),$ $\lambda_{2}=(2,1,0)$ and $\lambda_{3}=(1,1,1).$ We have $K_{\lambda_{1},0}^{B_{3}}(q)=q^{9}+q^{7}+q^{5},$ $K_{\lambda_{2},0}^{B_{3}}(q)=q^{8}+q^{7}+q^{6}+q^{5}+q^{4}$ and $K_{\lambda_{3},0}^{B_{3}}(q)=q^{6}+q^{4}+q^{2}.$ Then it is impossible to associate a statistic $\mathrm{ch}_{B_{n}}$ to the $11$ tableaux of type $B_{3},$ weight $0$ and shape $\lambda_{1},\lambda_{2}$ or $\lambda_{3}$ compatible with the cyclage graph structure given in Example \[cont\_ex\_ch\_B\] (that is, such that $\mathrm{ch}_{B_{n}}(T)=\mathrm{ch}_{B_{n}}(T^{\prime})+1$ if $T\rightarrow T^{\prime}$) and relevant for computing the corresponding Kostka-Foulkes polynomials. The situation is similar for type $D_{3},$ $\left| \lambda\right| =3$ and $\mu=(1,0,0).$
Explicit formulas for $K_{\lambda,\mu}(q)$
==========================================
Explicit formulas for $\left| \lambda\right| \leq3$
-----------------------------------------------------
In the sequel we suppose that $\lambda$ is a partition such that $\lambda_{\overline{1}}\geq0.$ We give below the matrix $K(q)=(K_{\lambda,\mu
}(q))$ with $\left| \lambda\right| \leq3$ associated to each root system $B_{n},C_{n}$ and $D_{n}.$ When $\left| \lambda\right| =\left| \mu\right|
,$ $K_{\lambda,\mu}(q)$ can be regarded as a Kostka-Foulkes polynomial for the root system $A_{n-1}$. Such polynomials have been already compute (see [@mac] p 329). So we only give the entries of $K(q)$ corresponding to a weight $\mu$ such that $\left| \mu\right| \leq2$. In the following matrices we have labelled the columns by $\lambda$ and the rows by $\mu$ and represent each partition by its Young diagram. The expressions for the Kostka-Foulkes polynomials are obtained by using Proposition \[prop\_degreeK\], Theorem \[Th\_mor\_expli\], Proposition \[prop\_xhi\] and Corollary \[cor\_xhi\_L\].
### $K(q)$-matrix for the root system $B_{n}$
$$\begin{array}
[c]{ccccccc}\vspace{0.2cm} &
\begin{tabular}
[c]{|l|l|l|}\hline
& & \\\hline
\end{tabular}
&
\begin{tabular}
[c]{|l|l}\hline
& \multicolumn{1}{|l|}{}\\\hline
& \\\cline{1-1}\end{tabular}
&
\begin{tabular}
[c]{|l|}\hline
\\\hline
\\\hline
\\\hline
\end{tabular}
&
\begin{tabular}
[c]{|l|l|}\hline
& \\\hline
\end{tabular}
&
\begin{tabular}
[c]{|l|}\hline
\\\hline
\\\hline
\end{tabular}
&
\begin{tabular}
[c]{|l|}\hline
\\\hline
\end{tabular}
\\\begin{tabular}
[c]{|l|l|}\hline
& \\\hline
\end{tabular}
\vspace{0.2cm} & q^{n} & q^{n-1} & 0 & 1 & 0 & 0\\\begin{tabular}
[c]{|l|}\hline
\\\hline
\\\hline
\end{tabular}
\vspace{0.2cm} & q^{n+1} & q^{n}+q^{n-1} & q^{n-2} & q & 1 & 0\\\begin{tabular}
[c]{|l|}\hline
\\\hline
\end{tabular}
\vspace{0.4cm} & q^{2}\times\tfrac{q^{2n}-1}{q^{2}-1} & q^{n}+q\times
\tfrac{q^{2n-1}-1}{q-1} & q\times\tfrac{q^{2n-2}-1}{q^{2}-1} & q^{n} & q^{n-1}& 1\\
\emptyset & q^{n+2}\times\tfrac{q^{2n-1}-1}{q-1} & q^{n+1}\times
\tfrac{q^{2n-1}-1}{q-1} & q^{n-1}\times\tfrac{q^{2n}-1}{q^{2}-1} & q^{2}\times\tfrac{q^{2n}-1}{q^{2}-1} & q\times\tfrac{q^{2n}-1}{q^{2}-1} & q^{n}\end{array}$$
### $K(q)$-matrix for the root system $C_{n}$
$$\begin{array}
[c]{cccccc}\vspace{0.2cm} &
\begin{tabular}
[c]{|l|l|l|}\hline
& & \\\hline
\end{tabular}
&
\begin{tabular}
[c]{|l|l}\hline
& \multicolumn{1}{|l|}{}\\\hline
& \\\cline{1-1}\end{tabular}
&
\begin{tabular}
[c]{|l|}\hline
\\\hline
\\\hline
\\\hline
\end{tabular}
&
\begin{tabular}
[c]{|l|l|}\hline
& \\\hline
\end{tabular}
&
\begin{tabular}
[c]{|l|}\hline
\\\hline
\\\hline
\end{tabular}
\\\begin{tabular}
[c]{|l|l|}\hline
& \\\hline
\end{tabular}
\vspace{0.2cm} & 0 & 0 & 0 & 1 & 0\\\begin{tabular}
[c]{|l|}\hline
\\\hline
\\\hline
\end{tabular}
\vspace{0.2cm} & 0 & 0 & 0 & q & 1\\\begin{tabular}
[c]{|l|}\hline
\\\hline
\end{tabular}
\vspace{0.4cm} & q\times\tfrac{q^{2n}-1}{q^{2}-1} & q\times\tfrac{q^{2n-2}-1}{q-1} & q^{2}\times\tfrac{q^{2n-4}-1}{q^{2}-1} & 0 & 0\\
\emptyset & 0 & 0 & 0 & q\times\tfrac{q^{2n}-1}{q^{2}-1} & q^{2}\times
\tfrac{q^{2n-2}-1}{q^{2}-1}\end{array}$$
### $K(q)$-matrix for the root system $D_{n}$
$$\begin{array}
[c]{cccccc}\vspace{0.2cm} &
\begin{tabular}
[c]{|l|l|l|}\hline
& & \\\hline
\end{tabular}
&
\begin{tabular}
[c]{|l|l}\hline
& \multicolumn{1}{|l|}{}\\\hline
& \\\cline{1-1}\end{tabular}
&
\begin{tabular}
[c]{|l|}\hline
\\\hline
\\\hline
\\\hline
\end{tabular}
&
\begin{tabular}
[c]{|l|l|}\hline
& \\\hline
\end{tabular}
&
\begin{tabular}
[c]{|l|}\hline
\\\hline
\\\hline
\end{tabular}
\\\begin{tabular}
[c]{|l|l|}\hline
& \\\hline
\end{tabular}
\vspace{0.2cm} & 0 & 0 & 0 & 1 & 0\\\begin{tabular}
[c]{|l|}\hline
\\\hline
\\\hline
\end{tabular}
\vspace{0.2cm} & 0 & 0 & 0 & q & 1\\\begin{tabular}
[c]{|l|}\hline
\\\hline
\end{tabular}
\vspace{0.4cm} & q^{2}\times\tfrac{q^{2n-2}-1}{q^{2}-1} & q^{n-1}+q\times\tfrac{q^{2n-3}-1}{q-1} & q^{n-2}+q\times\tfrac{q^{2n-4}-1}{q^{2}-1} &
0 & 0\\
\emptyset & 0 & 0 & 0 & q^{2}\times\tfrac{q^{2n-2}-1}{q^{2}-1} &
q^{n-1}+q\times\tfrac{q^{2n-2}-1}{q^{2}-1}\end{array}$$
**Remark:** For $n\geq4$ the partitions $\lambda$ and $\mu$ in the above matrix verify $\lambda^{\ast}=\lambda$ and $\mu^{\ast}=\mu.$ Hence by (\[K=K\*\]) we have $K_{\lambda,\mu}^{D_{n}}(q)=K_{\lambda^{\ast},\mu^{\ast}}^{D_{n}}(q)=K_{\lambda^{\ast},\mu}^{D_{n}}(q)=K_{\lambda,\mu
^{\ast}}^{D_{n}}(q).$
Explicit formulas for the root system $B_{2}=C_{2}$ and $\mu=0$
---------------------------------------------------------------
Note first that the roots systems $B_{2}$ and $C_{2}$ are identical. More precisely denote by $\Psi$ the linear map$$\Psi:\left\{
\begin{tabular}
[c]{c}$P_{B_{2}}^{+}\rightarrow P_{C_{2}}^{+}$\\
$(\lambda_{\overline{2}},\lambda_{\overline{1}})\longmapsto(\lambda
_{\overline{2}}+\lambda_{\overline{1}},\lambda_{\overline{2}}-\lambda
_{\overline{1}})$\end{tabular}
\right. .$$ Accordingly to (\[simple\_roots\]), the simple roots for the roots systems $B_{2}$ and $C_{2}$ are $\alpha_{0}^{B_{2}}=\varepsilon_{\overline{1}},\alpha_{1}^{B_{2}}=\varepsilon_{\overline{2}}-\varepsilon_{\overline{1}}$ and $\alpha_{0}^{C_{2}}=2\varepsilon_{\overline{1}},\alpha_{1}^{C_{2}}=\varepsilon_{\overline{2}}-\varepsilon_{\overline{1}}.$ Thus we have $\Psi(\alpha_{0}^{B_{2}})=\alpha_{1}^{C_{2}}$ and $\Psi(\alpha_{1}^{B_{2}})=\alpha_{0}^{C_{2}}.$ This implies the equality$$K_{(\lambda,\mu)}^{B_{2}}(q)=K_{\Psi(\lambda,\mu)}^{C_{2}}(q). \label{B2=C2}$$ So it is sufficient to explicit the Kostka-Foulkes polynomials for the root system $C_{2}.$
Let $\lambda=(\lambda_{\overline{2}},\lambda_{\overline{1}})$ be a generalized partition of length $2.$
1. If $\lambda\in P_{+}^{C_{2}}$ then $$K_{\lambda,0}^{C_{2}}(q)=\left\{
\begin{tabular}
[c]{l}$q^{\tfrac{\lambda_{\overline{2}}+\lambda_{\overline{1}}}{2}}\left(
\dfrac{q^{\lambda_{\overline{1}}+2}-1}{q^{2}-1}+q^{2}\times\dfrac
{q^{\lambda_{\overline{1}}+1}-1}{q-1}\times\dfrac{q^{\lambda_{\overline{2}}-\lambda_{\overline{1}}}-1}{q^{2}-1}\right) $ if $\lambda_{\overline{2}}$
and $\lambda_{\overline{1}}$ are even\vspace{0.1cm}\\
$q^{\tfrac{\lambda_{\overline{2}}+\lambda_{\overline{1}}}{2}+1}\left(
\dfrac{q^{\lambda_{\overline{1}}+1}-1}{q^{2}-1}+q\times\dfrac{q^{\lambda
_{\overline{1}}+1}-1}{q-1}\times\dfrac{q^{\lambda_{\overline{2}}-\lambda_{\overline{1}}}-1}{q^{2}-1}\right) $ if $\lambda_{\overline{2}}$ and
$\lambda_{\overline{1}}$ are odd\vspace{0.1cm}\\
$0$ otherwise.
\end{tabular}
\right. .$$
2. If $\lambda\in P_{+}^{B_{2}}$ then $$K_{\lambda,0}^{B_{2}}(q)=\left\{
\begin{tabular}
[c]{l}$q^{\lambda_{\overline{2}}}\left( \dfrac{q^{2\lambda_{\overline{1}}+2}-1}{q^{2}-1}+q^{2}\times\dfrac{q^{2\lambda_{\overline{1}}+1}-1}{q-1}\times\dfrac{q^{\lambda_{\overline{2}}-\lambda_{\overline{1}}}-1}{q^{2}-1}\right) $ if $\lambda_{\overline{2}}+\lambda_{\overline{1}}$ is
even\vspace{0.1cm}\\
$q^{\lambda_{\overline{2}}+1}\times\dfrac{q^{2\lambda_{\overline{1}}+1}-1}{q-1}\times\dfrac{q^{\lambda_{\overline{2}}-\lambda_{\overline{1}}+1}-1}{q^{2}-1}$ otherwise
\end{tabular}
\right. .$$
$1:$ Note first that $K_{\lambda,0}^{C_{2}}(q)=0$ if $\left| \lambda\right|
$ is odd since all the tableaux of weight $0$ and type $C_{2}$ must have a pair number of boxes. So we can suppose that $\lambda_{\overline{2}}$ and $\lambda_{\overline{1}}$ have the same parity. By Theorem \[Th\_mor\_expli\] we must have $$K_{\lambda,0}^{C_{2}}(q)=\sum_{r+2m=\lambda_{\overline{2}}}q^{r+m}\sum
_{\eta\in((\lambda_{\overline{1}})\otimes r)_{1}}c_{(\lambda_{\overline{1}}),r}^{\eta}K_{\eta,0}^{C_{1}}(q)-\sum_{r+2m=\lambda_{\overline{1}}-1}q^{r+m}\sum_{\eta\in((\lambda_{\overline{2}}+1)\otimes r)_{1}}c_{(\lambda
_{\overline{2}}+1),r}^{\eta}K_{\eta,0}^{C_{1}}(q)$$ where by abuse of notation the second sum is equal to $0$ if $\lambda
_{\overline{1}}=0.$ Now the $K_{\eta,0}^{C_{1}}(q)$’s are Kostka-Foulkes polynomials for the root system $C_{1}=A_{1}$ hence $K_{\eta,0}^{C_{1}}(q)=q^{\eta/2}.$ Moreover Lemma \[lem\_plu\_hp\] implies that $$B(\gamma)\otimes B(r)=\underset{p=0}{\overset{\min(\gamma,r)}{\cup}}B(\gamma+r-2p)$$ for any integers $\gamma,r.$ We obtain$$K_{\lambda,0}^{C_{2}}(q)=\sum_{r+2m=\lambda_{\overline{2}}}\sum_{p=0}^{\min(\lambda_{\overline{1}},r)}q^{r+m}\times q^{\tfrac{\lambda_{\overline
{1}}+r}{2}-p}-\sum_{r+2m=\lambda_{\overline{1}}-1}\sum_{p=0}^{r}q^{r+m}\times
q^{\tfrac{\lambda_{\overline{2}}+r+1}{2}-p}.$$ Indeed we have $\min(\lambda_{\overline{2}}-1,r)=r$ in the second sum since $r\leq\lambda_{\overline{1}}-1<\lambda_{\overline{2}}+1.$ This can be rewritten as$$\begin{gathered}
K_{\lambda,0}^{C_{2}}(q)=\sum_{\underset{r\equiv\lambda_{\overline{2}}\text{
}\operatorname{mod}2}{r=0}}^{\lambda_{\overline{1}}}\sum_{p=0}^{r}q^{r+\tfrac{\lambda_{\overline{2}}-r}{2}+\tfrac{\lambda_{\overline{1}}+r}{2}-p}+\sum_{\underset{r\equiv\lambda_{\overline{2}}\text{ }\operatorname{mod}2}{r=\lambda_{\overline{1}}+1}}^{\lambda_{\overline{2}}}\sum_{p=0}^{\lambda_{\overline{1}}}q^{r+\tfrac{\lambda_{\overline{2}}-r}{2}+\tfrac{\lambda_{\overline{1}}+r}{2}-p}-\sum_{\underset{r\equiv\lambda
_{\overline{1}}-1\text{ }\operatorname{mod}2}{r=0}}^{\lambda_{\overline{1}}-1}\sum_{p=0}^{r}q^{r+\tfrac{\lambda_{\overline{1}}-r-1}{2}+\tfrac
{\lambda_{\overline{2}}+r+1}{2}-p}\\
=q^{\tfrac{\lambda_{\overline{2}}+\lambda_{\overline{1}}}{2}}\left(
\sum_{\underset{r\equiv\lambda_{\overline{2}}\text{ }\operatorname{mod}2}{r=0}}^{\lambda_{\overline{1}}}\sum_{p=0}^{r}q^{r-p}+\sum_{\underset
{r\equiv\lambda_{\overline{2}}\text{ }\operatorname{mod}2}{r=\lambda
_{\overline{1}}+1}}^{\lambda_{\overline{2}}}\sum_{p=0}^{\lambda_{\overline{1}}}q^{r-p}-\sum_{\underset{r\equiv\lambda_{\overline{1}}-1\text{ }\operatorname{mod}2}{r=0}}^{\lambda_{\overline{1}}-1}\sum_{p=0}^{r}q^{r-p}\right) .\end{gathered}$$ Then the Proposition easily follows by distinguishing the two cases $\lambda_{\overline{2}}$ even and $\lambda_{\overline{2}}$ odd.
$2:$ This is an immediate consequence of $1$ and (\[B2=C2\]).
**Remark:**
$\mathrm{(i):}$ Similar formulas also exist for the root system $A_{2}.$ For any partition $\lambda=(a,b,0)$ we have$$K_{\lambda,0}^{A_{2}}(q)=\left\{
\begin{tabular}
[c]{l}$q^{a-b}\times\dfrac{q^{a+1}-1}{q-1}$ if $a\geq2b$\\
$q^{b}\times\dfrac{q^{a-b+1}-1}{q-1}$ otherwise.
\end{tabular}
\right. .$$
$\mathrm{(ii):}$ For a weight $\mu\neq0,$ the situation becomes more complex and simple formulas for the $K_{\lambda,\mu}(q)$ seem do not exist.
[99]{} <span style="font-variant:small-caps;">T-H. Baker,</span> *An* *insertion scheme for* $C_{n}$* crystals*, in M. Kashiwara and T. Miwa, eds., Physical Combinatorics, Birkhäuser, Boston, 2000, **191**: 1-48.
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|
---
abstract: 'We show the effects of the perturbation caused by a passing by star on the Kuiper belt objects (KBOs) of our Solar System. The dynamics of the Kuiper belt (KB) is followed by direct $N$-body simulations. The sampling of the KB has been done with $N$ up to $131,062$, setting the KBOs on initially nearly circular orbits distributed in a ring of surface density $\Sigma \sim r^{-2}$. This modelization allowed us to investigate the secular evolution of the KB upon the encounter with the perturbing star. Actually, the encounter itself usually leads toward eccentricity and inclination distributions similar to observed ones, but tends also to excite the low-eccentricity population ($e\aplt 0.1$ around $a\sim 40$$\mathrm{AU}$ from the Sun), depleting this region of low eccentricities. The following long-term evolution shows a “cooling" of the eccentricities repopulating the low-eccentricity area. In dependence on the assumed KBO mass spectrum and sampled number of bodies, this repopulation takes place in a time that goes from 0.5Myr to 100Myr. Due to the unavoidable limitation in the number of objects in our long-term simulations ($N \leq 16384$), we could not consider a detailed KBO mass spectrum, accounting for low mass objects, thus our present simulations are not reliable in constraining correlations among inclination distribution of the KBOs and other properties, such as their size distribution. However, our high precision long term simulations are a starting point for future larger studies on massively parallel computational platforms which will provide a deeper investigation of the secular evolution ($\sim 100\,$Myr) of the KB over its whole mass spectrum.'
author:
- |
D. Punzo$^{1}$ $^{2}$, R. Capuzzo-Dolcetta$^{1}$, S. Portegies Zwart$^{3}$\
$^{1}$ Dep. of Physics, Sapienza, University of Roma, P.le A. Moro 1, Roma, Italy\
$^{2}$ Kapteyn Institute, Rijksuniversiteit, Landleven 12, 9747AD Groningen, Netherlands\
$^{3}$ Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands
bibliography:
- 'KBO.bib'
title: The secular evolution of the Kuiper belt after a close stellar encounter
---
MNRAS Accepted 2014 August 11th. Received 2014 July 14th; in original form 2014 March 21th.
Kuiper belt: general; methods: numerical; planets and satellites: dynamical evolution and stability.
Introduction
============
The Solar System is hedged by a ring composed of a huge number of small bodies: the Edgeworth-Kuiper belt [@Jewitt] (hereafter briefly called Kuiper belt, or KB). The Kuiper belt bears the signature it the early evolution of the Solar System, and a contains records of the end-state of the accretion processes occurred in that region. Therefore, the knowledge of the history of the Kuiper belt objects (KBOs) is relevant to be able to develop a full consensus of the formation of the Solar System.
The majority of the KBOs are located between about 30$\mathrm{AU}$ and 90 $\mathrm{AU}$ from the Sun, but most are around the 2:3 resonance with Jupiter, at 39.5$\mathrm{AU}$ and at its 1:2 resonance, roughly around 48$\mathrm{AU}$. The total mass is estimated from $0.01$ to $0.1$ M$_\oplus$ [@Luu]. There are several, indirect, arguments suggesting that this is just a small fraction of its initial mass because most of it has been lost (see [@Kenyon]). The size distribution of the KBOs is, usually, assumed as a power law $dn/dR = A R^{-q}$, where $A$ and $q$ are constants. The $q$ exponent is estimated $\sim 4.0 \pm 0.5$ [@Ber; @Fraser]. For a more detailed description of the Kuiper belt we refer, e.g., to [@Luu].
The KB has a bimodal inclination distribution resulting of two separate populations [@Brown]. The *dynamically cold* population refers to objects moving on almost planar orbits with relatively low inclinations (up to about 10$^\circ$) respect to the ecliptic. On the other side, the dynamically hot population is characterized by highly inclined orbits (up to 40$^\circ$) with respect to the ecliptic. Note that these two populations are different from what we call, in this paper, the *low-eccentricity* population, which are objects on nearly circular orbits (orbital eccentricities $< 0.1$) and the *high-eccentricity* population (eccentricities $\gtrsim$ 0.1.
In Fig. \[fig:oss\] the eccentricities and inclinations are plotted as function of the semi-major axis for KBOs observed from the Minor Planet Center (MPC) [@mpc] which is the center of the Smithsonian Astrophysical Observatory (SAO) dedicated to tracking, monitoring, calculating and disseminating data from asteroids and comets.
The KBOs have been sub-categorized in three groups:
1. classical KBOs ($42$ $< a < 49$ , $\langle e \rangle
\simeq 0.09$, $\langle i \rangle \simeq 7^{\circ}$);
2. scattered KBOs ($a > 30$ , $\langle e \rangle \simeq 0.49$, $\langle i \rangle \simeq 14^{\circ}$ );
3. main resonant KBOs :
- 4:3 resonance ($a \simeq36,4$, $\langle e \rangle \simeq
0.22$, $\langle i \rangle \simeq 8^{\circ}$);
- 3:2 resonance, Plutino’s ($a \simeq39.4$, $\langle e
\rangle \simeq 0.36$,$\langle i \rangle \simeq 13^{\circ}$);
- 2:1 resonance ($a \simeq47.8$, $\langle e \rangle \simeq
0.14$, $\langle i \rangle \simeq 10^{\circ}$).,
where semimajor axes, $a$, are in $\mathrm{AU}$.
These sub-populations have been explain through a phase of planet migration and a phase of clearing of the environment during the evolution of the early Solar System [@Mal1; @Mal2]. In the latter phase the resonance population was formed by sweeping resonance capture in which the Jovian planets withstand considerable orbital migration as a result of encounters with residual planetesimals. While Neptune moved outwards, a small body like Pluto in an initially circular orbit could have been captured into the 3:2 resonance. The high orbital eccentricity would subsequently be induced by repeated orbital crossings with Neptune.
Many others studies have attempted to better understand the properties of the KBOs. [@Gomes] investigated how the outward migration of Neptune, as proposed by [@Mal1; @Mal2], could have scattered objects from $25$ $\mathrm{AU}$ onto high-$i$ orbits leading to the current classical Kuiper belt region. He concluded that the high-$i$ population was formed closer to the Sun and brought into the classical Kuiper belt during planetary migration, whereas the cold population represents a primordial, relatively undisturbed population. This also led to the speculation that other mechanisms, such as planetary migration, have been the cause of the correlation between inclinations and colors in the classical Kuiper belt rather than environmental effects like the collisions among the KBOs (see [@Dore]). Detailed discussions about the correlation of the inclination with the color, size and binary of the KBOs are given by [@Levi1; @Bru; @Noll; @Volk]. More recently a model, called the Nice model, has been proposed [@Levi], which argues that the giant planets migrated from an initial compact configuration into their present orbits, long after the dissipation of the initial protoplanetary gas disk. The Nice model seems to provide a acceptable explanation for the formation of the classical and scattered populations, and for the correlation between inclinations and colors (for more detail see [@Levi]). The Nice model, however, predicts a higher eccentricities in classical KBO orbits than is observed.
An interaction between a passing field star and the the Solar System could also be responsible for some of the orbital families observed in the KBO, which is the main topic of this paper.
The fly-by star perturbation and the $N$-body scheme
====================================================
An encounter between a passing star and the Solar System is quite likely, considering that the Solar System was probably formed in an open star cluster [@simon1]. The hypothesis of a closely passing star has been hypothesized before, and used to explain KBO families [@Ida1; @Ida2; @Ida3; @Melita; @Malberg]. The cost of such calculations, however, prevented earlier research on the secular evolution of the KBO by mean of high resolution simulations.
We focus our attention on the investigation of the effects of the long-term evolution of the Kuiper belt after a close stellar encounter on the structure of the Kuiper belt. We adopt a direct $N$-body treatment in which the mutual, pair-wise, interactions between KBOs, planets and stars are taken into account self-consistently. Due to the computational expense of this method we are limited to about $131,072$ total bodies.
We modeled the early Solar System as composed by the Sun, the eight major planets, Pluto and the Kuiper belt. Each object was considered a point-mass; we did not account for collisions. The KBOs were initially moving in circular orbits in a flat ring in the plane of the ecliptic. This corresponds to an initially cold population, without a $z-$component in their motion. We adopted a surface density $\Sigma
\propto r^{-2}$, where $r$ is the heliocentric distance (see [@Holman]).
We studied two possible configurations:
- \(i) model A, with a radial extension in the range from $42$ $\mathrm{AU}$ to $48$ $\mathrm{AU}$ and four different values of the total mass of the KB, $M = 3, 6, 1, 30$ M$_\oplus$;
- \(ii) model B, with a radial extension in the range from $42$ $\mathrm{AU}$ to $90$ $\mathrm{AU}$ and a total mass $M = 30$ M$_\oplus$.
The mass function of the KBOs was derived from the conversion in mass of the size ($R$) distribution, assuming a constant KBO density (i.e. $\rho \sim 10^3$ kg/m$^{3}$), which results in $dm/dR \propto \rho R^{-2.0 \pm 0.5}$ with a cut at $m_{min} = \frac{4}{3} \pi \rho R^3 \approx 7.0 \; 10^{-13} $M$_\oplus$ (corresponding to $R=1$ km) and $m_{max}\approx 7.0 \;
10^{-4} $M$_\oplus$ (corresponding to $R=10^3$ km). In Fig. \[fig:dens\] we present the surface density of the models A and B as a function of the heliocentric distance for KBOs with the same individual mass, $m = M / N$. Here $M$ is the total mass of the Kuiper belt and $N$ the number of KBOs. With this choice we have a good sampling of the KB without an exceedingly large number ($N> 10^6$) of particles.
We integrate the equations of motion by direct summation $N$-body codes running on Graphics Processing Units (GPUs). For the gravitational $N$-body problem these accelerators give a manifold speed increase with respect to code running on CPU [@nyland; @Simon; @Jeroen]. Parallel computers equipped more than one hundred GPUs have been utilized for various studies [@HiGPUs; @GPUcomp; @Berczik1; @Berczik2] have been run efficiently in parallel to provide the computational power necessary to perform direct many body simulations. Access to such large GPU-equipped supercomputers, however, is not easy, in particular when the computations required a considerable fraction of the available hardware. We therefore mainly ran our simulation on the Little Green Machine, a at the Sterrewacht Leiden built dedicated GPU-equipped supercomputer, specifically built for performing GPU-related calculations. Even with this machine, we had to limit the number of bodies to about a hundred thousand, but we performed several simulations (of models A and B) for each realization of the initial conditions in order to assure that the results of our calculations were not a statistical anomaly. Recently [@Tjarda] demonstrated that performing multiple simulations with the same initial conditions provide a statistically correct sampling of the real solutions. In these runs we varied the mass, impact parameter and the inclination of the incoming star.
$$\label{eq:init}
\begin{cases}
\begin{array}{cc}
M_{\star} = [0.5;1;2] {\text M}_{\odot}\\
x=500 & v_{x,\infty}\simeq -3, \\
y=b\cos\theta & v_{y,\infty}=0, \\
z=b\sin\theta & v_{z,\infty}=0,
\end{array}
\end{cases}$$
where $x,y,z$ are in $\mathrm{AU}$ and velocities in km/s. The system of reference was centered on the Sun, and the impact parameters $b$ and inclination $\theta$ characterize the orbit of the encountering star. The incoming star was placed in a ring of radius $b$ at a distance of 500$\mathrm{AU}$ in the $x$-direction parallel to the $yz$ plane. In Tab. \[tab:tabcond\] we present the initial conditions for our simulations. In order to have a full coverage of the parameter space, $v_{y,\infty}$ and $v_{z,\infty}$ should be varied as free parameters. However, a systematic set of $N$ body simulations is computational expensive (at least when considering $N$ large enough to guarantee a good sampling) forced us to reduce the investigation in the parameter space. Consequently, we considered that the most relevant thing to do was exploring the role of the initial $yz$ spatial coordinates. Actually, the variation of two free parameters are enough for exploring encounters with different strength (see Sect. \[resu\]). Of course, a more extended study of the other free parameters could allow a wider comprehension of the role of stellar encounters on the KB structure.
index $b$ $\theta$ $y(\mathrm{AU})$ $z(\mathrm{AU})$ index $b$ $\theta$ $y(\mathrm{AU})$ $z(\mathrm{AU})$
------- ----- ---------- ------------------ ------------------ ------- ------- ---------- ------------------ ------------------
1 140 90 0.000 140.000 33 200 30 173.205 100.000
2 140 100 -24.311 137.873 34 200 60 100.000 173.205
3 140 110 -47.883 131.557 35 200 70 68.404 187.939
4 150 30 129.904 75.000 36 200 75 51.764 193.185
5 150 60 75.000 129.904 37 200 80 34.730 196.962
6 150 90 0.000 150.000 38 200 90 0.000 200.000
7 150 100 -26.047 147.721 39 200 105 -51.764 193.185
8 150 110 -51.303 140.954 40 200 120 -100.000 173.205
9 150 120 -75.000 129.904 41 200 135 -141.421 141.421
10 150 150 -129.904 75.000 42 200 150 -173.205 100.000
11 160 90 0.000 160.000 43 212.5 60 106.250 184.030
12 160 100 -27.784 157.569 44 212.5 75 54.999 205.259
13 160 110 -54.723 150.351 45 212.5 90 0.000 212.500
14 170 90 0.000 170.000 46 212.5 105 -54.999 205.259
15 170 100 -29.520 167.417 47 212.5 120 -106.250 184.030
16 170 110 -58.143 159.748 48 212.5 135 -150.260 150.260
17 170 90 0.000 170.000 49 212.5 150 -184.030 106.250
18 170 100 -29.520 167.417 50 225 30 194.856 112.500
19 170 110 -58.143 159.748 51 225 60 112.500 194.856
20 175 30 151.554 87.500 52 225 75 58.234 217.333
21 175 60 87.500 151.554 53 225 90 0.000 225.000
22 175 70 59.854 164.446 54 225 105 -58.234 217.333
23 175 80 30.388 172.341 55 225 120 -112.500 194.856
24 175 90 0.000 175.000 56 225 135 -159.099 159.099
25 175 120 -87.500 151.554 57 225 150 -194.856 112.500
26 175 150 -151.554 87.500 58 237.5 60 118.750 205.681
27 180 70 61.564 169.145 59 237.5 75 61.470 229.407
28 180 80 31.257 177.265 60 237.5 90 0.000 237.500
29 180 90 0.000 180.000 61 237.5 105 -61.470 229.407
30 190 70 64.984 178.542 62 237.5 120 -118.750 205.681
31 190 80 32.993 187.113 63 237.5 135 -167.938 167.938
32 190 90 0.000 190.000 64 237.5 150 -205.681 118.750
Calculations were performed using the direct summation code [ HiGPUs]{} [@HiGPUs], which is publicly available via the Astronomical Multipurpose Software Environment (AMUSE) [@amuse1; @amuse2; @simon2].
This code uses its own kernels to implement at best a 6th-order Hermite’s integrator [@nitadori] with block time-steps [@ars] method.
We tested the accuracy of [ HiGPUs]{} in getting the results of interest here through comparison with two symplectic $N$-body codes, [NBSymple]{} [@NBSymple], which is based on a symplectic second and sixth order method for the time integration of the equations of motion and [HUAYNO]{} [@HUYANO], which uses recursive-Hamiltonian splitting to generate multiple-timestep integrators that conserve momentum to machine precision. The comparison indicates as fully reliable the simulations done with the (much faster) [HiGPUs]{} code.
All the simulations were performed using a softening parameter, $\epsilon$, in the pairwise Newtonian potential $U_{ij} \propto
\sqrt{r_{ij}^2+\epsilon^2}$, where $r_{ij}$ is the $i-th$ to $j-th$ particle distance. The $\epsilon$ value was set to $4\times 10^{-4}$ $\mathrm{AU}$, which is $\sim 1500$ times smaller than the initial average distance to the nearest neighbour in our sampling, and $\sim 60$ times bigger than the radius of Pluto. This choice guarantees the preservation of the newtonian behaviour of the interobject force while keeping under control spurious fluctuations over the mean field (see following Subsect. \[softening\]). The maximum time step for the hierarchical block time steps was $\sim 0.02$ $yr$. The energy conservation was checked along the system evolution by its fractional time variation defined as
$$\left|\frac{\Delta E}{E}\right| = \left| \frac{E(t)-E(0)}{E(0)}\right|.$$
At the end of the simulations it was always below the value $10^{-7}$, which is more than sufficient to assure that we statistically correctly sample the result of a true (converged) solution to the $N$-body problem [@Tjarda].
The role of softening {#softening}
---------------------
The real KB is likely composed of various thousands objects. The study of the secular evolution of the KB with a high-precision, direct summation, $N$-body code after the encounter with a passing-by star is out of reach with our available hardware. For this reason, to represent the KBOs we limited to values of $N$ just below $10^4$, taking as reference value $N=2^{13}-10 = 8182$ (ten bodies represent the incoming star, the Sun and the planets) which showed to be a good compromise between accuracy (resolution) and computational speed. Of course, to give physical reliability to our results obtained by such subsampling, we needed the introduction of a softening parameter ($\epsilon$, as described above) whose size must be calibrated. Actually, the role of softening parameter in the N-body simulation is a long, highly debated question. It is well known that a softening parameter in the Newtonian particle-particle force has the double role of i) avoiding the ultraviolet singularity in the closest interaction and ii) reducing the spurious granularity effects induced by the use of a sub sampled N-body set of particles to reproduce the evolution of a large stellar system. The choice of the softening length, $\epsilon$, is characterized by a proper balance between the width of the softening length, to be small enough to preserve the Newtonian behaviour and, at the same time, large enough to avoid spurious collisionally in the evolutionary behaviour of the system. To fulfil the second requirement above, $\epsilon$ is necessarily much larger than the average KBO radius but this is not a serious issue because the average close neighbour distance, $\langle d_{cn} \rangle$, in the simulated KB system is $\simeq 0.6$ $\mathrm{AU}$ $>>$ the average KB radius. On the other hand, the first requirement above (preserve Newtonian behaviour of the force) requires an $\epsilon$ sufficiently smaller than $\langle d_{cn} \rangle$. Actually, we tested the simulations with two values of the softening ($4\times 10^{-4}$ $\mathrm{AU}$ and $4\times 10^{-5}$ $\mathrm{AU}$ for $\epsilon$), which are both significantly smaller than $\langle d_{cn} \rangle$ which is $\simeq 0.6$ $\mathrm{AU}$ for $N=8182$. As additional, practical, confirmation that the range of $\epsilon$ explored corresponds to reliable results, we saw that overall results remain almost unchanged with the two different choices for the $\epsilon$ value. So, we feel quite confident that our $N$ body results are solid in stating about KB secular evolution after the stellar encounter.
Results {#resu}
=======
Here we report on the results of the simulations using our two initial conditions, model A and model B.
Model A
-------
In model A we adopted a radial extension of the KBO between 42$\mathrm{AU}$ to 48$\mathrm{AU}$ using a total mass of $M = 1$, 3, 6 and 30M$_\oplus$. The mass and encounter parameters of the incoming star are presented in Tab.\[tab:tabcond\]. The simulations are carried out until the perturbation induced by the passing star is negligible, even compared to the inter KBO forces (the gravitational contribute on the total force on a generic KBO due to the passing star is five magnitude lower respect the Sun and one magnitude comparing with the nearest KBO neighbour).
In Fig. \[fig:plane\] we give the distribution of the KB as obtained by model A. The figure presents the projection of the sampled system onto the $xy$ and the $yz$ planes $\sim 2500$yr after the closest approach between the Sun and the perturbing star. Some KBOs were scattered out to a heliocentric distances exceeding 200$\mathrm{AU}$, and with very high eccentricities, but the majority ($\sim 73\%$) of objects remains bound. Moreover the KBOs are distributed in over densities triggered by the passing of the star which are resonances due to the planets contribution.
In Figs. \[fig:modA1\] and \[fig:modA2\] we show the distributions for eccentricity and inclination as a function of the semi-major axis for the KBOs from model A. Here we varied the impact parameter parameters $b$ and inclination $\theta$ of the encounter. The mass of the passing-by star was 1M$_\odot$ for these simulations and the total mass of the KB was $30$ M$_\oplus$. These simulations were performed with $N = 8182$ KBOs and run up to $10^4$yr.
In the Figures we have identified three main regimes, which we colored red, green and blue, indicating the highly, intermediate and relatively little perturbed system, respectively.
{width="100.00000%"}
{width="100.00000%"}
In the highly perturbing encounter (red zone in figs.\[fig:modA1\] and \[fig:modA2\]) the KB is almost completely destroyed. In the moderately perturbing encounter (blue zone) the post-encounter KBO is characterized by that the majority of objects remain confined in the classical region but with slightly elevated eccentricities and inclinations. These distributions are most comparable to the observed eccentricities and inclinations in the classic (observed) regions. However, the resonance regions and the scattered region are notoriously depleted compared to the observations. In particular, the distribution in inclinations is too much concentrated around a mean value, whereas the observed inclinations are distributed more evenly between $0^\circ$ to $40^\circ$.
The distribution of eccentricities in the mildly perturbed encounters (green zone in figs.\[fig:modA1\] and \[fig:modA2\]) has almost vanished and some objects have scattered to very small semi-major axes. The general shape of the KBO, however, seems to follow the data more closely that those that result from the other more strongly perturbed interactions. The majority of bodies resides in the classical part of the KB with an extended tail of monotonically increasing eccentricities with the semi-major axis, indicating an almost constant periastron distance. On the down-side, however, the distribution of inclinations is distinctively different than the observations.
We compare the distributions in eccentricities and inclinations of the KBOs at $10^4$yr after the encounter changing the total mass of the KB with values $1,3,6$ and $30$ M$_\oplus$ and the sampling of the KBOs with $N$ in range \[8182, 131,062\] fixing the $b = 200$ $\mathrm{AU}$ and $\theta = 90^\circ$ parameters. This comparison is performed using the two-dimensional Kolmogorv-Smirnov tests [@recipes]. The tests give probabilities for eccentricity as well as inclination $\geq
91.2\%$. The K-S test is a measure of the difference in the two distributions. The high values of these K-S test is an indication that the distributions, obtained varying the principal parameters of the sampling of the KB, show very small differences. Therefore, a small variation of the initial conditions for the KB gives rise to only small changes in final distribution, and on the short time frame of the encounter, the effect of the passing star is considerably stronger than any internal dynamical effect inside the KB. The effect the passing star has on the KB is almost impulsive, and variations in the mass of the passing star strongly affects the eccentricities and inclinations of the KBOs. These distributions therefore provide a sensitive characterization to constrain the mass and orbital parameters of the incoming star.
It may be relevant noting that some of the consequences of the encounter of star with the KB can be reliably predicted by the much simpler [*test particle*]{} approach, i.e. neglecting the internal interactions between the KBOs. Actually, a comparison of our results with test-particle simulations [@Ida1; @Ida2; @Ida3; @Melita] show a certain level of similarities:
1. a stellar encounter pumps up strongly the eccentricities and inclinations of objects in the outer region of a planetesimal disk. Moreover, if the classical KBOs acquire high eccentricities their perihelia migrate to the inside.
2. a strong stellar encounter (corresponding to star passing close to the KB disk) may deplete the original, flat, KBO distribution up to 95$\%$. However, contrary to [@Ida3], we find that a strong depletion correspond to a full destruction of the Solar system structure. On the other hand, we found also reasonable initial encounter conditions leading to “intermediate" cases, where a significant depletion (at about $13\%$ level) is compatible with the observed distributions of eccentricities and inclinations.
3. it is not possible to populate the observed resonances reproducing exactly the overdensity in the eccentricity and inclination distributions invoking only a fly-by star perturbation (see [@Ida1]).
The strong effect of the incoming star is clearly depicted in Figs. \[fig:modA1\] and \[fig:modA2\]. Varying the mass of the encountering star cases a migration in both $b$ and $\theta$. The low-mass star (0.5M$_\odot$) gives rise to a shift to smaller values of $b$ and $\theta$, whereas a higher mass star (2M$_\odot$) causes a shift toward larger values of both parameters.
The early evolution of the system strongly depends on the initial conditions of the passing star. We therefore decided to run more simulations in the middle regime (green zone) in the range $\theta =
[70, 80, 90]^\circ$ and $b = [170, 180, 190, 200]$ $\mathrm{AU}$, and in a the second regime of $\theta = [90, 100, 110, 120]^\circ$ and $b = [140,
150, 160, 170]$ $\mathrm{AU}$. All the simulations show a characteristic tail to a monotonically increasing eccentricities; quite similar to the distribution of the eccentricities of the observed scattered KBOs. This tail is characteristic for the relatively close encounter with a stellar perturber, and we confirm the earlier made conjecture of such an encounter [@Ida1]. The distribution in inclination, however, is still to easily reproduced.
To validate our visual comparison we performed a statistical cross-comparison between the observational and computational data for each of the simulations in Tab.\[tab:tabcond\] using the Hotelling’s two sample $T^2$ test [@Hotelling], which is a generalization of the Student’s $t$ test, where
$$F = \frac{n_x + n_y - p - 1}{(n_x + n_y -2)p} T^2,$$
with $F$ the Fisher-Snededecor random variable and $T^2$ is defined as: $$T^2 = \left(\bar{X}-\bar{Y}\right)^T \left[S \left(S\frac{1}{n_x}+\frac{1}
{n_y}\right)\right]\left(\bar{X}-\bar{Y}\right),$$ where $S$ is the pooled sample covariance matrix of $X$ and $Y$, namely, $$S = \frac{(n_x -1)S_x + (n_y -1)S_y}{(n_x -1) + (n_y -1)}$$ where $S_x$ is the covariance matrix of the sample for $X$, $\bar{X}$ is the mean of the sample, and the sample for each random variable $x_i$ in $X$ has $n_x$ elements, and similarly $S_y$ is the covariance matrix of the sample for $Y$, $\bar{Y}$ is the mean of the sample, and the sample for each random variable $y_i$ in $Y$ has $n_y$ elements. The Hotelling’s test states that two population are indistinguishable if $$F \lesssim F_{tab}(p, n_x + n_y - 1 - p, \alpha),$$ where $p$ is the number of parameters, $\alpha$ the significance level of the test and $F_{tab}$ is the theoretical value of $F$-distribution. In Tab. \[tab:tabA\] we present the values for $F$. Each $F$-value is the comparison between two samples: the observed KBOs and the computational one. In order to calculate the $T^2$ variable we compute it using the JD2000 Ephemeris (Right Ascension and Declination coordinates) of the observed KBOs and on the coordinates of the computational KBOs converted in equatorial coordinates. We have normalized the values with $F_{tab}(2, N + n - 3, 0.10) = 9.49122$ (where $n_x$ is $N$ our number of KBOs parameter and $n_y$ is $n =
1593$ the number of observed KBOs [@mpc]). In the table (Tab. \[tab:tabA\]) we have subtracted one from the result to make the clearer distinction that negative values represent results for which the computational and the observational distributions are statistically indistinguishable. We did not perform the Hotelling’s test directly on the eccentricity and inclination distributions because the test can be calculated only on two samples that have a *normal* distributions. On the other hand, ($\alpha$,$\delta$) are not independent from ($e$,$i$) values, therefore a negative value in Tab. \[tab:tabA\] tells also information about the eccentricity and inclination distributions.
This test suggests that encounters with a large impact parameter $b$ or large inclination $\theta$ are favored (strictly speaking, the other runs are rejected on this statistical). Although this method does not makes a distinction in quality of the results, other than accepting or rejecting it, the area of parameter space that give small perturbations (blue zone) seems to be favored. These simulations show a cold and low-eccentricity population of KBOs.
In Fig. \[fig:escaper\] we present the fraction of KBOs that escaped the Solar System as a result of the stellar encounter. Based on these results we prefer the highly scattered regime, with a low value for $b$ and $\theta$ (red zone). In fact only in the highly perturbed regime there is a substantial loss of mass which can reconcile the difference of two magnitude between the total mass observed and the total mass predicted by Solar System formation models [@Luu].
These contradicting results let us to perform a second series of simulations in which we adopted a wider range of semi-major axis, this we called model B.
30 60 70 80 90 100 110 120 150
----- ------ ------ ----- ----- ------ ----- ------ ------ ------
225 -0.1 -0.7 - - -0.8 - - -0.9 -0.8
200 7.0 1.3 0.8 0.2 0.0 - - -0.7 -0.6
190 - - 1.6 0.9 0.1 - - - -
180 - - 2.0 1.3 0.8 - - - -
175 4.7 2.3 1.3 0.3 0.4 - - -0.6 0.0
170 - - - - 0.5 0.7 -0.4 - -
160 - - - - -0.5 0.1 -0.2 - -
150 2.3 1.1 - - -0.1 0.1 -0.1 -0.3 0.1
140 - - - - -0.5 0.0 0.5 - -
: Values of the $F$ indicator obtained with different parameters $b$ (listed in the left column) and $\theta$ (in the upper row) for the model A.[]{data-label="tab:tabA"}
![The 2-D surface distribution of the fraction of the KBO escaped after the gravitational encounter for the model A, at $t =
10^4$ $yr$, in function of the parameters $b$ and $\theta$. In the color map is reported for each color the mean percentage of escaper for that zone in the parameter space.[]{data-label="fig:escaper"}](fig7.pdf)
Model B
-------
In model B we adopted a wider radial extension of the KBO, 42 $\mathrm{AU}$ to 90 $\mathrm{AU}$ with a total mass of $M = 30$M$_\oplus$. The mass and encounter parameters of the incoming star are presented in Tab.\[tab:tabcond\]. The success of model A in reproducing the observed parameters for the KBO led us to limit our parameter search this more extended distribution to $b = [200;237.5]$ and $\theta =
[60^\circ;150^\circ]$. The results of the Hotelling test are presented in Tab.\[tab:tabB\].
60 75 90 105 120 135 150
------- ------ ------ ------ ------ ------ ------ ------
237.5 -0.6 -0.8 -0.8 -0.9 -0.8 -0.4 -0.7
225 -0.2 -0.8 -0.6 -0.7 -0.9 -0.2 -0.7
212.5 -0.3 -0.5 -0.7 -0.7 -0.8 -0.9 -0.4
200 0.1 -0.3 -0.7 -0.8 -0.5 -0.9 -0.4
: Values of the $F$ indicator for various values of the parameters $b$ (left column) and $\theta$ (upper row) for the model B.[]{data-label="tab:tabB"}
In Fig. \[fig:modelB\] we present the distributions for eccentricity and inclination for $b = 200$ $\mathrm{AU}$ and $\theta = 90^\circ$. We determine it as the best model which gives a much better match with the observed inclinations; with value ranging from roughly $0^\circ$ up to $40^\circ$ in the classical regime and up to $30^\circ$ in the scattered regime whereas the eccentricities of the scattered population and the high-eccentricity population in the classical region are consistent with the observational data. With these parameters the initial KB lost $\sim 13\%$ of its mass in the encounter, which is quite small compared to the predictions [@Luu].
The parameters for this particular encounter has trouble reproducing the low-eccentricity population; in fact, the minimum value of the eccentricities in the classical regime $~0.1$. For this reason we started a series of simulations in which we study the long-term secular evolution of the KB, on which we report in §\[secularevo\].
For comparison we highlighted, in Fig. \[fig:modelBA\] the eccentricities after the encounter for model A and model B for one particular encounter. For the range where the initial conditions of model A and B overlap, the post-encounter distributions in eccentricity and inclination also overlap. This supports the earlier argument that inter-KBO dynamics is not important during the encounter.
Long term evolution {#secularevo}
-------------------
The main result for model A and B has been that an encounter in the early history of the Solar System can reproduce the high-eccentricity KB population as well as the majority of the scattered population, but the currently observed low-eccentricity population and part of the resonant populations are absent after the encounter. We will now investigate if these missing populations can be regrown by the long-term evolution of the KB.
We adopt model A with a 1 encountering star with an impact parameter $b = 200$ $\mathrm{AU}$ and inclination $\theta = 90^\circ$ for this follow-up study. Ideally, we should have taken the best model B, but because the missing populations are reachable with the limited range in semi-major axes in model A, we decided that the benefit of the higher local resolution of this model outweighs the more extended width of model B. We restart the simulation at $10^4$yr after the encounter. For convenience we removed the encountering star, because it would cause numerical problems if we allowed it to continue to move further away from the Solar System, whereas it would no longer perturb the KB.
In Fig. \[fig:secular1\] we present the evolution of the eccentricities and semi-major at four moments in time. This illustrates the effect of the secular evolution of the KBOs, due to their self gravity and the influence of the planets. Whereas the encounter with the star completely removes the low-eccentricity population, the subsequent long-term evolution within the KB regrows this population. During the secular evolution the high eccentricity orbits contained between $37$ $\mathrm{AU}$ and $46$ $\mathrm{AU}$ “cool" to lower eccentricities. In this model we considered $8182$ KBOs ($N=8182$) with a total mass of 30 M$_{\oplus}$; each KBO, then, has the mass $3.7\times 10^{-3}$ M$_\oplus$.
An important test of the importance of mutual gravitational interactions between KBOs in determining the KB secular evolution has been done through the expedient of “switching off” the pair interaction. We saw that these simulations did not show the “cooling" of the KB populations, with eccentricities which remained too high compared to the observations. We were driven to conclude that the mutual interactions among the KBOs are responsible of the secular evolution to the partial repopulation of the low-eccentricity distribution after a stellar encounter.
In Tab. \[tab:sec\] we present a summary of results of several simulations at varying $N$ and the KBO radius, which correspond to a variation of the individual KBO mass. Initial conditions are those of model A. We noted that, as expected, the time needed to repopulate the KBO distribution ($T$ in the right most column of Tab. \[tab:sec\]) scales roughly as the two body relaxation time scale, which, in a virialized system, has the following dependence on $N$ and $m$ [@binney]:
$$t_{rel} \propto \frac{N}{ln(N)} \frac{1}{\sqrt{N\,m}}.
\label{eq:rel}$$
label $N$ $m\:($M$_\oplus)$ $R$ (km) $T$ (Myr)
------- --------- -------------------- ---------- -----------
1 $8182$ $3.7\times10^{-3}$ 1741 0.50
2 $8182$ $3.7\times10^{-4}$ 808 1.58
3 $8182$ $3.7\times10^{-5}$ 375 5.00
4 $4086$ $3.7\times10^{-3}$ 1741 0.77
5 $65526$ $3.7\times10^{-3}$ 1741 1.15
6 $65526$ $3.7\times10^{-4}$ 808 3.64
7 $81820$ $7.0\times10^{-7}$ 100 91.56
: Entries are: $N$, the number of KBOs; $m$, the mass of the single KBO, in units of Earth mass; $R$, the radius of the single KBO; $T$, the time-scale to repopulate the KBO distribution. Actually, for the simulations $\#$ 5-6-7 we checked only the initial cooling phase and then the actual values of the parameter $T$ were extrapolated using Eq.$\,$\[eq:rel\]. []{data-label="tab:sec"}
After establishing the initial conditions which we considered to produce the observed KB, we run one more simulations, with $N=16,374$ and a total mass of 30M$_{\oplus}$ (35% of the KBOs have a mass identical to Pluto, while the others have only one-fifth of this mass). The incoming 1 star has approaches the Sun with impact parameter $b = 200$ $\mathrm{AU}$ and inclination $\theta = 90^\circ$. We continue this simulation for 0.9Myr.
In Fig. \[fig:secular2\] we present the energy conservation and the total number of escapers in function of time. The distribution of the eccentricities at 0.9Myr after the encounter is shown in Fig.\[fig:secular3\].
Conclusions
===========
We investigated the effect of an encounter between a passing star on the morphology of the Kuiper belt, and its subsequent long-term evolution. Using the current morphology of the KB we constrained the parameter of the incoming star. The orbit of the encountering star, the planets and those of the KBOs were integrated directly, as was the subsequent evolution of the internal dynamics of the KB and planets. The initial conditions for the Solar System ware taken from [@Ito], and the Kuiper belt objects were distributed in a flat disk between 42 and 90 $\mathrm{AU}$ in the plane of the Ecliptic, and with a power law density distribution with exponent $-2$. The total mass of Kuiper belt ranged between 1 and 30 M$_\oplus$, and the mass of the incoming star was chosen to be 0.5 1.0 and 2.0.
We compared the morphology of the KB directly after the encounter with the passing star, and after a secular evolution of up to 0.9Myr. The best results, directly after the encounter, are obtained when the incoming star approached the ecliptic plane with an impact parameter of 170-220 $\mathrm{AU}$ and an inclination above the Ecliptic of $60^\circ$ to $120^\circ$. The lower (best) values of both $b$ and $\theta$ are for the 0.5, encountering star whereas the upper values correspond to the 2.0 intruder. We summarize these results in Tab. \[tab:conc\]. In Fig. \[fig:Mstar\_vs\_d\] we present the impact parameter and the angle $\theta$ of the incoming star as a function of its mass. A correlation between these parameters is evident and shows a degeneration in the parameters space. In fact, using different parameters is possible to reproduce an encounter with the same strength and find similar proprieties in the final KBOs distributions.
During this encounter about 13% of the Kuiper belt is lost from the Solar system. Actually, results do not show a depletion of the original flat distribution up to $\sim 99$ $\%$ as suggested by the observed total mass of the KB, evaluated in the range $0.3-0.1$ M$_\oplus$, and the mass estimation from Solar system formation model, $30-10$ M$_\oplus$, [@Luu] and match the eccentricity and inclination distributions with the observation at the same time. On the other hand, a better coverage of the initial conditions of the incoming star can very likely enhance the possibility of finding an [*intermediate*]{} case, where a strong depletion can be compatible with the observed distributions in eccentricities and inclinations.
The morphology of the high-eccentricity and scattered population of the KB are well represented directly after the encounter. The low-eccentricity population, around $\sim 40$ $\mathrm{AU}$ and with eccentricities $\aplt 0.1$ is almost completely absent directly after the encounter. This mismatch in the morphology can be resolved by taking the secular evolution of the Kuiper belt into account. The low-eccentricity population is reinstated within a million years. Our models did not show any particular correlation between the inclinations distributions and the mass of the KBOs. However, due to the limited number of objects in our simulations, we could run only almost single-mass particle simulations. For example in our best model the gap in mass between the two population is only a factor 5 and the ratio between the radius is 1.7. Due to this limitation it is not possible to constrain any significant correlations among inclination and other properties, such as the size distribution of the KBOs and the number of KBO binaries. In conclusion, the sampling limiting our model and the relatively short time-scale of our simulations cannot give reliable results on that (actually, our finest simulation involved 16384 KBOs and was carried up to 0.9 Myear).
We expect that a more sophisticated investigation of the long-term ($\sim 100\,$Myr) evolution of the KB, with a proper population over the whole KBO mass spectrum will show the “relaxation" of the eccentricities to low values as it happens in the case of mass monodisperse-particle simulations. Such detailed studies would thus provide important information about the final distribution of the KBOs, which will allow a complete comparison with observable such as the size-inclination relation, but unfortunately it is hard to achieve at the moment without the access to a very large GPUs cluster.
While the secular evolution repopulated the low-eccentricity population, it triggered the further KB causing the depletion of the resonance population, which was initiated by the passing star. This loss of the resonant population can be due to the insufficient sampling of the KB in our simulations. Alternatively, the early migration of the planets is driving the repopulation of the resonant families [@Mal2; @Ida1]. Such planetary reordering would be a natural consequence of the Nice model [@Levi]. Moreover, the resonances, that we have suddenly after the passage of the fly-by star, do not show an eccentricity and inclination distributions compatible with the observations.
M 0.5 1.0 2.0
-------------- ------- ------- -------
$b$ $170$ $200$ $220$
$v_{\infty}$ $3$ $3$ $3$
$\theta$ $60$ $90$ $120$
: The optimal encounter parameters ($b$ and $\theta$) obtained for a star with mass $M$ (in solar masses) approaching with velocity at infinity of 3 km/s the Solar System (model B). The units of $b$ and $\theta$ are those adopted in this paper.[]{data-label="tab:conc"}
![The impact parameter $b$ (top) and inclination $\theta$ (bottom) as a function of the mass of the incoming star for our most favorite model (see Tab. \[tab:conc\]). The dashed line gives a fit to the tree points to indicate the trend, which follows $b = 170
+ 45(M-0.5)$ for the impact parameter and $\theta = 65 +
40(M_\star-0.5)$ for the inclination.[]{data-label="fig:Mstar_vs_d"}](fig14.pdf "fig:"){width="50.00000%"} ![The impact parameter $b$ (top) and inclination $\theta$ (bottom) as a function of the mass of the incoming star for our most favorite model (see Tab. \[tab:conc\]). The dashed line gives a fit to the tree points to indicate the trend, which follows $b = 170
+ 45(M-0.5)$ for the impact parameter and $\theta = 65 +
40(M_\star-0.5)$ for the inclination.[]{data-label="fig:Mstar_vs_d"}](fig15.pdf "fig:"){width="50.00000%"}
Acknowledgments
===============
D. Punzo thanks the Leiden Observatory (University of Leiden) for a period of hospitality. This work was made possible also thanks to a financial contributions from the Dept. of Physics (Sapienza, University of Rome), from the Netherlands Research Council NWO (grants \#643.200.503, \#639.073.803 and \#614.061.608), from the Netherlands Research School for Astronomy (NOVA), and from the HPC-EUROPA2 project (project number: 1249) with the support of the European Commission - Capacities Area - Research Infrastructures. Most of the computations were carried out on the computers owned by the ASTRO research group (Dep. of Physics, Sapienza, Univ. of Roma) and on the Little Green Machine at Leiden University and on the Lisa cluster at SURFSara in Amsterdam and by. We finally thank M. Spera for his help in porting the code to various architectures.
|
---
abstract: 'Energy-dependent speeds of light have been considered an observable signature of quantum gravity effects. The two simplest dispersion relationships produce either linear or quadratic corrections, in particle energy, to the photon speed. The macroscopic limits of these theories – how objects with small energy per particle, but with large mass, behave – are not fully understood. We here briefly discuss some features of the macroscopic limit, that are necessary for understanding how detectors and emitters interact with the high-energy photons that probe spacetime.'
author:
- Simon DeDeo
- 'Chanda Prescod-Weinstein'
title: 'Macroscopic Objects in Theories with Energy-dependent Speeds of Light'
---
Introduction {#intro}
============
Theories that seek to unify the descriptions of General Relativity and Quantum Mechanics naturally incorporate the Planck energy-scale, $M_\mathrm{Pl}$, near which they make testable predictions. Energy (as opposed to the four-momentum vector) is a frame-dependent quantity and in a scattering experiment the most natural frame is that of the center-of-mass of the particles. Statements, then, that a particular quantum gravity effect might arise at energies close to the Planck scale are simplest to understand if the effect is the outcome of a local scattering process with a Planckian center-of-mass energy.
If we preserve Lorentz invariance in the standard fashion, however, very few astrophysical constraints after inflation will be competitive with either atomic physics or collider experiments. For example, the “GZK cutoff” processes [@Greisen:1966p2247; @Zatsepin:1966p2248], due to photo-pion production on CMB photons by cosmic-rays, have a center-of-mass energy of approximately $100~\mathrm{MeV}$. Energies approaching the highest center-of-mass frames found in cosmic-ray collisions will soon be achieved, and probed at far greater precision, by the Large Hadron Collider [@Ellis:2008p4163].
This has not been the end of the story. A variety of suggestions have been made for how various astrophysical tests might provide new constraints on quantum gravitational effects. One, an energy-dependent speed of light, $c(E)$, has received great attention as a testable consequence of quantum gravity [@AmelinoCamelia:1998p2181].
In this brief note, we examine the “macroscopic limit” of such theories, to determine how the constraints on macroscopic behavior might alter the theory on the one-particle level that has been the center of much attention. We discuss a toy model, which we call an “observer preference” theory, that can resolve the tension between $c(E)$ theories, standard macroscopic transformations, and laboratory results on the Doppler shift of light.
$c(E)$ without preferred frames {#cwpf}
===============================
It is possible to construct, for many situations, a consistent framework in which the speed of light may be energy-dependent and yet there be no detectable preferred-frame effects. A particularly important example is “doubly special relativity” (DSR) [@AmelinoCamelia:2001p2304; @AmelinoCamelia:2001p2416].
Early on in studies of DSR [@AmelinoCamelia:2002p9859] it was understood that the behavior of macroscopic objects – *i.e.*, objects composed of multiple particles – could in many cases not be predicted by a simple extrapolation of the single-particle DSR laws. This includes both the theories termed (by Ref. [@AmelinoCamelia:2002p9859]) “DSR1” [@AmelinoCamelia:2002p9781] and “DSR2” [@Magueijo:2002p11389]. It is possible have modified boost transformations that respect the Planck scale and have an ordinary macroscopic limit – “DSR3” is given as one example in Ref. [@AmelinoCamelia:2002p9859]; such theories require more elaboration to specify $c(E)$, which we do not undertake.
Here we approach the question from a different angle and investigate what happens if we allow $c(E)$ to vary, but maintain the standard, Lorentzian transformation laws of macroscopic objects. Such an approach is different from that of, for example, Ref. [@AmelinoCamelia:2002p9781], since rather than build up the two-particle (and higher) cases from the one-particle case, we impose a relationship in the large-$N$ limit. By describing how macroscopic objects perform as measuring devices, we can investigate the implications of unusual one-particle relationships between $c$ and $E$ for the multi-particle sector.
We begin by attempting to preserve the equivalence of inertial frames – that the outcome of any local experiment should not depend on an observer’s velocity. At the same time, we wish to preserve the metric nature of relativity and so accept the General Relativistic definition of inertia as established by, for example, the Pound-Rebka experiment: a particle that satisfies the unaccelerated geodesic equation is in such a frame.
Modifying transformation laws may make the notion of inertial frame ambiguous; one thread of recent work makes this issue explicit by introducing an energy-dependent metric [@Smolin:2008p2498]. If different particles carry around different metrics, their associated geodesic equations may differ, and thus will their notion of inertial frame. Two particles moving relative to each other will disagree – to order $E_{\mathrm{com}}/M_{\mathrm{Pl}}$ – about who is truly accelerating.
Yet the phenomenologist need not worry about such issues, because the only observationally relevant inertial frames are those associated with macroscopic objects: stars, galaxies and satellites. Later formulations of DSR (*e.g.*, that of Ref. [@Magueijo:2003p2402]) suggested that these frames should – roughly – share the same notion of inertiality and furthermore be connected by the standard Lorentz transformations [^1].
According to Ref. [@Magueijo:2003p2402], the relevant quantity for determining the deformation of the transformation properties of an object is not total energy, but energy per elementary particle. Thus, the modification of the transformation laws for an ordinary object – whose elementary particles appear as low-energy electrons and the constituents of protons and neutrons – should be at most of order $m_\mathrm{proton}/M_\mathrm{Pl}$ – or more than $10^{-3}$ less than for a TeV photon.
This suggests, then, that assuming quantities associated with such “macroscopic” objects that serve as clocks and rulers transform in the usual Lorentz fashion is a reasonable starting point. Restricting our study to the interaction of macroscopic objects with individual high-energy particles is one way to avoid some of the paradoxes and difficulties of Ref. [@Schutzhold:2003p473].
Given these preliminaries, we can now formulate an effective theory for the measurement, by macroscopic objects, of energy dependent speeds of light. If $dx$ is the distance travelled by a photon (as measured by an inertial, macroscopic observer), and $dt$ is the time that observer measures, an energy dependent speed can be written formally as: $$\label{ce}
\left(\frac{dx}{dt}\right)= c(n_{\mu} \bar{p}^\mu),$$ where $(t,x,y,z)$ (a 4-vector, $x^\mu$) are the local co-ordinates of a macroscopic, inertial observer, $n^\mu$ is her 4-velocity, and $c(x)$ is some function. Since these two vectors (as well as the differential $dx^\mu$) are associated with measurements made by observers using macroscopic objects, we take them to transform in the standard fashion.
Conversely, we write $\bar{p}^\mu$ as the 4-momentum of the photon. This momentum is written with an overbar to indicate that it may transform differently from the vectors associated with macroscopic observers (and thus, *e.g.*, that its index might be raised and lowered by a different metric.) Because of this, $n_\mu \bar{p}^\mu$ may not transform as a scalar and its value will be frame dependent.
For the sake of argument, we take $c(E)$ to be parametrized by slight departures at the Planck scale, *i.e.*, $$c(E)=1-\alpha\left(\frac{E}{M_{\mathrm{Pl}}}\right)^n,$$ where $\alpha$ is positive, and $n$ is positive (to have a hope of recovering the low-energy limit.) Our choice of units here amounts to setting the zero-energy speed of light, $c(0)$, and thus the parameter for macroscopic Lorentz transformations, equal to unity.
Given this choice, we can then ask if Eq. \[ce\] can be made consistent in all frames, given that $x^\mu$ (the coordinates of the macroscopic observer, frame $\mathcal{O}$) and $n^\mu$ (her 4-velocity) transform in the usual Lorentz fashion, but $\bar{p}^\mu$ may not.
Let us take a (macroscopic) primed frame, $\mathcal{O}^\prime$ to be moving in the negative $\hat{x}$ direction with velocity $v$ with respect to $\mathcal{O}$. In the classical case, we would expect the primed frame to observe a blueshift. Having no preferred frame allows that $E$ may go to $E^\prime$ in some strange fashion, as long as Eq. \[ce\] still holds. We then have
$$\label{newframe}
\frac{dx+v~dt}{dt+v~dx}=1-\alpha\left(\frac{E^\prime}{M_{\mathrm{Pl}}}\right)^n,$$
where the left-hand side is found by Lorentz transforming $dx^\mu$, and the right-hand side comes from requiring Eq. \[ce\] to hold in the $\mathcal{O}^\prime$ frame, with $E$ allowed to transform to $E^\prime$ in a fashion we shall attempt to determine below.
For Eq. \[newframe\] to hold, then, we find the “quantum gravity” Doppler-shift law – the relation between measured photon energy in boosted frames – to be $$\label{transform}
E^\prime = E\left[\frac{1-v}{1+v\left(1-\alpha\left[\frac{E}{M_{\mathrm{Pl}}}\right]^{n}\right)}\right]^{1/n}.$$
The relationship requires that a photon experience a redshift when observed in the moving frame, contrary to the classical result of a blueshift when the observer is moving towards the source. In a previous draft of this paper (arXiv:0811.1999v1), it was claimed that the $n$ equal to two case is consistent with classical redshift relations (given our assumptions above regarding macroscopic transformations); this is incorrect.
A Toy Observer-Preference Model
===============================
An “observer preference” model is a toy model that attempts to reconcile the results of the previous section, which appear to require the unusual Doppler shift formula of Eq. \[transform\], with observation, by allowing a subset of non-colocated observers to see the standard Doppler-shift relationship and relaxing the Eq. \[ce\] relationship to hold only for this observer class.
For example, a particle $A$ might emit a photon at point $x_A$ (and time $t_A$); at point $x_B$ (and time $t_B$), it might be absorbed by a second particle, $B$. We can enforce that particle $B$ measure the photon to have an energy $E_B$ related to $E_A$ by $$\label{standard}
E_B=E_A\sqrt{\frac{1+v_0}{1-v_0}},$$ where here we have assumed that particle $A$ has velocity $v_0$ in the direction of $B$ (*i.e.*, for positive $v_0$, the photon is blueshifted in $B$’s frame.)
Doing so will mean that a global transformation, from the $A$ frame to the $B$ frame, that assumes standard macroscopic behavior will produce inconsistent results (the observed velocity-energy relationship of the photon will not be fixed by Eq. \[ce\] in all frames.) On the other hand, it will be consistent with laboratory measurements of Doppler shifts.
In order to see the effect on light propagation times, we will need to specify the energy of the photon along its entire path as measured by the “observer class” – the collection of 4-vectors along the path that see the relationship of Eq. \[ce\] obtain. The only constraint on this class is that at the endpoints of the path, the observers match the velocity of the emitter (at point $A$) and absorber (at point $B$.)
As one choice of the arbitrary function that achieves this, we take $$\label{interpolation}
v(\tau) = v_0\left(1-\frac{\tau}{\tau_F}\right),$$ where $v(\tau)$ is the velocity of the observer class (in the reference frame of the absorber), $\tau$ is the (macroscopically measured) proper time along the photon path, $\tau_F$ is the proper time at the absorber, and $v_0$ is the velocity of the emitter.
At each point on the photon path, $$\label{obsclass}
\frac{dx_\tau}{dt_\tau}=c(E_\tau),$$ where the $\tau$ subscript indicates that we are in a frame comoving with observer at point $\tau$, and $E_\tau$ is the measured energy of the photon by this observer, which we take to be given by the standard Doppler shift relationship: $$E_\tau = E_A\sqrt{\frac{1+v^\prime}{1-v^\prime}},$$ where $v^\prime$ is the velocity of the emitter in the frame of the observer at $\tau$, given by the macroscopic relativistic formula $$v^\prime = \frac{v_0-v(\tau)}{1-v_0v(\tau)}.$$ Now requiring the velocity, fixed by the relationship Eq. \[obsclass\] in that frame only, to transform in the standard macroscopic fashion, we have $$\frac{dx}{dt}=\frac{v(\tau)+c(E_\tau)}{1+v(\tau)c(E_\tau)},$$ in the coordinate system of the absorber, which, to lowest non-zero order in $E$, and first order in $v_0$, is, for $n=1$, $$\frac{dx}{dt}=1-\alpha\left(\frac{E}{M_\mathrm{pl}}\right)\left(1-v_0\left(2-3\frac{\tau}{\tau_F}\right)\right)+\cdots,$$ and, for $n=2$, $$\frac{dx}{dt}=1-\alpha\left(\frac{E}{M_\mathrm{pl}}\right)^2\left(1-v_0\left(2-4\frac{\tau}{\tau_F}\right)\right)+\cdots.$$ These propagation speeds are different from those expected from the $c(E)$ relationship observed by $B$, who observes the photon blueshift in the standard fashion of Eq. \[standard\]. Only when the photon is at the absorber $B$ – and thus $dx/dt$ coincides with the frame of the observer – do the two quantities become equal. Different choices of observer class (Eq. \[interpolation\]) will produce different results; for example, a velocity observer class defined as a linear function of $\tau$, but in the frame of the emitter and not the absorber, produces a $dx/dt$ relationship that differs at $\mathcal{O}(v_0^3)$.
Using the fact that $$\tau = \int \sqrt{1-\left(\frac{dx}{dt}\right)^2} dt,$$ and that $\tau$ is a scalar quantity, we can solve for $\tau$ as a function of $t$ to find (to lowest order in $E$ and first order in $v_0$) $$\label{n1}
\frac{dx}{dt}=1-\alpha\frac{E}{M_\mathrm{pl}}\left(1-2v_0+3\frac{t}{t_F}v_0+\ldots\right)$$ in the $n=1$ case, and $$\label{n2}
\frac{dx}{dt}=1-\alpha\left(\frac{E}{M_\mathrm{pl}}\right)^2\left(1-2v_0+4\frac{t}{t_F}v_0+\ldots\right)$$ in the $n=2$ case.
This suggests that in some cases one could maintain standard macroscopic transformations and standard laboratory results, if one took an “observer preference” model. The cost, in this toy model, is that the $c(E)$ relationship holds only locally, and only for one particular member of an observer class defined by an emitter-absorber pair and an interpolation rule such as Eq. \[interpolation\].
Note that the equation of motion for a photon, observed at $B$ to have energy $E_B$, but emitted by a stationary observer, is different from that of Eqs. \[n1\] and \[n2\]. In particular, since all members of the observer class are stationary relative to each other, we have (for the $n$ equal one case, *e.g.*), and taking the photon energy to be $E\sqrt{(1+v_0)/(1-v_0)}$, $$\frac{dx}{dt} = 1-\alpha\frac{E}{M_\mathrm{Pl}}(1+v_0+\cdots)$$ which, in contrast to Eq. \[n1\], has no apparent acceleration term – showing explicitly that the photon path depends upon the relative velocity of the emitter and absorber (our derivations above are “frame free” in the sense that identical results are obtained if the absorber is taken to move towards the emitter.) Taken at face value, this leads to strangely non-local and potentially acausal effects: the path a photon takes depends upon the velocity of the observer that will, in the future, observe it.
One simple way to resolve these paradoxes of “action at a distance” is to simply declare the various observer classes ahead of time. Instead of the interpolation law of Eq. \[interpolation\] depending on the emitter and absorber alone, we could fill the universe with observers, making sure that $n^\mu$ chosen at every point on the path agrees with the frame of the observer at that point.
Such an instantaneous, dynamical fixing of frame is related to “dragged ether” models such as that of Ref. [@Afshordi:2008p671]; the forbidding of co-located observers with relative boosts is equivalent to the breakdown of the theory at stream-crossing [^2]. In general, before stream-crossing, tests of such models must sample different parts of the “ether stream” – easy on astrophysical scales, but far harder terrestrially.
Depending on the scale of the dragging – Michelson-Morely experiments do require the dragging to be operative on scales of order the Earth radius – they can be constrained by experiments that look for the Sagnac effect [@Anderson:1994p2871].
The relative velocities within the experimental apparatus must be large; the Michelson-Gale experiment [@Mineur:1927p2833] was the first sensitive test. Modern-day experiments with such properties include “round-the-world clocks” of Hafele and Keating [@Hafele:1972p3010] and studies using the Mössbauer effect to check time-dilations in rotating frames [@Turner:1964p3060].
Achieving high accuracy with experiments that must incorporate large relative velocities is difficult; precision Mössbauer studies sensitive to drag achieve internal motions of $10^{-6}c$ compared to the $10^{-3}c$ CMB-relative speeds sensitive to drift. In past work, Ref. [@Sudarsky:2002p2107] noted that many astrophysical constraints on theories that amount to preferred frames are already beaten by laboratory studies, describing it as “the ghost of Michelson-Morley coming back for revenge.” While we leave a detailed analysis, of how different ether-dragging models may be restricted by terrestrial experiments, to later work, the “ghost of Michelson-Gale” may amount to lighter constraints.
*Acknowledgments*. We thank Lee Smolin for helpful discussion. SD thanks the Perimeter Institute (Canada) and the Institute for the Physics and Mathematics of the Universe (Japan) for their hospitality while this work was undertaken. CPW thanks Sabine Hossenfelder, Jonathan Hackett, Joseph Henson and in particular Clifford Johnson for thought-provoking discussions. CPW is supported by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.
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[^1]: Such theories satisfy constraints on macroscopic-object Lorentz invariance by violating high-energy “Gallelian” intuitions: transformation laws (and thus, under a metric theory, gravitational couplings) may depend on the internal constituents of the object in question. A flea has a Planck-scale rest mass, but a transformation law very different from a Planck-energy electron, since the energy per particle of a flea is on the order of 1 GeV.
[^2]: The “dragged ether” we consider here is quite different from the “partially dragged ether” of Fresnel [@MllerPedersen:2000p2837]; the latter was a “pre-relativistic” theory invented to explain the aberration of starlight [@Ferraro:2005p2831] and can be tested (and ruled out) by the Michelson-Morley experiment alone.
|
---
abstract: 'In this paper, we present a dataset containing 9,973 tweets related to the MeToo movement that were manually annotated for five different linguistic aspects: relevance, stance, hate speech, sarcasm, and dialogue acts. We present a detailed account of the data collection and annotation processes. The annotations have a very high inter-annotator agreement (0.79 to 0.93 k-alpha) due to the domain expertise of the annotators and clear annotation instructions. We analyze the data in terms of geographical distribution, label correlations, and keywords. Lastly, we present some potential use cases of this dataset. We expect this dataset would be of great interest to psycholinguists, socio-linguists, and computational linguists to study the discursive space of digitally mobilized social movements on sensitive issues like sexual harassment.'
author:
- |
Akash Gautam[^1]$^{1}$, Puneet Mathur$^{2}$, Rakesh Gosangi$^{3}$, Debanjan Mahata$^{3}$,\
**Ramit Sawhney$^{4}$, Rajiv Ratn Shah$^{1}$,\
$^{1}$ MIDAS, IIIT-Delhi *{akash15011,rajivratn}@iiitd.ac.in*,\
$^{2}$ University of Maryland, College Park *puneetm@cs.umd.edu*,\
$^{3}$ Bloomberg, New York, U.S.A. *{rgosangi,dmahata}@bloomberg.net*,\
$^{4}$ Netaji Subhas Institute of Technology *ramits.co@nsit.net.in*,\
**
bibliography:
- 'ref.bib'
title: '\#MeTooMA: Multi-Aspect Annotations of Tweets Related to the MeToo Movement'
---
Introduction {#introduction .unnumbered}
============
Over the last couple of years, the MeToo movement has facilitated several discussions about sexual abuse. Social media, especially Twitter, was one of the leading platforms where people shared their experiences of sexual harassment, expressed their opinions, and also offered support to victims. A large portion of these tweets was tagged with a dedicated hashtag \#MeToo, and it was one of the main trending topics in many countries. The movement was viral on social media and the hashtag used over 19 million times[^2] in a year.
The MeToo movement has been described as an essential development against the culture of sexual misconduct by many feminists, activists, and politicians. It is one of the primary examples of successful digital activism facilitated by social media platforms. The movement generated many conversations on stigmatized issues like sexual abuse and violence, which were not often discussed before because of the associated fear of shame or retaliation. This creates an opportunity for researchers to study how people express their opinion on a sensitive topic in an informal setting like social media. However, this is only possible if there are annotated datasets that explore different linguistic facets of such social media narratives.
Twitter served as a platform for many different types of narratives during the MeToo movement [@hosterman2018twitter]. It was used for sharing personal stories of abuse, offering support and resources to victims, and expressing support or opposition towards the movement [@lopez2019one]. It was also used to allege individuals of sexual misconduct, refute such claims, and sometimes voice hateful or sarcastic comments about the campaign or individuals. In some cases, people also misused hashtag to share irrelevant or uninformative content. To capture all these complex narratives, we decided to curate a dataset of tweets related to the MeToo movement that is annotated for various linguistic aspects.
In this paper, we present a new dataset (MeTooMA[^3]) that contains 9,973 tweets associated with the MeToo movement annotated for relevance, stance, hate speech, sarcasm, and dialogue acts. We introduce and annotate three new dialogue acts that are specific to the movement: Allegation, Refutation, and Justification. The dataset also contains geographical information about the tweets: from which country it was posted.
We expect this dataset would be of great interest and use to both computational and socio-linguists. For computational linguists, it provides an opportunity to model three new complex dialogue acts (allegation, refutation, and justification) and also to study how these acts interact with some of the other linguistic components like stance, hate, and sarcasm. For socio-linguists, it provides an opportunity to explore how a movement manifests in social media across multiple countries.
----------------------------- ----------------------- --------------------------------------------------------
**Dataset** **\#Annotated Posts**
[@pandey2018distributional] 2500 *accusational, validation, sensational*
[@khatua2018sounds] 1024
[@schrading2015analysis] 18,336 [*abuse, non-abuse*]{}
[@chowdhury2019speak] 5119 *recollection, non-recollection*
[@sharifirad2019learning] 3240 [*indirect, sexism, casual sexism, physical sexism*]{}
MeTooMA 9,937
----------------------------- ----------------------- --------------------------------------------------------
Related Datasets {#sec:rel_work .unnumbered}
================
Table \[tab: summary\] presents a summary of datasets that contain social media posts about sexual abuse and annotated for various labels.
- [@pandey2018distributional] created a dataset of 2,500 tweets for identification of malicious intent surrounding the cases of sexual assault. The tweets were annotated for labels like *accusational, validation, sensational*.
- Khatua *et al* [@khatua2018sounds] collected 0.7 million tweets containing hashtags such as *\#MeToo*, *\#AlyssaMilano*, *\#harassed*. The annotated a subset of 1024 tweets for the following assault-related labels: assault at the workplace by colleagues, assault at the educational institute by teachers or classmates, assault at public places by strangers, assault at home by a family member, multiple instances of assaults, or a generic tweet about sexual violence.
- [@schrading2015analysis] created the Reddit Domestic Abuse Dataset, which contained 18,336 posts annotated for 2 classes, *abuse* and *non-abuse*.
- [@chowdhury2019speak] presented a dataset consisting of 5119 tweets distributed into *recollection* and *non-recollection* classes. The tweet was annotated as *recollection* if it explicitly mentioned a personal instance of sexual harassment.
- Sharifirad *et al* [@sharifirad2019learning] created a dataset with 3240 tweets labeled into three categories of sexism: *Indirect sexism, casual sexism, physical sexism*.
SVAC (Sexual Violence in Armed Conflict) is another related dataset which contains reports annotated for six different aspects of sexual violence: *prevalence*, *perpetrators*, *victims*, *forms*, *location*, and *timing*.
Unlike all the datasets described above, which are annotated for a single group of labels, our dataset is annotated for **five different linguistic aspects**. It also has **more annotated samples** than most of its contemporaries.
Dataset {#sec:dataset .unnumbered}
=======
Data Collection
---------------
We focused our data collection over the period of October to December 2018 because October marked the one year anniversary of the MeToo movement. Our first step was to identify a list of countries where the movement was trending during the data collection period. To this end, we used Google’s interactive tool named MeTooRisingWithGoogle[^4], which visualizes search trends of the term “MeToo” across the globe. This helped us narrow down our query space to 16 countries. We then scraped 500 random posts from online sexual harassment support forums to help identify keywords or phrases related to the movement [^5]. The posts were first manually inspected by the annotators to determine if they were related to the MeToo movement. Namely, if they contained self-disclosures of sexual violence, relevant information about the events associated with the movement, references to news articles or advertisements calling for support for the movement. We then processed the relevant posts to extract a set of uni-grams and bi-grams with high tf-idf scores. The annotators further pruned this set by removing irrelevant terms resulting in a lexicon of 75 keywords. Some examples include: \#Sexual Harassment, \#TimesUp, \#EveryDaySexism, assaulted, \#WhenIwas, inappropriate, workplace harassment, groped, \#NotOkay, believe survivors, \#WhyIDidntReport.
We then used Twitter’s public streaming API[^6] to query for tweets from the selected countries, over the chosen three-month time frame, containing any of the keywords. This resulted in a preliminary corpus of 39,406 tweets. We further filtered this data down to include only English tweets based on tweet’s *language* metadata field and also excluded short tweets (less than two tokens). Lastly, we de-duplicated the dataset based on the textual content. Namely, we removed all tweets that had more than 0.8 cosine similarity score on the unaltered text in tf-idf space with any another tweet. We employed this de-duplication to promote more lexical diversity in the dataset. After this filtering, we ended up with a corpus of 9,973 tweets.
Table \[tab: count\_tweet\_real\] presents the distribution of the tweets by country before and after the filtering process. A large portion of the samples is from India because the MeToo movement has peaked towards the end of 2018 in India. There are very few samples from Russia likely because of content moderation and regulations on social media usage in the country[^7]. Figure \[fig:heatmap\] gives a geographical distribution of the curated dataset.
***Due to the sensitive nature of this data, we have decided to remove any personal identifiers (such as names, locations, and hyperlinks) from the examples presented in this paper. We also want to caution the readers that some of the examples in the rest of the paper, though censored for profanity, contain offensive language and express a harsh sentiment.***
**Country** **\#Tweets**
---------------- -------------- -----------
India 20,112 5,082
USA 8,943 2,773
United Kingdom 4,350 1,334
France 1,120 347
Australia 542 153
South Africa 1,085 103
Japan 830 13
Kenya 696 15
UAE 540 51
New Zealand 248 38
Iran 325 7
Canada 324 24
Sweden 139 20
Spain 62 9
Austria 88 2
Russia 42 2
**Total** **39,406** **9,973**
: **Distribution of tweets by the country.**[]{data-label="tab: count_tweet_real"}
![**Choropleth world map recording tweet frequency.**[]{data-label="fig:heatmap"}](fvc.png){height="3.6cm"}
Annotation Task
---------------
We chose against crowd-sourcing the annotation process because of the sensitive nature of the data and also to ensure a high quality of annotations. We employed three domain experts who had advanced degrees in clinical psychology and gender studies. The annotators were first provided with the guidelines[^8] document, which included instructions about each task, definitions of class labels, and examples. They studied this document and worked on a few examples to familiarize themselves with the annotation task. They also provided feedback on the document, which helped us refine the instructions and class definitions. The annotation process was broken down into five sub-tasks: for a given tweet, the annotators were instructed to identify relevance, stance, hate speech, sarcasm, and dialogue act. An important consideration was that the sub-tasks were not mutually exclusive, implying that the presence of one label did not consequently mean an absence of any.
### Task 1: Relevance
Here the annotators had to determine if the given tweet was relevant to the MeToo movement. Relevant tweets typically include personal opinions (either positive or negative), experiences of abuse, support for victims, or links to MeToo related news articles. Following are examples of a *relevant* tweet:
> *Officer \[name\] could be kicked out of the force after admitting he groped a woman at \[place\] festival last year. His lawyer argued saying the constable shouldn’t be punished because of the \#MeToo movement. \#notokay \#sexualabuse.*
and an *irrelevant* tweet:
> *Had a bit of break. Went to the beautiful Port \[place\] and nearby areas. Absolutely stunning as usual. \#beautiful \#MeToo \#Australia \#auspol \[URL\].*
We expect this relevance annotation could serve as a useful filter for downstream modeling.
### Task 2: Stance
Stance detection is the task of determining if the author of a text is in favour or opposition of a particular target of interest [@augenstein2016stance; @mohammad2016semeval]. Stance helps understand public opinion about a topic and also has downstream applications in information extraction, text summarization, and textual entailment [@sobhani2017stance]. We categorized stance into three classes: Support, Opposition, Neither. Support typically included tweets that expressed appreciation of the MeToo movement, shared resources for victims of sexual abuse, or offered empathy towards victims. Following is an example of a tweet with a *Support* stance:
> *Opinion: \#MeToo gives a voice to victims while bringing attention to a nationwide stigma surrounding sexual misconduct at a local level.\[URL\]. This should go on.*
On the other hand, Opposition included tweets expressing dissent over the movement or demonstrating indifference towards the victims of sexual abuse or sexual violence. An example of an *Opposition* tweet is shown below:
> *The double standards and selective outrage make it clear that feminist concerns about power imbalances in the workplace aren’t principles but are tools to use against powerful men they hate and wish to destroy. \#fakefeminism. \#men.*
### Task 3: Hate Speech
Detection of hate speech in social media has been gaining interest from NLP researchers lately [@waseem2016hateful; @badjatiya2017deep]. Our annotation scheme for hate speech is based on the work of [@basile-etal-2019-semeval]. For a given tweet, the annotators first had to determine if it contained any hate speech. If the tweet was hateful, they had to identify if the hate was *Directed* or *Generalized*. Directed hate is targeted at a particular individual or entity, whereas Generalized hate is targeted at larger groups that belonged to a particular ethnicity, gender, or sexual orientation. Following are examples of tweets with *Directed* hate:
> *\[username\] were lit minus getting f\*c\*i\*g mouthraped by some drunk chick \#MeToo (no body cares because I’m a male) \[URL\]*
and *Generalized* hate:
> *For the men who r asking “y not then, y now?”, u guys will still doubt her & harrass her even more for y she shared her story immediately no matter what! When your sister will tell her childhood story to u one day, i challenge u guys to ask “y not then, y now?” \#Metoo \[username\] \[URL\] \#a\*\*holes.*
### Task 4: Sarcasm
Sarcasm detection has also become a topic of interest for computational linguistics over the last few years [@bamman2015contextualized; @rajadesingan2015sarcasm] with applications in areas like sentiment analysis and affective computing. Sarcasm was an integral part of the MeToo movement. For example, many women used the hashtag \#NoWomanEver to sarcastically describe some of their experiences with harassment[^9]. We instructed the annotators to identify the presence of any sarcasm in a tweet either about the movement or about an individual or entity. Following is an example of a sarcastic tweet:
> *\# was pound before it was a hashtag. If you replace hashtag with the pound in the \#metoo, you get pound me too. Does that apply to \[name\].*
### Task 5: Dialogue Acts
A dialogue act is defined as the function of a speaker’s utterance during a conversation [@mctear2016conversational], for example, question, answer, request, suggestion, etc. Dialogue Acts have been extensive studied in spoken [@ang2005automatic] and written [@kim2010classifying] conversations and have lately been gaining interest in social media [@zarisheva2015dialog]. In this task, we introduced three new dialogue acts that are specific to the MeToo movement: Allegation, Refutation, and Justification.
**Allegation**: This category includes tweets that allege an individual or a group of sexual misconduct. The tweet could either be personal opinion or text summarizing allegations made against someone [@hutchings2012commercial]. The annotators were instructed to identify if the tweet includes the hypothesis of allegation based on first-hand account or a verifiable source confirming the allegation. Following is an example of a tweet that qualifies as an Allegation:
> *More women accuse \[name\] of grave sexual misconduct...twitter seethes with anger. \#MeToo \#pervert.*
**Refutation**: This category contains tweets where an individual or an organization is denying allegations with or without evidence. Following is an example of a Refutation tweet:
> *She is trying to use the \#MeToo movement to settle old scores, says \[name1\] after \[name2\] levels sexual assault allegations against him.*
**Justification**: The class includes tweets where the author is justifying their actions. These could be alleged actions in the real world (e.g. allegation of sexual misconduct) or some action performed on twitter (e.g. supporting someone who was alleged of misconduct). Following is an example of a tweet that would be tagged as Justification:
> *I actually did try to report it, but he and of his friends got together and lied to the police about it. \#WhyIDidNotReport.*
![**Geographical distribution of various class labels.**[]{data-label="fig:mannual_annotation"}](country_share_2.png)
[|llll|]{} &**SAGE** & &**SAGE**\
f\*ck&3.36 &hate &3.21\
f\*cking &3.04 &lie &2.95\
hijab &2.84 &predators &2.92\
bullshit &2.77 &nuns &2.91\
blog &2.70 &grop &2.91\
&**SAGE** & &**SAGE**\
accuse &1.45 & organisation &0.57\
bob &1.45 & told &0.56\
flopping &1.40 &discuss &0.56\
aces &1.40 &violent &0.55\
corrupt &1.35 &shocked &0.51\
&**SAGE** & &**SAGE**\
fund & 0.80 &mocks &2.47\
reconciliation &0.66 &tweet &2.19\
diversity &0.62 &practice &2.19\
protect &0.62 &feminism &2.11\
welcome &0.59 &minister &2.11\
&**SAGE** & &**SAGE**\
baseless &3.63 & lol &2.74\
wild &3.59 &gonna &2.71\
center &3.46 &trouble &2.71\
denies &3.17 &ooh &2.41\
threatens &3.07 &xoxo &2.20\
Dataset Analysis {#sec:description .unnumbered}
================
This section includes descriptive and quantitative analysis performed on the dataset.
Inter-annotator agreement
-------------------------
We evaluated inter-annotator agreements using Krippendorff’s alpha (K-alpha) [@krippendorff2011computing]. K-alpha, unlike simple agreement measures, accounts for chance correction and class distributions and can be generalized to multiple annotators. Table \[tab:aggreeemnts\] summarizes the K-alpha measures for all the annotation tasks. We observe very strong agreements for most of the tasks with a maximum of 0.92 for the relevance task. The least agreement observed was for the hate speech task at 0.78. Per recommendations in [@artstein2008inter], we conclude that these annotations are of good quality. We chose a straightforward approach of majority decision for label adjudication: if two or more annotators agreed on assigning a particular class label. In cases of discrepancy, the labels were adjudicated manually by the authors. Table \[tab:class\_dist\] shows a distribution of class labels after adjudication.
**Task** **Krippendorff’s $\alpha$**
--------------- -----------------------------
Relevance 0.92
Stance 0.90
Hate speech 0.78
Sarcasm 0.80
Allegation 0.86
Refutation 0.83
Justification 0.79
: **Inter-annotator agreements for all the annotation tasks.**[]{data-label="tab:aggreeemnts"}
**Task** **Label** **\#Samples** **%**
--------------- --------------- --------------- -------
Relevance Relevant 7,249 72.8%
Stance Support 3,074 30.9%
Opposition 743 7.4%
Hate Speech Directed 419 4.21%
Generalized 281 2.8%
Sarcasm Sarcastic 220 2.2%
Dialogue Acts Allegation 578 5.78%
Justification 292 2.9%
Refutation 216 2.1%
: **Distribution of class labels for all tasks.**[]{data-label="tab:class_dist"}
![**Word cloud representation of the dataset: font size is proportional to the frequency of a term. The words are organized and color-coded based on the NRC sentiment lexicon: positive sentiment (green + bottom half), negative sentiment (red + top half).**[]{data-label="fig:nrc_sentiment"}](nrc_wordcloud.png)
Geographical Distribution
-------------------------
Figure \[fig:mannual\_annotation\] presents a distribution of all the tweets by their country of origin. As expected, a large portion of the tweets across all classes are from India, which is consistent with Table \[tab: count\_tweet\_real\]. Interestingly, the US contributes comparatively a smaller proportion of tweets to Justification category, and likewise, UK contributes a lower portion of tweets to the Generalized Hate category. Further analysis is necessary to establish if these observations are statistically significant.
Label Correlations
------------------
We conducted a simple experiment to understand the linguistic similarities (or lack thereof) for different pairs of class labels both within and across tasks. To this end, for each pair of labels, we converted the data into its tf-idf representation and then estimated Pearson, Spearman, and Kendall Tau correlation coefficients and also the corresponding $p$ values. The results are summarized in Table \[tab:corr\_coeff\]. Overall, the correlation values seem to be on a lower end with maximum Pearson’s correlation value obtained for the label pair *Justification - Support*, maximum Kendall Tau’s correlation for *Allegation - Support*, and maximum Spearman’s correlation for *Directed Hate - Generalized Hate*. The correlations are statistically significant ($p$ $<$ 0.05) for three pairs of class labels: *Directed Hate - Generalized Hate, Directed Hate - Opposition, Sarcasm - Opposition*. Sarcasm and Allegation also have statistically significant $p$ values for Pearson and Spearman correlations.
Keywords
--------
We used SAGE [@eisenstein2011sparse], a topic modelling method, to identify keywords associated with the various class labels in our dataset. SAGE is an unsupervised generative model that can identify words that distinguish one part of the corpus from rest. For our keyword analysis, we removed all the hashtags and only considered tokens that appeared at least five times in the corpus, thus ensuring they were representative of the topic. Table \[tab:sage\_res\] presents the top five keywords associated with each class and also their salience scores. Though *Directed* and *Generalized* hate are closely related topics, there is not much overlap between the top 5 salient keywords suggesting that there are linguistic cues to distinguish between them. The word [predators]{} is strongly indicative of *Generalized Hate*, which is intuitive because it is a term often used to describe people who were accused of sexual misconduct. The word [lol]{} being associated with *Sarcasm* is also reasonably intuitive because of sarcasm’s close relation with humour.
Sentiment Analysis
------------------
Figure \[fig:nrc\_sentiment\] presents a word cloud representation of the data where the colours are assigned based on NRC emotion lexicon [@Mohammad13]: green for positive and red for negative. We also analyzed all the classes in terms of Valence, Arousal, and Dominance using the NRC VAD lexicon [@vad-acl2018]. The results are summarized in Figure \[fig:nrc\_vad\]. Of all the classes, *Directed-Hate* has the largest valence spread, which is likely because of the extreme nature of the opinions expressed in such tweets. The spread for the dominance is fairly narrow for all class labels with the median score slightly above 0.5, suggesting a slightly dominant nature exhibited by the authors of the tweets.
Discussion {#sec:discuss .unnumbered}
==========
This paper introduces a new dataset containing tweets related to the \#MeToo movement. It may involve opinions over socially stigmatized issues or self-reports of distressing incidents. Therefore, it is necessary to examine the social impact of this exercise, the ethics of the individuals concerned with the dataset, and it’s limitations.
**Mental health implications:** This dataset open sources posts curated by individuals who may have undergone instances of sexual exploitation in the past. While we respect and applaud their decision to raise their voices against their exploitation, we also understand that their revelations may have been met with public backlash and apathy in both the virtual as well as the real world. In such situations, where the social reputation of both accuser and accused may be under threat, mental health concerns become very important[^10]. As survivors recount their horrific episodes of sexual harassment, it becomes imperative to provide them with therapeutic care [@fredriksen2014creating] as a safeguard against mental health hazards. Such measures, if combined with the integration of mental health assessment tools in social media platforms, can make victims of sexual abuse feel more empowered and self-contemplative towards their revelations.\
**Use of MeTooMA dataset for population studies:** We would like to mention that there have been no attempts to conduct population-centric analysis on the proposed dataset. The analysis presented in this dataset should be seen as a proof of concept to examine the instances of \#MeToo movement on Twitter. The authors acknowledge that learning from this dataset cannot be used as-is for any direct social interventions. Network sampling of real-world users for any experimental work beyond this dataset would require careful evaluation beyond the observational analysis presented herein. Moreover, the findings could be used to assist already existing human knowledge. Experiences of the affected communities should be recorded and analyzed carefully, which could otherwise lead to social stigmatization, discrimination and societal bias. Enough care has been ensured so that this work does not come across as trying to target any specific individual for their personal stance on the issues pertaining to the social theme at hand. The authors do not aim to vilify individuals accused in the \#MeToo cases in any manner. Our work tries to bring out general trends that may help researchers develop better techniques to understand mass unorganized virtual movements.\
**Effect on marginalized communities:** The authors recognize the impact of the \#MeToo movement on socially stigmatized populations like LGBTQIA+. The \#MeToo movement provided such individuals with the liberty to express their notions about instances of sexual violence and harassment[^11]. The movement acted as a catalyst towards implementing social policy changes to benefit the members of these communities[^12]. Hence, it is essential to keep in mind that any experimental work undertaken on this dataset should try to minimize the biases against the minority groups which might get amplified in cases of sudden outburst of public reactions over sensitive media discussions.\
**Limitations of individual consent:** Considering the mental health aspects of the individuals concerned, social media practitioners should vary of making automated interventions to aid the victims of sexual abuse as some individuals might not prefer to disclose their sexual identities or notions. Concerned social media users might also repeal their social media information if found out that their personal information may be potentially utilised for computational analysis. Hence, it is imperative to seek subtle individual consent before trying to profile authors involved in online discussions to uphold personal privacy.\
Use Cases {#use-cases .unnumbered}
=========
The authors would like to formally propose some ideas on possible extensions of the proposed dataset:
- The rise of online **hate speech** and its related behaviours like cyber-bullying has been a hot topic of research in gender studies [@djuric2015hate]. Our dataset could be utilized for extracting actionable insights and virtual dynamics to identify gender roles for analyzing sexual abuse revelations similar to [@yuce2014bridging].
- The dataset could be utilized by psycholinguistics for extracting contextualized lexicons to examine how influential people are portrayed on public platforms in events of mass social media movements [@field2019contextual]. Interestingly, such analysis may help linguists determine the **power dynamics of authoritative people** in terms of perspective and sentiment through campaign modelling.
- Marginalized voices affected by mass social movements can be studied through **polarization analysis** on graph-based simulations of the social media networks. Based on the data gathered from these nodes, community interactions could be leveraged to identify indigenous issues pertaining to societal unrest across various sections of the society[@rho2018fostering].
- **Challenge Proposal**: The authors of the paper would like to extend the present work as a challenge proposal for building computational semantic analysis systems aimed at online social movements. In contrast to already available datasets and existing challenges, we propose tasks on detecting hate speech, sarcasm, stance and relevancy that will be more focused on social media activities surrounding revelations of sexual abuse and harassment. The tasks may utilize the message-level text, linked images, tweet-level metadata and user-level interactions to model systems that are **F**air, **A**ccountable, **I**nterpretable and **R**esponsible (FAIR).
Research ideas emerging from this work should not be limited to the above discussion. If needed, supplementary data required to enrich this dataset can be collected utilizing Twitter API and *JSON* records for exploratory tasks beyond the scope of the paper.
Conclusion {#conclusion .unnumbered}
==========
In this paper, we presented a new dataset annotated for five different linguistic aspects: relevance, stance, hate speech, sarcasm, and dialogue acts. To our knowledge, there are no datasets out there that provide annotations across so many different dimensions. This allows researchers to perform various multi-label and multi-aspect classification experiments. Additionally, researchers could also address some interesting questions on how different linguistic components influence each other: e.g. does understanding one’s stance help in better prediction of hate speech?
In addition to these exciting computational challenges, we expect this data could be useful for socio and psycholinguists in understanding the language used by victims when disclosing their experiences of abuse. Likewise, they could analyze the language used by alleged individuals in justifying their actions. It also provides a chance to examine the language used to express hate in the context of sexual abuse.
In the future, we would like to propose challenge tasks around this data where the participants will have to build computational models to capture all the different linguistic aspects that were annotated. We expect such a task would drive researchers to ask more interesting questions, find limitations of the dataset, propose improvements, and provide interesting insights.
[^1]: equal contribution
[^2]: https://www.usatoday.com/story/news/2018/10/13/metoo-impact-hashtag-made-online/1633570002/
[^3]: The dataset can be found at [ https://doi.org/10.7910/DVN/JN4EYU]( https://doi.org/10.7910/DVN/JN4EYU).
[^4]: https://metoorising.withgoogle.com/
[^5]: We scraped data from the discussion forums on the websites of two non-profit organizations (pandys and isurvive), which provide support and resources to survivors of abuse.
[^6]: https://www.tweepy.org/
[^7]: https://time.com/5636107/metoo-russia-womens-rights/
[^8]: The annotation guidelines will be released as supplementary material to this publication.
[^9]: https://www.good.is/articles/maura-quint-twitter-sexual-assault
[^10]: https://www.thelancet.com/journals/lancet/article/PIIS0140-6736(18)30991-7/fulltext
[^11]: https://rewire.news/article/2018/10/09/for-lgbtq-youth-metoo-is-not-a-heteronormative-issue/
[^12]: https://www.reuters.com/article/us-lgbt-rights-twitter/mequeer-takes-twitter-by-storm-as-lgbt-community-cries-metoo-idUSKCN1L71WW
|
---
abstract: 'Recently, a whole class of evergy-preserving integrators has been derived for the numerical solution of Hamiltonian problems [@BIT0; @BIT00; @BIT1]. In the mainstream of this research [@BIT3], we have defined a new family of symplectic integrators depending on a real parameter $\alpha$ [@BIT4]. For $\alpha = 0$, the corresponding method in the family becomes the classical Gauss collocation formula of order $2s$, where $s$ denotes the number of the internal stages. For any given non-null $\alpha$, the corresponding method remains symplectic and has order $2s-2$: hence it may be interpreted as a $O(h^{2s-2})$ (symplectic) perturbation of the Gauss method. Under suitable assumptions, it can be shown that the parameter $\alpha$ may be properly tuned, at each step of the integration procedure, so as to guarantee energy conservation in the numerical solution. The resulting method shares the same order $2s$ as the generating Gauss formula, and is able to preserve both energy and quadratic invariants.'
author:
- Luigi Brugnano
- Felice Iavernaro
- Donato Trigiante
title: Energy and quadratic invariants preserving integrators of Gaussian type
---
[^1]
[ address=[Dipartimento di Matematica “U.Dini”, Università di Firenze, Viale Morgagni 67/A, 50134 Firenze (Italy).]{} ]{}
[ address=[Dipartimento di Matematica, Università di Bari, Via Orabona 4, 70125 Bari (Italy).]{} ]{}
[ address=[Dipartimento di Energetica “S.Stecco”, Università di Firenze, Via Lombroso 6/17, 50134 Firenze (Italy).]{} ]{}
Introduction
============
When dealing with the numerical integration of canonical Hamiltonian systems in the form $$\label{hamilode}
\left\{ \begin{array}{l} \dot y = J\nabla H(y) \equiv f(y), \\
y(t_0) = y_0 \in\RR^{2m}, \end{array} \right.
\qquad J=\pmatrix{rr} 0 & I \\ -I & 0 \endpmatrix \in \RR^{2m \times
2m},$$ ($I$ is the identity matrix of dimension $m$), two main lines of investigation may be traced in the current literature, having as objective the definition and the study of symplectic methods and energy-conserving methods, respectively. In fact, the symplecticity of the map and the conservation of the energy function are the most relevant features characterizing a Hamiltonian system.
From the very beginning of this research activity, high order symplectic formulae were already available within the class of Runge-Kutta methods [@Feng; @Suris; @SC], the Gauss collocation formulae being one noticeable example. One important implication of symplecticity of the discrete flow, for Gauss-Legendre methods, is the conservation of quadratic invariants. This circumstance makes the symplecticity property of the method particularly appealing in the numerical simulation of isolated mechanical systems in the form , since it provides a precise conservation of the total angular momentum during the time evolution of the state vector. As a further positive consequence, a symplectic method also conserves quadratic Hamiltonian functions (see the monographs [@HLW; @LR] for a thorough analysis of symplectic methods).
Conversely, if one excludes the quadratic case, energy-conserving methods were initially not known within classical integration methods. The unsuccessful attempts to derive energy-preserving Runge-Kutta methods for polynomial Hamiltonians, culminated in the general feeling that such methods could not even exist (see [@IZ] and [@M2AN]). A completely new approach is represented by [*discrete gradient methods*]{} which are based upon the definition of a discrete counterpart of the gradient operator so that energy conservation of the numerical solution is guaranteed at each step and whatever the choice of the stepsize of integration (see [@G; @MQR]).
More recently, the conservation of energy has been approached by means of the definition of the [*discrete line integral*]{}, in a series of papers (such as [@IP1; @IT3]), leading to the definition of [*Hamiltonian Boundary Value Methods (HBVMs)*]{} (see for example [@BIT0; @BIT00; @BIT1; @BIT2; @BIS]). They are a class of methods able to preserve, in the discrete solution, polynomial Hamiltonians of arbitrarily high degree (and, hence, a [*practical*]{} conservation of any sufficiently differentiable Hamiltonian. Such methods admit a Runge-Kutta formulation which reveals their close relationship with classical collocation formulae [@BIT3]. An infinity extension of HBVMs has also been proposed in [@BIT2] and [@Ha]. These limit methods may be interpreted as a generalization of the *averaged vector field method* defined in [@QMcL].
Attempts to incorporate both symplecticity and energy conservation into the numerical method will clash with two non-existence results. The first [@GM] refers to non-integrable systems, that is systems that do not admit other independent first integrals different from the Hamiltonian function itself. According to the authors’ words, it states that
> *If \[the method\] is symplectic, and conserved $H$ exactly, then it is the time advance map for the exact Hamiltonian system up to a reparametrization of time.*
The second negative result [@CFM] refers to B-series symplectic methods applied to general (not necessarily non-integrable) Hamiltonian systems:
> *The only symplectic method (as $B$-series) that conserves the Hamiltonian for arbitrary $H(y)$ is the exact flow of the differential equation.*
Despite these discouraging results, in [@BIT4] a new class of symplectic integrators of arbitrarily high-order has been proposed which, under some mild assumptions (see the next section), may share both features, in the sense specified in the theorem below. We prefer the use of the term “integrator” rather than method since, strictly speaking, our integrator may select a different symplectic formula from one integration step to the next, in order to enforce the energy conservation property. In what follows, we sketch the main ideas behind this approach. For further generalizations, as well as for a number of numerical evidences, we refer to [@BIT4]. We will begin with introducing a family of one-step methods $$\label{met_alpha} y_1(\alpha,h)=\Phi_h(y_0,\alpha)$$ ($h$ is the stepsize of integration), depending on a real parameter $\alpha$, with the following specifics:
1. for any fixed choice of $\alpha \not = 0$, the corresponding method is a symplectic Runge-Kutta method with $s$ stages and of order $2s-2$, which exactly conserves all quadratic invariants;
2. for $\alpha=0$ one gets the Gauss collocation method (of order $2s$);
3. for any choice of $y_0$ and in a given range of the stepsize $h$, there exists a value of the parameter, say $\alpha_0$, depending on $y_0$ and $h$, such that $H(y_1(\alpha_0,h))=H(y_0)$ (energy conservation).
The parametric method (\[met\_alpha\]) realizes a symplectic perturbation of the Gauss method of size $O(h^{2s-2})$. Under suitable assumptions, as the parameter $\alpha$ ranges in a small interval centered at zero, the value of the numerical Hamiltonian function $H(y_1)$ will match $H(y(t_0+h))$ thus leading to energy conservation. This result is formalized as follows:
\[theo1\] Under suitable assumptions, there exists a real sequence $\{\alpha_k\}$ such that the numerical solution defined by $y_{k+1}=\Phi_h(y_{k},\alpha_k)$, with $y_0$ defined in (\[hamilode\]), satisfies $H(y_k)=H(y_0)$.
One important remark is in order to clarify this statement and how it relates to the above non-existence results. Let us select the value of the parameter $\alpha=\alpha_0$, if any, in order to enforce the energy conservation between the two state vectors $y_0$ and $y_1$, as indicated at item 3 above[^2]: the map $y \mapsto \Phi_h(y,\alpha_0)$ is symplectic and, by definition, assures the energy conservation condition $H(y_1)=H(y_0)$. However, it is worth noticing that it would fail to provide a conservation of the Hamiltonian function if we changed the initial condition $y_0$ or the stepsize $h$. For example, in general for any $\hat y_0 \not =y_0$, we would obtain $H(\Phi_h(\hat y_0,\alpha_0)) \not = H(y_0)$: in this case we should change the value of the parameter $\alpha$ in order to recover the equality condition.[^3] Strictly speaking, the energy conservation property described in Theorem \[theo1\] weakens the standard energy conservation condition mentioned in the two non-existence results stated above and hence our methods are not meant to produce a counterexample of these statements.
Definition of the methods
=========================
Let $\{c_1<c_2<\dots<c_s\}$ and $\{b_1,\dots, b_s\}$ be the abscissae and the weights of the Gauss-Legendre quadrature formula in the interval $[0,1]$. We consider the Legendre polynomials $P_j(\tau)$ of degree $j-1$, for $j=1,\dots,s$, shifted and normalized in the interval $[0,1]$ so that $\int_0^1
P_i(\tau)P_j(\tau) \mathrm{d} \tau = \delta_{ij}$, for $i,j=1,\dots,s,$ ($\delta_{ij}$ is the Kronecker symbol), and the $s\times s$ matrix $\P = \left( P_j(c_i) \right)$. Our starting point is the following well-known decomposition of the Butcher array $A$ of the Gauss method of order $2s$ [@HW pp.77–84]: $$\label{A}
A= \P X_s \P^{-1},$$ where $X_s$ is defined as $$\label{Xs}
X_s = \pmatrix{cccc}
\frac{1}2 & -\xi_1 &&\\
\xi_1 &0 &\ddots&\\
&\ddots &\ddots &-\xi_{s-1}\\
& &\xi_{s-1} &0\\
\endpmatrix,
\qquad\mbox{with}\qquad \xi_j=\frac{1}{2\sqrt{4j^2-1}}, \qquad
j=1,\dots,s-1.$$ We now consider the matrix $X_s(\alpha)$ obtained by perturbing as follows: $$\label{Xs_alpha}
X_s(\alpha) = \pmatrix{cccc}
\frac{1}2 & -\xi_1 &&\\
\xi_1 &0 &\ddots&\\
&\ddots &\ddots &-(\xi_{s-1}+\alpha)\\
& &\xi_{s-1}+\alpha &0\\
\endpmatrix \equiv X_s + \alpha W_s,$$ where $\alpha$ is a real parameter, $W_s = (e_se_{s-1}^T -
e_{s-1}e_s^T)$, and, as usual, $e_j\in\RR^s$ is the $j$th unit vector. The family of methods (\[met\_alpha\]) we are interested in, is formally defined by the following tableau (see (\[A\])–(\[Xs\_alpha\])): $$\label{qgauss}
\begin{array}{c|c}\begin{array}{c} c_1\\ \vdots\\ c_s\end{array} & \A(\alpha) \\
\hline &b_1\, \ldots ~ b_s
\end{array} \qquad\mbox{with}\qquad \A(\alpha)\equiv \P X_s(\alpha)\P^{-1} = A + \alpha \P W_s
\P^{-1}.$$ Therefore $A(0)=A$ and, moreover, the following result holds true [@BIT4].
For any fixed value of $\alpha$, the Runge-Kutta method (\[qgauss\]) is symmetric and symplectic. For $\alpha=0$, the usual Gauss-Legendre method of order $2s$ is recovered. For any fixed $\alpha\ne0$, a method of order $2s-2$ is obtained.
If we can choose $\alpha\equiv\alpha_0$ so that the energy-conservation property be satisfied, then, one obtains a [*(symmetric), Energy and QUadratic Invariants Preserving (EQUIP) method*]{}, as specified in Theorem \[theo1\], of Gaussian type. Indeed, the conservation of quadratic invariants easily follows from the structure of the matrix (\[Xs\_alpha\]) defining the method. In conclusion, these methods will provide an exact conservation of all quadratic invariants, besides the Hamiltonian function. In more details, if we denote, as usual, $Y=(Y_1^T\dots
Y_s^T)^T$ the vector of the stages, $e=(1,\dots,1)^T\in\RR^s$, and defining the error function $g(\alpha,h)= H(y_1(\alpha,h))-
H(y_0)$, the nonlinear system, in the unknowns $Y_1,\dots,Y_s$ and $\alpha$, that is to be solved at each step for getting energy conservation, reads, for the given stepsize $h$, $$\label{concon} \left\{ \begin{array}{l} Y = e \otimes y_0 + h
(\A(\alpha) \otimes I) F(Y), \\
g(\alpha,h) =0.
\end{array}
\right.$$ Concerning the question about the existence of a solution of , we make the following assumptions:
- the function $g$ is analytical in a rectangle $[-\bar \alpha, \bar \alpha] \times [-\bar
h, \bar h]$ centered at the origin;
- let $d$ be the order of the error in the Hamiltonian function associated with the Gauss method applied to the given Hamiltonian system and the given state vector $y_0$, that is: $$g(0,h) = H(y_1(0,h))- H(y_0) = c_0 h^{d} + O(h^{d+1}),
\qquad c_0 \not = 0.$$ Then, we assume that for, any fixed $\alpha
\not = 0$ in a suitable neighborhood of the origin, $$g(\alpha,h)
= c(\alpha) h^{d-2} + O(h^{d-1}), \qquad c(\alpha) \not =0.$$
We observe that, excluding the case where the Hamiltonian $H(y)$ is quadratic (which would imply $g(\alpha,h) = 0$, for all $\alpha$), the error in the numerical Hamiltonian function associated with the Gauss method is expected to behave as $O(h^{2s+1})$. Consequently, $d\ge
2s$.
The following result then holds true [@BIT4].
\[implicit\] Under the assumptions ($\mathcal A_1$) and ($\mathcal A_2$), there exists a function $\alpha_0=\alpha_0(h)$, defined in a neighborhood of the origin $(-h_0,h_0)$, such that:
- $g(\alpha_0(h),h)=0$, for all $h\in(-h_0,h_0)$; (ii) $\alpha_0(h)=\mathrm{const}\cdot h^2 + O(h^3)$.
The next result concerns the order of convergence of the method (\[concon\]) (again, the proof can be found in [@BIT4]).
\[fastorder\] Consider the parametric method and suppose that the parameter $\alpha$ is actually a function of the stepsize $h$, according to what stated in Theorem \[implicit\]. Then, the resulting method has order $2s$.
Numerical tests concerning the new EQUIP methods of Gaussian type can be found in [@BIT4] and in the companion paper [@BIT5].
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[^1]: Work developed within the project “Numerical Methods and Software for Differential Equations”.
[^2]: To avoid any misunderstanding, we emphasize that the value $\alpha_0$ is now maintained constant, otherwise the map would fail to be symplectic.
[^3]: More in general, the sequence $\{\hat \alpha_k\}$ that will satisfy Theorem \[theo1\] starting at $\hat y_0$ will differ from the sequence $\{\alpha_k\}$. Such sequences will be defined as the solution of the nonlinear system (\[concon\]), as described in the next section.
|
---
abstract: 'We consider a cosmological model starting from (1) the $(1+3+6)$-dimensional space-times consisting of the outer space (the $3$-dimensional expanding section) and the inner space (the $6$-dimensional section) and reaching (2) the Friedmann model after the decoupling of the outer space from the inner space, and derive fluctuations of the background radiation appearing in the above $10$-dimensional space-times. For this purpose we first derive the fluid-dynamical perturbations in the above $10$-dimensional space-times, corresponding to two kinds of curvature perturbations (in the scalar mode) in the non-viscous case, and next study the quantum fluctuations in the scalar and tensor modes, appearing at the stage when the perturbations are within the horizon of the inflating outer space. Lastly we derive the wave-number dependence of fluctuations (the power spectrum) in the two modes, which formed at the above decoupling epoch and are observed in the Friedmann stage. It is found that this can be consistent with the observed spectra of the cosmic microwave background radiation.'
address:
author:
-
bibliography:
- 'sample.bib'
title: 'Fluctuations of the cosmic background radiation appearing in the $10$-dimensional cosmological model '
---
Introduction
============
In order to derive the observed fluctuations of cosmic microwave background radiation, we study the cosmological evolution of the $(1+3+6)$-dimensional space-times, in which it is assumed that our universe was born as an isotropic and homogeneous 10-dimensional space-times and evolved to the state consisting of the 3-dimensional inflating outer space and the 6-dimensional collapsing inner space. Our 4-dimensional Friedmann universe appeared after decoupling of the outer space from the inner space. This scenario is supported by the present super-string theory (Kim et al. [@kim1; @kim2] in a matrix model).
In a previous paper[@tom] we discussed the entropy production at the stage when the above inflation and collapse coexist, and showed how viscous processes help the increase of cosmological entropy. We also discussed the possibility that we satisfy, at the same time, the condition that the entropy in the Guth level[@guth] is obtained and the condition that the inner space decouples from the outer space. In the subsequent paper[@tom2] , we studied the evolution of cosmological perturbations in the non-viscous case, solving the equations for geometrical perturbations.
In this paper we treat the fluidal perturbations corresponding to the geometrical perturbations, the quantum fluctuations in the scalar and tensor modes, and the consistency with the observations of cosmic microwave background radiation (CMB) in the non-viscous case. In Sect. 2, we first review the background model and the perturbed quantities. In Sect. 3, we derive the perturbed fluid-dynamical equations, corresponding to the geometrical perturbations in the 10-dimensional space-times, and solve them. In Sect. 4, we consider the quantum fluctuations in the scalar and tensor modes at the stage when they were within the horizon of the outer space with the inflationary expansion, and derive the initial conditions for their perturbations appearing after this stage. In Sect. 5, we derive the spectra of perturbations in these two modes, and compare them with the observed ones. In Sect. 6, concluding remarks are given. In Appendix A, we derive the higher-order terms in the two curvature perturbations with respect to small wave-numbers in the outer space.
Review of the background model and the perturbed quantities
===========================================================
Our background model
--------------------
We consider a cosmological model starting from (1) the $(1+3+6)$-dimensional space-times consisting of the outer space (the $3$-dimensional expanding section) and the inner space (the $6$-dimensional section), and reaching (2) the Friedmann model after the decoupling of the outer space from the inner space, as shown schematically in Fig. 1.
![\[fig:bkr1\] Scale factors of outer and inner spaces in the 10-dimensional space-times and the Friedmann model. $t_{dec}$ and $t_A$ denote the decoupling epoch (when the Friedmann model starts) and the singular epoch of the inner space, respectively.](Fig1d.eps){width="8cm"}
### Background 10-dimensional model before the decoupling
The background 10-dimensional space-time is expressed in the form of a product of two homogeneous spaces $\rm{M_d}$ and $\rm{M_D}$ as $$\label{eq:a01}
ds^2 = -dt^2 + r^2(t)\ {}^d g_{ij} (x^k) \ dx^i dx^j + R^2(t)\ {}^Dg_{ab} (X^c) \ dX^a dX^b,$$ where ${}^dg_{ij}$ and ${}^Dg_{ab}$ are the metrics of the outer space $\rm{M_d}$ and the inner space $\rm{M_D}$ with constant curvatures $K_r$ and $K_R$, respectively. Here the dimensions of $\rm{M_d}$ and $\rm{M_D}$ are $d = 3$ and $D = 6$. The inner space $\rm{M_D}$ expands initially and collapses after the maximum expansion with $K_R = 1$, while the outer space $\rm{M_d}$ continues to expand with $K_r = 0$ or $-1$. Then the background metric is $$\label{eq:a02}
\begin{split}
g_{00} &= -1, \quad g_{01} = g_{0a} = g_{ia} = 0, \\
g_{ij} &= r^2\ {}^d g_{ij}, \quad g_{ab} = R^2\ {}^D g_{ab},
\end{split}$$ and the Ricci tensor is $$\label{eq:a03}
\begin{split}
R^0_0 &= - \left(d \frac{\ddot{r}}{r} + D \frac{\ddot{R}}{R}\right), \\
R^i_j &= -\delta^i_j \ \left[\left(\frac{\dot{r}}{r}\right)^. + \frac{\dot{r}}{r} \left(d \frac{\dot{r}}{r} +
D \frac{\dot{R}}{R}\right) + (d-1) \frac{K_r}{r^2}\right], \\
R^a_b &= -\delta^a_b \ \left[\left(\frac{\dot{R}}{R}\right)^. + \frac{\dot{R}}{R} \left(d \frac{\dot{r}}{r} +
D \frac{\dot{R}}{R}\right) + (D-1) \frac{K_R}{R^2}\right],
\end{split}$$ where $i, j = 1, ..., d, \ a, b = d+1, ..., d+D$, and an overdot denotes $d/dt$. At the singular stage when $R$ is near $0$, the curvature terms with $K_r/r^2$ and $K_R/R^2$ are negligible, compared with the main terms, and the curvatures can be treated approximately as $K_r = K_R = 0$. The background energy-momentum tensor is $$\label{eq:a04}
T^\mu_\nu = p \delta^\mu_\nu + (\rho + p) u^\mu u_\nu,$$ where $u^\mu$ is the fluid velocity, $\rho$ the energy density, and $p$ the pressure. Here $\rho$ and $p$ are the common photon density and pressure in both spaces. The fluid is extremely hot and satisfies the equation of state $p = \rho/n$ of photon gas, where $n = d + D = 9$. Einstein equations are expressed as $$\label{eq:a05}
R^\mu_\nu = - 8\pi \bar{G} (T^\mu_\nu - \frac{1}{2} \delta^\mu_\nu T^\lambda_\lambda),$$ where $\bar{G}$ is the $(1+d+D)$-dimensional gravitational constant. In the following, we set $8\pi \bar{G} = 1$. The background equation of motion for the matter is $$\label{eq:a06}
\frac{\dot{\rho}}{\rho+p} + d \frac{\dot{r}}{r} +D \frac{\dot{R}}{R} = 0.$$ The Einstein equations for $r$ and $R$ were solved numerically in the previous paper[@tom] and their behaviors were shown in Figs. 1 - 7 of \[3\]. At the early stage, the expansion of the total universe is nearly isotropic (i.e. $r \propto R$). At the later stage, the inner space collapses after the maximum expansion, and at the final stage we have an approximate solution $$\label{eq:a07}
r = r_0 \ \tau^\eta, \quad R = R_0 \ \tau^\gamma \quad (r_0, \ R_0 : const)$$ with$$\label{eq:a08}
\eta = \{1 - [D(n-1)/d]^{1/2} \}/n , \quad \gamma = \{1 + [d(n-1)/D]^{1/2} \}/n,$$ and $\tau = t_A - t$, where $t_A$ is the final time corresponding to $R = 0$. For $d = 3$ and $D= 6$, we have $$\label{eq:a09}
\gamma = - \eta = 1/3.$$ For the solutions (\[eq:a07\]), Eqs. (\[eq:a04\]) and (\[eq:a05\]) lead to $R^0_0 = 0$ and $T^0_0 - \frac{1}{2} T^\mu_\mu \propto \rho$, so that we have $$\label{eq:a010}
\rho = 0$$ at the final stage. The curvature tensor is singular, on the other hand, in the limit $\tau \rightarrow 0$, like that in the $4$-dimensional Kasner space-time.[@wald]
![\[fig:bkr2\] Ratios of physical sizes of perturbations with the wave-number $k$ to the Hubble length $1/H$. The ratio $r(\tau) H/k $ in the outer space of the 10-dimensional space-times ($\tau \equiv t - t_A$) is shown as the solid line on the left-hand side, and the ratio $a(t_f) H/k $ in the Friedmann model is shown as the solid line on the right-hand side.](Fig2a.eps){width="8cm"}
### Decoupling of the two spaces and the Friedmann stage
As $\tau$ decreases, the inner space contracts and finally the size reaches the Planck length at the decoupling epoch. We discussed the decoupling condition at this quantum-gravitational epoch and the entropy production to this epoch in the previous paper \[3\].
At present we cannot analyze the process of decoupling accurately, because no quantum theory of gravitation has been established yet. However it is expected that the inner space is so homogeneous and quietly evolves without violent phenomena. This is because in both the inner and outer spaces the perturbations are assumed to be caused by quantum fluctuations before the decoupling, grow gravitationally, and remain very small and at the linear stage, before the decoupling. Note here that gravitational instability in the outer and inner spaces was treated in the previous paper \[5\]. Thus the inner space separates quietly from the outer space and disappears, while in the outer space the Friedmann model appears after the decoupling
After this decoupling epoch it is assumed here that the outer space is separated from the inner space and described using the Friedmann model with the metric $$\label{eq:a010a}
ds^2 = -d{t_f}^2 + a^2 (t_f) [d\chi^2 + \sigma^2(\chi ) (d\theta^2 + \sin^2 \theta d\Omega^2)]$$ with the cosmic time $t_f$ and $\sigma (\chi) = \sin \chi, \chi, \sinh \chi$ for the curvature $+, 0, -$ in the space, and the scale-factor $a (t_f) \propto {t_f}^{1/2}$ at the radiation-dominated hot stage. At the decoupling epoch $t_{dec}$ and $(t_f)_{dec}$, the entropy is assumed to be conserved in the 10-dimensional space-time and the Friedmann model. The behavior of scale-factors is shown in Fig. 1.
### Horizon crossing
The ratio of physical sizes of perturbations with the wave-length $1/k$ to the Hubble length $1/H$ is $$\label{eq:a010b}
r(\tau) H/k \propto \tau^{-4/3} = (t_A -t)^{-4/3}$$ before the decoupling, where $r \propto \tau^{-1/3}$ and $H \propto \tau^{-1}$, and it increases with time.
After the decoupling epoch we have the ratio at the Friedmann stage, $$\label{eq:a010c}
a(t_f) H/k \propto {t_f}^{-1/2},$$ which decreases with time. The two ratios are nearly equal at the decoupling epoch.
These ratios are shown schematically in Fig. 2. They can take the value $1$ in both sides before and after the decoupling epoch, that is, we can have the horizon crossing in both sides. Quantum fluctuations are created at the epoch of $r(\tau) H/k < 1$ in the outer space of the 10-dimensional space-time, and the fluctuations are observed as the fluctuations of the background radiation at the epochs of $a(t_f) H/k < 1$ at the Friedmann stage.
Perturbed quantities
--------------------
The simplest treatment of perturbations of geometrical and fluidal quantities is to expand them using harmonics, and to find the gauge-invariant quantities. For the four-dimensional universe (in the Friedmann model) it was shown in Bardeen’s theory on perturbations[@bar]. In the multi-dimensional universe consisting of the outer and inner homogeneous spaces $\rm{M_d}$ and $\rm{M_D}$ with different geometrical structures, we can have no harmonics in the $(d + D)$-dimensional space. Abbott et al.[@abb] considered separate expansions in $\rm{M_d}$ and $\rm{M_D}$ using the harmonics defined in the individual spaces, classified the perturbations in $\rm{M_d}$ and $\rm{M_D}$ individually as scalar (S), vector (V), and tensor (T), and classified the 6 types of perturbations in $\rm{M_d} + \rm{M_D}$ into three modes: the scalar mode (including SS), the vector mode (including SV, VS, VV), and the tensor mode (including ST, TS). The left and right sides of signatures correspond to the types of perturbations in $\rm{M_d}$ and $\rm{M_D}$, respectively.
In this paper only scalar and tensor modes are considered in the 10-dimensional space-times. So quantities in these modes are shown here.
### The scalar mode
The metric perturbations are expressed as $$\label{eq:a011}
\begin{split}
g_{00} &= - (1 + 2 A \ q^{(0)} Q^{(0)}),\\
g_{0i} &= - r b^{(0)} q_i^{(0)} Q^{(0)},\quad g_{0a} = - R B^{(0)} q^{(0)} Q_a^{(0)},\\
g_{ij} &= r^2 [(1+2h_L q^{(0)} Q^{(0)})\ {}^dg_{ij} + 2h_T^{(0)}q_{ij}^{(0)} Q^{(0)}],\\
g_{ab} &= R^2 [(1+2H_L q^{(0)} Q^{(0)})\ {}^Dg_{ab} + 2H_T^{(0)}q^{(0)} Q_{ab}^{(0)}],\\
g_{ia} &= 2rR G^{(0)} q_i^{(0)} Q_a^{(0)},
\end{split}$$ where $q^{(0)}, q_i^{(0)}, q_{ij}^{(0)}$ and $Q^{(0)}, Q_a^{(0)}, Q_{ab}^{(0)}$ are scalar harmonics in $\rm{M_d}$ and $\rm{M_D}$, respectively, and $A, b^{(0)}, B^{(0)}, h_L, H_L, h_T^{(0)}, H_T^{(0)}, $ and $G^{(0)}$ are functions of $t$.
The perturbations of fluid velocities and the energy-momentum tensor are expressed as $$\label{eq:a012}
u^0 = 1 - A q^{(0)} Q^{(0)},\quad
u^i = \frac{v^{(0)}}{r} q^{(0)i} Q^{(0)},\quad
u^a = \frac{V^{(0)}}{R} q^{(0)} Q^{(0)a},$$ and $$\label{eq:a013}
\begin{split}
T^0_0 &= - \rho ( 1+ \delta \ q^{(0)} Q^{(0)}),\\
T^0_i &= r (\rho + p)(v^{(0)} - b^{(0)}) q_i^{(0)} Q^{(0)},\quad
T^0_a = R (\rho + p)(V^{(0)} - B^{(0)}) q^{(0)} Q_a^{(0)},\\
T^i_j &= p (1 +\pi_L q^{(0)} Q^{(0)}) \delta^i_j,\quad
T^a_b = p (1 +\Pi_L q^{(0)} Q^{(0)}) \delta^a_b,\\
T^i_a &= 0,
\end{split}$$ where we consider a perfect fluid, so that the anisotropic pressure terms vanish and we have $$\label{eq:a014}
\pi_L = \Pi_L = \delta.$$ For the metric perturbations in Eq. (\[eq:a011\]) the following gauge-invariant quantities are defined: $$\label{eq:a015}
\begin{split}
\Phi_h &= h_L + \frac{h_T^{(0)}}{d} + \frac{r}{k_r^{(0)}} \frac{\dot{r}}{r} b^{(0)} -
\frac{r^2}{k_r^{(0)2}}\frac{\dot{r}}{r} \dot{h}_T^{(0)},\\
\Phi_H &= h_L + \frac{H_T^{(0)}}{D} + \frac{R}{k_R^{(0)}} \frac{\dot{R}}{R} B^{(0)} -
\frac{R^2}{k_R^{(0)2}}\frac{\dot{R}}{R}\dot{H}_T^{(0)},
\end{split}$$ $$\label{eq:a016}
\begin{split}
\Phi_A^{(r)} &= A + \frac{r}{k_r^{(0)}}\dot{b}^{(0)} + \frac{r}{k_r^{(0)}} \left(\frac{\dot{r}}{r} +
D\frac{\dot{R}}{R}\right) b^{(0)} \\
&- \frac{r^2}{k_r^{(0)2}}\left[\ddot{h}_T^{(0)} + \left(2\frac{\dot{r}}{r} +
D\frac{\dot{R}}{R}\right) \dot{h}_T^{(0)}\right] + D\left(H_L + \frac{H_T^{(0)}}{D}\right),\\
\Phi_A^{(R)} &= A + \frac{R}{k_R^{(0)}} \dot{B}^{(0)} + \frac{R}{k_R^{(0)}} \left(d\frac{\dot{r}}{r} +
\frac{\dot{R}}{R}\right) B^{(0)} \\
&- \frac{R^2}{k_R^{(0)2}} \left[\ddot{H}_T^{(0)} + \left(d\frac{\dot{r}}{r} +
2\frac{\dot{R}}{R}\right) \dot{H}_T^{(0)}\right] + d\left(h_L + \frac{h_T^{(0)}}{d}\right).
\end{split}$$ The gauge-invariant quantities $\Phi_h$ and $\Phi_A^{(r)}$ in the outer space correspond to the gauge-invariant perturbations defined by Bardeen[@bar] in the $(1+3)$-dimensional usual universes, and $\Phi_H$ and $\Phi_A^{(R)}$ in the inner space are similar to the above quantities. $\Phi_h$ and $\Phi_H$ represent the curvature perturbations in both spaces.
The gauge-invariant quantities for fluid velocity and energy density perturbations are given by $$\label{eq:a017}
v_s^{(0)} = v^{(0)} - \frac{r}{k_r^{(0)}} \dot{h}_T^{(0)}, \quad
V_s^{(0)} = V^{(0)} - \frac{R}{k_R^{(0)}} \dot{H}_T^{(0)},$$ and $$\label{eq:a018}
\epsilon_m = \delta +\frac{n+1}{n}\left[d\frac{\dot{r}}{k_r^{(0)}} (v^{(0)}-b^{(0)})+
D\frac{\dot{R}}{k_R^{(0)}} (V^{(0)}-B^{(0)})\right] .$$ It should be noticed that $v_s^{(0)}, V_s^{(0)}$ and $\epsilon_m$ do not vanish, though $\rho = 0$ at the final stage as in Eq. (\[eq:a010\]). As a gauge-invariant quantity that has no counterpart in the usual universe, we have $$\label{eq:a019}
\Phi_G = G^{(0)} - \frac{1}{2} \frac{k_R^{(0)}}{k_r^{(0)}}\frac{r}{R} h_T^{(0)} -
\frac{1}{2} \frac{k_r^{(0)}}{k_R^{(0)}}\frac{R}{r} H_T^{(0)}.$$ Moreover the auxiliary quantities ($\Phi_6$ and $\Phi_7$) and $\tilde{\Phi}_G$ are defined by $$\label{eq:ag2}
\frac{\dot{r}}{r} \frac{\dot{R}}{R} \Phi_6 \equiv \frac{\dot{R}}{R} \left(h_L +\frac{h_T^{(0)}}{d}\right) -
\frac{\dot{r}}{r} \left(H_L +\frac{H_T^{(0)}}{D}\right),$$ $$\label{eq:a4a}
\Phi_7 \equiv (r/\dot{r}) \Phi_h - (R/\dot{R}) \Phi_H - \Phi_6,$$ and $$\label{eq:ag1}
\tilde{\Phi}_G \equiv \frac{rR}{k_r^{(0)} k_R^{(0)}} \Phi_G.$$
### The tensor mode
We have only metric perturbations given by $$\label{eq:a027}
\begin{split}
g_{00} &= - 1, \quad g_{0i} = g_{0a} = g_{ia} = 0, \\
g_{ij} &= r^2 ({}^d g_{ij} + 2h_T^{(2)}q_{ij}^{(2)} Q^{(0)}),\\
g_{ab} &= R^2 ({}^D g_{ab} + 2H_T^{(2)}q^{(0)} Q_{ab}^{(2)}),
\end{split}$$ and have no fluidal perturbations, where have we neglected anisotropic stresses. In this mode, $h_T^{(2)}$ and $H_T^{(2)}$ correspond to the TS and ST parts of curvature perturbations and they themselves are gauge-invariant.
More details about perturbations can be seen in the previous paper \[5\].
Evolution of fluidal perturbations in the scalar mode
======================================================
In the previous paper \[5\], we derived the equations for geometrical perturbations $\Phi_h, \Phi_H, \tilde{\Phi}_G$ and $\Phi_6$ in the 10-dimensional space-times, and found their behavior by solving them. In this section we derive the equations for gauge-invariant variables representing fluidal perturbations $\epsilon_m, v_s^{(0)}$ and $V_s^{(0)}$ from the equations $\delta T^\nu_{\mu; \nu} = 0$ with $\mu
= 0, i$ and $a$, respectively, and derive their behaviors, where the suffices $\nu, i$ and $a$ take the values $0 \sim d+D, 1 \sim d$ and $d+1 \sim d+D$, respectively, in the outer and inner spaces with dimensions $d$ and $D$, respectively, where $d = 3$ and $D = 6$. In the following, $v_s^{(0)}$ and $V_s^{(0)}$ are expressed as $v_s$ and $V_s$ for simplicity.
First we obtain the following equation for $\dot{\epsilon}_m$ from $\delta T^A_{0; A} = 0$ $$\label{eq:a1}
\begin{split}
\frac{n}{n+1} &[\dot{\epsilon}_m - \frac{1}{n} (d\frac{\dot{r}}{r}+D\frac{\dot{R}}{R}) \epsilon_m]\\
&= -d\{\dot{\Phi}_h +[D\frac{\dot{R}}{R}+ (d-2)\frac{\dot{r}}{r}]\Phi_h \} +[-\frac{k_r^{(0)}}{r}
+\frac{d}{k_r^{(0)}}(\ddot{r}-\frac{\dot{r}^2}{r}) ] v_s \\
& -D\{\dot{\Phi}_H +[d\frac{\dot{r}}{r}+ (D-2)\frac{\dot{R}}{R}]\Phi_H\} +[-\frac{k_R^{(0)}}{R}
+\frac{D}{k_R^{(0)}}(\ddot{R}-\frac{\dot{R}^2}{R}) ] V_s \\
&-2[d\frac{\dot{r}}{r}(\frac{k_R^{(0)}}{R})^2+D\frac{\dot{R}}{R}(\frac{k_r^{(0)}}{r})^2]\tilde{\Phi}_G,
\end{split}$$ where $n = d+D, p=\rho/n$, a dot denotes $d/dt$, $k_r^{(0)}$ and $k_R^{(0)}$ are the wave-numbers in the outer and inner spaces, respectively, and $r= r_0 \tau^{-1/3}, \
R=R_0 \tau^{1/3}, \ \tau = t_0 - t$, and $t_0$ denotes the epoch of $r \rightarrow \infty$ and $R = 0$.
Equations for $\dot{v}_s$ and $\dot{V}_s$ are obtained from $\delta T^A_{i; A} = 0$ and $\delta T^A_{a; A} = 0$, respectively, as $$\label{eq:a2}
\begin{split}
\dot{v}_s &+ (\frac{\dot{r}}{r}-\frac{D}{n}\frac{\dot{R}}{R}) v_s = -\frac{D}{n}
\frac{k_r^{(0)}}{k_R^{(0)}} \frac{\dot{R}}{r}V_s + \frac{\epsilon_m}{n+1}\frac{k_r^{(0)}}{r}\\
&- \frac{k_r^{(0)}}{r} [(d-2) \Phi_h + D\Phi_H + 2(\frac{k_R^{(0)}}{R})^2 \tilde{\Phi}_G
+\frac{n+1}{n}D\frac{\dot{R}}{R}\Phi_7],
\end{split}$$ and $$\label{eq:a3}
\begin{split}
\dot{V}_s &+ (\frac{\dot{R}}{R}-\frac{d}{n}\frac{\dot{r}}{r}) V_s = -\frac{d}{n}
\frac{k_R^{(0)}}{k_r^{(0)}} \frac{\dot{r}}{R}v_s + \frac{\epsilon_m}{n+1}\frac{k_R^{(0)}}{R}\\
&- \frac{k_R^{(0)}}{R} [(D-2) \Phi_H + d\Phi_h + 2(\frac{k_r^{(0)}}{r})^2 \tilde{\Phi}_G
-\frac{n+1}{n} d\frac{\dot{r}}{r}\Phi_7].
\end{split}$$ From the latter two equations we obtain $$\label{eq:a4}
\begin{split}
(\frac{r}{k_r^{(0)}} v_s - \frac{R}{k_R^{(0)}} V_s)^. &= 2(\Phi_h - \Phi_H)
+\frac{1}{n}(d\frac{\dot{r}}{r}+D\frac{\dot{R}}{R})(\frac{r}{k_r^{(0)}} v_s -
\frac{R}{k_R^{(0)}} V_s)\\
&-2[(\frac{k_r^{(0)}}{r})^2 - (\frac{k_R^{(0)}}{R})^2] \tilde{\Phi}_G
-\frac{n+1}{n}(d\frac{\dot{r}}{r}+D\frac{\dot{R}}{R})\Phi_7,
\end{split}$$ and integrating this above equation, we have $$\label{eq:a5}
\begin{split}
\frac{r}{k_r^{(0)}} v_s - \frac{R}{k_R^{(0)}} V_s &= -\tau^{1/9} \int d\tau \ \tau^{-1/9}
\{2(\Phi_h - \Phi_H) -2[(\frac{k_r^{(0)}}{r})^2 - (\frac{k_R^{(0)}}{R})^2] \tilde{\Phi}_G\\
&-\frac{n+1}{n}(d\frac{\dot{r}}{r}+D\frac{\dot{R}}{R})\Phi_7 \} \ \equiv \ A_0(\Phi_h, \Phi_H,
\tilde{\Phi}_G, \Phi_7).
\end{split}$$ From Eqs. (\[eq:a1\]) and (\[eq:a5\]), we obtain $$\label{eq:a6}
\frac{n}{n+1} \dot{\epsilon}_m = \frac{1}{n+1}(d\frac{\dot{r}}{r}+D\frac{\dot{R}}{R})
\epsilon_m - A_1 + C_h v_s + C_H V_s,$$ and from Eq. (\[eq:a5\]) $$\label{eq:a7}
V_s = \frac{k_R^{(0)}}{R} (\frac{r}{k_r^{(0)}} v_s - A_0),$$ where $$\label{eq:a8}
\begin{split}
A_1 &\equiv d\{\dot{\Phi}_h +[D\frac{\dot{R}}{R}+(d-2)\frac{\dot{r}}{r}]\Phi_h\} \\
&+D\{\dot{\Phi}_H +[d\frac{\dot{r}}{r}+(D-2)\frac{\dot{R}}{R}]\Phi_H\}
+2 [d \frac{\dot{r}}{r}(\frac{k_R^{(0)}}{R})^2 +D \frac{\dot{R}}{R}(\frac{k_r^{(0)}}{r})^2] \tilde{\Phi}_G,
\end{split}$$ $$\label{eq:a10}
C_h \equiv -\frac{k_r^{(0)}}{r} + \frac{d}{k_r^{(0)}} (\ddot{r} -\dot{r}^2/r),$$ $$\label{eq:a11}
C_H \equiv -\frac{k_R^{(0)}}{R} + \frac{D}{k_R^{(0)}} (\ddot{R} -\dot{R}^2/R).$$ Next, differentiating Eq. (\[eq:a6\]) with respect to $t$ and eliminating $v_s$ and $\dot{v}_s$ using Eqs. (\[eq:a2\]) and (\[eq:a6\]), we obtain the following equations for $\epsilon_m$ and $v_s$ $$\label{eq:a12}
\ddot{\epsilon}_m + D_0 \dot{\epsilon}_m + D_1 \epsilon_m = \frac{n+1}{n} E$$ and $$\label{eq:a13}
v_s = \frac{1}{\tilde{C}} \{ \frac{1}{n+1} [n\dot{\epsilon}_m -(d\frac{\dot{r}}{r}+
D\frac{\dot{R}}{R}) \epsilon_m] +A_1 + C_H \frac{k_R^{(0)}}{R} A_0\},$$ where $$\label{eq:a14}
\tilde{C} \equiv C_h + \frac{k_R^{(0)}}{k_r^{(0)}} \frac{r}{R} C_H,$$ $$\label{eq:a15}
D_0 \equiv \frac{D}{n} (\frac{\dot{r}}{r} -\frac{\dot{R}}{R}) - \frac{\dot{\tilde{C}}}{\tilde{C}},$$ $$\label{eq:a16}
D_1\equiv -\frac{1}{n} (d\frac{\dot{r}}{r} +D\frac{\dot{R}}{R})^. -\frac{1}{n}
(d\frac{\dot{r}}{r} +D\frac{\dot{R}}{R}) (\frac{\dot{r}}{r} -\frac{\dot{\tilde{C}}}{\tilde{C}})
- \frac{k_r^{(0)}}{r} \frac{\tilde{C}}{n},$$ $$\label{eq:a9}
A_2 \equiv (d-2) \Phi_h +D\Phi_H +2(\frac{k_R^{(0)}}{R})^2 \tilde{\Phi}_G + \frac{n+1}{n} D
\frac{\dot{R}}{R} \Phi_7.$$ $$\label{eq:a17}
E \equiv - (A_1 + \frac{k_R^{(0)}}{R} C_H A_0)(\frac{\dot{r}}{r} -\frac{\dot{\tilde{C}}}{\tilde{C}})
- \dot{A}_1 - k_R^{(0)} (\frac{C_H}{R} A_0)^{.} + (\frac{D}{n} \frac{\dot{R}}{R} A_0 - A_2)
\frac{k_r^{(0)}}{r} \tilde{C},$$
Outside the horizons
--------------------
At epoch $t_{dec}$ when the outer space and the inner space decouple, $\tau$ is assumed to be so small that $$\label{eq:a18}
x \equiv \frac{3}{4r_0} k_r^{(0)} \tau^{4/3} \ll 1$$ and $$\label{eq:a19}
y \equiv \frac{3}{2R_0} k_R^{(0)} \tau^{2/3} \ll 1.$$ Under these conditions the perturbations with wave-numbers $k_r^{(0)}$ and $k_R^{(0)}$ are outside the horizons in the outer and inner spaces, and we have the relations $$\label{eq:c3}
\Phi_h = (\tau/\tau_i)^{-8/3} (\Phi_h)_i + \Delta \Phi_h,$$ $$\label{eq:c4}
\Phi_H = (\tau/\tau_i)^{-4/3} (\Phi_H)_i + \Delta \Phi_H,$$ where $\Delta \Phi_h$ and $\Delta \Phi_H$ consist of higher-order terms $O (x^2)$ and $O (y^2)$ with respect to $x$ and $y$, respectively, which are shown in Appendix A, and $\tau_i$ is an arbitrary epoch at the stage when Eqs.(\[eq:a18\]) and (\[eq:a19\]) are satisfied.
Now let us derive $\epsilon_m, v_s$ and $V_s$ corresponding to the above curvature perturbations, neglecting higher-order terms, such as $\Delta \Phi_h, \Delta \Phi_H,
\tilde{\Phi}_G$ and $\Phi_6$. Here we must pay attention to $\Phi_7$.
Substituting Eqs.(\[eq:c3\]) and (\[eq:c4\]) into Eqs.(\[eq:a14\]) $\sim$ (\[eq:a17\]), we obtain $$\label{eq:a21}
\begin{split}
C_h &= - \frac{k_r^{(0)}}{r_0} \tau^{1/3} + \frac{r_0}{k_r^{(0)}} \tau^{-7/3}, \quad
C_H = - \frac{k_R^{(0)}}{R_0} \tau^{-1/3} - \frac{2R_0}{k_R^{(0)}} \tau^{-5/3}, \\
\tilde{C} &\simeq - \frac{r_0}{k_r^{(0)}} \tau^{-7/3}, \quad \dot{\tilde{C}}/\tilde{C} =
-{\tilde{C}}'/\tilde{C} = 7/(3\tau), \quad \Phi_7 \simeq 3\tau (\Phi_h + \Phi_H)
\end{split}$$ $$\label{eq:a22}
D_0 \simeq - 17/(9\tau), \quad D_1 \simeq 0,$$ Moreover, $$\label{eq:a23}
\begin{split}
A_0 & \simeq 3\tau(\Phi_h + \Phi_H), \quad A_1 = (3/\tau)(\Phi_h + 2\Phi_H), \\
A_2 & \simeq - \frac{1}{3} (17\Phi_h +2\Phi_H), \quad E \simeq 0 .
\end{split}$$ From Eqs. (\[eq:a12\]), (\[eq:a13\]) and (\[eq:a7\]), we obtain in the lowest-order $$\label{eq:a24}
\epsilon_m = 0,$$ $$\label{eq:a25}
v_s = 4 x \ \Phi_h ,$$ and $$\label{eq:a26}
V_s = - 2 y \ \Phi_H.$$ So, $\epsilon_m$ is of higher-orders ($\sim O(x^2), O(y^2)$). Here $\Phi_h$ and $\Phi_H$ are independent, because $\Phi_{hi}$ and $\Phi_{Hi}$ are given arbitrarily.
Inside the horizons
-------------------
At earlier epochs of $\tau \gg \tau_{dec}$, $x$ and $y$ are comparable with $1$ or larger than $1$. It was shown in Sect. 3 of \[5\] that in the case of $x \gg 1$ and $y\gg 1$ under the condition $$\label{eq:a26z}
\mu/x (\equiv [(k_R^{(0)}/{R(t)})/(k_r^{(0)}/{r(t)})]^2) \ll 1,$$ the perturbations show wavy behaviors, depending on the wave-number $k_r^{(0)}$ (in the outer space) as $exp(i\omega x)$, where $\omega$ is a constant. In the case of $y \gg 1$ and $x\gg 1$ under the condition $\mu/x \gg 1$, the perturbations show wavy behaviors, depending on the wave-number $k_R^{(0)}$ (in the inner space) as $exp(i\omega y)$. In the former case ($\mu/x \ll 1$) the waves (depending on $k_r^{(0)}$) appear mainly in the outer space but do not appear in the inner space. In the latter case ($\mu/x \gg 1$), on the other hand, the waves (depending on $k_R^{(0)}$) appear mainly in the inner space but do not appear in the outer space.
At the stage of $x \gg 1$ and $y \gg 1$, the perturbations are inside the horizons and can be created by quantum fluctuations in both the outer space and the inner space. For these perturbations we assume that the values of $k_r^{(0)}$ and $k_R^{(0)}$ are smoothly distributed around the average values ($\bar{k}_r^{(0)}$ and $\bar{k}_R^{(0)}$). Here we pay attention to the perturbations with $x \gg 1, \bar{y} (\equiv (3/2R_0)
\bar{k}_R^{(0)}) \gg 1$ and $\bar{\mu}/x \equiv
[(\bar{k}_R^{(0)}/R(t)]/[k_r^{(0)}/r(t)]^2 \ll 1$. In this case, the perturbations with large ${k}_r^{(0)}$ have wavy behaviors in the outer space and the perturbations with ${k}_R^{(0)} \approx \bar{k}_R^{(0)}$ in the inner space are negligible. In the following we study their wavy behaviors proportional to $\exp (i\omega x) $. This is because such perturbations will survive and may be connected with the present observational information through the CMB radiation, after the decoupling of the outer space from the inner space. The other perturbations including the components ($\propto \exp i \omega y$) in the inner space will be disturbed or erased when the inner space is decoupled and disapear.
For perturbations with $x \gg 1, \bar{y} \gg 1$ and $\bar{\mu}/x \ll 1$, it is found from Eqs. (\[eq:a10\]), (\[eq:a11\]), (\[eq:a14\]), (\[eq:a15\]) and (\[eq:a16\]) that $$\label{eq:a26a}
\begin{split}
C_h & \simeq -k_r^{(0)}/r, \quad |C_H | \simeq \bar{k}_R^{(0)}/R \ll |C_h|, \\
\tilde{C} & \simeq C_h, \quad \dot{\tilde{C}}/\tilde{C} \simeq -\dot{r}/r,
\end{split}$$ and $$\label{eq:a26b}
D_0 \simeq \frac{7}{9\tau}, \quad D_1 \simeq (\frac{k_r^{(0)}}{r})^2 \frac{1}{9}(1+\frac{15}{16} x^{-2}).$$ Here let us put $\epsilon_m$ and curvature perturbations $\Phi_h$ and $\Phi_H$ as $$\label{eq:a26c}
{\epsilon_m = {\epsilon_{m0}\exp i\omega x, \quad \Phi_h = \Phi_{h0} \exp i\omega x}, \quad
\Phi_H = \Phi_{H0} \exp i\omega x}.$$ Then we obtain for $x \gg 1$ $$\label{eq:a27}
\begin{split}
\ddot{\epsilon}_m &+D_0 \dot{\epsilon}_m +D_1 \epsilon_m \\
&\simeq (k_r^{(0)}/r)^2 [(\frac{1}{9}-\omega^2)\epsilon_{m0} +i \omega (2\epsilon_{m0,x}
-\frac{1}{3}\epsilon_{m0}/x)\\
&+ \epsilon_{m0,xx} -\frac{1}{3} \epsilon_{m0,x}/x + \frac{5}{48} \epsilon_{m0}/x^2] \
\exp i\omega x \\
&= (\frac{k_r^{(0)}}{r^2})^2 (\frac{1}{9} -\omega^2) \ \epsilon_{m0} \ \exp i\omega x
\ [1 + O(1/x)] \quad {\rm for} \ \omega \ne 1/3.
\end{split}$$ On the other hand, we get from Eqs. (\[eq:a5\]) and (\[eq:a8\]) $$\label{eq:a28}
\begin{split}
A_0 &= - \frac{3\tau/2}{i\omega x} \ (\Phi_h-\Phi_H)
\ [1 + O(1/x)],\\
A_1 &= - \frac{k_r^{(0)}}{r} i\omega (d\Phi_h + D\Phi_H) \ [1 + O(1/x)], \\
\dot{A}_1 &= - {A_1}' = - (\frac{k_r^{(0)}}{r})^2 \omega^2 (d\Phi_h + D\Phi_H)
\ [1 + O(1/x)], \\
A_2 &= (\Phi_h + 6 \Phi_H) \ [1 + O(1/x)].
\end{split}$$ Here it is noticed that $\tilde{\Phi}_G$ is of higher-order with respect to $1/x \ (\ll 1)$ and is neglected. From Eqs. (78) and (81) of \[5\], it is found that $\Phi_6 = (2, 6) \tau \Phi_h$ for $\omega = 1, 1/3$, respectively, so that $\Phi_7 = 0$ for $\omega = 1, 1/3$. Moreover, using the condition $\bar{\mu}/x \ll 1$, we find that $$\label{eq:a28a}
\frac{\bar{k}_R^{(0)}}{R} |C_H A_0| \ll |A_1|.$$ Then from Eqs. (\[eq:a17\]) and (\[eq:a28\]), we obtain $$\label{eq:a30}
E \simeq (\frac{k_r^{(0)}}{r})^2 [(1+3 \omega^2) \Phi_{h0} + 6 (1 + \omega^2)
\Phi_{H0}] \exp i\omega x \ [1 + O(1/x)].$$ So it is found from Eqs. (\[eq:a12\]), (\[eq:a27\]) and (\[eq:a30\]) that $$\label{eq:a31}
\epsilon_{m0} \simeq (\frac{1}{9} - \omega^2)^{-1} \frac{10}{9}
[(1+ 3 \omega^2) \Phi_{h0}
+ 6 (1+\omega^2) \Phi_{H0}] \ [1 + O(1/x)] \quad {\rm for} \ \omega \ne 1/3$$ and $$\label{eq:a32}
\epsilon_{m0} \simeq - i x \frac{40}{27} ( \Phi_{h0} + 5 \Phi_{H0}) \quad {\rm for}
\ \omega = 1/3.$$ In Sect. 3 of \[5\], it was found that the approximate wavy solutions of equations for curvature perturbations are given only for $\omega = 1$ and $1/3$, and in these cases the solutions have the following relations $$\label{eq:a33}
\Phi_{H0} = - \frac{1}{3} \Phi_{h0} \quad {\rm for} \ \omega = 1,$$ and $$\label{eq:a34}
\Phi_{H0} = \Phi_{h0} \quad {\rm for} \ \omega = 1/3.$$ So we have the following expressions for $\epsilon_{m0}$ $$\label{eq:a35}
\epsilon_{m0} = \ O(1/x) \ \Phi_{h0} \quad \ \omega = 1,$$ and $$\label{eq:a36}
\epsilon_{m0} \simeq - \frac{80}{9} ix \Phi_{h0} \quad {\rm for} \ \omega = 1/3.$$
Quantum fluctuations
====================
In Sect. 2 and the previous paper \[5\], the perturbations were classified into three modes. In this paper we treat only their scalar and tensor modes and consider the perturbations created by the quantum effect in the comparably later stage of the $10$-dimensional universe which is associated with the inflating outer space and collapsing inner space. Here Weinberg’s procedure is used for the quantization [@wein].
The scalar mode
---------------
At the stage of $x \gg 1$ and $\bar{y} \gg 1$, the length of perturbations in the outer space can be smaller than the horizon size and they may be caused by the quantum effect, while at the later stage of $x < 1$ the length of perturbations is larger than the horizon size and they are frozen. So we should first consider the quantum fluctuations at earlier epochs of $x \gg 1$ and $\bar{y} \gg 1$. Additionally, moreover, we assume that $\bar{\mu}/x \ll 1$, corresponding to the perturbations in the inner space with the average value ($\bar{k}_R^{(0)}$), which was described in Sect. 3. Then these perturbations appear mainly in the outer space, and hence we can treat the perturbations, as if they are in the $4$-dimensional space-time (consisting of the time $t$ and the outer space).
The energy density perturbation $\epsilon_m$ is expressed by Eq. (\[eq:a12\]) in connection with gravitational perturbations. This equation is also derived from the action principle as $$\label{eq:b1}
I = \int dt d^3 \bm{x} ({\cal L}_\epsilon + {\cal L}_g),$$ where ${\cal L}_\epsilon$ and ${\cal L}_g$ are the fluidal and gravitational parts of the total Lagrangian, and $\bm{x}$ is the coordinate in the outer space. Here ${\cal L}_\epsilon$ can be derived from Eq. (\[eq:a12\]) as follows. At the stage of $x \gg 1, \bar{y} \gg 1$ and $\bar{\mu}/x \ll 1$, we have $D_0 $ and $D_1$ in Eq. (\[eq:a26b\]), and then the fluidal part in the equation of motion is derived using the following Lagrangian $$\label{eq:b2}
{\cal L}_\epsilon = \frac{1}{2} r^{7/3} [(\frac{\partial \epsilon_m}{\partial t})^2 +
(\frac{1}{3r})^2 (\frac{\partial \epsilon_m}{\partial \bm{x}})^2],$$ where $r = r_0 (t_0-t)^{-1/3}$, and $\partial \epsilon_m/\partial \bm{x} =
i \bm{k}_r^{(0)} \epsilon_m$ for $\epsilon_m \propto \exp (i {\bm{k}_r^{(0)}} \bm{x})$.
On the other hand, $\epsilon_m$ can be expanded as $$\label{eq:b3}
\epsilon_m (\bm{x},t) = \int d \bm{k}_r^{(0)} [\epsilon_m (k_r^{(0)}, t) \exp (i\bm{k}_r^{(0)} \bm{x})
\ \alpha(\bm{k}_r^{(0)}) +\epsilon_m^* (k_r^{(0)}, t) \exp (-i\bm{k}_r^{(0)} \bm{x}) \
\alpha^* (\bm{k}_r^{(0)})],$$ and $\Phi_h$ and $\Phi_H$ also can be written as $$\label{eq:b4}
\begin{split}
\Phi_h (\bm{x},t) &= \int d \bm{k}_r^{(0)} [\Phi_h (k_r^{(0)}, t) \exp (i\bm{k}_r^{(0)} \bm{x})
\ \alpha(\bm{k}_r^{(0)}) +\Phi_h^* (k_r^{(0)}, t) \exp (-i\bm{k}_r^{(0)} \bm{x}) \
\alpha^* (\bm{k}_r^{(0)})],\\
\Phi_H (\bm{x},t) &= \int d \bm{k}_r^{(0)} [\Phi_H (k_r^{(0)}, t) \exp (i\bm{k}_r^{(0)} \bm{x})
\ \alpha(\bm{k}_r^{(0)}) +\Phi_H^* (k_r^{(0)}, t) \exp (-i\bm{k}_r^{(0)} \bm{x}) \
\alpha^* (\bm{k}_r^{(0)})],
\end{split}$$ where the reality of these fields requires to take the above forms. The interaction of the photon field with the gravitational field makes the commutation relation of $\alpha(\bm{k}_r^{(0)})$ and $\alpha^* (\bm{k}_r^{(0)})$ complicated, but they become simple at very early times.[@wein]
In many cases when quantum fluctuations have so far been treated in a system of a scalar (inflaton) field and the gravitational field, the quantization of the scalar field is first tried.[@wein] In the present case also when we consider a system of a photon scalar field and the gravitational field, we try first the quantization of the photon scalar field in the following.
The canonical conjugate to $\epsilon_m (\bm{x},t)$ is then $$\label{eq:b5}
\pi_m (\bm{x},t) = \partial {\cal L}_\epsilon / \partial (\frac{\partial \epsilon_m}{\partial t})
= r^{7/3} \frac{\partial \epsilon_m}{\partial t}.$$ The commutator of $\epsilon_m$ and $\pi_m$ is $$\label{eq:b6}
[\epsilon_m (\bm{x},t), \epsilon_m (\bm{y},t)] = 0, \quad [\epsilon_m (\bm{x},t),
\partial \epsilon_m (\bm{y},t)/\partial t] = i r^{-7/3} \delta^3 (\bm{x} - \bm{y}).$$ These commutation relations imply that $\alpha (\bm{k})$ and $\alpha^\star (\bm{k})$ behave as conventionally normalized annihilation and creation operators $$\label{eq:b7}
[\alpha (\bm{k}), \alpha (\bm{k}')] = 0, \ \quad [\alpha (\bm{k}), \alpha^* (\bm{k}')] =
\delta^3 (\bm{k} - \bm{k}'),$$ when $\epsilon_m (k_r^{(0)}, t)$ is normalized at $r \rightarrow 0$ as $$\label{eq:b8}
\epsilon_m (k_r^{(0)}, t) \propto [r(t)]^{-2/3} [k_r^{(0)}]^{-1/2} \exp (i \omega k_r^{(0)}
\int^t_{t_*} \frac{dt'}{r(t')}),$$ where $t_*$ is arbitrary and $\omega$ is a constant ($= 1$ or $1/3$). This expression of $\epsilon_m (k_r^{(0)}, t)$ is used as the initial condition for created fields of energy density $\epsilon_m$. Here we choose the quantum state during the inflation of the outer space under the simple assumption that the state of the universe is the vacuum state $|0 \rangle$, defined so that $$\label{eq:b9}
\alpha (\bm{k}) |0\rangle = 0 \quad {\rm and} \quad \langle 0|0 \rangle = 1.$$ This corresponds to the Bunch-Davies vacuum[@bunch] in the outer space within the $10$-dimensional universe. As described in Sect. 3, $\epsilon_m$ and the curvature perturbations as quantum fluctuations are proportional each other. So the behavior of $\epsilon_m$ in Eq. (\[eq:b8\]) is common to that of $\Phi_h$ and $\Phi_H$, and, using Eqs. (\[eq:a33\]) - (\[eq:a36\]), we obtain $$\label{eq:b10}
\Phi_H (k_r^{(0)}, t) = - \frac{1}{3} \Phi_h (k_r^{(0)}, t)
\propto x \ [r(t)]^{-2/3} [k_r^{(0)}]^{-1/2} \exp (i \omega k_r^{(0)} \int^t_{t_*} \frac{dt'}{r(t')})$$ for $\omega = 1$, and $$\label{eq:b10a}
\Phi_H (k_r^{(0)}, t) = \Phi_h (k_r^{(0)}, t)
\propto x^{-1} \ [r(t)]^{-2/3} [k_r^{(0)}]^{-1/2} \exp (i \omega k_r^{(0)} \int^t_{t_*} \frac{dt'}{r(t')})$$ for $\omega = 1/3$.
The tensor mode
---------------
In the tensor mode, there are two types (ST) and (TS), as described in Sect. 2 and \[5\]. (ST) has the $3$-dimensional scalar and the $6$-dimensional tensor, while (TS) has the $3$-dimensional tensor and the $6$-dimensional scalar. Here we take up (TS) with the amplitude $h_T^{(2)}$, and neglect (ST) with the amplitude $H_T^{(2)}$, which may not be connected with the observation in the $3$-dimensional outer space, after the decoupling of the inner space.
In \[5\], we studied the behavior of tensor perturbations $h_T^{(2)}$. They satisfy $$\label{eq:b11}
\ddot{h}_T^{(2)} +(d\frac{\dot{r}}{r}+D\frac{\dot{R}}{R}) \dot{h}_T^{(2)}+
[(\frac{k_r^{(2)}}{r})^2+ (\frac{k_R^{(0)}}{R})^2] h_T^{(2)} = 0.$$ Here we consider the case of $$\label{eq:b12}
(\frac{\bar{k}_R^{(0)}}{R})/(\frac{k_r^{(2)}}{r}) \ll 1,$$ where $k_r^{(2)}$ and $k_R^{(0)}$ are the wave-numbers in the outer and inner spaces, respectively, and $\bar{k}_R^{(0)}$ is the average wave-number in the inner space. Then we have $$\label{eq:b13}
\ddot{h}_T^{(2)} - 3\frac{\dot{r}}{r} \dot{h}_T^{(2)}+ (\frac{k_r^{(2)}}{r})^2 h_T^{(2)} = 0,$$ where we used the relation $R \propto 1/r \propto \tau^{1/3}$ and $d = D/2= 3$. This equation can be also derived from the action principle as $$\label{eq:b14}
I = \int dt d^3 x \ {\cal L}_t,$$ where $$\label{eq:b15}
{\cal L}_t = \frac{1}{2} r^{-3} [(\frac{\partial h_T^{(2)}}{\partial t})^2 + \frac{1}{r^2}
(\frac{\partial h_T^{(2)}}{\partial x^i})^2]$$ for $\partial h_T^{(2)} \propto \exp i \bm{k}_r^{(2)} \bm{x}$.
On the other hand, the amplitude $\partial h_T^{(2)}$ takes the form $$\label{eq:b16}
h_T^{(2)} (\bm{x},t) = \int d \bm{k}_r^{(2)} \ [h_T^{(2)} (k_r^{(2)}, t) \exp (i\bm{k}_r^{(2)} \bm{x})
\ \alpha(\bm{k}_r^{(2)}) + h_T^{(2)*} (k_r^{(2)}, t) \exp (-i\bm{k}_r^{(2)} \bm{x}) \
\alpha^* (\bm{k}_r^{(2)})],$$ and the canonical conjugate to $h_T^{(2)} (\bm{x},t)$ is then $$\label{eq:b17}
\pi_T (\bm{x},t) = \partial {\cal L}_t /(\partial h_T^{(2)}/{\partial t})
= r^{-3} \frac{\partial h_T^{(2)}}{\partial t}.$$ The commutator of $h_T^{(2)}$ and $\pi_T$ is $$\label{eq:b18}
[h_T^{(2)} (\bm{x},t), h_T^{(2)} (\bm{y},t)] = 0, \ [h_T^{(2)} (\bm{x},t),
\partial h_T^{(2)} (\bm{y},t)/\partial t] = i r^{3} \delta^3 (\bm{x} - \bm{y}).$$ These commutation relations imply that $\alpha (\bm{k})$ and $\alpha^* (\bm{k})$ behave as conventionally normalized annihilation and creation operators, in the same way as Eq. (\[eq:b6\]), when $h_T^{(2)} (k_r^{(2)}, t)$ is normalized at $r \rightarrow 0$ as $$\label{eq:b19}
h_T^{(2)} (k_r^{(2)}, t) \propto [r(t)]^2 [k_r^{(2)}]^{-1/2} \exp (i k_r^{(2)} \int^t_{t_*}
\frac{dt'}{r(t')}).$$ This expression of $h_T^{(2)} (k_r^{(2)}, t)$ is used as the initial condition of created fields in the tensor mode $h_T^{(2)}$. Here we choose the quantum state during the inflation of the outer space, so that the state of the universe may satisfy the relation in Eq. (\[eq:b9\]).
Spectra of fluctuations and their comparison with CMB observation
=================================================================
The scalar mode
---------------
The information about the perturbations which are created by the quantum fluctuations inside the horizon can be used to make an initial condition for the evolution of perturbations which re-enter the horizon after the long inflation. For this purpose, we use the quantities which are conserved outside the horizon. In the $4$-dimensional universe with $3$-dimensional space-section, we have a gauge-invariant curvature perturbation, represented as $$\label{eq:c1}
{\cal R}_4 \equiv \Phi_H,$$ which is a conserved quantity.[@bar] In the $10$-dimensional universe, on the other hand, we have the following two independent similar quantities as the candidates $$\label{eq:c2}
{\cal R}_h \equiv (\tau/\tau_{dec})^{8/3} \Phi_h \quad {\rm and} \quad {\cal R}_H \equiv
(\tau/\tau_{dec})^{4/3} \Phi_H,$$ where $\tau_{dec}$ represents the epoch when the inner space decouples from the outer space. For $x (\equiv (3/4r_0) k_r^{(0)} \tau^{4/3}) < 1$ and $y (\equiv (3/2R_0)
k_R^{(0)} \tau^{2/3}) < 1$, ${\cal R}_h$ and ${\cal R}_H$ are nearly constant, and so these can be regarded as quantities conserved outside the horizon.
As other candidates for conserved quantities, we may consider $$\label{eq:c5}
{\cal R}_v (\equiv \frac{\dot{r}}{k_r^{(0)}} v_s) \quad {\rm and} \quad
{\cal R}_V (\equiv \frac{\dot{R}} {k_R^{(0)}} V_s),$$ but, for $x \ll 1$, they are not independent of ${\cal R}_h$ and ${\cal R}_H$, $$\label{eq:c6}
{\cal R}_v = -\frac{23}{32} \Phi_h + \frac{19}{4} \Phi_H,$$ $$\label{eq:c7}
{\cal R}_V = -\frac{11}{32} \Phi_h + \frac{13}{4} \Phi_H$$ with respect to the main terms. For $x \gg 1$ and $y \gg 1$, moreover, we find that $v_s$ and $V_s$ are comparable with $\Phi_h$ and $\Phi_H$, respectively, and ${\cal R}_v$ and ${\cal R}_V$ are $\sim v_s/x$ and $\sim V_s/y$, respectively, which are small, compared with $\Phi_h$ and $\Phi_H$. This means that the roles of ${\cal R}_v$ and ${\cal R}_V$ are small, compared with those of ${\cal R}_h$ and ${\cal R}_H$, respectively. In this paper, therefore, we adopt ${\cal R}_h$ and ${\cal R}_H$ as the conserved quantities in the $10$-dimensional universe. Neither of them, however, is necessarily a conserved quantity which is directly connected at epoch $\tau_{dec}$ with $R_4$ in the $4$-dimensional universe.
Here we construct the $10$-dimensional gauge-invariant conserved quantity ${\cal R}_{10}$ using ${\cal R}_h$ and ${\cal R}_H$, by imposing the following two conditions :
\(1) ${\cal R}_{10} = {\cal R}_{4}$ at epoch ($\tau_{dec}$) of the decoupling of the outer space from the inner space, and
\(2) ${\cal R}_{10}$ is consistent with the spectral constraint given by the CMB observation.
As the first candidate of ${\cal R}_{10}$, we consider a linear combination of $\Phi_h$ and $\Phi_H$ as $$\label{eq:c8}
{\cal R}_{10} = \lambda_0 {\cal R}_H + \lambda_1 {\cal R}_h,$$ where constants $\lambda_0$ and $\lambda_1$ are determined so as to satisfy the above two conditions (1) and (2).
At epoch $\tau$ when $x \ (= (3/4r_0) k_r^{(0)} \tau^{4/3}) \gg 1$ and $y
(= (3/4r_0) k_R^{(0)} \tau^{2/3}) \gg 1$, $\Phi_h$ and $\Phi_H$ are created by quantum fluctuations and they are expressed using Eqs.(\[eq:b10\]) and (\[eq:b10a\]) as $$\label{eq:c11}
\begin{split}
\Phi_h &= \tau^{14/9} [k_r^{(0)}]^{1/2} \exp (ix) +\alpha \tau^{-10/9} [k_r^{(0)}]^{-3/2}
(4r_0/3)^2 \exp (ix/3), \\
\Phi_H &= -\frac{1}{3} \tau^{14/9} [k_r^{(0)}]^{1/2} \exp (ix) +\alpha \tau^{-10/9}
[k_r^{(0)}]^{-3/2} (4r_0/3)^2 \exp (ix/3),
\end{split}$$ where we used $r \propto \tau^{-1/3}$ and $\alpha$ is an arbitrary constant. Inserting Eq.(\[eq:c11\]) into Eq.(\[eq:c2\]), we obtain $$\label{eq:c12}
\begin{split}
{\cal R}_h &= {\tau_{dec}}^{-8/3} [k_r^{(0)}]^{1/2} [{\tau}^{38/9} \exp (ix) + \alpha
{\tau}^{14/9} (\frac{4r_0}{3k_r^{(0)}})^2 \exp (ix/3)] \\
&= [k_r^{(0)}]^{-8/3} {\tau_{dec}}^{-8/3} (4r_0 \ x/3)^{19/6} [\exp (ix) +
\alpha x^{-2} \exp (ix/3)] ,
\end{split}$$ $$\label{eq:c13}
\begin{split}
{\cal R}_H &= {\tau_{dec}}^{-4/3} [k_r^{(0)}]^{1/2} [-\frac{1}{3} {\tau}^{26/9}
\exp (ix) + \alpha {\tau}^{2/9} (\frac{4r_0}{3k_r^{(0)}})^2 \exp (ix/3)] \\
&= [k_r^{(0)}]^{-5/3} {\tau_{dec}}^{-4/3} (4r_0 \ x/3)^{13/6} [-\frac{1}{3} \exp (ix)
+ \alpha x^{-2} \exp (ix/3)] .
\end{split}$$ As $x$ decreases and becomes smaller than $1$, the $x$ dependence of ${\cal R}_h$ and ${\cal R}_H$ changes from the wavy behavior ($\propto \exp (ix) $ and $\exp (ix/3) $) to the constant ones. At epoch $\tau_{eq}$ with $x = 1$, we have therefore $$\label{eq:c12a}
{\cal R}_h (\tau_{eq}) = \Xi \ (\zeta_h + \alpha \zeta'_h),$$ $$\label{eq:c13a}
{\cal R}_H (\tau_{eq}) = \Xi \ \frac{x_{dec}}{x} (-\frac{1}{3} \zeta_H + \alpha \zeta'_H),$$ where $$\label{eq:c13b}
\Xi \equiv [k_r^{(0)}]^{-8/3} {\tau_{dec}}^{-8/3} (4r_0/3)^{19/6} ,$$ where $x_{dec} \equiv (3/4r_0) k_r^{(0)} {\tau_{dec}}^{4/3}$, and $\zeta_h, \zeta'_h,
\zeta_H$ and $\zeta'_H$ are constants. The exact values of these constants are determined by solving dynamical equations for $\Phi_h$ and $\Phi_H$ given in \[5\], but they are estimated to be $\approx 1$, because ${\cal R}_h$ and ${\cal R}_H$ are nearly constant for $x < 1$ and $y < 1$.
Now we assume that the CMB spectrum is determined at epoch $\tau_{eq}$ when $x = 1$ (indicating the horizon exit), and consider the $k_r^{(0)}$ dependence of ${\cal R}_{10}$ at this epoch. Here ${\cal R}_{10}$ at epoch $\tau_{eq}$ is expressed as $$\label{eq:c14}
{\cal R}_{10} = {\cal R}_{0} z^{-5/3} + {\cal R}_{1} z^{-8/3}$$ around the observed wave-number $(k_r^{(0)})_{obs}$, where $z \equiv
k_r^{(0)}/(k_r^{(0)})_{obs}$, $$\label{eq:c15}
{\cal R}_{0} \equiv \lambda_0 {\rm Re}(-\frac{1}{3} \zeta_H + \alpha \zeta'_H) \
(x_{dec} \ \Xi)_{z = 1} ,$$ $$\label{eq:c16}
{\cal R}_{1} \equiv \lambda_1 {\rm Re}( \zeta_h + \alpha \zeta'_h) \ \Xi_{z = 1},$$ where Re means the real part, and $\lambda_0$ and $\lambda_1$ are coefficients in Eq.(\[eq:c8\]).
The CMB observation shows that the $k_r^{(0)}$ dependence of ${\cal R}_{10}$ is $$\label{eq:c17}
z^{(-4+n_s)/2} = z^{-1.517}$$ for $(k_r^{(0)})_{obs} = 0.002$ Mpc$^{-1}$, where $n_s = 0.966$ according to the WMAP 7 year result[@kom]. The condition that Eq.(\[eq:c14\]) and Eq.(\[eq:c17\]) should be consistent in the neighborhood of $z = 1$ is $$\label{eq:c18}
z^{-5/3} + \delta_1 z^{-8/3} = (1+\delta_1) z^{-1.517},$$ where $\delta_1 \equiv {\cal R}_{1}/ {\cal R}_{0} = (x_{dec})^{-1} (\lambda_1/ \lambda_0)
{\rm Re}(\zeta_h + \alpha \zeta'_h) /{\rm Re}(-\frac{1}{3}\zeta_H + \alpha \zeta'_H)$. From the continuity of this equation and its first derivative at $z = 1$, it is found that $$\label{eq:c19}
\delta_1 = -0.130.$$ That is, the observational spectrum (\[eq:c17\]) can be reproduced when ${\cal R}_H$ is main and ${\cal R}_h$ is about $10 \%$ of the total ${\cal R}_{10}$.
The above definition of ${\cal R}_{10}$ satisfies the condition of continuity of Eq.(\[eq:c18\]) in the first derivative, but not in the second derivative. In order to satisfy also the condition of continuity in the second derivative, we consider the second candidate of ${\cal R}_{10}$ at $\tau_{eq}$ expressed as $$\label{eq:c8a}
{\cal R}_{10} = \lambda_0 {\cal R}_H + \lambda_1 {\cal R}_h
+ \lambda_2 [{\cal R}_h]^2/{\cal R}_H,$$ where constants $\lambda_0, \lambda_1$ and $\lambda_2$ are determined so as to satisfy the above two conditions (1) and (2). Here ${\cal R}_{10}$ at epoch $\tau_{eq}$ is rewritten as $$\label{eq:c14a}
{\cal R}_{10} = {\cal R}_{0} z^{-5/3} + {\cal R}_{1} z^{-8/3} + {\cal R}_{2} z^{-11/3},$$ where ${\cal R}_{0}$ and ${\cal R}_{1}$ are defined by Eqs. (\[eq:c15\]) and (\[eq:c16\]), and $$\label{eq:c15a}
{\cal R}_{2} \equiv \lambda_2 [{\rm Re}( \zeta_h + \alpha \zeta'_h)^2
/{\rm Re}(-\frac{1}{3}\zeta_H + \alpha \zeta'_H)] [(x_{dec})^{-1} \Xi]_{z = 1}.$$ Then from the condition that Eqs.(\[eq:c14a\]) and (\[eq:c17\]) should be consistent in the neighborhood of $z = 1$, we have $$\label{eq:c18a}
z^{-5/3} + \delta_1 z^{-8/3} + \delta_2 z^{-11/3} = (1+\delta_1+ \delta_2) z^{-1.517},$$ where $\delta_1 \equiv {\cal R}_{1}/ {\cal R}_{0} $ and $\delta_2 \equiv {\cal R}_{2}
/ {\cal R}_{0} $ . From the continuity of this equation and its first and second derivatives at $z = 1$, it is found that $$\label{eq:c19a}
\delta_1 = -0.260 \quad {\rm and} \quad \delta_2 = 0.0696.$$ Now let us define the power spectrum of curvature perturbations as[@kom; @felice] $$\label{eq:c20}
{\cal P}_s \equiv \frac{4\pi (k_r^{(0)})^3}{(2\pi)^3} |{\cal R}_{10}|^2.$$ Then for $R_{10}, \delta_1$ and $\delta_2$ in Eqs.(\[eq:c8a\]) and (\[eq:c19a\]), $$\label{eq:c22}
{\cal P}_s = \frac{4\pi [(k_r^{(0)})_{obs}]^3}{(2\pi)^3} |{\cal R}_{0}|^2 (1+\delta_1 + \delta_2)^2,$$ The WMAP 7 year normalization[@kom] gives $$\label{eq:c23}
({\cal P}_s)_{obs} = 2.42 \times 10^{-9}$$ on the scale $(k_r^{(0)})_{obs} = 0.002$ Mpc$^{-1}$. Then we obtain $$\label{eq:c24}
{\cal R}_{0} = \lambda_0 |{\cal R}_H (\tau_{eq})| = \frac{\sqrt{2}\pi}{1+\delta_1 +\delta_2}
{[{\cal P}_s /(k_r^{(0)})^3]_{obs}}^{1/2}.$$ On the other hand, we have $$\label{eq:c24a}
\begin{split}
\frac{\lambda_1}{\lambda_0} &= \delta_1 \cdot
x_{dec} {\rm Re}(-\frac{1}{3}\zeta_H + \alpha \zeta'_H)/{\rm Re}( \zeta_h + \alpha \zeta'_h) , \\
\frac{\lambda_2}{\lambda_0} &= \delta_2 \cdot
[x_{dec} {\rm Re}(-\frac{1}{3}\zeta_H + \alpha \zeta'_H)/{\rm Re}( \zeta_h + \alpha \zeta'_h)]^2,
\end{split}$$ where $x_{dec} = (\tau_{dec}/\tau_{eq})^{4/3} = (r_{eq}/r_{dec})^4 \ll 1$, and the factor ${\rm Re}(-\frac{1}{3}\zeta_H + \alpha \zeta'_H)/{\rm Re}( \zeta_h + \alpha \zeta'_h)$ is of the order of $1$. Since $\delta_2 \simeq (\delta_1)^2$, we have $\lambda_2/ \lambda_0
\simeq (\lambda_1/\lambda_0)^2$.
It is concluded that ${\cal R}_{10}$ is consistent with the observed spectra of CMB radiation under the condition of (\[eq:c24\]) and (\[eq:c24a\]).
Thus we could derive the condition that the parameters $\lambda_0, \lambda_1$ and $\lambda_2$ in $\mathcal{R}_{10}$ should satisfy for the consistency with the CMB observation. From their ratios the role of the curvature perturbation in the inner space is found to be larger than that in the outer space. This condition and its consequeces are concerned with the condition at the earlier stage and the initial condition of the universe, which should be expressed as a perturbation model with the theoretical model parameters, and the above three parameters should be related to the latter parameters. They may be influenced through $\mathcal{R}_{10}$ by the process of decoupling, which has not been discussed here, because its quantum-gravitational process cannot be treated at present. This situation in the observational aspect is compared with the situations in other inflation models, later in the subsection 5.3
The tensor mode
---------------
In the limit of $x \ (\equiv ({3}/{4r_0}) k_r^{(2)} \tau^{4/3}) \rightarrow 0$, the gauge-invariant perturbation $h_T^{(2)}$ tends to $a + b \ln \tau$, as seen from the analyses in Abbott et al.[@abb] and the previous paper \[5\], where $a$ and $b$ are constants. So, as the quantity ${\cal R}_t$ which is conserved outside the horizon, we adopt $$\label{eq:c25}
{\cal R}_t \equiv h_T^{(2)} [a + b \ln \tau_{dec}]/[a + b\ln \tau],$$ so that ${\cal R}_t$ leads to a constant in the limit of $x \rightarrow 0$.
At the epoch $\tau_{eq}$ of $x = 1$, we have the relation $$\label{eq:c26}
\tau \propto [k_r^{(2)}]^{-3/4},$$ so that $r^2 [k_r^{(2)}]^{-1/2} \propto \tau^{-2/3} [k_r^{(2)}]^{-1/2} = const$, and from Eq. (\[eq:b19\]) $$\label{eq:c27}
h_T^{(2)} (k_r^{(2)}, \tau_{eq}) = \lambda_t \cdot \exp (ix),$$ where $\lambda_t$ is a constant. Then it is found from Eq. (\[eq:c25\]) that $$\label{eq:c27a}
{\cal R}_t (\tau_{eq}) = \lambda_t [a + b \ln \tau_{dec}]
[a - \frac{3}{4} b \ln k_r^{(2)} + const]^{-1} \exp (ix).$$ As $x$ decreases and becomes $< 1$, the $x$ dependence of ${\cal R}_t$ changes from the wavy behavior to the stationary constant one. But the $k_r^{(2)}$ dependence does not change, so that the spectrum in the tensor mode have the form of $$\label{eq:c28}
[a - \frac{3}{4} b \ln k_r^{(2)} + const]^{-1}.$$ The corresponding power spectrum is $$\label{eq:c29}
{\cal P}_t \equiv \frac{4\pi (k_r^{(2)})^3}{(2\pi)^3} |{\cal R}_{t} (\tau_{eq})|^2.$$ The amplitude of ${\cal R}_t (\tau_{eq})$ should be determined, corresponding to the observation, which has not been given yet. At present, we have the condition $r \equiv {\cal P}_t / {\cal P}_s < 0.24$ for $k_r^{(0)} = k_r^{(2)} = 0.002$ Mpc$^{-1}$.[@kom]
Comparison with the spectral analyses in other inflation models
---------------------------------------------------------------
In the $4$-dimensional universe due to the Einstein theory, the quantity conserved outside the horizon is uniquely defined using one of curvature perturbations.[@bar] But in hypothetical inflation models with inflaton scalar fields (including the non-minimal coupling with the Ricci scalar), the values of parameters such as slow-roll parameters ($\epsilon, \ \eta$) and the number $N$ of inflationary e-folds,[@wein; @ll] and the coupling parameter $\xi$ in the scalar field equation[@fak; @sal; @kai; @komhut] are not unique. The observed spectral index $n_s \ (\approx 0.97)$ is, therefore, obtained by adjusting the above parameters $\epsilon, \eta, N$ and $\xi$.
In the $R + R^2$ modified gravitational theory, we have an inflation model associated with the de Sitter type solution which was derived first by Nariai and Tomita[@nt] and rederived later by Starobinsky.[@st] Mukhanov and Chibisov[@mc] derived the quantum fluctuations generated at the de Sitter stage, and it was found that the spectral index $n_s$ of these fluctuations can be expressed as $$\label{eq:c30}
n_s - 1 = -1/[ 1 + \frac{1}{2} \ln (k_{obs}/a H)] = - 1/(1 + \frac{1}{2} N),$$ where $k_{obs}$ is the observed wave-number, $N$ is the inflationary e-fold, and $a$ and $H$ are the scale factor and the Hubble constant at the epoch when the de Sitter expansion ends. This number $N$ is determined to be $70$, so that we may have $n_s \simeq 0.97$ (the observed value).
In the present case of a photon scalar field in the $10$-dimensional universe, the inflation of the outer space is unique, because the scale factor $r$ of the outer space is $\propto \tau^{-1/3}$ (in the non-viscous case). On the other hand, the conserved quantity is not unique, because there are two independent curvature perturbations $\Phi_h$ and $\Phi_H$ before the decoupling of the outer and inner spaces. It is, therefore, a key point to determine how to combine them in this case, to derive ${\cal R}_{10}$ (connecting the two epochs outside the horizon). To obtain the observed spectral index $n_s$, we made the examples of the combination of $\Phi_h$ and $\Phi_H$ as the conserved quantity, so that the theoretical spectral index $n_s$ may be consistent with the observed one.
Concluding remarks
==================
In this paper I showed the possibility of deriving the observed fluctuation of CMB radiation from the quantum fluctuations which appeared at the inflating stage of the outer space in the $10$-dimensional universe. In contrast to the rapid inflation in the inflaton scalar field, our inflation is a power type, but we have two independent curvature perturbations which make possible the consistency with the observed spectra.
For simplicity, on the other hand, I neglected the viscosity which may play important roles in dynamics between the outer space and the inner space. If we take the viscosity into account, not only much entropy is produced (as shown in the previous paper[@tom]), but also the severe condition such as $\lambda_2/\lambda_0 \simeq (\lambda_1/\lambda_0)^2
\ll 1$ for producing the observed CMB fluctuations in the scalar mode may be softened. The next step is to study the perturbations and quantum fluctuations to derive the condition, in the case with viscous processes due to the transport of $10$-dimensional gravitational waves.[@tom; @TI]
Higher-order terms $\Delta \Phi_h$ and $\Delta \Phi_H$ of curvature perturbations $\Phi_h$ and $\Phi_H$
=======================================================================================================
The higher-order terms $\Delta \Phi_h$ and $\Delta \Phi_H$ (with respect to $x$ and $y \ (\ll 1)$) in Eqs. (\[eq:c3\]) and (\[eq:c4\]) are derived from the part of the Einstein equations in the scalar mode $$\label{eq:aa1}
\delta R^0_i = - \delta T^0_i,$$ $$\label{eq:aa2}
\delta R^0_a = - \delta T^0_a,$$ where the perturbed components of energy-momentum tensors are $$\label{eq:aa3}
\delta T^0_i = r(\rho+p)(v_s^{(0)}- b^{(0)}) q_i Q,$$ $$\label{eq:aa4}
\delta T^0_a = R(\rho+p)(V_s^{(0)}- B^{(0)}) q Q_a.$$ Using the expressions of $\delta R^0_i$ and $\delta R^0_a$ in the Appendix of Abbott et al’s paper[@abb], we obtain $$\label{eq:aa5}
\begin{split}
-\frac{r}{k_r^{(0)}} (\rho+p) v_s^{(0)} &= -(d-1)\dot{\Phi}_h +[-dD\frac{\dot{R}}{R}-
(d-1)(d-2)\frac{\dot{r}}{r} ] \Phi_h \\
&- D\dot{\Phi}_H +[(2-d)D\frac{\dot{r}}{r}-D(D-1)\frac{\dot{R}}{R} ] \Phi_H \\
&- (\frac{k_R^{(0)}}{R})^2 \dot{\tilde{\Phi}}_G +[2(2-d)\frac{\dot{r}}{r}(\frac{k_R^{(0)}}{R})^2 -
D\frac{\dot{R}}{R}(\frac{k_r^{(0)}}{r})^2] \tilde{\Phi}_G \\
&+ \{\frac{1}{2}(\frac{k_R^{(0)}}{R})^2+(d-1) [\frac{\ddot{r}}{r} -(\frac{\dot{r}}{r})^2]
+D \frac{\dot{r}}{r} \frac{\dot{R}}{R} \} (-\Phi_6 + \frac{r}{\dot{r}}\Phi_h
-\frac{R}{\dot{R}}\Phi_H),
\end{split}$$ $$\label{eq:aa6}
\begin{split}
-\frac{R}{k_R^{(0)}} (\rho+p) V_s^{(0)} &= -(D-1)\dot{\Phi}_H +[-dD\frac{\dot{r}}{r}-
(D-1)(D-2)\frac{\dot{R}}{R} ] \Phi_H \\
&- d\dot{\Phi}_h +[(2-D)d\frac{\dot{R}}{R}-d(d-1)\frac{\dot{r}}{r} ] \Phi_h \\
&- (\frac{k_r^{(0)}}{r})^2 \dot{\tilde{\Phi}}_G +[2(2-D)\frac{\dot{R}}{R}(\frac{k_r^{(0)}}{r})^2 -
D\frac{\dot{r}}{r}(\frac{k_R^{(0)}}{R})^2] \tilde{\Phi}_G \\
&+ \{\frac{1}{2}(\frac{k_r^{(0)}}{r})^2+(D-1) [\frac{\ddot{R}}{R} -(\frac{\dot{R}}{R})^2]
+d \frac{\dot{r}}{r} \frac{\dot{R}}{R} \} (\Phi_6 + \frac{R}{\dot{R}}\Phi_H
-\frac{r}{\dot{r}}\Phi_h),
\end{split}$$ where a dot denotes $d/dt$, and $\Phi_h, \Phi_H, \tilde{\Phi}_G$ and $\Phi_6$ are defined in Eqs.(15), (33) and (36) of \[5\]. At the final stage of the inflating outer space and the collapsing inner space, we have $\rho = p = 0$, as shown in Eq. (10) of \[5\]. So, the left-hand sides of Eqs. (\[eq:aa5\]) and (\[eq:aa6\]) vanish.
Now let us consider the stage of $x \ll 1$ and $y \ll 1$, where $x$ and $y$ are defined in Eqs. (\[eq:a18\]) and (\[eq:a19\]). Then, the lowest-order terms in $\Phi_h$ and $\Phi_H$ with respect to $x$ and $y$ are expressed as $$\label{eq:aa7}
\Phi_h = (\tau/\tau_i)^{-8/3} \Phi_{hi} , \quad \Phi_H = (\tau/\tau_i)^{-4/3} \Phi_{Hi},$$ and $\Phi_6 = \tilde{\Phi}_G = 0$. To derive the next-order terms, let us put $$\label{eq:aa8}
\Phi_h = (\tau/\tau_i)^{-8/3} \Phi_{hi} + \Delta \Phi_h,$$ $$\label{eq:aa9}
\Phi_H = (\tau/\tau_i)^{-4/3} \Phi_{Hi} + \Delta \Phi_H.$$ From Eq.(32) of \[5\], we can derive $$\label{eq:aa10}
\tilde{\Phi}_G = \frac{3}{2} [-\tau^{-2/3} \tau_i^{8/3} \Phi_{hi} + \tau^{2/3} \tau_i^{4/3}
\Phi_{Hi}] \ln (\tau/\tau_i),$$ $$\label{eq:aa11}
(\tilde{\Phi}_G)' = \frac{3}{2} [-\tau^{-5/3} \tau_i^{8/3} \Phi_{hi} + \tau^{-1/3} \tau_i^{4/3}
\Phi_{Hi}] + [\tau^{-5/3} \tau_i^{8/3} \Phi_{hi} + \tau^{-1/3} \tau_i^{4/3}
\Phi_{Hi}] \ln (\tau/\tau_i),$$ where $\tau \equiv t_0 - t$ and a dash denotes $d/d\tau$. For $\Phi_6$, we have $$\label{eq:aa12}
\Phi_6' + \frac{1}{\tau}\Phi_6 = 3\tau (\Delta \Phi_h' + \frac{8/3}{\tau} \Delta \Phi_h
+\Delta \Phi_H' + \frac{4/3}{\tau} \Delta \Phi_H) +2[(k_r^{(0)}/r)^2 -(k_R^{(0)}/R)^2]
\tilde{\Phi}_G,$$ which is derived from Eq.(58) of \[5\]. Here we define the auxiliary quantities $X$ and $Y$ by $$\label{eq:aa13}
X \equiv \Delta \Phi_h' + \frac{8}{3\tau} \Delta \Phi_h \quad {\rm and} \quad
Y \equiv \Delta \Phi_H' + \frac{4}{3\tau} \Delta \Phi_H.$$ Then, from Eqs. (\[eq:aa5\]), (\[eq:aa8\]), (\[eq:aa9\]), (\[eq:aa11\]) and (\[eq:aa12\]), we obtain the following equations for $X$ and $Y$ $$\label{eq:aa14}
2X + 6Y = A,$$ and $$\label{eq:aa15}
3X + 5Y = B + \frac{2}{\tau^2} \Phi_6,$$ where $A$ and $B$ are expressed as $$\label{eq:aa16}
\begin{split}
A &= -3 (k_R^{(0)}/R_0)^2 \tau^{-1} {\tau_i}^{4/3} \Phi_{Hi} +
\{[-2 (k_R^{(0)}/R_0)^2 \tau^{-7/3} + 3(k_r^{(0)}/r_0)^2 \tau^{-1}] {\tau_i}^{8/3} \Phi_{hi} \\
&-3 (k_r^{(0)}/r_0)^2 \tau^{1/3} {\tau_i}^{4/3} \Phi_{Hi} \} \ln (\tau/\tau_i),
\end{split}$$ and $$\label{eq:aa17}
\begin{split}
B &= 3(k_r^{(0)}/r_0)^2 \tau^{-1} {\tau_i}^{8/3} \Phi_{hi}
+\{[3(k_r^{(0)}/r_0)^2 \tau^{-1} - \frac{3}{2} (k_R^{(0)}/R_0)^2\tau^{-7/3}] {\tau_i}^{8/3}
\Phi_{hi} \\
&+ [-5 (k_r^{(0)}/r_0)^2 \tau^{1/3} + \frac{3}{2} (k_R^{(0)}/R_0)^2\tau^{-1}] {\tau_i}^{4/3}
\Phi_{Hi} \} \ln (\tau/\tau_i).
\end{split}$$ Eliminating $\Phi_6$ from Eq.(\[eq:aa15\]) by use of (\[eq:aa12\]), we have $$\label{eq:aa18}
3X' + 5Y' + \frac{3}{\tau} (X + 3Y) = B' +\frac{3}{\tau} B + \frac{2}{\tau^2} C,$$ where $C$ is $$\label{eq:aa19}
C \equiv 2[(k_r^{(0)}/r)^2 - (k_R^{(0)}/R)^2] \tilde{\Phi}_G$$ with $\tilde{\Phi}_G$ defined by Eq.(\[eq:aa10\]).
Integrating Eqs. (\[eq:aa14\]) and (\[eq:aa18\]) with respect to $X$ and $Y$, we obtain $X$ and $Y$, expressed as $$\label{eq:aa19a}
\begin{split}
X & = \tau^{-1} \{[\frac{3}{2}(\frac{k_r^{(0)}}{r_0})^2 -\frac{37}{28} \tau^{-4/3}
(\frac{k_R^{(0)}}{R_0})^2] \ln (\tau/\tau_i) + [\frac{23}{24} (\frac{k_r^{(0)}}{r_0})^2 +
\frac{243}{392} \tau^{-4/3} (\frac{k_R^{(0)}}{R_0})^2] \} {\tau_i}^{8/3} \Phi_{hi} \\
&+ \tau^{1/3} \{[-12(\frac{k_r^{(0)}}{r_0})^2 +\frac{9}{4} \tau^{-4/3}
(\frac{k_R^{(0)}}{R_0})^2] \ln (\tau/\tau_i) + [\frac{243}{8} (\frac{k_r^{(0)}}{r_0})^2 -
\frac{9}{4} \tau^{-4/3} (\frac{k_R^{(0)}}{R_0})^2] \} {\tau_i}^{4/3} \Phi_{Hi},
\end{split}$$ and $$\label{eq:aa19b}
\begin{split}
Y & = \tau^{-1} \{\frac{3}{28} \tau^{-4/3}
(\frac{k_R^{(0)}}{R_0})^2 \ln (\tau/\tau_i) - [\frac{23}{72} (\frac{k_r^{(0)}}{r_0})^2 +
\frac{81}{392} \tau^{-4/3} (\frac{k_R^{(0)}}{R_0})^2] \} {\tau_i}^{8/3} \Phi_{hi} \\
&+ \tau^{1/3} \{[\frac{7}{2}(\frac{k_r^{(0)}}{r_0})^2 -\frac{3}{4} \tau^{-4/3}
(\frac{k_R^{(0)}}{R_0})^2] \ln (\tau/\tau_i) + [-\frac{81}{8} (\frac{k_r^{(0)}}{r_0})^2 +
\frac{1}{4} \tau^{-4/3} (\frac{k_R^{(0)}}{R_0})^2] \} {\tau_i}^{4/3} \Phi_{Hi}.
\end{split}$$ Integrating Eq.(\[eq:aa13\]) with respect to $\Delta \Phi_h$ and $\Delta \Phi_H$, moreover, we obtain their following expressions $$\label{eq:aa20}
\begin{split}
\Delta \Phi_h & = \{[\ln (\tau/\tau_i) +\frac{19}{72}] x^2 + [-\frac{37}{28} \ln (\tau/\tau_i)
+\frac{939}{14\times 56}] y^2\} (\tau/\tau_i)^{-8/3} \Phi_{hi} \\
&+ \{-\frac{16}{3} [\ln (\tau/\tau_i) +\frac{89}{6}] x^2 + [\frac{3}{8} \ln (\tau/\tau_i)
-\frac{1089}{64\times 49}] y^2\} (\tau/\tau_i)^{-4/3} \Phi_{Hi},
\end{split}$$ and $$\label{eq:aa21}
\begin{split}
\Delta \Phi_H & = \{-\frac{23}{54} x^2 + [\frac{1}{42} (\ln \tau/\tau_i)^2
-\frac{9}{112}] y^2 \} (\tau/\tau_i)^{-8/3} \Phi_{hi} \\
&+ \{[\frac{7}{3} \ln \tau/\tau_i -\frac{61}{8}] x^2 - [\frac{1}{12} \ln (\tau/\tau_i)
+\frac{1}{48}] y^2 \} (\tau/\tau_i)^{-4/3} \Phi_{Hi} .
\end{split}$$ For $x (\ll 1)$ and $y (\ll 1)$, therefore, $\Delta \Phi_h$ and $\Delta \Phi_H$ are small, compared with the main terms $(\tau/\tau_i)^{-8/3} \Phi_{hi}$ and $(\tau/\tau_i)^{-4/3} \Phi_{Hi}$.
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---
abstract: 'This report presents a very simple algorithm for overlaping community-detection in large graphs under constraints such as the minimum and maximum number of members allowed. The algorithm is based on the simulation of random walks and measures the entropy of each random walk to detect the discovery of a community.'
author:
- |
Luis Argerich$^{1}$\
\
$^1$University of Buenos Aires (U.B.A), CS Department F.I.U.B.A\
title: 'Entropy Walker, a Fast Algorithm for Small Community Detection in Large Graphs'
---
Introduction
============
Community detection in large graphs is getting attention as an important application of Social Network Analysis (SNA), the ability to detect closely knit communities opens several applications from targeting ads to recommender systems. In this work we try to derive a very simple and efficient algorithm for community detection based on a size parameter. Being able to specify the minimum and maximum size of communities to detect can be a critical factor in the SNA area, some networks tend to form very small and dense communities while other networks form larger groups. The first section of this report discusses some existing algorithms for community detection in social graphs, then we introduce the idea behind the entropy walker and present our algorithm. The final sections show some examples of the algorithm being used in some toy examples and analyzes the scaling of the method for large graphs.
Previous Work
=============
Several algorithms have been developed for community detection in large graphs. Clutsering methods based in k-means need to know in advance the number of communities to find in the network. In practice this is not possible as the number of communities is usually unknown and furthermore due to social interactions the number of communities in a network might change over time making it very hard to set up as a parameter.
The modularity optimization algorithm \[B08\] automatically detects the number of communities but it doesn’t allow for overlapping communities. This is also inpractical for Social Networks as most nodes will be members of several different social circles.
BigClam \[Lesk13\] is a fast algorithm to detect overlaping communities, it’s based in non-negative matrix factorization but it needs to know the number of communities to detect, as mentioned before this is an important limitation.
\[McA13\] presents an algorithm to find social circles in networks but is based on node parameters “features”, we would like to perform the extraction of communities based in network structure only.
The idea of random walks being used to detect communities is also used in the MCL algorithm \[vDon99\] however MCL can’t control the size of the communities being detected and it needs to perform operations on the complete matrix of the graph limiting its use to small and medium sized networks.
Description
===========
We define a “tour” as a random walk of length “s”. The basic idea of the algorithm is to perform several tours starting from random nodes and to detect communities based on the result of those tours. “s” should be longer than the minimum number of members that we want for a community and it serves as an upper bound for the maximum number of members in a community.
It is likely for a random walker to get “trapped” inside nodes of a community, going back and forth between them because there are more inter-community edges than edges that will take the walker outside of the community. Even if the random walker goes outside the community chances are it might come back.
The algorithm will filter the random walks that aren’t likely to have found a community calculating the entropy of the tour \[Sha48\]. Tours with high entropy are unlikely to contain a community because they visit mostly different nodes. They are probably paths or bridges between communities and might be of interest for some other applications.
The entropy is computed using the very popular Shannon formula:
$$H=\sum^n P_i*log(1/P_i)$$
Where $P_i$ is just the probability of the node in the tour, in other words its frequency in the tour over the sum of all node frequencies. A threshold parameter establishes the maximum entropy for a tour to be accepted as a fraction of the maximum possible entropy that can be computed assuming a random walk that never visits the same node more than once. We call this parameter $et$ for entropy threshold.
When $et$ is 1 all the tours are accepted, lowering $et$ increases the amount of rejected tours. The graph in figure 1 shows the percentage of accepted tours for different values of $et$ using the Food Network as an example.
![Number of tours per entropy threshold.[]{data-label="fig1"}](entropy.png){width="3in"}
The $et$ parameter can be tuned based on two different goals. One possibility is to use it to limit the total number of tours to store in memory for very large graphs, a second use, more logical, is to set how dense a community has to be to be considered. This second use that is data dependant is probably the recommended one.
This is an example of a very low entropy tour from the food network: $ [cream-egg-cream-milk-cream-butter-raisin-vanilla-butter-raisin-cream-butter-cream-vanilla-egg-butter-cream-butter-egg-milk-butter-cream-milk-egg-milk-raisin-milk-vanilla-milk-yogurt]$
And this is an example of a high entropy tour from the same network: $[thyme-tomato-turmeric-carrot-beef-vinegar-beef-garlic-lamb-onion-chicken-ginger-cilantro-coriander-mint-parsley-bread-bell_pepper-cayenne-garlic-lamb-cinnamon-ginger-cumin-ginger-honey-cinnamon-orange_juice-vanilla-raisin]$
We can see how the first tour can be converted in a community with the top ingredients being used for the same kind of dishes, the second tour has a wide array of ingredients and can’t be considered a community. Maybe a bridge between different communities. As we have mentioned extracting the high entropy tours from a network might also be an interesting application.
After accepting or rejecting a tour based on its entropy the algorithm will try to see if this tour is new or if it is similar to an already seen tour. Locality sensitive hashing (LSH) can be used to make similar tours hash to the same bucket avoiding the need to compare new tours with the existing ones. If LSH maps the tour to a bucket where a tour is already stored then both tours are merged adding the frequencies of the nodes present in both tours. This greatly reduces the number of tours that need to be stored in memory and avoids the problem of two very similar tours being detected as different communities.
In some applications the $n$ most frequent nodes in a tour can be used as the key to a hash function to determine the bucket number for the node. This is a simplification of LSH using only one minhash computed from the most frequent nodes in a tour. When this is not possible or doesn’t work standard LSH can be used.
Now we describe the parameters used in the algorithm:
The algorithm uses several parameters to fine-tune its behaviour:
0.25cm
The algorithm will perform $nt$ random tours and check the entropy of each tour. If the tour entropy is below the $et$ threshold then the tour will be stored in a hash table along with a counter merging the tour with the already existing one if the bucket is not empty. It’s easy to notice that this process can be parallelized and that several million tours can be performed efficiently. The memory cost to store the tours depends on the algorithm parameters.
When the $et$ (entropy threshold) parameter is low the algorithm with detect only a few very dense communities and tours with frequency 1 can be considered a community. When the $et$ parameter is higher the algorithm will check many tours and it might make sense to discard the tours with lower frequencies keeping the ones that have been repetedly matched.
A Centrality Measure
====================
It is known that MonteCarlo Random Walks can be used to compute PageRank and/or Eigenvector centrality, the procedure used to detect communities can be used to compute at the same time a centrality score for the network nodes. So the first conclusion is that node centrality can be computed at the same time as the community detection algorithm runs, just adding 1 to a counter every time a node is visited by a tour and then normalizing the cummulative score.
The effect of entropy filter is show in Fig2. We can see that some nodes produce peaks for entropy thresholds below 1.00, this means that the centrality of those nodes is higher in the entropy filtered sets compared to the plain random walks without filtering. These peaks can be detected computing the delta between the eigenvector centrality and the tour computed centrality. From these peaks we can detect nodes that are both central to the network and to the small communities where they belong, this gives an index of in-community centrality.
Testing the procedure on the Facebook Ego Network the peaks matched nodes that had a high degree of connections with the members in their communities.
![Centrality Score for Different Entropy Thresholds.[]{data-label="fig2"}](centrality.png){width="3.5in"}
Personalized Circles
====================
Something interesting to notice is that the algorithm can be run starting always from the same node, in the style of a personalized PageRank, when that happens we get as a result the social circles of a given user. This is in some way similar to the algorithm used by Twitter to recommend users to follow\[Gup12\] the difference is that instead of computing a score for each node we compute scores for each random walk (tour) performed by the simulation.
For example we can run the algorithm from the Tomato ingredient to see what goes well with Tomato:
Instantaneous delicious recipes!
Analysis
========
This section presents some analysis and graphs about the behaviour of the algorithm.
Growth of the number of communities for a fixed entropy threshold
-----------------------------------------------------------------
It is interesting to analyze the number of tours that the algorithm will keep in memory as the network grows larger for a constant fixed entropy threshold. We found that the number of tours analyzed does not grow as the size of the network and is strongly dependant on network structure.
![Number of Tours per number of Nodes.[]{data-label="fig3"}](tourspernodes.png){width="3.5in"}
With only a few nodes small communities are common in a graph with high clustering, as the network grows larger the number of small communities quickly goes down. This can be explained because a random walker has now more options and is less likely to get trapped inside a community. Then after more nodes are added a threshold is passed and small communities emerge again. This curious behaviour in the formation of small communities as the network grows larger resulted an interesting find and can be useful to refine generic models for network growth.
Relationship to Clustering
--------------------------
The emergence of small communities in large networks is strongly related to the clustering coefficient of the network. When the clustering coefficient is very los there are not enough edges to form dense communities so small communities will not form in random networks. In the same way if the clustering coefficient is too high then the random walker can visit almost any node from any node and thus will not get trapped inside a small community, the whole network is the only existing community.
The following graph shows the number of tours detected for a fixed entropy threshold depending on the clustering coefficient of networks synthetically generated using the Barabasi-Albert model\[Bar99\].
![Number of Tours per clustering coef.[]{data-label="fig4"}](toursperclustering.png){width="3.5in"}
As the clustering coefficient gets larger the number of nodes in a tour has to be increased to detect communities.
Results
=======
Results on the Food Network
---------------------------
In our example we run the algorithm against the Eastern Food Network composed by different ingredients using in the Eastern cuisine. The idea is that the algorithm should be able to find groups of ingredients that are frequently used together. Using $et$ at 0.75 and simulating 150.000 tours of 30 hops the algorithm processed a total of 8308 tours to find clusters with 5 to 10 nodes in less than 5 seconds and these were the top results.
The number between parentheses reflects the number of times the same community was detected, so the higher the number the stronger the community. We can see that the algorithm quickly detects the ingredients for most deserts or breakfast-type preparations. In total the algorithm detected 141 overlapping communities. The following result looks like a good recipe to try:
As a point of comparision we run the modularity optimization algorithm \[Blon08\] as implemented in Gephi and got the following communities:
\[lemon, egg, orange, almond, orangejuice, cream, raisin, cinnamon, honey, butter, milk, vanilla, walnut\] \[coriander, pepper, blackpepper, chicken, thyme, cayenne, cilantro, dill, cumin, bellpepper, chickenbroth, ginger, turmeric, carrot\] \[garlic, parsley, onion, lemonjuice, beef, lamb, tomato, cucumber, bread, oliveoil, mint, vinegar, yogurt, potato\]
As we can see the modularity algorithm does a very good job but it lists all the ingredients that are similar together and is not very helpful to detect smaller groups that go very well together, for example communities of 3 or 4 ingredients. The algorithm presented here would create the following top 10 communities of 3 ingredients:
The graph of the communities found by Gephi looks like this \[Figure2\]
![Eastern Ingredients.[]{data-label="fig5"}](food1.jpg){width="3.5in"}
As we can see the results help to create new recipes starting with ingredients that go well together frequently. Something interesting is that by allowing overlapping communities we can see that some ingredients are partially in different groups. For example ginger is used for both savory and deserts. The modularity algorithm is forced to choose only one cluster for ginger but in our algorithm we can find it in different communities.
Results on Large Social Networks
--------------------------------
We also run the algorithm in a very large dump of a Social Network with a total of about 5 million nodes. The algorithm runs in constant time regardless of the size of the graph as it always simulates a constant number of random walks, the only difference in runtime is due to the time needed to access the adjacency list of each node and that is independant of the clustering algorithm.
Besides the runtime analysis we weree curious to investigate what kind of small communities the algorithm would find in a large Social Network. We run a modularity clustering phase first and then the entropy walker algorithm.
![Modularity Clustering of the Social Network.[]{data-label="fig6"}](socnetwork.jpg){width="4in"}
After running the entropy walker algorithm we found that 100
![An accepted random walk inside a modularity class.[]{data-label="fig6"}](insidecluster.jpg){width="3.5in"}
We see that the entropy walker algorithm finds small dense communities inside the big communities created by the modularity algorithm.
![Shape of a random walk.[]{data-label="fig7"}](walk_zoom.jpg){width="3.5in"}
Figure 4 shows an accepted random walk inside a modularity class. Figure 5 shows the shape of one of the accepted random walk, we can see the community is actually a clique so the algorithm is finding cliques or structures similar to cliques for the parametrized size of components that depend on the length of the random walks.
The Streamming Model
====================
In a streaming model the graph is constantly updated via the addition and deletion of nodes and edges. In this model the algorithm can be kept running continuously producing “infinite” tours. As the graph is updated communities that were previously detected might disappear and new communities can emerge. An algorithm like the Count-Min Sketch \[Mutu05\] can be used to keep in memory a list of only the top $n$ communities discovered so far. If a new very tight community forms it will be eventually found by the algorithm several times entering the top $n$ ranking. Besides keeping the top $n$ communities the streaming model can be used to detect communities that pass the entropy filter and the count-min sketch can be used to only list those communities that have repeated a number of times. Several strategies to prune old communities from memory can be used.
Conclusions
===========
The entropy walker is a very simple algorithm, the core is just a montecarlo simulation of random walks in a graph. The algorithm uses two very simple tricks to be able to compute communities from these random walks, first it is able to keep or discard a tour by calculating its entropy reasoning that a tour that gets trapped inside a community will visit several times the same nodes resulting in a low-entropy tour. The second trick is the use of LSH and the ability to merge similar tours into a single one to reduce memory consumption and be able to detect the same community even if the nodes have been visited in different order and with different frequencies.
The algorithm can run very quickly consuming very little memory even for massive graphs, it can be kept running continusly in a streamming model where the graph is constantly updated, this setup is perfect for the anlysis of large Social Networks.
References
==========
|
---
abstract: 'Approximate Bayesian Computation (ABC) is a statistical learning technique to calibrate and select models by comparing observed data to simulated data. This technique bypasses the use of the likelihood and requires only the ability to generate synthetic data from the models of interest. We apply ABC to fit and compare insurance loss models using aggregated data. We present along the way how to use ABC for the more common claim counts and claim sizes data. A state-of-the-art ABC implementation in Python is proposed. It uses sequential Monte Carlo to sample from the posterior distribution and the Wasserstein distance to compare the observed and synthetic data.'
author:
- 'Pierre-Olivier Goffard and Patrick J. Laub'
title: Approximate Bayesian Computations to fit and compare insurance loss models
---
*MSC 2010*: 60G55, 60G40, 12E10.\
*Keywords*: Bayesian statistics, approximate Bayesian computation, likelihood-free inference, risk management.
Introduction {#sec:intro}
============
Over a fixed time period, an insurance company experiences a random number of claims called the *claim frequency*, and each claim requires the payment of a randomly sized compensation called the *claim severity*. The claim frequency is a counting random variable while the claim sizes are non-negative continuous random variables. Let us say that the claim frequency and the claim severity distributions are specified by the parameters ${\bm{\theta}}_{\mathrm{freq}}$ and ${\bm{\theta}}_{\mathrm{sev}}$ respectively, with ${\bm{\theta}}= ({\bm{\theta}}_{\mathrm{freq}}; {\bm{\theta}}_{\mathrm{sev}})$. For each time $s = 1, \dots, t$ the number of claims $n_s$ and the claim sizes ${\bm{u}}_s \coloneqq (u_{s,1}, u_{s,2}, \dots, u_{s,n_s})$ are distributed as $$n_s \sim p_N(n {\,;\,}{\bm{\theta}}_{\mathrm{freq}}) \quad \text{and} \quad
({\bm{u}}_s {\mid}n_s) \sim f_U({\bm{u}}{\,;\,}n, {\bm{\theta}}_{\mathrm{sev}}) .$$ We wish to fit these distributions, however, we assume that these independent and identically distributed ([i.i.d.]{}) values $\{(n_1, {\bm{u}}_1), \dots, (n_t, {\bm{u}}_t)\}$ are unobservable. Instead, we only have access to some real-valued *summaries* of the claim data at each time, denoted by $$\label{eq:aggregated_rv}
x_s = \Psi(n_s, {\bm{u}}_s) \quad \text{for } s=1,\dots,t.$$ The summaries could be the aggregated claims if $\Psi(n, {\bm{u}}) = \sum_{i=1}^n u_i$ or the maximum claims if $ \Psi(n, {\bm{u}}) = \max_{1 \le i \le n} u_i$. Our problem is to take some observations of these summaries ${\bm{x}}= (x_1, \dots, x_t)$ and find the ${\bm{\theta}}$ which best explains them for a given parametric model.
Such incomplete data situations arise in reinsurance, see the monograph of @albrecher2017reinsurance [Chapter I, Section 3]. For instance, within a global non-proportional reinsurance agreement, the reinsurance company covers the risk that the insurer’s total claim amount is in excess of a threshold $c>0$. The reinsurer is only observing its payout at each time period $
x_s = (\sum_{i = 1}^{n_s} u_{s,i} - c)_+.
$ Being able to infer the parameters of the claim frequency and the claim severity distributions would help the reinsurer to better understand the risk they have underwritten.
When the summary is the aggregated loss $\Psi(n, {\bm{u}}) = \sum_{i=1}^n u_i$, we effectively *decompound* the random sum. Traditionally, a decompounding method builds a non-parametric estimate of the claim severity distribution based on the observations of the aggregated sums, see @buchmann2003 or @bogsted2010decompounding. A popular application is the study of discretely observed compound Poisson processes, see for instance @vanes2007 [@coca2018efficient] and @gugushvili2018non where a Bayesian non-parametric approach is used.
A Bayesian approach to estimating ${\bm{\theta}}$ would be to treat ${\bm{\theta}}$ as a random variable and find (or approximate) the *posterior distribution* $\pi({\bm{\theta}}{\mid}{\bm{x}})$. Bayes’ theorem tells us that $$\label{eq:posterior_distribution}
\pi({\bm{\theta}}{\mid}{\bm{x}}) \propto p({\bm{x}}{\mid}{\bm{\theta}}) \, \pi({\bm{\theta}}),$$ where $p({\bm{x}}{\mid}{\bm{\theta}})$ is the *likelihood* and $\pi({\bm{\theta}})$ is the *prior distribution*. The prior represents our beliefs about ${\bm{\theta}}$ before seeing any of the observations and is informed by our domain-specific expertise. The posterior distribution is a very valuable piece of information that gathers our knowledge over the parameters. A point estimate ${\widehat}{{\bm{\theta}}}$ may be derived by taking the mean or mode of the posterior. For an overview on Bayesian statistics, we refer to the book of @gelman2013bayesian.
The posterior distribution \[eq:posterior\_distribution\] rarely admits a closed-form expression, so it is approximated by an empirical distribution of samples from $\pi({\bm{\theta}}{\mid}{\bm{x}})$. Posterior samples are typically obtained using Markov Chain Monte Carlo (MCMC), yet a requirement for MCMC sampling is the ability to evaluate (at least up to a constant) the likelihood function $p({\bm{x}}{\mid}{\bm{\theta}})$. When considering the definition of ${\bm{x}}$ in \[eq:aggregated\_rv\], we can see that there is little hope of finding an expression for the likelihood function even in simple cases ([e.g.]{}when the claim sizes are [i.i.d.]{}). If the claim sizes are not [i.i.d.]{}or if the number of claims influences their amount, then the chance that a tractable likelihood for ${\bm{x}}$ exists is extremely low. Even when a simple expression for the likelihood exists, it can be prohibitively difficult to compute (such as in a big data regime), and so a likelihood-free approach can be beneficial.
We advertise here a likelihood-free estimation method known as *approximate Bayesian computation* (ABC). This technique has attracted a lot of attention recently due to its wide range of applicability and its intuitive underlying principle. One resorts to ABC when the model at hand is too complicated to write the likelihood function but still simple enough to generate artificial data. Given some observations ${\bm{x}}$, the basic principle consists in iterating the following steps:
(i) generate a potential parameter from the prior distribution ${\bm{\theta}}^{\ast} \sim \pi({\bm{\theta}})$;
(ii) simulate ‘fake data’ ${\bm{x}}^{\ast}$ from the likelihood $({\bm{x}}^{\ast} {\mid}{\bm{\theta}}^{\ast}) \sim p({\bm{x}}{\mid}{\bm{\theta}})$;
(iii) if ${\lVert{} {\bm{x}}-{\bm{x}}^{\ast}\rVert} \leq \epsilon$, where $\epsilon > 0$ is small, then store ${\bm{\theta}}^{\ast}$,
where ${\lVert{} \,\cdot\,\rVert}$ denotes a distance measure and $\epsilon$ is an acceptance threshold. The algorithm provides us with a sample of ${\bm{\theta}}$’s whose distribution is close to the posterior distribution $\pi({\bm{\theta}}{\mid}{\bm{x}})$.
The ABC algorithm presented in this work allows us to consider a wide variety of $\Psi$ functions \[eq:aggregated\_rv\] without imposing common simplifying assumptions such as assuming the claim amounts are [i.i.d.]{}and independent from the claim frequency. In addition to parameter estimation, ABC allows us to perform model selection in a Bayesian manner. This direction is also investigated. For a comprehensive overview on ABC, we refer to the monograph of @SiFaBe18. In finance and insurance, ABC has been considered in the context of operational risk management [@peters2006bayesian] and for reserving purposes [@peters2010chain].
The rest of the paper is organized as follows. provides a gentle introduction to ABC algorithms. We start by presenting the ABC routines used on count data and continuous data, then show how to use ABC to fit an insurance loss model based on aggregated data. explains how to adapt the ABC algorithm to compare models by computing the a posteriori model probability of each competing model. The performance of our ABC implementation are illustrated on simulated data in \[sec:Simu\] and on a real world insurance data set in \[sec:RealExample\].
Model calibration {#sec:ABC_model_calibration}
=================
ABC is a method for approximating the posterior probability $\pi({\bm{\theta}}{\mid}{\bm{x}})$ without using the likelihood function. The implementation of ABC is tied to the nature of the data at hand. In our problem, the frequency data is discrete, the individual claim sizes are continuous and the aggregated data is a mixture of discrete and continuous (due to the atom at $0$). We take advantage of this fact to introduce ABC algorithms for discrete data in \[sub:abc\_frequency\], continuous data in \[sub:abc\_cont\], and mixed data in \[sub:abc\_mixed\]. The acceptance–rejection algorithm laid out in the introduction most often leads to considerable computing time, so \[subsec:abc\_smc\] explains how to speed up ABC using sequential Monte Carlo (SMC). \[subsec:illustration\] shows the validity of our ABC implementation on an illustrative example.
ABC for count data {#sub:abc_frequency}
------------------
Consider some count data $n_1,\ldots, n_t \in {\mathbb{N}_0}$ which are [i.i.d.]{}with the probability mass function ([p.m.f.]{}) $p_N(n {\mid}{\bm{\theta}})$; for example, the $n_s$’s could be claim frequencies. The likelihood of such data is $
p({\bm{n}}{\mid}{\bm{\theta}}) = \prod_{s = 1}^{t} p_N(n_s {\mid}{\bm{\theta}}),
$ where ${\bm{n}}= (n_1,\ldots, n_t)$. For common discrete distributions, such as the Poisson or negative binomial, the likelihood function is tractable and may be plugged into an MCMC sampling algorithm to produce samples from the posterior distribution $
\pi({\bm{\theta}}{\mid}{\bm{n}}) \propto p({\bm{n}}{\mid}{\bm{\theta}})\pi({\bm{\theta}}).
$ Alternatively, we can sample from the posterior $\pi({\bm{\theta}}{\mid}{\bm{n}})$ in a likelihood-free way by acceptance–rejection, which is detailed in \[alg:AR\_count\_data\].
**input** observations ${\bm{n}}= (n_1,\ldots, n_t)$ **generate** ${\bm{\theta}}_k \sim \pi({\bm{\theta}})$ **generate** ${\bm{n}}_k \sim p({\bm{n}}{\mid}{\bm{\theta}}_k)$ ${\bm{n}}_k = {\bm{n}}$ then **store** ${\bm{\theta}}_k$ **return** $\{{\bm{\theta}}_1, \dots, {\bm{\theta}}_K \}$ which are [i.i.d.]{}samples from $\pi({\bm{\theta}}{\mid}{\bm{n}})$
\[alg:AR\_count\_data\] gives samples from $\pi_{0}({\bm{\theta}}{\mid}{\bm{n}})$ which is exactly the desired posterior distribution $\pi({\bm{\theta}}{\mid}{\bm{n}})$: $$\pi_{0}({\bm{\theta}}{\mid}{\bm{n}})
\propto \pi({\bm{\theta}}) \int_{{\mathbb{N}_0}^t} {\mathbb{I}}_{\{ {\bm{n}}= {\widetilde}{{\bm{n}}} \}} \, p({\widetilde}{{\bm{n}}} {\mid}{\bm{\theta}}) {\mathop{}\!\mathrm{d}}{\widetilde}{{\bm{n}}}
= \pi({\bm{\theta}}) p({\bm{n}}{\mid}{\bm{\theta}}),$$ where $${\mathbb{I}}_{\{{\bm{n}}={\widetilde}{{\bm{n}}}\}} = \begin{cases}
1, & \text{ if }{\bm{n}}={\widetilde}{{\bm{n}}}, \\
0, & \text{ otherwise.}
\end{cases}$$ As we collect more data, the probability of seeing an exact match $\{{\bm{n}}={\widetilde}{{\bm{n}}}\}$ decreases exponentially. This, combined with a diffuse prior distribution, will result in a cumbersome waiting time before getting a posterior sample. A natural refinement is to require an exact correspondence between the samples sorted in ascending order. The acceptance rate may still be too low to be practical, and in this case an approximate match between the observed and fake data must be considered. We discuss this matter within the continuous data case in the following section.
ABC for continuous data {#sub:abc_cont}
-----------------------
Let $u_1,\ldots, u_n \in {\mathbb{R}}$ be an [i.i.d.]{}sample of continuous data with a probability density function ([p.d.f.]{}) denoted $f_U(u {\mid}{\bm{\theta}})$. An example of such data would be the claim sizes. With the notation ${\bm{u}}= (u_1,\ldots, u_n)$, we can write the likelihood as $
p({\bm{u}}\mid {\bm{\theta}}) = \prod_{i = 1}^n f_U(u_i \mid {\bm{\theta}}).
$ If the data is fitted to a standard probability model, say gamma or normal, then we can sample from the posterior distribution $
\pi({\bm{\theta}}{\mid}{\bm{u}}) \propto \pi({\bm{\theta}})p({\bm{u}}{\mid}{\bm{\theta}}),
$ with an MCMC scheme. If the likelihood is unavailable, then we can adapt \[alg:AR\_count\_data\] to the case of continuous data for which exact correspondence between observed and fake data is not possible. Synthetic samples are then accepted whenever they fall sufficiently close to the observed data. That is, if the dissimilarity between two samples, assessed by a norm ${\lVert{} \,\cdot\,\rVert}$ on ${\mathbb{R}}^n$, is smaller than some tolerance threshold $\epsilon > 0$, see \[alg:AR\_abc\_continuous\_data\].
**input** observations ${\bm{u}}= (u_1,\ldots, u_n)$, $\epsilon > 0$ threshold, ${\lVert{} \cdot\rVert}$ norm **generate** ${\bm{\theta}}_k \sim \pi({\bm{\theta}})$ **generate** ${\bm{u}}_k \sim p({\bm{u}}{\mid}{\bm{\theta}}_k)$ ${\lVert{} {\bm{u}}-{\bm{u}}_k\rVert} < \epsilon$ then **store** ${\bm{\theta}}_k$ **return** $\{ {\bm{\theta}}_1, \dots, {\bm{\theta}}_K \}$ which are approximately $\pi({\bm{\theta}}{\mid}{\bm{u}}) $ distributed
The procedure depicted in \[alg:AR\_abc\_continuous\_data\] allows us to sample from an approximation of the posterior distribution given by $$\label{eq:abc_posterior_cont}
\pi_{\epsilon}({\bm{\theta}}{\mid}{\bm{u}}) \propto \pi({\bm{\theta}})
\int_{{\mathbb{R}}^t} {\mathbb{I}}_{\{{\lVert{} {\bm{u}}-{\widetilde}{{\bm{u}}}\rVert} <\epsilon\}}\, p({\widetilde}{{\bm{u}}} {\mid}{\bm{\theta}}){\mathop{}\!\mathrm{d}}{\widetilde}{{\bm{u}}},$$ where $${\mathbb{I}}_{\{{\lVert{} {\bm{u}}-{\widetilde}{{\bm{u}}}\rVert} <\epsilon\}} = \begin{cases}
1, & \text{ if }{\lVert{} {\bm{u}}-{\widetilde}{{\bm{u}}}\rVert} <\epsilon, \\
0, & \text{ otherwise.}
\end{cases}$$ Distribution is called the *ABC posterior* and it has the desirable theoretical property of converging toward the standard posterior $\pi({\bm{\theta}}{\mid}{\bm{u}})$ as $\epsilon$ tends to $0$, see @rubio2013simple [@prangle2018rare] or @bernton2019approximate.
The ABC procedure suffers from the so-called curse of dimensionality [@Bl10]. Specifically, if one takes the Euclidean distance or some variation of it to measure the dissimilarity between observed and fake data then the odds of getting an acceptable match will plummet as the number of observations, [i.e.]{}the dimension of ${\bm{u}}$, increases. The dimensionality curse can be alleviated by replacing ${\bm{u}}\in {\mathbb{R}}^n$ with summary statistics $S({\bm{u}}) \in {\mathbb{R}}^d$, where $d<n$, in \[alg:AR\_abc\_continuous\_data\] (specifically, in line $5$ the norm becomes ${\lVert{} S({\bm{u}}) - S({\bm{u}}_k)\rVert}$). While the choice of the summary statistics $S:{\mathbb{R}}^n\mapsto {\mathbb{R}}^d$ is arbitrary, it is desirable to have $d \ll n$ while limiting the information loss. This is difficult. When the model at hand admits sufficient statistics then these should be taken. In fact, the only statistics which uphold the convergence of $\pi_{\epsilon}({\bm{\theta}}{\mid}{\bm{u}})$ to $\pi({\bm{\theta}}{\mid}{\bm{u}})$ as $\epsilon \to 0$ are sufficient statistics. Note that the summary statistics $S$ are not to be confused with the $\Psi$ summaries in \[sec:intro\]!
When dealing with frequency data (see \[sub:abc\_frequency\]), it is possible to define a map $S: {\mathbb{N}_0}^t \mapsto {\mathbb{R}}^d$ which allows us to reduce the dimension and adopt the ABC procedure for continuous data. Consider for instance, the case where the claim frequency are Poisson distributed and the map $S$ corresponds to the empirical mean.
Most often, we will not be able to find sufficient statistics. Many research papers have been dedicated to designing ad hoc summary statistics in the ABC literature, we refer to the survey of @BlNuPrSi13. The problem is that it always implies a loss of information along with a convergence toward the posterior distribution conditionally to the summary statistics instead of the true posterior. We illustrate the use of summary statistics in \[subsec:illustration\], but do not use this technique in the other examples.
@bernton2019approximate recommend the Wasserstein distance to measure the dissimilarity between two samples. The Wasserstein distance is deemed difficult to compute but for real-valued [i.i.d.]{}observations it reduces to $$\mathcal{W}_p({\bm{u}},{\widetilde}{{\bm{u}}}) =\frac 1n\sum_{k=1}^{n}\, \left| u_{(k)}-{\widetilde}{u}_{(k)} \right|^p, \text{ for } p\geq1,$$ where $u_{(1)}<\ldots<u_{(n)}$ and ${\widetilde}{u}_{(1)}<\ldots<{\widetilde}{u}_{(n)}$ denote the order statistics of the observed and synthetic data respectively. The use of the order statistics as summary statistics is not new, it was investigated for instance in the work of @sousa2009approximate and @FePr12. Now that we have reviewed the use of ABC in the case of discrete and continuous data, we turn to the case of mixed data which is of primary interest for the actuarial application at the center of this work.
ABC for mixed data {#sub:abc_mixed}
------------------
We return to the problem of fitting a model to aggregated insurance data. Recall that, for each time period, a random number $n \in {\mathbb{N}_0}$ of claims are filed. The claim frequencies form an [i.i.d.]{}sample from the [p.m.f.]{}$p_N(n {\mid}{\bm{\theta}}_{\mathrm{freq}})$. Given $n$, the associated claim sizes ${\bm{u}}= (u_1,\ldots, u_n)$ have a joint [p.d.f.]{}denoted by $f_{U|N}({\bm{u}}{\mid}n, {\bm{\theta}}_{\mathrm{sev}})$.
The distribution of the available information $x \coloneqq \Psi(n, {\bm{u}})$ is parametrized by ${\bm{\theta}}=({\bm{\theta}}_{\mathrm{freq}}, {\bm{\theta}}_{\mathrm{sev}})$ and admits a point mass $p_X(0 {\mid}{\bm{\theta}})$ at $0$. Zeros can occur if no claims are filed ($n=0$) which occurs with probability $p_N(0 {\mid}{\bm{\theta}}_{\mathrm{freq}})$, or because of censoring effects like in the non-proportional reinsurance treaty case, see \[sec:intro\]. The continuous part of $x$’s distribution is characterized by the conditional [p.d.f.]{}$$[1-p_X(0 {\mid}{\bm{\theta}})] \, f_{X|X>0}(x {\mid}{\bm{\theta}}) \quad \text{for } x>0.$$ For a data history ${\bm{x}}= (x_1,\ldots, x_t)$ of $t$ time periods, we separate the zeros from the non-negative data points, so $${\bm{x}}= ({\bm{x}}^0, {\bm{x}}^+) = (\underbrace{0,\ldots, 0 \vphantom{x^+_{t_0}}}_{t_0 \text{ zeros}}, \underbrace{x^+_1,\ldots,x_{t-t_0}^+}_{t-t_0 \text{ non-zeros}}) \,.$$ The likelihood function may be written as $$\begin{aligned}
\label{eq:likelihood_function_x_1}
p({\bm{x}}{\mid}{\bm{\theta}})\ = & \ p_X(0 {\mid}{\bm{\theta}})^{t_0} [1-p_X(0 {\mid}{\bm{\theta}})]^{t-t_0} \prod_{s = 1}^{t-t_0}f_{X|X>0}(x_s^+ {\mid}{\bm{\theta}}) \\
= & \ p_X(0 {\mid}{\bm{\theta}})^{t_0} [1-p_X(0 {\mid}{\bm{\theta}})]^{t-t_0} p({\bm{x}}^+ \mid\, {\bm{\theta}}).\nonumber\end{aligned}$$ To evaluate the conditional [p.d.f.]{}$f_{X|X>0}$ in \[eq:likelihood\_function\_x\_1\] we must consider all possible values of $n$ which often leads to an infinite series without closed-form expression, as illustrated in \[ex:aggregated\_amounts\].
\[ex:aggregated\_amounts\] Consider the case where we only observe the aggregate claim sizes $
x_s = \sum_{i = 1}^{n_s} u_{s,i}
$ for $s = 1$, $\dots$, $t$, [i.e.]{}, $\Psi$ is the sum operator. If the claim sizes are [i.i.d.]{}and independent from the claim frequency, which is common in the actuarial science literature, the conditional [p.d.f.]{}of $X$ taking positive values is $$\label{eq:cond_pdf_aggregated_claim_size}
f_{X|X>0}(x {\mid}{\bm{\theta}}) = \frac{1}{1-p_N(0 {\mid}{\bm{\theta}}_{\mathrm{freq}})} \sum_{n = 1}^{\infty}f_U^{(\ast n)}(x {\mid}{\bm{\theta}}_\mathrm{sev}) p_N(n {\mid}{\bm{\theta}}_{\mathrm{freq}}),$$ where $f_U^{(\ast n)}(x {\mid}{\bm{\theta}}_\mathrm{sev})$ denotes the $n$-fold convolution product of $f_U(x {\mid}{\bm{\theta}}_\mathrm{sev})$ with itself. A closed-form expression of \[eq:cond\_pdf\_aggregated\_claim\_size\] is available only in a few cases. For the remaining cases, quite some energy has been dedicated by actuarial scientists to finding convenient numerical approximations. Note that none of the aforementioned numerical routines would be suited to the multiple evaluations of the conditional [p.d.f.]{}required for Bayesian inference or maximum likelihood inference via some optimization algorithm. We begin our numerical illustration of the ABC method on some cases where a closed-form expression of \[eq:cond\_pdf\_aggregated\_claim\_size\] is available, as we will be able to sample from the true posterior via an MCMC simulation scheme. Point estimates may also be compared to frequentist estimators such as the maximum likelihood or the method of moment estimators. The latter has been used in a similar situation in the work of @GoJaMe19.
The lack of analytical expression for the likelihood function justifies the use of a likelihood-free inference method such as ABC. The distribution of $x$ is of mixed type which means we cannot directly apply \[alg:AR\_abc\_continuous\_data\] as we would lose the convergence toward the standard posterior distribution. To address this issue, we ask that the number of zeros in the synthetic samples ${\widetilde}{t}_0$ matches the number of zeros in the observed data $t_0$ and we treat the non-negative data points as [i.i.d.]{}continuous data. So, in \[alg:AR\_abc\_mixed\] we retain synthetic samples that belong to the set $$\label{eq:B_set_mixed}
\mathcal{B}_{\epsilon,{\bm{x}}} = \bigl\{ {\widetilde}{{\bm{x}}} \in {\mathbb{R}}^t {\,;\,}{\bm{x}}^0={\widetilde}{{\bm{x}}}^0\text{ and }{\lVert{} {\bm{x}}^+-{\widetilde}{{\bm{x}}}^+\rVert} < \epsilon \bigr\}.$$
**input** observations ${\bm{x}}= (x_1,\ldots, x_t)$, $\epsilon > 0$ threshold, ${\lVert{} \cdot\rVert}$ norm **generate** ${\bm{\theta}}_k \sim \pi({\bm{\theta}})$ **generate** ${\bm{x}}_k \sim p({\bm{x}}{\mid}{\bm{\theta}}_k)$ ${\bm{x}}_k \in \mathcal{B}_{\epsilon,{\bm{x}}}$, then **store** ${\bm{\theta}}_k$ **return** $\{ {\bm{\theta}}_1, \dots, {\bm{\theta}}_K \}$ which are approximately $\pi({\bm{\theta}}{\mid}{\bm{x}}) $ distributed
\[alg:AR\_abc\_mixed\] samples from the approximate posterior distribution $$\label{eq:abc_posterior_mixed}
\pi_{\epsilon}({\bm{\theta}}{\mid}{\bm{x}}) \propto \pi({\bm{\theta}})
\int_{{\mathbb{R}}^t} {\mathbb{I}}_{\mathcal{B}_{\epsilon,{\bm{x}}}}({\widetilde}{{\bm{x}}})\, p({\widetilde}{{\bm{x}}} {\mid}{\bm{\theta}}){\mathop{}\!\mathrm{d}}{\widetilde}{{\bm{x}}},$$ where $${\mathbb{I}}_{\mathcal{B}_{\epsilon,{\bm{x}}}}({\widetilde}{{\bm{x}}}) = \begin{cases}
1, & \text{ if } {\bm{x}}^0={\widetilde}{{\bm{x}}}^0\text{ and }{\lVert{} {\bm{x}}^+-{\widetilde}{{\bm{x}}}^+\rVert} < \epsilon, \\
0, & \text{ otherwise.}
\end{cases}$$
The following result shows the convergence of $\pi_{\epsilon}$ toward the standard posterior as we let $\epsilon$ approaching $0$.
\[prop:convergence\_result\] Suppose that $$\underset{({\widetilde}{{\bm{x}}},{\bm{\theta}}) \in \mathcal{B}_{\epsilon,{\bm{x}}}\times {\bm{\Theta}}}{\sup}\, p({\widetilde}{{\bm{x}}} {\mid}{\bm{\theta}}) < \infty,$$ for some $\epsilon >0$. Then, for each ${\bm{\theta}}\in {\bm{\Theta}}$, we have $$\pi_\epsilon({\bm{\theta}}{\mid}{\bm{x}}) \longrightarrow \pi({\bm{\theta}}{\mid}{\bm{x}}),\text{ as }\epsilon \rightarrow 0.$$
The modified prior $\pi_\epsilon({\bm{\theta}}{\mid}{\bm{x}})$ is defined as $$\label{eq:modified_posterior}
\pi_\epsilon({\bm{\theta}}{\mid}{\bm{x}})
=\frac{\pi({\bm{\theta}})\int_{{\mathbb{R}}^t} {\mathbb{I}}_{\mathcal{B}_{\epsilon,{\bm{x}}}}({\widetilde}{{\bm{x}}})\, p({\widetilde}{{\bm{x}}} {\mid}{\bm{\theta}}) {\mathop{}\!\mathrm{d}}{\widetilde}{{\bm{x}}}}{\int_{{\bm{\Theta}}}\pi({\bm{\theta}})\int_{{\mathbb{R}}^t} {\mathbb{I}}_{\mathcal{B}_{\epsilon,{\bm{x}}}}({\widetilde}{{\bm{x}}})\, p({\widetilde}{{\bm{x}}} {\mid}{\bm{\theta}}) {\mathop{}\!\mathrm{d}}{\widetilde}{{\bm{x}}} {\mathop{}\!\mathrm{d}}{\bm{\theta}}}
=\frac{\pi({\bm{\theta}})p_\epsilon({\bm{x}}{\mid}{\bm{\theta}})}{\int_{{\bm{\Theta}}}\pi({\bm{\theta}})p_\epsilon({\bm{x}}{\mid}{\bm{\theta}}) {\mathop{}\!\mathrm{d}}{\bm{\theta}}},$$ where $p_\epsilon({\bm{x}}{\mid}{\bm{\theta}})$ is an approximation of the likelihood $$\label{eq:quasi_likelihood}
p_\epsilon({\bm{x}}{\mid}{\bm{\theta}})=\frac{\int_{{\mathbb{R}}^t} {\mathbb{I}}_{\mathcal{B}_{\epsilon,{\bm{x}}}}({\widetilde}{{\bm{x}}})\, p({\widetilde}{{\bm{x}}} {\mid}{\bm{\theta}}) {\mathop{}\!\mathrm{d}}{\widetilde}{{\bm{x}}}}{\int_{{\mathbb{R}}^t} {\mathbb{I}}_{\mathcal{B}_{\epsilon,{\bm{x}}}}({\widetilde}{{\bm{x}}}) {\mathop{}\!\mathrm{d}}{\widetilde}{{\bm{x}}}}.$$ Since the data is [i.i.d.]{}, we rearrange the vectors ${\bm{x}}$ and ${\widetilde}{{\bm{x}}}$ to set aside the zeros in the data, so ${\bm{x}}= ({\bm{x}}^0,{\bm{x}}^+)$ and ${\widetilde}{{\bm{x}}} = ({\widetilde}{{\bm{x}}}^0,{\widetilde}{{\bm{x}}}^+)$, respectively. It allows us to write the indicator function in \[eq:quasi\_likelihood\] as the product $$\label{eq:product_indicator_function}
{\mathbb{I}}_{\mathcal{B}_{\epsilon,{\bm{x}}}}({\widetilde}{{\bm{x}}}) = {\mathbb{I}}_{\{{\bm{x}}^0={\widetilde}{{\bm{x}}}^0\}} \cdot {\mathbb{I}}_{\{ {\lVert{} {\bm{x}}^+-{\widetilde}{{\bm{x}}}^+\rVert} \leq \epsilon\}}.$$ Inserting \[eq:product\_indicator\_function\] into the quasi-likelihood \[eq:quasi\_likelihood\] leads to $$\begin{aligned}
p_\epsilon({\bm{x}}{\mid}{\bm{\theta}})
& = p_X(0 {\mid}{\bm{\theta}})^{t_0} [ 1-p_X(0 {\mid}{\bm{\theta}}) ]^{t-t_0} \frac{ \int_{{\mathbb{R}}^{t-t_0}} {\mathbb{I}}_{\{ {\lVert{} {\bm{x}}^+ - {\widetilde}{{\bm{x}}}^+\rVert} \leq \epsilon\}} p({\widetilde}{{\bm{x}}}^+ {\mid}{\bm{\theta}}) {\mathop{}\!\mathrm{d}}{\widetilde}{{\bm{x}}} }{ \int_{{\mathbb{R}}^{t-t_0}} {\mathbb{I}}_{\{ {\lVert{} {\bm{x}}^+ - {\widetilde}{{\bm{x}}}^+\rVert} \leq \epsilon\}} {\mathop{}\!\mathrm{d}}{\widetilde}{{\bm{x}}} } \nonumber \\
& \phantom{=} \mathclap{\underset{\epsilon\rightarrow 0}{\longrightarrow}}
\quad p_X(0 {\mid}{\bm{\theta}})^{t_0} [1-p_X(0 {\mid}{\bm{\theta}})]^{t-t_0} p({\bm{x}}^+ \mid\, {\bm{\theta}}) = p({\bm{x}}{\mid}{\bm{\theta}}),\label{eq:limit_Rubio_Johanssen}
\end{aligned}$$ where the limit in \[eq:limit\_Rubio\_Johanssen\] follows from applying Proposition 1 of @rubio2013simple, see also @bernton2019approximate [Proposition 2]. Taking the limit as $\epsilon$ tends to $0$ in \[eq:modified\_posterior\] yields the announced result.
Following up on the discussion in \[sub:abc\_cont\], we take the Wasserstein distance to evaluate the dissimilarities between the non-negative portions of the fake and observed data. When comparing the non-negative data points, a small $\epsilon$ leads to an accurate but potentially slow ABC algorithm. The combination of a small $\epsilon$ and a prior more diffuse than the posterior distribution makes ABC rejection sampling inefficient as acceptance almost never occurs. We therefore move from the acceptance–rejection simulation scheme to a Sequential Monte Carlo (SMC) scheme inspired by the work of @DMDoJa12.
ABC using SMC {#subsec:abc_smc}
-------------
Sequential Monte Carlo (ABC-SMC) is an ABC approach where a sequence of distributions is constructed by gradually decreasing tolerance $\epsilon$ through a sequence $(\epsilon_g)_{g\geq1}$. The ABC-SMC algorithm starts by sampling a finite number of parameter sets (particles) from the prior distribution and each intermediate distribution (called a generation) is obtained as a weighted sample approximated via a multivariate Kernel Density Estimator (KDE).
We start by setting the number of generation $G$ and the number of particles $K$. For the first generation $(g=1)$, the tolerance level is set to $\epsilon_1 = \infty$. Particles are proposed from the prior distribution ${\bm{\theta}}^{1}_k \sim \pi({\bm{\theta}})$ and retained if the synthetic data ${\bm{x}}_k \sim p({\bm{x}}{\mid}\theta^1_k)$ satisfies ${\bm{x}}_k \in \mathcal{B}_{\infty,{\bm{x}}}$. It goes on until $K$ particles are selected. Note that the condition ${\bm{x}}_k \in \mathcal{B}_{\infty,{\bm{x}}}$ simply means that the number of zeros in the fake data matches the number of zeros in the observed data. A first approximation of the posterior distribution follows from fitting a multivariate Kernel Density Estimator (KDE) $K_h$ to the first generation of particles $$ {\widehat}{\pi}_{\epsilon_1}({\bm{\theta}}{\mid}{\bm{x}}) = \frac{1}{K}\sum_{k = 1}^K K_h \bigl( {\lVert{} {\bm{\theta}}-{\bm{\theta}}_k^1 \rVert} \bigr),$$ where $h$ denotes the bandwidth. For a given generation $g>1$, we hold an approximation ${\widehat}{\pi}_{\epsilon_{g-1}}({\bm{\theta}}{\mid}{\bm{x}})$ of the posterior distribution based on the $(g-1)^{\text{th}}$ generation of particles. New particles ${\bm{\theta}}_k^g$ are proposed by sampling repeatedly from ${\widehat}{\pi}_{\epsilon_{g-1}}({\bm{\theta}}{\mid}{\bm{x}})$ until the synthetic data ${\bm{x}}_k \sim p({\bm{x}}{\mid}{\bm{\theta}}_k^g)$ satisfies ${\bm{x}}_k \in \mathcal{B}_{\infty,{\bm{x}}}$. It goes on until $K$ particles are selected, the synthetic data is also kept. An acceptance threshold $\epsilon_g$ is defined as the empirical quantile of order $\alpha\in(0,1)$ of the distances $||{\bm{x}}^+-{\bm{x}}_k^+||\text{, }k = 1,\ldots, K$. Each particle is assigned a weight $$w_k^g \propto\frac{\pi({\bm{\theta}}_k^g)}{{\widehat}{\pi}_{g-1}({\bm{\theta}}_k^g {\mid}{\bm{x}})} {\mathbb{I}}_{\mathcal{B}_{\epsilon_g,{\bm{x}}}}({\bm{x}}_k), \quad k = 1,\ldots,K,$$ which is used to update the posterior approximation to $${\widehat}{\pi}_{\epsilon_g}({\bm{\theta}}{\mid}{\bm{x}}) = \sum_{k = 1}^K w_k^g K_h( {\lVert{} {\bm{\theta}}-{\bm{\theta}}_k^g \rVert} ).$$ The pseudocode of the algorithm is provided in \[app:algo\], see \[alg:SMC\_abc\].
A common choice for the kernel is the multivariate Gaussian kernel with covariance matrix set to twice the empirical covariance matrix assessed over the cloud of weighted particles $\{({\bm{\theta}}^g_k,w^g_k)\}_{k = 1,\ldots, K}$, see @beaumont2009adaptive.
The behavior of the algorithm can be investigated by calculating the Effective Sample Size (ESS), defined in @DMDoJa12 as $$\text{ESS}^g = \Bigl[\sum_{k=1}^K(w_k^g)^2\Bigr]^{-1},\text{ }g = 1,\ldots, G.$$ The effective sample size ranges from $1$ to $N$ and indicates whether the algorithm is efficient in sampling from the targeted distribution. An ESS falling below a certain threshold, typically $N/2$ see @DMDoJa12, should trigger a resampling step. We close this section by illustrating the performance of our ABC implementation on an example where both the likelihood and sufficient summary statistics are available.
Illustrations on total claim amounts data {#subsec:illustration}
-----------------------------------------
Let the claim frequency be geometrically distributed $$n_1,\ldots, n_t {\overset{\mathrm{i.i.d.}}{\sim}}{\mathsf{Geom}}(p = 0.8),$$ with [p.m.f.]{}given by $p_N(n {\,;\,}p) = (1-p)p^n$, $n \in {\mathbb{N}_0}$. Assume that the claim amounts are exponentially distributed $$u_{s,1},\ldots, u_{s,n_s} {\overset{\mathrm{i.i.d.}}{\sim}}{\mathsf{Exp}}(\delta = 5), \quad s=1,\ldots, t.$$ with [p.d.f.]{}defined as $f(x {\,;\,}\delta) = (1/\delta){\mathrm{e}}^{-x/\delta}$, $x>0$, irrespective of the claim frequency. The available data is the aggregated claim sizes $$x_s = \sum_{k = 1}^{n_s} u_{s,k}, \quad s = 1,\ldots, t,$$ and we assume that $t= 100$ data points are available to conduct the inference. The likelihood function of the data is given by $$p({\bm{x}}{\mid}{\bm{\theta}}) = (1-p)^t \bigl(\frac{p}{\delta}\bigr)^{t-t_0} \exp\Bigl[-\frac{1-p}{\delta} \sum_{s = 1}^{t-t_0} x_s^+ \Bigr],$$ and allows us to sample from the standard posterior distribution via an MCMC scheme. This compound geometric-exponential model admits $t_0$ (the number of zeros in the data) and $\sum_{s = 1}^{(t-t_0)}x_s^+$ (sum of the non-negative data points) as sufficient statistics which in turn allows us to sample from an ABC posterior based on sufficient summary statistics. We set uniform priors $$p \sim {\mathsf{Unif}}(0, 1), \quad \delta \sim {\mathsf{Unif}}(0, 100)$$ over the parameters of the ${\mathsf{Geom}}(p)$–${\mathsf{Exp}}(\delta)$ we want to fit. We set the number of generation to $G=10$, the number of particles to $K = 1000$ and the order of the quantile to $\alpha = 0.5$ for the ABC sampler. \[fig:abc\_geom\_exp\] displays the histograms of the posterior samples produced via MCMC, ABC with sufficient statistics and ABC using the Wasserstein distance.
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The MCMC posterior sample has been obtained by using the dedicated function in the `PyMC3` Python library, see @salvatier2016probabilistic.
Model selection {#sec:ABC_model_selection}
===============
When it comes to modeling claim data, one has plenty of options for both the claim frequency and the claim sizes, see for instance the book of @klugman2012loss [Chapters V & VI]. A decision must be made to find the most suitable models among a set of candidates $\{1,\ldots,M\}$. The Bayesian approach to model selection and hypothesis testing consists in defining a categorical random variable $m$ with state space $\{1,\ldots,M\}$ and a priori distribution $\pi(m)$. The a posteriori model evidence is then given by $$ \pi(m {\mid}{\bm{x}}) = \frac{p({\bm{x}}{\mid}m)\pi(m)}{\sum_{\widetilde{m} = 1}^M p({\bm{x}}{\mid}\widetilde{m})\pi(\widetilde{m})}, \quad m \in \{1,\ldots,M\}.$$ One often compares two models, say $1$ and $2$, by computing the Bayes factors $B_{12} = \pi(2 {\mid}{\bm{x}})/\pi(1 {\mid}{\bm{x}})$. For an overview on Bayesian model selection and Bayes factor, we refer the reader to @kass1995bayes. The marginal likelihood of the data according to given model $m \in \{1,\ldots, M\}$ is defined by $$\label{eq:marginal_likelihood}
p({\bm{x}}{\mid}m) = \int_{\Theta_m} p({\bm{x}}{\mid}m,{\bm{\theta}})\pi({\bm{\theta}}{\mid}m) {\mathop{}\!\mathrm{d}}{\bm{\theta}}, \quad \text{for } m \in \{1,\ldots, M\},$$ where $\Theta_m$ denotes the parameter space of model $m$. The evaluation of \[eq:marginal\_likelihood\] is challenging from a computational point of view, even when the likelihood is available. The acceptance–rejection implementation of ABC proposed in @grelaud2009abc reduces to add a layer to the standard \[alg:AR\_abc\_mixed\] by first drawing a model from $\pi(m)$. The posterior probability of a model is then proportional to the number of times this model was selected, see \[alg:AR\_abc\_model\_selection\].
**generate** $m_k \sim \pi(m)$ **generate** ${\bm{\theta}}_k \sim \pi({\bm{\theta}}{\mid}m)$ **generate** ${\bm{x}}_k \sim p({\bm{x}}{\mid}m_k,\, {\bm{\theta}}_k)$ ${\bm{x}}_k \in \mathcal{B}_{\epsilon,{\bm{x}}}$ then **store** $(m_k,{\bm{\theta}}_k)$
The spirit of \[alg:AR\_abc\_model\_selection\] relates to the Monte Carlo approach to the computation of models’ marginal likelihood, see for instance @mcculloch1991bayesian. Namely, the model evidence is evaluated by $$p({\bm{x}}{\mid}m) \approx \frac 1K \sum_{k=1}^K p({\bm{x}}{\mid}m,{\bm{\theta}}_k),$$ where ${\bm{\theta}}_1,\ldots, {\bm{\theta}}_K \sim \pi({\bm{\theta}}{\mid}m)$. This procedure might be inefficient as most of the ${\bm{\theta}}_i$ have small likelihoods when the posterior is more concentrated than the prior distribution. Importance sampling strategies have been proposed to address this issue. The sequential Monte Carlo idea used in \[alg:SMC\_abc\] have been adapted in the works of @10.1093/bioinformatics/btp619 and @PrFeCoBiFr14 to improve the sampling efficiency. Our implementation is described hereafter.
We fix the number of generations $G$ and the number of particles $K$. When several models are competing, a particle is a combination of a model and its parameters.
For the first generation ($g=1$), for each particle $k = 1,\ldots, K$, a model $m^1_k$ is drawn from $\pi(m)$ with parameter ${\bm{\theta}}^1_k$ sampled from the prior distribution $\pi({\bm{\theta}}{\mid}m^1_k)$ until the synthetic data ${\bm{x}}_k\sim p({\bm{x}}|m^1_k,{\bm{\theta}}^1_k)$ satisfies ${\bm{x}}_k\in\mathcal{B}_{\epsilon_{1},{\bm{x}}}$, where $\epsilon_1=\infty$. A first approximation of the posterior model probability is given by $$ {\widehat}{\pi}_{\epsilon_1}(m {\mid}{\bm{x}}) = \frac{1}{K} \sum_{k = 1}^K{\mathbb{I}}_{\{m^1_k=m\}}.$$ A multivariate Kernel Density Estimator (KDE) $K_h$ with bandwidth $h$ is then fitted to the parameter values associated to each model with $$ {\widehat}{\pi}_{\epsilon_1}({\bm{\theta}}{\mid}m, {\bm{x}}) = \frac{1}{K} \sum_{k = 1}^K\frac{1}{{\widehat}{\pi}_{\epsilon_1}(m {\mid}{\bm{x}})}K_h({\lVert{} {\bm{\theta}}-{\bm{\theta}}_k^1 \rVert}){\mathbb{I}}_{\{m^1_k=m\}},\text{ }m\in \{1,\ldots, M\}.$$ At a given generation $g \in \{ 1,\ldots, G\}$ and for each model $m \in \{1,\ldots, M\}$, we hold an approximation of the posterior model evidence ${\widehat}{\pi}_{\epsilon_{g-1}}(m {\mid}{\bm{x}})$ and the posterior distribution of the parameters ${\widehat}{\pi}_{\epsilon_{g-1}}({\bm{\theta}}{\mid}m, {\bm{x}})$. New particles $(m_k^g, {\bm{\theta}}_k^g)$ are proposed by sampling from $\pi(m)$ and ${\widehat}{\pi}_{\epsilon_{g-1}}({\bm{\theta}}{\mid}m_k^g, {\bm{x}})$ until the synthetic data ${\bm{x}}_k \sim p({\bm{x}}{\mid}m^g_k, {\bm{\theta}}_k^g)$ satisfies ${\bm{x}}_k \in \mathcal{B}_{\epsilon_{g-1},{\bm{x}}}$. Sampling is performed repeatedly until $K$ particles are selected. The acceptance threshold $\epsilon_g$ becomes the empirical quantile of order $\alpha \in (0,1)$ of the distances ${\lVert{} {\bm{x}}^+-{\bm{x}}^+_k\rVert}$, $k = 1,\ldots K$. To each particle is assigned a weight given by $$w_k^g \propto \frac{\pi({\bm{\theta}}_k^g {\mid}m_k^g)}{{\widehat}{\pi}_{\epsilon_{g-1}}({\bm{\theta}}_k^g {\mid}m_k^g, \,{\bm{x}})} {\mathbb{I}}_{\mathcal{B}_{\epsilon_g,{\bm{x}}}}({\bm{x}}_k), \quad k = 1,\ldots,K.$$ The model probability is then updated $${\widehat}{\pi}_{\epsilon_{g}}(m {\mid}{\bm{x}}) = \sum_{k = 1}^K w_k^i {\mathbb{I}}_{\{m^g_k=m\}},$$ along with the posterior distribution of the parameters associated to each model $${\widehat}{\pi}_{\epsilon_{g}}({\bm{\theta}}{\mid}m, {\bm{x}}) = \sum_{k=1}^K\frac{w_k^g}{{\widehat}{\pi}_{\epsilon_{g}}(m {\mid}{\bm{x}})}K_h( {\lVert{} {\bm{\theta}}-{\bm{\theta}}_k^g \rVert}) \, {\mathbb{I}}_{\{m^g_k=m\}}, \text{ }m = 1,\ldots, M.$$ The algorithm is summarized in \[alg:SMC\_abc\_model\_choice\] of \[app:algo\].\
Our ABC implementation when evaluating posterior model probabilities is tested on a simple example where we aim at fitting individual claim sizes generated from a lognormal distribution $$u_{1},\ldots, u_{n} {\overset{\mathrm{i.i.d.}}{\sim}}{\mathsf{LogNorm}}(\mu = 0,\sigma = 1),$$ with associated [p.d.f.]{}$$\label{eq:weibull_pdf}
f(x {\,;\,}\mu,\sigma) = \frac{1}{x\sigma\sqrt{2\pi}} \exp\Bigl[-\frac{(\ln x - \mu )}{2\sigma^2} \Bigr], \quad x>0.$$ The lognormal model is compared to a gamma model ${\mathsf{Gamma}}(r, m)$ with [p.d.f.]{}$$\label{eq:gamma_pdf}
f(x {\,;\,}r,m) = \frac{{\mathrm{e}}^{-x/m}x^{r-1}}{m^r\Gamma(r)}, \quad x>0,$$ and a Weibull model ${\mathsf{Weib}}(r, m)$ with [p.d.f.]{}$$\label{eq:weibull_pdf}
f(x {\,;\,}k,\beta) = \frac{k}{\beta} {\bigl( \frac{x}{\beta} \bigr)}^{k-1} \exp\bigl[ -{\bigl(\frac{x}{\beta} \bigr)}^k \bigr], \quad x>0.$$ Uniform priors are set over the parameters of all the model: $$\mu \sim {\mathsf{Unif}}(-20,20), \text{ and } \sigma \sim {\mathsf{Unif}}(0,5),$$ for the lognormal model, $$r \sim {\mathsf{Unif}}(0,5), \text{ and } m \sim {\mathsf{Unif}}(0,100),$$ for the gamma model, and $$k \sim {\mathsf{Unif}}(\tfrac{1}{10},5), \text{ and } \beta \sim {\mathsf{Unif}}(0,100),$$ for the Weibull model. The likelihood function of the data ${\bm{u}}= u_1,\ldots, u_n$ may be computed for these loss models and the model probability can be estimated through the Sequential Monte Carlo sampler of the `PyMC3` library. The computation of model probabilities via ABC is more demanding than simply estimating parameters. Namely, the number of iterations must be larger to lead to an accurate model probability estimation. We therefore set the number of iterations to $G = 25$. The model evidences of all three models are reported in \[tab:model\_evidence\_individual\_claim\_sizes\] for samples of size ranging from $25$ to $200$.
------------- -------------------- ---------------------- ------------------- -- -------------------- ---------------------- -------------------
${\mathsf{Gamma}}$ ${\mathsf{LogNorm}}$ ${\mathsf{Weib}}$ ${\mathsf{Gamma}}$ ${\mathsf{LogNorm}}$ ${\mathsf{Weib}}$
sample size
25 0.42 0.18 0.40 0.51 0.15 0.34
50 0.25 0.64 0.11 0.31 0.51 0.18
75 0.04 0.95 0.01 0.15 0.79 0.07
100 0.01 0.99 0.00 0.07 0.91 0.02
150 0.00 1.00 0.00 0.01 0.99 0.00
200 0.00 1.00 0.00 0.00 1.00 0.00
------------- -------------------- ---------------------- ------------------- -- -------------------- ---------------------- -------------------
: Model evidence for individual claim sizes data simulated by a ${\mathsf{LogNorm}}(\mu = 0, \sigma = 1)$ model. The model evidences computed via ABC fare well compared to the model evidences computed by relying on the likelihood function.[]{data-label="tab:model_evidence_individual_claim_sizes"}
Further approximate Bayesian model evidence computations are proposed in \[sec:Simu\] and \[sec:RealExample\] when the data at hand is aggregated.
Simulation Study {#sec:Simu}
================
This section aims at studying the finite sample behavior of our ABC implementation on two case studies based on simulated data. In \[subsec:neg\_bin\_Weibull\], we assume that the claim sizes are independent from the claim frequency and that the insurer have access to the right truncated aggregated sum. In \[subsec:frequency\_dependent\_exponential\], we consider a model in which the average of the claim sizes depends on the number of claims and the insurer have access to the total claim sizes for each time period.
Our goal is to check whether our ABC sampling algorithm manage to return a posterior sample that concentrates around the true value when the model is well specified. Another question is how does the ABC posterior behave when the model is misspecified? The ABC posterior samples are compared, in that case, to the maximum likelihood estimates of the parameters.
Finally, we assume that the claim frequency data is available in addition to the aggregated data. The number of claims is then input directly in our ABC implementation to specify how many claim sizes should be generated for each time period. It reduces the computing time, and allow us to drop the parametric assumption over the claim frequency distribution and direct our focus on the claim amounts distribution.
In both examples, the number of generations for ABC is set to $G = 7$ and each consists of $K = 1000$ particles when only one model is considered and when the claim frequency is not available. Knowing the number of claims leads to a reduction in calculation time, which in turn allows us to bring the number of iterations to $G=10$.
Negative-Binomial Weibull model with truncation {#subsec:neg_bin_Weibull}
-----------------------------------------------
Let the claim frequency be negative binomial distributed $$n_1,\ldots, n_t {\overset{\mathrm{i.i.d.}}{\sim}}{\mathsf{NegBin}}(\alpha = 4,\, p = \tfrac23),$$ with [p.m.f.]{}$$p_N(n {\,;\,}\alpha,p) = \binom{\alpha+n-1}{n} \, p^\alpha(1-p)^n, \quad n\geq0,$$ while the claim sizes are Weibull distributed $$u_{s,1},\ldots, u_{s,n_s} {\overset{\mathrm{i.i.d.}}{\sim}}{\mathsf{Weib}}(k = \tfrac13 ,\, \beta = 1), \quad s=1,\ldots, t.$$ The available data is the aggregated claim size in excess of a threshold $c$, given by $$\label{eq:aggregated_data_test1}
x_s = \Bigl(\sum_{i = 1}^{n_s} u_{s,i} - c \Bigr)_+, \quad s=1,\ldots, t.$$ It corresponds to the data available to a reinsurance company within the frame of a global non-proportional treaty over a non-life insurance portfolio. The cases $t = 50$ and $t = 250$ are considered. The prior distributions over the four parameters are $$\label{eq:prior_assumptions_nbinom_weib}
\alpha \sim {\mathsf{Unif}}(0,10),\text{ } p \sim {\mathsf{Unif}}(\tfrac{1}{1000},1),\text{ } k \sim {\mathsf{Unif}}(\tfrac{1}{10},10), \text{ and } \beta \sim {\mathsf{Unif}}(0,20).$$
displays the ABC posterior samples when only using the aggregated data \[eq:aggregated\_data\_test1\].\
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The $p$ and $k$ posteriors are quite informative, whereas the scale parameters $\alpha$ and $\beta$ are skewed in opposite directions and seem to compensate for each other. We then include the claim frequencies $n_s$ in the input data of our ABC algorithm to see if this helps in getting posterior samples closer to the target. displays the ABC posterior samples of the claim sizes model when the claim frequency data is available in addition to the summaries \[eq:aggregated\_data\_test1\].
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The ABC posteriors are very strongly concentrated around the true values $k=\frac13$ and $\beta=1$ compared to that of \[sub:hist\_test1\_negbin\_weib\].
We now turn to the case where the model is misspecified. The same data simulated from a ${\mathsf{NegBin}}(\alpha=4, p=\frac23)$–${\mathsf{Weib}}(k=\frac13, \beta=1)$ model is used to fit a ${\mathsf{NegBin}}(\alpha, p)$–${\mathsf{Gamma}}(r, m)$ model. The prior distributions over the four parameters are uniform with $$\label{eq:prior_assumptions_nbinom_gamma}
\alpha \sim {\mathsf{Unif}}(0, 20), \quad p \sim {\mathsf{Unif}}(\tfrac{1}{1000}, 1), \quad r \sim {\mathsf{Unif}}(0, 10),\text{ and }m \sim {\mathsf{Unif}}(0, 20).$$ The true values for the gamma distribution parameters are replaced by the maximum likelihood estimators based on a large sample of Weibull distributed individual losses. displays the ABC posterior samples when only using the aggregated data \[eq:aggregated\_data\_test1\].
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\
The ABC posterior distributions are informative regarding $p$, $r$ and $m$, however the algorithm does not improve significantly the prior assumption over $\alpha$.
displays the ABC posterior samples for the parameter of the gamma distribution when the claim frequency data is available in addition to the summaries \[eq:aggregated\_data\_test1\].
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The posterior sample for $m$ does not seem to center around the maximum likelihood estimator. Note that the situation improves greatly when considering a larger sample, of size $500$ say. Also note that by fitting a gamma model on the individual losses, the mean *a posteriori* for $m$ is around $40$, which may explain why our ABC posterior somewhat miss the target.\
To perform model selection, we specify to our ABC algorithm the Weibull and the gamma distribution as competing models for the claim sizes and we set uniform priors as in \[eq:prior\_assumptions\_nbinom\_weib\] and \[eq:prior\_assumptions\_nbinom\_gamma\] over the parameters. The model evidences computed via ABC are reported in \[tab:model\_evidence\_abc\_test1\].
----- -- ------------------- ----------------------
Negative Binomial Observed Frequencies
50 0.51 0.88
250 0.44 1.00
----- -- ------------------- ----------------------
: Model evidence in favor of a ${\mathsf{Weib}}(k, \beta)$ model when compared against a ${\mathsf{Gamma}}(r, m)$ model for data simulated by a ${\mathsf{NegBin}}(\alpha = 4, p = \frac23)$–${\mathsf{Weib}}(k = \frac13, \beta = 1)$ model. The values should increase to 1 as the sample size increases.[]{data-label="tab:model_evidence_abc_test1"}
When only the summaries $x_s$ are available and the claim frequency is modeled by a negative binomial distribution then ABC cannot decide between the Weibull and the gamma distributions. When the claim counts $n_s$ are also available then ABC favors greatly the Weibull model for the claim sizes.
Dependence between the claim frequency and severity {#subsec:frequency_dependent_exponential}
---------------------------------------------------
Let the claim frequency be Poisson distributed $$n_1,\ldots, n_t {\overset{\mathrm{i.i.d.}}{\sim}}{\mathsf{Poisson}}(\lambda = 4),$$ with [p.m.f.]{}$$p_N(k {\,;\,}\lambda) =\frac{{\mathrm{e}}^{-\lambda} \lambda^k}{k!}, \quad k\geq0.$$ The claim sizes are assumed to be exponentially distributed with a scale parameter depending on the observed claim frequency with $$u_{s,1},\ldots, u_{s,n_s} {\mid}n_s {\overset{\mathrm{i.i.d.}}{\sim}}{\mathsf{Exp}}(\mu = \beta \, {\mathrm{e}}^{\delta n_s}), \text{ for } s=1,\ldots, t .$$ We denote this ${\bm{u}}_{s} \sim {\mathsf{DepExp}}(n_s {\,;\,}\beta,\delta)$, and take $\beta = 2$ and $\delta = 0.2$. The resulting conditional [p.d.f.]{}is $$f_U(x {\mid}n {\,;\,}\beta,\delta) = \frac{1}{\beta {\mathrm{e}}^{\delta n}} \exp\bigl(-\frac{x}{\beta {\mathrm{e}}^{\delta n}} \bigr), \quad x>0.$$ This dependence structure relates to the insurance ratemaking practice where premiums are computed using the average claim frequency and severity predicted by a generalized linear models (GLM). In the classical setting, the claim frequency is assumed to be Poisson distributed and the claim sizes are gamma distributed. The GLM are then fitted independently for the claim frequency and the claim severity, we refer to @renshaw_1994. Empirical studies, like the one conducted in @doi:10.1080/10920277.2011.10597626, have shown how the claim sizes may vary with the claim frequency. A standard practice is then to include the predicted claim frequency as a covariate within the claim sizes model, see for instance @SHI2015417. It then reduces to bump the expectation of the severity by a factor ${\mathrm{e}}^{\delta n_s}$. Our case study is inspired by @GARRIDO2016205 [Example 3.1]. The available data is the aggregated claim sizes $$\label{eq:aggregated_data_test_2}
x_s = \sum_{k = 1}^{n_s} u_{s,k}, \quad s = 1,\ldots, t.$$ We consider data histories of length $t = 50$ and $250$.
Uniform prior distributions are set over the model parameters as $$ \lambda \sim {\mathsf{Unif}}(0, 10), \text{ } \beta \sim {\mathsf{Unif}}(0, 20), \text{ and } \delta \sim {\mathsf{Unif}}(-1, 1).$$ displays the posterior samples of $\lambda$ the parameter of the Poisson distribution, $\beta$ the scale parameter of the exponential parameter and $\delta$ the frequency/severity correlation parameter.
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The algorithm does a tremendous job on this example even without including the claim count information of each time period.
displays the ABC posterior samples associated to the claim sizes distribution ${\mathsf{DepExp}}(n;\beta, \delta)$ when including the frequency information in addition to the summaries \[eq:aggregated\_data\_test\_2\].
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As already noted, the inclusion of the claim frequency information improves the ABC posterior samples.
Application to a real-world insurance data set {#sec:RealExample}
==============================================
We consider an open source insurance data set named `ausautoBI8999` consisting of $22,036$ settled personal injury insurance claims in Australia, the first five observations are displayed in \[tab:full\_data\].
Date Month Claim Severity
------------ ------- ----------------
1993-10-01 52 87.75
1994-02-01 56 353.62
1994-02-01 56 688.83
1994-05-01 59 172.80
1994-09-01 63 43.29
: `ausautoBI8999` personal injury claim data.[]{data-label="tab:full_data"}
The data is accessible from the `R` package `CASDatasets`, see @dutang2016casdatasets. The data is then aggregated monthly by reporting the number of claims along with the sum of all the compensations associated to each month, see \[tab:agg\_data\].
Month Claim Frequency Total Claim Severity
------- ----------------- ----------------------
49 149 1.55e+06
50 188 3.21e+06
51 196 4.81e+06
52 203 4.22e+06
53 226 5.27e+06
: Monthly aggregated data.[]{data-label="tab:agg_data"}
\[tab:aggregated\_data\]
Descriptive statistics for the claim sizes, claim frequencies and the aggregated claims sizes are reported in \[tab:desc\_stat\].
Statistics Claim Severity Claim Frequency Total Claim Severity
-- ------------ -- ---------------- ----------------- ----------------------
Count 2.20e+04 6.90e+01 6.90e+01
Mean 3.84e+04 3.19e+02 1.23e+07
Std 9.10e+04 1.09e+02 5.22e+06
Min 9.96e+00 9.40e+01 1.55e+06
25% 6.30e+03 2.31e+02 8.21e+06
50% 1.39e+04 3.12e+02 1.20e+07
75% 3.51e+04 3.81e+02 1.55e+07
Max 4.49e+06 6.06e+02 2.63e+07
: Descriptive statistics of the claim data.[]{data-label="tab:desc_stat"}
We are going to use ABC to fit and compare loss models using only the monthly aggregated data in \[tab:agg\_data\]. We would like to know whether the results differ from fitting the same loss models but using the individual claim sizes data in \[tab:full\_data\].
We start by studying the individual loss distribution. We fit a gamma, a lognormal and a Weibull model to the data shown in \[tab:full\_data\] using maximum likelihood estimation. The estimates of the parameters are given in \[tab:mle\_individual\_loss\] and will serve as benchmark for our ABC posterior samples.
Severity model Parameters MLE BIC
-- ---------------- ------------ --------- -----
$r$ 4.09e+0
$m$ 5.35e+3
$k$ 7.08e-1
$\beta$ 2.86e+4
$\sigma$ 9.56e+0
$\mu$ 1.46e+0
: Maximum likelihood estimates of a gamma, Weibull and lognormal distribution based on the individual claim sizes data.[]{data-label="tab:mle_individual_loss"}
The lognormal distribution seems to provide the best fit when looking at the values of the Bayesian Information Criteria (BIC). This result is visually confirmed by the quantile-quantile plots displayed in \[fig:qqplots\_aus\].
We then investigate the stationarity of the individual loss distribution by fitting the three loss models to the data associated to each time period separately. \[fig:aus\_gamma\_params,fig:aus\_Weibull\_params,fig:aus\_log\_norm\_params\] display the parameters of the gamma, Weibull and lognormal distribution respectively depending on the time period considered.
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; (-3, 1.9) node ; (2.5, 1.9) node ;
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The parameters of the Weibull and gamma distributions exhibit a high variability, see \[fig:aus\_Weibull\_params,fig:aus\_gamma\_params\], while the parameters of the lognormal distribution are more stable, see \[fig:aus\_log\_norm\_params\]. The model evidences, displayed in \[fig:model\_prob\_aus\], are computed using the Schwarz criterion that approximates the Bayes factor using the maximum likelihood estimators and the BIC.
The model probabilities mostly favor the lognormal model.
We use ABC to fit a ${\mathsf{NegBin}}(\alpha, p)$–${\mathsf{LogNorm}}(\mu,\sigma)$ model to the total claim severities data in \[tab:agg\_data\] which consists of $t=69$ summaries of the form $$\label{eq:aggregated_real_data}
x_s = \sum_{k = 1}^{n_s} u_{s,k}, \quad s = 1,\ldots, t.$$ We consider two sets of prior assumptions over the parameters:
1. $\alpha \sim {\mathsf{Unif}}(0, 20)$, $p \sim {\mathsf{Unif}}(\tfrac{1}{1000}, 1)$, $\begin{color}{MyBlue} \mu \sim {\mathsf{Unif}}(-10, 10) \end{color}$, and $\sigma \sim {\mathsf{Unif}}(0, 10)$,
2. $\alpha \sim {\mathsf{Unif}}(0, 20)$, $p \sim {\mathsf{Unif}}(\tfrac{1}{1000}, 1)$, $\begin{color}{MyRed} \mu \sim {\mathsf{Unif}}(0, 20) \end{color}$, and $\sigma \sim {\mathsf{Unif}}(0, 10)$.
Prior settings $1$ and $2$ only differ in the boundaries of the uniform distribution of $\mu$. We opt for a more intensive ABC calibration compared to that of section 4. The number of iterations is fixed at $G = 20$ when the claim frequencies are known and $G=15$ when they are not. The ABC posterior samples of the ${\mathsf{NegBin}}(\alpha, p)$–${\mathsf{LogNorm}}(\mu,\sigma)$ model using only the summaries $x_s$ in \[eq:aggregated\_real\_data\] are shown in \[sub:hist\_RD\_negbin\_lognormal\_priors\].
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; (-4.1, 1.8) node ; (-1.3, 1.8) node ; (0.9, 1.8) node ; (3.8, 1.8) node ;
The results with prior settings $1$ and $2$ are noticeably different. More specifically, the ABC posterior are tighter and more centered around the MLE estimates with prior $2$ at least when it comes to estimating the parameters $p$, $\mu$ and $\sigma$.
The ABC posterior samples when including the claim frequency information are shown in \[sub:hist\_RD\_freq\_lognormal\_priors\]. We keep the same prior assumptions over $\mu$ and $\sigma$.
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; (-4.5, 1.7) node ; (2, 1.7) node ;
Including the claim frequency data helps in making the results consistent from one prior setting to the other.
We now turn to the problem of selecting a model for the claim sizes, so we specify a negative binomial distribution ${\mathsf{NegBin}}(\alpha, p)$ with uniform prior distributions $$\alpha \sim {\mathsf{Unif}}(0, 20), \quad p \sim {\mathsf{Unif}}(0, 1)$$ to model the claim frequency and let our ABC algorithm pick a claim amounts models among the following:
- ${\mathsf{Weib}}(k, \beta)$ with prior distributions $$k \sim {\mathsf{Unif}}(\tfrac{1}{1000}, 1), \quad \beta \sim {\mathsf{Unif}}(0, 4 \times 10^4),$$
- ${\mathsf{Gamma}}(r,m)$ with prior distributions $$r \sim {\mathsf{Unif}}(0, 100), \quad \beta \sim {\mathsf{Unif}}(0, 1.5 \times 10^5),$$
- ${\mathsf{LogNorm}}(\mu,\sigma)$ with prior distributions $$\mu \sim {\mathsf{Unif}}(5, 10), \quad \sigma \sim {\mathsf{Unif}}(0, 3).$$
The bounds of the uniform distributions are set to reflect the variability of the parameters in \[fig:aus\_gamma\_params,fig:aus\_log\_norm\_params,fig:aus\_Weibull\_params\]. The model evidences are reported in \[tab:model\_evidence\_RD\].
-- ---------------------- -- ------- ----------- ---------
Gamma Lognormal Weibull
Negative Binomial 0.92 0.01 0.07
Observed Frequencies 0.00 0.49 0.51
-- ---------------------- -- ------- ----------- ---------
: ABC model evidence with the claim frequency and the aggregated claim sizes data.[]{data-label="tab:model_evidence_RD"}
We see that ABC strongly favors the gamma model when the claim frequency is assumed to have a negative binomial distribution. When including the claim count, ABC discards the gamma model but is unable to decide between the Weibull or the lognormal model. This result is of course a little disappointing but probably means that ABC would need more than $69$ observations to pick the right model.
Conclusion {#sec:conclusion}
==========
This paper is a case study of an ABC application in insurance. We showed how to use this method to calibrate insurance loss models with limited information (one data point per time period). The fact that the method does not require the knowledge of the likelihood function permits to go beyond the classical setting where independence is assumed between the claim frequency and the claim sizes.
An ABC routines essentially relies on two things: (i) an efficient sampling strategy and (ii) a reliable measure of dissimilarity between samples of data. We put together an ABC routine that implements a parallel sequential Monte Carlo sampler and uses the Wasserstein distance to compare the synthetic data to the observed one. The python code may be downloaded from the following GitHub repository <https://github.com/LaGauffre/ABCFitLoMo>.
ABC has become over the years a common practice in a variety of fields ranging from ecology to genetics. We believe that ABC could be also applied to a wide range of sophisticated models that arise in finance and insurance.
Acknowledgments {#acknowledgments .unnumbered}
===============
Patrick J. Laub conducted this research within the DAMI – Data Analytics and Models for Insurance – Chair under the aegis of the Fondation du Risque, a joint initiative by UCBL and BNP Paribas Cardif.
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Algorithmic details {#app:algo}
===================
**generate** ${\bm{\theta}}_k^1 \sim \pi({\bm{\theta}})$ **generate** ${\bm{x}}_k \sim p({\bm{x}}{\mid}{\bm{\theta}}_k^1)$
**compute** ${\widehat}{\pi}_{\epsilon_1}({\bm{\theta}}{\mid}{\bm{x}}) = \frac{1}{K}\sum_{k = 1}^K K_h( {\lVert{} {\bm{\theta}}-{\bm{\theta}}_k^1 \rVert} )$
**generate** ${\bm{\theta}}_k^g \sim {\widehat}{\pi}_{\epsilon_{g-1}}({\bm{\theta}}{\mid}{\bm{x}})$ **generate** ${\bm{x}}_k \sim p({\bm{x}}{\mid}{\bm{\theta}}_k^g)$
**set** $\epsilon_g = \text{Quantile}\bigl({\lVert{} {\bm{x}}^+-{\bm{x}}^+_1\rVert},\ldots,{\lVert{} {\bm{x}}^+-{\bm{x}}^+_K\rVert} {\,;\,}\alpha\bigr) $ **set** $w_k^g \propto\frac{\pi({\bm{\theta}}_k^g)}{{\widehat}{\pi}_{\epsilon_g}({\bm{\theta}}_k^g {\mid}{\bm{x}})} {\mathbb{I}}_{\mathcal{B}_{\epsilon_g,{\bm{x}}}}({\bm{x}}_k)$ **compute** ${\widehat}{\pi}_{\epsilon_g}({\bm{\theta}}{\mid}{\bm{x}}) = \sum_{k = 1}^K w_k^g K_h( {\lVert{} {\bm{\theta}}-{\bm{\theta}}_k^g \rVert} )$
**generate** $m_k^1 \sim \pi(m)$ **generate** ${\bm{\theta}}_k^1 \sim \pi({\bm{\theta}}{\mid}m_k^1)$ **generate** ${\bm{x}}_k \sim p({\bm{x}}{\mid}m_k^1, {\bm{\theta}}_k^1)$ **compute** ${\widehat}{\pi}_{\epsilon_{1}}(m {\mid}{\bm{x}}) = \frac{1}{K} \sum_{k = 1}^K{\mathbb{I}}_{\{m^1_k=m\}}$ **compute** ${\widehat}{\pi}_{\epsilon_{1}}({\bm{\theta}}{\mid}m,{\bm{x}}) = \frac{1}{K} \sum_{k = 1}^K\frac{1}{{\widehat}{\pi}_{\epsilon_{1}}(m {\mid}{\bm{x}})}K_h({\lVert{} {\bm{\theta}}-{\bm{\theta}}_k^1 \rVert} ){\mathbb{I}}_{\{m^1_k=m\}}$ **generate** $m_k^g \sim \pi(m)$ **generate** ${\bm{\theta}}_k^g \sim {\widehat}{\pi}_{\epsilon_{g-1}} ({\bm{\theta}}{\mid}m_k^g,\, {\bm{x}})$ **generate** ${\bm{x}}_k \sim p({\bm{x}}{\mid}m_k^g,\, {\bm{\theta}}_k^g)$ **set** $\epsilon_g = \text{Quantile}({\lVert{} {\bm{x}}^+-{\bm{x}}^+_1\rVert},\ldots,{\lVert{} {\bm{x}}^+-{\bm{x}}^+_K\rVert} {\,;\,}\alpha)$ **set** $w_k^g \propto\frac{\pi({\bm{\theta}}_k^g {\mid}m_k^g) }{{\widehat}{\pi}_{\epsilon_{g-1}} ({\bm{\theta}}_k^g {\mid}m_k^g, \,{\bm{x}})} {\mathbb{I}}_{\mathcal{B}_{\epsilon_g,{\bm{x}}}}({\bm{x}}_k)$ **compute** ${\widehat}{\pi}_{\epsilon_{g}}(m {\mid}{\bm{x}}) = \sum_{k = 1}^K w_k^g {\mathbb{I}}_{\{m^g_k=m\}}$ **compute** ${\widehat}{\pi}_{\epsilon_{g}}({\bm{\theta}}{\mid}m, {\bm{x}}) = \sum_{k=1}^K\frac{w_k^g}{{\widehat}{\pi}_{\epsilon_{g}}(m {\mid}{\bm{x}})}K_h({\lVert{} {\bm{\theta}}-{\bm{\theta}}_k^g \rVert} ){\mathbb{I}}_{\{m^g_k=m\}}$
|
---
address: 'Univ. of Michigan, 525 East University Avenue, Ann Arbor, MI-48109'
author:
- Manuel Blickle
date: 'June 11, 2000'
title: Cartier Isomorphism for Toric Varieties
---
[^1]
Introduction
============
In [@Cartier57], Cartier introduces an operation on the complex for varieties defined over a field of positive characteristic. In the case of a smooth variety this yields a complete description of the cohomology of the complex. To formulate the result let $X$ be a smooth variety over a perfect field then for all $a \geq 0$ the Cartier operator $${C: {{\mathcal{H}}}^a(F_*\Omega^\bullet_X) {\xrightarrow}{} \Omega^a_X}$$ is an isomorphism. Here $F: X {\xrightarrow}{} X$ denotes the Frobenius morphism on $X$ and ${{\mathcal{H}}}^a$ denotes the $a^{\text{th}}$ cohomology sheaf of $F_*\Omega^{\bullet}_X$. If the variety is not smooth, not much is known about the properties of the Cartier operator and the poor behaviour of the complex in this case makes its study difficult. If one substitutes the complex with the Zariski-complex the situation is better. For example, the Zariski differentials, though not locally free, are reflexive and there is a natural duality pairing between them.
We show how to extend the Cartier operator in a natural way to the Zariski differentials. Using a description of the Zariski-complex due to Danilov [@Danilov78] we show that this newly defined Cartier operator is an isomorphism for toric varieties. Moreover, it is induced by a split injection $$\xymatrix@1{ 0 \ar[r] &{{{\widetilde}{\Omega}}^a_X} \ar[r] &{F_*{{\widetilde}{\Omega}}^a_X.}}$$ As described in [@Anders] such a result yields the Bott vanishing theorem and the degeneration of the Hodge to spectral sequence for projective toric varieties.
Finally we will give an obstruction to the surjectivity of the Cartier map to show that in general it can not be an isomorphism.
The author would like to thank Karen Smith for invaluable discussions and the referee for a careful reading and valuable comments.
Notation and Generalities
-------------------------
Fix a perfect field $k$ of characteristic $p>0$. All schemes will be noetherian $k$-schemes unless otherwise stated. $X$ denotes a noetherian $k$-scheme. By an ${{\mathcal{O}}}_X$-module we always mean a coherent sheaf of ${{\mathcal{O}}}_X$-modules.
The *absolute Frobenius* of $X$ is the endomorphism $$F_X :X {\longrightarrow}X$$ which is the identity on the level of the topological space and the $p^{\text{th}}$ power map on the structure sheaf.
For $x \in {{\mathcal{O}}}_X$ and $\lambda \in k$ one has $F^*_X(\lambda
x)=\lambda^px^p=F_k(\lambda)F^*_X(x)$ where $F_k$ is just the Frobenius morphism for $k$. Thus $F_X$ is *not* a morphism of $k$-schemes unless $k={\mathbb{F}}_p$. Setting $X'=X\times_{F_k} k$ we get by the universal property of the fiber product that $F_X$ can be factored through $X'$. $$\xymatrix@=9pt{ {X} \ar[rrd]^{F}\ar@/_.5pc/[rrddd] \ar@/^1pc/[rrrrd]^{F_X} \\
&& {X'} \ar[rr]^{\pi_1}\ar[dd] && {X} \ar[dd] \\ \\
&& {k} \ar[rr]^{F_k} && {k} }$$ This map is called the relative Frobenius and we simply denote it by $F: X
{\xrightarrow}{} X'$. Note that as ${\mathbb{F}}_p$-schemes $X$ and $X'$ are isomorphic. For an ${{\mathcal{O}}}_X$-module ${\mathcal{M}}$ we denote the pushforward under $F$ by $F_*{\mathcal{M}}$.
For every map $\phi: X {\xrightarrow}{} Y$ of noetherian $k$-schemes we get a commutative diagram: $$\xymatrix{ {X} \ar^F[r] \ar_{\phi}[d] &{X'} \ar^{\phi'}[d] \\
{Y} \ar^F[r] &{Y'}
}$$ This follows by construction of the relative Frobenius as the unique solution to a mapping problem and the fact that we have a square like this for the absolute Frobenius. Thus, in particular, the functor $F_*$ commutes with $\phi_*$.
For a $k$-algebra $R$ we adopt the same notation: The (absolute) Frobenius $F_R: R {\xrightarrow}{} R$ is just the $p^{\text{th}}$ power map. Setting $R'=R \otimes_F
k$ we get similarly a relative ($k$-linear) Frobenius map $F: R' {\xrightarrow}{} R$. Again one sees that $R$ and $R'$ are isomorphic as rings, only their $k$-algebra structure is different. By $F_*M$ we denote the $R$-module $M$ viewed as an $R'$-module via the relative Frobenius.
The smooth case
---------------
In this section we will review the construction in the smooth case which we will extend later to the Zariski-complex on normal varieties.
The object of study is the algebraic complex $\Omega^{\bullet}_{X/k}=\Omega^\bullet_X$ (cf. [@EV]). The key observation is that the differential on $F_*\Omega^\bullet_X$ is ${{\mathcal{O}}}_{X'}$-linear by the Leibniz rule: $dF^*(x\otimes1)=dx^p=0$ for all $x\in {{\mathcal{O}}}_X$. This gives the cohomology objects of the de Rham complex ${{\mathcal{H}}}^a(F_*\Omega^\bullet_{X/S})$ the structure of ${{\mathcal{O}}}_{X'}$-modules.
Let $X$ be a noetherian $k$-scheme. There is a unique map of ${{\mathcal{O}}}_{X'}$-modules $$C^{-1}: \Omega^1_{X'}{\longrightarrow}{{\mathcal{H}}}^1(F_*\Omega^\bullet_X)$$ called the *inverse Cartier operator* such that for a local section $x\in {{\mathcal{O}}}_X$ one has $$C^{-1}(d(x\otimes 1)) = x^{p-1}dx \text{ \ in } {{\mathcal{H}}}^1.$$ For all $a \geq 0$ this $C^{-1}$ induces maps $$\bigwedge^a C^{-1}: \Omega^a_{X'} {\longrightarrow}{{\mathcal{H}}}^a(F_*\Omega^\bullet_X).$$
The following result completely describes the cohomology of the complex on a smooth $k$-scheme.
\[smooth:cartier\] Let $X$ be a smooth $k$-scheme. Then the inverse Cartier operator is an isomorphism for all $a \geq 0$. I.e. $${{\mathcal{H}}}^a(F_*\Omega^\bullet_X) \cong \Omega^a_{X'}.$$
\[Cart:DI\] The most striking application of the Cartier isomorphism is as the main ingredient of Deligne and Illusie’s algebraic proof of the Kodaira vanishing theorem. For $X$ smooth and under an additional assumption ($X$ lifts to characteristic $p^2$ and $p \geq \dim X$) they show in [@DI] that the inverse Cartier operator is induced by a map $$\bigoplus \Omega_{X'}^a[-a] {\xrightarrow}{} F_*\Omega^\bullet_X$$ in the derived category of ${{\mathcal{O}}}_{X'}$-modules. Taking hypercohomology one sees that this implies the degeneration of the Hodge to spectral sequence $H^a(\Omega_X^b) \Rightarrow {\mathbb{H}}^{a+b}(\Omega^\bullet_X)$. Such a result implies the Kodaira vanishing theorem as shown by Esnault and Viehweg in [@EVLog]. For details on this circle of ideas we refer the reader to the original article [@DI] and the excellent book [@EV].
Reflexive sheaves
-----------------
In our approach to generalize the Cartier operator reflexive sheaves play a central role. Here we review some properties and for completeness most of the proofs are given, too.
Let $X$ be a normal scheme and ${\mathcal{M}}$ be a ${{\mathcal{O}}}_X$-module. We denote the dual of ${\mathcal{M}}$ by ${\mathcal{M}}^* =
{\mathcal{H}om}_{{{\mathcal{O}}}_X}({\mathcal{M}},{{\mathcal{O}}}_X)$. There is a natural map from ${\mathcal{M}}$ to its double dual ${\mathcal{M}}^{**}$ which sends a section $s \in {\mathcal{M}}$ to the map $f_s$ whose value on the map $\lambda: {\mathcal{M}} {\xrightarrow}{} {{\mathcal{O}}}_X$ is $f_s(\lambda)=\lambda(s)$. The module ${\mathcal{M}}$ is called reflexive if the natural map to its double dual is an isomorphism. The module ${\mathcal{M}}^{**}$ is also called the reflexive hull of ${\mathcal{M}}$. It is reflexive [@EvGr] and universal with respect to this property: every map ${\mathcal{M}} {\xrightarrow}{} {\mathcal{N}}$ with ${\mathcal{N}}$ reflexive factors canonically through the natural map from ${\mathcal{M}}$ to ${\mathcal{M}}^{**}$.
Recall the following local version of the adjointness of $f^*$ and $f_*$ for a map of schemes $f: X {\xrightarrow}{} Y$.
\[localadj\] Let $f: X {\xrightarrow}{} Y$ be a map of schemes, ${\mathcal{F}}$ an ${{\mathcal{O}}}_X$-module and ${\mathcal{G}}$ an ${{\mathcal{O}}}_Y$-module. There is a natural isomorphism $$\Phi: f_* {\mathcal{H}om}_{{{\mathcal{O}}}_X}(f^*{\mathcal{G}},{\mathcal{F}}) {\xrightarrow}{\ \cong\ } {\mathcal{H}om}_{{{\mathcal{O}}}_Y}({\mathcal{G}},f_*{\mathcal{F}}).$$
To see there is a globally defined map let $\phi_U$ for $U\subseteq
Y$ be a local section of the left hand side. Composing the natural map ${\mathcal{G}}|_U {\xrightarrow}{} f_*f^*{\mathcal{G}}|_U=(f_*f^*{\mathcal{G}})|_U$ with $f_*(\phi_U):f_*f^*{\mathcal{G}}|_U {\xrightarrow}{} f_*{\mathcal{F}}|_U$ we get the desired local section of the right hand side. Since no choices were made this defines the global map $\Phi$. An inverse of $\Phi$ can be established similarly using the natural map $f^*f_* {\mathcal{F}} {\xrightarrow}{} {\mathcal{F}}$.
\[ReflExt\] Let $X$ be a normal scheme and let $i:U \hookrightarrow X$ the inclusion of an open set $U$ such that ${\operatorname{codim}}(X,X-U) \geq 2$. If ${\mathcal{M}}$ is a reflexive ${{\mathcal{O}}}_U$-module then $i_*{\mathcal{M}}$ is the unique reflexive ${{\mathcal{O}}}_X$-module which agrees with ${\mathcal{M}}$ on $U$.
First we show that $i_*{\mathcal{M}}$ is reflexive by showing that $i_*$ commutes with dualizing. For this we need the easily verifiable facts that for the given setting $i_* {{\mathcal{O}}}_U = {{\mathcal{O}}}_X$ and $i^*i_*{\mathcal{M}} = (i_* {\mathcal{M}})|_U = {\mathcal{M}}$. Now use the local adjointness Lemma \[localadj\] in the following calculation. $$\begin{split}
(i_*{\mathcal{M}})^* &= {\mathcal{H}om}_{{{\mathcal{O}}}_X}(i_*{\mathcal{M}},{{\mathcal{O}}}_X) =
{\mathcal{H}om}_{{{\mathcal{O}}}_X}(i_*{\mathcal{M}},i_*{{\mathcal{O}}}_U)\\
&= i_* {\mathcal{H}om}_{{{\mathcal{O}}}_U}(i^*i_*{\mathcal{M}},{{\mathcal{O}}}_U) \\
&= i_* {\mathcal{M}}^*
\end{split}$$ Thus $(i_* {\mathcal{M}})^{**} \cong i_*{\mathcal{M}}^{**} \cong i_*{\mathcal{M}}$ which shows that $i_* {\mathcal{M}}$ is reflexive.\
For the uniqueness part let ${\mathcal{N}}$ be a reflexive ${{\mathcal{O}}}_X$-module such that its restriction to $U$ is ${\mathcal{M}}$. A similar application of the adjointness lemma shows that $$\begin{split}
{\mathcal{N}}^* &= {\mathcal{H}om}_{{{\mathcal{O}}}_X}({\mathcal{N}},{{\mathcal{O}}}_X)
= i_*{\mathcal{H}om}_{{{\mathcal{O}}}_U}(i^* {\mathcal{N}},{{\mathcal{O}}}_U)\\
&= i_* ({\mathcal{N}}|_U)^* = i_*{\mathcal{M}}^*
\end{split}$$ and dualizing this equality we get ${\mathcal{N}}^{**} \cong i_* {\mathcal{M}}^{**}$ which shows that ${\mathcal{N}} \cong i_* {\mathcal{M}}$ since both are reflexive.
\[refl:flat\] Let $f:X {\xrightarrow}{} Y$ be a flat morphism of noetherian normal schemes and let ${\mathcal{M}}$ be a reflexive ${{\mathcal{O}}}_Y$-module. Then $f^* {\mathcal{M}}$ is a reflexive ${{\mathcal{O}}}_X$-module.
First we establish a natural map $f^*{\mathcal{M}}^* {\xrightarrow}{} (f^*{\mathcal{M}})^*$. Then we show by a local calculation that this is an isomorphism whenever $f$ is flat. Thus dualizing commutes with $f^*$ and therefore $f^*$ preserves reflexivity.
For the first part notice that quite generally, for any, not necessarily flat, $f$ and ${{\mathcal{O}}}_Y$-modules ${\mathcal{M}}$ and ${\mathcal{N}}$ we get: $$f^*{\mathcal{H}om}_{{{\mathcal{O}}}_Y}({\mathcal{M}},{\mathcal{N}}) {\xrightarrow}{} f^*{\mathcal{H}om}_{{{\mathcal{O}}}_Y}({\mathcal{M}},f_*f^*{\mathcal{N}})
{\xrightarrow}{\ \cong\ } f^*f_*{\mathcal{H}om}_{{{\mathcal{O}}}_X}(f^*{\mathcal{M}},f^*{\mathcal{N}})$$ The first map is composition with the natural map ${\mathcal{N}} {\xrightarrow}{}
f_*f^*{\mathcal{N}}$ and the second map is the local adjointness Lemma \[localadj\]. Composing this with the natural map $f^*f_*{\mathcal{H}om}_{{{\mathcal{O}}}_X}(f^*{\mathcal{M}},f^*{\mathcal{N}}) {\xrightarrow}{}
{\mathcal{H}om}_{{{\mathcal{O}}}_X}(f^*{\mathcal{M}},f^*{\mathcal{N}})$ gives the desired map.
To see that this is an isomorphism when $f$ if flat we reduce to checking on stalks. Thus it remains to show that for a flat map of local noetherian rings $R {\xrightarrow}{} S$ and finitely generated $R$-modules $M$ and $N$ $$S \otimes_R {\operatorname{Hom}}_R(M,N) {\xrightarrow}{\ \cong\ } {\operatorname{Hom}}_S(S \otimes_R M,S \otimes_R N). \tag{$*$} \label{local:eq}$$ which sends $s \otimes f$ to the map $s'\otimes m \mapsto s's \otimes f(m)$ is an isomorphism (this, of course, is the same as the map above). The case $M=R$ is easily verified since both sides are canonically isomorphic to $S \otimes_R
N$ and the above map is then the identity. Since tensor and ${\operatorname{Hom}}$ both commute with finite direct sums (\[local:eq\]) is an isomorphism for f.g. free $R$-modules. Now let $F {\xrightarrow}{} G {\xrightarrow}{} M {\xrightarrow}{} 0$ be a free presentation of $M$. Applying ${\operatorname{Hom}}$ and tensor (exact!) to this presentation in both orders we get a diagram: $$\xymatrix@=15pt{ 0 \ar[r] &{{\operatorname{Hom}}_R(M,N)\otimes S} \ar[r] \ar[d]
&{{\operatorname{Hom}}_R(G,N)\otimes S} \ar[r] \ar[d] &{{\operatorname{Hom}}_R(F,N)\otimes S} \ar[d] \\ 0
\ar[r] &{{\operatorname{Hom}}_S(M \otimes S,N \otimes S)} \ar[r] &{{\operatorname{Hom}}_S(G\otimes S,N \otimes
S)}\ar[r] &{{\operatorname{Hom}}_S(F\otimes S,N \otimes S)} }$$ The vertical maps to the right are isomorphisms since $F$ and $G$ are free. Thus by the 5-Lemma the left map is an isomorphism, too.
\[refl:fflat\] Let $f:X {\xrightarrow}{} Y$ be faithfully flat and ${\mathcal{M}}$ be an ${{\mathcal{O}}}_X$-module. Then ${\mathcal{M}}$ is reflexive if and only if $f^*{\mathcal{M}}$ is reflexive.
Cartier operator for normal schemes
===================================
For schemes that are not smooth, the complex is not very well behaved and it seems convenient to consider the Zariski-complex instead. In order to define the sheaves of Zariski differentials we assume that the variety $X$ is normal.
Zariski-complex
---------------
Let $X$ be a normal $k$-scheme and let $$i: U \hookrightarrow X$$ be the inclusion of the smooth locus. The *Zariski-complex* is the pushforward of the complex on $U$ and is denoted by ${{\widetilde}{\Omega}}^\bullet_X=i_*\Omega^\bullet_U$.
The objects of the Zariski-complex are just ${{\widetilde}{\Omega}}^a_X=i_*\Omega^a_U$ and are called the Zariski sheaves of differential $a$-forms. Normality of $X$ guarantees that the Zariski sheaves are reflexive by Lemma \[ReflExt\]. Equivalently, one can define ${{\widetilde}{\Omega}}^a_X$ as the unique reflexive ${{\mathcal{O}}}_X$-module which agrees with $\Omega^a_X$ on the smooth locus (i.e ${{\widetilde}{\Omega}}^a_X$ is the double dual or reflexive hull of $\Omega^a_X$ and the natural map $\Omega^a_X {\xrightarrow}{} i_*i^*\Omega^a_X = {{\widetilde}{\Omega}}^a_X$ is identified with the double dualizing map). With this in mind one can take any open smooth $U$ with codimension $X-U$ greater than or equal to two in the above definition. The differential on ${{\widetilde}{\Omega}}^\bullet_X$ is just the pushforward $i_*(d)$ of the ordinary differential on $\Omega^{\bullet}_U$. The Zariski sheaf of 1-forms ${{\widetilde}{\Omega}}^1_X$ satisfies a universal property similar to the universal property for the sheaf of Kähler differentials.
Every derivation $\delta:{{\mathcal{O}}}_X {\xrightarrow}{} {\mathcal{M}}$ to a reflexive ${{\mathcal{O}}}_X$-module ${\mathcal{M}}$ factors uniquely through $i_*d:{{\mathcal{O}}}_X {\xrightarrow}{} {{\widetilde}{\Omega}}^1_X$.
Observe that the derivation $i_*d$ is the compostition of the universal derivation $d:{{\mathcal{O}}}_X {\xrightarrow}{} \Omega^1_X$ with the double dualizing map $\Omega^1_X
{\xrightarrow}{} {{\widetilde}{\Omega}}^1_X$. By the universal properties of these maps the proposition is immediate.
Note that, if $X={\operatorname{Spec}}(R)$ is affine, the Zariski sheaf is the sheaf associated to the finitely generated $R$-module $\Gamma(X,{{\widetilde}{\Omega}}^a_X)=\Gamma(U,\Omega^a_U)$. The Zariski-complex shares many of the good properties of the complex on a smooth variety. For example, if $X$ is $n$-dimensional, the next proposition shows that the perfect pairing between $\Omega^a_U$ and $\Omega^{n-a}_U$ translates into a perfect pairing for the Zariski sheaves.
Let $X$ be a n-dimensional normal $k$-scheme. There is a natural isomorphism ${{\widetilde}{\Omega}}^a_X \rightarrow {\mathcal{H}om}_{{{\mathcal{O}}}_X}({{\widetilde}{\Omega}}^{n-a}_X,{{\widetilde}{\Omega}}^n_X)$.
Applying $i_*$ to the duality on the smooth locus and using the local version of the adjointness of $i_*$ and $i^*$ we get $$\begin{split}
{{\widetilde}{\Omega}}^a_X &= i_* \Omega^a_U = i_* {\mathcal{H}om}_{{{\mathcal{O}}}_U}(\Omega^{n-a}_U, \Omega^n_U) = i_* {\mathcal{H}om}_{{{\mathcal{O}}}_U}(i^*i_*\Omega^{n-a}_U, \Omega^n_U) \\
&={\mathcal{H}om}_{{{\mathcal{O}}}_X}(i_*\Omega^{n-a}_U,i_*\Omega^n_U)={\mathcal{H}om}_{{{\mathcal{O}}}_X}({{\widetilde}{\Omega}}^{n-a}_X, {{\widetilde}{\Omega}}^n_X)
\end{split}$$
### Stability under étale morphisms
A morphism of sheaves $f: Y {\xrightarrow}{} X$ is called *étale* if it is flat and unramified. Just as the sheaves of Kähler differentials are preserved under étale maps so are the Zariski sheaves.
\[Zariskietale\] Let $X$ be normal and $f: Y {\xrightarrow}{} X$ étale. Then $$f^*{{\widetilde}{\Omega}}^a_X \cong {{\widetilde}{\Omega}}^a_Y.$$
Note that by [@Milne Prop. 3.17(b)] $Y$ is normal thus the Zariski sheaves are defined on $Y$. Now $f^*{{\widetilde}{\Omega}}^a_X=f^*({\Omega^a_X}^{**})=(f^*\Omega^a_X)^{**}={{\widetilde}{\Omega}}^a_Y$ using Lemma \[refl:flat\] and $f^*\Omega^a_X \cong \Omega^a_Y$ by étaleness of $f$.
Another important observation is that the Frobenius commutes with étale pull back, i.e. $F_*f^*{\mathcal{M}} \cong f^*F_*{\mathcal{M}}$ for all ${{\mathcal{O}}}_X$-modules ${\mathcal{M}}$.
Let $X$ and $Y$ be $k$-varieties and $f: Y {\xrightarrow}{} X$ an étale morphism. Then $$\xymatrix{ Y \ar^F[r] \ar_f[d] & {Y'} \ar^f[d] \\
X \ar^F[r] & {X'}
}$$ is a fiber square, i.e. $Y \cong {Y'} \times_{X'} X$.
First assume that $f$ is finite and étale. Then by the above diagram and the universal property of the fiber product we have a natural map $Y {\xrightarrow}{} Y' \times_{X'} X $. To check this map is an isomorphism one reduces to the case where $X$ (and therefore $Y$) is affine. Since $Y' \times_{X'} X \cong Y \times_{F_X} X$ one further reduces to the case of the absolute Frobenius. Thus it remains to show that for a finite étale morphism of $k$-algebras $R {\xrightarrow}{} S$ we have $(F_S)_* S
\cong S \otimes_{F_R} (F_R)_*R$. This is done in [@HH90 page 50].\
In general an étale map can be factored into a finite ètale map followed by an open immersion: $f:Y {\xrightarrow}{} U {\xrightarrow}{} X$. It remains to check the statement for the open immersion $U {\xrightarrow}{} X$. As before one reduces to the case of the absolute Frobenius. As topological spaces $U \times_{F_X} X$ and $U$ are the same and a simple local calculation shows that they are isomorphic as $k$-schemes.
With this at hand we can apply [@Hartshorne Proposition III.9.3] and get:
\[etaleFrob\] Let ${\mathcal{M}}$ be an ${{\mathcal{O}}}_X$-module and $f: Y {\xrightarrow}{} X$ étale. Then $f^*F_*
{\mathcal{M}} \cong F_*f^* {\mathcal{M}}$.
Since similar in spirit to the above and used later we note here a Lemma which shows that $F_*$ commutes with completion. For this we consider the local case.
\[completeFrob\] Let $(R,m)$ be a local $k$-algebra, essentially of finite type over $k$. Then $\hat{R} \cong R \otimes_{R'} \hat{R}'$. Here, $\hat{R}$ denotes the $m$-adic completion of $R$.
As in the previous Lemma $R \otimes_{R'} \hat{R}' \cong R \otimes_{F_R}
\hat{R}$ and we reduce to the case of the absolute Frobenius. Now it follows from the definition of completion: $$R \otimes_{F_R} \hat{R} = R \otimes_{F_R} \varprojlim \frac{R}{m^i} \cong
\varprojlim \frac{R}{R \otimes_{F_R}m^i} = \varprojlim \frac{R}{(m^i)^{[p]}R} =
\hat{R}$$ In the last equality we used that the sequence $(m^i)^{[p]}$ is cofinal with $m^i$ and therefore they both have the same limit.
An easy calculation with the equation of the last Lemma shows that for any $R$-module $M$ $F_*$ commutes with completion, i.e. $F_*\hat{M} \cong \hat{R}' \otimes_{R'} F_*M$.
Cartier operator on Zariski differentials
-----------------------------------------
The Cartier isomorphism on the smooth scheme $U'$ gives after applying $i_*$ an isomorphism $$i_*C^{-1}:\ {{\widetilde}{\Omega}}^a_{X'} \xrightarrow{\
\cong\ \ } i_* {{\mathcal{H}}}^a(F_*\Omega^{\bullet}_U).$$ To obtain an analog of the Cartier operator for the Zariski sheaves it remains to construct natural maps $$\phi_a:\ {{\mathcal{H}}}^a(F_*{{\widetilde}{\Omega}}^{\bullet}_X) {\longrightarrow}i_* {{\mathcal{H}}}^a(F_*\Omega^{\bullet}_U)$$ for all $a \geq 0$. This is obtained with the help of the following general Lemma.
Let ${\mathcal{A}}^\bullet$ be a complex in an abelian category and $G$ a left exact functor of abelian categories. One has natural maps $$\phi_a: \ {{\mathcal{H}}}^a(G{\mathcal{A}}^\bullet) {\xrightarrow}{} G{{\mathcal{H}}}^a({\mathcal{A}}^\bullet)$$ for all $a \geq 0$.
To see this denote by ${\mathcal{Z}}^a$, ${\mathcal{B}}^a$ the cycles and boundaries of ${\mathcal{A}}^\bullet$ (resp. ${\mathcal{Z}}^a_G$, ${\mathcal{B}}^a_G$ for $G{\mathcal{A}}^\bullet$). By the left exactness of $G$ we get ${\mathcal{Z}}^a_G \cong
G{\mathcal{Z}}^a$. Applying $G$ to the sequence $0 {\xrightarrow}{} {\mathcal{Z}}^a {\xrightarrow}{} {\mathcal{A}}^a
{\xrightarrow}{} {\mathcal{B}}^{a+1} {\xrightarrow}{} 0$ and arranging with the equivalent sequence for $G{\mathcal{A}}^\bullet$ in the diagram $$\xymatrix{
{0} \ar[r] &{{\mathcal{Z}}^a_G} \ar@{=}[d] \ar[r] &{G{\mathcal{A}}^a} \ar@{=}[d] \ar[r] &{{\mathcal{B}}^{a+1}_G} \ar@{_(-->}[d]\ar[r] & {0} \\
{0} \ar[r] &{G{\mathcal{Z}}^a} \ar[r] &{G{\mathcal{A}}^a} \ar[r] &{G{\mathcal{B}}^{a+1}}
}$$ we see that (Snake Lemma) the vertical arrow to the right is an injection. Now, using the equivalent diagram for the short exact sequence defining cohomology $$\xymatrix{
{0} \ar[r] &{{\mathcal{B}}^a_G} \ar@{_(->}[d] \ar[r] &{{\mathcal{Z}}^a_G} \ar@{=}[d] \ar[r] &{{{\mathcal{H}}}^a(G{\mathcal{A}}^\bullet)} \ar@{-->}[d]^{\phi_a} \ar[r] &{0} \\
{0} \ar[r] &{G{\mathcal{B}}^a} \ar[r] &{G{\mathcal{Z}}^a} \ar[r] &{G{{\mathcal{H}}}^a({\mathcal{A}}^\bullet)}
}$$ together with the Snake Lemma gives the dotted vertical arrow to the right. Furthermore, $\phi_a$ is injective if and only if ${\mathcal{B}}^a_G {\xrightarrow}{}
G{\mathcal{B}}^a$ is an isomorphism and surjective if and only if the map $G{\mathcal{Z}}^a
{\xrightarrow}{} G{{\mathcal{H}}}^a({\mathcal{A}}^\bullet)$ is surjective.
Since $i_*$ is left exact we can apply this construction to the complex $F_*\Omega^\bullet_U$ and therefore obtain natural maps $\phi_a :
{{\mathcal{H}}}^a(i_*F_*\Omega^{\bullet}_U) {\longrightarrow}i_* {{\mathcal{H}}}^a(F_*\Omega^{\bullet}_U)$ for all $a \geq 0$. Since $F_*$ commutes with $i_*$ the first term is just ${{\mathcal{H}}}^a(F_*{{\widetilde}{\Omega}}^{\bullet}_X)$
By the Cartier isomorphism on the smooth locus ${{\mathcal{H}}}^a(F_*\Omega_U^\bullet)$ is isomorphic to $\Omega^a_{U'}$ which is a locally free and therefore reflexive ${{\mathcal{O}}}_{U'}$-module. By Lemma \[ReflExt\] $i_*{{\mathcal{H}}}^a(F_*\Omega_U^\bullet)$ is a reflexive ${{\mathcal{O}}}_{X'}$-module. Thus, by the universal property of the reflexive hull, the map $\phi_a$ factors through the natural map from ${{\mathcal{H}}}^a(F_*{{\widetilde}{\Omega}}_X^\bullet)$ to its double dual: $$\xymatrix{
{{{\mathcal{H}}}^a(F_*{{\widetilde}{\Omega}}_X^\bullet)} \ar^{\phi_a}[r] \ar[d]_{(\ )^{**}}
&{i_*{{\mathcal{H}}}^a(F_*\Omega_U^\bullet)} \\
{{{\mathcal{H}}}^a(F_*{{\widetilde}{\Omega}}_X^\bullet)^{**}} \ar_{\cong}[ru]
}$$ By the uniqueness part of Lemma \[ReflExt\] the diagonal map has to be an isomorphism since $\phi_a$ is an isomorphism on $U'$. If we use this natural isomorphism to identify $i_*{{\mathcal{H}}}^a(F_*\Omega_U^\bullet)$ with the double dual of ${{\mathcal{H}}}^a(F_*{{\widetilde}{\Omega}}_X^\bullet)$ the map $\phi_a$ becomes nothing but double dualizing. Composing $\phi_a$ with the inverse of $i_*C^{-1}$ we get:
For all $a \geq 0$ the Cartier isomorphism on $U$ induces a natural ${{\mathcal{O}}}_{X'}$-linear map $${\widetilde}{C}:\ {{\mathcal{H}}}^a(F_*{{\widetilde}{\Omega}}^{\bullet}_X) {\longrightarrow}{{\widetilde}{\Omega}}^a_{X'}.$$
By construction, ${\widetilde}{C}$ is an isomorphism if and only if the map $\phi_a$ is. This in turn is the case if and only if ${{\mathcal{H}}}^a(F_*{{\widetilde}{\Omega}}^\bullet_X)$ is reflexive. This observation is key for the proof of the next Proposition.
\[etaleCartier\] Let $f: Y {\xrightarrow}{} X$ étale. If the Cartier operator ${\widetilde}{C}_X$ on $X$ is an isomorphism then so is ${\widetilde}{C}_Y$ on $Y$. If in addition $f$ is finite, then the reverse implication also holds.
With the above observation we only have to show that if ${{\mathcal{H}}}^a(F_*{{\widetilde}{\Omega}}^\bullet_X)$ is reflexive then so is ${{\mathcal{H}}}^a(F_*{{\widetilde}{\Omega}}^\bullet_Y)$. Since an étale map is flat we can apply Lemma \[refl:flat\] to see that the reflexivity of ${{\mathcal{H}}}^a(F_*{{\widetilde}{\Omega}}^\bullet_X)$ implies that $f^*{{\mathcal{H}}}^a(F_*{{\widetilde}{\Omega}}^\bullet_X)$ is reflexive. By flatness $f^*$ commutes with taking cohomology and thus $$f^*{{\mathcal{H}}}^a(F_*{{\widetilde}{\Omega}}^\bullet_X)={{\mathcal{H}}}^a(f^*F_*{{\widetilde}{\Omega}}^\bullet_X).$$ By Corollary \[etaleFrob\] $f^*F_* {{\widetilde}{\Omega}}^a_X = F_*f^*{{\widetilde}{\Omega}}^a_X$ which by \[Zariskietale\] is just $F_* {{\widetilde}{\Omega}}^a_Y$.\
If $f$ is also finite then it is faithfully flat and an application of Corollary \[refl:fflat\] yields the reverse implication.
Cartier isomorphism for toric varieties
=======================================
In this section we give the proof that the Cartier map is an isomorphism for affine toric varieties over $k$. The main ingredient is a description of the Zariski-complex due to Danilov [@Danilov78]. An interesting aspect of the given proof is that it is analogous to Danilov’s proof of a Poincaré type lemma for toric varieties which says that the Zariski-complex is a resolution of the constant sheaf $k$. Thus, as in the smooth case, the Cartier isomorphism can be viewed as a characteristic $p$ analog of the Poincaré lemma, both determining the cohomology of the complex. Furthermore, using the results of the previous section on the stability of the Cartier isomorphism under étale morphisms we conclude that also for toroidal varieties the Cartier operator is an isomorphism.
Affine toric varieties
----------------------
First we recall some notation from toric geometry, for a more detailed introduction to toric geometry see [@FultonToric] or [@Danilov78]. Let $N \cong {\mathbb{Z}}^n$ be a lattice, $M=N^*$ the dual lattice and we denote $M
\otimes_{{\mathbb{Z}}} {\mathbb{Q}}$ by $M_{\mathbb{Q}}$. The identification of $N$ with ${\mathbb{Z}}^n$ allows us to consider the elements of $N$ as $n$-tuples of integers, similarly for $M$.
By a cone in $N$ we mean a subset $\sigma \subseteq N_{{\mathbb{Q}}}$ of the form $\sigma=\{r_1v_1+ \dotsb +r_sv_s|r_i \geq 0\}$ for some $v_i$ in $N$. The dual cone of $\sigma$ in $M$ is $\sigma^{\vee}=\{u \in M_{{\mathbb{Q}}}| (u,v) \geq 0, \text{
for all } v\in \sigma\}$ where $(\ ,\ )$ is induced from the natural pairing between the dual lattices $M$ and $N$. A face of $\sigma$ is any set $\sigma \cap u^{\bot}$ for some $u \in \sigma^{\vee}$ and is a cone in $N$. For cones $\tau$ and $\tau'$ in $M_{{\mathbb{Q}}}$ $(\tau-\tau')$ denotes the set of all $t-t'$ for $t \in \tau$ and $t' \in \tau'$.
Let $\sigma^{\vee}$ be a cone in $M_{{\mathbb{Q}}}$. Let $k$ be a perfect field and let $A=k[\sigma^{\vee} \cap M]$ be the affine semigroup ring corresponding to $\sigma$, i.e. $A$ is the $k$-subalgebra of $k[x_1,x^{-1}_1, \dots
,x_n,x_n^{-1}]$ generated by the monomials $x^m=x_1^{m_1}\cdot \ldots \cdot
x_n^{m_n}$ for $m \in \sigma^{\vee}$. $X=X_\sigma = {\operatorname{Spec}}(A)$ denotes the corresponding affine toric variety. Note that $A$ carries a natural $M$-grading where the generator $x^m$ corresponding to $m \in \sigma^{\vee}
\subseteq M$ of $A$ is given degree $m$. As is well known $X$ is normal [@FultonToric] and we have the notion of Zariski differentials defined above. As before we denote by $U$ its smooth locus. Since $X$ is affine ${{\widetilde}{\Omega}}^a_X$ is the sheaf associated to some $A$-module, which we will call ${{\widetilde}{\Omega}}^a_A$.
### Danilov’s description
There is a nice description of the modules ${{\widetilde}{\Omega}}^a_A$ due to Danilov [@Danilov78] which we will give next.
Let $V$ denote the vector space $M \otimes_{{\mathbb{Z}}} k$. For each codimension one face $\tau$ of $\sigma^{\vee}$ define a subspace $V_{\tau} \subset V$. $$V_{\tau}=(M \cap (\tau-\tau))\otimes_{{\mathbb{Z}}}k$$ For every $m \in \sigma^{\vee}$ we define a subspace $V_m \subseteq V$: $$V_m = \bigcap_{\tau \ni m} V_{\tau}$$ where $\tau$ is ranging over the codimension one faces of $\sigma^{\vee}$ containing $m$.
\[omegaA\] ${{\widetilde}{\Omega}}^a_A$ is isomorphic to the graded $A$-submodule $V^a_\sigma$ of the $M$-graded $A$-module $\bigwedge^a V \otimes_k A$ with degree $m \in M$ piece $$V^a_\sigma(m)=\bigwedge^a V_m \cdot x^m .$$ The isomorphism is given by the following degree preserving map of graded $A$-modules $$\xymatrix@R=0pt{
{{f_a:\ \ \ V_\sigma^a}} \ar[r] &{{{\widetilde}{\Omega}}^a_A} \\
{\qquad \qquad \scriptstyle (m_1\otimes1\wedge \dots \wedge m_a\otimes 1) \cdot x^m}
\ar@{|->}[r] &{\scriptstyle x^{m-m_1\dots-m_a}dx^{m_1}\wedge \dots \wedge
dx^{m_a}}}$$
For the proof see [@Danilov78 p. 110]. With this identification of ${{\widetilde}{\Omega}}_A^a$ with $V_\sigma^a$ the differential $d:{{\widetilde}{\Omega}}^a_A {\xrightarrow}{}
{{\widetilde}{\Omega}}^{i+1}_A$ on the piece of degree $m$ is given by wedging with $m
\otimes 1$. Thus the degree $m$ part of the complex ${{\widetilde}{\Omega}}^{\bullet}_X$ is just $$\xymatrix@C=.5cm{
0 \ar[r] & {\bigwedge^0 V_m} \ar[rr]^{m\otimes 1 \cdot} && {\bigwedge^1 V_m} \ar[rr]^{m\otimes 1 \wedge} && {\bigwedge^2 V_m} \ar[rr]^{m\otimes 1 \wedge} && {\hspace{12pt}\cdots\hspace{12pt}}}$$ The key observation is that this complex is exact whenever $m \otimes 1$ is a nonzero element of the vector space $V_m$.
If the characteristic of $k$ is zero this is the case if and only if $m \neq 0$. Thus the only degree on which the complex is not exact is degree $0 \in M$. Assuming that the cone $\sigma^\vee$ does not contain any linear subspace we see that $V_0=0$ and therefore in degree zero we just get $\bigwedge^0 V_0=k$. This was Danilovs proof of the Poincaré lemma for the Zariski-complex on an affine toric variety:
If the characteristic of $k$ is zero and $\sigma^{\vee}$ is a cone with vertex (i.e. $\sigma^{\vee}$ does not contain a linear subspace) then $$\xymatrix@C=0.5cm{ 0 \ar[r] & {k} \ar[r]^{} & {{{\widetilde}{\Omega}}^{\bullet}_A}}$$ is exact, i.e. ${{\widetilde}{\Omega}}^{\bullet}_A$ is a resolution of $k$.
Back to the case when the characteristic of $k$ is greater than zero. Then we get the following analog of \[smooth:cartier\] for affine toric varieties.
\[toriccartier\] If $k$ has characteristic $p > 0$ then the map $$\phi: \bigoplus {{\widetilde}{\Omega}}^{a}_{A'}[-a] {\longrightarrow}F_*{{\widetilde}{\Omega}}^{\bullet}_A$$ sending $\ (m'\otimes 1) \cdot x^m\ $ to $\ (m'\otimes 1) \cdot x^{pm}\ $ is a split injection and induces an isomorphism $C^{-1}: {{\widetilde}{\Omega}}^a_{A'} {\xrightarrow}{}
{{\mathcal{H}}}^a(F_*{{\widetilde}{\Omega}}^{\bullet}_A)$.
As already in the statement of this Theorem we make use of the identification in \[omegaA\] without further mentioning.
By definition $\phi$ is additive. Noting that $A' = A \otimes_{F_k} k$ and ${{\widetilde}{\Omega}}^a_{A'} = {{\widetilde}{\Omega}}^a_A \otimes_{F_k} k$ it is easily checked that $\phi$ is also $A'$-linear. Since $m \otimes 1 = 0$ if and only if $m \in pM$ we see that the cohomology lives exclusively in degrees $pm$ for $m \in M$; in these degrees the differential (wedging with $pm \otimes 1 = 0$) is the zero map. $\phi$ maps the graded piece of degree $m$ isomorphically to the graded piece of degree $pm$. Thus the image of $\phi$ is exactly the graded pieces of degree $pm$ for $m \in M$ which is the cohomology of $F_*{{\widetilde}{\Omega}}^{\bullet}_A$. Thus ${{\mathcal{H}}}^a(F_*{{\widetilde}{\Omega}}^{\bullet}_A)$ is isomorphic to the graded $A'$-submodule of degree $pM$ pieces of $F_*{{\widetilde}{\Omega}}^a_A$ and therefore $\phi$ is also split. Using \[omegaA\] one easily verifies that $d \circ \phi=0$, i.e $\phi$ is a map of complexes.
It remains to show that the above defined map actually is an inverse of the Cartier operator. Using the description of \[omegaA\] we see that for all $m
\in \sigma^{\vee}$ $\ \phi(dx^m \otimes 1)=x^{(p-1)m}dx^m$ which is just the defining property of the inverse Cartier map from which ${\widetilde}{C}$ was constructed.
Since checking that the Cartier operator ${\widetilde}{C}$ is an isomorphism is a local matter we get as an immediate Corollary:
Let $X$ be a toric variety over a perfect field $k$. Then the Cartier operator $${\widetilde}{C}:\ {{\mathcal{H}}}^a(F_*{{\widetilde}{\Omega}}^{\bullet}_X) {\longrightarrow}{{\widetilde}{\Omega}}^a_{X'}$$ is an isomorphism.
Theorem \[toriccartier\] for *affine* toric varieties is much stronger than what we just established for the toric case. Not only is the Cartier map an isomorphism but also it is induced by a split injection ${{\widetilde}{\Omega}}^a_{X'}
{\xrightarrow}{} F_* {{\widetilde}{\Omega}}^a_X$. This stronger statement can still be achieved for toric varieties in general (not necessarily affine). The key point is that one has to ensure that the map $\phi$ defined locally on the affine toric pieces as above patches to give a global map of sheaves.
For simplicity of notation we assume in the following discussion that $k={\mathbb{F}}_p$. Notice that the description of ${{\widetilde}{\Omega}}^a_X$ in Lemma \[omegaA\] localizes. I.e. if $\tau$ is a face of $\sigma$ then $X_\tau$ is an open subset of $X_\sigma$ and we have an inclusion ${{\widetilde}{\Omega}}^a_{X_\sigma} \subseteq
{{\widetilde}{\Omega}}^a_{X_\tau}$. This corresponds to the inclusion $V^a_\sigma \subseteq
V^a_\tau$. The reason for this is that the map $V^a_\sigma {\longrightarrow}{{\widetilde}{\Omega}}^a_{X_\sigma}$ is induced from the map $\bigwedge^a V {\xrightarrow}{}
{{\widetilde}{\Omega}}^a_{T_N}$ for *any* cone $\sigma$ in $N$, here $T_N$ is the $n$-dimensional torus.
Now let $X$ be the toric variety associated to a fan $\Sigma$ in $N$ (cf. [@FultonToric]). The torus $T_N$ is an open subset of $X$ and also of each affine toric piece $X_\sigma$ of $X$ for $\sigma \in \Sigma$. Thus ${{\widetilde}{\Omega}}^a_{X_\sigma} \subseteq {{\widetilde}{\Omega}}^a_{T_N}$ and the map $\phi_\sigma:{{\widetilde}{\Omega}}^a_{X_\sigma} {\xrightarrow}{} F_*{{\widetilde}{\Omega}}^a_{X_\sigma}$ is induced from the corresponding map for the torus $T_N$ ($T_N$ is the toric variety associated the cone $\{0\} \subseteq N$). This shows that the maps $\phi_\sigma$ agree on the intersections of their domains and thus give rise to a globally defined map $\phi$ as required. We just proved (well, sketched a proof of) the following Proposition.
Let $X$ be a toric variety over a perfect field $k$. Then the Cartier operator is an isomorphism and is induced by a split injection $$0 {\xrightarrow}{} \bigoplus_a {{\widetilde}{\Omega}}^a_{X'}[-a] {\xrightarrow}{} F_*{{\widetilde}{\Omega}}^\bullet_X.$$
The existence of a split injection inducing the Cartier operator was already shown by Buch, Thomsen, Lauritzen and Mehta in [@Anders] using the existence of natural liftings of characteristic $p$ toric varieties to toric varieties over ${\mathbb{Z}}/p^2{\mathbb{Z}}$, along with liftings of Frobenius. Their methods though do not yield any information on whether the Cartier operator is an isomorphism, but are sufficient to imply Bott vanishing and the degeneration of the Hodge to spectral sequence for projective toric varieties.
Toroidal varieties
------------------
With the help of Proposition \[etaleCartier\] we can extend the results from the previous section to a important class of varieties: toroidal embeddings.
A normal scheme $X$ locally of finite type over a field $k$ will be called *weakly toroidal* if for each point $x \in X$ there is a neighborhood $U(x)$ of $x$ and a diagram $$\xymatrix{ & {Y(x)} \ar_{f}[ld] \ar^g[rd] & \\
{Z(x)} & & {U(x)}}$$ where $Z(x)$ is an affine toric variety, $f$ is étale and $g$ is finite and étale.
This class of varieties includes the class of toroidal embeddings as defined in [@TE]. We have the following Theorem.
Let $X$ be weakly toroidal. Then the Cartier operator ${\widetilde}{C}$ on $X$ is an isomorphism.
To see that the Cartier operator on $X$ is an isomorphism is a local issue so we can assume that there is a diagram $$\xymatrix{ & {Y} \ar_{f}[ld] \ar^g[rd] & \\
{Z} & & {X}}$$ with $Z$ affine toric, both maps étale and $g$ also finite. On the toric variety $Z$ the Cartier operator is an isomorphism and thus by \[etaleCartier\] it is an isomorphism on $Y$, and again by \[etaleCartier\], it is an isomorphism on $X$.
An Obstruction to surjectivity
==============================
The last section shows that the Cartier operator is an isomorphism in the case of toroidal embeddings and it is natural to ask what is the exact class of varieties for which the Cartier operator is an isomorphism. It seems too much to hope that it would be an isomorphism for all normal varieties. And in fact there is a necessary and sufficient condition for ${\widetilde}{C}$ to be a surjection at the top level, i.e. for ${\widetilde}{C}: {{\mathcal{H}}}^d(F_*{{\widetilde}{\Omega}}^{\bullet}_X) {\xrightarrow}{}
{{\widetilde}{\Omega}}^d_X$ where $d$ is the dimension of $X$.
To simplify the notation we assume $k = {\mathbb{F}}_p$. Let $(R,m)$ be a normal Cohen-Macaulay local domain of dimension $d$, essentially of finite type over $k$. Then $R$ is called $F$-injective if the Frobenius $F: R {\xrightarrow}{} F_*R$ induces an injection on local cohomology $H^d_m(R) {\xrightarrow}{} H^d_m(F_*R)$. To relate this to the Cartier operator on $X={\operatorname{Spec}}R$ we denote the global sections of ${{\widetilde}{\Omega}}^d_X$ by $\omega_R$ which by normality is a canonical module for $R$. Since the Cartier operator $\Gamma(X,{{\mathcal{H}}}^d(F_*{{\widetilde}{\Omega}}^\bullet_X)) {\xrightarrow}{}
\omega_R$ is surjective if and only if its composition with the natural surjection $F_*\omega_R {\xrightarrow}{} \Gamma(X,{{\mathcal{H}}}^d(F_*{{\widetilde}{\Omega}}^\bullet_X)) {\xrightarrow}{} \omega_R$ is surjective we consider now the map $C: F_*\omega_R {\xrightarrow}{} \omega_R$ to which we also refer to as the Cartier map. This map is surjective iff its $m$-adic completion $\hat{C}: \hat{R} \otimes F_*\omega_R {\xrightarrow}{} \hat{R} \otimes
\omega_R$ is a surjective map of $\hat{R}$-modules by the faithfully flatness of completion. By the remark after Lemma \[completeFrob\] $F_*$ commutes with completion and by [@BrunsHerzog 3.3.5] $\ \hat{R} \otimes \omega_R$ is a canonical module for $\hat{R}$. Thus $\hat{R} \otimes \omega_R \cong
\omega_{\hat{R}}$ and $\hat{R} \otimes F_*\omega_R \cong F_*\omega_{\hat{R}}$. Let $E$ denote the injective hull of the residue field of $\hat{R}$ and take the Matlis dual of the completed Cartier map $\hat{C}$ to get $${\operatorname{Hom}}_{\hat{R}}(\omega_{\hat{R}},E) {\xrightarrow}{} {\operatorname{Hom}}_{\hat{R}}(F_*\omega_{\hat{R}},E).$$ Note that, since $F$ is a finite map $F_*\omega_{\hat{R}} \cong
{\operatorname{Hom}}_{\hat{R}}(F_*\hat{R},\omega_{\hat{R}})$ (cf. [@BrunsHerzog 3.3.7]). This together with the local duality theorem [@BrunsHerzog 3.5.8] shows that the last map is just $$H^d_{\hat{m}}(\hat{R}) {\xrightarrow}{} H^d_{\hat{m}}(F_*\hat{R})$$ where we can leave out the completion and get the map induced by the Frobenius $F: H^d_m(R) {\xrightarrow}{} H^d_m(F_*R)$. Summarizing we get the following Proposition.
Let $(R,m)$ be a local Cohen-Macaulay domain essentially of finite type over $k$. Then the Cartier map $C: F_*\omega_R {\xrightarrow}{} \omega_R$ is surjective if and only if $R$ is $F$-injective.
One implication of this Proposition was already noticed by Mehta and Shrinivas in [@MehtaSr] where they use the Cartier operator on the top spot to obtain information about the singularities of the variety.
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[^1]: To appear in Journal of Algebra
|
---
abstract: 'We present the first results from an optical reverberation mapping campaign executed in 2014, targeting the active galactic nuclei (AGN) MCG+08-11-011, NGC2617, NGC4051, 3C382, and Mrk374. Our targets have diverse and interesting observational properties, including a “changing look” AGN and a broad-line radio galaxy. Based on continuum-H$\beta$ lags, we measure black hole masses for all five targets. We also obtain H$\gamma$ and [He[ii]{}$\lambda 4686$]{} lags for all objects except 3C382. The [He[ii]{}$\lambda 4686$]{} lags indicate radial stratification of the BLR, and the masses derived from different emission lines are in general agreement. The relative responsivities of these lines are also in qualitative agreement with photoionization models. These spectra have extremely high signal-to-noise ratios (100–300 per pixel) and there are excellent prospects for obtaining velocity-resolved reverberation signatures.'
author:
- 'M. M. Fausnaugh, C. J. Grier, M. C. Bentz, K. D. Denney, G. De Rosa, B. M. Peterson, C. S. Kochanek, R. W. Pogge, S. M. Adams, A. J. Barth, Thomas G. Beatty, A. Bhattacharjee, G. A. Borman, T. A. Boroson, M. C. Bottorff, Jacob E. Brown, Jonathan S. Brown, M. S. Brotherton, C. T. Coker, S. M. Crawford, K.V. Croxall, Sarah Eftekharzadeh, Michael Eracleous, M. D. Joner, C. B. Henderson, T. W.-S. Holoien, Keith Horne, T. Hutchison, Shai Kaspi, S. Kim, Anthea L. King, Miao Li, Cassandra Lochhaas, Zhiyuan Ma, F. MacInnis, E. R. Manne-Nicholas, M. Mason, Carmen Montuori, Ana Mosquera, Dale Mudd, R. Musso, S. V. Nazarov, M. L. Nguyen, D. N. Okhmat, Christopher A. Onken, B. Ou-Yang, A. Pancoast, L. Pei, Matthew T. Penny, Radosław Poleski, Stephen Rafter, E. Romero-Colmenero, Jessie Runnoe, David J. Sand, Jaderson S. Schimoia, S. G. Sergeev, B. J. Shappee, Gregory V. Simonian, Garrett Somers, M. Spencer, D. A. Starkey, Daniel J. Stevens, Jamie Tayar, T. Treu, Stefano Valenti, J. Van Saders, S. Villanueva Jr., C. Villforth, Yaniv Weiss, H. Winkler, W. Zhu'
bibliography:
- '../../refs.bib'
title: Reverberation Mapping of Optical Emission Lines in Five Active Galaxies
---
Introduction
============
Understanding the interior structure of active galactic nuclei (AGN) has been a major goal of extragalactic astrophysics since their identification as cosmological objects [@Schmidt1963]. The current schematic structure of the central part of an AGN includes three main components: an accretion disk around a super-massive black hole (SMBH), a broad line region (BLR), and an obscuring structure at some distance beyond the BLR. This basic picture accounts for the large luminosities and prominent recombination/excitation lines observed in Seyfert galaxy and quasar spectra [@Burbidge1967; @Weedman1977], as well as the dichotomy between Type 1 and Type 2 objects [@Lawrence1991; @Antonucci1993].
While this model has qualitatively explained the observational properties of AGN, the details of AGN interior structure remain poorly understood. The basic physics of the accretion disk are probably linked to the magnetorotational instability [@Balbus1998], but it has not been possible to fully simulate an accretion disk and compare with observations [@Koratkar1999; @Yuan2014]. It is also unclear if the BLR simply consists of ambient gas near the SMBH, or if it is more directly connected with the accretion process. For example, broad-line emitting gas might correspond to inflowing gas from large scales that feeds the accretion disk, or a portion of the BLR gas may be the result of an outflowing wind driven by radiation pressure from the accretion disk [@Collin-Souffrin1987; @Murray1997; @Elvis2000; @Proga2004; @Proga2010; @Higginbottom2014; @Elitzur2016]. The BLR could instead correspond to the portion of the obscuring structure lying within the dust sublimation radius [@Netzer1993; @Simpson2005; @Gaskell2008; @Nenkova2008; @Mor2012]. Other models explore the possibility that the accretion disk, BLR, and obscuring structure are not distinct at all, but different observational aspects of a single structure bound to the central SMBH (e.g., @Elitzur2006 [@Czerny2011; @Goad2012]).
Reverberation mapping (RM, @Blandford1982 [@Peterson1993; @Peterson2014]) is an effective way of investigating these scenarios. RM exploits the intrinsic variability of AGN to investigate the matter distribution around the SMBH. The inner parts of the accretion disk emit in the far/extreme UV, providing ionizing photons that drive line emission from BLR gas. As the accretion disk stochastically varies, changes in the continuum flux are reprocessed as line emission by BLR gas after a time delay that corresponds to the light-travel time across the BLR. Measuring this time delay (or “lag”) provides a means of measuring the characteristic size-scale of the line-emitting gas. Similarly, the UV continuum (or X-rays) deposits a small fraction of the accretion luminosity in the outer parts of the accretion disk and obscuring structure. Continuum variations will therefore change the local temperature of these structures, which can drive variable emission at longer continuum wavelengths—the outer part of the accretion disk emits primarily in the optical and the obscuring structure emits in the IR. By measuring any lag between the primary UV signal and light echoes at longer wavelengths, it is possible to “map” the size of the accretion disk and obscuring structure.
Early RM experiments were able to measure or constrain the physical scales of the three primary components: the accretion disk is of order a few light days from the SMBH (e.g., @Wanders1997 [@Sergeev2005]), the BLR ranges from several light days to a few light months or light years, depending on the AGN luminosity (@Wandel1999 [@Kaspi2000; @Peterson2004; @Kaspi2005]), and the obscuring structure extends several light months or light years beyond the BLR [@Clavel1989; @Oknyanskij2001; @Suganuma2006]. More recent RM studies have provided additional details. The detection of continuum lags across the accretion disk provides information about the disk’s temperature gradient, and it appears that the disks are somewhat larger than the predictions from standard models (e.g., @Shappee2014 [@Edelson2015; @Fausnaugh2016; @McHardy2016]), as also found in microlensing studies of lensed quasars (e.g., @Morgan2010 [@Blackburne2011; @Mosquera2013]). Mid- to far-IR echoes from the obscuring structure have facilitated investigation of AGN dust properties, and suggest that the obscuring structure is clumpy and has a mixed chemical composition [@Kishimoto2007; @Vazquez2015].
RM of the BLR is of particular importance for AGN studies because velocity information in the broad-line profile combined with the observed time delay provides a well-calibrated estimate of the SMBH mass. Approximately 60 AGN have RM mass measurements [@Bentz2015], and this sample anchors the scaling relations used to infer the majority of SMBH masses throughout the universe (e.g., @McLure2004 [@Vestergaard2006; @Trakhtenbrot2012; @Park2013; @Mejia-Restrepo2016], and references therein). New insights into the BLR structure have also become available with velocity-resolved analyses (e.g., @Denney2010 [@Bentz2010; @Barth2015; @Valenti2015; @Du2016]). By combining information about the BLR time delay as a function of line-of-sight velocity, it is possible to distinguish among geometric and dynamical configurations, such as flattened versus spherical matter distributions and dynamics dominated by rotation, infall, or outflow [@Horne1994; @Horne2004; @Bentz2010; @Grier2013; @Pancoast2014a; @Pancoast2014b]. So far, only about 10 AGN have such detailed velocity-resolved results, but they suggest a wide range of dynamics and geometries.
In this work, we present the first results from an intensive RM campaign executed in 2014. This campaign had two primary goals: to measure SMBH masses in several objects with interesting or peculiar observational properties, and to expand the sample of AGN with velocity-resolved reverberation signatures. NGC5548 was also observed in this campaign as part of the multiwavelength AGN STORM project (@DeRosa2015 [@Edelson2015; @Fausnaugh2016; @Goad2016]). Ground-based spectroscopic results for this object are presented by @Pei2017. Here, we present the final data and initial analysis of other AGN from this campaign, reporting continuum and line light curves, continuum-line lag measurements, and SMBH masses for five objects. We detected variability in the H$\beta$, H$\gamma$ and [He[ii]{}$\lambda 4686$]{} emission lines for most objects, which we also use to explore the photoionization conditions in the BLR [@Korista2004; @Bentz2010]. These data are of exceptional quality and should allow us recover velocity-resolved reverberation signatures in future work.
In §2, we present our target AGN, observations, data reduction, and light curves. In §3, we explain our time-series analysis and report continuum-line lags. In §4 we measure the gas velocities and estimate SMBH masses. In §5 we discuss our results, and in §6 we summarize our findings. We assume a consensus cosmology with $H_0 =
70 {\rm\ km\ s^{-1}\ Mpc^{-1}}$, $\Omega_{\rm m} = 0.3$, and $\Omega_{\Lambda} = 0.7$.
Observations and Data Reduction
===============================
Targets
-------
In spring of 2014 we monitored 11 AGN over the course of a six-month RM campaign. The AGN were selected with the aim of expanding the database of RM SMBH masses, particularly for objects with diverse and peculiar observational characteristics. The second goal of our campaign was to investigate the dynamics and geometry of the BLR with velocity-resolved reverberation signatures, i.e., velocity-delay maps and dynamical models (see e.g. @Grier2013 [@Pancoast2014a]). Here, we focus on results related to SMBH masses, and we will pursue the velocity-resolved analysis in future work.
Figure\[fig:targets\] shows [*g*]{}-band light curves from the Las Cumbres Observatory (LCO) 1m network for nine of our targets (we discuss these data in detail in §2.3). Not shown are Akn120, which was dropped early in the campaign because of low variability, and NGC5548, for which the results are presented elsewhere (@Fausnaugh2016 [@Pei2017]). In order to estimate a black hole mass, we must measure a continuum–line lag. We have not been able to measure such a reverberation signal for Mrk668, NGC3227, CBS0074, and PG1244+026. These sources have lower signal-to-noise ratios (S/Ns) than the other objects (generally 30–70 per pixel, although NGC3227 was $\sim\! 90$ per pixel; see §2.5.3), and they display lower variability amplitudes. The fractional root-mean-square amplitude ($F_{\rm var}$ as defined in §2.5.3 below) is 0.012 for Mrk668, 0.037 for NGC3227, 0.010 for CBS0074, and 0.025 for PG1244+026. For Mrk668, the slow rate of change in the light curve also makes it impossible to measure short lags. For NGC3227, the light curve is problematic because of the limited sampling and large gaps; however, this object was also observed during a monitoring campaign in 2012, and we will combine the data from both campaigns in a future analysis. For CBS0074 and PG1244+026, we have not been able to obtain a sufficiently precise calibration of the spectra (see §2.2.2) to detect emission line variability.
We succeeded in measuring black hole masses for MCG+08-11-011, NGC2617, NGC4051, 3C382, and Mrk374. Table\[tab:targets\] lists the some of the important properties of these objects (several of which are measured in this study), and we provide additional comments as follows:
i. MCG+08-11-011 is a strong X-ray source for which spectral signatures of a relativistically-broadened Fe K$\alpha$ line have been observed with [*Suzaku*]{} [@Bianchi2010]. The Fe K$\alpha$ emission is believed to be emitted close to the inner edge of the accretion disk, and can potentially be used to measure the spin parameter of the black hole. Because the black hole mass and spin are to some extent degenerate when fitting the broad Fe K$\alpha$ profile, an independent mass estimate from RM can greatly assist with the spin measurement.
ii. NGC2617 was discovered by @Shappee2014 to be a “changing look” AGN. In 2013, after a large X-ray/optical outburst, follow-up spectroscopic observations showed the presence of broad lines, while archival spectra from 2003 show only a weak broad component of H$\alpha$. This means that the classification of NGC2617 changed from a Seyfert 1.9 to Seyfert 1.0 sometime in the intervening decade. Few optical “changing look” AGN are known, although systematic searches through long-term survey data (such as the SDSS, @LaMassa2015 [@MacLeod2016]) and targeted repeat spectroscopy [@Runnoe2016; @Runco2016; @Ruan2016] have recently expanded the sample size to approximately 20 objects, depending on how “changing look” AGN are defined. The absolute rate of this phenomenon is very uncertain, but these recent studies suggest that it may be relatively common over several decades, a time scale that long-term spectroscopic surveys are only beginning to probe. Velocity-resolved dynamical information is of special interest in an object such as this, since the presence of outflows or infall may provide clues about the physical mechanism behind the change in Seyfert category.
iii. NGC4051 has been the target of several optical and X-ray RM campaigns [@Shemmer2003; @Peterson2000; @Peterson2004; @Denney2009b; @Miller2010; @Turner2017]. However, the short H$\beta$ lag, comparable to the cadence of most monitoring campaigns, has led to mixed and inconsistent results. @Denney2009b found an H$\beta$ lag of $1.87 \pm 0.52$ days, roughly a factor of 2 smaller than previous studies. Because of the large change, as well as the lag’s small value compared to the monitoring cadence, we re-observed NGC4051 during the 2014 campaign to check this result. For one month of the campaign (2014 February 17 to 2014 March 16 UTC), we also increased the monitoring cadence of NGC4051 to twice nightly, in order to securely resolve the expected short H$\beta$ lag.
NGC4051 is also an archetypal narrow-line Seyfert 1 (NLS1), meaning that the width of its H$\beta$ line is $\lesssim 2\,000
{\rm\ km\ s}^{-1}$. There are two competing theories to explain the NSL1 phenomenon: high accretion rates or rotationally-dominated BLR dynamics seen nearly face-on. Both explanations can account for the narrow linewidths given the AGN luminosity. Insight into the structure of the BLR can help distinguish between these explanations, so there is considerable interest in reconstructing a velocity-delay map for this object.
iv. 3C382 is an FR II broad-line radio galaxy [@Osterbrock1975; @Osterbrock1976]. Few radio-loud AGN have RM mass measurements, although there are notable examples such as 3C390 [@Shapovalova2010; @Dietrich2012], 3C273 [@Kaspi2000; @Peterson2004], and 3C120 [@Peterson2004; @Grier2012]. These objects are typically more luminous than radio-quiet AGN, so they have large lags (of order months to years) that are difficult and expensive to measure. However, radio emission is thought to be associated with more massive black holes, which can be tested by anchoring radio-loud AGN to the RM mass scale. Radio jets can also provide an indirect estimate of the inclination of the BLR, if the BLR is a disky structure with the rotation axis aligned to that of the jet (@Wills1986). Several jet-orientation indicators exist for 3C382, and @Eracleous1995 estimated the BLR inclination in 3C382 using dynamical models of the double-peaked H$\alpha$ profile. Velocity-delay maps and dynamical models would provide an interesting comparison to these estimates.
v. We observed Mrk374 in an RM campaign from 2012, but the AGN did not display sufficient variability to measure emission line lags at that time. Although Mrk374 is our least variable source, we succeeded in measuring a line lag from the 2014 campaign, and we present the first RM-based black hole mass here.
{width="\textwidth"}
Spectra
-------
### Observations
We obtained spectra on an approximately daily cadence between 2014 January 04 and 2014 July 06 UTC using the Boller and Chivens CCD Spectrograph on the 1.3m McGraw-Hill telescope at the MDM Observatory. We used the 350 mm$^{-1}$ grating, yielding a dispersion of 1.33Åper pixel with wavelength coverage from 4300Å to 5600Å. We kept the position angle of the slit fixed to 0$^\circ$ for the entire campaign, with a slit width of 50 to minimize losses due to differential refraction and aperture effects caused by extended emission (i.e., the host-galaxy and narrow line region, @Peterson1995). Because of the large slit width, the spectroscopic resolution for point sources (such as the AGN) is limited by the image seeing. We discuss this in more detail in §4, but comparison with high-resolution observations suggest that the effective spectral resolution is approximately 7.0Å.
The two-dimensional spectra were reduced using standard [IRAF]{} tasks for overscan, bias, and flat-field corrections, and cosmic rays were removed using LA-cosmic [@vanDokkum2001]. We extracted one-dimensional spectra from a 150 window centered on a linear fit to the trace, and we derived wavelength solutions from comparison lamps taken in the evening and morning of all observing nights. We also corrected for zero-point shifts in the wavelength solutions (due to flexure in the telescope) by taking xenon lamp exposures just prior to each observing sequence. However, every AGN was observed for a series of three 20 minute exposures and the wavelength zero-point can drift over the course of this hour, especially at high airmass. We therefore tie the wavelength solution of the first exposure to the contemporaneous xenon lamp, and then apply shifts that align the [\[O[iii]{}\]$\lambda 5007$]{} emission line of subsequent exposures to that of the first. This procedure results in wavelength solutions accurate to 0.56Å, as measured from night-sky emission lines.
We applied relative flux calibrations using sensitivity curves derived from nightly observations of standard stars. For most of the campaign, we use Feige 34 [@Oke1990] to define the nightly sensitivity curve. However, this star began to set near dusk at the end of the campaign, so we tied our relative flux calibration to BD+33$^{\circ}$2642 [@Oke1990] for the final two weeks. The change in standard star could potentially result in a systematic change in the observed continuum slopes. However, BD+33$^{\circ}$2642 and Feige 34 were observed for a one-month overlap period before the transition, and the sensitivity curves derived from both stars agree well during this time period. Of the targets presented here, only 3C382 was observed during the final two weeks, and we did not find any anomalous changes in the spectral slope during this period. As a check on the relative flux calibration, we also looked for a “bluer when brighter” trend, caused by an increasing fraction of host-galaxy light when the AGN is in a faint state and/or intrinsic variations in the AGN spectral energy distribution (e.g., @Wilhite2005 [@Sakata2010]). We measured the spectral slope by fitting a straight line to each spectrum with the emission lines masked, and for all cases except the weakly varying Mrk 374, we found a significant anticorrelation between the mean flux and the spectral slope. Detecting the “bluer when brighter” effect lends additional confidence to our relative flux calibration.
We also obtained six epochs of observations with the 2.3m telescope at Wyoming Infrared Observatory (WIRO) and the WIRO Long Slit Spectrograph. The WIRO spectra were used to fill in gaps in the MDM monitoring, and we matched the spectrograph configuration to that of the MDM spectrograph as closely as possible. This includes a 50 slit at position angle 0$^\circ$ for all observations, and we used the same extraction/sky apertures as for the MDM observations. The wavelength calibrations and spectral slopes of the WIRO data agree well with the MDM observations, and we discuss the calibration of the WIRO data to the MDM flux scale in §2.5.1.
### Night-to-Night Flux Calibration
In order to account for variable atmospheric extinction and seeing, we employ the calibration algorithms introduced by @Fausnaugh2017. This approach is similar to the older method of @vanGroningen1992, but yields markedly better calibrations. We assume that the [\[O[iii]{}\]$\lambda 5007$]{} emission line is constant over the course of our campaign, and we transform the observed spectra so that their [\[O[iii]{}\]$\lambda 5007$]{} line profiles match those of the “photometric” nights (nights with clear conditions and stable seeing). We treat the WIRO and MDM spectra separately and inter-calibrate the two flux scales below (§2.5.1).
@Fausnaugh2017 discusses the details of our implementation and provides a [python]{} package ([ mapspec]{}[^1]) to build and apply a rescaling model to time-series spectra. For completeness, we briefly outline the procedure here:
i. First, we collected the spectra taken on photometric nights (as judged by the observers onsite) and estimated their [\[O[iii]{}\]$\lambda 5007$]{} line fluxes. The line fluxes were measured by subtracting a linearly interpolated estimate of the local continuum underneath the line and then integrating the remaining flux using Simpson’s method. We provide the wavelength regions of the integration and the continuum fit in Tables \[tab:windows\] and \[tab:con\_windows\]. We applied iterative 3$\sigma$ clipping to the line fluxes, where $\sigma$ is their root-mean-square (rms) scatter, in order to reject any outliers (due to slit losses or anomalies in the sky conditions). We then averaged the remaining flux measurements to estimate the true line flux. The measured [\[O[iii]{}\]$\lambda 5007$]{} line fluxes for each object are given in Table\[tab:targets\]. Table \[tab:targets\] also gives the number of photometric epochs used to determine these fluxes for each AGN (we usually took three spectra per epoch).
ii. We then combined the remaining photometric spectra into a reference spectrum using a noise-weighted average. In this step, any residual wavelength shifts were removed by aligning the [\[O[iii]{}\]$\lambda 5007$]{} line profiles using Markov Chain Monte Carlo (MCMC) methods—the spectra are shifted by the wavelength shift that minimizes the sum of the squares of residuals between the [\[O[iii]{}\]$\lambda 5007$]{} line profiles. Linear interpolation is used for wavelength shifts of fractional pixels.
iii. Due to changes in seeing, spectrograph focus, and small guiding errors, the spectral resolution of each observation is slightly different. To address this, we smooth the reference spectrum with a Gaussian kernel so that the [\[O[iii]{}\]$\lambda 5007$]{} linewidth matches the largest [\[O[iii]{}\]$\lambda 5007$]{} linewidth in the time series. The smoothed reference spectrum will define the final resolution of the calibrated spectra.
iv. The time-series spectra are then aligned to the reference by matching the [\[O[iii]{}\]$\lambda 5007$]{} line profiles, again in a least-squares sense using MCMC methods. The differences in line profiles are modeled by a flux rescaling factor, a wavelength shift, and a smoothing kernel. After rescaling, we combine spectra from a single night using a noise-weighted average.
Imaging
-------
Our spectroscopic observations are supplemented with broad-band imaging observations. Contributing telescopes were the 0.7m at the Crimean Astrophysical Observatory (CrAO), the 0.5m Centurian 18 at Wise Observatory (WC18, @Brosch2008), and the 0.9m at West Mountain Observatory (WMO). CrAO uses an AP7p CCD with a pixel scale of 176 and a $15' \times 15'$ field of view, WC18 uses a STL-6303E CCD with a pixel scale of 147 and a $75' \times 50'$ field of view, and WMO uses a Finger Lakes PL-3041-UV CCD with a pixel scale of 061 and a field of view of $21' \times 21'$. Fountainwood Observatory (FWO) also provided observations of NGC4051 with a 0.4m telescope using an SBIG 8300M CCD. The pixel scale of this detector is 035 and the field of view is $19' \times 15'$.All observations were taken with the Bessell [*V*]{}-band.
In addition, we obtained [*ugriz*]{} imaging with the LCO 1m network [@Brown2013], which consists of nine identical 1m telescopes at four observatories spread around the globe. These data were originally acquired as part of LCO’s AGN Key project [@Valenti2015]. The main goal is to search for continuum reverberation signals, which we will pursue in a separate study (Fausnaugh et al., in preparation). However, 3C382 and Mrk374, which are our faintest sources, had low variability amplitudes and poorer S/Ns, so we included the LCO [*g*]{}-band data in the continuum light curves of these objects. Each LCO telescope has the same optic system and detectors—at the time of the RM campaign, the detectors were SBIGSTX-16803 cameras with a field of view of $16' \times 16'$ and a pixel scale of $0\farcs 23$.
We analyzed the imaging data using the image subtraction software ([ISIS]{}) developed by @Alard1998. Images were first uploaded to a central repository and vetted by eye for obvious reduction errors or poor observing conditions. We then registered the images to a common coordinate system and constructed a high S/N reference frame by combining the best-seeing and lowest-background images. When combining, [ISIS]{} adjusts the images to a common seeing by convolving the point-spread function (PSF) of each image with a spatially variable kernel. Finally, we subtracted the reference frame from each image, again allowing [ISIS]{} to match the PSFs using its convolution routine. Reference images and subtractions for each telescope/filter/detection system were constructed separately—we discuss combining the photometric measurements in §2.5.2.
Mean and rms spectra
--------------------
Figures \[fig:mcg0811\]–\[fig:mrk374\] show the noise-weighted mean spectrum $$\begin{aligned}
\overline F(\lambda) =
\frac{\sum_{i=1}^{N_t}F(\lambda,t_i)/\sigma^2(\lambda,t_i)}{\sum_{i=1}^{N_t}1/\sigma^2(\lambda,t_i)}\end{aligned}$$ for each object using the MDM observations, where $F(\lambda,t_i)$ is the flux density at epoch $t_i$ and $\sigma(\lambda,t_i)$ is its uncertainty. Figures \[fig:mcg0811\]–\[fig:mrk374\] also show root-mean-square (rms) residual spectra, defined as $$\begin{aligned}
\sigma_{\rm rms}(\lambda) = \sqrt{\frac{1}{N_t -1} \sum_{i=1}^{N_t}
\left[F(\lambda,t_i) - \overline F (\lambda) \right]^2}\label{equ:rms}.\end{aligned}$$ By the Wiener-Khinchin theorem, this statistic is proportional to the integrated variability power at each wavelength, so $\sigma_{\rm rms}$ is free of constant contaminants such as host-galaxy and narrow emission line flux. However, the total variability power contains contributions from both intrinsic variations and from statistical fluctuations/measurement uncertainties. In order to separate these components, we use a maximum-likelihood method (cf. @Park2012a [@Barth2015; @DeRosa2015]). We solve for the intrinsic variability $\sigma_{\rm var}(\lambda)$ that minimizes the negative log-likelihood
$$\begin{aligned}
-2 \ln \mathcal{L} =
\sum_{i=1}^{N_t} \frac{\left[ F(\lambda,t_i) - \hat F(\lambda)\right]^2}
{\sigma^2(\lambda,t_i) + \sigma_{\rm var}^2(\lambda)} \nonumber\\
+ \sum_{i=1}^{N_t} \ln \left[\sigma^2(\lambda,t_i) + \sigma_{\rm var}^2(\lambda) \right]
\label{equ:rms_opt}\end{aligned}$$
where $\hat F(\lambda)$ is the “optimal average” weighted by $\sigma^2(t_i) + \sigma^2_{\rm var}$. We self-consistently fit for $\hat F(\lambda)$ while solving for $\sigma_{\rm var}(\lambda)$, and we show the estimate of $\sigma_{\rm var}(\lambda)$ with the red lines in Figures \[fig:mcg0811\]–\[fig:mrk374\]. In the limit that $\sigma(\lambda,t_i)\rightarrow 0$, it is clear that $\sigma_{\rm var}
$ is equivalent to $ \sigma_{\rm rms}$. For high S/N data such as these, $\sigma_{\rm var}(\lambda)$ is nearly equal to $\left[\sigma_{\rm rms}^2(\lambda) - \overline \sigma^2(\lambda)
\right]^{1/2}$, where $\overline \sigma^2(\lambda)$ is the average of the squared measurement uncertainties across the time-series: $$\begin{aligned}
\overline \sigma^2(\lambda) = \frac{1}{N}\sum_{i=1}^{N_t}\sigma^2(\lambda,t_i).\end{aligned}$$ The overall effect is to reduce the squared amplitude of the variability spectrum by the mean squared measurement uncertainty—in all objects except for Mrk374, this effect is negligible.
Light curves
------------
### Spectroscopic Light Curves
We extracted spectroscopic light curves for the wavelength windows listed in Table\[tab:windows\] for each AGN. We chose these windows based on visual inspection of the variable line profiles in the $\sigma_{\rm var}(\lambda)$ spectra, with the main goal of capturing the strongest variations in the lines. For 3C382, the component tentatively identified as [He[ii]{}$\lambda 4686$]{} is blue-shifted by almost 100Å relative to the systematic redshift, and if variable [He[ii]{}$\lambda 4686$]{} has a similar profile as the Balmer lines in this object, this component corresponds to the blue wing of the line.
The rest-frame 5100Å continuum, which is relatively free of emission/absorption lines, was estimated by averaging the flux density in the listed wavelength region. Emission-line fluxes were determined in the same way as for the [\[O[iii]{}\]$\lambda 5007$]{} line. First we subtracted a linear least-squares fit to the local continuum underneath the emission line. Wavelength regions for the continuum fits are given in Table \[tab:con\_windows\]. Then we integrated the remaining flux using Simpson’s method (we did not assume a functional form for the emission line). In cases where the broad H$\beta$ wing extends underneath \[O[iii]{}\]$\lambda$4959, we subtracted the narrow emission line (again with a local linear approximation of the underlying flux) before integrating the broad line. We did not attempt to separate the narrow components of H$\beta$ and H$\gamma$ from the broad components. These narrow components act as constant flux-offsets for the light curves.
The continuum estimates can lead to significant systematic uncertainties, because the continuum-fitting windows may be contaminated by broad-line wing emission, and the local linearly interpolated continuum may leave residual continuum flux to be included in the line profile. Both of these effects can introduce spurious correlations between the continuum and line light curves, which may biased the final lag estimates. Because we use the $\sigma_{\rm var}(\lambda)$ spectra to select the line and continuum windows, variability in the line wings probably does not have a large impact on our results, and we have found the the resulting light curves (and their lags) are robust to five to ten angstrom changes in the continuum and line windows. Larger shifts, especially as the continuum fitting windows move further from the lines, can result in significantly different lags (of order three times the statistical uncertainties). Full spectral decompositions may be able to address this issue in future studies (see @Barth2015 for a detailed discussion). We discuss these systematic uncertainties further in §4.
After we extracted line fluxes from the WIRO and MDM spectra, we combined the measurements by forcing the light curves to be on the same flux scale. We used the mean MDM [\[O[iii]{}\]$\lambda 5007$]{} line to define this scale, and multiplied the WIRO line fluxes so that the mean value matched that of MDM. A more sophisticated inter-calibration model would include an additive offset, to account for different amounts of host-galaxy starlight in the MDM and WIRO spectra. However, with the limited amount of WIRO data, additional calibration parameters cannot be well-constrained, and we found the simple multiplicative approach to be adequate. The required rescaling factors were 1.21 for MCG+08-11-011, 1.14 for NGC4051, 1.09 for 3C382, and 1.73 for Mrk374. Weather at WIRO prevented observations of NGC2617.
The statistical uncertainty on the continuum flux was estimated from the standard deviation within the wavelength region, $$\begin{aligned}
\sigma(t_j) = \sqrt{ \frac{1}{N_{\lambda}-1} \sum_{i = 1}^{N_{\lambda}}
\left[ F(\lambda_i,t_j) - \overline F(t_j)\right]^2},\end{aligned}$$ where $\overline F(t_j)$ is the evenly-weighted average flux density at epoch $t_j$. Uncertainties on the line light curves were estimated using a Monte Carlo approach: we perturbed the observed spectrum with random deviates scaled to the uncertainty at each wavelength, subtracted a new estimate of the underlying continuum (and the narrow \[O[iii]{}\]$\lambda4959$ line when appropriate), and re-integrated the line flux. The deviates were drawn from the multivariate normal distribution defined by the covariance matrix of the rescaled spectrum—these covariances can affect the statistical uncertainty by a factor of two or more (see @Fausnaugh2017 for more details). We repeated this procedure $10^3$ times and took the central 68% confidence interval of the output flux distributions as an estimate of the statistical uncertainty.
Because the integrated [\[O[iii]{}\]$\lambda 5007$]{} line flux is not explicitly forced to be equal from night to night, the scatter of the [\[O[iii]{}\]$\lambda 5007$]{} line light curve serves as an estimate of our calibration uncertainty [@Barth2015]. We extracted narrow [\[O[iii]{}\]$\lambda 5007$]{} line light curves in the same way as for the broad lines, and the results are shown in Figure \[fig:oiii\]. Several points are noticeably below the means of their light curves, particularly for NGC2617 and 3C382. These observations were taken in poor weather, and display significant scatter between the individual rescaled exposures prior to averaging. This suggests variable amounts of flux-losses between the AGN and extended [\[O[iii]{}\]$\lambda 5007$]{}/host-galaxy, due to variable seeing and large guiding errors that move the object in the slit. Although the rescaling model from §2.2.2 cannot correct this issue, the offsets of these points are not very large compared to the statistical uncertainties (no more than 3.1$\sigma$), and we opt to include them in the analysis. Since the effect due to spatially extended [\[O[iii]{}\]$\lambda 5007$]{} emission is relatively small even in very poor conditions, it will be unimportant in good conditions.
The fractional standard deviations of the narrow line light curves are given in Table \[tab:targets\] and range between 0.1% and 1.4%. These values only represent our ability to correct for extrinsic variations (such as weather conditions) in the observed spectra. Additional systematic uncertainties dominate the epoch-to-epoch uncertainties of the light curves, including (but not limited to) the nightly sensitivity functions, continuum subtraction, and additional spectral components such as Fe[ii]{} emission. The latter two issues are especially problematic for the [He[ii]{}$\lambda 4686$]{} light curves.
To account for these systematics, we rescaled the light curve uncertainties so that they approximate the observed flux variations from night to night. We selected three adjacent points $F(t_{j -1})$, $F(t_j)$, and $F(t_{j +1})$, linearly interpolated between $F(t_{j-1})$ and $F(t_{j+1})$, and measure $\Delta = [F(t_j) -
I(t_j)]/\sigma(t_{j})$ where $I(t_j)$ is the interpolated value at $t_j$ and $\sigma(t_j)$ is the statistical uncertainty on $F(t_j)$. The deviate $\Delta$ therefore measures the departure of the light curve from a simple linear model. We calculated $\Delta$ for $j = 2$ to $N_t-1$ (i.e, ignoring the first and last points), and we multiplied the statistical uncertainties $\sigma(t_j)$ by the mean absolute deviation (MAD) $\overline{ |\Delta|}$. We also imposed a a minimum value of 1.0 on these rescaling factors. Inspection of the distribution of $\Delta$ shows that the residuals are reasonably (but not perfectly) represented by a Gaussian with a similar MAD value. This method ensures that the uncertainties account for any systematics that the rescaling model cannot capture. We have ignored the uncertainty in the interpolation $I(t_j)$, so our method slightly overestimates the required rescaling factors. Monte Carlo simulations may be able to assess the importance of uncertainty in $I(t_j)$ for future work. The rescaling factors are given in Table\[tab:lc\_prop\] and are fairly small, generally running between 1.0 and 2.0, with a mean of 1.8 and a maximum of 3.42 for the H$\beta$ light curve in NGC4051. NGC4051 has the largest rescaling factors overall, which may be due to real short time-scale variability that departs from our simple linear model [@Denney2010]. We therefore also experimented with using the unscaled light curve uncertainties in our time-series analysis (§3) for this object. We found that our results do not sensitively depend on the scale of the uncertainties, although our Bayesian lag analysis (§3.2) indicates that the unscaled uncertainties are probably underestimated.
### Broad-Band Light Curves
Differential photometric light curves were extracted from the subtracted broad-band images using [ISIS]{}’s built-in photometry package. The software performs PSF photometry by fitting a model to the reference frame PSF and convolving this model with the kernel that was fitted during image subtraction. Because this transformation accounts for variable seeing, while the image subtraction has removed sources of constant flux, the output light curves cleanly isolate intrinsic variations of the AGN from contaminants such as host-galaxy starlight and seeing-dependent aperture effects. Any other constant systematic errors are also automatically subtracted out of the differential light curves. However, [ISIS]{} accounts for only the local Poisson uncertainty from photon-counting, while there are also systematic errors from imperfect subtractions (e.g., @Hartman2004). We addressed this problem in the same way as @Fausnaugh2016. We inspected the differential light curves of comparison stars, and rescaled the uncertainties by a time dependent factor to make the comparison star residuals consistent with a constant model. The reduced $\chi^2$ of the comparison star light curves is therefore set to one, which requires an average error rescaling factor of 1.0 to 5.0, depending on the object and the telescope. Since our targets are fairly bright, the formal ISIS uncertainties are very small and rescaling even by a factor of five results in uncertainties no greater than 3–6%. See §2.2 of @Fausnaugh2016 for more details.
We next calibrated the differential broad-band light curves to the flux scale of the spectroscopic continuum light curve. The inter-calibration procedure solves for a maximum-likelihood shift and rescaling factor for each differential light curve, forcing the [ *V*]{}-band photometry to match the rest-frame 5100Å continuum flux. The inter-calibration parameters account for the different detector gains/bias levels, telescope throughputs, and (to first-order) a correction for the wider bandpass and different effective wavelengths of the broad-band filters compared to the spectroscopic-continuum averaging window. An advantage of this procedure is that it does not require accurate knowledge of the image zeropoints (or color corrections), which would otherwise limit the overall precision when combining data from different telescopes. The model also minimizes systematic errors that can result in strong correlations between measurements from the same telescope.
Because observations from various telescopes are never simultaneous, it is necessary to interpolate the light curves when fitting the inter-calibration parameters. We followed @Fausnaugh2016 and modeled the time-series as a damped random walk (DRW), as implemented by the [JAVELIN]{} software [@Zu2011]. Although recent studies have shown that the power spectra of AGN light curves on short time scales may be somewhat steeper than a DRW [@Edelson2014; @Kasliwal2015], @Zu2013 found that the DRW is an adequate description of the time scales considered here (see also @Skielboe2015 [@Fausnaugh2016; @Kozlowski2016a; @Kozlowski2016b]). Our interpolation scheme and fitting procedure are identical to those described by @Fausnaugh2016.
### Light-curve Properties
The final light curves are shown in Figures \[fig:mcg0811\]–\[fig:mrk374\] and given in Tables 5–14. We characterize the statistical properties of the light curves in Table\[tab:lc\_prop\], reporting the median cadence, mean flux-level, and average S/N. We also measure the light curve variability using a technique similar to our treatment of the variability spectra $\sigma_{\rm var}(\lambda)$. In the presence of noise, it is necessary to separate the intrinsic variability from that due to measurement errors. We therefore define the intrinsic variability of the light curves as $\sigma_{\rm var}$ and solve for it by minimizing $$\begin{aligned}
-2 \ln \mathcal{L} = \sum_i^{N_t} \frac{\left[ F(t_i) - \hat
F\right]^2}{\sigma^2(t_i) + \sigma_{\rm var}^{2} } + \sum_i^{N_t} \ln
\left[\sigma^2(t_i) + \sigma_{\rm var}^{ 2} \right],
\label{equ:fracvar}\end{aligned}$$ where $F(t_i)$ is the flux at epoch $i$, $\sigma(t_i)$ is its uncertainty, and $\hat F$ is the optimal average flux (weighted by $\sigma^2(t_i) + \sigma^2_{\rm var}$). For small measurement uncertainties, the fractional variability $\sigma_{\rm var}/\hat F$ converges to the standard definition of the “excess variance” [@Rodriguez1997] $$\begin{aligned}
F_{\rm var} = \frac{1}{\overline F}\sqrt{\frac{1}{N -1} \sum_i^N
\left[F(t_i) - \overline F \right]^2 - \overline \sigma^2}\end{aligned}$$ where $\overline \sigma$ is the time-averaged measurement uncertainty of the light curve. We therefore define $F_{\rm var} = \sigma_{\rm
var}/\hat F$, and report these values in Table\[tab:lc\_prop\]. These values are slightly underestimated, since $\hat F$ is not corrected for constant components (such as host-galaxy starlight or narrow line emission). We also approximate the S/N of the variability as $$\begin{aligned}
{\rm (S/N)_{var}} = \frac{\sigma_{\rm var}}{\overline
\sigma \sqrt{ 2/N_{\rm obs}}}.\end{aligned}$$ The $\sqrt{2/N_{\rm obs}}$ term enters because the variance of $\overline \sigma$ is expected to approximately scale as that of a reduced $\chi^2$ distribution. However, this calculation assumes uncorrelated uncertainties, and a full analysis requires treatment of the red-noise properties of the light curve (see @Vaughan2003).
With the exception of the 3C382 H$\gamma$, the 3C382 [He[ii]{}$\lambda 4686$]{}, and the Mrk374 H$\gamma$ light curves, we detect variability in all of the other emission lines at greater than $\sim \! 10\sigma$. The variability amplitudes of MCG+08-11-011 and NGC2617 are especially strong ($F_{\rm var}\gtrsim10\%$). For NGC4051, the continuum has little fractional variability ($F_{\rm var} =2\%$), which may be caused by a high fraction of host-galaxy starlight. For MCG+08-11-011, NGC2617, and NGC4051, the median cadence is near 1 day for all light curves, and the mean S/N usually ranges from several tens to hundreds. In fact, the S/N in the spectra is even higher, reaching 100 to 300 per pixel in the continuum. Combined with the large variability amplitudes, it likely that we will be able to construct velocity-delay maps and dynamical models for these objects in future work.
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Time-series Measurements
========================
We measure lags between continuum and line light curves using two independent methods: traditional cross-correlation techniques and a Bayesian analysis using the [JAVELIN]{} software.
Cross-Correlation
-----------------
The cross-correlation procedure derives a lag from the centroid of the interpolated cross-correlation function (ICCF, @Gaskell1987), as implemented by @Peterson2004. For a given time delay, we shift the abscissas of the first light curve, linearly interpolate the second light curve to the new time coordinates, and calculate the correlation coefficient $r_{cc}$ between all overlapping data points. We then repeat this calculation but shift the second light curve by the negative of the given time delay and interpolate the first light curve. The two values of $r_{cc}$ are averaged together, and the ICCF is evaluated by repeating this procedure on a grid of time delays spaced by 0.1 days. All ICCFs are measured relative to the 5100Åcontinuum light curve (inter-calibrated with the broad-band measurements). For each line light curve, the maximum value $r_{\rm
max}$ of the ICCF is given in Table\[tab:lc\_prop\]. The lag is estimated with the ICCF centroid, defined as $\tau_{\rm cent} = \int
\tau r_{cc}(\tau)\,d\tau / \int r_{cc}(\tau)\,d\tau$ for values of $r_{cc} \geq 0.8 r_{\rm
max}$.
We estimate the uncertainty on $\tau_{\rm cent}$ using the flux randomization/random subset sampling (FR/RSS) method of @Peterson2004. This technique generates perturbed light curves by randomly selecting (with replacement) a subset of the data from both light curves and adjusting the fluxes by a Gaussian deviate scaled to the measurement uncertainties. The lag $\tau_{\rm cent}$ is calculated for $10^3$ perturbations of the data, and its uncertainty is estimated from the central 68% confidence interval of the resulting distribution. The ICCF and centroid distributions are shown in Figure\[fig:linelags\] for all objects and line light curves, and Table\[tab:linelags\] gives the median values and central 68% confidence intervals of these distributions. For completeness, we also report in Table\[tab:linelags\] the lag $\tau_{\rm peak}$ that corresponds to $r_{\rm max}$. Note that these lags have been corrected to the rest frame of the source. For 3C382, we do not find meaningful centroids in the ICCFs of the H$\gamma$ and [He[ii]{}$\lambda 4686$]{}light curves. This is because of the width of the autocorrelation function of the continuum and its poor correlation with the line light curves. We therefore do not include these lines for the rest of the ICCF analysis.
Long-term trends in the light curves can bias the resulting ICCF due to red-noise leakage [@Welsh1999]. We therefore experimented with detrending the light curves and/or restricting the baseline over which to calculate the ICCF. For MCG+08-11-011 these experiments had no effect, while for Mrk374 and 3C382 they eliminated any lag signal in the data. For NGC2617, we found that restricting the data to 6620$<$HJD$-$2450000$<$6730 improved the ICCF by narrowing the central peak, as shown in the top four panels of Figure \[fig:detrend\]. However, this restriction changed the ICCF centroid by only 0.01 days, a negligible amount. For NGC2617, the peaks in the H$\gamma$ and [He[ii]{}$\lambda 4686$]{} ICCFs at $\pm 25$ days are also obvious aliases, so we only report the lag based on the peak near 0 days. For NGC4051, we found that detrending the continuum and line light curves with a second-order polynomial improves the ICCF, as shown in the bottom four panels of Figure \[fig:detrend\]. The long-term continuum trend is very weak, but there is a strong positive trend in the line light curves that is dominated by the linear term. Subtracting this linear trend decreases the median of the centroid distribution from 4.92 days to 2.56 days, a change of 1.5$\sigma$. We adopt the smaller lag because of the quality of the detrended ICCF, and our Bayesian method (described below) finds a lag consistent with this smaller value.
JAVELIN
-------
We also investigated the line lags using a Bayesian approach, as implemented by the [JAVELIN]{} software [@Zu2011]. [ JAVELIN]{} explicitly models the reverberating light curves and corresponding transfer functions so as to find a posterior probability distribution of lags. We have already discussed [JAVELIN]{}’s assumption that light curves are reasonably characterized by a DRW (§2.5.2). [JAVELIN]{} also assumes that the transfer function is a simple top-hat that can be parameterized by a width, an amplitude, and a mean time delay. This assumption is not very restrictive, since it is difficult to distinguish among transfer functions in the presence of noise [@Rybicki1994; @Zu2011] and a top-hat is broadly consistent with expectations for physically-plausible BLR geometries (e.g., disks or spherical shells).
We ran [JAVELIN]{} models for each line using the 5100Åcontinuum as the driving light curve, and we used internal [ JAVELIN]{} routines to remove any linear trends from the light curves during the fit. The damping time scale (a parameter of the DRW model) for most AGN is several hundred days or longer [@Kelly2009; @Macleod2010], and our light curves are not long enough to meaningfully constrain this parameter. We therefore (arbitrarily) fixed the damping time scale to 200 days. We also tested several different damping time scales (from a few days to 500 days), and found that the choice of 200 days does not affect the best-fit lags—an exact estimate of the damping time scale is not necessary to reasonably interpolate the light curves [@Kozlowski2016b]. Table\[tab:linelags\] gives the median and 68% confidence interval of the posterior lag distributions, denoted as $\tau_{\tt
JAV}$. We also employed models that fit all light curves from a single object simultaneously, which maximizes the available information. These results are given in Table\[tab:linelags\] as $\tau_{\rm multi}$. Posterior distributions of $\tau_{\rm multi}$ are shown by the blue histograms in Figure\[fig:linelags\]. For the H$\gamma$ and [He[ii]{}$\lambda 4686$]{} light curves from 3C382, we were again unable to constrain any lag signal, and we drop these light curves from the rest of this analysis.
Results
-------
We generally find consistent results between the ICCF method and [ JAVELIN]{} models. The largest discrepancies are the H$\beta$ lags for NGC2617 ($\Delta \tau = 1.6\sigma$) and 3C382 ($\Delta \tau
=2.0\sigma$), but these differences are not statistically significant. In NGC2617, where the ICCF method detects a lag consistent with zero in the H$\gamma$ or [He[ii]{}$\lambda 4686$]{} light curves, [JAVELIN]{} finds a lag at reasonably high confidence: the percentiles for $\tau_{\rm multi}=0$ in the posterior lag distributions of H$\gamma$ and [He[ii]{}$\lambda 4686$]{} are 8.3% and 1.1%, which are 1.4$\sigma$ and 2.3$\sigma$ detections for Gaussian probability distributions, respectively. For Mrk374, an H$\gamma$ lag is detected at high significance using [JAVELIN]{} (we do not claim a lag detection for [He[ii]{}$\lambda 4686$]{} in this object, since the $\tau_{\rm multi} = 0$ percentile is 20%, only $0.2\sigma$ for a Gaussian probability distribution). The detection of these lags represents a significant advantage of the [JAVELIN]{} technique over traditional cross-correlation methods. We adopt the $\tau_{\rm
multi}$ as our final lag measurements, since the multi-line global fits provide well-constrained lags, properly treat covariances between the lags from different light curves, and utilize the maximum amount of information available in the data.
The analysis of NGC4051 is especially difficult because the light curves exhibit low-amplitude variations. The lags in this object are also expected to be small, based on the AGN luminosity [@Bentz2013] and a previous well-sampled RM experiment [@Denney2009b]. For H$\beta$, [JAVELIN]{} finds a definite lag near 2 days, consistent with the detrended ICCF approach. For H$\gamma$, the ICCF method finds a lag consistent with zero, while the single-line [JAVELIN]{} fit finds a lag of $4.87 \pm 0.18$ days and the multi-line fit finds a lag of $2.40\pm 0.80$ days (rest frame). The single-line fit results in a complicated multi-modal posterior distribution with smaller peaks at 15 and 25 days that are caused by aliasing. For example, the 25-day lag is probably caused by aligning the H$\gamma$ maximum near 6745 days with the local maximum in the continuum light curve at 6720 days (Figure\[fig:n4051\]). However, the multi-line fit shows a strong, dominant peak for H$\gamma$ at $2.40$ days (rest frame). A probable explanation is that the H$\beta$ light curve matches the overall shape of H$\gamma$, but has stronger features against which to estimate a continuum lag—fitting both light curves simultaneously can therefore establish an H$\gamma$ lag with higher confidence. The problem with the H$\gamma$ light curve appears in a more serious form in the [He[ii]{}$\lambda 4686$]{}light curve, and [JAVELIN]{} finds a lag consistent with zero for this line.
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Linewidths and $M_{\rm BH}$ Calculations
========================================
After determining the characteristic size of the BLR from the mean time delay, the next step is to calculate the characteristic line-of-sight velocity of the BLR gas, from which we can derive SMBH masses. The BLR velocity is estimated from the width of emission lines in the MDM spectra. However, it is important to use the linewidth of the variable component of the profile, since we measure the BLR radius from the variable line flux. For example, the variable profile of 3C382 is radically different (and much broader) than the time-averaged profile in the mean spectrum (Figure\[fig:3c382\]). We therefore measure and report in Table 16 linewidths both in the mean spectrum $\hat F(\lambda)$, and in the rms spectrum $\sigma_{\rm
var}(\lambda)$, but we use the latter for mass determinations.
There are two common choices for linewidth measurements: the full-width at half-maximum (FWHM) and the line dispersion $\sigma_L$ (the rms width of the line profile). There are advantages and disadvantages associated with both approaches—while the FWHM is simpler to measure, there are ambiguities for noisy or complicated line profiles such as the double-peaked H$\beta$ profiles in MCG+08-11-011, NGC2617, and 3C382. On the other hand, although $\sigma_L$ is well-defined for arbitrary line profiles, it depends more sensitively on continuum subtraction and blending in the line wings [@Denney2016; @Mejia-Restrepo2016]. @Peterson2004 find that velocities estimated with $\sigma_L$ produce a tighter virial relation, and @Denney2013 find that the masses determined from UV and optical lines agree better using $\sigma_{\rm
L}$. We therefore adopt $\sigma_L$ as a measure of the BLR velocity in this study. For completeness, we also give the FWHM in Table\[tab:v\_alt\].
Linewidth uncertainties are estimated using a bootstrapping method. For $10^3$ iterations on each object with $N$ nightly spectra, we randomly select $N$ observations with replacement, recompute the mean and rms spectrum, and remeasure the linewidths in the rms spectrum. The central 68% confidence interval of the resulting distributions are adopted as the formal uncertainty of the linewidth. This approach can only account for statistical uncertainties in the linewidths, which therefore represent lower limits on the uncertainties. There are additional systematic errors from the choice of wavelength windows that define the line profiles (Tables \[tab:windows\] and \[tab:con\_windows\]), as well as blending of the broad-line wings. The choice of wavelength windows and continuum subtraction is problematic for weak lines, lines with low variability, and lines with unusual profiles. In particular, our estimates for the [He[ii]{}$\lambda 4686$]{} line in NGC2617, NGC4051, 3C382, and all lines in Mrk374 are certainly affected. Furthermore, the blue wing of H$\beta$ and the red wing of [He[ii]{}$\lambda 4686$]{} overlap in MCG+08-11-011 and NGC2617, and it is likely that the [He[ii]{}$\lambda 4686$]{} velocity is severely underestimated (the effect on H$\beta$ is probably smaller, though it may not be negligible). Spectral decompositions may help with these problems in future analyses; for now, we note that the linewidth uncertainties are underestimated in these cases, and we provide a treatment for this issue below.
We correct the linewidth measurements for the instrument resolution by subtracting the rms width of the spectrograph’s line-spread-function (LSF) in quadrature from the observed value of $\sigma_L$. Previous studies have found that the width of the LSF for the MDM spectrograph is near 3.2 or 3.4Å (FWHM 7.6–7.9Å, @Denney2010 [@Grier2012]). Based on comparisons with high spectral resolution observations, where the LSF width is negligible, we find a LSF width of 2.97Å (FWHM $=6.99\,\AA$). This value was determined using the catalog of high-resolution [\[O[iii]{}\]$\lambda 5007$]{} measurements from @Whittle1992, which contains intrinsic [\[O[iii]{}\]$\lambda 5007$]{} linewidths for MCG+0-11-011 and NGC4051. The [\[O[iii]{}\]$\lambda 5007$]{} line of NGC 4051 is undersampled in the MDM spectra (the intrinsic FWHM is 190 km s$^{-1}$, or 3.16Å in the observed frame), and does not give a reliable estimate the instrumental broadening. However, the intrinsic [\[O[iii]{}\]$\lambda 5007$]{} FWHM in MCG+08-11-011 is 605 km s$^{-1}$, or 10.52Å in the observed frame, which is well resolved. The observed FWHM in the MCG+08-11-011 reference spectrum (before smoothing, see §2.2.2 and below) is 12.63Å, which implies that the FWHM of the LSF is 6.99Å (a rms width of 2.97Å). This value is close to but slightly smaller than previous estimates. The MDM LSF may not be perfectly stable in time, so we adopt 2.97Å as the rms width of the instrumental broadening in our observations.
An additional correction must be applied because we smooth our reference spectra to approximately match the nights with the worst spectroscopic resolution (see §2.2.2). The kernel widths for this smoothing procedure were 1.4Å for MCG+08-11-011, 1.5Å for NGC2617, 1.8Å for NGC4051, 1.7Å for 3C382, and 1.9Å for Mrk374 (the FWHM values are a factor of 2.35 larger). We also subtract these values in quadrature from the observed line dispersion. The final rest-frame linewidths and their uncertainties are given in Table\[tab:v\_alt\].
We measure the SMBH masses as $$\begin{aligned}
M_{\rm BH} = \langle f\rangle \frac{\sigma_L^2c\tau_{\rm multi}}{G}\end{aligned}$$ where $c$ is the speed of light, $G$ is the gravitational constant, and $\langle f\rangle $ is the virial factor. The virial factor accounts for the unknown geometry and dynamics of the BLR, and is determined by calibrating a sample of RM AGN to the $M_{\rm
BH}$-$\sigma_{*}$ relation (e.g., @Onken2004 [@Park2012b; @Grier2013b]). We use the most recent calibration by @Woo2015 of $\langle f\rangle = 4.47\pm 1.25$ with a scatter of $0.43\pm 0.03$dex (a factor of 2.7). Finally, it is convenient to define the virial product, $\sigma_L^2c\tau/G$, which is an observed quantity that is independent of the mass calibration.
We calculate the statistical uncertainties on the virial products through standard error propagation. As discussed above, there are significant systematic uncertainties on both the linewidths and the lags, which probably dominate the final error budget (see also §2.4). We estimate the systematic uncertainty using repeat RM measurements gathered from the literature. There are 17 H$\beta$-based measurements of the virial product in NGC5548 over the last 30 years (see @Bentz2015). The (log) standard deviation of these measurements is 0.16 dex, while the mean statistical uncertainty is 0.10 dex. Taking $\sigma_{\rm
sys}^2 = \sigma_{\rm rms}^2 - \sigma_{\rm stat}^2$, we estimate a systematic uncertainty floor of 0.13 dex. Experimentation with alternative line windows, continuum interpolations, and detrending procedures suggests that this value (a factor of about $1.3$) captures most of the variation in the virial products of our sample. We therefore adopt 0.13 dex as our estimate of the systematic uncertainty on each virial product, and add this value in quadrature to the statistical uncertainties for the virial products. For our final mass estimates, we also add in quadrature the the uncertainty in the mean value of $\langle f\rangle $ ($\sim\!0.12$ dex) and its intrinsic scatter (0.43 dex). The virial products, final masses, and total uncertainties are given in Table \[tab:masses\].
We discuss the consistency of virial products for the same object derived from different emission lines in §5.2, and we comment on the H$\beta$-derived masses of individual objects below.
i. MCG+08-11-011 is our most variable object. The black hole mass estimate is $\sim\!2.8 \times 10^7\, {\rm M_\odot}$, and the uncertainty is dominated by uncertainty in the virial factor $f$. @Bianchi2010 found evidence for a relativistically broadened Fe K$\alpha$ line in the X-ray spectrum of this object, but the available mass estimates at that time were uncertain by an order of magnitude (10$^7$–10$^8$M$_\odot$). The results presented here may help measure the spin of the black hole in future studies.
ii. The mass reported here for NGC2617 of $\sim\!3.2 \times
10^{7}\, {\rm M_{\odot}}$ is in good agreement with the single-epoch mass estimated by @Shappee2014 of $(4 \pm 1) \times 10^{7}\,
{\rm M_{\odot}}$, also using the H$\beta$ emission-line. NGC2617 is the second “changing look” AGN with a direct RM mass measurement. The other object is Mrk 590, which was observed to change from a Seyfert 1.5 to 1.0 to 1.9 over several decades [@Denney2014], and has a RM mass of $\sim\! 5\times
10^{7}\,{\rm M_{\odot}}$ [@Peterson2004]. In terms of their black hole masses, there is nothing extraordinary about either NGC 2617 or Mrk 590. Our luminosity-independent RM mass also allows us to estimate a more robust Eddington ratio ($\dot m_{\rm Edd} =
L_{\rm Bol}/L_{\rm Edd}$) than from the single-epoch mass. Assuming a bolometric correction of 10 for the 5100Å continuum luminosity, we find that $\dot m_{\rm Edd} = 0.01$, after correcting for host-galaxy starlight (see §5.1). This value is somewhat low, though not atypical, for Seyfert 1 galaxies.
iii. For NGC4051, our measurement of the H$\beta$ lag ($2.24 \pm
0.33$ days) is in good agreement with the estimate of $1.87\pm 0.52$ days by @Denney2009b. The measurement is challenging because of the low-amplitude continuum variations, variable host-galaxy contamination from aperture effects [@Peterson1995], and a secular trend in the line light curve.
Our estimate of the virial product $\sim\! 1.1 \times 10^5
$M$_\odot$ is also consistent at the 2$\sigma$ level with the estimate of $(3.0\pm 1.0) \times 10^5$M$_\odot$ from @Denney2010. The difference is primarily due to a decrease in the linewidth by about 400 ${\rm km\ s^{-1}}$ compared to the 2007 campaign. The line and continuum wavelength window definitions are somewhat different between the 2014 and 2007 campaigns, and we found that using the wavelength windows from Tables \[tab:windows\] and \[tab:con\_windows\] for the rms spectrum from 2007 reduces the difference to only $\sim\!100 {\rm \ km\
s^{-1}}$ (i.e., $\sigma_{\rm L}$ was about 20% larger in 2007 than in 2014). If we use the wavelength regions from @Denney2010, the measurement from 2014 increases by $\sim\!
120 {\rm\ km\ s^{-1}}$. This suggests that the virial product is somewhat smaller than that reported by @Denney2010, but the mild 2$\sigma$ discrepancy indicates that the systematic uncertainties are comparable to the formal uncertainties. The remaining 100–300 ${\rm\ km\ s^{-1}}$ difference is physical—comparing the rms line profiles between the two campaigns, we found that the core of the H$\beta$ line is much more variable in 2014 than it was in 2007, weighting $\sigma_{\rm L}$ to smaller values. The lag has only increased by 0.26 days (19%), so the virial product shows a net decrease. This might indicate a change in the geometry and/or dynamics of the BLR. The dynamical time is of order only two or three years at two light days from a $10^6$ M$_\odot$ black hole, so such a change cannot be ruled out [*a priori*]{}. A comparison of the velocity resolved reverberation signals between 2007 and 2014 is therefore especially interesting.
Our SMBH mass estimate of $\sim \! 4.7 \times 10^5$ ${\rm M_{\odot}}$ for NGC4051 is at the very low end of the SMBH scale, and there are only two other RM masses below $10^{6}$ $\rm{M_{\odot}}$: NGC 4395 [@Peterson2005; @Edri2012] and UGC 06728 [@Bentz2016].
iv. In 3C382 the black hole mass is about $9.6 \times 10^8$ ${\rm
M_{\odot}}$, and a large source of uncertainty is the H$\beta$ lag. The $\sim$52 day lag is driven by the gentle inflection in the line light curve observed near the middle of the spectroscopic campaign, which was also observed in the imaging data about one month before the MDM observations began. The uncertainties on the H$\beta$ line lag are therefore quite large. By RM standards, 3C382 is also at a moderate redshift ($z \sim 0.06$) and faint ([*V*]{} $\sim\! 15.4 $), putting it near the limit of feasibility for monitoring campaigns with a 1m-class telescope.
Several estimates of the BLR orientation exist for this object. Emission from the radio lobes in 3C382 dominates over that of the core, indicating that the system is viewed more edge on (@Wills1986 give the core-to-lobe ratio as $\sim\! 0.1$). However, @Eracleous1995 find an inclination of 45$^{\circ}$ from dynamical modeling of the double-peaked broad H$\alpha$ line and show that this estimate is consistent with the radio properties. Velocity-delay maps and dynamical modeling of this object would be an interesting test of this inclination measurement. Unfortunately, the width of the continuum autocorrelation function and the low S/N of the line light curves are poorly suited for these experiments. On the other hand, a moderately inclined disk is broadly consistent with the double-peaked rms H$\beta$ and H$\gamma$ line profiles, and velocity-binned mean time delays may still provide interesting constraints on the BLR structure.
v. Mrk374 is our least variable source. Although the H$\beta$ lag is detected at a statistically significant level, the uncertainty on the ICCF centroid is somewhat larger than for the other objects ($\sim\! 33\%$). The mass is $\sim\!2.09\times 10^7$ ${\rm M_{\odot}}$, and the dominant uncertainty is from the linewidth measurement—it is clear from Figure \[fig:mrk374\] that the variability of the lines is very small and that there is some ambiguity in where the line profile begins and ends. At a redshift of $\sim\! 0.04$, Mrk374 is one of our fainter sources ([*V*]{}$ = 15.0$ mag), and, similar to 3C382, it is near the practical limits of a monitoring campaign lead by a 1m-class telescope.
Discussion
==========
Radius-Luminosity Relation
--------------------------
![Radius–luminosity relation for the targets of this study, compared to the relation from @Bentz2013. Luminosities are estimated from the mean of the continuum light curves corrected for Galactic extinction. The solid black line shows the best-fit relation measured by @Bentz2013, and the dashed black lines show the dispersion around the best fit. Open circles show the luminosities corrected for host-galaxy starlight, which results in excellent agreement with the relation from @Bentz2013. \[fig:RL\]](RL){width="50.00000%"}
Figure\[fig:RL\] shows the H$\beta$ lags of our five objects as a function of luminosity, the so-called radius-luminosity ($R$–$L$) relation (@Kaspi2000 [@Kaspi2005; @Bentz2009; @Bentz2013]). To estimate the luminosities, we first take the mean of the 5100Ålight curve and correct for Galactic extinction using the extinction map of @Schlafly2011 and a @Cardelli1989 extinction law with $R_V = 3.1$. We then convert the flux to luminosity using the luminosity distances in Table\[tab:targets\]. In the case of NGC4051, which has a large peculiar velocity relative to the Hubble flow ($z \sim 0.002$), we use a Tully-Fischer distance of 17.1 Mpc [@Tully2008]. This distance is uncertain by about 20%, and improving this measurement is an important step to investigate any discrepancies of this object from the $R$–$L$ relation and to estimate its true Eddington ratio. For these purposes, an [*HST*]{} program has recently been approved to obtain a Cepheid distance to NGC4051 ([*HST*]{} GO-14697; PI Peterson).
The final values of $\lambda L_{5100{\text{\normalfont\AA}}}$ are reported in Table \[tab:targets\], along with the adopted Galactic values of $E(B-V)$. We find that our objects all lie close to, but slightly below (except for 3C382), the $R$–$L$ relation. The major systematic uncertainties are internal extinction in the AGN and host-galaxy contamination. Internal extinction may move the points farther from the $R$–$L$ relation, but this effect is expected to be small. On the other hand, host-galaxy contamination can be very significant, especially for low-luminosity objects.
In order to correct for host contamination, we model high-resolution images of the targets and isolate the host-galaxy flux. This has previously been done for NGC4051 [@Bentz2006; @Bentz2013], and MCG+08-11-011, NGC2617, and Mrk374 were recently observed with [*HST*]{} for this purpose ([*HST*]{} GO-13816; PI Bentz). We also retrieved archival optical WFPC2 imaging of 3C382 ([*HST*]{} GO-6967, PI Sparks), but the data are not ideal for image decompositions and we discuss the host-galaxy flux estimate for this object separately. A more detailed analysis of the [*HST*]{} GO-13816 data and image decompositions will be presented in future work (Bentz et al, in preparation). However, following the procedures described by @Bentz2013, we made preliminary estimates of the host-galaxy contributions in the MDM aperture ($15\farcs 0 \times 5\farcs 0$ aligned at position angle $0^{\circ}$). The results are given in Table\[tab:targets\] (uncertainties on these values are estimated at 10% and included in Figure\[fig:RL\]). Applying this correction shows that host-contamination accounts for the entire discrepancy between the observed luminosities and the $R$–$L$ relation. The largest deviation from the $R$–$L$ relation is Mrk374, but the offset is only slightly greater than the $1\sigma$ scatter of the relation.
3C382 resides in a giant elliptical galaxy and there may be a significant contribution from the host’s starlight—several stellar absorption features are visible in the mean spectrum in Figure \[fig:3c382\]. In the archival [*HST*]{} images, the galaxy nucleus is saturated, hindering our ability to robustly remove the AGN flux and isolate the host’s starlight. The main problem is that the Sersic index of the host-galaxy is degenerate with the saturated core and tends to drift toward unreasonably high values ($n\approx 7.6$) when fitting the image in the same way as @Bentz2013. Fixing the Sersic index to more typical values (between 2 and 4) leads to host fluxes in the MDM aperture between 2.2 and 2.7$\times 10^{-15}$ erg cm$^{-2}$ s$^{-1}$ Å$^{-1}$, about 77% of the observed luminosity ($\log \lambda L_{\rm host} = 44.04$ to 44.12 \[erg s$^{-1}$\], after correcting for Galactic extinction). This estimate can be checked using the equivalent-width (EW) of the prominent Mg absorption feature at 5200Å rest-frame (5460Å observed-frame). In our mean spectrum, we find an EW of 2.8Å. In typical elliptical galaxy spectra, we find the EW is about 6.7 to 7.3Å, depending on the continuum estimation and assumptions about the host-galaxy properties.[^2] This implies that the featureless AGN continuum dilutes the absorption feature by a factor of 2.4 to 2.6, so that the host galaxy contributes approximately 40% of the observed luminosity. This rough estimate is a factor of two lower than the result from image decomposition, but the two values span the range of host-contributions from the other objects in our sample (42% to 71% of the observed luminosity, see Table \[tab:targets\]). We therefore adopt a host correction of $(60 \pm 20)$% of the observed luminosity ($\log \lambda L_{\rm 5100{\text{\normalfont\AA}}} = 43.98 \pm 0.15$ \[erg s$^{-1}$\]), and we note that this estimate can easily be improved by obtaining unsaturated high resolution images. The host correction moves 3C382 away from the $R$–$L$ relation, just beyond the $1\sigma$ dispersion. However, considering the large uncertainties, there does not appear to be any evidence that 3C382 has an anomalous H$\beta$ lag for its luminosity.
Virialization of the BLR
------------------------
{width="\textwidth"}
With the measurement of BLR velocity dispersions at a range of radii, it is possible to test if the BLR is virialized. Virialized dynamics predict $V(r) \propto r^{-1/2}$, where the constant of proportionality depends on the SMBH mass and BLR inclination/kinematics. If the BLR is virialized, the virial products $\sigma_L^2c\tau/G$ derived from different line species should be consistent with each other, assuming similar geometries and dynamics for the line-emitting gas.
In Table\[tab:masses\], the maximum differences between $\log \sigma_L^2c\tau/G$ for each object are $3.3\sigma$ in MCG+08-11-011, $2.8\sigma$ in NGC2617, $1.2\sigma$ in NGC4051, and $0.4\sigma$ in Mrk374. For NGC4051 and Mrk374, these differences are not significant. For MCG+08-11-011 and NGC 2617, the H$\beta$ and [He[ii]{}$\lambda 4686$]{} virial products are marginally discrepant at about 2.5–3.3$\sigma$. We show these results Figure\[fig:virial\], which displays the linewidths $\sigma_{\rm L}$ as a function of lag $\tau_{\rm multi}$, and the relation $\sigma_{\rm L}\propto \tau_{\rm
multi}^{-1/2}$ normalized by the value for H$\beta$. In this figure, we have applied a 0.13 dex uncertainty to both the lag $\tau$ and line width $\sigma_{\rm L}$, representative of the characteristic systematic uncertainties. While the H$\gamma$ points generally agree with the H$\beta$ relation, the [He[ii]{}$\lambda 4686$]{} points have very large offsets.
There are many systematic issues that could account for these differences. As discussed in §4, the red wing of [He[ii]{}$\lambda 4686$]{} is blended with the blue wing of H$\beta$ in both MCG+08-11-011 and NGC2617. The [He[ii]{}$\lambda 4686$]{} velocity is therefore likely underestimated because we cannot follow its red wing underneath H$\beta$. The [He[ii]{}$\lambda 4686$]{} lags are also small compared to the monitoring cadence, and the lag is only marginally detected at 2.3$\sigma$ in NGC2617. Furthermore, the choice of line window and continuum interpolation can have a significant effect on the lag and linewidths. Finally, we must assume that the 5100Å continuum light curve is a suitable proxy for the ionizing flux variations at extreme UV wavelengths. In NGC5548, we found a $\sim\! 2$ day lag between the far UV and optical emission [@Edelson2015; @Fausnaugh2016]. If a similar lag exists in these objects, it would change the [He[ii]{}$\lambda 4686$]{} virial products by a significant amount (0.3–0.4 dex), while the change in the H$\beta$ virial products would be much smaller (0.05–0.11 dex). The effect of adding a 2 day UV-optical lag to the optical-line lags is shown in Figure \[fig:virial\], and the additional lag would reduce the discrepancies in the virial products to 1.3$\sigma$ for MCG+08-11-011 and 2.0$\sigma$ for NGC2617. These AGN have masses and luminosities similar to NGC5548, so the existence of a UV-optical lag of this magnitude is very likely. Although a UV-optical lag affects the virial product and the characteristic size of the BLR, it does not affect the final mass estimate because the virial factor $\langle f
\rangle$ is calibrated using the $M_{\rm BH}$–$\sigma_{*}$ relation (see @Fausnaugh2016 [@Pei2017]).
If the remaining discrepancies are real, they indicate different dynamics and geometries for the [He[ii]{}$\lambda 4686$]{} line-emitting gas compared to that of H$\beta$. This might be plausible, since [He[ii]{}$\lambda 4686$]{} is a high-ionization state line and may originate in very different physical conditions than the Balmer lines (for example, a disk wind). If [He[ii]{}$\lambda 4686$]{} has different dynamics than H$\beta$, it would be necessary to calibrate a different virial factor $\langle f \rangle$ for the [He[ii]{}$\lambda 4686$]{} line when calculating the SMBH masses. However, we cannot rule out systematic effects and it is unclear if the [He[ii]{}$\lambda 4686$]{} discrepancies are physical. If systematic issues do account for the discrepancies, then the dynamics of the BLRs in these AGN would be consistent with virialized motion, as has been found for other AGN [@Peterson2004].
The H$\beta$ light curves and line profiles have much higher S/N and very clear lags compared to both [He[ii]{}$\lambda 4686$]{} and H$\gamma$, resulting in more reliable black hole masses. If we combine the virial products in Table\[tab:masses\] using an error-weighted average, the virial relation changes little, as shown in Figure\[fig:virial\] with the dashed lines. We therefore take the H$\beta$ masses for our standard SMBH mass estimates.
{width="\textwidth"}
Photoionization Physics
-----------------------
Photoionization models make predictions about the structure of the BLR that can be tested with RM of multiple recombination lines. The locally optimally emitting cloud model [@Baldwin1995] provides a natural explanation for the general similarity of AGN spectra, and predicts radial stratification of the BLR—high-ionization state lines, such as He[ii]{}$\lambda$1640/4686 and C[iv]{}$\lambda
1549$, should be primarily emitted at smaller radii than low-ionization state lines such as H$\beta$ and Mg[ii]{}$\lambda
2798$. @Korista2004 [hereinafter KG04] use this model to predict that the responsivity of high-order Balmer lines should be greater than that of low-order lines (in the sense that ${\rm H\gamma > H\beta
> H\alpha}$). KG04 also predict that high-ionization state lines such as [He[ii]{}$\lambda 4686$]{} should have greater responsivity than all of the Balmer lines. Radial stratification of the BLR in NGC5548 was first observed by @Clavel1991, and has since been observed in several other objects (@Peterson2004 [@Grier2013]). In addition, the expected trends of responsivity with ionization state/species have been confirmed in 16 AGN by LAMP [@Bentz2010; @Barth2015].
We confirm these results for the four objects with multiple line light curves presented here. The [He[ii]{}$\lambda 4686$]{} lags in MCG+08-11-011 and NGC2617 are less than 2 days, while the H$\beta$ lags are 14.82 and 6.38 days, respectively, clearly indicating radial stratification. Furthermore, the fractional variability of the light curves, as measured by $F_{\rm
var}$ (Table\[tab:lc\_prop\]), is generally larger for H$\gamma$ than H$\beta$ (or comparable for NGC4051 and Mrk374), while $F_{\rm var}$ for [He[ii]{}$\lambda 4686$]{} is always much greater than for the Balmer lines (although it is only slightly higher in NGC 4051). This implies that the relative line responsivities are [He[ii]{}$\lambda 4686$]{} $\gg$ H$\gamma$ $>$ H$\beta$, in agreement with the photoionization models. We also find that the H$\gamma$ lags are slightly shorter than the H$\beta$ lags within the same object (except for NGC4051). KG04 show that shorter lags are a natural consequence of the higher responsivity of H$\gamma$ compared to H$\beta$.
The formal definition of the responsivity of an emission line is $$\begin{aligned}
\eta_{\rm line} = \frac{\Delta \log F_{\rm line}}{\Delta \log \Phi}\end{aligned}$$ where $F_{\rm line}$ is the line flux and $\Phi$ is the photoionizing flux (KG04). The parameter $\eta_{\rm line}$ is therefore a measure of how efficiently the BLR converts a [*change*]{} in the photoionizing flux into a [*change*]{} in line emission. The ionizing flux $\Phi$ cannot be observed directly because these photons are at far UV wavelengths ($<912$Å). Therefore, we cannot measure $\eta_{\rm
line}$ directly, but we can measure the relative responsivity $\eta_{\rm line1}/\eta_{\rm line2} = \Delta \log F_{\rm line1}/\Delta
\log F_{\rm line2}$.
We present rough measurements of the relative responsivity of H$\beta$, H$\gamma$, and [He[ii]{}$\lambda 4686$]{} in Figure\[fig:responsivity\]. For each object, we first removed the lags of each line from the corresponding light curve. We then matched observed points to the nearest day between the H$\beta$ light curves and H$\gamma$ or [He[ii]{}$\lambda 4686$]{}light curves. The ratio $\eta_{\rm line}/\eta_{\rm H\beta}$ then corresponds to the slope of a linear least-squares fit to the data in the $\log F_{\rm H\beta}$-$\log F_{\rm line}$ plane.
We find that $\eta_{\rm H\gamma}/\eta_{\rm H\beta}$ ranges from $0.74$ to $1.44$ and that $\eta_{\rm He{\sc II}}/\eta_{\rm H\beta}$ ranges between $0.73$ and $6.23$. NGC4051, with $\eta_{\rm He{\sc
II}}/\eta_{\rm H\beta} \sim 0.73$, is an outlier, probably caused by over-subtracting the continuum before integrating the line flux. For comparison, KG04 calculate $\eta_{\rm line}$ for a fiducial model of the BLR in NGC5548, which includes an empirically motivated but [*ad hoc*]{} parameterization of the ionizing flux. From their Table 1, $\eta_{\rm H\gamma}/\eta_{\rm H\beta}$ ranges between 1.03 and 1.07, depending on the flux state of the AGN, while $\eta_{\rm He{\sc
II}}/\eta_{\rm H\beta}$ ranges from 1.26 to 1.61. Thus, while our fits for $\eta_{\rm H\gamma}/\eta_{\rm H\beta}$ are in reasonable agreement with this fiducial model, the values of $\eta_{\rm He{\sc
II}}/\eta_{\rm H\beta}$ are much larger than the model’s prediction. The spread of $\eta_{\rm line}/\eta_{\rm H\beta}$ in our fits is also fairly large, which may indicate a diversity of photoionization conditions in the BLRs of different objects (perhaps due to harder or softer ionizing fluxes than assumed for NGC5548).
Our estimates of the relative responsivities are sensitive to the total flux of the line light curves. For example, the sublinear slopes for $\eta_{\rm H\gamma}/\eta_{\rm H\beta}$ in NGC4051 and Mrk374 could be explained by missing variable line flux, perhaps in the wings of the line during low-flux states, or excess constant flux from the narrow emission lines or host-galaxy starlight. On the other hand, large values of $\eta_{\rm He{\sc II}}/\eta_{\rm H\beta}$ might be explained by contamination by Fe[ii]{} lines or misestimation of the continuum.
Summary and Future Prospects
============================
We have presented the initial analysis of data from an intensive RM monitoring campaign carried out in the first half of 2014. We succeeded in measuring continuum-line lags for six targets, five of which are presented here. (For NGC5548, see @Pei2017.) Our main results are:
i. Four new SMBH masses, as well as a refined measurement for NGC 4051.
ii. In addition to measuring H$\beta$ lags for all five targets, we measure H$\gamma$ lags in four objects and [He[ii]{}$\lambda 4686$]{} lags in two objects.
iii. Using the [He[ii]{}$\lambda 4686$]{} lags (or their upper limits), we show that the BLR is radially stratified. Although the [He[ii]{}$\lambda 4686$]{} virial products are somewhat smaller than those derived from H$\beta$, systematic effects such as blending in the line wings and the choice of continuum interpolation may account for these discrepancies. The BLRs are otherwise consistent with virialized dynamics with $V(r) \propto r^{-1/2}$.
iv. We find that [He[ii]{}$\lambda 4686$]{} is more responsive than the Balmer lines, and that H$\gamma$ is more responsive than H$\beta$, in agreement with predictions from photoionization modeling.
Many modern RM experiments are focused on measuring velocity-resolved reverberation signatures, in order to investigate the geometry and dynamics of the BLR. There are only six AGN with published velocity-delay maps [@Ulrich1996; @Bentz2010; @Grier2013] and five AGN with direct BLR dynamical models (@Pancoast2014b; one AGN, Arp 151, has both). The data presented here are of exceptional quality and very well-calibrated—based on the cadence and S/N of these observations, we have an excellent prospect of recovering velocity-delay maps and dynamical models in three objects (MCG+08-11-011, NGC2617, and NGC4051). This will expand the sample of AGN with detailed BLR information by $\sim$30%, demonstrating the continuing importance of targeted and intensive monitoring campaigns.
M.M.F. acknowledges financial support from a Presidential Fellowship awarded by The Ohio State University Graduate School. NSF grant AST-1008882 supported M.M.F., G.D.R., B.M.P., and R.W.P., M.C.B. gratefully acknowledges support through NSF CAREER grant AST-1253702 to Georgia State University. K.D.D. is supported by an NSF AAPF fellowship awarded under NSF grant AST-1302093. C.S.K. is supported by NSF grant AST-1515876. K.H. acknowledges support from STFC grant ST/M001296/1. This material is based in part upon work supported by the National Science Foundation (NSF) Graduate Research Fellowship Program under Grant No. DGE-0822215, awarded to C.B.H. A.M.M. acknowledges the support of NSF grant AST-1211146. M.E. thanks the members of the Center for Relativistic Astrophysics at Georgia Tech, where he was based during the observing campaign, for their warm hospitality. J.S.S. acknowledges CNPq, National Council for Scientific and Technological Development, Brazil. J.T. acknowledges support from NSF grant AST-1411685. Work by S.V.Jr. is supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1343012. Work by W.Z. was supported by NSF grant AST-1516842. T.W.-S.H. is supported by the DOE Computational Science Graduate Fellowship, grant number DE-FG02-97ER25308. E.R.C. and S.M.C. gratefully acknowledge the receipt of research grants from the National Research Foundation (NRF) of South Africa. T.T. acknowledges support by the National Science Foundation through grant AST-1412315 “Collaborative Research: New Frontiers in Reverberation Mapping,” and by the Packard Foundation through a Packard Research Fellowship. D.J.S. acknowledges support from NSF grants AST-1412504 and AST-1517649. A.J.B. and L.P. have been supported by NSF grant AST-1412693. B.J.S. is supported by NASA through Hubble Fellowship grant HF-51348.001 awarded by the Space Telescope Science Institute that is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555.
This work makes use of observations with the NASA/ESA Hubble Space Telescope. MCB acknowledges support through grant [*HST*]{} GO-13816 from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. This work is based on observations obtained at the MDM Observatory, operated by Dartmouth College, Columbia University, Ohio State University, Ohio University, and the University of Michigan. This paper is partly based on observations collected at the Wise Observatory with the C18 telescope. The C18 telescope and most of its equipment were acquired with a grant from the Israel Space Agency (ISA) to operate a Near-Earth Asteroid Knowledge Center at Tel Aviv University. The Fountainwood Observatory would like to thank the HHMI for its support of science research for undergraduate students at Southwestern University. This research has made use of NASA’s Astrophysics Data System, as well as 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.
[^1]: <https://github.com/mmfausnaugh/mapspec>
[^2]: We used two different templates for the “standard” giant elliptical spectrum: observations of the E0 galaxy NGC 1407 used to construct empirical templates [@Kinney1996; @Denney2009a], and a synthetic stellar population model from @Bruzual2003 consisting of a single 11 Gyr population at solar metallicity.
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---
abstract: 'Given a single input rainy image, our goal is to visually remove rain streaks and the veiling effect caused by scattering and transmission of rain streaks and rain droplets. We are particularly concerned with heavy rain, where rain streaks of various sizes and directions can overlap each other and the veiling effect reduces contrast severely. To achieve our goal, we introduce a scale-aware multi-stage convolutional neural network. Our main idea here is that different sizes of rain-streaks visually degrade the scene in different ways. Large nearby streaks obstruct larger regions and are likely to reflect specular highlights more prominently than smaller distant streaks. These different effects of different streaks have their own characteristics in their image features, and thus need to be treated differently. To realize this, we create parallel sub-networks that are trained and made aware of these different scales of rain streaks. To our knowledge, this idea of parallel sub-networks that treats the same class of objects according to their unique sub-classes is novel, particularly in the context of rain removal. To verify our idea, we conducted experiments on both synthetic and real images, and found that our method is effective and outperforms the state-of-the-art methods.'
author:
- Ruoteng Li
- 'Loong-Fah Cheong'
- 'Robby T. Tan'
bibliography:
- 'egbib.bib'
title: 'Single Image Deraining using Scale-Aware Multi-Stage Recurrent Network'
---
Introduction
============
Rain, particularly heavy rain, can impair visibility considerably. Individual rain streaks mar object’s appearance with their specular highlights, refraction, scattering and blurring effects. Distant rain streaks accumulated along the line of sight degrade the visibility of a background scene by creating fog-like veiling effect. Moreover, as a result of the projection process, rain streaks in the image have different sizes and densities, depending on their distances to the camera; these different rain layers with different sizes and densities are confusedly overlaid upon each other in the image, severely aggravating the problem. All these can affect the performance of current computer vision algorithms, particularly those assuming clear visibility.
![ Comparison of our method with state-of-the-art deraining algorithms. **Top Left**: Input image. **Top Right**: Result of [@Yang_2017_CVPR]. **Bottom Left**: Result of [@Fu_2017_CVPR]. **Bottom Right**: Result of the proposed method. []{data-label="fig:Cover"}](fig/Example/garden.jpg "fig:"){width="0.495\linewidth"} ![ Comparison of our method with state-of-the-art deraining algorithms. **Top Left**: Input image. **Top Right**: Result of [@Yang_2017_CVPR]. **Bottom Left**: Result of [@Fu_2017_CVPR]. **Bottom Right**: Result of the proposed method. []{data-label="fig:Cover"}](fig/Example/garden-alamy.jpg "fig:"){width="0.495\linewidth"} ![ Comparison of our method with state-of-the-art deraining algorithms. **Top Left**: Input image. **Top Right**: Result of [@Yang_2017_CVPR]. **Bottom Left**: Result of [@Fu_2017_CVPR]. **Bottom Right**: Result of the proposed method. []{data-label="fig:Cover"}](fig/Example/Xueyang_garden.jpg "fig:"){width="0.495\linewidth"} ![ Comparison of our method with state-of-the-art deraining algorithms. **Top Left**: Input image. **Top Right**: Result of [@Yang_2017_CVPR]. **Bottom Left**: Result of [@Fu_2017_CVPR]. **Bottom Right**: Result of the proposed method. []{data-label="fig:Cover"}](fig/Example/Ours_garden.jpg "fig:"){width="0.495\linewidth"}
A few rain streaks removal techniques have been proposed in the past decade to eliminate the rain streaks and restore visibility. Video-based rain streak removal methods (e.g. [@Garg:2006][@Kim_2015_TIP]) focus on image recovery from video sequence by exploring temporal information and frequency properties of rain. Some single-image based rain removal methods regard the problem as blind signal separation problem (e.g. [@Li_2016_CVPR]). Recently, deep learning methods have been applied to this area and have demonstrated their advantages in recovering background scenes from rainy images [@Yang_2017_CVPR][@Fu_2017_CVPR][@DBLP:journals/corr/FuHDLP16]. While all the aforementioned methods have demonstrated some degree of success, it is fair to say that they have not been subject to the full force of the tropical heavy rain and been tested where the scenes contain a range of depths. Both these factors render the deraining problem much harder; not only do we need to deal with a diverse range of rainfall from slight drizzle to an almost solid curtain of water, the rain might appear thinner in the foreground but thicker in the more distant areas in the same image. To make deraining algorithms resilient to these more severe conditions, there are several limitations of current algorithms that need to be carefully addressed.
First of all, most of the existing rain streaks removal methods ([@Luo_SparseCoding][@Kang12Rain][@Li_2016_CVPR]) are developed based on an assumption that the rain streaks distributed on a captured image is sparse. However, in the real world, particularly in the case of heavy rain or even in the case of moderate rain, if the scene extends far enough in depth, the dense rain streaks accumulation makes this assumption invalid as shown in Fig. \[fig:RainExample\]. Second, rain streaks of different sizes overlapping each other can cause ambiguity or even unintelligibility for feature-based and learning-based methods as shown in Fig. \[fig:Overlap\] [@Yang_2017_CVPR][@Li_2016_CVPR]. Although the appearance of an individual rain streak follows a rain model as described in [@Garg:2006], the large number of streaks overlapping in various sizes and densities at best significantly expands the feature space to learn, and at worst produces a rain image comprising of several scales of phenomena interpenetrating one another, rendering the existing feature-based learning methods [@Kang12Rain][@chen2013generalized][@Luo_SparseCoding][@Li_2016_CVPR][@Yang_2017_CVPR] inefficient in correctly detecting rain streaks. Finally, the atmospheric veiling effect caused by light scattering process of both tiny and large rain droplets plays an important role in degrading the visibility of a rainy scene (Fig.\[fig:RainExample\]). However most deraining algorithms did not address this problem properly.
![An example of heavy rain image. **Green** window: The compounding result of rain streaks accumulation and veiling effect. **Yellow** window: Rain streaks of various size overlap on each other. **Blue** and **Red** window: The fog-like veiling effect varies according to the object depth. Further object has stronger veiling effect. []{data-label="fig:RainExample"}](fig/Example/RainExample.jpg){width="1.0\linewidth"}
Considering the aforementioned limitations, our goal is to develop a novel method that is capable of removing rain streaks and rain accumulation from a single image under various conditions, ranging from slight rain to heavy rain, and thus enhancing the visibility of the image. To achieve our goal, we introduce a multi-stage scale-aware convolutional neural network. Our main idea here is that different sizes of rain-streaks visually degrade the scene in different ways. Large nearby streaks obstruct larger regions and are likely to reflect specular highlights more prominently than smaller distant streaks [@Tamburo2014]. These different effects of different streaks have their own characteristics in their image features, and thus need to be treated differently. Thick rain streaks also tend to be lower in density and thus need larger spatio-temporal windows to properly analyze them. One might think that the different layers of a deep learning algorithm might be able to do this automatically, but when the differently-sized rain streaks are so inextricably mixed together, we contend that this is far from being the cases. To realize the different treatments effectively, we create parallel sub-networks that are trained and made aware of these different scales of rain streaks. To our knowledge, this idea of parallel sub-networks that treat the same class of objects according to their unique sub-classes is novel, particularly in the context of rain removal.
Our scale-aware multi-stage convolutional neural network consist of three parts. First, we adopt DenseNet [@Huang_2017_CVPR] as a backbone to extract general features. Next to it is parallel sub-networks, each of which is trained to estimate rain streaks intensity map at a scale. These parallel sub-networks are recurrent convolutional layers with shortcut connections [@He_2015_CVPR], which are iteratively refined to produce better rain streaks predictions. Then the input image subtracts the summed results of all the subnetworks and feeds forward to the next stage of parallel recurrent sub-networks. In each of the subsequent stages, the recurrent subnetworks predict the residual rain streaks based on the proceeding subtraction results from previous stage. Finally, the estimated rain streak maps will be combined with the input image to restore a clean background scene.
![ An example of real rain image. Rain streaks of different sizes may appear at the same time, and overlap each other. The enlarged windows demonstrate the different features of the rain streaks. []{data-label="fig:RainStreakComparison"}](fig/Example/overlap_diff_streak.jpg){width="1.0\linewidth"}
Our key contributions can be summarized as follows:
1. We introduce an end-to-end network to remove both the rain streaks and their accumulation effect.
2. The proposed scale-aware network addresses the overlapping rain streaks of different sizes and densities using parallel recurrent sub-networks. Each sub-network is trained to extract rain streak features at a certain range of rain-streak sizes, decomposing the task of learning various streak sizes into learning smaller ranges of sizes. The reason of doing this is twofold. Firstly, different size of rain streaks is visually different and thus has different image features as shown in Fig. \[fig:RainStreakComparison\]. Secondly, these differently-sized rain streaks manifest themselves at different densities — thick rain streaks occur more sparsely — and thus the scale of analysis for each subtask should be different.
3. Our method is able to remove the veiling effects created by atmospheric light scattering process of the accumulated rain droplets. A few CNN-based image enhancement methods (e.g. [@dehaze][@Yang_2017_CVPR]) require post-processing and suffer from darkened output results. Based on our introduced rain model, our new formulation and network architecture allow our network to do end-to-end training and to recover the sharp background with brighter and richer preserved details.
4. Our network outperforms the state-of-the-art deraining methods on both synthetic and real rain datasets.
In our investigation, we find that using DenseNet [@Huang_2017_CVPR] to extract general features improves the deraining quality significantly. Comparing the results with those of simple shallower convolutional networks, our experiments show the deeper network structure performs better.
Related Works
=============
There are a number of methods proposed to improve the visibility of rain images, and we can categorize them into video-based and single-image based methods.
#### Video Based Methods
Early rain streaks removal methods focus on rain removal from video. Garg and Nayar’s [@Garg:2006] assumes the background scene to be static and explores the temporal information of dynamic rain streaks to detect the streak location, where the intensity change is larger than a predefined threshold. Having detected the rain streaks, it further removes the rain streaks by taking the average intensity of the pixels taken from the previous and subsequent frames. This method is applicable only on static scenes. Bossu et al.’s [@Bossu2011] proposes a rain detection algorithm based on the concept of foreground-background separation. The foreground model is used to detect rain streaks by applying selection rules based on the photometric properties of rain, which assumes a raindrop is a moving object brighter than the background. A histogram of orientation of rain streaks (HOS) is used to reject those detected pixels that do not correspond to rain streaks. Kim et al.’s [@Kim_2015_TIP] utilizes optical flow and only needs three successive frames to detect rain and also differentiate rain from other moving objects. It obtains the initial rain map by comparing a frame with the warped image of the subsequent frame using optical flow. The initial map is decomposed into valid rain streaks and non-rain streaks. Similar method utilizing temporal information has been applied to image restoration under water [@DBLP:conf/iccv/TianN09].
![An example of heavy rain accumulation and results of two state-of-the-art rain removal methods. **Left**: Input rainy image. **Middle**: Results of [@Li_2016_CVPR]. Though this method removes some small-sized rain streaks, it cannot remove most of the long streaks effectively. **Right**: Results of [@Yang_2017_CVPR]. This method removes thinner rain streaks, but the thicker and wider rain streaks are left behind. []{data-label="fig:Overlap"}](fig/Example/overlapBox.jpg "fig:"){width="0.325\linewidth"} ![An example of heavy rain accumulation and results of two state-of-the-art rain removal methods. **Left**: Input rainy image. **Middle**: Results of [@Li_2016_CVPR]. Though this method removes some small-sized rain streaks, it cannot remove most of the long streaks effectively. **Right**: Results of [@Yang_2017_CVPR]. This method removes thinner rain streaks, but the thicker and wider rain streaks are left behind. []{data-label="fig:Overlap"}](fig/Example/overlap_lp.jpg "fig:"){width="0.325\linewidth"} ![An example of heavy rain accumulation and results of two state-of-the-art rain removal methods. **Left**: Input rainy image. **Middle**: Results of [@Li_2016_CVPR]. Though this method removes some small-sized rain streaks, it cannot remove most of the long streaks effectively. **Right**: Results of [@Yang_2017_CVPR]. This method removes thinner rain streaks, but the thicker and wider rain streaks are left behind. []{data-label="fig:Overlap"}](fig/Example/overlap_4iter.jpg "fig:"){width="0.325\linewidth"}
![An example of heavy rain accumulation and results of two state-of-the-art rain removal methods. **Left**: Input rainy image. **Middle**: Results of [@Li_2016_CVPR]. Though this method removes some small-sized rain streaks, it cannot remove most of the long streaks effectively. **Right**: Results of [@Yang_2017_CVPR]. This method removes thinner rain streaks, but the thicker and wider rain streaks are left behind. []{data-label="fig:Overlap"}](fig/Example/overlap_cr.jpg "fig:"){width="0.325\linewidth"} ![An example of heavy rain accumulation and results of two state-of-the-art rain removal methods. **Left**: Input rainy image. **Middle**: Results of [@Li_2016_CVPR]. Though this method removes some small-sized rain streaks, it cannot remove most of the long streaks effectively. **Right**: Results of [@Yang_2017_CVPR]. This method removes thinner rain streaks, but the thicker and wider rain streaks are left behind. []{data-label="fig:Overlap"}](fig/Example/overlap_lp_cr.jpg "fig:"){width="0.325\linewidth"} ![An example of heavy rain accumulation and results of two state-of-the-art rain removal methods. **Left**: Input rainy image. **Middle**: Results of [@Li_2016_CVPR]. Though this method removes some small-sized rain streaks, it cannot remove most of the long streaks effectively. **Right**: Results of [@Yang_2017_CVPR]. This method removes thinner rain streaks, but the thicker and wider rain streaks are left behind. []{data-label="fig:Overlap"}](fig/Example/overlap_4iter_cr.jpg "fig:"){width="0.325\linewidth"}
#### Single-Image Based Methods
For single-image rain streak removal, Kang et al.’s [@Kang12Rain] introduces a method that decomposes an input image into its low frequency component (structure layer) and a high-frequency component (texture layer). The high-frequency layer, which contains rain streaks, is used to extract rain streaks component and background details using sparse-coding based dictionary learning. Luo et al’s [@Luo_SparseCoding] proposes a discriminative sparse coding framework. Its objective function employs a dictionary, its background coefficients and rain layer coefficients. The goal of the objective function is to learn the background and rain layer by forcing the coefficient vector to be sparse.
Li et al’s [@Li_2016_CVPR] decomposes the rain image into rain-free background layer and rain streak layer by utilizing Gaussian Mixture Models (GMMs) as a prior of background and rain streaks layers. The GMM prior for the background is learned from natural images, while that for the rain layer is learned from the input rain image. Fu et al’s [@Fu_2017_CVPR] is a deep convolutional network that is based on Kang et al’s idea [@Kang12Rain]. The network receives high-frequency component of an input rainy image and learns the negative residual map of rain streaks. The output of the network is then added back to the low-frequency component of the input rainy image to restore the clean background. Yang et al’s [@Yang_2017_CVPR] is a CNN based method that learns to detect and remove the rain streaks simultaneously from a single image. The network uses a contextualized dilated network to learn a pool of features. From the features, a binary rain region mask is learnt to detect the rain streak location. Subsequently, the rain region mask is fed into the network to further learn rain streak intensity, and hence the clean image can be restored by subtracting the rain intensity map from input rain image.
Rain Models
===========
The widely used rain model describes the observed rain image $ \mathbf{O}$ as a linear combination of the rain-free background scene $ \mathbf{B} $ and the rain streak layer $ \mathbf{R} $ [@Luo_SparseCoding][@Li_2016_CVPR][@Huang_2012_ICM] :
$$\mathbf{O} = \mathbf{B} + \mathbf{R}.
\label{eq:LinearSuperposition}$$
The objective for any rain streak removal algorithm is to remove the rain streaks layer $\mathbf{R}$ from the rainy image $\mathbf{O}$ to obtain the background scene $ \mathbf{B}$.
Rain removal algorithms based on Eq. (\[eq:LinearSuperposition\]) assume the rain streaks sparse and utilize individual rain streak characteristics (e.g. [@Li_2016_CVPR][@Kim_2015_TIP]) to separate background and rain streak layers. Unfortunately, in the real world, rain appearance does not depend only on individual rain streaks, but also on the accumulation of multiple rain streaks in the space from the camera to the background scene, as shown in Fig. \[fig:RainExample\].
In an image, the projected appearance of the rain streaks will have different sizes and densities. The further away the streaks, the smaller the size and the denser the imaged rain streaks. If we assume rain streaks at the same depth as one layer, we divide the captured rainy scene into a clean background and multiple layers of rain streaks, each of which has approximately the same size and density. Based on this, we can generalize Eq. (\[eq:LinearSuperposition\]) to model a rainy scene: $$\mathbf{O} = \mathbf{B} + \sum_{i}^{n}\mathbf{R_i},
\label{eq:LinearSuperpositionMultiple}$$ where $n$ is the number of rain-streak layers along the line of sight to the background objects. $\mathbf{R_i}$ represents the pixel intensity of rain streaks at layer $i$.
According to [@Kaushal2017], rain droplets can cause light scattering and attenuation. Thus, the resultant visibility under moderate and heavy rain conditions are similar to those under haze and fog. The light scattering process contributes to the atmospheric veiling effect in a typical heavy rainy scene (see blue window in Fig. \[fig:RainExample\]). In this case, the purely additive rain model introduced in Eq. (\[eq:LinearSuperposition\]) does not fully capture the appearance of rain accumulation, and as a result, the existing rain removal methods based on Eq. (\[eq:LinearSuperposition\]) cannot handle it. To address this, we further generalize the rain model: $$\mathbf{O} = \boldsymbol{\alpha} \odot (\mathbf{B} + \sum_{i}^{n}\mathbf{R_i} ) + (\boldsymbol{1} - \boldsymbol{\alpha}) \odot \mathbf{A},
\label{eq:RainStreakAccumulationEquation}$$ where $\odot$ indicates element-wise multiplication, $\boldsymbol{\alpha}$ is a 2D map representing the transmittance introduced by the scattering process of rain droplets, and $\mathbf{A}$ is a 2D map representing the atmospheric light of the scene. This model is similar to that proposed in [@Yang_2017_CVPR], although we remove the binary mask in our model.
Deraining Method
================
\[h!\] {width="1.0\linewidth"}
Our scale-aware multi-stage recurrent deraining network is illustrated in Fig. \[fig:NetArch\]. Unlike the existing CNN-based deraining methods, we create multiple recurrent sub-networks to handle rain streaks at different sizes and densities. The veiling effect of rain accumulation, $(\boldsymbol{1}-\boldsymbol{\alpha})\mathbf{A}$, can be deemed as another layer and thus can be processed using another recurrent sub-network in parallel to those that handle rain streaks. However, since the underlying physical process generating the veiling effect is different from that of rain streaks (Fig. \[fig:RainExample\] Green), the network architecture treats this effect differently. For ease of discussion, we will first discuss our network that deals only with rain streaks, called ’**SMRNet**’ (Scale-aware Multi-stage Recurrent Network) and shown in Fig. \[fig:NetArch\](a). Then, we will discuss the integration of this network with the sub-network that deals with the veiling effect, which we call ’**SMRNet-veil**’, shown in Fig. \[fig:NetArch\](b).
SMRNet
------
Focusing on multiple layers and overlapping rain streaks, based on Eq. (\[eq:LinearSuperpositionMultiple\]), our goal is to estimate the rain-free background $\mathbf{B}$ and each rain streak intensity map $\mathbf{R_1}, \mathbf{R_2}, ..., \mathbf{R_n} $ given the input rainy image $\mathbf{O}$. Generally, we want to minimize $\parallel \mathbf{O} - \mathbf{B} - \sum_{i}^{n}\mathbf{R_i} \parallel_F^2$, where $\parallel \cdot \parallel_F$ is the Frobenius norm. This is a totally ill-posed problem, even when we regard there is only one layer of rain streaks. Early rain streak removal methods like [@Li_2016_CVPR][@chen2013generalized] use hand-crafted features or data-driven features for the priors of $\mathbf{B}$ and $\mathbf{R_i}$. However, in our method, the priors are learned by the network from the training data. In order to obtain the priors of $\mathbf{B}$ and $\mathbf{R_i}$ during the training phase, we add the estimation loss of $\mathbf{B}$ and $\mathbf{R_i}$ as in our objectives: $$\mathcal{L} = \mathcal{L_B} + \sum_{i}^{n} \mathcal{L}_{\mathcal{R}_i},
\label{eq:Lossfunction}$$ where $\mathcal{L_B}$ represents reconstruction loss for the background scene, and $\mathcal{L}_{\mathcal{R}_i}$ represents the loss for estimating the $i^{th}$ rain streak layer.
As illustrated in Fig. \[fig:NetArch\], our network first adopts DenseNet [@Huang_2017_CVPR] to extract rain image features $\mathbf{F}$. However, we remove the ’transition layers’ (1 conv layer followed by 1 pooling layer) between each ’Dense block’ from the DenseNet so that our network does not downsample the image. The features are then fed into a series of parallel recurrent convolutional sub-networks to learn rain intensity map $\mathbf{R_i}$ at different scales and densities. Since each recurrent sub-network focuses only on one type of rain streak feature $\mathbf{R_i}$, resources can be dedicated toward these selected rain streaks without having to accommodate the competing demands of the different rain streak types in the network with their different desiderata for representation, yielding an enhanced learning of foreground items. The recurrent module iterates four times in order to refine the estimation using the previous prediction combined with the feature representations. Then, the input image will subtract all the estimated rain intensity maps $\mathbf{R_i^{1}}$’s to obtain a temporary image $\mathbf{T_1}$ , which is a preliminary de-rained result. This preliminary result is often marred by dark streak artifacts and contains residual rain streaks. The dark streaks arise possibly due to “double removal” by two parallel sub-networks. The reasons for the residual rain streaks are manifold.
Firstly, even in a synthetic rain image, an image region may contain multiple rain streaks stacked right on top of one another. It is difficult to remove all the rain streaks in one go; instead, we are much more likely to see the removal of only the nearest and thickest rain streaks, upon which the further and finer streaks are revealed. Secondly, in the real world rain sequences, there might be effects not modelled in the training. For instance, the rainfall is simply heavier, or there are local variation of density and direction not related to the depth factor (e.g. the variation around the table in Fig. \[fig:Cover\]). These effects would pose difficulties for a single-stage network solution, even with its parallel sub-networks. In view of the preceding issues, we send the preliminary derained result to the next stage together with the feature representation $\mathbf{F}$ and the various estimated rain layers for further refinement. A predictor $\mathbf{R_i^{j}}$ in next stage can better remove the residual rain streaks because firstly the dominant signals have been removed. This argument can be understood in various senses: (1) in the case of coincident rain streaks mentioned before, the removal of the nearest (and thus brightest) rain streaks reveal the underlying fainter rain streaks; (2) local rain streaks that are inconsistent in density or direction with the global pattern are better detected and processed after the dominant global pattern is removed; (3) the rain streaks can also be dominated by the veiling effect to be discussed in the next subsection, and the latter’s removal in the first stage helps reveal rain streaks better.
The second reason for having these multiple stages is when we are faced with an unprecedented heavy rain not seen in the training. The first stage may only partially remove the rain streaks, having not seen such dense rain streak pattern before. However, the partially derained result at the end of the first stage amounts to an image of a lighter rainy scene, and we find that the successive stages can successfully remove the remaining rain streaks. The third reason is concerned with the removal of the dark streak artifacts. One can regard the concatenated predictions from the earlier stages as providing some form of explicit ’communication’ between each recurrent sub-network, leading to reduced ’duplicate work’. At the end of the network, the learned rain streaks from all the stages are concatenated together to aid the final clean image recovery.
![Examples of training data for SMR-Net (Top row left) and SMRNet-Veil (Top row right). **Middle** : Rain streak maps of the blue lake example (top left). **Bottom**: Rain streak maps of the indoor shelf (top right) example. From left to right are the corresponding maps for small-sized, mid-sized, and large-sized rain streaks respectively. (The rain streak map intensity is enhanced for visualization purpose) []{data-label="fig:Trainingdata"}](fig/Exp/TrainingData/rain-gt.jpg "fig:"){width="0.251\linewidth"} ![Examples of training data for SMR-Net (Top row left) and SMRNet-Veil (Top row right). **Middle** : Rain streak maps of the blue lake example (top left). **Bottom**: Rain streak maps of the indoor shelf (top right) example. From left to right are the corresponding maps for small-sized, mid-sized, and large-sized rain streaks respectively. (The rain streak map intensity is enhanced for visualization purpose) []{data-label="fig:Trainingdata"}](fig/Exp/TrainingData/rain-img.jpg "fig:"){width="0.251\linewidth"} ![Examples of training data for SMR-Net (Top row left) and SMRNet-Veil (Top row right). **Middle** : Rain streak maps of the blue lake example (top left). **Bottom**: Rain streak maps of the indoor shelf (top right) example. From left to right are the corresponding maps for small-sized, mid-sized, and large-sized rain streaks respectively. (The rain streak map intensity is enhanced for visualization purpose) []{data-label="fig:Trainingdata"}](fig/Exp/TrainingData/haze-gt.jpg "fig:"){width="0.223\linewidth"} ![Examples of training data for SMR-Net (Top row left) and SMRNet-Veil (Top row right). **Middle** : Rain streak maps of the blue lake example (top left). **Bottom**: Rain streak maps of the indoor shelf (top right) example. From left to right are the corresponding maps for small-sized, mid-sized, and large-sized rain streaks respectively. (The rain streak map intensity is enhanced for visualization purpose) []{data-label="fig:Trainingdata"}](fig/Exp/TrainingData/haze-img.jpg "fig:"){width="0.223\linewidth"}
![Examples of training data for SMR-Net (Top row left) and SMRNet-Veil (Top row right). **Middle** : Rain streak maps of the blue lake example (top left). **Bottom**: Rain streak maps of the indoor shelf (top right) example. From left to right are the corresponding maps for small-sized, mid-sized, and large-sized rain streaks respectively. (The rain streak map intensity is enhanced for visualization purpose) []{data-label="fig:Trainingdata"}](fig/Exp/TrainingData/rain-dense-enhance.jpg "fig:"){width="0.32\linewidth"} ![Examples of training data for SMR-Net (Top row left) and SMRNet-Veil (Top row right). **Middle** : Rain streak maps of the blue lake example (top left). **Bottom**: Rain streak maps of the indoor shelf (top right) example. From left to right are the corresponding maps for small-sized, mid-sized, and large-sized rain streaks respectively. (The rain streak map intensity is enhanced for visualization purpose) []{data-label="fig:Trainingdata"}](fig/Exp/TrainingData/rain-mid-enhance.jpg "fig:"){width="0.32\linewidth"} ![Examples of training data for SMR-Net (Top row left) and SMRNet-Veil (Top row right). **Middle** : Rain streak maps of the blue lake example (top left). **Bottom**: Rain streak maps of the indoor shelf (top right) example. From left to right are the corresponding maps for small-sized, mid-sized, and large-sized rain streaks respectively. (The rain streak map intensity is enhanced for visualization purpose) []{data-label="fig:Trainingdata"}](fig/Exp/TrainingData/rain-sparse-enhance.jpg "fig:"){width="0.32\linewidth"}
![Examples of training data for SMR-Net (Top row left) and SMRNet-Veil (Top row right). **Middle** : Rain streak maps of the blue lake example (top left). **Bottom**: Rain streak maps of the indoor shelf (top right) example. From left to right are the corresponding maps for small-sized, mid-sized, and large-sized rain streaks respectively. (The rain streak map intensity is enhanced for visualization purpose) []{data-label="fig:Trainingdata"}](fig/Exp/TrainingData/haze-dense-enhance.jpg "fig:"){width="0.32\linewidth"} ![Examples of training data for SMR-Net (Top row left) and SMRNet-Veil (Top row right). **Middle** : Rain streak maps of the blue lake example (top left). **Bottom**: Rain streak maps of the indoor shelf (top right) example. From left to right are the corresponding maps for small-sized, mid-sized, and large-sized rain streaks respectively. (The rain streak map intensity is enhanced for visualization purpose) []{data-label="fig:Trainingdata"}](fig/Exp/TrainingData/haze-mid-enhance.jpg "fig:"){width="0.32\linewidth"} ![Examples of training data for SMR-Net (Top row left) and SMRNet-Veil (Top row right). **Middle** : Rain streak maps of the blue lake example (top left). **Bottom**: Rain streak maps of the indoor shelf (top right) example. From left to right are the corresponding maps for small-sized, mid-sized, and large-sized rain streaks respectively. (The rain streak map intensity is enhanced for visualization purpose) []{data-label="fig:Trainingdata"}](fig/Exp/TrainingData/haze-sparse-enhance.jpg "fig:"){width="0.32\linewidth"}
Method
----------------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
Metric PSNR SSIM VIF FSIM PSNR SSIM VIF FSIM PSNR SSIM VIF FSIM PSNR SSIM VIF FSIM
ID [@Kang12Rain] 27.21 0.800 0.266 0.765 25.07 0.773 0.229 0.757 20.91 0.667 0.24 0.785 18.34 0.630 0.179 0.704
DSC [@Luo_SparseCoding] 30.02 0.893 0.548 0.918 28.37 0.860 0.463 0.882 24.44 0.762 0.415 0.836 21.77 0.786 0.427 0.874
LP [@Li_2016_CVPR] 32.02 0.925 0.524 0.936 29.41 0.895 0.430 0.901 24.54 0.807 0.409 0.876 20.42 0.750 0.365 0.854
Details Net [@Fu_2017_CVPR] 33.43 0.949 0.608 0.955 31.01 0.931 0.507 0.934 26.48 0.838 0.438 0.899 22.24 0.831 **0.458** **0.921**
JORDER [@Yang_2017_CVPR] 35.86 0.956 0.627 0.963 29.69 0.913 0.472 0.922 25.79 0.823 0.416 0.894 20.34 0.788 0.417 0.898
JORDER-R [@Yang_2017_CVPR] 36.02 0.934 0.516 0.937 28.25 0.889 0.396 0.899 25.16 0.801 0.352 0.878 20.20 0.768 0.357 0.880
Ours **36.23** **0.965** **0.636** **0.968** **32.82** **0.949** **0.540** **0.952** **29.66** **0.877** **0.490** **0.934** **25.29** **0.843** 0.417 0.884
\[table:SyntheticQuantitativeResult\]
Veil Module
-----------
Not only does the veiling effect, degrade visibility, its presence also hinders the complete removal of all the rain streaks effectively, as shown in Fig. \[fig:Overlap\]. For these reasons, we develop an additional module to specifically handle it, with the module placed in parallel with the recurrent rain streak sub-networks. Taking the rain accumulation transmittance $\boldsymbol{\alpha}$ from Eq. (\[eq:RainStreakAccumulationEquation\]) into consideration, the new data term takes on the form of $ \parallel \mathbf{O} - \boldsymbol{\alpha}(\mathbf{B} + \sum_{i}^{n}\mathbf{R_i}) - (\boldsymbol{1}-\boldsymbol{\alpha})\mathbf{A} \parallel_F^2$.
Rearranging Eq. (\[eq:RainStreakAccumulationEquation\]), $\mathbf{B}$ can be written as: $$\mathbf{B} = {\boldsymbol{1} \over \boldsymbol{\alpha}}(\mathbf{O} - \mathbf{A}) - \sum_{i}^{n}\mathbf{R_i} + \mathbf{A} .
\label{eq:AdjustedRainModel}$$ Based on Eq. (\[eq:AdjustedRainModel\]), the proposed network estimates $ 1 \over \boldsymbol{\alpha} $ instead of $\boldsymbol{\alpha}$ so that the clean background image $\mathbf{B}$ can be directly predicted by element-wise multiplication of $1 \over \boldsymbol{\alpha}$ and the input image $\mathbf{O}$. In this way, our network can be trained in an end-to-end manner. Hence, the corresponding loss function for SMRNet-veil model is: $$\mathcal{L} = \mathcal{L_B} + \mathcal{L}_{\boldsymbol{1} \over \boldsymbol{\alpha}} + \sum_{i}^{n} \mathcal{L}_{\mathcal{R}_i},
\label{eq:LossfunctionVeil}$$ where $\mathcal{L}_{{\boldsymbol{1}} \over {\boldsymbol{\alpha}}}$ indicates the loss for the $ \boldsymbol{1} \over \boldsymbol{\alpha} $ transmission map. In the training phase, we augment the training set with different values of $\mathbf{A}$ and assume these are known. In the test phase, we follow [@dehaze] and use the brightest pixel as $\mathbf{A}$.
Note that, in contrast to our method, previous methods [@dehaze; @Yang_2017_CVPR] estimate the transmittance map and background scene separately, requiring a different process or network to deal with the veiling effect, and thus cannot be trained end-to-end.
Experiment
==========
[@ ccccccc @]{} & Input & LP[@Li_2016_CVPR] & JORDER [@Yang_2017_CVPR] & DetailsNet [@Fu_2017_CVPR] & Ours & Ground Truth\
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Training Data
-------------
In view of the difficulty in obtaining ground truths for real rain images, we choose to render synthesized rain streaks on clean natural images to fulfill the training need of our network. For testing, we use both synthetic and real rain data. Based on the model in [@Garg:2006], we render synthesized rain streaks of multiple sizes and densities on the BSD300 [@BSD200] dataset; Eq. (\[eq:LinearSuperpositionMultiple\]) is used for the rendering for the training of SMRNet (see the left figure in Fig. \[fig:Trainingdata\]). These differently-sized rain streaks will be handled by different recurrent sub-networks. We divide these sizes into three ranges: ’small’, ’middle’, and ’large’, respectively in the range of (0,60\], (60,300\], and (300, 600\], where the size of a rain streak is measured by its occupied area (in pixel) on a rain image. In this experiment, we have synthesized 3300 rain images containing 11 different rain streak orientations.
For training SMRNet-Veil, we need depth information to render the veiling effect properly. Since the BSD300 dataset does not provide depth information, we use the NYU depth dataset [@Silberman:ECCV12] for this purpose. Specifically, we render the same rain streaks as before but with additional veiling effect generated according to Eq. (\[eq:RainStreakAccumulationEquation\]) (see the right figures in Fig. \[fig:Trainingdata\]). The transmittance $\alpha$ of a point $\mathbf{x}$ in the scene follows the free space light attenuation model [@Kaushal2017]: $$\alpha(\mathbf{x}) = \exp({- \beta d(\mathbf{x})}),
\label{eq:LightAttenuation}$$ where $d$ represents the depth of that point and the parameter $\beta$ is the attenuation factor. In generating different veiling effect, we set different values of $\beta$.
Results of Synthetic Rain Data
------------------------------
There are four synthetic datasets evaluated in our experiments. Table \[table:SyntheticQuantitativeResult\] shows the results of our method compared with other state-of-the-art rain streak removal methods. To quantitatively evaluate these methods, four evaluation metrics are used: Peak Signal-to-Noise Ratio (PSNR) [@PSNR], Structure Similarity Index (SSIM) [@SSIM], Visual Information Fidelity (VIF) [@VIF_Sheikh_2004], and Feature Similarity Index (FSIM) [@FSIM_zhang_2011].
The Rain12 dataset [@Yang_2017_CVPR][@Li_2016_CVPR] includes 12 synthesized rain images with only one type of rain streaks, which can be considered as light rain, as shown in Fig. \[fig:SyntheticRainResult\] (a). It is noteworthy that although our method does not focus on sparse light rain streak removal, its performance is still on par with the state-of-the-art performance. The Rain12S extends Rain12 dataset to include more adverse rain conditions, under which rain streaks have various sizes and densities as shown Fig. \[fig:SyntheticRainResult\] (b). From an inspection of the qualitative results, our method is able to remove all rain streaks of different sizes. In order to compare the generalization of these methods, we also render differently-sized streaks on 100 images from the COCO dataset [@DBLP:journals/corr/LinMBHPRDZ14] using the same rain rendering method (shown in Fig. \[fig:SyntheticRainResult\] (c)). Our method outperforms other methods because it can handle the thick and thin rain streaks at the same time, thus restoring a clearer background. Finally, in order to evaluate the veiling effect removal, we render synthesized rain streaks and atmospheric veils following Eq. (\[eq:RainStreakAccumulationEquation\]) on 12 images from the BSD300 dataset (different from those used in the training data; see Fig. \[fig:SyntheticRainResult\]).
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![ The performance of the networks with different number of recurrent modules[]{data-label="fig:CompareDifferentArch"}](fig/Example/CompareArch.jpg){width="0.98\linewidth"}
Results of Real Rain Data
-------------------------
Fig.\[fig:RealRainResult\] demonstrates the results of the proposed method and the recent state-of-the-art methods on real rain images. From the figure, one can see that our method is able to remove a range of different rain streaks, from dense and thin streaks to sparse and thick. However, JORDER [@Yang_2017_CVPR] cannot remove very thin nor very thick streaks (Fig. \[fig:RealRainResult\] (c)(d)). DetailsNet [@Fu_2017_CVPR] relies on the guided filter to separate high frequency signal from the real rain images. Low-frequency thick streaks in (d) cannot be fully extracted and therefore DetailsNet cannot remove them effectively. In the row (c), the rain streaks on the two sides of the image (zoom in to see clearly) are different and yet our method is able to remove all of them in one shot. For the cases with veiling effects (Fig.\[fig:RealRainResult\] (a,b,c,e)), SMRNet-veil is able to recover the clean background without suffering from blurred object boundaries.
Discussion
----------
### Evaluations on Parallel Modules
In order to better understand the effectiveness of the network architecture, we compare a series of networks with different number of parallel recurrent modules, which are used to estimate the differently sized rain streaks. We denote the network that has no recurrent module but directly estimates the clean background as **0 recurrent module** network. The proposed network with 3 parallel recurrent modules is denoted as **3 recurrent modules** network. We evaluate these networks’ performance at each epoch during training on the Rain100-COCO dataset. The performance results in PSNR metric are shown in Fig. \[fig:CompareDifferentArch\]. Due to the limited space, we include the ablation study of the effectiveness of recurrent module and the multiple stages in the supplementary material.
Conclusion
==========
In this paper, we proposed a scale-aware multi-stage recurrent network to solve rain streak and rain streak accumulation removal problem. The proposed network estimates rain streaks of different sizes and densities individually in order to reduce intra-class competition. We generated multiple synthetic rain dataset to train our network and evaluate our network on both synthesized rain datasets and real rain dataset. The results demonstrate that our method attain the state-of-the-art performance.
|
---
abstract: 'A comprehensive theory of the Weyl-Wigner formalism for the canonical pair angle-angular momentum is presented, with special emphasis in the implications of rotational periodicity and angular-momentum discreteness.'
author:
- 'I. Rigas'
- 'L. L. Sánchez Soto'
- 'A. B. Klimov'
- 'J. Řeháček'
- Z Hradil
title: Wigner function for twisted photons
---
Introduction
============
A quantum system has a dynamical symmetry group $G$ if its Hamiltonian is a function of the generators of $G$. In this case, the Hilbert space of the system splits into a direct sum invariant subspaces (carriers of the irreducible representations of $G$) and the discussion of any physical property can be restricted to one of these subspaces [@Barut:1987].
The existence of such a symmetry also allows for the explicit construction of a phase space for the system as the coadjoint orbit associated with an irreducible representation of $G$ [@Kostant:1970; @Kirillov:1976] (in fact, it turns out to be a symplectic manifold). In consequence, to every operator on Hilbert space we can associate a function on phase space, opening the way to formally representing quantum mechanics as a statistical theory on classical phase space. Various aspects of this formalism for basic quantum systems have been developed by a number of authors [@Weyl:1928; @Wigner:1932; @Moyal:1949; @Stratonovich:1956; @Agarwal:1970; @Berezin:1975; @Agarwal:1981; @Bertrand:1987; @Varilly:1989; @Atakishiyev:1998; @Brif:1998; @Benedict:1999].
There are, however, important differences with respect to a classical description. They come from the noncommuting nature of conjugate quantities, which precludes their simultaneous precise measurement and, therefore, imposes a fundamental limit to the accuracy with which we can determine a point in phase space. As a distinctive consequence of this, there is no unique rule by which we can associate a classical phase-space variable to a quantum operator and depending on the operator ordering, various functions can be defined. For example, the quantum state (i.e., the density matrix) of the system can be mapped onto a whole family of functions parametrized by a number $s$; the values $+ 1$, 0, and $-1$ corresponding to the Husimi $Q$, the Wigner $W$, and the Glauber-Sudarshan $P$ functions, respectively. These phase-space functions are known as quasiprobability distributions, as in quantum mechanics they play a role similar to that of genuine probability distributions in classical statistical mechanics (for reviews, see Refs. [@Balazs:1984; @Hillery:1984; @Lee:1995; @Schroek:1996]).
Apart from the description of the harmonic oscillator (for which $G$ is the Heisenberg-Weyl group and the corresponding phase space is the plane $\mathbb{R}^2$), this formalism has also been successfully applied to spin-like systems (or qubits in the modern parlance of quantum information), for which $G$ is the group SU(2) and the phase space is the two-dimensional Bloch sphere. However, one can rightly argue that this Wigner function, although describing a discrete system, is not defined in a discrete phase space. In fact, the growing interest in quantum information has fueled the search for discrete phase-space counterparts of the Wigner function (see Ref. [@Klimov:2008] for a complete and up-to-date review). The main advantage of such a representation consists in that even states from different irreducible representations can be pictured on the same phase space, which is basically a direct product of two-dimensional discrete tori.
There is still another “mixed” canonical pair: angle and angular momentum. Now, the symmetry group $G$ is noncompact and can be taken as the two-dimensional Euclidean group E(2), whereas the associated phase space is the discrete cylinder $\mathbb{Z} \times
\mathcal{S}_1$ ($\mathcal{S}_1$ denotes here the unit circle), since one of the variables is continuous and the other is discrete. Several interesting properties of a number of systems, such as molecular rotations, electron wave packets, Hall fluids, and light fields, to cite only a few examples, can be described in terms of this symmetry group [@Rigas:2008]. In quantum optics, it is the basic tool to deal with the orbital angular momentum of the so-called twisted photons [@Molina:2007; @Franke:2008], which have been proposed for applications in quantum experiments [@Vaziri:2002].
The construction of a proper Wigner function for this case is still under discussion. Although some interesting attempts have been published [@Mukunda:1979; @Bizarro:1994; @Mukunda:2005], they seem of difficult application to practical problems. Quite interesting group-theoretical approaches to this problem can be also found in Refs. [@Nieto:1998; @Plebanski:2000]. In this paper, we approach this interesting problem from the perspective of finite-dimensional systems and construct a *bona fide* Wigner function that fulfills all the reasonable requirements and is easy to handle and to interpret. We also discuss its applications to some relevant quantum states.
Wigner function for position-momentum {#sec:qpWig}
=====================================
In this section we briefly recall the relevant structures needed to set up the Wigner function for Cartesian quantum mechanics. This is to facilitate comparison with the angular case later on. For simplicity, we choose one degree of freedom only, so the associated phase space is the plane $\mathbb{R}^2$.
The canonical Heisenberg commutation relations between Hermitian coordinate and momentum operators ${\hat {q}}$ and ${\hat {p}}$ are (in units $\hbar = 1$) $$\label{eq:HWcom}
[{\hat {q}}, {\hat {p}}] = i \, ,$$ so that they are the generators of the Heisenberg-Weyl algebra. In the unitary Weyl form this is expressed as $$\label{eq:Weyl}
{\hat {U}} (p) {\hat {V}}(q) = {\hat {V}} (q) {\hat {U}}(p) \, e^{i q p} \, ,$$ where $${\hat {V}} (q) = \exp(- i q {\hat {p}} ) \, ,
\qquad
{\hat {U}} (p) = \exp(i p {\hat {q}} ) \, ,$$ are the generators of translations in position and momentum, respectively. In the Cartesian case, these exponentials can be entangled to define a displacement operator $$\label{eq:HWDisp1}
{\hat {D}} (q,p) = {\hat {U}} (p) {\hat {V}}(q) e^{- i q p/2} =
\exp[i(p {\hat {q}} - q {\hat {p}})] \, ,$$ with the parameters $(q,p)$ labelling phase-space points. However, this cannot be done for other canonical pairs, as we shall see.
The displacement operators form a complete trace-orthonormal set (in the continuum sense) in the space of operators acting on $\mathcal{H}$ (the Hilbert space of square integrable functions on $\mathbb{R}$): $$\label{eq:HWDispOrtho}
{\mathop{\mathrm{Tr}}\nolimits}[ {\hat {D}} (q, p) \, {\hat {D}}^\dagger (q^\prime, p^\prime) ] =
2 \pi \delta (q - q^\prime) \delta (p - p^\prime) \, .$$ Note that ${\hat {D}}^\dagger (q, p) = {\hat {D}} (-q, - p)$, while ${\hat {D}}(0,0) = {\hat {\openone}}$.
The mapping of the density matrix ${\hat {\varrho}}$ into a Wigner function defined on $\mathbb{R}^2$ is established in a canonical way: $$\begin{aligned}
\label{eq:Wigcan}
& W(q, p) = {\mathop{\mathrm{Tr}}\nolimits}[ {\hat {\varrho}} \,{\hat {w}}(q,p) ] \, , & \nonumber \\
& & \label{eq:HWWignerDef} \\
& {\hat {\varrho}} = \displaystyle
\frac{1}{(2\pi)^{2}} \int_{\mathbb{R}^{2}}{\hat {w}}(q,p) W(q,p) \, dq dp \, , &
\nonumber\end{aligned}$$ where the (Hermitian) Wigner kernel ${\hat {w}}$ (a particular instance of a Stratonovitch-Weyl quantizer) is the double Fourier transform of the displacement operator: $$\label{eq:HWkernelDef}
{\hat {w}} (q, p) = \frac{1}{(2\pi)^2} \int_{\mathbb{R}^2}
\exp[-i(p q^\prime -q p^\prime)] {\hat {D}} (q^\prime, p^\prime) \,
dq^\prime dp^\prime \, .$$ One can immediately check that the Wigner kernels are also a complete trace-orthonormal set. Furthermore, they transform properly under displacements $$\label{eq:HWKernelDisp}
{\hat {w}} (q, p) = {\hat {D}} (q,p) \,{\hat {w}} (0, 0) \,
{\hat {D}}^\dagger (q, p) \, ,$$ where $$\label{eq:Parity}
{\hat {w}}(0,0)=\int_{\mathbb{R}^{2}} {\hat {D}}(q, p) \, dq dp = 2 {\hat {P}} \, ,$$ and ${\hat {P}}$ is the parity operator.
The Wigner function in (\[eq:HWWignerDef\]) fulfills all the basic properties required for any good probabilistic description. First, due to the Hermiticity of ${\hat {w}} (q,p)$, it is real for Hermitian operators. Second, on integrating $W (q, p)$ over one variable, the probability distribution of the conjugate variable is reproduced $$\label{eq:HWProps2}
\int_\mathbb{R} W(q, p) \, dp = \langle q|{\hat {\varrho}}| q \rangle \, ,
\quad
\int_\mathbb{R} W(q, p) \, dq = \langle p|{\hat {\varrho}}| p \rangle \, .$$ Third, $W(q, p)$ is covariant, which means that for the displaced state ${\hat {\varrho}}^\prime = {\hat {D}}(q_0, p_0) \,{\hat {\varrho}} \,
{\hat {D}}^\dagger (q_0, p_0)$, one has $$\label{eq:HWProps3}
W_{{\hat {\varrho}}^\prime} (q, p) = W_{{\hat {\varrho}}} (q-q_0, p-p_0) \, ,$$ so that the Wigner function follows displacements rigidly without changing its form, reflecting the fact that physics should not depend on a certain choice of the origin.
Finally, the overlap of two density operators is proportional to the integral of the associated Wigner functions: $$\label{eq:HWProps4}
{\mathop{\mathrm{Tr}}\nolimits}( {\hat {\varrho}}_1 \,{\hat {\varrho}}_2 ) \propto
\int_{\mathbb{R}^2} W_1(q, p) W_2(q,p) \, dq dp \, .$$ This property (often known as traciality) offers practical advantages, since it allows one to predict the statistics of any outcome, once the Wigner function of the measured state is known.
Wigner function for discrete systems {#sec:Discrete}
====================================
Many quantum systems can be appropriately described in a finite-dimensional Hilbert space. The previous standard approach can be extended to these discrete systems, since they do have a dynamical symmetry group. However, in a continuous Wigner function for these systems, there is a lot of information redundancy. The goal of this section is to carry out a non-redundant discrete phase-space analysis for this case.
Let us consider a system living in a Hilbert space $\mathcal{H}_{d}$, of dimension $d$ (a qudit). It is useful to choose a computational basis $ | n \rangle $ ($n = 0, \ldots , d-1$) in $\mathcal{H}_{d}$ and introduce the basic operators [@Schwinger:1960] $$\label{CC}
{\hat {X}} | n \rangle = |n + 1 \rangle \, ,
\qquad
{\hat {Z}} | n \rangle = \omega(n) | n \rangle \, ,$$ where addition and multiplication must be understood modulo $d$ and, for simplicity, we use the notation $$\omega (m) \equiv \omega^{m} = \exp (i 2\pi m/d) \, ,$$ $\omega = \exp( i 2\pi/d)$ being a $d$th root of the unity. The operators ${\hat {X}}$ and ${\hat {Z}}$ generate a group under multiplication known as the generalized Pauli group [@Nielsen:2000] and obey $$\label{eq:ZXwXZ}
{\hat {Z}} {\hat {X}} = \omega \, {\hat {X}} {\hat {Z}} \, ,$$ which is the finite-dimensional version of the Weyl form (\[eq:Weyl\]) of the commutation relations.
The monomials $\{ \hat{Z}^{k} \hat{X}^{l}\}$ ($k,l = 0, 1, \ldots,
d-1$) form a basis in the space of all the operators acting in $\mathcal{H}_{d}$ [@Klimov:2005]. It seems then natural to introduce the unitary displacement operators $${\hat {D}}(k, l) = e^{i\phi (k,l)} {\hat {Z}}^{k} {\hat {X}}^{l} \, ,
\label{eq:desp}$$ where $\phi (k,l)$ is a phase. The unitarity condition imposes that $$\phi (k, l) + \phi (-k,-l) = - \frac{2\pi}{d} kl \, .
\label{ph_condition}$$ Different choices have been analyzed in the literature [@Vourdas:2007]; one of special relevance is $$\phi ( k, l ) = \frac{2 \pi}{d} 2^{-1} \, kl \, ,
\label{phi1}$$ where $2^{-1}$ is the multiplicative inverse of 2 in $\mathbb{Z}_{d}$ when d is prime and $2^{-1}=1/2$ for nonprime dimensions.
In this way, we have got a discrete phase space of the system as a $d\times d$ grid of points, in a such a way that the coordinate of each point $(k, l)$ define powers of $Z$ (“position”) and $X$ (“momentum”) and the whole phase space is isomorphic to a discrete two-dimensional torus.
The following mapping from the Hilbert space into the discrete phase space \[equivalent to (\[eq:Wigcan\])\] $$\begin{aligned}
& W(k, l) = {\mathop{\mathrm{Tr}}\nolimits}[ {\hat {\varrho}} \, {\hat {w}}(k, l)] \, , & \nonumber \\
& & \\
& {\hat {\varrho}} = \displaystyle
\frac{1}{d^{2}} \sum_{k,l} {\hat {w}}(k, l) W(k, l) \, , & \nonumber
\label{eq:Wigdis}\end{aligned}$$ is established in terms of the following (Hermitian) Wigner kernel $${\hat {w}}(k, l) = \frac{1}{d^{2}}
\sum_{m,n}\omega (kn-lm) \,{\hat {D}}(m,n) \, , \label{eq:kernel}$$ which is normalized, satisfies the overlap condition $${\mathop{\mathrm{Tr}}\nolimits}[ {\hat {w}}(k, l) {\hat {w}}(k^{\prime}, l^{\prime}) ] = d \,
\delta_{k, k^{\prime}} \, \delta_{l, l^{\prime}} \, ,$$ and it is explicitly covariant: $${\hat {w}} (k, l) = {\hat {D}} (k,l) \,{\hat {w}}(0,0) \,{\hat {D}}^\dagger (k, l) \, ,
\label{eq:par}$$ where $${\hat {w}}(0,0) = \frac{1}{d^{2}} \sum_{k,l} {\hat {D}} (k, l) \, .
\label{eq:pari}$$ It is interesting to note that the phase (\[phi1\]) for prime dimensions leads to ${\hat {w}}(0,0) = {\hat {P}}$, ${\hat {P}}$ being the parity operator. In view of these properties, one can easily conclude that the corresponding Wigner function $W (k, l)$ fulfills properties fully analogous as those for the continuous harmonic oscillator.
Wigner function for angle-angular momentum {#sec:WigPhiL}
==========================================
In this section, we consider the conjugate pair angle and angular momentum. To avoid the difficulties linked with periodicity, the simplest solution [@Louisell:1963; @Mackey:1963; @Carruthers:1968] is to adopt two angular coordinates, such as, e.g., cosine and sine, we shall denote by $\hat{C}$ and $\hat{S}$ to make no further assumptions about the angle itself. One can concisely condense all this information using the complex exponential of the angle $\hat{E} =
\hat{C} + i \hat{S}$, which satisfies the commutation relation $$\label{ELE}
[ \hat{E}, \hat{L} ] = \hat{E} \, ,$$ or, equivalently, $$\begin{aligned}
& [{\hat {C}}, {\hat {L}} ] = i {\hat {S}} ,
\qquad
[{\hat {S}}, {\hat {L}} ] = - i {\hat {C}} \, , & \nonumber \\
& & \\
& [{\hat {C}}, {\hat {S}} ] = 0 \, . & \nonumber\end{aligned}$$ In mathematical terms, this defines the Lie algebra of the two-dimensional Euclidean group E(2). Note also, that from the Baker-Campbell-Hausdorff formula, one gets $$\label{eq:ExpoCommute1}
e^{-i\phi{\hat {L}}} {\hat {E}} =
e^{i\phi} \, {\hat {E}} e^{- i\phi {\hat {L}}} \, ,$$ which is the unitary Weyl form of (\[ELE\]).
The action of ${\hat {E}}$ on the angular momentum basis is $$\label{E}
\hat{E} | \ell \rangle = | \ell - 1 \rangle \, ,$$ and, since the integer $\ell$ runs from $- \infty$ to $+ \infty$, ${\hat {E}}$ is a unitary operator whose normalized eigenvectors $$\label{phi_states}
| \phi \rangle = \frac{1}{\sqrt{2 \pi}}
\sum_{\ell \in \mathbb{Z}} e^{i \ell \phi} | \ell \rangle \, ,$$ form a complete basis $$\langle \phi | \phi^\prime \rangle =
\sum_{\ell \in \mathbb{Z}} \delta (\phi - \phi^\prime -2 \ell \pi )
= \delta_{2\pi} (\phi - \phi^\prime ) \, ,
\label{eq:PhiNorm1}$$ where $\delta_{2\pi}$ represents the periodic delta function (or Dirac comb) of period $2 \pi$.
As anticipated in the Introduction, the phase space is now the semi-discrete cylinder $\mathbb{Z} \times \mathcal{S}_1$. Following the ideas of Sec. \[sec:Discrete\], a displacement operator can be introduced as $$\label{eq:Displace1}
{\hat {D}} (\ell, \phi) = e^{i \alpha (\ell,\phi)} \,
{\hat {E}}^{-\ell} e^{-i\phi {\hat {L}}} \, ,$$ where $ \alpha(\ell,\phi)$ is a phase to be specified. Note that here there is no possibility to rewrite Eq. (\[eq:Displace1\]) as an entangled exponential, since the action of the operator to be exponentiated would not be well defined. The requirement of unitarity imposes now $$\label{eq:AlphaCondition}
\alpha(\ell, \phi) + \alpha(-\ell, - \phi) = \ell \phi \, .$$
As desired, the displacement operators form a complete trace-orthonormal set: $$\label{eq:DispOrtho}
{\mathop{\mathrm{Tr}}\nolimits}[ {\hat {D}} ( \ell , \phi ) {\hat {D}}^\dagger (\ell^\prime, \phi^\prime ) ]
= 2 \pi \, \delta_{\ell, \ell^\prime} \, \delta_{2\pi}(\phi -
\phi^\prime) \, ,$$ whose resemblance with relation (\[eq:HWDispOrtho\]) is evident.
We can introduce then the canonical mapping $$\begin{aligned}
& W (\ell, \phi) = {\mathop{\mathrm{Tr}}\nolimits}[ {\hat {\varrho}} \,{\hat {w}} ( \ell,\phi) ] \, , &
\nonumber \\
& & \label{eq:WigFunDef1} \\
& \displaystyle
{\hat {\varrho}} = \frac{1}{(2 \pi )^{2}} \,
{\sum_{{\ell} \in \mathbb{Z}}} {\int_{2\pi}}{\hat {w}}(\ell,\phi) W(\ell,\phi) \, d\phi \, , &
\nonumber\end{aligned}$$ where the Wigner kernel ${\hat {w}}$ is defined, in close analogy to the previous cases, as $$\label{eq:WigKerDef1}
{\hat {w}} (\ell, \phi) =
\frac{1}{(2\pi)^2}
{\sum_{{\ell^\prime} \in \mathbb{Z}}} {\int_{2\pi}}\exp[-i ( \ell^\prime \phi - \ell \phi^\prime)]
{\hat {D}} (\ell^\prime, \phi^\prime) \, d\phi^\prime \, .$$
The set of Wigner kernels constitutes a complete orthogonal Hermitian operator basis. In addition, they are explicitly covariant: $$\label{eq:WigKerDisp}
{\hat {w}} (\ell,\phi) = {\hat {D}}(\ell,\phi) \,
{\hat {w}}(0,0) \,{\hat {D}}^\dagger (\ell,\phi) \, ,$$ with $${\hat {w}} (0,0) = \frac{1}{(2\pi)^2} {\sum_{{\ell} \in \mathbb{Z}}} {\int_{2\pi}}{\hat {D}} (\ell,\phi) \, d\phi \, ,$$ although the interpretation of ${\hat {w}} (0,0)$ as the parity on the cylinder is problematic.
All these properties automatically guarantee that we have indeed a well-behaved Wigner function for this canonical pair.
Examples {#sec:Examples}
========
To work out the explicit form of the Wigner function for a given state, one first needs to specify the phase $\alpha(\ell,\phi)$ in Eq. (\[eq:AlphaCondition\]). For convenience, in this paper the choice $$\label{eq:AlphaExplicit}
\alpha (\ell,\phi) = - \ell \phi /2$$ shall be used, as it is linear in both arguments, and it appears to be the simplest function fulfilling the unitarity condition and the periodicity in $\phi$ [@Rigas:2008].
In this case, the Wigner kernel (\[eq:WigKerDef1\]) becomes $$\begin{aligned}
\label{eq:ExplicitKernel1}
{\hat {w}} (\ell,\phi) & = & \displaystyle
\frac{1}{(2\pi)^2} {\sum_{{\ell^\prime,\ell^{\prime \prime}} \in \mathbb{Z}}}
{\int_{2\pi}}e^{i \ell^{\prime} \phi^{\prime}/2} \,
e^{-i\ell^{\prime \prime} \phi^{\prime}} \nonumber \\
& \times & \displaystyle
e^{i(\ell \phi^{\prime} - \ell^{\prime} \phi)}
|\ell^{\prime \prime} \rangle \langle \ell^{\prime \prime} -
\ell^{\prime}| \,
d\phi^\prime \, . \end{aligned}$$ After some manipulations, we obtain $$\begin{aligned}
\label{eq:ExplictKernel3}
{\hat {w}} (\ell,\phi) & = & \displaystyle
\frac{1}{2\pi} {\sum_{{\ell^\prime} \in \mathbb{Z}}} e^{-2 i \ell^\prime \phi}
|\ell + \ell^\prime \rangle \langle \ell - \ell^\prime | \nonumber \\
& + & \displaystyle
\frac{1}{2\pi^2} {\sum_{{\ell^{\prime},\ell^{\prime \prime}} \in \mathbb{Z}}}
\frac{(-1)^{\ell^{\prime \prime}}}{\ell^{\prime \prime}+1/2}
e^{-(2 \ell^{\prime} + 1) i \phi}
\nonumber \\
& \times &
|\ell + \ell^{\prime \prime}+\ell^{\prime}+1\rangle
\langle\ell +\ell^{\prime \prime}-\ell^{\prime}| \, ,\end{aligned}$$ which coincides with the kernel derived by Plebanski and coworkers [@Plebanski:2000] in the context of deformation quantization.
Note that (\[eq:ExplictKernel3\]) splits into “even” and “odd” parts, depending on whether the matrix elements $\varrho_{\ell \ell^{\prime}} = \langle \ell| {\hat {\varrho}} |
\ell^{\prime}\rangle$ have $\ell \pm \ell^{\prime}$ even (first sum) or odd (second sum).
For an angular momentum eigenstate $| \ell_0 \rangle$, one immediately gets $$\label{eq:ExampleOAMstaleEll}
W_{| \ell_0 \rangle} (\ell,\phi) = \frac{1}{2\pi}
\delta_{\ell, \ell_0} \, ,$$ which is quite reasonable in this case: it is flat in $\phi$ and the integral over the whole phase space gives the unity, reflecting the normalization of $|\ell_0\rangle$.
For an angle eigenstate $|\phi_0 \rangle$, one has $$\label{eq:ExamlpeE-StatePhi2}
W_{| \phi_0 \rangle} (\ell, \phi) = \frac{1}{2\pi} \,
\delta_{2\pi}(\phi-\phi_0) \, .$$ Now, the Wigner function is flat in the conjugate variable $\ell$, and thus, the integral over the whole phase space diverges, which is a consequence of the fact that the state $|\phi_0\rangle$ is not normalizable.
The coherent states $| \ell_0, \phi_0 \rangle$ (parametrized by points on the cylinder) introduced in Ref. [@Kowalski:1996] (see also Refs. [@Gonzalez:1998; @Kastrup:2006] for a detailed discussion of the properties of these relevant states) are characterized by $$\begin{aligned}
\langle \ell | \ell_0 , \phi_0 \rangle & = &
\displaystyle
\frac{1}{\sqrt{\vartheta_3 \left ( 0 \big | \frac{1}{e} \right )}}
e^{-i \ell \phi_0} \, e^{-(\ell - \ell_0)^2/2} \, , \nonumber \\
\label{eq:ExampleCoh1}
& & \\
\langle \phi| \ell_0 , \phi_0 \rangle & = &
\displaystyle
\frac{e^{i \ell_0 (\phi - \phi_0)}}
{\sqrt{\vartheta_3 \left ( 0 \big | \frac{1}{e} \right )}}
\vartheta_3\left(\frac{\phi-\phi_0}{2} \Big | \frac{1}{e^2} \right) ,
\nonumber\end{aligned}$$ where $\vartheta_3$ denotes the third Jacobi theta function [@Mumford:1983].
The Wigner function for the state $|\ell_0,\phi_0\rangle$ splits as $$\label{eq:ExampleWCoh}
W_{|\ell_0,\phi_0 \rangle} (\ell,\phi) =
W^{(+)}_{|\ell_0,\phi_0\rangle} (\ell, \phi) +
W^{(-)}_{|\ell_0,\phi_0\rangle} (\ell, \phi) \, .$$ The “even” part turns out to be $$W^{(+)}_{|\ell_0,\phi_0\rangle} (\ell,\phi) =
\frac{1}{2 \pi \vartheta_3 \left (0 \big |\frac{1}{e} \right )}
e^{-(\ell -\ell_0)^2}
\vartheta_3 \left(\phi-\phi_0 \Big| \frac{1}{e} \right) \, .$$ This seems a sensible result, since it is a discrete Gaussian in the variable $\ell$, and for the continuous angle $\phi$ it is a Jacobi theta function, which plays the role of the Gaussian for circular statistics [@Rehacek:2008]. However, the “odd” contribution spoils this simple picture: $$\begin{aligned}
\nonumber
W^{(-)}_{|\ell_0,\phi_0\rangle} (\ell,\phi) &=&
\frac{e^{i(\phi-\phi_0)-1/2}}{2\pi^2 \vartheta_3
\left ( 0 \big | \frac{1}{e} \right )}
\vartheta_3\left( \phi- \phi_0 +i/2 \Big|\frac{1}{e} \right) \nonumber \\
& \times &
{\sum_{{\ell^\prime} \in \mathbb{Z}}} (-1)^{\ell^\prime - \ell + \ell_0}
\frac{e^{-\ell^\prime {}^2 - \ell^\prime}}{\ell^\prime + \ell_0 - \ell + 1/2} \, .
\label{eq:ExampleWoddCoh} \end{aligned}$$
![Plot of the Wigner function for a coherent state with $\ell_0 =0$ and $\phi_0 = 0$. The cylinder extends vertically from $\ell = -4 $ to $\ell = +4$. The two corresponding marginal distributions are shown.[]{data-label="fig:CoherentStandard"}](Figure1){width="0.80\columnwidth"}
In Fig. \[fig:CoherentStandard\], the Wigner function for the coherent state $| \ell_0 = 0, \phi_0 = 0\rangle$ is plotted on the discrete cylinder. A pronounced peak at $\phi=0$ for $\ell=0$ and slightly smaller ones for $\ell=\pm 1$ can be observed. The associated marginal distributions \[obtained from Eq. (\[eq:ExampleWCoh\]) by integrating over $\phi$ or by summing over $\ell$, respectively\] are also plotted. They are strictly positive, as correspond to true probability distributions.
![Unwrapped plot of the Wigner function for a coherent state with $\ell_0 =0$ and $\phi_0 = 0$. The plane extends from $\ell = -4 $ to $\ell = +4$ and from $\phi=-\pi$ to $\phi=\pi$.[]{data-label="fig:CoherentPico"}](Figure2){width="0.90\columnwidth"}
For quantitative comparisons, however, sometimes it may be convenient to “cut” this cylindrical plot along a line $\phi$=constant and unwrap it. This is shown in Fig. \[fig:CoherentPico\]. Here, the range of $\ell$ is from -4 to 4, while the angle is plotted between $-\pi$ to $\pi$.
A closer look at these figures reveals also a remarkable fact: for values close to $\phi=\pm \pi$ and $\ell= \pm 1$, the Wigner function takes negative values. Actually, a numeric analysis suggests the existence of negativities close to $\phi = \pm \pi$ for any odd value of $\ell$.
![Plot and marginal distributions of the Wigner function for an even superposition $|\ell_1 +_\theta\ell_2\rangle $ with $\ell_{1,2}=\pm 3$ for $\ell = -4 $ to $\ell = +4$.[]{data-label="fig:EvenTornillo"}](Figure3){width="0.80\columnwidth"}
![Plot and marginal distributions of the Wigner function for an even superposition $|\ell_1 +_\theta\ell_2\rangle $ with $\ell_1= 4, \ell_2=-3$ for $\ell = -4 $ to $\ell = +5$.[]{data-label="fig:OddTornillo"}](Figure4){width="0.80\columnwidth"}
As our last example, we address the superposition $$\label{eq:SuperDef}
|\Psi \rangle = \frac{1}{\sqrt{2}}
(|\ell_1\rangle + e^{i\phi_0} | \ell_2\rangle )$$ of two angular-momentum eigenstates with a relative phase $e^{i \phi_0}$. The analysis can be carried out for the superposition of any number of eigenstates, but (\[eq:SuperDef\]) is enough to display the relevant features.
The Wigner function splits again; now the “even” part reads as $$\begin{aligned}
\label{eq:ExampleSuperEven}
W_{ | \Psi \rangle}^{(+)} (\ell, \phi) & = &
\frac{1}{4\pi} \{ \delta_{\ell, \ell_1} + \delta_{\ell, \ell_2} \nonumber \\
& + &
2 \delta_{\ell_ 1+\ell_2, 2\ell} \, \cos[\phi_0 + (\ell_2 - \ell_1) \phi]
\} \, . \end{aligned}$$ For the “odd” part, the diagonal contributions vanish, and one has $$\begin{aligned}
W_{ |\Psi \rangle}^{(-)} (\ell, \phi) & = & \displaystyle
\frac{1}{\pi^2} \cos[\phi_0 + (\ell_2 - \ell_1) \phi] \nonumber \\
& \times & \displaystyle
\frac{(-1)^{\ell+(\ell_1+\ell_2 -1)/2}}{\ell_1 +\ell_2 - 2\ell}
\delta_{\ell_1 + \ell_2 = \mathrm{odd}} \, ,
\label{eq:ExamplesSuperOdd2}\end{aligned}$$ where $\delta_{\ell_1 + \ell_2 = \mathrm{odd}}$ indicates that the sum is nonzero only when $\ell_1 + \ell_2$ is odd.
In consequence, when $| \ell_1 - \ell_2|$ is odd, the interference term contains contributions for any $\ell$, damped as $1/\ell$. When $|\ell_1 - \ell_2|$ is an even number, the contribution (\[eq:ExamplesSuperOdd2\]) vanishes and we have three contributions: two flat slices coming from the states $| \ell_1\rangle$ and $| \ell_2\rangle$ and an interference term located at $\ell = (\ell_1 + \ell_2)/2$.
These features are illustrated in Figs. \[fig:EvenTornillo\] and \[fig:OddTornillo\]. The state $|\Psi \rangle $ is plotted for $\ell_2 = -3$ and $\ell_1 = 3$ and (Fig. \[fig:EvenTornillo\]) and $\ell_2 = -3$ and $\ell_1 = 4$ (Fig. \[fig:OddTornillo\]). Changing the relative phase $\phi_0$ results in a global rotation of the cylinder. In can be observed in Fig. \[fig:OddTornillo\] that the two rings at $\ell = -3$ and $\ell=4$ (as opposed to the rings at $\ell = \pm 3$ in Fig. \[fig:EvenTornillo\]), are not flat in $\phi$, but show a weak dependence on the angle due to the odd contributions added to the flat Kronecker deltas.
Concluding remarks
==================
In summary, we have carried out a full program for a complete phase-space description in terms a Wigner function for the canonical pair angle-angular momentum. An experimental demonstration in terms of optical beams is presently underway in our laboratory.
We acknowledge discussions with Hubert de Guise, Jose Gracia-Bondia, Hans Kastrup, Jakub Rembielinski and Krzysztof Kowalski. This work was supported by the Czech Ministry of Education, Project MSM6198959213, the Czech Grant Agency, Grant 202/06/0307, the Spanish Research Directorate, Grant FIS2005-06714, and the Mexican CONACYT, Grant 45705.
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---
abstract: 'We present error mitigation (EM) techniques for noisy intermediate-scale quantum computers (QC) based on density matrix purification and perturbative corrections to the target energy. We incorporate this scheme into the variational quantum eigensolver (VQE) and demonstrate chemically-accurate ground state energy calculations of various alkali metal hydrides using IBM quantum computers. Both the density matrix purification improvements and the perturbative corrections require only meager classical computational resources, and are conducted exclusively as post-processing of the measured density matrix. The improved density matrix leads to better simulation accuracy at each step of the variational optimization, resulting in a better input into the next optimization step without additional measurements. Adding perturbative corrections to the resulting energies further increases the accuracy, and decreases variation between consecutive measurements. These EM schemes allow for previously unavailable levels of accuracy over remote QC resources.[^1]'
author:
- 'T. D. Morris'
- 'Z. P. Parks'
- 'A. J. McCaskey'
- 'J. Jakowski'
- 'R. C. Pooser'
bibliography:
- 'references.bib'
title: Density matrix based perturbative corrections for improved quantum simulation accuracy
---
[*Introduction.—*]{} Noisy intermediate scale quantum (NISQ) computers can be used to test small algorithms in support of codesign efforts to improve the performance of scientific applications. Error mitigation (EM) is a necessary step for improving quantum computing results obtained from these devices. Recent studies have shown that NISQ devices can yield accurate results for a range of scientific simulation problems when EM is used to characterize systematic device noise and adjust expectation value data accordingly [@kandala2017; @kandala_error_2019; @endo_practical_2018; @temme_error_2017; @li_efficient_2017]. In particular, quantum chemistry calculations have reached chemical accuracy for an array of small molecules with minimal basis sets [@cao_quantum_2018; @kandala_hardware-efficient_2017; @Romero2018; @romero_strategies_2017; @yung_transistor_2014; @peruzzo_variational_2014; @hempel_quantum_2018; @Ryabinkin2018; @chembench]. EM has also been used to adjust resultant data in nuclear bound state calculations [@Dumitrescu2018], scalar field theory computations [@yeter-aydeniz_scalar_2018], and in Hamiltonian and quantum state learning applications [@KAH_inprep].
Although it has been used in many applications involving the variational quantum eigensolver, EM is applicable to many other algorithms, including in hybrid machine learning applications that train a quantum register to perform state preparation, quantum approximate optimization [@farhi_quantum_2014], or in quantum imaginary time evolution [@motta_quantum_nodate]. EM requires an assumption about the underlying error model, and the results depend on characterizing how well the machine approximates the model. We previously demonstrated that building assumptions about state preparation as well as the expected final density matrix into EM allows for higher accuracy [@chembench; @yeter-aydeniz_scalar_2018]. In this case EM is a characterization of how far the machine deviates from our assumptions. Here, we augment EM in two ways for quantum chemistry calculations. These modifications extend the reach of an array of molecules to chemical accuracy, even when programming a quantum computer remotely from the QASM level, where one has no gate or pulse level control (which would otherwise allow one to carry out more accurate gate and control-level characterization for use in EM [@Eugene-mitigability]). First, we take into account symmetries of the problem and modify our density purification scheme, enforcing spin symmetry, to achieve a more physical density matrix. Second, we apply perturbative corrections which are motivated by interpreting unitary operation as acting on the Hamiltonian, as opposed to the wavefunction. This interpretation is similar in spirit to the In-Medium Similarity Renormalization Group (IMSRG) theory [@IMSRG-16], which is a promising framework for solving the many-body problem. Within the IMSRG, the coupling of a subset of the Hilbert space to its complement is suppressed via a set of differential equations. When dealing with closed-shell ground states, this decreasing coupling enables computable corrections to energy. In this work, we demonstrate that an analagous perturbative correction, combined with energies arising from a symmetry-corrected, purified density matrix, can yield chemically accurate results for a wide range of molecules and system sizes.
[*Molecular hamiltonians and basis sets.—*]{}We start by considering a recently proposed “benchmark” class of molecules from the alkali-hydrides: H$_2$, LiH, NaH, KH [@chembench]. Computation of ground state energy, correlation corrections, and potential energy curves have become basic benchmark tasks in the NISQ computing era. The electronic structure calculations required to compute these phenomena correspond to a Hamiltonian with a nuclear repulsion term, a one-electron term which accounts for electronic kinetic energy and electrodynamic interactions with the inner core, and a two electron term that accounts for electronic interactions: $$\begin{aligned}
\label{eq:H_tot}
\hat{H}&= & H_0 + \sum_{p,q} h^{p}_{q} \cdot \hat p^\dagger \hat q
+ \frac{1}{4} \sum_{p,q,r,s} \bar g^{pq}_{sr} \cdot \hat{p}^\dagger \hat{q}^\dagger \hat{r} \hat{s},\end{aligned}$$
$$\begin{aligned}
\label{eq:H_tot}
H&= & H_0 + \sum_{p,q} h_{pq} a_p^\dagger a_q
+ \frac{1}{4} \sum_{p,q,r,s} g_{pqrs} a_p^\dagger a_q^\dagger a_s a_r,\end{aligned}$$
where $H_0$ is the nuclear repulsion, $p$, $q$, $r$, $s$ index molecular spin orbitals, $a_p^\dagger$, $a_q$ etc. are electron creation and annihilation operators, $h_{pq}$ are matrix elements of the core Hamiltonian, and $ g_{pqrs}$ are anti-symmetrized two-electron repulsion integrals $g_{pqrs} = \langle pq||rs\rangle$. We freeze all but the highest occupied and lowest occupied orbitals that arise from a hartree-fock calculation [@chembench]. Additionally, for a given single Slater determinant $|\phi \rangle$, one can always exactly rewrite the above hamiltonian in normal ordered form using Wick’s theorem: $$\begin{aligned}
\label{eq:H_N_tot}
H_N&= & E_0 + \sum_{p,q} f_{pq}:a_p^\dagger a_q:
+ \frac{1}{4} \sum_{p,q,r,s} \Gamma_{pqrs} :a_p^\dagger a_q^\dagger a_s a_r:,\notag \\\end{aligned}$$ where $$\begin{aligned}
E_0 &= H_0+\sum_i h_{ii}+\frac{1}{2}\sum_{ij}g_{ijij}\,, \\
f_{pq} &= h_{pq}+\sum_i g_{piqi} , \\
\Gamma_{pqrs} &= g_{pqrs} .\end{aligned}$$ Here $:a_p^\dagger \ldots a_q:$ indicates that the operator is normal-ordered, i.e. that $\langle \phi |:a_p^ \dagger \ldots a_q:|\phi\rangle=0$. Also here and for the remainder of this work, i,j,k,$\ldots$ (a,b,c,$\ldots$) are indices of (un)occupied orbitals of the respective $|\phi\rangle$.
As is usual in quantum chemical calculations, we neglect the contribution to electron correlation from the lowest energy core electrons (frozen core approximation) and only consider a subset of active orbitals in Eq. \[eq:H\_tot\] which leads to the effective Hamiltonian $$\begin{aligned}
\label{eq:H_tot-v2}
\hat{H}&= & H'_0 + \sum_{p,q} h'^{p}_{q} \cdot \hat p^\dagger \hat q
+ \frac{1}{4} \sum_{p,q,r,s} \bar g^{pq}_{sr} \cdot \hat{p}^\dagger \hat{q}^\dagger \hat{r} \hat{s},\end{aligned}$$ in which $H_{0}$ and core interaction integrals $h^{p}_{q}$ are replaced with their effective terms $$H'_{0} = E_{nucl} +\sum_a \big( {h}^a_a + \tfrac{1}{2} \sum_{b} \bar{g}^{ab}_{ab}\big),$$ and $$h'^p_q = {h}^{p}_q + \tfrac{1}{2} \sum_a \bar{g}^{a p }_{a q}$$ where $a$ and $b$ run over occupied frozen-core spin-orbitals which are not active, and the indices $p$, $q$, $r$, $s$ in the Eq. \[eq:H\_tot-v2\] run over active spin orbitals instead of all spin orbitals. Calculating the ground state energy of Eq. \[eq:H\_tot\] variationally amounts to specifying an ansatz wavefunction which is as close as possible to an eigenstate of $\hat{H}$. Encoding such a wavefunction into the quantum register of a computer allows one to calculate expectation values $\langle H \rangle$ by continually sampling from it. The most efficient representation takes into account the mapping of operators in $H$ onto the quantum gates that the hardware is capable of: $$\begin{split}
\label{eq:H_spin}
\hat{H} = \sum_{i,\alpha} h^i_\alpha \sigma^i_\alpha + \sum_{i,j,\alpha,\beta} h^{ij}_{\alpha \beta} \sigma^i_\alpha \sigma^j_\beta \\
+ \sum_{i,j,k,\alpha,\beta,\gamma} h^{ijk}_{\alpha \beta \gamma} \sigma^i_\alpha \sigma^j_\beta \sigma^k_\gamma + \ldots,
\end{split}$$ where $\sigma^i_\alpha \in \{\sigma_x,\sigma_y,\sigma_z,I\}$ correspond to the Pauli matrices applied to the $i^{th}$ qubit. This Hamiltonian is amenable to specific trial ansatz, namely the unitary coupled cluster, but we add additional terms to this wavefunction to improve the algorithmic accuracy (discussed further below).
[*Mapping to qubits.—*]{} In order to map the problem onto qubits, we use the standard Jordan-Wigner tranformation [@jordan_uber_1928].
![The three parameter UCC insired ansatz for 4 qubits and 2 electrons. This ansatz consists of 3 parameters. The parameter $\theta_0$ controls the double excitation from the doubly occupied lower spatial orbital into the doubly occupied higher orbital. The parameter $\theta_1$ ($\theta_2$) controls single excitation amplitude within $\alpha$ spin ($\beta$ spin) block.[]{data-label="fig:uccsdex"}](figs/better_ucc3_circuit.png){width="\columnwidth"}
For our experiments, we employ the unitary coupled cluster-inspired ansatz found in [@chembench] and shown in Figure \[fig:uccsdex\]. Simulations show that this ansatz is capable of reproducing the exact energy of any 2 electron 4 spin-orbital singlet states, and is thus particularly suitable for the modeling of ground states of each alkali-hydride within the frozen core approximation. We note that instead of tranforming the Hamiltonian to the Pauli representation, we instead use Pauli expectation values in order to construct the two body reduced density matrix, or 2-RDM, which is sufficient for evaluation of energy and key to our approach for error mitigation. [*Error Mitigation Techniques.—*]{}Several methods for EM in near-term quantum hardware have been proposed including Richardson extrapolation [@Benjamin-PRX_17-EM; @Gambetta-PRL_17-EM; @Benjamin-PRX_18-EM], the quasiprobability method [@Gambetta-PRL_17-EM; @Benjamin-PRX_18-EM], the quantum subspace expansion [@deJong-PRA_17], and ancilla qubit stabilizers [@Benjamin-PRL-19-EM]. Employing quantum algorithms that exploit symmetries of spin and conserve particle number during the VQE optimization leads to reduced search space, improved convergence and decreased error rates since only a fraction of the Hilbert space is sampled. However, in practice, the spin and electron number are not exactly conserved due to unwanted cross-talk and systematic noise (over and under rotations).
Density matrix purification schemes are essentially methods of projecting inexact, unphysical, or ensemble density matrices onto the the density matrix of closest pure state, and can thus correct for noise that violates number or spin symmetry conservation. We accomplish this via a procedure referred to in literature as McWeeney purification [@purif], where iterative application of the matrix polynomial ${\cal P} \to 3{\cal P}^2-2{\cal P}^3$, drives eigenvalues that were originally close to 0 or 1 to exactly 0 or 1.
Although this process is appropriate if one has the full A-body density matrix, as in our simulations, one can not appeal to this procedure for purifying a 2-RDM arising from a 3- or higher body system. In this case, purification must proceed via semi-definite constraints subject to N-representability as pursued in [@marginal]. This procedure was previously exploited to dramatically improve final energies in benchmark simulations [@chembench]. To further improve the quality of the measured 2-RDM, we impose spin symmetry on the measured density matrix elements before subjecting the 2-RDM to purification. This amounts to measuring only density matrix elements that conserve total spin; and in the 2 electron case only $S_z=0$ density matrix elements are allowed. Further, we force spin reflection symmetry of the 2-RDM to be obeyed by averaging elements with their appropriate spin reflection, resulting in a measured density matrix that is more physical prior to purification. We could also have only measured one spin reflection in order to cut down on measurements. Here and for the proceeding perturbative correction, we propagate statistical error by performing simple bootstrapping, where each measured circuit is resampled 10,000 times; our error analysis proceeds after each resampling [@Efron1979]. This procedure produces distribution average and errors that have been shown in previous work to be consistent with other forms of statistical error propagation [@chembench].
[*Perturbative correction.—*]{} Attempting to apply perturbation theory on beyond-single slater determinant wavefunctions is a very complicated, if feasible process. To do so with the ansaëtze typically employed in quantum computations would require so much classical computation as to likely make the benefit of quantum computing moot. It is however possible to formulate an accurate approximation of perturbation theory, where one works not with the wavefunction as it evolves, but instead the hamiltonian itself. We will take the unitary coupled cluster (UCC) wavefunction as a model, where the ground state is written as $$|\Psi\rangle = e^{T-T^\dagger}|\phi\rangle = U|\phi\rangle \,,$$ with $|\phi\rangle$ typically the Hartree-Fock (HF) determinant, and the amplitudes of $T$ are varied to minimize the energy, which now takes the form $$\label{eq:energy_ucc}
E = \langle\Psi|H|\Psi\rangle = \langle\phi|U^\dagger H U|\phi\rangle = \langle\phi|\bar{H}|\phi\rangle \,.$$ It can be seen from Eq. \[eq:energy\_ucc\], it is valid to think of the transformation as acting on the Hamiltonian to produce $\bar{H}$, whose matrix elements connecting $|\phi\rangle$ to higher excitations are suppressed, leaving $|\phi\rangle$ as the exact eigenstate of $\bar{H}$. This is the basic interpretation of how the IMSRG approaches the A-body diagonlization problem. Working now with $\bar{H}$ allows one to appeal to the simple formula for perturbation theory, $$\label{eq:mp2_form}
\Delta \bar{E}^{[2]} = \sum_{ia} \frac{|\bar{f}_{ia}|^2}{\bar{\epsilon}_i-\bar{\epsilon}_a}+ \frac{1}{4}\sum_{ijab}\frac{|\bar{\Gamma}_{ijab}|^2}{\bar{\epsilon}_i+\bar{\epsilon}_j-\bar{\epsilon}_a-\bar{\epsilon}_b} \,,$$ where all matrix elements are those of the transformed Hamiltonian, normal ordered with respect to $|\phi\rangle$ (we define the matrix elements and the energy denominators in the supplemental information). We have omitted, three- and higher- body pieces that could be induced by the transformation, their importance and analogy to traditional coupled cluster perturbative triples and higher rank approximations will be explored in a future work. As opposed to attempting to measure the matrix elements found in Eq. \[eq:mp2\_form\] on hardware,which would be exceedingly expensive, we instead approximate them with our measured 2-RDM. We do so by noticing that the derivative with respect to a given cluster amplitude can be approximately written in the following ways, $$\label{eq:derive_general}
\frac{\delta E}{\delta T_I} \approx 2\Re(\langle \phi | [A_I,\bar{H}]|\phi \rangle) \approx 2\Re(\langle\phi|e^{T^\dagger-T}[A_I,H]e^{T-T^\dagger}|\phi\rangle) \,,$$ where $A_I$ are typical cluster type excitations out of the occupied Slater determinant $|\phi\rangle$. As the solution is asymptotically approached, both sides must be analytically zero and approximately equal. We observe that the RHS of Eq. \[eq:derive\_general\] are nothing but the matrix elements encountered in the numerators of Eq. \[eq:mp2\_form\], whose expressions in terms of the bare hamiltonian and density matrix elements are presented in Eqs. S4,5, supplemental material [@Supplemental]. We also approximate the transformed energies $\bar{\epsilon}_i,\bar{\epsilon}_a$ using the measured 2-RDM by appealing to a connection between cluster amplitudes and certain density matrix elements (see Eqs. S6 and S7, [@Supplemental]). The purified 2-RDM can then be used to form an estimate of $\Delta \bar{E}^{[2]}$, that when added to the purified energies, provides a more robust ground-state energy estimate, termed in this work as RDM-PT2. Our correction depends only on the measured 2-RDM, and thus does not actually depend on using the UCC ansatz, which is corroborated by the fact that our simulated ansatz is not that of UCC theory.
All simulations on hardware consisted of 2-electrons correlated in the highest occupied and lowest unnoccupied orbitals, however, the RDM-PT2 correction is appropriate even for frozen orbitals. It simplifies to simple M[ø]{}ller-Plosset perturbation theory when the 2-RDM is that of a single slater determinant, and becomes a better approximation as the ansatz becomes perturbatively close to the true ground state. If the frozen orbitals represent only dynamical correlations, it is likely their contribution to the ground state energy can be captured perturbatively. One caveat is that this necessitates approximations to the 3-RDM, since $\bar{\Gamma}_{ijab}$ depends on the 3-RDM for more than 2-fermion systems (see Eq. S7, [@Supplemental]). Fortunately the 3-RDM elements can be approximated with a simple formula in terms of the 1- and 2- RDM matrix elements (Eq. S8, [@Supplemental]), which is derived in [@mazziotti2007reduced]. This allows for accounting for all orbitals, even for our largest system KH, which consisted of 20 electrons in 28 orbitals. This correction scales as o$^2$a$^2$v$^2$, where o is the number of frozen core orbitals, a is the number of active orbitals, and v is the number of frozen virtual orbitals. Thus our correction scales roughly the same as traditional CC theory truncated at singles and doubles.
[*Results.—*]{}Figure \[fig:h2\_opt\_eq\] shows the optimization of the H$_2$ molecule close to equilibrium geometry of $r=0.7$ angstroms, using the COBYLA optimizer found in scipy. Plotted are the differences in calculated energy from the exact energy for raw data with only readout error mitigation, pure energies generated from a physical purified density matrix, and perturbatively corrected RDM-PT2 energies. The optimizer searched for optimal parameters based on the purified energies, as the perturbative estimate is not variational and would not be appropriate for the search.
![Optimization of ansatz for H$_2$ at the equilibrium geometry of 0.7 angstroms. []{data-label="fig:h2_opt_eq"}](figs/h2_equilibrium_legend.png){width="0.95\columnwidth"}
In the top panel of Figure \[fig:h2\_dissociation\], the pure and RDM-PT2 energy differences measured on IBM quantum hardware are shown for four bond lengths of $r=$0.7,2.0,3.0,and 4.0. Below are the results from simulated results using the error model taken from the Qiskit module Aer using numbers reported as realistic on IBM Poughkeepsie. Final results and errors for each geometry were estimated by taking the average and standard deviation of the last 5 points of the COBYLA optimization, and adding it to statistical error in quadrature. For the simulated results from Aer, purified results already fall within the chemically accurate band of 1.5 milliHartree of the exact result, but RDM-PT2 always yields an improvement. For the calculations on quantum hardware, one sees a different story. At larger bond lengths, it was more difficult to achieve adequate agreement with exact energies with only purification. However, the RDM-PT2 energies were able to reach exact results, within error bars. It is important to note that if the approximately transformed energies are not used as the bond length increases, the perturbative estimate dramatically overestimates the needed correction, yielding overbound results. It will be interesting to see how this correction fairs in more complicated dissociation curves that involve double or triple bond breaking, where degeneracy often spoils perturbative corrections.
![The top panel shows the results of four different quantum simulations run on IBM poughkeepsie, the bottom shows results using the built in Qiskit error simulator with properties obtained from poughkeepsie. []{data-label="fig:h2_dissociation"}](figs/h2_dissociation_legend.png){width="\columnwidth"}
Figure \[fig:LiH\_eq\_unfrz\] shows the optimization on hardware of LiH at its equilibrium geometry. The top band shows chemical accuracy with respect the frozen space, while the bottom band is with respect to the full space. It is clear that the RDM-PT2 using only frozen orbitals and the RDM-PT2 where all orbitals are correlated both give very good agreement with their respective spaces. Figure \[fig:hydrides\] reflects the same analysis for all the hydrides, H$_2$, LiH, NaH, and KH. For the hydrides heavier than H$_2$, the frozen core energy deviations are shown with a slight left offset, while the full space deviations are slightly offset to the right. In all cases, it was possible to achieve final RDM-PT2 energies whose error bars fell within 1.5 mHa of the exact energy. Given for reference are the second order many-body perturbation theory energies with respect to a hartree-fock reference. Our perturbative corrections are not just reproducing naive perturbation theory, they achieve chemical accuracy whereas HF-PT2 energies do not.
![Equlibrium simulation of LiH, shown are the two bands corresponding to exact energies within the frozen and full space sto-3g spaces.[]{data-label="fig:LiH_eq_unfrz"}](figs/LiH_unfrz_legend.png){width="\columnwidth"}
![Final results obtained from hardware simulations run on IBM hardware for equilibrium geometries of frozen core hydride type molecules. Although purified results show great variability with regards to machine and individual run, the perturbative expression seems fairly robust.[]{data-label="fig:hydrides"}](figs/hydrides_legend.png){width="\columnwidth"}
[*Conclusion.—*]{}We remotely performed quantum computation of simple hydride molecules using the minimal basis set. Using the standard process of freezing all but the most important orbitals, these problems were simulable on quantum hardware available from IBM via the cloud. More importantly, we derived a new, but simple perturbative correction that can augment current error mitigation techniques to dramatically improve ground state measurements of variational quantum eigensolvers. In addition to bringing the small space calculations we performed into very good agreement with exact energies, the correction also allows for the unfreezing of dynamically-correlated orbitals. This opens up a path for gradual improvement of calculations as quantum computers grow into larger systems, a key requirement to eventually reach quantum advantage.
[*Acknowledgments*]{} This work was supported as part of the ASCR Quantum Testbed Pathfinder Program at Oak Ridge National Laboratory under FWP \#ERKJ332. This research used quantum computing system resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725. Oak Ridge National Laboratory manages access to the IBM Q System as part of the IBM Q Network.
In order to motivate the perturbative expansion about the ground state energy, we write down the energy expression for a differential of the energy with respect to a given cluster amplitude, $T_I$ with excitation operator $A_I$, which for reasons that will become apparent, we will call the “middle” energy differential $$\label{eq:middle_derive}
\Delta E_{M,I} = \langle\phi|e^{-(T-T^\dagger)-\Delta_I A_I}He^{T-T^\dagger+\Delta_I A_I}|\phi\rangle \,.$$ Additionally, we define two more differentials corresponding to the “inner” and “outer” exponentials, i.e. $$\label{eq:inner_derive}
\Delta E_{I,I} = \langle\phi|e^{-(T-T^\dagger)}e^{-\Delta_I A_I}He^{\Delta_I A_I}e^{T-T^\dagger+\Delta_I A_I}|\phi\rangle \,,$$ and $$\label{eq:outer_derive}
\Delta E_{O,I} = \langle\phi|e^{-\Delta_I A_I}e^{-(T-T^\dagger)}He^{T-T^\dagger}e^{\Delta_I A_I}|\phi\rangle \,.$$ We now observe that if one carefully expands all three above expressions, they are all identical up to expressions involving the three commutator expressions (and higher commutator powers of $T-T^\dagger$, which we neglect to discuss here as we expect they are negligible) $[T-T^\dagger,[\Delta_I A_I,H]]$,$[\Delta_I A_I[T-T^\dagger,H]]$, and $[H,[\Delta_I A_I,T-T^\dagger]]$. Since $A_I$ is one given excitation, and finite rank truncated Unitary Coupled Cluster is suspected to only be valid if the cluster amplitudes required are not large, it is likely that all three commutators are vanishingly small as the solution is approached. This claim would likely not be valid for large amplitudes, however in the cases inspected here it appears this is true given the success presented. Thus we make the approximation $\Delta E_{M,I}\approx\Delta E_{I,I}\approx\Delta E_{O,I}=\Delta E_{I}$, where we will now dispense with the subscript corresponding to “outer”, “middle”, and “inner”. It is then possible to take the differential limit, and make the identification of $\bar{f}_{ia}=\frac{1}{2} \frac{\delta E_{ia}}{\delta T_{ia}}$ and $\bar{\Gamma}_{ijab}=\frac{1}{2}\frac{\delta E_{ijab}}{\delta T_{ijab}}$ with our desired “transformed” matrix elements for both single and double excitations out of the Slater Determinant $|\phi\rangle$ and approximate . In order to evaluate these matrix elements, we use the expressions that depend only on the 2-RDM that come from Eq. \[eq:derive\_general\]. These are: $$\begin{aligned}
&\bar{f}_{ia} = \Re(\langle \phi | [a^\dagger_i a_a,\bar{H}]|\phi \rangle) \approx \Re(\langle\phi|U^\dagger[a^\dagger_i a_a,H]U|\phi\rangle)\notag \\
&= \sum_m \big( h_{im}\rho_{ma}-h_{am}\rho_{mi}\big) \notag \\
&+\frac{1}{2}\sum_{m,v,w}\big(g_{imvw}\rho_{vwam}-g_{amvw}\rho_{vwim}\big) \label{eq:onebodygrad}\end{aligned}$$
$$\begin{aligned}
&\bar{\Gamma}_{ijab} = \Re(\langle \phi | [a^\dagger_ia^\dagger_j a_b a_a,\bar{H}]|\phi \rangle) \notag\\
&\approx \Re(\langle\phi|U^\dagger[a^\dagger_ia^\dagger_j a_b a_a,H]U|\phi\rangle)\notag \\
&= (1-P_{ij})\sum_m h_{im}\rho_{mjab}
-(1-P_{ab})\sum_m h_{am}\rho_{ijmb} \notag \\
&+\frac{1}{2}\sum_{m,n}g_{ijmn}\rho_{mnab}
-\frac{1}{2}\sum_{m,n}g_{mnab}\rho_{ijmn}\notag \\
&-(1-P_{ij})\frac{1}{2}\sum_{m,n,v}g_{ivmn}\rho_{mnjabv}\notag\\
&+(1-P_{ab})\frac{1}{2}\sum_{m,n,v}g_{mnav}\rho_{ijvbmn}\label{eq:twobodygrad}\end{aligned}$$
The 3-body dependence can be reduced to two-body dependence with the following formula, which is just the “reducible” 3-RDM found in [@mazziotti2007reduced]: $$\begin{aligned}
\rho_{pqrstu} \approx \frac{1}{3}(1-P_{pr}-P_{qr})(1-P_{su}-P_{tu})\rho_{pqst}\rho_{ru} \label{eq:3RDM}\end{aligned}$$
In order to better approximate the transformed energy denominators, we appeal to the fact that for naive many-body perturbation theory power counting, the leading order off-diagonal density matrices can be related to the cluster amplitudes, i.e. $$\begin{aligned}
\rho_{ia}^{[1]} = T_{ia}^{[1]} \\
\rho_{ijab}^{[1]} = T_{ijab}^{[1]} \,.\end{aligned}$$ We make use of this to form a very rough approximation of the cluster amplitudes in order to approximate the diagonal energies by making corrections from the leading order contractions of the cluster amplitudes with the normal ordered hamiltonian. This gives transformed energies that again depend only on the original Hamiltonian and the measured two body density matrices as follows: $$\begin{aligned}
\epsilon_{i} = h_{ii}+\sum_{\substack{j}}g_{ijij}-\sum_a h_{ia}\rho_{ai}-\frac{1}{2}\sum_{\substack{j,a,b}}g_{ijab}\rho_{abij}\notag\\ \label{eq:holespe} \\
\epsilon_{a} = h_{aa}+\sum_{\substack{j}}g_{ajaj}+\sum_i h_{ai}\rho_{ia}+\frac{1}{2}\sum_{\substack{i,j,b}}g_{abij}\rho_{ijab}\notag\\ \label{eq:partspe}\end{aligned}$$
[^1]: This manuscript has been authored by UT-Battelle, LLC, under Contract No. DE-AC0500OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for the United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan.
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abstract: '0.6cm We propose a new discrete symmetry in the generation space of the fundamental fermions, consistent with the observed fermion mass spectrum. In the case of the quarks, the symmetry leads to the unique prediction of a flat CKM matrix at high energy. We explore the possibility that evolution due to quantum corrections leads to the observed hierarchical form of the CKM matrix at low energies.'
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-1in
23 cm 1.5 cm 15.4 cm 0.4 cm
RAL-94-072\
13 June 1994\
[Generation Permutation Symmetry and the Quark Mixing Matrix]{}
[P. F. Harrison\
Physics Department, Queen Mary and Westfield College\
Mile End Rd. London E1 4NS. UK ]{}
[and]{}
[W. G. Scott\
Rutherford Appleton Laboratory\
Chilton, Didcot, Oxon OX11 0QX. UK ]{}
[*To be published in Physics Letters B.*]{}
0.6cm The problem of the origin of the masses and the mixing angles of the fundamental fermions must surely be amongst the most urgent in particle physics today. Even accepting the standard mechanism for fermion mass generation through Yukawa couplings to one or more non-zero Higgs fields, the reason for the existence of three fermion generations together with the explanation for the observed pattern of the individual masses and mixing angles remains mysterious. One possible way forward is to gain experience by constructing and analysing a wide variety of plausibly motivated candidate mass matrices (or ansatze) in the hope that something convincing will eventually emerge. Amongst the best known and perhaps the most thoroughly analysed such ansatz is that due to Fritzsch [@FRITZCH]. The present proposal has more in common with the approach pioneered by Harari et al. [@HARARI].
In this paper we motivate and analyse a new ansatz for the fermion mass matrices, which we believe has unique a priori appeal by virtue of the principles underlying its construction. Our proposal owes something to the straightforward and oft-repeated observation that the fermion generations are in some (yet to be defined) sense duplicate copies one of the other. That is to say that, in spite of the large mass differences observed from generation to generation, it is natural to assume that the three generations exist fundamentally on an equal footing. In constructing our ansatz, we take this notion seriously and insist that, at the most fundamental level, there be no physical basis for prefering one generation over another, ie. in the Lagrangian the assignment of the generation labels ($i=1$-3) must be entirely arbitrary. Such a demanding requirement has much of the character of established invariance principles in physics, and naturally puts very severe constraints on the form that the mass matrices can take. Indeed these constraints are so severe that they can often appear at first sight to be in conflict with the experimental facts. We show in this paper, however, that this is not neccessarily the case. The indisputable a priori appeal of the above idea, taken together with the uniqueness and economy of its implementation, have provided much of the motivation to pursue this analysis.
We begin by noting that a principle of the sort outlined above is trivially satisfied by the charged-current weak interaction in any weak basis, as a consequence of the universality of the weak interaction. On the other hand, the evident large mass differences observed, from generation to generation, tell us that the Yukawa couplings in the physical basis, are quite definitely not universal. At this point, the [*only*]{} solution that we can see, consistent with the principle we have expounded above, requires that we postulate that in some weak basis the Yukawa couplings for a given fermion species exhibit an invariance under [*permutations*]{} of the generation indices. A candidate mass matrix fulfilling our requirement, which is also hermitian is: $$m=\left(\matrix{
a & b & b^{*} \cr
b^{*} & a & b \cr
b & b^{*} & a \cr
} \right)$$ where $a$ is real and $b$ is complex. Note that the diagonal mass terms are all identical (they are all equal to $a$) and that the off-diagonal (weak-generation-changing) amplitudes for the ‘clockwise’ transisitions ($1 \! \rightarrow \! 2$, $ \! 2 \rightarrow \! 3$ and $3 \! \rightarrow \! 1$) are also all identical (they are all equal to $b$) and the amplitudes for the ‘anticlockwise’ transistions ($ 1 \! \rightarrow \! 3$, $3 \! \rightarrow \! 2$, and $ 2 \! \rightarrow \! 1$) are all equal to $b^*$, so that no generation is preferred. A matrix of this form is sometimes referred to as a circulant [@MATHS]. It might be argued that the mass matrices are unlikely to be hermitian and that a general circulant matrix with $a$ complex and with unrelated complex numbers $b$ and $c$ representing different amplitudes for the clockwise and anticlockwise transisitions, would also satisfy our requirement. Nothing is to be gained, however, by postulating this general form since, on taking the hermitian square ($mm^{\dagger}$), we immediately recover the form eq.(1), and, as is well known, only the hermitian square of the mass matrix can influence the measured masses and mixing angles.
Suppose that we postulate a matrix of the above form for the hermitian square of the mass matrix for the charged leptons. The observed mass spectrum can be reproduced by setting: $$\begin{aligned}
a&=&(\tau /3) +(\mu /3) + (e/3) \nonumber \\
b&=&(\tau /3)\ \omega_1 + (\mu /3)\ \omega_2 + (e/3)\ \omega_3\end{aligned}$$ where $\tau$, $\mu$ and $e$ represent the masses-squared of the $\tau$-lepton, muon and electron respectively, and the $\omega_i$, $i=$ 1-3 are the usual complex cube-roots of unity. In this form, in the rank-1 limit ($\mu,e \rightarrow 0$) the above matrix reproduces the matrix proposed by Harari et al. [@HARARI]. The form of eq.(2) follows from the general result that the spectrum of the eigenvalues of a circulant matrix is given by the (discrete) Fourier transform of its trailing diagonal. The eigenvectors of a matrix of the form eq.(1) are: (1,1,1), (1,$\omega_2$,$\omega_3$), (1,$\omega_3$,$\omega_2$). These are of course just the momentum eigenstates for a three-point one-dimensional lattice satisfying periodic boundary conditions. An operator of the form eq.(1) (with $b$ real and negative) was employed by Feynman [@FEYN] to describe the low lying energy states of the tri-phenyl-cyclo-propanyl ion. We consider it very significant that the matrix operator defined by eq.(1) and eq.(2) has so much in common with the simple derivative operators representing the ordinary kinetic terms in the Lagrangian, which as a consequence of translational invariance may also be represented by (infinite) circulant matrices. It might also be worth noting that the form eq.(1) may equivalently be regarded as the $3 \times 3$ generalisation of the phenomenologically successful $2 \times 2$ effective-theory [@PAIS] used to describe the properties of the neutral kaon system, prior to the discovery of CP violation.
Turning now to the quark mass matrices one might be tempted to postulate mass matrices of the form eq.(1), but with different parameters $a$ and $b$, chosen in analogy with the case of the leptons above, so as to reproduce the observed mass spectrum for the up-type and down-type quarks respectively. But matrices of the form eq.(1) commute with each other for all values of $a$ and $b$, so that the mass matrices for the up-type and down-type quarks would be simultaneously diagonalisable and the quark mixing (CKM [@CKM]) matrix would then be the identity (or a trivial permutation matrix), in clear disagreement with experiment.
With these considerations in mind, we have investigated mass matrices of the somewhat more general form: $$m=\left(\matrix{
a & be^{i\phi_3} & b^{*}e^{-i\phi_2} \cr
b^{*}e^{-i\phi_3} & a & be^{i\phi_1} \cr
be^{i\phi_2} & b^{*}e^{-i\phi_1} & a \cr
} \right)$$ with $a$ and $b$ still given by eq.(2) and with $\phi_1+\phi_2+\phi_3 = 0$, so that the mass eigenvalues are unchanged. In eq.(3) the off-diagonal amplitudes are equal in magnitude but differ in phase, so that the matrix eq.(3) does not commute with the matrix eq.(1), nor does it commute with matrices of the form eq.(3) with different values for the phases. The eigenvectors of a matrix of the form eq.(3) are: ($1,e^{-i\phi_3},e^{i\phi_2}$), ($1,\omega_2e^{-i\phi_3},\omega_3e^{i\phi_2}$), ($1,\omega_3e^{-i\phi_3},\omega_2e^{i\phi_2}$). If we postulate matrices of the form eq.(3) for (the hermitian squares of) the mass matrices for the up-type and down-type quarks, and construct unitary matrices $U$ and $D$ comprising the respective mass-ordered normalised eigenvectors, we find that the CKM matrix ($V=U^{\dagger}D$) may then itself be written as a circulant: $$V=\left(\matrix{
p & q & r \cr
r & p & q \cr
q & r & p \cr
} \right) .$$ Observables depend only on the phase differences ($\Delta \phi_i$) between the corresponding amplitudes in the up-type and down-type mass matrices: $$\begin{aligned}
|p|^2 & = & (3+2{\rm Re}S)/9 \nonumber \\
|q|^2 & = & (3-{\rm Re}S+\sqrt{3}{\rm Im}S)/9 \\
|r|^2 & = & (3-{\rm Re}S-\sqrt{3}{\rm Im}S)/9 \nonumber\end{aligned}$$ with $S=e^{i\Delta \phi_1}+e^{i\Delta \phi_2}+e^{i\Delta \phi_3}$. The convention independent CP violation parameter $J_{CP}$ [@CECELIA] is given by: $$J_{CP}=\frac{1}{27} {\rm Im}(e^{i(\Delta \phi_2-\Delta \phi_1)}
+e^{i(\Delta \phi_3-\Delta \phi_2)}+e^{i(\Delta \phi_1-\Delta \phi_3)})$$ For example, if $\Delta \phi_1 = 0^o$ and $\Delta \phi_2 = 60^o$ (and hence $\Delta \phi_3 = -60^o$) then $S=2$ and $|p|= \sqrt{7}/3 \simeq 0.882$, $|q|=|r|=1/3 \simeq 0.333$ and $J_{CP}=1/(18\sqrt{3}) \simeq 0.032$. We see no way to justify such a choice of phases however.
At this point, we return again to the similarity we noted above, between the operator eq.(1) and the simple derivative operators representing the ordinary kinetic terms in the Lagrangian. Building on this observation, we now note that a close analogy exists between the operator eq.(3) and (the hermitian square of) a full gauge-covariant kinetic operator. The phases $\phi_i$ ($i=1$-3) play a role here analogous to that of the gauge potential. The freedom to change the absolute phases using any (common) arbitrary diagonal matrix of phase factors, is analogous to local gauge invariance. A gauge-field configuration corresponding to a constant field-strength (ie. a uniform field) is of particular interest to us here, because a uniform field is manifestly translationally invariant. We note that even in the case of a uniform field, the inherent translational invariance cannot be explicit in all of the components of the gauge potential at once, after a choice of gauge has been made. In the same way if we set: $$\Delta \phi_2-\Delta \phi_1 = \Delta \phi_3-\Delta \phi_2 =
\Delta \phi_1-\Delta \phi_3$$ corresponding to a uniform field (in the discrete generation space), then it must be that no generation is preferred, even though the up-type and the down-type mass matrices clearly cannot both be circulant. As far as observables are concerned, this last requirement eq.(7) (together with the requirement $\Delta \phi_1 + \Delta \phi_2 + \Delta \phi_3 = 0$, above) completely specifies our ansatz (eg.$\Delta \phi_1=0^o,\Delta \phi_2= \pm 120^o,\Delta \phi_3= \mp 120^o$), up to the sign of $J_{CP}$. The CKM matrix is flat in this case, ie. all elements have equal modulus $|p| = |q| = |r| = 1/\sqrt{3} \simeq 0.577$, and $J_{CP}$ is extremal, ie. $|J_{CP}|= 1/(6\sqrt{3}) \simeq 0.096$ [@CECELIA].
If the above matrices are relevant at all, they are relevant only at very high energy, eg. unification (GUT) energies, and have to be evolved down to the electro-weak (EW) scale in order to be compared with experiment. The leading-order evolution equations [@BARGER] for the quark Yukawa matrices in the Standard Model (SM) can be written (neglecting the influence of the charged leptons): $$\begin{aligned}
\dot{\alpha_u} & = &\frac{3}{2}\alpha_u^2
-\frac{3}{4}(\alpha_u\alpha_d+\alpha_d\alpha_u)
+3{\rm Tr}(\alpha_u+\alpha_d)\alpha_u-8\alpha_3\alpha_u
-\frac{9}{4}\alpha_2\alpha_u
-\frac{17}{20}\alpha_1\alpha_u \nonumber \\
\dot{\alpha_d} & = &\frac{3}{2}\alpha_d^2
-\frac{3}{4}(\alpha_u\alpha_d+\alpha_d\alpha_u)
+3{\rm Tr}(\alpha_u+\alpha_d)\alpha_d-8\alpha_3\alpha_d
-\frac{9}{4}\alpha_2\alpha_d
-\frac{5}{20}\alpha_1\alpha_d \\
\dot{\alpha_3} & = &-7\alpha_3^2 \nonumber \hspace{10mm}
\dot{\alpha_2} = -\frac{19}{6} \alpha_2^2 \hspace{10mm}
\dot{\alpha_1} = \frac{41}{10}\alpha_1^2\end{aligned}$$ where Tr denotes the matrix trace, the dot denotes differentiation with respect to $T=(1/2\pi) \ln (E/E_0)$ and $E/E_0$ is the running energy scale, expressed as a fraction of the starting energy. The hermitian squares of the up-type and the down-type Yukawa matrices are represented by $\alpha_u$ and $\alpha_d$ respectively, where a factor of $1/4\pi$ has been incorporated in the definition of $\alpha_u$ and $\alpha_d$ to simplify the form of the evolution equations, in analogy with the case of the gauge couplings. The corresponding equations for the gauge couplings ($\alpha_i$, $i=$ 1-3) are included for completeness.
There has been much progress in understanding the effects of evolution analytically [@GRZAD], but for simplicity the results presented here are based on a straightforward numerical integration of eq.(8), employing an appropriate (variable) stepsize. Suitable starting values for the gauge couplings are taken from the fits of Amaldi et al. [@AMALDI]. For a given set of starting values for the Yukawa couplings, we calculate the quark mass spectrum and the CKM matrix at the lower energy scale. There is considerable freedom in choosing starting values for the Yukawa couplings consistent with the observed mass spectrum at low energies due (in large part) to the well known quasi-fixed-point [@ROSS], implicit in the evolution equations, which tends to focus the top Yukawa coupling towards its fixed-point value at low energies, independent of its starting value. In spite of this, we find that assuming [*perturbative*]{} starting values for the individual Yukawa couplings (ie. $\alpha_u,\alpha_d {\raisebox{-.6ex}{$\stackrel{\textstyle <}{\sim}$}}1$), chosen to reproduce the observed quark mass spectrum, the predicted evolution is always too slow to yield a realistic CKM matrix at low energies. Evolving down over a reasonable range in $T$ (the GUT scale and the EW scale are about five units apart in $T$) the CKM matrix remains approximately flat; that is to say, all elements remain close to their starting value, $|V_{ij}| \simeq 1/\sqrt{3} \simeq 0.577$, to within deviations at the level of 20% or less.
However, with recent experimental results from LEP and from the Tevatron tending to favour large values for the top mass [@TOP], it is becoming increasingly clear that the Yukawa couplings may very well be [*non-perturbative*]{} at high energy. Whilst we do not expect perturbative evolution equations to be quantitatively valid in a non-perturbative regime, we have done what we can to investigate this possibility, by applying eq.(8) also in the case that the Yukawa couplings assume non-perturbative values (ie. $\alpha_u,\alpha_d {\raisebox{-.6ex}{$\stackrel{\textstyle >}{\sim}$}}1$). As one might expect, with larger starting values for the Yukawa couplings, the evolution proceeds more rapidly. The observed quark mass spectrum at low energy, can still be correctly reproduced, thanks to the quasi-fixed-point. We now find, however, that the CKM matrix, although starting out absolutely flat, rapidly develops a significant hierarchy which, for suffiently large starting values for the Yukawa couplings, is not-at-all unlike the familiar hierarchy [@WOLF] of CKM amplitudes observed experimentally. That said, we have not succeeded in finding any one complete set of starting values which reproduces the quark mass spectrum and the CKM matrix simultaneously in every detail, and in view of the strict inapplicabilty of eq.(8) in the non-perturbative domain, neither should we expect to, even in the case that our ansatz was perfectly correct. Instead we give here a sample set of starting values that can be seen to reproduce most of the quark masses correctly, together with the main features of the CKM matrix. The input values for the (diagonalised) Yukawa couplings at high energy are: $\alpha_u=(6.0 \times 10^{-2}, 2.0 \times 10^{9}, 7.0 \times 10^{11})$, $\alpha_d=(1.5 \times 10^{-1}, 5.0 \times 10^{0}, 4.5 \times 10^{1})$ leading to $\alpha_u=(4.4 \times 10^{-11}, 8.3 \times 10^{-2}, 8.8 \times 10^{-2})$, $\alpha_d=(2.8 \times 10^{-10}, 6.0 \times 10^{-8}, 6.8 \times 10^{-5})$ at the EW scale ($\Delta T =-5$). The evolved CKM matrix is as follows (only the moduli of the elements are given here; phases are of course convention dependent): $$V=\left(\matrix{
0.975 & 0.222 & 0.011 \cr
0.222 & 0.974 & 0.047 \cr
0.012 & 0.046 & 0.999 \cr
} \right)$$ with $|J_{CP}| = 1.06 \times 10^{-4}$. The result eq.(9) bears a striking resemblance to the experimentally observed CKM matrix and suggests to us that it is evolution (albeit non-perturbative and presently incalculable) which is responsible for the observed hierarchy in the CKM matrix at low energy. Whilst results obtained by applying perturbative equations in a non-perturbative domain are unsatisfactory, in that they clearly cannot be used to falsify any hypothesis at all, we maintain that they do serve a useful purpose here as an illustration of existing possibilities. The problem of non-perturbative evolution may not be forever intractable: exact non-perturbative evolution equations for coupling constants in pure gauge theories have already been discussed in the literature [@EXACT]. Certainly it cannot be said that this ansatz is ruled out by experiment. On the contrary, if the trends we see applying leading-order perturbative evolution equations are at all representative of the effects of complete non-perturbative evolution, then all the indications are that we are on the right track.
In conclusion, in spite of the difficulties we have emphasised, we find the apparently natural emergence of a CKM-like hierarchy entirely within the SM framework very impressive. The matrix operators we have proposed come as close as one might hope to generalising (to the discrete generation space) the continuum gauge-covariant operators already present in the SM Lagrangian. One might even speculate that it is some analogue of the pure-gauge kinetic term, constructed from the relevant invariants [@CECELIA], which (classically extremised) accounts for the hierarchy of quark masses. At the very least, we believe that we have demonstrated that this simple and appealing ansatz merits further investigation.
It is a pleasure to thank R. G. Roberts for helpful discussions and encouragement.
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|
---
abstract: 'Here we discuss the effects of strong gravity that can be observed in electromagnetic spectra of active galactic nuclei (AGN). According to the unification model of an AGN, there is a supermassive black hole ($10^7 - 10^9 M_\odot$) in its center, surrounded by an accretion disk that radiates in the X-ray band. Accretion disks could have different forms, dimensions, and emission, depending on the type of central black hole (BH), whether it is rotating (Kerr metric) or nonrotating (Schwarzschild metric). We modeled the emission of an accretion disk around supermassive BH using numerical simulations based on a ray-tracing method in the Kerr metric. A broad emission line Fe K$\alpha$ at 6.4 keV with asymmetric profile (narrow bright blue peak and a wide faint red wing) has been observed in a number of type 1 AGN. The effects of strong gravitational field are investigated by comparison between the modeled and observed iron K$\alpha$ line profiles. The results of our modeling show that the parameters of the Fe K$\alpha$ line emitting region have significant influence on the line profile and thus, allow us to determine the space-time geometry (metric) in vicinity of the central BH of AGN, and also can give us information about the plasma conditions in these regions.'
address:
- 'Astronomical Observatory, Volgina 7, 11160 Belgrade, Serbia'
- 'Alexander von Humboldt Fellow, presently at Max Planck Institute for Radioastronomy, Bonn, Germany'
author:
- 'Predrag Jovanovi'' c [^1]'
- 'Luka Č. Popovi'' c'
title: |
Observational Effects of Strong Gravity in Vicinity of\
Supermassive Black Holes
---
Introduction
============
It is now widely accepted that AGN derive their extraordinary luminosities (sometimes more than $10^4$ times higher than luminosities of “ordinary” galaxies) from energy release by matter accreting towards, and falling into, a central supermassive BH. The accretion disks around the central BH represent an efficient mechanism for extracting gravitational potential energy and converting it into radiation, giving us the most probable explanation for the main characteristics of AGN (high luminosity, compactness, jet formation, rapid time variation in radiation and the profile of the Fe K$\alpha$ spectral line). Thus, AGN are powerful sources of radiation in a wide spectral range: from $\gamma$ rays to radio waves [@kr99].
The most important feature of the X-ray radiation of AGN (which is generated in the innermost region around a central BH) is a broad emission line Fe K$\alpha$ at 6.4 keV that may have an asymmetric profile (narrow bright blue peak and wide faint red peak). It was discovered in Seyfert 1 galaxy MCG-6-30-15 [@tan95] and later on observed in a number of AGN. In some cases the line width corresponds to one third of speed of light, indicating that its emitters rotate with relativistic velocities. Therefore, the line is probably produced in a very compact region near the central BH of AGN and can provide us some essential information about the plasma conditions and the space-time geometry in vicinity of the BH [@pop03a].
Black holes have only three measurable parameters (not including the Hawking temperature): charge, mass and angular momentum [@yaq07]. In this paper we will pay attention mostly to angular momentum or spin of central supermassive BH of AGN, which is a property of the space-time metric.
![Illustrations of accretion disk (left) and the corresponding Fe K$\alpha$ line profiles (right) in the case of Schwarzschild (top) and Kerr metric with angular momentum parameter $a=0.998$ (bottom). The disk inclination is $i=35^\circ$ and its inner and outer radii are $R_{in}=R_{ms}$ and $R_{out}=20$ R$_{g}$, respectively.[]{data-label="fig1"}](fig1a.eps "fig:"){width="80.00000%"}\
![Illustrations of accretion disk (left) and the corresponding Fe K$\alpha$ line profiles (right) in the case of Schwarzschild (top) and Kerr metric with angular momentum parameter $a=0.998$ (bottom). The disk inclination is $i=35^\circ$ and its inner and outer radii are $R_{in}=R_{ms}$ and $R_{out}=20$ R$_{g}$, respectively.[]{data-label="fig1"}](fig1b.eps "fig:"){width="80.00000%"}
![The same as in Fig. \[fig1\] but for a highly inclined disk with $i=75^\circ$.[]{data-label="fig2"}](fig2a.eps "fig:"){width="80.00000%"}\
![The same as in Fig. \[fig1\] but for a highly inclined disk with $i=75^\circ$.[]{data-label="fig2"}](fig2b.eps "fig:"){width="80.00000%"}
![The same as in Fig. \[fig2\] but for the Fe K$\alpha$ line emitting region in form of narrow annulus with width $=1 R_{g}$, extending from: $R_{in}=10$ R$_{g}$ to $R_{out}=11$ R$_{g}$ (top), $R_{in}=30$ R$_{g}$ to $R_{out}=31$ R$_{g}$ (middle) and $R_{in}=50$ R$_{g}$ to $R_{out}=51$ R$_{g}$ (bottom).[]{data-label="fig3"}](fig3a.eps "fig:"){width="85.00000%"}\
![The same as in Fig. \[fig2\] but for the Fe K$\alpha$ line emitting region in form of narrow annulus with width $=1 R_{g}$, extending from: $R_{in}=10$ R$_{g}$ to $R_{out}=11$ R$_{g}$ (top), $R_{in}=30$ R$_{g}$ to $R_{out}=31$ R$_{g}$ (middle) and $R_{in}=50$ R$_{g}$ to $R_{out}=51$ R$_{g}$ (bottom).[]{data-label="fig3"}](fig3b.eps "fig:"){width="85.00000%"}\
![The same as in Fig. \[fig2\] but for the Fe K$\alpha$ line emitting region in form of narrow annulus with width $=1 R_{g}$, extending from: $R_{in}=10$ R$_{g}$ to $R_{out}=11$ R$_{g}$ (top), $R_{in}=30$ R$_{g}$ to $R_{out}=31$ R$_{g}$ (middle) and $R_{in}=50$ R$_{g}$ to $R_{out}=51$ R$_{g}$ (bottom).[]{data-label="fig3"}](fig3c.eps "fig:"){width="85.00000%"}
Numerical simulations
=====================
An accretion disk could have different forms, dimensions and emissivity, depending on the type of its central BH, whether it is rotating (Kerr metric) or nonrotating (Schwarzschild metric). We modeled emission of an accretion disk around supermassive BH using numerical simulations based on a ray-tracing method in a Kerr metric, taking into account only photon trajectories reaching the observer’s sky plane in the infinity [@pop03a; @pop03b; @pj06]. Using this method we are able to obtain colorful images of accretion disk as would be seen by a distant observer with powerful high-resolution telescope (see left panels of Figs. \[fig1\] - \[fig3\]). From such disk images we then calculate total observed flux $F_{obs}$ according to the following expression: $$F_{obs} \left( {E_{obs}} \right) = {\int\limits_{image} {\epsilon \left(
{r} \right)}} g^{4}\delta \left( {E_{obs} - gE_{0}} \right)d\Xi ,$$ where $\epsilon \left({r} \right)$ is the disk emissivity, $E_0$ is the line transition energy ($E_0^{Fe\ K\alpha}=6.4$ keV), $g=E_{obs}/E_{em}$ is the energy shift due to relativistic effects ($E_{obs}$ is the observed energy and $E_{em}$ is the emitted energy from the disk) and $d\Xi$ is solid angle subtended by the accretion disk on observer’s sky. The modeled line profile is then obtained by binning the flux into the energy shift ($g$) axis (see right panels of Figs. \[fig1\] - \[fig3\]). The iron K$\alpha$ line shape strongly depends on emissivity law of the disk $\epsilon \left({r} \right)$, so we assume the standard Shakura-Sunyaev disk model [@sha73], where accretion occurs via an optically thick and geometrically thin disk.
Results
=======
The effects of strong gravitational field on the Fe K$\alpha$ line profile have been investigated in order to compare the modeled and observed line profiles. To obtain modeled line profiles, it is necessary to define a number of parameters which describe the line emitting region in the disk, such as constraints for its size, the disk inclination angle, the mass of the central BH and its angular momentum. For the disk inclination we adopted the averaged value from the study of the Fe K$\alpha$ line profiles of 18 Seyfert 1 galaxies: $i=35^\circ$ (see [@pop06] and references therein). The inner radius $R_{in}$ of the disk can not be smaller than the radius of the marginally stable orbit $R_{ms}$, that corresponds to $R_{ms}=6R_g$ (gravitational radius $R_g=GM/c^2$, where $G$ is gravitational constant, $M$ is the mass of central BH, and $c$ is the velocity of light) in the Schwarzschild metric and to $R_{ms}=1.23R_g$ in the case of the Kerr metric with angular momentum parameter $a=0.998$. To select the outer radius $R_{out}$ of the disk, we take into account some recent investigations of the Fe K$\alpha$ line profile showing that it should be emitted from the innermost part of the disk which outer radius is within several tens of R$_g$ (see [@pop06] and references therein).
In order to study observational effects of strong gravity in vicinity of supermassive BH in the center of AGN, we analyzed three cases for the Fe K$\alpha$ line emitting region: (i) $R_{in}=R_{ms}$, $R_{out}=20$ R$_{g}$ and $i=35^\circ$ (in both Schwarzschild and Kerr metric), (ii) $R_{in}=R_{ms}$, $R_{out}=20$ R$_{g}$ and $i=75^\circ$ (in both Schwarzschild and Kerr metric) and (iii) $i=75^\circ$ and the line emitting region in Kerr metric with $a=0.998$ is in form of narrow annulus with width $=1 R_{g}$, located between: (iiia) $R_{in}=10$ R$_{g}$ and $R_{out}=11$ R$_{g}$, (iiib) $R_{in}=30$ R$_{g}$ and $R_{out}=31$ R$_{g}$ and (iiic) $R_{in}=50$ R$_{g}$ and $R_{out}=51$ R$_{g}$.
Illustrations of an accretion disk and the corresponding Fe K$\alpha$ line shapes in the first case for Schwarzschild and Kerr metric are presented in Fig. \[fig1\]. As one can see in Fig. \[fig1\], the red peak of the Fe K$\alpha$ line is brighter in case of almost maximally rotating BH, but at the same time it is also more embedded into the blue peak wing and therefore less separable from it. Consequently, angular momentum of the central BH has significant influence on the line shape which supports assumption that the line originates from the innermost part of accretion disk, close to the central BH. This fact can be used for estimation of angular momentum of central BH in observed AGN (see e.g. [@tan95]).
Fig. \[fig2\] contains illustrations of the line emitting regions and the corresponding line shapes in the case of highly inclined disk ($i=75^\circ$). Here, the line profiles are broader than in the first case, mostly due to higher inclination. As it can be seen in Fig. \[fig2\], in case of Kerr metric, the red peak of the line is again more embedded into its blue peak wing (as in the first case) and it confirms that this effect can be most likely attributed to angular momentum.
Results for the third analyzed case are presented in Fig. \[fig3\]. From this figure one can see how the Fe K$\alpha$ line profile is changing as the function of distance from central BH. When the line emitters are located at the lower radii of the disk, i.e. closer to the central BH, the lines are broader and the line profiles are more asymmetric (see Fig. \[fig3\]). If the line emission is originating at larger distances from the BH, the red peak of the line becomes brighter and line profile narrower and more symmetric. In majority of AGN, where the broad Fe K$\alpha$ line is observed[^2], its profile is more similar to the modeled profile as obtained under assumption that the line emitters are located close to the central BH. Therefore, comparisons between the observed and modeled Fe K$\alpha$ line profiles can bring us some essential information about strong gravitational field in vicinity of central supermassive BH of AGN.
Conclusions
===========
We performed numerical simulations based on a ray-tracing method in a Kerr metric in order to model the emission of accretion disk around supermassive BH of AGN. We also simulated the influence of a strong gravitational field on the Fe K$\alpha$ line, showing that these effects can be detected in the observed line shapes. According to the obtained results, angular momentum or spin of central supermassive BH of AGN has significant influence on the line profile. Therefore, the analysis of the high resolution observations of the Fe K$\alpha$ line could be used for determination of the space-time geometry (metric) in vicinity of the supermassive BH, supposed to be in heart of AGN.
This work is a part of the project (146002) “Astrophysical Spectroscopy of Extragalactic Objects” supported by the Ministry of Science of Serbia. L. Č. Popovi' c is supported by Alexander von Humboldt foundation through Fritz Thyssen Special Programme.
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[^1]: Corresponding authorE-mail: , Phone: +381113089068, Fax: +381112419553
[^2]: Note here that in some AGN only the narrow Fe K$\alpha$ line is observed, but it is supposed to be emitted in the disk corona that is located farther from the disk, and therefore, these relativistic effects cannot be detected in the line profile
|
---
abstract: 'We study quantum dynamics of Grover’s adiabatic search algorithm with the equivalent two-level system. Its adiabatic and non-adiabatic evolutions are visualized as trajectories of Bloch vectors on a Bloch sphere. We find the change in the non-adiabatic transition probability from exponential decay for short running time to inverse-square decay for long running time. The size dependence of the critical running time is expressed in terms of Lambert $W$ function. The transitionless driving Hamiltonian is obtained to make a quantum state follow the adiabatic path. We demonstrate that a constant Hamiltonian, approximate to the exact time-dependent driving Hamiltonian, can alter the non-adiabatic transition probability from the inverse square decay to the inverse fourth power decay with running time. This may open up a new way of reducing errors in adiabatic quantum computation.'
author:
- Sangchul Oh
- Sabre Kais
title: 'Non-Adiabatic Quantum Dynamics of Grover’s Adiabatic Search Algorithm'
---
#### Introduction–
Grover’s quantum search algorithm [@Grover97] is known to find a marked one out of $N$ entries with the $O(\sqrt{N})$ queries on a quantum computer, otherwise the $O(N)$ queries are needed on a classical computer. While it was initially designed to be implemented on a quantum circuit model, its adiabatic quantum computation version [@Farhi], called Grover’s adiabatic search algorithm, was also proposed and the equivalence between them was proved [@Vandam; @Aharonov; @Roland03].
Much attention has been paid to solving an instantaneous eigenvalue problem of a time-dependent Hamiltonian of an adiabatic quantum algorithm because the minimum gap of a system determines the validity of an adiabatic quantum evolution and thus its computational complexity [@Messiah; @Farhi]. The non-adiabatic transition to other states, i.e., the deviation from the adiabatic evolution, is the main concern in adiabatic quantum computation. To know in detail how the non-adiabatic transition decreases asymptotically with running time, the minimum gap of the instantaneous eigenvalues is not enough, so a time-dependent Schrödinger equation has to be solved.
In this paper, we study quantum dynamics of Grover’s adiabatic search algorithm with an equivalent two-level system to calculate its non-adiabatic transition probability. The adiabatic and non-adiabatic evolutions of a quantum state are represented by trajectories on a Bloch sphere. We show that the non-adiabatic transition probability changes from exponential decay for short running time to inverse square decay for long running time. The dependence of the critical running time on the problem size is written in terms of Lambert $W$ function. Finally, We show that a constant driving Hamiltonian could reduce significantly the non-adiabatic transition probability, which may speed up adiabatic quantum computation.
#### Hamiltonian of adiabatic search algorithm–
Let us start with introducing the time-dependent Hamiltonian for Grover’s adiabatic search algorithm [@Roland03; @Schaller]. The adiabatic quantum computation is based on the adiabatic theorem which states that if a time-dependent Hamiltonian changes slowly enough, then an eigenstate of an initial Hamiltonian, an input state, evolves to an eigenstate of a final Hamiltonian, an output state [@Farhi; @Messiah]. Grover’s search algorithm takes the input state as a superposition of all possible states ${|{\varphi_{\rm in}} \rangle} = \frac{1}{\sqrt{N}}\sum_{i=0}^{N-1}{|{i} \rangle}$ with $N$ entries. It is the ground state of the initial Hamiltonian $H_0 = \mathbf{I} -{|{\varphi_{\rm in}}\rangle\langle{\varphi_{\rm in}}|}
= \mathbf{I} - \frac{1}{N}\sum_{i,j}{|{i}\rangle\langle{j}|}$ where $\mathbf{I}$ is an $N\times N$ identity matrix. Note that $\sum_{i,j}{|{i}\rangle\langle{j}|}$ is a matrix with all entries 1 whose eigenvalues are 0 ($N-1$ multiples) and $N$ [@Horn]. The output or target state ${|{w} \rangle}$ to find is the ground state of the final (or problem) Hamiltonian $H_p = \mathbf{I} -{|{w}\rangle\langle{w}|}$. The slow change from the initial to final Hamiltonians can be done as $H(t) = f(s)H_0 + g(s)H_p$ where $s\equiv t/T$ is the dimensionless (or macroscopic) time [@Messiah; @Betz], $T$ is the running time acting as an adiabatic parameter, and a turn-off function $f(s)$ and turn-on function $g(s)$ satisfy $f(0) = g(1) =1$ and $f(1) = g(0) = 0$. The simplest choice of $f$ and $g$ is to interpolate $H_0$ and $H_p$ linearly, i.e., $f(s) = 1-s$ and $g(s) = s$.
#### Instantaneous eigenvalues and eigenstates–
Grover’s search algorithm is understood as a rotation from the input state ${|{\varphi_{\rm in}} \rangle}$ to the target state ${|{w} \rangle}$. This implies it is essentially a two-dimensional problem formed by two linearly-independent vectors ${|{\varphi_{\rm in}} \rangle}$ and ${|{w} \rangle}$. While in quantum circuit model the full rotation is done by $O(\sqrt{N})$ successive finite rotations, it is done by a continuous rotation in adiabatic quantum computation. The two vectors ${|{w} \rangle}$ and ${|{\varphi_{\rm in}} \rangle}$ are linearly independent but not orthogonal. An orthonormal basis is easily constructed from the matrix representation of $\mathbf{I} -H(s) = f(s){|{\varphi_{\rm in}}\rangle\langle{\varphi_{\rm in}}|} + g(s){|{w}\rangle\langle{w}|}$ whose only the $w$-th diagonal element is different. Thus, the time-dependent Hamiltonian for Grover’s adiabatic search algorithm is represented with the orthonormal basis $\{{|{w} \rangle},{|{w_\perp} \rangle}\}$ as $$\begin{aligned}
H(s)= \mathbf{I} - \frac{f}{N}
\left[ \begin{array}{cc}
1 +N\frac{g}{f} & \sqrt{N-1}\\[10pt]
\sqrt{N-1} & N-1
\end{array}\right]\,,
\label{Hamil_B}\end{aligned}$$ where ${|{w_\perp} \rangle} = \frac{1}{\sqrt{N-1}}\sum_{i\ne w}^{N} {|{i} \rangle} \,.$ Hamiltonian (\[Hamil\_B\]) is written in convenient form as $$\begin{aligned}
H(s) = \frac{(f+g)}{2}\,\mathbf{I}
-\frac{1}{2N}\left[\begin{array}{rr}
Z(s) & X(s) \\[10pt]
X(s) & -Z(s)
\end{array}\right]\,,
\label{Hamil_C}\end{aligned}$$ where $Z(s)\equiv 2f +N(g -f)$ and $X(s) \equiv 2f\sqrt{N-1}$. Since the first term in Eq. (\[Hamil\_C\]) is not relevant to dynamics, it will be dropped. The Hamiltonian is written as $$\begin{aligned}
H(s) =
-\frac{\hbar\omega(s)}{2}\left[\begin{array}{rr}
\cos\theta(s) & \sin\theta(s) \\[10pt]
\sin\theta(s) &-\cos\theta(s)
\end{array}\right]\,,
\label{Hamil_D}\end{aligned}$$ where the gap between the ground and excited states is given by $\hbar\omega(s) \equiv\frac{1}{N} \sqrt{Z^2 + X^2} = \sqrt{ (f-g)^2 + \frac{4}{N}fg}\,$. Here mixing angle $\theta$ is defined by $\tan\theta(s) \equiv X(s)/Z(s)$. While a different choice of $f$ and $g$ gives rise to a different energy gap, hereafter we consider only a linear interpolation case. Hereafter we set $\hbar=1$.
As in a textbook of quantum mechanics, the instantaneous eigenstates of $H(s){|{e_\pm(s)} \rangle} = e_\pm(s) {|{e_\pm(s)} \rangle}$ read $$\begin{aligned}
{|{e_{-}(s)} \rangle}
= \left[\begin{array}{cc}
\cos\tfrac{\theta}{2}\\[10pt]
\sin\tfrac{\theta}{2}
\end{array}\right]\,,\quad
{|{e_{+}(s)} \rangle}
= \left[ \begin{array}{rr}
-\sin\tfrac{\theta}{2}\\[10pt]
\cos\tfrac{\theta}{2}
\end{array}\right]\,.\end{aligned}$$ As represented by a Bloch vector in Fig. \[Fig1\], the input state ${|{\varphi_{\rm in}} \rangle}={|{e_{-}(0)} \rangle}$ is a vector with azimuthal angle $\tan\theta=(2-N)/2\sqrt{N-1}$. The target state ${|{w} \rangle} = {|{e_-(1)} \rangle}$ points to the north pole. Thus, like the Landau-Zener-Majorana-Stückelberg problem [@Landau; @Zener; @Majorana; @Stuckelberg], Grover’s adiabatic search algorithm is just a rotation of a single qubit driven by time-dependent Hamiltonian (\[Hamil\_D\]).
![(color online). Trajectories of a Bloch vector ${\bf r}(t)$ on a Bloch sphere for various running times (a) $T=10$, (b) $T=100$, (c) $T=300$. (d) The blue longitudinal line represents the adiabatic path. Here $N=4$ is set. If $N$ is large, an initial Bloch vector becomes closer to ${|{w_\perp} \rangle}$.[]{data-label="Fig1"}](F1.eps)
#### Quantum dynamics of adiabatic search algorithm–
To understand non-adiabatic effects, we solve numerically a time-dependent Schrödinger equation $$\begin{aligned}
i\hbar\frac{d}{dt}{|{\psi(t)} \rangle} = H_T(t) {|{\psi(t)} \rangle}\,,\end{aligned}$$ where a time-dependent Hamiltonian $H_T(t)$ is given by Eq. (\[Hamil\_D\]). As illustrated in Fig. \[Fig1\], a quantum state ${|{\psi(t)} \rangle} = \alpha(t){|{w} \rangle} +
\beta(t) {|{w_\perp} \rangle}$ is visualized by a Bloch vector ${\bf r}(t) \equiv {\langle {\psi(t)}|}\bm{\sigma}{|{\psi(t)} \rangle}$ with Pauli matrices $\sigma_k$ for $k=x,y,z$. In the adiabatic limit of $T\gg \sqrt{N}$, an evolved quantum state remains in the instantaneous ground state, that is, ${|{\psi(t)} \rangle} \simeq {|{e_{-}(t)} \rangle}$ up to the dynamical and geometric phase factors. So, the Bloch vector ${\bf r}_{\rm ad}(s) = {\langle {e_{-}(s)}|}\bm{\sigma}{|{e_{-}(s)} \rangle}$ travels to the north pole along the longitude line on a Bloch sphere.
The adiabatic path is a good approximation to the exact evolution if running time $T$ is large enough, that is, the Hamiltonian changes slowly enough. For finite running time $T$, however, a real path deviates from the adiabatic path as illustrated in Fig. \[Fig1\]. To see this in detail, we examine how a quantum state ${|{\psi(t)} \rangle}$ is deviated from the instantaneous eigenstate ${|{e_{-}(t)} \rangle}$ as adiabatic parameter $T$ is varied. The evolved state ${|{\psi(t)} \rangle}$ is written in terms of instantaneous eigenstates as ${|{\psi(s)} \rangle} = a(s){|{e_{-}(s)} \rangle} + b(s) {|{e_{+}(s)} \rangle}\,$. Fig. \[Fig\_dev\] plots the transition probability $P(s) = 1- |a(s)|^2$ of being in an instantaneous ground state ${|{e_{+}(s)} \rangle}$ for various running time $T$. For short running time $T$, as shown in Fig. \[Fig\_dev\] [(a)]{}, the maximum of $P(s)$ does not coincide with the location of the minimum energy gap. As depicted in Figs. \[Fig\_dev\] [(b), (c), and (d)]{}, $P(s)$ becomes smaller and more symmetric and reaches at its peak at $s=1/2$ as running time $T$ is increased.
#### Transition of non-adiabatic transition–
The non-adiabatic transition probability $P(1)$ at $s=1$ indicates the error of adiabatic quantum computation. The asymptotic form of $P(1)$ for the Landau-Zener-Majorana-Stückelberg problem is known to decrease exponentially [@Landau; @Zener; @Majorana; @Stuckelberg]. Suzuki and Okada [@Suzuki], however, calculated numerically the residual energy, the difference between the energy expectation $E(s)={\langle {\psi(s)}|} H(s){|{\psi(s)} \rangle}$ and the instantaneous ground energy $e_{-}(s)$, for a modified Landau-Zener-Majorana-Stükelberg problem. They showed the transition of the residual energy from exponential decay only for short running time to the inverse-square decay for long running time. The similar result was obtained by Rezakhani [*et al.*]{} [@Rezakhani]. Note $1/T^2$ decay was reported for the simulated annealing system by Santoro [*et al.*]{} [@Santoro] and for adiabatic quantum teleportation by Oh [*et al.*]{} [@Oh13]. As illustrated in Fig. \[Fig3\], we calculate numerically the non-adiabatic transition probability $P(1)$ as a function of running time $T$ and find $$P(1) \sim
\begin{cases}
\exp\left( -A\,T\right) & \text{for\, $T < T_c$}\\[10pt]
B/{T^2} & \text{for\, $T > T_c$}
\end{cases}\,.
\label{transition_prob}$$ The coefficients $A$, $B$, and the transition time $T_c$ depend on the system size $N$, as shown in Fig. \[Fig4\]. The numerical data show $A\sim \pi/4N$ and $B\sim 4/N$. The critical running time $T_c$ can be defined by a solution of the transcendental equation $e^{-A\,T} = B/T^2$ in Eq. (\[transition\_prob\]). It is given by $$T_c = -\frac{2}{A} W_{-1}\left(-\frac{A\sqrt{B}}{2}\right)
\sim \frac{8N}{\pi} W_{-1}\left(-\frac{\pi}{4\sqrt{N^3}}\right)\,,
\label{Tc_Lambert}$$ where $W_{-1}$ is the lower branch of the Lambert $W$ function [@Lambert; @Corless].
#### Transitionless driving–
When the time-dependent Schrödinger equation is transformed to the adiabatic frame, it is clearly seen why the non-adiabatic transition happens. Demirplak and Rice [@Demirplak], and Berry [@Berry] showed that a time-dependent Hamiltonian $H_D(t)$, called the counter or transitionless driving term, in addition to the original time-dependent Hamiltonian makes a quantum state follow the original adiabatic state exactly. The main idea is to make a driving Hamiltonian cancel the non-adiabatic term seen in the adiabatic frame. The driving Hamiltonian $H_{D}(t)$ for Hamiltonian (\[Hamil\_D\]) reads $$\begin{aligned}
H_D(t) = i\hbar\,\frac{\partial U^{\dag}(t)}{\partial t}\,U(t)
= -\hbar\,\frac{\dot{\theta}}{2}\,\sigma_y
\label{Hamil_Driving}\end{aligned}$$ where the unitary operator $U(t)$ is composed of instantaneous eigenstates ${|{e_{\pm}(t)} \rangle}$ $$\begin{aligned}
U(t) = \left[
\begin{array}{rr}
\cos\frac{\theta}{2} & -\sin\frac{\theta}{2}\\[10pt]
\sin\frac{\theta}{2} & \cos\frac{\theta}{2}
\end{array}
\right]\,,\end{aligned}$$ and $\dot{\theta} = \frac{d\theta}{ds}\frac{ds}{dt} = \frac{1}{T}\frac{d\theta}{ds}$. Note Pauli operator $\sigma_y$ is represented by $\sigma_y= -i{|{w}\rangle\langle{w_{\perp}}|} + i {|{w_{\perp}}\rangle\langle{w}|}$. For linear interpolation, one has $\dot{\theta}(t) = 2\frac{\sqrt{N-1}}{NT}\left[(1-2s)^2 + \frac{4}{N}(1-s)s\right]^{-1}$. As expected, the driving Hamiltonian goes to zero in the adiabatic limit, $T\gg 1$.
While the driving Hamiltonian $H_D(t)$ makes a quantum state evolve exactly along the longitudinal line (adiabatic path) regardless of $T$, it seems to be difficult to control the strength $\dot{\theta}$ even in linear interpolation case. So, we investigate whether an approximate but constant driving Hamiltonian, instead of the exact time-dependent driving Hamiltonian (\[Hamil\_Driving\]), could reduce some errors. We consider two constant driving Hamiltonians which are the minimum and maximum values of $H_D$, respectively $$\begin{aligned}
H_D^{\min} = -\frac{\hbar\sqrt{N-1}}{NT}\,\sigma_y\,,\;\;
H_D^{\max} = -\frac{\hbar\sqrt{N-1}}{T}\,\sigma_y\,.\end{aligned}$$ Fig. \[Fig5\] shows how the instantaneous eigenvalues change when the driving Hamiltonian $H_D(s)$ is added to $H(s)$. The role of $H_D$ is to make the gap at the avoided crossing wider. While the approximate driving Hamiltonian $H_D^{\rm min}$ seems to make a very little change in adiabatic energy levels and the trajectory as shown in Fig. \[Fig6\], it produces drastic change in the non-adiabatic transition probability for long running time, from $O(1/T^2)$ to $O(1/T^4)$ as depicted in Fig. \[Fig7\]. Let take a close look at it in connection with the adiabatic condition $$\begin{aligned}
T\gg \frac{\max_{s} |{\langle {e_{+}(s)}|}\frac{d H}{ds}{|{e_{-}(s)} \rangle}|}{\min_s\Delta E(s)^2}\,,
\label{Adiabatic_condition}\end{aligned}$$ where $\Delta E$ is the energy gap. For two Hamiltonians $H(s)$ and $H(s) + H_{\min}^D$ with $T=10$, while the numerators in Eq. (\[Adiabatic\_condition\]) are same, the denominators change slightly, to be more specific, from $0.01$ to $0.010396$. Although the right-hand side of the inequality (\[Adiabatic\_condition\]) changes very little, $P(1)$ for long running time changes from the inverse square to fourth power decays. Note that $H_D^{\rm min}$ also reduces $P(1)$ for short running time.
![(color online). Trajectories of Bloch vectors on a Bloch sphere when the quantum evolution is driven (a) by adiabatically or exactly $H_D(t)$, (b) by $H_D^{\min}$, (c) by $H_D^{\max}$, and (d) without driving. Here $N=4$ and $T=10$ are taken.[]{data-label="Fig6"}](F6.eps)
![(color online). Non-adiabatic transition probability $P(1)$ as a function of running time $T$ with $H_D^{\min}$ (blue) and without driving Hamiltonian (red). Here $N=10$ is taken.[]{data-label="Fig7"}](F7.eps)
#### Conclusion–
We studied quantum dynamics of Grover’s adiabatic search algorithm as a time-dependent two-level system. The transition from the non-adiabatic and adiabatic quantum evolutions were visualized by changes in trajectories of Bloch vectors on a Bloch sphere. We found a drastic change in the non-adiabatic transition probability from well-known exponential decay for short running time to the inverse-square decay for longer running time. The dependence of the critical running time on the problem size is obtained with Lambert $W$ function. We showed an approximate but constant driving Hamiltonian could reduce the non-adiabatic transition probability significantly which becomes the inverse fourth power decay for long running time. It would be interesting to see whether the results obtained in this paper could be applied to other quantum system, for example, a quantum Ising model [@Campo], or quantum optimization problems [@Boixo]. While our results was obtained by numerical calculations, it would be interesting to seek an exact analytic solution.
[99]{} Lov K. Grover, , [**79**]{}, 325 (1997). E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, Science [**292**]{}, 472 (2001). W. van Dam, M. Mosca, and U. Vazirani, Proceedings of the 42nd Annual Symposium on Foundations of Computer Science, p. 279-287 (2001). D. Aharonov, W. Van Dam, J. KEPME, Z. Landau, S. Lloyd, and O. Regev, SIAM J. Comput. [**37**]{}, 166 (2007). J. Roland and N. J. Cerf, [**68**]{}, 062311 (2003); [*ibid*]{}, 062312 (2003). A. Messiah, [*Quantum Mechanics*]{} (North-Holland, Amsterdam, 1963). G. Schaller, S. Mostame, R. Schützhold, [**73**]{}, 062307 (2006). R. A. Horn and C. R. Johnson, [*Matrix Analysis*]{} (Cambridge Univ. Press, Cambridge, 1990), p. 39. V. Betz and S. Teufel, in Lect. Notes Phys. [**690**]{}, 19 (2006). L. D. Landau, Physics of the Soviet Union [**2**]{}, 46 (1932). C. M. Zener, Proc. R. Soc. London Ser. A [**137**]{}, 696 (1932). E. Majorana, Nuovo Cimento [**9**]{}, 43 (1932). E. C. G. Stückelberg, Helv. Phys. Acta [**5**]{}, 369 (1932). S. Suzuki and M. Okada, in Lect. Notes Phys. [**679**]{}, 207 (2005). A. T. Rezakhani, A. K. Pimachev, and D. A. Lidar [**82**]{}, 052305 (2010). G. E. Santoro, R. Martoňák, E. Tosatti, and R. Car, Science [**295**]{}, 2427 (2002). S. Oh, Y.-P. Shim, J. Fei, M. Friesen, and X. Hu, [**87**]{}, 022332 (2013). J. H. Lambert, Acta Helvetica, Physico-mathematico-anatomico-13botanico-medica [ **3**]{}, 128 (1758). R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, Adv. in Comp. Math., [**5**]{} 329 (1996). M. Demirplak and S. A. Rice, J. Phys. Chem. A [**107**]{}, 9937 (2003). M. V. Berry, J. Phys. A: Math. Theor. [**42**]{}, 365303 (2009). A. del Campo, M. M. Rams, and W. H. Zurek, [**109**]{}, 115703 (2012). S. Boixo, T. Albash, F. M. Spedalieri, N. Chancellor, and D. A. Lidar, Nat. Commun. [**4**]{}, 3067 (2013).
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---
abstract: 'Electron scattering form factors from $^{12}$C have been studied in the framework of the particle-hole shell model. Higher configurations are taken into account by allowing particle-hole excitations from the 1$s$ and 1$p$ shells core orbits up to the 1$f$-2$p$ shell. The inclusion of the higher configurations modifies the form factors markedly and describes the experimental data very well in all momentum transfer regions.'
address: '$^1$Instituto de Física, Universidade Federal do Rio de Janeiro, C.P. 68528, 21941-972 Rio de Janeiro, RJ, Brazil'
author:
- 'F. A. Majeed$^{1,\footnote{\,\,Permenant address: Department of Physics, College of Science, Al-Nahrain University, Baghdad, IRAQ. \,\,\,Email:fouad@if.ufrj.br}}$'
title: 'Longitudinal and Transverse Form Factors from $^{12}$C[^1]'
---
Introduction
============
Shell model calculations, carried out within a model space in which the nucleons are restricted to occupy a few orbits are unable to reproduce the measured static moments or transition strengths without scaling factors. Inadequacies in the shell model wavefunctions are revealed by the need to scale the matrix elements of the one-body operators by effective charges to match the experimental data. However, the introduction of effective charges may bring the calculated transition strengths which are defined at the photon point, as well as, the form factors at the first maximum, closer to the measured values, but the non-zero momentum transfer ($q$) values might deviate appreciably from the measured values [@RA03].
Electron scattering at 200 MeV on $^{12}$C and $^{13}$C, have been studied by Sato [*et al.*]{} [@TS85]. The effect of higher configurations wavefunctions are included in the work of Bennhold [*et al.*]{} [@CB85]. Booten [*et al.*]{} [@JB94] investigated the higher configurations contributions on some $p$-shell nuclei. Coulomb form factors of C2 transitions in several selected $p$-shell nuclei are discussed by Radhi [*et al.*]{} [@RA01] taking into account core-polarization effects. Configuration mixing shell model has been recently used [@FA06] to study the isovector states of $^{12}$C in the framework of particle-hole theory. The calculations are quite successful and describe the experimental form factors very well for all momentum transfer regions.
The purpose of the present work is to include higher-energy configurations by allowing excitation from 1$s$ and 1$p$ shells core orbits up to the 1$f$-2$p$ shell. The configurations which include the higher configurations is called the extended space configurations. The ground state of $^{12}$C is taken to have closed 1$s$$_{1/2}$ and 1$p$$_{1/2}$ shells. The states expected to be most strongly excited from closed-shell nuclei are linearly combination of a configurations in which one nucleon has been raised to a higher shell, forming pure single-particle-hole state [@TW84]. This approximation is called Tamm-Dancoff approximation (TDA)[@TJ66]. The dominant dipole, quadrupole and multipole $T$=1 single particle-hole states of $^{12}$C are considered with the framework of the harmonic oscillator (HO) shell model. The Hamiltonian is diagnoalized in the space of the single-particle hole states, in the presence of the modified surface delta interaction (MSDI) [@PM77]. The space of the single-particle-hole states include all shells up to 2$p$$_{1/2}$ shell. Admixture of higher configurations is also considered. A comparison of the calculated form factors using this model with the available experimental data for the dominantly $T$=1 states are discussed.
Theory
======
The ground state of $^{12}$C is taken to have closed 1$s$$_{1/2}$ and 1$p$$_{3/2}$ shells, and is represented by $\Psi$$_{0}$. The particle-hole state formed by promoting one particle from the shell-model ground state. The particle-hole state of the total Hamiltonian is represented by $\Phi$$_{JM}$($ab$$^{-1}$) with labels (a) for particles with quantum numbers ($n$$_{a}$$\ell$$_{a}$$j$$_{a}$) and (b) for holes with quantum numbers ($n$$_{b}$$\ell$$_{b}$$j$$_{b}$). The state $\Phi$$_{JM}$($ab$$^{-1}$) indicating that a particle was vacated from $j$$_{b}$ and promoted to $j$$_{a}$.
The excited state wavefunction can be constructed as a linear combinations of pure basis $\Phi$$^{,s}$ as [@TW84] $$\Psi^{n}_{JM}=\sum_{ab}\chi^{J}_{ab^{-1}}\Phi_{JM}(ab^{-1}),$$ where the amplitude $\chi$$^{J}_{ab^{-1}}$ can be determined from a diagonalization of the residual interaction. By including the isospin $T$ [@TJ66], one now has to solve the secular equation $$\sum_{ab}[\langle{\acute{a}\acute{b}^{-1}}|H|ab^{-1}\rangle_{JMTT_z}-E_n\delta_{\acute{a}\acute{b}^{-1},
ab^{-1}}]\,\chi^{JT}_{ab^{-1}}=0.$$ The matrix element of the Hamiltonian is given by [@PM77] $$\begin{aligned}
\langle{\acute{a}\acute{b}^{-1}}|H|ab^{-1}\rangle_{JMTT_z}
=(e_{\acute{a}}-e_{\acute{b}})\,\delta_{{a\acute{a}},{b\acute{b}}}\nonumber\\
+\langle{\acute{a}\acute{b}^{-1}}|V|ab^{-1}\rangle_{JMTT_z},\end{aligned}$$ where $e$$_{\acute{a}}$-$e$$_{\acute{b}}$ is the unperturbed energy of the particle-hole pair obtained from energies in nuclei with A$\pm$1 particles.
The matrix element of the residual interaction $V$ is given by the MSDI with the strength parameters $A$$_{0}$=0.8 MeV, $A$$_{1}$=1.0 MeV, $B$=0.7 MeV and $C$=$-$0.3 MeV [@PM77]. $$\begin{aligned}
\langle{\acute{a}\acute{b}^{-1}}|V|ab^{-1}\rangle_{JMTT_z}
=-\sum_{\acute{J}\acute{T}}(2\acute{J}+1)(2\acute{T}+1)\nonumber\\\times\left\{
\begin{array}{ccc}
j_{\acute{a}} & j_{b} & \acute{J}\\
j_{a} & j_{\acute{b}} & J
\end{array}\right\}\left\{
\begin{array}{ccc}
\frac{1}{2} & \frac{1}{2} & T\\
\frac{1}{2} & \frac{1}{2} & \acute{T}
\end{array}\right\}\langle{\acute{a}\,b}|V|a\acute{b}\rangle_{\acute{J}\,\acute{T}}.\end{aligned}$$
The matrix elements of the multipole operators $T$$_{J}$ are given in terms of the single particle matrix elements by [@TW84] $$\left\langle\Psi_{J}\|T_{Jt_{z}}\|\Psi_{0}\right\rangle
=\sum_{ab}\chi^{Jt_{z}}_{ab^{-1}}\left\langle
a\|T_{Jt_{z}}\|b\right\rangle,$$ where $t$$_{z}$=1/2 for protons and -1/2 for neutrons. The amplitudes $\chi$$^{Jt_{z}}_{ab^{-1}}$ can be written in terms of the amplitudes $\chi$$^{JT}_{ab^{-1}}$ in isospin space as [@PM77] $$\begin{aligned}
\
\chi^{Jt_{z}}_{ab^{-1}}=(-1)^{T_{f}-T_{i}}\left[\left(
\begin{array}{ccc}
T_{f} & 0 & T_{i} \\
-T_{z} & 0 & T_{z} \\
\end{array}
\right)\sqrt{2}\ \frac{\chi^{JT=0}_{ab^{-1}}}{2}\right.\nonumber\\
\left.+2t_{z}\left(\begin{array}{ccc}
T_{f} & 0 & T_{i} \\
-T_{z} & 0 & T_{z} \\
\end{array}
\right)\sqrt{6}\
\frac{\chi^{JT=1}_{ab^{-1}}}{2}\right],\end{aligned}$$ where $$\
T_{z}=\frac{Z-N}{2}$$
The single particle matrix elements of the electron scattering operator $T$$^{\eta}_{J}$ are those of Ref.[@BA85] with $\eta$ selects the longitudinal ($L$), transverse electric ($E\ell$) and transverse magnetic ($M$) operators, respectively. Electron scattering form factors involving angular momentum transfer $J$ is given by [@BA85] $$\begin{aligned}
\
|F^{\eta}_{J}(q)|^{2}=\frac{4\pi}{Z^{2}(2J_{i}+1)}\ |\langle\Psi_{J_{f}}\|T^{\eta}_{Jt_{z}}\|\Psi_{J_{i}}\rangle\nonumber\\
\times|F_{c.m}(q)|^{2} \ |F_{f.s}(q)|^{2}\end{aligned}$$ where $J$$_{i}$= 0 and $J$$_{f}$=$J$ for closed shell nuclei and $q$ is the momentum transfer. The last two terms in Eq.(8) are the correction factors for the ($c.m.$) and the finite nucleon size ($f.s.$)[@BA85]. The total inelastic electron scattering form factor is defined as [@TJ66] $$|F_{J}(q,\theta)|^{2}=|F^{L}_{J}(q)|^{2}+\left[\frac{1}{2}+\tan^{2}(\theta/{2})\right]
|F^{Tr}_{J}(q)|^{2},$$ where $|F^{Tr}_{J}(q)|^{2}$ is the transverse electric or transverse magnetic form factors.
Results and Discussion
======================
The unperturbed energies for the single particle-hole states for both positive and negative parity states used in this work are adopted from our previous theoretical work (see Table 1 and 2 from Ref.[@FA06]). Higher configurations are included in the calculations when the ground state is considered as a mixture of the $|(1$$s$$_{1/2}$)$^{4}$$\,(1$$p$$_{3/2}$)$^{8}$$\rangle$ and $|(2$$s$$_{1/2}$)$^{4}$$\,(2$$p$$_{3/2}$)$^{8}$$\rangle$ configurations, such that the ground state wavefunction becomes $$\begin{aligned}
|\Psi_{00}\rangle=\gamma|\Psi_{00}(1s_{1/2})^4(1p_{3/2})^8\rangle\nonumber\\
+\delta|\Psi_{00}(2s_{1/2})^4(2p_{3/2})^8\rangle\end{aligned}$$
with $\gamma^{2}$+$\delta^{2}$=1, $\chi^{JT}_{ab_{1}^{-1}}$=$\gamma\chi^{JT}_{ab^{-1}}$ and $\chi^{JT}_{ab_{2}^{-1}}$=$\delta\chi^{JT}_{ab^{-1}}$
The excited states is also assumed as a mixture of the particle-hole configurations, $|$a$_{1}$$b^{-1}_{1}$$\rangle$, $|$a$_{2}$$b^{-1}_{2}$$\rangle$, $|$a$_{2}$$b^{-1}_{1}$$\rangle$ and $|$a$_{1}$$b^{-1}_{2}$$\rangle$, where $|$a$_{1}$$\rangle$=$|$a$\rangle$=$|$$n$$_{a}$$\,$$\ell$$_{a}$$\,$$j$$_{a}$$\rangle$, $|$a$_{2}$$\rangle$=$|$a$\rangle$=$|$$n$$_{a}$$+1\,$$\ell$$_{a}$$\,$$j$$_{a}$$\rangle$, $|$b$_{1}$$\rangle$=$|$b$\rangle$=$|$$n$$_{b}$$\,$$\ell$$_{b}$$\,$$j$$_{b}$$\rangle$ and $|$b$_{2}$$\rangle$=$|$b$\rangle$=$|$$n$$_{b}$$+1\,$$\ell$$_{b}$$\,$$j$$_{b}$$\rangle$.
The matrix element given in Eq.(5) is called the model space matrix element, where $a$ and $b$ are defined by the amplitudes given in Tables \[tab1\] and \[tab2\] for the negative and positive parity states, respectively.
The extended space matrix element becomes
$$\begin{aligned}
\left\langle\Psi_{J}\|T_{Jt_{z}}\|\Psi_{0}\right\rangle
=\sum_{a_{1}b_{1}}\chi^{Jt_{z}}_{a_{1}b_{1}^{-1}}\left\langle
a_{1}\|T_{Jt_{z}}\|b_{1}\right\rangle\nonumber\\
+\sum_{a_{1}b_{2}}\chi^{Jt_{z}}_{a_{1}b_{2}^{-1}}\left\langle
a_{1}\|T_{Jt_{z}}\|b_{2}\right\rangle\nonumber\\
+\sum_{a_{2}b_{1}}\chi^{Jt_{z}}_{a_{2}b_{1}^{-1}}\left\langle
a_{2}\|T_{Jt_{z}}\|b_{1}\right\rangle\nonumber\\
+\sum_{a_{2}b_{2}}\chi^{Jt_{z}}_{a_{2}b_{2}^{-1}}\left\langle
a_{2}\|T_{Jt_{z}}\|b_{2}\right\rangle,\end{aligned}$$
where $$\begin{aligned}
\chi^{Jt_{z}}_{a_{1}b_{1}^{-1}}=C_{1}\,\chi^{Jt_{z}}_{ab^{-1}},\nonumber\\
\chi^{Jt_{z}}_{a_{1}b_{2}^{-1}}=C_{2}\,\chi^{Jt_{z}}_{ab^{-1}},\nonumber\\
\chi^{Jt_{z}}_{a_{2}b_{1}^{-1}}=C_{3}\,\chi^{Jt_{z}}_{ab^{-1}},\nonumber\\
\chi^{Jt_{z}}_{a_{2}b_{2}^{-1}}=C_{4}\,\chi^{Jt_{z}}_{ab^{-1}},\end{aligned}$$
[lcccc]{}
Particle-hole & E($1$$^{-}$) & E($2_{1}$$^{-}$)& E($2_{2}$$^{-}$)& E($3$$^{-}$)\
configuration & 18.44 MeV & 19.88 MeV & 23.50 MeV & 18.87 MeV\
$|$a$b^{-1}$$\rangle$ & $\chi$$^{11}$ & $\chi$$^{21}$ & $\chi$$^{31}$ & $\chi$$^{31}$\
\
(1$p$$_{1/2}$)(1$p$$_{3/2}$)$^{-1}$ & 0.0473 & 0.0000 & 0.0000 & 0.0000\
(1$d$$_{5/2}$)(1$s$$_{1/2}$)$^{-1}$ & -0.1810 & 0.8314 & 0.0703 & 0.9993\
(2$s$$_{1/2}$)(1$s$$_{1/2}$)$^{-1}$ & 0.9739 & 0.5430 & 0.0834 & 0.0000\
(1$d$$_{1/2}$)(1$s$$_{1/2}$)$^{-1}$ & 0.1333 & -0.1054& 0.9936 & 0.0318\
(1$f$$_{7/2}$)(1$p$$_{3/2}$)$^{-1}$ & 0.0000 & 0.0442 & 0.0000 & 0.0165\
(2$p$$_{3/2}$)(1$p$$_{3/2}$)$^{-1}$ & 0.0008 & 0.0000 & 0.0222 & 0.0000\
(1$f$$_{5/2}$)(1$p$$_{3/2}$)$^{-1}$ & 0.0000 & -0.0636& 0.0147 &-0.0030\
(2$p$$_{1/2}$)(1$p$$_{3/2}$)$^{-1}$ & 0.0000 & 0.0000 & 0.0000 & 0.0000\
[lc]{}
Particle-hole & E($3$$^{+}$)=27.10 MeV\
configuration & $\chi$$^{31}$\
$|$a$b^{-1}$$\rangle$ &\
\
(1$p$$_{1/2}$)(1$p$$_{3/2}$)$^{-1}$ & 0.0000\
(1$d$$_{5/2}$)(1$s$$_{1/2}$)$^{-1}$ & -0.0475\
(2$s$$_{1/2}$)(1$s$$_{1/2}$)$^{-1}$ & 0.0000\
(1$d$$_{1/2}$)(1$s$$_{1/2}$)$^{-1}$ & 0.0000\
(1$f$$_{7/2}$)(1$p$$_{3/2}$)$^{-1}$ & 0.9461\
(2$p$$_{3/2}$)(1$p$$_{3/2}$)$^{-1}$ & -0.3201\
(1$f$$_{5/2}$)(1$p$$_{3/2}$)$^{-1}$ & -0.0020\
(2$p$$_{1/2}$)(1$p$$_{3/2}$)$^{-1}$ & 0.0000\
The values of the parameters C$^{,s}$ are given in Table \[tab3\]. The states $1$$^{-}$, $2_{1}$$^{-}$, $2_{2}$$^{-}$, $3$$^{-}$ and $3$$^{+}$ are found experimentally at 18.12 MeV, 19.50 MeV, 22.70 MeV, 18.60 MeV and 20.60 MeV respectively [@RR87]. We obtain the values 18.44 MeV, 19.88 MeV, 23.50 MeV, 18.87 MeV and 27.10 MeV for the states $1$$^{-}$, $2_{1}$$^{-}$, $2_{2}$$^{-}$, $3$$^{-}$ and $3$$^{+}$, respectively.
[ccccc]{}
$J$$^{\pi}$ & $C$$_{1}$ & $C$$_{2}$ & $C$$_{3}$ & $C$$_{4}$\
\
3$^{+}$ & 0.92 & -0.27 & -0.27 & 0.078\
2$_{1}^{-}$ & -0.92 & 0.27 & -0.27 & 0.078\
2$_{2}^{-}$ & -0.92 & 0.27 & -0.27 & 0.078\
The $1$$^{-}$ (18.12 MeV), C1+E1 form factor is shown in Fig.1. The amplitudes $\chi$$^{,s}$ reduced by a factor 1.3, to agree with the low $q$ data [@TW84]. This state is dominated by (2$s$$_{1/2}$)(1$s$$_{1/2}$)$^{-1}$ particle-hole configuration, as given in Table \[tab1\]. The single-particle matrix elements are calculated with the harmonic oscillator wavefunctions (HO) with oscillator parameter $b=1.64$ fm to agree with the elastic form factor determination [@TK85]. Our results are consistent with the previous calculation of Donnelly [@TW70] and slightly in better agreement with the experimental data for the momentum transfer region $q \leq$ 1.0 fm$^{-1}$.
The transverse magnetic form factor M2 for the excitation to the $2_{1}$$^{-}$, 19.50 MeV state is shown in Fig.2. The amplitudes have to be enhanced by a factor 1.2 to account for the experimental data. The calculations incorporate the single-particle wavefunctions of the (HO) potential with $b=1.64$ fm and a value of $\gamma$=0.95, to account for the ground state correlation. The data are very well explained for the momentum-transfer $q \leq$ 3.0 fm$^{-1}$.
Figure3, shows the transverse magnetic form factor M2 for the excitation to the $2_{1}$$^{-}$, 22.70 MeV state. The amplitudes have to be reduced by a factor 1.82 to fit the low-$q$ data. The single-particle wavefunctions are those of the (HO) potential with size parameter $b=1.50$ fm and a value of $\gamma$=0.97, to account for the ground state correlation. The experimental data are very well described throughout the momentum-transfer regions and the results are consistent with that of Hicks [*et al.*]{}, [@RS84].
The $3$$^{-}$ (18.60 MeV), is dominated by (1$d$$_{5/2}$)(1$s$$_{1/2}$)$^{-1}$ particle-hole configuration, as given in Table \[tab1\]. The only multipole that contributes to the scattering is the longitudinal C3 multipole as shown in Fig.4. The calculations incorporate the single-particle wavefunctions of the (HO) potential with $b=1.64$ fm and $\gamma$ takes the value 1.0 . The experimental data are very well explained for the momentum-transfer values $q \leq$ 3.0 fm$^{-1}$ and the results are consistent with that of Hicks [*et al.*]{}, [@RS84] and Yamaguchi [*et al.*]{}, [@YM71], where the form factor seems to be a pure longitudinal form factor.
Figure5, shows the transverse magnetic form factor for the excitation to the $3$$^{+}$, 20.60 MeV state. The dominated configuration is the (1$f$$_{7/2}$)(1$p$$_{3/2}$)$^{-1}$ particle-hole configuration, as given in Table \[tab2\]. The only multipole that contributes to the scattering is the magnetic M3 multipole. The amplitudes have to be reduced by factor of 5 to account for the experimental data. The calculations incorporate the single-particle wavefunctions of the (HO) potential with $b=1.64$ fm, and a value $\gamma$=0.7, to account for the ground state correlation. The data are very well explained throughout the momentum-transfer values $q \leq$ 3.0 fm$^{-1}$.
![Form factor for the C1+E1 transition to the ($1$$^{-}$, 1) 18.44 MeV state compared with the experimental data taken from Ref. [@TW70].](fig1.eps){width="40.00000%"}
![Transverse magnetic form factor for the M2 transition to the ($2_{1}$$^{-}$, 1) 19.88 MeV state compared with the experimental data taken from Ref. [@RS84].](fig2.eps){width="40.00000%"}
![Transverse magnetic form factor for the M2 transition to the ($2_{2}$$^{-}$, 1) 23.50 MeV state compared with the experimental data taken from Ref. [@RR87].](fig3.eps){width="40.00000%"}
![Longitudinal form factor for the C3 transition to the (3$^{-}$, 1) 18.87 MeV state compared with the experimental data taken from Ref. [@YM71].](fig4.eps){width="40.00000%"}
![Transverse magnetic form factor for the M3 transition to the (3$^{+}$, 1) 27.10 MeV state compared with the experimental data taken from Ref. [@RS84].](fig5.eps){width="40.00000%"}
Conclusions
===========
The inclusion of higher energy configurations in the particle-hole shell model calculation succeeded in describing the form factors for the negative and positive parity states. The amplitudes of the transitions to the negative-parity states considered in this work have to be reduced by a factor 1.3 and 1.82 for the states $1$$^{-}$ and $2_{2}$$^{-}$ while the amplitudes for the $2_{1}$$^{-}$ state need to be enhanced by factor of 1.2, to describe the low-q data. The amplitudes for $3$$^{+}$ need to be reduced by a factor of 5. This reduction may be attributed to higher order effects, such as 2p-2h excitations, or even more. Correlation in the ground state wavefunction by mixing more than one configuration are necessary to describe the data. The single-particle wavefunctions of the (HO) potential with size parameter $b=1.64$ fm chosen to reproduce the root mean square charge radius are adequate to describe the data, except for M2 (23.50 MeV) transition where the $b$ value has to be reduced by a factor 14%.
[99]{}
R. A. Radhi, A. Bouchebak, Nucl. Phys. A[** 716**]{}, 87 (2003). T. Sato, [*et al.*]{}, Z.Phys. A[** 320**]{}, 507 (1985). C. Bennhold, [*et al.*]{}, Phys. Rev. C[** 46**]{}, 2456 (1992). J. G. L. Booten, [*et al.*]{},Nucl. Phys. A[** 569**]{}, 510 (1994). R. A. Radhi, [*et al.*]{},Nucl. Phys. A[** 696**]{}, 442 (2001). F. A. Majeed, R. A. Radhi, Chin. Phys. Lett. Vol. [**23**]{}, No.10, 2699 (2006). T. W. Donnelly, I. Sick, Rev. Mod. Phys. Vol. [**56**]{}, (3), 461 (1984). T. deForest, Jr., J. D. Walecka, Adv. Phys. [**15**]{}, 1 (1966). P. J. Brussaard, P. W. M. Glaudemans, [*Shell-Model Applications in Nuclear Spectrscopy*]{} (Amsterdam: North Holland),(1977). B. A. Brown, [*et al.*]{}, Phys. Rev. C[**32**]{}, 1127 (1985). R. S. Hicks, [*etal.*]{}, Phys. Rev. C[** 36**]{}, 485 (1987). T. W. Donnelly, Phys. Rev. C[** 1**]{}, 833 (1970). T. Sato, [*et al.*]{}, Z. Phys. A[**320**]{}, 507 (1985). R. S. Hicks, [*et al.*]{}, Phys. Rev. C[** 30**]{}, 1 (1984). A. Yamaguchi, [*et al.*]{}, Phys. Rev. C[** 3**]{}, 1750 (1971).
[^1]: Support by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) (Brazil), and the Third World Academy of Science (TWAS) (Italy) for under grant of the scheme (TWAS-CNPq exchange programs for postdoctoral researchers).
|
---
abstract: |
Deformed Special Relativity (DSR) is a generalization of Special Relativity based on a deformed Minkowski space, i.e. a four-dimensional space-time with metric coefficients depending on the energy. We show that, in the DSR framework, it is possible to derive the value of the electron mass from the space-time geometry via the experimental knowledge of the parameter of local Lorentz invariance breakdown, and of the Minkowskian threshold energy $%
E_{0,em}$ for the electromagnetic interaction.
author:
- '[**Fabio Cardone**]{}$^{a,b}$[**, Alessio Marrani**]{}$^{c,d}$[** **]{}'
- |
[**and Roberto Mignani**]{}$^{b-d}$\
$a$ Dipartimento di Fisica\
Università dell’Aquila\
Via Vetoio\
67010 COPPITO, L’Aquila, Italy\
$b$ I.N.D.A.M. - G.N.F.M.\
$c$ Università degli Studi ”Roma Tre”\
Via della Vasca Navale, 84\
I-00146 ROMA, Italy\
$d$ I.N.F.N. - Sezione di Roma III
title: '[**The electron mass from Deformed Special Relativity**]{}'
---
Introduction
============
In the last years, two of the present authors (F.C. and R.M.) proposed a generalization of [*Special Relativity*]{} (SR) based on a”deformation” of space-time, assumed to be endowed with a metric whose coefficients depend on the energy of the process considered \[1\]. Such a formalism ([*Deformed Special Relativity*]{}, DSR) applies in principle to [*all*]{} four interactions (electromagnetic, weak, strong and gravitational) - at least as far as their nonlocal behavior and nonpotential part is concerned - and provides a metric representation of them (at least for the process and in the energy range considered) (\[1\]-\[4\], \[7\], \[21\] and \[24\]-\[26\]). Moreover, it was shown that such a formalism is actually afive-dimensional one, in the sense that the deformed Minkowski space is embedded in a larger Riemannian manifold, with energy as fifth dimension \[5\].
In this paper, we will show that the DSR formalism yields an expression of the electron mass $m_{e}$ in terms of the parameter $\delta $ of local Lorentz invariance (LLI) breakdown and of the threshold energy for the gravitational metric, $E_{0,grav}$ (i.e. the energy value under which the metric becomes Minkowskian). This allows us to evaluate $m_{e}$ from the (experimental) knowledge of such parameters.
The organization of the paper is as follows. In Sect. 2 we briefly introduce the concept of deformed Minkowski space, and give the explicit forms of the phenomenological energy-dependent metrics for the four fundamental interactions. The LLI breaking parameter $\delta _{int}$ for a given interaction is introduced in Sect. 3. In Sect. 4 we assume the existence of a stable fundamental particle interacting gravitationally, electromagnetically and weakly, and show (by imposing some physical requirements) that its mass value (expressed in terms of $\delta _{e.m.}$ and $E_{0,grav}$) is just the electron mass. Sect. 5 concludes the paper.
Deformed Special Relativity in four dimensions (DSR4)
=====================================================
Deformed Minkowski space-time
-----------------------------
The generalized (“deformed”) Minkowski space $\widetilde{M_{4}}$ (DMS4) is defined as a space with the same local coordinates $x$ of $M_{4}$ (the four-vectors of the usual Minkowski space), but with metric given by the metric tensor[^1] $$\begin{aligned}
\eta _{\mu \nu }(E) &=&diag\left(
b_{0}^{2}(E),-b_{1}^{2}(E),-b_{2}^{2}(E),-b_{3}^{2}(E)\right) =\smallskip
\nonumber \\
&& \nonumber \\
&&\stackrel{\text{{\footnotesize ESC off}}}{=}\delta _{\mu \nu }\left[
\delta _{\mu 0}b_{0}^{2}(E)-\delta _{\mu 1}b_{1}^{2}(E)-\delta _{\mu
2}b_{2}^{2}(E)-\delta _{\mu 3}b_{3}^{2}(E)\right] \text{ }\end{aligned}$$ ($\forall E\in R_{0}^{+}$), where the $\left\{ b_{\mu }^{2}(E)\right\} $ are dimensionless, real, positive functions of the energy \[1\]. The generalized interval in $\widetilde{M_{4}}$ is therefore given by ($x^{\mu
}=(x^{0},x^{1},x^{2},x^{3})=(ct,x,y,z)$, with $c$ being the usual light speed in vacuum) (ESC on) $$ds^{2}=b_{0}^{2}(E)c^{2}dt^{2}-(b_{1}^{2}(E)dx^{2}+b_{2}^{2}(E)dy^{2}+b_{3}^{2}(E)dz^{2})=\eta _{\mu \nu }(E)dx^{\mu }dx^{\nu }=dx\ast dx.$$
The last step in (2) defines the scalar product $\ast $ in the deformed Minkowski space $\widetilde{M_{4}}$ [^2]. It follows immediately that it can be regarded as a particular case of a Riemann space with null curvature.
Let us stress that, in this formalism, the energy $E$ is to be understood as the [*energy of a physical process*]{} measured by the detectors via their electromagnetic interaction in the usual Minkowski space. Moreover, $E$ is to be considered as a [*dynamical variable*]{} (on the same footing as the space-time coordinates), because it specifies the [*dynamical behavior*]{} of the process under consideration, and, via the metric coefficients, it provides us with a [*dynamical map*]{} - in the energy range of interest - of the interaction ruling the given process. Let’s recall that the use of [*momentum components as dynamical variables*]{} on the same foot of the space-time ones can be traced back to Ingraham \[8\]. Dirac \[9\], Hoyle and Narlikar \[10\] and Canuto et al. \[11\] treated mass as a dynamical variable in the context of scale-invariant theories of gravity.
Moreover - as already stressed in the Introduction - the 4-d. deformed Minkowski space can be [*naturally embedded*]{} in a 5-d. Riemann space, with energy as fifth metrical coordinate \[5\]. Curved 5-d. spaces have been considered by several Authors \[12\]. On this respect, the DSR formalism is a kind of generalized ([*non-compactified*]{}) Kaluza-Klein theory, and resembles, in some aspects, the so-called ”Space-Time-Mass” (STM) theory (in which the fifth dimension is the rest mass), proposed by Wesson \[13\] and studied in detail by a number of Authors \[14\].
Energy-dependent phenomenological metrics for the four interactions
-------------------------------------------------------------------
As far as the phenomenology is concerned, we recall that a [*local*]{} breakdown of Lorentz invariance may be envisaged for all four fundamental interactions (electromagnetic, weak, strong and gravitational) whereby [*one gets evidence for a departure of the space-time metric from the Minkowskianone*]{} (at least in the energy range examined). The experimental data analyzed were those of the following four physical processes:
- the lifetime of the (weakly decaying) $K_{s}^{0}$ meson \[15\];
- the Bose-Einstein correlation in (strong) pion production \[16\];
- the superluminal photon tunneling \[17\];
- the comparison of clock rates in the gravitational field of Earth \[18\].
A detailed derivation and discussion of the energy-dependent phenomenological metrics for all the four interactions can be found in Ref.s \[1\]-\[4\]. Here, we confine ourselves to recall their following basic features:
[**1)**]{} Both the [**electromagnetic**]{} and the [**weak**]{} metric show the same functional behavior, namely
$$\eta (E)=diag(1,-b^{2}(E),-b^{2}(E),-b^{2}(E)); \label{emweak1}$$
$$\begin{aligned}
b^{2}(E) &=&\left\{
\begin{array}{lll}
(E/E_{0})^{1/3}, & 0<E\leq E_{0} & \\
& & \\
1, & E_{0}<E &
\end{array}
\right. = \\
&& \nonumber \\
&& \nonumber \\
&=&1+\theta (E_{0}-E)\left[ \left( \frac{E}{E_{0}}\right) ^{1/3}-1\right]
,E>0, \label{emweak2}\end{aligned}$$
(where $\theta (x)$ is the Heavyside theta function) with the only difference between them being the threshold energy $E_{0}$, i.e. the energy value at which the metric parameters are constant, i.e. the metric becomes Minkowskian ($\eta _{\mu \nu }(E\geq E_{0})\equiv g_{\mu \nu
}=diag(1,-1,-1,-1)$); the fits to the experimental data yield
$$\begin{gathered}
E_{0,e.m.}=\left( 4.5\pm 0.2\right) \mu eV\,; \nonumber \\
\nonumber \\
E_{0,weak}=\left( 80.4\pm 0.2\right) GeV.\end{gathered}$$
Notice that for either interaction the metric is isochronous, spatially isotropic and [*”sub-Minkowskian”*]{}, i.e. it approaches the Minkowskian limit from below (for $E<E_{0}$). Both metrics are therefore Minkowskian for $E>E_{0,weak}\simeq 80GeV$, and then our formalism is fully consistent with electroweak unification, which occurs at an energy scale $\sim 100GeV$.
Let us recall that the phenomenological electromagnetic metric (3)-(5) was derived by analyzing the propagation of evanescent waves in undersized waveguides \[16\]. It allows one to account for the observed superluminal group speed in terms of a nonlocal behavior of the waveguide, just described by an effective deformation of space-time in its reduced part \[3\]. As to the weak metric, it was obtained by fitting the data on the meanlife of the meson $K_{s}^{0}$ (experimentally known in a wide energy range $(30\div
350GeV)$ \[14\]), thus accounting for its apparent departure from a purely Lorentzian behavior (\[1\], \[19\]).
[**2)**]{} For the [**strong**]{} interaction, the metric was derived \[2\] by analyzing the phenomenon of Bose-Einstein (BE) correlation for $\pi $-mesons produced in high-energy hadronic collisions \[16\]. Such an approach permits to describe the BE effect as the decay of a ”fireball” whose lifetime and space sizes are directly related to the metric coefficients $\left\{ b_{\mu
,strong}^{2}(E)\right\} $, and to avoid the introduction of ”ad hoc” parameters in the pion correlation function \[2\]. The strong metric reads $$\eta
_{strong}(E)=diag(b_{0,strong}^{2}(E),-b_{1,strong}^{2}(E),-b_{2,strong}^{2}(E),-b_{3,strong}^{2}(E));$$ $$\begin{aligned}
b_{1,strong}^{2}(E) &=&\left( \frac{\sqrt{2}}{5}\right) ^{2}; \nonumber \\
&& \nonumber \\
b_{2,strong}^{2}(E) &=&\left( \frac{2}{5}\right) ^{2},\forall E>0;\end{aligned}$$
$$\begin{aligned}
b_{0,strong}^{2}(E) &=&b_{3,strong}^{2}(E)=\left\{
\begin{array}{lll}
1, & 0<E\leq E_{0,strong} & \\
& & \\
(E/E_{0,strong})^{2}, & E_{0,strong}<E &
\end{array}
\right. = \nonumber \\
&& \\
&& \nonumber \\
&=&\smallskip 1+\theta (E-E_{0,strong})\left[ \left( \frac{E}{E_{0,strong}}%
\right) ^{2}-1\right] ,E>0\end{aligned}$$
with
$$E_{0,strong}=\left( 367.5\pm 0.4\right) GeV.$$
Let us stress that, in this case, contrarily to the electromagnetic and the weak ones, [*a deformation of the time coordinateoccurs*]{}; moreover, [*thethree-space is anisotropic*]{}[*,*]{} with two spatial parameters constant (but different in value) and the third one variable with energy like the time one.
[**3)**]{} The [**gravitational**]{} energy-dependent metric was obtained \[4\] by fitting the experimental data on the relative rates of clocks in the Earth gravitational field \[18\]. Its explicit form is[^3]:
$$\eta
_{grav.}(E)=diag(b_{0,grav.}^{2}(E),-b_{1,grav.}^{2}(E),-b_{2,grav.}^{2}(E),-b_{3,grav.}^{2}(E));
\label{grav1}$$
$$\begin{aligned}
b_{0,grav.}^{2}(E) &=&b_{3,grav.}^{2}(E)=\left\{
\begin{array}{lll}
1, & & 0<E\leq E_{0,grav.} \\
& & \\
\frac{1}{4}(1+E/E_{0,grav.})^{2}, & & E_{0,grav.}<E
\end{array}
\right. = \nonumber \\
&& \nonumber \\
&& \nonumber \\
&=&1+\theta (E-E_{0,grav.})\left[ \frac{1}{4}\left( 1+\frac{E}{E_{0,grav.}}%
\right) ^{2}-1\right] ,E>0\end{aligned}$$
with $$E_{0,grav.}=\left( 20.2\pm 0.1\right) \mu eV. \label{grav2}$$ Intriguingly enough, this is approximately of the same order of magnitude of the thermal energy corresponding to the $2.7^{o}K$ cosmic background radiation in the Universe[^4].
Notice that the strong and the gravitational metrics are [*over-Minkowskian*]{} (namely, they approach the Minkowskian limit from above ($%
E_{0}<E$), at least for their coefficients $b_{0}^{2}(E)=b_{3}^{2}(E)$).
LLI breaking factor and relativistic energy in DSR
==================================================
The breakdown of standard local Lorentz invariance (LLI) is expressed by the LLI breaking factor parameter $\delta $ \[19\]. We recall that two different kinds of LLI violation parameters exist: the [*isotropic*]{} (essentially obtained by means of experiments based on the propagation of e.m. waves, e.g. of the Michelson-Morley type), and the [*anisotropic*]{} ones (obtained by experiments of the Hughes-Drever type \[19\], which test the isotropy of the nuclear levels).
In the former case, the LLI violation parameter reads \[19\] $$\begin{aligned}
\delta &=&\left( \frac{u}{c}\right) ^{2}-1, \\
u &=&c+v, \nonumber\end{aligned}$$ where $c$ is, as usual, the speed of light [*in vacuo*]{}, $v$ is the LLI breakdown speed (e.g. the speed of the preferred frame) and $u$ is the new speed of light (i.e. the [*”maximal causal speed”*]{} in Deformed Special Relativity \[1\]). In the [*anisotropic*]{} case, there are different contributions $\delta ^{A}$ to the anisotropy parameter from the different interactions. In the HD experiment, it is $A=S,HF,ES,W$, meaning strong, hyperfine, electrostatic and weak, respectively. These correspond to four parameters $\delta ^{S}$ (due to the strong interaction), $\delta ^{ES}$ (related to the nuclear electrostatic energy), $\delta ^{HF}$ (coming from the hyperfine interaction between the nuclear spins and the applied external magnetic field) and $\delta ^{W}$ (the weak interaction contribution).
In our framework, we can define $\delta $ as follows: $$\delta _{int.}\equiv \frac{m_{in.,int.}-m_{in.,grav.}}{m_{in.,int.}}=1-\frac{%
m_{in.,grav.}}{m_{in.,int.}}, \label{delta}$$ where $m_{in.,int.}$ is the inertial mass of the particle considered with respect to the given interaction [^5]. In other words, we assume that the [*local*]{} deformation of space-time corresponding to the interaction considered, and described by the metric (1), gives rise to a [*local violation of the Principle of Equivalence*]{} for interactions different from the gravitational one. Such a departure, just expressed by the parameter $\delta _{int.}$, does constitute also a measure of the amount of LLI breakdown. In the framework of DSR, $\delta
_{int.}$ embodies the [*geometrical contribution to the inertial mass*]{}, thus discriminating between two different metric structures of space-time.
Of course, if the interaction considered is the gravitational one, the Principle of Equivalence strictly holds, i.e. $$m_{in.,grav.}=m_{g}, \label{EP}$$ where $m_{g}$ is the gravitational mass of the physical object considered, i.e. it is its [*”gravitational charge”*]{} (namely, its coupling constant to the gravitational field).
Then, we can rewrite (\[delta\]) as: $$\delta _{int.}\equiv \frac{m_{in.,int.}-m_{g}}{m_{in.,int.}}=1-\frac{m_{g}}{%
m_{in.,int.}}, \label{delta-int}$$ and therefore, when the particle is subjected [*only*]{} to gravitational interaction, it is
$$\delta _{grav.}=0$$
In DSR the relativistic energy, for a particle subjected to a given interaction and moving along $\widehat{x^{i}}$ , has the form \[1\]: $$\begin{aligned}
E_{int.} &=&m_{in.,int.}u_{i,int.}^{2}(E)\widetilde{\gamma }_{int.}(E)=
\nonumber \\
&& \nonumber \\
&=&m_{in.,int.}c^{2}\frac{b_{0,int.}^{2}(E)}{b_{i,int.}^{2}(E)}\left[
1-\left( \frac{v_{i}b_{i,int.}(E)}{cb_{0,int.}(E)}\right) ^{2}\right]
^{-1/2}, \label{En1}\end{aligned}$$ where $\underline{u}_{int.}(E)$ is the [*maximal causal velocity*]{} for the interaction considered (i.e. the analogous of the light speed in SR), given by (\[1\],\[21\]) $$\underline{u}_{int.}(E)\equiv \left( c\frac{b_{0,int.}(E)}{b_{1,int.}(E)},c%
\frac{b_{0,int.}(E)}{b_{2,int.}(E)},c\frac{b_{0,int.}(E)}{b_{3,int.}(E)}%
\right) . \label{u}$$ In the non-relativistic (NR) limit of DSR, i.e. at energies such that $$v_{i}\ll u_{i,int.}(E),$$ Eq. (\[u\]) yields the following NR expression of the energy corresponding to the given interaction: $$E_{int.,NR}=m_{in.,int.}u_{i,int.}^{2}(E)=m_{in.,int.}c^{2}\frac{%
b_{0,int.}^{2}(E)}{b_{i,int.}^{2}(E)}. \label{EnNR}$$
In the case of the gravitational metric (\[grav1\])-(\[grav2\]), we have $$\frac{b_{0,grav.}^{2}(E)}{b_{3,grav.}^{2}(E)}=1,\forall E\in R_{0}^{+}.$$ Therefore, for $i=3$ , Eq.s (\[En1\]) and (\[EnNR\]) become, respectively ($v_{3}=v$): $$E_{grav.}=m_{g}c^{2}\left[ 1-\left( \frac{v}{c}\right) ^{2}\right]
^{-1/2}=m_{g}c^{2}\gamma ,$$ $$E_{grav.,NR}=m_{g}c^{2},$$ namely, the gravitational energy takes its [*standard, special-relativistic* ]{}values.
This means that the special characterization (corresponding to the choice $%
i=3$) of Eq.s (\[En1\]) and (\[EnNR\]) within the framework of DSR relates the gravitational interaction with SR, which is - as well known - based on the electromagnetic interaction in its Minkowskian form.
The electron as a fundamental particle and its ”geometrical” mass
=================================================================
Let us now consider for $E$ the threshold energy of the gravitational interaction:
$$E=E_{0,grav.}$$
where $E_{0,grav.}$ is the limit value under which the metric $\eta _{\mu
\nu ,grav.}(E)$ becomes Minkowskian (at least in its known components). Indeed, from Eq.s (\[grav1\])-(\[grav2\]) it follows ( $\forall E\in
(0,E_{0,grav.}]$): $$\begin{aligned}
\eta _{\mu \nu ,grav.}(E)
&=&diag(1,-b_{1,grav.}^{2}(E),-b_{2,grav.}^{2}(E),-1)\stackrel{\text{%
{\footnotesize ESC off}}}{=} \\
&&\smallskip \\
&=&\delta _{\mu \nu }\left[ \delta _{\mu 0}-\delta _{\mu
1}b_{1,grav.}^{2}(E)-\delta _{\mu 2}b_{2,grav.}^{2}(E)-\delta _{\mu 3}\right]
.\end{aligned}$$ Notice that at the energy $E=E_{0,grav.}$ the electromagnetic metric (\[emweak1\])-(\[emweak2\]) is Minkowskian, too (because $%
E_{0,grav.}>E_{0,e.m.}$).
On the basis of the previous considerations, it seems reasonable to assume that the physical object (particle) $p$ with a rest energy (i.e. gravitational mass) just equal to the threshold energy $E_{0,grav.}$, namely $$E_{0,grav.}=m_{g,p}c^{2}, \label{E0grav}$$ must play a fundamental role for either e.m. and gravitational interaction. We can e.g. hypothesize that $p$ corresponds to the lightest mass eigenstate which experiences both force fields (i.e., from a quantum viewpoint, coupling to the respective interaction carriers, the photon and the graviton). As a consequence, $p$ must be [*intrinsically stable*]{}, due to the impossibility of its decay in lighter mass eigenstates, even in the case such a particle is subject to weak interaction, too (i.e. it couples to all gauge bosons of the Glashow-Weinberg-Salam group $SU(2)\otimes U(1)$, not only to its electromagnetic charge sector[^6]).
Since, as we have seen, for $E=E_{0,grav.}$ the electromagnetic metric is Minkowskian, too, it is natural to assume, for $p$: $$m_{in.,p,e.m.}=m_{in.,p}$$ namely [*its inertial mass is that measured with respect to the electromagnetic metric*]{}.
Then, due to the Equivalence Principle (see Eq. (\[EP\])), the mass of $p$ is characterized by $$p:\left\{
\begin{array}{c}
m_{in.,p,grav.}=m_{g,p} \\
\\
m_{in.,p,e.m.}=m_{in.,p}.
\end{array}
\right.$$
Therefore, for such a fundamental particle the SSLI breaking factor (\[delta-int\]) of the e.m. interaction becomes: $$\delta _{e.m.}=\frac{m_{in.,p}-m_{g,p}}{m_{in.,p}}=1-\frac{m_{g,p}}{m_{in.,p}%
}\Leftrightarrow m_{g,p}=m_{in.,p}\left( 1-\delta _{e.m.}\right) .
\label{mgp}$$ Replacing (\[mgp\]) in (\[E0grav\]) yields: $$\begin{aligned}
E_{0,grav.} &=&m_{in.,p}\left( 1-\delta _{e.m.}\right) c^{2}\Leftrightarrow
\nonumber \\
&& \nonumber \\
&\Leftrightarrow &m_{in.,p}=\frac{E_{0,grav.}}{c^{2}}\frac{1}{1-\delta
_{e.m.}}. \label{final1}\end{aligned}$$ Thus, the obtained result allows us to evaluate the inertial mass of $p$ from the knowledge of the electromagnetic LLI breaking parameter $\delta
_{e.m.}$ and of the threshold energy $E_{0,grav.}$ of the gravitational metric.
The lowest limit to the LLI breaking factor of electromagnetic interaction has been recently determined by an experiment based on the detection of a DC voltage across a conductor induced by the steady magnetic field of a coil \[22\]. The value found in \[22\] corresponds to $$1-\delta _{e.m.}\widetilde{=}4\cdot 10^{-11}. \label{DC}$$ Then, inserting the value (\[grav2\]) for $E_{0,grav.}$ [^7]and (\[DC\]) in (\[final1\]), we get $$m_{in.,p}=\frac{E_{0,grav.}}{c^{2}}\frac{1}{1-\delta _{e.m.}}\geq \frac{%
2\cdot 10^{-5}}{4\cdot 10^{-11}}\frac{eV}{c^{2}}=0.5\frac{MeV}{c^{2}}%
=m_{in.,e} \label{final2}$$ (with $m_{in,e}$ being the inertial electron mass), where the $\geq $ is due to the fact that in general the LLI breaking factor constitutes an [*upper limit* ]{} (i.e. it sets the scale [*under which* ]{}a violation of LLI is expected).
If experiment \[22\] [*does indeed provide evidence* ]{}for a LLI breakdown (as it seems the case, although further confirmation is needed), Eq. (\[final2\]) yields $$m_{in.,p}=m_{in.,e}.$$ We find therefore the amazing result that [*the fundamental particle* ]{}$p$[* is nothing but the electron* ]{}$e^{-}$[* (or its antiparticle* ]{}$%
e^{+}$ [^8][*).* ]{}The electron is indeed the lightest massive lepton (pointlike, non-composite particle) with electric charge, and therefore subjected to gravitational, electromagnetic and weak interactions, but unable to weakly decay due to its small mass. Consequently, $e^{-}$ ($e^{+}$) shares all the properties we required for the particle $p$, whereby it plays a fundamental role for gravitational and electromagnetic interactions.
Conclusions
===========
The formalism of DSR describes -among the others -, in geometrical terms (via the energy-dependent deformation of the Minkowski metric) the breakdown of Lorentz invariance at [*local*]{} level (parametrized by the LLI breaking factor $\delta _{int.}$). We have shown that within DSR it is possible - on the basis of simple and plausible assumptions - to evaluate the inertial mass of the electron $e^{-}$ (and therefore of its antiparticle, the positron $e^{+}$) by exploiting the expression of the relativistic energy in the deformed Minkowski space $\widetilde{M_{4}}(E)_{E\in R_{0}^{+}}$ , the explicit form of the phenomenological metric describing the gravitational interaction (in particular its threshold energy), and the LLI breaking parameter for the electromagnetic interaction $\delta _{e.m.}$ .
Therefore, [*the inertial properties of one of the fundamental constituents of matter and of Universe do find a ”geometrical” interpretation in the context of DSR, by admitting for local violations of standard Lorentz invariance*]{}.
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[^1]: In the following, we shall employ the notation ”ESC on” (”ESC off”) to mean that the Einstein sum convention on repeated indices is (is not) used.
[^2]: Notice that our formalism - in spite of the use of the word ”deformation” - has nothing to do with the ”deformation” of the Poincaré algebra introduced in the framework of quantum group theory (in particular the so-called $\kappa $-deformations) \[6\]. In fact, the quantum group deformation is essentially a modification of the commutation relations of the Poincaré generators, whereas in the DSR framework the deformation concerns the metrical structure of the space-time (although the Poincaré algebra is affected, too \[7\]).
[^3]: The coefficients $b_{1,grav.}^{2}(E)$ and $b_{2,grav.}^{2}(E)$ are presently [*undetermined*]{} at phenomenological level.
[^4]: It is worth stressing that the energy-dependent gravitational metric (10)-(12) is to be regarded as a [*local*]{} representation of gravitation, because the experiments considered took place in a neighborhood of Earth, and therefore at a small scale with respect to the usual ranges of gravity (although a large one with respect to the human scale).
[^5]: Throughout the present work, $"int."$ denotes a physically detectable fundamental interaction, which can be operationally defined by means a phenomenological energy-dependent metric of deformed Minkowskian type.
[^6]: For precision’s sake, it should be noticed that actually the physically consistently-acting gauge group of the (unbroken) Glashow-Weinberg-Salam electroweak theory is not $SU(2)_{T}\otimes U(1)_{Y}$, but rather $$\left( SU(2)\otimes U(1)\right) /Z_{2}\approx U(2),$$ where $T$ and $Y$ respectively stand for weak isospin and hypercharge symmetries, and $\otimes $ is the usual direct group product. This cosetting by the discrete symmetry $Z_{2}$ is due to the very field content of the actual electroweak theory, as rigorously explained in \[23\].
[^7]: Let us recall that the value of $E_{0,grav}.$ was determined by fitting the experimental data on the slowing down of clocks in the Earth gravitational field \[18\]. See also Ref. \[4\].
[^8]: Of course, this last statement does strictly holds only if the CPT Theorem mantains its validity in the DSR framework, too. Although this problem has not yet been addressed in general on a formal basis, we can state that it holds true in the case we considered, since we assumed that the energy value is $E=E_{0,grav.}$, corresponding to the Minkowskian form of both electromagnetic and gravitational metric.
|
---
abstract: 'With the growing popularity of microblogging services such as Twitter in recent years, an increasing number of users are using these services in their daily lives. The huge volume of information generated by users raises new opportunities in various applications and areas. Inferring user interests plays a significant role in providing personalized recommendations on microblogging services, and also on third-party applications providing social logins via these services, especially in cold-start situations. In this survey, we review user modeling strategies with respect to inferring user interests from previous studies. To this end, we focus on four dimensions of inferring user interest profiles: (1) *data collection*, (2) *representation* of user interest profiles, (3) *construction and enhancement* of user interest profiles, and (4) the *evaluation* of the constructed profiles. Through this survey, we aim to provide an overview of state-of-the-art user modeling strategies for inferring user interest profiles on microblogging social networks with respect to the four dimensions. For each dimension, we review and summarize previous studies based on specified criteria. Finally, we discuss some challenges and opportunities for future work in this research domain.'
author:
- Guangyuan Piao
- 'John G. Breslin'
bibliography:
- 'library.bib'
date: 'Received: date / Accepted: date'
title: 'Inferring User Interests in Microblogging Social Networks: A Survey'
---
=1
[example.eps]{} gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore
=1
Introduction {#intro}
============
Microblogging[^1] social networks such as Twitter[^2] and Facebook[^3] are being widely used in our daily lives. Twitter and Facebook have 328 million and 2 billion monthly active users[^4][^5], which shows the popularity of these services. The abundant information generated by users in OSNs creates new opportunities for inferring user interest profiles, which can be used for providing personalized recommendations to those users either on those OSNs or on third-party services allowing social login functionality[^6] from the same OSNs. Social login is a technology which allows visitors to a website to log in using their OSN accounts rather than having to register a new one[^7]. A recent survey showed that over 94% of 18-34 year olds have used social login via Twitter, Facebook, etc.[^8] With the continued widespread development of the social login functionality, inferring user interest profiles from their OSN activities plays a central role in many applications for providing personalized recommendations with the permission of those users, especially for cold-start users who have joined those services recently.
In the literature, there have been many studies that focused on inferring user interest profiles with different purposes such as providing personalized recommendations with respect to news [@Abel2011g; @Gao:2011:ITU:2052138.2052335], research articles [@Bolting2015; @Nishioka:2016:PVT:2910896.2910898], and Points Of Interest (POI) [@Abel2012a]. Despite the popularity of inferring user interests in OSNs, there is a lack of an extensive review on user modeling strategies for inferring user interest profiles in OSNs. To our knowledge, only one related short survey [@Abdel-Hafez2013] has been formally published. [@Abdel-Hafez2013] provided a general overview of user modeling in social media websites which includes all types of OSNs without focusing on a specific type. As a result, the details of user modeling techniques for microblogging websites were not presented in [@Abdel-Hafez2013]. For example, including OSNs such as Delicious[^9] and Flickr[^10] which are based on *folksonomies* (folks taxonomies) together with microblogging OSNs for a single survey presents some difficulties due to the volume of literature on *folksonomy*-based user modeling [e.g., @Hung2008; @Szomszor2008; @Abel2011c; @Mezghani:2012:UPM:2187980.2188230; @Carmagnola2008a to name a few]. In addition, the survey conducted by [@Abdel-Hafez2013] does not cover studies from recent years. In this survey, we focus in particular on user modeling strategies in microblogging OSNs in terms of several user modeling dimensions, and analyze over 50 studies including more recent ones (see Appendix \[appendix:works\] for details of the surveyed studies).
There has been a varied set of terms used to denote inferring user interests in the literature, such as “user (interest) modeling/profiling/detection”, “inferring/modeling/predicting user interests”. User modeling/profiling, as a broad term, may refer to different meanings without a specific definition. A general definition of *user profiling* given by [@Zhou2012] is “the process of acquiring, extracting and representing the features of users”. Similarly, in [@Brusilovsky2007], the *user model* is defined in the context of adaptive systems as “a representation of information about an individual user that is essential for an adaptive system to provide the adaptation effect”. Based on a specific definition of what the *features* and *information* are in these definitions by [@Zhou2012] and [@Brusilovsky2007], the corresponding user models/profiles and the process of obtaining them might be different.
[@Rich1979] along with [@Cohen1979] and [@Perrault1978], where the terms *user model* and *user modeling* can be traced back to, also pointed out the need for classifying your user model as it might refer to several different things without a proper definition. Three major dimensions were used in [@Rich1979] for classifying user models:
- Are they models of a canonical user or are they models of individual users?
- Are they constructed explicitly by the user themselves or are they abstracted by the system on the basis of the user’s behavior?
- Do they contain short-term or long-term information?
Explicit information denotes the information which requires direct input by users such as surveys or forms, which will impose an additional burden on the users. Figure \[explicit\] shows an example of collecting *explicit* information about user interests during sign up on Twitter for the first time.
{width="\textwidth"}
Definition of User Modeling in This Survey
------------------------------------------
In the context of research on inferring user interests on OSNs, most studies have focused on exploiting *implicit* information such as the posts of users in order to infer user interest profiles. Based on the classification criteria from [@Rich1979], user models discussed in this survey are about individual users constructed implicitly based on their activities. For the third criterion used in [@Rich1979], there is no clear cut option as both short- and long-term information have been used in different user modeling strategies in the literature. In addition, user models can refer to various types of information relevant for each user in the domain of OSNs. For example, they might contain basic information such as age, gender, country, etc., or keywords that represent their interests. In this paper, we focus particularly on user models with respect to user interests. Although several terms such as “user model" and “user profile" have been used interchangeably in the literature, here we formally define these terms as follows:
A *user model* is a (data) structure that is used to capture certain *characteristics* about an individual user, and a *user profile* is the actual representation in a given user model. The process of obtaining the user profile is called *user modeling*.
Given this definition of a user model and the classification criteria from [@Rich1979], user model in this survey aims to capture user *interests* with respect to an *individual* user *implicitly* based on *long-term* or *short-term* knowledge via a user modeling strategy, to derive the interest profile of that user.
Figure \[fig:2\] presents an overview of the modified user profile-based personalization process from [@Abdel-Hafez2013] and [@Gauch:2007:UPP:1768197.1768200]. We modified the process from [@Abdel-Hafez2013] in order to reflect different aspects of user modeling strategies proposed in previous studies in the context of OSNs in detail. For example, we focus on data collection from *user activities*, *social networks/communities* or *external data* of an OSN instead of *explicit* or *implicit* feedback as most previous studies have focused on exploiting *implicit* information for inferring user interests. The modified user profile-based personalization process consists of three main phases. The first step is collecting data which will be used for inferring user interests. Subsequently, user interest profiles are constructed based on the data collected. We use *primitive interests* [@Kapanipathi2014] to denote the interests directly extracted from the collected data. Those primitive interests can either be used as the final output of a profile constructor or can be further enhanced, e.g., based on background knowledge from Knowledge Bases (KBs) such as Wikipedia[^11]. The output of the profile constructor is user interest profiles represented based on a predefined representation of interest profiles, e.g., word-based user interest profiles. Finally, the constructed user profiles are evaluated, and can be used in specific applications such as recommender systems for personalized recommendations.
{width="\textwidth"}
In this paper, we mainly discuss four dimensions of the user modeling process: (1) *data collection*, (2) *representation* of user interest profiles, (3) *profile construction and enhancement*, and (4) the *evaluation* of the constructed user interest profiles. In summary, the contribution of this paper is threefold.
- First, we provide a detailed review of user modeling approaches on microblogging services in terms of the three phases in Figure \[fig:2\] with the following focuses:
1. *What information is used for inferring user interest profiles?*
2. *How are the user interest profiles represented?*
3. *How are the user interest profiles constructed?*
4. *How are the constructed user profiles evaluated?*
- Second, we summarize the approaches with respect to these focuses based on specified criteria to be specified later on.
- Finally, we discuss the challenges and opportunities based on the strengths and weaknesses of different approaches.
[|l|l|]{}
---------------------
**OSNs**
**(\# of studies)**
---------------------
: Online Social Networks used for previous studies.[]{data-label="osns"}
& **Examples**\
Twitter (47) &
----------------------------------------------------------------------------------------------------------------------------------------------
[@Chen2010], [@Lu2012], [@Kapanipathi2014; @Kapanipathi2011],
[@piao2016exploring; @Guangyuan2017; @Piao2017; @Piao2016b; @Piao2016d], [@Besel:2016:ISI:2851613.2851819; @Besel:2016:QSI:3015297.3015298],
[@Abel2011g; @Abel2011e; @Abel2012a; @Abel2011d; @Abel:2013:TUM:2540128.2540558], [@Siehndel:2012:TUP:2887379.2887395],
[@Michelson2010], [@Bhattacharya:2014:IUI:2645710.2645765],
[@Orlandi2012], [@Hannon2012], [@Jiang2015],
[@Budak2014], [@Faralli2015; @Faralli2017], [@Weng:2010:TFT:1718487.1718520],
[@Zarrinkalam2015a; @Zarrinkalam2016],
[@Narducci2013], [@Xu2011], [@GarciaEsparza:2013:CCT:2449396.2449402],
[@Nishioka:2016:PVT:2910896.2910898; @Nishioka:2015:ITU:2809563.2809601], [@Gao:2011:ITU:2052138.2052335],
[@Vu:2013:IMU:2505515.2507883], [@Phelan:2009:UTR:1639714.1639794], [@Penas2013],
[@Sang:2015:PFT:2806416.2806470], [@Karatay2015a],
[@Kanta2012], [@OBanion2012], [@Nechaev],
[@Lim:2013:ICT:2491055.2491078], [@Bolting2015],
[@AnilKumarTrikhaFattaneZarrinkalam], [@Spasojevic:2014:LLS:2623330.2623350], [@Jipmo2017]
----------------------------------------------------------------------------------------------------------------------------------------------
: Online Social Networks used for previous studies.[]{data-label="osns"}
\
Facebook (7) &
-----------------------------------------------------------------------------------------
[@Kang2016], [@Orlandi2012], [@Kapanipathi2011],
[@Narducci2013], [@Bhargava:2015:UMU:2678025.2701365], [@Ahn:2012:IUI:2457524.2457681],
[@Spasojevic:2014:LLS:2623330.2623350]
-----------------------------------------------------------------------------------------
: Online Social Networks used for previous studies.[]{data-label="osns"}
\
LinkedIn (2) & [@Kapanipathi2011], [@Spasojevic:2014:LLS:2623330.2623350]\
Google+(1) & [@Spasojevic:2014:LLS:2623330.2623350]\
Table \[osns\] provides a summary of OSNs used for the works discussed in this survey. As we can see from the table, Twitter has been widely used due to its popularity and the higher degree of openness. Other OSNs such as Facebook or LinkedIn[^12] need to gain the permissions of users to access their data. Therefore, users have to be recruited for conducting an experiment, which results in less studies using these OSNs. In contrast to other studies, the study from Klout[^13], Inc. [@Spasojevic:2014:LLS:2623330.2623350], which is a social media platform that aggregates and analyzes data from multiple OSNs, leveraged all the OSNs listed in Table \[osns\]. As different design choices can be made for user modeling with different purposes, Table \[purpose\] provides an overview of the purpose of user modeling in each study. As we can see from the table, the majority of the previous studies have been conducted with the purpose of predicting user interests followed by recommending different types of content such as news, URLs, publications, and tweets.
[|l|l|]{} & **Examples**\
----------------
Predicting
user interests
----------------
: Purposes of user modeling in OSNs from previous studies.[]{data-label="purpose"}
&
---------------------------------------------------------------------------------------------------------------
[@Kapanipathi2014], [@Kang2016],
[@Michelson2010], [@Budak2014],
[@Bhattacharya:2014:IUI:2645710.2645765], [@Besel:2016:ISI:2851613.2851819; @Besel:2016:QSI:3015297.3015298],
[@Orlandi2012], [@Narducci2013],
[@Bhargava:2015:UMU:2678025.2701365], [@GarciaEsparza:2013:CCT:2449396.2449402],
[@Vu:2013:IMU:2505515.2507883], [@Ahn:2012:IUI:2457524.2457681], [@Abel2011e]
[@Zarrinkalam2016], [@Ahn:2012:IUI:2457524.2457681],
[@Spasojevic:2014:LLS:2623330.2623350], [@Jipmo2017],
[@Faralli2017], [@Jiang2015],
[@Xu2011], [@Penas2013], [@Lim:2013:ICT:2491055.2491078]
---------------------------------------------------------------------------------------------------------------
: Purposes of user modeling in OSNs from previous studies.[]{data-label="purpose"}
\
-----------------
News
recommendations
-----------------
: Purposes of user modeling in OSNs from previous studies.[]{data-label="purpose"}
&
--------------------------------------------------------
[@Abel2011g], [@Gao:2011:ITU:2052138.2052335],
[@Zarrinkalam2015a], [@Sang:2015:PFT:2806416.2806470],
[@Kanta2012], [@OBanion2012]
--------------------------------------------------------
: Purposes of user modeling in OSNs from previous studies.[]{data-label="purpose"}
\
-----------------
URL
recommendations
-----------------
: Purposes of user modeling in OSNs from previous studies.[]{data-label="purpose"}
&
-------------------------------------------------------------------------------------
[@Chen2010], [@Abel2011d],
[@piao2016exploring; @Piao2016a; @Guangyuan2017; @Piao2017; @Piao2016b; @Piao2016d]
-------------------------------------------------------------------------------------
: Purposes of user modeling in OSNs from previous studies.[]{data-label="purpose"}
\
-----------------
Publication
recommendations
-----------------
: Purposes of user modeling in OSNs from previous studies.[]{data-label="purpose"}
& [@Nishioka:2016:PVT:2910896.2910898], [@Bolting2015]\
-----------------
Tweet
recommendations
-----------------
: Purposes of user modeling in OSNs from previous studies.[]{data-label="purpose"}
&
-------------------------------------------------------
[@Lu2012], [@Sang:2015:PFT:2806416.2806470],
[@Karatay2015a], [@AnilKumarTrikhaFattaneZarrinkalam]
-------------------------------------------------------
: Purposes of user modeling in OSNs from previous studies.[]{data-label="purpose"}
\
-----------------
Researcher
recommendations
-----------------
: Purposes of user modeling in OSNs from previous studies.[]{data-label="purpose"}
& [@Nishioka:2015:ITU:2809563.2809601]\
POI recommendations & [@Abel2012a]\
----------------------
User recommendations
and classifications
----------------------
: Purposes of user modeling in OSNs from previous studies.[]{data-label="purpose"}
& [@Faralli2015]\
----------------
Concealing
user interests
----------------
: Purposes of user modeling in OSNs from previous studies.[]{data-label="purpose"}
& [@Nechaev]\
[|p[10cm]{}|]{} **Data Collection**\
1. using user activities
2. using the social networks/communities of a user
3. using external data
\
**Representation of User Interest Profiles**\
1. keyword profiles
2. concept profiles
3. multi-faceted profiles
\
**Construction and Enhancement of User Interest Profiles**\
1. profile construction with weighting schemes
- heuristic approaches
- probabilistic approaches
2. profile enhancement
- leveraging hierarchical knowledge
- leveraging graph-based knowledge
- leveraging collective knowledge
3. temporal dynamics
- constraint-based approaches
- interest decay functions
\
**Evaluation**\
[Table \[conceptual\_framework\] is a conceptual framework for discussing user modeling strategies proposed in the related work and to act as a “guide” to the rest of this survey.]{} The rest of this paper is organized as follows. In Section 2, we discuss what kind of information has been collected for inferring user interests. Section 3 introduces various representations of user interest profiles proposed in the literature. In Section 4, we review how user profiles have been constructed based on different dimensions such as considering the temporal dynamics of user interests. In Section 5, we discuss how those constructed user profiles have been evaluated in the literature. Finally, we conclude the paper with some discussions of opportunities and challenges with respect to user modeling on microblogging OSNs in Section 6.
Data Collection
===============
Overview
--------
This section of the survey discusses the first stage of user modeling, which is the data collection. In the context of OSNs, there are various information sources for collecting data in order to infer user interest profiles such as user information including the tweets or profiles with respect to a user and information from that user’s social network. The information used for user modeling is important as it might directly affect later stages such as the representation and construction of user interest profiles, and the quality of final profiles. The discussion is carried out over the criteria of whether the information is collected from a *user’s activities* or the *social networks/communities* of that user from the target microblogging platform (where the target users come from) or *external data*. Given Twitter is the largest microblogging social networking platform and is the most used OSNs in the literature as depicted in Table \[osns\], here we mainly focus on inferring user interest profiles on Twitter.
### Using user activities
A straightforward way of inferring user interests for a target user is leveraging information from the user’s activities in OSNs. Take Twitter as an example, a user can have different activities such as posting, re-tweeting, liking or replying to a tweet. Users can also describe themselves in their profiles or follow other people on Twitter which might reveal their interests. Therefore, we can leverage these user activities to infer user interests. This could be analyzing data from the posts, profiles or following activities of users. For instance, we can assume that a user is interested in `Microsoft` if the user mentions `Microsoft` frequently in the tweets or is following the Twitter account `@Microsoft`. However, inferring user interests from their activities such as posting tweets or re-tweeting requires users to be active, which is not always the case. For example, [@Gong2015] reported that a significant portion of Twitter users are *passive ones* who keep following other users in order to consume information on Twitter but who do not generate any content.
### Using the social networks/communities of a user
Leveraging information from the social networks/communities of a user can be useful to infer user interest profiles, especially for *passive users* who have little activity but who keep following other users to receive information. In this case, the generated content such as the posts and the profiles of users in a user’s social network can be used for inferring that user’s interests. For example, if many followees of a user post tweets with respect to `Microsoft` frequently or belong to a common community related to `Microsoft`, we can assume that the user is interested in `Microsoft` as well.
### Using external data
The ideal length of a post on any OSN ranges between 60 to 140 characters for better user engagement[^14]. Analyzing microblogging services such as Twitter is challenging due to their nature of generating short, noisy texts. Understanding those short messages plays a key role in user modeling in microblogging services. To this end, previous studies have investigated leveraging external data such as the content of embedded links/URLs in a tweet, in order to enrich the short text for a better understanding of it. [@Haewoon2010a] showed that most of the topics on Twitter are about news which could also be found in mainstream news sites. In this regard, some researchers have proposed linking microblogs to news articles and exploring the content of news articles in order to understand short texts in microblogging services better.
Review
------
### Using user activities {#Using information inside the platform}
The posts generated by users are the most common source of information for inferring user interests. Take Twitter as an example, the tweets or retweets of users provide a great amount of data that might implicitly indicate what kinds of topics a user might be interested in. Therefore, using the post streams of target users for inferring user interest profiles has been widely studied in the literature regardless of the different manners for how user interests are represented. For instance, [@Kapanipathi2014] extracted Wikipedia entities from the tweet streams of users while [@Chen2010] extracted keywords from them. Inferring user interests based on users’ posts requires users to be active, i.e., continuously generating content. On the one hand, there is an increasing number of users leveraging OSNs to seek the information they need, e.g., one in three Web users look for medical information, and over half of surveyed users consume news in OSNs[^15] [@Sheth2016]. On the other hand, there is also a rise of passive users in OSNs. For example, two out of five Facebook users only browse information without active participation within the platform[^16] [@Besel:2016:ISI:2851613.2851819], and [@Gong2015] reported that a significant portion of Twitter users are *passive ones* who consume information on Twitter without generating any content. Therefore, it is also important to infer user interest profiles for those *passive users* in OSNs.
Some studies pointed out that exploring posts for inferring user interests is computationally ineffective and unstable due to the changing interests of users [@Besel:2016:QSI:3015297.3015298; @Faralli2015; @Faralli2017; @Besel:2016:ISI:2851613.2851819; @Nechaev]. Instead of analyzing posts to infer user interests, these studies proposed using the *followeeship* information of users, which can infer more stable user interest profiles as the relationships of common users tend to be stable [@Myers2014]. In this line of work, *topical followees* that can be mapped to Wikipedia entities often need to be identified, e.g., identifying the followee account `@messi10stats` on Twitter as `wiki:Lionel_Messi`. One of the problems with these approaches based on topical followees is that only a small portion of users’ followees are topical ones. The authors from [@Faralli2015] and [@Guangyuan2017] both showed that, on average, only 12.7% and 10% of followees of users in their datasets can be linked to Wikipedia entities. Therefore, a lot of information from followees that do not have corresponding Wikipedia entities is missed. For example, based on the topical-followees approach we cannot infer any interests for a user who is following `@Alice` who has a biography as *“User Modeling and Recommender Systems researcher”*.\
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**Pros and cons.** Analyzing user activities for inferring user interests collects data from users themselves which can reflect their interests better compared to inferring from their social networks which will be discussed later. However, it requires users actively generate content in order to infer their interests from their generated content such as tweets, retweets, and likes on Twitter. Although leveraging the *topical-followees* approach can be used for inferring user interests for passive users, the usage of followees’ information is limited.
### Using the social networks/communities of a user
To cope with some problems such as inferring user interest profiles for passive users, information from social networks such as tweets from followees or followers or posts from Facebook friends can be utilized for inferring user interests for *passive users* as well as *active ones*. All aforementioned activities used for inferring a user’s interests can be analyzed with respect to a user’s social network as well for inferring that user’s interests. For instance, [@Chen2010] and [@Budak2014] explored the tweets of target users and their followees to infer user interests. Although using posts generated by users is of great potential for mining user interests, it also faces some challenges due to the short and noisy nature of microblogs. Compared to the aforementioned topical-followees approach, information from the social networks of users such as their followees can provide much more information. Returning to the example of inferring user interests for a user who is following `@Alice` in the previous subsection, we can infer this user is interested in `User Modeling` and `Recommender Systems` based on the biography of `@Alice` - *“User Modeling and Recommender Systems researcher”*. In [@Guangyuan2017], the authors proposed leveraging *biographies* of followees to extract entities instead of mapping followees to Wikipedia entities, and showed the improvement of inferred user interest profiles in the context of URL recommendations.
*List membership*, which is a kind of “tagging” feature on Twitter, has been explored as well. A list membership is a topical list or community which can be generated by any user on Twitter, and the creator of the list can freely add other users to the topical list. For instance, a user `@Bob` might create a topical list named “Java” and add his followees who have been frequently tweeting about news on this topic. Therefore, if a user `@Alice` is following users who have been added into many topical lists related to the topic `Java`, it might suggest that `@Alice` is interested in this topic as well. [@Kim2010a] studied the usage of Twitter lists and confirmed that lists can serve as good groupings of Twitter users with respect to their characteristics based on a user study. Based on the study, the authors also suggested that the Twitter list can be a valuable information source in many application domains including recommendations. In this regard, several studies have exploited list memberships of followees to infer user interest profiles [@Hannon2012; @Bhattacharya:2014:IUI:2645710.2645765; @Piao2017].
User interests might be following global trends in some trends-aware applications such as news recommendations. To investigate it, [@Gao:2011:ITU:2052138.2052335] proposed interweaving global trends and personal user interests for user modeling. In addition to leveraging the tweets of a target user for inferring user interests, the authors constructed a trend profile based on all tweets in the dataset in a certain time period. Afterwards, the final user interest profile was built by combining the two profiles. The results showed that combined user interest profiles can improve the performance of news recommendations while the first profile based on personal tweets plays a more significant role in the combination.\
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**Pros and cons.** On the one hand, a lot of data can be collected from the social networks of users, which is useful in the case of when inferring user interest profiles for passive users who do not generate much content but who keep following other users. On the other hand, it is difficult to distinguish the activities of a user’s followees that are relevant to the interests of that user. For example, the followees of a user can tweet a wide range of topics that they are interested in, and the user is not always interested in all those topics.
### Using external data
One of the challenges of inferring user interests from OSNs is that the generated content is often short and noisy [@Bontcheva2014]. To better understand the short texts of microblogging services such as tweets, external information beyond the target platform has been explored on top of the information sources discussed in the previous subsections. For instance, [@Abel2011g; @Abel2011e; @Abel:2013:TUM:2540128.2540558] proposed linking tweets to news articles and extract the *primitive interests* of users based on their tweets as well as the content of related news articles. Several strategies were proposed in [@Abel2011e], which were later on developed as a Twitter-based User Modeling Service [TUMS, @Tao2012]. However, it requires maintaining up-to-date news streams from mainstream news providers such as CNN[^17] in order to link tweets to relevant news articles. Instead, [@Abel2011d] and [@Piao2016d] leveraged the content of the embedded URLs in tweets. [@Hannon2012] used a third-party service Listorious[^18], which is a service providing annotated tags of list memberships on Twitter, for inferring user interest profiles. Given a target user *u*, the authors construct *u*’s interest profile based on the tags of list memberships with respect to the user.
With the popularity of different OSNs, users nowadays tend to have multiple OSN accounts across various platforms [@Liu2013b]. In this context, some of the previous studies have investigated exploiting user interest profiles from other OSNs for cross-system user modeling. For instance, [@Orlandi2012] and [@Kapanipathi2011] presented user modeling applications that can aggregate different user interest profiles from various OSNs. However, the evaluation of aggregated user interest profiles has not been provided. [@Abel2012a] investigated cross-system user modeling with respect to POI, and showed that the aggregation of Twitter and Flickr user data yields the best performance in terms of POI recommendations compared to modeling users separately based on a single platform. The result is in line with another study by them which aggregated user interest profiles on social tagging systems such as Delicious[^19], StumbleUpon[^20], and Flickr [@Abel2013].
The work from Klout [@Spasojevic:2014:LLS:2623330.2623350], which allows their users to add multiple OSN identities on their services, showed many insights on aggregating user information from multiple information sources in different OSNs for inferring user interests. The authors pointed out that using user-generated content (UGC) alone leads to a high precision but low recall for topic recommendations, and therefore, other information sources such as the ones from followees are needed. They also observed that the overlap of a user’s interests from different OSNs is very small, which shows that a user may not reveal all his/her interests on any single OSN alone due to the different characteristics of OSNs. Therefore, aggregating users’ information in different OSNs leads to a better understanding of their interests [@Spasojevic:2014:LLS:2623330.2623350].\
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**Pros and cons.** Leveraging external data such as the content of embedded URLs in a tweet can provide a better understanding of short microblogs, and exploring information from other OSNs of users can reveal their interests better compared to exploring a single OSN. Nevertheless, analyzing external data requires an additional effort and it is not always available. In addition, external data can also have irrelevant content with respect to user interests and might introduce some noise.
Summary and discussion
----------------------
In this section, we reviewed different information sources that have been used for collecting data in order to infer user interest profiles. Table \[datacollection\] summarizes information sources used for inferring user interest profiles in the literature. As we can see from Table \[datacollection\], user activities have been used widely for inferring user interest profiles in microblogging social networks in previous studies.
Although there have been many information sources used for inferring user interests, the comparison of different data sources for inferring user interest profiles has been less explored. Some approaches have utilized different aspects of information of followees such as *topical followees, biographies*, or *list memberships* [e.g., @Besel:2016:ISI:2851613.2851819; @Besel:2016:QSI:3015297.3015298; @Hannon2012; @Bhattacharya:2014:IUI:2645710.2645765; @Guangyuan2017]. However, it has not been clearly shown in these studies if these approaches perform better than exploiting users’ posts. The usefulness of user interest profiles built from various information sources might be different depending on different applications. For instance, [@Chen2010] showed that user interest profiles based on the user’s own streams perform better than profiles based on followee streams in the context of URL recommendations on Twitter. However, those profiles based on followee streams might be more useful for recommending followees.
In addition, combining different information sources have shown its efficiency in a few studies [e.g., @Abel2012a; @Piao2017]. However, how to combine different information sources for inferring user interests, and whether there is a synergistic effect on application performance by the combination might require more study. For instance, user interests extracted from different data sources can be either aggregated into a single user interest profile [e.g., @Orlandi2012; @Abel2012a] or remain as separate profiles [@Piao2017] to measure the preference score of a candidate item for recommendations. Also, combining different data sources has mainly been studied for aggregating user interests from multiple OSNs. Instead, combining different data sources inside the target platform might be useful for inferring user interests as well, e.g., combining extracted user interests from different information sources of followees and users.
Representation of User Interest Profiles
========================================
Overview
--------
In this section, we provide an overview of how user interest profiles have been represented in the different approaches. Here we first provide an overview of user representations for personalized information access that was introduced in [@Gauch2007], and *multi-faceted profiles* which have been proposed in several studies in the literature. We then carry out the review based on three different types of representations in the context of inferring user interest profiles in OSNs in the literature, which include (1) *keyword profiles*, (2) *concept profiles*, and (3) *multi-faceted profiles*.
In [@Gauch2007], the authors defined three types of user representations for personalized information access:
- keyword profiles;
- concept profiles;
- semantic network profiles.
**Keyword profiles.** In this representation of user interest profiles, each *keyword* or a *group of keywords* can be used for representing a topic of interest. This approach was predominant in every adaptive information retrieval and filtering system and is still popular in these areas [@Brusilovsky2007]. When using each keyword for representing user interests, the importance of each word with respect to users can be measured using a defined weighting scheme such as TF$\cdot$IDF (Term Frequency $\cdot$ Inverse Document Frequency) from information retrieval [@Salton1986]. In the case of using groups of keywords for representing user interests, the user interest profiles can be represented as a probability distribution over some topics, and each topic is represented as a probability distribution over a number of words. The topics can be distilled using topic modeling approaches such as Latent Dirichlet Allocation (LDA) [@Blei2003], which is an unsupervised machine learning method to learn topics from a large set of documents.
**Concept profiles.** Concept-based user profiles are represented as conceptual nodes (concepts) and their relationships, and the concepts usually come from a pre-existing knowledge base [@Gauch2007]. They can be useful for dealing with the problems that keyword profiles have. For example, WordNet [@Miller1995] groups related words together in concepts called *synsets*, which has been proved useful for dealing with *polysemy* in other domains. For example, [@Stefani] used WordNet synsets for representing user interests in order to provide personalized website access instead of using keywords as they are often not enough for describing someone’s interests. Another type of concept is *entities with URIs* (Uniform Resource Identifiers). For instance, this involves using `dbr:Apple_Inc.` to denote the company `Apple`, which is disambiguated based on the context of the word *apple* in a text such as tweet and linked to knowledge bases such as Wikipedia or DBpedia [@Auer2007]. DBpedia is the semantic representation of Wikipedia and it has become one of the most important and interlinked datasets on the Web of Data, which indicates a new generation of technologies responsible for the evolution of the current Web from a Web of interlinked documents to a Web of interlinked data [@Heath2011]. To facilitate reading, we use DBpedia concepts to denote concepts from Wikipedia or DBpedia.
**Semantic network profiles.** This type of profile aims to address the polysemy problem of keyword-based profiles by using a weighted semantic network in which each node represents a specific word or a set of related words. This type of profile is similar to concept profiles in the sense of the representation of conceptual nodes and the relationships between them, despite the fact that the concepts in semantic network profiles are learned (modeled) as part of user profiles by collecting positive/negative feedback from users [@Gauch2007]. As most previous works have focused on implicitly constructing user interest profiles in microblogging services, this type of profile has not been used in the domain of user modeling in microblogging services.
**Multi-faceted profiles.** Based on these representation strategies, user interest profiles can include different aspects of user interests such as interests inferred from their tweets, profiles or list memeberships. These different aspects of user interests can be combined to construct a single user interest profile or maintained separately as several user interest profiles for a target user. Although it is common to use a single representation with respect to a user interest profile, the *polyrepresentation theory* [@Ingwersen1994] based on a cognitive approach indicates that the overlaps between a variety of aspects or contexts with respect to a user within the information retrieval process can decrease the uncertainty and improve the performance of information retrieval. Based on this theory, [@White:2009:PUI:1571941.1572005] studied polyrepresentation of user interests in the context of a search engine. The authors combined five different aspects/contexts of a user for inferring user interests, and showed that polyrepresentation is viable for user interest modeling.
Review
------
### Keyword profiles
Similar to other adaptive information retrieval and filtering systems, representing user interests using *keywords* or *groups of keywords* is popular in OSNs as well. For instance, [@Chen2010] and [@Bhattacharya:2014:IUI:2645710.2645765] represented user interest profiles by using vectors of weighted keywords from the tweets and the descriptions of list memberships of users, respectively. Despite the huge volume of information from UGC, extracting keywords from microblogs for inferring user interest profiles is challenging due to the nature of short and noisy messages [@Liao2012].
As an alternative approach, another special type of keyword such as *tags* and *hashtags*[^21] has been used for inferring user interest profiles. In contrast to the words mined from the short texts of microblogs, keywords from tags/hashtags might be more informative and categorical in nature. [@Abel2011g; @Abel2011d] investigated hashtag-based user interest profiles by extracting hashtags from the tweets of users, and [@Hannon2012] leveraged keywords from the tags of users’ list memberships for representing their interest profiles.
Topics distilled from topic modeling approaches such as LDA are also popular for representing user interest profiles. A topic has associated words with their probabilities with respect to the topic. For example, an information technology-related topic can have some top associated words such as “google, twitter, apple, web”. [@Weng:2010:TFT:1718487.1718520] used LDA to distill 50 topics and represented each user as a probability distribution over these topics. In [@Abel2011e; @Abel2011g; @Abel:2013:TUM:2540128.2540558], the authors also used topics for representing user interests where those topics were extracted by ready-to-use NLP (Natural Language Processing) APIs such as OpenCalais[^22].\
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**Pros and cons.** Keyword profiles are the simplest to build, and do not rely on external knowledge from a knowledge base. One of the drawbacks of the keyword-based user profiles is *polysemy*, i.e., a word may have multiple meanings which cannot be distinguished by using keyword-based representation. In addition, these keyword-based approaches lack semantic information and cannot capture relationships among these words, and the assumption of topic modeling approaches that a document has rich information is not the case for microblogs [@Zarrinkalam2015]. [@Spasojevic:2014:LLS:2623330.2623350] further pointed out that topic modeling approaches cannot provide a scalable solution for inferring topics for millions of users which include a great number of passive users.
### Concept profiles
To address some problems of keyword-based approaches, researchers have proposed leveraging *concepts* from KBs such as DBpedia for representing user interests. One of the advantages of leveraging KBs is that we can exploit the background knowledge of these concepts to infer user interests which might not be captured if using keyword-based approaches. For instance, a big fan of the `Apple` company would be interested in any brand-new products from `Apple` even the names of these products have never been mentioned in the user’s primitive interests [@Lu2012]. Concepts from various types of KBs have been leveraged for different purposes of user modeling, such as the ones from simple concept taxonomies with respect to news [@Kang2016], domain-specific KBs such as STW[^23], ACM CCS, and Medical Subject Headings[^24] (MeSH) [@Nishioka:2016:PVT:2910896.2910898; @Nishioka:2015:ITU:2809563.2809601; @Bolting2015], and cross-domain KBs such as DBpedia [@Lu2012; @piao2016exploring; @Guangyuan2017; @Piao2017; @Faralli2015; @Piao2016b; @Piao2016d; @Abel2011g; @Abel2011d; @Abel2011e]. In the following, we discuss some details of the representation strategy using DBpedia concepts which have been the most widely used for representing user interest profiles.
**Entity-based profiles.** This approach extracts entities from information sources such as a user’s tweets, and uses these entities to represent user interest profiles. Take the following real-word tweet as an example [@Michelson2010]:\
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“*\#Arsenal winger Walcott: Becks is my England inspiration: http://tinyurl.com/37zyjsc*”,\
\
there are four entities such as `dbr:Arsenal_F.C.`, and `dbr:Theo_Walcott` within the tweet, which can be used for constructing entity-based user interest profiles. However, this approach is difficult to infer more specific interests which might need to be represented by combining multiple related entities or interests that cannot be found in a knowledge base. To address this issue, some studies have proposed representing each topic of interest as a *conjunction of multiple entities*, which are correlated on Twitter in a certain timespan [@Zarrinkalam2015a; @Zarrinkalam2016]. These sets of entities for representing a topic of interest can be learned via unsupervised approaches in a similar manner to learning topics with topic modeling approaches for keyword-based profiles.
**Category-based profiles.** An alternative approach is using DBpedia *categories*, which represents more general user interests compared to using DBpedia *entities*. Returning to the example in the previous paragraph, the categories of the mentioned entities in that tweet such as `dbr:Category:English_Football_League` can be used for representing the topic of interests instead of those entities. One can also choose the level or depth of categories in a KB for representing user interest profiles or use all categories related to primitive interests. The top-level DBpedia categories can refer to general ones such as `dbr:Category:Sports` and `dbr:Category:Health` compared to the categories in a lower level such as `dbr:Category:English_Football_League`. For example, [@Michelson2010] and [@Nechaev] used top-level categories to represent user interest profiles while other studies [@Faralli2017; @Kapanipathi2014; @Flati2014 etc.] used hierarchical categories to represent user interest profiles. Figure \[twixonomy\] shows an example of category-based representation of user interests based on extracted entities from followees’ account names, which is called *Twixonomy* [@Faralli2017].
{width="\textwidth"}
**Hybrid representations.** Each aforementioned representation has its strengths and weaknesses. In terms of entity- or category-based representations, extracting entities with URIs is a fundamental step for constructing either *entity-* or *category-based* user interest profiles. However, the task of extracting entities is non-trivial [@Kapanipathi2014] due to the noisy, informal language of microblogs [@Ritter2011]. In addition, knowledge bases might be out-of-date for emerging concepts on microblogging services, and therefore cannot capture these concepts during the entity extraction process. To overcome the drawbacks of using a single interest format, *hybrid representations* based on various interest formats have been explored as well. Instead of using only entities or categories for representing user interests, hybrid approaches combine different interest formats for constructing user profiles [@piao2016exploring; @Guangyuan2017; @Piao2017; @Faralli2015; @Piao2016d; @Nishioka:2016:PVT:2910896.2910898; @OBanion2012]. For example, [@OBanion2012] used categories as well as entities to represent user interest profiles. [@Piao2016d; @Piao2016b] proposed a hybrid approach using both DBpedia entities and WordNet synsets for representing user interests in order to capture user interests that might be missed due to the problem with entity recognition in microblogs.\
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**Pros and cons.** On the one hand, concept-based approaches present the semantics between concepts and can leverage background knowledge about concepts for propagating user interest profiles. On the other hand, these approaches rely on pre-existing or pre-constructed KBs which might be not always available in or lack of coverage with respect to some domains.
### Multi-faceted profiles
Multi-faceted profiles model multiple aspects for a target user based on different information sources or using different representation strategies in order to derive a comprehensive view of that user. The assumption here is that different aspects of users may complement each other and improve the inferred user interest profiles.
[@Hannon2012] proposed a multi-faceted user profile which includes user interests from target users, their followees, and followers. Figure \[multi\] shows an example from [@Hannon2012] for representing user interests, where user interests are represented based on the tags of list memberships of users, followees, or followers provided by a third-party service. The figure shows that user interests inferred from different aspects can complement each other and lead to a better understanding of a target user. However, they did not evaluate the effectiveness of multi-faceted profiles in the context of personalized recommendations and left it as a future work.
{width="\textwidth"}
The authors in [@Lu2012] and [@Chen2010] both constructed two keyword-based user interest profiles for each user. In [@Chen2010], two keyword-based user interest profiles were built based on the tweets of users and those of their followees for recommending URLs on Twitter. The results in [@Chen2010] showed that using user interest profiles based on the tweets of users performs better than using those based on the tweets of their followees. [@Lu2012] proposed using DBpedia entities and the affinity of other users to construct two user interest profiles for recommending tweets on Twitter. For a given user, the first user profile was represented as a vector of DBpedia entities, which were extracted from the user’s tweets. Both of these studies did not investigate the synergistic effect of combining these two aspects compared to considering a single aspect of users. More recently, [@Piao2017] showed that leveraging concept-based profiles from the biographies and list memberships of followees can complement each other and improve the URL recommendation performance on Twitter.\
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**Pros and cons.** Multi-faceted profiles provide a comprehensive view of a user with respect to his/her interests and can improve recommendation performance. On the other hand, multiple information sources have to be explored for constructing multi-faceted profiles.
Summary and discussion
----------------------
In this section, we reviewed various ways of representing user interests such as using *keywords*, various types of *concepts*, and some multi-faceted approaches. Table \[representation\] shows a summary of different representations of user interests adopted by previous studies.
Those different representations of user interests might work differently depending on the application where these user profiles are used. For example, we usually have to construct item profiles in the same way as constructing user interest profiles in order to measure the similarity between them for providing recommendations. The entity-based representation strategies for user interests might be appropriate for recommending items with long content, e.g., news or URL recommendations as the content of them is usually long. In contrast, these representation strategies might not work well for recommending items with short descriptions such as tweets due to the difficulty of extracting entities from them. For example, the low recall of entities on Twitter has been reported in both [@Kapanipathi2014] and [@Piao2016d] using several state-of-the-art NLP APIs. In a recent study [@Manrique:2017:SDA:3106426.3109440], the authors also showed that 30% of the titles of a research article cannot extract any entity at all. Some hybrid approaches such as combining word- and concept-based representations might be useful in this case.
In addition, different facets should be considered carefully for constructing multi-faceted profiles in the context of item recommendations. Each facet of multi-faceted profiles can have different importance for the recommended items, and leveraging completely unrelated facets might introduce noise to the constructed profiles. For example, [@Piao2017] showed that different weights are required for different facets in order to achieve the best performance in URL recommendations on Twitter. [@Abel2013] showed that it is helpful to have sufficient overlap between different facets of multi-faceted profiles for tag recommendations in a cold start.
It is also worth noting that the structure of user interest profiles can be different even with the same user interest format. Take a category-based user interest profile as an example, it can be a *vector*, *taxonomy* or *graph* by retaining the hierarchical or general relationships among categories. Also, the final profile extracted from the same structure can be different. For instance, both user interest profiles proposed in [@Faralli2017] (see Figure \[twixonomy\]) and [@Kapanipathi2014] were represented as a *taxonomy* at first, but were used differently for the final representation of user interests. In [@Faralli2017], entities or categories in different levels were used separately as an interest vector for representing a user, e.g., using categories that were two hops away from the user’s primitive interests as the final interest profile. However, using a specific abstraction level of the category taxonomy for all users does not consider that different users might have different depths or expertise levels in terms of a topic of interests. In contrast, [@Kapanipathi2014] sorted all categories in the taxonomy of a user based on their weights for representing the user’s interest profile. The different usages of the category taxonomy indicate some opportunities and challenges. On the one hand, the taxonomy structure of user interests is flexible enough to extract different abstraction levels of user interests or an overview of them. On the other hand, it has not been investigated which type of user interest profile obtained from the taxonomy structure is better.
Construction and Enhancement of User Interest Profiles
======================================================
Overview
--------
So far we have focused our discussion on collecting data from various sources for inferring user interests, and different representations for interest profiles. In this section, we provide details on how user interest profiles of a certain representation can be constructed based on the collected data. The overview of the construction and enhancement of user interest profiles is carried out based on three criteria:
- profile construction with weighting schemes;
- profile enhancement;
- temporal dynamics of user interests.
Based on a defined representation of user interest profiles, a profile constructor aims to determine the weights of user interest formats such as words or concepts in user profiles with a certain *weighting scheme*. The weights of interest formats denote the importance of these interests with respect to a user. In Section \[Profile Construction\], we review different weighting schemes based on various information sources such as users’ posts or their followees, etc. Primitive interest profiles, e.g., entity-based user profiles, can be further enhanced by using background knowledge from knowledge bases. For instance, this can be achieved by inferring category-based user interest profiles on top of the extracted entities from the data collected. Section \[Profile Enhancement\] describes the approaches leveraging knowledge bases for enhancing primitive interest profiles.
User interests can change over time in OSNs. For instance, a user interest profile built during the last two weeks might be totally different from one built from two years ago. In Section \[Temporal Dynamics of User Interests\], we look at whether or not the temporal dynamics of user interests have been considered when constructing user interest profiles, and if yes, how they have been incorporated during the construction process.
Review
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### Profile Construction with Weighting Schemes {#Profile Construction}
The output of a profile constructor is a primitive user interest profile represented by weighted interests based on a predefined representation. A *weighting scheme* is a function or process to determine the weights of user interests.
**Heuristic approaches.** A common and simple weighting scheme is using the frequency of an interest $i$ (e.g., a keyword or an entity) to denote the importance of $i$ with respect to a user *u*, which can be formulated as below when the data source is *u*’s posts:
$$TF_u(w_i) = frequency\mbox{ }of\mbox{ }i\mbox{ }in\mbox{ }u's\mbox{ }posts.$$
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Despite its simplicity, this approach has been widely used in the literature, particularly in entity-based user interest representations [@Kapanipathi2014; @Abel2011e; @Tao2012]. Interests represented as concepts such as entities extracted from tweets might come with their confidence scores, and these scores can be incorporated into a weighting scheme. For instance, [@Jiang2015] used TF with the confidence scores of extracted entities from tweets as their weighting scheme. One problem with TF is that common words or entities which appear frequently in many users’ interest profiles and may not be important as user interests. TF$\cdot$IDF is another common weighting scheme to cope with this problem. The IDF score of $i$ with respect to a user *u* based on *u*’s tweets can be measured as below [@Chen2010]:
$$IDF_u(i) = log\left[\frac{\#\mbox{ }all\mbox{ }users}{\#\mbox{ }users\mbox{ }using\mbox{ }i\mbox{ }at\mbox{ }least\mbox{ }once}\right].$$
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Instead of using users for measuring the IDF score of an interest, IDF has been applied in other ways as well. For example, [@Nishioka:2016:PVT:2910896.2910898] applied IDF with randomly retrieved tweets from the streaming API of Twitter, and [@Gao:2011:ITU:2052138.2052335] applied IDF to value the specificity of an interest within a given period of time. It is worth noting that the IDF weighting can also be applied after the *profile enhancement* process [e.g., @Piao2016d; @Nishioka:2016:PVT:2910896.2910898].
More sophisticated approaches can be applied for weighting user interests. In [@Vu:2013:IMU:2505515.2507883], the authors compared different weighting schemes such as TF$\cdot$IDF, TextRank [@mihalcea-tarau:2004:EMNLP], and TI-TextRank which was proposed by the authors by combining TF$\cdot$IDF and TextRank. Based on a user study, the authors showed that TI-TextRank performs best for ranking keywords from the tweets of users.
In the context of OSNs, specific approaches have to be devised for constructing user interest profiles by exploiting their social networks such as followees on Twitter [@Chen2010; @Lu2012]. To this end, several methods have been proposed. For example, [@Chen2010] first retrieved a set of *high-interest words* for followees as follows in order to build a user profile based on followees’ tweets: First, keyword-based user interest profiles were created using the TF$\cdot$IDF weighting scheme based on the tweets of followees, which are called *self-profiles*. Next, for each *self-profile* for followees of *u*, they picked all words that have been mentioned at least once, and selected the top 20% of words based on their occurrences. In addition, the words that are not in other followees’ profiles were removed. Subsequently, the weight of each word in the set of *high-interest words* was measured as below:
$$\begin{split}
FTF_u(i) = \mbox{ }& \#\mbox{ }u's\mbox{ } followees\mbox{ } who\mbox{ } have\mbox{ } i\mbox{ }\\
\mbox{ }& as\mbox{ } one\mbox{ } of\mbox{ } their\mbox{ } high-interest\mbox{ } words.
\end{split}$$
\
Similar approaches of $FTF_u(i)$ were adopted in [@Piao2017] and [@Bhattacharya:2014:IUI:2645710.2645765] but by exploring the list memberships of followees instead of their tweets for extracting user interests.
An alternative approach for aggregating the weights of interests in the followees’ profiles is normalizing each followee’s profiles and then aggregating those normalized weights for building user interest profiles [@Piao2017; @Spasojevic:2014:LLS:2623330.2623350]. In [@Piao2017], the authors showed that this simple alternative approach performs better compared to $FTF_u(i)$ for weighting entities extracted from the list memberships of followees when using inferred user interest profiles for URL recommendations on Twitter. These approaches assume that each followee is equally important when aggregating their interest profiles for building the user interest profile of a target user. However, some followees’ profiles can be more important compared to others with respect to the target user. In [@Karatay2015a], the authors incorporated the relative ranking scores of social networks into their weighting scheme to weight the entities of users.
**Probablistic approaches.** The aforementioned approaches focus on interests such as entities appearing in users’ posts, however, not all the entities related to a post explicitly appear in that post. In this regard, some approaches extracted interests such as entities by measuring the similarity between a post and an entity. For instance, [@Lu2012] and [@Narducci2013] used the Explicit Semantic Analysis (ESA) [@Gabrilovich] algorithm, which is designed to compute the similarity between texts, for obtaining the weights of entities for each tweet of a user. Those weights of entities were then aggregated for constructing entity-based primitive interests of users. [@Ahn:2012:IUI:2457524.2457681] quantified the degree of an interest, i.e., a Facebook entity, based on two factors: (1) the familiarity with each social neighbor, and (2) the similarity between the topic distributions of a social content and an interest. *Social content* is the combined text of a post and its comments between users, and the topic distributions of it is obtained using LDA.
The weights of user interests have also been learned in unsupervised ways in the literature. For instance, [@Weng:2010:TFT:1718487.1718520] treated tweet histories of each user as a big document, and used LDA to learn topic distributions for each user. [@AnilKumarTrikhaFattaneZarrinkalam] and [@Zarrinkalam2017] also used LDA to infer topic distributions for each user in time intervals where a topic is a set of DBpedia entities. Similarly, user interest profiles were represented as topic vectors where each topic is a set of temporally correlated entities on Twitter in [@Zarrinkalam2015a]. To this end, an entity graph based on their temporal correlation as defined by the authors was constructed, and the topics in a time interval were extracted using some existing community detection algorithms such as the *Louvain* method [@Rotta:2011:MLS:1963190.1970376]. The Louvain method is a simple and efficient algorithm for community detection, and relies upon a heuristic for optimizing modularity which quantifies the density of the links inside of the communities as compared to the links between communities. Subsequently, each topic $z$ was transformed into a set of weighted entities using the *degree centrality* of an entity in the topic (community). Finally, they obtained the weight of a topic based on the weight of an entity *c* with respect to the topic and the frequency of *c* in *u*’s tweets. [@Budak2014] proposed a probabilistic generative model to infer user interest profiles which are represented as an interest probability distribution over ODP (Open Directory Project[^25]) categories. In their proposed approach, the authors considered three aspects such as (1) the posts of a target user, (2) the activeness of the user, and (3) the influence of friends. They assumed that time is divided into fixed time steps, and transformed the problem into inferring the probability of a user being interested in each of the interests, given a social network that evolves over time including posts and social network information. [@Sang:2015:PFT:2806416.2806470] also proposed a probabilistic framework for inferring user interest profiles. Differring from [@Budak2014], [@Sang:2015:PFT:2806416.2806470] assumed users have long- and short-term interest (topic) distributions. Long-term interests denote stable preferences of users while short-term interests denote user preferences over short-term topics of events in OSNs. However, they did not consider users’ social networks.
In contrast to the aforementioned approaches, which assume all tweets posted by users are related to their interests, [@Xu2011] proposed a modified author-topic model [@Rosen-Zvi:2004:AMA:1036843.1036902] for distinguishing interest-related and unrelated tweets when learning the topic distributions of users.
### Profile Enhancement {#Profile Enhancement}
One of the advantages of constructing primitive interest profiles using concepts such as entities is that they can be further enhanced by external knowledge to deliver the final interest profiles. The approaches used in the literature for enhancing primitive user interests have mainly leveraged *hierarchical*, *graph-based*, or *collective* knowledge.
**Leveraging hierarchical knowledge**. One line of approach for enhancing entity-based primitive interest profiles is apply an adapted *spreading activation* [@Collins1975] function on a hierarchical knowledge base. For example, [@Kapanipathi2014] proposed representing user interest profiles as Wikipedia categories based on a hierarchical knowledge base, which is a refined Wikipedia category system built by the authors. The user interest profiles were then constructed using the hierarchical knowledge base with the following two steps. First, Wikipedia entities in users’ tweets were extracted as their primitive interests. Second, these entities were used as activated nodes for applying an adapted spreading activation function on the hierarchical knowledge base in order to infer weighted categories for representing user interest profiles.
The spreading activation function proposed by [@Kapanipathi2014] can be applied to any case where a set of entities and a hierarchical knowledge base are available. Therefore, many studies that followed have adopted this function but with different approaches for extracting entities or with different hierarchical knowledge bases [@Besel:2016:QSI:3015297.3015298; @Besel:2016:ISI:2851613.2851819; @Guangyuan2017; @Nishioka:2016:PVT:2910896.2910898; @Bolting2015]. For instance, [@Nishioka:2016:PVT:2910896.2910898] extracted entities and applied the spreading activation function on STW, which is a hierarchical knowledge base from the economics domain. [@Bolting2015] investigated several spreading activation functions including the one proposed in [@Kapanipathi2014] with the ACM CCS concept taxonomy in the computer science domain. The results showed that using a basic spreading activation function provides the best user interest profiles compared to using other ones in the context of research article recommendations. [@Besel:2016:QSI:3015297.3015298; @Besel:2016:ISI:2851613.2851819] extracted entities by mapping followees’ Twitter accounts to Wikipedia entities, and used WiBi [@Flati2014] as their hierarchical knowledge base for applying the spreading activation function proposed in [@Kapanipathi2014]. Similarly, [@Faralli2015] also mapped followees’ Twitter accounts to Wikipedia entities, and used them as users’ primitive interests for propagation with WiBi. However, a simpler propagation strategy was adopted in [@Faralli2015]. In [@Faralli2017], the authors extended their previous work [@Faralli] and proposed a methodology to build *Twixonomy*, which is a Wikipedia category taxonomy. *Twixonomy* is built by using a graph pruning approach based on a variant of Edmonds optimal branching [@Edmonds]. The authors showed that the proposed approach can generate a more accurate taxonomy compared to the one proposed in [@Kapanipathi2014]. As we mentioned in Section \[Using information inside the platform\], one issue with these approaches mapping followees’ accounts to Wikipedia entities is that only a limited percentage of followees’ accounts can be mapped to corresponding entities. For example, [@Faralli2015] and [@Guangyuan2017] reported that only 12.7% and 10% of followees’ accounts can be mapped to Wikipedia entities. In this regard, [@Guangyuan2017] considered the use of followees’ *biographies* for extracting entities, and applied two different propagation strategies; one is the spreading activation function from [@Kapanipathi2014], and the other is an interest propagation strategy exploring the DBpedia knowledge graph which will be discussed later on [@piao2016exploring].
Instead of using refined hierarchical knowledge from Wikipedia, some studies have explored other types of hierarchical knowledge bases as well. [@Kang2016] proposed mapping news categories to tweets for constructing user interest profiles. The authors leveraged news categories from two popular news portals in South Korea (Naver News[^26] and Nate News[^27]) to build their category taxonomy. This taxonomy consists of 8 main categories and 58 sub-categories, and each category consists of all news articles in the two news corpuses. To assign categories to a tweet, each tweet and news category are represented as a term vector where the weights of terms are calculated using TF$\cdot$IDF first. As there might be a semantic gap between terms in social media and news portals, the authors leveraged Wikipedia to transform the term vectors of tweets and news categories into a same vector space. The top two news categories to each tweet based on the cosine similarity between their vectors, and these news categories of a user’s tweets are then aggregated to construct the final user interest profiles.
[@Jiang2015] leveraged external knowledge sources such as DBpedia, Freebase [@Bollacker2008], and Yago [@Suchanek2007a] for constructing a topic hierarchy tree, which is a hierarchical knowledge base consists of over 1,000 topics distributed in 5 levels. However, the details for obtaining the topic hierarchy tree were not discussed in their study. The topic hierarchy tree used in Klout service is also bootstrapped using Freebase and Wikipedia, which consists of 3 levels with 15, around 700, and around 9,000 concepts in each level, respectively [@Spasojevic:2014:LLS:2623330.2623350]. In [@Bhargava:2015:UMU:2678025.2701365], the authors manually built a category taxonomy based on Facebook Page categories and the Yelp[^28] category list. The category taxonomy in [@Bhargava:2015:UMU:2678025.2701365] consists of three levels with 8, 58, and 137 categories in each level, respectively. The authors used features such as entities, hashtags, and document categories which can be extracted from Facebook *likes* and UGC as users’ primitive interests, and then measured the confidence of each concept in the category taxonomy based on these features using the Semantic Textual Similarity system [@Han2013].
**Leveraging graph-based knowledge**. Instead of leveraging hierarchical knowledge, many studies have leveraged graph-based knowledge for enhancing user profiles. For example, [@Michelson2010] exploited Wikipedia categories directly for propagating a user’s primitive interests. The authors summed the scores of a category which appeared in multiple depths in the category graph. Differing from exploring the categories of a specified depth [@Michelson2010], [@Siehndel:2012:TUP:2887379.2887395] represented user interest profiles using 23 top-level categories of the root node `Category:Main_Topic_Classifications` in Wikipedia. The Wikipedia entities in users’ tweets were extracted as their *primitive interests*, and these entities were then propagated up to the 23 top-level categories with a discounting strategy for the propagation.
With the advent of large, cross-domain Knowledge Graphs (KGs) such as DBpedia, different approaches leveraging background knowledge from KGs have been investigated. A knowledge graph is a knowledge base which consists of an ontology and instances of the classes in the ontology [@Farber]. The difference between a hierarchical category taxonomy such as WiBi and a knowledge graph such as DBpedia is displayed in Figure \[fig:wibidbpedia\] [@Guangyuan2017]. As we can see from the figure, for an entity, DBpedia goes beyond just categories to provide related entities via the entity’s properties/edges. Depending on the propagation strategies for those entities in a user’s primitive interests, different aspects, e.g., *related entities*, *categories* or *classes* of the entities can be leveraged for the propagation. For example, [@Penas2013] enriched categories in users’ primitive interests using similar categories defined by the `categorySameAs` relationship in DBpedia. [@Abel2012a] proposed using background knowledge from DBpedia for propagating user interest profiles with respect to POI. The authors considered entities that were two hops away from a user’s primitive interests and that were related to places. However, this approach did not consider any discounting strategy for the weights of propagated user interests. In [@Orlandi2012], the authors leveraged DBpedia categories one hop away from of the entities in a user’s primitive interests using a discounting strategy for propagating user interests.
Although [@Orlandi2012] leveraged DBpedia as the knowledge base instead of Wikipedia, they still exploited categories only, which makes no difference between using DBpedia and Wikipedia. To investigate other aspects of DBpedia such as related entities and classes of primitive interests, [@piao2016exploring] studied three approaches such as *category-*, *class-*, and *property-based* propagation strategies. This study found that exploiting categories and related entities via different properties of primitive interests provides the best performance compared to using corresponding categories only in the context of URL recommendations on Twitter.
An alternative graph for propagating entity-based user interest profiles is the Wikipedia entity graph. Compared to the DBpedia graph, where the edges between two entities are predefined properties in an ontology, the edges in the Wikipedia entity graph denote the mentions of the other entities in a Wikipedia entity (article). [@Lu2012] exploited a Wikipedia entity graph to enhance the entity-based primitive interests. Different from exploiting Wikipedia categories, the intuition behind this approach is that if a user is interested in `IPhone`, the user might be interested in other products from `Apple`, instead of being interested in other mobile phones in the same category such as `Smartphones`. To this end, the authors used the ESA algorithm to extract entities from the tweets of users as their primitive interests, and then expanded these entities using a random walk on the Wikipedia entity graph.
In [@Jipmo2017], the authors assumed there are a set of interests $i \in I$, e.g., `Sports`, `Politics`, etc., which the user modeling system needs to measure the corresponding weights for each interest. After building a bag of entities based on the ones extracted from a user’s tweets, the relevance score of an interest $i$ is measured as below, which can be seen as a spreading activation approach with some constraints:
$$S_i^u = \sum_{a \in BOE_u} \frac{1}{min\{dist(a, c), c \in BOC_i\}}
\bigskip$$
where $BOE_u$ denotes the bag of entities extracted from $u's$ tweets, and $BOC_i$ denotes a set of categories containing the name of $i$ in their titles. For example, for an interest `sports`, $BOC_i$ consists of categories such as `Category:Sports by year, Category:Sports in France`, etc. $dist(a, c)$ refers to the length of the shortest directed path from $a$ to $c$ in the Wikipedia graph.
**Leveraging collective knowledge**. More recently, some studies proposed leveraging collective knowledge powered by the great amount of interest profiles of all users in a dataset, and enhancing a user profile with other related interests identified as frequent patterns in all profiles using frequent pattern mining (FPM). FPM was designed to find frequent patterns (itemsets or a set of items that appear together in a transaction dataset frequently). In the context of user modeling, previous studies have treated each user interest as an item, each interest profile as a transaction, and all user interest profiles as the transaction dataset [@Faralli2015; @AnilKumarTrikhaFattaneZarrinkalam]. [@AnilKumarTrikhaFattaneZarrinkalam] leverages frequent pattern mining techniques to identify topic sets. Here, a topic set consists of the topics frequently appear together in user profiles. Afterwards, the other topics in the topic sets that contain the topics in a user’s profile are added into that user’s profile as well.
Take an example from [@AnilKumarTrikhaFattaneZarrinkalam], a topic set identified via FPM might consist of two topics $z_1$ and $z_2$, where $z_1=\{\texttt{Mixtape, Hip\_hop\_music, Rapping, Kanye\_West, Jay-Z, Remix}\}$ and $z_2=\{\texttt{Lady\_Gaga, Song, Album, Concert, Canadia\_Hot\_100}\}$. $z_1$ refers to the topic about hip hop music produced by two American rappers `Jay-Z` and `Kanye_West` while $z_2$ represents the topic about `Lady_Gaga`’s concert in Canada. As these two topics frequently appear together in user interest profiles, the users who are interested in $z_1$ might be also interested in $z_2$ even $z_2$ is not in their primitive interests. In contrast to [@AnilKumarTrikhaFattaneZarrinkalam], [@Faralli2015] did not directly enhance user interest profiles with other interests that occur together frequently, but used FPM for user classification and recommendation. It is worth noting that both [@Faralli2015] and [@AnilKumarTrikhaFattaneZarrinkalam] used the FP-Growth algorithm [@Han2000] for frequent pattern mining in their studies.
### Temporal Dynamics of User Interests {#Temporal Dynamics of User Interests}
User interests in OSNs can change over time, and many studies have been conducted in order to investigate the temporal dynamics of user interests in OSNs. For example, [@Jiang2015] showed that the similarity of current user interest profiles with the profiles at the beginning of the observation period of their dataset is the lowest while the similarity of current profiles with the ones built in the last month is the highest. Similarly, [@Abel2011g] showed that a user interest profile built in an earlier week differs more from the current profile compared to one built recently. In order to incorporate the temporal dynamics of user interests into user modeling strategies, there are mainly two types of approaches: (1) *constraint-based* approaches, and (2) *interest decay functions*.
**Constraint-based approaches.** Constraint-based approaches extract user interest profiles based on specified constraints, e.g., using a *temporal constraint* to build user interest profiles based on their tweets posted in the last two weeks or using an *item constraint* to construct user profiles based on the last 100 tweets of the users. For example, [@Abel2011g] investigated several temporal constraints such as *long-* and *short-term*, and *weekend* in their user modeling strategies on Twitter for a news recommender system. *Long-term* profiles extract user interests from entire historical tweets of users while *short-term* profiles extract user interests from tweets posted within the last two weeks. They showed that long-term entity-based profiles outperform short-term ones in the context of news recommendations. User interests can be different within different time frames such as during the week or on the weekends. The experimental results in [@Abel2011g] also showed that entity-based interest profiles based on their tweets posted on weekends can outperform long-term profiles for recommending news on weekends.
Some interests of users such as professional interests are stable while other interests such as the ones related to a certain event can be temporary. A user modeling strategy can apply temporal dynamics selectively to different information sources based on their characteristics. This type of strategy has been adopted in practical user modeling systems such as the one in Klout [@Spasojevic:2014:LLS:2623330.2623350], in which a 90 day window is used for capturing the temporal dynamics of user interests for some temporal information sources, and an all-time window is used for more permanent sources such as professional interests.
[@Nishioka:2016:PVT:2910896.2910898] compared both constraint-based approaches and interest decay functions for constructing user interest profiles on Twitter in the context of publication recommendations. Differing from the results in the domain of news [@Abel2011g], results from [@Nishioka:2016:PVT:2910896.2910898] showed that a constraint-based approach constructing user interest profiles within a certain period performs better than using an interest decay function in the context of publication recommendations.
**Interest decay functions.** Constraint-based approaches include interests which meet predefined constraints, and exclude other interests completely. Instead of constructing user interest profiles in a certain period (e.g., short-term), or based on temporal patterns (e.g., weekends), interest decay functions aim at including all the interests of a user but decaying old ones. The intuition behind those interest decay functions is that a higher weight should be given to recent interests than old ones.
A popular type of interest decay function applies exponential decay to user interests. For example, the interest decay function from [@Orlandi2012] is defined as follows:
$$\label{eq:orlandi}
x(t)=x_0 \cdot e^{-t/\beta}$$
\
Here, $x(t)$ is the decayed weight at time *t*, and $x_0$ denotes the initial weight (at time $t=0$). This interest decay function also has an initial time window (7 days), and the interests in the time window are not discounted. The authors in [@Orlandi2012] set $\beta =360days$ and $\beta =120days$ for their experiment, and showed that using $\beta =360days$ performs better than using $\beta =120days$ in terms of an evaluation based on a user study. We use `decay(Orlandi)` to denote this approach in this study. A similar decay function was used in [@Bhargava:2015:UMU:2678025.2701365] and [@Nishioka:2016:PVT:2910896.2910898], where a weight for the last update was used instead of initial weight [@Bhargava:2015:UMU:2678025.2701365]. In [@OBanion2012], the authors also used an exponential decay function: $x(t) = x_0 \cdot 0.9^d$ where *d* is the difference in days between the current date and the date that a concept was mentioned.
[@Abel2011d] also proposed a time-sensitive interest decay function, which is denoted by `decay(Abel)` in this survey. The weight of an entity *e* with respect to a user *u* at a specific time is measured as below.
$$\label{eq:abel}
w(e, time, T_{tweets, u, e}) = \sum_{t \in T_{tweets, u, e}} (1 - \frac{|time-time(t)|}{max_{time}-min_{time}})^d$$
\
where $T_{tweets, u, e}$ denotes the set of tweets mentioning *e* that have been posted by *u*. *time*(*t*) denotes the timestamp of a given tweet *t*, and $max_{time}$ and $min_{time}$ denote the highest (youngest) and lowest (oldest) timestamp of a tweet in $T_{tweets, u, e}$. In addition, the parameter *d* determines the influence of the temporal distance [$d=4$ in @Abel2011d]. In contrast to the aforementioned exponential decay functions, this approach incorporates the age of an entity *e* at the recommendation time, and the time span of *e* with respect to *u*.
In order to compare different interest decay functions in the context of user modeling in OSNs, [@piao2016exploring] investigated three interest decay functions for constructing user interest profiles on Twitter including `decay(Abel)` and `decay(Orlandi)`. The other one is a modified interest decay function from [@Ahmed2011], which was used in advertisement recommendations on web portals (i.e., Yahoo![^29]). The modified interest decay function used in [@piao2016exploring] is defined as follows:
$$\label{eq:ahmed}
w_{ik}^t = \mu_{2week}w_{ik}^{t, week} + \mu_{2month}w_{ik}^{t, month} + \mu_{all}w_{ik}^{t, all}$$
\
where $\mu_{2week} = \mu$, $\mu_{2month} = \mu^2$ and $\mu_{all} = \mu^3$ where $\mu = e^{-1}$. This decay function combines three levels of abstractions where the decay of user interests in each abstraction is $\mu$ times the previous abstraction. We use `decay(Ahmed)` to denote this approach in this survey. [@piao2016exploring] conducted a comparative study of user interest profiles constructed based on the three aforementioned interest decay functions and the profiles based on *short-* and *long-term* periods. Those interest profiles were then evaluated in the context of URL recommendations. The results showed that using `decay(Ahmed)` and `decay(Orlandi)` have competitive performance in terms of URL recommendations, and perform better than using `decay(Abel)` as well as *short-* and *long-term* profiles which were constructed without any interest decay. In addition, the experimental results indicate that although the performance increases by giving a higher weight to recent user interests, it starts decreasing once the weight of recent interests is too high. That is, although applying the decay function to recent user interests increases the performance, we still need the old history in order to provide the best performance in the context of URL recommendations.
Instead of considering the temporal dynamics of user interests with respect to individual users, global trends in an OSN can be incorporated into a user modeling strategy. In [@Gao:2011:ITU:2052138.2052335], the authors combined user interests from tweets of a target user (user profiles) and of all users (trend profiles) for constructing user interest profiles. The TF weighting scheme is used for constructing user profiles. For trend profiles, they applied a time-sensitive TF$\cdot$IDF (t-TF$\cdot$IDF) weighting scheme to concepts:
$$w_{t-TF \cdot IDF}(I_j, c) = w_{TF \cdot IDF}(I_j, c) \cdot (1-\hat{\sigma}(c))$$
\
where $w_{TF \cdot IDF}(I_j, c)$ denotes the TF$\cdot$IDF score of a concept *c* in a given time interval $I_j$, and $\hat{\sigma}(c)$ denotes the normalized standard deviation of timestamps of tweets that refer to *c*. [@Kanta2012] further incorporated location-aware trends into the trend-aware user modeling approach in [@Gao:2011:ITU:2052138.2052335] to improve the performance of inferred user interest profiles in the context of news recommendations.
Summary and discussion
----------------------
This section reviewed a number of approaches for constructing and enhancing user interest profiles. Table \[construction\] summarizes the approaches discussed in this section in terms of the three dimensions: (1) weighting schemes for constructing primitive interests, (2) approaches for incorporating the temporal dynamics of user interests, and (3) profile enhancement methods.
As we can see from the table, many studies have incorporated the temporal dynamics of user interests in their user modeling strategies. Among interest decay functions, exponential decay functions such as `decay(Orlandi)` have been adopted widely. When incorporating the temporal dynamics of user interests, it is important to choose constraint-based approaches or interest decay functions based on the purpose of user modeling. For instance, when using inferred user interest profiles for recommending items such as news or URLs in OSNs, interest decay functions perform better than constraint-based approaches such as short- and long-term profiles [@piao2016exploring]. However, the results from [@Nishioka:2016:PVT:2910896.2910898] indicate that a constraint-based approach based on a certain period for profiling outperforms the one applying exponential decay for building user profiles in the context of a publication recommender system. One possible explanation is that user interests change differently with respect to different domains. For example, user interests should be adapted to their recent interests for news or URL recommendations, however, user interests with respect to research may not.
[@Jiang2015] also pointed out that users have two types of interests; (1) *stable interests* [which they call primary interests in @Jiang2015], and (2) secondary interests. The stable interests of a user are original preferences inherent to that user, such as programmers who like efficient algorithms or lawyers who like debate, etc. [@Jiang2015]. In contrast, secondary interests are temporary ones which closely follow hot topics or events in a specific timespan. This is in line with the user modeling strategy used in Klout [@Spasojevic:2014:LLS:2623330.2623350], which applies a short-term window for capturing user interests that are temporary and uses a long-term window for more stable user interests.
Different types of knowledge from various knowledge bases have been leveraged for enhancing the primitive interests of users. The diversity of KBs and the different structures of hierarchical KBs indicate the complexity of representing knowledge in KBs as well. Table \[topicTree\] summarizes the differences between hierarchical KBs used in the literature. For instance, the constructed Wikipedia category taxonomy in [@Kapanipathi2014] consists of 15 levels with 802,194 categories while the topic hierarchy tree built by [@Jiang2015] consists of 5 levels with over 1,000 topics. The topic hierarchy tree used in Klout has 3 levels which consists of 15 main categories, around 700 sub-categories, and around 9,000 entities [@Spasojevic:2014:LLS:2623330.2623350]. A concept taxonomy built manually by referring to external websites such as news portals or Facebook Page categories has less complexity compared to a taxonomy based on KBs such as Wikipedia. For example, the category taxonomy built based on news portals [@Kang2016] has 8 main categories and 58 sub-categories. The one built based on Facebook and Yelp categories [@Bhargava:2015:UMU:2678025.2701365] also has 8 and 58 categories for the top-2 levels with an additional 137 categories in its third level. We can observe that the hierarchical knowledge bases used in practice or built based on taxonomies used in practice tend to have a small number of levels (2-5). Applying a spreading activation function, even the same one, to those different taxonomies might have different results. There is a lack of comparison of different hierarchical knowledge bases and their effect in the context of inferring user interest profiles.
Furthermore, although some studies investigated the comparison between using different KBs such as Wikipedia categories and the DBpedia graph, there was no comparative study on exploiting the Wikipedia entity graph [@Lu2012], categories in other KBs such as ODP, and the DBpedia graph. In addition, despite the fact that different KBs might be useful in different domains [@Nguyen], enhancing user interests based on other KBs such as Wikidata [@Vrandecic2014], or BabelNet [@NavigliPonzetto:12aij] has not been fully explored.
**Study** **\# Levels** **\# Topics** **Details**
---------------------------------------- --------------- --------------- ------------------------------------------------------
[@Kapanipathi2014] 15 802,194 N/A
[@Jiang2015] 5 $\sim$1,000 N/A
[@Spasojevic:2014:LLS:2623330.2623350] 3 $\sim$1,0000 15 $\rightarrow$ $\sim$700 $\rightarrow$ $\sim$9,000
[@Kang2016] 2 66 8 $\rightarrow$ 58
[@Bhargava:2015:UMU:2678025.2701365] 3 203 8 $\rightarrow$ 58 $\rightarrow$ 137
Evaluation Approaches
=====================
Overview
--------
In this section, we describe evaluation approaches used for evaluating different user interest profiles that are generated by different user modeling strategies in the literature. User modeling is one of the main building blocks in many adaptive systems such as recommender systems. Many previous studies on the evaluation of adaptive systems suggested that it is important to evaluate different blocks separately in order to identify the problems in the adaptive systems [@Paramythis2010; @Brusilovsky2001]. [@Gena2007] provided a list of methods for evaluating adaptive systems, where some of them can be used for evaluating the quality of user modeling component as well. These evaluation methods include (1) *questionnaires*, (2) *interviews*, and (3) *logging use*.
**Questionnaires.** Questionnaires consist of pre-defined questions, which can be in different styles such as scalar or multi-choice, and ranked [@Gena2007]. In our context, this approach can be used for collecting users’ explicit feedback about their interest profiles for evaluation. To this end, this approach requires recruiting users for the experiment of building user interest profiles with their OSN accounts. At the end of the experiment, these users can provide feedback on user interest profiles constructed by different user modeling strategies.
**Interviews.** The second approach is used to collect users’ opinions and experiences, preferences and behavior motivations [@Gena2007] with respect to adaptive systems. Interviews can be used after building users’ interest profiles to gather their opinion such as satisfaction and accuracy about the inferred user interest profiles. Compared to questionnaires, interviews are more flexible but more difficult to be administered. Therefore, this method has not been exploited for evaluating user modeling strategies in the literature.
**Extrinsic evaluation (Logging use).** This approach uses the actions of users in the context of adaptive systems for evaluation, e.g., whether a user liked a recommend item in a recommender system. This can be considered an extrinsic way of evaluating user interest profiles in terms of the performance of applications where these profiles are applied. For example, one common approach is using constructed user interest profiles as an input to a recommender system, and adopting some well-established evaluation metrics of recommender systems for measuring the quality of user interest profiles indirectly. Manual analysis is sometimes used together with other evaluation approaches. In this case, the authors present some examples of user interest profiles built for several users (e.g., some representative users on Twitter such as *Barack Obama*), and discuss the quality of profiles with respect to these users.
Review
------
### Evaluation based on Questionnaires
A common approach for evaluating constructed user interest profiles is based on a user study with questionnaires. For example, [@Narducci2013] evaluated user interest profiles built for 51 users from Facebook and Twitter based on their feedback on two aspects: *transparency* and *serendipity* using a 6-point discrete rating scale. The first aspect aims to evaluate to what extent the keywords in the profile reflect personal interests, and the second one aims to measure to what extent the profile contains unexpected interesting topics. Similarly, [@Kapanipathi2014] recruited 37 users and built category-based user interest profiles based on their tweets on Twitter. Afterwards, the 37 users provided explicit feedback, e.g., Yes/Maybe/No with respect to the categories in those profiles. Similar approaches have been used in [@Bhattacharya:2014:IUI:2645710.2645765], [@Besel:2016:ISI:2851613.2851819; @Besel:2016:QSI:3015297.3015298], [@Budak2014], and [@Orlandi2012]. However, instead of recruiting volunteers for an experiment, the authors in [@Budak2014] first inferred user interest profiles for 500 randomly chosen users on Twitter, and emailed them using the email addresses in their profiles to get feedback about their inferred interests. Instead of using the feedback from target users for inferred user interest profiles, [@Kang2016] and [@Michelson2010] labeled user interests themselves or used recruited annotators.
Explicit feedback can be obtained in a system which has user interest profiles that can be modifed by users. For example, [@GarciaEsparza:2013:CCT:2449396.2449402] implemented a stream filtering system where users are represented based on 18 defined categories such as `Music` and `Sports`. For evaluation, the authors asked each participant to give explicit feedback on their profiles by deleting or adding categories that they felt were incorrect or missing.
In contrast to obtaining explicit feedback on inferred user interest profiles, a user study can be conducted on the performance of a specific application where those inferred user interest profiles play an important role. For example, [@Chen2010] conducted a user study with respect to a URL recommender system on Twitter, which is based on the inferred user interest profiles. Therefore, instead of directly giving feedback on the constructed user interest profiles, the users participating in the study were given URL recommendations, and they marked each URL as one of their interests or not. Similarly, [@Nishioka:2016:PVT:2910896.2910898] obtained explicit feedback from users on publication recommendations based on their interest profiles. These user studies can also be considered as extrinsic evaluation, which we will discuss in the next section, as they are not evaluating user interest profiles directly.\
\
**Pros and cons.** Evaluation approaches based on the explicit feedback of profiled users with respect to their interest profiles would arguably be the most direct and accurate way for evaluating those profiles. However, this also requires recruiting volunteers and imposes an extra burden for users, and therefore limits the number of participants for evaluation [e.g., 37 users were recruited for evaluation in @Kapanipathi2014].
### Extrinsic Evaluation
To evaluate the quality of inferred user interest profiles without imposing an extra burden on users, offline evaluation in terms of the performance of a specific application has been used. In this case, user interest profiles are used as an input to an application such as a news recommender system where these profiles play an important role. Afterwards, different profiles created by different user modeling strategies are compared in terms of the recommendation performance using each profile. The recommendation performance can be evaluated by well-established evaluation metrics for recommender systems such as *mean reciprocal rank* (MRR) which denotes at which rank the first item relevant to the user occurs on average, *success at rank N* (S@N), which stands for the mean probability that a relevant item occurs within the top-N recommendations, and well-known *precision* and *recall*. For a complete list of evaluation metrics and their details we refer the reader to [@Bellogn] and [@Herlocker:2004:ECF:963770.963772] respectively.
For instance, [@Abel2011g] evaluated three different user modeling strategies in terms of S@N and MRR in the context of news recommendations, and [@Spasojevic:2014:LLS:2623330.2623350] evaluated their user modeling strategy in terms of precision and recall in the context of topic recommendations on Klout. Similarly, [@Sang:2015:PFT:2806416.2806470] also evaluated user interest profiles in terms of news recommendations in addition to tweet recommendations. [@piao2016exploring; @Guangyuan2017; @Piao2017; @Piao2016b; @Piao2016d] evaluated different user modeling strategies in the context of URL recommendations on Twitter where the set of ground truth URLs is those shared by users on Twitter in the last two weeks. In [@Faralli2015], the authors evaluated user interest profiles in terms of user classifications and recommendations. For the classification task, the user interest profiles were used for classifying each user to the appropriate label, e.g., Starbucks fan. For the recommendation task, the authors evaluated the performance of leveraging different hierarchical levels of interests with respect to interest recommendations using itemset mining. In contrast to previous studies which have focused on inferring user interest profiles, [@Nechaev] focused on users’ privacy and evaluated different followee-suggestion strategies for concealing user interests which can be inferred from users’ activities in OSNs based on state-of-the-art user modeling strategies.\
\
**Pros and cons.** Extrinsic evaluation provides an offline setting for evaluating inferred user interest profiles. Therefore, it facilitates the evaluation process of different user modeling strategies as these strategies are evaluated based on a collected dataset (or logs). However, this approach does not directly evaluate the inferred user interest profiles, and lacks the opinions of users with respect to the inferred interest profiles.\
\
There are other evaluation approaches used in some studies besides the aforementioned two methods. For example, [@Abel2011e] compared the number of distinct entities and topics in user interest profiles for evaluating news-based enrichment of their tweets. In [@Faralli2017], the authors run two experiments to evaluate their approach of building interest taxonomies. First, they compared their approach against other approaches proposed for constructing user interest taxonomies using other gold standard taxonomies. Second, they provided samples of generated user interest profiles, and compared inferred Wikipedia categories with respect to several users based on different user modeling strategies. Similarly, [@Xu2011] evaluated their topic modeling approach by comparing it against other topic modeling methods in terms of *perplexity*, and then discussed some user interest profiles produced by different approaches. User interest profiles have also been used for specific applications such as followee, tweet, and news recommendations [@Weng:2010:TFT:1718487.1718520; @Chen2012b; @Hong:2013:CMM:2433396.2433467; @Phelan:2009:UTR:1639714.1639794], where user modeling strategies were not evaluated or compared to other alternatives.
Summary and discussion
----------------------
[|c|c|l|]{} & & **Examples**\
& &
---------------------------------------------------------------------------------------------------------------
[@Kapanipathi2014], [@Kang2016],
[@Michelson2010], [@Budak2014],
[@Bhattacharya:2014:IUI:2645710.2645765], [@Besel:2016:ISI:2851613.2851819; @Besel:2016:QSI:3015297.3015298],
[@Orlandi2012], [@Narducci2013],
[@Bhargava:2015:UMU:2678025.2701365], [@GarciaEsparza:2013:CCT:2449396.2449402],
[@Vu:2013:IMU:2505515.2507883], [@Ahn:2012:IUI:2457524.2457681],
[@Chen2010], [@Nishioka:2016:PVT:2910896.2910898]
---------------------------------------------------------------------------------------------------------------
: Evaluation approaches for constructed user interest profiles.[]{data-label="evaluation"}
\
& &
-------------------------------------------------------------------------------
[@Abel2011d; @Abel2011g; @Abel2012a; @Abel2011e], [@Chen2010],
[@Zarrinkalam2015a], [@Sang:2015:PFT:2806416.2806470],
[@Kanta2012], [@OBanion2012],
[@piao2016exploring; @Guangyuan2017; @Piao2017; @Piao2016b; @Piao2016d],
[@Lu2012], [@Sang:2015:PFT:2806416.2806470], [@Gao:2011:ITU:2052138.2052335],
[@Karatay2015a], [@AnilKumarTrikhaFattaneZarrinkalam],
[@Nishioka:2015:ITU:2809563.2809601], [@Bolting2015],
[@Zarrinkalam2016], [@Ahn:2012:IUI:2457524.2457681],
[@Spasojevic:2014:LLS:2623330.2623350], [@Jipmo2017],
[@Faralli2015], [@Nechaev]
-------------------------------------------------------------------------------
: Evaluation approaches for constructed user interest profiles.[]{data-label="evaluation"}
\
In this section, we reviewed different evaluation approaches that have been used in the literature for evaluating constructed user interest profiles. Table \[evaluation\] provides a summary of previous studies in terms of evaluation methods.
Evaluating user interest profiles based on a user study is important for understanding different aspects of user interests, e.g., abstraction levels of user interests. For example, [@Orlandi:2013:CCI:2568488.2568810] studied the specificity of user interests and evaluated it based on a user study, which showed that users prefer to give a higher score over non-specific entities. However, the extra effort of recruiting users and gaining feedback from them is time consuming, and limits the scale of users for evaluation. The evaluation in terms of the performance of a specific application has the advantage of its offline setting and using a relatively larger number of users compared to a user study. Both evaluation approaches can be used in an appropriate way for designing and evaluating user modeling strategies. For example, based on a user study on the specificity of user interests [@Orlandi:2013:CCI:2568488.2568810], we can design ways to incorporate the feedback from users’ preferences regarding non-specific entities into a user modeling strategy, and evaluate the strategy at a large scale in offline settings based on a collected dataset such as the one from Twitter.
One of the challenges of the offline evaluation in terms of the performance of a specific application is the lack of benchmarks that are freely available [@Faralli2015]. Despite the openness of some microblogging services such as Twitter, it is time consuming to collect all data used in different user modeling approaches, e.g., tweets, list memberships, biographies of followees/followers in addition to the information about users. In addition, different datasets with different user sizes might produce different results even using the same user modeling strategies for comparison. It is also important to evaluate different user interest profiles in the context of different applications beyond a specific one. For example, in [@Manrique:2017:SDA:3106426.3109440], the authors showed that user interest profiles based on different user modeling strategies perform differently in the context of recommending articles based only on titles, abstracts, and full texts. Although the study [@Manrique:2017:SDA:3106426.3109440] is in the context of research article recommendations, it is highly likely that different user interest profiles from microblogging services will have different levels of performance based on the applications in which these profiles are applied.
Conclusions and Future Directions
=================================
In previous sections, we reviewed the state-of-the-art approaches used in different user modeling stages for inferring user interest profiles, which is beneficial both for researchers who are interested in user modeling in the social networks domain as well as those researchers in some other domains. It is also useful for third-party application providers who aim to utilize user interest profiles via social login functionalities in terms of providing personalized services for their users.
In this final section, we conclude this paper in Section \[conclusions\] with respect to the four dimensions of inferring user interest profiles: (1) data collection, (2) representations of user interest profiles, (3) construction and enhancement of user interest profiles, and (4) the evaluation of the constructed profiles. In Section \[fd\], we first review what progress has been made to date since [@Abdel-Hafez2013], and then outline some opportunities and challenges for inferring user interests on microblogging social networks which we envision can inspire future directions in this research field.
Conclusions
-----------
To sum up, user activities such as the tweets posted by users are the most widely used information source for inferring user interests. However, many recent studies have started exploring other information sources such as the social networks of users as an alternative to user activities as the passive usage of OSNs is on the rise. Regarding the representations of user interest profiles, a clear tendency of leveraging concepts such as DBpedia entities or categories can be observed given their advantages of using background knowledge about those concepts from a KB. In addition to leveraging the hierarchical or graph-based knowledge of a KB for enriching user interests, several recent studies also have shown the effectiveness of leveraging collective knowledge for enriching user interest profiles [@Faralli2015; @AnilKumarTrikhaFattaneZarrinkalam]. With respect to incorporating the temporal dynamics of user interests, there is no single best method for inferring user interests with different purposes. Instead, one should choose constraint-based or interest decay functions based on the application needs, and the characteristics of items. For evaluating user interest profiles, both questionnaires and extrinsic evaluation strategies have been adopted at comparable levels of popularity.
Future Directions {#fd}
-----------------
In [@Abdel-Hafez2013], the authors proposed three future directions with respect to user modeling in OSNs, which requires (1) more dynamicity, (2) more enrichment, and (3) more comprehensiveness. On the one hand, we observe that there have been many efforts towards the second direction. These efforts include leveraging the collective knowledge powered by all users [@Faralli2015; @AnilKumarTrikhaFattaneZarrinkalam] for enriching the interest profiles of each user, and the comparison between different KBs for enriching user interests [@Guangyuan2017]. On the other hand, the first and third directions proposed by [@Abdel-Hafez2013] have not made much progress. For example, [@Abdel-Hafez2013] proposed incorporating more dynamicity with respect to user interest profiles with some assumptions such as different topics might decay with different speed, and the interest weights of each user can have different weights in different context. On top of the directions proposed by [@Abdel-Hafez2013] and the recent studies we reviewed in this paper, we further proposed several future directions which are related to:
- mining user interests;
- multi-faceted user interests;
- comprehensive user modeling;
- evaluation of user modeling strategies.
**Mining user interests.** To better infer user interests, researchers have proposed various approaches such as enriching short content, filtering noise in UGC, and exploring social networks. Many studies have adopted traditional weighting schemes from information retrieval such as TF or TF$\cdot$IDF to somehow filter the noise in UGC for mining user interests. However, some studies have shown that incorporating some special characteristics of the services (e.g., temporal dynamics, short content) into the design of a weighting scheme can improve the quality of user interest profiles. For example, TI-TextRank which combines TF$\cdot$IDF and TextRank performs better than either of them on their own as a weighting scheme for user modeling on Twitter. In this regard, more weighting schemes adapted towards microblogging services should be investigated, e.g., combining different weighting schemes used in the literature. Furthermore, mining interest-related items from data sources such as posts [e.g., @Xu2011] can be useful as microblogging services have multiple usages such as information seeking, sharing and social networking [@Java2007].
In addition, more sophisticated approaches for understanding the semantics of UGC are required. For example, for those approaches that rely on extracted entities for inferring user interest profiles, extracting entities from microblogs is a fundamental step which is challenging by itself. Only a few studies have considered the uncertainty (confidence) of the extracted entities, which we think might impact the overall quality of the primitive interests of users as well as the enhanced ones. Moreover, most approaches have extracted explicitly mentioned entities based on NLP APIs such as Tag.Me[^30], Aylien[^31], OpenCalais, etc. However, there can be many entities implicitly mentioned in tweets. In [@Perera2016], the authors showed that over 20% of mentions of movies are implicit references, e.g., a tweet referring the movie *Gravity* - “ISRO sends probe to Mars for less money than it takes Hollywood to make a movie about it”. It shows that advanced methods for extracting entities, such as the one proposed in [@Perera2016], have great potential to improve the quality of user modeling. Also, considering the context of a microblog might be useful when extracting entities instead of just considering the single microblog of a user. The context might refer to some previous microblogs posted by the user, or other microblogs with the same hashtag in the microblogging service. For example, [@Shen:2013:LNE:2487575.2487686] showed that the quality of entity extraction can be improved by incorporating user interests as contextual information. Furthermore, promising results from recent studies [@Faralli2015; @AnilKumarTrikhaFattaneZarrinkalam] indicate that leveraging collective knowledge via frequent pattern mining approaches is also effective in inferring implicit user interests.
**Multi-faceted user interests.** There exists various aspects/views of users based on different dimensions of user modeling such as the data source, representation level, and temporal dynamics of user interests. Although many studies represent an individual user using a single user interest profile, we believe that multi-faceted user interest profiles should be given more attention as some previous studies have also shown their efficiency compared to a single model. It is not necessary to maintain several user interest profiles for a single user, but a single model can also be built with relevant information from different aspects, and a view/aspect made for the user based on the information needs for different applications. GeniUS [@Gao2012d] is a good example in this regard, which is a user modeling library that stores concept-based user interest profiles using the RDF[^32] format (a W3C recommendation) with widely used ontologies such FOAF [@Brickley2012], SIOC[^33], and WI[^34]. In GeniUS, user interest profiles are represented as DBpedia entities and enriched by background knowledge such as the type (domain) of an entity from DBpedia. Therefore, the constructed profile is flexible enough to retrieve its sub-profiles with respect to specific domains (e.g., `Music`), which is useful for recommending domain-specific items. The idea is that, for example, we only need your music-related interest profile in the context of music recommendations. The results in [@Gao2012d] indicate that domain-specific profiles clearly outperform the whole user profiles for domain-specific tweet recommendations in terms of six different domains. Although GeniUS only considers different views of users in terms of topical domains, the same idea can be extended to other views. For instance, different user profiles can be extracted dynamically with different approaches for incorporating temporal dynamics, e.g., retrieving short-term profiles for recommending tweets during an event, which might be more useful compared to using long-term profiles. Also, multiple user interest profiles in terms of representation level using different interest formats have been used in other domains such as personal assistants [@Guha2015], which can be useful for user modeling in microblogging services as well. In [@Guha2015], several user interest profiles based on different representations such as keywords and Freebase entities were constructed.
**Comprehensive user modeling.** In the previous survey on user modeling [@Abdel-Hafez2013], the authors also suggested that more comprehensive user modeling strategies should be investigated by considering different dimensions of user modeling together. Many of the previous studies have ignored some of the dimensions such as temporal dynamics [e.g., @Phelan:2009:UTR:1639714.1639794]. Investigating the synergistic effect of different dimensions is important for developing better user modeling strategies, which is crucial for the performance of applications. To this end, several research questions should be answered such as “which combinations of different approaches in each dimension can provide the best user interest profiles” or “does a dimension really matter in the context of the combination for providing the best performance?”. For example, [@Piao2016d] showed that a rich representation of user interests (using WordNet synsets and DBpedia entities) and enriching short content with the text of embedded URLs are the most important factors followed by temporal dynamics in the context of URL recommendations on Twitter. However, enhancing user interest profiles has little effect when we have a rich representation or enriched content of microblogs. Similar results have been observed in the context of inferring research interests of users based on their publications [@Manrique:2017:SDA:3106426.3109440]. The results in [@Manrique:2017:SDA:3106426.3109440] indicate that enhancing primitive interests can improve the performance when only short texts (e.g., titles) are available but not in the case when longer texts (e.g., full texts of publications) are available. We believe that these studies are good starting points for some future works, e.g., using different user interest profiles for different data sources instead of using a single representation of an individual user for the combination.
In addition, other user modeling dimensions which have been proposed in other domains can be considered in the social media domain as well. For example, a *scrutable* user model proposed in the context of teaching, which aims to let users have the right and possibility to have access to and control their user profiles [@Holden1999; @Carmagnola2011; @Kay2006], can be a promising dimension to be incorporated into user modeling strategies in OSNs and merits further investigation and evaluation.
**Evaluation of user modeling strategies.** As we mentioned in Section 5.3, the lack of common benchmarks and datasets hinders comparison with other approaches, which ends up with several studies directly comparing to results reported in previous studies [@Faralli2015]. This does not reflect a correct comparison due to the difference of datasets in terms of platforms as well as user sizes. However, it is also challenging due to the regulations of microblogging services such as Twitter[^35], and the differences in data sources used in each study. Another possible direction is providing all proposed approaches as user modeling libraries that are publicly available, in the same way as GeniUS and TUMS[^36], so that other researchers can easily reimplement the approaches proposed in previous studies for comparison.
It is also important to evaluate inferred user interest profiles in terms of multiple tasks or different settings to understand the strengths and weaknesses of different user interest profiles. For instance, [@Nishioka:2015:ITU:2809563.2809601] showed that considering the temporal dynamics of user interests has a positive influence on a computer science dataset but not on a medicine dataset. [@Manrique:2017:SDA:3106426.3109440] showed that different user modeling strategies work differently for different types of texts that are available in the context of research article recommendations. In this regard, evaluating the performance of different user modeling strategies based on different datasets or settings can provide a clear understanding of when to use what types of user profiles, which is important for researchers in different domains as well as third-party application providers with different types of content to be personalized. A recent work by [@Tommasso2018] provides a user interests dataset which is useful in this context. It includes half million Twitter users with an average of 90 multi-domain preferences per user on music, books, etc., where those preferences are extracted from multiple platforms based on the messages of those Twitter users who also use Spotify[^37], Goodreads[^38], etc.
Finally, previous studies have adopted accuracy and ranking metrics such as precision, recall, and MRR for the extrinsic evaluation of inferred user interest profiles. However, non-accuracy metrics such as serendipity, novelty, and diversity have received increasing attention in recommender systems [@Bellogn; @Kaminskas:2016:DSN:3028254.2926720]. Therefore, it is worth investigating the effect of different user modeling strategies and their inferred interest profiles in the context of recommender systems in terms of those non-accuracy metrics.
This publication has emanated from research conducted with the financial support of Science Foundation Ireland (SFI) under Grant Number SFI/12/RC/2289 (Insight Centre for Data Analytics). Thanks for the anonymous reviewers and the editor for their constructive feedback to improve this work.
Author Biographies {#author-biographies .unnumbered}
==================
**Guangyuan Piao** (<https://parklize.github.io>) is a Ph.D. student at the Insight Centre for Data Analytics (formerly DERI) at the National University of Ireland Galway. He received his B.Sc. in Computer Science from Jilin University, China, and received his M.Eng. degree in Information and Industrial Engineering from Yonsei University, South Korea. His main research interests include User Modeling, Recommender Systems, and Knowledge Graph. His current research focuses on semantics-aware user modeling and recommender systems leveraging knowledge graphs and latent semantics.\
**John G. Breslin** ([www.johnbreslin.com](www.johnbreslin.com)) is a Senior Lecturer in Electrical and Electronic Engineering at the College of Science and Engineering at the National University of Ireland Galway, where he is Director of the TechInnovate / AgInnovate programmes. John has taught electronic engineering, computer science, innovation and entrepreneurship topics during the past two decades. He is also a Co-Principal Investigator at the Insight Centre for Data Analytics, and a Funded Investigator at Confirm Smart Manufacturing and VistaMilk. He has written 190 peer-reviewed academic publications (h-index of 37, 5500 citations, best paper awards from DL4KGS, SEMANTiCS, ICEGOV, ESWC, PELS), and co-authored the books “The Social Semantic Web” and “Social Semantic Web Mining”. He co-created the SIOC framework, implemented in hundreds of applications (by Yahoo, Boeing, Vodafone, etc.) on at least 65,000 websites with 35 million data instances.
The List of Surveyed Works {#appendix:works}
==========================
Search Strategy
---------------
In order to draw up a list of search terms, the basic terms are extracted from primary articles are retrieved. After that, other search terms are obtained iteratively based on the keywords that were used interchangeably within the retrieved articles. Overall, the final list of terms used for searching articles is presented in Table \[terms\]. These search terms (ST) are used for constructing sophisticated search strings. For example, the search string can be constructed as ST1 AND ST3 while ST1 is a compound term from Term1 and Term2 (e.g., inferring user interests). Initial searches with these search terms for titles and abstracts from electronic databases can obtain many relevant articles but may not be sufficient [@Kitchenham2004]. In this regard, additional article candidates are obtained by checking the reference list from primary studies that are relevant, and searching relevant journals and conference proceedings. [@Abdel-Hafez2013] provided a review of user modeling in social media websites in 2013, which includes some approaches with respect to inferring user interests in the context of microblogging social networks. In addition to those approaches mentioned in [@Abdel-Hafez2013], we also review recent user modeling approaches for inferring user interests.
Term1 Term2
----- --------------------------------- --------------------------------
ST1 inferring, modeling, predicting (user) interests
ST2 user (interest) modeling, profiling, detection
ST3
: Search terms used in the search strategy of this survey.[]{data-label="terms"}
Selection Criteria
------------------
In order to assess and select relevant articles from primary studies, inclusion and exclusion criteria should be defined based on the research questions [@Kitchenham2004]. The inclusion criteria are as follows:
1. Published in English from 2004.
2. Studies on microblogging social networks.
3. Focus on user modeling strategies for inferring user interest profiles.
On the other hand, exclusion criteria can be defined as follows:
1. Studies that were not peer-reviewed or published.
2. Studies related to user modeling but not focus on microblogging social networks.
3. Studies related to user modeling, but not focus on inferring user interests.
Finally, inclusion or exclusion decisions are made for the fully obtained articles and those papers that only meet our criteria are selected. As a result, 51 articles are selected in this survey. These articles are distributed from 2010 to 2018, and the majority of them were published in conferences or workshops such as WI, UMAP, CIKM, and ECIR.
Surveyed Studies
----------------
The surveyed 51 works are retrieved from different journals, conferences, and workshops, mainly in the user modeling, recommender systems, and Web related fields as follows:
1. Journals
- ACM SIGAPP Applied Computing Review: [@Besel:2016:QSI:3015297.3015298]
- Web Semantics: Science, Services and Agents on the World Wide Web: [@Faralli2017]
- Social Network Analysis and Mining: [@Faralli2015]
- Information Systems: [@Kang2016]
- Procedia Computer Science: [@Jiang2015]
2. Conference proceedings
- **WI** (IEEE/WIC/ACM International Conference on Web Intelligence and Intelligent Agent Technology): [@Zarrinkalam2015a; @Xu2011; @Gao:2011:ITU:2052138.2052335; @Penas2013; @Ahn:2012:IUI:2457524.2457681]
- **UMAP** (Conference on User Modeling Adaptation and Personalization): [@Abel2011g; @Hannon2012; @Narducci2013]
- **CIKM** (ACM International Conference on Information and Knowledge Management): [@Vu:2013:IMU:2505515.2507883; @Piao2016b; @Sang:2015:PFT:2806416.2806470]
- **ECIR** (European Conference on Information Retrieval): [@Zarrinkalam2016; @Guangyuan2017; @AnilKumarTrikhaFattaneZarrinkalam]
- **ISWC** (International Conference on Semantic Web): [@Siehndel:2012:TUP:2887379.2887395; @Abel2011e]
- **IUI** (International Conference on Intelligent User Interfaces): [@Bhargava:2015:UMU:2678025.2701365; @GarciaEsparza:2013:CCT:2449396.2449402]
- **RecSys** (ACM Conference on Recommender Systems): [@Bhattacharya:2014:IUI:2645710.2645765; @Phelan:2009:UTR:1639714.1639794]
- **SEMANTiCS** (International Conference on Semantic Systems): [@piao2016exploring; @Orlandi2012]
- **HT** (ACM Conference on Hypertext and Social Media): [@Piao2017]
- **SIGIR** (International ACM Conference on Research and Development in Information Retrieval): [@Chen2010]
- **AAAI** (AAAI Conference on Artificial Intelligence): [@Lu2012]
- **KDD** (Knowledge Discovery and Data Mining): [@Spasojevic:2014:LLS:2623330.2623350]
- **IJCAI** (International Joint Conference on Artificial Intelligence): [@Abel:2013:TUM:2540128.2540558]
- **ICWE** (International Conference on Web Engineering): [@Abel2012a]
- **WebSci** (International Web Science Conference): [@Abel2011d]
- **ESWC** (Extended Conference on Semantic Web): [@Kapanipathi2014]
- **EKAW** (International Conference on Knowledge Engineering and Knowledge Management): [@Piao2016d]
- **ICSC** (IEEE International Conference on Semantic Computing): [@Bolting2015]
- **SAC** (ACM Symposium on Applied Computing): [@Besel:2016:ISI:2851613.2851819]
- **WSDM** (ACM International Conference on Web Search and Data Mining): [@Weng:2010:TFT:1718487.1718520]
- **JCDL** (Joint Conference on Digital Libraries): [@Nishioka:2016:PVT:2910896.2910898]
- **i-KNOW** (International Conference on Knowledge Technologies and Data-driven Business): [@Nishioka:2015:ITU:2809563.2809601]
- **SPIM** (International Conference on Semantic Personalized Information Management: Retrieval and Recommendation): [@Kapanipathi2011]
- **OpenSym** (International Symposium on Open Collaboration): [@Lim:2013:ICT:2491055.2491078]
- **ADMA** (Advanced Data Mining and Applications): [@Jipmo2017]
3. Workshop proceddings
- **AND** (Workshop on Analytics for Noisy Unstructured Text Data): [@Michelson2010]
- **Micropost** (Workshop on Making Sense of Microposts): [@Karatay2015a]
- **SMAP** (Workshop on Semantic and Social Media Adaptation and Personalization): [@Kanta2012]
- **RSWeb** (Workshop on Recommender Systems and the Social Web): [@OBanion2012]
- **BlackMirror** (Workshop on Re-coding Black Mirror): [@Nechaev]
4. Others
- Tech Report: [@Budak2014].
[^1]: <https://en.wikipedia.org/wiki/Microblogging>
[^2]: <https://twitter.com/>
[^3]: <https://www.facebook.com/>
[^4]: <https://www.omnicoreagency.com/twitter-statistics/>
[^5]: <https://www.omnicoreagency.com/facebook-statistics/>
[^6]: <https://en.wikipedia.org/wiki/Social_login>
[^7]: <https://hbr.org/2011/10/social-login-offers-new-roi-fr>
[^8]: <http://www.gigya.com/blog/why-millennials-demand-social-login/>
[^9]: <https://del.icio.us/>
[^10]: <https://www.flickr.com/>
[^11]: [www.wikipedia.org](www.wikipedia.org)
[^12]: <https://www.linkedin.com/>
[^13]: <https://klout.com/>
[^14]: <https://goo.gl/j97H1R>
[^15]: <http://bit.ly/pewsnsnews>
[^16]: <http://www.corporate-eye.com/main/facebooks-growing-problem-passive-users/>
[^17]: <http://edition.cnn.com/>
[^18]: <http://listorious.com>, not available at the time of writing.
[^19]: <https://www.delicious.com>
[^20]: <https://www.stumbleupon.com>
[^21]: <https://en.wikipedia.org/wiki/Hashtag>
[^22]: <http://www.opencalais.com/>
[^23]: <http://zbw.eu/stw>
[^24]: <https://www.nlm.nih.gov/mesh/>
[^25]: <https://en.wikipedia.org/wiki/DMOZ>
[^26]: <http://news.naver.com/>
[^27]: <http://news.nate.com//>
[^28]: <https://www.yelp.com/>
[^29]: <https://yahoo.com/>
[^30]: <https://tagme.d4science.org/tagme/>
[^31]: <https://aylien.com/>
[^32]: <https://www.w3.org/RDF/>
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---
abstract: 'It has been recently conjectured by Selem and Wilczek [@Sel06] the existence of a $ss-[\bar u \bar d]$ meson due to strong correlations between the two light antiquarks. We make a detailed study of this system within a dynamical quark model which has proven to be successful in reproducing the most important features of low-energy hadron phenomenology. Our results, obtained within a parameter-free calculation, show that the antidiquark component of the $ss \bar u \bar d$ system indeed entails the stronger attraction, and drives its energy much lower than the $\overline{N}\Xi$ threshold, but still above the $\overline{K^0}\,{K^*}^-$ or $\overline{{K^*}^0}\,K^-$ thresholds. We have also studied the $cc \bar u \bar d$ and $bb \bar u \bar d$ systems. Exotic mesons are only expected to exist in the limit of large mass for the two-quark subsystem, $bb\bar u \bar d$, since the calculated mass is below the $\overline{B^0}\,{B^*}^-$ or $\overline{{B^*}^0}\,B^-$ thresholds.'
address: 'Grupo de Física Nuclear and IUFFyM, Universidad de Salamanca, E-37008 Salamanca, Spain'
author:
- 'J. Vijande, A. Valcarce, and K. Tsushima'
title: 'Dynamical study of ${\bf QQ-\bar u \bar d}$ mesons'
---
Pacs: 12.39.-x, 14.40.-n, 14.65.-q
It has been recently re-emphasized [@Sel06; @Jaf05; @Jaf03] the potential importance of strong diquark (antidiquark) correlations in hadronic physics [@Lip71; @Kar06]. Theoretically the idea of diquark (antidiquark) correlations inside hadrons is a consequence of [*color cancellation*]{}. The disturbance produced by the color charges of two quarks in empty space can be halved by bringing them together into a single ${\bf\bar{3}}$ representation of the color $SU(3)$ group. If this is joined with the more favorable spin-singlet state and Fermi statistics, the quarks must be in the antisymmetric ${\bf\bar{3}}$ representation of flavor $SU(3)$. Besides, one should expect that any effect disrupting the correlations will induce a repulsive force. Among such effects we may quote the presence of an additional diquark or a spectator quark. Such ideas suggest that the easiest way of constructing low-energy exotics could be based on strongly correlated diquarks (antidiquarks) as building-blocks.
However, these arguments are rather qualitative, and merely based on the group theoretical structure of QCD. To study quantitatively whether or not such strong correlations between the light quarks (antiquarks) are indeed present, one needs QCD-based dynamical studies such as lattice QCD, although this eventually must be checked by experiments. In view of present status of the lattice QCD simulations [@Las06], it is still meaningful to use phenomenological models which contain the main features of QCD, once the model parameters are calibrated and constrained by as many observables as possible. Here, we perform such a consistent, parameter-free dynamical calculation for the study of the correlations between light antiquarks, and investigate the possible existence of the exotic meson, $ss-[\bar{u}\bar{d}]$, which has been recently conjectured by Selem and Wilczek [@Sel06]. For completeness, we have also analyzed the $cc \bar u \bar d$ and $bb \bar u \bar d$ systems.
The present study has been done within the framework of a constituent quark model which has been successfully applied to study the baryon spectra and the baryon-baryon interaction [@Rep05]. This model has been generalized to include also strange ($s$), charm ($c$), and beauty ($b$) flavors, and it has also been shown to give a reasonable description of the meson spectra [@Vij05]. The description of experimental data gets improved when four-quark $(qq\bar q \bar q)$ components are also considered [@Vij05b; @Vij06]. The model parameters have been strongly constrained by the study of different hadron observables, what represents an advancing feature compared to studies based on models designed [*adhoc*]{} for a particular problem.
The model is based on the assumption that the $u, d$ and $s$ constituent quarks acquire their masses due to the spontaneous breaking of the original $SU(3)_{L}\otimes SU(3)_{R}$ chiral symmetry at some momentum scale, which is one of the most important nonperturbative phenomena for low energy hadron structure. In this domain of momenta quarks are quasi-particles with constituent masses interacting through scalar (sigmas, OSE) and pseudoscalar (pions, OPE; kaons, OKE; and etas, OEE) boson-exchange potentials. Note that for the case of heavy quarks, $c$ and $b$, boson-exchange potentials are not present in the model [@Vij06], since chiral symmetry is badly broken already at the level of the current quark masses. Beyond the chiral symmetry breaking scale one expects the dynamics being governed by QCD perturbative effects. They are taken into account through the one-gluon-exchange (OGE) potential, a standard color Fermi-Breit interaction. Finally, any model imitating QCD should incorporate confinement (CON). Lattice calculations in the quenched approximation for heavy quarks show that the confining interaction is linearly dependent on the interquark distance. The presence of sea quarks, apart from valence quarks (unquenched approximation), suggests a screening effect on the potential when increasing the interquark distance. Creation of light-quark pairs out of vacuum in between the quarks becomes energetically preferable, resulting in a complete screening of quark color charges at large distances. String breaking has been definitively confirmed through lattice calculations [@SESAM] in coincidence with the quite rapid crossover from a linear rising to a flat potential well established in $SU(2)$ Yang-Mills theories [@Est99]. Explicit expressions for the interaction potentials derived from the nonrelativistic reduction of the Lagrangian density in the static approximation, and a more detailed discussion of the model can be found in Ref. [@Vij05].
For the description of the most general $QQ\bar{u}\bar{d}$ ($Q=s, c,$ or $b$) system we introduce the Jacobi coordinates, $$\begin{aligned}
\text{ \ \ \ \ \ \ \ \ \ \ }\vec{x} &=&\vec{r}_{1}-
\vec{r}_{2},
\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\vec{y
}=\vec{r}_{3}-\vec{r}_{4}, \nonumber \\
\vec{z} &=&\frac{m_{1}\vec{r}_{1}+m_{2}\vec{
r}_{2}}{m_{1}+m_{2}}-\frac{m_{3}\vec{r}_{3}+m_{4}\vec{r
}_{4}}{m_{3}+m_{4}},
\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ }\vec{R}=
\frac{\sum m_{i}\vec{r}_{i}}{\sum m_{i}}, \end{aligned}$$ where $1$ and $2$ ($3$ and $4$) stand for quarks (antiquarks). The ground state energy of the four-body problem can be estimated by a variational method using a trial wave function that includes all possible color-flavor-spin components relevant to a given configuration. For each component, $\mid \phi_i>$, such a basis wave function will be a tensor product of color ($c_i$), flavor ($f_i$), spin ($\chi_i$) and spatial ($R_i$) parts, $$\mid \phi_i>=\mid c_i(1234)> \otimes \mid f_i(1234)> \otimes
\mid \chi _i(1234)> \otimes \mid R_i(1234)>.$$ The most general spatial wave function can be expressed as a combination of six scalar quantities, $$\mid R_i(1234)>=R_i(\vec x^{\, 2},\vec y^{\, 2},\vec z^{\, 2},
\vec{x}\cdot\vec{y},\, \vec{x}\cdot\vec{z},\,
\vec{y}\cdot\vec{z}).$$ The variational spatial wave function is taken to be a linear combination of generalized Gaussians, $$\mid R_i(1234)>=
\sum_{j=1}^{n} \beta_{i}^{(j)} R_i^{(j)}=
\sum_{j=1}^{n} \beta_{i}^{(j)}
e^{-a^{(j)}_i \vec x^{\,2}-b^{(j)}_i \vec y^{\,2}-c^{(j)}_i \vec
z^{\,2}-d^{(j)}_i \vec x\cdot \vec y -e^{(j)}_i \vec
x\cdot \vec z -f^{(j)}_i \vec y\cdot \vec z},
\label{wave}$$ where $n$ is the number of terms to expand the spatial wave function of each color-flavor-spin component, and $a^{(j)}_i, b^{(j)}_i, ..., f^{(j)}_i$ are the variational parameters.
With respect to the color wave function, $\mid c_i(1234)>$, one can couple the two quarks $(1,2)$ and the two antiquarks $(3,4)$ to a color singlet state in different ways:
$$\begin{aligned}
&\mid &1_{13},1_{24}> \,\,\, , \,\,\, \mid 8_{13},8_{24}> \,\, ; \label{eq5} \\
&\mid &1_{14},1_{23}> \,\,\, , \,\,\, \mid 8_{14},8_{23}> \,\, ; \label{eq6} \\
&\mid &\overline{3}_{12},3_{34}> \,\,\, , \label{eq7}
\,\,\, \mid 6_{12},\overline{6}_{34}> \,\, .\end{aligned}$$
The couplings in Eqs. (\[eq5\]) and (\[eq6\]) are convenient for asymptotic meson-meson channels (or meson-meson molecules) while those in Eq. (\[eq7\]) are more appropriate for tetraquark bound states. With our choice of the Jacobi coordinates the color basis in Eq. (\[eq7\]) is more suitable to deal with the Pauli principle in an easier way.
The spin part can be written as,
$$\mid \chi_i(1234)> =\left[ (12)_{S_{12}}(34)_{S_{34}}\right] _{S},$$
where the spin of the two quarks is coupled to $S_{12}$ and that of the antiquarks to $S_{34}$.
Concerning the flavor part, $\mid f_i(1234)>$, since the heavy quarks (those with flavor $s$, $c$ or $b$) have isospin zero, they do not contribute to the total isospin. Therefore one can classify the tetraquark wave function by the isospin of the light quarks $I=0,1$. Taking into account all degrees of freedom, the Pauli principle must be satisfied for each subsystem of identical quarks (antiquarks). It restricts the quantum numbers of the basis states, that justifies to use the $[(QQ)(\bar u \bar d)]$ coupling.
Using the wave functions described above, we search for a variational solution for the Hamiltonian. The color, flavor and spin parts are integrated out and the coefficients $\beta_i^{(j)}$ of the spatial wave function are obtained by solving the system of linear equations, $$\sum_i \sum_{j=1}^n \beta_i^{(j)}
\, [\langle R_{i'}^{(k)}|\,H\,|R_i^{(j)}
\rangle - E\,\langle
R_{i'}^{(k)}|R_i^{(j)}\rangle \delta_{i,i'} ] = 0
\qquad \qquad {\rm for\,\, all}\,\, k,i',$$ once the eigenvalues $E(a^{(j)}_i, b^{(j)}_i, ..., f^{(j)}_i)$ are obtained by a minimization procedure with respect to the variational parameters. The stable tetraquark states are identified by comparing the obtained eigenvalues with the corresponding physical thresholds. If they are above the threshold they would be very broad objects, very hard to detect experimentally.
In a realistic model tetraquarks will not overpopulate the meson spectra, in fact they may complement two-quark components and, indeed, they seem to be necessary in order to understand the rich meson phenomenology [@Vij05b; @Vij06]. This is due, on the one hand, to the constituent mass of the quarks, and on the other one, to the finite spectra generated by screened confining potentials [@Vij03]. Only positive parity tetraquark states, those that do not need internal orbital angular momentum between the constituents, may appear in the low-energy region of the meson spectra and they could mix with $q\bar q $ states with the same quantum numbers. Negative parity four-quark states need a unit of orbital angular momentum what means an average excitation energy of 800$-$900 MeV [@Nir06]. These ideas have been recently used to explain the abnormal number of low-energy scalar-isoscalar mesons [@Vij05b] and also the unexpected low masses of positive parity ($0^+$ and $1^+$) open-charm mesons [@Vij06]. They are perfect examples of the way how the enlargement of the Fock space may help in the understanding of meson phenomenology. As explained in these works, only those states with exotic quantum numbers may appear as pure four-quark resonances on the meson spectra. Unfortunately, the present uncertainties on the experimental data concerning exotic channels prevents, for the moment, to extract a definitive conclusion about its existence [@Szc03].
Let us concentrate on the particular meson state, $ss - [\bar{u}\bar{d}]$, conjectured in Ref. [@Sel06]. It has the property of the $[\bar{u}\bar{d}]$ subsystem being an [*antidiquark*]{} state, which means the two antiquarks are in a color (${\bf 3}_c$), flavor ($I_{[\bar{u}\bar{d}]}=0$), and spin ($S_{[\bar{u}\bar{d}]}=0$) antisymmetric state. It requires a completely symmetric radial wave function for the two antiquarks to satisfy the Fermi statistics. As antidiquark component $[\bar u\bar d]$ should be in a relative $S-$wave, one can neglect the crossing terms in the trial radial wave function, those depending on the scalar product of different Jacobi coordinates in Eq. (\[wave\]). In order to obtain a color singlet wave function, the two $s$ quarks must be in a color antisymmetric, ${\bf \bar 3}_c$, state. Being flavor symmetric, the corresponding spin wave function may be either in (i) an antisymmetric, $S_{ss}=0$, state that would require the anti-natural radial antisymmetric wave function to describe the ground state of the system, or in (ii) a symmetric spin state, $S_{ss}=1$, that would combine with a natural symmetric radial wave function. Therefore, the conjectured meson with the presence of the antidiquark would be described by a $J^\pi=1^+$ $(L=0,S=1)$ state with isospin $I=0$. We can summarize the quantum numbers of the antidiquark component of the $ss\bar u \bar d$ system in the following way, $$|[{\bf 3}_c, S=0, I=0]_{\bar{u}\bar{d}},
|[{\bf \bar 3}_c, S=1, I=0]_{ss}; (S=1,I=0) \rangle.
\label{diqu}$$ A full calculation of the $J^\pi=1^+$ $(L=0,S=1)$ state with strangeness $-2$ would require also to consider other vectors in the Hilbert space. In particular, the same state could also be constructed from a different vector, where the two antiquarks would not be an antidiquark state while it still has a completely symmetric radial wave function,
$$|[{\bf \bar 6}_c, S=1, I=0]_{\bar{u}\bar{d}},
|[{\bf 6}_c, S=0, I=0]_{ss}, (S=1,I=0) \rangle.
\label{nodiqu}$$
This vector, which will be referred to as the [*nondiquark*]{} component of the $ss \bar u \bar d$ system, should be considered in the calculation of the four-quark state without requiring the antidiquark configuration, whereas, it will not be included if only the antidiquark configuration is imposed. For the [*full*]{} calculation, both the antidiquark and nondiquark configurations will be included with the corresponding configuration mixing.
In Table \[t1\] we present our results for the antidiquark configuration, nondiquark configuration, and full calculation for the $ss \bar u \bar d$ system. The same calculation has been repeated for the $cc \bar u \bar d$ and $bb \bar u \bar d$ systems, and the results are presented in Tables \[t2\] and \[t3\], respectively.
The first important conclusion that can be extracted from the results of Tables \[t1\], \[t2\] and \[t3\] is that the energy of the antidiquark configuration is always the lowest. It is interesting to note how the pseudoscalar force acting between the light quarks is responsible for that, since the results for the antidiquark and nondiquark configurations are almost degenerate if only the confinement and one-gluon exchange are retained. The reason for this stems on the different symmetry for both components in color-spin and flavor-spin spaces. While both are symmetric in color-spin space, the antidiquark (nondiquark) component is symmetric (antisymmetric) in flavor-spin space. Therefore, if strong diquark correlations were dictated by QCD for light quarks, the dynamical explanation could not rely on the simple one-gluon exchange dynamics, but it would need meson-exchange forces between the constituent quarks. The similar effect has been also observed in the case of baryon spectra, where pseudoscalar meson exchanges between the constituent quarks are able to revert the relative position in the energy spectra of the nucleon Roper resonance, with a dominant flavor-spin symmetric wave function, with respect to negative parity states, with a flavor-spin antisymmetric wave function [@Gar01]. It is also interesting to notice that the antidiquark and nondiquark components are not exactly degenerate when only the confining interaction is taken into account. This can be easily understood by looking at Table \[t4\], where we present the contribution of the interaction between $QQ$, $V_{12}$, $\bar u\bar d$, $V_{34}$, and $Q \bar n$ ($n=u,d$), $V_{13}$, for the $QQ\bar u \bar d$ system as a function of the mass of $Q$, $m_Q$. The minimization procedure modifies the variational parameter $a^{(j)}_i$ in Eq. (\[wave\]) for the $\vec x^{\, 2}$ coordinate due to the smaller size of the $QQ$ subsystem when the mass $m_Q$ increases. As a consequence it gives a smaller contribution to the energy of the system. In other words, the dependence on the mass of the quark is introduced into the calculation through the variational parameters.
As predicted by Selem and Wilczek [@Sel06], the mixing between the antidiquark and nondiquark components of the wave function diminishes when increasing the mass of the heavy quarks (see Table \[t5\]), in such a way that for the $b$ quark case the nondiquark component gives almost no contribution to the ground state energy of the system. This effect, interpreted as the less capacity of the spin of a heavy quark to disrupt the correlation of the diquark (antidiquark), comes from the $1/(m_im_j)$ ($m_{i,j}$: constituent quark masses) dependence of the one-gluon exchange interaction, which is responsible for the mixing between these components. Therefore, the mixing decreases with increasing the mass of the heavy quark. A similar evidence, the less capacity of the spin of the heavy quarks to disrupt the system, has also been observed in the spin-orbit splitting of the $\Lambda-, \Lambda^+_c-$ and $\Lambda_b-$hypernuclei [@Tsu98], where $s, c$ and $b$ quarks exclusively carry the total spin of the $\Lambda, \Lambda^+_c$ and $\Lambda_b$, respectively, and $u$ and $d$ quarks are coupled to a isospin zero and spin zero diquark in each baryon.
Regarding the possibility of observing these systems, the results obtained are always far above their corresponding lowest two-meson thresholds, as indicated in Tables \[t1\], \[t2\], and \[t3\], being the only exception the $bb\bar u\bar d$ system.
Experimentally the possibility to detect a $QQ \bar u \bar d$ meson relies on two different aspects. First of all the rate of production of $QQ \bar u \bar d$, and second the existence of decay modes that can provide a unique signature. For the production at hadronic or $e^+e^-$ colliders one needs to produce two pairs of charm or bottom quarks. These pairs should be close spatially and the quarks within each quark-antiquark pair should have small relative momenta in order to combine in a two-quark, $cc$ or $bb$, state. Finally, these two-quark states should pick up an antidiquark $[\bar u \bar d]$ to form the desired $QQ -[\bar u \bar d]$ system. The production rates for the case of charm quarks have been estimated in Refs. [@Gel03] and [@Moi96]. The signal of the strong antidiquark correlation would come from decay channels preferring to keep the antidiquark structure. So, instead decaying by splitting into a two-meson system, it would proceed through a two baryon system as it would be $\overline{N} \Xi_{cc}$ for the charm case and $\overline{N} \Xi_{bb}$ for the bottom case. The absence of a dynamical enhancement of the antidiquark component would open the decay into two mesons. For the $ss \bar u \bar d$ system, if the antidiquark component is strong enough so as to force it to decay into a two baryon system via $(ss-[\bar u \bar d]) \to \overline N \Xi$, one can expect a natural decay in an $S-$wave, which needs a $J^\pi=1^-$ state for the system. As the energy excitation for a unit of orbital angular momentum costs about 800$-$900 MeV [@Nir06], this would make the system to be in the continuum above the $\overline{N} \Xi$ threshold, therefore being broad and difficult to detect. The dynamical enhancement of the diquark component is one of the possible reasons to explain the decay of the $\Lambda(3/2^-)(1520)$ 45% of the time to $N\overline{K}$ channels in order to retain the diquark structure of the $\Lambda$ inside the $N$. Finally, let us remark that we only support the possible existence of $QQ\bar u \bar d$ states for $Q=b$ and with more uncertainty for $Q=c$, but never for $Q=s$. In the bottom case the predicted state should be very narrow and easy to observe (if produced) since it is far below the two-meson and two-baryon physical thresholds.
To summarize, we have made a dynamical, parameter-free, calculation for the $QQ\bar u \bar d$ system $(Q=s,c,b)$ within a [*realistic*]{} constituent quark model. We have found that the antidiquark configuration of these systems always gives a lower energy than the nondiquark configuration. The mixing between the antidiquark and nondiquark states is due to the one-gluon exchange potential, and because of its $1/(m_i m_j)$ dependence, it decreases when increasing the heavy quark mass. As a consequence one can expect that the conjectured mesons $QQ-[\bar u
\bar d]$ could be stable for $Q=b$, but we do not find any reason why these systems should be bound for $Q=s$. Moreover, for light quarks, $Q=s$, there is no dynamical reason why the antidiquark component should be favored compared to the nondiquark one (within an uncorrelated quark model). If the existence of any such systems with two light quarks, $Q=s$, and therefore with a strongly correlated light antidiquark would be confirmed via the postulated baryon-antibaryon final channel, it would mean that a dynamical mechanism responsible for the correlations has not been considered in our simple realizations of models for QCD (based on uncorrelated quarks), and the question for the existence of exotic systems, such as the pentaquark, should be addressed in a corresponding manner. On the other hand, the present results may imply that the pentaquark with a heavy $\bar{b}$ quark, $[u d][u d]\bar{b}$ with a negative parity, may have a chance to be stable, if the predicted repulsion between diquarks is not strong enough to destroy the system. This has a merit of further study within the same model.
Acknowledgements
================
This work has been partially funded by Ministerio de Ciencia y Tecnología under Contract No. FPA2004-05616, by Junta de Castilla y León under Contract No. SA-104/04, and by Generalitat Valenciana under Contract No. GV05/276.
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Antidiquark Nondiquark Full
-- ------------------------------------------------------------------ ------------- ------------ ---------------- --
$V_{CON}+V_{OGE}+V_{OPE}+V_{OSE}+V_{OEE}$ 1705 1974 1696 (97.59 %)
$V_{CON}+V_{OGE}+V_{OPE}+V_{OSE}$ 1644 1965
$V_{CON}+V_{OGE}+V_{OPE}$ 1843 2105
$V_{CON}+V_{OGE}$ 2092 2083
$V_{CON}$ 2520 2479
$\overline{N}\, \Xi$ threshold 2257
$\overline{K^0}\,{K^*}^-$ or $\overline{{K^*}^0}\,K^-$ threshold 1386$-$1390
: Calculated energies and physical thresholds, in MeV, for the $ss \bar u \bar d$ system. “Full” stands for the results calculated including both configurations, antidiquark and nondiquark. Inside the brackets is the percentage of the antidiquark component in the full calculation. \[tab1\] []{data-label="t1"}
Antidiquark Nondiquark Full
-- -------------------------------------------- ------------- ------------ ---------------- --
$V_{CON}+V_{OGE}+V_{OPE}+V_{OSE}+V_{OEE}$ 3929 4207 3927 (99.48 %)
$V_{CON}+V_{OGE}+V_{OPE}+V_{OSE}$ 3858 4210
$V_{CON}+V_{OGE}+V_{OPE}$ 3906 4229
$V_{CON}+V_{OGE}$ 4169 4197
$V_{CON}$ 4631 4644
$\overline{N}\, \Xi_{cc}$ threshold 4460
$D^+\,{D^*}^0$ or ${D^*}^+\,D^0$ threshold 3875$-$3876
: Same as Table \[t1\] for the $cc \bar u \bar d$ system.[]{data-label="t2"}
Antidiquark Nondiquark Full
-- ------------------------------------------------------------------ ------------- ------------ ----------------- --
$V_{CON}+V_{OGE}+V_{OPE}+V_{OSE}+V_{OEE}$ 10426 10797 10426 (99.95 %)
$V_{CON}+V_{OGE}+V_{OPE}+V_{OSE}$ 10355 10801
$V_{CON}+V_{OGE}+V_{OPE}$ 10403 10822
$V_{CON}+V_{OGE}$ 10673 10787
$V_{CON}$ 11154 11234
$\overline{N}\, \Xi_{bb}$ threshold
$\overline{{B^*}^0}\,B^-$ or $\overline{B^0}\,{B^*}^-$ threshold 10604
: Same as Table \[t1\] for the $bb \bar u \bar d$ system.[]{data-label="t3"}
-- ------------ ------------- ------------ ------------- ------------ --
Antidiquark Nondiquark Antidiquark Nondiquark
$<V_{12}>$ +458 $-$263 +241 $-$176
$<V_{34}>$ +527 $-$284 +513 $-$264
$<V_{13}>$ +259 +633 +225 +536
-- ------------ ------------- ------------ ------------- ------------ --
: Expectation value, in MeV, of different contributions of the confining interaction for the different components of the $QQ\bar u \bar d$ system and for two different values of the mass of the quark in the two-quark subsystem.[]{data-label="t4"}
$m_Q$ $M_{QQ\bar u \bar d}$ P($QQ-[\bar u \bar d]$)
-- ------- ----------------------- ------------------------- --
313 1431 96.09
555 1696 97.59
755 1980 98.18
1255 2815 99.11
1555 3352 99.37
: Probability, in %, of the antidiquark component, $QQ-[\bar u \bar d]$, as a function of the mass of the quark, in MeV, in the two-quark subsystem. We also give the mass of the $QQ\bar u \bar d$ system in MeV.[]{data-label="t5"}
|
---
abstract: 'After discussion of observational constraints on the nature of the MHD wind coupling between the Crab Pulsar and the Crab Nebula, the theory of transverse relativistic shock structure is reviewed and applied to the interpretation of the wisps in the Nebula as the manifestation of the distributed wind termination shock structure, energetically dominated by heavy ions, accelerated in the rotational equator of the pulsar to energies comparable to the total voltage across the pulsar’s open field lines and carrying a current comparable to the Goldreich- Julian current. New results on the variability of the shock structure are presented, which show that the gyrating ion bunches emit outwardly traveling finite amplitude compressional waves, in agreement with recent ground based observations. The implications of the theory for X-ray, $\gamma$-ray and high energy neutrino emission are briefly discussed, as are the problems of low magnetic energy density in the upstream wind and the origin of the Nebular radio emission. A brief discussion of other plerions leads to the conclusion that much more detailed observations are needed before these systems can be modeled with the same sophistication as can be done for the Crab Nebula.'
author:
- Jonathan Arons
title: ON THE COUPLING OF ROTATION POWERED PULSARS TO PLERIONIC NEBULAE
---
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Introduction
============
Pulsars and soft gamma ray repeaters (SGRs), known to be neutron stars with varying levels of confidence, are unresolved stellar sources. Rotation powered pulsars lose most of their energy in an invisible form; SGRs may also lose much of their energy in the same manner. Sometimes, this energy loss leaves its signature in the form of unpulsed nonthermal emission from surrounding nebulae - the positional identification of such nebulae with SGRs is the main argument for identifying the SGR phenomenon with neutron stars (Kulkarni, this meeting). The nebular nonthermal radiation is synchrotron and inverse Compton emission from relativistic particles and fields injected by the embedded pulsar.
The Crab Nebula is the most famous example of a plerion which forms a box calorimeter around its pulsar. Other well known examples include Vela-X (Bock [*et al.*]{} 1998) and 3C58 ([*e.g.*]{} Helfand [*et al.*]{} 1994), although in this latter nebula no pulsar has been specifically identified. There are about 15 plerions identified from radio images, while recent advances in X-ray imaging have found clear evidence for X-ray nebulae (presumably synchrotron nebulae) around pulsars (Harrus [*et al.*]{} 1996).
When the pulsar’s energy loss is relatively small and the pulsar’s space velocity is high, calorimetric nebulae enclosing the pulsar change their morphology and assume a cometary form. This change occurs when the space velocity of the pulsar exceeds the expansion velocity the plerion would have if the pulsar were stationary with respect to the interstellar medium. If the pulsar’s true age equals its characteristic age, then one expects to see cometary morphology or other distortions when $$t_{char} \equiv \frac{P}{2 \dot{P}} > \frac{10^4}{v_{100}^{5/3} P^{2/3}}
\left( \frac{I_{45}}{n_{ISM}} \right)^{1/3} \; {\rm years} ;
\label{eq:comet-age}$$ younger objects should be well embedded inside their nebulae. Here the rotation period $P$ is in seconds, $I_{45}$ is the moment of inertia measured in units of $10^{45}$ cgs, $v_{100}$ is the pulsar’s space velocity in units of 100 km/s, and the interstellar particle number density is in units of cm$^{-3}$. A subset of such “plerionic” nebulae forms within binary systems, when the pulsar’s outflow energy interacts with either the companion star or with the mass loss from that star (Arons and Tavani 1993).
The calorimetric nebulae surrounding young pulsars are of particular interest, since understanding the physics of nebular excitation in these systems can yield reasonably unambiguous constraints on the physics of the underlying pulsar, without the geometric confusion introduced by the more complex flows around older pulsars. The Crab Nebula is still the only system in which the quality and quantity of observational information enables meaningful physical progress at a fundamental level.
The Crab Nebula: The Interaction of PSR 0531+21 with the World
==============================================================
This remnant of SN1054 emits nebular radio, IR, optical, X- and $\gamma$-rays ($\varepsilon < 100$ MeV), all of which is synchrotron emission; higher energy photons probably are the Inverse Compton emission of the same electrons (and positrons) that emit the synchrotron radiation (de Jager and Harding 1992). The synchrotron lifetimes of the particles which emit photons at near-IR and shorter wavelengths are less than the age of the nebula, thus requiring a continuous source of power, a requirement fulfilled by the central pulsar, whose spin down luminosity $\dot{E}_R \approx 5 \times 10^{38}$ ergs/s exceeds the total nebular luminosity (primarily in X-rays and $\gamma$-rays) by about an order of magnitude - the pulsar has $\sim$ 10% efficiency in converting rotational energy loss into instantaneous particle acceleration power in the Nebula. Furthermore, this power gets delivered to particles whose maximum energy, as judged by the particle energies needed to radiate 100 MeV synchrotron photons, is $ \geq 10^{15.5}$ eV, comparable to the total voltage drop across the pulsar’s open field lines. The radiative lifetime of these 3 PeV electrons and positrons is quite short, $\sim $ months; X-ray emitters lose energy in a few years. Thus the high energy nebular emission provides a window into the pulsar’s energetics right now, demanding $10^{38.5} - 10^{39}$ electrons and positrons per second from the pulsar, assuming the particles are accelerated only once. The shrinkage of the nebular image with increasing photon energy shows that acceleration indeed does occur only once, and must be substantially complete at radii no larger than the X-ray torus seen in ROSAT images, [*i.e.*]{}, at projected radii less than 0.3 - 0.6 pc from the pulsar. The gamma ray source may well be smaller than the X-ray torus, as is suggested by the hard X-ray image of Pelling [*et al.*]{} (1987). If the acceleration process varies on time scales comparable to or longer than the radiative lifetime of the $\gamma$-rays (for example), then the gamma ray flux from the nebula will vary, since the short lived particles’ energy density and emissivity then follows the time variable acceleration physics. EGRET data have suggested that some gamma ray variability does occur on a time scale of months to years (de Jager [*et al.*]{} 1996).
By contrast, the radio and far IR emitting particles have synchrotron lifetimes greater than the nebular age. Therefore, their emission depends on the history of the nebula, representing a convolution of the pulsar’s efficiency as a provider of accelerated particle energy, particle number and magnetic field with the expansion history of the nonthermal bag. The expansion history depends on the density and geometry of the external medium that confines the relativistic particles and fields. Averaging over the nebula’s history, the lower energy particles in the nebula show that the pulsar has provided roughly $10^{40}$ synchrotron radiating particles/s. The discrepancy between this average injection rate and the instantaneous injection rate to the X-ray torus has not been resolved - the pulsar might have been a substantially more active provider of relativistic particles in earlier epochs, or it might provide an additional source of low energy particles today which is not associated with the X-ray torus. The nonthermal emission requires nonthermal nebular distributions of particles in energy space - “power laws”, in the simplest representation of the data.
Images are essential to understanding the physics (radio, optical, UV, X-Ray; hard X-ray and $\gamma$-ray if we could get them): for starters, the shrinkage of the nebular image with increasing photon energy shows that the acceleration site is at or near the pulsar, not distributed throughout the nebula, at least for the higher energy particles. Detailed imagery (radio, near IR, visual, soft X-ray) reveals the fine structure of the region where the pulsar rotational energy loss appears to be delivered to the nebula. As has been known since their discovery by Lampland (1921), the NE-SW direction from the pulsar (the short axis of Nebula) shows “wisps”, time variable (months to days) surface brightness enhancements which are always present, between 5 and 30 arc seconds from the pulsar (0.05 to 0.3 pc projected radius). If one assumes the nebula to be “optically thick” to the relativistic ram pressure $\dot{E}_R /4 \pi r^2 c$ of the unseen outflow, balancing this ram pressure with the total nebular pressure ($p_{neb} \sim 10^{-8}$ dynes/cm$^2$) yields a termination radius for the unseen outflow right in the middle of the wisp region (Rees and Gunn 1974), thus suggesting very strongly that the wisps are an observational signature of the coupling between the pulsar and the nebula. Recent ground based optical studies have shown these variations appear to be waves in brightness traveling outwards with speed $\leq 0.5c$ (Tanvir [*et al*]{} 1997), a result consistent with the preliminary results of the ongoing HST imaging campaign (J. Hester and J. Graham, personal communications). These variations are not correlated with timing glitches and other rotational noise features of the pulsar, thus suggesting the variability is an intrinsic feature of the mechanism which couples the unseen pulsar energy outflow to the nebula, rather than being a passive consequence of variability in the pulsar’s spindown.
The most efficient hypothesis is to assume the wisp region is the particle acceleration zone, in addition to being the region where the pulsar outflow energy becomes coupled to the nebular plasma. In this context “acceleration” means the conversion of the outflow energy, whatever it is, into the spectra of electrons and positrons which are injected into the nebula. As in the acceleration of cosmic rays by supernova remnant shocks rather than by supernovae themselves, such a hypothesis avoids the problem of the adiabatic losses which plague mechanisms which rely on accelerating the observed particle spectra within the pulsar’s magnetosphere (e.g., Tademaru 1973). Granted that the wisp variability time scale is months, that the $\gamma$-ray emitting particles have radiative loss times on the same order, and that EGRET may well have seen some gamma ray variability on month to year time scales, I am much attracted by the hypothesis that the variable wisps are the direct signature of the acceleration of particles to gamma ray emitting energies, and thus are the site of the high energy electron and positron acceleration in the Crab Nebula.
Wind Outflow From Crab Pulsar
=============================
Figure \[fig:cartoon\] shows a cartoon of the flow geometry near the Crab Pulsar. The X-ray morphology implies an outflowing “disk wind” in the equator, possibly associated with a corrugated equatorial return current sheet, which terminates at or within the X-ray torus - in most interpretations, the termination point identified depends on which feature the interpreter adopts as a termination shock, implicitly assumed to be infinitesimally thin on the scales of observational angular resolution. The polar outflow appears in the HST imagery as arcs both concave and convex toward the pulsar (the thin wisp and the second, time stationary strand of wisp 1), interpreted as possible shocks in a polar “jet” by Hester [*et al*]{} (1995). Within the context of the fully MHD wind theories advanced by Kennel and Coroniti (1984a,b), hoop stress in the toroidal magnetic field at high latitude may compress the flow axially, contributing to the polar X-ray enhancements - the bolometric synchrotron surface brightness scales $\propto B^4$ (Woltjer 1958, Gallant and Arons 1994). If so, the arc features identified by Hester [*et al.*]{} might be interpretable as parts of a polar termination shock’s structure, in a manner similar to the theory of the equatorial wind’s termination described below. However, such speculations are for future investigation, and will not be discussed further here.
It seems hard to avoid the conclusion that the outflowing energy feeding the X-ray torus has the character of a relativistic MHD wind. The most widely accepted models of the electrodynamics of pulsars’ polar caps require an electric current along polar field lines with density $J_\parallel \approx B/P$, which yields a total current $I_\parallel = 2 \mu \Omega_*^2 /c$ - here $\Omega_* = 2\pi /P$ and $\mu$ is the magnetic moment. If the polar current couples to the wind/nebula (an open circuit on light cylinder scales), the current induced $B$ makes a considerable contribution to pulsar torque, a theoretical possibility supported by the fact that observed torques don’t depend significantly on obliquity (Lyne and Manchester 1988). In such open circuited models, the number of electric current carrying particles shot into the nebula per unit time is $\dot{N}_R = I_\parallel /Ze = 2 \mu \Omega_*^2 /Zec$, where $Z$ is the atomic number of the current carrying particles. Z=1 if the polar current is electrons, as is the case when the obliquity between the magnetic moment and the rotation axis is less than $90^o$, the geometry believed to be appropriate for the Crab Pulsar ([*e.g.*]{} Romani 1996).
Note that such models require another particle outflow from the neutron star to supply the required return current! For open circuited models, this outflow probably is particles of the opposite charge sign extracted from an “auroral ring” around the polar flux tube, [*ejected along the field lines that map into the rotational equator of the PSR*]{}, there to flow out in the (corrugated) equatorial current sheet. The return current in open circuited theory is still a cartoon - its dynamics has yet to be explored. The results of shock theory applied to the equatorial wind strongly suggest this return current in the Crab to be heavy ions, accelerated to the full potential drop available (see below).
For the Crab, the number of electrons and positrons required to feed the X-ray source tells us that at least in the equatorial wind, the pulsar’s loss rate in pairs is $$\dot{N}_\pm \sim 10^{38.5-39} \; {\rm s}^{-1} \gg
\dot{N}_R ({\rm Crab}) \simeq 10^{34.2} \; {\rm s}^{-1}.
\label{eq:pair-lossrate}$$ It is well known that in pulsar flows, a total outflow rate of quasi-neutral plasma in excess of $\dot{N}_R$ is a necessary condition for applicability of relativistic MHD as the underlying theory (Arons 1979). Therefore, the simplest theory for the equatorial outflow is that the spindown energy loss is carried by a relativistic MHD wind, with electrons and positrons as the main constituents by number. This wind might also contain a minority population of heavy ions, which may or may not be energetically significant, and may also carry part of the energy flow in the Poynting flux of the wound up electromagnetic fields.
The rotational energy lost in nonradiative fields, pairs, heavy ions carried by a MHD wind through the radiationless cavity at $r < 0.1$ pc from the pulsar has its energy conservation described by $$\begin{aligned}
\dot{E}_R & = & r^2 \int d\Omega \left\{ \frac{c}{4\pi} {\bf E} \times {\bf B}
+ {\bf v}_{wind} \gamma_{wind} \left[(n_+ + n_- ) m_\pm c^2
+ n_i m_i c^2 \right] \right\} \cdot {\bf \hat{r}} \nonumber \\
& = & c\beta_{wind} \gamma_{wind} \dot{N}_i m_i c^2
\left( 1 + \frac{m_\pm}{m_i} \frac{n_+ + n_-}{n_i}\right) (1 + \sigma),
\label{eq:energy-cons}\end{aligned}$$ with the solid angle integration carried out over the sector of interest - an equatorial sector with total latitudinal opening angle on the order of $20^o$, if the wind feeding the observed X-ray torus has straight streamlines. The parameters $\bullet \; m_i\dot{N}_i$ = mass loss rate in ions $\bullet \; (n_+ + n_-)/n_i = \dot{N}_\pm / \dot{N}_i$ = ratio of pair number loss rate to ion number loss rate $\bullet \; \gamma_1$ = the bulk flow Lorentz factor (or the velocity $v_1$) $\bullet \; \sigma$ = ratio of Poynting flux to kinetic energy flux in the wind characterize the wind’s properties.
MHD wind theory with $\sigma \ll 1$ gives a “natural” explanation of the Crab Nebula’s dynamics[^1](Rees and Gunn 1974, Kennel and Coroniti 1984): $\bullet$ The deceleration of the post shock, low $\sigma$ pulsar wind ($v_r \propto 1/r^2$) compresses the magnetic field ($B \propto r$) until the magnetic energy reaches equipartition with the shocked relativistic plasma energy, whence deceleration and compression ceases - the model neatly explains equipartition as a dynamical effect; $\bullet$ The fit to the observed expansion velocity requires $\sigma << 1$ (caveat - Kennel and Coroniti neglected possible inertial loading by the filaments); $\bullet$ If the wind termination shock is assumed to create power law distributions of pairs, with upper and lower cutoffs determined by the jump conditions and the particle spectral index at the shock constrained by the final fit to the data, the global optical, X- and gamma-ray spectrum of the Crab can be reproduced by the model, once synchrotron cooling in the flow is properly incorporated (Kennel and Coroniti 1984b).
But, radio emission from the Nebula is entirely left out! The inferred particle injection rate ([*now*]{}, since rapid synchrotron losses make the X-rays a calorimeter for the current injection rate) is an order of magnitude smaller than the rate of injection of radio emitting particles, averaged over the life of the Nebula. Thus, the main stored component of the relativistic energy in the Nebula was neglected. The non-spherical morphology was not quantitatively addressed. Conceivably, the problem of radio emitting particle injection is related to the strong latitudinal asymmetries revealed by HST and other imaging.
Physics of Relativistic Shock Structure/Dissipation/Acceleration
================================================================
As remarked above, dynamic pressure balance puts the termination of the pulsar’s unseen outflow at $r_s \sim 0.1 - 0.2$ pc (Rees and Gunn 1974). Shock dissipation is the most likely wind termination mechanism in the MHD theory. A shock in a MHD outflow from the Crab pulsar must be transverse: $\angle ({\bf B}, {\bf v}) \approx \pi/2 - 10^{-9} \Rightarrow$ diffusive Fermi acceleration has little relevance to the conversion of flow energy into power law distributions of particles downstream. The phenomenological shock model with the shock regarded as infinitesimally thin and located at $r_s \approx r_{\rm wisp}$ simply requires the conversion of $\sim 10$% of the flow energy into the spectra of pairs with distribution functions immediately downstream of the shock $N_{\rm injected} (\gamma) \propto \gamma^{-s}, \; s \approx 2.2$; it doesn’t explain how the system achieves this efficiency.
My contributions to the subject come partly under this heading. The results, based on linear instability theory, particle-in-cell simulations and a modicum of quasi-linear theory, were mostly published some time ago (Alsop and Arons 1988, Langdon [*et al.*]{} 1988, Hoshino and Arons 1991, Gallant [*et al.*]{} 1992, Hoshino [*et al.*]{} 1992). Those interested in the detailed support of most of the assertions made in this section should consult the papers referenced.
Imagine what happens when the incoming flow “collides” with the magnetic step formed by the shock. The particles [*all*]{} reflect coherently, and start gyrating coherently within the shock front, now considered more realistically as a transition region of finite thickness in the flow. The coherently gyrating particles radiate cyclotron and synchrotron waves with fundamental frequencies $\omega_1 = eB/mc\gamma_1$, as well as large amounts of power at the harmonics $\omega_l = l \omega_1$, including the high harmonics $l \gg 1$. The basic mechanism is azimuthal bunching of the ring distributions in momentum space set up by the coherent reflection from the magnetic step, whose shape is self-consistently maintained by the particle rings in momentum space. The pairs radiate extraordinary modes, with $\omega_1 \geq (eB_1/m_\pm c \gamma_1) \sigma^{-1/2} \sim 10^{-3} \; s^{-1}$. The numerical value assumes the upstream magnetic field to be $B_1 \sim 10^{-4.5}$ Gauss, the upstream flow Lorentz factor to be $\gamma_1 \sim 10^6$ and $\sigma \sim 10^{-2.5}$, all values taken from the Kennel and Coroniti model or from the Gallant and Arons (1994) model described below. Coherent gyration of heavy ions as they encounter the magnetic step excites transverse magnetosonic waves which propagate with properties mainly determined by the pairs (if the pairs are numerically in the majority, as turns out to be the case), with frequencies $\omega \geq ZeB_1/m_i c \gamma_1 \sim 10^{-6.5} \; {\rm s}^{-1}$, a gyration time of months[^2].
Cyclotron reabsorption of the extraordinary modes at the shock’s leading edge thermalizes the pairs to a relativistic Maxwellian distribution with downstream temperature $T_\pm \approx \gamma_1 m_\pm c^2$ - the mean free path for extraordinary mode emission and absorption in the pair plasma is much smaller than the flow scale length, leading to the establishment of local thermodynamic equilibrium for essentially the same reason that the emission and absorption of virtual photons (Coulomb collisions) establishes LTE in a collisional nonrelativistic plasma - in the relativistic case, the pairs and their waves form a local [*hohlraum*]{}. The relativistic cyclotron instability in the ions simply serves to establish a level of electromagnetic fluctuations (corresponding to real photons in this case) at the thermal level far faster than would occur if two body encounters were the only means of creating the fluctuating electromagnetic field. The simulations show that when pair thermalization is complete, the radiation level also corresponds to LTE (in the Rayleigh-Jeans limit, as is expected for these classical investigations).
From the perspective of the pairs, the magnetosonic waves emitted by the more slowly developing relativistic ion cyclotron instability are an external source of energy, which can upset their thermal equilibrium. The simulations show that the magnetosonic waves have a nonthermal spectrum, basically corresponding to $1/f$ noise. These waves are preferentially cyclotron absorbed by the more mobile pairs, first at ion harmonics $l \sim m_i /Z m_\pm$, then, as the pairs gain energy and detune from the high harmonics, from waves in the power law spectrum with successively lower frequencies, until acceleration stops for pairs whose energy equals that of an upstream ion, for which the cyclotron frequency equals that of the ions that drive the acceleration.
Indeed, a simple application of quasi-linear theory to this process (Arons, unpublished) shows that the acceleration rate of an electron or a positron in a spectrum of linearly polarized magnetosonic waves $$U_k = r_{Li} \frac{(\delta B)^2}{4\pi} (kr_{Li})^{-2},$$ the spectrum exhibited by the simulations, is $$\frac{\dot{\gamma}}{\gamma} = \frac{0.017}{n_0^3}
\left(\frac{\delta B}{B} \right)^2 \Omega_{Li}.$$ Here $k$ is the wavenumber, $U_k dk$ is the wave energy density in the interval ($k, k+dk$), $r_{Li} = c/\Omega_{Li}$ is an ion’s Larmor radius and $\Omega_{Li}$ its relativistic Larmor frequency, and $$n_0 = \left(\frac{c^2 + v_A^2}{c_s^2 + v_A^2} \right)^{1/2},$$ is the index of refraction of low frequency, small amplitude magnetosonic waves in the pairs, with $c_s$ the relativstic sound speed in the pairs and $v_A$ the Alfven speed in the relativistically hot pairs. Typically $n_0 \sim \sqrt{2} - \sqrt{3}$. Formal applicability of quasi-linear theory requires $\delta B /B \ll 1$, while the simulations show that $\delta B /B >2$ for parameters of interest. Nevertheless, the simulations show that the acceleration rate is indeed Fermi-like, with $\dot{\gamma} / \gamma \sim \Omega_{Li}$ - quasi-linear theory yields the correct scaling of the rate for the resonant process even when the fluctuation amplitudes are large, although its estimate of the numerical value is less reliable.
Subjected to the nonthermal heating of this resonant absorption process, with losses being simply outflow of pairs from the region where the ions lose their energy to the pairs, the relativistic Maxwellian pairs downstream from the pair shock develop a nonthermal distribution $$N_\pm (\gamma) \propto \gamma^{-2}, \;
\gamma_1 < \gamma < (m_i/Zm_\pm) \gamma_1.$$ For lower energies, the particle spectrum remains that of a relativistic Maxwellian. The efficiency of energy transfer from the ions to the pairs is $$\varepsilon_a = \frac{{\rm nonthermal \; pair \; energy}}
{{\rm total \; upstream \; flow \; energy}} \approx 10 - 20 \% ,$$ a result known solely from simulation. These acceleration results obtain when the ions provide the largest component of the upstream flow energy, and are remarkably like those inferred from application of ideal MHD shock theory to the Crab Nebula, with power law populations of pairs in the post shock flow.
Wisps as Internal Shock Structure
=================================
The physics of these shocks implies a model for the coupling of the equatorial wind outflow from the Crab pulsar. Suppose the equatorial flow to be composed of $e^\pm$ pairs, heavy ions and wound up magnetic fields in an unknown mixture, all flowing out from the pulsar at super-Alfvenic speed. The shock thermalizes the pairs to a relativistic Maxwellian distribution within the leading edge of the shock structure - this thermalization region has radial thickness $\sim r_{L\pm} \sim 10 $ AU, unobservably thin to all but VLBI radio observations. The ions have much larger gyration radius in the compressed magnetic field supported by the shock heated pairs ($r_{Li} \sim 0.3 \; A/Z$ pc). Relativistic cyclotron instability of the gyrating ions generates large amplitude, long wavelength ($\lambda \sim 0.2 /l$ pc, $l \geq 1$), [*compressional*]{}, linearly polarized magnetosonic waves in the heated pairs. Cyclotron absorption of these waves causes gradual nonthermal acceleration of the pairs over a length $\sim$ several ion Larmor radii[^3]. The ions follow a coherent orbit for a couple of gyration cycles, becoming progressively more disorganized as the instability broadens their momentum distribution and drains their energy into magnetosonic waves. However, in the first few cycles of coherent gyrational flow, the turning points in the ion orbits are coherently spaced with separations $\sim r_{Li}$. Since the radial outflow (and inflow) momentum of the gyrating ions must be deposited in the magnetized pairs as the ions gyrate, the turning points correspond to compressions in the magnetic field and pair plasma. Such compressions correspond to surface brightness enhancements separated in radius by the ion Larmor radius scale, with bolometric synchrotron emissivity $\propto B^4$. The compressions also couple the ion momentum to the propagating magnetosonic waves of the pair plasma, since the cyclotron instability makes the ion reflection process time dependent in the ion drift frame. Thus the compressions created by the reflected ions should travel outwards in the pair flow frame with the magnetosonic speed of the nonlinear waves.
The properties of such compressions have a not unreasonable similarity to the observed properties of the wisps, suggesting that the wisps are the observable manifestation of the [*internal structure*]{} of an (energetically) ion dominated shock terminating the equatorial wind from the Crab pulsar. If this hypothesis is correct, the whole shock structure is spread across the sky, turning the Crab Nebula into a laboratory for the relativistic shock physics believed to be central to a wide variety of high energy astrophysical systems.
Gallant and Arons (1994) decided to test the configuration space aspects of the model outlined above by constructing a quantitative [*steady*]{} flow theory, assuming the flow to be confined to a sector of a sphere within latitude $\sim \pm 10^o$ of the pulsar’s rotational equator. The ion flow was modeled as a [*laminar*]{} stream of particles with no momentum dispersion, gyrating in the magnetic field embedded in a shock heated, Maxwellian pair fluid whose flow was modeled as adiabatic - no attempt was made to model either the nonthermal particle acceleration or the variability observed in the simulations, and by construction the model creates compressions which are stationary in space - the pairs flow through these standing waves. This model and its quantitative results, when applied to the I-band snapshot of van den Bergh and Pritchet (1988), are illustrated in Figure \[fig:ga-model\].
Among the model’s highlights are a simple explanation of the NW-SE brightness asymmetry of the wisps as being due to the Doppler boost in the mildly relativistic pair flow in the ion gyration region. The parameters inferred from the best fit of the model to the main wisps 1 and 2 in the NW and the faint wisp in the SE suggest this steady, reflected ion flow model has not unreasonable correspondence to van den Bergh and Pritchett’s observations of wisp separation, brightness and shape, which overdetermine the model:
----------------------------------------------------------------------------------
$\sigma \approx 3 \times 10^{-3}, $
$\gamma_1 \approx 4 \times 10^6 \approx 0.3 Ze \Phi_{open} /m_i c^2, $
$B_1 \approx 3 \times 10^{-5} \; {\rm Gauss},$
$\dot{N}_\pm \approx 10^{38} \; {\rm s}^{-1}, $
$m_i \dot{N}_i \approx 2 m_\pm \dot{N}_\pm \approx 10^{-15} \; {\rm M_\odot /yr}
\approx 50,000 \; {\rm metric \; tons /s}$,
$Z \dot{N}_i \approx 3 \times 10^{34} \; {\rm s}^{-1}
\approx {\rm Goldreich-Julian \; return \; current}$.
----------------------------------------------------------------------------------
Here $\Phi_{open} = \sqrt{\dot{E}_R /c} = 4 \times 10^{16}$ Volts is the total electric potential drop across the open magnetosphere. The fit to the data also fixes the ion Larmor radius and the tip angle of the equatorial outflow to the line of sight: $r_{Li} \approx 0.15$ pc and $\angle$(LOS, equatorial wind) $\approx 35^o$. The number of pairs flowing out in the equatorial wind is close to, but somewhat less than what was inferred in the Kennel and Coroniti model as the particle supply needed for the X-ray source, and the ion current is in good agreement with what one would expect if the equatorial wind carries the return current.
While the favorable comparison of the Gallant and Arons model to a single, relatively low resolution optical snapshot of the Crab gives some credence to the basic idea that the wind is energetically dominated by very high rigidity ions, the neglect of time dependence is a serious flaw in modeling the equatorial outflow. Variability of the equatorial wisps has been known since Lampland’s original discovery, seen most spectacularly in the high resolution “movie” created from the HST campaign now in progress and in the ground based optical observations reported by Tanvir [*et al.*]{} (1997). Both sets of data show that the wisps are outwardly propagating structures, behaving like spherical or cylindrical waves which lose coherence over several tenths of a parsec as they propagate away from the pulsar.
The kinetic simulations of ion dominated relativistic shocks in plane parallel geometry published by Hoshino [*et al.*]{} (1992) (see also Hoshino 1998) clearly show the shock structure to be time dependent, with a large amount of short wavelength power in the magnetic field. The basic relativistic cyclotron instability of the shock structure implies variability of brightness enhancements on the ion gyration time scale, with faster variability imposed on the basic structure by the higher harmonics. Such variability would be uncorrelated with pulsar timing variations, as seems to be the case in all the observations of wisp variability. However, the original simulations did not give a clear answer to whether the variability is in the form of oscillations of the shock structure around a mean position, or in the form of “radiation” of finite amplitude magnetosonic waves into the surrounding nebula.
An investigation of the time dependent theory (Spitkovsky and Arons, in prep) shows that the relativistic cyclotron instability of the ion ring formed immediately downstream of the thin shock in the pairs [*does*]{} launch outwardly running waves in the magnetic field, density and temperature of the pairs (contrary to the criticism of this model advanced by Tanvir [*et al.*]{} 1997, who assumed that shock variability corresponds to shock oscillation around a mean position), with fundamental period comparable to the ion Larmor time $t_{Li} = 2 \pi c/\Omega_{ci2} = m_i c \gamma_1 /Ze (3B_2) \approx 1.5$ months.
These calculations use a “hybrid” approach to modeling the shock structure. Particle-in-cell ions are injected into hot (Maxwellian) pairs, modeled as an ideal, relativistic MHD fluid with magnetic field transverse to the flow. The pair fluid is still modeled as adiabatic, with cyclotron absorption of the ion waves and nonthermal pair acceleration neglected. Some illustrative results from a plane parallel simulation are shown in Figure \[fig:hybrid\], which shows the compressions in the $B$ field that form as the ion ring breaks down into time dependent bunches in gyrophase. The compressions in the pairs and $B$ field observed in the code travel downstream with speed $v_{\rm wave} \approx 0.8c$; theoretically, amplitude waves should travel with speed[^4] $$v_{\rm wave} = \frac{v_{pairs} + v_{ms}}{1 + \frac{v_{pairs} v_{ms}}{c^2}}
= \frac{\frac{c}{3} + \frac{c}{\sqrt{2}}}{1 + \frac{\frac{c}{2}
\frac{c}{\sqrt{2}}}{c^2}} = 0.84c.$$
Here $v_{ms}$ is the wave group speed in the proper frame of the pair fluid. The numerical value is peculiar to the plane parallel geometry used in this particular calculation (mainly to test the code). These preliminary results suggest that the interpretation of the wisp region as the equatorial shock structure remains viable.
Speculations and Conclusions
============================
These dynamic models of distributed shock structure continue to show promise in the interpretation of the energy transfer between the Crab pulsar and its nebula. A complete theory requires addressing a variety of other issues, which one can only do at the order of magnitude level at present.
X-rays and Gamma Rays
---------------------
The plane parallel shock structure and acceleration calculations, and quasi-linear theory applied to those simulations, yield the maximum energy of the power law formed in the downstream pairs to be $E_{\pm , max} \approx (E_{ion})_{upstream} \approx 0.3 Ze\Phi_{open}
\approx 10^{16}$ eV. The same calculations yield an acceleration time to extend the power law to the maximum energy to be $t_a ( E_{\pm , max}) \approx \Omega_{Li2}^{-1} \approx 1.5$ months. The resulting spectrum rolls off above the photon energy $\varepsilon_2 \approx 0.3 (\hbar e B_2 /m_\pm c)
(m_i /Z m_\pm)^2 \gamma_1^2 \approx 20$ Mev, using the Gallant and Arons parameters, a value quite close to the rolloff energy of the variable $\gamma$ ray component of the spectrum reported by de Jager [*et al.*]{} (1996). The synchrotron loss time at these highest energies is $t_s (E_{\pm,max}) \sim 4$ months, and the size of the synchrotron source in the 10-100 MeV region then should be $R_s \leq v_{pairs}t_s \simeq 2 \times 10^{17} \; {\rm cm}$. Furthermore, the spatially progressive nature of the particle acceleration suggests that the size of the source at the highest energies will stop shrinking on scales smaller than about $10''$, and that interior to several arc seconds from the pulsar, the higher energy emission should show a “hole” in the surface brightness, with the hole size [*decreasing*]{} with decreasing photon energy, until the leading edge shock in the pairs is reached.
Because the shock structure is unsteady, the particle acceleration also varies. When the synchrotron loss time greatly exceeds the variability time scale of the accelerator introduced by the large amplitude magnetosonic waves, the radiation physics averages over the variable particle acceleration. At the highest photon energies, however, the synchrotron loss time is comparable to the fundamental magnetosonic wave time scale, suggesting that the 50-100 MeV source varies with fractional luminosity changes $$\frac{\delta L_{\gamma, synch}}{L_{\gamma , synch}} \approx
\frac{t_a ( E_{\pm , max})}{t_s( E_{\pm , max})} \approx 0.4.$$ At these photon energies, the inverse Compton source overlaps the synchrotron source in energy space. Since the inverse Compton source must have size comparable to the optical Nebula (de Jager and Harding 1992), there will not be variations of the inverse Compton radiation on these short time scales, with the result that the 50-100 MeV emission will vary less than the synchrotron component alone. Nevertheless, these predicted variations on the several month time scale should be marginally observable in the EGRET data. Also, a substantial, improvement of angular resolution in hard X-rays will soon allow new probes of the Crab’s inner workings, when the HESSI satellite with its $2''$ imaging at several hundred keV is launched.
The Radio Nebula
----------------
Optical, X- and $\gamma$-ray emission diagnoses the coupling physics “today”. The radio emission measures the integral of the pulsar’s input over history - most of the stored relativistic energy is in $B$ fields and radio emitting particles ($\sim 10^{50}$ ergs). Averaged over the whole Nebula, the radio emitting spectrum has the form $N_\pm(\gamma) \propto \gamma^{-1.5}, \; 10^{2.5} < \gamma < 10^4$; indeed, detailed spectral index maps (Bietenholz and Kronberg 1992) show this particle distribution to be remarkably homogeneous.
The wind termination shock models constructed to explain the wind (Kennel and Coroniti 1984a,b, Gallant and Arons 1994) don’t yield $N(\gamma) \propto \gamma^{-s}, \; s \sim 1.5$ at energies small compared to $10^6 m_\pm c^2$. Also, the pair injection rate inferred in the equatorial wind is between a few and 10 per cent of the average injection rate needed to explain the radio Nebula.
This discrepancy poses the following conundrum. Spectral continuity suggests one mechanism accelerates the synchrotron emitting particles, yet the shock jump conditions applied to the equatorial wind clearly show the particle spectrum must be “cut off” below $10^6 m_\pm c^2$ (Kennel and Coroniti 1984b); in the kinetic theory of the shock acceleration physics, the particle spectrum remains Maxwellian below $\gamma_1 m_\pm c^2$ (Hoshino [*et al.*]{} 1992), which yields the same low frequency emissivity as does a sharply cut off distribution, $f_\nu \propto \nu^{1/3}$, not the observed $f_\nu \propto \nu^{-0.25}$. It is possible that additional acceleration physics within the shock structure beyond cyclotron resonant absorption of the magnetosonic waves can lead to a nonthermal low energy spectrum like that observed - magnetic pumping is an interesting candidate. But, modifications of the acceleration physics will not alleviate the discrepancy between the rate of pair injection into the equatorial torus and the average injection rate of radio emitting electrons.
The discovery of the “polar jet” (Hester [*et al*]{} 1995) suggests that wind outflow at latitudes $|\lambda | > 10^o$, which could fill most of the solid angle around the pulsar, might provide the source of the larger number of particles feeding the radio source. If the acceleration physics is the same, perhaps spectral continuity is not a surprise. However, one would be quite surprised to find the wind outside of the magnetic equator to contain an energetically dominant component of heavy ions, for the electrodynamical reasons described earlier. Since the energetic dominance of the ions is essential to the cyclotron resonance acceleration explanation of the nonthermal pairs in the equatorial flow, invoking the same acceleration physics at higher latitudes doesn’t look to me to be the right theoretical path. One is left with the possibility that the higher latitude acceleration physics is different, or that radio electrons are accelerated in the outer Nebula (for example, by Fermi I acceleration at the shock just outside the synchrotron Nebula, whose presence was inferred by Sankrit and Hester 1996), and that spectral continuity is just a coincidence. Observations of the spectra of other plerions with sufficient spectral coverage (radio, IR, optical, X-rays) would greatly help in directing the course of theory - if most plerions show substantial spectral discontinuities between radio and higher frequencies, then the Crab’s spectral continuity clearly would be coincidental. But if the spectra all show continuity similar to the Crab’s, then a solution must be sought in terms of a more unified injection/acceleration scheme than is implied by current theory.
High Energy Neutrinos from the Crab Nebula
------------------------------------------
In principle, observation of high energy neutrinos from the Crab would be a direct test of the ion doped wind model (and of other ideas concerning ultra high energy ion acceleration by the pulsar, such as the outer gap construction of Bednarek and Protheroe 1997). The following numerical example illustrates the possibilities.
Suppose the pulsar injects heavy ions at the Goldreich and Julian rate, $\dot{N}_i = 10^{34.5}/Z \; {\rm s}^{-1}$, each with energy $\gamma_i m_i c^2 = \eta Ze \Phi_{open} = 4 \times 10^{16} Z\eta$ eV. The ions’ Larmor radii in the nebula are $r_{Li} = \gamma_i m_i c^2 /ZeB_{neb} = \eta \Phi_{open}/B_{neb}
\approx 0.15 \eta (10^{-4} \; {\rm Gauss} /B_{neb})$ pc. The ions drift out of the Nebula with drift velocity across $B$ $\sim c r_{Li} /R_{neb} \sim 0.1 c$, a speed about 10 times the hydrodynamic expansion velocity of the nebula. Then the number of ions contained in the Nebula is $N_i = \dot{N}_i (R_{neb} /c)(R_{neb} /r_{Li}) \sim 6 \times 10^{43} Z^{-1}.$ With a cross section for $\pi^\pm$ production on the order of 10 mb, and with $\sim 1 \; M_\odot$ of thermal material within the nebula (contained in the emission line filaments, corresponding to an average gas density of a few protons/cc), the high energy neutrino luminosity of the Crab Nebula should be at least $10^{29}$ high energy neutrinos/s. This is a lower limit, since the heavy ions continue to interact with the invisible gas confining the visible synchrotron nebula as they to wander outwards through the unknown magnetic field in the inertially confining material. At 2 kpc distance, the neutrino flux from the Crab should be about $5 \times 10^{-15}$ neutrinos/cm$^2$-s, substantially above background at the $\sim 10^{16}$ ev energy suggested by this elementary monoenergetic model. At these high energies, the count rate expected in the proposed high energy neutrino observatories is rather low. However, the use of the Crab Nebula as an ion calorimeter through the dynamics of the wisps does not tell us anything about the downstream spectrum of injected ions, and theory of ion acceleration at the pulsar is too primitive to be of much help. If the ions have a sufficiently steep injection spectrum, the production rate of lower energy neutrinos might be much higher.
Why is $\sigma \ll 1$?
----------------------
One of the most surprising conclusions of the Rees and Gunn (1974) model of the Crab is that $\sigma \ll 1$ in the wind upstream of the termination shock, a result which has persisted in all of this model’s descendents. Since the pulsar’s magnetosphere is magnetically dominated ($\sigma \gg 1$), and $\sigma$ is conserved in simple ideal MHD wind models, the fate of the pulsar’s magnetic energy has attracted quite a bit of theoretical attention. Suggestions which appear to have some possibility of success include dissipation of the magnetic field in a striped MHD wind by tearing modes (Coroniti 1990); dissipation of the magnetic field in a striped MHD wind because of insufficient current carriers to support the stripes (Michel 1994); and conversion of MHD flow to dissipative “vacuum” waves in the wind zone (Melatos and Melrose 1996). All of these ideas depend upon most of the magnetic field in the wind having a wave-like structure, either as standing oscillations in the fluid frame (Coroniti, Michel), or as large amplitude waves which propagate with respect to the plasma in the fluid frame, with structure dominated by displacement current even though the waves are subluminous (see also Melatos’ paper in these proceedings). If any of these thoughts is on the right track, the low $\sigma $ problem requires giving up the applied mathematical pleasures of studying the aligned rotator and axisymmetric ideal MHD winds - indeed, one has to give up ideal MHD! None of the proposed ideas, however, has been developed to the point of usefully confronting theory with the elaborate HST pictures or other high resolution observations.
Other Models of the Wisps and the Pulsar-Nebula Coupling
--------------------------------------------------------
I would be remiss if I neglected discussing some of the other ideas around for the interpretation of the Crab’s wisps and what they have to tell us about the pulsar-nebula connection. Woltjer (1958) is the first to suggest the idea that the wisps might be damped magnetosonic waves, perhaps driven by Baade’s star, the then mysterious object suspected of having something to do with the energization of the Crab Nebula. Scargle (1969) and Barnes and Scargle (1973) rediscovered this idea, and proposed the wisps to be magnetosonic waves in a relativistic electron-heavy ion plasma, launched by upstream variations of the vacuum magnetic dipole radiation then thought to carry the pulsar’s spin down energy. They attributed the time variability of the magnetic dipole radiation timing glitches, not to intrinsic instability in the termination of the strong wave, while the nebular relativistic electron spectrum was attributed to Landau damping (“Barnes damping”) of the waves in the Nebular plasma. Unfortunately, the wisp variability does not correlate with glitches, and Barnes damping does not lead to the observed particle spectrum. Nevertheless, the ideas expressed by these authors have a clear relationship to the model I outlined above.
Hester [*et al.*]{} (1995, and personal communications), has expressed the opinion that the wisps are thermal instabilities in the post-shock outflow from the pulsar, with the shock itself either unobserved or attributed to one or another of the time variable features seen in the HST pictures. In this case, the observed outflow velocity of the wisp features is the fluid flow speed. The main flaw in this view is that within the Kennel and Coroniti model, from which the suggestion derives, the cooling time is too long ($\sim $ 10 years) for the particles which mainly contribute to the pressure, much longer than the weeks to months needed to explain the variations. Chedia [*et al*]{} (1997) suppose the wisps to be drift waves in a low energy pair plasma (whose provenance is not otherwise explained). excited by a $\gamma \sim 10^6$ ion beam from the pulsar - these authors assume no shock forms, which is contrary to the known dynamics of a relativistic ion beam in the magnetized plasma. Begelman (1998b) hypothesizes the wisps to be travelling surface waves excited by the interaction between an axisymmetric equatorial outflow and a higher latitude outflow traveling with a different four velocity, a model which is not very specific about observational consequences that could discriminate it from other ideas. Presumably still more suggestions will be forthcoming as the quality of the observations continues to improve.
Other Plerions
==============
One may well ask whether other filled image SNR (“plerions”, composite SNR) have the same physics. Unfortunately, none of them have been sufficiently well studied to bring the kind of physical modeling described here to bear. We saw at this meeting the striking advances in X-ray detections of plerions, which, when coupled with existing and new radio observations, allow us to begin asking simple physical questions, such as whether spectral “breaks” between radio and X-ray observations tell us anything about the age of the system, assuming a single mechanism injects a single power law distribution at all energies into the plerion. But until the optical and IR spectral regions are filled in, until detections are made at energies high enough to allow one to follow possible variability of the acceleration mechanism, and until high angular resolution imaging allows one to study the variable features associated with the acceleration physics, one will be left with only the wonderful example of the Crab as the test of physical theories of pulsars as particle accelerators. The chance that this system’s physics does not typify all the pulsar-plerion pairs known or to be found, and by extension might not typify the excitation of nonthermal activity by other central compact objects (such as the jets driven by black holes in AGN), underlines the need to advance observations of a larger number of plerions across the whole spectrum (especially of the younger, calorimetric systems), with the highest possible angular imaging and with sufficient temporal coverage to follow the variations of the fine scale structure which are so revealing of the compact object-surrounding world interaction marvelously exhibited by the Crab. It [*does*]{} behoove us theorists to turn some of the insights laboriously gleaned from the Crab into predictions for other plerions, especially the young ones with calorimetric morphology. I’m sure there will be plenty of surprises for all of us.
I am particularly indebted to Jeff Hester for energetic discussions which clarified my understanding of the flow geometry around the Crab Pulsar, as well as many other entertaining aspects of astrophysics and of life, and to Anatoly Spitkovsky for illuminating collaboration and discussion. The research described here has been supported by NSF grant AST 9528271 and by NASA grant NAG 5-3073, and in part by the generosity of California’s taxpayers.
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[^1]: Recently Begelman (1998) proposed that MHD kink instability of toroidal magnetic fields allows one to construct a wind fed model of the Crab with $\sigma \sim 1$ in the wind, a conclusion which depends on the assumption, unstated in his paper, that the kinked magnetic fields coagulate into patches whose filling factor in the nebula is small, and within which most of the field energy annihilates. Such coagulation is not a known consequence of the kink instability (quite well studied in the low $\beta$ plasmas in fusion devices and the solar corona), and the virial theorem suggests such coagulation to be unlikely. The observed uniformity of the radio spectral index (Bietenholtz and Kronberg 1992) shows that the proposed annihilation must have little radiative consequences for the radio emission in the Nebula, which is energetically surprising. If a kink instability does occur, a more likely consequence of kinking and reconnection would be to fill the Nebula with magnetic loops whose filling factor is on the order of unity, in which case Rees and Gunn’s original arguments for low $\sigma$ in the wind are unaltered. Nevertheless, the fine fibered structure observed by Scargle (1969) and by Hester [*et al.*]{} (1995) in the optical emission from the Nebula suggests some mechanism for complicating the magnetic structure on a fine scale is at work, for which Begelman’s kink instability is a candidate.
[^2]: In the actual application to the Crab, the ions gyrate in a $B$ field already compressed by a factor of two to three above its upstream value by the preliminary shock in the pairs, which increases the gyration frequency by the same factor and yields an ion gyration time of 1-2 months.
[^3]: This length is the distance required to bring the highest energy pairs in the spectrum to their maximum energy $\sim \gamma_1 m_i c^2$, at which energy they radiate 100 MeV gamma rays. The nonthermal spectrum of infrared and optically emitting particles is established within a distance $ \sim 10 r_{L\pm} \approx 100$ AU downstream of the leading edge shock in the pairs. Therefore, one needs better than 10 mas angular resolution to allow detection of the initially thermal O/IR synchrotron spectrum.
[^4]: The adiabatic index of the pair fluid used in the simulation was $\Gamma = 3/2$, as would be the case if the pairs were heated only in the plane orthogonal to ${\bf B}$.
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abstract: 'Molecular gas in the merging starburst galaxy NGC 3256 has been imaged with the Submillimeter Array at a resolution of $1'''' \times 2''''$ (170 $\times$ 340 pc at 35 Mpc). This is the first interferometric imaging of molecular gas in the most luminous galaxy within $z$=0.01. There is a large disk of molecular gas ($r > 3$ kpc) in the center of the merger with a strong gas concentration toward the double nucleus. The gas disk having a mass of $3\times 10^{9}$ in the central 3 kpc rotates around a point between the two nuclei that are 850 pc apart on the sky. The molecular gas is warm and turbulent and shows spatial variation of the intensity ratio between CO isotopomers. High-velocity molecular gas is discovered at the galactic center. Its velocity in our line of sight is up to 420 offset from the systemic velocity of the galaxy; the terminal velocity is twice as large as that due to the rotation of the main gas disk. The high-velocity gas is most likely due to a molecular outflow from the gas disk, entrained by the starburst-driven superwind in the galaxy. The molecular outflow is estimated to have a rate of 10 and to play a significant role in the dispersal or depletion of molecular gas from the galactic center. A compact gas concentration and steep velocity gradient are also found around each of the twin nuclei. They are suggestive of a small gas disk rotating around each nucleus. If these are indeed mini-disks, their dynamical masses are $10^{9}$ within a radius of 170 pc.'
author:
- 'Kazushi Sakamoto, Paul T. P. Ho, and Alison B. Peck'
title: 'Imaging Molecular Gas in the Luminous Merger NGC 3256 : Detection of High-Velocity Gas and Twin Gas Peaks in the Double Nucleus '
---
Introduction
============
The luminous infrared galaxy NGC 3256 has an infrared luminosity of $10^{11.56}$ at the distance of 35.4 Mpc [1=170 pc; @Sanders03], which makes it the most luminous galaxy within $z$=0.01 [@Sargent89]. The vast luminosity, largely emitted in the far-IR, comes from within the central $\lesssim$ 20 of the galaxy [@Smith96]. The center of the merging galaxy hosts a starburst seen across a wide range of wavelengths [e.g., @Graham84; @Doyon94]. High-resolution imaging with the Hubble Space Telescope (HST) revealed hundreds of bright young clusters in the galactic center [@Zepf99; @Alonso-Herrero02].
The extreme starburst is expected to blow interstellar gas out of the system and may leave a gas-depleted elliptical galaxy [@Graham84]. Indeed, an outflow of interstellar medium, or superwind, has been detected. Its evidence includes LINER-like optical line ratios as well as large line widths (up to 6000 ) off the nucleus [@Moran99], blueshifted absorption and emission lines [@Heckman00; @Lipari00; @Lipari04], and the pattern of optical polarization (i.e., reflection by entrained dust) around the galaxy [@Scarrott96]. These observations suggest that the galaxy wind extends out to several kiloparsecs.
NGC 3256 is in the late stage of galaxy merging [@VV59; @Toomre77]. There is a pair of long tidal tails seen in the optical (see Fig. \[fig.opt\]) and in emission, extending 40 kpc on each side [@deVaucouleurs61; @English03]. Such morphology suggests a prograde-prograde merger of two gas-rich spiral galaxies of similar size [@Toomre72; @White79]. It also suggests that we view the system with a low inclination angle [@Feast78; @English03]. The radial light profile of the galaxy in the $K$-band is close to but has not achieved the de Vaucouleurs profile (i.e., $I(r) \propto \exp(-k r^{1/4})$ ) of elliptical galaxies [@Moorwood94; @Rothberg04], unlike the majority of merger remnants surveyed by @Rothberg04. This suggests that the merging of the stellar systems in NGC 3256 is not yet complete.
Likely corresponding to the incomplete merger, two nuclei with 5 (850 pc on the sky) separation have been detected in the near infrared, radio, and X-rays in the starbursting center of the merger [@Moorwood94; @Norris95; @Lira02]. They are aligned in the north-south direction, and the southern nucleus is highly obscured, rendering it invisible in the optical. @Neff03 suggested a low luminosity AGN in each nucleus from comparison of radio and X-ray observations, while @Jenkins04 found no strong evidence for an AGN in X-ray data alone.
There is abundant molecular gas ($10^{10}$ ) in the galactic center around the double nucleus, probably feeding the starburst [@Sargent89; @Casoli91; @Aalto91a; @Garay93]. NGC 3256 is one of the first galaxies in which unusually large / intensity ratio characteristics of luminous mergers was observed; it was attributed to the merger and starburst environment [@Aalto91b; @Casoli92b].
The observations so far are broadly in agreement with the previous observations of other luminous and ultraluminous mergers and with the galaxy evolution models involving merging. They suggest that a galaxy collision and subsequent merger bring most of gas in the progenitor galaxies to the merger center, cause a burst(s) of star formation as well as galaxy wind driven by the starburst, and eventually make an elliptical galaxy, sometimes with a phase of luminous nuclear activity [@Schweizer86; @Sanders96; @Genzel00; @Sanders04; @Veilleux05 and references threin].
The proximity of a source of this luminosity and the near face-on configuration make NGC 3256 an ideal target to test and refine the scenario of merger-induced starburst and galaxy evolution. Moreover, NGC 3256 is arguably the southern-sky counterpart of Arp 220, the archetype of the infrared-luminous merging galaxies [@Soifer84]. The two galaxies not only share the title of the most luminous galaxy within their respective distances but also the close binary nuclei, since Arp 220, which is twice as distant as and four times more luminous than NGC 3256, has two nuclei only 0.3 kpc apart on the sky. Thus they are most likely near the final stage of galaxy merger when the merger-induced evolution is expected to be most rapid and prominent. Only the southern location of NGC 3256 (Dec. = $-44\arcdeg$) has delayed one of the most important observations required to better understand the merger-starburst evolution — high-resolution observations of its molecular gas.
We have made high-resolution observations of the cold interstellar medium in NGC 3256 in 1.3 mm using the Submillimeter Array (SMA)[^1]. Our observations aim to uncover the distribution, kinematics, and physical properties of the cold molecular gas and dust in the merger-starburst environment of NGC 3256 with a high resolution and sensitivity. Of particular interest are the distribution and dynamics of gas around the starbursting double nucleus and how the molecular gas is affected by the starburst. The new data of 1 resolution are from the first interferometric observations of molecular gas in this galaxy. They provide more than 10-fold improvement in spatial resolution over previous single-dish observations made with $\gtrsim 20{\mbox{$''$}}$ beams.
We begin by introducing our SMA observations (§\[s.observations\]), then portray the overall distribution and kinematics of molecular gas in the central 9 kpc (§\[s.overall\]), report the detection of gas concentrations associated with the two nuclei and gas kinematics suggestive of rotation around each (§\[s.double\]), report the discovery of high velocity gas and model it as a molecular outflow from the starburst (§\[s.high-velocity-gas\]), and characterize the properties of molecular gas in the galactic center on the basis of intensity ratios of CO isotopomers (§\[s.gas-properties\]). Our observations are discussed in the context of the galaxy merger and merger-induced starburst in §\[s.discussion\], and summarized in §\[s.conclusions\].
In this paper, we refer to the north and south nucleus as N and S, respectively, and adopt the following positions for them on the basis of radio and X-ray observations [@Neff03; @Norris95; @Lira02]; $\alpha (N) =10^{\rm h}27^{\rm m}51\fs23$, $\delta (N) =-43\arcdeg54\arcmin14\farcs0$ and $\alpha (S) =10^{\rm h}27^{\rm m}51\fs22$, $\delta (S) =-43\arcdeg54\arcmin19\farcs2$ in J2000. The systemic velocity of the galaxy in the literature is $V_{\rm sys}$ (LSR, radio) $\approx 2775$ [@Casoli91; @Aalto91a; @Garay93; @English03].
SMA Observations and Data reduction \[s.observations\]
======================================================
We observed NGC 3256 at 1.3 mm using the Submillimeter Array (SMA), which consists of eight 6 m-diameter antennas at the summit of Mauna Kea, Hawaii [@Ho04]. We chose the center position of our observations to be $\alpha=10^{\rm h}27^{\rm m}51\fs22$, $\delta=-43\arcdeg54\arcmin19\farcs2$ (J2000), which is centered between the two nuclei. The primary beam of the SMA antennas has the FWHM size of 52 (= 9 kpc) at 230 GHz. The extended and compact array configurations were used for our observations in February 20th and April 4th 2004, respectively, with all eight antennas participating on both nights. They provided projected baselines ranging from 6 m to 179 m. The total integration time on the galaxy was 6.9 hours in two tracks.
The receivers were tuned to simultaneously observe the three CO lines, (2–1) in the upper side band (USB) and (2–1) and (2–1) in the lower sideband (LSB). The center frequency of the USB was 227.720 GHz, and the LSB was 10 GHz lower. Each sideband had 2 GHz of bandwidth and a spectral resolution of 0.81 MHz. The weather was excellent on both nights. The 225 GHz zenith opacity measured at the neighboring Caltech Submillimeter Observatory was 0.05 and 0.08 for the first and second nights, respectively. The double-sideband system temperature toward the galaxy was 160 K in median, even though the galaxy was observed between the elevation of 14 and 26.
We observed the quasar J1037295 once every 20 min for gain calibration. The flux density of the quasar was estimated to be 0.86 Jy and 0.84 Jy for LSB and USB, respectively, from comparison with Mars, for which we assumed the brightness temperature of 200 K. The system passband was calibrated by observing Jupiter, Saturn, and a few bright quasars. The elevation-dependent attenuation by the atmosphere was corrected by using the system temperature. The precision of our gain calibration was assessed by comparing, between the two tracks, the total flux detected in the same range of hour angle on the common baselines in the two array configurations. The flux agreed within 10%. We therefore assign, conservatively, $\pm 10$% for the random error of our flux measurements reported in this paper. The error in our absolute flux scaling, however, can be larger due to unknown or uncharacterized systemic effects.
The SMA data were reduced using MIR, which is an IDL version of MMA [@Scoville93], MIRIAD [@Sault95], and the NRAO AIPS package [@Bridle94]. After the standard passband and gain calibration, the spectral channels that did not have emission lines were combined to make continuum data of 0.33 GHz bandwidth in each sideband. Channels containing the high-velocity emission that we discovered were excluded from the continuum data. The continuum was subtracted from spectral line data in the $uv$ plane. Images of the line and continuum emission were then made by Fourier transforming the respective set of visibilities. Various spatial and spectral resolutions were achieved by changing $uv$-weighting and channel binning. Among our datasets are the lowest-resolution one made with natural weighting and 20 resolution to detect faint extended emission, and the highest resolution maps made with the super-uniform weighting and 10 resolution. Intermediate resolution maps with the robust $uv$ weighting are also used. The rms noise in the natural weighting data cubes is $20 \pm 2$ mJy . The SMA maps presented in this paper are not corrected for the primary beam attenuation, except for Fig \[fig.cor-2s\] (d). The attenuation is approximately Gaussian and is a factor of 2 for a source 26 from the map center. The correction for it is made in all flux and spectrum measurements, though it is rather small for most emission that we detect. Velocities in this paper are measured with respect to the local standard of rest (LSR) and are defined in the ratio convention.
We compared our data cubes with single-dish observations in the literature, and found that our interferometric observations recovered almost all of the flux in the region. The observations at the 15 m Swedish-ESO-Submillimeter-Telescope (SEST) gave the total (2–1) flux of (3.2 – 3.6)$\times 10^{3}$ Jy and (2–1) flux of (1.3 – 1.5)$\times 10^2$ Jy in their beam at the galactic center [@Casoli91; @Aalto91a; @Garay93][^2]. We measured total (2–1) and (2–1) flux of 4.1$\times 10^3$ Jy and 1.4$\times 10^2$ Jy from our natural-weighting data corrected for the primary beam attenuation and convolved to the SEST resolution, which we assumed to be 23 in FWHM at 230 GHz. The high fraction of flux recovery is consistent with the compact size of the CO emission, whose half-power diameter is estimated to be about 10 by @Aalto91a. The 20% larger flux in at the SMA than at the SEST may be due to a larger-than-usual error in the SMA calibration for this source at very low elevations, and may be also due to pointing and calibration uncertainties in the single-dish observations[^3]. The single-dish observations must also have slightly underestimated the CO flux by over-subtracting the emission as a linear baseline. This is because the CO line is wider than previously believed as we see below. However, this leads to only 1–2% underestimation of the single-dish CO flux. In any case, our relative calibration between the CO lines and continuum should be more accurate than their absolute calibration, because they were simultaneously observed with the same receivers and went through the same signal path and reduction procedure. Regarding missing flux in our higher resolution datasets, our highest-resolution data cube contains 54% of the total flux in our natural-weighting data.
Overall Gas Distribution and Kinematics \[s.overall\]
=====================================================
spatial distribution of molecular gas and dust \[s.overall-spatial\]
---------------------------------------------------------------------
The (2–1) emission as shown in Fig. \[fig.naturalmap\] is highly concentrated toward the galactic center. It has a half-maximum radius of about 5 (1 kpc) in the galactic plane (Fig. \[fig.12co.iring\]). The compact CO morphology is consistent with the SEST mapping result of @Aalto91a. The CO central peak is elongated in the north-south direction and connects the double nucleus of 5 separation. There are two peaks in this structure in the (2–1) map also shown in Fig. \[fig.naturalmap\]. Each peak is spatially associated with one of the two nuclei. We discuss these double peaks in more detail in the following sections using higher resolution images. The gas distribution extends at least to the radius of 3 kpc in our data, and it may be extended beyond our field of view. In the outer area, there are arcs or hints of spiral arms in the molecular gas. Some of them correspond to spiral arms or dust lanes in the optical images in the literature. For example, the horizontal feature near the northeast edge of the map, around the offset of $\Delta$R.A.=10 and $\Delta$Dec.=+15, corresponds to a spiral arm emanating from the galactic center [see the image of @Moorwood94 Fig. 2(b)].
The 1.3 mm continuum and (2–1) emission are also detected in the galactic center (see Fig. \[fig.naturalmap\]). The spatial distribution of the continuum is quite similar to that of (2–1) emission. There are peaks at or near the two nuclei, surrounded by extended emission. A weaker peak is also seen 6 east of the nucleus N, as is seen in (2–1) and (2–1). The shape of the (2–1) emission looks somewhat different from those of other emission mentioned above. However, this is likely due at least partly to the low signal-to-noise ratio in the data.
The total (2–1) fluxes within the galactocentric radii ($r_{g}$) of 10 and 20are $3.4\times 10^3$ Jy and $5.2\times 10^3$ Jy , respectively, as shown in Fig. \[fig.12co.iring\]. The total fluxes of (2–1) and (2–1) emission within the galactocentric radius of 10 are $1.5 \times 10^2$ Jy and 31 Jy , respectively. The total flux density of the 1.3 mm (222.7 GHz) continuum in the same area is 79 mJy. Each line flux is integrated over the same 600 range from 2530 to 3130 , and each flux measurement for both line and continuum was made in the same way as explained in the caption of Fig. \[fig.12co.iring\]. Each flux has a 10% scaling uncertainty as noted in §\[s.observations\].
gas mass estimate \[s.gas-mass-estimate\]
------------------------------------------
The mass of molecular gas ($M_{\rm mol}$) in the region is estimated in various ways from the line emission. The molecular gas mass of $2\times 10^{10}$ for $r_{g} \leq 10{\mbox{$''$}}= 1.7$ kpc is obtained from the conversion factor $N_{\rm H_2}/I_{\rm CO(1-0)}= 3.0\times 10^{20}$ (K )$^{-1}$, which is based on the virial analysis of molecular clouds in the disk of our Galaxy [@Scoville87; @Solomon87]. We assume that the CO(1–0) and CO(2–1) lines have the same brightness temperature and emitting area. This mass estimate and the ones in the rest of this section include the contribution of helium. The $M_{\rm mol}$ in the same region is estimated to be lower, $9\times 10^{9}$ and $2\times 10^{9}$ , from the conversion factors derived from $\gamma$-ray observations to count H-nuclei; the two estimates are from the conversion factor of $1.56\times 10^{20}$ (K )$^{-1}$ assumed to be constant over the Galaxy [@Hunter97] and $0.4\times 10^{20}$ (K )$^{-1}$ at $r_{g}= 2$ kpc in our Galaxy [@Strong04], respectively. Another estimate is from the (2–1) emission and the prescription of @Stutzki90 and @Wild92. They suggested that this optically thin line should have nearly a constant emissivity per molecule in the range of temperature and density relevant for starburst galaxies such as M82. NGC 3256, also being a starburst, is probably in the class of galactic nuclei to which the method is applicable. The observed (2–1) flux gives the molecular gas mass of the $1.4\times 10^{9}$ for $r_{g} \leq 1.7$ kpc for the assumed abundance of $[{\mbox{C$^{18}$O}}] /[{\mbox{H$_2$}}]= 10^{-6.8}$ [@Frerking82]. The mass of atomic gas within this radius is estimated to be $M_{\rm atom}$=(0.4–1.9)$\times 10^9$ by scaling the mass estimate by @English03, who obtained the mass of HI absorbing gas in the central 23.
The 1.3 mm continuum is also used for the mass estimate in the following way. First, about a half of the 1.3 mm flux density, or 45 mJy, is attributed to dust emission by subtracting the contributions from synchrotron and free-free emission from the observed flux density. The synchrotron emission at 1.3 mm is estimated to be 15 mJy by extrapolating a power law (of a spectral index of $-0.76$) obtained by fitting the radio flux measurements between 160 MHz and 5 GHz in the NASA Extragalactic Database[^4]. The free-free flux density is estimated to be 19 mJy from the region model of @Roy05, who derived properties of the regions in the galactic center by observing radio recombination lines in NGC 3256. After subtracting these contributions, the remaining emission of 45 mJy must be the thermal emission from dust in the interstellar medium. We used the dust emissivity of @Draine84 and the dust temperature of 44 K [@Smith96] to obtain the gas (molecular plus atomic) mass of $2.6 \times 10^9$ for $r_{g} \leq 1.7$ kpc. The mass is reduced by half if we adopt the dust opacity coefficient of @Hildebrand83. An assumption made in these estimates is that the gas to dust ratio and dust properties are the same in our Galaxy and NGC 3256.
The large scatter among the mass estimates is not surprising for a starburst nucleus. Many assumptions that can not be easily verified are inevitably made, including those on metallicity, molecular abundance, dust properties, and the properties of the molecular gas. For example, the high pressure in the interstellar medium as implied by the superwind from the starburst probably helps to bind molecular clouds. This increases the emissivity of the clouds and leads to an overestimation of gas mass if the conversion factor derived for selfgravitationally bound clouds is applied [@Bryant96; @Oka98]. Independent of these assumptions, dynamical mass sets a stringent constraint since the gas mass can not exceed it. As we see in the next section (§\[s.overall-kinematics\]), the dynamical mass for the central region is estimated to be $1.2\times 10^{10}$ for $r_{g} \leq 1.7$ kpc.
We adopt the total mass of neutral gas in the central region of $M_{\rm gas}(r_g < 1.7 \mbox{ kpc}) \sim 2.9 \times 10^9 {\mbox{$M_\odot$}}$ which is the logarithmic average of the three mass estimates from (2–1), (2–1), and dust emission. For each of the three methods, we geometrically averaged the estimates given above, except the one based on that exceeded the dynamical mass. We assign our estimate a factor of 2 error in each direction, on the basis of the range of the three estimates.
kinematics of the main gas disk \[s.overall-kinematics\]
---------------------------------------------------------
The velocity field of the (2–1) emission in Fig. \[fig.naturalmap\] shows a spider pattern that is typical of gas disks in spiral galaxies. The kinematical major axis is at a position angle of 90 in the vicinity of the double nucleus, and appears to be 70 in the outer regions. The outer position angle is consistent with those from previous spectroscopic observations in CO(2–1) [@Aalto91a] and Br$\gamma$ [@Moorwood94]. It also agrees with the isophotal fitting in the $K$-band by @Rothberg04, in which the major axis has a nearly constant position angle of $\approx$60between the sky-plane radius of 2 kpc and 6 kpc. The overall sense of rotation, i.e., east side being redshifted and west blueshifted, is the same as that of tidal tails [@English03]. The apparent 20 shift in the kinematical major axis could be due to streaming motion of the gas or warp of the main gas disk. The velocity field seen at this spatial scale (i.e., kpc scale) is relatively undistorted for a merging galaxy.
The cold gas component appears to have kinematically merged (or relaxed) and formed a disk in the kpc scale. We therefore fitted the velocity map in Fig. \[fig.naturalmap\] using the AIPS task [GAL]{} to derive the kinematical parameters of the gas disk; the accompanying intensity map was used for data weighting. The intensity weighting makes our parameters more appropriate for the inner part ($r \lesssim 10''$) of the CO disk, reducing the effect of possible warp or non-circular motion in the outer disk. The resulting dynamical center is between the two nuclei N and S. In order to assess the robustness of the fitting results by altering the effect of beam smearing, we did the same fitting using the moment maps made with the uniform weighting, which gave a better spatial resolution of $3\farcs3 \times 1\farcs3$ while retaining 87% of the flux in the natural weighting ($4\farcs8 \times 1\farcs9$) data. The dynamical center, inclination, position angle of the major axis, and the systemic velocity agreed within 08, 8, 2, and 9 , respectively. The average parameters are $\alpha_{\rm M}=10^{\rm h}27^{\rm m}51\fs25$, $\delta_{\rm M}=-43\arcdeg54\arcmin15\farcs7$ (J2000), $i_{\rm M} = 45\arcdeg$, $P.A._{\rm M} = 94\arcdeg$, and $V_{\rm sys, M} = 2791$ , respectively, where the subscript M is for the main (or merged) disk. One half of the abovementioned difference between the two fitting results gives each average value an estimate of uncertainty, though it is probably underestimated because any systematic effect such as non-circular or non-coplanar motion of gas could bias our fitting results.
The parameters we estimated are in reasonable agreement with those from previous observations. Specifically, the inclination and position angle are consistent with those by @Feast78 and others. The systemic velocity is consistent with those by @Casoli91, @Aalto91a, and many others. We use in this paper $V_{\rm sys} = 2775$ taken from the literature. No more than 10% error is introduced in the velocity related parameters derived in the following — such as dynamical mass, momentum, and energy — by the slight difference (16 ) from our fitted value. Lastly, but most notably, the dynamical center between the two nuclei is consistent with that measured from velocity field by @Lipari00. The parameters we derived from the millimeter-wave CO line are not affected nor biased by dust obscuration, which is known to be large in the southern part of the main disk. (For example, @Kotilainen96 estimated $A_{V} \sim 10.7$ mag for line emission toward the S nucleus.) As cautioned, the decoupled dynamical systems we suggest to exist around the double nucleus (see below) may have affected the parameters to some extent. We expect this effect to be small, because local distortion of the velocity field is not obvious around the double nucleus in the low resolution maps.
The dynamical mass within the radius of 10 $=$ 1.7 kpc of the main gas disk is estimated to be $M_{\rm dyn}$(M, $r\leq 1.7$ kpc) $= (1.2\pm 0.4) \times 10^{10}$ from the rotation velocity $V(r=10\arcsec)\sin i_{\rm M} = 125 \pm 20$ and the inclination (45$\pm$ 4) obtained from the kinematical fit. (See Fig. \[fig.high-res-pv\] (c) for the rotation velocity.) The fraction of gas in the dynamical mass is in the range of 0.1–0.5 for our estimate of gas mass and the central value of the dynamical mass.
Double Nucleus \[s.double\]
============================
gas distribution around the double nucleus
------------------------------------------
The high-resolution map in Fig. \[fig.cor-2s\] reveals compact gas peaks near or around the nuclei N and S. The southern peak is more compact than the northern one, which is more extended than the synthesized beam. The two gas peaks are connected with a bridge of CO emission. There are also arcs or partial spiral arms as well as secondary peaks that were already hinted in the lower resolution maps. The CO distribution traces some of the conspicuous dust lanes in the optical, as shown in Figure \[fig.coonhst\] that compares the CO contours with the $I$-band HST image of @Zepf99. Most notably, the following features correspond to dark lanes in the HST image: the arc-like dust lane about 4–8 east of the northern nucleus, the gas peak about 5 east of the southern nucleus, and the gas peak around the southern nucleus itself as well as the westward tail emanating from the peak. The ‘tail’ feature of molecular gas actually trails the S nucleus, judging from the clockwise rotation of the galaxy inferred from the spiral patterns in the HST image.
The (2–1) integrated intensity averaged within a 2-diameter aperture centered on each nucleus is 44 Jy for both of the nuclei. This corresponds to a surface density of neutral gas $1.3\times 10^3$ on the sky plane if we use the same scale as our mass estimate in the previous section and ignore the correction for missing flux. It also corresponds to a visual extinction of $A_{V} \sim 60$ mag for a uniform slab in front of the stars. The actual extinction toward the nuclei is obviously much smaller than this because at least near-IR images show both nuclei. @Kotilainen96 estimated $A_V \sim 2.4$ mag and $\sim 5.3$ mag for the continuum emission from N and S nucleus, respectively. The reasons for these smaller extinctions are most likely that stars and gas are mixed, that the gas and dust has a smaller scale height than the stars, and that the molecular gas consists of clumps.
It is notable, however, that the S nucleus suffers more extinction than the N nucleus despite the same gas surface densities inferred in the directions of the two nuclei. The absorbing column density suggested from the actual extinction is a factor of 2.2 larger for the S nucleus than for the N nucleus. This discrepancy suggests a geometrical effect. The S nucleus is probably more deeply embedded in gas and dust or located behind the main gas disk.
The majority of (2–1) emission in the galactic center comes from the main gas disk, despite the peaks around the two nuclei in the integrated intensity map. The areas within 1 from the two nuclei have only 8 % of the (2–1) flux observed in the $r_{\rm g} \leq 10$ region, which is the area within 10 $=$ 1.7 kpc from the midpoint of the double nucleus in the plane of the main gas disk. This fraction doubles, but is still small, if we apply a uniform correction for the flux resolved out in the highest resolution map.
gas kinematics and dynamical masses
-----------------------------------
### disturbances in the main disk
The high-resolution mean velocity map in Fig. \[fig.cor-2s\] shows the same overall rotation of the main gas disk as the lower resolution data. In addition it reveals small scale disturbances in the disk. The most notable of these outside the double nucleus is the noncircular motion associated with the spiral feature about 4–8 east of the N nucleus. The merged gas disk is certainly disturbed at the few 100 pc scale. However, we do not see in the main disk a double-peaked line profile that would have suggested two distinct components overlapping in our line of sight. It appears that most of the gas from the merger progenitors has coalesced in the main gas disk, except perhaps in close proximity to the two nuclei.
### possible mini-disks around the double nucleus \[s.mini-disks\]
On, and very close to, each nucleus, the velocity map indicates a steep velocity gradient in the east-west direction across the nucleus. The steep velocity gradients are confirmed in the position-velocity (PV) diagrams shown in Fig. \[fig.high-res-pv\]. The full velocity width within 1 of each nucleus is about 250 and 310 for the N and S nucleus, respectively. Our data do not show a clear sign of velocity offsets, i.e., relative motion, between the main gas disk and each nucleus.
We model the velocity gradient as rotation of gas around each nucleus. The Keplerian dynamical mass within 1 (= 170 pc) of each nucleus is $M_{\rm dyn}$(N, $r\leq 170$ pc) $= 6\times 10^{8} \sin^{-2} i_{\rm N}$ and $M_{\rm dyn}$(S, $r\leq 170$ pc) $=9\times 10^{8} \sin^{-2} i_{\rm S}$ in this model, where $i_{\rm N}$ and $i_{\rm S}$ are the inclinations of the spin axes with respect to our line of sight. The uncertainty in these dynamical masses is probably as large as a factor of 2 reflecting the uncertainty in the radii of the peak velocities. For $i_{\rm N} \sim i_{\rm S} \sim 45\arcdeg$, the dynamical masses are approximately $1\times 10^9$ and $2\times 10^9$ for N and S, respectively.
The dynamical mass for the N nucleus is consistent with (i.e., reasonably larger than) the optical estimate, $10^8$ within 40 pc, based on the HST STIS spectra across the nucleus [@Neff03; @Lipari04]. The mass of neutral gas within 1 radius in the suggested disk is about $1\times 10^8 \cos i_{\rm N}$ for the N disk (and $1\times 10^8 \cos i_{\rm S}$ for the S) if we scale the gas mass estimated in §\[s.gas-mass-estimate\] with (2–1) flux. The gas-to-dynamical mass ratio therefore does not prohibit the mini-disk model. The total mass of the two nuclei is at least 15 % of the dynamical mass within a 10 (1.7 kpc) radius from the dynamical center of the merger.
In the mini-disk model, the suggested rotation axis of each nucleus is roughly parallel to that of the main gas disk, and the sense of velocity gradient is about the same in the three disks with eastward being redshifted and westward blueshifted. This is consistent with the prograde-prograde merger that has been suggested for the galaxy. In the merger configuration, the spin axes of the progenitors and their orbital rotation axis are roughly parallel with each other and all rotation is in the same direction. Thus the dense gas disk to form around the core of each progenitor and the large gas disk to form from merged disk gas will have spin vectors in roughly the same direction and orientation, as in NGC 3256, if the transfer of angular momentum is negligible between the spin and orbital motion of the mini disks.
Obviously one has to be careful about the interpretation of the velocity structure in the main gas disk because of the unfortunate configuration that the two nuclei are aligned along the minor axis of the main disk. This configuration makes it difficult to separate the rotation of the main disk and that around each nucleus for a prograde-prograde system. We therefore made a position-velocity diagram across the midpoint M of the N and S nuclei. The PV diagram shown in Fig. \[fig.high-res-pv\] (c) still shows the steep velocity gradient that we saw on the nuclei N and S. There is, however, also a component with a shallower velocity gradient that almost linearly rises to the velocity offset of 120 at the offset of 10. We infer that this component with the shallow velocity gradient represents the main gas disk and that the central component with a steep velocity gradient is the contamination from the N and S nuclei. The contamination is quite possible because the midpoint M is only 25 from each nucleus while our spatial resolution in the north-south direction is 2.
This interpretation is supported by the PV diagram through the two nuclei and their midpoint (Fig. \[fig.high-res-pv\] (d)). It suggests wider line widths at the two nuclei than at their midpoint. Additional supporting evidence for the shallow rise of the rotation curve of the main disk is that the isovelocity contours in the mean velocity map (Figs. \[fig.naturalmap\] and \[fig.cor-2s\]) are roughly parallel to the minor axis within 5 from the dynamical center of the main disk. This indicates a near rigid-body rotation of the main disk in the region. It is consistent with the shallow velocity gradient in Fig. \[fig.high-res-pv\] (c) but not with a steeply rising rotation curve that approaches near its maximum velocity at a radius of 1.
Another cautionary remark is due for the outflow of ionized gas that @Lipari00 suggested to emanate from the N nucleus. We regard it as unlikely to be the cause of the steep velocity gradient of molecular gas across the nucleus. This is because the reported outflow is several kpc large and has an outflow axis in the position angle of 150 – 160. The flow axis is not aligned with the velocity gradient of the molecular gas but roughly perpendicular to it. Thus, although a starburst-driven wind can entrain molecular gas, the one reported from the optical observations would not cause the observed CO velocity gradient. Our estimate of the dynamical mass of the nuclei is therefore not affected by the outflow. No report of an outflow has been made for the S nucleus so far, although, if it exists, it would not be easily detected through the large extinction.
A qualitative argument supports the suggested gas disks around the two nuclei, though observational confirmation of them requires data at a higher resolution. If near-IR and radio peaks N and S are indeed the remnant nuclei of the merger progenitors that currently orbit in (or in and out of) the main gas disk, then collisional gas can follow each stellar nucleus to make a peak around it only if the gas is gravitationally bound to the nucleus. A massive remnant nucleus sinks toward the dynamical center because of dynamical friction. The gas around the nucleus tend to be left behind if not bound to the nucleus, because gas clouds are less massive than the nucleus and also because of hydrodynamical friction between the gas around the nucleus and that in the main gas disk. For cold molecular gas bound to each nucleus, the stable configuration is a rotationally supported disk around the nucleus. In this picture, some of the gas around each nucleus can be captured from the main gas disk.
### comparison with other galaxies
Small gas disks around merger nuclei separated by less than 1 kpc have been observed in Arp 220 [@Sakamoto99; @Mundell01]. Such a mini disk was also suggested in the center of M83 and attributed to a minor merger [@Sakamoto04]. In Arp 220, the dynamical mass of each disk is at least $1$–$2 \times 10^{9}$ within a radius of 100 pc and the two nuclei are separated by 0.3 kpc on the sky.
Another case of some similarity with NGC 3256 is the luminous infrared galaxy NGC 6240, which is a merger with two nuclei separated by 750 pc on the sky. @Tecza00 found a steep velocity gradient across each nucleus in the stellar velocity field measured with a CO absorption band, and inferred a pair of rotating [*stellar*]{} systems at the two nuclei with a dynamical mass for each of 2–8 $\times 10^9$ within a radius of 200 pc.
The pairs of compact mass concentrations in the two major mergers Arp 220 and NGC 6240 are comparable in size and mass with that in NGC 3256. NGC 3256 may be a new member of the (yet) small group of merging galaxies that show massive and kinematically-decoupled rotating disks in their central kpc.
### alternative models
Lastly, there could be a totally different interpretation of our high-resolution observations. Namely, one could assume that the putative nuclei N and S are not really remnant nuclei of the progenitors but just bright star forming regions and that they are there because there are compact concentrations of gas for star formation [c.f., @Eckart01]. This alternative would need a model to explain why there are two gas peaks in this galaxy and why there appears to be large velocity gradients across the gas clumps. We have not attempted such modeling. If both N and S have a low-luminosity AGN as suggested by @Neff03, then they are most likely galactic nuclei.
We also did not explore the opposite model by @Lipari00 that NGC 3256 has a third merger nucleus at 6 east of the N nucleus. The location is close to the CO spiral feature that we noted with its non-circular motion. However, the non-circular motion appears to be associated with the spiral rather than localized on the suggested nucleus. The line width at the position, at $-6$ in Fig. \[fig.high-res-pv\] (a), is less than half of the ones at the N and S nuclei. It is not significantly larger than the 100 line widths seen at other locations in the turbulent main gas disk (see Figs. \[fig.high-res-pv\] (b) and (c)). The third nucleus, if exists, appears to play a minor role in gas dynamics.
High-velocity gas \[s.high-velocity-gas\]
==========================================
observational signatures
------------------------
We have discovered high velocity molecular gas in the galactic center. This is most clearly seen in the position-velocity diagram in Figure \[fig.majpv\] as the high velocity component at the offset of 0. The high velocity gas is seen in both blueshifted and redshifted velocities and is more prominent in the latter. The PV diagram is along the major axis of the galaxy and is made by integrating the 6 wide area along the line of nodes. The double nucleus is within this 6-wide ‘slit’ and is also at the offset of 0. The data cube used for this PV diagram has 47 $\times$ 18 (FWHM) and 10 resolutions and has 92 % of the flux in the natural-weighting dataset. In this dataset, the redshifted gas is detected up to the velocity of 3120 , which is about 350 from the systemic velocity of the galaxy. The blueshifted gas at the galactic center is detected down to the velocity of 2580 , or about 200 from the systemic velocity, though an arm-like feature about 15 west of the double nucleus has velocities down to about 2540 . We searched in the range of 2400 – 3320 for emission at even more extreme velocities, but did not detect any at the abovementioned resolutions. The redshifted component, however, appears to have a faint tail up to the velocity of 3200 in a lower-resolution dataset as we see below. We did not detect the high velocity gas in and .
The high velocity gas is located [*between*]{} the two nuclei. This is seen in Figure \[fig.blue-red-gas\] that shows (2–1) maps made by averaging the velocities of the redshifted and blueshifted high velocity gas. In both maps, the high velocity gas is not peaked on either of the two nuclei but between them. The peak signal-to-noise ratio of the high velocity gas in the maps is 12.8 and 4.4 for the redshifted and blueshifted gas, respectively. Any error in continuum subtraction can hardly affect this emission, because the maximum flux density of the continuum is less than a third of the peak flux density of the fainter, blueshifted high velocity gas. The peak position of the redshifted high velocity gas is $\alpha =10^{\rm h}27^{\rm m}51\fs22$ and $\delta =-43\arcdeg54\arcmin16\farcs3$ (J2000) according to a Gaussian fit. The emission has a deconvolved size of $4\farcs6 \times 2\farcs5$ (P.A.=161). The emission is too weak, and possibly too extended, to be detected in our uniform weighing maps used in the previous section.
The high velocity gas appears almost at the same position in the redshifted and blueshifted velocities. Although the blueshifted emission is too weak to precisely determine its centroid position, the peak positions in the two maps in Figure \[fig.blue-red-gas\] agree within 13. In addition, we did not find a systematic velocity gradient within the redshifted high velocity gas. To check this, we made a mean velocity map of the high velocity gas to see velocity gradient in any particular direction, and also inspected channel maps to see if there was a systematic shift of the emission centroid. We did not detect a sign of velocity gradient in either of them. This can constrain the parameters in models for the high velocity gas (e.g., rotation curve for a rotation model and outflow geometry for an outflow model).
The integrated (2–1) flux of the high velocity gas is 3 % of the total (2–1) flux in the 5-diameter area at the galactic center. This is measured from the spectrum in Figure \[fig.hvgspec\]. The spectrum is extracted at the midpoint of the double nucleus from our natural weighted data convolved to the 5 resolution. The emission line spans from 2530 to 3200 when we use the first null on each side to determine the line edge. The full width at zero intensity (FWZI) of the emission is thus 670 . The CO flux in the red component above 2990 is 23 Jy in the central 5, that in the blue component below 2610 is 5 Jy , and the total flux in the line is 936 Jy .
molecular outflow model
-----------------------
### supporting evidence for outflow
We suggest that the high velocity gas is a bipolar outflow of molecular gas from the main gas disk. This is mainly because the galaxy is known to have a superwind extending several kpc out of the central starburst [@Scarrott96; @Moran99; @Heckman00; @Lipari00] and the flow of hot gas can entrain cold molecular gas. It is also because alternative explanations of the high velocity gas seem less likely as we see below. The location of the high velocity gas makes the main gas disk the most likely place from which the molecular wind emanates.
Among the existing observations of the superwind, @Heckman00 detected toward the central few arcsecond of NGC 3256 a doublet of D absorption line blueshifted by about 300 with respect to the systemic velocity of the galaxy. In our CO observations, the maximum velocity offsets are $+420$ and $-240$ with respect to the systemic velocity, according to the FWZI line width. The velocity offsets of the and CO lines are in good agreement in magnitude. Both lines trace neutral gas; the former traces atomic gas and the latter molecular gas. Thus the agreement supports the outflow model of the high velocity molecular gas.
### alternatives
Before going further into the outflow model, we briefly check its alternatives. First, rotation can cause high velocity. It would have to be the one around the dynamical center of the merger, where the high velocity gas is. The dynamical mass needed for the high velocity would be at least $3\times 10^{9}$ within a radius of 110 pc (=065) if we use the CO line width for velocity and the nominal offset between the blueshifted and redshifted CO peaks for size. The dynamical mass is larger than those of the two nuclei (§\[s.mini-disks\]) despite the lack of a $K$-band peak between them. In addition, the large mass does not fit the rotation curve that we inferred in §\[s.mini-disks\] to be nearly linearly rising to the radius of 10. The dynamical mass can be smaller and consistent with the observations if the extent of the high velocity gas is smaller, but a smaller gas disk means a higher mass density and a higher CO intensity in it. For example, a disk with a 10 pc radius would require the dynamical mass, mass density, and brightness temperature of $3\times 10^{8}$ , $5\times 10^{4}$ , and 100 K, respectively. An AGN may realize these conditions, but none has been found between the two nuclei. Thus we regard the rotation model less likely than the outflow model.
Other alternatives include locally large velocity dispersion, gas infall from out of the main disk, and an outflow driven by an AGN jet. The velocity dispersion does not seem to have a local source of turbulence, unless there is a special mechanism related to the merger gas dynamics. The gas infall from both (i.e., approaching and receding) sides toward the dynamical center seems too much a coincidence. No AGN jet has been seen in the region. Note that all the alternative models except the jet-driven outflow can not explain the blueshifted absorption, though the CO and features could be of different origins.
### outflow parameters
In the outflow model, we can estimate the following parameters from our observations.
[*Velocity: —*]{} If the outflow is well collimated along the rotation axis of the main gas disk then the maximum flow velocity corrected for inclination would be 600 for the receding flow and 350 for the approaching flow. However the outflow may have a wide opening angle. The inclination correction in that case would be smaller or even negligible.
[*Mass: —*]{} The mass of molecular gas involved in the high-velocity flow is computed to be $1.4\times 10^7$ and $0.3\times 10^7$ for the receding and approaching gas, respectively, from the (2–1) flux in the velocity ranges where the high velocity gas is distinguishable from the disk gas. These estimates are based on the (2–1)-to-gas mass scaling factor that we adopted for the galactic center (§\[s.gas-mass-estimate\]). Their uncertainties are larger than that for the disk gas because we do not know whether and how the (2–1) emissivity changes in the outflow gas. The contribution from atomic gas, which was a factor of 0.3 in the gas mass estimated in §\[s.gas-mass-estimate\], is excluded from the above numbers because atomic-to-molecular mass ratio in the outflow is likely different from that in the disk gas. The numbers given here suggest that [*molecular gas of the order of $10^7$ is being ejected from the central kiloparsec at velocities $\gtrsim 200$ in the molecular outflow.*]{} There is likely a larger amount of molecular gas in the outflow at smaller velocities.
[*Momentum and Kinetic Energy: —*]{} The momentum and kinetic energy of the molecular outflow are estimated using the same mass scaling and integration on the spectrum in Fig. \[fig.hvgspec\]. The red and blue high velocity gas have momentum of $8\times 10^{42}$ kg and $1\times 10^{42}$ kg , respectively, in the direction of our line of sight. The kinetic energy of the redshifted high-velocity gas is $1\times 10^{48}$ J, without correction for the inclination of the flow, while the blueshifted gas has one tenth of that energy. The unknown contribution of outflow gas slower than 200 becomes smaller for the momentum and kinetic energy because of the $(\Delta v)^{1}$ and $(\Delta v)^{2}$ weights used for these moments.
[*Timescale : —*]{} We estimate the characteristic timescale for the redshifted outflow with more than 200 velocity offset to be 2 Myr from the deconvolved size of the high velocity emission and the outflow velocity. This is not the age of the outflow, which should be comparable to the age of the starburst and is probably an order of magnitude larger. The timescale is rather the crossing time, for the high-velocity gas, of the $\lesssim 500$ pc region where the high velocity emission is detectable. For symmetry, we assume the same timescale for the blueshifted high velocity gas.
[*Fluxes and Mechanical Luminosity: —*]{} We can estimate the mass flux, momentum flux, and mechanical luminosity of the high velocity molecular outflow using the timescale and the parameters estimated above. The mass flux of molecular gas in the high velocity outflow is 9 , the momentum flux injected to the outflow is $1\times 10^{29}$ N, and the kinetic luminosity of the outflow gas is $2\times 10^{34}$ W. These values are for the high velocity molecular component. Contributions from atomic and ionized gas and those from slower velocity flow increase these numbers. For comparison, the average rates of $17^{+20}_{-9}$ , $10^{29.1 \pm0.5}$ N, and $10^{34.6\pm0.6}$ W have been reported for a sample of infrared luminous galaxies ($10^{11.36 \pm 0.4}$ ) from observations [@Rupke05].
The uncertainties in the parameters derived here are worth summarizing. The parameters involving gas mass inherit the uncertainties of the mass estimate, with an additional uncertainty from the possible variation of gas properties between the disk and the outflow. Those involving velocity have uncertainty due to the unknown outflow geometry. We did not apply an inclination correction, in effect assuming a wide opening angle of the flow. Still, velocity-related parameters are estimated from velocity components along our line of sight. If we apply the correction for a well-collimated outflow 45 inclined to our line of sight, then momentum and kinetic energy increase by a factor of 1.4 and 2, respectively. The momentum flux and the kinetic luminosity also increase by a factor of 1.4 and 2, respectively, since the crossing time does not change with the correction. As noted, the parameters do not include the contribution of the outflow gas whose low line-of-sight velocity does not distinguish the gas from the disk gas.
### energy and gas consumption budgets
The energy of the outflow can be supplied by a portion of the starburst. If we equally divide the far-IR luminosity of the galaxy among each of the two nuclei and the main disk, then the mechanical luminosity released from the starburst in the main disk is $3\times10^{35}$ W assuming a 10 Myr-old continuous starburst that has the Salpeter initial mass function (IMF) in the mass range of 1–100 [@Leitherer99]. Thus the molecular outflow is energetically possible with a wide margin. It can be caused by passing $\lesssim 10$ percent of the mechanical energy from the disk starburst to molecular gas; the rest can be used to drive the superwind of hot gas.
The star formation rate needed to generate the starburst luminosity is 10 under the same assumptions on the starburst. The rates are therefore of the same order of magnitude for the consumption of molecular gas by star formation and for the dispersal of molecular gas by the outflow. This comparison, however, should be viewed with caution because the star formation rate estimated from luminosity, as well as the gas outflow rate, has a large uncertainty. The former rate can change by a factor of 10 with different assumptions about the IMF, because high mass stars generate most luminosity while low mass stars usually determine most of the gas consumption.
Perhaps a more robust number to describe the significance of the molecular outflow is the depletion time of molecular gas solely by the outflow. It is estimated to be 70 ($=2/0.03$) Myr from the fraction of high velocity gas in the central 5, 0.03, and the time scale of the outflow, 2 Myr, under the simplistic assumption that the high velocity gas does not return[^5]. The timescale is comparable to the typical timescale of a starburst, again implying the significant role of the molecular outflow in the starburst. This timescale of gas depletion or dispersal by the outflow is estimated without explicitly using the CO-to- conversion factor. Hence the uncertainty in the depletion time due to the conversion factor is only indirect, through any variation of the scaling factor between the outflow and disk gas. We caution that the timescale is derived to measure the significance of the molecular outflow rather than to predict the evolution of the system, for which one needs to consider such factors as the time evolution of star formation, that of the outflow, and further gas accretion within the galaxy.
To summarize, our observations of the molecular outflow agree with the optical absorption-line surveys toward starburst galaxies [@Heckman00; @Rupke05] in that a starburst-driven superwind can be as important as the star formation itself in the gas consumption budget of a luminous infrared galaxy. @Rupke05 obtained a median value of 0.33 for the mass entrainment efficiency (i.e., the rate of mass loss due to superwind normalized by the star formation rate) in luminous infrared galaxies, to which NGC 3256 belongs. The efficiency for the molecular outflow in NGC 3256 is 1 if a third of the starburst in the galaxy contributes to the observed outflow, or 0.3 if the entire starburst contributes. Thus the entrainment efficiency for molecular gas in NGC 3256 agrees, in the order of magnitude, with the median efficiency estimated for atomic gas.
### geometry
We finally comment on the spatial configuration of the starburst in the galactic center. The spatial distribution of massive star formation is peaked toward the N nucleus as seen in \[\] and Br$\gamma$ [@Moorwood94], Pa$\alpha$ [@Alonso-Herrero02], H$\alpha$ and diffuse X-ray in the 0.3–10 keV range [@Lira02]. The S nucleus also shows a peak in Br$\gamma$, though its flux corrected for extinction is estimated to be 4 times smaller than that of the N nucleus [@Kotilainen96]. In contrast, the region between the two nuclei does not have a peak in these tracers of massive star formation, even though there are regions and super star clusters spreading between and around the two nuclei [@Alonso-Herrero02]. It is therefore somewhat puzzling to find a molecular outflow between the two nuclei and not from each nucleus, in particular from the nucleus N. Unfortunately the optical observations that indicated the presence of a superwind do not tell us the exact geometry of the wind and whether there are multiple superwinds in the region.
Clues to solve the puzzle may be that the base of the superwind likely has a size of 1 kpc or larger and that the two nuclei as well as the star forming clusters in the region have been moving with respect to the dynamical center of the system. @Heckman90 measured the pressure profile in the center of NGC 3256. They found that the central 1–2 kpc diameter region has very high pressure ($10^{-9.5}$ Pa) with a shallow decline outwards, and that the pressure gradient steepens outside the region. As they interpreted the profile, this is expected for a superwind with a base diameter of 1–2 kpc, within which static thermal pressure dominates [@Chevalier85; @Tomisaka86]. The starburst region, i.e., the region producing the high pressure in this case, thus encompasses the two nuclei. The two nuclei are about 500 pc from the dynamical center of the system and probably have a velocity of about 100 with respect to the dynamical center. Thus their crossing time across the central kpc is about 10 Myr. It is comparable to the typical timescale of a superwind. The two dominant energy sources thus move while a superwind develops, effectively making the region of energy injection larger than that of the two nuclei. Once a kiloparsec-size region of hot gas is formed by the moving energy sources, its evolution as a superwind is governed by the large scale structure of the ISM in which the hot gas is embedded. It is therefore conceivable that a main superwind centered around the dynamical center of the system blows out in the direction perpendicular to the main gas disk. This solution to the puzzle, however, seems to have a much room for improvement.
Molecular Gas Properties \[s.gas-properties\]
=============================================
temperature and filling factor \[s.gas\_temp\]
----------------------------------------------
Molecular gas in the center of NGC 3256 is warm. This can be seen in the map of peak brightness temperature shown in Fig. \[fig.cor-2s\]. The peak temperature in the map, in excess of 10 K, is high considering that it is averaged over the 170 pc $\times$ 340 pc beam. The map is made by searching for the maximum brightness temperature in the velocity direction of the data cube at each sky position. The peak brightness temperature gives a lower limit on the temperature of molecular gas at each position, except for low intensity regions dominated by noise. It is a lower limit because of beam dilution — the filling factor of gas in the beam can be much smaller than unity — and also because the emission may be optically thin. In particular, the map tends to give lower temperatures to regions with large line widths if the gas physical temperature and surface density are the same among regions. Despite these difficulties in reading the map, the map is indicative of higher gas temperature around the N nucleus and the tail region extending west from the S nucleus. As we noted, the N nucleus is the most active star forming region in the galactic center. The tail region corresponds to the tail feature seen in diffuse X ray emission [@Lira02 Fig. 12]. The small temperature depression on top of the peak at the N nucleus and unremarkable temperature at the S nucleus are at least partly due to the dilution caused by the large line widths there.
To put the beam diluted brightness temperature into perspective, we note that the peak value of 13 K in NGC 3256 [$L_{8-1000 \micron}=10^{11.56}$ ; @Sanders03] is a factor of 3 lower than that in the ultraluminous infrared galaxy Arp 220 ($10^{12.21}$ ) observed with the same line at almost the same linear resolution [@Sakamoto99]. It is also four times higher than the peak temperature of 3 K obtained in the starburst nucleus of M83 ($10^{10.10}$ ) by convolving the CO(2–1) observations of @Sakamoto04 to match in linear resolution. All three galaxies are moderately inclined with respect to our line of sight ($i \sim$ 25 – 45) and hence are expected to be similar in the degree of dilution due to galactic rotation. This comparison implies that molecular gas in galaxies with higher luminosity tends to be warmer or to have higher area filling factors or both when compared at 200 pc scale.
gas cloud properties and line excitation
----------------------------------------
### line ratio observations
NGC 3256 has a high / intensity ratio. Figure \[fig.cospectra\] compares the spectra of the three CO lines obtained with a simulated 30 beam (FWHM) centered on the midpoint of the double nucleus. The flux densities integrated from 2590 to 2970 are $4.62\times 10^{3}$, $1.37\times 10^{2}$, and $26.1$ in unit of Jy . The errors of these integrated intensities are 10% due to flux calibration and $\pm2.6$ Jy due to noise directly estimated from the maps. The flux calibration error, however, cancels out in a line ratio from our simultaneous observations. Thus the error in the line ratio is dominated by noise. We obtain integrated temperature ratios of $R_{12/13} (2-1) \equiv I({\mbox{$^{12}$CO}}\, 2-1) / I({\mbox{$^{13}$CO}}\, 2-1) = 30.9 \pm 0.6$, $R_{12/18} (2-1) \equiv I({\mbox{$^{12}$CO}}\, 2-1) / I({\mbox{C$^{18}$O}}\, 2-1) = 161 \pm 15$, and $R_{13/18} (2-1) \equiv I({\mbox{$^{13}$CO}}\, 2-1) / I({\mbox{C$^{18}$O}}\, 2-1) = 5.2 \pm 0.5$, where $I(\mbox{line}) \equiv \int \! T_{\rm b}(\mbox{line})dv$. Our $R_{12/13}$(2–1) ratio agrees with those from previous single-dish observations (see Fig. \[fig.coratio\]). The $R_{12/13}$(2–1) ratio averaged over 5 kpc in NGC 3256 is about three times higher than the ones observed in the central kpc of our Galaxy, $10\pm 1$ [@Sawada01], central 300 pc of the starburst galaxies IC 342 and NGC 253, $\sim 9$ [@Meier00 and Fig. \[fig.coratio\]], and in the central 500 pc of M82, 12 [@Mao00]. It is also higher than the $R_{12/13}$(1–0) values of $\approx 11$ observed in the centers of nearby spiral galaxies [@Sage91; @Aalto91b; @Paglione01]. On the other hand, the high $R_{12/13}$(2–1) ratio is within the range of values observed among luminous mergers, $\gtrsim 20$ [e.g., @Casoli92b; @Aalto95; @Glenn01].
The intensity ratios are not uniform in the galaxy. Our data indicate a trend that the $R_{12/13}$(2–1) ratio increases with the beam size (Fig. \[fig.coratio\]), suggesting higher ratios in the outer parts of the galaxy. The same trend is seen in $R_{12/18}$(2–1) with less significance. The $R_{13/18}$(2–1) ratio has too large uncertainty to detect its radial variation. The two nuclei have the lowest $R_{12/13}$(2–1) value of 17 in our measurements. One can infer that the ratio is locally low at the two nuclei by looking at the integrated intensity maps in Fig. \[fig.naturalmap\] where (2–1) shows peaks at the nuclei while (2–1) does not. The radial gradient of the / intensity ratio in the J=2–1 transition is opposite to that in the J=1–0 emission. The $R_{12/13}$(1–0) ratio at the center of the galaxy is high, 26–40 [@Aalto91a; @Casoli91; @Becker91; @Garay93], but it decreases to 11 at 43 NE of the galactic center [@Aalto95].
### gas model for the nucleus and the main disk
The two-component model of molecular gas suggested by @Aalto95 seems to work for our higher central $R_{12/13}$(2–1) value than in the centers of less luminous galaxies. In the model, molecular gas in a starburst nucleus has a significant amount of warm and turbulent envelope gas with moderate opacity ($\tau$\[(1–0)\] $\sim 1$ and $\tau$\[(1–0)\] $\ll 1$) around cloud cores of high optical depths ($\tau$\[(1–0)\] $\gg 1$ and $\tau$\[(1–0)\] $\sim 1$). The large contribution (i.e., area filling factor) of the warm envelope gas makes the intensity-weighted optical depth of emission moderate (1) and makes the $R_{12/13}$(1–0) ratio higher than in core-dominated quiescent gas. The warm gas with high area-filling factor noted in §\[s.gas\_temp\] fits the profile of the envelope gas. The dense and high-opacity cores, on the other hand, are supported by the (2–1)/(1–0) ratio close to unity, 1.3, in the center of NGC 3256 [@Casoli92b] and by the observations of dense-gas tracers such as HCN [@Casoli92a]. For the $R_{12/13}$ ratio in the J=2–1 transition, we note that the transition tends to have a higher opacity than the J=1–0 one. Thus we expect higher opacities in the envelope gas or, equivalently, a larger filling factor of optically thick cores. This and the presence of envelope gas make the $R_{12/13}$(2–1) ratio high at the galactic center but smaller than $R_{12/13}$(1–0), as observed.
The model needs to be slightly modified in the outer disk to accommodate the opposite radial trends in $R_{12/13}$(1–0) and $R_{12/13}$(2–1). For the outer disk, the same argument as above would lead to a lower $R_{12/13}$(2–1) value in the disk than in the nucleus. It is because, according to the model, the disk emission is dominated by cloud cores. Such a radial trend of the J=2–1 ratio contradicts our observations. One way to alleviate this is to assume that the CO excitation is subthermal in the disk cloud cores and that the cores have lower opacities in the J=2–1 transition than they are expected to have from their J=1–0 opacities if the cores were in LTE. In other words, many cloud cores in the merger disk may look like low-opacity envelopes in the J=2–1 lines. As a result, in the J=2–1 lines, the bulk of emission in the galaxy has moderate (in ) and moderate to low (in isotopes) optical depths throughout, with opacities increasing toward the center. The radial gradient of the $R_{12/18}$(2–1) ratio and the insignificant gradient of the $R_{18/13}$(2–1) ratio are consistent with this. The subthermal excitation in the disk could be due to lower gas density there than in the nucleus, large velocity dispersion in the merged gas disk, and lower fractional abundance of CO with respect to . The higher \[\]/\[\] abundance ratio expected from the fresh supply of low- gas from peripheries of the merger progenitors [@Casoli92b; @Henkel93] would also increase the $R_{12/13}$(2–1) ratio in low opacity clouds. Our conceptual model for the radial gradient of gas properties in NGC 3256 can be better constrained with more data. In particular, additional high-resolution data in other CO transitions, such as CO(1–0) or CO(3–2) would be most helpful to estimate the excitation conditions.
Our observations in terms of radial gradient of the $R_{12/13}$(2–1) ratio are consistent with those in the Antennae galaxy. @Zhu03 found that $R_{12/13}$(2–1) was about twice as high in the overlap region of the merger than in the centers of the colliding galaxies. Their galactic center ratios, $13\pm3$ and $16\pm4$ for each galaxy, are marginally higher than the ones we quoted for local starburst nuclei. The higher $R_{12/13}$(2–1) ratio in the overlap region, however, may be because the region has a starburst and resembles merger nuclei. @Aalto95 mentioned a decreasing $R_{12/13}$(2–1) gradient from the nucleus outwards in the polar ring galaxy NGC 660 and an increasing gradient in the merging galaxy NGC 2146. Gas properties and excitation conditions of CO may be diverse among mergers in their outer disks.
Discussion \[s.discussion\]
============================
Our observations have provided a wealth of new information on the molecular interstellar medium in the center of NGC 3256. We can place them in the general context of the galaxy merger and that of merger-induced starburst. They also point areas for further studies.
The main gas disk is what is expected from the dissipative nature of the interstellar medium and what has been predicted from numerical simulations [e.g., @Mihos96; @Barnes96]. In the simulations, gas from the disks of progenitor galaxies collides and eventually coalesces into a single disk rotating around the dynamical center of the merger, with a strong concentration toward the center. Such a merged gas disk has been observed in a number of luminous and ultraluminous merging galaxies [e.g., @Scoville97; @Downes98; @Bryant99; @Tacconi99]. In line with those models and observations, the main gas disk of NGC 3256 has the majority of molecular gas in the system and the overall gas distribution peaks toward the center of the merger. The main gas disk is certainly supported by rotation, with the center of rotation located between the two nuclei. Presumably reflecting the violent past of the system, however, the disk shows signs of disturbances in the forms of short spiral arms and non-circular motions.
Our observations reinforce the notion that properties of molecular gas in this merger as well as other luminous mergers are different from those in our Galaxy in the sense that the former contains more of warm, tenuous (i.e., high area-filling factor), and turbulent molecular gas [@Aalto95; @Scoville97; @Downes98]. This must be due to the galaxy merging and to the starburst in the center of the merger. Our observations also suggest that gas properties change in spatial scales from over a kiloparsec to as small as a few 100 pc, which is our highest resolution. Examples of such spatial variation include the highest brightness temperature gas associated with the N nucleus that hosts the most active star formation in the galaxy, the local minima of (2–1)/(2–1) ratio on each of the two nuclei, and the global radial gradient of the intensity ratio in the main gas disk. While these gas properties are probably the results of the merger and the starburst, the gas properties are also expected to influence the star formation from molecular gas. Therefore the study of gas properties at high resolutions helps to model the evolution of a starburst. We noted a trend of higher CO brightness temperature in more luminous galaxies. Statistical studies to probe trends like this and trends in line ratios will help characterize the molecular gas in mergers and starbursts.
The [*molecular*]{} outflow from a merger has little observations in luminous or ultraluminous infrared galaxies of $\log(L_{\rm IR}/{\mbox{$L_\odot$}}) \geq 11.5$, though surveys of atomic and ionized gas have shown that most luminous and ultraluminous infrared galaxies have a superwind [@Heckman90; @Heckman00; @Martin05; @Rupke05]. The recent detection by @Takano05 of high velocity gas in NH$_3$ absorption toward Arp 220 may be one of the rare observations of molecular outflow from a luminous infrared merger, although other interpretations are possible for the absorbing gas. In the model of superwind, an intense starburst deposits a large amount of kinetic energy from stellar winds and supernovae into the interstellar medium. The energy is thermalized in the ISM to generate hot gas of $10^{6}$–$10^{7}$ K that can not be contained in the galactic potential and blows out of the galactic disk. Although molecular gas can not be hot enough to blow out by itself, it is expected to be entrained by the wind of hot gas. There are examples of such entrainment of molecular gas in nearby starburst galaxies of lower luminosities, e.g., in M82 [@Nakai87; @Seaquist01; @Walter02]. It is therefore not surprising to detect a molecular outflow from such a luminous starburst as NGC 3256 in sensitive observations. The outflow parameters estimated in NGC 3256, i.e., the maximum outflow velocity, mass flux, momentum flux, and mechanical luminosity, are in order-of-magnitude agreement with those obtained from the survey of atomic Na absorption in luminous and ultraluminous infrared galaxies by @Rupke05. In particular, we reiterate the importance of gas outflow in the gas budget of the galactic center; molecular outflow can be nearly as significant as star formation in gas consumption. It is almost certain that we detected only a portion of the molecular outflow in NGC 3256, limited by our sensitivity. Observations at higher sensitivity and resolution would shed more light on the mechanism for molecular gas entrainment and the effects of the evacuation of molecular gas on the starburst evolution. Such observations will also help testing the alternative interpretations of the high velocity gas.
Regarding the double nucleus, we suggest that the gas peaks around and the steep velocity gradient across each nucleus are most likely due to a mini disk rotating around the nucleus, although the mini disks certainly merit further observations for confirmation. There is a striking similarity among NGC 3256, Arp 220, and NGC 6240 in terms of their twin (possible) remnant nuclei of a few 100 pc sizes and $10^{9}$ masses within the projected separation of 1 kpc. In the earlier phase of galaxy collision, numerical simulations suggest that a sizable fraction of gas in each galaxy can be funneled to the nucleus of each galaxy, forming a massive rotating core of gas and that of stars through starburst [@Noguchi88; @Barnes96; @Mihos96]. Observations of interacting galaxies find such starbursting gas concentrations in many, though not all, galaxies [@Aalto97; @Wilson00; @Yun01; @Kaufman02; @Evans02]. As the merger proceeds, stellar systems will merge from outside, leaving nuclear stellar cores until the final stages [@Funato92; @Boylan-Kolchin04]. Gas from the merger progenitors will also coalesce first from outside to form a large merged disk, which is most likely the main gas disk in NGC 3256. In the mean time, the nuclear gas trapped deep inside the gravitational potential of each progenitor likely remains within the nucleus. The pairs of massive nuclei in the three merging galaxies are plausibly the remnants of such massive cores of stars and gas. (Gas was presumably striped from the stellar cores in NGC 6240.) The massive remnant nuclei should strongly perturb the larger, merged gas disk as they sink toward the dynamical center of the system owing to dynamical friction. Recently @Escala05 suggested from numerical simulations of a binary massive black hole in a merger that cold gas in the central merged disk can be a significant source of dynamical friction and that the gas disk is affected in return forming a gap or spirals or both around the binary. The situation may be similar in NGC 3256. The gas concentration around each nucleus should eventually contact the gas around the other nucleus or in the main disk, loses angular momentum, and probably falls toward the dynamical center of the system to become a part of the main gas disk. The complicated dynamics of gas, stars, and presumably massive black holes, remains to be studied along with the evolution of starburst in the perturbed region. Our observations show that NGC 3256, along with Arp 220, is one of the best targets for such studies.
Conclusions \[s.conclusions\]
=============================
We have made the first interferometric observations of molecular gas and dust in the center of the southern luminous merger NGC 3256, improving the spatial resolution over previous single-dish observations by an order of magnitude. Our main findings are the following:
1. There is a large disk of molecular gas ($r > 3$ kpc) with a strong concentration toward the double nucleus. The velocity field of the disk suggests rotation around a point between the two nuclei. The mass of neutral gas in the central 20 ($r \leq 1.7$ kpc) is estimated to be $3 \times 10^9 {\mbox{$M_\odot$}}$. Our observations suggest that the molecular gas in the merger system has mostly merged and settled into this main disk at the radii of 1–5 kpc.
2. In the central kiloparsec, our highest resolution ($1\arcsec \times 2\arcsec =$ 170 pc $\times$ 340 pc) image shows that molecular gas peaks near or at each of the two nuclei. There is also a steep velocity gradient across each nucleus, suggestive of a small gas disk rotating around the nucleus. The dynamical mass of each nucleus is estimated from the gas velocity to be $10^{9}$ within a radius of 1 (= 170 pc). The suggested configuration — two mini disks around the binary nuclei along with a larger main disk — is similar to that in Arp 220.
3. We discovered high velocity molecular gas of $ 10^{7}$ at the galactic center between the two nuclei. It is up to about 400 from the systemic velocity of the galaxy. The terminal velocity is about two times larger than that due to rotation of the main gas disk. We have modeled the high velocity gas as a molecular outflow, entrained by the superwind that the starburst in the central kiloparsec is known to have created. Outflow parameters, including the outflow rate of 10 , are estimated from the data. They suggest that the dispersal of molecular gas from the starburst region can be as significant as the gas consumption by star formation in the gas budget.
4. Our simultaneous observations of the (2–1), (2–1), and (2–1) emission provided accurate measurements of intensity ratio among those lines. The / intensity ratio in the $J$=2–1 transition is higher at the nucleus than in the centers of our Galaxy and nearby starburst galaxies. Our observations are consistent with the model that the intense starburst in the merger nucleus is taking place in warm turbulent gas with a high filling factor and moderate opacity. Positive radial gradients of / and / ratios are observed in the $J$=2–1 transitions. They may reflect subthermal excitation in the merger disk.
We appreciate the SMA staff for their skilled help before and during the observations. We are also grateful to Barry Rothberg for kindly providing us with his near-IR image of the galaxy, and to Kohji Tomisaka, Keiichi Wada, and Junichiro Makino for their helpful comments. The presentation of the paper is improved by the comments from the referee. The blue image in Fig. \[fig.opt\] is based on photographic data obtained using The UK Schmidt Telescope. The UK Schmidt Telescope was operated by the Royal Observatory Edinburgh, with funding from the UK Science and Engineering Research Council, until 1988 June, and thereafter by the Anglo-Australian Observatory. Original plate material is copyright (c) the Royal Observatory Edinburgh and the Anglo-Australian Observatory. The plates were processed into the present compressed digital form with their permission. The Digitized Sky Survey was produced at the Space Telescope Science Institute under US Government grant NAG W-2166. This research has made use of the NASA/IPAC Extragalactic Database (NED), and of NASA’s Astrophysics Data System Bibliographic Services. This research is based in part on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Institute. STScI is operated by the association of Universities for Research in Astronomy, Inc. under the NASA contract NAS 5-26555.
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[^1]: The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics, and is funded by the Smithsonian Institution and the Academia Sinica.
[^2]: Only the (2–1) flux of @Aalto91a is cited here because their (2–1) flux is more than twice larger than the other two observations.
[^3]: The size of the SEST beam given in the papers ranges from 22 at 1.3 mm (=230.6 GHz) to 25 at 230 GHz. The conversion factor from temperature to flux density scales with the square of the beam size for a compact source. Thus the single-dish flux values given above increase by 18% if we adopt the largest beam size.
[^4]: The NASA/IPAC Extragalactic Database (NED) is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
[^5]: We can not easily assess whether the molecular gas escapes the galaxy or not, without information about the current height of molecular gas above the main disk, the shape of the gravitational potential, and structure of the superwind. If the gas returns, then the timescale is the one during which all the gas experiences the out-of-plane trip at least once. The molecular gas may well be dissociated and ionized in the hot wind during the trip.
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abstract: 'In this paper, a new fractional step method is proposed for simulating stiff and nonstiff chemically reacting flows. In stiff cases, a well-known spurious numerical phenomenon, i.e. the incorrect propagation speed of discontinuities, may be produced by general fractional step methods due to the under-resolved discretization in both space and time. The previous random projection method has been successfully applied for stiff detonation capturing in under-resolved conditions. Not to randomly project the intermediate state into two presumed equilibrium states (completely burnt or unburnt) as in the random projection method, the present study is to randomly choose the time-dependent advance or stop of a reaction process. Each one-way reaction has been decoupled from the multi-reaction kinetics using operator splitting and the local smeared temperature due to numerical dissipation of shock-capturing schemes is compared with a random one within two limited temperatures corresponding to the advance and its inverse states, respectively, to control the random reaction. The random activation or deactivation in the reaction step is thus promising to correct the deterministic accumulative error of the propagation of discontinuities. Extensive numerical experiments, including model problems and realistic reacting flows in one and two dimensions, demonstrate this expectation as well as the effectiveness and robustness of the method. Meanwhile, for nonstiff problems when spatial and temporal resolutions are fine, the proposed random method recovers the results as general fractional step methods, owing to the increasing possibility of activation with diminishing randomness by adding a shift term.'
address: 'Chair of Aerodynamics and Fluid Mechanics, Department of Mechanical Engineering, Technical University of Munich'
author:
- 'Jian-Hang Wang'
- Shucheng Pan
- 'Xiangyu Y. Hu'
- 'Nikolaus A. Adams'
bibliography:
- 'split-random.bib'
title: A Split Random Reaction Method for Stiff and Nonstiff Chemically Reacting Flows
---
Chemically reacting flows, Stiff source terms, Nonequilibrium kinetics, Fractional step methods, Operator splitting, Wrong propagation speed of discontinuities
Introduction
============
One of the main numerical challenges for chemically reacting flows is that the chemical kinetics often includes reactions with widely varying time scales, which may be orders of magnitude faster than the fluid dynamical time scale [@bao2000random; @bao2001random; @bao2002random]. Consider a combustion problem, where the chemical reaction, i.e. the burning process, may be much faster than the gas flow for example. This leads to severe problems of numerical stiffness due to the source terms representing reactions [@yee2013spurious]. When the chemical scales are not resolved numerically in time and space (using a grid size larger than the width of the reaction zone), it is not only impossible to capture the detailed structure of the reaction zone (such as the von Neumann spike), but also might calculate a spurious solution with the incorrect propagation of discontinuities and nonphysical states, even though standard dissipative numerical methods that were developed for non-reacting flows with good performance are employed.
The latter numerical phenomenon is well-known and has been an active area of research in the past three decades. It was first observed by Colella *et al.* [@colella1986theoretical] in 1986 who considered both the reactive Euler equations and a simplified system obtained by coupling the inviscid Bergers equation with a single reaction equation. LeVeque & Yee [@leveque1990study] showed that a similar spurious propagation phenomenon can happen even with scalar equations, by properly defining a model problem with a stiff source term. By analysis of such a simple scalar problem, they found that the propagation error is mainly due to numerical dissipation contained in the scheme, which smears the discontinuity front and activates the source term in a nonphysical manner. To overcome this difficulty, a natural strategy is to avoid any numerical dissipation in the scheme [@bao2002random; @zhang2014equilibrium] or to use sufficiently fine mesh. By using a front-tracking approach such as the ghost fluid/level-set method [@nguyen2002fully; @bourlioux1991theoretical; @bourlioux1992theoretical] or the local grid/timestep refinement [@jeltsch1999error; @bihari1999multiresolution], the correct propagation speed of the reactive front may be obtained. The random choice method proposed by Chorin in [@chorin1976random; @chorin1977random] had been successfully used in [@colella1986theoretical; @majda1990numerical] for the solution of under-resolved detonation waves, which is based on the exact solution of the Riemann problem at randomly chosen locations within the computational cells and does not introduce any viscosity. In [@deng2017new], Deng *et al.* introduced a hybrid reconstruction scheme named MUSCL-THINC-BVD to reduce numerical dissipation around discontinuities significantly to a tolerable level for the examined model experiments.
However, in wider areas resolution of fine scale is not always realistic due to expensive computational costs, unless one is interested in the detailed structure of a detonation wave. The best one can hope is to capture the speed of the discontinuity as well as other global features of the fluid dynamics [@bao2002random]. Also, since numerical dissipation/viscosity is an essential feature of modern shock-capturing schemes with considerable popularity, there is another category of works focused on accepting the diffused profiles by shock-capturing schemes and then make careful use of the averaged information for the correct ignition of source terms in the following reaction step. Engquist [@engquist1991robust] presented a simple temperature extrapolation method, which uses an extrapolated temperature from outside the shock profile to activate the chemical source term. This approach is easily extended to multi-dimensions, but it does not work well in insufficient spatial resolutions. In [@berkenbosch1998detonation], Berkernbosch suggested introducing a suitable ignition temperature which is considerably lower than any temperature actually found in the reaction zone of a resolved detonation. Helzel [@helzel2000modified] proposed a modified fractional step method for under-resolved detonation waves, in which the exact Riemann solution is required to determine where burning should occur. Tosatto & Vigevano [@tosatto2008numerical] proposed a MinMax method, based on a two-value variable reconstruction within each cell, where the appropriate maximum and minimum values of the unknown are considered within the local neighbouring cells. In [@wang2012high] Wang *et al.* proposed a new high-order finite-difference method utilizing the idea of Harten ENO subcell resolution method for stiff source terms with a single reaction and in [@yee2013spurious] well-balanced high-order nonlinear filter schemes were added to the subcell resolution method for reacting flows, effectively delaying the onset of wrong speed of propagation in coarse grids and moderate stiff source terms. When the grid is refined, a counter-intuitive spurious behavior (see [@yee2013spurious; @zhang2015short]) with incorrect shock location was observed. All these methods are confronted with difficulties in the extension to either high-dimensional or multi-species/multi-reaction kinetics based reacting flows. Zhang *et al.* [@zhang2014equilibrium] reported their equilibrium state method with the idea of replacing the cell average representation with a two-equilibrium-state reconstruction. The two equilibrium states are locally defined in each transition cell, making its extension to high dimensions straightforward. They also extended the method to a simple multi-reaction system by treating the two one-way reactions totally independent. Unfortunately, realistic nonequilibrium chemical kinetics with multiple finite-rate reversible reactions has not been discussed in any literature so far. In [@bao2000random; @bao2001random; @bao2002random], Bao & Jin introduced a random projection method for the reaction step by replacing the ignition temperature with a uniformly distributed random variable. Although the random projection method cannot avoid the introduction of numerical dissipation by shock-capturing schemes, it can eliminate the effect of any numerical dissipation, even with a 1st-order shock-capturing scheme, owing to its random nature. The method was strictly proved using a scalar problem and successfully applied to various model problems of 1D or 2D reactive Euler equations. With the presumption of two time-independent equilibrium states of totally burnt and unburnt gases (regardless of the detailed reaction process), the method is only suitable for under-resolved stiff cases. Here we further discuss the fractional step method using an arbitrary shock-capturing scheme to capture stiff detonation waves in under-resolved conditions. More generally, the main goal of this study is to simulate chemically reacting flows with real-world multi-species multi-reaction nonequilibrium chemistry in a unified manner, regardless of the stiff/nonstiff source terms or the under-/well-resolved conditions in grid and timestep. For the convection step, any modern shock-capturing scheme can be used to solve the homogeneous conservation laws apart from the ordinary differential equations (ODEs) for reaction source terms. Following the convection step, the zero-dimensional ODEs based on the present local smeared state in each cell/point is to be solved in the reaction step of fractional step methods. The idea of random projection method implies the reaction being activated or deactivated in one reaction step has no direct correlation with the final correct shock location, unless the activation (in both scalar and Euler problems) or deactivation (only in scalar problems) constantly occurs without restriction. The accumulative error will grow with time and leads to spurious propagation of discontinuities in a long run. On the contrary, if the activation or deactivation can occur alternatively and randomly according to a certain possibility, the correct shock location can be obtained with temporal convergence. Unlike Bao & Jin’s random projection method, the activation and deactivation of chemical reactions in our proposed method will not be projected into two prescribed equilibrium states, as a *priori*, but two time-dependent states corresponding to advancing the reaction in one timestep forward and making the reaction stand still, respectively. The criterion to the progress of a reaction is by comparing the local smeared temperature with a randomized temperature depending on the advance state and the state of the inverse of advance. In this way, every reaction step contains an effect of the predictor-corrector algorithm (predictor is the advance state and corrector draws the predictor back to the current state) for the correct and controllable propagation of the reacting front. Besides, by adding a shift term into the random temperature sampling when the resolution is improving, the chosen random temperature tends to be below the mean value of the two limited temperatures and thus activation of the reaction is increasingly possible to happen as the deterministic methods always do. That is, the proposed method recovers the solution of a general fractional step method in nonstiff cases when the spatial and temporal resolutions are fine to resolve the reaction scales. Consequently, the method is promising for both stiff and nonstiff problems in under-resolved and resolved conditions.
On the other hand, different from the famous ODEs solvers such as the implicit solver VODE [@brown1989vode], explicit CHEMEQ2 [@mott2001chemeq2], scale-separated MTS/HMTS [@gou2010dynamic] and the recent quasi-steady-state approximation based ERENA [@morii2016erena], the present random ODEs solver basically takes the advantages of the Split Single Reaction Integrator (SSRI) [@nguyen2009mass] for chemical kinetics in both mass conservation and preserving the positivity of mass fractions. Using analytical solutions in SSRI or the approximate exact solution in our development, almost unconditional stability can be a promise for the present ODEs solver. Therefore, even when the timestep is large and under-resolved for small chemical time scales, the ODEs solver is still able to work effectively and also paves the way for subsequent randomization of each reaction. Not limited to model problems with simplified kinetics reported in previous literature, operator splitting upon the reaction system makes the proposed random method applicable for real-world reacting flows with complicated nonequilibrium chemistry involving multiple species and reactions, e.g. the hydrogen-air combustion kinetics.
The paper is organized as follows. In Section \[Formulation\], we introduce the concerned reactive Euler equations with chemical reaction source terms. A standard fractional step method to solve the Euler system is outlined by operator splitting into the convection step and reaction step. In the reaction step, a new ODEs solver, as the generalization of SSRI, is developed to approximate the exact solution with advantages of exact mass conservation and strict definite positivity as well as almost unconditional stability. Based on the split reaction-by-reaction ODEs solver for general chemical kinetics, individual random reaction between advancing and stopping its process can be realized to correct the deterministic spurious propagation of discontinuities in stiff and under-resolved conditions. Next in Section \[section2\], by comparing with other standard methods, we examine the pure ODEs solver and the split random reaction method as a new fractional step method for capturing stiff detonations, respectively, by extensive classical model examples and realistic reacting flows in both 1D and 2D numerically. Conclusions will be drawn in the last section. More information about the ODEs solver and reaction mechanism used in numerical tests are provided in the final appendices.
Formulation {#Formulation}
===========
We have a first glance at the mathematical model of the time-dependent reacting flows involving nonequilibrium chemical kinetics, i.e. reactive Euler equations with chemical source terms. Assuming the flow is compressible, inviscid and in two dimensions for simplicity, the multi-species Euler equations coupled with reaction source terms take the form $$\label{Euler_eq}
\begin{aligned}
U_t + F(U)_x + G(U)_y = S(U),
\end{aligned}$$ where $$\label{Euler_eq1}
\begin{aligned}
U=
\begin{pmatrix}
\rho\\
\rho u\\
\rho v\\
\rho e_t\\
\rho y_1\\
\rho y_2\\
\cdots\\
\rho y_{N_s-1}\\
\end{pmatrix},
F(U)=
\begin{pmatrix}
\rho u\\
\rho u^2 + p\\
\rho u v\\
(\rho e_t + p) u\\
\rho u y_1\\
\rho u y_2\\
\cdots\\
\rho u y_{N_s-1}\\
\end{pmatrix},
G(U)=
\begin{pmatrix}
\rho v\\
\rho u v\\
\rho v^2 + p\\
(\rho e_t + p) v\\
\rho v y_1\\
\rho v y_2\\
\cdots\\
\rho v y_{N_s-1}\\
\end{pmatrix},
S(U)=
\begin{pmatrix}
0\\
0\\
0\\
0\\
\dot{\omega_1}\\
\dot{\omega_2}\\
\cdots\\
\dot{\omega_{N_s-1}}\\
\end{pmatrix}
\end{aligned}$$ are vectors of the conserved variables, advection flux in the x- or y-direction and source terms, respectively, with $\dot{\omega_i}$ representing the rate of change of species $i$ in the reactive gas mixture due to the chemical kinetics consisting of $N_r$ reactions and $N_s$ species. Furthermore, $e_t = e + \frac{1}{2}(u^2+v^2)$ is the specific total energy including the specific internal energy $e$. To the closure of the system, the equation of state (EoS) for the chemically reactive mixture should be added. Thus the density $\rho$, pressure $p$ and temperature $T$ of the gas mixture can be explicitly connected by $$\label{EoS}
\begin{aligned}
p = \rho \sum^{N_s}_{i=1} y_i \frac{R_u}{W_i} T,
\end{aligned}$$ with $y_i$ and $W_i$ denoting the mass fraction and molecular weight of the $i$-th species, respectively, and $R_u$ being the universal gas constant.
The above conservation laws of mass, momentums and energy with source terms are usually solved numerically in a fractional step manner, i.e. based on operator splitting, we have a set of partial differential equations (PDEs) for the homogeneous fluid transport dynamics $$\label{S_c}
\begin{aligned}
S_c:\quad U_t + F(U)_x + G(U)_y = 0
\end{aligned}$$ assuming the chemical reactions are frozen and mass fractions of all species are transported during the pure convection process, apart from the system of ODEs in the chemical kinetics $$\label{S_r}
\begin{aligned}
S_r:\quad \frac{dy_i}{dt} = \frac{\dot{\omega_i}}{\rho},\quad i=1,\dots,N_s,
\end{aligned}$$ under adiabatic and constant-volume conditions with fixed total density and constant specific internal energy. The first-order accurate Lie splitting scheme [@mclachlan2002splitting] (also known as Godunov splitting [@toro2013riemann]) or the second-order Strang splitting [@strang1968construction] can be employed to approximate the solution from the discrete time level $n$ to $n+1$ with a timestep of $\Delta t$, in the following forms $$\label{Lie}
\begin{aligned}
U^{n+1} = S_r^{(\Delta t)} \circ S_c^{(\Delta t)} U^{n},
\end{aligned}$$ or $$\label{Strang}
\begin{aligned}
U^{n+1} = S_c^{(\frac{\Delta t}{2})} \circ S_r^{(\Delta t)} \circ S_c^{(\frac{\Delta t}{2})} U^{n}.
\end{aligned}$$ In many practical cases nearly identical results are obtained with both splitting schemes [@deiterding2003parallel]. Regardless of the selection of operator splitting schemes, the method of fractional steps decouples the physical processes of hydrodynamic transport and chemical reaction, i.e. a convection step and a reaction step from a computational viewpoint. Accordingly, for the convection operator $S_c$, any modern shock-capturing methods especially some high-order low-dissipation schemes such as WENO-JS5 [@jiang1996efficient], WENO-CU6 [@hu2010adaptive] and the recently proposed TENO6 [@fu2016family] with local/global Lax-Friedrich flux splitting can be adopted. Also, in the reaction step, for $S_r$, any ODEs solver such as VODE, CHEMEQ2 and MTS, etc., can be conveniently implemented as a “black-box”, intaking $\{y_1,\dots,y_{N_s}\}^{n}$ and outputting $\{y_1,\dots,y_{N_s}\}^{n+1}$ with several case-dependent constant inputs such as $e$, $\rho$, $T$, etc. Besides, local sub-stepping/cycling can be presumed or executed adaptively in the ODEs solver.
Despite using high-order shock-capturing schemes in $S_c$, numerical dissipation or viscosity is inherently existing. The captured discontinuities in the discrete space will therefore be smeared instead of sharp jumps, which indicates the predicted properties in such smeared locations/areas of the flowfield are numerically averaged properties rather than physically realistic ones. It is the nonphysical properties in the smeared discontinuities that further induce the incorrect (too early) ignition of chemical reactions by pointwisely evaluating the source terms and finally lead to a spurious solution with a bifurcating wave pattern and wrong propagation speed of the reacting front. On the other hand, with more or less numerical viscosity, modern high-resolution shock-capturing schemes benefit in good robustness and accuracy, being widely accepted for solving homogeneous conservation laws in practice. From this aspect, it is highly desirable to develop methods for reacting flows that, instead of avoiding the numerical viscosity, make correct use of it.
Therefore, attention has to be paid from $S_c$ to $S_r$: we introduce and improve SSRI for solving ODEs of the nonequilibrium chemical kinetics so that the multi-reaction system can be decoupled into a series of single reaction steps. Then we introduce the idea of random projection into the ODEs solver in order to realize the random ignition of reactions. In our development, each reaction process will be randomly advanced one timestep forward (activation) or be ceased (deactivation) instead of being projected into two prescribed equilibrium states (completely burnt and unburnt). In this way, the randomization of reactions can be achieved for the general real-world nonequilibrium kinetics of multiple finite-rate reactions, no matter the source terms are stiff or nonstiff and the numerical discretization in space and time is under-resolved or resolved, in a unified manner. Hereafter, we term the randomized and reaction-by-reaction ODEs solver for the nonequilibrium chemistry, to be Split Random Reaction Method (SRR) in the reaction step $S_r$, independent of the convection operator $S_c$.
Split reaction-by-reaction ODEs solver for chemical kinetics
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In a common nonequilibrium chemical kinetics accounting for the ODEs in Eq. , chemical production rates are derived from a reaction mechanism that consists of $N_s$ species and $N_r$ reactions $$\sum_{i=1}^{N_s} \nu_{ji}^f X_i \Longleftrightarrow \sum_{i=1}^{N_s} \nu_{ji}^b X_i, \quad j=1,\dots,N_r,$$ where $\nu_{ji}^f$ and $\nu_{ji}^b$ are the stoichiometric coefficients of species $i$ appearing as a reactant and as a product in reaction $j$. The net production rate of species $i$ in Eqs. and is usually the summation of the production rate from each single elementary reaction as $$\dot{\omega_i} = W_i \sum_{j=1}^{N_r} (\nu_{ji}^b-\nu_{ji}^f) \left[ k_j^f \prod_{l=1}^{N_s} \left[\frac{\rho_l}{W_l}\right]^{\nu_{jl}^f} - k_j^b \prod_{l=1}^{N_s} \left[\frac{\rho_l}{W_l}\right]^{\nu_{jl}^b} \right]$$ with $k_j^f$ and $k_j^b$ denoting the forward and backward reaction rate of each chemical reaction. Note that reactions are reversible here for the sake of generality.
In SSRI, Nguyen *et al.* successfully utilize operator splitting in a reaction-by-reaction manner to decouple the above multi-reaction system in order to achieve definite positivity and mass conservation during the temporal integration. However, only simple one-way reactions with constant rates and two or three reactants at most are considered, using an analytical exact solution. For reactions with more than three reactants or the stoichiometric coefficients of reactants are larger than one, which indicates the overall order of the reaction is usually higher than two, analytical solutions are explicitly unavailable or difficult to derive. Alternatively, numerical solutions which require root-finding algorithms result in additional computational costs.
Following the idea of SSRI, we also decouple the multi-reaction system by operator splitting at first, taking the Lie splitting for example, which means during a given timestep, we traverse all the reactions by visiting each reaction separately and successively. That is, one simply needs to consider the effect of one reaction on the mass production or consumption of species involved in this reaction and then move on to the next one till the completeness of traversal $$\label{R1st}
\begin{aligned}
S_r: \quad R_{1st}^{(\Delta t)} = R_{N_r}^{(\Delta t)} \circ R_{N_r-1}^{(\Delta t)} \circ \cdots \circ R_2^{(\Delta t)} \circ R_1^{(\Delta t)},
\end{aligned}$$ where each $R_j$ corresponds to a single reaction channel, independent of all other reactions. The reaction-by-reaction idea agrees with the physical reality that in a microscopic scale, one molecule/atom can only experience one reaction or event with others or by itself solely at one time instance, which is the case in the stochastic simulation of chemical kinetics [@gibson1999efficient]. Unsurprisingly in a macroscopic scale, the reactions involving large numbers of species molecules/atoms can be treated as simultaneously occurring processes. In the original SSRI, the second-order accurate Strang splitting is adopted and the traversal goes forward first from the fastest reaction to the lowest one for half a timestep and goes backward in a reverse direction afterwards for the rest half timestep. Here we take the traversal order not according to reaction rates but to the number of index in the reaction mechanism that we adopt, which is more general but simpler without loss of the convergence rate, i.e. $$\label{R2nd}
\begin{aligned}
S_r: \quad R_{2nd}^{(\Delta t)} & = R_1^{(\frac{\Delta t}{2})} \circ R_2^{(\frac{\Delta t}{2})} \circ \cdots \circ R_{N_r-1}^{(\frac{\Delta t}{2})} \circ R_{N_r}^{(\frac{\Delta t}{2})} \circ
R_{N_r}^{(\frac{\Delta t}{2})} \circ R_{N_r-1}^{(\frac{\Delta t}{2})} \circ \cdots \circ R_2^{(\frac{\Delta t}{2})} \circ R_1^{(\frac{\Delta t}{2})} \\
& = \overline{ R_{1st}^{(\frac{\Delta t}{2})} } \circ R_{1st}^{(\frac{\Delta t}{2})},
\end{aligned}$$ where $\overline{ R_{1st} } $ is the inverse operator of $R_{1st}$. Accordingly for each $R_j$, we have $$\label{Rj}
\begin{aligned}
R_j:\quad & \sum_{i=1}^{N_s} \nu_{ji}^f S_i \Longleftrightarrow \sum_{i=1}^{N_s} \nu_{ji}^b S_i, \\
&\frac{dy_i}{dt} = \frac{\dot{\omega_i}^j}{\rho}, \quad i=1,\dots,N_s, \\
&\dot{\omega_i}^j = W_i (\nu_{ji}^b-\nu_{ji}^f) \left[ k_j^f \prod_{l=1}^{N_s} \left[\frac{\rho_l}{W_l}\right]^{\nu_{jl}^f} - k_j^b \prod_{l=1}^{N_s} \left[\frac{\rho_l}{W_l}\right]^{\nu_{jl}^b} \right].
\end{aligned}$$ We now rewrite the ODEs in Eq. in the following form [@mott2001chemeq2] $$\label{qss}
\frac{dy_i}{dt} = q_i^j - p_i^j y_i, \quad i=1,\dots,N_s,$$ where $q_i^j \geq 0$ is the production rate and $p_i^j y_i^j \geq 0$ is the loss rate for the $i^{th}$ species through reaction $j$.
Following the operator splitting of reactions, we continue to split the reversible reaction, e.g. reaction $j$ if applicable, apart into the forward reaction and backward reaction as $$\label{Rfb}
R_j^{(\Delta t)} = R_{j,b}^{(\Delta t)} \circ R_{j,f}^{(\Delta t)}$$ such that the species involved will either gain mass or lose mass through the one-way forward/backward reaction from Eq. , i.e. $$\label{pq}
\begin{aligned}
\text{if gain mass}: q_i^j \geq 0, \, p_i^j y_i = 0, \\
\text{else lose mass}: q_i^j = 0, \, p_i^j y_i \geq 0,
\end{aligned}$$ with the simplified $$\label{pq1}
\begin{aligned}
q_i^j &= \frac{W_i}{\rho} \nu_{ji}^b \left[ k_j^f \prod_{l=1}^{N_s} \left[\frac{\rho_l}{W_l}\right]^{\nu_{jl}^f} \right], \, p_i^j y_i = 0 \quad \text{for product species}, \\
q_i^j &= 0, \, p_i^j y_i = \frac{W_i}{\rho} \nu_{ji}^f \left[ k_j^f \prod_{l=1}^{N_s} \left[\frac{\rho_l}{W_l}\right]^{\nu_{jl}^f} \right] \quad \text{for reactant species} \\
\end{aligned}$$ in a forward reaction for example. It is clear that a backward reaction can be thought of as a forward one inversely if we exchange the reactants and products. Also, an irreversible reaction can be treated as a reversible one with a backward reaction rate being equal to zero such that the idea of splitting is still applicable.
Since each elementary reaction has been numerically decoupled from the rest and each reversible reaction again has been split into two oppositely unidirectional reactions, one finally merely ought to solve a single reaction equation as $$\label{one-way}
a A + b B + \cdots \longrightarrow x X + y Y + \cdots$$ in every operation. Mass conservation and positivity of mass fractions, the two highly significant requirements for either accuracy or stability of the numerical integration, can be carefully and properly treated.
In some simple cases for the reaction Eq. from wide applications, with the following forms $$\label{one-way-simple}
\begin{aligned}
A \longrightarrow \text{products}, \\
\text{or} \quad A + B \longrightarrow \text{products}, \\
\text{or} \quad 2A \longrightarrow \text{products}, \\
\text{or} \quad A + B + C \longrightarrow \text{products}, \\
\text{or} \quad 3A \longrightarrow \text{products}, \\
\end{aligned}$$ one may easily find their analytical solutions, see \[appendix1\]. It is thus natural to employ the analytical solutions rather than numerical solutions, with the advantages of avoiding introducing any numerical scheme error and being unconditionally stable [@nguyen2009mass]. However, as previously stated, for the general form of Eq. (usually with a higher overall order than two) whose analytical solution is explicitly unavailable or difficult to derive, a more convenient alternative is to perform quasi-steady-state (QSS) methods to obtain the approximate exact solution.
The QSS methods are based on the exact solution of Eq. if $p_i^j$ and $q_i^j$ are constant [@jay1997improved; @verwer1995explicit], i.e. $$\label{exact_solution}
y_i^{n+1} = y_i^n e^{-p_i^j \Delta t} + \frac{q_i^j}{p_i^j}(1-e^{-p_i^j \Delta t}), \quad i=1,\dots,N_s.$$ However, in practice $p_i^j$ and $q_i^j$ inherently depend on $\{y_1,\dots,y_{N_s}\}$ from Eq. or and Eq. provides an approximate solution if one assumes $p_i^j$ and $q_i^j$ are fixed during the timestep. The present SRR method is based on this plain approximate exact solution without invoking traditional time-integration schemes such as the Euler scheme with a poor stability [@nguyen2009mass]. Consequently, the QSS-based SRR method is almost unconditionally stable, which means the timestep size is not limited to the characteristic time sizes of chemical species and thus a larger timestep rendering less computational efforts is possible.
The plain QSS approximation adopted here in the SRR method is first-order accurate. But given that fluid dynamic calculations are seldom accurate to better than a few percent, any requirement of the chemical integrator to calculate the species concentrations more accurately than a few tenths of a percent is usually extensive. And the chemical integrator may be relatively low-order [@mott2001chemeq2].
### treatment for mass conservation
If we straightforwardly employ the approximate solution of QSS in Eq. for all the species through a reaction, we will have $$\label{sum_exact_solution}
\begin{aligned}
\sum_{i=1}^{N_s} y_i^{n+1} & = \sum_{i=1}^{N_s} \left( y_i^n e^{-p_i^j \Delta t} + \frac{q_i^j}{p_i^j}(1-e^{-p_i^j \Delta t}) \right) \\
& \neq 1.
\end{aligned}$$ It is obvious to see that mass conservation is not preserved. To cure this problem and utilize the excellent stability of the QSS approximation, instead of advancing $y^n$ to $y^{n+1}$ for all the species involved, one can choose to only advance $y_k^n$ to $y_k^{n+1}$ of a reactant species $k$ by Eq. and update other $\{y_{i,i\neq k}\}^{n+1}$ by the law of mass conservation of a single reaction equation in Eq. . This merit of knowing the exact net gain or loss of mass of other species originates from the operation upon only one reaction decoupled from others in both the present method and the original SSRI. Therefore, for the reactant $k$, combining Eqs. and we have $$\label{reactant_k}
y_k^{n+1} = y_k^n e^{-p_k^j \Delta t}$$ and for the rest species including other reactants and all the products in the reaction $j$, taking species $i$ for example, its change of mass fraction $\Delta y_i=y_i^{n+1} - y_i^{n}$ should obey $$\label{delta}
\frac{\Delta y_i/W_i}{ \nu_{ji}^b-\nu_{ji}^f } = \frac{\Delta y_k/W_k}{ \nu_{jk}^b-\nu_{jk}^f },$$ (which is essentially the conservation of the number of particles involved in a reaction system,) giving the below update $$\label{mass_delta}
\begin{aligned}
y_i^{n+1} &= y_i^{n} + \Delta y_i \\
&= y_i^{n} + \frac{\nu_{ji}^b-\nu_{ji}^f}{\nu_{jk}^b-\nu_{jk}^f} \frac{W_i}{W_k} \Delta y_k.
\end{aligned}$$ It is easy to prove that $\sum_{i=1}^{N_s} \Delta y_i = 0 $ which is equivalent to $\sum_{i=1}^{N_s} y_i =1$ for mass conservation.
### Positivity-preserving treatment
Since we only need to consider a single one-way reaction (forward or backward reaction) after two splitting procedures, the mass loss of reactants are exactly known and the non-negative mass fraction should be promised for the reactant species which are suffering mass loss. Without loss of generality, considering the forward reaction of the $j^{th}$ reaction and assuming that reactant $k$ with $\nu_{jk}^b=0$ is imposed by the QSS approximation in Eq. , we further look into another reactant species, e.g. $i$ with $\nu_{ji}^b=0$, and we combine Eqs. and to obtain $$\label{mass_delta1}
\begin{aligned}
y_i^{n+1} %&= y_i^{n} + \frac{\nu_{ji}^f}{\nu_{jk}^f} \frac{W_i}{W_k} y_k^n(e^{-p_k^j\Delta t}-1) \\
&= y_i^{n} - \frac{\nu_{ji}^f}{\nu_{jk}^f} \frac{W_i}{W_k} y_k^n + \frac{\nu_{ji}^f}{\nu_{jk}^f} \frac{W_i}{W_k} y_k^n e^{-p_k^j\Delta t}.
\end{aligned}$$ Recalling Eq. for reactants $i$ and $k$, we have $$\label{pq2}
\frac{p_i^j y_i}{p_k^j y_k} = \frac{\nu_{ji}^f}{\nu_{jk}^f} \frac{W_i}{W_k}.$$ Then rearrange Eq. and substitute it into Eq. we can obtain $$\label{ynew}
\begin{aligned}
y_i^{n+1} %&= y_i^{n} - y_i^{n}\frac{p_i^j}{p_k^j} + \frac{\nu_{ji}^f}{\nu_{jk}^f} \frac{W_i}{W_k} y_k^n e^{-p_k^j\Delta t} \\
&= y_i^{n}\frac{p_k^j-p_i^j}{p_k^j} + \frac{\nu_{ji}^f}{\nu_{jk}^f} \frac{W_i}{W_k} y_k^n e^{-p_k^j\Delta t}.
\end{aligned}$$ With the aid of Eq. , it is readily to see that we can guarantee the positivity of $y_i^{n+1}$, i.e. $y_i^{n+1} \geq 0$, by choosing $p_k^j \geq p_i^j$ since the second right-hand term is always non-negative. Therefore, for this reaction, in order to preserve the positivity of species mass fractions, especially for the reactants involved, the reactant $k$ using the QSS approximation should satisfy $$p_k^j = max\{p_i^j\} \quad \text{among all the reactant species in reaction \,} j.$$ Regarding the positivity preserving for the choosen reactant $k$, according to Eq. , it is naturally satisfied owing to the positivity of the exponential function.
The original SSRI and its improved counterpart in this study both can perform sufficiently well for the pure system of ODEs in chemical kinetics as a stand-alone solver. Randomization of this ODEs solver in the next subsection is not designed for integrating the ODEs accurately and individually, but mainly aimed at cancelling the effect of the harmful but unavoidable introduction of numerical dissipation resulting from the hydrodynamic solver $S_c$ using shock-capturing schemes into the reaction step $S_r$, i.e. the early ignition.
Finite randomization of chemical reactions
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Bao & Jin [@bao2000random; @bao2001random; @bao2002random] first proposed the idea of random projection into the ODEs solver instead of the deterministic projection which strictly obeys the time-dependent integration based on the local smeared information around the discontinuities. They also theoretically proved the random projection method gives basically first-order convergence for the scalar problem. For both scalar problems and Euler equations with stiff source terms, their random projection method is numerically demonstrated to be of excellent performance in obtaining the correct propagation of shocks and reacting fronts in under-resolved spatial and temporal discretizations.
After two steps of operation splitting upon the ODEs system in $S_r$, one only needs to consider the randomization of a single one-way reaction from time point $t_n$ to $t_{n+1}$ for an interval $\Delta t$. In Bao & Jin’s formulation, temperature will be a randomized variable instead of its local value to determine the progress (completely burnt or not) of the entire reaction system, by comparing with a pre-known ignition temperature, $T_{ign}$. A upper and lower limit of temperature are needed, i.e. $T_u$ and $T_b$ (corresponding to the two equilibrium states of the initial combustible gas mixture being completely burnt and unburnt) as a *priori*. Therefore, in such cases the equilibrium states are presumed and distributed before and behind the discontinuity as initial conditions, having not taken into account the far more complicated time-dependent finite-rate nonequilibrium chemistry without defined equilibrium states. By the above split reaction method, we advance the current state vector $\{y_1,\dots,y_{N_s}\}$ through a single one-way reaction indexed by the subscript $j$ for generality, as in Eq. , as $$\label{y+}
\begin{aligned}
\{y_1,\dots,y_{N_s}\}^{+} = R_j^{(\Delta t)} \{y_1,\dots,y_{N_s}\},
\end{aligned}$$ where $\{y_1,\dots,y_{N_s}\}^{+}$ represents the advance in time by one operation $R_j$ (i.e. $R_j^f$ or $R_j^b$ after splitting the reversible reaction in Eq. ). Thus, we can obtain the change of mass fractions for the species involved in this reaction, i.e. $$\label{dy}
\begin{aligned}
\{\Delta y_1,\dots,\Delta y_{N_s}\}_j = \{y_1,\dots,y_{N_s}\}^{+} - \{y_1,\dots,y_{N_s}\}.
\end{aligned}$$ An inverse operation from time level $n$ back for a timestep $\Delta t$ is therefore upon the current state vector, giving $$\label{y-}
\begin{aligned}
\{y_1,\dots,y_{N_s}\}^{-} = \{y_1,\dots,y_{N_s}\} - \{\Delta y_1,\dots,\Delta y_{N_s}\}_j.
\end{aligned}$$ It is to be noted that during either advance or its inverse operation, any mass fraction of species involved should be inside \[0,1\] and once a species’ mass fraction exceeds the range (usually larger than one because the positivity-preserving QSS approximation prevents negative mass fractions), all the mass fractions should be rescaled properly according to Eq. . For the two limited states with superscripts $+$ and $-$, two limited temperature $T^{+}$ and $T^{-}$ can be derived according to the EoS in Eq. with the help of the basic thermodynamic relation which is implicit about temperature, $$\label{thermodynamic}
\begin{aligned}
& h-e = \frac{p}{\rho}, \\
& p = p(y_1,\dots,y_{N_s}, T), \\
& h = h(y_1,\dots,y_{N_s}, T), \\
\end{aligned}$$ where $\rho$ and $e$ are fixed during the constant-volume adiabatic reaction and $h$ represents the specific enthalpy. If we assume the present reaction is exothermic, $T^{+}$ will be a high temperature and $T^{-}$ will be a low temperature, with the local temperature $T$ falling between the two limits, i.e. $T^{-}<T<T^{+}$, and *vice versa*. $T^{+}$ will thus be naturally imagined as the $T_b$ in the original random projection method while $T^{-}$ corresponds to $T_u$. Given the two limited values of temperature, we can assemble the local random temperature by $$\label{random_T}
\begin{aligned}
T^{*} = T^{-} + \theta_n ( T^{+} - T^{-} ),
\end{aligned}$$ where $\theta_n$ is a random real number between 0 and 1 and $T^{*}$ is the randomized local temperature with $min\{T^{-},T^{+}\}<T^{*}<max\{T^{-},T^{+}\}$ and $T^{*}\neq T$ in general. Regarding the generation of random number $\theta_n$, Bao & Jin suggested the van der Corput’s sampling scheme since it produces an equidistributed sequence on the interval \[0,1\], and among all known uniformly distributed sequences the deviation of van der Corput’s sequence is minimal [@hammersley2013monte]. Besides, we have also tested the in-built random number generator in Fortran 95, trivial distinctions were detected except for the different degree of statistical noise/fluctuation.
Provided the random temperature $T^{*}$, the single unidirectional reaction $j$ can be controlled by $$\label{random_projection}
\begin{aligned}
P_j^{(\Delta t)}: \quad
\{y_1,\dots,y_{N_s}\}_j=
\begin{cases}
\{y_1,\dots,y_{N_s}\}^{+}, & if \: T>T^{*}, \\
\{y_1,\dots,y_{N_s}\}, & otherwise,
\end{cases}
\end{aligned}$$ which indicates the reaction can be activated only if the local temperature is sufficiently high; otherwise, the reaction is to be ceased and the reacting front stops developing for this moment. This is the mechanism of preventing a too fast detonation wave. Having considered reaction $j$ by random projection to either the advance state or the current state, the updated state vector $\{y_1,\dots,y_{N_s}\}_j$ will be taken in as the initial state, as $\{y_1,\dots,y_{N_s}\}$ in Eq. , for the next reaction $j+1$ in a new operation till the end of the multi-reaction system.
The random process from the current state to a new state in the forward direction of time or not plays a similar role as the predictor-corrector algorithm. Since the random temperature $T^{*}$ and the beforehand predicted local temperature $T$ both lie between the two temperature limits, activation and deactivation both can happen for enough times in a long-term period of time. Thus the accumulative propagation of the discontinuity over many time steps converges to the correct position, taking into account the possibilities of both moving forward and standing still, as proved in [@bao2000random]. On the contrary, with traditional deterministic ODEs solvers, once the early triggering of the chemical reaction occurs , the reacting front will be forced to move one grid point forward. But no mechanism in such solvers is invented to halt this moving forward, thus a faster and faster shock will develop unrestrictedly to a spurious one.
Inserting Eq. into the split reaction method in Eqs. and , the present SRR method, denoted by $P$, is more than an ODEs solver, having the following form $$\label{SRR1}
\begin{aligned}
P_{1st}^{(\Delta t)} = P_{N_r}^{(\Delta t)} \circ P_{N_r-1}^{(\Delta t)} \circ \cdots \circ P_2^{(\Delta t)} \circ P_1^{(\Delta t)}
\end{aligned}$$ corresponding to the Lie’s reaction-by-reaction splitting or $$\label{SRR2}
\begin{aligned}
P_{2nd}^{(\Delta t)} =
P_1^{(\frac{\Delta t}{2})} \circ P_2^{(\frac{\Delta t}{2})} \circ \cdots \circ P_{N_r-1}^{(\frac{\Delta t}{2})} \circ P_{N_r}^{(\frac{\Delta t}{2})} \circ
P_{N_r}^{(\frac{\Delta t}{2})} \circ P_{N_r-1}^{(\frac{\Delta t}{2})} \circ \cdots \circ P_2^{(\frac{\Delta t}{2})} \circ P_1^{(\frac{\Delta t}{2})},
\end{aligned}$$ corresponding to the Strang splitting. It thus transforms the state vector of species mass fractions by $$\label{SRR2}
\begin{aligned}
\{y_1,\dots,y_{N_s}\}^{n+1} = P^{(\Delta t)} \{y_1,\dots,y_{N_s}\}^{n}
\end{aligned}$$ through the entire multi-reaction system of chemical kinetics.
Due to the randomization of integrating the reaction system in $P$, the present SRR method can overcome the disadvantage of numerical dissipation introduced by the convection term, $S_c$. So when reacting flows are of interest to solve in many applications, SRR is very likely to be suitable, especially for stiff cases in under-resolved conditions. If only an ODEs system, such as a zero-dimensional ignition problem, is under consideration, the above reaction-by-reaction ODEs solver or the original SSRI is sufficient to provide deterministic solutions with good accuracy and robustness.
Last but not the least, in nonstiff cases when the spatial and temporal resolutions are fine to resolve the reaction area (usually at least tens of points are required in the reacting front [@jones2016passive] and the time interval $\Delta t$ is also very small according to the CFL condition), the present SRR method will gradually reduce to a deterministic ODEs solver if we shift the sampling interval of random temperature in Eq. by $$\begin{aligned}
T^{**} =
\begin{cases}
T^{*} - \frac{1}{2}(T^{+} - T^{-})
(1-f), & if \: f<1, \\ % \left| \frac{T^{+}-T^{-} }{T^{++}-T^{--}} \right|)
T^{*}, & otherwise,
\end{cases}
\end{aligned}$$ where $$\begin{aligned}
f = N \left| \frac{T^{+}-T^{-} }{T^{++}-T^{--}+\epsilon} \right|
\end{aligned}$$ with $T^{++}$ representing the temperature corresponding to a state in $N$ timesteps forward (e.g. $N=5$) and $T^{--}$ corresponding to its inverse state according to Eqs. and and $\epsilon$ is a small positive number. Thus $f$ is a dynamic measure for the resolution of the concerned reaction. When $f$ is large, e.g. $f > 1$, random projection plays an important role for the under-resolved stiff case. When the resolution is fine enough, $f$ is small and $T^{*}$ tends to shift downwards for up to a half bandwidth of $\left[ T^{-},T^{+}\right]$ to be lower than $T$ (linearly approximated to be $\left( T^{+}+T^{-} \right)/2$) such that activation will happen for an increasing possibility according to Eq. . The random reaction reduces to a deterministic process with consistency in non-stiff cases. However, for the original random projection method, its relying on two presumed equilibrium states (including $T_b$ and $T_u$) essentially conflicts with the finite-rate nonequilibrium kinetics when the time scale is resolved and stiffness tends to diminish.
Due to the reduced randomness between activation and deactivation, the proposed SRR method can also cope with nonstiff problems while the original random projection method is merely suitable for under-resolved stiff cases.
Numerical results and discussion {#section2}
================================
In this section, we have three parts of numerical experiments: the first subsection validates the split reaction-by-reaction ODEs solver based on either analytical solutions if available or the plain QSS approximation for the zero-dimensional reaction operator, ignoring the fluid transport. The following two parts consider the coupled fluid dynamics with chemical kinetics by using simplified model kinetics and real-world finite-rate kinetics, respectively. Both 1D and 2D problems are taken into account, showing the dimensional independence of the present method.
Reaction-split ODEs solver for chemical kinetics
------------------------------------------------
### Michaelis-Menten test
The first case concerns the Michaelis-Menten system [@higham2008modeling] with four species through three reactions as $$\begin{aligned}
S_1 + S_2 \xrightarrow{k_1} S_3,\\
S_3 \xrightarrow{k_2} S_1 + S_2,\\
S_3 \xrightarrow{k_3} S_2 + S_4,
\end{aligned}$$ where the rate constants $k_1$, $k_2$ and $k_3$ are $10^6$, $10^{−4}$ and $10^{−1}$, respectively. We can see the second reaction is the reverse counterpart of the first. The initial concentration data from [@wilkinson2011stochastic; @higham2008modeling] are $5 \times 10^{-7}$ for $S_1$ and $2 \times 10^{-7}$ for $S_2$ with void $S_3$ and $S_4$. For this case, analytical solutions are provided for each reaction, see \[appendix1\], and we easily compare the convergence rates of the reaction splitting schemes of Lie and Strang, respectively. Reactions are simulated until $t=50$. In Table \[test1\_convergence\], the $L_1$ and $L_{\infty}$ error norms of species $S_1$ and $S_4$ are detailed, showing the expected convergence rate, i.e. 1st order for Lie splitting and 2nd order for Strang splitting.
\[test1\_convergence\]
### Hydrogen-air ignition delay test {#h2combustion}
For this case, we apply the reaction-split solver for more complicated chemical kinetics. The hydrogen ignition in air considers not only temperature-dependent reversible reactions but also third-body reactions, making the approximate solution to each reaction is practically preferred. Herein the mechanism of H$_2$-air combustion is from O’Conaire *et al.* [@o2004comprehensive], consisting of nine species (including the inert $\text{N}_2$) with twenty-three reversible reactions (equivalently forty-six one-way reactions), as listed in \[appendix2\]. This mechanism has exhibited good prediction for the ignition delay time in [@zhukov2012verification]. All the temperature-dependent reaction rates are calculated using the Arrhenius law $$\label{Arrhenius}
k_r = A T^B \text{exp}(-T_{ign}/T),$$ where the subscript $r$ denotes $f$ for forward reactions or $b$ for backward reactions and $T$ is the temperature. The parameters $A$, $B$ and $T_{ign}$ for the forward rate of each reaction are often given in the mechanism. When parameters are not provided associatedly, the backward rate needs to be calculated from the equilibrium constant $K_{eq}$ and $k_f$ by assuming the corresponding reaction to be in chemical equilibrium, i.e. $K_{eq} = k_f/k_b$, where $$K_{eq} = \left( \frac{1\text{atm}}{R_uT}\right)^{\sum_{i=1}^{N_s}(\nu_i^b-\nu_i^f)} \text{exp}\left( {-\sum_{i=1}^{N_s} (\frac{h_i}{R_i T}-\frac{s_i}{R_i})(\nu_i^b-\nu_i^f)} \right)$$ including the species gas constant $R_i$, specific enthalpy $h_i$ and specific entropy $s_i$ (to be approximated by thermodynamical polynomials as in [@mcbride2002nasa]). The third-body effect is accounted for by the summation of the third-body collision efficiencies times the corresponding molar densities of species.
The ignition delay problem is a zero-dimensional homogeneous case in space since we assume a constant-volume and adiabatic environment. Initially the reactive H$_2$-air mixture is at a pressure of 1 atm, and in the molar ratio $2:1:3.76$ for $\text{H}_2:\text{O}_2:\text{N}_2$. Nitrogen is inert for the mechanism, and thus acts as a diluent. The initial temperature of the mixture is highly important for hydrogen ignition induction. All simulations end at $t=1\times 10^{-3}$s.
We firstly vary the initial temperature $T_0$ from $950$ K to $1400$ K with an equal interval of $50$ K. A fixed timestep of $1 \times 10^{-8}$s is applied, in which condition Lie splitting is sufficiently accurate. With an increasing initial temperature, the reaction rates are usually accelerated; thus the ignition delay time, corresponding to the time instance when the mixture temperature ascends most rapidly with time, generally decreases. We compare the ignition delay times predicted by the present solver with the experimental data and the CHEMKIN [@kee1989chemkin] results from Ref. [@zhukov2012verification] (see its Fig. 3) in Fig. \[test2\_delay\_time\]. We can see that, in spite of varying setups, the QSS-based reaction-split method exhibits good predictions for the ignition induction of hydrogen using the present mechanism, especially in the high initial temperature range. In Fig. \[mass\_fraction\], we compare the computed mass fractions with CHEMEQ2 at an initial temperature of $1000$ K, good agreement being reached especially at the ignition time. By setting the initial temperature at $1000$ K and $1200$ K, respectively, we consider the mass conservation resulted from the reaction-split method (abbreviated as QRS) and CHEMEQ2 in Fig. \[sum\_mass\_fraction\]. It is readily to see that QRS can always preserve the mass conservation, whereas the CHEMEQ2 results show that total mass loss or gain occurs obviously around the ignition time when species concentrations vary most dramatically.
![Ignition delay times with different initial temperatures[]{data-label="test2_delay_time"}](ignition_delay_time.pdf)
![Time histories of mass fractions of H and $\text{H}_2\text{O}$[]{data-label="mass_fraction"}](mass_fraction.pdf)
![Time histories of the sum of mass fractions; ’1000’ $\sim$ $T_0=1000$ K, ’1200’ $\sim$ $T_0=1200$ K[]{data-label="sum_mass_fraction"}](sum_mass_fraction.pdf)
Reactive Euler equations with simplified model kinetics
-------------------------------------------------------
In this part, we consider reactive Euler equations coupled with simplified model kinetics in several stiff detonation problems. In severe stiff cases, the Arrhenius form of reaction rates in Eq. also can be expressed in the Heaviside form as $$\label{Heaviside}
k_r =
\begin{cases}
A T^B, & T \geq T_{ign}, \\
0, & T < T_{ign}. \\
\end{cases}$$ The EoS in Eq. for the model problems is also simplified by $$p = (\gamma-1) \left( \rho e - q_1 \rho y_1 - q_2 \rho y_2 - \cdots - q_{N_s} \rho y_{N_s} \right)$$ and $T=p/\rho$. Numerical experiments cover single reaction to multi-reaction system in 1D and 2D detonation problems. In our computation, the AUSM+ scheme [@liou1996sequel] is employed together with MUSCL reconstruction using a TVD Minmod limiter [@leveque1992numerical] in the convection step; the reaction step adopts the SRR method or merely the reaction-split solver as a deterministic method.
EXAMPLE 1 (A Chapman-Jouguet (CJ) Detonation). The first case considers the simplest reacting model, which has been studied in [@zhang2014equilibrium], with only one reaction and two mutually dependent species $$\label{case1_model}
\begin{aligned}
A \longrightarrow B,
\end{aligned}$$ where $A$ represents the fuel being burnt by the one-way reaction and the mass fraction of the product can be directly given by $y_B=1-y_A$.
The parameters for the reaction model and species properties are $$\label{case1_para}
\begin{aligned}
\left( \gamma ,q_A, q_B \right) &= \left( 1.4, 25, 0 \right), \\
\left( A, B, T_{ign} \right) &= \left( 16418, 0.1, 15 \right).
\end{aligned}$$ The initial condition to generate the detonation wave consists of two parts in only one spatial dimension, with piecewise constants given by $$\label{case1_initial}
\begin{aligned}
\left( p,T,u,y_A,y_B \right) =
\begin{cases}
\left( 21.435,12.75134,2.899,0, 1 \right), & x<10, \\
\left( 1,1,0,1,0 \right), & x \geq 10. \\
\end{cases}
\end{aligned}$$ The left part gas is at the burnt equilibrium state and it is moving at a speed $u_{CJ}$ relative to the stationary unburnt gas of the right part. In fact, for any given initial state on the right, the initial CJ state on the left can be obtained in theory [@bao2000random; @yee2013spurious; @zhang2014equilibrium]. This problem is solved on the interval $\left[0,30\right]$. The left-end boundary condition is the inflow condition with fixed identical constants as the initial data on the left; the boundary condition for the right end is extrapolation from the mirror image points inside the domain. The exact solution is simply a CJ detonation wave moving to the right and we obtain the reference ’exact’ solution by the deterministic method using a resolved grid ($\Delta x = 0.0025$) and a tiny timestep of $\Delta t=0.0001$. We compare the results given by SRR and the deterministic method, respectively, using two sets of grid ($\Delta x=0.25, 0.025$) and timestep ($\Delta t = 0.01, 0.001$). Figure \[example1\] shows the computed pressure, density, temperature and mass fraction. Clearly, the proposed random method can capture the correct propagation of the detonation wave with both coarse and fine grids, while the deterministic method produces the spurious solutions in the same under-resolved conditions, i.e. a weak detonation wave propagates faster than the theoretical detonation speed of $D_{CJ}=7.124$ in this case [@zhang2014equilibrium]. Besides, since a coarser grid with a larger timestep indicates the stiffness is more severe, the deterministic method produces far more nonphysical weak detonation wave compared to our SRR or the reference solution. Also to be noted, the location of mass fraction on the coarse grid may be few grid points away from the exact location due to random effect, but such a deviation does not grow in time [@bao2000random], essentially unlike the error accumulation of the deterministic method.
![Example 1 one reaction, CJ detonation at $t=1.5$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example1"}](1_pressure1.pdf "fig:") ![Example 1 one reaction, CJ detonation at $t=1.5$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example1"}](1_pressure2.pdf "fig:")\
![Example 1 one reaction, CJ detonation at $t=1.5$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example1"}](1_density1.pdf "fig:") ![Example 1 one reaction, CJ detonation at $t=1.5$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example1"}](1_density2.pdf "fig:")\
![Example 1 one reaction, CJ detonation at $t=1.5$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example1"}](1_temperature1.pdf "fig:") ![Example 1 one reaction, CJ detonation at $t=1.5$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example1"}](1_temperature2.pdf "fig:")\
![Example 1 one reaction, CJ detonation at $t=1.5$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example1"}](1_fraction1.pdf "fig:") ![Example 1 one reaction, CJ detonation at $t=1.5$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example1"}](1_fraction2.pdf "fig:")\
EXAMPLE 2 (A Strong Detonation). This example considers a reacting model, which has been studied in [@zhang2014equilibrium], with one reaction and three species $$\label{case2_model}
\begin{aligned}
2 \text{H}_2 + \text{O}_2 \longrightarrow 2 \text{H}_2\text{O}.
\end{aligned}$$
The parameters for the reaction kinetics and species properties are $$\label{case2_para}
\begin{aligned}
\left( \gamma ,q_{\text{H}_2}, q_{\text{O}_2}, q_{\text{H}_2\text{O}},W_{\text{H}_2}, W_{\text{O}_2}, W_{\text{H}_2\text{O}} \right) &= \left( 1.4, 300, 0, 0, 2, 32, 18 \right), \\
\left( A, B, T_{ign} \right) &= \left( 10^6, 0, 2 \right).
\end{aligned}$$ The initial condition of piecewise constants is given by $$\label{case2_initial}
\begin{aligned}
\left( p,T,u,y_{\text{H}_2},y_{\text{O}_2},y_{\text{H}_2\text{O}} \right) =
\begin{cases}
\left( 20,10,8, 0,0,1 \right), & x<2.5, \\
\left( 1,1,0,\frac{1}{9},\frac{8}{9},0 \right), & x \geq 2.5. \\
\end{cases}
\end{aligned}$$ The left part gas is at the burnt equilibrium state and it is moving at a speed larger than $u_{CJ}$ relative to the stationary unburnt gas of the right part so that a strong detonation wave is to occur. This problem is solved on the interval $\left[0,50\right]$.
The exact solution consists of a detonation wave, followed by a contact discontinuity and a shock, all moving to the right. Similarly, we obtain the reference solution by the deterministic method using a resolved grid and a tiny timestep, and then compare the results by SRR and the deterministic method using a very coarse grid and another finer grid with proper timesteps, as explained in Fig. \[example2\]. Note that in the deterministic method, we adopt both the Arrhenius model and Heaviside model for the chemical kinetics. It is readily to see the proposed SRR method can capture all discontinuities effectively, while the deterministic method produces the spurious solutions in the same under-resolved conditions. In particular, using the Heaviside model, the deterministic method produces more severely incorrect solution due to its greater stiffness compared to the Arrhenius model (see the right column of Fig. \[example2\]).
![Example 2 one reaction, strong detonation at $t=1$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; green cross line $\sim$ deterministic solution with Heviside kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example2"}](2_pressure1.pdf "fig:") ![Example 2 one reaction, strong detonation at $t=1$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; green cross line $\sim$ deterministic solution with Heviside kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example2"}](2_pressure2.pdf "fig:")\
![Example 2 one reaction, strong detonation at $t=1$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; green cross line $\sim$ deterministic solution with Heviside kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example2"}](2_density1.pdf "fig:") ![Example 2 one reaction, strong detonation at $t=1$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; green cross line $\sim$ deterministic solution with Heviside kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example2"}](2_density2.pdf "fig:")\
![Example 2 one reaction, strong detonation at $t=1$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; green cross line $\sim$ deterministic solution with Heviside kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example2"}](2_temperature1.pdf "fig:") ![Example 2 one reaction, strong detonation at $t=1$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; green cross line $\sim$ deterministic solution with Heviside kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example2"}](2_temperature2.pdf "fig:")\
![Example 2 one reaction, strong detonation at $t=1$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; green cross line $\sim$ deterministic solution with Heviside kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example2"}](2_fraction1.pdf "fig:") ![Example 2 one reaction, strong detonation at $t=1$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; green cross line $\sim$ deterministic solution with Heviside kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example2"}](2_fraction2.pdf "fig:")\
EXAMPLE 3 (A Strong Detonation). This case considers a multi-step reaction mechanism with two one-way reactions and five species $$\label{case3_model}
\begin{aligned}
&1) \qquad \text{H}_2 + \text{O}_2 \longrightarrow 2 \text{OH}, \\
&2) \qquad 2 \text{OH} + \text{H}_2 \longrightarrow 2 \text{H}_2\text{O},
\end{aligned}$$ with $\text{N}_2$ as a dilute catalyst. Similar examples have been studied in [@bao2002random].
The parameters for the reaction model and species properties are $$\label{case3_para}
\begin{aligned}
\left( \gamma ,q_{\text{H}_2}, q_{\text{O}_2}, q_{\text{OH}}, q_{\text{H}_2\text{O}}, q_{\text{N}_2}\right) &= \left( 1.4, 0, 0, -20, -100, 0 \right), \\
\left( W_{\text{H}_2}, W_{\text{O}_2}, W_{\text{OH}}, W_{\text{H}_2\text{O}}, W_{\text{N}_2}\right) &= \left( 2, 32, 17, 18, 28 \right), \\
\left( A^1, B^1, T_{ign}^1 \right) &= \left( 10^5, 0, 2 \right), \\
\left( A^2, B^2, T_{ign}^2 \right) &= \left( 2\times10^4, 0, 10 \right). \\
\end{aligned}$$ The initial condition of piecewise constants is given by $$\label{case3_initial}
\begin{aligned}
\left( p,T,u,y_{\text{H}_2}, y_{\text{O}_2}, y_{\text{OH}}, y_{\text{H}_2\text{O}}, y_{\text{N}_2}\right) =
\begin{cases}
\left( 40,20,10, 0, 0, 0.17, 0.63, 0.2 \right), & x<2.5, \\
\left( 1,1,0,0.08,0.72,0,0,0.2 \right), & x \geq 2.5. \\
\end{cases}
\end{aligned}$$ The left part gas is at the burnt equilibrium state and it is moving at a speed larger than $u_{CJ}$ relative to the stationary unburnt gas of the right part so that a strong detonation wave is to occur. This problem is solved on the interval $\left[0,50\right]$.
The exact solution consists of a detonation wave, followed by a contact discontinuity and a shock, all moving to the right. Figure \[example3\] presents the computational conditions and results obtained accordingly. All waves are captured with the correct speeds by the SRR method, in good agreement with the reference solution. However, the deterministic method obviously fails using the Heaviside model with the same under-resolved grids and timesteps. This is also because the stiffness of the Heaviside model as an infinite-rate reaction model is more severe and the deterministic method is poor to deal with stiffness unless both the space and time scales are resolved. Besides, the error of the spurious weak detonation by the deterministic method using the Arrhenius model grows with time, although its difference from the correct one is not very apparent at the present time point.
![Example 3 two reactions, strong detonation at $t=3$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; green cross line $\sim$ deterministic solution with Heviside kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example3"}](3_pressure1.pdf "fig:") ![Example 3 two reactions, strong detonation at $t=3$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; green cross line $\sim$ deterministic solution with Heviside kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example3"}](3_pressure2.pdf "fig:")\
![Example 3 two reactions, strong detonation at $t=3$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; green cross line $\sim$ deterministic solution with Heviside kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example3"}](3_density1.pdf "fig:") ![Example 3 two reactions, strong detonation at $t=3$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; green cross line $\sim$ deterministic solution with Heviside kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example3"}](3_density2.pdf "fig:")\
![Example 3 two reactions, strong detonation at $t=3$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; green cross line $\sim$ deterministic solution with Heviside kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example3"}](3_temperature1.pdf "fig:") ![Example 3 two reactions, strong detonation at $t=3$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; green cross line $\sim$ deterministic solution with Heviside kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example3"}](3_temperature2.pdf "fig:")\
![Example 3 two reactions, strong detonation at $t=3$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; green cross line $\sim$ deterministic solution with Heviside kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example3"}](3_fraction1.pdf "fig:") ![Example 3 two reactions, strong detonation at $t=3$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; green cross line $\sim$ deterministic solution with Heviside kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example3"}](3_fraction2.pdf "fig:")\
EXAMPLE 4 (A Strong Detonation). This case considers a more complicated multi-step reaction model with three one-way reactions and five species involved $$\label{case4_model}
\begin{aligned}
&1) \qquad \text{H}_2 \longrightarrow 2 \text{H}, \\
&2) \qquad 2\text{H} + \text{O}_2 \longrightarrow 2 \text{OH}, \\
&3) \qquad 2 \text{OH} + \text{H}_2 \longrightarrow 2 \text{H}_2\text{O},
\end{aligned}$$ without $\text{N}_2$ here. The model is extended from the above two-reaction example, but with three distinct reaction rates (fast, medium and slow, respectively) to enlarge the stiffness due to multiple timescales.
The parameters for the reaction model and species properties are $$\label{case4_para}
\begin{aligned}
\left( \gamma ,q_{\text{H}_2}, q_{\text{O}_2}, q_{\text{OH}}, q_{\text{H}_2\text{O}}, q_{\text{H}}\right) &= \left( 1.4, 0, 0, -20, -100, 10 \right), \\
\left( W_{\text{H}_2}, W_{\text{O}_2}, W_{\text{OH}}, W_{\text{H}_2\text{O}}, W_{\text{H}}\right) &= \left( 2, 32, 17, 18, 1 \right), \\
\left( A^1, B^1, T_{ign}^1 \right) &= \left( 10^7, 0, 1.5 \right), \\
\left( A^2, B^2, T_{ign}^2 \right) &= \left( 10^5, 0, 2 \right), \\
\left( A^3, B^3, T_{ign}^3 \right) &= \left( 10^3, 0, 10 \right). \\
\end{aligned}$$ The initial condition of piecewise constants is given by $$\label{case4_initial}
\begin{aligned}
\left( p,T,u,y_{\text{H}_2}, y_{\text{O}_2}, y_{\text{OH}}, y_{\text{H}_2\text{O}}, y_{\text{H}}\right) =
\begin{cases}
\left( 40,20,10, 0, 0, 0.17, 0.72, 0.11 \right), & x<2.5, \\
\left( 1,1,0,0.2,0.8,0,0,0 \right), & x \geq 2.5. \\
\end{cases}
\end{aligned}$$ The left part gas is at the burnt equilibrium state and it is moving at a speed larger than $u_{CJ}$ relative to the stationary unburnt gas of the right part so that a strong detonation wave is to occur. This problem is solved on the interval $\left[0,50\right]$.
![Example 4 three reactions, strong detonation at $t=1.5$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example4"}](4_pressure1.pdf "fig:") ![Example 4 three reactions, strong detonation at $t=1.5$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example4"}](4_pressure2.pdf "fig:")\
![Example 4 three reactions, strong detonation at $t=1.5$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example4"}](4_density1.pdf "fig:") ![Example 4 three reactions, strong detonation at $t=1.5$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example4"}](4_density2.pdf "fig:")\
![Example 4 three reactions, strong detonation at $t=1.5$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example4"}](4_temperature1.pdf "fig:") ![Example 4 three reactions, strong detonation at $t=1.5$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example4"}](4_temperature2.pdf "fig:")\
![Example 4 three reactions, strong detonation at $t=1.5$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example4"}](4_fraction1.pdf "fig:") ![Example 4 three reactions, strong detonation at $t=1.5$: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution with Arrhenius kinetics; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.25$, $\Delta t=0.01$; right column $\sim$ $\Delta x=0.025$, $\Delta t=0.001$.[]{data-label="example4"}](4_fraction2.pdf "fig:")\
The exact solution shares the same wave pattern with the former example while the wave profiles differ greatly due to the change in the kinetics model. Figure \[example4\] presents the computational conditions and computed results. All waves are captured with the correct speeds by the SRR method numerically, in good agreement with the reference solution with a location of the detonation wave at $x \approx 17$. However, the deterministic method obviously fails using the Arrhenius model with the same under-resolved grids and timesteps, by yielding a too fast weak detonation located at $x=40$. And the incorrect weak detonation wave by the deterministic method using the Heaviside model has already run out of the domain at $t=1.5$ (thus not shown in the plots).
EXAMPLE 5 (A CJ Detonation in 2D). This 2D case extends EXAMPLE 1 to model the radially symmetric point-source explosion, where $A$ in Eq. is amplified by $10000$ times to approximate the infinitely fast reaction with extreme stiffness. Similar tests have been studied in [@bao2002random; @helzel2000modified].
With radial symmetry, 1/4 part of the explosion is convenient to take into use as in space, $\left[0,50\right] \times \left[0,50\right]$. The hot-spot area of the initial high-temperature high-pressure burnt gas is a circle with radius 10 and the reactive unburnt gas takes the outside. Initial condition is the same as in Example 1 except the initial velocity of the circle area is adjusted to along the radial direction, i.e. $$\label{case5x_initial1}
\begin{aligned}
\left(u,v\right) =
\begin{cases}
\left( 2.899x/r,2.899y/r \right), & r<10, \\
\left( 0,0 \right), & r \geq 10, \\
\end{cases}
\end{aligned}$$ where $r=\sqrt{x^2+y^2}$.
In our computations, a coarse grid ($200 \times 200$) and a finer grid ($2000 \times 2000$) are employed referring to Example 1. Corresponding timesteps are $\Delta t=1 \times 10^{-2}$ and $1 \times 10^{-3}$, respectively. Unfortunately, we cannot obtain the reference solution by the deterministic method with a further refined grid for this 2D case. With the finer grid, the deterministic method still gives the obviously spurious solution at $t=1.5$, see the left column of Fig. \[example5\], in that a nonphysical weak detonation wave is generated and the reacting front is no more circular. In contrast, our SRR method can capture the shape and location of the CJ detonation front accurately, see the right column of the figure, by observing the radial velocity vector in the pressure contour even in the low resolution and the self-similarly circular outwards-developing detonation fronts in black/white lines of two resolutions at different times. The line-marked locations calculated by the random method in two resolutions agree excellently with each other and thus a grid convergence to the exact solution is reasonable to expect for the proposed SRR method. Besides, with ignorable curvature effects [@aslam1999detonation; @short2016steady] as the detonation radius is large and the under-resolved reaction zone is infinitesimal, the calculated speed of the detonation front approaches the 1D theoretical speed of $D_{CJ}=7.1247$ as in Example 1.
\
EXAMPLE 6 (A Strong Detonation in 2D). The present case considers the same multi-step reaction mechanism as in EXAMPLE 3 except that $q_{\text{OH}}$ in Eq. changes into $-50$. This is also a multi-dimensional case used to prove the dimension-independent nature of the proposed method, unlike the original random projection method which requires a dimension-by-dimension scanning for local projection. The test is also studied in [@zhang2014equilibrium].
The initial condition of piecewise constants in the $\left[0,6\right] \times \left[0,2\right]$ 2D domain consists of $$\label{case6_initial1}
\begin{aligned}
\left( p,T,u,v\right) =
\begin{cases}
\left( 40,20,10, 0 \right), & x<0.5, \\
\left( 1,1,0,0 \right), & x \geq 0.5, \\
\end{cases}
\end{aligned}$$ $$\label{case6_initial2}
\begin{aligned}
\left( y_{\text{H}_2}, y_{\text{O}_2}, y_{\text{OH}}, y_{\text{H}_2\text{O}}, y_{\text{N}_2} \right) =
\begin{cases}
\left( 0,0,0.17,0.63,0.2 \right), & x<0.5, \\
\left( 0,0,0.17,0.63,0.2 \right), & x \geq 0.5, y \geq 1.2, \\
\left( 0.08,0.72,0,0,0.2 \right), & x \geq 0.5, y < 1.2, \\
\end{cases}
\end{aligned}$$ as shown in Fig. \[schematic\]. We can see that the computational domain is composed of three parts (zone A, B and C) with shock and contact surface. Both zone A and B are filled with burnt gas and zone C is filled with the reactive unburnt gas.
In our computations, a uniformly distributed coarse grid ($300 \times 100$) and a refined grid ($3000 \times 1000$) are employed. Corresponding timesteps are $\Delta t=5 \times 10^{-4}$ and $5 \times 10^{-5}$, respectively. The reference solution is obtained by the deterministic method using the fine grid and tiny timestep. The comparison of the SRR method and deterministic method on capturing stiff detonation waves is based on the under-resolved grid and timestep. In Fig. \[example6\], it is readily to see at $t=0.1$ the spurious solution given by the deterministic method on the coarse grid contains a too fast weak detonation wave, which has passed half of the domain. However, the correct detonation waves from the SRR method on the same resolution and the deterministic method on a fine grid agree with each other excellently and fall far behind the spurious weak detonation wave. Good agreement of the self-similar propagation of the detonation wave from $t=0.1$ to $0.3$ is also can be seen in the mass fraction contour given by the reference solution and the under-resolved SRR solution, respectively. The slight difference between the two correct solutions lies in some small around-shock statistical fluctuations due to the random nature of the method [@bao2000random].
![Example 6 2D case, two reactions, strong detonation at $t=0.1$: top $\sim$ reference solution; middle $\sim$ deterministic solution with Arrhenius kinetics; bottom $\sim$ SRR solution; in the mass fraction contour, locations of the detonation front at $t=0.1, 0.2, 0.3$ are additionally marked by setting $y_{\text{O}_2}=0.5$ in white solid lines.[]{data-label="example6"}](pressure_example6.pdf "fig:") ![Example 6 2D case, two reactions, strong detonation at $t=0.1$: top $\sim$ reference solution; middle $\sim$ deterministic solution with Arrhenius kinetics; bottom $\sim$ SRR solution; in the mass fraction contour, locations of the detonation front at $t=0.1, 0.2, 0.3$ are additionally marked by setting $y_{\text{O}_2}=0.5$ in white solid lines.[]{data-label="example6"}](density_example6.pdf "fig:")\
![Example 6 2D case, two reactions, strong detonation at $t=0.1$: top $\sim$ reference solution; middle $\sim$ deterministic solution with Arrhenius kinetics; bottom $\sim$ SRR solution; in the mass fraction contour, locations of the detonation front at $t=0.1, 0.2, 0.3$ are additionally marked by setting $y_{\text{O}_2}=0.5$ in white solid lines.[]{data-label="example6"}](temperature_example6.pdf "fig:") ![Example 6 2D case, two reactions, strong detonation at $t=0.1$: top $\sim$ reference solution; middle $\sim$ deterministic solution with Arrhenius kinetics; bottom $\sim$ SRR solution; in the mass fraction contour, locations of the detonation front at $t=0.1, 0.2, 0.3$ are additionally marked by setting $y_{\text{O}_2}=0.5$ in white solid lines.[]{data-label="example6"}](massfraction_example6.pdf "fig:")\
Reactive Euler equations with real-world nonequilibrium kinetics
----------------------------------------------------------------
In this subsection, we try to validate the SRR method in capturing stiff detonation waves governed by the reactive Euler equations coupled with real-world chemical nonequilibrium kinetics, in which the much more complicated reaction mechanism will introduce multiple temperature-dependent reactions with distinct timescales. To our knowledge, both the two test cases below are reported for the first time, taking into account the detailed hydrogen-air combustion mechanism as in Subsection \[h2combustion\]. Two different scenarios with the CJ detonation and strong detonation wave, respectively, are simulated in 1D or 2D domain, regardless of the dimensional independence property of the proposed method.
The convection operator adopts an ordinary shock capturing scheme as in the former subsection, and the reaction step is solved by the proposed SRR method and the popular CHEMEQ2 integrator as the deterministic method to make a comparison. In particular, reaction splitting in the SRR method is based on the 2nd-order Strang’s scheme to reduce splitting errors.
EXAMPLE 7 (A Realistic CJ Detonation). The setup of this case consists of two parts divided by a shock moving to the right in a 1D domain of length $L=4$m: the left part is post-shock and filled with high-temperature high-pressure burnt gas while the right part is pre-shock and filled with reactive unburnt gas in one atmosphere pressure and room temperature, see details in Table \[example7\_IC\]. The theoretical CJ detonation states for the unburnt gas can be generated using the NASA Chemical Equilibrium Analysis (CEA) program [@gordon1994computer] and according to the CJ condition [@bao2000random; @yee2013spurious; @zhang2014equilibrium], i.e. $$D_{CJ} = u_{CJ} + (\gamma p_b/\rho_b)^{1/2},$$ we adopt $u_b=800\text{m/s} \approx u_{CJ}$ for the initial velocity of the burnt gas, to generate a CJ detonation wave sweeping the stationary unburnt gas. The shock is initially located at $x=0.5$m. Boundary condition for the left/right end is simply extrapolation from the mirror image points inside the domain. All simulations stop at $t=1.2 \times 10^{-3}$s.
The exact solution is a steady self-similar CJ detonation wave travelling from left to right, in similar with the model problem of Example 1. We obtain the reference exact solution by the deterministic method using a very fine grid with 10000 points and a fixed tiny timestep of $\Delta t = 5 \times 10^{-9}$s. Two sets of under-resolved grid and timestep are considered, i.e. $\Delta x=0.08\text{m}, \Delta t=1 \times 10^{-6}\text{s}$ and $\Delta x=0.02\text{m}, \Delta t=2.5 \times 10^{-7}\text{s}$, respectively.
We can see that in Fig. \[example7\_1\] at the given time: although the resolution of the grid and timestep is far lower than the resolved solution, the SRR method predicts the properties of the flowfield in quite good agreement with the reference solution, including the location of the detonation wave and the variable profiles. The obtained profiles tend to converge to the reference solution with the increase of the resolution (and the decrease of stiffness), which also indicates the proposed method can recover nonstiff problems by reducing to the deterministic reference solution under high resolutions, as stated previously. In contrast, using the same under-resolved grid and timestep, the deterministic method yields the spurious nonphysical weak detonation ahead of the shock and the flowfield profiles are totally changed in an incorrect way. In Fig. \[example7\_2\], wave propagation at different times is presented by looking into the pressure distribution. Despite the deviation by few grid points, the SRR method can always capture the correct wave location while the error in the location of reaction front by the deterministic method is deteriorating in the form of a too fast weak detonation wave. Note that the von Neumann spike inside the reaction zone of the reference solution can be calculated only by very fine resolution both in space and time.
\[example7\_IC\]
post-shock gas pre-shock gas
---------------------------- -------------------- ---------------
pressure (Pa) 1481999.362037 101325
temperature (K) 2941.677242 298
velocity (m/s) ($\approx u_{CJ}$) 0
mass fraction
$y_{\text{H}}$ 0.000247 0
$y_{\text{O}}$ 0.001617 0
$y_{\text{H}_2\text{O}}$ 0.225404 0
$y_{\text{OH}}$ 0.014915 0
$y_{\text{O}_2}$ 0.013336 0.226362
$y_{\text{H}_2}$ 0.002429 2.852103E-2
$y_{\text{H}_2\text{O}_2}$ 2.601600E-6 0
$y_{\text{HO}_2}$ 1.857550E-5 0
$y_{\text{N}_2}$ 0.742031 0.745117
: Initial condition for Example 7 9-species 23-reaction hydrogen-air CJ detonation
![Example 7 9-species 23-reaction hydrogen-air CJ detonation at $t=1.2 \times 10^{-3}$s: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution by CHEMEQ2; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.08\text{m}$, $\Delta t=1 \times 10^{-6}\text{s}$; right column $\sim$ $\Delta x=0.02\text{m}$, $\Delta t=2.5 \times 10^{-7}\text{s}$.[]{data-label="example7_1"}](example6_p1.pdf "fig:") ![Example 7 9-species 23-reaction hydrogen-air CJ detonation at $t=1.2 \times 10^{-3}$s: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution by CHEMEQ2; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.08\text{m}$, $\Delta t=1 \times 10^{-6}\text{s}$; right column $\sim$ $\Delta x=0.02\text{m}$, $\Delta t=2.5 \times 10^{-7}\text{s}$.[]{data-label="example7_1"}](example6_p2.pdf "fig:")\
![Example 7 9-species 23-reaction hydrogen-air CJ detonation at $t=1.2 \times 10^{-3}$s: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution by CHEMEQ2; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.08\text{m}$, $\Delta t=1 \times 10^{-6}\text{s}$; right column $\sim$ $\Delta x=0.02\text{m}$, $\Delta t=2.5 \times 10^{-7}\text{s}$.[]{data-label="example7_1"}](example6_d1.pdf "fig:") ![Example 7 9-species 23-reaction hydrogen-air CJ detonation at $t=1.2 \times 10^{-3}$s: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution by CHEMEQ2; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.08\text{m}$, $\Delta t=1 \times 10^{-6}\text{s}$; right column $\sim$ $\Delta x=0.02\text{m}$, $\Delta t=2.5 \times 10^{-7}\text{s}$.[]{data-label="example7_1"}](example6_d2.pdf "fig:")\
![Example 7 9-species 23-reaction hydrogen-air CJ detonation at $t=1.2 \times 10^{-3}$s: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution by CHEMEQ2; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.08\text{m}$, $\Delta t=1 \times 10^{-6}\text{s}$; right column $\sim$ $\Delta x=0.02\text{m}$, $\Delta t=2.5 \times 10^{-7}\text{s}$.[]{data-label="example7_1"}](example6_t1.pdf "fig:") ![Example 7 9-species 23-reaction hydrogen-air CJ detonation at $t=1.2 \times 10^{-3}$s: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution by CHEMEQ2; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.08\text{m}$, $\Delta t=1 \times 10^{-6}\text{s}$; right column $\sim$ $\Delta x=0.02\text{m}$, $\Delta t=2.5 \times 10^{-7}\text{s}$.[]{data-label="example7_1"}](example6_t2.pdf "fig:")\
![Example 7 9-species 23-reaction hydrogen-air CJ detonation at $t=1.2 \times 10^{-3}$s: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution by CHEMEQ2; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.08\text{m}$, $\Delta t=1 \times 10^{-6}\text{s}$; right column $\sim$ $\Delta x=0.02\text{m}$, $\Delta t=2.5 \times 10^{-7}\text{s}$.[]{data-label="example7_1"}](example6_h1.pdf "fig:") ![Example 7 9-species 23-reaction hydrogen-air CJ detonation at $t=1.2 \times 10^{-3}$s: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution by CHEMEQ2; black solid line $\sim$ reference solution; left column $\sim$ $\Delta x=0.08\text{m}$, $\Delta t=1 \times 10^{-6}\text{s}$; right column $\sim$ $\Delta x=0.02\text{m}$, $\Delta t=2.5 \times 10^{-7}\text{s}$.[]{data-label="example7_1"}](example6_h2.pdf "fig:")\
![Example 7 9-species 19-reaction hydrogen-air CJ detonation at $t=0.4, 0.8, 1.2 \times 10^{-3}$s: purple square line $\sim$ SRR solution; red circle line $\sim$ deterministic solution by CHEMEQ2; black solid line $\sim$ reference solution; both solutions $\sim$ $\Delta x=0.02\text{m}$, $\Delta t=2.5 \times 10^{-7}\text{s}$.[]{data-label="example7_2"}](p_prog.pdf)
EXAMPLE 8 (A Realistic Strong Detonation in 2D). The setup of this case consists of two parts divided by a shock travelling to the right in a 2D domain of $[0,3]\text{m} \times [0,1]\text{m}$, as in Fig. \[schematic2\]: the left red part is post-shock and filled with high-temperature high-pressure burnt gas while the right blue part is pre-shock and filled with reactive unburnt gas in one atmosphere pressure and room temperature. Geometry of the post-shock burnt gas part follows $$\{|y-0.5|>0.25, x<0.5 \} \cup \{|y-0.5| \leq 0.25, x-0.25<y<1.25-x \},$$ and the unburnt gas occupies the rest of domain before the initial shock. Initial states are identical with those in Example 6 except the x-velocity of the post-shock part is increased to $u_b=2000\text{m/s}>u_{CJ}$, to create a strong detonation wave. The boundary condition for the left/right end is simply extrapolation from the mirror image points inside the domain and the top/bottom boundary is considered as a slip wall. All simulations stop at $t=1 \times 10^{-3}$s.
We obtain the reference exact solution by the deterministic method using a very fine grid with $3000 \times 1000$ points and a fixed tiny timestep of $\Delta t = 2.5 \times 10^{-8}$s. In comparison, a set of under-resolved uniform grid and timestep is considered, i.e. $150 \times 50, \Delta t=2.5 \times 10^{-7}$s (we found using the linearly scaled $\Delta t=5 \times 10^{-7}$s corresponding to the $150 \times 50$ grid appears too large to integrate the ODEs system by CHEMEQ2 stably without any parameter tuning). From Fig. \[example8\_1\], it is clearly to see the density distributions along with locations of the detonation wave at different times in three solutions. In comparison with the reference solution, the SRR method computes the reasonable locations of the reacting front at all times. Due to the considerably low resolution used in the SRR method, detailed characteristics presented in the reference solution such as the triple points, slip lines, small vortices and peak values of density are diffused while the overall flowfield including the profile of reacting front has been correctly captured. In stark contrast, for the deterministic method with the same resolution, a developing spurious weak detonation wave can be easily detected with a maximum error of nearly 10% of the domain length in only 1 millisecond. It not only validates the wider effectiveness of the proposed method but also implies even tiny numerical dissipation is potential to be dangerous in a long-term development of reacting flows for ordinary shock-capturing schemes in under-resolved conditions.
![Schematic for the 2D domain in Example 8.[]{data-label="schematic2"}](example7.pdf)
![Example 8 the density distribution and the detonation front location at different times: left $\sim$ reference solution; middle $\sim$ deterministic solution by CHEMEQ2; right $\sim$ SRR solution; the location of the reacting front is marked by the white solid line with $y_{\text{H}_2\text{O}}=0.1$. []{data-label="example8_1"}](2s.pdf "fig:")\
![Example 8 the density distribution and the detonation front location at different times: left $\sim$ reference solution; middle $\sim$ deterministic solution by CHEMEQ2; right $\sim$ SRR solution; the location of the reacting front is marked by the white solid line with $y_{\text{H}_2\text{O}}=0.1$. []{data-label="example8_1"}](4s.pdf "fig:")\
![Example 8 the density distribution and the detonation front location at different times: left $\sim$ reference solution; middle $\sim$ deterministic solution by CHEMEQ2; right $\sim$ SRR solution; the location of the reacting front is marked by the white solid line with $y_{\text{H}_2\text{O}}=0.1$. []{data-label="example8_1"}](6s.pdf "fig:")\
![Example 8 the density distribution and the detonation front location at different times: left $\sim$ reference solution; middle $\sim$ deterministic solution by CHEMEQ2; right $\sim$ SRR solution; the location of the reacting front is marked by the white solid line with $y_{\text{H}_2\text{O}}=0.1$. []{data-label="example8_1"}](8s.pdf "fig:")\
![Example 8 the density distribution and the detonation front location at different times: left $\sim$ reference solution; middle $\sim$ deterministic solution by CHEMEQ2; right $\sim$ SRR solution; the location of the reacting front is marked by the white solid line with $y_{\text{H}_2\text{O}}=0.1$. []{data-label="example8_1"}](10s.pdf "fig:")\
Conclusions
===========
A new fractional step method for simulating chemically reacting flows, especially for capturing stiff detonation waves in under-resolved conditions has been developed. Two procedures based on operator splitting are included: for the convection part of the reactive Euler equations, any standard shock-capturing scheme is free to utilize; for the reaction step the multi-species multi-reaction ODEs system in the source terms is further split to solve in a reaction-by-reaction manner, from which exact mass conservation, strict positivity preserving and almost unconditional stability are guaranteed. Unlike deterministic methods that integrate the ODEs directly or the random projection method that requires two presumed equilibrium states, each reaction in the reaction system either proceeds a timestep forward or stops according to a local random temperature in the proposed method. Chemical reaction, e.g. ignition, in the smeared discontinuities due to numerical viscosity in the shock-capturing method is therefore a random process , rather than a deterministic one with growing error accumulation. A wide range of numerical experiments including not only simple model kinetics but also real-world nonequilibrium chemistry such as the temperature-dependent finite-rate hydrogen-air combustion are considered in 1D and 2D flows, demonstrating the proposed method can effectively predict the correct propagation of discontinuities as well as the overall flowfield information in under-resolved conditions. Besides, the diminishing randomness by adding a shift term to generate a random temperature below its local smeared value enables the regression of the proposed random method into a deterministic method in terms of nonstiff cases with fine resolutions in space and time. Also, its dimensional independence makes further 3D extension of the proposed method straightforward.
Acknowledgements {#acknowledgements .unnumbered}
================
The financial support from the EU Marie Sk[ł]{}odowska-Curie Innovative Training Networks (ITN-ETN) (Project ID: 675528-IPPAD-H2020-MSCA-ITN-2015) for the first author is gratefully acknowledged.
Analytical solutions to some simple forms of a one-way reaction equation {#appendix1}
========================================================================
For the simplest form of a reaction in Eq. , $$\begin{aligned}
A \longrightarrow \text{products}, \\
%\text{or} \quad A + B \longrightarrow \text{products}, \\
%\text{or} \quad 2A \longrightarrow \text{products}, \\
%\text{or} \quad A + B + C \longrightarrow \text{products}, \\
%\text{or} \quad A + 2B \longrightarrow \text{products}, \\
\end{aligned}$$ we simply have an ODE for the molar concentration $\left[ A \right]$, as $$\label{A1}
\frac{d \left[ A \right]}{dt} = - k \left[ A \right],$$ with $k$ being the rate constant and initial value of $\left[ A \right]_0$ at $t = t_0$. The above ODE written in the expression of molar concentration is equivilent to Eq. using density and mass fraction since $$\left[ A \right] = \frac{\rho_A}{W_A} = \frac{\rho y_A}{W_A}.$$ The solution to Eq. by seperation of varibles is $$\left[ A \right] = \left[ A \right]_0 e^{-k (t-t_0)}.$$
For the reaction form $$\label{A+B}
\begin{aligned}
A + B \longrightarrow \text{products}, \\
\end{aligned}$$ we have the ODEs system as $$\label{A,B}
\begin{aligned}
\frac{d \left[ A \right]}{dt} = - k \left[ A \right]\left[ B \right], \\
\frac{d \left[ B \right]}{dt} = - k \left[ A \right]\left[ B \right]. \\
\end{aligned}$$ This also means that $$\begin{aligned}
d \left[ A \right] = d \left[ B \right]
\end{aligned}$$ holds for any time interval $dt$ and thus $$\label{A=B}
\begin{aligned}
\left[ A \right] - \left[ A \right]_0 = \left[ B \right] - \left[ B \right]_0.
\end{aligned}$$ Substituting relation into Eq. , we have $$\begin{aligned}
\frac{d \left[ A \right]}{dt} = - k \left[ A \right] ( \left[ A \right] + \Delta_{AB} ),
\end{aligned}$$ where $ \Delta_{AB} = \left[ B \right]_0 - \left[ A \right]_0 $, leading to the solution of $\left[ A \right]$ as $$\begin{aligned}
\left[ A \right] =
\begin{cases}
\frac{\Delta_{AB}}{ \frac{\left[B\right]_0}{\left[A\right]_0} e^{\Delta_{AB} k(t-t_0)} - 1 }, \quad & \text{if} \, \Delta_{AB} \neq 0, \\
\frac{1}{ k(t-t_0) + \frac{1}{ \left[ A \right]_0 } }, \quad & \text{otherwise}. \\
\end{cases}
\end{aligned}$$
For reaction $$\begin{aligned}
2A \longrightarrow \text{products}, \\
\end{aligned}$$ it is a special case for reaction and the solution is $$\begin{aligned}
\left[ A \right] &= \frac{1}{ k(t-t_0) + \frac{1}{ \left[ A \right]_0 } }. \\
\end{aligned}$$
For a more complicated third-order reaction $$\begin{aligned}
A + B + C \longrightarrow \text{products}, \\
\end{aligned}$$ we also ultilize the relations $$\begin{aligned}
\left[ A \right] - \left[ A \right]_0 = \left[ B \right] - \left[ B \right]_0 = \left[ C \right] - \left[ C \right]_0
\end{aligned}$$ and perform seperation of varibles to get $$\begin{aligned}
\frac{d \left[ A \right]}{\left[ A \right] ( \left[ A \right] + \Delta_{AB} ) ( \left[ A \right] + \Delta_{AC} )} = - k dt .
\end{aligned}$$ Finally, we can only have the implicit solution for $\left[ A \right]_0 \neq \left[ B \right]_0 \neq \left[ C \right]_0$ in general, obeying $$\begin{aligned}
\left( \frac{\left[A\right]}{\left[A\right]+\Delta_{AC}} \frac{\left[C\right]_0}{\left[A\right]_0} \right)^{\frac{1}{\Delta_{CB}\Delta_{AC}} }
-
\left( \frac{\left[A\right]}{\left[A\right]+\Delta_{AB}} \frac{\left[B\right]_0}{\left[A\right]_0} \right)^{\frac{1}{\Delta_{CB}\Delta_{AB}} }
= e^{-k(t-t_0)}.
\end{aligned}$$
Only when $\left[ A \right]_0 = \left[ B \right]_0 = \left[ C \right]_0$ or the special reaction $$\begin{aligned}
3A \longrightarrow \text{products}, \\
\end{aligned}$$ the explicit analytical solution exists, i.e. $$\begin{aligned}
\left[ A \right] = \sqrt{\frac{1}{\frac{1}{\left[A\right]_0^2} + 2k(t-t_0)}}.
\end{aligned}$$
After the determination of the new state of the reactant species $\left[ A \right]$, states of the remaining species including all the products and other reactants can be updated by the law of mass conservation in Eq. .
Reaction mechanism for hydrogen-air combustion {#appendix2}
==============================================
ID Elementary reaction $A$ $B$ $E_a$
------- ------------------------------------------------------------------------------------------------- ---------- ------- --------
1,2 $\text{H} + \text{O}_2 \Longleftrightarrow \text{OH} + \text{O}$ 1.91e+14 0.0 16.44
3,4 $\text{H}_2 + \text{O} \Longleftrightarrow \text{H} + \text{OH}_2 $ 5.08e+04 2.67 6.292
5,6 $\text{H}_2 + \text{OH} \Longleftrightarrow \text{H} + \text{H}_2\text{O} $ 2.16e+08 1.51 3.43
7,8 $\text{O} + \text{H}_2\text{O} \Longleftrightarrow \text{OH} + \text{OH} $ 2.97e+06 2.02 13.4
9,10 $\text{H}_2 + \text{M} \Longleftrightarrow \text{H} + \text{H} + \text{M} $ 4.57e+19 -1.4 105.1
11,12 $\text{O} + \text{O} + \text{M} \Longleftrightarrow \text{O}_2 + \text{M} $ 6.17e+15 -0.5 0.0
13,14 $\text{H} + \text{O} + \text{M} \Longleftrightarrow \text{OH} + \text{M} $ 4.72e+18 -1.0 0.0
15,16 $\text{H} + \text{OH} + \text{M} \Longleftrightarrow \text{H}_2\text{O} + \text{M} $ 4.50e+22 -2.0 0.0
17,18 $\text{H} + \text{O}_2 + \text{M} \Longleftrightarrow \text{H}\text{O}_2 + \text{M} $ 3.48e+16 -0.41 -1.12
19,20 $\text{H} + \text{O}_2 \Longleftrightarrow \text{H}\text{O}_2 $ 1.48e+12 0.60 0.0
21,22 $\text{H} + \text{HO}_2 \Longleftrightarrow \text{H}_2 + \text{O}_2 $ 1.66e+13 0.0 0.82
23,24 $\text{H} + \text{HO}_2 \Longleftrightarrow \text{OH} + \text{OH} $ 7.08e+13 0.0 0.3
25,26 $\text{HO}_2 + \text{O} \Longleftrightarrow \text{OH} + \text{O}_2 $ 3.25e+13 0.0 0.0
27,28 $\text{OH} + \text{HO}_2 \Longleftrightarrow \text{H}_2\text{O} + \text{O}_2 $ 2.89e+13 0.0 -0.5
29,30 $\text{H}\text{O}_2 + \text{H}\text{O}_2 \Longleftrightarrow \text{H}_2\text{O}_2 + \text{O}_2$ 4.20e+14 0.0 11.98
31,32 $\text{H}\text{O}_2 + \text{H}\text{O}_2 \Longleftrightarrow \text{H}_2\text{O}_2 + \text{O}_2$ 1.30e+11 0.0 -1.629
33,34 $\text{H}_2\text{O}_2 + \text{M} \Longleftrightarrow \text{OH} + \text{OH} + \text{M} $ 1.27e+17 0.0 45.5
35,36 $\text{H}_2\text{O}_2 \Longleftrightarrow \text{OH} + \text{OH} $ 2.95e+14 0.0 48.4
37,38 $\text{H}_2\text{O}_2 + \text{H} \Longleftrightarrow \text{H}_2\text{O} + \text{OH} $ 2.41e+13 0.0 3.97
39,40 $\text{H}_2\text{O}_2 + \text{H} \Longleftrightarrow \text{H}_2 + \text{HO}_2 $ 6.03e+13 0.0 7.95
41,42 $\text{H}_2\text{O}_2 + \text{O} \Longleftrightarrow \text{OH} + \text{HO}_2 $ 9.55e+06 2.0 3.97
43,44 $\text{H}_2\text{O}_2 + \text{OH} \Longleftrightarrow \text{H}_2\text{O} + \text{HO}_2 $ 1.00e+12 0.0 0.0
45,46 $\text{H}_2\text{O}_2 + \text{OH} \Longleftrightarrow \text{H}_2\text{O} + \text{HO}_2 $ 5.80e+14 0.0 9.56
References {#references .unnumbered}
==========
|
---
abstract: 'We study the behavior of blocks in flat families of finite-dimensional algebras. In a general setting we construct a finite directed graph encoding a stratification of the base scheme according to the block structures of the fibers. We furthermore show that all block structures are determined by “atomic” ones living on the components of a Weil divisor. A byproduct is that the number of blocks of fibers defines a lower semicontinuous function on the base scheme. We prove that this semicontinuity also holds in more general settings. We furthermore discuss how to obtain information about the simple modules in the blocks by generalizing and establishing several properties of decomposition matrices by Geck and Rouquier.'
author:
- Ulrich Thiel
bibliography:
- 'references.bib'
title: 'Blocks in flat families of finite-dimensional algebras'
---
=1
Introduction {#introduction .unnumbered}
============
It is a classical fact in ring theory that a non-zero noetherian ring $A$ can be decomposed as a direct product $A= \prod_{i=1}^n B_i$ of indecomposable rings $B_i$. Such a decomposition is unique up to permutation and isomorphism of the factors. Let us denote by $\msf{Bl}(A)$ the set of the $B_i$, called the of $A$. The decomposition of $A$ into blocks induces a decomposition $A\tn{-}\msf{Mod} = \bigoplus_{i=1}^n B_i\tn{-}\msf{Mod}$ of the category of (left) $A$-modules. In particular, a simple $A$-module is a simple $B_i$-module for a unique block $B_i$ and so we get an induced decomposition $\msf{Irr} A = \coprod_{i=1}^n \msf{Irr} B_i$ of the set of simple modules. Let us denote by $\msf{Fam}(A)$ the set of the $\msf{Irr} B_i$, called the of $A$. The blocks and families of a ring are important invariants which help to organize and simplify its representation theory. The aim of this paper is to investigate how these invariants vary in a flat family of finite-dimensional algebras.
More precisely, we consider a algebra $A$ over an integral domain $R$, i.e., $A$ is finitely generated and flat as an $R$-module. This yields a family of algebras parametrized by $\Spec(R)$ consisting of the (or ) $$\label{specialization_definition}
A(\fp) \dopgleich \msf{k}(\fp) \otimes_R A \simeq A_\fp/\fp_\fp A_\fp \;,$$ where $\msf{k}(\fp) = \msf{Frac}(R/\fp)$ is the residue field of $\fp \in \Spec(R)$ in $R$ and $A_\fp$ is the localization of $A$ in $\fp$. Note that the fiber $A(\fp)$ is a finite-dimensional $\msf{k}(\fp)$-algebra. Now, the primary goal would be to describe for any $\fp$ the blocks of $A(\fp)$, e.g., the number of blocks, and to describe the simple modules in each block, e.g., the number of such modules and their dimensions. It is clear that there will be no general theory giving the precise solutions to these problems for arbitrary $A$. For example, we can take the group ring $A = \bbZ \mrm{S}_n$ of the symmetric group. The fibers of $A$ are precisely the group rings $\bbQ \mrm{S}_n$ and $\bbF_p \mrm{S}_n$ for all primes $p$, and the questions above are still unanswered. Nonetheless, and this is the point of this paper, there are some general phenomena, some patterns in the behavior of blocks and simple modules along the fibers, which are true quite generally. The best situation turns out to be when $R$ is noetherian and normal, and the generic fiber $A^K$ is a split $K$-algebra, where $K$ is the fraction field of $R$ (we will shortly address the case of non-split generic fiber). This setting includes many examples in representation theory like Brauer algebras, Hecke algebras, and (restricted) rational Cherednik algebras. We show (see Theorem \[block\_stratification\_thm\], Lemma \[fine\_block\_stratification\], and Corollary \[normal\_split\_corollary\]) that in this case we can always construct a finite directed graph encoding the block structures of all fibers and giving a complete overview about what happens to blocks under specialization. In Figure \[brauer\_graph\_example\] below we give an example of such a graph in case of a generic Brauer algebra.
& & &\
& & & & & &\
& & & & &
The partitions on the top of each vertex describe the generic block structure on the zero locus of the ideal at the bottom by showing which blocks of the generic fiber $A^K$ will “glue” when specializing to the corresponding zero locus (we will make this precise in §\[blocks\_of\_localizations\]). If at any given vertex we remove the zero loci at all vertices below, we obtain a locally closed subset on which the block structure is always equal to the one described by the vertex. So, this graph encodes a stratification of the base scheme, in this case the two-dimensional scheme $\Spec(\bbZ \lbrack \boldsymbol{\delta} \rbrack)$. We want to point out that it is central for us to work with (affine) schemes. For example in Figure \[brauer\_graph\_example\] we have one vertex with zero locus $(2)$, i.e., we consider the Brauer algebra in characteristic two. Now, we do not only have the case $\boldsymbol{\delta} \in \lbrace 0,1 \rbrace = \bbF_2$, which is described by the two vertices below $(2)$, but we also have a generic characteristic two case, described by the generic point of $\bbF_2 \lbrack \boldsymbol{\delta} \rbrack$, and this is really different from the case of specialized $\boldsymbol{\delta}$. There is one further aspect visible in Figure \[brauer\_graph\_example\]. Namely, in the middle row of the graph we have four subschemes of codimension one on which the block structure is different from the generic one, i.e., the one of $A^K$. And the block structures on these components have an “atomic” character, i.e., any other block structure is obtained by gluing “atomic” ones. We can thus say that the block structures are governed by “atomic” block structures living on the components of a Weil divisor of the base scheme. This Weil divisor should be considered as a sort of new of $A$. Note that the values occurring in this discriminant in Figure \[brauer\_graph\_example\] are precisely the parameters where the Brauer algebra is not semisimple anymore (the precise parameters have been determined by Rui [@Rui] for all $n \in \bbN$). In Lemma \[cellular\_alg\_blgen\_decgen\] we prove why this must be the case.
Now, as already mentioned, our aim is clearly not to derive new results about Brauer algebras. Our intention is to show that the kind of behavior just described is actually a very general phenomenon. It also holds for group algebras, Hecke algebras, (restricted) rational Cherednik algebras, etc.—we can always draw such a graph with “atomic” block structures on a Weil divisor.
We can of course collapse the above graph by just considering the number of blocks and not their actual block structure in comparison to the generic one. What we obtain is a stratification of the base scheme by the number of blocks of the fibers. In other words, the map $\Spec(R) \rarr \bbN$, $\fp \mapsto \#\msf{Bl}(A(\fp))$, is lower semicontinuous. We show that this property in fact also holds in cases where we do not have split generic fiber—as long as we restrict to a “nice” enough subset of $\Spec(R)$. More precisely, we show in Corollary \[finite\_type\_alg\_closed\_setting\] that $\msf{Max}(R) \rarr \bbN$, $\fm \mapsto \#\msf{Bl}(A(\fm))$, is lower semicontinuous whenever $R$ is a finite type algebra over an algebraically closed field, where $\msf{Max}(R)$ is the subset of closed points of $\Spec(R)$. This establishes the lower semicontinuity of blocks (in closed points) for example also for quantized enveloping algebras of semisimple Lie algebras at roots of unity, enveloping algebras of semisimple Lie algebras in positive characteristic, and quantized function algebras of semisimple groups at roots of unity. More generally, this also applies to Hopf PI triples as introduced by Brown–Goodearl [@Brown-Goodearl-Quantum-Groups] (see also Brown–Gordon [@BG-Ramification] and Gordon [@Gordon:Representations-of-semisimple-Lie-in-pos]), where questions about blocks in closed points have been raised and studied. We note that the number of blocks will in general not be lower semicontinuous on the whole of $\Spec(R)$, see Example \[ken\_example\].\
In §\[semicontinuity\_of\_blocks\] we discuss the construction of the block graph and the corresponding stratification as illustrated above. The main results here are Theorem \[block\_stratification\_thm\], Lemma \[fine\_block\_stratification\], Corollary \[normal\_split\_corollary\], Lemma \[maximal\_atomic\_codim\_1\], and Corollary \[finite\_type\_alg\_closed\_setting\]. Before we start, we review a few standard facts in §\[notations\], essentially to fix notations. We have also included an appendix with some more elementary results we use throughout the paper. Even though partially standard, we feel that there are several results which are not mentioned in the literature. In §\[blocks\_via\_central\_characters\] we establish a relationship between blocks of fibers and reductions of central characters. The main result here is Theorem \[maintheorem\_blex\_descr\] which essentially says that once we know the central characters of simple $A^K$-modules, we can compute the whole block graph. This fact is very useful for explicit computations. In §\[blocks\_and\_dec\_maps\] we address questions about the simple modules in a block. The main tool here are the decomposition matrices introduced by Geck and Rouquier. In Theorem \[brauer\_rec\] we show that they satisfy Brauer reciprocity in a quite general setting in which it was not known to hold before. In §\[preservation\] we contrast the preservation of simple modules with the preservation of blocks under specialization, and show in Theorem \[decgen\_blgen\_inclusion\_main\_theorem\] that preservation of simple modules implies preservation of blocks. It is an interesting question to ask when the converse holds. We show in Example \[decgen\_eq\_blgen\_counterex\] that in general we do not have an equivalence, but in Lemma \[cellular\_alg\_blgen\_decgen\] we establish one context where this is true (this context includes the Brauer algebras and explains why our Weil divisor is given by the non-semisimple parameters). Finally, in §\[brauer\_graph\] we generalize the concept of Brauer graphs and show how these relate to blocks. In §\[open\_problems\] we mention some open problems we encountered. In §\[notes\] we give a short overview on results in the literature which address our questions in some or the other form.\
Acknowledgements {#acknowledgements .unnumbered}
----------------
I would like to thank Cédric Bonnafé for many helpful discussions about this topic, for showing and explaining me the relevant part of the manuscript [@BR-cellules] with Raphaël Rouquier, and for providing Example \[decgen\_eq\_blgen\_counterex\]. The manuscript [@BR-cellules] is certainly one of the main motivations for this paper, see §\[notes\]. I would furthermore like to thank Gwyn Bellamy, Ken Brown, Meinolf Geck, and Gunter Malle for commenting on parts of a preliminary version of this article. Moreover, I thank Ken Brown for providing Example \[ken\_example\]. I was partially supported by the DFG SPP 1489.
Notations
=========
To fix notations and to recall some standard facts, we begin with a short review of basic block theory. For us, a is always a ring with identity and a is always a *left* module unless we explicitly say it is a *right* module.
Block decompositions {#block_decompositions}
--------------------
For a ring $A$ we denote by $\msf{Idem}(A)$ the set of non-zero idempotents of $A$ and by $\mathscr{E}(A)$ we denote the set of finite sets $\lbrace e_i \rbrace_{i \in I}$ of pairwise orthogonal non-zero idempotents satisfying $1 = \sum_{i \in I} e_i$. We similarly define the sets $\msf{Idem}_p(A)$ and $\mathscr{E}_p(A)$ using primitive idempotents. If $e$ is an idempotent, then $Ae$ is a projective left ideal of $A$, and this yields a bijection between $\mathscr{E}(A)$ and direct sum decompositions of the $A$-module $A$ into non-zero left ideals up to permutation of the summands. The idempotent $e$ is primitive if and only if the $A$-module $Ae$ is indecomposable.
Let us now concentrate on idempotents in the center $Z \dopgleich \msf{Z}(A)$ of $A$. To simplify notations, we set $\msf{Idem}_c(A) \dopgleich \msf{Idem}(Z)$, $\msf{Idem}_{cp}(A) \dopgleich \msf{Idem}_p(Z)$, $\mathscr{E}_c(A) \dopgleich \mathscr{E}(Z)$, and $\mathscr{E}_{cp}(A) \dopgleich \mathscr{E}_p(Z)$. Primitive idempotents of $Z$ are also called idempotents of $A$. If $c$ is a central idempotent of $A$, then $Ac = cA$ is a two-sided ideal of $A$ and at the same time a ring with identity element equal to $c$ (hence not a subring). This yields a bijection between $\mscr{E}_c(A)$ and direct sum decompositions of the ring $A$ into non-zero two-sided ideals of $A$ up to permutation of the summands, and such decompositions are in turn in bijection with direct product decompositions of the ring $A$ into non-zero rings up to permutation and isomorphism of the factors. A central idempotent $c$ is centrally-primitive if and only if $Ac$ is an indecomposable ring. It is a standard fact—and the starting point of block theory—that if $\mscr{E}_{cp}(A)$ is not empty, then it contains exactly one element, namely $\msf{Idem}_{cp}(A)$ itself, and that any central idempotent of $A$ is a sum of a subset of $\msf{Idem}_{cp}(A)$. We then say that $A$ has a , call the centrally-primitive idempotents of $A$ also the , and call the corresponding rings $Ac$ the of $A$. In this case we prefer to write $\msf{Bl}(A) \dopgleich \msf{Idem}_{cp}(A)$. To avoid pathologies we set $\msf{Bl}(0) \dopgleich \emptyset$ for the zero ring $0$.
Families of simple modules {#families}
--------------------------
Let $\mscr{C} \dopgleich \lbrace c_i \rbrace_{i \in I} \in \mscr{E}_c(A)$ be *some* decomposition, not necessarily a block decomposition. Let $B_i \dopgleich Ac_i$. If $V$ is a non-zero $A$-module, then $V = \bigoplus_{i \in I} c_i V$ as $A$-modules and each summand $c_i V$ is a $B_i$-module. In this way we obtain a decomposition $A\tn{-}\msf{Mod} = \bigoplus_{i \in I} B_i\tn{-}\msf{Mod}$ of module categories, which also restricts to a decomposition of the category of finitely generated modules. If a non-zero $A$-module $V$ is under this decomposition obtained from a $B_i$-module, then $V$ is said to to $B_i$. This is equivalent to $c_i V = V$ and $c_j V = 0$ for all $j \neq i$. An indecomposable, and thus any simple, $A$-module clearly belongs to a unique $B_i$. We thus get a decomposition $\msf{Irr} A = \coprod_{i \in I} \msf{Irr} B_i$ of the set of (isomorphism classes of) simple modules. We call the sets $\Irr (A,B_i) \dopgleich \msf{Irr} B_i$ the of $A$ and denote the set of $\mscr{C}$-families by $\msf{Fam}_{\mscr{C}}(A)$. Note that we have a natural bijection $$\mscr{C} \overset{\sim}{\longrightarrow} \msf{Fam}_{\mscr{C}}(A)$$ given by $c_i \mapsto \Irr B_i$. In case $\mscr{C}$ is actually a block decomposition, we call the $\mscr{C}$-families simply the of $A$ and set $\msf{Fam}(A) \dopgleich \msf{Fam}_{\mscr{C}}(A)$. Recall that any central idempotent of $A$ is a sum of a subset of the block idempotents of $A$. Hence, for general $\mscr{C}$ as above the families are a finer partition of $\msf{Irr}A$ than the $\mscr{C}$-families, i.e., any $\mscr{C}$-family is a union of families.
Linkage relation and noetherian rings {#linkage_relation}
-------------------------------------
For a general ring $A$ the on $\msf{Idem}_p(A)$ is the relation $\sim$ defined by $e \sim e'$ if and only if there is $f \in \msf{Idem}_p(A)$ and non-zero $A$-module morphisms $Af \rarr Ae$ and $Af \rarr Ae'$. The equivalence classes of the equivalence relation generated by $\sim$ are called the . It is now a standard fact that if $\lbrace e_i \rbrace_{i \in I} \in \mscr{E}_p(A)$ and $\lbrace c_i \rbrace_{i \in I'}$ is the set of linkage class sums, then $\lbrace c_i \rbrace_{i \in I'} \in \mscr{E}_{cp}(A)$, so $A$ has a block decomposition. In this case the indecomposable projective modules $Ae$ and $Ae'$ belong to the same block if and only if $e$ and $e'$ lie in the same linkage class. If $A \neq 0$ is noetherian, then $\mscr{E}_p(A) \neq \emptyset$, so noetherian rings have a block decomposition.
Semiperfect rings {#semiperfect_rings}
-----------------
Another class of rings having block decompositions are rings. Recall that one of the many properties characterizing a ring $A$ as semiperfect is that every finitely generated (or, equivalently, just every simple) $A$-module has a projective cover. Another characterization is that there exists a decomposition $\lbrace e_i \rbrace_{i \in I} \in \mscr{E}(A)$ with $e_i$ being , i.e., $\msf{End}_A(Ae_i)$ is local. A module with local endomorphism ring is also called as this property is stronger than being indecomposable. Hence, by the preceding paragraph, a semiperfect ring has a block decomposition. Recall that every artinian ring, and thus every finite-dimensional algebra over a field, is semiperfect. Assume now that $A$ is semiperfect. If $e_i$ is a primitive idempotent of $A$, it is already and $Ae_i$ is the projective cover of its head, which is a simple $A$-module. In fact, if $\lbrace e_i \rbrace_{i \in I} \in \mscr{E}_{p}(A)$, then there is a subset $I' \subs I$ such that $( P_i )_{i \in I'}$ is a system of representatives of the isomorphism classes of projective indecomposable $A$-modules, and their heads $S_i \dopgleich \msf{Hd}(P_i) = P_i/\msf{Rad}(P_i)$ give a system of representatives of the isomorphism classes of simple $A$-modules. The projective class group $\msf{K}_0(A) \dopgleich \msf{K}_0(A\tn{-}\msf{proj})$ is a free abelian group with basis formed by the isomorphism classes of the $( P_i )_{i \in I'}$ and we have a natural isomorphism $\msf{K}_0(A) \simeq \msf{G}_0(A) \dopgleich \msf{K}_0(A\tn{-}\msf{mod})$ mapping $P_i$ to $S_i$.
Base change of blocks {#notations_base_change_blocks}
---------------------
Let $\phi:R \rarr S$ be a morphism of commutative rings. If $V$ is an $R$-module, we write $V^S \dopgleich \phi^*V \dopgleich S \otimes_R V$ for the scalar extension of $V$ to $S$ and by $\phi_V:V \rarr V^S$ we denote the canonical map $v \mapsto 1 \otimes v$.
\[phi\_injective\_lemma\] In each of the following cases the map $\phi_V:V \rarr V^S$ is injective:
\[phi\_injective\_lemma:proj\] $\phi$ is injective and $V$ is $R$-projective.
\[phi\_injective\_lemma:faithfully\_flat\] $\phi$ is faithfully flat.
\[phi\_injective\_lemma:localiz\] $\phi$ is the localization morphism for a multiplicatively closed subset $\Sigma \subs R$ and $V$ is $\Sigma$-torsion-free.
The first case follows from [@Bou-Algebra-1-3 II, §5.1, Corollary to Proposition 4], the second follows from [@Bou-Commutative-Algebra-1-7 I, §3.5, Proposition 8(i,iii)], and the last case follows from the fact that $\phi$ is flat in conjunction with [@Bou-Commutative-Algebra-1-7 I, §2.2, Proposition 4].
If $A$ is an $R$-algebra, then the $S$-module $A^S$ is naturally an $S$-algebra and the map $\phi_A:A \rarr A^S$ is a ring morphism. Moreover, if $V$ is an $A$-module, then the underlying $S$-module of $A^S \otimes_A V$ is simply $V^S$. Our aim is to study the behavior of blocks under the morphism $\phi_A:A \rarr A^S$. Clearly, if $e \in A$ is an idempotent, also $\phi_A(e) \in A^S$ is an idempotent, and if $e$ is central, so is $\phi_A(e)$ by the elementary fact that $$\phi_A(\msf{Z}(A)) \subs \msf{Z}(A^S) \;.$$
To describe some properties of $\phi_A$ with respect to idempotents and blocks, we introduce the following notations.
We say that $\phi_A$ is:
if $\phi_A(e) \neq 0$ for any non-zero idempotent $e$ of $A$,
if $\phi_A(c) \neq 0$ for any non-zero central idempotent $c$ of $A$,
if for each idempotent $e' \in A^S$ there is an idempotent $e \in A$ with $\phi_A(e) = e'$,
if for each central idempotent $c' \in A^S$ there is a central idempotent $c \in A$ with $\phi_A(c) = c'$,
if $\phi_A$ induces a bijection between the *isomorphism classes* of primitive idempotents of $A$ and the isomorphism classes of primitive idempotents of $A^S$,
if $\phi_A$ induces a bijection between the centrally-primitive idempotents of $A$ and the centrally-primitive idempotents of $A^S$.
Recall that *central* idempotents are isomorphic if and only if they are *equal*, so we do not have to consider isomorphism classes in the definition of “block bijective”. Note that in case $\phi_A$ is idempotent stable, respectively central idempotent stable, it induces a map between the sets of decompositions $\mscr{E}(A)$ and $\mscr{E}(A^S)$, respectively between $\mscr{E}_c(A)$ and $\mscr{E}_c(A^S)$, as defined in §\[block\_decompositions\]. The following lemma reveals two opposing situations in which $\phi_A$ is idempotent stable (and thus central idempotent stable). We denote by $\msf{Rad}(A)$ the Jacobson radical of $A$.
\[idempotent\_stable\] If $\msf{Ker}(\phi_A) \subs \msf{Rad}(A)$, then $\phi_A$ is idempotent stable. This holds in the following two cases:
\[idempotent\_stable:inj\] $\phi_A$ is injective (see Lemma \[phi\_injective\_lemma\]),
\[idempotent\_stable:surj\] $\phi$ is surjective, $\msf{Ker}(\phi) \subs \msf{Rad}(R)$, and $A$ is finitely generated as an $R$-module.
If $e \in A$ is an idempotent contained in $\msf{Rad}(A)$, then by a well-known characterization of the Jacobson radical (see [@CR-Methods-1 5.10]) we conclude that $e^\dagger = 1-e \in A^\times$ is a unit, and since $e^\dagger$ is also an idempotent, we must have $e^\dagger = 1$, implying that $e = 0$. If $\phi_A$ is injective, the condition clearly holds. In the second case we have $
\msf{Ker}(\phi_A) = \msf{Ker}(\phi) A \subs \msf{Rad}(R) A \subs \msf{Rad}(A)$, where the last inclusion follows from [@Lam-First-Course-91 Corollary 5.9].
Now, suppose that $\phi_A$ is idempotent stable and that both $A$ and $A^S$ have block decompositions. Let $\lbrace c_i \rbrace_{i \in I}$ be the block idempotents of $A$ and let $\lbrace c_j' \rbrace_{j \in J}$ be the block idempotents of $A^S$. Since $\phi_A$ is idempotent stable, we have $\msf{Bl}_\phi(A^S) \dopgleich \phi_A(\lbrace c_i \rbrace_{i \in I}) \in \mscr{E}_c(A^S)$. We call the $\phi_A(c_i)$ the of $A^S$ and call the corresponding families as defined in §\[families\] the of $A^S$, denoted $\msf{Fam}_\phi(A^S)$. As explained in §\[families\] each $\phi$-block $\phi_A(c_i)$ is a sum of a subset of the block idempotents of $A^S$ and the $\phi$-families are coarser than the families in the sense that each $\phi$-family is a union of $A^S$-families. In particular, we have $$\label{get_more_blocks_equation}
\# \msf{Bl}(A) = \# \msf{Bl}_\phi(A^S) \leq \# \msf{Bl}(A^S) \;.$$ The following picture illustrates this situation: $$\begin{tikzcd}[column sep=small]
\underset{c_{1_1}'}{\bullet} \underset{c_{1_2}'}{\bullet} \cdots \underset{c_{1_{m_1}}'}{\bullet} & \underset{c_{2_1}'}{\bullet} \underset{c_{2_2}'}{\bullet} \cdots \underset{c_{2_{m_2}}'}{\bullet} & \cdots & \underset{c_{n_1}'}{\bullet} \underset{c_{n_2}'}{\bullet} \cdots \underset{c_{n_{m_n}}'}{\bullet} & A^S\tn{-blocks} \\
\underset{\phi_A(c_1)}{\bullet} \arrow{u} \arrow[end anchor=230]{u} \arrow[end anchor=310]{u} & \underset{\phi_A(c_2)}{\bullet} \arrow{u} \arrow[end anchor=230]{u} \arrow[end anchor=310]{u} & \cdots & \underset{\phi_A(c_n)}{\bullet} \arrow{u} \arrow[end anchor=230]{u} \arrow[end anchor=310]{u} & \phi\tn{-blocks} \\
\underset{c_1}{\bullet} \arrow[mapsto]{u}{\phi_A} & \underset{c_2}{\bullet} \arrow[mapsto]{u}{\phi_A} & \cdots & \underset{c_n}{\bullet} \arrow[mapsto]{u}{\phi_A} & A\tn{-blocks}
\end{tikzcd}$$
In Appendix \[appendix\_blocks\_base\_change\] we have collected several further facts about base change of blocks. We will use these results in the sequel.
Semicontinuity of blocks {#semicontinuity_of_blocks}
========================
In this section we construct the stratifications and graphs mentioned in the introduction. The main results are Theorem \[block\_stratification\_thm\], Lemma \[fine\_block\_stratification\], Corollary \[normal\_split\_corollary\], Lemma \[maximal\_atomic\_codim\_1\], and Corollary \[finite\_type\_alg\_closed\_setting\].
Throughout this paragraph, we assume that $A$ is a finite flat algebra over an integral domain $R$ with fraction field $K$.
Blocks of localizations {#blocks_of_localizations}
-----------------------
Before we consider blocks of specializations, we first take a look at blocks of localizations of $A$ as these are much easier to control and are still strongly related to blocks of specializations as we will see in the next paragraph.
It follows from Corollary \[finite\_flat\_int\_block\_dec\] that $A$ and any localization $A_\fp$ for $\fp \in \Spec(R)$ has a block decomposition, even if $A$ is not necessarily noetherian. Since the canonical map $\phi_\fp: A_\fp \rarr A^K$ is injective by Lemma \[phi\_injective\_lemma\], we have the notion of $\phi_\fp$-blocks and $\phi_\fp$-families of $A^K$ as defined in §\[notations\_base\_change\_blocks\]. To shorten notations, we call them the and , and write $\msf{Fam}_\fp(A^K)$ for the $\fp$-families. Recall that we have a natural bijection $$\msf{Bl}(A_\fp) \simeq \msf{Fam}_\fp(A^K) \;.$$ There is also the following more concrete view of $\fp$-blocks. Let $(c_i)_{i \in I}$ be the block idempotents of $A^K$. If $c \in A_\fp$ is any block idempotent, we know from §\[block\_decompositions\] that there is $I' \subs I$ with $c = \sum_{i \in I'} c_i$ in $A^K$. Hence, to any block idempotent of $A_\fp$ we can associate a subset of $I$, and if we take all block idempotents of $A_\fp$ into account, we get a partition $\gamma_A(\fp)$ of the set $I$, from which we can recover the block idempotents of $A_\fp$ by taking sums of the $c_i$ over the members of $\gamma_A(\fp)$. Hence, we get a map $$\gamma_A:\Spec(R) \rarr \msf{Part}(I)$$ to the set of partitions of the set $I$. If $\fq \subs \fp$, then we have an embedding $A_\fp \hookrightarrow A_\fq$ and by the same argumentation as above the block idempotents of $A_\fp$ are obtained by summing up block idempotents of $A_\fq$. Hence, the map $\gamma_A$ is actually a morphism of posets if we equip $\Spec(R)$ with the partial order $\leq$ defined by $\fp \leq \fq$ if $\fq \subs \fp$ (i.e., $\msf{V}(\fp) \subs \msf{V}(\fq)$) and we equip $\msf{Part}(I)$ with the partial order $\leq$ defined by $\mscr{P} \leq \mscr{Q}$ if $\mscr{P}$ is a coarser partition than $\mscr{Q}$, i.e., the members of $\mscr{P}$ are unions of members of $\mscr{Q}$. We consider the image $\Gamma_A$ of $\gamma_A$ as a sub-poset of $\msf{Part}(I)$ with the induced order $\leq$ and call its elements the of $A$. To $\mscr{P} \in \Gamma_A$ we attach the $$\Gamma_A(\mscr{P}) \dopgleich \gamma_A^{-1}(\mscr{P}) \subs \Spec(R) \;,$$ and the $$\label{skeleton_def}
\Gamma_A^\leq(\mscr{P}) \dopgleich \bigcup_{\mscr{P}' \leq \mscr{P}} \Gamma_A(\mscr{P}') \;.$$ We clearly have a finite decomposition $$\label{block_stratification}
\Spec(R) = \coprod_{\mscr{P} \in \Gamma_A} \Gamma_A(\mscr{P})$$ and the relation $$\label{stratum_from_skeleton}
\Gamma_A(\mscr{P}) = \Gamma_A^\leq(\mscr{P}) \setminus \bigcup_{\mscr{P}' <\mscr{P}} \Gamma_A^\leq(\mscr{P'}) \;,$$ The set $\Gamma_A^\leq$ of all skeleta is naturally in bijection with $\Gamma_A$ since from any skeleton $\Gamma_A^\leq(\mscr{P})$ we can recover $\mscr{P}$ as the unique maximal local block structure in the points of $\Gamma_A^\leq(\mscr{P})$. Moreover, we have $\mscr{P}' \leq \mscr{P}$ if and only if $\Gamma_A^\leq(\mscr{P}') \subs \Gamma_A^\leq(\mscr{P})$. Hence, when considering $\Gamma_A^\leq$ as a poset ordered by inclusion, then in fact $\Gamma_A \simeq \Gamma_A^\leq$ as posets. The of $A$ is the finite directed graph defined by the poset $\Gamma_A$ together with the skeleta $\Gamma_A^\leq(\mscr{P})$ as vertex labels. This graph encodes all all information about local block structures of $A$. In Figure \[brauer\_graph\_example\] we have given an example of such a graph in the case of a Brauer algebra as mentioned in the introduction.
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The poset $\Gamma_A$ clearly has a unique maximal element, namely the block structure $\gamma_A(\bullet)$ of $A$ in the generic point $\bullet$ of $\Spec(R)$, i.e., $\gamma_A(\bullet) = \lbrace \lbrace i \rbrace \mid i \in I \rbrace$ is the block structure of the generic fiber $A^K = A_\bullet$. The deviation of block structures from the generic one thus takes place on the set $$\msf{BlEx}^{\mrm{loc}}(A) \dopgleich \lbrace \fp \in \Spec(R) \mid \gamma_A(\fp) < \gamma_A(\bullet) \rbrace = \bigcup_{\mscr{P} \in \msf{Max}(\Gamma_A)} \Gamma_A^\leq(\mscr{P}) \;.$$ We call this set the of $A$ for reasons to become apparent soon. The generic local block structure occurs precisely on the set $$\msf{BlGen}^{\mrm{loc}}(A) \dopgleich \Spec(R) \setminus \msf{BlEx}^{\mrm{loc}}(A) = \lbrace \fp \in \Spec(R) \mid \gamma_A(\fp) = \gamma_A(\bullet) \rbrace \;.$$
Our aim is to show that the skeleta are in fact *closed* subsets of $\Spec(R)$ and that (\[block\_stratification\]) is a stratification of the scheme $\Spec(R)$. In Figure \[brauer\_graph\_example\] we can see this in the example already. The key ingredient in proving this is the following general proposition, which is essentially due to Bonnafé and Rouquier [@BR-cellules Proposition C.2.11]. We give a slightly more general version here.
\[br\_gen\_lemma\] Let $R$ be an integral domain with fraction field $K$, let $A$ be a finite flat $R$-algebra, and let $\mscr{F} \subs A^K$ be a finite set. Then $$\msf{Gen}_A(\mscr{F}) \dopgleich \lbrace \fp \in \Spec(R) \mid \mscr{F} \subs A_\fp \rbrace$$ is a [neighborhood]{.nodecor} of the generic point of $\Spec(R)$. If $A$ is finitely presented flat, then $\msf{Gen}_A(\mscr{F})$ is an [open]{.nodecor} subset of $\Spec(R)$, and if moreover $R$ is a Krull domain, the complement $\msf{Ex}_A(\mscr{F})$ of $\msf{Gen}_A(\mscr{F})$ in $\Spec(R)$ is a [reduced Weil divisor]{.nodecor}, i.e., it is either empty or pure of codimension one with finitely many irreducible components.
Let us first assume that $A$ is actually $R$-free. For an element $\alpha \in K$ we define $I_\alpha \dopgleich \lbrace r \in R \mid r\alpha \in R \rbrace$. This is a non-zero radical ideal in $R$, and it has the property that $\alpha \in R_{\fp }$ if and only if $I_\alpha \nsubseteq \fp $. To see this, suppose that $\alpha \in R_{\fp }$. Then we can write $\alpha = \frac{r}{x}$ for some $x \in R \setminus \fp $. Hence, $x \alpha = r \in R$ and therefore $x \in I_\alpha$. Since $x \notin \fp $, it follows that $I_\alpha \nsubseteq \fp $. Conversely, if $I_\alpha \nsubseteq \fp $, then there exists $x \in I_\alpha$ with $x \notin \fp $. By definition of $I_\alpha$ we have $x \alpha \gleichdop r \in R$ and since $x \notin \fp $, we can write $\alpha = \frac{r}{x} \in R_{\fp }$. Now, let $(a_1,\ldots,a_n)$ be an $R$-basis of $A$. Then we can write every element $f \in \sF$ as $f = \sum_{i=1}^n \alpha_{f,i} a_i$ with $\alpha_{f,i} \in K$. Let $$I \dopgleich \prod_{{f \in \sF, \; i =1,\ldots,n}} I_{\alpha_{f,i}} \unlhd R \;.$$ By the properties of the ideals $I_\alpha$ we have the following logical equivalences: $$\begin{array}{rcl}
(\sF \subs A_{\fp }) & \Longleftrightarrow & (\alpha_{f,i} \in R_{\fp } \quad \forall f \in \sF, \ i =1,\ldots,n) \\ &\Longleftrightarrow& (I_{\alpha_{f,i}} \not\subs \fp \quad \forall f \in \sF, \ i=1,\ldots,n) \\ & \Longleftrightarrow & (I \not\subs \fp ) \;,
\end{array}$$ the last equivalence following from the fact that $\fp $ is prime. Hence, $$\msf{Ex}_A(\sF) = \Spec(R) \setminus \msf{Gen}_A(\sF) = \msf{V}(I) = \bigcup_{f \in \mscr{F}, \; i=1,\ldots,n} \msf{V}(I_{\alpha_{f,i}}) \;,$$ implying that $\msf{Gen}_A(\sF)$ is an open subset of $\Spec(R)$.
Next, still assuming that $A$ is $R$-free, suppose that $R$ is a Krull domain. To show that $\msf{Ex}_A(\mscr{F})$ is either empty or pure of codimension $1$ in $\Spec(R)$ with finitely many irreducible components, it suffices to show this for the closed subsets $\msf{V}(I_\alpha)$. If $\alpha \in R$, then $I_\alpha = R$ and therefore $\msf{V}(I_\alpha) = \emptyset$. So, let $\alpha \notin R$. Let $\msf{V}(I_\alpha) = \bigcup_{\lambda \in \Lambda} \msf{V}(\fq_\lambda)$ be the decomposition into irreducible components. Note that this decomposition is unique and contains every irreducible component of $\msf{V}(I_\alpha)$ since $\msf{V}(I_\alpha)$ is a sober topological space. The inclusion $\msf{V}(I_\alpha) \sups \msf{V}(\fq_\lambda)$ is equivalent to $I_\alpha = \sqrt{I_\alpha} \subs \sqrt{\fq_\lambda} = \fq_\lambda$. Since an irreducible component is a maximal proper closed subset, we see that the $\fq_\lambda$ are the minimal prime ideals of $\Spec(R)$ containing $I_\alpha$. Let $\fq = \fq_\lambda$ for an arbitrary $\lambda \in \Lambda$. We will show that $\msf{ht}(\fq) = 1$. Since $I_\alpha \subs \fq$, we have seen above that $\alpha \notin R_{\fq}$. As $R$ is a Krull domain, also $R_{\fq}$ is a Krull domain by [@Mat-Commutative Theorem 12.1]. By [@Bou-Commutative-Algebra-1-7 VII, §1.6, Theorem 4] we have $$R_{\fq} = \bigcap_{\substack{ \fq' \in \Spec(R_\fq) \\ \msf{ht}(\fq') = 1}} (R_\fq)_{\fq'} = \bigcap_{\substack{ \fq' \in \Spec(R) \\ \fq' \subs \fq \\ \msf{ht}(\fq') = 1}} R_{\fq'} \;.$$ Since $\alpha \notin R_{\fq}$, this shows that there exists $\fq' \in \Spec(R)$ with $\fq' \subs \fq$, $\msf{ht}(\fq') = 1$ and $\alpha \notin R_{\fq'}$. The last property implies $I_\alpha \subs \fq'$ and now the minimality in the choice of $\fq$ implies that $\fq' = \fq$. Hence, $\msf{ht}(\fq) = 1$ and this shows $\msf{V}(I_\alpha)$ is pure of codimension $1$. Since $I_\alpha \neq 0$, there is some $0 \neq r \in I_\alpha$. This element is contained in all the height one prime ideals $\fq_\lambda$. As $R$ is a Krull domain, a non-zero element of $R$ can only be contained in finitely many height one prime ideals (see [@Huneke-Swanson-Integral-Closure 4.10.1]), so $\Lambda$ must be finite.
Now, assume that $R$ is an arbitrary integral domain and that $A$ is finite flat. Then Grothendieck’s generic freeness lemma [@Grothendieck:EGA-4-2 Lemme 6.9.2] shows that there exists a non-zero $f \in R$ such that $A_f$ is a free $R_f$-module. Note that $\Spec(R_f)$ can be identified with the distinguished open subset $\msf{D}(f)$ of $\Spec(R)$. We obviously have $$\msf{Gen}_{A_f}(\mscr{F}) = \msf{Gen}_A(\mscr{F}) \cap \msf{D}(f) \;.$$ By the arguments above, $\msf{Gen}_{A_f}(\mscr{F})$ is an open subset of $\msf{D}(f)$, and thus of $\Spec(R)$. This shows that $\msf{Gen}_A(\mscr{F})$ is a neighborhood in $\Spec(R)$.
Next, let $R$ be arbitrary and assume that $A$ is finitely presented flat. It is a standard fact (see [@stacks-project Tag 00NX]) that the assumptions on $A$ imply that $A$ is already finite locally free, i.e., there exist a family $(f_i)_{i \in I}$ of elements of $R$ such that the standard open affines $\msf{D}(f_i)$ cover $\Spec(R)$ and $A_{f_i}$ is a finitely generated free $R_{f_i}$-module for all $i \in I$. Since $\Spec(R)$ is quasi-compact, see [@GorWed10-Algebraic-geomet Proposition 2.5], we can assume that $I$ is finite. Again note that $\Spec(R_{f_i})$ can be identified with $\msf{D}(f_i)$ and that $$\label{pure_codim_proof_1}
\msf{Gen}_{A_{f_i}}(\mscr{F}) = \msf{Gen}_A(\mscr{F}) \cap \msf{D}(f_i) \;.$$ By the above, the set $\msf{Gen}_{A_{f_i}}(\mscr{F})$ is open and since the $\msf{D}(f_i)$ cover $\Spec(R)$, it follows that $\msf{Gen}_A(\mscr{F})$ is open. Now, suppose that $R$ is a Krull domain. Similarly as in (\[pure\_codim\_proof\_1\]) we have $$\msf{Ex}_{A_{f_i}}(\mscr{F}) = \msf{Ex}_A(\mscr{F}) \cap \msf{D}(f_i) \;.$$ Suppose that $\msf{Ex}_A(\mscr{F})$ is not empty and let $Z$ be an irreducible component of $\msf{Ex}_A(\mscr{F})$. There is an $i \in I$ with $Z \cap \msf{D}(f_i) \neq \emptyset$. The map $T \mapsto \ol{T}$ defines a bijection between irreducible closed subsets of $\msf{D}(f_i)$ and irreducible closed subsets of $\Spec(R)$ which meet $\msf{D}(f_i)$, see [@GorWed10-Algebraic-geomet §1.5]. This implies that $Z \cap \msf{D}(f_i)$ is an irreducible component of $\msf{Ex}_A(\mscr{F}) \cap \msf{D}(f_i) = \msf{Ex}_{A_{f_i}}(\mscr{F})$. It follows from the above that $Z \cap \msf{D}(f_i)$ is of codimension $1$ in $\msf{D}(f_i)$. Hence, $Z$ is of codimension $1$ in $\Spec(R)$ by [@stacks-project Tag 02I4]. All irreducible components of $\msf{Ex}_A(\mscr{F})$ are thus of codimension $1$ in $\Spec(R)$. Since each set $\msf{Ex}_{A_{f_i}}(\mscr{F})$ has only finitely many irreducible components and since $I$ is finite, also $\msf{Ex}_A(\mscr{F})$ has only finitely many irreducible components.
For $\fp \in \Spec(R)$ let us denote by $\mscr{B}_A(\fp) \subs A^K$ the set of block idempotents of $A_\fp$. Clearly, $\mscr{B}_A(\fp)$ and $\gamma_A(\fp)$ are in bijection by taking sums of the $c_i$ over the subsets in $\gamma_A(\fp)$. Note that $\mscr{B}_A(\fp)$ is constant on $\Gamma_A(\mscr{P})$ for any $\mscr{P}$. We can thus define $\msf{Gen}_A(\mscr{P}) \dopgleich \msf{Gen}_A(\mscr{B}_A(\fp))$ where $\fp \in \Gamma_A(\mscr{P})$ is arbitrary. We are now ready to prove our first main result.
\[block\_stratification\_thm\] Suppose that $A$ is [finitely presented]{.nodecor} as an $R$-module. Then the map $\gamma_A:\Spec(R) \rarr \Gamma_A$ is [lower semicontinuous]{.nodecor}, i.e., each $\Gamma_A^\leq(\mscr{P})$ is [closed]{.nodecor} in $\Spec(R)$. In particular, $\Gamma_A(\mscr{P})$ is [open]{.nodecor} in $\Gamma_A^\leq(\mscr{P})$, thus [locally closed]{.nodecor} in $\Spec(R)$.
Since $\Spec(R) = \coprod_{\mscr{P} \in \Gamma_A} \Gamma_A(\mscr{P})$, we have $\Spec(R) \setminus \Gamma_A^\leq(\mscr{P}) = \bigcup_{\mscr{P}' \not\leq \mscr{P}} \Gamma_A(\mscr{P}')$. Let $\mscr{P}' \not\leq \mscr{P}$ and $\fp' \in \msf{Gen}_A(\mscr{P}')$. Then $\mscr{P}' \leq \gamma_A(\fp')$. But this implies that $\gamma_A(\fp') \not\leq \mscr{P}$ since otherwise $\mscr{P}' \leq \gamma_A(\fp') \leq \mscr{P}$. Hence, $\msf{Gen}_A(\mscr{P}') \subs \Spec(R) \setminus \Gamma_A^\leq(\mscr{P})$. Conversely, we clearly have $\Gamma_A(\mscr{P}') \subs \msf{Gen}_A(\mscr{P}')$. This shows that $$\Spec(R) \setminus \Gamma_A^\leq(\mscr{P}) = \bigcup_{\mscr{P}' \not\leq \mscr{P}} \Gamma_A(\mscr{P}') = \bigcup_{\mscr{P}' \not\leq \mscr{P}} \msf{Gen}_A(\mscr{P}')$$ is open, so $$\Gamma_A^\leq(\mscr{P}) = \bigcap_{\mscr{P}' \not\leq \mscr{P}} \msf{Ex}_A(\mscr{P}')$$ is closed. Using (\[stratum\_from\_skeleton\]) we now see that $\Gamma_A(\mscr{P})$ is locally closed.
Theorem \[block\_stratification\_thm\] implies in particular that $\msf{BlGen}^{\mrm{loc}}(A)$ is a *dense open* subset of $\Spec(R)$.
\[blex\_is\_weil\_div\] Suppose that $A$ is [finitely presented]{.nodecor} as an $R$-module and that $R$ is a [Krull domain]{.nodecor}. Then $\msf{BlEx}^{\mrm{loc}}(A)$ is a reduced Weil divisor.
This follows directly from Proposition \[br\_gen\_lemma\] since $\msf{BlEx}^{\mrm{loc}}(A) = \msf{Ex}_A(\mscr{B}_A(\bullet))$.
\[fp\_projective\_remark\] We note that $A$ is finitely presented flat if and only if it is finite projective, see [@Lam-Lectures-Modules-Rings-99 Theorem 4.30] or [@stacks-project Tag 058R]. Hence, we could have equally assumed that $A$ is finite projective in Theorem \[block\_stratification\_thm\] but we preferred the seemingly more general notion.
We assume for the rest of this paragraph that $R$ is *noetherian* (which implies that $A$ is finitely presented as an $R$-module).
\[stratum\_closure\] For any $\mscr{P} \in \Gamma_A$ we have $\ol{\Gamma_A(\mscr{P})} \subs \Gamma_A^\leq(\mscr{P})$. In particular, the partition (\[block\_stratification\]) is a [stratification]{.nodecor} of the scheme $\Spec(R)$.
Since $\Spec(R)$ is noetherian, the locally closed set $\Gamma_A(\mscr{P})$ has only finitely many irreducible components $Z_1,\ldots,Z_n$. We then have $\ol{\Gamma_A(\mscr{P})} = \bigcup_{i=1}^n \ol{Z_i}$. If $\xi_i$ denotes the generic point of $Z_i$ (note that any irreducible locally closed set has a unique generic point), then $\xi_i$ is also the generic point of $\ol{Z_i}$. Since $\xi_i \in Z_i \subs \Gamma_A(\mscr{P})$, we have $\gamma_A(\xi_i) = \mscr{P}$, so $\xi_i \in \Gamma_A^\leq(\mscr{P})$. We thus obtain $\ol{\Gamma_A(\mscr{P})} = \bigcup_{i=1}^n \msf{V}(\xi_i) \subs \Gamma_A^\leq(\mscr{P})$.
In general it is not true that we have equality $\ol{\Gamma_A(\mscr{P})} = \Gamma_A^\leq(\mscr{P})$, so the stratification (\[block\_stratification\]) is in general not a so-called *good* stratification. For example, in Figure \[brauer\_graph\_example\] we have $\mscr{P}' \dopgleich \gamma_A( (3) ) = \lbrace \lbrace 1,2,3 \rbrace, \lbrace 4 \rbrace \rbrace < \lbrace \lbrace 1,2 \rbrace, \lbrace 3 \rbrace, \lbrace 4 \rbrace \rbrace = \gamma_A((2)) \gleichdop \mscr{P}$, so $(3) \in \Gamma_A^\leq(\mscr{P})$, but $(3)$ is not contained in $\ol{\Gamma_A(\mscr{P})} = \msf{V}((2))$. The problem here is that the skeleton $\Gamma_A^\leq(\mscr{P})$ has an irreducible component on which the maximal local block structure is strictly smaller than the maximal one on the entire skeleton. To overcome this defect, we construct a refinement of the stratification which gives a much better overview of how the local block structures are formed.\
For any two-element subset $\lbrace i,j \rbrace \subs I$ we define the corresponding as $$\Gamma_A^\leq(\lbrace i,j \rbrace) \dopgleich \lbrace \fp \in \Spec(R) \mid c_i \tn{ and } c_j \tn{ lie in the same block of } A_\fp \rbrace \;.$$ It is not hard to see that $$\Gamma_A^\leq(\lbrace i,j \rbrace) = \msf{Ex}_A(c_i) \cap \msf{Ex}_A(c_j) \cap \bigcap_{ \substack{I' \subs I \\ i,j \notin I'}} \msf{Ex}_A( c_i + \sum_{k \in I'} c_k ) \cap \msf{Ex}_A( c_j + \sum_{k \in I'} c_k ) \;,$$ so $\Gamma_A^\leq(\lbrace i,j \rbrace)$ is *closed* in $\Spec(R)$ by Proposition \[br\_gen\_lemma\]. We denote by $\Xi_A^{(1)}$ the set of irreducible components of gluing loci and we call these the . On each $Z \in \Xi_A^{(1)}$ there is clearly a unique maximal local block structure $\gamma_A(Z)$, namely the one in the generic point of $Z$. We denote by $\msf{At}(\Gamma_A)$ the set of these block structures and call them . The important point is now that for any $\fp \in \Spec(R)$ we can determine $\gamma_A(\fp)$ simply by determining the atomic gluing loci containing $\fp$. More precisely, we have $\gamma_A(\fp) = I/\!\!\sim_\fp$, where $\sim_\fp$ is the equivalence relation on $I$ generated by $i \sim_\fp j$ if and only if $\fp \in \Gamma_A^\leq( \lbrace i,j \rbrace)$. From this it is clear that $$\label{block_structure_as_meet}
\gamma_A(\fp) = \bigwedge_{ \substack{Z \in \Xi_A^{(1)} \\ \fp \in Z } } \gamma_A(Z) \;,$$ where $\wedge$ denotes the *meet* in the lattice $\msf{Part}(I)$, i.e., $\mscr{P} \wedge \mscr{P}'$ for $\mscr{P},\mscr{P}' \in \msf{Part}(I)$ is the finest partition of $I$ being coarser than both $\mscr{P}$ and $\mscr{P}'$, and this is obtained by joining members with non-empty intersection. Hence, any local block structure of $A$ is a meet of atomic local block structures (whence, the prefix “atomic”). But we note that not all meets must actually occur as block structures since atomic gluing loci might have empty intersection. The point is that once we know the atomic gluing loci and their maximal local block structures, we essentially know the complete local block graph by analyzing intersections of atomic gluing loci. To this end, we inductively define the sets $\Xi_A^{(r)}$ for $r \in \bbN$ as being the set of irreducible components of intersections of any two elements of $\Xi_A^{(r-1)}$, with $\Xi_A^{(1)}$ being defined above already as the set of atomic gluing loci. This yields an increasing sequence $\Xi_A^{(1)} \subs \Xi_A^{(2)} \subs \ldots$ of irreducible closed subsets of $\Spec(R)$ which will eventually become stationary since $\Spec(R)$ is noetherian. We add $\Spec(R)$ to this maximal set and denote the resulting set by $\Xi_A$. We consider it as a poset ordered by inclusion. Let $\msf{At}(\Xi_A) \dopgleich \Xi_A^{(1)}$ and for $Z \in \Xi_A$ we denote by $\msf{At}(Z)$ the set of all $T \in \msf{At}(\Xi_A)$ containing $Z$. For any $Z \in \Xi_A$ we define $$\Xi_A(Z) \dopgleich Z \setminus \bigcup_{Z' < Z} Z' \;.$$ Note that $\Xi_A(\Spec(R)) = \msf{BlGen}(A)$ and that $\msf{BlEx}(A) = \bigcup_{T \in \msf{At}(\Xi_A)} T = \bigcup_{Z \in \Xi_A} Z$.
\[fine\_block\_stratification\] We have $\Spec(R) = \coprod_{Z \in \Xi_A} \Xi_A(Z)$, and this is a [good]{.nodecor} stratification of the scheme $\Spec(R)$. Moreover, for any $Z \in \Xi_A$ we have $$\label{skeleton_gen_point_meet}
\gamma_A(Z) = \bigwedge_{T \in \msf{At}(Z)} \gamma_A(T) \;.$$ and $$\Xi_A(Z) \subs \Gamma_A(\gamma_A(Z)) \;.$$ Hence, for any $\mscr{P} \in \Gamma_A$ we have $$\Gamma_A(\mscr{P}) = \coprod_{ \substack{Z \in \Xi_A \\ \gamma_A(Z) = \mscr{P}} } \Xi_A(Z) \;,$$ so the stratification of $\Spec(R)$ by the $\Xi_A(Z)$ is a refinement of the stratification (\[block\_stratification\]).
Suppose that $\Xi_A(Z) \cap \Xi_A(Z') \neq \emptyset$ with $Z \neq Z'$. Then there is $\fp \in Z \cap Z'$. Let $Z''$ be an irreducible component of $Z \cap Z'$ containing $\fp$. If $Z'' = Z$, then $Z \cap Z' = Z$, so $Z < Z'$. But this contradicts $\fp \in \Xi_A(Z') \subs Z' \setminus Z$. Hence, $Z'' < Z$. But since $Z'' \in \Xi_A$ by construction of $\Xi_A$ and $\fp \in \Xi_A(Z) \subs Z \setminus Z''$, this is again a contradiction. Hence, we have a partition $\Spec(R) = \coprod_{Z \in \Xi_A} \Xi_A(Z)$. It is clear that each $\Xi_A(Z)$ is locally closed in $\Spec(R)$. By definition we have $\Xi_A(Z) = Z \setminus \bigcup_{Z' < Z} Z'$, thus $Z = \Xi_A(Z) \cup \bigcup_{Z' < Z} Z'$. From this we obtain inductively that $Z = \bigcup_{Z' \leq Z} \Xi_A(Z')$. Since the closure of $\Xi_A(Z)$ is obviously equal to $Z$ (recall that $Z$ is irreducible), it follows that $\Spec(R) = \coprod_{Z \in \Xi_A} \Xi_A(Z)$ is a good stratification. That the local block structure in the generic point of $Z \in \Xi_A(Z)$ is given by (\[skeleton\_gen\_point\_meet\]) follows directly from (\[block\_structure\_as\_meet\]) and $\Xi_A(Z) \subs \Gamma_A(\gamma_Z(A))$ is now clear.
\[maximal\_atomic\_codim\_1\] Suppose that $R$ is normal. Then the maximal atomic gluing loci are the irreducible components of the local block divisor $\msf{BlEx}^\mrm{loc}(A)$ and are thus of codimension one in $\Spec(R)$.
We have $\msf{BlEx}^{\mrm{loc}}(A) = \bigcup_{T \in \msf{At}(\Xi_A)} T = \bigcup_{T \in \Max(\Xi_A)} T$. It is now clear that the $T \in \Max(\Xi_A)$ are precisely the irreducible components of $\msf{BlEx}^{\mrm{loc}}(A)$. The claim thus follows from Corollary \[blex\_is\_weil\_div\].
Again we can consider $\Xi_A$ as a directed graph and attach the corresponding maximal local block structure $\gamma_A(Z)$ to each vertex $Z$. We call this the of $A$. It not only gives us complete information about local block structures of $A$ but at once also an overview of how these block structures are formed. In Figure \[brauer\_graph\_example\_inside\] we repeat the example from the introduction.
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The nice fact about the gluing loci and their bock structures is that we can describe them rather explicitly, see Theorem \[maintheorem\_blex\_descr\].\
Instead of considering a refinement of the stratification (\[block\_stratification\]) we will now construct an interesting coarsening. For the moment we can drop the assumption about $R$ being noetherian and just assume that $A$ is finitely presented flat as an $R$-module. For $n \in \bbN$ we define $$\label{bl_leqn_loc}
\msf{Bl}^{\tn{loc}}_{\leq n}(A) \dopgleich \lbrace \fp \in \Spec(R) \mid \#\msf{Bl}(A_\fp) \leq n \rbrace \;.$$ It is clear that $$\label{bl_leqn_closed}
\msf{Bl}^{\tn{loc}}_{\leq n}(A) = \bigcup_{ \substack{\mscr{P} \in \Gamma_A \\ \#\mscr{P}\leq n }} \Gamma_A(\mscr{P}) \;,$$ so $\msf{Bl}^{\tn{loc}}_{\leq n}(A)$ is closed in $\Spec(R)$ by Theorem \[block\_stratification\_thm\]. This means nothing else than the map $\Spec(R) \rarr \bbN$, $\fp \mapsto \#\msf{Bl}(A_\fp)$, being [lower semicontinuous]{.nodecor}. Consequently, $$\label{bl_n_loc}
\msf{Bl}_n^{\tn{loc}}(A) \dopgleich \msf{Bl}_{\leq n}^{\tn{loc}}(A) \setminus \msf{Bl}_{\leq n-1}^\tn{loc}(A) = \lbrace \fp \in \Spec(R) \mid \#\msf{Bl}(A_\fp) = n \rbrace$$ is locally closed in $\Spec(R)$ and we have a partition $$\label{block_number_partition}
\msf{Bl}^{\tn{loc}}_{\leq n}(A) = \coprod_{n \in \bbN} \msf{Bl}_n^{\tn{loc}}(A) \;.$$ Note that $$\msf{BlEx}^{\mrm{loc}}(A) = \lbrace \fp \in \Spec(R) \mid \#\gamma_A(\fp) < \#\gamma_A(\bullet) \rbrace = \msf{Bl}_{\leq \gamma_A(\bullet)-1}^{\mrm{loc}}(A) \;.$$ If $R$ is noetherian, it follows from Lemma \[stratum\_closure\] that $$\label{bln_closure}
\ol{\msf{Bl}^{\tn{loc}}_{ n}(A)} \subs \bigcup_{m<n} \msf{Bl}^{\tn{loc}}_{m}(A) \;,$$ so the partition (\[block\_number\_partition\]) is in fact a stratification of $\Spec(R)$. Again, in general it will not be a good stratification, however.
Blocks of specializations
-------------------------
We finally turn to our actual problem, namely blocks of specializations of $A$. Compared to blocks of localizations there is in general no possibility to compare the actual block structures of specializations. We can, however, compare numerical invariants in general and thus define for arbitrary $A$: $$\msf{Bl}_{\leq n}(A) \dopgleich \lbrace \fp \in \Spec(R) \mid \#\msf{Bl}(A(\fp)) \leq n \rbrace \;,$$ $$\msf{Bl}_{n}(A) \dopgleich \msf{Bl}_{\leq n}(A) \setminus \msf{Bl}_{\leq n-1}(A) = \lbrace \fp \in \Spec(R) \mid \#\msf{Bl}(A(\fp)) = n \rbrace \;,$$ $$\beta(A) \dopgleich \msf{max}\lbrace \#\msf{Bl}(A(\fp)) \mid \fp \in \Spec(R) \rbrace \;,$$ $$\msf{BlEx}(A) \dopgleich \msf{Bl}_{\leq \beta(A)-1}(A) \;,$$ $$\msf{BlGen}(A) \dopgleich \Spec(R) \setminus \msf{BlEx}(A) \;.$$ In general, these invariants will be distinct from the corresponding ones for blocks of localizations, see Example \[ken\_example\]. This is why we attached the superscript “loc” to these invariants in the preceding paragraph. There is, however, a rather general setting where blocks of specializations are naturally identified with blocks of localizations, namely when $R$ is normal and $A^K$ splits. In this case not only the above sets are equal to their local versions but we can also compare the actual block structures of specializations and all results from the preceding paragraph are actually also results about blocks of specializations (we can thus remove the superscript “loc” and the prefix “local” everywhere under these assumptions). The key ingredient to establish this natural correspondence is the next proposition. To formulate it more generally, we use the property *block-split* introduced in Definition \[block\_split\_def\] but note that the reader might just simply replace it by the more special property *split*. Moreover, we recall that a local integral domain $R$ is called if its henselization $R^h$ is again an *integral* (local) domain. This is equivalent to the normalization of $R$ being again *local* (see [@Raynaud:Henselian IX, Corollaire 1]). This clearly holds if $R$ is already *normal*. Examples of non-normal unibranch rings are the local rings in ordinary cusp singularities of curves.
\[unibranched\_block\_bijective\] Let $R$ be an integral domain and let $A$ be a finite flat $R$-algebra with [block-split]{.nodecor} generic fiber $A^K$ (e.g., if $A^K$ splits). Let $\fp \in \Spec(R)$ and suppose that $R_\fp$ is unibranch (e.g., if $R_\fp$ is normal). Then the quotient morphism $A_\fp \twoheadrightarrow A(\fp)$ is block bijective.
By assumption, $R_\fp$ and its henselization $R_\fp^h$ are integral domains. Since $A$ is $R$-flat, it follows that $A_\fp = R_\fp \otimes_R A$ is $R_\fp$-flat and that $A_\fp^h \dopgleich R_\fp^h \otimes_{R_\fp} A_\fp$ is $R_\fp^h$-flat. Hence, both $A_\fp$ and $A_\fp^h$ have block decompositions by Lemma \[finite\_flat\_int\_block\_dec\]. Let $\fp_\fp^h$ be the maximal ideal of $R_\fp^h$. The henselization morphism $R_\fp \rarr R_\fp^h$ is local and faithfully flat by [@Gro67-Elements-de-geom Théorème 18.6.6(iii)]. We now have a commutative diagram $$\begin{tikzcd}
A_\fp \arrow[hookrightarrow]{r} \arrow[twoheadrightarrow,swap]{d} & A_\fp^h \arrow[twoheadrightarrow]{d} \\
A(\fp) = A_\fp/\fp_\fp A_\fp \arrow[hookrightarrow]{r} & A_\fp^h/\fp_\fp^h A_\fp^h
\end{tikzcd}$$ of idempotent stable morphisms. We know from Lemma \[semiperfect\_std\_settings\]\[semiperfect\_std\_settings:hensel\] and Lemma \[idempotent\_surjective\_block\_bijective\] that $A_\fp^h \rarr A_\fp^h/\fp_\fp^h A_\fp^h$ is block bijective. Since $A$ has block-split generic fiber and $R_\fp \rarr R_\fp^h$ is a faithfully flat morphism of integral domains, we can use Theorem \[faithfully\_flat\_ext\_block\_bij\] to deduce that $A_\fp \rarr A_\fp^h$ is block bijective. In [@Gro67-Elements-de-geom Théorème 18.6.6(iii)] it is proven that $R_\fp/\fp_\fp \simeq R_\fp^h/\fp_\fp^h $. Hence, the map $A_\fp/\fp_\fp A_\fp \rarr A_\fp^h/\fp_\fp^h A_\fp^h$ is an isomorphism and so in particular block bijective. We thus have $$\#\msf{Bl}(A_\fp^h) = \# \msf{Bl}(A_\fp) \leq \# \msf{Bl}(A(\fp)) = \#\msf{Bl}(A_\fp^h/\fp_\fp^h A_\fp^h) = \# \msf{Bl}(A_\fp^h)$$ by equation (\[get\_more\_blocks\_equation\]). Hence, $\# \msf{Bl}(A_\fp) = \# \msf{Bl}(A(\fp))$, so $A_\fp \twoheadrightarrow A(\fp)$ is block bijective.
\[normal\_split\_corollary\] Suppose that $R$ is normal and $A^K$ splits. Then $A_\fp \twoheadrightarrow A(\fp)$ is block bijective for all $\fp \in \Spec(R)$. Hence, all results from §\[blocks\_of\_localizations\] can be used to study blocks of specializations of $A$.
\[normal\_split\_corollary\_details\] Assume that $R$ is normal and that $A^K$ splits. Then, the map $\Spec(R) \rarr \bbN$, $\fp \mapsto \#\msf{Bl}(A(\fp))$, is lower semicontinuous and $\Spec(R) = \coprod_{n \in \bbN} \msf{Bl}_{n}(A)$ is a partition into locally closed subsets. Moreover, $\beta(A) = \#\msf{Bl}(A^K)$ and $\msf{BlEx}(A)$ is a reduced Weil divisor in $\Spec(R)$. If $R$ is also noetherian, then $\Spec(R) = \coprod_{n \in \bbN} \msf{Bl}_{n}(A)$ is a stratification.
Even though this setting is restrictive, we still include a lot of important examples in representation theory like Brauer algebras, Hecke algebras, restricted rational Cherednik algebras, etc. But clearly there are also many interesting examples which are not included, like restricted quantized enveloping algebras, mainly because they do not have a split generic fiber. However, even in these cases our results can be applied when we restrict to a certain subset of $\Spec(R)$. We discuss a general strategy.
Assume that $R'$ is an integral extension of $R$ which is also an integral domain. Let $K'$ be the fraction field of $R'$ and let $\psi:\Spec(R') \twoheadrightarrow \Spec(R)$ be the morphism induced by $R \subs R'$. The scalar extension $A' \dopgleich R' \otimes_{R} A$ is again a finitely presented flat $R'$-algebra (using Remark \[fp\_projective\_remark\]). For any $\fp \in \Spec(R)$ and any $\fp' \in \Spec(R')$ lying over $\fp$ we have a diagram
$$\label{prop_P1_morphisms}
\begin{tikzcd}
& A_{\fp'}' \arrow[twoheadrightarrow]{d} \\
A(\fp) = A_\fp/\fp_\fp A_\fp \arrow[hookrightarrow]{r} & A'(\fp') = A'_{\fp'}/{\fp'}_{\fp'} A'_{\fp'}
\end{tikzcd}$$
and it then follows from (\[get\_more\_blocks\_equation\]) that $$\label{main_thm_proof_block_lesseq}
\#\msf{Bl}(A(\fp)) \leq \#\msf{Bl}(A'(\fp')) \geq \#\msf{Bl}(A'_{\fp'}) \;.$$ Let $X$ be a set contained in $$X_{R'}(A) \dopgleich \lbrace \fp \in \Spec(R) \mid \#\msf{Bl}(A(\fp)) = \#\msf{Bl}(A'(\fp')) = \#\msf{Bl}(A'_{\fp'}) \tn{ for all } \fp' \in \psi^{-1}(\fp) \rbrace \;.$$ We have seen in Corollary \[normal\_split\_corollary\] that in case $R$ is normal and $A^K$ splits we can choose $R=R'$ and have $X = \Spec(R)$. In general $X$ will be a proper subset of $\Spec(R)$ and we have to choose $R'$ appropriately to enlarge it a bit more. Let us first concentrate on what we can say when restricting to $X$. We introduce the following restricted versions of our invariants: $$\label{bl_leqn_blowdown}
\msf{Bl}_{\leq n}^X(A) \dopgleich \msf{Bl}_{\leq n}(A) \cap X = \psi(\msf{Bl}_{\leq n}^{\mrm{loc}}(A')) \cap X \;,$$ $$\msf{Bl}_n^X(A) \dopgleich \msf{Bl}_{n}(A) \cap X = \psi(\msf{Bl}_{n}^{\mrm{loc}}(A')) \cap X \;,$$ $$\beta^X(A) \dopgleich \msf{max}\lbrace \#\msf{Bl}(A(\fp)) \mid \fp \in X \rbrace \;,$$ $$\msf{BlEx}^X(A) \dopgleich \msf{Bl}^X_{\leq \beta^X(A)-1} \;,$$ $$\msf{BlGen}^X(A) \dopgleich X \setminus \msf{BlEx}^X(A) \;.$$
\[bln\_spec\_strat\] The map $X \rarr \bbN$, $\fp \mapsto \#\msf{Bl}(A(\fp))$, is lower semicontinuous on $X$ and $X = \coprod_{n \in \bbN} \msf{Bl}_n(A)$ is a partition into locally closed subsets. Moreover, $$\label{beta_invariant_estimate}
\beta^X(A) \leq \#\msf{Bl}(A^{K'}) \;.$$ If $R$ is noetherian, then $X = \coprod_{n \in \bbN} \msf{Bl}^X_n(A)$ is a [stratification]{.nodecor} of $X$.
Since $\psi$ is a closed morphism and $\msf{Bl}_{\leq n}^{\mrm{loc}}(A')$ is closed in $\Spec(R')$ by (\[bl\_leqn\_closed\]), it follows that $\psi(\msf{Bl}_{\leq n}^{\mrm{loc}}(A'))$ is closed in $\Spec(R)$, hence $\msf{Bl}^X_{\leq n}(A)$ is closed in $X$ by (\[bl\_leqn\_blowdown\]). Since $\msf{Bl}^X_{n}(A) = \msf{Bl}^X_{\leq n}(A) \setminus \msf{Bl}^X_{\leq n-1}(A)$, it is clear that $\msf{Bl}^X_n(A)$ is locally closed in $X$. We have shown in (\[bln\_closure\]) that $\ol{\msf{Bl}_n^{\mrm{loc}}(A')} = \bigcup_{m \leq n} \msf{Bl}_m^{\mrm{loc}}(A')$. Hence, since $\psi$ is closed, we obtain $$\ol{\msf{Bl}^X_n(A)} = \psi(\ol{\msf{Bl}_n^{\mrm{loc}}(A')}) \cap X = \bigcup_{m \leq n} \psi(\msf{Bl}_m^{\mrm{loc}}(A')) \cap X = \bigcup_{m \leq n} \msf{Bl}^X_m(A)$$
Note that in (\[beta\_invariant\_estimate\]) we could only bound $\beta^X(A)$ above by $\msf{Bl}(A^{K'})$, and not by $\msf{Bl}(A^K)$. In fact, we will see in Example \[ken\_example\] that we may indeed have $\beta^X(A) > \#\msf{Bl}(A^K)$ in general. This is an important difference to blocks of localizations where we always have the maximal number of blocks in the generic point. In the following lemma we describe a situation where we have $\beta^X(A) = \#\msf{Bl}(A^{K'})$. We recall that $X$ being *very dense* means that the embedding $X \hookrightarrow \Spec(R)$ is a *quasi-homeomorphism*, i.e., the map $Z \mapsto Z \cap X$ is a bijection between the closed (equivalently, open) subsets of the two spaces. This notion was introduced by Grothendieck [@Grothendieck:EGA-4-3 §10].
\[x\_very\_dense\_beta\] Suppose that $X$ is [very dense]{.nodecor} in $\Spec(R)$, that $R$ is noetherian, and that $\psi$ is finite. Then $\beta^X(A) = \#\msf{Bl}(A^{K'})$, thus $\msf{BlEx}^X(A) = \psi(\msf{BlEx}^{\mrm{loc}}(A')) \cap X$. If moreover $R'$ is normal and $R$ is universally catenary, then $\msf{BlEx}^X(A)$ is a reduced Weil divisor in $X$.
The assumption imply that $R'$ is noetherian, too. We know from Theorem \[block\_stratification\_thm\] that $\msf{BlGen}^{\mrm{loc}}(A')$ is a non-empty open subset of $\Spec(R')$. In particular, it is constructible. Since $\Spec(R)$ is quasi-compact, also $\psi$ is quasi-compact by [@GorWed10-Algebraic-geomet Remark 10.2.(1)]. It thus follows from Chevalley’s constructibility theorem, see [@GorWed10-Algebraic-geomet Corollary 10.71], that $\psi(\msf{BlGen}^{\mrm{loc}}(A'))$ is constructible in $\Spec(R)$. Since $X$ is very dense in $\Spec(R)$, we conclude that $\psi(\msf{BlGen}^{\mrm{loc}}(A')) \cap X \neq \emptyset$ by [@Grothendieck:EGA-4-3 Proposition 10.1.2]. Hence, there is $\fp \in X$ and $\fp' \in \msf{BlGen}^{\mrm{loc}}(A')$ with $\psi(\fp') = \fp$. But then we have $\#\msf{Bl}(A(\fp)) = \#\msf{Bl}(A'(\fp')) = \#\msf{Bl}(A^{K'})$, so $\beta^X(A) = \#\msf{Bl}(A^{K'})$. Now, assume that $R'$ is normal and $R$ is universally catenary. We know that $\msf{BlEx}^{\mrm{loc}}(A')$ is either empty or pure of codimension one in $\Spec(R')$ by Corollary \[blex\_is\_weil\_div\]. In [@Huneke-Swanson-Integral-Closure Theorem B.5.1] it is shown that the extension $R \subs R'$ satisfies the dimension formula, hence $\psi(\msf{BlEx}^{\mrm{loc}}(A'))$ is either empty or pure of codimension one. Since $X$ is very dense in $\Spec(R)$, the same is also true for $X \cap \psi(\msf{BlEx}^{\mrm{loc}}(A')) =\msf{BlEx}^X(A)$.
\[finite\_type\_alg\_closed\_setting\] Suppose that $R$ is a finite type algebra over an algebraically closed field. Let $X$ be the set of closed points of $\Spec(R)$. Then the map $X \rarr \bbN$, $\fm \mapsto \#\msf{Bl}(A(\fm))$, is lower semicontinuous and $X = \coprod_{n \in \bbN} \msf{Bl}_n^X(A)$ is a stratification of $X$. Moreover, $\beta^X(A) = \#\msf{Bl}(A^{\ol{K}})$, where $\ol{K}$ is an algebraic closure of $K$. If $R$ is also universally catenary, then $\msf{BlEx}^X(A)$ is a reduced Weil divisor in $X$.
Let $K'$ be a finite extension of $K$ such that $A^{K'}$ splits (this is always possible, see [@CR-Methods-1 Proposition 7.13]) and let $R'$ be the integral closure of $R$ in $K'$. Now, $\#\msf{Bl}(A'(\fp')) = \#\msf{Bl}(A'_{\fp'})$ for all $\fp' \in \Spec(R)$ by Proposition \[unibranched\_block\_bijective\]. Since $R$ is a finite type algebra over an algebraically closed field $k$, the residue field in a closed point $\fm$ of $\Spec(R)$ is just $k$. Hence, the specialization $A(\fm)$ is a finite-dimensional algebra over an algebraically closed field, thus splits and we therefore have $\#\msf{Bl}(A(\fm)) = \#\msf{Bl}(A'(\fm'))$ for any $\fm' \in \psi^{-1}(\fm)$ by Lemma \[split\_center\_lemma\]. Hence, $X \subs X_{R'}(A)$. The claim about semicontinuity and the stratification thus follows from Corollary \[bln\_spec\_strat\]. It is shown in [@GorWed10-Algebraic-geomet Proposition 3.35] that $X$ is very dense in $\Spec(R)$. Since $R$ is a finite type algebra over a field, it is japanese, so $\psi$ is a finite morphism. Hence, $\beta^X(A) = \#\msf{Bl}(A^{K'}) = \#\msf{Bl}(A^{\ol{K}}$ by Lemma \[x\_very\_dense\_beta\]. Also the claim that $\msf{BlEx}^X(A)$ is a reduced Weil divisor if $R$ is universally catenary follows from Lemma \[x\_very\_dense\_beta\].
\[ken\_example\] The following example due to K. Brown shows that in the setting of Lemma \[finite\_type\_alg\_closed\_setting\] we may indeed have $\beta^X(A) > \#\msf{Bl}(A^K)$ so that the map $\fp \mapsto \#\msf{Bl}(A(\fp))$ will not be lower semicontinuous on the whole of $\Spec(R)$. Let $k$ be an algebraically closed field of characteristic zero, let $X$ be an indeterminate over $k$, let $R \dopgleich k \lbrack X^n \rbrack$ for some $n > 1$, and let $A \dopgleich k \lbrack X \rbrack$. Let $C_n$ be the cyclic group of order $n$. We fix a generator of $C_n$ and let it act on $X$ by multiplication with a primitive $n$-th root of unity. Then $R = k \lbrack X \rbrack^{C_n}$, so $A$ is free of rank $n$ over $R$. Moreover, $\Frac(A) = k(X)$ is a Galois extension of degree $n$ of $K \dopgleich \Frac(R)$ by [@Ben-Polynomial-invariants Proposition 1.1.1], so in particular $K \neq k(X)$ since $n>1$. By [@Goodearl-Warfield Ex. 6R] we have $$A^K = A \otimes_R K = A\lbrack (R \setminus \lbrace 0 \rbrace)^{-1}\rbrack = \Frac(A) = k(X) \;,$$ so the $K$-algebra $A^K = \msf{Z}(A^K)$ is *not* split (and thus also not block-split by Lemma \[split\_center\_lemma\]). It is clear that $$\#\msf{Bl}(A^K) = 1 \;.$$ Now, let $\fm \dopgleich (X^n-1) \in \msf{Max}(R)$. Then $\msf{k}(\fp) = k$ and since $k$ is algebraically closed, we have $A(\fm) = A/\fm A \simeq k^n$ as $k$-algebras. In particular, $$\#\msf{Bl}(A(\fm)) = n > 1 = \#\msf{Bl}(A^K) \;.$$
Finally, we want to provide a setting where our base ring is not necessarily normal but we still get a global result on $\Spec(R)$.
\[split\_fibers\_results\] Suppose that $A$ has split fibers, i.e., $A(\fp)$ splits for all $\fp \in \Spec(R)$. Then the map $\Spec(R) \rarr \bbN$, $\fp \mapsto \#\msf{Bl}(A(\fp))$, is lower semicontinuous and $\Spec(R) = \coprod_{n \in \bbN} \msf{Bl}_n(A)$ is a partition into locally closed subsets. Moreover, $\beta(A) = \#\msf{Bl}(A^K)$. If $R$ is also universally catenary, japanese, and noetherian, then $\msf{BlEx}(A)$ is a reduced Weil divisor in $\Spec(R)$.
Let $R'$ be the integral closure of $R$ in $K$. Then $\#\msf{Bl}(A'(\fp')) = \#\msf{Bl}(A'_{\fp'})$ for all $\fp' \in \Spec(R')$ by Proposition \[unibranched\_block\_bijective\]. Since $A(\fp)$ splits, we moreover have $\#\msf{Bl}(A(\fp)) = \#\msf{Bl}(A'(\fp'))$ for all $\fp \in \Spec(R)$ $\fp' \in \psi^{-1}(\fp)$ by Lemma \[split\_center\_lemma\]. Hence, $X_{R'}(A) = \Spec(R)$. The claim about semicontinuity and the partition follows from Corollary \[bln\_spec\_strat\]. Now, assume that $R$ is universally catenary, japanese, and noetherian. Since $R$ is japanese, it follows by definition that $\psi$ is finite. The claim about $\msf{BlEx}(A)$ being a reduced Weil divisor now follows from \[x\_very\_dense\_beta\].
Blocks via central characters {#blocks_via_central_characters}
=============================
The main result in this section is Theorem \[maintheorem\_blex\_descr\] which gives an explicit description of the gluing loci introduced in §\[blocks\_of\_localizations\] via zero loci of central characters of simple modules of the generic fiber. This allows us (in principle) to construct the whole (atomic) block graph once we know the central characters. Parts of the argumentation are due to Bonnafé and Rouquier [@BR-cellules Appendice C].
Müller’s theorem
----------------
The central ingredient to establish a relationship between blocks and central characters is the general Lemma \[mueller\_theorem\_idempotents\] below, which is usually referred to as . We were not able to find a proof of it in this generality in the literature, so we include a proof here but note that this is known. The main ingredient is an even more general result by B. Müller [@Mul-Localization-in-non-commu-0] about the fibration of cliques of prime ideals in a noetherian ring over its center, see Lemma \[clique\_reduction\_bijection\]. We will recall only a few basic definitions from the excellent exposition in [@Goodearl-Warfield §12] and refer to *loc. cit.* for more details.
Throughout this paragraph, we assume that $A$ is a noetherian ring.
If $\fp, \fq$ are prime ideals of $A$, we say that there is a from $\fp$ to $\fq$, written $\fp \leadsto \fq$, if there is an ideal $\fa$ of $A$ such that $\fp \cap \fq \supsetneq \fa \sups \fp \fq$ and $(\fp \cap \fq)/\fa$ is non-zero and torsion-free both as a left $(A/\fp)$-module and as a right $(A/\fq)$-module. The bimodule $(\fp \cap \fq)/\fq$ is then called a between $\fq$ and $\fp$. The equivalence classes of the equivalence relation on $\Spec(A)$ generated by $\leadsto$ are called the of $A$. We write $\msf{Clq}(A)$ for the set of cliques of $A$ and $\msf{Clq}(\fp)$ for the unique clique of $A$ containing $\fp$. For the proof of Lemma \[mueller\_theorem\_idempotents\] we will need a few preparatory lemmas.\
We call the supremum of lengths of chains of prime ideals in a $A$ the of $A$. The following lemma is standard.
\[cliques\_zero\_dim\] Suppose that $A$ is noetherian and of classical Krull dimension zero. Then there is a canonical bijection $$\begin{array}{rcl}
\msf{Bl}(A) & \overset{\sim}{\longrightarrow} & \msf{Clq}(A) \\
c & \longmapsto & X_c \dopgleich \lbrace \fm \in \Max(A) \mid c^\dagger \in \fm \rbrace \;,
\end{array}$$ where $c^\dagger = 1-c$. If moreover $A$ is [commutative]{.nodecor}, then the cliques are singletons, i.e., there is a unique $\fm_c \in \Max(A)$ with $c^\dagger \in \fm_c$. Hence, in this case we have $\msf{Bl}(A) \simeq \msf{Max}(A) \simeq \Spec(A)$.
The first assertion is proven in [@Goodearl-Warfield Corollary 12.13]. In a commutative noetherian ring the cliques are singletons (see [@Goodearl-Warfield Exercise 12F]), and this immediately implies the second assertion.
\[annihilator\_of\_torsionfree\_mod\_p\] Let $\fp $ be a prime ideal of a noetherian ring $A$ and let $V$ be a non-zero $A$-module with $\fp \subs \msf{Ann}(V)$. If $V$ is torsion-free as an $(A/\fp )$-module, then $\fp = \msf{Ann}(V)$.
Suppose that $\fp \subsetneq \msf{Ann}(V)$. Then $\msf{Ann}(V)/\fp $ is a non-zero ideal of the noetherian prime ring $A/\fp $ and thus contains a regular element $\ol{x}$ by [@Jat86-Localization-in- Corollary 2.3.11]. But then $\ol{x}V = 0$, contradicting the assumption that $V$ is a torsion-free $(A/\fp )$-module.
\[cliques\_elementary\_notes\] The following holds:
\[cliques\_elementary\_notes:lift\] If $\fp $ and $\fq$ are prime ideals of $A$ and if $\fb$ is an ideal of $A$ with $\fb \subs \fp \cap \fq$ such that $\fp /\fb \leadsto \fq/\fb$ in $A/\fb$, then $\fp \leadsto \fq$ in $A$.
\[cliques\_elementary\_notes:reduction\] Let $\fp $ and $\fq$ be two prime ideals of $A$ with $\fp \leadsto \fq$ and let $\fb$ be an ideal of $A$. If there exists a linking ideal $\fa $ from $\fp $ to $\fq$ with $\fb \subs \fa $, then $\fp /\fb \leadsto \fq/\fb$ in $A/\fb$.
We can write a linking ideal from $\fp /\fb$ to $\fq/\fb$ as $\fa /\fb$ for an ideal $\fa $ containing $\fb$. By definition, we have $$(\fp \cap \fq)/\fb = (\fp /\fb) \cap (\fq/\fb) \supsetneq \fa /\fb \sups (\fp /\fb) \cdot (\fq/\fb) = (\fp \fq)/\fb \;,$$ implying that $\fp \cap \fq \supsetneq \fa \sups \fp \fq$. Moreover, we have $$\left( ( \fp \cap \fq)/\fb \right) / \left( \fa/\fb \right) \cong (\fp \cap \fq)/\fa$$ as $(A/\fb)$-bimodules. By definition, $(\fp \cap \fq)/\fa $ is torsionfree as a left module over the ring $$(A/\fb)/(\fp /\fb) \cong A/\fp \;.$$ Similarly, it follows that $(\fp \cap \fq)/\fa $ is torsionfree as a right module over the ring $A/\fq$. Hence, $\fa $ is a linking ideal from $\fp $ to $\fq$.
We have $$\fp /\fb \cap \fq/\fb = (\fp \cap \fq)/\fb \supsetneq \fa /\fb \sups (\fp \fq + \fb)/\fb = (\fp /\fb) \cdot (\fq/\fb) \;.$$ Since $$\left( (\fp \cap \fq)/\fb \right) / \left( \fa /\fb \right) \cong (\fp \cap \fq)/\fa \;, \quad (A/\fb)/(\fp /\fb) \cong A/\fp \;, \quad (A/\fb)/(\fq/\fb) \cong A/\fq \;,$$ it follows that $\fa /\fb$ is a linking ideal from $\fp /\fb$ to $\fq/\fb$.
\[cliques\_central\_reduction\] Let $\fp $ and $\fq$ be distinct prime ideals of a noetherian ring $A$ with $\fp \leadsto \fq$. If $\fz$ is a centrally generated ideal of $A$ with $\fz \subs \fp $ or $\fz \subs \fq$, then $\fz \subs \fp \cap \fq$ and $\fp /\fz \leadsto \fq/\fz$ in $A/\fz$.
This is proven in [@Mul85-Affine-Noetheria] but we also give a proof here for the sake of completeness. First note that since $\fz$ is centrally generated and $\fp \leadsto \fq$, it follows from [@Goodearl-Warfield Lemma 12.15] that already $\fz \subs \fp \cap \fq$. Let $\fa $ be a linking ideal from $\fp $ to $\fq$. We claim that $\fz$ is contained in $\fa $. To show this, suppose that $\fz$ is not contained in $\fa $. Then $(\fa + \fz)/\fa $ is a non-zero submodule of $(\fp \cap \fq)/\fa $ which is torsionfree as a left $(A/\fp )$-module and as a right $(A/\fq)$-module. In conjunction with the fact that $\fz$ is centrally generated it now follows from Lemma \[annihilator\_of\_torsionfree\_mod\_p\] that $$\fp = \msf{Ann}( _A( (\fa + \fz)/\fa ) ) = \msf{Ann}( ((\fa + \fz)/\fa )_A) = \fq,$$ contradicting the assumption $\fp \neq \fq$. Hence, we must have $\fz \subs \fa $ and it thus follows from Lemma \[cliques\_elementary\_notes\]\[cliques\_elementary\_notes:reduction\] that $\fp /\fz \leadsto \fq/\fz$.
\[clique\_reduction\_bijection\] Let $\fz$ be a centrally generated ideal of a noetherian ring $A$. Let $\fp $ be a prime ideal of $A$ with $\fz \subs \fp $. Then all prime ideals in $\msf{Clq}(\fp )$ contain $\fz$ and the map $$\begin{array}{rcl}
\msf{Clq}(\fp ) & \longmapsto & \msf{Clq}(\fp /\fz) \\
\fq & \longmapsto & \fq/\fz
\end{array}$$ is a bijection between a clique of $A$ and a clique of $A/\fz$.
It follows immediately from [@Goodearl-Warfield Lemma 12.15] that all prime ideals in $\msf{Clq}(\fp )$ contain $\fz$. If $\fq \in \msf{Clq}(\fp )$, then there exists a chain $\fp = \fp _0, \fp _1, \ldots, \fp _{r-1}, \fp _r = \fq$ of prime ideals of $A$ with $\fp _i \leadsto \fp _{i+1}$ or $\fp _{i+1} \leadsto \fp _i$ for all indices $i$. An inductive application of Lemma \[cliques\_central\_reduction\] shows now that $\fp _i/\fz \leadsto \fp _{i+1}/\fz$ or $\fp _{i+1}/\fz \leadsto \fp _{i}/\fz$ for all $i$. Hence, $\fp /\fz$ and $\fq/\fz$ lie in the same clique of $A/\fz$ so that the map $\msf{Clq}(\fp ) \rarr \msf{Clq}(\fp /\fz)$ is well-defined. On the other hand, similar arguments and Lemma \[cliques\_elementary\_notes\]\[cliques\_elementary\_notes:lift\] show that if $\fq/\fz \in \msf{Clq}(\fp /\fz)$, then also $\fq \in \msf{Clq}(\fp )$, so that we also have a well-defined map $\msf{Clq}(\fp /\fz) \rarr \msf{Clq}(\fp )$. It is evident that both maps defined are pairwise inverse thus proving the first assertion. The second assertion is now obvious.
\[mueller\_theorem\_idempotents\] Let $A$ be a ring with center $Z$ such that $Z$ is noetherian and $A$ is a finite $Z$-module. If $\fz$ is a centrally generated ideal of $A$ such that $A/\fz A$ is of classical Krull dimension zero, then the inclusion $(Z+\fz)/\fz \hookrightarrow A/\fz A$ is block bijective. In other words, the block idempotents of $A/\fz A$ are already contained in the central subalgebra $(Z+\fz)/\fz$.
Let $\ol{A} \dopgleich A/\fz$ and let $\ol{Z} \dopgleich (Z+\fz)/\fz$. Then $\ol{A}$ is a finitely generated $\ol{Z}$-module since $A$ is a finitely generated $Z$-module. Hence, $\ol{Z} \subs \ol{A}$ is a finite centralizing extension and now it follows from going up in finite centralizing extensions [@McR-NN-rings Theorem 10.2.9] that the classical Krull dimension of $\ol{Z}$ is equal to that of $\ol{A}$, which is zero by assumption. Hence, by Lemma \[cliques\_zero\_dim\] we have $\msf{Bl}(\ol{Z}) \simeq \msf{Clq}(\ol{Z})$ and $\msf{Bl}(\ol{A}) \simeq \msf{Clq}(\ol{A})$. Since $\#\msf{Bl}(\ol{Z}) \leq \#\msf{Bl}(\ol{A})$, the claim is thus equivalent to the claim that over each clique of $\ol{Z}$, there is just one clique of $\ol{A}$. So, let $X,Y \in \msf{Clq}(\ol{A})$ be two cliques. We pick $\fM/\fz \in X$ and $\fN/\fz \in Y$ with $\fM,\fN$ maximal ideals of $A$. Assume that $X$ and $Y$ lie over the same clique of $\ol{Z}$. Since $\ol{Z}$ is commutative, we know from Lemma \[cliques\_zero\_dim\] that all cliques are singletons and so the assumption implies that $\fM/\fz$ and $\fN/\fz$ lie over the same maximal ideal of $\ol{Z}$, i.e., $$(\fM/\fz) \cap \left((Z + \fz)/\fz\right) = (\fN/\fz) \cap \left((Z + \fz)/\fz\right) \;,$$ hence $$\fM \cap (Z+\fz) = \fN \cap (Z+\fz) \;.$$ Since $Z \subs Z+\fz$, we thus get $$\fM \cap Z = \fM \cap Z \cap (Z+\fz) = \fN \cap Z \cap (Z+\fz) = \fN \cap Z \;.$$ Now, Müller’s theorem [@Goodearl-Warfield Theorem 13.10] implies that $\fM$ and $\fN$ lie in the same clique of $A$. An application of Lemma \[clique\_reduction\_bijection\] thus implies that $\fM/\fz$ and $\fN/\fz$ lie in the same clique of $A/\fz$, so $X=Y$.
Blocks as fibers of a morphism
------------------------------
We assume that $A$ is a finite flat algebra over a *noetherian* integral domain $R$.
By Lemma \[ff\_center\_ext\] the morphism $$\Upsilon: \Spec(Z) \rarr \Spec(R) \;,$$ induced by the canonical morphism from $R$ to the center $Z$ of $A$ is finite, closed, and surjective. The center $Z$ of $A$ is naturally an $R$-algebra and so we can consider its fibers $$Z(\fp) = \msf{k}(\fp) \otimes_R Z/\fp Z = Z_\fp/\fp_\fp Z_\fp$$ in prime ideals $\fp$ of $R$. On the other hand, the image of $Z_\fp = \msf{Z}(A_\fp)$ under the canonical (surjective) morphism $A_\fp \twoheadrightarrow A(\fp)$ yields a central subalgebra $$\msf{Z}_\fp(A) \dopgleich (Z_\fp + \fp_\fp A_\fp)/\fp_\fp A_\fp$$ of $A(\fp)$. In general this subalgebra is *not* equal to the center of $A(\fp)$ itself. We have a surjective morphism $$\label{center_reduction_map}
\varphi_\fp : Z(\fp) \twoheadrightarrow \msf{Z}_\fp(A)$$ of finite-dimensional $\msf{k}(\fp)$-algebras. This morphism is in general *not* injective—it is if and only if $$\fp_\fp A_\fp \cap Z_\fp = \fp_\fp Z_\fp \Leftrightarrow \Rad(A_\fp) \cap Z_\fp = \Rad(Z_\fp) \;.$$ Nonetheless, we have the following result.
\[center\_reduction\_block\_bijective\] The map $\varphi_\fp:Z(\fp) \rarr \msf{Z}_\fp(A)$ in (\[center\_reduction\_map\]) is block bijective.
Since $\varphi_\fp$ is surjective, the induced map $^a\varphi_\fp: \Spec(\msf{Z}_\fp(A)) \rarr \Spec(Z(\fp))$ is injective, so $\#\Bl(\msf{Z}_\fp(A)) \leq \#\Bl(Z(\fp))$ by Lemma \[cliques\_zero\_dim\]. Now we just need to show that $\varphi_\fp$ does not map any no non-trivial idempotent to zero. Since $R_\fp$ is noetherian, also $A_\fp$ is noetherian. The Artin–Rees lemma [@Mat-Commutative Theorem 8.5] applied to the $R_\fp$-module $A_\fp$, the submodule $Z_\fp$ of $A_\fp$, and the ideal $\fp_\fp$ of $R_\fp$ shows that there is an integer $k \in \bbN_{>0}$ such that for any $n > k$ we have $$\fp_\fp^n A_\fp \cap Z_\fp = \fp_\fp^{n-k}(( \fp_\fp^k A_\fp) \cap Z_\fp) \;.$$ In particular, there is $n \in \bbN_{>0}$ such that $\fp_\fp^n A_\fp \cap Z_\fp \subs \fp_\fp Z_\fp$. Now, let $\ol{e} \in Z(\fp) = Z_\fp/\fp_\fp Z_\fp$ be an idempotent with $\varphi_\fp(\ol{e}) = 0$. By assumption, $\ol{e} \in \Ker(\varphi_\fp) = (\fp_\fp A_\fp \cap Z_\fp)/\fp_\fp Z_\fp $. Hence, if $e \in Z_\fp$ is a representative of $\ol{e}$, we have $e \in \fp_\fp A_\fp \cap Z_\fp$. We have $e^n \in \fp_\fp^n A_\fp \cap Z_\fp \subs \fp_\fp Z_\fp$, so already $\ol{e} = 0$.
\[mueller\_maintheorem\] For any $\fp \in \Spec(R)$ there are canonical bijections $$\label{mueller_bijection}
\msf{Bl}(A(\fp)) \simeq \msf{Bl}(\msf{Z}_\fp(A)) \simeq \msf{Bl}(Z(\fp)) \simeq \Upsilon^{-1}(\fp) \;.$$ The first bijection $\msf{Bl}(A(\fp)) \simeq \msf{Bl}(\msf{Z}_\fp(A))$ is induced by the embedding $\msf{Z}_\fp(A) \hookrightarrow A(\fp)$. In other words, all block idempotents of $A(\fp)$ are already contained in the central subalgebra $\msf{Z}_\fp(A)$ of $A(\fp)$. The second bijection is the bijection from Lemma \[center\_reduction\_block\_bijective\]. The last bijection $\msf{Bl}(Z(\fp)) \simeq \Upsilon^{-1}(\fp)$ maps a block idempotent $c$ of $Z(\fp)$ to the (by the theorem unique) maximal ideal $\fm_c$ of $Z$ lying above $\fp$ such that $c^\dagger \in (\fm_c+\fp_\fp Z_\fp)/\fp_\fp Z_\fp$, where $c^\dagger = 1-c$.
The first bijection follows directly from Lemma \[mueller\_theorem\_idempotents\] applied to $A_\fp$ and the centrally generated ideal $\fz \dopgleich \fp_\fp A_\fp$. Let $\Upsilon_\fp:\Spec(Z_\fp) \rarr \Spec(R_\fp)$ be the morphism induced by the canonical map $R_\fp \rarr Z_\fp$. Recall from Lemma \[ff\_center\_ext\] that $R_\fp \subs Z_\fp$ is a finite extension so that $\Upsilon_\fp$ is surjective. We have $$\begin{aligned}
\Upsilon_{\fp }^{-1}(\fp _{\fp }) & = \lbrace \fQ \in \Spec(Z_{\fp }) \mid \fQ \cap R_{\fp } = \fp _{\fp } \rbrace \\&= \lbrace \fQ \in \Spec(Z_{\fp }) \mid \fp _{\fp } \subs \fQ \rbrace \\ &= \lbrace \fQ \in \Spec(Z_{\fp }) \mid \fp _{\fp }Z_{\fp } \subs \fQ \rbrace \\
& \simeq \Spec(Z(\fp)) \;.\end{aligned}$$ In the second equality we used the fact that $R_{\fp } \rarr Z_{\fp }$ is a finite morphism and $R_{\fp }$ is local with maximal ideal $\fp _{\fp }$. The identification with $\Spec(Z(\fp))$ is canonical since $Z(\fp) = Z_\fp/\fp_\fp Z_\fp$. The morphism $\Theta_\fp: \Spec(Z_{\fp }) \rarr \Spec(Z)$ induced by the localization map $Z \rarr Z_\fp$ is injective by [@Eis-Commutative-Algebra Proposition 2.2(b)]. We claim that this map induces $\Upsilon^{-1}_\fp(\fp_\fp) \simeq \Upsilon^{-1}(\fp)$. If $\fQ \in \Upsilon_{\fp }^{-1}(\fp _{\fp })$, then clearly $(\fQ \cap Z) \cap R = \fQ \cap R \subs R \cap \fp _{\fp } = \fp $ and therefore $\Theta_\fp$ induces an injective map $\Upsilon_{\fp }^{-1}(\fp _{\fp }) \rarr \Upsilon^{-1}(\fp )$. If $\fQ \in \Upsilon^{-1}(\fp )$, then, since $\fQ \cap R = \fp $, we have $\fQ \cap (R \setminus \fp ) = \emptyset$ so that $\fQ_{\fp } \in \Spec(Z_{\fp })$ and clearly $\fp _{\fp } \subs \fQ_{\fp }$, implying that $\fQ_{\fp } \in \Upsilon_{\fp }^{-1}(\fp _{\fp })$. The map $\Upsilon_{\fp }^{-1}(\fp _{\fp }) \rarr \Upsilon^{-1}(\fp )$ is thus bijective. Hence, we have a canonical bijection $\Spec(Z(\fp)) \simeq \Upsilon^{-1}(\fp)$. Now, recall from Lemma \[cliques\_zero\_dim\] that $\Spec(Z(\fp)) \simeq \msf{Bl}(Z(\fp))$.
Blocks and gluing loci via central characters
---------------------------------------------
We assume that $R$ is *noetherian* and *normal*, and that $A$ is a finite flat $R$-algebra with *split* generic fiber $A^K$.
Recall from Corollary \[normal\_split\_corollary\] that the quotient map $A_\fp \twoheadrightarrow A(\fp)$ induces $\msf{Bl}(A_\fp) \simeq \msf{Bl}(A(\fp))$, so together with Theorem \[mueller\_maintheorem\] we have a canonical bijection $$\label{bl_ap_ups_bij}
\msf{Bl}(A_\fp) \simeq \Upsilon^{-1}(\fp) \;.$$ Recall from §\[blocks\_of\_localizations\] that $\msf{Fam}_\fp(A^K)$ is the partition of $\Irr A^K$ induced by the blocks of $A_\fp$ and that we naturally have $\msf{Bl}(A_\fp) \simeq \msf{Fam}_\fp(A^K)$. Altogether, we now have canonical bijections $$\label{pfam_fiber_bij}
\msf{Fam}_\fp(A) \simeq \msf{Bl}(A_\fp) \simeq \Upsilon^{-1}(\fp) \simeq \msf{Bl}(A(\fp)) \;.$$ Since $A$ has split generic fiber $A^K$, we have a central character $\Omega_S:\msf{Z}(A^K) \rarr K$ for every simple $A^K$-module $S$. Recall that $\Omega_S(z)$ is the scalar by which $z \in \msf{Z}(A^K)$ acts on $S$. Since $R$ is normal, the image of the restriction of $\Omega_S$ to $\msf{Z}(A) \subs \msf{Z}(A^K)$ is contained in $R \subs K$. We thus get a well-defined $R$-algebra morphism $$\Omega_S':\msf{Z}(A) \rarr R \;.$$ It is a classical fact that $S,T \in \Irr A^K$ lie in the same family if and only if $\Omega_S' = \Omega_T'$. We can thus label the central characters of $A^K$ as $\Omega_\mathcal{F}$ with $\mathcal{F}$ a family (block) of $A^K$. Using Theorem \[mueller\_maintheorem\] this description generalizes modulo $\fp$ so that we get an explicit description of the $\fp$-families, and thus of the block stratification.
\[maintheorem\_blex\_descr\] Under the bijection $\Upsilon^{-1}(\fp) \simeq \msf{Fam}_\fp(A)$ from (\[pfam\_fiber\_bij\]) the $\fp$-family of a simple $A^K$-module $S$ corresponds to $\Ker \Omega_S^\fp$. Hence, two simple $A^K$-modules $S$ and $T$ lie in the same $\fp$-family if and only if $\Omega_S'(z) \equiv \Omega_T'(z) \ \msf{mod} \ \fp$ for all $z \in \msf{Z}(A)$. So, if $z_1,\ldots,z_n$ is an $R$-algebra generating system of $\msf{Z}(A)$ and $\mscr{F},\mscr{F}'$ are two distinct $A^K$-families, then the corresponding gluing locus is given by $$\Gamma_A(\lbrace \mscr{F},\mscr{F}' \rbrace) = \msf{V}( \lbrace \Omega_\mscr{F}(z_i) - \Omega_{\mscr{F}'}(z_i) \mid i =1,\ldots,n \rbrace) \;.$$
Considering the explicit form of the bijection given in Theorem \[mueller\_maintheorem\] we see that the bijection (\[bl\_ap\_ups\_bij\]) maps a block idempotent $c$ of $A_\fp$ to the (by the theorem unique) maximal ideal $\fQ_c$ of $Z$ lying above $\fp$ and satisfying $c^\dagger \in (\fQ_c)_\fp$. Let $c_\fQ$ be the block idempotent of $A_\fp$ corresponding to $\fQ \in \Upsilon^{-1}(\fp)$.
For $S \in \Irr A^K$ let $\Omega_S^{\fp }:Z \rarr R/\fp $ be the composition of $\Omega_S'$ and the quotient morphism $R \rarr R/\fp $. It is clear that $\Ker(\Omega_S^{\fp }) \in \Upsilon^{-1}(\fp )$. Note that $\Omega_S'(z) \equiv \Omega_T'(z) \ \msf{mod} \ \fp$ for all $z \in \msf{Z}(A)$ if and only if $\Omega_S^{\fp } = \Omega_T^{\fp }$. We have an exact sequence $$0 \longrightarrow \Ker(\Omega_S') \longrightarrow Z \overset{\Omega_S'}{\longrightarrow} R \longrightarrow 0$$ of $R$-modules. Since $\Omega_S'$ is an $R$-algebra morphism, the canonical map $R \rarr Z$ is a section of $\Omega_S'$ and therefore $Z = R \oplus \Ker(\Omega_S')$ as $R$-modules. Similarly, we have $Z = R \oplus \Ker(\Omega_T')$. Since $\Ker(\Omega_S') \subs \Ker(\Omega_S^{\fp })$ and $\Ker(\Omega_T') \subs \Ker(\Omega_T^{\fp })$, this implies that $\Omega_S^\fp = \Omega_T^\fp$ if and only if $\Ker(\Omega_S^\fp) = \Ker(\Omega_T^\fp)$.
Now, suppose that $\Ker(\Omega_S^{\fp }) = \Ker(\Omega_T^{\fp })$. Denote this common kernel by $\fQ$. Clearly, $\fQ \in \Upsilon^{-1}(\fp )$. We know that the corresponding block idempotent $c_{\fQ}$ of $A_{\fp }$ has the property that $c_{\fQ}^\dagger \in \fQ_{\fp }$. Since $\Ker(\Omega_S') \subs \Ker(\Omega_S^{\fp }) = \fQ = \Ker(\Omega_T^{\fp }) \sups \Ker(\Omega_T')$, this certainly implies that $c_{\fQ}^\dagger S= 0 = c_{\fQ}^\dagger T$. Hence, $S$ and $T$ lie in the same $\fp $-family.
Conversely, suppose that $S$ and $T$ lie in the same $\fp$-family. We can write the corresponding block idempotent of $A_\fp$ as $c_{\fQ}$ for some $\fQ \in \Upsilon^{-1}(\fp )$. By definition, $c_{\fQ}^\dagger S = 0 = c_{\fQ}^\dagger T$. We know that $c_{\fQ}^\dagger \in \fQ_{\fp }$ and $c_{\fQ} \notin \fQ_{\fp }$ and therefore $\Ker( (\Omega_S')_{\fp }) = \fQ_{\fp } = \Ker( (\Omega_T')_{\fp }) $. Hence, $\fQ \subs \Ker(\Omega_S') \subs \Ker(\Omega_S^{\fp })$ and $\fQ \subs \Ker(\Omega_T') \subs \Ker(\Omega_T^{\fp })$. Since $\fQ, \Ker(\Omega_S^{\fp }), \Ker(\Omega_T^{\fp }) \in \Upsilon^{-1}(\fp )$ and all prime ideals in $\Upsilon^{-1}(\fp )$ are incomparable, we thus conclude that $\Ker(\Omega_S^{\fp }) = \Ker(\Omega_T^{\fp })$.
The equation for the gluing locus is now clear.
Blocks and decomposition maps {#blocks_and_dec_maps}
=============================
To obtain information about the actual members of the $A(\fp)$-families we use decomposition maps as introduced by Geck and Rouquier [@Geck-Rouquier-Dec] (see also [@Geck-Pfeiffer] and [@Thiel-Dec]). For the theory of decomposition maps we need the following (standard) assumption:
$A$ is finite free with split generic fiber and for any non-zero $\fp \in \Spec(R)$ there is a discrete valuation ring $\sO$ with maximal ideal $\fm$ in $K$ dominating $R_\fp$ such that the canonical map $\msf{G}_0(A(\fp)) \rarr \msf{G}_0(A^\sO(\fm))$ of Grothendieck groups is an isomorphism.
Here, $\msf{G}_0$ denotes the Grothendieck group, i.e., the zeroth $\msf{K}$-group of the category of finitely generated modules. We call a ring $\sO$ as above a in $\fp$. We refer to [@Thiel-Dec] for more details. The following lemma lists two standard situations in which the above assumptions hold. Part \[dec\_theory\_assumptions\_lemma:dedekind\] is obvious and part \[dec\_theory\_assumptions\_lemma:noeth\] was proven in [@Thiel-Dec Theorem 1.22].
\[dec\_theory\_assumptions\_lemma\] A finite free $R$-algebra $A$ with split generic fiber satisfies the above assumptions in the following two cases:
\[dec\_theory\_assumptions\_lemma:dedekind\] $R$ is a Dedekind domain.
\[dec\_theory\_assumptions\_lemma:noeth\] $R$ is noetherian and $A$ has split fibers.
If $\sO$ is a perfect $A$-gate in $\fp$, then there is a group morphism $$\msf{d}_A^{\fp,\sO}:\msf{G}_0(A^K) \rarr \msf{G}_0(A(\fp))$$ between Grothendieck groups generalizing reduction modulo $\fp$. In case $R$ is normal, it was proven by Geck and Rouquier [@Geck-Rouquier-Dec] that this map is independent of the choice of $\sO$ and in this case we just write $\msf{d}_A^\fp$. We note that in case $R$ is noetherian and $A$ has split fibers, any decomposition map in the sense of Geck and Rouquier can be realized by a perfect $A$-gate, see [@Thiel-Dec Theorem 1.22].
Brauer reciprocity
------------------
An important tool for relating decomposition maps and blocks is the so-called which we prove in Theorem \[brauer\_rec\] below in our general setup (this was known to hold before only in special settings). Recall that the for a finite-dimensional algebra $B$ over a field $F$ is the $\bbZ$-linear pairing $\langle \cdot, \cdot \rangle_B:\msf{K}_0(B) \times \msf{G}_0(B) \rarr \bbZ$ uniquely defined by $$\label{intertwining_form_def}
\langle \lbrack P \rbrack, \lbrack V \rbrack \rangle \dopgleich \dim_F \Hom_B(P,V)$$ for a finite-dimensional projective $B$-module $P$ and a finite-dimensional $B$-module $V$, see [@Geck-Rouquier-Dec §2]. Here, $\msf{K}_0(B)$ is the zeroth $\msf{K}$-group of the category of finite-dimensional projective $B$-modules. The intertwining form is always non-degenerate, see Lemma \[intertwining\_nondeg\]. Due to the non-degeneracy of $\langle \cdot,\cdot \rangle_{A^K}$ there is at most one adjoint $$\msf{e}_A^{\fp,\sO}:\msf{K}_0(A(\fp)) \rarr \msf{K}_0(A^K)$$ of $\msf{d}_A^{\fp,\sO}:\msf{G}_0(A^K) \rarr \msf{G}_0(A(\fp))$ with respect to $\langle \cdot,\cdot \rangle_{A(\fp)}$, characterized by the relation $$\label{brauer_rec_adjoint_rel}
\langle \msf{e}_A^{\fp,\sO}(\lbrack \ol{P} \rbrack), \lbrack V \rbrack \rangle_{A^K} = \langle \lbrack \ol{P} \rbrack, \msf{d}_A^{\fp,\sO}(\lbrack V \rbrack) \rangle_{A(\fp)} \;.$$ for all finitely generated $A^K$-modules $V$ and all finitely generated projective $A(\fp)$-modules $\ol{P}$, see Lemma \[intertwining\_nondeg\]. Brauer reciprocity is about the existence of this adjoint.
\[brauer\_rec\] The (unique) adjoint $\msf{e}_A^{\fp,\sO}$ of $\msf{d}_A^{\fp,\sO}$ exists. Moreover, the diagram $$\label{brauer_rec_diagram}
\begin{tikzcd}
\msf{K}_0(A^K) \ar{r}{\msf{c}_{A^K}} & \msf{G}_0(A^K) \ar{d}{\msf{d}_A^{\fp,\sO}} \\
\msf{K}_0(A(\fp)) \ar{r}[swap]{\msf{c}_{A(\fp)}} \ar{u}{\msf{e}_A^{\fp,\sO}} & \msf{G}_0(A(\fp))
\end{tikzcd}$$ commutes, where the horizontal morphisms are the canonical ones () mapping a class $\lbrack P \rbrack$ of a projective module $P$ to its class $\lbrack P \rbrack$ in the Grothendieck group. If $R$ is normal, the morphism $\msf{e}_A^{\fp,\sO}$ does not depend on the choice of $\sO$ and we denote it by $\msf{e}_A^\fp$.
Since $\langle \cdot, \cdot \rangle_{A^K}$ is non-degenerate by Lemma \[intertwining\_nondeg\], it follows that $\msf{d}_A^{\fp,\sO}$ has at most one adjoint $\msf{e}_A^{\fp,\sO}$, characterized by equation (\[brauer\_rec\_adjoint\_rel\]), see [@Scheja-Storch-Alg-2 Satz 78.1]. By assumption there is a perfect $A$-gate $\sO$ in $\fp$. Let $\fm$ be the maximal ideal of $\sO$. Since $A^K$ splits by assumption, Corollary \[algebra\_dvr\_semiperfect\] implies that $A^\sO$ is semiperfect. The morphism $\msf{K}_0(A^\sO) \rarr \msf{K}_0(A^\sO(\fm))$ induced by the quotient map $A^\sO \twoheadrightarrow A^\sO(\fm)$ is thus an isomorphism by lifting of idempotents. Furthermore, by assumption the morphism $\msf{d}_A^{\fp,\fm}:\msf{G}_0(A(\fp)) \rarr \msf{G}_0(A^\sO(\fm))$ is an isomorphism and then the proof of Theorem \[faithfully\_flat\_ext\_block\_bij\] shows that the canonical morphism $\msf{e}_A^{\fp,\fm}:\msf{K}_0(A(\fp)) \rarr \msf{K}_0(A^\sO(\fm))$ is also an isomorphism. We can thus define a morphism $\msf{e}_A^{\fp,\sO}:\msf{K}_0(A(\fp)) \rarr \msf{K}_0(A^K)$ as the following composition $$\begin{tikzcd}
\msf{K}_0(A(\fp)) \arrow{r}{\simeq} \arrow[bend right=15]{rrr}[swap]{\msf{e}_A^{\fp,\sO}} & \msf{K}_0(A^\sO(\fm)) \arrow{r}{\simeq} & \msf{K}_0(A^\sO) \arrow{r} & \msf{K}_0(A^K)
\end{tikzcd}$$
We will now show that $\msf{e}_A^{\fp,\sO}$ is indeed an adjoint of $\msf{d}_A^{\fp,\sO}$. The arguments in the proof of [@CR-Methods-1 18.9] can, with some refinements, be transferred to our more general situation and this is what we will do. Let $\ol{P}$ be a finitely generated projective $A(\fp)$-module and let $V$ be a finitely generated $A^K$-module. Since $\msf{K}_0(A^\sO) \simeq \msf{K}_0(A^\sO(\fm))$, there exists a finitely generated projective $A^{\sO}$-module $P$ such that $(\msf{e}_{A}^{\fp,\fm})^{-1}(\lbrack P/\fm P \rbrack) = \lbrack \ol{P} \rbrack$ and then we have $\msf{e}_A^{\fp,\sO}(\lbrack \ol{P} \rbrack) = \lbrack P^K \rbrack$. Let $\wt{V}$ be an $A^{\sO}$-lattice in $V$. Then by definition of $\msf{d}_A^{\fp,\sO}$, see [@Thiel-Dec Corollary 1.14], we have $\msf{d}_A^{\fp,\sO}(\lbrack V \rbrack) = (\msf{d}_A^{\fp,\fm})^{-1}(\lbrack \wt{V}(\fm) \rbrack)$. We denote by $\ol{V}$ a representative of $\msf{d}_A^{\fp,\sO}(\lbrack V \rbrack)$. Since $P$ is a finitely generated projective $A^{\sO}$-module, we can write $P \oplus Q = (A^{\sO})^n$ for some finitely generated projective $A^\sO$-module $Q$ and some $n \in \bbN$. Since $\Hom_{A^{\sO}}$ is additive, we get $$\begin{aligned}
\Hom_{A^{\sO}}(P, \wt{V}) \oplus \Hom_{A^{\sO}}(Q,\wt{V}) & = \Hom_{A^{\sO}}(P \oplus Q, \wt{V}) = \Hom_{A^{\sO}}( (A^{\sO})^n, \wt{V}) \\ &= ( \Hom_{A^{\sO}}( A^{\sO}, \wt{V}) )^n \simeq \wt{V}^n \;.\end{aligned}$$ This shows that $\Hom_{A^{\sO}}(P,\wt{V})$ is a direct summand of $\wt{V}^n$ and as $\wt{V}^n$ is $\sO$-free, we conclude that $\Hom_{A^{\sO}}(P,\wt{V})$ is $\sO$-projective and thus even $\sO$-free since $\sO$ is a discrete valuation ring. Since $P$ is a finitely generated projective $A^\sO$-module, it follows from Lemma \[base\_ring\_change\_of\_hom\] that there is a canonical $K$-vector space isomorphism $$K \otimes_{\sO} \Hom_{A^{\sO}}(P,\wt{V}) \simeq \Hom_{A^K}(P^K,V)$$ and a canonical $\msf{k}(\fm)$-vector space isomorphism $$\msf{k}(\fm) \otimes_{\sO} \Hom_{A^{\sO}}(P,\wt{V}) \simeq \Hom_{A^{\sO}(\fm)}(P/ \fm P, \wt{V}/\fm \wt{V}) \;.$$ Combining all results and the fact that both $\msf{e}_A^{\fp,\sO}$ and $\msf{d}_A^{\fp,\sO}$ preserve dimensions by construction, we can now conclude that $$\begin{aligned}
\langle \msf{e}_A^{\fp,\sO}(\lbrack \ol{P} \rbrack), \lbrack V \rbrack \rangle_{A^K} &= \dim_K \Hom_{A^K}( P^K, V) = \dim_{\sO} \Hom_{A^{\sO}}( P, \wt{V}) \\ &= \dim_{\msf{k}(\fm)} \Hom_{A^{\sO}(\fm)}( P/\fm P, \wt{V}/\fm \wt{V}) = \dim_{\msf{k}(\fp)} \Hom_{A(\fp)}( \ol{P}, \ol{V}) \\ &= \langle \lbrack \ol{P} \rbrack, \msf{d}_A^{\fp,\sO}(\lbrack V \rbrack) \rangle_{A(\fp)} \;.\end{aligned}$$ Proving the commutativity of diagram (\[brauer\_rec\_diagram\]) amounts to proving that $\msf{c}_{A(\fp)}( \lbrack \ol{P} \rbrack ) = \msf{d}_A^{\fp,\sO} \circ \msf{c}_{A^K} \circ \msf{e}_A^{\fp,\sO} ( \lbrack \ol{P} \rbrack)$ for every finitely generated projective $A(\fp)$-module $\ol{P}$. To prove this, note that the diagram $$\begin{tikzcd}
\msf{K}_0(A^{\sO}(\fm)) \arrow{r}{\msf{c}_{A^{\sO}(\fm)}} & \msf{G}_0(A^{\sO}(\fm)) \\
\msf{K}_0(A(\fp)) \arrow{r}[swap]{\msf{c}_{A(\fp)}} \arrow{u}{\msf{e}_A^{\fp,\fm}} & \msf{G}_0(A(\fp)) \arrow{u}[swap]{\msf{d}_A^{\fp,\fm}}
\end{tikzcd}$$ commutes. As above we know that there exists a finitely generated projective $A^{\sO}$-module $P$ such that $(\msf{e}_A^{\fp,\fm})^{-1}(\lbrack P/\fm P \rbrack) = \lbrack \ol{P} \rbrack$ and $\msf{e}_A^{\fp,\sO}(\lbrack \ol{P} \rbrack) = \lbrack P^K \rbrack$. Since $P$ is a finitely generated projective $A^{\sO}$-module and $A$ is a finite $\sO$-module, it follows that $P$ is also a finitely generated projective $\sO$-module. As $\sO$ is a discrete valuation ring, we conclude that $P$ is actually $\sO$-free of finite rank. Hence, $P$ is an $A^{\sO}$-lattice in $P^K$ and therefore $$\begin{aligned}
\msf{d}_A^{\fp,\sO} \circ \msf{c}_{A^K} \circ \msf{e}_A^{\fp,\sO} ( \lbrack \ol{P} \rbrack) &= \msf{d}_A^{\fp,\sO} ( \lbrack P^K \rbrack) = (\msf{d}_{A}^{\fp,\fm})^{-1} ( \lbrack P/\fm P \rbrack) = (\msf{d}_{A}^{\fp,\fm})^{-1} \circ \msf{c}_{A(\fm)}( \lbrack P/\fm P \rbrack) \\ &= \msf{c}_{A(\fp)} \circ (\msf{e}_{A}^{\fp,\fm})^{-1}( \lbrack P/\fm P \rbrack) = \msf{c}_{A(\fp)} ( \lbrack \ol{P} \rbrack). \end{aligned}$$ If $R$ is normal, then the independence of $\msf{e}_A^{\fp,\sO}$ from the choice of $\sO$ follows from the independence of $\msf{d}_A^{\fp,\sO}$ from the choice of $\sO$ and the fact that $\msf{d}_A^{\fp,\sO}$ has at most one adjoint.
Preservation of simple modules vs. preservation of blocks {#preservation}
---------------------------------------------------------
In [@Thiel-Dec] we studied the set $$\msf{DecGen}(A) \dopgleich \left\lbrace \fp \in \Spec(R) \mid \msf{d}_A^{\fp,\sO} \tn{ is trivial for any }A\tn{-gate in }\fp \right\rbrace \;.$$ where $\msf{d}_A^{\fp,\sO}$ being means that it induces a bijection between simple modules. We have proven in [@Thiel-Dec Theorem 2.3] that $\msf{DecGen}(A)$ is open if $R$ is noetherian and $A$ has split fibers. Using Brauer reciprocity we thus deduce that in this case the locus of all $\fp$ such that $\msf{e}_A^{\fp,\sO}$ is trivial for any $\sO$ is an open subset of $\msf{Spec}(R)$.
If $\fp \in \msf{DecGen}(A)$, then the simple modules of $A^K$ and $A(\fp)$ are “essentially the same”, in particular their dimensions are the same. This is why explicit knowledge about $\msf{DecGen}(A)$ is quite helpful to understand the representation theory of the fibers of $A$, see [@Thiel-Dec]. So far, we do not have an explicit description of $\msf{DecGen}(A)$, however. Brauer reciprocity enables us to prove the following relation between decomposition maps and blocks.
\[decgen\_blgen\_inclusion\_main\_theorem\] We have an inclusion $$\label{decgen_blgen_inclusion}
\msf{DecGen}(A) \subs \msf{BlGen}(A) \;.$$
Let $\fp \in \Spec(R)$ be non-zero. By assumption there is a perfect $A$-gate $\sO$ in $\fp$. If $\fp \in \msf{DecGen}(A)$, then by definition $\msf{d}_A^{\fp,\sO}$ is trivial, so the matrix $\msf{D}_A^{\fp,\sO}$ of this morphism in bases given by isomorphism classes of simple modules of $A^K$ and $A(\fp)$, respectively, is equal to the identity matrix when ordering the bases appropriately. It now follows from Brauer reciprocity, Theorem \[brauer\_rec\], that $\msf{C}_{A(\fp)} = \msf{C}_{A^K}$ in appropriate bases, where $\msf{C}_{A(\fp)}$ is the matrix of the Cartan map $\msf{c}_{A(\fp)}$ and $\msf{C}_{A^K}$ is the matrix of the Cartan map $\msf{c}_{A^K}$. Due to the linkage relation explained in §\[semiperfect\_rings\], the families of $A^K$ and of $A(\fp)$ are determined by the respective Cartan matrices. Since $\msf{C}_{A(\fp)} = \msf{C}_{A^K}$, it follows that $\#\msf{Bl}(A(\fp)) = \#\msf{Bl}(A^K)$, so $\fp \in \msf{BlGen}(A)$.
Suppose that $A$ has split fibers and that $R$ is noetherian. Then the fact that $\#\msf{Bl}(A(\fp)) = \#\msf{Bl}(Z(\fp))$ by Theorem \[mueller\_maintheorem\] together with the Lemma \[num\_blocks\_rad\_dim\] yields the following equivalence: $$\label{blgen_equivalent_condition_with_rad}
\fp \in \msf{BlGen}(A) \Longleftrightarrow \dim_K (Z^K + \Rad(A^K)) = \dim_{\msf{k}(\fp)} (Z(\fp) + \Rad(A(\fp))) \;.$$ Let $\sO$ be a perfect $A$-gate in $\fp$. This exists by Lemma \[dec\_theory\_assumptions\_lemma\]\[dec\_theory\_assumptions\_lemma:noeth\]. Suppose that $\fp \in \msf{DecGen}(A)$. In [@Thiel-Dec Theorem 2.2] we have proven that this implies that $$\dim_K \Rad(A^K) = \dim_{\msf{k}(\fp)} \Rad(A(\fp))\;.$$ Let $X \dopgleich Z + J$, where $J \dopgleich \Rad(A^K) \cap A^\sO$. The arguments in [@Thiel-Dec] show that $X$ is an $A^\sO$-lattice of $Z^K + \Rad(A^K)$ and that the reduction in the maximal ideal $\fm$ of $\sO$ is equal to $Z^\sO(\fm) + \Rad(A^\sO(\fm))$. We thus have $\dim_K(Z^K + \Rad(A^K)) = \dim_{\msf{k}(\fm)}(Z^\sO(\fm) + \Rad(A^\sO(\fm)))$. Since $A(\fp)$ splits, the $\msf{k}(\fm)$-dimension of $Z^\sO(\fm) + \Rad(A^\sO(\fm))$ is equal to the $\msf{k}(\fp)$-dimension of $Z(\fp) + \Rad(A(\fp))$. Hence, we have $\fp \in \msf{BlGen}(A)$ by (\[blgen\_equivalent\_condition\_with\_rad\]). This yields another proof of the inclusion $\msf{DecGen}(A) \subs \msf{BlGen}(A)$ in case $A$ has split fibers.
\[decgen\_eq\_blgen\_counterex\] The following example due to C. Bonnafé shows that in the generality of Theorem \[decgen\_blgen\_inclusion\_main\_theorem\] we do not have equality in (\[decgen\_blgen\_inclusion\]). Let $R$ be a discrete valuation ring with fraction field $K$ and uniformizer $\pi$, i.e., $\fp \dopgleich (\pi)$ is the maximal ideal of $R$. Denote by $k \dopgleich R/\fp$ the residue field in $\fp$. Let $$A \dopgleich \left\lbrace \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Mat_2(R) \mid b,c \in \fp \right\rbrace \;.$$ This is an $R$-subalgebra of $\Mat_2(R)$ and it is $R$-free with basis $$\label{decgen_eq_blgen_counterex_gens}
e \dopgleich E_{11} , \; f \dopgleich E_{22} , \; x \dopgleich \pi E_{12} , \; y \dopgleich \pi E_{21},$$ where $E_{ij} = (\delta_{i,k}\delta_{j,l})_{kl}$ is the elementary matrix. Clearly, $A^K = \Mat_{2}(K)$, so the generic fiber of $A$ is split semisimple. In particular, $A^K$ has just one block, and this block contains just one simple module we denote by $S$. Now, consider the specialization $\ol{A} \dopgleich A(\fp) = A/\fp A$. We know from Corollary \[algebra\_dvr\_semiperfect\] that the quotient map $A \twoheadrightarrow \ol{A}$, $a \mapsto \ol{a}$, is block bijective, so we must have $\#\msf{Bl}(A(\fp)) \leq \msf{Bl}(A^K)$ and therefore $\# \msf{Bl}(A(\fp)) = 1$, so $\fp \in \msf{BlGen}(A)$. Let $\ol{J}$ be the $k$-subspace of $\ol{A}$ generated by $\ol{x}$ and $\ol{y}$. This is in fact a two-sided ideal of $\ol{A}$ since it is stable under multiplication by the generators (\[decgen\_eq\_blgen\_counterex\_gens\]). Moreover, we have $\ol{x}^2 = 0 = \ol{y}^2$, so $\ol{J}$ is a nilpotent ideal of $\ol{A}$. Hence, $\dim_k \Rad(\ol{A}) \geq 2$. The number of simple modules of $\ol{A}$ is by [@Lam-First-Course-91 Theorem 7.17] equal to $\dim_k \ol{A}/(\Rad(\ol{A}) + \lbrack \ol{A},\ol{A} \rbrack)$, so $\#\Irr \ol{A} \leq 2$ since $\dim_k \ol{A} = \dim_K A^K = 4$. The two elements $\ol{e}$ and $\ol{f}$ are orthogonal idempotents and so the constituents of the two $\ol{A}$-modules $\ol{A} \ol{e}$ and $\ol{A}\ol{f}$ are non-isomorphic. So, we have $\# \Irr \ol{A} \geq 2$ and due to the aforementioned we conclude that $\#\Irr \ol{A} = 2$. Let $\ol{S}_1$ and $\ol{S}_2$ be these two simple modules. Since $R$ is a discrete valuation ring, reduction modulo $\fp$ yields the well-defined decomposition map $\msf{d}_A^\fp:\msf{G}_0(A^K) \rarr \msf{G}_0(A(\fp))$, see [@Thiel-Dec Corollary 1.14]. It is an elementary fact that the all simple $\ol{A}$-modules must be constituents of $\msf{d}_A^\fp(\lbrack S \rbrack) = \lbrack S/\fp S \rbrack$. Since $\dim_K S = 2$, the only possibility is that $\msf{d}_A^\fp(\lbrack S \rbrack) = \lbrack \ol{S}_1 \rbrack + \lbrack \ol{S}_2 \rbrack$ and $\dim_k \ol{S}_i = 1$. In particular, $\fp \notin \msf{DecGen}(A)$, so $\fp \in \msf{BlGen}(A) \setminus \msf{DecGen}(A)$. Finally, we note that $\ol{A}$ also splits since $\#\Irr(\ol{A}) = 2$ implies by the above formula that $\dim_k \Rad(\ol{A}) = 2$ and we have $\dim_k \ol{A} = \dim_k \Rad(\ol{A}) + \sum_{i=1}^2 (\dim_k \ol{S}_i)^2$, so $\ol{A}$ is split by [@Lam-First-Course-91 Corollary 7.8].
\[blgen\_decgen\_complement\_diagonal\] Assume that the $A^K$-families are singletons, and that $\#\Irr A(\fp) \leq \# \Irr A^K$ for all $\fp \in \Spec(R)$. Then $$\msf{BlGen}(A) \setminus \msf{DecGen}(A) = \lbrace \fp \in \Spec(R) \mid \msf{D}_A^\fp \tn{ is diagonal but not the identity } \rbrace \;.$$
Since the $A^K$-families are singletons, we have $\#\msf{Irr}(A^K) = \#\msf{Bl}(A^K)$. We clearly have $\#\Irr A(\fp) \geq \#\msf{Bl}(A(\fp))$ for all $\fp \in \Spec(R)$. Assume that $\fp \in \msf{BlGen}(A)$. Then we have $\#\Irr A(\fp) \geq \#\msf{Irr}(A^K)$, so $\#\Irr A(\fp)) = \#\Irr A^K$ by our assumption. Hence, the decomposition matrix $\msf{D}_A^\fp$ is quadratic. By Theorem \[dec\_block\_compat\] the $\fp$-families are equal to the Brauer $\fp$-families. Since $\fp \in \msf{BlGen}(A)$ and the $A^K$-families are singletons, it follows that $\msf{D}_A^\fp$ is a diagonal matrix. The claim is now obvious.
\[cellular\_alg\_blgen\_decgen\] Let $R$ be a noetherian integral domain with fraction field $K$ and let $A$ be a cellular $R$-algebra of finite dimension such that $A^K$ is semisimple. Then $\msf{DecGen}(A) = \msf{BlGen}(A)$.
First of all, specializations of $A$ are again cellular by [@Graham-Lehrer-Cellular 1.8]. Moreover, it follows from [@Graham-Lehrer-Cellular Proposition 3.2] that $A$ has split fibers, so $A$ satisfies Lemma \[dec\_theory\_assumptions\_lemma\]\[dec\_theory\_assumptions\_lemma:noeth\] and therefore our basic assumption in this paragraph. Let $\Lambda$ be the poset of the cellular structure of $A^K$. Since $A^K$ is semisimple, each cell module $M_\lambda$ has simple head $S_\lambda$ and $\#\Irr A^K = \#\Lambda$. Let $\fp \in \Spec(R)$. The poset for the cellular structure of $A(\fp)$ is again $\Lambda$. Denote by $M_\lambda^\fp$ the corresponding cell modules of $A(\fp)$. There is a subset $\Lambda'$ of $\Lambda$ such that $M_\lambda^\fp$ has simple head $S_\lambda^\fp$ for all $\lambda \in \Lambda'$ and that these heads are precisely the simple $A(\fp)$-modules. In particular, we have $\#\Irr A(\fp) \leq \#\Irr A^K$. Now, assume that $\fp \in \msf{BlGen}(A)$. By Lemma \[blgen\_decgen\_complement\_diagonal\] we just need to show that the decomposition matrix $\msf{D}_A^\fp$, which is square by the proof of Lemma \[blgen\_decgen\_complement\_diagonal\], cannot be a non-identity diagonal matrix. By [@Graham-Lehrer-Cellular Proposition 3.6] we know that $\lbrack M_\lambda: S_\lambda \rbrack = 1$ and $\lbrack M_\lambda^\fp:S_\lambda^\fp \rbrack = 1$. By construction, it is clear that $\msf{d}_A^\fp(\lbrack M_\lambda \rbrack) = \lbrack M_\lambda^\fp \rbrack$. Hence, if $\msf{d}_A^\fp(\lbrack S_\lambda \rbrack) = n_\lambda \lbrack S_\lambda^\fp \rbrack$, we have $n_\lambda = \lbrack M_\lambda^\fp:S_\lambda^\fp \rbrack = 1$. Hence, $\msf{D}_A^\fp$ is the identity matrix, so $\fp \in \msf{BlGen}(A)$.
The Brauer graph {#brauer_graph}
----------------
Geck and Pfeiffer [@Geck-Pfeiffer] have introduced the so-called Brauer $\fp$-graph of $A$ in our general context but assuming that $A^K$ is semisimple so that the $A^K$-families are singletons. For general $A$ this definition seems not to be the correct one. We introduce the following generalization of this concept.
Suppose that $R$ is normal so that we have unique decomposition maps. The of $A$ is the graph with vertices the simple $A^K$-modules and an edge between $S$ and $T$ if and only if in the $A^K$-family of $S$ there is some $S'$ and in the $A^K$-family of $T$ there is some $T'$ such that $\msf{d}_A^\fp(\lbrack S' \rbrack)$ and $\msf{d}_A^\fp(\lbrack T' \rbrack)$ have a common constituent. The connected components of this graph are called the of $A$.
If the $A^K$-families are singletons, we have an edge between $S$ and $T$ if and only if $\msf{d}_A^\fp(\lbrack S \rbrack)$ and $\msf{d}_A^\fp(\lbrack T \rbrack)$ have a common constituent, so this indeed generalizes the Brauer $\fp$-graph from [@Geck-Pfeiffer] for $A^K$ semisimple. Our final theorem shows that decomposition maps are compatible with $\fp$-families and $A(\fp)$-families, and relates the Brauer $\fp$-families to the $\fp$-families.
\[dec\_block\_compat\] Assume that $R$ is normal. The following holds:
\[dec\_block\_compat:a\] A finite-dimensional $A^K$-module $V$ belongs to a $\fp$-block of $A$ if and only if $\msf{d}_A^\fp(\lbrack V \rbrack)$ belongs to a block of $A(\fp)$.
\[dec\_block\_compat:b\] Two finite-dimensional $A^K$-modules $V$ and $W$ lie in the same $\fp$-block if and only if $\msf{d}_A^\fp(\lbrack V \rbrack)$ and $\msf{d}_A^\fp(\lbrack W \rbrack)$ lie in the same block of $A(\fp)$.
If $\mathcal{F} \in \msf{Fam}_\fp(A)$ is a $\fp$-family, then $$\msf{d}_A^\fp(\mathcal{F}) \dopgleich \lbrace T \mid T \tn{ is a constituent of } \msf{d}_A^\fp(\lbrack S \rbrack) \tn{ for some } S \in \mathcal{F} \rbrace$$ is a family of $A(\fp)$, and all families of $A(\fp)$ are obtained in this way.
The Brauer $\fp$-families are equal to the $\fp$-families.
By assumption there is a perfect $A$-gate $\sO$ in $\fp$. Let $\fm$ be the maximal ideal of $\sO$. We have the following commutative diagram of canonical morphisms which are all idempotent stable: $$\begin{tikzcd}
A_\fp \arrow[hookrightarrow]{r} \arrow[twoheadrightarrow]{d} & A^\sO \arrow[hookrightarrow]{r} \arrow[twoheadrightarrow]{d} & A^K \\
A(\fp) \arrow[hookrightarrow]{r} & A^\sO(\fm)
\end{tikzcd}$$ Since $R$ is assumed to be normal, it follows from Proposition \[unibranched\_block\_bijective\] that $A_\fp \twoheadrightarrow A(\fp)$ is block bijective. By assumption the morphism $\msf{d}_A^{\fp,\fm}: \rG_0(A(\fp)) \rarr \rG_0(A^\sO(\fm))$ is an isomorphism and therefore $A(\fp) \hookrightarrow A^\sO(\fm)$ is block bijective by Theorem \[faithfully\_flat\_ext\_block\_bij\]. Furthermore, by assumption the generic fiber $A^K$ is split and therefore $A^\sO \twoheadrightarrow A^\sO(\fm)$ is block bijective by Corollary \[algebra\_dvr\_semiperfect\]. Because of (\[get\_more\_blocks\_equation\]) it thus follows that $A_\fp \hookrightarrow A^\sO$ is block bijective.
Now, let $V$ be a finite-dimensional $A^K$-module and let $\wt{V}$ be an $A^\sO$-lattice of $V$. Suppose that $V$ belongs to an $A_\fp$-block of $A^K$. Since $A_\fp \hookrightarrow A^\sO$ is block bijective, the $A_\fp$-blocks of $A^K$ coincide with the $A^\sO$-blocks of $A^K$ and therefore $V$ belongs to an $A^\sO$-block of $A^K$. Since $\wt{V}$ is $\sO$-free, it follows from Lemma \[block\_extension\_compatibility\] that $\wt{V}$ belongs to a block of $A^\sO$. Again by Lemma \[block\_extension\_compatibility\] and the fact that $A^\sO \twoheadrightarrow A^\sO(\fm)$ is block bijective, it follows that $\wt{V}/\fm \wt{V}$ belongs to a block of $A^\sO(\fm)$. Since $A(\fp) \hookrightarrow A^\sO(\fm)$ is block bijective, Lemma \[block\_extension\_compatibility\] shows that $\msf{d}_A^\fp(\lbrack V \rbrack)$ belongs to a block of $A(\fp)$.
Conversely, suppose that $\msf{d}_A^\fp(\lbrack V \rbrack)$ belongs to a block of $A(\fp)$. Then $\wt{V}/\fm \wt{V}$ belongs to a block of $A^\sO(\fm)$ and therefore $\wt{V}$ belongs to a block of $A^\sO$ by Lemma \[block\_extension\_compatibility\]. But then $V$ belongs to an $A^\sO$-block of $A^K$ and thus to an $A_\fp$-block of $A^K$ by Lemma \[block\_extension\_compatibility\].
This follows now from part \[dec\_block\_compat:a\].
Fix a $\fp$-family $\mathcal{F}$ of $A^K$. If $S \in \mathcal{F}$, then $\msf{d}_A^\fp(\lbrack S \rbrack)$ belongs to an $A(\fp)$-block by \[dec\_block\_compat:a\] and therefore all constituents of $\msf{d}_A^\fp(\lbrack S \rbrack)$ belong to a fixed family $\ol{\mathcal{F}}_S$. If $S' \in \mathcal{F}$ is another simple module, then by \[dec\_block\_compat:b\] the constituents of $\msf{d}_A^\fp(\lbrack S' \rbrack)$ also lie in $\ol{\mathcal{F}}_S$. Hence, $\msf{d}_A^\fp(\mathcal{F})$ is contained in a fixed $A(\fp)$-family $\ol{\mathcal{F}}$. Let $T \in \ol{\mathcal{F}}$ be arbitrary. Due to the properties of decomposition maps there is some $S \in \Irr A^K$ such that $T $ is a constituent of $\msf{d}_A^\fp(\lbrack S \rbrack)$. Since $T$ and $\msf{d}_A^\fp(\lbrack S \rbrack)$ lie in the same $A(\fp)$-block by \[dec\_block\_compat:a\] and \[dec\_block\_compat:b\], we must have $S \in \mathcal{F}$ by \[dec\_block\_compat:b\]. Hence, $\ol{\mathcal{F}} = \msf{d}_A^\fp(\mathcal{F})$ is an $A(\fp)$-family. Since every simple $A(\fp)$-module is a constituent of $\msf{d}_A^\fp(\lbrack S \rbrack)$ for some simple $A^K$-module $S$, it is clear that any $A(\fp)$-family is of the form $\msf{d}_A^\fp(\mathcal{F})$ for a $\fp$-family $\mathcal{F}$.
Let $S$ and $T$ be simple $A^K$-modules contained in the same Brauer $\fp$-family, i.e., in the $A^K$-family of $S$ there is some $S'$ and in the $A^K$-family of $T$ there is some $T'$ such that $\msf{d}_A^\fp(\lbrack S' \rbrack)$ and $\msf{d}_A^\fp(\lbrack T' \rbrack)$ have a common constituent. It follows from part \[dec\_block\_compat:b\] that $S'$ and $T'$ lie in the same $\fp$-family of $A^K$. Since $S'$ is in the same $A^K$-family as $S$, it is also in the same $\fp$-family as $S$ because the $\fp$-families are unions of $A^K$-families. Similarly, $T'$ is in the same $\fp$-family as $T$. Hence, $S$ and $T$ lie in the same $\fp$-family.
Conversely, suppose that $S$ and $T$ lie in the same $\fp$-family. We have to show that they lie in the same Brauer $\fp$-family. Let $(S_i)_{i=1}^n$ be a system of representatives of the isomorphism classes of simple $A^K$-modules and let $(U_j)_{j=1}^m$ be a system of representatives of the isomorphism classes of simple $A(\fp)$-modules. Let $\sQ \dopgleich (Q_i)_{i=1}^n$ with $Q_i$ being the projective cover of $S_i$, and let $\sP \dopgleich (P_j)_{j=1}^m$ with $P_j$ being the projective cover of $U_j$. Let $\msf{C}_{A(\fp)}$ be the matrix of the Cartan map $\msf{c}_{A(\fp)}$ with respect to the chosen bases, and similarly let $\msf{C}_{A^K}$ be the matrix of $\msf{c}_{A^K}$. Furthermore, let $\msf{D}_A^\fp$ be the matrix of $\msf{d}_A^\fp$ with respect to the chosen bases. Since $\msf{C}_{A(\fp)} = \msf{D}_A^\fp \msf{C}_{A^K} (\msf{D}_A^\fp)^{\rT}$ by Brauer reciprocity, Theorem \[brauer\_rec\], we have $$\label{brauer_p_fam_thm_equ}
(\msf{C}_{A(\fp)})_{p,q} = (\msf{D}_A^\fp \msf{C}_{A^K} (\msf{D}_A^\fp)^{\rT})_{p,q} = \sum_{k,l=1}^n (\msf{D}_A^\fp)_{p,k} (\msf{C}_{A^K})_{k,l} (\msf{D}_A^{\fp})_{q,l}$$ for all $p,q$. Let $U$ be a constituent of $\msf{d}_A^\fp(\lbrack S \rbrack)$ and let $V$ be a constituent of $\msf{d}_A^\fp(\lbrack T \rbrack)$. Since $S$ and $T$ lie in the same $\fp$-family of $A^K$, both $\msf{d}_A^\fp(\lbrack S \rbrack)$ and $\msf{d}_A^\fp(\lbrack T \rbrack)$ lie in the same block of $A(\fp)$ by \[dec\_block\_compat:b\], and therefore $U$ and $V$ lie in the same family of $A(\fp)$. As the families of $A(\fp)$ are equal to the $\sP$-families of $A(\fp)$ by §\[semiperfect\_rings\], there exist functions $f: \lbrack 1,r \rbrack \rarr \lbrack 1,m \rbrack$, $g: \lbrack 1,r-1 \rbrack \rarr \lbrack 1,m \rbrack$ with the following properties: $U_{f(1)} = U$, $U_{f(r)} = V$, and for any $j \in \lbrack 1,r-1 \rbrack$ both $P_{f(j)}$ and $P_{f(j+1)}$ have $U_{g(j)}$ as a constituent. We can visualize the situation as follows: $$\begin{tikzcd}[column sep=tiny]
P_{f(j)} & & P_{f(j+1)} \\
U_{f(j)} \arrow{u} & U_{g(j)} \arrow{ul} \arrow{ur} & U_{f(j+1)} \arrow{u}
\end{tikzcd}$$ where an arrow $U \rarr P$ signifies that $U$ is a constituent of $P$. For any $j \in \lbrack 1,r-1 \rbrack$ we have $(\msf{C}_{A(\fp)})_{g(j),f(j)} \neq 0$ and so it follows from (\[brauer\_p\_fam\_thm\_equ\]) that there are indices $k(j)$ and $l(j)$ such that $$(\msf{D}_A^\fp)_{g(j),k(j)} \neq 0\;, \quad (\msf{C}_{A^K})_{k(j),l(j)} \neq 0 \;, \quad (\msf{D}_A^\fp)_{f(j),l(j)} \neq 0 \;.$$ Similarly, since $(\msf{C}_{A(\fp)})_{g(j),f(j+1)} \neq 0$, there exist indices $k'(j)$ and $l'(j)$ such that $$(\msf{D}_A^\fp)_{g(j),k'(j)} \neq 0\;, \quad (\msf{C}_{A^K})_{k'(j),l'(j)} \neq 0 \;, \quad (\msf{D}_A^\fp)_{f(j+1),l'(j)} \neq 0 \;.$$ This can be visualized as follows: $$\begin{tikzcd}[column sep=tiny]
\msf{d}_A^\fp(\lbrack S_{l(j)} \rbrack) \arrow[dashed,-]{rr} & & \msf{d}_A^\fp(\lbrack S_{k(j)} \rbrack) & & \msf{d}_A^\fp(\lbrack S_{k'(j)} \rbrack) \arrow[dashed,-]{rr} & & \msf{d}_A^\fp(\lbrack S_{l'(j)} \rbrack) \\
U_{f(j)} \arrow{u} & & & U_{g(j)} \arrow{ul} \arrow{ur} & & & U_{f(j+1)} \arrow{u}
\end{tikzcd}$$ Here, the dashed edges in the upper row signify that the respective simple $A^K$-modules lie in the same $A^K$-family. Since $U_{f(1)} = U$ and $U_{f(r)} = V$, this shows that $S$ and $T$ lie in the same Brauer $\fp$-family of $A^K$.
Questions and open problems {#open_problems}
===========================
Are all atomic gluing loci already maximal atomic gluing loci? A related question is: if a gluing locus $Z$ is properly contained in another locus $Z'$, is then $Z$ an irreducible component of $Z'$?
Are the block structures in all vertices of the atomic block graph distinct?
Are the intersections of atomic gluing loci always irreducible?
What graph theoretic properties can be proven about the atomic block graph?
Are there “nice” conditions on $A$ implying that the irreducible components of $\msf{BlEx}(A)$ are normal or even smooth?
How can we explicitly describe the complement $\msf{BlGen}(A) \setminus \msf{DecGen}(A)$? Which conditions ensure that it is empty?
Notes
=====
The behavior of blocks under specialization has been studied in several situations already. All of our results are well-known in modular representation theory of finite groups since the work of R. Brauer and C. Nesbitt [@Brauer-Nesbitt]. Our Corollary \[normal\_split\_corollary\] and Theorem \[dec\_block\_compat\] generalize results by S. Donkin and R. Tange [@Donkin-Tange-Brauer] about algebras over Dedekind domains. Our results about lower semicontinuity of the number of blocks generalize a result by P. Gabriel [@Gabriel-finite-rep-type-open] to mixed characteristic and non-algebraically closed settings, see also the corresponding result by I. Gordon [@Gordon:Representations-of-semisimple-Lie-in-pos]. In general, K. Brown and I. Gordon [@BG-Ramification; @BG-Ramification2] used Müller’s theorem [@Mul-Localization-in-non-commu-0] to study blocks under specialization. Theorem \[mueller\_maintheorem\] has been treated in a more special setting by K. Brown and K. Goodearl [@Brown-Goodearl-Quantum-Groups]. The codimension one property in Corollary \[normal\_split\_corollary\_details\] and Theorem \[maintheorem\_blex\_descr\] were proven by C. Bonnafé and R. Rouquier [@BR-cellules] in a more special setting. Their work is without doubt one of the main motivations for this paper. Blocks and decomposition matrices of generically semisimple algebras over discrete valuation rings have been studied by M. Geck and G. Pfeiffer [@Geck-Pfeiffer], and more generally by M. Chlouveraki [@Chlouveraki:2009aa]. Brauer reciprocity has been studied more generally by M. Geck and R. Rouquier [@Geck-Rouquier-Dec], and by M. Neunhöffer [@Neunhoeffer-PhD]. M. Neunhöffer and S. Scherotzke [@Neunhoeffer-Scherotzke] have shown generic triviality of $\msf{e}_A^\fp$ over Dedekind domains.
Base change of blocks {#appendix_blocks_base_change}
=====================
In this appendix we collect several facts about base change of blocks. Some results here should also be of independent interest.
Existence of block decompositions
---------------------------------
\[block\_dec\_reflection\] Let $\phi:R \rarr S$ be a morphism of commutative rings and let $A$ be an $R$-algebra. Suppose that $\phi_A:A \rarr A^S$ is central idempotent stable. If $A^S$ has a block decomposition, also $A$ has a block decomposition.
If $A$ does not contain any non-trivial central idempotent, then $A$ is indecomposable and thus has a block decomposition. So, assume that $A$ is not indecomposable and let $c$ be a non-trivial central idempotent. Then $A = Ac \oplus Ac^\dagger$. We can now continue this process to get finer and finer decompositions of $A$ as a ring. Since $\phi_A$ is central idempotent stable, we get decompositions of the same size of $A^S$. As $A^S$ has a block decomposition, this process has to end after finitely many steps. We thus arrive at a ring decomposition of $A$ with finitely many and indecomposable factors, hence, at a block decomposition of $A$.
\[finite\_flat\_int\_block\_dec\] A non-zero finite flat algebra over an integral domain has a block decomposition.
Let $R$ be an integral domain with fraction field $K$, let $\phi:R \hookrightarrow K$ be the embedding, and let $A$ be a finite flat $R$-algebra. Since $A$ is $R$-torsion-free, it follows from Lemma \[phi\_injective\_lemma\]\[phi\_injective\_lemma:localiz\] that $\phi_A$ is injective and so $\phi_A$ is idempotent stable by Lemma \[idempotent\_stable\]\[idempotent\_stable:inj\]. Since $\phi_A^*A = A^K$ is a finite-dimensional algebra over a field, it has a block decomposition. Hence, $A$ has a block decomposition by Lemma \[block\_dec\_reflection\].
The important point of the corollary above is that we do *not* have to assume $R$ to be noetherian—otherwise $A$ is noetherian and we already know it has a block decomposition.
Block compatibility of scalar extension of modules
--------------------------------------------------
Recall the decomposition of the module category of a ring $A$ relative to a decomposition in $\mscr{E}_c(A)$ described in §\[families\]. We have the following compatibility.
\[block\_extension\_compatibility\] Let $\phi:R \rarr S$ be a morphism of commutative rings and let $A$ be an $R$-algebra. Suppose that $\phi_A$ is central idempotent stable and let $V$ be a non-zero $A$-module. In any of the following cases the $A$-module $V$ belongs to the block $c_i$ if and only if the $A^S$-module $V^S$ belongs to the $\phi$-block $\phi_A(c_i)$:
$\phi$ is injective and $V$ is $R$-projective.
$\phi$ is faithfully flat.
$R$ is local or a principal ideal domain and $V$ is $R$-free.
As $c_jV$ is a direct summand of $V$, it follows that we have a canonical isomorphism $\msf{ext}_V^S(c_j V) \simeq \phi_A^*(c_jV)$. By definition, it is clear that $\msf{ext}_V^S(c_jV) = \phi_A(c_j)\phi_A^*V$ and so we have a canonical isomorphism $\phi_A^*(c_j V) \simeq \phi_A(c_j)\phi_A^*V $ of $A^S$-modules for all $j$. The claim thus holds if we can show that no non-zero direct summand $V'$ of $V$ is killed by $\phi_A^*$, i.e., $\phi_A^*V' \neq 0$. But this is implied by the assumptions in each case. Namely, in the first two cases it follows from Lemma \[phi\_injective\_lemma\] that $\phi_V$ is injective, which implies that $\phi_{V'}$ is also injective, so $\phi_A^*V'$ cannot be zero for non-zero $V'$. In the third case neither $\phi$ nor $\phi_V$ need to be injective, so this needs extra care. First of all, since $V$ is assumed to be $R$-free, the assumptions on $R$ imply that a direct summand $V'$ of $V$, which a priori is only $R$-projective, is already $R$-free, too. In case $R$ is local, this follows from Kaplansky’s theorem [@Kaplansky] and in case $R$ is a principal ideal domain, this is a standard fact. Now, if $V$ is $R$-free with basis $( v_\lambda )_{\lambda \in \Lambda}$, then it is a standard fact (see [@Bou-Algebra-1-3 II, §5.1, Proposition 4]) that $\phi_A^*V$ is $S$-free with basis $(\phi_V(v_\lambda))_{\lambda \in \Lambda}$. This shows that $\phi_A^*V \neq 0$ for any non-zero $R$-free $A$-module $V$. This applied to direct summands of $V$, which are $R$-free as shown, proves the claim.
Field extensions {#blocks_field_extensions}
----------------
Throughout this paragraph let $A$ be a finite-dimensional algebra over a field $K$. From (\[get\_more\_blocks\_equation\]) we know that $\#\msf{Bl}(A) \leq \#\msf{Bl}(A^L)$ for any extension field $L$ of $K$.
\[block\_split\_def\] We say that $A$ is if $\#\msf{Bl}(A) = \#\msf{Bl}(A^L)$ for any extension field $L$ of $K$.
Our aim is to show the following lemma.
\[split\_center\_lemma\] If $\msf{Z}(A)$ is a split $K$-algebra (e.g., if $A$ itself splits), then $A$ is block-split. The converse holds if $K$ is perfect.
The first assertion of the lemma is essentially obvious since $\msf{Z}(A)$ is semiperfect and therefore $$\label{bl_irr_equation}
\#\msf{Bl}(A) = \#\msf{Bl}(\msf{Z}(A)) = \msf{rk}_\bbZ \ \msf{K}_0(\msf{Z}(A)) = \#\msf{rk}_\bbZ \ \msf{G}_0(\msf{Z}(A)) = \#\Irr \msf{Z}(A) \;,$$ where the second equality follows from the fact that idempotents in a commutative ring are isomorphic if and only if they are equal, see [@Lam-First-Course-91 Ex. 22.2]. The same equalities of course also hold for $\msf{Z}(A)^L = \msf{Z}(A^L)$, where $L$ is an extension field of $K$. Hence, if $\msf{Z}(A)$ is split, then $A$ is block-split. If $A$ itself is split, it is a standard fact that its center splits, so $A$ is block-split.\
We will prove the converse (assuming that $K$ is perfect) from a more general point of view as the results might be of independent interest and we re-use some of them in the last section. First of all, the field extension $K \subs L$ induces natural group morphisms $$\msf{d}_A^L:\msf{G}_0(A) \rarr \msf{G}_0(A^L) \quad \tn{and} \quad
\msf{e}_A^L:\msf{K}_0(A) \rarr \msf{K}_0(A^L) \;.$$ Without any assumptions on the field $K$ we have the following property.
\[d\_e\_injective\] The morphisms $\msf{d}_A^L$ and $\msf{e}_A^L$ are injective.
Let $(S_i)_{i \in I}$ be a system of representatives of the isomorphism classes of simple $A$-modules. For each $i$ let $(T_{ij})_{j \in J_i}$ be a system of representatives of the isomorphism classes of simple $A^L$-modules which occur as constituents of $S_i^L$. Then by [@Lam-First-Course-91 Proposition 7.13] the set $(T_{ij})_{i \in I, j \in J_i}$ is a system of representatives of the isomorphism classes of simple $A^L$-modules. Hence, the matrix $\msf{D}_A^L$ of $\msf{d}_A^L$ in bases given by the isomorphism classes of simple modules is in column-echelon form, has no zero columns, and no zero rows. In particular, $\msf{d}_A^L$ is injective.
For each $i \in I$ let $P_i$ be the projective cover of $S_i$ and for each $j \in J_i$ let $Q_{ij}$ be the projective cover of $T_{ij}$. By the above, $(Q_{ij})_{i \in I, j \in J_i}$ is a system of representatives of the isomorphism classes of projective indecomposable $A^L$-modules. We claim that in the direct sum decomposition of the finitely generated projective $A^L$-module $P_i^L$ into projective indecomposable $A^L$-modules only the $Q_{ij}$ with $j \in J_i$ occur. With the same argument as above, this implies that $\msf{e}_A^L$ is injective. So, let us write $P_i^L = \bigoplus_{\lambda \in \Lambda} U_\lambda$ for (not necessarily non-isomorphic) projective indecomposable $A^L$-modules $U_\lambda$. The $U_\lambda$ are the up to isomorphism unique projective indecomposable $A^L$-modules occurring as direct summands of $P_i^L$. As the radical is additive by [@Lam-First-Course-91 Proposition 24.6(ii)], we have $\Rad(P_i^L) = \bigoplus_{\lambda \in \Lambda} \Rad(U_\lambda)$, so $$S_i^L = (P_i/\Rad(P_i))^L = P_i^L/\Rad(P_i)^L = \bigoplus_{\lambda \in \Lambda} U_\lambda/(\Rad(P_i)^L \cap U_\lambda) \;.$$ Moreover, we have $\Rad(P_i)^L \subs \Rad(P_i^L)$. This follows from the fact that $\Rad(A)^L \subs \Rad(A^L)$ by [@Lam-First-Course-91 Theorem 5.14] and the fact that $\Rad(P_i) = \Rad(A) P_i$ and $\Rad(P_i^L) = \Rad(A^L) P_i^L$ by [@Lam-First-Course-91 Theorem 24.7] since $P_i$ and $P_i^L$ are projective. For each $\lambda \in \Lambda$ the radical of $U_\lambda$ is a proper submodule of $U_\lambda$ and therefore $$\Rad(P_i)^L \cap U_\lambda \subs \Rad(P_i^L) \cap U_\lambda = \Rad(U_\lambda) \subsetneq U_\lambda \;.$$ Hence, the head of $U_\lambda$ is a constituent of $U_\lambda/(\Rad(P_i^L) \cap U_\lambda)$, and since all constituents of the latter are constituents of $S_i^L$, we must have $\msf{Hd}(U_\lambda) \simeq S_{ij_{\lambda}}$ for some $j_\lambda \in J_i$ by the above. This implies that $U_\lambda = Q_{ij_\lambda}$, thus proving the claim.
\[d\_iso\_iff\_e\_iso\] The following holds:
The morphism $\msf{d}_A^L$ is an isomorphism if and only if it induces a bijection between isomorphism classes of simple modules. Similarly, the morphism $\msf{e}_A^L$ is an isomorphism if and only if it induces a bijection between isomorphism classes of projective indecomposable modules.
\[d\_iso\_iff\_e\_iso:reflect\] If $\msf{d}_A^L$ is an isomorphism, so is $\msf{e}_A^L$. The converse holds if $K$ is perfect.
For the proof of Lemma \[d\_iso\_iff\_e\_iso\] we will need the following well-known elementary lemma that is also used in the last section. Recall from (\[intertwining\_form\_def\]) the intertwining form $\langle \cdot,\cdot \rangle_A$ of $A$.
\[intertwining\_nondeg\] Let $P$ be a projective indecomposable $A$-module and let $V$ be a finitely generated $A$-module. Then $$\langle \lbrack P \rbrack, \lbrack V \rbrack \rangle_A = \lbrack V:\msf{Hd}(P) \rbrack \cdot \dim_K \End_A(\msf{Hd}(P)) \;,$$ where $\msf{Hd}(P) = P/\Rad(P)$ is the head of $P$. In particular, $\langle \cdot, \cdot \rangle_A$ is non-degenerate.
We first consider the case $V=\msf{Hd}(P)$. Let $f \in \Hom_A(P,\Hd(P))$ be non-zero. Since $\Hd(P)$ is simple, this morphism is already surjective and thus induces an isomorphism $P/\Ker(f) \cong \Hd(P)$. But as $\Rad(P)$ is the unique maximal submodule of $P$, we must have $\Ker(f) = \Rad(P)$ and thus get an induced morphism $\Hd(P) \rarr \Hd(P)$. This yields a $K$-linear morphism $\Phi: \Hom_A(P,\Hd(P)) \rarr \End_A(\Hd(P))$. On the other hand, if $f \in \End_A(\Hd(P))$, then composing it with the quotient morphism $P \rarr P/\Rad(P) = \Hd(P)$ yields a morphism $P \rarr \Hd(P)$. In this way we also get a $K$-linear morphism $\Psi:\End_A(\Hd(P)) \rarr \Hom_A(P,\Hd(P))$. By construction, $\Phi$ and $\Psi$ are pairwise inverse, hence $\langle \lbrack P \rbrack, \lbrack \msf{Hd}(P) \rbrack \rangle_A = \dim_K \Hom_A(P,\msf{Hd}(P)) = \dim_K \End_A(\Hd(P))$ as claimed.
Now, suppose that $V$ is a simple $A$-module not isomorphic to $\msf{Hd}(P)$. We can write $P = Ae$ for some primitive idempotent $e \in A$. Since $A$ is artinian, $e$ is already local and now it follows from [@Lam-First-Course-91 21.19] that $\Hom_A(Ae, V)$ is non-zero if and only if $V$ has a constituent isomorphic to $\Hd(Ae)$. This is not true by assumption, and therefore $\Hom_A(P,V) = 0$, so $\langle \lbrack P \rbrack, \lbrack V \rbrack \rangle_A = 0$.
Finally, for $V$ general we have $
\lbrack V \rbrack = \sum_{S \in \Irr A} \lbrack V : S \rbrack \lbrack S \rbrack$ in $\msf{G}_0(A)$. By the above we get $$\begin{aligned}
\langle \lbrack P \rbrack, \lbrack V \rbrack \rangle_A &= \sum_{S \in \Irr A} \lbrack V:S \rbrack \langle \lbrack P \rbrack, \lbrack S \rbrack \rangle_A = \lbrack V:\Hd(P) \rbrack \langle \lbrack P \rbrack, \lbrack \Hd(P) \rbrack \rangle_A \\ &= \lbrack V : \Hd(P) \rbrack \cdot \dim_K \End_A(\Hd(P)) \end{aligned}$$ as claimed. It follows that the Gram matrix $\mscr{G}$ of $\langle \cdot,\cdot \rangle$ with respect to the basis $(\msf{P}(S))_{S \in \Irr A}$ of $\msf{K}_0(A)$ and the basis $(S)_{S \in \Irr A}$ of $\msf{G}_0(A)$ is diagonal with positive diagonal entries. The determinant of $\mscr{G}$ is thus a non-zero divisor on $\bbZ$ and since both $\msf{K}_0(A)$ and $\msf{G}_0(A)$ are $\bbZ$-free of the same finite dimension, it follows that $\langle \cdot,\cdot \rangle_A$ is non-degenerate, see [@Scheja-Storch-Alg-2 Satz 70.5].
We use the same notations as in the proof of Lemma \[d\_e\_injective\]. Since $_AA$ is a projective $A$-module, there is a decomposition $_AA = \bigoplus_{i \in I} P_i^{r_i}$ for some $r_i \in \bbN$. Using Lemma \[intertwining\_nondeg\] we see that $$\begin{aligned}
\dim_K \Hd(P_j) & = \langle \lbrack _AA \rbrack, \lbrack \Hd(P_j) \rbrack \rangle_A = \sum_{i \in I} r_i \langle \lbrack P_i \rbrack, \lbrack \Hd(P_j) \rbrack \rangle_A = r_j \langle \lbrack P_j \rbrack, \lbrack \Hd(P_j) \rbrack \rangle_A \\
& = r_j \dim_K \End_A(\Hd(P_i)) \;.\end{aligned}$$ Hence, $r_i = \frac{n_i}{m_i}$, where $n_i \dopgleich \dim_K S_i$ and $m_i \dopgleich \dim_K \End_A(S_i)$. In particular, $$\label{d_iso_iff_e_iso:equ}
\dim_K A = \sum_{i \in I} \frac{n_i}{m_i} \dim_K P_i \;.$$
Now, suppose that $\msf{d}_A^L$ is an isomorphism. Then clearly $\#\Irr A = \#\Irr A^L$. The properties of the matrix $\msf{D}_A^L$ of the morphism $\msf{d}_A^L$ derived in the proof of Lemma \[d\_e\_injective\] immediately imply that $\msf{D}_A^L$ is diagonal. Since it is invertible with natural numbers on the diagonal, it must already be the identity matrix, i.e., $\msf{d}_A^L$ induces a bijection between the isomorphism classes of simple modules. In particular, $(S_i^L)_{i \in I}$ is a system of representatives of the isomorphism classes of simple $A^L$-modules. The properties of the matrix $\msf{E}_A^L$ of $\msf{e}_A^L$ derived in the proof of Lemma \[d\_e\_injective\] now imply that we must have $P_i^L \simeq Q_i^{s_i}$ for some $s_i \in \bbN$. We argue that $s_i = 1$. This shows that $\msf{e}_A^L$ is an isomorphism inducing a bijection between the isomorphism classes of projective indecomposable modules. In the same way we deduced equation (\[d\_iso\_iff\_e\_iso:equ\]) we now get $$\label{d_iso_iff_e_iso:equ2}
\dim_K A = \dim_L A^L = \sum_{i \in I} \frac{n_i'}{m_i'} \dim_L Q_i$$ with $$n_i' = \dim_L \Hd(Q_i) = \dim_L S_i^L = \dim_K S_i = n_i$$ and $$m_i' = \dim_L \End_{A^L}(\Hd(Q_i)) = \dim_L \End_{A^L}(S_i^L) = \dim_K \End_K(S_i) = m_i \;,$$ using the fact that $L \otimes_K \End_A(S_i) \simeq \End_{A^L}(S_i^L)$, see Lemma [@Rei-Maximal-orders-0 Theorem 2.38]. Since $\dim_L Q_i \leq \dim_L P_i^L = \dim_K P_i$, equations (\[d\_iso\_iff\_e\_iso:equ\]) and (\[d\_iso\_iff\_e\_iso:equ2\]) imply that $\dim_L Q_i = \dim_K P_i$, so $Q_i = P_i^L$.
Conversely, suppose that $\msf{e}_A^L$ is an isomorphism. With the properties of the matrix $\msf{E}_A^L$ of $\msf{e}_A^L$ established in the proof of Lemma \[d\_e\_injective\] we see similarly as above that $\msf{e}_A^L$ already induces a bijection between the projective indecomposable modules. In particular, $P_i^L \simeq Q_i$. Due to the properties of the matrix $\msf{D}_A^L$ of $\msf{d}_A^L$ established in the proof of Lemma \[d\_e\_injective\] the only constituent of $S_i^L$ is $T_i$. Since $P_i$ is the projective cover of $P_i$, we have a surjective morphism $\varphi:P_i \twoheadrightarrow S_i$ with $\Ker(\varphi) = \Rad(P_i)$. Scalar extension induces a surjective morphism $\varphi^L:P_i^L \twoheadrightarrow S_i^L$ with $\Ker(\varphi^L) = \Ker(\varphi)^L = \Rad(P_i)^L \subs \Rad(P_i^L)$. It thus follows from [@CR-Methods-1 Corollary 6.25(i)] that $P_i^L$ is the projective cover of $S_i^L$. Now, we assume that $K$ is perfect. Then by [@CR-Methods-1 Theorem 7.5] all simple $A$-modules are separable, so $S_i^L = T_i^{s_i}$ for some $s_i$. Since projective covers are additive, we get $P_i^L = Q_i^{s_i}$. As $P_i^L = Q_i$, this implies that $s_i=1$, so $S_i^L = T_i$ is simple. Hence, $\msf{d}_A^L$ induces a bijection between the isomorphism classes of simple modules.
With the same arguments as in the proof of Lemma \[d\_iso\_iff\_e\_iso\] we can show that the converse in Lemma \[d\_iso\_iff\_e\_iso\]\[d\_iso\_iff\_e\_iso:reflect\] still holds when we only assume that all simple $A$-modules are separable, i.e., they remain semisimple under field extension. This holds for example when $A$ splits or if $A$ is a group algebra (over any field). We do not know whether it holds more generally.
Let $Z \dopgleich \msf{Z}(A)$. Suppose that $L$ is an extension field of $K$ with $\# \msf{Bl}(A) = \#\msf{Bl}(A^L)$. By (\[bl\_irr\_equation\]) we know that $\#\Irr Z = \#\Irr Z^L$. The arguments in the proof of Lemma \[d\_e\_injective\] thus imply that the matrix $\msf{D}_A^L$ of the morphism $\msf{d}_Z^L:\msf{G}_0(Z) \rarr \msf{G}_0(Z^L)$ must be a diagonal matrix. We claim that it is the identity matrix. Since this holds for any $L$, it means that the simple modules of $Z$ remain simple under any field extension, so $Z$ splits. Our assumption implies that $\#\msf{Idem}_p(Z) = \#\msf{Idem}_p(Z^L)$, so every primitive idempotent $e \in Z$ remains primitive in $Z^L$. This shows that $\msf{e}_A^L:\msf{K}_0(Z) \rarr \msf{K}_0(Z^L)$ induces a bijection between projective indecomposable modules. In particular, it is an isomorphism. Now, Lemma \[d\_iso\_iff\_e\_iso\] shows that also $\msf{d}_A^L$ is an isomorphism. Since its matrix $\msf{D}_A^L$ is invertible with natural numbers on the diagonal, it must be the identity.
In the proof of Lemma \[split\_center\_lemma\] we have deduced that for a commutative finite-dimensional $K$-algebra $Z$ the condition $\msf{rk}_\bbZ \ \msf{K}_0(Z) = \msf{rk}_\bbZ \ \msf{K}_0(Z^L)$ already implies that $\msf{e}_Z^L$ induces a bijection between projective indecomposable modules. This follows from the fact that idempotents in a commutative ring are isomorphic if and only if they are equal. This is not true for a non-commutative ring $A$. Here, we can have $\msf{rk}_\bbZ \ \msf{K}_0(A) = \msf{rk}_\bbZ \ \msf{K}_0(A^L)$ but still a primitive idempotent $e \in A$ can split into a sum of *isomorphic* orthogonal primitive idempotents of $A^L$. Then the matrix $\msf{E}_A^L$ of $\msf{e}_A^L$ is diagonal but not the identity.
Let us record the following additional fact:
\[num\_blocks\_rad\_dim\] If $\msf{Z}(A)$ splits, then $$\#\msf{Bl}(A) = \dim_K \msf{Z}(A) - \dim_K \Rad(\msf{Z}(A)) = \dim_K \msf{Z}(A) - \dim_K (\msf{Z}(A) \cap \Rad(\msf{Z}(A)) )\;.$$
This follows immediately from (\[bl\_irr\_equation\]) and the fact that that $\Rad(\msf{Z}(A)) = \msf{Z}(A) \cap \Rad(A)$ since $\msf{Z}(A) \subs A$ is a finite normalizing extension, see [@Lor-Finite-normalizing Theorem 1.5].
Faithfully flat extensions {#ff_extensions_bb}
--------------------------
We will need the following general result.
\[faithfully\_flat\_ext\_block\_bij\] Let $\phi: R \hookrightarrow S$ be a faithfully flat morphism of integral domains and let $A$ be a finite flat $R$-algebra. Let $K$ and $L$ be the fraction field of $R$ and $S$, respectively. If $\#\msf{Bl}(A^K) = \#\msf{Bl}(A^L)$, then the morphism $\phi_A:A \rarr A^S$ is block bijective.
Recall from Corollary \[finite\_flat\_int\_block\_dec\] that both $A$ and $A^S$ have block decompositions. The map $\phi_A: A \rarr A^S$ is injective by Lemma \[phi\_injective\_lemma\]\[phi\_injective\_lemma:faithfully\_flat\] since $\phi$ is faithfully flat. Hence, $\phi_A$ is idempotent stable by Lemma \[idempotent\_stable\]\[idempotent\_stable:inj\] and therefore $\# \msf{Bl}(A) \leq \#\msf{Bl}(A^S)$ by (\[get\_more\_blocks\_equation\]). We thus have to show that $\# \msf{Bl}(A) \geq \#\msf{Bl}(A^S)$. We split the proof of this fact into several steps.
The case $R=K$ and $S=L$ holds by assumption. Now, assume that still $R=K$ but that $S$ is general as in the theorem. Since $A$ is $R$-flat, the extension $A^S$ is $S$-flat and thus $S$-torsionfree. Hence, the map $A^S \rarr A^L$ is injective by Lemma \[phi\_injective\_lemma\]\[phi\_injective\_lemma:localiz\]. In particular, it is idempotent stable by Lemma \[idempotent\_stable\]\[idempotent\_stable:inj\] and so $\#\msf{Bl}(A^S) \leq \#\msf{Bl}(A^L)$ by (\[get\_more\_blocks\_equation\]). In total, we have $\#\msf{Bl}(A) \leq \# \msf{Bl}(A^S) \leq \# \msf{Bl}(A^L) = \#\msf{Bl}(A^K) = \#\msf{Bl}(A)$. Hence, $\# \msf{Bl}(A) = \# \msf{Bl}(A^S)$.
Finally, let both $R$ and $S$ be general as in the theorem. Let $\Sigma \dopgleich R \setminus \lbrace 0 \rbrace$ and $\Omega \dopgleich S \setminus \lbrace 0 \rbrace$. Then $K = \Sigma^{-1}R$ and $L = \Omega^{-1}S$. Set $T \dopgleich \Sigma^{-1}S$. Since $R$ and $S$ are integral domains, we can naturally view all rings as subrings of $L$ and so we get the two commutative diagrams $$\begin{tikzcd}[column sep=small, row sep=small]
& L & & & A^L \\
& T \arrow[hookrightarrow]{u} & & & A^T \arrow{u} \\
K \arrow[hookrightarrow]{ur} \arrow[hookrightarrow, bend left]{uur} & & S \arrow[hookrightarrow]{ul} \arrow[hookrightarrow, bend right]{uul} & A^K \arrow[bend left]{uur} \arrow{ur} & & A^S \arrow{ul} \arrow[bend right]{uul} \\
& R \arrow[hookrightarrow]{ur} \arrow[hookrightarrow]{ul} & & & A \arrow{ur} \arrow{ul}
\end{tikzcd}$$ the right one being induced by the left one. All morphisms in the left diagram are clearly injective. We claim the same holds for the right diagram. We have noted at the beginning that the map $A \rarr A^S$ is injective. Since $A$ is $R$-flat, it is $R$-torsionfree and so the map $A \rarr A^K$ is injective by Lemma \[phi\_injective\_lemma\]\[phi\_injective\_lemma:localiz\]. We have argued above already that the map $A^S \rarr A^L$ is injective. Since $S \hookrightarrow T$ is a localization map, the induced scalar extension functor is exact so that $A^T$ is a flat $T$-module. In particular, $A^T$ is $T$-torsionfree and so $A^T \rarr A^L$ is injective by \[phi\_injective\_lemma\]\[phi\_injective\_lemma:localiz\]. The map $A^K \rarr A^L$ is injective by Lemma \[phi\_injective\_lemma\]\[phi\_injective\_lemma:proj\]. Due to the commutativity of the diagram, the remaining maps must be injective, too. We can thus view all scalar extensions of $A$ naturally as subsets of $A^L$. We claim that $$\label{a_eq_ak_cap_as}
A = A^K \cap A^S$$ as subsets of $A^L$. Because of the commutative diagram above, this intersection already takes place in $A^T$. Consider $A^K$ as an $R$-module now. We have a natural identification $$\phi^*(A^K) = S \otimes_R A^K = S \otimes_R (\Sigma^{-1} A) = (\Sigma^{-1} S) \otimes_R A = T \otimes_R A = A^T$$ as $S$-modules by [@Bou-Commutative-Algebra-1-7 II, §2.7, Proposition 18]. Note that the map $A^K \rarr A^T$ in the diagram above is the map $\phi_{A^K}$, when considering $A^K$ as an $R$-module. The $R$-submodule $A$ of $A^K$ is now identified with $\phi_{A^K}(A)$ and $\msf{ext}_{A^K}^S(A)$ is the $S$-submodule of $A^T$ generated by $A \subs A^T$, which is precisely $A^S$. Since $\phi$ is faithfully flat, it follows from [@Bou-Commutative-Algebra-1-7 I, §3.5, Proposition 10(ii)] applied to the $R$-module $A^K$ and the submodule $A$ that $$A = \phi_{A^K}(A) = \phi_{A^K}(A^K) \cap \msf{ext}_{A^K}^S(A) = A^K \cap A^S$$ inside $A^T$. Let $(c_i)_{i \in I} \in \sE_{\mrm{cp}}(A^S)$ and let $(d_j)_{j \in J} \in \sE_{\mrm{cp}}(A^K)$. By assumption the morphism $A^K \rarr A^L$ is block bijective, which means that $(d_j)_{j \in J} \in \sE_{\mrm{cp}}(A^L)$. Since $A^S \rarr A^L$ is idempotent stable, there exists by the arguments preceding (\[get\_more\_blocks\_equation\]) a partition $(J_i)_{i \in I}$ of $J$ such that the non-zero central idempotent $c_i$ can in $A^L$ be written as $c_i = \sum_{j \in J_i} d_j$. But this shows that $c_i \in A^K \cap A^S$, hence $c_i \in A$ and so $(c_i)_{i \in I} \in \sE_{\mrm{cp}}(A)$ by (\[a\_eq\_ak\_cap\_as\]). Hence, $\# \msf{Bl}(A) = \# \msf{Bl}(A^S)$.
Reductions
----------
Now, we consider a situation which in a sense is opposite to the one considered in the last paragraph, namely we consider the quotient morphism $\phi:R \twoheadrightarrow R/\fm \gleichdop S$ for a local commutative ring $R$ with maximal ideal $\fm$ and a finitely generated $R$-algebra $A$. By Lemma \[idempotent\_stable\]\[idempotent\_stable:surj\] the morphism $\phi_A:A \twoheadrightarrow A^S \simeq A/\fm A \gleichdop \ol{A}$ is idempotent stable. The question whether $\phi_A$ is idempotent surjective is precisely the question whether idempotents of $\ol{A}$ can be lifted to $A$, and this is a classical topic in ring theory. The following lemma is standard, we omit the proof.
\[idempotent\_surjective\_block\_bijective\] If $\phi_A:A \twoheadrightarrow \ol{A}$ is idempotent surjective, it is primitive idempotent bijective and block bijective.
The next theorem was proven by M. Neunhöffer [@Neunhoeffer-PhD Proposition 5.10].
\[neunhoeffer\_semiperfect\] The morphism $\phi_A:A \twoheadrightarrow \ol{A}$ is idempotent surjective if and only if $A$ is semiperfect.
We recall two standard situations of idempotent surjective reductions.
\[semiperfect\_std\_settings\] In the following two cases the morphism $\phi_A:A \twoheadrightarrow \ol{A}$ is idempotent surjective:
\[semiperfect\_std\_settings:complete\] $R$ is noetherian and $\fm$-adically complete.
\[semiperfect\_std\_settings:hensel\] $R$ is henselian.
For a proof of the first case, see [@Lam-First-Course-91 Proposition 21.34]. For a proof of the second case assuming that $A$ is *commutative*, see [@Raynaud:Henselian I, §3, Proposition 2]. To give a proof for non-commutative $A$ let $\ol{e} \in \ol{A}$ be an idempotent. Let $k \dopgleich R/\fm$ and let $\ol{B} \dopgleich k \lbrack \ol{e} \rbrack$ be the $k$-subalgebra of $\ol{A}$ generated by $\ol{e}$. Since $\ol{A}$ is a finite-dimensional $k$-algebra, also $\ol{B}$ is finite-dimensional. Moreover, $\ol{B}$ is commutative. Let $e \in A$ be an arbitrary element with $\phi_A(e) = \ol{e}$. Let $B \dopgleich R \lbrack e \rbrack$, a commutative subalgebra of $A$. Note that $\ol{B} = B/\fm B$. Since $A$ is a finitely generated $R$-module, the Cayley–Hamilton theorem implies that $B$ is a finitely generated $R$-algebra. Now, by the commutative case, the map $\phi_B:B \twoheadrightarrow \ol{B}$ is idempotent surjective and so there is an idempotent $e' \in B \subs A$ with $\phi_A(e') = \phi_B(e') = \ol{e}$. This shows that $\phi_A$ is idempotent surjective.
The next theorem was again proven by M. Neunhöffer [@Neunhoeffer-PhD Proposition 6.2]. It is one of our key ingredients in proving Brauer reciprocity for decomposition maps in a general setting.
\[neunhoeffer\_theorem\] Suppose that $R$ is a valuation ring with fraction field $K$ and that $A$ is a finite flat $R$-algebra with split generic fiber $A^K$. If $\wh{R} \otimes_R A$ is semiperfect, where $\wh{R}$ is the completion of $R$ with respect to the topology defined by a valuation on $K$ defining $R$, then also $A$ is semiperfect.
\[algebra\_dvr\_semiperfect\] Suppose that $R$ is a *discrete* valuation ring and that $A$ is a finite flat $R$-algebra with split generic fiber. Then $A$ is semiperfect. In particular, $\phi_A:A \twoheadrightarrow \ol{A}$ is primitive idempotent bijective and block bijective.
Since $R$ is a discrete valuation ring, its valuation topology coincides with its $\fm$-adic topology so that the topological completion $\wh{R}$ is $\wh{\fm}$-adically complete, where $\fm$ denotes the maximal ideal of $R$ and $\wh{\fm}$ denotes the maximal ideal of $\wh{R}$. Hence, $\wh{R} \otimes_R A$ is semiperfect by Lemma \[semiperfect\_std\_settings\]\[semiperfect\_std\_settings:complete\] and Theorem \[neunhoeffer\_semiperfect\]. Now, Theorem \[neunhoeffer\_theorem\] shows that $A$ is also semiperfect.
One part of Corollary \[algebra\_dvr\_semiperfect\], the fact that idempotents lift, was also stated earlier by C. Curtis and I. Reiner [@CR-Methods-1 Exercise 6.16] in an exercise in the special case where $A^K$ is assumed to be semisimple. The semisimplicity assumption was later removed by J. Müller in his PhD thesis [@Muller:PhD Satz 3.4.1] using the Wedderburn–Malcev theorem (this can be applied without perfectness assumption on the base field if $A^K$ splits since then $A^K/\Rad(A^K)$ is separable, see [@Curtis-Reiner-Associative-Algebras Theorem 72.19]).
Three further elementary facts
==============================
\[flat\_faithfully\_flat\] A finitely generated module $M$ over an integral domain $R$ is flat if and only if it is faithfully flat. In particular, if $M \neq 0$, we have $0 \neq \msf{k}(\fp) \otimes_R M = M(\fp)$ for all $\fp \in \Spec(R)$.
We can assume that $M \neq 0$. Since $M$ is flat, it is torsion-free and so the localization map $M \rarr M_\fp$ is injective, see Lemma \[phi\_injective\_lemma\]\[phi\_injective\_lemma:localiz\]. Hence, $M_\fp \neq 0$. Since $M$ is a finitely generated $R$-module, also $M_\fp$ is a finitely generated $R_\fp$-module and now Nakayama’s lemma implies that $0 \neq M_\fp/\fp_\fp M_\fp = \msf{k}(\fp) \otimes_R M$. Hence, $M$ is faithfully flat by [@Mat-Commutative Theorem 7.2].
\[ff\_center\_ext\] Let $A$ be a finite flat algebra over an integral domain $R$. Then the structure map $R \rarr A$, $r \mapsto r \cdot 1_A$, is injective. Hence, we can identify $R \subs \msf{Z}(A)$. If $R$ is noetherian, the induced map $\Upsilon: \Spec(\msf{Z}(A)) \rarr \Spec(R)$ is finite, closed, and surjective.
It follows from Lemma \[flat\_faithfully\_flat\] that $A$ is already faithfully flat. Let $\phi:R \rarr A$ be the structure map. This is an $R$-module map and applying $- \otimes_R A$ yields a map $$A \simeq R \otimes_R A \overset{\phi \otimes_R A}{\longrightarrow} A \otimes_R A$$ of right $A$-modules, mapping $a$ to $1 \otimes a$. This map has an obvious section mapping $a \otimes a'$ to $aa'$, hence it is injective. Since $A$ is faithfully flat, the original map $\phi$ has to be injective, too. As the image of $\phi$ is contained in the center $Z$ of $A$, the structure map is actually an injective map $R \hookrightarrow Z$. Now, assume that $R$ is noetherian. Since $A$ is a finitely generated $R$-module, also $Z$ is a finitely generated $R$-module. Hence, $R \subs Z$ is a finite ring extension and now it is an elementary fact that $\Upsilon$ is closed and surjective.
The following lemma about base change of homomorphism spaces is well known but we could not find a reference in this generality (see [@Bou-Commutative-Algebra-1-7 II, §5.3] for a proof in case of a *commutative* base ring).
\[base\_ring\_change\_of\_hom\] Let $A$ be an algebra over a commutative ring $R$ and let $\phi:R \rarr S$ be a morphism into a commutative ring $S$. Let $V$ and $W$ be $A$-modules. If $V$ is finitely generated and projective as an $A$-module, then there is a canonical $S$-module isomorphism. $$S \otimes_R \Hom_A(V,W) \simeq \Hom_{A^S}(V^S,W^S) \;.$$
We can define a map $\gamma:S \otimes_R \Hom_A(V,W) \rarr \Hom_{A^S}(V^S,W^S)$ by mapping $s \otimes f$ with $s \in S$ and $f \in \Hom_A(V,W)$ to $s_r \otimes f$, where $s_r$ denotes right multiplication by $s$. It is a standard fact that this is an $S$-module morphism, see [@Rei-Maximal-orders-0 (2.36)]). Recall that $\Hom_A(-,W)$ commutes with *finite* direct sums by [@Bou-Algebra-1-3 II, §1.6, Corollary 1 to Proposition 6]. This shows that the canonical isomorphism $\Hom_A(A,W) \simeq W$ induces a canonical isomorphism $\Hom_A(A^n,W) \simeq W^n$ for any $n \in \bbN$ and now we conclude that there is a canonical isomorphism $$S \otimes_R \Hom_A(A^n,W) \simeq S \otimes_R W^n \simeq (S \otimes_R W)^n \simeq \Hom_{A^S}( (A^S)^n, W^S) \;,$$ which is easily seen to be equal to $\gamma$. The assertion thus holds for finitely generated *free* $A$-modules. Now, the assumption on $V$ allows us to write without loss of generality $A^n = V \oplus X$ for some $A$-module $X$. It is not hard to see that we get a commutative diagram $$\begin{tikzcd}[column sep=0.5ex]
S \otimes_R \Hom_A(A^n,W) \arrow{rrrrrrrrrrrrrrr}{\simeq} \arrow{d}[swap]{\simeq} &&&&&&&&&&&&&&& \left( S \otimes_R \Hom_A(V,W) \right) \arrow{d} & \oplus & \left( S \otimes_R \Hom_A(X,W) \right) \arrow{d} \\
\Hom_{A^S}( (A^S)^n, W^S) \arrow{rrrrrrrrrrrrrrr}{\simeq} &&&&&&&&&&&&&&& \left( \Hom_{A^S}(V^S,W^S) \right) & \oplus & \left( \Hom_{A^S}(X^S, W^S) \right)
\end{tikzcd}$$ where the horizontal morphisms are obtained by the projections and the vertical morphisms are the morphisms $\gamma$ in the respective situation. The commutativity of this diagram implies that the morphism $S \otimes_R \Hom_A(V,W) \rarr \Hom_{A^S}(V^S,W^S)$ also has to be an isomorphism.
|
---
abstract: 'We propose three models for the traffic of vehicles within a network formed by sites (cities, car-rental agencies, parking lots, etc.) and connected by two-way arteries (roads, highways), that allow forecasting the vehicular flux in a sequence of $n$ consecutive steps, or units of time. An essential approach consists in using, as an *a priori* information, previous observations and measurements. The formal tools used in our analysis consists in: (1) associating a digraph to the network where the edges correspond to arteries and the vertices with loops represent the sites. (2) From an initial set of numbers, that are the distribution of vehicles within the network, we construct a matrix that we transform into a stochastic matrix (SM) by normalizing the rows, whose entries are now transition probabilities. This matrix becomes the generator of the evolution of the traffic flow. And (3), we use the Perron-Frobenius theory for a formal analysis. We investigate three models: (a) a closed four-site network having a conserved number of vehicles; (b) to this network we add an influx and an outflux of vehicles to characterize an open system; asymptotically, $n \rightarrow \infty$, the SM raised to the power $n$ goes to a unique stationary matrix. And (c), we construct a nonlinear model because the formal structure permits the existence of several ($L$) stationary states for the distribution of vehicles at each site, that alternate cyclically with time. Each state represents the traffic for $L$ different moments. These models were used to analyze the traffic in a sector of the city of Tigre, located in the province of Buenos Aires, Argentina. The results are presented in a following paper.'
author:
- 'D. Otero$^{1}$, D. Galetti$^{2}$, and S. S. Mizrahi$^{3}$'
title: 'Modeling a vehicular traffic network. Part I'
---
Introduction
============
Understanding the structure and systematic of the flow of vehicles along a network of arteries is a crucial point for planning the construction of connections (roads and highways) linking sites, as is the need to forecast the vehicular flow for traffic control. For that aim the observation and the gathering of data are essential practices that have to go along with theoretical approaches. Mathematical modeling and numerical simulations are essential procedures to grasp at the traffic flow and jam problems in order to recommend solutions for the urban and inter-city vehicular mobility. Methods have been developed since the 1940 decade, see for instance the seminal papers [@pipes1953; @lighthill1955] and references therein. Further studies were done in several places uninterruptedly and many articles and reports were published on the subject, among which we cite, for instance, the Refs. [@bando1994; @bando19951; @bando19952; @kerner1997; @helbing1998; @nagatani1998; @deangelis1999; @aw2000; @helbing2001; @nagatani2002; @bellomo2002; @greenberg2004; @kisselev2004; @gasser2004; @orosz2005; @li2006; @siebel2006; @guanghan2009; @bressan2011; @bressan2012; @goh2012; @forstall2013; @bressan2014; @canec2015; @bressan2016; @piccoli; @morein], and the books [@kerner2004; @manhke2009].
The success of a model depends essentially on its capability to describe the present traffic dynamics as well as to forecast its trend. There are approaches that make use of fluid dynamics and differential equations [@bando19951; @helbing1998] whereas others utilize the $n$-step evolution of a stochastic matrix [@manhke2009; @ching2006; @woess-PF]. Here we consider the vehicular circulation occurring essentially inter-sites, and by sites we mean towns, parking lots, car-rental agencies, etc. In this study we do not describe the features of the vehicles, as mean size, mean speed, stopping distance nor the characteristics of the arteries as length, width and number of lanes. It consists in schematizing a traffic network imaged by a digraph, as shown in Fig. \[fig01\], that could be scaled up to higher complexity. The mathematical elements we utilize are essentially the digraph methods and the Perron-Frobenius theory for stochastic matrices, where each $n$-step entry is a Markov chain. We first present two linear models: (1) a network consisting of four sites, each one connected to all the others by two-way arteries. The total number of circulating vehicles is assumed to be constant in time. (2) Thereafter, we extend that network introducing an input of new vehicles and an output of old ones; see the digraph in Fig. \[fig03\]. To each digraph one associates a square matrix $\mathbb{A}$ whose entries contain the information obtained empirically, namely, the number of vehicles established by counts. To forecast the distribution of the vehicles at future moments in the network we normalize the rows of $\mathbb{A}$ to 1, thus getting a matrix $\mathbb{M}$. Its entries are the fractions of vehicles in sites and roads. In this way, the predictions will be probabilistic and we assume that the daily change of the distribution obeys an evolution law based on an $n$-step process – $n$ standing for the discretized time. This approach constitutes an adaptation of a model proposed in Ref. [@harary1966] for human migrations. We present the theory, work out illustrative numerical examples and analyze the results. Additionally, we extend our analysis by examining a third model, (3) a nonlinear $n$-step process; by non-linearity we mean that some entries of matrix $\mathbb{M}$ depend on $n$. With a specific choice of periodic functions for the entries we verify the existence of several stationary régimes (instead of a single one as it occurs in the linear models), such that the vehicle distributions change cyclically instead of remaining constant. This nonlinear approach enables more detailed predictions about the traffic dynamics because it permits to slice, for instance, a 24-hour day into several sub-periods of traffic observation in the stationary régimes.
The present theoretical study was used to analyze the urban traffic in a selected sector of Tigre, a city located in the province of Buenos Aires, Argentina. The raw data are counts of circulating vehicles as recorded by cameras positioned in several intersections. The methodology, analysis and comparison between raw data and the utilized model are presented in a following paper, to be referred as Part II.
Model I: Inter-site circulation of vehicles between without input or output.
============================================================================
The first model consists of four sites, to which we attribute the letters $A, B, C, D$, connected by arteries (roads, highways, etc.) and it contains a fixed number of vehicles; in short it is a conservative network. Pictorially we represent the network by a digraph, as shown in Fig. \[fig01\],
![[]{data-label="fig01"}](Digrafosistfechado.jpg){height="2.4in" width="3.0in"}
where each *loop*, associated with a *vertex*, corresponds to a site that accommodates a certain number of vehicles, whereas each *edge* is a link between the vertices that contains a number of vehicles in transit from one site to the other along the direction of the arrows. To the digraph of Fig. \[fig01\] one associates the array shown in Table \[A0\],
– $A$ $B$ $C$ $D$ Sum of the lines entries
---------------------------- ------------- ------------- ------------- ------------- --------------------------
$A$ $R_{1}$ $P_{1}$ $P_{5}$ $Q_{4}$
$B$ $Q_{1}$ $R_{2}$ $P_{2}$ $P_{6}$
$C$ $Q_{5}$ $Q_{2}$ $R_{3}$ $P_{3}$
$D$ $P_{4}$ $Q_{6}$ $Q_{3}$ $R_{4}$
Sum of the columns entries ${\ T}_{1}$ ${\ T}_{2}$ ${\ T}_{3}$ ${\ T}_{4}$
–
: []{data-label="A0"}
the sum of the entries of each row, $U_{i}$, and the sum of the entries of each column, $T_{i}$, are presented in Table \[A0\], and in the last row one finds the sums for the total number of vehicles $Y$ in the network.
The core of the array of Table \[A0\] is a $4\times 4$ matrix $$\mathbb{A}=\left(
\begin{array}{cccc}
R_{1} & P_{1} & P_{5} & Q_{4} \\
Q_{1} & R_{2} & P_{2} & P_{6} \\
Q_{5} & Q_{2} & R_{3} & P_{3} \\
P_{4} & Q_{6} & Q_{3} & R_{4}\end{array}\right) , \label{A1}$$ whose entries are the number vehicles observed and counted in an *ad hoc* interval of time or an average over several previous observations.
Assuming that we want to forecast the evolution of the distribution of vehicles in the network at a daily basis, for instance, we adopt the causal interpretation based on the hypothesis that the vehicles distribution evolves according to a $n$-step process: (*a*) The entries in the main diagonal in matrix (\[A1\]), $A_{ii}=R_{i}$, stand for the number of vehicles at site $i$; (*b*) the off diagonal entries $A_{ij}$ ($i\neq j$) are for the number of vehicles that left site $i$ and are on their way to site $j$. For a period of 24 hours the sum of the entries of row $i$, $U_{i}=\sum_{j}A_{ij}$ represents the number of vehicles that remained in or left the site $i$ for the other sites. Complementarily, $T_{j}=\sum_{i}A_{ij}$ represents the number of vehicles that are in site $j$ plus those that have departed from the other sites and are on their way to arrive at it. So, we may consider that matrix (\[A1\]) represents a continuous observation for a 24-hour period, or an average over sub-periods of observations, for instance, six minutes sample at every two hours.
The Stochastic Matrix
======================
By normalizing the rows of matrix (\[A1\]) we construct a stochastic matrix (SM) to be adopted as the generator of the evolution that could describe and forecast the number of vehicles in the sites plus those in transit. Writing the parameters for the first row as $$r_{1}=\frac{R_{1}}{U_{1}},\quad p_{1}=\frac{P_{1}}{U_{1}},\quad p_{5}=\frac{P_{5}}{U_{1}},\quad q_{4}=\frac{Q_{4}}{U_{1}}, \label{normline}$$ the same goes for the other rows, which are used to construct the transition probability matrix $$\mathbb{M}=\left(
\begin{array}{cccc}
r_{1} & p_{1} & p_{5} & q_{4} \\
q_{1} & r_{2} & p_{2} & p_{6} \\
q_{5} & q_{2} & r_{3} & p_{3} \\
p_{4} & q_{6} & q_{3} & r_{4}\end{array}\right)\ . \label{A2}$$ The sum of the (non-negative) entries of each row is $1$ whereas the sum of the entries of each column is not necessarily $1$, however, if additionally the sum of the entries of each column happens also to be $1$ then the matrix is said to be *doubly stochastic*. In more realistic instances the entries of a SM, as (\[A2\]), can be constructed using empirical data collected from previous observations and $$\mathbb{U}^{\mathrm{T}}\left( 0\right) =\left(
\begin{array}{cccc}
U_{1}\left( 0\right) & U_{2}\left( 0\right) & U_{3}\left( 0\right) &
U_{4}\left( 0\right)\end{array}\right) , \label{A22}$$ represents an initial state for the distribution of vehicles; the superscript $\mathrm{T}$ stands for transposition. The $n$-step process evolves of state (\[A22\]) as $$\mathbb{U}^{\mathrm{T}}(n)= \mathbb{U}^{\mathrm{T}}(0)\mathbb{M}^{n} \ ,
\label{A24}$$ that is presumed to forecast the distribution of vehicles at the $n$-th moment.
Properties of stochastic matrices (SM)
--------------------------------------
**(1)** The sum of the components of the vector $\mathbb{U}(n)$ is a conserved quantity, $$\sum_{i}U_{i} {\left( n\right) }=\sum_{i}U_{i}{\left( 0\right) } \quad
n=1,2,3,...\ . \label{A25}$$ because $\mathbb{M}^{n}$ is a SM [@harary1966].
**(2)** The SM $\mathbb{M}\geq 0$, of dimension $N \times N$, has eigenvalues $\lambda _{1}=1$ and $\left\vert \lambda
_{2}\right\vert ,\left\vert \lambda _{3}\right\vert ,...,\left\vert \lambda
_{N}\right\vert <1$, therefore $\lim_{n\rightarrow \infty }\left\vert
\lambda _{k}\right\vert ^{n}=0$, for $k\neq 1$. Eigenvalue $\lambda _{1}$ is known in the literature as the Perron-Frobenius (PF) eigenvalue [@ching2006].
**(3)** If all the eigenvalues of $\mathbb{M}$ have multiplicity 1 with linearly independent eigenvectors $\mathbb{X}^{\left( 1\right) }$, $\mathbb{X}^{\left( 2\right) }$, ..., $\mathbb{X}^{\left( N\right) }$, we construct the matrix $$\mathbb{Q}=\left(
\begin{array}{cccc}
\mathbb{X}^{\left( 1\right) } & \mathbb{X}^{\left( 2\right) } & \cdots &
\mathbb{X}^{\left( N\right) }\end{array}\right) , \label{A3}$$ and write the decomposition of $\mathbb{M}$ and its powers as $$\mathbb{M}=\mathbb{QDQ}^{-1},\quad \text{and\quad }\mathbb{M}^{n}=\mathbb{QD}^{n}\mathbb{Q}^{-1}, \label{A8}$$ where $\mathbb{D}$ is a diagonal matrix whose entries are the eigenvalues and $\mathrm{Tr}\mathbb{M}^n = \sum_{k=1}^{N}\left(\lambda _{k}\right)^n$ since the eigenvalue equation $\mathbb{M}^{n}\mathbb{X}^{\left( k\right) }=\left( \lambda _{k}\right) ^{n}\mathbb{X}^{\left( k\right) }$ holds. The spectral decomposition of $\mathbb{M}^{n}$ in terms of the stationary state and the decaying modes is $$\mathbb{M}^{n}=\mathbb{C}_{1}+\sum_{l=2}^{N}\left( \lambda _{l}\right) ^{n}\mathbb{C}_{l}. \label{B28}$$ The matrix $\mathbb{C}_{1}$ (associated with the PF eigenvalue) is a stationary SM; the other matrices $\mathbb{C}_{l}$, $l>1$, are not stochastic, having however the property $\mathrm{Tr}\left( \mathbb{C}_{l}\right) =1$. Since $\left\vert \lambda
_{l}\right\vert <1$, then $\lim_{n\rightarrow \infty }\mathbb{M}^{n}=\mathbb{C}_{1}$, which is asymptotically irreversible, $\mathrm{\det }\left( \mathbb{C}_{1}\right) =0$. The eigenvalues $\lambda _{l} $ can be associated with characteristic relaxation times: we write $\lambda _{l}=\mathrm{sgn}\left( \lambda _{l}\right) $ $\left\vert \lambda _{l}\right\vert
$, where $\mathrm{sgn}\left( \cdot \right) $ is the sign of the argument. Therefore, we define the characteristic decay time as $T_{l}=-\left( \ln \left\vert \lambda _{l}\right\vert \right) ^{-1}$, such that Eq. (\[B28\]) can be written as $$\mathbb{M}^{n}=\mathbb{C}_{1}+\sum_{l=2}^{N}\left[ \mathrm{sgn}\left(
\lambda _{l}\right) \right] ^{n}\mathbb{C}_{l}\exp \left( -\frac{n}{T_{l}}\right) . \label{B29}$$ Thereafter, one can write the evolution of the vector $\mathbb{U}^{\mathrm{T}}\left( 0\right) $, $\mathbb{U}^{\mathrm{T}}\left( n\right) =\mathbb{U}^{\mathrm{T}}\left( 0\right) \mathbb{M}^{n}$ as $$\mathbb{U}^{\mathrm{T}}\left( n\right) =\mathbb{U}^{\mathrm{T}}\left(
0\right) \mathbb{C}_{1}+\sum_{l=2}^{N}\left[ \mathrm{sgn}\left( \lambda
_{l}\right) \right] ^{n}\left( \mathbb{U^{\mathrm{T}}}\left( 0\right)
\mathbb{C}_{l}\right) \exp \left( -\frac{n}{T_{l}}\right) , \label{B30}$$ where $\mathbb{U}^{\mathrm{T}}\left( 0\right) \mathbb{C}_{1}=\lim_{n\rightarrow \infty }\mathbb{U}^{\mathrm{T}}\left( n\right) $ is the asymptotic distribution of vehicles and $\left[ \mathrm{sgn}\left( \lambda _{l}\right) \right] ^{n}\times \left( \mathbb{U^{\mathrm{T}}}\left( 0\right) \mathbb{C}_{l}\right) \exp \left( -n/T_{l}\right) $ are ($N-1$) partial distributions in the transient regimes and the matrices can have negative entries, although the entries of the vectors $\mathbb{U}^{\mathrm{T}}\left( 0\right)
\mathbb{C}_{1} $ and $\mathbb{U}^{\mathrm{T}}\left( n\right) $ are positive. We illustrate the theory through an example. Let us consider the SM $$\mathbb{M}=\left(
\begin{array}{ccc}
1/2 & {0} & 1/2 \\
1/4 & 1/2& 1/4\\
4/10& 3/10& 3/10\end{array}\right) \ , \label{C1}$$ whose eigenvectors and eigenvalues are
\[C2\] $$\begin{aligned}
\mathbb{X}^{\left( 1\right) } &=&\left(
\begin{array}{r}
{0.577350} \\
{0.577350} \\
{0.577350}\end{array}\right) ,\quad {\ \lambda }_{1}{\ =1},\quad \text{{\ stationary mode,}}
\label{C1a} \\
\mathbb{X}^{\left( 2\right) } &=&\left(
\begin{array}{r}
{0.576331} \\
-{0.802915} \\
-{0.152215}\end{array}\right) ,\quad {\ \lambda }_{2}{\ =0.367945},{\ \quad }\text{{\ decaying
mode,}} \label{C2c} \\
\mathbb{X}^{\left( 3\right) } &=&\left(
\begin{array}{r}
{0.660268} \\
{0.039495} \\
-{0.749991}\end{array}\right) ,\quad {\ \lambda }_{3}{\ =-0.067945,\quad }\text{{\ decaying mode.}}
\label{C2e}\end{aligned}$$
The matrix $\mathbb{M}$ is regular because $\mathbb{M}^{2}$ is already positive (all entries are positive). We construct the matrix $\left( \begin{array}{ccc}
\mathbb{X}^{\left( 1\right) } & \mathbb{X}^{\left( 2\right) } & \mathbb{X}^{\left( 3\right) } \end{array}\right)$, $$\mathbb{Q}=\left(
\begin{array}{rrr}
{0.577350} & {0.576331} & {0.660268} \\
{0.577350} & -{0.802915} & {0.039495} \\
{0.577350} & -{0.152215} & -{0.749991}\end{array}\right) , \label{C4}$$ and $\mathbb{D}=\mathrm{Diag}[1.0,0.367945,-0.067945]$. The matrix $\mathbb{M}^{n}=\mathbb{QD}^{n}\mathbb{Q}^{-1}=\left(
\begin{array}{ccc}
\mathbb{Y}^{\left( 1,n\right) } & \mathbb{Y}^{\left( 2,n\right) } & \mathbb{Y}^{\left( 3,n\right) }\end{array}\right)$ so that the three columns are $$\mathbb{Y}^{\left( 1,n\right) }=\underbrace{\frac{{\ 11}}{{\ 27}}\left(
\begin{array}{c}
{\ 1} \\
{\ 1} \\
{\ 1}\end{array}\right) }_{\text{stationary mode}}+\underbrace{\left(
\begin{array}{r}
{0.304793} \\
{-0.424622} \\
{-0.326908}\end{array}\right) \left( {\lambda _{2}}\right) ^{n}+\left(
\begin{array}{r}
{0.287800} \\
{0.017215} \\
{-0.326908}\end{array}\right) \left( {\ \lambda _{3}}\right) ^{n}}_{\text{ decaying modes}}
\label{C11}$$ $$\mathbb{Y}^{\left( 2,n\right) }=\underbrace{\frac{{\ 2}}{{\ 9}}\left(
\begin{array}{c}
1 \\
1 \\
1\end{array}\right) }_{\text{stationary mode}}+\underbrace{\left(
\begin{array}{r}
{-0.544452} \\
{0.758503} \\
{0.143795}\end{array}\right) \left( {\ \lambda _{2}}\right) ^{n}+\left(
\begin{array}{r}
{0.322230} \\
{0.019275} \\
{-0.366017}\end{array}\right) \left( {\ \lambda _{3}}\right) ^{n}}_{\text{decaying modes}}
\label{C12}$$ $$\mathbb{Y}^{\left( 3,n\right) }=\underbrace{\frac{{\ 10}}{{\ 27}}\left(
\begin{array}{c}
1 \\
1 \\
1\end{array}\right) }_{\text{stationary mode}}+\underbrace{\left(
\begin{array}{r}
{0.239659} \\
{-0.333880} \\
{-0.063296}\end{array}\right) \left( {\ \lambda _{2}}\right) ^{n}+\left(
\begin{array}{r}
{-0.610029} \\
{-0.036490} \\
{0.692926}\end{array}\right) \left( {\ \lambda _{3}}\right) ^{n}}_{\text{decaying modes}}\ .
\label{C13}$$ We further write the spectral decomposition of matrix (\[C1\]) in terms of the modes or regimes $\mathbb{M}^{n}\mathbb{=C}_{1}+\mathbb{C}_{2}\left( {\
\lambda _{2}}\right) ^{n}+\mathbb{C}_{3}\left( {\ \lambda _{3}}\right) ^{n}$, where the three matrices and their determinants are $$\mathbb{C}_{1}=\left(
\begin{array}{ccc}
11/27 & 2/9 & 10/27 \\
11/27 & 2/9 & 10/27 \\
11/27 & 2/9 & 10/27\end{array}\right) ;\quad \det \mathbb{C}_{1}=0, \label{C15}$$ $$\mathbb{C}_{2}=\left(
\begin{array}{rrr}
{\ 0.304793} & {\ -0.544452} & {\ 0.239659} \\
{\ -0.424622} & {\ 0.758503} & {\ -0.333880} \\
{\ -0.326908} & {\ 0.143795} & {\ -0.063296}\end{array}\right) ;\quad \det \mathbb{C}_{2}=1.\,\allowbreak 076\,11\,\times 10^{-7}\text{,} \label{C16}$$ $$\mathbb{C}_{3}=\left(
\begin{array}{rrr}
{\ 0.287800} & {\ 0.322230} & {\ -0.610029} \\
{\ 0.017215} & {\ 0.019275} & {\ -0.036490} \\
{\ -0.326908} & {\ -0.366017} & {\ 0.692926}\end{array}\right) ;\quad \det \mathbb{C}_{3}=3.\,\allowbreak 245\,95\times 10^{-13}.
\label{C17}$$ While matrix $\mathbb{C}_{1}$ does not have an inverse, $\mathbb{C}_{2}$ and $\mathbb{C}_{3}$ do have, but they are not stochastic. The decay characteristic times associated with the modes $2$ and $3$ are calculated as $T_{k}=-\left( \ln \left\vert \lambda _{k}\right\vert \right) ^{-1}$, resulting in the values $T_{2}= 1.000178$ and $T_{3}=0.371878$, so mode $3$ decays faster than mode $2$, while $T_{1}=\infty $ for mode $1$. Thus we can write $$\mathbb{M}^{n}=\mathbb{C}_{1}+\mathbb{C}_{2}\exp \left( -\frac{n}{T_{2}}\right) +\mathbb{C}_{3}\exp \left( -\frac{n}{T_{3}}\right) . \label{C18}$$ An initial state vector, for instance, $$\mathbb{U}^{\mathrm{T}(0)}=\left(
\begin{array}{ccc}
100 & 160 & 300\end{array}\right) , \label{C18a}$$ will evolve as $$\mathbb{U}^{\mathrm{T}}\left( n\right) =\mathbb{U}^{\mathrm{T}}\left(
0\right) \mathbb{C}_{1}+\left( \mathbb{U^{\mathrm{T}}}\left( 0\right)
\mathbb{C}_{2}\right) \exp \left( -\frac{n}{T_{2}}\right) +\left( \mathbb{U^{\mathrm{T}}}\left( 0\right) \mathbb{C}_{3}\right) \exp \left( -\frac{n}{T_{3}}\right) . \label{C19}$$ Thereafter, one can follow the evolution of the three components of the vector (\[C19\]):
\[C20\] $$\begin{aligned}
\mathbb{U}^{\mathrm{T}}\left( 0\right) \mathbb{C}_{1} &=&\left(
\begin{array}{ccc}
228 & 125 & 207\end{array}\right) , \label{C20a} \\
\mathbb{U^{\mathrm{T}}}\left( 0\right) \mathbb{C}_{2} &=&\left(
\begin{array}{ccc}
-136 & 110 & -48\end{array}\right) , \label{C20b} \\
\mathbb{U^{\mathrm{T}}}\left( 0\right) \mathbb{C}_{3} &=&\left(
\begin{array}{ccc}
-67 & -74 & 141\end{array}\right) , \label{C20c}\end{aligned}$$
where $\lim_{n\rightarrow \infty }\mathbb{U}^{\mathrm{T}}\left( n\right) =\mathbb{U}^{\mathrm{T}}\left( 0\right) \mathbb{C}_{1} $ is the asymptotic component and $\mathbb{U^{\mathrm{T}}}\left( 0\right) \mathbb{C}_{l}\times
\exp \left( -n/T_{l}\right) $, $l=2,3$, are the components of the transient regimes that present negative numbers in some entries, although all the components of vectors $\mathbb{U}^{\mathrm{T}}\left( n\right) $ and $\mathbb{U}^{\mathrm{T}}\left( 0\right) \mathbb{C}_{1} $, Eqs. (\[C19\]) and (\[C20a\]), are positive. The sum of the components of $\mathbb{U}^{\mathrm{T}}\left( 0\right) \mathbb{C}_{1}$ is $560$ which corresponds to the conserved sum of the entries of the vector (\[C18a\]), whereas the sums of the components of the vectors (\[C20b\]) and (\[C20c\]), $\sum_{i=1}^{3}\left( \mathbb{U^{\mathrm{T}}}\left( 0\right) \mathbb{C}_{2}\right) _{i}=-74$ and $\sum_{i=0}^{3}\left( \mathbb{U^{\mathrm{T}}}\left( 0\right) \mathbb{C}_{3}\right) _{i}=0$ respectively, do not need to be necessarily positive, that we interpret as virtual distributions. As $n$ increases the components $\left( \mathbb{U^{\mathrm{T}}}\left( 0\right) \mathbb{C}_{l}\right) \exp \left( -n/T_{l}\right) $ go to zero due to the exponential decay factor and only the asymptotic vector, Eq. (\[C20a\]), survives.
**(4)** About the trace operation $$\lim_{n\rightarrow \infty }\mathrm{Tr}\left[ \mathbb{M}^{n}\right]
=1+\lim_{n\rightarrow \infty }\sum_{k=2}^{N}\left( \lambda _{k}\right)
^{n}=1. \label{B1}$$ The same holds true for doubly stochastic matrices.
**(5)** At the limit $n\rightarrow \infty $ the stationary matrix is $$\mathbb{C}_{1}:=\lim_{n\rightarrow \infty }\mathbb{M}^{n}=
\left(
\begin{array}{ccccc}
m_{1} & m_{2} & \cdots & m_{N-1} & m_{N} \\
m_{1} & m_{2} & \cdots & m_{N-1} & m_{N} \\
\vdots & \vdots & & \vdots & \vdots \\
m_{1} & m_{2} & \cdots & m_{N-1} & m_{N} \\
m_{1} & m_{2} & \cdots & m_{N-1} & m_{N}\end{array}\right) \label{B2}$$
with the values of the entries depending on matrix $\mathbb{M}$ and $\sum_{j=1}^{N}m_{j}=1$; as all the rows are the same the eigenvalues are $\left\{1,0,0,...,0\right\} _{N}$ and the matrix is idempotent, $\mathbb{C}_{1}^2 = \mathbb{C}_{1}$. Although the matrix $\mathbb{M}$ determines the stationary matrix $\mathbb{C}_{1}$, the inverse is not possible, the knowledge of $\mathbb{C}_{1}$ does not permit the full determination of $\mathbb{M}$. This is the essence of the irreversible evolution. Writing one row of matrix (\[B2\]) as the vector $\mathbb{Z}^{\mathrm{T}} =\left(\begin{array}{ccccc}
m_{1} & m_{2} & \cdots & m_{N-1} & m_{N} \end{array}\right)$, it can be noted that it is a stationary distribution because $\mathbb{Z}^{\mathrm{T}}\mathbb{M}=\mathbb{Z}^{\mathrm{T}}$ (eigenvalues 1), or $\mathbb{C}_{1}\mathbb{M} =\mathbb{C}_{1}$. From Example 1, Eqs. (\[C1\]) and (\[C15\]), it can be verified that $ \left( \begin{array}{ccc} 11/27 & 2/9 & 10/27 \end{array} \right)
\mathbb{M} =\left( \begin{array}{ccc} 11/27 & 2/9 & 10/27 \end{array}\right)$.
A generic vector $\mathbb{X}_{0}^{\mathrm{T}}=\left( \begin{array}{ccccc}
x_{1} & x_{2} & \cdots & x_{N-1} & x_{N}\end{array}\right) $ evolves as $\mathbb{X}_{n}^{\mathrm{T}}=\mathbb{X}_{0}^{\mathrm{T}}\mathbb{M}^{n}$ and asymptotically becomes $\mathbb{X}_{\infty }^{\mathrm{T}}=\mathbb{X}_{0}^{\mathrm{T}}\mathbb{C}_{1} $; thus it “looses memory” about its components but keeps the sum $\sum_{i=1}^{N}x_{i}=Y$ as a recollection, i.e., $\mathbb{X}_{0}^{\mathrm{T}} \Rightarrow \mathbb{X}_{\infty }^{\mathrm{T}}
=Y\mathbb{Z}^{\mathrm{T}}$.
**(6)** In the numerical examples to be worked out below the eigenvalues of the matrices have multiplicity $1$, nevertheless the math and the physical analysis can be extended to the situation where eigenvalues have multiplicity higher than $1$. As an illustration about this point we consider here an interesting behavior that affects the time decay in the case of occurrence of a degenerate eigenvalue (in comparison with the situation of no degeneracy). Pointedly, within a closed network, it causes a delay in the time the flux of vehicles takes to evolve to the stationary matrix $\lim_{n \rightarrow
\infty}\mathbb{M}^n$. In this case, the diagonalization of the matrix $\mathbb{M}$ – the decomposition (\[A8\]) – that produces the diagonal matrix $\mathbb{D}$ is not anymore possible because there are fewer linearly independent eigenvectors than the dimension $N$ of the matrix $\mathbb{M}$. Notwithstanding, we can write the decomposition in a form similar to (\[A8\]), $\mathbb{M}=\mathbb{RFR}^{-1}$, which is the so-called *Jordan canonical form*, where the matrix $\mathbb{R}$ is different from $\mathbb{Q}$, and the matrix $\mathbb{F}$, besides having the eigenvalues of $\mathbb{M}$ in the diagonal line, will contain, additionally, in the adjacent line parallel to the main diagonal, the number $1$ in the entries of the blocks containing the degenerate eigenvalues, while the other entries are filled with zeros. The theory is due to the french mathematician Camille Jordan. For a rigorous mathematical presentation of the matter we recommend the reader to consult, for instance, Ref. [@horn2013].
The decomposition of a stochastic matrix in the Jordan form can be cast as $$\mathbb{M}=\mathbb{RDR}^{-1} + \mathbb{L},\quad \text{and} \quad
\mathbb{M}^{n}=\mathbb{R} \mathbb{D}^{n}\mathbb{R}^{-1} +f(n)\mathbb{L},
\label{A12}$$ where $\mathbb{D}$ is still a diagonal matrix containing the eigenvalues, $\mathbb{L}=\mathbb{R}\mathbb{O}\mathbb{R}^{-1}$ with $\mathbb{O}$ the matrix containing $1$’s (and $0$’s) in a secondary diagonal line of matrix $\mathbb{F}$ and $f(n)$ is a function only of $n$. We illustrate this case through an example.
**** Let us consider the doubly stochastic matrix $$\begin{aligned}
\mathbb{M} &=&\left(
\begin{array}{ccc}
2/5 & 1/2 & 1/10 \\
3/10 & 3/10 & 2/5 \\
3/10 & 1/5 & 1/2\end{array}\right) ;\quad \det \mathbb{M} = 1/100\ , \label{A133}
$$whose eigenvectors and eigenvalues are $$\left(
\begin{array}{c}
1 \\
1 \\
1\end{array}\right) \leftrightarrow \chi_{PF} = 1,\quad \left( \begin{array}{r}
-2 \\
1 \\
1\end{array}\right) \leftrightarrow \chi_{2}=\frac{1}{10},\ \text{multiplicity 2}\ ,
\label{A14}$$ $\chi_{PF}$ is the Perron-Frobenius eigenvalue and $\chi_{2}$ is degenerate, with only one linearly independent eigenvector. The decomposition of matrix (\[A133\]) in the Jordan form is $$\mathbb{M} =\mathbb{RFR}^{-1}=\left(
\begin{array}{rrr}
1 & 0 & 1 \\
1 & 5/2 & -1/2 \\
1 & -5/2 & -1/2\end{array}\right) \left(
\begin{array}{rrr}
1 & 0 & 0 \\
0 & \chi_2 & 0 \\
0 & 1 & \chi_2\end{array}\right) \left(
\begin{array}{rrr}
1/3 & 1/3 & 1/3 \\
0 & 1/5 & -1/5 \\
2/3 & -1/3 & -1/3\end{array}\right)
\label{A15}$$ where the matrix $\mathbb{F}$ is not diagonal although the eigenvalues remain in the main diagonal. We can rewrite the matrix (\[A15\]) as $$\mathbb{M} = \mathbb{RDR}^{-1}+f\left( 1\right) \mathbb{L}
= \mathbb{C}_{1}\allowbreak +\chi _{2}\mathbb{C}_{2} +\chi
_{2}\mathbb{C}_{3}
\label{A16}$$ where the matrices $\mathbb{C}_{i}$ are $$\mathbb{C}_{1}=\frac{1}{3}\left(
\begin{array}{ccc}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1\end{array}\right) ,\quad \mathbb{C}_{2}=\left(
\begin{array}{rrr}
2/3 & -1/3 & -1/3 \\
-1/3 & 2/3 & -1/3 \\
-1/3 & -1/3 & 2/3\end{array}\right) ,\quad \mathbb{C}_{3}=\left(
\begin{array}{rrr}
0 & 2 & -2 \\
0 & -1 & 1 \\
0 & -1 & 1\end{array}\right) \ .
\label{A17}$$ For an $n$-step process we have $
\mathbb{M}^{n}=\mathbb{C}_{1}+\left( \chi _{2}\right) ^{n}\mathbb{C}_{2}+n\left( \chi _{2}\right) ^{n}\mathbb{C}_{3}.$ Setting $\chi_{2} = \exp \left( -1/T_2 \right) $ we get $$\mathbb{M}^{n}=\mathbb{C}_{1}+\exp \left({-\frac{n}{T_{2}}}\right)\mathbb{C}_{2}
+ n\exp\left({-\frac{n}{T_{2}}}\right) \mathbb{C}_{3}
\label{A18}$$ with the characteristic decay time $T_{2}=-\left( \ln \chi _{2}\right)
^{-1}=\left( \ln 10\right) ^{-1}\approx 0.43$ for the exponential decay modes. Due to the factor $n$ that multiplies the exponential in the third term, it is going to decay at a lower pace than the second one, thus dominating the evolution of $\mathbb{M}^{n}$ toward the stationary matrix $\mathbb{C}_{1}$ as $n\rightarrow \infty$.
Entropy
-------
The complexity of the flow of vehicles in a network, as the one represented by the digraph (\[fig01\]), has a measure that can be evaluated by calculating the entropy at each moment $n$. Since $\sum_{j=1}^{N}\left( \mathbb{M}^{n}\right) _{ij}=1$ ($N$ stands for the dimension of the square matrix), for the SM we define the partial entropy associated to each row $i$ as $$S_{i}\left( n\right) =-\sum_{j=1}^{N}\left( \mathbb{M}^{n}\right) _{i,j}\ln
\left( \mathbb{M}^{n}\right) _{i,j}\qquad \mathrm{for\quad }i=1,...,N;
\label{Ent1}$$ noting that if no vehicle migrates from one site to the other, then $\left(
\mathbb{M}^{n}\right) _{ij}=\delta _{ij}$ and $S_{i}\left( n\right) =0$. We can also define a measure of the global complexity as an arithmetic mean over all the sites, $$G\left( n\right) =\frac{1}{N}\sum_{i=1}^{N}S_{i}\left( n\right) =-\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{N}\left( \mathbb{M}^{n}\right) _{ij}\ln \left(
\mathbb{M}^{n}\right) _{ij}\ , \label{Ent2}$$ outlining a *global mean entropy*.
Model I: Illustration with numbers
==================================
In the four sites network (digraph of Fig. \[fig01\]) we assume that there are, for instance, initially, $10\ 000$, $2\ 500$, $11\ 000$ and $13\ 000$ vehicles; the matrix (\[A1\]) with numbers in its entries is diagonal and is more conveniently expressed as the vector (\[A22\]), $$\mathbb{U}^{\mathrm{T}}\left( 0\right) =\left(
\begin{array}{cccc}
10000 & 2500 & 11000 & 13000\end{array}\right) , \label{B02}$$ and the total number of vehicles in the network is $$Y=\sum_{i=1}^{4}U_{i}\left( 0\right) =36\ 500. \label{B1b}$$ At a given moment the vehicles begin to circulate through the arteries and after few days (or at several moments of one day) of observation the average number of the vehicles that remain circulating (or parked) within each site (the vertices) are assumed *ad hoc* to be $3\ 000$, $1\ 000$, $4\ 000$, and $4\ 500$, while the remaining vehicles are on their way traveling to the other sites, characterized in the digraph by the values associated to the edges. We put the available information in a matrix form $$\mathbb{A}\left( 1\right) =\left(
\begin{array}{rrrr}
3000 & 1500 & 2500 & 3000 \\
500 & 1000 & 500 & 500 \\
3000 & 1500 & 4000 & 2500 \\
4000 & 1500 & 3000 & 4500\end{array}\right) \ , \label{B0}$$ where in the diagonal entries one finds the number of vehicles in each site and the off-diagonal entries stand for the number of those traveling from one site (line $i$) to another (column $j$), and $\left( \mathbb{A}\left(
1\right) \right) = 12\ 500$ is the number of vehicles still at the sites. The number of vehicles in each row ($U_{k}\left( 0\right) $) are given by the vector ([B02]{}).
The Stochastic Matrix
---------------------
We assume that the matrix (\[B0\]) is the seed that permits to construct another matrix, in a suited form, to be used to forecast the circulation of the vehicles in future moments. This hypothesis is framed formally by normalizing each row. The result is the SM $$\mathbb{M}=\left(
\begin{array}{cccc}
3/10 & 3/20 & 1/4 & 3/10 \\
1/5 & 2/5 & 1/5 & 1/5 \\
3/11 & 3/22 & 4/11 & 5/22 \\
4/13 & 3/26 & 3/13 & 9/26\end{array}\right)\ , \label{D1}$$ whose eigenvalues are approximately $$\begin{tabular}{|c||c|c|c|c|}
\hline
$k$ & $1$ & $2$ & $3$ & $4$ \\ \hline
\multicolumn{1}{|l||}{$\lambda _{k}$} & \multicolumn{1}{l|}{${\ 1.0}$} &
\multicolumn{1}{l|}{${\ 0.27}$} & \multicolumn{1}{l|}{${\ 0.13}$} &
\multicolumn{1}{l|}{${\ 0.01}$} \\ \hline
\end{tabular}\ . \label{D1.1}$$ The eigenvalue $\lambda _{1}=1.0$ is the PF while the others are quite smaller than $1$, this is an indication that under this kind of evolution the traffic should stabilize after very few steps. As $\det \mathbb{M}=\left({2600}\right)^{-1}<1$, then $\lim_{n\rightarrow \infty }$$\det \left(
\mathbb{M}^{n}\right) =0$ and $\lim_{n\rightarrow \infty }\mathrm{Tr}\left(
\mathbb{M}^{n}\right) =1$.
Now we calculate the number of vehicles in each site plus those that left (in a period of 24 hours). The prediction of the distribution for the following days is given by Eq. (\[A24\]), and the total number of vehicles is a conserved quantity, $\lim_{n\rightarrow \infty }\sum_{i}U_{i}\left( n\right) =36\,500 = Y$. The distribution changes with $n$ until the stabilization of the flow is attained swiftly in less than $4$ steps. This trend could be guessed due to the wide difference between the PF eigenvalue ($\lambda _{1}$=1) and $\lambda _{2}$, see the frame (\[D1.1\]). The vehicular circulation presents three characteristic decay times towards the stationary distribution $$\begin{tabular}{|c||c|c|c|c|}
\hline
$i$ & ${\ 1}$ & ${\ 2}$ & ${\ 3}$ & ${\ 4}$ \\ \hline
\multicolumn{1}{|l||}{$T_{i}$} & \multicolumn{1}{l|}{${\ \infty }$} &
\multicolumn{1}{l|}{${\ 0.76}$} & \multicolumn{1}{l|}{${\ 0.49}$}
& \multicolumn{1}{l|}{${\ 0.22}$} \\ \hline
\end{tabular}\ ,
\label{C3}$$ where $T_2$ dictates the trend of the decay. In Fig. \[fig031\] we plot the sequences of the distributions $U_{i}^{\mathrm{T}}(n)$ that stabilize swiftly to asymptotic ones. For $\mathbb{U}^{\mathrm{T}}{\ (0)}=\left(
\begin{array}{cccc}
{10000} & {\ 2500} & {\ 11000} & {\ 13000}\end{array}
\right) $ we get $\mathbb{U}^{\mathrm{T}}{\ (\infty )
\approx \mathbb{U}^{\mathrm{T}}{\ (6)} =}\left( \begin{array}{cccc}
{10097} & {\ 6659} & {\ 9702} & {\ 10042}\end{array}
\right) $.
![[]{data-label="fig031"}](TrafficUn.jpg){height="2.94in" width="3.94in"}
Entropy
-------
The values of the global mean entropy, (\[Ent2\]), are given in Table \[D4\],
${\ n}$ $1$ $2$ $3$ $4$ $5$ $6$
--------- ----- ----- ----- ----- ----- ----- -- --
: []{data-label="D4"}
and in Fig. \[fig010\] the diamond shaped marks represent the calculated values of $G\left( n\right) $ of Table \[D4\]; the solid line only links the points.
![[]{data-label="fig010"}](TrafficGRn.jpg){height="3.0in" width="4.2in"}
The stabilization of the distribution of vehicles in each site, and the stationarity of the flow is reflected in the behavior of the global mean entropy which increases swiftly and then stabilizes at a maximum value, around $1.37$, that we could call the “thermalization” of the traffic.
Model II: Inter-site traffic with input (source) and output (sink) of vehicles
==============================================================================
We now introduce two physical modifications in the inter-site traffic model studied in the preceding sections: (a) there is a daily input of new vehicles from site $E$ to city $A$, that are integrated into the network and (b) there is also an output of old vehicles that are taken off the circuit from the same site $A$ and are stocked in the site $F$. These new features are drawn in the digraph with additional edges and loops, labeled as $v_{1}$ and $v_{2}$, $r_{5}$ and $r_{6},$ respectively, as shown in Fig. \[fig03\].
![[]{data-label="fig03"}](Trafficinputoutput2.jpg){height="2.44in" width="2.6in"}
In this digraph the letters $A$, $B$, $C$, $D$, $E$ and $F$ label the vertices, whereas the $r_{i}$’s are for the loops and $q_{i}$, $p_{i}$, $s_1$ and $v_1$ are for the edges. One expresses the network by the array in Table \[A5\],
Vertices $A$ $B$ $C$ $D$ $E$ $F$ Sum of the line entries
--------------------------- ------------- ------------- ------------- ------------- ------------- ------------- -------------------------
$A$ $R_{1}$ $P_{1}$ $P_{5}$ $Q_{4}$ $0$ $S_{1}$
$B$ $Q_{1}$ $R_{2}$ $P_{2}$ $P_{6}$ $0$ $0$
$C$ $Q_{5}$ $Q_{2}$ $R_{3}$ $P_{3}$ $0$ $0$
$D$ $P_{4}$ $Q_{6}$ $Q_{3}$ $R_{4}$ $0$ $0$
$E$ ${\ V}_{1}$ $0$ $0$ $0$ $R_{5}$ $0$
$F$ $0$ $0$ $0$ $0$ $0$ $R_{6}$
Sum of the column entries ${\ Z}_{1}$ ${\ Z}_{2}$ ${\ Z}_{3}$ ${\ Z}_{4}$ ${\ Z}_{5}$ ${\ Z}_{6}$
: []{data-label="A5"}
whose core is the matrix (\[A6\]) $$\mathbb{B}=\left(
\begin{array}{cccccc}
R_{1} & P_{1} & P_{5} & Q_{4} & 0 & S_{1} \\
Q_{1} & R_{2} & P_{2} & P_{6} & 0 & 0 \\
Q_{5} & Q_{2} & R_{3} & P_{3} & 0 & 0 \\
P_{4} & Q_{6} & Q_{3} & R_{4} & 0 & 0 \\
V_{1} & 0 & 0 & 0 & R_{5} & 0 \\
0 & 0 & 0 & 0 & 0 & R_{6}\end{array}\right) , \label{A6}$$ whose entries contain the number of counted vehicles in an *ad hoc* interval of time, or by averaging from previous observations. The matrix (\[A1\]) is a submatrix of matrix (\[A6\]).
As the time goes on the distribution of vehicles, in matrix (\[A6\]), changes due to their circulation. The sum of the entries of row $i$, $W_{i}\left( 0\right)
=\sum_{j}B_{ij}$, gives the number of vehicles that are in site $i$ plus those that left it having as destination all the other sites, excluding site $F$, which is a depository of the vehicles removed from circulation. The total number of vehicles in the network, $\overline{Y}$, is conserved. The numbers $W_{i}\left( 0\right) $ can be cast as a vector, $$\mathbb{W}^{\mathrm{T}}\left( 0\right) =\left(
\begin{array}{cccccc}
W_{1}\left( 0\right) & W_{2}\left( 0\right) & W_{3}\left( 0\right) &
W_{4}\left( 0\right) & W_{5}\left( 0\right) & W_{6}\left( 0\right)\end{array}\right) .$$
The stochastic matrix
---------------------
The dynamical evolution is ruled by a stochastic matrix associated to (\[A6\]) which is constructed by normalizing the entries in each row, resulting in $$\mathbb{N}=\left(
\begin{array}{cccccc}
r_{1} & p_{1} & p_{5} & q_{4} & 0 & s_{1} \\
q_{1} & r_{2} & p_{2} & p_{6} & 0 & 0 \\
q_{5} & q_{2} & r_{3} & p_{3} & 0 & 0 \\
p_{4} & q_{6} & q_{3} & r_{4} & 0 & 0 \\
v_{1} & 0 & 0 & 0 & r_{5} & 0 \\
0 & 0 & 0 & 0 & 0 & 1\end{array}\right) \quad , \label{A6a}$$ where the first row is $$r_{1}=\frac{R_{1}}{W_{1}},\quad p_{1}=\frac{P_{1}}{W_{1}},\quad p_{5}=\frac{P_{5}}{W_{1}},\quad q_{4}=\frac{Q_{4}}{W_{1}},\quad s_{1}=\frac{S_{1}}{W_{1}}; \label{A6b}$$ and the same goes for the other rows. The evolution of an initial vector is calculated as $$\mathbb{W}^{\mathrm{T}}\left( n\right) =\mathbb{W}^{\mathrm{T}}\left(
0\right) \mathbb{N}^{n}. \label{A6c}$$
Model II: Illustrating the model with numbers
=============================================
To illustrate the model we consider the matrix (\[A6a\]) $$\mathbb{B}\left( 1\right) =\left(
\begin{array}{cccccc}
{\ 3000} & {\ 1500} & {\ 2500} & {\ 3000} & {\ 0} & {\ 50} \\
{\ 500} & {\ 1000} & {\ 500} & {\ 500} & {\ 0} & {\ 0} \\
{\ 3000} & {\ 1500} & {\ 4000} & {\ 2500} & {\ 0} & {\ 0} \\
{\ 4000} & {\ 1500} & {\ 3000} & {\ 4500} & {\ 0} & {\ 0} \\
{\ 100} & {\ 0} & {\ 0} & {\ 0} & {\ 1000} & {\ 0} \\
{\ 0} & {\ 0} & {\ 0} & {\ 0} & {\ 0} & {\ R}_{6}\end{array}\right) , \label{A7}$$ where we kept the same values of the entries of the $4\times 4$ matrix ([B0]{}) but being now enlarged, with dimensions $6\times 6$, and with new non-null entries, $B_{16}=50$, $B_{51}=100$, $B_{55}=1000$; the entry $B_{66}=R_{6}$ is an arbitrary number, that we set as $0$ since it is not important within the dynamics. Summing the entries in each row of matrix (\[A7\]) we have the vector $$\mathbb{W}^{\mathrm{T}}\left( 0\right) =\left(
\begin{array}{cccccc}
{\ 10050} & {\ 2500} & {\ 11000} & {\ 13000} & {\ 1100} &0 \end{array}\right) \label{A9}$$ or as a diagonal matrix $\mathbb{B}\left( 0\right) = \mathrm{Diag}[10050,2500,11000,13000,1100,0]$ and the total number of vehicles is $\overline{Y}=37\, 650$. Comparing with Model I, the increment in the number of vehicles in vector (\[A9\]) is small relatively to those in the string (\[B02\]), namely an increment of $1\, 150$ to the previous $36\, 500$.
The Stochastic Matrix
---------------------
From the matrix (\[A7\]) we construct the stochastic matrix $$\mathbb{N}=\left(
\begin{array}{cccccc}
300/1005 & 150/1005 & 250/1005 & 300/1005 & 0
& 5/1005 \\
1/5 & 2/5 & 1/5 & 1/5 & 0 & 0 \\
3/11 & 3/22 & 4/11 & 5/22 & 0 & 0 \\
4/13 & 3/26 & 3/13 & 9/26 & 0 & 0 \\
1/11 & 0 & 0 & 0 & 10/11 & 0 \\
0 & 0 & 0 & 0 & 0 & 1\end{array}\right)\quad , \label{A11}$$ whose eigenvalues are, approximately, $$\begin{tabular}{|c||c|c|c|c|c|c|}
\hline
${\ k}$ & ${\ 1}$ & ${\ 2}$ & ${\ 3}$ & ${\ 4}$ & ${\ 5}$ & ${\ 6}$ \\ \hline
${\ \lambda }_{k}$ & ${\ 1.0}$ & ${\ 0.999}$ & ${\ 0.909}$ & ${\
0.266}$ & ${\ 0.132}$ & ${\ 0.011}$ \\ \hline
\end{tabular}\ . \label{A11.1}$$ The first eigenvalue is the PF while the second and third ones are close to $1$, making the traffic to take much more time to attain a stationary circulation than in the former model where the traffic goes stationary in 3 or 4 steps; these eigenvalues can be compared to those in frame (\[D1.1\]).
The distribution of vehicles evolves as $\mathbb{W}^{\mathrm{T}}\left(
n\right) =\mathbb{W}^{\mathrm{T}}\left( 0\right) \mathbb{N} ^{n}$, with $\mathbb{W}^{\mathrm{T}}\left( 0\right)$ from Eq. (\[A9\]). For $n=2^{12}$ we get $\mathbb{W}^{\mathrm{T}}\left( 2^{12}\right) =\left(
\begin{array}{cccccc} {\ 37} & {\ 24} & {\ 36} & {\ 37} & {\ 0} & {37516}\end{array} \right) $. The last entry stands for the $37\ 516$ vehicles that went out of circulation, at an average daily fraction $1/201$ of all vehicles, stockpiled in site $F$, whereas the number of vehicles still in circulation diminishes continuously. The fifth entry stands for the supply of new vehicles into the network: there were initially $1\ 000$ vehicles at site $E$, with a daily supply into the network $ABCD$ of $1/11$ fraction of vehicles still in the yard $E$ (the supply lasted few days). We note that after $2^{12}$ steps only a tiny fraction of vehicles (134), $0.36\%$, continue circulating within the network $ABCD$. In Fig. \[fig04\] we plot the evolution of the entries of $\mathbb{W}\left( n\right)$, considering in the abscissa $k\equiv \left( \log _{2}n\right) +1$, for $n=2^0,2^1,...,2^{12}$, such that $k=1,2,..,13$.
![[]{data-label="fig04"}](TrafficWk.jpg){height="3.0in" width="4.0in"}
Thus, after an initial increase of the number of vehicles circulating within the network $ABCD$ there is a stabilization, see Fig. \[fig08\], then begins a slow decrease in their number, however after consuming a quite long time compared to the time it takes to attain equilibration in the model without source and sink.
![[]{data-label="fig08"}](TrafficYdiss.jpg){height="2.7in" width="3.6in"}
We note that for the digraph in Fig. \[fig01\] the asymptotic equilibrium of the evolution is attained nearly after four steps, $n=4$, whereas for the network with source and sink, the digraph in Fig. \[fig03\], the tendency to the stationary distribution, $\mathbb{W}\left( 2^{12} \right)$, occurs after $2^{12}$ steps, or for $\left( \mathbb{N}\right) ^{2^{12}}$. In short, in the former network model the stationarity is reached at a rate linear in time, while here it occurs at an exponential rate. The characteristic decay times $\tilde{T}_{k}=-\left( \ln \left\vert \lambda _{k}\right\vert \right) ^{-1}$ are $$\begin{tabular}{|c||c|c|c|c|c|c|}
\hline
${\ k}$ & ${\ 1}$ & ${\ 2}$ & ${\ 3}$ & ${\ 4}$ & ${\ 5}$ & ${\ 6}$ \\ \hline
& & & & & & \\
${\ \tilde{T}}_{k}$ & ${\ \infty }$ & \multicolumn{1}{l|}{${\ 726.20}$} &
\multicolumn{1}{l|}{${\ 10.50}$} & \multicolumn{1}{l|}{${\ 0.76}$} &
\multicolumn{1}{l|}{${\ 0.49}$} & \multicolumn{1}{l|}{${\ 0.22}$}
\\ \hline
\end{tabular}~, \label{A13}$$ to be compared to the characteristic times in the frame (\[C3\]). Since $\tilde{T}_{2}/T_{2}\approx 961$, $\tilde{T}_{3}/T_{3}\approx
\allowbreak 13.9$, $\tilde{T}_{4}/T_{4}\approx 3.4$, we perceive the huge ratio of the characteristic times $\tilde{T}_{2}/T_{2}$ meaning that the introduction of a source and a sink in the $ABCD$ network, for the specific chosen numbers, contribute to an average time (to attain the stationary state) that is about three orders of magnitude higher than the network without source and sink.
Entropy
-------
The global mean entropy, Eq. (\[Ent2\]), for $N=6$, at times $n=2^{k}$ with $k=0,1,2,...,12$ is plotted in Fig. \[fig06\]. Compared to Fig. \[fig010\]
![[]{data-label="fig06"}](TrafficGk.jpg){height="2.6in" width="3.4in"}
it shows a quite different trend, the entropy increases, attains a maximum value and then begins a monotonic decrease until zero, when all the vehicles accumulate at the site $F$, as they are being continuously collected.
Nonlinear model: multiple stationary states and cyclic changes
==============================================================
More realistically one cannot expect that the traffic flow could be described strictly by a linear model, even with sources and sinks in the network, since it leads to only one stationary state. Thus adopting a nonlinear approach seems more realistic because it introduces non-trivial changes. In this model the entries of a stochastic matrix contain functions of $n$ as an intrinsic variable, and there is a freedom to choose functions and parameters in order to emulate a flow displaying several stationary regimes in one single day, for instance.
For the sake of illustration we consider a two-site network ($A$ and $B$) and two arteries connecting them, with no sinks or sources. We assume that the stochastic matrix has dimension 2 and we write it as $$\mathbb{M} \left( n \right) =\left(
\begin{array}{cc}
g_{1}\left( n\right) & 1-g_{1}\left( n\right) \\ \\
g_{2}\left( n\right) & \ 1-g_{2}\left( n\right)\end{array}\right) \ ,
\label{L1}$$ whose entries are $n$-dependent and remain non-negative for any positive integer $n$, as long as $0 \leq g_{k}\left( n\right) \leq 1$. Considering that the functions $g_{k}\left( n\right) $ change periodically with $n$, we set $g_{1}\left( n\right) = a_1 + b_1\sin \left( \frac{2\pi n}{L}\right) $ and $g_{2}\left( n\right) = a_2 + b_2\cos \left( \frac{2\pi n}{L}\right) $, with $a_1=a_2=1/2$, $b_1=b_2=-1/2$, and, as it will be seen below, the integer $L$ stands for the multiplicity of the stationary states. For the sake of illustration we adopt $L=3$, which implies three asymptotic matrices of an $n$-step evolution. This property for nonlinear stochastic matrices enlarges the descriptive possibilities of physical systems in comparison with the linear models. With those settings the matrix (\[L1\]) is written as $$\mathbb{M} \left( n \right) =\left(
\begin{array}{cc}
\frac{1}{2}\left( 1-\sin \frac{2\pi n}{3}\right) & \frac{1}{2}\left( 1+\sin
\frac{2\pi n}{3}\right) \\ \\
\frac{1}{2}\left( 1-\cos \frac{2\pi n}{3}\right) & \frac{1}{2}\left( 1+\cos
\frac{2\pi n}{3}\right)\end{array}\right)\ , \label{L2}$$ and is 3-cycle, meaning that for each element of the set $n=\{3k,3k+1,3k+2\}$ (excluding $n=0$) the matrix $\mathbb{M}\left( n\right)$ repeats itself, as shown in the Table \[Tnl1\].
$k$ $n=3k$ $n=3k+1$ $n=3k+2$
----------- ----------- ----------- -----------
$0$ $0$ $1$ $2$
$1$ $3$ $4$ $5$
$2$ $6$ $7$ $8$
$\vdots $ $\vdots $ $\vdots $ $\vdots $
: []{data-label="Tnl1"}
The stochastic matrices and decay modes eigenstates and eigenvalues are given in Table \[Tnl2\], and their expansion in the several modes and their decay times are given in Table \[Tnl3\].
---------------------------------------------------------------------------------------------------------------------------------------------------------------
$n$-step stochastic matrices Decay mode eigenstate and eigenvalue
--------------------------------------------------------------------- -----------------------------------------------------------------------------------------
$\mathbb{M}\left( 3k+1\right) \doteq \mathbb{M}_{1}=\left( $\frac{1}{2\sqrt{4+\sqrt{3}}}\left(
\begin{array}{cc} \begin{array}{c}
\frac{1}{2}-\frac{\sqrt{3}}{4} & \frac{1}{2}+\frac{\sqrt{3}}{4} \\ -\left( \sqrt{3}+2\right) \\
\frac{3}{4} & \frac{1}{4}\end{array}\right) $ 3\end{array}\right) \leftrightarrow -\frac{1}{4}\left( \sqrt{3}+1\right) =\lambda _{1}$
$\mathbb{M}\left( 3k+2\right) \doteq \mathbb{M}_{2}=\left( $\frac{1}{2\sqrt{4-\sqrt{3}}}\left(
\begin{array}{cc} \begin{array}{c}
\frac{1}{2}+\frac{\sqrt{3}}{4} & \frac{1}{2}-\frac{\sqrt{3}}{4} \\ \sqrt{3}-2 \\
\frac{3}{4} & \frac{1}{4}\end{array}\right) $ 3\end{array}\right) \leftrightarrow \frac{1}{4}\left( \sqrt{3}-1\right) =\lambda _{2}$
$\mathbb{M}\left( 3k\right) \doteq \mathbb{M}_{3}=\left( $\left(
\begin{array}{cc} \begin{array}{c}
\frac{1}{2} & \frac{1}{2} \\ 1 \\
0 & 1\end{array}\right) ,\quad k\neq 0$ 0\end{array}\right) \leftrightarrow \frac{1}{2}=\lambda _{3} $
---------------------------------------------------------------------------------------------------------------------------------------------------------------
: []{data-label="Tnl2"}
The PF eigenstates for the three matrices in Table \[Tnl2\] are the same, $\frac{1}{\sqrt{2}}\left( \begin{array}{c} 1 \\ 1 \end{array} \right)$.
--------------------------------------------------------------------
$T_{i} = -\left( \ln \left\vert \lambda _{i}\right\vert \right)
^{-1}$
-- -----------------------------------------------------------------
--------------------------------------------------------------------
: []{data-label="Tnl3"}
The $\mathbb{C}_{l,i}$ matrices are given in Table \[Tnl4\].
---------------------------------------------------------------------------------------
$i$ $\mathbb{C}_{1,i}$ $\mathbb{C}_{2,i}$
----- ------------------------------------ --------------------------------------------
$1$ $\frac{1}{\sqrt{3}+5}\left( $\frac{1}{\sqrt{3}+5}\left(
\begin{array}{cc} \begin{array}{cc}
3 & \sqrt{3}+2 \\ \sqrt{3}+2 & -\left( \sqrt{3}+2\right) \\
3 & \sqrt{3}+2\end{array}\right) $ -3 & 3\end{array}\right) $
$2$ $\frac{1}{5-\sqrt{3}}\left( $\frac{1}{5-\sqrt{3}}\left(
\begin{array}{cc} \begin{array}{cc}
3 & 2-\sqrt{3} \\ 2-\sqrt{3} & -\left( 2-\sqrt{3}\right) \\
3 & 2-\sqrt{3}\end{array}\right) $ -3 & 3\end{array}\right) $
$3$ $\left( $\left(
\begin{array}{cc} \begin{array}{rr}
0 & 1 \\ 1 & -1 \\
0 & 1\end{array}\right) $ 0 & 0\end{array}\right) $
---------------------------------------------------------------------------------------
: []{data-label="Tnl4"}
We now assume that initially the number of vehicles in the two cities, $A$ and $B$, are $a_0$ and $b_0$, represented by the vector $\left( \begin{array}{cc} a_0 & b_0 \end{array} \right)$. For predicting the number of vehicles at three different moments of a day we set the rules in Table \[Tnl55\]
--------------------------------------------------------------------------------------------------------------------------------------------------------------
$i$ $r-th$ DAY evolution, $r=1,2,3,...$ $n$
----- ------------------------------------------------------------------------------------------------------------------------------ -------------------------
$1$ $\left( $3\left( r-1 \right)+1$
\begin{array}{cc}
a_{r}^{\left( 1\right) } & b_{r}^{\left( 1\right) }\end{array}\right) =\left(
\begin{array}{cc}
a_{0} & b_{0}\end{array}\right) \mathbb{M}_{1}\left( \mathbb{M}_{2}\mathbb{M}_{3}\mathbb{M}_{1}\right) ^{r-1}$
$2$ $\left( $3\left( r-1 \right)+2$
\begin{array}{cc}
a_{r}^{\left( 2\right) } & b_{r}^{\left( 2\right) }\end{array}\right) =\left(
\begin{array}{cc}
a_{0} & b_{0}\end{array}\right) \mathbb{M}_{1}\mathbb{M}_{2}\left( \mathbb{M}_{3}\mathbb{M}_{1}\mathbb{M}_{2}\right) ^{r-1}$
$3$ $\left( $3r$
\begin{array}{cc}
a_{r}^{\left( 3\right) } & b_{r}^{\left( 3\right) }\end{array}\right) =\left(
\begin{array}{cc}
a_{0} & b_{0}\end{array}\right) \left( \mathbb{M}_{1}\mathbb{M}_{2}\mathbb{M}_{3}\right) ^{r}$
--------------------------------------------------------------------------------------------------------------------------------------------------------------
: []{data-label="Tnl55"}
and by moment we mean some, arbitrarily chosen, interval of time for counting the number of vehicles passing by an established mark – an intersecting artery – or the traffic flux, (a sampling method) at three different hours of a day, for instance, morning, noon and evening, to be compared with the predictions of the model.
The properties of stochastic matrices permit to calculate their products, presented in Table \[Tnl55\], as
\[matprod\] $$\begin{aligned}
\mathbb{M}_{1}\left( \mathbb{M}_{2}\mathbb{M}_{3}\mathbb{M}_{1}\right)
^{r-1} &=& \mathbb{H}_{1,1}+\mathbb{H}_{2,1}\left( -1\right) ^{r-1}e^{-\frac{r-1}{\tau _{1}}}\ , \\
\mathbb{M}_{1}\mathbb{M}_{2}\left( \mathbb{M}_{3}\mathbb{M}_{1}\mathbb{M}_{2}\right) ^{r-1} &=& \mathbb{H}_{1,2}+\mathbb{H}_{2,2}\left( -1\right)
^{r-1}e^{-\frac{r-1}{\tau _{1}}}\ , \\
\left( \mathbb{M}_{1}\mathbb{M}_{2}\mathbb{M}_{3}\right) ^{r} &=& \mathbb{H}_{1,3}+\mathbb{H}_{2,3}\left( -1\right) ^{r}e^{-\frac{r}{\tau _{1}}}\ , \end{aligned}$$
with $r=1,2,...$, and since the eigenvalues of the three matrices (\[matprod\]) coincide, $\lambda_1 = -(1/16)$, the decay time is $\tau_1 =
-\left(\ln\left(1/16\right)\right)^{-1} \approx 0.36$. The $\mathbb{H}$ matrices are
$$\begin{aligned}
\mathbb{H}_{1,1} &=&\frac{1}{34}\left(
\begin{array}{cc}
21-3\sqrt{3} & 13 +3\sqrt{3} \\
21-3\sqrt{3} & 13+3\sqrt{3}\end{array}\right) \approx \left(
\begin{array}{cc}
0.46\, & 0.54\, \\
0.46\, & 0.54\,\end{array}\right) \\
\mathbb{H}_{2,1} &=&\frac{1}{17}\left(
\begin{array}{rc}
-\left( \frac{11}{4}\sqrt{3}+2\right) & \frac{11}{4}\sqrt{3}+2 \\
\frac{3}{2}\left( \sqrt{3}+\frac{3}{2}\right) & -\frac{3}{2}\left( \sqrt{3}+\frac{3}{2}\right)\end{array}\right) \approx \left(
\begin{array}{rr}
-0.40\, & 0.40\, \\
0.29\, & -0.29\,\end{array}\right)\end{aligned}$$
$$\begin{aligned}
\mathbb{H}_{1,2} &=&\frac{1}{17}\left(
\begin{array}{cc}
9+3\sqrt{3} & 8-3\sqrt{3} \\
9+3\sqrt{3} & 8-3\sqrt{3}\end{array}\right) \approx \left(
\begin{array}{cc}
0.84\, & 0.16\, \\
0.84\, & 0.16\,\end{array}\right) \\
\mathbb{H}_{2,2} &=&\frac{1}{272}\left(
\begin{array}{rr}
-25+3\sqrt{3} & 25-3\sqrt{3} \\
3\left( \sqrt{3}+3\right) & -3\left( \sqrt{3}+3\right)\end{array}\right) \approx \left(
\begin{array}{rr}
-0.07 & 0.07 \\
0.05 & -0.05\end{array}\right)\end{aligned}$$
$$\begin{aligned}
\mathbb{H}_{1,3} &=&\frac{1}{34}\left(
\begin{array}{rr}
3\sqrt{3}+9 & 25-3\sqrt{3} \\
3\sqrt{3}+9 & 25-3\sqrt{3}\end{array}\right) \approx \left(
\begin{array}{rr}
0.42 & 0.58\, \\
0.42 & 0.58\,\end{array}\right) \\
\mathbb{H}_{2,3} &=&\frac{1}{34}\left(
\begin{array}{rr}
25-3\sqrt{3} & -25+3\sqrt{3} \\
-\left( 3\sqrt{3}+9\right) & 3\sqrt{3}+9\end{array}\right) \approx \left(
\begin{array}{rr}
0.58 & -0.58 \\
-0.42 & 0.42\end{array}\right)\ .\end{aligned}$$
Thus, asymptotically the average number of vehicles associated with each city (whose distribution will repeat daily) at the three moments of one day are
$$\begin{aligned}
\left(
\begin{array}{cc}
a_{\infty }^{\left( 1\right) } & b_{\infty }^{\left( 1\right) }\end{array}\right) &=&\lim_{r\rightarrow \infty }\left(
\begin{array}{cc}
a_{0} & b_{0}\end{array}\right) \mathbb{M}_{1}\left( \mathbb{M}_{2}\mathbb{M}_{3}\mathbb{M}_{1}\right) ^{r-1}=\left(
\begin{array}{cc}
a_{0} & b_{0}\end{array}\right) \mathbb{H}_{1,1} \notag \\
&=& \left( a_{0}+b_{0}\right)\ \left(
\begin{array}{rr}
\frac{3}{34}\left( 7-\sqrt{3}\right) & \frac{1}{34}\left( 3\sqrt{3}+13\right)\end{array}\right) \notag \\
&\approx & \left( a_{0}+\,b_{0}\right)\ \left(
\begin{array}{cc}
0.46 & 0.54\end{array}\right) \ , \label{Rnl3}\end{aligned}$$
$$\begin{aligned}
\left(
\begin{array}{cc}
a_{\infty }^{\left( 2\right) } & b_{\infty }^{\left( 2\right) }\end{array}\right) &=&\lim_{k\rightarrow \infty }\left(
\begin{array}{cc}
a_{0} & b_{0}\end{array}\right) \mathbb{M}_{1}\mathbb{M}_{2}\left( \mathbb{M}_{3}\mathbb{M}_{1}\mathbb{M}_{2}\right) ^{r}=\left(
\begin{array}{cc}
a_{0} & b_{0}\end{array}\right) \mathbb{H}_{1,2} \notag \\
&=& \left( a_{0}+b_{0}\right) \left(
\begin{array}{rr}
\frac{3}{17}\left( \sqrt{3}+3\right) & \frac{1}{17}\left( 8-3\sqrt{3}\right)\end{array}\right) \notag \\
&\approx & \left( a_{0}+\,b_{0}\right) \left(
\begin{array}{cc}
0.84 & 0.16\end{array}\right) \ , \label{Rnl4}\end{aligned}$$
$$\begin{aligned}
\left(
\begin{array}{cc}
a_{\infty }^{\left( 3\right) } & b_{\infty }^{\left( 3\right) }\end{array}\right) &=&\lim_{k\rightarrow \infty }\left(
\begin{array}{cc}
a_{0} & b_{0}\end{array}\right) \left( \mathbb{M}_{1}\mathbb{M}_{2}\mathbb{M}_{3}\right) ^{r}=\left(
\begin{array}{cc}
a_{0} & b_{0}\end{array}\right) \mathbb{H}_{1,3} \notag \\
&=& \left( a_{0}+b_{0}\right) \left(
\begin{array}{cc}
\frac{3}{34}\left( \sqrt{3}+3\right) & \frac{1}{34}\left( 25-3\sqrt{3}\right)\end{array}\right) \notag \\
&\approx & \left( a_{0}+\,b_{0}\right) \left(
\begin{array}{cc}
0.42 & 0.58\end{array}\right)\ . \label{Rnl5}\end{aligned}$$
In these last three equations (third lines) the numbers in each pair $(0.46,0,54)$, $(0.84,0.16)$ and $(0.42,0.58)$ relate to the fractions, of all vehicles, $a_0+b_0$, associated with each city at different moments. For instance, the first pair (morning monitoring) means that $46\%$ of all the vehicles are either in city $A$ or driving toward city $B$, while $54\%$ are in $B$ or driving toward $A$. The same holds for the two other pairs, although being for noon and evening respectively. These fractions change periodically and continuously, as schematized in Fig. \[cyclic\], where the arrows indicate the direction of the flow of vehicles. The results do not depend on the initial values $a_0$ and $b_0$ individually, but only on their sum, so some information is lost. The percentages depend only on the chosen parameters of the stochastic matrix.
![[]{data-label="cyclic"}](asimpciclico3.jpg){height="2.4in" width="3.2in"}
The flow of vehicles change cyclically, ruled by the stationary matrices $\mathbb{H}_{1,i}$ and $\mathbb{H}_{2,i}$ as displayed in Fig. [cyclicH]{}.
![[]{data-label="cyclicH"}](cyclicH.jpg){height="1.2in" width="1.6in"}
The global mean entropy also presents asymptotic regular cyclical changes as shown in Fig. \[NLentropy\], which is in line with Figs. \[cyclic\] and \[cyclicH\].
![[]{data-label="NLentropy"}](EntropiaNL.jpg){height="2.0in" width="3.0"}
As so, this particular nonlinear model leads to three stationary matrices (three fixed points in the language of dynamical systems) characterizing a 3-cycle continuous change of the vehicular traffic, that we assumed as being three moments of observation every day. This approach can be extended to $L$ stationary states at $L$ different moments. The model is scalable by increasing the number of sites and arteries that connect them, and the obtainment of numerical results depends only on computational capabilities. To construct a model that is more kin to the real traffic within a network it is advisable to insert the dynamics of input and output of vehicles (as considered in the previous model), thus turning it into a hybrid model.
Summary and conclusions
=======================
We presented three network models to picture the vehicular traffic between sites that could be cities, parking lots or car-rental agencies, and arteries (highways, roads) that connect the sites for the circulation of vehicles. We opted to use the mathematical formalism based on stochastic matrices to simulate the evolution in time of the distribution of vehicles. By doing a convenient decomposition of the dynamically evolved stochastic matrix in several modes we have separated the stationary matrix from the transient ones, and for these we have defined characteristic decay times. The first model considered a network without sources and sinks for the circulation of vehicles (a closed system) and using a numerical example we verified that after very few steps of evolution the vehicular distribution attains the stationary state. The second model consists of the same previous network accreted with a source and a sink, i.e., two additional sites and arteries. One extra site contains a certain number of new vehicles that are inserted daily into the network at a given rate, and the other extra site is a depository of old and crashed vehicles removed from the network, at an also established rate. Depending on numerical values chosen as entries of the SM’s, the main differences that result from the former model are: (1) very long decay times to attain the stationary state, where all the vehicles will go eventually to the depository site; (2) the evolution of the global mean entropy begins by increasing, then it attains a maximum value that decreases steeply, reaching zero soon after. We believe that this model could help urban planners to establish what could be the ideal injection of vehicles within a network in order to avoid an excessive density that could saturate the free circulation capacity within the arteries.
The third model, for two sites and two arteries connecting them, is nonlinear. Comparing the properties with the linear models, the differences are quite noteworthy: instead of a single stationary state, now multi-stationary states are possible and each mode has its own relaxation time although the PF eigenvalue ($\lambda_{PF} = 1$) is always present, even for the $n$-step evolution. We chose specific sinusoidal functions of $n$ in the entries of the SM, and asymptotically one gets $L$ stationary states. Therefore, the model admits the possibility to describe a particular situation for the traffic flow: $L$ different number of vehicles at each site, changing cyclically in one day. We consider that this model is more realistic to describe the possible variations of the traffic flow at different moments of the day.
Finally, we recall that all three models are scalable, i.e., depending on the computational resources, the network can be extended to a large number of vertices and edges. A following paper (Part II) makes use of the formalism and models discussed here to analyze of a real situation, the urban traffic within a sector of Tigre, a city located in the province of Buenos Aires, Argentina. The recorded traffic by video cameras was made available by the traffic controllers.
SSM thanks the CNPq, a Federal Brazilian Agency, for financial support.
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abstract: 'We propose an optical means to realize a spin hall effect (SHE) in neutral atomic system by coupling the internal spin states of atoms to radiation. The interaction between the external optical fields and the atoms creates effective magnetic fields that act in opposite directions on “electrically" neutral atoms with opposite spin polarizations. This effect leads to a Landau level structure for each spin orientation in direct analogy with the familiar SHE in semiconductors. The conservation and topological properties of the spin current, and the creation of a pure spin current are discussed.'
author:
- 'Xiong-Jun Liu$^{a}$[^1], Xin Liu$^{b,c}$, L. C. Kwek$^{a,d}$ and C. H. Oh$^{a}$[^2]'
title: Optically Induced Spin Hall Effect in Atoms
---
Information devices based on spin states of particles require a lot less power consumption than equivalent charge based devices [@spintronics1]. To implement practical spin-based logical operations, a basic underlying theory, i.e. spin hall effect (SHE) has been widely studied for the creation of spin currents in semiconductors [@zhang; @niu; @hall1; @hall2]. Nearly all current publications on SHE involve some form of spin-orbit coupling, including the interaction between charged particles in semiconductors and external electric field. The physics of SHE in semiconductors is: in the presence of spin-orbit coupling, the applied electric field leads to a transverse motion (perpendicular to the electric field), with spin-up and spin-down carriers moving oppositely to each other, creating a transverse spin current. However, spin current can also be generated by interacting optical fields with charged particles in semiconductors [@optical1; @optical2], even in absence of spin-orbit coupling [@liu].
In this letter, we show how SHE can be induced by optical fields in neutral atomic system. Quantum states of atoms can be manipulated by coupling their internal degrees of freedom (atomic spin states) to radiation, making it possible to control atomic spin propagation through optical methods. We consider here an ensemble of cold Fermi atoms interacting with two external light fields (Fig. 1). The ground ($|g_\pm,\pm\frac{1}{2}\rangle$) and excited ($|e_\pm,\pm\frac{1}{2}\rangle$) states are hyperfine angular momentum states (atomic spins) with their total angular momenta $F_g=F_e=1/2$.
![Fermi atoms with four-level internal hyperfine spin states interacting with two light fields. This can be experimentally realized with alkali atoms, such as $^6$Li atoms ($2^2S_{1/2}(F=1/2)\longleftrightarrow2^2P_{1/2}(F'=1/2)$) [@li].[]{data-label="fig1"}](fig_1.eps){width="0.7\columnwidth"}
The transitions from $|g_-,-\frac{1}{2}\rangle$ to $|e_+,\frac{1}{2}\rangle$ and from $|g_+,\frac{1}{2}\rangle$ to $|e_-,-\frac{1}{2}\rangle$ are coupled respectively by a $\sigma_+$ light with the Rabi-frequency $\Omega_2=\Omega_{20}\exp(i(\bold k_2\cdot\bold r+l_2\vartheta))$ and by a $\sigma_-$ light with the Rabi-frequency $\Omega_1=\Omega_{10}\exp(i(\bold k_1\cdot\bold r+l_1\vartheta))$, where $\bold k_{1,2}=k_{1,2}\bold{\hat e}_z$ are the wave-vectors and $\vartheta=\tan^{-1}({y}/{x})$. $l_1$ and $l_2$ indicate that $\sigma_+$ and $\sigma_-$ photons are assumed to have the orbital angular momentum $\hbar l_1$ and $\hbar l_2$ along the $+z$ direction, respectively [@angular]. For simplicity, we replace the notations $|\alpha_\pm,\pm\frac{1}{2}\rangle$ by $|\alpha_\pm\rangle \ (\alpha=e,g)$. The $\bold r$-representation atomic wave function is denoted by $\Phi_\alpha(\bold r,t)$. It is helpful to introduce the slowly-varying amplitudes of atomic wave-functions by (setting $\omega_{g_\pm}=0$): $\Psi_{g_\pm}=\Phi_{g_\pm}, \Psi_{e_+}=\Phi_{e_+}(\bold
r,t)e^{-i(\bold k_2\cdot\bold r-(\omega_{e_+}-\Delta_1)t)},
\Psi_{e_-}=\Phi_{e_-}(\bold r,t)e^{-i(\bold k_1\cdot\bold
r-(\omega_{e_-}-\Delta_2)t)}$, where $\hbar\omega_\alpha$ is the energy of the state $|\alpha\rangle$, $\Delta_{1,2}$ are corresponding detunings. The total Hamiltonian $H=H_0+H_1+H_2$ of the system reads: $$\begin{aligned}
\label{eqn:Hamiltonian1a}
H_0&=&\sum_{\alpha=e_{\pm}g_{\pm}}\int
d^3r\Psi_{\alpha}^*\bigr(-\frac{\hbar^2}{2m}\nabla^2+V(\bold
r)\bigr) \Psi_{\alpha},\nonumber\\
H_1&=&\hbar\Delta_1\int
d^3r\Psi^*_{e_+}S_{e_+e_+}\Psi_{e_+}\nonumber\\
&&+\hbar\int
d^3r(\Psi^*_{e_+}\Omega_{10}e^{il_1\vartheta}S_{1+}\Psi_{g_-}+h.a.),\\
H_2&=&\hbar\Delta_2\int d^3r\Psi^*_{e_-}S_{e_-e_-}\Psi_{e_-}\nonumber\\
&&+\hbar\int
d^3r(\Psi^*_{e_-}\Omega_{20}e^{il_2\vartheta}S_{2+}\Psi_{g_+}+h.a.),\nonumber\end{aligned}$$ where the atomic operators $S_{e_{\pm}e_{\pm}}=|e_{\pm}\rangle\langle e_{\pm}|$, $S_{1+}=|e_+\rangle\langle g_-|, S_{2+}=|e_-\rangle\langle g_+|$, $S^{\dag}_{1+}=S_{1-}$, $S^{\dag}_{2+}=S_{2-}$, and $V(\bold r)$ is an external trap potential. The collisions (s-wave scattering) between cold Fermi atoms are negligible. The interaction part of the Hamiltonian can be diagonalized with a local unitary transformation: $\tilde{H}_I=U(\bold r)H_IU^{\dag}(\bold
r)=U(\bold r)(H_1+H_2)U^{\dag}(\bold r)$ where $$\begin{aligned}
\label{eqn:unit1}
U(\bold r)={\left[ \begin{matrix} U_1& 0\\
0& U_2\end{matrix} \right]}\end{aligned}$$ with $$\begin{aligned}
\label{eqn:unit2}
U_j={\left[ \begin{matrix} \cos\theta_j& \sin\theta_je^{il_j\vartheta(\bold r)}\\
-\sin\theta_je^{-il_j\vartheta(\bold r)}& \cos\theta_j\end{matrix}
\right]}, \ \ j=1,2.\nonumber\end{aligned}$$ Under this transformation the four eigenstates of interaction Hamiltonian can be obtained as $\bigr[|\psi_+\rangle,
|\psi_-\rangle\bigr]^{{\small T}}=U_1\bigr[|e_+\rangle,
|g_-\rangle\bigr]^{T}$ and $\bigr[|\phi_+\rangle,
|\phi_-\rangle\bigr]^{\small T}=U_2\bigr[|e_-\rangle,
|g_+\rangle\bigr]^{T}$. The mixing angles $\theta_j$ are defined by $\tan\theta_{1,2}=E_{\psi,\phi}^-/|\Omega_{1,2}|$, where the eigenvalues of $|\psi_{\pm}\rangle$ and $|\phi_{\pm}\rangle$ can be calculated by: $E_{\psi,\phi}^{\pm}=(\Delta_{1,2}\pm\sqrt{\Delta_1^2+4|\Omega_{1,2}|^2})/2$.
In order to suppress the spontaneous emission of the excited states, we consider the large detuning case, i.e. $\Delta_j^2\gg\Omega_{j0}^2$. By calculating all values up to the order of $\Omega_{j0}^2/\Delta_j^2$, one can verify that $\tan^2\theta_j=\Omega^2_{j0}/(\Delta_j^2-\Omega_{j0}^2)$, and $E^+_{\psi,\phi}\gg E^-_{\psi,\phi}$. We then invoke the adiabatic condition so that the population of the higher levels $|\psi_+\rangle$ and $|\phi_+\rangle$ is adiabatically eliminated. Moreover, the total system is confined to the ground eigenstates $$\begin{aligned}
\label{eqn:state1}
|\Psi\rangle=\cos\gamma|S_\downarrow\rangle+\sin\gamma|S_\uparrow\rangle,\end{aligned}$$ where to facilitate further discussions, we have put the effective spin states [@liu]: $|S_\downarrow\rangle=|\psi_-\rangle$, $|S_\uparrow\rangle=|\phi_-\rangle$ with their $z$-component effective spin polarizations $S_z^{\uparrow}\approx\hbar/2$ and $S_z^{\downarrow}\approx-\hbar/2$. The parameter $\gamma$ describes the probability of an atom in states $|S_\downarrow\rangle$ and $|S_\uparrow\rangle$, determined through the initial condition. One can also see the population of excited states $|e_\pm\rangle$ is very small, therefore the atomic decay can be neglected in the present situation.
Under adiabatic condition, the local transformation $U(\bold r)$ leads to a diagonalized $SU(2)$ gauge potential: $(e/c)\bold
A_\alpha=-i\hbar\langle S_\alpha|\bold\nabla|S_\alpha\rangle, \
(\alpha=\downarrow,\uparrow)$. Accordingly, the effective (scalar) trap potentials read $V_\alpha(\bold r)=V(\bold
r)-\hbar|\Delta_j|^{-1}\Omega^2_{j0}-(2m)^{-1}|\hbar\langle
S_\alpha|\bold\nabla|S_\alpha\rangle|^2$ with $j=1$ (for $\alpha=\downarrow$) and $j=2$ (for $\alpha=\uparrow$). Specially we have $\bold A_\downarrow=-\bold A_\uparrow=\hbar
lce^{-1}\frac{\Omega_0^2}{\Delta^2}(x\hat e_y-y\hat e_x)/\rho^2$ and $V_{eff}(\bold r)=V_{\uparrow,\downarrow}(\bold r)=V(\bold
r)-\hbar\Omega^2_{0}/\Delta-\hbar^2l^2\Omega_0^4/(2m\Delta^4\rho^2)$ with $\rho=\sqrt{x^2+y^2}$, when we choose $\Delta_1=\Delta_2=\Delta$, $\Omega_{10}=\Omega_{20}=\Omega_0$ and $l_1=-l_2=l$, i.e. the angular momenta of the two light fields are opposite in direction. This result is intrinsically interesting: by coupling the atomic spin states to radiation, we find the atomic system can be described as an ensemble of charged particles with opposite spins experiencing magnetic fields in opposite directions but subject to the same electric field. With the help of gauge and trap potentials, we rewrite the Hamiltonian effectively as $$\begin{aligned}
\label{eqn:Hamiltonian2}
H&=&\int
d^3r\Psi_{s_\downarrow}^{*}\bigr[\frac{1}{2m}(\hbar\partial_k+i\frac{e}{c}A_k)^2\bigr]
\Psi_{s_\downarrow}\nonumber\\
&&+\int
d^3r\Psi_{s_\uparrow}^{*}\bigr[\frac{1}{2m}(\hbar\partial_k-i\frac{e}{c}A_k)^2\bigr]
\Psi_{s_\uparrow}\\&&+\int
d^3r(V_\downarrow|\Psi_{s_\downarrow}|^2+V_\uparrow|\Psi_{s_\uparrow}|^2)),\nonumber\end{aligned}$$ where $\Psi_{s_\downarrow}(\bold r)=\cos\gamma\langle\bold
r|S_\downarrow\rangle$ and $\Psi_{s_\uparrow}(\bold
r)=\sin\gamma\langle\bold r|S_\uparrow\rangle$ are spin wave functions in $\bold r$-representation.
Before calculating the spin current, we study the properties of spin currents in our model. We first consider a conservation law. The general spin density in the present system is calculated using $\vec S(\bold r,t)=\Psi_{s_\downarrow}^{*}\vec
S_\downarrow\Psi_{s_\downarrow}+\Psi_{s_\uparrow}^*\vec
S_\uparrow\Psi_{s_\uparrow}$ with $\vec
S_{\uparrow\downarrow}=(S_x^{\uparrow\downarrow},S_y^{\uparrow\downarrow},S_z^{\uparrow\downarrow})$. Moreover, the spin current density $\vec J_k(\bold r,
t)=(J_k^{s_x},J_k^{s_y},J_k^{s_z})$ is defined by $$\begin{aligned}
\label{eqn:spincurrent1}
\vec J_k&=&-\frac{i\hbar}{m}\vec S_{\downarrow}(\Psi_{s_\downarrow}^*D_{1k}\Psi_{s_\downarrow}-\Psi_{s_\downarrow}D^*_{1k}\Psi_{s_\downarrow}^*)\nonumber\\
&&-\frac{i\hbar}{m}\vec
S_{\uparrow}(\Psi_{s_\uparrow}^*D_{2k}\Psi_{s_\uparrow}-\Psi_{s_\uparrow}D^*_{2k}\Psi_{s_\uparrow}^*),\end{aligned}$$ where $D_{1k}=\partial_k+i\frac{e}{c}A_{k}$ and $D_{2k}=\partial_k-i\frac{e}{c}A_{k}$ are the covariant derivative operators. By a straightforward calculation, we verify the following continuity equation $$\begin{aligned}
\label{eqn:continuity}
\partial_t\vec S(\bold r,t)+\partial_k\vec J_k=A_k\sigma_z\hat{\bold
e}_z\times\vec J_k(\bold r, t),\end{aligned}$$ where $\sigma_z$ is the usual Pauli matrix. It is easy to see that the right hand side of Eq. (\[eqn:continuity\]) equals zero for the $s_z$-component spin current. Thus the spin current $J_k^{s_z}$ is conserved. But there is no spin-orbit coupling in the present atomic system. Thus, it is easily verified that the orbit-angular momentum current is also conserved. These results underlines the conservation law for $s_z$-component of total angular momentum current of atoms. Secondly, it is interesting that the two spin functions in the effective Hamiltonian (\[eqn:Hamiltonian2\]) are not independent: there is a nontrivial effective coupling mediated by an effective electromagnetic field. This implies that the spin current in our model could have a nontrivial hidden topology. We next discuss the topological properties of $J_k^{s_z}$. We note that $\Psi_{s_\alpha}=n_a^{1/2}\zeta_\alpha$, where the complex $\zeta_\alpha=|\zeta_\alpha|e^{-i\varphi_\alpha}$ with $|\zeta_\downarrow|^2+|\zeta_\uparrow|^2=1$, and $n_a$ is the total atomic density which is assumed to be constant in our case. The $s_z$-component of eq. (\[eqn:spincurrent1\]) can be recast as $$\begin{aligned}
\label{eqn:spincurrent2}
J_k^{s_z}&=&-i\frac{n_a\hbar^2}{2m}(\zeta_\downarrow\partial_k\zeta_\downarrow^*-
\zeta_\downarrow^*\partial_k\zeta_\downarrow+\zeta_\uparrow^*\partial_k\zeta_\uparrow-
\zeta_\uparrow\partial_k\zeta_\uparrow^*)-\nonumber\\
&&-\frac{n_a\hbar^2e}{mc}A_k.\end{aligned}$$ Furthermore, we introduce the unit vector field, $\vec\lambda=(\bar\zeta, {\vec\sigma}\zeta)$, where $\zeta=(\zeta_\downarrow,\zeta_\uparrow^*)^T$ and $\vec\sigma=(\sigma_x,\sigma_y,\sigma_z)$. It then follows that $\lambda_1=\zeta_\downarrow^*\zeta_\uparrow^*+\zeta_\downarrow\zeta_\uparrow,
\lambda_2=i(\zeta_\downarrow\zeta_\uparrow-\zeta_\downarrow^*\zeta_\uparrow^*)$ and $\lambda_3=|\zeta_\downarrow|^2-|\zeta_\uparrow|^2$. Using these definitions we have $$\begin{aligned}
\label{eqn:spincurrent3}
(\bold\nabla\times{\bold
J}^{s_z})_m=-\frac{n_a\hbar^2e}{mc}B_m-\frac{n_a\hbar^2}{2m}\epsilon_{mkl}\vec\lambda
\cdot(\partial_k\vec\lambda\times\partial_l\vec\lambda).\end{aligned}$$ The contribution $\vec\lambda\cdot(\partial_k\vec\lambda\times\partial_l\vec\lambda)$ provides a topological term in the spin current induced by optical fields. Note each value of $\vec\lambda$ represents a point in the two-dimensional sphere $S^2$. The variation of $\vec\lambda$ depends on the relative density of the two spin components $\gamma(\bold r)$ and the sum of the phase spreadings $\varphi_\uparrow(\bold r)+\varphi_\downarrow(\bold r)$. It is easily verified that $\vec\lambda$ can cover the entire surface $S^2$, when the parameter distributions in the interaction region satisfy: $\varphi_1+\varphi_2: 0\rightarrow2n_1\pi$ and $\gamma:
0\rightarrow n_2\pi/2$ with integers $n_1,n_2\geq1$. We then obtain a map degree between the unit vector field $\vec\lambda$ and the spatial vectors $F:\vec\lambda\rightarrow\bold r/r$ so that the closed-surface integral $$\oint_s(\bold\nabla\times{\bold J}^{s_z})\cdot d\bold
S\sim\oint_sds_m\epsilon_{mkl}\frac{\bold
r}{r}\cdot(\partial_k\frac{\bold r}{r}\times\partial_l\frac{\bold
r}{r})\sim4\pi n,$$ where $n$ is the winding number. In physics, such mapping corresponds to the formation of a local inhomogeneity in the densities of the spin-up and spin-down atoms. The Hamiltonian (\[eqn:Hamiltonian2\]) can also be regarded as a system of $two$-$flavor$ oppositely charged particles interacting with $one$ external effective magnetic field. Such model has a wide range of applications to, e.g. two-band superconductivity [@super], etc. It is interesting to note that, under certain conditions liquid metallic hydrogen, as an example, might allow for the coexistence of superconductivity with both electronic and protonic Cooper pairs [@cooper]. Faddeev et al. have also discovered a series of nontrivial topological properties in such systems [@faddeev]. Based on this technique, we have developed an optical way to realize $two$-$flavor$ artificially charged system, which could provide a deeper understanding of the essential physical mechanisms in cold atomic systems.
For practical application, an important goal is to create a pure spin current injection, where the massive current is zero. To this end, we propose using a columnar spreading light fields that [@angular] $\Omega_{0}(\bold r)=f\rho$ with $f>0$. Furthermore, we set the $x$-$y$ harmonic trap $V(\bold
r)=\frac{1}{2}m\omega^2_\perp(\vec\rho+x_0\bold{\hat e}_x)^2$ centered at $\vec\rho=-x_0\bold{\hat e}_x$, where the frequency is tuned to $\omega_\perp^2=(1+\frac{\hbar
l^2f^2}{2m\Delta^3})\frac{2\hbar f^2}{m\Delta}$. It should be emphasized that the (positive) potential $V(\bold r)$ can typically be obtained with a blue-detuned dipole trap method [@trap] for instance. The uniform magnetic and electric fields corresponding to the gauge and scalar potentials are $$\begin{aligned}
\label{eqn:field1} \bold B_\downarrow(-\bold
B_\uparrow)=\frac{\hbar l c}{e}\frac{f^2}{\Delta^2}\hat e_z,
\bold E=-\bigr(1+\frac{\hbar
l^2f^2}{4m\Delta^3}\bigr)\frac{2\hbar f^2x_0}{e\Delta}\hat e_x,\end{aligned}$$ along the $z$ ($-z$) and $x$ directions respectively. Atoms in different spin states $|S_\alpha\rangle$ experience the opposite magnetic fields $\bold B_\alpha$ but the same electric field $\bold E$. This leads to a Landau level structure for each spin orientation. To calculate the spin currents explicitly, one needs to obtain the eigenstates of the present system. However, the present forms of gauge and scalar potentials in the Hamiltonian (\[eqn:Hamiltonian2\]) cannot be diagonalized easily; consequently, we need to resort to perturbation theory to calculate the spin currents. The Hamiltonian for a single atom in state $|S_{\alpha}\rangle$ can be written as: $H^{\alpha}=H^{\alpha}_0+H'$ ($\alpha=\downarrow,\uparrow$), where the perturbation part $H'=-eEx$ and $$\begin{aligned}
\label{eqn:hamiltonian3}
H^{\alpha}_0=\frac{\hbar^2eB}{2mc}(R_{\alpha}^2+P_{\alpha}^2)+\frac{1}{2m}p_z^2\end{aligned}$$ with $R_{\uparrow\downarrow}=(\frac{c}{eB})^{1/2}(p_x-\frac{e}{c}A^{\uparrow\downarrow}_x),
P_{\uparrow\downarrow}=(\frac{c}{eB})^{1/2}(p_y-\frac{e}{c}A^{\uparrow\downarrow}_y)$ and $B=|\bold B_{\uparrow,\downarrow}|$. One can verify that $[R_{\alpha},P_{\beta}]=i\hbar\delta_{\alpha\beta}$, so the eigenfunction $\mu^{\alpha}_{nk}(R)$ of $H^{\alpha}_0$ is Hermite polynomial with the Landau level $E_{nk,\alpha}=(n+1/2)\hbar\omega+\hbar k^2/2m$ and $\omega=eB/mc$. For the weak field case, the spin/massive current carried by an atom can be calculated perturbatively to the first order correction on the state $|\mu^{\alpha}_{nk}\rangle$ $$\begin{aligned}
\label{eqn:spincurrent4}
(j^y_{s_z,m})_{nk,\alpha}&=&\langle\mu_{nk}^\alpha|j^y_{s_z,m}|\mu_{nk}^{\alpha}\rangle+\nonumber\\
&&+\bigl(\sum_{n'\alpha'}\frac{\langle\mu^{\alpha'}_{n'k}|H'|\mu^{\alpha}_{nk}
\rangle\langle\mu^{\alpha}_{nk}|j^y_{s_z,m}|\mu^{\alpha'}_{n'k}\rangle}{E_{n,\alpha}-E_{n',\alpha'}}\nonumber\\
&&+h.a.\bigl),\end{aligned}$$ where the spin current operator is $(j^y_{s_z})_{\alpha}=\frac{\hbar}{2}(S^{\alpha}_zv^{\alpha}_y+v^{\alpha}_yS^{\alpha}_z)$ with $v^{\alpha}_y=[y,H^{\alpha}]/ih=(eB/m^2c)^{1/2}P_{\alpha}$ and the massive current operator $j_{m}=mv_y$. It is easy to see that $\langle\mu^{\alpha}_{nk}|j^y_{s_z,m}|\mu^{\alpha'}_{n'k}\rangle=0$ when $n'\neq n\pm1$, thus only the terms with $n'=n\pm1$ contribute in the above equation.
If the spatial spreads of the atoms in $x$ and $z$ directions are $L_x$ and $L_z$, respectively, the average current density for the total system is given by $J^y_{s_z,m}=\frac{1}{L_xL_z}\int
dE\bigr((j^{(0)}_{s_z,m})_{nk,\alpha}
+(j^{(1)}_{s_z,m})_{nk,\alpha}\bigr)f(E)$ with $f(E)$ as the Fermi distribution function. For $^6$Li atoms we may consider the initial condition that $\sin^2\gamma=\cos^2\gamma=1/2$, i.e. the atoms have the equal probability in state $|S_\downarrow\rangle$ and $|S_\uparrow\rangle$, as in the usual optical trap [@trap1]. Rewriting the perturbation part as $H'=i\hbar
E(\frac{ec}{B})^{1/2}\frac{\partial}{\partial R}$ and substituting this result into Eq (\[eqn:spincurrent4\]) and finally we get $$\begin{aligned}
\label{eqn:spincurrent7}
J^y_{s_z}=n_a(\hbar+\frac{\hbar^2l^2f^2}{4m\Delta^3})\frac{\Delta
x_0}{l},\end{aligned}$$ whereas the massive current $J_m$ is zero. In fact, under present interaction of external effective electric and magnetic fields, the atoms with opposite velocities in $y$ direction have opposite spin polarizations (see Fig.2). Thus the massive current vanishes whereas a pure spin current is obtained. This result allows us to create conserved spin currents without using atomic beams.
![(Color online) Under the interaction of effective electric and magnetic fields induced by light fields, atoms in state $|S_\downarrow\rangle$ and $|S_\uparrow\rangle$ have opposite momenta in $y$ direction.[]{data-label="fig1"}](fig_2.eps){width="0.8\columnwidth"}
To observe SHE in present cold atomic system, a spin-sensitive measurement [@niu1] can be used. For example, the technique of atom chip [@atomchip] can be used to implement the spin current described in the present model. The created spin current can lead to spin accumulations on opposite sides along $y$ direction of the atom chip. Experimentally, one could detect such spatially separated spin polarizations using magnetic resonance force microscopy (MRFM) [@mrfm] for instance. Another means to detect the spin accumulation can also be achieved with fluorescence. For $^6$Li atoms, one may perform resonant Raman transitions from the accumulated spin state $S_z=-1/2$ to $2^2P_{1/2} (F=3/2, m=-3/2)$ or from $S_z=1/2$ to $2^2P_{1/2}
(F=3/2, m=3/2)$ ($D_1$ line) and then observe the fluorescence. As long as the spin accumulations are separated to say larger than $5-10$ microns, the two separated images can be resolved in an experiment.
One should also discuss the adiabatic condition employed in above calculations. Atomic motion may lead to transitions between ground eigenstates and excited ones, e.g. the element of transition between $|\psi_+\rangle$ and $|\psi_-\rangle$ can be calculated by $\tau_\pm=|\langle\psi_+|\partial_t|\psi_-\rangle|=|\bar{\bold
v}\cdot\nabla\theta_1(\bold
r)+\frac{1}{2}l_1\sin2\theta_1\bar{\bold
v}\cdot\nabla\vartheta(\bold r)|$ where $\bar{\bold v}$ is the average resulting velocity in spin currents. The adiabatic condition requires $\tau_\pm\ll|E_\psi^+-E_\psi^-|$. For a numerical evaluation one typically set the parameters $\Delta\sim10^8$s$^{-1}$, $l<10^4, f=5\times10^{10}$(s$\cdot$ m)$^{-1}$, $x_0 \approx 2.0 \mu m$. We then find the velocity $\bar{v}<1.0$ m/s and $\tau_\pm/|E_\psi^+-E_\psi^-|\sim10^{-3}\ll1$, which guarantees the validity of the adiabatic condition. In the case of $^6$Li atomic system, the $x$-$y$ trap potential $V(\bold r)$ can be achieved through $D_2$ transition (from $2^2S_{1/2}$ to $2^2P_{3/2}$) with a blue detuning [@li; @trap]. With former parameters and by tuning the trap frequency to $\omega_\perp\approx 728.5$Hz, one achieves the uniform magnetic and electric fields in Eq. (\[eqn:field1\]). Furthermore, if we employ an atomic system with the atomic density $n_a \approx1.0
\times10^{10}$cm$^{-3}$, one can find the spin current $J^y_{s_z}\approx 1.322 \times10^{-5}$eV/cm$^{2}$. On the other hand, we note that for all practical purposes, the light fields have a finite cross section $S_{xy}$, which may have a boundary effect on the calculation of spin currents. For the cold Fermi atomic gas, the spatial scale of the interaction region is about $0.1$ mm [@trap1], then such boundary effect can be neglected since $lf^2S_{xy}/\Delta^2\sim10^2\gg 1$. This means the effective magnetic flux induced by the light fields can support a sufficiently large degeneracy at each Landau level.
In conclusion we proposed an optical means to realize a new type of spin hall effect in the neutral atomic system. The spin current created in this way is conserved and could possess interesting topological properties. The present atomic system is equivalent to a [*two-flavor*]{} artificially charged system, providing a direct analogy between the dynamics of electrons in solid systems, e.g. two-band superconductors [@super; @cooper], and the behavior of cold atoms in optical potentials. The effect also provides understanding of the basic physical mechanisms of SHE with a wide range of applications in cold atomic systems.
We thank M. Barrett, R. Ballagh and J. Fuchs for fruitful discussions and helpful comments. This work is supported by NUS academic research Grant No. WBS: R-144-000-172-101, US NSF under the grant DMR-0547875, and NSF of China under grants No.10275036.
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[^1]: Electronic address: phylx@nus.edu.sg
[^2]: Electronic address: phyohch@nus.edu.sg
|
---
abstract: 'We consider a “Scalar-Maxwell-Einstein-Gauss-Bonnet” theory in four dimension, where the scalar field couples non-minimally with the Gauss-Bonnet (GB) term. This coupling with the scalar field ensures the non topological character of the GB term. In such higher curvature scenario, we explore the effect of electromagnetic field on scalar field collapse. Our results reveal that the presence of a time dependent electromagnetic field requires an anisotropy in the background spacetime geometry and such anisotropic spacetime allows a collapsing solution for the scalar field. The singularity formed as a result of the collapse is found to be a curvature singularity which may be point like or line like depending on the strength of the anisotropy. We also show that the singularity is always hidden from exterior by an apparent horizon.'
author:
- |
Narayan Banerjee[^1]\
Department of Physical Sciences,\
Indian Institute of Science Education and Research Kolkata,\
Mohanpur Campus, Nadia, West Bengal 741246, India.\
Tanmoy Paul[^2]\
Department of Theoretical Physics,\
Indian Association for the Cultivation of Science,\
2A $\&$ 2B Raja S.C. Mullick Road,\
Kolkata - 700 032, India.\
title: '**Electromagnetic effect on anisotropic scalar field collapse in higher curvature gravity**'
---
Introduction
============
Relativistic astrophysics have gone through extensive developments over a few decades, following the discovery of high energy phenomena in the universe such as gamma ray bursts. Interesting physical properties can be emerged from compact stellar objects like neutron stars where the effect of strong gravity fields and hence general relativity are seen to play a fundamental role. The high strength of gravitational field is also present in the end stage of a continual gravitational collapse of a massive star. This collapsing phenomena, dominated by the force of gravity, is fundamental in black hole physics and have received increasing attention in the past decades. The first systematic analysis of gravitational collapse in general relativity was given way back in 1939 by Oppenheimer and Snyder [@oppenheimer] (see also the work by Datt [@datt]). Subsequent developments in the study of gravitational collapse have been comprehensively explored by Joshi[@joshi1; @joshi2].\
Scalar fields have been of great interest in the gravity sector for its own reasons. The various forms of scalar field potential are good enough for cosmological requirements such as playing the role of the driver of the present or the early accelerated expansion of the universe. A suitable scalar potential can often mimic different equations of state of fluid distribution. Thus scalar fields, although hardly motivated by other branches of physics for its raison de etre, always enjoyed a lot of attention in gravitational physics.\
Scalar fields are also quite popular in the context of collapsing spacetime geometry. The collapsing phenomena of a massless scalar field was discussed by Christodoulou [@1]. The possibilities of end product of a scalar field collapse, whether a naked singularity or a black hole, has also been explored in [@2]. The variants of scalar field collapse and its consequences are demonstrated in [@4; @5; @6; @7; @8; @cai1; @cai2; @9; @10; @17; @18] (see also [@soumya; @nb4; @13; @14; @15; @16]).\
The implementation of electromagnetic field in cosmological and astrophysical processes is an attractive research area in theoretical physics. Many investigations in this direction are devoted to understand the interaction between electromagnetic and gravitational fields. Bekenstein [@new1] extended the work from neutral to charged case by generalizing Oppenheimer-Volkoff equations [@new2] regarding the force balance of a star. Since then a considerable amount of work has been done in this scenario. Rosales et al. [@new3] figured out that electric charge plays the same role as that of anisotropy in the collapse, when the radial pressure is less than the tangential pressure. Thorne studied [@thorne] cylindrically symmetric gravitational collapse with magnetic field and concluded that magnetic field can prevent the collapse of cylinder before singularity formation. Ardavan and Partovi [@ardavan] investigated dust solution of the field equations with electromagnetic field and found that the electrostatic force is balanced by gravitational force during collapse of charged dust. Stein-Schabes [@stein] investigated that charged matter collapse may produce naked singularity instead of a black hole. Germani and Tsagas [@germani] discussed the collapse of magnetized dust in Tolman–Bondi model. Recently, Herrera and his collaborators [@herrera; @prisco] have discussed the role of electromagnetic field on the structure scalars and dynamics of self-gravitating objects. Sharif and his collaborators [@sharif1; @sharif2; @sharif3] have extended this work for cylindrical and plane symmetries.\
Among the present emphasis in gravitational physics, a theoretical search for an alternative and may be more fundamental theories of gravity is of a central attraction. The most usual way to modify Einstein’s theory of gravity in a four dimensional context is to add higher curvature terms which should arise naturally from the requirement of general coordinate invariance. Such corrections to Einstein’s gravity have their natural origin in a fundamental theory like “String Theory”. In this context F(R) [@paul1; @s1; @s2; @24; @25; @26], Gauss-Bonnet (GB) [@s1; @s2; @27; @28; @29; @maeda; @mann] or more generally Lanczos-Lovelock [@lanczos; @lovelock] gravity are some of the candidates in higher curvature gravitational theory. Higher curvature terms become extremely relevant at the regime of large curvature. The spacetime curvature inside a collapsing star gradually increases as the collapse continues and becomes very large near the final state of the collapse. Thus for a collapsing geometry where the curvature becomes very large near the final state of the collapse, the higher curvature terms are expected to play a crucial role. Motivated by this idea, the collapsing scenarios in the presence of higher curvature gravity have been discussed in [@paul2; @rituparno; @nb1; @nb3].\
In the present work, we investigate the possible effects of electromagnetic field in a scalar field collapse in the presence of higher curvature like Gauss-Bonnet gravity. The advantage of Gauss-Bonnet (GB) gravity is that the equations of motion do not contain any higher derivative terms (higher than two) of the metric and thus leads to ghost free solution. The particular questions that we addressed in this paper are the following,
1. What are the possible effects of electromagnetic field on scalar field collapse in the presence of Gauss-Bonnet gravity?
2. What is the end product of the collapse, a black hole or a naked singularity?
In order to address the above questions, we consider a “Scalar-Maxwell-Einstein-Gauss-Bonnet” theory in four dimension where the scalar field is coupled non-minimally to the GB term. It may be mentioned that without the ’non-minimal’ coupling, the GB contribution in the action does not contribute nontrivially to the field equations in less than 5 dimensions. The presence of electromagnetic field forces us to consider an anisotropic spacetime model which is discussed in section 2. In section 3, we obtain the exact solution for the metric. Section 4 and 5 address the visibility of the singularity produced as a result of the collapse and a matching of the solution with an exterior spacetime respectively. We end the paper with some concluding remarks in section 6.\
The model
=========
To explore the effect of electromagnetic field on scalar field collapse in presence of higher curvature like Gauss-Bonnet (GB) gravity, we consider a “Scalar-Maxwell-Einstein-Gauss-Bonnet” theory in four dimensions where the GB term is coupled with the scalar field. This coupling guarantees the non topological character of the GB term. The action for this model is given by, $$\begin{aligned}
S = \int d^4x \sqrt{-g} \bigg[\frac{R}{2\kappa^2} - \frac{1}{2}g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi - V(\Phi) + \frac{1}{8}\xi(\Phi)G
- \frac{1}{4}F_{\mu\nu}F^{\mu\nu}\bigg]
\label{action}\end{aligned}$$
where $g$ is the determinant of the metric, $R$ is the Ricci scalar, $1/(2\kappa^2)=M_{p}^2$ is the four dimensional squared Planck mass, $G=R^2-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\alpha\beta}
R^{\mu\nu\alpha\beta}$ is the GB term, $\Phi$ denotes the scalar field also endowed with a potential $V(\Phi)$. The coupling between scalar field and GB term is symbolized by $\xi(\Phi)$ in the action. The last term in the action denotes the electromagnetic field lagrangian where $F_{\mu\nu}$ is the electromagnetic field tensor and is defined by : $F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$, $A_{\mu}$ is the electromagnetic four-potential.\
To obtain the gravitational field equation, we need to determine the energy-momentum tensor for the scalar field ($\Phi$) and for the electromagnetic field ($A_{\mu}$) respectively. These stress tensors have the following expressions : $$\begin{aligned}
T_{\mu\nu}(\Phi)&=&\frac{2}{\sqrt{-g}} \frac{\delta}{\delta g^{\mu\nu}}\bigg[\sqrt{-g}\bigg(\frac{1}{2}g^{\alpha\beta}\partial_{\alpha}\Phi
\partial_{\beta}\Phi + V(\Phi)\bigg)\bigg]\nonumber\\
&=&\bigg[\partial_{\mu}\Phi\partial_{\nu}\Phi - \frac{1}{2}g_{\mu\nu}\partial_{\alpha}\Phi\partial^{\alpha}\Phi - g_{\mu\nu}V(\Phi)\bigg]
\label{em tensor 1}
\end{aligned}$$
for the scalar field $\Phi$ and
$$\begin{aligned}
T_{\mu\nu}(A)&=&\frac{2}{\sqrt{-g}} \frac{\delta}{\delta g^{\mu\nu}}\bigg[\frac{1}{4}\sqrt{-g}F_{\alpha\beta}F^{\alpha\beta}\bigg]\nonumber\\
&=&\bigg[g^{\alpha\beta}F_{\mu\alpha}F_{\nu\beta} - \frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\bigg]
\label{em tensor 2}\end{aligned}$$
for the electromagnetic field. These above expressions of energy-momentum tensor along with the variation of the action with respect to $g^{\mu\nu}$ leads to the gravitational field equation as follows : $$\begin{aligned}
&\frac{1}{\kappa^2}&G_{\mu\nu} + R_{\mu\nu\alpha\beta}\nabla^{\alpha}\nabla^{\beta}\xi - R_{\mu\nu}\Box\xi
+ R_{\alpha\nu}\nabla_{\mu}\nabla^{\alpha}\xi + R_{\mu\beta}\nabla^{\beta}\nabla_{\nu}\xi - \frac{1}{2}\nabla_{\mu}\nabla_{\nu}\xi\nonumber\\
&-&\frac{1}{2}g_{\mu\nu}\bigg(2R_{\alpha\beta}\nabla^{\alpha}\nabla^{\beta}\xi - R\Box\xi\bigg)
= \bigg(\partial_{\mu}\Phi\partial_{\nu}\Phi - \frac{1}{2}g_{\mu\nu}\partial_{\alpha}\Phi\partial^{\alpha}\Phi - g_{\mu\nu}V(\Phi)\bigg)\nonumber\\
&+&\bigg(g^{\alpha\beta}F_{\mu\alpha}F_{\nu\beta} - \frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\bigg)~~~~,
\label{gravitational equation}
\end{aligned}$$
where $G_{\mu\nu}$ is the Einstein tensor and $\Box$ ($=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$) symbolizes the d’Alembertian operator. It may be noticed that the above equation of motion does not contain any derivative of the metric components higher than two. Similarly the scalar field and the electromagnetic field equations are given by : $$\begin{aligned}
\Box\Phi - V'(\Phi) + \frac{1}{8}\xi'(\Phi)G - \frac{1}{2}F_{\mu\nu}F^{\mu\nu} = 0
\label{scalar equation}
\end{aligned}$$
and
$$\begin{aligned}
\nabla_{\mu}\bigg(\partial^{\mu}A^{\nu} - \partial^{\nu}A^{\mu}\bigg) = 0~~~,
\label{electromagnetic equation}
\end{aligned}$$
where a prime denotes the derivative with respect to the scalar field $\Phi$. It is well known that Einstein-Gauss-Bonnet gravity in four dimensions reduces to standard Einstein gravity, the additional terms actually cancel each other. In the present case, the non-minimal coupling with the scalar field assists the contribution from the GB term survive. It is easy to see, in all the field equations above, that a constant $\xi$ (essentially no coupling) would immediately make the GB contribution trivial.\
The aim here is to construct a model for a continual collapse. A non static metric ansatz for the interior is taken that fits our purpose. We also consider that the scalar field as well as the electromagnetic field (or gauge field) are homogeneous in space. Under such condition, it can be shown that an anisotropy is essential in order to sustain an electromagnetic field (see Appendix - I). As a candidate of anisotropic model, here we consider the following Bianchi-I metric for interior spacetime, $$\begin{aligned}
ds^2 = -dt^2 + e^{[2\alpha(t) + 2\sigma(t)]} \bigg(dr^2 + r^2d\theta^2\bigg) + e^{[2\alpha(t) - 4\sigma(t)]}dz^2~~~~~.
\label{ansatz2}
\end{aligned}$$
It is evident that $e^{\alpha+\sigma}$ and $e^{\alpha-2\sigma}$ are the scale factor along radial direction and along $z$ direction respectively. Hence the Hubble parameter along radial ($H_r$) and $z$ ($H_z$) direction are defined as follows : $$\begin{aligned}
H_r&=&\dot{\alpha} + \dot{\sigma}~,\nonumber\\
H_z&=&\dot{\alpha} - 2\dot{\sigma}~~~~~~.
\label{hubble}
\end{aligned}$$
Therefore due to the introduction of the gauge field $A_{\mu}(t)$, the spatial isotropy is broken and the deviation from isotropy is controlled by $\sigma(t)$. However the metric in eqn.(\[ansatz2\]) clearly indicates that $\frac{\partial}{\partial\theta}$, $\frac{\partial}{\partial z}$ are the two killing vector fields for the interior spacetime. Therefore the interior geometry possesses a cylindrical symmetry with $z$ as the longitudinal direction which implies that the anisotropy is generated along the $z$ direction. Hence the component of $A_{\mu}(t)$ can be taken as : $$\begin{aligned}
A_{\mu}(t) = \bigg(0, 0, 0, v(t)\bigg)~~~~~~~~~.
\nonumber
\end{aligned}$$
With these above components of $A_{\mu}(t)$, eqn.(\[electromagnetic equation\]) turns out to be : $$\begin{aligned}
\frac{d}{dt}\bigg[e^{\alpha + 4\sigma} \dot{v}\bigg] = 0~~~~~~~,
\label{electromagnetic equation anisotropy1}
\end{aligned}$$
which can be solved to yield
$$\begin{aligned}
\dot{v}(t) = C e^{[-\alpha(t) - 4\sigma(t)]}~~~~~~~~,
\label{electromagnetic equation anisotropy2}
\end{aligned}$$
where $C$ is the constant of integration and an overdot represents the derivative with respect to time ($t$). Using the metric presented in eqn.(\[ansatz2\]) along with the above solution of $\dot{v}(t)$, eqn.(\[gravitational equation\] can be simplified and takes the following form : $$\begin{aligned}
\dot{\alpha}^2 = \dot{\sigma}^2 + \frac{\kappa^2}{3}\bigg[V(\Phi) + \frac{\dot{\Phi}^2}{2} + \frac{C^2}{2}e^{-4\alpha-4\sigma}\bigg]
- \kappa^2\dot{\xi}\big(\dot{\alpha} - 2\dot{\sigma}\big)\big(\dot{\alpha} + \dot{\sigma}\big)^2~~~~,
\label{grav eqn anisotropy1}
\end{aligned}$$
$$\begin{aligned}
\ddot{\sigma}&=&-3\dot{\alpha}\dot{\sigma} + \frac{\kappa^2}{3}C^2e^{-4\alpha-4\sigma} - \kappa^2\ddot{\xi} \big(\dot{\alpha}\dot{\sigma}
+ \dot{\sigma}^2\big)\nonumber\\
&-&\kappa^2\dot{\xi}\bigg[\dot{\alpha} \big(3\dot{\sigma}^2+\ddot{\alpha}\big) + \dot{\sigma}\big(\ddot{\alpha}+2\ddot{\sigma}\big)
+ 3\dot{\alpha}^2\dot{\sigma}\bigg]~~~~~~~~,
\label{grav eqn anisotropy2}
\end{aligned}$$
$$\begin{aligned}
\ddot{\alpha}= -3\dot{\alpha}^2 + \kappa^2\bigg[V(\Phi) + \frac{C^2}{6}e^{-4\alpha-4\sigma}\bigg] + \frac{\kappa^2}{2}\ddot{\xi}
\big(-\dot{\alpha}^2 + \dot{\sigma}^2\big)\nonumber\\
+ \frac{\kappa^2}{2}\dot{\xi}\bigg[-5\dot{\alpha}^3 + \dot{\alpha}\big(9\dot{\sigma}^2-2\ddot{\alpha}\big) + 4\dot{\sigma}^3
+ 2\dot{\sigma}\ddot{\sigma}\bigg]~~~~~~~~~.
\label{grav eqn anisotropy3}
\end{aligned}$$
Similarly the scalar field equation of motion (see eqn.(\[scalar equation\])) leads to the following form (recall that the scalar field depends only on the coordinate $t$),
$$\begin{aligned}
\ddot{\Phi}&=&-3\dot{\alpha}\dot{\Phi} - V'(\Phi) + 3\xi'(\Phi)\big(\dot{\alpha}+\dot{\sigma}\big)\nonumber\\
&\bigg[&\dot{\alpha}^3
- \dot{\alpha}^2\dot{\sigma} + \dot{\alpha}\big(-2\dot{\sigma}^2+\ddot{\alpha}\big) - \dot{\sigma}\big(\ddot{\alpha}+2\ddot{\sigma}\big)\bigg]~~~.
\label{scalar eqn anisotropy} \end{aligned}$$
It is evident that due to presence of Gauss-Bonnet term, cubic as well as quartic powers of $\dot{\alpha}$ and $\dot{\sigma}$ appear in the above equations. This indicates the non triviality of the Gauss-Bonnet term in presence of the coupling function $\xi(\Phi)$ even in four dimension.
Exact solutions : anisotropic collapsing model
==============================================
In this section, we present an analytic solution of the field equations (eqn.(\[grav eqn anisotropy1\]) to eqn.(\[scalar eqn anisotropy\])) and in order to do this, we consider a string inspired model [@nojiri_55] as follows, $$\begin{aligned}
V(\Phi) = V_0 e^{-2\Phi/\Phi_0}~~~~~,
\nonumber\end{aligned}$$ and $$\begin{aligned}
\xi(\Phi) = \xi_0 e^{2\Phi/\Phi_0}~~~~~~,\end{aligned}$$
where $V_0$, $\xi_0$ and $\Phi_0$ are the parameters of the model. With these forms of $V(\Phi)$ and $\xi(\Phi)$, eqn.(\[grav eqn anisotropy1\]) to eqn.(\[scalar eqn anisotropy\]) turn out be $$\begin{aligned}
\dot{\alpha}^2&=&\dot{\sigma}^2 + \frac{\kappa^2}{3}\bigg[V_0 e^{-2\Phi/\Phi_0} + \frac{\dot{\Phi}^2}{2}
+ \frac{C^2}{2}e^{-4\alpha-4\sigma}\bigg]\nonumber\\
&-&\frac{2\kappa^2\xi_0}{\Phi_0}e^{2\Phi/\Phi_0}\dot{\Phi}\big(\dot{\alpha} - 2\dot{\sigma}\big)\big(\dot{\alpha} + \dot{\sigma}\big)^2~~~~,
\label{new1}\end{aligned}$$
$$\begin{aligned}
\ddot{\sigma}&=&-3\dot{\alpha}\dot{\sigma} + \frac{\kappa^2}{3}C^2e^{-4\alpha-4\sigma} - \kappa^2\xi_0e^{2\Phi/\Phi_0}
\big(\frac{2}{\Phi_0}\ddot{\Phi}+\frac{4}{\Phi_0^2}\dot{\Phi}^2\big)\big(\dot{\alpha}\dot{\sigma}
+ \dot{\sigma}^2\big)\nonumber\\
&-&\frac{2\kappa^2\xi_0}{\Phi_0}e^{2\Phi/\Phi_0}\dot{\Phi}\bigg[\dot{\alpha} \big(3\dot{\sigma}^2+\ddot{\alpha}\big)
+ \dot{\sigma}\big(\ddot{\alpha}+2\ddot{\sigma}\big)
+ 3\dot{\alpha}^2\dot{\sigma}\bigg]~~~~~~~~,
\label{new2}\end{aligned}$$
$$\begin{aligned}
\ddot{\alpha}&=&-3\dot{\alpha}^2 + \kappa^2\bigg[V_0e^{-2\Phi/\Phi_0} + \frac{C^2}{6}e^{-4\alpha-4\sigma}\bigg]
+ \kappa^2\xi_0e^{2\Phi/\Phi_0}\big(\frac{\ddot{\Phi}}{\Phi_0}+\frac{2\dot{\Phi}^2}{\Phi_0^2}\big) \nonumber\\
&\big(&-\dot{\alpha}^2 + \dot{\sigma}^2\big)
+ \frac{\kappa^2\xi_0}{\Phi_0}e^{2\Phi/\Phi_0}\dot{\Phi}\bigg[-5\dot{\alpha}^3
+ \dot{\alpha}\big(9\dot{\sigma}^2-2\ddot{\alpha}\big) + 4\dot{\sigma}^3
+ 2\dot{\sigma}\ddot{\sigma}\bigg]
\label{new3}\end{aligned}$$
and $$\begin{aligned}
\ddot{\Phi}&=&-3\dot{\alpha}\dot{\Phi} + \frac{2V_0}{\Phi_0}e^{-2\Phi/\Phi_0}
+ \frac{6\xi_0}{\Phi_0}e^{2\Phi/\Phi_0}\big(\dot{\alpha}+\dot{\sigma}\big)\nonumber\\
&\bigg[&\dot{\alpha}^3
- \dot{\alpha}^2\dot{\sigma} + \dot{\alpha}\big(-2\dot{\sigma}^2+\ddot{\alpha}\big) - \dot{\sigma}\big(\ddot{\alpha}+2\ddot{\sigma}\big)\bigg]
\label{new4}\end{aligned}$$
respectively. Here we are interested on the collapsing solutions where the volume of the two cylinder (recall that the interior spacetime has a cylindrical symmetry) decreases monotonically with time. Keeping this in mind, the above four equations (eqn.(\[new1\]), eqn.(\[new2\]), eqn.(\[new3\]), eqn.(\[new4\])) are solved for $\alpha(t)$, $\sigma(t)$, $\Phi(t)$ and the solutions are the following: $$e^{\alpha(t)} \propto (t_0-t)^{\alpha_0},
\label{sol_alpha}$$
$$\begin{aligned}
e^{\sigma(t)} \propto (t_0-t)^{\sigma_0},
\label{sol_sigma}\end{aligned}$$
and
$$\Phi(t) = \Phi_0 \ln {\bigg[\frac{1}{\kappa}\big(t_0 - t\big)\bigg]}
\label{sol_scalar}$$
where $t_0$ is a constant of integration. The constants $\alpha_0$, $\sigma_0$ and $C$ $\big($appeared in the solution of electromagnetic field, see eqn.(\[electromagnetic equation anisotropy2\])$\big)$ are related to $V_0$, $\xi_0$ (taken as greater than zero, which is consistent with the local astronomical tests [@astronomy]) and $\Phi_0$ through the following four relations, $$\begin{aligned}
\alpha_0 + \sigma_0 = \frac{1}{2},
\label{condition_1}\end{aligned}$$ $$\begin{aligned}
\alpha_0^2 = \sigma_0^2 + \frac{\kappa^2}{3}\big(V_0\kappa^2+\frac{1}{2}\Phi_0^2\big) + \frac{\kappa^2}{6}C^2
+ \frac{\xi_0}{2}\big(2\sigma_0-\alpha_0\big),
\label{condition_2}
\end{aligned}$$ $$\begin{aligned}
\frac{\kappa^2}{3}C^2 = \sigma_0\big(3\alpha_0-1\big)\big(1+\xi_0\big),
\label{condition_3}
\end{aligned}$$ and $$\begin{aligned}
\alpha_0&=&3\alpha_0^2 - \kappa^4V_0 - \frac{\kappa^2}{6}C^2\nonumber\\
&+&\xi_0\big(5\alpha_0^3-9\alpha_0\sigma_0^2-3\alpha_0^2-4\sigma_0^3-3\sigma_0^2\big).
\label{condition_4}\end{aligned}$$
Eqn.(\[sol\_sigma\]) depicts that the exponent of $e^{\sigma(t)}$ (effectively $\sigma_0$) determines the strength of anisotropy of the spacetime. Further it may be observed from eqn.(\[condition\_3\]) that for $C\neq0$, the anisotropy factor $\sigma_0$ cannot be zero. These reflect the fact that the presence of the time dependent electromagnetic field calls for an anisotropy in the spacetime geometry. However for $C=0$ (i.e in the absence of the electromagnetic field), the spacetime either becomes isotropic ($\sigma_0=0$) or possesses a certain anisotropy with $\sigma_0=1/6$. Later we discuss the possible consequences of such situations on the collapsing phenomena. The solutions of $\alpha(t)$, $\sigma(t)$ (in eqn.(\[sol\_alpha\]), eqn.(\[sol\_sigma\])) immediately lead to the evolution of scale factor along radial and longitudinal directions as, $$\begin{aligned}
a_r(t)&=&e^{[\alpha(t)+\sigma(t)]}\nonumber\\
&=&B^{(r)}_0(t_0 - t)
\label{sol_scale1}
\end{aligned}$$ and $$\begin{aligned}
a_z(t)&=&e^{[\alpha(t)-2\sigma(t)]}\nonumber\\
&=&B^{(z)}_0(t_0 - t)^{\frac{1}{2}-3\sigma_0}
\label{sol_scale2}\end{aligned}$$
respectively where $B^{(r)}_0$ and $B^{(z)}_0$ are integration constants. To derive the above two expressions, we use eqn.(\[condition\_1\]). The expression of $a_r(t)$ (see eqn.(\[sol\_scale1\])) clearly reveals that $ra_r(t)$ decreases monotonically with time. Therefore, the volume of the cylinder of the scalar field collapses with time and goes to zero at $t\rightarrow t_0$, giving rise to a finite time zero proper volume singularity. On the other hand, the evolution of the scale factor along longitudinal direction $a_z(t)$ depends on the anisotropy factor $\sigma_0$. For $\sigma_0 < 1/6$, $a_z(t)$ decreases monotonically with time and goes to zero at $t\rightarrow t_0$, while the condition $\sigma_0 > 1/6$ entails that $a_z(t)$ continually increases and as a result, diverges at $t\rightarrow t_0$. Therefore the singularity appeared at $t\rightarrow t_0$ is a point singularity for $\sigma_0 < 1/6$ while for the other condition, the collapse ends to a line singularity. This directs us to argue that the nature of the singularity depends entirely on the strength of anisotropy of the spacetime with the limiting situation as defined by $\sigma_0=1/6$. For such limiting case, $a_z(t)$ becomes constant (finite) which in turn leads the collapse to a “finite line singularity”. Further recall from eqn.(\[condition\_3\]) that this limiting condition corresponds to $C=0$. Therefore the final fate of the collapsing scalar field in absence of the electromagnetic field is depicted by such “finite line singularity”.\
In order to investigate whether the singularity is a curvature singularity or just an artifact of coordinate choice, one must look into the behaviour of Kretschmann curvature scalar ($K = R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}$) at $t\rightarrow t_0$. For the metric presented in eqn.(\[ansatz2\]), $K$ has the following expression,\
$$K = 4\bigg(\frac{\dot{a}_r}{a_r}\bigg)^4 + 8\bigg(\frac{\dot{a}_r\dot{a}_z}{a_ra_z}\bigg)^2 + 4\bigg[2\bigg(\frac{\ddot{a}_r}{a_r}\bigg)
+ \bigg(\frac{\ddot{a}_z}{a_z}\bigg)\bigg]^2
\label{curvature_scalar1}$$
Using the solutions of $a_r(t)$ and $a_z(t)$ (see eqn. (\[sol\_scale1\]) and eqn.(\[sol\_scale2\])), the above expression of $K$ can be simplified as,\
$$\begin{aligned}
K = \frac{4}{\big(t_0 - t\big)^4}\bigg[\frac{1}{16} + \frac{1}{2}(\alpha_0-2\sigma_0)^2 + \bigg((\alpha_0-2\sigma_0)^2 - (\alpha_0-2\sigma_0)
-\frac{1}{2}\bigg)^2\bigg]
\label{curvature_scalar2}\end{aligned}$$
It is clear from eqn.(\[curvature\_scalar2\]) that the Kretschmann scalar diverges at $t\rightarrow t_0$ and thus the collapsing cylinder discussed here ends up in a curvature singularity.\
From eqn.(\[sol\_scale1\]) and eqn.(\[sol\_scale2\]), we obtain the plot (figure (\[plot scale\])) of $a_r(t)$, $a_z(t)$ versus $t$.
![$a_r(t)$, $a_z(t)$ versus $t$[]{data-label="plot scale"}](scale.eps){width="3.0in" height="2.0in"}
Figure (\[plot scale\]) clearly demonstrates that $a_r(t)$ decreases with time linearly and goes to zero as $t$ tends to $t_0$. On the other hand $a_z(t)$ decreases almost uniformly until $t$ approaches a value close to $t_0$, where it hurries towards a zero proper volume singularity. This collapsing behaviour of $a_z(t)$ is shown in the dashed curve where $\sigma_0$ is taken as $0.03$. On the other hand, the green solid curve depicts the diverging character of $a_z(t)$ for $\sigma_0 = 0.3$.\
Visibility of the singularity
=============================
The visibility of curvature singularity to an exterior observer depends on the formation of an apparent horizon. The condition for such a surface is given by $$\begin{aligned}
g^{\mu\nu} Z_{,\mu} Z_{,\nu}\bigg|_{r_{ah},t_{ah}} = 0
\label{app hor 1}\end{aligned}$$
where $Z$ is the proper radius of the two cylinder, given by $ra_r(t)$ in the present case, $r_{ah}$ and $t_{ah}$ being the comoving radial coordinate and time of formation of the apparent horizon respectively. Using the form of $g^{\mu\nu}$ presented in eqn.(\[ansatz2\]), above expression can be simplified and turns out to be, $$\begin{aligned}
r_{ah}^2 \dot{a}_r(t_{ah})^2 = 1
\label{app hor 2}\end{aligned}$$
where we use $a_r(t)=e^{[\alpha(t)+\sigma(t)]}$. Due to the solution of $a_r(t)$, eqn.(\[app hor 2\]) takes the following form : $$\begin{aligned}
\big[t_0 - t_{ah}\big] = \frac{1}{4}B^{(r)}_0 r_{ah}^2~~~~~~~~.
\label{app hor 3}\end{aligned}$$ The above expression clearly demonstrates that $t_{ah}$ is less than $t_0$ (i.e. $t_{ah} < t_0$). Therefore the formation of apparent horizon lags behind than the formation of singularity. Thus, the curvature singularity discussed here is always covered from an exterior observer by the apparent horizon. At this stage, it may be mentioned that the singularity formed is not a central singularity, it is formed at any value of $r$ within the distribution. Such a singularity in general relativity is always covered by a horizon [@31_NS].\
Matching of the interior spacetime with an exterior geometry
============================================================
To complete the model, the interior spacetime geometry of the collapsing scalar field cylindrical cloud (recall that the interior geometry is cylindrically symmetric) needs to be matched to an exterior geometry. For the required matching, the Israel conditions are used, where the metric coefficients and extrinsic curvatures (first and second fundamental forms respectively) are matched at the boundary of the cylinder [@nolan]. At this stage, it deserves mention that the Gauss-Bonnet term (controlled by the coupling function $\xi(\Phi)$) generates an effective energy momentum tensor which can not be zero (since it arises effectively from spacetime curvature) at the exterior and hence the presence of Gauss-Bonnet gravity spoils the matching of the collapsing interior spacetime with a vacuum exterior geometry. Such spoiling of matching (with a vacuum exterior) due to the presence of Gauss-Bonnet gravity can also be found in the previous literature [@paul2]. In addition, the energy momentum tensor carried by the electromagnetic field will further lead to an inconsistency if the interior collapsing cloud is matched with a vacuum exterior. For instance, since vacuum has zero electromagnetic (em) field, such a matching would lead to a discontinuity in the em field, which means a delta function in the gradient of the em field. As a consequence, there will appear square of a delta function in the stress-energy, which is definitely an inconsistency. Keeping these in mind, here we match the interior geometry with a generalized cylindrically symmetric exterior spacetime [@nolan; @kompaneets; @jordan] at the boundary hypersurface $\Sigma$ given by $r = r_0$. The metric inside and outside of $\Sigma$ are given by, $$\begin{aligned}
ds_{-}^2 = -dt^2 + e^{[2\alpha(t) + 2\sigma(t)]} \bigg(dr^2 + r^2d\theta^2\bigg) + e^{[2\alpha(t) - 4\sigma(t)]}dz^2
\label{inside metric}\end{aligned}$$ and $$\begin{aligned}
ds_{+}^2 = e^{2(\Upsilon-\Psi)}\big(-dT^2+d\rho^2\big) + R^2e^{-2\Psi}d\theta^2 + e^{2\Psi}\big(dz+Wd\theta\big)^2
\label{outside metric}\end{aligned}$$ respectively, where $T,\rho,\theta$ and $z$ are the exterior coordinates and $\Upsilon$, $\Psi$, $R$, $W$ are functions of $T$ and $\rho$. Therefore $\frac{\partial}{\partial\theta}$ and $\frac{\partial}{\partial z}$ are the killing vector fields of the exterior spacetime which yields a cylindrical symmetry in the exterior. The same hypersurface $\Sigma$ can alternatively be defined by the exterior coordinates as $T = T(t)$ and $\rho = \rho(t)$. Then the metrics on $\Sigma$ from inside and outside coordinates turn out to be, $$\begin{aligned}
ds_{-,\Sigma}^2 = -dt^2 + e^{[2\alpha(t) + 2\sigma(t)]} r_0^2d\theta^2 + e^{[2\alpha(t) - 4\sigma(t)]}dz^2
\nonumber\end{aligned}$$ and $$\begin{aligned}
ds_{+,\Sigma}^2 = e^{2(\Upsilon_{\Sigma} - \Psi_{\Sigma})}\big(-\dot{T}^2 + \dot{\rho}^2\big) dt^2
+ R_{\Sigma}^2e^{-2\Psi_{\Sigma}}d\theta^2 + e^{2\Psi_{\Sigma}}\big(dz + W_{\Sigma}d\theta\big)^2
\nonumber\end{aligned}$$ where $\Upsilon_{\Sigma}(t)$ $\big(=\Upsilon(T(t),\rho(t))\big)$, $\Psi_{\Sigma}(t)$, $R_{\Sigma}(t)$ and $W_{\Sigma}(t)$ are the respective functions defined on $\Sigma$ and dot represents $\frac{d}{dt}$. Matching the first fundamental form on $\Sigma$ (i.e. $ds_{-,\Sigma}^2 = ds_{+,\Sigma}^2$) yields the following conditions : $$\begin{aligned}
e^{2(\Upsilon_{\Sigma} - \Psi_{\Sigma})}\big(\dot{T}^2 - \dot{\rho}^2\big) = 1~~~,
\label{con 1}\end{aligned}$$
$$\begin{aligned}
e^{\Psi_{\Sigma}(t)}&=&a_z(t)\nonumber\\
&=&B^{(z)}_0(t_0 - t)^{\frac{1}{2}-3\sigma_0}~~~,
\label{con 2}\end{aligned}$$
$$\begin{aligned}
R_{\Sigma}(t)&=&r_0 a_r(t)a_z(t)\nonumber\\
&=&r_0 B^{(r)}_0 B^{(z)}_0(t_0 - t)^{\frac{3}{2}-3\sigma_0}
\label{con 3}\end{aligned}$$
and $$\begin{aligned}
W_{\Sigma}(t) = 0.
\label{con 4}\end{aligned}$$
In order to match the second fundamental form, we calculate the normal of the hypersurface $\Sigma$ from inside ($\vec{n}^{-} = n^{-}_t$, $n^{-}_r$, $n^{-}_{\theta}$, $n^{-}_z$) and outside ($\vec{n}^{+} = n^{+}_T$, $n^{+}_{\rho}$, $n^{+}_{\theta}$, $n^{+}_z$) coordinates as follows : $$\begin{aligned}
n^{-}_t = 0~~,~~~~~~~~n^{-}_r = a(t)~~,~~~~~~~n^{-}_{\theta} = n^{-}_z = 0\label{inside normal}\end{aligned}$$ and $$\begin{aligned}
n^{+}_T = \frac{e^{(\Upsilon_{\Sigma} - \Psi_{\Sigma})}\dot{\rho}}{\sqrt{\dot{T}^2-\dot{\rho}^2}}~~,\nonumber
\end{aligned}$$ $$\begin{aligned}
n^{+}_{\rho} = \frac{e^{(\Upsilon_{\Sigma} - \Psi_{\Sigma})}\dot{T}}{\sqrt{\dot{T}^2-\dot{\rho}^2}}~~,\nonumber
\end{aligned}$$ $$\begin{aligned}
n^{+}_{\theta} = n^{+}_z = 0.
\label{outside normal}\end{aligned}$$
The above expressions of $\vec{n}^{-}$ and $\vec{n}^{+}$ leads to the extrinsic curvature of $\Sigma$ from interior and exterior coordinates respectively, and are given by, $$\begin{aligned}
K_{tt}^- = 0~~,~~~~~~~~~K_{\theta\theta}^- = r_0a_r(t)~~,~~~~~~~K_{zz}^- = 0
\label{inside extrinsic}\end{aligned}$$ (all the other components of $K_{\mu\nu}^{-}$ are zero) from interior metric, and $$\begin{aligned}
K_{tt}^+ = e^{(\Upsilon_{\Sigma}-\Psi_{\Sigma})} \sqrt{\dot{T}^2 - \dot{\rho}^2}
\bigg[\big(\Psi_{\rho}\dot{T}-\Psi_T\dot{\rho}\big) - \big(\Upsilon_{\rho}\dot{T}-\Upsilon_T\dot{\rho}\big)\bigg]~~,
\nonumber\end{aligned}$$ $$\begin{aligned}
K_{\theta\theta}^+ = \frac{R_{\Sigma}e^{-(\Upsilon_{\Sigma}+\Psi_{\Sigma})}}{\sqrt{\dot{T}^2 - \dot{\rho}^2}}
\bigg[\big(R_{\rho}\dot{T}-R_T\dot{\rho}\big) - R_{\Sigma}\big(\Psi_{\rho}\dot{T}-\Psi_T\dot{\rho}\big)\bigg]~~~~,
\nonumber\end{aligned}$$ $$\begin{aligned}
K_{zz}^+ = \frac{e^{-(\Upsilon_{\Sigma}-3\Psi_{\Sigma})}}{\sqrt{\dot{T}^2 - \dot{\rho}^2}}
\bigg[\Psi_{\rho}\dot{T}-\Psi_T\dot{\rho}\bigg]~~~~,
\nonumber\end{aligned}$$ $$\begin{aligned}
K_{z\theta}^+&=&K_{\theta z}^+\nonumber\\
&=&\frac{e^{-(\Upsilon_{\Sigma}-3\Psi_{\Sigma})}}{\sqrt{\dot{T}^2 - \dot{\rho}^2}} \bigg[W_{\rho}\dot{T}-W_T\dot{\rho}\bigg]
\label{outside extrinsic}\end{aligned}$$
(all the other components of $K_{\mu\nu}^{+}$ are zero) from exterior metric, where the subscription denotes the respective derivative on the hypersurface $\Sigma$, such as $R_T = \frac{\partial R}{\partial T}\bigg|_{\Sigma}$.\
The equality of the extrinsic curvatures at $\Sigma$ from both sides is therefore equivalent to the following conditions :\
$$\begin{aligned}
\bigg[R_{\rho}\dot{T} - R_T\dot{\rho}\big)\bigg]
\frac{R_{\Sigma}e^{-(\Upsilon_{\Sigma}+\Psi_{\Sigma})}}{\sqrt{\dot{T}^2 - \dot{\rho}^2}}&=&r_0 a_r(t)\nonumber\\
&=&r_0B^{(r)}_0(t_0 - t)~~~~~,
\label{con 5}\end{aligned}$$
$$\begin{aligned}
\bigg[\Upsilon_{\rho}\dot{T} - \Upsilon_T\dot{\rho}\bigg] = 0~~~~,
\label{con 6}\end{aligned}$$
$$\begin{aligned}
\bigg[\Psi_{\rho}\dot{T} - \Psi_T\dot{\rho}\bigg] = 0~~~~,
\label{con 7}\end{aligned}$$
and $$\begin{aligned}
\bigg[W_{\rho}\dot{T} - W_T\dot{\rho}\bigg] = 0.
\label{con 8}\end{aligned}$$
Eqn.(\[con 5\]) can be further simplified by using the conditions obtained in eqn.(\[con 1\]), eqn.(\[con 2\]), eqn.(\[con 3\]) and finally we obtain the following expression $$\begin{aligned}
\bigg[R_{\rho}\dot{T} - R_T\dot{\rho}\bigg]&=&\frac{e^{2\Upsilon_{\Sigma}}}{a_z(t)}\nonumber\\
&=&\frac{e^{2\Upsilon_{\Sigma}}}{B^{(z)}_0(t_0 - t)^{\frac{1}{2}-3\sigma_0}}
\label{con 9}\end{aligned}$$
The above four relations along with eqn.(\[con 1\]) to eqn.(\[con 4\]) completely specify the matching at the boundary of the collapsing scalar field with an exterior cylindrically symmetric geometry.\
Conclusion
==========
We consider a “Scalar-Maxwell-Einstein-Gauss-Bonnet” theory in four dimensions where the scalar field couples non-minimally with the Gauss-Bonnet (GB) term. This coupling with the scalar field guarantees the non topological character of the GB term. In this higher curvature theory, we examine the possible effects of the electromagnetic field on scalar field collapse.\
The presence of electromagnetic field requires an anisotropic metric. We consider a special Bianchi-I metric $\big($which possesses a cylindrical symmetry, the radial scale factor ($a_r(t)$) is different form the longitudinal scale factor ($a_z(t)$)$\big)$ as a candidate of an anisotropic model. With the aforementioned metric, an exact solution is obtained for the spacetime geometry, which clearly reveals that the radius of a two cylinder decreases monotonically with time. Therefore, the volume of the cylinder of the scalar field collapses and goes to zero at a finite time ($t_0$) leading to a zero proper volume singularity. From the behaviour of Kretschmann scalar, it is found that the singularity formed as a result of the collapse is a finite time curvature singularity.\
On the other hand, the evolution of the longitudinal scale factor indicates that for $\sigma_0 < 1/6$, $a_z(t)$ decreases with time and goes to zero at $t \rightarrow t_0$ while the condition $\sigma_0 > 1/6$ makes $a_z(t)$ an increasing function of time and as a consequence, diverges at $t \rightarrow t_0$. The parameter $\sigma_0$ is essentially determined by the Gauss-Bonnet coupling (with the scalar field) $\xi_0$ and the parameters $V_0$, $\Phi_0$. However such collapsing or diverging behaviours of $a_z(t)$ demonstrate that the singularity we discussed here is point like or line like depending on the condition whether $\sigma_0 < 1/6$ or $\sigma_0 > 1/6$ respectively. Moreover, it may be mentioned that the parameter $\sigma_0$ actually regulates the strength of the spacetime anisotropy. Therefore it can be argued that in the present context, the pattern of the singularity (point like or line like) is controlled by the strength of anisotropy of the spacetime with the limiting situation is defined by $\sigma_0=1/6$. For such limiting case, $a_z(t)$ becomes constant (finite) which in turn leads the collapse to a “finite line singularity”. Further this limiting condition corresponds to $C=0$ (see eqn.(\[condition\_3\])). Therefore the final state of the scalar field collapse in absence of the electromagnetic field is demonstrated by such “finite line singularity”.\
The visibility of curvature singularity to an exterior observer depends on apparent horizon. The formation of apparent horizon is investigated and it turns out that the apparent horizon forms before the collapsing cloud hits to singularity. Therefore the curvature singularity is hidden from exterior by an apparent horizon. Here, it deserves mentioning that the singularity is independent of the radial coordinate $r$ and it is covered by a horizon. This result is consistent with the result obtained by Joshi [*et al*]{} [@31_NS] that unless one has a central singularity, it can not be a naked singularity. It is interesting to note that the result obtained in the present work in the presence of Gauss-Bonnet term is completely consistent with the corresponding GR result. Such consistency between Gauss-Bonnet gravity and Einstein’s GR is also in agreement with [@paul2].\
Finally, we match the interior collapsing spacetime geometry with a generalized cylindrically symmetric exterior geometry at the boundary of the cloud ($\Sigma$). For this matching, the Israel junction conditions are used where the metric coefficients and extrinsic curvatures are matched on $\Sigma$.\
Appendix - I: Situation of isotropic spacetime {#appendix---i-situation-of-isotropic-spacetime .unnumbered}
==============================================
The non static isotropic metric ansatz is taken as, $$\begin{aligned}
ds^2 = -dt^2 + a^2(t) \bigg[dr^2 + r^2d\theta^2 + dz^2\bigg]
\label{ansatz1}
\end{aligned}$$ with $a(t)$ is the scale factor of the spacetime characterized by the coordinates $t$ ($=x^0$), $r$ ($=x^1$), $\theta$ ($=x^2$) and $z$ ($=x^3$) where $t$ is the timelike one. Moreover the scalar field and the electromagnetic field are considered to be dependent only on $t$. Therefore $F_{\mu\nu}$ has three non zero independent components : $F_{01}$, $F_{02}$ and $F_{03}$. With these non zero components of $F_{\mu\nu}$, we obtain various components of $T_{\mu\nu}(A)$ from eqn.(\[em tensor 2\]) and are given by, $$\begin{aligned}
T_{00}&=&\frac{1}{2}\bigg[F_{01}F^{01} + F_{02}F^{02} + F_{03}F^{03}\bigg]\nonumber\\
T_{11}&=&-\frac{1}{2}a^2\bigg[F_{01}F^{01} - F_{02}F^{02} - F_{03}F^{03}\bigg]\nonumber\\
T_{22}&=&-\frac{1}{2}a^2\bigg[-F_{01}F^{01} + F_{02}F^{02} - F_{03}F^{03}\bigg]\nonumber\\
T_{33}&=&-\frac{1}{2}a^2\bigg[-F_{01}F^{01} - F_{02}F^{02} + F_{03}F^{03}\bigg]\nonumber\\
T_{10}&=&T_{20} = T_{30} = 0\nonumber\\
T_{12}&=&-a^2 F_{01}F^{02}~~~~~,~~~~T_{13} = -a^2 F_{01}F^{03}~~~~,~~~T_{23} = -a^2 F_{02}F^{03}
\label{1}
\end{aligned}$$
Using the above expressions of $T_{\mu\nu}(A)$, the non diagonal components of gravitational equation are simplified to the following form :\
$$\begin{aligned}
F_{01}F^{02} = F_{01}F^{03} = F_{02}F^{03} = 0
\label{2}
\end{aligned}$$
which has the solution as $F_{01} = F_{02} = F_{03} = 0$. Thus a spatially flat isotropic spacetime cannot support the time dependent electromagnetic field. However a Bianchi-I spacetime, although it is spatially flat, can sustain the gauge field by virtue of its anisotropy.
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[^1]: E-mail address: narayan@iiserkol.ac.in
[^2]: E-mail address: pul.tnmy9@gmail.com
|
---
author:
- |
Shigeru Furuichi$^1$[^1] and Minghua Lin$^2$[^2]\
$^1$[Department of Computer Science and System Analysis,]{}\
[College of Humanities and Sciences, Nihon University,]{}\
[3-25-40, Sakurajyousui, Setagaya-ku, Tokyo, 156-8550, Japan]{}\
$^2$[Department of Mathematics and Statistics,]{}\
[University of Regina, Regina, Saskatchewan, S4S 0A2, Canada]{}
title: A matrix trace inequality and its application
---
In this short paper, we give a complete and affirmative answer to a conjecture on matrix trace inequalities for the sum of positive semidefinite matrices. We also apply the obtained inequality to derive a kind of generalized Golden-Thompson inequality for positive semidefinite matrices.
[**Keywords :** ]{} Matrix trace inequality, positive semidefinite matrix, majorization and Golden-Thompson inequality
15A39 and 15A45
Introduction
============
We give some notations. The set of all $n \times n$ matrices on the complex field $\mathbb{C}$ is represented by $M(n,\mathbb{C})$. The set of all $n \times n$ Hermitian matrices is also represented by $M_h(n,\mathbb{C})$. Moreover the set of all $n \times n$ nonnegative (positive semidefinite) matrices is also represented by $M_+(n,\mathbb{C})$. Here $X\in M_+(n,\mathbb{C})$ means we have $\langle \phi \vert X \vert \phi \rangle \geq 0$ for any vector $\vert \phi \rangle \in \mathbb{C}^n$.
The purpose of this short paper is to give the answer to the following conjecture which was given in the paper [@Furu1].
[**([@Furu1])**]{}\[con\] For $X,Y\in M_+(n,\mathbb{C})$ and $p\in \mathbb{R}$, the following inequalities hold or not?
- $Tr[(I + X +Y +Y^{1/2}XY^{1/2})^p] \leq Tr[(I+X+Y+XY)^p]$ for $p \geq 1$.
- $Tr[(I + X +Y +Y^{1/2}XY^{1/2})^p] \geq Tr[(I+X+Y+XY)^p]$ for $0 \leq p \leq 1$.
We firstly note that the matrix $I+X+Y+XY=(I+X)(I+Y)$ is generally not positive semidefinite. However, the eigenvalues of the matrix $(I+X)(I+Y)$ are same to those of the positive semidefinite matrix $(I+X)^{1/2}(I+Y)(I+X)^{1/2}$. Therefore the expression $Tr[(I+X+Y+XY)^p]$ always makes sense.
We easily find that the equality for (i) and (ii) in Conjecture \[con\] holds in the case of $p=1$. In addition, the case of $p=2$ was proven by elementary calculations in [@Furu1].
Putting $T=(I+X)^{1/2}$ and $S=Y^{1/2}$, Conjecture \[con\] can be reformulated by the following problem, because we have $Tr[(I+X+Y+XY)^p]=Tr[(T^2+T^2S^2)^p]=Tr[(T^2(I+S^2))^p]=Tr[(T(I+S^2)T)^p]=Tr[(T^2+TS^2T)^p].$
\[prob02\] For $T,S\in M_+(n,\mathbb{C})$ and $p\in \mathbb{R}$, the following inequalities hold or not?
- $ Tr[(T^2+ST^2S)^p] \leq Tr[(T^2+TS^2T)^p]$ for $p \geq 1$.
- $ Tr[(T^2+ST^2S)^p] \geq Tr[(T^2+TS^2T)^p]$ for $0 \leq p \leq 1$.
Main results
============
To solve Problem \[prob02\], we use the concept of the majorization. See [@MO] for the details on the majorization. Here for $X\in M_h(n,\mathbb{C})$, $\lambda^{\downarrow }(X)=\left(\lambda_1^{\downarrow} (X),\cdots,\lambda_n^{\downarrow} (X) \right)$ represents the eigenvalues of the Hermitian matrix $X$ in decreasing order, $\lambda_1^{\downarrow} (X) \geq \cdots \geq \lambda_n^{\downarrow} (X)$. In addition $x \prec y$ means that $x=(x_1,\cdots,x_n)$ is majorized by $y=(y_1,\cdots,y_n)$, if we have $$\sum_{j=1}^kx_j \leq \sum_{j=1}^k y_j\quad (k=1,\cdots,n-1)$$ and $$\sum_{j=1}^nx_j =\sum_{j=1}^n y_j.$$
We need the following lemma which can be obtained as a consequence of Ky Fan’s maximum principle.
[**(p.35 in [@Bha])**]{} \[Lind-lem\] For $A,B \in M_h(n,\mathbb{C})$ and any $k=1,2,\cdots,n$, we have $$\sum_{j=1}^k \lambda_{j}^{\downarrow}(A+B) \leq \sum_{j=1}^k \lambda_{j}^{\downarrow}(A)+\sum_{j=1}^k \lambda_{j}^{\downarrow}(B).$$
Then we have the following theorem.
\[the-01\] For $S,T\in M_+(n,\mathbb{C})$, we have $$\label{maj_ineq00}
\lambda^{\downarrow }(T^2+ST^2S) \prec \lambda^{\downarrow }(T^2+TS^2T)$$
[*Proof*]{}: For $S,T\in M_+(n,\mathbb{C})$, we need only to show the following $$\label{maj_ineq01}
\sum_{j=1}^{k} \lambda_{j}^{\downarrow} (T^2+ST^2S) \leq \sum_{j=1}^{k} \lambda_{j}^{\downarrow} (T^2+TS^2T)$$ for $k=1,2,\cdots,n-1$, since we have $$\sum_{j=1}^n \lambda_j ^{\downarrow} (T^2+ST^2S) =\sum_{j=1}^n \lambda_j ^{\downarrow} (T^2+TS^2T),$$ which is equivalent to $Tr[T^2+ST^2S]=Tr[T^2+TS^2T]$.
By Lemma \[Lind-lem\], we have $$\label{Lind-ineq01}
2 \sum_{j=1}^k \lambda_{j}^{\downarrow}(X) \leq
\sum_{j=1}^k \lambda_{j}^{\downarrow}\left( X+Y \right)+\sum_{j=1}^k \lambda_{j}^{\downarrow}\left( X-Y \right).$$ for $X,Y\in M_h(n,\mathbb{C})$ and any $k=1,2,\cdots,n$.
For $X\in M(n,\mathbb{C})$, the matrices $XX^*$ and $X^*X$ are unitarily similar so that we have $ \lambda_j^{\downarrow}(XX^*) =\lambda_j^{\downarrow}(X^*X) $. Then we have the following inequality: $$\begin{aligned}
2 \sum_{j=1}^{k} \lambda_{j}^{\downarrow} \left( T^2+TS^2T \right) &=&
\sum_{j=1}^{k} \lambda_{j}^{\downarrow}\left( T^2+TS^2T \right) +\sum_{j=1}^{k} \lambda_{j}^{\downarrow}\left( T^2+TS^2T \right) \\
&=& \sum_{j=1}^{k} \lambda_{j}^{\downarrow}\left( (T+iTS)(T-iST)\right) +\sum_{j=1}^{k} \lambda_{j}^{\downarrow}\left( (T-iTS)(T+iST)\right) \\
&=& \sum_{j=1}^{k} \lambda_{j}^{\downarrow}\left( (T-iST)(T+iTS)\right) +\sum_{j=1}^{k} \lambda_{j}^{\downarrow}\left( (T+iST)(T-iTS)\right) \\
&=& \sum_{j=1}^{k} \lambda_{j}^{\downarrow}\left( T^2+ST^2S+i\left( T^2S-ST^2 \right)\right) +
\sum_{j=1}^{k} \lambda_{j}^{\downarrow}\left( T^2+ST^2S-i\left( T^2S-ST^2 \right) \right) \\
&\geq& 2 \sum_{j=1}^{k} \lambda_{j}^{\downarrow} \left( T^2+ST^2S\right),\end{aligned}$$ for any $k=1,2,\cdots,n-1$, by using the inequality (\[Lind-ineq01\]) for $X=T^2+ST^2S$ and $Y=i(T^2S-ST^2)$. Thus we have the inequality (\[maj\_ineq01\]) so that the proof is completed.
From Theorem \[the-01\], we have the following corollary.
\[the01\] For $T,S\in M_+(n,\mathbb{C})$ and $p\in \mathbb{R}$, the following inequalities hold.
- $ Tr[(T^2+ST^2S)^p] \leq Tr[(T^2+TS^2T)^p]$ for $p \geq 1$.
- $ Tr[(T^2+ST^2S)^p] \geq Tr[(T^2+TS^2T)^p]$ for $0 \leq p \leq 1$.
[*Proof :*]{} Since $f(x)=x^p,(p\geq 1)$ is convex function and $f(x)=x^p,(0\leq p \leq 1)$ is concave function, we have the present corollary thanks to Theorem \[the-01\] and a general property of majorization (See p.40 in [@Bha]).
As mentioned in Introduction, Corollary \[the01\] implies the following corollary by putting $T=(I+X)^{1/2}$ and $S=Y^{1/2}$.
\[con-cor\] For $X,Y\in M_+(n,\mathbb{C})$ and $p\in \mathbb{R}$, the following inequalities hold.
- $Tr[(I + X +Y +Y^{1/2}XY^{1/2})^p] \leq Tr[(I+X+Y+XY)^p]$ for $p \geq 1$.
- $Tr[(I + X +Y +Y^{1/2}XY^{1/2})^p] \geq Tr[(I+X+Y+XY)^p]$ for $0 \leq p \leq 1$.
Thus Conjecture \[con\] was completely solved with an affirmative answer.
An application
==============
In this section, we give a kind of one-parameter extension of the famous Golden-Thompson inequality [@Gol; @Tho] for positive semidefinite matrices, applying the obtained result in the previous section. For this purpose, we denote the generalized exponential function by $\exp_{\nu}(X) \equiv \left( I +\nu X \right) ^{\frac{1}{\nu}}$ for $\nu \in (0,1]$ and $X\in M(n,\mathbb{C})$ such that $Tr[(I +\nu X)^{\frac{1}{\nu}}] \in \mathbb{R}$. In addition, we use the following inequalities proved in [@Furu2].
[**([@Furu2])**]{} \[GT\_ineq\_gen\_before\] For $X,Y\in M_+(n,\mathbb{C})$, and $\nu \in (0,1]$, we have
- $$\label{GT_ineq_gen_EQ00}
Tr[ \exp_{\nu}(X+Y) ] \leq Tr[ \exp_{\nu}(X+Y+\nu Y^{1/2}XY^{1/2}) ].$$
- $$\label{GT_ineq_gen_EQ}
Tr[\exp_{\nu}(X+Y+\nu XY)] \leq Tr[\exp_{\nu}(X) \exp_{\nu}(Y)].$$
As mentioned in the below of Conjecture \[con\], the expression of the left hand side in (\[GT\_ineq\_gen\_EQ\]) makes also sense, since we have $Tr[\exp_{\nu}(X+Y+\nu XY)] = Tr[\left\{ (I+\nu X)^{1/2} (I+\nu Y) (I+\nu X)^{1/2} \right\}^{\frac{1}{\nu}}] \geq 0$.
From (i) of Corollary \[con-cor\] and Lemma \[GT\_ineq\_gen\_before\], we have the following proposition.
\[gen-GT\] For $X,Y\in M_+(n,\mathbb{C})$ and $\nu \in (0,1]$, we have $$\label{gen-GT-ineq}
Tr[ \exp_{\nu}(X+Y) ] \leq Tr[\exp_{\nu}(X) \exp_{\nu}(Y)].$$
[*Proof*]{}: The right hand side of (\[GT\_ineq\_gen\_EQ00\]) is bounded from the above by applying (i) of Corollary \[con-cor\] and putting $X_1=\nu X$, $Y_1=\nu Y$ and $p=\frac{1}{\nu}$: $$\begin{aligned}
Tr\left[\exp_{\nu}(X+Y+\nu Y^{1/2}XY^{1/2})\right] &=& Tr\left[ \left\{ I+\nu (X+Y+\nu Y^{1/2}XY^{1/2}) \right\}^{\frac{1}{\nu}}\right] \\
&=& Tr \left[ (I+X_1+Y_1+Y_1^{1/2}X_1Y_1^{1/2})^p \right]\\
&\leq & Tr \left[ (I+X_1+Y_1+X_1Y_1)^p \right]\\
&=& Tr \left[ \left\{ I+\nu (X+Y+\nu XY) \right\}^{\frac{1}{\nu}} \right]\\
&=& Tr\left[ \exp_{\nu}(X+Y+\nu XY) \right],\end{aligned}$$ which is the left hand side of (\[GT\_ineq\_gen\_EQ\]). Thus we have the present proposition thanks to Lemma \[GT\_ineq\_gen\_before\].
Note that the inequality (\[gen-GT-ineq\]) can be regarded as a kind of one-parameter extension of the Golden-Thompson inequality for positive semidefinite matrices $X$ and $Y$.
Ackowledgements {#ackowledgements .unnumbered}
===============
We would like to thank the anonymous reviewer for providing valuable comments to improve the manuscript. The first author (S.F.) was supported in part by the Japanese Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement of Young Scientists (B), 20740067
[99]{} S.Furuichi,A mathematical review of the generalized entropies and their matrix trace inequalities, Proceedings of WEC2007, pp.840-845 (2007). A.W.Marshall and I.Olkin, Inequalities: Theory of majorization and its applications, Academic Press, 1979. R.Bhatia, Matrix Analysis, Springer, 1997. S.Golden, Lower bounds for the Helmholtz function, Phys. Rev., Vol.137(1965), pp.B1127-B1128. C.J.Thompson, Inequality with applications in statistical mechanics, J.Math.Phys., Vol.6(1965), pp.1812-1813.
S.Furuichi, Trace inequalities in nonextensive statistical mechanics, Linear Alg.Appl., Vol.418(2006), pp.821-827.
[^1]: E-mail:furuichi@chs.nihon-u.ac.jp
[^2]: E-mail:lin243@uregina.ca
|
---
abstract: 'Multi-view Geometry is reviewed from an Algebraic Geometry perspective and multi-focal tensors are constructed as equivariant projections of the Grassmannian. A connection to the principal minor assignment problem is made by considering several flatlander cameras. The ideal of the quadrifocal variety is computed up to degree 8 (and partially in degree 9) using the representations of ${\operatorname{GL}}(3)^{\times 4}$ in the polynomial ring on the space of $3 \times 3 \times 3 \times 3$ tensors. Further representation-theoretic analysis gives a lower bound for the number of minimal generators.'
address: 'Department of Mathematics and Statistics, Auburn University, Auburn, AL '
author:
- Luke Oeding
bibliography:
- '/Users/oeding/Dropbox/BibTeX\_bib\_files/main\_bibfile.bib'
title: The quadrifocal variety
---
Introduction and background
===========================
The multi-view variety
----------------------
Multi-view Geometry is a branch of Computer Vision [@HartleyZisserman]. An important task in Computer Vision is to efficiently reconstruct the 3-dimensional scene from the 2-dimensional projections. Typically, one first estimates the multi-focal tensor associated to the $n$ views using correspondences arising from one object seen in multiple images. From the multi-focal tensor one reconstructs the camera matrices. After the camera matrices are known, one uses the point correspondences to triangulate the 3D points.
In the standard pinhole camera model, the projection of 3D world points to multiple 2D images is represented by a collection of $3\times 4$ matrices $(A_{1},\dots,A_{n})$ and the mapping $$\label{multi-view1}
\begin{matrix}
{\mathbb{P}}^{3} & \to& {\mathbb{P}}^{2}\times \dots \times {\mathbb{P}}^{2}
\\{}
[x] &\mapsto & ([A_{1}x],\dots,[A_{n}x]).
\end{matrix}$$ For cameras in general position the multi-view mapping defines a 3 dimensional subvariety of the Cartesian product of projective spaces called the *multi-view* variety. Aholt, Sturmfels, and Thomas demonstrated rich algebraic geometry arising from this construction in [@AST]. They utilized a certain Hilbert scheme to describe the multi-view variety, its defining ideal, and further algebraic properties. Their theoretical techniques included Borel fixed monomial ideals, a universal Gröbner basis, degeneration to a special monomial ideal, and more. This work catalyzed a new area coined “Algebraic Vision” by Sameer Agarwal and Rekha Thomas. The reader may wish to consult the following other examples of recent work in this field [@AAT; @ALST_epipolar2; @ALST_epipolar1; @AAT; @AholtOeding; @JKSW_rigid_multiview].
Moduli spaces and quadrifocal tensors
-------------------------------------
Suppose the entries of the camera matrices are not known, but are considered as parameters. By moding out by the projective rescaling in each camera plane we have a moduli space of camera matrices. The algebraic varieties of multi-focal tensors are models for these moduli spaces, and we want to know their basic algebraic properties. Bifocal tensors are just $3\times 3$ matrices of rank 2, defined by the $3\times 3$ determinant. Chris Aholt and the author resolved the long-standing open question of describing the ideal of trifocal tensors [@AholtOeding], building on work of Alzati and Tortora [@Alzati-Tortora]. They also computed its algebraic degree and Hilbert polynomial using Maple, Macaulay2 [@M2], and Bertini [@Bertini].=-1
In this paper we will be mainly concerned with the *quadrifocal variety*. One may record the correspondences induced from one 3D point seen in 4 images by the 81 special $4\times 4$ minors of stacked camera matrix $A= (A_{1}^{\top} | A_{2}^{\top} | \cdots | A_{n}^{\top})$ that only use one column from each of the first four blocks of $A$. These coordinates are linear in each block, yielding a $3\times 3\times 3\times 3$ *quadrifocal* tensor. The quadrifocal variety is the Zariski closure in ${\mathbb{P}}^{80}$ of the set of quadrifocal tensors.
We seek a complete description of the polynomial defining equations of the quadrifocal variety. The main result of the present article is a first step in this direction.
\[thm:main\] Let $I_{d}$ denote the degree $d$ piece of the ideal of the quadrifocal variety.
$I_{d}$ is zero for $d<3$.
$I_{3}$ is $600$-dimensional.
$I_{4}$ is $48,600$-dimensional but contains no minimal generators.
$I_{5}$ is $1,993,977$-dimensional and contains at least $1,377$ minimal generators.
$I_{6}$ is $54,890,407$-dimensional and contains at least $37,586$ minimal generators.
$I_{7}$ is $1,140,730,128$-dimensional and likely contains no minimal generators.
$I_{8}$ is $18,940,147,947$-dimensional and contains at least $162,000$ minimal generators.
$I_{9}$ is $\geq 223,072,284,455$-dimensional and contains at least $3,087,000$ minimal generators.
The star refers to the fact that some of our computations were done using random points of the quadrifocal variety, so the dimensions reported are only upper bounds, but the lower bounds hold *with high probability*. In Section \[sec:Rep\] we give an invariant description of all these equations and computational evidence that the equations reported here are likely *not* a minimal set of generators.
Until now, it was only known that a quadrifocal tensor must adhere to 51 non-linear constraints [@ShashuaWolf]. Indeed, the quadrifocal variety has codimension $51$, so there must be at least $51$ equations, but our results show that it is very far from being a complete intersection. For instance, there are 600 cubic minimal generators. In Section \[sec:Contract\] we give a simple description of these equations via contractions. From the contraction description we see that the cubic equations are a consequence of the fact that every contraction of a quadrifocal tensor is a homography tensor [@Shashua00homographytensors]. Additional equations are needed to take the set of tensors having that property alone and cut it down to the quadrifocal locus.
Multi-focal tensors in general
------------------------------
If $\pi$ is a partition of $4$, the entries of a multi-focal tensor of profile $\pi$ are given by the minors of $A$ that use $\pi_{i}$ columns from block $i$. Multi-focal tensors record correspondences between multiple images. An *epipole* is the image of one camera’s focal point seen in another view. Correspondences between pairs of image points (in 2 views) leads to the bi-focal tensor, or *fundamental matrix*. Whereas a point-point-line correspondences (in 3 views) are encoded by a *trifocal tensor* [@PF2], and correspondences of quadruples of image points (in 4 views) are encoded by a *quadrifocal tensor* [@ShashuaWolf]. See [@HartleySchaffalitzky1; @HartleySchaffalitzky2; @ShashuaKaminski; @WolfShashua; @ThirthalaPollefeys] for applications of quadrifocal tensors.
Each multi-focal tensor can be determined by observing some minimal number of correspondences in multiple images. From 7 point correspondences in 2 images one can reconstruct the bifocal tensor (fundamental matrix). 5 point correspondences in 3 or 4 images suffice to determine the trifocal and quadrifocal tensors [@HartleyZisserman Parts III&IV]. For a summary of these and other minimal problems in Computer Vision, see [@minimal], and in particular one can check [@5pt4; @5pt5; @5pt6; @4pt1; @4pt2] for recent algorithms for the relative pose problems in 3 and 4 views.
Since algorithms for determining the relative pose of 2 cameras exist (for instance [@MartinecPajdla]) one may ask what is the advantage of considering more than 2 views at a time. The first advantage is that fewer 3D points need to be identified for more than 2 views, which can be useful when many points in one view become outliers for another. In the 3 view case, the trifocal tensor can be used in the structure from motion problem when the cameras move along a straight line (see [@Chan2013627; @VidalHartley; @4pt3]). Other advantages include greater stability, and the possibility to avoid certain unstable or critical configurations. In the 4 view case, while the quadrifocal tensor is more difficult to construct, one may avoid iterative algorithms which are not guaranteed to converge [@HartleyZisserman Ch. 1]. Moreover, unlike in the 3 view case where one view plays a special role, in the 4 view case all 4 images can play the same role, and it is expected that this symmetry can be exploited. For 3D to 2D projections and 4 or more views there are no $n$-focal tensors if $n>4$, hence we only consider the 2, 3 and 4 view cases.
Outline
-------
For the reader’s convenience we collect notation in Section \[sec:notation\]. We give a uniform presentation of multi-focal tensors from the viewpoint of Algebraic Geometry via equivariant projections of a Grassmannian in Section \[sec:Review\], which connects our work to [@FaugerasMourrain; @HartleySchaffalitzky1]. Section \[sec:Principal\] addresses the case of different dimensional cameras and contains a connection between the multi-focal variety and the variety of principal minors of square matrices, connecting this work to [@LinSturmfels]. We discuss contractions and homography tensors in Section \[sec:Contract\], and state Proposition \[prop:600\] describing the 600 bihomogeneous cubics that are the minimal generators of the quadrifocal ideal in the lowest degree. We use symmetry-enhanced calculations to determine the quadrifocal ideal up to degree $8$ and partially compute the ideal in degree 9 in Section \[sec:Rep\]. In addition, we use representation theory and computations utilizing `SchurRings` to determine necessary minimal generators of the quadrifocal ideal. We conjecture that these necessary minimal generators also suffice.
Notation {#sec:notation}
========
Let $U$ and $V$ denote vector spaces, which we always consider the complex numbers, denoted $\mathbb{C}$, to be our ground field. The direct sum of $U$ and $V$ is denoted $U\oplus V$, and their tensor product is denoted $U\o V$. The $k$-th exterior power of $V$, or the $k$-mode skew-symmetric tensors is denoted $\bw{k}V$. Its elements are linear combinations of $k$-fold wedge products $v_{1}\wedge v_{2}\wedge \cdots\wedge v_{n}$ with $v_{i}\in V$. We discuss alternating tensors in more depth in Section \[sec:ext\]. The vector space of symmetric $d$-mode tensors is denoted $S^{d}V$. If we choose a basis $x_{1},\ldots, x_{m}$ of $V$ we may consider $S^{d}V$ as the vector space of homogeneous degree $d$ polynomials on $x_{1},\ldots, x_{m}$. The symmetric algebra on $V$ is denoted $Sym^{\bullet}V = \bigoplus_{d\geq 0}S^{d}V$, and is isomorphic to the polynomial ring ${\mathbb{C}}[x_{1},\ldots,x_{n}]$. The tensor algebra on $V$ is denoted $V^{\otimes } = \bigoplus_{d\geq 0} V^{\otimes d}$.
The vector space dual, denoted $V^{*}$, is the space of all linear functionals $V \to {\mathbb{C}}$. After bases are chosen for $U$ and $V$, $U^{*}\o V$ may be thought of as the space of matrices representing linear mappings $U\to V$. If $M \in U^{*}\o V$ is a matrix $\bw{k}M \in \bw{k}U^{*}\o \bw{k}V$ may be thought of as the matrix whose entries are the $k\times k$ minors of $M$. The general (special) linear group of all invertible (determinant 1) linear transformations of $V$ is denoted ${\operatorname{GL}}(V)$ (resp. ${\operatorname{SL}}(V)$).
We use the notation $\pi\vdash d$ to denote a partition $\pi = (p_{1},p_{2},\ldots,p_{n})$ with $\sum_{i}p_{i} = d$. We let $S_{\pi}V$ to denote the corresponding Schur module, which we think of as an explicit representative of an irreducible submodule of the $d$-fold tensor product $V^{\o d}$. Given vector spaces $V_{1}, V_{2}, V_{3}, V_{4}$ and multi-partition $\pi = (\pi_{1},\pi_{2},\pi_{3},\pi_{4})$ with $\pi \vdash d$ we use the shortened notations $S_{\pi}V$ and $S_{\pi_{1}}S_{\pi_{2}}S_{\pi_{3}}S_{\pi_{4}}$ for $S_{\pi_{1}}V_{1} \o S_{\pi_{2}}V_{2} \o S_{\pi_{3}}V_{3} \o S_{\pi_{4}}V_{4}$.
The projective space of all lines through the origin in $V$ is denoted ${\mathbb{P}}V$. If $v\in V$ is nonzero the line through $v$ is denoted $[v]$. If $X \subset {\mathbb{P}}U $ and $Y \subset {\mathbb{P}}V$ are algebraic varieties their Cartesian product, denoted $X\times Y$ is a subvariety of ${\mathbb{P}}(U\oplus V)$. The cone over the projective $X\subset {\mathbb{P}}V$, denoted $\widehat{X}$ is an affine subvariety of $V$.
Multifocal tensors sometimes need a mixture of indices and multi-indices. For instance, fundamental matrices are indexed by a pair of double indices: the notation $(F_{\{i,j\},\{k,l\}})$ indicates a matrix with rows indexed by the double index $\{i,j\},$ and columns indexed by the double index $\{k,l\}$, while $(T_{i,j,\{k,l\}})$ denotes a $3$-mode tensor with the first two modes indexed by $i$ and $j$ respectively, with the third mode indexed by the double index $\{k,l\}$.
Epipoles, fundamental matrices, trifocal and quadrifocal tensors {#sec:Review}
================================================================
The multi-view setup
--------------------
Multiple view geometry arises when one considers many images taken of the same scene, from (possibly) different viewpoints and is beautifully presented in [@HartleyZisserman]. The following introduction is an invariant view inspired by the ideas in [@HartleyZisserman Ch. 17]. Let $A_{j}$ denote $3 \times 4$ camera matrices (non-degenerate) for $1\leq j\leq n$, with row spaces equal to $V_{j}$, a three-dimensional vector space.
Let $W$ denote a 4-dimensional vector space, whose projectivization represents the 3-dimensional “projective world”. For fixed camera matrices $A_{j}$, the multi view map (which also appeared in [@AST]) is $$\label{multi-view}
\xymatrix{{\mathbb{P}}W \ar[rr]^{\hspace{-3em}(A_{1},\dots,A_{n}) }&& {\mathbb{P}}V_{1}\times\dots \times {\mathbb{P}}V_{n} }
.$$
Now we wish to treat the camera matrices as variable or as having indeterminate entries. The map in is the same if we replace the matrices $A_{j}$ with scalar multiples of themselves. So, we should consider our space of parameters for cameras to be $${\mathbb{P}}(W^{*}\o V_{1}) \times \dots \times {\mathbb{P}}(W^{*} \o V_{n}) \qquad\text{$n$-camera space}
.$$ We note here that if different camera models are taken, this paradigm may be easily altered to accommodate such changes by altering the spaces in which the cameras are modeled.
The Grassmannian
----------------
Faugeras and Mourrain studied multi-view geometry from the point of view of the Grassmann algebra, see [@FaugerasMourrain; @HartleySchaffalitzky1]. We also adapt that approach as it provides a uniform treatment and a convenient way to organize many of our computations.
### Exterior products {#sec:ext}
Recall if $V$ is a vector space with basis $u_{1},\dots,u_{n}$, we construct the exterior powers of $U$, denoted $\bw{k}U$ by considering the alternating (or wedge) product $\wedge$ and forming the vector space of $k$-vectors (length $k$ wedge products) with basis consisting of pure $k$-vectors $\left \{u_{i_{1}},\dots,u_{i_{k}}\mid 1\leq i_{1}<i_{2}<\dots<i_{k}\leq n \right\}$. Thus $\bw{k} U$ has dimension $\binom{n}{k}$ if $0\leq k\leq n$ and 0 otherwise. It is straightforward to see that $v_{1}\wedge\dots\wedge v_{k} \neq 0$ if and only if the vectors $\{v_{1},\ldots,v_{k}\}$ are linearly independent in $U$. Consider two non-zero $k$-vectors $v_{1}\wedge\dots \wedge v_{k}$ and $w_{1}\wedge\dots \wedge w_{k}$ and the underlying vector spaces $E:=\textrm{span}\{v_{1},\dots, v_{k}\}$ and $F:=\textrm{span}\{w_{1},\dots, w_{k}\}$. It is straightforward to check that $$v_{1}\wedge\dots\wedge v_{k} = \lambda ( w_{1}\wedge\dots \wedge w_{k}) \quad \textrm{for some} \quad \lambda \neq0 \quad \textrm{if and only if} \quad \bw{k}E = \bw{k}F,$$ and equality holds when $\lambda$ is the determinant of the change of basis between $E$ and $F$.
We denote by ${\mathbb{P}}\bw{k} U$ the projective space consisting of lines through the origin in $\bw{k} U$, which we may consider as the set of classes $[\omega] = \left\{\lambda \omega \mid \lambda \in{\mathbb{C}}\setminus \{ 0\}, \omega \in \bw{k}U \setminus \{ 0\} \right \}$.
This motivates the definition of the Grassmannian (in its minimal embedding). Let ${\operatorname{Gr}}(k,U)$ denote the set of $k$-planes in $U$. The rational map: $$\begin{matrix}
{\operatorname{Gr}}(k,U) &\to& {\mathbb{P}}\bw{k} U\\
M & \mapsto & \bw{k}M
\end{matrix}$$ is an embedding (in fact, the embedding is a minimal rational embedding). The usual Plücker embedding is a slight variant of this construction, which we will review next in the context of multiple view geometry.
### From multiple views to the Grassmannian
It is natural to consider the following $4\times 3n$ blocked matrix, which will present a convenient way to keep track of external constraints on the multi-view setup $$M = \left(\begin{array}{c|c|c|c} A_{1}^{\top} & A_{2}^{\top}&\dots & A_{n}^{\top}\end{array}\right) \in (W^{*}\o V_{1})\oplus \dots\oplus (W^{*}\o V_{n}) = W^{*}\o (\textstyle \bigoplus_{j}V_{j})
.$$ The non-degeneracy condition is that each matrix $A_{j}$ in an $n$-tuple $([A_{1}^{\top}],[A_{2}^{\top}],\dots,[A_{n}^{\top}])$ must have full rank, which occurs in an open set; so the row space of $M$ parameterizes the (Grassmannian variety) 4-dimensional subspaces of a $3n$-dimensional space.
The maximal minors of $M$ give coordinates (the Plücker coordinates) on the Grassmannian ${\operatorname{Gr}}(4,3n)$. These minors are also known as *multilinear coordinates*.
In invariant language this parameterization is the following map $$\label{plucker}
\begin{array}{rcl}\varphi\colon
W^{*}\o (\bigoplus_{j}V_{j}) &\longrightarrow &\bw{4}W^{*}\o \bw{4}(\bigoplus_{j}V_{j}) \cong \bw{4}(\bigoplus_{j}V_{j})
\\ M &\longmapsto & \bw{4}M
\end{array}
.$$ The image of $\varphi$ (the Zariski closure of the image of an open set) is isomorphic to the cone over the Grassmannian of 4-dimensional subspaces of $\bigoplus_{j}V_{j}$, which is the row space of $M$. In other words $$\textstyle
Im(\varphi) =
\widehat{{\operatorname{Gr}}}(4, \bigoplus_{j}V_{j}) \subset \bw{4}(\bigoplus_{j}V_{j}).$$ Notice that because $W$ is 4-dimensional, $\bw{4}W$ is one-dimensional, and thus passing to the maximal minors of the concatenated matrix $M$ removes the dependency on the world points represented by ${\mathbb{P}}W$.
It is well known that the dimension of the Grassmannian ${\operatorname{Gr}}(r,{\mathbb{C}}^{N})$ is $r(N-r)$. In our example $\dim({\operatorname{Gr}}(4,3n)) = 4(3n-4)$.
If we restrict to camera space, we have to consider the image of the map up to the $n$-dimensional torus action which records the projective ambiguity in each of the $n$ cameras. We can restrict the target of the map to the appropriate *GIT quotient*: $$\textstyle
{\mathbb{P}}(W^{*}\o V_{1}) \times \dots \times {\mathbb{P}}(W^{*} \o V_{n}) \to \widehat{Gr}(4, \bigoplus_{j}V_{j}) /\!/ ({\mathbb{C}}^{*})^{n} \subset \bw{4}(\bigoplus_{j}V_{j})/\!/ ({\mathbb{C}}^{*})^{n}.$$ From this we obtain the dimension of the GIT quotient (see [@HartleyZisserman § 17.5] and [@AST § 6]) $$\dim \widehat{Gr}(4, \bigoplus_{j}V_{j}) /\!/ ({\mathbb{C}}^{*})^{n} \quad=\quad 4(3n-4) +1 - n \quad =\quad 11n-15.$$
Here is a classical formula for the degree of the Grassmann manifold, (see [@MukaiGrassmannian], [@FultonINT Ex. 14.7.11] or [@DolgachevAG § 10.1.2, eq. (10.6)]) $$\deg{\operatorname{Gr}}(r,n) \quad = \quad
\frac{
1!2!\cdots r! \dim({\operatorname{Gr}}(r,n))!
}
{
(n-r)!(n-r+1)! \cdots n!
}
.$$ For example $$\deg({\operatorname{Gr}}(4,6)) = 14, \qquad
\deg({\operatorname{Gr}}(4,9)) = 1662804, \qquad
\deg({\operatorname{Gr}}(4,12)) = 1489877926680.$$ One hope is that a better understanding of the GIT quotient of the Grassmannian might allow us to find the degree of the multi-focal tensor varieties.
Symmetry
--------
For $j \in \{1,2,3,4\}$ let $A_{j}$ be a $3 \times 4$ (non-degenerate) camera matrix (an element of $W^{*}\o V_{j}$), with blocking $A_{j} = (B_{j}| {\bf x}_{j})$. On each matrix $A_{j}$ we have an action of ${\operatorname{GL}}(V_{j}) \cong {\operatorname{GL}}(3)$ acting by change of coordinates in the camera plane. The action is $$\begin{gathered}
\left({\operatorname{GL}}(V_{1})\times {\operatorname{GL}}(V_{2})\times {\operatorname{GL}}(V_{3})\times {\operatorname{GL}}(V_{4}) \right)
\times
\left( (W^{*}\o V_{1}) \oplus (W^{*}\o V_{2}) \oplus (W^{*}\o V_{3}) \oplus (W^{*}\o V_{4}) \right)\\
\to \left( (W^{*}\o V_{1}) \oplus (W^{*}\o V_{2}) \oplus (W^{*}\o V_{3}) \oplus (W^{*}\o V_{4}) \right)\\
(g_{1},g_{2},g_{3},g_{4}),(A_{1},A_{2},A_{3},A_{4}) \mapsto (g_{1}A_{1},g_{2}A_{2},g_{3}A_{3},g_{4}A_{4}),\end{gathered}$$ where we take the action of each $g_{j}$ to be a change of basis in the row space of $A_{j}$. Because the matrices $A_{j}$ are assumed to be full rank, we can, without loss of generality, act by an element of $GL(3)^{\times 4}$ and assume that $B_{j}= Id_{3}$ and move the 4-tuple $(A_{1}^{\top} | A_{2}^{\top} | A_{3}^{\top} | A_{4}^{\top})$ to $$\label{Amatrix}
A \cong \left( \begin{array}{c|c|c|c}
\begin{smallmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
x_{1,1}& x_{1,2} & x_{1,3}
\end{smallmatrix}
&
\begin{smallmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
x_{2,1}& x_{2,2} & x_{2,3}
\end{smallmatrix}
&
\begin{smallmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
x_{3,1}& x_{3,2} & x_{3,3}
\end{smallmatrix}
&
\begin{smallmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
x_{4,1}& x_{4,2} & x_{4,3}
\end{smallmatrix}
\end{array}\right)
.$$
There is an action of ${\operatorname{GL}}(W)$ acting on simultaneously on all of the column spaces of $A_{j}$ (which are all equal to $W$). While this action doesn’t turn out to be useful (it is trivialized in the tensorial map), the action of $\mathfrak{S}_{4}$ permuting the matrices $A_{j}$, also permutes the indices in the image of the tensorial map, and preserves the set of quadrifocal tensors.
Multilinearity spaces
---------------------
The $4\times 4$ minors of $M$ come in several classes, which have invariant descriptions. By construction, $ \bw{4}(\bigoplus_{j}V_{j})$ is a vector space with a natural ${\operatorname{GL}}(3n)$-action, but we can further view this as a ${\operatorname{GL}}( \bw{4}(\bigoplus_{j}V_{j}))$-action (each ${\operatorname{GL}}(V_{j})$ acting by invertible linear change of coordinates in $V_{j}$), and there is a natural inclusion of $G:=\prod_{j}{\operatorname{GL}}(V_{j})$, which may be thought of as block diagonal (with the proper choice of basis) inside ${\operatorname{GL}}(3n)$. Moreover, we may view $G$ as a product of a torus ${\mathbb{T}}^{n}:=\prod_{j}T_{j}\cong ({\mathbb{C}}^{*})^{n}$ ($T_{j}\cong{\mathbb{C}}^{*}$ acting by scaling block $j$ of $M$) and $\prod_{j}{\operatorname{SL}}(V_{j})$. In summary $$G := \prod_{j}T_{j} \times \prod_{j}{\operatorname{SL}}(V_{j}) \subset {\operatorname{GL}}(\bw{4}\bigoplus V_{j}).$$
Thus, we may consider ${\operatorname{Gr}}(4,\bigoplus V_{j})$ as a $G$-variety. On the other hand, on the GIT quotient, the torus action is trivialized, so we will consider $G':=\prod_{j}{\operatorname{SL}}(V_{j})$ acting on ${\operatorname{Gr}}(4, \bigoplus V_{j})\subset \bw{4}\bigoplus V_{j}$ and on the GIT quotient $ \widehat{Gr}(4, \bigoplus_{j}V_{j}) /\!/ ({\mathbb{C}}^{*})^{n} \subset \bw{4}(\bigoplus_{j}V_{j})/\!/ ({\mathbb{C}}^{*})^{n}$. The effect of trivializing the torus action is that we may identify every $V_{j}$ with its dual, and this induces an identification of every irreducible representation $S_{\pi}V_{j}$ with its dual $S_{\pi}V_{j}$.
Now $G$ acts on $\bw{4}(\bigoplus_{j}V_{j})$ and it has a decomposition into irreducible $G$-modules: $$\begin{matrix}
\bw{4}\left(\bigoplus_{j}V_{j}\right) &= &
\left( \bigoplus_{i\neq j} \bw{3}V_{i}\o V_{j} \right) \oplus
\left( \bigoplus_{i < j} \bw{2}V_{i}\o \bw{2} V_{j} \right) \\
&&\oplus
\left( \bigoplus_{i\not\in \{j,k\}, j<k} \bw{2}V_{i}\o V_{j} \o V_{k} \right) \oplus
\left( \bigoplus_{i<j<k<l}V_{i}\o V_{j} \o V_{k} \o V_{l} \right) .\end{matrix}$$ The 4 non-isomorphic module classes and their descriptions are listed in Table \[table1\].
$$\begin{array}{|rrrl|c|l|}
\hline
&&&\hspace{-5em}\text{Columns used} &\text{Invariant description} & \text{Space} \\
\hline
i_{1}&i_{2}&i_{3}&j & \bw{3}V_{i}\o V_{j} \;\;\cong\;\; V_{j} &\text{\emph{epipole space}} \\
i_{1}&i_{2}&j_{1}&j_{2}& \bw{2}V_{i}\o \bw{2} V_{j} \;\;\cong\;\; V_{i}^{*}\o V_{j}^{*} & \text{\emph{fundamental matrix space}}\\
i_{1}&i_{2}&j&k& \bw{2}V_{i}\o V_{j}\o V_{k} \;\;\cong\;\; V_{i}^{*}\o V_{j}\o V_{k} &\text{\emph{trifocal space}}\\
i&j&k&l& V_{i}\o V_{j}\o V_{k}\o V_{l} & \text{\emph{quadrifocal space}}\\
\hline
\end{array}$$
Now we will consider the projection of the cone over the Grassmannian $\widehat{Gr}(4,\bigoplus_{j}V_{j})$ to each type of multi-linearity space. The images of the projections are respectively the single view, the epipolar variety, the trifocal variety, and the quadrifocal variety. The fiber of the projection over a general point is the product of the ignored camera planes and the torus acting on the utilized camera planes. So it is interesting to consider the minimal number of cameras in each case. Moreover, because the projections on the level of vector spaces are equivariant, the images are automatically invariant (with respect to the appropriate group).
Suppose now that there are at most $4$ cameras. For each $i \in \{1,2,3,4\}$ let the set $\{e^{i}_{1},e^{i}_{2},e^{i}_{3}\}$ denote a basis of $V_{i}$, which also provides an ordered basis on ${\mathbb{C}}^{12}\cong V_{1}\oplus V_{2}\oplus V_{3}\oplus V_{4}$. In the remainder of this section we consider the parameterization of epipoles Sec. \[epi\], fundamental matrices Sec. \[fun\], trifocal tensors Sec. \[tri\], and quadrifocal tensors Sec. \[quad\], all up to the $G$-action using the matrix $A$ in . These results are summarized in Table \[table2\].
$$\begin{array}{|c|c|}
\hline & \\[-10pt]
S = \begin{psmallmatrix}
( x_{1,1}-x_{2,1}) &
(-1)( x_{1,2}-x_{2,2})&
( x_{1,3}-x_{2,3})
\end{psmallmatrix}^{\top}
&\text{\emph{$(1,2)$-epipole}} \\
&\\[-10pt]
\hline &\\[-10pt]
F = \begin{psmallmatrix}0&
x_{1,3}-x_{2,3}&
-x_{1,2}+x_{2,2}\\
-x_{1,3}+x_{2,3}&
0&
x_{1,1}-x_{2,1}\\
x_{1,2}-x_{2,2}&
-x_{1,1}+x_{2,1}&
0\\
\end{psmallmatrix}
&\text{\emph{$(1,2)$-fundamental matrix}}\\
&\\[-10pt] \hline&\\[-10pt]
\begin{array}{ll}
\left. \begin{smallmatrix}T_{i,i,\{k,l\}} &=& -x_{1,i}+x_{2,i} \\
T_{i,k,\{i,l\}} &=& -x_{1,i}+x_{3,i} \\
T_{k,i,\{i,l\}} &=& x_{2,i}-x_{3,i} ,
\end{smallmatrix} \right\}& \begin{array}{l}k, l \text{ distinct, } \\i \text{ distinct from } k \,\&\, l,\end{array} \\
\left.\begin{smallmatrix} T_{i,j,\{k,l\}} &=& 0 \end{smallmatrix}\hspace{3.2em} \right\}& \text{ else}.
\end{array}
&\text{\emph{$(1,2,3^{*})$-trifocal tensor}}\\
&\\[-10pt]
\hline
&\\[-10pt]
\begin{array}{ll}
\left. \begin{smallmatrix}
Q_{i,i,k,l} &=& (-1)^{i} (x_{1,i}-x_{2,i}) &=& -Q_{i,i,l,k},\\
Q_{i,k,i,l} &=& (-1)^{i} (x_{1,i}-x_{3,i}) &=& -Q_{i,l,i,k}, \\
Q_{i,k,l,i} &=& (-1)^{i} (x_{1,i}-x_{4,i}) &=& -Q_{i,l,k,i}, \\
Q_{k,i,i,l} &=& (-1)^{i} (x_{2,i}-x_{3,i}) &=& -Q_{l,i,i,k},\\
Q_{k,i,l,i} &=& (-1)^{i} (x_{2,i}-x_{4,i}) &=& -Q_{l,i,k,i},\\
Q_{k,l,i,i} &=& (-1)^{i} (x_{3,i}-x_{4,i}) &=& -Q_{l,k,i,i}
\end{smallmatrix} \right\}& \begin{array}{l}k< l \text{, } \\ i \text{ distinct from } k \,\&\, l, \end{array} \\
\left.\begin{smallmatrix}Q_{i,j,k,l} &=& 0 \end{smallmatrix}\hspace{9em} \right\} & \text{ else}.
\end{array}
& \text{\emph{$(1,2,3,4)$-quadrifocal tensor}}\\
\hline
\end{array}$$
Epipoles {#epi}
--------
For a pair of cameras, we consider the projection of ${\operatorname{Gr}}(4,6) = {\operatorname{Gr}}(4,V_{1}\oplus V_{2})\subset {\mathbb{P}}\bw{4}(V_{1}\oplus V_{2}) = {\mathbb{P}}^{14}$ to epipolar space ${\mathbb{P}}(V_{1}\o \bw{3}V_{2}) = {\mathbb{P}}^{2}$. The target space is (naturally isomorphic to) the projective plane, and the map subjects onto ${\mathbb{P}}^{2}$. The image is naturally ${\operatorname{GL}}(V_{1})\times {\operatorname{GL}}(V_{2})$-invariant, the action of ${\operatorname{GL}}(V_{1})$ being trivial and the 2-dimensional torus acts by a weight of $(3,1)$.
The image of the projection is the space of epipoles in view 1 imposed by view 2. Thus the epipoles may be recovered from the multi-view setup via a projection from the Grassmannian. To get an expression of an epipole in coordinates consider just two cameras and the matrix $$A \cong \left( \begin{array}{c|c}
\begin{smallmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
x_{1,1}& x_{1,2} & x_{1,3}
\end{smallmatrix}
&
\begin{smallmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
x_{2,1}& x_{2,2} & x_{2,3}
\end{smallmatrix}
\end{array}\right)
.$$ An element $S$ of single camera space has the following form in the Plücker coordinates associated to the determinants of the matrices constructed from one column from the first block of $A$ the three columns of the second block: $$S = \sum_{1\leq p \leq 3} S_{p,\{1,2,3\}}e^{1}_{p}\wedge e^{2}_{1}\wedge e^{2}_{2} \wedge e^{2}_{3}
\quad \in \quad \bw{3}V_{2}\o V_{1} \cong V_{1}
.$$ Applying this to $A$ we get the coordinates of the epipole $$\begin{pmatrix}
S_{1,\{1,2,3\}}(A) \\
S_{2,\{1,2,3\}}(A) \\
S_{3,\{1,2,3\}}(A)
\end{pmatrix}
=
\begin{pmatrix}
( x_{1,1}-x_{2,1})\\
(-1)( x_{1,2}-x_{2,2})\\
( x_{1,3}-x_{2,3})
\end{pmatrix}
.$$
Fundamental matrices {#fun}
--------------------
Again for a pair of cameras, we may consider the projection of ${\operatorname{Gr}}(4,6) = {\operatorname{Gr}}(4,V_{1}\oplus V_{2})\subset {\mathbb{P}}\bw{4}(V_{1}\oplus V_{2}) = {\mathbb{P}}^{14}$ to fundamental matrix space ${\mathbb{P}}(V_{1}^{*}\o V_{2}^{*}) = {\mathbb{P}}^{8}$. The target space may be interpreted as the projectivization of a space of $3 \times 3$ matrices, and caries the natural action of ${\operatorname{GL}}(V_{1})\times {\operatorname{GL}}(V_{2})$. We might call the image of the projection the *variety of fundamental matrices*, which is also naturally ${\operatorname{GL}}(V_{1})\times {\operatorname{GL}}(V_{2})$-invariant. Because the vector spaces $V_{1}$ and $V_{2}$ play symmetric roles, this image is also naturally $\mathfrak{S}_{2}$ invariant. It is well known that the matrices in the image of the projection have a one-dimensional kernel and the image variety is just the (degree 3) determinantal hypersurface.
Note that $\widehat{Gr}(4,6)$ is 9-dimensional, the GIT quotient $\widehat{{\operatorname{Gr}}(4,6)}/\!/ ({\mathbb{C}}^{*})^{2}$ is 7-dimensional, and thus birational to the projective variety of $3 \times 3$ matrices of rank $\leq 2$, the variety of fundamental matrices. Also note that the 2-dimensional torus acts by a weight of $(2,2)$.
An element $F$ of fundamental matrix space has the following form in Plücker coordinates: $$F = \sum_{1\leq i,j,k,l \leq 3,\; i<j, \;k<l} F_{\{i,j\},\{k,l\}}e^{1}_{i}\wedge e^{1}_{j}\wedge e^{2}_{k} \wedge e^{2}_{l}
\quad \in \quad \bw{2}V_{i}\o \bw{2} V_{j} \cong V_{i}^{*}\o V_{j}^{*}
.$$ For a fixed pair of distinct indices $i,j$, a fundamental matrix is gotten by applying $F$ to a blocked $4\times 12$ camera matrix $A$. In particular, $F(A)$ may be thought of as a vector whose coordinates are determinants of the $4\times 4$ submatrices of $A$ obtained by taking two columns from each from blocks $i$ and $j$ of $A$.
Now apply $F$ to $A$ in . The (1-2) fundamental matrix associated to $A$ is described by coordinates $ F_{\{i,j\},\{k,l\}}$, and we can represent this as the matrix $$F(A) =
\begin{pmatrix}0&
x_{1,3}-x_{2,3}&
-x_{1,2}+x_{2,2}\\
-x_{1,3}+x_{2,3}&
0&
x_{1,1}-x_{2,1}\\
x_{1,2}-x_{2,2}&
-x_{1,1}+x_{2,1}&
0\\
\end{pmatrix},$$ where the $(p,q)$ entry is $ F_{\{i,j\},\{k,l\}}$ such that $\{p,i,j\} = \{q,k,l\} =\{1,2,3\}$.
The trifocal variety {#tri}
--------------------
For a triple of cameras, we consider the projection of ${\operatorname{Gr}}(4,9) = {\operatorname{Gr}}(4,V_{1}\oplus V_{2} \oplus V_{3})\subset {\mathbb{P}}\bw{4}(V_{1}\oplus V_{2} \oplus V_{3}) = {\mathbb{P}}^{\binom{9}{4}-1}$ to trifocal space ${\mathbb{P}}(V_{1}\o V_{2}\o V_{3}^{*}) = {\mathbb{P}}^{26}$. The target space may be interpreted as the projectivization of a space of $3 \times 3 \times 3$ tensors, and caries the natural action of ${\operatorname{GL}}(V_{1})\times {\operatorname{GL}}(V_{2})\times {\operatorname{GL}}(V_{3})$. The image of the projection is called the *trifocal variety*, which is also naturally ${\operatorname{GL}}(V_{1})\times {\operatorname{GL}}(V_{2})\times {\operatorname{GL}}(V_{3})$-invariant. Because the vector spaces $V_{1}$ and $V_{2}$ play symmetric roles, this image is also naturally $\mathfrak{S}_{2}$ invariant. Non-trivial permutations involving $V_{3}$ will not preserve this trifocal variety but produce an isomorphic copy it.
The trifocal variety is 18-dimensional ([@HartleyZisserman p. 368]). The GIT quotient $\widehat{{\operatorname{Gr}}(4,9)}/\!/ ({\mathbb{C}}^{*})^{3}$ is 18-dimensional and thus is birational to the trifocal variety. In this case, the 3-dimensional torus acts by a weight of $(1,1,2)$.
For the trifocal tensor, consider 3 cameras in special position producing $$A \cong \left( \begin{array}{c|c|c}
\begin{smallmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
x_{1,1}& x_{1,2} & x_{1,3}
\end{smallmatrix}
&
\begin{smallmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
x_{2,1}& x_{2,2} & x_{2,3}
\end{smallmatrix}
&
\begin{smallmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
x_{3,1}& x_{3,2} & x_{3,3}
\end{smallmatrix}
\end{array}\right)
.$$ An element $T$ of trifocal space has the following form in Plücker coordinates: $$T = \sum_{1\leq i,j,k,l \leq 3,\; k<l}
T_{i,j,\{k,l\}} e^{1}_{i}\wedge e^{2}_{j}\wedge e^{3}_{k} \wedge e^{3}_{l}
\quad \in \quad
V_{1}\o V_{2}\o \bw{2}V_{3} \cong V_{1}\o V_{2}\o V_{3}^{*}
.$$ A trifocal tensor is gotten by applying $T$ to $A$. In particular, $T(A)$ may be thought of as a vector whose coordinates are determinants of the $4\times 4$ submatrices of $A$ obtained by taking one column from each of the first two blocks of $A$ and two columns of from the third block of $A$.
The locus of trifocal tensors is the ${\operatorname{GL}}(3)\times{\operatorname{GL}}(3)\times {\operatorname{GL}}(3)$-orbit of the 9-dimensional linear space whose coordinates are given by $T_{i,j,\{k,l\}}$ satisfying the following conditions: $$\begin{array}{ll}
\left. \begin{array}{l}T_{i,i,\{k,l\}}(A) = -x_{1,i}+x_{2,i} \\
T_{i,k,\{i,l\}}(A) = -x_{1,i}+x_{3,i} \\
T_{k,i,\{i,l\}}(A) = x_{2,i}-x_{3,i} ,
\end{array} \right\}& k \text{ and } l \text{ distinct, and } i \text{ distinct from } k,l, \\
\begin{array}{l} T_{i,j,\{k,l\}}(A)=0 \end{array} & \text{else}.
\end{array}$$
The quadrifocal variety {#quad}
-----------------------
For a quadruple of cameras, we consider the projection of ${\operatorname{Gr}}(4,12) = {\operatorname{Gr}}(4,V_{1}\oplus V_{2} \oplus V_{3}\oplus V_{4})\subset {\mathbb{P}}\bw{4}(V_{1}\oplus V_{2} \oplus V_{3}\oplus V_{4}) = {\mathbb{P}}^{\binom{12}{4}-1}$ to quadrifocal space ${\mathbb{P}}(V_{1}\o V_{2}\o V_{3}\o V_{4}) = {\mathbb{P}}^{80}$. The target space may be interpreted as the projectivization of a space of $3 \times 3 \times 3 \times 3$ tensors, and caries the natural action of ${\operatorname{GL}}(V_{1})\times {\operatorname{GL}}(V_{2})\times {\operatorname{GL}}(V_{3})\times {\operatorname{GL}}(V_{4})$. The image of the projection is called the *quadrifocal variety*, which is also naturally ${\operatorname{GL}}(V_{1})\times {\operatorname{GL}}(V_{2})\times {\operatorname{GL}}(V_{3})\times {\operatorname{GL}}(V_{4})$-invariant. Because the vector spaces $V_{1},V_{2}, V_{3}$ and $V_{4}$ play symmetric roles, this image is also naturally $\mathfrak{S}_{4}$ invariant.
The quadrifocal variety is 29-dimensional ([@HartleyZisserman p. 423]), and the GIT quotient $\widehat{{\operatorname{Gr}}(4,12)}/\!/ ({\mathbb{C}}^{*})^{4}$ is 29-dimensional, and thus birational to the quadrifocal variety. In this case, the 4-dimensional torus acts by a weight of $(1,1,1,1)$.
An element $Q$ of quadrifocal space has the following form in Plücker coordinates: $$Q = \sum_{1\leq i,j,k,l \leq 3}Q_{i,j,k,l}e^{1}_{i}\wedge e^{2}_{j}\wedge e^{3}_{k} \wedge e^{4}_{l}
\quad \in \quad V_{i}\o V_{j}\o V_{k}\o V_{l}
.$$ A quadrifocal tensor is gotten by applying $Q$ to a blocked $4\times 12$ camera matrix $A$. In particular, $Q(A)$ may be thought of as a vector whose coordinates are determinants of the $4\times 4$ submatrices of $A$ obtained by taking one column from each of the 4 blocks of $A$.
For the quadrifocal tensor, consider 4 cameras in special position. $$A \cong \left( \begin{array}{c|c|c|c}
\begin{smallmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
x_{1,1}& x_{1,2} & x_{1,3}
\end{smallmatrix}
&
\begin{smallmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
x_{2,1}& x_{2,2} & x_{2,3}
\end{smallmatrix}
&
\begin{smallmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
x_{3,1}& x_{3,2} & x_{3,3}
\end{smallmatrix}
&
\begin{smallmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
x_{4,1}& x_{4,2} & x_{4,3}
\end{smallmatrix}
\end{array}\right)
.$$
\[lem:quadForm\] The locus of quadrifocal tensors is the ${\operatorname{GL}}(3)\times{\operatorname{GL}}(3)\times {\operatorname{GL}}(3)\times {\operatorname{GL}}(3)$-orbit of the 12-dimensional linear space of tensors whose coordinates are given by $Q_{i,j,k,l}$ satisfying the following conditions: $$\begin{array}{ll}
\left. \begin{array}{l}
Q_{i,i,k,l}(A) =(-1)^{i} (x_{1,i}-x_{2,i}) = -Q_{i,i,l,k}(A),\\
Q_{i,k,i,l}(A) =(-1)^{i} (x_{1,i}-x_{3,i}) = -Q_{i,l,i,k}(A), \\
Q_{i,k,l,i}(A) = (-1)^{i} (x_{1,i}-x_{4,i}) = -Q_{i,l,k,i}(A), \\
Q_{k,i,i,l}(A) = (-1)^{i} (x_{2,i}-x_{3,i}) =-Q_{l,i,i,k}(A),\\
Q_{k,i,l,i}(A) = (-1)^{i} (x_{2,i}-x_{4,i}) =-Q_{l,i,k,i}(A),\\
Q_{k,l,i,i}(A) = (-1)^{i} (x_{3,i}-x_{4,i}) = -Q_{l,k,i,i}(A)
\end{array} \right\}& k< l \text{ and } i \text{ distinct from } k,l, \\
\begin{array}{l} Q_{i,j,k,l}(A)=0 \end{array} & \text{else}.
\end{array}$$
Arbitrary dimensional cameras: a connection to principal minors {#sec:Principal}
===============================================================
It is curious to study the case when $W$ and $V_{i}$ arbitrary dimensions (see [@BertoliniBesanaTurrini]). For instance, when each $V_{i}$ has dimension two, we might imagine the cameras to be flatlanders’ cameras. When the $V_{i}$ have dimension more than 3, we might consider an image plane that records higher dimensional data such as temperature or color.
Principal minors
----------------
In the case $V_{i}\cong{\mathbb{C}}^{2}$ we find it very interesting that seemingly unrelated work regarding principal minors fits into this framework [@HoltzSturmfels; @LinSturmfels; @BorodinRains; @Oeding_principal; @oeding_thesis; @GriffTsat1; @GriffTsat2].
Suppose $W$ has dimension $m$ and that we have $m$ copies of $V_{i}$, each with dimension 2. Consider again the matrix $M = \left(\begin{array}{c|c|c|c} A_{1}^{\top} & A_{2}^{\top}&\dots & A_{m}^{\top}\end{array}\right)$, now as a $m\times 2m$ matrix consisting of $m$ blocks of size $m\times 2$. By a left action of ${\operatorname{GL}}(W)$ we may assume that each block of $M$ is of the form $A_{i}^{\top} = \begin{pmatrix}e_{i}& b_{i}\end{pmatrix}$, where $e_{i}$ is the $i$-th standard basis vector of $W$ and $b_{i}$ is arbitrary. Let $B$ denote the $m\times m$ matrix with columns $b_{i}$.
It is straightforward to see that the maximal minors of $M$ correspond to the minors of $B$ and moreover that the minors of $M$ that use precisely one column from each block of $M$ correspond to the principal minors of $B$.
In turn, this identification naturally gives the space $V_{1}\o\dots\o V_{m}$ the interpretation as the space of all $2^{m}$ principal minors of an $m\times m$ matrix (the $0\times 0$ minor may be assumed to be $1$ in this construction).
The case $m=4$ was studied by [@BorodinRains; @LinSturmfels], who discovered the minimal defining equations of the ideal of relations among principal minors of a generic $4\times 4$ matrix, and found that the algebraic variety coincides with the main component of the singular locus of the $2\times 2\times 2\times 2$ hyperdeterminant.
Weyman and Zelevinski [@WeyZel_Sing] analyzed the singular locus of the hyperdeterminant. In the $2\times2\times2\times 2$ case, there are 8 components of the singular locus: $$\nabla_{\text{cusp}} \cup \nabla_{\text{node}}(\emptyset) \cup \nabla_{\text{node}}(\{i,j\}) \quad 1\leq i< j \leq 4.$$ Moreover, according to Holweck, [@Holweck_SingGrass] the “main” component $\nabla_{\text{node}}(\emptyset)$ also has the interpretation as the projective dual of the secant line variety: $$\nabla_{\text{node}}(\emptyset) = \sigma_{2}({\mathbb{P}}^{1}\times {\mathbb{P}}^{1}\times {\mathbb{P}}^{1}\times {\mathbb{P}}^{1})^{\vee} \subset {\mathbb{P}}({V_{1}\o V_{2}\o V_{3}\o V_{4}})^{*}
.$$ The variety of principal minors of $4\times 4$ matrices is the projection $${\operatorname{Gr}}(4,V_{1}\oplus V_{2} \oplus V_{3} \oplus V_{4}) \to {\mathbb{P}}({V_{1}\o V_{2}\o V_{3}\o V_{4}})
.$$ The dual variety of ${\operatorname{Gr}}(4,8)$ is *not* one of the 3 cases which is normal [@Holweck_SingGrass], however it does have many other nice properties, such as a finitely generated invariant ring, finitely many orbits in the null cone, and a set of normal forms that depend on 8 parameters, [@CDZG]. It would be nice to see how to exploit these coincidences.
In the case that the matrix $B$ is assumed to be symmetric, [@HoltzSturmfels] re-invigorated a classical study going back to Cayley of relations amongst principal minors of symmetric matrices. (See [@LinSturmfels] and [@Oeding_principal] for a historical discussion.) In particular, the ideal of relations of the principal minors of a generic symmetric $3 \times 3$ matrix is generated by a single equation known as Cayley’s $2\times 2\times 2$ hyperdeterminant, denoted $h$ for this discussion. In the $4\times 4$ case Holtz and Sturmfels discovered that the $\mathfrak{S_{4}}\ltimes \prod_{i=1..4}{\operatorname{SL}}(V_{i})$-orbit of $h$ generates the ideal of relations, and conjectured that a similar orbit of equations, which they called the hyperdeterminantal module, generates the ideal of relations among principal minors of a symmetric matrix. This conjecture was solved (set-theoretically) by the author [@Oeding_principal; @oeding_thesis]. We note that in the framework of this paper, imposing that the matrix $B$ be symmetric naturally imposes a restriction from the Grassmannian ${\operatorname{Gr}}(m,2m)$ to the Lagrangian Grassmannian $\textrm{LGr}(m,2m)$, (see [@oeding_thesis]).
In the case that the matrix $B$ is assumed to be skew-symmetric, one considers the principal Pfaffians. It turns out that the relations among principal Pfaffians are precisely the equations of the orthogonal Grassmannian, which are analogous to the Plücker relations and were known classically, (see [@LM02] for a modern treatment in invariant language).
Higher dimensional images
-------------------------
It would be interesting to consider, for instance, a camera with fixed position continuously viewing a scene in time in our paradigm as $W$ a $5$-dimensional vector space and $V_{i}$ each as $4$-dimensional vector spaces (3 space dimensions and one time). This seems to be a promising approach to understanding this and a wide variety of generalizations.
Contractions and homography tensors {#sec:Contract}
===================================
According to Shashua and Wolf [@ShashuaWolf] a contraction of a quadrifocal tensor is a homography tensor [@Shashua00homographytensors] in the other 3 views. Moreover, this property defines the set of homography tensors. Similarly, the contraction of a homography tensor is an LLC mapping (a rank 2 matrix relating the epipoles in two views). We will give an invariant description of this process and show how to get 600 independent internal constraints on quadrifocal tensors.
Consider $V_{1}\o V_{2}\o V_{3}\o V_{4}$ and suppose we choose a basis of $V_{4}$, $\{x,y,z\}$. Then we may write $Q\in V_{1}\o V_{2}\o V_{3}\o V_{4}$ as $$Q(x,y,z) =xQ_{1} + y Q_{2} + z Q_{3},$$ where $Q_{i}$ are the standard $3 \times 3 \times 3$ slices of the $3 \times 3 \times 3 \times 3$ tensor. By abuse of notation, we also think of $Q(x,y,z)$ as a function from ${\mathbb{C}}^{3}$ to $V_{1}\o V_{2}\o V_{3}$, and let $x,y,z$ act as variables.
The following results come from [@ShashuaWolf] and [@Shashua00homographytensors].
If $Q$ is a quadrifocal tensor then $Q(x,y,z)$ is a homography tensor for every value of $x,y,z$.
For our purposes, the above result can be used to define the notion of “homography tensor.” Now choose a basis $a,b,c$ of $V_{3}$. If $H \in V_{1}\o V_{2}\o V_{3}$, then we may write $H(a,b,c) = aH_{1}+bH_{2} + cH_{3}$. In slightly different language [@AholtOeding Prop. 7.2] we showed that the Zariski closure of the homography tensors is an irreducible variety (we called it $\textrm{P-Rank}^{2,2,2}$), defined (at least set-theoretically) by 30 cubic equations.
If $H$ is a homography tensor then $H(a,b,c)$ is an LLC (Linear Line Complex) mapping for every value of $a,b,c$.
For our purposes, an LLC is a skew-symmetric $3 \times 3$ matrix, which necessarily has (even) rank $\leq 2$. Thus for all values of $a,b,c$ the matrix $H(a,b,c)$ must satisfy the constraint $\det(H(a,b,c)) \equiv 0$. In particular, every coefficient in the cubic polynomial in $a,b,c$ must vanish. This condition gives a basis of the ten-dimensional space of cubics. Note, the coefficient of $a^{3}$ is the determinant of the first slice, and the coefficients on $b^{3}$ and $c^{3}$ are respectively the determinants of the second and third slices. This space also has the interpretation of $\bw{3}V_{1}^{*}\o \bw{3}V_{2}^{*}\o S^{3}V_{3}^{*}$ as a $G$-module.
We can apply the same method to quadrifocal tensors. Namely if $Q$ is a quadrifocal tensor, $Q(x,y,z)$ must satisfy all of the internal trifocal constraints for all values of $x,y,z$. In particular, $Q(x,y,z)(a,b,c)$ must be an epipolar matrix (whose entries are bi-homogeneous quadrics), and we must have the bi-homogeneous sextic polynomial (of bi-degree (3,3)) $\det(Q(x,y,z)(a,b,c)) \equiv 0$ for all values of $x,y,z,a,b,c$.
The space of bi-degree (3,3) sextics is $100$-dimensional, and the coefficients of the expression $\det(Q(x,y,z)(a,b,c))$ provide a basis of a 100-dimensional space of cubics that vanish on the quadrifocal variety. Note, the coefficient of $x^{3}a^{3}$ is the determinant of the first slice, and there are 8 other monomials that are the product of two cubes, the coefficients of which correspond to the determinants of the 8 other slices. This space also has the interpretation as the ${\operatorname{GL}}(3)^{\times 4}$-module $\bw{3}V_{1}\o \bw{3}V_{2}\o S^{3}V_{3} \o S^{3}V_{4}$.
If we interchange the roles of $V_{1},V_{2},V_{3},V_{4}$, and apply the same construction we obtain 6 non-isomorphic modules of the same format $\bw{3}V_{i}\o \bw{3}V_{j}\o S^{3}V_{k} \o S^{3}V_{l}$. In particular, we find a space of 600 cubic polynomials in the ideal of the quadrifocal variety, 54 of which are determinants of $3 \times 3$ slices of a $3 \times 3 \times 3 \times 3$ tensor.
Let $G:=\mathfrak{S}_{4}\ltimes {\operatorname{GL}}(3)^{\times 4}$. Since the quadrifocal variety is $G$-invariant, we can describe its ideal as a $G$ module. For convenience, when the $\mathfrak{S}_{4}$ symmetry is present, we write $S_{\pi_{1}}S_{\pi_{2}} S_{\pi_{3}} S_{\pi_{4}}$ for the direct sum of $S_{\pi_{1}}V_{1}^{*}\o S_{\pi_{2}}V_{2}^{*}\o S_{\pi_{3}}V_{3}^{*}\o S_{\pi_{4}}V_{4}^{*}$ and all non-redundant copies of it obtained by permuting the indices.
The above discussion implies the following:
\[prop:600\] Suppose $Q$ is a quadrifocal tensor. Then the 600 polynomials forming a basis of the $G$-module $S_{3}S_{3}S_{1,1,1}S_{1,1,1}$ vanish on $Q$.
Because one of the contractions of a trifocal tensor also form a homography, we know that 10 cubic polynomials vanish on the set of trifocal tensors. In [@AholtOeding] we showed that this condition cuts out a subset of the $3 \times 3 \times 3$ tensors consisting of 4 irreducible algebraic varieties. In order to distinguish the trifocal variety, more equations are needed.
\[thm:AO\] The ideal of the trifocal variety in $V_{1}\o V_{2}\o V_{3}^{*}$ is generated by 10 cubic, 81 quintic, and 1980 sextic polynomials. These are represented by the following ${\operatorname{GL}}(3)^{\times 3}$ modules:
$\begin{matrix}
\M_3 &=& \bw{3}V_{1}^{*}\o \bw{3}V_{2}^{*}\o S^{3}V_{3}, \\
\\
\M_{5}&=& S_{221}V_{1}^{*}\o S_{221}V_{2}^{*}\o S_{311}V_{3} &\oplus& S_{221}V_{1}^{*}\o S_{221}V_{2}^{*}\o S_{221}V_{3}, \\
\\
\M_{6}&=
& S_{411}V_{1}^{*}\o S_{33}V_{2}^{*} \o S_{222}V_{3} &\oplus
& S_{33}V_{1}^{*}\o S_{411}V_{2}^{*}\o S_{222} V_{3} &\oplus
& S_{33}V_{1}^{*}\o S_{222}V_{2}^{*}\o S_{411}V_{3} \\&\oplus
& S_{222}V_{1}^{*}\o S_{33}V_{2}^{*} \o S_{411}V_{3} &\oplus
& S_{33}V_{1}^{*}\o S_{33}V_{2}^{*} \o S_{222} V_{3} &\oplus
& S_{33}V_{1}^{*}\o S_{222}V_{2}^{*}\o S_{33}V_{3} \\&\oplus
& S_{222}V_{1}^{*}\o S_{33}V_{2}^{*} \o S_{33} V_{3} &\oplus
& S_{33}V_{1}^{*}\o S_{321}V_{2}^{*} \o S_{321}V_{3} &\oplus
& S_{321}V_{1}^{*}\o S_{33}V_{2}^{*}\o S_{321}V_{3} .
\end{matrix}$
In the next section we work to obtain a similar statement for quadrifocal tensors.
Computational results for the quadrifocal ideal {#sec:Rep}
===============================================
The goal of this section is to give a description of the lowest degree part of the vanishing ideal for the quadrifocal variety in terms of $G$-modules. First recall that the polynomial ring $\bigoplus_{d} S^{d}({V_{1}^{*}\o V_{2}^{*}\o V_{3}^{*}\o V_{4}^{*}})$ has a graded isotopic decomposition (see [@Landsberg-Manivel04], for instance) with respect to $G = \mathfrak{S}_{4}\ltimes {\operatorname{GL}}(3)^{\times 4}$: $$S^{d}({V_{1}^{*}\o V_{2}^{*}\o V_{3}^{*}\o V_{4}^{*}}) = \bigoplus_{\pi}S_{\pi}V^{*} \o {\mathbb{C}}^{m_{\pi}}
,$$ with Schur modules $S_{\pi}V^{*}:=S_{\pi_{1}}V_{1}^{*} \o S_{\pi_{2}}V_{2}^{*} \o S_{\pi_{3}}V_{3}^{*} \o S_{\pi_{4}}V_{4}^{*}$ and multiplicity space $ {\mathbb{C}}^{m_{\pi}} $. The multiplicity space ${\mathbb{C}}^{m_{\pi}}$ has a basis given by linear combinations of fillings of shape $\pi$. We write $(S_{\pi_{1}}S_{\pi_{2}}S_{\pi_{3}}S_{\pi_{4}})\o {\mathbb{C}}^{m}$ to indicate the $G$ module (occurring with multiplicity $m$) gotten by summing over all non-redundant permutations of the partitions indexing $S_{\pi}V^{*}$.
Let $I_{d}$ denote the degree $d$ piece of the ideal of the quadrifocal variety, and let $G = \mathfrak{S}_{4} \ltimes {\operatorname{GL}}(3)^{\times 4}$. The symmetry assisted computations of $I_{d}$ up to degree $d\leq 9$ using random points of the quadrifocal variety are reported in Table \[toDeg9\]. For $5\leq d \leq 9$ the dimensions of $I_{d}$ and the list necessary modules of minimal generators in Table \[toDeg9\] are correct with high probability.
[|c|r|l|r|]{}
--------
graded
piece
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: The dimension and isotypic description of the ideal of the quadrifocal variety up to degree 9.[]{data-label="toDeg9"}
&
--------------
$\dim I_{d}$
--------------
: The dimension and isotypic description of the ideal of the quadrifocal variety up to degree 9.[]{data-label="toDeg9"}
&
-----------------------
necessary $G$ modules
of minimal generators
-----------------------
: The dimension and isotypic description of the ideal of the quadrifocal variety up to degree 9.[]{data-label="toDeg9"}
&
-------------------
dimension of
necessary mingens
-------------------
: The dimension and isotypic description of the ideal of the quadrifocal variety up to degree 9.[]{data-label="toDeg9"}
\
$I_{2}$ &0 &$\begin{array}{rl}\M_{2}=&0\end{array}$ &0\
$I_{3}$ &600 &$\begin{array}{rl}\M_{3}= &S_{3}S_{3}S_{1,1,1}S_{1,1,1}\end{array}$ &600\
$I_{4}$ &48,600& $\begin{array}{rl}\M_{4}=&0\end{array}$ & 0\
$I_{5}$ & 1,993,977 & $\begin{array}[t]{rl}\M_{5}=& S_{3,1,1}S_{3,1,1}S_{3,1,1}S_{3,1,1}\\ &\oplus S_{2,2,1}S_{2,2,1}S_{2,2,1}S_{2,2,1}\end{array}$ & 1,377\
$I_{6}$ & 54,890,407 & $\begin{array}[t]{rl}
\M_{6}=& S_{4,1,1}S_{3,3}S_{2,2,2}S_{2,2,2}\o {\mathbb{C}}^{2}\\
&\oplus S_{3,3}S_{3,3}S_{2,2,2}S_{2,2,2}\o {\mathbb{C}}^{2}\\
&\oplus S_{3,2,1}S_{3,2,1}S_{2,2,2}S_{2,2,2}\\
&\oplus S_{2,2,2}S_{2,2,2}S_{2,2,2}S_{2,2,2}\o {\mathbb{C}}^{2}\\
&\oplus S_{6}S_{3,3}S_{3,3}S_{2,2,2}
\end{array}$ &37,586\
$I_{7}$ &1,140,730,128 &$ \begin{array}{rl}\M_{7}=& 0 \end{array}$& 0\
$I_{8}$ & 18,940,147,947 & $\begin{array}[t]{rl}\M_{8}= & S_{4,4}S_{4,4}S_{4,4}S_{4,2,2} \otimes {\mathbb{C}}^{2}\end{array}$ & 162,000\
$I_{9}$ & $\geq$ 223,072,284,455 & $ \begin{array}[t]{rl}\M_{9}\geq & S_{5, 4} S_{5, 4} S_{5, 4}S_{4, 3, 2} \\ & \oplus S_{5, 4} S_{5, 4} S_{5, 4} S_{5, 2, 2}\end{array}$ &$ \geq$ 3,087,000\
*Description of computation:* The qualifier “with high probability” refers to the fact that we computed the ideal on random subsets of points from the quadrifocal variety, so in principle we could have chosen a set of points in special position and would have over counted the dimension of the ideal. The set of points in such special position, however, being of lower dimension, has measure zero in the quadrifocal variety, so we say the results hold with high probability. The proof techniques we use for the degree $\leq 4$ computations, however are valid unconditionally.
The content of this computation is two-fold. First we computed (in Maple) the entire ideal degree by degree, up to degree 8 and partially in degree 9, making use of the isotypic decomposition of the polynomial ring. Second, we checked for representation-theoretic certificates for necessity of minimal generators using `SchurRings` in Macaulay2. In the ancillary files associated with the arXiv version of this manuscript we provide a minimal set of data necessary to check our work. This includes basic maple scripts, fillings that yield a basis of each isotypic component, and the resulting modules in the ideal. Here is a summary of these computations.
Every irreducible $G$-module has the property that it is the vector space spanned by the $G$-orbit of a so-called *highest weight vector* (see [@LandsbergTensorBook]). We used Young symmetrizers to obtain a basis of each multiplicity space ${\mathbb{C}}^{m_{\pi}}$ in the isotypic decomposition of the polynomial ring above. Then we computed the subspace of ${\mathbb{C}}^{m_{\pi}}$ that vanished on the quadrifocal tensors. This algorithm was used and outlined in [@AholtOeding; @BatesOeding] for example. Since we learned this algorithm from the paper of Landsberg and Manivel [@Landsberg-Manivel04], we call this algorithm the *Landsberg-Manivel-algorithm* or the *LM-algorithm* for short.
The output of this symmetry-enhanced polynomial interpolation computation is a $G$-module description of the ideal in each degree. The dimensions of these modules give the beginning of the Hilbert function of the quadrifocal ideal, and are also reported in Table \[toDeg9\]. We will use notation of `SchurRings`: The ring $Sym^{\bullet}( {V_{1}^{*}\o V_{2}^{*}\o V_{3}^{*}\o V_{4}^{*}}) $ is regarded as a tower of rings. The variables in each ring are represented as $s_{\pi}$, (respectively $t_{\pi}$, $u_{\pi}$, $v_{\pi}$), for partitions $\pi$. The correspondence between the two notations is $${m} s_{\pi_{1}}t_{\pi_{2}}u_{\pi_{3}}v_{\pi_{4}} \leftrightarrow S_{\pi_{1}}V_{1}^{*} \o S_{\pi_{2}}V_{2}^{*} \o S_{\pi_{3}}V_{3}^{*} \o S_{\pi_{4}}V_{4}^{*} \o {\mathbb{C}}^{m}.$$
In degree $3$ our application of the LM-algorithm found the following modules: [$$I_{3}=
({s}_{(1,1,1)} {t}_{(1,1,1)} {u}_{3}+\blue({s}_{(1,1,1)} {t}_{3}+{s}_{3} {t}_{(1,1,1)}\blue) {u}_{(1,1,1)}) {v}_{3}+(\blue({s}_{(1,1,1)} {t}_{3}+{s}_{3} {t}_{(1,1,1)}\blue) {u}_{3}+{s}_{3} {t}_{3}
{u}_{(1,1,1)}) {v}_{(1,1,1)}.$$]{} To save space, we only record those modules up to the $\mathfrak{S}_{4}$-action: $$\M_{3}:={s}_{3} {t}_{3} {u}_{(1,1,1)} {v}_{(1,1,1)}$$ This module corresponds to the same $600$ polynomials in Proposition \[prop:600\]. Non-vanishing of polynomials on random points of a variety implies non-vanishing for the entire variety, so this $600$ dimensional vector space of cubics are the only cubics vanishing on the quadrifocal variety. Therefore, the $d\leq 3$ computations hold with no “high probability” qualifier.
The modules we found (by the LM-algorithm) in degree 4 are (up to the $\mathfrak{S}_{4}$-action): $$I_{4}=({s}_{4} {t}_{4}+\blue({s}_{4}+{s}_{(3,1)}\blue) {t}_{(3,1)}) {u}_{(2,1,1)} {v}_{(2,1,1)}$$
Because $81*600 = 48600$, we guess that all the equations in $I_{4}$ come from linear combinations of products of linear forms with the $600$ cubics in $I_{3}$ and thus we would guess that there are no new generators in degree 4. We would like to prove this using representation theory.
The multiplication in the ring $S^{\bullet} ({V_{1}^{*}\o V_{2}^{*}\o V_{3}^{*}\o V_{4}^{*}})$ is just the usual polynomial multiplication. It is not easy to determine the isotopic version of multiplication. However, we can get a lower bound on the modules of minimal generators in our ideal. The basic idea is the following. The multiplication in the ring $S^{\bullet}({V_{1}^{*}\o V_{2}^{*}\o V_{3}^{*}\o V_{4}^{*}})$ is also the restriction of the multiplication in the tensor ring $({V_{1}^{*}\o V_{2}^{*}\o V_{3}^{*}\o V_{4}^{*}})^{\otimes}$. The tensor product of two representations of the form $S_{\pi}V^{*}$ is obtained by an iteration of the Littlewood-Richardson rule.
Using the package `SchurRings` [@SchurRings] we computed the tensor product $$I_{3}\o( {V_{1}^{*}\o V_{2}^{*}\o V_{3}^{*}\o V_{4}^{*}})$$ and found that it exactly coincides with $I_{4}$. This is an indication that there are probably no new generators in degree $4$, however it could be that the modules resulting in $I_{3}\o {V_{1}^{*}\o V_{2}^{*}\o V_{3}^{*}\o V_{4}^{*}}$ are not actually in $S^{4}({V_{1}^{*}\o V_{2}^{*}\o V_{3}^{*}\o V_{4}^{*}})$. The only way that there could be new minimal generators in degree 4 is if there were already some syzygies amongst the degree 3 equations. At least in degree 4 it is possible to look at the highest weight vectors and check that they are in the ideal generated by the 600 cubics in Macaulay2. On the other hand, we can argue in a less computationally intensive way by using the following special case of a more general idea from [@RaicuProducts], which was employed in [@RaicuGSS; @OedingRaicu].
\[removeBox\] Suppose $F_{\pi}:=(F_{\pi_{1}},F_{\pi_{2}},F_{\pi_{3}},F_{\pi_{4}})$ is a filling using the ordered alphabets $(\A_{1},\A_{2},\A_{3},\A_{4})$ giving a nonzero realization of the module $S_{\pi}V^{*}$ in $S^{d}({V_{1}^{*}\o V_{2}^{*}\o V_{3}^{*}\o V_{4}^{*}})$. If $F_{\mu}:=(F_{\mu_{1}},F_{\mu_{2}},F_{\mu_{3}},F_{\mu_{4}})$ is a filling giving a nonzero realization of the module $S_{\mu}V^{*}$ in $S^{d-1}({V_{1}^{*}\o V_{2}^{*}\o V_{3}^{*}\o V_{4}^{*}})$ and $F_{\mu}$ may be obtained from $F_{\pi}$ by respectively deleting the last used letter in each of the alphabets $(\A_{1},\A_{2},\A_{3},\A_{4})$, then $S_{\pi}V^{*}$ is in the ideal generated by $S_{\mu}V^{*}$.
Now we apply Lemma \[removeBox\] to see that $S_{211}S_{211}S_{31}S_{31}$ is in the ideal generated by $S_{111}S_{111}S_{3}S_{3}$. In this case everything occurs with multiplicity one, so our work is much easier. The filling $$\young(14,2,3)\o \young(14,2,3)\o \young(123,4)\o \young(123,4)$$ produces a realization of a copy of $S_{31}S_{31}S_{211}S_{211}$ in the ideal of the quadrifocal variety. Notice that by removing the fourth letter we obtain the filling $$\young(1,2,3)\o \young(1,2,3)\o \young(123)\o \young(123)
,$$ which produces a nonzero copy of $S_{111}S_{111}S_{3}S_{3}$.
The same argument may be applied to $S_{4}S_{4}S_{211}S_{211}$, with the same result. Therefore, none of the modules in degree 4 are minimal generators of the ideal of the quadrifocal variety.
In degree 5 we note that $1929501-600*3321 = 1,377$. This means that the degree 3 equations cannot generate all of the ideal in degree 5, and there must be at least $1,377$ new minimal generators in degree 5. Moreover, if there are no degree 2 syzygies amongst the degree 3 equations then the space of minimal generators in degree 5 would be precisely $1,377$-dimensional. In principle one could try to use a degree-limited Gröbner basis computation in Macaulay2 to check if each highest weight vector of each of the modules in $I_{5}$ is actually in $\langle I_{3}\rangle_{5}$, but memory limitations become a problem.
However, we can argue by comparing multiplicities of isotypic components. In degree 5 the LM-algorithm produced the following modules (up to the $\mathfrak{S}_{4}$-action): [$$\begin{gathered}
I_{5} = \red({s}_{5} {t}_{5}+\blue({s}_{5}+2 {s}_{(4,1)}\blue) {t}_{(4,1)}+\blue({s}_{5}+{s}_{(4,1)}+{s}_{(3,2)}\blue) {t}_{(3,2)}+\blue(3 {s}_{(4,1)}+7 {s}_{(3,1,1)}\blue) {t}_{(3,1,1)}\red) {u}_{(3,1,1)} {v}_{(3,1,1)}\\
+\green(\red(\blue({s}_{5}+2{s}_{(4,1)}\blue) {t}_{(4,1)}+{s}_{(4,1)} {t}_{(3,2)}+\blue(2 {s}_{5}+2 {s}_{(4,1)}+2 {s}_{(3,2)}\blue) {t}_{(3,1,1)}\red) {u}_{(3,1,1)}\\
+\red({s}_{5} {t}_{5}+\blue({s}_{5}+2 {s}_{(4,1)}\blue){t}_{(4,1)}+\blue({s}_{5}+{s}_{(4,1)}+{s}_{(3,2)}\blue) {t}_{(3,2)}
+\blue({s}_{(4,1)}+{s}_{(3,1,1)}\blue) {t}_{(3,1,1)}
+{s}_{(2,2,1)} {t}_{(2,2,1)}\red)
{u}_{(2,2,1)}\green) {v}_{(2,2,1)}.\end{gathered}$$ ]{}
Using `SchurRings` we found that the only modules left in the difference between $I_{5}$ and $I_{4}\otimes {V_{1}\o V_{2}\o V_{3}\o V_{4}}$ are $$\M_{5}:={s}_{(3,1,1)} {t}_{(3,1,1)} {u}_{(3,1,1)} {v}_{(3,1,1)}+{s}_{(2,2,1)} {t}_{(2,2,1)} {u}_{(2,2,1)} {v}_{(2,2,1)}.$$ From this computation we know that these two summands must be among the minimal generators. We do not know if there are any other minimal generators in degree 5.
While this `ShurRings` computation ignores the structure of the multiplicity spaces in the ideals, to first approximation it tells if there is any representation-theoretic reason for modules to be among the minimal generators.
In degree 6 the LM-algorithm produced the following modules (up to the $\mathfrak{S}_{4}$-action) $I_{6}=$ [$$\begin{gathered}
\red{(}s_{6}t_{6}+\blue{(}s_{6}+2s_{(5,1)}\blue{)}t_{(5,1)}+\blue{(}s_{6}+2s_{(5,1)}+2s_{(4,2)}\blue{)}t_{(4,2)}+\blue{(}3s_{(5,1)}+3s_{(4,2)}+10s_{(4,1,1)}\blue{)}t_{(4,1,1)}\red{)}u_{(4,1,1)}v_{(4,1,1)}\\
+\red(\blue{(}s_{6}+s_{(5,1)}+s_{(4,2)}\blue{)}t_{(4,1,1)}u_{(4,1,1)}+s_{(4,1,1)}t_{(4,1,1)}u_{(3,3)}\red)v_{(3,3)}\\
+\green(\red(\blue{(}s_{6}+3s_{(5,1)}\blue{)}t_{(5,1)}+\blue{(}s_{6}+3s_{(5,1)}+3s_{(4,2)}\blue{)}t_{(4,2)}+\blue(2s_{6}+7s_{(5,1)}+7s_{(4,2)}+12s_{(4,1,1)}\blue{)}t_{(4,1,1)}\red)u_{(4,1,1)}\\
+\blue{(}s_{(5,1)}+s_{(4,2)}+2s_{(4,1,1)}\blue{)}t_{(4,1,1)}u_{(3,3)}\\
+\red(s_{6}t_{6}+\blue(2s_{6}+6s_{(5,1)}\blue{)}t_{(5,1)}+\blue(2s_{6}+6s_{(5,1)}+6s_{(4,2)}\blue{)}t_{(4,2)}+\blue(3s_{6}+10s_{(5,1)}+10s_{(4,2)}+16s_{(4,1,1)}\blue{)}t_{(4,1,1)}\\
+\blue{(}s_{6}+2s_{(5,1)}+2s_{(4,2)}+4s_{(4,1,1)}+s_{(3,3)}\blue{)}t_{(3,3)}\\
+\blue(3s_{6}+12s_{(5,1)}+12s_{(4,2)}+16s_{(4,1,1)}+3s_{(3,3)}+24s_{(3,2,1)}\blue{)}t_{(3,2,1)}\red)u_{(3,2,1)}\green)v_{(3,2,1)}\\
+\green(\red{(}s_{(5,1)}t_{(5,1)}+\blue{(}s_{(5,1)}+s_{(4,2)}\blue{)}t_{(4,2)}
+\blue{(}s_{6}+3s_{(5,1)}+3s_{(4,2)}+6s_{(4,1,1)}\blue{)}t_{(4,1,1)}\red)u_{(4,1,1)}\\
+\red{(}s_{(4,1,1)}t_{(4,1,1)}+s_{6}t_{(3,3)}\red)u_{(3,3)}+\red(\blue{(}s_{6}+3s_{(5,1)}\blue{)}t_{(5,1)}+\blue{(}s_{6}+3s_{(5,1)}+3s_{(4,2)}\blue{)}t_{(4,2)}\\
+\blue{(}s_{6}+4s_{(5,1)}+4s_{(4,2)}+5s_{(4,1,1)}\blue{)}t_{(4,1,1)}+\blue{(}s_{(5,1)}+s_{(4,2)}+s_{(4,1,1)}\blue{)}t_{(3,3)}
\\
+\blue{(}s_{6}+4s_{(5,1)}+4s_{(4,2)}+6s_{(4,1,1)}+3s_{(3,3)}+8s_{(3,2,1)}\blue{)}t_{(3,2,1)}\red)u_{(3,2,1)}\\
+\red{(}s_{6}t_{6}+2s_{(5,1)}t_{(5,1)}+\blue{(}s_{6}+s_{(5,1)}+2s_{(4,2)}\blue{)}t_{(4,2)}\\
+\blue{(}2s_{(5,1)}+2s_{(4,2)}+2s_{(4,1,1)}\blue{)}t_{(4,1,1)}+\blue{(}s_{(5,1)}+3s_{(4,1,1)}+3s_{(3,3)}\blue{)}t_{(3,3)}\\
+\blue{(}s_{(5,1)}+s_{(4,2)}+s_{(4,1,1)}+4s_{(3,2,1)}\blue{)}t_{(3,2,1)}+3s_{(2,2,2)}t_{(2,2,2)}\red)u_{(2,2,2)}\green)v_{(2,2,2)}
. \end{gathered}$$ ]{} Applying the same `SchurRings` test we find the following necessary minimal generators: [$$\begin{gathered}
\M_{6}:=
\green({s}_{6} {t}_{(3,3)} {u}_{(3,3)}+\red(\blue(2 {s}_{(4,1,1)}+2 {s}_{(3,3)}\blue) {t}_{(3,3)}+{s}_{(3,2,1)} {t}_{(3,2,1)}+2 {s}_{(2,2,2)} {t}_{(2,2,2)}\red) {u}_{(2,2,2)}\green) {v}_{(2,2,2)}
.\end{gathered}$$ ]{} These modules could not come from any of the lower degree parts of the ideal, so they are among the minimal generators in degree 6, however there could be more minimal generators that we haven’t accounted for if there were syzygies among the lower degree generators.
The space of degree 7 polynomials $S^{7}({V_{1}^{*}\o V_{2}^{*}\o V_{3}^{*}\o V_{4}^{*}})$ decomposes as a sum of 288 isotypic components (up to $\mathfrak{S}_{4}$ symmetry) with dimension of multiplicity spaces as large as 301. Thus applying the LM-algorithm directly in degree 7 was more challenging and we were forced to exploit a parallelism, following a process similar to that in [@OedingSam]. We ran the LM-algorithm in a separate instance of Maple for each of the 288 isotypic components. This allowed us to distribute the computation. Running in parallel the degree 7 computation took approximately 12 hours on our two servers with respectively 40 and 24 processors. We found 201 modules occurring non-trivially the ideal. The `SchurRings` test yielded no new necessary minimal generators.
The parallelized LM-algorithm for degree 8 involved 619 Schur modules with multiplicity spaces up to 608-dimensional. Parts of this computation took approximately 20 days on our servers. We found 453 modules occurring in $I_{8}$ with nonzero multiplicity. The `ShurRings` test only produced one module occurring in $I_{8}$ with greater multiplicity than can occur in $I_{7}\o {V_{1}^{*}\o V_{2}^{*}\o V_{3}^{*}\o V_{4}^{*}}$, namely $$\M_{8}:=
2s_{(4,4)}t_{(4,4)}u_{(4,4)}v_{(4,2,2)}
.$$ That this module is not in the ideal generated by $\M_{3}+\M_{5}+\M_{6}$ can be seen for shape reasons alone. The Schur module $\M_{8}$ is indexed by a multi-partition containing three partitions of shape $(4,4)$, but no module the list of known minimal generators is indexed by quadruple of partitions with a subset of 3 of them that each fit in a box of dimensions $2\times 4$. Therefore $\M_{8}$ must occur with multiplicity 2 in the minimal generators in the ideal.
In degree 9 there are 1205 isotypic components, with multiplicity spaces as high as 2226-dimensional. After approximately one month of computational time on our servers we were able to evaluate 1158 of these isotypic components on the quadrifocal variety, 951 of which occurred non-trivially in the ideal. The computations that finished were those modules for which the multiplicity was low (under around 300).
The main obstruction to completing the degree 9 computation is a lack of available machines able to perform Maple computations. More specifically, if a given isotypic component has multiplicity $m$ in the polynomial ring, we must populate an $m\times m$ matrix with $m^{2}$ evaluations (from the Young symmetrizer algorithm), once to verify we have found a basis of the space, and a second time to evaluate this basis on points of the quadrifocal variety. Each of these evaluations could be done in a separate instance. So in the case of multiplicity 2226, we must perform approximately 10 million evaluations, which could potentially be done on a separate processor if were to take fuller advantage of the parallel nature of this problem. One of the largest multiplicity spaces in degree 9 we were able to handle was 456 dimensional, and the computation in multiplicity 2226 case is approximately 23 times larger than this. Since this scale of resources is not currently available to us, it seems unlikely that we will be able to complete the degree 9 computation.
Though we were only able to partially compute $I_{9}$, we were able to obtain two new modules in degree 9 that must be among the minimal generators of the quadrifocal ideal: $$\widetilde \M_{9}:=s_{(5, 4)} t_{(5, 4)} u_{(5, 4)} v_{(4, 3, 2)} +
s_{(5, 4)} t_{(5, 4)} u_{(5, 4)} v_{(5, 2, 2)}
.$$ These two modules occur with multiplicity 3 in $I_{9}$, but instances of these modules coming from $I_{8}\o{V_{1}^{*}\o V_{2}^{*}\o V_{3}^{*}\o V_{4}^{*}}$ occurred with multiplicity 2. Therefore, these modules must occur with multiplicity at least 1 among the minimal generators of $I$.
The dimension counts in Table \[toDeg9\] are a straightforward application of a hook length formula, which is also implemented in `SchurRings` for example.
We wondered if Table \[toDeg9\] might be a complete list of minimal generators for the quadrifocal ideal. We performed further experiments with modules in degrees 9, 10, and 11 having low multiplicity. We found that $S_{5, 5} S_{5, 5} S_{5, 5}S_{4, 3, 3}$ occurred with multiplicity one greater in $I_{10}$ than what could come from the part of $I_{9}$ that we were able to compute. This was the only new module we found in low multiplicity in degree 10. Further experiments suggest that there may be many more new modules of minimal generators with low multiplicity in degree 11. At present we are not confident enough to form a conjecture as to what happens in higher degree, nor in which degree the last set of minimal generators must occur.
In a previous draft we had incorrectly computed the number of equations in the ideal because of an unfortunate programming error.
Acknowledgements {#acknowledgements .unnumbered}
================
The author would like to thank Bernd Sturmfels for suggesting this problem and Claudiu Raicu for help with `SchurRings`. Fédéric Holweck also provided useful comments. The author is also grateful to the S. Korean National Institute for Mathematical Sciences (NIMS) and the Simons Institute for the Theory of Computing for their generous support while this work was carried out. The author also thanks an anonymous referee, whose suggestions improved the exposition of this manuscript.
Part of the computational work reported in this work was performed on the Auburn CASIC High Performance Computing Cluster. The author is grateful for the availability and access to this resource.
|
---
abstract: 'The life-cycle of a partial differential equation (PDE) solver is often characterized by three development phases: the development of a stable numerical discretization; development of a correct (verified) implementation; and the optimization of the implementation for different computer architectures. Often it is only after significant time and effort has been invested that the performance bottlenecks of a PDE solver are fully understood, and the precise details varies between different computer architectures. One way to mitigate this issue is to establish a reliable performance model that allows a numerical analyst to make reliable predictions of how well a numerical method would perform on a given computer architecture, before embarking upon potentially long and expensive implementation and optimization phases. The availability of a reliable performance model also saves developer effort as it both informs the developer on what kind of optimisations are beneficial, and when the maximum expected performance has been reached and optimisation work should stop. We show how discretization of a wave-equation can be theoretically studied to understand the performance limitations of the method on modern computer architectures. We focus on the roofline model, now broadly used in the high-performance computing community, which considers the achievable performance in terms of the peak memory bandwidth and peak floating point performance of a computer with respect to algorithmic choices. A first principles analysis of operational intensity for key time-stepping finite-difference algorithms is presented. With this information available at the time of algorithm design, the expected performance on target computer systems can be used as a driver for algorithm design.'
address:
- 'Seismic Laboratory for Imaging and Modeling (SLIM),The University of BritishColumbia'
- 'Earth Science and Engineering department,Imperial college, London'
author:
- Mathias Louboutin
- Michael Lange
- 'Felix J. Herrmann'
- Navjot Kukreja
- Gerard Gorman
bibliography:
- 'bib\_geophys1.bib'
title: 'Performance prediction of finite-difference solvers for different computer architectures'
---
Finite-differences ,HPC ,Modelling ,Multi-physics ,Performance ,Wave-equation
Introduction
============
The increasing complexity of modern computer architectures means that developers are having to work much harder at implementing and optimising scientific modelling codes for the software performance to keep pace with the increase in performance of the hardware. This trend is driving a further specialisation in skills such that the geophysicist, numerical analyst and software developer are increasingly unlikely to be the same person. One problem this creates is that the numerical analyst makes algorithmic choices at the mathematical level that define the scope of possible software implementations and optimizations available to the software developer. Additionally, even for an expert software developer it can be difficult to know what are the right kind of optimisations that should be considered, or even when an implementation is “good enough” and optimisation work should stop. It is common that performance results are presented relative to a previously existing implementation, but such a relative measure of performance is wholly inadequate as the reference implementation might well be truly terrible. One way to mitigate this issue is to establish a reliable performance model that allows a numerical analyst to make reliable predictions of how well a numerical method would perform on a given computer architecture, before embarking upon potentially long and expensive implementation and optimization phases. The availability of a reliable performance model also saves developer effort as it both informs the developer on what kind of optimisations are beneficial, and when the maximum expected performance has been reached and optimisation work should stop.
Performance models such as the roofline model by [@williams2009roofline] help establish statistics for best case performance — to evaluate the performance of a code in terms of hardware utilization (e.g. percentage of peak floating point performance) instead of a relative speed-up. Performance models that establish algorithmic optimality and provide a measure of hardware utilization are increasingly used to determine effective algorithmic changes that reliably increase performance across a wide variety of algorithms [@asanovic2006landscape]. However, for many scientific codes used in practice, wholesale algorithmic changes, such as changing the spatial discretization or the governing equations themselves, are often highly invasive and require a costly software re-write. Establishing a detailed and predictive performance model for the various algorithmic choices is therefore imperative when designing the next-generation of industry scale codes.
We establish a theoretical performance model for explicit wave-equation solvers as used in full waveform inversion (FWI) and reverse time migration (RTM). We focus on a set of widely used equations and establish lower bounds on the degree of the spatial discretization required to achieve optimal hardware utilization on a set of well known modern computer architectures. Our theoretical prediction of performance limitations may then be used to inform algorithmic choice of future implementations and provides an absolute measure of realizable performance against which implementations may be compared to demonstrate their computational efficiency.
For the purpose of this paper we will only consider explicit time stepping algorithms based on a second order time discretization. Extension to higher order time stepping scheme will be briefly discussed at the end. The reason we only consider explicit time stepping is that it does not involve any matrix inversion, but only scalar product and additions making the theoretical computation of the performance bounds possible. The performance of other classical algorithm such as matrix vector products or FFT as described by [@Patterson] has been included for illustrative purposes.
Introduction to stencil computation
===================================
A stencil algorithm is designed to update or compute the value of a field in one spatial location according to the neighbouring ones. In the context of wave-equation solver, the stencil is defined by the support (grid locations) and the coefficients of the finite-difference scheme. We illustrate the stencil for the Laplacian, defining the stencil of the acoustic wave-equation (Eq. \[eqn:Acou\]), and for the rotated Laplacian used in the anisotropic wave-equation (Eq. \[eqn:TTI\], \[eqn:DiffOp\]) on Fig. \[fig:stencil\] - \[fig:stencilr\]. The points coloured in blue are the value loaded while the point coloured in red correspond to a written value.
![Stencil for the acoustic and anisotropic wave-equation for different orders of discretization. A new value for the centre point (red) is obtained by weighted sum of the values in all the neighbour points (blue). a) 2nd order laplacian, b) second order rotated Laplacian, c) 16th order Laplacian, d) 16th order rotated Laplacian[]{data-label="fig:stencil"}](./Figures/acoustic_2_ls.pdf "fig:"){width="\textwidth"} a) \[fig:acou2\]
![Stencil for the acoustic and anisotropic wave-equation for different orders of discretization. A new value for the centre point (red) is obtained by weighted sum of the values in all the neighbour points (blue). a) 2nd order laplacian, b) second order rotated Laplacian, c) 16th order Laplacian, d) 16th order rotated Laplacian[]{data-label="fig:stencil"}](./Figures/tti_2_ls.pdf "fig:"){width="\textwidth"} b) \[fig:ani2\]
![Stencil for the acoustic and anisotropic wave-equation for different orders of discretization. A new value for the centre point (red) is obtained by weighted sum of the values in all the neighbour points (blue). a) 2nd order laplacian, b) second order rotated Laplacian, c) 16th order Laplacian, d) 16th order rotated Laplacian[]{data-label="fig:stencil"}](./Figures/acoustic_16_ls.pdf "fig:"){width="\textwidth"} c) \[fig:acou16\]
![Stencil for the acoustic and anisotropic wave-equation for different orders of discretization. A new value for the centre point (red) is obtained by weighted sum of the values in all the neighbour points (blue). a) 2nd order laplacian, b) second order rotated Laplacian, c) 16th order Laplacian, d) 16th order rotated Laplacian[]{data-label="fig:stencil"}](./Figures/tti_16_ls.pdf "fig:"){width="\textwidth"} d) \[fig:ani16\]
![Stencil for the 16th order acoustic and anisotropic wave-equation with distance to centre highlighting a) Laplacian, b) rotated Laplacian[]{data-label="fig:stencilr"}](./Figures/acoustic_16_radius.pdf "fig:"){width="\textwidth"} a) \[fig:acour\]
![Stencil for the 16th order acoustic and anisotropic wave-equation with distance to centre highlighting a) Laplacian, b) rotated Laplacian[]{data-label="fig:stencilr"}](./Figures/tti_16_radius.pdf "fig:"){width="\textwidth"} b) \[fig:anir\]
The implementation of a time stepping algorithm for a wavefield $u$, solution of the acoustic wave-equation (Eq. \[eqn:Acou\]) is straightforward from the representation of the stencil. We do not include the absorbing boundary conditions (ABC) as depending on the choice of implementation it will either be part of the stencil or be decoupled and treated separately.
\[alg:ts\]
In Algorithm \[alg:ts\], $(X,Y,Z)$ is the set of all grid positions in the computational domain, $(x,y,z)$ are the local indices ,$(x_i, y_i, z_i)$ are the indices of the stencil positions for the centre position $(x,y,z)$ and $n_t$ is the number of time steps and $q$ is the source term decoupled from the stencil. In the following we will concentrate on the stencil itself, as the loops in space and time do not impact the theoretical performance model we introduce. The roofline model is solely based on the amount of input/output (blue/red in the stencils) and arithmetic operations (number of sums and multiplication) required to update one grid point, and we will prove that the optimal reference performance is independent of the size of the domain (number of grid points) and of the number of time steps.
**Notes on parallelization:**
Using a parallel framework to improve an existing code is one of the most used tool in the current stencil computation community. It is however crucial to understand that this is not an algorithmic improvement from the operational intensity. We will prove that the algorithmic efficiency of a stencil code is independent of the size of the model, and will therefore not be impacted by a domain-decomposition like parallelization via OpenMP or MPI. The results shown in the following are purely dedicated to help the design of a code from an algorithmic point of view, while parallelization will only impact the performance of the implemented code by improving the hardware usage.
Roofline Performance Analysis {#performance-analysis}
=============================
The roofline model is a performance analysis framework designed to evaluate the floating point performance of an algorithm by relating it to memory bandwidth usage [@williams2009roofline]. It has proved to be very popular because it provides a readily comprehensible performance metric to interpret runtime performance of a particular implementation according to the achievable optimal hardware utilization for a given architecture [@Williams2008].
This model has been applied to real-life codes in the past to analyze and report performance including oceanic climate models [@epicoco2014roofline], combustion modeling [@chan2013software] and even seismic imaging [@andreolli2014genetic]. It has also been used to evaluate the effectiveness of implementation-time optimizations like autotuning [@datta2009auto], or cache-blocking on specific hardware platforms like vector processors [@sato2009performance] and GPUs [@kim2011performance]. Tools are available to plot the machine-specific parameters of the roofline model automatically [@RooflineModelToolkit]. When more information about the target hardware is available, it is possible to refine the roofline model into the cache-aware roofline model which gives more accurate predictions of performance [@CacheAwareRoofline]. The analysis presented here can be extended to the cache-aware roofline model but in order to keep it general, we restrict it to the general roofline model.
The roofline model has also been used to compare different types of basic numerical operations to predict their performance and feasibility on future systems [@barba2013will], quite similar to this paper. However, in this paper, instead of comparing stencil computation to other numerical methods, we carry out a similar comparison between numerical implementations using different stencil sizes. This provides an upper-bound of performance on any hardware platform at a purely conceptual stage, long before the implementation of the algorithm.
Other theoretical models to predict upper-bound performance of generic code on hypothetical hardware have been built [@lai2013performance; @wahib2014scalable; @ECMModel; @ActiveLearning] but being too broad in scope, can not be used to drive algorithmic choice like choosing the right discretization order. Some of these models have also been applied to stencil codes [@ECMModelStencil; @datta2009optimization], however the analysis was of a specific implementation and could not be applied in general. There are many tools to perform performance prediction at the code-level [@KernCraft; @PerfBoundsCodeNarayanan; @exasat; @rahman2011understanding]. However, any tool that predicts performance based on a code is analyzing the implementation and not the algorithm in general. Although performance modeling is a deep and mature field, most work is restricted to modeling the performance of specific implementations in code. @KahanScalar makes a comparison quite similar to the one we do here where two algorithmic choices for the same problem are being compared with a performance model.
In this section we demonstrate how one creates a roofline model for a given computer architecture, and derives the operational intensity for a given numerical algorithm. This establishes the theoretical upper-bound for the performance of a specific algorithm on that architecture. A general roofline performance analysis consists of three steps:
- The memory bandwidth, bytes per second, and the peak number of floating point operations per second (FLOPS) of the computer architecture are established either from the manufacturers specification or through measurement using standard benchmarks.
- The operational intensity (OI) of the algorithm is established by calculating the ratio of the number of floating point operations performed to memory traffic, FLOPs per byte. This number characterizes the algorithmic choices that affect performance on a computer system. In combination with the measured memory bandwidth and peak performance of a computer architecture, this provides a reliable estimate of the maximum achievable performance.
- The solver is benchmarked in order to establish the achieved performance. A roofline plot can be created to illustrate how the achieved performance compares to the maximum performance predicted by the roofline for the algorithms OI. This establishes a measure of optimality of the implementation, or alternatively the maximum possible gain from further optimization of the software.
Establishing the Roofline {#roofline-model}
-------------------------
The roofline model characterises a computer architecture using two parameters: the maximum memory bandwidth, $B_{peak}$, in units of $bytes/s$; and the peak FLOPS achievable by the hardware, $F_{peak}$. The maximally achievable performance $F_{ac}$ is modelled as:
$$F_{ac} = \min\left(\mathcal{I} B_{peak}, F_{peak}\right),
\label{perf-limits}$$
where $\mathcal{I}$ is the OI.
As illustrated in Fig. \[fig:ExampleRoof\] this limitation defines two distinct regions:
- [**Memory-bound**]{}: The region left of the ridge point constitutes algorithms that are limited by the amount of data coming into the CPU from memory. Memory-bound codes typically prioritise caching optimizations, such as data reordering and cache blocking.
- [**Compute-bound**]{}: The region right of the ridge point contains algorithms that are limited by the maximum performance of the arithmetic units in the CPU and thus defines the maximum achievable performance of the given architecture. Compute-bound codes typically prioritise vectorization to increase throughput.
![Roofline diagram showing the operational intensity of three well-known algorithms as reported by @williams2009roofline: sparse matrix-vector multiplication (SpMV), stencil computation and 3D Fast Fourier Transform (3DFFT). The hardware limits are taken from @Andreolli2015 and the compute-limited area is highlighted through shading.[]{data-label="fig:ExampleRoof"}](Figures/RoofSimpleEx.pdf){width="1.000\hsize"}
It is worth noting that changing from single to double-precision arithmetic halves the OI because the volume of memory that must be transferred between the main memory and the CPU is doubled. The peak performance will be impacted as well, since the volume of data and the number of concurrently used floating point units (FPU) changes. As commonly employed by industry, we assume single precision arithmetic for the examples presented here, but it is straightforward to [extend]{} to double precision.
@Andreolli2015 illustrates an example of deriving the theoretical performance for a system that consists of two Intel Xeon E5-2697 v2 (2S-E5) with 12 cores per CPU each running at 2.7 Ghz without turbo mode. Since these processors support 256-bit SIMD instructions they can process eight single-precision operations per clock-cycle (SP FP). Further, taking into account the use of Fused Multiply-Add (FMA) operations (two per cycle), this yields [ $$\begin{aligned}
F_{peak} &= 8 (SP FP) \times 2 (FMA) \times 12 (cores) \times 2 (CPUs)
\times 2.7 \text{Ghz} \\
&= 1036.8\ \text{GFLOPS.}
\end{aligned}$$]{} Clearly, this assumes full utilization of two parallel pipelines for Add and Multiply operations.
A similar estimate for the peak memory bandwidth $F_{peak}$ can be made from the memory frequency ($1866 \ GHz$), the number of channels ($4$) and the number of bytes per channel ($8$) and the number of CPUs ($2$) to give $F_{peak} =
1866 \times 4 \times 8 \times 2 = 119\ GByte / s$.
It is important to note here that there is an instruction execution overhead that the above calculations did not take into account and therefore these theoretical peak numbers are not achievable ($\simeq 80\%$ is achievable in practice [@Andreolli2015]). For this reason, two benchmark algorithms, STREAM TRIAD for memory bandwidth [@McCalpin1995; @McCalpin2007] and LINPACK for floating point performance [@Dongarra:1987:LBE:647970.742568], are often used to measure the practical limits of a particular hardware platform. These algorithms are known to achieve a very high percentage of the peak values and are thus indicative of practical hardware limitations.
Performance Model
-----------------
The key measure to using the roofline analysis as a guiding tool for algorithmic design decisions and implementation optimization is the operational intensity, $\mathcal{I}$, as it relates the number of FLOPs to the number of bytes moved to and from RAM. $\mathcal{I}$ clearly does not capture many important details about the implementation such as numerical accuracy or time to solution. Therefore, it is imperative to look at $\mathcal{I}$ in combination with these measures when making algorithmic choices.
Here we analyze the algorithmic bounds of a set of finite-difference discretizations of the wave-equation using different stencils and spatial orders. We therefore define algorithmic operational intensity $\mathcal{I}_{alg}$ in terms of the total number of FLOPs required to compute a solution, and we assume that our hypothetical system has a cache with infinite size and no latency inducing zero redundancy in memory traffic [@Williams2008]. This acts as a theoretical upper bound for the performance of any conceivable implementation.
We furthermore limit our theoretical investigation to analysing a single time step as an indicator of overall achievable performance. This assumption allows us to generalize the total number of bytes in terms of the number of spatially dependant variables (e.g. wavefields, physical properties) used in the discretized equation as $\mathcal{B}_{global} = 4 N (l + 2 s)$, where $l$ is the number of variables whose value is being loaded, $s$ is the number of variables whose value is being stored, $N$ is the number of grid points and $4$ is the number of bytes per single-precision floating point value. The term $2 s$ arises from the fact that most computer architectures will load a cache line before it gets overwritten completely. However, some computer architectures, such as the Intel Xeon Phi, have support for stream stores, so that values can be written directly to memory without first loading the associated cache line, in which case the expression for the total data movement becomes $\mathcal{B}_{global}
= 4 N (l + s)$. It is important to note here that limiting the analysis to a single time step limits the scope of the infinite caching assumption above.
Since we have assumed a constant grid size $N$ across all spatially dependant variables, we can now parametrize the number of FLOPs to be computed per time step as $\mathcal{F}_{total}(k) = N \mathcal{F}_{kernel}(k)$, where $\mathcal{F}_{kernel}(k)$ is a function that defines the number of flops performed to update one grid point in terms of the stencil size $k$ used to discretize spatial derivatives. Additional terms can be added corresponding to source terms and boundary conditions but they are a small proportion of the time step in general and are neglected here for simplicity. This gives us the following expression for OI as a function of $k$, $\mathcal{I}_{alg}(k)$:
$$\mathcal{I}_{alg}(k) = \mathcal{F}_{total}(k) / \mathcal{B}_{global} = \frac{\mathcal{F}_{kernel}(k)}{4(l+s)}.
\label{eqn_oi}$$
Operational intensity for finite-differences {#the-roofline-model-for-pde-solvers}
============================================
We derive a general model for the operational intensity of wave-equation PDEs solvers with finite-difference discretizations using explicit time stepping and apply it to three different wave-equation formulations commonly used in the oil and gas exploration community: an acoustic anisotropic wave-equation; vertical transverse isotropic (VTI); and tilted transversely isotropic (TTI) [@liu2009stable]. The theoretical operational intensity for the 3D discretized equations will be calculated as a function of the finite-difference stencil size $k$, which allows us to make predictions about the minimum discretization order required for each algorithm to reach the compute-bound regime for a target computer architecture. For completeness we describe the equations in Appendix \[PDE\].
Stencil operators
-----------------
As a baseline for the finite-difference discretization, we consider the use of a 1D symmetric stencil of size $k$, which uses $k$ values of the discretized variable to compute any spatial derivatives enforcing a fixed support for all derivatives. Other choices of discretization are possible, such as choosing the stencil for the first derivative and applying it iteratively to obtain high order derivatives. Our analysis will still be valid but require a rewrite of the following atomic operation count. The number of FLOPs used for the three types of derivatives involved in our equation are calculated as:
- first order derivative with respect to $x_i$ ($\frac{du}{dx_i}$): $(k + 1)\textrm{ mult } + (k - 1)\textrm{ add }= 2k\ FLOPs$
- second order derivative with respect to $x_i$ ($\frac{d^2u}{dx^2_i}$): $(k + 1)\textrm{ mult } + (k - 1)\textrm{ add } = 2k\ FLOPs$
- second order cross derivative with respect to $x_i, x_j$ ($\frac{d^2u}{dx_i dx_j}$): $(k^2 - 2k)\textrm{ mult } + (k^2 - 2k - 1)\textrm{ add } = 2k^2 -4k -1\ FLOPs$
where in 3D, $x_i \text{ for } i = 1,2,3$ correspond to the three dimensions $x,y,z$ and $u$ is the discretized field.
Equation $\frac{du}{dx_i}$ $\frac{d^2u}{dx^2_i}$ $\frac{d^2u}{dx_i dx_j}$ mult add duplicates
----------------- ------------------- ----------------------- -------------------------- ------ ----- ------------
Acoustic: 0 $3\times2k$ 0 3 5 $- 4$
VTI: $2\times($ 0 $3\times2k$ 0 5 5 $- 2)$
TTI: $2\times($ 0 $3\times2k$ $3\times( 2k^2 - 4k -1)$ 44 17 $- 8)$
: Derivation of $FLOPs$ per stencil invocation for each equation.
\[tab:flops-3d\]
Computing the total wavefield memory volume $B_{global}$ for each equation we have $4\times4N\ bytes$ for Acoustic (load velocity, two previous time steps and write the new time step), $9\times4N\ bytes$ for VTI (load velocity, two anisotropy parameters, two previous time steps for two wavefields and write the new time step for the two wavefields) and $15\times4N\ bytes$ for TTI (VTI plus 6 precomputed cos/sin of the tilt and dip angles). Eq. \[eqn\_oi\] allows us to predict the increase of the operational intensity in terms of $k$ by replacing $B_{global}$ by its value. The OI $\mathcal{I}_{alg}(k)$ for the three wave-equations is given by:
- Acoustic anisotropic: $\mathcal{I}_{alg}(k) = \frac{3k}{8} + \frac{1}{4}$,
- VTI: $\mathcal{I}_{alg}(k) = \frac{k}{3} + \frac{4}{9}$,
- TTI: $\mathcal{I}_{alg}(k) = \frac{k^2}{5} - \frac{k}{5} + \frac{5}{3}$,
and plotted as a function of $k$ on Fig. \[fig:OI\_stencil\_size\]. Using the derived formula for the algorithmic operational intensity in terms of stencil size, we can now analyze the optimal performance for each equation with respect to a specific computer architecture. We are using the theoretical and measured hardware limitations reported by @Andreolli2015 to demonstrate how the main algorithmic limitation shifts from being bandwidth-bound at low $k$ to compute-bound at high $k$ on a dual-socket Intel Xeon in Fig. \[fig:RoofEqns\_xeonA\] - \[fig:RoofEqns\_xeonT\] and an Intel Xeon Phi in Fig. \[fig:RoofEqns\_phiA\] - \[fig:RoofEqns\_phiT\].
It is of particular interest to note from Fig. \[fig:RoofEqns\_xeonA\] that a $24^{th}$ order stencil with $k=25$ provides just enough arithmetic load for the 3D acoustic equation solver to become compute-bound, while $k=25$ falls just short of the compute-bound region for the VTI algorithm. On the other hand a $6^{th}$ order stencil with $k=7$ is enough for the TTI algorithm to become compute-bound due to having a quadratic slope with respect to $k$ (Fig. \[fig:OI\_stencil\_size\]) instead of a linear slope.
At this point, we can define $\mathcal{I}_{min}$, which is the minimum OI required for an algorithm to become compute-bound on a particular architecture, as the x-axis coordinate of the ridge point in Fig. \[fig:RoofEqns\_xeonA\] - \[fig:RoofEqns\_xeonT\] and \[fig:RoofEqns\_phiA\] - \[fig:RoofEqns\_phiT\]. Note that the ridge point x-axis position changes between the two different architectures. This difference in compute-bound limit shows that a different spatial order discretization should be used on the two architecture to optimize hardware usage. As reported by @Andreolli2015 the $\mathcal{I}_{min}$ as derived from achievable peak rates is $9.3\ FLOPs/byte$ for the Intel Xeon and $10.89\ FLOPs/byte$ for the Intel Xeon Phi. This entails that while the acoustic anisotropic wave-equation and VTI are memory bound for discretizations up to $24^{th}$ order, the TTI equation reaches the compute bound region with even a $6^{th}$ order discretization.
![Increase in algorithmic OI with increasing stencil sizes on a dual-socket Intel Xeon E5-2697 v2 [@Andreolli2015] for a 3D acoustic kernel. The $24^{th}$ order stencil is coincident with the ridge point — the transition point from memory-bound to compute-bound computation.[]{data-label="fig:RoofEqns_xeonA"}](Figures/RoofAc3D_xeon.pdf){width="0.9500\hsize"}
![Increase in algorithmic OI with increasing stencil sizes on a dual-socket Intel Xeon E5-2697 v2 [@Andreolli2015] for a 3D VTI kernel. Similarly to the acoustic model, the $24^{th}$ order stencil is coincident with the ridge point.[]{data-label="fig:RoofEqns_xeonV"}](Figures/RoofVTI3D_xeon.pdf){width="0.9500\hsize"}
![Increase in algorithmic OI with increasing stencil sizes on a dual-socket Intel Xeon E5-2697 v2 [@Andreolli2015] for a 3D TTI kernel. The $6^{th}$ order stencil is already compute-bound.[]{data-label="fig:RoofEqns_xeonT"}](Figures/RoofTTI3D_xeon.pdf){width="0.9500\hsize"}
![Increase in algorithmic OI with increasing stencil sizes on a Intel Xeon Phi 7120A co-processor [@Andreolli2015] for a 3D acoustic kernel. Unlike the Xeon E5-2697, the $30^{th}$ order stencil is the smallest one to be compute-bound (vs $24^{th}$ order).[]{data-label="fig:RoofEqns_phiA"}](Figures/RoofAc3D_phi.pdf){width="0.9500\hsize"}
![Increase in algorithmic OI with increasing stencil sizes on a Intel Xeon Phi 7120A co-processor [@Andreolli2015] for a 3D VTI kernel. $32^{nd}$ is the minimum compute-bound stencil. It is not equivalent to the acoustic on this architecture.[]{data-label="fig:RoofEqns_phiV"}](Figures/RoofVTI3D_phi.pdf){width="0.9500\hsize"}
![Increase in algorithmic OI with increasing stencil sizes on a Intel Xeon Phi 7120A co-processor [@Andreolli2015] for a 3D TTI kernel. The $6^{th}$ order stencil is already compute-bound similarly to the Xeon E5-2697.[]{data-label="fig:RoofEqns_phiT"}](Figures/RoofTTI3D_phi.pdf){width="0.9500\hsize"}
From the analytical expression derived we can now generalize the derivation of minimum OI values by plotting the simplified expressions for $\mathcal{I}_{alg}(k)$ against known hardware OI limitations, as shown in Fig. \[fig:OI\_stencil\_size\]. We obtain a theoretical prediction about the minimum spatial order required for each algorithm to provide enough arithmetic load to allow implementations to become compute-bound. Most importantly, Fig. \[fig:OI\_stencil\_size\] shows that the TTI wave-equation has a significantly steeper slope of $\mathcal{I}(k)$, which indicates that it will saturate a given hardware for a much smaller spatial discretization than the acoustic wave or the VTI algorithm.
Moreover, assuming a spatial discretization order of $k-1$, we can predict that on the Intel Xeon CPU we require a minimum order of $24$ for the acoustic wave solver, $26$ for VTI and $6$ for TTI. On the Nvidia GPU, with a slightly lower hardware $\mathcal{I}$, we require a minimum order of $22$ for the acoustic wave solver, $24$ for VTI and $6$ for TTI, while even larger stencils are required for the Intel Xeon Phi accelerator: a minimum order of $28$ for the acoustic wave solver, $30$ for VTI and $6$ for TTI. This derivation demonstrates that overall very large stencils are required for the acoustic anisotropic solver and VTI to fully utilize modern HPC hardware, and that even TTI requires at least order $6$ to be able to computationally saturate HPC architectures with a very high arithmetic throughput, like the Intel Xeon Phi.
![Increase in OI with stencil size $k$ and machine-specific minimum OI values for all three hardware architectures considered in this paper.[]{data-label="fig:OI_stencil_size"}](Figures/OI_stencil_size.pdf){width="0.7500\hsize"}
Example: MADAGASCAR modelling kernel {#rsf-modelling-kernel}
====================================
We demonstrate our proposed performance model and its flexibility by applying it on a broadly used and benchmarked modelling kernel contained in Madagascar [@Madagascar]. We are illustrating the ease to extend our method to a different wave-equation and by extension to any PDE solver. The code implements the 3D anisotropic elastic wave-equation and is described in [@Weiss2013]. We are performing our analysis based on the space order, hardware and runtime described in [@Weiss2013]. The governing equation considered is:
$$\begin{aligned}
&\rho \frac{d^2 u_i}{dt^2} = \frac{d \sigma_{ij}}{dx_j} + F_i, \\
&\sigma_{ij} = c_{ijkl}\epsilon_{kl}, \\
&\epsilon_{kl} = \frac{1}{2}[\frac{u_l}{dx_k} + \frac{u_k}{dx_l}],\\
&u_i(.,0) = 0, \\
&\frac{d u_i(x,t)}{dt}|_{t=0} = 0.\\
\end{aligned}
\label{PDErsf}$$
where $\rho$ is the density, $u_i$ is the $i^{th}$ component of the three dimensional wavefield displacement ($i=1,2,3$ for $x,y,z$), $F$ is the source term,$\epsilon$ is the strain tensor, $\sigma$ is the stress tensor and $c$ is the stiffness tensor. The equation is discretized with an $8^{th}$ order star stencil for the first order derivatives and a second order scheme in time and solves for all three components of $u$. Eq. \[PDErsf\] uses Einstein notations meaning repeated indices represent summation:
$$\begin{aligned}
\frac{d \sigma_{ij}}{dx_j} &= \sum_{j=1}^3 \frac{\sigma_{ij}}{dx_j}, \\
c_{ijkl}\epsilon_{kl} &= \sum_{k=1}^3 \left( \sum_{l=1}^3 c_{ijkl}\epsilon_{kl} \right).
\end{aligned}
\label{Einstein}$$
From this equation and knowing the finite-difference scheme used we can already compute the minimum required bandwidth and operational intensity. We need to solve this equation for all three components of the wave $u$ at once as we have coupled equations in $\epsilon$ and $u$. For a global estimate of the overall memory traffic, we need to account for loading and storing $2 \times 3N$ values of the displacement vector and loading $N$ values of $\rho$. In case the stiffness tensor is constant in space the contribution of $c_{ijkl}$ is $64$ independently of $N$, which yields an overall data volume of $\mathcal{B}_{global} = 4N(6 + 1) + 64 \simeq 28N\ Bytes$. In the realistic physical configuration of a spatially varying stiffness tensor, we would estimate loading $64N$ values of $c_{ijkl}$, leaving us with a data volume of $B_{global} =4N (6 + 1 + 64) = 284N\ Bytes$. Finally we consider symmetries in the stiffness tensor are taken into account reducing the number of stiffness values to load to $21N$ and leading to a data volume of $B_{global} = (6 + 1 + 21)\times4N = 112N\ Bytes$.
The number of valuable FLOPs performed to update one grid point can be estimated by:
- 9 first derivatives ($\partial_k u_l \text{, for all } k,l = 1,2,3$) : $9 \times (8$ mult [$+\ 7$ add$) = 135\ FLOPs$]{}
- 9 sums for $\epsilon_{kl}$ ($9 \times 9$ adds) and $9 \times 8$ mult for $\sigma_{ij}$ = $153\ FLOPs$
- 9 first derivatives $\partial_j\sigma_{ij}$ and 9 sums = $144\ FLOPs$
- 3 times 3 sums to update $u_i$ = $9\ FLOPs$.
The summation of all four contributions gives a total of 441 operations and by dividing by the memory traffic we obtain the operational intensity $\mathcal{I}_{stiff}$ for variable stiffness and $\mathcal{I}_{const}$ for constant stiffness:
$$\begin{aligned}
\mathcal{I}_{stiff} &= \frac{441N}{112N} = 3.93, \\
\mathcal{I}_{const} &= \frac{441N}{28N} = 15.75.
\end{aligned}
\label{OIrsf}$$
Using the OI values derived above we can now quantify the results presented by @Weiss2013 by interpreting their runtime results with respect to our performance measure. The achieved GFLOPS have been obtained on the basis of 1000 time steps with $8^{th}$ order spatial finite-differences and $2^{nd}$ order temporal finite-differences. We interpret Fig. 11a) of @Weiss2013 to give a run time of approximately $53$ seconds and a domain size of $N=225^3$. We obtain with this parameter the following achieved performances:
$$\begin{aligned}
F &= \frac{N^3 F_{kernel} N_{t}}{W}, \\
&= \frac{225^3\times441\times1000}{53}, \\
&= 94.8\text{GFLOPS},
\end{aligned}$$
where $N_t$ is the number of time steps, and $W$ is the run time.
Fig. \[RSF\_roof\_GPU\] shows the calculated performance in relation to our predicted algorithmic bounds $\mathcal{I}_{stiff}$ and $\mathcal{I}_{const}$. The use of a constant stiffness tensor puts the OI of the considered equation in the compute-bound region for the benchmarked GPU architecture (NVIDIA GTX480). Assuming a spatially varying stiffness tensor, we can calculate an achieved hardware utilization of $40.5\%$ based on the reported results, assuming an achievable peak memory bandwidth of $150.7\ GByte/s$, as reported by @Konstantinidis2015 and a maximum achievable performance of $150.7\
GByte/s \times 1.5528\ FLOPs/Byte = 234\ GFLOPS$. Assuming $80\%$ [@Andreolli2015] of peak performance is achievable, the roofline model suggests that there is still potential to double the performance of the code through software optimization. It is not possible to draw such a conclusion from traditional performance measures such as timings or scaling plots. This highlights the importance of a reliable performance model that can provide an absolute measure of performance in terms of the algorithm and the computer architecture.
![Roofline model for the 3D elastic anisotropic kernel from [@Weiss2013] on a 480-core NVIDIA GTX480 GPU (with hardware specification from @Konstantinidis2015).[]{data-label="RSF_roof_GPU"}](Figures/Roofrsf_gpu.pdf){width="1.000\hsize"}
Cost-benefit analysis {#cost}
=====================
So far we discussed the design of finite-difference algorithms purely from a performance point of view without regard to the numerical accuracy and cost-to-solution. Now we discuss the impact of the discretization order on the achieved accuracy of the solution and how that, in turn, affects the wall clock time required for computation. To do so, we look at the numerical requirements of a time-stepping algorithm for the wave-equation. More specifically we concentrate on two properties, namely dispersion and stability, in the acoustic case. This analysis is extendable to more advanced wave-equations such as VTI and TTI with additional numerical analysis. The dispersion criteria and stability condition for the acoustic wave-equation is given by [@CFL; @dispersion]:
$$\label{eq:stability}
\begin{aligned}
&{v_{max}dt\over h}\le \sqrt{a_1\over a_2} \ \text{CFL condtion, stability} \\
&h\le {v_{min}\over pf_{max}} \ \text{dispersion criterion},
\end{aligned}$$
where:
- is the sum of the absolute values of the weights of the finite-difference scheme for the second time derivative of the wavefield; ${\partial^2u \over \partial t^2}$
- is the sum of the absolute values of the weights of the finite-difference approximation of $\nabla^2u$;
- is the maximum velocity;
- is the maximum frequency of the source term that defines the minimum wavelength for a given minimum velocity $\lambda_{min} = \frac{v_{min}}{f_{max}}$;
- is the number of grid points per wavelength. The number of grid points per wavelength impacts the amount of dispersion (different wavelengths propagating at different velocities) generated by the finite-difference scheme. The lower the number, the higher the dispersion will be for a fixed discretization order.
These two conditions define the computational setup for a given source and physical model size. Knowing that $a_2$ increases with the spatial discretization order, Eq. \[eq:stability\] shows that higher discretization orders require a smaller time-step hence increasing the total number of time steps for a fixed final time and grid size. However, higher order discretizations also allow to use less grid points per wavelength (smaller $p$). A smaller number of grid points per wavelengths leads to a smaller overall computational domain as a fixed physical distance is represented by a coarser mesh and as the grid spacing has been increased, the critical time-step (maximum stable value) is also increased. Overall, high order discretizations have better computational parameters for a predetermined physical problem. From these two considerations, we can derive an absolute cost-to-solution estimation for a given model as a function of the discretization order for a fixed maximum frequency and physical model size. The following results are not experimental runtimes but estimations of the minimum achievable runtime assuming a perfect performance implementation. We use the following setup:
- We fix the physical model size as 500 grid point in all three directions for a second order discretization (minimum grid size).
- The number of grid points per wavelength is $p=6$ for a second order spatial discretization and $p=2$ for a 24th order discretization and varies linearly for intermediate orders.
- The number of time steps is 1000 for the second order spatial discretization and computed according to the grid size/time step for other spatial orders.
The hypothetical numerical setup (with $a_1=4$, second order time discretization) is summarized in Table \[tab:stab\]. We combine the estimation of a full experimental run with the estimated optimal performance and obtain an estimation of the optimal time-to-solution for a fixed physical problem. The estimated runtime is the ratio of the total number of GFLOPs (multiply $\mathcal{F}_{kernel}$ by the number of grid points and time steps) to the maximum achievable performance for this OI. Table \[tab:cost\] shows the estimated runtime assuming peak performance on two systems: a dual-socket Intel Xeon E5-2697 v2 and an Intel Xeon Phi 7120A co-processor.
Order $a_2$ $p$ $h$ $dt$ $N$ $n_t$
------------ ------- ----- ----- -------- ---------- -------
2nd order 12 6 1 0.5774 1.25e+08 1000
6th order 18.13 5 1.2 0.5637 7.24e+07 1024
12th order 21.22 4 1.5 0.6513 3.70e+07 887
18th order 22.68 3 2 0.8399 1.56e+07 688
24th order 23.57 2 3 1.2359 4.63e+06 468
: Cost-to-solution computational setup summary.[]{data-label="tab:stab"}
[width=1]{}
Order $\mathcal{I}_{alg}(k)$ GFLOPs GFLOPS Xeon GFLOPS Phi Runtime Xeon Runtime Phi
------- ------------------------ ----------- ------------- ------------ -------------- -------------
2nd 1.375 2.75e+03 137.5 275 20s 10s
6th 2.875 3.414e+03 287.5 575 12s 6s
12th 5.125 2.691e+03 512.5 1025 6s 3s
18th 7.375 1.266e+03 737.5 1475 2s 1s
24th 9.625 3.337e+02 962.5 1925 1s 1s
: Cost-to-solution estimation for several spatial discretizations on fixed physical problem.[]{data-label="tab:cost"}
We see that by taking advantage of the roofline results in combination with the stability conditions, we obtain an estimate of the optimal cost-to-solution of an algorithm. It can be seen that higher order stencils lead to better hardware usage by lowering the wall-time-to-solution. These results, however, rely on mathematical results based on homogeneous velocity. In the case of an heterogenous model, high order discretizations may result in inaccurate, even though stable and non dispersive, solutions to the wave-equation. The choice of the discretization order should then be decided with more than just the performance in mind.
Conclusions
===========
Implementing an optimising solver is generally a long and expensive process. Therefore, it is imperative to have a reliable estimate of the achievable peak performance, FLOPS, of an algorithm at both the design and optimised implementation stages of development.
The roofline model provides a readily understandable graphical tool, even for a non-specialist, to quickly assess and evaluate the computational effectiveness of a particular implementation of an algorithm. We have shown how the roofline model can be applied to finite-difference discretizations of the wave-equation commonly used in the geophysics community. Although the model is quite simple, it provides a reliable estimate of the peak performance achievable by a given finite-difference discretization regardless of the implementation. Not only does this aid the algorithm designer to decide between different discretization options but also gives solver developers an absolute measure of the optimality of a given implementation. The roofline model has also proved extremely useful in guiding further optimization strategies, since it highlights the limitations of a particular version of the code, and gives an indication of whether memory bandwidth optimisations, such as loop blocking techniques, or [FLOPs]{} optimisations, such as SIMD vectorisation, are likely to improve results.
However, one should always be mindful of the fact that it does not provide a complete measure of performance and should be complemented with other metrics, such as time to solution or strong scaling metrics, to establish a full understanding of the achieved performance of a particular algorithmic choice and implementation.
Acknowledgements
================
This work was financially supported in part by the Natural Sciences and Engineering Research Council of Canada Collaborative Research and Development Grant DNOISE II (CDRP J 375142-08) and the Imperial College London Intel Parallel Computing Centre. This research was carried out as part of the SINBAD II project with the support of the member organizations of the SINBAD Consortium.
Wave-equations {#PDE}
==============
In the following equations $u$ is the pressure field in the case of acoustic anisotropic while $p,r$ are the split wavefields for the anisotropic case. We denote by $u(.,0)$ and respectively $p,r$ the value of $u$ for all grid points at time $t=0$. The physical parameters are $m$ the square slowness, $\epsilon,
\delta$ the Thomsen parameters and $\theta,\phi$ the tilt and azimuth. The main problem with the TTI case is the presence of transient functions ($cos$, $sin$) known to be extremely expensive to compute (typically about an order of magnitude more expensive than an add or multiply). Here we will assume these functions are precomputed and come from a look-up table, thus only involving memory traffic In the acoustic anisotropic case the governing equations are:
$$\begin{aligned}
&m \frac{d^2 u(x,t)}{dt^2} - \nabla^2 u(x,t) =q, \\
&u(.,0) = 0, \\
&\frac{d u(x,t)}{dt}|_{t=0} = 0.
\end{aligned}
\label{eqn:Acou}$$
In the anisotropic case we consider the equations describe in [@liu2009stable]. More advanced formulation have been developed however this equation allow an explicit formulation on the operational intensity and simple stencil expression. It is the formulation we are also using in our code base. In the VTI case the governing equations are:
$$\begin{aligned}
&m \frac{d^2 p(x,t)}{dt^2} - (1+2\epsilon)D_{xx} p(x,t) - \sqrt{(1+2\delta)} D_{zz} r(x,t) =q, \\
&m \frac{d^2 r(x,t)}{dt^2} - \sqrt{(1+2\delta)}D_{xx} p(x,t) - D_{zz} r(x,t) =q, \\
&p(.,0) = 0, \\
&\frac{d p(x,t)}{dt}|_{t=0} = 0, \\
&r(.,0) = 0, \\
&\frac{d r(x,t)}{dt}|_{t=0} = 0.
\end{aligned}
\label{eqn:vVTI}$$
For TTI the governing equations are:
$$\begin{aligned}
&m \frac{d^2 p(x,t)}{dt^2} - (1+2\epsilon)(G_{\bar{x}\bar{x}} +G_{\bar{y}\bar{y}} )p(x,t) - \sqrt{(1+2\delta)}G_{\bar{z}\bar{z}} r(x,t) = q, \\
&m \frac{d^2 r(x,t)}{dt^2} - \sqrt{(1+2\delta)}(G_{\bar{x}\bar{x}} +G_{\bar{y}\bar{y}} ) p(x,t) - G_{\bar{z}\bar{z}} r(x,t) = q, \\
&p(.,0) = 0, \\
&\frac{d p(x,t)}{dt}|_{t=0} = 0, \\
&r(.,0) = 0, \\
&\frac{d r(x,t)}{dt}|_{t=0} = 0,
\end{aligned}
\label{eqn:TTI}$$
where the rotated differential operators are defined as
$$\begin{aligned}
G_{\bar{x}\bar{x}} = & cos(\phi)^2 cos(\theta)^2 \frac{d^2}{dx^2} +sin(\phi)^2 cos(\theta)^2 \frac{d^2}{dy^2}+ \\
& sin(\theta)^2 \frac{d^2}{dz^2} + sin(2\phi) cos(\theta)^2 \frac{d^2}{dx dy} - sin(\phi) sin(2\theta) \frac{d^2}{dy dz} -cos(\phi) sin(2\theta) \frac{d^2}{dx dz} \\
G_{\bar{y}\bar{y}} = & sin(\phi)^2 \frac{d^2}{dx^2} +cos(\phi)^2 \frac{d^2}{dy^2} - sin(2\phi)^2 \frac{d^2}{dx dy}\\
G_{\bar{z}\bar{z}} = & cos(\phi)^2 sin(\theta)^2 \frac{d^2}{dx^2} +sin(\phi)^2 sin(\theta)^2 \frac{d^2}{dy^2}+ \\
&cos(\theta)^2 \frac{d^2}{dz^2} + sin(2\phi) sin(\theta)^2 \frac{d^2}{dx dy} + sin(\phi) sin(2\theta) \frac{d^2}{dy dz} +cos(\phi) sin(2\theta) \frac{d^2}{dx dz}. \\
\end{aligned}
\label{eqn:DiffOp}$$
|
---
abstract: 'Electron spin resonance (ESR) spectroscopy is an important tool to characterize the ground state of conduction electrons and to measure their spin-relaxation times. Observing ESR of the itinerant electrons is thus of great importance in graphene and in single-wall carbon nanotubes (SWCNTs). Often, the identification of CESR signal is based on two facts: the apparent asymmetry of the ESR signal (known as a Dysonian lineshape) and on the temperature independence of the ESR signal intensity. We argue that these are insufficient as benchmarks and instead the ESR signal intensity (when calibrated against an intensity reference) yields an accurate characterization. We detail the method to obtain the density of states from an ESR signal, which can be compared with theoretical estimates. We demonstrate the success of the method for K doped graphite powder. We give a benchmark for the observation of ESR in graphene.'
author:
- Péter Szirmai
- Gábor Fábián
- Balázs Dóra
- János Koltai
- Viktor Zólyomi
- Jenő Kürti
- 'Norbert M. Nemes'
- László Forró
- Ferenc Simon
title: 'Density of states deduced from ESR measurements on low-dimensional nanostructures; benchmarks to identify the ESR signals of graphene and SWCNTs'
---
Introduction
============
Electron spin resonance (ESR) has proven to be an important method in identifying the ground state of strongly correlated electron systems. ESR helped e.g. to identify the ordered spin-density wave ground state in the Bechgaard salts [@TorrancePRL1982] and for carbonaceous materials, ESR was key to discover the AC$_{60}$ (A=K, Rb, Cs) fulleride polymer [@JanossyPRL1994].\
A natural expectation is that ESR can be applied for single-wall carbon nanotubes (SWCNTs) [@IijimaNAT1993] and graphene [@NovoselovSCI2004], which are the two novel members of the carbon nanostructure family. The ESR literature on graphene is yet restricted to a single report [@ForroPSSB2009]. Although there exists larger literature on the SWCNTs, the situation is yet unclear. In general, the ESR signal on itinerant electrons yields a direct measurement of the spin-relaxation time (often called as spin-decoherence time), $T_1$, through the relation: $T_1=1/\gamma \Delta B$, where $\Delta B$ is the homogeneous ESR line-width and $\gamma/2 \pi=28.0\,\text{GHz/T}$ is the electron gyromagnetic ratio. $T_1$ is the central parameter which characterizes the usability of the materials for spintronics. This explains the motivation of the ESR studies on graphene and SWCNTs.\
One important question is whether the ESR signal of the itinerant (i.e. the conduction electrons) can be observed at all. It was argued on a theoretical basis [@DoraPRL2008] that it cannot be observed due to the Tomonaga-Luttinger liquid ground state of the metallic SWCNTs [@BockrathNAT; @KatauraNAT2003; @PichlerPRL2004]. It seemed that the only way to explore the local magnetism in SWCNTs is to spin label it either by means of $^{13}$C nuclei [@SingerPRL2005] or by an electron spin label [@SimonPRL2006]. The literature situation on the SWCNT ESR studies is conflicting, and it is reviewed herein without any judgement on validity. Petit *et al.* [@PetitPRB1997] reported the observation of the ESR signal of itinerant electrons. Salvetat *et al.* [@SalvetatPRB2005] reported that the ESR signal occuring around $g \approx 2$ is caused by defects in the SWCNTs. Likodimos *et al.* [@likodimos] reported that a similar signal is related to the itinerant electrons with a possible antiferromagnetic order at low temperature. Corzilius *et al.* [@CorziliusPSSB2008] reported the observation of the itinerant electron ESR in SWCNT samples prepared by chemical vapor deposition.
Often, the identification of the itinerant electron ESR signal is based on two facts: the asymmetry of the ESR lineshape (also known as a Dysonian) and the temperature independence of the ESR signal intensity. The Dysonian lineshape also occurs for localized spins (e.g. for paramagnetic impurities) which are embedded in a metal thus this property cannot be used for the above identification. This is discussed as Eq. 3.6 in the seminal paper of Feher and Kip as the “slowly diffusing magnetic dipole case” [@FeherKip]. The temperature independence of the ESR intensity could be observed for localized paramagnetic spins when they are embedded in a metal with increasing conductivity, $\sigma$ with decreasing temperature; then the microwave penetration depth $\lambda=\sqrt{\frac{2}{\mu_0 \omega \sigma}}$ (here $\mu_0$ is the permeability of the vacuum and $\omega$ is the frequency of the microwaves).
There has been remarkable progress in the quest for the intrinsic ESR signal in SWCNTs using samples made of nanotubes separated according to their metallicity [@ArnoldSeparation]. However, both kinds of samples, i.e. those made of purely metallic or semiconducting nanotubes shows similar ESR signals [@KuzmanyPSSB2010], thus the situation remains unresolved.
A parallel situation happened for high $T_c$ superconductors: soon after their discovery [@BednorzMueller1986] several reports claimed to have observed the “intrinsic” ESR signal in these compounds. Later it turned out for all studies that the signal of parasitic phases (which happen to have strong paramagnetic signals), the so-called green and brown-phases were observed. Later, spin labeling (e.g. Gd substituting Y in YBa$_2$Cu$_3$O$_{7-\delta}$) turned out to be successful to study the electronic structure [@JanossyGdYBCO].
The ESR signal of itinerant electrons in the SWCNTs is expected to have i) a $g$-factor near 2, ii) a line-width, $\Delta B$ smaller than 1 mT, and iii) a signal intensity corresponding to the low density of states (DOS) with no temperature dependence. All properties present a significant hindrance for the signal identification since most impurity in carbon have $g\approx 2$, a maximum 1 mT line-width, and the Curie spin-susceptibility of even a small amount of impurity overwhelms the small Pauli susceptibility of the itinerant electrons. Since nothing is known about the $g$-factor and the line-width *a priori*, only the magnitude of the calibrated ESR signal when compared to the theoretical estimates of the Pauli spin-susceptibility provides a clear-cut ESR signal identification in graphene or SWCNTs.
Here, we outline the method to determine the calibrated ESR signal intensity and the resulting DOS in one- and two-dimensional carbon. The method is demonstrated for K doped graphite powder which is regarded as a model system of biased graphene [@GrueneisPRB2009]. A good agreement is obtained between the theoretical and expeirmental DOS for the KC$_8$ doped graphite system. We note, that a similar program was applied successfully when the ESR signals of Rb$_3$C$_{60}$ [@JanossyPRL1993] and MgB$_{2}$ [@SimonPRL2001] were discovered. We give benchmarks which can be used to decide whether the ESR of the itinerant electrons is observed in graphene.
Experimental
============
We used commercial graphite powder (Fischer Scientific) and potassium (99.95 % purity: Sigma-Aldrich) for the intercalation experiments. The graphite powder (3 mg) was mixed with 3 mg MnO:MgO powder (Mn concentration 1.5 ppm) and ground in a mortar. MgO separates the graphite powder pieces, which enables the penetration of exciting microwave and its Mn content acts as an ESR intensity standard. The mixture was vacuum annealed at 500 $\mathrm{^{\circ}C}$ for 1 h in an ESR quartz tube and inserted into an Ar glove-box without air exposure. Alkali doping was performed by heating the ESR quartz tube containing the graphite powder and potassium for 29 hours using the standard temperature gradient method in Ref. [@Dresselhaus2002] to obtain Stage I, i.e. KC$_8$ intercalated graphite. ESR measurements were performed with a JEOL X-band spectrometer at room temperature.
Results and discussion
======================
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First, we discuss spin-susceptibility, $\chi_s$, calculated from the ESR signal in different dimensions. ESR spectroscopy measures the net amount of magnetic moments, which is an extensive thermodynamic variable, i.e. proportional to the sample amount. The corresponding intensive variable, which characterizes the material is the spin-susceptibility, $\chi_s$ which reads as: $$\begin{split}
\chi_s=\mu_0\cdot\frac{\sum m}{B_{\text{res}}\cdot V_D} \hspace{1.5cm} \mathrm{(SI)}\\
\chi_s=\frac{\sum m}{B_{\text{res}} \cdot V_D} \hspace{1.5cm}\mathrm{(Gaussian)}
\end{split}$$ where $m$ is the magnetic moment, $B_{res}$ is the magnetic field of the resonance, $V_D$ is the volume in $D$ dimension ($D=2;3$), and $\mu_0$ is the permeability of the vacuum. Clearly, the unit of $\chi_s$ depends on the dimension $D$.
$\chi_s$ is either due to the Curie spin-susceptibility for non-interacting spins or the Pauli spin-susceptibility for itinerant electrons in a metal. The relevant expressions are given in Table \[susceptibilities\]. Therein, $A_c$/$V_c$ denotes the unit area/volume, $g$ is the $g$-factor, $\mu_{\text{B}}$ is the Bohr moment and $k_{\text{B}}$ is the Boltzmann constant. $S$ is the spin state of the non-interacting spins and $\varrho(\varepsilon_{\text{F}})$ is the DOS at the Fermi level in units of $\text{states} /\text{eV}\cdot unit$. Here, $unit$ refers to the unit chosen, e.g. for C$_{60}$ fulleride salts, the unit could be 60 carbon atoms. Then the DOS is larger but so is the unit volume which cancels in the result. For graphene, the two atom basis is used as $unit$.
The ESR intensity of a metal can be calibrated against a Curie spin system with known amount of spins. This leads to the comparison of the Pauli and the Curie spin-susceptibilities:
$$\begin{split}
\frac{I_{\text{ESR}}(\text{Pauli})}{I_{\text{ESR}}(\text{Curie})}=\frac{\sum m_{\displaystyle \text{Pauli}}}{\sum m_{\displaystyle \text{Curie}}}=\left(\frac{g_{\text{Pauli}}}{ g_{ \text{Curie}}}\right)^2\times\hspace{1.5cm}\\
\noalign{\smallskip}
\frac{4}{3}S(S+1)\cdot k_{\text{B}}T\varrho(\varepsilon_{\text{F}})\frac{B_{\mathrm{res}}(\text{Pauli})}{B_{\mathrm{res}}(\text{Curie})}\cdot
\!\frac{ \left(\frac{\displaystyle V_D}{\displaystyle V_c(D)}\right)(\text{Pauli})}{\displaystyle \left(\frac{\displaystyle V_D}{\displaystyle V_c(D)}\right)(\text{Curie})}
\label{chi_Curie_div_Pauli}
\end{split}$$
where $I_{\text{ESR}}$ denotes the ESR signal. $V_D$ and $V_c(D)$ are the volume of the sample and the unit cell in $D$ dimensions, respectively. Note that $V_D/V_c(D)(\text{Pauli})=N(\text{ Pauli})$ is the number of units in the metallic sample and $V_D/V_c(D)(\text{Curie})=N(\text{Curie})$ is the number of Curie spins. Eq. (\[chi\_Curie\_div\_Pauli\]) is correct for both SI and Gaussian units and is independent of the choice of $unit$, as expected.
For $S=1/2$ and $g_{\text{Pauli}},g_{\text{Curie}}\approx 2$, Eq. (\[chi\_Curie\_div\_Pauli\]) simplifies to: $$\begin{split}
\frac{I_{\text{ESR}}(\text{Pauli})}{I_{\text{ESR}}(\text{Curie})}= k_{\text{B}}T
\varrho(\varepsilon_{\text{F}}) \frac{N(\text{ Pauli})}{N( \text{ Curie})}.
\label{chi_C_div_P_simple}
\end{split}$$
![\[GraphDoped\] ESR spectrum of saturated K doped graphite powder sample at $T=$300 K. The inset shows a zoom on the ESR spectrum showing the presence of the six lines of the $\mathrm{Mn^{2+}}$ hyperfine structure. The solid curve is a fit.](GraphDoped){width="\linewidth" height="0.7\linewidth"}
Mn:MgO Graphite
----------------------------------------- -------------------------- ----------
$\left\langle S\cdot(S+1)\right\rangle$ 9/4
$M_{\mathrm{mol}}$ \[g/mol\] 40 12
$m$ \[mg\] 3 3
$C_{\mathrm{spin}}$ 1.5 ppm 1
$I_{\mathrm{ESR}}$ $6\cdot 5.3\cdot10^{-3}$ 205
: \[calib2\] Parameters of the $\chi_s$ calibration of KC$_8$.
We present the case of KC$_8$ as an example of the ESR intensity calibration. In Fig. \[GraphDoped\], we show the ESR signal of the mixture of MnO:MgO and saturated K doped graphite. Parameters of the calibration are given in Table \[calib2\]: $C_{\mathrm{spin}}$ is the spin concentration and the effective $\left\langle S\cdot(S+1)\right\rangle_{\mathrm{Mn^{2+}}}=9/4$ as only the $-1/2 \rightarrow 1/2$ transition is observed from the 5 Zeeman transitions of the Mn$^{2+}$ ($S=5/2$) [@AbragamBleaneyBook].
The sample content gives: $N(\text{Pauli})/N( \text{Curie})\approx3.33$ and Eq. (\[chi\_Curie\_div\_Pauli\]) yields $\varrho(\varepsilon_{\text{F}})\approx0.34(2)\;\mathrm{states/(eV\cdot C\:atom)}$, in good agreement with $\varrho(\varepsilon_{\text{F}})=0.327\;\mathrm{states/(eV\cdot C\:atom)}$ obtained by specific heat measurements [@Dresselhaus2002].
In the following, we analyze the case of graphene. There, $A_c = 5.24$ Å$^2/(\text{unit cell})$ is the graphene elementary cell and the DOS, at $T=0$ and $\Gamma=0$ ($\Gamma$ is the damping parameter), reads as a function of the chemical potential $\mu$ [@NovoselovRMP2009]: $$\begin{split}
\varrho(\mu)=\frac{2A_c \mu}{\pi \hbar^2 v^2_{\text{F}}}.
\label{graphene_DOS}
\end{split}$$ Here, $v_{\text{F}}\approx 10^{6}\,\text{m/s}$ is the Fermi velocity. Consequently, $\varrho(\mu)=\mu \cdot 0.0770\,\text{states/(eV}^2\cdot \text{unit cell})$ if $\mu$ is measured in eV. Thus Eq. (\[chi\_C\_div\_P\_simple\]) (in two dimensions) at room temperature reads:
$$\begin{split}
\frac{\sum m_{\displaystyle \text{gr}}}{\sum m_{\displaystyle \text{Curie}}}=\underbrace{0.026 \cdot 0.0770}_{\approx 1/500}
\cdot \mu[\text{eV}] \frac{N(\text{gr})}{N(\text{Curie})}
\label{chi_graphene_div_Curie}
\end{split}$$
$N(\text{gr})$ is the number of graphene unit cells in the sample.
Finally, we assess the feasibility of ESR spectroscopy on graphene. ESR spectrometer performance is given by the limit-of-detection (LOD$_0$) i. e. the number of $S=1/2$ Curie magnetic moments at room temperature which are required for a signal-to-noise ratio of $S/N =10$ for $\Delta B =0.1$ mT linewidth, and 1 s/spectrum-point time constant. For modern spectrometers LOD$_0=10^{10}$ spins/0.1 mT. To calculate the LOD for a broadened ESR line, LOD$(\Delta B)$, we introduce a function to track the effect of broadening:
$$\begin{split}
f(\Delta B)=\left\{
\begin{array}{lr}
\frac{\displaystyle \Delta B}{\displaystyle 0.1 \,\text{mT}} & \hspace{0.5 cm}\text{if}\hspace{0.5 cm} \Delta B\leq 1\,\text{mT}\\
\noalign{\smallskip}
\frac{\displaystyle \Delta B^2}{\displaystyle 0.1\,\text{mT}^2} & \hspace{0.5 cm}\text{if}\hspace{0.5cm} \Delta B> 1\,\text{mT}\\
\end{array}
\right.
\label{line_width_function}
\end{split}$$
This function is 1 if $\Delta B=0.1\,\text{mT}$ and it is 10 if $\Delta B=1\,\text{mT}$ which is the usual maximum modulation amplitude. For line-widths above this value, the function grows quadratically, which describes that the amplitude of the derivative ESR signal drops quadratically. Using this function: LOD$(\Delta B)=$LOD$_0\cdot f(\Delta B)$. Comparison with Eq. (\[chi\_graphene\_div\_Curie\]) yields that numerically ($\mu$ in eV units) $$\begin{split}
\text{LOD}(\text{gr})=500/\mu \cdot \text{LOD}_0 \cdot f(\Delta B)
\label{LOD_graphene}
\end{split}$$ is the LOD for graphene. We could conclude that $$\begin{split}
A_{\text{lb}}(\text{gr})=500/\mu \cdot \text{LOD}_0 \cdot f(\Delta B) \cdot A_c
\label{Alb_graphene}
\end{split}$$ which gives a lower bound for the area of the graphene sheet which enables the ESR measurement. Assuming a $\Delta B=0.1$ mT and a shift in chemical potential by gate bias of $\sim$0.2 eV we estimate $A_{\text{lb}}(\text{gr})\approx1.3\,\mathrm{mm^2}$.
Summary
=======
In summary, we detailed the method of obtaining the calibrated ESR intensity and the DOS in carbonaceous materials. We argue that a similar analysis is required for the identification of the ESR signal of itinerant electrons in SWCNT and graphene.
Acknowledgements
================
Work supported by the OTKA Grant Nr. K 81492, and Nr. K72613, by the ERC Grant Nr. ERC-259374-Sylo, the Marie Curie ERG project CARBOTRON, and by the New Széchenyi Plan Nr. TÁMOP-4.2.1/B-09/1/KMR-2010-0002. BD acknowledges the Bolyai programme of the Hungarian Academy of Sciences. The Swiss NSF and its NCCR “MaNEP” are acknowledged for support.
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---
abstract: 'The precise location of the water ice condensation front (Ôsnow lineÕ) in the protosolar nebula has been a debate for a long time. Its importance stems from the expected substantial jump in the abundance of solids beyond the snow line, which is conducive to planet formation, and from the higher ÔstickinessÕ in collisions of ice-coated dust grains, which may help the process of coagulation of dust and the formation of planetesimals. In an optically thin nebula, the location of the snow line is easily calculated to be around 3 AU, subject to brightness variations of the young Sun. However, in its first 5 to 10 million years, the solar nebula was optically thick, implying a smaller snowline radius due to shielding from direct sunlight, but also a larger radius because of viscous heating. Several models have attempted to treat these opposing effects. However, until recently treatments beyond an approximate 1+1D radiative transfer were unfeasible. We revisit the problem with a fully self-consistent 3D treatment in an axisymmetric disk model, including a density-dependent treatment of the dust and ice sublimation. We find that the location of the snow line is very sensitive to the opacities of the dust grains and the mass accretion rate of the disk. We show that previous approximate treatments are quite efficient at determining the location of the snow line if the energy budget is locally dominated by viscous accretion. Using this result we derive an analytic estimate of the location of the snow line that compares very well with results from this and previous studies. Using solar abundances of the elements we compute the abundance of dust and ice and find that the expected jump in solid surface density at the snow line is smaller than previously assumed. We further show that in the inner few AU the refractory species are also partly evaporated, leading to a significantly smaller solid state surface density in the regions where the rocky planets were formed.'
address:
- 'Astronomical institute Utrecht, Utrecht University, P.O. Box 80000, NL-3508 TA Utrecht, The Netherlands'
- 'Max Planck Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany'
- 'Astronomical Institute ‘Anton Pannekoek’, Science Park 904, NL-1098 XH Amsterdam, The Netherlands'
- 'Department of Astrophysics/IMAPP, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands'
author:
- 'M. Min'
- 'C.P. Dullemond'
- 'M. Kama'
- 'C. Dominik'
title: 'The thermal structure and the location of the snow line in the protosolar nebula: axisymmetric models with full 3-D radiative transfer'
---
accretion; Solar nebula; radiative transfer; planetary formation
Introduction
============
The efficiency of the formation of rocky planets and the cores of gas giant planets depends sensitively on the amount of solids present in each location of the solar nebula. In most models of planet formation the solar nebula has been assumed to resemble the structures proposed by and @1981IAUS...93..113H. In this model, and subsequent incarnations of it, the surface density of the gas is approximately given by $$\label{eq:gas mmsn}
\Sigma^{\mathrm{MMSN}}_{\mathrm{gas}}(r) = 1700 \left(\frac{R}{1\;\mathrm{AU}}\right)^{-3/2}\quad \mathrm{gram/cm}^2$$ [see @2006plfo.book..129T] where the acronym MMSN stands for “minimum mass solar nebula”, a name by which this model is often identified. In the inner regions (small values of $R$) of the nebula the dust grains are refractory in nature, while beyond some radius $R_{\mathrm{ice}}$ water ice is present, most likely in the form of ice mantels surrounding the refractory grains. In the Hayashi MMSN model this location is assumed to be $R_{\mathrm{ice}}=2.7$ AU, which is consistent for temperatures expected in an optically thin nebula. Often this water ice sublimation boundary radius is called the ‘snow line’. Since ices may contain a considerable mass compared to the mass in refractory grains, the total surface density was assumed to jump at $R=R_{\mathrm{ice}}$: $$\label{eq:mmsn}
\Sigma^{\mathrm{MMSN}}_{\mathrm{solid}}(r) = 7.1 F_{\mathrm{ice}} \left(\frac{R}{1\;\mathrm{AU}}\right)^{-3/2}\quad \mathrm{gram/cm}^2$$ with $$F_{\mathrm{ice}} =\left\{ \begin{matrix}
1, & \hbox{for} & R< R_{\mathrm{ice}} \\
4.2, & \hbox{for} & R> R_{\mathrm{ice}}
\end{matrix}\right.$$ [@2006plfo.book..129T]
This model serves in many planet formation models to compute the midplane gas and solid density and the isolation mass of planetary embryos. However, it has been shown that the outcomes of planet formation models depends critically on the value of $R_{\mathrm{ice}}$ and the strength of the ice jump $F_{\mathrm{ice}}$ . Also dust coagulation and planet formation and migration models are very sensitive to these parameters . Simply taking the values to be $R_{\mathrm{ice}}=2.7$AU and $F_{\mathrm{ice}}=4.2$ (for $R>R_{\mathrm{ice}}$) is probably too simplistic and may lead to major errors in predictions of the progress of the planet formation process.
Besides the amount of solid mass available also the strength of the bonds between dust grains is of importance to their aggregation behavior. The presence of ice coatings on grains can significantly increase their ’stickiness’, allowing grains to coagulate more easily. Perhaps even more importantly the bonds formed are stronger preventing aggregates that formed to be destroyed again by collisions.
Over the last decade or so, there has been enormous progress in astronomy in the understanding of the structure and evolution of ‘protoplanetary disks’, i.e. the dust+gas disks surrounding many very young stars. These disks very likely give a good impression of what our own solar nebula looked like 4.567 billion years ago, and the knowledge gained in that field can help getting a better handle on the structure of the solar nebula and the location of the ice condensation front. Indeed, modeling tools that were originally meant for modeling protoplanetary disks and comparing model predictions to observations have been used to make models of the solar nebula and the snow line. For instance, the 1+1D disk structure model of , in which the equations are first solved vertically (1D) and then connected radially (1+1D), has been used by @2001ApJ...554..391H to study water in the solar nebula, in particular concerning the issue of the D/H ratio in meteorites. @2005ApJ...627L.153D made a model of the surface density of the solar nebula disk as a function of radial coordinate and accretion rate. He used this model to study the ice condensation front [@2005ApJ...620..994D]. This model includes both the effect of irradiation of the disk by the young sun and the heating of the disk near the midplane through viscous dissipation of potential energy (i.e. accretional heating). Like in the Hersant study, these models are 1+1D models, i.e. they are 1-D vertical models of density and temperature as a function of $z$. The Davis models include radiative transfer using a variable eddington factor method. @2006ApJ...640.1115L also modeled this, using an updated version of the @1997ApJ...490..368C two-layer disk model, i.e. slightly simpler than the 1+1D modeling of Davis. Recently, @2009Icar..200..672D combined complex models of ice formation and viscous evolution of the disk with a simple radiative transfer approximation to compute the evolution of the solid surface density.
While these modeling efforts have improved the understanding of the distribution of ices in the solar nebula substantially, they still rely on rather dramatic simplifications of the treatment of radiative transfer. The reason for this is very understandable: it is quite a numerical challenge to model multi-dimensional radiative transfer in protoplanetary disks that are very optically thick and actively accreting (and hence producing heat near the midplane). However, in recent years multi-dimensional radiative transfer tools, in particular those based on the Monte Carlo method, have improved dramatically in speed. Extreme optical depths are now not necessarily a problem anymore. We have developed a code, MCMax, that can now handle extreme optical depths efficiently, even if many of the photon packages originate from the optically thickest regions of the disk . In addition, we have adjusted the code to fully take into account the density dependend sublimation state of both the refractory as well as the icy material . It is therefore a right time to revisit the problem of the snow line with the full force of multi-dimensional radiative transfer and investigate (a) where is the ice line as a function of time and (b) what is the gas and solid surface density distribution in the disk as a function of time. This is the goal of this paper.
Model
=====
Our model combines the radiative transfer equipment and basic disk model setup described in . We add here: a self-consistent and automatic computation of the radial structure, viscous heating of the disk by accretion and a self-consistent treatment of water ice and dust evaporation.
Right at this point we already note that we will replace ‘time’ $t$ with ‘accretion rate’ $\dot M$ and study only the inner 20AU of the disk. The reason is that the long term evolution of protoplanetary disks depends very strongly on processes such as photoevaporation of the [*outer*]{} disk regions [see e.g. @2009ApJ...690.1539G]. The inner disk regions, say inward of 20 AU, are regulated entirely by the ‘feeding’ of matter by the outer disk. How this feeding depends on time requires a detailed study of the complex processes happening at large radii. But given a recipe of feeding, i.e. an accretion rate as a function of time $\dot M(t)$, the inner disk can be assumed to be entirely determined. In particular if we avoid issues such as ionization degree and possible instabilities in the disk [see e.g. @2006ApJ...648..484H] and assume that no mass pile-up can happen anywhere in the inner disk (this assumption must be relaxed in future work), then the stationary disk structure inward of 20 AU is entirely determined by the stellar parameters and the accretion rate $\dot M$. For any given star our model thus becomes a [*1-parameter model*]{}. So instead of having $t$ as our parameter, we will have $\dot M$ as our parameter. Generally, a high mass accretion rate corresponds to early times in the evolution of the solar nebula, while low mass accretion rates correspond to later times. Alternatively, in the episodic accretion scenario, low mass accretion rates represent the typical state, which is punctuated by relatively short episodes with high mass accretion rates.
The main improvement of our study over earlier work is that we treat the flow of radiation through the disk in a fully self-consistent manner, assuming that dust opacities are everywhere dominant over the gas opacities. Our model is axially symmetric, i.e. all variables depend on radial coordinate $r$ and vertical coordinate $z$, but do [*not*]{} depend on longitudinal coordinate $\phi$. The movement of photons is, however, fully 3-D, i.e. a photon can move also outside of the $r,z$-plane and has all the 3-D freedom of motion. In our Monte Carlo scheme [based on @2001ApJ...554..615B] we split the total input luminosity $L$ into $N$ ‘photon packages’, each carrying a luminosity $L_{\mathrm{package}}=L/N$. The total luminosity includes the stellar luminosity $L_{*}$ and the accretional heating luminosity $L_{\mathrm{visc}}$, i.e. $L=L_{*}+L_{\mathrm{visc}}$. As these photon packages travel through the disk they update the local temperature of the dust according to the scheme by @2001ApJ...554..615B. Since our model is axisymmetric, these updates have to only be done in 2-D, i.e. the temperature only depends on (and will only be updated in) $r,z$, no matter what is the longitudinal location $\phi$ of the photon package. So while the radiative transfer is 3-D, the resulting structure is 2-D.
The main difference between models doing radiative transfer in 1+1D and our model doing full 3D is in the radial energy diffusion. In the 1+1D model the energy from the star is intercepted by the disk and further diffusion is assumed to be vertical. This approximation is expected to be accurate for the regions of the disk close to the midplane where the very high densities do not allow much radial diffusion. However, radial energy diffusion is important in the surface layers and, more importantly, in the innermost regions of the disk, directly illuminated by the central star. Our model naturally takes this into account and allows us to see in which parts of the disk the vertical diffusion approximation is valid. Also, 3D radiative transfer is the only way to properly treat the complex shape of the dust condensation front close to the star .
For the central star in our model we take the Sun, i.e. the mass $M_\star=M_{{\odot}}$, the luminosity $L_\star=L_{{\odot}}$, and the stellar radius $R_\star=R_{{\odot}}$. The spectrum of the Sun is approximated by a blackbody of 5777K. Although the parameters of the early Sun were likely different from this, we do not expect much difference on the results in this paper.
Treatment of viscous accretion
------------------------------
We assume that viscous heating is described by the so-called $\alpha$-disk model . In this model it is assumed that the energy produced locally by viscous heating is proportional to the local gas pressure through the proportionality constant $\alpha$. The total energy locally created per unit volume is then given by $$\label{eq:gamma}
\Gamma_\mathrm{viscous}=\frac{9}{4}\alpha P(z)\Omega(R),$$ where $P(z)$ is the gas pressure, and $\Omega(R)$ is the Keplerian angular velocity at radius $R$ from the central star. If we define a uniform accretion rate $\dot{M}$ throughout the disk we have for the total energy released per unit surface area of the disk per unit time at a radius $R$ from the Sun is $$\label{eq:tot energy}
F_\mathrm{viscous}=\frac{3GM_\star\dot{M}}{4\pi}R^{-3}\left[1-\left(\frac{R_\star}{R}\right)^{1/2}\right],$$ where $G$ is the gravitational contstant, and $M_\star$ and $R_\star$ are the mass and radius of the central star. Combining Eqs. \[eq:gamma\] and \[eq:tot energy\] we can solve for the surface density of the disk for any given value of $\alpha$. In this paper we will consider two values of $\alpha$, $0.1$ and $0.01$. From Eq. (\[eq:tot energy\]) it is clear that the total amount of energy released is independent of the structure of the disk. This speeds up convergence since there is no feedback from the disk structure on the total energy that is released.
We basically treat the energy released by viscous stress in the same way as the energy released by the central star. Thus a fraction of the photon packages released in the Monte Carlo radiative transfer process is now released from the inner regions of the disk. Where previous studies that take into account radiation from viscous stress rely on assumptions on the radiative transfer in the optically thick inner regions, we treat the radiative transfer fully consistent in the framework of Monte Carlo radiative transfer as outlined by . The total accretional heating luminosity of the disk is given by the integral of Eq. (\[eq:tot energy\]) from the stellar surface, $R_\star$, to the outer radius, $R_\mathrm{out}$. Assuming $R_\mathrm{out}>>R_\star$ this leads to $$L_\mathrm{visc}=\frac{1}{2}\,\frac{GM_\star\dot{M}}{R_\star}.$$ Note that a large fraction of this energy is most likely released inside the dust sublimation radius and thus can easily escape the disk. This energy does not influence the dust temperature and is thus not taken into account in our computation. In our method the energy produced through viscous heating is directly deposited from the gas onto the dust grains. This assumes a strong coupling between gas and dust, which is an appropriate assumption in the regions where the densities of both gas and dust are high. In the regions where a large fraction of the dust is evaporated due to high temperatures, we limit the amount of energy deposited on the grains by viscous heating by taking into account the number of collisions between the gas and the dust and assuming a gas molecule can at maximum transfer its total kinetic energy onto the grain in such a collision. This limits the total energy that can be transferred to the dust per unit time. We find that indeed in the optically thick regions this limit exceeds the available energy by many orders of magnitude. However, in the inner regions, this limit prevents the small amount of dust grains available to be overloaded with accretional energy from the dense gas surrounding it. The dust grains are assumed to radiate the energy they absorbed with a wavelength distribution according to the local dust temperature after which the regular procedure of Monte Carlo radiative transfer is proceeded.
In the dusty regions of the disk the optical depth from the midplane, where most energy is released, to the outside is extremely large. Therefore, a photon package undergoes a large number of interaction steps before it leaves the system (on the order of several million interactions). If all these interaction steps would be computed separately, the computation time required would be on the order of a minute per photon package, putting strong constraints on the number of photon packages that can be used. introduced an analytical solution for the random walk regime of the radiative transfer that can be applied in the optically thick regions and reduces computation times by orders of magnitude. This algorithm therefore allows the use of a large number of photon packages and thus to obtain proper statistics everywhere.
------------- ----------- ----- -------- ----------- ------------- ------------------ --------------
Silicates FeS C-dust Water ice Gas to $F_\mathrm{ice}$ $\kappa_R$
solid ratio \[cm$^2$/g\]
only CO 47% 14% - 39% 97 1.63 572
only C-dust 24% 7% 20% 49% 49 1.97 544
mixture 32% 10% 13% 45% 66 1.84 508
------------- ----------- ----- -------- ----------- ------------- ------------------ --------------
The dust and ice composition and opacities
------------------------------------------
In order to estimate the abundances of the various dust components we use here the Solar abundances of the elements as determined by @1998SSRv...85..161G. The dust mixture is then constructed as follows:
- Since iron sulfide is the dominant form of (solid) sulfur in meteorites in the solar system it is first assumed that all the available sulfur is in the form of FeS. This takes approximately half the available number of Fe-atoms.
- Then we assume all Mg and Si go into silicates together with the remaining Fe-atoms. This creates silicates with an average composition of roughly MgFe$_{0.5}$SiO$_{3.5}$. We assume this to be in the form of amorphous silicates. This takes approximately 15% of the available O-atoms.
- Next we put a fraction $w$ of the available C-atoms in the form of carbonaceous dust grains. The remaining C-atoms are assumed to form CO. This takes away up to $\sim 50$% of the available number of O-atoms depending on the value of $w$.
- The remaining O-atoms are assumed to form water ice everywhere in the disk where the temperature allows it.
From the above procedure we also can compute the gas to dust ratio. We find that the resulting gas-to-dust ration can deviate up to a factor of 2 from the canonical value 100. The question remains what the most realistic value of the carbon abundance in dust, $w$, is. We take as a standard the value $w=0.5$, based on in-situ measurements of the dust in comet Halley . However, the carbon fraction in dust could have been higher in the early Solar nebula since the interstellar solid carbon abundance is much higher. Also, in the inner regions, at temperatures above $\sim1000$K, carbon grains will be destroyed by combustion so lower values of $w$ are to be expected. The dust/ice/gas mixture resulting from this procedure for different choices of $w$ are summarized in Table \[tab:abundances\].
In order to compute the opacities of the dust species as a function of wavelength we need to know the refractive indices of the grains. For the silicates we use the measurements by and , for the iron sulfide we take the laboratory data from @1994ApJ...423L..71B, for the carboneceous material we take the measurements from , and for the water ice we take the data from @1984ApOpt..23.1206W. In order to convert the refractive index data to opacities we use the method by to simulate irregularly shaped particles. The particle sizes are assumed to follow the so-called MRN distribution [@1977ApJ...217..425M]. We thus assume that no significant grain growth has occurred with respect to the grains in the interstellar medium. We will come back to this assumption in the discussion of the results.
Treatment of evaporation and condensation
-----------------------------------------
The temperature at which a species evaporates depends on the vapor pressure of the material. We treat the evaporation and condensation of the dust species following . In order to converge the structure of the inner regions had to put constraints on the vertical and radial gradients of the fraction of the material in the gas phase. This is to avoid instabilities caused by the sudden jump in opacity at the inner edge of the disk. Since the ice is located in the inner regions of the disk, the ill conditioned convergence is no issue here. Therefore, for the water ice we do not put any constraints on the gradients of the condensed fraction. It turns out that convergence is reached relatively fast and no instabilities like those reported by are found for the ice sublimation zone. Although the density structure does not change significantly anymore after $\sim30$ iterations, we ran all models up to 100 iterations to make sure also the slowly changing structures are converged.
To simplify the evaporation of the refractory dust species we take the evaporation parameters for amorphous olivine for the entire dust mixture of silicates, iron sulfide and carboneceous material. For the iron sulfide it might be argued that this is not a correct assumption since the evaporation temperature of FeS is around 680K. However, we assume that above this temperature the iron condenses as metallic iron, contributing similarly to the opacity of the grains. The evaporation characteristics of both ice and olivine are based on sublimation data from @1994ApJ...421..615P.
Convection {#sec:convection}
----------
In the implementation of radiative transfer and the computation of heat balance in the disk we employ, the effect of convection is not taken into account. However, we will show that convection is an important way of transporting energy generated in the midplane of the disk by accretion to the surface. Therefore, we have implemented a rough treatment of convection in the radiative transfer code to test the effects on our results. In the computations below, convection is not taken into account unless stated otherwise.
Convection becomes an important way of transporting heat when the radiative temperature gradient becomes too large. According to the Schwarzschild criterion for convection this happens when $$\label{eq:convection}
\frac{d\ln T}{d\ln P}>\nabla_\mathrm{ad},$$ where $T$ and $P$ are the gas density and pressure, and $\nabla_\mathrm{ad}$ is the adiabatic limit which we take to be that of a diatomic gas: $\nabla_\mathrm{ad}=2/7$. Our rough implementation to correct the obtained temperature structure for convection is simply as follows. We start at the surface of the disk, where the temperature gradient is very small and work our way to the midplane. At each location the criterion according to Eq. \[eq:convection\] is determined. If the temperature gradient is too steep the temperature gradient is adjusted such that Eq. \[eq:convection\] becomes an equality. In some cases this causes the temperature structure close to midplane to be adjusted such that the midplane temperature is reduced. This way we can model the effects of cooling of the midplane by convection although we should stay aware of the approximate nature of the implementation.
Our implementation of convection is very similar to the one by @1998ApJ...500..411D with the difference that we do not consider the turbulent flux, and we take the convection efficiency, their parameter $\zeta$, to be unity. In the relatively less dense regions of the disk transfer of energy through radiation dominates and the criterion Eq. (\[eq:convection\]) is fulfilled. In this case we do not alter the temperature structure. However, in those regions of the disk where radiative transfer cannot fulfill Eq. (\[eq:convection\]) we do adjust the temperature structure to be adiabatic. This means that convection becomes important only in the more massive disks where radiative energy transfer is difficult.
Results
=======
Location of the snow line as a function of $\dot M$
---------------------------------------------------
In Fig. \[fig:snow lineCarbon0.5a0.01\] we display the locations in the disk where water ice can exist for various values of the mass accretion rate. It is clear that the location of the snow-line depends strongly on the value of $\dot{M}$. This is caused by two main effects. The first is that when $\dot{M}$ is increased, the energy released by viscous stress is increased. Secondly, a larger value of $\dot{M}$ causes the disk to be more massive (see also Fig. \[fig:surfacedens\]) and thus the greenhouse effect, keeping the energy at the midplane, is much stronger (as explained below, see also Eq. (\[eq:midplaneT1\])) and the midplane temperature is increased further.
Going vertically up in the disk, the effect from viscous heating becomes less important and the snow line moves in significantly. This continues up to the point where radiation from the Sun becomes the dominant energy source and causes the snow line to move out again. The fact that the location of the snow line in the upper, optically thin region of the disk is not constant is caused by the pressure dependent evaporation temperature of the ice in combination with a decreasing density when going higher up in the disk. This shape of the ice region is consistent with the findings from @2005ApJ...620..994D but the snow line in our computations lies significantly further out. The reason for this is most likely predominantly caused by the different opacity of the grains which is assumed. @2005ApJ...620..994D use for the opacity the analytical expression from @1986ApJ...308..883R. As outlined below the most important parameter determining the midplane temperature is the Rosseland mean opacity at 160K divided by the gas to solid ratio, since this determines how efficiently the energy created by viscous stress is kept in the midplane at the location of the ice evaporation front. In the study by @2005ApJ...620..994D this parameter is $\kappa_R/f=2.7\,$cm$^2$/g, while in our study, taking a realistic dust grain model, we find that $\kappa_R/f$ lies in the range $5.9$ to $11.1\,$cm$^2$/g (see Table \[tab:abundances\]). Thus in our model the energy generated by viscous heating is more efficiently captured in the midplane regions.
In Figs. \[fig:snow lineCarbon0.01\] and \[fig:snow lineCO0.10\] we show the extreme cases of the range of dust parameters we studied. Fig. \[fig:snow lineCarbon0.01\] shows the case where all carbon atoms are locked into solid carbon, i.e. the opacity is at its maximum. In this case the greenhouse effect, heating the midplane efficiently is at its maximum. The other extreme, presented in Fig. \[fig:snow lineCO0.10\], shows the case where all carbon is in CO, i.e. the opacity is minimum, and the turbulence is very high, $\alpha=0.1$. This high value of $\alpha$ implies that one can get the same mass accretion rate with a smaller surface density. Thus for a given value of $\dot{M}$ the surface density drops and the greenhouse effect is smaller. In this case the energy can easily escape the midplane and the midplane temperature is low leading to a snow line close to the star.
Analytical estimate for the location of the snow line in the midplane
---------------------------------------------------------------------
We take the analytic estimate of the midplane temperature from @1990ApJ...351..632H. This only includes viscous heating in the midplane and results in a midplane temperature, assuming high optical depths, of $$\label{eq:midplaneT1}
T^4=\frac{9\Sigma_\mathrm{gas} \kappa_R G M_\star \dot{M}}{128\pi \sigma R^3 f},$$ where $\kappa_R$ is the Rosseland mean opacity of the gas/ice mixture and $f$ is the gas to solid ratio such that the total optical depth through the disk is equal to $\tau_\mathrm{tot}=\kappa_R\Sigma_\mathrm{gas}/f=\kappa_R\Sigma_\mathrm{solid}$. The values of $\kappa_R$ and $f$ for the models we computed are listed in Table \[tab:abundances\]. The surface density of the gas, $\Sigma_\mathrm{gas}$, can be expressed in terms of $T$ and $\alpha$ using Eqs. (\[eq:gamma\]) and (\[eq:tot energy\]). The integral of Eq. (\[eq:gamma\]) in vertical direction is equal to the total energy given in Eq. (\[eq:tot energy\]). Assuming $R>>R_\star$ and taking the temperature to be the midplane temperature at each height in the disk we get [see also @2007prpl.conf..555D] $$\label{eq:surfdens}
\Sigma_\mathrm{gas}=\frac{\mu m_p \dot{M}}{3\pi \alpha k_b T}\sqrt{\frac{G M_\star}{R^3}},$$ where $\mu$ is the average molecular mass and $m_p$ is the proton mass. Substituting Eq. (\[eq:surfdens\]) into Eq. (\[eq:midplaneT1\]) we get $$\label{eq:midplaneT2}
T^5=\frac{3 \mu m_p \dot{M}^2 \kappa_R}{128\pi^2 \alpha k_b \sigma f} \left[\frac{G M_\star}{R^3}\right]^\frac{3}{2} .$$
Taking the above we find that the snow line is located at $$\label{eq:Restimate}
R_\mathrm{ice}=\left[C\cdot
\frac{
\dot{M}^2
\kappa_R}{f \alpha}
\right]^\frac{2}{9},$$ where $\kappa_R$ is the Rosseland mean opacity of the gas/ice mixture computed at 160K (see Table \[tab:abundances\]), and $C$ is given by $$C=\frac{3\mu m_p \left(G M_\star\right)^{3/2}}{128\pi^2k_b\sigma T_\mathrm{ice}^5}=1.7\cdot10^{22}\,\mathrm{cm^{5/2}\,s^2/g}.$$ We find that this formula agrees very well with our findings for the range of parameters considered here (within a few percent). The formula becomes less accurate when the snow line is close to the star, and clearly the equation fails when the radiation of the Sun is the dominant energy source in the midplane. However, this is never the case in the models considered here. For easy computation, the value of $C=3.5\cdot10^{14}$ can be used in Eq. \[eq:Restimate\] when $\dot{M}$ is given in $M_{{\odot}}/$yr and $\kappa_R$ is given in cm$^2/$g to give the location of the snow line $R_\mathrm{ice}$ in AU.
Using Eq. \[eq:Restimate\] provides a way to scale our results to those obtained by @2005ApJ...620..994D taking into account the correct values of $\kappa_R$, $f$, and $\dot{M}$. Doing this we find that the agreement is excellent, and thus the differences are indeed due to the different values of $\kappa_R/f$. We also compared our results to those computed by @2007ApJ...654..606G. The values for the location of the snowline as computed using Eq. (\[eq:Restimate\]) are compared to those obtained by @2005ApJ...620..994D and @2007ApJ...654..606G in table \[tab:compare\]. Note that @2005ApJ...620..994D adopted a different description for the viscosity of the disk, which explains partly the differences. The overall agreement with the approximated and numerically computed values implies that estimates of the location of the snow line using approximate methods are in general correct when appropriate opacity values are used.
[lccc]{} $\dot{M}$ \[$M_{\odot}/$yr\] & $\kappa_R/(f\alpha)$ \[cm$^2$/g\] & $R_\mathrm{ice}$ \[AU\] (Eq. \[eq:Restimate\]) & $R_\mathrm{ice}$ \[AU\] (numerical)\
\
$8.06\cdot10^{-6}$ & $\sim270$ & 32.3 & $\sim37$\
$1.75\cdot10^{-6}$ & $\sim270$ & 16.4 & $\sim15.5$\
$1.13\cdot10^{-7}$ & $\sim270$ & 4.8 & $\sim3.6$\
$2.53\cdot10^{-10}$ & $\sim270$ & 0.3 & $\sim0.6$\
\
$1\cdot10^{-7}$ & $27.7$ & 2.8 & $\sim3.4$\
$1\cdot10^{-8}$ & $27.7$ & 1.0 & $\sim1.4$\
$1\cdot10^{-9}$ & $27.7$ & 0.4 & $\sim0.6$\
\
$1\cdot10^{-6}$ & 770 & 16.1 & $\sim16$\
$1\cdot10^{-7}$ & 770 & 5.8 & $\sim6$\
$1\cdot10^{-8}$ & 770 & 2.1 & $\sim2.2$\
$1\cdot10^{-9}$ & 770 & 0.7 & $\sim0.8$\
\
$1\cdot10^{-6}$ & 59 & 9.1 & $\sim9$\
$1\cdot10^{-7}$ & 59 & 3.3 & $\sim3.2$\
$1\cdot10^{-8}$ & 59 & 1.2 & $\sim1.2$\
$1\cdot10^{-9}$ & 59 & 0.4 & $\sim0.5$\
Solid surface density distribution as a function of $\dot M$
------------------------------------------------------------
In Fig. \[fig:surfacedens\] we show the surface density as a function of radius for different values of the mass accretion rate. Both the total surface density (top panel) as well as the surface density in solids (lower panel) are shown. The sudden drop in the total surface density at the location where the solids start to become important is an effect of the sudden increase in opacity, causing a steep temperature gradient which translates into a steep surface density gradient through Eqs. \[eq:gamma\] and \[eq:tot energy\]. The surface density in solids is adjusted iteratively until the temperature nowhere in the disk exceeds the sublimation temperature. This means in the inner regions solid matter is removed until the greenhouse effect, keeping the energy locked in the midplane regions of the disk, drops to a level where the temperature is low enough. For very high accretion rates, a lot of matter has to be removed to let the large amount of energy produced in the midplane escape. This balance between condensation and evaporation results in a thermostat keeping the midplane temperature at the silicate evaporation temperature over a significant part of the inner disk. When the accretion rate is reduced, more matter can condense. However, at the same time reducing the mass accretion rate reduces the total surface density and thus the amount of matter that can condense. Both these effects play a role in determining the surface density in the solid phase.
At the radius of water ice sublimation we have a gradual increase in the surface density in the solid phase. We find that the increase in the solid phase due to water ice is $F_\mathrm{ice}=1.84$ for the case where half of the carbon atoms is in the solid phase. This increase is quite small compared to the $F_\mathrm{ice}=4.2$ taken currently in the literature. We have to take into account here that we have only considered water ice condensation. If we take in addition CH$_4$ and NH$_3$ ice, which are the most important ice species after water ice, we gain about a factor 1.5 [see e.g. @2009Icar..200..672D] so we get $F_\mathrm{ice}\sim2.8$. Note however, that these ices condense at lower temperatures, so somewhat further out then the water-ice condensation radius we compute here.
The surface density is in principle a function of the mass accretion rate, the value of $\alpha$ and the opacity of the grains. However, in the innermost region the dust evaporates to reduce the greenhouse effect such that the midplane temperature drops below the dust sublimation temperature. In this regime the solid surface density is adjusted to reach a given vertical optical depth, $\kappa_R\Sigma_\mathrm{gas}/f$, roughly following Eq. (\[eq:midplaneT1\]). The maximum vertical optical depth that can exist before the midplane gets too hot is thus only a function of the total energy released in the midplane, and thereby only of $\dot{M}$ (see Eq. \[eq:tot energy\]). The amount of mass that can exist in this region is robustly determined through the mass accretion rate and the opacity of the grains and is independent of the gas density or the description of the accretion mechanism. Therefore, in this region we can easily scale the solid surface density as a function of $\dot{M}$ we find in this study to different values of the opacity of the grains when, for example, they grow to larger sizes.
Effects of convection
---------------------
When studying the temperature structure in the disk in detail we find that often the temperature gradient is steeper than the adiabatic temperature gradient (see Eq.\[eq:convection\]). In these cases the midplane will cool by convection. If we correct for the convective cooling in the way discussed in section \[sec:convection\], the midplane temperature decreases in some cases significantly. We find that this only plays a role when the midplane temperature is dominated by accretion and in addition, is most important when the temperature is above $\sim500\,$K. Thus, the location of the snow line is only mildly affected. For the case of $\dot{M}=10^{-6}\,M_{{\odot}}/$yr we find that due to the decreased midplane temperature the fraction of condensed solids can increase locally by a factor of 5 with respect to the case where convection is ignored. For low accretion rates convection becomes unimportant at almost all radii and the disk cools through radiation [see also @1998ApJ...500..411D]. The reason for the dependence of the importance of convection on the surface density of the disk is simply due to the fact that less massive disks can easily cool through radiation, so convection is not needed to lose the energy from the midplane. In the more massive disks it is very difficult for the radiation to escape from the midplane and convective cooling is much more efficient. Our treatment of convection is somewhat simplified and more advanced methods, also including turbulent fluxes, should be implemented in the future to get a clear picture of the exact effects.
In Fig. \[fig:convection\] we plot the surface density in the solid phase for computations with and without convective cooling of the midplane. It is clear that for the high accretion rates convective cooling is important to gain extra solids in the region where the terrestrial planets are (i.e. in the inner few AU). When convective cooling is ignored we find that there is never enough mass in the solids around 1AU to create the Earth. However, with convective cooling we find that there is much more mass. Note that the amount of mass in the solids can be increased even further for the high accretion phases by growing the grains to somewhat larger sizes, thereby decreasing the opacity and thus the midplane temperature. We conclude that convective cooling is an important mechanism to take into account when modeling the solid density distribution in the early Solar nebula.
Discussion
==========
Implications for planet formation
---------------------------------
In this section we compare our results with constraints from the minimum mass solar nebula (MMSN). We will use here the ’classical’ MMSN as mentioned above. There is still quite some debate on what the minimum mass is needed to form our solar system [see e.g. @2005ApJ...627L.153D; @2007ApJ...671..878D; @2009ApJ...698..606C], and most of these models find the need for a more massive solar nebula than the classical one we use here. Therefore, our choice of the MMSN is conservative, and the computations could be repeated to find stronger constraints by using other models. In addition, planet formation might be more dynamic, i.e. planets could accrete mass from different parts of the disk through dynamic interactions.
According to the MMSN (Eq. \[eq:mmsn\]) in order to form the Earth one needs a solid surface density of 7.1g/cm$^2$ at 1AU. As can be seen directly from Fig. \[fig:convection\] this surface density in solids is never achieved in our standard model. This is because for the high accretion models the midplane gets too hot, so solids have to evaporate to let the energy escape, while for the low accretion models there simply is not enough mass at all. However, as discussed above, if we take into account convective cooling of the midplane the amount of condensed solids can increase by a factor of 5. This way the surface density at 1AU is only slightly below the requirement from the MMSN. Inside 1AU the surface density in solids is never even close to the MMSN, which might pose serious problems for the formation scenario of Venus.
Jupiter is generally assumed to form outside the snow line. Thus, when Jupiter formed at its current position, we have as an additional constraint that $R_\mathrm{ice}<5.2\,$AU. For this the mass accretion rate needs to be smaller than $\dot{M}<8\cdot10^{-8}\,M_{{\odot}}/$yr. Since the midplane temperature at the snow line should be smaller than 160K convective cooling is not important, so we can use our standard model. At this mass accretion rate the solid surface density is approximately $\Sigma_\mathrm{solid}\approx 2\,$g/cm$^2$, which is slightly lower than the mass needed according to Eq. \[eq:mmsn\]. However, typical giant planet formation models require up to 3 to 4 times the value of the MMSN [see e.g @1996Icar..124...62P].
We thus find that our computations put constraints on the scenario of the formation of the Solar system in the context of the MMSN. However, as we already noted above, the MMSN might be an optimistic estimate, and in reality a much larger mass could be required. A possible solution to still meet the requirements of planet formation in that case would be to lower the opacity of the grains significantly by, for example, growing them to larger sizes.
Dust properties in the hot midplane regions
-------------------------------------------
We find that at high mass accretion rates there is a significant region in the disk where the midplane temperature is above 1000K. This is out to $\sim$1.7AU in the case where $\dot{M}=10^{-6}\,M_{{\odot}}$/yr, and out to $\sim$0.6AU in the case of $\dot{M}=10^{-7}\,M_{{\odot}}$/yr. This is including convective cooling. Thus we find that a considerable region in the early Solar system had conditions where thermal processing of the solid state material was very efficient. Also, at these temperatures grain coagulation is influenced by *sintering* [see e.g. @2003Icar..164..139P]. Sintering is the process where the bonds between grains are molten together by the impact of grain collisions or by a long exposure to high temperatures. The structures formed under these conditions can be extremely open aggregates when the strong bonds created do not allow restructuring or compaction of the aggregates. @2003Icar..164..139P compute that for aggregates composed of silica spheres exposure to temperatures around $\sim1250$K is needed for efficient sintering. This is comparable to the crystallization temperature of silica found by . Since both sintering and crystallization are processes that are caused by the atoms in the lattice wanting to move to the minimum energy state, it is not unlikely that the temperatures of these processes are similar. In that case sintering of Fe/Mg silicates will happen at slightly lower temperatures, i.e. around $\sim1000$K. The collision speeds in the midplane regions of the disk are too low to cause collisional sintering.
As mentioned before, grain growth might help to increasing the solid surface density by reducing the opacity of the particles and in this way decreasing the temperature in the disk. A low opacity reduces the greenhouse effect and thus allows more grains to condense before the midplane temperature gets too high. In fact, when the Rosseland mean opacity computed at the dust evaporation temperature of 1500K reduces by a given factor, the maximum allowed surface density increases by this factor. In order to test to what sizes we have to grow the grains in order to significantly increase the solid surface density we performed computations of the Rosseland mean opacity at 1500K for larger grain sizes. We increased the entire size distribution by a given factor $\gamma$ which means that we do not only grow large particles, but efficiently remove the small particles as well. Thus, where the MRN distribution we used so far runs from $0.005\,\mu$m to $0.25\,\mu$m, the new size distribution contains grains with sizes from $(\gamma\times0.005)\,\mu$m to $(\gamma\times0.25)\,\mu$m. For these larger grain sizes the effects of anisotropic scattering become significant. The argumentation on the maximum total optical depth were made under the assumption of isotropic scattering. When anisotropic scattering is important, the same argumentation can be made when scaling the scattering cross section by a factor $(1-g)$ [see @1978usaf.book.....I] where $0<g<1$ is the anisotropy parameter. For values of $g$ close to unity, most radiation is scattered in forward direction which for radiative transfer purposes is effectively a strong reduction of the effect of scattering. Values of $g$ close to zero are more equivalent to isotropic scattering.
The solid curve in Fig. \[fig:grain growth\] shows the increase in the maximum allowed solid surface density as a function of the scale factor $\gamma$. For example, we see that by increasing the grain size distribution by a factor 1000 the maximum allowed surface density increases by a factor $\sim$44. This is because at these grain sizes the Rosseland mean opacity (corrected for anisotropic scattering) is reduced by a factor of $\sim$44. The dotted curve in Fig. \[fig:grain growth\] shows the same but now for very fluffy aggregates. The opacities here are computed using the Aggregate Polarizability Mixing Rule . It is clear that compared to the grain growth computed using compact grains (the solid curve) the fluffy aggregates have to grow to much larger sizes in order to reduce the opacity sufficiently to allow more solid mass to condense. If indeed sintering causes rigid bonds, prohibiting compaction of the formed aggregates, such fluffy aggregates are not an unlikely outcome of the aggregation process.
An interesting feature for the compact grains (and to a lesser extend for the fluffy aggregates) in Fig. \[fig:grain growth\] is that the opacity of the grains first increases, which causes more mass to evaporate. Only when the particles are sufficiently large the opacity decreases and the solid surface density can increase. This is caused by the increasing scattering efficiency when the grains grow from interstellar grain sizes to several microns in size. It implies that in order to form large particles, a barrier has to be taken. The complex behavior of grain growth under these conditions asks for a more detailed study. For the fluffy aggregates this effect is much weaker. This is because for these aggregated structures the scattering is heavily forward peaked which makes the effect of scattering on the radiative transfer very small.
Another possibility to decrease the opacity of the grains in the hot regions is to take away the species that contribute most to the opacity. We took a mixture of silicates, iron sulfide and carboneceous grains. It is well known that iron sulfide is only stable at temperatures below 680K [@1994ApJ...421..615P]. This means that the iron sulfide opacity source is most likely not there, though it can be argued that metallic iron will take over. Removing iron sulfide from the mixture we decrease the Rosseland mean opacity at 1500K by a factor $\sim$1.2, so this would allow only somewhat more mass to form. The dominant species of opacity in our computations is carbon. If we were to remove all the carboneceous species but leave the iron sulfide we reduce the Rosseland mean opacity at 1500K by a factor $\sim$3. Indeed, we find that the surface density in the inner region for our model without carboneceous solids is a factor of 3 higher that our standard model. Removing both the carboneceous species and the iron sulfide reduces the Rosseland mean opacity at 1500K by a factor $\sim$20. Though we did not perform the exact radiative transfer computations we can deduce that this would allow $\sim20$ times more mass in silicate grains to condense in the hot inner regions of the disk. This is assuming no other species adding opacity, like for instance metallic iron, form in these parts of the disk. Since metallic iron is a very strong opacity source (even slightly stronger than iron sulfide) even a small amount would already increase the opacity again to approximately the same level as with iron sulfide and carbon present.
In conclusion we find that the solid surface density in the region controlled by evaporation due to viscous heating is completely determined by the opacity of the grains and thus heavily depends on the local grain size and chemical composition.
Coagulation timescale in the thermostat region
----------------------------------------------
Another important effect of the opacity-temperature control system is that the reduced number density of small dust grains will have a significant influence on the coagulation time scale in that region. The time scale at which coagulation can proceed depends strongly on the initial number density of grains at which the aggregation process has to start. Smaller particle densities increase the mean free path for particles and, consequently, the collision time for a dust particle to find another one to stick to and grow. This effect is extended through the further growth phases. have shown that this effect extends the time scales for removing small grains from the disk by coagulation approximately inversely proportional to the initial number density. This means that that coagulation will be strongly slowed down in this region, and that the time scales over which opacity dominates the number of grains that can be present in the thermostat region might be determined by the mass accretion time scale of the disk rather than the coagulation time scale. We believe that this effect can have significant influence on coagulation and disk evolution models. Further studies on this issue are necessary and will be the subject of a follow-up paper.
Conclusions
===========
We successfully computed the solid density structure and snow line in the early Solar nebula for various parameters of the mass accretion rate and grain composition. This was done using full 3-D radiative transfer taking into account both the energy released by viscous heating as well as radiation from the Sun. The density structure of both the gas and the solids are computed self consistently using an $\alpha$ description for the viscosity and detailed evaporation physics for the dust and ice. We find that:
- The location of the snow line is determined by the opacity of the grains and the mass accretion rate. We show that approximate radiative transfer methods to compute the location of the snow line are quite accurate.
- At high mass accretion rates the midplane temperature in the inner few AU can exceed the silicate evaporation temperature. This causes a balance between condensation and evaporation of matter acting like a thermostat keeping the midplane at the silicate evaporation temperature over a significant region. The solid mass in the inner regions is dramatically reduced by this effect and models computing the solid state surface density of the early Solar system should take this into account.
- For our standard set of parameters the surface density in the solid state in the region of the terrestrial planets is below the MMSN for all values of the mass accretion rate. This poses serious constraints on the scenarios of planet formation. It could point to a significantly lower opacity value at the time of planet formation, which can be achieved by increasing the average grain size or significantly change the chemical composition.
- At high accretion rates the increased midplane temperature of the disk due to viscous heating causes the midplane temperatures to be $>1000\,$K in a relatively large part of the disk. Under these conditions thermal processing of dust material as well as sintering of aggregates becomes important.
- We find that for the high mass accretion rates the effects of convective cooling of the midplane are important to take into account.
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---
abstract: 'Compact stars consisting of massless quark matter and fermionic dark matter are studied by solving the Tolman-Oppenheimer-Volkoff equations for two fluids separately. Dark matter is further investigated by incorporating inter-fermionic interactions among the dark matter particles. The properties of stars made of quark matter particles and self-interacting and free dark matter particles are explored by obtaining their mass-radius relations. The regions of stability for such a compact star are determined and it is demonstrated that the maximum stable total mass of such a star decreases approximately linearly with increasing dark matter fraction.'
author:
- Payel Mukhopadhyay
- 'Jürgen Schaffner-Bielich'
date: 'November 1, 2015 '
title: Quark stars admixed with dark matter
---
Introduction
============
A quark star is a hypothetical compact star and consists of self-bound strange quark matter (SQM) [@Witten:1984abc; @Itoh:1970abc; @Farhi:1984abc; @Weber:2005abc; @Ivanenko:1965abc]. The existence of quark stars is controversial and its equation of state is also uncertain.
One of the popular models for the equation of state of the quark star is the so called MIT Bag model [@Chodos:1974abc]. The model is often used for describing cold and massless (strange) quark matter [@Haensel:1986abc; @Alcock:1986abc]. Standard values for the MIT bag constant are around *$B^{1/4}$* = 145 MeV as follows from fits to hadron masses [@Haensel:1986abc], which results in maximum masses of about 2.0$ M_{\odot}$ at a radius of about 11 km [@Witten:1984abc; @Haensel:1986abc] , which are actually very close to the ones of realistic neutron star models.
Dark matter stars are modelled in our work as a free or self-interacting fermion gas at zero temperature. The possible candidates for dark matter particles are a type of fermion predicted in extensions of the standard model including supersymmetric particles, the neutralino, the gravitino and the axino [@Baltz:2004abc]. In our discussion, we consider dark matter to be made of fermionic particles with a mass of 100 GeV, the classical WIMP mass scale. We assume that the dark matter particles cannot self-annihilate as in asymmetric dark matter (ADM) [@Kaplan:1992abc; @Nussinov:1985abc].
Self-annihilating WIMP dark matter with masses above a few GeV accreted onto neutron stars may trigger a conversion of most of the star into a strange star [@Perez:2010abc] or the accreted dark matter may significantly affect the kinematical properties of the compact star [@Perez:2012abc]. Constraints on the properties of dark matter candidates can be obtained from stars which can accrete asymmetric dark matter in its lifetime and then collapse into a neutron star [@Kouvaris:2011abc]. Constraints on the mass of dark matter candidates can also be obtained by the possible collapse of compact stars due to dark matter accretion [@Goldman:2013qla; @Kouvaris:2015rea]. The cooling process of compact objects can be affected by the capture of dark matter which can annihilate the star [@Lavallaz:2010abc]. Recent studies have been done to explore compact stars with non self-annihilating dark matter to analyze the gravitational effects of dark matter on the stellar matter under intense conditions [@Li:2012abc; @Leung:2011abc; @Sandin:2009abc; @Xiang:2014abc]. In these studies, masses of dark matter in the GeV range has been assumed. Studies have also been performed to investigate the compact objects formed due the admixture of neutron star matter and dark matter [@Laura:2015abc] leading to the possibilities for new stable solutions of compact stars with planet like masses. To study of the effect of dark matter on compact objects is therefore of great interest. If quark stars do exist in nature, they can also accumulate dark matter and hence their properties might change. This accumulation will lead to various changes in the mass-radius relation of a quark star which is studied in this work.
This paper is organised as follows: in section 2, we shortly discuss about the two fluid TOV equations. In section 3, we discuss the equations of state for both quark matter and fermionic dark matter and discuss the general scaling solutions for these stars. In section 4, we present the numerically obtained results (mass-radius relations) for quark matter stars by solving the TOV equations. In section 5, TOV equations are solved for dark matter composed of both strongly self-interacting and free fermions and their corresponding mass-radius relations are obtained. Section 6 is dedicated to the numerical solutions of two fluid TOV equations namely, dark matter and quark matter which are coupled together only by gravity. We demonstrate that the maximum mass of the quark star admixed with dark matter reduces due to the presence of dark matter and decreases in a linear fashion in case in strongly self-interacting dark matter fermions while for free dark matter, the maximum mass remains almost unaffected. Finally, in section 7, we summarize our findings and discuss our results.
Throughout the paper, we use natural units where *c*=$\hbar$=1, *c* being the speed of light and $\hbar$ is the reduced Planck’s constant.
Two fluid Tolmann-Oppenheimer-Volkoff equations
===============================================
Since our aim is to see the properties of a quark star admixed with dark matter, we need the TOV equations for two fluids admixed with each other. There will be a hydrostatic equilibrium condition for each of the two fluids and the fact that there is only gravitational interaction between them will be encoded in the metric describing the system. The two fluid TOV equations that we use here are [@Sandin:2009abc; @Laura:2015abc]:
$$\begin{split}
\emph{\(\frac{dp_{1}}{dr}\)} &=-\emph{\(\frac{GM(r)\rho_{1}(r)}{r^{2}}\)}\left(1+\emph{\(\frac{p_{1}(r)}{\rho_{1}(r)}\)}\right)\times \\
& \left(1+4\pi r^{3} \emph{\(\frac{(p_{1}(r)+p_{2}(r))}{M(r)}\)}\right)\left(1-2G\emph{\(\frac{M(r)}{r}\)}\right)^{-1}
\end{split}$$
$$\begin{split}
\emph{\(\frac{dp_{2}}{dr}\)} &=-\emph{\(\frac{GM(r)\rho_{2}(r)}{r^{2}}\)}\left(1+\emph{\(\frac{p_{2}(r)}{\rho_{2}(r) }\)}\right)\times\\
&\left(1+4\pi r^{3} \emph{\(\frac{(p_{1}(r)+p_{2}(r))}{M(r)}\)}\right)\left(1-2G\emph{\(\frac{M(r)}{r}\)}\right)^{-1}
\end{split}$$
$$\emph{\(\frac{dM_{1}}{dr}\)}=4\pi r^{2} \rho_{1}(r)$$
$$\emph{\(\frac{dM_{2}}{dr}\)}=4\pi r^{2} \rho_{2}(r)$$
$$M(r)=M_{1}(r)+M_{2}(r)$$
Here *M(r)* represents the total mass at radius *r*, *$p_{1}$*, *$p_{2}$*, $\rho_{1}$ and $\rho_{2}$ are the pressures and densities of the fluids 1 and 2 respectively. We could separate out the hydrostatic equilibrium condition for the two stars into equations (1) and (2) because the interaction acts only through gravity and nothing else. The gravitational interaction is taken into account because of the fact that the mass that is considered in the equation *M(R)* is the total mass of both the fluids at radius *r* which means each of the fluid attracts the other gravitationally. The equations for the conservation of mass for the two fluids remains the same as that for individual fluids.
For solving the two fluid TOV equations, we need proper boundary conditions. *$M_{1}(0)$* and [$M_{2}(0)$]{} must be equal to zero at *r*=0. Central pressures for the two fluids are calculated from the central densities given as the initial condition using the respective equation of states for the two fluids. Then the two TOV equations are solved together simultaneously and we obtain either *$R_{1}$* or *$R_{2}$* as the radius of the complete star depending on which fluid ends up having a larger radius. The radius of the individual fluids occur at those points where the individual pressures drop down to zero.
Equations of state for quark matter and free and self-interacting dark matter
=============================================================================
The equations for state for quark matter is discussed using MIT Bag model. The EOS for free dark matter particles along with strongly self-interacting dark matter particles is briefly described using statistical mechanics of free and self-interacting fermions. Scaling relations for quark stars and dark matter stars is also discussed.
Equation of state for quark matter
-----------------------------------
The MIT Bag equation of state [@Haensel:1986abc; @Alcock:1986abc] is taken as the equation of state for quark matter in our work. In this model, the quarks are assumed to be made of free fermions constrained within a bag with a vaccumm pressure that keeps the particles within the bag. The MIT Bag equation of state is: $$p = \emph{\(\frac{1}{3}\)}(\epsilon -4B)$$ Here, *p* denotes the pressure, $\epsilon$ denotes the energy density and *B* is the Bag constant whose standard accepted values are around *B*^1/4^ = 145 MeV or *B*^1/4^= 200 MeV [@Haensel:1986abc]. Note that the equation of state for a cold gas of interacting massless quarks within perturbative quantum chromodynamics can be approximated by a similar form of equation of state as the MIT Bag model [@Fraga:2001abc].
Equation of state for free and self-interacting dark matter fermions
---------------------------------------------------------------------
Dark matter will be assumed to be made of fermions of mass 100 GeV. For a study of compact fermionic stars, we refer to some recent papers [@Sagert:2006abc; @Macher:2005abc; @Silbar:2004abc]. The equation of state for a gas of free fermions can be calculated via explicit expressions for energy density ($\epsilon$) and pressure (*p*) [@Haensel:2007abc] $$\label{xx}
\small
\begin{split}
\epsilon &=\emph{\(\frac{1}{\pi^{2}}\)} \int_{0}^{k_{F}} k^2 \sqrt{m_f^2 + k^2} dk\\
&=\emph{\(\frac{m_f^4}{8\pi^{2}}\)}[(2z^3+z)\sqrt{1+z^2}-sinh^{-1}(z)]
\end{split}$$ $$\label{xx}
\small
\begin{split}
p &=\emph{\(\frac{1}{3\pi^{2}}\)} \int_{0}^{k_{F}} \emph{\(\frac{k^4}{\sqrt{m_f^2 + k^2}}\)} dk\\
&=\emph{\(\frac{m_f^4}{24\pi^{2}}\)}[(2z^3-3z)\sqrt{1+z^2}+3sinh^{-1}(z)]
\end{split}$$ where *z*=$k_{F}$/$m_{f}$ is the dimensionless Fermi momentum.
Similarly, the interactions between the fermions is modelled by considering the simplest two-body interactions between fermions. The repulsion amongst the fermions constituting the dark matter star has been modelled by considering the interaction energy density to be proportional to *$n^2$* [@Gaurav:2006abc; @Pratik:2009abc; @Rainer:2010abc] to the lowest order approximation, where *$n$* is the number density of fermions. The resulting equation of state has been calculated in reference [@Gaurav:2006abc]:
$$\label{xx}
\small
\begin{split}
\emph{\(\frac{\epsilon}{m_f^{4}}\)} &= \emph{\(\frac{1}{8\pi^{2}}\)}[(2z^3+z)\sqrt{1+z^2}-sinh^{-1}(z)] \\& + [\left(\emph{\(\frac{1}{3\pi^{2}}\)}\right)^2 y^2z^6]
\end{split}$$
$$\label{xx}
\small
\begin{split}
\emph{\(\frac{p}{m_f^{4}}\)} &=\emph{\(\frac{1}{24\pi^{2}}\)}[(2z^3-3z)\sqrt{1+z^2}+3sinh^{-1}(z)]\\ &+[\left(\emph{\(\frac{1}{3\pi^{2}}\)}\right)^2 y^2z^6]
\end{split}$$
where *z*=$k_{F}$/$m_{f}$ is again the dimensionless Fermi momentum and *y* is the dimensionless interaction strength. Also, *y*=$m_{f}$/$m_{I}$ where *$m_I$* is the scale of interaction. The mass of the fermions $m_{f}$ used for self-interacting and free dark matter has been taken as 100 GeV and it is assumed that they don’t self annihilate [@Kaplan:1992abc; @Nussinov:1985abc]. The value of the interaction strength *y* determines whether the self-interactions are weak or strong. For example, neutralinos, that form a candidate for WIMP dark matter and are in the mass range of 100 GeV [@Bottino:2005abc] can have weak self-interactions with *y* $\sim$ 0.1 or they may be strongly self-interacting with *y* $\sim$ $10^3$ where the strong interaction scale corresponds to $\Lambda_{QCD}$ $\simeq$ 200 MeV. In our discussions, we would focus on two situations, one for free fermionic dark matter matter with *y*=0 and the other for strongly self-interacting dark matter with *y*=$10^3$. For a reference of self-interacting fermionic stars and the corresponding interaction strengths we refer to [@Gaurav:2006abc].
Scaling relations for quark matter and dark matter
---------------------------------------------------
We generally scale dimensional quantities to dimensionless ones in order to represent any arbitrary mass configuration of a star in a single graph. From equation (6), it is clear that if we scale the energy density and pressure values by four times the Bag Constant (4*B*), the EOS reduces to a dimensionless form [@Witten:1984abc; @Haensel:1986abc] the corresponding total mass and radius of the star would then be scaled by $\sqrt{4B}$. Similarly , for the dark matter fermionic particles, it is a natural choice to scale the pressure and energy density by the fermion mass $m_{f}^4$ which will again make the equations dimensionless. The scaling relations for quark matter are $\epsilon^{\prime}_{quark}$ = $\epsilon_{quark}$/(4*B*), *$p^{\prime}_{quark}$* = *$p_{quark}$*/(4*B*), *$M^{\prime}_{quark}$* = *$M_{quark}$*/(2*$\sqrt B$*), *$R^{\prime}_{quark}$* = *$R_{quark}$*/(2*$\sqrt B$*). The corresponding relations for fermions are $\epsilon^{\prime}_{f}$ = $\epsilon_{f}$/*$m_{f}^4$*, *$p^{\prime}_{f}$* = *$p_{f}$*/*$m_{f}^4$*, *$M^{\prime}_{f}$* = *$M_{f}$*/*a*, *$R^{\prime}_{f}$* = *$R_{f}$*/*b* where *a* = *$M_{p}^{3}$*/*$m_{f}^{2}$* and *b* = *$M_{p}$*/*$m_{f}^{2}$* where *$M_{p}$* is the Planck Mass ( *G* = *$M_{p}^{-2}$* ). For detailed derivations of the scaling relations we refer to [@Pratik:2009abc].
Solving TOV equations for quark matter star
============================================
Numerical solutions to the mass-radius relations of quark stars can be found in literature [@Andreas:2015abc]. In our nomenclature *$M_{quark}$* and *$R_{quark}$* represents the mass and radius of the quark star respectively. The curve is shown in Fig. 1 for two different Bag values.
{height="7.5cm" width="3.5in"}
Upto a certain point mass increases with the radius reaching a maximum value of mass at a certain value of radius after which the mass starts decreasing, the star starts becoming unstable from this point.The maximum stable mass for $\textit{B}^{1/4}$ = 145 MeV is about 2.01 $ M_{\odot}$ and the corresponding radius of around 11 km. While $\textit{B}^{1/4}$=200 MeV gives a maximum mass of about 1.06 $M_{\odot}$ with the radius being around 5.8 km. Quark stars are incompressible stars and form a self bound system [@Itoh:1970abc; @Ivanenko:1965abc].
Solving TOV equation numerically for free and Strongly self-interacting dark matter particles
=============================================================================================
Solutions for free fermionic dark matter
-----------------------------------------
We first consider dark matter made of free fermionic particles with 100 GeV mass. Single fluid TOV equations are solved taking (7) and (8) as the equation of state. *$M_{dark}$* and *$R_{dark}$* represents the mass and radius of the dark matter star composed of free fermions respectively. The resulting mass-radius curve is plotted in Fig. 2.
{height="7.5cm" width="3.5in"}
From the graph, we see that the mass at first increases with a decrease in radius for increasing central energy density values, reaches a maximum and then starts decreasing. Stellar configurations to the right side of the maximum masse are stable whereas those on the left side are unstable. The maximum stable mass for the dark matter star made of free fermions comes out to be 6.27$\times$ $10^{-5}$ $M_{\odot}$ with a radius of 0.81 meters.
Solutions for strongly self-interacting dark matter
---------------------------------------------------
The TOV equations are solved for strongly self-interacting dark matter particles (*y* = $10^3$) of mass 100 GeV. The equation of state used are (9) and (10). *$M_{int}$* and *$R_{int}$* represents the mass and radius of the dark matter star composed of strongly self-interacting fermions respectively. The resulting mass-radius curve is plotted in Fig 3.
{height="7.5cm" width="3.5in"}
For strong self-interaction, the mass and radius are much larger compared to free fermions and hence the maximum mass and the minimum radius is about 1000 times larger. From the curve it is observed that for very low central densities of dark matter particles, i.e the tail of the graph, the rate of increase of mass with decreasing radius is much higher as compared to the free dark matter particle case discussed in the previous subsection. The maximum mass and the minimum radius for the self-interacting dark matter star turns out to be 2.67 $\times$ $10^{-2}$ $M_{\odot}$ and 0.189 km respectively, larger than for the non interacting case due to repulsive forces between the dark matter particles.
Solution of TOV equation for an admixture of quark matter and dark matter
===========================================================================
Nomenclature used is *$M_{quark}$* and *$R_{quark}$* for the mass and radius of quark matter, *$M_{dark}$* and *$R_{dark}$* for the mass and radius of the star composed of free dark matter particles, *$M_{int}$* and *$R_{int}$* for strongly self-interacting dark matter star. $\epsilon_{0,quark}$, $\epsilon_{0,dark}$ and $\epsilon_{0,intdark}$ represents the central energy densities of quark matter, dark matter made of free fermions and dark matter made of strongly self-interacting fermions respectively.
Solution for combination of quark matter and free dark matter particles
-----------------------------------------------------------------------
{height="7.5cm" width="3.5in"}
The two fluid TOV equations (1), (2), (3) and (4) are solved for a mixture of quark matter with MIT Bag model by taking the bag value to be *$B^{1/4}$*= 145 MeV and dark matter composed of free fermionic particles of mass 100 GeV. We start with the initial given central energy densities for the two components and compute the corresponding central pressures using the EOS for the respective fluids (eqn. (6) for quark matter and (7) and (8) for dark matter).
{height="7.5cm" width="3.5in"}
{height="7.5cm" width="3.5in"}
{height="7.5cm" width="3.5in"}
Mass (*$M_{quark}$*) vs. central energy density of quark matter ($\epsilon_{0,quark}$) are plotted for three different values of central density of dark matter ($\epsilon_{0,dark}$) each kept constant at a time (See Fig.4). From the plot, it is clear that as the central density of dark matter is increased in the mixture, the maximum mass of the quark matter still reaches to 2.005 $ M_{\odot}$ but now at higher central densities ($\epsilon_{0,quark}$) of quark matter after which the quark matter becomes unstable and the star would collapse. This behaviour can be explained via the fact that as ($\epsilon_{0,dark}$) increases, then within the stable branch of dark matter, the allowed mass of dark matter inside the quark star also increases which contributes to a greater gravitational pull, so, a much higher central quark energy density ($\epsilon_{0,quark}$) is needed to support the maximum possible mass against the greater gravitational pull. The maximum stable mass of the quark component (*$M_{max,quark}$*) is almost the same as pure quark star (2.01 $M_{\odot}$) because the maximum possible value of the dark matter mass is 6.27 . $10^{-5}$ $M_{\odot}$ (Section 5.A) , which is much less than 2.01 $M_{\odot}$, to cause a notable reduction in the quark matter mass. Fig. 5 shows the plot for the radius (*$R_{quark}$*) vs. the central energy density ($\epsilon_{0,quark}$) of quark matter for different values of $\epsilon_{0,dark}$. The figure shows that the maximum stable radius of the quark matter is independent of the amount of free fermionic dark matter present in the admixed star.
{height="7.5cm" width="3.5in"}
After observing that the maximum possible mass of quark matter is hardly reduced in the presence of free fermionic dark matter particles of various central densities, it is essential to determine which configurations of the admixed star are stable. The plots for the profile of dark matter component is obtained by keeping $\epsilon_{0,quark}$ fixed and slowly varying $\epsilon_{0,dark}$ (Fig. 6 and 7). It is seen that the dark matter masses and radii are the same for varying $\epsilon_{0,quark}$ which is expected since dark matter is more compact than quark matter and is not affected very much by the presence of quarks.
Figs. 4, 5, 6 and 7 allow us to analyse the stability of the entire configuration. Since we realise from fig. 6 and 7 that $\epsilon_{0,dark}$ for which dark matter mass hits a maxima is the same for all $\epsilon_{0,quark}$, we at first mark those points where the quark matter becomes unstable i.e hits the maximum mass by doing the plots done in figures 4 and 5 for different $\epsilon_{0,dark}$. As we slowly increase $\epsilon_{0,dark}$ , the dark matter content inside the admixed star keeps on increasing and the radius of the dark matter keeps on decreasing. After a sufficiently large $\epsilon_{0,dark}$ , the dark matter mass content hits a maximum after which the dark matter mass decreases with further increase in $\epsilon_{0,dark}$. This is then the unstable branch for the dark matter . After this critical value of $\epsilon_{0,dark}$, no dark matter configurations are stable and hence quark matter and dark matter can’t exist together since the dark matter would collapse into a black hole. Hence we expect that up to a certain value of $\epsilon_{0,dark}$ , the quark matter stable mass increases to reach 2.005 $M_{\odot}$ for sufficiently high $\epsilon_{0,quark}$, and after a critical $\epsilon_{0,dark}$, dark matter itself becomes unstable which leads to instability of the entire admixture of the dark matter and quark matter.\
Next we study the configurations in the $\epsilon_{0,quark}$ - $\epsilon_{0,dark}$ plane. At first, $\epsilon_{0,dark}$ is kept fixed and $\epsilon_{0,quark}$ is slowly increased. The stable boundary is marked in the contour plot (Fig.8) by the maximum stable quark matter mass for increasing $\epsilon_{0,dark}$ which gives the line inclined at an angle in the contour plot. The sequences continue up to the points where the dark matter mass reaches its maxima. Above this value of $\epsilon_{0,dark}$, all configurations become unstable since dark matter itself becomes unstable. This leads to the boundary line that is almost parallel to the x-axis.\
For a quark star admixed with dark matter made of free gas of fermionic particles, the maximum possible mass of the stable configuration is approximately $M_{total}$ $\sim$ 2.01 $M_{\odot}$ with a dark matter content of around 0.63 $\times$ $10^{-4}$ $M_{\odot}$ which has a small radius of about 0.80 meters while the quark matter extends much further to a radius of around 11 km.
Solution for combination of quark matter and strongly self-interacting dark matter
----------------------------------------------------------------------------------
The two fluid TOV equations (eqns. (1), (2), (3), (4)) are solved now for massive dark matter fermions taken to be strongly self-interacting using the model discussed in section 5.B. The interaction strength *y* is taken to be $10^{3}$. The presence of self-interactions causes the maximum stable mass of a dark star to be increased from about $10^{-4}$ *$M_{\odot}$* for the free fermionic case to about $10^{-2}$ *$M_{\odot}$* for the strongly self-interacting case (Section 5.A and 5.B).
{height="7.5cm" width="3.5in"}
{height="7.5cm" width="3.5in"}
The plot for the mass of the quark component (*$M_{quark}$*) vs. the central energy density $\epsilon_{0,quark}$ of the quark component for different values of $\epsilon_{0,intdark}$ (Fig. 9) reveals that the maximum stable mass of the quark matter decreases with increasing central energy density of dark matter ($\epsilon_{0,intdark}$) within the stable branch of dark matter, though the decrease is very moderate. The maximum stable mass of the quark component at $\epsilon_{0,intdark}$ = $10^5$ MeV/$fm^3$ is 1.995 $M_{\odot}$ and this mass reduces to 1.937 $M_{\odot}$ at $\epsilon_{0,intdark}$ = $ 2 \times 10^6 $ MeV/$fm^3$. It is also observed just as in the previous section that the maximum quark mass ($M_{quark,max}$) is attained at a much higher value of $\epsilon_{0,quark}$, the reason being the same as described in free fermion case. The corresponding plot for the radius of the quark component vs. $\epsilon_{0,quark}$ for two different $\epsilon_{0,intdark}$ is shown in Fig. 10 which shows that the maximum radius of quark component also decreases with an increase in $\epsilon_{0,intdark}$ .
{height="7.5cm" width="3.5in"}
{height="7.5cm" width="3.5in"}
{height="7.5cm" width="3.3in"}
.
Keeping $\epsilon_{0,quark}$ constant and obtaining $M_{int}$ vs. $\epsilon_{0,intdark}$ and $R_{int}$ vs. $\epsilon_{0,intdark}$ gives the profile for dark matter present in the admixture (Fig. 11 and 12). It is evident from Figs. 11 and 12 that the dark matter mass and radius profile does not change much with increasing $\epsilon_{0,quark}$ since dark matter is much more compact than quark matter and its particles are also much more massive to be significantly affected by quark matter particles. Figs. 9, 10, 11 and 12 allow us to determine the stability of the quark matter star admixed with self-interacting dark matter. For the first two plots showing the dependence of the mass and radius of the quark matter vs. $\epsilon_{0,quark}$ tells us up to which point the quark star configuration would remain stable by noting the point of maxima of the mass and the radius. The next two plots , Fig. 11 and 12 allows us to determine up to which point the dark matter remains stable for varying $\epsilon_{0,quark}$ . It is evident from the graphs that the dark matter parameter profile is almost independent of $\epsilon_{0,quark}$. So, the $\epsilon_{0,intdark}$ at which the dark matter becomes unstable is the same for all $\epsilon_{0,quark}$.
The contour plot showing the dependence of the total mass of the entire star ($M_{total}$) on the dimensionless central energy densities of the two fluids (Fig. 13) reflects the decrease in the maximum stable total mass with increase in $\epsilon_{0,intdark}$.
The region of stability is marked in the contour diagram . The shape of the boundary is similar to the free case discussed before. The upper branch of the boundary line is an indicator of the $\epsilon_{0,intdark}$ after which the the dark matter component becomes unstable for a given $\epsilon_{0,quark}$. As a quick check, in the contour diagram, the plot converges to the appropriate mass limit for low $\epsilon_{0,intdark}$. For low $\epsilon_{0,intdark}$ say, $10^5$ MeV/$fm^3$, the maximum stable mass is $\sim$ 2.0 $M_{\odot}$ at a radius of about 11 km showing the convergence to pure quark star limit.
{height="7.5cm" width="3.5in"}
Fig. 14 shows the maximum total mass vs. the fraction of dark matter which can be fitted by a linear fit. The slope of the fit comes out to be about -3.62.
The reason for the decrease of the maximum total mass with increasing dark matter content is the increased gravitational force due to extra dark matter content which causes a collapse of the star. Using the linear fit in the Fig. 14, the equation for the dependence of the maximum total stable mass of the admixed star on the dark matter fraction present is given as:
$$\small
\frac{M_{tot,max}}{M_{\odot}} = 2.004- 3.62 \emph{\(\frac{M_{int}}{M_{tot,max}}\)}$$
or,
$$\small
\frac{M_{tot,max}}{M_{\odot}} = 2.004- 3.62 f$$
where *f* is the fraction of self-interacting dark matter in the admixed star at the maximum stable total mass. The maximum allowed strongly self-interacting dark matter content is about 2.64 $\times$ $10^{-2}$ $M_{\odot}$ at a maximum total stable mass of about 1.95 $M_{\odot}$ which gives a maximum limit on the possible dark matter fraction $f_{max}$ $\simeq$ 0.014. Radio timing observations of the pulsar J0348+0432 and phase-resolved optical spectroscopy of its white-dwarf companion lead to a precise pulsar mass measurement of 2.01 $\pm$ 0.04 $M_{\odot}$ which is by far the highest yet measured with this precision [@Demorest:2010abc; @Antoniadis:2013abc]. The maximum stable dark admixed quark star mass (1.95 $M_{\odot}$) falls slightly lower than the error limit of the highest measured pulsar mass.
Summary and Discussions
=======================
Pure quark matter is studied by using the MIT Bag model [@Alcock:1986abc]. Dark matter stars are studied here by considering the dark matter to be made up of fermionic particles of mass 100 GeV [@Kaplan:1992abc; @Nussinov:1985abc] with the assumption that these particles do not self-annihilate. The maximum stable mass of the dark matter star composed of strongly self-interacting particles is about 2.7$\times$ $10^{-2}$ $M_{\odot}$ at a radius of about 0.19 km while for free fermions, the mass is about 6.0$\times$ $10^{-5}$ $M_{\odot}$ at a radius of around 1 meter.\
The complete dimensional two-fluid TOV equations equations are solved to study the behaviour of admixed quark matter with dark matter. First, the equations are solved for a mixture of quark matter and free dark matter. The maximum stable mass of the admixed star is almost the same as that for a pure quark star for increasing dark matter fraction within the star. As the content of dark matter is gradually increased in the admixed star, the dark matter reaches its maximum stable configuration after which no admixed star configuration remains stable since the dark matter component collapses to form a black hole. For a quark star admixed with dark matter made of free gas of particles, the maximum possible mass of the stable configuration is approximately $M_{total}$ $\sim$ 2.01 $M_{\odot}$ with a dark matter content of around 0.63 $\times$ $10^{-4}$ $M_{\odot}$. A reduction in the maximum stable total mass is noted in case of a quark star admixed with dark matter star composed of strongly self-interacting fermions. The decrease was from about 2.01 $ M_{\odot}$ for zero dark matter content inside the star to about 1.95 $ M_{\odot}$ for the maximum allowed mass of strongly interacting dark matter in the star. The maximum dark matter content is around 2.64 $\times$ $10^{-2}$ $M_{\odot}$ at a maximum stable total mass of about 1.95 $M_{\odot}$. The maximum stable total mass in case of strongly self-interacting dark matter is seen to reduce linearly with increasing dark matter fraction in the star. The maximum accretion rate of dark matter by the quark star can be estimated to be about *$\frac{M_{int,max}}{\tau}$* $\sim$ 2.03$\times$ $10^{-12}$ $M_{\odot}$ per year, where $\tau$ $\sim$ 1.3 $\times$ $10^{10}$ years is the estimate for the age of the universe and $M_{int,max}$ is the maximum possible self-interacting dark matter content in the quark star. If the accretion rate is higher than this, the quark star will collapse .\
Acknowledgements
================
This work started started as a summer project of P.M at the Goethe University in Frankfurt and is supported by Deutscher Akademischer Austausch Dienst (DAAD). P.M thanks the Institute of Theoretical Physics for their hospitality. We thank Andreas Zacchi, Rainer Stiele and Chhanda Samanta for helpful discussions and a critical reading of the manuscript.\
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abstract: 'We propose a kind of novel topological quantum state of semimetals in a quasi-one-dimensional (1D) crystals BaMX$_3$ (M = V, Nb or Ta; X = S or Se) family by using symmetry analysis and first principles calculation. We find that in BaVS$_3$ the valence and conduction bands are degenerate in the $k_z=\pi/c$ plane ($c$ is the lattice constant along $\hat{z}$ axis) of the Brillouin Zone (BZ). These nodal points form a node-surface and they are protected by a nonsymmorphic crystal symmetry consisting of a two-fold rotation about the $\hat{z}$ axis and a half-translation along the same $\hat{z}$ axis. The band degeneracy in the node-surface is lifted in BaTaS$_3$ by including strong spin-orbit coupling (SOC) of Ta. The node-surface is reduced into 1D node-lines along the high-symmetry paths $k_x=0$ and $k_x$ = $\pm{\sqrt{3}}k_y$ on the $k_z=\pi/c$ plane. These node-lines are robust against SOC and guaranteed by the symmetries of $P6_3/mmc$ space group. These node-line states are entirely different from previous proposals which are based on the accidental band touchings. We also propose a useful material design for realizing topological node-surface and node-line semimetals.'
address:
- '$^{1}$ Department of Physics, Shaoxing University, Shaoxing 312000, China'
- '$^{2}$ Department of Materials, Nanjing University, Nanjing, China'
- '$^{3}$ Department of Physics, Harbin Institute of Technology, Harbin'
- '$^{4}$ School of Physics and Engineering, Sun Yat-sen University, Guangzhou 510275, China'
- '$^{5}$ Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China'
- '$^{6}$ Collaborative Innovation Center of Quantum Matter, Beijing 100190, China'
author:
- 'Qi-Feng Liang$^{1}$'
- Jian Zhou$^2$
- Rui Yu$^3$
- 'Zhi Wang$^{4}$'
- 'Hongming Weng$^{5, 6}$'
bibliography:
- 'refs.bib'
title: '[Node-Surface and Node-Line Fermions From Nonsymmorphic Lattice Symmetries]{}'
---
Introduction
============
Searching for new topological states of matters becomes an active field since the discovery of topological insulators (TI)[@Hasan_Kane_RMP_2010; @Qi_Zhang_RMP_2011]. Recently, much attention has been draw to the topological semimetal (TSM) due to the stimulation of successful design[@Weng_PRX2015] and experimental observations of Weyl semimetal (WSM) in transition-metal monophosphides [@Xu_2015; @LvBQ_PRX2015]. WSM are topological metallic states in which the Fermi surfaces are consisted of discrete two-fold-degenerate Weyl points. These Weyl points are topologically robust since they carry nontrivial chiral charges. The idea of TSMs has also been extended to node-line semimetal (NLS) hosting one-dimensional (1D) contacts of conduction and valence bands. [@burkov] The node-line fermions have been proposed in realistic materials such as Graphene-network[@WengHM_PRB2015], Cu$_3$N[@Kim_PRL2015], Cu$_3$PdN[@YuRui_PRL2015] and Ca$_3$P$_2$[@XieCava_APLM2015]. Experimental characterization of topological properties of NLSs was carried out for PbTaSe$_2$[@BianG_arxiv2015]. [One of the important topological properties of NLSs is that they support “drumhead" like flat surface band [@burkov; @WengHM_PRB2015] which may potentially enhance the superconductivity transition temperature[@Heikkila_arxiv2015]]{}.
In previous proposals, [@WengHM_PRB2015; @Kim_PRL2015; @YuRui_PRL2015] ring-like node-line contacts are fragile against spin-orbit coupling (SOC). Including SOC lifts the node-line contacts and drives the systems into topological insulators or WSM. Therefore, searching for stable NLSs robust to SOC [@FangC_PRB2015] is of great interest and importance. Recently, materials design principles involving a layer-stacking process of topological insulators and magnetic insulators are given for WSM [@Burkov_PRL2011] and NLSs[@Phillips_PRB2014]. Alternatively, in the present work we provide a new route to realize such exotic TSM state by arranging one-dimensional ionic chains in parallel. By using first principles calculations and symmetry consideration, we show that the quasi-one-dimensional crystal BaVS$_3$ exhibits a two-dimensional (2D) touching of valence and conduction bands, namely a node-surface, at the $k_z=\pi/c$ plane with $c$ being the $\hat{z}$-axis lattice constant. In a cousin compound BaTaS$_3$, the large SOC of Ta atom lifts the 2D degeneracy and reduces it into 1D node-lines which are robust to SOC. The node-lines found in the present work locate at the high-symmetry paths $k_x=0$ and $k_x=\pm{\sqrt{3}}k_y$ on the $k_z=\pi/c$ plane, demonstrating their purely symmetrical origin that is different from the accidental-band-degeneracy mechanism of previous node-lines[@BianG_arxiv2015; @FangC_PRB2015].
Crystal structure
=================
![ (Color online) (a) Unit cell of Hexagonal BaMX$_3$. Large (green), mediate (yellow) and small (brown) spheres denote the Ba, M and X atoms. (b) Splitting of $d$-orbitals under the crystal fields. The crystal field of MX$_3$-octahedron splits the five-fold $d$ orbitals into two-fold $e_g$ and three-fold $t_{2g}$ orbitals. The crystal-field of the hexagonal lattice further splits the three-folded $t_{2g}$ orbitals into two-folded $e'_g$ and $a_g$ orbitals. $a_g$ has the character of d$_{3z^2-r^2}$ orbital shown in the inset. (c) The Brillouin Zone of BaMX$_3$. Shadowed planes are the projected 2D Brillouin zone to different crystal facets. \[fig:crystal\] ](fig1-2.png){width="50.00000%"}
The BaMX$_3$ (M = V, Nb or Ta; X = S or Se) is a group of quasi-one-dimensional crystals adopting hexagonal structure with the space group of *P*6$_3$/*mmc* (No. 194)[@Gardner_BVS1969; @Gardner_BTaS1969] as shown in Fig. \[fig:crystal\]. M atoms are surrounded by octahedron of X atoms and these octahedrons form linear chains along the $\hat{z}$-axis by sharing common surfaces. Those chains are lined up and arranged into a trigonal lattice in the $x-y$ plane with Ba atoms filling the space between the chains. As the inter-chain distance is much larger than the intra-chain distance, these materials are considered as quasi-one-dimensional crystals. Each unit cell contains two formula units of BaMX$_3$ and thus has two M atoms, as labeled by A and B, respectively. Under the crystal field of the surrounding MX$_3$ octahedra, the $d$-orbitals of the two M atoms are split into two-fold $e_g$ orbital and three-fold $t_{2g}$ orbitals. The three-fold $t_{2g}$ orbitals is further split into a two-fold $e'_g$ orbitals and a single $a_g$ orbital by the triangle-crystal field of the ionic chain array \[see in Fig.\[fig:crystal\] (b) for this evolution of $d$-orbitals\]. The character of the wave-function mostly composed of $a_g$ manifold is $d_{3z^2-r^2}$ atomic orbital as shown in the inset of Fig.\[fig:crystal\] (b). With this quasi-1D crystal structure, the first principles calculations of BaMX$_3$’s electronic structure are performed by using the Vienna *ab initio* simulation package (VASP) [@VASP] with generalized gradient approximation [@PBE] in the projector augmented-wave method[@PAW_Blochl]. The Hubbard $U$ is simulated through Dudarev’s method [@Dudarev_LDAU] by setting $(U-J)_{\mathrm{V}}= 5.0$ eV and $(U-J)_{\mathrm{Ta}}= 2.0$ eV. Slightly changing of $U-J$ value will not change the conclusion of this work. Tight-binding Hamiltonians are constructed based on the maximally localized Wannier functions (MLWFs)[@Vanderbilt_RMP], and from these Hamiltonians surface band structures are calculated for slabs of BaMX$_3$.
Electronic Structures
=====================
Node-Surface in BaVS$_3$
------------------------
Firstly, we investigate the electronic structure of BaVS$_3$ in which the effect of SOC can be safely ignored. It is in a paramagnetic phase at room temperature. [It is reported that BaVS$_3$ undergoes a structure phase transition at 240 K and then enters into an insulating state through a metal-insulator phase transition at 70K with the nature of insulating state still being in debate. Here we are only interested in the high temperature paramagnetic state.]{} The corresponding band structure is shown in Fig. \[fig:nosoc\](a). By analyzing the separated contributions from different $d$-orbitals, we find the electronic states at the fermi-surface are mainly consisted of $a_g$ orbitals. Notably, one can find that the conduction and valence bands exactly stick together at the high-symmetry paths $\mathrm{A-L-H-A}$, while at other directions $\mathrm{\Gamma-M-K-\Gamma-A}$, they are split \[see in Fig.\[fig:crystal\](c) for the definitions of high-symmetry points\]. The dispersions of these bands are small in the $k_z=0$ and $k_z=\pi/c$ planes but large along the $k_z$ direction, indicating the quasi-one-dimensional nature of BaVS$_3$ crystal. By inspecting the band structure along a path off the high symmetry directions, namely $\mathrm{H-{AL}/{2}}$ with $\mathrm{{AL}/{2}}$ being the middle point of path $A-L$, one finds the degeneracy remains and he may guess the degeneracy take place throughout the $k_z=\pi/c$ plane. The conjecture is readily confirmed by the 3D plotting of band structure in Fig.\[fig:nosoc\] (b). We fix the value of $k_z$ being $\pi/c$ but vary $k_x$ and $k_y$. It is clearly seen that the conduction and valence bands are exactly overlapping at the $k_z=\pi/c$ plane, which is the node-surface with band touching points, or equivalently the Dirac nodal points, in it. Slightly deviating off this plane, $i.e.$, $k_z=0.9\pi/c$, the degeneracy is split. The nodal points are not on the same energy level and the Fermi level cuts the node-surface only at a continuous 1D-line on the $k_z=\pi/c$ plane, which connects the hole and electron pockets of Fermi surface indicated with different colors shown in Fig.\[fig:nosoc\](c).
![ (Color online) Node-surface in BaVS$_3$ under room-temperature paramagnetic phase with negligible spin-orbit coupling. (a) Band dispersions of BaVS$_3$. The orbital nature are represented by circle (red), square (green) and triangle (blue) symbols for the $d_{3z^2-r^2}$,$d_{x^2-y^2}$ and $d_{xy}$ orbitals. (b) 3D plot of the bands in $k_z=\pi/c$ (left) and $k_z=0.9\pi/c$ (right) plane. (c) Fermi-surface of BaVS$_3$. Dashed lines highlight the contact boundary between two partition of the fermi surface around $k_z=\pi/c$. \[fig:nosoc\] ](fig2-2.png){width="50.00000%"}
To the best of our knowledge, it is the first time that a node-surface with conduction and valence band-touching nodes is revealed in a realistic material. From band-theory point of view, one would expect that the two $a_g$ orbitals in BaVS$_3$ unit cell may form bonding and anti-bonding bands, and the two valence electrons provided by the V$^{+4}$ ions will fully occupy the bonding band and leave the anti-bonding one empty, making the material a band insulator. This naive band-theory picture, however, fails in the present case due to the nonsymmorphic crystal symmetry contained in the $P6_3/mmc$ group [@Parameswaran_NatPhys2015]. The nonsymmorphic symmetry is a crystal symmetry which can not be decomposed into the product of a lattice translation and a point-group operation.
It has long been known[@Konig_PRB] that nonsymmorphic symmetries keep energy bands to “stick together" at the boundary of BZ. The node-surface presented in our work therefore is a faithful demonstration of this physics as checked by the symmetry analysis below.
The participated symmetries to protect the degeneracy at $k_z=\pi/c$ plane are the time-reversal symmetry ${\mathcal{T}}=\mathcal{K}$ with $\mathcal{K}$ being the complex conjugation (in case without spin-oribt cpupling), space inversion $\mathcal{I}$ and the skew axial symmetry $\mathcal{S}_z$= {$C_{2{z}}|\mathbf{T}_z={\mathbf{c}}/{2}$}, a nonsymmorphic symmetry combining a half-vector translation along the $\hat{c}$ axis and a two-fold rotation about the $\hat{z}$ axis. From these symmetries, two compound symmetries, $\mathcal{C}=\mathcal{IT}$ and $\mathcal{S}=\mathcal{IS}_z$ are constructed. $\mathcal{C}$ is preserved at any $\mathbf{k}$-point of BZ, while $\mathcal{S}$ is respected at the $k_z=0$ and $k_z=\pi/c$ planes. Applying $\mathcal{S}$ twice on the lattice can bring the system back to its starting point, one has $\mathcal{S}^2=1$ and the corresponding eigen-values of $\mathcal{S}$ are $\pm 1$, which can be used to label the eigen-states of the Hamiltonian. Due to the anti-commutation of $\mathcal{C}$ and $\mathcal{S}$ on $k_z=\pi/c$ plane \[see Appendix AI for the derivation\], the action of $\mathcal{C}$ switches each eigen-state to its degenerate partner of opposite $\mathcal{S}$-label, ensuring the two-fold degeneracy of energy bands on the entire $k_z=\pi/c$ plane as found in Fig. \[fig:nosoc\] (b). The details of the symmetry consideration and the derivation of the actions of the symmetry operators are summarized in Appendix AI.
![ (Color online) Node-lines in BaTaS$_3$ with strong spin-orbit coupling. (a) Band structure of BaTaS$_3$. The orbital characteristic of a$_g$ band is highlighted by circle (red) symbols. (b) 2D plot of the bands in $k_z=\pi/c$ plane. The gradually varied colors indicate the splitting of the bands. The darkest lines along $A-L$ paths are node-lines denoted in the bottom contour plot by dashed lines. (c) Full view of Fermi surface of BaTaS$_3$. The contact points between the two partitions of the fermi surface around $k_z=\pi/c$ are highlighted by solid circles (white). \[fig:soc\] ](fig3-2.png){width="50.00000%"}
Symmetry Guaranteed Nodal-Lines in BaTaS$_3$
--------------------------------------------
If SOC is included, the above discussion is not valid and the node-surface of $k_z=\pi/c$ plane will not persist. Luckily for BaVS$_3$, the effective SOC splitting in V’s $a_g$ bands is vanishingly small (about $1\sim2$ meV) and thus the conduction and valence bands can still be considered as degenerate at room temperature paramagnetic phase. However, for heavier M = Nb, and Ta, the SOC is large and its effect has to be taken into account. In Fig.\[fig:soc\](a), we show the band structure of BaTaS$_3$ with SOC turned on. The inclusion of SOC lifts the degeneracy of bands at a general $\mathbf{k}$-point in the the node-surface $k_z=\pi/c$ plane, as seen from the splitting of energy bands along $\mathrm{L-H-A}$ paths. Surprisingly, one can still find that along a special path $\mathrm{A-L}$, the energy bands remain four-fold degenerate (counting both spin and orbital degrees of freedom), forming a so called node-line structure with Dirac nodes along it. In Fig.\[fig:soc\](b) we also plot a 3D band structure in $k_z=\pi/c$. Gradual colors are used to indicate the energy splitting in the conduction and valence bands. In this plot and its contour projection onto the bottom $k_z=\pi/c$, three node-lines related by $C_{3z}$ rotation symmetry are found along the high-symmetry paths $k_x=0$ and $k_x=\pm{\sqrt{3}}k_y$.
The existence of node-lines also changes the Fermi surface around the $k_z=\pi/c$ plane as shown in Fig.\[fig:soc\](c). Due to the non-zero band dispersion of the node-lines, the Fermi level cuts these lines at six discrete points. In the 3D Fermi surface plot, it is readily found that the Fermi lines connecting two different colors in the spinless case \[Fig.\[fig:nosoc\](c)\] now evolve into six separated contacting points on the $A-L$ paths. Comparing with the Fermi-surface of BaVS$_3$ in Fig.\[fig:nosoc\](c), the additional ellipsoid-shaped Fermi surface surrounding the $\Gamma$ point is from the two $e'_g$ bands as clearly shown in the band structure in Fig.\[fig:soc\](a).
Now let us discuss about the symmetry protection of the four-fold degenerate node-lines in presence of SOC. The derivations of the actions of symmetry operators and their associated commutation relations are given in Appendix AII. In the spinful case, time reversal operator is expressed as $\mathcal{T}=is_y\mathcal{K}$ and thus $\mathcal{T}^2=-1$. The skew axis $\mathcal{S}_z$ now also acts on the spin space, $\mathcal{S}_z:(s_x,s_y,s_z)\mapsto(-s_x,-s_y,s_z)$. Applying it twice rotates the spin by $2\pi$, giving a minus sign for a spin-1/2 system. Since acting $\mathcal{S}=\mathcal{IS}_z$ twice on the real space brings the system back to its origin, one finds that the square of $\mathcal{S}$ becomes $\mathcal{S}^2=-1$ at its invariant plane $k_z=\pi/c$. Notice that $\mathcal{I}$ does not act on spin space. Then the eigen-states of Bloch Hamiltonian can be labeled by the eigen-values $\pm i$ of $\mathcal{S}$, $\mathcal{S}|\phi^{\pm}(\mathbf{k})\rangle=\pm i|\phi^{\pm}(\mathbf{k})\rangle$. Unlike the spinless case, applying $\mathcal{C}$ on $|\phi^{\pm}(\mathbf{k})\rangle$ translates it to its Kramer partner with the same $\mathcal{S}$-label. Thus, the two Kramer pairs with opposite $\mathcal{S}$-label are not related and generally the bands should be two-fold degenerate. Therefore, extra symmetries are needed to provide the protection of the node-lines. We find the mirror symmetry $\mathbf{M}_x$ plays this role. $\mathbf{M}_x$ acts both on the real space and spin space. It takes an anti-commutation relation with $\mathcal{S}$ on the intersection line of its invariant plane $k_x=0$ and the invariant plane of $\mathcal{S}$, $k_z=\pi/c$ . Because of $\{\mathbf{M}_x, \mathcal{S}\}=0$, applying $\mathbf{M}_x$ on $|\phi^{\pm}(\mathbf{k})\rangle$ will translate the state to a degenerate state with opposite $\mathcal{S}$-label. Therefore, with the help of $\mathbf{M}_x$, the two Kramer pairs of opposite $\mathcal{S}$-eigenvalues are now related by $\mathbf{M}_x$ at the high-symmetry path $k_x=0$ on $k_z=\pi/c$, proving the existence of the four-fold degenerate node-lines.
The node-lines bring about surface states on the surface of BaTaS$_3$, which is another demonstration of the nontrivial topology of the node-line. In Fig.\[fig:edge\](a) we plot the surface electronic structure of a 20-unit-cell-thick BaTaS$_3$ slab with {-110} facet. From the figure, one can see segments of thick (red) lines inside local “band gap" which are mainly contributed by the surface layers. When the surface bands immerse into bulk bands, the hybridization between the surface and the bulk bands will smear out the surface contribution. By using recursive Green’s function method, we calculated the surface Green’s function at the {-110} surface and the obtained density of states are plotted in Fig.\[fig:edge\](b) for Fermi energy $E_F$ = 0.0, 0.18 and 0.27 eV. Bright lines high-lighting the surface states’ contribution can be easily seen in these figures. On the other hand, drumhead-like flat surface bands observed in Graphene-network[@WengHM_PRB2015] and Cu$_3$PdN [@YuRui_PRL2015], however, is not found here due to the large dispersions of the node-lines.
The observation of surface states at the slab’s surface is unexpected because the symmetries that protect the node-lines are broken at the surfaces. This is understood from the nontrivial Berry phase $\pi$ associated with each node-line[@BianG_arxiv2015]. In order to verify this point, we construct an effective four-band Hamiltonian for BaTaS$_3$ in which the two $a_g$ orbitals of Ta are considered \[see in Appendix B for the construction of the Hamiltonian\]. The low-energy Hamiltonian around a point on the node-line $\mathbf{k}_0=(0,k_{y},\pi/c$) is written as, $$\begin{aligned}
H_{\mathbf{q}}=\epsilon(\mathbf{k})+q_z\tau_y s_x+q_z\tau_x+q_x\tau_ys_z,
\label{eq:Hq_main}\end{aligned}$$ with $\mathbf{q}=(q_x,q_y,q_z)$ being small deviation from $\mathbf{k}_0$. $\tau$ denotes the orbital degree of freedom and the dependency of $H_{\mathbf{q}}$ on $q_y$ is contained in $\epsilon(\mathbf{k})$ based on the symmetry consideration. As $\epsilon(\mathbf{k})$ is not important for the solutions of (\[eq:Hq\_main\]), we omit it in the following discussion. The effective model (\[eq:Hq\_main\]) has a symmetry $\tau_xs_z$ whose eigenvectors build up a unitary matrix that block-diagonalizes $H_{\mathbf{q}}$. By using an isospin-rotation and re-scaling of $q_z$, one can finally rewrite the Hamiltonian in the nonzero sub-block into the form ${H}_{\tilde{\mathbf{q}}}=\left[\begin{array} {cc} 0 & \tilde{q}_z+i\tilde{q}_x \\ \tilde{q}_z-i\tilde{q}_x & 0\\ \end{array}\right]$. When $\mathbf{\tilde{q}}$ transverses along a closed loop circling the node-line, the wavefunction of the occupied state accumulates a nonzero Berry phase $\pi$ (mod $2\pi$), demonstrating the nontrivial topology of the node-line.
![ (Color online) Band structure of a 20-unit-cell-thick slab of BaTaS$_3$ with {-110} faced.(a) Surface bands highlighted by the thickness and color of the lines scaled by the weight of surface contribution to the eigenstates. (b) Surface local density of states at different Fermi energies, E$_F$ = 0.0 eV (left), 0.18 eV (middle) and 0.27 eV (right). \[fig:edge\] ](fig4-2.png){width="50.00000%"}
Parallel Ionic Chains: Material design for Node-surface and Node-line States
----------------------------------------------------------------------------
Our work provides a principle for novel materials design: creating TSMs through arrangement of ionic chains in parallel. Previously, Balents *et al.* [@Burkov_PRL2011; @burkov] and Phillips *et al.*[@Phillips_PRB2014] have considered a layer-stacking approach to construct TSMs. In their scheme, the use of TI layers provides Dirac points from the TI layers’ surface states \[see in Fig.\[fig:design\](b)\]. When the TI layers are stacked along the $\hat{z}$ axis, the Dirac points transverses along the $k_z$ direction and forms a node-line in the 3D BZ. The node-line will persist or be split into discrete Weyl points [@Burkov_PRL2011] dependent on the details of the inter-layer couplings and the magnetization of the magnetic layers[@Phillips_PRB2014].
![ (Color online) Material design for TSMs. (a) (top) parallel arrangement of 1D ionic chains. Inset shows the band structure of a single ionic chains which contains two crossing point at the boundary of BZ, $k_z$=$\pm\pi/c$. (bottom) The two-fold crossing point forms a 2D node-surface degeneracy when the chains are arranged into lattice in the $x$-$y$ plane. (b) (top) Stacking of TI layers to realize a topological semi-metal. Inset shows the 2D Dirac cone band structure of the TI surface state. (bottom) The Dirac point forms a node-line when the layers are stacked up in the $z$ direction. \[fig:design\] ](fig5-2.png){width="50.00000%"}
Alternatively, here we propose to construct TSMs by parallel-arrangement of ionic chains. The method is schematically shown in Fig. \[fig:design\](a). In BaMX$_3$, the individual MX$_3$ chain also respects the skew symmetry $\mathcal{S}_z$, which guarantees an energy band crossing at the BZ’s boundary, $k_z=\pm\pi/c$. The parallel-arrangement of the ionic chains then introduces inter-chain coupling. Due to the inter-chain coupling, the crossing point translates along the $k_x$ and $k_y$ directions and builds up a 2D band touching at the 3D BZ’s boundary \[see the bottom of Fig.\[fig:design\](b)\]. The inclusion of hopping parameters along $x-y$ plane ususually breaks the degeneracy of the Dirac point at a general $\mathbf{k}$ point where $\mathcal{S}_z$ is broken. Luckily for the present case this nonsymmorphic symmetry is respected at the whole $k_z=\pi/c$ plane and the node-surface persists. When SOC is switched on, the node-surface degeneracy is lifted generally. However, mirror planes containing the $\hat{z}$ axis are presented in the $P6_3/mmc$ group. The node-surface reduces into node-lines at the high-symmetry paths where the mirror planes intersect with node-surface. This is a new route of realizing TSMs with peculiar symmetry-guaranteed node-surface or node-lines.
Discussions and Conclusions
===========================
We notice that the general physics of band-sticking in nonsymmorphic crystals has already been pointed out by Parameswaran *et al.* [@Parameswaran_NatPhys2015] and some first principles calculations have been done on the family of BaMX$_3$[@Mattheiss_SSC1995; @Whangbo_JSSC2002; @JiangXF_PRB2004]. However, the topological features of such 2D degeneracy of conduction and valence bands are revealed for the first time, to the best of our knowledge. The node-lines induced by strong SOC from node-surface without SOC is quite different from those proposed previously[@WengHM_PRB2015; @Kim_PRL2015; @YuRui_PRL2015; @XieCava_APLM2015], in which the node-lines are only stable in the absence of SOC. Some proposals claim the stability of node-lines with the inclusion of SOC, such as those in SrIrO$_3$ [@FangC_PRB2015] and PbTaSe$_2$ [@BianG_arxiv2015]. These node-lines are off the high-symmetry paths and originate from accidental band touching and no guarantee of their existence. However, the node-lines in the present case are four-fold degenerate and guaranteed by happening at the high-symmetry paths.
We also notice that by decreasing the temperature, BaVS$_3$ experiences firstly a structure phase transition from the room temperature hexagonal phase to the low temperature orthogonal phase, and then a paramagnetic-to-anti-ferromagnetic phase transition under which the material becomes a gapped state. In BaTaS$_3$ and BaTaSe$_3$[@Donohue_JSSC1974; @Ohtani_MRB2004], semi-conductor-metal transition are observed at low temperature, however, no clear signature of structure or magnetic transition is detected and the nature of semiconductor-metal transition is still under debate. Whether this phase-transition is topological in nature [@Parameswaran_NatPhys2015] or not and what is the relation between the observed phase transitions and the node-surface and node-line discovered in the present work need further investigations.
In summary, we have found a 2D node-surface at the $k_z=\pi/c$ plane in the band structure of BaVS$_3$ with negligible SOC ( about 1$\sim$2 meV). Such novel TSM is protected by the non-symmorphic symmetry $\mathcal{S}_z$. In its family compound BaTaS$_3$ with strong SOC, node-surface evolves into node-lines along three high symmetry paths on the $k_z=\pi/c$ plane. The symmetries that protect the node-lines are the nonsymmorphic skew axial symmetries $\mathcal{S}_z$, mirror symmetry $\mathbf{M}_x$, inversion symmetry $\mathcal{I}$ and time-reversal symmetry $\mathcal{T}$. Two-orbital effective model are constructed to compute the nontrivial Berry phase associated with each node-line. The physics of the node-lines originated from the nonsymmorphic symmetry is analyzed. Surface states are observed in a slab of BaTaS$_3$ even though the large dispersions of node-lines and the breaking of crystal symmetries at the surface. Our work paves a way to design materials with peculiar energy band degeneracy through arrangement of ionic chains in parallel.
*Acknowledgements—* The work is supported by National Natural Foundation of China (NFSC) (Grants No.11574215, No. 11274359 and No. 11422428). Q.F.L acknowledges the support from SRF for ROCS, SEM. H. M. W is also supported by the National 973 program of China (Grants No. 2011CBA00108 and No. 2013CB921700), and the “Strategic Priority Research Program (B)" of the Chinese Academy of Sciences (Grant No. XDB07020100).
Appendix
========
Symmetry Analysis
-----------------
### The spinless case
Let us prove that in a crystal preserving the time-reversal-symmetry $\mathcal{T}$, inversion symmetry $\mathcal{I}$ and a two-fold skew-axis along $\hat{z}$, $\mathcal{S}_z:(x,y,z,t)\mapsto (-x,-y,z+\frac{1}{2},t)$, the energy bands at $k_z=\pi/c$ plane will be two-fold degenerated if the spin effect is excluded. By using these three symmetries, one can compose two compound symmetries, $\mathcal{C}=\mathcal{IT}$ and $\mathcal{S}=\mathcal{IS}_z$. In the real space, $\mathcal{C}$ and $\mathcal{S}$ act as, $$\begin{aligned}
\mathcal{C}:(x,y,z,t)\mapsto(-x,-y,-z,-t)\nonumber\\
\mathcal{S}:(x,y,z,t)\mapsto(x,y,-z-\frac{1}{2},t).\nonumber\\
\label{eq:CS}\end{aligned}$$ In momentum space, one easily finds that all the $\mathbf{k}$ points are invariant points under the action of $\mathcal{C}$. On the other hand, $\mathcal{S}$ translates $(k_x,k_y,k_z)$ to $(k_x,k_y,-k_z)$, which means $k_z=0$ and $k_z=\pi/c$ are the two invariant planes of $\mathcal{S}$. Applying $S$ twice, the coordinates in the real space return back and one finds $\mathcal{S}^2=1$, which means the eigen-values of $\mathcal{S}$ are $\pm 1$ and one can use them to label the Bloch states $|\psi(\mathbf{k})\rangle$ of the system’s Hamiltonian by $\mathcal{S}|\psi^{\pm}(\mathbf{k})\rangle=\pm|\psi^{\pm}(\mathbf{k})\rangle$.
We find that under the action of $\mathcal{C}$, each eigen-state $|\psi^{\pm}(\mathbf{k})\rangle$ is switched to a degenerate state of an opposite $\mathcal{S}$-label. Before showing this, we first prove that $\mathcal{C}$ and $\mathcal{S}$ take an anti-commutation relation at $k_z=\pi/c$. By using Eq.(\[eq:CS\]), one obtain, $$\begin{aligned}
\mathcal{SC}:(x,y,z,t)\mapsto(-x,-y,z-\frac{1}{2},-t)\nonumber\\
\mathcal{CS}:(x,y,z,t)\mapsto(-x,-y,z+\frac{1}{2},-t),
\label{eq:CS_SC}\end{aligned}$$ from which one finds in $\mathbf{k}$-space, $$\begin{aligned}
\mathcal{CS}=\mathbf{T}_{\mathbf{c}}\mathcal{SC}=e^{i{k_z}{c}}\mathcal{SC},
\label{eq:CS-SC}\end{aligned}$$ with $\mathbf{T}_{\mathbf{c}}$ being the translation along the $\hat{z}$ axis by one primary vector $\mathbf{c}$. Therefore the anti-commutator of $\mathcal{C}$ and $\mathcal{S}$ becomes zero, $\{\mathcal{C},\mathcal{S}\}=0$ at $k_z=\pi/c$. With this anti-commutation relation, one can prove that $\mathcal{C}$ relates the eigenstates to their degenerate partners of opposite $\mathcal{S}$-label as follows, $$\begin{aligned}
\mathcal{S}\mathcal{C}|\psi^{\pm}(\mathbf{k})\rangle=e^{-i\frac{\pi}{c}c}\mathcal{C}\mathcal{S}|\psi^{\pm}(\mathbf{k})\rangle=\mp\mathcal{C}|\psi^{\pm}(\mathbf{k})\rangle.
\label{eq:label}\end{aligned}$$ Then on the $k_z=\pi/c$ plane, one proves that each band is two-fold degenerated. We should stress here that for the case of no spin, actually the inversion $\mathcal{I}$ is not necessary for the degeneracy at $k_z=\pi/c$. We can use an alternative compound symmetry $\mathbf{\Theta}=\mathcal{TS}_z$ to protect the degeneracy. It is easy to check that it takes $\mathbf{\Theta}^2=-1$ at $k_z=\pi/c$ plane, which ensures a Krammer degeneracy with the two related states labeled by eigen-values $\pm i$ of $\mathbf{\Theta}$. However, in the following spinful case, $\mathbf{\Theta}$ is not enough to protect the node-lines. For the purpose of consistence with the spinful case below, we use the same compound symmetries $\mathcal{IS}_z$ and $\mathcal{IT}$ in the spinless case.
### the effect of spin-orbit coupling
Including the spin degree of freedom and the effect of SOC generally splits the four-fold degeneracy (counting both spin and orbit degree of freedom) at $k_z=\pi/c$. In this spinful case, $\mathcal{S}_z$ also acts on the spin space, $\mathcal{S}_z:(s_x,s_y,s_z)\mapsto(-s_x,-s_y,s_z)$. The square of $\mathcal{S}_z$ then rotates the spins by $2\pi$, contributing a minus sign for the spin$-\frac{1}{2}$ system. Therefore the square of $\mathcal{S}=\mathcal{IS}_z$ becomes $\mathcal{S}^2=-1$ and $\mathcal{S}$ now has eigen-values $\pm i$, which again can be used to label the eigen-states of Bloch Hamiltonian, $\mathcal{S}|\phi^{\pm}(\mathbf{k})\rangle=\pm i|\phi^{\pm}(\mathbf{k})\rangle$. Unlike the spinless case, the action of $\mathcal{C}=\mathcal{IT}$ on the eigen-states $|\phi^{\pm}(\mathbf{k})\rangle$ only switches it to a partner state of the same $\mathcal{S}$-label, even though the anti-commutation relation of $\mathcal{C}$ and $\mathcal{S}$ retains. This point is easily checked as, $$\begin{aligned}
\mathcal{S}\mathcal{C}|\phi^{\pm}(\mathbf{k})\rangle&=&e^{-ik_zc}\mathcal{C}\mathcal{S}|\phi^{\pm}(\mathbf{k})\rangle\nonumber \\
&=& -\mathcal{C}\left(\pm i|\phi^{\pm}(\mathbf{k})\rangle \right)\nonumber\\
&=& \pm i \mathcal{C}|\phi^{\pm}(\mathbf{k})\rangle.
\label{eq:label_soc}\end{aligned}$$ Eq.(\[eq:label\]) is used in the first line of Eqs.(\[eq:label\_soc\]) and the last equality in Eq.(\[eq:label\_soc\]) is obtained based on that time reversal operator $\mathcal{T}=i s_y\mathcal{K}$ contained in $\mathcal{C}$ conjugates $i$ to $-i$.
As $\mathcal{C}$ only related the eigen-state to its partner with the same $\mathcal{S}$-label, we only obtain two Krammers pairs that can not be related by symmetry $\mathcal{S}$ and $\mathcal{C}$ only. To explain the four-degenerated node-lines of BaTaS$_3$, we need other symmetries. We find it is the mirror symmetry $\mathbf{M}_x:(x,y,z)\mapsto(-x,y,z)$ that provides the needed protection for these node-lines. The implication is that the high symmetry path $k_x=0$ at $k_z=\pi/c$ is just the intersecting line of the invariant plane $k_x=0$ of $M_x$ and the invariant plane $k_z=\pi/c$ of $\mathcal{S}$. On this invariant line, action of $\mathbf{M}_x$ will transform the eigen-state $|\phi^{\pm}(\mathbf{k})\rangle$ to a partner of an opposite $\mathcal{S}$-label, which then related the two Krammer pairs and ensures a four-fold degenerate node-line. Before clarifying this point, we first prove that $\mathbf{M}_x$ anti-commutes with $\mathcal{S}$. It is easily checked that in the real space, action of $\mathcal{S}\mathbf{M}_x$ and $\mathbf{M}_x\mathcal{S}$ on the coordinate lead to the same result. The anti-commutation relation of $\mathbf{M}_x$ and $\mathcal{S}$ comes from their action on the spin space. In spin space, $\mathcal{S}$ reflects $s_x$ and $s_y$ but keeps $s_z$ invariant, from which one can express its action by $\exp\left(i\pi/2\hat{s}_z\right)=i\hat{s}_z$. Similarly, the action of $\mathbf{M}_z$ on the spin is expressed by $\exp\left(i\pi/2\hat{s}_x\right)=i\hat{s}_x$. It is the action of $\mathbf{M}_x$ and $\mathcal{S}$ on the spin space that ensure the anti-commutation relation, $\{\mathcal{S},\mathbf{M}_x\}=0$. By applying $\mathbf{M}_x$ on the eigenstate $|\phi^{\pm}(\mathbf{k})\rangle$, one then finds, $$\begin{aligned}
\mathcal{S}\mathbf{M}_x|\phi^{\pm}(\mathbf{k})\rangle&=&-\mathbf{M}_x\mathcal{S}|\phi^{\pm}(\mathbf{k})\rangle \nonumber\\
&=&-\mathbf{M}_x\left(\pm i|\phi^{\pm}(\mathbf{k})\rangle\right)\nonumber\\
&=&\mp i\mathbf{M}_x|\phi^{\pm}(\mathbf{k})\rangle.
\label{eq:label_m}\end{aligned}$$ Eqs.(\[eq:label\_soc\]) and (\[eq:label\_m\]) finally finish the proving that at $k_x=0$ on the $k_z=\pi/c$ plane there must be a four-fold-degenerated node-line guaranteed by symmetries $\mathcal{S}=\mathcal{IS}_z$, $\mathcal{C}=\mathcal{IT}$ and $\mathbf{M}_x$. The node-lines at $k_x=\pm\sqrt{3}k_y$ on the $k_z=\pi/c$ are then resulted from the $C_{3z}$ rotation.
Effective Hamiltonian
---------------------
There are two active d${3z^2-r^2}$ orbits hosted on Ta atom in the unit cell and they compose the electronicstates around the fermi surface. When SOC is turned off, we can use these two orbits, denoted by A and B, to construct a $2\times2$ Hamiltonian at each $\mathbf{k}$-point in the momentum space, $$\begin{aligned}
\hat{H}_0(\mathbf{k})=\left[
\begin{array}{cc}
H_{AA}(\mathbf{k}) & H_{AB}(\mathbf{k}) \\
H_{AB}^{*}(\mathbf{k}) &H_{BB}(\mathbf{k}) \\
\end{array}
\right].\label{eq:H_0}\end{aligned}$$ Here $|\eta,\mathbf{k}\rangle=\frac{1}{\sqrt{N}}\sum_n e^{i \mathbf{k} \cdot\mathbf{r}_{n\eta}}\psi(\mathbf{r}-\mathbf{r}_{n\eta})$ are the two basis functions with $\eta=A,B$ denoting the orbits. $\psi(\mathbf{r}-\mathbf{r}_{n\eta})$ is the local $d3z^2-r^2$ wave function centered on atom site $\mathbf{r}_{n\eta}$ in the n$^{th}$ unit cell. For the spinful Hamiltonian, We need to take the spin degree of freedom into consideration and thus define four basis functions, $|\eta,\mathbf{k},\sigma\rangle$ with $\sigma=\uparrow $ or $\downarrow$ denoting the spin direction. The Hamiltonian is then enlarged to a $4\times4$ size, $$\begin{aligned}
H(\mathbf{k})=\left[
\begin{array} {cccc}
H_{AA}^{\uparrow\uparrow}(\mathbf{k}) &H_{AB}^{\uparrow\uparrow}(\mathbf{k}) &H_{AA}^{\uparrow\downarrow}(\mathbf{k}) &H_{AB}^{\uparrow\downarrow}(\mathbf{k}) \\
&H_{BB}^{\uparrow\uparrow}(\mathbf{k}) &H_{BA}^{\uparrow\downarrow}(\mathbf{k}) &H_{BB}^{\uparrow\downarrow}(\mathbf{k}) \\
& & H_{AA}^{\downarrow\downarrow}(\mathbf{k}) &H_{AB}^{\downarrow\downarrow}(\mathbf{k}) \\
\dag & & & H_{BB}^{\downarrow\downarrow}(\mathbf{k})
\label{eq:H}
\end{array}
\right].\nonumber\\\end{aligned}$$
The time-reversal operator $\mathcal{T}$ now becomes $\mathcal{T}=i\hat{s}_y\mathcal{K}$ with $\hat{s}_y=\left[\begin{array}{cc} 0& i\\-i& 0 \end{array}\right]$ being the Pauli matix acting on the spin space. It reads, $$\begin{aligned}
\mathcal{T}|\eta,\mathbf{k},\sigma \rangle & = & \mathrm{sng}(\sigma) |\eta,-\mathbf{k},\bar{\sigma} \rangle \nonumber\\
\mathcal{T}H(\mathbf{k}) & = & H(-\mathbf{k})\mathcal{T}.
\label{eq:T_psi2}\end{aligned}$$ with the sign function sng$(\sigma)$ being = $+$1 ($-$1) for spin $\uparrow$ ($\downarrow$) and $H(\mathbf{k})=e^{-i\mathbf{k}\cdot\mathbf{r}}H(\mathbf{r})e^{i\mathbf{k}\cdot\mathbf{r}}$. For the entries of Hamiltonian (\[eq:H\]), one finds, $$\begin{aligned}
&H&_{\eta_1\eta_2}^{\sigma_1\sigma_2} (\mathbf{k}) = \langle \eta_1,\mathbf{k},\sigma_1| H(\mathbf{k}) | \eta_2,\mathbf{k}, \sigma_2\rangle \nonumber \\
& =& -\mathrm{sgn}(\sigma_2) \langle\eta_1,\mathbf{k},{\sigma}_1| H(\mathbf{k}) \mathcal{T} | \eta_2,-\mathbf{k}, \bar{\sigma}_2\rangle \nonumber \\
&= & -\mathrm{sgn}(\sigma_2) \langle\eta_1,\mathbf{k},{\sigma}_1|\mathcal{T} \left[H(\mathbf{-k}) | \eta_2,-\mathbf{k}, \bar{\sigma}_2\rangle \right] \nonumber \\
&=& \mathrm{sgn}(\sigma_2)\mathrm{sgn}(\sigma_1) \langle\mathcal{T} (\eta_1,-\mathbf{k},\bar{\sigma}_1)|\mathcal{T} \left[H(-\mathbf{k}) | \eta_2,-\mathbf{k}, \bar{\sigma}_2\rangle \right] \nonumber \\
&=& \mathrm{sgn}(\sigma_1)\mathrm{sgn}(\sigma_2) \left[\langle\eta_2,-\mathbf{k},\bar{\sigma}_2| H(-\mathbf{k})\right] | \eta_1,-\mathbf{k}, \bar{\sigma}_1\rangle \nonumber \\
&=& \mathrm{sgn}(\sigma_1)\mathrm{sgn}(\sigma_2) H_{\eta_2\eta_1}^{\bar{\sigma}_2\bar{\sigma}_1} (-\mathbf{k}).
\label{eq:T_spin}\end{aligned}$$ To obtain the second and forth lines of the above equation, we use the transformation (\[eq:T\_psi2\]) of wave function for the right and left ket, respectively. The transformation of Hamiltoina under $\mathcal{T}$ is used to obtain the third line. To get the fifth line, we use the property of time-reversal symmetry $\langle \mathcal{T}\phi|\mathcal{T}|\psi\rangle=\langle\psi|\phi\rangle$.
The $P6_3/mmc$ group also contains an inversion symmetry $\mathcal{I}:(x,y,z)\mapsto(-x,-y,-z)$. ${\mathcal{I}}$ does not flip the spin and orbit indices, and the wave functions and Hamiltonian are transformed as, $$\begin{aligned}
{\mathcal{I}}|\eta,\mathbf{k},\sigma \rangle & = & |\eta,-\mathbf{k}, {\sigma} \rangle \nonumber\\
{\mathcal{I}}H(\mathbf{k}) & = & H(-\mathbf{k}){\mathcal{I}}.
\label{eq:I_psi}\end{aligned}$$ Applying Eq. (\[eq:I\_psi\]) one finds the consequence of applying $\mathcal{I}$ onto the elements of Hamiltonian (\[eq:H\]), $$\begin{aligned}
H_{\eta_1\eta_2}^{\sigma_1\sigma_2}(k_x,k_y,k_z)=H_{ {\eta}_1 {\eta}_2}^{\sigma_1\sigma_2}(-k_x,-k_y,-k_z).
\label{eq:I_spin}\end{aligned}$$
Applying Eqs.(\[eq:T\_spin\]) and (\[eq:I\_spin\]) successively one obtains $$H_{\eta_1\eta_2}^{\sigma_1\sigma_2}(k_x,k_y,k_z)=\mathrm{sng}(\sigma_1)\mathrm{sng}({\sigma}_2)H_{\eta_2\eta_1}^{\bar{\sigma}_2\bar{\sigma}_1}(k_x,k_y,k_z) ,
\label{eq:IT}$$ which reduces the Hamiltonian (\[eq:H\]) to the below form, $$\begin{aligned}
H({\mathbf{k}})=\left[
\begin{array} {cccc}
a({\mathbf{k}}) & f({\mathbf{k}}) & 0 &g({\mathbf{k}}) \\
&b({\mathbf{k}}) & -g({\mathbf{k}}) & 0 \\
& & a({\mathbf{k}}) & f^{*}({\mathbf{k}})\\
\dag & & & b({\mathbf{k}})
\end{array}
\right],
\label{eq:H_IT}\end{aligned}$$ where $a({\mathbf{k}})$ and $b({\mathbf{k}})$ are real functions, and $f({\mathbf{k}})$ and $g({\mathbf{k}})$ are complex functions.
Hamiltonian (\[eq:H\_IT\]) is mathematically equivalent to a linear combination of five Dirac matrices $\Gamma_{a}$ together with the unit matrix $\Gamma_0$. The eigenvalues of Hamiltonian (\[eq:H\_IT\]) become four-fold degenerate only if the five coefficients of the Dirac matrices become zero, that is to say $a(\mathbf{k})-b(\mathbf{k}) = f(\mathbf{k}) = g(\mathbf{k}) =0$, which, however, does not hold at a general $\mathbf{k}$ point. Below we prove that this four-fold-degenerating condition readily fulfills at three high-symmetries paths, $k_x=0$ and $k_x=\pm{\sqrt{3}}k_y$ on the $k_z=\pi/c$ plane with the help of $\mathcal{S}_z$ and a mirror symmetry $\textbf{M}_x:(x,y,z)\mapsto(-x,y,z)$.
In real space, the skew axial symmetries $\mathcal{S}_z$ flips the orbit indices. While in spin space, it keeps the $s_z$ component invariant but reverse $s_x$ and $s_y$, indicating that its action can be represented by $i\hat{s}_z$. Therefore the wave function and Hamiltonian transform as, $$\begin{aligned}
\mathcal{S}_z|\eta,\mathbf{k},\sigma \rangle & = & ie^{-i\frac{k_zc}{2}}\mathrm{sng}(\sigma) |\bar{\eta},-k_x,-k_y,k_z,{\sigma} \rangle \nonumber\\
\mathcal{S}_zH(\mathbf{k}) & = & H(-k_x,-k_y,k_z)\mathcal{S}_z.
\label{eq:Sz_psi1}\end{aligned}$$
With Eq.(\[eq:Sz\_psi1\]), one finds the below relation for Bloch Hamiltonian (\[eq:H\_IT\]), $$\begin{aligned}
H_{\eta_1\eta_2}^{\sigma_1\sigma_2}(\mathbf{k}) & = & \langle \eta_1,\mathbf{k},\sigma_1| {H}(\mathbf{k}) | \eta_2,\mathbf{k},\sigma_2\rangle \nonumber\\
& = & \langle \eta_1,\mathbf{k},\sigma_1| \hat{\mathcal{S}_z}^{-1} \hat{\mathcal{S}_z}{H}(\mathbf{k})\hat{\mathcal{S}_z}^{-1} \mathcal{S}_z| \eta_2,\mathbf{k},\sigma_2\rangle \nonumber\\
& = & \langle \eta_1,\mathbf{k},\sigma_1| \hat{\mathcal{S}_z}^{-1} {H}(\tilde{\mathbf{k}}) \hat{\mathcal{S}_z} | \eta_2,\mathbf{k},\sigma_2\rangle \nonumber\\
& = & \langle \bar{\eta}_1,\tilde{\mathbf{k}},\sigma_1| {H}(\tilde{\mathbf{k}}) | \bar{\eta}_2,\tilde{\mathbf{k}},\sigma_2\rangle \nonumber\\
& = & \mathrm{sng}(\sigma_1) \mathrm{sng}(\sigma_2)H_{\bar{\eta}_1\bar{\eta}_2}^{\sigma_1\sigma_2} (-k_x,-k_y,k_z),
\label{eq:S_aa}\end{aligned}$$ with $\tilde{\mathbf{k}}=(-k_x,-k_y,k_z)$. Then a constraint for the spinful Hamiltonian (\[eq:H\_IT\]) is obtained, $$H_{\eta_1\eta_2}^{\sigma_1\sigma_2}(k_x,k_y,k_z)=\mathrm{sng}(\sigma_1)\mathrm{sng}(\sigma_2)H_{\bar{\eta}_1\bar{\eta}_2}^{\sigma_1\sigma_2}(-k_x,-k_y,k_z).
\label{eq:Skew_XZ}$$ Combining Eqs.(\[eq:I\_spin\]) and (\[eq:Skew\_XZ\]), one arrives at the below important relation of Hamiltonian posed by symmetry $\mathcal{S}=\mathcal{IS}_z$, $$H_{\eta_1\eta_2}^{\sigma_1\sigma_2}(k_x,k_y,k_z)=\mathrm{sgn}(\sigma_1)\mathrm{sgn}(\sigma_2)H_{\bar{\eta}_1\bar{\eta}_2}^{{\sigma}_1{\sigma}_2}(k_x,k_y,-k_z).
\label{eq:Skew_final}$$
Applying the transformation $H_{\eta_1\eta_2}^{\sigma_1\sigma_2}(\mathbf{k}+\mathbf{G})$ = $e^{i\mathbf{G}\cdot\mathbf{r}_{\eta_1\eta_2}}H_{\eta_1\eta_2}^{\sigma_1\sigma_2}(\mathbf{k})$ to Eq.(\[eq:Skew\_final\]) for $k_z=\pi/c$, one finds $$\begin{aligned}
&&H_{AB}^{\sigma_1\sigma_2}(k_x,k_y,\pi/c) \nonumber \\
&=&\mathrm{sgn}(\sigma_1)\mathrm{sgn}(\sigma_2)H_{BA}^{{\sigma}_1{\sigma}_2}(k_x,k_y,-\pi/c) \nonumber \\
&=& \mathrm{sgn}(\sigma_1)\mathrm{sgn}(\sigma_2) e^{i\frac{2\pi}{c}\frac{c}{2}}H_{BA}^{{\sigma}_1{\sigma}_2}(k_x,k_y,\pi/c),
\label{eq:cons1}\end{aligned}$$ so that $$\begin{aligned}
H_{AB}^{\uparrow\downarrow}(k_x,k_y,\pi/c)&=&H_{BA}^{\uparrow\downarrow}(k_x,k_y,\pi/c) \nonumber\\
H_{AB}^{\uparrow\uparrow}(k_x,k_y,\pi/c)&=&-H_{BA}^{\uparrow\uparrow}(k_x,k_y,\pi/c),
\label{eq:fg}\end{aligned}$$ proving in Hamiltonian (\[eq:H\_IT\]) that $$\begin{aligned}
g(k_x,k_y,\pi/c)=0,~~\mathbf{Re}\left[f(k_x,k_y,\pi/c)\right]=0.
\label{eq:g_ref}\end{aligned}$$ Applying Eq. (\[eq:Skew\_final\]) to $H_{AA}^{\uparrow\uparrow}(\mathbf{k})$, one similarly obtains $$\begin{aligned}
H_{AA}^{\uparrow\uparrow}(k_x,k_y,\pi/c)=H_{BB}^{\uparrow\uparrow}(k_x,k_y,\pi/c),
\label{eq:cons2}\end{aligned}$$ which proves $$\begin{aligned}
a(k_x,k_y,\pi/c) = b(k_x,k_y,\pi/c).
\label{eq:ab}\end{aligned}$$ Therefore, in the presence of $\mathcal{I}, \mathcal{T}$ and $\mathcal{S}_z$, most elements of Hamiltonian (\[eq:H\_IT\]) vanishes at $k_z=\pi/c$ plane except imaginary party of $H_{AB}^{\uparrow\downarrow}(k_x,k_y,\pi/c)$ and a diagonal element $\epsilon(\mathbf{k})=[a(k_x,k_y,\pi/c)+b(k_x,k_y,\pi/c)]/{2}$.
It is possible to prove that $\mathbf{Im}\left[f(0,k_y,\pi/c)\right]=0$ at the high-symmetry path $k_x=0$ on $k_z=\pi/c$ plane. To do so, we need extra symmetries, which in the present case is a mirror symmetry $\mathbf{M}_x$ with the mirror plane lying at the $y$-$z$ plane. In real space the mirror symmetry $\mathbf{M}_x$ keeps the orbit indices invariant and inverses the $x$-component of coordinates. While in spin space, it keeps the spin component $s_x$ invariant but reverse $s_y$ and $s_z$, indicating its action on the spin space represented by $i\hat{s}_x$. Therefore the wave function and Hamiltonian transform as, $$\begin{aligned}
\mathbf{M}_x|\eta,\mathbf{k},\sigma \rangle & = & i|{\eta},-k_x,k_y,k_z,\bar{\sigma} \rangle \nonumber\\
\mathbf{M}_xH(\mathbf{k}) & = & H(-k_x,k_y,k_z)\mathbf{M}_x.
\label{eq:Sz_psi2}\end{aligned}$$ which indicates, $$\begin{aligned}
H_{\eta_1\eta_2}^{\sigma_1\sigma_2}(k_x,k_y,k_z)= H_{{\eta}_1 {\eta}_2}^{\bar{\sigma}_1\bar{\sigma}_2}(-k_x,k_y,k_z).
\label{eq:mx_spin}\end{aligned}$$ With Eq.(\[eq:mx\_spin\]), one gets $H_{AB}^{\uparrow\uparrow}(0,k_y,k_z)= H_{AB}^{\downarrow\downarrow}(0,k_y,k_z)$ and $H_{AB}^{\uparrow\downarrow}(0,k_y,k_z)= H_{AB}^{\downarrow\uparrow}(0,k_y,k_z)$, which proves $$\begin{aligned}
\mathbf{Im}\left[f(0,k_y,k_z)\right]=0, ~\mathbf{Re}\left[g(0,k_y,k_z)\right]=0.
\label{eq:f_reg}\end{aligned}$$ For entry $H_{AA}^{\uparrow\uparrow}$, $\mathbf{M}_x$ also sets $H_{AA}^{\uparrow\uparrow}(0,k_y,k_z)=H_{AA}{\downarrow\downarrow}(0,k_y,k_z)$ which results into the constraint,
By summarizing the above discussion, we prove that there exists a node-line at the $k_x=0$ path on $k_z=\pi/c$ plane for BaMX$_3$ when SOC is turned on. For other two node-lines at $k_x=\pm\sqrt{3}k_y$ (see in Fig. \[fig:soc\](b)), they can be produced by using the $\hat{C}_3$ symmetry of the system, which rotates the $k_x=0$ node-line to the $k_x=\pm\sqrt{3}k_y$ directions.
We then expand the Hamiltonian around a point on the node-line $\mathbf{k}=(0,k_y,\pi/c)+(q_x,q_y,q_z)$ with $(q_x,q_y,q_z)$ being small deviations from the expanding point $\mathbf{k}_0=(0,k_y,\pi/c)$. From the above Eqs. (\[eq:g\_ref\]), (\[eq:ab\]) and (\[eq:f\_reg\]) one finds, $$\begin{aligned}
\frac{\partial g}{\partial q_x}\Big|_{\mathbf{k}_0}&=&0,~\frac{\partial g}{\partial q_y}\Big|_{\mathbf{k}_0}=0,~\mathbf{Re}\frac{\partial g}{\partial q_z}\Big|_{\mathbf{k}_0}=0,\nonumber\\
\mathbf{Re}\frac{\partial f}{\partial q_x}\Big|_{\mathbf{k}_0}&=&0,~\frac{\partial f}{\partial q_y}\Big|_{\mathbf{k}_0}=0,~\mathbf{Im}\frac{\partial f}{\partial q_z}\Big|_{\mathbf{k}_0}=0 \nonumber\\
\frac{\partial a}{\partial q_i}\Big|_{\mathbf{k}_0}&=&\frac{\partial b}{\partial q_i}\Big|_{\mathbf{k}_0}, (i=x,y,z)
\label{eq:H_expand}\end{aligned}$$ from which the low energy Hamiltonian is obtained, $$\begin{aligned}
H_{\mathbf{q}}=\epsilon(\mathbf{k})+\alpha q_z\tau_y s_x+ q_z\tau_x+q_x\tau_ys_z
\label{eq:Hq}\end{aligned}$$ As $\epsilon(\mathbf{k})$ is not important for the solution of Hamiltonian (\[eq:H\_expand\]), we omit it in the following discussions. The Hamiltonian (\[eq:Hq\]) takes a symmetry $\tau_xs_y$ and by using the unitary matrix $U$ which diagonalizes $\tau_xs_y$, $$\begin{aligned}
U=\frac{1}{\sqrt{2}}\left[
\begin{array} {cccc}
1 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 \\
0 & -i & 0 & i \\
-i & 0 & i & 0 \\
\end{array}
\right],
\label{eq:U}\end{aligned}$$ one can transform the Hamiltonian $H_{\mathbf{q}}$ into a block-diagonalized matrix $\tilde{H}_{\mathbf{q}}$, $$\begin{aligned}
\tilde{H}_{\mathbf{q}}=\epsilon(\mathbf{k})+\frac{1}{\sqrt{2}}\left[
\begin{array} {cccc}
\alpha q_z & q_z+iq_x & 0 & 0 \\
q_z-iq_x & -\alpha q_z & 0 & 0 \\
0 & 0 & -\alpha q_z & q_z+iq_x \\
0 & 0 & q_z-iq_x & \alpha q_z \\
\end{array}
\right].\nonumber \\
\label{eq:H_q2}\end{aligned}$$ For each nonzero sub-block of Hamiltonian (\[eq:H\_q2\]), the diagonal terms are further moved to the off-diagonal entries through an isospin-rotation about the $\tilde{\tau}_y$ axis $\exp(i{\pi}/{8}\tilde{\tau}_y)$. After a rescale of $q_z$ by $1/(\alpha^2+1)$ , one obtains a simple form of $2\times2$ Hamiltonian ${H}_{\tilde{\mathbf{q}}}=\left[\begin{array} {cc} 0 & \tilde{q}_z+i\tilde{q}_x \\ \tilde{q}_z-i\tilde{q}_x & 0\\ \end{array}\right]$ for each sub-block. The occupied state has an eigen-value of $-\sqrt{\tilde{q}_x^2+\tilde{q}_z^2}$ with the corresponding eigen-vector being ${\sqrt{2}}/{2}\left[\begin{array} {c} 1 \\ -\exp(-i\theta) \end{array}\right]$. When $\mathbf{\tilde{q}}$ transverses along a closed loop circling the node-line, the accumulated Berry phase is computed to $\pi$ as follows, $$\begin{aligned}
\mathbf{\Phi}_B&=&-i\lim_{N\rightarrow\infty}\sum_{j=0}^{N-1}\log\langle j|j+1\rangle\nonumber \\
&=& -i\lim_{N\rightarrow\infty}\sum_{j=0}^{N-1}\log\left(\frac{1+\exp[i(\theta_{j+1}-\theta_j)]}{2}\right)\nonumber \\
&=&-i\int_0^{2\pi}\frac{id\theta}{2}=\pi,
\label{eq:Berry}\end{aligned}$$ where the loop is discretized into N successive points labeled by $j$ $(j=0,1,2,\ldots,N-1)$ with $N=0$. The Berry phase $\mathbf{\Phi}_B$ here is only meaningful for modulo $2\pi$ as $|N\rangle=e^{i2m\pi}|0\rangle$ is also well defined for condition $N=0$, which will increase $\mathbf{\Phi}_B$ by $2m\pi$.
Tight-Binding Model
-------------------
![ (color on line) Hopping parameters of the tight-binding model for the (a) spin-independent and (b) spin-dependent hopping processes. The gray and golden triangles denote the two orbits of the model and the bonds linking them present the corresponding hopping parameters. \[fig:hopping\] ](fig6.png){width="50.00000%"}
![ (Color online) Fitting of the ab initio band structure by the ting-binding model for spinless case of BaVS$_3$ (top) and the spinful case of BaTaS$_3$ (bottom). Fitting parameters are t$_{z}$ = -0.52 eV, t$_{xy}^{ab}$ = -0.032 eV and t$_{xy}$ = -0.056 eV for BaVS$_3$ and t$_{z}$ = -0.90 eV, t$_{xy}^{ab}$ = -0.040 eV, t$_{xy}$ = -0.055 eV and t$_{soc}$= 0.062 eV for BaTaS$_3$. \[fig:tbfit\] ](fig7.png){width="50.00000%"}
In this section we construct a tight-binding model that is fully consisted with the symmetry constraints discussed above. The active orbits are the two local $dz^2$ states centered on the Ta atoms and they provide the four local basis states when spin degree of freedom is included. Four hopping parameters are given to describe the $dz^2$ bands near the fermi surface and their definitions are presented in Fig.\[fig:hopping\](a) schematically. The two types of Ta atoms are denoted by triangles pointing at different orientations. The hopping process along the TaS$_3$ chains is denoted by $t_z$. In the $x-y$ plane, hopping process take places between the same Ta -sublattice is defined as $t_{xy}$ while the hopping between different sublattice in the slant direction is defined as $t_{xy}^{ab}$. There are also spin-dependent hoppings when the spin orbit coupling is taken into consideration.
With these hopping parameters we are able to construct a $4\times4$ Hamiltonian $H_{TB}(\mathbf{k})$ as, $$\begin{aligned}
H_{TB}(\mathbf{k})=\left[\begin{array}{cccc}
g(\mathbf{k}) & u(\mathbf{k})+v(\mathbf{k})& 0 & 0 \\
& g(\mathbf{k}) & 0 & 0 \\
& & g(\mathbf{k}) & u^*(\mathbf{k}) +v^*(\mathbf{k}) \\
\dag & & & g(\mathbf{k}) \\
\end{array}\right], \nonumber \\
\label {eq:H_tb}\end{aligned}$$ where $$\begin{aligned}
g(\mathbf{k})&=&2t_{xy}\left[ \cos\mathbf{k}\cdot\mathbf{a}+\cos\mathbf{k}\cdot\mathbf{b}
+\cos\mathbf{k} \cdot (\mathbf{a}+\mathbf{b})\right] \nonumber\\
&=&2t_{xy}\left[2\cos\frac{k_xa_0}{2}\cos\frac{\sqrt{3}k_ya_0}{2}+\cos{k_xa_0}\right] \nonumber\\
v(\mathbf{k})&=&2it_{soc}\sin\frac{\mathbf{k}_z\mathbf{c}}{2}[ \sin\mathbf{k}\cdot\mathbf{a}+\sin\mathbf{k}\cdot\mathbf{b}
-\sin\mathbf{k} \cdot (\mathbf{a}+\mathbf{b})] \nonumber\\
&=&2it_{soc}\sin\frac{k_zc}{2}\left[2\sin\frac{k_xa_0}{2}\cos\frac{\sqrt{3}k_ya_0}{2}-\sin{k_xa_0}\right] \nonumber\\
u(\mathbf{k})&=&t^{ab}_{xy}\cos\frac{\mathbf{k}_z\cdot\mathbf{c}}{2}\left[ \cos\mathbf{k}\cdot\mathbf{a}+\cos\mathbf{k}\cdot\mathbf{b}
+\cos\mathbf{k} \cdot (\mathbf{a}+\mathbf{b})\right] \nonumber\\
&+&2t_z\cos\frac{\mathbf{k}_z\cdot\mathbf{c}}{2} \nonumber\\
&=&2t^{ab}_{xy}\cos\frac{k_zc}{2}\left[2\cos\frac{k_xa_0}{2}\cos\frac{\sqrt{3}k_ya_0}{2}+\cos{k_xa_0}\right] \nonumber \\
&+& 2t_z\cos\frac{k_zc}{2} \nonumber \\
\label{eq:diag_6}\end{aligned}$$ with $\hat{a}=(\frac{1}{2},-\frac{\sqrt{3}}{2},0)a_0$ and $\hat{b}=(\frac{1}{2},\frac{\sqrt{3}}{2},0)a_0$. Here $a_0$ is the latticelattice constant in the $x-y$ plane. It is easily verified that at high symmetry paths $k_x=0$ and $k_x$ = $\pm{\sqrt{3}}k_y$ on the $k_z=\pi/c$ plane, $u(\mathbf{k})=v(\mathbf{k})=0$, indicating that there exist three node-lines.
We also fit the *ab initio* band structure shown in the main text with this TB-model and the results are shown in Fig. \[fig:tbfit\]. With suitable parameters, the *ab initio* band structures are fitted qualitatively well. As the *ab initio* band structures are strongly hybridized to the other valence bands, we are more focused on the bands along the path $\mathrm{A-L-H-A}$.
|
---
abstract: |
Bayesian inference is developed for matrix-variate dynamic linear models (MV-DLMs), in order to allow missing observation analysis, of any sub-vector or sub-matrix of the observation time series matrix. We propose modifications of the inverted Wishart and matrix $t$ distributions, replacing the scalar degrees of freedom by a diagonal matrix of degrees of freedom. The MV-DLM is then re-defined and modifications of the updating algorithm for missing observations are suggested.
*Some key words:* Bayesian forecasting, dynamic models, inverted Wishart distribution, state space models.
author:
- 'K. Triantafyllopoulos[^1]'
title: 'Missing observation analysis for matrix-variate time series data'
---
Introduction
============
Suppose that, in the notation of West and Harrison (1997, Chapter 16), the $p\times r$ matrix-variate time series $\{y_t\}$ follows a matrix-variate dynamic linear model (MV-DLM) so that $$\label{model}
y_t'=F_t'\Theta_t+\epsilon_t' \quad \textrm{and} \quad \Theta_t =
G_t \Theta_{t-1}+\omega_t,$$ where $F_t$ is a $d\times r$ design matrix, $G_t$ is a $d\times d$ evolution matrix and $\Theta_t$ is a $d\times p$ state matrix. Conditional on a $p\times p$ covariance matrix $\Sigma$, the innovations $\epsilon_t$ and $\omega_t$ follow, respectively, matrix-variate normal distributions (Dawid, 1981), i.e. $$\epsilon_t|\Sigma\sim N_{r\times p} (0,V_t,\Sigma)\quad \textrm{and}
\quad \omega_t|\Sigma\sim N_{d\times p} (0,W_t,\Sigma).$$ This is equivalent to writing $\textrm{vec}(\epsilon_t)|\Sigma\sim
N_{rp}(0,\Sigma\otimes V_t)$ and $\textrm{vec}(\omega_t)|\Sigma\sim N_{dp}(0,\Sigma\otimes W_t)$, where $\textrm{vec}(.)$ denotes the column stacking operator of a matrix, $\otimes$ denotes the Kronecker product of two matrices and $N_{rp}(.,.)$ denotes the multivariate normal distribution.
We assume that the innovation series $\{\epsilon_t\}$ and $\{\omega_t\}$ are internally and mutually uncorrelated and also they are uncorrelated with the assumed initial priors $$\label{priors1}
\Theta_0|\Sigma\sim N_{d\times
p}(m_0,P_0,\Sigma)\quad\textrm{and}\quad \Sigma \sim IW_p
(n_0,n_0S_0),$$ for some known $m_0$, $P_0$, $n_0$ and $S_0$. Here $\Sigma\sim
IW_p(n_0,n_0S_0)$ denotes the inverted Wishart distribution with $n_0$ degrees of freedom and parameter matrix $n_0S_0$. The covariance matrices $V_t$ and $W_t$ are assumed known; usually $V_t=I_r$ (the $r\times r$ identity matrix) and $W_t$ can be specified using discount factors as in West and Harrison (1997, Chapter 6). Alternatively, $W_t=W$ may be considered time-invariant and it can be estimated from the data using the EM algorithm (Dempster [*et al.*]{}, 1977; Shumway and Stoffer, 1982). With the above initial priors (\[priors1\]) the posterior distribution of $\Theta_t|\Sigma,y_1,\ldots,y_t$ is a matrix-variate normal distribution and the posterior distribution of $\Sigma|y_1,\ldots,y_t$ is an inverted Wishart distribution with degrees of freedom $n_t=n_{t-1}+1$ and a parameter matrix $n_tS_t$, which are calculated recurrently (West and Harrison, 1997, Chapter 16).
Missing data in time series are typically handled by evaluating the likelihood function (Jones, 1980; Ljung, 1982; Shumway and Stoffer, 1982; Harvey and Pierse, 1984; Wincek and Reinsel, 1984; Kohn and Ansley, 1986; Ljung, 1993; Gómez and Maravall, 1994; Luceño, 1994; Luceño, 1997). In the context of model (\[model\]) a major obstacle in inference is when a sub-vector or sub-matrix $\widetilde{y}_t$ of $y_t$ is missing at time $t$. Then the scalar degrees of freedom of the inverted Wishart distribution of $\Sigma|y_1,\ldots,y_t$, are incapable to include the information of the observed part of $y_t$, but to exclude the influence of the missing part $\widetilde{y}_t$. For example consider $p=2$ and $r=1$ or $y_t=[y_{1t}~y_{2t}]'$ and suppose that at time $t$, $y_{1t}$ is missing ($\widetilde{y}_t=y_{1t}$), while $y_{2t}$ is observed. Let $n_{t-1}$ denote the degrees of freedom of the inverted Wishart distribution of $\Sigma|y_1,\ldots,y_{t-1}$. One question is how one should update $n_t$, since the information at time $t$ is partial (one component observed and one missing). Likewise, given this partial information at time $t$, another question is how to estimate the off-diagonal elements of $\Sigma$, which leads to the estimation of the covariance of $y_{1t}$ and $y_{2t}$.
In this paper, introducing several degrees of freedom that form a diagonal matrix, we propose modifications to the inverted Wishart and matrix $t$ distributions. We prove the conjugacy between these distributions and we discuss modifications in the recursions of the posterior moments in the presence of missing data. This approach does not require to order all missing observations in one matrix (Shumway and Stoffer, 1982; Luceño, 1997) and therefore it can be applied for sequential purposes as new data are observed.
Matrix-variate dynamic linear models {#mvdlm}
====================================
Modified inverted Wishart distribution {#c4s2}
--------------------------------------
Suppose that $\Sigma$ is a $p\times p$ random covariance matrix, $S,R$ are $p\times p$ covariance matrices and $N$ is a $p\times p$ diagonal matrix with positive diagonal elements. Let tr$(.)$, etr$(.)$ and $|.|$ denote the trace, the exponent of the trace and the determinant of a square matrix, respectively. The density of the inverted Wishart distribution is given by $$\label{eq00}
p(\Sigma)=c|R|^{(k-p-1)/2} |\Sigma|^{-k/2}\textrm{etr}\left(
-\frac{1}{2} R\Sigma^{-1} \right),$$ from which it is deduced that $$\label{eq0}
\int_{\Omega} |\Sigma|^{-k/2}\textrm{etr}\left( -\frac{1}{2}
R\Sigma^{-1} \right)\,d\Sigma=c^{-1}|R|^{-(k-p-1)/2},$$ with $\Omega=\{\Sigma\in \re^{p\times p}:\Sigma>0\}$, $c^{-1}=2^{(k-p-1)p/2} \Gamma_p \{(k-p-1)/2\}$, and $k>2p$, where $\Gamma_p(.)$ is the multivariate gamma function.
The function $$\label{eq1}
p(\Sigma)=c |\Sigma |^{-\left\{ v+ \textrm{tr}
\left(N\right)/(2p)\right\} } \textrm{etr} \left( -\frac{1}{2}
N^{1/2} SN^{1/2}\Sigma^{-1} \right),$$ where $c$ does not depend on $\Sigma$, is a density function.
If the following bijective transformation is applied $$\label{tr1}
R=N^{1/2}SN^{1/2}\quad \mbox{and} \quad
k=2v+\frac{\textrm{tr}(N)}{p},$$ then (\[eq1\]) is directly obtained from (\[eq00\]).
From the above bijection and the Wishart integral, we can see that the normalizing constant $c$ is $$c=c_0|S|^{\left\{
2v+\textrm{tr}\left(N\right)/p-p-1\right\}/2} \left( \prod_{j=1}^p
n_j\right)^{\left\{2v+\textrm{tr}\left(N\right)/p-p-1\right\}/2},$$ where $$c_0^{-1}=2^{\left\{2v+\textrm{tr}\left(N\right)/p-p-1\right\}p/2}
\Gamma_p \left\{
\frac{2v+\textrm{tr}\left(N\right)/p-p-1}{2}\right\},$$ for $N=\textrm{diag}(n_1,\ldots ,n_p)$ and $n_i>0$ $(i=1,\ldots,p)$.
Density (\[eq1\]) proposes a modification of the inverted Wishart distribution in order to incorporate a diagonal matrix of degrees of freedom. The modification consists of a bijective transform of the two distributions. We will then say that $\Sigma$ follows the [*modified inverted Wishart*]{} distribution and we will write $\Sigma\sim MIW_p(S,N,v)$, where $v$ is a scalar hyperparameter. Note that when $n_1=\cdots =n_p=n$ and $v=p$, the above distribution reduces to an inverted Wishart distribution with $n$ degrees of freedom.
With $k$ and $R$ as defined in equation (\[tr1\]), the mean of $\Sigma$ is $$E(\Sigma)=\frac{R}{k-2p-2}=\left\{\frac{\textrm{tr}\left(N\right)}{p}+2v-2p-2
\right\}^{-1}N^{1/2}SN^{1/2},$$ for $p^{-1}\textrm{tr}(N)>2p-2v+2$. The next result gives the distribution of a $MIW$ matrix conditional on a normal matrix.
\[th21\] Let $Y$ be an $r\times p$ random matrix that follows a matrix normal distribution, conditional on $\Sigma$, and $\Sigma$ a $p\times p$ covariance random matrix that follows a modified inverted Wishart distribution, written $Y|\Sigma\sim N_{r\times p}(m,P,\Sigma)$ and $\Sigma\sim MIW_p(S,N,v)$ respectively, for some known quantities $m$, $P$, $S$, $N$, and $v$. Then, the conditional distribution of $\Sigma$ given $Y$, is $$\Sigma|Y\sim MIW_p(S^*,N^*,v),$$ where $N^{*1/2}S^{*}N^{*1/2}=(Y-m)'P^{-1}(Y-m) +N^{1/2}SN^{1/2}$ and $N^{*}=N+rI_p$.
Form the joint distribution of $Y$ and $\Sigma$ and write $$\begin{aligned}
p(\Sigma|Y)&\propto &p(Y,\Sigma)=p(Y|\Sigma)p(\Sigma)\nonumber\\
&\propto &|\Sigma|^{-\left\{ v+r/2+\textrm{tr}(N)/(2p) \right\}
}\textrm{etr}\bigg[-\frac{1}{2}
\{ (Y-f)'Q^{-1}(Y-f)\nonumber\\
&&+N^{1/2} SN^{1/2}\}\Sigma^{-1} \bigg],\label{eq:app:joint}\end{aligned}$$ which is sufficient for the proof with the definition of $S^{*}$ and $N^{*}$.
In the context of Proposition \[th21\] the joint distribution of $Y$ and $\Sigma$ is referred to as joint normal modified inverted Wishart distribution with notation $Y,\Sigma\sim NMIW_{r\times
p,p}(m,P,S,N,v)$, for $m$, $P$, $S$, $N$, and $v$ as defined in Proposition \[th21\]. The next result gives the marginal distribution of $Y$. First we give some background material on the matrix $t$ distribution.
Let $X$ be an $r\times p$ random matrix. Then, the matrix $t$ distribution is defined by $$\label{eqa6}
p(X)=c|Q+(X-M)'P^{-1}(X-M)|^{-(k+r+p-1)/2},$$ with $$c=\frac{\Gamma_p\{(k+r+p-1)/2\}|Q|^{(k+p-1)/2}|P|^{-p/2}}{\pi^{rp/2}\Gamma_p\{(k+p-1)/2\}},$$ where $M$ is an $r\times p$ matrix, $P$ a $r\times r$ covariance matrix, $Q$ a $p\times p$ covariance matrix, and $k$ any positive real number.
\[th3\] Let $Y$ be an $r\times p$ random matrix that follows a matrix normal distribution conditional on $\Sigma$, and $\Sigma$ be a $p\times p$ covariance random matrix that follows a modified inverted Wishart distribution, written $Y|\Sigma\sim N_{r\times
p}(f,Q,\Sigma)$, and $\Sigma\sim MIW_p(S,N,v)$ respectively, for known quantities $f$, $Q$, $S$, $N$, and $v$. Then, the marginal distribution of $Y$ is $$\label{eq81}
p(Y)=c |N^{1/2}SN^{1/2}+(Y-f)'Q^{-1}(Y-f)|
^{-\left\{2v+\textrm{tr}\left(N\right)/p+d-p-1\right\}/2},$$ which by analogy of the $MIW$ distribution, is a modification of the matrix $t$ distribution and it is written as $MT(f,Q,S,N,v)$.
The joint distribution of $Y$ and $\Sigma$ is given by equation (\[eq:app:joint\]). Hence, the marginal distribution of $Y$ is $$p(Y)=\int_{\Omega}p(Y,\Sigma)\,d\Sigma,$$ where $\Omega=\{\Sigma\in \re^{p\times p}:\Sigma>0\}$. Set $R=(Y-f)'Q^{-1}(Y-f)+N^{1/2} SN^{1/2}$ and $k=2v+r+\textrm{tr}(N)/p$ and from equation (\[eq0\]) we have equation (\[eq81\]).
The distribution of Proposition (\[th3\]) can be derived from the matrix $t$ distribution (see equation (\[eqa6\])). The normalizing constant $c$ of (\[eq81\]) is obtainable from (\[eqa6\]) as $$c=\frac{\pi^{pr/2}\Gamma_p\{ (k+p-1)/2\} }{\Gamma_p\{ (k+r+p-1)/2\}
}|S|^{(k+p-1)/2}\left(\prod_{j=1}^p
n_j\right)^{(k+p-1)/2}|Q|^{-p/2},$$ where $N=\textrm{diag}(n_1,\ldots,n_p)$ and $k=2v-2p+\textrm{tr}(N)/p$. Note that if all the diagonal elements of $N$ are the same (i.e. $n_1=\cdots =n_p=n$) and $v=p$, then the above distribution reduces to a matrix $t$ distribution with $n$ degrees of freedom.
Finally we give the marginal distribution of $\Sigma$. Consider the following partition of $\Sigma$, $S$, and $N$ $$\Sigma=\left[\begin{array}{cc} \ \Sigma_{11} & \Sigma_{12}
\\ \Sigma_{12}' & \Sigma_{22}
\end{array}\right], \quad S=\left[\begin{array}{cc} \ S_{11} &
S_{12} \\ S_{12}' & S_{22}
\end{array}\right], \quad N=\left[\begin{array}{cc} \ N_1 &
0 \\ 0' & N_2
\end{array}\right],$$ where $\Sigma_{11}$, $S_{11}$ and $N_{11}$ have dimension $q\times
q$, for some $1\leq q<p$. The next result gives the marginal distribution of $\Sigma_{11}$.
\[th2\] If $\Sigma\sim MIW_p(S,N,v)$, under the above partition of $\Sigma$ the distribution of $\Sigma_{11}$ is $\Sigma_{11}\sim
MIW_q(S_{11},N_{11},v_1)$, where $v_1=v-p+q+2^{-1}p^{-1}\textrm{tr}(N)-2^{-1}q^{-1}\textrm{tr}(N_1)$.
The proof suggests the adoption of transformation (\[tr1\]) together with the partition of $R$ in (\[eq00\]) as $$R=\left[\begin{array}{cc} R_{11} & R_{12}\\
R_{12}' & R_{22}\end{array}\right].$$ Using marginalization properties of the inverted Wishart distribution, upon noticing $$N^{1/2}SN^{1/2}=\left[\begin{array}{cc} \ N_1^{1/2}S_{11}N_1^{1/2}
&
N_1^{1/2}S_{12}N_2^{1/2}\\
N_2^{1/2}S_{12}'N_1^{1/2} &
N_2^{1/2}S_{22}N_2^{1/2}\end{array}\right],$$ we get $\Sigma_{11}\sim MIW_q(S_{11},N_1,v_1)$, with $v_1$ as required.
A similar result can be obtained for $\Sigma_{22}$. Consequently, if we write $\Sigma=\{\sigma_{ij}\}$ $(1\leq i,j\leq p)$ and $N=\textrm{diag}(n_1,\ldots,n_p)$, then the diagonal variances $\sigma_{ii}$ follow modified inverted Wishart distributions, $\sigma_{ii}\sim MIW_1(s_{ii},n_i,v_i)$, where $v_i=v-p+1+2^{-1}p^{-1}\textrm{tr}(N)-2^{-1}n_i$. These in fact are inverted gamma distributions $\sigma_{ii}\sim
IG(v_i+n_i/2-1,n_is_{ii}/2)$. Note that if $n_1=\cdots =n_p=n$ and $v=p$, then we have that $\sigma_{ii}\sim IG(n/2,ns_{ii}/2)$ (the inverted gamma distribution used in West and Harrison (1997) when $p=1$).
We close this section with a brief discussion on an earlier study proposing the incorporation of several degrees of freedom for inverted Wishart matrices (Brown [*et al.*]{}, 1994). This approach is based on breaking the degrees of freedom on blocks and requiring for each block the marginal density of the covariance matrix to follow an inverted Wishart distribution. However, in that framework the conjugacy between the normal and that distribution is lost and as a result the proposed estimation procedure may be slow and probably not suitable for time series application. Relevant inferential issues of that approach are discussed in Garthwaite and Al-Awadhi (2001). Our proposal of the $MIW$ distribution retains the desired conjugacy and it leads to relevant modifications of the matrix $t$ distribution, which provides the forecast distribution. Furthermore, the $MIW$ density leads to fast computationally efficient algorithms, which are suitable for sequential model monitoring and expert intervention (Salvador and Gargallo, 2004). Finally, according to Proposition \[th2\], the marginal distributions of $MIW$ matrices are also $MIW$, which means that several degrees of freedom are included in the marginal models too, something that is not the case in the approach of Brown [*et al.*]{} (1994).
Matrix-variate dynamic linear models revisited {#c4s3}
----------------------------------------------
We consider model (\[model\]), but now we replace the initial priors (\[priors1\]) by the priors $$\label{priors}
\Theta_0|\Sigma\sim N_{d\times
p}(m_0,P_0,\Sigma_0)\quad\textrm{and}\quad \Sigma_0 \sim MIW_p
(S_0,N_0,p),$$ for some known $m_0$, $P_0$, $S_0$ and $N_0$. Practically we have replaced the inverted Wishart prior by the $MIW$ and so, for each $t=1,\ldots,T$, we use $p$ degrees of freedom $n_{1t},\ldots,n_{pt}$ in order to estimate $\Sigma|y^t$, where $y^t$ denotes the information set, comprising of observed data $y_1,\ldots,y_t$. The next result provides the posterior and forecast distributions of the new MV-DLM.
\[th4\] One-step forecast and posterior distributions in the model (\[model\]) with the initial priors (\[priors\]), are given, for each $t$, as follows.
\(a) Posterior at $t-1:\qquad \Theta_{t-1},\Sigma|y^{t-1}\sim
NMIW_{d\times p,p} (m_{t-1},P_{t-1},S_{t-1},N_{t-1},p)$,\
for some $m_{t-1}$, $P_{t-1}$, $S_{t-1}$ and $N_{t-1}$.
\(b) Prior at $t:\qquad\Theta_t,\Sigma|y^{t-1}\sim
NMIW_{d\times p,p} (a_t,R_t,S_{t-1},N_{t-1},p)$,\
where $a_t=G_tm_{t-1}$ and $R_t=G_tP_{t-1}G_t'+W_t$.
\(c) One-step forecast at $t$:$\qquad y_t'|\Sigma,y^{t-1}\sim
N_{r\times p} (f_t',Q_t,\Sigma)$,\
with marginal:$\qquad y_t'|y^{t-1}\sim
MT_{r\times p} (f_t',Q_t,S_{t-1},N_{t-1},p)$,\
where $f_t'=F_t'a_t$ and $Q_t=F_t'R_tF_t+V_t$.
\(d) Posterior at $t:\qquad \Theta_t,\Sigma|y^t\sim
NMIW_{d\times p,p} (m_t,P_t,S_t,N_t,p)$,\
with $$\begin{gathered}
m_t=a_t+A_te_t',\quad
P_t=R_t-A_tQ_tA_t',\\
N_t=N_{t-1}+rI_p, \quad N_t^{1/2}S_tN_t^{1/2}=N_{t-1}^{1/2}S_{t-1}
N_{t-1}^{1/2}+e_tQ_t^{-1}e_t',\\
A_t=R_tF_tQ_t^{-1}, \quad \mbox{and} \quad e_t=y_t-f_t.\end{gathered}$$
The proof of this result follows immediately from Propositions \[th21\] and \[th3\]. For $t=1$, (a) coincides with the priors (\[priors\]). From Proposition \[th3\], the marginal posterior of $\Theta_t|y^t$ is $\Theta_t|y^t\sim MT_{d\times
p}(m_t,P_t,S_t,N_t,p)$. Thus the above proposition gives a recursive algorithm for the estimation and forecasting of the system for all $t=1,\ldots,T$.
Proposition \[th4\] gives a generalization of the updating recursions of matrix-variate dynamic models (West and Harrison, 1997, Chapter 16). The main difference of the two algorithms is that the scalar degrees of freedom $n_t$ of the standard recursions are replaced by $N_t$ in the above proposition and that the inverted Wishart distribution is replaced by the modified inverted Wishart distribution (in order to account for the matrix of degrees of freedom). As a result the classical Bayesian updating of West and Harrison (1997) is obtained as a special case of the distributional results of Proposition \[th4\], by setting $N_t=n_tI_p=\textrm{diag}(n_t,\ldots,n_t)$ $(t=0,1,\ldots,T)$, where $n_t$ represent the scalar degrees of freedom of the inverted Wishart distribution of $\Sigma_t|y^t$ and $n_0$ is the initial degrees of freedom.
Missing observations {#c4s4}
====================
In this section we consider missing observations at random. Our approach is based on excluding any missing values of the calculation of the updating equations (state and forecast distributions) thus excluding the unknown influence of these unobserved variables. This approach is explained for univariate dynamic models in West and Harrison (1997, Chapters 4,10).
The univariate dynamic linear model with unknown observational variance is obtained from model (\[model\]) for $p=r=1$. In this case the posterior recursions of $m_t$, $P_t$ and $S_t$ of West and Harrison (1997, Chapter 4) follow from Proposition \[th4\] as a special case. Now suppose that at time $t$ the scalar observation $y_t$ is missing so that $y^t=y^{t-1}$. It is then obvious that the posterior distribution of $\Theta_t$ equals its prior distribution (since no information comes in to the system at time $t$). Then we have $m_t=a_t$, $P_t=R_t$, $S_t=S_{t-1}$ and $N_t=n_t=n_{t-1}=N_{t-1}$. To incorporate this into the updating equations of the posterior means and variances, we can write $m_t=a_t-A_te_tu_t$, $P_t=R_t-A_tA_t'Q_tu_t$, $n_tS_t=n_{t-1}S_{t-1}+e_t^2u_t/Q_t$ and $n_t=n_{t-1}+u_t$, where $u_t$ is zero, if $y_t$ is missing and $u_t=1$, if $y_t$ is observed. So when $p=1$ the inclusion of $u_t$ in the posterior recursions leads to identical analysis as in West and Harrison (1997) and in references therein. The introduction of $u_t$ in the recursions automates the posterior/prior updating in the presence of missing values and it motivates the case for $p,r\geq 1$.
Moving to the multivariate case, first we consider model (\[model\]) as defined in the previous section with $r=1$. Assume that we observe all the $p\times 1$ vectors $y_{i}$, $i=1,\ldots ,t-1$. At time $t$ some observations are missing (sub-vectors of $y_t$, or the entire $y_t$). To distinguish the former from the latter case we have the following definition.
\[df41\] A partial missing observation vector is said to be any strictly sub-vector of the observation vector that is missing. If the entire observation vector is missing it is referred to as full missing observation vector.
Considering the MV-DLM (\[model\]), it is clear that in the case of a full missing vector we have $$\label{eq20}
\Theta_t,\Sigma|y^t\sim NMIW_{d\times p,p} (m_t,P_t,S_t, N_t,p),$$ where $m_t=a_t$, $P_t=R_t$, $S_t=S_{t-1}$, $N_t=N_{t-1}$, since no information comes in at time $t$. This equation relates to the standard posterior distribution of West and Harrison (1997) by setting $N_t=\textrm{diag}(n_t,\ldots,n_t)$, for a scalar $n_t>0$ and evidently reducing the $MIW$ distribution by a $IW$ distribution. If one starts with a prior $N_0=\textrm{diag}(n_0,\ldots,n_0)$, and assuming that at some time $t$, there is a full missing vector $y_t$, then it is clear that the posterior (\[eq20\]) equals to the posterior of $\Theta_t,\Sigma|y^t$ using the standard recursions (West and Harrison, 1997). Any differences between the two algorithms is highlighted only by observing partial missing vectors and this has been the motivation of the new algorithm.
Define a $p\times p$ diagonal matrix $U_t=\mbox{diag}(i_{1t},\ldots ,i_{pt})$ with $$i_{jt}=\left\{ \begin{array}{cc} 1 & \textrm{if $y_{jt}$ is
observed},\\ 0 & \textrm{if $y_{jt}$ is missing,}
\end{array}\right.$$ for all $1\le j\le p$, where $y_t=[y_{1t}~\cdots ~y_{pt}]'$.
Then, the posterior distribution (\[eq20\]) still applies with recurrences $$\begin{gathered}
m_t=a_t+A_te_t'U_t\label{eq21}\\
P_t=R_t-A_tA_t'Q_tu_t\label{eq22}\\
N_t=N_{t-1}+U_t\label{eq23}\\
N_t^{1/2}S_tN_t^{1/2}=N_{t-1}^{1/2}S_{t-1}
N_{t-1}^{1/2}+U_te_tQ_t^{-1}e_t'U_t,\label{eq24}\end{gathered}$$ where $u_t=\mbox{tr}(U_t)/p$. Some explanation for the above formulae are in order.
First note that if no missing observation occurs $U_t=I_p$, $u_t=1$ and we have the standard recurrences as in Proposition \[th4\]. On the other extreme (full missing vector), $U_t=0$, $u_t=0$ and we have equation (\[eq20\]). Consider now the case of partial missing observations. Equation (\[eq23\]) is the natural extension of the single degrees of freedom updating, see West and Harrison (1997, Chapter 16). For equation (\[eq21\]) note that the zero’s of the main diagonal of $U_t$ convey the idea that the corresponding to the missing values elements of $m_t$ remain unchanged and equal to $a_t$. For example, consider the case of $p=2$, $d=2$ and assume that you observe $y_{1t}$, but $y_{2t}$ is missing. Then $$m_t=a_t+\left[\begin{array}{cc} \ A_{1t}(y_{1t}-f_{1t}) & 0\\
A_{2t}(y_{1t}-f_{1t}) & 0\end{array}\right],$$ where $A_t=[A_{1t}~A_{2t}]'$. The zero’s on the right hand side reveal that the second column of $m_t$ is the same as the second column of $a_t$. Similar comments apply for equations (\[eq22\]) and (\[eq24\]).
Considering the case of $r\geq 2$, we define $U_{kt}$ to be the diagonal matrix $U_{kt}=\mbox{diag}(i_{1k,t},\ldots,i_{pk,t})$ with $$i_{jk,t}=\left\{ \begin{array}{cc} 1 & \textrm{if $y_{jk,t}$ is
observed},\\ 0 & \textrm{if $y_{jk,t}$ is missing,}
\end{array}\right.$$ where $y_t=\{y_{jk,t}\}$, $(j=1,\ldots,p;k=1,\ldots,r)$.
Then, the moments of equation (\[eq20\]) can be updated via $$\begin{gathered}
m_t=a_t+A_te_t'\prod_{k=1}^rU_{kt}, \quad P_t=R_t-A_tQ_tA_t'u_t,
\quad
N_t=N_{t-1}+\sum_{k=1}^rU_{kt}\\
N_t^{1/2}S_tN_t^{1/2}=N_{t-1}^{1/2}S_{t-1}
N_{t-1}^{1/2}+\left(\prod_{k=1}^rU_{kt}\right)e_tQ_t^{-1}
e_t'\left(\prod_{k=1}^rU_{kt}\right),\end{gathered}$$ where $u_t=\mbox{tr}(\prod_{k=1}^rU_{kt})/p$. Similar comments as in the case of $r=1$ apply. Definition \[df41\] is trivially extended in the case when observations form a matrix ($r\geq 2$).
We illustrate the proposed methodology by considering simulated data, consisting of 100 bivariate time series $y_1,\ldots,y_{100}$, generated from a local level model $y_t=[y_{1t}~y_{2t}]'=\psi_t+\epsilon_t$ and $\psi_t=\psi_{t-1}+\zeta_t$, where $\psi_0$, $\epsilon_t$ and $\zeta_t$ are all simulated from bivariate normal distributions. The correlation of $\epsilon_{1t}$ and $\epsilon_{2t}$ is set to $0.8$, while the elements of $\zeta_t$ are uncorrelated. This model is a special case of model (\[model\]) with $\Theta_t'F_t=\psi_t$ and $G_t=I_2$. Figure \[fig1\] (solid line) shows the simulated data; the gaps in this figure indicate missing values at times $t=24,43,60,75,86$. At times $t=24,43,86$, $y_{t2}$ is only missing (partial missing vectors), at time $t=75$, $y_{t1}$ is only missing (partial missing vector) and at time $t=60$, both $y_{t1},y_{t2}$ are missing (full missing vector). For this data set, we compare the performance of recursions (\[eq21\])-(\[eq24\]) with that of the classic or old recursions of West and Harrison (1997), which assume that when there is at least one missing value we set $U_t=0$ and $u_t=0$. For example using the old recursions, for $t=24$ one would set $U_{24}=0$ and $u_{24}=0$, losing the “partial” information of $y_{24,1}=-3.739$, which is observed. On the other hand, the new recursions would suggest for $t=24$ to set $$U_{24}=\left[\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right]
\quad \textrm{and} \quad u_{24}=1/2.$$
Figure \[fig1\] shows the one-step forecast mean of $\{y_t\}$ using the new recursions (dashed line) and the old recursions (dotted/dashed line). We observe that the new method produces a clear improvement in the forecasts as the old recursions provide poor forecasts, especially in the low panel of Figure \[fig1\] (for $\{y_{1t}\}$). What is really happening in this case is that, under the old recursions, the missing values of $y_{2t}$ affect the recursions for $y_{1t}$, since the observed information at $y_{1t}$ is wrongly “masked” or “ignored” for the points of time when $y_{2t}$ is missing. On the other hand, the new recursions use the explicit information from each sub-vector of $y_t$ and thus the new recursions result in a notably more accurate forecast performance. This is backed by the mean square standardized forecast error vector, which for the new recursions is $[1.300~1.825]'$, while for the old recursions is $[1.545~2.182]'$. Under the old recursions we can not obtain an estimate of the covariance between an observed $y_{1t}$ and a missing $y_{2t}$. However, this is indeed obtained under the proposed new recursions and so the respective correlations at points of time where there are gaps are $0.633$ (at $t=24$), $0.779$ (at $t=43$), $0.812$ (at $t=75$) and $0.809$ (at $t=86$); the mean of these correlations is $0.792$, which is close to the real $0.8$ under the simulation experiment.
Acknowledgements {#acknowledgements .unnumbered}
================
I am grateful to Jeff Harrison for useful discussions on the topic of missing data in time series. I would like to thank a referee for helpful comments.
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[^1]: Department of Probability and Statistics, University of Sheffield, Sheffield S3 7RH, UK, email: [k.triantafyllopoulos@sheffield.ac.uk]{}
|
---
abstract: 'Transient computing has become popular in public cloud environments for running delay-insensitive batch and data processing applications at low cost. Since transient cloud servers can be revoked at any time by the cloud provider, they are considered unsuitable for running interactive application such as web services. In this paper, we present VM deflation as an alternative mechanism to server preemption for reclaiming resources from transient cloud servers under resource pressure. Using real traces from top-tier cloud providers, we show the feasibility of using VM deflation as a resource reclamation mechanism for interactive applications in public clouds. We show how current hypervisor mechanisms can be used to implement VM deflation and present cluster deflation policies for resource management of transient and on-demand cloud VMs. Experimental evaluation of our deflation system on a Linux cluster shows that microservice-based applications can be deflated by up to 50% with negligible performance overhead. Our cluster-level deflation policies allow overcommitment levels as high as 50%, with less than a 1% decrease in application throughput, and can enable cloud platforms to increase revenue by 30%.'
author:
- Alexander Fuerst
- 'Ahmed Ali-Eldin'
- Prashant Shenoy
- Prateek Sharma
bibliography:
- 'sample.bib'
title: 'Cloud-scale VM Deflation for Running Interactive Applications On Transient Servers'
---
Introduction {#sec:intro}
============
Transient computing is becoming commonplace in cloud environments. Today, all major cloud providers such as Amazon, Azure, and Google offer transient cloud servers in the form of preemptible instances that can be unilaterally revoked during periods of high server demand. Transient computing resources enable cloud providers to increase revenue by offering idle servers at significant discounts (often 7-10X cheaper) while retaining the ability to reclaim them during periods of higher demand.
While transient cloud servers have become popular due to their discounted prices, their revocable nature has meant that users typically limit their use for running disruption-tolerant jobs such as batch or data processing tasks. They have traditionally not been used for online web services due to potential downtimes that occur when the underlying servers are revoked.
In this paper, we present virtual machine (VM) deflation as an alternative mechanism for reclaiming resources from transient cloud servers. We argue that VM deflation is more attractive than outright preemption for applications, since they continue to run, albeit more slowly, under resource pressure rather than being terminated. Deflation simplifies application design since they no longer need to implement fault tolerance approaches such as checkpointing to handle server preemptions. Deflation also expands the classes of applications that are suitable to run on transient cloud servers—even web services can utilize such servers since downtimes from preemptions are no longer a risk; with the exception of mission critical web workloads, less critical web applications that are willing to tolerate occasional slowdowns can run on such servers at a much lower cost than on traditional cloud servers.
The notion of resource deflation was first proposed as a cascade deflation approach [@deflation-eurosys19] that collaboratively reclaimed resources from the application, the OS, and the hypervisor. Cascade deflation requires cooperation from the OS and the application and is impractical in public clouds that treat VMs as “black boxes.” Instead, a hypervisor-only approach to deflation that requires no support from the application or OS is better suited to Infrastructure as a Service (IaaS) public clouds—the key focus of our work.
By fractionally reclaiming resources from applications instead of outright preemption, VM deflation reduces the risk of downtimes for interactive applications, with a modest decrease in application performance. In designing and implementing our hypervisor-only deflation approach, our paper makes the following contributions.
We demonstrate the feasibility of using VM deflation as a resource reclamation mechanism in public clouds using real CPU, memory, disk, and network traces from two top-tier cloud providers (Azure and Alibaba). Our analysis shows that cloud VMs running interactive applications have substantial slack and can withstand deflation of 30-50% of their allocated resources with less than a 1% performance impact.
We then show how current hypervisor mechanisms such as hot-plug and throttling can be used to implement VM deflation. We also present several cluster-wide policies for VM deflation-based resource reclamation. Our policies present different tradeoffs and capabilities while attempting to minimize the performance impact of VM deflation.
We implement a prototype of our VM deflation mechanisms and policies on a virtualized Linux cluster and evaluate its efficacy using realistic web applications as well as other workloads. We also conduct a trace-driven evaluation of our policies using VM-level workloads from a cloud provider. Our results show that:
1. The resource utilization of cloud VMs is low, which makes deflation a viable technique for transient resources.
2. Deflation can be implemented with hypervisor and guest-OS level overcommitment. These deflation mechanisms can reclaim large amounts of resources in a black-box manner, with minimal performance degradation. For interactive microservice based applications, even 50% deflation results in negligible reduction in performance.
3. Our cluster-level deflation policies make deflation an effective technique for increasing cluster overcommitment (the ratio of committed VM allocations to cluster hardware availability) by up to 50%; nearly eliminates the risk of preemptions; and results in less than 1% drop in application throughput.
The rest of this paper is structured as follows. Section \[sec:background\] presents background on transient computing and deflation. Section \[sec:feasibility\] presents our feasibility analysis of VM deflation in public clouds. Section \[sec:mechanisms\] and \[sec:policies\] present VM deflation mechanisms and cluster-wide deflation policies, respectively. Section \[sec:impl\] and \[sec:eval\] present our implementation and experimental results. Finally, Section \[sec:related\] and \[sec:conclusions\] present related work and our conclusions.
Background {#sec:background}
==========
In this section, we provide background on transient cloud computing, and VM deflation.
[**Transient computing.**]{} Our work assumes a cloud data center where applications run on traditional (“on-demand”) servers or transient servers. Both types of servers are provisioned using virtual machines, and cloud applications run inside such VMs. Cloud offerings such as Amazon spot Instances [@warning-time], Google Preemptible VMs [@preemptible], and Azure batch VMs [@azure-batch] are examples of transient servers. Transient cloud servers represent surplus capacity that is offered at discounted rates but these resources can be reclaimed under resource pressure (e.g., higher demand for on-demand servers).
Batch-oriented applications are particularly well suited for transient computing. Such applications tend to be both delay and disruption tolerant and can handle longer completion times. In the event of a preemption, they can simply be restarted from the beginning or restarted from a checkpoint if the application is amenable to periodic checkpointing. Consequently, transient cloud servers have become popular for running large batch workloads at a substantial discount over using on-demand servers [@flint-eurosys16].
[**Deflation.**]{} While current transient servers implement resource reclamation in the form of preemptions—where the VM is unilaterally revoked by the cloud provider—our work explores the use of VM deflation as an alternative approach for resource reclamation under pressure. Although deflation frees up fewer resources than preemption (which frees up all of the VM resources), it enables applications to continue execution and eliminates application downtimes due to preempted servers [@deflation-eurosys19]. Our hypothesis is that occasional performance degradation, rather than termination and downtime, is more acceptable to many interactive and web applications, except the most critical ones, making transient computing feasible for a broader class of applications.
Since modern hypervisors allow resource allocation of resident VMs to be increased or decreased dynamically, VM deflation can be realized using current hypervisor mechanisms, such as ballooning [@waldspurger2002memory], hotplugging, changing CPU shares, etc. While any of the existing techniques can be used to implement VM deflation mechanisms, the challenge lies in the design of judicious policies on [*when*]{} and [*what*]{} to deflate and by [*how much*]{}, while minimizing the impact of deflation on application performance. We note that while VM deflation mechanisms are similar to elasticity (e.g., vertical scaling) mechanisms, our goal is to focus on cluster-wide deflation policies for resource reclamation, a different problem than elastic scaling as discussed in Section \[sec:related\].
Figure \[fig:defl-over\] gives an overview of our deflation system—the cluster manager implements the global VM deflation and placement policies (Section \[sec:policies\]) and places new VMs onto servers. The hypervisor implements local deflation policies (also in Section \[sec:policies\]), and uses VM deflation mechanisms (Section \[sec:mechanisms\]). The hypervisor also sends notifications to the application manager (such as a load balancer), which can help applications respond to deflation.
![Overview of our deflation system.[]{data-label="fig:defl-over"}](Figs_deflat-full-system.pdf){width="30.00000%"}
Feasibility of Deflation in Public Clouds {#sec:feasibility}
=========================================
Before presenting our deflation techniques, we examine the efficacy and feasibility of deflating public cloud applications. We use publicly-available resource usage traces from two top-tier cloud providers, Azure [@resourcecentral-sosp] and Alibaba [@alibaba-trace]. The goal of our analysis is to understand the feasibility of deflating CPU, memory, disk, and network allocations of real cloud applications, and specifically interactive web applications, under time-varying workloads that they exhibit. We seek to answer two key research questions through our feasibility analysis: (1) How much slack is present in cloud VMs and by how much can these VMs be safely deflated without any performance impact? (2) How does workload class and VM size impact the deflatability of VMs?
![Application behavior under different levels of deflation.[]{data-label="fig:deflation-model"}](Figs_util-curve2.pdf){width="1.8in"}
Application Behavior under Deflation
------------------------------------
We first present an abstract model to capture the performance behavior of an application under different amounts of resource deflation. Figure \[fig:deflation-model\] illustrates this behavior. We assume that an application running inside a cloud VM will have a certain amount of slack—unused CPU and memory resources. Reclaiming these unused resources represented by the slack will typically have negligible performance impact on the application since they are surplus resources; the behavior in this operating region is depicted by the horizontal portion of the performance curve labelled slack in Figure \[fig:deflation-model\]. Once all of the slack has been reclaimed by deflating the VM, any further deflation will actually impact performance. We assume that initially this performance impact is linear with increasing amounts of VM deflation. For some applications, this behavior can even be sub-linear, which means that a certain reduction in allocated resources yields proportionately less performance slowdown. For less elastic applications, however, the impact can be super-linear. In either case, beyond a certain point—represented by the knee of the curve—the performance drops precipitously, implying that allocated resources are insufficient for satisfactory performance.
This abstract model captures the three regions with varying performance impacts on applications due to deflation. Clearly, deflating slack is the simplest approach since it usually has little or no performance impact. When additional resources need to be reclaimed, the deflation policy should ensure that such deflation minimizes the performance impact and does not push application performance beyond the knee of the curve.
Figure \[fig:util-all\] depicts this behavior for three different applications. As can be seen, different applications have different amounts of slack (with SpecJBB not exhibiting any slack at all in this example), and the size of the linear performance degradation region also varies from application to application. The figure illustrates the need to take application’s characteristics into account when reclaiming its allocated resources using deflation.
![Application performance when all resources (CPU, memory, I/O) are deflated in the same proportion. []{data-label="fig:util-all"}](graphs_no-spark-utils-all.pdf){width="33.00000%"}
![Deflation can result in underallocated resources.[]{data-label="fig:underalloc"}](Figs_underalloc.pdf){width="25.00000%"}
Usage-based Feasibility Analysis
--------------------------------
### CPU Deflation
We analyze VM traces of CPU utilization in the Azure dataset to quantify their deflation capability. The dataset, which includes data from 2 million VMs, provides CPU utilization time series for each VM at 5-minute granularity. Importantly for us, each VM trace is partitioned into one of three classes—interactive, delay-insensitive, and unknown—depending on the type of application resident in the VM. We analyze all three classes of VM traces but pay particular attention to interactive applications, which tend to be dominated by web-based services. To analyze the impact of deflation, we assume that the CPU allocation of the VM is reduced by a certain percentage and calculate the percentage of time for which the *maximum* CPU usage over each interval in the original trace exceeds this value. We observe that as long as the CPU utilization is below this deflated allocation, there will be no performance impact on the application. However, during periods where the utilization exceeds the allocation under deflation (i.e., underallocation), the application will experience a slowdown.
As shown in Figure \[fig:underalloc\], the resource utilization and deflation determine how much time a VM is underallocated. The total amount of under-allocation (area of the utilization curve above the deflated allocation) is the decrease in application throughput. We want to quantify the slack in the VMs under different levels of deflation such that there is no performance impact on the application.
Figure \[fig:bp-thresh\] shows a box plot of the fraction of time spent by VMs above the deflated resource allocation (i.e., underallocated) for all 2 million VMs. Even at high deflation levels (50%), the median VM spends 80% of the time below the deflated allocation. This result indicates that even high deflation levels of as much as 50% do not lead to significant resource bottlenecks for applications.
Since the Azure dataset labels each VM trace with the class of application hosted by the VM, we break down the overall result in Figure \[fig:bp-over-thresh\] by application type. Figure \[fig:bp-over-thresh\] depicts a box plot of the fraction of time that VMs of different application classes exceed their deflated allocations under different levels of deflation. The figure shows that interactive applications, which include web workloads, tend to have lower overall utilization and hence more slack than delay insensitive batch workloads (presumably since they are over provisioned to handle unexpected peak loads). Consequently, interactive application VMs are more amenable to deflation of their surplus (slack) capacity. Thus, for any given deflation level, interactive VMs see significantly *less* impact in terms the CPU usage exceeding the deflated allocation. The percentage of time when the interactive VMs get impacted ranges from 1% to 15%, as deflation percentage is varied from 10% to 50%. In contrast, batch jobs see 1% to 30% impact. This result shows that interactive applications and web workloads can be subjected to deflation just like, and perhaps more so, than delay-insensitive batch applications.
Figure \[fig:bp-mem\] examines whether the VM size has an impact on its ability to be deflated. Based on the trace we partition VMs into 3 groups – small VMs with 2 GB RAM or lower, medium VMs with up to 8 GB RAM, and large VMs with more than 8GB RAM, and examine the percentage of time the VM CPU usage exceeds the deflated allocation within each group. The figure shows that VM size has no direct correlation to the deflatability of a VM, and all VMs see a similar performance impact under different deflation levels regardless of VM size. The result implies that VMs of all sizes are more or less equally amenable to deflation.
Finally, Figure \[fig:bp-p95\] examines the deflatability of VMs for VMs with different peak loads. We compute the $95^{th}$ percentile of CPU usage for all VMs and partition VMs into four classes; those with low peak utilization of less than 33%, those with moderate peak load between 33% and 66% peak utilization, those with higher load between 66% and 80% utilization and finally, the rest with high peak loads above 80%. As shown in the figure, higher peak loads implies that VMs see greater impact when deflated since the peak will exceed the deflated allocation for longer durations. Interestingly, for deflation levels of up to 20%, all VMs, except the ones with peak load exceeding 80%, have enough slack to see minimal impact. The figure generally indicates that the peak load, represented by a high percentile of the utilization distribution is a coarse indicator of the “deflatability”’ of the VM; VMs with lower peak loads are more amenable to deflation.
![Fraction of time (i.e. probability) of CPU usage of VMs being higher than different deflation targets.[]{data-label="fig:bp-thresh"}](AzureGraphs_percent-over-thresh-box-total-vector.pdf){width="2in"}
{width="\linewidth"}
\[fig:bp-over-thresh\]
{width="\linewidth"}
\[fig:bp-mem\]
{width="\linewidth"}
\[fig:bp-p95\]
### Memory and I/O Deflation
We also analyze the memory, disk, and network deflation feasibility based on Alibaba’s resource traces [@alibaba-analysis] [@alibaba-trace], that provide a time series of resource utilization for their internal container-based interactive services. Note that VM-based applications have a higher deflation potential because they are overprovisioned and must include additional resources for the guest OS; thus this container-level analysis of Alibaba’s cloud applications provides a very conservative (lower-bound) estimate of the actual deflation potential.
**Memory.** We analyze the memory usage of the applications under different deflation levels in Figure \[fig:ali-mem\]. Interestingly, as shown, the fraction of time that the application spends above different deflation thresholds is generally high. At first glance, this might suggest that the high memory utilization leaves little slack to deflate memory (e.g., even at 10% memory deflation, the applications would spend more than 70% time underallocated).
However, further analysis of the memory usage traces indicates that this is not really the case. First, the Alibaba memory traces provide the [*total*]{} memory usage and do not provide a fine-grain breakdown of memory usage, such as such as working set size, page-cache and disk-buffer pages. Over 90% of the applications in Alibaba trace are JVM-based services that overallocate memory (for the heap) to reduce the garbage collection overhead. As is well known, modern applications and operating systems aggressively used unallocated RAM for purposes of caching and buffering. Hence, the total memory usage shown in Figure \[fig:ali-mem\] is not a true measure of deflation potential of applications.
Conventional wisdom holds that application performance will be affected when the memory is deflated below its *working set* size, and deflation of other memory used for caching or garbage collection should have a lesser impact on performance. In fact, our experiments have shown that, even when memory is deflated *below* the working set size, the performance degradation, while noticeable, is not serve. For instance, Figure \[fig:util-all\] shows the resilience of Memcached, a highly memory-dependent application. Figure \[fig:jbb-mem\] shows that even SpecJBB (which is representative of the JVM-based applications that comprise the trace) can have its memory deflated by up to 30% without significant drop in performance.
To further analyze the true memory deflation potential, we we use the memory-bus bandwidth used by the different applications as a proxy metric for memory usage. As shown in Figure \[fig:ali-mem-band\], we see that the memory bandwidth usage is very low, with the mean memory bandwidth utilization across all containers being less than one-tenth of one percent, while the maximum is only 1%. This indicates that the applications are not reading/writing to the RAM in proportion to their memory allocations, and that the memory deflatability should be significantly higher than what is indicated by Figure \[fig:ali-mem\] alone.
**Disk and Network.** Finally, we examine the deflatability of disk and network bandwidth in Figures \[fig:ali-disk\] and \[fig:ali-net\] using the Alibaba trace. We see that the usage of both I/O resources is very low. The boxplot of application’s disk bandwidth that rises above various deflation thresholds is given in Figure \[fig:ali-disk\]. The percentage of time the actual disk bandwidth usage rises above various deflated allocations is low, indicating there is ample room to deflate the allocated I/O bandwidth. Even at a high deflation level of 50%, containers are underallocated less than 1% of the time. Network usage (sum of normalized incoming and outgoing traffic) is also low: in Figure \[fig:ali-net\] we can see that even this combined network bandwidth is not impacted by even at high (70%) deflation levels, only suffering underallocation 1% of their lifetime. Below 50% deflation, the impact is near-zero and cannot be plotted.
Our analysis shows that low-priority VMs can be shrunk to fit incoming VMs without preemption. Deflation allows providers to continue offering high-priority traditional VMs, and sell unused server space for low-priority VMs that can be deflated. This allows consumers to still have fully-resourced VMs available for a variety of applications. Because the average resource utilization is low, it makes sense for cloud providers to offer low-priority VMs.
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Deflatable Virtual Machines {#sec:mechanisms}
===========================
In this section we describe how VM deflation mechanisms can be implemented using existing hypervisor mechanisms.
VM Deflation Mechanisms
-----------------------
VM deflation requires the ability to dynamically shrink the resources allocated to the VM. Modern hypervisors expose interfaces to determine the current resource allocation of a VM and to dynamically modify this allocation. A cluster or cloud management framework can use these hypervisor APIs to implement VM deflation mechanisms.
Our system implements two classes of deflation mechanisms—[*transparent*]{} mechanisms, which transparently shrink the VM’s resource allocation, and [*explicit*]{} mechanisms, where the deflation is performed in a manner that is visible to the guest OS, (and by extension, to the applications and the application cluster manager). In the former case, the guest OS and applications are unaware of the deflation and the VM simply runs “slower” than prior to deflation. In the latter case, since deflation is visible to the guest OS and/or applications, they can take explicit measures, if wanted, to deal with deflation. We describe each mechanism and a hybrid approach that exploits the key benefits of both approaches.
Transparent VM Deflation
------------------------
Since hypervisors offer virtualized resources to virtual machines, they can also *overcommit* these resources by multiplexing virtual resources onto physical ones. Transparent VM deflation is implemented using these hypervisor overcommitment mechanisms. For example, the hypervisor allows virtual CPUs (vCPUs) of the VM to be mapped onto dedicated physical CPU cores. Such an allocation can be deflated by remapping the vCPUs onto a smaller number of physical cores using the hypervisor’s CPU scheduler. Thus the guest OS and applications inside the VM still see the same number of vCPUs, but these vCPUs run slower.
In the case of memory, hypervisors allocate an amount of physical memory to a VM and multiplexes the VM’s virtualized memory address-space onto physical memory, via two-dimensional paging. Memory deflation thus involves dynamically reducing the physical memory allocated to a VM.
In the case of network, one or more logical network interfaces of a VM are mapped onto one or more physical NICs and a certain bandwidth of the physical NICs is allocated to each vNIC by the hypervisor. Network deflation involves reducing the physical NIC bandwidth allocated to the VM. Finally, in the case of local disks, the I/O bandwidth allocated to each VM can be throttled. With the above hypervisor level transparent techniques, the VM and applications are oblivious of the deflation, which is done at the hypervisor level outside of the VM. The VM may get scheduled at a lower frequency or have less physical memory, etc. Our deflation framework has been implemented in KVM and Linux using Linux’s cgroups facility. By running KVM VMs inside of cgroups, we can control the physical resources available for the VM to use. For deflating CPUs, we use CPU bandwidth control by setting the CPU shares of the deflatable VM. The memory footprint of a deflatable VM is controlled by restricting the VM’s physical memory allocation by setting the memory limit in the memory cgroup. Similarly for disk and network I/O, we use the respective I/O cgroups to set bandwidth limits.
Explicit Deflation via Hotplug
------------------------------
Modern virtualization environments now support the ability to explicitly hot plug (and unplug) resources from running guest operating systems. Explicit deflation mechanisms use these hot unplug techniques to reduce the VM’s allocation in a manner that is visible to the guest OS and the applications. In the case of CPU, if a VM has $n$ vCPUs allocated to it, its CPU resources are reclaimed by unplugging $k$ out of $n$ vCPUs. Hot plugging and unplugging requires guest OS support, since it must reschedule/rebalance processes and threads to a smaller or larger number of cores. Thus, the deflation is visible to the guest OS and applications. In the case of memory, we use memory unplugging to inform the OS and applications of the resource pressure, which allows them to return unused pages, shrink caches, etc. Explicit unplugging of NICs and disks is generally unsafe, and we rely on the transparent hypervisor-level mechanisms for these.
Hot unplugging has a safety threshold—unplugging too many resources (e.g., too much memory) beyond this safety threshold can cause OS or application failures. Furthermore, hot unplug can only be done in coarse-grained units. For example, it is not possible to unplug 1.5 vCPUs.
Hybrid Deflation Mechanisms
---------------------------
Both transparent and explicit deflation have advantages and disadvantages. Explicit deflation—by virtue of being visible, allows the OS and applications to gracefully handle resource deflation. However, deflation can only be done in coarse-grained units and has a safety threshold. Transparent deflation can be done in more fine-grained slices and has a much broader deflation range than explicit deflation. It does not require any guest OS support but can impose a higher performance penalty since the OS and applications do not know that they are deflated.
Our hybrid deflation technique combines both mechanisms to exploit the advantages of each. Initially, a VM is deflated using explicit deflation until its safety threshold is reached for each resource. From this point, transparent deflation is used for further resource reclamation to extract the maximum possible resources from the VM under high resource pressure. Figure \[fig:hybrid-code\] presents the high-level pseudo-code of our hybrid deflation approach. The key challenge is to determine the hot unplug safety threshold so as to switch over from explicit to transparent deflation.
``` {.numberLines .python language="Python" numbers="left" frame="single" basicstyle="\scriptsize\sffamily"}
def deflate_hybrid(target):
hotplug_val = max(get_hp_threshold(), round_up(target))
deflate_hotplug(hotplug_val)
deflate_multiplexing(target)
```
For deflating CPUs, we first set the hotplug target by rounding up the target number of vCPUs (line 2 in Figure \[fig:hybrid-code\]). Then the cgroups based CPU multiplexing deflation can deflate the VM the rest of the way. The hotplug operation may not always succeed in removing all the CPUs requested—the guest OS unplugs the CPU only if it is safe to do so. If the number of reclaimed CPUs via hotplug is less than the number requested, then the multiplexing-based CPU deflation takes up the slack. When deflating memory, we set the hotplug threshold by using the guest OS’s resident set size (RSS)—since unplugging memory beyond the RSS results in guest swapping, and we presume that it is safe to unplug as long as the VM has more memory than the current RSS value.
Our hybrid deflation mechanisms can be used to reclaim significant amounts of CPU, memory, and I/O resources from applications. When deflating memory, hybrid deflation allows the guest OS to hot-unplug unused memory, which can improve performance, as shown in Figure \[fig:jbb-mem\]. The figure shows the mean response time with the SpecJBB 2015 benchmark, and we see that the performance with both transparent and hybrid deflation is largely unaffected up to 40% deflation, and hybrid deflation improves performance by about 10%. Additional results with CPU deflation and with other applications are presented later in Section \[sec:eval\].
![Performance of SpecJBB 2015 with transparent and hybrid memory deflation.[]{data-label="fig:jbb-mem"}](graphs_jbb-hv-hybr.pdf){width="30.00000%"}
Cluster Deflation Policies {#sec:policies}
==========================
In this section, we describe how the mechanisms discussed in the previous section can be used to implement cluster-level deflation policies. We assume a cloud resource management framework that multiplexes physical servers in the cluster across two pools of VMs: non-deflatable higher-priority VMs and deflatable lower-priority VMs. When there is surplus capacity in the cluster, the cloud manager allocates these resources to lower priority VMs (without deflating them). When demand from higher-priority VM causes resource pressure, resources from lower priority VMs are reclaimed using deflation and re-assigned to higher priority VMs. Below, we describe *policies* for doing so that determine how much each VM is actually deflated by, and under what conditions. Our policies assume the worst-case linear correlation between deflation and performance, as shown by Figures \[fig:util-all\] and \[fig:underalloc\]. Which policy to apply we leave up to cloud providers as they have different trade-offs and capabilities that we discuss in Section \[sec:clust-policy-eval\]. The policies we propose are implemented at the level of a physical server. That is, the deflation of a VM is determined by the “local” conditions and the resource profiles of co-located VMs.
Server-level Deflation Policies {#subsec:server-deflation}
-------------------------------
Our system uses three policies for deflation–proportional, priority-based and deterministic—that we describe below.
### Proportional Deflation
In the simplest case, we assume that all VMs that fall into two broad classes: high-priority non-deflatable VMs (aka on-demand), and low-priority deflatable VMs. A server may host VMs of both classes.
Proportional deflation involves deflating each low priority VM in proportion to its original maximum size. More formally, suppose we need to reclaim $R$ amount of a particular resource (CPU, memory, etc.) from $n$ deflatable VMs, and suppose $M_i$ is the original undeflated allocation of that resource allocated to VM $i$. The proportional deflation policy reclaims $x_i$ amount from each VM $i$: $$x_i = M_i - \alpha_1 \cdot M_i,
\label{eq:simple-prop}
\vspace*{-3pt}$$ where $\alpha_1$ is determined by the constraint that $\sum x_i = R$, and is given by $\alpha_1 = 1-({R}/{\sum_i^n M_i})$. Intuitively, we want VMs to deflate in proportion to their size, to avoid excessively deflating small VMs. Note that a new incoming VM may be deflatable, and is included in the pool of $n$ deflatable VMs, and can thus start its execution in a deflated mode under high resource pressure conditions.
This simple proportional deflation policy forms the basis of more sophisticated policies for addressing various cluster management requirements. For instance, some VMs may have a “limit” to their deflatability or QoS minimum requirements if deflated by more than, say, 80%. Applications can provide these requirements to the cluster on provisioning. The cluster manager enforces the minimum resource allocation ($m_i$) with proportional deflation policy, and reclaim resources from each VM using the following relation: $$x_i = (M_i - m_i) - \alpha_2 \cdot ({M_i-m_i}) \\
\label{eq:min-prop}
\vspace*{-3pt}$$
The proportional deflation is performed for each resource (CPU, memory, disk bandwidth, network bandwidth) individually. Enforcing the minimum resource allocation limits can minimize application performance degradation, but can reduce the overcommitment (and possibly revenue) of cloud platforms.
### Priority-based Deflation
Since the impact of deflation is application dependent, a cloud platform can offer multiple classes of deflatable VMs. These priority levels influence how much each VM is deflated by, and can be offered by cloud providers at different prices. These priority classes can be chosen by the user based on their price sensitivity and application characteristics.
The proportional deflation policy (Equation \[eq:simple-prop\]) can be extended to incorporate priorities through a weighted proportional deflation framework. Let $\pi_i\in (0,1)$ be the priority level of VM-$i$. Then, $$x_i = M_i - \alpha_3 \cdot \pi_i \cdot M_i ,
\label{eq:prio-simple}
\vspace*{-3pt}$$ where low $\pi_i$ values indicate lower priority and higher deflatability.
VM priorities can also be applied to determine the minimum resource allocation levels ($m_i$) of the VMs. Intuitively, VMs with a higher priority ($\pi_i$) have a lower deflation tolerance, and thus larger $m_i$ values. For instance, cloud platforms can determine the VM’s minimum resource allocation level as: $m_i = \pi_i \cdot M_i$, and we can then extend the minimum-level-aware deflation (Equation \[eq:min-prop\]) with weighted proportional deflation:
$$x_i = (M_i - \pi_i M_i) - \alpha_4 \cdot \pi_i ({M_i-\pi_i M_i})
\label{eq:prio-all}
\vspace*{-3pt}$$
### Deterministic Deflation
With the above proportional deflation policies, a VM’s deflation level is determined dynamically based on the local resource pressure on the server. In some cases, cloud platforms and applications may require a more deterministic deflation policy, that only deflates VMs to a pre-specified level. VM priorities can be used for determining the deflation levels of VMs—with higher priorities ($\pi_i$) indicating lower deflation. In this case, deflation is binary: either the deflatable VMs are allocated 100% of their resource allocation ($M_i$), or $\pi_i\cdot M_i$. In case of multiple deflatable VMs on a server, VMs are deflated in decreasing order of $\pi_i$’s until sufficient resources are reclaimed to run the new VM.
**Reinflation:** Both our proportional and priority-based policies can also reinflate previously deflated VMs when additional resources become available. When $R_{\text{free}}$ additional resources have become available, we reinflate VMs proportionally by setting $R = -R_{\text{free}}$ in equations \[eq:simple-prop\], \[eq:min-prop\], \[eq:prio-simple\], \[eq:prio-all\], and effectively run the proportional deflation backwards in all the cases. For deterministic deflation, the highest priority VMs are reinflated first.
Deflation-aware VM Placement {#subsec:vm-placement}
----------------------------
The initial placement of VMs onto physical servers also affects their deflation. Conventionally, for non-deflatable VMs, bin-packing based techniques are used by cluster managers to place VMs onto the “right” server in order to minimize fragmentation and total number of servers required. This is often solved through multi-dimensional bin-packing lens. The VM’s CPU, memory, disk and network resource needs as well as the resources available on each server are multi-dimensional vectors. Policies such as best-fit or first-fit can be used to choose a specific server. We use the notion of “fitness” for placing VMs onto a server. Similar to [@tetris], we use the [*cosine similarity*]{} between the demand vector and the availability vector to determine fitness: $ \text{fitness}(\mathbf{D}, \mathbf{A_j}) = \frac{\mathbf{A_j} \cdot \mathbf{D}}{|\mathbf{A_j}||\mathbf{D}|}$. Here, $\mathbf{D}$ is the demand vector of the new VM, and $\mathbf{A_j}$ is the resource availability vector of server $j$. If $A_j = 0$, i.e. there are no available resources, a small value $\epsilon$ can be added to it, or the server can be removed from consideration, to prevent division by 0. The availability vector is given by $A_j = \text{Total}_j - \text{Used}_j + (\text{deflatable}_j / \text{overcommitted}_j)$, where $\text{deflatable}_j$ is the maximum amount of resources that can be reclaimed by deflation and $\text{overcommitted}_j$ is the extent of the deflation already done. By evaluating all severs and considering their level of overcommitment, this approach prefers servers with lower overcommitment, and thus achieves better load balancing.
### Placement With Cluster Partitions.
The above VM placement approach results in VMs of different priority levels sharing physical servers. This “mixing” can be beneficial and improve overall cluster utilization, since lower priority VMs can be deflated to make room for higher priority VMs. However, increasing the number of co-located deflated VMs can potentially result in higher performance interference (aka noisy neighbor effect).
While performance interference can be mitigated through stronger hypervisor and hardware-level isolation techniques, it can also be addressed by VM placement. The key idea is to *partition* the cluster into multiple priority pools, and only place VMs in their respective priority pools. Within a pool, we use the bin-packing approach for deflatable VMs and continue to use either proportional or deterministic deflation policies on the individual servers. The size of the different pools can be based on the typical workload mix.
Thus, higher priority VMs will generally run on servers with lower overcommitment and lower risk of performance interference, and lower priority VMs face higher risk of overcommitment. This approach also allows cloud operators to limit and control the distribution of overcommittment of different servers, which reduces the risk of severe performance degradation due to overcommitment.
A possible downside of cluster partitions is that if a partition becomes “full” even after deflating all its VMs to their maximum limits, new VMs may have to be rejected using the admission control mechanism. This can reduce cluster overcommitment and revenue.
### Pricing Considerations
Our work assumes that deflatable VMs are priced differently from traditional on-demand VMs. Similar to preemptible VMs, a cloud provider may choose to offer deflatable VMs at fixed discounted prices (e.g., at 60-80% discount). The cloud provider may also price deflatable VMs based on priority levels, where the priority level determines the proportion by which VM can be deflated and also the discount in the price. Finally, the cloud provider may use variable pricing where the deflatable VM is billed based on the actual allocation of resources over time, with lower prices charged during periods of deflation. The different pricing policies, when combined with placement and server-level deflation policies, result in different levels of application performance, cluster utilization, and revenue. These tradeoffs are presented in the evaluation section.
Implementation {#sec:impl}
==============
We have implemented all the deflation mechanisms and policies discussed in Sections \[sec:mechanisms\]-\[sec:policies\] as well as deflation-aware web applications, as part of a deflation-aware cluster manager framework. Our system is comprised of two main components (see Figure \[fig:defl-over\]). A centralized cluster manager implements and invokes the VM placement policies and generally controls the global-state of the system. In addition, we run local deflation controllers that run on each server. These local controllers control the deflation of VMs by responding to resource pressure, by implementing the proportional deflation policies described in \[sec:policies\]. Both the centralized cluster manager and the local-controllers are implemented in about 4,000 lines of Python and communicate with each other via a REST API.
[**Deflation Mechanisms.**]{} Our prototype is based on the KVM hypervisor [@kivity2007kvm], and uses the libvirt API for running VMs and for dynamic resource allocation required for deflation. Our hybrid resource deflation mechanisms presented in \[sec:mechanisms\] are implemented by the per-server local controller. CPU and memory hot-plugging (and unplugging) are performed via QEMU’s agent-based hotplug. Hotplug commands are first passed to the user-space QEMU agent, which then forwards them to the guest OS kernel. Thus, the guest OS is made aware of the deflation attempt, and knows the unplug is not due to hardware-failure, and allows the hotplug to be “virtualization friendly”. For example, if the guest kernel cannot safely unplug the requested amount of memory, the hot unplug operation is allowed to return unfinished. In this case, the memory reclaimed through hot plug will be lower, but the safety of the operation is maintained.
For hypervisor level multiplexing of resources, we run KVM VMs inside cgroups containers, which allows us to multiplex resources. For CPU multiplexing, we adjust the cgroups cpu shares of the VM through libvirt’s cgroups API. For transparent memory deflation, we adjust the VM’s physical memory usage by setting the memory usage of the cgroup (). Disk and network bandwidth are also dynamically adjusted via libvirt API’s.
**Deflation Policies.** The server-level deflation policies are implemented by a local deflation controller on each server, which maintains and manager all aspects of the server’s resource allocation state, and determines deflation amounts of different VMs. Each server updates the central master about all changes in server utilization after every deflation event. New VMs are placed on servers using a three-step approach. First, the centralized cluster manager finds the “best” server for the VM based on the VM size and utilizations of all servers. The second step involves the server computing the deflation required to accommodate the new VM. If this violates any resource constraint, then the server rejects the VM. Finally, the actual deflation is performed and the VM is launched.
**Deflation-aware Web Cluster:** When running web clusters on deflatable VMs, the load balancer can be made deflation aware for improved performance. The load balancer can adjust the number of requests sent to a VM based on its deflation level. We implement a deflation-aware load-balancing policy in HAProxy [@haproxy]. We have modified HAProxy’s Weighted Round Robin algorithm by dynamically changing the weights assigned to the different servers based on the current deflation level, which adjusts the number of requests sent to each server based on the “true” resource availability. The load balancer changes are implemented in Python and Kotlin in a total of 300 LOCs and are wrapped in a Docker container.
Experimental Evaluation {#sec:eval}
=======================
In this section, using testbed experiments and simulation, we show the performance of deflatable VMs, and focus on answering the following questions:
1. What is the performance of interactive applications when deployed on deflatable VMs?
2. What is the impact of deflation policies on cluster utilization, application throughput, and cloud revenue?
Evaluation Environment
----------------------
### Web-based interactive applications.
![The micro-service architecture of the social network application used in our evaluation (Courtesy of [@gan2019open]).[]{data-label="fig:death"}](Figs_socialNet_arch.png){width="45.00000%"}
We use two interactive applications to evaluate deflation on real-world web workloads:
**Wikipedia:** We replicate the German Wikipedia on our local testbed. We choose the German Wikipedia as it is the second most popular Wikipedia in terms of number of views—with more than 720000 page views per hour—, and the fourth most in terms of number of articles—with more 2.25 Million articles [@WikiStats]. We setup a KVM VM with MediaWiki, MySQL database, Apache HTTP webserver, and Memcached. Our workload generator randomly selects from the top 500 largest pages (page sizes ranging from 0.5–2.2 MB).
**DeathStarBench** is a recently released benchmark that implements different applications using the microservice architecture [@gan2019open]. We evaluate the benchmark’s social networking application, which consists of 30 microservices (Figure \[fig:death\]) built using Redis, Memcached, MongoDB, RabbitMQ, Nginx, Jaeger, and other custom made services that provide the required functionality. We run each micro-service runs in a separate Docker container using using Docker swarm. We use a workload generator based on wrk2 [^1] for evaluating the overall application performance.
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### Cluster-level simulation framework.
To analyze various cluster-level deflation policies, we have developed a trace-driven discrete event simulation framework that allows us to understand the impact on application and cloud-level metrics. The simulation framework is written in Python in about 2000 lines of code, and implements our VM deflation and pricing policies, and allows large-scale simulations with different policy and workload combinations. We use the Azure VM-level dataset to determine the starting and stopping times of VMs, their size (aka resource vectors), and CPU utilization history. We also use the VM metadata such as VM category (batch, interactive, unknown), and the 95-th percentile CPU utilization to determine priority levels for our priority-based deflation policies. The simulation framework allows us to determine the deflation levels of VMs, preemptions in the cluster, and also correlate VM’s dynamic resource allocation with its CPU utilization time-series to determine the performance impact of deflation. For the simulation-based cluster-level experiments, we primarily focus on the effect of deflation on the cloud provider. This complements our application-focused performance evaluation done using web services in the next subsection, as well as prior work on deflation [@deflation-eurosys19] that looked at performance of distributed applications under deflation.
Given our focus on deflatability of interactive applications, we assume that the interactive VMs in the trace are deflatable, while the unknown and batch VMs are non-deflatable (“on-demand”). This translates to roughly 50% of the VMs being deflatable. We consider each VM’s CPU core count and memory size for bin-packing as well as all deflation policies. We determine VM priorities based on their 95-th percentile CPU usage and use 4 priority levels. We show results on a randomly sampled trace of 10,000 VMs, which require a cluster of 40 servers each with 48 CPUs and 128 GB RAM. For simulating varying degrees of cluster overcommitment, we first find the minimum cluster size capable of running all VMs without any preemptions or admission-controlled rejections. We then vary and increase the overcommitment by reducing the number of servers and use the same VM-trace throughout for all the experiments. We do not look at the impact on individual application performance in a cluster settings for two reasons 1) the cluster level impact of deflation was examined in [@deflation-eurosys19] and 2) we want to focus on the effect of our deflation policies on large-scale cluster management.
VM deflation of Web services
----------------------------
Our first set of experiments aim to measure the effect of *transparent* deflation on the performance of different types of web services, and how the reduction in resource allocation can be mitigated by well-engineered web applications.
**Multi-tiered Applications.** In order to evaluate the effects of deflation on the QoS of multi-tiered services, we use the German Wikipedia replica running on a VM with 30 CPU cores, and 16 GB of memory. We subject it to a mean load of 800 requests/s selected randomly from the 500 largest pages. We set the request time out period to 15 seconds, and consider that requests that take longer are dropped, or no longer interesting to the users. We progressively deflate the VM’s CPU for this CPU-bound application. Figure \[fig:wikiviolin\] shows a violin plot of the distribution of the response times of the requests at each deflation level, with the y-axis in log-scale. As shown, the response time does not increase significantly until the deflation increases above 70%—even though the average CPU usage at 50% deflation is 100%. We find that the average response time for the application with no deflation is 0.3s, with 50% deflation is 0.45s, and with 80% deflation is 0.6s—which is $2\times$ the undeflated response time. The 99th percentile response time is 6.8s for no deflation, and increases by only 43% to 9.74s even at 80% deflation. We also find that, even when deflated to a single core, the application did not crash when serving a load of 800 req/s. This leads us to believe that many well architected web services tolerate deflation well, with a disproportionately small performance penalty. This observation is further reinforced by Figure \[fig:wikiloss\], which shows the percentage of requests served with different deflation settings. Similar to our previous result, we see that noticeable request loss rates occur only after 70% deflation.
**Micro-service based Applications.** We next evaluate the impact of transparent deflation on micro-service based applications. Figure \[fig:death\] shows the architecture of the social networking application described previously. The application microservices can be classified based on their functionality into three logical classes that are similar to multi-tiered applications, namely, frontend microservices, logic microservices, and finally, caching and storage microservices. In the social networking service used, there are three frontend microservices, 15 logic microservices, and 12 backend microservices. In our deflation experiment, we deflate all microservices except for the databases, i.e., we deflate all frontend and logic microservices, and the four memcached microservices from the backend, deflating a total of 22 microservices out of 30. We start by allocating a maximum limit of 2 cores per microservice, and a minimum of 0.05 CPUs for each container. Each container is allocated 800MB of memory. We use the workload generator to generate 500 requests per second, and deflate the 22 microservices by 30%, 50%, 60% and 65%. Figure \[fig:uservices\] shows the median, 90th%, and 99th% response times in milliseconds. We again see that the service can be deflated by up to 50% with no performance losses. Beyond this level, the degradation in QoS and response time is more abrupt than the the multi-tiered Wikipedia case, likely due to the higher communication- and coordination-intensive nature of the application.
Deflation-aware Web Load Balancing
----------------------------------
Next, we evaluate the effect of *explicit* deflation on clustered web services. To do so, we compare the performance of using vanilla HAProxy [@haproxy] with our modified deflation-aware HAProxy. We run three replicas of the German Wikipedia application behind HAProxy. Each instance starts with 10 vCPU cores, and 10 GB of memory. We assume that two of these instances are running on deflatable VMs , and the third runs on a non-deflatable VM.
We generate an average load of 200 requests/s and deflate the two deflatable VMs equally. Our deflation-aware load balancer attempts to masks the impact of deflation by changing the server weights based on the deflated number of vCPUs, causing more requests to be sent to the third undeflated replica. Figure \[fig:lbc\] shows the average and 90th percentile response times for the unmodified and deflation-aware load balancers. We see that the deflation-aware load balancer yields 15 to 40% lower tail latency at high deflation levels of 40 to 80% when compared to vanilla load balancing; mean response times are also lower or comparable as shown in the figure.
![Our deflation-aware load balancer yields lower response times even at high deflation levels.[]{data-label="fig:lbc"}](graphs_lb-compare2.pdf){width="38.00000%"}
Impact Of Cluster Deflation Policies
------------------------------------
\[sec:clust-policy-eval\] We now evaluate the effect of VM deflation at a cluster level using trace-driven simulations. We are interested in the differences with current transient server offerings that rely on preemptions, and the impact of the different deflation policies on cluster overcommitment, VM performance, and cloud revenue.
![Failure probability with deflation remains very low even for high cluster overcommitment.[]{data-label="fig:preemptions"}](Figs_num-preemptions-10000-vector.pdf){width="35.00000%"}
### Eliminating Preemptions.
VM deflation is intended to eliminate preemptions, which are detrimental to interactive applications because they cause downtimes. Currently, cloud operators preempt low-priority VMs when there is high resource pressure, which increases at high cluster overcommitment levels. Figure \[fig:preemptions\] shows the failure probability for low-priority VMs under different overcommitment levels. Failure probability represents the probability of failure to reclaim sufficient resources from deflatable VMs due to “too much” overcommitment; for traditional preemptible instances, it is same as preemption probability. Even at 70% overcommitment, the failure probability is below 1% for proportional deflation, compared to 35% for preemptible VMs. From a provider standpoint, this implies that they can reclaim the desired amount of resources via deflation with $>$0.99 probability. The priority-based and deterministic deflation policies have higher failure probability than proportional but still below preemptible VMs. More broadly, this result shows that a judicious choice of overcommitment level (of as much as 50%) allows the provider to eliminate preemptions and use deflation to reclaim the necessary resources under resource pressure.
**Result:** *Deflatable VMs have very low probability of resource reclamation failure even when the overcommitment is as high as 50%*
### Throughput.
![Decrease in throughput of deflatable VMs is low even at high overcommitment.[]{data-label="fig:tputloss"}](Figs_throughput-loss-10000-vector.pdf){width="35.00000%"}
While deflation can eliminate preemptions, it comes with an important tradeoff: the reduction in resource allocation due to overcommitment can reduce application performance and throughput. We examine the effect of deflation on VM performance at a cluster level, using the CPU-traces of the Azure VMs. Note that a VM’s deflation is *dynamic* and based on the time-varying resource pressure conditions as VMs are launched and terminated. At a given point in time, the performance depends on the deflation and the VM’s resource utilization. Thus if the VM is deflated when its resource (CPU) utilization is low, then we are reclaiming unused resources (i.e., slack), and there should be no drop in throughput. The loss in throughput only occurs when a VM is deflated below its CPU usage, and is proportional to the total underutilization (area under the curve of Figure \[fig:underalloc\] in Section \[sec:feasibility\]. Based on this principle, Figure \[fig:tputloss\] shows the decrease in throughput for the different deflation policies at varying overcommitment levels.
We see negligible reduction in throughput below 40% overcommitment, and a 1% reduction at 50% overcommitment. Even at 80% overcommitment, the loss in throughput is below 5% for all deflation policies. We note that this is fundamentally due to the low utilization of VMs of the Azure VMs (especially interactive VMs), as was shown earlier in Figure \[fig:bp-over-thresh\]. Additionally, the average VM deflation is *not* equal to the cluster overcommitment but is significantly lower. Our cluster was provisioned for the *peak* load, and furthermore, deflatable VMs significantly improve the bin-packing efficiency by allowing the cluster manager to slightly adjust VM allocations to make room for new VMs that would have otherwise not fit and required an additional server.
The priority-based and deterministic deflation policies take into account the VM’s anticipated utilization levels by using their 95 percentile CPU usage to determine the deflation priority and the minimum allocation levels. Thus, high utilization VMs are deflated less, which reduces their loss in throughput compared to simple proportional deflation. Thus, we see that adding priorities can reduce the loss in throughput by an order of magnitude. When we place VMs into dedicated cluster partitions based on their priority (as described in Section \[subsec:vm-placement\]), Figure \[fig:tputloss\] also shows that incorporating partitioning does not significantly impact throughput loss. Cluster-partitioning is thus a viable technique that can be used by cloud operators to minimize the risk of performance interference among deflatable VMs of different priorities.
Interestingly, deterministic deflation, which deflates VMs in their priority order, has the lowest decrease in throughput. This is because the proportional deflation policies (both the simple and priority-based proportional) result in deflation of *all* VMs, even though the magnitude of deflation of each VM is small. Thus, even high priority deflatable VMs are deflated, and their throughput will decrease if their CPU utilization is higher than the deflated allocation. With deterministic deflation, the lower priority VMs (with lower 95 percentile CPU usage) are penalized more, but the average cluster-wide throughput loss is reduced.
**Result:** *Deflatable VMs allow clusters to be overcommited by 80%, and keep the performance degradation to less than 5%.*
**Impact on Quality of Experience.** The low average loss in throughput represents a low risk of QoS violations, since performance is affected only when the application’s peak usage coincides with deflation. However, end-users of interactive applications may observe a perceivable drop in their quality of experience due to the jitter and the longer response times during deflation. Ultimately, evaluating the user experience with deflation requires user studies similar to [@skype-study], and is a potential candidate for future work. End-users can be alerted with a “degraded mode” warning during periods of high deflation, similar to downtime indicators for popular web services. Finally, we note that distributed applications can also run on a mix of non-deflatable and deflatable VMs with different priorities (similar to [@spotweb-hpdc19]), and reduce the risk of QoS violations even further.
### Cloud Revenue.
![Increase in cloud revenue due to deflatable VMs.[]{data-label="fig:revenue"}](Figs_revenue-vector.pdf){width="35.00000%"}
We have seen how deflatable VMs can minimize preemptions and have negligible impact on performance of interactive applications. Since deflation allows for increased overcommitment, it provides cloud platforms the opportunity to increase their revenue on low-priority resources. Figure \[fig:revenue\] shows the increase in revenue from the low-priority (i.e., deflatable) resources, at different cluster overcommitment levels for different combinations of deflation and pricing policies. For ease of exposition, we assume that the static price of deflatable VMs is $0.2\times$ the on-demand price—corresponding to the discounts offered by current transient cloud servers such as EC2 spot instances, Google Preemptible VMs, and Azure Low-priority Batch VMs. For VMs with different deflation priorities, we set their price equal to the priority—i.e, priority-level 0.5 has price $0.5\times$ the on-demand price, etc. We also evaluate variable allocation-based pricing which considers the actual resource allocation over time, and again price resources linearly (i.e, VMs pay half price when at 50% allocation).
Figure \[fig:revenue\] shows that as the cluster overcommitment increases, the revenue with static-pricing VMs increases, and the cloud platform can increase revenue by 15% at 60% overcommitment. Having priority-based differentiated pricing significantly increases the revenue, since higher priority VMs pay more. The priority-based pricing (when used with priority-based deflation) increases the revenue per server by $2\times$ compared to simple static pricing.
Interestingly, the revenue with allocation-based pricing scheme, which charges VMs what they were actually allocated, does not increase with increasing overcommitment. This is because at low overcommitment levels, VMs are not deflated and thus pay “full price”, and as the overcommitment increases, there are more VMs running per server, but they are highly deflated, and thus the total revenue remains the same.
**Policy Comparison:** *Deflation policies have different tradeoffs. Proportional deflation minimizes resource reclamation failure, but provides lower revenues. Priority-based deflation and pricing increases revenue, but also increases failure probability.*
Related Work {#sec:related}
============
VM deflation draws upon many related techniques and systems.
**Systems for handling transient server revocation** use a combination of fault tolerance and resource allocation to mitigate the performance and cost effects of preemptions. Prior work has focused on system [@spotcheck; @spoton] and application [@flint; @exosphere; @marathe2014exploiting; @pado-eur17; @proteus-eur17; @conductor] support for handling preemptions. We believe that deflatable VMs minimize the need for such middleware, and can avoid the performance, development, and deployment costs associated with preemption.
**Resource overcommitment mechanisms** have been extensively studied and optimized to allow for more efficient virtualized clusters. Memory overcommitment typically relies on a combination of hypervisor and guest OS mechanisms, and has received significant attention [@waldspurger2002memory; @amit2014vswapper; @singleton]. Memory ballooning is another memory overcommitment technique with generally inferior performance to hotplug [@fraser-ballooning-hotplug; @liu2015hotplug]. Hotplug can also be used for reducing energy consumption [@zhang2014dimmer], since unused but powered-on RAM draws a significant amount of energy. CPU hotplugging can also be used to mitigate lock-holder preemption problems in overcommitted vCPUs [@ding2014gleaner; @ouyang2013preemptable]. Burstable VMs [@ec2-burstable; @bhuvan-burstable] also offer dynamic resource allocation, but are the “inverse” of deflatable VMs. The resource allocation is high by default for deflatable VMs and only reduced during resource pressure, whereas burstable VMs have low allocation by default and only ocassionally can be “inflated” to higher allocations. Furthermore, burstable VMs have been restricted to CPU and I/O bursting, whereas deflatable VMs also adjust memory.
**Resource consolidation** using dynamic resource allocation [@borg] and VM migration [@wood2009sandpiper] is common to increase cluster utilization. VMWare’s distributed resource scheduler [@vmware-drs] uses per-VM reservations (minimum limits) and shares for dynamically allocating resources—similar to our resource-pressure based local deflation policies. Many approaches for performance-sensitive resource allocation among co-located VMs have been suggested [@liu2014reciprocal; @moldable-vms; @zhou2010vmctune; @stopgap-elastic; @elasticity-driver-vee15], but they assume some application performance model, which our work does not. VM memory allocations can be set using working-set estimation [@zhang2016iballoon; @chiang2013working; @zhao2009dynamic], utility-maximizing [@hines2011-ginko], or market-based approaches [@agmon2014ginseng; @nom-vee]. As noted earlier, deflation was first proposed in [@deflation-eurosys19] but required OS and application cooperation, while we focus on a hypervisor-only deflation approach.
**Vertical scaling with performance differentiation** for a single server under resource pressure due to increasing application load and server overbooking has been well studied in the past [@lakew2015performance; @Padala:2009; @rao2013qos]. All previous work we are aware of tackles the problem of performance differentiation for a single server. Our work focuses on cluster-wide performance optimization when resources are deflated across the whole cluster. Application performance models and workload prediction is a key component of elastic scaling [@gong2010press; @nguyen2013agile; @padala2007adaptive; @shen2011cloudscale; @ali2012adaptive]. In contrast, deflation is a black-box, application agnostic, and reactive technique for handling resource pressure. Our deflatable VMs use a combination of overcommitment mechanisms that are adapt to application resource usage, and we consider the simultaneous deflation of *all* resources. Deflation also exposes an explicit performance tradeoff, whereas elastic scaling approaches typically only reclaim unused resources.
Conclusions
===========
\[sec:conclusions\]
In this paper we proposed the notion of deflatable VMs for running low-priority interactive applications. Deflatable VMs allow applications to continue running on transient resources, while minimizing the risk of preemptions and the associated downtimes. Our VM deflation mechanisms and cluster-level deflation policies reduce the performance overhead of applications and allow cloud platforms to increase cluster overcommitment and revenue. The performance of deflatable VMs is within 10% of their undeflated allocation—making them a viable alternative to current cloud transient VMs.
**Acknowledgments.** We wish to thank all the anonymous reviewers and our shepherd Renato Figueiredo, for their insightful comments and feedback. This research was supported by NSF grants 1836752, 1763834, and 1802523.
[^1]: https://github.com/giltene/wrk2
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---
abstract: 'This article examines large time behaviour and the second eigenvalue problem for Markovian mean-field interacting particle systems with jumps. Our first main result is on the time required for convergence of the empirical measure process of the particle system to its invariant measure; we show that, there is a constant $\Lambda \geq 0$ such that, when there are $N$ particles in the system and when time is of the order $\exp\{N(\Lambda+O(1))\}$, the process has mixed well and is very close to its invariant measure. We then obtain large-$N$ asymptotics of the second eigenvalue of the generator associated with the empirical measure process when it is reversible with respect to its invariant measure. We show that its absolute value scales like $\exp\{-N\Lambda\}$. The main tools used in establishing our results are the large deviation properties of the empirical measure process from its large-$N$ limit. As an application of the study of large time behaviour, we also show convergence of the empirical measure of the system of particles to a global minimum of a certain entropy function when particles are added over time in a controlled fashion. The controlled addition of particles is analogous to the cooling schedule associated with the search for a global minimum of a function using the simulated annealing algorithm.'
address:
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Department of Electrical Communication Engineering\
Indian Institute of Science\
Bangalore 560012, India\
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Department of Electrical Communication Engineering\
and Robert Bosch Centre for Cyber Physical Systems\
Indian Institute of Science\
Bangalore 560012, India\
author:
-
-
bibliography:
- 'ims-template.bib'
title: 'Large Time Behaviour and the Second Eigenvalue Problem for Finite State Mean-Field Interacting Particle Systems'
---
Introduction {#section:introduction}
============
In this paper, we study large time behaviour and the second eigenvalue problem for Markovian mean-field interacting particle systems with jumps. Our motivation is to provide an understanding of metastable phenomena in engineered systems such as load balancing networks [@aghajani-etal-17; @aghajani-ramanan-19; @mukhopadhyay-etal-16; @mitzhenmaker-00; @graham-00], wireless local area networks [@bhattacharya-kumar-17; @benaim-leboudec-08; @bordenave-etal-12; @kumar-etal-06; @ramaiyan-etal-08; @bianchi-98], and in natural systems involving grammar acquisition, sexual evolution [@panageas-vishnoi-16; @panageas-etal-16], epidemic spread [@leonard-90; @djehiche-kaj-95], etc. These systems are briefly described in Section \[section:examples\].
Before we describe our main contributions, let us describe the setting of our mean-field interacting particle system.
The setting
-----------
Let there be $N$ particles. Each particle has a state associated with it which comes from a finite set $\mathcal{Z}$; the state of the $n$th particle at time $t$ is denoted by $X_n^N(t) \in \mathcal{Z}$, where the superscript $N$ indicates that there are $N$ particles in the system. The empirical measure of the system of particles at time $t$ is defined by $$\begin{aligned}
\mu_N(t) \coloneqq \frac{1}{N} \sum_{n=1}^N \delta_{X_n^N(t)} \in M_1(\mathcal{Z}),\end{aligned}$$ where $\delta_{\cdot}$ denotes the Dirac measure. Here, $M_1(\mathcal{Z})$ denotes the space of probability measures on $\mathcal{Z}$ equipped with the topology of weak convergence. Note that $\mu_N(t)$ is a random measure, i.e., an $M_1(\mathcal{Z})$-valued random variable. Each particle has a set of allowed transitions; to define this, let $(\mathcal{Z}, \mathcal{E})$ be a directed graph with the interpretation that whenever $(z,z^\prime) \in \mathcal{E}$, a particle in state $z$ is allowed to move from $z$ to $z^\prime$. To specify the interaction among particles and evolution of the states of the particles over time, for each $(z,z^\prime) \in \mathcal{E}$, we are given a function $\lambda_{z,z^\prime}: M_1(\mathcal{Z}) \to [0, \infty)$. We consider the generator $\Psi^N$ acting on functions $f$ on $\mathcal{Z}^N$ by
$$\begin{aligned}
\Psi^N f(\mathbf{z}^N) = \sum_{n=1}^N \sum_{z_n^\prime: (z_n, z_n^\prime) \in \mathcal{E}} \lambda_{z_n, z_n^\prime}(\overline{\mathbf{z}^N}) (f(\mathbf{z}^N_{n,z_n,z_n^\prime}) - f(\mathbf{z}^N));\end{aligned}$$
here $\overline{\mathbf{z}^N} = \frac{1}{N}\sum_{n=1}^N \delta_{z_n} \in M_1(\mathcal{Z})$ denotes the empirical measure associated with the configuration $\mathbf{z}^N \in \mathcal{Z}^N$, and $\mathbf{z}^N_{n,z_n,z_n^\prime}$ denotes the resultant configuration of the particles when the $n$th particle changes it state from $z_n$ to $z_n^\prime$.
We make the following assumptions on the model:
1. The graph $(\mathcal{Z}, \mathcal{E})$ is irreducible. \[assm:a1\]
2. The functions $\lambda_{z,z^\prime}(\cdot)$, $(z,z^\prime) \in \mathcal{E}$, are Lipschitz on $M_1(\mathcal{Z} )$ and there exist positive constants $c, C $ such that $c \leq \lambda_{z,z^\prime}(\xi) \leq C$ for all $(z,z^\prime) \in \mathcal{E}$ and all $\xi \in M_1(\mathcal{Z})$. \[assm:a2\]
Let $D([0, \infty), \mathcal{Z}^N)$ denote the space of $\mathcal{Z}^N$-valued functions on $[0, \infty)$ that are right continuous with left limits (càdlàg), equipped with the Skorohod-$J_1$ topology (see [[@ethier-kurtz Chapter 3]]{}). Note that, under the Lipschitz assumption \[assm:a2\] on the transition rates, the $D([0, \infty), \mathcal{Z}^N)$-valued martingale problem for $\Psi^N$ is well posed (see [[@ethier-kurtz Exercise 15, Section 4.1]]{}); therefore, given an initial configuration of particles $(X_n^N(0), 1 \leq n \leq N) \in \mathcal{Z}^N$, we have a Markov process $\left( (X_n^N(t), 1 \leq n \leq N), t \geq 0\right) $ on the space $D([0,\infty),\mathcal{Z}^N)$. To describe the process in words, a particle in state $z$ at time $t$ moves to state $z^\prime$ at rate $\lambda_{z,z^\prime}(\mu_N(t))$, independent of everything else; i.e., the evolution of the state of a particle depends on the states of the other particles via the empirical measure of states of all the particles, hence the name mean-field interaction.
We now describe another Markov process associated with the particle system. Consider the mapping $$\begin{aligned}
\left( (X_n^N(t), 1 \leq n \leq N), t \geq 0\right) \mapsto \left( \frac{1}{N} \sum_{n=1}^N \delta_{X^N_n(t)}, t \geq 0\right) = (\mu_N(t), t \geq 0)\end{aligned}$$ that maps the evolution of the states of all the particles to the empirical measure process $(\mu_N(t), t \geq 0)$. This process is also Markovian with state space $M_1^N(\mathcal{Z})$ – the elements of $M_1(\mathcal{Z})$ that can arise as empirical measures of $N$-particle configurations. Its generator $L^N$ acting on functions $f$ on $M_1^N(\mathcal{Z})$ is given by $$\begin{aligned}
L^N f(\xi) = N \sum_{(z,z^\prime) \in \mathcal{E}} \xi(z) \lambda_{z,z^\prime}(\xi) \left[f\left( \xi + \frac{\delta_{z^\prime}}{N} - \frac{\delta_z}{N}\right) - f(\xi) \right].\end{aligned}$$
The mean-field limit
--------------------
It often turns out that we can study various properties of our particle system by examining its macroscopic behaviour encoded in the process $\mu_N$. Towards this, we often approximate the process $\mu_N$, for large $N$, using a deterministic dynamical system. This approximation is often referred to as the mean-field limit.
Under assumptions \[assm:a1\] and \[assm:a2\], one can show the following law of large numbers [@mckean-67; @gartner-88; @sznitman-91; @benaim-leboudec-08]. Suppose that the initial conditions $\{\mu_N(0)\}_{N \geq 1}$ converge weakly to a deterministic measure $\nu \in M_1(\mathcal{Z})$. Then for any fixed $T > 0$, the empirical measure process $(\mu_N(t), 0 \leq t \leq T)$ converges in $D([0,T],M_1(\mathcal{Z}))$ to the solution to the ODE $$\begin{aligned}
\dot{\mu}(t) = \Lambda_{\mu(t)}^* \mu(t), \, 0 \leq t \leq T, \, \mu(0) = \nu,
\label{eqn:MVE}\end{aligned}$$ where, for any $\xi \in M_1(\mathcal{Z})$, $\Lambda_{\xi}$ denotes the $|\mathcal{Z}|\times |\mathcal{Z}|$ rate matrix when the empirical measure is $\xi$ (i.e. $\Lambda_\xi(z,z^\prime)= \lambda_{z,z^\prime}(\xi)$ for $(z,z^\prime) \in \mathcal{E}$, $\Lambda_{\xi}(z,z^\prime) = 0$ when $(z,z^\prime) \notin \mathcal{E}$, and $\Lambda_{\xi}(z,z) = -\sum_{z^\prime \neq z} \lambda_{z,z^\prime}(\xi)$ for $z \in \mathcal{Z}$), $\Lambda^*_\xi$ denotes its transpose, and $D([0,T],M_1(\mathcal{Z}))$ denotes the space of $M_1(\mathcal{Z})$-valued càdlàg functions on $[0,T]$ equipped with the Skorohod-$J_1$ topology (we assume that all paths are left continuous at $T$). The above ODE is referred to as the McKean-Vlasov equation. The assumption \[assm:a2\] on the Lipschitz continuity of the transition functions $\lambda_{z,z^\prime}$ implies that the above ODE is well-posed. Therefore, for large values of $N$, the above mean-field convergence result suggests that one can approximate the trajectory of the empirical measure process $\mu_N$ using the McKean-Vlasov equation, and hence, one can study properties of the process $\mu_N$ from that of the ODE (\[eqn:MVE\]).
One can also show convergence in the stationary regime. Take for instance the situation when the McKean-Vlasov equation has a unique globally asymptotically stable equilibrium. Since $\mu_N$ is a Markov process on a finite set and is irreducible (by assumption \[assm:a1\]), there exists a unique invariant probability measure for the process $\mu_N$ which we denote by $\wp_N$. (Note that this is a probability measure on $M_1(\mathcal{Z})$.) Let $\xi^* \in M_1(\mathcal{Z})$ denote the unique globally asymptotically stable equilibrium of the McKean-Vlasov equation (\[eqn:MVE\]). One can then show that $\wp_N$ converges weakly to $\delta_{\xi^*}$ as $N \to \infty$, the point mass at $\xi^*$ (see [@benaim-leboudec-08; @bordenave-etal-12]). This also suggests that we can understand the stationary behaviour of the $N$-particle system by examining the limiting behaviour of the deterministic McKean-Vlasov equation, i.e. we can effectively “interchange" the limits $N \to \infty$ and $t \to \infty$ of $\mu_N(t)$ to understand its stationary behaviour, when the McKean-Vlasov equation has a unique globally asymptotically stable equilibrium. Settings when this is not the case are what we study in this paper.
Motivation, contributions, and outline of the paper
---------------------------------------------------
We now describe the questions that we address in this paper and highlight our main results.
Suppose that the limiting McKean-Vlasov equation (\[eqn:MVE\]) has multiple $\omega$-limit sets (multiple stable equilibria and/or limit cycles). If we focus on a fixed time interval $[0,T]$, let the number of particles $N \to \infty$ and let the initial conditions $\mu_N(0)$ converge weakly to a deterministic limit $\nu$, then the mean-field convergence suggests that the empirical measure process tracks the solution to the McKean-Vlasov equation (\[eqn:MVE\]) over $[0,T]$ starting at $\nu$. If we then let $t \to \infty$, the solution to the McKean-Vlasov equation goes to an $\omega$-limit set of (\[eqn:MVE\]) depending on the initial condition $\nu$. On the other hand, for a large but fixed $N$, the process would track the McKean-Vlasov equation with very high probability, and as time becomes large, would thus enter a neighbourhood of the $\omega$-limit set corresponding to the initial condition $\nu$; however, since $N$ is finite, the process can exit the basin of attraction of this $\omega$-limit set. It is then likely to remain in a neighbourhood of another $\omega$-limit set for a large amount of time before transiting to the next one, and so on. These are examples of *metastable phenomena,* and it turns out that the sojourn times in the basin of attraction of an $\omega$-limit set are of the order $\exp\{O(N)\}$, as we shall soon see. One of the aims of our paper is to quantify such metastable phenomena, namely
1. the mean time spent by the process near an $\omega$-limit set,
2. the probability of reaching a given $\omega$-limit set before reaching another one,
3. the probability of traversing a given set of $\omega$-limit sets in a particular order,
and so on. These quantifications help predict the performance of engineering systems, some of which we will describe in Section \[section:examples\]. We study the aforementioned metastability questions in Section \[section:large\_time\_behaviour\]. Such large time phenomena for diffusion processes with a small noise parameter have been studied in the past by Freidlin and Wentzell [@freidlin-wentzell] under the “general position condition" (see [[@freidlin-wentzell Sections 6.4-6.6]]{}). Hwang and Sheu [@hwang-sheu-90] studied large time behaviour for diffusion processes under a more general setup. The key in both these works is the large deviation properties of the small noise diffusion processes over finite time durations, which have been established in [[@freidlin-wentzell Chapter 5]]{}. In this paper, we extend the analysis to Markov mean-field jump processes, specifically $(\mu_N(\cdot))_{N \geq 1}$. The tools that we use in our paper are similar to those used by Hwang and Sheu, along with the large deviation properties of our process $\mu_N$ over finite time durations, which have been established by L[é]{}onard [@leonard-95] and Borkar and Sundaresan [@borkar-sundaresan-12].
Our study of the large time behaviour of the process $\mu_N$ (i.e., (i)-(iii) above) enables us to show our first main result: there exists a constant $\Lambda \geq 0$ such that, if time is of the order $\exp\{N(\Lambda+O(1))\}$, then the process $\mu_N$ is very close to its invariant measure (see Theorem \[thm:conv\]). Conversely, when $\Lambda > 0$ and when time is of the order $\exp\{N(\Lambda - \delta) \}$ with $\delta > 0$ arbitrarily small, the process may not have reached a neighbourhood of the “most stable" equilibria, with nontrivial probability in the exponential scale (see Theorem \[thm:mixing\]). A similar result for the mean-field discrete-time setting but without the specification of the constant $\Lambda$ was established by Panageas and Vishnoi [@panageas-vishnoi-16]. Let us reemphasise that our setting is a continuous time setting. Further, our desire to identify the constant $\Lambda$ demands that we must study the large deviation asymptotics in some detail in the continuous time setting. The proof of our theorem is similar to that of [[@hwang-sheu-90 Theorem 2.1, Part I]]{}, where similar results are established for convergence to the invariant measure for small noise diffusions.
The next question that we want to understand is the asymptotics of the second eigenvalue of the generator $L^N$ of the Markov process $\mu_N=(\mu_N(t), t \geq 0)$ when it is reversible with respect to its invariant measure. This study is motivated by the fact that, for a fixed $N$, the convergence speed of the process $\mu_N$ to its invariant measure (over time) can be understood by studying the modulus of the second eigenvalue of the generator of $\mu_N$. We show that, for the same constant $\Lambda$ above, the modulus of the second eigenvalue scales like $\exp\{-N\Lambda\}$ (see Theorem \[thm:eval\_problem\]). It turns out that $\Lambda$ can be positive only when there are metastable states in the limiting dynamics. In such situations, one expects slower convergence to the invariant measure for large values of $N$. On the other hand, $\Lambda$ can be $0$, for example, when the limiting dynamics (\[eqn:MVE\]) has a unique globally asymptotically stable equilibrium; in this special case, convergence of $\mu_N$ to its invariant measure does not suffer from the slowing down phenomenon associated with positive $\Lambda$. In fact, Panageas and Vishnoi [@panageas-vishnoi-16] and Panageas et al. [@panageas-etal-16] show that the mixing time is $O(\log N)$ in the discrete-time setting. Kifer [@kifer-90] considers a more restrictive discrete-time model, which does not cover the mean-field model, and identifies the constant analogous to $\Lambda$ [@kifer-90 Theorem 4.3]. The restriction is that the state space of $\mu_N$ is the same for each $N$ and that a certain uniform finite duration large deviation principle should hold with the rate function satisfying a continuity property. One can view our result as an extension of Kifer’s [@kifer-90 Theorem 4.3] to the continuous time *mean-field* setting, where the state space of the Markov process $\mu_N$ changes with $N$. Hwang and Sheu [@hwang-sheu-90] establish a result similar to ours on the scaling of the second eigenvalue of a reversible small noise diffusion process, and our method of proof is inspired by their approach.
As an application of the large time behaviour of $\mu_N$, we also study convergence of the empirical measure process to a global minimum of a natural ‘entropy’ function when particles are injected over time at a specific rate reminiscent of the simulated annealing algorithm’s cooling schedule, $N(t) = \lfloor \frac{\log (2+t)}{c^* + \delta}\rfloor$ for a suitable $c^*$ and any $\delta > 0$. The convergence to a global minimum holds for all starting points (see Theorem \[thm:conv-globalmin\]). Such approaches may be of use in situations where a population growth schedule is applied in order to engineer the mean-field system’s move to a desired equilibrium point as time $t \to \infty$. One can also use this approach to study numerically the most likely region in which the process $\mu_N$ spends time for large values of $N$, under stationarity. Again, our proof is inspired by the analysis of the simulated annealing algorithm in [[@hwang-sheu-90 Part III]]{}.
The rest of the paper is organised as follows. In Section \[section:ldp\_finite\], we discuss a known large deviation principle for the empirical measure process $\mu_N$ over a finite time horizon. This result plays an important role in the study of large time behaviour of $\mu_N$ and the large deviation principle for the invariant measure $\{\wp_N\}_{N\geq 1}$. We then study the large time behaviour of the process $\mu_N$ in Section \[section:large\_time\_behaviour\], and prove our first main result on the proximity of the law of $\mu_N$ to its invariant measure. In Section \[section:eval\_problem\], we study the asymptotics of the second eigenvalue of the generator of the process $\mu_N$ in the reversible case. Finally, in Section \[section:conv\_global\_minimum\], we study the convergence of the empirical measure process to a global minimum of the aforementioned ‘entropy’ function when particles are injected into the system at a suitable rate.
Examples {#section:examples}
--------
The mean-field interacting particle system that we have described can be used to model many interesting phenomena that arise in various domains such as physics, engineering, biology, etc. In this section, we shall describe some applications that are relevant to communication networks and shall point to the related literature that study these applications via mean-field models. We remark that, as far as the applicability of the mean-field model is concerned, the examples and the related literature that we have mentioned below are by no means exhaustive.
The first example is load balancing in networks. We describe the simplest model, the power of two choices, studied by Mitzenmacker [@mitzhenmaker-00]. Here, each particle is a single server $M/M/1$ queue, and the state represents the number of customers waiting in the queue. In load balancing, one is interested in routing the incoming customers to an appropriate queue so as to minimise the average delay experienced by a customer. The obvious way to do this is to route the customer to a queue with the least number of waiting customers. But, since there are a large number of queues, polling all of them and finding the ones with the least number of customers is expensive. So a simple alternative is to pick a queue at random and route the incoming customer to that queue, which is studied in [@graham-00]. It turns out that, if we pick two queues at random and route the customer to the least loaded queue between the two (with ties broken uniformly at random), the delay decreases dramatically. This algorithm demonstrates the power of two choices, and the evolution of the state of each queue under this algorithm can be described using the mean-field model which has been used to analyse the delay performance [@mitzhenmaker-00]. For related problems on load balancing in networks, see Mukhopadhyay et al. [@mukhopadhyay-etal-16] who study heterogenous servers, Aghajani et al. [@aghajani-etal-17; @aghajani-ramanan-19] who study non-Markovian queues, etc., and the references therein. Note that one important difference with our setting is that the state space of a queue is countably infinite in this class of problems. The finite state space model arises in the above settings when the buffers are finite and packets arriving at a fully buffered queue are lost.
Another example arises in the modelling of a wireless local area network (WLAN). Here, each particle is a wireless node trying to access a common medium, and the state of a particle represents the aggressiveness with which a packet transmission is attempted. The nodes interact with each other via the medium access control (MAC) protocol implemented in the system. Whenever a wireless node encounters a collision due to a transmission from another node, it changes its state to a less aggressive one, and whenever it succeeds, it changes its state to a more aggressive one. Therefore, the evolution of the state of a node depends on the empirical measure of the states of all the nodes, as in our mean-field model. This model was first proposed by Bianchi [@bianchi-98] and has proved to be very successful in analysing the performance of the MAC protocol; other works that focus on the WLAN application include: Bordenave et al. [@bordenave-etal-12] who studied a two time scale mean-field interacting particle system with a fast varying background process to model partial interference among nodes, Kumar et al. [@kumar-etal-06] who used the mean-field model to study the performance of WLANs using a fixed-point analysis, Ramaiyan et al. [@ramaiyan-etal-08] and Bhattacharya and Kumar [@bhattacharya-kumar-17] who looked at the problem of short term unfairness using the aforementioned fixed-point analysis, etc. Note that our model is a continuous-time modification of the discrete-time models in the above papers. Yet the continuous time model provides accurate predictions on the discrete-time model; see [[@borkar-sundaresan-12 page 4]]{}. For a continuous-time model, see Boorstyn et al. [@boorstyn-etal-87].
Other applications that use the mean-field model include analysis and control of spread of epidemics in networks [@benaim-leboudec-08; @akhil-etal-19; @leonard-90; @djehiche-kaj-95], dynamic routing in circuit-switched networks [@anantharam-91], scheduling in cellular systems [@manjerkar-etal-14], game-theoretic modelling and analysis of behaviour of agents in societal networks [@reiffers-sundaresan; @li-etal-15], etc.
Large deviations over finite time durations {#section:ldp_finite}
===========================================
In this section, we present a large deviation principle for the process $\mu_N$ over finite time durations. This result will be used later to study the large-time behaviour of $\mu_N$ and the rate of convergence of $\mu_N$ to its invariant measure.
Fix $T>0$. We introduce some notations. Let $p_{\nu_N}^{(N)}$ denote the solution to the $D([0,T], M_1(\mathcal{Z}))$-valued martingale problem for $L^N$, i.e., the law of the empirical measure process $(\mu_N(t), 0 \leq t \leq T)$, and let $p_{\nu_N,T}^{(N)}$ denote the law of the terminal-time empirical measure $\mu_N(T) \in M_1(\mathcal{Z})$, with initial condition $ \mu_N(0) = \nu_N $. Let $\mathcal{AC}[0,T]$ denote the space of absolutely continuous $M_1(\mathcal{Z})$-valued paths on $[0,T]$ (in particular they are differentiable for almost all $t \in [0,T]$; see [[@leonard-95 Definition 3.1]]{}). Define $$\begin{aligned}
\tau^*(u) \coloneqq \left\{
\begin{array}{lll}
\infty & \text{ if } u < -1 \\
1 & \text{ if } u = -1 \\
(u+1) \log (u+1) - u& \text{ if } u > -1,
\end{array}
\right.\end{aligned}$$ which is the Fenchel-Legendre transform of $\tau(u) = e^u-u-1, u \in \mathbb{R}$. We recall the following large deviation principle (LDP) for the sequence $\{p_{\nu_N}^{(N)}\}_{N \geq 1}$ on $D([0,T], M_1(\mathcal{Z}))$ (see [[@leonard-95 Theorem 3.1]]{}, [[@borkar-sundaresan-12 Theorem 3.2]]{}). See [[@dembo-zeitouni Section 1.2]]{} for the definition of LDP and a good rate function.
Suppose that the initial conditions $\nu_N \to \nu$ in $M_1(\mathcal{Z})$. Then the sequence of probability measures $\{p_{\nu_N}^{(N)}, N\geq 1\}$ on the space $D([0,T], M_1(\mathcal{Z}))$ satisfies the LDP with a good rate function $S_{[0,T]}(\mu|\nu)$. Moreover, if $S_{[0,T]}(\mu|\nu) < \infty$, then $\mu \in \mathcal{AC}[0,T],\, \mu(0) = \nu$ and there exists a family of rate matrices $L(t), 0 \leq t \leq T$ such that $\mu$ is the unique solution to $$\begin{aligned}
\dot{\mu}(t) = L(t)^* \mu(t), \, 0 \leq t \leq T, \, \mu(0) = \nu,\end{aligned}$$ and $$\begin{aligned}
& S_{[0,T]}(\mu|\nu) = \int_{[0,T]} \sum_{(i,j)\in \mathcal{E}} \mu(t)(i) \lambda_{i,j}(\mu(t)) \tau^*\left( \frac{l_{i,j}(t)}{\lambda_{i,j}(\mu(t))} -1 \right) dt.\end{aligned}$$ \[thm:finite-duration-ldp\]
We can interpret the rate function $S_{[0,T]}$ as follows. Starting at $\nu_N$, the process $\mu_N$ is likely to be in the neighbourhood of the solution to the McKean-Vlasov equation (\[eqn:MVE\]) with initial condition $\nu$ (with very high probability). In order for the process $\mu_N$ to be in the neighbourhood of some other path, we need to apply a control given by the rate matrix $L$; $S_{[0,T]}(\mu|\nu)$ is the cost of this control. In particular, since the solution to the McKean-Vlasov equation starting at $\nu$ has zero-cost (i.e. $S_{[0,T]}(\mu_{\nu}|\nu) = 0$ where $\mu_\nu$ denotes the solution to (\[eqn:MVE\]) starting at $\nu$), the limiting behaviour that $\mu_N(\cdot) \xrightarrow{P} \mu_\nu(\cdot)$ in $D([0,T],M_1(\mathcal{Z}))$ as $N \to \infty$ follows.
Here is an outline of the proof of Theorem \[thm:finite-duration-ldp\]: one looks at a system of non-interacting particles where the transition rates of a particle do not depend on the empirical measure, and considers the corresponding empirical measure process over $[0,T]$. Since at most one particle can jump at a given point of time, the measure $p_{\nu_N}^{(N)}$ is absolutely continuous with the measure corresponding to the above non-interacting system on $D([0,T],M_1(\mathcal{Z}))$. One can then write the Radon-Nikodym derivative using the Girsanov formula and show continuity properties of the same. An application of an extension of Sanov’s theorem (see [[@dawson-gartner-87 Theorem 3.5]]{}) tells us that the non-interacting particle system obeys the LDP on $D([0,T],M_1(\mathcal{Z}))$. The above theorem then follows by an application of Varadhan’s integral lemma (see [[@dembo-zeitouni Theorem 4.3.1]]{}). This approach has been carried out for a system of interacting diffusions in [@dawson-gartner-87] and for jump processes in [@leonard-95; @borkar-sundaresan-12].
We now recall a theorem that gives the large deviation principle for the sequence $\{p_{\nu_N,T}^{(N)}\}_{N \geq 1}$ on $M_1(\mathcal{Z})$. This can be obtained from the above theorem by an application of the contraction principle to the coordinate projection map $D([0,T], M_1(\mathcal{Z})) \ni \mu \mapsto \mu(T)$ (see [@dembo-zeitouni Theorem 4.2.1], [@borkar-sundaresan-12 Theorem 3.3]).
Suppose that the initial conditions $\nu_N \to \nu$ in $M_1(\mathcal{Z})$. Then the sequence of probability measures $\{p_{\nu_N,T}^{(N)}\}_{N \geq 1}$ on the space $M_1(\mathcal{Z})$ satisfies the LDP with the good rate function $$\begin{aligned}
S_T(\xi|\nu) \coloneqq \inf \{S_{[0,T]}(\mu|\nu): &\, \mu(0) = \nu, \mu(T) = \xi, \mu \in \mathcal{AC}[0,T]\}.\end{aligned}$$ Moreover, the infimum is attained, i.e., there exists a path $\hat{\mu} \in \mathcal{AC}[0,T] $, and a rate matrix $L (t) = (l_{ij}(t), (i,j) \in \mathcal{E})$ such that $$\begin{aligned}
\frac{d\hat{\mu}(t)}{dt} =L^*(t) \hat{\mu}(t),\, \hat{\mu} (0)=\nu,\, \hat{\mu} (T) = \xi,\end{aligned}$$ and $S_{[0,T]}(\hat{\mu}|\nu) = S_T(\xi|\nu)$. \[thm:ldp\_terminal\_time\]
Here, $S_T(\xi|\nu)$ can be interpreted as the minimum cost of passage from the profile $\nu$ to the profile $\xi$ in time $T$, among all paths from $\nu$ to $\xi$ in time $T$.
We also have the following *uniform* LDP for the sequence $\{p_{\nu_N}^{(N)}\}_{ N \geq 1}$ (see [[@borkar-sundaresan-12 Corollary 3.1]]{}) when the initial condition is allowed to lie in a compact set.
For any compact set $K \subset M_1(\mathcal{Z})$, any closed set $F \subset D([0,T], M_1(\mathcal{Z}))$, and any open set $G \subset D([0,T], M_1(\mathcal{Z}))$, we have $$\begin{aligned}
\limsup_{N \to \infty} \frac{1}{N} \log \sup_{\nu \in K} p_\nu^{(N)} \{\mu_N \in F\} \leq - \inf_{\nu \in K } \inf_{\mu \in F} S_{[0,T]} (\mu|\nu),\end{aligned}$$ and $$\begin{aligned}
\liminf_{N \to \infty} \frac{1}{N} \log \sup_{\nu \in K} p_\nu^{(N)} \{\mu_N \in G\} \geq - \sup_{\nu \in K} \inf_{\mu \in G} S_{[0,T]} (\mu|\nu).\end{aligned}$$ \[cor:uniform\_ldp\]
For a proof of the above, see [[@dembo-zeitouni Corollary 5.6.15]]{}. Note that, since the space $M_1(\mathcal{Z})$ is compact, we may take $K = M_1(\mathcal{Z})$ in the above corollary.
Large time behaviour {#section:large_time_behaviour}
====================
In the study of large-time behaviour of $\mu_N$, an important role is played by the Freidlin-Wentzell quasipotential $V : M_1(\mathcal{Z}) \times M_1(\mathcal{Z}) \to [0, \infty)$ defined by $$\begin{aligned}
V(\nu,\xi) \coloneqq \inf \{S_{[0,T]}(\mu|\nu):\mu(T)= \xi, T > 0\},\end{aligned}$$ i.e., $V(\nu, \xi)$ denotes the minimum cost of transport from $\nu$ to $\xi$ in an arbitrary but finite time.
We say that $\nu \sim \xi$ ($\nu$ is equivalent to $\xi$) if $V(\nu, \xi) = 0$ and $V(\xi, \nu) = 0$. It is easy to see that $\sim$ defines an equivalence relation on $M_1(\mathcal{Z})$. To study the large time behaviour of the process $\mu_N$, we make the following assumptions on the McKean-Vlasov equation (\[eqn:MVE\]) (see [@freidlin-wentzell Chapter 6, Section 2, Condition A]):
1. There exists a finite number of compact sets $K_1, K_2, \ldots, K_l$ such that
- For each $i = 1,2, \ldots l$, $\nu_1, \nu_2 \in K_i$ implies $\nu_1 \sim \nu_2$.
- For each $i \neq j$, $\nu_1 \in K_i$ and $\nu_2 \in K_j$ implies $\nu_1 \nsim \nu_2$.
- Every $\omega$-limit set of the dynamical system (\[eqn:MVE\]) lies completely in one of the compact sets $K_i$.
\[assm:b1\]
It can be shown that $V(\nu_1, \nu_2) = 0$ whenever $\nu_1,\nu_2 \in K_i$ for any $1 \leq i \leq l$. We can therefore define $$\begin{aligned}
V(K_i, K_j) \coloneqq \inf\{S_{[0,T]}(\mu|\nu):\nu \in K_i, \mu(T) \in K_j\},\end{aligned}$$ which is interpreted as the minimum cost of going from $K_i$ to $K_j$. We also define the minimum cost of going from $K_i$ to $K_j$ without touching the other compact sets $K_k, k \neq i,j$ by $$\begin{aligned}
\tilde{V}(K_i, K_j) \coloneqq \inf & \{S_{[0,T]}(\mu|\nu):\nu \in K_i, \mu(t) \notin \cup_{k\neq i, j} K_k \\
& \text{ for all } 0 \leq t \leq T, \mu(T) \in K_j, T>0\}.\end{aligned}$$
Preliminary results
-------------------
It turns out that, under assumption \[assm:b1\], the large time behaviour of the process $\mu_N$ can be studied via a discrete time Markov chain whose state space is the union of small neighbourhoods of the compact sets $K_i, 1 \leq i \leq l$. To study this chain, we introduce some notation. Let $L = \{1,2,\ldots, l\}$. Given $0 < \rho_1 < \rho_0$, let $\gamma_i$ (resp. $\Gamma_i$) denote the $\rho_1$-open neighbourhood (resp. $\rho_0$-open neighbourhood) of $K_i$. Let $\gamma = \cup_{i=1}^l \gamma_i$, $\Gamma = \cup_{i=1}^l \Gamma_i$, and $C =M_1(\mathcal{Z}) \setminus \overline{\Gamma}$. For a set $A \subset M_1(\mathcal{Z})$ and $\delta>0$, let $[A]_\delta$ denote the $\delta$-open neighbourhood of $A$, and for a subset $W \subset L$, abusing notation, let $[W]_\delta$ denote the $\delta$-open neighbourhood of $\cup_{i \in W} K_i$. For each $n \geq 1$, we define the sequence of stopping times: $\tau_0\coloneqq 0$, $\sigma_n \coloneqq \inf\{t>\tau_{n-1}: \mu_N(t) \in C\}$, $\tau_n \coloneqq \inf\{t > \sigma_n : \mu_N(t) \in \gamma\}$, and define $Z^N_n \coloneqq \mu_N(\tau_n)$, which is a discrete time Markov chain on the state space $\gamma \cap M^N_1(\mathcal{Z})$. For a measurable set $A \in M_1(\mathcal{Z})$, we define the stopping time $\tau_A \coloneqq \inf\{t > 0: \mu_N(t) \notin A\}$, which denotes the time exit from the set $A$. Finally, for a subset $W \subset L$, we define the stopping time $\hat{\tau}_W \coloneqq \inf\{t >0 : \mu_N(t) \in \cup_{i \in W}\gamma_i\}$, and $\bar{\tau}_W \coloneqq \inf \{t > 0: \mu_N(t) \in \cup_{i \in L \setminus W} \gamma_i \}$, which denote the time of entry into the $\rho_1$-neighbourhood of $W$ and the time of entry into the $\rho_1$-neighbourhood of $L \setminus W$, respectively.
We now state some results on the behaviour of the exit time from certain sets, which will be used in the paper subsequently. These results are known in the case of both Markov jump processes as well as diffusion processes; see [[@borkar-sundaresan-12 Appendix]]{}, and [[@freidlin-wentzell Chapter 6, Section 2]]{}. The main ingredients that are used in proving these results are (i) the strong Markov property of the $\mu_N$ process, (ii) Theorem \[thm:finite-duration-ldp\] and Corollary \[cor:uniform\_ldp\] on the LDP for finite time durations, and (iii) the joint continuity of the terminal time rate function $S_T(\cdot|\cdot)$ (see [[@borkar-sundaresan-12 Lemma 3.3]]{}). In the following results, $P_\nu$ denotes the solution to the $D([0,\infty),M_1(\mathcal{Z}))$-valued martingale problem for $L^N$ with initial condition $\nu \in M_1^N(\mathcal{Z})$ and $E_\nu$ denotes the corresponding expectation; note that in $P_\nu$ and $E_\nu$ we suppress the dependence on $N$ for ease of readability.
Let $K \subset M_1(\mathcal{Z})$ be a compact set such that all points in $K$ are equivalent to each other. Then, given $\varepsilon> 0$, there exist $\delta > 0$ and $N_0 \geq 1$ such that for all $N \geq N_0$ and $\nu \in [K]_{\delta}\cap M^N_1(\mathcal{Z})$, $$\begin{aligned}
E_\nu \tau_{[K]_{\delta}} \leq \exp\{N\varepsilon\}.\end{aligned}$$ \[lemma:fw17\]
Let $K \subset M_1(\mathcal{Z})$ be a compact set and $G$ be a neighbourhood of $K$. Then, given $\varepsilon > 0$, there exist $\delta>0$ and $N_0 \geq 1$ such that for all $\nu \in \overline{[K]_{\delta}} \cap M_1^N(\mathcal{Z})$ and $N \geq N_0$ $$\begin{aligned}
E_\nu \left( \int_0^{\tau_G} {1_}{\{\mu_N(t) \in \overline{[K]_\delta}\}} dt \right) \geq \exp\{-N\varepsilon\}.\end{aligned}$$ \[lemma:fw18\]
Let $K \subset M_1(\mathcal{Z})$ be a compact set that does not contain any $\omega$-limit set of (\[eqn:MVE\]) entirely. Then, there exist positive constants $c, T_0$ and $N_0 \geq 1$ such that for all $T \geq T_0$, $N \geq N_0$ and any $\nu \in K \cap M_1^N(\mathcal{Z})$, we have $$\begin{aligned}
P_{\nu}( \tau_K \geq T ) \leq \exp\{-Nc(T-T_0)\}.\end{aligned}$$ \[lemma:fw19\]
Under the conditions of Lemma \[lemma:fw19\], there exist $ C>0$ and $N_0 \geq 1$ such that for all $\nu \in K \cap M_1^N(\mathcal{Z})$ and $N \geq N_0$, $$\begin{aligned}
E_\nu \tau_K \leq C.\end{aligned}$$ \[cor:fw19\]
Recall the definition of the discrete time Markov chain $Z^N$ on $\gamma \cap M_1^N(\mathcal{Z})$. The next lemma gives upper and lower bounds on the one-step transition probabilities of the chain $Z^N$. These estimates play an important role in the study of large-time behaviour of the process $\mu_N$, as we shall see in the sequel.
\[lemma:bsa6\] Given $\varepsilon >0$, there exist $\rho_0 >0$ and $N_0 \geq 1$ such that, for any $\rho_2 < \rho_0$, there exists $\rho_1 < \rho_2$ such that for any $\nu \in [K_i]_{\rho_2} \cap M_1^N(\mathcal{Z})$ and $N \geq N_0$, the one-step transition probability of the chain $Z^N$ satisfies $$\begin{aligned}
\exp\{-N(\tilde{V}(K_i, K_j)+\varepsilon)\} \leq P(\nu, \gamma_j) \leq \exp\{-N(\tilde{V}(K_i, K_j)-\varepsilon)\}.
\label{eqn:tpm_zn}\end{aligned}$$
The key ingredient in the proof of the above lemma is Corollary \[cor:uniform\_ldp\] on the uniform large deviation principle on bounded sets. For the lower bound, one constructs a specific trajectory from $\nu$ to $K_j$ and examines its cost. For the upper bound, one uses the strong Markov property at the hitting time of $[L]_{\rho_1}$ and the uniform large deviation principle. For details, the reader is referred to proof of [[@borkar-sundaresan-12 Lemma A.6]]{} for the case of Markov jump processes, and proof of [@freidlin-wentzell Lemma 2.1, page 152] for the case of small noise diffusions.
Behaviour near attractors indexed by subsets of L
-------------------------------------------------
We now recall some results on the behaviour of the process $\mu_N$ near a small neighbourhood of attractors indexed by a given subset of $L$. Let $W \subset L$. A $W$-graph is a directed graph on $L$ such that (i) each element of $L \setminus W$ has exactly one outgoing arrow and (ii) there are no closed cycles in the graph. We denote the set of $W$-graphs by $G(W)$. For a $W$-graph $g$, define $\tilde{V}(g) = \sum_{(m\to n) \in g} \tilde{V}(K_m, K_n)$. Note that, using the estimate (\[eqn:tpm\_zn\]), $\tilde{V}$ can be used to estimate the probability that the process $\mu_N$ traverses through a sequence of neighbourhoods in the order specified by the graph $g$.
For $i \in L\setminus W$ and $j \in W$, let $G_{i,j}(W)$ denote the set of $W$-graphs in which there is a sequence of arrows leading from $i$ to $j$. Define $$\begin{aligned}
I_{i,j}(W) \coloneqq \min\{\tilde{V}(g):g \in G_{i,j}(W)\} - \min\{\tilde{V}(g): g \in G(W)\}.\end{aligned}$$ We recall the following result on the probability that the first entry of $\mu_N$ into a neighbourhood of a set $W \subset L$ takes place via a given compact set $K_j$, starting from a neighbourhood of $K_i$.
Let $W \subset L$, and let $i \in L \setminus W$ and $j \in W$. Given $\varepsilon > 0$, there exist $\rho > 0$ and $N_0 \geq 1 $ such that for any $\rho_1 \leq \rho$, $\nu \in \gamma_i \cap M^N_1(\mathcal{Z})$ and $N \geq N_0$, we have $$\begin{aligned}
\exp\{-N(I_{i,j}(W)+\varepsilon)\} \leq P_\nu(\mu_N(\hat{\tau}_W) \in \gamma_j) \leq \exp\{-N(I_{i,j}(W)-\varepsilon)\}.\end{aligned}$$ \[lemma:hs15\]
The proof of [[@freidlin-wentzell Lemma 3.3, page 159]]{} holds verbatim, by making use of the estimates in Lemma \[lemma:bsa6\].
Our next step is to understand the mean entry time $E_\nu \hat{\tau}_W$. For this, we need the following estimate on the stopping time $\tau_1$; see [[@hwang-sheu-90 Lemma 1.3, Part I]]{} for a similar estimate for small noise diffusion processes.
Given $\varepsilon >0$, there exist $\rho_1 >0$ and $N_0 \geq 1$ such that, for any $\nu \in \gamma \cap M_1^N(\mathcal{Z})$ and $N \geq N_0$, we have $$\begin{aligned}
E_\nu \tau_1 \leq \exp\{N\varepsilon\}.\end{aligned}$$ \[lemma:hs13\]
With a sufficiently small $\rho_1 > 0$ to be chosen later, let $\rho_0 = 2 \rho_1$ so that $[K_i]_{\rho_0}$ does not intersect with $[K_j]_{\rho_0}$ for all $j \neq i$. Note that, for any $\nu \in \gamma$, $$\begin{aligned}
E_{\nu} \tau_1 = E_\nu \sigma_0 +E_\nu (\tau_1 - \sigma_0).\end{aligned}$$ Consider the first term. By Lemma \[lemma:fw17\], there exist $\rho > 0$ and $N_0 \geq 1$ such that for all $\rho_1 \leq \rho$, $\nu \in \gamma \cap M_1^N(\mathcal{Z})$ and $N \geq N_0$, we have $$\begin{aligned}
E_\nu \sigma_0 \leq \exp\{N\varepsilon/2\}.\end{aligned}$$ Let $F = M_1(\mathcal{Z}) \setminus \gamma$. By the strong Markov property, the second term is $$\begin{aligned}
E_\nu (\tau_1 - \sigma_0) = E_{\mu_N(\sigma_0)} (\tau_{F}).\end{aligned}$$ Therefore, it suffices to estimate $E_{\nu^\prime} \tau_F$ for $\nu^\prime \in F$. Since the compact set $F$ does not contain any $\omega$-limit set, by Corollary \[cor:fw19\], there exist a constant $C >0$ and $N_1 \geq N_0$ such that for any $\nu^\prime \in F \cap M_1^N(\mathcal{Z})$ $$\begin{aligned}
E_{\nu^\prime} \tau_{F} \leq C.\end{aligned}$$ This completes the proof of the lemma.
Define $$\begin{aligned}
I_i(W) \coloneqq &\min\{\tilde{V}(g): g \in G(W)\} - \min\{\tilde{V}(g): g \in G( W \cup \{i\}) \text{ or } \\& g \in G_{i,j}(W \cup \{j\}), i \neq j, j \in L\setminus W\} \\end{aligned}$$ The next lemma is about the mean entry time into a neighbourhood of a given set $W \subset L$ starting from a neighbourhood of $K_i$; see [[@hwang-sheu-90 Lemma 1.6, Part I]]{}\] for a similar estimate on small noise diffusion processes.
\[lemma:hs16\] Let $W \subset L$, and let $i \in L \setminus W$. Given $\varepsilon > 0$, there exist $\rho > 0$ and $N_0 \geq 1 $ such that for any $\rho_1 \leq \rho$, $\nu \in \gamma_i \cap M^N_1(\mathcal{Z})$ and $N \geq N_0$, we have $$\begin{aligned}
\exp\{N(I_i(W)-\varepsilon)\} \leq E_\nu \hat{\tau}_W \leq \exp\{N(I_i(W)+\varepsilon)\}.\end{aligned}$$
We first prove the upper bound. Note that, by the strong Markov property, we have $$\begin{aligned}
E_\nu \hat{\tau}_W = E_\nu \tau_v \leq \sum_{m=1}^\infty E_\nu \left({1_}{v = m} \times m \sup_{\nu^\prime \in \gamma} E_{\nu^\prime} \tau_1 \right),\end{aligned}$$ where $v$ is the hitting time of the chain $Z_n^N$ on the set $W$. Using Lemma \[lemma:hs13\] and the upper bound on $E_\nu v$ derived in [[@freidlin-wentzell Lemma 3.4, page 162]]{}, for sufficiently small $\rho_1$ and sufficiently large $N$, we have that $$\begin{aligned}
E_\nu \hat{\tau}_W \leq \exp\{N(I_i(W) +\varepsilon)\}\end{aligned}$$ holds for all $\nu \in \gamma_i \cap M_1^N(\mathcal{Z})$. For the lower bound, Lemma \[lemma:fw18\] implies that, for all sufficiently small $\rho_1$ and sufficiently large $N$, we have that $$\begin{aligned}
E_\nu \tau_1 \geq \exp\{-N\varepsilon\}\end{aligned}$$ holds for all $\nu \in \gamma$. Also, $$\begin{aligned}
E_\nu \hat{\tau}_W = E_\nu \tau_v \geq \sum_{m=1}^\infty E_\nu \left(1_{v = m} \times m \inf_{\nu^\prime \in \gamma} E_{\nu^\prime} \tau_1 \right),\end{aligned}$$ hence, using the lower bound on $E_\nu v$ derived in [[@freidlin-wentzell Lemma 3.4, page 162]]{}, we get $$\begin{aligned}
E_\nu \hat{\tau}_W \geq \exp\{N(I_i(W) - \varepsilon)\}\end{aligned}$$ for all $\nu \in \gamma_i \cap M_1^N(\mathcal{Z})$ and sufficiency large $N$.
Cycles
------
We now define the notion of cycles, which helps us to describe the most probable way in which the process $\mu_N$, for large $N$, traverses neighbourhoods of various compact sets $K_i$, and the time required to go from one to another. Define $\tilde{V}(K_i) \coloneqq \min_{j \neq i} \tilde{V}(K_i, K_j)$. We say that $i \to j$ if $\tilde{V}(K_i) = \tilde{V}(K_i,K_j)$. Note that, using the estimates (\[eqn:tpm\_zn\]) on the transition probability of the discrete time Markov chain $Z^N$, we see that the indices that attain the minimum above are the most likely sets that will be visited by the process $\mu_N$, for large enough $N$, starting from a neighbourhood of $K_i$. For $i, j \in L$, we say that $i \Rightarrow j$ if there exists a sequence of arrows leading from $i$ to $j$, i.e., there exists $i_1, i_2 , \ldots, i_n$ in $L$ such that $i \to i_1 \to i_2 \to \cdots \to i_n \to j$. Again, the above sequence of arrows from $i$ to $j$ is one among the *locally* most likely sequences in which the process traverses from a neighbourhood of $K_i$ to that of $K_j$ for large $N$.
A cycle $\pi$ is a subgraph of $L$ satisfying
1. $i \in \pi$ and $i \Rightarrow j$ implies $j \in \pi$.
2. For any $i \neq j$ in $\pi$, we have $i \Rightarrow j $ and $j \Rightarrow i$.
It can be shown that there exists a cycle (see the proof of [@hwang-sheu-90 Lemma 1.9, Part I]). We now define cycle of cycles. Let $L_0 = L$. Define $$\begin{aligned}
L_1 \coloneqq \{\pi: \pi \text{ is a cycle in } L\} \cup \{i \in L: i \text{ is not in any cycle}\}.\end{aligned}$$ For $\pi_1, \pi_2 \in L_1$, $\pi_1 \neq \pi_2$, define $$\begin{aligned}
\hat{V}(\pi_1) \coloneqq \max \{\tilde{V}(K): K \in \pi_1\},\end{aligned}$$ $$\begin{aligned}
\tilde{V}(\pi_1, \pi_2) \coloneqq \hat{V}(\pi_1) + \min \{\tilde{V}(K_1, K_2)-\tilde{V}(K_1): K_1 \in \pi_1, K_2 \in \pi_2\},\end{aligned}$$ and $$\begin{aligned}
\tilde{V}(\pi_1) \coloneqq \min\{\tilde{V}(\pi_1, \pi_2): \pi_2 \in L_1, \pi_2 \neq \pi_1\}.\end{aligned}$$ We say that $\pi_1 \to \pi_2$ if $ \tilde{V}(\pi_1) = \tilde{V}(\pi_1, \pi_2)$, and we say that $\pi_1 \Rightarrow \pi_2$ if there is a sequence of arrows leading from $\pi_1$ to $\pi_2$. This gives a cycle of cycles, which we call 2-cycles.
Let us now define the hierarchy of cycles. Having defined $(m-1)$-cycles and the sets $L_0, L_1, \ldots, L_{m-2}$, we define $m$-cycles as follows. Note that $$\begin{aligned}
L_{m-1} = \{\pi^{m-1}: &\pi^{m-1} \text{ is an } (m-1)\text{-cycle}\} \\
& \cup \{\pi^{m-2} \in L_{m-2} : \pi^{m-2} \text{ is not in any } (m-1)\text{-cycle}\}.\end{aligned}$$ For $\pi^{m-1} \in L_{m-1}$, define $$\begin{aligned}
\hat{V}(\pi^{m-1}) \coloneqq \max \{\tilde{V}(\pi^{m-2}): \pi^{m-2} \in \pi^{m-1}\},\end{aligned}$$ $$\begin{aligned}
\tilde{V}(\pi_1^{m-1}, \pi_2^{m-1}) \coloneqq \hat{V}(\pi_1^{m-1}) + & \min \{\tilde{V}(\pi_1^{m-2}, \pi_2^{m-2})-\tilde{V}(\pi_1^{m-2}): \\
& \pi_1^{m-2} \in \pi_1^{m-1}, \pi_2^{m-2} \in \pi_2^{m-1}\},\end{aligned}$$ and $$\begin{aligned}
\tilde{V}(\pi_1^{m-1}) \coloneqq \min\{\tilde{V}(\pi_1^{m-1}, \pi_2^{m-1}): \pi_2^{m-1} \in L_{m-1}, \pi_2^{m-1} \neq \pi_1^{m-1}\}.\end{aligned}$$ We say that $\pi_1^{m-1} \to \pi_2^{m-1}$ if $\tilde{V}(\pi_1^{m-1}) = \tilde{V}(\pi_1^{m-1}, \pi_2^{m-1}) $. We have
An $m$-cycle $\pi^m$ is a subgraph of $L_{m-1}$ satisfying
1. For $\pi_1^{m-1}, \pi_2^{m-1} \in L_{m-1}$, $\pi_1^{m-1} \in \pi^m$ and $\pi_1^{m-1} \Rightarrow \pi_2^{m-1}$ implies $\pi_2^{m-1} \in \pi^m$.
2. For any $\pi_1^{m-1}, \pi_2^{m-1} \in \pi^m$, we have $\pi_1^{m-1} \Rightarrow \pi_2^{m-1}$ and $\pi_2^{m-1}\Rightarrow \pi_1^{m-1}$.
If we continue this way, for some $m \geq 1$, the set $L_m$ will eventually be a singleton, at which point we stop.
We now state some results on the mean exit time from a cycle and the most probable cycle the process $\mu_N$ visits upon exit from a given cycle. For convenience, the set of elements of $L$ constituting a $k$-cycle $\pi^k$ (through the hierarchy of cycles) is also denoted by $\pi^k$. Also, for $W \subset L$, we define $\gamma_W = \cup_{i \in W} \gamma_i$.
Let $\pi^k$ be a $k$-cycle and $K_i \in \pi^k$. Let $W = L \setminus \pi^k$. Given $\varepsilon > 0$, there exist $\rho >0$ and $N_0 \geq 1$ such that for all $\rho_1 \leq \rho$, $\nu \in \gamma_i \cap M_1^N(\mathcal{Z})$ and $N \geq N_0$, we have $$\begin{aligned}
\exp \{N(\tilde{V}(\pi^k) - \varepsilon)\} \leq E_\nu \hat{\tau}_W \leq \exp \{N(\tilde{V}(\pi^k) + \varepsilon)\}.\end{aligned}$$ \[cor:hs110\]
Let $\pi_1^k, \pi_2^k$ be $k$-cycles, $\pi_1^k \neq \pi_2^k$, and $K_i \in \pi_1^k$. Let $W = L \setminus \pi_1^k$. Given $\varepsilon > 0$, there exist $\rho >0$ and $N_0 \geq 1$ such that for all $\rho_1 \leq \rho$, $\nu \in \gamma_i \cap M_1^N(\mathcal{Z})$ and $N \geq N_0$, we have $$\begin{aligned}
\exp \{-N(\tilde{V}(\pi_1^k, \pi_2^k) - \tilde{V}(\pi_1^k) + \varepsilon)\} & \leq P_\nu (\mu_N(\hat{\tau}_W) \in \gamma_{\pi_2^k}) \\
&\leq \exp \{-N(\tilde{V}(\pi_1^k, \pi_2^k) - \tilde{V}(\pi_1^k) - \varepsilon)\}.\end{aligned}$$ \[cor:hs111\]
Note that Corollary \[cor:hs110\] follows from Lemma \[lemma:hs16\] and the fact that $I_i(W) = \tilde{V}(\pi^k)$ (which is shown in [@hwang-sheu-90 Corollary A.4, Appendix]). Corollary \[cor:hs111\] is a consequence of Lemma \[lemma:hs15\] along with the fact that $\min\{I_{i,j}(W) : i \in \hat{\pi}^k\} = \tilde{V}(\pi^k, \hat{\pi}^k) - \tilde{V}(\pi^k)$ (see [@hwang-sheu-90 Corollary A.6, Appendix]). Similar estimates as in Corollaries \[cor:hs110\] and \[cor:hs111\] in the case of small noise diffusion processes have been shown in [[@hwang-sheu-90 Corollary 1.10, Part I]]{} and [[@hwang-sheu-90 Corollary 1.11, Part I]]{}, respectively.
We also need the following lemmas that provide estimates on the probabilities of exit within certain times from given cycles.
Let $\pi_1^k, \pi_2^k$ be $k$-cycles and let $\pi_1^k \to \pi_2^k$. Then, given $\varepsilon >0$, there exist $\delta >0$, $\rho >0$ and $N_0 \geq 1$ such that for all $\rho_1 \leq \rho$, $\nu \in \gamma_{\pi_1^k} \cap M_1^N(\mathcal{Z})$ and $N \geq N_0$, we have $$\begin{aligned}
P_\nu \left(\bar{\tau}_{\pi_1^k} \leq \exp\{N(\tilde{V}(\pi_1^k)-\delta)\}, \mu_N(\bar{\tau}_{\pi_1^k}) \in \gamma_{\pi_2^k} \right) \geq \exp\{-N\varepsilon\}.\end{aligned}$$ \[lemma:hitting\_place\]
Let $\pi^k$ be a $k$-cycle. Then, given $\varepsilon > 0$, there exists $\rho > 0$ such that for all $\rho_1 \leq \rho$, we have $$\begin{aligned}
\lim_{N \to \infty} \sup_{\nu \in \gamma_{\pi^k} \cap M_1^N(\mathcal{Z})} P_\nu\left( \exp\{N(\tilde{V}(\pi^k) - \varepsilon)\} \leq \bar{\tau}_{\pi^k} \leq \exp\{N(\tilde{V}(\pi^k) + \varepsilon)\} \right) =1.\end{aligned}$$ Furthermore, given $\varepsilon > 0$, there exist $\delta >0$, $\rho > 0$ and $N_0 \geq 1$ such that for all $\rho_1 \leq \rho$, $N \geq N_0$ and $\nu \in \gamma_{\pi^k} \cap M^N_1(\mathcal{Z})$, we have $$\begin{aligned}
P_\nu\left( \bar{\tau}_{\pi^k} < \exp\{N(\tilde{V}(\pi^k) - \delta)\} \right) \leq \exp\{-N\varepsilon\}, \text{ and} \\
P_\nu\left( \bar{\tau}_{\pi^k} > \exp\{N(\tilde{V}(\pi^k) + \delta)\} \right) \leq \exp\{-N\varepsilon\}.\end{aligned}$$ \[lemma:hitting\_time\]
Lemma \[lemma:hitting\_place\] can be proved using Lemma 3.3 and [[@freidlin-wentzell Chapter 6, Theorem 6.2]]{}, and Lemma \[lemma:hitting\_time\] can be proved using the same arguments used in the proof of [[@freidlin-wentzell Chapter 6, Theorem 6.2]]{}. Similar estimates as in Lemmas \[lemma:hitting\_place\] and \[lemma:hitting\_time\] in the case of small noise diffusion processes have been shown in [[@hwang-sheu-90 Lemma 2.1, Part I]]{} and [[@hwang-sheu-90 Lemma 2.2, Part I]]{}, respectively.
Let $\pi^k$ be a $k$-cycle and assume that $\tilde{V}(\pi^k) > 0$. Given $\varepsilon > 0$, there exist $\delta>0, \rho >0$ and $N_0 \geq 1$ such that for all $\rho_1 \leq \rho, \nu \in M_1^N(\mathcal{Z})$ and $N \geq N_0$, we have $$\begin{aligned}
P_{0,\nu}(\bar{\tau}_{\pi^k} \leq \exp\{N(\hat{V}(\pi^k)+\delta\}) \leq \exp\{-N(\tilde{V}(\pi^k) - \hat{V}(\pi^k) - \varepsilon)\}.\end{aligned}$$ \[lemma:exit\_time\_vhat\]
We proceed via the steps in the proof of [[@hwang-sheu-90 Lemma 2.1, Part III]]{}. Let $\pi^{k-1} \in \pi^k$ be a $(k-1)$-cycle such that $\tilde{V}(\pi^{k-1}) = \hat{V}(\pi^k)$. With $\rho_1>0$ to be chosen later, for each $n \geq 1$, define the minimum of $\bar{\tau}_{\pi^k}$ and successive entry and exit times from a $\rho_1$-neighbourhood of $\pi^{k-1}$ as follows: $$\begin{aligned}
\hat{\theta}_0 & \coloneqq \inf\{ t >0 : \mu_N(t) \in [\pi^{k-1}]_{\rho_1}\} \wedge \bar{\tau}_{\pi^k}, \\
\bar{\theta}_n & \coloneqq \inf\{t > \hat{\theta}_{n-1}: \mu_N(t) \in [L \setminus \pi^{k-1}]_{\rho_1} \} \wedge \bar{\tau}_{\pi^k} , \\
\hat{\theta}_{n+1}& \coloneqq \inf\{t > \bar{\theta}_n : \mu_N(t) \in [\pi^{k-1}]_{\rho_1} \} \wedge \bar{\tau}_{\pi^k}.\end{aligned}$$ With $\delta > 0$ to be chosen later, using the strong Markov property, for any $\nu \in [\pi^k]_{\rho_1} \cap M_1^N(\mathcal{Z})$, we have $$\begin{aligned}
P_{\nu}&(\bar{\tau}_{\pi^k} \leq \exp\{N(\hat{V}(\pi^k)+\delta)\}) = P_\nu (\hat{\theta}_0 = \bar{\tau}_{\pi^k}, \bar{\tau}_{\pi^k} \leq \exp\{N(\hat{V}(\pi^k)+\delta)\}) \nonumber \\
+ & P_\nu \left(\hat{\theta}_0 < \bar{\tau}_{\pi^k}, \bigcup_{n \geq 1} \left\{\bar{\tau}_{\pi^k} = \bar{\theta}_n, \bar{\tau}_{\pi^k} \leq \exp\{N(\hat{V}(\pi^k)+\delta)\}, \bar{\tau}_{\pi^k} \geq \hat{\theta}_{n-1} \right\} \right) \nonumber \\
+ & P_\nu \left(\hat{\theta}_0 < \bar{\tau}_{\pi^k}, \bigcup_{n\geq 1} \left\{\bar{\tau}_{\pi^k} = \hat{\theta}_n, \bar{\tau}_{\pi^k} \leq \exp\{N(\hat{V}(\pi^k)+\delta)\}, \bar{\tau}_{\pi^k} \geq \bar{\theta}_n \right\} \right).
\label{eqn:temp10}\end{aligned}$$ We now upper bound each of the terms in \[eqn:temp10\].Consider the first term. It can be shown using Corollary \[cor:hs111\] and [@hwang-sheu-90 Corollary A.6, Appendix] that, there exist $\rho_1>0$ and $\delta > 0$ such that for any $\nu \in [\pi^k]_{\rho_1}$ and sufficiently large $N$, we have $$\begin{aligned}
P_\nu(\hat{\theta}_0 = \bar{\tau}_{\pi^k}) \leq \exp\{-N(\tilde{V}(\pi^k) - \hat{V}(\pi^k) - \varepsilon)\}.\end{aligned}$$
Consider the second term in \[eqn:temp10\]. For any $\nu_1 \in [\pi^{k-1}]_{\rho_1} \cap M_1^N(\mathcal{Z})$, the probability of the unionised event can be upper bounded by $$\begin{aligned}
P_{\nu_1} & \left(\bigcup_{n \geq 1}\left\{\bar{\tau}_{\pi^k} = \bar{\theta}_n, \bar{\tau}_{\pi^k} \leq \exp\{N(\hat{V}(\pi^k)+\delta)\}, \bar{\tau}_{\pi^k} \geq \hat{\theta}_{n-1} \right\} \right)\\
& \leq P_{\nu_1} \left(\bigcup_{n=1}^M \left\{\bar{\tau}_{\pi^k} = \bar{\theta}_n, \bar{\tau}_{\pi^k} \leq \exp\{N(\hat{V}(\pi^k)+\delta)\}, \bar{\tau}_{\pi^k} \geq \hat{\theta}_{n-1} \right\} \right) \\
& \, \, \, \, + P_{\nu_1} \left( \bigcup_{n \geq M+1} \left\{\bar{\tau}_{\pi^k} = \bar{\theta}_n, \bar{\tau}_{\pi^k} \leq \exp\{N(\hat{V}(\pi^k)+\delta)\}, \bar{\tau}_{\pi^k} \geq \hat{\theta}_{n-1} \right\} \right) \\
& \leq P_{\nu_1} (\bar{\tau}_{\pi^k} = \bar{\theta}_n \text{ and } \bar{\tau}_{\pi^k} \geq \hat{\theta}_{n-1} \text{ for some } n \leq M) \\
& \, \, \, \, + P_{\nu_1} (\hat{\theta}_M \leq \exp\{N(\hat{V}(\pi^k)+\delta)\} \text{ and } \hat{\theta}_M \leq \bar{\tau}_{\pi^k})\\
& \leq P_{\nu_1}(\hat{\theta}_M = \bar{\tau}_{\pi^k}) + P_{\nu_1} (\hat{\theta}_M \leq \exp\{N(\hat{V}(\pi^k)+\delta)\} \text{ and } \hat{\theta}_M \leq \bar{\tau}_{\pi^k}).\end{aligned}$$ Again, the first term above can be bounded by $$\begin{aligned}
P_{\nu_1}(\hat{\theta}_M \leq \bar{\tau}_{\pi^k}) \leq \exp\{-N(\tilde{V}(\pi^k) - \hat{V}(\pi^k) - \varepsilon)\},\end{aligned}$$ for all $\nu_1 \in [\pi^{k-1}]_{\rho_1} \cap M_1^N(\mathcal{Z})$ and sufficiently large $N$. The second term can be bounded by $\exp\{-NM\}$ for large enough $M$, by the same argument used in the proof of [@hwang-sheu-90 Lemma 1.7, Part I]. Choosing $M$ sufficiently large, the above implies that the second term in (\[eqn:temp10\]) is bounded by $\exp\{-N(\tilde{V}(\pi^k) - \hat{V}(\pi^k) - \varepsilon)\}$. A similar argument gives the same bound for the third term in (\[eqn:temp10\]).
We also remark that, using the estimates (\[eqn:tpm\_zn\]) of the transition probabilities of the discrete time Markov chain $Z^N$, we can study large deviations for the process $\mu_N$ in the stationary regime. Since $\mu_N$ is a Markov process on a finite state space, and since the graph $(\mathcal{Z},\mathcal{E})$ of allowed particle transitions is irreducible (Assumption \[assm:a1\]), there exists a unique invariant probability measure $\wp_N$ on $M_1(\mathcal{Z})$ for the process $\mu_N$. We state the following result:
Assume \[assm:a1\], \[assm:a2\] and \[assm:b1\]. Then, the sequence of invariant measures $\{\wp_N\}_{N \geq 1}$ satisfies the large deviation principle on $M_1(\mathcal{Z})$ with good rate function $s$ given by $$\begin{aligned}
s(\xi) = \min_{1 \leq i \leq l} \{W(i) + V(K_i, \xi)\} - \min_{1 \leq j \leq l} W(j),
$$ where $$\begin{aligned}
W(i) = \min_{g \in G(i)} \sum_{(m,n) \in G} \tilde{V}(m,n).\end{aligned}$$ \[thm:invariant-measure-gase\]
Convergence to the invariant measure
------------------------------------
In this section, we prove our first main result on the convergence of $\mu_N$ to its invariant measure. Let $i_0 \in L$ be such that $ \min\{\tilde{V}(g): g \in G(i_0)\} = \min\{\tilde{V}(g): g \in G(i), i \in L \} $. We anticipate that $K_{i_0}$ is one of the most stable compact sets (among possibly others). This is because Theorem \[thm:invariant-measure-gase\] tells us that the rate function that governs the LDP for $\{\wp_N\}_{N \geq 1}$ vanishes on $K_{i_0}$. Hence, for a large but fixed $N$, over large time intervals, one expects that there is positive probability (in the exponential scale) for the process $\mu_N$ to be in a small neighbourhood of $K_{i_0}$.
Define $$\begin{aligned}
\Lambda \coloneqq \min\{\tilde{V}(g): g \in G(i), i \in L \} - \min\{\tilde{V}(g): g \in G(i, j), i, j \in L, i \neq j\}.\end{aligned}$$ Let $P_T(\nu, \cdot ) = P_{\nu}(\mu_N(T) \in \cdot )$ denote the transition probability kernel associated with the process $\mu_N$. Note that we suppress the dependence on $N$ for ease of readability. We first show a lower bound for the transition probability $P_T(\nu_1, K_{i_0} ) $ of reaching a small neighbourhood of $K_{i_0}$ when $T$ is of the order $\exp\{N(\Lambda-\delta_0)\}$ for some $\delta_0>0$.
Given $\varepsilon >0$, there exist $\delta_0 > 0$, $\rho > 0$ and $N_0 \geq 1$ such that for all $\rho_1 \leq \rho$, $N \geq N_0$, $\nu \in M_1^N(\mathcal{Z})$, we have $$\begin{aligned}
P_{T_0}(\nu, \gamma_{i_0}) \geq \exp\{-N\varepsilon\},
\label{eqn:lb_tpm_i0}\end{aligned}$$ where $T_0 =\exp\{N(\Lambda - \delta_0)\}$. Furthermore, there exist $\nu_0 \in M_1(\mathcal{Z})$ and $\beta >0$ such that for all $N \geq N_0$ and $\nu \in [\nu_0]_{\rho_1} \cap M_1^N(\mathcal{Z})$ $$\begin{aligned}
P_{T_0}(\nu, \gamma_{i_0}) \leq \exp\{-N\beta\}.
\label{eqn:ub_tpm}\end{aligned}$$ \[thm:mixing\]
We follow the steps in Hwang and Sheu [[@hwang-sheu-90 Part I, Theorem 2.3]]{}. With $\rho > 0$ to be chosen later, we first show that (\[eqn:lb\_tpm\_i0\]) holds for all $\nu \in \gamma \cap M_1^N(\mathcal{Z})$. Towards this, let $m$ be the smallest integer such that $L_{m+1}$ is a singleton. For $0 \leq k \leq m$, let $\pi_0^k \in L_k$ be the $k$-cycle containing $i_0$. Let $V_k = \max \{\tilde{V}(\pi^k) : \pi^k \subset \pi_0^{k+1}, \pi^k \neq \pi_0^k\}$. Using [@hwang-sheu-90 Lemma A.10, Appendix], we have $\Lambda = \max\{V_k: 0 \leq k \leq m\}$.
Fix $j \in L$ and consider $\nu \in [K_j]_{\rho}$. Let $\pi_1^m \in L_m$ be such that $K_j \in \pi_1^m$. If $\pi_1^m \neq \pi_0^m$, then we have $\pi_1^m \Rightarrow \pi_0^m$, that is, there exists $\pi_2^m, \pi_3^m, \ldots, \pi_n^m =\pi_0^m, n \leq l$ such that $\pi_1^m \to \pi_2^m \to \pi_3^m \to \cdots \to \pi_n^m = \pi_0^m$. Therefore, with $\delta$ to be chosen later, by the strong Markov property (we use the standard notation $E_\nu(A; B)$ for $E_\nu(1_A 1_B)$ where $A$ and $B$ are measurable sets), $$\begin{aligned}
P_\nu (\hat{\tau}_{\pi^m_0} & \leq n \exp\{N(V_m - \delta)\}) \geq E_\nu ( \bar{\tau}_{\pi^m_1} \leq \exp\{N(V_m -\delta )\}, \mu_N(\bar{\tau}_{\pi^m_1}) \in \pi_2^m; \\
& E_{\mu_N(\bar{\tau}_{\pi^m_1})}(\bar{\tau}_{\pi_2^m} \leq \exp\{N(V_m -\delta )\},\mu_N(\bar{\tau}_{\pi^m_2}) \in \pi_3^m; \\
& \cdots E_{\mu_N(\bar{\tau}_{\pi^m_{n-2}})}( \bar{\tau}_{\pi^m_{n-1}} \leq \exp\{N(V_m -\delta )\},\mu_N(\bar{\tau}_{\pi^m_{n-1}}) \in \pi_0^m)\\
& \cdots )).\end{aligned}$$ Since $V(\pi_i^m) \leq V_m$ for all $1 \leq i \leq n$, the above becomes
$$\begin{aligned}
P_\nu (\hat{\tau}_{\pi^m_0} & \leq n \exp\{N(V_m - \delta)\}) \\
& \geq E_\nu ( \bar{\tau}_{\pi^m_1} \leq \exp\{N(\tilde{V}(\pi_1^m) -\delta )\}, \mu_N(\bar{\tau}_{\pi^m_1}) \in \pi_2^m; \\
& E_{\mu_N(\bar{\tau}_{\pi^m_1})}(\bar{\tau}_{\pi_2^m} \leq \exp\{N(\tilde{V}(\pi_2^m)-\delta )\},\mu_N(\bar{\tau}_{\pi^m_2}) \in \pi_3^m; \\
& \cdots E_{\mu_N(\bar{\tau}_{\pi^m_{n-2}})}( \bar{\tau}_{\pi^m_{n-1}} \leq \exp\{N(\tilde{V}(\pi_{n-1}^m) -\delta )\},\mu_N(\bar{\tau}_{\pi^m_{n-1}}) \in \pi_0^m)\\
& \cdots )).\end{aligned}$$
By Lemma \[lemma:hitting\_place\], there exist $\rho >0$, $\delta > 0$ and $N_0 \geq 1 $ such that each of the above probabilities is at least $\exp\{-N\varepsilon/l\}$ for sufficiently large $N$, i.e. we have $$\begin{aligned}
P_\nu (\hat{\tau}_{\pi^m_0} \leq n \exp\{N(V_m - \delta))\}) \geq \exp\{-N n \varepsilon/l\} \geq \exp\{-N\varepsilon\},\end{aligned}$$ On the other hand, if $K_j$ is such that $K_j \in \pi_0^m$, the above holds trivially. Therefore, there exist $\delta_1 > 0$ and $N_1 \geq 1$ such that for all $\nu \in \gamma \cap M_1^N(\mathcal{Z})$ and $N \geq N_1$, we have $$\begin{aligned}
P_\nu (\hat{\tau}_{\pi^m_0} \leq \exp\{N(V_m - \delta_1)\}) \geq \exp\{-N \varepsilon\}.\end{aligned}$$ We now use the above bound to show (\[eqn:lb\_tpm\_i0\]). Let $T = \exp\{N(\Lambda - \delta_1)\}, T_m = \exp\{N(V_m - \delta_1)\}$ and $T_{m-1} =\exp\{N(V_{m-1}-\delta_1)\}$. Then, for any $\nu \in \gamma \cap M_1^N(\mathcal{Z})$ and $N \geq N_1$, we have $$\begin{aligned}
P_\nu (\mu_N(T) \in \gamma_{i_0})& \geq E_\nu (\hat{\tau}_{\pi^m_0} \leq T_m; E_{\mu_N(\hat{\tau}_{\pi^m_0})}(\mu_N(T-\hat{\tau}_{\pi^m_0}) \in \gamma_{i_0})) \nonumber \\
& \geq \inf_{\substack{\nu \in [\pi_0^m]_{\rho}\cap M_1^N(\mathcal{Z}) \\ T-T_m \leq t \leq T}} P_\nu(\mu_N(t) \in \gamma_{i_0}) P_\nu (\hat{\tau}_{\pi^m_0}\leq T_m) \nonumber \\
& \geq \inf_{\substack{\nu \in [\pi_0^m]_{\rho}\cap M_1^N(\mathcal{Z}) \\ T-T_m \leq t \leq T}} P_\nu(\mu_N(t) \in \gamma_{i_0}) \exp\{-N\varepsilon\}.
\label{eqn:temp1}\end{aligned}$$ To get a lower bound for the above infimum, fix $\nu \in [\pi_0^m]_\rho \cap M_1^N(\mathcal{Z})$ and $T-T_{m} \leq t \leq T$. Define the stopping time $\theta \coloneqq \inf\{s> t-T_{m-1} : \mu_N(s) \in [\pi_0^m]_{\rho}\}$. Then, for a large $T^*$ (not depending on $N$) to be chosen later, we have $$\begin{aligned}
P_\nu &(\mu_N(t) \in \gamma_{i_0}) \nonumber \\
&\geq E_\nu (\theta \leq t-T_{m-1}+T^* , \bar{\tau}_{\pi_0^m} > T; E_{\mu_N(\theta)}(\mu_N(t-\theta)) \in \gamma_{i_0}) \nonumber \\
& \geq P_\nu(\theta \leq t-T_{m-1}+T^* , \bar{\tau}_{\pi_0^m} > T) \inf_{\substack{\nu^\prime \in [\pi_0^m]_{\rho}\cap M_1^N(\mathcal{Z}) \\ T_{m-1}-T^*\leq t \leq T_{m-1}}} P_{\nu^\prime} (\mu_N(t) \in \gamma_{i_0}).
\label{eqn:temp2}\end{aligned}$$ Note that $$\begin{aligned}
P_\nu(\theta \leq t-T_{m-1}+T^* & , \bar{\tau}_{\pi_0^m} > T) \\
&= P_\nu(\bar{\tau}_{\pi^m_0} > T) - P_\nu(\theta > t-T_{m-1}+T^* , \bar{\tau}_{\pi_0^m} > T).\end{aligned}$$ By Lemma \[lemma:hitting\_time\], since $\Lambda \leq \tilde{V}(\pi^m_0)$, we have $$\begin{aligned}
P_\nu(\bar{\tau}_{\pi^m_0} > T) \geq P_\nu(\bar{\tau}_{\pi^m_0} > \exp\{N(\tilde{V}(\pi^m_0) - \delta)\}) \to 1\end{aligned}$$ as $N \to \infty$. For the second term, note that $$\begin{aligned}
P_\nu& (\theta > t-T_{m-1}+T^* , \bar{\tau}_{\pi_0^m} > T) & \\
& = P_\nu (\mu_N(s) \notin [\pi_0^m]_\rho \text{ for all } t-T_{m-1}\leq s \leq t-T_{m-1}+T^*, \bar{\tau}_{\pi_0^m} > T) \\
& = P_\nu (\mu_N(s) \notin \gamma \text{ for all } t-T_{m-1}\leq s \leq t-T_{m-1}+T^*, \bar{\tau}_{\pi_0^m} > T) \\
& \leq P_\nu (\mu_N(s) \notin \gamma \text{ for all } t-T_{m-1}\leq s \leq t-T_{m-1}+T^*).\end{aligned}$$ The second equality follows since $\mu_N(s) \notin [\pi_0^m]_\rho$ and $\bar{\tau}_{\pi_0^m} > T$ implies that we have exited $[\pi_0^m]_\rho$ and we have not yet entered a neighbourhood of any other attractor, which is the same as saying $\mu_N(t) \notin \gamma$ and $\bar{\tau}_{\pi_0^m} >T$. By the Markov property, the above probability equals $$\begin{aligned}
E_\nu & \left( E_{\mu_N(t-T_{m-1})}(\mu_N(s) \notin \gamma \text{ for all } s \in [t-T_{m-1}, t-T_{m-1}+T^*]) \right)\\
& \leq \sup_{\nu^\prime \in F} P_{\nu^\prime} (\tau_F \geq T^*),\end{aligned}$$ where $F = M_1(\mathcal{Z}) \setminus \gamma$. By Lemma \[lemma:fw19\], $T^*$ can be chosen large enough (not depending on $N$) that the above probability is at most $1/2$. Therefore, (\[eqn:temp2\]) becomes $$\begin{aligned}
\inf_{\substack{\nu \in [\pi_0^m]_\rho \cap M_1^N(\mathcal{Z}) \\ T-T_m \leq t \leq T}} P_\nu(\mu_N(t) \in \gamma_{i_0}) \geq \frac{1}{2} \inf_{\substack{\nu^\prime \in [\pi_0^m]_{\rho} \cap M_1^N(\mathcal{Z}) \\ T_{m-1}-T^* \leq t \leq T_{m-1}}} P_{\nu^\prime} (\mu_N(t) \in \gamma_{i_0}),\end{aligned}$$ and (\[eqn:temp1\]) becomes $$\begin{aligned}
P_\nu (\mu_N(T) \in \gamma_{i_0}) \geq \frac{1}{2} \exp\{-N\varepsilon\} \inf_{\substack{\nu^\prime \in [\pi_0^m]_{\rho} \cap M_1^N(\mathcal{Z}) \\ T_{m-1}-T^* \leq t \leq T_{m-1}}} P_{\nu^\prime} (\mu_N(t) \in \gamma_{i_0}),\end{aligned}$$ for sufficiently large $N$ and $\nu \in \gamma \cap M_1^N(\mathcal{Z})$. Repeating the above argument $m$ times, we see that there exists $N_2 \geq 1$ such that for all $\nu \in \gamma$ and $N \geq N_2$, we have $$\begin{aligned}
P_\nu(\mu_N(T) \in \gamma_{i_0}) & \geq \left(\frac{1}{2}\right)^m\exp\{-Nm\varepsilon\} \inf_{\substack{\nu^\prime \in [\pi_0^1]_{\rho}\cap M_1^N(\mathcal{Z}) \\ T_0 -T^*\leq t \leq T_0}} P_{\nu^\prime} (\mu_N(t) \in \gamma_{i_0}) \\
& \geq \left(\frac{1}{2}\right)^{m}\exp\{-N(m+1)\varepsilon\} \inf_{\substack{\nu^\prime \in [K_0]_{\rho} \cap M_1^N(\mathcal{Z})\\ T_0 -T^*\leq t \leq T_0}} P_{\nu^\prime} (\mu_N(t) \in \gamma_{i_0}) \\
& \geq \left(\frac{1}{2}\right)^{m+1}\exp\{-N(m+1)\varepsilon\},\end{aligned}$$ where $T_0 = \exp\{N(V_0 - m\delta)\}$. Thus, we conclude that there is $N_3 \geq 1$, $\delta_3>0$ and $\rho > 0$ such that for all $\nu \in \gamma \cap M_1^N(\mathcal{Z})$ and $N \geq N_3$, we have $$\begin{aligned}
P_\nu(\mu_N(T) \in \gamma_{i_0}) \geq \exp\{-N(m+3)\varepsilon\},\end{aligned}$$ where $T = \exp\{N(\Lambda-\delta_3)\}$. This establishes (\[eqn:lb\_tpm\_i0\]) for all $\nu \in \gamma \cap M_1^N(\mathcal{Z})$. For any $\nu \in M_1^N(\mathcal{Z})\setminus \gamma$, from Lemma \[lemma:fw19\], there exists $T^\prime$ large enough and $N_4 \geq N_3$ such that $P_\nu(\tau_{M_1(\mathcal{Z}) \setminus \gamma} \leq T^\prime) \leq \frac{1}{2}$ for all $N \geq N_4$. Therefore, we have $$\begin{aligned}
P_\nu(\mu_N(T) \in \gamma_{i_0}) & \geq E_\nu (\tau_{M_1(\mathcal{Z}) \setminus \gamma} \leq T^\prime, P_{\mu_N(\tau_F)} (\mu_N(T-T^\prime) \in \gamma_{i_0})) \\
& \geq \frac{1}{2} \inf_{\nu^\prime \in \gamma} P_{\nu^\prime}(\mu_N(T-T^\prime) \in \gamma_{i_0}) \\
& \geq \frac{1}{2}\exp\{-N(m+3)\varepsilon\}.\end{aligned}$$ Thus, we have established (\[eqn:lb\_tpm\_i0\]) for any $\nu \in M_1^N(\mathcal{Z})$.
We now turn to (\[eqn:ub\_tpm\]). Since $\Lambda = \max\{V_k, 0 \leq k \leq m\}$, there exists a $k$ such that $V_k = \Lambda$. From the definition of $V_k$, we see that there exists $\pi^k \in L_k$ such that $$\begin{aligned}
\tilde{V}(\pi^k) = \Lambda, \pi^k \subset \pi_0^{k+1}, \text{ and }\pi^k \neq \pi^k_0.\end{aligned}$$ where $\pi_0^{k+1}$ is the $(k+1)$-cycle that contain $K_{i_0}$. Therefore, Lemma \[lemma:hitting\_time\] implies that, for some $\beta > 0$, for some $\delta_4 < \delta_3$ and an appropriately chosen $\rho > 0$, with $T = \exp\{N(\Lambda-\delta_3)\} = \exp\{N(\tilde{V}(\pi^k) - \delta_3)\}$, we have $$\begin{aligned}
P_\nu(\mu_N(T) \in \gamma_{i_0}) \leq P_\nu(\bar{\tau}_{\pi^k} \leq T) \leq \exp\{-N\beta\},\end{aligned}$$ for any $\nu \in [\pi^k]_{\rho} \cap M_1^N(\mathcal{Z})$ and sufficiently large $N$. This completes the proof of the theorem.
The above theorem immediately gives a lower bound on $P_T(\nu, \xi)$ for any $\xi$ in a small neighbourhood of $K_{i_0}$, over time durations of order $\exp\{N(\Lambda-\delta)\}$ for some $\delta > 0$. Let us make this precise.
Under the conditions of Theorem \[thm:mixing\], for all $\nu \in M_1^N(\mathcal{Z})$, $\xi \in \gamma_{i_0} \cap M_1^N(\mathcal{Z})$ and $N$ sufficiently large, we have $$\begin{aligned}
P_{T_0}(\nu, \xi) \geq \exp\{-2N\varepsilon\}.\end{aligned}$$ \[cor:lowerboundtransition\]
Given $\varepsilon > 0$, let $\rho, N_0$ and $T_0$ be as in the statement of Theorem \[thm:mixing\]. Choose $t$ large enough (not depending on $N$) and $\rho^\prime < \rho$ such that for all $\rho_1 \leq \rho^\prime$ we have $S_t(\nu_1|\nu_2) \leq \varepsilon/2$ for all $\nu_1, \nu_2 \in \gamma_{i_0}$. This is possible by the joint continuity of the rate function $S_t(\cdot|\cdot)$ and the fact that $V(\nu_1, \nu_2) = 0$ whenever $\nu_1, \nu_2 \in K_{i_0}$. Therefore, using the large deviation lower bound, there exists $N_2 \geq N_1$ such that $$\begin{aligned}
P_t(\nu_1, \nu_2) \geq \exp\{-N(S_t(\nu_2 | \nu_1) + \varepsilon/2)\} \geq \exp\{-N\varepsilon\},\end{aligned}$$ for all $\nu_1, \nu_2 \in \gamma_{i_0} \cap M_1^N(\mathcal{Z})$ and $N \geq N_2$. Therefore, by Theorem \[thm:mixing\], for $\nu \in M_1^N(\mathcal{Z}), \xi \in \gamma_{i_0} \cap M_1^N(\mathcal{Z})$ and $N \geq N_2$, we have $$\begin{aligned}
P_{T_0}(\nu, \xi) & = \sum_{\nu_2 \in \gamma_{i_0} \cap M_1^N(\mathcal{Z})} P_{T_0-t}(\nu_1, \nu_2) P_t(\nu_2, \xi) \\
& \geq P_{T_0-t}(\nu_1, \gamma_{i_0}) \inf_{\nu_2 \in \gamma_{i_0} \cap M_1^N(\mathcal{Z})} P_t(\nu_2, \xi) \\
& \geq \exp\{-2N \varepsilon\}.\end{aligned}$$
We now prove our first main result, namely convergence of $\mu_N$ to the invariant measure. Intuitively, the result says that, over a time duration $\exp\{N(\Lambda+\delta)\}$ for any $\delta>0$, the process is very close to its invariant measure $\wp_N$. Define $\tilde{L}_0 \coloneqq \{i \in L : W(K_i) = 0\}$, i.e, $\tilde{L}_0$ denotes the set of minimisers of the rate function $s$ (see Theorem \[thm:invariant-measure-gase\]).
Given $\delta > 0$, there exist $\varepsilon > 0$ and $N_0 \geq 1 $ such that for all $\nu \in M_1^N(\mathcal{Z})$ and $N \geq N_0$ $$\begin{aligned}
\left| E_\nu (f(\mu_N(T) ))- \langle f, \wp_N \rangle \right| \leq \|f\|_\infty \exp\{-\exp(N\varepsilon)\},\end{aligned}$$ where $T = \exp\{N(\Lambda+\delta)\}$ and $f \in B(M_1(\mathcal{Z}))$. \[thm:conv\]
We follow the steps in Hwang and Sheu [[@hwang-sheu-90 Part I, Theorem 2.5]]{}. Let $\varepsilon > 0$, and let $T_0, \delta_0, \rho, \rho_1$ and $N_0 \geq1$ be as in the statement of Theorem \[thm:mixing\]. Note that, for any $\nu \in M_1^N(\mathcal{Z})$, $\xi \notin [\tilde{L}_0]_{\rho_1}$ and for some fixed $t > 0$, $$\begin{aligned}
P_{T_0}(\nu,\xi)& = \sum_{\nu^\prime \in [K_{i_0}]} P_{T_0-t}(\nu, \nu^\prime) P_t(\nu^\prime, \xi)\\
& \geq \exp\{-2N\varepsilon\} \inf_{\nu^\prime \in [K_{i_0}]} P_t(\nu^\prime, \xi) \\
& \geq \exp\{-2N\varepsilon\} \exp\{-N\sup_{\nu^\prime \in [K_{i_0}]} S_t(\xi|\nu^\prime)\}\end{aligned}$$ where the first inequality follows from Corollary \[cor:lowerboundtransition\] and the second from the uniform LPD (Corollary \[cor:uniform\_ldp\]). Hence, we can find a function $U : M_1(\mathcal{Z}) \to [0, \infty)$ such that $U(\xi) = 0$ for $\xi \in [\tilde{L}_0]_{\rho_1}$ and $$\begin{aligned}
P_{T_0}(\nu,\xi) \geq c_N \exp\{-NU(\xi)\}
\label{eqn:temp8}\end{aligned}$$ holds for all $\nu \in M_1^N(\mathcal{Z})$, $\xi \notin [\tilde{L}_0]_{\rho_1}$ and sufficiency large $N$; here $c_N$ is such that $$\begin{aligned}
\pi_N(\xi) = c_N\exp\{-NU(\xi)\}\end{aligned}$$ is a probability measure on $M_1^N(\mathcal{Z})$. Define $Q_{T_0}(\nu ,\cdot) \coloneqq P_{T_0}(\nu, \cdot)/ \pi(\cdot)$. We have, for any $\nu_1, \nu_2 \in M_1^N(\mathcal{Z})$ and sufficiently large $N$, $$\begin{aligned}
E_{\nu_1}&(f(\mu_N(T_0))) - E_{\nu_2}(f(\mu_N(T_0))) \\
& = \sum_{\xi \in M_1^N(\mathcal{Z})}P_{T_0}(\nu_1,\xi) f(\xi) - \sum_{\xi \in M_1^N(\mathcal{Z})}P_{T_0}(\nu_2,\xi) f(\xi) \\
& = \sum_{\xi \in M_1^N(\mathcal{Z})}Q_{T_0}(\nu_1,\xi) f(\xi) \pi_N(\xi)- \sum_{\xi \in M_1^N(\mathcal{Z})}Q_{T_0}(\nu_2,\xi) f(\xi) \pi_N(\xi) \\
& = \sum_{\xi \in M_1^N(\mathcal{Z})}(Q_{T_0}(\nu_1,\xi)- \exp\{-2N\varepsilon\})f(\xi) \pi_N(\xi) \\
& \, \, \, \, \, \, \, \, -\sum_{\xi \in M_1^N(\mathcal{Z})}(Q_{T_0}(\nu_2,\xi) - \exp\{-2N\varepsilon\}) f(\xi) \pi_N(\xi) \\
& \leq (1-\exp\{-2N\varepsilon\}) (\sup_\xi f(\xi) - \inf_\xi f(\xi)),\end{aligned}$$ where the last inequality follows from (\[eqn:temp8\]) and the fact that $Q_{T_0}(\cdot, \cdot) \geq 1$. Therefore, we have that $$\begin{aligned}
\sup_{\nu_1, \nu_2} | E_{\nu_1}&(f(\mu_N(T_0))) - E_{\nu_2}(f(\mu_N(T_0))) | \leq (1-\exp\{-2N\varepsilon\}) \| f\|_\infty.\end{aligned}$$ Continuing this procedure $k$ times, and by using the Markov property, we get $$\begin{aligned}
\sup_{\nu_1, \nu_2} |E_{\nu_1}&(f(\mu_N(kT_0))) - E_{\nu_2}(f(\mu_N(kT_0))) | \leq (1-\exp\{-2N\varepsilon\})^k \| f\|_\infty,\end{aligned}$$ and hence, we have $$\begin{aligned}
\sup_{\nu} | E_{\nu}&(f(\mu_N(kT_0))) - \langle f, \wp_N \rangle | \leq (1-\exp\{-2N\varepsilon\})^k \| f\|_\infty.\end{aligned}$$ Choose $k = \exp\{N(\delta_0+\delta)\}$, then we have $kT_0 = \exp\{N(\Lambda + \delta)\}$ and the above becomes $$\begin{aligned}
\sup_{\nu} | E_{\nu}&(f(\mu_N(kT_0))) - \langle f, \wp_N \rangle | \leq \exp\{-\exp(N(-2\varepsilon+\delta_0 + \delta))\}.\end{aligned}$$ We can choose $\varepsilon$ small enough such that the quantity $-2\varepsilon+ \delta > 0$, and hence for some $\varepsilon^\prime > 0$, we have $$\begin{aligned}
\sup_{\nu} | E_{\nu}&(f(\mu_N(T))) - \langle f, \wp_N \rangle | \leq \exp\{-\exp(N\varepsilon^\prime)\},\end{aligned}$$ for sufficiently large $N$, where $T = \exp\{N(\Lambda+\delta)\}$. This establishes the result.
Asymptotics of the second eigenvalue for reversible processes {#section:eval_problem}
=============================================================
In this section, our goal is to understand the convergence rate of $\mu_N$ to its invariant measure for a fixed $N$. For this purpose, we shall assume that the Markov process $\mu_N$ is reversible. That is, the operator $L^N$ is self-adjoint in $L^2(\wp_N)$ and it admits a spectral expansion; let $0 =\lambda_1^N > -\lambda_2^N \geq -\lambda_3^N \ldots$ denote its eigenvalues in the decreasing order, and let $u_1^N \equiv 1, u_2^N, u_3^N,\ldots$ denote their corresponding eigenfunctions. The spectral expansion enables us to write, for any $f \in B(M_1(\mathcal{Z}))$, $$\begin{aligned}
E_\nu f(\mu_N(t)) = \langle f, \wp_N\rangle + \sum_{k \geq 1} e^{-t\lambda_k^N} ( f, u_k^N ) u_k^N(\nu),
\label{eqn:spectral_expansion}\end{aligned}$$ where $( \cdot, \cdot )$ denotes the inner product in $L^2(\wp_N)$. Therefore, the convergence rate of $E_\nu f(\mu_N(t))$ to its stationary value $\langle f, \wp_N \rangle$ is determined by the leading term in the above sum, which is the second eigenvalue $\lambda_2^N$. Hence, to understand convergence of $\mu_N$ to its invariant measure, we study the asymptotics of the second eigenvalue $\lambda_2^N$.
We first need the following lemma that estimates the probability that the process $\mu_N$ is outside a small neighbourhood of the set $\cup_{i=1}^l K_i$. This can be shown using Theorem \[thm:conv\] with deals with the convergence to the invariant measure and Theorem \[thm:invariant-measure-gase\] which addresses large deviations of the invariant measure $\{\wp_N\}_{N \geq 1}$.
Fix $\rho_1 >0$ and let $B$ be the $\rho_1$-neighbourhood of $\cup_{i \in L} K_i$. Given $\varepsilon > 0$, there exist $\delta >0$ and $N_0 \geq 1$ such that for each $\nu \in M_1^N(\mathcal{Z})$ and $N \geq N_0$, we have $$\begin{aligned}
P_\nu \left(\mu_N(T) \in M_1^N(\mathcal{Z}) \setminus B\right) \leq \exp\{-N\delta\},\end{aligned}$$ where $T = \exp\{N(\Lambda+ \varepsilon)\}$.
We are now ready to prove our next main result on the asymptotics of the second eigenvalue $\lambda_2^N$.
$$\begin{aligned}
\lim_{N \to \infty} \frac{1}{N} \log \lambda_2^N = -\Lambda.\end{aligned}$$
\[thm:eval\_problem\]
(Lower bound): Suppose that there exists a subsequence $\{N_k\}_{k \geq 1}$ such that $$\begin{aligned}
\log \lambda_2^{N_k} < -N_k(\Lambda + \varepsilon)
\label{eqn:temp-assumption}\end{aligned}$$ for some $\varepsilon > 0$. We will show that this contradicts $\int (u_2^{N_k}(\nu))^2 \wp_N(d\nu) =1 $ for sufficiently large $k$. Fix $\rho >0 $ and define $B \coloneqq \cup_{i=1}^l [K_i]_{\rho}$. Then, using the lower semicontinuity of the rate function $S_t(\cdot|\cdot)$ and Corollary \[cor:uniform\_ldp\] on uniform LDP, we see that for sufficiently large $t$, there exists $\delta_1 >0$ such that $\inf\{S_t(\xi|\nu):\xi, \nu \in B^c\} = \delta_1 > 0$. Therefore, for any $\nu \in B^c \cap M_1^N(\mathcal{Z})$ and any $\delta_2>0$, there exists $N_0 \geq 1$ such that for all $N \geq N_0$, $$\begin{aligned}
P_\nu(\mu_N(t) = \nu) \leq \exp\{-N(S_t(\nu|\nu) - \delta_2)\} \leq \exp\{-N(\delta_1 + \delta_2)\}.\end{aligned}$$ On the other hand, (\[eqn:spectral\_expansion\]) implies that, $$\begin{aligned}
P_\nu(\mu_N(t) = \nu) & = E_\nu(1_{\nu}(\mu_N(t))) \\
&\geq e^{-\lambda_2^Nt} (u_2^N(\nu))^2 \wp_N(\nu),\end{aligned}$$ so that $$\begin{aligned}
\int_{B^c} |u_2^N|^2 \wp_N (d \nu) \leq \exp\{-N(\delta_1 + \delta_2)\}
\label{eqn:temp11}\end{aligned}$$ for all $N \geq N_0$. To bound the integral over $B$, by Theorem \[thm:conv\], with $T = \exp\{N(\Lambda+\varepsilon/2)\}$, there exist $\delta_3>0$ and $N_1 \geq N_0$ such that for all $N \geq N_1$, $$\begin{aligned}
\left|E_\nu f(\mu_N(T)) - \langle f, \wp_N \rangle \right| \leq \|f\|_\infty \exp\{-\exp(N\delta_3)\},\end{aligned}$$ for any $f \in B(M_1(\mathcal{Z}))$. On the other hand, from (\[eqn:spectral\_expansion\]), for any $\nu \in B \cap M_1^N(\mathcal{Z})$, with $f = 1_\nu$, we have
$$\begin{aligned}
\left| E_\nu f(\mu_N(T)) - \langle f, \wp_N \rangle \right| & = \sum_{i \geq 2} \exp\{-\lambda_i^NT\} \langle f, \wp_N(\nu) \rangle u_i^{N}(\nu) \\
& \geq \exp\{-\lambda_2^{N}T\} (u_2^{N}(\nu))^2 \wp_N(\nu),\end{aligned}$$
so that, by our assumption (\[eqn:temp-assumption\]), there exists a $k_0 \geq 1$ such that $$\begin{aligned}
u_2^{N_k}(\nu))^2 \wp_{N_k}(\nu) & \leq \exp\{\lambda_2^{N_k}T\} \exp\{-\exp(N_k\delta_3)\} \\
& \leq \exp\{2\exp(-N_k(\Lambda+\varepsilon)) \exp(N_k(\Lambda+\varepsilon/2))\} \exp\{-N_k\delta_3\}\end{aligned}$$ for all $k \geq k_0$. Since $|M_1^{N_k}(\mathcal{Z})| \leq (N_k+1)^{|\mathcal{Z}|}$ for all $k$, the above implies that, for some $\delta_4 > 0$, $$\begin{aligned}
\int_B (u_2^{N_k}(\nu))^2 \wp_{N_k}(d\nu) \leq \exp\{-N_k\delta_4\}
\label{eqn:temp12}\end{aligned}$$ for all $k \geq k_0$. Therefore, (\[eqn:temp11\]) and (\[eqn:temp12\]) implies that, for some $\delta > 0$, $$\begin{aligned}
\int_{M_1(\mathcal{Z})} (u_2^{N_k}(\nu))^2 \wp_{N_k}(d\nu) \leq \exp\{-N_k\delta\}\end{aligned}$$ for all sufficiently large $k$, which is a contradiction to $\int (u_2^{N_k}(\nu))^2 \wp_{N_k} (d\nu) =1$ for all sufficiently large $k$.
(Upper bound): Suppose that there exists a subsequence $\{N_k\}_{k \geq 1}$ such that $\log \lambda_2^N > N_k(-\Lambda + \varepsilon)$ for some $\varepsilon>0$. Let $\nu_0, \delta_0<\varepsilon/2, \rho, N_0$ be as in Theorem \[thm:mixing\]. Then, with $f(\nu) =1_{[K_{i_0}]_{\rho/2}}(\nu)$ and $T = \exp\{N(\Lambda-\delta_0/2)\}$, (\[eqn:ub\_tpm\]) implies that $$\begin{aligned}
E_\nu f(\mu_N(T)) = P_\nu(\mu_N(T) \in [K_{i_0}]_{\rho/2}) \leq \exp\{-N\beta\}\end{aligned}$$ for all $N \geq N_0$ and $\nu \in [\nu_0]_{\rho/2 } \cap M_1^N(\mathcal{Z})$. Also, by Theorem \[thm:invariant-measure-gase\], for any $\delta >0$, there exists $N_1 \geq N_0$ such that for all $N \geq N_1$, we have $$\begin{aligned}
\langle f, \wp_N \rangle = \wp_N([K_{i_0}]_{\rho/2}) \geq \exp\{-N\delta\}.\end{aligned}$$ This is possible since $\inf_{\xi \in [K_{i_0}]_{\rho/2}}s(\xi) =0$. Therefore, for all $N \geq N_1$, $$\begin{aligned}
\int_{M_1(\mathcal{Z})} |E_\nu(f(\mu_N(T))) & - \langle f, \wp_N \rangle |^2 \wp_N(d\nu) \\
&\geq \int_{[\nu_0]_{\rho/2}} \left|E_\nu(f(\mu_N(T))) - \langle f, \wp_N \rangle \right|^2 \wp_N(d\nu) \\
& \geq \wp_N([\nu_0]_{\rho/2}) (\exp\{-N\beta\} - \exp\{-N\delta\}) \\
&\geq \wp_N([\nu_0]_{\rho/2}) \exp\{-N\delta_1\}, \text{ for some } \delta_1 > 0 \\
&\geq \exp\{-N\delta_2\}, \text{ for some } \delta_2 > 0,\end{aligned}$$ where the last inequality follows by Theorem \[thm:invariant-measure-gase\]. On the other hand, for any function $f$ with $\int |f|^2 d\wp_N \leq 1$, we have $$\begin{aligned}
\int_{M_1(\mathcal{Z})} |E_\nu(f(\mu_N(T))) & - \langle f, \wp_N \rangle |^2 \wp_N(d\nu) \\
& = \int_{M_1(\mathcal{Z})} \sum_{k\geq 2} e^{-2\lambda_k^NT} \langle f, u_2^N\rangle u_2^N(\nu)^2 \wp_N(d\nu) \\
&\leq \exp\{-2\lambda_2^NT\} \int_{M_1(\mathcal{Z})} |f|^2 d\wp_N \\
&\leq \exp\{-2\lambda_2^NT\}.\end{aligned}$$ Therefore, we have $ \exp\{-2\lambda_2^{N}T\} \geq \exp\{-N\delta_2\}$ whenever $N \geq N_1$. By our assumption, we see that $$\begin{aligned}
\exp\{-2\exp(-N_k(\Lambda-\varepsilon)) \exp(N_k(\Lambda-\delta_0))\} \geq \exp\{-N_k\delta_1\}\end{aligned}$$ for sufficiently large $k$, which is a contradiction since $\delta_0 < \varepsilon/2$.
Using the above theorem, we see that, if $\Lambda>0$, then as $N$ becomes large, it takes longer for the process $\mu_N$ to be close to its invariant measure. This particularly means that metastable states reduce the rates of convergence of $\mu_N$ to its invariant measure. On the other hand, if there is a unique global attractor of the limiting McKean-Vlasov equation (\[eqn:MVE\]), then we see that $\Lambda = 0$, and convergence rate of $\mu_N$ to its invariant measure does not suffer from such a slowing down phenomenon.
Convergence to the global minimum {#section:conv_global_minimum}
=================================
In this section, our goal is to increase the number of particles $N$ over time so as to obtain, with very high probability, convergence of the empirical measure process to a global minimum of the rate function $s$ that governs the LDP for the sequence of invariant measure $\{\wp_N\}_{N\geq 1}$.
Fix $c>0$. Let $N_0 = \min\{n \in \mathbb{N} : \exp\{nc\} - 2 \geq 0 \}$, $t_{N_0} = 0$, and for each $N > N_0$, let $t_N = \exp\{Nc\}-2$. For each $N \geq N_0$ define the generator $L^N_t$ acting on bounded measurable functions on $M_1(\mathcal{Z})$ by $$\begin{aligned}
L_t^Nf (\xi) \coloneqq \sum_{(z,z^\prime) \in E} N_t \xi(z) \lambda _{z,z^\prime}(\xi) \left[ f\left(\xi+\frac{e_{z^\prime}}{N_t}- \frac{e_z}{N_t}\right) - f(\xi) \right], \, t\in [t_{N}, t_{N+1}).\end{aligned}$$ where $N_t = N$ for $t \in [t_N, t_{N+1})$. Let $z_0 \in \mathcal{Z}$ be a fixed state and let $\nu \in M_1^{N_0}(\mathcal{Z})$. We say that a probability measure $P_{0,\nu}$ on $D([0,\infty),M_1(\mathcal{Z}))$ is a solution to the martingale problem for $\{L^N\}_{N \geq N_0}$ with initial condition $\nu$ if $P_{0,\nu}(\bar{\mu}: \bar{\mu}(0) = \nu)=1$, for each $N \geq N_0$, the restriction of $P_{0,\nu}$ on $D([t_N, t_{N+1}), M_1^{N}(\mathcal{Z}))$ is a solution to the $D([t_N, t_{N+1}), M_1^{N}(\mathcal{Z}))$-valued martingale problem for $L^N$, and $$\begin{aligned}
P_{0,\nu}\left(\bar{\mu}: \bar{\mu}(t_{N+1}) = \frac{N}{1+N}\bar{\mu}(t_{N+1}^-) + \frac{1}{N+1} \delta_{z_0}\right) =1.\end{aligned}$$ Again, by the boundedness assumption on transition rates \[assm:a2\], for each $\nu \in M_1^{N_0}(\mathcal{Z})$, there exists a unique probability measure $P_{0,\nu}$ that solves the martingale problem for $\{L^N\}_{N \geq 1}$ with initial condition $\nu$. Let $\bar{\mu}$ be the process on $D([0,\infty),M_1(\mathcal{Z}))$ whose law is $P_{0,\nu}$. To describe the process in words, we start with $N_0$ particles and follow the mean-field interaction described in Section \[section:introduction\], except that at each time instant $t_N, N > N_0$, we add a new particle whose state is set to $z_0$.
We anticipate that if $c$ is small then $N_t$ is so large that the fluid limit kicks in too quickly over time and the process $\bar{\mu}$ converges (over time) to a local minimum of $s$ with positive probability depending on the initial condition $\bar{\mu}(0)$. When $c$ is sufficiently large, we anticipate that there is enough time for exploration and therefore we will converge to a global minimum of $s$. Recall that the set of global minimisers of $s$ is denoted by $\tilde{L}_0$. Our interest in this section is in finding a constant $c^*$ such that for all $c > c^*$ and $\nu \in M_1^{N_0}(\mathcal{Z})$, we have, $$\begin{aligned}
P_{0,\nu} (\bar{\mu}(t) \text{ lies in a neighbourhood of }\tilde{L}_0) \to 1
\label{eqn:conv_globalmin}\end{aligned}$$ as $t \to \infty$.
We use the results in the previous sections to identify the constant $c^*$. Since $N_t \to \infty$ as $t \to \infty$, for a fixed $T > 0$ and large enough $t$, the large deviation properties of the process $\{\bar{\mu}(s), t \leq s \leq t+T \}$ from the limiting dynamics (\[eqn:MVE\]) starting at an arbitrary $\bar{\mu}(t)$ can be obtained similar to the LDP of the process $\mu_N$ studied in Theorem \[thm:finite-duration-ldp\] and Corollary \[cor:uniform\_ldp\]. Therefore, the results in the previous sections on the large time behaviour for the process $\{\mu_N(t), t\geq 0\}$ are also valid for $\{\bar{\mu}(t) , t \geq 0\}$ when time $t$ is large enough; we make these precise now.
Let $\pi_1^k$ and $\pi_2^k$ be $k$-cycles and suppose that $\pi_1^k \to \pi_2^k$ and $\tilde{V}(\pi_1^k)/c < 1$. Then, given $\varepsilon > 0$, there exist $\delta>0$ and $\rho>0$ such that for all $\rho_1 < \rho$, there is $t^* >0$ such that $$\begin{aligned}
P_{t,\nu}(\bar{\tau}_{\pi^k_1} \leq t + t^{(\tilde{V}(\pi_1^k) -\delta)/c}, \bar{\mu}(\bar{\tau}_{\pi_1^k}) \in \gamma_{\pi_2^k}) \geq t^{-\varepsilon/c}\end{aligned}$$ holds uniformly for all $\nu \in [\pi_1^k]_{\rho_1} \cap M_1^{N_{t}}(\mathcal{Z})$ and $t \geq t^*$. \[lemma:hs22\]
The condition $\tilde{V}(\pi_1^k) /c <1$ in the above lemma ensures that during the time duration $[t, t^{\tilde{V}(\pi_1^k)/c}]$, for large enough $t$, the number of particles does not change so that Lemma \[lemma:hitting\_place\] for the process $\mu_N$ is applicable for the process $\bar{\mu}$.
Let $\pi^k$ be a $k$-cycle and suppose that $\tilde{V}(\pi^k)/c < 1$. Then, given $\delta > 0$ such that $(\tilde{V}(\pi^k)+\delta)/c < 1$, there exist $\varepsilon > 0$ and $\rho>0$ such that for all $\rho_1 < \rho$, there is $t^* > 0$ such that $$\begin{aligned}
P_{t,\nu}(\bar{\tau}_{\pi^k} & < t + t^{(\tilde{V}(\pi^k) -\delta)/c}) \leq t^{-\varepsilon/c}, \text{ and }\\
P_{t,\nu}(\bar{\tau}_{\pi^k}& > t + t^{(\tilde{V}(\pi^k) +\delta)/c}) \leq t^{-\varepsilon/c}\end{aligned}$$ holds uniformly for all $\nu \in [\pi^k]_{\rho_1} \cap M_1^{N_{t}}(\mathcal{Z})$ and $t \geq t^*$. \[lemma:hs23\]
Let $\pi^k$ be a $k$-cycle and suppose that $\hat{V}(\pi^k)/c < 1$. Given $\varepsilon>0$, there exist $\delta \in (0, c-\hat{V}(\pi^k))$ and $\rho>0$ such that for all $\rho_1 \leq \rho$, there is $t^* >0$ such that $$\begin{aligned}
P_{t,\nu}(\bar{\tau}_{\pi^k}\leq t + t^{(\hat{V}(\pi^k)+\delta)/c}) \leq t^{-(\tilde{V}(\pi^k) - \hat{V}(\pi^k)-\varepsilon)/c}\end{aligned}$$ holds uniformly for all $\nu \in [\pi^k]_{\rho_1} \cap M_1^{N_{t}}(\mathcal{Z})$ and $t \geq t^*$. \[lemma:hs24\]
Recall the definition of the sets $L$ and $C$ from Section \[section:large\_time\_behaviour\].
\[lemma:conv-fw19\] Given $\rho_0 > 0$ and $\rho_1 <\rho_0$ and their associated sets $L$ and $C$, given $v>0$, there exist $T^* > 0$ and $t^* >0 $ such that $$\begin{aligned}
P_{t,\nu}(\hat{\tau}_L \geq t + T^*) \leq t^{-v/c}\end{aligned}$$ holds uniformly for all $\nu \in C \cap M_1^{N_{t}}(\mathcal{Z})$ and $t \geq t^*$.
To answer the question on the convergence of $\bar{\mu}$ to the global minimum of $s$, we define the following quantities, analogous to what is done in Hwang and Sheu [@hwang-sheu-90]. Let $m$ be such that $L_{m+1}$ is a singleton (denote it by $\{\pi^{m+1}\}$). Define $$\begin{aligned}
A_m \coloneqq \{\pi^m \in L_m:\tilde{V}(\pi^m) = \hat{V}(\pi^{m+1})\}.\end{aligned}$$ Inductively define, for each $\pi^{k+1} \in L_{k+1}$, $$\begin{aligned}
A_{k}(\pi^{k+1}) \coloneqq \{\pi^k \in \pi^{k+1} : \tilde{V}(\pi^k) = \hat{V}(\pi^{k+1})\},\end{aligned}$$ and for each $k \geq 1$, define $$\begin{aligned}
A_k \coloneqq \bigcup_{\pi^{k+1} \in A_{k+1}} A_k(\pi^{k+1}).\end{aligned}$$ Also, for each $\pi^k \in L_k$, define $$\begin{aligned}
c_{k-1}(\pi^k) \coloneqq \left\{\begin{array}{ll}
0, \text{ if } \{ \pi^{k-1} \in \pi^k : \pi^{k-1} \notin A_{k-1}(\pi^k) \} = \emptyset, \\
\max\{\tilde{V}(\pi^{k-1}):\pi^{k-1} \notin A_{k-1}(\pi^k), \pi^{k-1} \in \pi^k\}, \text{ otherwise},
\end{array}
\right.\end{aligned}$$ and for each $k \geq 1$, define $$\begin{aligned}
c_{k-1} \coloneqq \max\{c_{k-1}(\pi^k), :\pi^k \in A_k\}.\end{aligned}$$ Finally, define $$\begin{aligned}
c^* \coloneqq \max\{c_k, 0 \leq k \leq m\}.\end{aligned}$$ Similar to [@hwang-sheu-90 Lemma A.11, Appendix], we can show that $A_0 = \tilde{L}_0$, the set of minimisers of the rate function $s$ that governs the LDP for the invariant measure $\{\wp_N\}_{N\geq 1}$. We now have the following theorem on the convergence of $\bar{\mu}$ to the global minimum of the rate function $s$.
For $c > c^*$ and any $\rho_1 > 0$, $$\begin{aligned}
P_{0,\nu} (\bar{\mu}(t) \in [\tilde{L}_0]_{\rho_1}) \to 1\end{aligned}$$ as $t \to \infty$, uniformly for all $\nu \in M_1^{N_0}(\mathcal{Z})$. \[thm:conv-globalmin\]
It suffices to show that, for any $\delta > 0$ with $(c^*+\delta)/c<1$, there exist $\varepsilon > 0$, $\rho_1>0$ and $t^* > 0$ such that $$\begin{aligned}
P_{t, \nu} (\bar{\mu}(t + t^{(c^*+\delta)/c}) \in [\tilde{L}_0]_{\rho_1}) \geq 1- t^{-\varepsilon/c}\end{aligned}$$ for all $t > t^*$ and $\nu \in M_1^{N_{t}}(\mathcal{Z})$. Define the stopping time $$\begin{aligned}
\theta \coloneqq \inf\{s>t: \bar{\mu}(s) \in [L]_{\rho_1}\}.\end{aligned}$$ By Lemma \[lemma:conv-fw19\], for any $M >0$, there exists $T^*>0$ such that for all $\nu \in M_1^{N_0}(\mathcal{Z})$ and large enough $t$, we have $$\begin{aligned}
P_{t, \nu} (\theta > t + T^*) \leq t^{-M/c}.\end{aligned}$$ By the strong Markov property, we have $$\begin{aligned}
P_{t, \nu}&( \bar{\mu}(t + t^{(c^*+\delta)/c})\in [\tilde{L}_0]_{\rho_1}) \nonumber \\
& \geq E_{t,\nu} (\theta \leq t + T^*; E_{\theta, \bar{\mu}(\theta)} (\bar{\mu}(t+t^{(c^*+\delta)/c}) \in [\tilde{L}_0]_{\rho_1})) \nonumber \\
& \geq \inf_{\substack{t \leq t_1 \leq t+T^* \\ \nu_1 \in [L]_{\rho_1}}} P_{t_1, \nu_1} (\bar{\mu}(t+t^{(c^*+\delta)/c}) \in [\tilde{L}_0]_{\rho_1}) (1-t^{-M/c}).
\label{eqn:temp7}\end{aligned}$$ To bound the first term above, fix a $t_1$ such that $t \leq t_1 \leq t+T^*$ and $\nu_1 \in [L]_{\rho_1}$. Define the stopping time $\theta_m \coloneqq \inf \{t>t_1: \bar{\mu}(t) \in [A_m]_{\rho_1}\}$. We have $$\begin{aligned}
P_{t_1, \nu_1} &(\bar{\mu}(t+t^{(c^*+\delta)/c}) \in [\tilde{L}_0]_{\rho_1}) \nonumber \\
& \geq E_{t_1, \nu_1} (\theta_m < t+t^{(c^*+\delta/2)/c}; E_{\theta_m, \bar{\mu}(\theta_m)}(\bar{\mu}(t+t^{(c^*+\delta)/c}) \in [\tilde{L}_0]_{\rho_1})) \nonumber \\
& \geq \inf_{t \leq t_2 \leq t+t^{(c^*+\delta/2)/c}, \nu_2 \in [A_m]_{\rho_1}} P_{t_2, \nu_2}(\bar{\mu}(t+t^{(c^*+\delta)/c}) \in [\tilde{L}_0]_{\rho_1}) \nonumber \\
& \, \, \, \, \, \, \, \, \, \, \times P_{t_1, \nu_1}(\theta_m \leq t+t^{(c^*+\delta/2)/c}). \label{eqn:temp6}\end{aligned}$$ We first bound the second term $P_{t_1, \nu_1}(\theta_m \leq t+t^{(c^*+\delta/2)/c})$. Note that, by Lemma \[lemma:hs22\], for any $M_1>0$, there exists $\delta_1>0$ such that $$\begin{aligned}
P_{t_1, \nu_1}(\theta_m > t_1+t_1^{(c_m-\delta_1)/c}) \leq 1-t_1^{-M_1/c}\end{aligned}$$ for sufficiently large $t$. Let $T_1 = t_1 + t_1^{(c_m-\delta_1)/c}$, and define the stopping time $\hat{\theta} \coloneqq \inf\{t>T_1: \bar{\mu}(t) \in [L]_{\rho_1} \}$. Again, by Lemma \[lemma:conv-fw19\], there exists a large enough $T^*$ such that $P_{T_1, \nu}(\hat{\theta} > T_1+T^*) \leq T_1^{-M/c}$ for all $\nu \in M_1^{N_{T_1}}(\mathcal{Z})$. Therefore, using the strong Markov property, we have $$\begin{aligned}
P_{t_1, \nu_1}&(\theta_m > t+t^{(c^*+\delta/2)/c}) \nonumber \\
& \leq E_{t_1, \nu_1}(\theta_m \geq \hat{\theta}, \hat{\theta}< T_1+T^*; E_{\hat{\theta}, \bar{\mu}(\hat{\theta})}(\theta_m >t+t^{(c^*+\delta/2)/c})) \nonumber \\
& \, \, \, \, \, \, \, \, + P_{t_1, \nu_1}(\hat{\theta} > T_1+T^*) \nonumber \\
& \leq P_{t_1, \nu_1}(\theta_m > T_1) \sup_{\substack{T_1 \leq t \leq T_1 + T^* \\ \nu \in [L]_{\rho_1}}} P_{t,\nu}(\theta_m > t+t^{(c^*+\delta/2)/c}) +t_1^{-M/c} \nonumber\\
& \leq (1-t_1^{-M_1/c} ) \sup_{\substack{T_1 \leq t \leq T_1 + T^* \\ \nu \in [L]_{\rho_1}}} P_{t,\nu}(\theta_m > t+t^{(c^*+\delta/2)/c}) +t_1^{-M/c}. \label{eqn:temp5}\end{aligned}$$ We now focus on $P_{t,\nu}(\theta_m > t+t^{(c^*+\delta/2)/c})$ for a fixed $t \in [T_1, T_1+T^*]$ and $\nu \in [L]_{\rho_1}$, and repeat the above steps; this will introduce a multiplication factor of $(1-T_1^{-M_1/c})$ along with $$\begin{aligned}
\sup_{\substack{T_2 \leq t \leq T_2 + T^* \\ \nu \in [L]_{\rho_1}}} P_{t,\nu}(\theta_m > t+t^{(c^*+\delta/2)/c}),\end{aligned}$$ where $T_2 = T_1 + T_1^{(c_m-\delta_1)/c}$, in the first term in (\[eqn:temp5\]), and an addition of $t_1^{-M/c}$ in the second term. Therefore, repeating the above steps $r \sim t_1^{\delta/2c}$ times, we get $$\begin{aligned}
P_{t_1, \nu_1}(\theta_m > t+t^{(c^*+\delta/2)/c}) \leq \prod_{n=0}^r (1-T_n^{*{-M_1/c}}) + rt_1^{-M/c},\end{aligned}$$ where $T_0^* = t_1$, and $$\begin{aligned}
T_{n+1}^* = T_n^* + T_n^{*{(c_m-\delta_1)/c}} + T^*.\end{aligned}$$ Note that, $$\begin{aligned}
\prod_{n=0}^r (1-T_n^{*{-M_1/c}}) & \leq \exp\left\{-\sum_{n=0}^r T_n^{*{-M_1/c}} \right\} \nonumber \\
& = \exp\left\{ -\sum_{n=0}^r T_n^{*{-M_1/c - (c_m-\delta_1)/c}}(T_{n+1}^* - T_n^*) \right\} \nonumber \\
& \leq \exp\left\{- \int_{T_0^*}^{T_r^*} u^{-(M_1/c) - (c_m-\delta_1)/c} du\right\} \nonumber \\
& = \exp \left\{ - \left( T_r^{*{1-(c_m+M_1-\delta_1)/c}} - t_1^{1-(c_m+M_1-\delta_1)/c} \right)\right\}. \label{eqn:temp14}\end{aligned}$$ Since $T_n \geq t_1$ for all $n \geq 1$, we see that $T_r^* \geq t_1 + r t_1^{(c_m-\delta_1)/c} \sim t_1 + t_1^{(c_m-\delta_1+\delta/2)/c}$. Therefore, $$\begin{aligned}
- & \left(T_r^{*{1-(c_m+M_1-\delta_1)/c}} - t_1^{1-(c_m+M_1-\delta_1)/c} \right) \\
& \leq -\left( (t_1+t_1^{(c_m-\delta_1+\delta/2)/c})^{1-(c_m+M_1-\delta_1)/c} - t_1^{1-(c_m+M_1-\delta_1)/c}\right) \\
& \leq -\left( t_1^{1-(c_m+M_1-\delta_1)/c} \left( 1+t_1^{(c_m-\delta_1+\delta/2)/c-1}\right)^{1-(c_m+M_1-\delta_1)/c} -1\right) \\
& \leq -c^\prime \left(t_1^{1-(c_m+M_1-\delta_1)/c} t_1^{(c_m-\delta_1+\delta/2)/c-1} \right) \\
& = -c^\prime t_1^{(\delta/2 - M_1)/c},\end{aligned}$$ for some constant $c^\prime >0$ and large enough $t_1$. Hence, (\[eqn:temp14\]) becomes $$\begin{aligned}
\prod_{n=0}^r (1-T_n^{*^{-M_1/c}}) \leq \exp\{-c^\prime t_1^{(\delta/2-M_1)/c}\}.\end{aligned}$$ We choose $M_1 = \delta/4$; the above and (\[eqn:temp5\]) then implies $$\begin{aligned}
P_{t_1, \nu_1}&(\theta_m > t+t^{(c^*+\delta/2)/c}) \leq \exp\{-c^\prime t_1^{\delta/4c}\} + t_1^{-(M-\delta/2)/c},\end{aligned}$$ and this implies that, for any $M^\prime > 0$, $$\begin{aligned}
P_{t_1,\nu_1} (\theta_m > t+t^{(c^*+\delta/2)/c}) \leq t^{-M^\prime/c}
\label{eqn:temp13}\end{aligned}$$ for sufficiently large $t$, $t \leq t_1 \leq t+T^*$ and for all $\nu \in [L]_{\rho_1}$.
We now bound the first term in (\[eqn:temp6\]), $P_{t_2,\nu_2}(\bar{\mu}(t+t^{(c^*+\delta)/c}) \in [\tilde{L}_0]_{\rho_1})$ where $t \leq t_2 \leq t + t^{(c^*+\delta/2)/c}$ and $\nu_2 \in [A_m]_{\rho_1}$. Let $\pi_0^m \in A_m$ be the $m$-cycle such that $\nu_2 \in [\pi_0^m]_{\rho_1}$. Define the following quantities: $$\begin{aligned}
\tilde{t}_0 \coloneqq t+t^{(c^*+\delta)/c} - t^{(c_{m-1}(\pi^m_0)+\delta)/c}, \text{ and } \\
\tilde{t}_1 \coloneqq t+t^{(c^*+\delta)/c} - t^{(c_{m-1}(\pi^m_0)+\delta/2)/c}.\end{aligned}$$ Define the stopping time $\theta \coloneqq \inf \{t>\tilde{t}_0: \bar{\mu}(t) \in [\pi^m_0]_{\rho_1}\}$, if $c^* > c_{m-1}(\pi^m_0)$ and $\theta = t_2$ otherwise. By the strong Markov property, $$\begin{aligned}
P_{t_2,\nu_2}& (\bar{\mu}(t+t^{(c^*+\delta)/c}) \in [\tilde{L}_0]_{\rho_1}) \nonumber \\
& \geq E_{t_2,\nu_2}(\theta \leq \tilde{t}_1; E_{\theta, \bar{\mu}(\theta)}(\bar{\mu}(t+t^{(c^*+\delta)/c}) \in [\tilde{L}_0]_{\rho_1} )\nonumber \\
& \geq P_{t_2, \nu_2} (\theta \leq \tilde{t}_1) \inf_{\tilde{t}_0 \leq t_3 \leq \tilde{t}_1, \nu_3 \in [\pi_0^m]_{\rho_1}} P_{t_3,\nu_3}(\bar{\mu}(t+t^{(c^*+\delta)/c}) \in [\tilde{L}_0]_{\rho_1}).
\label{eqn:temp9}\end{aligned}$$ We first estimate $P_{t_2, \nu_2}(\theta \leq \tilde{t}_1)$ when $c^* > c_{m-1}(\pi_0^m)$ (if this is not the case, then by definition of $\theta$, we have $P_{t_2, \nu_2}(\theta \leq \tilde{t}_1)=1$) . Note that $$\begin{aligned}
P_{t_2,\nu_2} (\theta > \tilde{t}_1) & = P_{t_2,\nu_2} (\bar{\mu}(t) \notin [\pi^m_0]_{\rho_1} \text{ for all } \tilde{t}_0 \leq t \leq \tilde{t}_1) \\
& \leq P_{t_2,\nu_2} (\bar{\mu}(t) \notin [L]_{\rho_1} \text{ for all } \tilde{t}_0 \leq t \leq \tilde{t}_1) + P_{t_2, \nu_2}(\bar{\tau}_{\pi^m_0} \leq \tilde{t}_1).\end{aligned}$$ Lemma \[lemma:hs23\] implies that $$\begin{aligned}
P_{t_2, \nu_2}(\bar{\tau}_{\pi^m_0} \leq \tilde{t}_1) \leq t^{-\delta/c}\end{aligned}$$ for large $t$ and small enough $\rho_1 > 0$. Also, with this $\rho_1$, by using Lemma \[lemma:conv-fw19\], we see that $$\begin{aligned}
P_{t_2,\nu_2} (\bar{\mu}(t) \notin [L]_{\rho_1} \text{ for all } \tilde{t}_0 \leq t \leq \tilde{t}_1) \leq t^{-M_1/c}\end{aligned}$$ for large $t$, where $M_1$ can be chosen as large as we want. This shows that there exists $\varepsilon_1>0$ such that $$\begin{aligned}
P_{t_2,\nu_2} (\theta \leq \tilde{t}_1) \geq 1-2t^{-\varepsilon_1/c}\end{aligned}$$ uniformly for all $\nu_2 \in [\pi_0^m]_{\rho_1}$ and large enough $t$. Hence, from (\[eqn:temp13\]), (\[eqn:temp9\]) and (\[eqn:temp6\]), we get $$\begin{aligned}
P_{t_1,\nu_1}&(\bar{\mu}(t+t^{(c^*+\delta)/c}) \in [\tilde{L}_0]_{\rho_1}) \\
& \geq (1-t^{-M^\prime/c}) (1-2t^{-\varepsilon_1/c}) \\
& \, \, \, \, \, \, \, \, \times \inf_{\substack{t_2 \geq \tilde{t}_0, \\ \nu_2 \in [\pi_0^m]_{\rho_1} \\ \pi_0^m \in A_m \\ \tilde{\delta} \in [\delta/4, \delta]}} P_{t_2,\nu_2}(\bar{\mu}(t_2+t_2^{(c_{m-1}(\pi^m_0)+\tilde{\delta})/c}) \in [\tilde{L}_0]_{\rho_1})\end{aligned}$$ and therefore, for some $\varepsilon>0$, we have $$\begin{aligned}
\inf_{\substack{t \leq t_1 \leq t+T^*, \\ \nu_1 \in [L]_{\rho_1}}}&P_{t_1,\nu_1}(\bar{\mu}(t+t^{(c^*+\delta)/c}) \in [\tilde{L}_0]_{\rho_1}) \\
& \geq (1-t^{-\varepsilon/c}) \inf_{\substack{t_2 \geq \tilde{t}_0 \\ \nu_2 \in [\pi^m_0]_{\rho_1} \\ \pi_0^m \in A_m \\ \tilde{\delta} \in [\delta/4, \delta]}} P_{t_2,\nu_2}(\bar{\mu}(t_2+t_2^{(c_{m-1}(\pi^m_0)+\tilde{\delta})/c}) \in [\tilde{L}_0]_{\rho_1}).\end{aligned}$$ We now focus on the second term. This probability inside the infimum can be lower bounded using similar steps above starting with (\[eqn:temp9\]); instead of the random variable $\theta$, we consider the hitting time of a suitable $(m-1)$-cycle. Continuing this procedure $m$ times, we eventually reach $A_0$. Therefore, we can show $$\begin{aligned}
\inf_{\substack{t \leq t_1 \leq t+T^* \\ \nu_1 \in [\tilde{L}_0]_{\rho_1}}}&P_{t_1,\nu_1}(\bar{\mu}(t+t^{(c^*+\delta)/c}) \in [\tilde{L}_0]_{\rho_1}) \geq (1-t^{-\varepsilon/c})^{m+1},\end{aligned}$$ and the result now follows from (\[eqn:temp7\]).
We now show that the conclusion of Theorem \[thm:conv-globalmin\] fails if we choose $c < c^*$. Given $c < c^*$, let $\pi^k \in L_k$ be such that $\hat{V}(\pi^k) \leq c < \tilde{V}(\pi^k)$; this is possible from the definition of $c^*$. Note that $\tilde{L}_0 \cap \pi^k = \emptyset$. The below result shows that the exit time from a neighbourhood of $\pi^k$ is infinite with positive probability, and this in particular implies that (\[eqn:conv\_globalmin\]) fails.
Let $\pi^k$ be a $k$-cycle such that $\hat{V}(\pi^k) \leq c < \tilde{V}(\pi^k)$. There exist $\varepsilon \in (0, \tilde{V}(\pi^k)-c)$, $c^\prime >0, \rho_1>0$ and $t^*>0$ such that for all $\nu \in [\pi^k]_{\rho_1}\cap M_1^{N_{t}}(\mathcal{Z})$ and $t \geq t^*$, we have $$\begin{aligned}
P_{t, \nu}(\bar{\tau}_{\pi^k} <\infty) \leq c^\prime t^{1-(\tilde{V}(\pi^k) - \varepsilon)/c}.\end{aligned}$$
We proceed via the steps in Hwang and Sheu [@hwang-sheu-90]. Let $T_0 = t$, and define, for all $n \geq 1$, $$\begin{aligned}
T_{n+1} \coloneqq T_n + T_n^{\hat{V}(\pi^k)/c}, \text{ and}\\
T^*_{n+1} \coloneqq T_n + \frac{1}{2} T_n^{\hat{V}(\pi^k)/c}.\end{aligned}$$ (In the above definitions, we assume that $\hat{V}(\pi^k) > 0$; if this is not the case, then we replace $T_n^{\hat{V}(\pi^k)/c}$ in the above definitions by a sufficiently large constant, and the following arguments will go through.) We have, for any $r \geq 1$, $$P_{t, \nu} (\bar{\tau}_{\pi^k} <T_r) = P_{t, \nu} (\bar{\tau}_{\pi^k} <T_{r-1}) + P_{t, \nu} (T_{r-1} \leq \bar{\tau}_{\pi^k} <T_r).
\label{eqn:temp3}$$ To bound the second term, define the stopping time $\theta \coloneqq \inf\{t>T^*_{r-1}:\bar{\mu}(t) \in [L]_{\rho_1}\}$ where $\rho_1$ is to be chosen later. Then, $$\begin{aligned}
P_{t, \nu} (T_{r-1} \leq \bar{\tau}_{\pi^k} <T_r) & = P_{t, \nu} (T_{r-1} \leq \bar{\tau}_{\pi^k} <T_r, \theta \leq T_{r-1}^*+T^* ) \nonumber \\
&\, \, \, \, \, \, \, \, \, \, + P_{t, \nu} (T_{r-1} \leq \bar{\tau}_{\pi^k} <T_r, \theta > T_{r-1}^*+T^*) ,
\label{eqn:temp4}\end{aligned}$$ where $T^*$ is such that the second term above is upper bounded by $T_{r-1}^{*^{-M/c}}$ for some $M>0$ to be chosen later (this is possible by Lemma \[lemma:conv-fw19\]). To bound the first term, note that $$\begin{aligned}
P_{t, \nu} (T_{r-1} \leq &\bar{\tau}_{\pi^k} <T_r, \theta \leq T_{r-1}^*+T^* ) \\
& \leq P_{t, \nu} (\theta\leq \bar{\tau}_{\pi^k} <T_r, \theta \leq T_{r-1}^*+T^* ) \\
& \leq E_{t,\nu} (\bar{\mu}(\theta) \in [\pi^k]_{\rho_1}, \theta \leq T_{r-1}^*+T^* ; E_{\theta, \bar{\mu}(\theta)}(\bar{\tau}_{\pi^k} < T_r)) \\
& \leq T_{r-1}^{*-(\tilde{V}(\pi^k) - \hat{V}(\pi^k) - \varepsilon)/c}\end{aligned}$$ holds for sufficiently large $t$ and small enough $\rho_1$. Here, the second inequality follows by the strong Markov property and the third from Lemma \[lemma:hs24\]. Choose $M$ sufficiently large, so that (\[eqn:temp3\]), (\[eqn:temp4\]) and the above implies $$\begin{aligned}
P_{t,\nu}(\bar{\tau}_{\pi^k} < T_r) \leq P_{t,\nu}(\bar{\tau}_{\pi^k} < T_{r-1}) + 2T_{r-1}^{*^{-(\tilde{V}(\pi^k) - \hat{V}(\pi^k) -\varepsilon)/c}}.\end{aligned}$$ Therefore, we have $$\begin{aligned}
P_{t,\nu}(\bar{\tau}_{\pi^k} < T_r) &\leq 2\sum_{n=0}^r T_n^{*^{-(\tilde{V}(\pi^k) - \hat{V}(\pi^k) -\varepsilon)/c}} \\
& \leq c_1^\prime \sum_{n=0}^r T_n^{-(\tilde{V}(\pi^k) - \hat{V}(\pi^k) -\varepsilon)/c} \\
& = c_1^\prime \sum_{n=0}^r T_n^{-(\tilde{V}(\pi^k) - \varepsilon)/c} (T_{n+1} - T_n)\\
& \leq c_1^\prime \int_{t}^{T_r} u^{-(\tilde{V}(\pi^k)-\varepsilon)/c} du,\end{aligned}$$ where $c_1^\prime $ is a positive constant. Choose $\varepsilon $ such that $\tilde{V}(\pi^k) - \varepsilon > c$ so that the above implies $$\begin{aligned}
P_{t,\nu}(\bar{\tau}_{\pi^k} < T_r) &\leq c_1^\prime \int_{t}^ \infty u^{-(\tilde{V}(\pi^k)-\varepsilon)/c} du \\
& \leq c^\prime t^{1-(\tilde{V}(\pi^k)-\varepsilon)/c},\end{aligned}$$ where $c^\prime$ is a positive constant. Let $r \to \infty$, and the result follows since $T_r \to \infty$.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank Laurent Miclo for fruitful discussions. This work was supported by the Indo-French Centre for Applied Mathematics. The first author was also supported by the IISc-Cisco Research Fellowship grant.
|
---
abstract: 'Beam-beam simulations predict that PEP-II luminosity can be increased by operating the horizontal betatron tune near and above a half-integer resonance. However, effects of the resonance and its synchrotron sidebands significantly enhance betatron and chromatic perturbations which tend to reduce dynamic aperture. In the study, chromatic variation of horizontal tune near the resonance was minimized by optimizing local sextupoles in the Interaction Region. Dynamic aperture was calculated using tracking simulations in LEGO code. Dependence of dynamic aperture on the residual orbit, dispersion and $\beta$ distortion after correction was investigated.'
author:
- 'Y. Cai, Y. Nosochkov, SLAC, Menlo Park, CA 94025, USA'
title: |
TRACKING SIMULATIONS NEAR\
HALF-INTEGER RESONANCE AT PEP-II [^1]
---
[SLAC–PUB–9812\
May 2003\
]{}
[**Tracking Simulations Near Half-Integer Resonance at PEP-II [^2]**]{}
[**Abstract** ]{}
\
INTRODUCTION
============
PEP-II [@cdr] has been operating at the betatron tune $\nu_x/\nu_y$ close to $24.569/23.639$ in the High Energy Ring (HER) and $38.649/36.564$ in the Low Energy Ring (LER). These working points were selected experimentally for a reliable machine performance, good luminosity and beam lifetime. However, the beam-beam simulations predict that luminosity can be increased by operating betatron tune very close and above the half-integer resonance. Fig. \[fig:lumin\] shows the LER tune diagram with synchro-betatron resonances up to the 4th order and a contour plot of the single bunch luminosity. Calculation of luminosity was done using the beam-beam code developed at SLAC [@bbeam] which has been recently upgraded to the three dimensional version.
The difficulty of operating close to half-integer resonance comes from enhancement of the resonance effects on betatron motion. It is well known that perturbation of $\beta$ function created by focusing errors depends on tune $\nu$ as $$\frac{\Delta\beta}{\beta}(s)=\frac{1}{2\sin2\pi\nu}
\oint \beta(l) \Delta K_1(l) \cos2\phi(s,l) dl,
\label{eqn:dbeta}$$ where $\mu$ is phase advance, $\phi(s,l)\!=\!\pi\nu\!-\!\left|\mu(s)\!-\!\mu(l)\right|$, and $\Delta K_1$ is a focusing error created mainly by quadrupole field imperfections, horizontal orbit at sextupoles, and momentum error. Close to half-integer resonance, growth of $\Delta\beta/\beta$ comes from the resonance term $[\sin2\pi\nu]^{-1}$ which behaves as $1/\Delta\nu$ when distance to the resonance is as small as $\Delta\nu\!\ll\!1/2\pi$. On the other hand, orbit and dispersion are not excited by the half-integer resonance.
For significant enhancement of luminosity, fractional value of horizontal tune should be in the range of $[\nu_x]\!\approx\!.51$. At this working point, enhancement of $\Delta\beta_x/\beta_x$ in HER and LER due to the resonance term in Eqn. \[eqn:dbeta\] would be a factor of 6.7 and 12.8, respectively, compared to the present tune. Without compensation, the large $\beta$ growth may significantly increase amplitude dependent non-linear aberrations and reduce dynamic aperture and beam lifetime.
![Single bunch luminosity scan \[$10^{30}$ cm$^{-2}$s$^{-1}$\].[]{data-label="fig:lumin"}](WPAG013f1.eps){width="82mm"}
More resonance effects are generated by the synchrotron sidebands of the half-integer resonance: $2\nu_x\!+\!m\nu_s\!=\!n$, where $\nu_s$ is a synchrotron tune, and $m, n$ are integers. In the LER, where $\nu_s\!=\!0.025$, the 1st and 2nd synchro-betatron resonances occur at $[\nu_x]\!=\!.5125$ and .525, while in HER with $\nu_s\!=\!0.045$ the 1st sideband is at $[\nu_x]\!=\!.5225$. Tracking simulations will show that the sidebands have a strong effect on dynamic aperture, therefore working tune should be chosen reasonably far from them. In addition, variation of tune with synchrotron momentum oscillations should be minimized to avoid crossing with these resonances.
Optimization of PEP-II lattice near half-integer resonance and analysis of dynamic aperture are discussed below. The optics with $\beta_x^*/\beta_y^*\!=\!50/1.25$ cm at the Interaction Point (IP) is used.
LATTICE OPTIMIZATION
====================
PEP-II has two tuning sections which can be locally adjusted to change betatron tune without affecting the rest of machine optics. Initially, only these sections were modified to move the horizontal tune closer to half-integer, and vertical tune to $[\nu_y]\!\approx\!.61$ as suggested by beam-beam analysis. But tracking simulations showed that dynamic aperture was not sufficiently large with machine errors and synchrotron momentum oscillations of up to $\pm8\sigma_p$, where $\sigma_p$ is the [*rms*]{} value of relative momentum spread $\frac{\Delta p}{p}$ in the beam. Analysis of chromaticity indicated that non-linear variation of horizontal tune with momentum needs to be further reduced to avoid crossing with the synchro-betatron resonances.
In PEP-II, the most contribution to non-linear chromaticity is generated in the final quadrupole doublets near IP. This chromaticity is compensated by the Interaction Region (IR) sextupoles located in the same phase with the doublets. Variation of strength of these sextupoles allows to compensate quadratic dependence of tune on $\frac{\Delta p}{p}$, and a small adjustment of sextupole phase advance helps reduce the higher order variation.
Minimum of the second order chromaticity was achieved by reducing strengths of the IR sextupoles correcting horizontal chromaticity. Further improvement in LER resulted from reduction of horizontal phase advance between the IR horizontal sextupoles and IP by $5^\circ$. For correction of the machine linear chromaticity, strength of the global sextupoles was increased to compensate for the weaker IR sextupoles. Because the adjusted IR sextupoles in LER have a non-zero design orbit, the reduced sextupole strength created a feed-down effect of linear focusing and coupling. This small perturbation was compensated by a slight adjustment of the IR magnet strengths.
The optimized horizontal tune for momentum range of $-10\sigma_p\!<\!\frac{\Delta p}{p}\!<\!10\sigma_p$ is shown in Fig. \[fig:htunex\], \[fig:ltunex\], where the working point is $\nu_x/\nu_y\!=\!24.51/23.61$ in HER and $38.518/36.61$ in LER. The straight dash lines depict the half-integer synchrotron sidebands. In HER, a positive linear chromaticity $\xi\!=\!+1$ was used in Fig. \[fig:htunex\] to counteract the negative slope of non-linear tune variation. Due to the large synchrotron tune, it was possible to place the HER working point below the 1st synchrotron sideband without crossing with the resonance lines. In the LER, synchrotron tune is a factor of 2 smaller while energy spread is 25% larger, therefore the closest to half-integer working point was chosen between the 1st and 2nd sidebands.
![HER horizontal tune vs. $\frac{\Delta p}{p}$ at $\nu_x\!=\!24.51$.[]{data-label="fig:htunex"}](WPAG013f2.eps){width="82mm"}
![LER horizontal tune vs. $\frac{\Delta p}{p}$ at $\nu_x\!=\!38.518$.[]{data-label="fig:ltunex"}](WPAG013f3.eps){width="82mm"}
DYNAMIC APERTURE SIMULATIONS
============================
Calculations of dynamic aperture were performed using tracking simulations in LEGO code [@lego]. First, dependence of aperture on betatron tune near half-integer resonance was investigated for lattice without magnet errors, but with synchrotron momentum oscillations of $\pm8\sigma_p$. The resultant horizontal tune scan is shown in Fig. \[fig:hscanx\], \[fig:lscanx\] for HER and LER, where dynamic aperture is normalized by the [*rms*]{} size of a fully coupled beam.
![HER dynamic aperture without magnet errors vs. $\nu_x$ at $\nu_y\!=\!23.61$. Synchrotron sidebands: 1) $2\nu_x\!-\!\nu_s\!=\!49$, 2) $2\nu_x\!-\!2\nu_s\!=\!49$.[]{data-label="fig:hscanx"}](WPAG013f4.eps){width="82mm"}
![LER dynamic aperture without magnet errors vs. $\nu_x$ at $\nu_y\!=\!36.61$. Synchrotron sidebands: 1) $2\nu_x\!-\!\nu_s\!=\!77$, 2) $2\nu_x\!-\!2\nu_s\!=\!77$, 3) $2\nu_x\!-\!3\nu_s\!=\!77$.[]{data-label="fig:lscanx"}](WPAG013f5.eps){width="82mm"}
The HER horizontal dynamic aperture vanishes in the vicinity of the main half-integer resonance $2\nu_x\!=\!49$ and its 1st sideband. In LER, the strongly affecting resonances are $2\nu_x\!+\!m\nu_s\!=\!77$ with $m\!=\!0,-1,-2,-3$. Naturally, the working tune should be chosen reasonably far from these resonances. The following scenarios for more and less aggressive horizontal tune $\nu_x$ near half-integer were investigated in the simulations:
1. HER: 24.51, LER: 38.518.
2. HER: 24.529, LER: 38.529.
Dynamic aperture scan versus vertical tune was performed for the first scenario of $\nu_x$ and the range of $[\nu_y]$ from .55 to .64. It showed that dynamic aperture gradually reduces as $[\nu_y]$ becomes closer to $[\nu_x]$ and the working point approaches the crossing of half-integer and coupling resonances. Based on this scan, the vertical fractional tune of $[\nu_y]\!=\!.61$ was chosen for these simulations. A lower $\nu_y$ may be considered for further luminosity enhancement.
Secondly, tracking simulations with field errors, misalignment and $\frac{\Delta p}{p}\!=\!\pm8\sigma_p$ synchrotron momentum oscillations were performed for the selected working points. For statistics, ten different settings (“seeds”) of random machine errors were used in each tracking. Perturbation of beam orbit, linear chromaticity, betatron tune and vertical dispersion was compensated using realistic correction schemes in LEGO. Since distortion of $\beta$ function becomes more sensitive to focusing errors near half-integer resonance, a special correction of $\frac{\Delta \beta}{\beta}$ was implemented in LEGO. It uses MICADO method to find the most effective quadrupoles to minimize $\beta$ perturbation.
Due to the greater effect of errors near half-integer resonance, a better machine correction is needed to maintain acceptable dynamic aperture. To verify tolerance to various errors, simulations were performed for different levels of machine correction. It has been confirmed that beam orbit should be decreased for an acceptable dynamic aperture. The better orbit correction reduces the feed-down focusing errors in sextupoles as well as residual dispersion in the machine. On the other hand, correction of vertical dispersion did not significantly affect dynamic aperture in the observed range of residual [*rms*]{} $\Delta\eta_y$ from $\sim$70 to 5 mm.
As expected, the simulations confirmed that compensation of $\Delta
\beta/\beta$ is required in the first scenario, where $\nu_x$ is closer to the resonance. Typically, $\Delta \beta/\beta$ was corrected to the [*rms*]{} level of $<\!5\%$. In the second scenario, at $[\nu_x]\!=\!.529$, correction of $\beta$ function was less important, although it helped to improve cases with small aperture. Table 1 summarizes the observed approximate levels of [*rms*]{} orbit and $\Delta\beta_x/\beta_x$ for acceptable dynamic aperture.
------------------------------ ---------- ---------- ---------- ----------
**.529** **.510** **.529** **.518**
orbit (mm) 1 1 0.4 0.2
$\Delta \beta_x/\beta_x$ (%) 25 5 25 5
------------------------------ ---------- ---------- ---------- ----------
: Tolerances on [*rms*]{} orbit and $\Delta\beta_x/\beta_x$.
\[tab:toler\]
The resultant dynamic aperture in HER at $\nu_x/\nu_y\!=\!24.51/23.61$ for a good level of correction is shown in Fig. \[fig:haper51\] at the injection point. Stable particle motion corresponds to the area inside the dash lines which represent 10 different seeds of random machine errors. The solid half-ellipse, shown for reference, is the $10\sigma$ size of a fully coupled beam at injection with emittance $\epsilon_x\!=\!48$ nm and $\epsilon_y\!=\!\epsilon_x/2$. In this simulation, the residual [*rms*]{} orbit, dispersion and $\beta$ distortions after correction were: $0.27/0.31$ mm, $\Delta\eta\!=\!31/8$ mm and $\Delta\beta/\beta\!=\!1.9/1.5$% in $x/y$ planes, respectively. Linear chromaticity was set to +1 to minimize non-linear tune variation with momentum.
![HER dynamic aperture at $\nu_x/\nu_y\!=\!24.51/23.61$.[]{data-label="fig:haper51"}](WPAG013f6.eps){width="60mm"}
The LER dynamic aperture at $\nu_x/\nu_y\!=\!38.518/36.61$ is shown in Fig. \[fig:laper518\] at the injection point. The solid half-ellipse corresponds to $10\sigma$ size of a fully coupled beam at injection with emittance $\epsilon_x\!=\!24$ nm. The residual [*rms*]{} orbit, dispersion and $\beta$ distortions after correction were: $0.20/0.23$ mm, $\Delta\eta\!=\!23/9$ mm and $\Delta\beta/\beta\!=\!2.4/2.5$% in $x/y$ planes, respectively. Linear chromaticity was set to zero in this case.
![LER dynamic aperture at $\nu_x/\nu_y\!=\!38.518/36.61$.[]{data-label="fig:laper518"}](WPAG013f7.eps){width="60mm"}
Implementation of the tune near half-integer resonance has been recently performed at PEP-II. The working point was successfully moved to $\nu_x/\nu_y\!=\!24.52/23.63$ in HER and $38.52/36.57$ in LER. After the necessary machine adjustments luminosity has been improved by $\sim$15% to the new record of $6.1\!\cdot\!10^{33}$ cm$^{-2}$s$^{-1}$.
CONCLUSION
==========
Beam-beam simulations performed for PEP-II predicted an enhancement of luminosity for a betatron tune near a half-integer resonance. Horizontal fractional tune of .52 has been recently implemented at PEP-II and $\sim$15% luminosity gain has been achieved. Particle tracking simulations showed that an improved machine correction is needed for acceptable dynamic aperture at $[\nu_x]\!=\!.51$ in HER and .518 in LER. This requires a minimization of non-linear chromaticity and tighter correction of orbit and $\beta_x$ distortions. At $[\nu_x]\!=\!.529$, a looser orbit correction may be used while compensation of $\Delta\beta_x$ may not be necessary.
[9]{}
PEP-II Conceptual Design Report, SLAC–418, 1993. Y. Cai, [*et al.*]{}, Phys. Rev. ST Accel. Beams 4, 011001 (2001). Y. Cai, [*et al.*]{}, SLAC–PUB–7642, 1997.
[^1]: Work supported by Department of Energy contract DE–AC03–76SF00515.
[^2]:
|
---
author:
- 'Akhmet’ev P.M.'
title: 'A remark on the Hopf invariant for spherical 4-braids'
---
Introduction
============
An approach by J.Wu describes homotopy groups $\pi_{n}(S^2)$ of the standard 2-sphere as isotopy classes of spherical $n+1$–strand Brunnian braids, for more details, see f.ex. [@B-M-V-W], Sequence (1.1). This approach is not possible for $n=3$ in the case of $4$-strand braids.
The homotopy group $\pi_3(S^2)$ in an infinite cyclic group, detected by the Hopf invariant $$\begin{aligned}
\label{hopfinv}
H: \pi_3(S^2) \mapsto \Z. \end{aligned}$$ An element of $\pi_3(S^2)$ is represented by a mapping $l:S^3 \mapsto S^2$, which is considered up to homotopy. The Hopf invariant $H(l)$ is well-defined as the integer linking number of two oriented curves $l^{-1}(a)$, $l^{-1}(b)$, where $a,b \in S^2$ be a pair of regular points of $l$. The Hopf invariant is very important for applications.
The goal of the paper is to modify the definition of Brunnian spherical 4-component braids, and to define the Hopf invariant as a function of isotopy classes of spherical braids, which are Brunnian in a new strong sense. The Hamiltonian provides an elegant method for generating simple geometrical examples of complicated braids and links, as is presented in [@B]. Let us formulate the following problems:
$\bullet$ Derive applications of higher-order winding numbers to generate Hamiltonian motion of 4 vortex in two dimensions on the sphere. For 3 vortex on the plane this is done in [@B].
$\bullet$ To unify the approach [@V] Ch.3 to $\pi_{\ast}(S^2)$ with the Wu’s approach. $\bullet$ To investigate integer lifts of the generator of $\pi_4(S^2)$ (the Arf invariant) by the Wu’s approach.
The paper is organized as following. In Section 2 we recall required definitions concerning first-order stage of the construction and determine the linking numbers of spherical 4-component braids. In Section 3 the Hopf invariant for 4-component spherical braids is well-defined. This is a second-order particular defined invariant: to define this invariant we should assume that the all linking numbers (there are two) of components of a spherical braid are equal to zero. Main results are formulated in Theorems $\ref{hopf}$, $\ref{main}$. In Section 4 we give proofs of the main results.
In a private letter (August 2013) prof. Viktor Ginzbugr (about a draft of the paper): “The subject is certainly interesting...”. I am grateful to him for the interest.
The results was presented at International Conference “Nonlinear Equations and Complex Analysis” in Russia (Bashkortostan, Bannoe Lake) during the period since March 18 (arrival day) till March 22 (departure day), 2013.
Linking numbers for spherical braids
====================================
By a spherical (ordered) $n$–braid we mean a collection of embeddings of the standard circles $$f:\bigcup_{i=1}^n S^1_i \subset S^2 \times S^1,$$ where the composition of this embedding with the standard projection $S^2 \times S^1 \to S^1$ on the second factor in the target space, restricted to an arbitrary component $S^1_i$, $i=1,\dots,n$ is the identity mapping $S^1_i \to S^1$. The space of all ordered spherical $n$-braids up to isotopy is denoted by $Br_n$. It is well-known that $Br_n$ is a group.
For a fixed value $t \in S^1$, a braid $f \in Br_n$ intersects the level $S^2 \times t$ by an (ordered) collection of $n$ points $\{z_1(t), \dots z_n(t)\}$. Let assume that $n=4$. Denote by $$g=g(f): S^{1}_{1} \cup S^1_{2} \cup S^1_{3} \subset S^2 \times S^1,$$ the $3$-component braid, obtained from $f$ by eliminating of the last component $S^1_4$.
Let us identify the sphere $S^2$ with the Riemann sphere, or with the complex projective line $\hat \C$. For a braid $f$ let us consider the collection of M$\ddot{\rm{o}}$bius transformations, which transforms the points $z_{1},z_{2},z_{3}$ into $0,1,\infty$ correspondingly: $$F(z;t) = \frac{(z-z_{1}(t))(z_{2}(t)-z_{3}(t))}{(z - z_{3}(t))(z_{2}(t)-z_{1}(t))}.$$ The image $F(f)$ is a 4-strand braid with the constant components $\{z_{1}(t), z_{2}(t), z_{3}(t)\} = \{0,1,\infty\}$. Denote this braid by $$\begin{aligned}
\label{Mobius}
F(f) = f^{norm}.\end{aligned}$$ The 3-strand braid $g$, constructed from $f^{norm}$ is the constant braid at the points $\{0,1,\infty\}$. The last component $f^{norm}_4(S^1)$ of $F(f)$ is represented by a closed path $z_4(t) \in \hat \C \setminus \{0,1,\infty\}, t \in \R^1/2\pi$.
For a given (ordered) 4-component braid $f$ let us define the linking number $Lk(f)$, $$\begin{aligned}
\label{link}
Lk: Br_4 \to \Z. \end{aligned}$$ Consider the following 1-form $$\begin{aligned}
\label{omega0}
\omega_0 = \frac{1}{2 \pi \i} \frac{dz}{z}. \end{aligned}$$ By definition we get $$d \log(z) = \frac{1}{2 \pi \i} \frac{dz}{z},$$ where $\log(z)$ is given by the formula: $$\log(z) = (2\pi \i)^{-1} \int \frac{dz}{z},$$ assuming that $\log(1)=0$, as a multivalued complex function.
Define $Lk(f)$ by the formula: $$Lk(f) = \Re{\int_{0}^{2\pi} \frac{d z_4(t)}{z_4(t)}} = \int_{f^{norm}_4} \omega_0,$$ where $\Re$ is the real part of the integral. By construction, $Lk(f)$ is the winding number, i.e. the integer number of rotations of the path $z_4(t)$ with respect to the origin and the infinity in $\hat\C$.
The permutation group $\Sigma(4)$ of the order $24$ acts on the space of ordered spherical braids: $$\begin{aligned}
\label{action}
\Sigma(4) \times Br_4 \to Br_4.
\end{aligned}$$ The image of an ordered braid $f$ by a transposition $\sigma:(1,2,3,4)\mapsto(\sigma_1,\sigma_2,\sigma_3,\sigma_4)$ is well-defined by the corresponding re-ordering of components of $f$. Let us investigate the orbit of the linking numbers $Lk(f)$ with respect to $(\ref{action})$. Simply say, we investigate how many independent linking numbers of components of braids are well-defined?
Let us consider the following exact sequences of groups: $$\begin{aligned}
\label{first}
0 \longrightarrow \A_4 \longrightarrow \Sigma_4 \longrightarrow \Z/2 \longrightarrow 0,\end{aligned}$$ $$\begin{aligned}
\label{second}
0 \longrightarrow \Z/2 \times \Z/2 \longrightarrow \A_4 \longrightarrow \Z/3 \longrightarrow 0.\end{aligned}$$ The subgroup $\A_4 \subset \Sigma_4$ in the sequence $(\ref{first})$ is represented by permutations, which preserve signs (equivalently, which is decomposed into an even number of elementary transpositions). The subgroup $\Z/2 \times \Z/2 \subset \A_4$ in the sequence $(\ref{second})$ is generated by the permutations $\{(1,2)(3,4); (1,3)(2,4); (1,4)(2,3)\}$.
Let us consider $2$-primary subgroup $K \subset \Sigma$ of the order $8$, which is defined as the extension of the subgroup $\Z/2 \times \Z/2$ from the sequence $(\ref{second})$, been included in the sequence $(\ref{first})$. An epimorphism $$\begin{aligned}
\label{epi}
\theta = (\theta_1,\theta_2): K \to \Z/2 \times \Z/2,\end{aligned}$$ is defined as follows: $\theta_1(\sigma)=1$ (the group $\Z/2$ is in the multiplicative form), if $\sigma$ preserves a (non-ordered) partition $(1,3)(2,4)$, and $\theta_1(\sigma)=-1$, otherwise. Therefore $\theta_1$ is an epimorphism with the kernel $\Z/2 \times \Z/2$ from the left subgroup of the sequence $(\ref{second})$. The epimorpism $\theta_2(\sigma)$ is determined by the sign of a permutation $\sigma$, this is the restriction of the right epimorphism in the sequence $(\ref{first})$ to the subgroup $K \subset \Sigma_3$.
\[linking\]
–1. The function $(\ref{link})$ is invariant with respect to the action $(\ref{action})$ (the re-numbering of components) by an arbitrary permutation, which in the kernel of $\theta$ in $(\ref{epi})$, and is skew-invariant for the action by a permutation, which is in the kernel of $\theta_1$ (the composition of $\theta$ with the projection on the first factor), but not in the kernel of $\theta_2$ (the composition of $\theta$ with the projection on the second factor).
–2. Denote by $\tilde{f} \in Br_4$ the ordered braid, which is obtained from $f \in Br_4$ by the action $(\ref{action})$ by the element $(1,2)$ ($\theta((1,2)) \in \Z/2 \times \Z/2$ is the product of the generators). There exists an ordered braids $f \in Br_4$, for which the linking numbers $Lk(f)$, $Lk(\tilde{f})$ are arbitrary integers.
From Lemma one may deduce the following corollary.
\[winding\]
–1. For an arbitrary braid $f \in Br_4$ the linking number $Lk(f)$ is well-defined as the differences of the winding number of the component $2$ between the components $1$ and $3$ with the winding number of the component $4$ between the components $1$ and $3$.
–2. For a braid $\tilde{f} \in Br_4$, where $f \in Br_4$ is an arbitrary, $\tilde{f}$ is defined in Lemma $\ref{linking}$, the linking number $Lk(\tilde{f})$ is well-defined as the winding number of the component $2$ between the components $1$ and $3$ with the winding number of the component $4$ between the components $2$ and $3$.
–3. An arbitrary well-defined homomorphism $Br_4 \to \Z$, which is a function of the windings numbers between components is a linear combination of $Lk(f)$ and $Lk(\tilde{f})$.
Corollary $(\ref{winding})$ motivates the following definition.
Let $f \in Br_4$ be a (ordered) spherical braid. Define the total linking number $LK(f) \in \Z \oplus \Z$ by the following formula: $$LK(f) = Lk(f) \oplus Lk(\tilde f).$$ The total linking number is a well-defined homomorphism $$LK: Br_4 \mapsto \Z \oplus \Z.$$
Hopf invariant of braids
========================
Let $f \in Br_4$ be a (ordered) spherical braid with the trivial total linking number: $LK(f)=0$. Such braids generate the subgroup in the group $Br_4$, denote this subgroup by $Brunn_4 \subset Br_4$. Let us remark that this subgroup coincides no with the subgroup of Brunnian braids $Brun_4$, defined in [@B-M-V-W], sequence 1.1.
\[hopf\] There exists a well-defined homomorphism $$\begin{aligned}
\label{Hopf}
H: Brunn_4 \to \Z, \end{aligned}$$ called the Hopf invariant. The homomorphism $(\ref{Hopf})$ is invariant with respect to the action $(\ref{action})$ by an arbitrary permutation, which in the kernel of $\theta$ in $(\ref{epi})$, and is skew-invariant with respect to the action by a permutation, which is in the kernel of $\theta_1$ (the composition of $\theta$ with the projection on the first factor), but not in the kernel of $\theta_2$ (the composition of $\theta$ with the projection on the second factor).
Higher homotopy groups are described from the spherical braids groups with non-ordered components in the [@B-M-V-W], Sequence (1.1).
Definition of the Hopf invariant {#definition-of-the-hopf-invariant .unnumbered}
--------------------------------
Let $f \in Brun_4$ be an arbitrary. Consider the braid $f^{norm}$, given by $(\ref{Mobius})$. Recal, for the braid $f^{norm}$ the braid $g \in Br_3$, which consists of the straits $(1)$,$(2)$,$(3)$ of $f^{norm}$, is the constant braid at the points $0,1,\infty$ in $\hat \C$ correspondingly. Consider the strait (4) of the braid $f^{norm}$. This strait is represented by an oriented closed path $i: S^1 \to \hat C \setminus \{0 \cup 1 \cup \infty\}$. This path determines a cycle, which is an oriented boundary, because of the condition $LK(f^{norm})=0$. (Evidently, $LK(f^{norm})=LK(f)$, because the group of M$\ddot{\rm{o}}$bius transformations is connected.)
Consider the inclusions $$I_0: \hat\C \setminus \{0 \cup 1 \cup \infty \} \subset \hat\C \setminus \{1 \cup \infty\},$$ $$I_{\infty}: \hat\C \setminus \{0 \cup 1 \cup \infty\} \subset \hat\C \setminus \{0 \cup 1\},$$ $$I_1: \hat\C \setminus \{0 \cup 1 \cup \infty\} \subset \hat\C \setminus \{0 \cup \infty\}.$$ Because $H_1( \hat\C \setminus \{1 \cup \infty\};\Z) = \pi_1( \hat\C \setminus \{1 \cup \infty\})$, for the homomorphism $$I_{0,\sharp}: \pi_1( \hat\C \setminus \{0 \cup 1 \cup \infty\}) \to \pi_1( \hat\C \setminus \{0 \cup 1 \cup \infty\})$$ we get $I_{0,\sharp}([i])=0$. Analogously $I_{\infty,\sharp}([i])=0$, $I_{1,\sharp}([i])=0$.
There exist the following 3 maps of copies of the standard 2-disk $$e_{0}: D^2_0 \to \hat C \setminus \{1 \cup \infty\}, \quad e_0 \vert_{\partial D^2} = i,$$ $$e_{\infty}: D^2_{\infty} \to \hat C \setminus \{0 \cup 1\}, \quad e_{\infty} \vert_{\partial D^2} = i,$$ $$e_{1}: D^2_1 \to \hat C \setminus \{0 \cup \infty\}, \quad e_1 \vert_{\partial D^2} = i.$$
Consider a 2-sphere, which is represented by a gluing $D^2_0 \cup_{\partial} D^2_{\infty}$ of the disks $D^2_0$, $D^2_{\infty}$ along the common boundary, which is identified with the circle $S^1_4$. Denote this sphere by $S^2_1$. Analogously define spheres $S^2_0=D^2_{\infty} \cup_{\partial} D^2_{1}$, $S^2_{\infty}=D^2_{1} \cup_{\partial} D^2_{0}$.
Consider the following commutative diagram of inclusions: $$\begin{aligned}
\label{0,1}
\begin{array}{ccc}
\hat \C \setminus \{0 \cup \infty \cup 1\} & \subset & \hat \C \setminus \{0 \cup \infty\} \\
\cap & & \cap \\
\hat \C \setminus \{\infty \cup 1\} & \subset & \hat \C \setminus \{\infty\} \\
\end{array}\end{aligned}$$
Consider the mappings $e_0: D_0^2 \to \hat \C \setminus \{1 \cup \infty\}$, $e_1: D_1^2 \to \hat \C \setminus \{0 \cup \infty\}$ to the left bottom and to the right upper spaces of the diagram $(\ref{0,1})$ correspondingly. The mapping $e_0 \cup_{\partial} e_1: S^2_{\infty} \to \hat \C \setminus \{\infty\}$ is well defined by gluing of the two mappings $e_0$, $e_1$ along the common mapping $i$ of the boundaries. Consider the standard 3-ball $D^3_{\infty}$ (with corners along the curve $S^1_4$) with the boundary $\partial D^3_{\infty} = S^2_{\infty}$. The mapping $e_0 \cup_{\partial} e_1$ can be extended to the mapping $$\begin{aligned}
\label{dinfty}
d_{\infty}: D^3_{\infty} \to \hat C \setminus \{\infty\}.\end{aligned}$$ The target space of this mapping is the right bottom space of the diagram $(\ref{0,1})$. Because the target space of the mapping $d_{\infty}$ is contractible, the mapping $d_{\infty}$ is well-defined up to homotopy. By the analogous constructions the following mappings $$\begin{aligned}
\label{d1}
d_{1}: D^3_{1} \to \hat \C \setminus \{1\},\end{aligned}$$ $$\begin{aligned}
\label{d0}
d_{0}: D^3_{0} \to \hat \C \setminus \{0\}\end{aligned}$$ are well-defined.
The mappings $(\ref{dinfty})$, $(\ref{d1})$, $(\ref{d0})$ determine the mapping $$\begin{aligned}
\label{mapping}
h=h(f): S^3 \to S^2 \end{aligned}$$ as follows. Take a 3-sphere $S^3$, which is catted into 3 balls $D^3_{\infty}, D^3_1, D^3_0$ along the common circle $S^1_4 \subset S^3$. The sphere $S^3$ is represented as the join $S^1_4 \ast S^1_a$ of the two standard circle. On the circle $S^1_a$ take 3 points $x_0, x_{1}, x_{\infty} \in S^1_a$. The subsets $S^1_4 \ast [x_0,x_1] \subset S^3$, $S^1_4 \ast [x_1,x_{\infty}] \subset S^3$, $S^1_4 \ast [x_{\infty},x_0] \subset S^3$ are 3 copies of 3D disks, which are glued along corresponding subdomains in its boundaries.
Let us identify $D^3_{\infty} \cong S^1_4 \ast [x_0,x_1]$, $D^3_{0} \cong S^1_4 \ast [x_1,x_{\infty}]$, $D^3_{1} \cong S^1_4 \ast [x_{\infty},x_0]$. The boundary $\partial D^3_{\infty}$ is identified with the balls $S^1_4 \ast \{0\} \cong D^2_0$, $S^1_4 \ast \{1\} \cong D^2_1$, which are glued along the common boundary $S^1_4$. The boundary $\partial D^3_{0}$ is identified with the balls $S^1_4 \ast \{1\} \cong D^2_1$, $S^1_4 \ast \{\infty\} \cong D^2_{\infty}$, which are identify along the same boundary $S^1_4$. The boundary $\partial D^3_{1}$ is identified with the balls $S^1_4 \ast \{\infty\} \cong D^2_{\infty}$, $S^1_4 \ast \{0\} \cong D^2_{0}$, which are identified along the same boundary $S^1_4$. The mappings $d_0$, $d_1$, $d_{\infty}$ on the corresponded balls are well-defined by the formulas $(\ref{d0})$,$(\ref{d1})$,$(\ref{dinfty})$ correspondingly. This mappings define the mapping $(\ref{mapping})$ on the 3-sphere.
\[defHopf\] The Hopf invarian $H(f)$ for a braid $f \in Brunn_4$ in the formula $(\ref{Hopf})$ is defined as the Hopf invariant of the mapping $h$ by the formula $(\ref{hopfinv})$. The mapping $h=h(f)$ is explicitly defined from the braid $f$ by the formula $(\ref{mapping})$.
A formula to calculate the Hopf invariant {#a-formula-to-calculate-the-hopf-invariant .unnumbered}
-----------------------------------------
Let us introduces an explicit formula to calculate the Hopf invariant for a braid $f \in Brunn_4$. Consider the complex plane $\C$. The 4-th strain of the braid $f^{norm}$ determines a curve on the plane without two points $\{0,1\}$, which will be denoted by $$\begin{aligned}
\label{pathgamma}
\gamma: S^1 \to \C \setminus \{0 \cup 1\}.\end{aligned}$$ Let us consider the complex 1-form $(\ref{omega0})$. Define a complex 1-form $$\begin{aligned}
\label{omega1}
\omega_1 = \frac{1}{2 \pi \i} \frac{dz}{z-1}. \end{aligned}$$ Define a real (multivalued) function $\lambda_0$ by integration along the path $\gamma(t)$, $t \in [0,t] \subset S^1$ of the real part of the form $(\ref{omega0})$ as following: $$\begin{aligned}
\label{lambda0}
\lambda_0(t) = \Re {\int_0^t \omega_0}.\end{aligned}$$ Define a real (multivalued) function $\lambda_1$ by integration along the path of the real part of the form $(\ref{omega1})$ as following: $$\begin{aligned}
\label{lambda1}
\lambda_1(t) = \Re {\int_0^t \omega_1}.\end{aligned}$$ To take the multivalued functions $(\ref{lambda0})$, $(\ref{lambda1})$ well-defined, assume that the path $\gamma$ starts at the point $2 \in \C$: $\lambda_0(0)=2$, $\lambda_1(0)=2$.
Define a closed 1-form $\psi(t)$ along a curve $\gamma(t) \in \C \setminus \{0 \cup 1\}$ by the following formula: $$\begin{aligned}
\label{form}
\psi(t) = \lambda_0(t) \omega_1 - \lambda_1(t) \omega_0.\end{aligned}$$
Let us consider a function, which is well-defined as the real part of the integral $$\begin{aligned}
\label{int}
\Psi(T) = \Re{\int_{0}^T \psi(t) d\gamma}, \quad t \in [0,T] \subset S^1. \end{aligned}$$
\[main\] The Hopf invariant of a braid $f \in Brun_4$ in the formula $(\ref{Hopf})$, which is defined by Definition $\ref{defHopf}$, is calculated by the formula: $$\begin{aligned}
\label{m}
H(f)=\Psi(2\pi) = \frac{1}{2}\Re{ \int_{0}^{2\pi} \psi(t) d\gamma}, \end{aligned}$$ where $\gamma$ is the closed path, determined by the 4-th straight $S^1_4$ of the braid $f^{norm}$ by the formula $(\ref{pathgamma})$.
From Theorem $\ref{main}$ we get a corollary.
\[cor\]
–1. The Hopf invariant $(\ref{Hopf})$ is an epimorphism.
–2. Assume that a braid $f \in Brunn_4$ is such that the braid $f^{norm}$ is represented by a commutator of the straight $(4)$ with straights $(1)$ and $(2)$ (such a braid is called the Borromean rings). Then $H(f)=\pm1$, where the sign in the formula depends on the sign of the commutator.
### Proof of Corollary $\ref{cor}$ {#proof-of-corollary-refcor .unnumbered}
It is sufficient to proof –2. The right side of the formula $(\ref{form})$ coincides with the formula (28) [@B], which is simplified for the considered example. For the Borromean ring the formula $\ref{m}$ is non-trivial. The factor $\frac{1}{2}$ in the right side of the formula provides $H(f)=1$ for the right Borromean rings. Corollary is proved.
Proof of Theorem $\ref{main}$
=============================
Proofs of Lemma $\ref{linking}$, Corollary $\ref{winding}$, and of Theorem $\ref{hopf}$ are clear. Let us proof Theorem $\ref{main}$. Consider the mapping $h: S^3 \to S^2 = \hat C$, which is defined by the formula $(\ref{mapping})$. Take volume forms $\Omega_0, \Omega_1 \in \Lambda^2(S^2)$, each form is an ill–supported form at the point $0$, $1$ correspondingly. The Hopf invariant $(\ref{Hopf})$ is calculated by the formula: $$\begin{aligned}
\label{integral}
H(f) = \frac{1}{2} \int_{S^3} h^{\ast}(\Omega_0) \wedge \beta_1 + h^{\ast}(\Omega_1) \wedge \beta_0, \end{aligned}$$ where $x \in S^3$, $h^{\ast}(\Omega_0) \in \Lambda^2(S^3)$ is the pull-back of $\Omega_0 \in \Lambda^2(S^2)$ by $h: S^3 \to S^2$, $\beta_0 \in \Lambda^1(S^3)$ is an arbitrary 1-form, such that $d(\beta_0)=h^{\ast}(\Omega_0)$, $\beta_1 \in \Lambda^1(S^3)$ is defined analogously to the definition of $\beta_0$.
Evidently, the 1-forms $\beta_0$ in the integral $(\ref{integral})$ is represented up to a coboundary such that $\beta_0 = 0$ inside the ball $D^3_0$. This follows from the fact that the curve $h^{-1}(0)$ is outside the ball $D_0^3$. Analogously, we may assume that $\beta_1 = 0$ in the ball $D^3_1$. Then we get the following simplification of $(\ref{integral})$: $$H(f) = \frac{1}{2} \int_{D_{\infty}^3} h^{\ast}(\Omega_0) \wedge \beta_1 + h^{\ast}(\Omega_1) \wedge \beta_0,$$ assuming that $\beta_1$ is inside $D_{\infty}^3 \cup D_0^3$, $\beta_0$ is inside $D_{\infty}^3 \cup D_1^3$.
In the ball $D_{\infty}^3 \cup D_0^3$ the 3-form $h^{\ast}(\Omega_0) \wedge \beta_1$ is exact, we get $\alpha_1 \in \Lambda^2(D_{\infty}^3 \cup D_0^3)$, $d\alpha_1 = h^{\ast}(\Omega_0) \wedge \beta_1$. Moreover, we may put $\alpha_1 = \beta_0 \wedge \beta_1$.
In the ball $D_{\infty}^3 \cup D_1^3$ the 3-form $h^{\ast}(\Omega_1) \wedge \beta_0$ is exact, we get $\alpha_0 \in \Lambda^2(D_{\infty}^3 \cup D_1^3)$, $d\alpha_0 = h^{\ast}(\Omega_1) \wedge \beta_0$. Moreover, we may put $\alpha_0 = \beta_1 \wedge \beta_0 = -\beta_0 \wedge \beta_1$.
Apply the 3D Gauss-Ostrogradsky formula, we get $$\int_{D_{\infty}^3} h^{\ast}(\Omega_0) \wedge \beta_1 = \int_{D^2_1} \beta_0 \wedge \beta_1 ,$$ $$\int_{D_{\infty}^3} h^{\ast}\Omega_1 \wedge \beta_0 = - \int_{D^2_{0}} \beta_0 \wedge \beta_1 .$$
The 2-form $\beta_0 \wedge \beta_1 \in \Lambda^2(D^2_1)$ is exact. Because in the disk $D^2_1$ the 0-form $\lambda_1$ is well defined, and $d\lambda_1 = \beta_1$, we get: $d(\lambda_1 \beta_0) = - \beta_0 \wedge \beta_1$.
Analogously, the 2-form $\beta_0 \wedge \beta_1 \in \Lambda^2(D^2_{0})$ is exact. Because in the disk $D^2_{0}$ the 0-form $\lambda_0$ is well defined, and $d\lambda_0 = \beta_0$, we get: $d(\lambda_0 \beta_1) = \beta_0 \wedge \beta_1$.
Apply the 2D Green formula we get: $$\int_{D^2_1} \beta_0 \wedge \beta_1 = -\int_{\gamma} \lambda_1 \beta_0,$$ $$\int_{D^2_0} \beta_0 \wedge \beta_1 = \int_{\gamma} \lambda_0 \beta_1,$$
The integral $(\ref{integral})$ is simplified as $$H(f) = \frac{1}{2} \int_{\gamma} \lambda_0 \beta_1 - \lambda_1 \beta_0.$$ This formula coincides with the formula $(\ref{m})$. Theorem $\ref{main}$ is proved.
[99911122]{}
V. Bardakov, R. Mikhailov, J. Wu, V. Vershinin, [*Brunnian braids on surfaces*]{}, Alg. Geom. Top., 12 (2012), 1607–1648, arXiv: 0909.3387
Mitchell A Berger, [*Hamiltonian dynamics generated by Vassiliev invariants*]{}, J. Phys. A: Math. Gen. 34 (2001) 1363–1374.
V. A. Vassiliev, *[Complements of discriminants of smooth maps: topology and applications]{}, 2-d extended edition, Translations of Math. Monographs, 98, AMS, Providence, RI, (1994) 268 pp.; расширенный русский перевод: В. А. Васильев, *[ Топология дополнений к дискриминантам]{}, Фазис, Москва, (1997) 536 с.**
$$$$ $$$$
Troitsk, Moscow region, IZMIRAN $\qquad \qquad \qquad \qquad$
pmakhmet@mi.ras.ru $\qquad \qquad \qquad \qquad$ $$$$
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---
abstract: 'ANAIS (Annual modulation with NaI Scintillators) is a project aiming to set up at the new facilities of the Canfranc Underground Laboratory (LSC), a large scale NaI(Tl) experiment in order to explore the DAMA/LIBRA annual modulation positive result using the same target and technique. Two 12.5kg each NaI(Tl) crystals provided by Alpha Spectra took data at the LSC in the ANAIS-25 set-up. The comparison of the background model for the ANAIS-25 prototypes with the experimental results is presented. ANAIS crystal radiopurity goals have been achieved for $^{232}$Th and $^{238}$U chains, but a $^{210}$Pb contamination out-of-equilibrium was identified, whose origin has been studied. The high light collection efficiency obtained with these prototypes allows to anticipate an energy threshold of the order of 1 keVee. A new detector, with improved performances, was received in March 2015 and very preliminary results are shown.'
author:
- 'J. Amaré, S. Cebrián, C. Cuesta[^1], E. García, C. Ginestra, M. Martínez[^2], M. A. Oliván, Y. Ortigoza, A. Ortiz de Solórzano, C. Pobes[^3], J. Puimedón, M. L. Sarsa, J. A. Villar, and P. Villar'
bibliography:
- 'LRT15\_CCuesta\_ANAIS.bib'
title: Background analysis and status of the ANAIS dark matter project
---
[ address=[Laboratorio de Física Nuclear y Astropartículas, Universidad de Zaragoza, Calle Pedro Cerbuna 12, 50009 Zaragoza, Spain\
Laboratorio Subterráneo de Canfranc, Paseo de los Ayerbe s/n, 22880 Canfranc Estación, Huesca, Spain]{} ]{}
The ANAIS experiment
====================
The ANAIS (Annual modulation with NaI Scintillators) project is intended to search for dark matter annual modulation with ultrapure NaI(Tl) scintillators at the Canfranc Underground Laboratory (LSC) in Spain. The motivation of the ANAIS experiment is to provide a model-independent confirmation of the annual modulation positive signal reported by the DAMA collaboration [@DAMAphaseI] using the same target and technique. To achieve this goal, ANAIS detectors should have similar performance to DAMA/LIBRA ones in terms or threshold and background, but as we will report below, reducing the threshold below 2keVee will imply a good sensitivity to the DAMA/LIBRA singled out region in the WIMP parameter space even in the case of higher background.
The total NaI(Tl) active mass will be divided into modules, each consisting of a 12.5kg NaI(Tl) crystal encapsulated in copper and optically coupled to two photomultipliers (PMTs) working in coincidence. Several modules, accounting for around 100 kg, will be set-up at LSC along next months. The shielding for the experiment consists of 10cm of archaeological lead, 20cm of low activity lead, 40cm of neutron moderator, an anti-radon box (to be continuously flushed with boil-off nitrogen), and an active muon veto system made up of plastic scintillators designed to cover top and sides of the whole ANAIS set-up. The hut that will house the experiment at the hall B of LSC (under 2450m.w.e.) is already operative, shielding materials and electronic chain components are prepared for mounting. Different PMT models were tested in order to choose the best option in terms of light collection and background. The Hamamatsu R12669SEL2 was selected, and all the units are available at the LSC.
The main challenge has been the achievement of the required low background, in particular the development of crystals having a potassium content at the ppb level. A modular approach has been followed, from the ANAIS-0 module to the ANAIS-25 and ANAIS-37 set-ups. The ANAIS-0 module, which consists in a 9.6kg ultrapure NaI(Tl) crystal, 4“x4”x10", made by Saint-Gobain was first operated. It was thoroughly studied to characterize ANAIS background, optimize events selection, design the calibration method, test the acquisition code and electronics, and determine the optimum configuration of PMTs and light guides. The main results have been published in [@ANAISbulk; @ANAISbkg; @anais40K; @ANAISom]. Two prototypes of 12.5kg mass (named D0 and D1), made by Alpha Spectra with ultrapure NaI powder took data at the LSC from December 2012 to March 2015 for a general performance and background assessment. We will refer in the following to this set-up as ANAIS-25. The ANAIS-37 set-up combines the ANAIS-25 modules with a new module also built from Alpha Spectra, using improved protocols for purification and growing of the powder and crystal, designed in view of the ANAIS-25 results.
ANAIS-25
========
The main goals for this set-up were to measure the crystal internal contamination, determine light collection efficiency, fine tune the data acquisition and test the filtering and analysis protocols. ANAIS-25 set-up consisted of two modules of 12.5kg each provided by Alpha Spectra, Inc. Colorado. The modules are cylindrical, 4.75$"$ diameter and 11.75$"$ length, with quartz windows for PMTs coupling. A Mylar window in the lateral face allows for low energy calibration. Two types of photomultiplier were tested: One module coupled to two Hamamatsu R12669SEL2 and the other coupled to Hamamatsu R11065SEL, later replaced by the R12669SEL2 model too. The modules were surrounded by 10cm of archaeological lead plus 20cm of low activity lead shielding at the Canfranc Underground Laboratory. ANAIS-25 modules took data from December 2012 to March 2015.
The first feature to be remarked is the excellent light collection as it can be seen in Table \[tab:ASphekev\], especially with the Ham. R12669SEL2 which is the PMT model chosen to be used at the ANAIS experiment. This light collection has a good impact in both resolution and energy threshold.
---- ----------------- --------------
D0 Ham. R12669SEL2 15.6$\pm$0.2
D1 Ham. R11065SEL 12.6$\pm$0.1
D1 Ham. R12669SEL2 15.2$\pm$0.1
---- ----------------- --------------
: Light collection efficiencies (phe./keV) for ANAIS-25 detectors, derived from the 22.6keV line ($^{109}$Cd calibration).[]{data-label="tab:ASphekev"}
Background contributions have been thoroughly analyzed. A detailed study of cosmogenic radionuclide production in NaI(Tl) derived from these data can be found at references [@ANAIScosmo; @ANAIScosmoLRT]. The preliminary results of ANAIS-25 were published in [@ANAISricap13] and Table \[tab:A25cont\] shows the results of the activities determined for the main crystal contaminations: $^{40}$K content has been measured performing coincidence analysis between 1461keV and 3.2keV energy depositions in different detectors [@anais40K] and the activities from $^{210}$Pb and $^{232}$Th and $^{238}$U chains have been determined on the one hand, by quantifying Bi/Po sequences, and on the other, by comparing the total alpha rate with the low energy depositions attributable to $^{210}$Pb, which are fully compatible. These results give a moderate contamination of $^{40}$K, above the initial goal of ANAIS (20ppb of K) but acceptable, a high suppression of $^{232}$Th and $^{238}$U chains but a high activity of $^{210}$Pb at the mBq/kg level. The origin of such contamination was identified and has been addressed by Alpha Spectra (see next section).
------------------------- ----------------- ------ -------------------
1.25$\pm$0.11 (41ppb K) 0.010$\pm$0.002 3.15 0.0020$\pm$0.0008
------------------------- ----------------- ------ -------------------
: Internal contamination measured in the ANAIS-25 crystals.[]{data-label="tab:A25cont"}
A preliminary study has been carried out by using low energy events populations from internal $^{40}$K and $^{22}$Na. The K-shell electron binding energy following electron capture in $^{40}$K (3.2keV) and $^{22}$Na (0.9 keV) can be tagged by the coincidence with a high energy $\gamma$ ray (1461keV and 1274keV respectively). In Figure \[fig:A25trigger\] are shown both populations tagged by the high energy gamma, together with the events effectively triggering our acquisition. From Figure \[fig:A25trigger\], it can be concluded that triggering at 1keVee is clearly achieved in ANAIS-25 detectors and then, an energy threshold of the order of 1keVee is at reach. The main issue to reach the 1keVee threshold is the effective removal of PMT origin events, which are dominating the background below 10keVee. Following the work done in [@ANAISbulk] we have designed specific filtering protocols for ANAIS-25 detectors, being triggering and filtering efficiencies still under study. A preliminary spectrum, after filtering and correcting by the efficiencies of the cuts, determined with low energy events from a Cd-109 calibration, is shown in Figure \[fig:A25le\].
![ANAIS-25 D0 coincident events at low energy for $^{40}$K (left), and for $^{22}$Na (right).[]{data-label="fig:A25trigger"}](40K.png "fig:"){width="45.00000%"} ![ANAIS-25 D0 coincident events at low energy for $^{40}$K (left), and for $^{22}$Na (right).[]{data-label="fig:A25trigger"}](22Na.png "fig:"){width="45.00000%"}
![Preliminary filtered background spectra corrected by triggering and filtering efficiencies for ANAIS-25 detectors D0 and D1 (filtering procedures still being optimized).[]{data-label="fig:A25le"}](Bkg.png){width="50.00000%"}
The background model of the ANAIS-25 modules has been developed following the same procedure than for the previous prototype ANAIS-0 [@ANAISbkg]. The background sources considered in the model include: activities from external components such as PMTs, copper encapsulation, quartz windows, silicone pads, and archaeological lead; contribution from radon of the air filling the inner volume of the shielding, as one hundredth of the measured external air Rn activity since the volume inside the shielding is flushed with boil-off nitrogen; intrinsic activities from the NaI(Tl) crystals (as reported on Table \[tab:A25cont\]) and concerning $^{129}$I, since there is a broad range of activity values in iodine compounds (depending on the ore origin) and a direct quantification was not possible, concentration of the isotope was assumed to be the same as estimated by DAMA/LIBRA ($^{129}$I/$^{nat}$I=(1.7$\pm$0.1)$\cdot$10$^{13}$) [@DAMAapparatus]; and cosmogenic origin activities in the NaI(Tl) crystals as quantified in [@ANAIScosmo]. The contribution of these background sources has been assessed by Monte Carlo simulation using the Geant4 code. Figure \[fig:A25sim\] compares the energy spectra summing all the simulated contributions described above with the measured data for ANAIS-25 detectors, considering anticoincidence data. A good agreement is obtained at high energy, but in the very low energy region some contribution seemed to be missing. Since upper limits on radionuclide activity have been used for several components, the background seems to be overestimated in some energy regions. Inclusion of cosmogenics was essential to reproduce coincidence data. It was found that the inclusion in the model of an additional activity of $\sim$0.2mBq/kg of $^{3}$H in the NaI crystals significantly improves the agreement with data at low energy, as shown in Figure \[fig:A25sim2\]. This value is about twice the upper limit set for DAMA/LIBRA crystals ($<$0.09mBq/kg [@DAMAapparatus]), but lower than the saturation activity which can be deduced from the production rate at sea level of $^{3}$H in NaI, calculated in [@mei] as explained in [@ANAIScosmoLRT]. Figure \[fig:A25sim2\] summarizes the different contributions from the explained background model of ANAIS-25 detectors, for anticoincidence data, to the rate in the region from 1 to 10keV.
![Comparison of the energy spectra summing all the simulated contributions (before and after adding cosmogenics) with the measured raw data for ANAIS-25 D0, considering anticoincidence data. Two different energy ranges are shown.[]{data-label="fig:A25sim"}](A25simle.png "fig:"){height="0.3\textheight"} ![Comparison of the energy spectra summing all the simulated contributions (before and after adding cosmogenics) with the measured raw data for ANAIS-25 D0, considering anticoincidence data. Two different energy ranges are shown.[]{data-label="fig:A25sim"}](A25simhe.png "fig:"){height="0.3\textheight"}
![Comparison of the very low energy spectra of simulations with the measured data for ANAIS-25 detectors, , after filtering and considering anticoincidence, including $^{3}$H in the model.[]{data-label="fig:A25sim2"}](A25simvle.png){width="55.00000%"}
ANAIS-37
========
Alpha Spectra new module consists in a 12.5kg crystal made with a more purified powder, grown under improved conditions in order to prevent radon contamination, according to Alpha Spectra information. The crystal was encapsulated following similar protocols and using same materials to those required for ANAIS-25 detectors. An aluminized Mylar window was also built to allow low energy calibrations of the module. The crystal was received on the 6$^{th}$ of March, 2015 and Ham. R12669SEL2 PMTs were coupled to this new module at LSC clean room (see Figure \[fig:A37\]). Profiting from the ANAIS-25 setup, this new crystal was very fast commissioned and data taking started the 11$^{th}$ of March, 2015. We will refer in the following to this set-up as ANAIS-37, and it consists in three modules, 12.5kg mass each. The new module (D2) is placed in between the two ANAIS-25 modules (D0 and D1) to maximize the coincidence efficiency for the potassium determination (see Figure \[fig:A37setup\]).
![New NaI(Tl) module that is part of the ANAIS-37 set-up.[]{data-label="fig:A37"}](ANAIS37.png){width="70.00000%"}
![Left, picture taken during installation of ANAIS-37 set-up. Right, schematic drawing of the ANAIS-37 experimental layout at LSC consisting of 10cm archaeological lead plus 20cm low activity lead, all enclosed in a PVC box continuously flushed with boil-off nitrogen and active vetoes anti-muons.[]{data-label="fig:A37setup"}](A37setup2.png "fig:"){height="0.25\textheight"} ![Left, picture taken during installation of ANAIS-37 set-up. Right, schematic drawing of the ANAIS-37 experimental layout at LSC consisting of 10cm archaeological lead plus 20cm low activity lead, all enclosed in a PVC box continuously flushed with boil-off nitrogen and active vetoes anti-muons.[]{data-label="fig:A37setup"}](A37setup.png "fig:"){height="0.25\textheight"}
Very preliminary results corresponding to 50 days of live-time are presented here. The total alpha rate of the new module D2 is determined through pulse shape analysis. A corresponding alpha rate of 0.58$\pm$0.01mBq/kg has been observed, which is a factor 5 lower than alpha rate in ANAIS-25 modules (3.15mBq/kg). We can conclude that effective reduction of Rn entrance in the growing and/or purification at Alpha Spectra has been achieved. The potassium content of the new D2 crystal has been analyzed using the same technique applied to previous prototypes. Bulk $^{40}$K content is estimated by searching for the coincidences between 3.2keV energy deposition in one detector (following EC) and the 1461keV gamma line escaping from it and being fully absorbed in the other detector. Efficiency of the coincidence was estimated using Geant4. We can conclude that the new D2 crystal has a potassium content of 44$\pm$4ppb compatible with that obtained with previous Alpha Spectra crystals (shown in Table \[tab:A25cont\]).
The light collection efficiency has been estimated following the same procedure as in the previous modules. As a result, 15.4$\pm$0.1phe./keV are obtained with this new module, also compatible with that obtained with previous Alpha Spectra modules and the Ham. R12669SEL2 PMTs (shown in Table \[tab:ASphekev\]).
Sensitivity
===========
Prospects of the sensitivity to the annual modulation in the WIMP mass - cross-section parameter space are shown in Figure \[fig:sens\] for 100kg configuration and 5 years of data taking. The analysis window considered is from 1 to 6keVee. The background assumed is the one measured in ANAIS-25 (corrected by the efficiencies of the cuts applied to remove PMT events and the trigger efficiency, and shown in Figure \[fig:A25le\]), but the $^{210}$Pb activity measured in the new module D2, i.e. the contribution of 2.57mBq/kg of $^{210}$Pb has been subtracted to the background measured at ANAIS-25. Further reduction from anticoincidence measurement, dependent on the matrix assumed, is expected. The most conservative approach to derive these prospects has been followed, but even in this case there is a considerable discovery potential of dark matter particles as responsible of the DAMA/LIBRA signal. An energy threshold lower than 2keVee is a crucial issue in this moment for the ANAIS experiment in order to guarantee enough sensitivity to test the DAMA/LIBRA result.
![Prospects of the sensitivity to the annual modulation for 100kg total detection mass and presently achieved background (module D2) in the framework of the ANAIS experiment. Five years data taking have been assumed and an energy window from 1 to 6keVee. These prospects correspond to a detection limit at 90% CL with a critical limit at 90% CL, according to sensitivity estimates proposed in [@sensitivityPlots].[]{data-label="fig:sens"}](prospects.png){width="50.00000%"}
This work was supported by the Spanish Ministerio de Economía y Competitividad and the European Regional Development Fund (MINECO-FEDER) (FPA2011-23749), the Consolider-Ingenio 2010 Programme under grants MULTIDARK CSD2009-00064 and CPAN CSD2007-00042, and the Gobierno de Aragón (Group in Nuclear and Astroparticle Physics, ARAID Foundation and C. Cuesta predoctoral grant). C. Ginestra and P. Villar are supported by the MINECO Subprograma de Formación de Personal Investigador. We also acknowledge LSC and GIFNA staff for their support.
[^1]: Present address: Department of Physics, Center for Experimental Nuclear Physics and Astrophysics, University of Washington, Seattle, WA, USA
[^2]: Present address: Università di Roma La Sapienza, Piazzale Aldo Moro 5, 00185 Roma, Italy
[^3]: Present address: Instituto de Ciencia de Materiales de Aragón, Universidad de Zaragoza-CSIC, Calle Pedro Cerbuna 12, 50009 Zaragoza, Spain
|
---
abstract: |
One way of studying a relational structure is to investigate functions which are related to that structure and which leave certain aspects of the structure invariant. Examples are the automorphism group, the self-embedding monoid, the endomorphism monoid, or the polymorphism clone of a structure. Such functions can be particularly well understood when the relational structure is countably infinite and has a first-order definition in another relational structure which has a finite language, is totally ordered and homogeneous, and has the Ramsey property. This is because in this situation, Ramsey theory provides the combinatorial tool for analyzing these functions – in a certain sense, it allows to represent such functions by functions on finite sets.
This is a survey of results in model theory and theoretical computer science obtained recently by the authors in this context. In model theory, we approach the problem of classifying the reducts of countably infinite ordered homogeneous Ramsey structures in a finite language, and certain decidability questions connected with such reducts. In theoretical computer science, we use the same combinatorial methods in order to classify the computational complexity for various classes of infinite-domain constraint satisfaction problems. While the first set of applications is obviously of an infinitary character, the second set concerns genuinely finitary problems – their unifying feature is that the same tools from Ramsey theory are used in their solution.
address:
- |
Laboratoire d’Informatique (LIX), CNRS UMR 7161\
École Polytechnique\
91128 Palaiseau\
France
- |
Équipe de Logique Mathématique\
Université Denis Diderot - Paris 7\
UFR de Mathématiques\
75205 Paris Cedex 13, France
author:
- Manuel Bodirsky
- Michael Pinsker
title: Reducts of Ramsey structures
---
Introduction
============
*“I prefer finite mathematics much more than infinite mathematics. I think that it is much more natural, much more appealing and the theory is much more beautiful. It is very concrete. It is something that you can touch and something you can feel and something to relate to. Infinity mathematics, to me, is something that is meaningless, because it is abstract nonsense.”*
(Doron Zeilberger, February 2010)
*“To the person who does deny infinity and says that it doesn’t exist, I feel sorry for them, I don’t see how such view enriches the world. Infinity may be does not exist, but it is a beautiful subject. I can say that the stars do not exist and always look down, but then I don’t see the beauty of the stars. Until one has a real reason to doubt the existence of mathematical infinity, I just don’t see the point.”*
(Hugh Woodin, February 2010)
Sometimes, infinite mathematics is not just beautiful, but also *useful*, even when one is ultimately interested in finite mathematics. A fascinating example of this type of mathematics is the recent theorem by Kechris, Pestov, and Todorcevic [@Topo-Dynamics], which links Ramsey classes and topological dynamics. A class of finite structures $\mathcal C$ closed under isomorphisms, induced substructures, and with the joint embedding property (see [@HodgesLong]) is called a *Ramsey class* [@RamseyClasses; @NesetrilSurvey] (or *has the Ramsey property*) if for all $P,H \in \mathcal C$ and every $k \geq 2$ there is a $S \in \mathcal C$ such that for every coloring of the copies of $P$ in $S$ with $k$ colors there is a copy $H'$ of $H$ in $\mathcal C$ such that all copies of $P$ in $H'$ have the same color. This is a very strong requirement — and certainly from the finite world. Proving that a class has the Ramsey property can be difficult [@NesetrilSurvey], and Ramsey theory rather provides a tool box than a theory to answer this question.
Kechris, Pestov, and Todorcevic [@Topo-Dynamics] provide a characterization of such classes in topological dynamics, connecting Ramsey classes with *extreme amenability* in (infinite) group theory, a concept from the 1960s [@Granirer]. The result can be used in two directions. One can use it to translate deep existing Ramsey results into proofs of extreme amenability of topological groups (and this is the main focus of the already cited article [@Topo-Dynamics]). One can also use it in the other direction to obtain a more systematic understanding of Ramsey classes. A key insight for this direction is the result of Nešetřil (see [@RamseyClasses]) which says that Ramsey classes $\mathcal C$ have the *amalgamation property*. Hence, by [Fraïssé]{}’s theorem, there exists a countably infinite homogeneous and $\omega$-categorical structure $\Gamma$ such that a finite structure is from $\mathcal C$ if and only if it embeds into $\Gamma$. The structure $\Gamma$ is unique up to isomorphism, and is called the *[Fraïssé]{} limit* of $\mathcal C$. Now let $\mathcal D$ be any amalgamation class whose [Fraïssé]{} limit $\Delta$ is bi-interpretable with $\Gamma$. By the theorem of Ahlbrandt and Ziegler [@AhlbrandtZiegler], two $\omega$-categorical structures are first-order bi-interpretable if and only if their automorphism groups are isomorphic as *(abstract) topological groups*. In addition, the above-mentioned result from [@Topo-Dynamics] shows that whether or not $\mathcal D$ is a Ramsey class only depends on the automorphism group $\text{Aut}(\Delta)$ of $\Delta$; in fact, and much more interestingly, it only depends on $\text{Aut}(\Delta)$ viewed as a topological group (which has cardinality $2^\omega$). From this we immediately get our first example where [@Topo-Dynamics] is used in the second direction, with massive consequences for finite structures: the Ramsey property is preserved under first-order bi-interpretations. We will see another statement of this type (Proposition \[prop:addingConstantsPreservesRamsey\]) and more concrete applications of such statements later (in Section \[sect:minimalfunctions\], Section \[sect:interpret\], and Section \[sect:csp\]).
#### **Constraint Satisfaction.**
Our next example where infinite mathematics is a powerful tool comes from (finite) computer science. A *constraint satisfaction problem* is a computational problem where we are given a set of variables and a set of constraints on those variables, and where the task is to decide whether there is an assignment of values to the variables that satisfies all constraints. Computational problems of this type appear in many areas of computer science, for example in artificial intelligence, computer algebra, scheduling, computational linguistics, and computational biology.
As an example, consider the <span style="font-variant:small-caps;">Betweenness</span> problem. The input to this problem consists of a finite set of variables $V$, and a finite set of triples of the form $(x,y,z)$ where $x,y,z \in V$. The task is to find an ordering $<$ on $V$ such that for each of the given triples $(x,y,z)$ we have either $x<y<z$ or $z<y<x$. It is well-known that this problem is NP-complete [@Opatrny; @GareyJohnson], and that we therefore cannot expect to find a polynomial-time algorithm that solves it. In contrast, when we want to find an ordering $<$ on $V$ such that for each of the given triples $(x,y,z)$ we have $x<y$ or $x<z$, then the corresponding problem can be solved in polynomial time.
Many constraint satisfaction problems can be modeled formally as follows. Let $\Gamma$ be a structure with a finite relational signature. Then the *constraint satisfaction problem for $\Gamma$*, denoted by $\operatorname{CSP}(\Gamma)$, is the problem of deciding whether a given primitive positive sentence $\phi$ is true in $\Gamma$. By choosing $\Gamma$ appropriately, many problems in the above mentioned application areas can be expressed as $\operatorname{CSP}(\Gamma)$. The <span style="font-variant:small-caps;">Betweenness</span> problem, for instance, can be modeled as $\operatorname{CSP}(({\mathbb Q}; {\text{\it Betw}}))$ where $\mathbb Q$ are the rational numbers and ${\text{\it Betw}}= \{(x,y,z) \in {\mathbb Q}^3 \; | \; x<y<z \vee z<y<x \}$.
Note that even though the structure $\Gamma$ might be infinite, the problem $\operatorname{CSP}(\Gamma)$ is always a well-defined and *discrete* problem. Since the signature of $\Gamma$ is finite, the complexity of $\operatorname{CSP}(\Gamma)$ is independent of the representation of the relation symbols of $\Gamma$ in input instances of $\operatorname{CSP}(\Gamma)$. The task is to decide whether there *exists* an assignment to the variables of a given instance, and we do not have to exhibit such a solution. Therefore, the computational problems under consideration are finitistic and concrete even when the domain of $\Gamma$ is, say, the real numbers.
There are many reasons to formulate a discrete problem as $\operatorname{CSP}(\Gamma)$ for an infinite structure $\Gamma$. The advantages of such a formulation are most striking when $\Gamma$ can be chosen to be $\omega$-categorical. In this case, the computational complexity of $\operatorname{CSP}(\Gamma)$ is fully captured by the *polymorphism clone* of $\Gamma$; the polymorphism clone can be seen as a higher-dimensional generalization of the automorphism group of $\Gamma$. When studying polymorphism clones, we can apply techniques from universal algebra, and, as we will see here, from Ramsey theory to obtain results about the computational complexity of $\operatorname{CSP}(\Gamma)$.
#### **Contributions and Outline.**
In this article we give a survey presentation of a technique how to apply Ramsey theory when studying automorphism groups, endomorphism monoids, and polymorphism clones of countably infinite structures with a first-order definition in an ordered homogeneous Ramsey structure in a finite language – such structures are always $\omega$-categorical. We present applications of this technique in two fields. Let $\Delta$ be a countable structure with a first-order definition in an ordered homogeneous Ramsey structure in a finite language. In model theory, our technique can be used to classify the set of all structures $\Gamma$ that are first-order definable in $\Delta$. In constraint satisfaction, it can be used to obtain a complete complexity classification for the class of all problems CSP$(\Gamma)$ where $\Gamma$ is first-order definable in $\Delta$. We demonstrate this for $\Delta = ({\mathbb Q}; <)$, and for $\Delta = (V;E)$, the countably infinite random graph.
Reducts {#sect:reducts}
=======
One way to classify relational structures on a fixed domain is by identifying two structures when they *define* one another. The term “define” will classically stand for “first-order define”, i.e., a structure $\Gamma_1$ has a first-order definition in a structure $\Gamma_2$ on the same domain iff all relations of $\Gamma_1$ can be defined by a first-order formula over $\Gamma_2$. When $\Gamma_1$ has a first-order definition in $\Gamma_2$ and vice-versa, then two structures are considered equivalent *up to first-order interdefinability*.
Depending on the application, other notions of definability might be suitable; such notions include syntactic restrictions of first-order definability. In this paper, besides first-order definability, we will consider the notions of *existential positive definability* and *primitive positive definability*; in particular, we will explain the importance of the latter notion in theoretical computer science in Section \[sect:csp\].
The structures which we consider in this article will all be countably infinite, and we will henceforth assume this property without further mentioning it. A structure is called *$\omega$-categorical* if all countable models of its first-order theory are isomorphic. We are interested in the situation where all structures to be classified are *reducts* of a single countable $\omega$-categorical structure in the following sense (which differs from the standard definition of a reduct and morally follows e.g. [@RandomReducts]).
Let $\Delta$ be a structure. A *reduct* of $\Delta$ is a structure with the same domain as $\Delta$ all of whose relations can be defined by a first-order formula in $\Delta$.
When all structures under consideration are reducts of a countably infinite base structure $\Delta$ which is $\omega$-categorical, then there are natural ways of obtaining classifications up to first-order, existential positive, or primitive positive interdefinability by means of certain sets of functions. In this section, we explain these ways, and give some examples of classifications that have been obtained in the past. In the following sections, we then observe that these results have actually been obtained in a more specific context than $\omega$-categoricity, namely, where the structures are reducts of an *ordered Ramsey* structure $\Delta$ which has a finite relational signature and which is *homogeneous* in the sense that every isomorphism between finite induced substructures of $\Delta$ can be extended to an automorphism of $\Delta$. We further develop a general framework for proving such results in this context.
We start with first-order definability. Consider the assignment that sends every structure $\Gamma$ with domain $D$ to its automorphism group $\operatorname{Aut}(\Gamma)$. Automorphism groups are closed sets in the convergence topology of all permutations on $D$, and conversely, every closed permutation group on $D$ is the automorphism group of a relational structure with domain $D$. The closed permutation groups on $D$ form a complete lattice, where the meet of a set of groups is given by their intersection. Similarly, the set of those relational structures on $D$ which are *first-order closed*, i.e., which contain all relations which they define by a first-order formula, forms a lattice, where the meet of a set $S$ of such structures is the structure which has those relations that are contained in all structures in $S$. Now when $\Gamma$ is a countable $\omega$-categorical structure, then it follows from the proof of the theorem of Ryll-Nardzewki (see [@HodgesLong]) that its automorphism group $\operatorname{Aut}(\Gamma)$ still has the first-order theory of $\Gamma$ encoded in it. And indeed we can, up to first-order interdefinability, recover $\Gamma$ from its automorphism group as follows: For a set ${\mathcal F}$ of finitary functions on $D$, let $\operatorname{Inv}({\mathcal F})$ be the structure on $D$ which has those relations $R$ which are *invariant under ${\mathcal F}$*, i.e., those relations that contain $f(r_1,\ldots,r_n)$ (calculated componentwise) whenever $f\in {\mathcal F}$ and $r_1,\ldots,r_n\in R$.
\[thm:groups-fo\] Let $\Delta$ be $\omega$-categorical. Then the mapping $\Gamma \mapsto \operatorname{Aut}(\Gamma)$ is an antiisomorphism between the lattice of first-order closed reducts of $\Delta$ and the lattice of closed permutation groups containing $\operatorname{Aut}(\Delta)$. The inverse mapping is given by ${\mathcal G}\mapsto\operatorname{Inv}({\mathcal G})$.
This connection between closed permutation groups and first-order definability has been exploited several times in the past in order to obtain complete classifications of reducts of $\omega$-categorical structures. For example, let $\Delta$ be the order of the rational numbers – we write $\Delta={(\mathbb{Q};<)}$. Then it has been shown in [@Cameron5] that there are exactly five reducts of $\Delta$, up to first-order interdefinability, which we will define in the following.
On the permutation side, let $\leftrightarrow$ be the function that sends every $x\in \mathbb{Q}$ to $-x$. For our purposes, we can equivalently choose $\leftrightarrow$ to be any permutation that inverts the order $<$ on $\mathbb{Q}$. For any fixed irrational real number $\alpha$, let $\circlearrowright$ be any permutation on $\mathbb{Q}$ with the property that $x< y< \alpha < u< v$ implies ${\circlearrowright}{(u)}< {\circlearrowright}(v)<{\circlearrowright}(x)<{\circlearrowright}(y)$, for all $x,y,u,v\in \mathbb{Q}$. We will consider closed groups *generated* by these permutations: For a set of permutations ${\mathcal F}$ and a closed permutation group ${\mathcal G}$, we say that ${\mathcal F}$ generates ${\mathcal G}$ iff ${\mathcal G}$ is the smallest closed group containing ${\mathcal F}$.
On the relational side, for $x_1,\ldots,x_n\in \mathbb{Q}$ write $\overrightarrow{x_1\ldots x_n}$ when $x_1<\ldots<x_n$. Then we define a ternary relation ${\text{\it Betw}}$ on $\mathbb{Q}$ by ${\text{\it Betw}}:=\{(x,y,z)\in \mathbb{Q}^{3}\;|\; \overrightarrow{xyz}\vee \overrightarrow{zyx}\}$. Define another ternary relation ${\text{\it Cycl}}$ by ${\text{\it Cycl}}:=\{(x,y,z) \in\mathbb{Q}^{3}\;|\; \overrightarrow{xyz}\vee \overrightarrow{yzx}\vee \overrightarrow{zxy}\}$. Finally, define a $4$-ary relation ${\text{\it Sep}}$ by $$\begin{aligned}
\{(x_1,y_1,x_2,y_2) \in \mathbb{Q}^{4} \; | \; &
\overrightarrow{x_1x_2y_1y_2} \vee \overrightarrow{x_1y_2y_1x_2} \vee
\overrightarrow{y_1x_2x_1y_2} \vee \overrightarrow{y_1y_2x_1x_2} \\
\vee & \overrightarrow{x_2x_1y_2y_1} \vee
\overrightarrow{x_2y_1y_2x_1} \vee \overrightarrow{y_2x_1x_2y_1} \vee \overrightarrow{y_2y_1x_2x_1}\} \,.
\end{aligned}$$
\[thm:cameron5\] Let $\Gamma$ be a reduct of ${(\mathbb{Q};<)}$. Then exactly one of the following holds:
- $\Gamma$ is first-order interdefinable with ${(\mathbb{Q};<)}$; equivalently, $\operatorname{Aut}(\Gamma)={\operatorname{Aut}({(\mathbb{Q};<)})}$.
- $\Gamma$ is first-order interdefinable with $(\mathbb{Q};{\text{\it Betw}})$; equivalently, $\operatorname{Aut}(\Gamma)$ equals the closed group generated by $\operatorname{Aut}({(\mathbb{Q};<)})$ and $\leftrightarrow$.
- $\Gamma$ is first-order interdefinable with $(\mathbb{Q};{\text{\it Cycl}})$; equivalently, $\operatorname{Aut}(\Gamma)$ equals the closed group generated by $\operatorname{Aut}({(\mathbb{Q};<)})$ and $\circlearrowright$.
- $\Gamma$ is first-order interdefinable with $(\mathbb{Q};{\text{\it Sep}})$; equivalently, $\operatorname{Aut}(\Gamma)$ equals the closed group generated by $\operatorname{Aut}({(\mathbb{Q};<)})$ and $\{\leftrightarrow, \circlearrowright\}$.
- $\Gamma$ is first-order interdefinable with $(\mathbb{Q};=)$; equivalently, $\operatorname{Aut}(\Gamma)$ equals the group of all permutations on $\mathbb{Q}$.
Another instance of the application of Theorem \[thm:groups-fo\] in the classification of reducts up to first-order interdefinability has been provided by Thomas [@RandomReducts]. Let $G=(V;E)$ be the random graph, i.e., the up to isomorphism unique countably infinite graph which is homogeneous and which contains all finite graphs as induced subgraphs. It turns out that up to first-order interdefinability, $G$ has precisely five reducts, too.
On the permutation side, observe that the graph $\bar{G}$ obtained by making two distinct vertices $x,y\in V$ adjacent iff they are not adjacent in $G$ is isomorphic to $G$; let $-$ be any permutation on $V$ witnessing this isomorphism. Moreover, for any fixed vertex $0\in V$, the graph obtained by making all vertices which are adjacent with $0$ non-adjacent with $0$, and all vertices different from $0$ and non-adjacent with $0$ adjacent with $0$, is isomorphic to $G$. Let $\operatorname{sw}$ be any permutation on $V$ witnessing this fact.
On the relational side, define for all $k\geq 2$ a $k$-ary relation $R^{(k)}$ on $V$ by $$\begin{aligned}
R^{(k)}:=\{(x_1,\ldots,x_k)\;|\;&\text{ all } x_i\text{ are distinct},\\ &\text{ and the number of edges on }\{x_1,\ldots,x_k\}\text{ is odd}\}.\end{aligned}$$
\[thm:thomas5\] Let $\Gamma$ be a reduct of the random graph $G=(V;E)$. Then exactly one of the following holds:
- $\Gamma$ is first-order interdefinable with $G$; equivalently, $\operatorname{Aut}(\Gamma)=\operatorname{Aut}(G)$.
- $\Gamma$ is first-order interdefinable with $(V;R^{(3)})$; equivalently, $\operatorname{Aut}(\Gamma)$ equals the closed group generated by $\operatorname{Aut}(G)$ and $\operatorname{sw}$.
- $\Gamma$ is first-order interdefinable with $(V;R^{(4)})$; equivalently, $\operatorname{Aut}(\Gamma)$ equals the closed group generated by $\operatorname{Aut}(G)$ and $-$.
- $\Gamma$ is first-order interdefinable with $(V;R^{(5)})$; equivalently, $\operatorname{Aut}(\Gamma)$ equals the closed group generated by $\operatorname{Aut}(G)$ and $\{\operatorname{sw}, -\}$.
- $\Gamma$ is first-order interdefinable with $(V;=)$; equivalently, $\operatorname{Aut}(\Gamma)$ equals the group of all permuations on $V$.
In a similar fashion, the reducts of several prominent $\omega$-categorical structures $\Delta$ have been classified up to first-order interdefinability by finding all closed supergroups of $\operatorname{Aut}(\Delta)$. Examples are:
- The countable homogeneous $K_n$-free graph, i.e., the unique countable homogeneous graph which contains precisely those finite graphs which do not contain a clique of size $n$ as induced subgraphs, has 2 reducts up to first-order interdefinability (Thomas [@RandomReducts]), for all $n\geq 3$.
- The countable homogeneous $k$-hypergraph has $2^k+1$ reducts up to first-order interdefinability (Thomas [@Thomas96]), for all $k\geq 2$.
- The structure $(\mathbb{Q};<,0)$, i.e., the order of the rationals which in addition “knows” one of its points, has 116 reducts up to first-order interdefinability (Junker and Ziegler [@JunkerZiegler]).
All these examples have in common that the structures have a high degree of symmetry in the sense that they are homogeneous in a finite language – intuitively, one would expect the automorphism group of such a structure to be rather large. And indeed, Thomas conjectured in [@RandomReducts]:
\[conj:thomas\] Let $\Delta$ be a countable relational structure which is homogeneous in a finite language. Then $\Delta$ has finitely many reducts up to first-order interdefinability.
It turns out that all the examples above are not only homogeneous in a finite language; in fact, they all have a first-order definition in (in other words: are themselves reducts of) an *ordered Ramsey structure* which is homogeneous in a finite language. Functions on such structures, in particular automorphisms of reducts, can be analyzed by the means of Ramsey theory, and we will outline a general method for classifying the reducts of such structures in Sections \[sect:ramseyclasses\] to \[sect:decidability\].
We now turn to analogs of Theorem \[thm:groups-fo\] for syntactic restrictions of first-order logic. A first-order formula is called *existential* iff it is of the form $\exists x_1\ldots\exists x_n.\ \phi$, where $\phi$ is quantifier-free. It is called *existential positive* iff it is existential and does not contain any negations. Now observe that similarly to permutation groups, the *endomorphism monoid* $\operatorname{End}(\Delta)$ of a relational structure $\Delta$ with domain $D$ is always closed in the pointwise convergence topology on the space of all functions from $D$ to $D$, and that every closed transformation monoid ${\mathcal M}$ acting on $D$ is the endomorphism monoid of the structure $\operatorname{Inv}({\mathcal M})$, i.e., the structure with domain $D$ which contains those relations which are invariant under all functions in ${\mathcal M}$. Note also that the set of closed transformation monoids on $D$, ordered by inclusion, forms a complete lattice, and that likewise the set of all existential positive closed structures forms a complete lattice. The analog to Theorem \[thm:groups-fo\] for existential positive definability is an easy consequence of the homomorphism preservation theorem (see [@HodgesLong]) and goes like this:
\[thm:monoids-expos\] Let $\Delta$ be $\omega$-categorical. Then the mapping $\Gamma\mapsto \operatorname{End}(\Gamma)$ is an antiisomorphism between the lattice of existential positive closed reducts of $\Delta$ and the lattice of closed transformation monoids containing $\operatorname{Aut}(\Delta)$. The inverse mapping is given by ${\mathcal M}\mapsto\operatorname{Inv}({\mathcal M})$.
All the closed monoids containing the group of all permutations on a countably infinite set $D$ (which equals the automorphism group of the empty structure $(D;=)$) have been determined in [@BodChenPinsker], and their number is countably infinite. Therefore, every structure has infinitely many reducts up to existential positive interdefinability. In general, it will be impossible to determine all of them, but sometimes it is already useful to determine certain closed monoids, as in the following theorem about endomorphism monoids of reducts of the random graph from [@RandomMinOps]. We need the following definitions. Since the random graph $G=(V;E)$ contains all countable graphs, it contains an infinite clique. Let $e_E$ be any injective function from $V$ to $V$ whose image induces such a clique in $G$. Similarly, let $e_N$ be any injection from $V$ to $V$ whose image induces an independent set in $G$.
\[thm:randomMinimalMonoids\] Let $\Gamma$ be a reduct of the random graph $G=(V;E)$. Then at least one of the following holds.
- $\operatorname{End}(\Gamma)$ contains a constant operation.
- $\operatorname{End}(\Gamma)$ contains $e_E$.
- $\operatorname{End}(\Gamma)$ contains $e_N$.
- $\operatorname{Aut}(\Gamma)$ is a dense subset of $\operatorname{End}(\Gamma)$ (equipped with the topology of pointwise convergence).
Theorem \[thm:randomMinimalMonoids\] states that for reducts $\Gamma$ of the random graph, either $\operatorname{End}(\Gamma)$ contains a function that destroys all structure of the random graph, or it contains basically no functions except the automorphisms. This has the following non-trivial consequence. A theory $T$ is called *model-complete* iff every embedding between models of $T$ is elementary, i.e., preserves all first-order formulas. A structure is said to be model-complete iff its first-order theory is model-complete.
\[cor:mc\] All reducts of the random graph are model-complete.
It is not hard to see (cf. [@RandomMinOps]) that an $\omega$-categorical structure $\Gamma$ is model-complete if and only if $\operatorname{Aut}(\Gamma)$ is dense in the monoid of self-embeddings of $\Gamma$. Now let $\Gamma$ be a reduct of $G$, and let ${\mathcal M}$ be the closed monoid of self-embeddings of $\Gamma$; we will show that $\operatorname{Aut}(\Gamma)$ is dense in ${\mathcal M}$. We apply Theorem \[thm:randomMinimalMonoids\] to ${\mathcal M}$ (which, as a closed monoid containing $\operatorname{Aut}(G)$, is also an endomorphism monoid of a reduct $\Gamma'$ of $G$). Clearly, $\Gamma'$ and $\Gamma$ have the same automorphisms, namely those permutations in ${\mathcal M}$ whose inverse is also in ${\mathcal M}$. Therefore we are done if the last case of the theorem holds. Note that ${\mathcal M}$ cannot contain a constant operation as all its operations are injective. So suppose that ${\mathcal M}$ contains $e_N$ – the argument for $e_E$ is analogous. Let $R$ be any relation of $\Gamma$, and $\phi_R$ be its defining quantifier-free formula; $\phi_R$ exists since $G$ has quantifier-elimination, i.e., every first-order formula over $G$ is equivalent to a quantifier-free formula. Let $\psi_R$ be the formula obtained by replacing all occurrences of $E$ by *false*; so $\psi_R$ is a formula over the empty language. Then a tuple $a$ satisfies $\phi_R$ in $G$ iff $e_N(a)$ satisfies $\phi_R$ in $G$ (because $e_N$ is an embedding) iff $e_N(a)$ satisfies $\psi_R$ in $G$ (as there are no edges on $e_N(a)$) iff $e_N(a)$ satisfies $\psi_R$ in the substructure induced by $e_N[V]$ (since $\psi_R$ does not contain any quantifiers). Thus, $\Gamma$ is isomorphic to the structure on $e_N[V]$ which has the relations defined by the formulas $\psi_R$; hence, $\Gamma$ is isomorphic to a structure with a first-order definition over the empty language. This structure has, of course, all injections as self-embeddings, and all permutations as automorphisms, and hence is model-complete; thus, the same is true for $\Gamma$.
It follows from [@tcsps-journal Proposition 19] that all reducts of the linear order of the rationals $(\mathbb Q;<)$ are model-complete as well. This is remarkable, since similar structures do not have this property – for example, $(\mathbb Q;<,0)$ is first-order interdefinable with the structure $(\mathbb Q;<,[0,\infty))$ which is not model-complete.
We now turn to an even finer way of distinguishing reducts of an $\omega$-categorical structure, namely up to *primitive positive interdefinability*. This is of importance in connection with the constraint satisfaction problem from the introduction, as we will describe in more detail in Section \[sect:csp\]. We call a formula *primitive positive* iff it is existential positive and does not contain disjunctions. A *clone* on domain $D$ is a set of finitary operations on $D$ which contains all projections (i.e., functions of the form $(x_1,\ldots,x_n)\mapsto x_i$) and which is closed under composition. A clone ${\mathcal C}$ is *closed* (also called *locally closed* or *local* in the literature) iff for each $n\geq 1$, the set of $n$-ary functions in ${\mathcal C}$ is a closed subset of the space $D^{D^n}$, where $D$ is taken to be discrete. The closed clones on $D$ form a complete lattice with respect to inclusion – the structure of this lattice has been studied in the universal algebra literature (see [@GoldsternPinsker], [@Pin-morelocal]). Similarly, the set of relational structures with domain $D$ which are *primitive positive closed*, i.e., which contain all relations which they define by primitive positive formulas, forms a complete lattice. For a structure $\Gamma$, we define $\operatorname{Pol}(\Gamma)$ to consist of all finitary operations on the domain of $\Gamma$ which preserve all relations of $\Gamma$, i.e., an $n$-ary function $f$ is an element of $\operatorname{Pol}(\Gamma)$ iff for all relations $R$ of $\Gamma$ and all tuples $r_1,\ldots,r_n\in R$ the tuple $f(r_1,\ldots,r_n)$ is an element of $R$. It is easy to see that $\operatorname{Pol}(\Gamma)$ is always a closed clone. Observe also that $\operatorname{Pol}(\Gamma)$ is a generalization of $\operatorname{End}(\Gamma)$ to higher (finite) arities.
\[thm:clones-pp\] Let $\Delta$ be $\omega$-categorical. Then the mapping $\Gamma\mapsto \operatorname{Pol}(\Gamma)$ is an antiisomorphism between the lattice of primitive positive closed reducts of $\Delta$ and the lattice of closed clones containing $\operatorname{Aut}(\Delta)$. The inverse mapping is given by ${\mathcal C}\mapsto\operatorname{Inv}({\mathcal C})$.
It turns out that even for the empty structure $(X;=)$, the lattice of primitive positive closed reducts is probably too complicated to be completely described – the lattice has been thoroughly investigated in [@BodChenPinsker].
The structure $(X;=)$ (where $X$ is countably infinite), and therefore all countably infinite structures, have $2^{\aleph_0}$ reducts up to primitive positive interdefinability.
Fortunately, it is sometimes sufficient in applications to understand only parts of this lattice. We will see examples of this in Section \[sect:csp\].
Ramsey Classes {#sect:ramseyclasses}
==============
While Theorems \[thm:groups-fo\], \[thm:monoids-expos\] and \[thm:clones-pp\] provide a theoretical method for determining reducts of an $\omega$-categorical structure $\Delta$ by transforming them into sets of functions on $\Delta$, understanding these infinite objects could turn out difficult without further tools for handling them. We will now focus on structures which have the additional property that they are reducts of an *ordered Ramsey structure* that is homogeneous in a finite relational signature; such structures are $\omega$-categorical since homogeneous structures in a finite language are $\omega$-categorical and since reducts of $\omega$-categorical structures are $\omega$-categorical. This is less restrictive than it might appear at first sight: we remark that it could be the case that all homogeneous structures with a finite relational signature are reducts of ordered homogeneous Ramsey structures with a finite relational signature (that is, we do not know of a counterexample). It turns out that in this context, certain infinite functions can be represented by finite ones, making classification projects more feasible.
\[defn:orderd\] A structure is called *ordered* iff it has a total order among its relations.
Let $\tau$ a relational signature. For $\tau$-structures ${\mathcal S},{\mathcal H},{\mathcal P}$ and an integer $k \geq 1$, we write ${\mathcal S}\rightarrow ({\mathcal H})^{\mathcal P}_k$ iff for every $k$-coloring $\chi$ of the copies of ${\mathcal P}$ in ${\mathcal S}$ there exists a copy ${\mathcal H}'$ of ${\mathcal H}$ in ${\mathcal S}$ such that all copies of ${\mathcal P}$ in ${\mathcal H}'$ have the same color under $\chi$.
A class ${\mathcal C}$ of finite $\tau$-structures which is closed under isomorphisms, induced substructures, and with the joint embedding property (see [@HodgesLong]) is called a *Ramsey class* iff it is closed under substructures and for all $k\geq 1$ and all ${\mathcal H}, {\mathcal P}\in {\mathcal C}$ there exists ${\mathcal S}$ in ${\mathcal C}$ such that ${\mathcal S}\rightarrow ({\mathcal H})^{\mathcal P}_k$.
A relational structure is called *Ramsey* iff its *age*, i.e., the set of finite structures isomorphic to a finite induced substructure, is a Ramsey class.
Examples of Ramsey structures are the dense linear order $(\mathbb{Q};<)$ and the *ordered random graph* $(V;E,<)$, i.e., the Fraïssé limit of the class of finite ordered graphs. We remark that the random graph itself is not Ramsey, but since it is a reduct of the ordered random graph, the methods we are about to expose apply as well.
We will now see that one can find regular patterns in the behavior of any function acting on an ordered Ramsey structure which is $\omega$-categorical.
Let $\Gamma$ be a structure. The *type* $\operatorname{tp}(a)$ of an $n$-tuple $a \in \Gamma$ is the set of first-order formulas with free variables $x_1,\dots,x_n$ that hold for $a$ in $\Gamma$.
We recall the classical theorem of Ryll-Nardzewski about the number of types in $\omega$-categorical structures.
\[thm:RN\] The following are equivalent for a countable structure $\Gamma$ in a countable language.
- $\Gamma$ is $\omega$-categorical, i.e., any countable model of the theory of $\Gamma$ is isomorphic to $\Gamma$.
- $\Gamma$ has for all $n\geq 1$ only finitely many different types of $n$-tuples.
We also mention that moreover, as a well-known consequence of the proof of this theorem, two tuples in a countable $\omega$-categorical structure have the same type if and only if there is an automorphism of $\Gamma$ which sends one tuple to the other.
A *type condition* between two structures $\Gamma_1,\Gamma_2$ is a pair $(t_1,t_2)$, where each $t_i$ is a type of an $n$-tuple in $\Gamma_i$. A function $f:\Gamma_1{\rightarrow}\Gamma_2$ *satisfies* a type condition $(t_1,t_2)$ if for all $n$-tuples $(a_1,\ldots,a_n)$ of type $t_1$, the $n$-tuple $(f(a_1),\ldots,f(a_n))$ is of type $t_2$.
A *behavior* is a set of type conditions between two structures. A function *has behavior $B$* if it satisfies all the type conditions of the behavior $B$. A behavior $B$ is called *complete* iff for all types $t_1$ of tuples in $\Gamma_1$ there is a type $t_2$ of a tuple in $\Gamma_2$ such that $(t_1,t_2)\in B$.
A function $f: \Gamma_1 {\rightarrow}\Gamma_2$ is *canonical* iff it has a complete behavior. If $F\subseteq \Gamma_1$, then we say that $f$ is *canonical on $F$* if its restriction to $F$ is canonical.
Observe that the function $\leftrightarrow$ of Theorem \[thm:cameron5\] is canonical for the structure $(\mathbb{Q};<)$. The function $\circlearrowright$ is not, but it is canonical on each of the intervals $(-\infty,\alpha)$ and $(\alpha,\infty)$. For the random graph, the function $-$ of Theorem \[thm:thomas5\] is canonical, while $\operatorname{sw}$ is canonical on $V\setminus\{0\}$. Also, $\operatorname{sw}$ is canonical as a function from $(V;E,0)$ to $(V;E)$, where $(V;E,0)$ denotes the structure obtained from $(V;E)$ by adding a new constant symbol for the element $0$ by which we defined the function $\operatorname{sw}$. Moreover, the constant function and $e_E, e_N$ of Theorem \[thm:randomMinimalMonoids\] are canonical on $(V;E)$. We will now show that it is no coincidence that canonical functions are that ubiquitous.
Let $\Delta$ be a structure. A property $P$ *holds for arbitrarily large finite substructures of $\Delta$* iff for all finite substructures $F\subseteq \Delta$ there is a copy of $F$ in $\Delta$ for which $P$ holds.
The following observation is just an easy application of the definition of a Ramsey class, but crucial in understanding functions on ordered Ramsey structures.
\[lem:canonicalOnArbitrarilyLargeFinite\] Let $\Delta$ be ordered Ramsey and $\omega$-categorical, and let $f: \Delta{\rightarrow}\Delta$. Then $f$ is canonical on arbitrarily large finite substructures.
The proof goes along the following lines: Let $F$ be any finite substructure of $\Delta$. Then the function $f$ induces a mapping from the tuples in $\Delta$ to the set of types in $\Delta$ (each tuple is sent to the type of its image under $f$). If we restrict this mapping to tuples of length at most the size of $F$, then since $\Delta$ is $\omega$-categorical, the range of this restriction is finite by Theorem \[thm:RN\], and thus is a $k$-coloring of tuples for some finite $k$. Now apply the Ramsey property once for every type of tuple that occurs in $F$ – see [@BodPinTsa] for details. We remark that this lemma would be false if one dropped the order assumption, which implies that coloring induced substructures and coloring tuples in $\Delta$ are one and the same thing.
The motivation for working with ordered Ramsey structures is the rough idea that all “important” functions can be assumed to be canonical. While this is simply false when stated boldly like this, it is still true for some functions when the idea is further refined, as we will show in the following. Observe that if $\Delta$ is $\omega$-categorical, then for each $n\geq 1$ there are only finitely many possible type conditions for $n$-types over $\Delta$ (Theorem \[thm:RN\]). Suppose that $\Delta$ has in addition a finite language and *quantifier elimination*, i.e., every first-order formula in the language of $\Delta$ is equivalent to a quantifier-free formula over $\Delta$; this follows in particular from homogeneity in a finite language. Then, if $n(\Delta)$ is the largest arity of its relations, then a function $f:\Delta{\rightarrow}\Delta$ is canonical iff for every type $t_1$ of an $n(\Delta)$-tuple in $\Delta$ there is a type $t_2$ in $\Delta$ such that $f$ satisfies the type condition $(t_1,t_2)$. In other words, the complete behavior of $f$ is already determined by its behavior on $n(\Delta)$-types. Hence, a canonical function on $\Delta$ is essentially a function on the $n(\Delta)$-types of $\Delta$ – a finite object.
Let $f,g: \Delta{\rightarrow}\Delta$. We say that $f$ *generates* $g$ over $\Delta$ iff $g$ is contained in the smallest closed monoid containing $f$ and $\operatorname{Aut}(\Delta)$. Equivalently, for every finite subset $F$ of $\Delta$, there exists a term $\beta\circ f\circ\alpha_1\circ f\circ\alpha_2\circ\cdots\circ f\circ \alpha_n$, where $\beta,\alpha_i\in\operatorname{Aut}(\Delta)$, which agrees with $g$ on $F$.
\[lem:generatesCanonical\] Let $\Delta$ be a structure in a finite language which is ordered, Ramsey, and homogeneous. Let $f: \Delta{\rightarrow}\Delta$. Then $f$ generates a canonical function $g:\Delta{\rightarrow}\Delta$.
Let $(F_i)_{i\in\omega}$ be an increasing sequence of finite substructures of $\Delta$ such that $\bigcup_{i\in\omega} F_i=\Delta$. By Lemma \[lem:canonicalOnArbitrarilyLargeFinite\], for each $i\in\omega$ we find a copy $F_i'$ of $F_i$ in $\Delta$ on which $f$ is canonical. Since there are only finitely many possibilities of canonical behavior, one behavior occurs an infinite number of times; thus, by thinning out the sequence, we may assume that the behavior is the same on all $F_i'$. By the homogeneity of $\Delta$, there exist automorphisms $\alpha_i$ of $\Delta$ sending $F_i$ to $F_i'$, for all $i\in\omega$. Also, since the behavior on all the $F_i'$ is the same, we can inductively pick automorphisms $\beta_i$ of $\Delta$ such that $\beta_{i+1}\circ f\circ\alpha_{i+1}$ agrees with $\beta_i\circ f\circ\alpha_i$ on $F_i$, for all $i\in\omega$. The union over the functions $\beta_i\circ f\circ \alpha_i: F_i{\rightarrow}\Delta$ is a canonical function on $\Delta$.
The identity function $\operatorname{id}: \Delta{\rightarrow}\Delta$ is generated by $f$ and is canonical.
The problem with the preceding lemma is the second proof, which makes it trivial. What we really want is that $f$ generates a canonical function $g$ which represents $f$ in a certain sense – it should be possible to retain specific properties of $f$ when passing to the canonical functions. For example, we could wish that if $f$ violates a certain relation, then so does $g$; or, if $f$ is not an automorphism of $\Delta$, we will look for a canonical function $g$ which is not an automorphism of $\Delta$ either.
We are now going to refine our method, and fix constants $c_1,\dots,c_n$ such that $f \notin \operatorname{Aut}(\Delta)$ is witnessed on $\{c_1,\dots,c_n\}$. We then consider $f$ as a function from $(\Delta,c_1,\dots,c_n)$ to $\Delta$, where $(\Delta,c_1,\dots,c_n)$ denotes the expansion of $\Delta$ by the constants $c_1,\dots,c_n$. It turns out that $f$ is canonical on arbitrarily large substructures of $(\Delta,c_1,\dots,c_n)$, and that it generates a canonical function $g: (\Delta,c_1,\dots,c_n) \rightarrow \Delta$ which agrees with $f$ on $c_1,\dots,c_n$; in particular, $g$ is not an automorphism of $\Delta$, and the problem of triviality in Proposition 20 no longer occurs. In order to do this, we must assure that $(\Delta,c_1,\ldots,c_n)$ still has the Ramsey property. This leads us into topological dynamics.
Topological Dynamics {#sect:topologicaldynamics}
====================
We have seen in the previous section that our approach crucially relies on the fact that when an ordered homogeneous Ramsey structure is expanded by finitely many constants, the expansion is again Ramsey (it is clear that the expansion is again ordered and homogeneous). To prove this, we use a characterization in topological dynamics of those ordered homogeneous structures which are Ramsey.
Recall that a *topological group* is an (abstract) group $G$ together with a topology on the elements of $G$ such that $(x,y) \mapsto xy^{-1}$ is continuous from $G^2$ to $G$. In other words, we require that the binary group operation and the inverse function are continuous.
A topological group is *extremely amenable* iff any continuous action of the group on a compact Hausdorff space has a fixed point.
Kechris, Pestov and Todorcevic have characterized the Ramsey property of the age of an ordered homogeneous structure by means of extreme amenability in the following theorem.
\[thm:KPT\] Let $\Delta$ be an ordered homogeneous relational structure. Then the age of $\Delta$ has the Ramsey property iff $\operatorname{Aut}(\Delta)$ is extremely amenable.
This theorem can be applied to provide a short and elegant proof of the following.
\[prop:addingConstantsPreservesRamsey\] Let $\Delta$ be ordered, Ramsey, and homogeneous, and let $c_1,\ldots,c_n\in \Delta$. Then $(\Delta,c_1,\ldots,c_n)$ is Ramsey as well.
When $\Delta$ is ordered, Ramsey, and homogeneous, then $\operatorname{Aut}(\Delta)$ is extremely amenable. Note that the automorphism group of $(\Delta,c_1,\ldots,c_n)$ is an open subgroup of $\operatorname{Aut}(\Delta)$. The proposition thus follows directly from the following fact – confer [@BodPinTsa].
Let $G$ be an extremely amenable group, and let $H$ be an open subgroup of $G$. Then $H$ is extremely amenable.
Minimal Functions {#sect:minimalfunctions}
=================
The results of the preceding section provide a tool for “climbing up” the lattice of closed monoids containing the automorphism group of an ordered Ramsey structure which is homogeneous and has a finite language.
\[defn:minimalClone\] Let ${\mathcal C}, {\mathcal D}$ be closed clones. Then ${\mathcal D}$ is called *minimal above ${\mathcal C}$* iff ${\mathcal D}\supseteq{\mathcal C}$ and there are no closed clones between ${\mathcal C}$ and ${\mathcal D}$.
Observe that transformation monoids can be identified with those clones which have the property that all their functions depend on only one variable. Hence, Definition \[defn:minimalClone\] also provides us with a notion of a minimal closed monoid above another closed monoid.
It follows from Theorem \[thm:clones-pp\] and Zorn’s Lemma that if $\Delta$ is an $\omega$-categorical structure in a finite language, then every closed clone containing $\operatorname{Pol}(\Delta)$ contains a minimal closed clone above $\operatorname{Pol}(\Delta)$. Similarly, as a consequence of Theorem \[thm:monoids-expos\], every closed monoid containing $\operatorname{End}(\Delta)$ contains a minimal closed monoid.
For closed permutation groups, minimality can be defined analogously. Then Theorem \[thm:groups-fo\] implies that for $\omega$-categorical structures $\Delta$ in a finite language, every closed permutation group containing $\operatorname{Aut}(\Delta)$ contains a minimal closed permutation group above $\operatorname{Aut}(\Delta)$.
Clearly, if a closed clone ${\mathcal D}$ is minimal above ${\mathcal C}$, then any function $f\in{\mathcal D}\setminus{\mathcal C}$ generates ${\mathcal D}$ with ${\mathcal C}$ (i.e., ${\mathcal D}$ is the smallest closed clone containing $f$ and ${\mathcal C}$) – similar statements hold for monoids and groups. In the case of clones and monoids and in the setting of reducts of ordered Ramsey structures which are homogeneous in a finite language, we can standardize such generating functions. This is the contents of the coming subsections.
Minimal unary functions {#subsect:minimalUnary}
-----------------------
Adapting the proof of Lemma \[lem:generatesCanonical\], with the use of the Proposition \[prop:addingConstantsPreservesRamsey\], one can show the following.
\[lem:generatesCanonicalWithConstants\] Let $\Delta$ be ordered, Ramsey, homogeneous, and of finite language. Let $f: \Delta{\rightarrow}\Delta$, and let $c_1,\ldots,c_n\in \Delta$. Then $f$ together with $\operatorname{Aut}(\Delta)$ generates a function which agrees with $f$ on $\{c_1,\ldots,c_n\}$ and which is canonical as a function from $(\Delta,c_1,\ldots,c_n)$ to $\Delta$.
Let $\Gamma$ be a finite language reduct of a structure $\Delta$ which is ordered, Ramsey, homogeneous, and of finite language, and let ${\mathcal N}$ be a minimal closed monoid containing $\operatorname{End}(\Gamma)$. Then, setting $n(\Gamma)$ to be the largest arity of the relations of $\Gamma$, we can pick constants $c_1,\ldots,c_{n(\Gamma)}\in \Gamma$ and a function $f\in {\mathcal N}\setminus\operatorname{End}(\Gamma)$ such that $f{\notin}\operatorname{End}(\Gamma)$ is witnessed on $\{c_1,\ldots,c_{n(\Gamma)}\}$. By the preceding lemma, $f$ and $\operatorname{Aut}(\Delta)$ generate a function $g$ which behaves like $f$ on $\{c_1,\ldots,c_{n(\Gamma)}\}$ and which is canonical as a function from $(\Delta,c_1,\ldots,c_{n(\Gamma)})$ to $\Delta$. This function $g$, together with $\operatorname{End}(\Gamma)$, generates ${\mathcal N}$. Since there are only finitely many choices for the type of the tuple $(c_1,\ldots,c_{n(\Gamma)})$ and for each choice only finitely many behaviors of functions from $(\Delta,c_1,\ldots,c_{n(\Gamma)})$ to $\Delta$, we get the following.
\[prop:finiteMinimalReducts\] Let $\Gamma$ be a finite language reduct of a structure $\Delta$ which is ordered, Ramsey, homogeneous, and of finite language. Then the number of minimal closed monoids above $\operatorname{End}(\Gamma)$ is finite, and each such monoid is generated by $\operatorname{End}(\Gamma)$ plus a canonical function $g:(\Delta,c_1,\ldots,c_{n(\Gamma)}){\rightarrow}\Delta$, for constants $c_1,\ldots,c_{n(\Gamma)}\in\Gamma$.
Since for every relation $R$ of $\Gamma$ we can add its negation to the language, we get the following
\[cor:finiteMinimalSelfEmbeddings\] Let ${\mathcal M}$ be the monoid of self-embeddings of a finite-language structure $\Gamma$ which is a reduct of a structure $\Delta$ which is ordered, Ramsey, homogeneous, and of finite language. Then the number of minimal closed monoids above ${\mathcal M}$ is finite, and each such monoid is generated by ${\mathcal M}$ and a canonical function $g:(\Delta,c_1,\ldots,c_{n(\Gamma)}){\rightarrow}\Delta$.
The following is an example for the random graph $G=(V;E)$. Since $G$ is model-complete, its monoid of self-embeddings is just the topological closure ${\langle \operatorname{Aut}(G) \rangle}$ of $\operatorname{Aut}(G)$ in the space $V^V$. Therefore, the minimal closed monoids above the monoid of self-embeddings of $G$ are just the minimal closed monoids above ${\langle \operatorname{Aut}(G) \rangle}$.
\[thm:randomMinimalUnary\] Let $G=(V;E)$ be the random graph. The minimal closed monoids containing ${\langle \operatorname{Aut}(G) \rangle}$ are the following:
- The monoid generated by a constant operation with $\operatorname{Aut}(G)$.
- The monoid generated by $e_E$ with $\operatorname{Aut}(G)$.
- The monoid generated by $e_N$ with $\operatorname{Aut}(G)$.
- The monoid generated by $-$ with $\operatorname{Aut}(G)$.
- The monoid generated by $\operatorname{sw}$ with $\operatorname{Aut}(G)$.
Minimal higher arity functions {#subsect:MinimalHigherArity}
------------------------------
We now generalize the concepts from unary functions and monoids to higher arity functions and clones.
\[defn:typesOn Products\] Let $\Delta$ be a structure. For $1\leq i\leq m$ and a tuple $x$ in the power $\Delta^m$, we write $x_i$ for the $i$-th coordinate of $x$. The *type* of a sequence of tuples $a^1,\ldots,a^n\in \Delta^m$, denoted by $\operatorname{tp}(a^1,\ldots,a^n)$, is the cartesian product of the types of $(a^1_i,\ldots,a^n_i)$ in $\Delta$.
With this definition, the notions of *type condition*, *behavior*, *complete behavior*, and *canonical* generalize in complete analogy from functions $f:\Gamma_1{\rightarrow}\Gamma_2$ to functions $f:\Gamma_1^m{\rightarrow}\Gamma_2$, for structures $\Gamma_1, \Gamma_2$. It can be shown that for ordered structures, the Ramsey property is not lost when going to products; an example of a proof can be found in [@BodPinTsa].
\[prop:ORPL\] Let $\Delta$ be ordered and Ramsey, and let $m\geq 1$. Let moreover a number $k\geq 1$, an $n$-tuple $(a^1,\ldots,a^n) \in\Delta^m$, and finite $F_i\subseteq\Delta$ be given for $1\leq i\leq m$. Then there exist finite $S_i\subseteq\Delta$ with the property that whenever the $n$-tuples in $S_1{\times}\cdots{\times}S_m$ of type $\operatorname{tp}(a^1,\ldots,a^n)$ are colored with $k$ colors, then there are copies $F_i'$ of $F_i$ in $S_i$ such that the coloring is constant on $F_1'{\times}\cdots{\times}F_m'$.
We remark that Proposition \[prop:ORPL\] does not hold in general if $\Delta$ is not assumed to be ordered – an example for the random graph can be found in [@RandomMinOps]. Similarly to the unary case (Proposition \[prop:finiteMinimalReducts\]), one gets the following.
\[prop:canonicalMinimalClones\] Let $\Gamma$ be a finite language reduct of a structure $\Delta$ which is ordered, Ramsey, homogeneous and of finite language. Then every minimal closed clone above $\operatorname{Pol}(\Gamma)$ is generated by $\operatorname{Pol}(\Gamma)$ and a canonical function $g:(\Delta,c_1,\ldots,c_{k})^m{\rightarrow}\Delta$, where $m\geq 1$, $k\geq 0$, and $c_1,\ldots,c_{k}\in \Delta$. Moreover, $m$ only depends on the number of $n(\Gamma)$-types in $\Gamma$ (and not on the clone), and $k$ only depends on $m$ and $n(\Gamma)$, and the number of minimal closed clones above $\operatorname{Pol}(\Gamma)$ is finite.
In the case of minimal closed clones above an endomorphism monoid, the arity of the generating canonical functions can be further reduced as follows.
\[prop:canonicalMinimalClonesAboveEnd\] Let $\Gamma$ be a finite language reduct of a structure $\Delta$ which is ordered, Ramsey, homogeneous and of finite language. Then every minimal closed clone above $\operatorname{End}(\Gamma)$ is generated by $\operatorname{End}(\Gamma)$ and a canonical function $g:(\Delta,c_1,\ldots,c_{n(\Gamma)}){\rightarrow}\Delta$, or by $\operatorname{End}(\Gamma)$ and a canonical function $g:(\Delta,c_1,\ldots,c_m)^m{\rightarrow}\Delta$, where $m$ only depends on the number of $2$-types in $\Gamma$ (and not on the clone). In particular, the number of minimal closed clones above $\operatorname{End}(\Gamma)$ is finite.
Using this technique, the minimal closed clones containing the automorphism group of the random graph $G=(V;E)$ have been determined. In the following, let $f: V^2 \rightarrow V$ be a binary operation; we now define some possible behaviors for $f$. We say that $f$ is
- *of type $p_1$* iff for all $x_1,x_2,y_1,y_2 \in V$ with $x_1 \neq x_2$ and $y_1 \neq y_2$ we have $E(f(x_1,y_1),f(x_2,y_2))$ if and only if $E(x_1,x_2)$;
- *of type $\max$* iff for all $x_1,x_2,y_1,y_2 \in V$ with $x_1 \neq x_2$ and $y_1 \neq y_2$ we have $E(f(x_1,y_1),f(x_2,y_2))$ if and only if $E(x_1,x_2)$ or $E(y_1,y_2)$;
- *balanced in the first argument* iff for all $x_1,x_2,y \in V$ with $x_1 \neq x_2$ we have $E(f(x_1,y),f(x_2,y))$ if and only if $E(x_1,x_2)$;
- *balanced in the second argument* iff $(x,y) \mapsto f(y,x)$ is balanced in the first argument;
- *$E$-dominated in the first argument* iff for all $x_1,x_2,y \in V$ with $x_1 \neq x_2$ we have that $E(f(x_1,y),f(x_2,y))$;
- *$E$-dominated in the second argument* iff $(x,y) \mapsto f(y,x)$ is $E$-dominated in the first argument.
The *dual* of an operation $f(x_1,\ldots,x_n)$ on $V$ is defined by $-f(-x_1,\ldots,-x_n)$.
\[thm:minimalRandomClones\] Let $G=(V;E)$ be the random graph, and let ${\mathcal C}$ be a minimal closed clone above ${\langle \operatorname{Aut}(G) \rangle}$. Then ${\mathcal C}$ is generated by $\operatorname{Aut}(G)$ together with one of the unary functions of Theorem \[thm:randomMinimalUnary\], or by $\operatorname{Aut}(G)$ and one of the following canonical operations from $G^2$ to $G$:
- a binary injection of type $p_1$ that is balanced in both arguments;
- a binary injection of type $\max$ that is balanced in both arguments;
- a binary injection of type $\max$ that is $E$-dominated in both arguments;
- a binary injection of type $p_1$ that is $E$-dominated in both arguments;
- a binary injection of type $p_1$ that is balanced in the first and $E$-dominated in the second argument;
- the dual of one of the last four operations.
In [@BodPin-Schaefer], the technique of canonical functions was applied again to climb up further in the lattice of closed clones above $\operatorname{Aut}(G)$ – we will come back to this in Section \[sect:csp\].
Another example are the minimal closed clones containing all permutations of a countably infinite base set $X$. Observe that the set ${\mathcal S}_X$ of all permutations on $X$ is the automorphism group of the structure $(X;=)$ which has no relations.
\[thm:minimalAboveS\] The minimal closed clones containing ${\langle {\mathcal S}_X \rangle}$ on a countably infinite set $X$ are:
- The closed clone generated by ${\mathcal S}_X$ and any constant operation;
- The closed clone generated by ${\mathcal S}_X$ and any binary injection.
Observe that any constant operation and any binary injection on $X$ are canonical operations for the structure $(X;=)$.
We end this section with a last example which lists the minimal closed clones containing the self-embdeddings of the dense linear order $(\mathbb{Q};<)$. As with the random graph and the empty structure, since $(\mathbb{Q};<)$ is model-complete it follows that the monoid of self-embeddings of $(\mathbb{Q};<)$ is just the closure of $\operatorname{Aut}((\mathbb{Q};<))$ in $\mathbb{Q}^\mathbb{Q}$.
Let $\operatorname{lex}$ be a binary operation on $\mathbb{Q}$ such that $\operatorname{lex}(a,b) < \operatorname{lex}(a',b')$ iff either $a < a'$ or $a = a'$ and $b < b'$, for all $a,a',b,b'\in \mathbb{Q}$. Observe that $\operatorname{lex}$ is canonical as a function from $\mathbb{Q}^2$ to $\mathbb{Q}$. Next, let $\operatorname{pp}$ be an arbitrary binary operation on $\mathbb{Q}$ such that for all $a,a',b,b'\in \mathbb{Q}$ we have $\operatorname{pp}(a, b)\leq \operatorname{pp}(a', b')$ iff one of the following cases applies:
- $a \leq 0$ and $a \leq a'$;
- $0 < a$, $0 <a'$, and $b \leq b'$.
The name of the operation $\operatorname{pp}$ stands for “projection-projection”, since the operation behaves as a projection to the first argument for negative first argument, and a projection to the second argument for positive first argument. Observe that $\operatorname{pp}$ is canonical if we add the origin as a constant to the language. Finally, define the *dual* of an operation $f(x_1,\ldots,x_n)$ on $\mathbb{Q}$ by ${\leftrightarrow}(f({\leftrightarrow}(x_1),\ldots,{\leftrightarrow}(x_n)))$.
\[thm:minimalDLOClones\] Let $(\mathbb{Q};<)$ be the order of the rationals, and let ${\mathcal C}$ be a minimal closed clone above ${\langle \operatorname{Aut}((\mathbb{Q};<)) \rangle}$. Then ${\mathcal C}$ is generated by $\operatorname{Aut}((\mathbb{Q};<))$ together with one of the following operations:
- a constant operation;
- the operation $\leftrightarrow$;
- the operation $\circlearrowright$;
- the operation $\operatorname{lex}$;
- the operation $\operatorname{pp}$;
- the dual of $\operatorname{pp}$.
Decidability of Definability {#sect:decidability}
============================
We turn to another application of the ideas of the last sections. Consider the following computational problem for a structure $\Gamma$: Input are quantifier-free formulas $\phi_0,\ldots,\phi_n$ in the language of $\Gamma$ defining relations $R_0,\ldots,R_n$ on the domain of $\Gamma$, and the question is whether $R_0$ can be *defined* from $R_1,\ldots,R_n$. As in Section \[sect:reducts\], “defined” can stand for “first-order defined” or syntactic restrictions of this notion. We denote this computational problem by ${\operatorname{Expr}_{ep}}(\Gamma)$ and ${\operatorname{Expr}_{pp}}(\Gamma)$ if we consider existential positive and primitive positive definability, respectively.
For *finite* structures $\Gamma$ the problem ${\operatorname{Expr}_{pp}}(\Gamma)$ is in co-NEXPTIME (and in particular decidable), and has recently shown to be co-NEXPTIME-hard [@Willard-cp10]. For infinite structures $\Gamma$, the decidability of ${\operatorname{Expr}_{pp}}(\Gamma)$ is not obvious. An algorithm for primitive positive definability has theoretical and practical consequences in the study of the computational complexity of CPSs (which we will consider in Section \[sect:csp\]). It is motivated by the fundamental fact that expansions of structures $\Gamma$ by primitive positive relations do not change the complexity of $\operatorname{CSP}(\Gamma)$. On a practical side, it turns out that hardness of a CSP can usually be shown by presenting primitive positive definitions of relations for which it is known that the CSP is hard. Therefore, a procedure that decides primitive positive definability of a given relation might be a useful tool to determine the computational complexity of CSPs.
Using the methods of the last sections, one can show decidability of ${\operatorname{Expr}_{ep}}(\Gamma)$ and ${\operatorname{Expr}_{pp}}(\Gamma)$ for certain infinite structures $\Gamma$. The following uses the same terminology as in [@MacphersonSurvey].
We say that a class $\mathcal C$ of finite $\tau$-structures (or a $\tau$-structure with age $\mathcal C$) is *finitely bounded* if there exists a finite set of finite $\tau$-structures $\mathcal F$ such for all finite $\tau$-structures $A$ we have that $A \in \mathcal C$ iff no structure from $\mathcal F$ embeds into $A$.
\[thm:decidability\] Let $\Delta$ be ordered, Ramsey, homogeneous, and of finite language, and let $\Gamma$ be a finite language reduct of $\Delta$. Then ${\operatorname{Expr}_{ep}}(\Gamma)$ and ${\operatorname{Expr}_{pp}}(\Gamma)$ are decidable.
Examples of structures $\Delta$ that satisfy the assumptions of Theorem \[thm:decidability\] are ${{(\mathbb{Q};<)}}$, the Fraïssé limit of ordered finite graphs (or tournaments [@RamseyClasses]), the Fraïssé limit of finite partial orders with a linear extension [@RamseyClasses], the homogeneous universal ‘naturally ordered’ $C$-relation [@BodirskyPiguet], just to name a few. CSPs for structures that are definable in such structures are abundant in particular for qualitative reasoning calculi in Artificial Intelligence.
We want to point out that that decidability of primitive positive definability is already non-trivial when $\Gamma$ is trivial from a model-theoretic perspective: for the case that $\Gamma$ is the structure $(X; =)$ (where $X$ is countably infinite), the decidability of ${\operatorname{Expr}_{pp}}(\Gamma)$ has been posed as an open problem in [@BodChenPinsker]. Theorem \[thm:decidability\] solves this problem, since $(X;=)$ is isomorphic to a reduct of the structure $({\mathbb Q}; <)$, which is clearly finitely bounded, homogeneous, ordered, and Ramsey.
The proof of Theorem \[thm:decidability\] goes along the following lines, and is based on the results of the last sections. We outline the algorithm for ${\operatorname{Expr}_{pp}}(\Gamma)$; the proof for ${\operatorname{Expr}_{ep}}(\Gamma)$ is a subset. So the input are formulas $\phi_0,\ldots,\phi_n$ defining relations $R_0,\ldots,R_n$, and we have to decide whether $R_0$ has a primitive positive definition from $R_1,\ldots, R_n$. Let $\Theta$ be the structure which has $R_1,\ldots,R_n$ as its relations. By Theorem \[thm:clones-pp\], $R_0$ is not primitive positive definable from $R_1,\ldots, R_n$ if and only if there is a finitary function $f\in\operatorname{Pol}(\Theta)$ which violates $R_0$. By the ideas of the last section, such a polymorphism can be chosen to be canonical as a function from $(\Delta,c_1,\ldots,c_k)^m$ to $\Delta$, where $c_i\in\Delta$. Such canonical functions are essentially finite objects since they can be represented as functions on types. Therefore, the algorithm can then check for a given canonical function whether it is a polymorphism of $\Theta$ and whether it violates $R_0$. Also, $k$ and $m$ can be calculated from the input, and so there are only finitely many complete behaviors to be checked. Finally, the additional assumption that $\Delta$ be finitely bounded allows the algorithm to check whether a function on types really comes from a function on $\Delta$. We refer to [@BodPinTsa] for details.
Interpretability {#sect:interpret}
================
Many $\omega$-categorical structures can be derived from other $\omega$-categorical structures via first-order interpretations. In this section we will discuss the fact already mentioned in the introduction that bi-interpretations can be used to transfer the Ramsey property from one structure to another. A special type of interpretations, called *primitive positive interpretations*, will become important in Section \[sect:csp\]. The definition of interpretability we use is standard, and follows [@HodgesLong].
When $\Delta$ is a structure with signature $\tau$, and $\delta(x_1,\dots,x_k)$ is a first-order $\tau$-formula with the $k$ free variables $x_1,\dots,x_k$, we write $\delta(\Delta^k)$ for the $k$-ary relation that is defined by $\delta$ over $\Delta$.
A relational $\sigma$-structure $\Gamma$ has a *(first-order) interpretation* in a $\tau$-structure $\Delta$ if there exists a natural number $d$, called the *dimension* of the interpretation, and
- a $\tau$-formula $\delta(x_1, \dots, x_d)$ – called *domain formula*,
- for each $k$-ary relation symbol $R$ in $\sigma$ a $\tau$-formula $\phi_R(\overline x_1, \dots, \overline x_k)$ where the $\overline x_i$ denote disjoint $d$-tuples of distinct variables – called the *defining formulas*,
- a $\tau$-formula $\phi_=(x_1,\dots,x_d,y_1,\dots,y_d)$, and
- a surjective map $h: \delta(\Delta^d) \rightarrow \Gamma$ – called *coordinate map*,
such that for all relations $R$ in $\Gamma$ and all tuples $\overline a_i \in \delta(\Delta^d)$ $$\begin{aligned}
(h(\overline a_1), \dots, h(\overline a_k)) \in R \;
& \Leftrightarrow \;
\Delta \models \phi_R(\overline a_1, \dots, \overline a_k) \; , \text{ and } \\
h(\overline a_1)=h(\overline a_2)
& \Leftrightarrow \; \Delta \models \phi_=(\overline a_1,\overline a_2) \; .\end{aligned}$$
If the formulas $\delta$, $\phi_R$, and $\phi_=$ are all primitive positive, we say that $\Gamma$ has a *primitive positive interpretation* in $\Delta$; many primitive positive interpretations can be found in Section \[sect:csp\]. We say that $\Gamma$ is *interpretable in* $\Delta$ *with finitely many parameters* if there are $c_1,\dots,c_n \in \Delta$ such that $\Gamma$ is interpretable in the expansion of $\Delta$ by the singleton relations $\{c_i\}$ for all $1 \leq i \leq n$. First-order *definitions* are a special case of interpretations: a structure $\Gamma$ is *(first-order) definable* in $\Delta$ if $\Gamma$ has an interpretation in $\Delta$ of dimension one where the domain formula is logically equivalent to true.
\[lem:interpret\] If $\Delta$ is an $\omega$-categorical structure, then every structure $\Gamma$ that is first-order interpretable in $\Delta$ with finitely many parameters is $\omega$-categorical as well.
The following nicely describes interpretability between structures in terms of the (topological) automorphism groups of the structures.
Let $\Delta$ be an $\omega$-categorical structure with at least two elements. Then a structure $\Gamma$ has a first-order interpretation in $\Delta$ if and only if there is a continuous group homomorphism $f: \operatorname{Aut}(\Delta) \to \operatorname{Aut}(\Gamma)$ such that the image of $f$ has finitely many orbits in its action on $\Gamma$.
Note that if $\Gamma_2$ has a $d$-dimensional interpretation $I$ in $\Gamma_1$, and $\Gamma_3$ has an $e$-dimensional interpretation $J$ in $\Gamma_2$, then $\Gamma_3$ has a natural $ed$-dimensional interpretation in $\Gamma_1$, which we denote by $J \circ I$. To formally describe $J \circ I$, suppose that the signature of $\Gamma_i$ is $\tau_i$ for $i = 1,2,3$, and that $I = (d,\delta,(\phi_R)_{R \in \tau_2}, \phi_=,h)$ where $d$ is the dimension, $\delta$ the domain formula, $\phi_=$ and $(\phi_R)_{R \in \tau_2}$ the interpreting relations, and $h$ the coordinate map. Similarly, let $J = (e,\gamma,(\psi_R)_{R \in \tau_3}, \psi_=,g)$. We use the following.
Let $\Gamma_1,\Gamma_2, I$ as in the preceding paragraph. Then for every first-order $\tau_2$-formula $\phi(x_1,\dots,x_k)$ there is $\tau_1$-formula $$\phi^I(x^1_1,\dots,x^d_1,\dots,x^1_k,\dots,x_k^d)$$ such that for all $a_1,\dots,a_k \in \delta((\Gamma_1)^d)$ $$\Gamma_2 \models \phi(h(a_1),\dots,h(a_k)) \;
\Leftrightarrow \; \Gamma_1 \models \phi^I(a_1,\dots,a_k) \; .$$
We can now define the interpretation $J \circ I$ as follows: the domain formula $\eta$ is $\gamma^I$, and the defining formula for $R \in \tau_3$ is $(\psi_R)^I$. The coordinate map is from $\eta((\Gamma_1)^{ed}) \rightarrow \Gamma_3$, and defined by $$(a^1_1,\dots,a^d_1,\dots,a^1_e,\dots,a^d_e) \; \mapsto \; g(h(a^1_1,\dots,a^d_1),\dots,h(a^1_e,\dots,a^d_e)) \; .$$
Two interpretations of $\Gamma$ in $\Delta$ with coordinate maps $h_1$ and $h_2$ are called *homotopic* if the relation $\{(\bar x,\bar y) \; | \; h_1(\bar x) = h_2(\bar y) \}$ is definable in $\Delta$. The *identity interpretation* of a structure $\Gamma$ is the 1-dimensional interpretation of $\Gamma$ in $\Gamma$ whose coordinate map is the identity. Two structures $\Gamma$ and $\Delta$ are called *bi-interpretable* if there is an interpretation $I$ of $\Gamma$ in $\Delta$ and an interpretation $J$ of $\Delta$ in $\Gamma$ such that both $I \circ J$ and $J \circ I$ are homotopic to the identity interpretation (of $\Gamma$ and of $\Delta$, respectively).
\[thm:bi-interpret\] Two $\omega$-categorical structures $\Gamma$ and $\Delta$ are bi-interpretable if and only if $\operatorname{Aut}(\Gamma)$ and $\operatorname{Aut}(\Delta)$ are isomorphic as topological groups.
As a consequence of this result and Theorem \[thm:KPT\] we obtain the following.
\[cor:interpret-ramsey\] For ordered bi-interpretable $\omega$-categorical homogeneous structures $\Gamma$ and $\Delta$, one has the Ramsey property if and only if the other one has the Ramsey property.
We give an example. This corollary can be used to deduce that an important structure studied in temporal reasoning in artificial intelligence has the Ramsey property. For the relevance of this fact in constraint satisfaction, see Section \[sect:csp\].
We have already mentioned that the age of $({\mathbb Q}; <)$ has the Ramsey property. Let $\Gamma$ be the structure whose elements are pairs $(x,y) \in {\mathbb Q}^2$ with $x<y$, representing *intervals*, and which contains all binary relations $R$ over those intervals such that the relation $\{(x,y,u,v) \; | \; ((x,y),(u,v)) \in R\}$ is first-order definable in $({\mathbb Q}; <)$. Hence, $\Gamma$ has a 2-dimensional interpretation $I$ in $({\mathbb Q}; <)$, whose coordinate map $h_1$ is the identity map on $D := \{(x,y) \in {\mathbb Q}^2 \; | \; x<y\}$.
The structure $\Gamma$ is known under the name *Allen’s Interval Algebra* in artificial intelligence. We claim that its age has the Ramsey property. Using the homogeneity of ${(\mathbb{Q};<)}$, it is easy to show that $\Gamma$ is homogeneous as well. By Corollary \[cor:interpret-ramsey\], it suffices to show that $\Gamma$ and $({\mathbb Q}; <)$ are bi-interpretable. We first show that $({\mathbb Q};<)$ has an interpretation $J$ in $\Gamma$. The coordinate map $h_2$ of $J$ maps $(x,y) \in D$ to $x$. The formula $\phi_=(a,b)$ is $R_0(a,b)$ where $R_0$ is the binary relation $\{((x,y),(u,v)) \; | \; x=u\}$ from $\Gamma$. The formula $\phi_<(a,b)$ is $R_1(a,b)$ where $R_1$ is the binary relation $\{((x,y),(u,v)) \; | \; x<u\}$.
We prove that $J \circ I$ is homotopic to the identity interpretation of $({\mathbb Q};<)$ in $({\mathbb Q};<)$. This holds since the relation $\{(x,y,u) \in {\mathbb Q}^3 \; | \; h_2(h_1(x,y))=u\}$ has the first-order definition $x=u$ in $({\mathbb Q}; <)$. To show that $I \circ J$ is homotopic to the identity interpretation, observe that the relation $\{(a,b,c) \in D^3 \; | $ $h_1(h_2(a),h_2(b))=c\}$ has the first-order definition $R_0(a,c) \wedge R_3(b,c)$ in $\Gamma$, where $R_0$ is the binary relation from $\Gamma$ as defined above, and $R_3$ is the binary relation $\{((x,y),(u,v))\in\Gamma^2 \; | \; x=v \}$ from $\Gamma$. This shows that $\Gamma$ and $({\mathbb Q}; <)$ are bi-interpretable.
Complexity of Constraint Satisfaction {#sect:csp}
=====================================
In recent years, a considerable amount of research concentrated on the computational complexity of $\operatorname{CSP}(\Gamma)$ for *finite* structures $\Gamma$. Feder and Vardi [@FederVardi] conjectured that for such $\Gamma$, the problem $\operatorname{CSP}(\Gamma)$ is either in P, or NP-complete[^1]. This conjecture has been fascinating researchers from various areas, for instance from graph theory [@HellNesetrilSurvey] and from finite model theory [@FederVardi; @KolaitisVardi; @AtseriasBulatovDawar]. It has been discovered that complexity classification questions translate to fundamental questions in universal algebra [@JBK; @IMMVW], so that lately also many researchers in universal algebra started to work on questions that directly correspond to questions about the complexity of CSPs.
For arbitrary infinite structures $\Gamma$ it can be shown that there are problems $\operatorname{CSP}(\Gamma)$ that are in NP, but neither in P nor NP-complete, unless P=NP. In fact, it can be shown that for every computational problem $\mathcal P$ there is an infinite structure $\Gamma$ such that $\mathcal P$ and $\operatorname{CSP}(\Gamma)$ are equivalent under polynomial-time Turing reductions [@BodirskyGrohe]. However, there are several classes of infinite structures $\Gamma$ for which the complexity of $\operatorname{CSP}(\Gamma)$ can be classified completely.
In this section we will see three such classes of computational problems; they all have the property that
- every problem in this class can be formulated as $\operatorname{CSP}(\Gamma)$ where $\Gamma$ has a first-order definition in a *base structure* $\Delta$;
- $\Delta$ is ordered homogeneous Ramsey with finite signature.
For all three classes, the classification result can be obtained by the same method, which we describe in the following two subsections.
Climbing up the lattice {#sect:climb}
-----------------------
Clearly, if we add relations to a structure $\Gamma$ with a finite relational signature, then the CSP of the structure thus obtained is computationally at least as complex as the CSP of $\Gamma$. On the other hand, when we add a primitive positive definable relation to $\Gamma$, then the CSP of the resulting structure has a polynomial-time reduction to $\operatorname{CSP}(\Gamma)$. This is not hard to show, and has been observed for finite domain structures in [@JeavonsClosure]; the same proof also works for structures over an infinite domain.
\[lem:pp-reduce\] Let $\Gamma = (D; R_1,\dots,R_l)$ be a relational structure, and let $R$ be a relation that has a primitive positive definition in $\Gamma$. Then the problems $\operatorname{CSP}(\Gamma)$ and $\operatorname{CSP}(D; R,R_1,\dots,R_l)$ are equivalent under polynomial-time reductions.
When we study the CSPs of the reducts $\Gamma$ of a structure $\Delta$, we therefore consider the lattice of reducts of $\Delta$ which are closed under primitive positive definitions (i.e., which contain all relations that are primitive positive definable from the reduct), and describe the border between tractability and NP-completeness in this lattice. We remark that the reducts of $\Delta$ have, since we expand them by all primitive positive definable relations, infinitely many relations, and hence do not define a CSP; however, we consider $\Gamma$ tractable if and only if all structures obtained from $\Gamma$ by dropping all but finitely many relations have a tractable CSP. Similarly, we consider $\Gamma$ hard if there exists a structure obtained from $\Gamma$ by dropping all but finitely many relations that has a hard CSP. With this convention, it is interesting to determine the *maximal tractable* reducts, i.e., those reducts closed under primitive positive definitions which do not contain any hard relation and which cannot be further extended without losing this property.
Recall the notion of a *clone* from Section \[sect:reducts\]. By Theorem \[thm:clones-pp\], the lattice of primitive positive closed reducts of $\Delta$ and the lattice of closed clones containing $\operatorname{Aut}(\Delta)$ are antiisomorphic via the mappings $\Gamma\mapsto\operatorname{Pol}(\Gamma)$ (for reducts $\Gamma$) and ${\mathcal C}\mapsto \operatorname{Inv}({\mathcal C})$ (for clones ${\mathcal C}$). We refer to the introduction of [@BodChenPinsker] for a detailed exposition of this well-known connection. Therefore, the maximal tractable reducts correspond to *minimal tractable* clones, which are precisely the clones of the form $\operatorname{Pol}(\Gamma)$ for a maximal tractable reduct.
The proof strategy of the classification results presented in Sections \[ssect:equality\], \[ssect:temporal\], and \[ssect:schaefer\] is as follows. We start by proving that certain reducts $\Gamma$ have an NP-hard CSP. How to show this, and how to find those ‘basic hard reducts’ will be the topic of the next subsection. Let $R$ be one of the relations from those hard reducts. If $R$ does not have a primitive positive definition in $\Gamma$, then Theorem \[thm:clones-pp\] implies that $\Gamma$ has a polymorphism $f$ that does not preserve $R$. We are now in a similar situation as in Section \[sect:minimalfunctions\]. Introducing constants, we can show that $f$ generates an operation $g$ that still does not preserve $R$ but is canonical with respect to the expansion of $\Gamma$ by constants. There are only finitely many canonical behaviours that $g$ might have, and therefore we can start a combinatorial analysis. In the three classifications that follow, this strategy always leads to polymorphisms that imply that CSP$(\Gamma)$ can be solved in polynomial time.
Primitive positive interpretations, and adding constants
--------------------------------------------------------
Surprisingly, in all the classification results that we present in Sections \[ssect:equality\], \[ssect:temporal\], and \[ssect:schaefer\], there is a single condition that implies that a CSP is NP-hard. Recall that an interpretation is called *primitive positive* if all formulas involved in the interpretation (the domain formula, the formulas $\phi_R$ and $\phi_=$) are primitive positive. The relevance of primitive positive interpretations in constraint satisfaction comes from the following fact, which is known for finite domain constraint satisfaction, albeit not using the terminology of primitive positive interpretations [@JBK]. In the present form, it appears first in [@BodirskySurvey].
\[thm:pp-interpret-hard\] Let $\Gamma$ and $\Delta$ be structures with finite relational signatures. If there is a primitive positive interpretation of $\Gamma$ in $\Delta$, then there is a polynomial-time reduction from $\operatorname{CSP}(\Gamma)$ to $\operatorname{CSP}(\Delta)$.
All hardness proofs presented later can be shown via primitive positive interpretations of Boolean structures (i.e., structures with the domain $\{0,1\}$) with a hard CSP. In fact, in all such Boolean structures the relation $\operatorname{NAE}$ defined as $$\operatorname{NAE}= \{0,1\}^3 \setminus \{(0,0,0),(1,1,1)\}$$ is primitive positive definable. This fact has not been stated in the original publications; however, it deserves to be mentioned as a unifying feature of all the classification results presented here. It is often more convenient to interpret other Boolean structures than $(\{0,1\}; \operatorname{NAE})$, and to then apply the following Lemma. An operation $f: D^k \rightarrow D$ is called *essentially a permutation* if there exists an $i$ and a bijection $g: DÊ\rightarrow D$ so that $f(x_1,\dots,x_k)=g(x_i)$ for all $(x_1,\dots,x_k) \in D^k$.
\[lem:nae\] Let $\Delta$ be a structure that interprets a Boolean structure $\Gamma$ such that all polymorphisms of $\Gamma$ are essentially a permutation. Then the structure $(\{0,1\}; \operatorname{NAE})$ has a primitive positive interpretation in $\Delta$, and $\operatorname{CSP}(\Delta)$ is NP-hard.
Since the polymorphisms of $\Gamma$ preserve the relation $\operatorname{NAE}$, and by the well-known finite analog of Theorem \[thm:clones-pp\] (due to [@Geiger] and independently, [@BoKaKoRo]), $\operatorname{NAE}$ is primitive positive definable in $\Gamma$. When $\phi$ is such a primitive positive definition, by substituting all relations in $\phi$ by their defining relations in $\Delta$ we obtain an interpretation of $(\{0,1\}; \operatorname{NAE})$ in $\Delta$. Hardness of $\operatorname{CSP}(\Delta)$ follows from the NP-hardness of $\operatorname{CSP}((\{0,1\}; \operatorname{NAE}))$ (this problem is called <span style="font-variant:small-caps;">Not-all-3-equal-3Sat</span> in [@GareyJohnson]) and Theorem \[thm:pp-interpret-hard\].
Typical Boolean structures $\Gamma$ such that all polymorphisms of $\Gamma$ are essentially a permutation are the structure $(\{0,1\}; \{ (t_1,t_2,t_3,t_4) \in \{0,1\} \; | \; t_1+t_2+t_3+t_4=2\}$, the structure $(\{0,1\}; \operatorname{1IN3})$, or the structure $(\{0,1\}; \operatorname{NAE})$ itself.
Sometimes it is not possible to give a primitive positive interpretation of the structure $(\{0,1\}; \operatorname{NAE})$ in $\Gamma$, but it is possible after expanding $\Gamma$ with constants. Under an assumption about the endomorphism monoid of $\Gamma$, however, introducing constants does not change the computational complexity of $\Gamma$. More precisely, we have the following.
\[thm:constants-hard\] Let $\Gamma$ be an $\omega$-categorical structure with a finite relational signature such that $\operatorname{Aut}(\Gamma)$ is dense in $\operatorname{End}(\Gamma)$. Then for any finite number of elements $c_1,\dots,c_k$ of $\Gamma$ there is a polynomial-time reduction from $\operatorname{CSP}((\Gamma, \{c_1,\},\dots,\{c_k\}))$ to $\operatorname{CSP}(\Gamma)$.
Reducts of equality {#ssect:equality}
-------------------
One of the most fundamental classes of $\omega$-categorical structures is the class of all reducts of $(X;=)$, where $X$ is an arbitrary countably infinite set. Up to isomorphism, this is exactly the class of countable structures that are preserved by all permutations of their domain. The other two classes of $\omega$-categorical structures that we will study here both contain this class.
We go straight to the statement of the complexity classification in terms of primitive positive interpretations. This is essentially a reformulation of a result from [@ecsps] which has been formulated without primitive positive interpretations. It turns out that when $\Gamma$ is preserved by the operations from one of the minimal clones above the clone generated by all the permutations of $X$, then CSP$(\Gamma)$ can be solved in polynomial time, and otherwise CSP$(\Gamma)$ is NP-hard.
Let $\Gamma$ be a reduct of $(X;=)$. Then exactly one of the following holds.
- $\Gamma$ has a constant endomorphism. In this case, $\operatorname{CSP}(\Gamma)$ is trivially in P.
- $\Gamma$ has a binary injective polymorphism. In this case, $\operatorname{CSP}(\Gamma)$ is in P.
- All relations with a first-order definition in $(X;=)$ have a primitive positive definition in $\Gamma$. Furthermore, the structure $(\{0,1\}; \operatorname{NAE})$ has a primitive positive interpretation in $\Gamma$, and $\operatorname{CSP}(\Gamma)$ is NP-complete.
It has been shown in [@ecsps] that $\operatorname{CSP}(\Gamma)$ is in P when $\Gamma$ has a constant or a binary injective polymorphism. Otherwise, by Theorem \[thm:minimalAboveS\], every polymorphism of $\Gamma$ is generated by the permutations of $X$. Hence, every relation $R$ with a first-order definition in $(X;=)$ is preserved by all polymorphisms of $\Gamma$, and it follows from Theorem \[thm:clones-pp\] that every relation is primitive positive definable in $\Gamma$.
This holds in particular for the relation $E_6$ defined as follows. $$\begin{aligned}
E_6 = \{(x_1,x_2,y_1,y_2,z_1,z_2) \in X^6\; | \; &
(x_1=x_2 \wedge y_1 \neq y_2 \wedge z_1 \neq z_2) \\
& \vee \;
(x_1 \neq x_2 \wedge y_1 = y_2 \wedge z_1 \neq z_2) \\
& \vee \; (x_1 \neq x_2 \wedge y_1 \neq y_2 \wedge z_1 = z_2) \}
\end{aligned}$$ We now show that the structure $(\{0,1\}; \operatorname{1IN3})$ has a primitive positive interpretation in $(X;E_6)$, which by Lemma \[lem:nae\] also shows that $(\{0,1\}; \operatorname{NAE})$ has a primitive positive interpretation in $(X;E_6)$ and that $\operatorname{CSP}(\Gamma)$ is NP-hard.
The dimension of the interpretation is 2, and the domain formula is *‘true’*. The formula $\phi_{\operatorname{1IN3}}(x_1,x_2,y_1,y_2,z_1,z_2)$ is $E_6(x_1,x_2,y_1,y_2,z_1,z_2)$, and The formula $\phi_=(x_1,x_2,y_1,y_2)$ is $$\begin{aligned}
\exists a_1,a_2,u_1,u_2,u_3,u_4,z_1,z_2. \; & a_1=a_2 \wedge
E_6(a_1,a_2,u_1,u_2,u_3,u_4) \\
\wedge \; & E_6(u_1,u_2,x_1,x_2,z_1,z_2)
\wedge E_6(u_3,u_4,z_1,z_2,y_1,y_2) .\end{aligned}$$ Note that the primitive positive formula $\phi_=(x_1,x_2,y_1,y_2)$ is equivalent to $x_1=x_2 \Leftrightarrow y_1=y_2$. The map $h$ maps $(a_1,a_2)$ to $1$ if $a_1=a_2$, and to $0$ otherwise.
Note that both the constant and the binary injective operation are canonical as functions over $(X;=)$.
Reducts of the dense linear order {#ssect:temporal}
---------------------------------
An extension of the result in the previous subsection has been obtained in [@tcsps-journal]; there, the complexity of the CSP for all reducts of $({\mathbb Q};<)$ has been classified. By a theorem of Cameron, those reducts are (again up to isomorphism) exactly the structures that are *highly set-transitive* [@Cameron5], i.e., structures $\Gamma$ such that for any two finite subsets $A,B$ with $|A|=|B|$ of the domain there is an automorphism of $\Gamma$ that maps $A$ to $B$.
The corresponding class of CSPs contains many computational problems that have been studied in Artificial Intelligence, in particular in temporal reasoning [@vanBeek; @Nebel; @BroxvallJonsson], but also in scheduling [@and-or-scheduling] or general theoretical computer science [@Opatrny; @GalilMegiddo]. The following theorem is a consequence of results from [@tcsps-journal]. Again, we show that the hardness proofs in this class are captured by interpreting Boolean structures with few polymorphisms via primitive positive interpretations with finitely many parameters; this has not appeared in [@tcsps-journal], so we provide the proof. The central arguments in the classification follow the reduct classification technique based on Ramsey theory that we present in this survey; see Figure \[fig:tcsp\] for an illustration of the bottom of the lattice of reducts of $({\mathbb Q};<)$, and the border of tractability for such reducts.
![An illustration of the classification result for temporal constraint languages. Double-circles mean that the corresponding operation has a dual generating a distinct clone which is not drawn in the figure. For the definition of mi, min, mx, and ll, see [@tcsps-journal].[]{data-label="fig:tcsp"}](main-2.pdf)
\[thm:tcsp\] Let $\Gamma$ be a reduct of $({\mathbb Q};<)$. Then exactly one of the following holds.
- $\Gamma$ has one out of 9 binary polymorphisms (for a detailed description of those see [@tcsps-journal]), and $\operatorname{CSP}(\Gamma)$ is in P.
- $\operatorname{Aut}(\Gamma)$ is dense in $\operatorname{End}(\Gamma)$, and the structure $(\{0,1\}; \operatorname{NAE})$ has a primitive positive interpretation with finitely many parameters in $\Gamma$. In this case, $\operatorname{CSP}(\Gamma)$ is NP-complete.
Before we derive Theorem \[thm:tcsp\] from what has been shown in [@tcsps-journal], we would like to point to Figure \[fig:tcsp\] for an illustration of the clones that correspond to maximal tractable reducts. The diagram also shows the constraint languages that just contain one of the important relations ${\text{\it Betw}}$ (introduced in the introduction), ${\text{\it Cycl}}$, ${\text{\it Sep}}$ (${\text{\it Cycl}}$ and ${\text{\it Sep}}$ already appeared in Section \[sect:reducts\]), $E_6$ (which appeared earlier in this section), $T_3$, and $-T_3$. Here, $T_3$ stands for the relation $$\{ (x,y,z) \in \mathbb Q^3\; | \; (x=y<z) \vee (x=z<y) \} \; ,$$ and when $R \subseteq {\mathbb Q}^k$, then $-R$ denotes $\{(-t_1,\dots,-t_k) \; | \; (t_1,\dots,t_k) \in R\}$.
The importance of those relations comes from the fact (shown in [@tcsps-journal]) that unless $\Gamma$ has one out of the 9 binary polymorphisms mentioned in Theorem \[thm:tcsp\] then there is a primitive positive definition of at least one of the relations ${\text{\it Betw}}$, ${\text{\it Cycl}}$, ${\text{\it Sep}}$, $E_6$, $T_3$, or $-T_3$.
It has been shown in [@tcsps-journal] that unless $\Gamma$ has a constant endomorphism, $\operatorname{Aut}(\Gamma)$ is dense in $\operatorname{End}(\Gamma)$. We have already seen that there is a primitive positive interpretation of $(\{0,1\}; \operatorname{NAE})$ in structures isomorphic to $({\mathbb Q}; E_6)$. Now suppose that $T_3$ is primitive positive definable in $\Gamma$. We give below a primitive positive interpretation of the structure $(\{0,1\}; \operatorname{1IN3})$ in $\Delta = ({\mathbb Q}; T_3,0)$. Hence, there is also a primitive positive definition of $(\{0,1\}; \operatorname{1IN3})$ in the expansion of $\Gamma$ by the constant $0$. Expansions by constants do not change the computational complexity of $\operatorname{CSP}(\Gamma)$ since $\operatorname{Aut}(\Gamma)$ is dense in $\operatorname{End}(\Gamma)$. Thus, Lemma \[lem:nae\] shows NP-hardness of $\operatorname{CSP}(\Gamma)$, and that $(\{0,1\}; \operatorname{NAE})$ has a primitive positive interpretation in $(\Gamma,0)$.
The interpretation of $(\{0,1\}; \operatorname{1IN3})$ in $\Delta$
- has dimension 2;
- the domain formula $\delta(x_1,x_2)$ is $T_3(0,x_1,x_2)$;
- the formula $\phi_{\operatorname{1IN3}}(x_1,x_2,y_1,y_2,z_1,z_2)$ is $$\exists u. \; T_3(u,x_1,y_1) \wedge T_3(0,u,z_1) \; ;$$
- the formula $\phi_=(x_1,x_2,y_1,y_2)$ is $T_3(0,x_1,y_2)$;
- the coordinate map $h: \delta(\Delta^2) \rightarrow \{0,1\}$ is defined as follows. Let $(b_1,b_2)$ be a pair of elements of $\Delta$ that satisfies $\delta$. Then exactly one of $b_1, b_2$ must have value $0$, and the other element is strictly greater than $0$. We define $h(b_1,b_2)$ to be $1$ if $b_1=0$, and to be $0$ otherwise.
To see that this is the intended interpretation, let $(x_1,x_2),(y_1,y_2),(z_1,z_2) \in \delta(\Delta^2)$, and suppose that $t:=(h(x_1,x_2),h(y_1,y_2),h(z_1,z_2))=(1,0,0) \in \operatorname{1IN3}$. We have to verify that $(x_1,x_2,y_1,y_2,z_1,z_2)$ satisfies $\phi_{\operatorname{1IN3}}$ in $\Delta$. Since $h(x_1,x_2)=1$, we have $x_1 = 0$, and similarly we get that $y_1,z_1 > 0$. We can then set $u$ to $0$ and have $T_3(u,x_1,y_1)$ since $0=u=x_1<y_1$, and we also have $T_3(0,u,z_1)$ since $0=u<z_1$. The case that $t=(0,1,0)$ is analogous. Suppose now that $t=(0,0,1) \in \operatorname{1IN3}$. Then $x_1,y_1 > 0$, and $z_1=0$. We can then set $u$ to $\min(x_1,y_1)$, and therefore have $T_3(u,x_1,y_1)$, and $T_3(0,u,z_1)$ since $0=z_1<u$. Conversely, suppose that $(x_1,x_2,y_1,y_2,z_1,z_2)$ satisfies $\phi_{\operatorname{1IN3}}$ in $\Delta$. Since $T_3(0,u,z_1)$, exactly one out of $u,z_1$ equals $0$. When $u=0$, then because of $T_3(u,x_1,y_1)$ exactly one out of $x_1,y_1$ equals $0$, and we get that $(h(x_1,x_2),h(y_1,y_2),h(z_1,z_2)) \in \{(0,1,0),(1,0,0)\} \subseteq \operatorname{1IN3}$. When $u>0$, then $x_1>0$ and $y_1>0$, and so $(h(x_1,x_2),h(y_1,y_2),h(z_1,z_2)) = (0,0,1) \in \operatorname{1IN3}$.
An interpretation of $(\{0,1\}; \operatorname{1IN3})$ in $({\mathbb Q}; -T_3,0)$ can be obtained in a dual way.
Next, suppose that ${\text{\it Betw}}$ is primitive positive definable in $\Gamma$. We will give a primitive positive interpretation of $(\{0,1\}; \operatorname{NAE})$ in $({\mathbb Q}; {\text{\it Betw}}, 0)$. Hence, when ${\text{\it Betw}}$ has a primitive positive definition in $\Gamma$, then by Theorem \[thm:constants-hard\] (since $\operatorname{Aut}(\Gamma)$ is dense in $\operatorname{End}(\Gamma)$) and Lemma \[lem:nae\] we obtain NP-hardness of $\operatorname{CSP}(\Gamma)$.
The dimension of the interpretation is one, and the domain formula is $x \neq 0$, which is clearly equivalent to a primitive positive formula over $({\mathbb Q}; {\text{\it Betw}},0)$. The map $h$ maps positive points to $1$, and all other points from ${\mathbb Q}$ to $0$. The formula $\phi_=(x_1,y_1)$ is $$\exists z. \; {\text{\it Betw}}(x_1,0,z) \wedge {\text{\it Betw}}(z,0,y_1)$$ Note that the primitive positive formula $\phi_=$ is over $({\mathbb Q}; {\text{\it Betw}}, 0)$ equivalent to $(x_1>0 \Leftrightarrow y_1 > 0)$. Finally, $\phi_{\operatorname{NAE}}(x_1,y_1,z_1)$ is $$\exists u. \; {\text{\it Betw}}(x_1,u,y_1) \wedge {\text{\it Betw}}(u,0,z_1) \; .$$
If ${\text{\it Sep}}$ has a primitive positive definition in $\Gamma$, then the statement follows easily from the previous argument since ${\text{\it Betw}}(x,y,z)$ has a 1-dimensional primitive positive interpretation in $({\mathbb Q}; {\text{\it Sep}})$ (the formula $\phi_{\text{\it Betw}}(x,y,z)$ is $\exists u. {\text{\it Sep}}(u,x,y,z)$).
Finally, if ${\text{\it Cycl}}$ is primitive positive definable in $\Gamma$, we give a 3-dimensional primitive positive interpretation of the structure $(\{0,1\}; R,\neg)$ where $R = \{0,1\}^3 \setminus \{(0,0,0)\}$ and $\neg = \{(0,1),(1,0)\}$. The idea of the interpretation is inspired by the NP-hardness proof of [@GalilMegiddo] for the ‘Cyclic ordering problem’ (see [@GareyJohnson]).
The dimension of our interpretation is three, and the domain formula $\delta(x_1,x_2,x_3)$ is $x_1 \neq x_2 \wedge x_2 \neq x_3 \wedge x_3 \neq x_1$, which clearly has a primitive positive definition in $({\mathbb Q}; {\text{\it Cycl}})$. The coordinate map $h$ sends $(x_1,x_2,x_3)$ to $0$ if ${\text{\it Cycl}}(x_1,x_2,x_3)$, and to $1$ otherwise. Let $\phi(x_1,x_2,x_3,y_1,y_2,y_3)$ be the formula $$\begin{aligned}
& {\text{\it Cycl}}(x_1,y_1,x_2) \wedge {\text{\it Cycl}}(y_1,x_2,y_2) \wedge {\text{\it Cycl}}(x_2,y_2,x_3) \\
& {\text{\it Cycl}}(y_2,x_3,y_3) \wedge {\text{\it Cycl}}(x_3,y_3,x_1) \wedge {\text{\it Cycl}}(y_3,x_1,y_1) \; .\end{aligned}$$ When $(a_1,\dots,a_6)$ satisfies $\phi$, we can imagine $a_1,\dots,a_6$ as points that appear clockwise in this order on the unit circle. In particular, we then have that ${\text{\it Cycl}}(a_1,a_3,a_5)$ holds if and only if ${\text{\it Cycl}}(a_2,a_4,a_6)$ holds. The formula $\phi_=(x_1,x_2,x_3,y_1,y_2,y_3)$ is $$\begin{aligned}
\exists u^1_1,\dots,u^4_3. \; & \phi(x_1,x_2,x_3,u^1_1,u^1_2,u^1_3) \wedge \\
& \bigwedge_{i=1}^3 \phi(u^i_1,u^i_2,u^i_3,u^{i+1}_1,u^{i+1}_2,u^{i+1}_3) \wedge \phi(u^4_1,u^4_2,u^4_3,y_1,y_2,y_3) \, ,\end{aligned}$$ which is equivalent to $$\delta(x_1,x_2,x_3) \wedge \delta(y_1,y_2,y_3) \wedge ({\text{\it Cycl}}(x_1,x_2,x_3) \Leftrightarrow {\text{\it Cycl}}(y_1,y_2,y_3)) \, ;$$ this is tedious, but straightforward to verify, and we omit the proof.
The formula $\phi_\neg(x_1,x_2,x_3,y_1,y_2,y_3)$ is $\phi_=(x_1,x_2,x_3,z_1,z_3,z_2)$.
The formula $\phi_{R}(x_1,x_2,x_3,y_1,y_2,y_3,z_1,z_2,z_3)$ is $$\begin{aligned}
\exists a,b,c,d,e,f,g,h,i,j,k,l,m,n. \; & {\text{\it Cycl}}(a,c,j) \wedge {\text{\it Cycl}}(b,j,k) \wedge {\text{\it Cycl}}(c,k,l) \\
\wedge \; & {\text{\it Cycl}}(d,f,j) \wedge {\text{\it Cycl}}(e,j,l) \wedge {\text{\it Cycl}}(f,l,m) \\
\wedge \; & {\text{\it Cycl}}(g,i,k) \wedge {\text{\it Cycl}}(h,k,m) \wedge {\text{\it Cycl}}(i,m,n) \\\wedge \; & {\text{\it Cycl}}(n,m,l)
\wedge \phi_=(x_1,x_2,x_3,a,b,c) \\
\wedge \; & \phi_=(y_1,y_2,y_3,d,e,f) \wedge \phi_=(z_1,z_2,z_3,g,h,i)\end{aligned}$$ The proof that for all tuples $\bar a_1, \bar a_2, \bar a_3 \in
{\mathbb Q}^3$ $$\begin{aligned}
(h(\overline a_1), h(\overline a_3), h(\overline a_3)) \in R \;
& \Leftrightarrow \;
({\mathbb Q}; {\text{\it Cycl}}) \models \phi_{R}(\overline a_1, \overline a_2, \overline a_3)\end{aligned}$$ follows directly the correctness proof of the reduction presented in [@GalilMegiddo].
Reducts of the random graph {#ssect:schaefer}
---------------------------
The full power of the technique that is developed in this paper can be used to obtain a full complexity classification for all reducts of the random graph $G=(V;E)$ [@BodPin-Schaefer]. Again, the result can be stated in terms of primitive positive interpretations – this is not obvious from the statement of the result in [@BodPin-Schaefer], therefore we provide the proofs.
\[thm:schaefer\] Let $\Gamma$ be a reduct of the countably infinite random graph $G$. Then exactly one of the following holds.
- $\Gamma$ has one out of 17 at most ternary canonical polymorphisms (for a detailed description of those see [@BodPin-Schaefer]), and $\operatorname{CSP}(\Gamma)$ is in P.
- $\Gamma$ admits a primitive positive interpretation of $(\{0,1\}; \operatorname{1IN3})$. In this case, $\operatorname{CSP}(\Gamma)$ is NP-complete.
It has been shown in [@BodPin-Schaefer] that $\Gamma$ has one out of 17 at most ternary canonical polymorphisms, and $\operatorname{CSP}(\Gamma)$ is in P, or one of the following relations has a primitive positive definition in $\Gamma$: the relation $E_6$, or the relation $T$, $H$, or $P^{(3)}$, which are defined as follows. The $4$-ary relation $T$ holds on $x_1,x_2,x_3,x_4 \in V$ if $x_1,x_2,x_3,x_4$ are pairwise distinct, and induce in $G$ either
- a single edge and two isolated vertices,
- a path with two edges and an isolated vertex,
- a path with three edges, or
- a complement of one of the structures stated above.
To define the relation $H$, we write $N(u,v)$ as a shortcut for $E(u,v) \wedge u \neq v$. Then $H(x_1,y_1,x_2,y_2,x_3,y_3)$ holds on $V$ if $$\begin{aligned}
& \bigwedge_{i,j \in \{1,2,3\}, i \neq j, u \in \{x_i,y_i\},v \in \{x_j,y_j\}} N(u,v) \\
\wedge & \; \big(((E(x_1,y_1) \wedge N(x_2,y_2) \wedge N(x_3,y_3))
\\
& \vee \; (N(x_1,y_1) \wedge E(x_2,y_2) \wedge N(x_3,y_3)) \label{eq:rel} \\
& \vee \; (N(x_1,y_1) \wedge N(x_2,y_2) \wedge E(x_3,y_3)) \big)\; .\end{aligned}$$ The ternary relation $P^{(3)}$ holds on $x_1,x_2,x_3$ if those three vertices are pairwise distinct and do not induce a clique or an independent set in $G$.
Suppose first that $T$ is primitive positive definable in $\Gamma$. Let $R$ be the relation $\{ (t_1,t_2,t_3,t_4) \in \{0,1\} \; | \; t_1+t_2+t_3+t_4=2\}$. We have already mentioned that all polymorphisms of $(\{0,1\}; R)$ are essentially permutations. To show that $(\{0,1\}; \operatorname{NAE})$ has a primitive positive interpretation in $\Gamma$, we can therefore use Lemma \[lem:nae\] and it suffices to show that there is a primitive positive interpretation of the structure $(\{0,1\}; R)$ in $(V;T)$. For a finite subset $S$ of $V$, write $\# S$ for the parity of edges between members of $S$. Now we define the relation $L \subseteq V^6$ as follows. $$\begin{aligned}
L := \big \{ x \in V^6 \; | \; & \text{the entries of $x$ are pairwise distinct, and }\\
&\#\{x_1,x_2,x_3\}=\#\{x_4,x_5,x_6\} \big \} \end{aligned}$$ It has been shown in [@BodPin-Schaefer] that the relation $L$ is pp-definable in $(V;T)$. We therefore freely use the relation $L$ (and similarly $\neq$, the disequality relation) in primitive positive formulas over $(V;T)$.
Our primitive positive interpretation of $(\{0,1\}; R)$ has dimension three. The domain formula $\delta(x_1,x_2,x_3)$ is $x_1 \neq x_2 \; \wedge \; x_1 \neq x_3 \; \wedge \; x_2 \neq x_3$. The formula $\phi_R(x^1_1,x^1_2,x^1_3,\dots,x^4_1,x^4_2,x^4_3)$ of the interpretation is $$\begin{aligned}
\exists y_1,y_2,y_3,y_4. \; & T(y_1,\dots,y_4) \\
\wedge & L(x^1_1,x^1_2,x^1_3,y_2,y_3,y_4) \\
\wedge & L(x^2_1,x^2_2,x^2_3,y_1,y_3,y_4) \\
\wedge & L(x^3_1,x^3_2,x^3_3,y_1,y_2,y_4) \\
\wedge & L(x^4_1,x^4_2,x^4_3,y_1,y_2,y_3)\end{aligned}$$ The formula $\phi_=$ is $L(x_1,x_2,x_3,y_1,y_2,y_3)$. Finally, the coordinate map sends a tuple $(a_1,a_2,a_3)$ for pairwise distinct $a_1,a_2,a_3$ to $1$ if $P^{(3)}(a_1,a_2,a_3)$, and to $0$ otherwise.
Next, suppose that $H$ is primitive positive definable in $\Gamma$. We give a 2-dimensional interpretation of $(\{0,1\}; \operatorname{1IN3})$ in $\Gamma$. The domain formula is *‘true’*. The formula $\phi_=(x_1,x_2,y_1,y_2)$ is $$\begin{aligned}
\exists z_1,z_2,u_1,u_2,v_1,v_2. \; & H(x_1,x_2,u_1,u_2,z_1,z_2) \wedge N(u_1,u_2) \\
\wedge \; & H(z_1,z_2,v_1,v_2,y_1,y_2) \wedge N(v_1,v_2) \, .\end{aligned}$$ This formula is equivalent to a primitive positive formula over $\Gamma$ since $N(x,y)$ is primitive positive definable by $H$. The formula $\phi_{\operatorname{1IN3}}(x_1,x_2,y_1,y_2,z_1,z_2)$ is $$\begin{aligned}
\exists x_1',x_2',y_1',y_2',& z_1',z_2'. \,
H(x_1',x_2',y_1',y_2',z_1',z_2') \\
\wedge \; & \phi_=(x_1,x_2,x_1',x_2')
\wedge \phi_=(x_1,x_2,x_1',x_2')
\wedge \phi_=(x_1,x_2,x_1',x_2') \, .\end{aligned}$$ The coordinate map sends a tuple $(x_1,x_2)$ to $1$ if $E(x_1,x_2)$ and to $0$ otherwise.
Finally, suppose that $P^{(3)}$ has a primitive positive definition in $\Gamma$. We give a 2-dimensional primitive positive interpretation of $(\{0,1\}; \operatorname{NAE})$. For $k\geq 3$, let $Q^{(k)}$ be the $k$-ary relation that holds for a tuple $(x_1,\dots,x_k) \in V^k$ iff $x_1,\dots,x_k$ are pairwise distinct, and $(x_1,\dots,x_k){\notin}P^{(k)}$. It has been shown in [@BodPin-Schaefer] that the relation $Q^{(4)}$ is primitive positive definable by the relation $P^{(3)}$. Now, the formula $\phi_=(x_1,x_2,y_1,y_2)$ is $\exists z_1,z_2. Q^{(4)}(x_1,x_2,z_1,z_2) \wedge Q^{(4)}(z_1,z_2,y_1,y_2)$. The formula $\phi_{\operatorname{NAE}}(x_1,x_2,y_1,y_2,z_1,z_2)$ is $$\begin{aligned}
\exists u,v,w. \; & P^{(3)}(u,v,w) \wedge Q^{(4)}(x_1,x_2,u,v) \\
\wedgeÊ\; & Q^{(4)}(y_1,y_2,v,w) \wedge Q^{(4)}(z_1,z_2,w,u) \, .\end{aligned}$$ The coordinate map sends a tuple $(x_1,x_2)$ to $1$ if $E(x_1,x_2)$ and to $0$ otherwise.
Concluding Remarks and Further Directions
=========================================
We have outlined an approach to use Ramsey theory for the classification of reducts of a structure, considered up to existential positive, or primitive positive interdefinability. The central idea in this approach is to study functions that preserve the reduct, and to apply structural Ramsey theory to show that those functions must act regularly on large parts of the domain. This insight makes those functions accessible to combinatoral arguments and classification.
Our approach has been illustrated for the reducts of $(\mathbb Q; <)$, and the reducts of the random graph $(V;E)$. One application of the results is complexity classification of constraint satisfaction problems in theoretical computer science. Interestingly, the hardness proofs in those classifications all follow a common pattern: they are based on primitive positive interpretations. In particular, we proved complete complexity classifications without the typical computer science hardness proofs – rather, the hardness results follow from mathematical statements about primitive positive interpretability in $\omega$-categorical structures.
There are many other natural and important $\omega$-categorical structures besides $(\mathbb Q; <)$ and $(V;E)$ where this approach seems promising. We have listed some of the simplest and most basic examples in Figure \[fig:table\]. In this table, the first column specifies the ‘base structure’ $\Delta$, and we will be interested in the class of all structures definable in $\Delta$. The second column lists what is known about this class, considered up to first-order interdefinability. The third column describes the corresponding Ramsey result, when $\Delta$ is equipped with an appropriate linear order. The fourth column gives the status with respect to complexity classification of the corresponding class of CSPs. The fifth class indicates in which areas in computer science those CSPs find applications.
Reducts of First-order Reducts Ramsey Class CSP Dichotomy Application, Motivation
------------------------------------ -------------------------------------------------------------------- ------------------------------------------------- --------------- -------------------------------
$(X; =)$ Trivial Ramsey’s theorem Yes Equality Constraints
$(\mathbb Q; <)$ Cameron [@Cameron] Ramsey’s theorem Yes Temporal Reasoning
$(V; E)$ Thomas [@RandomReducts] Nešetřil + Rödl [@NesetrilRoedl] Yes Schaefer’s theorem for graphs
Homogeneous universal poset ? Nešetřil + Rödl [@PosetRamsey] ? Temporal Reasoning
Homogeneous C-relation Adeleke, Macpherson, Neumann [@AdelekeMacPherson; @AdelekeNeumann] Deuber, Miliken ? Phylogeny Reconstruction
Countable atomless Boolean algebra ? Graham, Leeb, Rothschild (see [@Topo-Dynamics]) ? Set Constraints
Allen’s Interval Algebra ? This paper, Section \[sect:interpret\] ? Temporal Reasoning
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[^1]: By Ladner’s theorem [@ladner], there are infinitely many complexity classes between P and NP, unless P=NP.
|
---
abstract: 'The mathematical modeling of generics in Java and other similar nominally-typed object-oriented programming languages is a challenge. In this short paper we present the outline of a novel order-theoretic approach to modeling generics, in which we also elementarily use some concepts and tools from category theory. We believe a combined order-theoretic and category-theoretic approach to modeling generics holds the keys to overcoming much of the adversity found when analyzing features of generic OO type systems.'
author:
- |
Moez A. AbdelGawad\
Informatics Research Institute, SRTA-City, Alexandria, Egypt\
`moez@cs.rice.edu`
title: |
Java Generics: An Order-Theoretic Approach\
(Abridged Outline)[^1]
---
Introduction
============
Generics have been added to Java so as to increase the expressiveness of its type system [@JLS05; @JLS18; @Bracha98; @Corky98; @Thorup99]. Generics in Java and other mainstream nominally-typed OOP languages similar to it such as C\# [@CSharp2015], Scala [@Odersky14], C++ [@CPP2017], and Kotlin [@Kotlin18], however, include some features—such as variance annotations (*e.g.*, Java wildcards), $F$-bounded generics, and Java erasure—that have been hard to analyze and reason about so far [@Torgersen2004; @MadsTorgersen2005; @Cameron2007; @Cameron2008; @Summers2010; @Tate2011]. As a result, the type systems of mainstream nominally-typed OOP languages, which are built based on current mathematical models of generics, are overly complex[^2], thereby hindering the progress of these type systems.
Further, support of some features in Java generics has a number of irregularities or “rough edges.” These include type variables that can have upper $F$-bounds but cannot have lower bounds (let alone lower $F$-bounds), wildcard type arguments that can have an upper-bound or a lower-bound but not both, and Java erasure—a feature prominent in Java and Java-based OOP languages such as Scala and Kotlin—that is usually understood, basically, as being “outside the type system.”
In this short paper we outline a new order-theoretic approach to modeling Java generics, and we report on our progress in developing this approach. The main contribution of the approach is demonstrating how concepts and tools from order theory can significantly simplify analyzing and reasoning about subtle features of OO generics, including the ones mentioned above.
Fundamentally, in the approach we use the nominal *subclassing* relation (as a partial ordering between classes[^3]) together with some novel order-theoretic tools to construct the generic nominal *subtyping* relation (also a partial ordering, between parameterized types) and the *containment* relation (a third partial ordering, between generic type arguments). These three ordering relations lie at the heart of mainstream generic OO type systems. Using order theoretic tools, as well as some concepts and tools from category theory, we further analyze these three relations and the relationship between them. Consequently, we demonstrate the value of the approach by exploring extensions of generic OO type systems that are naturally suggested by such analysis.
Description
===========
#### Constructing The Generic Subtyping Relation
The first step in the order-theoretic approach to modeling generics is defining two operators that construct ordering relations (*i.e.*, posets)
. In particular, the first operator, called *ppp* and denoted $\ltimes$, takes two input posets and a subset of the first poset and constructs a *partial poset product* [@AbdelGawad2018a; @Davey2002] of the two input posets. The second operator, called $wc$ (for *wildcards*) and denoted $\triangle$, takes as input a bounded poset (*i.e.*, one with top and bottom elements) and constructs a “triangle-shaped” poset—corresponding to wildcard type arguments—that is roughly composed of three copies of the input poset. See [@AbdelGawad2018b] for more details on $\ltimes$ and $\triangle$.
The formal definition of *ppp* is the order-theoretic counterpart of the definition of partial Cartesian product of graphs presented in [@AbdelGawad2018a; @AbdelGawad2018b], while the formal definition of the wildcards operator *wc* is presented in [@AbdelGawad2018b]. It is worthy to note that if the input poset of *wc* is a chain (*i.e.*, a “straight edge”), then $wc$ will construct an exact triangle-shaped output poset. The poset constructed by *wc* is “triangle-shaped” due to the existence of three variant subtyping rules for generic wildcard types in generic OOP (where covariant subtyping, roughly, is modeled by the left side of the triangle, contravariant subtyping is modeled by the right side, while the base of the triangle models invariant subtyping). See [@AbdelGawad2018b; @AbdelGawad2017a] for details and examples.
Next, given a finite subclassing relation $C$,[^4] operators *ppp* and *wc* are used to construct, iteratively, the infinite subtyping relation $S$ between ground parameterized types[^5]. In particular, given $S_{i}$, a finite approximation of $S$, *wc* constructs the corresponding wildcard type arguments, ordered by containment. Given $C$ and the constructed arguments, *ppp* then
pairs generic classes
with these arguments and adds types corresponding to non-generic classes
to construct the poset $S_{i+1}$,
ordered by subtyping, that next approximates $S$.[^6]
#### \[sec:The-Erasure-Connection\]The Erasure Galois Connection and Nominal Subtyping
Erasure—where, intuitively, type arguments of a parameterized type are “erased”—is a feature prominent in Java and Java-based OO languages such as Scala and Kotlin, but that can be also defined and made explicit in other generic nominally-typed OOP languages. In the order-theoretic approach to modeling generics, erasure is modeled as a mapping from types to classes.[^7]
Also, in the order-theoretic approach to modeling generics the ‘most general wildcard instantiation’ of a generic class is called the *free type* corresponding to the class. For example, a generic class [`C`]{} with one type parameter has the type [`C<?>`]{} as its corresponding free type[^8].
By maintaining a clear separation between classes ordered by subclassing, on one hand, and types ordered by subtyping, on the other, the construction of the subtyping relation using the subclassing relation (as presented earlier, using order-theoretic tools) allows us to observe that a Galois connection [@Davey2002] exists between the two fundamental relations in generic nominally-typed OOP. This connection between subclassing and subtyping is expressed formally using the erasure and free type mappings.
In particular, if $E$ is the erasure mapping that maps a parameterized type to the class used to construct the type, and if $FT$ is the free type mapping that maps a class to its most general wildcard instantiation, then the connection between subclassing and subtyping states that for all parameterized types $t$ and classes $c$ we have $$E\left(t\right)\leq c\Longleftrightarrow t<:FT\left(c\right)\label{eq:EGC}$$ where $\leq$ is the subclassing relation between classes and $<:$ is the subtyping relation between parameterized types.[^9]^,^[^10]
It should be noted that the erasure Galois connection expresses a fundamental property of generic nominally-typed OOP, namely, that subclassing (*a.k.a.*, type inheritance) is *the* source of subtyping in generic nominally-typed OOP. In other words, the property states, in one direction, that type inheritance is *a* source of subtyping (*i.e.*, subclassing causes subtyping between parameterized types) and, in the other direction, that type inheritance is the *only* source of subtyping in generic nominally-typed OOP (*i.e.*, subtyping between parameterized types comes from nowhere else other than subclassing). This property of generic nominally-typed OOP—stated as ‘inheritance is *the* source of subtyping’—corresponds to the ‘inheritance *is* subtyping’ property of *non*-generic nominally-typed OOP [@NOOPsumm; @InhSubtyNWPT13].
#### Extending Generics: Interval Types and Doubly $F$-bounded Generics
The construction of the generic subtyping relation using tools from order theory suggests how generics in nominally-typed OOP languages can be extended in two specific directions.
First, the approach suggests how wildcard type arguments can simultaneously have lower bounds and upper bounds, thereby defining *interval type arguments* (as a generalization of wildcard type arguments) and defining *interval types* (as a generalization of wildcard types—which are parameterized types with wildcard type arguments). In particular, interval types and the subtyping relation between them can simply be constructed by replacing the *wc* operator (that we presented earlier) with a novel operator *int* ($\Updownarrow$) that constructs interval type arguments (and the containment relation between them) from an input subtyping relation.[^11] The subtyping relation is then iteratively constructed from the subclassing relation, again as the solution of a recursive poset equation.[^12]
Second, the approach suggests how to define *doubly $F$-bounded generics* (*dfbg*, for short), ** where a type variable can have both an upper $F$-bound and a lower $F$-bound.[^13] See [@AbdelGawad2018e] for more details on *dfbg*.
Considering *dfbg* led us to distinguish between *valid type arguments*, which satisfy declared type parameter bounds, and *admittable type arguments*, which do not necessarily satisfy bounds, and to thus define *valid parameterized types*, whose type arguments are valid, and *admittable parameterized types*, whose type arguments are admittable but may not be valid (such as the admittable but invalid Java type [`Enum<Object>`]{}), and, accordingly, to define *valid subtyping relations*, between valid parameterized types, and *admittable subtyping relations*, between two admittable parameterized types that are not necessarily valid.
#### Induction, Coinduction, (Co)Inductive Types and Mutual (Co)Induction in Generic OOP
In logic, *coinductive* reasoning can, intuitively, be summarized as asserting that a statement is proven to be true if there is *no* (finite or “good”) reason for the statement to *not* hold [@Kozen2016; @AbdelGawad2019a]. While analyzing *dfbg* in [@AbdelGawad2018e], we use a coinductive logical argument to prove that checking the validity of type arguments inside some particular bounds-declarations of generic classes is unnecessary. Also, in [@Tate2011] Tate et al. conclude that Java wildcards are some form of coinductive ** bounded existentials.[^14]
Combined, these factors motivated us to consider, in some depth, the status of (co)inductive types in our order-theoretic approach [@AbdelGawad2019b], which led us to define the notions of $F$-subtypes and $F$-supertypes of a generic class $F$.[^15] The value of defining these notions is illustrated in the definition of *dfbg*, where a type variable, say [`T`]{}, with both a lower $F$-bound and an upper $F$-bound ranges over a set of $F$-supertypes and $F$-subtypes specified by the bounds of [`T`]{}. See [@AbdelGawad2018e; @AbdelGawad2019b] for more details.
Further, the mutual dependency between the containment relation (as an ordering of generic type arguments) and the subtyping relation (as an ordering of parameterized types), in addition to the fact that classes in OO programs, including generic classes, are frequently defined mutually-recursively[^16], led us to also define an order-theoretic notion of *mutual* (*co*)*induction* to allow studying least and greatest fixed point solutions of mutually-recursive definitions[^17] in an order-theoretic context [@AbdelGawad2019].
#### \[sec:Category-Theory\]Using Category Theory in Modeling Generics
Category theory can be viewed simply as a (major) generalization of order theory [@Fong2018; @Priestley2002; @spivak2014category]. In particular, each poset can be viewed, canonically, as a (*thin*, *small*) ** category [@Fong2018].
As such, some concepts and tools from category theory, such as adjunctions, monads, $F$-(co)algebras, initial algebras (*e.g.*, co-free types), final coalgebras (*e.g.*, free types), and operads, can be used to generalize the order-theoretic model of generics, and to situate it in the context of category theory.
A more detailed account of the use of category theory in our approach is presented in [@AbdelGawad2019h].
Discussion
==========
In this short paper we presented the outline of an order-theoretic model of generic nominally-typed OOP. This model demonstrates that in generic nominally-typed OOP:
$\bullet$The subtyping relation between parameterized types can be constructed solely from the subclassing (*i.e.*, inheritance) relation between classes using order-theoretic tools,
$\bullet$Erasure can be modeled as a map (*i.e.,* a homomorphism) from parameterized types ordered by subtyping to classes ordered by subclassing,
$\bullet$Wildcard type arguments can be modeled as intervals over the subtyping relation,[^18]
$\bullet$Generic classes can be modeled as type generators over the subtyping relation[^19],
$\bullet$The complex [`open`]{} and [`close`]{} operations (*i.e.*, capture conversion; see [@JLS18 $\mathsection$5.1.10, p.113]) are *not* needed in the definition of the subtyping relation between ground parameterized types[^20], since the relation can be defined exclusively using the containment relation between generic type arguments, and
$\bullet$Upper $F$-bounded type variables—*e.g.*, type variable [`T`]{} in the class declaration [`class Enum<T extends Enum<T>>`]{}—range over $F$-subtypes (modeled as coinductive types of the type generators modeling generic classes), while lower $F$-bounded type variables range over $F$-supertypes (modeled as inductive types).
Moreover, the model hints that:
$\bullet$Circular (*a.k.a.*, infinite, or infinitely-justified) subtyping relations can be modeled by an order-theoretic coinductive interpretation [@KennedyDecNomVar07] of the subtyping relation, and
$\bullet$Mutually-recursive definitions in OOP (*e.g.*, of classes, and of the subtyping and containment relations) can be modeled by mutually-(co)inductive mathematical objects.
Additionally we observe that, by incorporating nominal subtyping, the presented order-theoretic model of generics crucially depends on the *finite* inheritance relation between classes[^21]. On the other hand, extant models of generic OOP—which capture conversion and bounded existentials are prominent characteristics of—are inspired by structural (*i.e.*, non-nominal) models of polymorphic functional programming. Those models thus largely ignore the nominal subclassing relation—explicitly declared by OO software developers—when interpreting the generic subtyping relation and other features of generic OOP. Influenced by their origins in functional programming, those models depend instead on concepts and tools[^22] developed for structural typing and structural subtyping.
On account of these observations we believe the order-theoretic model of generics is a significantly simpler and more intuitive model of generic OOP than extant models, and that it is more in the spirit of nominally-typed OO type systems than those models are.
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[^1]: A less-structured, but more-detailed outline of the order-theoretic approach to modeling generics is presented in [@AbdelGawad2019g]. The two outline papers, together with [@AbdelGawad2019h] which outlines the use of category theory in the approach, succinctly summarize the main points of the detailed articles [@AbdelGawad2016a; @AbdelGawad2016c; @AbdelGawad2017a; @AbdelGawad2017b; @AbdelGawad2018a; @AbdelGawad2018b; @AbdelGawad2018c; @AbdelGawad2018e; @AbdelGawad2019a; @AbdelGawad2019b; @AbdelGawad2019]. As motivating and illustrating as examples may be, to shorten the three outline papers we intentionally elide most code examples as well as examples of subclassing and subtyping relations that illustrate the construction of the generic subtyping relation and other features of the approach. To aid readers interested in examples or more details however, the outline articles always cite one detailed article (or more) while discussing each piece of the approach.
[^2]: Check, for example, [@GenericsFAQWebsite], or sections of the Java language specification that specify crucial parts of its generic type system, *e.g.*, [@JLS18 $\mathsection$4.5 & $\mathsection$5.1.10].
[^3]: In this work Java interfaces and Scala traits are treated as abstract classes. In this paper the term ‘class’ thus refers to classes and other similar type-constructing constructs. Also, in other OOP literature parameterized types are sometimes called *object types*, *class types*, *reference types*, *generic types*, ** or just *types*.
[^4]: In $C$ it is assumed that a generic class takes one type argument, and that if a generic class extends (*i.e.*, inherits from, or, is a subclass of) another generic class then the superclass is passed the parameter of the subclass as the superclass type argument (*e.g.*, as in the declaration [`class D<T> extends C<T>`]{}, where [`T`]{}, the type parameter of [`D`]{}, is used “as is” as the type argument of superclass [`C`]{}). While we do not expect any significant complications when these simplifying assumptions are relaxed, we keep a discussion of how these assumptions can be relaxed to future work.
It is worthy to mention that the second assumption, in some sense, models the most general case (of a type argument passed to the superclass) and that a more complex inheritance relation (such as [`class D<T> extends C<E<T>>`]{}) only restricts the set of *valid* subtyping relations between instantiations of the subclass (*e.g.*, [`D`]{}) and those of its superclasses (*e.g.*, [`C`]{}). (See later discussion of *valid* versus *admittable* subtyping relations).
[^5]: Ground parameterized types are ones with no type variables. Such types are infinite in number due to the possibility of having arbitrary-depth nesting of type arguments [@AbdelGawad2017a]. Subtyping between these types is the basis for defining the full subtyping relation that includes type variables [@FJ/FGJ; @TAPL].
[^6]: This iterative construction process constructs the (least fixed point) solution of the recursive poset equation $$S=C\ltimes_{C_{g}}\triangle\left(S\right)$$ where $S$ is the subtyping relation, $C$ is the subclassing/inheritance relation, and $C_{g}$ is the subset of generic classes in $C$. See [@AbdelGawad2018b] for more details.
[^7]: To model *erased* *types*, the erasure mapping is composed with a notion of a *default type* that maps each generic class to some corresponding parameterized type (*i.e.*, a particular instantiation of the class). We keep further discussion of *default type arguments* and *default types* to future work.
[^8]: A *non-*generic class is mapped to the only type it constructs—a type typically homonymous to the class—as its corresponding free type.
[^9]: For example, in Java the statement $$\mathtt{LinkedList\le List\Longleftrightarrow LinkedList\negthickspace<\negthickspace String\negthickspace>\;<:\;List\negthickspace<?\negthickspace>},$$ where $t$, in Equation (\[eq:EGC\]) on page , is instantiated to type [`LinkedList<String>`]{} and $c$ is instantiated to class [`List`]{}, asserts that class [`LinkedList`]{} being a subclass of [`List`]{} is equivalent to (*i.e.*, if and only if, or implies and is implied by) [`LinkedList<String>`]{} being a subtype of the free type [`List<?>`]{}, which is a true statement in Java.
[^10]: Based on the strong relation between category theory and order theory—see below—the Galois connection between subclassing and subtyping is called *JEA*, ** the Java erasure adjunction.
[^11]: The formal definition of operator *int* is presented in [@AbdelGawad2018c]. Unlike the *wc* operator, operator *int* does not require the input poset to be bounded, *i.e.*, it does not assume the existence of a greatest type [`Object`]{} and a least type [`Null`]{} in the input (subtyping) relation.
[^12]: Namely, the equation $S=C\ltimes_{C_{g}}\Updownarrow\left(S\right)$. See [@AbdelGawad2018c] for more details.
[^13]: It is worthy to note that the definition of *dfbg* got inspiration from functions in real analysis. See [@AbdelGawad2018e] for more details.
[^14]: Given their historical origins [@Knaster1928; @Tarski1955], induction and coinduction—and accordingly (co)inductive mathematical objects—are naturally best studied in lattice theory, which is a sub-field of order theory.
[^15]: A parameterized type [`Ty`]{} is an $F$-subtype of class $F$ iff [`Ty <: F<Ty>`]{}. Dually, type [`Ty`]{} is an $F$-supertype of class $F$ iff [`F<Ty> <: Ty`]{}. The names of these two notions come from category theory (see later discussion), where $F$-subtypes correspond to $F$-coalgebras while $F$-supertypes correspond to $F$-algebras of a generic class $F$ (called a functor $F$ in category theory, and a generator—or a constructor—$F$ in lattice theory and order theory).
[^16]: *E.g.*, assuming the absence of primitive types in Java, the definitions of classes [`Object`]{} and [`Boolean`]{} are mutually dependent on each other (since class [`Boolean`]{} extends [`Object`]{}, and, without primitive type [`bool`]{}, the fundamental [`equals()`]{} method in [`Object`]{} returns a [`Boolean`]{}).
[^17]: Which are common in OOP but also in programming in general.
[^18]: In particular, intervals with upper bound [`Object`]{} or lower bound [`Null`]{}.
[^19]: I.e., as mathematical functions that take in type arguments, ordered by containment, and produce parameterized types, ordered by subtyping.
[^20]: Ground parameterized types constitute the full set over which type variables of generic classes range.
[^21]: Since it is always explicitly declared using class *names*, inheritance/subclassing is an inherently nominal relation.
[^22]: Such as existentials, abstract datatypes, and the opening/closing of type “packages.”
|
---
abstract: 'For two positive definite integral ternary quadratic forms $f$ and $g$ and a positive integer $n$, if $n\cdot g$ is represented by $f$ and $n\cdot dg=df$, then the pair $(f,g)$ is called a [*representable pair by scaling $n$*]{}. The set of all representable pairs in ${\text{gen}}(f)\times {\text{gen}}(g)$ is called a genus-correspondence. In [@Jagy], Jagy conjectured that if $n$ is square free and the number of spinor genera in the genus of $f$ equals to the number of spinor genera in the genus of $g$, then such a genus-correspondence respects spinor genus in the sense that for any representable pairs $(f,g), (f'',g'')$ by scaling $n$, $f'' \in {\text{spn}}(f)$ if and only if $g'' \in {\text{spn}}(g)$. In this article, we show that by giving a counter example, Jagy’s conjecture does not hold. Furthermore, we provide a necessary and sufficient condition for a genus-correspondence to respect spinor genus.'
address:
- 'Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea'
- 'Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea'
author:
- 'Jangwon Ju and Byeong-Kweon Oh'
title: 'genus-correspondences respecting spinor genus'
---
[^1]
Introduction
============
For a positive definite integral ternary quadratic form $$f(x,y,z)=ax^2+by^2+cz^2+pyz+qzx+rxy \ \ (a,b,c,p,q,r \in {{\mathbb Z}}),$$ it is quite an old problem determining the set $Q(f)$ of all positive integers $k$ such that $f(x,y,z)=k$ has an integer solution. If the class number of $f$ is one, then one may easily compute the set $Q(f)$ by using, so called, the local-global principle. However, if the class number of $f$ is bigger than $1$, then determining the set $Q(f)$ exactly seems to be quite difficult, except some very special ternary quadratic forms. If the integer $k$ is sufficiently large, then the theorem of Duke and Schulze-Pillot in [@ds] implies that if $k$ is primitively represented by the spinor genus of $f$, then $k$ is represented by $f$ itself.
Recently, W. Jagy proved in [@Jagy] that for any square free integer $k$ that is represented by a sum of two integral squares, it is represented by any ternary quadratic form in the spinor genus $x^2+y^2+16kz^2$. To prove this, he introduced, so called [*a genus-correspondence*]{}, and proved some interesting properties on the genus-correspondence. To be more precise, let ${\text{gen}}(f)$ (${\text{spn}}(f)$) be the set of genus (spinor genus, respectively) of $f$, for any ternary quadratic form $f$. Let $f$ and $g$ be positive definite integral ternary quadratic forms, and assume that there is a positive integer $n$ such that $$\label{gen-co}
\text{$n\cdot g$ is represented by $f$} \qquad \text{and} \qquad n\cdot dg=df.$$ In this article, we denote such a pair $(f,g)$ by a [*representable pair by scaling $n$*]{}. Note that $n\cdot f$ is also represented by $g$ for any representable pair $(f,g)$ by scaling $n$. As stated in [@Jagy], W. K. Chan proved that for any ternary quadratic form $f'\in {\text{gen}}(f)$, there is a ternary quadratic form $g'\in {\text{gen}}(g)$ such that $(f',g')$ is a representable pair by scaling $n$, and conversely for any $\tilde g \in {\text{gen}}(g)$, there is an $\tilde f \in {\text{gen}}(f)$ such that $(\tilde f,\tilde g)$ is also a representable pair by scaling $n$. Jagy defined the set of representable pairs by scaling $n$ by a [*genus-correspondence*]{} and proved some properities on a genus-correspondence. He also conjectured that if $n$ is square free and the number of spinor genera in the genus of $f$ equals to the number of spinor genera in the genus of $g$, then such a genus-correspondence [*respects spinor genus*]{} in the sense that for any representable pairs $(f,g), (f',g')$ by scaling $n$, where $f' \in {\text{gen}}(f)$ and $g' \in {\text{gen}}(g)$, $$\label{respect}
f' \in {\text{spn}}(f) \qquad \text{if and only if} \qquad g' \in {\text{spn}}(g).$$
In this article, we give an example such that Jagy’s conjecture does not hold. In fact, the concept of “genus-correspondence" in [@Jagy] is a little bit ambiguous. We modify the notion of a genus-correspondence as follows: For a positive integer $n$, let $\mathfrak C$ be a subset of ${\text{gen}}(f)\times {\text{gen}}(g)$ such that each pair in $\mathfrak C$ is a representable pair by scaling $n$. We say $\mathfrak C$ is a genus-correspondence if for any $f' \in {\text{gen}}(f)$, there is an $g' \in {\text{gen}}(g)$ such that $(f',g') \in \mathfrak C$, and conversely, for any $\tilde g\in {\text{gen}}(g)$, there is an $\tilde f \in {\text{gen}}(f)$ such that $(\tilde f,\tilde g) \in \mathfrak C$. Note that the set of all representable pairs by scaling $n$ is a genus-correspondence. We show that without assumption that $n$ is square free, there is a genus-correspondence respecting spinor genus if the number of spinor genera in ${\text{gen}}(f)$ is equal to the number of spinor genera in ${\text{gen}}(g)$.
In Section 5, we discuss when Jagy’s original conjecture is true. We provide a necessary and sufficient condition for the genus-correspondence consisting of all representable pairs by scaling $n$ in ${\text{gen}}(f)\times {\text{gen}}(g)$ to respect spinor genus under the assumption that $n$ is square free and the number of spinor genera in ${\text{gen}}(f)$ is equal to the number of spinor genera in ${\text{gen}}(g)$.
The subsequent discussion will be conducted in the better adapted geometric language of quadratic spaces and lattices. The term [*lattice*]{} will always refer to a [*non-classic integral*]{} ${{\mathbb Z}}$-lattice on an $n$-dimensional positive definite quadratic space ${{\mathbb Q}}$. Here a ${{\mathbb Z}}$-lattice $L$ is said to be non-classic integral if the norm ideal $\mathfrak{n}(L)$ of $L$ is contained in ${{\mathbb Z}}$. The discriminant of a lattice $L$ is denoted by $dL$ and the number of (proper) spinor genera in ${\text{gen}}(L)$ is denoted by $g(L)$. For any rational number $a$, $L^a$ is the lattice whose bilinear map $B$ is scaled by $a$.
Let $L= {{\mathbb Z}}x_1 + {{\mathbb Z}}x_2 + \cdots + {{\mathbb Z}}x_k$ be a ${{\mathbb Z}}$-lattice of rank $k$. We write $$L \simeq (B(x_i,x_j)).$$ The right hand side matrix is called a matrix presentation of $L$. If $L$ admits an orthogonal basis $\{x_1,x_2,\dots,x_k\}$, then we simply write $$L \simeq \langle Q(x_1),Q(x_2),\dots,Q(x_k) \rangle.$$ Throughout this paper, we say a ${{\mathbb Z}}$-lattice $L$ is [*primitive*]{} if the norm ideal $\mathfrak{n}(L)$ is exactly ${{\mathbb Z}}$. For a prime $p$, the group of units in ${{\mathbb Z}}_p$ is denoted by ${{\mathbb Z}}_p^{\times}$. Unless confusion arises, we will simply use $\Delta_p$ to denote a non-square element in ${{\mathbb Z}}_p^{\times}$, when $p$ is odd.
We denote by $\langle a,b,c,s,t,u \rangle$ for the ternary ${{\mathbb Z}}$-lattice with a matrix presentation $$\begin{pmatrix}a&u&t\\u&b&s\\t&s&c\end{pmatrix},$$ for convenience. For any ${{\mathbb Z}}$-lattice $L$, the equivalence class containing $L$ up to isometry is denoted by $[L]$. For any integer $a$, we say that $\frac a2$ is divisible by a prime $p$ if $p$ is odd and $a \equiv 0~(\text{mod}~p)$, or $p=2$ and $a\equiv0~(\text{mod}~4)$.
Any unexplained notations and terminologies can be found in [@ki] or [@om].
Watson’s transformations on the set of spinor genera
====================================================
Let $L$ be a non-classic integral ${{\mathbb Z}}$-lattice on a quadratic space $V$. For a prime $p$, we define $$\Lambda_p(L)= \{ x \in L : Q(x + z) \equiv Q(z) \ (\text{mod} \ p) \mbox{ for all $z \in L$}\}.$$ Let $\lambda_p(L)$ be the primitive lattice obtained from $\Lambda_p(L)$ by scaling $V=L\otimes \mathbb Q$ by a suitable rational number. Recall that a ${{\mathbb Z}}$-lattice $L$ is called primitive if $\mathfrak n(L)={{\mathbb Z}}$. For general properties of $\Lambda_p$-transformation, see [@lambda] and [@lambda2].
For $L' \in {\text{gen}}(L)$ ($L' \in {\text{spn}}(L)$) and any prime $p$, one may easily show that $\lambda_p(L') \in {\text{gen}}(\lambda_p(L))$ ($\lambda_p(L') \in {\text{spn}}(\lambda_p(L))$, respectively). It is well known that as a map, $$\label{lambda5}
\lambda_p:{\text{gen}}(L)\longrightarrow{\text{gen}}(\lambda_p(L))$$ is surjective. Furthermore, $\lambda_p({\text{spn}}(L))={\text{spn}}(\lambda_p(L))$. If we define ${\text{gen}}(L)_S$ the set of all spinor genera in ${\text{gen}}(L)$, then the map $$\lambda_p : {\text{gen}}(L)_S \longrightarrow {\text{gen}}(\lambda_p(L))_S$$ given by ${\text{spn}}(L')\mapsto {\text{spn}}(\lambda_p(L'))$ for any ${\text{spn}}(L')\in{\text{gen}}(L)_S$ is well-defined and surjective. In particular, $g(L)\geq g(\lambda_p(L))$ for any prime $p$.
Henceforth, $L$ is always a positive definite non-classic integral ternary ${{\mathbb Z}}$-lattice.
\[H-type defn\] For a ${{\mathbb Z}}$-lattice $L$ and a prime $p$, if $g(L)=g(\lambda_p(L))$, then we say the lattice $L$ is of $H$-type at $p$.
From the definition, if $L$ is of $H$-type at $p$, then so is $L'$ for any $L' \in {\text{gen}}(L)$.
\[odd\] Let $L$ be a primitive ternary ${{\mathbb Z}}$-lattice and let $p$ be an odd prime. Assume that after scaling by a unit in ${{\mathbb Z}}_p$ suitably, $$L_p \simeq \langle1,p^\alpha{{\epsilon}}_1,p^\beta{{\epsilon}}_2\rangle,$$ where $\alpha,\beta (\alpha\leq\beta)$ are nonnegative integers and ${{\epsilon}}_1,{{\epsilon}}_2 \in \{1,\Delta_p\}$. If $L$ is not of $H$-type at $p$, then the pairs $(\alpha,\beta)$, $({{\epsilon}}_1,{{\epsilon}}_2)$ satisfy one of the conditions in Table 1.
[ll]{}\
$(\alpha, \beta)$ & $(\epsilon_1, \epsilon_2)$\
$(1,2)$&$(1,1)$\
$(1,2)$&$(\Delta,1)$\
$(2,k),~(k \geq 3)$&$(1,1)$\
$(2,2k+1),~(k\geq1)$&$(1,\Delta)$\
By 102:7 of [@om], we know that $$g(L)=[J_{{{\mathbb Q}}} : P_DJ_{{{\mathbb Q}}}^L] \qquad \text{and} \qquad g(\lambda_p(L)) = [J_{{{\mathbb Q}}} : P_DJ_{{{\mathbb Q}}}^{\lambda_p(L)}],$$ where $D$ is the set of positive rational numbers. Clearly, $\theta(O^+(\lambda_p(L)_q))=\theta(O^+(L_q))$ for any prime $q\neq p$. Now one may easily check that if the pairs $(\alpha,\beta)$, $({{\epsilon}}_1,{{\epsilon}}_2)$ do not satisfy one of the conditions in Table 1, then $\theta(O^+(\lambda_p(L)_p))=\theta(O^+(L_p))$. This implies that $g(L)=g(\lambda_p(L))$.
\[even\] Let $L$ be a primitive ternary ${{\mathbb Z}}$-lattice. If $L$ is not of $H$-type at $2$, then there is an $\eta \in {{\mathbb Z}}_2^{\times}$ such that $$(L^{\eta})_2\simeq\langle1,2^\alpha{{\epsilon}}_1,2^\beta{{\epsilon}}_2\rangle,$$ where $\alpha, \beta (\alpha \leq \beta)$ are nonnegative integers and ${{\epsilon}}_1,{{\epsilon}}_2\in{{\mathbb Z}}_2^{\times}$, and the pairs $(\alpha,\beta)$, $({{\epsilon}}_1,{{\epsilon}}_2)$ satisfy one of the conditions in Table 2.
The proof is quite similar to the above lemma. For the computation of the spinor norm map, see [@local].
[ll||ll]{}\
$(\alpha, \beta)$ & $(\epsilon_1, \epsilon_2)$ & $(\alpha, \beta)$ & $(\epsilon_1, \epsilon_2)$\
$(0,4)$ &$\epsilon_1 \equiv \epsilon_2 \equiv 1~ (4)$ & $(5,6)$ &$2\epsilon_1+\epsilon_2 \in Q(\langle1,2\epsilon_1 \rangle)$\
$(1,6)$ &$\epsilon_2 \in Q(\langle 1, 2\epsilon_1 \rangle)$ & $(5,7)$ & $\epsilon_1\epsilon_2\equiv1~(4)$\
$(2,2)$ &$\epsilon_1=1,~\epsilon_2\equiv3~(4)$& $(5,8)$ &$\epsilon_2\equiv2\epsilon_1+5~(8)$\
$(2,4)$ & $\epsilon_1\equiv1~(4)$& $(5,9)$ & $\epsilon_1\epsilon_2\equiv 1~(4)$\
$(2,6)$ & $\epsilon_1\equiv1~(4)$ & $(5,2k),~(k\geq 5)$ &$1+2\epsilon_1\nequiv \epsilon_2~(8)$\
$(2,2k-1),~(k\geq 4)$ &${{\epsilon}}_1\equiv2{{\epsilon}}_2+3~(8)$& $(5,2k+1),~(k\geq 5)$ & $1+2\epsilon_1\nequiv \epsilon_1\epsilon_2~(8)$\
$(2,2k),~(k\geq 4)$ & $\epsilon_1\equiv1~(4)$ & $(6,7)$ & $5\notin Q(\langle\epsilon_1,2\epsilon_2\rangle)$\
$(3,6)$ & $\epsilon_2\equiv1~(8)$ & $(6,9)$ & $5\notin Q(\langle\epsilon_1,2\epsilon_2\rangle)$\
$(4,4)$ & $\epsilon_1\equiv \epsilon_2\equiv1~(4)$& $(6,2k-1),~(k\geq 6)$ &$\epsilon_1\nequiv5~(8)$\
$(5,5)$ & $\epsilon_2\equiv3\epsilon_1+6~(8)$ & $(6,2k),~(k\geq6)$ & $\epsilon_1,\epsilon_2\nequiv5~(8)$ and\
&&&${{\epsilon}}_1\nequiv {{\epsilon}}_2 \Rightarrow {{\epsilon}}_1 ~\text{or}~{{\epsilon}}_2 \equiv 1 ~(8)$\
\[H-type rmk\] Let $p$ be a prime and let $L$ be a primitive ternary ${{\mathbb Z}}$-lattice. As a function from ${\text{gen}}(L)_S$ to ${\text{gen}}(\lambda_p(L))_S$, $\lambda_p$ is a $2^a$ to one function for some $a=0,1$ or $2$.
Note that if $L$ is not of $H$-type at $p$, then $$|\theta(O^+(\lambda_p(L_p)))|=\begin{cases} 4\cdot |\theta(O^+(L_p))| &\text{if}~p=2,~ (\alpha,\beta)=(2,4) \ \text{and}~{{\epsilon}}_1\equiv{{\epsilon}}_2\equiv1~(4),\\
2\cdot|\theta(O^+(L_p))| &\text{otherwise}.\end{cases}$$ Suppose that $\lambda_p({\text{spn}}(L))={\text{spn}}(M)$ with $\lambda_p(L)=M$. For any ${\text{spn}}(M')\in {\text{gen}}(M)_S$, there is a split rotation $\Sigma\in J_V$ such that $M'=\Sigma M$. Since $$\lambda_p(\Sigma L)=\Sigma\lambda_p(L)=\Sigma M=M',$$ we have $\lambda_p({\text{spn}}(\Sigma L))={\text{spn}}(M')$. Note that $L'\in{\text{spn}}(L'')$ if and only if $\Sigma L'\in{\text{spn}}(\Sigma L'')$, for any $L',L''\in{\text{gen}}(L)$. Therefore $\vert\lambda_p^{-1}({\text{spn}}(M))\vert$ is independent of the choices of $M \in {\text{gen}}(\lambda_p(L))$. The lemma follows from this and the fact that $\lambda_p$ is surjective and the number of spinor genera in any genus of a ternary quadratic form is a power of $2$.
$\Gamma_p$-transformations on the set of spinor genera
======================================================
Let $V$ be a (positive definite) ternary quadratic space and let $L$ be a primitive ternary ${{\mathbb Z}}$-lattice on $V$. Let $p$ be a prime such that $$L_p \simeq \begin{pmatrix} 0&\frac12\\ \frac12&0\end{pmatrix} \perp \langle \epsilon \rangle,~ \text{where}~ \epsilon \in {{\mathbb Z}}_p^{\times}.$$ For any nonnegative integer $m$, let $\mathcal G_{L,p}(m)$ be a genus on $W$ such that each ${{\mathbb Z}}$-lattice $M \in \mathcal G_{L,p}(m)$ satisfies $$M_p \simeq \begin{pmatrix} 0&\frac12\\ \frac12&0\end{pmatrix} \perp \langle \epsilon p^m \rangle \quad \text{and} \quad M_q \simeq (L^{p^m})_q \ \text{ for any $q \ne p$}.$$ Here $W=V$ if $m$ is even, and $W=V^p$ otherwise.
For a nonnegative integer $m$, let $N \in \mathcal{G}_{L,p}(m+1)$ be a primitive ternary ${{\mathbb Z}}$-lattice. By Weak Approximation Theorem, there exists a basis $\{x_1, x_2, x_3 \}$ for $N$ such that $$(B(x_i,x_j))\equiv\begin{pmatrix}0&\frac12\\ \frac12&0\end{pmatrix}\perp \langle p^{m+1} \delta\rangle \ (\text{mod} \ p^{m+2}),$$ where $\delta$ is an integer not divisible by $p$. We define two sublattices of $N$ $$\aligned
&\Gamma_{p,1}(N) = {{\mathbb Z}}px_1 + {{\mathbb Z}}x_2+ {{\mathbb Z}}x_3, &\Gamma_{p,2}(N) = {{\mathbb Z}}x_1 + {{\mathbb Z}}px_2+ {{\mathbb Z}}x_3.
\endaligned$$ Note that $\Gamma_{p,i}(N)$ for $i=1,2$ depends on the choices of basis for $N$. However, the set $\{\Gamma_{p,1}(N), \Gamma_{p,2}(N)\}$ of sublattices of $N$ is independent of the choices of basis for $N$. In fact, these two sublattices are unique sublattices of $N$ with index $p$ whose norm is $p{{\mathbb Z}}$. We say that a ${{\mathbb Z}}$-lattice $M$ is [*a $\Gamma_p$-descendant of $N$*]{} if $M \simeq \Gamma_{p,i}^{\frac1p}(N)$ for some $i=1,2$.
\[exchange\] Let $p,q$ be distinct primes and let $N\in\mathcal{G}_{L,p}(m+1)$ for some nonnegative integer $m$.
1. If $M$ is a $\Gamma_p$-descendant of $N$, then $\lambda_q(M)$ is a $\Gamma_p$-descendant of $\lambda_q(N)$.
2. Assume that $N\in \mathcal{G}_{L',q}(m'+1)$ for some nonnegative integer $m'$. Then any $\Gamma_q$-descendant of a $\Gamma_p$-descendant of $N$ is a $\Gamma_p$-descendant of some $\Gamma_q$-descendant of $N$.
Note that if $p,q$ are distinct primes, then $(\Gamma_{p,i}(N))_q=N_q$ and $(\Lambda_p(N))_q=N_q$. The lemma follows directly from this.
In [@graph], we defined a multi-graph $\mathfrak{G}_{L,p}(m)$ and proved some properties of this graph. For those who are unfamiliar with the notations, we introduce the definition of this multi-graph briefly: the set of vertices in ${{\mathfrak{G}_{L,p}(m)}}$ is the set of equivalence classes in $\mathcal G_{L,p}(m)$, say, $\{[M_1], [M_2], \ldots , [M_h] \}$. The set of edges is exactly the set of equivalence classes in $\mathcal G_{L,p}(m+1)$, say, $ \{[N_1], [N_2],\ldots,[N_k] \}$. For each equivalence class $[N_w] \in\mathcal G_{L,p}(m+1)$, two vertices contained in the edge $[N_w]$ are defined by $[\Gamma_{p,1}(N_w)^{\frac 1p}]$ and $[\Gamma_{p,2}(N_w)^{\frac1p}]$ that are defined above. Note that both lattices are contained in $\mathcal G_{L,p}(m)$. Hence this graph might have loops or multiple edges.
Two vertices $[T_i], [T_j] \in \mathfrak{G}_{L,p}(0)$ are connected by an edge if and only if there are ${{\mathbb Z}}$-lattices $T_i' \in [T_i]$ and $T_j' \in [T_j]$ such that $T_i'$ and $T_j'$ are connected by an edge in the graph $Z(T,p)$ which is defined in [@sp1]. If two lattices $T_i ,T_j \in \mathcal G_{L,p}(0)$ are spinor equivalent, then both $[T_i]$ and $[T_j]$ are contained in the same connected component. Moreover, the set of vertices in each connected component of $\mathfrak{G}_{L,p}(0)$ consists of at most two spinor genera, and it consists of only one spinor genus if and only if $\bold{j}(p) \in P_D J_{{{\mathbb Q}}}^K$, where $D$ is the set of positive rational numbers and $$\bold{j}(p) = (j_{q}) \in J_{{\mathbb Q}}\quad \text{such that $j_p =p$ and $j_q = 1$ for any prime $q \ne p$}.$$ We say that $\mathfrak{G}_{L,p}(0)$ is of $O$-type if the set of vertices in the connected component of the graph $\mathfrak{G}_{L,p}(0)$ consists of only one spinor genus, and it is of $E$-type otherwise.
Now, we consider the general case. For any positive integer $m$, we say that a graph $\mathfrak G_{L,p}(m)$ is of $E$-type if $m$ is even and $\mathfrak G_{L,p}(0)$ is of $E$-type, and it is of $O$-type otherwise.
Assume that $\mathfrak G_{L,p}(m)$ is of $E$-type and $M \in \mathcal G_{L,p}(m)$. Since the map $$\lambda_p^{\frac m2} : {\text{spn}}(T) \to {\text{spn}}(\lambda_p^{\frac m2}(T))$$ is surjective for any $T \in \mathcal G_{L,p}(m)$, there is a ${{\mathbb Z}}$-lattice $M' \in \mathcal G_{L,p}(m)$ such that $M' \not \in {\text{spn}}(M)$ and $[M']$ is connected to $[M]$ by a path by Lemma 3.5 in [@graph]. Note that $g(\mathcal G_{L,p}(m))=g(\mathcal G_{L,p}(m'))$ if and only if $m\equiv m' \pmod 2,$ where $g(\mathcal G_{L,p}(m))$ is the number of spinor genera contained in $\mathcal G_{L,p}(m)$. In particular, $g(\mathcal G_{L,p}(m))=g(\mathcal G_{L,p}(0))$ for any even $m$. So, every ${{\mathbb Z}}$-lattice $M'$ satisfying the above condition is contained in a single spinor genus. From the existence of such a ${{\mathbb Z}}$-lattice $M'$, we may define $$\text{Cspn}(M)=\begin{cases} \text{spn}(M) \quad &\text{if $\mathfrak G_{L,p}(m)$ is of $O$-type,}\\
\text{spn}(M) \cup {\text{spn}}(M') \quad &\text{otherwise}.
\end{cases}$$ In Lemma 3.10 of [@graph], we proved that the set of all vertices in the connected component of $\mathfrak G_{L,p}(m)$ containing $[M]$ is the set of equivalence classes in $\text{Cspn}(M)$.
\[coco\] For an integer $m\geq0$, let $[N] \in \mathcal G_{L,p}(m+1)$ be an edge of the graph $\mathfrak G_{L,p}(m)$. Then the set of all edges in the connected component of $\mathfrak G_{L,p}(m)$ containing $[N]$ is the set of all classes in $\text{Cspn}(N)$.
It suffices to show that the set of edges in the connected component of $\mathfrak G_{L,p}(m)$ containing $[N]$ is exactly the set of vertices in the connected component of $\mathfrak G_{L,p}(m+1)$ containing the vertex $[N]$ by Lemma 3.10 of [@graph]. Note that if $N_1$ and $N_2$ are different $\Gamma_p$-descendant of $K$ for some $K \in \mathcal G_{L,p}(m+2)$, then $\lambda_p(K)$ is a $\Gamma_p$-descendant of both $N_1$ and $N_2$. This implies that every class in $\text{Cspn}(N)$ is contained in the set of edges in the connected component of $\mathfrak G_{L,p}(m)$ containing $[N]$. Conversely, assume that $[N']$ is contained in the set of edges in the connected component of $\mathfrak G_{L,p}(m)$ containing $[N]$. Without loss of generality, we may assume that there is a ${{\mathbb Z}}$-lattice $M$ that is a $\Gamma_p$-descendant of both $N$ and $N'$. If $m=0$ or $m\ge 1$ and $\lambda_p(N) \ne \lambda_p(N')$, then there is a ${{\mathbb Z}}$-lattice $K$ whose $\Gamma_p$-descendants are both $N$ and $N'$ by Lemmas 3.2 and 3.3 of [@graph], that is, as vertices, $[N]$ and $[N']$ are contained in the edge $[K]$. Now suppose that $m\ge1$ and $\lambda_p(N)= \lambda_p(N')$. Then in this case, there is a ${{\mathbb Z}}$-lattice $S \in \mathcal G_{L,p}(m+1)$ such that $\lambda_p(S)\ne \lambda_p(N)$ and one of $\Gamma_p$-descendants of $S$ is $M$ (see Lemma 3.3 of [@graph]). Hence there are edges containing $\{[N], [S]\}$ and $\{[S], [N']\}$ in the graph $\mathfrak G_{L,p}(m+1)$. This completes the proof.
genus-correspondences
=====================
Let $n$ be a positive integer. Let $M$ be a ternary $\mathbb{Z}$-lattice on a quadratic space $V$ and let $N$ be a $\mathbb{Z}$-lattice on $V^n$. Assume that there is a representation $$\phi : M^n \to N \ \ \text{such that} \ \ [N:\phi(M^n)]=n.$$ Then clearly, $N^n$ is also represented by $M$. For any ${{\mathbb Z}}$-lattice $M_1 \in {\text{gen}}(M)$, since $(M_1^n)_p \simeq (M^n)_p {{\rightarrow}}N_p$ for any prime $p$, there is a ${{\mathbb Z}}$-lattice $N_1 \in {\text{gen}}(N)$ that represents $(M_1)^n$. Conversely, for any ${{\mathbb Z}}$-lattice $N' \in {\text{gen}}(N)$, there is a ${{\mathbb Z}}$-lattice $M' \in {\text{gen}}(M)$ such that $(M')^n {{\rightarrow}}N'$ (see [@Jagy]). For $M_1 \in {\text{gen}}(M)$ and $N_1 \in {\text{gen}}(N)$ such that $(M_1)^n {{\rightarrow}}N_1$, the pair $([N_1],[M_1]) \in {\text{gen}}(N)/\sim\times {\text{gen}}(M)/\sim$ is called [*a representable pair by scaling $n$*]{}. A subset $\mathfrak C \subset {\text{gen}}(N)/\sim\times {\text{gen}}(M)/\sim$ consisting of representable pairs by scaling $n$ is called [*a genus-correspondence*]{} if for any $N' \in {\text{gen}}(N)$, there is an $M' \in {\text{gen}}(M)$ such that $([N'],[M']) \in \mathfrak C$, and vice versa. We say a genus-correspondence $\mathfrak C$ [*respects spinor genus*]{} if for any two $([N_1],[M_1]), ([N_2],[M_2]) \in \mathfrak C$, $$N_1 \in {\text{spn}}(N_2) \qquad \text{if and only if} \qquad M_1 \in {\text{spn}}(M_2).$$ Concerning this, Jagy conjectured in [@Jagy] that if $n$ is square free and $g(N)=g(M)$, then any genus-correspondence respects spinor genus. However, the following example shows that the conjecture is not true.
\[exam3\] [Let $N_1=\langle 12 \rangle \perp \begin{pmatrix} 15&5\\5&135 \end{pmatrix}$ and $M_1=\langle 1,~20,~80 \rangle$. Then one may easily check that $g(M_1)=g(N_1)=2$, $dN_1 = 15\cdot dM_1$ and $M_1^{15}$ is represented by $N_1$. The genus of $N_1$ consists of the following $12$ lattices up to isometry: $$\begin{array}{llll}
\!\!\!\!\! &N_1=\langle 12,15,135,5,0,0\rangle,\!\!\!\! &N_2=\langle3,7,1200,0,0,1 \rangle,\! \!\!&N_3= \langle 3,60,140,20,0,0 \rangle, \\
\!\!\!\!\! &N_4=\langle3,27,300,0,0,1 \rangle, \!\! \!\!&N_5=\langle 27,27,40,10,10,3\rangle,\! \!\!&N_6=\langle12,28,83,12,4,-4 \rangle,\\
\!\!\!\!\! &N_7=\langle12,28,75,0,0,4 \rangle, \!\! \!\! &N_8=\langle15,35,48,0,0,5 \rangle, \!\!\! &N_9=\langle7,12,300,0,0,2 \rangle, \\
\!\!\! \!\! &N_{10}=\langle12,43,60,20,0,6 \rangle, \!\! \! \! &N_{11}=\langle 8,12,303,4,-2,4 \rangle,\! \!\! &N_{12}=\langle12,35,60,10,0,0 \rangle.
\end{array}$$ Note that up to isometry, $$\text{spn}(N_1)=\{N_i : 1\le i\le6\}\quad \text{and} \quad \text{spn}(N_7)=\{ N_i : 7\le i\le12\}.$$ The genus of $M_1$ consists of the following $6$ lattices up to isometry: $$\begin{array}{llll}
&\hspace{-5mm}M_1=\langle1,20,80,0,0,0 \rangle, &M_2=\langle5,16,20,0,0,0 \rangle, &M_3=\langle4,20,25,10,0,0 \rangle, \\
&\hspace{-5mm}M_4=\langle4,5,80,0,0,0 \rangle, &M_5=\langle9,9,20,0,0,1 \rangle, &M_6=\langle4,20,21,0,2,0 \rangle.
\end{array}$$ Note that up to isometry, $$\text{spn}(M_1)=\{M_i : 1\le i\le3\} \quad \text{and}\quad \text{spn}(M_4)=\{ M_i : 4\le i\le6\}.$$ Define a genus-correspondence $\mathfrak{S}$ as follows: $$\aligned
\mathfrak{S}=\{&([N_1],[M_1]),([N_9],[M_1]),([N_3],[M_2]),([N_7],[M_2]),\\
&([N_5],[M_3]),([N_{11}],[M_3]),([N_2],[M_4]),([N_8],[M_4]),\\
&([N_6],[M_5]),([N_{10}],[M_5]),([N_4],[M_6]),([N_{12}],[M_6])\}.
\endaligned$$ Then one may easily check that $\mathfrak{S}$ does not respect spinor genus.]{}
In the remaining, we show that if we take a genus-correspondence suitably, then it respects spinor genus under the assumption that $g(M)=g(N)$. We do not assume that $n$ is square free for a while.
\[spn representable\] For ternary $\mathbb{Z}$-lattices $N$ and $M$, assume that $([N],[M])$ is a representable pair by scaling $n$. Then for any $N'\in\text{spn}(N)$, there is a $\mathbb{Z}$-lattice $M'\in\text{spn}(M)$ such that $([N'],[M'])$ is a representable pair by scaling $n$. Conversely, for any $M''\in\text{spn}(M)$ there is a $\mathbb{Z}$-lattice $N''\in\text{spn}(N)$ such that $([N''],[M''])$ is a representable pair by scaling $n$.
Since $([N],[M])$ is a representable pair by scaling $n$, $\sigma(M^n)\subseteq N$, for some isometry $\sigma\in O(V)$. Let $N'\in\text{spn}(N)$. Then there are $\sigma'\in O(V)$ and $\Sigma\in J'_V$ such that $N'=\sigma'\Sigma N$. If we define $M'=\sigma'\Sigma\sigma(M)=\sigma'\sigma(\sigma^{-1}\Sigma\sigma) M \in \text{spn}(M)$, then $$(M')^n= \sigma'\Sigma\sigma (M^n) \subseteq \sigma'\Sigma N=N'.$$ The converse can be proved similarly.
A bipartite graph with partitions $U$ and $V$ of vertices and with $E$ of edges is denoted by $\mathfrak G(U,V,E)$, or simply $\mathfrak G(U,V)$. For each vertex $u \in U$ of the bipartite graph $\mathfrak G(U,V,E)$, we define $\mathcal N(u)=\{ v : uv \in E\}$. For a vertex $v \in V$, $\mathcal N(v)$ is defined similarly. The graph $\mathfrak G(U,V,E)$ is called [*$(a,b)$-regular*]{} if $\mathcal N(u)=a$ for any $u \in U$, and $\mathcal N(v)=b$ for any $v \in V$.
For two bipartite graphs $\mathfrak G(U,V,E)$ and $\mathfrak G(V,W,E')$, we define a [*juxtaposition bipartite graph*]{} of two bipartite graphs, denoted by $\mathfrak G_V(U,W,\tilde E)$, as follows; $U$ and $W$ are partitions of vertices and there is an edge $uw \in \tilde E$ for $u \in U$ and $w \in W$ if and only if there is a vertex $v \in V$ such that $uv \in E$ and $vw \in E'$.
For a representable pair $([N],[M])$ by scaling $n$, we define a bipartite graph $$\mathfrak{G}(N,M)=\mathfrak G({\text{gen}}(N)_S,{\text{gen}}(M)_S)$$ such that two vertices $\text{spn}(N')\in\text{gen}(N)_S$ and $\text{spn}(M')\in\text{gen}(M)_S$ are connected by an edge if and only if there are lattices $N''\in \text{spn}(N')$ and $M''\in\text{spn}(M')$ such that $([N''],[M''])$ is a representable pair by scaling $n$.
\[regular\] Let $N$ and $M$ be two ${{\mathbb Z}}$-lattices such that $([N],[M])$ is a representable pair by scaling $n$. Then for some positive integers $u,v$ such that $ug(N)=vg(M)$, the graph $\mathfrak G(N,M)$ is $(u,v)$-regular. In particular, if $g(N)=g(M)$, then the graph $\mathfrak{G}(N,M)$ is a regular bipartite graph.
Let $\text{spn}(N')\in\text{gen}(N)_S$ and $\text{spn}(M')\in\text{gen}(M)_S$ be two vertices the graph $\mathfrak{G}(N,M)$ such that $\text{spn}(M')\in \mathcal{N}(\text{spn}(N'))$. By Lemma \[spn representable\], we may assume that $(N',M')$ is a representable pair by scaling $n$, that is, there is a representation $\phi \in O(V)$ such that $\phi((M')^n)\subset N'$. Let $\text{spn}(N'')$ be another vertex in $\text{gen}(N)_S$. Choose a split rotation $\Sigma\in J_V$ such that $N''=\Sigma N'$. Since $$\phi(\phi^{-1}\Sigma\phi\Sigma^{-1}(\Sigma(M')^n))\subset \Sigma(N'),$$ $(\Sigma N',\phi^{-1}\Sigma\phi\Sigma^{-1}(\Sigma(M')))$ is a representable pair by scaling $n$. Furthermore, since $\phi^{-1}\Sigma\phi\Sigma^{-1} \in J_V'$, we have $\text{spn}(\Sigma M') \in \mathcal{N}(\text{spn}(N''))$. Note that for any two lattices $M',M''\in \text{gen}(M)$, $M'\in\text{spn}(M'')$ if and only if $\Sigma M'\in\text{spn}(\Sigma M'')$. Therefore $$| \mathcal{N}(\text{spn}(N'))|= |\mathcal{N}(\text{spn} (N''))|.$$ Similarly, we also have $| \mathcal{N}(\text{spn}(M'))|= |\mathcal{N}(\text{spn} (M''))|$ for any $M', M'' \in {\text{gen}}(M)$. The lemma follows from this.
\[exist\] Let $N$ and $M$ be two ${{\mathbb Z}}$-lattices such that $([N],[M])$ is a representable pair by scaling $n$. If $g(N)=g(M)$, then there is a genus-correspondence respecting spinor genus.
We may assume that $${\text{gen}}(N)_S =\{ {\text{spn}}(N_i) : i=1,2,\dots,g\}\ \ \text{and} \ \ {\text{gen}}(M)_S =\{ {\text{spn}}(M_i) : i=1,2,\dots,g\}.$$ Since the graph $\mathfrak{G}(N,M)$ defined above is a regular bipartite graph, there is a perfect matching by Hall’s marriage theorem. Hence, without loss of generality, we may assume that each $([N_i],[M_i])$ is a representable pair by scaling $n$. We define a genus-correspondence $\mathfrak{S}$ as follows: for $([N'],[M'])\in\text{gen}(N)/\sim\times\text{gen}(M)/\sim$, $([N'],[M'])\in\mathfrak{S}$ if and only if $([N'],[M'])$ is a representable pair by scaling $n$ and there is an $i \ (1\le i\le g)$ such that $N'\in {\text{spn}}(N_i)$ and $M' \in {\text{spn}}(M_i)$. Then by Lemma \[spn representable\], $\mathfrak{S}$ is a genus-correspondence respecting spinor genus.
Genus-correspondence respecting spinor genus
============================================
From now on, we assume that $n$ is a square free positive integer. Let $N$ and $M$ be ternary ${{\mathbb Z}}$-lattices such that the pair $([N], [M])$ is a representable pair by scaling $n$. In this section, We find a necessary and sufficient condition for the genus-correspondence ${\text{gen}}(N)/\sim\times{\text{gen}}(M)/\sim$ to respect spinor genus under the assumption that $g(M)=g(N)$.
\[important\] Let $p$ be a prime and let $N$ and $M$ be primitive ternary ${{\mathbb Z}}$-lattices. Then $([N],[M])$ is a representable pair by scaling $p$ if and only if $M^p \simeq \Lambda_p(N)$ or $M$ is a $\Gamma_p$-descendant of $N$.
Without loss of generality, we may assume that $M^p$ is a sublattice of $N$ with index $p$. Then there is a basis $\{x_1,x_2,x_3\}$ for $N$ such that $M^p={{\mathbb Z}}x_1+{{\mathbb Z}}x_2+{{\mathbb Z}}(px_3)$. Hence there are integers $a,b,c,s,t,u$ such that $$(B(x_i,x_j))=\begin{pmatrix}pa&\frac{pu}2&\frac t2\\\frac{pu}2&pb&\frac s2\\\frac t2&\frac s2&c\end{pmatrix}.$$ Hence if $s\equiv t\equiv 0\pmod p$, then clearly, $M^p=\Lambda_p(N)$. If $s$ or $t$ is not divisible by $p$, then a Jordan decomposition of $N_p$ has an isotropic $\frac 12{{\mathbb Z}}_p$-modular component. Furthermore, $M^p$ is a sublattice of $N$ with index $p$ whose norm is $p{{\mathbb Z}}$. Therefore $M$ is a $\Gamma_p$-descendant of $N$. Note that the converse is almost trivial.
\[step-r\] For two ternary ${{\mathbb Z}}$-lattices $N$ and $M$, assume that $([N],[M])$ is a representable pair by scaling $n$. For any prime $p$ dividing $n$, there is a ${{\mathbb Z}}$-lattice $N(p)$ such that $([N],[N(p)])$ is a representable pair by scaling $p$, and $([N(p)],[M])$ is a representable pair by scaling $\frac np$.
By assumption, we may assume that $M^n$ is a sublattice of $N$ with index $n$. Choose a basis $\{x_1,x_2,x_3\}$ for $N$ such that $M^n={{\mathbb Z}}x_1+{{\mathbb Z}}x_2+{{\mathbb Z}}(nx_3)$. Define $N(p)=({{\mathbb Z}}x_1+{{\mathbb Z}}x_2+{{\mathbb Z}}(px_3))^{\frac1p}$. Then one may easily show that $\mathfrak n(N(p))={{\mathbb Z}}$ and $N(p)$ satisfies all conditions given above.
\[step-r2\] Let $a,b$ be positive integers such that $ab$ is square free. Let $([N],[L])$ and $([L],[M])$ be representable pairs of ternary ${{\mathbb Z}}$-lattices by scaling $a$ and $b$, respectively. Then the graph $\mathfrak G(N,M)$ is exactly same to the juxtaposition bipartite graph $\mathfrak G_{{\text{gen}}(L)_S}({\text{gen}}(N)_S,{\text{gen}}(M)_S)$.
It suffices to show that each set of edges for both graphs is same, which follows directly from the above lemma.
Let $([N],[M])$ be a representable pair of ternary ${{\mathbb Z}}$-lattices by scaling $p$, where $p$ is a prime. Then $\lambda_p(N) \simeq M$ or $M$ is a $\Gamma_p$-descendent of $N$ by Lemma \[important\]. If the former holds, then the bipartite graph $\mathfrak G(N,M)$ is $(1,1)$-regular or $(1,2)$-regular by Lemma \[H-type rmk\]. Furthermore, it is $(1,1)$-regular if and only if $N$ is of $H$-type at $p$. Note that $(1,4)$-regularity is impossible in our situation.
Now, assume that the latter holds. Then there is a ${{\mathbb Z}}$-lattice $L$ and a nonnegative integer $m$ such that $N \in \mathcal G_{L,p}(m+1)$ and $M \in \mathcal G_{L,p}(m)$. If the graph $\mathfrak G_{L,p}(m+1)$ is of $E$-type, then the bipartite graph $\mathfrak G(N,M)$ is $(1,2)$-regular by Lemma \[coco\], and if the graph $\mathfrak G_{L,p}(m)$ is of $E$-type, then the bipartite graph $\mathfrak G(N,M)$ is $(2,1)$-regular. Finally, if both $\mathfrak G_{L,p}(m+1)$ and $\mathfrak G_{L,p}(m)$ are of $O$-type, then the bipartite graph $\mathfrak G(N,M)$ is $(1,1)$-regular. Note that both $\mathfrak G_{L,p}(m+1)$ and $\mathfrak G_{L,p}(m)$ cannot be of $E$-type simultaneously. We say $N$ is of $(E,O)$-type ($(O,E)$-type) at $p$ if the graph $\mathfrak G_{L,p}(m+1)$ ($\mathfrak G_{L,p}(m)$, respectively) is of $E$-type. Finally, we say $N$ is of $(O,O)$-type if both $\mathfrak G_{L,p}(m+1)$ and $\mathfrak G_{L,p}(m)$ are of $O$-type.
Let $([N],[M])$ be a representable pair of ternary ${{\mathbb Z}}$-lattices by scaling $n$, where $n$ is a square free positive integer. Without loss of generality, we assume that $M^n$ is a sublattice of $N$ with index $n$. Choose a basis $\{x_1,x_2,x_3\}$ for $N$ such that $M^n={{\mathbb Z}}x_1+{{\mathbb Z}}x_2+{{\mathbb Z}}(nx_3)$. Let $n_2$ be a product of primes $p$ dividing $n$ such that the rank of a $\frac12{{\mathbb Z}}_p$-modular component in a Jordan decomposition of $N_p$ is two, and let $n_1$ be the integer satisfying $n=n_1n_2$. Let $n_2(e)$ be a product of primes $q$ dividing $n_2$ such that ${\text{ord}}_q(4\cdot dM) \equiv 0 \pmod 2$, and let $n_2(o)$ be the integer satisfying $n_2=n_2(e)n_2(o)$. Define a ternary ${{\mathbb Z}}$-lattice $$L_{N,M}=({{\mathbb Z}}x_1+{{\mathbb Z}}x_2+{{\mathbb Z}}(n_1n_2(e)x_3))^{\frac1{n_1n_2(e)}}.$$ Note that pair $([N],[L_{N.M}])$ ($([L_{N,M}],[M])$) is a representable pair by scaling $n_1n_2(e)$ ($n_2(o)$, respectively). Let $g_{N,M}=g(L_{N,M})$ be the number of (proper) spinor genera in the genus of $L_{N,M}$.
\[resol\] Let $n$ be a square free positive integer and let $([N],[M])$ be a representable pair by scaling $n$. Then, any connected component of the bipartite graph $\mathfrak G(N,M)$ is a complete $K_{\alpha,\beta}$-graph, where $$\alpha=\frac{g(M)}{g_{N,M}} \qquad \text{and} \qquad \beta=\frac{g(N)}{g_{N,M}}.$$
Without loss of generality, we assume that $M^n$ is a sublattice of $N$ with index $n$. Let $n_1n_2(e)=p_1p_2\dots p_s$ and $n_2(o)=q_1q_2\dots q_t$, where each $p_i$ and $q_j$ is a prime. By Lemma \[step-r2\], the graph $\mathfrak G(N,M)$ is a juxtaposition of the graphs $\mathfrak G(N,L_{N,M})$ and $\mathfrak G(L_{N,M},M)$, where $L_{N,M}$ is a ${{\mathbb Z}}$-lattice defined above. Let $\{x_1,x_2,x_3\}$ be a basis for $N$ such that $L_{N,M}^{n_1n_2(e)}={{\mathbb Z}}x_1+{{\mathbb Z}}x_2+{{\mathbb Z}}(n_1n_2(e)x_3)$. Define $$N(i)=({{\mathbb Z}}x_1+{{\mathbb Z}}x_2+{{\mathbb Z}}(p_1p_2\dots p_ix_3))^{\frac1{p_1p_2\dots p_i}}.$$ Then the graph $\mathfrak G(N,L_{N,M})$ is a juxtaposition of the graphs $$\mathfrak G(N,N(1)), \mathfrak G(N(1),N(2)), \dots, \mathfrak G(N(s-1),L_{N,M}),$$ all of which are either a $(1,1)$-regular graph or a $(1,2)$-regular graph. Therefore the graph $\mathfrak G(N,L_{N,M})$ is a $\left(1,\frac {g(N)}{g_{N,M}}\right)$-regular graph. Similarly, one may easily check that the graph $\mathfrak G(L_{N,M},M)$ is a $\left(\frac{g(M)}{g_{N,M}},1\right)$-regular graph. The theorem follows from these two observations.
\[resol2\] Let $n$ be a square free positive integer and let $([N],[M])$ be a representable pair by scaling $n$. Assume that $g(N)=g(M)$. Then $g(N)=g_{N,M}$ if and only if the genus-correspondence ${\text{gen}}(N)/\sim\times{\text{gen}}(M)/\sim$ respects spinor genus.
Note that the genus-correspondence ${\text{gen}}(N)/\sim\times{\text{gen}}(M)/\sim$ respects spinor genus if and only if the graph $\mathfrak G(N,M)$ is $(1,1)$-regular. Hence the corollary follows directly from the above theorem.
Recall that we are assuming that $n$ is a square free positive integer and $([N],[M])$ is a representable pair by scaling $n$. Now we further assume that $g(N)=g_{N,M}=g(M)$, that is, the genus-correspondence ${\text{gen}}(N)/\sim\times{\text{gen}}(M)/\sim$ respects spinor genus by Theorem \[resol\]. Assume that $${\text{gen}}(N)_S=\{{\text{spn}}(N_i):i=1,2,\dots,g\} \ \ \text{and}\ \ {\text{gen}}(M)_S=\{{\text{spn}}(M_i):i=1,2,\dots,g\},$$ and there is a unique edge containing ${\text{spn}}(N_i)$ and ${\text{spn}}(M_i)$ in the graph $\mathfrak{G}(N,M)$ for any $i=1,2,\dots,g$.
Let $k$ be a positive integer. Under the assumptions given above, ${\text{spn}}(N_i)$ represents $nk$ if and only if ${\text{spn}}(M_i)$ represents $k$, for any $i=1,2,\dots,g$.
First assume that $n=p$ is a prime. Then $\lambda_p(N)\simeq M$ or $M$ is a $\Gamma_p$-descendant of $N$ by Lemma \[important\]. If the former holds, one may easily show that $N$ represents $pk$ if and only if $M$ represents $k$. If the latter holds, then also one may easily show that $N$ represents $pk$ if and only if at least one of $\Gamma_p$-descendants of $N$ represents $k$. Hence the lemma follows directly from this.
Let $n=p_1p_2\cdots p_r$, where each $p_i$ is a prime. We may assume that $M^n$ is a sublattice of $N$ with index $n$. Let $\{x_1,x_2,x_3\}$ be a basis for $N$ such that $M^n={{\mathbb Z}}x_1+{{\mathbb Z}}x_2+ {{\mathbb Z}}(nx_3)$. Define $$N(i)=({{\mathbb Z}}x_1+{{\mathbb Z}}x_2+{{\mathbb Z}}(p_1p_2\dots p_ix_3))^{\frac1{p_1p_2\dots p_i}}.$$ Then the graph $\mathfrak G(N,M)$ is a juxtaposition of the graphs $$\mathfrak G(N,N(1)), \mathfrak G(N(1),N(2)), \dots, \mathfrak G(N(r-1),M).$$ Since $g(N)=g_{N,M}=g(M)$, we know that each graph $\mathfrak G(N(i),N(i+1))$ is a $(1,1)$-regular graph. Hence the lemma follows directly from the induction on $r$.
A set $S=\{c_1,c_2,\dots,c_g\}$ of integers is said to be a complete system of spinor exceptional integers for ${\text{gen}}(L)$, for some ternary ${{\mathbb Z}}$-lattice $L$, if for any subset $U \subset S$, there is a unique ${\text{spn}}(L') \in {\text{gen}}(L)_S$ such that every integer in $U$ is represented by ${\text{spn}}(L')$ and every integer in $S-U$ is not represented by ${\text{spn}}(L')$. For details, see [@bh].
\[complete sys\] Under the same assumptions given above, suppose that there is a complete system $\{c_1,c_2,\dots,c_g\}$ of spinor exceptional integers for ${\text{gen}}(M)$. Then $\{nc_1,$ $nc_2, \cdots,nc_g\}$ is a complete system of spinor exceptional integers for ${\text{gen}}(N)$.
The corollary follows directly form the above lemma.
The following example was first introduced by Jagy in [@Jagy].
[Let $n$ be a square free integer that is represented by a sum of two integral squares. Define ternary ${{\mathbb Z}}$-lattices $$N=\langle1,1,16n\rangle \ \ \text{and}\ \ M=\langle1,1,16\rangle.$$ Then one may easily check that $([N],[M])$ is a representable pair by scaling $n$ and $g(N)=g_{N,M}=g(M)=2$. Note that $\{1\}$ is a complete system of spinor exceptional integers for ${\text{gen}}(M)$. By Corollary \[complete sys\], we know that $\{n\}$ is a complete system of spinor exceptional integers for ${\text{gen}}(N)$. In fact, Jagy proved in [@Jagy] that $n$ is represented by any ${{\mathbb Z}}$-lattice in ${\text{spn}}(N)$. However one may easily show that it holds even if $n$ is not square free.]{}
[abcd]{}
J. W. Benham and J. S. Hsia, [*On spinor exceptional representations*]{}, Nagoya Math. J. $\mathbf{87}$(1982), 247–260.
W. K. Chan and A.G. Earnest, [*Discriminant bounds for spinor regular ternary quadratic lattices*]{}, J. London Math. Soc. (2) $\mathbf{69}$(2004), 545–561.
W. K. Chan, B.-K. Oh, [*Finiteness theorems for positive definite n -regular quadratic forms*]{}, Trans. Amer. Math. Soc. $\mathbf{355}$(2003), 2385–2396.
W. Duke, and R. Schulze-Pillot, [*Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids*]{}, Invent. Math. $\mathbf{99}$(1990), 49-?57.
A. G. Earnest and J. S. Hsia, [*Spinor norms of local integral rotations, [II]{}*]{}, Pacific J. Math. $\mathbf{61}$(1975), 71–86.
J. S. Hsia and M. Jöchner, [*Almost strong approximations for definite quadratic spaces*]{} Invent. Math. $\mathbf{129}$(1997), 471-487.
W. C. Jagy, [*Integral positive ternary quadratic forms*]{}, Quadratic and higher degree forms, Dev. Math. $\mathbf {31}$(2013), 169–179.
J. Ju, I. Lee and B.-K. Oh, [*A generalization of Watson transformation and representations of ternary quadratic forms*]{}, J. Number Theory $\mathbf {167}$(2016), 202–231.
Y. Kitaoka, [*Arithmetic of quadratic forms*]{}, Cambridge University Press, 1993.
O. T. O’Meara, [*Introduction to quadratic forms*]{}, Springer Verlag, New York, 1963.
R. Schulze-Pillot, [*Darstellung durch definite ternare quadratische Formen und das Bruhat- Tits-Gebaude der Spingruppe*]{}, Dissertation U, Göttingen 1979.
R. Schulze-Pillot, [*Darstellung durch spinorgeschlechter ternärer quadratischer formen*]{} , J. Number Theory $\mathbf{12}$(1980), 529–540.
[^1]: This work of the second author was supported by the National Research Foundation of Korea (NRF-2014R1A1A2056296).
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---
abstract: 'We use the measured X-ray luminosity function (XLF) of high-mass X-ray binaries (HMXBs) in nearby star-forming galaxies to constrain the common envelope (CE) mechanisms, which play a key role in governing the binary evolution. We find that the XLF can be reproduced quite closely under both CE mechanisms usually adopted, i.e., the $\alpha_{\rm CE}$ formalism and the $\gamma$ algorithm, with a reasonable range of parameters considered. Provided that the parameter combination is the same, the $\gamma$ algorithm is likely to produce more HMXBs than the $\alpha_{\rm CE}$ formalism, by a factor of up to $\sim$ 10. In the framework of the $\alpha_{\rm CE}$ formalism, a high value of $\alpha_{\rm CE}$ is required to fit the observed XLF, though it does not significantly affect the global number of the HMXB populations. We present the detailed components of the HMXB populations under the $\gamma$ algorithm and compare them with those in Zuo et al. and observations. We suggest the distinct observational properties, as well as period distributions of HMXBs, may provide further clues to discriminate between these two types of CE mechanisms.'
author:
- 'Zhao-Yu Zuo$^{1,3}$ and Xiang-Dong Li$^{2,3}$'
title: 'Common Envelope Mechanisms: Constraints from the X-ray Luminosity Function of High Mass X-ray Binaries'
---
Introduction
============
The common envelope (CE) evolution is among the most important and least well-constrained processes in binary evolution. It is commonly thought to occur if the mass transfer is dynamically unstable. The result is that the accreting star spirals into the envelope of the donor star [see @iben93; @taam00; @webbink08; @taam10; @Ivanova13 for reviews]. The orbital energy and angular momentum of the accreting star are then transferred into the CE via an as of yet unknown mechanism. This may end with a stellar merger or, if the binary can survive, a binary with a much shorter orbital period. The CE evolution is critical in the formation of various kinds of compact binaries.
There have been extensive three dimensional hydrodynamical simulations [e.g., @rl96; @stc98; @stb00; @Fryxell00; @OShea05; @Fryer06; @Passy12; @rt12]. However the physics of CE evolution still remains poorly understood, primarily due to a mix of various kinds of physical processes operating over a large range of timescales and length scales during the CE phase. Due to the difficulties in modeling the detailed CE evolution, population synthesis simulations commonly resort to simplified and parameterized descriptions to relate the post- and pre-CE orbital parameters [@ty79]. One such parametrization dictates the CE phase in terms of a simple energy budget [known as the $\alpha_{\rm CE}$ formalism, @heuvel76; @Webbink84; @ls88; @iben93] and the other in terms of the angular momentum budget [named as the $\gamma$ algorithm, @nvyp00; @nt05]. Both approaches have the power to account for some specific classes of post-CE binaries (PCEBs), such as cataclysmic variables [CVs, @king88], subdwarf B binaries [@mr03; @han02; @han03], low-mass X-ray binaries [@cc06], and other compact objects thought to have suffered a merger, which are probably responsible for gamma-ray bursts [@fryer99; @Thone] and Type Ia supernova [SNe Ia, @it84; @bbr05; @ruiter11; @meng11; @my12; @wh12]. There is an energetic debate over the two approaches in the literature.
High-mass X-ray binaries (HMXBs) are good examples of the effect of CE evolution. Luminous HMXBs usually have experienced CE evolution so that they have close orbits, which lead to a high wind-capture rate by the compact star. Tight orbits also help the binary survive the SN kick during the formation of the compact star. However, if the initial binary orbit is not large enough, CE evolution may lead to mergers, reducing the HMXB production. Thus, the populations and the specific characteristics (for example the orbital period distribution) of HMXBs can be used to probe the physical interactions during the CE phase.
The formation of HMXBs involves several evolutionary pathways [@bv00; @Linden10 see also Tauris & van den Heuvel 2006]. Beginning with two relatively massive stars ($\gtrsim\,10\,M_{\odot}$), the more massive primary evolves and commences mass transfer to the secondary. The mass transfer can be either dynamically stable or unstable. In the latter case, CE evolution occurs that greatly shrinks the binary orbit. The resultant binary consists of the primary’s core and the secondary. The primary’s core then collapses to form a neutron star (NS) or black hole (BH). An HMXB appears when the compact star is able to accrete from the secondary by capture of the stellar wind or Roche lobe overflow (RLOF). Note that the secondary can be on the main-sequence (MS) or a (super)giant star. In some cases the second mass transfer may also lead to a CE phase, during which the envelope of the secondary is stripped, leaving a naked helium core. This leads to the formation of HMXBs with Wolf-Rayet companions.
HMXBs have some unique statistical characteristics [for catalogs, see @lph05; @lph06]. One of the most striking features is that their X-ray luminosity function (XLF) follows a universal power law form over a broad X-ray luminosity range, from $\sim 10^{35}$ to $\sim 10^{40} \rm \,erg\,s^{-1}$. This was first discovered by @ggs03, and further confirmed recently by @mineo12 [hereafter MGS12 for short]. The XLF has been shown to follow a power law with a single slope of $\sim$ 1.6, without any significant feature near the critical Eddington luminosity of an NS or a stellar mass BH. Additionally, the collective luminosity of HMXB populations scales with the star formation rate (SFR) as $L_{\rm X} (\rm erg\,s^{-1}) \approx 2.6\cdot10^{39}\times \rm SFR (M_{\odot}\,yr^{-1})$.
In the present work, we apply the updated evolutionary population synthesis (EPS) techniques to model the XLF of HMXBs, taking into account both the $\alpha_{\rm CE}$ algorithm and $\gamma$ algorithm (with different choices of $\alpha_{\rm CE}$ and $\gamma$, respectively), to describe the CE evolution. By comparing the observational sample with our theoretical expectations, we try to discriminate or constrain the effects of the two CE mechanisms.
This paper is organized as follows. In §2 we describe the population synthesis method and the input physics for X-ray binaries (XRBs) in our model. The calculated results and discussions are presented in §3. Our conclusions are in §4.
MODEL DESCRIPTION
=================
We use the EPS code developed by @Hurley00 [@Hurley02] and recently updated by @zuo14a to calculate the expected number and the X-ray luminosity of HMXBs. In the present code, the model for compact object formation has been significantly revised by taking into account the formation of NSs through electron capture supernovae [ECS, @Podsiadlowski04] and the fallback process for both delayed and direct BH formation during core collapse [@fk01]. The prescriptions for the wind mass loss rates of massive stars [@vink01 see also Belczynski et al. 2010] and the compact remnant masses [@fryer12 see also Belczynski et al. 2012] are adopted in the code. We also update the criteria for CE occurrence as described below.
The CE Phase
------------
During the binary evolution, the mass ratio ($q=M_{\rm donor}/M_{\rm accretor}$) is a crucial factor determining the stability of mass transfer. If it is larger than a critical value, $q_{\rm crit}$, the mass transfer is dynamically unstable and a CE phase follows [@p76]. The ratio $q_{\rm crit}$ varies with the evolutionary state of the donor star at the onset of RLOF and the mass loss mechanisms during the mass transfer [@hw87; @w88; @prp02; @ch08]. In this study, we adopt an updated $q_{\rm crit}$ for Hertzsprung gap donor star, recently calculated by @shao12 [also see the Appendix A in Zuo, Li & Gu 2014a for more details]. If the primary is on the first giant branch (FGB) or the asymptotic giant branch (AGB), we use $$q_{\rm crit}=[1.67-x+2(\frac{M_{\rm c1}}{M_1})^5]/2.13$$ where $M_{\rm c1}$ is the core mass of the donor star, and $x$=d ln$R_1/$d ln$M$ is the mass-radius exponent of the donor star. If the mass donor star is a naked helium giant, $q_{\rm crit}=0.784$ [see @Hurley02 for more details].
### The $\alpha_{\rm CE}$ formalism
In the energy budget approach, the CE evolution is parameterized in terms of the orbital energy and binding energy as $E_{\rm bind} \equiv \alpha_{\rm CE}
\triangle E_{\rm orb}$ [@Webbink84; @webbink08], where the parameter $\alpha_{\rm CE}$ describes the efficiency of converting the orbital energy (**$E_{\rm orb}$**) into the kinetic energy, which is used to eject the envelope, and $E_{\rm bind}$ is the binding energy of the envelope. The CE evolution is governed by the following equation [@Kiel06]: $$\alpha_{\rm CE}[\frac{GM_{\rm c}M_{2}}{2 a_{\rm f}}-\frac{GM_{\rm
c}M_{2}}{2 a_{\rm i}}]=-\frac{GM_1M_{\rm env}}{R_{L_1}\lambda},$$ which yields the ratio of final (post-CE) and initial (pre-CE) orbital separations as $$\frac{a_{\rm f}}{a_{\rm i}}= \frac{M_{\rm c}M_{2}}{M_{1}}
\frac{1}{M_{\rm c}M_2/M_1+2M_{\rm env}/(\alpha_{\rm CE} \lambda
R_{\rm L1})},$$ where $G$ is the gravitational constant, $M_{\rm c}$ the helium-core mass of the primary star (of mass $M_1$), $M_2$ the mass of the secondary star, $R_{L_1}$ the RL radius of the primary star, $M_{\rm env}$ the mass of the primary’s envelope, $a_i$ and $a_f$ denote the initial and final orbital separations, respectively, and $\lambda$ is a parameter related to the stellar mass-density distribution.
The $\lambda$ value depends on the structure and evolution of the donor star. However, in previous studies, it was usually adopted as constant ($\sim 0.5$) for simplicity [@Hurley02; @zuo08]. Here we calculate the values of $\lambda$ from detailed stellar models including the contribution from the internal (and ionization) energies within the envelope [@zuo14a also see Xu & Li 2010 and Loveridge et al. 2011].
We consider three constant, global values of $\alpha_{\rm CE}$. For our basic model, we use $\alpha_{\rm CE}=0.5$ [@zuo14a]. We also consider two other extreme values of $\alpha_{\rm CE}=1.0$ and 0.1 since $\alpha_{\rm CE}$ is expected to be no more than unity if we consider the internal energies in calculating $E_{\rm bind}$. Different CE efficiencies for the first and second CE episodes are also examined to test its effect on the XLFs. Models with different values of $\alpha_{\rm CE}$ are denoted as A01A01, A05A05, A10A10, A01A05, and A05A01, respectively, where the two digits following each letter correspond to the values of $\alpha_{\rm CE}$ during the first and second CE episodes, respectively.
Alternatively, recent studies on WD binaries show that $\alpha_{\rm CE}$ may be a function of binary parameters rather than constant [@pw07; @zgn00; @marco11; @dkk12], although the final relationship has not yet been well developed. Following @marco11, we adopt $$\alpha_{\rm CE}=0.05\times q^{1.2},$$ where $q$ is the ratio of the donor’s mass to the accretor’s mass at the time of the CE interaction, and this model is denoted as AqAq.
### The $\gamma$ algorithm
In the angular momentum budget approach, the CE interaction is parameterized in terms of $\gamma$, the ratio of the fraction of angular momentum lost, and the fraction of mass loss: $$\frac{\triangle J}{J}=\gamma \frac{M_{\rm env}}{M_1+M_2}$$ where $\triangle J$ is the change of the total angular momentum ($J$) during the CE phase. Implicitly assuming the conservation of energy, the orbital separation after the CE is then given by $$\frac{a_{\rm f}}{a_{\rm i}}=(\frac{M_1}{M_{\rm c}})^2(\frac{M_{\rm c}+M_2}{M_1+M_2})
[1-\gamma (\frac{M_{\rm env}}{M_1+M_2})]^2$$
This description was first suggested by @nvyp00 in their investigation of the formation of double WD binaries. They found that when the energy approach is applied to describe the first CE phase, a negative value of $\alpha_{\rm CE}$ is required, which is clearly unphysical. Among the possible solutions leading to the known close double WDs, @nt05 found that $0.5<\gamma<3$ for the first (putative) CE phase, and $1<\gamma<4$ for the second CE phase. They noted that a value of $\gamma$ between 1.5 and 1.75 can account for all known observed PCEBs, including double WDs, pre-CVs, and sdB plus MS binaries.
For the $\gamma$ algorithm, we consider several constant, global values of $\gamma$ from 1.7 to 1.0, as well as different $\gamma$ values for the first and second CE episodes in our calculation. These models are denoted as G17G17, G15G15, G13G13, G10G10, G10G17 and G17G10 where the two digits following each letter correspond to the values of $\gamma$ during the first and second CE episodes, respectively.
In the study, we first compare the two mechanisms under the same assumptions, as described below. The parameter combination is kept the same as in @zuo14a, where the best-fit model in the $\alpha_{\rm CE}$ formalism is achieved. In this case, only values of $\gamma$ and $\alpha_{\rm CE}$ are changed to see their effects on the XLF. Then we manage to determine the best-fit model in the $\gamma$ algorithm by varying all the key parameters, and see their effects on the XLF (see Table 1). Finally, the two mechanisms are compared under each best-fit model (i.e., model A05A05 vs. model M1).
\[tab:m7\]
Model $P(q_0)$ IMF $f$ $\eta_{\rm Edd, BH}$ $\sigma_{\rm kick}$ winds
------- ------------------- ---------- ----- ---------------------- --------------------- ------- --
M1 $\propto q_0^{0}$ Salpeter 0.5 20 110 STD
M2 $\propto q_0^{0}$ Salpeter 0.5 20 110 WEAK
M3 $\propto q_0^{0}$ Salpeter 0.8 20 110 STD
M4 $\propto q_0^{0}$ Salpeter 0.5 100 110 STD
M5 $\propto q_0^{1}$ Salpeter 0.5 20 110 STD
M6 $\propto q_0^{0}$ MT87 0.5 20 110 STD
M7 $\propto q_0^{0}$ Salpeter 0.5 20 190 STD
M8 $\propto q_0^{0}$ Salpeter 0.5 20 265 STD
: Parameters adopted for each model under the $\gamma$ algorithm. Here $q_0$ is the initial mass ratio, IMF is the initial mass function, $f$ binary fraction, $\eta_{\rm Edd, BH}$ - the factor of super-Eddington accretion rate allowed for BH XRBs, $\sigma_{\rm kick}$ is the dispersion of kick velocity, $\eta_{\rm bol, BH(NS)}$ is the bolometric correction factor for BH(NS) XRBs, STD is the standard stellar winds while WEAK represents the standard wind mass loss rate reduced to 50%, MT87 represents the IMF of @mt87. In the best-fit model of $\gamma$ algorithm (M1), the parameters are as follows: SFH=50 Myr, $\alpha=0$, $\eta_{\rm Edd, BH}$=20, $f=0.5$, $\sigma_{\rm kick} = 110 \,\rm km\,s^{-1}$, $\eta_{\rm bol, BH}$=0.2, $\eta_{\rm bol, NS}$=0.1 and Salpeter IMF.
Input Parameters
----------------
We follow the evolution of a large number of binary systems, initially consisting of two zero-age MS stars. As HMXBs in the MGS12 samples reside in nearby star-forming galaxies, we adopt a constant SFR for 50 Myr and a fixed subsolar metallicity ($0.5\,Z_{\odot}$, where $Z_{\odot}=0.02$) accordingly [see @zuo14a for details]. Since the observed average XLF has already been normalized, we choose a Salpeter initial mass function (IMF) and set the mass range as $0.1-100\,\rm M_{\odot}$ for the normalization in order to be in parallel with MGS12[^1]. We evolve $10^6$ primordial systems[^2] and set up the same grid of initial parameters (primary mass, secondary mass and orbital separation) as in @Hurley02.
For the initial secondary’s mass ($M_2$), a power law distribution of $P(q_0)\propto q_0^{\alpha}$ is assumed, where $q_0\equiv M_2/M_1$. In our basic model, a flat distribution is assumed, i.e., $\alpha=0$. We adopt a logarithmically flat distribution of initial orbital separations $\ln a$ [@Hurley02].
We assume a binary fraction $f=0.5$ and that all binaries are initially in a circular orbit. For the SN kicks imparted on an NS, we assume a Maxwellian distribution with $\sigma_{\rm kick}=
110\, \rm km\,s^{-1}$ [@zuo14d]. For compact objects formed with partial mass fallback, the natal kicks are decreased by a factor of (1-$f_{\rm b}$) where $f_{\rm b}$ is the fraction of the stellar envelope that falls back after the SN explosion.
{width="6in"}
X-ray luminosity and source type
--------------------------------
We adopt the same procedures to compute the $0.5-8$ keV X-ray luminosity for MS/super-giant (SG) HMXBs and Be-XRBs as in @zuo14a. For wind accretion, we use the classical @Bondi44 formula to calculate the mass transfer rate to the compact star. In the case of RLOF, we discriminate transient and persistent sources using the criteria in @l01 [i.e., Eq. 36 therein] for MS and red giant (RG) donor. The corresponding X-ray luminosity is calculated as follows: $$\begin{aligned}
&&L_{\rm X, 0.5-8 keV}\nonumber\\
&&=\left\{
\begin{array} { ll}
\eta_{\rm bol}\eta_{\rm out}L_{\rm Edd}&\ \rm transients\ in\ outbursts, \\
\eta_{\rm bol}\min(L_{\rm bol},\eta_{\rm Edd}L_{\rm Edd})&\ \rm persistent\
systems,
\end{array}
\right.\end{aligned}$$ where $\eta_{\rm bol}$ is the bolometric correction factor converting the bolometric luminosity ($L_{\rm bol}$) to the $0.5-8$ keV X-ray luminosity, ranging between $\sim 0.1$ and $\sim 0.8$ [@bel08]; $L_{\rm bol} \simeq 0.1\dot{M}_{\rm acc}c^2$ where $\dot{M}_{\rm acc}$ is the average mass accretion rate and $c$ is the velocity of light. The critical Eddington luminosity $L_{\rm Edd} \simeq 4\pi GMm_{\rm
p}c/\sigma_{T}=1.3 \times 10^{38}m$ergs$^{-1}$ (where $\sigma_{T}$ is the Thomson cross section, $m_{\rm p}$ the proton mass, and $m$ the accretor mass in the units of solar mass). We introduce the ‘Begelman’ factor $\eta_{\rm Edd}$ to allow super-Eddington luminosities. We fix $\eta_{\rm Edd, NS}=5$ for NS XRBs [@zuo14a]; for BH XRBs, $\eta_{\rm Edd, BH}$ is set as a free parameter in the study. For transient sources, the outburst luminosity is taken as a fraction ($\eta_{\rm out}$) of the critical Eddington luminosity. We take $\eta_{\rm out}=0.1$ and 1 for NS(BH) transients with orbital period $P_{\rm orb}$ less and longer than 1 day (10 hr), respectively [@chen97; @Garcia03; @bel08]. For Be-XRBs we employ a phenomenological definition as in @zuo14a [also see Belczynski & Ziolkowski, 2009]. Technically, we randomly select 25% [$f_{\rm Be}=0.25$, @s88; @z02; @mg05] of NS binaries hosting a ($3.0\,M_{\odot}-20.0\,M_{\odot}$) B/O star to be Be-XRBs, and estimate their numbers. The X-ray luminosity of a Be-XRB is calculated using the empirical relation (Eq. 11) in @dll06, which is based on the data compiled by @rp05. Considering the duration of type I outbursts in Be-XRBs [$\sim 0.2-0.3 P_{\rm orb}$, @reig11], we adopt an upper value of the duty cycle $DC_{\rm max}=0.3$ to calculate the source numbers.
Results
=======
{width="6in"}
{width="6in"}
We first compare the results in the $\alpha_{\rm CE}$ formalism and $\gamma$ algorithm under the same input parameters: SFH=50 Myr, $\alpha=0$, $\eta_{\rm Edd, BH}$=100, $f=0.5$, $\sigma_{\rm kick} = 110 \,\rm km\,s^{-1}$, $\eta_{\rm bol, BH}$=0.6, $\eta_{\rm bol, NS}$=0.3 and Salpeter IMF [@zuo14a]. For each CE episode, models are designed by changing only one parameter each time to test its effect. Figure 1 compares the simulated XLFs with different treatments of the CE phase. Clearly, under the same parameter combination the $\gamma$ algorithm can produce more (up to one order of magnitude) HMXBs than the $\alpha_{\rm CE}$ formalism. In the framework of the $\alpha_{\rm CE}$ formalism, though all models can fit the observed XLF quite closely in most of the luminosity range (i.e., $10^{35}-\sim 10^{39}\,\rm
ergs\,s^{-1}$), a high value of $\alpha_{\rm CE}$ seems more preferable. This is mainly due to the sparseness of short period RLOF HMXBs in the case of smaller $\alpha_{\rm CE}$ [compare with the right panel of Figure 1 in @zuo14a], the progenitors of which coalesce during the binary evolution, especially in the first CE phase (see models A05A01 and A01A01). In the case of $\gamma$ algorithm, the normalization of the simulated XLFs is rather sensitive to the value of $\gamma$, especially in the first CE phase (see models G10G17 and G17G17 or models G10G10 and G17G10). Smaller values of $\gamma$ give a better fit to the observed XLF, not only in the normalization, but also in the overall shape. Considering that many parameters may considerably influence the XLF [@zuo14a], further thorough parameter studies are needed to determine the best-fit model in the $\gamma$ algorithm.
{width="3in"} {width="3in"}
The key parameters we vary include: the binary fraction $f$, the super-Eddington factor $\eta_{\rm Edd}$, the bolometric correction factor $\eta_{\rm bol}$, the mass ratio, the IMF, the natal kick distribution, the wind mass loss rates and the value of $\gamma$. Some parameters affect only the normalization, such as $f$ and $\eta_{\rm bol}$; some affect only the shape, for example, $\eta_{\rm Edd}$; while others affect both. We perform a suite of EPS models and find that the best-fit model in the $\gamma$ algorithm can be achieved when parameters are adopted as follows: SFH=50 Myr, $\alpha=0$, $\eta_{\rm Edd, BH}$=20, $f=0.5$, $\sigma_{\rm kick} = 110 \,\rm km\,s^{-1}$, $\eta_{\rm bol, BH}$=0.2, $\eta_{\rm bol, NS}$=0.1, Salpeter IMF and $\gamma=1.0$ (i.e., model M1). We also examine other values of $\gamma$ , and find that the results are not better than in the case of $\gamma=1.0$ (especially when $\gamma \gtrsim\,1.5$). In order to show the dependences of the XLF on the parameters, we also change these parameters one by one. The details are listed in Table \[tab:m7\].
Figure 2 clearly shows that the parameters act in different ways. Several parameters have only minor effects, i.e., the wind mass loss rate (model M2) and the initial mass ratio distribution of the secondary star (model M5). Some (e.g. models M3 and M6) mainly increase the number of HMXB populations. An increase of the binary fraction (model M3) gives more XRBs, hence an overall shift of the XLF. A flatter IMF (model M6) reflects more massive stars, hence more compact objects that may result in XRBs. An increase of the dispersion velocity $\sigma_{\rm kick}$ (models M7 and M8) means that the natal kicks of higher magnitude are chosen more frequently from the Maxwellian distribution, hence more disruptions of binaries during the SN explosions. This decreases the number of potential HMXBs, and meanwhile changes the shape of the XLF. We note the large uncertainties in $\sigma_{\rm kick}$, $f$, and $\eta_{\rm bol}$ make it difficult to tightly constrain the value of $\gamma$. The apparent luminosity ‘knee’ of XLFs is weakened if we restrict the super-Eddington factor to 20 (compare model M4 with others), implying that the maximum super-Eddington luminosity allowed is likely $\sim 20$ in the case of $\gamma$ algorithm. To sum up, in the framework of the $\gamma$ algorithm, the observed XLF can also be reconstructed within the reasonable range of the parameters adopted.
In order to explore the nature of HMXBs in the case of $\gamma$ algorithm, we also examine the detailed observational properties (i.e., orbital period, the current mass $M_2$ of the donor star, etc.) of the simulated HMXBs, and compare them with those in @zuo14a [i.e, $\alpha_{\rm CE}$ formalism] and observations. Shown in Figure 3 are the detailed components of the simulated XLF (left) and the accretion modes in XRBs (right) and in Figure 4 are the $P_{\rm orb}-L_{\rm X}$ (left) and $P_{\rm orb}-M_2$ (right) distributions in model M1. It is clear that under the $\gamma$ algorithm BH-He XRBs dominate in the low luminosity range (i.e., $L_{\rm X}<\sim10^{37}\rm\,erg\,s^{-1}$) of the XLF while this is not the case in the $\alpha_{\rm CE}$ formalism, where BH-MS XRBs instead dominate [@zuo14a]. Unfortunately, due to the limited instrument capabilities available, most of the extragalactic X-ray sources remain unresolved. We still do not clearly know their nature (for example, the spectral type of the donor star and the type of the compact star), especially the sources in low luminosities. We suggest further check with higher-precision observations is still needed in the future. The orbital period distribution is also distinct from that in @zuo14a, with a much larger population of relatively short period (less than several tens of days) systems . This is more clearly revealed in Figure 5 for the normalized orbital period $P_{\rm orb}$ distribution in models A05A05 (left) and M1 (right). We can see that short period HMXB population keeps growing under the $\gamma$ algorithm, while most HMXBs under the $\alpha_{\rm CE}$ formalism are produced within the first 20 Myrs. These distinct observational properties of HMXBs, as well as different period distributions may provide further clues to discriminate between the two models.
{width="5in"}
{width="5in"}
We note the discrepancy in the BH-He HMXB population between models is solely a result of different treatments on CE, in which the $\gamma$ algorithm predicts a survival, while the $\alpha_{\rm CE}$ formalism predicts, a merger instead. The progenitors of BH-He HMXBs always have the following features. First, the primary stars are very massive, $\sim 30-80\,M_{\odot}$, so they can form BHs in a mild (with low/no kicks) way, which will not disrupt the system. Second, the companion stars are relatively less massive, i.e., $\sim 10-35\,M_{\odot}$. The orbits of the binaries are not too wide, of the order of tens to hundreds of $R_{\odot}$. These conditions guarantee that the primary can overfill its RL rapidly and transfer mass in a dynamically stable way (because the mass ratio is not too extreme[^3]). After that, the primary evolves to a BH, and the rejuvenated secondary star expands and fills its RL. However, due to the large mass ratio, the mass transfer this time is unstable, and a CE is triggered. The $\alpha_{\rm CE}$ formalism always leads to binary mergers due to the huge amount of binding energy in the giant envelope. In the case of $\gamma$ algorithm, the orbital evolution is determined by the mass ratio $q=M_{\rm donor}/M_{\rm accretor}$ and the core mass fraction $\mu=M_{\rm c}/M_{\rm donor}$. From Eq. 6 in @nt05, it is easy to deduce that the binary orbit not only shrinks (but still different from that in the $\alpha_{\rm CE}$ formalism), but also expands (see also Figure 3 therein). This expansion of the orbit not only avoids binary mergers, but also delays the XRB formation significantly. This is also why, under the same assumptions the $\gamma$ algorithm can produce more HMXBs than the $\alpha_{\rm CE}$ formalism and the HMXBs can keep emerging after 20 Myr in the case of the $\gamma$ algorithm.
To illustrate the formation and evolution of a typical BH-He HMXB, we present one example evolutionary sequence for $M_1$, $M_2$, $P_{\rm orb}$, and $L_{\rm X}$ under the $\gamma$ algorithm in Figure 6. We consider a primordial binary system in a $\sim 91.44\,R_{\odot}$ circular orbit. The initial stellar masses are 35.493 and 12.532 $\,M_{\odot}$ for the primary and secondary, respectively. The primary evolves first, and fills its RL on the HG (at 5.5483 Myr). The mass transfer proceeds rapidly as it evolves across the HG until the end of CHeB, at which point (5.5598$\,$Myr) it becomes an 11.069$\,M_{\odot}$ HeMS star with a 34.451$\,M_{\odot}$ (rejuvenated) MS star in a $109.626\,R_{\odot}$ orbit. Shortly after that, the naked helium star evolves across the HeHG and collapses at 6.2418 Myr, leaving a 7.617$\,M_{\odot}$ BH with an MS companion in a 167.93 $\,R_{\odot}$ orbit. Subsequently, the MS star evolves to expand and fills its RL on the HG, and then the binary enters into the CE stage (10.5754 Myr). At this time, the mass ratio is $q\sim4.3$ and the core mass fraction $\mu\sim0.3$, and the orbit shrinks slightly from 134.38$\,R_{\odot}$ to 118.55$\,R_{\odot}$, as calculated from Eq. 6 in @nt05. At the end of the CE, the envelope of the giant star is expelled, leaving a 10.58$\,M_{\odot}$ HeMS star. The stellar wind from the HeMS star is then accreted by the BH, resulting in a BH-HeMS XRB. At last, the HeMS evolves to explode as an SN (11.28 Myr), which results in a 7.348$\,M_{\odot}$ BH and disrupts the binary system.
Our findings are generally consistent with other previous studies concerning the CE evolution. For example, in the case of the $\alpha_{\rm CE}$ formalism, a high value of $\alpha_{\rm CE}$ is required to account for the observed WDMS PCEBs [$\alpha_{\rm CE}\gtrsim 0.1$, @dkw10], the shape of the delay-time distribution and the birth rate of SNe Ia for the double-degenerate systems [@my12], and the displacements of HMXBs [$\alpha_{\rm CE}\sim 0.8-1.0$, @zuo14b], while a lower value of $\alpha_{\rm CE}$ may be excluded [@my12; @zuo14b]. An exception is from @f13 where a low value of $\alpha_{\rm CE} \sim 0.1$ is preferred, most likely due to the oversimplified treatments for the binding energy parameter, where $\lambda$ is adopted as one overall, while this is not the case for massive stars [$\lambda \sim 0.1$, @xu10]. It is interesting to note that to create double WDs, the standard $\alpha_{\rm CE}$ formalism is also possible if the first mass transfer between an RG and an MS star can be stable and non-conservative. This leads to a modest widening of the orbit, with an effect similar to the $\gamma$ algorithm [@woods12]. In the framework of the $\alpha_{\rm CE}$ formalism, our simulations are also comparable to previous studies concerning HMXB populations [@pv96; @tts98; @Linden10]. The major formation pathways of HMXBs in @zuo14a are consistent with the results obtained by @Linden10. The predicted observational properties of HMXBs (such as the orbital period distributions) are also similar. The number of HMXBs is also found to be not very sensitive to the value of $\alpha_{\rm CE}$. However, it seems that neither the $\alpha_{\rm CE}$ formalism nor the $\gamma$ algorithm can account for all the specific classes of observed PCEBs [@meng11; @my12]. Moreover, even within the framework of the $\alpha_{\rm CE}$ formalism, different studies often give controversial results on the possible range of $\alpha_{\rm CE}$ and its dependence on other parameters [see @zgn00; @marco11; @davis12; @tn13 also Ivanova et al. 2013 and references therein]. Our work suggests that in the case of HMXBs, both the $\alpha_{\rm CE}$ formalism and the $\gamma$ algorithm are possible to reproduce the observed XLF. In the framework of $\alpha_{\rm CE}$ formalism, a high value of $\alpha_{\rm CE}$ is needed, although a constant value is not required. We also show the distinct observational properties, such as the period distribution of HMXBs, that may serve as possible keys to understanding the CE evolution and to discriminate between different CE models.
SUMMARY
=======
We have used an EPS code to model the XLF of HMXBs with a range of theoretical models describing the CE phase. Our study shows that the observed XLF can be reproduced quite closely under both CE mechanisms. Provided that the same parameter combination is chosen, the $\gamma$ algorithm seems to produce more HMXBs than the $\alpha_{\rm CE}$ formalism, by a factor of up to $\sim 10$. Additionally, in the framework of the $\alpha_{\rm CE}$ formalism, a high value of $\alpha_{\rm CE}$ around $\sim 0.5-1.0$ better fits the observed XLF. We present the detailed properties of HMXB populations under the $\gamma$ algorithm, and find that the simulated HMXBs have a much larger population of short period (less than about several tens of days) BH-He systems than in the $\alpha_{\rm CE}$ formalism, which may serve as clues to discriminate between the two kinds of models. Our work motivates further high-resolution X-ray and optical observations of HMXB populations in nearby star-forming galaxies.
We thank the anonymous referee for helpful suggestions that enabled us to improve the manuscript. This work was supported by the National Natural Science Foundation (under grant numbers 11103014, 11003005, 11133001, 11333004, and 10873008), the Research Fund for the Doctoral Program of Higher Education of China (under grant number 20110201120034), the National Basic Research Program of China (973 Program 2009CB824800), the Strategic Priority Research Program of CAS under grant No. XDB09010200, and the Fundamental Research Funds for the Central Universities.
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[^1]: We have improperly chosen an IMF of @Kroupa01 and set the mass range as $5.0\,M_{\odot} - +\infty$ for the normalization in @zuo14a. The predicted HMXBs were overestimated by a factor of $\sim 3$, but the results and basic conclusions of the paper remain largely unchanged [see @zuo14d for details].
[^2]: We also change the number of the binary systems by a factor of eight, and find no significant difference in the final results.
[^3]: If the mass ratio is extreme, a CE is triggered, followed by a second CE between the newly formed BH and an expanding giant, instead resulting in much tighter BH-HeMS XRBs instead. However, their population is relatively minor in this case.
|
LYCEN 2003-36\
October 15th, 2003\
****
Calibration of the EDELWEISS Cryogenic Heat-and-ionisation Germanium Detectors\
for Dark Matter Search
The EDELWEISS Collaboration:\
O. Martineau$^{1}$, A. Benoît$^{2}$, L. Bergé$^{3}$, A. Broniatowski$^{3}$, L. Chabert$^{1}$, B. Chambon$^{1}$, M. Chapellier$^{4}$, G. Chardin$^{5}$, P. Charvin$^{5,6}$, M. De Jésus$^{1}$, P. Di Stefano$^{1}$, D. Drain$^{1}$, L. Dumoulin$^{3}$, J. Gascon$^{1}$, G. Gerbier$^{5}$, E. Gerlic$^{1}$, C. Goldbach$^{7}$, M. Goyot$^{1}$, M. Gros$^{5}$, J.P. Hadjout$^{1}$, S. Hervé$^{5}$, A. Juillard$^{3}$, A. de Lesquen$^{5}$, M. Loidl$^{5}$, J. Mallet$^{5}$, S. Marnieros$^{3}$, N. Mirabolfathi$^{6}$, L. Mosca$^{5,6}$, X.-F. Navick$^{5}$, G. Nollez$^{7}$, P. Pari$^{4}$, C. Riccio$^{5,6}$, V. Sanglard$^{1}$, L. Schoeffel$^{5}$, M. Stern$^{1}$, L. Vagneron$^{1}$
[$^{1}$Institut de Physique Nucléaire de Lyon-UCBL, IN2P3-CNRS, 4 rue Enrico Fermi, 69622 Villeurbanne Cedex, France\
$^{2}$Centre de Recherche sur les Très Basses Températures, SPM-CNRS, BP 166, 38042 Grenoble, France\
$^{3}$Centre de Spectroscopie Nucléaire et de Spectroscopie de Masse, IN2P3-CNRS, Université Paris XI, bat 108, 91405 Orsay, France\
$^{4}$CEA, Centre d’Études Nucléaires de Saclay, DSM/DRECAM, 91191 Gif-sur-Yvette Cedex, France\
$^{5}$CEA, Centre d’Études Nucléaires de Saclay, DSM/DAPNIA, 91191 Gif-sur-Yvette Cedex, France\
$^{6}$Laboratoire Souterrain de Modane, CEA-CNRS, 90 rue Polset, 73500 Modane, France\
$^{7}$Institut d’Astrophysique de Paris, INSU-CNRS, 98 bis Bd Arago, 75014 Paris, France ]{}
[**Abstract**]{}
Several aspects of the analysis of the data obtained with the cryogenic heat-and-ionisation Ge detectors used by the EDELWEISS dark matter search experiment are presented. Their calibration, the determination of their energy threshold, fiducial volume and nuclear recoil acceptance are detailed.
Introduction
============
The dominant ($\sim$ 90%) component of the mass budget of the Universe may consist in Weakly Interacting Massive Particles (WIMPs), which could be the Lightest Supersymmetric Particles (neutralinos in most models)[^1]. WIMPs would be present at the galactic scale as a halo of mass typically ten times larger than the visible part of the galaxy. The EDELWEISS collaboration has developped heat-and-ionisation Ge detectors [@xfn] to measure recoils induced by elastic scattering of galactic WIMPs on a target nucleus. Constraints on the spin-independent WIMP-nucleon cross-section in the framework of the Minimal SuperSymmetric Model (MSSM) have been derived from the nuclear recoil rate measured with the EDELWEISS detectors [@ed2000], [@ed2002]. We present in this paper the experimental details of these measurements. In Section \[detectors\], we describe briefly the experimental setup and the method of detection of an energy deposit in the target. We then show the calibration procedure for the heat and ionisation signals (Section \[channels\]), the trigger threshold determination (Section \[seuil\]) and the tagging of the nuclear recoils (Section \[zoneneu\]). We finally present an original method to determine the fiducial volume of the detectors (Section \[volfid\]).
The EDELWEISS detectors {#detectors}
=======================
Experimental setup
------------------
The experimental setup of the EDELWEISS-I experiment is described in [@ed2000] and [@ed2002]. We simply recall that up to three detectors can be housed in a low background dilution cryostat working at a regulated temperature (27 mK in [@ed2000] and 17 mK in [@ed2002]). The EDELWEISS detectors are made of a germanium absorber (target for the incident particles) equiped with a thermal sensor and with metallic electrodes for charge collection. The simultaneous measurement of both phonons and charges created by a single interaction is therefore possible.\
The main characteristics of the detectors studied in this article are given in Table \[det\]. For all of these detectors, the absorber is a $\sim$320 g Ge cylindrical crystal ($\sim$70 mm diameter and 20 mm thickness). Their edges have been beveled at an angle of 45$^o$ (Fig. \[detect\]). The electrodes for ionisation measurement are made of 100 nm Al layers sputtered on the surfaces after etching. The top electrode is divided in a central part and a guard ring, electrically decoupled for radial localization of the charge deposition. The bottom electrode is the common reference. For the GGA1 and GGA3 detectors (GSA1 and GSA3), a 60 nm hydrogenated amorphous germanium (silicon) layer was deposited under the electrodes in order to reduce the charge collection problems associated with events where the energy is deposited close to the detector surface. It has indeed been shown that the probability that charge carriers be collected on the same-sign electrode during the diffusion phase which preceeds the charge collection (dead layer problem) is reduced for this type of detectors \[5, 6\].\
The thermal sensor consists of a Neutron Transmutation Doped germanium crystal (NTD)[^2], close to the metal-insulator transition. It is glued on a sputtered gold pad near the edge of the bottom Al electrode (Fig. \[detect\]). The resistance of the DC-polarized GeAl6 sensor was chosen to be $\sim$ 3 M$\Omega$ for GeAl6 ($T_{running}\sim$27 mK), and ranged from 3 M$\Omega$ to 6 M$\Omega$ at 17 mK for the other detectors. Reliable electrical contacts and heat links have been achieved by the ultrasonic bonding of gold wires (diameter 25 $\mu$m) on gold pads. The thickness of these pads has been chosen to minimize the production of dislocations in the absorber caused by the bonding. A thermal analysis of the detectors will be published in Ref. [@N03].
Detection method
----------------
The rise in temperature due to an energy deposit in the absorber gives rise to a variation $\Delta{R}$ of the thermal sensor resistance. When the sensor is polarized by a constant current $I$, $\Delta{R}$ then induces a voltage fluctuation $\Delta{V}$ across the resistor, which corresponds to the heat signal: $$\label{vchal}
\Delta{V}=\Delta{R}\times{I}$$ The ionisation signal is obtained by collection of the electron-hole pairs created by the interaction in the germanium crystal polarized through a bias voltage applied to the electrodes. A low bias voltage[^3] is required to limit the heating of the cristal due to the drift of the charge carriers, known as the Neganov-Luke effect [@lneg].\
\
The energy $E_{R}$ deposited by a particle interacting in the detector can be determined by subtracting the Neganov-Luke effect from the heat signal : $$\label{erec}
E_{R}=\left(1+\frac{V}{\varepsilon_{\gamma}}\right)E_{H}-\frac{V}{\varepsilon_{\gamma}}E_{I}$$ where V is the bias voltage and $\varepsilon_{\gamma}=3$ V the mean electron-hole pair creation potential in germanium for $\gamma$-ray interactions (electron recoils). The variables $E_H$ and $E_I$ stand respectively for the heat and ionisation signal amplitudes calibrated for $\gamma$-ray interactions following the procedure described in Section \[calib\].\
\
We define the quenching variable Q as: $$\label{qdef}
Q=\frac{E_I}{E_R}$$ This variable is of particular interest in the case of WIMP search since nuclear and electronic recoils correspond to different ionisation efficencies. As $E_I$ and $E_H$ are calibrated using $\gamma$-rays, $Q=1$ for electronic interactions by definition. In the case of nuclear recoils (such as those that would be produced by WIMP interactions), this ratio is much lower: $Q\sim0.3$. The simultaneous measurement of heat and ionisation therefore provides an event-by-event identification of the type of recoils and thus gives an efficient method to reject the dominant $\gamma$-ray background. The precise definition of the rejection criteria is discussed in Section \[zoneneu\].
Calibration and resolution of heat and ionisation signals {#channels}
=========================================================
Calibration of heat and ionisation channels {#calib}
-------------------------------------------
The ionisation signal $E_{I}$ is calibrated using a $^{57}$Co source that can be inserted in the liquid He bath of the cryostat to a distance of $\sim$ 10 cm from the detectors, with only a $\sim$ 0.5 cm thick copper shielding layer between the source and the detectors. The 122 and 136 keV peaks are clearly visible on the spectra (Fig. \[reso\]c), allowing a precise calibration of the ionisation signal. The linearity of the signal amplitude has been verified using the 46.52 keV line from $^{210}$Pb (Fig. \[reso\]b) in the detector environment and the 8.98 and 10.37 keV lines from the decay of cosmic-ray induced long life isotopes $^{65}$Zn and $^{68}$Ge in the detector. The calibration factor is observed to be stable within a fraction of percent over periods of months. Because of the parasite capacitance between the centre and guard electrodes, a charge fully collected on an electrode also induces a signal on the other. This cross-talk of a few percents is purely linear and remains constant in time for a given detector. It can thus be easily corrected off-line (Fig. \[bipbrut\]). The ionisation signal $E_{I}$ is defined as the sum of the guard ring and center electrode signal amplitudes after correction of the cross-talk and calibration of the two channels.\
The heat signal amplitude $E_{H}$ is periodically calibrated using the same $^{57}$Co source. In contrast with ionisation, the heat signal appears to be very sensitive to long term drifts of the NTD temperature. It may for example vary by a few percent during several hours after transfers of cryogenic fluids. Between two $^{57}$Co calibrations, the heat signal is therefore monitored on a continuous basis using the data from the low-background physics runs themselves by setting the average value of the $Q$ ratio to 1 for electron recoils. The 46.52 keV line from $^{210}$Pb and the 8.98 and 10.37 keV lines associated with cosmogenesis activation of $^{65}$Zn and $^{68}$Ge in the detector (Fig. \[reso\]a) are used to check the quality of the calibration of the heat signal.\
It should be stressed again at this point that the heat and ionisation signals are calibrated using $\gamma$-ray sources, which induce electron recoils. The $E_I$ and $E_H$ values thus correspond to the actual energy deposit for this type of interactions only, and are therefore expressed in keV electron equivalent (keV$_{ee}$).
Resolution of heat and ionisation channels
------------------------------------------
For each detector, the baseline resolutions of the heat and the two ionisation channels are regularly controlled through runs with an automatic random trigger. These runs show that the noises of the three channels are not correlated. The ionisation baseline resolution can therefore be written as : $$\label{bline}
{\left(\sigma^0_{I}\right)}^2={\left(\sigma_{center}^0\right)}^2+{\left(\sigma_{guard}^0\right)}^2$$ The $^{57}$Co calibrations give a measurement of the resolutions for the ionisation and heat signals at 122 keV. Typical values obtained for the detectors studied here are given in Table \[resotab\].\
\
We parametrize the heat and ionisation signals resolutions at a given electron-equivalent energy E as : $$\label{sigt}
\sigma_{I,H}(E)=\sqrt{\left(\sigma_{I,H}^{0}\right)^2+\left(a_{I,H}E\right)^2}$$ where the factors $a_{I}$ and $a_{H}$ are deduced from the resolution of the ionisation and heat signals at 122 keV. The resolutions of the 10.37 and 46.52 keV peaks observed in low-background physics runs fit well with the expressions $\sigma_{I,H}(E)$ from Eq. (\[sigt\]) (Fig. \[reso\]d). It can be noted that the resolutions at $E_{I}\sim$10 keV$_{ee}$ -an energy below which most of the WIMPs signal is expected- is dominated by the baseline resolutions $\sigma_{I}^0$ and $\sigma_{H}^0$.\
Finally, the recoil energy resolution can be computed from the heat and ionisation signal resolutions using Eq. (\[erec\]). The noises of both signals being uncorrelated, this resolution can be written as: $$\label{resrec}
\sigma_{E_{R}}=\sqrt{\left(1+\frac{V}{\varepsilon_{\gamma}}\right)^2\left(\sigma_{E_{H}}\right)^2+\left(\frac{V}{\varepsilon_{\gamma}}\right)^2\left(\sigma_{E_{I}}\right)^2}$$ In the case of GeAl6, and for the bias voltage applied during the low-background physics run ($V$=6 V), the resolution values displayed in Table \[resotab\] lead to $\sigma_{E_{R}}\sim$ 8 keV FWHM around 30 keV. This value is reduced to 4 keV FWHM in the condition of the low-background physics run recorded with GGA1 ($V$=4 V) [@ed2002].
Threshold {#seuil}
=========
The ionisation and heat channel data are continuously digitized and filtered at a rate of 200 kHz and 2kHz, respectively. When a filtered ionisation value exceeds a fixed threshold value, data samples in all detectors are stored to disk. The trigger is defined by requiring a minimum threshold on the absolute value of any of the filtered ionisation channels. For each event, the list of all detectors having triggered is stored as a bit pattern.
The ionisation threshold value, $E_{I,th}$ is defined as the ionisation energy (in keV$_{ee}$) at which the trigger efficiency reaches 50%. It is the most important parameter governing the recoil energy dependence of the efficiency. Its value is measured using two different techniques: one is based on the Compton plateau observed with a $\gamma$-ray source, and the other on coincidence neutron data.
In the first one, a $\gamma$-ray spectra is recorded using a source producing a important Compton plateau, such as $^{60}$Co or $^{137}$Cs. Monte Carlo simulations indicate that the shape of the plateau above 10 keV can be linearly extrapolated to lower energy. The efficiency as a function of $E_I$, $\epsilon(E_I)$, is thus obtained by dividing the measured rate by the straight line extrapolated from the rate above 10 keV. The resulting $\epsilon(E_I)$ data is fitted by a integral of a gaussian (erf), yielding the experimental value of $E_{I,th}$. However, this method is limited by the large data sample necessary to obtain a significant number of events in the threshold region.
The second technique was made possible by the simultaneous operation of three detectors with a $^{252}$Cf neutron source (and thus could not be applied to the GeAl6 detector). Neutron scattering induces a large number of coincidence events where at least two detectors are hit. The upper pannel of Fig. \[picseuil\] shows the $E_I$ distribution recorded in one detector with the condition that any of the other two detector triggered (unfilled histogram). Despite that the detector under study is not requested in the trigger pattern, the peak at $E_I$=0 due to baseline noise is not overwhelmingly large, due to the importance of the coincident rate. An unbiased sample of events with $E_I$ $>2$ keV is thus obtained. When in this sample it is further requested that the detector under study be present in the trigger pattern, the shaded histogram is obtained. The ratio of the two distributions shown in the lower pannel of Fig. \[picseuil\] correspond to the efficiency $\epsilon(E_I)$. This interpretation is valid in the region close to $E_{I,th}$ and above because in that energy range the contribution of the peak due to baseline events is negligible and because the slope of the unbiased distribution is reasonnably small compared to the experimental resolution on $E_I$. Indeed, applying this method to a distribution $N(E_I)$ proportionnal to $\exp(-E_I/\tau)$ and smeared with an experimental r.m.s. resolution $\sigma$, this method would result in a shift of $-\sigma^2/\tau$ of the deduced value of $E_{I,th}$ relative to the true value. In the present case, where the range of exponential slopes and resolution are 3 $<$ $\tau$ $<$ 8 keV and 1 $<$ 2.35$\sigma$ $<$ 2 keV, the shift should not exceed 0.2 keV.
Both Compton and neutron coincidence techniques give consistent ionisation threshold measurements. The coincidence measurements are the most precise, as the neutron source has the advantage of yielding a maximum rate at the lowest energy, and in addition, the quenching of ionisation for nuclear recoils ensure that the stability of the measurement can be tested by imposing a cut on the heat signal $E_H$ without affecting the ionisation signals with $E_I$ above $\sim E_H/2$. The measured $E_{I,th}$ values for the different ionisation channels of the detectors under study are listed in the last column of Table \[resotab\].
Nuclear recoil band {#zoneneu}
===================
Figure \[rerneu\] shows a ($E_R$, $Q$) distribution from the data recorded with a $^{252}$Cf source emitting $\gamma$-rays and neutrons. Experimentally, the $Q$ variable appears to follow a gaussian distribution at the $\sim{2}\sigma$ level for both nuclear and electron recoils populations (Fig. \[qdis\]). We therefore parametrize the region of 90% acceptance for the nuclear recoils by the following cut: $$\label{defqn}
|Q-<Q_n>|\leq{1.65\sigma_{Q_n}}$$ where $<Q_n>$ and $\sigma_{Q_n}$ are the average value and the standard deviation of the $Q$ distribution for nuclear recoils, both variables being determined for each detector from $^{252}$Cf calibration data under the same experimental conditions as the low-background physics runs.
Neutron line
------------
The neutron line is the average $Q$ value for the nuclear recoils population. It is parametrized from $^{252}$Cf calibration data by : $$\label{qneuav}
<Q_n>(E_R)=a\left(E_{R}\right)^b$$ The $a$ and $b$ values resulting from the fit of the experimental data for each EDELWEISS detector are statistically consistent with the values $a=0.16$ and $b=0.18$ quoted in [@heid]. The biases on the determination of $<Q_n>$ due to experimental calibration uncertainties, heat quenching effects [@sicane], and multiple scatterings are globally taken into account with this measurement.
Electron and nuclear recoils zones standard deviations
------------------------------------------------------
The standard deviation of the electronic and nuclear recoil distributions, respectively noted $\sigma_{Q_{\gamma}}$ and $\sigma_{Q_n}$, can be calculated with Eqs. (\[erec\]) and (\[qdef\]) by propagation of the experimental values $\sigma_{I}$ and $\sigma_{H}$: $$\begin{aligned}
\label{qgamma}
\sigma_{Q_{\gamma}}(E_R)&=&\frac{(1+V/3)}{E_{R}}\sqrt{{\sigma^2_{I}}+\sigma^2_{H}} \\
\label{qneu0}
\sigma^0_{Q_n}(E_R)&=&\frac{1}{{E_{R}}}\sqrt{\left(1+\frac{V}{3}<Q_n>\right)^2{\sigma^2_{I}}+\left(1+\frac{V}{3}\right)^2<Q_n>^2{\sigma^2_{H}}}\end{aligned}$$ In the case of $^{60}$Co, $^{252}$Cf calibrations and low-background physics runs, the experimental values of $\sigma_{Q_{\gamma}}$ at high energy are significantly larger (up to $\sim$+30% at 122 keV) than those calculated from the resolutions given in Table \[resotab\] with Eqs. (\[bline\]), (\[sigt\]) and (\[qgamma\]) (Fig. \[sigc\]a). A dependance of the heat signal amplitude on the position of the interaction provides an explanation for this discrepency. This hypothesis is consistent with the $\sim1$% heat signal amplitude difference observed between center and guard events in $^{57}$Co calibrations. We therefore enlarge the $a_{H}$ coefficient in Eq. (\[sigt\]) so that the analytic expression given in Eq. (\[qgamma\]) for $\sigma_{Q_{\gamma}}(E_R)$ actually follows the experimental distribution for $^{60}$Co, $^{252}$Cf and low-background physics runs. We have checked that 90% of the experimental events then fall inside the electron recoil zone defined in this way.\
Even after correcting the $a_H$ value, the nuclear recoils $Q$ distribution of $^{252}$Cf calibration data is broader at high energy than what is expected from Eq. (\[qneu0\]) (Fig. \[sigc\]b). Atomic scattering processes [@lindhard], fluctuations in the number of charges created by a nuclear recoil [@fano] and multiple scattering (see Section \[multscat\]) are in particular expected to give an intrinsic width to the $Q$ distribution for nuclear recoils and thus explain this behavior. The experimental $\sigma_{Q_{n}}$ dependance on recoil energy is properly described when a constant $C$ is quadratically added to the term associated with the experimental resolution. The equation (\[qneu0\]) is thus re-written as follows: $$\begin{aligned}
\label{qneu1}
%\sigma_{Q_n}(E_R)=\frac{1}{{E_{R}}}\sqrt{\left(1+\frac{V}{3}<Q_n>\right)^2{\sigma^2_{I}}+\left(1+\frac{V}{3}\right)^2<Q_n>^2{\sigma^2_{H}}+C^2}
\sigma_{Q_n}(E_R)=\sqrt{\sigma^0_{Q_n}(E_R)+C^2}\end{aligned}$$ Typical values of $C\sim$0.040 are determined for each EDELWEISS detector by fitting the experimental $\sigma_{Q_n}$ points using Eq. (\[qneu1\]). With this definition, we have checked for each detector that 90% of nuclear recoils induced by $^{252}$Cf calibrations are inside the nuclear recoil zone defined in Eq. (\[defqn\]).
Effect of multiple scattering {#multscat}
-----------------------------
The nuclear recoil zone is determined through neutron calibrations, for which the proportion of multiple interactions is around 40% between 20 and 200 keV. This is of particular importance because in contrast to neutrons, WIMPs are expected to interact only once in the detector, and the $Q$ variable is in this case larger than when the same energy is deposited in multiple nuclear interactions, as can be deduced from Eq. (\[qneuav\]).\
We therefore evaluated quantitatively the effect of multiple interactions using a GEANT [@geant] simulation of $^{252}$Cf calibrations of the EDELWEISS detectors. The $Q$ variable has been calculated for the simulated nuclear events by associating with Eq. (\[qneuav\]) an ionisation signal of amplitude $e_{I}=0.16\left(e_{R}\right)^{1.18}$ to an energy deposit $e_{R}$ in a single interaction, and summing each individual $e_I$ to obtain the total $E_I$ energy for a given neutron. The effect of multiple interactions has then been evaluated with these simulated data by smearing the resulting $Q$ distribution with the experimental resolution given in Eq. (\[qneu0\]), and then by comparing the distributions obtained when selecting or not single interactions events (Fig. \[simurer\]).\
Although multiple interactions tend to lower $<Q_n>$, this effect remains weak, and the $Q$ distribution associated with single interactions events is only slightly narrower and completely included in the wider band. The nuclear recoils zone determined through $^{252}$Cf calibrations has therefore been conservatively used for the low-bakground physics run analysis.
Analysis energy range
---------------------
Equations (\[qgamma\]) and (\[qneu0\]) predict that the discrimination between electronic and nuclear interactions is deteriorated at low energies (see also Fig. \[rerneu\]). Rejection of the $\gamma$-ray background at a given level therefore defines a lower bound for the analysis energy range.\
Secondly, the detection efficiency has to be as close to 100% as possible in the analysis window in order to insure a good quality for the data set. The trigger threshold is therefore another factor which has to be taken into account for the definition of the analysis lower energy bound. For both 2000 [@ed2000] and 2002 [@ed2002] runs, the choice of the analysis lower bound has mainly been driven by this last factor. The threshold values of 5.7 and 3.5 keV$_{ee}$ for the ionisation signal indeed correspond respectively to recoil energies of 30 and 20 keV for a 100% detection efficiency, and 90% efficiency when the nuclear recoil zone is taken into account.\
Extensive $\gamma$-rays and neutron calibrations are performed before the physics data taking is initiated in order to fix the lowest recoil energy value corresponding to acceptable levels of $\gamma$-ray background rejection and detection efficiency. This ensures that the lower limit of the analysis window is not influenced by the possible presence of events in the final data set. The definition of the upper bound of the analysis window is described in [@ed2002].
Fiducial volume {#volfid}
===============
Modelisation of the collection process {#deffid}
--------------------------------------
The segmentation of the upper charge collection electrode in a central part and a guard ring leads to the definition of a fiducial volume. This volume is shielded against a significant amount of the radioactivity of the detector environment by the peripherical volume, as shown in [@ed2000]. To allow for the experimental resolution on the ionisation signals, the fiducial cut is defined as corresponding to a fraction of 3/4 of the charge collected on the center electrode. In order to give a robust and precise estimation of the detector volume associated with this fiducial cut, it is necessary to relate a given ratio of the two ionisation signal amplitudes to a given volume inside the detector. This is not a straightforward process: first, for non-WIMP interactions, multiple interactions have to be taken into account, and furthermore, interactions between charges may play a crucial role in the collection process. In particular, the important proportion of events with a charge signal shared between the two channels observed in each detector for $^{60}$Co calibrations (see e.g. Fig. 3) hints to the importance of these charge interactions processes.\
In order to test their influence on the determination of the fiducial volume, we choose to model the collection process with the simplified phenomenological description of charge collection given in Ref. [@penn], associated with the hypothesis of a plasma effect before charge drift. We will see that, even if some of our results cannot be explained in the framework of this very simplified model (Section \[resfid\]), it provides a good empirical tool to determine the fiducial volume and estimate systematic errors on its value (Section \[valfid\]). The model used here assumes the distribution of the charges in a sphere with uniform density, extending to a maximal radius $r_b$ before the charge is fully collected. Charges are distributed among the two electrodes depending on the position of the interaction relative to the surface corresponding to the separation between drift lines going to the center and guard rings. Here, we assume for simplicity that this surface is parametrized by a cylinder of radius $R_C$ (Fig. \[vfid\]). For an interaction at the radius $R>R_C+r_b$ in the crystal, the whole charge is fully associated with the guard ring. If $R<R_C-r_b$, then the charge has to be associated with the center electrode. Finally, if $R_C+r_b>R>R_C-r_b$, then the charge is splitted among the two electrodes, with a relative proportion associated with the center electrode corresponding to the fraction of the sphere inside the cylinder of radius $R_C$. For given values of $r_b$ and $R_C$, the fiducial volume is determined in this model by the following expression of the fiducial radius: $$\label{rfid}
R_{fid}=R_C+2\cos\left({\frac{13\pi}{9}}\right)r_b$$ A fraction of 1/4 of the total volume of a sphere of radius $r_b$ centered on $R_{fid}$ is inside the cylinder of radius $R_{C}$. In the framework of our model, interactions inside the cylinder of radius $R_{fid}$ thus correspond to a charge collection equal or greater than 3/4 of the total charge.
Validity and limits of the modelisation {#resfid}
---------------------------------------
In order to test its ability to reproduce the distribution of charge amplitudes, ionisation signals are simulated in the framework of this simple model, using the program GEANT [@geant] for $^{60}$Co and $^{252}$Cf calibrations, as described in Section \[multscat\]. The parameters $r_b$ and $R_C$ of the simulated data are then adjusted to match the experimental distribution of the $Y$ variable on a given energy range, the $Y$ variable being defined as the normalized difference of the ionisation signals: $$Y=\frac{E_{guard}-E_{center}}{E_{guard}+E_{center}}$$ The result of this optimisation is shown in Fig. \[super\] in the case of a $^{60}$Co calibration of the GeAl6 detector under 6.3 V bias voltage. The shape of the simulated distribution closely follows that of the experimental data, while a simulation using an alternative model (linear distribution of the charge, detailed in [@ltd9]) clearly exhibits a different pattern.\
\
We have also studied the evolution of the ($r_b$, $R_C$) parameters as a function of bias voltage for the GeAl6 detector [@these]. $R_C$ should not depend on the value and sign of the bias voltage, since it is related to the static field distribution only, while $r_b$ should increase with decreasing bias voltage: as the field increases, the less time there is for diffusion processes. The values of the parameters $r_b$ and $R_C$ determined for $^{60}$Co and $^{252}$Cf calibrations of the GeAl6 detector versus the applied field are displayed in Fig. \[rbrc\]. The $r_b$ and $R_C$ values follow the expected behavior. Moreover, the mean measured value of $R_C$ ($<R_C>=24.45\pm0.05$ mm) is statistically compatible with the value $R_{electro}=24.4$ mm expected from numerical calculations of the electric potential inside this detector (see Table \[det\]).\
The very large $r_b$ values are a clear sign that macroscopic charge extension perpendicular to the drift direction occurs before the charge collection is completed. However the data does not support that this expansion is driven by the plasma effect invoqued in Ref. [@penn]: a charge cloud size of the order or above a millimeter is indeed not compatible with results of studies on the dead layer \[5, 16\]. Furthermore, Fig. \[rbrc\] shows that $r_b(-)<<r_b(+)$ and that the values of $r_b$ for $^{60}$Co and $^{252}$Cf calibrations do not differ significantly for a same bias voltage. These two experimental results are also in strong disagreement with the predictions derived from the hypothesis of a plasma effect: firstly, the observed asymmetry for $r_b$ values between positive and negative bias voltage does not find any explanation in the framework of the plasma model, and secondly, the plasma effect should be weaker in the case of $^{60}$Co calibrations than for $^{252}$Cf (and thus $r_b$ values much smaller), since $\gamma$-rays induce much lower charge densities than neutrons. These are strong indications that the simple model presented here does not provide a proper description of the dynamics of the charge drift and collection. Charge repulsion during drift, not taken into account here, could for example play an important role in the collection process. A more detailed study, with dedicated detectors, has been initiated in the EDELWEISS collaboration in the aim of better understanding the collection process [@alex].
Measurement of the fiducial volume {#valfid}
----------------------------------
Our results clearly point out the limits of the modelisation presented in Section \[deffid\]. Still, it has to be stressed that this model reproduces correctly the distribution of charges among the electrodes (Fig. \[super\]), which represents the net effect of the charge collection process. It is therefore sufficient to give a precise determination of the fiducial volume and evaluate possible systematic errors, before a better, physically motivated model replaces it.\
We have calculated $R_{fid}$ with Eq. (\[rfid\]) and the $r_b$ and $R_C$ values determined from $^{252}$Cf calibrations under the same bias voltage as that of the low-background physics run for each detector. For all detectors except GeAl10, the $R_C$ values are compatible with those expected from the geometry of the electrodes and from numerical simulations of the electric potential inside the detectors. The values determined for $R_{fid}$ for the detectors are summarized in Table \[vfidt\]. The systematic error associated with the uncertainty on the exact mechanism producing the charge expansion is evaluated by taking the difference between the fiducial volume value deduced using the linear model and the one presented in Section \[deffid\]. Despite the poor description of the charge distribution by the linear model (see Fig. \[super\]), this difference is only 1%. The variation of the energy range used to determine the values of $r_b$ and $R_C$ through comparison of the experimental and simulated $Y$ distributions proved to be a minor contribution to this systematic error.\
\
An alternative evaluation of the fiducial volume is the fraction of cosmic activation events at 8.98 and 10.37 keV (see Fig. \[reso\]) selected by the fiducial volume cut. Such events are expected to be evenly spread inside the detector, and are observed at rates varying between 3 to 15 events per detector per day. In the few days following a neutron calibration, the 10.37 keV rate is also enhanced due to $^{71}Ge$ activation ($T_{1/2}$ = 2.7 d), a population that is also expected to be evenly spread inside the detector. The measured fractions, directly interpreted as $V_{fid}$ values, are listed in the last column of Table \[vfidt\]. They are compatible within statistics with the values derived from the neutron calibration data and the collection process modelisation. The cosmic activation data is however less precise due to statistics, but this measurement is a good cross-check for the determination of the fiducial volume, and validates the use of the model presented in Section \[deffid\] to determine the fiducial volume.
Conclusion
==========
We have described in the present work the calibration aspects of the data analysis in the EDELWEISS experiment. In particular, the nuclear recoil zone and fiducial volume have been estimated using several methods, allowing to define a conservative value of these important parameters. A simple parametrization allows us to reproduce accurately the distribution of the charges between the centre and guard electrodes associated with $^{60}$Co and $^{252}$Cf calibrations, making possible the systematic studies necessary to establish the robustness of the determination of the fiducial volume of the detectors.
[99]{} L. Bergström, Rep. Prog. Phys [**63**]{}, 793 (2000). X.F. Navick [*et al.*]{}, NIM A [**444**]{}, 361 (2000). A. Benoît [*et al.*]{}, Phys. Lett. B [**479**]{} 8 (2000). A. Benoît [*et al.*]{}, Phys. Lett. B [**545**]{} 43 (2002). P. Luke [*et al.*]{}, IEEE Trans. Nucl. Sci. 41 (4) (1994) 1074.\
T. Shutt [*et al.*]{}, NIM A [**444**]{}, 340 (2000).\
T. Shutt [*et al.*]{}, in Proc. [*9$^{th}$ Int. Workshop on Low Temperature Detectors*]{}, AIP conference proceedings [**605**]{}, 513 (2001). XF. Navick [*et al.*]{}, to be published. B. Neganov and V. Trofimov, USSR patent No 1037771, Otkrytia i izobreteniya [**146**]{}, 215 (1985).\
P.N. Luke, J. Appl. Phys. [**64**]{}, 6858 (1988). P. Di Stefano [*et al.*]{}, Astropart. Phys. [**14**]{}, 329 (2001). E. Simon [*et al.*]{}, NIM A [**507**]{}, 643 (2003). R. Brun [*et al.*]{}, [*GEANT3*]{}, CERN report DD/EE/84-1 (1987). J. Lindhard [*et al.*]{}, Mat. Phys. Medd. Dan. Vid. Selsk [**10**]{}, 1 (1963). T. Yamaya [*et al.*]{}, NIM [**159**]{}, 181 (1979). M.J. Penn [*et al.*]{}, in Proc. [*6$^{th}$ Int. Workshop on Low Temperature Detectors*]{}, NIM A [**370**]{}, 215 (1996). O. Martineau [*et al.*]{}, in Proc. [*9$^{th}$ Int. Workshop on Low Temperature Detectors*]{}, AIP conference proceedings [**605**]{}, 505 (2001). O. Martineau, Recherche de WIMPs par l’expérience EDELWEISS: caractérisation des détecteurs et analyse des données, PhD thesis, Université Lyon I (2002) (in french).\
Available at http://edelweiss.in2p3.fr/pub/fichiers/theses.html. A. Broniatowski [*et al.*]{}, in Proc. [*9$^{th}$ Int. Workshop on Low Temperature Detectors*]{}, AIP conference proceedings [**605**]{}, 521 (2001).\
A. Broniatowski, to be published in LTD10 proceedings (Genova, July 2003).
-------- -------- --------------- ---------- ----------- ---------------
Label Mass $R_{electro}$ Vol. NTD Amorphous $T_{running}$
(g) (mm) (mm$^3$) layer (mK)
GeAl6 321.62 24.4 4.0 none 27
GeAl9 325.43 24.0 5.6 none 17
GeAl10 323.91 24.0 5.6 none 17
GGA1 318.50 24.0 1.64 Ge 17
GGA3 324.40 24.0 5.6 Ge 17
GSA1 313.68 24.0 5.6 Si 17
GSA3 297.03 24.0 5.6 Si 17
-------- -------- --------------- ---------- ----------- ---------------
: \[det\] *Main parameters for the EDELWEISS detectors studied in this article. “$R_{electro}$” refers to the radius value of the cylindrical volume associated with charge collection on the center electrode. These values are calculated through electrostatic simulation of the detector, taking into account the actual electrodes geometry. The existence of an amorphous Ge or Si layer under the electrodes is also mentionned. “$T_{running}$” is the value of the regulated cryostat temperature while running.*
\
\
---------- -------------- -------------- -------------- -------------- -------------- -------------- --------------
Center Guard Heat Ion. Heat Center Guard
Detector (keV$_{ee}$) (keV$_{ee}$) (keV$_{ee}$) (keV$_{ee}$) (keV$_{ee}$) (keV$_{ee}$) (keV$_{ee}$)
GeAl6 2.0 1.4 2.2 2.8 3.5 6.0 4.0
GeAl9 1.2 1.4 0.5 2.6 3.3 4.3 4.9
GeAl10 1.1 1.3 0.4 3.0 3.5 3.3 4.3
GGA1 1.3 1.3 1.3 2.8 3.5 3.5 3.5
GGA3 1.3 1.5 0.4 3.1 2.7 2.9 3.9
GSA1 1.2 1.4 0.6 3.1 2.8 3.5 3.4
GSA3 1.1 1.3 1.4 3.3 3.3 3.0 3.4
---------- -------------- -------------- -------------- -------------- -------------- -------------- --------------
: \[resotab\] *Typical values obtained in keV$_{ee}$ for the full width half maximum resolution for heat and ionisation signals at 0 and 122 keV for the detectors studied in this article. The precision on these measurements are $\pm0.1$ keV at 0 keV and $\sim\pm$0.2 keV at 122 keV. Also given here are the threshold values for the center and guard channels. The precision is $\pm$0.1 keV for both channels, except for GeAl6 where it is $\pm$0.5 keV.*
\
\
---------- ------- ------------- -------------- -------------- -------------- --------------
Detector Bias $r_b$ $R_C$ $R_{fid}$ $V_{fid}$ Activation
(V) (mm) (mm) (mm) (%) $V_{fid}$(%)
GeAl6 +6.34 $4.3\pm0.2$ $24.5\pm0.3$ $23.0\pm0.3$ $54.6\pm1.4$ $50\pm3$
GeAl9 +2.00 $6.1\pm0.2$ $23.5\pm0.4$ $21.4\pm0.4$ $47.4\pm2.0$ $53\pm4$
GeAl10 -3.00 $2.3\pm0.2$ $21.9\pm0.4$ $21.1\pm0.4$ $46.0\pm1.7$ $50\pm4$
GGA1 -4.00 $1.6\pm0.1$ $24.6\pm0.2$ $24.1\pm0.2$ $60.1\pm1.1$ $57\pm3$
GGA3 -4.00 $1.4\pm0.1$ $24.1\pm0.2$ $23.6\pm0.2$ $57.7\pm0.7$ $60\pm5$
GSA1 -4.00 $1.5\pm0.1$ $24.2\pm0.2$ $23.7\pm0.2$ $58.3\pm0.8$ $61\pm4$
GSA3 -4.00 $1.5\pm0.1$ $23.9\pm0.2$ $23.3\pm0.2$ $56.2\pm0.8$ $61\pm5$
---------- ------- ------------- -------------- -------------- -------------- --------------
: \[vfidt\] *Values of various parameters for the EDELWEISS bolometers determined from $^{252}$Cf calibrations under the given bias voltage. The error bars correspond to statistical errors. The systematic error on $V_{fid}$ is $\sim$1%. “Activation” refers to the fraction of 8.98 and 10.34 keV events recorded with the fiducial volume cuts ($E_{center}>3E_{guard}$).*
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{width="8cm" height="8cm"} {width="8.8cm" height="8.8cm"}
{width="12cm" height="14cm"}
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{width="8cm" height="8cm"} {width="8cm" height="8cm"}
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[^1]: See e.g. [@berg] for a review.
[^2]: The NTD thermal sensors have been produced by Torre and Mangin for GeAl6 and by Haller-Beeman Associates for the other detectors.
[^3]: During the data takings, the bias voltage applied to the top electrode varied from $\pm$ 3 V to $\pm$ 9 V depending on the detector.
|
---
author:
- |
V.A. Baskov, A.V. Koltsov, A.I. L’vov, A.I. Lebedev, L.N. Pavlyuchenko, , E.V. Rzhanov, S.S. Sidorin, G.A. Sokol\
P.N. Lebedev Physical Institute, Leninsky prospect 53, Moscow 119991, Russia\
Email:
- |
S.V. Afanasiev, A.I. Malakhov\
Joint Institute for Nuclear Research, Joliot-Curie 6, Dubna 141980, Moscow region, Russia
- |
A.S. Ignatov, V.G. Nedorezov\
Institute for Nuclear Research, 60-letiya Oktyabrya prospekt 7a, Moscow 117312, Russia
title: 'Studies of eta-mesic nuclei at the LPI electron synchrotron'
---
Introduction: $\eta$-mesic nuclei {#introduction-eta-mesic-nuclei .unnumbered}
=================================
$\eta$-mesic nuclei, i.e. nuclear systems $_\eta A$ having the $\eta$-meson bound in a nuclear orbit by strong interaction with $A$ nucleons, have been predicted long ago [@haider86; @liu86] — soon after recognizing the attractive character of the $\eta N$ interaction at low energies [@bhalerao85]. Observations and investigations of these exotic systems would be very valuable for understanding meson-baryon interactions in free space and in nuclei and for studies of properties of hadrons in the dense nuclear matter.
[r]{}[0.5]{} {width="40.00000%"}
The $\eta$-meson, together with pions and kaons, belongs to the SU(3) octet of pseudoscalar mesons and has, therefore, a similar $q\bar q$ space structure. In contrast to the pion, however, the pseudoscalar coupling of $\eta$ to the nucleon is empirically rather small [@tiator94]. Nevertheless the amplitude of $\eta N$ $s$-wave scattering is not as small as that for $\pi N$ scattering because of the contribution of the $s$-wave resonance $S_{11}(1535)$ which is actually a chiral partner of the nucleon — the lowest lying baryon with the opposite parity to the nucleon. This resonance has the mass slightly above the $\eta N$ threshold, $m_\eta+m_N = 1486$ MeV, and owing to its very strong coupling to the $\eta N$ channel \[with the branching ratio ${\rm Br}\;(S_{11}(1535)\to\eta N) \simeq 55\%$\] strongly enhances all interactions in this channel. A nice illustration of this feature is provided by Mainz data [@mcnicoll10] on the total cross section of $\eta$ photoproduction off protons. A huge near-threshold enhancement shown in Fig. \[eta-x-sect\] is just a manifestation of the $S_{11}(1535)$ resonance excited in the reaction $\gamma p\to
S_{11}(1535)\to\eta p$.
The $S_{11}(1535)$ resonance strongly contributes to the low-energy $\eta N$ scattering and, in particular, makes the threshold value of the $\eta N$ scattering amplitude (i.e. the $\eta N$ scattering length $a_{\eta N}$) positive. In the framework of a dynamical resonance model for the coupled channels $\pi N$, $\eta N$ and $\pi\pi
N$, Bhalerao and Liu [@bhalerao85] found a\_[N]{} = 0.28 + i 0.19 fm. \[a-etaN-BL\] The positive value of ${\rm Re\,}a_{\eta N}$ means an effective attraction between $\eta$ and $N$, so that one can expect that several nucleons could jointly bind $\eta$ to a nuclear orbit. The first-order static-limit on-shell optical potential of $\eta$ in the nuclear matter at zero energy $E_\eta^{\rm kin}=0$ is equal to U(r) = -2 a\_[N]{}(r) ( + ), what gives \[together with Eq. (\[a-etaN-BL\])\] $U = -34 -i\;23$ MeV at normal nuclear matter density $\rho=\rho_0 = 0.17~\rm fm^{-3}$. The imaginary part of the potential describes a local absorption rate $\Gamma = -2\,{\rm Im}\,U$ of $\eta$ in the nuclear substance.
With the above strength of the $\eta A$ potential, $\eta$-mesic nuclei $_\eta A$ are expected to exist for all $A\ge 10$ [@haider02; @haider09]. Actually, due to a sharp (cusp) energy dependence of the $\eta N$ scattering amplitude near threshold, Fermi motion of nucleons and $\eta$ reduces the optical potential \[especially its imaginary part\], and this makes $\eta$-mesic nuclei to exist only for $A\ge 12$. For binding energies and widths of the lightest $\eta$-mesic nuclei Haider and Liu predicted [@haider02; @haider09] E\_= -1.19 [MeV]{}, && \_= 7.34 [MeV for\^[12]{}\_[ ]{}C]{}, E\_= -3.45 [MeV]{}, && \_= 10.76 [MeV for\^[16]{}\_[ ]{}O]{}, E\_= -6.39 [MeV]{}, && \_= 13.20 [MeV for\^[26]{}\_[ ]{}Mg]{}. \[eq:EB-Liu\] Note, however, that a stronger $\eta N$ scattering amplitude was inferred in some other analyses. For example, using a $K$-matrix model for coupled channels $\pi N$, $\eta N$, $\gamma N$ and $\pi\pi N$, Green and Wycech [@green97; @green05] found from fit to available data a\_[N]{} = (0.910.06) + i (0.270.02) fm. With such a big strength of $\eta N$ interaction lighter $\eta$-mesic nuclei could also exist.
As an example of different predictions for binding energies and widths of $\eta$-mesic nuclei we mention very elaborated calculations [@oset02a; @oset02b; @oset02c], in which a model for meson-baryon interaction with dynamically generated resonances was build using a unitarized chiral perturbation theory for coupled channels $\pi N$, $\eta N$, $K\Lambda$, $K\Sigma$ and $\pi\pi N$ and then self-energies of all the particles in the nuclear matter were evaluated consistently. This approach leads to the $\eta N$ scattering length $a_{\eta N} =
0.264 + i\, 0.245~\rm fm$ close to that obtained in Eq. (\[a-etaN-BL\]). The resulting $\eta A$ potential is, however, found stronger owing to nonlinear dressing effects: $U = -54 -i\, 29$ MeV at normal nuclear density. Also stronger are $\eta$-meson bindings found in [@oset02c]: E\_= -9.71 [MeV]{}, && \_= 35.0 [MeV for\^[12]{}\_[ ]{}C]{}, E\_= -12.57 [MeV]{}, && \_= 33.4 [MeV for\^[24]{}\_[ ]{}Mg]{}. \[eq:EB-Oset\]
Bindings with equally large widths arise also in calculations [@jido02; @nagahiro05; @jido08] that use a chiral doublet model and treat $\eta A$ and $S_{11}(1535)A$ attraction as a result of partial restoration of chiral symmetry in the dense nuclear matter leading to reduction of the $S_{11}(1535){-}N$ mass gap. It is clear that experimental data on energies and widths of $\eta$-mesic nuclei are needed to test these and many other models and calculations.
Signature for eta-mesic nuclei produced in photoreactions {#signature-for-eta-mesic-nuclei-produced-in-photoreactions .unnumbered}
=========================================================
A mechanism of $\eta$-mesic nuclei formation and decay in the photoreaction + A N’ + \_[ ]{}(A-1)N’ + + N + (A-2) \[reac:piN\] is shown in Fig. \[diagrams-piN\]a. A fast nucleon $N'$ ejected forward at the first stage of the reaction, i.e. in the subprocess + N’ N’ + \_[slow]{}, \[reac:eta\] escapes the nucleus, whereas a slow $\eta$ is captured by remaining $A-1$ nucleons to a bound state. At $E_\gamma \sim 800{-}900$ MeV, a minimal momentum transfer to $\eta$ in the reaction (\[reac:eta\]) is not large (less than $70~{\rm MeV}/c$). That is why the total cross section of $\eta$-mesic nuclei formation off light nuclei (like carbon or oxygen implied in the following) turns out to be a few $\mu$b [@kohno89; @lebedev89; @lebedev91; @lebedev95; @tryasuchev99; @tryasuchev01], i.e. $\simeq 2{-}7\%$ of the total cross section $\sigma_{\gamma
A}^\eta$ of inclusive $\eta$ photoproduction, with the exact value strongly dependent on the assumed strength of the optical potential $U$.
![a) $\eta$-mesic nuclei formation and decay with the emission of back-to-back $\pi N$ pairs. b) Background creation of back-to-back $\pi N$ pairs by unbound $\eta$.[]{data-label="diagrams-piN"}](fig-diagram-piN.eps){width="80.00000%"}
Energies $E[{}_\eta(A-1)]$ of the produced $\eta$-mesic nuclei can, in principle, be determined through missing mass measurements in the reaction $(\gamma,p)$ using tagged photons $\gamma$ and a magnetic spectrometer for $N'=p$. Indirectly, the same energy E\[\_(A-1)\] = E\_+ E\_[A-1]{} = E\_[N]{} + E\_[A-2]{} \[eq:E-eta-(A-1)\] can also be found from the observed energy of a correlated back-to-back $\pi N$ pair produced at the second stage of the reaction (\[reac:piN\]) where the captured $\eta$ meson annihilates through the subprocess N N. \[etaNpiN\] The energy excitation of $(A-2)$ in (\[eq:E-eta-(A-1)\]) is not a fixed value. It rather depends on whether an $s$-shell or $p$-shell nucleon $N$ is knocked out in the process (\[etaNpiN\]). Therefore a distribution of the experimental observable $E_{\pi N}$ has appropriately a bigger width than the width of the $\eta$-mesic nucleus.
Neglecting binding and Fermi motion of nucleons and $\eta$, we have the following kinematical characteristics of the ejected correlated $\pi N$ pairs (as for energies, momenta and velocities): && s = E\_+ E\_N = m\_+m\_N = 1486 MeV, && E\_\^[kin]{} = 313 [MeV]{},E\_N\^[kin]{} = 94 [MeV]{}, p\_= p\_N = 431 [MeV]{}/c, && \_= 0.95,\_N=0.42. \[kinema-piN\] A simple simulation that takes into account the Fermi motion of nucleons and $\eta$ as well as binding of these particles reveals that fluctuations around these ideal parameters are substantial (see Fig. \[simulation-piN\]) \[specifically, we used in this simulation the $\eta$-meson binding energy of 10 MeV with the width 25 MeV; for nucleons, we assumed a Fermi-gas distribution with binding energies distributed between 5 and 30 MeV\]. In particular, the angle $\theta_{\pi N}$ between the emitted pion and nucleon may not be so close to $180^\circ$, and a subtraction of background events with $\theta_{\pi N} \ne 180^\circ$ used sometimes in practice should be done cautiously. A shift of the peak down to 1486 MeV in the distribution of the total energy $E_{\pi N}=E_\pi+E_N$ seen in Fig. \[simulation-piN\] is related with binding of both the $\eta$-meson (by 10 MeV) and the nucleon (by 15 MeV).
![Simulation of $\pi N$ pairs emitted in $\eta$-mesic nuclei decays. Shown are distributions over kinetic energies of the particles, their total energy, velocities, and the $\pi N$ relative angle.[]{data-label="simulation-piN"}](fig-Tpi.ps "fig:"){width="33.00000%"} ![Simulation of $\pi N$ pairs emitted in $\eta$-mesic nuclei decays. Shown are distributions over kinetic energies of the particles, their total energy, velocities, and the $\pi N$ relative angle.[]{data-label="simulation-piN"}](fig-TN.ps "fig:"){width="33.00000%"} ![Simulation of $\pi N$ pairs emitted in $\eta$-mesic nuclei decays. Shown are distributions over kinetic energies of the particles, their total energy, velocities, and the $\pi N$ relative angle.[]{data-label="simulation-piN"}](fig-Etot.ps "fig:"){width="33.00000%"}
![Simulation of $\pi N$ pairs emitted in $\eta$-mesic nuclei decays. Shown are distributions over kinetic energies of the particles, their total energy, velocities, and the $\pi N$ relative angle.[]{data-label="simulation-piN"}](fig-betapi.ps "fig:"){width="33.00000%"} ![Simulation of $\pi N$ pairs emitted in $\eta$-mesic nuclei decays. Shown are distributions over kinetic energies of the particles, their total energy, velocities, and the $\pi N$ relative angle.[]{data-label="simulation-piN"}](fig-betaN.ps "fig:"){width="33.00000%"} ![Simulation of $\pi N$ pairs emitted in $\eta$-mesic nuclei decays. Shown are distributions over kinetic energies of the particles, their total energy, velocities, and the $\pi N$ relative angle.[]{data-label="simulation-piN"}](fig-costheta.ps "fig:"){width="33.00000%"}
[r]{}[0.4]{} {width="35.00000%"}
Notice that $\pi N$ pairs with the characteristics (\[kinema-piN\]) do not necessary originate from $\eta$-mesic nuclei decays. They can also be produced by slow etas in a background nonresonance process shown in Fig. \[diagrams-piN\]b. The resonance and nonresonance processes correspond to a resonance (Breit-Wigner) and nonresonance part of the full propagator \[i.e. the Green function $G({\bm r}_1,
{\bm r}_2; E_\eta)$\] of the $\eta$-meson moving in the optical potential $U(r)$. Jointly, these parts generate a complicated spectrum of $E_\eta$ similar to that obtained in a toy model with a square-well potential [@lvov98; @sokol99]. Shown in Fig. \[spectral-function\] is the spectral function in that model, S(E\_) = ([r]{}\_1) ([r]{}\_2) |G([r]{}\_1, [r]{}\_2; E\_)|\^2 d[r]{}\_1 d[r]{}\_2, that characterizes near-threshold energy distribution of the propagated etas as well as the near-threshold energy dependence of the yield of $\pi N$ pairs produced by these $\eta$. Bound states of the $\eta$-meson give pronounced peaks in the yield of the $\pi N$ pairs at subthreshold energies $E_\eta$. Generally, observation of a relatively narrow resonance peak in the spectrum of $E_\eta$ in the region $E_\eta < m_\eta$ is mandatory for claiming an observation of $\eta$-mesic nuclei at all. We refer to recent works by Haider and Liu [@haider10a; @haider10b] where a deeper and more elaborated consideration is given in relation with a recent experiment.
Since $\eta$ is isoscalar, the $\pi N$ pairs produced in the subprocess (\[etaNpiN\]) have isospin $\frac12$ and hence the following isotopic contents \[for $\eta$-mesic nuclei with $A\gg 1$\]: (N) ={
[lll]{} 1/3 & [for]{} & \^+n,\
1/6 & [for]{} & \^0p,\
1/6 & [for]{} & \^0n,\
1/3 & [for]{} & \^-p.
. \[piN-modes\] From these, the channel $\pi^+n$ was chosen for detection in our experiment.
Previous searches for $\eta$-mesic nuclei {#previous-searches-for-eta-mesic-nuclei .unnumbered}
=========================================
Searches for $\eta$-mesic nuclei began very soon after their predictions [@haider86] followed by suggestions [@liu86; @kohno89; @lebedev89; @lebedev91; @kohno90] to seek these novel high-energy nuclear excitations in missing-mass experiments using the inclusive reactions $(\pi^+,p)$ and $(\gamma,p)$.
The first two experiments have been done along this line in 1988 at Brookhaven [@chrien88] and Los Alamos [@lieb88a; @lieb88b]. In both experiments, a $\pi^+$ beam was used and several targets (Li, C, O and Al) were examined. The inclusive $(\pi^+,p)$ reaction \^+ + A \_(A-1) + p was studied in [@chrien88] with a magnetic spectrometer, whereas the Los Alamos experiment had also an additional $4\pi$ BGO crystal ball for detecting charged paticles ejected in the subprocess (\[etaNpiN\]) of $\eta$-mesic nuclei decays to $\pi N$ pairs in coincidence with the forward proton $p$.
The Brookhaven experiment did not find a theoretically expected signal [@liu86] — a relatively narrow peak of a predicted strength in the missing mass spectrum. The team working at Los Alamos did report a preliminary evidence for a wanted peak for the $^{16}$O target but this report was not confirmed (published) since then.
It was recognized in the following that the above obtained negative or incomplete results do not necessarily mean that the predicted $\eta$-mesic nuclei do not exist. It was possible that the binding energies and especially the widths of the $\eta$ bound states were theoretically underestimated. This point of view was supported by many-body calculations [@chiang91] taking into account some effects disregarded in the first theoretical works [@haider86; @liu86], in particular — dressing, binding and collisional decays of the $S_{11}(1535)$ resonance in the dense nuclear matter. The analysis of [@chiang91] was later extended and revised [@oset02a; @oset02b; @oset02c] (in particular, dressing of mesons was also included) with the main conclusion survived that $\eta$-mesic nuclei widths are bigger than those found in [@haider86; @liu86].
The next experiment has been performed at the Lebedev Physical Institute in Moscow/Troitsk [@sokol99; @sokol00] (see also a summary in [@sokol08]). It was triggered [@sokol94; @lebedev95a] by a suggestion [@sokol91] to seek $\eta$-mesic nuclei through observing decay products of $\eta$-mesic nuclei, namely two correlated back-to-back particles, a pion and a nucleon, ejected in the process of annihilation of captured $\eta$-mesons in the nucleus, Eq. (\[etaNpiN\]). It was hoped that a background for the two very energetic particles (the pion and the nucleon) ejected in decays of $\eta$-mesic nuclei transversely to the beam would be lower than that for ejection of forward protons in the inclusive processes. Besides, it was hoped that background conditions in photon-induced reactions would be generally better than those in pion-induced ones.
Studies of the reaction + \^[12]{}[C]{} (\_[ ]{}\^[11]{}[Be]{} [ or ]{} \_[ ]{}\^[11]{}[C]{}) + N \^+ + n + X + N \[reac-C:piN\] done in the middle of 1990’s at the LPI electron synchrotron indeed showed a signal of an enhanced production of the correlated back-to-back $\pi^+n$ pairs ejected transversely to the photon beam when the photon energy exceeded the $\eta$-meson photoproduction threshold. Energy resolution of the experimental setup was, however, not sufficient to resolve a peak similar to that shown in Fig. \[spectral-function\] and to determine whether the observed correlated pairs were produced by bound or unbound intermediate etas.
After the works [@sokol99; @sokol00] gaining and using information on the decay products became mandatory for experiment planning and data analysis in all further searches for $\eta$-mesic nuclei.
In 2004 an evidence for the $\eta$-mesic nucleus $_\eta^3$He formed in the reaction + \^3[He]{} \_\^3[He]{} \^0 + p + X has been reported from Mainz [@pfeiffer04]. A resonance-like structure was observed in a contribution to the cross section from back-to-back $\pi^0p$ pairs found after a background subtraction. A later study [@pheron12] revealed, however, that the background has a rather complicated structure, so that the conclusions of Ref. [@pfeiffer04] cannot be confirmed. At the moment their statement is that the existence of the $\eta$-mesic nucleus $_\eta^3$He is not yet established.
One more attempt to find $\eta$-mesic nuclei by detecting their $\pi^-p$ decay products has recently been done at the JINR nuclotron [@afanasiev11]. The reaction studied was d + \^[12]{}[C]{} (\_[ ]{}\^[11]{}[Be]{} [ or ]{} \_[ ]{}\^[11]{}[C]{}) + N\_1 + N\_2 \^- + p + X + N\_1 + N\_2. \[reac:Dubna\] The measured effective mass spectra of the correlated back-to-back $\pi^-p$ pairs show a presence of resonance-like peaks lying slightly below the threshold energy $m_\eta+m_N=1486$ MeV. However, a consistent interpretation of these peaks was not yet obtained.
To date the strongest evidence for the existence of $\eta$-mesic nuclei came from the precision COSY-GEM experiment [@budzanowski09]. Following ideas of the work [@hayano99] borrowed in turn from previous experience in studying deeply-bound pionic states in nuclei, the reaction p + \^[27]{} \^3[He]{} + \^[25]{}\_[ ]{}[Mg]{} \^3[He]{} + p + \^- + X \[reac:COSY-GEM\] of a recoilless formation of the $\eta$-mesic nuclei $^{25}_{~\eta}\rm
Mg$ was explored and the mass of this $\eta$-mesic nucleus was determined through precision missing-mass measurements in $(p,
{}^3{\rm He})$. A clear peak was found in the missing mass spectrum that corresponds to the binding energy $-13.13\pm 1.64$ MeV and the width $10.22\pm 2.98$ MeV of the formed $\eta$-mesic nucleus. An upper limit of $\approx 0.5$ nb was found for the cross section of the $\eta$-mesic nucleus formation.
Recently Haider and Liu argued [@haider10a; @haider10b] that the observed peak in (\[reac:COSY-GEM\]) is shifted down from the genuine binding energy of $\eta$ because of interference of the resonance and nonresonance mechanisms of the reaction (similar to those shown in Fig. \[diagrams-piN\]). This very interesting effect signifies that the genuine $\eta$ binding in ${}_{~\eta}^{25}{\rm Mg}$ is $\approx -8$ MeV with the width $\approx 19$ MeV.
On the two-nucleon decay mode of $\eta$-mesic nuclei {#on-the-two-nucleon-decay-mode-of-eta-mesic-nuclei .unnumbered}
====================================================
[r]{}[0.4]{} {width="35.00000%"}
{width="40.00000%"}
The main novelty in our present research is exploring a new possibility for searching for $\eta$-mesic nuclei, namely through observation of their two-nucleon decay mode arising owing to the two-nucleon absorption of the captured $\eta$ in the nucleus, NN NN, \[2N-absorption\] see Fig. \[diagram-NN\]. Ejected in this process correlated back-to-back nucleons of the $NN$ pairs have very high energies ($E_N^{\rm kin}\simeq \frac12 m_\eta = 274$ MeV) and momenta ($p_N\simeq 770~{\rm MeV}/c$), so that they are to be visible (especially in coincidence) at the background of other particles emitted in photoreactions at $E_\gamma\sim 800$ MeV and thus should provide a bright signature for the $\eta$-mesic nucleus formation.
The $NN$ pair production in decays of $\eta$ in the nuclear matter was considered among other channels by Chiang, Oset and Liu [@chiang91] in terms of the self-energy of $S_{11}(1535)$ that includes a contribution of $S_{11}(1535)N\to NN$. A more direct and rather transparent evaluation of this process has been done by Kulpa and Wycech [@kulpa98b] who used available experimental data on the inverse reactions $pp\to pp\eta$, $pn\to pn\eta$ and $pn\to d\eta$ and then converted them into the rate of (\[2N-absorption\]). In terms of the imaginary part $W_{NN}$ of the optical potential $U$, this rate was found to be proportional to $\rho^2$, being $W_{NN}=3.4$ MeV at central nuclear density. This is only about 15% of $W_N \sim 23$ MeV related with the absorption of $\eta$ by one nucleon. Nevertheless such a small fraction of $NN$ can be quite visible experimentally because of a specific isotopic contents of the $\pi N$ and $NN$ pairs.
The matter is that $\gtrsim 90\%$ of these $NN$ pairs are proton plus neutron because the cross section of $pp\to pp\eta$ (and $nn\to
nn\eta$) is by order or magnitude less than that of $pn\to pn\eta$ (plus $pn\to d\eta$), see Fig. \[fig-etaNN\] where pertinent Uppsala-Celsius [@calen96; @calen97; @calen98a; @calen98b] and COSY [@smyrski00; @moskal09] data are shown (and see also, e.g., [@baru03] for theoretical explanations). This difference can be traced to isospin factors and Fermi statistics signs in the dominating pion-exchange mechanism of the reaction $NN\to NN\eta$ shown in Fig. \[diagrams-NNeta\]. If the experimental setup detects one charged and one neutral particle from the pairs, it detects $\sim90\%$ of $NN$ and only $\sim33\%$ of $\pi N$. Then count rates of the setup would not be so different for $pn$ and $\pi^+n$ pairs. That seems to be exactly what we see in our experiment.
![Pion-exchange mechanism of $NN\to NN\eta$. Isospin factors, which accompany the $\pi NN$ coupling $g$ and the $\pi N\to\eta N$ amplitude $T$, and the Fermi-statistics signs (both shown in this Figure) jointly determine the big difference between the cross sections of $pp\to pp\eta$ and $pn\to pn\eta$ (plus $pn\to d\eta$). Antisymmetrization of the initial state and initial/final state interactions are not shown.[]{data-label="diagrams-NNeta"}](fig_pp.ps "fig:"){width="50.00000%" height="13ex"} ![Pion-exchange mechanism of $NN\to NN\eta$. Isospin factors, which accompany the $\pi NN$ coupling $g$ and the $\pi N\to\eta N$ amplitude $T$, and the Fermi-statistics signs (both shown in this Figure) jointly determine the big difference between the cross sections of $pp\to pp\eta$ and $pn\to pn\eta$ (plus $pn\to d\eta$). Antisymmetrization of the initial state and initial/final state interactions are not shown.[]{data-label="diagrams-NNeta"}](fig_pn.ps "fig:"){width="50.00000%" height="13ex"}
![Simulation of $NN$ pairs emitted in decays of $\eta$-mesic nuclei. Shown are distributions over the kinetic energy and velocity of one of the nucleons, the total energy of the pair and the relative angle.[]{data-label="simulation-NN"}](figNN-TN.ps "fig:"){width="36.00000%"} ![Simulation of $NN$ pairs emitted in decays of $\eta$-mesic nuclei. Shown are distributions over the kinetic energy and velocity of one of the nucleons, the total energy of the pair and the relative angle.[]{data-label="simulation-NN"}](figNN-Etot.ps "fig:"){width="36.00000%"}\
![Simulation of $NN$ pairs emitted in decays of $\eta$-mesic nuclei. Shown are distributions over the kinetic energy and velocity of one of the nucleons, the total energy of the pair and the relative angle.[]{data-label="simulation-NN"}](figNN-bN.ps "fig:"){width="36.00000%"} ![Simulation of $NN$ pairs emitted in decays of $\eta$-mesic nuclei. Shown are distributions over the kinetic energy and velocity of one of the nucleons, the total energy of the pair and the relative angle.[]{data-label="simulation-NN"}](figNN-costheta.ps "fig:"){width="36.00000%"}
Neglecting binding effects and effects of Fermi motion of nucleons and $\eta$, we have the following kinematical characteristics of the correlated $NN$ pairs (i.e. energies, momenta, velocities) ejected in $\eta$-mesic nuclei decays: E\_[N\_1]{}\^[kin]{} = E\_[N\_2]{}\^[kin]{} = 12 m\_= 274 [MeV]{}, p\_[N\_1]{} = p\_[N\_2]{} = 767 [MeV]{}/c, \_[N\_1]{} = \_[N\_2]{} = 0.63. \[kinema-NN\] Actually, the Fermi motion and binding leads to fluctuations around these ideal parameters as a simple simulation reveals, see Fig. \[simulation-NN\]. Note that the angular correlation in the $NN$ pairs is stronger than that in the $\pi N$ pairs — owing to higher momenta of particles in $NN$.
The first studies of the photoreaction + \^[12]{}[C]{} (\_[ ]{}\^[11]{}[Be]{} [ or ]{} \_[ ]{}\^[11]{}[C]{}) + N p + n + X + N \[reac:NN\] have recently been done at the LPI synchrotron and we report below on the obtained results.
Experimental setup at LPI {#experimental-setup-at-lpi .unnumbered}
=========================
[r]{}[0.3]{} {width="30.00000%"}
Our experiment was performed at the bremsstrahlung photon beam of the 1.2-GeV electron synchrotron of the Lebedev Physical Institute. Photons were produced with an electron beam of intensity $I_e \simeq 10^{12}
~\rm s^{-1}$ and the duty factor $\simeq 10\%$. The energy of the beam was usually $E_e = E_{\gamma\;\rm max} = 850~\rm MeV$ (i.e. above the $\eta$ photoproduction threshold off free nucleons, $E_{\eta\;\rm
thr}=708~\rm MeV$); additional measurements of subthreshold backgrounds have been done at $E_e = E_{\gamma\;\rm max} = 650~\rm
MeV$.
The experimental setup included two time-of-flight arms (two scintillation telescopes — C and N arms) for detecting in coincidence charged and neutral particles (back-to-back pairs), see Fig. \[exp-setup\]. These arms were both positioned at $90^\circ{-}90^\circ$ with respect to the beam axis in order to minimize background.
The C-arm used for detection of charged particles is a plastic TOF spectrometer for charged pions and protons. It consists of a start detector T1 ($20\times 20\times 2~\rm cm^3$), a stop detector T2 ($50\times 50\times 5~\rm cm^3$) and three energy losses detectors $\Delta E_1$, $\Delta E_2$ and $\Delta E_3$ ($40\times 40\times 2~\rm
cm^3$ each). A 4 mm lead (Pb) plate was used in some runs for TOF calibrations with ultrarelativistic electrons/positrons produced in the lead plate with high-energy photons emitted from the target owing to production and decays of neutral pions.
The N-arm is a plastic TOF spectrometer for neutrons. It consists of a veto counter A ($50\times 50\times 2~\rm cm^3$) and four plastic blocks — the neutron detectors N1, N2, N3 and N4 ($50\times 50\times
10~\rm cm^3$ each). Again, a 4 mm lead plate was used in some runs for TOF calibrations. The efficiency of the N-arm for neutrons of energies above 50 MeV was $\approx 30\%$.
In both arms each volume of scintillator counters/blocks was viewed from corners with 4 phototubes. The time-of-flight bases in the C and N arms were 1.4 m and the time resolution was $\simeq
200$ ps ($1\sigma$). The target was a carbon cylinder of the 10 cm length along the beam axis. Its diameter was 4 cm, i.e. slightly more than the diameter of the collimated photon beam (3 cm). The distance between the target and the start detector T1 was 0.7 m.
Mostly, the setup was the same as in our previous work [@sokol00; @sokol08] but a few useful changes have been made:
- $\Delta E_i$ detectors have been placed after the time-of-flight interval T1-T2. This enabled us to have a better $\pi^\pm/p$ separation and time resolution.
- A transverse size of the start detector T1 was cut off according to required geometry. This reduced a background load of the C-arm.
- A thickness of the start detector was also reduced in order to improve time resolution.
- All unnecessary layers of absorbers used previously to suppress radiative backgrounds have been removed from the time-of-flight interval, with the effect of reducing the $e^+/e^-$ background created by photons from $\pi^0$ decays.
[r]{}[0.4]{} {width="35.00000%"}
General tests of the setup, including preliminary time calibrations of the arms, have been done in special runs, in which the quasifree reaction $\gamma p\to\pi^+n$ inside carbon nuclei was observed. In such runs the two arms of the setup have been moved to the angles $50^\circ{-}50^\circ$ where the high count rate enabled one to do the calibrations quickly. Lead convertors used in these runs provided reliable ultrarelativistic reference points $\beta=1$ for particle’s velocities $\beta_C$ and $\beta_N$ measured in the C- and N-arms. A two-dimensional $\beta_C{-}\beta_N$ plot on Fig. \[2-dim-bb\] illustrates this procedure.
The calibration done provided a linear scale of velocities in the range $\beta = 0.6{-}1$ with errors of about 3% ($1\sigma$). We have checked the linearity of the scale by using cosmic rays and setting different distances between detectors.
Results and comparison with simulations {#results-and-comparison-with-simulations .unnumbered}
=======================================
Measurement runs have mostly been done in 2009 at two maximal beam energies: $E_{\gamma\;\rm max} = 650$ MeV and 850 MeV. The on-line trigger was a coincidence of particles in the C- and N-arms within a time gate of 50 ns.
For further off-line analysis events were selected with an additional condition of sufficiently long ranges of the charged particles, E\_i > E\_i\^[thr]{} \[eq:selection\] with experimentally adjusted thresholds $E_i^{\rm thr}$. In this way low-energy particles in the C-arm were rejected.
A two-dimensional histogram in the variables $\Delta E{-}\beta_C$, where $\Delta E$ is the minimal energy loss in the $\Delta E_i$ detectors, E=\_i(E\_i), is shown in Fig. \[2-dim-bE\] for the beam energy $E_{\gamma\;\rm
max} = 850$ MeV. Results of simulations using the Intra Nuclear Cascade (INC) model [@pschenichnov97] in the GEANT-3 framework are shown in Fig. \[2-dim-bE-INC\] for comparison. The INC model takes into account production of various mesons and baryon resonances, their free propagation in the nuclear matter, and then various $2\to 2$ collisional reactions including $\eta N\to\pi N$. This model successfully describes many photoreactions in wide kinematical ranges as was demonstrated, beyond [@pschenichnov97], in simulations of the GRAAL experiment at energies 500–1500 MeV [@ignatov08]. Binding effects for $\eta$ and reactions like $\eta NN\to NN$ were not included into the model, so one can try to find effects arising due to formation and decay of $\eta$-mesic nuclei through characteristic deviations of the model predictions from the experimental data.
The simulation shows that the selection (\[eq:selection\]) of particles with sufficiently long ranges distinguishes very well protons (as particles with $\beta_C \leq 0.7$) and pions (as particles with $\beta_C \geq 0.7$): the overlap is less than 1%.
![Two-dimensional $\Delta E{-}\beta_C$ distribution, the INC model.[]{data-label="2-dim-bE-INC"}](fig-2dim-bE.eps){width="60.00000%"}
![Two-dimensional $\Delta E{-}\beta_C$ distribution, the INC model.[]{data-label="2-dim-bE-INC"}](fig-2dim-bE-INC.eps){width="60.00000%"}
Considering one-dimensional spectra over $\beta_C$ of events selected according to the condition (\[eq:selection\]) of sufficiently long ranges and imposing the additional cut-off $0.3 <\beta_N < 0.7$ for neutron velocities, we find rather interesting structures in the spectra. Shown in Fig. \[bC\] are experimental data (blue areas) together with results of the INC simulation (pink hatched areas). Separately shown are INC predictions for the number of protons and charged pions in the C-arm. There is a qualitative agreement of the INC simulation with the experimental data for the case of the subthreshold beam energy, $E_e=650$ MeV. Meanwhile, in the case of $E_e=850$ MeV there is a clear excess of the experimentally observed events over the simulation results in two velocity regions closely corresponding to the kinematics of $\eta$-mesic nuclei decays with emission of $\pi N$ and $NN$ correlated pairs, Eqs. (\[kinema-piN\]) and (\[kinema-NN\]).
Knowing from the INC simulations that the ”normal” (without $\eta$-mesic nuclei) dynamics of the considered reaction does not yield a sufficient amount of protons and pions with the velocities of about $\beta_C\sim 0.7$, we interpret the found anomaly at $\beta_C\sim 0.7$ as a result of production of low-energy $\eta$-mesons followed by their two-nucleon annihilation.
The energy resolution of the experimental setup is not sufficient to say whether an essential part of these $\eta$-mesons is produced in the bound state, but theoretical arguments discussed in above make such a statement plausible.
Concerning the excess of pions with $\beta_C\simeq 0.95$, this feature is in agreement with our measurements reported earlier [@sokol99; @sokol00; @sokol08]. It can be interpreted as an evidence for one-nucleon annihilation of produced low-energy $\eta$-mesons (bound or unbound).
Electron/positron peaks shown in Fig. \[bC\] originate from calibration runs with the lead plate inserted. They were not included into simulations made.
![Velocity distribution of charged particles selected according to the criterion $\Delta E_i > 0$ (for all $i=1,2,3$) at $E_e=650$ and 850 MeV. A well visible excess of events over the INC simulation is seen at the right panel — in the case of the beam energy exceeding the $\eta$-photoproduction threshold — in both velocity regions corresponding to the expected velocities of the $\pi N$ and $NN$ decay products of $\eta$-mesic nuclei.[]{data-label="bC"}](fig-bC-650.eps){width="45.00000%"}
The observed proton peak in the $\beta_C$ distribution is very unusual because it corresponds to $pn$ pairs with very high kinetic energies $T_p\sim T_n\sim 200{-}300$ MeV and transverse momenta $p_p\sim
p_n\sim 400{-}800$ MeV/c. One should keep in mind that photons which produce such pairs have quite a modest energy $650~{\rm MeV} < E_\gamma
< 850$ MeV. Ordinary photoproduction reactions do not give nucleons with such a high energy and momentum. Creation and annihilation of intermediate low-energy $\eta$-mesons seems to be the only explanation to these events.
Assuming that the observed access events are mainly related with formation and isotropic decays of $\eta$-mesic nuclei with $A=11$, we can estimate their photoproduction cross section. The number of photons of the energies $E_\gamma = 650{-}850$ MeV that hit the carbon target in experimental runs was evaluated via comparison of the total yield of charged pions detected by a single C-arm of the setup with predictions of INC for that yield, thus giving the result $N_\gamma
\simeq 1.36\times 10^{11}$. Taking into account the solid angle of the C-arm telescope ($\Omega_C = 0.027$ sr), efficiencies of detectors, a geometric efficiency of the $N$-arm of the setup ($\sim 18\%$ as found from theoretically expected angular distributions of particles of the correlated pairs), we arrived at the following cross section of $\eta$-mesic nucleus formation: (+ [ ]{}\^[12]{}[C]{}\_A + X) 10 b. \[x-section\] We write it as an upper limit because part of the observed events can be related with unbound etas. This number is consistent with available theoretical estimates (typically, a few $\mu$b).
Conclusions {#conclusions .unnumbered}
===========
The new obtained data confirm the main features of the $\pi N$ signal of formation and decay of $\eta$-mesic nuclei off the carbon target in the photoreaction found in our previous work.
A new signature for formation and decay of $\eta$-mesic nuclei, the back-to-back $pn$ pairs, was explored. For the first time an experimental evidence was found that the yield of such pairs in the region of $\beta_C\sim 0.6{-}0.7$ is quite large and therefore is also suitable for searching for $\eta$–mesic nuclei.
Assuming that the observed excess of events is related with $\eta$-mesic nuclei, an estimate of the total cross section of formation of $\eta$-nuclei in the photoreaction off carbon have been obtained, see Eq. (\[x-section\]).
We have plans to carry out a more precise experiment, with a better energy resolution, at the deuteron beam of the JINR nuclotron.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in part by the RFBR grants 08-02-00648-a and 10-02-01433-a. A nice work of the accelerator group of the LPI synchrotron and its leader G.G. Subbotin is highly appreciated.
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|
---
abstract: 'The Casimir effect is a force arising in the macroscopic world as a result of radiation pressure of vacuum fluctuations. It thus plays a key role in the emerging domain of nano-electro-mechanical systems (NEMS). This role is reviewed in the present paper, with discussions of the influence of the material properties of the mirrors, as well as the geometry dependence of the Casimir effect between corrugated mirrors. In particular, the lateral component of the Casimir force and restoring torque between metal plates with misaligned corrugations are evaluated.'
author:
- 'Cyriaque Genet$^1$, Astrid Lambrecht$^2$, and Serge Reynaud$^2$'
title: The Casimir effect in the nanoworld
---
Introduction {#intro}
============
The Casimir force was predicted in $1948$ by H.B.G. Casimir as an attractive force between two perfectly reflecting, plane and parallel mirrors in vacuum [@Casimir48]. The force has been measured in different experiments with an increasing control of the experimental conditions . This has been considered as an important aim which should allow an accurate comparison between theoretical predictions and experimental observations [@Milonni94; @LamoreauxResource99; @Reynaud01; @GenetIAP02; @LambrechtPoincare]. These advances have been reviewed in a number of papers, for example [@LambrechtPoincare; @Bordag01; @Milton05] and in a special issue of the New Journal of Physics [@NJP06].
Meanwhile, it has been realized that the Casimir force was a dominant force at micron or sub-micron distances, and then clearly an important aspect in the domain of micro- and nano-oscillators (MEMS, NEMS) [@BuksPRB2001; @ChanScience2001; @ChanPRL2001] now emerging from modern nanofabrication techniques [@EkinciRSI2005]. If the Casimir force has been primarly considered as a source of stiction between mobile parts, it is now recognized as an essential source of actuation to be used in the design of MEMS and NEMS.
In both fundamental and technological contexts, it is extremely important to take into account the real experimental situations which largely differ from the ideal conditions considered by Casimir. We review below some theoretical tools which have shown their efficiency for a general formulation of the Casimir effect, accounting for the material properties of the interacting plates as well as for the effect of non planar boundary geometries.
Idealized Casimir force {#sec:1}
=======================
The Casimir force and energy between two perfectly reflecting, plane and parallel mirrors immersed in quantum vacuum have the following forms $$\begin{aligned}
F_{\rm Cas} = \frac{\pi^{2}\hbar c}{240}\frac{A}{L^{4}} \ \ , \ \
E_{\rm Cas} = - \frac{\pi^{2}\hbar c}{720}\frac{A}{L^{3}}. \label{FEcas}\end{aligned}$$ These expressions correspond to an attractive force $F_{\rm Cas}$ and a binding energy $E_{\rm Cas}$. Remarquably, they depend only on geometrical quantities, the area $A$ of the mirrors and their distance $L$ ($A\gg L^{2}$), and fundamental constants, the Planck constant $\hbar$ and the speed of light $c$.
Imperfect reflection {#sec:3}
====================
Experiments are performed with real metallic mirrors which good reflectors only at frequencies below their plasma frequency $\omega_{\rm P}$ which depends on the properties of the conduction electrons in the metal. The effect of imperfect reflection on the Casimir force and energy has been recognized long time ago [@Lifshitz56; @Schwinger78] though it has been described with good accuracy only recently [@Lamoreaux99; @Lambrecht00; @KlimPRA00]. We recall below the scattering theory of the Casimir force which has been developed and used to this aim [@Jaekel91; @GenetPRA03; @LambrechtNJP06].
We begin with perfectly plane and parallel mirrors, separated by a distance $L$. The two mirrors form a Fabry-Perot cavity and the fluctuations of the intracavity fields propagating back and forth along the cavity axis can be calculated in terms of the fluctuations of the incoming free-space fields. The field modes are characterized by their frequency $\omega$, transverse wavevector ${\bf k}$ with components $k_{x},k_{y}$ in the plane of the mirrors, and by their polarization $p$. Time invariance of the problem, as well as transverse spatial translation invariance (along $x$ and $y$) ensure that the frequency, the transverse wavevector and the polarization are conserved quantities throughout the scattering process on the cavity. The scattering couples only the free vacuum modes with opposite signs for the component $k_{z}$ of the wavevector along the longitudinal $z$ axis of the cavity. We write $r_{\bf k}^{p}[\omega]$ the reflection amplitude of the mirror $i=1,2$ as seen from the inner side of the cavity. These amplitudes obey general physical properties of causality, unitarity and high frequency transparency.
The spectral density of the vacuum intracavity fields is changed with respect to that of free-fields outside the cavity. The ratio of energy inside the cavity to energy outside the cavity is fixed, for a given mode, by the following function $$\begin{aligned}
g_{\bf k}^{p}[\omega] = \frac{1-|\rho_{\bf k}^{p}[\omega]|^{2}}{|1-\rho_{\bf k}^{p}[\omega]|^{2}} \ \ , \ \ \rho_{\bf k}^{p}[\omega] = r_{\bf k}^{p}[\omega]_{1}r_{\bf k}^{p}[\omega]_{2}e^{2ik_{z}L}.\end{aligned}$$ This statement constitues a theorem which has been demonstrated for lossless as well as lossy mirrors [@GenetPRA03; @Barnett96]. It does not depend on the state of the fields and is therefore valid for vacuum fluctuations as well as for thermal fluctuations, assuming thermal equilibrium. We do not discuss here the issue of thermal dependence of the Casimir effect (see for example the recent review [@Brevik06]) and restrict our attention to the zero temperature limit.
The force is the difference in radiation pressure between inner and outer faces of the mirrors, integrated over all the modes. Using analyticity properties, the force and energy may be written as integrals over imaginary frequencies $\omega =i\xi$ $$\begin{aligned}
F&=&\frac{\hbar A}{\pi} \sum_{p}\int\frac{{\rm d}^{2}{\bf k}}{4\pi^{2}}\int_{0}^{\infty}{\rm d}\xi \kappa[i\xi]\frac{\rho_{\bf k}^{p}[i\xi]}{1-\rho_{\bf k}^{p}[i\xi]} \ \ , \nonumber \\
E&=&\frac{\hbar A}{2\pi}\sum_{p}\int\frac{{\rm d}^{2}{\bf k}}{4\pi^{2}}\int_{0}^{\infty}{\rm d}\xi \ln \left(1-\rho_{\bf k}^{p}[i\xi]\right). \label{FEscatt}\end{aligned}$$ $\kappa[i\xi]=\sqrt{{\bf k}^{2}+\xi^{2} / c^{2}}$ is the longitudinal component of the wavevector evaluated for imaginary frequencies.
The expressions (\[FEscatt\]) are regular for any physical model of the reflection amplitudes. High frequency transparency of any real mirror ensures that the integrals are convergent, and free from the divergences usually associated with the infinitness of vacuum energy. They reproduce the Lifshitz expression for the Casimir force [@Lifshitz56; @Schwinger78] when assuming that the metal plates have large optical thickness with reflection amplitudes given by the Fresnel laws on the vacuum-bulk interface $$\begin{aligned}
r_{\bf k}^{\rm TE}[i\xi]&=&-\frac{\sqrt{\xi^{2}\left(\varepsilon[i\xi]-1\right)+c^{2}\kappa^{2}}-c\kappa}{\sqrt{\xi^{2}\left(\varepsilon[i\xi]-1\right)+c^{2}\kappa^{2}}+c\kappa} \ \ , \nonumber \\
r_{\bf k}^{\rm TM}[i\xi]&=&-\frac{\sqrt{\xi^{2}\left(\varepsilon[i\xi]-1\right)+c^{2}\kappa^{2}}-c\kappa\varepsilon[i\xi]}{\sqrt{\xi^{2}\left(\varepsilon[i\xi]-1\right)+c^{2}\kappa^{2}}+c\kappa\varepsilon[i\xi]}. \label{Fresnel}\end{aligned}$$ Here $\varepsilon[i\xi]$ is the dielectric function describing a optical response of the material inside the mirrors. Taken together, relations (\[FEscatt\]) and (\[Fresnel\]) reproduce the Lifshitz expression [@Lifshitz56]. They are known to tend to the original Casimir expression in the limit $\varepsilon \rightarrow \infty$ which produces perfectly reflecting mirrors [@Schwinger78].
We may emphasize at this point that relations (\[FEscatt\]) are more general than Lifshitz expression which, incidentally, were not written originally in terms of reflection amplitudes [@Katz77]. They are valid for example for non-local optical responses of the mirrors provided the reflection amplitudes are substituted by their possibly more complicated expressions. The only limitation, discussed below, is associated with the assumption of specular scattering.
Finite conductivity corrections {#sec:4}
===============================
We now review the corrections to the Casimir expression coming from the finite conductivity of the bulk material. Here, these corrections are deduced from relations (\[FEscatt\]), assuming Fresnel laws (\[Fresnel\]) for a local optical response of the bulk material. This function may be given by a simple description of the conduction electrons in terms of a plasma model $$\begin{aligned}
\varepsilon[i\xi]=1+\frac{\omega_{\rm P}^{2}}{\xi^2},\end{aligned}$$ characterized by a plasma frequency $\omega_{\rm P}$ and wavelength $\lambda_{\rm P}\equiv 2\pi c/\omega_{\rm P}$. It may be given by a more realistic representation based upon tabulated optical data and which includes the contribution of interband electrons [@Lambrecht00].
The corrections to the Casimir effect are conveniently represented in terms of factors measuring the reduction of the force and energy with respect to the ideal limit of perfect mirrors $$\begin{aligned}
F&=&\eta_{\rm F}F_{\rm Cas} \ \ , \ \ \eta_{\rm F} < 1 \ \ \textrm{and} \nonumber \\
E&=&\eta_{\rm E}E_{\rm Cas} \ \ , \ \ \eta_{\rm E} < 1.\end{aligned}$$ The results of the calculations are plotted on Fig.(\[fig:1\]) for Au-covered mirrors. They are shown as $\eta_{\rm F}$ varying versus the ratio of the cavity length $L$ to the plasma wavelength $\lambda_{\rm P}$.
For metals used in recent experiments, the plasma wavelength lies around $0.1\mu$m ($136$nm for Au and Cu). At large distances $L\gg\lambda_{\rm P}$, the ideal Casimir formula is recovered ($\eta_{\rm F}\rightarrow 1$), as expected. At short distances, a significant reduction of the force is obtained, with a change in the power law for the variation of the force with distance. This change can be understood as the result of the Coulomb interaction of surface plasmons at the two vacuum interfaces [@GenetAFLdB04; @Henkel04]. This interpretation may be actually generalized to arbitrary distances at the price of a full electromagnetic treatment of the plasmonic as well as ordinary photonic modes [@Intravaia05; @Intravaia07]. The plasma model is sufficient for a first description of the variation of the force with distance but it is not sufficient for a precise comparison.
First, the relaxation of the conduction electrons has to be accounted for. Then, interband transitions are reached for metals like Au, Cu or Al for photon energies of a few eV and their effect on the optical response has to be taken into account for evaluating the Casimir force at short (sub-micron) distances. This can be done by using tabulated optical data which are integrated using causality relations [@Lambrecht00]. The result of the corresponding evaluation is shown on Fig.(\[fig:1\]). It is worth stressing that calculations are sensitive to the existing differences in optical data between different tabulated sets [@Pirozhenko06]. This means that an imperfect knowledge of the optical properties of the mirrors used in the experiment is a source of uncertainty in the experiment-theory comparison. Ideally, if the aim is to have a reliable theoretical evaluation of the Casimir force to be compared with experiments, it is necessary to measure the reflection amplitudes *in situ*.
Silicon slab mirrors {#sec:5}
====================
As stressed in the introduction, the relevance of the Casimir effect on nanosystems calls for a precise understanding not only of the influence of material optical properties on the Casimir force, but also of the influence of geometrical parameters, such as the thickness of the coatings [@IannuzziPNAS2004; @LisantiPNAS2005] or the thickness of the mirrors themselves. In this context, structures made of silicon, the reference material used in nano-fabrication processes, are particularly interesting to study [@LambrechtEPL2007; @Chen07].
The reflection amplitude corresponding to a slab of finite thickness $D$ is different from the bulk expression and is given through a Fabry-Perot formula $$\begin{aligned}
r_{\bf k}^{p}[i\xi]_{\rm slab}&=&r_{\bf k}^{p}[i\xi]\frac{1-e^{-2\delta}}{1-(r_{\bf k}^{p}[i\xi])^{2}e^{-2\delta}} \ \ , \nonumber \\
\delta &=&\frac{D}{c}\sqrt{\xi^{2}(\varepsilon[i\xi]-1)+c^{2}\kappa^{2}}. \label{Rslab}\end{aligned}$$ $r_{\bf k}^{p}[i\xi]$ is the bulk reflection amplitude given by (\[Fresnel\]). Using these reflection amplitudes for calculating the Casimir force between two Si slabs, interesting behaviours have been noted [@LambrechtEPL2007] which differ from the situation of metallic mirrors. In particular, it was shown that the material thickness has a stronger influence on the Casimir force for Si slabs than for Au slabs. For Si, the force decreases as soon as the slab separation $L$ is larger than the slab thickness $D$, as seen on Fig.(\[fig:2\]).
In contrast to metals which become perfect reflectors in the limit of zero frequency, Si is a semiconductor with a finite transverse plasma frequency $\omega_{0}$ corresponding to a cut-off wavelength $\lambda_{0}=2\pi c / \omega_{0}\sim 286$nm. For cavity length $L$ smaller than this cut-off wavelength, Si tends to become transparent. The associated optical thickness $\delta$ given in Eq.(\[Rslab\]) is large, so that the Si slab behaves like a bulk Si mirror with low reflectivity at high frequency. The Casimir force is then much smaller than the perfect reflection limit of Eq.(\[FEcas\]). On the other hand, at low frequencies $\omega\ll\omega_{0}$, one will have $\delta\ll 1$ together with $c\kappa\rightarrow 0$, low frequencies being predominant at large distances. In this latter case, the slab is transparent again, and the Casimir force between two Si slabs is decreased when $L\geq D$. This result can have interesting consequences for nanostructures as it opens a way to control the magnitude of the Casimir force and possibly eliminate an unwanted Casimir source of stiction. From a fundamental point of view, it also offers a new solution to study the comparison between experiment and theory of the Casimir force [@Chen07]
Geometry and the Casimir effect {#sec:6}
===============================
Geometry effects are expected to lead to a rich variety of behaviours in the Casimir physics [@Balian7778; @Plunien86; @Balian0304]. Recent advances make it possible to explore this interplay, both from experimental and theoretical point of views. This also offers new possibilities for tailoring the Casimir force through specific designs [@EmigEPL03].
Force and energy evaluations between non planar mirrors are commonly obtained using the so-called proximity-force approximation (PFA) [@Derjaguin68; @Langbein71]. This approximation amounts to an averaging of plane-plane contributions over the distribution of local interplate separations defined by the chosen geometry. For the energy, the PFA leads to $$\begin{aligned}
E_{\rm PFA} = \int\frac{{\rm d}^{2}{\bf r}}{A}E_{\rm PP}\left(\ell)\right) \ , \
\ell \equiv L-h_{1}({\bf r})-h_{2}({\bf r}), \label{PFArough}\end{aligned}$$ with $h_{1}({\bf r})$ and $h_{2}({\bf r})$ the surface profiles of each mirrors. Such profiles can be described by their spectra evaluated over the surface $A$ of the mirrors $$\begin{aligned}
\int \frac{{\rm d}^{2} {\bf r}}{A}h_{i}({\bf r})h_{j}({\bf r})=\int\frac{{\rm d}^{2}{\bf k}}{(2\pi)^{2}}h_{i}[{\bf k}]h_{j}[-{\bf k}] \ , \ i,j=1,2 \end{aligned}$$ with $h_{i}[{\bf k}]$ the Fourier transform of $h_{i}({\bf r})$, and by the associated correlation lengths $\ell_{\rm C}$. When they are smaller than the other length scales, the amplitudes of deformations can be considered as perturbations. A second order expansion in the profiles can thus be performed leading to $$\begin{aligned}
E_{\rm PFA} = E_{\rm PP}+\frac{1}{2}\frac{\partial^{2}E_{\rm PP}}{\partial L^{2}}\int \frac{{\rm d}^{2} {\bf r}}{A}(h_{1}({\bf r})+h_{2}({\bf r}))^{2}. \label{PFAgen}\end{aligned}$$ The trivial first-order term has been discarded, assuming that the deformations have zero spatial averages $\int {\rm d}^{2}{\bf r}h_{i=1,2}({\bf r}) / A =0$.
The evaluation of the effect of geometry through the PFA, based on a summation procedure over local contributions assumes some additivity property of the Casimir effect, whereas the Casimir force is known not to be additive. The PFA can only be accurate for surfaces which can be considered as nearly plane with respect to other scales such as the separation distance $L$ [@GenetEPL03]. For example, it allows one to calculate the Casimir force in the plane-sphere (PS) configuration as $$\begin{aligned}
F_{\rm PS}=\frac{2\pi R}{A}E_{\rm PP}, \ \ \textrm{with} \ \ L\ll R, \label{PFAPS}\end{aligned}$$ where $E_{\rm PP}$ is the Casimir energy in the plane-plane (PP) geometry. Most recent experiments are performed in the plane-sphere geometry which is much simpler to control than the plane-plane configuration. The PFA is here expected to be valid provided the radius $R$ of the sphere is much larger than the distance $L$ of closest approach.
But the PFA certainly fails for describing more general surface profiles. As far as plate deformations are concerned, it can only be valid in the limit $\ell_{\rm C}\gg L$ which corresponds to a trivial specular description of the reflection process on the surfaces [@MaiaNetoPRA05]. For the general case, a description of non specular scattering process on mirrors is available for analyzing the connection between geometry and the Casimir effect [@MaiaNetoPRA05]. An expression for the Casimir energy between parallel mirrors with arbitrary surface profiles has been derived in [@LambrechtNJP06; @RodriguesPRA2007] $$\begin{aligned}
E=\hbar\int\limits_{0}^{\infty}\frac{{\rm d}\xi}{2\pi}{\rm Tr} \ {\rm ln}\left(1-\rm{R}_{1}\left(i\xi\right)e^{-K\left(i\xi\right)L}\rm{R}_{2}\left(i\xi\right)e^{-K\left(i\xi\right)L}\right)\ \ \label{formule}\end{aligned}$$ This expression is based on non-specular reflection matrices $\rm{R}_{1}$ and $\rm{R}_{2}$ associated to each mirror. While the operator $e^{-K\left(i\xi\right)L}$ corresponds to propagation of the field between the two mirrors, and is diagonal in the plane-wave basis with elements given by $K(i\xi) =\sqrt{{\bf k}^2+\xi^2 / c^{2}}$, the two matrices $\rm{R}_{1}$ and $\rm{R}_{2}$ are non-diagonal on plane-waves. This corresponds to a mixing of polarizations and wavevectors, due to non-specularity diffraction on the gratings formed by the profiles on the surfaces of the mirrors.
As it is reviewed below, this formula (\[formule\]) has been used to evaluate the effect of surface roughness [@MaiaNetoPRA05] or corrugations on the Casimir force [@RodriguesEPL2006; @RodriguesPRL2006]. Analytical expressions have been derived through a perturbative treatment, with the roughness or the corrugation amplitudes taken as the smallest length scales involved in the problem. The effect of the optical response of the metal has been included in these calculations. It is worth stressing that this formula has a wider range of validity. It can in principle describe structured plates with large corrugation amplitudes, as well as material properties not limited to a simple plasma model. The only task for a quantitative evaluation of the Casimir force or energy is to obtain the actual form of the reflection operators $\rm{R}_{1}$ and $\rm{R}_{2}$ to be inserted into Eq.(\[formule\]).
Roughness correction {#sec:7}
====================
A correction to the Casimir force that must be accounted for is the effect of surface roughness, intrinsic to any real mirror. This effect is analyzed in recent experiments through procedures based on the PFA [@BordagPLA95; @KlimPRA99]. The general formula (\[formule\]) has been used to go beyond this approximation [@MaiaNetoPRA05]. As already stressed, the roughness amplitude must be the smallest length scale for perturbation theory to hold. Meanwhile, the plasma wavelength, the mirror separation and the roughness correlation length may have arbitrary relative values with respect to each other.
We remind that the roughness profiles are defined with respect to reference mirror planes separated by the mean distance $L$. We assume that profiles have zero averages and show no cross-correlations. We also suppose that the area $A$ of each plate is large enough to include many correlation areas ($A\gg \ell_{\rm C}^{2}$), so that surface averages are identical to statistical averages. Up to second order in the profiles, the correction to the Casimir energy may thus be written as follows $$\begin{aligned}
\delta E_{\rm PP}=\int\frac{{\rm d}^{2}{\bf k}}{(2\pi)^{2}}G_{\rm rough}[{\bf k}]\sigma[{\bf k}]. \label{beyondPFArug}\end{aligned}$$ Here $\sigma[{\bf k}]$ corresponding to the roughness spectrum added over the two plates. $G_{\rm rough}[{\bf k}]$ is a spectral sensitivity to roughness of the Casimir energy. Due to cylindrical symmetry with respect to rotations in the transverse plane, it only depends on $k=|{\bf k}|$. This dependence reveals that the roughness correction does not only depend on the root-mean-square (rms) roughness, but also on the spectral distribution of the roughness. Fig.(\[fig:3\]) displays $G_{\rm rough}[k]$ normalized by $E_{\rm PP}$ as it has been calculated for Au-covered mirrors and for various interplate distances.
The rich behaviours of $G_{\rm rough}[k]$ as a function of the length scales is discussed in [@MaiaNetoEPL05].
What we want to stress here is that this function describes deviations from the PFA. The width of the roughness spectrum $\sigma[{\bf k}]$ is indeed fixed by the inverse of the correlation length $\ell_{\rm C}$. When this spectrum is contained in the region where $G_{\rm rough}[k]$ remains close to its secular limit $G_{\rm rough}[0]$, we can approximate Eq.(\[beyondPFArug\]) as proportional to the rms roughness $$\begin{aligned}
\delta E_{\rm PP}\simeq G_{\rm rough}[0]\langle h_{1}^{2}+h_{2}^{2}\rangle. \label{PFArug}\end{aligned}$$ This corresponds effectively to the PFA expression, as the consequence of a theorem which was proved in [@MaiaNetoPRA05] $$\begin{aligned}
G_{\rm rough}[k\rightarrow 0]= \frac{1}{2}\frac{\partial^{2}E_{\rm PP}}{\partial L^{2}}.
\label{PFtheorem}\end{aligned}$$ Equation (\[PFtheorem\]) is nothing but a properly stated “Proximity Force Theorem”. It can however not be confused with the “Proximity Force Approximation” (\[PFArug\]) which is a good approximation only for smooth enough mirrors, that is also for large enough roughness correlation lengths $\ell_{\rm C}$.
In the general case, the PFA result (\[PFArug\]) underestimates the effect of roughness. When performing the theory-comparison, one has therefore to carefully assess the roughness correction by measuring the roughness spectra *in situ* and using the roughness sensitivity function as given in [@MaiaNetoPRA05; @MaiaNetoEPL05]. The PFA can only be used if $\ell_{\rm C}$ has been proven to be large enough or, in a looser way, when the roughness correction has been estimated to have a negligible value.
Lateral force between corrugated plates {#sec:8}
=======================================
As the roughness effect remains a small correction to the Casimir force, it seems difficult to measure the deviation from PFA regime and check its agreement with theory. Fortunately, there exists an experimental configuration showing more promising perspectives as a potential probe of the non-trivial interplay between the Casimir effect and geometry.
This configuration corresponds to periodically corrugated metallic plates placed face to face in vacuum, so that a lateral component of the Casimir force arises due to the breaking of the transverse translational invariance [@Golestanian]. A recent experiment has demonstrated the feasibility of a lateral force measurement at separation distances of the order of $\sim 100$nm [@ChenPRL02]. Since it would disappear in the absence of corrugation, the lateral force should not be considered as a small correction to the otherwise dominant normal Casimir force, as it was the case for the study of roughness. As we will see below, the deviation from PFA indeed appears as a factor in front of the lateral force, so that a precise measurement of this force would test in a crucial manner the interplay between Casimir effect and geometry [@RodriguesPRL2006]. As the experiments are performed at short distances, it cannot be described with the assumption of perfect reflection, where analytical results are available [@EmigPRA03; @EmigPRL05]. Again, the general scattering formula (\[formule\]) shows the ability to give an estimation for the lateral force for arbitrary relative values of the length scales $\lambda_{\rm C}$, $\lambda_{\rm P}$ and $L$, provided the corrugation amplitudes $a_{i=1,2}$ remain the smallest length scales of the problem.
We consider two metallic mirrors, both sinusoidally corrugated along one dimension, with the same corrugation wavelength $\lambda_{C}$, separated by a distance $L$ and facing each other with a relative spatial mismatch $b$ between the corrugation crests -see Fig.(\[fig:4\]).
The profiles $h_{i=1,2}({\bf r})$, ${\bf r}=(x,y)$, of the two uniaxial (along $y$) corrugated mirrors are defined by the two functions $h_{1} = a_{1}\cos\left(k_{\rm C}x\right)$ and $h_{2}=a_{2}\cos\left(k_{\rm C}\left(x-b\right)\right)$ with $k_{\rm C}= 2\pi / \lambda_{\rm C}$ the wavevector associated to the corrugation wavelength $\lambda_{\rm C}$. We take both profiles with zero spatial averages. At the second order in the corrugations, cross-terms of the form $a_{1}a_{2}$ appear which contribute to the lateral force because the energy depends on the transverse mismatch $b$.
This fact, a consequence of the correlation between the two corrugation profiles, induces a contrast with the case of roughness where the effect was associated with quadratic terms $h_{i=1,2}^{2}$. It implies that the evaluation of the lateral force only involves first-order non-specular amplitudes calculated on each mirror separately. The full calculation gives the second-order correction to the Casimir energy induced by the corrugations $$\begin{aligned}
\delta E_{\rm PP}=A\frac{a_{1}a_{2}}{2}\cos (k b) G_{\rm C}[k].
\label{secorder}\end{aligned}$$ The function $G_{\rm C}[{\bf k}]$ is given in [@RodriguesPRL2006] and does only depend on the modulus $k$ of ${\bf k}$. Here again, the PFA regime is recovered in the $k\rightarrow 0$ limit, as a consequence $$\begin{aligned}
G_{\rm C}[k\rightarrow 0]=\frac{1}{2} \frac{\partial^{2}E_{\rm PP}}{\partial L^{2}}.\end{aligned}$$ This theorem is ensured, for any specific model of the material medium, by the fact that $G_{\rm C}$ is given for $k\rightarrow 0$ by the specular limit of non-specular reflection amplitudes [@MaiaNetoPRA05].
In order to compare with experiments, we consider the expression of the lateral force in the plane-sphere configuration. It is derived from the plane-plane configuration using the PFA, reliable as long as $L\ll R$. In fact, there is no interplay between curvature and corrugation provided $RL\gg\lambda_{\rm C}^{2}$, a condition met in the experiment reported in [@ChenPRL02].
From Eq.(\[PFAPS\]), the lateral force in the plane-sphere geometry is eventually given as [@RodriguesPRL2006] $$\begin{aligned}
F_{\rm PS}^{\rm lat}=-\frac{\partial }{\partial b}E_{\rm PS}^{\rm lat}=\pi a_{1}a_{2}kR\sin (kb )
\int_{\infty}^{L}{\rm d}L^{\prime}G[k,L^{\prime}].\end{aligned}$$ The force is plotted in Fig.(\[fig:5\]) as a function of $k$, with length scales $\lambda_{\rm C}$, $\lambda_{\rm P}$ and $L$ fitting the experimental values [@ChenPRL02]. As the corrugation amplitudes are not small enough in the experiment to meet the perturbation condition, the theory and experiment can unfortunately not be compared directly. The plot on Fig.(\[fig:5\]) nevertheless shows the interesting fact that the length scales taken from the experiment, with $k$ indicated by the vertical dashed line, clearly fall outside the PFA sector in the perturbative calculation. For related implications, we refer the reader to the discussions in [@RodriguesPRA2007].
It appears clearly on the figure that the PFA overestimates the magnitude of the lateral force for arbitrary $k$. We also note that the PFA prediction for the force scales as $k$ when $k$ increases from zero. At larger values of $k$ in contrast, the lateral force decreases. This is due to the one-way propagation factor separating the two first-order non-specular reflections at each plate, given as a decresing exponential $e^{-kL}$ in the high $k$ limit [@RodriguesPRL2006]. It follows that there is a maximal force when $k$ is varied. It corresponds to $k=9\times 10^{-3}$nm$^{-1}$ with the other length scales corresponding to the experiment. The ratio $L / \lambda_{\rm C} = 1 / \pi$ is thus falling outside the PFA sector which confirms that a lateral force measurement is an excellent candidate for probing deviations from the PFA.
Torque {#sec:9}
======
Another interesting observable for exploring the non-trivial geometry dependence of the Casimir energy is the torque arising when the corrugations of the two plates are misaligned. With this angular mismatch between the corrugations, rotational symmetry is broken and induces a restoring torque between the plates.
The calculations are quite similar to those which were done for aligned corrugated surfaces, in particular because the same non-specular reflection coefficients are used to describe each plate. The second-order correction is still given by the sensitivity function $G_{\rm C}[{\bf k}]$ which does only depend on the modulus of the corrugation wavevector ${\bf k}$. The difference with the lateral force case lies only in the fact that the corrugation profiles $h_{i=1,2}({\bf r})=a_{i}\cos ({\bf k}_{i}\cdot {\bf r} - kb_{i})$ corresponds to different corrugation wavevectors ${\bf k}_{i=1,2}$ having however the same modulus $k=2\pi /\lambda_{\rm C}$. The angular mismatch between ${\bf k}_{1}$ and ${\bf k}_{2}$ is given by the angle $\theta$. The parameters $b_{i}$ represent lateral displacements with respect to the configuration with a line of maximum height at the origin. We assume that the corrugation $h_{2}$ is restricted to a rectangular section of area $L_{x}L_{y}$ centered at $x=b_{2},y=0$ and much larger than $L^{2}$ so that diffraction at the borders can be neglected. With these assumptions, and in the limit of long corrugation lines $kL_{y}\gg 1$ with $L_{x}$ smaller or of the order of $L_{y}$, the energy correction per unit area is given in [@RodriguesEPL2006] as $$\begin{aligned}
\frac{\delta E_{\rm PP}}{L_{x}L_{y}}=\frac{a_{1}a_{2}}{2}G_{\rm C}[k]\cos (kb)\frac{\sin (kL_{y}\theta /2)}{kL_{y}\theta /2}.
\label{torque}\end{aligned}$$ The spatial coefficient $b=b_{2}\cos \theta -b_{1}$ is the relative lateral displacement along the direction ${\bf k}_{1}$. As expected by symmetry, this correction is invariant under the transformation $\theta \rightarrow - \theta$ and $\theta\rightarrow \pi -\theta$ due to the fact that the corrugation lines have no orientation. The case $\theta =0$ corresponds to the result of pure lateral displacement discussed in the preceding section.
The scale of the energy variation with $b$ and $\theta$ is set by the parameter $ \lambda_{\rm C} / L_{y}$. In fact, if plate $2$ is released after a rotation of $\theta >\lambda_{\rm C} / L_{y}$, its subsequent motion is a combination of rotation and lateral displacements. Rotation is favored over lateral displacements for $\theta <\lambda_{\rm C} / L_{y}$ (see Fig.(1) in [@RodriguesEPL2006]). The torque $\tau = - \partial \delta E_{\rm PP} / \partial\theta $ is evaluated in [@RodriguesEPL2006] for corrugated Au mirrors, with corrugation amplitudes $a_{1}=a_{2}=14$nm, corrugation length $L_{y}=24\mu$m and separated by a distance of $L=1\mu$m. It is maximum at $\theta = 0.66 \lambda_{\rm C} / L$ and is plotted in Fig.(\[fig:6\]) as a function of $k$. It starts increasing linearly with $k$ in the $k\rightarrow 0$ PFA sector and for the same reason as the lateral force, it decreases exponentially in the high-$k$ limit.
As is clear on Fig.(\[fig:6\]), the PFA overestimates the magnitude of the torque by a factor of the order of $2$ at the peak value of the torque. The discrepancy even increases with $k$, since smaller values of $k$ correspond to smoother surfaces. The conditions are gathered up towards a direct experimental evidence of a non-trivial effect of geometry.
Fig.(\[fig:6\]) also displays the torque when evaluated between perfect metallic corrugated mirrors [@EmigPRA03]. The corresponding deviation with respect to the calculation given by Eq.(\[torque\]) stresses that at a separation distance of $L=1\mu$m, the optical response of the metal must be accounted for in an accurate evaluation of the torque. The perfect conductor limit is reached only if the plasma wavelength $\lambda_{\rm P}$ is the smallest length scales (apart from the corrugation amplitudes) of the problem.
Conclusion
==========
New perspectives for studying the interplay between Casimir effect and geometry are today clearly visible. The theoretical formalism is better and better mastered, so that a rich variety of configurations can be studied. Meanwhile, novel experimental capabilities are available, allowing one to address challenging questions. Proposals have been recently made for measuring the torque between birefringent dielectric disks [@MundayPRA2005]. A measurement between metallic corrugated mirrors seems to be more easily accessible, with the torque turning out to be up to three orders of magnitude higher than the torque between dielectric plates, for comparable separation distance. At the same time, alternative routes are explored in order to probe quantum vacuum geometrical effects [@RodriguezPRL2007]. Cold atoms techniques also look like particularly promising, as they should allow one to see deviations from the PFA on the lateral component of the Casimir-Polder force, with a Bose-Einstein condensate used as a local probe trapped close to a corrugated surface [@DalvitPRL08]. These trends suggest that demonstrations of non-trivial effects of geometry should be within reach.
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|
---
author:
- |
Zehao Dou$^1$, Xiang Yan$^2$, Dongge Wang$^1$ and Xiaotie Deng$^1$\
[$^1$ Peking University, `{zehaodou, dgwang96, xiaotie}@pku.edu.cn`]{}\
[$^2$ Shanghai Jiao Tong University, `xyansjtu@163.com`]{}
bibliography:
- 'iclr2020\_conference.bib'
title: Finding Mixed Strategy Nash Equilibrium for Continuous Games through Deep Learning
---
Introduction
============
Nash equilibrium [@nash1950equilibrium] is one of the most important solution concepts in game scenario with multiple rational participants. It plays an important role in theoretical analysis of games to guide rational decision-making processes in multi-agent systems. With the recent success of machine learning applications in games, it attracts even more research interests on applying machine learning technique for unsolved game theory problems, for example, computation of Nash equilibrium for multi-player games. In this paper, we focus on games with continuous action spaces, which include the famous application for Generative Adversarial Networks (GANs) [@goodfellow2014generative], as well as many important game types such as the colonel blotto game [@gross1950continuous], Cournot competition [@varian1996intermediate]. We develop a solution significantly improves the status-quo.
There have been several successful approaches to compute Nash equilibrium for multi-player (mostly 2-player) continuous game [@raghunathan2019game; @balduzzi2018mechanics]. These works seek Nash equilibria corresponding to pure strategies, in which each player takes a specific action to achieve its best payoff given other players’ actions. A major concern for such a solution concept is its possible non-existence. As a result, the convergences to a Nash equilibrium for these approaches were proven under the assumption for the existence of a pure strategy Nash equilibrium, which can hardly be checked in practice, and their applicability is limited to specific types of games. On the contrary, it is known that mixed strategy Nash equilibria always exist under mild conditions. And note that any pure strategy Nash equilibrium is also a mixed strategy Nash equilibrium, which means the latter one is a much more desired solution concept.
However, a key challenge that obstructs the study of computing a mixed strategy Nash equilibrium, especially for a continuous game, lies on how to design an efficient method to represent the mixed strategy. To be precise, a pure strategy can be represented by a single variable choosing from some region. But as a distribution on each player’s action space, a mixed strategy with respect to the player is defined in a (subspace of) real space $\mathbb{R}$. More generally, exact representation for a mixed strategy of a player usually requires many variables in a continuous space. In addition, the corresponding probability distribution may not have a density function in closed-form.
To address this challenge, we introduce a pushforward measure technique. It is a common tool in measure theory to transfer a measure to some specific measure space [@bogachev2007measure]. Specific to a continuous game, the probability distribution corresponding to a mixed strategy is obtained via a mapping parameterized by neural nets from a multi-dimensional uniform distribution.
With this pushforward representation, we generalize the Gradient-based Nikaido-Isoda (GNI) function, defined in [@raghunathan2019game], to handle mixed strategy Nash equilibria. The original GNI function can be viewed as a measure for the distance between any joint strategy profile and a Nash equilibrium after applying the payoff functions of players. With proper generalization and modification, we develop its mixed strategy version as a proper measure for a Nash equilibrium. We prove that the distance becomes zero if and only if a stationary mixed Nash equilibrium is obtained. Then we apply the gradient descent algorithm to the general GNI function, which converges to a stationary mixed Nash equilibrium under the convexity assumptions on the payoff functions.
Finally, we compare our method with baseline algorithms in numerical experiments. Our approach shows effective convergence property in all the randomly generated quadratic games, general blotto games and GAMUT games, which outperforms other baselines.
Background and Problem Description
==================================
The discrete action space Nash equilibrium computation has been most widely studied in the literatures. Most well-known being the Lemke–Howson algorithm [@lemke1964equilibrium] for solving the bimatrix game. The state-of-art work in theoretical computer science of Tsaknakis and Spirakis provided a solution of $1/3$ approximation in polynomial time [@tsaknakis2007optimization]. Surprisingly, an empirical work [@fearnley2015empirical] shows it performs well against practical game solving methods for the bimatrix game.
However, continuous action space game computation is widely used in practice. But few methods are known for the general Nash equilibrium computation. Several recent effort to develop computational method of Nash equilibrium for multi-player (mostly 2-player) continuous game [@raghunathan2019game; @balduzzi2018mechanics] have been restricted to pure strategies.
Game-theoretical approach has had useful applications to machine learning such as the optimization of GAN network training [@daskalakis2017training; @gidel2018variational] and adjustment on the gradient descent method [@balduzzi2018mechanics]. However they are limited to pure strategy Nash equilibrium. We are the first work to study the mixed strategy continuous game Nash equilibrium computation. Our work is motivated by the utilization of the Nikaido-Isoda (NI) function for loss function minimization [@uryas1994relaxation; @raghunathan2019game]. We start to establish a theoretical formulation of the extend mixed strategy continuous action space Nash equilibrium as a result of the minimization on a functional variation-based Nikaido-Isoda function.
Continuous Game Nash Equilibrium
--------------------------------
$$\label{eqn:ne}
\begin{aligned}
&\text{Find}~~{\mathbf{x}}^*=(x_{1}^*,x_{2}^*,\cdots,x_{N}^*)\\
&\text{s.t.}~~ x_{i}^* = \arg\min_{{\mathbf{x}}\in\mathbb{R}^{n}:{\mathbf{x}}_{-i}={\mathbf{x}}_{-i}^*}f_{i}({\mathbf{x}})
\end{aligned}$$
Here $N$ denotes the number of players, and $x_{i}\in\mathbb{R}^{n_{i}}$ the strategy of the $i$-th player where $n_i$ is the dimension of his action space. Let $n=\sum_{i=1}^{N}n_{i}$, and ${\mathbf{x}}=(x_1,x_2,\cdots,x_N)\in\mathbb{R}^{n}$ denotes the joint pure strategy among all players while ${\mathbf{x}}_{-i}=(x_{1},\cdots,x_{i-1},x_{i+1},\cdots,x_{N})\in\mathbb{R}^{n-n_{i}}$ the joint pure strategy among players except $i$. $f_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ denotes the utility function (cost) of $i$-th player. A solution ${\mathbf{x}}^*$ to (\[eqn:ne\]) is called a pure strategy Nash equilibrium.
Nikaido-Isoda (NI) Function
---------------------------
In the paper ([@nikaido1955note]), Nikaido-Isoda (NI) function is introduced as: $$\label{ni}
\phi({\mathbf{x}}) = \sum_{i=1}^{N}\left(f_{i}({\mathbf{x}})-\inf_{\hat{{\mathbf{x}}}\in \mathbb{R}^{n}:\hat{{\mathbf{x}}}_{-i}={\mathbf{x}}_{-i}}f_{i}(\hat{{\mathbf{x}}})\right)\triangleq \sum_{i=1}^{N}\phi_{i}({\mathbf{x}})$$ From the Equation (\[ni\]), we know $\phi({\mathbf{x}})\geqslant 0$ for $\forall {\mathbf{x}}\in \mathbb{R}^{n}$, and $\phi({\mathbf{x}})=0$ is the global minimum of NI function which can only be achieved at a Nash equilibrium (NE). Therefore, a common algorithm of computing NE points is minimizing the NI function above. However, it is a huge difficulty to handle the global infimum. On the one hand, global infimum can not be obtained in finite time. On the other hand, the infimum can be unbounded below in some games, for example the two-player bi-linear games, where $f_{1}({\mathbf{x}}) = x_1^{T}Mx_2 = -f_{2}({\mathbf{x}})$. All of the facts above show us the shortcomings of NI function, and in order to rectify them, [@raghunathan2019game] introduces the following Gradient-based Nikaido-Isoda (GNI) function.
Gradient-based Nikaido-Isoda (GNI) Function
-------------------------------------------
If we calculate local infimum in the NI function $\phi({\mathbf{x}})$ instead of global infimum, the time complexity and unbounded infimum are no longer shortcomings. In precise, given the local radius $\lambda$, local infimum can be approximated by steepest descent direction, and we get the following GNI function: $$V({\mathbf{x}}; \lambda) = \sum_{i=1}^{N}\big(f_{i}({\mathbf{x}})-f_{i}(x_{1}, \cdots, x_{i-1}, x_{i} - \lambda \nabla_{i}f_{i}({\mathbf{x}}), x_{i+1},\cdots,x_{N})\big)$$ By minimizing $V({\mathbf{x}},\lambda)$, a stationary Nash point ${\mathbf{x}}^*$, where $\nabla_{x_i}f_i({\mathbf{x}}^*)=0$ for $\forall i$, can be approximated efficiently. Furthermore, if all the utility functions $f_{i}$ are convex, then the stationary Nash points (SNP) obtained are actually Nash Equilibrium (NE).
(MC-GNI) Gradient-based Nikaido-Isoda Function of Mixed Strategy on Continuous Games
====================================================================================
Theoretical Analysis of MC-GNI
==============================
Experiments
===========
To evaluate the practical performance of our approach, we apply it to three types of games, two-player quadratic games, general blotto games, and GAMUT games, the most popular games for evaluation of Nash equilibrium algorithms. In all the experiments, we set the local radius $\lambda = 1e-3$ and we use gradient descent as our optimization method with step size $\rho = 1e-2$ and momentum $\kappa = 0.9$. The network architecture we use for the pushforward functions $g_{\theta}$ is a 6-layer fully connected neural network with the size of each layer as: 20, 40, 160, 160, 40, 20. The size of its output layer is the dimension of each player’s action space. From forward to backward, the activation function we use is: $\tanh, \tanh, \tanh$, ReLU, $\tanh, \tanh$.
We mainly compare our approach with two recent studies, gradient descent for GNI function [@raghunathan2019game] (gradGNI in short), and Symplectic Gradient Adjustment algorithm [@balduzzi2018mechanics] (SGA in short), as they outperformed other existing algorithms applicable to continuous game settings. For all these methods, we either follow the standard hyper-parameters mentioned in the original papers, or the ones resulting in the best convergence.
Two-player Quadratic Game
-------------------------
The two-player quadratic game is defined by the the players’ payoff functions $f_i$ ($i=1,2$): $$\label{eqn:quadratic}
f_i({\mathbf{x}}) = {\mathbf{x}}^T Q_i{\mathbf{x}}+ r_i^T {\mathbf{x}},$$ where $Q_i\in \mathbb{R}^{(n_1+n_2)\times (n_1+n_2)}$, $r_i\in\mathbb{R}^{n_1+n_2}$, ${\mathbf{x}}=(x_1,x_2)$ and $x_i\in \mathbb{R}^{n_i}$. In our experiments, we choose $n_1 = n_2 \in \{3, 5, 10\}$. For each pair of $n_i$, we randomly generate 100 instances for the matrix $Q_i$ and $r_i$ for $i=1,2$. Each item in each matrix $Q_i$ and each vector $r_i$ follows the uniform distribution on $[0,1]$ independently.
We show the converging process of all algorithms for one game instance ($n_1=n_2=3$) in Fig. \[fig:quadratic\] as an example. As we can see, our approach effectively converges to a stationary Nash equilibrium point. While the gradGNI approach also converges in this instance, its result has a larger local regret. In other words, it obtains a worse approximation to Nash equilibrium, which coincides with the essential difference between pure strategy and mixed strategy. The MC-GNI approach searches for the equilibrium in the mixed strategy space, which includes the pure strategy space that the gradGNI approaches searches in. On the other hand, the SGA approach diverges in this game instance. We further take the average of the final local regret after 2000 iterations for all the 100 instances, summarized in Tab. \[tab:result\]. All the algorithms show consistency as the dimension of action space increases, and MC-GNI outperforms others regardless of the randomness of game structures.
General Blotto Game
-------------------
We next consider the general blotto game, which differs from previous games in the action space of each player for which further constraints apply.
In a blotto game, player $1$ and $2$ (sometimes known as two colonels) have a budget of resource $X_1$, $X_2$ respectively. W.l.o.g we set $X_1 \leq X_2$. There are $m$ battlefields in total. In each battlefield $j$, when two players allocate $x_{1j},x_{2j}$ resource on it, the payoff of player $i$ is: $$\label{eqn:blotto_payoff}
U_{ij} = f(x_{ij}-x_{-ij}), \mbox{ where } f(\chi)=\tanh{(\chi)},$$ where $-i$ denotes the player other than player $i$. Each player’s payoff across all $m$ battlefields is the sum of the payoffs across the individual battlefields. For each player $i$, a feasible pure strategy $x_i=(x_{i1},\dots,x_{im}) \in \mathbb{R}_{+}^{m}$ must also satisfies $\sum_{j=1}^{m} x_{ij} \leq X_{i}$. Here we adopt the generalized blotto game proposed by [@golman2009general] with continuous payoff functions. The payoff functions in vanilla blotto game [@gross1950continuous] is discontinuous, for which our method as well as baselines fails. In our experiments, we set $m \in \{3, 5, 10\}$. For each $m$, we randomly generate 100 instance for the budget $X_i$, following the uniform distribution on $[0,1]$ independently. We show the converging process of all algorithms for one game instance ($m=3$) in Fig. \[fig:blotto\] as an example. All the algorithms converges for this game, and both the gradGNI and SGA approaches converges faster and more smoothly comparing with our MC-GNI. However, similar to the quadratic game, their final results have larger local regret. This coincides with the fact that the mixed strategy is a better solution concept than the pure strategy, especially in blotto games. We further take the average of the final local regret after 2000 iterations for all the 100 instances, summarized in Tab. \[tab:result\]. All the algorithms show consistency as the dimension of action space increases, and MC-GNI outperforms others regardless of the randomness of game structures.
GAMUT Games
-----------
Finally, we apply our method on the game instance generated by the comprehensive GAMUT suite of game generators designated for testing game-theoretic algorithms [@nudelman2004run]. GAMUT includes a group of random distributions, based on each of which the payoff of each player for each pure strategy profile can be drown independently. In precise, we extend the quadratic game to a multi-player version, where $r_i=0$, and 100 game instances with 4 players are generated. For each instance, one of the distributions from the GAMUT set is selected, and each item in each matrix $Q_i$ is sampled according to it independently.
We show the converging process of all algorithms for one game instance in Fig. \[fig:gamut\]. Both MC-GNI and SGA converge, but SGA has a much worse final result than our MC-GNI. And this time, gradGNI diverges. Furthermore, we take the average of the final local regert after 2000 iterations for all the 100 instances, shown in Table \[tab:result\].
From these different games, we know that our MC-GNI converges and performs better than two baselines in all of the three games, which shows the effectiveness and efficiency of our MC-GNI model. As the first algorithm to compute the mixed strategy Nash equilibrium of games with continuous action space, we believe that the technique we introduced here will enable new optimization researches of many exciting interaction domains of algorithmic game theory and deep learning.
MC-GNI (our model) gradGNI SGA
---------------------- ---------------------------------- --------------------- --------------------
Quadratic ($n_i=3$) **$\mathbf{(1.63\pm1.20)}$e-3** ($1.01\pm0.03$)e-1 $2.59\pm0.17$
Quadratic ($n_i=5$) **$(\mathbf{2.84\pm1.95})$e-3** ($2.95\pm0.19$)e-1 $3.92\pm 0.22$
Quadratic ($n_i=10$) **$\mathbf{(3.76\pm 3.02)}$e-3** ($1.47\pm0.08$)e-1 $2.54\pm0.09$
Blotto ($m=3$) **($\mathbf{6.32\pm4.97}$)e-6** ($2.62\pm0.38$)e-5 ($5.26\pm0.91$)e-5
Blotto ($m=5$) **($\mathbf{4.52\pm3.09}$)e-6** ($1.10\pm0.06$)e-5 ($1.21\pm0.18$)e-5
Blotto ($m=10$) **($\mathbf{3.62\pm2.39}$)e-6** ($7.60\pm 0.49$)e-6 ($5.94\pm0.26$)e-6
GAMUT ($n_i=3$) **($\mathbf{4.95\pm0.42}$)e-3** ($4.80\pm0.81$)e-1 ($0.94\pm0.13$)e-1
GAMUT ($n_i=5$) **($\mathbf{8.90\pm0.79}$)e-3** ($1.52\pm0.27$)e-1 ($2.59\pm0.60$)e-1
GAMUT ($n_i=10$) **($\mathbf{1.54\pm0.86}$)e-2** ($1.84\pm0.48$)e-1 ($1.76\pm0.32$)e-1
: Comparison results.[]{data-label="tab:result"}
|
---
abstract: 'This Letter presents *ab initio* calculations of the magneto-thermoelectric power (MTEP) and of the spin-Seebeck coefficient in MgO based tunnel junctions with Fe and Co leads. In addition, the normal thermopower is calculated and gives for pure Fe and Co an quantitative agreement with experiments. Consequently, the calculated values in tunnel junctions are a good estimation of upper limits. In particular, spin-Seebeck coefficients of more than $100 \mu V/K$ are possible. The MTEP ratio exceed several 1000% and depends strongly on temperature. In the case of Fe leads the MTEP ratio diverges even to infinity at certain temperatures. The spin-Seebeck coefficient as a function of temperature shows a non-trivial dependence. For Fe/MgO/Fe even the sign of the coefficient changes with temperature.'
author:
- Michael Czerner
- Michael Bachmann
- Christian Heiliger
title: 'Spin caloritronics in magnetic tunnel junctions: *Ab initio* studies'
---
The emerging research field of spin caloritronics [@bauer10] combines the spin-dependent charge transport with energy or heat transport. In comparison to thermoelectrics the spin degree of freedom is considered as well. The influence of a temperature gradient on a spin-dependent current and vice versa was pointed out by Johnson and Silbsee [@silsbee87]. Since then a number of effects are discussed on the nanometer scale like thermal spin-transfer torque [@hatami07], magneto-thermoelectric power (MTEP) in metallic multilayers [@gravier06], thermally excited spin-currents [@tsyplyatyev06], magneto-Peltier cooling [@hatami09].
Recently the spin-Seebeck effect was experimentally discovered by Uchida et al. [@uchida08] in a NiFe alloy. However, the interpretation of the measured effect is rather complicated. Thereby, the spins have different electrochemical potentials $\mu^\uparrow$ and $\mu^\downarrow$ due to a temperature gradient $\Delta T$ across the sample. The spin-Seebeck coefficient is defined as $$S_s=\frac{\mu^\uparrow-\mu^\downarrow}{\Delta T}
\label{eq:sS}$$ There are in principle two effects that give rise to a spin voltage under an applied temperature gradient. The effect measured by Uchida et al. was recently explained by a spin pumping effect at the contact between the ferromagnet and the normal electrode [@xiao10].
The other effect is the analogue to the classical charge Seebeck effect. The origin of this effect is a different asymmetry of the density of states (DOS) around the Fermi energy in both spin channels. The asymmetry of the DOS is the main reason for a thermopower (or Seebeck voltage) in classical thermoelectrics. Introducing a magnetic material with different asymmetry in the DOS for both spins lead to different Seebeck coefficients for both spins $S^\uparrow$ and $S^\downarrow$. Both spin channels can be seen as a thermocouple leading to the spin-Seebeck coefficient $$S_s=S^\uparrow-S^\downarrow
\label{eq:Ss}$$ For the classical thermopower a charge is spatially separated whereas as for the spin-Seebeck effect both spins are unequally occupied at the same position. Therefore, spin relaxation processes will destroy this effect if the sample size is larger then typical spin-diffusion lengths. Consequently, half metallic materials are promising. Nevertheless, the understanding of the spin caloritronic effects also for normal magnetic metals is of fundamental interest. Surprisingly, Uchida et al. [@uchida08] measured a spin-Seebeck coefficient although the sample size is quite larger than the spin-diffusion length. Therefore, this measured effect has another origin as already pointed out above.
Due to these two different effects there is a confusion about the nomenclature. In particular, “spin-Seebeck effect” is used for both effects. The analogue of the charge Seebeck effect is given by the different Seebeck coefficients in both spin channels. Therefore, it is also possible to call this effect spin-dependent Seebeck effect. However, this is again confusing with respect to $S^\uparrow$ and $S^\downarrow$ which are the spin-dependent Seebeck coefficients. Therefore, throughout this letter we will use the nomenclature spin-Seebeck effect meaning the analogue to the charge Seebeck effect.
The effect of magneto-thermoelectric power (MTEP) is the dependence of the normal charge Seebeck coefficients on the relative magnetic orientation $\theta$ of both magnetic layers. The MTEP ratio is given by $$\frac{S(0^\circ)-S(\theta)}{\min (|S(0^\circ)|,|S(\theta)|)}
\label{eq:MTEP}$$ Gravier et al. [@gravier06] measured for $\theta=180^\circ$ a MTEP ratio of about 30% in all-metallic junctions.
In this letter we investigate the spin-Seebeck effect (SSE) and the magneto-thermoelectric power (MTEP) in magnetic tunnel junctions. Thereby, we use ballistic transport that is in particular without spin-diffusion effects. For MTEP this is only a minor approximation because for the thermoelectric power (charge Seebeck coefficient) the electric charges are spatially separated which makes this effect robust. In the case of the SSE spin flip scattering destroys the effect leading to a vanishing spin-voltage if the sample size is larger than the spin diffusion length. Consequently, our investigations aim to give an upper limit of what is possible. This means that our calculated spin-Seebeck coefficients are basically only valid next to the barrier.
For a large Seebeck coefficient a strong asymmetry within the density of states is advantageous. In addition, for the spin-Seebeck a strong asymmetry within the spin channels is necessary. The latter is fulfilled for MgO based tunnel junctions with Fe or Co leads that show a very high tunnel magneto resistance (TMR) ratio [@gradhand08]. In such junctions MgO acts as an symmetry filter having a large transmission probability for $\Delta_1$ states only. With respect to these states Fe or Co is half-metallic having $\Delta_1$ states only in the majority spin channel. Therefore, one can expect also a high spin-Seebeck effect in MgO based tunnel junctions. A disadvantage of Fe or Co leads is that they are only half-metallic with respect to specific states. This means that the spin diffusion length is rather small in comparison to real half-metals.
Our method for the transport calculations is based on the Green’s function formalism implemented in the Korringa-Kohn-Rostoker method [@heiliger08]. In this method non-collinear alignment of the magnetic layers can be considered to calculate the transport properties at an arbitrary relative angle between the magnetizations of the leads. The potentials are calculated self-consistently within a supercell approach for the parallel alignment of the magnetic moments of the magnetic layers. Due to the relatively thick MgO barrier of 6 monolayers both magnetic layers are decoupled. Therefore, the other magnetic orientations are obtained by rotating the potentials of the parallel alignment without an additional self-consistent cycle. For the calculation of the energy dependent transmission probability semi-infinite leads are considered by self energies. For both calculations the atomic sphere approximation is used and the cut-off for the angular momentum is 3. The energy dependent transmission probability $T(E)$ is used to calculate the moments $$L_n=\frac{2}{h} \int T(E) (E-\mu)^n (-\partial_E f(E,\mu,T)) dE
\label{eq:L_n}$$ where $f(E,\mu,T)$ is the Fermi occupation function at a given energy $E$, electrochemical potential $\mu$, and temperature $T$. The conductance $G$ and the Seebeck coefficient $S$ are given by [@ouyang09] $$G=e^2 L_0 \ \ \ \ \ \ S=-\frac{1}{e T} \frac{L_1}{L_0}
\label{eq:G_S}$$ For a better convergence with respect to the energy mesh we apply a very small bias voltage of $1meV$ to avoid sharp resonances in $T(E)$. By using spin-dependent transmission probabilities $T^\uparrow(E)$ and $T^\downarrow(E)$ the spin-dependent Seebeck coefficients $S^\uparrow$ and $S^\downarrow$ are calculated. Eventually we use these spin-dependent Seebeck coefficients to obtain the spin-Seebeck coefficient using Eq. (\[eq:Ss\]). By using $T(E)=T^\uparrow(E)+T^\downarrow(E)$ we calculate the charge Seebeck coefficient. The temperature dependence is included in the occupation function only.
First we calculate the Seebeck coefficient for pure Fe and Co, where Co has the same bcc structure like Fe. Fig. \[S\_pure\] shows the calculated results as a function of temperature. Experimental values are in the $\mu V /K$ range for pure Fe [@blatt67] and for a NiFe alloy [@uchida08] which means that our results have the correct order of magnitude. However, the details of the temperature dependence of Seebeck coefficient in pure Fe [@blatt67] is quite different to ours. Origins of these discrepancies are the not known quality of the samples and, in particular for high temperatures, missing inelastic contribution within the theory. To stress this point we are only investigating the temperature dependence due to changes in the occupation function. Nevertheless, our method is suitable to calculate the Seebeck and consequently the (ideal) spin-Seebeck coefficients in the right order of magnitude.
Next we investigate the MTEP in the tunnel junctions as a function of the relative magnetization of both magnetic layers to each other. For this purpose, we look at symmetric tunnel junctions with Fe and Co leads, where Co has again the same structure like Fe. The magnetic layers are 20 monolayers, MgO has 6 monolayers, and the junction is connected to reservoirs represented by Cu in a bcc-Fe structure. The positions of the atoms are ideal to get only the influence of the magnetic material and not of different relaxations in addition. It is well known that the interface structure and therefore also relaxation at the interface can influence the transport characteristics. Therefore, we plan to do investigations of the influence of different interfaces on the spin-Seebeck coefficient and MTEP in the future. Fig. \[MTEP\] upper panel shows the Seebeck coefficient as a function of the relative angle of the magnetization for a temperature of 300 K. The angular dependence show an almost constant Seebeck coefficient up to about $120^\circ$. There is a drastic change at angles close to the anti-parallel alignment. Fig. \[MTEP\] middle panel shows the temperature dependence of the MTEP ratio at anti-parallel alignment. There is a huge MTEP effect that can be much larger than for all-metallic junctions which show an experimental value of about 30% [@gravier06]. However, the temperature dependence is non-trivial including divergence at certain temperatures and change of sign. In addition, there is a large difference between the two magnetic materials.
Note that the Seebeck coefficient can be also negative. Therefore, it is not obvious which magnetic alignment causes the divergences of the MTEP ratio for the Fe/MgO/Fe junctions. Consequently, we show in Fig. \[MTEP\] lower panel the temperature dependence of the Seebeck coefficient for parallel and anti-parallel alignment. This viewgraph shows that the first two divergences for negative MTEP ratio are caused by a vanishing Seebeck coefficient for anti-parallel alignment. In contrast, the divergence at high temperature is due to a vanishing Seebeck coefficient in the parallel alignment.
For magnetic tunnel junctions the calculation of transport parameters can be rather tedious due to the rich structure of $T(E)$ around the Fermi level. Even for one particular energy for $T(E)$ the k point mesh for the integration within the first Brillouin zone has to be very dense typically tens of thousands [@waldron07]. Consequently, convergence studies with respect to the number of k points and the number of energy points have to be carried out. The latter is in particular important for small temperatures. For this purpose Fig. \[conv\] shows the MTEP of Fe/MgO/Fe for different k point and energy meshes. The qualitative behavior of MTEP is basically independent of the number of k points. Only the position where the second divergence of the MTEP occurs changes slightly. The influence of the different energy meshes on the MTEP is similar. Main differences are at very small temperatures and the position of the third divergence of the MTEP. In both cases the position of the divergences changes their position due to the relative small slope of the Seebeck coefficients. A small change in the Seebeck coefficient shifts the point where the Seebeck coefficient vanishes and therefore the position of the divergence. Nevertheless the qualitative behavior is nearly unchanged. For Figs. \[MTEP\], \[sS\] and \[T\_E\] we actually use the larger k point mesh with 160,000 k points and the dense energy mesh with a distance between the energy points of 0.68 meV.
In Fig. \[sS\] we present the spin-Seebeck coefficients for Fe/MgO/Fe and Co/MgO/Co as a function of temperature. The absolute values are comparable to the classical Seebeck coefficients. However, note that these values are not robust and can be seen only as an upper limit. The temperature dependence for Fe/MgO/Fe is quite complicated with sign change of the slope with temperature.
The temperature dependencies of the MTEP and of the spin-Seebeck coefficient can be understood by looking at the energy dependent transmission probability. Features of these transmission probabilities on the other hand can be understood by looking at electronic structure at the interface between the magnetic material and the barrier [@heiliger06]. We will not further discuss the electronic states but we will discuss in Fig. \[T\_E\] how $T(E)$ can explain the temperature dependence seen in Fig. \[MTEP\] and Fig. \[sS\].
For this purpose we show in Fig. \[T\_E\] the spin-dependent Seebeck coefficients and transmission probabilities for Fe/MgO/Fe. First, we start our discussion with the majority spin. In this case $T^\uparrow(E)$ is a smooth function showing two peaks one above and one below the Fermi level. The positions of the peaks are asymmetric to the Fermi level. Eq. (\[eq:L\_n\]) shows that the Seebeck coefficient is basically the center of the mass of $T(E) \partial_E f(E,\mu,T)$ divided by temperature. The contributing states within the integral are centered around the Fermi level and the width is increasing with increasing temperature. Consequently, starting from 0K the peak in the transmission above the Fermi level contribute to the Seebeck coefficient shifting the center of mass to higher energies which leads to the increase of the Seebeck coefficient. This is shown in the lower panel of Fig. \[T\_E\]. When the temperature is large enough that the peak below the Fermi level contributes to the Seebeck coefficient the Seebeck coefficient starts to decrease. For very high temperatures both peaks contribute equally to the Seebeck coefficient leading to a center of mass closely to the Fermi level and a vanishing Seebeck coefficient. In a similar way the dependence of the Seebeck coefficient for the minority spin can be understood although $T^\downarrow(E)$ has a more complicated structure.
In summary, we calculated the spin-Seebeck coefficient and the magneto-thermoelectric power for MgO based tunnel junctions with Fe and Co leads. Spin-Seebeck values of up to $150 \mu V/K$ are possible which is similar to the value of the normal charge Seebeck coefficient. The calculated values can be seen as an upper limit of what is possible in experiments. Due to spin diffusion the spin-voltage will be strongly reduced with distance from the barrier. Nevertheless, our calculation shows what is the maximum possible difference in spin chemical potentials next to the barrier. Besides the absolute values we predict a non-trivial temperature dependence of the spin-Seebeck coefficient that changes drastically by going to other magnetic material. In particular, for Fe/MgO/Fe the sign of the slope of the Seebeck coefficient changes with temperature whereas for Co/MgO/Co the sign of the slope is the same for all temperatures. This means that different materials have different optimal working temperatures. The MTEP ratio can be several 1000% in tunnel junctions. In particular, the non-trivial temperature dependence show even a divergence at certain temperatures. Consequently, in future work not only the material has to be analyzed in detail but also the temperature dependence.
We thank G.E.W. Bauer for useful discussion and acknowledge support from DFG SPP 1386 and DFG grant HE 5922/1-1.
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abstract: 'The Large Magellanic Cloud (LMC) has $\sim$60 confirmed supernova remnants (SNRs). Because of the known distance, 50 kpc, the SNRs’ angular sizes can be converted to linear sizes, and their X-ray observations can be used to assess X-ray luminosities ($L_X$). We have critically examined the LMC SNRs’ sizes reported in the literature to determine the most plausible sizes. These sizes and the $L_X$ determined from *XMM-Newton* observations are used to investigate their relationship in order to explore the environmental and evolutionary effects on the X-ray properties of SNRs. We find: (1) Small LMC SNRs, a few to 10 pc in size, are all of Type Ia with $L_X>10^{36}$ ergs s$^{-1}$. The scarcity of small core-collapse (CC) SNRs is a result of CCSNe exploding in the low-density interiors of interstellar bubbles blown by their massive progenitors during their main sequence phase. (2) Medium-sized (10-30 pc) CC SNRs show bifurcation in $L_X$, with the X-ray-bright SNRs either in an environment associated with molecular clouds or containing pulsars and pulsar wind nebulae and the X-ray-faint SNRs being located in low-density interstellar environments. (3) Large (size$>$30 pc) SNRs show a trend of $L_X$ fading with size, although the scatter is large. The observed relationship between $L_X$ and sizes can help constrain models of SNR evolution.'
author:
- 'Po-Sheng Ou (歐柏昇)'
- 'You-Hua Chu (朱有花)'
- Pierre Maggi
- 'Chuan-Jui Li (李傳睿)'
- 'Un Pang Chang (曾遠鵬)'
- 'Robert A. Gruendl'
title: 'X-ray Luminosity and Size Relationship of Supernova Remnants in the LMC'
---
[UTF8]{}[bsmi]{}
Introduction {#sec:intro}
============
Most supernova remnants (SNRs), regardless of progenitor types, exhibit some kind of X-ray emission. Thermal emission can arise from shocked interstellar medium (ISM) and/or SN ejecta, while relativisic electrons interacting with amplified magnetic field can produce non-thermal (synchrotron) emission. In the cases of core-collapse (CC) SNRs, there may exist additional X-ray emission from pulsars and pulsar wind nebulae (PWNe). See @vink2012 for a comprehensive review of X-ray emission from SNRs.
To make a statistical study of X-ray emission of SNRs, we need a large sample of SNRs with known distances. The Galactic sample of SNRs is quite incomplete because of heavy absorption in the Galactic plane, and the distances to individual SNRs are often very uncertain. The Large Magellanic Cloud (LMC), on the other hand, has small internal and foreground absorption column densities [@schlegel1998], and hosts a large sample of SNRs all at essentially the same known distance 50 kpc[^1] [@pietrzynski2013]. At least 59 SNRs have been confirmed and a few additional SNR candidates have been suggested [@maggi2016; @bozzetto2017]. This large sample of LMC SNRs is ideal for systematic and statistical studies of X-ray emission from SNRs.
Recently, @maggi2016 analyzed [*XMM-Newton*]{} observations of the 59 confirmed SNRs in the LMC, deriving physical properties of the X-ray-emitting plasma from spectral fits. Because of the known distance, it is possible to determine the X-ray luminosity of each SNR. In the meantime, @bozzetto2017 [hereafter Bo2017] measured the sizes of the 59 LMC SNRs using X-ray, radio and optical images. Intrigued by these results, we have examined the relationship between X-ray luminosity and size of LMC SNRs in order to explore evolutionary effects and environmental impacts on X-ray properties of SNRs.
This paper reports our investigation of the relationship between X-ray luminosity and size of SNRs in the LMC. In Section 2, we discuss the physical meaning of SNR sizes measured at optical or X-ray wavelengths, examine the SNR sizes reported in the literature, and assess the most reliable sizes that represent the SNRs’ full extent. In Section 3, we plot X-ray luminosities against sizes for LMC SNRs and note intriguing features in the distribution of SNRs in this plot. In Section 4, we discuss the physical reasons behind the distribution of SNRs in the plot of X-ray luminosity versus size. Finally, a summary is given in Section 5.
 
Sizes of LMC SNRs {#sec:LxSize}
=================
Both X-ray and optical images have been used to measure SNR sizes, but it should be noted that while X-ray and optical H$\alpha$ emission both originate from post-shock gas, they arise under different physical conditions. Generally speaking, X-ray emission comes from hot gas with temperatures $\gtrsim 10^6$K, while H$\alpha$ emission originates from ionized gas at $\sim 10^4$K; therefore, an SNR size measured in X-rays may differ from that measured in H$\alpha$.
Measurements of X-ray and H$\alpha$ sizes of SNRs can also differ because of different instrumental sensitivities. For example, the *XMM-Newton* observations of the LMC SNRs detect volume emission measures ($EM_V \equiv \int n_en_HdV$, where $n_e$ is the electron density, $n_H$ is the hydrogen density, and $V$ is the emitting volume) of 10$^{54}$ – 10$^{60}$ cm$^{-3}$ [@maggi2016]. For a spherical volume of highly ionized interstellar gas (${n_e}/{n_H} \sim 1.2$ for a typical helium to hydrogen number ratio of 0.1), the rms density derived from the volume emission measure is $$\langle n_H^2 \rangle^{1/2}=\bigg(\frac{5~EM_V}{ \pi f D^3}\bigg)^{1/2},$$ where $D$ is the diameter of the X-ray emitting gas, and $f$ is the volume filling factor. For SN ejecta dominated by heavy elements, ${n_e}/{n_H}$ should be greater than 1.2, and hence the rms hydrogen density in equation (1) is the upper limit, which is about (0.001–3) $f^{-1/2}$ cm$^{-3}$ for the LMC SNRs as derived from the *XMM-Newton* measurements. Meanwhile, narrow-band H$\alpha$ CCD images can easily detect emission measures ($EM_{\ell} \equiv \int n_en_Hd\ell$, where $\ell$ is the emitting path length) down to 10–20 cm$^{-6}$ pc, and for an emitting path length of 5 pc the electron density needs to be at least 1.4–2 cm$^{-3}$. Thus, for SNRs running into a dense medium with densities $>$ 1 H-atom cm$^{-3}$, the shocked gas can be detected in both X-ray and optical wavelengths, while those running into a medium with densities $\ll$1 H-atom cm$^{-3}$ may be visible in X-rays but not in optical.
Another factor that can affect the measurements of SNR sizes is the wavelength-dependent confusion from the background. SNRs may be located adjacent to HII regions or superposed on a complex background, in which case the boundary of an SNR can be diagnosed by sharp filamentary morphology, enhanced \[\] line emission, high-velocity components in optical emission lines, nonthermal radio emission, and diffuse X-ray emission [@chu1997]. When more than one of the above diagnostics are detected, the SNR boundary can be more reliably measured. However, if X-ray emission is the only diagnostic detected and the SNR emission is superposed on a large-scale diffuse X-ray emission, the background confusion can prevent accurate measurements of the SNR size.
Several publications have reported sizes of the LMC SNRs, but there are often discrepancies between their measurements. @badenes2010 [hereafter Ba2010] used mainly X-ray images from *Chandra* or *XMM-Newton* to determine the SNR sizes, and adopted previous optical and radio measurements when high-resolution X-ray images were not available. @desai2010 [hereafter De2010] considered optical and X-ray images, and measured SNR sizes based on the extent of diffuse X-ray emission or filamentary H$\alpha$ shell structure. Bo2017 considered optical, X-ray and radio images. @maggi2016 also listed SNR sizes, but they only gave the maximal diameters in X-ray; thus these sizes are often much larger than the ones reported by the above three references. Below we compare the SNR sizes reported by Ba2010, De2010, and Bo2017.
Bo2017 has the largest and most complete SNR sample, and is thus chosen to be the reference for comparisons. Figure 1 compares SNR sizes reported by De2010 and Bo2017 in the left panels, and Ba2010 and Bo2017 in the right panels. The upper panels plot SNR sizes from one source versus another, while the lower panels plot the ratios of SNR sizes from two sources.
De2010 and Bo2017 both used primarily optical and X-ray images for the SNR size measurements, but there are still discrepancies greater than 16% and up to 50%. The discrepancies are caused by the following reasons: (1) The SNR sizes can be measured only in X-rays and the surface brightness varies significantly, such as N23, or the background is complex, such as the Honeycomb and 0532-67.5; in such cases the discrepancy in size measurements can be as high as 50%. (2) The SNR is superposed on an region or a superbubble, whose H$\alpha$ emission can confuse the size measurements, such as N157B and N186D. (3) The SNR size is measured without simultaneously considering optical, X-ray, and radio images that show wavelength-dependent distribution of emission, such as 0534$-$69.9, DEML238, DEML299, and J0550$-$6823. (4) The irregular shape of an SNR can cause subjective size measurements to differ by up to $\sim$20%, such as N86. For these discrepant objects, we examine their H$\alpha$, \[\], X-ray, and 24 $\mu$m images (in Appendix A), consider radio and kinematic properties available in the literature, and make new measurements (described in Appendix B).
The comparisons between Ba2010 and Bo2017 sizes, right panels of Figure 1, show numerous large discrepancies. These discrepancies are caused by the larger uncertainties in Ba2010 sizes that were compiled from previous measurements based on mainly X-ray images and some optical images. As mentioned above and detailed in Appendices A–B, multi-wavelength examination of an SNR provides the most comprehensive picture of its physical structure and boundaries, and size measurements based on only one single wavelength may not reflect the SNR’s true extent.
Relationship between sizes and $L_X$ {#sec:LxSize}
====================================
The LMC provides an ideal sample of SNRs for us to study the relationship between their X-ray luminosities ($L_X$) and sizes. The size of an SNR may be intuitively thought to reflect its evolutionary stage, because an SNR expands as the shock wave propagates outward and a large SNR would be older than a small SNR, if the ambient interstellar densities are similar. However, the ambient ISM does have a wide variety of physical properties and conditions, and the relationship between size and evolution can be quite complex. Through the relationship between $L_X$ and size we hope to investigate effects of ambient environment and evolution on the SNRs’ X-ray luminosities.
SNRs rarely show round, symmetric shell structures with well-defined sizes. To assign a “size” to an irregular SNR, we adopt the average of its major and minor axes. Such SNR sizes determined from data of Ba2010, De2010, and Bo2017 are tabulated in Appendix C. As discussed at length in Section 2, De2010 and Bo2017 sizes were determined primarily with optical and X-ray images of SNRs and are in agreement for most cases. For 83% of the SNRs that have Bo2017 and De2010 sizes differ by less than 16%, we adopt their average sizes from Bo2017. For the SNRs with larger discrepancies between De2010 and Bo2017, we discuss individual objects and determine their average sizes in Appendix B, and list their adopted sizes in the table in Appendix C.
The LMC sample of SNRs have been studied in X-rays with *XMM-Newton* by @maggi2016. They fit the X-ray spectra with the package XSPEC [@arnaud1996] using a combination of collisional ionization equilibrium (CIE) models and non-equilibrium ionization (NEI) models, derived their X-ray fluxes in the 0.3 to 8 keV band, and computed their $L_X$ for an LMC distance of 50 kpc [@pietrzynski2013]. These observed (absorbed) $L_X$ are listed in the last column of the Table in Appendix C.
Using the $L_X$ and average sizes in Appendix C, we plot the $L_X$ versus size for the LMC SNRs in Figure 2. We have also made the same $L_X$–size plot with unabsorbed $L_X$ and present it in Figure 3. (These unabsorbed $L_X$ are from the same model fits that produced the absorbed $L_X$ published by @maggi2016). The distribution of the SNRs are qualitatively similar to that in Figure 2. Note that we did not include SN 1987A (size$=0.45$ pc, $L_X=2.7\times 10^{36}$ ergs s$^{-1}$) because its inclusion will leave vast empty space on the left and compress all the data points on the right in Figure 2 and 3.
At first glance, the $L_X$ – size plot for LMC SNRs shows a scattered diagram; however, if the sizes are divided into three ranges: $<$10 pc as “small”, 10–30 pc as “medium”, and $>$30 pc as “large”, it is possible to see interesting trends in each size range:\
(1) For sizes a few to 10 pc, only a small group of SNRs exist with $L_X$ of a few $\times$ 10$^{36}$ ergs s$^{-1}$, and all of them are of Type Ia. For comparison, we add the Galactic SNRs with sizes a few to 10 pc in Figure 3; the data are taken from the Chandra Supernova Remnant Catalog[^2]. Interestingly, Tycho and Kepler SNRs, two small Type Ia SNRs in our Galaxy, are also located in the similar part of $L_X$–size plot as the small LMC Type Ia SNRs. In contrast, the Galactic CC SNR Cas A is an order of magnitude more luminous than these small Type Ia SNRs, and the $\sim$100-year old Galactic Type Ia SNR G1.9+0.3 is smaller and significantly fainter [@reynolds2008; @borkowski2010; @borkowski2013].\
(2) For sizes 10–30 pc, there is a bifurcation in the distribution of SNRs. The X-ray-bright ones have $L_X > 10^{36}$ ergs s$^{-1}$ and the X-ray-faint ones have $L_X < 10^{35}$ ergs s$^{-1}$ for sizes below $\sim$20 pc, and these two groups converge to a few $\times$10$^{35}$ ergs s$^{-1}$ towards 30 pc size. It is worth noting that the X-ray-faint medium-sized SNRs are mostly CC SNRs.\
(3) For sizes $>$30 pc, while $L_X$ exhibits a wide range, the majority of the SNRs appear to show a general trend of $L_X$ decreasing with size.
It also appears that the Type Ia SNRs show smaller scatter in $L_X$ for any given size than the CC SNRs, especially in the medium size range (Figures 2 and 3). The scatter in $L_X$ reflects the ambient interstellar density: Type Ia SNe occur in diffuse medium with moderate densities, while CC SNe can take place near dense molecular clouds or in a very low-density environment produced by energy feedback from massive stars. Because of the smaller scatter in $L_X$, the smooth variations of Type Ia SNRs’ $L_X$ versus size may demonstrate the SNR evolution.
The dashed line in the lower right corner of Figure 2 corresponds to a constant surface brightness of $10^{29}$ ergs s$^{-1}$ arcsec$^{-2}$, which represents the typical detection limit of the *XMM-Newton* observations used by @maggi2016. Consequently, no SNRs are located beneath this dashed line.
Discussion {#sec:Discussion}
==========
We have examined the physical structures and environments of SNRs in the three size ranges in order to understand the physical significance of their distributions in the $L_X$–size plot. The discussion in this section is ordered according to the SNR sizes.
Small Known LMC SNRs Are Dominated by Type Ia
---------------------------------------------
It is striking that the small LMC SNRs, with sizes a few to 10 pc, are all Type Ia SNRs with $L_X$ of a few $\times$ 10$^{36}$ ergs s$^{-1}$. (Note that SN 1987A is outside the size range under discussion.) For comparison, we show that the Galactic Type Ia SNRs Kepler and Tycho are both located in a similar region as the young LMC Type Ia SNRs. The small range of $L_X$ for small Type Ia SNRs and the scarcity of small CC SNRs can be explained as follows.
Type Ia SNe are usually considered to explode in a tenuous and uniform ISM [e.g., @badenes2005]. On the other hand, CC SNe usually explode inside interstellar bubbles blown by the fast stellar winds of their massive progenitors during the main sequence phase [@castor1975; @weaver1977]. Interstellar bubble interiors have very low densities, and hence CC SNe inside bubbles are called “cavity explosions”. It is conceivable that the interstellar environments of Type Ia and CC SNe have very different density profiles.
Density profiles of ambient medium strongly affect the evolution of an SNR’s $L_X$. In a classical model of a SN explosion in a uniform ISM, the resulting SNR goes through free expansion phase, Sedov phase (i.e., adiabatic phase), and radiative phase [@woltjer1972]. The Sedov phase starts when the swept-up ISM mass is several times the SN ejecta mass [e.g., @dwarkadas1998]. The $L_X$ of an SNR during the Sedov phase can be calculated [e.g., @hamilton1983]. To illustrate the evolution of $L_X$ for different ambient densities, we plot $L_X$ against age and size in Figure 4.
 
For a Type Ia SNR in a partially neutral ISM, only the ionized interstellar gas can be swept up by the shock. Thus, for a uniform density of $\sim$1 H-atom cm$^{-3}$ and a neutral fraction of $\eta$, the Sedov phase will start when the swept-up ionized gas reaches 1.4 $M_\odot$ in mass, corresponding to a radius of 2.4$(1-\eta)^{-1/3}$ pc. This radius is 5.2 pc if $\eta$ = 0.9, and 3 pc if $\eta$ = 0.5. These sizes are comparable to the young Balmer-dominated Type Ia SNRs in the LMC, 0509$-$67.5 and 0519$-$69.0; thus, it is likely that these young Type Ia SNRs are entering the Sedov phase. However, the interstellar density is so much lower than the SN ejecta density that their X-ray emission is still dominated by that produced by the reverse shock into the SN ejecta. This is evidenced in the SN ejecta abundance revealed in the X-ray spectra of these small Balmer-dominated Type Ia SNRs, although the X-ray emission shows a shell morphology [@warren2004; @kosenko2010]. The larger Type Ia SNRs, such as DEML71 and 0548$-$70.4 with sizes in the 20-30 pc range, must be in the Sedov phase already. Furthermore, their forward shock and reverse shock have traveled farther apart, and their X-ray emission shows the forward shock in an interstellar shell well resolved from the reverse shock in the SN ejecta [@hughes2003; @hendrick2003].
X-ray emission from reverse shocks is the cause of the high $L_X$ of small Type Ia SNRs. The small scatter of these young bright Type Ia SNRs in the $L_X$ vs size plot reflects their similar ages, the relative uniformity of SNe Ia (in term of nucleosynthesis and explosion energy), and the modest effect the progenitors have on changing their immediate surrounding. The smallest Galactic Type Ia SNR G1.9+0.3 has a low $L_X$ because it is so young ($<$200 yr) that the reverse shock has only gone through very little of the SN ejecta [@reynolds2008; @borkowski2014].
For CC SNRs whose SNe exploded in cavities of wind-blown bubbles, due to the extremely low density within the bubbles ($\sim 10^{-4}-10^{-2}$ H-atom cm$^{-3}$), the X-ray emission from shocked gas would be too faint to be detected at a young age; only when the SNR’s forward shock hits the dense shell/wall of a bubble will the X-ray luminosity jump up several orders of magnitude [@dwarkadas2005]. As shown by @naze2001, main sequence O stars have interstellar bubbles of sizes 15–20 pc. By the time a massive star explodes as a CC SN, its main-sequence bubble has grown larger, and hence the SNR shock goes through the low-density bubble interior without producing detectable X-ray emission until it hits the bubble shell wall at radius of 10 pc or larger.
For illustration, considering a spherical interstellar bubble with a radius of 10 pc and assuming a simplistic extreme case (upper limit) of average density of 0.01 H-atom cm$^{-3}$ in the bubble interior, we can calculate the total mass in the bubble interior to be $\lesssim$ 1 $M_\odot$; thus, when the SNR shock reaches the bubble wall, it has swept up only $\sim$1 $M_\odot$, much lower than the CCSN ejecta mass, a few to a few tens $M_\odot$; thus, the Sedov phase has not been reached. The bubble shell consists of swept-up ISM that was originally distributed in the bubble cavity. Assuming the bubble was blown in a diffuse ISM with density of 1 H-atom cm$^{-3}$, the total mass in the bubble shell would be 100 $M_\odot$; therefore, the SNR reaches the Sedov phase when the forward SNR shock traverses the bubble shell.
During the free-expansion phase, the SNR shock is not significantly decelerated and it remains fast until it hits the bubble wall. Assuming a constant shock velocity of 10,000 km s$^{-1}$, it only takes 1000 years for the SNR to grow to a radius of 10 pc. Consequently, SNRs inside interstellar bubbles not only emit very faintly in X-rays, but also expand very rapidly to reach the dense shell wall. Such “cavity explosions” explain the absence of small CC SNRs in the $L_X$–size plot. Cavity explosions are also responsible for the discrepancies between ionization ages and dynamical ages of LMC SNRs, such as N132D, N63A, and N49B [@hughes1998].
We have plotted the young CC SNR Cas A in Figure 3 for comparison. Cas A is small in size and luminous in X-rays. These properties are caused by its interaction with a dense circumstellar medium, i.e., material ejected by the SN progenitor [@fesen2001]. Circumstellar bubbles are often observed around Wolf-Rayet stars and luminous blue variables (LBVs), and circumstellar bubbles are smaller than interstellar bubbles [@chu2003]. Cas A SN must have exploded in a circumstellar bubble.
X-ray-Bright and X-ray-Faint Medium-Sized SNRs
----------------------------------------------
The medium-sized LMC SNRs show clear bifurcation in their $L_X$. In the X-ray-bright group with $L_X \ge 10^{36}$ ergs s$^{-1}$, only one is of Type Ia, and the other seven are CC SNRs. Among these X-ray-bright CC SNRs, four are interacting with molecular clouds, as CO emission was detected near the SNRs N23, N49, and N132D [@banas1997; @park2003] and H$_2$ absorption is detected in *Spitzer* IRS observations towards N63A (Segura-Cox et al. 2018, in preparation). None of these four X-ray-bright CC SNRs show sharp H$\alpha$ shell structure enclosing the diffuse X-ray emission, indicating that the forward SNR shocks are still in the low-density interiors of bubbles. In the cases of N23 and N132D, where no prominent shocked cloudlets are seen, the X-ray emission does show limb-brightening, indicating that the ambient medium is dense enough to produce detectable X-ray emission but not optical H$\alpha$ emission, and this ambient medium may correspond to the conduction layer in a bubble interior [@weaver1977]. As N23 and N132D are both associated with molecular clouds, their bubble shells and conduction layers must have higher densities, which contribute to the bright X-ray emission. In the cases of N49 [@bilikova2007; @park2012] and N63A [@warren2003], it is clear that dense cloudlets, possibly associated with the molecular clouds, have been shocked and contribute to the X-ray emission. The other three X-ray-bright CC SNRs possess bright PWNe: 0540$-$69.3 [@gotthelf2000], N157B [@wang1998], and 0453$-$68.5 [@gaensler2003]. Pulsars and PWNe are powerful sources of nonthermal X-ray emission and provide additional X-ray emission to boost their SNRs’ total $L_X$. Note that the PWN of 0453$-$68.5 is not particularly dominating, but its X-ray image show a limb-brightened sharp shell that indicates that the shock has already reached the bubble shell. While 0453$-$68.5 has a PWN, it is the SNR shock impact on the dense bubble shell giving rise to $L_X$.
The X-ray-faint medium-sized SNRs are mostly associated with CC SNe. Among the three X-ray-faint CC SNRs smaller than 20 pc, 0536$-$69.2 and \[HP99\]483 are not detected in optical, and the Honeycomb SNR shows only a small patch of honeycomb-like nebulosity resulting from SNR shocking a piece of shell wall [@chu1995; @meaburn2010]. The absence of sharp optical shells enclosing the diffuse X-ray emission indicates a low-density ISM around these SNRs. The Honeycomb SNR has hit a small piece of dense gas and hence it has the highest $L_X$ among these three, but still a couple orders of magnitude fainter than the SNRs interacting with molecular clouds. The X-ray-faint SNRs with sizes 20–30 pc all show optical shell structure enclosing their diffuse X-ray emission, and they have higher $L_X$ than the smaller ones, except J0449$-$6920, whose *XMM-Newton* observation was too shallow to make accurate measurements. These CC SNRs may represent cavity explosions whose SNR shocks have just reached the bubble shell walls. The SNRs N11L and N120 have just reached the bubble shell, but the bubble shell densities are not as high as those of N23 and N132D.
Fading of X-rays in Large SNRs
------------------------------
Among the large (size $>$30 pc) LMC SNRs, a general trend of decreasing $L_X$ for larger SNRs can be seen, but for any given size, the differences in $L_X$ can be up to one order of magnitude.
As an SNR sweeps up more interstellar gas, the shock velocity decreases and when it goes much below $\sim$300 km s$^{-1}$, the post-shock temperature will be below 10$^6$ K, too low to generate X-ray-emitting gas. The hot gas in SNR interior cools, and the X-ray emission diminishes.
The scatter in $L_X$ may be caused by the differences in ambient gas densities ($n_0$) and the SN explosion energies ($E$). To evaluate the effects of these two factors, we consider a spherical SNR of radius $R$, whose X-ray emission originates from shocked ISM in a shell. Its $L_X$ is $\propto$ (emitting volume) $\times$ (density)$^2$ $\times$ (emissivity). As (emitting volume) $\times$ (density) is proportional to the total interstellar mass within radius $R$, it is $\propto$ $R^3n_0^2$. The emissivity is a slow function of temperature for photon energies below 2 keV [@hamilton1983]. Since the large old SNRs are likely at low X-ray emitting temperatures, a few $\times$10$^6$ K at most, we will treat the emissivity as a constant, and $L_X$ $\propto$ $R^3n_0^{2}$.
The total kinetic energy in the SNR shell scales with the explosion energy, so $E \propto R^3n_0 v^2$. The large old SNRs have low expansion velocities of a few $\times 10^2$ km s$^{-1}$, so we will also approximate the expansion velocity as a constant. Thus, $L_X$ $\propto$ $E n_0$ [^3]. The effects of the ambient density and the SN explosion energy are about equally important. However, the ranges of the ambient gas densities and the SN explosion energies are quite different. The ambient interstellar density can range from 0.01 to a few hundred H-atom cm$^{-3}$, about 4 orders of magnitude, while the SN explosion energies are mostly clustered around 10$^{51}$ ergs with extreme values differing by no more than 3 orders of magnitude [e.g., @woosley1986].
Hence, the large scatter in $L_X$ for SNRs with the same size is more likely caused by the detailed differences in the ambient gas densities, and the SN explosion energy plays a lesser role in raising the scatter in $L_X$.
Summary
=======
The LMC is at a known distance of 50 kpc, and thus the linear sizes of SNRs in the LMC can be determined from their angular size measurements [@desai2010; @bozzetto2017], and their X-ray luminosities can be determined from *XMM-Newton* X-ray observations [@maggi2016], allowing a unique opportunity to examine the relationship between $L_X$ and size of SNRs. We have critically compared LMC SNR sizes reported by different authors in the literature and adopted the most reasonable sizes to investigate how $L_X$ vary with sizes among the LMC sample of SNRs.
We find that the $L_X$ – size relationship for LMC SNRs can be divided into small, medium, and large size ranges:\
(1) The small LMC SNRs with sizes a few to 10 pc are all young Type Ia SNRs with $L_X$ a few times 10$^{36}$ ergs s$^{-1}$. The apparent scarcity of small CC SNRs may be caused by their “cavity explosions”, as massive progenitors of CCSNe have blown interstellar bubbles and the SN explosions take place in the very low-density interiors of the bubbles.\
(2) The medium-sized SNRs, with sizes 10–30 pc, show bifurcation in their $L_X$ with an order of magnitude difference in $L_X$. The X-ray-bright CC SNRs either are in an environment associated with molecular clouds or have pulsars and pulsar-wind nebulae emitting nonthermal X-ray emission.\
(3) The large SNRs, with sizes greater than $\sim$30 pc, show a general trend of fading $L_X$ at large sizes. As these sizes are larger than the normal interstellar bubbles blown by massive stars, the large SNRs have swept up the bubble material and extended into the diffuse ISM. As the SNR shocks sweep up more ISM, the shock velocity slows down. When the post-shock velocities are too low to produce X-ray-emitting material, the hot plasma in SNR interiors cool and reduce the X-ray emission.
This project is supported by Taiwanese Ministry of Science and Technology grant MOST 104-2112-M-001-044-MY3.
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Images of the SNRs with Large Discrepancies between different Size Measurements
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Descriptions of size determinations for individual SNRs
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*N86*.– The shape of the SNR is irregular because of the breakout structure in the north, through which the hot gas flows out from the SNR shell [@williams1999]. For N86 in Figure 2, we have used both sizes of De2010 (75.0 pc, including the breakout) and Bo2017 (61.5 pc, not including the breakout) in the $L_X$ – size plot to illustrate the uncertainty in the size. Only low-resolution *ROSAT* X-ray images are available for N86; high-resolution *Chandra* and *XMM-Newton* observations will help refine the size determination.
*N186D*.– The size of SNR N186D in H$\alpha$ cannot be unambiguously measured, because N186D is projected on the rim of the superbubble N186E. Through the analysis of velocity fields in N186D, @laval1989 determined the SNR size to be $\sim$40 pc, which is consistent with the size of the \[\]-enhanced shell (see Figure 5 in Appendix A). Based on these considerations, we adopt the size 36.8 pc given by De2010.
*N23*.– X-ray emission of N23 is enhanced in the southeast side, likely due to a denser ambient medium [@williams1999]. The size reported by De2010, $18 \times 12$ pc, corresponds to only the X-ray-brightest region. @maggi2016 and Bo2017 included the fainter X-ray emission from the northwest side and reported a larger SNR size, 23.6 pc, which is more accurate and hence adopted in the $L_X$–size plot in Figure 2.
*SNR 0532-67.5*.– This SNR may be associated with the OB association LH75 [@chu1997]. This SNR has no optical counterpart, indicating that it is in a low-density medium, possibly caused by the fast stellar winds and SN explosions from LH75. The size of this SNR can be measured only in X-rays. There is bright X-ray emission in a $\sim 40 \times 20$ pc region, and a fainter and larger X-ray arc connected with the bright region. As in the case of the SNR N23, we include both the bright and faint X-ray emission regions in the size estimate, and adopt a size of 67.5 pc as the size of SNR 0532-67.5.
*SNR 0534-69.9*.– The optical images of this SNR show only a faint filament associated with the brightest X-ray emission region. *Chandra* observations show the SNR clearly in X-rays, although the southern rim is much fainter than the rest of the SNR. We have measured and adopted the full extent of the SNR shown in X-rays, about 33.5 pc, similar as the size measured by @maggi2016, which is larger than those reported by De2010 and Bo2017.
*DEM L238*.– Comparing H$\alpha$ and *Chandra* images, the X-ray emitting region is larger than the optical shell. We adopt the full extent of the SNR, 47.5 pc.
*Honeycomb*.– The Honeycomb SNR is near the 30 Doradus complex, and to the south of the superbubble 30 Dor C. This region has a very complex star formation history and chaotic nebular morphology. The lack of bright ionized gas region suggests an evolved environment with low ISM densities. The optical morphology of the SNR is very irregular, consisting of many cells instead of a simple shell [@chu1995; @meaburn2010], leading to large uncertainty in the determination of SNR size. For the Honeycomb SNR, we have used both sizes of De2010 (15 pc) and Ba2010 (25.5 pc) in the $L_X$ – size plot to illustrate the uncertainty in the size.
*N157B*.– The environment of N157B in H$\alpha$ is very complex because this SNR is superposed on the HII region of the OB association LH99 [@chu1997], and dissected by a foreground dark cloud. The most reliable measurement of the SNR size is through the analysis of gas kinematics using long-slit high-dispersion spectroscopic observations, 25$\times$18 pc [@chu1992]. This SNR boundary has been confirmed by sharp filaments revealed by *HST* images as shown in Figure 5. The size 21.8 pc given by De2010 is taken from @chu1992.
*DEML299*.– This SNR is inside a large optical shell. The size reported by De2010 corresponds to the large optical shell. The X-ray emission actually extends from the shell cavity to the southwest, indicating an outflow. The \[\]/H$\alpha$ ratio is enhanced in the shell structure and in the superposed filaments of supergiant shell LMC-2. The SNR is clearly in a very complex environment. We include all the diffuse X-ray emission region and \[\] enhanced filaments, and measure a size of 100$\times$50 pc. A smaller SNR size, $\sim$55 pc, has been reported by @warth2014 and @maggi2016 based on the diffuse X-ray emission and a surrounding \[\]-enhanced filament. The large discrepancy between these two size measurements illustrate the difficulty in determining SNR sizes in a complex environment confused by other energetic feedback processes from massive stars. We adopt both 73.5 and 56.5 pc in Table 1 (Appendix C) and Figure 2.
*J0550-6823*.– While the diameter of the optical shell is only $\sim$68 pc, there is X-ray and radio emission extending over 90 pc in the east-west direction; therefore, we adopt the size of 90$\times 68$ pc by @bozzetto2012.
Sizes and X-ray luminosities of LMC SNRs
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[ccccccccc]{} J0448-6700 & \[HP99\] 460 & 55.0 & 59.2 &60.8&& 60.8 & 0.46\
J0449-6920 & –& 33.2 &30.0& 28.8 & & 28.8 & 0.07\
J0450-7050 & SNR 0450-70.9& 89.2 & 97.5 & 109.4 & & 109.4 & 0.59\
J0453-6655 & SNR in N4 & 63.0& 64.5 & 60.6& & 60.6 & 1.17\
J0453-6829 & SNR 0453-68.5 & 30.0& 30.0& 30.4& & 30.4 & 13.85\
J0454-6713 & SNR 0454-67.2 & 44.2 & 37.5& 32.5 & & 32.5 & 1.58\
J0454-6626 & N11L & 21.8 & 18.0& 20.4 & & 20.4 & 0.63\
J0455-6839 & N86 & 87.0 & 75.0 & 61.5 & Y & 61.5–75.0 & 1.42\
J0459-7008 & N186D & 37.5 & 36.8 & 29.0 & Y & 36.8 & 1.09\
J0505-6753 & DEM L71 &18.0 & 20.2 & 18.6 && 18.6 & 44.59\
J0505-6802 & N23 & 27.8 & 15.0 & 23.6& Y & 23.6 & 26.25\
J0506-6541 & DEM L72 & 102.0 &83.2 & 96.2&& 96.2 & 0.53\
J0506-7026 & \[HP99\] 1139 & 82.5 & – & 42.5 && 42.5 & 1.44\
J0508-6902 & \[HP99\] 791 & – & – & 67.0&& 67.0 & 0.37\
J0508-6830 & –& – & – & 30.8& & 30.8 &0.09\
J0509-6844 & N103B & 7.0 & 7.5 & 7.0 && 7.0 & 51.7\
J0509-6731 & SNR 0509-67.5 & 7.25 &8.4 & 7.6 && 7.6 & 16.51\
J0511-6759 & –& – & – & 55.5& & 55.5 & 0.16\
J0512-6707 & \[HP99\] 483 & – & – & 12.5&& 12.5 & 0.09\
J0513-6912 & DEM L109 & 53.8 & 57.8 & 55.6 && 55.6 & 0.51\
J0514-6840 & – & – & – &55.0&& 55.0 & 0.4\
J0517-6759 & – & – & – & 66.8 && 66.8 & 0.24\
J0518-6939 & N120 & 33.5 & 21.8 & 23.4 && 23.4 & 0.88\
J0519-6902 & SNR 0519-69.0 & 7.8 & 8.2 & 8.6 && 8.6 & 34.94\
J0519-6926 & SNR 0520-69.4 & 43.5 & 33.8 & 31.2 && 31.2 & 2.69\
J0521-6543 & DEM L142 & – & 40.5 & 34.5 && 34.5 & –\
J0523-6753 & SNR in N44 & 57.0 & 52.5 & 57.5 && 57.5 &0.9\
J0524-6624 & DEM L175a & 58.5 & 51.8 & 36.25 && 36.2&–\
J0525-6938 & N132D & 28.5 & 26.2& 25.5 && 25.5 & 315.04\
J0525-6559 & N49B & 42.0 & 36.0 & 38.8 && 38.8 & 38.03\
J0526-6605 & N49 & 21.0 & 21.0 & 18.8 && 18.8 & 64.37\
J0527-6912 & SNR 0528-69.2 & 36.8 & 35.2 & 35.0 && 35.0 & 1.99\
J0527-6550 & DEM L204 & 75.8 & 67.5& 76.2 && 76.2 & –\
J0527-6714 & SNR 0528-6716 & – & – & 54.0 && 54.0 & 0.58\
J0527-7104 & \[HP99\] 1234 & 49.0 & – & 70.2 && 70.2 & 0.25\
J0528-6727 & DEM L205 & – & – & 55.0 & & 55.0 & 0.21\
J0529-6653 & DEM L214 & 25.0 & – & 33.1 && 33.1 & 1.04\
J0530-7008 & DEM L218 & 53.2 & 47.2 & 49.4 && 49.4 &0.72\
J0531-7100 & N206 & 48.0 & 45.0& 45.0 & & 45.0 & 2.55\
J0532-6732 & SNR 0532-67.5 & 63.0 & 67.5 & 45.0 & Y & 67.5 & 2.48\
J0533-7202 & – & – & – & 45.0 && 45.0 & 0.57\
J0534-6955 & SNR 0534-69.9 & 28.5 & 23.2 & 28.8 & Y & 33.5 & 6.33\
J0534-7033 & DEM L238 & 45.0 & 40.5 & 47.5 & Y & 47.5 & 1.55\
J0535-6916 & SN 1987A & 0.5 & $>$1.5 & 0.45 && 0.45 & 27.39\
J0535-6602 & N63A & 16.5 & 19.5 & 18.5 && 18.5 & 185.68\
J0535-6918 & Honeycomb & 25.5 & 15.0 & 18.8 & Y & 15.0–25.5 & 0.4\
J0536-6735 & DEM L241 & 33.8 & 36.0 & 34.0 & & 34.0 & 3.84\
J0536-7039 & DEM L249 & 45.0 & 37.5 & 39.2 && 39.2 & 1.43\
J0536-6913 & SNR 0536-69.2 & 120 & – & 16.5 & & 16.5 & 0.22\
J0537-6628 & DEM L256 & 51.0 & 48.0 & 46.9 && 46.9 & 0.32\
J0537-6910 & N157B & 25.5 & 21.8 & 31.5 & Y & 21.8 & 15.0\
J0540-6944 & SNR in N159 & 19.5 & 27.0& 26.2 && 26.2 & 0.43\
J0540-6920 & SNR 0540-69.3 & 15.0 & 18.0& 15.6 && 15.6 & 87.35\
J0541-6659 & \[HP99\] 456& – & – & 71.5 && 71.5 & 0.77\
J0543-6858 & DEM L299 & 79.5 & 73.5 & 56.5 & Y & 56.5–73.5 & 1.68\
J0547-6943 & DEM L316B & 21.0 & 46.5 & 45.0 && 45.0 & 1.47\
J0547-6941 & DEM L316A & 14.0 & 30.0& 30.0 && 30.0 & 1.26\
J0547-7025 & SNR 0548-70.4& 25.5 & 28.5 & 28.1 && 28.1 & 2.94\
J0550-6823 & – & 78.0 & 65.2& 81.9 & Y & 81.9 & 1.22\
[^1]: Note that due to the LMC’s inclination of 18-23 degrees in the line of sight, the error in the distance and linear size can be uncertain by up to 10% [@subramanian2010], and the luminosity can be uncertain by 20%. These uncertainties, however, do not affect the general conclusions of these paper.
[^2]: See http://hea-www.cfa.harvard.edu/ChandraSNR/.
[^3]: Note that this is in interesting contrast with the radio luminosity, which scales as $L_{radio}$ $\propto$ $E^{1.3} n_0^{0.45}$ or $L_{radio}$ $\propto$ $E^{1.45} n_0^{0.3}$ depending on the magnetic field amplification mechanism by the shock [@chomiuk2009].
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abstract: |
A parton and hadron cascade model, PACIAE, is employed to investigate the net charge transfer fluctuation within $|\eta|$=1 in $Au+Au$ collisions at $\sqrt{s_{NN}}$=200 GeV. It is turned out that the observable of net charge transfer fluctuation, $\kappa$ , in hadronic final state (HM) is nearly a factor of 3 to 5 larger than that in initial partonic state (QGM). However, only twenty percent of the net charge transfer fluctuation in the QGM can survive the hadronization.\
[PACS numbers: 25.75.Dw, 24.85.+p, 24.10.Lx]{}
author:
- 'Dai-Mei Zhou$^1$, Xiao-Mei Li$^2$, Bao-Guo Dong$^2$, and Ben-Hao Sa$^{3,2,1,4}$ [^1]'
title: 'Discrepancy between hadron matter and quark-gluon matter in net charge transfer fluctuation'
---
Recently the net charge transfer fluctuation has been proposed in [@jeon1; @jeon2] as a signal of Quark-Gluon-Plasma (QGP) phase transition expected to be existing in the relativistic nucleus-nucleus collisions. Following [@quig] the observable $$\kappa(\eta)=D_u(\eta)/(dN_{ch}/d{\eta})$$ is employed to describe the net charge transfer fluctuation. In the above equation the net charge transfer deviation, $D_u$, reads $$D_u(\eta)=<u(\eta)^2>-<u(\eta)>^2,$$ where the net charge transfer, $u$, is defined by $$u(\eta)=[Q_F({\eta})-Q_B({\eta})]/2,$$ where the $Q_F({\eta})$ ($Q_B({\eta})$) is referred to the net charge in forward (backward) region of $\eta$ and the $N_{ch}$ stands for the charge multiplicity accounted according to the charge of particle. The $\kappa$ is argued to be a measure of the local unlike-sign charge correlation length [@jeon2] and the charge correlation length in QGP phase (in quark-gluon matter, QGM) is expected to be much smaller than the one in hadronic matter (HM) because the charge unit is 1/3 and 1 in QGM and HM [@jeon3; @bass1; @asak], respectively.
In [@jeon2] a neutral cluster model was used first and the hadronic transport models of HIJING [@wang1], RQMD [@sorg1], and UrQMD [@bass2] were employed then to study the net charge transfer fluctuation. The results from above hadronic transport models could be summarized as follows: 1. The discrepancy among them is not obvious from each other. 2. The $\kappa(\eta=1)$ calculated in interval of $|\eta|<1$ is equivalent to the net charge fluctuation at $\eta=1$ and is close to the STAR datum [@star1] of $\sim0.27\pm0.02$ (cited from [@jeon2] directly) in $Au+Au$ collisions at $\sqrt{s_{NN}}$=130 GeV. 3. The $\kappa(\eta)$ does not strongly depend on the centrality.
A parton and hadron cascade model, PACIAE, is employed in this letter investigating the net charge transfer fluctuation, $\kappa$, within $|\eta|<1$ both in the early partonic stage (QGM) and in the hadronic final state (HM) in $Au+Au$ collisions at $\sqrt{s_{NN}}$=200 GeV. As expected the later results are quite close to the results in HIJING, RQMD, and UrQMD. However the former results are smaller than the later one by a factor of 3 to 5. Unfortunately, the $\kappa$ in QGM seems to be hard to survive the hadronization.
As the simplified version of PACIAE model has been published in [@sa1] and a nice bit of detailed description has been given in [@sa2], here we just give a brief introduction for PACIAE model. In the PACIAE model a nucleus-nucleus collision is decomposed into nucleon-nucleon collisions. The nucleons in a nucleus is distributed randomly according to Wood-Saxon distribution. A nucleon-nucleon collision is described by the Lowest-Leading-Order (LLO) pQCD parton-parton hard interactions with parton distribution function in a nucleon and by the soft interactions considered empirically, that is so called “multiple mini-jet production” in HIJING model [@wang1]. However, in PACIAE model that is performed by PYTHIA model [@sjo1] with string fragmentation switched-off. Therefore, the consequence of nucleus-nucleus collision is a configuration of $q$ ($\bar q$), diquark (anti-diquark), and $g$, besides the spectator nucleons and beam remnants [@sjo1]. The diquark (anti- diquark) is forced to split into $qq$ ($\bar q\bar q$) randomly.
So far we have introduced the partonic initialization of nucleus-nucleus collision in PACIAE model, what follows is then parton evolution (scattering). To the end, the 2$\to$2 LLO pQCD differential cross section [@comb] is used. Of course, that must be regularized first by introducing the color screen mass. The total cross section of parton $i$ bombarding with $j$ could then be calculated via a integral over the squared momentum transfer in a subprocess $ij\to kl$ and a summation over partons $k$ and $l$. With above differential and total cross sections the parton scattering can be simulated by Monte Carlo method. As of now, only 2$\to$2 processes are involved, among them there are six elastic and three inelastic processes [@comb].
As for the hadronization we first assume that the partons begin to hadronize when the interactions among them have been ceased (freeze-out). They could hadronize by either fragmentation model [@ff1; @and1] or coalescence model [@biro1; @csiz]. What the fragmentation models included here are the Field-Feynman model, i. e. Independent Fragmentation (IF) model [@ff1] and Lund string fragmentation model [@and1]. However, the program built in [@sjo1] is employed for the implementation of fragmentation model. On the contrary, we do write a program for coalescence model ourselves.
The hadron evolution (hadronic rescattering) is modeled as usual two body collisions and is copied directly from a hadron and string cascade model LUCIAE [@sa3]. There is no need to say more about the hadronic rescattering referring to [@sa3] if necessary.
Since we are not aimed to reproduce the experimental date but to study the physics of net charge transfer fluctuation, we do not adjust model parameters at all. In the calculations the IF model [@ff1] is adopted for hadronization and the net charge transfer fluctuation is counted in the interval of $|\eta|<$1. The simulated results by default PACIAE model are indicated with “HM w/ QGM” (HM with QGP assumption), since the hadronic final state is evolved from partonic initial state. If the simulation is ended up at the stage of partonic scattering and the net charge transfer fluctuation is counted over partons only, the results will be referred to as “QGM”. In that calculation it is assumed that the gluon does not contribute to the net charge but it does contribution to charge multiplicity by 2/3 as assumed in [@jeon4; @sa4]. If the simulation is ended up at the stage of partonic scattering and both the partons and beam remnants (hadrons) are counted in net charge transfer fluctuation the corresponding results are then symboled as “QGM w/ remnant”. It should be mentioned here that the spectator nucleons do not affect the net charge transfer fluctuation in $|\eta|<$1. A calculation where the string fragmentation in PYTHIA is switched-on and followed directly by the hadronic rescattering is referred to as “HM”, since in this simulation only hadronic transport is taken into account, like in HIJING, RQMD, UrQMD, and JPCIAE [@sa5].
Fig. \[ctrfl1\] gives the observable of net charge transfer fluctuation, $\kappa$, as a function of pseudorapidity, $\eta$, in the simulations of “HM”, “HM w/ QGM”, and “QGM” (solid circles, squares, and triangles, respectively) in 0-5 % most central $Au+Au$ collisions at $\sqrt{s_{NN}}$=200 GeV. One sees in this figure that the $\kappa$ of “HM w/ QGM” reproduces nicely the STAR datum and the $\kappa$ of “HM” is a bit larger than the STAR datum at $\eta$=1. The trend of $\kappa$ varying with $\eta$, both in “HM” and “HM w/ QGM”, is similar to the ones in HIJING, RQMD, UrQMD (cf. Fig. 5 in [@jeon2]). On the contrary, the $\kappa$ of “QGM” keeps nearly constant, like the charge fluctuation as function of rapidity interval in QGM in thermal model [@jeon3] and in transport model [@sa4]. It is interesting to see that the $\kappa$ in “HM” is larger than $\kappa$ in “QGM” by a factor of 3 to 5 from upper $\eta$ to the lower $\eta$. The discrepancy between $\kappa$ in “HM” and in “HM w/ QGM ” amounts $\sim$ 20% in the average, that means that the probability of net charge transfer fluctuation in QGM surviving hadronization can be estimated to be $\sim$ 20% either.
In Fig. \[ctrfl2\] the centrality dependence of $\kappa(\eta)$ is given both for “HM” and “QGM” (solid and open symbols, respectively) for 0-5, 30-40, and 70-80% central $Au+Au$ collisions at $\sqrt{s_{NN}}$=200 GeV. Both of the $\kappa(\eta)$ in “HM” and in “QGM” do not strongly depend on the centrality which is consistent with the results in hadronic transport models HIJING, RQMD, and UrQMD (cf. Fig. 5 in [@jeon2]).
We compare the $\kappa(\eta)$ in “QGM” to the one in “QGM w/ remnant” in Fig. \[ctrfl3\] for 0-5% most central $Au+Au$ collisions at $\sqrt{s_{NN}}$=200 GeV. One sees in this figure that the influence of beam remnants upon the $\kappa(\eta)$ in “QGM” amounts $\sim$35% in average. This influence does not change the status of big difference between $\kappa$ in “HM” and in “QGM” shown in Fig. \[ctrfl1\].
In summary, a parton and hadron cascade model, PACIAE, is employed investigating the net charge transfer fluctuation within $|\eta|$=1 both in partonic initial state and in hadronic final state for a range centralities of $Au+Au$ collisions at $\sqrt{s_{NN
}}$=200 GeV. In the hadronic final state ($\kappa$ in “HM”) the observable of net charge transfer fluctuation, $\kappa$, turns out to be nearly a factor of 3 to 5 larger than the $\kappa$ in partonic initial state ($\kappa$ in “QGM”). However, the $\kappa$ in “QGM” is hard to survive the hadronization, the survival probability amounts $\sim$ twenty percent.
Finally, the financial support from NSFC (10475032) in China are acknowledged.
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[^1]: E-mail: sabh@iris.ciae.ac.cn
|
---
abstract: 'Recently, a quantum anomalous Hall insulator (QAHI)/superconductor heterostructure has been realized and shows half-quantized conductance plateaus in two-terminal conductance measurements \[Q. L. He *et al.*, Science [**357**]{}, 294 (2017)\]. The half-quantized conductance plateaus are considered as a solid evidence of chiral Majorana edge modes. However, there is a strong debate over the origin of the half-quantized conductance plateaus. In this work, we propose a Josephson junction based on the QAHI/superconductor heterostructure to identify the existence of chiral Majorana edge modes. We find that the critical Josephson current dramatically increases to a peak value when a half-quantized conductance plateau $\sigma_{12}=e^2/2h$ is showing up for the $N=1$ chiral topological superconductor phase with a single chiral Majorana mode. Furthermore, we show that the critical Josephson current of the $N=1$ chiral topological superconductor exhibits an $h/e$-period oscillation and is robust to disorder, in contrast to the behaviors of conventional two-dimensional electron gas systems. We also estimate experimentally relevant parameters and believe that the supercurrent can be observed in experiments.'
author:
- 'Chui-Zhen Chen'
- James Jun He
- 'Dong-Hui Xu'
- 'K. T. Law'
title: 'Emergent Josephson current of $N=1$ chiral topological superconductor in quantum anomalous Hall insulator/superconductor heterostructures'
---
[^1]
[*Introduction*]{}— A Majorana fermion is its own antiparticle [@Wilczek; @Kitaev1] and can appear as quasiparticle excitations in condensed matter physics. The search for these exotic quasiparticle excitations has been one of the central subjects in condensed matter physics, for the reason that they obey non-Abelian statistics and have potential applications in quantum computations [@RG; @Ivanov; @Fujimoto; @STF; @Alicea2; @Kitaev2; @Nayak; @HasanREV; @QiREV; @FlensbergREV; @BeenakkerREV]. It is predicted that superconductors or superfluids with $p_x+i p_y$ paring symmetry and the $\nu=5/2$ fractional quantum Hall state can host Majorana fermions [@Nayak; @Stone2006; @Gurarie2005; @RG; @Moore1991]. Majorana fermion can also exit when an $s$-wave superconductor is attached to topological insulators or semiconductors with spin-orbit coupling [@Sau; @ORV; @Fu08; @Alicea2010; @ZhouTong; @Jia2016; @Kouwenhoven].
![(Color online). Schematic plot of a quantum anomalous Hall insulator (QAHI) with $C=1$ and $C=0$ under an $s$-wave superconductor as a topological superconductor with $N=2$, $N=1$ and $N=0$ Majorana modes (black and solid lines), respectively. (a)-(c) The Josephson junction of QAHI/superconductor heterostructure with $N=2$, $N=1$ and $N=0$, respectively. For the $N=1$ phase, the Josephson current can be carried by a chiral Majorana mode \[black line in (b)\], which can exist between two chiral topological superconductors, while there are no Andreev bound states in the junction for (a) both $N=2$ and (c) $N=0$ phases. As a result, the critical supercurrent will dramatically increase to a peak value when QAHI/superconductor heterostructure sweeps from $N=2$ to $N=0$ phases. (d)-(f) Two-terminal conductance measurement of QAHI/superconductor heterostructure with $N=2$, $N=1$ and $N=0$, respectively. When the QAH/SC heterostructure is in the $N=1$ phase, an incoming fermionic mode from lead $1$ \[blue cuboid in (e)\] is split into two chiral Majorana modes \[red and black lines in (e)\], giving rise to the half-quantized conductance plateau observed in experiment [@Qinglin]. \[fig1\] ](Fig1.pdf){width="3.3in"}
Quantum anomalous Hall effect was proposed as quantum Hall effect without Landau levels by F. D. Haldane in 1988 [@Haldane]. Recently, the quantum anomalous Hall effect was observed [@Chang2013] in Cr-doped (Bi,Sb)$_2$Te$_3$ thin films experimentally as theoretically predicted previously [@Dai]. When a quantum anomalous Hall insulator (QAHI) with Chern number $C=1$ in proximity to an $s$-wave superconductor, the system is topologically equivalent to a chiral topological superconductor (CTSC) with Chern number $N=2$, assuming that the induced superconducting paring is infinitesimal. For a finite superconducting gap, by tuning chemical potential or external magnetic field, a new $N = 1$ CTSC phase with a single chiral Majorana edge mode (CMEM) emerges when one of the two CMEMs in the $N = 2$ CTSC is annihilated [@Qi_QAHSC]. Later, theoretical studies showed that the $N = 1$ CTSC can give rise to a half-quantized longitudinal conductance plateau [@Chung; @WangJing].
Remarkably, half-quantized conductance plateaus (HQCPs), i.e. $\sigma_{12}=e^2/2h$, were observed by the two-terminal conductance measurements in a very recent experiment [@Qinglin]. The experimental setup consists of a superconductor island (Nb) on the top of QAHI (Cr-doped (Bi,Sb)$_2$Te$_3$ thin film) as shown in Fig. \[fig1\](e) [@Qinglin]. These HQCPs are regarded as a hallmark of the existence CTSCs with a single CMEM. However, there is a strong debate over the origin of the HQCPs [@Ji2017; @Huang2017; @Lian2017; @Law2017; @Yu-Hang2018], because they can also possibly be attributed to trivial reasons. For example, if Nb in the middle part is in a metallic phase instead of a superconducting phase, the system becomes a QAHI-metal-QAHI junction. Because each of the two QAHIs contributes a quantized conductance of $e^2/h$, the conductance of the QAHI-metal-QAHI junction has a half-quantized value $e^2/2h$. As a result, other evidences to verify the existence of CTSC with a single CMEM is desirable.
In this work, we propose to measure the Josephson current in a superconductor-QAHI-superconductor junction to identify the existence of CMEMs. The Josephson junction consists of a QAHI coupled to two $s$-wave superconductors on the top of two ends \[see Figs. \[fig1\](a)-(c)\]. When the QAHI/superconductor heterostructure is in the $N=2$ phase in Fig. \[fig1\](a), two CMEMs are not well spatially separated and the Josephson junction is equivalently connected by a chiral fermionic mode. The Josephson current is negligible, since there are no Andreev bound states between two superconductors. On the contrary, if the QAHI/superconductor heterostructure is in the $N=1$ CTSC phase in Fig. \[fig1\](b), the chiral fermionic mode in QAHI is spatially separated into two CMEMs at the interfaces of QAHI and CTSC \[see solid black and red arrows in Fig. \[fig1\](b)\]. In this case, the Josephson current $I_c \sim e/hE_T$ is carried by the CMEM \[black line in Fig. \[fig1\](b)\] and decays as $1/L$, with the Thouless energy $E_T\propto1/L$ and the circumference $L$. For comparison, a similar setup measuring two-terminal conductance of a QAHI/superconductor heterostructure is shown in Figs. \[fig1\](d)-(f) where an $s$-wave superconductor is placed on the top of the central region of a QAHI. HQCPs emerge in this setup when $N=\pm1$ CTSC phases are realized in the central region \[see Fig. \[fig1\](e)\].
In the following, we first introduce the model Hamiltonian for the QAHI/superconductor heterostructure, and then calculate the Josephson current as well as the two-terminal conductance by using the recursive Green’s function method. Importantly, we find two critical Josephson current peaks when the two HQCPs show up at $N=\pm1$ CTSC phases, respectively. This provides important transport features to identify the existence of a single CMEM. At last, we show that the critical Josephson current exhibits an $h/e$-period oscillation and is robust to disorder, in contrast to the behaviors of conventional two-dimensional electron gas systems. We also estimate experimentally relevant parameters and believe that the Josephson current can be observed experimentally.
[*Model Hamiltonians*]{}— The effective Hamiltonian of Cr-doped (Bi,Sb)$_2$Te$_3$ thin film can be written as [@Dai; @WangJing3] $$\begin{aligned}
\!\!\!\!\!\!\!\!\!\!\!
H \!&=&\!\hbar v_F (k_y\sigma_x\tau_z \!-\!k_x \sigma_y\tau_z) \!+\! m({\bf k}) \tau_x \!+\! M'_z \sigma_z \!+\! V({\bf r})\end{aligned}$$ in the basis of a four-component electron operator $\Psi_{\bf k}=[\psi_{{\bf k}t \uparrow}, \psi_{{\bf k}t \downarrow},\psi_{{\bf k}b \uparrow},\psi_{{\bf k}b\downarrow}]$ with the momentum ${\bf k}$. Here $t$ ($b$) denotes the top (bottom) layer of topological insulator surfaces and $\uparrow$ ($\downarrow$) is the spin index. The Pauli matrices $\sigma_{x,y,z}$ and $\tau_{x,z}$ are defined in spin and layer spaces, respectively. The onsite disorder is $V({\bf r})=diag\{V_{t,\uparrow},V_{t,\downarrow},V_{b,\uparrow},V_{b,\downarrow}\}$ with $V_{t/b,\uparrow/\downarrow}$ uniformly distributed in $[-W/2,W/2]$ and the disorder strength $W$. $M'_z = M_z + M_0$ represents the spin splitting in the $z$ direction [@WangJing3], where $M_0$ is the spin splitting without magnetic field $B$ and $M_z=\mu_M B$ with a proportionality constant $\mu_M$. $m({\bf k}) = m_0 - m_1 k^2$ describes the effective coupling between the top layer and the bottom layer. In the clean limit, the system is a QAHI with Chern number $C=sgn(M'_z)$ if $|M'_z|>|m_0|$, while it becomes a normal insulator with zero Chern number when $|M'_z|<|m_0|$. In our numerical simulations, we discretize the model Hamiltonian into a tight-binding model and add the Peirls substitution $\phi_{n,m}=\frac{e}{\hbar}\int^{n}_{m}{\bf A\cdot}d{\bf r}$ to the hopping term between ${\bf r}_n$ and ${\bf r}_m$ lattice sites. We choose Landau gauge for the vector potential ${\bf A}=(-By,0)$ with the coordinate $y$. The Fermi velocity is $v_F=1$, $m_1=1$, $m_0 = -0.03$, $M_0=0.17$ and $\mu_M= 10^{4}$.
![(Color online). (a) A comparative plot of the critical Josephson current $I_c$ and two-terminal conductance $\sigma_{12}$ versus the magnetization $M_z$. During the magnetic field scanning, the QAHI/supercondutor heterostructure shows a various of topological phases with Chern number $N$. It is oblivious that there is a $I_c$ peak for each of the two half quantized plateaus $\sigma_{12}=e^2/2h$, originating from $N=\pm1$ chiral topological superconductor phases, respectively. (b) The Josephson current distribution for $N=1$ chiral topological superconductor with $M_z=0.07$ and size $W\times L=160\times320$. \[fig2\] ](Fig2.pdf){width="3.5in"}
In proximity to an $s$-wave superconductor, the Bogoliubov-de Gennes (BdG) Hamiltonian of the QAHI/superconductor heterostructure is given by $$\begin{aligned}
H_{\text{BdG}} &=& \left(
\begin{array}{cc}
H(\mathbf{k})-\mu & \Delta \\
\Delta^{\dagger} & -H^{\ast}(\mathbf{-k}) +\mu \\
\end{array}
\right)
\\
\Delta &=& \left(
\begin{array}{cc}
\Delta_{t}i\sigma_y & 0 \\
0 & \Delta_{b}i\sigma_y \\
\end{array}
\right) \exp(i\Phi_s). \nonumber\end{aligned}$$ in the basis of $\Phi_{\bf k} = [\Psi_{\bf k},\Psi^{\dagger}_{\bf -k}]^{T}$. Here $\Delta_{t}=0.12$ and $\Delta_{b}=0$ are the induced pairing potentials on the top and bottom layers, respectively. $\Phi_s$ is the phase factor of $\Delta$ and the chemical potential $\mu=0$ .
[*Emergent Josephson current in chiral TSC*]{}— In proximity to an $s$-wave superconductor, a $C=1$ QAH state can regarded as an $N=2$ CTSC state and it can show two new phases ($N=\pm1$) with a single CMEM by tuning the magnetization $M_z$, as shown by the green color region in Fig. \[fig2\](a). When the condition $(-\sqrt{\Delta_t^2+4m_0^2}-\Delta_t)/2<M'_z<(-\sqrt{\Delta_t^2+4m_0^2}+\Delta_t$)/2 is satisfied [@Qi_QAHSC; @Chung; @WangJing], the system enters the $N=1$ CTSC phase. Since a $C=1$ QAHI is topologically equivalent to an $N=2$ CTSC phase, there must be a CMEM at the boundary between the QAHI and the $N=1$ CTSC as depicted in Fig. \[fig1\](b). As a result, when a chiral fermionic mode in the QAH is injecting into the $N=1$ CTSC, it splits into two branches of CMEMs. One branch of the CMEMs tunneling into the $N=1$ CTSC, the other branch of the CMEMs is trapped as Andreev bound states between the two superconductors \[see black line in Fig. \[fig1\](b)\]. These Andreev bound states will give rise to Josephson current when two superconductors phases are different. Similarly, in Fig. \[fig1\](e), a chiral fermionic mode splits into two CMEMs at the boundary between the QAHI and the $N=1$ CTSC. One CMEM tunnels through the $N=1$ CTSC phase while the other is reflected, giving rise to a HQCP [@Chung; @WangJing].
Next, we numerically evaluated the Josephson current and the two-terminal conductance by the recursive Green’s function method [@Datta1996; @Furusaki; @PatrickGF]. The main results are shown in Fig. \[fig2\](a). We can see that the critical Josephson current $I_c$ has a peak value for each of two HQCPs $\sigma_{12}=e^2/2h$. This is the central conclusion of this work. The Josephson current is concentrated on the two edges of the sample when the system is in the $N=1$ CTSC phase with $M_z = 0.07$ \[see Fig. \[fig2\](b)\]. For the $N=2$ and $N=0$ CTSCs, because there are no Andreev bound states between two superconductors, the critical Josephson current is almost zero. On the other hand, two $N=1$ superconductors will trap Andreev bound states in the QAH region, resulting in critical Josephson current $I_c$ peak we discussed above. Furthermore, we find that the critical Josephson current $I_c$ peaks are strongly oscillated, due to periodic change of the magnetic flux in the QAH region.
![(Color online). The critical Josephson current $I_{c}$ with a period of $h/e$ versus the magnetic flux $\Phi[h/2e]$ for different magnetizations (a) $M_z=0.05$ to $0.07$ at the temperature $T=\Delta/500$; and for different temperatures (b) $T=\Delta/150$ to $\Delta/2000$ at $M_z=0.07$. \[fig3\] ](Fig3.pdf){width="3.3in"}
Now let’s come to investigate the oscillation of the Josephson current in the $N=1$ CTSC phase. In Fig. \[fig3\], the critical Josephson current $I_c$ oscillates with a period of $h/e$ as a function of the magnetic flux $\Phi$ for different magnetizations $M_z$ [@note1]. This is consistent with previous results of the three-dimensional topological insulator based Josephson junction [@Shapiro]. Since the chiral edge mode has a $\pi$ Berry phase, the $I_c$ has a peak value at $\Phi=h/2e$ instead of $\Phi=0$. Therefore, the $h/e$-period oscillation of Josephson current is the key feature of the one-dimensional chiral edge mode [@Zyuzin; @Ma; @Shapiro; @Liu2017], which was also discovered in the quantum Hall insulator based Josephson junction [@Zyuzin; @Ma; @Liu2017], and is very distinct from the behaviors of conventional two-dimensional electron gas systems. Moreover, upon decreasing the temperature from $T=\Delta/200$ to $T=\Delta/2000$, we find that the magnitude of current oscillation becomes more pronounced and hardly varies with temperature after $T=\Delta/1000=1.2\times10^{-4}$, because the Thouless energy $E_T\approx10^{-3}\gg T$.
![(Color online). (a) Plots of the disorder-averaged critical Josephson current $\bar{I}_{c}$ and its fluctuation $\delta I_{c}$ (dash lines) as a function of the magnetic phase flux $\Phi[h/2e]$ at $M_z=0.07$ and the temperature $T=\Delta/500$. (b) The critical Josephson current $I_{c}$ versus the magnetization $M_z$ for various disorder strength. Other model parameters are the same as Fig. \[fig2\](a). We average over 100 disorder configurations in (a) and one configuration in (b). \[fig4\] ](Fig4.pdf){width="3.5in"}
Since the QAHI is achieved by doping topological insulator thin films with magnetic impurities, we study disorder effects on the critical Josephson current in the following. We plot the disorder-averaged critical Josephson current $\bar{I}_c$ in Fig. \[fig4\]. In Fig. \[fig4\].(a), $\bar{I}_c$ remains stable when $W=0.1$ due to the fact that the chiral edge mode is robust to moderate disorder. We can see that the current fluctuation $\delta I_c$ becomes comparable to the amplitude of $h/e$ periodic oscillation for $W=0.5$ \[see Fig. \[fig4\](a)\]. This means that the $h/e$-period oscillation starts to be destroyed, because the chiral propagating edge mode in QAH region can be scattered to the opposite side assisted by the disorder induced bulk states. However, the critical Josephson current $I_c$ peak is very sustainable and clear in Fig. \[fig4\](b) even at $W=0.5$, as long as the Andreev bound states remain between two superconductors. Therefore, we conclude that the critical Josephson current $I_c$ peak is more stable than $h/e$-period oscillation, and they are both robust to the moderate disorder because the chiral edge states are topologically protected.
The critical Josephson current can be estimated by $I_c\approx \frac{e\Gamma^2}{\hbar} E_T=\frac{e\Gamma^2}{\hbar} \frac{\hbar v_F}{2L}$ with the Thouless energy $E_T$ , interface transparency $\Gamma$, the Fermi velocity $v_F$ and the circumference $L$ of the CMEM trapped between two superconductors [@Bardeen1972]. When $L=1 -2\mu m$, the critical Josephson current is $I_c=50 - 100$nA with $v_F=5\times10^5 m/s$ [@Xue2010] and $\Gamma=0.5$. Therefore, we believe that the critical Josephson current $I_c$ peak can be observed in experiments.
[*Discussion and Conclusion*]{}— In the above analysis, we show that the critical Josephson current $I_c$ has a robust peak value for each of the two HQCPs $\sigma_{12}=e^2/2h$ due to the existence of the single CMEM in the $N=\pm1$ CTSCs. If the $N=1$ CTSC in Fig. \[fig1\](e) is so strongly disordered that it becomes a gapless Majorana metal, and the gapless Majorana metal can still lead to a HQCP [@Huang2017]. However, in this circumstance, the Josephson junction in Fig. \[fig1\](b) can not sustain a Josephson current peak. That’s because the Andreev bound states cannot be trapped between two gapless Majorana metals and thus the Josephson current is extremely small and strongly fluctuated. In conclusion, we find that a comparative study of the Josephson current and the two-terminal conductance can provide a more reliable evidences for the existence of single CMEM in the QAHI/superconductor heterostructure.
After submitting this paper to Physical Review B in July 2018, we noticed a study of a similar experimental setup in arXiv:1810.01891 [@Chang-An2018].
[*Acknowledgement.*]{}— We thank Kang L. Wang, Qing Lin He and Jie Liu for illuminating discussions. The authors acknowledge the support of HKRGC through C6026-16W, the Croucher Foundation and the Tai-chin Lo Foundation. D.-H.X. is supported by the National Natural Science Foundation of China (Grant No. 11704106), the Scientific Research Project of Education Department of Hubei Province (Grant No. Q20171005) and the Chutian Scholars Program in Hubei Province.
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[^1]: phlaw@ust.hk
|
---
abstract: '[**High-temperature superconductivity in the iron-based materials emerges from, or sometimes coexists with, their metallic or insulating parent compound states. This is surprising since these undoped states display dramatically different antiferromagnetic (AF) spin arrangements and N$\rm \acute{e}$el temperatures. Although there is general consensus that magnetic interactions are important for superconductivity, much is still unknown concerning the microscopic origin of the magnetic states. In this review, progress in this area is summarized, focusing on recent experimental and theoretical results and discussing their microscopic implications. It is concluded that the parent compounds are in a state that is more complex than implied by a simple Fermi surface nesting scenario, and a dual description including both itinerant and localized degrees of freedom is needed to properly describe these fascinating materials.**]{}'
author:
- Pengcheng Dai
- Jiangping Hu
- Elbio Dagotto
title: 'Magnetism and its microscopic origin in iron-based high-temperature superconductors'
---
Introduction
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Soon after the discovery of high critical temperature (high-$T_c$) superconductivity in copper oxides [@bednorz], neutron scattering studies revealed that the parent compounds of these superconductors have an antiferromagnetic (AF) ground state with a simple collinear spin structure (Fig. 1a) [@vaknin; @tranquada88]. Because the associated AF spin fluctuations may be responsible for electron pairing and superconductivity [@scalapino; @dagotto; @palee], over the past 25 years a tremendous effort has focused on characterizing the interplay between magnetism and superconductivity in these materials [@fujita]. In the undoped state, the parent compounds of copper oxide superconductors are Mott insulators and have exactly one valence fermion with spin 1/2 for each copper atom, leading to robust electronic correlations and localized magnetic moments [@dagotto; @palee]. Superconductivity emerges after introducing charge carriers that suppress the static AF order. Although the strong Coulomb repulsion in the parent compounds is screened by the doped charge carriers, the electronic correlations are certainly important for the physics of the doped cuprates, particularly in the underdoped regime [@palee].
Consider now the iron-based superconductors [@johnston; @stewart; @paglione10]. Several parent compounds of these materials, such as LaFeAsO, BaFe$_2$As$_2$, NaFeAs, and FeTe, are not insulators but semimetals [@kamihara; @rotter; @cwchu; @mawkuen2]. In these cases, electronic band structure calculations have revealed that their Fermi Surfaces (FS) are composed of nearly cylindrical hole and electron pockets at the $\Gamma(0,0)$ and $M(1,0)/M(0,1)$ points, respectively [@mazin2011n; @hirschfeld]. The high density of states resulting from the extended momentum space with nearly parallel FS between the hole and electron pockets leads to an enhancement of the particle-hole susceptibility. This suggests that FS nesting among those pockets could induce spin-density-wave (SDW) order at the in-plane AF wavevector ${\bf Q}_{AF} = (1,0)$ with a collinear spin structure (Fig. 1b) [@dong], much like the FS-nesting induced SDW in pure chromium [@fawcett]. Neutron scattering experiments on LaFeAsO [@cruz], BaFe$_2$As$_2$ [@qhuang], and NaFeAs [@slli09] have reported results compatible with the theoretically predicted AF spin structure, albeit with an ordered magnetic moment smaller than expected from first-principles calculations [@mazin08]. In addition, quasiparticle excitations between the hole and electron FS can induce $s^{\pm}$-wave superconductivity [@mazin2011n; @hirschfeld; @kuroki08; @chubukov; @fwang09]. One of the consequences of this superconducting state is that the imaginary part of the dynamic susceptibility, $\chi^{\prime\prime}(Q,\omega)$ should have a sharp peak, termed spin resonance in copper oxide superconductors [@eschrig], at ${\bf Q}_{AF} = (1,0)$ below $T_c$ [@maier; @korshunov]. This prediction is also confirmed by inelastic neutron scattering (INS) experiments in iron-based superconductors such as hole-doped Ba$_{1-x}$K$_x$Fe$_2$As$_2$ [@christianson; @chenglinzhang; @castellan], electron-doped BaFe$_{2-x}T_x$As$_2$ ($T=$Co, Ni) [@lumsden; @schi09; @dsinosov09; @jtpark; @clester; @hfli; @hqluo12], and FeTe$_{1-x}$Se$_x$ [@hamook; @qiu09; @lumsden2]. Finally, angle resolved photoemission spectroscopy (ARPES) experiments find that the general characterization of the FS and the superconducting order parameter are consistent with the band structure calculations and with isotropic $s$-wave superconducting gaps [@prichard]. Therefore, at first sight it may appear that antiferromagnetism in the iron-based materials originates from FS nesting of itinerant electrons, superconductivity must have a $s^{\pm}$-wave symmetry for related reasons, and electron correlations or local moments do not play an important role for magnetism and superconductivity [@mazin2011n].
However, although the parent compounds of iron pnictide superconductors have metallic ground states consistent with band structure calculations, there are reasons to believe that electron correlations could be sufficiently strong to produce an “incipient” Mott physics [@si2008; @qmsi09], where local moments are as important as itinerant electrons for magnetic, transport, and superconducting properties in these materials [@cfang; @cxu]. In fact, the $s^{\pm}$ pairing symmetry is also naturally derived in multi-orbital $t-J$-type models [@seo2008; @Fang2011] and recent diagonalization calculations [@nicholson11] have shown that the AF state, as well as the $A_{1g}$ $s$-wave pairing state, evolve smoothly from weak to strong coupling, suggesting that the physics of the pnictides could also be rationalized based on short length scale concepts not rooted in weak-coupling nesting. After all, in the context of the copper oxide superconductors, weak coupling studies of the one-orbital Hubbard model also led to the correct checkerboard AF state and $d$-wave pairing, showing that these problems can be attacked from a variety of view points. In addition, the newly discovered $A_y$Fe$_{2-x}$Se$_2$ ($A=$ K, Rb, Cs, Tl) iron-chalcogenide superconductors [@jgguo; @mhfang] do [*not*]{} exhibit hole pockets [@Wang_122Se; @Zhang_122Se; @Mou_122Se], but have strong AF ordered insulating phases with extremely high Néel transition temperatures [@wbao1; @fye]. Such a strong magnetism and high superconducting transition temperature ($T_c\approx 33$ K) cannot be explained by FS nesting since this is based on the enhancement of the particle-hole susceptibility due to an extended momentum space with nearly parallel Fermi surfaces, i.e. it applies only to particle and hole FS’s and not to purely electronic Fermi pockets.
Since iron-based superconductors have six electrons occupying the nearly degenerate $3d$ Fe orbitals, the system is intrinsically multi-orbital and therefore it is technically difficult to define and study a simple microscopic Hamiltonian to describe the electronic properties of these materials and characterize the strength of the electronic correlations. From optical conductivity measurements [@qazilbash09], it has been argued that electronic correlations in Fe pnictides are weaker than in underdoped copper-oxides but are stronger than those of Fermi liquid metals, contrary to the conclusion based on local density approximation calculations [@mazin2011n]. Therefore, it is important to determine whether magnetism in Fe-based materials arises from weakly correlated itinerant electrons [@mazin2011n], as in the case of the SDW in chromium [@fawcett], or whether it requires some degree of electron correlations [@HG], or if magnetism is dominated by the contributions of quasi-localized moments induced by incoherent electronic excitations [@qmsi09] such as in the AF insulating state of Cu oxides [@palee].
In this review, recent experimental and theoretical progress in the study of iron-based superconductors is summarized, with focus on the undoped parent compounds. In section II, the magnetically ordered states in nonsuperconducting iron pnictides, iron chalcogenides, and iron selenides are discussed. Section III describes the effect of electron and hole doping on static AF order and their associated spin excitations. In section IV, we provide several examples where deviations from the simple SDW FS nesting picture are prominent. Finally in section V, we present our perspective on the importance of electron correlations in these materials.
Magnetic order arrangements and spin waves in the parent compounds
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Although the overall crystal structures and chemical formulas of the copper-oxide superconductors can be quite different, their parent compounds are all AF Mott insulators characterized by the Cu spin structure shown in Fig. 1a, where the tetragonal or pseudo-tetragonal unit cells have a nearest-neighbor Cu-Cu spacing with $a\approx b\approx 3.8$ Å. In the notation of reciprocal lattice units (rlu) $(2\pi/a,2\pi/b,2\pi/c)$, the AF Bragg peaks occur at the in-plane ordering wave vectors ${\bf Q}_{AF}=(\pm 1/2+m,\pm 1/2+n)$, where $m,n=0,\pm 1, \pm 2, \cdots$ rlu, shown as red circles in Fig. 1e [@vaknin; @tranquada88]. Time-of-flight INS experiments [@coldea; @headings] have mapped out spin waves of the insulating La$_2$CuO$_4$ throughout the Brillouin zone and found no evidence for spin-wave broadening at high energies. The dispersions of spin waves are well described by a Heisenberg Hamiltonian with nearest-neighbor (NN) exchange coupling $J_1=111.8\pm 4$ meV and next-nearest-neighbor (NNN) exchange $J_2=-11.4\pm 3$ meV [@coldea]. Therefore, the dominant magnetic exchange coupling in La$_2$CuO$_4$ is the NN magnetic interaction and the higher-order interactions amount to only $\sim$10% of the total magnetic energy with a bandwidth of $\sim$320 meV (Fig. 1i).
![Antiferromagnetic structure and spin-wave dispersions for the insulating copper oxide La$_2$CuO$_4$ and the parent compounds of iron-based superconductors BaFe$_2$As$_2$, FeTe, and $A_y$Fe$_{1.6+x}$Se$_2$. The chemical unit cells are marked light green. The dark and light brown As/Te/Se atoms indicate their vertical positions above and below the Fe-layer, respectively. (a) The AF structure of La$_2$CuO$_4$, where the chemical unit cell is marked light green. (b) The collinear AF structure of nonsuperconducting iron pnictides in the FeAs-layer, where spins are aligned anti-parallel along the orthorhombic $a_o$-axis [@cruz; @qhuang; @slli09]. (c) The bi-collinear AF structure of FeTe [@webao08; @slli08]. (d) The block AF order of the insulating $A_y$Fe$_{1.6+x}$Se$_2$, where the $\sqrt{5}\times\sqrt{5}$ superlattice structure is marked by solid line with lattice parameter $a_s=8.663$ Å and the orthorhombic lattice cell is shaded green [@wbao1; @fye]. The iron vacancies are marked as yellow squares. (e) The wave vector dependence of the AF order in the $(H,K)$ plane of the reciprocal space for La$_2$CuO$_4$ [@coldea]; (f) BaFe$_2$As$_2$ [@qhuang]; (g) Fe$_{1.05}$Te [@webao08; @slli08]; and (h) the insulating $A_y$Fe$_{2-x}$Se$_2$ [@wbao1; @fye]. (i) Spin-wave dispersions along two high symmetry directions for La$_2$CuO$_4$ [@coldea]. The overall energy scale of spin waves for copper oxides is about $320$ meV and spin waves are instrumental resolution limited. (j) Spin-wave dispersions for BaFe$_2$As$_2$ and they broad considerably for energies above $\sim$100 meV [@lharriger]. (k) Spin-wave dispersions for Fe$_{1.05}$Te, and spin waves are very broad for energies above 30 meV [@lipscombe]. (l) Spin waves for the insulating Rb$_{0.89}$Fe$_{1.58}$Se$_2$ [@mywang11]. In spite of dramatically different dispersions for various iron-based materials, their spin wave overall energy scales are similar and about 220 meV, less than that of the insulating copper oxides. Twinning is considered.](Fig1R2)
Four years after the initial discovery of superconductivity in LaFeAsO$_{1-x}$F$_x$ [@kamihara], there are now three major families of iron-based superconductors: the iron pnictides [@johnston; @stewart], iron chalcogenides [@mawkuen2; @mhfang08], and alkaline iron selenides [@jgguo; @mhfang]. The parent compounds of the pnictides, such as $A$FeAsO ($A=$ La, Ce, Sm, Pr, etc.), $A$Fe$_2$As$_2$ ($A=$ Ba, Sr, Ca), and NaFeAs, all have the same collinear AF structure as shown in Fig. 1b, with a small ordered moment ($<1\ \mu_B$/Fe) and N$\rm \acute{e}$el temperature $T_N\le 200$ K [@cruz; @qhuang; @cwchu]. The AF spin moments are aligned along the weak orthorhombic unit cell $a$-axis direction ($a\approx5.62$, $b\approx 5.57$ Å). In reciprocal space, the AF Bragg peaks occur at in-plane ordering wave vectors ${\bf Q}_{AF}=(\pm 1+m, n)$ and at ${\bf Q}_{AF}\approx (m, \pm 1+n)$ due to twinning (red circles in Fig. 1f), consistent with the $\Gamma(0,0)\leftrightarrow M(1,0)/M(0,1)$ FS nesting picture [@mazin2011n]. However, although the calculated FS of the chalcogenides Fe$_{1+y}$Te$_{1-x}$Se$_x$ is similar to that of iron pnictides [@subedi], surprisingly its parent compound Fe$_{1+y}$Te actually has a bi-collinear spin structure (Fig. 1c) [@webao08; @slli08]. Here, the AF Bragg peaks occur at ${\bf Q}_{AF}=(\pm 1/2+m,\pm 1/2+n)$ (Fig. 1g) in the pseudo-tetragonal notation ($a\approx b\approx 5.41$ Å), suggesting that FS nesting cannot induce such AF order. Finally, the parent compounds of the alkaline iron selenide $A$Fe$_{1.6+x}$Se$_2$ superconductors are insulators [@jgguo; @mhfang] and form a $\sqrt{5}\times\sqrt{5}$ block AF structure as shown in Fig. 1d with a large ordered moment ($\sim$$3\ \mu_B$/Fe) along the $c$-axis and $T_N\approx 500$ K [@wbao1; @fye]. In reciprocal space, defined using the pseudo-tetragonal unit cell of iron pnictides ($a\approx b\approx 5.41$ Å), the block AF Bragg peaks appear at ${\bf Q}_{AF}=(\pm 0.2+m,\pm 0.6+n)$ and $(\pm 0.6+m,\pm 0.2+n)$ combining left and right chiralities (red circles in Fig. 1h).
Since the parent compounds of iron-based superconductors can have different AF spin structures and either metallic or insulating ground states [@johnston; @stewart; @jgguo; @mhfang], the microscopic origin of the AF order cannot be induced by a simple FS nesting. If magnetism is relevant for high-$T_c$ superconductivity, then it would be important to determine magnetic exchange couplings for different classes of Fe-based superconductors and compare the results with those of the copper-oxides [@coldea]. For pnictides, INS experiments have mapped out spin waves on single crystals of CaFe$_2$As$_2$ [@diallo09; @jzhao], SrFe$_2$As$_2$ [@raewings], and BaFe$_2$As$_2$ [@lharriger] throughout the Brillouin zone. Although there are still debates concerning whether spin waves in these materials can be described by a pure itinerant picture [@diallo09; @raewings] or require local moments [@jzhao; @lharriger], the overall spin-wave energy scales are around 220 meV. Therefore, magnetic exchange couplings in iron pnictides are clearly smaller than those of copper oxides (Figs. 1i and 1j). Although spin waves are broadened at high energies, the spin-wave dispersion curves (Fig. 1j) can still be described by a Heisenberg Hamiltonian with strong anisotropic NN exchange couplings ($J_{1a}\gg J_{1b}$) and fairly large NNN exchange coupling ($J_2$) [@jzhao; @lharriger]. This large in-plane magnetic exchange coupling anisotropy has been interpreted as due to possible electronic nematic phase and/or orbital ordering [@jzhao; @lharriger]. Table I compares the effective magnetic exchange couplings of the Fe-based systems studied thus far against those of the insulating copper-oxide La$_2$CuO$_4$.
For the chalcogenides Fe$_{1+y}$Te, the commensurate bi-collinear AF spin structure in Fig. 1c becomes incommensurate for concentration $y>0.12$ [@rodriguez]. The overall spin-waves energy scale (Fig. 1k) is similar to those of the iron pnictides. Although the large static ordered moment of $\sim$$2\ \mu_B$/Fe in Fe$_{1+y}$Te [@webao08; @slli08] suggests that local moments may be important, spin waves are rather broad in energy and difficult to fit using a Heisenberg Hamiltonian with only NN and NNN exchange couplings [@lipscombe]. By including third-neighbor (NNNN) exchange couplings, a Heisenberg Hamiltonian can fit the spin-wave dispersion with an anisotropic ferromagnetic NN exchange couplings and strong AF NNN exchange coupling (Table I). In a separate INS experiment on Fe$_{1.1}$Te, the total integrated Fe magnetic moment was found to increase with increasing temperature from 10 K to 80 K [@zaliznyak]. These results suggest that in the temperature range relevant for superconductivity, there is a remarkable redistribution of the magnetism arising from both itinerant and localized electrons.
![The electronic phase diagrams and the evolution of FS’s, static AF order, and spin excitations upon electron or hole doping to BaFe$_2$As$_2$. (a) The AF and superconducting phase diagram for hole-doped Ba$_{1-x}$K$_x$Fe$_2$As$_2$. In the underdoped regime, there is a region of coexisting AF order and superconductivity [@ewiesenmayer11]. Incommensurate spin excitations appear for $x\ge 0.4$ [@castellan] and persist till $x=1$ at KFe$_2$As$_2$ [@chlee11]. (b) Phase diagram for electron-doped BaFe$_{2-x}$Ni$_x$As$_2$ [@hluo12]. The long commensurate AF order changes into short-range incommensurate AF order near $x=0.092$. The inset shows the transverse incommensurate AF order. Superconductivity in the electron-doped materials only extends to $x\approx 0.25$. (c) Schematics of FS’s correspond to 35% hole-doped BaFe$_2$As$_2$ [@chenglinzhang] with possible nesting vectors marked with arrows. The $d_{xz}$, $d_{yz}$, and $d_{xy}$ orbitals for different Fermi surfaces are colored as red, green, and blue, respectively. (d) FS’s of BaFe$_2$As$_2$ with orbital characters [@graser10]. (e) Fermi surfaces of 8% electron-doped BaFe$_2$As$_2$ [@chenglinzhang]. For all three cases, FS’s are plotted with zero wave vector transfers along the $c$-axis. (f) Longitudinally elongated spin excitations at $E=20$ meV seen in the optimally hole-doped Ba$_{0.67}$K$_{0.33}$Fe$_2$As$_2$ [@chenglinzhang]. (g) Transversely elongated spin waves at $E=20$ meV for BaFe$_2$As$_2$ [@lharriger]. (h) Transversely elongated spin excitations at $E=20$ meV for BaFe$_{1.9}$Ni$_{0.1}$As$_2$ [@jtpark; @msliu12]. (i) Energy dependence of $\chi^{\prime\prime}(\omega)$ for BaFe$_2$As$_2$ (blue solid line) and BaFe$_{1.9}$Ni$_{0.1}$As$_2$ below (red dashed line) and above (red solid circles) $T_c$ in absolute units of $\mu_B^2$eV$^{-1}$f.u.$^{-1}$. The sharp peak near $E\approx 8$ meV below $T_c$ is the neutron spin resonance coupled directly to superconductivity [@lumsden; @schi09; @dsinosov09; @jtpark; @clester; @hfli; @hqluo12]. ](Fig2R2)
In the case of insulating $A$Fe$_{1.6+x}$Se$_2$, spin waves have an acoustic mode and two optical modes separated by spin gaps (Fig. 1l) [@mywang11]. In contrast to iron pnictide $A$Fe$_2$As$_2$ [@diallo09; @jzhao; @raewings; @lharriger] and iron chalcogenide Fe$_{1+y}$Te [@lipscombe; @zaliznyak], spin waves in insulating $A$Fe$_{1.6+x}$Se$_2$ can be well-described by a Heisenberg Hamiltonian with NN, NNN, and NNNN exchange couplings [@mywang11]. Comparing effective exchange couplings for different iron-based materials (Table I), it is clear that the NN exchange couplings are quite different, but the NNN exchange couplings are AF and rather similar. In addition, spin waves for iron-based materials are much broader at high energies. This is different from the insulating copper oxides, where the NN exchange coupling dominates the magnetic interactions and spin waves are instrumental resolution limited throughout the Brillouin zone [@coldea; @headings]. These results suggest that itinerant electrons play a role in spin waves of metallic iron-based materials.
Parent compounds $T_N$ (K) $SJ_{1a}$ (meV) $SJ_{1b}$ (meV) $SJ_{2a}$ (meV) $SJ_{2b}$ (meV) $SJ_{3}$ (meV)
------------------------------------------------ ------------ ----------------- ----------------- ----------------- ----------------- ----------------
La$_2$CuO$_4$, Ref. [@coldea] $317\pm 3$ $111.8\pm4$ $111.8\pm4$ $-11.4\pm 3$ $-11.4\pm 3$ 0
CaFe$_2$As$_2$, Ref. [@jzhao] $\sim$170 $49.9\pm 9.9$ $-5.7\pm4.5$ $18.9\pm 3.4$ $18.9\pm 3.4$ 0
BaFe$_2$As$_2$, Ref. [@lharriger] $\sim$138 $59.2\pm2.0$ $-9.2\pm 1.2$ $13.6\pm 1$ $13.6\pm 1$ 0
Fe$_{1.05}$Te, Ref. [@lipscombe] $\sim$68 $-17.5\pm 5.7$ $-51.0\pm 3.4$ $21.7\pm 3.5$ $21.7\pm 3.5$ $6.8\pm 2.8$
Rb$_{0.89}$Fe$_{1.58}$Se$_2$, Ref. [@mywang11] $\sim$475 $-36\pm 2$ $15\pm 8$ $12\pm 2$ $16\pm 5$ $9\pm 5$
: \[tab:5/tc\] Comparison of effective magnetic exchange couplings for parent compounds of copper-based and iron-based superconductors obtained by fitting spin waves with a Heisenberg Hamiltonian with NN ($J_{1a}, J_{1b}$), NNN ($J_{2a},J_{2b}$), and NNNN ($J_3$). The N$\rm \acute{e}$el temperatures for different materials are also listed.
The effects of hole and electron doping on the magnetic correlations and excitations
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As discussed before [@johnston; @stewart; @paglione10], superconductivity in Fe-based materials can be induced via electron/hole doping, pressure, and isoelectronic substitution. Figures 2a and 2b show the electronic phase diagrams of hole and electron doping on BaFe$_2$As$_2$, respectively. In the undoped state, BaFe$_2$As$_2$ exhibits simultaneous structural and magnetic phase transitions below $\sim$138 K, changing from the high-temperature paramagnetic tetragonal phase to the low-temperature orthorhombic phase with the collinear AF structure (Fig. 1b) [@qhuang]. Upon electron-doping BaFe$_2$As$_2$ by partially replacing Fe by Co or Ni to form BaFe$_{2-x}T_x$As$_2$, the static AF order is suppressed and superconductivity emerges [@johnston; @stewart; @paglione10]. From systematic transport and magnetic measurements of single crystals [@nni08; @jhchu09], the phase diagram for BaFe$_{2-x}$Co$_x$As$_2$ was established, where the single structural/magnetic phase transition in BaFe$_2$As$_2$ splits with increasing Co-doping. Neutron diffraction experiments on BaFe$_{2-x}$Co$_x$As$_2$ [@lester09] confirm that the commensurate AF order appears below the structural transition temperature and superconductivity coexists with AF order for $0.06\leq x\leq 0.102$. Neutron scattering measurements on BaFe$_{2-x}$Co$_x$As$_2$ with coexisting AF order and superconductivity reveal that the intensity of AF Bragg peaks actually decreases below $T_c$ without changing the spin-spin correlation lengths [@pratt09; @adchristianson09]. While these results indicate that the static AF order competes with superconductivity, it remains unclear whether the long-range AF order truly coexists microscopically with superconducting regions [@mywang; @mywang11b]. Recently, for electron-doped samples near optimal superconductivity it has been shown that the commensurate static AF order changes into transversely incommensurate short-range AF order that coexists and competes with superconductivity (see inset in Fig. 2b) [@dkpratt11; @hluo12]. Taking the temperature dependence of the orthorhombic lattice distortion of BaFe$_{2-x}$Co$_x$As$_2$ into account [@snandi10], the AF order, structure, and superconductivity phase diagrams for BaFe$_{2-x}T_x$As$_2$ are shown in Fig. 2b.
Although the superconducting transition temperature for hole-doped Ba$_{1-x}$K$_x$Fe$_2$As$_2$ can reach up to $T_c=38$ K [@rotter] as compared to the $T_c\approx 25$ K for electron-doped BaFe$_{2-x}T_x$As$_2$ [@johnston; @stewart], those materials are much less studied because of the difficulty in growing high-quality single crystals. The initial transport and neutron scattering experiments on powder samples indicated a gradual suppression of the concurrent structural and magnetic phase transitions with increasing K-doping. For the underdoped regime $0.2\leq x\leq 0.4$, commensurate AF order appears to microscopically coexist with superconductivity [@hchen09]. Subsequent neutron scattering and muon spin rotation ($\mu$SR) measurements on single crystals grown in Sn-flux suggested mesoscopic separation of the AF and superconducting phases [@jtpark09]. However, recent neutron [@savci], X-ray scattering, and $\mu$SR work [@ewiesenmayer11] on high-quality powder samples confirm the microscopic coexistence of the commensurate AF order with superconductivity in the underdoped region between $0.2\leq x\leq 0.3$ and the suppression of the orthorhombic phase below $T_c$ (Fig. 2a). Since currently there is no neutron diffraction work on high-quality single crystals of Ba$_{1-x}$K$_x$Fe$_2$As$_2$ grown using FeAs-flux, it is unclear if there are also short-range incommensurate AF order in Ba$_{1-x}$K$_x$Fe$_2$As$_2$ near optimal superconductivity.
The appearance of static incommensurate AF order along the transverse direction of the collinear AF ordering wave vector ${\bf Q}_{AF}=(\pm 1,0)$ in BaFe$_{2-x}T_x$As$_2$ suggests that such order arises from the electron doping effect of FS nesting [@dkpratt11; @hluo12]. Based on a five-orbital tight-binding model, fitted to the density functional theory (DFT) band structure for BaFe$_2$As$_2$ [@graser10], there should be five FS pockets with different orbital contributions in the two-dimensional reciprocal space at ${\bf Q}_z=0$ (Fig. 2d). The intraorbital, but interband, scattering process between $\Gamma(0,0)\leftrightarrow M(1,0)$ shown in Fig. 2d favors the transversely lengthened vertices [@jhzhang10]. This momentum anisotropy is compatible with the experimentally observed elliptically shaped low-energy spin excitations in superconducting BaFe$_{2-x}T_x$As$_2$ [@jtpark; @clester; @hfli; @hqluo12] and spin waves in BaFe$_2$As$_2$ (Fig. 2g) [@lharriger]. Upon electron-doping to enlarge the electron pockets near $M(1,0)/(0,1)$ and shrink the hole pockets near $\Gamma(0,0)$, the mismatch between the electron and hole Fermi pockets becomes larger (Figs. 2e), resulting in a more transversely elongated ellipse in the low-energy magnetic response (Fig. 2h). Indeed, this is qualitatively consistent with the doping evolution of the low-energy spin excitations [@jtpark; @hqluo12; @msliu12].
For hole-doped Ba$_{1-x}$K$_x$Fe$_2$As$_2$, one should expect enlarged hole Fermi pockets near $\Gamma(0,0)$ and reduced electron pockets near $M(1,0)/(0,1)$, as shown in Fig. 2c. Based on first principles calculations, spin excitations for optimally hole-doped Ba$_{1-x}$K$_x$Fe$_2$As$_2$ at $x=0.4$ should have longitudinally elongated ellipses [@jtpark], and gradually evolve into incommensurate magnetic scattering (elastic and/or inelastic) with increasing $x$ due to poor nesting between the hole and electron Fermi pockets [@castellan]. INS experiments on single crystal Ba$_{0.67}$K$_{0.33}$Fe$_2$As$_2$ [@chenglinzhang] indeed confirm that the low-energy spin excitations are longitudinally elongated ellipses that are rotated 90$^\circ$ from that of the electron-doped BaFe$_{2-x}T_x$As$_2$ (Fig. 2f) [@jtpark; @clester; @hfli; @hqluo12]. Furthermore, INS measurements on powder samples of Ba$_{1-x}$K$_x$Fe$_2$As$_2$ reveal that spin excitations change from commensurate to incommensurate for $x\ge 0.4$, although their exact line shape and incommensurability in reciprocal space are unknown [@castellan]. Finally, INS experiments on hole-overdoped KFe$_2$As$_2$ found incommensurate spin fluctuations along the longitudinal direction (inset in Fig. 2a), again consistent with the FS nesting picture [@chlee11]. Figure 2a shows the electronic phase diagram of hole-doped Ba$_{1-x}$K$_x$Fe$_2$As$_2$ based on the present understanding of these materials.
Although FS nesting is compatible with a number of experimental observations for the evolution of spin excitations in electron/hole-doped iron-based superconductors, there are several aspects of the problem where such a scenario cannot be reconciled with experiments. In a recent INS experiment on optimally electron-doped BaFe$_{1.9}$Ni$_{0.1}$As$_2$, magnetic excitations throughout the Brillouin zone have been measured in absolute units and compared with spin waves for AF BaFe$_2$As$_2$ [@msliu12]. In the fully localized (insulating) case, the formal Fe$^{2+}$ oxidation state in BaFe$_2$As$_2$ would give a $3d^6$ electronic configuration and Hund’s rules would yield $S=2$. The total fluctuating moments should be $\left\langle m^2\right\rangle=(g\mu_B)^2
S(S+1)=24\ \mu_B^2$ per Fe assuming $g=2$ [@clester; @msliu12]. For spin waves in the insulating Rb$_{0.89}$Fe$_{1.58}$Se$_2$, the total moment sum rule appears to be satisfied [@mywang11]. The fluctuating moments for BaFe$_2$As$_2$ and BaFe$_{1.9}$Ni$_{0.1}$As$_{2}$ are $\left\langle
m^2\right\rangle=3.17\pm 0.16$ and $3.2\pm 0.16\ \mu_B^2$ per Fe(Ni), respectively [@msliu12]. While these values are considerably smaller than those of the fully localized case, they are much larger than expected from the fully itinerant SDW using the random phase approximation [@hpark12]. A calculation combining DFT and dynamical mean field theory (DMFT) suggests that both the band structure and the local moment aspects (e.g. Hunds coupling) of the iron electrons are needed for a good description of the magnetic responses [@msliu12]. Figure 2i shows the energy dependence of $\chi^{\prime\prime}(\omega)$ for BaFe$_2$As$_2$ and BaFe$_{1.9}$Ni$_{0.1}$As$_{2}$, and it is clear that the impact of electron doping and superconductivity are limited to spin excitation energies below 100 meV. These results suggest that high-energy spin excitations are likely to arise from the local moments instead of FS nesting effects.
Deviations from the simple SDW Fermi surface nesting picture
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After the early research efforts on Fe-based superconductors [@johnston; @stewart; @paglione10], recent experimental and theoretical investigations are providing a more refined perspective of these materials. Below, several selected examples will be discussed, supplementing those presented in the neutrons sections.
[*Strength of electronic correlations*]{}. The strength of electronic correlations are often characterized via the ratio between the on-site Hubbard repulsion coupling $U$ and the bandwidth $W$ of the hole or electron carriers. Early on, it was assumed that pnictides were in the weak-interaction limit $U/W\ll 1$. However, recent investigations revealed that the electronic correlations induce large enhancements between the effective and bare electronic masses, signaling that correlation effects cannot be neglected. For instance, Haas-van Alphen experiments for KFe$_2$As$_2$ unveiled discrepancies between the band-theory calculated and observed FS’s, including a large electronic mass enhancement 3-7 caused by band narrowing [@tterashima10]. Similar ratios for the overdoped Tl$_2$Ba$_2$CuO$_{6+\delta}$ copper-oxides have been reported [@pmcrourke], suggesting that the undoped parent compounds of the pnictides resemble the overdoped copper oxides.
Additional insight is provided by optical conductivity experiments, since the ratio $R$ between the experimentally measured kinetic energy and that of band-theory calculations can be measured and contrasted against other compounds [@qazilbash09]. $R\approx 1$ signals a good metal such as Ag. LaFePO presents a ratio $R\approx 0.5$ which is borderline between weak and moderate coupling. However, pnictides such as BaFe$_{2-x}T_x$As$_2$ are characterized by an even stronger correlation that induces a ratio $R\approx 0.3$ which is similar to results for overdoped La$_{2-x}$Sr$_x$CuO$_4$, widely considered to be a “correlated metal”. Other studies have arrived to similar conclusions with regards to the correlation strength [@nakamura; @dsinosov11]. In agreement with experiments, DFT+DMFT predicts a mass enhancement $m^\ast/m_{band}\sim$2-3 for BaFe$_{2-x}T_x$As$_2$ and $\sim$7 for FeTe [@zpyin10]. Moreover, ARPES studies of NaFeAs revealed band reconstructions in the magnetic state involving bands well below the FS [@che10], contrary to a weak coupling picture.
![Summary of the phase diagram of multiorbital Hubbard models and the electronic state of Fe near the FS. (a) Sketch of the phase diagram of a typical multiorbital Hubbard model in the undoped limit, varying the on-site same-orbital repulsion $U$ and the ratio between the Hund coupling $J_H$ and $U$. Highlighted is a region dubbed “physical region” where the properties of the model are in good agreement with experiments. Note the location of this region in the intermediate magnetic-metallic phase, with magnetic order at ${\bf Q}_{AF}=(1,0)$, at similar distance from the paramagnetic state and from the insulator orbitally-ordered state [@qluo10]. (b) Sketch of the DOS illustrating the phenomenon of FS orbital order, which is a weight redistribution at the FS of the states associated with the $xz$ and $yz$ $d$-orbitals. Even though the integral over energy gives similar values for both orbitals, at the FS there are drastic differences that influence on several properties such as transport [@daghofer10; @daghofer12]. (c) Sketch of the anisotropy found in transport experiments for detwinned Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$. Note that this anisotropy is present at temperatures substantially larger than $T_N$ [@irfisher11]. (d) Orbitals of relevance for the discussion of the Fe-based superconductors and their splitting at the FS. (e) Sketch of the ARPES results of Ref. [@shimojima11], illustrating the absence of a nesting partner for one of the hole pockets. The material still displays a nearly uniform superconducting gap at this and all the other hole and electron pockets. (f) Sketch of the magnetic moment at wave vector ${\bf Q}_{AF}=(1,0)$ for two models. On the left is the result for a traditional model of pnictides, with the $xz$, $yz$, and $xy$ $d$-orbitals active at the FS. This model displays magnetic order in a broad range of couplings from very weak to strong. On the right, results of a model with the same FS but totally different orbital composition. While at small $U$ there is no order, at larger couplings this model converges to the same ${\bf Q}_{AF}=(1,0)$ order [@nicholson]. ](Fig3R2)
Hubbard model investigations provide additional insight on this subject. When compared with similar efforts for the cuprates, the study of Hubbard models for the pnictides is far more challenging because several Fe orbitals are needed. For this reason, many efforts are restricted to mean-field Hartree-Fock approximations. For the undoped three-orbital Hubbard model, employing the $d_{xz}$, $d_{yz}$, and $d_{xy}$ orbitals of relevance at the FS, a sketch of a typical mean-field phase diagram varying $U$ and the Hund coupling $J_H$ [@qluo10] is in Fig. 3a. Three regimes are identified: a small $U$ phase where the state is paramagnetic, followed with increasing $U$ by an intermediate regime simultaneously metallic and magnetic [@qluo10], and finally a large-$U$ phase where a gap in the density-of-states (DOS) is induced leading to an insulator (with concomitant orbital order). Comparing the theoretical predictions for the magnetic moment in the ${\bf Q}_{AF}=(1,0)$ wave-vector channel against neutrons, and the one-particle spectral function $A({\bf k},\omega)$ against ARPES, the intermediate-coupling region dubbed “physical region” in yellow in Fig. 3a represents qualitatively the undoped BaFe$_{2-x}T_x$As$_2$ compounds [@qluo10]. In this regime, $U/W\sim$0.3-0.4, and similar results were reported for the two- and five-orbital models [@mDaghofer]. Note that Hartree-Fock usually produces critical couplings smaller than they truly are because of the neglect of quantum fluctuations. In fact, recent investigations beyond Hartree-Fock [@kkubo11] suggest that the relevant $U$ may be larger than those found in Hartree-Fock [@qluo10] by approximately a factor two. The study of effective low-energy Hamiltonians starting from first-principles calculations also led to the conclusion that $U/W$ is between 0.5 and 1.0 for the pnictides depending on the particular compound [@arita-imada]. Thus, the regime of relevance is neither very weak coupling nor strong coupling but the more subtle, and far less explored, intermediate region. Previous efforts converged to similar conclusions [@Johannes09]. This is also compatible with the notion that the parent compound is close to a Mott insulator [@si2008; @qmsi09]. In the “physical region” the ratio $J_H/U$ is approximately 1/4 [@qluo10], as in other estimations [@zpyin10], highlighting the importance of $J_H$ in these materials that are sometimes referred to as Hund metals [@hund-metal]. Finally, it is very important to remark that the above described analysis of $U/W$ holds for pnictides but the recent discovery of the alkaline iron selenides [@jgguo; @mhfang] has opened a new chapter in this field and it is conceivable that for these materials $U/W$ will be larger than in pnictides explaining, for example, the large values of the iron moments.
[*Role of the orbital degree of freedom*]{}. The “physical region” in Fig. 3a is not only close to the paramagnetic regime, but also similarly close to the insulator, which in the mean-field approximation is also orbitally ordered [@mDaghofer]. The potential relevance of the orbital degree of freedom in pnictides has been discussed [@lv10; @wgyin10]. The orbital can be of relevance not only in its long-range-ordered form, but also via its coupling to the spin and its influence near the FS. In fact, polarized ARPES experiments on BaFe$_2$As$_2$ [@shimojima] reported that at the FS there was an asymmetry between the populations of the $d_{xz}$ and $d_{yz}$ orbitals. Theoretical studies showed that this effect indeed occurs in the ${\bf Q}_{AF}=(1,0)$ magnetic state, and it is linked to an orbital-dependent reduction in the DOS at the FS [@daghofer10], sketched in Fig. 3b, phenomenon dubbed “Fermi surface orbital order”.
This effect, while not sufficiently strong to induce long-range order as in manganites, can still severely influence the properties of the material. Consider for example the transport anisotropy observed in detwinned BaFe$_{2-x}T_x$As$_2$ single-crystals [@irfisher11; @matanatar], sketched in Fig. 3c. At low temperatures the difference between the $a$-axis (spins antiparallel, Fig. 1b) and $b$-axis (spin parallel, Fig. 1b) directions can be rationalized based on the magnetic state, since the different spin arrangements along the $a$ and $b$ break rotational invariance [@xtzhang]. However, both in the undoped case and particularly in the lightly-doped regime, the asymmetry persists well above the N$\rm \acute{e}$el temperature, $T_N$, into a new temperature scale $T^\ast$ that may be associated with the onset of nematic order [@cfang; @cxu], similarly as in some ruthenates and copper oxides [@efradkin]. ARPES experiments on the same materials [@myi] reported a $d_{xz}$ and $d_{yz}$ band splitting (Fig. 3d) that occurs above $T_N$ in the same region where transport anisotropies were found. Although the splitting is too small to be a canonical long-range orbital order, it reveals the importance of fluctuations above the critical temperatures. Optical spectra studies also unveiled anisotropies in the spectra persisting up to 2 eV, incompatible with SDW scenarios [@Nakajima]. Note that the discussion on this subject is still fluid. While neutron diffraction investigations showed that $T_N$ actually substantially increases as the pressure needed to detwin the crystals increases, potentially explaining the observed resistivity anisotropies [@dhital], magnetic torque measurements without external pressure revealed clear evidence for electronic nematicity [@kasahara]. Recent calculations addressing transport indeed find an important role of the orbital states above $T_N$ [@rmfernandes]. The orbital degree of freedom, closely entangled to the spin and the lattice, may lead to a more complex “normal” state than anticipated from weak coupling particularly because of the FS orbital order [@daghofer10]. In fact, neutron scattering shows that although the low-energy magnetic dispersion changes substantially when crossing critical temperatures, the higher energy features remain the same over a large doping and temperature range [@msliu12], suggesting that spin, orbital, and lattice are closely entangled. Establishing who is the “driver” and who is the “passenger” may define an important area of focus of future research.
[*Local moments at room temperature*]{}. Another deviation from a simple weak coupling picture is the observation of local magnetic moments at room temperature. Within the SDW scenario, magnetic moments are formed upon cooling simultaneously with the development of long-range magnetic order. But recent Fe X-ray emission spectroscopy experiments unveiled the existence of local moments in the room-temperature paramagnetic state [@hgretarsson]. In fact, with the only exception of FeCrAs, for all the pnictides and chalcogenides investigated a sizable room temperature magnetic moment was found. This includes LiFeAs, that actually does not order magnetically at any temperature [@cwchu], and $A$Fe$_{1.6+x}$Se$_2$ with a regular arrangement of Fe vacancies (Fig. 1d). These observed local moments are similar in magnitude to those reported in the low-temperature neutron scattering experiments reviewed in previous sections. Similar conclusions to those of [@hgretarsson] were reached in a study of $3s$ core level emission for CeFeAsO$_{0.89}$F$_{0.11}$ [@bondino] and also in LDA+DMFT investigations [@hansmann].
[*Polarized ARPES results and orbital composition*]{}. While research using ARPES techniques applied to pnictides has already been reviewed [@prichard], some intriguing recent results addressing the influence of nesting are included in our discussion. Using bulk-sensitive laser ARPES on BaFe$_2$(As$_{0.65}$P$_{0.35}$)$_2$ and Ba$_{0.6}$K$_{0.4}$Fe$_2$As$_2$, an orbital-independent superconducting gap magnitude was found for the hole-pockets FS’s [@shimojima11]. These results are incompatible with nesting where the FS nested portions must have a robust component of the same orbital to be effective. Actually, the red hole pocket shown in the sketch in Fig. 3e, that experimentally displays a robust and nearly wave-vector-independent superconducting gap similar to those found in the other hole pockets, does not have a matching electron pocket with the same orbital composition and, thus, it cannot develop its superconductivity via a nesting pairing mechanism [@mazin2011n]. Perhaps inter-orbital pairing [@amoreo] or orbital fluctuations could be relevant to explain this paradox. Recent theoretical work [@nicholson] addressed the importance of orbital composition via two models: one with nested electron- and hole-pocket Fermi surfaces with the standard orbital composition of pnictide models, and another with the same FS shape but with electron and hole pockets having totally different orbital compositions. As sketched in Fig. 3f, the former develops magnetic order at smaller values of $U$ than the latter. However, with sufficiently large $U$ both have magnetic ground states with the same wavevector ${\bf Q}_{AF}=(1,0)$ (Fig. 3f). At large $U$ it is clear that the ${\bf Q}_{AF}=(1,0)$ order can be understood within a local picture, based on the similar magnitude of the super-exchange interactions between NN and NNN spins using a simple Heisenberg model.
[*Additional experimental results*]{}. De Haas-van Alphen studies [@bjarnold] in non-superconducting BaFe$_2$P$_2$, the end member of the superconducting series BaFe$_2$(As$_{1-x}$P$_x$)$_2$, indicate that the differences in the pairing susceptibility varying $x$ are caused by increases in $U$ and $J_H$ rather than improved geometric nesting. Moreover, ARPES studies of LiFeAs, without long-range magnetic order at low temperatures, report a strong renormalization of the band structure by a factor $\sim$3 and the absence of nesting [@Borisenko]. Yet, at $T_c=18$ K [@cwchu] LiFeAs still becomes superconducting suggesting that nesting is not necessary for superconductivity to develop. Similarly, ARPES experiments on superconducting $A$Fe$_{1.6+x}$Se$_2$ [@yzhang; @tqian; @Dxmou] revealed the absence of the hole-like FS’s necessary for the $\Gamma(0,0)\leftrightarrow M(1,0)$ $s^{\pm}$-wave superconductivity. Also note that related materials such as LaFePO with a well-nested FS also do not order magnetically. Why weak coupling arguments would work in some cases and not others? Finally, scanning tunneling microscopy (STM) experiments [@tmchuang] on Ca(Fe$_{1-x}$Co$_x$)$_2$As$_2$ shows an exotic “nematic” electronic structure, similar to those found for strongly coupled copper-oxides.
[*Additional theoretical results*]{}. In fluctuation-exchange approximation studies it was concluded that the nesting results are not robust against the addition of self-energy corrections [@arita09]. Other calculations have suggested that magnetic order in pnictides is neither fully localized nor fully itinerant: the $J_H$ coupling forms the local moments, while the particular ground state is selected by itinerant one-electron interactions [@Johannes09]. Moreover, studies of a spin-fermion model for the pnictides [@lv10; @wgyin10; @shliang11] revealed the crucial role played by the Hund’s rule coupling and suggested that the Fe superconductors are closer kin to manganites, where similar spin-fermion models were extensively studied [@dagotto-cmr], than to copper-oxides with regards to their diverse magnetism and incoherent normal-state electron transport.
Conclusions
===========
Recent studies of Fe-based superconductors are unveiling a perspective of these exciting materials that is far richer than previously anticipated. While in the early days weak coupling approaches seemed sufficient to understand these compounds, several recent efforts, reviewed in part here, suggest that understanding the physics of these materials may require more refined concepts, better many-body theoretical calculations, and additional sophisticated experiments for a more in-depth rationalization of their properties. In fact, evidence is building that pnictides and chalcogenides inhabit the mostly unexplored “intermediate” region of Hubbard $U/W$ couplings, which is neither very weak coupling, where FS nesting concepts apply, nor strong coupling, where localized spins provide a good starting point as it occurs in the undoped copper oxides. The situation is qualitatively summarized in Fig. 4 where a crude sketch of a plausible phase diagram for a generic undoped Hubbard model is provided varying temperature $T$ and $U/W$ at, e.g., a fixed $J_H/U$ such as 1/4. In weak coupling, first a critical value of $U$ must be crossed before magnetic order develops at low temperatures. In this region, nesting works properly. As $U$ increases, $T_N$ first increases, reaches a broad maximum, and then eventually in the regime of localized spins $T_N$ starts to decrease since it becomes regulated by the Heisenberg superexchange that scales as $1/U$. Above $T_N$ a “crossover” temperature that roughly grows like $U$ is shown separating regions with and without “preformed” local moments. Since the pnictides have local moments at room temperature, then a tentative location for these materials is provided by the dashed line. However, whether this line coincides with the maximum $T_N$ or is shifted to the left or the right is too early to say, but it cannot be too far from optimal otherwise local moments would be absent, if far left, or an insulator should be found at low temperatures, if far right. Theoretical mean-field estimates reviewed here using the multiorbital Hubbard model find that $U/W$$\sim$$0.3-0.5$ could work for pnictides. However, for chalcogenides and alkaline iron selenides, and also after including quantum fluctuations, the ratio $U/W$ may increase further, and it may reach the $U/W\sim1$ threshold widely consider to mark the starting point for a strong coupling description. Note also that the sketch in Fig. 4 is based on our knowledge on the one-orbital Hubbard model and a proper multiorbital analysis will lead to an even richer phase diagram. In fact, a critical $U$ for the transition between the magnetic metallic state and the magnetic insulating state at low temperatures should also be present, but it is not shown in the sketch for simplicity: this transition should occur at a $U$ larger than the pnictides dashed line since these materials are metallic at low temperatures.
In summary, the Fe-based superconductors continue surprising us with their exotic properties that do not fit into the simple limits of weak or strong coupling $U$. Additional experimental and theoretical efforts are needed to unveil the secrets of this intriguing family of materials.
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[**Acknowledgments**]{} We thank Leland W. Harriger for preparing the figures shown in this manuscript. We are also grateful to Thomas A. Maier for calculating the Fermi surfaces of BaFe$_2$As$_2$ shown in Fig. 2d. P.D. is supported by the U.S. NSF DMR-1063866, OISE-0968226, and by U.S. DOE, BES, under Grant No. DE-FG02-05ER46202. Work at Institute of Physics is supported by the Ministry of Science and Technology of China 973 program (2012CB821400). E.D. is supported by the U.S. DOE, BES, Materials Sciences and Engineering Division, and by the U.S. NSF DMR-11-04386.
[**Author contributions**]{}
P.D. and E.D. wrote the experimental and theoretical portions of the article, respectively. J.P.H. revised the article. All authors discussed the outline of the article.
[**Additional information**]{} The authors declare no competing financial interests.
[**Reprints and permissions**]{} information is available online at http://ngp.nature.com/reprintsandpermissions/.
Correspondence and requests for materials should be addressed to P.D. (e-mail: pdai@utk.edu) or E.D. (e-mail: edagotto@utk.edu).
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abstract: 'This contribution is an attempt to try to understand the matter-antimatter asymmetry in the universe within the [*spin-charge-family-theory*]{} [@norma; @pikanorma] if assuming that transitions in non equilibrium processes among instanton vacua and complex phases in mixing matrices are the sources of the matter-antimatter asymmetry, as studied in the literature [@gross; @rubakovshaposhnikov; @dinekusenko; @tapeiling] for several proposed theories. The [*spin-charge-family-theory*]{} is, namely, very promising in showing the right way beyond the [*standard model*]{}. It predicts families and their mass matrices, explaining the origin of the charges and of the gauge fields. It predicts that there are, after the universe passes through two $SU(2)\times U(1)$ phase transitions, in which the symmetry breaks from $SO(1,3) \times SU(2) \times SU(2) \times U(1) \times SU(3)$ first to $SO(1,3) \times SU(2) \times U(1) \times SU(3)$ and then to $SO(1,3) \times U(1) \times SU(3)$, twice decoupled four families. The upper four families gain masses in the first phase transition, while the second four families gain masses at the electroweak break. To these two breaks of symmetries the scalar non Abelian fields, the (superposition of the) gauge fields of the operators generating families, contribute. The lightest of the upper four families is stable (in comparison with the life of the universe) and is therefore a candidate for constituting the dark matter. The heaviest of the lower four families should be seen at the LHC or at somewhat higher energies.'
address: ' Department of Physics, FMF, University of Ljubljana, Jadranska 19, 1000 Ljubljana'
author:
- 'N. S. Mankoč Borštnik'
title: 'Can the matter-antimatter asymmetry be easier to understand within the “spin-charge-family-theory”, predicting twice four families and two times $SU(2)$ vector gauge and scalar fields? '
---
\#1)\#2 \#1\]\#2 \#1[\_[1.5pt ]{}]{}
Keywords: Unifying theories, Kaluza-Klein theories, dark matter, new families, matter-antimatter asymmetry, higher dimensional spaces, instanton vacua, P and CP noninvariance.
Introduction
============
The [*theory unifying spin and charges and predicting families*]{} ([*spin-charge-family-theory*]{}) assumes that spinors carry in $d \ge 4$ ($d= 1 + 13$ is studied) only two kinds of the spin. The Dirac kind $\gamma^{a}$ manifests after several appropriate breaks of the starting symmetry as the spin and all the charges. The second kind called $\{ \gamma^a$ ($\{ \gamma^a, \tilde{\gamma}^b\}_{+}=0$) generates families. Accordingly there are in $d \ge 4$, besides the vielbeins, also the two kinds of the spin connection fields, which are the gauge fields of the corresponding operators $S^{ab}$ and $\tilde{S}^{ab}$. Those connected with $S^{ab}$ manifest in $d=(1+3)$ as the vector gauge fields, while those connected with $\tilde{S}^{ab}$ manifest as the scalar fields and determine on the tree level the mass matrices.
Let me make a short review of the so far made predictions of the [*spin-charge-family-theory*]{}:
- The [*spin-charge-family-theory*]{} has the explanation for the appearance of the internal degrees of freedom – the spin and the charges while unifying them under the assumption that the universe went through several phase transitions which cause the appropriate breaks of the starting symmetry. Then the fact that the right handed (with respect to SO(1,3)) fermions are weak chargeless, while the left handed ones carry the weak charge emerges, as well as that there exist leptons (singlets with respect to the colour charge) and quarks (triplets with respect to the colour charge) [@norma; @pikanorma].
- The theory explains the appearance of massless families at the low energy regime under the assumption that there are breaks which leave the massless fermions of only one handedness [@hnd]. Assuming that breaks of symmetries affect the whole internal space — the space defined by both kinds of the Clifford algebra objects — it predicts in the energy regime close below $10^{16}$ GeV eight massless families. The manifested symmetry is (assumed to be) at this stage $SO(1,3) \times SO(4) \times U(1) \times SU(3)$. The next break of the symmetry of the universe to $SO(1,3) \times SU(2) \times
U(1) \times SU(3)$ leaves four families massless [@pikanorma], while the vacuum expectation values of superposition of the starting fields which manifest in $(1+3)$ as scalar fields, make the upper four families and the corresponding gauge fields massive. After the electroweak break also the lower four families become massive due to the vacuum expectation values of superposition of the starting fields, together with the weak bosons.
- The theory predicts the fourth family, which will be observed at the LHC or at somewhat higher energies [@gmdn], and the fifth stable family (with no mixing matrix elements couplings to the lower four families in comparison with the age of the universe), the baryons and neutrinos of which are the candidates to form the dark matter.
- The masses of this fifth family members are according to the so far made rough estimations [@pikanorma; @gmdn] larger than a few TeV and smaller than $10^{10}$ TeV. The members of the family have approximately the same mass, at least on the tree level [@normaproc2010talk].
- The studies [@gn] of the history of the stable fifth family members in the evolution of the universe and of their interactions with the ordinary matter in the DAMA’s and the CDMS’s experiments done so far lead to the prediction that the masses of the fifth family members, if they constitute the dark matter, are a few hundred TeV, independent of the fifth family fermion-antifermion asymmetry. The Xe experiment looks like to be in disagreement, but careful analyses show that one should wait for further data [@discussGN] to make the final conclusion.
The lightest fifth family baryon is, in the case that all the quarks have approximately (within a hundred GeV) the same mass [@gn], the fifth family neutron, due to the attractive electromagnetic interaction. The difference in the weak interaction can be for large enough masses neglected.
- The fermion asymmetry in the [*approach*]{} has not yet really been studied.
- The studies [@gn] of the evolution of this stable fifth family members rely on my rough estimations [@gn] of the behaviour of the coloured fifth family objects (single quarks and antiquarks or coloured pairs of quarks or of antiquarks) during the colour phase transition. These estimations namely suggest that the coloured objects either annihilate with the anti-objects or they form colourless neutrons and antineutrons and correspondingly decouple from the plasma soon after the colour phase transition starts, due to the very strong binding energy of the fifth family baryons (with respect to the first family baryons) long enough before the first family quarks start to form the baryons. These estimations should be followed by more accurate studies.
- The so far done studies suggest strongly that the number density of the fifth family neutrinos (of approximately the same mass as the fifth family quarks and leptons), which also contribute to the dark matter, is pretty much reduced due to the neutrino-antineutrino annihilation closed below the electroweak break. The weak annihilation cross section is expected to play much stronger role for neutrinos than for strongly bound fifth family quarks in the fifth family neutron (due to the huge binding energy of the fifth family quarks), what also remains to be proved.
- The estimations [@gmdn] of the properties of the lower four families on the tree level call for the calculations beyond the tree level, which should hopefully demonstrate, that the loop corrections (in all orders) bring the main differences in the properties of the family members. These calculations are in progress [@AN].
Although we can say that the [*spin-charge-family-theory*]{} looks very promising as the right way to explain where do the assumptions of the [*standard model*]{} originate, there are obviously many not yet studied, or at least far from being carefully enough studied open problems. Many a problem is common to all the theories, like the first family baryon asymmetry, which I am going to discuss within the [*spin-charge-family-theory*]{} in this contribution. Some of the problems are common to all the theories assuming more than so far observed $(1+3)$ dimensions, in particular the [*spin-charge-family-theory*]{} shares some problems with all the Kaluza-Klein-like theories. We are trying to solve them first on toy models [@hnd].
The main new step in the [*spin-charge-family-theory*]{} — the explanation of the appearance of families by assuming that both existing kinds of the Clifford algebra objects should be used to treat correctly the fermion degrees of freedom — limits very much the properties of families and their members. The simple starting action in $d= (1+13)$, which in $d=(1+3)$ demonstrates the mass matrices, namely fixes to high extent the fermion properties after the breaks of symmetries. Therefore this proposal might soon be studied accurately enough to show whether it is the right theory or not.
This contribution is an attempt to try to understand what can the [*spin-charge-family-theory*]{} say about the fermion-antifermion asymmetry when taking into account the proposals of the references [@rubakovshaposhnikov; @dinekusenko; @tapeiling] (and of the works cited therein). These works study the soliton solutions of non Abelian gauge fields with many different vacua and evaluate fermion number nonconservation due to possible transitions among different vacua in non equilibrium processes during the phase transitions through which the universe passed. In such processes fermion (and also antifermion) currents are not conserved since $CP$ is not nonconserved. To the $CP$ nonconservation also the complex matrix elements determining the transitions among families contribute and consequently influence the first family fermion-antifermion asymmetry.
Since the [*spin-charge-family-theory*]{} predicts below the unification scale of all the charges two kinds of phase transitions (first from $SO(1,3) \times SU(2) \times SU(2) \times U(1) \times SU(3)$ to $SO(1,3) \times SU(2) \times U(1) \times SU(3)$, in which the upper four families gain masses and so do the corresponding vector gauge fields, and then from $SO(1,3) \times SU(2) \times U(1) \times SU(3)$ to $SO(1,3) \times U(1) \times SU(3)$, in which the lower four families and the corresponding gauge fields gain masses), in which besides the vector gauge fields also the scalar gauge fields (the gauge fields of $\tilde{S}^{ab}$ and also of $S^{ab}$ with the scalar index with respect to $(1+3)$) contribute, the fermion-antifermion asymmetry might very probably have for the stable fifth family an opposite sign than for the first family.
It might therefore be that the existence of two kinds of four families, together with two kinds of the vector gauge fields and two kinds of the scalar fields help to easier understanding the first family fermion-antifermion asymmetry.
Although I am studying the fermion asymmetry, together with the discrete symmetries, in the [*spin-charge-family-theory*]{} for quite some time (not really intensively), this contribution is stimulated by the question of M.Y. Khlopov [@MYN], since he is proposing the scenario, in which my stable fifth family members should manifest an opposite fermion asymmetry than the first family members, that is antifermion-fermion asymmetry. While in the case that the fifth family members have masses around 100 TeV or higher and the neutron is the lightest baryon and neutrino the lightest lepton [@gn] the fifth family baryon asymmetry plays no role (since in this case the fifth family neutrons and neutrinos as well as their antiparticles interact weakly enough among themselves and with the ordinary matter that the assumption that they constitute the dark matter is in agreement with the observations). Maxim [@m] claims that the fifth family members with the quark masses not higher than 10 TeV are also the candidates for the dark matter, provided that $\bar{u}_5 \bar{u}_5 \bar{u}_5$ is the lightest antibaryon and that there is an excess of antibaryons over the baryons in the fifth family case.
A short overview of the theory unifying spin and charges and explaining families {#approach}
================================================================================
In this section I briefly repeat the main ideas of the [*spin-charge-family-theory*]{}. I kindly ask the reader to learn more about this theory in the references [@norma; @pikanorma] as well as in my talk presented in this proceedings and in the references therein.
I am proposing a simple action in $d=(1+13)$-dimensional space. Spinors carry two kinds of the spin (no charges).\
i. The Dirac spin, described by $\gamma^a$’s, defines the spinor representation in $d=(1+ 13)$. After the break of the starting symmetry $SO(1,13)$ (through $SO(1,7) \times
SO(6)$) to the symmetry of the [*standard model*]{} in $d=(1+3)$ ($SO(1,3)\times U(1)\times SU(2)\times SU(3)$) it defines the hyper charge ($U(1)$), the weak charge ($SU(2)$, with the left handed representation of $SO(1,3)$ manifesting naturally the weak charge and the right handed ones appearing as the weak singlets) and the colour charge ($SU(3)$).\
ii. The second kind of the spin [@norma], described by $\tilde{\gamma}^a$’s ($\{\tilde{\gamma}^a, \tilde{\gamma}^b\}_{+}= 2 \, \eta^{ab}$) and anticommuting with the Dirac $\gamma^a$ ($\{\gamma^a, \tilde{\gamma}^b\}_{+}=0$), defines the families of spinors.\
Accordingly spinors interact with the two kinds of the spin connection fields and the vielbeins.
We have $$\begin{aligned}
&& \{ \gamma^a, \gamma^b\}_{+} = 2\eta^{ab} =
\{ \tilde{\gamma}^a, \tilde{\gamma}^b\}_{+},\quad
\{ \gamma^a, \tilde{\gamma}^b\}_{+} = 0,\nonumber\\
&&S^{ab}: = (i/4) (\gamma^a \gamma^b - \gamma^b \gamma^a), \quad
\tilde{S}^{ab}: = (i/4) (\tilde{\gamma}^a \tilde{\gamma}^b
- \tilde{\gamma}^b \tilde{\gamma}^a),\quad \{S^{ab}, \tilde{S}^{cd}\}_{-}=0.
\label{snmb:tildegclifford}\end{aligned}$$ The action $$\begin{aligned}
S \, &=& \int \; d^dx \; E\;{\mathcal L}_{f} +
\nonumber\\
& & \int \; d^dx \; E\; (\alpha \,R + \tilde{\alpha} \, \tilde{R})\,,
\end{aligned}$$ $$\begin{aligned}
{\mathcal L}_f &=& \frac{1}{2}\, (E\bar{\psi} \, \gamma^a p_{0a} \psi) + h.c.\,,
\nonumber\\
p_{0a } &=& f^{\alpha}{}_a p_{0\alpha} + \frac{1}{2E}\, \{ p_{\alpha}, E f^{\alpha}{}_a\}_-,
\nonumber\\
p_{0\alpha} &=& p_{\alpha} -
\frac{1}{2} S^{ab} \omega_{ab \alpha} -
\frac{1}{2} \tilde{S}^{ab} \tilde{\omega}_{ab \alpha}\,,
\nonumber\\
R &=& \frac{1}{2} \, \{ f^{\alpha [ a} f^{\beta b ]} \;(\omega_{a b \alpha, \beta}
- \omega_{c a \alpha}\,\omega^{c}{}_{b \beta}) \} + h.c. \;,
\nonumber\\
\tilde{R} &=& \frac{1}{2}\, f^{\alpha [ a} f^{\beta b ]} \;(\tilde{\omega}_{a b \alpha,\beta} -
\tilde{\omega}_{c a \alpha} \tilde{\omega}^{c}{}_{b \beta}) + h.c.\;,
\label{wholeaction}\end{aligned}$$ manifests ($f^{\alpha [a} f^{\beta b]}= f^{\alpha a} f^{\beta b} - f^{\alpha b} f^{\beta a}$) after the break of symmetries all the known gauge fields and the scalar fields, and the mass matrices. To see the manifestation of the covariant momentum and the mass matrices we rewrite formally the action for a Weyl spinor in $d=(1+13)$ as follows $$\begin{aligned}
{\mathcal L}_f &=& \bar{\psi}\gamma^{m} (p_{m}- \sum_{A,i}\; g^{A}\tau^{Ai} A^{Ai}_{m}) \psi
+ \nonumber\\
& & \{ \sum_{s=7,8}\; \bar{\psi} \gamma^{s} p_{0s} \; \psi \} + \nonumber\\
& & {\rm the \;rest},
\label{faction}\end{aligned}$$ where $m=0,1,2,3$ with $$\begin{aligned}
\tau^{Ai} = \sum_{a,b} \;c^{Ai}{ }_{ab} \; S^{ab},
\nonumber\\
\{\tau^{Ai}, \tau^{Bj}\}_- = i \delta^{AB} f^{Aijk} \tau^{Ak}.
\label{tau}\end{aligned}$$ All the charges and the spin of one family are determined by $S^{ab}$, with $S^{ab}$ as the only internal degree of freedom of one family (besides the family quantum number, determined by $\tilde{S}^{ab}$), manifesting after the breaks at the low energy regime as the generators of the observed groups (Eq. (\[tau\])) $U(1), SU(2)$ and $SU(3)$, for $A=1,2,3$, respectively.
The breaks of the starting symmetry from $SO(1,13)$ to the symmetry $SO(1,7) \times SU(3) \times U(1)$ and further to $SO(1,3) \times SU(2) \times SU(2) \times U(1) \times SU(3) $ are assumed to leave all the low lying families of spinors massless. There are eight such massless families ($2^{8/2-1}$) before further breaks.
Accordingly the first row of the action in Eq. (\[faction\]) manifests the effective [*standard model*]{} fermions part of the action before the weak break, while the second part manifests, after the appropriate breaks of symmetries (when $\omega_{ab \sigma}$ and $\tilde{\omega}_{ab \sigma}$, $\sigma \in (5,6,7,8),$ fields gain the nonzero vacuum expectation values on the tree level) the mass matrices.
The generators $\tilde{S}^{ab}$ take care of the families, transforming each member of one family into the corresponding member of another family, due to the fact that $\{S^{ab}, \tilde{S}^{cd}\}_{-}=0$ (Eq.(\[snmb:tildegclifford\])).
Using the technique [@snmb:hn02hn03] and analysing the vectors as the eigenvectors of the [*standard model*]{} groups we present vectors in the space of charges and spins in terms of projectors and nilpotents as can be learned in Appendix, in the references [@norma; @pikanorma] and also in my talk in the Proceedings of Bled workshop 2010.
I present in Table \[Table I.\] the eightplet (the representation of $SO(1,7)$ of quarks of a particular colour charge ($\tau^{33}=1/2$, $\tau^{38}=1/(2\sqrt{3})$), and $U(1)$ charge ($\tau^{4}=1/6$) and on Table \[Table Il.\] the eightplet of the corresponding (colour chargeless) leptons.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
i $$ $|^a\psi_i>$ $\Gamma^{(1,3)}$ $ S^{12}$ $\Gamma^{(4)}$ $\tau^{13}$ $\tau^{23}$ $Y$ $Q$
--- ---------------- --------------------------------------------------------------------------------- ------------------ ---------------- ---------------- ---------------- ---------------- ---------------- ----------------
${\rm Octet},\;\Gamma^{(1,7)} =1,\;\Gamma^{(6)} = -1,$
${\rm of \; quarks}$
1 $ u_{R}^{c1}$ $ \stackrel{03}{(+i)}\,\stackrel{12}{(+)}| 1 $\frac{1}{2}$ 1 0 $\frac{1}{2}$ $\frac{2}{3}$ $\frac{2}{3}$
\stackrel{56}{(+)}\,\stackrel{78}{(+)}
||\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]} $
2 $u_{R}^{c1}$ $\stackrel{03}{[-i]}\,\stackrel{12}{[-]}|\stackrel{56}{(+)}\,\stackrel{78}{(+)} 1 $-\frac{1}{2}$ 1 0 $\frac{1}{2}$ $\frac{2}{3}$ $\frac{2}{3}$
||\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]}$
3 $d_{R}^{c1}$ $\stackrel{03}{(+i)}\,\stackrel{12}{(+)}|\stackrel{56}{[-]}\,\stackrel{78}{[-]} 1 $\frac{1}{2}$ 1 0 $-\frac{1}{2}$ $-\frac{1}{3}$ $-\frac{1}{3}$
||\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]}$
4 $ d_{R}^{c1} $ $\stackrel{03}{[-i]}\,\stackrel{12}{[-]}| 1 $-\frac{1}{2}$ 1 0 $-\frac{1}{2}$ $-\frac{1}{3}$ $-\frac{1}{3}$
\stackrel{56}{[-]}\,\stackrel{78}{[-]}
||\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]} $
5 $d_{L}^{c1}$ $\stackrel{03}{[-i]}\,\stackrel{12}{(+)}|\stackrel{56}{[-]}\,\stackrel{78}{(+)} -1 $\frac{1}{2}$ -1 $-\frac{1}{2}$ 0 $\frac{1}{6}$ $-\frac{1}{3}$
||\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]}$
6 $d_{L}^{c1} $ $\stackrel{03}{(+i)}\,\stackrel{12}{[-]}| -1 $-\frac{1}{2}$ -1 $-\frac{1}{2}$ 0 $\frac{1}{6}$ $-\frac{1}{3}$
\stackrel{56}{[-]}\,\stackrel{78}{(+)}
||\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]} $
7 $ u_{L}^{c1}$ $\stackrel{03}{[-i]}\,\stackrel{12}{(+)}| -1 $\frac{1}{2}$ -1 $\frac{1}{2}$ 0 $\frac{1}{6}$ $\frac{2}{3}$
\stackrel{56}{(+)}\,\stackrel{78}{[-]}
||\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]}$
8 $u_{L}^{c1}$ $\stackrel{03}{(+i)}\,\stackrel{12}{[-]}|\stackrel{56}{(+)}\,\stackrel{78}{[-]} -1 $-\frac{1}{2}$ -1 $\frac{1}{2}$ 0 $\frac{1}{6}$ $\frac{2}{3}$
||\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]}$
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: \[Table I.\] The 8-plet of quarks - the members of $SO(1,7)$ subgroup of the group $SO(1,13)$, belonging to one Weyl left handed ($\Gamma^{(1,13)} = -1 = \Gamma^{(1,7)} \times \Gamma^{(6)}$) spinor representation of $SO(1,13)$. It contains the left handed weak charged quarks and the right handed weak chargeless quarks of a particular colour $(1/2,1/(2\sqrt{3}))$. Here $\Gamma^{(1,3)}$ defines the handedness in $(1+3)$ space, $ S^{12}$ defines the ordinary spin (which can also be read directly from the basic vector, both vectors with both spins, $\pm \frac{1}{2}$, are presented), $\tau^{13}$ defines the third component of the weak charge, $\tau^{23}$ the third component of the $SU(2)_{II}$ charge, $\tau^{4}$ (the $U(1)$ charge) defines together with the $\tau^{23}$ the hyper charge ($Y= \tau^4 + \tau^{23}$), $Q= Y + \tau^{13}$ is the electromagnetic charge. The reader can find the whole Weyl representation in the ref. [@Portoroz03].
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
i $$ $|^a\psi_i>$ $\Gamma^{(1,3)}$ $ S^{12}$ $\Gamma^{(4)}$ $\tau^{13}$ $\tau^{23}$ $Y$ $Q$
--- ------------ --------------------------------------------------------------------------------- ------------------ ---------------- ---------------- ---------------- ---------------- ---------------- ------
${\rm Octet},\;\Gamma^{(1,7)} =1,\;\Gamma^{(6)} = -1,$
${\rm of \; quarks}$
1 $ \nu_{R}$ $ \stackrel{03}{(+i)}\,\stackrel{12}{(+)}| 1 $\frac{1}{2}$ 1 0 $\frac{1}{2}$ $0$ $0$
\stackrel{56}{(+)}\,\stackrel{78}{(+)}
||\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{(+)}\;\;\stackrel{13\;14}{(+)} $
2 $\nu_{R}$ $\stackrel{03}{[-i]}\,\stackrel{12}{[-]}|\stackrel{56}{(+)}\,\stackrel{78}{(+)} 1 $-\frac{1}{2}$ 1 0 $\frac{1}{2}$ $0$ $0$
||\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]}$
3 $e_{R}$ $\stackrel{03}{(+i)}\,\stackrel{12}{(+)}|\stackrel{56}{[-]}\,\stackrel{78}{[-]} 1 $\frac{1}{2}$ 1 0 $-\frac{1}{2}$ $-1$ $-1$
||\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]}$
4 $ e_{R} $ $\stackrel{03}{[-i]}\,\stackrel{12}{[-]}| 1 $-\frac{1}{2}$ 1 0 $-\frac{1}{2}$ $-1$ $-1$
\stackrel{56}{[-]}\,\stackrel{78}{[-]}
||\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]} $
5 $e_{L}$ $\stackrel{03}{[-i]}\,\stackrel{12}{(+)}|\stackrel{56}{[-]}\,\stackrel{78}{(+)} -1 $\frac{1}{2}$ -1 $-\frac{1}{2}$ 0 $-\frac{1}{2}$ $-1$
||\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]}$
6 $e_{L} $ $\stackrel{03}{(+i)}\,\stackrel{12}{[-]}| -1 $-\frac{1}{2}$ -1 $-\frac{1}{2}$ 0 $-\frac{1}{2}$ $-1$
\stackrel{56}{[-]}\,\stackrel{78}{(+)}
||\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]} $
7 $ \nu_{L}$ $ \stackrel{03}{[-i]}\,\stackrel{12}{(+)}| -1 $\frac{1}{2}$ -1 $\frac{1}{2}$ 0 $-\frac{1}{2}$ $0$
\stackrel{56}{(+)}\,\stackrel{78}{[-]}
||\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]}$
8 $\nu_{L}$ $\stackrel{03}{(+i)}\,\stackrel{12}{[-]}|\stackrel{56}{(+)}\,\stackrel{78}{[-]} -1 $-\frac{1}{2}$ -1 $\frac{1}{2}$ 0 $-\frac{1}{2}$ $0$
||\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]}$
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: \[Table Il.\] The 8-plet of leptons - the members of $SO(1,7)$ subgroup of the group $SO(1,13)$, belonging to one Weyl left handed ($\Gamma^{(1,13)} = -1 = \Gamma^{(1,7)} \times \Gamma^{(6)}$) spinor representation of $SO(1,13)$. It contains the colour chargeless left handed weak charged leptons and the right handed weak chargeless leptons. The rest of notation is the same as in Table \[Table Il.\].
In both tables the vectors are chosen to be the eigenvectors of the operators of handedness $\Gamma^{(n)}$, the generators $\tau^{13}, \, \tau^{23}, \,\tau^{33}$ $ \tau^{38}$, $Y= \tau^{4} + \tau^{23}$ and $Q= Y + \tau^{13}$. They are also eigenvectors of the corresponding $\tilde{S}^{ab}$, $\tilde{\tau}^{Ai}, A=1,2,3$ and $\tilde{Y}, \tilde{Q}$. One easily sees that the right handed vectors (with respect to $SO(1,3)$ ) are weak ($SU(2)_{I}$) chargeless and are doublets with respect to the second $SU(2)_{II}$, while the left handed are weak charged and singlets with respect to $SU(2)_{II}$.
The generators $\tilde{S}^{ab}$ transform each member of a family into the same member of other $2^{\frac{8}{2}-1}$ families. The eight families of the first member of the eightplet of quarks from Table \[Table I.\], for example, that is of the right handed $u$-quark of the spin $\frac{1}{2}$, are presented in the left column of Table \[Table II.\]. The corresponding right handed neutrinos, belonging to eight different families, are presented in the right column of the same table. The $u$-quark member of the eight families and the $\nu$ members of the same eight families are generated by $\tilde{S}^{cd}$, $c,d \in \{0,1,2,3,5,6,7,8\}$ from any starting family.
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$I_R$ $u_{R}^{c1}$ $ \stackrel{03}{[+i]}\,\stackrel{12}{(+)}|\stackrel{56}{(+)}\,\stackrel{78}{[+]}|| $\nu_{R}$ $ \stackrel{03}{[+i]}\,\stackrel{12}{(+)}|\stackrel{56}{(+)}\,\stackrel{78}{[+]}||
\stackrel{9 \;10}{(+)}\:\; \stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]}$ \stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{(+)}\;\;\stackrel{13\;14}{(+)}$
---------- -------------- ------------------------------------------------------------------------------------- ----------- -------------------------------------------------------------------------------------
$II_R$ $u_{R}^{c1}$ $ \stackrel{03}{[+i]}\,\stackrel{12}{(+)}|\stackrel{56}{[+]}\,\stackrel{78}{(+)}|| $\nu_{R}$ $ \stackrel{03}{(+i)}\,\stackrel{12}{[+]}|\stackrel{56}{(+)}\,\stackrel{78}{[+]}||
\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]}$ \stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{(+)}\;\;\stackrel{13\;14}{(+)}$
$III_R$ $u_{R}^{c1}$ $ \stackrel{03}{(+i)}\,\stackrel{12}{[+]}|\stackrel{56}{(+)}\,\stackrel{78}{[+]}|| $\nu_{R}$ $ \stackrel{03}{(+i)}\,\stackrel{12}{[+]}|\stackrel{56}{[+]}\,\stackrel{78}{(+)}||
\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]}$ \stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{(+)}\;\;\stackrel{13\;14}{(+)}$
$IV_R$ $u_{R}^{c1}$ $ \stackrel{03}{(+i)}\,\stackrel{12}{[+]}|\stackrel{56}{[+]}\,\stackrel{78}{(+)}|| $\nu_{R}$ $ \stackrel{03}{[+i]}\,\stackrel{12}{(+)}|\stackrel{56}{[+]}\,\stackrel{78}{(+)}||
\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]}$ \stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{(+)}\;\;\stackrel{13\;14}{(+)}$
$V_R$ $u_{R}^{c1}$ $ \stackrel{03}{(+i)}\,\stackrel{12}{(+)}|\stackrel{56}{(+)}\,\stackrel{78}{(+)} || $\nu_{R}$ $ \stackrel{03}{(+i)}\,\stackrel{12}{(+)}|\stackrel{56}{(+)}\,\stackrel{78}{(+)} ||
\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]}$ \stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{(+)}\;\;\stackrel{13\;14}{(+)}$
$VI_R$ $u_{R}^{c1}$ $ \stackrel{03}{(+i)}\,\stackrel{12}{(+)}|\stackrel{56}{[+]}\,\stackrel{78}{[+]}|| $\nu_{R}$ $ \stackrel{03}{(+i)}\,\stackrel{12}{(+)}|\stackrel{56}{[+]}\,\stackrel{78}{[+]}||
\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]}$ \stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{(+)}\;\;\stackrel{13\;14}{(+)}$
$VII_R$ $u_{R}^{c1}$ $\stackrel{03}{[+i]}\,\stackrel{12}{[+]}|\stackrel{56}{(+)}\,\stackrel{78}{(+)}|| $\nu_{R}$ $\stackrel{03}{[+i]}\,\stackrel{12}{[+]}|\stackrel{56}{(+)}\,\stackrel{78}{(+)}||
\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]}$ \stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{(+)}\;\;\stackrel{13\;14}{(+)}$
$VIII_R$ $u_{R}^{c1}$ $ \stackrel{03}{[+i]}\,\stackrel{12}{[+]}|\stackrel{56}{[+]}\,\stackrel{78}{[+]}|| $\nu_{R}$ $ \stackrel{03}{[+i]}\,\stackrel{12}{[+]}|\stackrel{56}{[+]}\,\stackrel{78}{[+]}||
\stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{[-]}\;\;\stackrel{13\;14}{[-]}$ \stackrel{9 \;10}{(+)}\;\;\stackrel{11\;12}{(+)}\;\;\stackrel{13\;14}{(+)}$
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: \[Table II.\] Eight families of the right handed $u_R$ quark with the spin $\frac{1}{2}$, the colour charge $\tau^{33}=1/2$, $\tau^{38}=1/(2\sqrt{3})$ and of the colourless right handed neutrino $\nu_R$ of the spin $\frac{1}{2}$ are presented in the left and in the right column, respectively. $S^{ab}, a,b \in \{0,1,2,3,5,6,7,8\}$ transform $u_{R}^{c1}$ of the spin $\frac{1}{2}$ and the chosen colour $c1$ to all the members of the same colour: to the right handed $u_{R}^{c1}$ of the spin $-\frac{1}{2}$, to the left $u_{L}^{c1}$ of both spins ($\pm \frac{1}{2}$), to the right handed $d_{R}^{c1}$ of both spins ($\pm \frac{1}{2}$) and to the left handed $ d_{L}^{c1}$ of both spins ($\pm \frac{1}{2}$). They transform equivalently the right handed neutrino $\nu_R$ of the spin $\frac{1}{2}$ to the right handed $\nu_R$ of the spin ($-\frac{1}{2}$), to $\nu_L$ of both spins, to $e_R$ of both spins and to $e_L$ of both spins. $\tilde{S}^{ab}, a,b \in \{0,1,2,3,5,6,7,8\}$ transform a chosen member of one family into the same member of all the eight families.
Let us present also the quantum numbers of the families from Table \[Table II.\]. In Table \[Table IV.\] the handedness of the families $\tilde{\Gamma}^{(1+3)}(= -4i \tilde{S}^{03} \tilde{S}^{12})$, $\tilde{S}^{03}_{L}, \tilde{S}^{12}_L$, $\tilde{S}^{03}_{R}, \tilde{S}^{12}_R$ (the diagonal matrices of $SO(1,3)$ ), $\tilde{\tau}^{13}$ (of one of the two $SU(2)_{I}$), $\tilde{\tau}^{23}$ (of the second $SU(2)_{II}$) are presented.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$i$ $\tilde{\Gamma}^{(1+3)}$ $\tilde{S}^{03}_{L}$ $ \tilde{S}^{12}_L$ $\tilde{S}^{03}_{R}$ $\tilde{S}^{12}_R$ $\tilde{\tau}^{13}$ $\tilde{\tau}^{23}$ $\tilde{\tau}^{4}$ $\tilde{Y}'$ $\tilde{Y}$ $\tilde{Q}$
----- -------------------------- ---------------------- --------------------- ---------------------- -------------------- --------------------- --------------------- -------------------- ---------------- ---------------- -------------
$1$ $-1$ $ - \frac{i}{2}$ $ \frac{1}{2}$ $0$ $0$ $ \frac{1}{2}$ $0$ $-\frac{1}{2}$ $0$ $-\frac{1}{2}$ $0$
$2$ $-1$ $ -\frac{i}{2}$ $ \frac{1}{2}$ $0$ $0$ $-\frac{1}{2}$ $0$ $-\frac{1}{2}$ $0$ $-\frac{1}{2}$ $-1$
$3$ $-1$ $ \frac{i}{2}$ $ - \frac{1}{2}$ $0$ $0$ $ \frac{1}{2}$ $0$ $-\frac{1}{2}$ $0$ $-\frac{1}{2}$ $0$
$4$ $-1$ $ \frac{i}{2}$ $ - \frac{1}{2}$ $0$ $0$ $-\frac{1}{2}$ $0$ $-\frac{1}{2}$ $0$ $-\frac{1}{2}$ $-1$
$5$ $1 $ $0$ $0$ $\frac{i}{2}$ $ \frac{1}{2}$ $0 $ $ \frac{1}{2}$ $-\frac{1}{2}$ $\frac{1}{2}$ $0$ $0$
$6$ $1 $ $0$ $0$ $\frac{i}{2}$ $ \frac{1}{2}$ $0 $ $-\frac{1}{2}$ $-\frac{1}{2}$ $-\frac{1}{2}$ $-1$ $-1$
$7$ $1 $ $0$ $0$ $-\frac{i}{2}$ $- \frac{1}{2}$ $0 $ $ \frac{1}{2}$ $-\frac{1}{2}$ $\frac{1}{2}$ $0$ $0$
$8$ $1 $ $0$ $0$ $- \frac{i}{2}$ $-\frac{1}{2}$ $0 $ $-\frac{1}{2}$ $-\frac{1}{2}$ $-\frac{1}{2} $-1$ $-1$
$
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: \[Table IV.\] The quantum numbers of each member of the eight families presented in Table \[Table II.\] are presented: The handedness of the families $\tilde{\Gamma}^{(1+3)}= -4i \tilde{S}^{03} \tilde{S}^{12}$, the left and right handed $SO(1,3)$ quantum numbers $\tilde{S}^{03}_{L}, \tilde{S}^{12}_L$, $\tilde{S}^{03}_{R}, \tilde{S}^{12}_R$ (of $SO(1,3)$ group in the $\tilde{S}^{mn}$ sector), $\tilde{\tau}^{13}$ of $SU(2)_{I}$, $\tilde{\tau}^{23}$ of the second $SU(2)_{II}$, $\tilde{\tau}^4$, $\tilde{Y}'= \tilde{\tau}^{23} -
\tilde{\tau}^4 \, \tan\tilde{\theta}_2$, taking $\tilde{\theta}^2=0$, $\tilde{Y}
=\tilde{\tau}^{4} + \tilde{\tau}^{23}$, $\tilde{Q}= \tilde{\tau}^{4} + \tilde{S}^{56}$. See also the ref. [@normaproc2010talk].
We see in Table \[Table IV.\] that four of the eight families are singlets with respect to one of the two $SU(2)$ ($SU(2)_{I}$) groups determined by $\tilde{S}^{ab}$ and doublets with respect to the second $SU(2)$ ($SU(2)_{II}$), while the remaining four families are doublets with respect to the first $SU(2)_{I}$ and singlets with respect to the second $SU(2)_{II}$. When the first break appears, to which besides the vielbeins also the spin connections contribute, we expect that if only one of the two $SU(2)$ subgroups of $SO(1,7) \times U(1)$ breaking into $SO(1,3) \times SU(2)\times U(1)$ contributes in the break [@normaproc2010talk], namely that of the charges $\tilde{\tau}^{2 i}$, together with $\tilde{N}^{i}_{-} $, there will be four families massless and mass protected after this break, namely those, which are singlets with respect to $\vec{\tilde{\tau}}^{2}$ and with respect to $\tilde{N}^{i}_{-} $ (Table \[Table IV.\]), while for the other four families the vacuum expectation values of the scalars (particular combinations of vielbeins $f^{\sigma}{}_{s}$, and spin connections $\tilde{\omega}_{abs}, s \in \{5,8\}$) will take care of the mass matrices on the tree level and beyond the tree level.
Discrete symmetries of the theory unifying spin and charges and explaining families {#cpt}
-----------------------------------------------------------------------------------
Let us define the discrete operators of the parity ($ P$) and of the charge conjugation ($C$). $$\begin{aligned}
\label{cp}
P &=& \gamma^0 \, \gamma^8 \, I_{x}\, , \nonumber\\
C &=& \Pi_{Im\, \gamma^a}\, \gamma^a \, K\,.\end{aligned}$$ $K$ means complex conjugation, while in our choice of matrix representation of the $\gamma^a$ matrices $\Pi_{Im\, \gamma^a}\, \gamma^a \,= \gamma^2 \gamma^5 \gamma^7 \gamma^9 \gamma^{11} \gamma^{13}$.
One can easily check that $P$ transforms the $ u_{R}^{c1}$ from the first row in Table \[Table I.\] into the $ u_{L}^{c1}$ of the seventh row in the same table. The $CP$ transforms the fermion states of table \[Table I.\] into the corresponding states of antifermions: $ u_{R}^{c1}$ from the first row in table \[Table I.\] with the spin $\frac{1}{2}$, weak chargeless and of the colour charge ($(\frac{1}{2}, \frac{1}{2 \sqrt{3}})$) into a right handed antiquark $ \bar{u}_{R}^{\bar{c1}}$, weak charged and of the colour charge ($(-\frac{1}{2}, -\frac{1}{2 \sqrt{3}})$) as presented in table \[Table anti\].
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i $$ $|^a\psi_i>$ $\Gamma^{(1,3)}$ $ S^{12}$ $\Gamma^{(4)}$ $\tau^{13}$ $\tau^{23}$ $Y$ $Q$
--- ---------------------------- --------------------------------------------------------------------------------- ------------------ ---------------- ---------------- ---------------- ---------------- ---------------- ----------------
${\rm Octet},\;\Gamma^{(1,7)} =-1,\;\Gamma^{(6)} = 1,$
${\rm of \; antiquarks}$
1 $ \bar{u}_{R}^{\bar{c1}}$ $ \stackrel{03}{[-i]}\,\stackrel{12}{[-]}| 1 $-\frac{1}{2}$ -1 $-\frac{1}{2}$ $0$ $-\frac{1}{6}$ $-\frac{2}{3}$
\stackrel{56}{[-]}\,\stackrel{78}{(+)}
||\stackrel{9 \;10}{[-]}\;\;\stackrel{11\;12}{(+)}\;\;\stackrel{13\;14}{(+)} $
2 $\bar{u}_{R}^{\bar{c1}}$ $\stackrel{03}{(+i)}\,\stackrel{12}{(+)}| 1 $ \frac{1}{2}$ -1 $-\frac{1}{2}$ $0$ $-\frac{1}{6}$ $-\frac{2}{3}$
\stackrel{56}{[-]}\,\stackrel{78}{(+)}
||\stackrel{9 \;10}{[-]}\;\;\stackrel{11\;12}{(+)}\;\;\stackrel{13\;14}{(+)}$
3 $\bar{d}_{R}^{\bar{c1}}$ $\stackrel{03}{[-i]}\,\stackrel{12}{[-]}| 1 $-\frac{1}{2}$ -1 $ \frac{1}{2}$ $0$ $-\frac{1}{6}$ $\frac{1}{3}$
\stackrel{56}{(+)}\,\stackrel{78}{[-]}
||\stackrel{9 \;10}{[-]}\;\;\stackrel{11\;12}{(+)}\;\;\stackrel{13\;14}{(+)}$
4 $ \bar{d}_{R}^{\bar{c1}} $ $\stackrel{03}{(+i)}\,\stackrel{12}{(+)}| 1 $\frac{1}{2}$ -1 $\frac{1}{2}$ $0$ $-\frac{1}{6}$ $\frac{1}{3}$
\stackrel{56}{(+)}\,\stackrel{78}{[-]}
||\stackrel{9 \;10}{[-]}\;\;\stackrel{11\;12}{(+)}\;\;\stackrel{13\;14}{(+)} $
5 $\bar{d}_{L}^{\bar{c1}}$ $\stackrel{03}{(+i)}\,\stackrel{12}{[-]}| -1 $-\frac{1}{2}$ 1 $0$ $\frac{1}{2}$ $\frac{1}{3}$ $\frac{1}{3}$
\stackrel{56}{(+)}\,\stackrel{78}{(+)}
||\stackrel{9 \;10}{[-]}\;\;\stackrel{11\;12}{(+)}\;\;\stackrel{13\;14}{(+)}$
6 $\bar{d}_{L}^{\bar{c1}} $ $\stackrel{03}{[-i])}\,\stackrel{12}{(+)}| -1 $\frac{1}{2}$ 1 $0$ $\frac{1}{2}$ $\frac{1}{3}$ $\frac{1}{3}$
\stackrel{56}{(+)}\,\stackrel{78}{(+)}
||\stackrel{9 \;10}{[-]}\;\;\stackrel{11\;12}{(+)}\;\;\stackrel{13\;14}{(+)} $
7 $ \bar{u}_{L}^{\bar{c1}}$ $\stackrel{03}{(+i)}\,\stackrel{12}{[-]}| -1 $-\frac{1}{2}$ 1 $0$ $-\frac{1}{2}$ $-\frac{2}{3}$ $-\frac{2}{3}$
\stackrel{56}{[-]}\,\stackrel{78}{[-]}
||\stackrel{9 \;10}{[-]}\;\;\stackrel{11\;12}{(+)}\;\;\stackrel{13\;14}{(+)}$
8 $\bar{u}_{L}^{\bar{c1}}$ $\stackrel{03}{[-i]}\,\stackrel{12}{(+)}| -1 $ \frac{1}{2}$ 1 $0$ $-\frac{1}{2}$ $-\frac{2}{3}$ $-\frac{2}{3}$
\stackrel{56}{[-]}\,\stackrel{78}{[-]}
||\stackrel{9 \;10}{[-]}\;\;\stackrel{11\;12}{(+)}\;\;\stackrel{13\;14}{(+)}$
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: \[Table anti\] The 8-plet of antiquarks to the quarks obtained from Table \[Table I.\] by the $CP$ ($= \gamma^2 \gamma^5 \gamma^7
\gamma^9 \gamma^{11} \gamma^{13}K \gamma^0 \gamma^8 \, I_{x}$) conjugation.
The fermion-antifermion asymmetry within the theory unifying spin and charges and explaining families {#matterasymmetry}
=====================================================================================================
As said in the abstract, I shall here follow the ideas from the references [@gross; @rubakovshaposhnikov; @dinekusenko; @tapeiling]. The difference from the studies there in here is, as explained, in the number of families (there are two decoupled groups of four families and consequently two stable families), in the number of gauge fields contributing to the phase transitions and in the types of the gauge fields contributing to phase transitions.
Let us assume that the fermion-antifermion asymmetry is zero, when the expanding universe cools down to the temperature below the unification scale of the spin and the charges that is to the temperature below, let say, $10^{16}$ TeV, when there are eight massless families, manifesting the symmetry $SO(1,3) \times SU(2) \times SU(2) \times U(1) \times SU(3)$, and distinguishing among themselves in the quantum numbers defined by $\tilde{S}^{ab}$.
Then we must investigate, how much do the following processes contribute to the fermion-antifermion asymmetry in non equilibrium thermal processes in the expanding universe:
- The nonconservation of currents on the quantum level due to the triangle anomalies [@tapeiling; @rubakovshaposhnikov; @dinekusenko], which are responsible for $P$ and $CP$ nonconservation $$\begin{aligned}
\label{nonconsrvedcurrents}
\partial^{m} \, j^{A i \,\alpha (i)}_{ m} &=& \frac{(g^{A})^2}{8 \pi^2} \; \frac{1}{2} \,
\varepsilon_{mnpr}\, F^{Ai\, mn} F^{Ai \, pr}.
\end{aligned}$$ Here $j^{A i \,\alpha }_{ m} $ stays for the currents of fermions (and antifermions), which carry a particular charge denoted by a charge group $A$, in our case $A=4$ means the $U(1)$ charge originating in $SO(6)$, $A=3$ means the $SU(3)$ (colour) charge, $A=2_I$ means the weak $SU(2)_I$ charge of the left handed doublets, while $A=2_{II}$ stays for the $SU(2)_{II}$ charge of the right handed singlets before the $SU(2)_{II}$ break, $A=1$ stays for the actual $U(1)$ charge (the [*standard model*]{} like hyper charge after the $SU(2)_{II}$ break and the electromagnetic one after the weak break).
In my case also the fields, which look like scalar fields in $d=(1+3)$, $\tilde{A}^{\tilde{A}i}_{s}$, $s,t \in{5,6,\cdots}$, and to which the fermions are coupled, contribute.
All the fermions and antifermions, which are coupled to a particular gauge field $A^{Ai}_{m}$ and in my case also $\tilde{A}^{\tilde{A}i}_{s}$ contribute to the current $$\begin{aligned}
\label{current}
j^{A i \,\alpha (i)}_m = \psi^{Ai\, \alpha (i)\dagger}\, \gamma^0 \gamma^m \,\psi^{Ai \, \alpha (i)}.
\end{aligned}$$ $(i) \in \{1,8\}$ enumerates families, in my case twice four families which are distinguishable by the quantum numbers originating in $\tilde{S}^{ab}$, namely, after the break of $SU(2)_I$ the lower four families, which are doublets with respect to $\tilde{N}^{i}_{+}$ and $\tilde{\tau}^{I\,i}$ and singlets with respect to $\tilde{N}^{i}_{-}$ and $\tilde{\tau}^{II\,i}$, stay massless, while the upper four families are doublets with respect to $\tilde{N}^{i}_{-}$ and $\tilde{\tau}^{IIi}$ and singlets with respect to $\tilde{N}^{i}_{+}$ and $\tilde{\tau}^{Ii}$. After the electroweak break all the eight families become massive, but the upper four families have no mixing matrix elements since the way of breaking leaves all the $\omega_{msa}$ and $\tilde{\omega}_{msa}$, with $m=0,1,2,3; s=5,6,\cdots$, equal to zero. $\alpha$ distinguishes the multiplets in each family, in my case of the two $SU(2)$ gauge groups $\alpha$ distinguishes the $SU(2)_I$ doublets, that is one colour singlet and one colour triplet, and the $SU(2)_{II}$ doublets, again one colour singlet and one colour triplet. $A^{Ai}_{m}$ are the corresponding gauge fields, with tensors $F^{A}_{mn} = \tau^{A i} \,F^{A i}_{mn}$ and $F^{A i}_{mn}= A^{A i}_{n,m} - A^{A i}_{m,n} +
g^A \, f^{Ai j k} \, A^{A j}_{m}\, A^{A k}_{n} $. (The scalar fields $\tilde{A}^{\tilde{A}i}_{s}$ define tensors $\tilde{F}^{\tilde{A}i\, st}= \tilde{A}^{\tilde{A}i}_{t,s} -
\tilde{A}^{\tilde{A}i}_{s,t} + g^{\tilde{}A} \, f^{A\, ijk} \tilde{A}^{\tilde{A}j}_{s}\,
\tilde{A}^{\tilde{A}k}_{t}$.)
The nonconserved currents affect the fermions and antifermions. (In the later case the $\tau^{A i}$ are replaced by $\bar{\tau}^{A i}$, both fulfilling the same commutation relations $\{\tau^{A i}, \tau^{B j} \}_{-} =
i \delta^{AB} \, f^{A \,ijk} \, \tau^{A k} $, $\{\bar{\tau}^{A i}, \bar{\tau}^{B j} \}_{-} =
i \delta^{AB} \, f^{A \,ijk} \, \bar{\tau}^{A k}$). One obtains $\bar{\tau}^{A i}$ from $\tau^{A i}$ by the $CP$ transformation $P= \gamma^0 \gamma^8 \,I_{x}$, while $C= \prod_{ Im \gamma^a}\,\gamma^a \, K$ (See \[cpt\]).
- The nonconservation of the fermion numbers originating in the complex phases of the mixing matrices of the two times $4 \times 4$ mass matrices for each member of a family, after the two successive breaks causes two phase transitions when the symmetry $SU(2) \times SU(2) \times U(1)$ breaks first to $SU(2) \times U(1)$ and finally to $U(1)$ and the two types of gauge fields manifest their masses while the two groups of four with the mixing matrices decoupled families gain nonzero mass matrices in the first break the upper four families and in the second break the lower four families.
I am following here the references [@gross; @rubakovshaposhnikov; @dinekusenko; @tapeiling]. The nonconservation of currents may be expected whenever the non-Abelian gauge fields manifest a non trivial structure of vacua, originating in the [*instanton*]{} solutions of the Euclidean non-Abelian gauge theories in $(1+3)-$dimensional space, that is in $A^{A}_{m}$, which fulfil the boundary condition $\lim_{r \to \infty} \;\tau^{Ai}\,A^{Ai}_{m} = U^{-1} \partial_m U$, summed over $i$ for a particular gauge group $A$ (and similarly might be that the fields $ \lim_{\rho \to \infty} \;\tilde{\tau}^{\tilde{A}i}\,\tilde{A}^{\tilde{A}i}_{s}=
U^{-1} \partial_s U$, with $r=\sqrt{(x^0)^2 + \vec{x}^2}$ and $\rho = \sqrt{\sum_s}\, (x^s)^2$, for a particular $\tilde{A}$ , contribute as well, where the effect of the triangle anomalies in the case of scalar gauge fields depending on $x^{\sigma}, \sigma = 5,6,7,8$ and the corresponding meaning of the winding numbers distinguishing among the different vacua in this case might be negligible and should be studied). The vacua distinguish among themselves in the topological quantum numbers $n_{A}$ ($ n_{\tilde{A}}$), determined by a particular choice of $U$ $$\begin{aligned}
\label{na}
n_{A} = \frac{(g^{A})^2 }{16 \pi^2} \, \int d^4 x \, \varepsilon_{mnpr}\, Tr (F^{Amn} F^{Apr})=
\frac{(g^{A})^2}{32 \pi^2} \, \int d^{4} x \partial_{m} K^{A\, m}, \end{aligned}$$ where $K^{A}_m= \sum_{i}4 \varepsilon_{mnpr} \,( A^{Ai}_{n} \partial_{p} A^{A}_{r}+
\frac{2}{3}\, g^A f^{Aijk} A^{Ai}_{n} A^{Aj}_{p}A^{Ak}_{r}).$ (Similarly also the topological quantum number $n_{\tilde{A}}$ might be non negligible.)
Instanton solutions fulfilling the boundary condition $\lim_{r \to \infty}\; \tau^{Ai}\, A^{Ai}_{m} = U^{-1} \partial_m U$ for a particular gauge group $A$ (or $\lim_{\rho \to \infty}\; \tilde{\tau}^{\tilde{A}i}\,\tilde{A}^{\tilde{A}i}_{s} =
U^{-1} \partial_m U$ for a particular $\tilde{A}$), each with its own $U$ for a particular $A$ (or $\tilde{A}$), connect vacua $|n_{A}>$ with different winding numbers [^1] $n_{A}$ (and correspondingly for $n_{\tilde{A}}$). The true vacuum $|\theta^{A}>$ is for each $A$ (let it count also $\tilde{A}$) in a stationary situation a superposition of the vacua, determined by the time independent gauge transformation [@gross] ${\cal T}$, ${\cal T} |\theta^{A}> = e^{i\theta^{A}} |\theta^{A}>$, where $\theta^{A}$ is a parameter, which weights the contribution of a vacuum to the effective Lagrange density ${\cal L}_{eff}= {\cal L} + \sum_{A} \, \frac{\theta^{A}}{16 \pi^2} \,
F^{Ai \,mn} \frac{1}{2} \varepsilon_{mnpr}
F^{Ai\, pr}$, for a particular gauge field. ${\cal T}$ acts as the raising operator for the handedness (chirality). The second term of the effective Lagrange density ${\cal L}_{eff}$ violates parity $P$ and then also $CP$. The vacuum state with the definite handedness has also a definite topological quantum number. In the presence of the massless fermions all the vacua $|\theta^{A}>$, for each $A$, are equivalent.
The fermion currents (Eq.(\[current\])) are not conserved in processes, for which the gauge fields are such that the corresponding winding number $n_A$ of Eq. (\[na\]) is nonzero. Correspondingly also the fermion (and antifermion numbers), carrying the corresponding charge, are not conserved $$\begin{aligned}
\label{nafermion}
\Delta n_{A i \,\alpha (i)}= n_{A}. \end{aligned}$$ The fermion number of all the fermions interacting with the same non-Abelian gauge field with nonzero winding number, either of a vector or of a scalar type (whose contribution should be studied and hopefully understood), changes in such processes for the [*same amount*]{}: Any member of a family, interacting with the particular field and therefore also the corresponding members of each family, either a quark or a lepton member of doublets, change for the same amount, before the breaks or after the breaks (in my case first from $SO(1,3) \times SU(2) \times SU(2) \times U(1) \times SU(3)$ to $SO(1,3) \times SU(2) \times U(1) \times SU(3)$ and finally to $SO(1,3) \times U(1) \times SU(3)$) of the symmetries.
For a baryon three quarks are needed. It is the conservation of the colour charge which requires that the lepton number and the baryon number ought to be conserved separately as long as the charge group is a global symmetry. The transformations, which allow rotations of a lepton to a quark or opposite, conserve the fermion number, but not the lepton and not the baryon number.
Instanton solutions of the non-Abelian gauge fields, which connect different vacua (see the refs. [@tapeiling], page 481, and [@rubakovshaposhnikov], page 6), are characterized by the highest value of the [*instanton*]{} field between the two vacua, that is by the [*sphaleron*]{} energy.
The question arises, can the [*instanton*]{} solutions be responsible for the baryon asymmetry of the universe? The authors of the papers [@rubakovshaposhnikov; @dinekusenko] discuss and evaluate the probability for tunnelling from one vacuum to the other at low energy regime and also at the energies of [*sphalerons*]{}. When once the system of gauge fields is in one vacuum the probability for the transition to another vacuum depends not only on the [*sphalerons*]{} height (energy) but also on the temperature. If the temperature is low, then the transition is negligible. At the temperature above the phase transition (the authors [@rubakovshaposhnikov] discuss the electroweak phase transition starting at around $100$ GeV, while in my case there is also the $SU(2)_{II}$ phase transition at around $10^{16}$ GeV or slightly below) when the fermions are massless and the expansion rate of the universe is much slower that the rate of nonconservation of the fermion number, and in the case of non equilibrium processes in phase transitions, the fermion number nonconservation can be large. The authors conclude that more precise evaluations (treating several models) of the probability that in a non thermal equilibrium phase transition and below it the fermion number would not be conserved due to transitions to vacua with different winding numbers in the amount as observed for the (first family) baryon number excess in the universe are needed.
What can be concluded about the fermion number asymmetry, caused by the transitions of gauge fields to different vacua, in my case, where at energies above the $ SU{2}_{II}$ phase transition there are eight families of massless fermions, with the charges manifesting the symmetries first of $SU(2)_I \times SU(2)_{II} \times U(1)$ and correspondingly with the two kinds of the vector gauge $SU(2)$ fields which both might demonstrate the vacua with different winding numbers? In addition also the scalar gauge fields might contribute with their even more rich vacua (if they do that at all). The phase transitions caused first by the break of the symmetry $SU(2)_I \times SU(2)_{II} \times U(1)$ to $SU(2)_{II} \times U(1)$, when the upper four families gain masses (and the corresponding gauge vector fields become massive) and then by the final break to $U(1)$, with the $\tilde{S}^{ab}$ sector causing the masses in both transitions and may be also taking care of the richness of vacua with different winding numbers, might show up after a careful study as a mechanism for generating the fermion-antifermion (or the antifermion-fermion) asymmetry. Although I do not yet see, how do the non equilibrium processes in the first order phase transitions decide about the excess of either fermions or of antifermions.
So, is it in my case possible that the two successive non equilibrium phase transitions leave the excess of antifermions in the case of the upper four families and the excess of fermions in the lower four families? Or there is a negligible excess of either fermions or antifermions in the upper four families? We saw in the ref. [@gn] that an excess of either fermions or antifermions is not important for massive enough (few $100$ TeV) stable fifth family members. The excess of fermions over antifermions is certainly what universe made a choice of for the lower four families, whatever the reason for this fact is. Can this be easier understood within the [*spin-charge-family-theory*]{}? All these need a careful study.
The fermion number nonconservation originates also in the complex phases of the mass mixing matrices of each of the two groups of four family members. It might be that the vacua, triggered by [*instanton*]{} solutions of the gauge vector and scalar fields, and the mass matrices, determined on the tree level by the vacuum expectation values of the scalar gauge fields in the $\tilde{S}^{ab}$ sector, are connected (since in the [*instanton*]{} case also the scalar fields, the gauge fields of charges originating in $\tilde{S}^{ab}$ might exhibit the [*instanton*]{} solutions).
Conclusion
==========
In this contribution I pay attention to the origin of baryon asymmetry of our universe within the [*spin-charge-family-theory*]{} under the assumption that the asymmetry is caused [**i.**]{} by the [*instanton*]{} solutions of the non-Abelian gauge fields which determine vacua with different winding numbers and [**ii.**]{} by the complex matrix elements of the mixing matrices.
The [*spin-charge-family-theory*]{} namely assumes besides the Dirac Clifford algebra objects also the second ones $\tilde{\gamma^a}$ as a necessary mechanism (or better a mathematical tool) which should be used in order that we consistently describe both: spin and charges, as well as families. The second kind is namely responsible for generating families, defining the equivalent representations with respect to the Dirac one. Correspondingly there are besides the two kinds of the vector gauge fields, the $SU(2)_{I}$ and $SU(2)_{II}$, also the scalar gauge fields, the two $SU(2)$ from $SO(4)$ and the two $SU(2)$ from $SO(1,3)$, the superposition of the gauge fields of $\tilde{S}^{ab} (= \frac{i}{4} (\tilde{\gamma^a} \tilde{\gamma^b} - \tilde{\gamma^b}\tilde{\gamma^a}))$, which might contribute to vacua with different winding numbers (what has to be studied). The scalar fields, originating in the $\tilde{S}^{ab}$ charges, are responsible with their vacuum expectation values (and in loop corrections) for the mass matrices of fermions after the breaks of symmetries.
The theory predicts twice four families (which differ in the family quantum numbers in the way that the upper four families are doublets with respect to $\tilde{\tau}^{II \, i}$ and $\tilde{N}^{i}_{-}$, while the lower four families are doublets with respect to $\tilde{\tau}^{I \, i},$ and $\tilde{N}^{i}_{+}$) which all are massless above the last two phase transitions.
What should be clarified in the [*spin-charge-family-theory*]{} is whether the predicted twice four families (rather than once three families of the [*standard model*]{}) and the fact that there are gauge fields belonging to two kinds of generator ($S^{ab}$ and $\tilde{S}^{ab}$) make the baryon number asymmetry easier to be understood within these two phenomena — the [*instanton*]{} responsibility for the fermion number nonconservation and the complex matrix elements of the mixing matrices responsibility for the fermion number nonconservation.
The manifestation of the [*instanton*]{} gauge vector and scalar fields in the determination of the properties of the vacuum might be correlated with the vacuum expectation values of the scalar fields defining the mass matrices of twice the four families. Both manifestations appear in possibly non equilibrium phase transitions of the expanding universe, which cause breaking of particular symmetries and also the fermion number nonconservation. In this contribution I just follow the way suggested by the ref. [@rubakovshaposhnikov] and by the authors cited in this reference, while taking into account the requirement of the [*spin-charge-family-theory*]{}. The fermion number nonconservation obviously ended in the excess of (what we call) fermions for the lower four families, while for the upper four families we have to see whether there is the excess of either the stable fifth family fermions or antifermions. To answer these questions a careful study is needed. It even might be that there was at the non equilibrium phase transitions the same excess of antifermions for the upper four families as it is of fermions for the lower four families, while later the complex matrix elements in the mixing matrices change this equality drastically. But yet it must be understood the origin of both sources of the fermion number nonconservation.
Some useful relations {#sabprop}
=====================
The following Cartan subalgebra set of the algebra $S^{ab}$ (for both sectors) is chosen: $$\begin{aligned}
S^{03}, S^{12}, S^{56}, S^{78}, S^{9 \;10}, S^{11\;12}, S^{13\; 14}\nonumber\\
\tilde{S}^{03}, \tilde{S}^{12}, \tilde{S}^{56}, \tilde{S}^{78}, \tilde{S}^{9 \;10},
\tilde{S}^{11\;12}, \tilde{S}^{13\; 14}.
\label{cartan}\end{aligned}$$ A left handed ($\Gamma^{(1,13)} =-1$) eigen state of all the members of the Cartan subalgebra $$\begin{aligned}
&& \stackrel{03}{(+i)}\stackrel{12}{(+)}|\stackrel{56}{(+)}\stackrel{78}{(+)}
||\stackrel{9 \;10}{(+)}\stackrel{11\;12}{(-)}\stackrel{13\;14}{(-)} |\psi \rangle = \nonumber\\
&&\frac{1}{2^7}
(\gamma^0 -\gamma^3)(\gamma^1 +i \gamma^2)| (\gamma^5 + i\gamma^6)(\gamma^7 +i \gamma^8)||
\nonumber\\
&& (\gamma^9 +i\gamma^{10})(\gamma^{11} -i \gamma^{12})(\gamma^{13}-i\gamma^{14})
|\psi \rangle .
\label{start}\end{aligned}$$ represent the $u_R$-quark with spin up and of one colour.
$ \tilde{S}^{ab} $ generate families from the starting $u_R$ quark In particular $\tilde{S}^{03}(= \frac{i}{2}
[\stackrel{03}{\tilde{(+i)}} \stackrel{12}{\tilde{(+)}} +
\stackrel{03}{\tilde{(-i)}} \stackrel{12}{\tilde{(+)}} +
\stackrel{03}{\tilde{(+i)}} \stackrel{12}{\tilde{(-)}}+
\stackrel{03}{\tilde{(-i)}} \stackrel{12}{\tilde{(-)}}])$ applied on a right handed $u_R$-quark with spin up and a particular colour generate a state which is again a right handed $u$-quark of the same colour. $$\begin{aligned}
\stackrel{03}{\tilde{(-i)}}\stackrel{12}{\tilde{(-)}} &&
\stackrel{03}{(+i)}\stackrel{12}{(+)}| \stackrel{56}{(+)} \stackrel{78}{(+)}||
\stackrel{9 10}{(+)} \stackrel{11 12}{(-)} \stackrel{13 14}{(-)}=\nonumber\\
&&\stackrel{03}{[\,+i]} \stackrel{12}{[\,+\,]}| \stackrel{56}{(+)} \stackrel{78}{(+)}||
\stackrel{9 10}{(+)} \stackrel{11 12}{(-)} \stackrel{13 14}{(-)},\end{aligned}$$ where $$\begin{aligned}
\stackrel{ab}{(\pm i)} &=&
\frac{1}{2}\, ( \gamma^a \mp \gamma^b),
\stackrel{ab}{(\pm 1)} = \frac{1}{2} \,( \gamma^a \pm i\gamma^b),\nonumber\\
\stackrel{ab}{[\pm i]}& =& \frac{1}{2} (1 \pm \gamma^a \gamma^b), \quad
\stackrel{ab}{[\pm 1]} = \frac{1}{2} (1 \pm i \gamma^a \gamma^b), \nonumber\\
\stackrel{ab}{\tilde{(\pm i)}} &=&
\frac{1}{2} (\tilde{\gamma}^a \mp \tilde{\gamma}^b), \quad
\stackrel{ab}{\tilde{(\pm 1)}} =
\frac{1}{2} (\tilde{\gamma}^a \pm i\tilde{\gamma}^b), \nonumber\\
\stackrel{ab}{\tilde{[\pm i]}} &=& \frac{1}{2} (1 \pm \tilde{\gamma}^a \tilde{\gamma}^b), \quad
\stackrel{ab}{\tilde{[\pm 1]}} = \frac{1}{2} (1 \pm i \tilde{\gamma}^a \tilde{\gamma}^b).
\label{deftildefun}\end{aligned}$$
We present below some useful relations which are easy to derive [@pikanorma].
$$\begin{aligned}
\label{relations}
\stackrel{ab}{(k)} \stackrel{ab}{(k)}& =& 0, \; \stackrel{ab}{(k)} \stackrel{ab}{(-k)}
= \eta^{aa} \stackrel{ab}{[k]}, \; \stackrel{ab}{[k]} \stackrel{ab}{[ k]} = \stackrel{ab}{[k]}, \nonumber\\
\stackrel{ab}{[k]} \stackrel{ab}{[-k]} &=& 0,
\stackrel{ab}{(k)} \stackrel{ab}{[ k]} = 0,\quad \;
\stackrel{ab}{[k]} \stackrel{ab}{( k)} = \stackrel{ab}{(k)}, \nonumber\\
\stackrel{ab}{(k)} \stackrel{ab}{[-k]} &=& \stackrel{ab}{(k)}\, ,
\quad \; \stackrel{ab}{[k]} \stackrel{ab}{(-k)} =0.
$$
$$\begin{aligned}
\stackrel{ab}{\tilde{(k)} } \stackrel{ab}{(k)}& =& 0,
\quad \;
\stackrel{ab}{\tilde{(-k)}} \stackrel{ab}{(k)}=
-i \eta^{aa} \stackrel{ab}{[k]},\nonumber\\
\stackrel{ab}{\tilde{( k)}} \stackrel{ab}{[k]}&=&
i \stackrel{ab}{(k)},\quad
\stackrel{ab}{\tilde{( k)}} \stackrel{ab}{[-k]} = 0.
\label{graphbinomsfamilies}\end{aligned}$$
$$\begin{aligned}
N^{\pm}_{+} &=& N^{1}_{+} \pm i \,N^{2}_{+} =
- \stackrel{03}{(\mp i)} \stackrel{12}{(\pm )}\,, \quad N^{\pm}_{-}= N^{1}_{-} \pm i\,N^{2}_{-} =
\stackrel{03}{(\pm i)} \stackrel{12}{(\pm )}\,,\nonumber\\
\tilde{N}^{\pm}_{+} &=& - \stackrel{03}{\tilde{(\mp i)}} \stackrel{12}{\tilde{(\pm )}}\,, \quad
\tilde{N}^{\pm}_{-}= \stackrel{03} {\tilde{(\pm i)}} \stackrel{12} {\tilde{(\pm )}}\,,\nonumber\\
\tau^{1\pm} &=& (\mp)\, \stackrel{56}{(\pm )} \stackrel{78}{(\mp )} \,, \quad
\tau^{2\pm}= (\mp)\, \stackrel{56}{(\mp )} \stackrel{78}{(\mp )} \,,\nonumber\\
\tilde{\tau}^{1\pm} &=& (\mp)\, \stackrel{56}{\tilde{(\pm )}} \stackrel{78}{\tilde{(\mp )}}\,,\quad
\tilde{\tau}^{2\pm}= (\mp)\, \stackrel{56}{\tilde{(\mp )}} \stackrel{78}{\tilde{(\mp )}}\,.\end{aligned}$$
[99]{}
N.S. Mankoč Borštnik, Phys. Lett. [**B 292**]{} (1992) 25-29, Int. J. Theor. Phys. [**40**]{} (2001) 315-337, hep-ph/0711.4681, 94-113, J. Math. Phys. [**34**]{} (1993) 3731-3745, Mod. Phys. Lett. [**A 10**]{} (1995) 587-595. A. Borštnik Bračič, N.S. Mankoč Borštnik, hep-ph/0512062, Phys. Rev. [**D 74**]{} (2006) 073013-16. C.G. Callan, R.F. Dashen, D.J. Gross, Phys. Lett. [**63 B**]{} (1976) 334. V.A. Rubakov, M.E. Shaposhnikov, Phys.Usp. [**39**]{} (1996) 461-502, arxiv:hep-ph/9603208v2. M. Dine, A. Kusenko, Rev.Mod.Phys. [**76**]{} (2004) 1, arxiv:hep-ph/0303065v3. Ta-Pei Cheng, Ling-Fong Li, Gauge theory of elementary particle physics, Clarendon Press, Oxford (1986) N.S. Mankoč Borštnik, H. B. Nielsen, Phys. Lett. [**B 633**]{} (2006) 771-775, hep-th/0311037, hep-th/0509101, arXiv:1001.4679v3. G. Bregar, M. Breskvar, D. Lukman, N.S. Mankoč Borštnik, New J. of Phys. [**10**]{} (2008) 093002, hep-ph/07082846. N.S. Mankoč Borštnik, to appear in the Proceedings to the $13^{th}$ Bled workshop “What Comes Beyond the Standard Models”, Bled 2010. G. Bregar, N.S. Mankoč Borštnik, Phys. Rev. [**80**]{} (2009) 083534, arXiv:0907.0196v1. The work in progress. A. Hernandez-Galeana, N.S. Mankoč Borštnik, to appear in the Proceedings to the $13^{th}$ Bled workshop “What Comes Beyond the Standard Models”, Bled 2010. M.Yu. Khlopov, N.S. Mankoč Borštnik, to appear in the Proceedings to the $13^{th}$ Bled workshop “What Comes Beyond the Standard Models”, Bled 2010. M.Yu. Khlopov et al, to appear in the Proceedings to the $13^{th}$ Bled workshop “What Comes Beyond the Standard Models”, Bled 2010. N.S. Mankoč Borštnik, J. of Math. Phys. [**34**]{} (1993) 3731, N.S. Mankoč Borštnik, H.B. Nielsen, [*J. of Math. Phys.*]{} [**43**]{} (2002) 5782, hep-th/0111257, J. of Math. Phys. [**44**]{} (2003) 4817, hep-th/0303224. A. Borštnik bračič, N.S. Mankoč Borštnik, hep-ph/0401043, p. 31-57.
[^1]: In the ref. [@tapeiling; @gross] the [*instanton*]{} field $\tau^{Ai}\, A^{Ai}_{m}= \frac{r^2}{r^2 + \lambda^2} \,U^{-1} \partial_m U$, with $U =
\frac{x^0 + i\vec{\sigma}^{A} \cdot \vec{x}}{r}$, $r^2 = (x^0)^2 + \vec{x}^2,$ is presented. Operators $\vec{\sigma}^{A}$ are the three Pauli matrices, used to denote the $SU(2)$ gauge group in this case: $\vec{\tau}^{A}=
\frac{\vec{\sigma}^{A}}{2}$, $\{\tau^{Ai}, \tau^{Aj}\}_{-} = i \varepsilon^{ijk} \tau^{Ak}$. The corresponding action $ \int d^{4}x\, \frac{1}{2} \varepsilon_{mnpr} \, F^{Ai\, mn} F^{Ai pr}= \frac{8 \pi^2}{g^2}$, while $U =
\frac{x^0 + i\vec{\sigma}^{A} \cdot \vec{x}}{r}$, defines the $n=1$ vacuum state.
|
---
abstract: 'We describe a novel experimental setup that combines the advantages of both laser-induced fluorescence and cavity ring-down techniques. The simultaneous and correlated measurement of the ring-down and fluorescence signals yields absolute absorption coefficients for the fluorescence measurement. The combined measurement is conducted with the same sample in a single, pulsed laser beam. The fluorescence measurement extends the dynamic range of a stand-alone cavity ring-down setup from typically three to at least six orders of magnitude. The presence of the cavity improves the quality of the signal, in particular the signal-to-noise ratio. The methodology, dubbed cavity-enhanced laser-induced fluorescence (CELIF), is developed and rigorously tested against the spectroscopy of 1,4-bis(phenylethynyl)benzene in a molecular beam and density measurements in a cell. We outline how the method can be utilised to determine absolute quantities: absorption cross sections, sample densities and fluorescence quantum yields.'
author:
- 'Scott E. Sanders'
- 'Oliver R. Willis'
- 'N. Hendrik Nahler'
- Eckart Wrede
title: 'Absolute absorption and fluorescence measurements over a dynamic range of 10$^\textbf{6}$ with cavity-enhanced laser-induced fluorescence'
---
Introduction {#sec:intro}
============
Laser-induced fluorescence (LIF) has become a well established spectroscopic technique since the advent of the laser.[@Donovan2007; @Kinsey1977; @Zare2012] It is an indirect absorption technique as the spontaneously emitted photons are recorded as signal after incident laser light has been absorbed by a species. Most commonly, the wavelength integrated fluorescence is detected leading to a fluorescence excitation spectrum that is equivalent to an absorption spectrum. Additional information can be obtained from the wavelength-dispersed fluorescence. In gerneral, LIF possesses intrinsically low noise allowing the detection of very low concentrations in confined volumes. High spatial resolution is usually achieved by right-angle collection of the fluorescent light with a large-aperture lens. An absolute measurement of absorbance requires detailed knowledge of the fluorescence process and careful calibration of the detection system (fluorescence quantum yield, geometrical setup, spectral response of the detector, *etc.*). Using amplified photodetectors, laser-induced fluorescence can be measured linearly over a large dynamic range from single-photon counting to saturation of the photodetector. The measurement is virtually background free if stray light from the incident laser is suppressed effectively, *e.g.* with an optical filter. If the fluoresecence lifetime is long compared to the laser pulse length, stray light can also be discriminated against by time gating of the signal. If the fluorescence lifetime exceeds the time between collisions, quenching of the fluorescence can occur, reducing the signal potentially to a level which makes a measurement impossible. In addition, quenching, predissociation and other non-radiative processes complicate a straightforward relationship between signal and concentration, due to possible dependences of these processes on the excited state. Nevertheless, LIF measurements have been carried out in different media from collision-free environments, *e.g.* molecular beams, to liquids and solids.[@Brecher1976]
Over the last two decades, cavity ring-down spectroscopy (CRDS) has become a well-established and widely applied spectroscopic technique.[@Berden2009; @O'Keefe1988] CRDS is based on the Beer-Lambert law and performs a direct absorption measurement. Consequently, fluorescence of the sample is not a detection requirement. In a CRD experiment, light enters a cavity formed by two highly reflective mirrors (typically $R > 0.999$). During each pass a small fraction of light leaks out through the mirrors and is detected as an exponential decay behind the exit mirror. The inverse of the decay rate of the signal is referred to as ring-down time, which for an empty cavity is given by $$\tau_0 = \frac{d}{c (1 - R)},$$ where $d$ is the distance between the mirrors of reflectivity $R$ and $c$ is the speed of light. Depending on the reflectivity of the mirrors, several thousand round trips can be achieved. With cavity lengths of the order of 1m the effective pathlengths are of the order of many kilometers. The absorption of light by a sample inside the cavity causes an additional loss and consequently a shorter ring-down time. The reduced ring-down time, $\tau$, is directly linked to the absorption coefficient $$\label{eq:alpha_CRD}
\alpha = \sigma \rho = \frac{1}{c} \left(\frac{1}{\tau} - \frac{1}{\tau_0}\right),$$ where $\sigma$ is the absorption cross section and $\rho$ is the number density of the sample, leading to a photon loss per pass of $\alpha d$. As the ring-down time is the measurable, the technique is immune to power fluctuations of the incident laser source leading to an absolute and self-calibrated measurement.
Eq. \[eq:alpha\_CRD\] assumes that the entire cavity is filled by the sample which is the case in a typical cell experiment. If the sample volume is localized, *e.g.* inside a molecular beam, then the right-hand side of eq. \[eq:alpha\_CRD\] needs to be multiplied by the ratio of the cavity and sample lengths, $d/s$, to account for the higher density in the smaller sample volume.
In cavity ring-down spectroscopy, the absorbance is measured over the pathlength through the sample (integrated column density) and, in contrast to LIF, is not spatially resolved. Similar to LIF, CRDS has been applied to gas-phase, liquid and solid samples. The direct absorption measurement removes the issue of fluorescence quenching but introduces the problem of large, unwanted losses in liquid and solid samples (scattering and reflection). CRDS measurements are not background free as they are based on the detection of a change in signal. The dynamic range is defined by the minimal and maximal detectable change in ring-down time. In a typical CRDS setup, the dynamic range spans three orders of magnitude.[@Lehmann2008] The sensitivity of LIF is generally superior for spatially confined samples whereas CRDS can match or exceed this sensitivity for larger samples that fill the entire length of the cavity.
In this study, we present a novel, direct combination of the CRD and LIF techniques using a single, pulsed laser beam that we name cavity-enhanced laser-induced fluorescence or CELIF. The CRD and LIF signals are recorded simultaneously on a shot-to-shot basis. For the first time the fluorescence signal, cavity ring-down intensity and cavity ring-down time are cross-correlated in a way that significantly enhances both techniques. The CRD measurement provides the absolute calibration of the LIF signal. The combined techniques lower the detection limit of the CRD measurement by several orders of magnitude.
Previously, several groups have used different combinations of CRD and LIF, not necessarily using a single laser beam, to measure, *e.g.*, fluorescence quantum yields and quenching rates.[@Spaanjaars1997; @Hagemeister1999; @Bahrini2006; @Tokaryk2007] Richman *et al.* detected fluorophor-doped aerosols within a cavity by their fluorescence signal.[@Richman2005] Furthermore, CRD spectroscopy, instead of Rayleigh scattering, has recently been used to calibrate density measurements in flames via LIF.[@Dreyer2001; @Luque2004; @Lamoureux2010] None of these previous studies used the cross-correlation of the CRD and LIF signals as presented here.
In this paper, we describe and fully characterize the novel CELIF technique in detail. We outline how absolute quantities, such as fluorescence quantum yields, can be directly extracted from the single beam CELIF measurement by cross-normalization of the LIF and CRD signals. Absorption coefficients are accessible with a single method and measurements can be carried out over a dynamic range spanning more than six orders of magnitude. The ring-down cavity rejects the majority of the laser light and stretches the laser pulse in time. The amount of light inside the cavity, generating the LIF signal, is measured by the integrated cavity ring-down signal. This measurement is used for a very robust shot-to-shot normalization of the LIF signal against the light intensity, leading to a much enhanced signal-to-noise ratio. In comparison to a single-pass LIF measurement, CELIF greatly reduces saturation and power broadening due to the much lower photon densities and effectively eliminates stray light due to the transversal mode structure of the cavity.
Section \[sec:experiment\] describes the methodology and how the integrated CRD signal is used for the shot-to-shot normalization of the LIF signal with respect to the fluctuating intensity of a pulsed laser. Absolute absorption coefficients determined from the CRD measurement can then be used to cross calibrate the LIF signal. In this work, CELIF was implemented using a pulsed molecular beam setup and a cell experiment. Section \[sec:results\] presents measurements that scrutinize the methodology, particularly the increased dynamic range. In section \[sec:discussion\], we discuss the characteristics of the method including its enhanced limit of detection and how absolute fluorescence quantum yields can be determined. In the conclusions, we outline the power of the CELIF technique and its most suitable applications.
Experiment {#sec:experiment}
==========
The fundamental idea of this setup is the combination of CRDS and LIF in a single, pulsed laser beam. A schematic layout of the setup is shown in fig. \[fig:exp\]. It is a straightforward combination of a classical LIF and pulsed CRDS setup: the sample is intersected by the laser beam that is confined in the cavity and the laser-induced fluorescence is collected at right angles. In our setup the wavelength-integrated fluorescence is recorded. Without loss of generality, the technique can be used in a setup where the dispersed fluorescence spectrum is measured. In principle, the technique should be widely applicable to any gas-phase, liquid and even solid samples which have been used in previous CRDS studies as long as the fluorescence light can be extracted from the sample volume.
In the following subsection we derive how in our combined CRD/LIF setup the cavity ring-down signal is used to normalize and absolutely calibrate the LIF signal.
CELIF methodology {#sec:method}
-----------------
In a general LIF measurement, the time-integrated LIF signal, $S{^\text{LIF}}$, is proportional to the light intensity, $I{^\text{LIF}}$, that has interacted with the sample within the LIF probe volume: $$\label{eq:SLIF}
S{^\text{LIF}}(\lambda) = \alpha(\lambda) \cdot \Gamma(\lambda) \cdot g \cdot I{^\text{LIF}},$$ where $\lambda$ is the excitation wavelength, $\Gamma$ is the fluorescence quantum yield and $g$ is a geometry dependent factor of the detection system. In principle, $g$ is also a function of $\lambda$ as the fluorescence spectrum may depend on the excited state. However, this dependence can only be considered if the dispersed fluorescence spectrum is recorded as a function of excitation wavelength. In order to obtain the absorption coefficient, $\alpha$, from a fluorescence excitation measurement, $S{^\text{LIF}}$ and $I{^\text{LIF}}$ need to be measured. The factor $g$ is an instrument function that is not readily available but can be determined via a meticulous external calibration. The fluorescence quantum yield, $\Gamma$, is generally unknown and needs to be measured or predicted from theory in a separate study.
Fundamentally, CELIF is a LIF measurement where the simultaneous CRD measurement is used for the normalization and the calibration of the LIF signal. In the following, we derive how the time-integrated CRD signal is correlated to $I{^\text{LIF}}$ and how it is subsequently used to provide the normalization of $S{^\text{LIF}}$ to eliminate shot-to-shot fluctuations in the laser intensity from eq. \[eq:SLIF\]. We then describe how the absolute absorption coefficient determined via the ring-down time measurement (eq. \[eq:alpha\_CRD\]) is used for the absolute calibration of the normalized LIF signal such that in a CELIF measurment prior knowledge of $g$ and $\Gamma$ is not required.
However, there are differences in the measurement of the fluorescence signal between a CELIF and a typical LIF experiment. Only a fraction of the laser light is transmitted through the cavity entrance mirror and is subsequently interacting with the sample. As a consequence, the initial fluorescence is very small in comparison to single-pass LIF. The light confined in the cavity undergoes up to several thousand round trips, interacting with the sample twice on each of them. Each of these interactions induces further fluorescence so that the integrated fluorescence leads to an appreciable LIF signal. Effectively, the cavity stretches the short laser pulse into the exponential decay characteristic for the cavity.
Without loss of generality, we base the following derivation on a fixed excitation wavelength and on a sample distribution that is symmetric with respect to a LIF probe volume that is placed at the center of the cavity. The photon loss per single pass is ${\mathcal{L}}= \sigma \rho s $, with the sample length, $s$, given that ${\mathcal{L}}\ll 1$. The light intensity at the center after entering the cavity is $$\label{eq:ILIF0}
I_0{^\text{LIF}}= I_\text{L} T {\mathcal{F}}\left( 1 - \frac{{\mathcal{L}}}{2}\right),$$ where $I_\text{L}$ is the laser intensity incident on the cavity entrance mirror and $T{\mathcal{F}}$ is the fraction of light that can be resonantly coupled into the cavity. $T$ is the transmission of the mirror and ${\mathcal{F}}$ is the fraction of the laser bandwidth that is resonant with the mode structure of the cavity.
In addition to the sample loss, a fraction of light leaks out of the cavity upon reflection at the mirror. The light intensity at the center of the cavity after $i$ single passes is $$I_i{^\text{LIF}}= I_0{^\text{LIF}}\left[ \left(1-{\mathcal{L}}\right) R\right]^i,$$ where $R$ is the reflectivity of the cavity mirrors. Using the summation rule for geometric series, the summed light intensity, $I_n{^\text{LIF}}$, that has crossed the LIF probe volume after $n$ single passes and the integrated intensity in the limit $n\to\infty$, $I{^\text{LIF}}$, are: $$\begin{aligned}
I_n{^\text{LIF}}&= I_0{^\text{LIF}}\sum_{i=0}^n \left[ \left(1-{\mathcal{L}}\right) R\right]^i \nonumber\\
&= I_0{^\text{LIF}}\frac{{1 - \left[(1 -{\mathcal{L}}) R\right]^{n+1}}}{1 - (1 -{\mathcal{L}}) R}\\
\label{eq:ILIFint}
I{^\text{LIF}}&= \lim_{n\to\infty} I_n{^\text{LIF}}= \frac{I_0{^\text{LIF}}}{1 - (1 -{\mathcal{L}}) R}\end{aligned}$$ Using eq. \[eq:ILIF0\] and the approximations $\mathcal{L}\ll 1$ and $T\approx 1 - R$ shows that $I{^\text{LIF}}\approx I_\text{L}\mathcal{F}$. This means that, through the repeated use of the decaying light pulse inside the cavity, the amount of light that interacts with the sample is the amount incident on the entrance mirror that is resonant with the mode structure of the cavity. We will discuss the value of ${\mathcal{F}}$ for our setup in section \[sec:CELIF\_characteristics\].
Similarly, we can derive the integrated CRD intensity, $I{^\text{CRD}}$, at the exit mirror taking into account that, following the initial single pass, light is only collected after every round trip: $$\begin{aligned}
I_0{^\text{CRD}}&= I_\text{L} T^2 {\mathcal{F}}\left( 1 - {\mathcal{L}}\right)\label{eq:ICRD0}\\
I_n{^\text{CRD}}&= I_0{^\text{CRD}}\sum_{j=0}^{n/2} \left[ \left(1-{\mathcal{L}}\right) R\right]^{2j} \nonumber \\
&= I_0{^\text{CRD}}\frac{{1 - \left[(1 -{\mathcal{L}})^2 R^2\right]^{n+1}}}{1 - (1 -{\mathcal{L}})^2 R^2}\\
I{^\text{CRD}}&= \lim_{n\to\infty} I_n{^\text{CRD}}= \frac{I_0{^\text{CRD}}}{1 - (1 -{\mathcal{L}})^2 R^2}\label{eq:ICRD}\end{aligned}$$
In the limit of an empty cavity (${\mathcal{L}}= 0$) and $T=1-R$, the measured, time-integrated light intensity imparting on the detector is $$\label{eq:transmission}
I{^\text{CRD}}= I_\text{L} T{\mathcal{F}}/2$$ where $T{\mathcal{F}}/2$ is the cavity transmission function. The comparison of the integrated LIF and CRD intensities yields: $$\label{eq:ILIF_accurate}
I{^\text{LIF}}= I{^\text{CRD}}\frac{[1+(1-{\mathcal{L}})R](1-{\mathcal{L}}/2)}{T(1-{\mathcal{L}})}$$ In the limit of small loss ${\mathcal{L}}\ll 1$ and additionally $R \approx 1$, $I{^\text{LIF}}$ can be approximated as $$\label{eq:ILIF_approx}
I{^\text{LIF}}\approx I{^\text{CRD}}\frac{1+R}{T} \approx I{^\text{CRD}}\frac{2}{T}$$ As an example, for a relatively poor mirror reflectivity of $R = 0.998$ and a large loss of ${\mathcal{L}}= 0.001$, which would be amenable to a single-pass absorption measurement, the relative error of this approximation is smaller than $10^{-3}$. This error is dominated by the mirror reflectivity and reduces to $5\cdot 10^{-5}$ for $R = 0.9999$.
Therefore, we define the normalized CELIF signal as $$\label{eq:CELIF_norm}
S{^\text{CELIF}}= \frac{S{^\text{LIF}}}{I{^\text{CRD}}}.$$ Note that $S{^\text{CELIF}}$ is a relative quantity (unitless in our case) that can be calibrated to equal the absolute absorption coefficient, $\alpha$, derived from the measurement of ring-down times. The calibrated CELIF absorption coefficient is $$\label{eq:alpha_LIF}
\alpha = \sigma\rho = {\mathcal{K}}\cdot S{^\text{CELIF}},$$ where ${\mathcal{K}}$ is the proportionality factor that can be determined from a simultaneous LIF and CRD measurement, provided the absorption leads to a sufficient reduction in ring-down time (equations \[eq:alpha\_CRD\] and \[eq:alpha\_LIF\]): $$\label{eq:cal_LIF}
{\mathcal{K}}= \frac{I{^\text{CRD}}}{S{^\text{LIF}}} \frac{1}{c} \left(\frac{1}{\tau} - \frac{1}{\tau_0}\right).$$
This method provides an absolute calibration of the LIF signal. Therefore the limited dynamic range of the CRD method can be extended towards the generally lower detection limit of the LIF method. The combination of LIF and CRDS into CELIF enables the measurement of an absolute absorption coefficient over the combined dynamic range.
Equations \[eq:alpha\_LIF\] and \[eq:cal\_LIF\] hold true for an absorption measurement at a fixed wavelength. Generally, ${\mathcal{K}}$ is a function of the excitation wavelength (fluorescence quantum yield, mirror transmission) and the fluorescence spectrum (spectral response of the detection system). A single-pass LIF measurement needs to account for the wavelength dependencies of the quantum yield and the detection system. In addition, a calibrated CELIF spectrum requires knowledge of the wavelength-dependent exit mirror transmission. In section \[sec:yield\] we describe a procedure that uses the simultaneously measured ring-down times and Rayleigh scattering collected with the LIF detection setup to determine the wavelength dependence of $T$ based on our unchanged setup. With two sets of UV and visible mirrors we found the transmission to vary by less than 5% over a wavelength range of 1 nm which will allow to approximate the transmission as constant in many applications. We note, however, that the common approximation $T = 1 - R$, with $R(\lambda)$ derived from an empty-cavity ring-down scan, could not be generally applied over the usable wavelength range of each set of mirrors.
Apparatus
---------
A general CELIF apparatus is shown in fig. \[fig:exp\]. It consists of a standard ring-down cavity including beam-shaping, mode-matching optics and a suitably fast photodetector (x-axis in figure). A typical LIF detection system, including a collimation lens and photodetector, is added at right angles to the cavity axis, preferably at the center of the sample (y-axis in figure). Two experimental setups were used in this study. Setup 1 introduced the sample via an unskimmed molecular beam (z-axis in figure) whereas setup 2 was used for cell measurements where the entire cavity was filled with the sample gas.
![Experimental setup. The sample volume is situated at the center of the ring-down cavity and the fluorescence is collected at right angles.[]{data-label="fig:exp"}](Fig01){width="8.6cm"}
Setup 1 was based on our CRD spectrometer that was used to study the torsional motions of jet-cooled 1,4-bis(phenylethynyl)benzene (BPEB), the details of which are described in ref. . Briefly, solid BPEB was sublimated in a heated oven attached to the front of a pulsed solenoid valve (Parker, General Valve Series 9). The gaseous BPEB was picked-up by the argon carrier gas in the channel of the oven and cooled in the subsequent supersonic expansion. The BPEB sample density could be controlled over more than three orders of magnitude by varying the oven temperature. The cavity axis crossed the molecular beam approximately 5 mm downstream of the oven orifice. The $\text{S}_1 \leftarrow\text{S}_0$ transition of BPEB is very strong[@Beeby2002; @CrossSectionNote] and was excited over a wavelength range of 317–321.5 nm. Over this range, the reflectivity of the cavity mirrors (Layertec, center wavelength 330 nm) varied from 99.8 to 99.9% leading to empty-cavity ring-down times of $1.2-2.5~\mu$s (cavity length $\approx 84$ cm). The doubled output of the dye laser (Sirah Cobra-Stretch, pumped by Continuum Surelite I-10, 10 Hz repetition rate) ranged from 30 to $100~\mu$J with a bandwidth of 0.045 cm$^{-1}$ and a pulse length of $\approx 5$ ns. The LIF collection optics imaged the full probe volume (overlap of molecular and laser beams) onto the photodetector.
With setup 2 we carried out N$_2$ Rayleigh scattering (at 583.5 nm) and acetone fluorescence (at 313 nm) measurements to verify the methodology as described in section \[sec:method\]. We note that the same detector setup was used for the Rayleigh scattering and fluorescence measurements. In general, CELIF signal is referred to signal collected by the LIF detection setup normalized by the CRD intensity. Therefore CELIF signal in the course of this paper refers to Rayleigh scattering or fluorescence signals. The gas pressure of the filled cavity (length $\approx 81$ cm) was monitored over a range of $0.01-1000$ mbar with capacitance manometers (Leybold CR090/CTR100 and MKS 626A) to cross reference the spectroscopically determined sample densities. For the Rayleigh scattering measurements, empty-cavity ring-down times in excess of 40 $\mu$s were achieved with the corresponding set of mirrors ($R>99.99\%$). For the fluorescence measurements, the same set of mirrors was used as in setup 1 and the drop-off in their reflectivity to about 99.7% shortened the ring-down time to $\approx 800$ ns. The output of the (doubled) dye laser (Quanta-Ray PDL-2, pumped by Continuum Minilite II, 10 Hz repetition rate) was $\approx 300~\mu$J at 583.5 nm and $\approx 100~\mu$J at 313 nm, both with a bandwidth of 0.3 cm$^{-1}$. The LIF optics imaged a 3 mm long probe volume onto the photodetector.
In both setups we use two identical photomultipliers (Hamamatsu, H7732-10 module with R928 tube), the signals of which were simultaneously recorded using a two-channel digitizer (National Instruments NI PCI-5124, 12-bit for setup 1 and AlazarTech ATS460, 14-bit for setup 2).
Data acquisition and analysis {#sec:daq}
-----------------------------
For each laser shot, the LIF transient and the CRD transient are measured simultaneously as shown in fig. \[fig:transients\] (a) and (b), respectively. The CRD transient follows the typical exponential decay (fig. \[fig:transients\] (b)) from which the ring-down time, $\tau$, is extracted by a non-linear least squares fit. In combination with the empty-cavity ring-down time, $\tau_0$, the absorption coefficient, $\alpha$, is determined as in a typical CRD experiment. In the following text, a CRD measurement or spectrum is based on the ring-down times according to eq. \[eq:alpha\_CRD\].
We also integrate the CRD transient to extract $I{^\text{CRD}}$ (*cf.* eq. \[eq:ICRD\]) which is proportional to the light intensity inside the ring-down cavity (eq. \[eq:ILIF\_approx\]). The LIF transient which follows the ring-down decay is integrated to yield the LIF signal, $S{^\text{LIF}}$. According to eq. \[eq:CELIF\_norm\] the CELIF signal, $S{^\text{CELIF}}$, is obtained through a shot-to-shot normalization of $S{^\text{LIF}}$ with respect to $I{^\text{CRD}}$.
As alluded to earlier, CELIF is a LIF measurement where, without further calibration, only relative quantities are obtained. In CELIF this absolute calibration is provided by the absorption coefficient, $\alpha$, from the simultaneous CRD measurement, see eq. \[eq:alpha\_LIF\]. This calibration is particularly robust as both the LIF and CRD measurements use the same laser photons and sample molecules.
Results {#sec:results}
=======
![Simultaneously recorded (a) LIF and (b) CRD transients of the transition at 319.44 nm (*cf.* fig. \[fig:BPEBspectra\]) of BPEB. The LIF transient follows the same exponential decay as the CRD transient due to the negligible sub-ns fluorescence lifetime of the excited state[@Fujiwara2008] compared to the 1.80 $\mu$s ring-down time. The same integration limits were used to derive $S{^\text{CELIF}}$, see eq. \[eq:CELIF\_norm\]. Simultaneously recorded (c) Rayleigh scattering (measured with the LIF detector) and (d) CRD transients of BPEB at 321.0 nm. The vertical scale in panel (c) is magnified to show the presence of Rayleigh scattering. The absolute noise is equivalent to the baseline noise in panel (a).[]{data-label="fig:transients"}](Fig02 "fig:"){width="8.6cm"}\
Based on our previous CRD work,[@Greaves2006] we chose BPEB as our model system. One of its characteristic properties is a fluorescence lifetime of $<1$ ns following the $\text{S}_1 \leftarrow \text{S}_0$ electronic excitation.[@Fujiwara2008] The left column of fig. \[fig:transients\] shows samples of simultaneously recorded LIF and CRD transients on resonance. As the fluorescence lifetime is negligible compared to the 1–2 $\mu$s ring-down times, the LIF transient follows the exponential decay of the ring-down signal. Based on this fact, we chose the same integration boundaries for the LIF signal ($S{^\text{LIF}}$) and ring-down intensity ($I{^\text{CRD}}$), indicated by the hashed areas in the figure, to derive the normalized CELIF signal ($S{^\text{CELIF}}$). In the more general case, where the fluorescence lifetime is no longer small in comparison to the ring-down time, both the entire LIF and CRD transients need to be integrated to ensure correct normalization.
![(a) Dependence of the LIF signal ($S{^\text{LIF}}$) on the incident laser intensity as measured by the ring-down detector ($I{^\text{CRD}}$). Data has been recorded using the fluorescence of acetone at the excitation wavelength of 313 nm at two pressures of 0.1 mbar ($\circ$) and 0.3 mbar ($\bullet$) leading to losses per pass of $(1.42\pm 0.02)\cdot 10^{-3}$ and $(4.31\pm 0.04)\cdot 10^{-3}$ respectively (linear least-squared fit). Error bars are significantly smaller than the symbol size. (b) Pressure-dependent N$_2$ Rayleigh scattering at 583.5 nm. The photon loss is derived from the ring-down times of CRD measurement (eq. \[eq:alpha\_CRD\]) and the CELIF data (shifted for clarity) is calibrated according to eq. \[eq:alpha\_LIF\] at 1 bar. Each data point is based on an average of 311 laser shots in a 10 mbar pressure range. The error bars represent the spread of the data. The solid lines are the linear least-squared fits to the data.[]{data-label="fig:powerdep"}](Fig03 "fig:"){width="8.6cm"}\
In order to assess detection limits and signal-to-noise ratios, we also measured off-resonance transients of BPEB at 321 nm shown in the right column of fig. \[fig:transients\]. As BPEB has an extended $\pi$-system along its major axis leading to a large polarizability, we can detect its Rayleigh scattering in a seeded molecular beam (fig. \[fig:transients\]c) with our CELIF setup. The corresponding CRD transient (fig. \[fig:transients\]d) did not cause a reduction in ring-down time with respect to the empty-cavity demonstrating the higher sensitivity of the CELIF setup.
To confirm the validity of eqs \[eq:ILIF\_accurate\] to \[eq:alpha\_LIF\], we measured the linearity of the LIF signal with respect to the laser intensity, as measured via the integrated CRD intensity, $I{^\text{CRD}}$, and the dependence of the CELIF signal, $S{^\text{CELIF}}$, on the sample density, $\rho$. Fig. \[fig:powerdep\](a) shows the fluorescence signal of acetone following laser excitation at 313 nm as a function of laser intensity. The cavity was filled with 0.1 and 0.3 mbar of acetone respectively. Each data point represents the average over 5000 laser shots. The standard errors in laser intensity and LIF signal are smaller than the size of the symbols. The linear least-squares fits demonstrate the validity of the shot-to-shot normalization using eq. \[eq:CELIF\_norm\] and show the absence of any saturation effects in the measurement range.
The linearity of the CELIF signal with respect to the sample density of a filled cavity was confirmed with N$_2$ Rayleigh scattering at 583.5 nm (fig. \[fig:powerdep\]b) in the pressure range 0.1–1000 mbar. The simultaneously measured ring-down times provide an absolute measurement for the absorption and therefore the photon loss per pass using eq. \[eq:alpha\_CRD\]. The Rayleigh scattering signal obtained by the CELIF setup was calibrated to match the CRD photon loss at a N$_2$ pressure of 1 bar according to eq. \[eq:alpha\_LIF\]. The slopes of the independent linear least squared fits to the data in fig. \[fig:powerdep\]b are indistinguishable proving the validity of eqs \[eq:CELIF\_norm\]–\[eq:cal\_LIF\].
Moreover, we found that during these experiments with variable cavity pressures, movement of the mirrors could be observed by the ring-down measurement. Only after we improved the mechanical stability of our cavity and mirror mounts, we were able to set up a cavity unaffected by the change from vacuum to atmospheric pressure. This was verified by comparison of the CRD density measurement to the reading of the capacitance manometer. However, the simultaneous CELIF measurement is immune to these cavity misalignments as these are fully compensated for by the shot-to-shot normalization procedure leading to higher quality data.
Figure \[fig:BPEBspectra\] shows a series of simultaneously recorded CRD and CELIF spectra of jet-cooled BPEB as a function of relative sample density $\rho / \rho_0$. The dynamic range of this particular CRD experiment is rather poor as shown in the left column. The main limitations are set by the small sample volume (diameter of the molecular beam), the shot-to-shot variation of sample concentration and the comparatively low reflectivity of the ring-down mirrors. In contrast, the CELIF measurement covers a much larger dynamic range as shown in the right column. With a change in sample density, the amplification of the photodetector was adapted to avoid saturation of the detection system. However, the series of CELIF measurements was internally calibrated each time the amplification of the photodetector was changed and we were able to follow the relative sample density, $\rho / \rho_0$, over a range of more than three orders of magnitude. The very low baseline noise in the CELIF measurements is evident in panels (b) and (d). Even in panel (f) where the sample density is reduced by three orders of magnitude the CELIF baseline noise is comparable to the CRD baseline noise at the highest sample concentration, panel (a).
![Comparison of simultaneously recorded CRD (left column) and CELIF (right column) spectra of jet-cooled BPEB at different relative concentrations, $\rho/\rho_0$. The CRD spectra are shown to the same scale as the absolute noise does not change with concentration. The absolute photon loss per pass of the CELIF spectrum (b) was calibrated to the simultaneously recorded CRD spectrum (a). The CELIF spectra (d) and (f) were calibrated with respect to spectrum (b). For details of the BPEB spectroscopy see ref. .[]{data-label="fig:BPEBspectra"}](Fig04){width="8.6cm"}
Discussion {#sec:discussion}
==========
Our work has shown that recorded CELIF signals are linearly related to absorbances derived from CRD measurements. In this single laser beam setup, the cavity ring-down part provides the absolute scale for the LIF measurement without the requirement of any external calibration. With this combination of the two techniques, the high sensitivity of LIF greatly extends the accessible absorbance measurement range of CRDS on an absolute scale due to a greatly improved signal-to-noise ratio. The presence of the cavity introduces changes to the LIF technique that we will discuss in the following sections.
Sources of noise and error
--------------------------
Fig. \[fig:BPEBspectra\] clearly shows the improvement in quality of the CELIF spectra compared to the CRD spectra. We will first discuss the sources of noise (or error) in both techniques. Shot-to-shot fluctuations in the sample density affect both techniques similarly as the same molecules are probed.
For the CRD technique, mode fluctuations inside the cavity lead to increased noise on the ring-down transient, which, together with the electronic noise of the detector and the digitization error, increases the error in the exponential fit. All these sources of noise/error contribute to the shot-to-shot fluctuations in the measured ring-down time and hence the absorption coefficient, $\alpha$. A changing mirror alignment, *e.g.* due to pressure changes inside the cell, will lead to a systematic error in $\alpha$, as we observed in our early measurements of the pressure dependence of the N$_2$ Rayleigh scattering (*cf.* fig. \[fig:powerdep\] (b)). Saturation of the photodetector (non-linear amplification) may also introduce a systematic error as the ring-down transient may be distorted. Although, this can easily be avoided by adjusting the CRD signal level via the laser pulse energy and the PMT voltage.
In contrast, mode fluctuations or poor mirror alignment do not affect the CELIF measurement as only the integrated LIF and CRD signals, $S{^\text{LIF}}$ and $I{^\text{CRD}}$ respectively, are analyzed and not the shape of their transients. Thus, the noise/error in the integrals is only due to the noise of the detectors and the digitization errors. Saturation of the photodetectors needs to be avoided as described above. The strength of the CELIF technique is the precise determination of the laser intensity in the cavity, $I{^\text{LIF}}$, via the integrated ring-down intensity, $I{^\text{CRD}}$. The subsequent normalization, $S{^\text{LIF}}/I{^\text{CRD}}$, minimizes the effect of laser shot-to-shot fluctuations on $S{^\text{CELIF}}$. Overall, the noise in $S{^\text{LIF}}$ and $I{^\text{CRD}}$ is much lower than the noise/error in the determination of the ring-down time leading to the observed, low baseline noise in the CELIF spectra.
General characteristics of CELIF measurements {#sec:CELIF_characteristics}
---------------------------------------------
The cavity invokes stringent conditions on the shape and spectral composition of the beam interacting with the sample. Well designed and aligned cavity setups can be excited in a single transversal mode, TEM$_{00}$, leading to a well defined Gaussian beam waist at the center. This confinement effectively eliminates any stray light from an empty cavity, greatly reducing the background signal of the CELIF measurement. In contrast to a single-pass LIF setup, baffles and optical filters to suppress stray light are not necessary. This allows the detection of the whole fluorescence spectrum, including fluorescence on the excitation wavelength. As we have demonstrated in fig. \[fig:powerdep\], very clean measurements of Rayleigh scattering are possible.
The mirror reflectivity and cavity length define the longitudinal mode structure supported by the cavity. The effect of the mode structure on a CRD measurement has been discussed in detail by Zalicki and Zare[@Zalicki1995] and needs to be taken into account similarly for CELIF. If we briefly consider a mode-matched cw cavity setup, the laser intensity inside the cavity will build up until the equilibrium is reached. This intensity (resonant with a cavity mode) is by a factor of the finesse higher than the incident laser intensity. However, in our setups we use non-fourier-limited laser pulses, the spectral bandwidths of which span in the order of ten cavity modes. As the laser pulse length is comparable to the round-trip time, the cavity mode structure is not fully formed. The cavity transmission function, $T{\mathcal{F}}/2$ (*cf.* eq. \[eq:transmission\]) of a comparable case using a square temporal laser pulse is discussed in Ref. , from which we conclude that $0.5 < {\mathcal{F}}< 1$ for our cavities.
As outlined in section \[sec:method\], through the repeated use of the light pulse inside the cavity, the integrated light intensity creating the LIF signal in a CELIF experiment is $I_\text{L}{\mathcal{F}}$. In a single-pass, pulsed LIF experiment the fluorescence signal is created by the full intensity of the probe laser, $I_\text{L}$. Therefore, the total light intensity in both techniques differs only by the factor $T{\mathcal{F}}/(1-R)$, *cf.* eqs \[eq:ILIF0\] and \[eq:ILIFint\]. Note that ${\mathcal{F}}$ cancels in the normalization of the LIF signal by the integrated CRD intensity in eq. \[eq:ILIF\_accurate\]. In terms of number of photons interacting with the sample, CELIF and single-pass, pulsed LIF are comparable. However, the photon flux per sample pass is several orders of magnitude lower for CELIF resulting in much reduced power broadening of spectral lines.
The amount of molecules that is excited in any given time interval is proportional to the number of laser photons present in the cavity. Consequently, the time evolution of the LIF signal is the temporal convolution of the ring-down and fluorescence decays. The presence of the cavity effectively stretches the initial laser pulse in time, as characterized by the ring-down decay. Thus, like in a CRDS experiment, the ring-down time defines the temporal resolution of the CELIF setup with which the sample evolution can be monitored. Brown *et al.* have shown how, in a CRD experiment, the time evolution of the sample density can be extracted from a (non-exponential) ring-down decay using a forward convolution.[@Brown2000a] If sample densities change on the timescale of the ring-down time, similar techniques can be applied to map the temporal evolution of the sample density using CELIF.
Limit of detection and extension of dynamic range
-------------------------------------------------
The dynamic range of absorbance measurements is at one end determined by the limit of detection (LOD) defined by the noise level and at the other end by saturation. It is important to note that in CRD measurements the noise level stays almost constant across the entire dynamic range, see fig. \[fig:powerdep\]b. The lowest absorbance that can be measured needs to cause a signal change on the ring-down trace greater than the overall noise on the trace. At the other end, CRD measurements are not valid any more when large absorbances lead to very short ring-down decay times. In a very carefully set up pulsed CRD measurement, the dynamic range in absorbance may cover three orders of magnitude. A typical pulsed CRD measurement in the UV spectral range spans two orders of magnitude in dynamic range.[@Wang2000]
![Comparison of limits of detection (LOD) for CELIF ($\bullet$) and CRD ($\circ$) measurements of jet-cooled BPEB. The signal-to-noise ratio is the baseline-corrected signal divided by standard deviation of the baseline noise ($\sigma_\text{b}$). The horizontal axis shows the baseline-corrected signal expressed as the photon loss per trip. The typical LOD of $3\sigma_\text{b}$ is indicated by the horizontal line. The data was extracted from simultaneously recorded transients at 319.69 nm (*cf.* fig. \[fig:BPEBspectra\]).[]{data-label="fig:det-limit"}](Fig05){width="8.6cm"}
In a LIF experiment, the LOD is defined by the baseline noise of the detection system. At very low signals, when the photodetector amplification is high, the baseline noise is amplified as well. The absolute noise typically increases with signal as can be seen in the upper trace of fig. \[fig:powerdep\]b.
We quantified the limits of detection (LOD) of simultaneously recorded CELIF and CRD signals following the recommendations in ref. . The gross analyte signal, $S_\text{t}$, was determined from the strongest absorption in fig. \[fig:BPEBspectra\] (319.69 nm). The system blank, $S_\text{b}$, was recorded far off resonance at 320.98 nm and comprises the fluctuation in nozzle intensity, Rayleigh scattering and the noise of the detection system. Data is recorded as a function of sample density and averaged over 2500 laser shots. The noise of the system blank, $\sigma_\text{b}$, is determined by the standard deviation. Figure \[fig:det-limit\] shows the net signal over the noise, $(S_\text{t} - S_\text{b}) / \sigma_\text{b}$, as a function of the net signal, $S_\text{t} - S_\text{b}$, for both CELIF and CRDS. The commonly accepted limit of detection of $3\sigma_\text{b}$ is indicated by the horizontal line in the plot. Any data point above this line is a detected signal with at least a 99.7% confidence limit. The CRD measurement at 52 ppm with $(S_\text{t} - S_\text{b}) / \sigma_\text{b}=3$ corresponds to the strongest absorption line in the spectrum fig. \[fig:BPEBspectra\]c. This LOD corresponds to a minimum detectable absorption coefficient of $\alpha_\text{min}=6\cdot 10^{-7}$ cm$^{-1}$.
In the CRD measurement, $\sigma_\text{b}$ is independent of sample concentration leading to the linear dependence seen in fig. \[fig:det-limit\]. As in single-pass LIF or CRD measurements, the CELIF signal increases linearly with sample concentration. Due to the large Rayleigh scattering cross section of BPEB, in these measurements the system blank $S_\text{b}$ is proportional to the sample density. This means that at high concentrations $\sigma_\text{b}$ becomes dominated by the shot-to-shot fluctuations in sample density leading to the flattening of the signal-to-noise ratio in fig. \[fig:det-limit\].
In our measurements, the accessible range of sample concentrations is limited by the vapor pressure and the temperature control of the sample oven (low concentrations) and the finite sample volume and the thermal stability of the molecule (high concentrations). Higher sample concentrations than shown in fig. \[fig:det-limit\] could have been detected by both CELIF and CRDS. Even if the CRD decay is too short to be measured, the fluorescence can be linearly recorded until saturation caused by optical and sample density sets in. This implies that very strong and weak transitions can be measured in the same scan where the calibration is maintained across the recorded spectrum.
Our lowest CELIF signal-to-noise value of 17 indicates that considerably lower concentrations are accessible. Based on the fluctuations of the signal blank, that is dominated by BPEB Rayleigh scattering, we estimate the CELIF LOD as 0.1 ppm or $\alpha_\text{min}=1.5\cdot 10^{-9}$ cm$^{-1}$, which constitutes at least a factor of 400 improvement. For our systems, signal levels were sufficient to analyze integrated LIF traces. However, for samples with lower absorbances or fluorescence quantum yields, photon counting can be used to further lower the detection limit (see the following subsection).
We examined the noise in the CRD and CELIF signals using the N$_2$ Rayleigh scattering as a function of N$_2$ pressure in a filled cavity as shown in fig. \[fig:powerdep\]b. For the CRD measurement the noise is independent of the N$_2$ pressure as expected.[@Berden2009] Both the noise of the integrated signals on the LIF, $\sigma(S{^\text{LIF}})$, and the CRD detector, $\sigma(I{^\text{CRD}})$, contribute to the noise of the CELIF measurement, $\sigma(S{^\text{CELIF}})$, according to eq. \[eq:CELIF\_norm\]. We found that $\sigma(S{^\text{LIF}})$ increases with signal, according to a typical PMT response, while $\sigma(I{^\text{CRD}})$ is almost constant, leading overall to the observed increase of $\sigma(S{^\text{CELIF}})$ with pressure, *i.e.* signal.
In a filled cavity at high pressure, the signal-to-noise ratios of the CELIF and CRD measurements are comparable whereas at low pressures the CELIF ratio is two times larger. Compared to this, in the BPEB molecular beam measurements the signal-to-noise ratio and limit of detection are three orders of magnitude better for CELIF. CRD measures the integrated-column density along the cavity axis whereas CELIF, like LIF, images the density in a localised probe volume. In the N$_2$ Rayleigh scattering experiment, the CRD sample length is about 200 times longer than the length of the LIF probe volume. For the CRD measurement, this considerable increase in sample length almost compensates for the higher sensitivity of the CELIF measurement.
In light of this, CELIF is—like LIF—best applied to localized samples, *e.g.* found in molecular beams, in flames or at interfaces. The strength of CRD lies in the long effective path length, *e.g.* of a filled cavity. In case of a localized sample volume the effective path length through the sample can be reduced by orders of magnitude such that this crucial advantage of CRD is lost, as demonstrated in fig. \[fig:det-limit\]. Even for a filled cavity measurement, CELIF improves the signal-to-noise ratio in comparison with CRD, particularly at low sample concentrations, as seen in fig. \[fig:powerdep\]b.
Absolute quantities from LIF measurements
-----------------------------------------
So far, we have discussed how, from the CRD measurement, we can directly extract the absorption coefficient, $\alpha$, which in turn can be used to provide an absolute calibration for the CELIF measurement. This requires a measurement range in which both CRD and CELIF measurements provide a non-zero $\alpha$, *cf.* eqs \[eq:alpha\_CRD\] and \[eq:cal\_LIF\]. As in conventional CRD measurements, for a known sample length the knowledge of either the sample density or the absorption cross section will allow the absolute measurement of the other. However, in these BPEB measurements using a molecular beam, the sample density cannot easily be determined. We have since measured absolute sample densities of the deuterated mercapto radical (SD) in a dilute molecular beam where the absorption cross section is known.[@Mizouri2013] Employing photon counting, the lowest detected sample density was $1.1\cdot 10^{5}$ cm$^{-3}$ corresponding to an $\alpha_\text{min}=7.9\cdot 10^{-11}$ cm$^{-1}$. The SD and BPEB measurements used a similar CELIF setup and detection wavelength. This further improved CELIF LOD is by a factor of 7500 superior to the CRD measurements presented here. We can therefore conclude that CELIF extends the dynamic range of absorbance measurements compared to a sole CRD measurement by at least three orders of magnitude.
LIF is a popular technique for quantitative measurements of species in flames due to its high sensitivity, spatial resolution and non-invasive nature. However, for absolute calibration, separate measurements like Rayleigh scattering and, more recently, CRD spectroscopy[@Dreyer2001; @Luque2004; @Lamoureux2010] were used. Variations in optical setup and sample composition between the sequential LIF and CRD measurement may affect the calibration. In contrast, our single-beam CELIF method uses the simultaneous and correlated LIF and CRD measurements of the same sample with the same laser pulse to give a robust and consistent calibration.
Consideration of fluorescence lifetimes
---------------------------------------
The CELIF method is equally applicable to short and long fluorescence lifetimes compared to the laser pulse length. Short fluorescence lifetimes impose challenges on single-pass, pulsed LIF. The fluorescence signal is obscured by the stray light of the excitation pulse and the Rayleigh scattered light from the sample and cannot easily be discriminated against by gated detection. Stray-light-free signal may only be sampled over a limited temporal range leading to a large noise on the digitized signal. The comparison of fluorescence excitation spectra of BPEB with a fluorescence lifetime of $\tau_\text{F} \approx 500$ ps[@Fujiwara2008] obtained with CELIF and single-pass LIF under similar conditions is shown in fig. \[fig:CELIFvLIF\]. For the single-pass LIF measurement, the laser beam was expanded such that the probe volume was approximately 30 times larger than for CELIF in order to increase the signal-to-noise ratio and to limit saturation. The LIF signal was normalized on a shot-to-shot basis against the laser intensity recorded on a pyro detector at the exit window. Stray light was suppressed using a long-pass filter (Semrock, 341 nm blocking edge BrightLine) in front of the LIF photomultiplier. The observed noise on the LIF baseline is $\sim$25 times larger than in the CELIF spectrum. This demonstrates the difficulty measuring the 500 ps fluorescence free from stray light using a 5 ns laser pulse. As discussed earlier, in CELIF measurements stray light is not supported by the cavity. Although we occasionally observe a small initial peak due to stray light at the very start of the CRD and LIF transients, both these peaks can be completely removed by appropriate gating of the signal. Furthermore, the remaining long LIF signal can be digitized with hundreds of sample points reducing the digitization noise significantly. In order to separate the stray light from fast fluorescence signal, ps lasers and fast signal digitization need to be employed as demonstrated by the fluorescence lifetime measurements of BPEB by Fujiwara *et al.*[@Fujiwara2008]
For short fluorescence lifetimes, the LIF transient will follow the CRD transient. In this case, only a part of the transients needs to be integrated, *e.g.* using the same limits for both the CRD and LIF signals, to ensure the correct CELIF normalization, see fig. \[fig:transients\]. For long fluorescence lifetimes, the only necessary change to gain valid CELIF spectra is the full integration of both transients. In principle, by deconvoluting the LIF transient by the CRD transient fluorescence lifetimes can be extracted. Long ring-down times reduce the number of photons per unit time in the cavity and, if combined with long fluorescence lifetimes, can lead to signal levels that require photon counting.
![Comparison of normalized BPEB spectra from (a) single-pass LIF and (b) CELIF. The LIF probe volume was approximately 30 times larger than for CELIF and both spectra were recorded with similar sample densities.[]{data-label="fig:CELIFvLIF"}](Fig06){width="8.6cm"}
Fluorescence yields {#sec:yield}
-------------------
Several groups have used combinations of CRDS with LIF in order to measure quantum yields or quenching rates. Spaanjaars *et al.* combined a CRD measurement with LIF in a single setup—although not strictly a single beam experiment as the probe laser was split into two beams that crossed the cylindrical burner at different angles—to extract relative predissociation rates of OH in a flame.[@Spaanjaars1997] The simultaneous measurement of the absorption via CRD and the fluorescence from similar probe volumes allowed an accurate calibration of the relative predissociation rates and quantum yields. Bahrini *et al.* measured absorption and fluorescence excitation spectra of CaBr and CaI with a CELIF type setup to obtain relative quantum yields within individual vibrational bands. Unfortunately, it is not clear how the LIF and CRD measurements were calibrated with respect to each other, in particular, as only some spectra were measured concurrently.[@Bahrini2006] The experiments by Hagemeister *et al.* deployed a similar experimental approach to the one described here to measure relative single-vibronic level fluorescence quantum yields of tropolone and tropolone-water clusters.[@Hagemeister1999] However, in order to extract accurate relative quantum yields knowledge of the wavelength dependent mirror transmission is required, see eq. \[eq:ILIF\_approx\].
In the following we describe how CELIF can be used to measure absolute fluorescence quantum yields in a self-calibration scheme with Rayleigh scattering. Considering equations \[eq:SLIF\] and \[eq:ILIF\_approx\]–\[eq:alpha\_LIF\], the absorption coefficient is $$\alpha(\lambda) = \frac{T(\lambda)}{2g\cdot\Gamma(\lambda)}\cdot S{^\text{CELIF}}.$$ Performing a Rayleigh scattering measurement (where $\Gamma=1$) with the CELIF setup, the Rayleigh scattering coefficients, $\alpha_\text{R}=\sigma_\text{R}\rho$, can be extracted from the simultaneous CRD measurement or from known Rayleigh cross sections and sample densities. The fraction $\alpha_\text{R}/S{^\text{CELIF}}_\text{R}$ is the calibration factor ${\mathcal{K}}_\text{R}(\lambda)=T(\lambda)/2g$. The absolute fluorescence quantum yield, $\Gamma_\text{F}$, is then obtained from a subsequent CELIF measurement using the above calibration, $$\label{eq:qy_absolute}
\Gamma_\text{F}(\lambda) = {\mathcal{K}}_\text{R}(\lambda)\cdot \frac{S{^\text{CELIF}}_\text{F}(\lambda)}{\alpha{^\text{CRD}}(\lambda)},$$ where the absorption coefficient, $\alpha{^\text{CRD}}$, is determined from the ring-down time. Strictly, the above equation neglects the wavelength dependence of $g$ caused by the potentially different response of the detector and the collection optics to the Rayleigh scattered light and the fluorescence spectrum.
Compared to a CRD measurement, CELIF can be used over a wider range of wavelengths using the same set of cavity mirrors. The sensitivity of CRD cruicially depends on the large effective path length given by the high mirror reflectivity. Following the derivation of eqs \[eq:ILIF\_approx\] and \[eq:CELIF\_norm\], the CELIF signal is largely independent of the mirror reflectivity and as a consequence the useful wavelength range of the mirrors is extended. In particular, once the wavelength-dependent calibration factor $K_\text{R}(\lambda)$ is established, the CELIF measurement is independent of the mirror characteristics.
In summary, in a well characterized LIF setup, where the geometric factor of the detection system, $g$, is known, the three variables that determine the LIF signal are the cross section, $\sigma$, the sample density, $\rho$, and the fluorescence quantum yield, $\Gamma$. As outlined above, knowledge of $\alpha=\sigma\rho$ allows the determination of $\Gamma$. Likewise, the sample density can be measured based on a transition for which absorption and fluorescence are known.
CELIF implementation
--------------------
The benefits of a CELIF setup can easily be gained on an existing CRD experiment. To extend such a setup with LIF detection, the only additions to the system are collection optics, a photodetector and a recording channel. This straightforward and fairly low-cost addition increases the dynamic range of the previous CRD experiment by multiple orders of magnitude. In turn, a new stand-alone LIF setup requires very careful calibration to measure absolute quantities where the challenge remains that most of these calibrations are based on separate measurements where sample and laser beam are similar but not identical. Adding a cavity and CRD detection to the LIF setup provides an *in situ* absolute calibration of the LIF measurement.
Conclusions {#sec:conclusions}
===========
We have demonstrated how a conventional, pulsed CRD setup can be extended by fluorescence detection in a straightforward manner that combines the advantages of both the CRD and LIF techniques. The CELIF technique uses the same laser beam and sample in the cavity. Its simultaneous absorption (CRD) and fluorescence (LIF) detection allows a rigorous absolute calibration of the LIF measurement at sample densities that lead to a measurable reduction in ring-down time. This calibration can subsequently be applied across the entire LIF dynamic range. From the calibrated LIF signal, absolute quantities such as the fluorescence quantum yield, absorption cross section and sample density can be extracted. In our experiment, we have shown how the limited dynamic range of the CRD measurement can be extended by at least three orders of magnitude towards lower absorbances, where the change in ring-down time is too small to be detected.
CELIF very elegantly overcomes two main obstacles of single-pass LIF. The cavity very effectively suppresses stray light allowing detection on the laser excitation wavelength. The probe laser pulse is stretched in time and the resulting decrease in laser intensity reduces saturation. Our measurements show how CELIF improves the signal-to-noise ratio when compared to our single-pass LIF spectrum. We believe that CELIF is most suited for localized sample volumes, such as molecular beams or surfaces, where CRDS cannot fulfill its full potential due to the small absorption path length.
In our respective research areas, we apply CELIF to molecular spectroscopy in supersonic beams and dynamics at surfaces. Other research fields where absolute spectroscopic quantities such as cross sections and quantum yields are required include astrochemistry, atmospheric chemistry and plasma physics/chemistry. For the more recently evolved research area of cold and ultra-cold molecules, the measurement of absolute densities of molecules in a trap is important. For example, in order to control the outcome of chemical reactions at sub-Kelvin temperatures with external fields, samples densities will need to be increased to $10^{10}$ cm$^{-3}$.[@Carr2009] The extended dynamic range of the CELIF technique is perfectly suited to follow the time evolution of absolute trap densities.
We believe that CELIF is an elegant, easy to implement and cost effective way to gain these absolute quantities over a large dynamic range and we hope that it finds applications in many research fields.
Acknowledgements
================
We thank Neil Ord, Sian Matthews, Nicholas Andrews and Rebecca Smith for their various contributions throughout the project and Kelvin Appleby for electronic support. We thank EPSRC for the DTA studentships to support SES and ORW, the Royal Society for the University Research Fellowship for NHN, and Durham University for infrastructure support via the HEFCE Science Research Investment Fund.
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---
author:
- Josep Martí
- 'Pedro L. Luque-Escamilla'
- 'Gustavo E. Romero'
- 'Juan R. Sánchez-Sutil'
- 'Álvaro J. Muñoz-Arjonilla'
bibliography:
- 'references.bib'
date: 'Received xxxx, 2015; accepted xxxx, xxxx'
title: 'Real-time evolution of a large-scale relativistic jet '
---
[Astrophysical jets are ubiquitous in the Universe on all scales, but their large-scale dynamics and evolution in time are hard to observe since they usually develop at a very slow pace.]{} [We aim to obtain the first observational proof of the expected large-scale evolution and interaction with the environment in an astrophysical jet. Only jets from microquasars offer a chance to witness the real-time, full-jet evolution within a human lifetime, since they combine a $^\backprime$short$^\prime$, few parsec length with relativistic velocities.]{} [The methodology of this work is based on a systematic recalibraton of interferometric radio observations of microquasars available in public archives. In particular, radio observations of the microquasar [GRS 1758$-$258]{} over less than two decades have provided the most striking results.]{} [Significant morphological variations in the extended jet structure of [GRS 1758$-$258]{} are reported here that were previously missed. Its northern radio lobe underwent a major morphological variation that rendered the hotspot undetectable in 2001 and reappeared again in the following years. The reported changes confirm the Galactic nature of the source. We tentatively interpret them in terms of the growth of instabilities in the jet flow. There is also evidence of surrounding cocoon. These results can provide a testbed for models accounting for the evolution of jets and their interaction with the environment.]{}
Introduction
============
Astrophysical jets are observed in a variety of environments [@2005AdSpR..35..908D] including young stellar and Herbig-Haro (HH) objects , planetary nebulae [@1998AJ....116.1357S], microquasars , active galactic nuclei (AGN), and distant quasars [@1984RvMP...56..255B]. These outflows are triggered when the magnetic field taps the rotational energy from a central compact object or disk [@1982MNRAS.199..883B; @89879520150101]. This mechanism is characterized well because the mass accretion and ejection events evolve fast enough to be appropriately sampled with multi-wavelength observations of both microquasars and AGN [@2000Sci...289.2317G; @2002Natur.417..625M], where these processes occur on time scales from hours or days to years, respectively. However, the long-term evolution of the jet flow far away from the central engine is not understood as well. The large sizes of relativistic jets in AGN make them evolve on characteristic times of $\sim10^6$ yr [@2008MNRAS.388..625S], while the low velocities of HH jets imply shorter characteristic times of $\sim 10^3$ yr [@2001ApJ...551..347D]. Extended jets in both types of systems will then appear almost frozen during a human lifetime.
In this context, only microquasar jets that combine relatively short lengths and relativistic velocities offer a chance to study large-scale jet dynamics and its interaction with the environment in almost real time. Here, we have payed attention to which is believed to be a microquasar associated with a low-mass X-ray binary system. Although its Galactic nature has never been confirmed because of the lack of optical and infrared spectra of the companion star [@2014ApJ...797L...1L], [GRS 1758$-$258]{} appears as a very bright and persistent hard X-ray source towards the Galactic centre region . This source has strong spectral similarities in X-rays with the classical black hole candidate [@1999ApJ...525..901M]. It displays double-sided radio jets [@1992ApJ...401L..15R; @1994AIPC..304..413M] whose arc-minute extension implies a parsec-scale linear size if located at the distance of the Galactic centre, hereafter assumed to be 8.5 kpc. In this Letter, we revisit the huge archive of [GRS 1758$-$258]{} observations at radio wavelengths obtained with the Very Large Array (VLA) interferometer of the National Radio Astronomy Observatory (NRAO). This provides us with a unique set of highly sensitive maps with an angular resolution that is well suited to exploring the evolution of the [GRS 1758$-$258]{} jets over more than a decade.
Data analysis and results
=========================
We data-mined the public archives of the VLA interferometer hosted by NRAO. The angular size and morphology of the arcminute [GRS 1758$-$258]{} jets are well sampled by using the 6 cm wavelength and the C-configuration of the array. Two intermediate frequency bands 50 MHz wide were available from the VLA correlator products. A total of four useful observing projects between 1992 and 2008 could be retrieved with this instrumental setup (see first block of Table \[log\]).
All downloaded projects were individually recalibrated by using the AIPS software package of NRAO, taking special care to remove corrupt visibilities in both the target and the calibrator sources. The phase calibrator was , while the visibility amplitude was tied to the known flux densities of and . Some older projects needed to be transformed from the B1950.0 to the J2000.0 reference system, by using the AIPS task UVFIX, in order to make the multi-epoch map comparison easier. Radio maps were created and deconvolved using the CLEAN algorithm as provided within the IMAGR task of AIPS. Their respective synthesized beams were individually determined first and later averaged. The resulting averaged elliptical Gaussian beam was finally used to convolve the clean components of each observing epoch in a second run of the IMAGR task.
In Fig. \[sequence\], we present the final sequence of 6 cm maps showing the appearance of the [GRS 1758$-$258]{} during the years 1992, 1997, 2001, and 2008 on arcminute scales. The different panels can be considered as approximate matching-beam maps with similar point spread functions. Having the same angular resolution, they are therefore suitable for meaningful comparison of the jet morphological variations visible in different epochs. In all cases, natural weighting of the interferometric visibilities (i.e. a +5 value of the IMAGR ROBUST parameter) was applied to maximize sensitivity to extended emission. The AIPS task DBCON was used to concatenate short time slots of different VLA monitoring projects into a single observing epoch.
In Table \[obsdata\] we compile the positions, deconvolved angular sizes, flux densities coming from the northern hotspot component that dominates the lobe emission, and its radio luminosities in the 0.1-$10^5$ GHz range, assuming a $-0.7$ typical spectral index for non-thermal radio emission. We additionally include the minimum energy content and magnetic field assuming equipartition of energy between relativistic particles and magnetic field, using the @pacholczyk1970radio formulation. An estimate of the relativistic electron density needed to account for the observed radio luminosity is also given.
Finally, a few additional archive projects exist in the C, D and CD configurations (see 2nd block of Table \[log\]). Although there was not high enough quality to provide new individual frames in Fig. \[sequence\], they were selected to enhance the sensitivity to extended emission in deep imaging that is discussed below.
--------- --------- ---------------- ----------- ------------
Project VLA Observation On-source Central
code config. date time (s) Julian day
AM345 C 1992 Mar 21 1770
1992 Apr 09 6040 2448719
1992 Apr 11 2610
AM560 C 1997 Aug 03 2440
1997 Aug 05 2440
1997 Aug 08 2440
1997 Aug 11 2160
1997 Aug 14 2170 2450674
1997 Aug 15 1860
1997 Aug 18 2450
1997 Aug 20 2430
1997 Aug 24 2420
AR458 C 2001 Jul 08 473
2001 Jul 24 533
2001 Aug 4 673
2001 Aug 9 603
2001 Aug 16 453 2452132
2001 Aug 26 413
2001 Aug 30 403
2001 Sep 07 533
AS930 C 2008 Apr 01 4840
2008 Apr 07 4840 2454563
2008 Apr 12 4840
AM345 D 1992 Sep 26-27 5690 2448892
AM428 CD 1993 Oct 3-4 6460 2448899
AR476 C 2002 Oct 15 1453
2002 Oct 16 1513 2452582
2002 Nov 11 1623
2002 Dec 2 1533
AR523 C 2004 Apr 30 953 2453126
2004 May 05 953 2453131
--------- --------- ---------------- ----------- ------------
: Log of VLA 6 cm observations used in this work.[]{data-label="log"}
{width="16.0cm"}
Discussion
==========
In 1992, the jet’s northern lobe in Fig. \[sequence\] ended in a sort of bow-shaped working surface with a conspicuous component at its vertex that we interpret as the terminal hotspot. Comparison with the 1997 frame, where the hotspot is brighter and easily recognized, implies a noticeable shift of $13^{\prime\prime} \pm 1^{\prime\prime}$ on a time interval of 5.4 yr. This value is consistent with previous, very conservative upper limits that supported the idea of a continuously powered stationary jet flow . When assuming the hotspot identification is correct, the 2[${\rlap.}^{\prime \prime}$]{}4 $\pm$ 0[${\rlap.}^{\prime \prime}$]{}2 yr$^{-1}$ associated proper motion translates into a velocity of the projected jet head of ($0.32 \pm 0.03)c$, where $c$ is the speed of light. In the 2001 frame, the northern jet lobe was observed to evolve into two elongated fragments that were nearly parallel to the jet direction. The conspicuous hotspot of 1997 completely lost its shape, and only traces of its extended emission remained at its position after four years. Seven years later, in 2008, a new hotspot had clearly reappeared at some time in between. Had the 2008 northern hotspot existed in 2001, it should have been detected well above the rms noise. Moreover, from Table \[obsdata\] the position offset between the 1997 and 2008 hotspots is estimated to be 1[${\rlap.}^{\prime \prime}$]{}9 $\pm$ 0[${\rlap.}^{\prime \prime}$]{}7. Thus, we are observing a newly formed structure. On the other hand, the southern counter-jet is also visible in Fig. \[sequence\], but it did not display a structure that was as well organized as the northern jet. Only hints of precession previously noticed by are evident in 1997, but they are remarkably absent in the even more sensitive 2008 frame. This is additional evidence that the [GRS 1758$-$258]{} jet evolution is real.
The low declination of the source limits the achievable dynamic ranges of the Fig. \[sequence\] maps. Therefore, difference maps were computed to assess the reliability of the observed structural variations better. Our two more sensitive observing epochs were used for this purpose. Figure \[zoom\_hotspot\] shows a zoomed view of morphology differences in the northern hotspot, together with the subtraction of the 1997 clean component model from the 2008 visibility data. Residuals emerge at the $4\sigma$ to $5\sigma$ level. Beyond the field of view shown here, they are also aligned along the jet position angle, suggesting changes that affect the whole jet flow.
![ Close up of the northern radio lobe of [GRS 1758$-$258]{} as observed with the VLA at 6 cm in 1997 (left) and 2008 (middle), when the hotspot was fainter. The coloured horizontal bar provides a linear intensity scale in $\mu$Jy beam$^{-1}$. The bottom left corner bar gives the angular scale. North is up and east left. The right panel shows the residual difference between the two epochs. Its colour vertical bar is scaled in units of the rms noise in the difference map (10 $\mu$Jy beam$^{-1}$). The synthesized beam in all panels is the same as in Fig.\[sequence\] (bottom right ellipse). []{data-label="zoom_hotspot"}](fig2.eps){width="10.0cm"}
The first consequence of the observed phenomena is an upper limit to the [GRS 1758$-$258]{} distance. This comes from the time scale ($\tau \sim 11$ yr) of major structural changes. The northern jet flow was fully renewed between 1997 and 2008, as shown in Figs. \[sequence\], and \[zoom\_hotspot\]. Given its angular size ($\theta \sim 60^{\prime\prime}$), causality arguments dictate that the maximum possible distance lies within the boundaries of the Milky Way ($\sim c \tau / \theta \sim 12$ kpc). Having the jet inclined with respect to the line-of-sight would reduce this value further. Therefore, the Galactic origin of [GRS 1758$-$258]{} becomes fully confirmed for the first time.
---------- ----------------------------------------------------- -------------------------------------------------------------------------- ------------------- --------------------------------- ----------------------- ---------------------------- --------------------------- ----------------------------
Epoch R. A. Dec. Flux Deconvolved Radio Minimum Magnetic Relativistic
(J2000.0) (J2000.0) density angular size luminosity energy field e- density
$18^h 01^m$ $-25^{\circ} 43^{\prime}$ (mJy) (arc-second$^2$) (erg s$^{-1}$) (erg) (Gauss) (cm$^{-3}$)
1992 12[${\rlap.}^{s}$]{}8 $\pm$ 0[${\rlap.}^{s}$]{}1 $48^{\prime\prime} \pm 1^{\prime\prime}$ $0.37 \pm 0.03$ $ (18 \pm 3) \times ( \leq 9 )$ $1.0 \times 10^{31}$ $\leq 3.3 \times 10^{44}$ $\geq 4.0 \times 10^{-5}$ $ \geq 7.6 \times 10^{-4}$
1997 13[${\rlap.}^{s}$]{}00 $\pm$ 0[${\rlap.}^{s}$]{}02 35[${\rlap.}^{\prime \prime}$]{}6 $\pm$ 0[${\rlap.}^{\prime \prime}$]{}7 $ 0.26 \pm 0.02$ $ (15 \pm 2) \times (6 \pm 1)$ $7.1 \times 10^{30}$ $1.9 \times 10^{44}$ $ 4.7 \times 10^{-5}$ $ 1.0 \times 10^{-3}$
2001$^a$ 12[${\rlap.}^{s}$]{}9 $\pm$ 0[${\rlap.}^{s}$]{}1 $33^{\prime\prime} \pm 1^{\prime\prime}$ $ 0.29 \pm 0.04$ $ (25 \pm 4) \times (4 \pm 2)$ $ 7.9 \times 10^{30}$ $ 2.1 \times 10^{44}$ $ 4.6 \times 10^{-5}$ $ 1.2 \times 10^{-3}$
13[${\rlap.}^{s}$]{}13 $\pm$ 0[${\rlap.}^{s}$]{}03 $52^{\prime\prime} \pm 1^{\prime\prime}$ $ 0.26 \pm 0.04$ $(24 \pm 6) \times (3 \pm 2)$ $ 7.1 \times 10^{30}$ $ 1.6 \times 10^{44}$ $ 5.1 \times 10^{-5}$ $ 1.1 \times 10^{-3}$
2008 13[${\rlap.}^{s}$]{}11s $\pm$ 0[${\rlap.}^{s}$]{}03 36[${\rlap.}^{\prime \prime}$]{}8 $\pm$ 0[${\rlap.}^{\prime \prime}$]{}6 $0.18 \pm 0.02$ $(13 \pm 2) \times (8 \pm 1)$ $ 4.9 \times 10^{30}$ $ 1.6 \times 10^{44} $ $ 3.9 \times 10^{-5}$ $ 7.4 \times 10^{-4}$
---------- ----------------------------------------------------- -------------------------------------------------------------------------- ------------------- --------------------------------- ----------------------- ---------------------------- --------------------------- ----------------------------
The physical interpretation of these morphological changes is uncertain, although their disruptive appearance suggests that they may be due to the onset of hydrodynamic instabilities [@1997atas.conf...17B] such as Kelvin-Helmholtz (KH) or Rayleigh-Taylor (RT) instabilities. The stability condition essentially depends on the ratio $\eta = n_j / n_a$ with $n_j$ and $n_a$ the density of the jet and the ambient medium, respectively. If $\eta < 1$, the jet may be unstable. When assuming that [GRS 1758$-$258]{} jet is baryonic and that most of the mass flux is in the form of a thermal plasma (e.g., about 90%), the relativistic particles responsible for the non-thermal radio emission, whose density of $\sim 10^{-3}$ cm$^{-3}$ has been estimated from equipartition (see Table \[obsdata\]), account for the remaining 10%. Thus, the actual jet density is tentatively estimated as $n_j \sim 10^{-2}$ cm$^{-3}$.
Given that microquasars are typically located in much less dense environments than the canonical interstellar medium , we assume $n_a \sim 0.1$ cm$^{-3}$. This yields a density contrast in the vicinity of the terminal hotspot of $\eta \sim 10^{-1}$. Therefore, the jet could be prone to undergoing hydrodynamic instabilities, and its disruption may occur if the KH or RT modes have enough time to grow up to a length scale comparable to the jet radius $r_j$. These time scales may be calculated as $t_{KH} \sim (2 r_j /c) \eta^{1/2}$ and $t_{RT} \sim (2 r_j /c)(2\eta /3)^{1/2}$, respectively. When assuming that the filaments of radio emission converging to the hotspot in Fig. \[zoom\_hotspot\] are actually tracing the jet flow, $r_j \sim 0.1$ pc for [GRS 1758$-$258]{} at its estimated distance. Taking this value, growth time scales of destructive RT and KH instabilities are found to be very similar and to last several months.
Considering the uncertainties involved in this calculation, it thus appears conceivable from the physical point of view that the [GRS 1758$-$258]{} jets undergo RT and/or KH instabilities that completely reorganize their collimated outflow on a yearly time scale, as suggested by the multi-epoch observations reported in this work. In the maps presented in Fig. \[sequence\], we have merged data sets scattered over no more than a two-months span, so that they are safely below the limit for avoiding an excessive smearing of the imaged structures.
We emphasize that changes in the microquasar central engine cannot be responsible for the morphological disruption of the outer jet, since the cooling time $t_c$ of relativistic electrons by synchrotron radiation in the jet head is too long. Specifically, $t_c[{\rm s}] \sim 5 \times 10^8 \gamma^{-1} B^{-2}$, where $\gamma$ is the Lorentz factor of the electrons and $B$ the magnetic field in Gauss. For $B \sim 10^{-5}$ G and $\gamma \sim 10^6$, we get $t_c \sim 3$ Myr! It is clear that the structure must be destroyed and the electrons then diffuse into the interstellar medium, escaping from the region where they were confined. This plasma leaving the hotspot can form a cocoon structure around the radio lobes, creating cavities or bubbles, as seen in the environment of Galactic and extragalactic sources of relativistic jets with continued activity [@2006MNRAS.370.1513K; @2005Natur.436..819G; @2006ApJ...644L...9W; @2010Natur.466..209P]. In this context, it makes sense to wonder if this cocoon structure also exists in [GRS 1758$-$258]{}. To shed light on this issue, we decided to concatenate all observational projects in Table \[log\] to produce the deepest radio map available for this microquasar (see Fig. \[cocoon\]). The total on-source integration time of this huge data set amounts to about $19^h$, allowing us to reach a background rms noise of 6 $\mu$Jy beam$^{-1}$ with natural weight. In this map, bridges of extended emission emanate from both radio lobes and almost surround the whole bipolar jet complex with an elliptical shape. The faintest cocoon edges appear at a $4\sigma$ level and cover more than 50% of the elliptical perimeter. Such diffuse features reveal, for the first time, the expected cocoon-like structure around [GRS 1758$-$258]{}. This is a new similarity between microquasars and radio galaxies that has never been observed before.
New, more sensitive multi-wavelength observations will provide a benchmark for validating our current ideas of the processes involving collimated flows in astrophysical contexts, which otherwise could not be appropriately sampled in time. For instance, detailed changes in jet structures and fainter cocoon shells created by jet relict particles should be revealed by new radio interferometers, such as EVLA, LOFAR, or SKA.
![ Natural-weight 6 cm map of [GRS 1758$-$258]{} and its surrounding cocoon diffuse emission obtained by combining all VLA data sets (see Table \[log\]). The horizontal bar in the top right corner gives the angular scale. The dashed red ellipse sketches the edges of a previously unseen, cocoon-like structure around this microquasar. The synthesized beam of 10[${\rlap.}^{\prime \prime}$]{}6 $\times$ 5[${\rlap.}^{\prime \prime}$]{}6, with position angle 6[${\rlap.}^{\circ}$]{}7, is displayed by the bottom left ellipse. Contours shown correspond to $-3$, 3, 4, 5, 6, 7, 8, 9, 10, 11 ,12, 14, 16, 18, 20, 22, 24, and 26 times the rms background noise of 6 $\mu$Jy beam$^{-1}$. Three unrelated compact radio sources also appear in this map. []{data-label="cocoon"}](fig4.eps){width="7.5cm"}
This work was supported by grant AYA2013-47447-C3-3-P from the Spanish Ministerio de Economía y Competitividad (MINECO), and by the Consejería de Economía, Innovación, Ciencia y Empleo of Junta de Andalucía under excellence grant FQM-1343 and research group FQM-322, as well as FEDER funds. GER is a member of CONICET. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
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---
abstract: 'The increasing penetration of distributed energy resources poses numerous reliability issues to the urban distribution grid. The topology estimation is a critical step to ensure the robustness of distribution grid operation. However, the bus connectivity and grid topology estimation are usually hard in distribution grids. For example, it is technically challenging and costly to monitor the bus connectivity in urban grids, e.g., underground lines. It is also inappropriate to use the radial topology assumption exclusively because the grids of metropolitan cities and regions with dense loads could be with many mesh structures. To resolve these drawbacks, we propose a data-driven topology estimation method for MV and LV distribution grids by only utilizing the historical smart meter measurements. Particularly, a probabilistic graphical model is utilized to capture the statistical dependencies amongst bus voltages. We prove that the bus connectivity and grid topology estimation problems, in radial and mesh structures, can be formulated as a linear regression with a least absolute shrinkage regularization on grouped variables (*group lasso*). Simulations show highly accurate results in eight MV and LV distribution networks at different sizes and 22 topology configurations using PG&E residential smart meter data.'
author:
- 'Yizheng Liao, Yang Weng, Guangyi Liu, Ram Rajagopal, [^1]'
bibliography:
- 'ref.bib'
title: Urban MV and LV Distribution Grid Topology Estimation via Group Lasso
---
Introduction
============
A core mission of building Smart Cities is providing sustainable and economical energy. To achieve this goal, distributed energy resources (DERs), such as photovoltaic (PV) devices, energy storage devices, and electric vehicles, have been deeply integrated into the distribution grids to provide sustainable energy and reduce electricity cost. Such a trend will continue in the future deepening the DERs penetration further. While offering new opportunities, the increasing DER penetration triggers reliability risks to the operation of distribution systems. The distributed generation and the bidirectional power flow can cause the installed protective devices and operation systems to become insufficient. For example, the local grid may become unstable with the presence of even a small-scale of DERs [@dey2010urban]. Also, the voltage unbalance and transformer overload may occur due to the frequent plug-in electric vehicles in the low voltage grid [@clement2010impact]. For the future distribution grid with deeply penetrated DERs, better grid monitoring tools (for islanding and line work hazards) are needed for system operation, where topology information (for one or more new buses) is a prerequisite.
In transmission grids, the grid topology is usually available to system operators. The topology error caused by the infrequent reconfiguration can be identified by the topology processor and state estimation[@huang2012electric; @abur2004power; @Lugtu80]. Unfortunately, in medium voltage (MV) and low voltage (LV) distribution grids, a topology can frequently change and make existing methods with limited performance. Furthermore, in many major metropolitan areas and industrial parks, MV/LV distribution grid branches are immense and mostly underground. For example, in New York City, over 94,000 miles of distribution lines are underground[@rudin2012machine]. Thus, the installation of topology identification devices in urban distribution grids is time-consuming and expensive. Even worse, the methods proposed for overhead transmission grid become infeasible in underground distribution grids due to the frequent topology reconfiguration [@brown2008impact].
These critical challenges discussed above cause many previous assumptions on topology estimation to become invalid. For example,[@hayes2016event; @arghandeh2015topology; @baran2009topology] require the locations of switches or admittance matrices and look for the most likely topology from a collection of configurations. In [@deka2015structure] and [@sharon2012topology], the impedances are also needed if only partial measurements are available. These requirements become unsuitable in metropolitan distribution systems because switch connectivity statuses and admittance matrices are often unavailable. Also, in many field applications, these details may be outdated or inaccurately recorded due to unreported power engineering activities, e.g., manual outage restoration in the substations. Furthermore, many DERs in distribution grids are not owned or operated by the utilities. Thus, their operation information may be inaccessible to the utilities. [@li2013blind] and [@anwar2016estimation] use the DC approximation and SCADA data to estimate the grid topology. However, the branches in distribution grid usually have non-negligible resistance. Furthermore, [@peppanen2016distribution; @deka2016estimating; @deka2016tractable] are designed for radial networks only, but many LV distribution grids are mesh in metropolitan districts and in regions with high load densities (e.g., industrial parks) [@rudin2012machine; @allan2013reliability; @diaz2002application]. Many MV distribution grids have mesh structure and are operated with a radial topology. Several utilities, such as Taipower, Florida Power Company, Hong Kong Electric Company, Singapore Power, and Korea Electric Power Cooperation, have operated mesh (closed-loop) MV distribution grids in their service zones [@chen2004feasibility; @kim2013advanced; @jeon2016underground; @pagel2000energizing; @teo1995principles]. Recent studies [@celli2004meshed; @de2014investigation] show that MV grid with mesh operational topology is more reliable and efficient with high penetration of DERs. [[@cavraro2017voltage] proposes a maximum posterior probability approach to identify mesh operational topology from a candidate pool. In [@talukdar2017learning], the distribution grid is formulated as a linear dynamic system and a Wiener filtering-based method is proposed to recover the radial and mesh structures.]{} At last, [@cavraro2015data; @deka2016estimating; @yuan2016inverse] require the phasor measurement units (PMUs), which are not widely available in current distribution grids.
The increasing investment in Advanced Metering Infrastructure (AMI) provides a new opportunity to utilize the historical data to solve new problems, such as estimating the underground distribution line connectivity in urban areas[@dey2010urban]. Hence, we only use household smart meter data in this paper, which include voltage magnitude, real power, and reactive power. Our goal is helping these buses to locate their connectivity to each other and the backbone local grid.
Mathematically, we firstly represent an MV/LV distribution system in a probabilistic graphical model. Then, we propose a method that estimates the connectivity of a bus. [By exploiting the linear relationship between nodal voltages and injected currents, this algorithm uses the historical data of voltage phasors to fit a linear regression with the $L_1$ penalty on grouped variables, which is known as the *group lasso* problem. Based on the bus connectivity estimation, we extend the group lasso approach to reconstruct network topology when multiple buses have uncertain connectedness. Furthermore, the voltage phase angle is usually unavailable in distribution grids due to the lack of PMU deployment. To address this issue, we utilize two approximations in distribution grids and extend the proposed algorithm to only use voltage magnitude to recover distribution grid topology.]{} Compared with existing approaches, our generalized grouping-based method shows several advantages. Firstly, our method can estimate a mesh grid with a limited amount of data. Secondly, our algorithm has no error propagation because the bus connectivity is estimated independently [@bolognani2013identification; @liao2015distribution]. Thirdly, the computational complexity of the proposed algorithm is linear in term of data length. Finally, our approach has a reliable performance with noisy measurements.
Our data-driven algorithm is validated by the simulations of two IEEE distribution test cases [@teng2002modified; @kersting2001radial] and six European MV and LV distribution grids [@pretticodistribution] with 18 network configurations. We also utilize Pacific Gas and Electric Company (PG&E) residential smart meter measurements and emulated rooftop PV generation data [@dobos2014pvwatts] from National Renewable Energy Laboratory (NREL) for simulations. The numerical results show that our method outperforms recent works, especially in mesh systems [@bolognani2013identification; @liao2015distribution]. Compared with our previous work [@liao2016urbanpes], we firstly prove that the incremental change of voltage measurements can also be used to reconstruct the distribution grid topology. Secondly, unlike [@liao2016urbanpes], we only apply the regularized linear regression on a subgroup of variables. Thirdly, we validate our algorithm on more network topologies and configurations with real data.
For the remainder of this paper, the MV/LV distribution grid model, its graphical model representation, and the data-driven topology estimation problem are presented in Section \[sec:model\]. We prove that the bus connectivity can be efficiently estimated by a linear regression with $L_1$ regularization in Section \[sec:main\]. In Section \[sec:joint\], we formulate the grid topology reconstruction process as a convex optimization problem. Section \[sec:num\] validates the performance of proposed methods using multiple grids and real data. Section \[sec:con\] gives the conclusions.
MV/LV Distribution Grid Model {#sec:model}
=============================
An MV/LV distribution grid is composited by buses and branches. To embed the smart meter information into the topology estimation problem, for an $M$-bus system, we build a probabilistic graphical model $\mathcal{G}=(\mathcal{M},\mathcal{E})$ with a set of vertices $\mathcal{M}=\{1,2,\cdots,M\}$ and a set of unidirectional edges $\mathcal{E} = \{e_{ik},i,k \in \mathcal{M}\}$. In our graphical model $\mathcal{G}$, a vertex is represented by a random variable $V_i$ and corresponds to a bus. If the measurements at bus $i$ and $k$ exist a statistical dependence, then an edge $e_{ik}$ is in the edge set $\mathcal{E}$. The visualization of the distribution system and its graphical model representation are presented in Fig. \[fig:cyber\_physical\]. For bus $i$, the voltage measurement at time $t$ is $v_i[t] = |v_i[t]|e^{j\theta_i[t]} \in \complex$. The units of the magnitude $|v_i[t]| \in \reals$ and phase angle $\theta_i[t] \in \reals$ are per unit and degree, respectively. Bus $1$ is assumed to be the slack bus with constant magnitude and phase angle. All measurements are noiseless and in the steady state. In Section \[sec:num\], we will discuss the cases with noisy measurements.
![The representation of a physical network and its corresponding graphical model $\mathcal{G}$.[]{data-label="fig:cyber_physical"}](cyber_physical){width="0.7\linewidth"}
In many MV and LV distribution grids, voltage measurements have an irregular probability distribution. To better formulate the bus connectivity estimation problem, the incremental changes in the voltage measurements are used to estimate the grid topology [@chen2016quickest]. The incremental voltage change at bus $i$ is $\Delta v_i[t] = v_i[t] - v_i[t-1]$. $\Delta v_i[1] = 0$ when $t=1$. Since bus $1$ is the slack bus, $\Delta v_1[t] = 0$ for all $t$. $\Delta V_i$ represents the random variable of voltage change.
![A modified mesh $8$-bus system [@teng2002modified]. The bus connectivity within the red dashed box is unknown. The dashed branches are added to create mesh structures.[]{data-label="fig:new_8bus"}](new_8bus){width="0.6\linewidth"}
With the system modeling above, the problem we want to address in this paper is defined as follows:
- Problem: data-driven bus connectivity and grid topology estimation based on bus voltage incremental changes
- Given: the time-series voltage incremental measurements $\Delta v_i[t], t = 1,\cdots,T, i \in \mathcal{M}$ and a grid with partially known topology, as shown in Fig. \[fig:new\_8bus\]
- Find: (1) the bus connectivity; (2) the unknown grid topology $\mathcal{E}$.
Bus Connectivity Estimation with Group Lasso {#sec:main}
============================================
Problem Formulation {#sec:formulation}
-------------------
In our graphical model, bus voltage incremental changes are modeled as random variables. Using chain rules, the joint probability $\P(\Delta \mathbf{V}_\mathcal{M})$ can be expressed as $$\begin{aligned}
&\P(\Delta V_2,\Delta V_3,\cdots,\Delta V_M) \nonumber \\
=& \P(\Delta V_2)\P(\Delta V_3|\Delta V_2)\cdots \P(\Delta V_M|\Delta V_2,\cdots,\Delta V_{M-1}). \label{eq:joint}\end{aligned}$$ Bus 1 is omitted because it is the slack bus with constant voltage. If the slack bus does not have a constant voltage, we model the slack bus voltage incremental change as a random variable ($\Delta V_1$) and include it in (\[eq:joint\]). In a distribution grid, adjacent buses are highly correlated [@liao2015distribution]. Therefore, we can approximate the joint probability $\P(\Delta \mathbf{V}_\mathcal{M})$ as $$\label{eq:dist}
\P(\Delta \mathbf{V}_\mathcal{M}) \simeq \prod_{i=2}^M\P(\Delta V_i|\Delta\mathbf{V}_{\mathcal{N}(i)}),$$ where $\mathcal{N}(i)$ denotes the neighbor set that includes the adjacent buses of bus $i$, i.e., $\mathcal{N}(i) = \{k \in \mathcal{M} | e_{ik}\in \mathcal{E}\}$. If this approximation holds, finding the bus connectivity is equivalent to finding the adjacent buses. The existing works [@peppanen2016distribution; @deka2016estimating; @deka2016tractable], are restricted to only find parent nodes because of the radial topology assumption. However, an MV/LV distribution grid topology in urban area can be radial or mesh. Our goal is proposing a method that is suitable for both types. Next, we will take a two-stage proof to show that $\Delta V_i$ only has statistical dependency with its adjacent buses under an appropriate assumption. We will also show why the approximation of $\P(\Delta\mathbf{V}_\mathcal{M})$ in (\[eq:dist\]) holds. In the rest of this paper, the set complement, i.e., $\mathcal{X}\backslash \mathcal{Y} = \{i\in \mathcal{X}, i \notin \mathcal{Y}\}$, is represented by the operator $\backslash$.
\[ass:ass1\] In distribution grid,
1. the incremental change of the current injection $\Delta I$ at each non-slack bus is independent, i.e., $\Delta I_i \perp \Delta I_k$ for all $i \neq k$.
2. the incremental changes of the current injection $\DI$ and bus voltage $\DV$ follow Gaussian distribution with zero means and non-zero variances.
Fig. \[fig:IP\_indept\] illustrates the pairwise mutual information of incremental changes in bus current injection. The mutual information $I(X, Y)$ is a measure of the statistical dependence between two random variables $X$ and $Y$. When the mutual information is zero, these two random variables are independent, i.e., $X \perp Y$ [@cover2012elements]. In Fig. \[fig:IP\_indept\], the relatively small mutual information means that one can approximate that the current injections are independent with some approximation errors. This assumption has also been adopted in other works, such as [@deka2015structure; @deka2017topology]. [To further validate the independency of current injection increment $\Delta I$, we illustrate the average autocorrelation of current injection increment in *LV\_suburban* system and IEEE 123-bus system. The error bar is one standard deviation. We can observe that in both LV and MV distribution grids, the autocorrelation of $\Delta I$ drops significantly as the lag increases. This observation proves that the current injection increment is independent over time.]{}
![[Mutual information of pairwise current injection increment $\Delta I$ and power injection increment $\Delta P$ in the IEEE 123-bus system.]{}[]{data-label="fig:IP_indept"}](IP_indept){width="0.9\linewidth"}
![[Average autocorrelation of current injection increment $\Delta I$ of *LV\_suburban* system and IEEE 123-bus system. The error bar is one standard deviation.]{}[]{data-label="fig:current_autocorr"}](current_autocorr){width="0.9\linewidth"}
In some distribution grid topology estimation works, the injected power increment independence is adopted, instead of injected current increment independence. In distribution grid, the end-user load models depend on many factors, such as load type, time frame, and voltage balance [@moller2016probabilistic; @collin2014development; @schneider2011multi]. In Fig. \[fig:IP\_indept\], we also illustrate the mutual information of pairwise power injection increment $\Delta P$ in the IEEE 123-bus system. The histograms of mutual information of $\Delta I$ and $\Delta P$ are similar. Therefore, for the data sets we used in this paper, both independence assumptions are held. We prefer the assumption of current injection independence because it simplifies the proof of following theorems and lemmas.
As the first stage of proof, we will show that $\P(\Delta \mathbf{V}_\mathcal{M}) = \prod_{i=2}^M\P(\Delta V_i|\Delta\mathbf{V}_{\{\mathcal{N}(i)\cup \mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i)\}})$, where $\mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i)$ denotes the set that includes the adjacent buses’ indices of bus $i$’s neighbors (buses that are two-hop away from bus $i$), i.e., $\mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i) = \{k \in \mathcal{M}|e_{kl} \in \mathcal{E}, l \in \mathcal{N}(i)\}$.
\[lemma: linear\_indept\] Let $X_1$, $X_2$, and $Y$ be Gaussian random variables, where $X_1$ and $X_2$ are independent. Then, given the following equation, $$a_1 X_1 + b_1 X_2 = Y$$ where $a_1, b_1\in \reals$, $X_1$ and $X_2$ are conditionally independent given $Y=y$ if $a_1 = 0$, $b_1 = 0$, or $a_1 = b_1 = 0$. (see Appendix \[sec:lemma1\_proof\] for proof.)
When $\mathbf{X}_1,\mathbf{X}_2$, and $\mathbf{Y}$, are Gaussian random vectors, Lemma \[lemma: linear\_indept\] can be extended to prove that $\mathbf{a}_1^T\mathbf{b}_1 = \mathbf{0}$ is the necessary condition that $\mathbf{X}_1 \perp \mathbf{X}_2 | \mathbf{Y}=\mathbf{y}$, where $\mathbf{a}_1^T\mathbf{X}_1 + \mathbf{b}_1^T\mathbf{X}_2 = \mathbf{Y}$.
\[lemma:linear\_indept2\] Let $X_1$, $X_2$, $Y$, $Z$ be Gaussian random variables, where $X_1$ and $X_2$ are independent. Then, given the following equations, $$\begin{aligned}
c_1 Y + d_1 Z &=& X_1 \label{eq:lemma2_1}\\
c_2 Y + d_2 Z &=& X_2,\label{eq:lemma2_2}
\end{aligned}$$ where $c_1, d_1, c_2, d_2 \in \reals$, $X_1$ and $X_2$ are conditionally independent given $Z=z$ if $c_1 = 0$ or $c_2 = 0$. (see Appendix \[sec:lemma2\_proof\] for proof.)
When $\mathbf{X}_1,\mathbf{X}_2,\mathbf{Y}, \mathbf{Z}$ are Gaussian random vectors, Lemma \[lemma:linear\_indept2\] can be extended to prove that $\mathbf{c}_1^T\mathbf{c}_2 = \mathbf{0}$ and $\mathbf{c}_1 \neq \mathbf{c}_2 \neq 0$ are the necessary condition that $\mathbf{X}_1 \perp \mathbf{X}_2 | \mathbf{Z}=\mathbf{z}$, where $\mathbf{c}_1^T\mathbf{Y} + \mathbf{d}_1^T\mathbf{Z} = \mathbf{X}_1$ and $\mathbf{c}_2^T\mathbf{Y} + \mathbf{d}_2^T\mathbf{Z} = \mathbf{X}_2$.
\[thm:two\_step\_dependency\] In a distribution grid, the incremental voltage change of bus $i$ and the incremental voltage changes of all other buses that are not in $\{\mathcal{N}(i) \cup \mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i)\}$ are conditionally independent, given the incremental voltage changes of buses in $\{\mathcal{N}(i) \cup \mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i)\}$, i.e., $\Delta V_i \perp \left\{\Delta V_k, k \notin \{i,\mathcal{N}(i), \mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i)\} \right\} |\{\Delta V_q, q \in \mathcal{N}(i) \cup \mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i)\}$.
![An example used to show Theorem \[thm:two\_step\_dependency\].[]{data-label="fig:6bus_loop"}](6bus_loop){width="0.45\linewidth"}
We will use a simple example to show the conditional independence. A formal proof is given in Appendix \[sec:thm1\_proof\]. Using the circuit equation $\mathbf{Y}\Delta\mathbf{V} = \Delta\mathbf{I}$, the system in Fig. \[fig:6bus\_loop\] is expressed as: $$\thickmuskip=-2mu
\begin{bmatrix}
y_{11} & -y_{12} & -y_{13} & 0 & 0 & 0\\
-y_{12} & y_{22} & 0 & -y_{24} & 0 & 0\\
-y_{13} & 0 & y_{33} & -y_{34} & -y_{35} & 0 \\
0 & -y_{24} & -y_{34} & y_{44} & 0 & 0\\
0 & 0 & -y_{35} & 0 & y_{55} & -y_{56} \\
0 & 0 & 0 & 0 & -y_{56} & y_{66}
\end{bmatrix}
\begin{bmatrix}
\Delta V_1 \\ \Delta V_2 \\ \Delta V_3 \\ \Delta V_4 \\ \Delta V_5 \\ \Delta V_6
\end{bmatrix}
=
\begin{bmatrix}
\Delta I_1 \\ \Delta I_2 \\ \Delta I_3 \\ \Delta I_4 \\ \Delta I_5 \\ \Delta I_6
\end{bmatrix}$$ where $y_{ik} = y_{ki}$ denotes the deterministic admittance between bus $i$ and $k$, $y_{ii} = \sum_{k=1,i\neq k}^6y_{ik}+\frac{1}{2}b_i$ for $i = 2,\cdots, 6$, and $b_i$ is the shunt admittance at bus $i$. For bus 1, which connects with the slack bus, $y_{11} = y_{01}+\sum_{k=1,i\neq k}^6y_{ik}+\frac{1}{2}b_i$, where $y_{01} \neq 0$ is the admittance of the branch that connects bus $1$ and the slack bus $0$. If $y_{ik} = 0$, there is no branch between bus $i$ and $k$.
For bus $2$, the neighbor set $\mathcal{N}(2) = \{1,4\}$ and the two-hop neighbor set $\mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(2) = \{3\}$. Given $\Delta V_1 = \Delta v_1$, $\Delta V_3 = \Delta v_3$, and $\Delta V_4 = \Delta v_4$, we have following equations: $$\begin{aligned}
\DI_1 &=& y_{11}\Dv_1 - y_{12}\DV_2 - y_{13}\Dv_3 \label{eq:v1}\\
\DI_2 &=& -y_{12}\Dv_1 + y_{22}\DV_2 - y_{24}\Dv_4 \label{eq:v2} \\
\DI_3 &=& -y_{13}\Dv_1 + y_{33}\Dv_3 - y_{34}\Dv_4 - y_{35}\DV_5 \label{eq:v3}\\
\DI_4 &=& -y_{24}\DV_2 - y_{34}\Dv_3 + y_{44}\Dv_4 \label{eq:v4} \\
\DI_5 &=& -y_{35}\Dv_3 + y_{55}\DV_5 - y_{56}\DV_6 \label{eq:v5}\\
\DI_6 &=& -y_{56}\DV_5 + y_{66}\DV_6 \label{eq:v6}\end{aligned}$$
To prove the conditional independency of $\DV$, we firstly need to check if the independency among $\DI$ still holds, given $\DV_1, \DV_3$ and $\DV_4$. Let’s consider $X_1 = \DI_2, X_2 = \DI_3$, $\mathbf{Y} = [\DV_2,\DV_5]$, and $\mathbf{Z} = [\DV_1,\DV_3,\DV_4]$. Using Lemma \[lemma:linear\_indept2\], we know that $\DI_2 \perp \DI_3$ given $\{\DV_1,\DV_3,\DV_4\}$. Therefore, $\DV_2$ and $\DV_5$ are conditionally independent, according to (\[eq:v2\]) and (\[eq:v3\]).
To prove the conditional independence between $\DV_2$ and $\DV_6$, we combine (\[eq:v3\]) and (\[eq:v6\]) and have the following equation: $$\DI_6 - \frac{y_{56}}{y_{35}}\DI_3 = \frac{y_{56}}{y_{35}}(y_{13}\Dv_1 - y_{33}\Dv_3 + y_{34}\Dv_4) + y_{66}\DV_6.$$ Applying Lemma \[lemma:linear\_indept2\], we prove that $\DI_2$ and $\DI_3+\DI_6$ are conditionally independent, given $\{\DV_1,\DV_3,\DV_4\}$. Therefore, $\DV_2$ and $\DV_6$ are conditionally independent given $\{\DV_1,\DV_3,\DV_4\}$. We can extend this approach to other pairs of buses and prove that Theorem \[thm:two\_step\_dependency\] holds for this example.
![[Conditional correlation between buses in IEEE 123-bus system. The circle indicates the neighbors of bus $i$. The crossing indicates the two-hop neighbor of bus $i$. The square without markers represents the bus pair that are more than two-hop away.]{}[]{data-label="fig:bus_corr"}](bus_corr){width="\linewidth"}
With Theorem \[thm:two\_step\_dependency\], we can show that $\P(\Delta \mathbf{V}_\mathcal{M}) = \prod_{i=2}^M\P(\Delta V_i|\Delta\mathbf{V}_{\mathcal{N}(i)\cup \mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i)})$. This observation is similar to the results in [@deka2017topology].
[In Fig. \[fig:bus\_corr\], we show the conditional correlations of voltage increments between each bus pair in IEEE 123-bus distribution system using the real load data from PG&E. The distribution grid configuration and simulation setup are described in Section \[sec:num\]. In Fig. \[fig:bus\_corr\], the color in a square represents the absolute conditional correlation of voltage increments of two buses. As discussed in [@hastie2015statistical], if the voltage increments of two buses are independent, their conditional correlation is zero (dark color). In Fig. \[fig:bus\_corr\], the circle refers to the bus neighbors and the crossing indicates the two-hop neighbors. If a square without any marker, it means the pair of buses is more than two-hop away. As illustrated in Fig. \[fig:bus\_corr\], the conditional correlation between the voltages of two-hop neighbors is higher than the conditional correlations of other bus pairs, but it is still much lower than the conditional correlation between two neighbors. The diagonal bus pairs have conditional correlation of 1 because it is the self correlation. Based on this observation, we make the following assumption:]{}
\[ass:ass2\] In a distribution grid, given $\DV_{\mathcal{N}(i)}$, the conditional correlations between $\DV_i$ and $\Delta \mathbf{V}_{\mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i)}$ are relatively small.
With Assumption \[ass:ass2\], we can simplify $\P(\Delta \mathbf{V}_{\mathcal{M}})$ to depend on the voltages of neighbors. [As a highlight, unlike existing methods in [@yuan2016inverse; @deka2017topology], the assumption of our method is inspiring by real data observation. In Section \[sec:num\], we use numerical simulation to demonstrate that this approximation does not degrade the performance of topology estimation.]{}
\[lemma:one\_step\_cond\_indept\] In a distribution grid, the voltage change of bus $i$ and the voltage changes of all other buses that are not connected with bus $i$ are conditionally independent, given the voltage changes of the neighbors of bus $i$, i.e., $\Delta V_i \perp \left\{\Delta V_k, k \in \mathcal{M}\backslash \{\mathcal{N}(i),i\} \right\}|\Delta \mathbf{V}_{\mathcal{N}(i)}$.
With Lemma \[lemma:one\_step\_cond\_indept\] as the second stage of the proof, (\[eq:dist\]) holds with equality, i.e., $\P(\Delta \mathbf{V}_\mathcal{M}) = \prod_{i=2}^M\P(\Delta V_i |\Delta\mathbf{V}_{\mathcal{N}(i)})$. Therefore, the voltage incremental change at each bus only depends on $\Delta V$ of its neighbors. In next subsections, we will propose how to find $\mathcal{N}(i)$ using $\Delta V$. Also, we will show the robustness where only $\Delta |V|$ is available.
Bus Connectivity Reconstruction via Linear Regression
-----------------------------------------------------
![Histogram of $\Delta|v[t]|$ of four buses in IEEE 123-bus system.[]{data-label="fig:dv"}](dv){width="0.8\linewidth"}
For bus $i$, $\Delta \mathbf{V}_{\mathcal{M}\backslash\{i\}}$ is a collection of all nodal voltages in the graphical model $G$ beside $V_i$. As discussed in Assumption \[ass:ass1\], we assume $\Delta\mathbf{V}_\mathcal{M}$ to be a Gaussian random vector, which has been empirically shown in Fig. \[fig:dv\]. The conditional distribution of $\Delta V_i$ given $\Delta \mathbf{V}_{\mathcal{M}\backslash\{i\}}$ is also a Gaussian random variable. Therefore, based on the probability density function of Gaussian distribution, $\Delta V_i$ is a linear equation of $\Delta \mathbf{V}_{\mathcal{M}\backslash\{i\}}$ and an error term $\epsilon_{\mathcal{M}\backslash\{i\}}$, i.e., $$\label{eq:lin_reg}
\Delta V_i = \Delta \mathbf{V}_{\mathcal{M}\backslash\{i\}}^H\boldsymbol{\beta}^{(i)} + \epsilon_{\mathcal{M}\backslash\{i\}},$$ where $\boldsymbol{\beta}^{(i)}$ denotes the parameter vector and $H$ denotes the transpose operator. The error term $\epsilon_{\mathcal{M}\backslash\{i\}}$ is a Gaussian variable with a zero mean and a variance of $\text{Var}(\Delta V_i | \Delta\mathbf{V}_{\mathcal{M}\backslash\{i\}})$. It is also independent with $\Delta \mathbf{V}_{\mathcal{M}\backslash\{i\}}$ [@meinshausen2006high]. Because of Lemma \[lemma:one\_step\_cond\_indept\], we know that $\Delta V_i$ and $\Delta V_k$ are conditionally dependent if there is an edge $e_{ik}$. The non-zero coefficient in $\boldsymbol{\beta}^{(i)}$ indicates that two nodes are statistically correlated. Hence, the bus adjacency identification problem is equivalent to a linear regression problem. We can use the parameter estimate $\widehat{\boldsymbol{\beta}}^{(i)}$ to find the neighbors of bus $i$.
From a physical perspective, we can also show that the nonzero coefficients in the parameter vector $\boldsymbol{\beta}^{(i)}$ indicate the bus connectivity. Specifically, at bus $i$, the increments of current injection and nodal voltages have the following relationship: $$\begin{aligned}
\Delta I_i &=& \Delta V_iy_{ii} - \sum_{k \in \mathcal{N}(i)}\Delta V_ky_{ik}, \nonumber \\
\Delta V_i &=& \sum_{k \in \mathcal{N}(i)} \frac{y_{ik}}{y_{ii}}\Delta V_k + \frac{\Delta I_i}{y_{ii}}.\label{eq:v_linear}\end{aligned}$$ with $y_{ii} = \sum_{k \in \mathcal{N}(i)}y_{ik} +\frac{1}{2}b_i$. Compared with (\[eq:lin\_reg\]), we find that if bus $k$ connects with bus $i$, i.e., $k \in \mathcal{N}(i)$, the $k$-th element of $\boldsymbol{\beta}^{(i)}$ is $y_{ik}/y_{ii}$. If bus $d$ is not an element of the set $\mathcal{N}(i)$, the $d$-th element of $\boldsymbol{\beta}^{(i)}$ is zero. The reason is that $y_{id} = y_{di} = 0$ and the voltage changes are conditionally independent, as Lemma \[lemma:one\_step\_cond\_indept\]. The variation introduced by $\Delta I_i$ is captured by $\epsilon_{\mathcal{M}\backslash\{i\}}$. If we assume $\Delta V$ has a zero mean, $\epsilon_{\mathcal{M}\backslash\{i\}}$ also has a zero mean and follows a Gaussian distribution. These results are consistent with our previous discussion. [In some cases, $\Delta I_i$ may be correlated with $\sum \DV_k$. But in our simulation in Section \[sec:num\], we find that the variation of $\Delta I_i$ is much smaller than the variation of $\sum\DV_k$. Hence, we approximate $\Delta I_i$ as the noise term of linear regression in (\[eq:lin\_reg\]).]{}
Many distribution grids are not fully connected. The graphical model $\mathcal{G}$ is sparse and many elements in $\boldsymbol{\beta}^{(i)}$ are zero. A well-known regularization to ensure the sparsity in a linear regression is $L_1$ norm. This formulation is known as *Lasso* [@tibshirani1996regression]. In lasso, the objective function is the sum of squared errors with a constraint on the sum of the absolute values of parameters ($L_1$ norm), i.e.: $$\label{eq:Lasso}
\widehat{\boldsymbol{\beta}}^{(i)} = \arg\min_{\boldsymbol{\beta}^{(i)}} \sum_{t=1}^T\left(\Delta v_i[t]-\sum_{\substack{k=2 \\ k \neq i}}^M \Delta v_k[t]\boldsymbol{\beta}^{(i)}_k\right)^2 + \lambda\|\boldsymbol{\beta}^{(i)}\|_1,$$ where the regularization parameter $\lambda$ is non-negative, $\|\boldsymbol{\beta}^{(i)}\|_1$ denotes the regularization term, and $\|.\|_1$ denotes $L_1$ norm. If $\lambda = 0$, (\[eq:Lasso\]) becomes an ordinary least squares problem. The objective function of (\[eq:Lasso\]) is convex and can be solved by many well-known methods [@efron2004least; @friedman2010regularization]. When solving the lasso problem in (\[eq:Lasso\]), the bus connectivity $\widehat{\mathcal{N}}(i)$ can be estimated by finding the indices of non-zero elements of $\widehat{\boldsymbol{\beta}}^{(i)}$. How to choose $\lambda$ is discussed in Section \[sec:lambda\_path\].
Bus Neighbors Estimation via Group Lasso {#sec:group_lasso}
----------------------------------------
In the previous section, we have formulated the bus connectivity estimation problem as a lasso problem. However, this formulation is difficult to solve by utilizing many well-known lasso solvers because these approaches only solve lasso problem with real numbers. In power systems, voltage and admittance are complex numbers. To address this issue, we propose a group lasso approach that converts a complex number lasso formulation to a real number lasso problem.
For two arbitrary complex numbers $x$ and $y$, their product $z = xy$ is expressed as $$\begin{aligned}
\re(z) &=& \re(x)\re(y) - \im(x)\im(y), \\
\im(z) &=& \re(x)\im(y) + \im(x)\re(y).\end{aligned}$$ Thus, the linear equation in (\[eq:lin\_reg\]) can be rewritten as $$\begin{aligned}
\begin{bmatrix}
\re(\Delta V_i) \\
\im(\Delta V_i)
\end{bmatrix}
&=& \sum_{\substack{k=2 \\ k \neq i}}^M
\begin{bmatrix}
\re(\Delta V_k) & -\im(\Delta V_k) \\
\im(\Delta V_k) & \re(\Delta V_k)
\end{bmatrix}
\begin{bmatrix}
\re(\boldsymbol{\beta}^{(i)}_k) \\
\im(\boldsymbol{\beta}^{(i)}_k) \\
\end{bmatrix}, \nonumber \\
\mathbf{Z}_i &=& \sum_{\substack{k=2 \\ k \neq i}}^M \mathbf{X}_k\boldsymbol{\gamma}^{(i)}_k =\mathbf{X}\boldsymbol{\gamma}^{(i)}\label{eq:group_lin}.\end{aligned}$$ In (\[eq:group\_lin\]), we transform a complex linear equation to a real linear equation. We can apply the $L_1$ constraint to the new parameter vector $\boldsymbol{\gamma}^{(i)}$, which becomes an ordinary lasso problem.
Solving the linear regression in (\[eq:group\_lin\]) with $L_1$ penalty $\|\boldsymbol{\gamma}^{(i)}\|_1$ results a sparse estimate $\widehat{\boldsymbol{\gamma}}^{(i)}$. However, we cannot guarantee that both $\re(\boldsymbol{\beta}^{(i)}_k)$ and $\im(\boldsymbol{\beta}^{(i)}_k)$ are zero or nonzero at the same time. If bus $k$ is not connected with bus $i$, $\boldsymbol{\beta}^{(i)}_k$ is zero in (\[eq:Lasso\]). Thus, both $\re(\boldsymbol{\beta}^{(i)}_k)$ and $\im(\boldsymbol{\beta}^{(i)}_k)$ are zeros. To enforce the sparsity on both real and imaginary parts of $\boldsymbol{\beta}$, in (\[eq:group\_lin\]), we need to apply sparsity constraint to a group of elements in $\boldsymbol{\gamma}^{(i)}$ such that all elements within a group will be zero if one of them is zero. This problem formulation is known as *Group Lasso* [@yuan2006model]. Particularly, we can estimate $\boldsymbol{\gamma}^{(i)}$ as follows: $$\label{eq:group_lasso}
\widehat{\boldsymbol{\gamma}}^{(i)} = \arg\min_{\boldsymbol{\gamma}^{(i)}} \sum_{t=1}^T \left\|\mathbf{z}_i[t] - \sum_{\substack{k=2 \\ k \neq i}}^M \mathbf{x}_k[t]\boldsymbol{\gamma}^{(i)}_k\right\|_2^2 + \lambda \sum_{\substack{k=2 \\ k \neq i}}^M \|\boldsymbol{\gamma}^{(i)}_k\|_2.$$ Unlike the lasso formulation in (\[eq:Lasso\]), in (\[eq:group\_lasso\]), we use $L_2$ norm because it enforces the entire vector $\boldsymbol{\gamma}_k^{(i)}$ to be zero or nonzero. See [@hastie2015statistical] for more details on group lasso.
We can construct $\widehat{\boldsymbol{\beta}}^{(i)}$ from $\widehat{\boldsymbol{\gamma}}^{(i)}$ and find the non-zero elements in $\widehat{\boldsymbol{\beta}}^{(i)}$. Alternatively, if bus $k$ is not a neighbor of bus $i$, both elements in $\widehat{\boldsymbol{\gamma}}^{(i)}$ are zero. Algorithm \[alg:alg1\] summarizes the steps of proposed bus connectivity estimation algorithm.
$\Delta v_i[t]$ for $t = 1,\cdots,T$ For bus $i$, solve the group lasso problem in (\[eq:group\_lasso\]) and estimate the parameter vector $\widehat{\boldsymbol{\gamma}}^{(i)}$. Compute $\widehat{\boldsymbol{\beta}}^{(i)}$ as $\widehat{\boldsymbol{\beta}}^{(i)}_k = \widehat{\boldsymbol{\gamma}}^{(i)}_{k,1} + j\widehat{\boldsymbol{\gamma}}^{(i)}_{k,2}$. Find $\widehat{\mathcal{N}}(i)$ as $\widehat{\mathcal{N}}(i) = \{k|\widehat{\boldsymbol{\beta}}^{(i)}_k \neq 0\}$
\[alg:alg1\]
Bus Connectivity Estimation using Voltage Magnitude Only
--------------------------------------------------------
In distribution grids, voltage angles $\theta$ are hard to acquire because PMUs are not widely available. When only the change of voltage magnitude $\Delta|V_i|$ is available, $\mathbf{X}_k$ and $\mathbf{Z}_i$ become $\Delta |V_k|$ and $\Delta |V_i|$ respectively. Also, $\boldsymbol{\gamma}_k^{(i)}$ reduces to a scalar. The objective function of group lasso problem in (\[eq:group\_lasso\]) becomes $$\begin{aligned}
&& \sum_{t=1}^T \left(\Delta | v_i[t]| - \sum_{\substack{k=2\\k \neq i}}^M \Delta | v_k[t]|\gamma^{(i)}_k\right)^2 + \lambda \sum_{\substack{k=2 \\ k \neq i}}^M \|\gamma^{(i)}_k\|_2 \nonumber \\
&=& \sum_{t=1}^T \left(\Delta | v_i[t]| - \sum_{\substack{k=2 \\ k \neq i}}^M \Delta | v_k[t]|\gamma^{(i)}_k\right)^2 + \lambda \|\boldsymbol{\gamma}^{(i)}\|_1, \label{eq:V_lasso}\end{aligned}$$ where $\Delta |v_i[t]| = |v_i[t]| - |v_i[t-1]|$ and $\Delta |v_i[1]| = 0$. In (\[eq:V\_lasso\]), $\|\gamma^{(i)}_k\|_2$ is equivalent to $|\gamma^{(i)}_k|$ and $\sum_k \|\gamma^{(i)}_k\|_2 = \sum_k |\gamma^{(i)}_k| = \|\boldsymbol{\gamma}^{(i)}\|_1$, where $\boldsymbol{\gamma}^{(i)} \in \reals^{M-2}$. Hence, with $\Delta|V|$ only, we can reconstruct the bus connectivity using the ordinary lasso. [Unlike transmission grid, the $R/X$ ratio is large in distribution grids. Because of non-negligible branch resistance, the voltage measurements have larger variation. As proved in [@weng2016distributed], in distribution grid, the statical correlations among bus voltage magnitudes are more significant than those among bus voltage phase angles. Therefore, when bus voltage phase data are unavailable, we can still achieve high accuracy of topology estimation using voltage magnitude. In Section \[sec:num\], multiple simulation results show that using $\Delta |V|$ can provide accurate distribution grid topology estimation.]{} A detailed discussion of (\[eq:V\_lasso\]) is given in Appendix \[sec:V\_lasso\].
Choice of the Regularization Parameter $\lambda$ {#sec:lambda_path}
------------------------------------------------
The choice of $\lambda$ is critical in lasso and group lasso problems because it affects the number of non-zero coefficients in $\boldsymbol{\beta}^{(i)}$ and number of non-zero vector $\boldsymbol{\gamma}_k^{(i)}$. When $\lambda$ is small, the penalty term has no effect and the solution is close to the ordinary least squares (OLS) solution. When $\lambda$ is large, some coefficients of $\widehat{\boldsymbol{\beta}}^{(i)}$ or some vectors $\widehat{\boldsymbol{\gamma}}_k^{(i)}$ are zeros. A well-known criterion to choose the parameter $\lambda$ is Bayesian information criterion (BIC). For bus $i$, the BIC is defined as
$$\label{eq:BIC_i}
\text{BIC}_i(\lambda) = \frac{\text{RSS}_i(\lambda)}{\widetilde{T}\widehat{\sigma}^2} + \frac{\ln\widetilde{T}}{\widetilde{T}}\times \hat{k},$$
where $\hat{k}$ denotes the number of non-zero elements in $\widehat{\boldsymbol{\beta}}^{(i)}$ or the number of non-zero vector $\widehat{\boldsymbol{\gamma}}_k^{(i)}$, and $\widehat{\sigma}^2$ denotes the empirical variance of the residual [@zou2007degrees]. $\widetilde{T}$ is $2T$ for the problem in (\[eq:group\_lasso\]) or $T$ for the problem in (\[eq:V\_lasso\]). The residual sum of squares (RSS) is defined as $$\text{RSS}_i(\lambda) = \sum_{t=1}^T \left\|\mathbf{z}_i[t] - \sum_{\substack{k=2 \\ k \neq i}}^M \mathbf{x}_k[t]\widehat{\boldsymbol{\gamma}}^{(i)}_k\right\|_2^2.$$ We select the $\lambda$ that minimizes $\text{BIC}_i(\lambda)$ as the optimal tuning parameter for bus $i$. The selection process requires to solve the problem in (\[eq:group\_lasso\]) or (\[eq:V\_lasso\]) multiple times. Thankfully, many lasso solvers, such as the least angle regression (LAR) [@efron2004least; @yuan2006model], can solve the lasso problem with multiple $\lambda$s at once. Therefore, this selection process can be completed without any additional computation. For (\[eq:joint\_OR\]) and (\[eq:joint\_AND\]), we can use the same approach to choose $\lambda$.
Fig. \[fig:lambda\_path\] shows the path of BIC in each step of LAR algorithm. At each step, the LAR algorithm chooses a $\lambda$ and computes the corresponding coefficients $\widehat{\boldsymbol{\gamma}}^{(i)}$. Then, it decreases $\lambda$ and repeats the process above. Therefore, at Step 1, $\lambda$ has the largest value and all coefficients are zero. For the last step, all coefficients are non-zero. In Fig. \[fig:lambda\_path\], we can observe that the proposed scheme finds the sparse coefficient vector. Notice that in Fig. \[fig:lambda\_path\], we do not pick up $\lambda$ that yields the minimum BIC because the estimated coefficient vector has no sparsity, e.g., all elements in the coefficient estimate are non-zero. Hence, we choose $\lambda$ that reduces BIC significantly with high sparsity.
![BIC at each step for computing the bus connectivity in IEEE 123-bus system. The circle represents the BIC at each step. The red crossing represents the corresponding BIC of the selected $\lambda$.[]{data-label="fig:lambda_path"}](lambda_path){width="\linewidth"}
Grid Topology Estimation via Group Lasso {#sec:joint}
========================================
In the previous section, we have used group lasso to estimate the bus connectivity. However, a sub-network contains multiple unknown branches. In this section, we will extend the presented method from a bus to a network.
The neighbor set $\mathcal{N}(i)$ can be found by solving the group lasso problem of bus $i$. [Using the neighbor set of each bus, we can find the unknown branch between two buses. In [@meinshausen2006high], two rules are proved to find the unknown edges in graphical models with a guarantee of false alarm rate: AND rule and OR rule. Specifically, if bus $i$ and $k$ are adjacent, bus $i$ is a neighbor of bus $k$ and vice versa.]{} In (\[eq:group\_lasso\]), we find the neighbors of a single bus. Therefore, an edge between bus $i$ and $k$, $e_{ik}$, is estimated twice since $k \in \mathcal{N}(i)$ and $i \in \mathcal{N}(k)$. A simple approach is combining them, i.e., $\widehat{e_{ik}}^\text{AND} = \widehat{\boldsymbol{\beta}}^{(i)}_k \wedge \widehat{\boldsymbol{\beta}}^{(k)}_i$, where $\wedge$ denotes the logical “and”. We can apply this AND rule to every pair of buses within the unknown subnetwork and recover the topology at last.
Using the AND rule, an edge can only be recovered if both $\widehat{\boldsymbol{\beta}}^{(i)}_k$ and $\widehat{\boldsymbol{\beta}}^{(k)}_i$ are nonzero. If either of them has an error, this edge will be not included. Therefore, the AND rule has a high probability to miss the true edge. To minimize the number of missing edges, we propose the OR rule, $ \widehat{e}_{ik}^\text{OR} = \widehat{\boldsymbol{\beta}}^{(i)}_k \vee \widehat{\boldsymbol{\beta}}^{(k)}_i$, where $\vee$ is the logical “or” operator.
[Although both AND and OR rules have been statistically proven to estimate unknown edges [@meinshausen2006high], they are not been proven to satisfy the power system constraint that the estimated graph is a connected network.]{} Some buses may create an independent graph and are isolated from the main grid. To overcome this [power system constraint]{}, we merge both rules together and propose the AND-OR rule. Unlike the AND or OR rule, this new rule has multiple steps. We firstly use the AND rule to estimate the edge set $\widehat{\mathcal{E}}_\text{AND}$. Then, we diagnose and adjust $\widehat{\mathcal{E}}_\text{AND}$ by physical law. In distribution grid, we assume that the differences between bus voltage magnitudes have a strong impact on the direction of power flow [@coffrin2014linear]. Therefore, each load bus is expected to have a neighbor bus with higher voltage magnitude on average because most load buses absorb powers. If none of the bus $i$’s neighbors has higher voltage magnitude, bus $i$ is possibly isolated from the main grid. We can find a new neighbor for bus $i$ by apply the modified OR rule, i.e.: [ $$\label{eq:AND_OR_rule}
\widehat{e}_{ik}^\text{AND-OR} = (\widehat{\E}(|V_k|) > \widehat{\E}(|V_i|)) \wedge (\widehat{\boldsymbol{\beta}}^{(i)}_k \vee \widehat{\boldsymbol{\beta}}^{(k)}_i),$$ ]{} where $\widehat{\E}$ denotes the sample mean. The estimated edge set is $\widehat{\mathcal{E}}_\text{AND-OR} = \widehat{\mathcal{E}}_\text{AND} \cup \{\widehat{e}_{ik}^\text{AND-OR}\}$. Please note that we use $|V|$, not $\Delta |V|$, to enforce the AND-OR rule. The steps of AND-OR rule are summarized in Algorithm \[alg:and-or\]. We will show that the AND-OR rule provides more accurate and robust estimates than either the AND or OR rule in Section \[sec:num\].
Topology estimate of the AND rule $\widehat{\mathcal{E}}_\text{AND}$
\[alg:and-or\]
Grid Topology Estimation via Group Lasso {#grid-topology-estimation-via-group-lasso}
----------------------------------------
While the AND-OR rule is robust in reconstructing distribution grid topology, this process requires finding the bus connectivity of each bus at first. This leads to high computational time for large systems, due to solving multiple optimization problems. Also, it has not utilized the interactions between bus measurements. To improve the algorithm, we will formulate the grid topology estimation as a single optimization problem via group lasso.
Let’s start with a simple case. We assume only voltage magnitude data $\Delta |V|$ are available. In a fully connected $3$-bus system, we can express the voltage relationships using the following linear system, $$\begin{aligned}
\Delta |\mathbf{V}| &=& \mathbf{P}\widetilde{\boldsymbol{\beta}},\label{eq:joint_linear_sys} \\
\begin{bmatrix}
\Delta | V_1| \\
\Delta | V_2| \\
\Delta | V_3|
\end{bmatrix}
&=&
\begin{bmatrix}
\Delta | V_2| & 0 & 0\\
0 & \Delta | V_1| & 0 \\
\Delta | V_3| & 0 & 0\\
0 & 0 & \Delta | V_1| \\
0 & \Delta | V_3| & 0 \\
0 & 0 & \Delta | V_2|
\end{bmatrix}^T
\begin{bmatrix}
\widetilde{\beta}^{(1)}_2 \\
\widetilde{\beta}^{(2)}_1 \\
\widetilde{\beta}^{(1)}_3 \\
\widetilde{\beta}^{(3)}_1 \\
\widetilde{\beta}^{(2)}_3 \\
\widetilde{\beta}^{(3)}_2
\end{bmatrix}. \nonumber\end{aligned}$$ Estimating the distribution grid topology by the AND or OR rule is equivalent to solving the linear system in (\[eq:joint\_linear\_sys\]) with different penalties. Specifically, for an $M$-bus system, the following optimization problem is equivalent to the OR rule: $$\label{eq:joint_OR}
\widehat{\boldsymbol{\beta}}^\text{OR} = \arg\min_{\widetilde{\boldsymbol{\beta}}}\sum_{t=1}^T\left\|\Delta | \mathbf{v}[t]| - \mathbf{P}[t]\widetilde{\boldsymbol{\beta}}\right\|_2^2 + \lambda\|\widetilde{\boldsymbol{\beta}}\|_1,$$ where $\Delta | \mathbf{v}[t]| \in \reals^{(M-1)\times 1}$, $\mathbf{P}[t] \in \reals^{(M-1)\times (M-1)(M-2)}$, and $\widetilde{\boldsymbol{\beta}} \in \reals^{(M-1)(M-2)\times 1}$. With the $L_1$ norm penalty, any element in $\widetilde{\boldsymbol{\beta}}$ can be zero. Therefore, (\[eq:joint\_OR\]) is equivalent to the OR rule. When $M$ is large, the dimension of $\mathbf{P}[t]$ is high. However, since $\mathbf{P}[t]$ is sparse and only contains $M$ unique values, we can store and process it efficiently. In Section \[sec:compute\], we will further simplify $\mathbf{P}[t]$.
For the AND rule, $\widetilde{\beta}^{(i)}_k$ and $\widetilde{\beta}^{(k)}_i$ are either zero or non-zero simultaneously. Therefore, we can use the group lasso to solve problem in (\[eq:joint\_linear\_sys\]). In details, we can solve the following optimization problem: $$\label{eq:joint_AND}
\widehat{\boldsymbol{\beta}}^\text{AND} = \arg\min_{\widetilde{\boldsymbol{\beta}}}\sum_{t=1}^T\left\|\Delta | \mathbf{v}[t]| - \mathbf{P}[t]\widetilde{\boldsymbol{\beta}}\right\|^2 + \lambda \sum_{\substack{i,k=2 \\ k \neq i}}^M \|(\widetilde{\beta}^{(i)}_k,\widetilde{\beta}^{(k)}_i)\|_2.$$ In (\[eq:joint\_AND\]), we group all bus pairs into a penalty term to decide the pairwise connectivity. Thus, (\[eq:joint\_AND\]) is equivalent to the AND rule. To apply the AND-OR rule, we can follow the same step shown in Algorithm. \[alg:and-or\].
Simulation and Results {#sec:num}
======================
We firstly use IEEE $8$-bus and $123$-bus networks [@teng2002modified; @kersting2001radial], with additional branches to create mesh structures, to validate the performance of our lasso-based approach on mesh networks. Fig. \[fig:new\_8bus\] illustrates the modified mesh $8$-bus system. Also, we justify our methods on six European representative distribution systems with different topologies, which include MV and LV distribution grids in urban (*LV\_urban*, *MV\_urban*, *MV\_two\_substations*, *Urban*), suburban (*LV\_suburban*) and rural (*MV\_rural*) areas[@pretticodistribution]. To explore the impact of loops, we add several branches in LV and MV grids and generate *LV\_suburban\_mesh* and *MV\_urban\_mesh* systems. *Urban* system is a large-scale distribution grid that includes MV and LV networks. It has about 13,000 customers, 126 MV/LV substations, and 3237 branches. Most urban and suburban branches in these networks are underground. The European system topology and details are available in Appendix \[sec:eu\_network\]. In each network, the feeder or substation is selected as the slack bus.
The smart meter hourly load readings from PG&E are used in all simulations. Since PG&E data set does not have the reactive power, we emulate $q_i[t]$ according to a random lagging power factor $pf_i[t]$, e.g., $pf_i[t] \sim \Unif(0.85,0.95)$. For the load profile in MV grid, we aggregate the load profiles of 10 to 300 residents, depending on the load capacity. The hourly voltage measurements $v_i[t]$ are obtained by MATPOWER[@Zimmerman10] and $N = 8760$ measurements are computed at each bus.
Estimation Error of Bus Connectivity
------------------------------------
For bus $i$, we define the connectivity error as $$\text{Error}(i) = \underbrace{\sum_{k \in \mathcal{N}(i)} \indic{k \notin \widehat{\mathcal{N}}(i)}}_{\text{false estimation}} + \underbrace{\sum_{k \in \widehat{\mathcal{N}}(i)} \indic{k \notin \mathcal{N}(i)}}_{\text{missing}},$$ where $\widehat{\mathcal{N}}(i)$ denotes the neighbor set estimate using (\[eq:group\_lasso\]) or (\[eq:V\_lasso\]) and $\indic{}$ denotes the indicator function. The first part represents the number of missing neighbors and the second part represents the number of incorrect neighbors.
System Lasso Group Lasso
------------------------- ------- -------------
123-bus 10 0
123-bus with loops 10 1
123-bus with PV 6 0
123-bus with loops & PV 6 0
: Total Branch Error of Lasso (\[eq:Lasso\]) and Group Lasso (\[eq:group\_lasso\]) using $\Delta V$
\[tab:complex\_lasso\]
Table \[tab:complex\_lasso\] shows the total branch estimation error of lasso (\[eq:Lasso\]) and group lasso (\[eq:group\_lasso\]) using $\Delta V$. Without grouping the real and imaginary parts, the lasso method has worse performance because of the inconsistency between real and imaginary parts of the complex estimate. By adding grouping constraint, our method achieves nearly perfect results.
Our simulations show that for 8-bus networks, with or without loops, our algorithm presented in Section \[sec:main\] achieves zero error using $\Delta |V|$. Fig. \[fig:nodeMissing123\] shows the error at each bus for 123-bus networks, with or without loops. We can observe that most buses have zero error by using $\Delta |V|$. While identifying all connectivities in a $123$-bus system is excessive in practice, our method can find all connectivities successfully except two or three buses. For the European representation network, we observe the similar performance as IEEE systems.
![Errors of bus connectivity reconstruction for different networks using $\Delta | V|$ only.[]{data-label="fig:nodeMissing123"}](nodeMissing123){width="\linewidth"}
For the bus connectivity estimation with integrated DERs, our simulations show that the proposed approach finds the buses’ neighbors without any error in 8-bus system. In 123-bus networks, as shown in Fig. \[fig:nodeMissing123\], our approach has nearly perfect performance.
Network Topology Reconstruction Error Rate
------------------------------------------
In this section, we discuss the performance on grid topology reconstruction. We use the error rate (ER) as the performance evaluation metric, which is defined as $$\begin{aligned}
\text{ER} &=& \frac{1}{|\mathcal{E}|}\left(\underbrace{\sum_{e_{ij} \in \widehat{\mathcal{E}}} \indic{e_{ij} \notin \mathcal{E}}}_{\text{false estimation}} + \underbrace{\sum_{e_{ij} \in \mathcal{E}} \indic{e_{ij} \notin \widehat{\mathcal{E}}}}_{\text{missing}}\right)\times 100\%
$$ where $\widehat{\mathcal{E}}$ denotes the edge set estimates, $|\mathcal{E}|$ is the size of $\mathcal{E}$, and $\indic{.}$ is the indicator function. The first and second terms represent the number of falsely estimated branches and the number of missing branches, respectively.
In Table \[tab:mismatch\_group\], we summarize the error rates of 123-bus systems and six EU representative distribution grids with different topology configurations and decision rules. When the system is integrated with PMUs, the group lasso method reconstructs most MV and LV systems with zero error. When only voltage magnitude is available, the performances of OR and AND rules are degraded. But the AND-OR rule still achieves nearly perfect performance in most test cases. For *Urban* system, which contains both MV and LV grids, the majority of the error is due to the missing edges. Only four branches are incorrectly estimated. These missing edges are ones that connect MV/LV transformers. For many utilities, the location information of these transformers is known. With the locational information, the error rate is reduced to $0.6\%$. A similar improvement can be obtained for *Urban* with all switches closed. To validate our algorithm on large-scale distribution grids without transformers, we create an artificial grid (*LV\_large*) by combing 31 *LV\_suburban\_mesh* grids. From Table \[tab:mismatch\_group\], we observe that the error does not scale by the size of grid.
In some distribution grids, the voltage of slack bus may not be constant. As discussed in Section \[sec:formulation\], we can model the slack bus voltage as a random variable ($\Delta V_1$) and include it in (\[eq:joint\]). To validate this case, we perform a simulation on a LV grid in *Urban* system. In this simulation, the feeder is selected as the slack bus and its voltage is determined by the upper MV grid. Hence, the voltage measurements of slack bus are not constant. By utilizing the proposed algorithm, the error rate does not change when the slack bus voltage is not constant.
---------------------- -------- -------------- -------------- --------------
System Total AND OR AND-OR
Branch $\Delta |V|$ $\Delta |V|$ $\Delta |V|$
$8$-bus 7 0% 14.29% 0%
$8$-bus 10 20% 10% 0%
3 loops
$123$-bus 122 4.07% 2.44% 0%
$123$-bus 124 4.07% 1.63% 0%
2 loops
*LV\_urban* 13 0% 0% 0%
*LV\_suburban* 114 4.42% 0.88% 0%
*LV\_suburban\_mesh* 129 2.33% 8.53% 0.78%
15 loops
*MV\_urban* 34 0% 5.88% 0%
*MV\_urban* 35 0% 0% 0%
switch 34-35
1 loop
*MV\_urban* 35 2.86% 5.71% 2.86%
switch 23-35
1 loop
*MV\_urban* 35 2.86% 5.71% 0%
switch 13-35
1 loop
*MV\_urban* 37 0% 0% 0%
3 switches
3 loops
*MV\_urban\_mesh* 44 0% 4.55% 0%
10 loops
*MV\_two\_stations* 46 4.35% 8.7% 0%
*MV\_two\_stations* 47 4.25% 12.77% 0%
switch 14-37
1 loop
*MV\_two\_stations* 47 0% 0% 0%
switch 24-48
1 loop
*MV\_two\_stations* 48 0% 6.25% 0%
2 switches
2 loops
*MV\_rural* 116 0% 11.30% 0%
*MV\_rural* 119 0% 11.02% 0%
3 switches
3 loops
*Urban* 3237 19.62% 10.72% 6.98%
*Urban* 3242 19.56% 10.61% 6.97%
all switches
5 loops
*LV\_large* 4030 0.99% 5.98% 0.12%
465 loops
---------------------- -------- -------------- -------------- --------------
: Network Topology Reconstruction Error Rate without DERs
\[tab:mismatch\_group\]
![Comparison of ER amongst three methods using $\Delta |V|$ in IEEE 123-bus systems with radial and mesh structures.[]{data-label="fig:lasso_compare2_all"}](lasso_compare2_all){width="0.8\linewidth"}
The performance comparison amongst our lasso-based algorithm, a correlation-based algorithm [@bolognani2013identification], and an information theory-based algorithm [@liao2015distribution] is illustrated in Fig. \[fig:lasso\_compare2\_all\]. The error rates are averaged over $100$ iterations. The proposed method consistently recovers the topology with nearly $0\%$ error rate. This result is comparable with [@liao2015distribution], while the detection ability of [@bolognani2013identification] drops. The approach in [@liao2015distribution] is excluded in the mesh network comparison because it can only be applied to radial networks.
Networks with DER Integration
-----------------------------
The penetration of DERs has grown significantly during last decade and will keep increasing in the future. To evaluate the proposed algorithm with integrated DERs, we install several rooftop photovoltaic (PV) systems in the distribution networks. The profile of hourly power generation is obtained from NREL PVWatts Calculator, an online simulator that estimates the PV power generation based on weather history of PG&E service zone and the physical parameters of a $5$kW PV panel in LV grids or a $20$kW PV panel in MV grids[@dobos2014pvwatts]. The power factor is fixed as $0.90$ lagging, which satisfies the regulation of many U.S. utilities [@ellis2012review] and IEEE standard [@ieee2014guide].
System Total PV AND OR AND-OR
--------------------- ---------- -------- -------- -------- --
8-bus 8 14.29% 14.29% 0%
8-bus 8 10.00% 10.00% 0%
3 loops
123-bus 12 2.44% 0.81% 0%
123-bus 12 4.07% 0% 0%
2 loops
*LV\_suburban* 10 0.88% 8.85% 0%
*LV\_suburban* 20 2.65% 12.39% 0.88%
*LV\_suburban* 33 4.42% 7.96% 0.88%
*MV\_urban* 7 2.78% 2.78% 0%
*MV\_urban* 7 0% 2.86% 0%
switch 34-35
1 loop
*MV\_urban* 7 0% 2.70% 0%
3 switches
3 loops
*MV\_two\_stations* 10 4.35% 4.35% 0%
*MV\_two\_stations* 10 0% 2.08% 0%
2 switches
2 loops
*MV\_rural* 20 5.17% 12.07% 0.86%
*MV\_rural* 20 11.86% 15.25% 2.52%
3 switches
3 loops
*Urban* 300 19.96% 10.10% 9.02%
*LV\_large* 300 1.41% 3.05% 0.42%
465 loops
: Network Topology Reconstruction Error Rate with Rooftop PV System using $\Delta |V|$
\[tab:mismatch\_renewable\]
The error rates of grid topology reconstruction with the rooftop PVs integration are presented in Table \[tab:mismatch\_renewable\]. OR rule and AND rule have performance degradation. The error rate of the AND-OR rule is still the lowest one and most network topologies can be recovered perfectly. Additionally, we can observe that different levels of PV penetration do not have a significant impact on the algorithm performance. For *Urban*, with the prior knowledge of transformer locations, the AND-OR rule misses $22$ branches amongst $3237$ branches and has no false estimation error. The error rate reduces to $0.68\%$. For *LV\_large* system, the error rate of AND-OR rule is similar to the case without DERs in Table \[tab:mismatch\_group\].
Computational Complexity {#sec:compute}
------------------------
The least angle regression (LAR) has a computational complexity of $\mathcal{O}(M^3+TM^2)$, where $T$ denotes the number of observation and $M$ denotes the grid size. Since the complexity is dominated by $M$, finding the bus connectivity takes a long time in a large-scale system. Fortunately, for a particular bus in the distribution grid, the number of neighbors is relatively small compared with the grid size [@pagani2011towards]. Therefore, to estimate the connectivity of a particular bus, we select $K$ buses from an $M$-bus system and then apply (\[eq:V\_lasso\]) only to these $K$ buses. The complexity for bus connectivity estimation reduces to $\mathcal{O}(K^3+TK^2)$. For the entire system, the complexity is $\mathcal{O}(M(K^3+TK^2))$, which is linear in term of the system size $M$. Also, the dimension of $P[t]$ in (\[eq:joint\_OR\]) and (\[eq:joint\_AND\]) becomes $(K-1)\times(K-1)(K-2)$, which is smaller than the full model and independent of the system size. For the systems we analyze in this paper, the average number of neighbor per bus is $2$, and the maximum is $10$. We choose $K=\sqrt{M}$ to ensure that the true neighbors are contained within these $K$ buses.
In this paper, we use the mutual information as the measure to choose the $K$ most relative buses. Specifically, in [@liao2015distribution], the authors prove that if two buses are connected, their mutual information is higher than the pair of buses that are not connected. Therefore, to solve the lasso problem efficiently, firstly, we compute the pairwise mutual information. Secondly, for bus $i$, we find the top $K$ buses that have the largest mutual information with bus $i$. Thirdly, we apply group lasso to these $K$ buses and find the neighbors of bus $i$.
![Total computational time of the correlation-based algorithm and our method with different bus selections. We use $\Delta |V|$ and IEEE $123$-bus system with mesh structures.[]{data-label="fig:timeComp"}](timeComp){width="0.8\linewidth"}
The average computational time of our lasso-based method and the correlation-based algorithm is summarized in Fig. \[fig:timeComp\]. We use LAR method to solve the problem in (\[eq:V\_lasso\]) and apply AND-OR rule to find the topology. Fig. \[fig:timeComp\] shows our method is consistently faster than [@bolognani2013identification]. Also, by selecting $K$ most relative buses, the proposed algorithm is faster by a factor of 12 and achieves the same accuracy. For the $8$-bus system with mesh structures in Fig. \[fig:new\_8bus\], our method only uses $0.3$ seconds to recover the topology. In [@weng2016distributed], the authors extend the information theory-based algorithm, which is designed for radial network, to mesh grids. The extended algorithm requires over $100$ seconds to estimate the topology of mesh $8$-bus network. Hence, our approach is faster than other ones in mesh systems. For *Urban* system with $K=30$, the average computational time is less than $270$ seconds, making it useful for semi-real time applications.
Sensitivity Analysis
--------------------
In this subsection, we will discuss how data accuracy, data length, load pattern, and data resolution affect the algorithm performance.
### Sensitivity to Data Accuracy
Our method relies on the smart meter measurements. Hence, it is important to know whether the existing meters’ accuracy is sufficient for topology estimation. In the U.S., ANSI C12.20 standard (Class 0.5) requires the smart meters to have an error less than $\pm 0.5\%$ [@ansc12; @zheng2013smart]. Table \[tab:noise\] shows the average error rate with different noise levels over 20 iterations. The AND-OR rule outperforms other two rules in all levels of noise. For the 123-bus system with loops and *MV\_urban*, the AND-OR rule consistently reconstructs the entire network without any error. The accuracy of *LV\_suburban* is reduced when the noise level is high.
---------------- ------- -------- ------- ------- ------ ------
Noise Level 0.1% 0.5% 0.1% 0.5% 0.1% 0.5%
123-bus 8.94% 10.28% 2.44% 2.44% 0% 0%
*LV\_suburban* 7.96% 10.84% 4.42% 8.36% 0% 4.2%
*MV\_urban* 5.41% 5.41% 0% 0% 0% 0%
all switches
closed
---------------- ------- -------- ------- ------- ------ ------
: Error Rate With Different Noise Levels using $\Delta|V|$
\[tab:noise\]
In some countries, the utilities pre-process the voltage measurements, e.g., round up the float data to integers. These types of data processing can create identical measurements, e.g., $110$ volts, for a majority of time, making our algorithm relative poor due to the loss of a statistical relationship. However, our method can give recommendations to what data resolution is needed to utilize smart meter beyond billing purpose only.
In distribution grids, some switches may change statuses to protect circuits temperately. If we include the measurements collected during the temperate switch changes, our algorithm may have a decrease in accuracy. To overcome this issue, we can apply the method discussed in [@liao2016urban] to identify and remove these measurements. Then, we use the rest data to estimate grid topology. Also, as shown previously, our algorithm can estimate a large-scale grid in a few minutes. Thus, we can apply this algorithm to data sets acquired at consecutive time slots and check if the results are identical.
### Sensitivity to Data Length
To understand the impact of data set size, we validate the proposed algorithm by using measurements from 2 up to 360 days. Fig. \[fig:data\_duration\] shows the error rates of different networks with various data lengths. For the 123-bus loopy network, we can see that with around 100 days’ measurements ($100\times24 = 2400$ data points), the AND-OR rule achieves zero error. For *LV\_suburban\_mesh* and *MV\_urban\_mesh*, only ten days’ data are required to achieve perfect estimation.
![Error rates of the AND-OR rule with different data lengths using $\Delta |V|$.[]{data-label="fig:data_duration"}](data_duration){width="0.8\linewidth"}
### Sensitivity to Load Pattern
To understand our algorithm’s sensitivity to load pattern, we validate the proposed algorithm on the “ADRES-Concept” Project load profile [@Einfalt11; @VUT16]. This data set contains real and reactive powers profile of 30 houses in Upper-Austria. The data were sampled every second over 7 days in summer and 7 days in winter.
In Fig. \[fig:EU\_winter\_summer\], we compare the error rates using winter and summer load profiles individually. The voltage profiles are obtained by using IEEE 123-bus test case and assuming that the connectivity between bus 78 and bus 102 is unknown. From Fig. \[fig:EU\_winter\_summer\], we observe that the data coming from different seasons do not impact the algorithm performance. Also, our algorithm converges faster to zero error.
![Comparison of error rate with data from different seasons using $\Delta|V|$.[]{data-label="fig:EU_winter_summer"}](EU_winter_summer){width="\linewidth"}
### Sensitivity to Data Resolution
Fig. \[fig:EU\_time\] illusrates the performance of AND-OR rule under different sampling frequencies. When the sampling period is 1 minute, we need about 2 hours’ voltage profile to have perfect estimation of the entire system. The frequency of distribution grid reconfiguration is range from hours to weeks [@jabr2014minimum; @dorostkar2016value]. Therefore, the proposed method is suitable for the existing system and real-time operation. If the sampling period is 30 minutes, our algorithm needs about 50 hours’ measurements. This data requirement is still less than the current network reconfiguration frequency. Also, Fig. \[fig:EU\_time\] shows that the estimation time can be reduced by using high sampling frequency.
![Comparison of error rate with different data resolutions using $\Delta|V|$.[]{data-label="fig:EU_time"}](EU_time){width="\linewidth"}
Conclusion {#sec:con}
==========
A data-driven algorithm of bus connectivity and grid topology estimation is presented in MV and LV distribution grids. Comparing with past studies, our method does not need the admittance matrices or switch location information. Only the smart meter data (voltage magnitude profile) are utilized for topology estimation. Also, unlike many past studies, our method can estimate not only radial systems but also mesh networks. We prove that, representing a distribution grid as a graphical model, the grid topology can be efficiently recovered by group lasso. We validate the proposed algorithm on eight MV and LV distribution networks with 21 network configurations using real data from PG&E and NREL. With or without DER penetration, our algorithm estimates the topology of a large-scale MV/LV grid with over 95% accuracy in a short period of computational time. Finally, we analyze the algorithm performance under different noise levels, data resolutions, data duration, and load patterns. The results indicate that our method can provide robust estimation in various scenarios and outperform other existing methods.
Acknowledgement
===============
The authors would like to acknowledge Dr. Giuseppe Prettico from European Commission Joint Research Centre for sharing the European Representative Distribution Networks. We would also like to thank Vienna University of Technology - Institute of Energy Systems and Electrical Drives for providing the ADRES-Concept data set. The first author would like to thank Dr. Junjie Qin from Stanford University for discussions and Stanford Leavell Fellowship for the financial support. Part of this project is supported by State Grid Corporation technology project (SGRIJSKJ(2016)800).
Proof of Lemma \[lemma: linear\_indept\] {#sec:lemma1_proof}
----------------------------------------
We will prove Lemma \[lemma: linear\_indept\] using a counterexample. Assuming $a_1 \neq 0$ and $b_1 \neq 0$, given $Y = y$, we have the following equation $$\begin{aligned}
a_1 X_1 + b_1 X_2 &=& y, \\
X_1 &=& (y- b_1 X_2)/a_1.\end{aligned}$$ Since $a_1$ and $b_1$ are non-zeros, $X_1$ always depends on $X_2$. Therefore, to have $X_1$ and $X_2$ conditionally independent, at least one coefficient needs to be zero, e.g., $a_1 = 0$, $b_1 = 0$, or $a_1 = b_1 = 0$.
Proof of Lemma \[lemma:linear\_indept2\] {#sec:lemma2_proof}
----------------------------------------
Assuming $c_1 = 0$, (\[eq:lemma2\_1\]) becomes $d_1 Z= X_1$. Given $Z = z$, $X_1 = d_1 z$ is a constant. Therefore, $X_1$ and $X_2$ are conditionally independent. When $c_2 = 0$ and $Z=z$, $X_2$ becomes a constant and therefore, $X_1$ and $X_2$ are conditionally independent.
When $c_1 = 0$ and $c_2 = 0$, both $X_1$ and $X_2$ become constants. Therefore, they are not random variables and we cannot determine statistical dependency.
Therefore, if $c_1 = 0$ or $c_2 = 0$, $X_1$ and $X_2$ are conditionally independent given $Z=z$.
Proof of Theorem \[thm:two\_step\_dependency\] {#sec:thm1_proof}
----------------------------------------------
Let’s firstly recall the following relationship between currents and voltages in a grid with $M$ buses. For bus $i$, $$\label{eq:iv_p}
\Delta I_i = \Delta V_iy_{ii} - \sum_{k \in \mathcal{N}(i)}\Delta V_ky_{ik},$$ with $y_{ii} = \sum_{k \in \mathcal{N}(i)}y_{ik} +\frac{1}{2}b_i$. Given $\Delta V_k = \Delta v_k$ for all $k \in \mathcal{N}(i) \cup \mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i)$, the equation above becomes $$\label{eq:iv_p_rewrite}
\Delta I_i + \sum_{k \in \mathcal{N}(i)}\Delta v_ky_{ki} = \Delta V_iy_{ii}.$$ This equation has only has two random variables, $\DV_i$ and $\DI_i$. We can rewrite the equation above as $$\label{eq:bus_i}
\DI_i = a_i\DV_i + b_i,$$ where $a_i$ and $b_i$ are constants.
For bus $l$ in $\mathcal{N}(i)$, given $\DV_k = \Dv_k$ for all $k \in \mathcal{N}(i) \cup \mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i)$, we have a similar equation $$\label{eq:bus_l}
\Delta I_l + \sum_{k \in \mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i)}\Delta v_ky_{ki} + \DV_iy_{il} = \Delta v_ly_{ll}.$$ In (\[eq:bus\_l\]), the only unknown voltage variable is $\DV_i$ and therefore, $\DI_i$ and $\DI_l$ are conditionally dependent given $\Delta \mathbf{V}_{\mathcal{N}(i) \cup \mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i)}$ for all $l \in \mathcal{N}(i)$.
For bus $p$ in $\mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i)\backslash\mathcal{N}(i)$, given $\DV_k = \Dv_k$ for all $k \in \mathcal{N}(i) \cup \mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i)$, we the following equation [ $$\Delta v_py_{pp} = \Delta I_p + \sum_{\substack{k \in \mathcal{N}(i) \cup \mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i) \\ k \neq p}}\Delta v_ky_{kp} + \sum_{\substack{r \in \\\mathcal{M}\backslash\{\mathcal{N}(i) \cup \mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i) \cup \{i\}\}}} \DV_ry_{rp} \label{eq:bus_p}$$ ]{} [We can rewrite (\[eq:bus\_p\]) as $$\label{eq:bus_p_rewrite}
\Delta I_p =
\sum_{\substack{r \in \\ \mathcal{M}\backslash\{\mathcal{N}(i) \cup \mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i) \cup \{i\}\}}} \DV_r \widetilde{\mathbf{a}}_r + \widetilde{b}$$ If assuming the random vector $\mathbf{Y}$ in Lemma \[lemma:linear\_indept2\] as $\mathbf{Y} = [\DV_i, \DV_{\mathcal{M}\backslash\{\mathcal{N}(i) \cup \mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i) \cup \{i\}\}}]^T$, using (\[eq:iv\_p\_rewrite\]) and (\[eq:bus\_p\_rewrite\]), we can apply Lemma \[lemma:linear\_indept2\] to show that $\DI_i$ and $\DI_p$ are conditionally independent given $\Delta \mathbf{V}_{\mathcal{N}(i) \cup \mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i)}$ for all $p \in \mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i)\backslash\mathcal{N}(i)$.]{} Similarly, for buses that are more than two steps away from bus $i$, we can observe that their incremental current injection $\DI_q$ and $\DI_i$ are conditionally independent, i.e., $\DI_i \perp \DI_q|\DV_{\mathcal{N}(i) \cup \mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i)}$ for $q \in \mathcal{M}\backslash\{\mathcal{N}(i) \cup \mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i)\}$.
[ For any bus $q$ that is two more hops away from bus $i$, i.e., $q \in \mathcal{M}\backslash\{\mathcal{N}(i) \cup \mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}(i) \cup \{i\}\}$, the nodal equation is $$\label{eq:bus_q}
\DI_q = y_{qq}\DV_q + \sum_{k \in \mathcal{M}\backslash\{i,q,\mathcal{N}(i)\}} y_{qk} \DV_k.$$ As demonstrated in the example of Theorem \[thm:two\_step\_dependency\], the voltage at bus $q$ can be written as a summation of current injections and the voltage at bus $q$, i.e., $$\label{eq:bus_q_i}
\DI_q = c_q\DV_q + \sum_{k \in \mathcal{M}\backslash\{i,q,\mathcal{N}(i)\}} \mathbf{d}_k \DI_k,$$ where $c_q$ and $\mathbf{d}$ are constants. As we have proved earlier, $\DI_i$ and $\DI_q$ are conditionally independent, given $\DV_{\mathcal{N}(i) \cup \mathcal{N}^{ \textup{\uppercase\expandafter{\romannumeral2}}}{i}}$. Also, $\DI_i$ and $\DI_k$ are conditionally independent because $\DI_k$ are current injections of buses that are two or more hops away. Therefore, $\DI_i$ and $\DI_q + \sum_{k \in \mathcal{M}\backslash\{i,q,\mathcal{N}(i)\}}\DI_k$ are conditionally independent. Using (\[eq:bus\_i\]) and (\[eq:bus\_q\]), $\DV_i$ and $\DV_q$ are conditionally independent. This proof holds for every bus $q$ that is more than two hops away from bus $i$. ]{}
Proof of (\[eq:V\_lasso\]) {#sec:V_lasso}
--------------------------
In this section, we will show that with the increment of voltage magnitude $\Delta|V|$, the linear relationship expressed in (\[eq:v\_linear\]) still holds. Therefore, we can still use the lasso method to find the bus connectivity.
Since $y_{ik}$ and $\Delta V_i$ are all complex numbers, we can express them in polar form, i.e., $y_{ik} = |y_{ik}|\exp{j\phi_{ik}}$ and $\Delta V_i = \Delta |V_i|\exp{j\theta_i}$. Then, letting $\epsilon = \Delta I_i/y_{ii} $, (\[eq:v\_linear\]) becomes $$\begin{aligned}
&& \Delta |V_i|\exp{j\theta_i} \\
&=& \sum_{k \in \mathcal{N}(i)} \frac{|y_{ik}|}{|y_{ii}|}\exp{j(\phi_{ik}-\phi_{ii})}\Delta |V_k|\exp{j\theta_k} + \epsilon\\
&=& \sum_{k \in \mathcal{N}(i)}\Delta |V_k|\frac{|y_{ik}|}{|y_{ii}|} \exp{j(\phi_{ik}-\phi_{ii}+\theta_k)} + \epsilon.\end{aligned}$$ Reorganizing the equation above, we have $$\begin{aligned}
\Delta |V_i| &=& \sum_{k \in \mathcal{N}(i)}\Delta |V_k|\frac{|y_{ik}|}{|y_{ii}|} \exp{j(\phi_{ik}-\phi_{ii}+\theta_k-\theta_i)} \\
&+& \epsilon\exp{-j\theta_i} \\
&=& \sum_{k \in \mathcal{N}(i)} \Delta |V_k| \gamma_{ik} + \tilde{\epsilon}.\end{aligned}$$ If bus $i$ and $k$ are not connected, $y_{ik} = |y_{ik}| = 0$. Therefore, $\gamma_{ik} = 0$ since the exponential term cannot yield zero. The lasso problem in (\[eq:V\_lasso\]) is an approximation of the equation above because we assume $\gamma_{ik}$ is a real number. However, this assumption does not affect the results because $|\gamma_{ik}|=0$ is the only solution for non-connected branch pairs.
European Representative Distribution Networks {#sec:eu_network}
---------------------------------------------
In this section, we will briefly summarize the five representative distribution networks used in Section \[sec:num\]. For more details, please refer to [@pretticodistribution]. The topology maps below are duplicated from [@pretticodistribution] with several modifications.
*LV\_urban* and *LV\_suburban* systems represent the low voltage networks in urban and suburban areas respectively. *LV\_suburban\_mesh* system in Fig. \[fig:LV\_semiurban\_mesh\] is a modified grid from *LV\_suburban* system by adding additional branches to create loops. For all networks, the nodal voltage is 400V and all branches are underground. Bus 1 connects the LV grid to the substation with a 20kV/0.4kV transformer. *LV\_large* network is an artificial distribution grid by combining 31 *LV\_surban\_mesh* LV distribution grids. This network includes 3534 buses and 4030 branches. Bus 1 of 20 *LV\_surban\_mesh* grids are connected at a common slack bus Bus 0. The rest 11 *LV\_surban\_mesh* grids are connected at the end buses (e.g., Bus 50 and Bus 114) of the first 20 grids.
![Suburban Low Voltage Network (*LV\_suburban*). The dashed lines indicate the branches that form suburban mesh low voltage network (*LV\_suburban\_mesh*). All branches have the impedance $0.0019+j0.001\Omega$.[]{data-label="fig:LV_semiurban_mesh"}](LV_semiurban_mesh){width="\linewidth"}
*MV\_urban* and *MV\_two\_substations* systems represent the medium voltage networks in urban and suburban areas. The bus voltage is 20kV and most branches are underground in these grids. Bus 1 connects with the HV/MV substation. Today, many MV distribution grids have mesh structures but radial operational topologies. Recently, several papers have shown that the closed-loop MV distribution grids can reduce power losses, provide a better voltage profile, and improve the power quality and service reliability [@chen2004feasibility; @kim2013advanced]. Several utilities, such as Taipower, Florida Power Company, Hong Kong Electric Company, Singapore Power, and Korea Electric Power Cooperation, have operated mesh MV distribution grids in their service zone [@chen2004feasibility; @pagel2000energizing; @teo1995principles; @jeon2016underground]. Also, as studied by [@celli2004meshed; @de2014investigation], for distribution grids with high DER penetration, MV distribution grids with mesh operational topology will be more reliable and efficient. In this study, our goal is to understand the performance of the proposed algorithm under both existing and future grid structures. Therefore, we validate our algorithm on MV grids with closed switches and closed-loop structure. *MV\_urban\_mesh* system in Fig. \[fig:MV\_urban\_mesh\] are modified from *MV\_urban* grid by adding additional branches.
![Urban Medium Voltage Network (*MV\_urban*). The red dashed lines indicate the branches and form urban mesh medium voltage network (*MV\_urban\_mesh*). All branches have the impedance $0.0844 + j0.0444\Omega$.[]{data-label="fig:MV_urban_mesh"}](MV_urban_mesh){width="\linewidth"}
*MV\_rural* grid is a rural medium voltage network with four switches. Compared with urban MV networks, the distance of each branch in the rural network is much longer. To understand the impact of switch status, we close these switches and analyze the method performance on the loopy networks.
*Urban* system is an urban distribution grid, which includes MV feeders, LV feeders, MV/LV substations, and HV/MV substation. Most low voltage branches and all medium voltage branches are underground. To understand the impact of switch status, we close all switches and validate our method on the mesh networks. Table \[tab:rx\] summarizes the $X/R$ ratios of all networks used in this paper. This table indicates that our algorithm works for typical medium and low voltage grids. Table \[tab:avg\_degree\] shows the average and maximum number of neighbors per bus. These results show that the approach using $K$ top relative bus is valid.
Grid Minimum Average Maximum
------------------------ --------- --------- ---------
123-bus 0.49 1.66 2.36
*LV\_urban* 0.19 0.25 0.57
*LV\_suburban* 0.53 0.54 0.57
*MV\_urban* 0.52 0.52 0.53
*MV\_two\_substations* 0.52 0.52 0.53
*MV\_rural* 0.92 0.93 0.93
*Urban* 0.07 0.39 3.48
: $X/R$ Ratio of the networks[]{data-label="tab:rx"}
Grid Average Maximum
------------------------ --------- ---------
123-bus 1.98 4
*LV\_urban* 1.85 4
*LV\_suburban* 1.98 3
*MV\_urban* 2.11 5
*MV\_two\_substations* 2.04 3
*MV\_rural* 2.03 9
*Urban* 2.00 10
: Average and Maximum Numbers of Neighbors[]{data-label="tab:avg_degree"}
[^1]: Y. Liao and R. Rajagopal are with Department of Civil and Environmental Engineering, Stanford University, Stanford, CA, 94305 USA e-mail: {yzliao,ramr}@stanford.edu. Y. Weng is with School of Electrical, Computing, and Energy Engineering, Arizona State University, Tempe, AZ, 85287 USA e-mail: yang.weng@asu.edu. G. Liu is with State Grid GEIRI North America, Santa Clara, CA, 95054 USA e-mail: guangyiliu2010@gmail.com.
|
---
abstract: 'We investigate competition between two phase transitions of the second kind induced by the self-attractive nonlinearity, *viz.*, self-trapping of the leaky modes, and spontaneous symmetry breaking (SSB) of both fully trapped and leaky states. We use a one-dimensional mean-field model, which combines the cubic nonlinearity and a double-well-potential (DWP) structure with an elevated floor, which supports leaky modes (quasi-bound states) in the linear limit. The setting can be implemented in nonlinear optics and BEC. The order in which the SSB and self-trapping transitions take place with the growth of the nonlinearity strength depends on the height of the central barrier of the DWP: the SSB happens first if the barrier is relatively high, while self-trapping comes first if the barrier is lower. The SSB of the leaky modes is characterized by specific asymmetry of their radiation tails, which, in addition, feature a resonant dependence on the relation between the total size of the system and radiation wavelength. As a result of the SSB, the instability of symmetric modes initiates spontaneous Josephson oscillations. Collisions of freely moving solitons with the DWP structure admit trapping of an incident soliton into a state of persistent shuttle motion, due to emission of radiation. The study is carried out numerically, and basic results are explained by means of analytical considerations.'
author:
- 'Krzysztof B. Zegadlo$^{1}$, Nir Dror$^{2}$, Marek Trippenbach$^{3}$, Miroslaw A. Karpierz$^{1}$ and Boris A. Malomed$^{2}$'
title: 'Spontaneous symmetry breaking of self-trapped and leaky modes in quasi-double-well potentials'
---
Introduction
============
Usually, the ground state (GS) of quantum-mechanical systems exactly follows the symmetry of the underlying Hamiltonian [@LL], while excited states may realize different representations of the same symmetry (a different situation is exemplified by the Jahn-Teller effect in molecules, which makes the GS of the electron subsystem spatially asymmetric, thus breaking the symmetry of the respective Hamiltonian [@RE]). In particular, for the double-well potential (DWP), which is dealt with in the present work, the GS wave function is spatially even, while the first excited state is odd. This is not necessarily true in many-body settings. In that context, the mean-field description of atomic Bose-Einstein condensates (BECs) is provided by the Gross-Pitaevskii equation (GPE) [@BEC], which includes the cubic term accounting for attractive forces between colliding atoms. Essentially the same is the nonlinear Schrödinger equation (NLSE) modeling the propagation of optical signals in Kerr-nonlinear media [NLS]{}. If the self-focusing nonlinearity is strong enough, it gives rise to the phase transition in the form of *spontaneous symmetry breaking* (SSB) of the GS [@book]. In its simplest manifestation, which is provided by the DWP, the SSB implies that one well traps a larger atomic density or field power than the other. This effect also implies the breakup of the basic principle of quantum mechanics, according to which the GS cannot be degenerate, as the SSB gives rise to a pair of two mutually symmetric GSs in the DWP, with the maximum of the wave function found in either potential well (as mentioned above, the Jahn-Teller effect gives rise to a qualitatively similar situation). The same DWP setting admits a symmetric state coexisting with the asymmetric ones, but, above the SSB point, the symmetric wave function no longer represents the GS, being unstable against symmetry-breaking perturbations. In the course of the spontaneous transition from the unstable symmetric state to a stable asymmetric one, the choice between the two mutually degenerate asymmetric states is determined by random perturbations, which push the system to build the maximum of the wave function in the left or right potential well. The SSB is a ubiquitous phenomenon, with well-known manifestations in nonlinear optics, BEC, superfluidity, superconductivity, ferromagnetism, etc. [book]{}.
The concept of the SSB in nonlinear systems of the NLS type was, plausibly, introduced for the first time in 1979 by E. B. Davies [@Davies], who addressed a nonlinear extension of the Schrödinger equation for a pair of quantum particles with an isotropic interaction potential. In this context, the SSB was predicted as the breaking of the rotational symmetry in the GS. Another early work, which predicted the SSB in a relatively simple form, addressed the *self-trapping model*, based on a system of linearly coupled ordinary differential equations including the self-attractive cubic terms [@Scott].
In the effectively one-dimensional geometry, the SSB can be studied in the framework of the scaled NLSE/GPE with potential $H(x)$ of the DWP type, for the amplitude of the electromagnetic wave, or the single-particle wave function, $\psi \left( x,z\right) $: $$i\frac{\partial \psi }{\partial z}=-\frac{1}{2}\frac{\partial ^{2}\psi }{%
\partial x^{2}}-\left\vert \psi \right\vert ^{2}\psi +H(x)\psi ,
\label{NLSE}$$where $z$ is the propagation distance in optics, or time in the GPE. This equation can be reduced to a system of coupled ordinary differential equations for two amplitudes, $u_{1,2}(z)$, by means of the *tight-binding approximation* [@tight], which replaces $\psi (x,z)$ by a linear superposition of two stationary wave functions, $\phi $, corresponding to the states trapped separately in either potential well, with their centers located at $x=\pm a$ [@Ananikian]:$$\psi \left( x,z\right) =u_{1}(z)\phi \left( x-a\right) +u_{2}(z)\phi \left(
x+a\right) . \label{12}$$
The analysis of the SSB in BEC and similar models based on Eq. (\[NLSE\]) was initiated in Refs. [@Milburn] and [@Smerzi]. In this case, in the framework of the mean-field approximation, the symmetry breaking is the phase transition of the second kind (alias the supercritical bifurcation, which does not admit hysteresis [@Joseph]). Further, GPE (\[NLSE\]) was extended by adding an extra (free) spatial coordinate, which transforms the DWP into a two-dimensional dual-core structure [@Warsaw]. In such a setting, the self-attractive nonlinearity gives rise to matter-wave solitons, which self-trap in the free direction [@soliton]. The SSB destabilizes symmetric solitons and replaces them by asymmetric ones, provided that the norm of the wave function (which determines the effective strength of the intrinsic nonlinearity) exceeds a critical value [Warsaw]{}. In the latter case, the mean-field symmetry breaking is a phase transition of the first kind (alias a subcritical bifurcation [@Joseph]), which includes hysteresis. The subcritical transition is typical to solitons in dual-core waveguides with the Kerr self-focusing [soliton-dual-core]{}. The same type of the transition may be featured by CW (continuous-wave) states in dual-core systems with non-Kerr nonlinearities [@Snyder].
In addition to the analysis of static symmetric and asymmetric modes, dynamical regimes, most typically in the form of oscillations of the mean-field wave function between two wells of the DWP structure, were analyzed too. Following the analogy with Josephson oscillations of the wave function of Cooper-paired electrons in superconducting tunnel junctions [Ustinov]{}, the possibility of oscillations in *bosonic Josephson junctions* was predicted [@junction]. The simplest dynamical model of the Josephson oscillations in bosonic systems was derived by means of the tight-binding approximation ([@superconductor]).
Experimental manifestations of the SSB have been observed in both BEC and photonics. Self-trapping of a macroscopically asymmetric state of the atomic condensate of $^{87}$Rb atoms, loaded into the DWP, as well as Josephson oscillations in that setting, were reported in Ref. [@Markus] (in that case, the effective nonlinearity is self-repulsive, therefore the respective SSB occurs not in the symmetric GS, but rather in the antisymmetric first excited state). The SSB of laser beams coupled into an effective transverse DWP created in the self-focusing photorefractive medium was demonstrated in Ref. [@photo]. Other experimentally observed SSB effects in optics are spontaneously established asymmetric regimes of operation of coupled lasers [@lasers; @NatPhot; @NaturePhot2; @Chili], and breaking of chiral symmetry in metamaterials [@Kivshar].
In addition to usual bound states, one may work with quasi-localized modes in potentials which do not admit complete trapping in linear quantum mechanics, but give rise to leaky bound states, alias quasi-bound ones. The combination of such a potential and self-attractive nonlinearity makes it possible to transform the leaky states into truly bound ones [Carr1,Carr2]{}. This possibility, in turn, suggests another setting, which is the subject of the present work: DWP structures embedded into a potential barrier. In the linear limit, this structure support solely symmetric leaky modes, that may be transformed into self-trapped ones with the help of the cubic self-attraction. The main feature of the system which, to the best of our knowledge, was not explored before, is competition between two different mean-field phase transitions of the second kind, driven by the nonlinearity: the SSB and transition to the self-trapping. Realization of the competition in stationary states of the DWP system is the main subject of the present work. We demonstrate that, depending on parameters of the DWP structure and nonlinearity strength, either transition may happen first, with the growth of the nonlinearity. Another essential problem addressed in the paper is a dynamical one, namely, Josephson oscillations in the DWP structure, initiated by the instability of the symmetric mode, and collisions of free solitons with the structure.
It is relevant to mention that, in terms of the BEC realizations, the present setting represents macroscopic quantum states, with the phase transitions between them being quasi-classical ones, considered in the framework of the mean-field approximation. The validity of this approximation is usually justified by the large number of atoms in the condensate [@BEC]. The consideration of a few-body state in the DWP can give rise to quantum phase transitions, such as those in the Lipkin-Meshkov-Glick model, which applies in this case [@LMG]. As suggested by a recent analysis of the three-dimensional many-body quantum gas with repulsive binary interactions, which is pulled to the center by a potential $\sim -1/r^{2}$ [@Grisha], the quantum phase transition may produce results similar to but different from their mean-field counterparts [@HS]. In particular, the GS predicted by the mean-field may be replaced by a metastable state in the quantum many-body theory [@Grisha]. In any case, the consideration of truly quantum phase transitions in the DWP structure is a subject for a separate work.
The subsequent presentation is structured as follows. The model is elaborated in Section II. Results of the analysis of symmetric and spontaneously emerging asymmetric trapped and leaky modes in it are summarized in Section III. Detailed results are obtained in a numerical form, and their basic features are explained by means of an analytical approach. Both the trapped and leaky modes undergo the SSB transition with the increase of the norm, the symmetric modes getting unstable above the transition point. The nonlinear evolution of the unstable modes, which features Josephson oscillations, is studied by means of systematic simulations in Section IV. A related possibility is capture of incident solitons by the DWP structure into shuttle states. This possibility is studied in a systematic form in Section V. The paper is concluded by Section VI.
The model
=========
The underlying dynamical model, based on Eq. (\[NLSE\]), gives rise to stationary modes with propagation constant $k$ (in BEC, $-k$ is the chemical potential), $$\psi \left( x,z\right) =e^{ikz}u\left( x\right) , \label{psi}$$where real modal functions $u(x)$ satisfy the equation
$$-ku+\frac{1}{2}u^{\prime \prime }+u^{3}=H\left( x\right) u.
\label{stationaryNLSE}$$
Solutions with $k>0$ represent self-trapped localized states, while $k<0$ corresponds to delocalized leaky modes, which do not vanish at $x\rightarrow
\pm \infty $. The states of these two types are characterized, respectively, by convergent and divergent norms, $N=\int_{-\infty }^{+\infty
}u^{2}(x)dx$ (proportional to the total power of the light beam in optics, or the total number of atoms in BEC), and Hamiltonian (energy),$$E=\int_{-\infty }^{+\infty }\left[ \frac{1}{2}\left( u^{\prime }\right) ^{2}-%
\frac{1}{2}u^{4}+2H(x)u^{2}\right] dx. \label{H}$$
The DWP can be readily implemented in the experiment. In optics, waveguides with this structure are fabricated with the help of the implanting technique [@Opt-DWP], while in BEC the DWP setting can be created by means of electromagnetic fields [@BEC-DWP]. In the present work, calculations are reported for the rectangular DWP profile with the elevated floor:
$$H\left( x\right) =\left\{
\begin{array}{ll}
A, & ~\mathrm{at}~~|x|<0.5, \\
2, & ~\mathrm{at}~~3<|x|<7, \\
0, & ~\mathrm{elsewhere},%
\end{array}%
\right. \label{potential}$$
where $A>0$ is the height of the inner potential barrier, see Fig. [potential-fig]{}, while height $H_{\max }=2$ of the outer barriers is fixed by scaling. Values of lengths adopted in Eq. (\[potential\]) adequately represent the generic situation, as demonstrated by additional numerical results (not shown here in detail). Indeed, it is shown below that the symmetry-breaking and self-trapping transitions, and the competition between them, crucially depend on the tunneling transparency of the central barrier and the nonlinearity strength, i.e., the barrier height, $A$, and total norm, $N$. These are two control parameters which are subject to the variation in the subsequent analysis. For the same reason, the main findings are not sensitive to a particular shape of the DWP. In particular, the rectangular form of the DWP, adopted in Eq. (\[potential\]), which may be essential for some dynamical effects, such as temporal scaling in the relaxation of perturbations [@Azbel], produces results which are essentially the same as generated by smooth DWP profiles (for the self-trapping of the leaky modes in a single potential well, this property was known before [@Carr1]). As concerns the necessity of having the elevated potential floor, it may make the experimental creation of the structure easier, as the floor" is naturally built by overlap of fringes of two potential barriers which determine the DWP (in previously reported experimental realizations of the DWP in BEC [@Markus], the bottom level of the potential had to be depressed, because those DWPs trapped condensates with the repulsive interactions, on the contrary to the case of the self-attraction, considered herein).
![The symmetric double-well potential for leaky modes (quasi-bound states), defined as per Eq. (\[potential\]).[]{data-label="potential-fig"}](potential.pdf)
Obviously, in the absence of the self-attractive cubic term, potential ([potential]{}) cannot support any bound state in the respective linear model, while weakly delocalized quasi-bound states are possible. Indeed, a straightforward estimate of the tunneling coefficient for the tall barriers separating the inner and outer parts of structure (\[potential\]) yields $$T\simeq \exp \left( -\sqrt{2H_{\max }-q^{2}}W\right) \approx \allowbreak
4.4\times 10^{-4}, \label{T}$$where $H_{\max }=2$ and $W=4$ are the height and width of the potential barriers, as per Eq. (\[potential\]), and $q\equiv \pi /l=\pi /6$ is the wavenumber of the lowest quasi-bound state in the potential box of width $%
l=6 $.
For the study of collisions of moving solitons with the DWP structure, the height of the inner barrier is fixed to be $A=0.5$, while the height, $H_{0}$, and width, $W_{0}$, of the outer barriers will be varied, to allow clearer observations of different collision scenarios: $$H\left( x\right) =\left\{
\begin{array}{ll}
0.5, & ~\mathrm{at}~~|x|<0.5, \\
H_{0}, & ~\mathrm{at}~~3<|x|<3+W_{0}\equiv \Lambda /2, \\
0, & ~\mathrm{elsewhere}.%
\end{array}%
\right. \label{P2}$$
Detailed consideration of the SSB in the leaky modes will require an explicit calculation of small-amplitude radiation tails attached to those modes outside of the barriers, i.e., at $|x|>7$, see Eq. (\[potential\]). For this purpose, the DWP is embedded into a broad free-space domain, $%
|x|<L/2$, with zero boundary conditions (b.c.): $$u(|x|=L/2)=0. \label{bc}$$
Spontaneous symmetry breaking (SSB) of leaky and trapped modes
==============================================================
The structure of symmetric and asymmetric modes
-----------------------------------------------
Numerical solutions of Eq. (\[stationaryNLSE\]) were obtained by means of the shooting and Newton-matrix methods. While the former one makes it possible to find all relevant solutions independently of an input trial function, the latter method can be applied to obtain solution with high accuracy, provided that the initial guess is taken not too far from the final result. For the setting addressed in this paper, the combination of both algorithms is the most efficient way of obtaining stationary solutions.
As is typical for the SSB in systems with self-focusing nonlinearity, it was found that the GS is spatially symmetric below the bifurcation point ($k<k_{%
\mathrm{bif}}$) and asymmetric above it, at $k>k_{\mathrm{bif}}$. The symmetric state exists at $k>k_{\mathrm{bif}}$ too, but in that case it is not a GS, and is no longer stable. As mentioned above, all mean-field phase transitions exhibited by the present system are of the second kind, featuring no hysteresis or bistability between symmetric and asymmetric modes.
Generic examples of unstable symmetric and stable asymmetric states of both trapped ($k>0$) the leaky ($k_{\mathrm{bif}}<k<0$) types are displayed in Figs. \[symmetric and asymmetric modes\](a) and (b), respectively. In the latter case, the delocalized tails of the leaky mode are extremely small, with amplitude $$u_{\mathrm{rad}}^{(\max )}\approx 1.2\times 10^{-4}. \label{rad}$$Taking into regard the value of the amplitude of the delocalized mode at its center, $U_{0}\approx 0.40$, tunneling coefficient (\[T\]) predicts amplitude $u_{\mathrm{rad}}^{(\max )}\sim TU_{0}\simeq 1.8\times 10^{-4}$, in reasonable agreement with its numerically found counterpart (\[rad\]). For given $k$, the spatial period of the tail in the free space is expected to be $\lambda =\pi \sqrt{2/|k|}\approx \allowbreak 15$ for $k=-0.085$ in Fig. \[symmetric and asymmetric modes\](b), while the numerical solutions demonstrates a close value, $\lambda \approx 13.5$ (a small deviation from the predicted value may be explained by the proximity of the tail to the outer barrier).
The formally diverging contribution of the tails to the total norm of the leaky mode is negligible, $N_{\mathrm{rad}}\simeq \left( L/2\right) \left[
u_{\mathrm{rad}}^{(\max )}\right] ^{2}$, where $L\approx 20$ is the total size of the free-space part of the configuration displayed in Fig. [symmetric and asymmetric modes]{}(b). Indeed, using estimate (\[rad\]) for the tail’s amplitude yields $N_{\mathrm{rad}}\sim 10^{-7}$, therefore the leaky modes have definite values of the norm, as indicated in the caption to Fig. \[symmetric and asymmetric modes\](b).
For the sake of direct comparison between leaky and trapped modes, in Fig. \[symmetric and asymmetric modes\](c) we display stationary states with the same propagation constant as in Fig. \[symmetric and asymmetric modes\](b), but in the case when they are confined by impenetrable (infinitely tall) outer potential barriers in Eq. (\[potential\]). It is seen that, similar to the situation shown in Fig. \[symmetric and asymmetric modes\](b), the smaller and larger values of $N$ correspond to the broken asymmetric and unbroken symmetric modes, respectively. However, in the infinitely deep potential box the profiles of the wave functions are, naturally, narrower and taller.
The symmetric modes displayed in Fig. \[symmetric and asymmetric modes\] feature split peaks, due to the fact that the inner potential barrier in Eq. (\[potential\]) is relatively high. On the other hand, the same potential structure with an essentially smaller barrier’s height $A$ supports single-peak symmetric modes, see Fig. \[unstable evolution\] below.
SSB of radiation tails in leaky modes
-------------------------------------
As mentioned above, the SSB of trapped modes is a known effect, which was previously studied in other forms [@book; @NaturePhot2]. A new phenomenon reported here is the SSB of leaky modes, which include nonvanishing tails extending into the free space outside of the DWP structure. Even if the tails have small amplitudes, it is interesting to analyze their structure in asymmetric modes, as this issue was not considered previously. To this end, we here focus on the symmetric and asymmetric states in the setting based on the DWP (\[potential\]) with $A=0.5$ and $\Lambda =14$, embedded into a broad domain of size $L=128$, see Eqs. (\[potential\]), (\[P2\]), and (\[bc\]). In this case, the asymmetric modes are found at $k\geq -0.100$.
A characteristic example of *asymmetric tails*, i.e., left and right ones with unequal amplitudes, is shown in Fig. \[tails\](a) for $k=-0.075$. Further, separately calculated total norms of the right and left tails, in regions $\Lambda /2<x<L/2$ and $-\Lambda /2<x<-L/2$, along with the norm of the tails in the coexisting unstable symmetric leaky mode, are displayed, as functions of $k$, in Fig. \[tails\](b). This dependence exhibits two notable features. First, the asymmetry between the right and left tails emerges at $k=-0.100$ and gradually increases with the increase of $k$ (i.e., decrease of $|k|$), even if each tail’s norm vanishes in the limit of $k$$\rightarrow 0$ (when the transition to the self-trapped mode takes place, and the tails vanish). This feature is illustrated by Fig. \[tails\](c), which displays the asymmetry measure vs. $k$:$$\theta (k)\equiv \frac{\int_{\Lambda /2}^{L/2}u^{2}(x;k)dx-\int_{-\Lambda
/2}^{-L/2}u^{2}(x;k)dx}{\int_{\Lambda /2}^{L/2}u^{2}(x;k)dx+\int_{-\Lambda
/2}^{-L/2}u^{2}(x;k)dx}~. \label{theta_tail}$$Second, the dependence of the tails’ norms on $k$ shows strong oscillations, which is explained by the commensurability-incommensurability transitions between the wavelength of the radiation tail and the total size of the free-space domains, $L/2-\Lambda /2$. Indeed, the radiation wavenumber given by the free-space dispersion relation for linearized equation ([stationaryNLSE]{}), $q=\sqrt{-2k}$, determines the the radiation half-wavelength, $\pi /q$, which, in the case of the commensurability, satisfies relation $(\pi /q)n=L/2-\Lambda /2$, with $n=1,2,3,...$ . Thus, maxima of the radiation amplitude are expected at discrete values of the propagation constant, $$k_{n}=-2\left[ \pi n/\left( L-\Lambda \right) \right] ^{2}. \label{k}$$As shown in Fig. \[tails\](b), Eq. (\[k\]) quite accurately predicts positions of the tail-norm peaks for $n=2,3,4,5,6,7,$ and $8$, for $%
L-\Lambda =114$, which corresponds to the present case \[$n=1$ yields $%
k_{1}\approx -1.5\times 10^{-3}$, for which the tail’s amplitudes are too small to discern the corresponding maximum, while Eq. (\[k\]) with $n=9$ predicts $k_{9}\approx -0.123$, for which asymmetric modes do not exist).
SSB diagrams
------------
Getting back to the consideration of the SSB for the entire system, systematic results are presented by means of plots $N(k)$ and $E(N)$ \[the Hamiltonian is defined by Eq. (\[H\])\] for symmetric and asymmetric modes, which are displayed in Figs. \[bifurcation diagrams\](a-c) and (d-f) for three different values of height $A$ of the inner barrier of potential structure (\[potential\]). Note that the $N(k)$ curves obey the Vakhitov-Kolokolov criterion [@VK; @VK2], which is necessary but not sufficient for the stability of modes supported by the self-attractive nonlinearity (it does not catch the instability of the symmetric modes coexisting with asymmetric ones). It is relevant to compare these plots with their counterparts,$$N_{\mathrm{sol}}=2\sqrt{2k},~E_{\mathrm{sol}}=-(1/3)\left( 2k\right) ^{3/2}%
\text{,} \label{sol}$$for the NLS solitons in the free space, given by Eq. (\[psi\]) with$$u_{\mathrm{sol}}(x)=\sqrt{2k}\,\mathrm{sech}\left( \sqrt{2k}x\right) ,
\label{soliton}$$which are displayed by dashed curves in Figs. \[bifurcation diagrams\](a-f).
The SSB in the families of trapped and leaky states is characterized by the asymmetry ratio,
$$\Theta \equiv N^{-1}\left[ \int_{0}^{+\infty }u^{2}(x)dx-\int_{-\infty
}^{0}u^{2}(x)dx\right] \label{theta}$$
\[cf. a similar definition for the tails, given by Eq. (\[theta\_tail\])\], which is shown as a function of $N$ in Figs. \[bifurcation diagrams\](g-i). These plots clearly identify the SSB-onset points, at which the symmetric mode gets destabilized, and simultaneously a stable asymmetric one emerges. In accordance with what is said above, the bifurcation is of the supercritical type [@Joseph], i.e., the emerging branches of the asymmetric states immediately go forward". Conclusions about the stability and instability of the solution branches displayed in Fig. \[bifurcation diagrams\] were produced by means of the well-known method [@VK2] based on numerical computation of (in)stability eigenvalues (imaginary parts of eigenfrequencies) for small perturbations added to the stationary solutions, using the respective linearized (Bogoliubov - de Gennes) equations. In particular, the instability of those symmetric states which coexist with asymmetric ones is always represented by a single pair of purely imaginary eigenfrequencies.
The fact that the asymmetric modes, when they exists, have smaller values of $E$ for given $N$, hence they realize the system’s GS \[see Figs. [bifurcation diagrams]{}(d-f)\], can be easily understood: having their center shifted from the layer occupied by the positive potential ($|x|<0.5$) to a region where $H=0$ \[$0.5<|x|<3$, see Figs. \[potential-fig\] and [symmetric and asymmetric modes]{}\], they obviously reduce the integral value of $E$. The same argument explains why, for given $k$, the asymmetric modes feature smaller $N$: having smaller overlap with the region of $H>0$, they need a smaller norm to compensate the shift of $k$ towards negative values induced by the positive potential, see Eq. (\[stationaryNLSE\]).
Note that the families of states displayed in Figs. \[bifurcation diagrams\](a-c) comprise both $k<0$ and $k>0$, i.e., the leaky and trapped modes alike. In particular, the SSB bifurcation occurs at $k>0$ in panels \[bifurcation diagrams\](a,b), and at $k<0$ in (c) (in the latter case, the SSB sets in at $k=-0.100$, as shown for the same system in Fig. [tails]{}). A noteworthy fact is that the system features the competition of the two different mean-field phase transitions driven by the increase of $N$, i.e., the strength of the self-attraction: the transition from the quasi-bound (leaky) state to the self-trapped one, which was previously found in single-well elevated potentials [@Carr1; @Carr2], and the SSB in the DWP structure. Thus, in the cases shown in Figs. \[bifurcation diagrams\](a,b) the self-trapping transition happens first (at smaller $N$), while in Fig. \[bifurcation diagrams\](c) the SSB takes place prior to the onset of the self-trapping.
The values of the propagation constant and norm at the SSB bifurcation point are shown, as a function of the inner-barrier’s height $A$ \[see Eq. ([potential]{})\], in Fig. \[bifurcation point\]. In panel (\[bifurcation\_k\]), the boundary between the SSB occurring with the delocalized and trapped modes ($k_{\mathrm{bif}}<0$ and $k_{\mathrm{bif}}>0$, respectively) is located at $A\approx 0.30$, the same value corresponding to the boundary designated by the square symbol in panel \[bifurcation\_N\]). That is, the SSB happens first (at smaller $N$) at $A>0.30$, while the transition to the self-trapping precedes the SSB at $A<0.30$.
The fact that the SSB happens with the trapped and leaky modes, respectively, at small and large $A$, as seen in Fig. \[bifurcation point\], is easy to explain: small $A$ implies strong linear coupling between the wave functions in the two barely separated wells, hence very large $N$ is required to induce the SSB, being far greater than the value of $N$ needed for the onset of the self-trapping, which is determined by the fixed parameters of the outer barrier in the DWP structure (\[potential\]). On the contrary, large $A$ implies weak linear coupling between the strongly separated wells, hence the respective strength of the nonlinearity (measured by $N$), required for the SSB, is much smaller than the value necessary for the commencement of the self-trapping. These arguments clearly suggest that the same sequence of the SSB and self-trapping phase transitions should take place in generic DWP structures.
These arguments can be cast in a more definite form, if the central barrier in Eq. (\[potential\]) is approximated by $H_{\mathrm{central}}(x)=A\delta
(x)$, and the outer barriers are made impenetrable, similar to what is shown in Fig. \[symmetric and asymmetric modes\](c). These conjectures replace the present model by the one for an infinitely deep potential box split by delta-functional barrier, which is the simplest model of the SSB [NaturePhot2,Chili]{}. In the limit of large $A$, the latter model predicts the following value of the norm at the SSB bifurcation point,$$N_{\mathrm{bif}}\approx 8\pi ^{2}/\left( 3l^{2}A\right) \label{analyt}$$(see caption to Fig. 2 in Ref. [@NaturePhot2]), where $l/2$ is the coordinate of the point at which the wave function vanishes (half-width of the infinitely deep box). In particular, one can identify $l\simeq 9$ for $%
A=0.5$ in Fig. \[symmetric and asymmetric modes\](b), hence Eq. ([analyt]{}) yields $N_{\mathrm{bif}}~\simeq 0.65$, while the respective numerical value in Fig. \[bifurcation point\](b) is $N_{\mathrm{bif}%
}~\approx 0.62$, which implies a reasonable agreement for the present (not really large) value of $A$.
Evolution of unstable symmetric states
======================================
Conclusions concerning the stability and instability of the symmetric and asymmetric modes, presented in Fig. \[bifurcation diagrams\], were verified, in addition to the computation of eigenvalues for small perturbations, by direct simulations, performed by dint of the finite-difference algorithm. The instability development of unstable symmetric states was catalyzed by adding small initial symmetry-breaking perturbations to them. This was done for the unstable symmetric states with both $k>0$ and for $k<0$, i.e., self-trapped and leaky ones. Typical examples are displayed in Fig. \[unstable evolution\], in which the unstable symmetric modes feature single- and double (split)-peak shapes at small and large values of the inner-barrier’s height, $A=0.05$ and $A=0.5$, respectively (in the top and bottom rows of the figure).
As mentioned above, the instability of symmetric states is accounted for by pure imaginary eigenfrequencies, hence the originally developing instability is not oscillatory. However, the nonlinearity makes the unstable dynamics oscillatory, as seen in Fig. \[unstable evolution\]. In other words, the unstable symmetric modes spontaneously develop bosonic Josephson oscillations. Close to the instability onset, the effective oscillation period is very large (as it diverges precisely at the onset point), gradually decreasing deeper into the instability region. The dynamical symmetry breaking induced by the weak and moderate instability is incomplete, leading to periodic oscillations between the original symmetric state and a new asymmetric one, as observed in Fig. \[unstable evolution\](a-e). Stronger instability causes complete symmetry breaking, replacing the symmetric state by an irregularly vibrating asymmetric mode, as seen in Fig. \[unstable evolution\](f).
It is relevant to note the difference between the oscillatory regimes generated by the instability of self-trapped and leaky modes. Indeed, while Fig. \[unstable evolution\](c) demonstrates that the shape of the oscillating mode is sharp in the former case, the shape is fuzzy in Fig. \[unstable evolution\](e) because it involves an essential radiation component in the case when the underlying unstable mode is a leaky one.
Collisions of free solitons with the quasi-double-well potential structure {#sec:CollisionResults}
==========================================================================
In addition to the analysis of the stationary states performed above, it is also relevant to consider collisions of free solitons with the elevated DWP structure. For this purpose, the initial soliton was created as the tilted (moving) version of the static one given by Eqs. (\[psi\]) and ([soliton]{}),$$\psi \left( x,z\right) =\sqrt{2k}\exp \left( i\left( k-c^{2}/2\right)
z+icx\right) \text{\textrm{sech}}\left( \sqrt{2k}\left( \left(
x-x_{0}\right) -cz\right) \right) , \label{tilt}$$where $c$ is a real tilt (velocity), and $x_{0}$ is the initial position of the soliton, chosen far enough from the localized potential structure (we here take $x_{0}=-15$). Generic findings are produced here for incident solitons with $k=3,$ the corresponding norm being $$N=2\sqrt{2k}\approx 4.90, \label{N}$$according to Eq. (\[sol\]), other values of $N$ giving similar results.
Figure \[ThetaVsHeightMapping\] presents a parameter chart for three different outcomes of the collisions, produced by varying the height of the outer barriers, $H_{0}$ in Eq. (\[P2\]), and tilt $c$ in Eq. (\[tilt\]). In the region designated as (1), i.e., for $c$ small and/or $H_{0}$ large enough, the incident soliton bounces back the left outer barrier, as shown in Fig. \[FundCollision1\] for $H_{0}=0.6$ and $c=0.8$. At larger $c$, in a relatively narrow region (2) of Fig. \[ThetaVsHeightMapping\], the soliton gets captured inside the potential structure, which resembles the previously explored possibility of capturing an incident Bragg-grating soliton by a cavity formed by two locally repulsive defects [@cavity]. Similar to that setting, the trapped soliton performs shuttle motion between the outer and inner potential barriers, as shown in Fig. \[FundCollision2\], for $H_{0}=0.6$ and $c=1.15$. The shuttle dynamical regime is an essential addition to the stationary states revealed by the analysis presented above. At still larger $c$, but yet staying within the boundaries of the narrow region (2), the shuttle motion of the trapped soliton becomes irregular, see Fig. \[FundCollision5\] for $H_{0}=1.3$ and $c=1.8$.
![The chart of outcomes of collisions of free solitons, launched with tilt $c$ \[see Eq. (\[tilt\])\], with potential structure (\[P2\]), in the plane of $\left( H_{0},c\right) $. The norm of the incident soliton is fixed as per Eq. (\[N\]). In this figure, the width of the outer potential barriers is $W_{0}=4$. In region (1), the soliton bounces back from the left outer barrier. Region (2) refers to trapping the soliton inside the potential structure, where it performs shuttle motion. In region (3), the incident soliton passes the structure. The dashed line is the analytical prediction produced by Eq. ([c\_cr]{}).[]{data-label="ThetaVsHeightMapping"}](ThetaVsHeightMapping.pdf){width="3.2in"}
In region (3), the initial tilt, $c$, is large enough for the soliton to pass the potential structure, see an example in Fig. \[FundCollision3\] for $H_{0}=0.6$ and $c=1.5$. Another option admitted by this scenario is shown in Fig. \[FundCollision4\], where the soliton passes the left outer barrier, bounces back from the inner one, and escapes in the reverse direction, passing the left outer barrier again. Naturally, this collision pattern is common for lower $H_{0}$, when the outer barriers are *lower* than the inner one, which plays the role of a strong “bouncer". Furthermore, at $c>1.85$, the incident soliton splits into two fragments, one escaping and the other one staying in a chaotically evolving trapped state, see an example in Fig. \[FundCollision6\], for $H_{0}=1.2$ and $%
c=1.92$.
Collision scenarios were also explored by varying width $W_{0}$ of the outer barriers, while keeping their height constant, $H_{0}=1$, as well as characteristics of the inner barrier and the distance between the barriers, see Eq. (\[P2\]). The respective results, for the same incident soliton as used above ($k=3$, $N=4.90$), are summarized in Fig. [ThetaVsWidthMapping]{}, where regions (1)-(3) have the same meaning as their counterparts in Fig. \[ThetaVsHeightMapping\].
In the latter case, the results may be classified into three outcomes of the collision, depending on $W_{0}$. The first outcome occurs at $0<W_{0}<0.8$. It is characterized by a rapidly growing region of the shuttle motion of the trapped soliton \[region (2)\], and *sharp transitions* between the three evolution scenarios, $(1)\leftrightarrow (2)$, $(2)\leftrightarrow (3)$. The second outcome, which was observed in the range of $0.8<W_{0}<1.8$ (in this case, $W_{0}$ is, roughly, close to the width of the incident soliton), is distinguished by *gradual transitions* between the scenarios. That means that, for certain values of $c$, the soliton does not fully bounce from the barrier, nor penetrates it, but rather splits into two segments, one of which escapes, while the other remains trapped. In this case, both the upper and lower boundaries of region (2) represent tilts at which the soliton is split into equal fragments. An example of such an outcome is shown in Fig. \[SplitSolitonCollision\], for $W_{0}=1.2$ and $c=1.66$. The third outcome, which is observed at $W_{0}>1.8$, is distinguished by the fact that the variation of $W_{0}$ almost does not affect the soliton’s motion. In contrast to the second outcome, and similar to the first one, the respective transitions between the regions are sharp.
![The same as in Fig. \[ThetaVsHeightMapping\], but varying the width of the outer potential barriers, $W_{0}$, while their height is fixed, $H_{0}=1$. The dashed curve shows the analytical prediction for the boundary of area (1) given by Eq. (\[parabola\]). []{data-label="ThetaVsWidthMapping"}](ThetaVsWidthMapping.pdf){width="3.2in"}
![Outcomes of the collisions, for $W_{0}=1.2$ and $c=1.66$. In this borderline example, the soliton hits the right (second) outer potential barrier and splits into two fragments, one escaping and the other one staying trapped in the potential structure.[]{data-label="SplitSolitonCollision"}](fundcollisionmum3wo1p2theta1p66.pdf){width="3.2in"}
It is easy to explain the parabolic boundary of region (1) in Fig. [ThetaVsHeightMapping]{}, as well as the boundary of the same region in Fig. \[ThetaVsWidthMapping\], using the perturbation theory for NLS solitons, which treats them as quasi-particles with mass $N$ [@RMP]. Indeed, the kinetic energy of the soliton is $E_{\mathrm{kin}}=(N/2)c^{2}$, while the height of the outer potential barrier for the quasi-particle, $E_{0}$, can be obtained from the third term in expression (\[H\]), assuming that the soliton’s center is located at the midpoint of the potential barrier:$$E_{0}=H_{0}\int_{-W_{0}/2}^{+W_{0}/2}u_{\mathrm{sol}}^{2}\left( x^{\prime
}\right) dx^{\prime }=H_{0}N\tanh \left( \frac{1}{4}W_{0}N\right) ,
\label{E0}$$where $u_{\mathrm{sol}}$ is the solitons’s profile (\[soliton\]), $%
x^{\prime }\equiv x-\left( 3+W_{0}/2\right) $ \[see Eq. (\[P2\])\], and the result is expressed in terms of the soliton’s norm, as per Eq. (\[sol\]). Next, equating $E_{\mathrm{kin}}$ to $E_{0}$ predicts that the boundary between the rebound and passage corresponds to$$c_{\mathrm{cr}}=\sqrt{2H_{0}\tanh \left( \frac{1}{4}W_{0}N\right) }.
\label{parabola}$$In particular, the respective prediction for dependence $c_{\mathrm{cr}%
}(H_{0})$ corresponding to the case displayed in Fig. [ThetaVsHeightMapping]{}, with $W_{0}=4$ and $N$ fixed as per Eq. (\[N\]), simplifies to$$c_{\mathrm{cr}}=\sqrt{2H_{0}}. \label{c_cr}$$Figure \[ThetaVsHeightMapping\] demonstrates that Eq. (\[c\_cr\]) predicts the parabolic boundary of region (1) quite accurately, a discrepancy at large $H_{0}$ being explained by the fact that the collision with the tall barrier causes a deformation of the soliton’s shape. Further, the full analytical expressions (\[parabola\]) predicts the boundary of the same region in Fig. \[ThetaVsWidthMapping\] well enough too.
As concerns the trapping regime in area (2) of Figs. [ThetaVsHeightMapping]{} and \[ThetaVsWidthMapping\], it is explained by the fact that, while slowly passing the left barrier, and then passing the inner one, the soliton with the initial tilt slightly exceeding value ([parabola]{}) suffers radiation losses due to its deceleration and acceleration. The losses cause a drop in the kinetic energy below the level necessary for clearing the right barrier. A feature which relates the trapping and splitting to the leaky modes considered above is tunneling of the radiation across the outer potential barriers, which can be seen, in particular, in Figs. \[FundCollisions\](b,e,f).
Conclusions
===========
We have extended the known possibility of the stabilization of leaky modes in quasi-trapping potentials by means of the self-attractive nonlinearity. Unlike the previously studied single-well potential, we have introduced the spatially symmetric DWP (double-well potential) with the elevated floor, embedded into the potential barrier. The setting can be implemented in nonlinear optical waveguides and BEC. The new possibility offered by this system is the competition of two phase transitions of the second kind, described in the mean-field approximation: the onset of the self-trapping of the leaky modes, and the SSB (spontaneous symmetry breaking) of both true bound states and leaky modes, under the action of the self-attractive nonlinearity. With the increase of the norm of the wave field (which determines the nonlinearity strength), the former or latter transition happens first if the central barrier of the DWP structure is, respectively, low or tall. These conclusions are generic, as they do not depend on details of the DWP structure. New effects are revealed by the consideration of the SSB of the leaky modes: asymmetry of radiation tails, which are parts of these modes, and the commensurability-incommensurability interplay between the radiation wavelength and the total size of the system, into which the DWP is embedded. Systematic results have been produced in the numerical form, and their basic features were explained with the help of analytical considerations. The simulations demonstrate that unstable symmetric modes initiate Josephson oscillations. Collisions of freely moving solitons with the DWP structure were studied in a systematic form too, revealing various generic outcomes of the collisions, boundaries between which were explained in the analytical form. In particular, in addition to the stationary states with the unbroken and broken symmetry, the collisions reveal the dynamical mode, in the form of a soliton which performs persistent shuttle motion in the DWP structure.
Relevant possibilities for the extension of the analysis reported in this work may be offered by two-component systems, as well as by a two-dimensional generalization of the present setting. On the other hand, the results produced by the competition of the mean-field phase transitions suggest that it may be interesting too to consider quantum phase transitions in a many-body bosonic state with attractive inter-particle interactions, loaded into the DWP.
Acknowledgment
==============
M.T. and B.A.M. acknowledge partial support from the National Science Center of Poland in the framework of the HARMONIA Program, No. 2012/06/M/ST2/00479. K.B.Z. acknowledges support from the National Science Center of Poland through Project ETIUDA No. 2013/08/T/ST2/00627.
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abstract: 'The present paper is devoted to study 2-local derivations on W-algebra $W(2,2)$ which is an infinite-dimensional Lie algebras with some out derivations. We prove that all 2-local derivations on the W-algebra $W(2,2)$ are derivation. We also give a complete classification of the 2-local derivation on the so called thin Lie algebra and prove that it admits a lots of 2-local derivations which are not derivations.'
author:
- '**Xiaomin Tang**'
- |
Xiaomin Tang$^{1,2,}$ [^1]\
[1. School of Mathematical Science, Heilongjiang University, Harbin, 150080, P. R. China, ]{}\
[2. School of Mathematical Science, Harbin Engineering University, Harbin, 150001, P. R. China]{}
title: '2-Local derivations on the W-algebra $W(2,2)$'
---
**Key words**: W-algebra $W(2,2)$, thin Lie algebra, derivation, 2-local derivation.
**Mathematics Subject Classification**: 17A32, 17B30, 17B10.
Introduction
============
In 1997, Šemrl [@Sem] introduced the notion of 2-local derivations on algebras. Namely, for an associative algebra $\mathcal{L}$, a map $\Delta : \mathcal{L} \to
\mathcal{L}$ (not necessarily linear) is called a *2-local derivation* if, for every pair of elements $x,y \in \mathcal{L},$ there exists a derivation $\Delta_{x,y} : \mathcal{L} \to \mathcal{L}$ (depending on $x, y$) such that $\Delta_{x,y} (x) = \Delta(x)$ and $\Delta_{x,y}(y) = \Delta(y).$
The concept of 2-local derivation is actually an important and interesting property for an algebra. For a given algebra $\mathcal{L}$, the main problem concerning these notions is to prove that they automatically become a derivation or to give examples of 2-local derivations of $\mathcal{L},$ which are not derivations. Recently, several papers have been devoted to similar notions and corresponding problems for Lie algebras $\mathcal{L}$. In [@AyuKudRak; @ChenWang] the authors prove that every 2-local derivation on a semi-simple Lie algebra $\mathcal{L}$ is a derivation and that each finite-dimensional nilpotent Lie algebra, with dimension larger than two admits 2-local derivation which is not a derivation. In [@ayu2019] the authors study 2-local derivations on some infinite-dimensional Lie algebras, i.e., they that all 2-local derivations on the Witt algebra as well as on the positive Witt algebra are (global) derivations, and give an example of infinite-dimensional Lie algebra with a 2-local derivation which is not a derivation. In [@ayu2020; @zhao2020] the authors prove that every 2-local derivation on some class of generalized Witt algebras (or their Borel subalgebras) is a derivation.
As we see that the Lie algebras whose every 2-local derivation is a derivation almost all have a common quality, that is any derivation of these Lie algebras is inner. We naturally want to know what form of the 2-local derivation has if the Lie algebra has some out derivations? In the present paper we study 2-local derivations on the infinite-dimensional Lie algebra $W(2,2)$ and so called thin Lie algebra $\mathfrak{T}$. Note that both $W(2,2)$ and $\mathfrak{T}$ all have some out derivations. We prove that every 2-local derivation on W-algebra $W(2,2)$ is a derivation and the tin Lie algebra $\mathfrak{T}$ admits many 2-local derivations which are not derivations.
In Section 2 we give some preliminaries concerning W-algebra $W(2,2)$. In Section 3 we prove that every 2-local derivations on W-algebra $W(2,2)$ are automatically derivations. In Section 4 we complete describe the 2-local derivation on the so-called thin Lie algebra and show that it admits 2-local derivations which are not derivations.
Throughout this paper, we denote by $\mathbb{Z}$, $\mathbb{N}$, $\mathbb{Z}^*$ and $\mathbb{C}$ the sets of all integers, positive integers, nonzero integers and complex numbers respectively. All algebras are over $\mathbb{C}$.
Preliminaries
=============
In this section we give some necessary definitions and preliminary results.
A derivation on a Lie algebra $\mathcal{L}$ is a linear map $D:\mathcal{L}\rightarrow \mathcal{L}$ which satisfies the Leibniz law, that is, $$D([x,y])=[D(x),y]+[x, D(y)]$$ for all $x,y\in \mathcal{L}.$ The set of all derivations of $\mathcal{L}$ with respect to the commutation operation is a Lie algebra and it is denoted by $Der(\mathcal{L}).$ For all $a\in \mathcal{L}$, the map ${\rm ad} (a)$ on $\mathcal{L}$ defined as ${\rm ad} (a)x=[a,x],\ x\in\mathcal{L}$ is a derivation and derivations of this form are called *inner derivation*.
Recall that a map $\Delta: \mathcal{L}\rightarrow \mathcal{L}$ (not liner in general) is called a *2-local derivation* if for every $x,y\in \mathcal{L},$ there exists a derivation $\Delta_{x,y}:\mathcal{L}\rightarrow \mathcal{L}$ (depending on $x,y$) such that $\Delta(x)=\Delta_{x,y}(x)$ and $\Delta(x)=\Delta_{x,y}(y)$. For a 2-local derivation on $\mathcal{L}$ and $k\in \mathbb{C}$, $x\in \mathcal{L}$, we have $$\label{recall1}
\Delta(kx)=\Delta_{x,kx}(kx)=k\Delta_{x,kx}(x)=k\Delta(x).$$
The W-algebra $W(2,2)$ is an infinite-dimensional Lie algebra with the $\mathbb{C}$-basis $$\{L_m, I_m| m\in \mathbb{Z} \}$$ and the Lie brackets are given by $$\begin{aligned}
&&[L_m,L_n]=(m-n)L_{m+n},\\
&&[L_m,I_n]=(m-n)I_{m+n}, \\
&&[I_m,I_n]=0, \ \forall m,n\in \mathbb{Z}.\end{aligned}$$ A class of central extensions of $W(2,2)$ first introduced by [@JiangDong] in their recent work on the classification of some simple vertex operator algebras, and then some scholars studied the theory on structures and representations of $W(2,2)$ or its central extensions, see [@Chenhj; @GJP; @Jiangzhangwei; @Rad; @tangw22; @Wangy] and so forth. We now recall and establish several auxiliary results.
\[lemma\_1\] (see [@GJP]) Denote by ${\rm{Der}} (W(2, 2))$ and by ${\rm{Inn}} (W(2, 2))$ the space of derivations and the space of inner derivations of $W(2, 2)$ respectively. Then $${\rm{Der}} (W(2, 2))={\rm{Inn}} (W(2, 2))\oplus\mathbb{C}D,$$ where $D$ is an outer derivation defined by $D(L_m)=0$, $D(I_m)=I_m$ for all $m \in \mathbb{Z}$.
\[lemma\_2\] Let $\Delta$ be a 2-local derivation on the W-algebra $W(2,2)$. Then for every $x,y\in W(2,2)$, there exists a derivation $\Delta_{x,y}$ of $W(2,2)$ such that $\Delta(x)=\Delta_{x,y}(x)$, $\Delta(y)=\Delta_{x,y}(y)$ and it can be written as $$\label{tangtang6}
\Delta_{x,y}={\rm ad} (\sum_{k\in \mathbb{Z}}\left(a_k(x,y)L_k+b_k(x,y)I_k\right))+\lambda(x,y)D$$ where $\lambda, a_k, b_k (k\in \mathbb{Z})$ are complex-valued functions on $W(2,2)\times W(2,2)$ and $D$ is given by Lemma \[lemma\_1\].
By Lemma \[lemma\_1\], obviously the derivation $\Delta_{x,y}$ can be written as the form of (\[tangtang6\]).
2-Local derivations on $W(2,2)$
===============================
Now we shall give the main result concerning 2-local derivations on $W(2,2)$.
\[thm-tang\] Every 2-local derivation on the W-algebra $W(2,2)$ is a derivation.
For the proof of this Theorem we need several Lemmas. For a 2-local derivation $\Delta: W(2,2)\rightarrow W(2,2)$ and $x,y\in \mathcal{L},$ below we always use the symbol $\Delta_{x,y}$ for the derivation of $W(2,2)$ satisfying $\Delta(x)=\Delta_{x,y}(x)$ and $\Delta(x)=\Delta_{x,y}(y)$; and $D$ for the out derivation of $W(2,2)$ given by Lemma \[lemma\_1\].
\[mainlem\] Let $\Delta$ be a 2-local derivation on $W(2,2)$. Take any but fixed $y\in W(2,2)$.
(i) For a given $i\in \mathbb{Z}$, if $\Delta (L_i)=0$ then $$\label{txm1}
\Delta_{L_i,y}={\rm ad} \left(a_i(L_i,y)L_i+b_i(L_i,y)I_i\right)+\lambda(L_i,y)D;$$
(ii) If $\Delta (I_0)=0$ then for any $y\in W(2,2)$ we have $$\label{txm2}
\Delta_{I_0,y}={\rm ad} (a_0(I_0,y)L_0+\sum_{k\in \mathbb{Z}} b_k(I_0,y)I_k)$$ where $\lambda, a_k, b_k (k \in \mathbb{Z})$ are complex-valued functions on $W(2,2)\times W(2,2)$.
By Lemma \[lemma\_2\], we can assume that $$\begin{aligned}
\Delta_{L_i,y}={\rm ad} (\sum_{k\in \mathbb{Z}}(a_k(L_i,y)L_k+b_k(L_i,y)I_k))+\lambda(L_i,y)D, \label{zhong2tang}\\
\Delta_{I_0,y}={\rm ad} (\sum_{k\in \mathbb{Z}}(a_k(I_0,y)L_k+b_k(I_0,y)I_k))+\lambda(I_0,y)D \label{zhong3tang}\end{aligned}$$ for some complex-valued functions $\lambda, a_k, b_k (k \in \mathbb{Z})$ on $W(2,2)\times W(2,2)$.
\(i) When $\Delta(L_{i})=0$, in view of (\[zhong2tang\]) we obtain $$\begin{aligned}
\Delta(L_{i})&=&\Delta_{L_i, y}(L_{i})\\
&=&[\sum_{k\in \mathbb{Z}}(a_k(L_i,y)L_k+b_k(L_i,y)I_k), L_i]+\lambda(L_i,y)D(L_i)\\
&=& \sum_{k\in \mathbb{Z}}((k-i)a_k(L_i,y)L_{k+i}+(k-i)b_{k}(L_i,y)I_{k+i})=0.\end{aligned}$$ From the above equation, one has $(k-i)a_k(L_i,y)=(k-i)b_{k}(L_i,y)=0$ for all $k\in \mathbb{Z}$, which deduces $a_k(L_i,y)=b_k(L_i,y)=0$ for all $i\in \mathbb{Z}$ with $k\neq i$. Then Equation (\[zhong2tang\]) becomes (\[txm1\]), as deserved.
\(ii) When $\Delta(I_{0})=0$, then it follows from (\[zhong3tang\]) that $$\begin{aligned}
\Delta(I_{0})&=&\Delta_{I_0, y}(I_{0})\\
&=&[\sum_{k\in \mathbb{Z}}(a_k(I_0, y)L_k+b_k(I_0, y)I_k), I_0]+\lambda(I_0, y)D(I_0)\\
&=& \sum_{k\in \mathbb{Z}}ka_k(I_0, y)L_{k}+\lambda(I_0, y)I_0=0.\end{aligned}$$ Then we have $\lambda(I_0, y)=0$ and $ka_k(I_0, y)=0$ for all $k\in \mathbb{Z}$, i.e., $a_k(I_0, y)=0$ for all $k\in \mathbb{Z}^*$. This with (\[zhong3tang\]) implies that (\[txm2\]) holds. The proof is completed.
\[lem13\] Let $\Delta$ be a 2-local derivation on $W(2,2)$ such that $\Delta(L_{0})=\Delta(L_{1})=0.$ Then $$\begin{aligned}
\label{zhong1}
\Delta(L_{i})=0, \ \ \forall i\in \mathbb{Z}.\end{aligned}$$
In view of $\Delta(L_{0})=\Delta(L_{1})=0$, by using Lemma \[mainlem\] we can assume that $$\begin{aligned}
\Delta_{L_0,y}={\rm ad} (a_0(L_0,y)L_0+b_0(L_0,y)I_0)+\lambda(L_0,y)D, \label{zhong21}\\
\Delta_{L_1,y}={\rm ad} (a_1(L_1,y)L_0+b_1(L_1,y)I_1)+\lambda(L_1,y)D \label{zhong22}\end{aligned}$$ for all $y\in W(2,2)$, where $\lambda, a_k, b_k (k \in \mathbb{Z})$ are complex-valued functions on $W(2,2)\times W(2,2)$. Let $i\in\mathbb{Z}$ be a fixed index. Then by taking $y=L_i$ in (\[zhong21\]) and (\[zhong22\]) respectively we get $$\begin{aligned}
\Delta(L_i)&=&\Delta_{L_0,L_i}(L_i)=[a_0(L_0,L_i)L_0+b_0(L_0,L_i)I_0, L_i]+\lambda(L_0,L_i)D(L_i),\\
&=& -ia_0(L_0,L_i)L_i-ib_0(L_0,L_i)I_i\end{aligned}$$ and $$\begin{aligned}
\Delta(L_i)&=&\Delta_{L_1,L_i}(L_i)=[a_1(L_1,L_i)L_1+b_1(L_1,L_i)I_1, L_i]+\lambda(L_1,L_i)D(L_i),\\
&=& (1-i)a_0(L_0,L_i)L_{i+1}+(1-i)b_0(L_0,L_i)I_{i+1}.\end{aligned}$$ By the above two equations, it follows that $$ia_0(L_0,L_i)L_i+ib_0(L_0,L_i)I_i+(1-i)a_0(L_0,L_i)L_{i+1}+(1-i)b_0(L_0,L_i)I_{i+1}=0,$$ which implies $a_0(L_0,L_i)=b_0(L_0,L_i)=0$. It concludes that $\Delta(L_i)=0$. We finish the proof.
\[lem16\] Let $\Delta$ be a 2-local derivation on $W(2,2)$ such that $\Delta(L_{i})=0$ for all $i\in \mathbb{Z}$. Then for any $x=\sum_{t\in \mathbb{Z}} (\alpha_tL_t+\beta_t I_t)\in W(2,2)$, we have $$\label{guol}
\Delta(x)=\Delta(\sum_{k\in \mathbb{Z}} (\alpha_tL_t+\beta_t I_t))= \mu_x \sum_{t\in \mathbb{Z}} \beta_t I_t$$ where $\mu_x$ is a complex number depending on $x$.
For $x=\sum_{t\in \mathbb{Z}} (\alpha_tL_t+\beta_t I_t)\in W(2,2)$, since $\Delta (L_i)=0$ for any $i\in \mathbb{Z}$, from Lemma \[mainlem\] we have $$\begin{aligned}
\Delta(x)&=&\Delta_{L_i,x}(x)\\
&=&[a_i(L_i,x)L_i+b_i(L_i,x)I_i, x]+\lambda(L_i,x)D(x)\\
&=&\sum_{t\in \mathbb{Z}}(i-t)(\alpha_t a_i(L_i,x)L_{i+t}+(\beta_t a_i(L_i,x)+\alpha_t b_i(L_i,x)) I_{i+t} )+ \lambda(L_i,x) \sum_{t\in \mathbb{Z}} \beta_t I_t.\end{aligned}$$ By taking enough diffident $i\in \mathbb{Z}$ in the above equation and, if necessary, let these $i$’s to be large enough, we obtain that $\Delta(x)=\lambda(L_i,x) \sum_{t\in \mathbb{Z}} \beta_t I_t.$ Note that $\mu_x\doteq\lambda(L_i,x)$ is a constant since it is independent on $i$.
\[tangtang666\] Let $\Delta$ be a 2-local derivation on $W(2,2)$ such that $\Delta(I_{0})=0$ and $\Delta(L_{i})=0$ for all $i\in \mathbb{Z}$. Then for any $p\in \mathbb{Z}^*$ and $y\in W(2,2)$, there are $\xi_p^y,\eta_p^y\in \mathbb{C}$ such that $$\begin{aligned}
\Delta_{L_{p}+I_{2p}, y}= {\rm ad} (\xi_p^yL_p+\eta_p^y I_p+ \xi_p^y I_{2p}).\label{xi2}\end{aligned}$$
For $p\in \mathbb{Z}^*$, by $\Delta(L_{i})=0$ for all $i\in \mathbb{Z}$ and Lemma \[lem16\] we have $$\label{tXXz}
\Delta(L_{p}+I_{2p})=\mu_{L_{p}+I_{2p}} I_{2p},$$ where $\mu_{L_{p}+I_{2p}}\in \mathbb{C}$ is given by (\[guol\]). In view of $\Delta(I_{0})=0$ and Lemma \[txm1\] we know that $$\begin{aligned}
&&\Delta(L_{p}+I_{2p})\\
&=&\Delta_{I_0,L_{p}+I_{2p}}(L_{p}+I_{2p})\\
&=&[a_0(I_0,L_{p}+I_{2p})L_0+\sum_{k\in \mathbb{Z}} b_k(I_0,L_{p}+I_{2p})I_k, L_{p}+I_{2p}]\\
&=& -pa_0(I_0,L_{p}+I_{2p})(L_{p}+2I_{2p})+\sum_{k\in \mathbb{Z}}(k-p)b_k(I_0,L_{p}+I_{2p})I_{k+p}.\end{aligned}$$ This, together with (\[tXXz\]), gives that $-pa_0(I_0,L_{p}+I_{2p})=0$ and $-2pa_0(I_0,L_{p}+I_{2p})=\mu_{L_{p}+I_{2p}}$, i.e., we get $\mu_{L_{p}+I_{2p}}=0$. It follows by (\[tXXz\]) that $$\Delta(L_{p}+I_{2p})=0. \label{xi1}$$ Next, for every $y\in W(2,2)$, by Lemma \[lemma\_2\] we can assume that $$\label{tangtang66}
\Delta_{L_p+I_{2p},y}={\rm ad} (\sum_{k\in \mathbb{Z}}(a_k(L_p+I_{2p},y)L_k+b_k(L_p+I_{2p},y)I_k))+\lambda(L_p+I_{2p},y)D.$$ From (\[xi1\]) and (\[tangtang66\]), one has $$\begin{aligned}
&&\Delta(L_{p}+I_{2p})\\
&=&\Delta_{L_p+I_{2p},y}(L_{p}+I_{2p})\\
&=&[\sum_{k\in \mathbb{Z}}(a_k(L_p+I_{2p},y)L_k+b_k(L_p+I_{2p},y)I_k), L_{p}+I_{2p}]+\lambda(L_p+I_{2p},y)I_{2p}\\
&=&\sum_{k\in \mathbb{Z}}a_k(L_p+I_{2p},y)((k-p)L_{k+p}+(k-2p)I_{k+2p})\\
&&\ \ \ \ +\sum_{k\in \mathbb{Z}}(k-p)b_k(L_p+I_{2p},y)I_{k+p}+\lambda(L_p+I_{2p},y)I_{2p}=0.\end{aligned}$$ From this, it is easy to see that $(k-p)a_k(L_p+I_{2p},y)L_{k+p}=0$ for all $k\in \mathbb{Z}$ and so that $a_k(L_p+I_{2p},y)=0$ for all $k\neq p$. Using this conclusion we observe the coefficient of $I_{3p}$ in the above equation, then one has $$(p-2p)a_p(L_p+I_{2p},y)+(2p-p)b_{2p}(L_p+I_{2p},y)=0,$$ which implies $a_p(L_p+I_{2p},y)=b_{2p}(L_p+I_{2p},y)$. Furthermore, by observing the coefficient of $I_{k}, k\neq 3p$ in the above equation we get $\lambda(L_p+I_{2p},y)=0$ and $(k-p)b_k(L_p+I_{2p},y)=0$ for all $k\neq p, 2p$, i.e., $b_k(L_p+I_{2p},y)=0$ for all $k\neq p, 2p$. Finally, by denoting $\xi_p^y=a_p(L_p+I_{2p},y)$ and $\eta_p^y=b_p(L_p+I_{2p},y)$ we finish the proof.
\[lem-mel6\] Let $\Delta$ be a 2-local derivation on $W(2,2)$ such that $\Delta(L_{0})=\Delta(L_{1})=\Delta(I_{0})=0.$ Then $\Delta(x)= 0$ for all $x\in W(2,2)$.
Take any but fixed $x=\sum_{t\in \mathbb{Z}} (\alpha_tL_t+\beta_t I_t)\in W(2,2)$, where $(\alpha_t)_{t\in \mathbb{Z}}, (\beta_t)_{t\in \mathbb{Z}}$ are both sequences which contain only finitely many nonzero entries.
Since $\Delta(L_{0})=\Delta(L_{1})=0$, it follows by Lemma \[lem13\] that $$\begin{aligned}
\label{zhong166}
\Delta(L_{i})=0, \ \ \forall i\in \mathbb{Z}.\end{aligned}$$ This, together with Lemma \[lem16\], gives $$\label{guol6}
\Delta(x)=\Delta(\sum_{k\in \mathbb{Z}} (\alpha_tL_t+\beta_t I_t))= \mu_x \sum_{t\in \mathbb{Z}} \beta_t I_t$$ for some $\mu_x\in \mathbb{C}$. Now, for any $p\in \mathbb{Z}^*$, by (\[zhong166\]) and $\Delta(I_{0})=0$, we obtain by Lemma \[tangtang666\] that $$\begin{aligned}
\Delta_{L_{p}+I_{2p}, x}= {\rm ad} (\xi_p^xL_p+\eta_p^x I_p+ \xi_p^x I_{2p})\label{xi26}\end{aligned}$$ for some $\xi_p^x,\eta_p^x\in \mathbb{C}$. Therefore, from (\[xi26\]) one has $$\begin{aligned}
\Delta(x)&=&\Delta_{L_p+I_{2p},x}(x)\nonumber\\
&=&[\xi_p^xL_p+\eta_p^x I_p+ \xi_p^x I_{2p}, \sum_{t\in \mathbb{Z}} (\alpha_tL_t+\beta_t I_t)]\nonumber\\
&=&\sum_{t\in \mathbb{Z}}((p-t)\xi_p^x \alpha_tL_{p+t}+(p-t)\xi_p^x \beta_tI_{p+t}) \label{final}\\
&&\ \ \ \ +\sum_{t\in \mathbb{Z}}((p-t)\eta_p^x \alpha_t I_{p+t}+(2p-t)\xi_p^x \alpha_t I_{2p+t}.\nonumber\end{aligned}$$ Next the proof is divided into three cases according to the situations of $(\alpha_t)_{t\in \mathbb{Z}}, (\beta_t)_{t\in \mathbb{Z}}$.
[**Case i.** ]{} $(\beta_t)_{t\in \mathbb{Z}}$ is a zero sequence, i.e., $x=\sum_{t\in \mathbb{Z}} \alpha_tL_t$. Then by (\[guol6\]), it is easy to see that $\Delta(x)=0$.
[**Case ii.** ]{} $(\alpha_t)_{t\in \mathbb{Z}}$ is a zero sequence, i.e., $x=\sum_{t\in \mathbb{Z}} \beta_tI_t$. Then by (\[guol6\]) and (\[final\]) we have $$\Delta(x)= \mu_x \sum_{t\in \mathbb{Z}} \beta_t I_t=\sum_{t\in \mathbb{Z}}(p-t)\xi_p^x \beta_tI_{p+t}$$ for all $p\in \mathbb{Z}$. By taking enough diffident $p$ in the above equation and, if necessary, let these $p$’s to be large enough, we obtain that $\Delta(x)=0$.
[**Case iii.**]{} Both $(\alpha_t)_{t\in \mathbb{Z}}$ and $(\beta_t)_{t\in \mathbb{Z}}$ are not zero sequences. Hence there is a nonzero term $\alpha_{t_0}L_{t_0}$ in $x=\sum_{t\in \mathbb{Z}} (\alpha_tL_t+\beta_t I_t)$ for some $t_0\in \mathbb{Z}$. Take two integers $p=p_1$ and $p=p_2$ in (\[final\]) such that $p_i-t_0\neq 0, i=1,2$, then by $(p_i-t_0)\xi_{p_i}^x \alpha_{t_0}L_{p_i+t_0}=0$ in (\[final\]) we have $\xi_{p_i}^x=0$. Then by (\[guol6\]) and (\[final\]) we have $$\Delta(x)= \mu_x \sum_{t\in \mathbb{Z}} \beta_t I_t=\sum_{t\in \mathbb{Z}}(p_i-t)\eta_{p_i}^x \alpha_t I_{p_i+t}, \ \ i=1,2.$$ By taking $p_1$ and $p_2$ in the above equation such that $p_1, p_2, p_1-p_2$ are large enough, we see that $\Delta(x)=0$. The proof is completed.
Now we are in position to prove Theorem \[thm-tang\].
**Proof of Theorem \[thm-tang\] :** Let $\Delta$ be a 2-local derivation on $W(2,2)$. Take a derivation $\Delta_{L_0,L_1}$ such that $$\Delta(L_0)=\Delta_{L_0,L_1}(L_0)\ \ \text{and} \ \ \Delta(L_1)=\Delta_{L_0,L_1}(L_1).$$ Set $\Delta_1=\Delta-\Delta_{L_0,L_1}.$ Then $\Delta_1$ is a 2-local derivation such that $\Delta_1(L_0)=\Delta_1(L_1)=0.$ By lemma \[lem13\], $\Delta_1(L_i)=0$ for all $i\in\mathbb{Z}.$ From this with Lemma \[lem16\], we have $\Delta_1(I_0)=\mu_{I_0}I_0$ for some $\mu_{I_0}\in \mathbb{C}$. Now we set $\Delta_2=\Delta_1-\mu_{I_0}D$. Then $\Delta_2$ is a 2-local derivation such that $$\begin{aligned}
\Delta_2(L_{0})=\Delta_1(L_{0})-\mu_{I_0}D(L_{0})=0-0=0,\\
\Delta_2(L_{1})=\Delta_1(L_{1})-\mu_{I_0}D(L_{1})=0-0=0,\\
\Delta_2(I_{0})=\Delta_1(I_{0})-\mu_{I_0}D(I_{0})=\mu_{I_0}I_{0}-\mu_{I_0}I_{0}=0.\end{aligned}$$ By lemma \[lem-mel6\], it follows that $\Delta_2=\Delta-\Delta_{L_0,L_1}-\mu_{I_0}D\equiv0.$ Thus $\Delta=\Delta_{L_0,L_1}+\mu_{I_0}D$ is a derivation. The proof is completed. $\Box$
2-local derivation on the thin Lie algebra
==========================================
Let us consider the following (see [@Kha]) so-called [*thin Lie algebra*]{} $\mathfrak{T}$ with a basis $\{e_n: n\in \mathbb{N}\}$, which is defined by the following table of multiplications of the basis elements: $$[e_1,e_n]=e_{n+1},\ \ \ n\geq 2,$$ and other products of the basis elements being zero. In this section, we study the 2-local derivation on the thin Lie algebra and prove that it admits a lots of 2-local derivations which are not derivations. Recall that the authors in [@ayu2019] give a special example of 2-local derivations on $\mathfrak{T}$. The following lemma is given by [@ayu2019] with a slight difference.
\[thm12\] Any derivation $\delta$ on the algebra thin Lie algebra $\mathfrak{T}$ if of the form $\delta\doteq \delta_{\alpha,\beta}^{(n,m)}$ which satisfies $$\begin{aligned}
&& \delta_{\alpha,\beta}^{(n,m)}(e_1)=\sum\limits_{i=1}^n\alpha_ie_i, \label{a231}\\
&& \delta_{\alpha,\beta}^{(n,m)}(e_j)=(j-2)\alpha_1e_j+\sum\limits_{i=2}^m\beta_{i}e_{i+j-2},\ \ \ j\geq2, \label{a232}\end{aligned}$$ where $n,m-1\in \mathbb{N}$ and $\alpha=(\alpha_1, \cdots, \alpha_n)\in \mathbb{C}^n, \beta=(\beta_2, \cdots, \beta_m)\in \mathbb{C}^{m-1}$.
Let $\delta$ be a derivation on $\mathcal{L}.$ We set $\delta(e_1)=\sum\limits_{i=1}^n\alpha_ie_i,\ \ \delta(e_2)=\sum\limits_{i=1}^m\beta_ie_i,$ where $\alpha_i, \beta_j\in \mathbb{C},$ $i=2, \cdots, n,\ j=1, \cdots,m $ and $n,m\in \mathbb{N}.$ Then we have $\delta(e_3)=\delta([e_1,e_2])=[\delta(e_1),e_2]+[e_1,\delta(e_2)]=\alpha_1e_3+\sum\limits_{i=1}^n\beta_{i}e_{i+1}$. From this, one has $0=\delta([e_2,e_3])=[\delta(e_2),e_3]+[e_2, \delta(e_3)]=\beta_1e_4$, and so that $\beta_1=0$. This means that (\[a232\]) replacing $\delta_{\alpha,\beta}^{(n,m)}$ by $\delta$ holds for $j=2$. We assume that (\[a232\]) holds for $j(\ge 2)$. Further, We have $$\begin{aligned}
\delta(e_{j+1})&=&\delta([e_1,e_j])=[\delta(e_1),e_j]+[e_1,\delta(e_j)]\\
&=&[\sum\limits_{i=1}^n\alpha_ie_i,e_j]+[e_1, (j-2)\alpha_1e_j+\sum\limits_{i=2}^m\beta_{i}e_{i+j-2}]\\
&=&(j-1)\alpha_1e_{j+1}+\sum\limits_{i=2}^m\beta_{i}e_{i+j-1},\end{aligned}$$ which proves that (\[a232\]) holds For $j+1$. By induction on $j$ we know that (\[a232\]) holds. Conversely, it is easy to check that a linear map $\delta$ on $\mathfrak{T}$ satisfying (\[a231\]) and (\[a232\]) is a derivation. Denote this derivation $\delta$ by $\delta_{\alpha, \beta}^{(n,m)}$. The proof is completed.
Now we give a complete classification of the 2-local derivation on $\mathfrak{T}$ as follow.
\[th-exp\] Every 2-local derivation $\Delta$ on the thin Lie algebra $\mathfrak{T}$ is of the form $$\Delta=\delta_{\alpha,\beta}^{(s,t)}+\Omega_{\theta, \lambda}^{(q,m)}$$ for some $s,t-1,m-1\in \mathbb{N}$, $\lambda\in \mathbb{C}$, and $\alpha=(\alpha_1, \cdots, \alpha_s)\in \mathbb{C}^s, \beta=(\beta_2, \cdots, \beta_t)\in \mathbb{C}^{t-1}$, $\theta=(\theta_2, \cdots, \theta_m)\in \mathbb{C}^{m-1}$ and $q\in \{t\in \mathbb{Z}: t>2 \}$, where $\delta_{\alpha,\beta}^{(s,t)}$ is given by Lemma \[thm12\] and $\Omega_{\theta, \lambda}^{(q,m)}:\mathfrak{T}\rightarrow \mathfrak{T}$ is a map that satisfies for any $x=\sum_{i=1}^p k_ie_i\in \mathfrak{T}$, $$\label{ex3}
\Omega_{\theta, \lambda}^{(q,m)}(x) = \begin{cases}
\sum\limits_{i=2}^p\sum\limits_{j=2}^mk_i\theta_{j}e_{i+j-2}, & \text{if $k_1\neq 0$,}\\
\lambda k_q e_q, & \text{if $x=k_q e_q$ for some $q$ with $2<q\le p$,}\\
0, & \text{ others}
\end{cases}$$
Suppose that $\Delta$ is a 2-local derivation on the thin Lie algebra $\mathfrak{T}$. Let $\widetilde{\Delta}=\Delta-\Delta_{e_1,e_2}$. Then $\widetilde{\Delta}$ is also a 2-local derivation on the thin Lie algebra $\mathfrak{T}$ satisfying $\widetilde{\Delta}(e_1)=\widetilde{\Delta}(e_2)$=0.
Take any but fixed $x=\sum\limits_{i=1}^pk_ie_i\in \mathfrak{T}$. If $x=0$, then by (\[recall1\]) we know $\widetilde{\Delta}(x)=0$. Hence below we always assume that $x\neq 0$, i.e., $k_p\neq 0$ for some $p\in \mathbb{N}$.
For the derivation $\widetilde{\Delta}_{e_1, x}$, as $\widetilde{\Delta}_{e_1, x}(e_1)=\widetilde{\Delta}(e_1)=0$, it follows by Lemma \[thm12\] that $$\begin{aligned}
&&\widetilde{\Delta}_{e_1, x}(e_1)=0, \\
&&\widetilde{\Delta}_{e_1, x}(e_j)=\sum\limits_{i=2}^m\beta_{i}^xe_{i+j-2} , \ \forall j\ge 2,\end{aligned}$$ for some $m\in \mathbb{N}$ with $m\ge 2$ and $\beta_{i}^x\in \mathbb{C}, i=2, \cdots, m$ with $\beta_{m}^x\neq 0$. Therefore we have $$\begin{aligned}
\label{ximage1}
\widetilde{\Delta}(x)&=&\widetilde{\Delta}_{e_1, x}(x)=k_1\nonumber\widetilde{\Delta}_{e_1, x}(e_1)+\cdots+k_p\nonumber\widetilde{\Delta}_{e_1, x}(e_p)\\
&=&k_2\beta_{2}^xe_2+(k_2\beta_{3}^x+k_3\beta_{2}^x)e_3 +\cdots \\
&&\ \ +(k_{p-1}\beta_{m}^x+k_p\beta_{m-1}^{x})e_{p+m-3}+k_p\beta_{m}^xe_{p+m-2}\nonumber\\
&=& \beta_{2}^x \sum\limits_{i=2}^pk_ie_i+\beta_{3}^x \sum\limits_{i=2}^pk_ie_{i+1}+\cdots+\beta_{m}^x \sum\limits_{i=2}^pk_ie_{i+m-2}\nonumber\end{aligned}$$
For the derivation $\widetilde{\Delta}_{e_2, x}$, by $\widetilde{\Delta}_{e_2, x}(e_2)=\widetilde{\Delta}(e_2)=0$ and Lemma \[thm12\], we have $$\begin{aligned}
&&\widetilde{\Delta}_{e_2, x}(e_1)=\sum_{i=1}^n \alpha_i^x e_i, \\
&& \widetilde{\Delta}_{e_2, x}(e_j)=(j-2)\alpha_1^x e_j , \ \forall j\ge 2,\end{aligned}$$ for some $n\in \mathbb{N}$ and $\alpha_{i}^x\in \mathbb{C}, i=1, \cdots, n$ with $\alpha_{n}^x\neq 0$. From this, we get $$\begin{aligned}
\label{ximage2}
\widetilde{\Delta}(x)&=&\widetilde{\Delta}_{e_2, x}(x)\nonumber\\
&=&k_1(\alpha_{1}^xe_1+\cdots+\alpha_{n}^xe_n) \\
&&\ \ +k_3\alpha_{1}^xe_3+2k_4\alpha_{1}^xe_4+\cdots+(p-2)k_p\alpha_{1}^xe_p.\nonumber\end{aligned}$$ Next, according to the situations of coefficients $k_1, \cdots, k_p$ in $x=\sum\limits_{i=1}^pk_ie_i$, the proof is divided into the following cases.
[**Case 1.**]{} When $k_1\neq 0$. By comparing (\[ximage1\]) with (\[ximage2\]), we have $k_1\alpha_{1}^xe_1=0$ and so that $\alpha_{1}^x=0$. Therefore, (\[ximage2\]) becomes $$\widetilde{\Delta}(x)=k_1(\alpha_{2}^xe_2+\cdots+\alpha_{n}^xe_n).$$ This, together with (\[ximage1\]), gives that $n=p+m-2$ and $$\label{tangzz}
\begin{cases}
k_1\alpha_2^x=&k_2\beta_2^x, \\
k_1\alpha_3^x=&k_2\beta_3^x+k_3\beta_2^x,\\
k_1\alpha_4^x=&k_2\beta_4^x+k_3\beta_3^x+k_4\beta_2^x,\\
\vdots &\vdots\\
k_1\alpha_n^x=&k_p\beta_m^x.
\end{cases}$$ Note that $k_1\neq 0$, if we given a sequence of numbers $\beta_2^x, \cdots, \beta_m^x$ then we can get a sequence of numbers $\alpha_2^x, \cdots, \alpha_n^x$ satisfying (\[tangzz\]). Hence in this case we let $\Delta(x)$ be of the form (\[ximage1\]), namely, by denoting $\theta_j=\beta_{j}^x, j=2, \cdots, m$ we have $$\widetilde{\Delta}(x)=\theta_{2} \sum\limits_{i=2}^pk_ie_i+\theta_{3}
\sum\limits_{i=2}^pk_ie_{i+1}+\cdots+\theta_{m} \sum\limits_{i=2}^pk_ie_{i+m-2}=\sum_{i=2}^p\sum_{j=2}^mk_i\theta_{j}e_{i+j-2}.$$
[**Case 2.**]{} When $k_1=0$. By (\[ximage2\]) we have $$\label{zzz1}
\widetilde{\Delta}(x)=\alpha_{1}^x(k_3e_3+2k_4e_4+\cdots+(p-2)k_pe_p).$$ From this we see that if $p=2$ or $\alpha_{1}^x =0$ then $\widetilde{\Delta}(x)=0$. Assume that $p\ge 3$ and $\alpha_{1}^x \neq 0$. On the other hand, by (\[zzz1\]) and (\[ximage1\]) we see that $\alpha_{1}^x(p-2)k_pe_p=k_p\beta_{m}^xe_{p+m-2}$ and so that $p=p+m-2$. In other words, $m=2$. Therefore, (\[ximage1\]) becomes $$\label{zzz2}
\widetilde{\Delta}(x)=\beta_{2}^x (k_2e_2+\cdots+k_pe_p).$$
[**Subcase 2.1**]{} When $k_2\neq 0$. Then by (\[zzz1\]) and (\[zzz2\]) one has $\beta_{2}^x k_2e_2=0$ which deduces $\beta_{2}^x=0$. Hence by (\[zzz2\]) we have $\widetilde{\Delta}(x)=0$.
[**Subcase 2.2**]{} When $k_2= 0$. In view of (\[zzz1\]) and (\[zzz2\]), we get $$\label{complete1}
\widetilde{\Delta}(x)=\alpha_{1}^x(k_3e_3+2k_4e_4+\cdots+(p-2)k_pe_p)=\beta_{2}^x (k_3e_3+k_4e_4+\cdots+k_pe_p).$$ If there are two coefficients $k_s, k_t$, $3\le s<t\le p$ in (\[complete1\]) such that $k_s k_t\neq 0$, then we have $\alpha_{1}^x(s-2)=\beta_{2}^x$ and $\alpha_{1}^x(t-2)=\beta_{2}^x$. This yields $\alpha_{1}^x=0$ and then $\widetilde{\Delta}(x)=0$. If there exist only one $k_q\neq 0$ for some $3\le q \le p$, i.e., $x=k_q e_q$, then we have by (\[complete1\]) that $\widetilde{\Delta}(x)=\lambda k_qe_q$ by denoting $\lambda\doteq\beta_{2}^x$. If all $k_j's$ are equal to $0$, then $\widetilde{\Delta}(x)=0$.
Now, by summarizing the above processes we get $\widetilde{\Delta}=\Omega_{\theta, \lambda}^{(q,m)}$ for some appropriate $\theta, \lambda, q,m$. Note that $\widetilde{\Delta}=\Delta-\Delta_{e_1,e_2}$. Let the derivation $\Delta_{e_1,e_2}$ be of the form $\delta_{\alpha,\beta}^{(n,s)}$ for some appropriate $\alpha,\beta,n,s$ in Lemma \[thm12\], then we complete the proof.
By Theorem \[th-exp\], we know the thin Lie algebra admits a lots of 2-local derivations which are not derivations. We give two examples as follows.
Let $\Delta=\delta_{\alpha,\beta}^{(s,t)}+\Omega_{\theta, \lambda}^{(q,m)}:\mathfrak{T}\rightarrow \mathfrak{T}$ with $m=2$ and $\alpha=0$, $\beta=0$, $\theta=1$, $\lambda=0$, that is $$\Delta(\sum\limits_{i=1}^pk_ie_i) =
\begin{cases}
\sum\limits_{i=2}^pk_ie_i, & \text{if } k_1\neq 0,\\
0, & \text{if } k_1=0.
\end{cases}$$ The authors in [@ayu2019] have shown that such $\Delta$ is a 2-local derivation on $\mathfrak{T}$ but it is not a derivation.
Let $\delta_{\alpha,\beta}^{(s,t)}=\delta_{\alpha,\beta}^{(s,t)}+\Omega_{\theta, \lambda}^{(q,m)}:\mathfrak{T}\rightarrow \mathfrak{T}$ with $\delta_{\alpha,\beta}^{(s,t)}=0$, $m=q=3$ and $\theta=(1,1)$, $\lambda=2$, that is $\Delta=\Omega_{(1,1), 2}^{(3,3)}$. Exactly we have $$\label{yb4}
\Delta(\sum\limits_{i=1}^pk_ie_i) =
\begin{cases}
\sum\limits_{i=2}^pk_ie_i+\sum\limits_{i=2}^pk_ie_{i+1}, & \text{if } k_1\neq 0,\\
2k_3 e_3, & \text{if } \sum\limits_{i=1}^pk_ie_i=k_3 e_3,\\
0, & \text{if } k_1=0.
\end{cases}$$ Then by theorem \[th-exp\] it is easy to see that $\Delta$ is a $2$-local derivation. We will see that $\Delta$ is not a derivation. In fact, let $x=e_1+e_2$ and $y=-e_1-e_2+2e_3.$ Then we have $\Delta(x)=e_2+e_3$, $\Delta(y)=-e_2+e_3+2e_4$ and $\Delta(x+y)=\Delta(2e_3)=4e_3\neq \Delta(x)+\Delta(y)=2e_3+2e_4$. So, $\Delta$ is not additive, and therefore is not a derivation.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is supported in part by National Natural Science Foundation of China (Grant No. 11771069) and the fund of Heilongjiang Provincial Laboratory of the Theory and Computation of Complex Systems.
[22]{}
Sh.A Ayupov, K.K. Kudaybergenov, B. Yusupov, *2-Local derivations on generalized Witt algebras. Linear Multilinear Algebra*, inpress, doi: 10.1080/03081087.2019.1708846. Sh.A Ayupov, B. Yusupov, *2-Local derivations of infinite-dimensional Lie algebras*. J. Algebra Appl., ID: 2050100 (2020). Sh.A Ayupov, K.K. Kudaybergenov, I.S. Rakhimov, *2-Local derivations on finite-dimensional Lie algebras*, Linear Algebra Appl. **474** (2015), 1-11. Z. Chen, D. Wang, *2-Local automorphisms of finite-dimensional simple Lie algebras,* Linear Algebra Appl. **486**, 335-344 (2015). H. Chen, J. Li, *Left-symmetric algebra structures on the W-algebra $W (2, 2)$*, Linear Algebra Appl. **437** (2012), 1821-1834. S. Gao, C. Jiang, Y. Pei, *Derivations, central extensions and automorphisms of a Lie algebra*, Acta Math. Sin. **52** (2009), 281-288. W. Jiang, W. Zhang. *Verma modules over the W (2, 2) algebras*, J. Geom. Phys. **98**(2015), 118-127. K. Khakimdjanova, Yu. Khakimdjanov, Sur une classe d’algebres de Lie de dimension infinie, Commun. Algebra, Vol.**29**(1) (2001), 177-191. G. Radobolja. *Subsingular vectors in Verma modules, and tensor product of weight modules over the twisted Heisenberg-Virasoro algebra and $W (2, 2)$ algebra*, J. Math. Phys. **54** (2013), 071701. P. Šemrl, Local automorphisms and derivations on $B(H),$ Proc. Amer. Math. Soc., **125** (1997), 2677-2680. X. Tang, *Biderivations, linear commuting maps and commutative post-Lie algebra structures on W-algebras*, Commun. Algebra **45** (2017), 5252-5261. Y. Wang, Q. Geng, Z. Chen, *The superalgebra of $W (2, 2)$ and its modules of the intermediate series*, Commun. Algebra **45** (2017), 749-763. Y. Zhao, Y. Chen, K. Zhao, *2-local Derivations on Witt Algebras,* J. Algebra Appl., inpress, doi: 10.1142/S0219498821500687 (2020). W. Zhang, C. Dong, *W-algebra $W(2,2)$ and the vertex operator algebra $L(\frac{1}{2},0)\otimes L(\frac{1}{2},0))$,* Commun. Math. Phys. **285** (2009), 991-1004.
[^1]: Corresponding author, E-mail: tangxm@hlju.edu.cn
|
---
abstract: 'The best gravitational lenses for detecting distant galaxies are those with the largest mass concentrations and the most advantageous configurations of that mass along the line of sight. Our new method for finding such gravitational telescopes uses optical data to identify projected concentrations of luminous red galaxies (LRGs). LRGs are biased tracers of the underlying mass distribution, so lines of sight with the highest total luminosity in LRGs are likely to contain the largest total mass. We apply this selection technique to the Sloan Digital Sky Survey and identify the 200 fields with the highest total LRG luminosities projected within a $3\farcm5$ radius over the redshift range $0.1 \leq z \leq 0.7$. The redshift and angular distributions of LRGs in these fields trace the concentrations of non-LRG galaxies. These fields are diverse; 22.5% contain one known galaxy cluster and 56.0% contain multiple known clusters previously identified in the literature. Thus, our results confirm that these LRGs trace massive structures and that our selection technique identifies fields with large total masses. These fields contain $2-3$ times higher total LRG luminosities than most known strong-lensing clusters and will be among the best gravitational lensing fields for the purpose of detecting the highest redshift galaxies.'
author:
- 'Kenneth C. Wong, Ann I. Zabludoff, S. Mark Ammons, Charles R. Keeton, David W. Hogg, and Anthony H. Gonzalez'
bibliography:
- 'lrgbeams.bib'
title: A NEW APPROACH TO IDENTIFYING THE MOST POWERFUL GRAVITATIONAL LENSING TELESCOPES
---
INTRODUCTION {#sec:intro}
============
Gravitational lensing by galaxy clusters is an important tool for studying the high-redshift universe. Galaxies at redshifts $1 \lesssim z \lesssim 3$ can be magnified into extended arcs, enabling studies of these sources at spatial resolutions beyond what is feasible in similar unlensed objects [e.g., @brammer2012; @frye2012; @livermore2012; @sharon2012; @yuan2012]. Lensing by foreground galaxy clusters can also magnify very high-redshift ($z \gtrsim 7$) sources into detectability, allowing us to measure their physical properties [e.g., @kneib2004; @pello2004; @schaerer2005; @richard2006; @richard2008; @stark2007; @bradley2008; @bradley2012; @zheng2009; @zheng2012; @laporte2011; @bouwens2012; @hall2012; @coe2013] and making them ideal targets for spectroscopic follow-up [e.g., @bradac2012]. Such studies are particularly important for characterizing objects on the faint end of the galaxy luminosity function at these redshifts, as even the deepest [*HST*]{} observations in blank fields require too large a time investment to probe to such depths.
The lensing power of foreground clusters depends on a variety of physical properties. The total mass of the cluster is very important, as the lensing strength depends on the surface mass density of the lens. @wong2012 found that distributing the mass among multiple cluster-scale halos along the line of sight (LOS) can increase the lensing cross section compared to having the same mass in a single rich cluster. This effect results from interactions among the multiple lens potentials, boosting the magnification in the field. Analysis of the Millennium [@springel2005] and Millennium XXL [@angulo2012] simulations shows that lines of sight with large total masses may contain multiple massive ($\gtrsim 10^{14} M_{\odot}$) halos and produce some of the highest lensing cross sections in the universe (K. D. French et al. 2013, in preparation). Individual halo properties, including concentration, ellipticity, orientation, and redshift also affect lensing cross sections [e.g., @bartelmann1995; @meneghetti2003; @wong2012].
State-of-the-art lensing analyses focus on fields identified by a single massive cluster [e.g., @postman2012]. Even X-ray surveys [e.g., @bohringer2000; @vikhlinin2009] and @sunyaev1972 effect surveys [e.g., @vanderlinde2010; @marriage2011; @williamson2011] for great lensing fields are biased toward lines of sight with a dominant cluster-scale halo because the signal is not proportional to projected mass. In other words, a line of sight with a single massive cluster looks identical in X-ray or SZ observations to a similar cluster with additional smaller projected halos whose masses may not be sufficient to have a detectible hot X-ray gas component. In X-ray or SZ observations, the scaling of the signal with halo mass is faster than linear [e.g., @bonamente2008; @vikhlinin2009], so a line of sight with multiple structures will have a lower signal than if the same total mass were concentrated in a single cluster. Additional lower mass halos, which may not have detectable hot gas components, may be missed entirely in the field. Thus, these studies do not necessarily select for the largest total mass and/or the most advantageous mass configuration. [*The gravitational lenses explored to date may not in fact be the best directions on the sky to look.*]{}
We explore a new optical selection technique to identify the best lines of sight (hereafter referred to as “beams") for gravitational lensing. By selecting fields that have the greatest total luminosity in luminous red galaxies [LRGs; e.g., @eisenstein2001], which are biased tracers of the underlying matter distribution [@zehavi2005; @li2006; @ho2009; @white2011] and detectable to high redshifts, we are likely to find beams with the largest single massive halos (galaxy clusters) and with chance alignments of multiple group and cluster-scale halos. This technique requires a wide-field multi-band photometric dataset with accurate redshifts, photometric or spectroscopic. In essence, we are using fewer, but more biased, tracers of the mass along the LOS than methods like the Cluster Red Sequence technique [CRS; @gladders2000] and the Gaussian Mixture Brightest Cluster Galaxy algorithm [GMBCG; @hao2009; @hao2010] that exploit the relationship between halo mass and red galaxy counts within the halo [@lin2004b]. @zitrin2012 derived mass models of clusters in the @hao2010 GMBCG sample, including some of the most powerful lenses (Einstein radius $> 30$). Like X-ray and SZ surveys, these approaches may not be sensitive to multiple projected halos, as smaller structures (i.e., poor clusters) may be hard to identify as galaxy overdensities in color-magnitude space. In contrast, even individual LRGs can be indicative of cluster-scale structures [@ho2009].
The Sloan Digital Sky Survey (SDSS), with its large sky coverage and photometric LRG selection out to $z \sim 0.7$, is ideal for identifying the best lensing beams using LRGs. Most arc-producing lensing clusters are at intermediate redshift [$0.3 \lesssim z \lesssim 0.8$; @bartelmann1998; @gladders2003; @hennawi2007], although higher-redshift lensing clusters have been found [e.g., @huang2009; @gonzalez2012]. The SDSS is deep enough to probe a volume-limited sample of very bright LRGs ($M_{i} - 5\mathrm{log}_{10}(h) \lesssim -22.5$) out to $z \sim 0.7$. The 5-band optical photometry provides LRG selection, luminosities, colors, and photometric redshifts for over $10^{6}$ galaxies [e.g., @padmanabhan2005; @padmanabhan2007; @ross2011].
We present our beam selection method and apply it to the SDSS, identifying the 200 beams with the highest LRG luminosity concentrations and therefore likely to contain the largest total masses projected within a radius of $3\farcm5$. The LRG photometric redshift distributions show that many of these beams have multiple structures along the line of sight. Follow-up galaxy spectroscopy in the first fields selected using this method has revealed a diversity of structures, including chance alignments of multiple cluster-scale halos and total masses $\gtrsim 2\times10^{15} h^{-1} M_{\odot}$ (S. M. Ammons et al. 2013, in preparation).
We describe our method of selecting massive beams in § \[sec:selection\]. In § \[sec:results\], we apply it to the SDSS, list the highest-ranked beams and their properties, and discuss applications of this method to future surveys. We summarize our main conclusions in § \[sec:conclusions\]. Throughout this paper, we assume a $\Lambda$CDM cosmology with $\Omega_{m} = 0.274$, $\Omega_{\Lambda} = 0.726$, and $H_{0} = 100~h$ km s$^{-1}$ Mpc$^{-1}$ with $h = 0.71$. All magnitudes given are on the AB magnitude system [@oke1983].
SELECTION OF MASSIVE BEAMS {#sec:selection}
==========================
Our approach to selecting lines of sight with large total masses is based on using LRGs as indicators of massive halos. LRGs are strongly clustered and are biased tracers of massive structure [e.g., @zehavi2005; @li2006; @ho2009; @white2011]. They are among the most luminous galaxies in optical light [$L \gtrsim L^{*}$; @ho2009] and thus are visible to large distances. LRGs show little variation in their SEDs [@eisenstein2003; @cool2008], making them easy to identify through their optical colors in broadband imaging data. They have been surveyed over large regions of the sky, making them useful probes of the evolution of large scale structure over a cosmologically interesting volume [e.g. @eisenstein2001; @padmanabhan2005]. Projected concentrations of LRGs on the sky are therefore indicative of either an extremely rich galaxy cluster or a superposition of multiple group and cluster-scale halos, given that each individual LRG is likely to occupy an overdense region.
Our selection technique makes use of this relationship between LRGs and massive structures, identifying beams that have the highest total LRG luminosity. The stellar mass-to-light ($M_{*}/L$) ratios of LRGs are strongly correlated with their rest-frame colors [@bell2001], which, given their homogeneous SEDs, implies that they have similar $M_{*}/L$ ratios. Indeed, @kauffmann2003 find that the $M_{*}/L$ ratio of galaxies flattens at high luminosities with smaller scatter for redder rest-frame optical wavelengths, and that the most luminous galaxies have the highest $M_{*}/L$ ratios [see also @zaritsky2006]. As a result, the optical/near-IR luminosities of LRGs can be used to estimate their stellar masses.
Relating the LRG luminosity to the mass of its host halo is complicated by the relatively flat slope and substantial scatter of the stellar-to-halo mass (SHM) relation for halo masses above $\sim 10^{12} M_{\odot}$ [e.g., @mandelbaum2006; @yang2008; @conroy2009; @more2009; @behroozi2010; @behroozi2012; @moster2010; @leauthaud2012]. While this scatter is smaller than for the luminosity to halo mass relation [e.g., @yang2008; @cacciato2009], stellar mass is still not a precise tracer of halo mass for individual massive galaxies above this threshold. There is a scaling between the mass and luminosity of a galaxy cluster [e.g., @lin2003; @lin2006; @tinker2005; @cacciato2013a; @cacciato2013b], although central galaxies contribute fractionally less to the stellar mass for larger halo masses [@lin2004a; @gonzalez2007; @leauthaud2012; @lidman2012], further suggesting that individual LRGs may not give a good estimate of halo mass. On the other hand, this effect should be mitigated when estimating the total mass in a particular field by integrating over all LRGs in the field and in redshift space. Furthermore, many of these galaxies are satellite galaxies of higher mass clusters, which increases the total LRG luminosity in the most massive halos [@white2011; @behroozi2012]. Therefore, lines of sight containing high total LRG luminosities are likely to have large total masses, either distributed in multiple, projected cluster halos or dominated by a single massive cluster. The former configurations are more challenging to identify through other selection methods.
Simple number counts of LRGs also can be useful, as there is a relationship between number of LRGs and halo mass. However, the relation has large scatter for individual clusters [0.21 dex for $M \sim 10^{15} M_{\odot}$ clusters; @ho2009]. Using total luminosity is likely to be a better tracer of total mass due to the SHM relation, despite its shallow slope at high masses. In addition, the small number of LRGs in clusters leads to large Poisson errors, whereas we are unlikely to miss the brightest galaxies that contribute the most to the total luminosity. We perform a simple test that demonstrates that using total LRG luminosity provides a better contrast to the field galaxy population than simple number counts (see Appendix \[app:nsort\]). For completeness, we list there the additional beams that would have been selected using number counts instead.
RESULTS & DISCUSSION {#sec:results}
====================
In this section, we apply our massive beam selection technique to the SDSS. The SDSS is currently the survey that has the best characteristics for our selection technique. The latest data release includes imaging of roughly a third of the sky in five optical broadband filters. The depth of the photometric observations is sufficient to detect and classify LRGs within $0.1 \leq z \leq 0.7$ [@padmanabhan2005], where we expect a large number of lensing clusters to lie. The SDSS also includes spectroscopic redshifts for roughly a third of the LRGs. We examine the LRG redshift distributions in a comparison sample of known lensing clusters, comparing these fields to the 200 best beams in the SDSS as ranked by their integrated LRG luminosity. Our top beams have higher total LRG luminosity and potentially more mass than even these known lensing clusters. Roughly $75\%$ of our beams contain known galaxy clusters, confirming the power of our selection technique. We also discuss possible applications of this technique to current and future surveys.
Defining the LRG Sample {#subsec:lrg}
-----------------------
We select our sample of LRGs from the SDSS Data Release 9 [DR9; @ahn2012]. We identify LRGs using a modified version of the photometric selection criteria of @padmanabhan2005 [@padmanabhan2007]. The criteria consist of two separate cuts, denoted “Cut I" and “Cut II", which are designed to select LRGs at $z \lesssim 0.4$ and $z \gtrsim 0.4$, respectively. The details of the LRG selection are given in Appendix \[app:selection\].
The photometric redshifts in the DR9 catalog are computed using the method in @csabai2003. We limit our sample to LRGs at redshifts $0.1 \leq z \leq 0.7$. At $z \lesssim 0.1$, the Cut I criteria are too permissive, resulting in a large fraction of interlopers and causing biases in the photometric redshifts when compared to spectroscopic redshifts. At $z \gtrsim 0.7$, most objects are at the faint edge of our sample, resulting in larger photometric errors. We do not have enough spectroscopically observed objects to assess the quality of the photometric redshifts beyond this point.
We find good agreement of the photometric redshifts with the DR9 spectroscopic redshifts between $0.1 \leq z \leq 0.7$ (Figure \[fig:zvz\]). We define the photometric redshift accuracy to be the normalized median absolute deviation, $\sigma_{z}/(1+z) \equiv 1.48 \times \mathrm{median}(|\Delta z| / (1+z))$. For objects with $0.1 \leq z_{phot} \leq 0.7$, we calculate $\sigma_{z} / (1+z) = 0.017$, with catastrophic outlier rates of 4.2% with $|\Delta z| / (1+z) > 0.05$ and 0.3% with $|\Delta z| / (1+z) > 0.1$. The photometric redshifts are unbiased to within $\overline{\Delta z} / (1+z) \leq 0.01$ for most of the redshift range probed. There is a slight bias at the $\overline{\Delta z} / (1+z) \leq 0.02$ level at $0.6 \leq z \leq 0.7$, which affects less than 10% of our sample. The subsample of objects at the faint end of our magnitude range show the same behavior as the full sample and do not contain additional biases.
To improve the redshift accuracy of our sample, we replace the LRG photometric redshifts and errors with SDSS spectroscopic redshifts and errors where available, which is roughly for one-third of the LRG sample. The redshift uncertainties and outlier rates given in Figure \[fig:zvz\] are thus upper limits. Hereafter, when referring to an LRG’s redshift and its uncertainty, we mean the spectroscopic redshift when available and the photometric redshift otherwise.
In deriving the LRG luminosities, we account for K-corrections and luminosity evolution using an elliptical galaxy template generated by evolving a @bruzual2003 stellar population synthesis model. @delucia2006 find that massive elliptical galaxies in dense environments have roughly solar metallicities and stellar populations with a median formation redshift of $z \sim 2.5$ for $M_{*} \gtrsim 10^{11} M_{\odot}$, with higher formation redshifts for more massive systems. Similar formation redshifts for massive ellipticals are supported by observational studies [e.g., @vandokkum2003; @treu2005a; @treu2005b; @perezgonzalez2008]. There is evidence that massive early-type galaxies are well-characterized by a @salpeter1955 initial mass function [e.g., @auger2010; @treu2010] or even more bottom-heavy IMFs [e.g., @cappellari2012; @conroy2012; @spiniello2012; @vandokkum2012]. Therefore, we generate the template SED assuming an instantaneous burst of star formation at $z = 3$ with a Salpeter IMF and solar metallicity.
We perform K-corrections using the template SED at the age of the galaxy at its observed redshift, and the model is then passively evolved to $z = 0$. All galaxy luminosities are normalized to $z = 0$ quantities to ensure a fair comparison of their luminosities. We use a single model instead of fitting templates to the observed photometry [e.g., @eisenstein2001], as LRGs are typically red, quiescent galaxies with little recent star formation and very homogeneous SEDs [@eisenstein2003; @cool2008]. Furthermore, we are interested in corrections to the rest-frame $i$-band, which for the template fitting method can only be determined beyond $z \sim 0.2$ by extrapolating the fits to redder rest-frame wavelengths than are covered by the SDSS photometry. We choose the $i$-band because it is less affected by extinction and tends to trace stellar mass, and thus total mass, better than bluer filters as a result of being less sensitive to recent star formation.
We also only include objects with derived absolute magnitudes within the broad range $-24.7 < M_{i} - 5\mathrm{log}_{10}(h) < -21$ to eliminate objects with spurious photometric redshifts or aberrant inferred luminosities, while retaining the most luminous LRGs. We visually inspect objects at the bright end of this range to ensure that we include the brightest LRGs in our sample. Our final sample contains 1,151,117 LRGs, of which 361,438 have spectroscopic redshifts.
Characteristics of the LRG Sample {#subsec:lrgchar}
---------------------------------
Here, we investigate the properties of our final SDSS LRG sample. The redshift and $i$-band luminosity (after accounting for K-corrections and luminosity evolution) distributions of our sample are shown in Figure \[fig:samplehist\].
In the left panel of Figure \[fig:samplehist\], the spike in the LRG redshift histogram at $z \sim 0.35$ and trough at $z \sim 0.4$ arise from the combination of several effects. $z \sim 0.35$ is roughly the redshift at which the 4304 Å G-band absorption feature is redshifted into the SDSS $r$-band, which can masquerade as the 4000 Å break in color-redshift space. As a result, the photometric selection criteria select an excess of galaxies around this redshift. This bias also affects our spectroscopic subsample and in spectroscopic LRG samples with similar photometric selection criteria [e.g., @zehavi2005]. Secondly, the transition between galaxies selected by the Cut I and Cut II criteria is roughly at $z \sim 0.4$. Cut I has an apparent magnitude cut at $r < 19.7$, whereas Cut II has a cut at $i < 20$. This results in a sharp decrease in the number of Cut I-selected LRGs around $z \sim 0.4$ because we hit the $r < 19.7$ magnitude limit. Meanwhile, Cut II, while probing fainter objects, is optimized to select LRGs at $z \sim 0.5$ and is less efficient at $z \sim 0.4$. This results in a deficit of LRGs around $z \sim 0.4$, accentuating the $z \sim 0.35$ peak.
In addition to these selection effects, there is a small photometric redshift bias in the range $0.3 \lesssim z \lesssim 0.4$ at the $|\Delta z| / (1+z) < 0.01$ level that pulls galaxies toward $z \sim 0.35$, despite the fact that the photometric redshifts remain unbiased overall. The spectroscopic LRG subsample is not affected by this problem. Around this redshift, LRGs transition nearly orthogonally in color-color space, leading to a degeneracy that makes the photometric redshifts less precise, as was also noted by @padmanabhan2005. This bias, while still small, conspires with the other effects to “sharpen" the spike at $z \sim 0.35$, which was already present due to the selection effects discussed above. This feature is not in the photometric redshift distribution of the @padmanabhan2005 LRG sample due to known biases in their photometric redshifts that drive some objects with true redshifts near $z \sim 0.35$ to photometric redshifts of $z \sim 0.4$, smoothing out the feature. For $\sim62\%$ of the galaxies with photometric redshifts in the range $0.3 \leq z \leq 0.4$ and a measured spectroscopic redshift, the absolute difference between the photometric and spectroscopic redshift is smaller than the photometric redshift error given in the SDSS catalog.
In Figure \[fig:magvol\], we plot the distribution of LRG absolute $i$-band magnitudes as a function of redshift. The redshift axis has been rescaled so that it is linear in the enclosed comoving volume within a survey area of $\pi$ steradians. The sharp edge at the lower right where LRG selection is truncated represents the apparent magnitude cut at $i < 20$ for the Cut II objects. The sharpness of this cutoff is due to our single-model method of handling K-corrections and luminosity evolution. Applying a template-fitting method to the observed photometry [e.g., @eisenstein2001] would result in scatter about this cutoff. The “sawtooth" feature at $z \sim 0.4$, which is also in the LRG redshift-luminosity distribution of @padmanabhan2007, results from the $r < 19.7$ apparent magnitude cut for the Cut I objects and corresponds to the trough in the redshift distribution (see Figure \[fig:samplehist\]). This visualization shows that our LRG selection is approximately volume-limited out to $z = 0.4$ for $M_{i} - 5\mathrm{log}_{10}(h) \lesssim -21$ and to our upper redshift limit of $z = 0.7$ for $M_{i} - 5\mathrm{log}_{10}(h) \lesssim -22.5$, with the caveat that our selection is less efficient around $z \sim 0.4$.
LRG Properties of Known Lensing Cluster Fields {#subsec:known}
----------------------------------------------
We examine the LRG properties of a sample of known strong lensing clusters from @hennawi2008 and the Cluster Lensing and Supernova Survey with Hubble [CLASH; @postman2012] that lie within our chosen redshift range and overlap the sky coverage of our SDSS LRG sample. Our goal is to identify the best lensing fields using our new selection technique, and this set of known lensing clusters provides a calibration to which we can compare the LRG properties of our new beams (§ \[subsec:beams\]). If our beams have more total LRG luminosity ($\sim$mass) than known strong lenses, as well as multiple lensing planes in some cases, it is likely that they comprise a better sample of strong lenses. @hennawi2008 select their sample from the SDSS using the CRS selection method to identify clusters between $0.1 \lesssim z \lesssim 0.6$. We only use those systems labeled by @hennawi2008 as “definite" or “tentative" lensing clusters from visual identification of lensed arcs. The CLASH sample is a mostly X-ray selected sample of 20 massive clusters with an additional five known lensing clusters. Abell 2261 is in both samples, but we treat it as a part of the CLASH sample here.
We identify LRGs in these fields within an aperture of $3\farcm5$ (as we do for our SDSS beams in § \[subsec:beams\]) and show their redshift distributions in Figure \[fig:compzhist\]. In many fields, even the massive lensing cluster is marked by only a few LRGs. By selecting beams from SDSS that have a larger total luminosity in LRGs than these fields, we maximize the chance of finding mass concentrations that can act as powerful lenses. While these comparison fields tend to be single mass concentrations, several show non-negligible mass concentrations projected along the line of sight that are unassociated with the main cluster. Thus, the lens modeling of these known clusters should account for LOS mass concentrations. Such effects have not been explicitly treated in most past analyses, but can influence the inferred mass model [e.g., @hoekstra2011].
Selection of SDSS Beams {#subsec:beams}
-----------------------
To find beams with high total LRG luminosity, we search a fixed angular radius around each LRG in our sample. In principle, this selection can be performed with an arbitrary search radius. For our search, we choose an aperture of radius $3\farcm5$. Chance projections of the most massive clusters ($\sim 10^{15} M_{\odot}$) tend to benefit from interactions among their lens potentials [@wong2012], even in regions beyond their Einstein radii. In particular, the boost at intermediate magnifications ($\mu \sim 3-10$) due to lensing interactions at larger radii is critical in increasing the detectability of very high-redshift lensed galaxies. Beyond 3.5’, the strength of these interactions fall off to the point where the halos can be treated as independent lensing fields. Furthermore, typical ground-based near-infrared detectors that are well suited for follow-up observations of lensed high-redshift galaxies have fields of view roughly this size or larger.
For each beam centered on an LRG, we tabulate the total number of LRGs within the aperture, as well as the integrated rest-frame $i$-band luminosity of those LRGs as a proxy for total mass in the beam. We rank all the beams centered on an LRG in descending order by the total LRG luminosity in the beam. Overlapping beams containing the same LRGs but that are centered on different LRGs are further ranked by the luminosity of the LRG at the beam center. This choice makes it more likely that in beams with single dominant clusters, we select the central galaxy, which is often the most luminous [@lin2004a], though not always [e.g., @vonderlinden2007; @coziol2009; @skibba2011; @hikage2012]. Starting from the beam with the highest total luminosity and moving down this ranked list, we construct our list of top beams. If a beam is centered on an LRG that has already been included as part of a previous beam, we skip over that beam, but allow LRGs within it to be counted in beams further down the list. This method makes it possible for an LRG to be part of multiple beams, but minimizes overlap among beams in dense regions. There will always be some beams adjacent to or overlapping one another whose centers are separated by slightly more than the selection radius. While these beams could be counted as a single field for follow-up purposes, we count them as separate beams for consistency. In our final catalog of the top 200 beams, there are 28 that overlap another beam in the top 200.
This method can find beams containing dense concentrations of LRGs (e.g., clusters), but may not necessarily be centered on a cluster center. This can occur if an LRG in the outer parts of a cluster has other LRGs within $3\farcm5$ of it that are not within $3\farcm5$ of the more central LRGs of that cluster. This is not a flaw in the methodology as we are not specifically looking for fields centered on a single dense cluster. Rather, our selection is more likely to find lines of sight containing multiple mass concentrations. We do not account for the boundaries of the survey region when performing our beam selection. This is conservative because we can only underestimate the total luminosity of LRGs in a given field by ignoring these edge effects.
We compare the total luminosity and total number of LRGs in our full sample of beams to that of the comparison sample of lensing clusters from @hennawi2008 and CLASH in Figure \[fig:nlhist\]. Both the @hennawi2008 and CLASH samples tend to have lower total LRG luminosity and number counts compared to the SDSS beams, suggesting that our beams at the extreme tail of these distributions contain larger total masses.
We present a list of the 200 best beams as ranked by their total LRG luminosity in Table \[tab:beams\]. We choose a sample size of 200 because this is roughly the beam rank above which our beams exceed the total LRG luminosities of massive lensing clusters. These top 200 beams have total LRG luminosities $2-3$ times greater than the average total LRG luminosity of the comparison sample. We do recover five of the Hennawi/CLASH clusters in our sample, although our beams may be centered on different coordinates that include more LRGs in the field. We find that $\sim60\%$ of our selected beams overlap with the top 200 beams selected by LRG number counts. For completeness, we provide a separate list of beams in Appendix \[app:nsort\] that would have been in the top 200 (or had an equal number of LRGs to beams in our top 200) if we had chosen to sort by LRG number counts instead.
The redshift distributions of the LRGs in the top 200 beams are shown in Figure \[fig:zhist\]. The beams show a wide diversity of configurations, including beams dominated by a single massive peak, as well as beams with multiple structures along the line of sight. The latter configurations indicate chance alignments of galaxy clusters or groups in these fields, which can lead to advantageous lensing configurations for magnifying very high-redshift ($z \gtrsim 7$) galaxies [@wong2012]. Even single LRGs can trace massive lensing clusters (see Figure \[fig:compzhist\]).
We show color images of our top 20 beams in Figure \[fig:images\] to give a sense of the nature of these fields. The images are taken from the SDSS DR9 SkyServer[^1] and show 7$\times$7 fields centered on the coordinates of each beam to roughly match our circular selection aperture. The contours overplotted on the images trace the projected galaxy overdensities as determined from the full catalog of primary SDSS photometric galaxies brighter than $r = 22$. The contours generally trace the LRGs, confirming that the LRGs mark the densest structures in these fields. Like the redshift distributions of these beams, the angular distributions show a large diversity. Some beams are dominated by a single concentration of galaxies, whereas others have multiple clumps projected on the sky.
We perform a search in the NASA/IPAC Extragalactic Database (NED) for galaxy clusters that have been identified by past studies within our selected beams. We find that 22.5% of our beams contain one known cluster, and 56.0% contain multiple clusters, confirming the high mass concentrations projected along these lines of sight. We list the number of known clusters in each beam in Table \[tab:beams\] and present a list of these clusters in Appendix \[app:clusters\].
We visually inspect the SDSS SkyServer images of each of our top 200 fields. The majority show large concentrations of spheroidal galaxies indicative of cluster-scale structures. The other beams could be chance alignments of smaller group-scale halos, or could contain a number of misclassified objects masquerading as LRGs. While we have limited the number of misclassifications with our selection cuts (see Appendix \[app:selection\]), there are inevitably a small fraction of objects that contaminate our LRG catalog. We will describe the first results from our spectroscopic follow-up survey of several of these beams in S. M. Ammons et al. (2013, in preparation).
Possible Errors in Beam Selection {#subsec:errors}
---------------------------------
We use a Monte Carlo method to calculate the uncertainty on the total LRG luminosity of each of our top beams. We begin by searching the SDSS DR9 for all primary photometric objects classified as galaxies within our beams. Objects fainter than $r = 22$ are removed, as this is roughly where the morphological star/galaxy separation breaks down[^2]. We replace the photometric redshift and its associated error with a spectroscopic redshift and error where available.
For each beam, we apply Gaussian errors to the quantities defined in our LRG selection criteria (e.g. apparent magnitude, redshift; see Appendix \[app:selection\]) over 1000 Monte Carlo trials. For a given trial, we re-determine which galaxies in the beam satisfy the selection criteria and compute the total LRG luminosity. The final distribution of 1000 total LRG luminosities for each beam therefore accounts for both the intrinsic errors in the galaxy properties themselves, as well as galaxies falling into or out of the LRG selection cuts. Because the resulting luminosity distributions are non-Gaussian in general, we report median values in Table \[tab:beams\]. The error bars associated with these median values correspond to the 16% and 84% quantiles of the distribution for each beam. In Figure \[fig:beamstats\], we plot the observed total luminosity for each beam, along with the median of the Monte Carlo trials with the associated error range.
In general, the medians of the luminosity distributions generated through the Monte Carlo trials tend to be lower than the observed values. This is expected, because these beams were selected to be the top-ranked beams by total LRG luminosity. Given the scatter in the total luminosity, the selected beams are likely those that are among the best and also happen to scatter upwards in observed total LRG luminosity. Also, while this scatter affects the detailed beam rankings relative to the “true" rankings, Figure \[fig:beamstats\] shows that even among the top 200, there is a noticeable decrease in the median integrated LRG luminosity when ordered by the observed values, indicating that the observed values reliably rank the beams. One may also suspect that our top beams are biased toward low mass-to-light (M/L) ratios. For the initial beams that we have followed-up spectroscopically (S. M. Ammons et al. 2013, in preparation), we compare their M/L ratios to those of a sample of our comparison lensing clusters (§ \[subsec:known\]) using virial masses from the literature [@mantz2010; @zitrin2011; @coe2012]. We find that our beams have roughly comparable M/L ratios, suggesting that a potential bias in M/L is not a dominant effect.
We show the 20-beam central moving average of the median total LRG luminosities of the Monte Carlo trials in Figure \[fig:beamstats\]. While the values trend downward with beam rank as expected, the steepness of the trend noticeably decreases after the first $20-30$ beams. To test the robustness of our selected beams, we calculate statistics for a larger sample of the top 1000 beams. When re-ranked by the median total LRG luminosities over 200 trials per beam, $\sim70-75$% of the beams within the top $N$ beams, where $N \leq 200$, remain within the top $N$ beams. Only five of the top 200 beams fall out of the top 400 when re-ranked in this manner. Thus, most of the top 200 beams are robust to these measurement uncertainties.
As a result of the flux-limited nature of our sample (see Figure \[fig:magvol\]), the reported LRG luminosities in our beams are probably biased low given that we are not as sensitive to lower-luminosity LRGs at higher redshifts ($z \gtrsim 0.4$). Furthermore, galaxies at higher redshifts that could otherwise scatter into the LRG selection cuts are likely to have larger magnitude errors due to their faintness, which can exclude them from being classified as LRGs based on the $r$-band magnitude error cut (see Appendix \[app:selection\]). Thus, these biases potentially affect beams with high-redshift structures more than ones with the bulk of LRGs at lower redshift.
Potential Applications to Other Surveys {#subsec:future}
---------------------------------------
Our selection method can be applied not just to current surveys like the SDSS, but also to other ongoing and planned wide-area photometric and spectroscopic surveys. Ideally, we want a deeper survey than SDSS to probe a volume-limited sample of LRGs to higher redshift ($z \sim 1$) so as to include any cluster-scale lenses at $0.7 \lesssim z \lesssim 1$. However, any further observing time would be better utilized by expanding the survey area rather than going much deeper in redshift, as the number density of massive cluster-scale halos is decreasing at higher redshift [e.g., @tinker2008] and the lensing geometry for magnifying $z \geq 7$ galaxies is becoming unfavorable.
Surveys with goals that include detection of weak gravitational lensing, with their wide field of view, deep accurate multi-band photometry [particularly for cosmic magnification/convergence studies, e.g., @vanwaerbeke2010; @hildebrandt2011; @ford2012], and accurate photometric redshifts, are promising for our selection method. Current examples include the Kilo-Degree Survey [KiDS; @dejong2013] and the complementary VISTA Kilo-degree Infrared Galaxy Survey (VIKING), as well as the $3\pi$ survey with the Panoramic Survey Telescope and Rapid Response System [Pan-STARRS; @kaiser2004]. The ongoing Dark Energy Survey (DES) will provide deep coverage in the southern sky, complementary to the SDSS coverage. Planned wide-field instruments and observatories, such as the Hyper Suprime-Cam on Subaru and the Large Synoptic Survey Telescope (LSST), will probe large areas of the sky to unprecedented depths and thus produce excellent datasets for our method. Space-based missions such as Euclid [@laureijs2011] and the Wide-Field Infrared Survey Telescope (WFIRST) will generate deep high-resolution photometry over a large fraction of the sky.
Spectroscopic surveys have the advantage of more robust LRG selection and more accurate redshifts (and therefore luminosities). A spectroscopic survey with sufficient resolution can also provide information on the physical clustering of the LRGs, which can lead to higher-order estimates of the underlying matter distribution. Because LRGs are ideal targets for studies of the baryon acoustic oscillation (BAO) feature [e.g., @eisenstein2005], the goals of BAO surveys match up well with searches for massive beams using our selection method. Spectra from the ongoing Baryon Oscillation Spectroscopic Survey [BOSS; @dawson2013], a fraction of which are included in the DR9 sample used here, will provide deep spectroscopic data across large areas of the sky and specifically target LRGs. The upcoming Big Baryon Oscillation Spectroscopic Survey (BigBOSS) will also achieve these goals and probe to deeper redshifts.
CONCLUSIONS {#sec:conclusions}
===========
We present a new method of selecting lines of sight (“beams") that contain large total masses and are likely to be the most powerful gravitational lenses for magnifying very high-redshift galaxies. These fields are important for studying the first galaxies, as their extreme lensing strengths can magnify the background source population into detectability. We select fields based on the total luminosity in luminous red galaxies [LRGs; e.g., @eisenstein2001] along the line of sight.
We identify the 200 lines of sight in the SDSS DR9 that have the highest total LRG luminosities projected within a radius of $3\farcm5$ and within $0.1 \leq z \leq 0.7$, a key redshift range for lensing high-redshift galaxies. The total luminosities of LRGs in these fields ($\sim 10^{11.85} - 10^{12.1} h^{-2} L_{\odot}$), which can include $4-18$ total LRGs, are $2-3$ times larger than those of most known strong lensing galaxy clusters from the @hennawi2008 and CLASH [@postman2012] surveys, suggesting that they contain larger total masses. In those beams with multiple LRG peaks in redshift space, those peaks can be individually as rich as known lensing clusters, which may only be traced by a few LRGs.
The distribution of LRGs in these fields, both along the line-of-sight and in projection on the sky, show a diversity of structure. The LRGs trace both the redshift and angular concentrations of non-LRG galaxies. Some beams are dominated by a single mass peak, while others contain multiple mass peaks distributed along the line of sight, which can maximize the source plane area that is highly magnified for the purpose of detecting high-redshift galaxies [@wong2012]. Visual inspection of the fields reveals many beams with obvious galaxy clusters, confirming that the LRGs do trace dense structures. 22.5% of the top 200 beams contain one known cluster and 56.0% contain multiple known clusters previously identified in the literature. The rest of the beams may contain new groups and clusters.
Our analysis of the uncertainties in the integrated LRG luminosities of these beams shows that while the detailed rankings are susceptible to fluctuations, the top 200 beams generally comprise fields that have large concentrations of massive galaxies. $70-75\%$ of the beams remain in the top 200 when an extended sample of the top 1000 beams is sorted by the median LRG luminosity derived by our Monte Carlo error analysis, which accounts for uncertainties in the individual LRG luminosities and for galaxies falling into or out of our selection criteria.
Our follow-up galaxy spectroscopy in a subset of these beams has revealed multiple massive halos and large total masses ($\gtrsim 2\times10^{15} h^{-1} M_{\odot}$), confirming the power of using LRGs as tracers of massive structure (S. M. Ammons et al. 2013, in preparation). We are modeling the mass distributions in these beams from the spectroscopy alone, but will eventually combine that analysis with a strong lensing analysis of the arcs detected in these fields. The large total masses and multiple projected massive clusters make these beams likely to be among the best gravitational lenses known. Future science applications include the detection of faint lensed $z \geq 7$ galaxies that can be followed-up spectroscopically, high spatial resolution studies of strongly lensed galaxies at $z \sim 1-3$, weak lensing studies, and improved detections of $\gamma-$ray sources and supernovae at cosmological distances.
We thank Decker French for her contributions to this project. We are particularly indebted to Daniel Eisenstein and Nikhil Padmanabhan for useful discussions regarding the LRG selection criteria. We also thank Leon Baruah, Michael Blanton, Marcello Cacciato, Shirley Ho, Daniel Marrone, Jeremiah Ostriker, Ashley Ross, Jeremy Tinker, and Adi Zitrin for helpful discussions and input. This work was supported by NSF grants AST-0908280 and AST-1211385 and NASA grants ADAP-NNX10AD476 and ADAP-NNX10AE88G. This work performed in part under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. A.I.Z. thanks the Max Planck Institute for Astronomy and the Center for Cosmology and Particle Physics at New York University for their hospitality and support during her stays there. 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.
SORTING BY NUMBER OF LRGS VS TOTAL LRG LUMINOSITY {#app:nsort}
=================================================
In selecting beams likely to contain large total masses as traced by LRGs, we choose to use total LRG luminosity rather than simple number counts, despite the fact that there are additional uncertainties associated with this method. This choice arises from the expectation that LRG luminosity traces stellar mass, which is related to halo mass. While there have not been quantitative studies of the relationship between halo mass and total LRG luminosity at the halo masses we are interested in, we run a simple test here to see whether total LRG luminosity or number counts in galaxy clusters provides a better contrast relative to the field population, i.e., a stronger measure of overdensity. We can then invoke the known relationship between halo mass and LRG number counts [@ho2009] to suggest that either total LRG luminosity or number counts is a better tracer of halo mass. We use early-type galaxy luminosity functions to compare the expected galaxy luminosity and number counts in beams containing a massive cluster and field galaxies to beams with field galaxies alone.
Adopting the $i$-band luminosity function (LF) of @bernardi2003 to represent the field population of LRGs, we calculate the number of galaxies within $0.1 \leq z \leq 0.7$ and brighter than $M_{i} - 5\mathrm{log}_{10}(h) \leq -21$ expected in a circular field of view of radius $3\farcm5$. We then take the cluster galaxy $i$-band luminosity function of @popesso2005 to represent the population of LRGs in clusters. We use the @popesso2005 LF constraints for a cluster-centric radius $r \leq 2.0~h^{-1}$ Mpc with local background subtraction, although our results are qualitatively similar using their results for different cluster-centric radii. We combine the @popesso2005 bright-end and faint-end components into a single LF, although in practice, the faint-end component contributes very little at the luminosities we are interested in. Both the field and cluster samples are based on $z < 0.3$ LFs from SDSS data, so evolution in the LF is not taken into account, but both the field and cluster galaxy LFs will have smaller normalizations at higher redshifts.
We integrate the cluster LF down to the chosen limiting magnitude to determine a normalization factor, then normalize the cluster LF for a range of cluster richness, defined as the number of LRGs in the cluster. Since the @popesso2005 analysis ignored the contribution of the brightest cluster galaxy (BCG) when computing the LF, we add its contribution separately. We identify the luminosity at which the integrated LF is equal to one galaxy and double the differential number counts in the LF bins brighter than that luminosity. This effectively modifies the integral of the LF such that there is one additional galaxy in the cluster with the same average luminosity as its brightest (non-BCG) member. This is conservative in the sense that we will underestimate the luminosity of the BCG since there is a known magnitude gap between the two brightest galaxies in clusters [e.g., @more2012].
For a range of cluster richness, we compare number counts and total luminosity for the field plus cluster to those for the field alone. We define the number of LRGs in the field to be $$\label{eq:nfield}
N_{field} = \left( \frac{\pi R^{2}}{4\pi~\mathrm{ster}} \right) \\
\int_{z_{min}}^{z_{max}} (c/H(z)) (1+z)^{2} D_{A}(z)^2 dz \\
\int_{L_{min}}^{\infty} \phi_{field}(L) dL,$$ where $R$ is the beam radius, $c/H(z)$ is the Hubble distance at redshift $z$, $D_{A}$ is angular diameter distance, and $\phi_{field}(L)$ is the luminosity function for the field. The total luminosity for galaxies in the field is then given by $$\label{eq:lfield}
L_{field} = \left( \frac{\pi R^{2}}{4\pi~\mathrm{ster}} \right) \\
\int_{z_{min}}^{z_{max}} (c/H(z)) (1+z)^{2} D_{A}(z)^2 dz \\
\int_{L_{min}}^{\infty} \phi_{field}(L) L dL.$$ For the cluster galaxies, we define $N_{cluster}$ to be the number of non-BCG galaxies in the cluster. The total cluster richness is therefore $N_{cluster} + 1$, and the total luminosity of the cluster galaxies as a function of richness is $$\label{eq:lcluster}
L_{cluster}(N_{cluster} + 1) = \\
\frac{N_{cluster} \int_{L_{min}}^{\infty} (1 + \mathcal{H}(L - L_{BCG})) \phi_{cluster}(L) L dL}{\\
\int_{L_{min}}^{\infty} \phi_{cluster}(L) dL},$$ where $\phi_{cluster}(L)$ is the luminosity function for the cluster with arbitrary normalization. The factor of $N_{cluster} / \int_{L_{min}}^{\infty} \phi_{cluster}(L) dL$ serves to normalize the total luminosity for the given richness. The $1 + \mathcal{H}(L - L_{BCG})$ term, where $\mathcal{H}$ is the Heaviside step function, accounts for the luminosity of the BCG. $L_{BCG}$ is determined by solving the equation $$\label{eq:lbcg}
\int_{L_{BCG}}^{\infty} \phi_{cluster}(L) dL = \\
\frac{\int_{L_{min}}^{\infty} \phi_{cluster}(L) dL}{N_{cluster}}$$ for $L_{BCG}$. Note that the arbitrary normalization of the cluster LF cancels on both sides of Equation \[eq:lbcg\].
We assume $R = 3\farcm5$, $z_{min} = 0.1$, $z_{max} = 0.7$, and test a range of luminosity cuts from $-22.5 \leq M_{min} -5\mathrm{log}_{10}(h) \leq -21$, where $M_{min} = -2.5\mathrm{log}_{10}(L_{min}/L_{\odot})$. We then define the “richness contrast" to be $$\label{eq:ncontrast}
C_{N}(N_{cluster}+1) = \frac{N_{field} + N_{cluster}+1}{N_{field}},$$ and the “luminosity contrast" to be $$\label{eq:lcontrast}
C_{L}(N_{cluster}+1) = \frac{L_{field}+L_{cluster}(N_{cluster}+1)}{L_{field}}.$$
These contrasts represent the relative LRG number and luminosity, respectively, of a beam containing a cluster and field galaxies to a beam containing field galaxies alone. In general, while both richness and total luminosity are strongly correlated, the luminosity contrast is stronger than the number contrast, increasing with greater cluster richness. As an example, for a richness of $N_{cluster} + 1$ = 12 and $L_{min}$ corresponding to a limiting magnitude of $M_{i} - 5\mathrm{log}_{10}(h) = -21$, $C_{L}$ is 16% greater than $C_{N}$ . @reid2009 find the LRG occupation number for $\gtrsim 10^{15} M_{\odot}$ halos to be $\sim 3-5$ LRGs, although their LRG selection was limited to more luminous LRGs than our sample. We test a range of limiting absolute magnitudes, $-21 \leq M_{i} - 5\mathrm{log}_{10}(h) \leq -22.5$, and find that the contrast for both methods becomes much greater for brighter limiting magnitudes.
For this test, we attempt to be conservative where possible. The magnitude gap for BCGs, which we ignore, would favor the luminosity method more by increasing the cluster luminosity. The field early-type LF is based on morphological+spectral PCA criteria, and almost certainly includes more galaxies than would pass our LRG selection criteria. This effect serves to reduce both contrasts, but given our finding that the luminosity contrast is stronger than the richness contrast, the luminosity contrast will be more strongly affected.
In Table \[tab:nbeams\], we list the $3\farcm5$-radius beams containing $\geq 11$ LRGs that do not overlap with the top 200 luminosity-sorted beams in Table \[tab:beams\]. We present these beams for completeness, as these beams would have been in the top 200 (or had an equal number of LRGs to beams in our top 200) if we had chosen to sort by LRG number counts instead.
SDSS LRG SELECTION CRITERIA {#app:selection}
===========================
In this Appendix, we explain our LRG catalog selection criteria. We base our selection of SDSS DR9 LRGs on the criteria originally defined in @padmanabhan2005, with some minor modifications. The LRG selection uses two separate selection criteria (denoted “Cut I" and “Cut II" for the $z \lesssim 0.4$ and $z \gtrsim 0.4$ samples, respectively) to define the full sample. The SQL queries for the two cuts are given in Figures \[fig:sql1\] and \[fig:sql2\]. The two cuts are not mutually exclusive, so we remove duplicate objects after combining the two samples into a single catalog.
SELECT
g.objID,
g.run,
g.rerun,
g.camcol,
g.field,
g.fieldID,
g.obj,
g.ra,
g.dec,
g.b,
g.dered_g,
g.dered_r,
g.dered_i,
g.extinction_r,
g.petroMag_r,
g.psfMag_r,
g.ModelMag_r,
g.ModelMagErr_r,
g.psfMag_i,
g.ModelMag_i,
g.ModelMagErr_i,
g.deVRad_r,
g.petroR50_r,
g.flags,
p.z,
p.zErr,
f.quality,
f.psfWidth_r
FROM
Galaxy as g, Photoz as p, Field as f
WHERE p.objID = g.objID
and f.fieldID = g.fieldID
and f.quality = 3
and f.psfWidth_r < 2
and (g.dered_g - g.dered_r) < 3.0
and (g.dered_r - g.dered_i) < 1.5
and g.extinction_r < 0.2
and g.modelmagerr_r < 0.2
and (g.petroMag_r + (2.5 * LOG10(2 * pi() * POWER(g.petroR50_r,2.0)))) < 24.2
and ABS(g.b) > 30
and ABS((g.dered_r - g.dered_i) - ((g.dered_g - g.dered_r) / 4.0) - 0.18) < 0.2
and (g.petroMag_r - g.extinction_r) <
(13.6 + ((0.7 * (g.dered_g - g.dered_r)) + (1.2 * (g.dered_r - g.dered_i - 0.18))) / 0.3)
and (g.petroMag_r - g.extinction_r) < 19.7
and (g.psfMag_r - g.ModelMag_r) > 0.3
and (g.flags & 0x0000000000000002) = 0
and (g.flags & 0x0000000000000004) = 0
and (g.flags & 0x0000000000000008) = 0
and (g.flags & 0x0000000000040000) = 0
and (g.flags & 0x0000000000100000) = 0
and (g.flags & 0x0000000002000000) = 0
and (g.flags & 0x0000000004000000) = 0
and (g.flags & 0x0000000070000000) != 0
\[fig:sqlcut1\]
SELECT
g.objID,
g.run,
g.rerun,
g.camcol,
g.field,
g.fieldID,
g.obj,
g.ra,
g.dec,
g.b,
g.dered_g,
g.dered_r,
g.dered_i,
g.extinction_r,
g.petroMag_r,
g.psfMag_r,
g.ModelMag_r,
g.ModelMagErr_r,
g.psfMag_i,
g.ModelMag_i,
g.ModelMagErr_i,
g.deVRad_r,
g.petroR50_r,
g.flags,
p.z,
p.zErr,
f.quality,
f.psfWidth_r
FROM
Galaxy as g, Photoz as p, Field as f
WHERE P.objID = g.objID
and f.fieldID = g.fieldID
and f.quality = 3
and f.psfWidth_r < 2
and (g.dered_g - g.dered_r) < 3.0
and (g.dered_r - g.dered_i) < 1.5
and g.extinction_r < 0.2
and g.modelmagerr_r < 0.2
and (g.petroMag_r + (2.5 * LOG10(2 * pi() * POWER(g.petroR50_r,2.0)))) < 24.2
and ABS(g.b) > 30
and ((g.dered_r - g.dered_i) - ((g.dered_g - g.dered_r) / 8)) > 0.55
and (g.dered_g - g.dered_r) > 1.4
and g.dered_i < (18.3 + (2 * ((g.dered_r - g.dered_i) - (g.dered_g - g.dered_r) / 8)))
and g.dered_i < 20
and (g.psfMag_i - g.ModelMag_i) > 0.2 * (21 - g.dered_i)
and g.deVRad_r > 0.2
and (g.flags & 0x0000000000000002) = 0
and (g.flags & 0x0000000000000004) = 0
and (g.flags & 0x0000000000000008) = 0
and (g.flags & 0x0000000000040000) = 0
and (g.flags & 0x0000000000100000) = 0
and (g.flags & 0x0000000002000000) = 0
and (g.flags & 0x0000000004000000) = 0
and (g.flags & 0x0000000070000000) != 0
\[fig:sqlcut2\]
We modify the original LRG selection criteria, which selected LRGs for clustering studies that had different requirements than our applications. We add cuts to exclude fields with poor quality or that have an effective $r$-band PSF width $\geq 2$, indicating poor seeing. We remove objects with $r$-band extinction $\geq 0.2$ or $r$-band magnitude errors $\geq 0.2$, as we are less confident in these objects’ absolute magnitudes (N. Padmanabhan, private communication). We reduce the galactic latitude criterion from $b \leq 45^{\circ}$ to $b \leq 30^{\circ}$, which strikes a balance between excluding fields with a high-density of stars that could complicate follow-up observations and including as much survey area as possible. This cut eliminates less than 20% of the total sky coverage of DR9. The rest of our color and photometric flag cuts remove most spurious stellar contaminants. The photometry flags are standard quality cuts to remove spurious objects (A. Ross, private communication). Based on a visual inspection of subsets of objects in our sample, we also cut objects with the “subtracted" or “deblended\_as\_psf" flags set, which removes $< 1$% of our sample but cuts out objects contaminated by a nearby bright star. We also cut objects with the “deblend\_pruned" flag set, which removes a tiny fraction ($< 100$ objects) of our sample with apparent blending issues.
We use similar criteria to get the spectroscopic sample of LRGs, requiring in addition that the “zwarning" flag is zero and the redshift error is $< 10^{-3}$. We then match the spectroscopic and photometric samples, replacing the photometric redshift and its error with the spectroscopic redshift and error where applicable. We apply cuts on the object redshift only after combining the catalogs, as an object with a photometric redshift outside of our target redshift range ($0.1 \leq z \leq 0.7$) may have a spectroscopic redshift within that range.
We then calculate absolute magnitudes for the LRGs, accounting for K-corrections and passive evolution, and apply our absolute $i$-band magnitude cut. We extend the original @padmanabhan2005 absolute magnitude cut ($-21 < M_{i} - 5\mathrm{log}_{10}(h) < -24$) to include more luminous galaxies up to $-24.7$ based on visual inspection. While extending this cut does introduce contaminants, it also includes the most luminous LRGs, which contribute the most to the total luminosity of a beam. We check through visual inspection that these contaminants do not affect our top 200 beams.
KNOWN CLUSTERS IN TOP 200 BEAMS {#app:clusters}
===============================
We present the list of known clusters in our top 200 beams in Table \[tab:clusters\]. These clusters were identified in a search of the NASA/IPAC Extragalactic Database[^3] (NED) as galaxy clusters within $3\farcm5$ of our beam centers[^4]. The clusters in each beam are ordered by angular offset from the beam center.
[^1]: http://skyserver.sdss3.org/dr9/
[^2]: See www.sdss3.org/dr9/imaging/other\_info.php\#stargalaxy
[^3]: http://ned.ipac.caltech.edu
[^4]: Z. Wen reports in a private communication that most of our LRG-selected fields contain at least one cluster identified in his larger @wen2012 sample, which was not available through NED at the time of our analysis.
|
---
abstract: 'In this paper, we construct odd unimodular lattices in dimensions $n=36,37$ having minimum norm $3$ and $4 s=n-16$, where $s$ is the minimum norm of the shadow. We also construct odd unimodular lattices in dimensions $n=41,43,44$ having minimum norm $4$ and $4 s=n-24$.'
author:
- 'Masaaki Harada[^1]'
title: Construction of Some Unimodular Lattices with Long Shadows
---
[**Dedicated to Professor Yasumasa Nishiura on His 60th Birthday**]{}
Introduction
============
Shadows of odd unimodular lattices appeared in [@CS98] (see also [@SPLAG p. 440]), and shadows play an important role in the study of odd unimodular lattices. Let $L$ be an odd unimodular lattice in dimension $n$. The [*shadow*]{} $S(L)$ of $L$ is defined to be $S(L)= L_0^* \setminus L$, where $L_0$ denotes the even sublattice of $L$ and $L_0^*$ denotes the dual lattice of $L_0$. Note that the norm of a vector of $S(L)$ is congruent to $n/4$ modulo $2$ [@CS98].
We define $$\sigma(L)=4 \min(S(L)),$$ where $\min(S(L))$ denotes the minimum norm of $S(L)$. Elkies [@E95a] began the investigation of odd unimodular lattices $L$ with long shadows, that is, large $\sigma(L)$. It was shown in [@E95a] that $\sigma(L) \le n$ and ${\mathbb{Z}}^n$ is the only odd unimodular lattice $L$ with $\sigma(L)=n$, up to isomorphism. For the case $\sigma(L)<n$, we may assume that $L$ has minimum norm at least $2$ (see [@E95b; @NV03]). Elkies [@E95b] determined all such lattices for the case $\sigma(L)=n-8$. For the next case $\sigma(L)=n-16$, Nebe and Venkov [@NV03] showed that if there is an odd unimodular lattice $L$ with minimum norm $\min(L) \ge 3$ then $n \le 46$. Moreover, in this case, it is not known whether there is such a lattice with minimum norm $3$ or not for only $n=36,37,38,39,41,42,43$ (see [@Ga07 p. 148] and Table \[Tab:1\]). It follows from [@NV03 Table 1] that there is no odd unimodular lattice $L$ with $\min(L) \ge 4$ for the case $\sigma(L)=n-16$ (see also [@Gau07 Table 2]).
For the case $\sigma(L)=n-24$, Gaborit [@Ga07] showed that if there is an odd unimodular lattice $L$ with $\min(L) = 4$ then $n \le 47$ (see also [@Gau07 Table 2]). Moreover, in this case, it is not known whether there is such a lattice or not for only $n=37,41,43,44,45$ (see [@Ga07 p. 148] and Table \[Tab:2\]).
The aim of this paper is to provide the existence of some unimodular lattices with long shadows whose existences were previously not known. These lattices are constructed from self-dual ${\mathbb{Z}}_k$-codes ($k=4,5$). The paper is organized as follows. In Section \[Sec:2\], we give definitions and some basic results on unimodular lattices and self-dual ${\mathbb{Z}}_k$-codes. In Section \[Sec:36\], we give conditions to construct odd unimodular lattices $L$ in dimension $36$ with $\min(L) =3$ and $\sigma(L)=20$ using self-dual ${\mathbb{Z}}_4$-codes by Construction A (Proposition \[prop:36con\]). By finding a self-dual ${\mathbb{Z}}_4$-code satisfying these conditions, the first example of such a lattice is constructed. This dimension is the smallest dimension of an odd unimodular lattice $L$ with $\min(L) =3$ and $\sigma(L)=n-16$ whose existence was previously not known. A new odd unimodular lattice $L$ in dimension $36$ with $\min(L)=4$ and $\sigma(L)=12$ is constructed as a neighbor of the above lattice. In Section \[Sec:Ot\], by considering self-dual ${\mathbb{Z}}_k$-codes ($k=4,5$), we construct an odd unimodular lattice $L$ in dimension $n$ with $$\begin{aligned}
(n,\min(L),\sigma(L))=&
(37,3,n-16),
(41,4,n-24),
(43,4,n-24), \text{ and}
\\ &
(44,4,n-24).\end{aligned}$$ The current state of knowledge about the existence of unimodular lattices $L$ in dimension $n$ with $(\min(L),\sigma(L))=
(3,n-16)$ and $(4,n-24)$ is listed in Tables \[Tab:1\] and \[Tab:2\], respectively. All computer calculations in this paper were done by [Magma]{} [@Magma].
Preliminaries {#Sec:2}
=============
Unimodular lattices
-------------------
A (Euclidean) lattice $L \subset {\mathbb{R}}^n$ in dimension $n$ is [*unimodular*]{} if $L = L^{*}$, where the dual lattice $L^{*}$ of $L$ is defined as $\{ x \in {{\mathbb{R}}}^n \mid (x,y) \in {\mathbb{Z}}\text{ for all }
y \in L\}$ under the standard inner product $(x,y)$. A unimodular lattice $L$ is [*even*]{} if the norm $(x,x)$ of every vector $x$ of $L$ is even and [*odd*]{} otherwise. An even unimodular lattice in dimension $n$ exists if and only if $n \equiv 0 \pmod 8$, while an odd unimodular lattice exists for every dimension. The minimum norm $\min(L)$ of $L$ is the smallest norm among all nonzero vectors of $L$. Two lattices $L$ and $L'$ are [*isomorphic*]{}, if there exists an orthogonal matrix $A$ with $L' = L \cdot A =\{xA \mid x \in L\}$. The automorphism group of $L$ is the group of all orthogonal matrices $A$ with $L = L \cdot A$.
The theta series $\theta_{L}(q)$ of $L$ is the formal power series $
\theta_{L}(q) = \sum_{x \in L} q^{(x,x)}.
$ The kissing number is the second nonzero coefficient of the theta series, that is, the number of vectors of minimum norm in $L$. Conway and Sloane [@CS98] showed that when the theta series of an odd unimodular lattice $L$ in dimension $n$ is written as $$\label{eq:theta}
\theta_L(q)=
\sum_{j =0}^{\lfloor n/8\rfloor} a_j\theta_3(q)^{n-8j}\Delta_8(q)^j,$$ the theta series of the shadow $S(L)$ is written as $$\label{eq:theta-S}
\theta_{S(L)}(q)= \sum_{j=0}^{\lfloor n/8\rfloor}
\frac{(-1)^j}{16^j} a_j\theta_2(q)^{n-8j}\theta_4(q^2)^{8j},
$$ where $\Delta_8(q) = q \prod_{m=1}^{\infty} (1 - q^{2m-1})^8(1-q^{4m})^8$ and $\theta_2(q), \theta_3(q)$ and $\theta_4(q)$ are the Jacobi theta series [@SPLAG].
Self-dual ${\mathbb{Z}}_k$-codes and Construction A
---------------------------------------------------
Let ${\mathbb{Z}}_{k}$ be the ring of integers modulo $k$, where $k$ is a positive integer greater than $1$. A [*${\mathbb{Z}}_{k}$-code*]{} $C$ of length $n$ is a ${\mathbb{Z}}_{k}$-submodule of ${\mathbb{Z}}_{k}^n$. We shall exclusively deal with the case $k=4$. Two ${\mathbb{Z}}_k$-codes are [*equivalent*]{} if one can be obtained from the other by permuting the coordinates and (if necessary) changing the signs of certain coordinates. The [*dual*]{} code $C^\perp$ of $C$ is defined as $C^\perp = \{ x \in {\mathbb{Z}}_{k}^n\ | \ x \cdot y = 0$ for all $y \in C\}$, where $x \cdot y = x_1 y_1 + \cdots + x_n y_n$ for $x=(x_1,\ldots,x_n)$ and $y=(y_1,\ldots,y_n)$. A code $C$ is [*self-dual*]{} if $C=C^\perp.$ Let $C$ be a self-dual ${\mathbb{Z}}_k$-code of length $n$. Then the following lattice $$A_{k}(C) = \frac{1}{\sqrt{k}}
\{(x_1,\ldots,x_n) \in {\mathbb{Z}}^n \mid
(x_1 \bmod k,\ldots,x_n \bmod k)\in C\}$$ is a unimodular lattice in dimension $n$. This construction of lattices is called Construction A.
Self-dual ${\mathbb{Z}}_4$-codes {#Subsec:Z4}
--------------------------------
Let $C$ denote a ${\mathbb{Z}}_4$-code of length $n$. The [*Euclidean weight*]{} of a codeword $x=(x_1,\ldots,x_n)$ of $C$ is $n_1(x)+4n_2(x)+n_3(x)$, where $n_{\alpha}(x)$ denotes the number of components $i$ with $x_i=\alpha$ $(\alpha=1,2,3)$. The [*minimum Euclidean weight*]{} $d_E(C)$ of $C$ is the smallest Euclidean weight among all nonzero codewords of $C$. Every ${\mathbb{Z}}_4$-code $C$ of length $n$ has two binary codes $C^{(1)}$ and $C^{(2)}$ associated with $C$: $$C^{(1)}= \{ c \bmod 2 \mid c \in C \} \text{ and }
C^{(2)}= \left\{ c \bmod 2 \mid c \in {\mathbb{Z}}_4^n, 2c\in C \right\}.$$ The binary codes $C^{(1)}$ and $C^{(2)}$ are called the [*residue*]{} and [*torsion*]{} codes of $C$, respectively. If $C$ is self-dual, then $ C^{(1)}$ is a binary doubly even code with $C^{(2)} = {C^{(1)}}^{\perp}$ [@Z4-CS]. It is easy to see that $\min\{d(C^{(1)}),4d(C^{(2)}))\} \le d_E(C)$, where $d(C^{(i)})$ denotes the minimum weight of $C^{(i)}$ $(i=1,2)$. In addition, $d_E(C) \le 4d(C^{(2)})$ (see [@H10]). Also, $A_4(C)$ has minimum norm $\min\{4,d_E(C)/4\}$ (see [@Z4-BSBM]). Therefore, if $A_4(C)$ has minimum norm $3$ (resp. $4$), then $C$ must have minimum Euclidean weight $12$ (resp. at least $16$) and $C^{(2)}$ has minimum weight at least $3$ (resp. at least $4$).
Two ${\mathbb{Z}}_4$-codes differing by only a permutation of coordinates are called permutation-equivalent. Any self-dual ${\mathbb{Z}}_4$-code $C$ of length $n$ with residue code of dimension $k_1$ is permutation-equivalent to a code $C'$ with generator matrix in standard form $$\label{eq:g-matrix}
\left(\begin{array}{ccc}
I_{k_1} & A & B_1+2B_2 \\
O &2I_{n-2k_1} & 2D
\end{array}\right),$$ where $A$, $B_1$, $B_2$, $D$ are $(1,0)$-matrices, $I_k$ denotes the identity matrix of order $k$, and $O$ denotes the zero matrix [@Z4-CS]. In this paper, when we give a generator matrix of $C$, we consider a generator matrix in standard form (\[eq:g-matrix\]) of $C'$, which is permutation-equivalent to $C$, then we only list the $k_1 \times (n-k_1)$ matrix $
\left(\begin{array}{cc}
A & B_1+2B_2
\end{array}\right)
$ to save space. Note that $\left(\begin{array}{ccc}
O &2I_{n-2k_1} & 2D
\end{array}\right)$ in (\[eq:g-matrix\]) can be obtained from $\left(\begin{array}{ccc}
I_{k_1} & A & B_1+2B_2
\end{array}\right)$ since $C'^{(2)} = {C'^{(1)}}^{\perp}$.
Dimension $n=36$ and $\sigma(L)=n-16, n-24$ {#Sec:36}
===========================================
In this section, we give conditions to construct odd unimodular lattices $L$ in dimension $n=36$ with $\min(L) =3$ and $\sigma(L)=n-16$ from self-dual ${\mathbb{Z}}_4$-codes by Construction A. By finding a self-dual ${\mathbb{Z}}_4$-code satisfying these conditions, the first example of such a lattice is constructed. A new optimal odd unimodular lattice $L$ with $\sigma(L)=n-24$ is also constructed from the above lattice.
Let $L_{36}$ be an odd unimodular lattice in dimension $36$ having minimum norm at least $3$. Using (\[eq:theta\]) and (\[eq:theta-S\]), it is easy to determine the possible theta series $\theta_{L_{36}}(q)$ and $\theta_{S(L_{36})}(q)$ of $L_{36}$ and its shadow $S(L_{36})$: $$\begin{aligned}
\theta_{L_{36}}(q) =&
1
+ (960 - \alpha)q^3
+ (42840 + 4096 \beta)q^4
\\ & \hspace{4cm}
+ (1882368 + 36 \alpha - 98304 \beta)q^5
+ \cdots,
\\
\theta_{S(L_{36})}(q) =&
\beta q + (\alpha - 60 \beta) q^3
+ (3833856 - 36 \alpha + 1734 \beta)q^5
+ \cdots,\end{aligned}$$ respectively, where $\alpha$ and $\beta$ are nonnegative integers. Then, the following lemma is immediate.
\[lem:36\] Let $L$ be an odd unimodular lattice in dimension $36$ having minimum norm $3$. Then, the kissing number of $L$ is at most $960$, and the equality holds if and only if $\sigma(L)=20$.
Now, we give a method for construction of unimodular lattices $A_4(C)$ with $\min(A_4(C))=3$ and $\sigma(A_4(C))=20$, using self-dual ${\mathbb{Z}}_4$-codes $C$. It is known that a binary $[36,k,3]$ code exists only if $k \le 30$ (see [@Brouwer-Handbook]). Hence, the dimension of the residue code of a self-dual ${\mathbb{Z}}_4$-code of length $36$ and minimum Euclidean weight $12$ is at least $6$.
\[prop:36con\] Let $C$ be a self-dual ${\mathbb{Z}}_4$-code of length $36$ such that $C^{(1)}$ is a binary doubly even $[36,6,16]$ code and $C^{(2)}$ has minimum weight $3$. Then $A_4(C)$ is a unimodular lattice with $\min(A_4(C))=3$ and $\sigma(A_4(C))=20$.
Let $B$ be a binary doubly even $[36,6,16]$ code such that $B^\perp$ has minimum weight $3$. The weight enumerator of $B$ is written as: $$W_{B}(y)=
1
+a y^{16}
+b y^{20}
+c y^{24}
+d y^{28}
+e y^{32}
+(2^6-1-a-b-c-d-e) y^{36},$$ where $a,b,c,d$ and $e$ are nonnegative integers. Since $B^\perp$ contains a codeword of weight $3$, $2^6-1-a-b-c-d-e=0$. By the MacWilliams identity, the weight enumerator of $B^\perp$ is given by: $$\begin{aligned}
W_{B^\perp}(y)=
&1
+ \frac{1}{8}
(- 216 + 4a + 3b + 2c + d ) y
+ (378 -6a - 6b - 5c - 3d ) y^2
\\&
+ \frac{1}{8}
(- 24024 + 388a +403b + 386c+ 273d ) y^3 +
\cdots.
$$ Since $B^\perp$ has minimum weight $3$, we have $$c= 270 - 6 a - 3 b \text{ and }
d=-324 + 8 a + 3 b.$$ Thus, the weight enumerators of $B$ and $B^\perp$ are written using $a$ and $b$: $$\begin{aligned}
W_{B}(y)=&
1
+ a y^{16}
+ b y^{20}
+ (270 -6a - 3b)y^{24}
+ (-324 + 8a + 3b) y^{28}
\\ &
+ (117 -3a - b) y^{32},
\\
W_{B^\perp}(y)=&
1 + (- 1032 + 32a + 8b ) y^3
+ ( 17649 -448 a - 128 b) y^4 + \cdots,\end{aligned}$$ respectively. Then, only $(a,b)=(27,36)$ satisfies the condition that the above coefficients in $W_{B}(y)$ and $W_{B^\perp}(y)$ are nonnegative integers. Hence, the weight enumerators of $B$ and $B^\perp$ are uniquely determined as $$\begin{aligned}
\label{eq:WE36}
W_{B}(y)=&
1+ 27 y^{16} + 36 y^{20},
\\
\label{eq:WE36d}
W_{B^\perp}(y)=&
1 + 120 y^3 + 945y^4 + 5832 y^5 + 30576 y^6 + 130680 y^7 + \cdots,\end{aligned}$$ respectively. As described in Section \[Subsec:Z4\], we have $$\label{eq:dE}
\min\{d(C^{(1)}),4d(C^{(2)}))\} \le d_E(C) \le 4d(C^{(2)}).$$ Hence, $d_E(C)=12$ and $A_4(C)$ has minimum norm $3$. Let $e_i$ denote the $i$-th unit vector $(\delta_{i,1},\delta_{i,2},\ldots,\delta_{i,36})$, for $i=1,2,\ldots,36$, where $\delta_{ij}$ is the Kronecker delta. By (\[eq:WE36d\]), there are $120$ codewords of weight $3$ in $C^{(2)}$, and we denote the set of the $120$ codewords by $C^{(2)}_3$. Then, $A_4(C)$ contains the following set of vectors of norm $3$: $$\{\pm e_{j_1} \pm e_{j_2} \pm e_{j_3} \mid
\{j_1,j_2,j_3\} \in S\},$$ where $S=\{\operatorname{supp}(x) \mid x \in C^{(2)}_3\}$, and $\operatorname{supp}(x)$ denotes the support of $x$. Hence, there are at least $960$ vectors of norm $3$ in $A_4(C)$. By Lemma \[lem:36\], the result follows.
It was shown in [@PST] that there are four inequivalent binary $[36,7,16]$ codes containing the all-one vector ${\mathbf{1}}$. Such a code and its dual code have the following weight enumerators: $$\begin{aligned}
&1+ 63 y^{16}+ 63 y^{20}+ y^{36},
\\
&
1
+ 945 y^{4}
+ 30576 y^{6}
+ 471420 y^{8}
+ 3977568 y^{10}
+ \cdots,\end{aligned}$$ respectively. Let $B_{36,7,1}$ denote the binary $[36,7,16]$ code containing ${\mathbf{1}}$ with automorphism group of order $1451520$ which is the symplectic group $Sp(6,2)$. Using an approach used in [@HLM Section 4], we verified that $B_{36,7,1}$ contains only one doubly even $[36,6,16]$ subcode $B_{36,6}$ such that $B_{36,6}^\perp$ has minimum weight $3$ up to equivalence. The weight enumerators of $B_{36,6}$ and $B_{36,6}^\perp$ are given by (\[eq:WE36\]) and (\[eq:WE36d\]), respectively. We verified by [Magma]{} that $B_{36,6}$ has automorphism group of order $51840$ containing the symplectic group $Sp(4,3)$ as a subgroup of index $2$.
Starting from a given binary doubly even code $B$, a method for construction of all self-dual ${\mathbb{Z}}_4$-codes $C$ with $C^{(1)}=B$ was given in [@Z4-PLF Section 3]. Using this method, we construct a self-dual ${\mathbb{Z}}_4$-code $C_{36}$ with $C_{36}^{(1)}=B_{36,6}$ explicitly. By Proposition \[prop:36con\], $A_4(C_{36})$ is the desired unimodular lattice with $\sigma(A_4(C_{36}))=20$, and we have the following:
There is a unimodular lattice $L$ in dimension $36$ having minimum norm $3$ with $\sigma(L)=20$.
We verified by [Magma]{} that the unimodular lattice $A_4(C_{36})$ has automorphism group of order $1698693120$. For the code $C_{36}$, we give a generator matrix in standard form (\[eq:g-matrix\]), by only listing the $6 \times 30$ matrix: $$\left(\begin{array}{cc}
A & B_1+2B_2
\end{array}\right)
=
\left(\begin{array}{cc}
000010001011100111100010 &130113\\
011001000101000011101001 &231133\\
100010010111110001110111 &000230\\
110011100011001100111111 &032133\\
011110101100111001011001 &321113\\
111111111111110000000000 &222203
\end{array}\right).$$ There are three unimodular lattices containing the even sublattice of $A_4(C_{36})$. We denote the two unimodular lattices rather than $A_4(C_{36})$ by $N_{36}$ and $N'_{36}$. It follows from the theta series of $A_4(C_{36})$ and $S(A_4(C_{36}))$ that both $N_{36}$ and $N'_{36}$ have minimum norm $4$ and kissing number $42840$, thus, $\sigma(N_{36})=\sigma(N'_{36})=16$. We verified by [Magma]{} that the two lattices are isomorphic. We also verified by [Magma]{} that $N_{36}$ has automorphism group of order $849346560$, which is different to those of two previously known unimodular lattices with minimum norm $4$ and kissing number $42840$ in [@lattice-datebase].
Let $C_{36,0}$ be the subcode of $C_{36}$ consisting of codewords of Euclidean weight divisible by $8$. Then $C_{36,0}$ is a subcode of index $2$ in $C_{36}$ (see [@DHS01 Lemma 3.1]). By Proposition 3.8 in [@DHS01], there is a self-dual ${\mathbb{Z}}_4$-code $D_{36}$ containing $C_{36,0}$ with $A_4(D_{36}) = N_{36}$. For the code $D_{36}$, we give a generator matrix in standard form (\[eq:g-matrix\]), by only listing the $6 \times 30$ matrix: $$\left(\begin{array}{cc}
A & B_1+2B_2
\end{array}\right)
=
\left(\begin{array}{cc}
0001101111001000100110& 0131310\\
1101111000101001010010& 1233122\\
0111100011110010100011& 0121203\\
1110100110001110111011& 3133300\\
0101001101100001101101& 1203301\\
0000011111111111111110& 0022113\\
1111111111111100000000& 2220232
\end{array}\right).$$ We verified that the residue code $D_{36}^{(1)}$ is equivalent to $B_{36,7,1}$.
The remaining three binary $[36,7,16]$ codes containing ${\mathbf{1}}$ have automorphism groups of orders $10752$, $1920$ and $672$ [@PST]. We denote these codes by $B_{36,7,2}$, $B_{36,7,3}$ and $B_{36,7,4}$, respectively. We verified that $B_{36,7,2}$, $B_{36,7,3}$ and $B_{36,7,4}$ contain $1,2$ and $1$ doubly even $[36,6,16]$ subcodes such that the dual codes have minimum weight $3$, respectively, up to equivalence, and the four codes and $B_{36,6}$ are inequivalent to each other. By Proposition \[prop:36con\], the four inequivalent codes rather than $B_{36,6}$ also give examples of unimodular lattices $L$ with $\min(L)=3$ and $\sigma(L)=20$.
Other cases {#Sec:Ot}
===========
Dimension $n=37$ and $\sigma(L)=n-16$
-------------------------------------
Let $L_{37}$ be an odd unimodular lattice in dimension $n=37$ having minimum norm at least $3$. We give the possible theta series of $L_{37}$ and its shadow $S(L_{37})$: $$\begin{aligned}
\theta_{L_{37}}(q) =&
1
+ (1184 - \alpha)q^3
+ (37962 - 2 \alpha + 2048 \beta) q^4
\\ & \hspace{4cm}
+ (1758240 + 36 \alpha - 45056 \beta)q^5
+ \cdots,
\\
\theta_{S(L_{37})}(q) =&
\beta q^{5/4}
+ (2 \alpha - 59 \beta)q^{13/4}
+ (8486912 - 70 \alpha + 1674 \beta)q^{21/4}
+ \cdots,\end{aligned}$$ respectively, where $\alpha$ and $\beta$ are nonnegative integers. It turns out that $L_{37}$ has minimum norm $3$ and kissing number $1184$ if and only if $\sigma(L_{37})=21$.
It is known that a binary $[37,k,3]$ code exists only if $k \le 31$ (see [@Brouwer-Handbook]). Hence, the dimension of the residue code of a self-dual ${\mathbb{Z}}_4$-code of length $37$ and minimum Euclidean weight $12$ is at least $6$. When $n=37$, we have a weaker result than Proposition \[prop:36con\].
Let $B$ be a binary doubly even $[37,6]$ code such that the dual code has minimum weight $3$. Then the weight enumerator $W_B(y)$ of $B$ is given by: $$\begin{aligned}
\label{eq:WE371}
W_B(y) &= 1 + 20 y^{16} + 42 y^{20} + y^{24} \text{ or}, \\
\label{eq:WE372}
W_B(y) &= 1+ y^{12} + 17 y^{16} + 45 y^{20}. \end{aligned}$$
The proof is similar to that of Proposition \[prop:36con\]. Let $B$ be a binary doubly even $[37,6]$ code such that $B^\perp$ has minimum weight $3$. The weight enumerator $W_B(y)$ is written as: $$\begin{gathered}
W_{B}(y)=
1
+ a y^4
+ b y^8
+ c y^{12}
+ d y^{16}
+ e y^{20}
+ f y^{24}
+ g y^{28}
\\
+ h y^{32}
+ (2^6-1-a - b - c - d - e - f - g - h) y^{36},\end{gathered}$$ where $a,b,c,d,e,f,g$ and $h$ are nonnegative integers. Then the weight enumerator of $B^\perp$ is given by: $$\begin{aligned}
W_{B^\perp}(y)=&
1
+ \frac{1}{8}
(- 271 + 8a + 7 b + 6 c + 5 d + 4 e + 3 f + 2 g + h )y
\\&
+ \frac{1}{8}
(4761 -24 a - 49 b - 66 c - 75 d - 76 e - 69 f - 54 g - 31 h )y^2
\\&
+ \frac{1}{8}
(- 50295 + 1256 a + 959 b + 830 c + 805 d + 820 e + 811 f
\\ &
\qquad
+ 714 g + 465 h )y^3 + \cdots.\end{aligned}$$ From the condition that $B^\perp$ has minimum weight $3$, we have $$\begin{aligned}
g&= 455 - 28a - 21b - 15c - 10d - 6e - 3f,\\
h&= -639 + 48a + 35b + 24c + 15d + 8e + 3f.\end{aligned}$$ Thus, the weight enumerators are written as: $$\begin{aligned}
W_{B}(y)=&
1 + a y^4 + b y^8 + c y^{12} + d y^{16} + e y^{20}
+ fy^{24}
\\ &+ (455- 28a - 21b - 15c - 10d - 6e - 3f) y^{28}
\\ &+ (-639 + 48a + 35b + 24c + 15d + 8e + 3f ) y^{32}
\\ & + (247 -21a - 15b -10c - 6d - 3e - f )y^{36},
\\
W_{B^\perp}(y)=&
1 + (- 2820 + 448a + 280b + 160c + 80d + 32e + 8f )y^3
+ \cdots.
$$ Then, only $$(a,b,c,d,e,f)=
(0, 0, 0, 20, 42, 1)\text{ and } (0, 0, 1, 17, 45, 0)$$ satisfy the condition that the coefficients in $W_B(y)$ are nonnegative integers. These cases correspond to (\[eq:WE371\]) and (\[eq:WE372\]), respectively.
For (\[eq:WE371\]) and (\[eq:WE372\]), $W_{B^\perp}(y)$ is given by: $$\begin{aligned}
\label{eq:WE371d}
W_{B^\perp}(y)&= 1 + 132 y^3 + 1072 y^4 + 6705 y^5 + 36324 y^6 + \cdots, \\
\label{eq:WE372d}
W_{B^\perp}(y)&= 1+ 140 y^3 + 1080 y^4 + 6633 y^5 + 36252 y^6 + \cdots,\end{aligned}$$ respectively.
\[cor:37con\] Let $C$ be a self-dual ${\mathbb{Z}}_4$-code of length $37$ such that $C^{(1)}$ is a binary doubly even $[37,6,12]$ code and $C^{(2)}$ has minimum weight $3$. Then $A_4(C)$ is a unimodular lattice with $\min(A_4(C))=3$ and kissing number at least $1120$.
By (\[eq:dE\]), $C$ has minimum Euclidean weight $12$. It follows from (\[eq:WE372d\]) that $A_4(C)$ has at least $1120$ vectors of norm $3$.
As a subcode of some binary maximal doubly even code of length $37$, we found a doubly even $[37,6,12]$ code $B_{37}$ such that $B_{37}^\perp$ has minimum weight $3$. The weight enumerators of $B_{37}$ and $B_{37}^\perp$ are given by (\[eq:WE372\]) and (\[eq:WE372d\]), respectively. We verified by [Magma]{} that $B_{37}$ has automorphism group of order $120$ which is the symmetric group of degree $5$. Corollary \[cor:37con\] only guarantees that $A_4(C)$ has minimum norm $3$ and kissing number at least $1120$ for a self-dual ${\mathbb{Z}}_4$-code $C$ with $C^{(1)}=B_{37}$. Using the method in [@Z4-PLF Section 3], we found a self-dual ${\mathbb{Z}}_4$-code $C_{37}$ such that $C_{37}^{(1)}=B_{37}$ and the kissing number of $A_4(C_{37})$ is exactly $1184$. Therefore, we have the following:
There is a unimodular lattice $L$ in dimension $37$ having minimum norm $3$ with $\sigma(L)=21$.
We verified by [Magma]{} that the unimodular lattice $A_4(C_{37})$ has automorphism group of order $7864320$. For the code $C_{37}$, we give a generator matrix in standard form (\[eq:g-matrix\]), by only listing the $6 \times 31$ matrix: $$\left(\begin{array}{cc}
A & B_1+2B_2
\end{array}\right)
=
\left(\begin{array}{cc}
0101001010110011110000011 & 003121\\
0111100111010100000110110 & 323200\\
1010001100010011111110101 & 313301\\
0000001110001100111101100 & 033031\\
1110011111111010100000000 & 000132\\
1001110010000111111111000 & 220010
\end{array}\right).$$
Dimension $n=41$ and $\sigma(L)=n-24$
-------------------------------------
Let $L_{41}$ be an odd unimodular lattice in dimension $41$ having minimum norm $4$. We give the possible theta series of $L_{41}$ and its shadow $S(L_{41})$: $$\begin{aligned}
\theta_{L_{41}}(q) =&
1
+ (15170 + 128\alpha )q^4
+ (1226720 - 1792\alpha - 524288\beta) q^5
+ \cdots,
\\
\theta_{S(L_{41})}(q) =&
\beta q^{1/4}
+ (\alpha - 79\beta) q^{9/4}
+ (104960 - 55\alpha + 3040\beta)q^{17/4}
+ \cdots,\end{aligned}$$ respectively, where $\alpha$ and $\beta$ are nonnegative integers. It turns out that $L_{41}$ has kissing number $15170$ if and only if $\sigma(L_{41})=17$.
Let $C_{41}$ be the ${\mathbb{Z}}_4$-code with generator matrix in standard form (\[eq:g-matrix\]), where $\left(\begin{array}{cc}
A & B_1+2B_2
\end{array}\right)$ is listed in Figure \[Fig:41\]. We verified that $C_{41}$ is a self-dual ${\mathbb{Z}}_4$-code of minimum Euclidean weight $16$ such that $A_4(C_{41})$ has kissing number $15170$. Hence, we have the following:
There is a unimodular lattice $L$ in dimension $41$ having minimum norm $4$ with $\sigma(L)=17$.
$$\left(\begin{array}{cc}
A & B_1+2B_2
\end{array}\right)
=
\left(\begin{array}{cc}
10111111111001000010001& 101031013\\
11001100111111100111000& 112233010\\
01101100001100110110100& 001332012\\
11010101000111001010100& 023102310\\
11111010110101011110111& 033211301\\
10000101001101101010101& 132003201\\
11100110011010001101111& 210211330\\
01110001111111010000111& 230321312\\
01101110010011001111011& 232022022
\end{array}\right)$$
The residue code $C_{41}^{(1)}$ is a binary doubly even $[41,9,12]$ code with a trivial automorphism group and weight enumerator: $$1
+ y^{12}
+ 89 y^{16}
+ 288 y^{20}
+ 108 y^{24}
+ 23 y^{28}
+ 2 y^{32}.$$
Dimension $n=43$ and $\sigma(L)=n-24$
-------------------------------------
Let $L_{43}$ be an odd unimodular lattice in dimension $43$ having minimum norm $4$. We give the possible theta series of $L_{43}$ and its shadow $S(L_{43})$: $$\begin{aligned}
\theta_{L_{43}}(q) =&
1
+ (9030 + 32 \alpha)q^4
+ (941184 - 320\alpha - 131072\beta)q^5
+ \cdots,
\\
\theta_{S(L_{43})}(q) =&
\beta q^{3/4}
+ (\alpha - 77\beta)q^{11/4}
+ (660480 - 53 \alpha + 2883 \beta)q^{19/4}
+ \cdots,\end{aligned}$$ respectively, where $\alpha$ and $\beta$ are nonnegative integers. It turns out that $L_{43}$ has kissing number $9030$ if and only if $\sigma(L_{43})=19$.
Let $C_{43}$ be the ${\mathbb{Z}}_4$-code with generator matrix in standard form (\[eq:g-matrix\]), where $\left(\begin{array}{cc}
A & B_1+2B_2
\end{array}\right)$ is listed in Figure \[Fig:43\]. We verified that $C_{43}$ is a self-dual ${\mathbb{Z}}_4$-code of minimum Euclidean weight $16$ such that $A_4(C_{43})$ has kissing number $9030$. Hence, we have the following:
There is a unimodular lattice $L$ in dimension $43$ having minimum norm $4$ with $\sigma(L)=19$.
$$\left(\begin{array}{cc}
A & B_1+2B_2
\end{array}\right)
=
\left(\begin{array}{cc}
01010011110100111 &0033202030013\\
01100001111010111 &3212001202013\\
10100011110100010 &1122313032032\\
00001111111011100 &1100211211331\\
01101101000000110 &3300133330012\\
00011011001111100 &2213322021123\\
00111011110010110 &1230201103220\\
00100000101001101 &0121022213003\\
11000010110100011 &1012310010101\\
01011101010110101 &3321023022100\\
10001011010111000 &3021302111320\\
00001000111100101 &3231010323110\\
11111111111111000 &0222002002232
\end{array}\right)$$
The residue code $C_{43}^{(1)}$ is a binary doubly even $[43,13,12]$ code with a trivial automorphism group and weight enumerator: $$1
+ 29 y^{12}
+ 1067 y^{16}
+ 3498 y^{20}
+ 3010 y^{24}
+ 569 y^{28}
+ 18 y^{32}.$$
Dimension $n=44$ and $\sigma(L)=n-24$
-------------------------------------
Let $L_{44}$ be an odd unimodular lattice in dimension $44$ having minimum norm $4$. We give the possible theta series of $L_{44}$ and its shadow $S(L_{44})$: $$\begin{aligned}
\theta_{L_{44}}(q) =&
1
+ (6600 + 16 \alpha) q^4
+ (811008 - 128 \alpha- 65536 \beta) q^5
+ \cdots,
\\
\theta_{S(L_{44})}(q) =&
\beta q
+(\alpha - 76 \beta) q^3
+ (1622016 - 52 \alpha + 2806 \beta) q^5
+ \cdots,\end{aligned}$$ respectively, where $\alpha$ and $\beta$ are nonnegative integers. It turns out that $L_{44}$ has kissing number $6600$ if and only if $\sigma(L_{44})=20$.
Let $C_{44}$ be the ${\mathbb{F}}_5$-code of length $44$ whose generator matrix is $$\left(
\begin{array}{ccc@{}c}
\quad & {\Large I_{22}} & \quad &
\begin{array}{cc}
A & B \\
-B^T & A^T
\end{array}
\end{array}
\right),$$ where $A$ and $B$ are $11 \times 11$ negacirculant matrices with the first rows $$(1, 0, 0, 0, 0, 3, 3, 0, 1, 0, 3) \text{ and }
(0, 1, 1, 0, 2, 1, 0, 0, 0, 2, 3),$$ respectively, that is, $A$ and $B$ have the following form $$\left( \begin{array}{ccccc}
r_1&r_2&r_3& \cdots &r_{11} \\
-r_{11}&r_1&r_2& \cdots &r_{10} \\
-r_{10}&-r_{11}&r_1& \cdots &r_{9} \\
\vdots &\vdots & \vdots && \vdots\\
-r_2&-r_3&-r_4& \cdots&r_1
\end{array}
\right),$$ and $A^T$ denotes the transposed matrix of $A$. Since $AA^T+BB^T=4I_{11}$, $C_{44}$ is self-dual. In addition, it can be checked that $C_{44}$ has the following weight enumerator: $$1+ 924 y^{12}+ 1848 y^{13}
+ 19272 y^{14} +91168 y^{15}
+\cdots.$$ We verified by [Magma]{} that the unimodular lattice $A_5(C_{44})$ has minimum norm $4$ and kissing number $6600$. Hence, we have the following:
There is a unimodular lattice $L$ in dimension $44$ having minimum norm $4$ with $\sigma(L)=20$.
Our computer search failed to discover a unimodular lattice with long shadow using a self-dual ${\mathbb{Z}}_4$-code for this case and the remaining cases that the existences are not known.
Summary
-------
As a summary, we list the number $\#$ of known non-isomorphic unimodular lattices $L$ in dimension $n$ with $\min(L)=3$ and $\sigma(L)=n-16$ (resp. $\min(L)=4$ and $\sigma(L)=n-24$) in Table \[Tab:1\] (resp. Table \[Tab:2\]). Both tables update the two tables given in [@Ga07 p. 148]. We remark that the existence of a unimodular lattice $L$ in dimension $37$ having minimum norm $4$ is still unknown for any $\sigma(L)$ (see [@lattice-datebase]).
\[Tab:1\]
[c|c|l||c|c|l]{} $n$ & $\#$ & & $n$ & $\#$ &\
23& $1$ & (see [@NV03]) &35& $\ge 1$ & [@NV03]\
24& $1$ & (see [@NV03]) &36& $\ge 1$ & $A_4(C_{36})$ in Section \[Sec:36\]\
25& $0$ & (see [@NV03]) &37& $\ge 1$ & $A_4(C_{37})$ in Section \[Sec:Ot\]\
26& $1$ & (see [@NV03]) &38& ? &\
27& $2$ & (see [@NV03]) &39& ? &\
28& $36$ & (see [@NV03]) &40& $\ge 1$ & (see [@Ga07])\
29& $\ge 1$ & [@NV03] &41& ? &\
30& $\ge 1$ & [@NV03] &42& ? &\
31& $\ge 1$ & [@NV03] &43& ? &\
32& $\ge 1$ & [@NV03] &44& 0 & [@NV03]\
33& $\ge 1$ & [@NV03] &45& 0 & [@NV03]\
34& $\ge 1$ & [@NV03] &46& 1 & [@NV03]\
\[Tab:2\]
[c|c|l||c|c|l]{} $n$ & $\#$ & & $n$ & $\#$ &\
32 & 5 & [@CS98] &42 & $\ge 1$ & (see [@Ga07])\
36 & $\ge 3$ & [@lattice-datebase], $A_4(D_{36})$ in Section \[Sec:36\] &43 & $\ge 1$ & $A_4(C_{43})$ in Section \[Sec:Ot\]\
37 & ? & &44 & $\ge 1$ & $A_5(C_{44})$ in Section \[Sec:Ot\]\
38 & $\ge 1$ & (see [@Ga07]) &45 & ? &\
39 & $\ge 1$ & (see [@Ga07]) &46 & $\ge 1$ & (see [@Ga07])\
40 & $\ge 1$ & (see [@Ga07]) &47 & $\ge 1$ & (see [@Ga07])\
41 & $\ge 1$ & $A_4(C_{41})$ in Section \[Sec:Ot\] &&&\
[**Acknowledgment.**]{} The author would like to thank the anonymous referee for helpful suggestions that improved Proposition \[prop:36con\] and Corollary \[cor:37con\]. This work was supported by JST PRESTO program.
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[^1]: Department of Mathematical Sciences, Yamagata University, Yamagata 990–8560, Japan, and PRESTO, Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332–0012, Japan. email: mharada@sci.kj.yamagata-u.ac.jp
|
---
abstract: |
Final results of the measurement of $R = \sigma
(e^+e^- \rightarrow hadrons)/\sigma(e^+e^- \rightarrow \mu^+
\mu^-)$ in the energy region from 2 to 5 GeV with the upgraded Beijing Spectrometer (BESII) at the Beijing Electron Positron Collider (BEPC) are presented. Preliminary results of the inclusive momentum spectra and second binomial moment measured with the $R$ scan data at 2.2, 2.6, 3.0, 3.2, 4.6 and 4.8 GeV are reported.
address:
- 'Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100039, P.R.C.'
- 'University of Michigan, Ann Arbor, MI48109, USA'
author:
- 'W.B. Yan, W.G. Li, Z.G. Zhao , Representing BES Collaboration'
title: 'Final $R$-value results from 2-5 GeV from BES and QCD test with $R$ scan data'
---
Introduction
============
The lowest order cross section for $e^+e^-\rightarrow\gamma^*\rightarrow \mbox{hadrons}$ in units of the lowest-order QED cross section for $e^+e^- \rightarrow \mu^+\mu^-$ is defined as $R$, namely $R=\sigma(e^+e^- \rightarrow \mbox{hadrons})/\sigma(e^+e^-\rightarrow
\mu^+\mu^-)$, where $\sigma (e^+e^- \rightarrow \mu^+\mu^-)
= \sigma^0_{\mu \mu}=4\pi \alpha^2(0) / 3s$. Presently the uncertainty in $R$ in the energy region below 5 GeV dominates the uncertainties in both $\alpha(M^2_{Z})$, the QED running coupling constant evaluated at the Z pole, and $a_{\mu}^{SM}$, the value of $(g-2)_{\mu}$ based on the Standard Model calculation [@rreview_zg].
Hadron production from $e^+e^-$ annihilation is one of the most valuable testing grounds for Quantum Chromodynamics (QCD). Particularly, it is interesting and important to analyze low energy $e^+e^-$ collision data, for example, the inclusive momentum spectrum, defined as $\xi = - \ln (2p/\sqrt{s})$, where $p$ and $\sqrt{s}$ are the momentum of the charged particles and center-of-mass (c.m.) energy respectively, with today’s knowledge. A purely analytical approach giving quantitative predictions for $\xi$ is the QCD calculation using the so-called Modified Leading Logarithmic Approximation (MLLA) [@mlla] under the assumption of Local Parton Hadron Duality (LPHD) [@lphd], which expresses the limiting spectrum for hadrons as
$$\begin{aligned}
\frac{1}{\sigma_{had}} \frac{d \sigma}{d \xi} = K_{LPHD} \times
f_{MLLA} (\xi,\Lambda_{eff})
\label{express}\end{aligned}$$
where $K_{LPHD}$ is an overall normalization factor describing hadronization and $f$ is a complex function of $\xi$ and effective scale parameter $\Lambda_{eff}$ [@mlla]. Eq. 1. is valid in the range of $0 \leq \xi \leq \ln
(0.5\sqrt{s} / \Lambda_{eff})$.
Another example is the second binomial moments, which is a measure of the strength of hadron-hadron correlations and a sensitive probe for higher order QCD or non-perturbative effect. It is defined as $R_2
= {\langle n_{ch}(n_{ch}-1) \rangle}/{\langle n_{ch} \rangle}^2$, where $n_{ch}$ is the charged particle multiplicity. According to the next leading order QCD calculation (NLO), $R_2$ is given by
$$\begin{aligned}
R_2 = \frac{11}{8}(1 - c \sqrt{\alpha_s(\sqrt{s})})
\label{r2exp}\end{aligned}$$
with $c = 0.55 (0.56)$ for five (three) active flavors. There has been a long standing discrepancy between the value of $R_2$ calculated by NLO and that measured with $e^+e^-$, $\mu^+p$ and $\nu_{\mu}p$ experiments. In addition, there is relatively little data in the energy region below 5 GeV to compare with QCD calculations.
This paper presents the final results of the $R$ values measured with BESII [@bes2] at BEPC in the energy region from 2-5 GeV. We also report preliminary results on the inclusive momentum spectra, the momentum distribution of charged particles, and second binomial moments $R_2$ obtained from the analysis of $R$ scan data.
$R$ values in 2-5 GeV
=====================
Refs. [@besr1; @besr2] describe in detail the $R$ scan performed by BES at 91 energy points between 2-5 GeV, and the experimental study of the background, particularly the beam associated background. The triggers and the determination of the trigger efficiency; the measurement of luminosity; the hadronic event selection and background subtraction; the determination of the detection efficiency for hadronic events; and the initial state radiative correction can be also found in Refs. [@besr1; @besr2].
![(a) A compilation of measurements of $R$ in the cm energy range from 1.4 to 5 GeV. (b) $R$ values from this experiment in the resonance region between 3.75 and 4.6 GeV.[]{data-label="fig:besr"}](besrnew.eps "fig:"){width="16pc" height="16pc"} -0.8cm
-0.4 cm
The final $R$ values measured by BES in this experiment are displayed in Fig. \[fig:besr\], together with those measured by MarkI [@markI], $\gamma\gamma 2$ [@gamma2] and Pluto [@pluto]. The $R$ values from BESII have an average uncertainty of about 6.6%, which represents a factor of two to three improvement in precision in the 2 to 5 GeV energy region. These improved measurements have a significant impact on the global fit to the electroweak data and the determination of the SM prediction for the mass of the Higgs particle [@bolek]. In addition, they are expected to provide an improvement in the precision of the calculated value of $a_{\mu}^{SM}$, and test the QCD sum rules down to 2 GeV [@dave; @martin; @kuehn].
Test of QCD models with $R$ scan data
=====================================
$\xi$ spectrum
--------------
The measured $\xi$ spectrum at five different energies between 2.6 and 4.8 GeV are shown in Figure \[comxi\]. The errors in the spectrum include errors from hadronic event selection and uncertainties from the event generators.
![Measured $\xi$ spectrum (solid dot) at 2.6, 3.0, 3.2, 4.6 and 4.8 GeV. Solid curves are the fitting of the limiting spectrum. The dotted line is an extrapolation of the fitted result.[]{data-label="comxi"}](comxi.eps "fig:"){width="16pc" height="15pc"} -0.8cm
-0.4 cm
Momentum spectrum
-----------------
The momentum spectra at the five energy points are shown in Figure \[dnall\], together with those measured at higher energy up to 130 GeV in other experiments. The momentum spectra show that hadron production at very small momentum $p \leq 0.1$ GeV is approximately energy independent. This behavior has been explained in Ref. [@q0momentum] to be due to the coherent emission of low energetic (i. e. long wavelength) gluons by the total color current. Correspondingly the number of produced hadrons at small momentum is approximately constant.
![Charged particle momentum spectrum[]{data-label="dnall"}](dnall.eps "fig:"){width="16pc" height="15pc"} -0.8cm
-0.4 cm
Second binomial moment
----------------------
Based on the multiplicity measured, we can obtain the second binomial moment $R_2$. The results are displayed in Figure \[r2bes\], together with both NLO calculations and published data at higher energies up to 100 GeV from $e^+e^-$, $\mu^+p$ and $\nu_{\mu}p$ experiments [@emcr2; @wa21r2]. Our measured $R_2$, although with large errors, are consistent with those of other measurements done at higher energies.
It’s interesting to see that $R_2$ predicted by LO QCD are significantly higher than the measured data, while the NLO calculation comes closer to the data, although disagreement ($\sim 0.07$ in $R_2$) remains sizeable.
![Energy dependence of second binomial moments $R_2$[]{data-label="r2bes"}](r2bes.eps "fig:"){width="16pc" height="16pc"} -0.8cm
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|
---
abstract: 'We perform a detailed investigation of total lifetimes for the doubly heavy baryons $\Xi_{QQ''}$, $\Omega_{QQ''}$ in the framework of operator product expansion over the inverse heavy quark mass, whereas, to estimate matrix elements of operators obtained in OPE, approximations of nonrelativistic QCD are used.'
---
=-18mm =-22mm
\#1[0= 0=0 1= 1=1 0>1 \#1 / ]{}
[**Lifetimes of doubly heavy baryons**]{}\
A.K. Likhoded$^{a)}$, A.I. Onishchenko$^{b)}$\
\
[*Protvino, Moscow region, 142284 Russia*]{}\
[b) Institute for Theoretical and Experimental Physics]{}\
[*Moscow, B. Cheremushkinskaja, 25, 117259 Russia\
Fax: 7 (095) 123-65-84*]{}
Introduction
============
At present a number of powerful techniques based on Operator Product Expansion (OPE) and effective field theories have been developed. These tools allow one consistently to include into consideration various nonperturbative contributions, written in terms of a few number of universal quantities. The coefficients (Wilson coefficients) in front of these operators are generally expanded in series over the QCD coupling constant, inverse heavy quark mass and/or relative velocity of heavy quarks inside the hadron. The accuracy, obtained in such calculations, can be systematically improved, and it is limited only by the convergence properties of the mentioned series. The described approach have been already widely used for making the precise predictions in the heavy quark sector of Standard Model (SM), such as decays, distributions and partial width asymmetries involving the CP violation[^1] for the heavy hadrons. The sensitivity of Wilson coefficients to virtual corrections caused by some higher-scale interactions makes this approach to be invaluable in searching for a “new” physics at forthcoming experiments.
The approach under discussion has been successfully used in the description of weak decays of the hadrons containing a single heavy quark, as carried out in the framework of Heavy Quark Effective Theory (HQET) [@HQET], in the annihilation and radiative decays of heavy quarkonia $Q\bar Q$, where one used the framework of non-relativistic QCD (NRQCD) [@NRQCD], and in the weak decays of long-lived heavy quarkonium with mixed flavours $B_c^+$ [@Beneke] [^2]. The experimental data on the weak decays of heavy hadrons can be used for the determination of basic properties of weak interactions at a fundamental level, in particular, for the extraction of CKM matrix elements. The same approach is also valid for the baryons containing two heavy quarks.
In addition to the information extracted from the analysis of hadrons with a single heavy flavor, the baryons with two heavy quarks, $QQ^\prime q$, provide a way to explore the nonspectator effects, where their importance is increased. Here we would like to note, that in the case of systems with two heavy quarks, the hypothesis on the quark-hadron duality is more justified, and, so, the results of OPE-based approach turn out to be more reliable. For these baryons we can apply a method, based on the combined HQET-NRQCD techniques [@HQET; @NRQCD; @Beneke], if we use the quark-diquark picture for the bound states. The expansion in the inverse heavy quark mass for the heavy diquark $QQ^\prime$ is a straightforward generalization of these techniques in the mesonic decays of $B_c$ [@NRQCD; @Beneke], with the difference that, instead of the color-singlet systems, we deal with the color–anti-triplet ones, with the appropriate account for the interaction with the light quark. First estimates of the lifetimes for the doubly heavy baryons $\Xi_{cc}^{\diamond}$ and $\Xi_{bc}^{\diamond}$ were recently performed in [@DHD1; @DHD2]. Using the same approach, but different values of parameters[^3] a repetition of our results for the case of doubly charmed baryons was done in [@Guberina]. The spectroscopic characteristics of baryons with two heavy quarks and the mechanisms of their production in different interactions were discussed in refs. [@DHS1; @DHS2; @DHS3; @DHSR] and [@prod], respectively.
In this paper, we present the calculation of lifetimes for the doubly heavy baryons as well as reconsider the previous estimates with a use of slightly different set of parameters adjusted in the consideration of lifetime data for the observed heavy hadrons and improved spectroscopic inputs. As we made in the description of inclusive decays of the $\Xi_{cc}^{\diamond}$ and $\Xi_{bc}^{\diamond}$-baryons, we follow the papers [@Beneke; @vs], where all necessary generalizations to the case of hadrons with two heavy quarks and other corrections are discussed. We note, that in the leading order of OPE expansion, the inclusive widths are determined by the mechanism of spectator decays involving free quarks, wherein the corrections in the perturbative QCD are taken into account. The introduction of subleading terms in the expansion over the inverse heavy quark masses[^4] allows one to take into account the corrections due to the quark confinement inside the hadron. Here, an essential role is played by both the motion of heavy quark inside the hadron and chromomagnetic interactions of quarks. The important ingredient of such corrections in the baryons with two heavy quarks is the presence of a compact heavy diquark, which implies that the square of heavy quark momentum is enhanced in comparison with the corresponding value for the hadrons with a single heavy quark. The next characteristic feature of baryons with two heavy quarks is the significant numerical impact on the lifetimes by the quark contents of hadrons, since in the third order over the inverse heavy quark mass, $1/m_Q^3$, the four-quark correlations in the total width are enforced in the effective lagrangian due to the two-particle phase space of intermediate states (see the discussion in [@vs]). In this situation, we have to add the effects of Pauli interference between the products of heavy quark decays and the quarks in the initial state as well as the weak scattering involving the quarks composing the hadron. Due to such terms we introduce the corrections depending on spectators and involving the masses of light and strange quarks in the framework of non-relativistic models with the constituent quarks, because they determine the effective physical phase spaces, strongly deviating from the naive estimates in th decays of charmed quarks. We take into account the corrections to the effective weak lagrangian due to the evolution of Wilson coefficients from the scale of the order of heavy quark mass to the energy, characterizing the binding of quarks inside the hadron [@vs].
The paper is organized as follows. In agreement with the general picture given above, we describe the scheme for the construction of OPE for the total width of baryons containing two heavy quarks with account of corrections to the spectator widths in Section 2. The procedure for the estimation of non-perturbative matrix elements of operators in the doubly heavy baryons is considered in Section 3 in terms of non-relativistic heavy quarks. Section 4 is devoted to the numerical evaluation and discussion of parameter dependence of lifetimes of doubly heavy baryons. We conclude in section 5 by summarizing our results.
Description of the method
=========================
In this section we describe the approach used for the calculation of total lifetimes for the doubly heavy baryons, originally formulated in [@DHD1; @DHD2], together with some new formulae, required for the evaluation of nonspectator effects in the decays of other[^5] baryons in the family of doubly heavy baryons, not considered previously.
The optical theorem along with the hypothesis of integral quark-hadron duality, leads us to a relation between the total decay width of heavy quark and the imaginary part of its forward scattering amplitude. This relationship, applied to the $\Xi_{QQ'}^{(*)}$-baryon total decay width $\Gamma_{\Xi_{QQ'}^{(*)}}$, can be written down as: $$\Gamma_{\Xi_{QQ'}^{(*)}} =
\frac{1}{2M_{\Xi_{QQ'}^{(*)}}}\langle\Xi_{QQ'}^{(*)}|{\cal T}
|\Xi_{QQ'}^{(*)}\rangle ,
\label{1}$$ where the $\Xi_{QQ'}^{(*)}$ state in Eq. (\[1\]) has the ordinary relativistic normalization, $\langle \Xi_{QQ'}^{(*)}|\Xi_{QQ'}^{(*)}\rangle = 2EV$, and the transition operator ${\cal T}$ is determined by the expression $${\cal T} = \Im m\int d^4x~\{{\hat T}H_{eff}(x)H_{eff}(0)\},$$ where $H_{eff}$ is the standard effective hamiltonian, describing the low energy interactions of initial quarks with the decays products, so that $$H_{eff} = \frac{G_F}{2\sqrt 2}V_{q_3q_4}V_{q_1q_2}^{*}[C_{+}(\mu)O_{+} +
C_{-}(\mu)O_{-}] + h.c.$$ where $$O_{\pm} = [\bar q_{1\alpha}\gamma_{\nu}(1-\gamma_5)q_{2\beta}][\bar
q_{3\gamma}\gamma^{\nu}(1-\gamma_5)q_{4\delta}](\delta_{\alpha\beta}\delta_{
\gamma\delta}\pm\delta_{\alpha\delta}\delta_{\gamma\beta}),$$ and $$C_+ = \left [\frac{\alpha_s(M_W)}{\alpha_s(\mu)}\right ]^{\frac{6}{33-2f}},
\quad
C_- = \left [\frac{\alpha_s(M_W)}{\alpha_s(\mu)}\right ]^{\frac{-12}{33-2f}},\\$$ where f is the number of flavors, $\{\alpha,\beta,\gamma,\delta \}$ run over the color indeces.
Under an assumption, that the energy release in the heavy quark decay is large, we can perform the operator product expansion for the transition operator ${\cal T}$ in Eq.(\[1\]). In this way we obtain series of local operators with increasing dimensions over the energy scale, wherein the contributions to $\Gamma_{\Xi_{QQ'}^{(*)}}$ are suppressed by the increasing inverse powers of the heavy quark masses. This formalism has already been applied to calculate the total decay rates for the hadrons, containing a single heavy quark [@bigi] (for the most early work, having used similar methods, see also [@vs; @GRT]) and hadrons, containing two heavy quarks [@DHD1; @DHD2]. As was already pointed in [@DHD1], the expansion, applied here, is simultaneously in the powers of both inverse heavy quark masses and the relative velocity of heavy quarks inside the hadron. Thus, this fact shows the difference between the description for the doubly heavy baryons and the consideration of both the heavy-light mesons (the expansion in powers of $\frac{\Lambda_{QCD}}{m_Q}$) and the heavy-heavy mesons [@Beneke] (the expansion in powers of relative velocity of heavy quarks inside the hadron, where one can apply the scaling rules of nonrelativistic QCD [@4]).
The operator product expansion explored has the form: $${\cal T} = \sum_{i=1}^2 C_1(\mu)\bar Q^iQ^i + \frac{1}{m_{Q^i}^2}C_2(\mu)\bar
Q^i g\sigma_{\mu\nu}G^{\mu\nu}Q^i
+ \frac{1}{m_{Q^i}^3}O(1) \label{4}$$
The leading contribution in Eq.(\[4\]) is determined by the operators $\bar
Q^iQ^i$, corresponding to the spectator decay of $Q^i$-quarks. The use of motion equation for the heavy quark fields allows one to eliminate some redundant operators, so that no operators of dimension four contribute. There is a single operator of dimension five, $Q^i_{GQ} = \bar Q^i g \sigma_{\mu\nu}
G^{\mu\nu} Q^i$. As we will show below, significant contributions come from the operators of dimension six $Q_{2Q^i2q} = \bar Q^i\Gamma q\bar q\Gamma^{'}Q^i$, representing the effects of Pauli interference and weak scattering for doubly heavy baryons. Furthermore, there are also other operators of dimension six $Q_{61Q^i} = \bar Q^i \sigma_{\mu\nu}\gamma_{l} D^{\mu}G^{\nu l}Q^i$ and $Q_{62Q^i} = \bar Q^i D_{\mu} G^{\mu\nu}\Gamma_{\nu}Q^i$, which are suppressed in comparison with $Q_{2Q^i2q}$ [@vs]. In what follows, we do not calculate the corresponding coefficient functions for the latter two operators, so that the expansion is certainly complete up to the second order of $\frac {1}{m}$, only.
Further, the different contributions to OPE are given by the following: $$\begin{aligned}
{\cal T}_{\Xi_{cc}^{++}} &=& 2 {\cal T}_{35c} + {\cal
T}_{6,PI}^{(1)},\nonumber\\ {\cal T}_{\Xi_{cc}^{+}} &=& 2 {\cal
T}_{35c} + {\cal T}_{6,WS}^{(2)},\nonumber\\ {\cal
T}_{\Omega_{cc}^{+}} &=& 2 {\cal T}_{35c} + {\cal
T}_{6,PI}^{(3)},\nonumber\\ {\cal T}_{\Xi_{bc}^{+}} &=& {\cal
T}_{35b} + {\cal T}_{35c} + {\cal T}_{6,PI}^{(4)} + {\cal
T}_{6,WS}^{(4)},\nonumber\\ {\cal T}_{\Xi_{bc}^{0}} &=& {\cal
T}_{35b} + {\cal T}_{35c} + {\cal T}_{6,PI}^{(5)} + {\cal
T}_{6,WS}^{(5)},\nonumber \\ {\cal T}_{\Omega_{bc}^{0}} &=& {\cal
T}_{35b} + {\cal T}_{35c} + {\cal T}_{6,PI}^{(6)} + {\cal
T}_{6,WS}^{(6)},\nonumber \\ {\cal T}_{\Xi_{bb}^{0}} &=& 2 {\cal
T}_{35b} + {\cal T}_{6,WS}^{(7)},\nonumber\\ {\cal
T}_{\Xi_{bb}^{-}} &=& 2 {\cal T}_{35b} + {\cal
T}_{6,PI}^{(8)},\nonumber \\ {\cal T}_{\Omega_{bb}^{-}} &=& 2
{\cal T}_{35b} + {\cal T}_{6,PI}^{(9)},\nonumber\end{aligned}$$ where the $35$-labelled terms account for the operators of dimension three $O_{3Q^i}$ and five $O_{GQ^i}$, the $6$-marked terms correspond to the effects of Pauli interference and weak scattering. The explicit formulae for these contributions have the following form: $${\cal T}_{35b} = \Gamma_{b,spec}\bar bb - \frac{\Gamma_{0b}}{m_b^2}[2P_{c1} +
P_{c\tau 1} + K_{0b}(P_{c1} + P_{cc1}) + K_{2b}(P_{c2} + P_{cc2})]O_{Gb},
\label{5}$$ $${\cal T}_{35c} = \Gamma_{c,spec}\bar cc - \frac{\Gamma_{0c}}{m_c^2}[(2 +
K_{0c})P_{s1} + K_{2c}P_{s2}]O_{Gc}, \label{6}$$ where $$\Gamma_{0b} = \frac{G_F^2m_b^5}{192{\pi}^3}|V_{cb}|^2\qquad , \Gamma_{0c} =
\frac{G_F^2m_c^5}{192{\pi}^3}$$ with $K_{0Q} = C_{-}^2 + 2C_{+}^2,~K_{2Q} = 2(C_{+}^2 - C_{-}^2)$, and $\Gamma_{Q,spec}$ denotes the spectator width (see [@bigi; @9; @10; @11]): $$P_{c1} = (1-y)^4,\quad P_{c2} = (1-y)^3,$$ $$\begin{aligned}
P_{c\tau 1} &=& \sqrt{1-2(r+y)+(r-y)^2}[1 - 3(r+y) + 3(r^2+y^2) - r^3 - y^3
-4ry + \nonumber\\
&& 7ry(r+y)] + 12r^2y^2\ln\frac{(1-r-y+\sqrt{1-2(r+y)+(r-y)^2})^2}{4ry}\end{aligned}$$ $$P_{cc1} = \sqrt{1-4y}(1 - 6y + 2y^2 + 12y^3)
24y^4\ln\frac{1+\sqrt{1-4y}}{1-\sqrt{1-4y}}$$ $$P_{cc2} = \sqrt{1-4y}(1 + \frac{y}{2} + 3y^2)
- 3y(1-2y^2)\ln\frac{1+\sqrt{1-4y}}{1-\sqrt{1-4y}}$$ where $y = \frac{m_c^2}{m_b^2}$ and $r = m_{\tau}^2/m_b^2$. The functions $P_{s1} (P_{s2})$ can be obtained from $P_{c1} (P_{c2})$ by the substitution $y = m_s^2/m_c^2$. In the $b$-quark decays, we neglect the value $m_s^2/m_b^2$ and suppose $m_s = 0$.
The calculation of both the Pauli interference effect for the products of heavy quark decays with the quarks in the initial state and the weak scattering of quarks, composing the hadron, depends on the quark contents of baryons and results in: $$\begin{aligned}
{\cal T}_{6,PI}^{(1)} &=& 2 {\cal T}_{PI,u\bar d}^c \\
{\cal T}_{6,WS}^{(2)} &=& 2 {\cal T}_{WS,cd} \\
{\cal T}_{6,PI}^{(3)} &=& 2 {\cal T}_{PI,u\bar d}^{c'} + 2 \sum_{l} {\cal
T}_{PI,\nu_l \bar l}^c \\
{\cal T}_{6,PI}^{(4)} &=& {\cal T}_{PI,u\bar d}^c + {\cal T}_{PI,s\bar c}^b +
{\cal T}_{PI,d\bar u}^b + \sum_l{\cal T}_{PI,l\bar\nu_l}^b\\
{\cal T}_{6,WS}^{(4)} &=& {\cal T}_{WS,bu} + {\cal T}_{WS,bc}\\
{\cal T}_{6,PI}^{(5)} &=& {\cal T}_{PI,s\bar c}^b + {\cal T}_{PI,d\bar u}^b +
{\cal T}_{PI,d\bar u}^{'b} + \sum_l{\cal T}_{PI,l\bar\nu_l}^b\\
{\cal T}_{6,WS}^{(5)} &=& {\cal T}_{WS,cd} + {\cal T}_{WS,bc} \\
{\cal T}_{6,PI}^{(6)} &=& {\cal T}_{PI,u\bar d}^{c'} + \sum_{l} {\cal
T}_{PI,\nu_l \bar l}^c + {\cal T}_{PI,s\bar c}^b + {\cal T}_{PI,d\bar u}^b +
\sum_l{\cal T}_{PI,l\bar\nu_l}^b + {\cal T}_{PI,s\bar c}^{'b}\\
{\cal T}_{6,WS}^{(6)} &=& {\cal T}_{WS,bc} + {\cal T}_{WS,cs}\\
{\cal T}_{6,WS}^{(7)} &=& 2 {\cal T}_{WS,bu} \\
{\cal T}_{6,PI}^{(8)} &=& 2 {\cal T}_{PI,d\bar u}^{'b} \\
{\cal T}_{6,PI}^{(9)} &=& 2 {\cal T}_{PI,s\bar c}^{'b}\end{aligned}$$ so that $$\begin{aligned}
{\cal T}_{PI,s\bar c}^b &=&
-\frac{G_F^2|V_{cb}|^2}{4\pi}m_b^2(1-\frac{m_c}{m_b})^2\nonumber\\
&& ([(\frac{(1-z_{-})^2}{2}- \frac{(1-z_{-})^3}{4})
(\bar b_i\gamma_{\alpha}(1-\gamma_5)b_i)(\bar
c_j\gamma^{\alpha}(1-\gamma_5)c_j) + \nonumber\\
&& (\frac{(1-z_{-})^2}{2} -
\frac{(1-z_{-})^3}{3})(\bar b_i\gamma_{\alpha}\gamma_5 b_i)(\bar
c_j\gamma^{\alpha}(1-\gamma_5)c_j)] \label{16}
\\&& [(C_{+} - C_{-})^2 +
\frac{1}{3}(1-k^{\frac{1}{2}})(5C_{+}^2+C_{-}^2+6C_{-}C_{+})]+ \nonumber\\
&& [(\frac{(1-z_{-})^2}{2} - \frac{(1-z_{-})^3}{4})(\bar
b_i\gamma_{\alpha}(1-\gamma_5)b_j)(\bar c_j\gamma^{\alpha}(1-\gamma_5)c_i) +
\nonumber\\
&& (\frac{(1-z_{-})^2}{2} - \frac{(1-z_{-})^3}{3})(\bar
b_i\gamma_{\alpha}\gamma_5b_j)(\bar
c_j\gamma^{\alpha}(1-\gamma_5)c_i)]k^{\frac{1}{2}}(5C_{+}^2+C_{-}^2+6C_{-}C_{+}
)),\nonumber\\
{\cal T}_{PI,\tau\bar\nu_{\tau}}^b &=&
-\frac{G_F^2|V_{cb}|^2}{\pi}m_b^2(1-\frac{m_c}{m_b})^2\nonumber\\
&& [(\frac{(1-z_{\tau})^2}{2} - \frac{(1-z_{\tau})^3}{4})(\bar
b_i\gamma_{\alpha}(1-\gamma_5)b_j)(\bar c_j\gamma^{\alpha}(1-\gamma_5)c_i) +
\label{17}\\
&& (\frac{(1-z_{\tau})^2}{2} - \frac{(1-z_{\tau})^3}{3})(\bar
b_i\gamma_{\alpha}\gamma_5b_j)(\bar
c_j\gamma^{\alpha}(1-\gamma_5)c_i)],\nonumber\\
{\cal T}_{PI,d\bar u}^{b'} &=&
-\frac{G_F^2|V_{cb}|^2}{4\pi}m_b^2(1-\frac{m_d}{m_b})^2\nonumber\\
&& ([(\frac{(1-z_{-})^2}{2}- \frac{(1-z_{-})^3}{4})
(\bar b_i\gamma_{\alpha}(1-\gamma_5)b_i)(\bar
d_j\gamma^{\alpha}(1-\gamma_5)d_j) + \nonumber\\
&& (\frac{(1-z_{-})^2}{2} -
\frac{(1-z_{-})^3}{3})(\bar b_i\gamma_{\alpha}\gamma_5 b_i)(\bar
d_j\gamma^{\alpha}(1-\gamma_5)d_j)] \label{18}
\\&& [(C_{+} + C_{-})^2 +
\frac{1}{3}(1-k^{\frac{1}{2}})(5C_{+}^2+C_{-}^2-6C_{-}C_{+})]+ \nonumber\\
&& [(\frac{(1-z_{-})^2}{2} - \frac{(1-z_{-})^3}{4})(\bar
b_i\gamma_{\alpha}(1-\gamma_5)b_j)(\bar d_j\gamma^{\alpha}(1-\gamma_5)d_i) +
\nonumber\\
&& (\frac{(1-z_{-})^2}{2} - \frac{(1-z_{-})^3}{3})(\bar
b_i\gamma_{\alpha}\gamma_5b_j)(\bar
d_j\gamma^{\alpha}(1-\gamma_5)d_i)]k^{\frac{1}{2}}(5C_{+}^2+C_{-}^2-6C_{-}C_{+}
)),\nonumber\\
{\cal T}_{PI,s\bar c}^{b'} &=&
-\frac{G_F^2|V_{cb}|^2}{16\pi}m_b^2(1-\frac{m_s}{m_b})^2\sqrt{(1-4z_{-})}
\nonumber\\
&& ([(1-z_{-})(\bar b_i\gamma_{\alpha}(1-\gamma_5)b_i)(\bar
s_j\gamma^{\alpha}(1-\gamma_5)s_j) + \nonumber\\
&&\frac{2}{3}(1+2z_{-})(\bar b_i\gamma_{\alpha}\gamma_5 b_i)(\bar
s_j\gamma^{\alpha}(1-\gamma_5)s_j)] \label{181}
\\&& [(C_{+} + C_{-})^2 +
\frac{1}{3}(1-k^{\frac{1}{2}})(5C_{+}^2+C_{-}^2-6C_{-}C_{+})]+ \nonumber\\
&& [(1-z_{-})(\bar
b_i\gamma_{\alpha}(1-\gamma_5)b_j)(\bar s_j\gamma^{\alpha}(1-\gamma_5)s_i) +
\nonumber\\
&&\frac{2}{3}(1+2z_{-})(\bar
b_i\gamma_{\alpha}\gamma_5b_j)(\bar
s_j\gamma^{\alpha}(1-\gamma_5)s_i)]k^{\frac{1}{2}}(5C_{+}^2+C_{-}^2-6C_{-}C_{+}
)),\nonumber\\
{\cal T}_{PI,u\bar d}^c &=&
-\frac{G_F^2}{4\pi}m_c^2(1-\frac{m_u}{m_c})^2\nonumber\\
&& ([(\frac{(1-z_{-})^2}{2}- \frac{(1-z_{-})^3}{4})
(\bar c_i\gamma_{\alpha}(1-\gamma_5)c_i)(\bar
u_j\gamma^{\alpha}(1-\gamma_5)u_j) + \nonumber\\
&& (\frac{(1-z_{-})^2}{2} -
\frac{(1-z_{-})^3}{3})(\bar c_i\gamma_{\alpha}\gamma_5 c_i)(\bar
u_j\gamma^{\alpha}(1-\gamma_5)u_j)] \label{19}
\\&& [(C_{+} + C_{-})^2 +
\frac{1}{3}(1-k^{\frac{1}{2}})(5C_{+}^2+C_{-}^2-6C_{-}C_{+})]+ \nonumber\\
&& [(\frac{(1-z_{-})^2}{2} - \frac{(1-z_{-})^3}{4})(\bar
c_i\gamma_{\alpha}(1-\gamma_5)c_j)(\bar u_j\gamma^{\alpha}(1-\gamma_5)u_i) +
\nonumber\\
&& (\frac{(1-z_{-})^2}{2} - \frac{(1-z_{-})^3}{3})(\bar
c_i\gamma_{\alpha}\gamma_5c_j)(\bar
u_j\gamma^{\alpha}(1-\gamma_5)u_i)]k^{\frac{1}{2}}(5C_{+}^2+C_{-}^2-6C_{-}C_{+}
)),\nonumber\\
{\cal T}_{PI,u\bar d}^{c'} &=&
-\frac{G_F^2}{4\pi}m_c^2(1-\frac{m_s}{m_c})^2\nonumber\\
&& ([\frac{1}{4}(\bar c_i\gamma_{\alpha}(1-\gamma_5)c_i)(\bar
s_j\gamma^{\alpha}(1-\gamma_5)s_j) +
\frac{1}{6}(\bar c_i\gamma_{\alpha}\gamma_5 c_i)(\bar
s_j\gamma^{\alpha}(1-\gamma_5)s_j)] \label{191}
\\&& [(C_{+} - C_{-})^2 +
\frac{1}{3}(1-k^{\frac{1}{2}})(5C_{+}^2+C_{-}^2+6C_{-}C_{+})]+ \nonumber\\
&& [\frac{1}{4}(\bar
c_i\gamma_{\alpha}(1-\gamma_5)c_j)(\bar s_j\gamma^{\alpha}(1-\gamma_5)s_i) +
\nonumber\\
&&\frac{1}{6}(\bar
c_i\gamma_{\alpha}\gamma_5c_j)(\bar
s_j\gamma^{\alpha}(1-\gamma_5)s_i)]k^{\frac{1}{2}}(5C_{+}^2+C_{-}^2+6C_{-}C_{+}
)),\nonumber\\
{\cal T}_{PI,\nu_{\tau}\bar\tau}^c &=&
-\frac{G_F^2}{\pi}m_c^2(1-\frac{m_s}{m_c})^2\nonumber\\
&& [(\frac{(1-z_{\tau})^2}{2} - \frac{(1-z_{\tau})^3}{4})(\bar
c_i\gamma_{\alpha}(1-\gamma_5)c_j)(\bar s_j\gamma^{\alpha}(1-\gamma_5)s_i) +
\label{192}\\
&& (\frac{(1-z_{\tau})^2}{2} - \frac{(1-z_{\tau})^3}{3})(\bar
c_i\gamma_{\alpha}\gamma_5c_j)(\bar
s_j\gamma^{\alpha}(1-\gamma_5)s_i)],\nonumber\\
{\cal T}_{WS,bc} &=&
\frac{G_F^2|V_{cb}|^2}{4\pi}m_b^2(1+\frac{m_c}{m_b})^2(1-z_{+})^2[(C_{+}^2 +
C_{-}^2 +
\frac{1}{3}(1 - k^{\frac{1}{2}})(C_{+}^2 - C_{-}^2))\nonumber\\
&&(\bar b_i\gamma_{\alpha}(1
- \gamma_5)b_i)(\bar c_j\gamma^{\alpha}(1 - \gamma_5)c_j) + \label{20}\\
&& k^{\frac{1}{2}}(C_{+}^2 - C_{-}^2)(\bar b_i\gamma_{\alpha}(1 - \gamma_5)b_j)
(\bar c_j\gamma^{\alpha}(1 - \gamma_5)c_i)],\nonumber\\
{\cal T}_{WS,bu} &=&
\frac{G_F^2|V_{cb}|^2}{4\pi}m_b^2(1+\frac{m_u}{m_b})^2(1-z_{+})^2[(C_{+}^2 +
C_{-}^2 +
\frac{1}{3}(1 - k^{\frac{1}{2}})(C_{+}^2 - C_{-}^2))\nonumber\\
&&(\bar b_i\gamma_{\alpha}(1
- \gamma_5)b_i)(\bar u_j\gamma^{\alpha}(1 - \gamma_5)u_j) + \label{21}\\
&& k^{\frac{1}{2}}(C_{+}^2 - C_{-}^2)(\bar b_i\gamma_{\alpha}(1 - \gamma_5)b_j)
(\bar u_j\gamma^{\alpha}(1 - \gamma_5)u_i)],\nonumber\\
{\cal T}_{WS,cd} &=&
\frac{G_F^2}{4\pi}m_c^2(1+\frac{m_d}{m_c})^2(1-z_{+})^2[(C_{+}^2 + C_{-}^2 +
\frac{1}{3}(1 - k^{\frac{1}{2}})(C_{+}^2 - C_{-}^2))\nonumber\\
&&(\bar c_i\gamma_{\alpha}(1
- \gamma_5)c_i)(\bar d_j\gamma^{\alpha}(1 - \gamma_5)d_j) + \label{22}\\
&& k^{\frac{1}{2}}(C_{+}^2 - C_{-}^2)(\bar c_i\gamma_{\alpha}(1 - \gamma_5)c_j)
(\bar d_j\gamma^{\alpha}(1 - \gamma_5)d_i)],\nonumber\end{aligned}$$ $$\begin{aligned}
{\cal T}_{PI,d\bar u}^b &=& {\cal T}_{PI,s\bar c}^b~(z_{-}\to 0)\\
{\cal T}_{PI,e\bar\nu_e}^b &=& {\cal T}_{PI,\mu\bar\nu_{\mu}}^b =
{\cal T}_{PI,\tau\bar\nu_{\tau}}^b~(z_{\tau}\to 0),\\
{\cal T}_{PI,\nu_e\bar e}^c &=& {\cal T}_{PI,\nu_{\mu}\bar\mu}^c =
{\cal T}_{PI,\nu_{\tau}\bar\tau}^c~(z_{\tau}\to 0),\end{aligned}$$ where the following notation has been used: $$\begin{aligned}
\mbox{in Eq.}~(\ref{16})&& z_{-} = \frac{m_c^2}{(m_b-m_c)^2},\quad
k = \frac{\alpha_s(\mu)}{\alpha_s(m_b-m_c)},\nonumber\\ \mbox{in
Eq.}~(\ref{17})&& z_{\tau} =
\frac{m_{\tau}^2}{(m_b-m_c)^2},\nonumber\\ \mbox{in
Eq.}~(\ref{18})&& z_{-} = \frac{m_c^2}{(m_b-m_d)^2},\quad k =
\frac{\alpha_s(\mu)}{\alpha_s(m_b-m_d)},\nonumber\\ \mbox{in
Eq.}~(\ref{181})&& z_{-} = \frac{m_c^2}{(m_b-m_s)^2},\quad k =
\frac{\alpha_s(\mu)}{\alpha_s(m_b-m_s)},\nonumber\\ \mbox{in
Eq.}~(\ref{19})&& z_{-} = \frac{m_s^2}{(m_c-m_u)^2},\quad k =
\frac{\alpha_s(\mu)}{\alpha_s(m_c-m_u)},\nonumber\\ \mbox{in
Eq.}~(\ref{191})&& k =
\frac{\alpha_s(\mu)}{\alpha_s(m_c-m_s)},\nonumber\\ \mbox{in
Eq.}~(\ref{192})&& z_{\tau} =
\frac{m_{\tau}^2}{(m_c-m_s)^2},\nonumber\\ \mbox{in
Eq.}~(\ref{20})&& z_{+} = \frac{m_c^2}{(m_b+m_c)^2},\quad k =
\frac{\alpha_s(\mu)}{\alpha_s(m_b+m_c)},\nonumber\\ \mbox{in
Eq.}~(\ref{21})&& z_{+} = \frac{m_c^2}{(m_b+m_u)^2},\quad k =
\frac{\alpha_s(\mu)}{\alpha_s(m_b+m_u)},\nonumber\\ \mbox{in
Eq.}~(\ref{22})&& z_{+} = \frac{m_s^2}{(m_c+m_d)^2},\quad k =
\frac{\alpha_s(\mu)}{\alpha_s(m_c+m_d)}.\nonumber\end{aligned}$$ In the evolution of coefficients $C_{+}$ and $C_{-}$, we have taken into account the threshold effects, connected to the heavy quark masses.
In expressions (\[5\]) and (\[6\]), the scale $\mu$ has been taken approximately equal to $m_c$. In the Pauli interference term, we suggest that the scale can be determined on the basis of the agreement of the experimentally known difference between the lifetimes of $\Lambda_c$, $\Xi_c^{+}$ and $\Xi_c^{0}$ with the theoretical predictions in the framework described above[^6]. In any case, the choice of the normalization scale leads to uncertainties in the final results. Theoretical accuracy can be improved by the calculation of next-order corrections in the powers of QCD coupling constant.
The coefficients of leading terms, represented by operators $\bar bb$ and $\bar
cc$, are equivalent to the widths fot the decays of free quarks and are known in the next-to-leading logarithmic approximation of QCD [@12; @13; @14; @15; @16], including the mass corrections in the final state with the charmed quark and $\tau$-lepton [@16] in the decays of $b$-quark and with the strange quark mass for the decays of $c$-quark. In the numerical estimates, we include these corrections and mass effects, but we neglect the decay modes suppressed by the Cabibbo angle, and also the strange quark mass effects in $b$ decays.
The expressions for the contribution of operator $\sum_{i=1}^2O_{GQ^i}$ are known in the leading logarithmic approximation. The expressions for the contributions of operators with the dimension 6 have been calculated by us with account of masses in the final states together with the logarithmic renormalization of the effective lagrangian for the non-relativistic heavy quarks at energies less than the heavy quark masses.
Thus, the calculation of lifetimes for the baryons $\Xi_{QQ'}^{\diamond}$ is reduced to the problem of evaluating the matrix elements of operators, which is the subject of next section.
Matrix elements in NRQCD approximation.
=======================================
By using the equations of motion, the matrix element of operator $\bar Q^jQ^j$ can be expanded in series over the powers of ${1}/{m_{Q^j}}$: $$\begin{aligned}
\langle \Xi_{QQ'}^{\diamond}|\bar
Q^jQ^j|\Xi_{QQ'}^{\diamond}\rangle _{norm} = 1 - \frac{\langle
\Xi_{QQ'}^{\diamond}|\bar
Q^j[(i\boldsymbol{D})^2-(\frac{i}{2}\sigma
G)]Q^j|\Xi_{QQ'}^{\diamond}\rangle_{norm}}{2m_{Q^j}^2} +
O(\frac{1}{m_{Q^j}^3}).\end{aligned}$$ Thus, we have to estimate the matrix elements of operators from the following list only: $$\begin{aligned}
&& \bar Q^j(i\boldsymbol{ D})^2Q^j,\quad (\frac{i}{2})\bar
Q^j\sigma GQ^j,\quad \bar Q^j\gamma_{\alpha}(1-\gamma_5)Q^j\bar
q\gamma^{\alpha}(1-\gamma_5)q,\nonumber\\ && \bar
Q^j\gamma_{\alpha}\gamma_5Q^j\bar
q\gamma^{\alpha}(1-\gamma_5)q,\quad \bar
Q^j\gamma_{\alpha}\gamma_5Q^j\bar
Q^k\gamma^{\alpha}(1-\gamma_5)Q^k,\\ && \bar
Q^j\gamma_{\alpha}(1-\gamma_5)Q^j\bar
Q^k\gamma^{\alpha}(1-\gamma_5)Q^k.\nonumber\end{aligned}$$ The meaning of each term in the above list, was already discussed by us in the previous papers on the decays of doubly heavy baryons [@DHD1; @DHD2], so we omit it here.
Further, employing the NRQCD expansion of operators $\bar Q Q$ and $\bar Qg\sigma_{\mu\nu}G^{\mu\nu}Q$, we have $$\begin{aligned}
\bar QQ &=& \Psi_Q^{\dagger}\Psi_Q -
\frac{1}{2m_Q^2}\Psi_Q^{\dagger}(i\boldsymbol{ D})^2\Psi_Q +
\frac{3}{8m_Q^4}\Psi_Q^{\dagger}(i\boldsymbol{ D})^4\Psi_Q
-\nonumber\\ &&
\frac{1}{2m_Q^2}\Psi_Q^{\dagger}g\boldsymbol{\sigma}\boldsymbol{
B}\Psi_Q - \frac{1}{4m_Q^3}\Psi_Q^{\dagger}(\boldsymbol{
D}g\boldsymbol{ E})\Psi_Q + ... \label{32}\\ \bar
Qg\sigma_{\mu\nu}G^{\mu\nu}Q &=&
-2\Psi_Q^{\dagger}g\boldsymbol{\sigma}\boldsymbol{ B}\Psi_Q -
\frac{1}{m_Q}\Psi_Q^{\dagger}(\boldsymbol{ D}g\boldsymbol{
E})\Psi_Q + ... \label{33}\end{aligned}$$ Here the factorization at scale $\mu$ ($m_{Q} > \mu > m_{Q}v_{Q}$) is supposed. We have omitted the term of $\Psi_Q^{\dagger}\boldsymbol{\sigma}
(g\boldsymbol{E} \times \boldsymbol{ D})\Psi_Q$, corresponding to the spin-orbital interactions, which are not essential for the basic state of baryons under consideration. The field $\Psi_Q$ has the standard non-relativistic normalization.
Now we would like to make some comments on the difference between the descriptions of interactions of the heavy quark with the light and heavy heavy ones. As well known, in the doubly heavy subsystem there is an additional parameter which is the relative velocity of quarks. It introduces the energy scale equal to $m_Q v$. Therefore, the Darwin term ($\boldsymbol{ D}\boldsymbol{ E}$) in the heavy subsystem stands in the same order of inverse heavy quark mass in comparison with the chromomagnetic term ($\boldsymbol{\sigma}\boldsymbol{ B}$) (they have the same power in the velocity $v$). This statement becomes evident if we apply the scaling rules of NRQCD [@4]: $$\Psi_Q\sim (m_Qv_Q)^{\frac{3}{2}},\quad|\boldsymbol{ D}|\sim
m_Qv_Q,\quad gE\sim m_Q^2v_Q^3,\quad gB\sim m_Q^2v_Q^4,\quad g\sim
v_Q^{\frac{1}{2}}.$$ For the interaction of heavy quark with the light one, there is no such small velocity parameter, so that the Darwin term is suppressed by the additional factor of $k/m_Q\sim
\Lambda_{QCD}/m_Q$.
Further, the phenomenological experience with the potential quark models shows, that the kinetic energy of quarks practically does not depend on the quark contents of system, and it is determined by the color structure of state. So, we suppose that the kinetic energy is equal to $T = m_dv_d^2/2 + m_lv_l^2/2$ for the quark-diquark system, and it is $T/2 = m_{b}v_{b}^2/2 +
m_{c}v_{c}^2/2$ in the diquark (the color factor of 1/2). Then $$\frac{\langle
\Xi_{QQ'}^{\diamond}|\Psi_Q^{\dagger}(i\boldsymbol{D})^2\Psi_Q|\Xi_{QQ'}^{
\diamond} \rangle }{2M_{\Xi_{QQ'}^{\diamond}}m_Q^2}\simeq
v_Q^2\simeq
\frac{2m_qT}{(m_q+m_{Q'}+m_{Q})(m_{Q'}+m_{Q})}+\frac{m_{Q'}T}{m_Q(m_Q+m_{Q'
})},$$ $$\frac{\langle
\Xi_{QQ'}^{\diamond}|\Psi_{Q'}^{\dagger}(i\boldsymbol{D})^2\Psi_{Q'}|\Xi_{QQ'}^
{
\diamond} \rangle }{2M_{\Xi_{QQ'}^{\diamond}}m_{Q'}^2}\simeq
v_{Q'}^2\simeq
\frac{2m_qT}{(m_q+m_{Q'}+m_{Q})(m_{Q'}+m_{Q})}+\frac{m_QT}{m_{Q'}(m_Q+m_{Q'})},$$ where the diquark terms dominate certainly. Applying the quark-diquark approximation and relating the matrix element of chromomagnetic interaction of diquark with the light quark to the mass difference between the exited and ground states $M_{\Xi_{QQ'}^{\diamond *}} - M_{\Xi_{QQ'}^{\diamond}}$, we have $$\begin{aligned}
\frac{\langle \Xi_{cc}^{\diamond}|\bar cc|\Xi_{cc}^{\diamond}\rangle
}{2M_{\Xi_{cc}^{\diamond}}} &=& 1 -
\frac{1}{2}v_c^2 -
\frac{1}{3}\frac{M_{\Xi_{cc}^{\diamond *}}-M_{\Xi_{cc}^{\diamond}}}{m_c}
- \frac{5 g^2}{18 m_c^3}|\Psi (0)|^2 + ... \nonumber\\
&\approx& 1 - 0.073 -0.025 - 0.009 +\ldots \\
\frac{\langle \Omega_{cc}^{\diamond}|\bar cc|\Omega_{cc}^{\diamond}\rangle
}{2M_{\Omega_{cc}^{\diamond}}} &=& 1 -
\frac{1}{2}v_c^2 -
\frac{1}{3}\frac{M_{\Omega_{cc}^{\diamond *}}-M_{\Omega_{cc}^{\diamond}}}{m_c}
- \frac{5 g^2}{18 m_c^3}|\Psi (0)|^2 + ... \nonumber\\
&\approx& 1 - 0.078 -0.025 - 0.009 +\ldots \\
\frac{\langle \Xi_{bc}^{\diamond}|\bar
cc|\Xi_{bc}^{\diamond}\rangle }{2M_{\Xi_{bc}^{\diamond}}} &=& 1 -
\frac{1}{2}v_c^2 + \frac{g^2}{3m_bm_c^2}|\Psi^d (0)|^2 -
\frac{1}{6m_c^3}g^2|\Psi^d (0)|^2 + \ldots
\nonumber\\ &\approx& 1 - 0.098 + 0.006 - 0.010\ldots \\
\frac{\langle \Omega_{bc}^{\diamond}|\bar
cc|\Omega_{bc}^{\diamond}\rangle }{2M_{\Omega_{bc}^{\diamond}}} &=& 1 -
\frac{1}{2}v_c^2 + \frac{g^2}{3m_bm_c^2}|\Psi^d (0)|^2 -
\frac{1}{6m_c^3}g^2|\Psi^d (0)|^2 + \ldots
\nonumber\\ &\approx& 1 - 0.099 + 0.006 - 0.010\ldots \\
\frac{\langle \Xi_{bb}^{\diamond}|\bar bb|\Xi_{bb}^{\diamond}\rangle
}{2M_{\Xi_{bb}^{\diamond}}} &=& 1 -
\frac{1}{2}v_b^2 -
\frac{1}{3}\frac{M_{\Xi_{bb}^{\diamond *}}-M_{\Xi_{bb}^{\diamond}}}{m_b}
- \frac{5 g^2}{18 m_b^3}|\Psi (0)|^2 + ... \nonumber\\
&\approx& 1 - 0.021 -0.003 - 0.002 +\ldots \\
\frac{\langle \Omega_{bb}^{\diamond}|\bar bb|\Omega_{bb}^{\diamond}\rangle
}{2M_{\Omega_{bb}^{\diamond}}} &=& 1 -
\frac{1}{2}v_b^2 -
\frac{1}{3}\frac{M_{\Omega_{bb}^{\diamond *}}-M_{\Omega_{bb}^{\diamond}}}{m_b}
- \frac{5 g^2}{18 m_b^3}|\Psi (0)|^2 + ... \nonumber\\
&\approx& 1 - 0.021 -0.003 - 0.002 +\ldots\end{aligned}$$ The numerical values of parameters used in the calculations above are given in the next section. Our presentation here is less detailed than in previous papers [@DHD1; @DHD2]. However, we hope, that the interested reader can find there all needed details.
Analogous expressions may be obtained for the matrix elements of operator $Qg\sigma_{\mu\nu}G^{\mu\nu}Q$ $$\begin{aligned}
\frac{\langle \Xi_{cc}^{\diamond}|\bar
cg\sigma_{\mu\nu}G^{\mu\nu}c|\Xi_{cc}^{\diamond}\rangle
}{2M_{\Xi_{cc}^{\diamond}}m_c^2} &=&
-\frac{4}{3}\frac{(M_{\Xi_{cc}^{\diamond *}} -
M_{\Xi_{cc}^{\diamond}})}{m_c}
- \frac{7g^2}{9m_c^3}|\Psi^d (0)|^2 \approx -0.124, \\
\frac{\langle \Xi_{bc}^{\diamond}|\bar
cg\sigma_{\mu\nu}G^{\mu\nu}c|\Xi_{bc}^{\diamond}\rangle
}{2M_{\Xi_{bc}^{\diamond}}m_c^2} &=& \frac{4g^2}{3m_b
m_c^2}|\Psi^d (0)|^2 - \frac{g^2}{3m_c^3}|\Psi^d (0)|^2 \approx
0.005, \\
\frac{\langle \Xi_{bb}^{\diamond}|\bar
bg\sigma_{\mu\nu}G^{\mu\nu}b|\Xi_{bb}^{(\diamond)}\rangle
}{2M_{\Xi_{bb}^{\diamond}}m_b^2} &=&
-\frac{4}{3}\frac{(M_{\Xi_{bb}^{\diamond *}} -
M_{\Xi_{bb}^{\diamond}})}{m_b} - \frac{7g^2}{9m_b^3}|\Psi^d (0)|^2 \approx
-0.189, \\
\langle \Omega_{QQ'}|\bar
cg\sigma_{\mu\nu}G^{\mu\nu}c|\Omega_{QQ'}\rangle &=&
\langle \Xi_{QQ'}^{\diamond}|\bar
cg\sigma_{\mu\nu}G^{\mu\nu}c|\Xi_{QQ'}^{\diamond}\rangle\end{aligned}$$ The permutations of quark masses lead to the required expressions for the operators of $\bar bb$ and $\bar
bg\sigma_{\mu\nu}G^{\mu\nu}b$.
For the four quark operators, determining the Pauli interference and the weak scattering, we use the estimates in the framework of non-relativistic potential model [@DHD1; @DHD2]: $$\begin{aligned}
\langle \Xi_{cc}^{\diamond}|(\bar c\gamma_{\mu}(1-\gamma_5)c)(\bar
q\gamma^{\mu}(1-\gamma_5)q)|\Xi_{cc}^{\diamond}\rangle &=& 12(m_c+m_q)\cdot
|\Psi^{dl}(0)|^2,\\
\langle \Xi_{cc}^{\diamond}|(\bar c\gamma_{\mu}\gamma_5 c)(\bar
q\gamma^{\mu}(1-\gamma_5)q)|\Xi_{cc}^{\diamond}\rangle &=& 8(m_c+m_q)\cdot
|\Psi^{dl}(0)|^2,\\
\langle \Omega_{cc}|(\bar c\gamma_{\mu}(1-\gamma_5)c)(\bar
s\gamma^{\mu}(1-\gamma_5)s)|\Omega_{cc}\rangle &=& 12(m_c+m_s)\cdot
|\Psi^{dl}(0)|^2,\\
\langle \Omega_{cc}|(\bar c\gamma_{\mu}\gamma_5 c)(\bar
s\gamma^{\mu}(1-\gamma_5)s)|\Omega_{cc}\rangle &=& 8(m_c+m_s)\cdot
|\Psi^{dl}(0)|^2,\\
\langle \Xi_{bc}^{\diamond}|(\bar b\gamma_{\mu}(1-\gamma_5)b)(\bar
c\gamma^{\mu}(1-\gamma_5)c)|\Xi_{bc}^{\diamond}\rangle &=&
8(m_b+m_c)\cdot |\Psi^{d}(0)|^2,\\ \langle
\Xi_{bc}^{\diamond}|(\bar b\gamma_{\mu}\gamma_5b)(\bar
c\gamma^{\mu}(1-\gamma_5)c)|\Xi_{bc}^{\diamond}\rangle &=&
6(m_b+m_c)\cdot |\Psi^{d}(0)|^2,\\ \langle
\Xi_{bc}^{\diamond}|(\bar b\gamma_{\mu}(1-\gamma_5)b)(\bar
q\gamma^{\mu}(1-\gamma_5)q)|\Xi_{bc}^{\diamond}\rangle &=&
2(m_b+m_l)\cdot |\Psi^{dl}(0)|^2,\\ \langle
\Xi_{bc}^{\diamond}|(\bar b\gamma_{\mu}\gamma_5b)(\bar
q\gamma^{\mu}(1-\gamma_5)q)|\Xi_{bc}^{\diamond}\rangle &=& 0,\\
\langle \Xi_{bc}^{\diamond}|(\bar c\gamma_{\mu}(1-\gamma_5)c)(\bar
q\gamma^{\mu}(1-\gamma_5)q)|\Xi_{bc}^{\diamond}\rangle &=&
2(m_c+m_l)\cdot |\Psi^{dl}(0)|^2,\\ \langle
\Xi_{bc}^{\diamond}|(\bar c\gamma_{\mu}\gamma_5c)(\bar
q\gamma^{\mu}(1-\gamma_5)q)|\Xi_{bc}^{\diamond}\rangle &=& 0,\\
\langle \Omega_{bc}|(\bar b\gamma_{\mu}(1-\gamma_5)b)(\bar
c\gamma^{\mu}(1-\gamma_5)c)|\Omega_{bc}\rangle &=&
8(m_b+m_c)\cdot |\Psi^{d}(0)|^2,\\ \langle
\Omega_{bc}|(\bar b\gamma_{\mu}\gamma_5b)(\bar
c\gamma^{\mu}(1-\gamma_5)c)|\Omega_{bc}\rangle &=&
6(m_b+m_c)\cdot |\Psi^{d}(0)|^2,\\ \langle
\Omega_{bc}|(\bar b\gamma_{\mu}(1-\gamma_5)b)(\bar
s\gamma^{\mu}(1-\gamma_5)s)|\Omega_{bc}\rangle &=&
2(m_b+m_s)\cdot |\Psi^{dl}(0)|^2,\\ \langle
\Omega_{bc}|(\bar b\gamma_{\mu}\gamma_5b)(\bar
s\gamma^{\mu}(1-\gamma_5)s)|\Omega_{bc}\rangle &=& 0,\\
\langle \Omega_{bc}|(\bar c\gamma_{\mu}(1-\gamma_5)c)(\bar
s\gamma^{\mu}(1-\gamma_5)s)|\Omega_{bc}\rangle &=&
2(m_c+m_s)\cdot |\Psi^{dl}(0)|^2,\\ \langle
\Omega_{bc}|(\bar c\gamma_{\mu}\gamma_5c)(\bar
s\gamma^{\mu}(1-\gamma_5)s)|\Omega_{bc}\rangle &=& 0,\\
\langle \Xi_{bb}^{\diamond}|(\bar b\gamma_{\mu}(1-\gamma_5)b)(\bar
q\gamma^{\mu}(1-\gamma_5)q)|\Xi_{bb}^{\diamond}\rangle &=& 12(m_b+m_q)\cdot
|\Psi^{dl}(0)|^2,\\
\langle \Xi_{bb}^{\diamond}|(\bar b\gamma_{\mu}\gamma_5 b)(\bar
q\gamma^{\mu}(1-\gamma_5)q)|\Xi_{bb}^{\diamond}\rangle &=& 8(m_b+m_q)\cdot
|\Psi^{dl}(0)|^2.\end{aligned}$$ The color structure of wave functions leads to the relations $$\langle \Xi_{QQ'}^{\diamond}|(\bar Q_iT_{\mu}Q_k)(\bar
q_k\gamma^{\mu}(1-\gamma_5)q_i)|\Xi_{QQ'}^{\diamond}\rangle
=
-\langle \Xi_{QQ'}^{\diamond}|(\bar QT_{\mu}Q)(\bar
q\gamma^{\mu}(1-\gamma_5)q)|\Xi_{QQ'}^{\diamond}\rangle ,$$ where $T_{\mu}$ is an arbitrary spinor matrix.
Numerical estimates
===================
Performing the numerical calculations of lifetimes for the doubly heavy baryons, we have used the following set of parameters: $$\begin{aligned}
&&m_s = 0.2~\mbox{GeV}\quad m_l = 0.~GeV\quad m_s^{*} =
0.45~\mbox{GeV}\quad m_l^{*} = 0.3~\mbox{GeV}\nonumber
\label{parameters}\\ &&\quad\quad |V_{cs}| = 0.9745 \quad |V_{bc}|
= 0.04 \quad T = 0.4~\mbox{GeV}
\\ &&\quad\quad\quad m_c = 1.55~\mbox{GeV}\quad m_b = 5.05~\mbox{GeV}\end{aligned}$$ The numerical values of diquark wavefunctions at the origin for baryons under consideration are collected in Table \[WF\]. The masses of doubly heavy baryons may be found in Table \[Bmasses\].
$\Xi_{cc}^{++}$ $\Xi_{cc}^{+}$ $\Omega_{cc}^{+}$ $\Xi_{bc}^{+}$ $\Xi_{bc}^{0}$ $\Omega_{bc}^{0}$ $\Xi_{bb}^{0}$ $\Xi_{bb}^{-}$ $\Omega_{bb}^{-}$
----------------------------------- ----------------- ---------------- ------------------- ---------------- ---------------- ------------------- ---------------- ---------------- -------------------
$\Psi^d (0)$, GeV$^{\frac{3}{2}}$ 0.150 0.150 0.150 0.205 0.205 0.205 0.380 0.380 0.380
: The values of diquark wavefunctions for the doubly heavy baryons at the origin.[]{data-label="WF"}
$\Xi_{cc}^{++}$ $\Xi_{cc}^{+}$ $\Omega_{cc}^{+}$ $\Xi_{bc}^{+}$ $\Xi_{bc}^{0}$ $\Omega_{bc}^{0}$ $\Xi_{bb}^{0}$ $\Xi_{bb}^{-}$ $\Omega_{bb}^{-}$
-------------- ----------------- ---------------- ------------------- ---------------- ---------------- ------------------- ---------------- ---------------- -------------------
$M$, GeV 3.478 3.478 3.578 6.82 6.82 6.92 10.093 10.093 10.193
$M^{*}$, GeV 3.61 3.61 3.71 - - - 10.193 10.193 10.293
: The masses of doubly heavy baryons $M$, and $M^{*}$ stands for the mass of the baryon with lowest excited state of light quark-diquark system.[]{data-label="Bmasses"}
The wavefunctions as well as masses for the considered baryons are taken from [@DHS1; @DHS2], where their estimates in the non-relativistic model with the Buchmüller-Tye potential were done. The $b$-quark mass is obtained from the requirement, that for any given value of $c$-quark mass the theoretically computed $B_d$-meson lifetime equals to experimentally measured value. This matching condition leads to the following approximate relation $$m_b = m_c + 3.5~\mbox{GeV}. \label{mb-mc}$$ The $c$-quark mass is determined from the analogous matching procedure for the $B_c$-meson lifetime [@Onishchenko]. The $m_q^{*}$-values in Eq. (\[parameters\]) represent the constituent masses for the corresponding light quarks, used by us in estimations of hadronic matrix elements.[^7] For the value of light quark-diquark function at the origin we assume $$|\Psi^{dl} (0) |^2 = \frac{2}{3}\frac{f_D^2 M_D
k^{-\frac{4}{9}}}{12},$$ where $f_D = 170~\mbox{MeV}$. This expression obtained by performing the steps similar to [@Rujula; @Cortes] for the derivation of hyperfine splitting in the light quark-diquark system. The factor $k^{-\frac{4}{9}}$ accounts for the low energy logarithmic renormalization of $f_D$ constant. We have used this equation for all doubly heavy baryons, considered in this paper. Even though we use this relation to compute central values of lifetimes, the precise values of wavefunction parameters are under question, so in the presented results we have allowed for variations.
The renormalization scale $\mu$ is chosen in the following way: $\mu_1 = m_b$ and $\mu_2 = m_c$ in the estimates of Wilson coefficients $C_{+}(\mu)$ and $C_{-}(\mu)$ for the effective four-fermion weak lagrangian at low energies with the $b$ and $c$-quarks, correspondingly. For nonspectator effects, which are the Pauli interference and weak scattering of valence quarks, the renormalization scale $\mu$ is obtained from the fit of theoretical predictions for the lifetime differences of baryons $\Lambda_c$, $\Xi_c^{+}$ and $\Xi_c^{0}$ over the experimental data.
In Table \[cc\] we present the results of calculations for the doubly charmed baryons. Together with the total lifetimes of these baryons we have shown the relative spectator and nonspectator contributions.
$\Xi_{cc}^{++}$ $\Xi_{cc}^{+}$ $\Omega_{cc}^{+}$
-------------------------- ----------------- ---------------- -------------------
$\sum c\to s$, ps$^{-1}$ 3.104 3.104 3.104
PI, ps$^{-1}$ -0.874 - 0.621
WS, ps$^{-1}$ - 1.776 -
$\tau$, ps 0.45 0.20 0.27
: The lifetimes of doubly charmed baryons together with the relative spectator and nonspectator contributions to the total widths.[]{data-label="cc"}
From this Table we see the importance of nonspectator effects, producing huge differences in the values of lifetimes. The analogous results for other doubly heavy baryons can be found in Tables \[bc\] and \[bb\].
$\Xi_{bc}^{+}$ $\Xi_{bc}^{0}$ $\Omega_{bc}^{0}$
-------------------------- ---------------- ---------------- -------------------
$\sum b\to c$, ps$^{-1}$ 0.632 0.632 0.631
$\sum c\to s$, ps$^{-1}$ 1.511 1.511 1.509
PI, ps$^{-1}$ 0.807 0.855 0.979
WS, ps$^{-1}$ 0.653 0.795 1.713
$\tau$, ps 0.28 0.26 0.21
: The lifetimes of $(bcq)$-baryons together with the relative spectator and nonspectator contributions to the total widths.[]{data-label="bc"}
$\Xi_{bb}^{0}$ $\Xi_{bb}^{-}$ $\Omega_{bb}^{-}$
-------------------------- ---------------- ---------------- -------------------
$\sum b\to c$, ps$^{-1}$ 1.254 1.254 1.254
PI, ps$^{-1}$ - -0.0130 -0.0100
WS, ps$^{-1}$ 0.0189 - -
$\tau$, ps 0.79 0.80 0.80
: The lifetimes of $(bbq)$-baryons together with the relative spectator and nonspectator contributions to the total widths.[]{data-label="bb"}
A small comment concerns with the corrections to the spectator decays of heavy quarks, caused by the motion of heavy quarks inside the hadron and interactions with the light degrees of freedom. The corrections due to the quark-gluon operators of dimension 5 are numerically small [@vs]. The most important terms come from the kinetic energy of heavy quarks.
In Figs. \[ccu\]-\[bbs\] we have shown the dependence of baryons lifetimes from the values of light quark-diquark wavefunctions at the origin. We see quite a different behaviour withthe increase of $|\Psi^{dl} (0)|$-parameter.
Here, we would like to note, that in this paper we do not give a detail discussion of nonspectator effects on the total lifetimes and semileptonic branching ratios of doubly heavy baryons and promise to fill this gap in one of our subsequent papers [@Onishchenko1].
Finally, concerning the uncertainties of the presented estimates, we note that they are mainly determined by the following:
1\) The $c$-quark mass is poorly known, but constrained by the fits to the experimental data, discussed above, can lead to the uncertainty $\frac{\delta\Gamma}{\Gamma}\approx 15\%$ in the case of doubly charmed baryons and $\frac{\delta\Gamma}{\Gamma}\approx
10\%$ for the case of $bcq$ - baryons.
2\) The uncertainties in the values of diquark and light quark-diquark wavefunctions lead to $\frac{\delta\Gamma}{\Gamma}\approx 30\%$ in the case of doubly charmed baryons and $\frac{\delta\Gamma}{\Gamma}\approx 15\%$ for the $bcq$ - baryons.
Thus, the estimated uncertainty in predictions for the lifetimes of doubly heavy baryons is close to $25\%$ in the case of $(bcq)$ - baryons, of order of $45\%$ in the case of doubly charmed baryons and less then $5\%$ in the case of $(bbq)$ - baryons.
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Conclusion
==========
In the present paper we have performed a detail investigation and numerical estimates for the lifetimes of doubly heavy baryons. The used approach is based on OPE expansion of total widths for the corresponding hadrons, and it is combined with the formalism of effective fields theories developed previously. In this way, we have accounted for the both perturbative QCD and mass corrections to the Wilson coefficients of operators. The nonspectator effects, presented by Pauli interference and weak scattering, and their influence on the total lifetimes are considered. The obtained results show the significant role played by them in the description of lifetimes of doubly heavy baryons.
The authors thank V.V.Kiselev for fruitful discussions and remarks concerning the presentation of results.
This work is in part supported by the Russian Foundation for Basic Research, grants 96-02-18216 and 96-15-96575. The work of A.I. Onishchenko was supported by International Center of Fundamental Physics in Moscow and Soros Science Foundation.
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[^1]: For review see [@al].
[^2]: The first experimental observation of the $B_c$-meson was recently reported by the CDF Collaboration [@CDFbc]; see ref.[@thbc] for a theoretical review of $B_c$-meson physics before the observation.
[^3]: See comments on the difference in the numerical values of lifetimes of doubly charmed baryons in [@Onishchenko].
[^4]: It was shown in [@bigi] that the first order $1/m_Q$-correction is absent, and the corrections begin with the $1/m_Q^2$-terms.
[^5]: Others mean those of not considered in [@DHD1; @DHD2].
[^6]: A more extended description is presented in [@DHD1].
[^7]: See [@DHD1] for details.
|
---
abstract: |
We use the UKIDSS Ultra-deep survey (UDS), currently the deepest panoramic near infra-red survey, together with deep Subaru optical imaging to measure the clustering, number counts and luminosity function of galaxies at $z\sim 2$ selected using the BzK selection technique. We find that both star-forming (sBzK) and passive (pBzK) galaxies, to a magnitude limit of $K_{AB} <
23$, are strongly clustered. The passive galaxies are the most strongly clustered population, with scale lengths of $r_0 =
15.0^{+1.9}_{-2.2}$h$^{-1}$Mpc compared with $r_0 =
6.75^{+0.34}_{-0.37}$h$^{-1}$Mpc for star-forming galaxies. The direct implication is that passive galaxies inhabit the most massive dark-matter halos, and are thus identified as the progenitors of the most massive galaxies at the present day. In addition, the pBzKs exhibit a sharp flattening and potential turn-over in their number counts, in agreement with other recent studies. This plateau cannot be explained by the effects of incompleteness. We conclude that only very massive galaxies are undergoing passive evolution at this early epoch, consistent with the downsizing scenario for galaxy evolution. Assuming a purely passive evolution for the pBzKs from their median redshift to the present day, their luminosity function suggests that only $\sim 2.5 \%$ of present day massive ellipticals had a pBzK as a main progenitor.
author:
- |
W. G. Hartley$^{1}$[^1], K. P. Lane$^{1}$, O. Almaini$^{1}$, M. Cirasuolo$^{2}$, S. Foucaud$^{1}$, C. Simpson$^{3}$, S. Maddox$^{1}$, I. Smail$^{4}$, C. J. Conselice$^{1}$, R. J. McLure$^{2}$, J. S. Dunlop$^{5}$\
$^{1}$School of Physcis and Astronomy, University of Nottingham, University Park, Nottingham NG7 2RD\
$^{2}$SUPA[^2] Institute for Astronomy, University of Edinburgh, Royal Observatory, Edinburgh EH9 3HJ\
$^{3}$Astrophysics Research Institute, Liverpool John Moores University, Twelve Quays House, Egerton Wharf, Birkenhead CH41 1LD\
$^{4}$Institute for Computational Cosmology, Department of Physics, Durham University, Durham DH1 3LE\
$^{5}$Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Rd., Vancouver, B.C., V6T 1Z1, Canada\
bibliography:
- 'mn-jour.bib'
- 'papers\_cited\_by\_KL.bib'
title: 'The clustering and abundance of star-forming and passive galaxies at $z \sim 2$'
---
\[firstpage\]
Infrared: galaxies – Cosmology: large-scale structure – Galaxies: High Redshift – Galaxies: Evolution – Galaxies: Formation.
Introduction
============
There is growing evidence to support the view that the most massive objects in the Universe were the first to assemble and complete their star formation (@Kodama_etal:2004 [@Thomas_etal:2005; @DeLucia_etal:2004; @Bundy_etal:2006; @Stott_etal:2007]); this phenomenon has become known as [*downsizing*]{} [@Cowie_etal:1999]. A number of key issues remain unresolved however. There are indications that the build up of the galaxy colour bimodality occurs around $z\sim2$ (e.g. @Cirasuolo_etal:2007), but the precise evolutionary path from the distant Universe to the present day is still undetermined. The mechanism that terminates the major episode of star-formation is poorly understood [@Benson_etal:2005], and it is now clear that massive galaxies undergo significant size evolution from $z\sim2$ to the present day (e.g. @Cimatti_etal:2008 and references therein).
Galaxy clustering is an important tool for investigating these populations, since the amplitude of clustering on large scales ($>1$Mpc) can provide a measurement of the dark matter halo mass [@Mo_and_White:1996; @Sheth_and_Tormen:1999]. In principle, therefore, one can relate galaxy populations from the distant past to the present day by tracing the evolution of dark matter halos within the context of a framework for structure formation.
A full exploration of these issues will require large spectroscopic surveys of IR-selected galaxies over a representative volume of the distant Universe. Such surveys are currently at an early stage, so recent work has focused on the photometric colour selection of passive vs. star-forming galaxies at the crucial $z\sim 2$ epoch. The two key methods to date are the BM/BX selection [@Erb_etal:2003], which is an extension of the Lyman-break dropout technique, and the BzK technique [@Daddi_etal:2004] which is based on $K$-band selection. Recent studies have shown that the BM/BX technique is reasonably efficient at selecting actively star-forming galaxies at $z\sim 2$, but largely misses the passive galaxy population at these epochs [@Quadri_etal:2007; @Grazian_etal:2007]. In contrast, the BzK method appears to be the most complete of the broad-band techniques for the selection of both star-forming and passive galaxies [@Daddi_etal:2004; @VanDokkum_etal:2006; @Grazian_etal:2007]. Based on initial $K$-band selection, in principle it does not suffer the same heavy biases caused by dust or the age of the stellar populations.
An alternative is to use a larger set of filters over a range of wavelengths and infer a galaxy’s redshift from comparison of the calculated magnitudes with a set of templates (@Cirasuolo_etal:2007 and references therein). This work makes use of such photometric redshifts and in principle one can derive the stellar age of a galaxy using such templates. However due to the uncertainties in determining stellar ages from the template fits, and the desire for comparison with the literature, the bulk of the present analysis is based on simple BzK selection.
The BzK technique was first developed using the K20 survey [@Cimatti_etal:2002] by [@Daddi_etal:2004], and preferentially selects star-forming and passive galaxies in the redshift range $1.4 < z < 2.5$ by using the B-, z- and K$_s$-band broadband filters. Recent work has shown that the number counts of these populations differ markedly [@Kong_etal:2006; @Lane_etal:2007], with star-forming galaxies being far more abundant at all magnitudes. The abundance of sBzK galaxies allowed [@Kong_etal:2006] (henceforth K06) to perform a detailed clustering analysis, segregated by limiting K-band luminosity in the range $18.5 < K_{vega} < 20$. They found that the clustering of sBzKs is strongly dependent on $K$-band luminosity. [@Hayashi_etal:2007] studied the clustering of sBzKs over a smaller area (180 arcmin$^{2}$) but to much greater depth (K$_{AB} < 23.2$) and confirmed this strong luminosity dependence. The sBzK population therefore appear to inhabit a range of halo masses, and are therefore likely to be the progenitors of a wide range of present day galaxies.
There have been fewer studies of [*passive*]{} galaxy clustering at high redshift, in part because of their relatively low surface density. K06 found early evidence that pBzKs are more strongly clustered than sBzKs, although the limited depth and area of this survey ($320 {\rm arcmin}^2$ to K$_{Vega}=20$) gives rise to significant statistical uncertainly, particularly given the relatively low surface density of pBzKs at these magnitudes ($0.38$ arcmin$^{-2}$). [@Blanc_etal:2008] measured the clustering and number counts of BzK-selected galaxies to the same depth as K06, but over a much wider area (0.71 deg$^2$ total between two fields). Their BzK clustering amplitudes are smaller than those of K06 but consistent at the $1\sigma$ level. Analysis of deeper survey data over a similar area is required to confirm these results, and we present such an analysis in this work.
[@Lane_etal:2007] (henceforth L07) used the Early Data Release (EDR) from the UKIDSS UDS to investigate the number counts and overlap in colour-selected galaxies, including those selected by the BzK technique. They found that the flattening in pBzK number counts observed by K06 extends to become a plateau at the EDR depth of $K_{AB} < 22.5$ and noted that if this were to continue to turn over it could imply an absence of high-redshift passive galaxies at low luminosities.
In this work we investigate the number counts and clustering of BzK-selected galaxies over a substantially greater depth and area than any previous study. Section 2 describes the data and galaxy number counts. In Section 3 we present an angular clustering analysis, which is extended by de-projection to infer the real-space correlation lengths. In Section 4 we use photometric redshifts to derive a luminosity function for these populations, and compare with the present-day galaxy luminosity function. A discussion and conclusions are presented in Section 5. Throughout this paper we assume a flat $\Lambda$CDM cosmology with $\Omega_m = 0.3$ and $h = 0.73$.
Source identification and number counts
=======================================
Data set and sample definitions
-------------------------------
![BzK colour-colour diagram for sources in the UDS DR1. The sBzK and pBzK selection regions are marked accordingly. Also visible are the passive galaxy track identified in [@Lane_etal:2007], and the stellar locus used to match our photometric system to that of [@Daddi_etal:2004] (see text). We show mean error-bars for the galaxies in the BzK selection regions.[]{data-label="bzk"}](PSfiles/UDS_DR1_BzK_v1.3_daddifilters.ps){width="240pt"}
The UKIRT Infrared Deep Sky Survey (UKIDSS) has been underway since spring 2005 and comprises 5 sub-surveys covering a range of areas and depths [@Lawrence_etal:2007]. This survey has been made possible by the advent of the UKIRT Wide-Field Camera [@Casali_etal:2007].
We base the present study on data from the deepest component of UKIDSS, the Ultra Deep Survey (Almaini et al. in preparation). This aims to reach final depths of $K_{AB} = 25.0, H_{AB} = 25.4, J_{AB} =
26.0$ ($5\sigma$, point source, 2”) over an area of 0.8 deg$^2$. The size of the UDS field significantly reduces the effects of cosmic variance and on this scale is the deepest near-infrared survey to date. For this work we use the UDS DR1 release [@Warren_etal:2007], which reaches $5\sigma$ (point source) depths of K$_{AB} = 23.5$ and J$_{AB} = 23.7$. For details of the completeness estimation, image stacking, mosaicing and catalogue extraction procedures see [@Foucaud_etal:2007] and Almaini et al. (in preparation). In addition to these data, deep $B,V,R,i^{\prime},z^{\prime}$ imaging is also available from Subaru Suprimecam to limiting depths of $ B_{AB} = 28.4, ~V_{AB} = 27.8, ~R_{AB} = 27.7,
~i_{AB}^{\prime} = 27.7$ and $ z_{AB}^{\prime} = 26.7$ [@Furusawa_etal:2008]. The UDS and Subaru survey areas are not entirely coincident, which reduces our usable area to 0.63 deg$^2$ in this analysis.
Construction of the BzK colour-colour diagram
---------------------------------------------
The original BzK selection technique was introduced in @Daddi_etal:2004 based on photometry in the Bessel $B$-band, Gunn $z$-band and $K_s$-bands as defined on the FORS1, FORS2 and ISAAC instruments at the Very Large Telescope (VLT). To correct our colours to these filters we used the stellar track of K06, who in turn used published empirical stellar spectra from @Pickles:1998 and synthetic stellar spectra from @Lejeune_etal:1997, convolved with the filter responses of the VLT instruments used in @Daddi_etal:2004. These derived colours provide a convenient reference for matching stars on the well-defined BzK stellar locus. We use the adjusted stellar locus of K06 in the following way.
The stellar locus can be split into a main branch and a “knee” feature, clearly visible in the lower part of figure \[bzk\]. The intersection of these branches provides a fixed point in the BzK plane from which we can derive a quantitative adjustment. Using a least squares fit to each section, we derived the following offset in $z-K$ and $B-z$ colours to this fixed point to convert to the photometric system of @Daddi_etal:2004: $$(z-K)_{Daddi} = (z^{\prime}-K)_{UDS} - 0.26$$ $$(B-z)_{Daddi} = (B-z^{\prime})_{UDS} + 0.06$$
This correction is then applied to all sources in the BzK plane. We henceforth refer to BzK photometry defined in this way.
The standard BzK definitions of @Daddi_etal:2004 were then used to construct our BzK sample. K-band sources ($> 5 \sigma$) from the UDS were used as the primary catalogue, with optical magnitudes extracted directly from the Subaru imaging data after careful matching of the Subaru and UDS astrometric frames. Aperture magnitudes were then extracted using a 2-arcsec diameter. Only sources outside the contaminated halos of saturated optical stars were used.
![Differential number counts of passive and star-forming BzK galaxies with apparent K-band magnitude, compared with the full-sample of $K$-selected galaxies. The error-bars shown are standard Poisson errors on the counts. Two sets of values are shown for pBzKs: the standard selection of pBzKs (filled, red triangles); and the worst case sample described in the text (open, green triangles). Numerical values are reproduced in table \[tab-counts\]. Literature values from K06 and [@Blanc_etal:2008] are also shown (dot-dashed lines and dashed lines respectively). The plateau identified in previous works (K06, L07, @Blanc_etal:2008) and subsequent turn-over prior to our magnitude limit are apparent in the pBzK number counts.[]{data-label="counts"}](PSfiles/DR1_bzk_K_number_counts_worstcase.ps){width="240pt"}
All K-band sources with $K_{AB} < 23.5$ were used for BzK selection unless both the B- and z$^{\prime}$-band magnitudes were fainter than the $3
\sigma$ limits measured on those images ($0.29\%$ of sources outside of contaminated regions), since these cannot be constrained within the BzK plane.
Overall this procedure results in the selection of 15177 sBzKs (21.7% of the full sample) and 742 pBzKs (1.06 % of the sample), 11551 and 702 respectively at $K_{AB} < 23.0$.
Number counts
-------------
![$(z - K)_{AB}$ colour versus K-band magnitude. The dashed line is our completeness limit of $K_{AB} = 23.5$ for the full sample, while the solid line shows how the completeness limit in z affects $(z - K)$ colour. We begin to be incomplete above $(z - K) > 2.94$. The impact of this incompleteness is discussed in the text.[]{data-label="kzk"}](PSfiles/Kvszk.ps){width="240pt"}
[|l|l|l|l|l|]{} K bin&all&sBzK&pBzK&Worst\
centre&sources& & &case pBzK\
17.41&2.832&0.198&-&-\
17.91&2.803&0.676&-&-\
18.41&2.957&0.198&-&-\
18.91&3.203&0.500&-&-\
19.41&3.380&1.403&0.676&0.676\
19.91&3.556&2.025&1.429&1.429\
20.41&3.720&2.504&1.914&1.923\
20.91&3.872&2.982&2.181&2.185\
21.41&3.998&3.315&2.386&2.394\
21.91&4.121&3.592&2.386&2.394\
22.41&4.229&3.778&2.377&2.431\
22.91&4.285&3.849&2.158&2.394\
23.41&4.258&3.536&1.560&2.158\
23.91&3.942&0.801&-&-\
\[tab-counts\]
The differential K-band number counts are illustrated in Figure \[counts\], and tabulated in Table 1, for both the full sample of galaxies and those selected as passive and star-forming BzKs. We find a steep rise in the counts of star-forming sBzKs toward fainter magnitudes. In contrast, pBzK number counts exhibit an apparent flattening at $K\sim 21$, consistent with the findings of K06 and L07. Since these galaxies are sampled over a relatively narrow redshift range, this may imply that the passive population consists largely of luminous galaxies at this early epoch (see Section 4).
To investigate the reality of the turn-over, we note that the galaxies around and beyond this feature have very faint B-band magnitudes, with a substantial fraction (62%) below the $3\sigma$ detection limit in this band ($B_{lim} = 28.4$). A non-detection in the $B$-band does not affect their classification, however, and merely pushes them red-ward in B-z colour in the BzK diagram illustrated in Figure 1. The more worrying class are the objects which are below the $3\sigma$ limit in $z'$ [*and*]{} $B$, which cannot be assigned BzK classification (164 objects). Figure \[kzk\] shows that even within our K-band limit, we become incomplete above $(z - K) \simeq 3$. We study the effect of this incompleteness by considering the extreme case in which all of the objects with B- and $z^{\prime}$-band magnitudes fainter than the $3\sigma$ limits (28.4 and 26.7 respectively) are pBzKs. In this ’worst case’ sample the number counts still exhibit a plateau, as before, but with the absence of a turn-over.
At our conservative limit of $K_{AB}< 23$ our completeness is $> 95$% for compact galaxies (Almaini et al., in prep.), so the feature is unlikely to be due to $K$-band incompleteness unless there is unprecedented size-evolution in the pBzK population towards fainter magnitudes compared to the general population. The extreme compactness of passive galaxies at these redshifts observed by [@Cimatti_etal:2008] suggests that of the two classes, the sBzKs should suffer more from such incompleteness. There is no evidence from their number counts to suggest that they are adversely affected in this way, so we expect that the pBzKs are likewise unaffected below this magnitude.
Photometric errors are another source of concern. Adjusting the boundaries for pBzK selection by the mean photometric errors for sources close to the boundaries ($0.1$ magnitudes), we found that such errors had no noticeable impact on the decline in the pBzK population. Additionally, since the pBzK region of the colour-colour plane is sparsely populated, photometric errors are likely to scatter fainter objects [*into*]{} pBzK selection, rather than the opposite. The faint pBzK counts are thus likely to be slightly [*overestimated*]{} because of photometric errors.
We conclude that the flattening in the pBzK population is likely to be real and the subsequent turn-over is probable but not certain. We note that the turnover corresponds to absolute magnitude $ M_K = -23.6$ at z = 1.4, which is close to the value of $L^*$ determined for the K-band luminosity function at these redshifts [@Cirasuolo_etal:2007]. Taken at face value, these results are suggestive of a sharp decline in the number density of pBzK galaxies towards lower luminosities, consistent with expectation from downsizing. As correctly pointed out in [@Grazian_etal:2007] and predicted by [@Daddi_etal:2004], this is not necessarily equivalent to a true decline in the number density of passive galaxies, since the efficiency and completeness of the pBzK technique is largely untested at such faint magnitudes. In particular, there is some evidence that passive galaxies can also be found among the redder sBzK galaxies (at large values of $z-K$), with a possible incompleteness as high as $34\%$ [@Grazian_etal:2007]. We find that randomly adding an additional $34$% of galaxies to the pBzK sample from this part of the diagram is indeed sufficient to remove the apparent turn-over, since these are predominantly the fainter sBzK galaxies.
The sBzK number counts are significantly higher than those of K06, [@Blanc_etal:2008] and [@Imai_etal:2008] (not plotted); while the pBzK number counts are lower. The UDS field is more than 6 times larger than that of the K20 survey and 4 times larger than the one used in [@Imai_etal:2008]. The combined field used in [@Blanc_etal:2008] is of a similar size to the UDS, cosmic variance is the most likely explanation for the difference in number counts in this case.
Clustering properties
=====================
Angular clustering
------------------
The 2-point angular correlation function, $w(\theta)$ is defined by the joint probability of finding two galaxies in solid angle elements $\delta\Omega_1$ and $\delta\Omega_2$ at a given separation [@Peebles:1980], $$\delta P = n^2 \delta\Omega_1 \delta\Omega_2 (1 + w(\theta_{12})).$$
![The angular correlation function for our BzK-selected galaxy samples. The best-fitting power laws are shown, with slopes fixed to the fiducial value of $\delta=0.8$. The pBzK galaxies are very strongly clustered, much more so than sBzKs, indicating that they occupy the most massive dark matter halos at their epoch.[]{data-label="wth"}](PSfiles/clustering_deconv_new_BzK.ps){width="240pt"}
To estimate the correlation function we use the estimator of [@Landy_and_Szalay:1993],
$$w(\theta) = \frac{DD - 2DR + RR}{RR}$$
where DD, DR and RR are the counts of data-data, data-random and random-random pairs respectively at angular separation $\theta$, normalised by the total number possible. Although this estimator is relatively robust against systematic errors, there remains a small bias due to the finite field size, which is corrected for an integral constraint by a constant, C. We follow the method of [@Roche_and_Eales:1999] by using the random–random counts to estimate the size of this bias:
$$C=\frac{\Sigma N_{RR}(\theta)\theta^{-0.8}}{\Sigma N_{RR}(\theta)},$$
where the sums extend to the largest separations within the field.
The sBzK and pBzK samples were selected to a limit of $K_{AB} <
23$. We adopted this conservative magnitude limit to ensure the minimum contamination by spurious sources. We fit a single power law of the form $w(\theta) = A(\theta^{-\delta} - C)$ to the corrected data over the separation range $0.01 - 0.1$ degrees, fixing the slope to the fiducial $\delta=0.8$ and minimising the $\chi^2$. The errors on the measurements are a combination of those due to shot noise (estimated by bootstrap resampling) and an estimate of those due to cosmic variance. The estimates for cosmic variance were found by splitting the field into 4 and computing the variance of $w(\theta)$.
In order to fit the clustering reliably, we wish to avoid small-scale excesses due to multiple galaxy occupation of a single halo. The lower bound was chosen to correspond to $\sim 0.9$Mpc (co-moving) at z$\sim 2$ for this reason. The upper bound is a conservative estimate of the limit to which our data have enough signal for a reliable fit to be obtained.
For sBzK galaxies the amplitude was found to be $A=1.79^{+0.17}_{-0.17} \times 10^{-3}$ (deg.$^{0.8}$)and for the pBzK population $A = 6.37^{+1.58}_{-1.54} \times 10^{-3}$ (deg.$^{0.8}$). The clustering amplitude of the pBzKs is inconsistent with the clustering of sBzK galaxies at the 3-sigma level. Figure \[wth\] shows the clustering measurements corrected for the integral constraint for the sBzK and pBzK galaxies.
De-projected clustering amplitude
---------------------------------
The real space clustering and projected clustering are linked by the relativistic Limber equation [@Limber:1954]. If the redshift distribution of a sample is known, the Limber equation can be inverted and the correlation length, $r_0$, can be calculated in a robust manner [@Peebles:1980; @Magliocchetti_and_Maddox:1999]
To estimate the redshift distribution we use photometric redshifts based on the Subaru and UDS bands previously introduced, with the addition of Spitzer data taken as part of the SWIRE survey [@Lonsdale_etal:2003]. The method is described fully in [@Cirasuolo_etal:2007], and consists of minimising the $\chi^2$ of synthetic galaxy templates. The distributions are shown in figure \[nz\]. The photometric redshift distribution was then used directly in the inverted Limber’s equation during the calculation of $r_0$. The values obtained in this manner are r$_0 = 15.8^{+2.0}_{-2.2}h^{-1}$ Mpc and $7.69^{+0.40}_{-0.41}h^{-1}$ Mpc for $K < 23$ pBzKs and sBzKs respectively. The quoted errors are due to the error in the fit to the clustering amplitude and therefore take into account the shot noise and cosmic variance.
![Photometric redshift distributions for passive (left) and star-forming (right) BzK-selected galaxies (solid line histogram). The over-plotted dashed lines are distributions with photometric errors de-convolved (see text).[]{data-label="nz"}](PSfiles/BzK_nz_deconv.eps){width="240pt"}
The photometric redshifts are subject to errors ($\sigma/(1+z) = 0.095$ and $0.105$ for pBzKs and sBzKs respectively), however, and assuming that the true redshift distribution is highly peaked, these errors are likely to broaden the measured distribution. A broader distribution will result in a larger inferred clustering scale length. It is therefore important that we take such errors into account. We do so by assuming the errors are Gaussian, and then deconvolve the errors from the redshift distribution using a Fourier-based Wiener filter. A more detailed description of this method is provided in the appendix. The resulting de-convolved redshift distributions are shown in figure \[nz\].
The scaling lengths recovered using the corrected redshift distribution are as follows: $15.0^{+1.9}_{-2.2}h^{-1}$ Mpc and $6.75^{+0.34}_{-0.37}h^{-1}$ Mpc for pBzKs and sBzKs respectively.
However, we note that our photometric redshift code is largely untested for pBzK and sBzK galaxies. This work will be refined by the use of an ongoing ESO Large Programme using VIMOS and FORS2 on the VLT to target one sixth of the UDS DR1 galaxies with photometric redshifts $> 1$.
We also confirm previous claims for a dependence of sBzK clustering on apparent magnitude [@Hayashi_etal:2007]. Figure \[sBzK\] and Table \[sBzKtab\] show the values for sBzKs with varying limiting magnitude. The dependence is much stronger at magnitudes of $K < 22$, indicating a strong correlation between halo mass and sBzK luminosity for these objects.
![The dependence of clustering strength of sBzK-selected galaxies on limiting K$_{AB}$-band magnitude. Our measurements (open squares) are shown together with literature values from [@Blanc_etal:2008] and [@Hayashi_etal:2007] (open triangle and asterisks respectively). Our values confirm the magnitude dependence of sBzK clustering strength.[]{data-label="sBzK"}](PSfiles/Lit_sBzK_K_deconv.ps){width="240pt"}
[|l|l|l|]{} K$_{AB,lim}$&N&r$_0$\
20.0&92&$39.0^{+11.8}_{-15.9}$\
20.5&250&$16.4^{+6.5}_{-10.0}$\
21.0&689&$12.6^{+2.7}_{-3.2}$\
21.5&1724&$8.71^{+1.49}_{-1.68}$\
22.0&3789&$5.00^{+0.49}_{-0.55}$\
22.5&7199&$6.23^{+0.64}_{-0.69}$\
23.0&11551&$6.75^{+0.34}_{-0.37}$\
\[sBzKtab\]
BzK galaxies selected at$1.4 < z < 2.5$
----------------------------------------
The BzK selection was defined by [@Daddi_etal:2004] to isolate galaxies in the redshift range $1.4 < z < 2.5$. Clearly from figure \[nz\], and as expected by [@Daddi_etal:2004], there are contaminating objects from outside of this range. Using our photometric redshifts we can asses how successful the BzK selection technique is in reproducing the clustering of objects within the desired range.
Following the same method outlined for the full samples, we compute clustering amplitudes find correlation lengths of $11.1^{+1.7}_{-1.8}h^{-1}$ Mpc and $5.46^{+0.36}_{-0.37}h^{-1}$ Mpc for $1.4 < z < 2.5$ pBzKs and sBzKs respectively. These values are slightly lower than those for the full samples and indicate that the high redshift tails in the full sample are at least as highly clustered as those within the $1.4 < z < 2.5$ range. However, the conclusions drawn from the full sample are still valid, namely that the pBzK galaxies are significantly more strongly clustered than the sBzK galaxies.
Luminosity function
===================
{width="\textwidth"}
A luminosity function was constructed for our BzK-selected galaxies in the same way as detailed in [@Cirasuolo_etal:2007], by using the $1/V_{max}$ method [@Schmidt:1968]. Figure \[lum\] shows the luminosity function for the BzK galaxies with photometric redshifts in the range $1.4<z<2.5$ (points with error-bars), compared with all K-selected galaxies in the same range (solid line). Also plotted is the $z=0$ luminosity function from [@Kochanek_etal:2001]. It is clear that sBzK galaxies sample a wide range in luminosity, while the pBzK population are dominated by bright galaxies with $M_K>-23$. At this epoch, however, the bright end of the luminosity function is still dominated by star-forming objects, consistent with [@Cirasuolo_etal:2007] who found that the galaxy colour bimodality is weak at these redshifts. The pBzK galaxies are nevertheless likely to be among the most massive systems, since the mass-to-light ratio will be significantly lower for the actively star-forming systems.
Under the assumption that the pBzKs passively evolve to the present day (with minimal merging) one can estimate their contribution to the bright end of the present-day luminosity function. The implied evolution is modelled from $z\sim 1.60$ (the median value for pBzKs) to $z=0$ assuming the spectral evolution models of [@Bruzual_and_Charlot:2003]. For simplicity we chose a solar metalicity model and a Salpeter initial mass function [@Salpeter:1955] and assume only passive evolution from initial formation bursts at redshifts $z_f=3$ and $z_f=10$. This results in a Johnson $K$-band absolute magnitude evolution in the range $0.96 < \Delta M_{K} < 1.21$. Taking the mean value we can estimate the minimum number of present day bright ($K < -23.4$) galaxies that could previously have been pBzKs. Under our assumptions we find that $2.5\%$ of such galaxies can be explained by passively evolved pBzKs. It is assumed that the remainder is made up of the descendants of bright sBzKs and merger remnants from within and between the two classes.
Discussion and conclusions
==========================
We present the number counts, clustering and luminosity function of galaxies at $z\sim 2$ selected using the BzK selection criteria. The pBzK galaxies show a marked flattening in their number counts which cannot be explained by the effects of incompleteness, and a possible turn over at faint magnitudes. We conclude that, at this epoch, it is generally luminous, massive galaxies that are undergoing passive evolution. This is consistent with the down-sizing scenario, in which the most massive galaxies are formed first and are the first to evolve onto the red sequence [@Kodama_etal:2004].
The angular clustering of the passive galaxy sample is very strong, approximately $4$ times the amplitude of the sBzK population. This is in part due to their relatively narrow redshift distribution (K06), but a large difference remains after de-projection to the real-space correlation length. We find $r_0 = 15.0^{+1.9}_{-2.2}h^{-1}$Mpc and $r_0 = 6.75^{+0.34}_{-0.37}h^{-1}$Mpc for the pBzK and sBzK galaxies respectively. Our value for the correlation length of sBzKs is almost twice that found by [@Hayashi_etal:2007], who used a sample with similar limiting magnitude but with smaller field size ($180$ arcmin$^2$, compared with $\sim 2250$ arcmin$^2$ for the UDS). Our sample consists of more than 10 times the number of sBzKs than that of [@Hayashi_etal:2007] and in addition we make fewer assumptions regarding their redshift distribution. Further surveys reaching depths of $K_{AB,lim} \sim 23$ are required to fully account the effects of cosmic variance, however.
We also confirm that the scale-length for the clustering of sBzK galaxies is dependent on apparent magnitude, consistent with the work of [@Hayashi_etal:2007] (figure \[sBzK\]). This dependence is far more significant at magnitudes below $K_{AB} \sim 22$ indicating a strong correlation between such galaxies and the mass of the hosting halo.
In addition to this luminosity dependence there is a clear enhancement at small scales in the sBzK clustering. This enhancement is indicative of multiple sBzK galaxies occupying a single dark matter halo. The scale at which this turn-off occurs can provide an indication of the size, and hence mass, of the hosting dark matter halos. The turn-off occurs at $\sim 0.01$ deg, which corresponds to $\sim 0.3$ Mpc at z$=1.65$. Halos of this size have masses between $10^{11}$ and $10^{12} M_{\sun}$ [@Mo_and_White:2002]. A full consideration of this enhancement within the halo occupation distribution framework will be presented in a future paper. The host halo mass can also be derived from comparing the de-projected clustering scale length with models of dark matter halo clustering evolution. Using models based on [@Mo_and_White:2002] we find a typical halo mass of $\sim 6\times10^{11} M_{\sun}$.
Applying the model of clustering evolution to the pBzKs we can qualitatively conclude that they are inhabitants of the most massive halos at their epoch (in excess of $10^{13} M_{\odot}$); halos which will eventually become massive groups and clusters by the present day. The evolutionary path that pBzKs take with redshift will also be the subject of future work, using further colour selection techniques.
Our conclusion based on the $r_0$ measurement is strengthened by their luminosity function. Even under the strict assumption of purely passive evolution, the descendants of pBzKs occupy the bright end of the luminosity function. Such galaxies are group and cluster dominant galaxies in the local universe. The brightest ($K < 21$) sBzKs have clustering scale lengths comparable to, or greater than, that which we have found for the pBzKs. This finding indicates that such galaxies will also become group and cluster dominant galaxies by $z=0$. The implication is that we are indeed witnessing the epoch at which the build up of the red sequence begins.
Acknowledgments {#acknowledgments .unnumbered}
===============
OA, IRS and RJM acknowledge the support of the Royal Society. SF, MC, KPL and WGH acknowledge the support of STFC. We are grateful to Xu Kong for help in matching photometric filters and also to Kaz Sekiguchi and Hisanori Furusawa for the Subaru data used in this study. We also extend out gratitude to the staff at UKIRT for their tireless efforts to ensure that the UKIDSS surveys are a success. We would also like to thanks the anonymous referee for their thorough reading and useful comments.
Appendix {#appendix .unnumbered}
========
In this Appendix we describe the Fourier method used to deconvolve photometric redshift errors from the n(z) distribution, which can then be used to invert the Limber equation. The errors in the photometric redshift distribution are assumed to be Gaussian and it is also assumed that the measured distribution is simply the convolution of these errors with the true redshift distribution. Under these assumptions it should be possible to deconvolve the errors from the measured distribution, using the fact that a convolution is simply a multiplication in the Fourier domain. Dividing the Fourier transform of the measured distribution by that of the Gaussian errors, should then give us an estimate of the true distribution.
In practice, a relatively large number of terms in the discreet Fourier transform are required to reproduce the redshift distribution accurately, and as the the Fourier transform of a Gaussian is also a Gaussian, small levels of high frequency noise can be amplified greatly to give a spurious result. One way to avoid such a problem is to use a Wiener filter:
$$W(k) = \frac{1}{H(k)}\times\left(\frac{H(k)^2}{H(k)^2+\frac{1}{SNR}}\right).$$
Where H(k) is the Fourier transform of the Gaussian errors and SNR the signal to noise ratio. In the limit of SNR being infinite, this filter tends to $1/H(k)$ as in the simple deconvolution above. SNR is estimated by fitting the power spectrum of the redshift distribution to a function of the form $a\times10^{-b.k} + c$, with c identified as being the noise level. The resultant redshift distributions are shown in figure \[nz\].
\[lastpage\]
[^1]: E-mail: ppxwh1@nottingham.ac.uk
[^2]: Scottish Universities Physics Alliance
|
---
abstract: 'The finite time end of entanglement between two decohering qubits can be modified by local, unitary actions performed during the decoherence process. Depending on the time when such action is taken, the end can be speeded up or slowed down, or even averted all together. This phenomenon offers practical applications for the stabilization of entangled quantum states. Details concerning hastening or delaying the finite time end of entanglement are presented for two qubits which decay spontaneously into statistically independent reservoirs.'
author:
- 'A. R. P. Rau$^{1*}$, Mazhar Ali$^2$ and G. Alber$^2$'
title: 'Hastening, delaying, or averting sudden death of quantum entanglement'
---
Entanglement is a key feature of the quantum physics of more than one particle. From its historical beginnings [@EPR] to the current practical interest in it as a core resource for the field of quantum information sciences [@Nielsen], this property of quantum systems continues to fascinate and to shed new light onto the nature of our quantum world. The fields of quantum computing, quantum cryptography and key distribution [@QKD], and quantum teleportation[@teleport] all rely on having entangled states of two qubits. Since each qubit is inevitably subject to decoherence and decay processes, no matter how much they may be screened from the external environment, it is important to consider possible degradation of any initially established entanglement. In particular, there has been increasing discussion of what has been called “sudden death", a finite time when the entanglement disappears even under decoherence mechanisms which may be only asymptotic in time [@end; @Eberly; @Almeida]. Clearly, such finite time disappearance of entanglement can seriously affect its application in any of the above fields.
It is well known in the context of spontaneous emission processes [@spontaneous] or delayed choice experiments [@delayed], for example, that characteristic quantum phenomena and effects of decoherence can be influenced significantly by suitable actions, such as measurements. Therefore, it would be of interest if A(lice) and B(ob), the two members of the entangled pair, can take suitable individual actions when faced with the prospect of loss of entanglement to postpone that end. Some studies on changing the initial state into an equivalently entangled but more robust state have been carried out [@initialization]. In this Letter, we deal with the more direct question that, even given an initial state and a set up which will end in disentanglement at finite time, can they themselves take suitable actions later to change the fate of their entanglement. We answer in the affirmative. In particular, it is shown that simple local unitary operations can alter the time of disentanglement. This is even possible if these local operations are separated spacelikely so that they are not connected by any causal relation. The operations we consider can either hasten or delay that time depending on its time of application. A suitable window for this application can even avert completely the finite or sudden death. In that case, entanglement will persist and decay only asymptotically just as do the decoherences for both qubits. While our discussion will be for two qubits, it is clear that similar results will apply also to other systems such as qubit-qutrit [@Ann] and qutrit-qutrit [@Lastra; @Derkacz] where also questions of the finite end of entanglement have been considered. We will present such results elsewhere.
We consider the model of two two-level atoms and “amplitude damping" in the form of spontaneous (pure exponential) decay into statistically independent reservoirs from the excited to the ground state being the only postulated dynamics [@Eberly]. Whatever the initial state, whether pure or mixed, and whether entangled (non-separable) or not, the final state reached in asymptotic time is clearly one of both atoms in the ground state, that is, a product state of the two ground-state atoms/qubits with no entanglement. It is also a pure state with zero entropy. The much discussed model [@Eberly] considers mixed states with a density matrix of the form
$$\rho(t) = \frac{1}{3} \left(
\begin{array}{cccc}
a(t) & 0 & 0 & 0 \\
0 & b(t) & z(t) & 0 \\
0 & z(t) & c(t) & 0 \\
0 & 0 & 0 & d(t)
\end{array}
\right).
\label{eqn1}$$
The coefficients $(a, b, c, d, z)$ may be considered real, and Tr $\rho=(a+b+c+d)/3 =1$. The four states are described as usual by $(|++\rangle, |+-\rangle, |-+\rangle, |--\rangle)$ with Alice and Bob’s states shown as the first and second entries, respectively, in the ket, and $+/-$ denoting excited/ground state. The initial condition chosen, of $b(0)=c(0)=z(0)=1$, along with the only evolution, that + decays to $-$ at a steady rate $\exp(-\Gamma t/2)$ in amplitude, keeps $b(t)=c(t)$ throughout.
The form of $\rho(t)$ in Eq. (\[eqn1\]) is preserved by time evolution. The off-diagonal density matrix element is given by $\dot{z}(t)=-\Gamma z(t)$ (an overhead dot indicates differentiation with respect to time) and the diagonal ones by $$\frac {d}{dt}
\left(
\begin{array}{c}
\rho_{++} \\
\rho_{+-} \\
\rho_{-+} \\
\rho_{--}
\end{array}
\right) = \left(
\begin{array}{cccc}
-2\Gamma & 0 & 0 & 0 \\
\Gamma & -\Gamma & 0 & 0 \\
\Gamma & 0 & -\Gamma & 0 \\
0 & \Gamma & \Gamma & 0
\end{array}
\right) \left(
\begin{array}{c}
\rho_{++} \\
\rho_{+-} \\
\rho_{-+} \\
\rho_{--}
\end{array}
\right).
\label{eqn2}$$ No elaborate derivation is necessary, these equations having an obvious structure dictated by the “decay" from + to “feed" into $-$. Their solutions are also immediate. In terms of a logarithmic, dimensionless time parameter, $\gamma = \exp(-\Gamma t/2)$, we have
$$\begin{aligned}
\rho_{++}(t) =a(t) & = & a(0)\gamma^4 \nonumber \\
\rho_{+-}(t) =\rho_{-+} (t) =b(t) & = & [b(0)+a(0)]\gamma^2-a(0)\gamma^4 \nonumber \\
\rho_{--}(t) =d(t) \!& = &\!\! 3+a(0)(\gamma^4 \!- \!\gamma^2)\!-\!\![3\!-\!d(0)]\gamma^2 \nonumber \\
z(t) & = & z(0)\gamma^2.
\label{eqn3}\end{aligned}$$
Were $(b, c, z)$ to be the only non-zero elements in Eq. (\[eqn1\]), we would have an entangled pure state $(|+-\rangle+|-+\rangle)$. The coefficients would all decay with a factor $\gamma^2$, and so would the entanglement only asymptotically. The additional choice of either $(a(0)=1, d(0)=0)$ or $(a(0)=0, d(0)=1)$ gives a mixed state which is non-separable. Both choices give the same entropy, defined as $-\sum \rho_i \ln \rho_i$ in terms of the eigenvalues of $\rho$. This value is $\ln (3/4^{1/3})$ at $t=0$ and decreases to 0 asymptotically when the system is in the pure state $|--\rangle$. But their evolution of entanglement or separability is very different [@Eberly; @Zubairy].
Various measures of entanglement all coincide in their conclusion that the second choice of $d(0)=1$ leads to non-separability only asymptotically at infinite time whereas the first choice with $a(0)=1$ leads to a finite time for the end of entanglement, the “sudden death" [@Eberly]. Concurrence [@Wootters] having been discussed in previous papers, we choose [*negativity*]{} as a more easily measurable quantity in terms of the partially transposed density matrix [@Horodecki] as an indicator of non-separability.
The [*negativity*]{} is defined as the sum of the absolute values of all the negative eigenvalues of the partially transposed density matrix [@vidal] for a quantum state. For the $2 \otimes 2$ system, there can be at most one such possible negative eigenvalue [@sanpera]. Viewing Eq. (\[eqn1\]) in terms of $2 \times 2$ blocks and transposing the off-diagonal blocks to define such a partial transpose, its eigenvalues contain one that can possibly be negative. This eigenvalue, or alternatively, six times that value, is given by $a(t)+d(t)-\sqrt{[a(t)-d(t)]^2+4z^2(t)}$, and it can take negative values so long as $ad<z^2$. When $a(0)=0$, since $a(t)$ from Eq. (\[eqn3\]) remains zero at all times, the system retains non-separability for all finite t. However, for the choice $(d(0)=0, a(0)=1)$, the system starts as non-separable or entangled but when $a(t)d(t)$ crosses $z^2(t)$ during the subsequent evolution, the entanglement is lost. It can be seen that the time $t_0$ at which this happens is
$$\Gamma t_0 = \ln (1/\gamma_0^2)=\ln (1+1/\sqrt{2}).
\label{eqn4}$$
This is the time of “sudden death" [@Eberly]. At this point, we have $a=6-4\sqrt{2}, b=2\sqrt{2}-2, d=1, z=2-\sqrt{2} $.
Previous papers have examined the evolution of entanglement for different initial choices of the above parameters [@Eberly; @Zubairy]. The evolution of Werner states [@Werner], with a form slightly different from the one in Eq. (\[eqn1\]) with non-zero entries in the other two corners as well, has also been studied [@Zubairy]. It has also been noted that different “initializations", wherein an initial given state such as in Eq. (\[eqn1\]) is switched to another with equivalent entanglement, can lead to a change in the time of non-separability [@initialization]. A recent paper has collected compactly necessary and sufficient conditions for this under both amplitude and phase damping [@Huang].
Another observation, with multimode radiation fields, is that spontaneous emission can also lead to a revival of entanglement from a separable configuration [@multimode]. In three-level atoms or qutrits, finite end of entanglement for pairs and abrupt changes in lower bounds on entanglement have been noted [@Lastra], as also quantum interference between different decay channels creating an asymptotic entanglement [@Derkacz]. The finite end phenomenon has also been noted for mixed qubit-qutrit states [@Ann] and it seems to be a generic phenomenon for all entangled-pair systems.
We turn, however, to a different question, whether, given the qubit-qubit system above and initial conditions that lead to separability in finite time, a suitable intervention may alter that time. Such a question is clearly even of practical interest because, as noted in [@Roszak], “finite end may affect the feasibility of solid-state based quantum computing". Therefore, a simple intervention that prolongs the entanglement resource can be of broad interest. Indeed, the above discussion and, especially, the asymmetry noted between the two choices of whether it is $a$ or $d$ that is initially zero, suggests a way for such an intervention. When $a(t)=0$, the non-separability in the mixed state because of the presence of $d$ simply continues as the states that are entangled “decay down" to enhance $d$. The other situation of $a(0)=1$ and, therefore, a non-zero $a(t)$ is quite different, because it feeds into the entangled sector “from above". At a crucial point/time when $ad>z^2$, which, as is especially clear from the partial transposed density matrix (see Eq. (\[eqn1\])), has to do with the $(|++\rangle, |--\rangle)$ sector, it is this feed which swamps the entangled sector and makes the mixed state separable.
The above diagnosis of which aspect of the asymmetry between $a$ and $d$ is responsible for the separability suggests a “switch" between them. Such a switch, which leaves the other coefficients $(b,c,z)$ unchanged, amounts to interchanging + and $-$ for both qubits. This is a local unitary transformation that both Alice and Bob can easily implement, by individual $\sigma_x$ operations for spins or laser coupling of the excited and ground states for two-level atoms. Consider the same initial condition as before, with $(a(0)=1, d(0)=0)$, which leads to sudden death. Before the time $t_0$ corresponding to the end of entanglement, consider such local unitary operations that merely interchange $a$ and $d$. If this is done at the time $t_A$ when $a=d$, which happens when ${\rm exp}(-\Gamma t_A) \equiv \gamma_A^2 =3/4$, clearly there will be no effect upon the subsequent evolution, the end still coming at time $t_0$. See Figs. \[fig1\] and \[fig2\]. If the switch is made at any time intermediate between $t_A$ and $t_0$ (see FIG. \[fig2\]), separability occurs earlier, a minimum being at the switch time $\Gamma t_1 \approx 0.357$.
More interestingly, a switch earlier than $t_A$ prolongs the entanglement as shown in the figures. Moreover, switch times before $t_B$ with $\Gamma t_B \approx 0.1293$ avoid the finite time end all together, leading to separability only asymptotically. As a practical matter, therefore, when Alice and Bob separate at $t=0$ and know that they face an end to their entanglement at $t_0$, they can agree beforehand to make the local unitary switch between + and $-$ at a certain time as desired to alter that end.
In FIG \[fig1\], [*negativity*]{} is plotted against the parameter $\Gamma t$. The solid line corresponds to a situation when no switch is made and the sudden death happens at $\Gamma t_0 \approx 0.5348$. If the switch is made at $\Gamma t_1 \approx 0.357$ (dotted-dashed line), the sudden death reaches a minimum value of $\Gamma t_< \approx 0.48$. Any switch made earlier than $t_A$ leads to a delay in sudden death in comparison with $t_0$. Two instances are shown in the upper curves. If the switch is made at $\Gamma t \approx 0.223$ (dashed line), the [*negativity*]{} comes to an end at $\Gamma t \approx 0.716$. Any switch made earlier than $\Gamma t_B \approx 0.1293$ avoids a finite end, leading only to asymptotic decay of entanglement.
FIG \[fig2\] displays the time of sudden death $t_{end}$ against the time of switching $t_{sw}$. The earlier the switch is made than $\Gamma t_A \approx 0.2877$, the more the end of entanglement is delayed. Sudden death is avoided completely when the switch takes place earlier than $\Gamma t_B \approx 0.1293$.
Interestingly, switching + and $-$ at only one end, that is, either Alice or Bob makes the local unitary $\sigma_x$ transformation, also alters the end of entanglement. Now, $a$ and $c$ in the density matrix in Eq. (\[eqn1\]) are interchanged, as also $b$ and $d$, while $z$ moves to the corners of the anti-diagonal. The roles of $(a,d)$ and $(b,c)$ in Eq. (\[eqn3\]) are interchanged and we find that sudden death is hastened or delayed depending on the time of switching but it is now no longer averted indefinitely. FIG. \[fig3\] shows the results to be contrasted with those in FIG. \[fig2\]. A maximum, but still finite, delay is obtained for the earliest switch at $\Gamma t_{sw}=0$, its value $\Gamma t_{end} = \ln (3+\sqrt{5})/2 \approx 0.9624$ being a little less than double that of $\Gamma t_0$. A simple, analytical expression describes the curve in FIG. \[fig3\]. With $x = \exp (-\Gamma t_{sw}), y = \exp (-\Gamma t_{end})$, we have
$$y(x) = \frac{3-\sqrt{9-24 x+20 x^2}}{2(2-x)}.
\label{eqn5}$$
The value of $\Gamma t_0$ in Eq. (\[eqn4\]) corresponds to the root $y=x=2-\sqrt{2}$.
In summary, we have shown that a simple local unitary operation that can be carried out on both qubits of an entangled pair changes the subsequent evolution of their entanglement. For mixed states under conditions which lead to a loss of that entanglement at finite time, termed sudden death, such an operation can either hasten or delay that death, depending on the time at which it is carried out. There is a critical time before which the operation can even completely avert the sudden death of entanglement. When the local transformation is done at only one of the qubits, sudden death is hastened or delayed but not averted completely.
One of us (ARPR) thanks the Theoretische Quantenphysik group at the Technische Universität Darmstadt, for its hospitality during the course of this work. M. Ali acknowledges financial support by the Higher Education Commission, Pakistan, and the Deutscher Akademischer Austausch Dienst.
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Email: arau@phys.lsu.edu
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|
---
author:
- 'D. Hucul'
- 'I. V. Inlek'
- 'G. Vittorini'
- 'C. Crocker'
- 'S. Debnath'
- 'S. M. Clark'
- 'C. Monroe'
bibliography:
- '2x1\_2.bib'
title: Modular Entanglement of Atomic Qubits using both Photons and Phonons
---
**Quantum entanglement is the central resource behind applications in quantum information science, from quantum computers [@MikeAndIke] and simulators of complex quantum systems [@NatPhysQSIM] to metrology [@Metrology] and secure communication [@MikeAndIke]. All of these applications require the quantum control of large networks of quantum bits (qubits) to realize gains and speedups over conventional devices. However, propagating quantum entanglement generally becomes difficult or impossible as the system grows in size, owing to the inevitable decoherence from the complexity of connections between the qubits and increased couplings to the environment. Here, we demonstrate the first step in a modular approach [@musiqcpaper] to scaling entanglement by utilizing a hierarchy of quantum buses on a collection of three atomic ion qubits stored in two remote ion trap modules. Entanglement within a module is achieved with deterministic near-field interactions through phonons [@blatt:2008], and remote entanglement between modules is achieved through a probabilistic interaction through photons [@DuanRMP]. This minimal system allows us to address generic issues in synchronization and scalability of entanglement with multiple buses, while pointing the way toward a modular large-scale quantum computer architecture that promises less spectral crowding and less decoherence. We generate this modular entanglement faster than the observed qubit decoherence rate, thus the system can be scaled to much larger dimensions by adding more modules.**
Small modules of qubits have been entangled through native local interactions in many physical platforms, such as trapped atomic ions through their Coulomb interaction [@blatt:2008], Rydberg atoms through their electric dipoles [@Saffman2010; @Grangier2010], nitrogen-vacancy centers in diamond through their magnetic dipoles [@dolde:2013], and superconducting Josephson junctions through capacitive or inductive couplings [@neeley:2010; @dicarlo:2010]. However, each of these systems is confronted with practical limits to the number of qubits that can be reliably controlled, stemming from inhomogeneities, the complexity and density of the interactions between the qubits, or quantum decoherence. Scaling beyond these limits can be achieved by invoking a second type of interaction that can extend the entanglement to other similar qubit modules. Such an architecture should therefore exploit both the local interactions within the qubit modules, and also remote interactions between modules (an example architecture is shown in Fig 1). Optical interfaces provide ideal buses for this purpose [@CZKM97; @DLCZ], as optical photons can propagate over macroscopic distances with negligible loss. Several qubit systems have been entangled through remote optical buses, such as atomic ions [@moehring:2007], neutral atoms [@nolleke:2013], and nitrogen-vacancy centers in diamond [@bernien:2013].
{width="6.5"}
In the experiment reported here, we juxtapose local and remote entanglement buses utilizing trapped atomic ion qubits, balancing the requirements of each interface within the same qubit system. The observed entanglement rate within and between modules is faster than the observed entangled qubit decoherence rate. Surpassing this threshold implies that this architecture can be scaled to much larger systems, where entanglement is generated faster than coherence is lost.
The qubits in this experiment are defined by the two hyperfine ‘clock’ states, ${{\left| {F = 0, m_F = 0} \right\rangle}} \equiv {{\left| {0} \right\rangle}}$ and ${{\left| {F = 1, m_F = 0} \right\rangle}} \equiv {{\left| {1} \right\rangle}}$, which are separated by $\omega_0 = 2\pi \times 12.64282$ GHz in the ${}^{2}S_{1/2}$ manifold of trapped ${}^{171}$Yb$^{+}$ atoms. Laser cooling, optical pumping, and readout occur via standard state-dependent fluorescence techniques [@olmschenk:2007]. The qubits are trapped in two independent modules separated by $\sim$1 meter as shown in Fig. \[fig:network and dream\]a. (The ion traps, light collection optics, and interferometer could in principle be part of a modular, scalable architecture as shown in Fig. \[fig:network and dream\]b.)
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In order to generate remote entanglement between atoms in physically separated ion trap modules, we synchronously excite each atom with a resonant fast laser pulse [@moehring:2007]. A fraction of the resulting spontaneously emitted light is collected into an optical fiber, with each photon’s polarization ($\sigma^+$ or $\sigma^-$) entangled with its parent atom due to atomic selection rules (Fig. \[fig:network and microwaves\]a). Each photon passes through a quarter-wave plate that maps circular to linear polarization ($\sigma^+ \rightarrow H$ and $\sigma^- \rightarrow V$), and then the two photons interfere on a 50/50 beam-splitter, where detectors monitor the output (see Fig. \[fig:network and dream\]a and Methods Summary) [@matsukevich:2008]. We select the two-photon Bell states of light ${{\left| {HV} \right\rangle}} +e^{i\phi_D}{{\left| {VH} \right\rangle}}$, where $\phi_D$ is $0$ or $\pi$ depending on which pair of detectors registers the photons [@simon:2003]. Finally, a series of microwave pulses transfers the atoms into the $\{ {{\left| {0} \right\rangle}}, {{\left| {1} \right\rangle}} \}$ basis (Fig. \[fig:network and microwaves\]b), ideally resulting in the heralded entangled state of the two remote atomic qubits ${{\left| {01} \right\rangle}} + e^{i\phi_{AB}} {{\left| {10} \right\rangle}}$.
The intermodule phase is given by $$\phi_{AB} = \phi_D + \Delta \omega_{AB} t + k c\Delta \tau + k \Delta x + \Delta\phi_T.
\label{eqn:phase}$$ In this equation, the phase evolves with the difference in qubit splittings between module A and B, $\Delta \omega_{AB} = \omega_{0,A}-\omega_{0,B} \approx 2\pi \times 2.5$ kHz, owing to controlled Zeeman shifts [@olmschenk:2007]. The stable geometric phase factors $kc\Delta \tau < 10^{-2}$ and $k \Delta x < 10^{-2}$ result from the difference in excitation time $\Delta \tau < 100$ ps and difference in path length $\Delta x < 3$ cm between each atom and the beam-splitter. Here $c$ is the speed of light and $k \sim 0.33$ m$^{-1}$ is the wavenumber associated with the energy difference of the photon decay modes (here, the energy difference between $\sigma^+$ and $\sigma^-$ photons). The final contribution is the stable phase difference of the microwave transfer pulses $\Delta \phi_T$ across the modules.
In previous experiments, entanglement between remote atom spins at rates of 0.002 sec$^{-1}$ was accomplished using atom-photon frequency entanglement [@olmschenk:2009], and at rates of 0.026 sec$^{-1}$ using atom-photon polarization entanglement [@matsukevich:2008]. Here, we dramatically increase the single photon collection efficiency by using high numerical aperture microscope objectives and detecting two out of four Bell states of light emitted by the atoms to achieve a heralded entanglement rate of 4.5 sec$^{-1}$ (see Methods Summary).
Given a heralded photon coincidence event, we verify entanglement between ion trap modules by measuring atomic state populations and coherences following standard 2-qubit tomography protocols [@sackett:2000]. We measure an average entangled Bell state fidelity of $0.78 \pm 0.03$. Imperfect mode matching at the beam-splitter contributes $0.08 \pm 0.02$ to the infidelity. The measured atom-photon polarization entanglement is 0.92 per ion trap which contributes 0.15 to the remote entangled state infidelity. We attribute the atom-photon polarization infidelity to spatially inhomogeneous rotations of the photon polarization or polarization-dependent loss. Combining imperfect ion-photon polarization entanglement with imperfect mode matching at the beam-splitter yields an expected fidelity of $0.79 \pm 0.02$, consistent with observation.
Since the phase of the entangled state evolves in time (2nd term of Eq. \[eqn:phase\]), the remote atomic entanglement coherence time can be measured with Ramsey spectroscopy. Unlike a Ramsey experiment with a single atom, this measurement is not sensitive to long-term stability of the local oscillator [@olmschenk:2007; @chwalla:2007]. We measure the remote entangled state coherence time by repeating the above experiment with constant transfer pulse phase $\Delta \phi_T$ while varying the Ramsey zone delay before a final $\pi/2$ microwave rotation. We utilize a spin echo pulse in the middle of the Ramsey zone delay to account for slow magnetic field gradient drifts, and measure an entanglement coherence time of 1.12 seconds, well in excess of the required time to create remote entanglement between modules (Fig \[fig:lifetime\]c).
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In addition to using a photonic interconnect between ion traps, we use the Coulomb-coupled transverse phonon modes of the atoms to create entanglement within one module (see Fig. \[fig:network and microwaves\]c). Off-resonant laser beams drive stimulated Raman transitions between the qubit levels and impart spin-dependent forces detuned from the phonon modes. Following conventional Coulomb gate protocols [@molmer:1999; @blatt:2008], after a certain time the motion returns to its original state (see Methods Summary), and the four two-qubit basis states are ideally mapped to the following entangled states $$\begin{split}
{{\left| {00} \right\rangle}} &\rightarrow {{\left| {00} \right\rangle}} -ie^{-i\phi_{A}}{{\left| {11} \right\rangle}} \\
{{\left| {11} \right\rangle}} &\rightarrow {{\left| {11} \right\rangle}} -ie^{i\phi_{A}}{{\left| {00} \right\rangle}} \\
\end{split}
\quad \quad
\begin{split}
{{\left| {01} \right\rangle}} &\rightarrow {{\left| {01} \right\rangle}}-i{{\left| {10} \right\rangle}} \\
{{\left| {10} \right\rangle}} &\rightarrow {{\left| {10} \right\rangle}}-i{{\left| {01} \right\rangle}},
\end{split}
\label{eqn:molmer}$$ where $\phi_{A}$ is the intramodular phase from this optical Raman process in module A [@lee:2005]. This phase depends on the relative optical phase of two non-copropagating lasers. Using the above gate operation on two Doppler-cooled atoms within a module ($\bar{n}\sim3$), we create the state ${{\left| {00} \right\rangle}} - ie^{-i\phi_{A}}{{\left| {11} \right\rangle}}$ with a fidelity of $0.85 \pm 0.01$, excluding detection error, as shown in Fig. \[fig:2x1\]a,b.
We now describe the integration of both photonic and phononic buses to generate entangled 3-particle states. The three atoms are first prepared in the state ${{\left| {\psi_1 \psi_2} \right\rangle}}_A {{\left| {\psi_3} \right\rangle}}_B = {{\left| {00} \right\rangle}}_A{{\left| {0} \right\rangle}}_B$ with atoms 1 and 2 in module A and the remote atom 3 in module B (see Fig. \[fig:network and dream\]a). After heralding entanglement between atom 2 in module A and atom 3 in module B using photons, we re-initialize atom 1 to the state ${{\left| {0} \right\rangle}}_A$ with an individual addressing optical pumping beam, and then we entangle atoms 1 and 2 within module A using phonons. Ideally, this produces the state $$\begin{aligned}
{{\left| {\psi_1 \psi_2} \right\rangle}}_A {{\left| {\psi_3} \right\rangle}}_B = \Big ( {{\left| {00} \right\rangle}}_{A}-ie^{-i\phi_{A}}{{\left| {11} \right\rangle}}_{A} \Big ){{\left| {1} \right\rangle}}_{B} \nonumber \\
+e^{i\phi_{AB}}\Big( {{\left| {01} \right\rangle}}_{A} - i{{\left| {10} \right\rangle}}_{A} \Big ) {{\left| {0} \right\rangle}}_{B}
\label{eq:tripartite}\end{aligned}$$ In the above state, the parity of any pair of atoms is correlated with the spin state of the third atom. We take advantage of this property to probe the parity of atoms 1 and 2 in module A, and correlate it with the state of remote atom 3 in module B. After making photon and phonon connections between the atoms, we apply a $\pi/2$ Raman rotation to atoms 1 and 2 with a variable phase $\phi$ followed by state detection of all three atoms. When the remote atom is measured in state ${{\left| {\psi_3} \right\rangle}}_B = {{\left| {1} \right\rangle}}$, the spin parity of atoms 1 and 2 in module A is $\Pi = \Pi_c \cos (\phi_{A} - 2\phi)$. When the remote atom is measured in state ${{\left| {\psi_3} \right\rangle}} = {{\left| {0} \right\rangle}}_B$, the atoms in module A should be mapped to a state with zero average parity, regardless of the phase of the $\pi/2$ Raman rotation. We observe this correlation with a remote entangled state generation rate of $\sim$4 sec$^{-1}$ as shown in Fig. \[fig:2x1\]c,d. The fidelity of detecting the state ${{\left| {00} \right\rangle}}_A-ie^{-i\phi_{A}}{{\left| {11} \right\rangle}}_A$ of atoms 1 and 2 conditioned on detecting the remote atom 3 in the state ${{\left| {1} \right\rangle}}_B$ is $0.63 \pm 0.03$.
Scaling this architecture to many modules can vastly simplify the complexity of phases to be tracked and controlled. For $N \gg 1$ modules each with $n \gg 1$ qubits and $m \ll n$ optical ports at each module, the number of overall phases is reduced by a factor of $1/N + (m/n)^2$ compared to that for a fully connected set of $nN$ qubits [@musiqcpaper]. Of course in a modular architecture there may be overheads associated the reduced connectivity, but it will be useful to have flexibility in this tradeoff.
The intermodule phase $\phi_{AB}$ in the experiment is easily controlled by setting the phase difference of microwave rotations between the two modules. The intramodule phase $\phi_A$ is determined by the optical phase difference of the two Raman lasers and is passively stable for a single entangling experiment for typical gate times of order 100 $\mu s$. Tracking and controlling the optical phases between many entangled pairs in spatially separated modules at different times can be accomplished by utilizing “phase insensitive" gates [@lee:2005].
Scaling this system will also require mitigating crosstalk within modules. For example, when generating photons for intermodular entanglement, laser scatter and radiated light will disturb neighboring qubits within a module. This may require the use of different species of atoms as photonic and memory qubits. Quantum information could then be transferred from the photonic qubits to the memory qubits via the Coulomb bus [@schmidt:2005]. The second (photonic) species can also be used for intermittent sympathetic cooling [@barrett:2003].
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These experiments demonstrate a first step toward a modular architecture using multiple quantum buses to generate entanglement. This modular architecture can be expanded to include many modules and an optical cross connect switch to create a flexible, reconfigurable photonic network between modules (Fig. \[fig:network and dream\]b) and thus be made fault tolerant for the execution of extended quantum circuits [@musiqcpaper]. Modular architectures may be used as the backbone of a quantum repeater network [@briegel:1998] and of a quantum network of clocks [@komar:2013]. The experiments here suggest a figure of merit for a quantum repeater network with maximum separation between nodes: the coherent entanglement distance $D_{\text{ent}} = d_q R \tau$, where the physical qubit separation $d_q$ is multiplied by the entanglement rate $R$ and the entangled state coherence time $\tau$. This figure of merit indicates the maximum entanglement distance between modules of a quantum network with a positive output entanglement rate. The experiments presented here give $D_{\text{ent}} = 1$ m $\times$ 4.5 sec$^{-1}$ $\times$ 1.12 sec $\approx$ 5 meters, orders of magnitude larger than previous experiments in any platform. The coherent entanglement distance in this experiment can be lengthened by increasing the remote entanglement rate and entangled state coherence time. In addition, the development of low-loss UV fibers or the efficient down-conversion of photons to telecommunication wavelengths can increase the qubit separation without affecting the entanglement rate and enable long distance quantum repeater networks [@pelc:2010].
Methods Summary
===============
In this experiment, ion trap module A is a segmented, four blade design useful for holding chains of trapped atoms. A trap drive frequency of 37.15 MHz is used to achieve secular transverse frequencies of $\sim$2.4 MHz. Module B is a four rod Paul trap that confines a single atom. This trap is driven at 37.72 MHz to achieve secular frequencies of $\sim$1.5 MHz.
In order to generate remote entanglement between atoms in physically separated ion traps, we optically pump both atoms to the ${{\left| {0,0} \right\rangle}}$ state. A picosecond laser pulse resonant with the ${}^{2}S_{1/2}\rightarrow {}^{2}P_{1/2}$ transition excites trapped atoms in different modules. The atoms spontaneously emit photons of which $\sim$10 % are collected by a large NA = 0.6 single atom microscope objective, resulting in the entangled photon-polarization, atom-spin state $\frac{1}{2}({{\left| {1,1} \right\rangle}}{{\left| {\sigma^{-}} \right\rangle}}- {{\left| {1,-1} \right\rangle}}{{\left| {\sigma^{+}} \right\rangle}})^{\otimes 2}$. The emitted photons pass through $\lambda/4$ waveplates to convert the photon polarization to linear horizontal (H) or linear vertical (V) resulting in the atom photon state $({{\left| {1,1} \right\rangle}}{{\left| {V} \right\rangle}}-i{{\left| {1,-1} \right\rangle}}{{\left| {H} \right\rangle}})^{\otimes 2}$. Each objective is mode matched to a single-mode optical fiber which delivers the photons to an interferometer with a 50/50 beam-splitter as the central element. The interferometer effects a Bell state measurement of the photon state. We detect two out of the four possible Bell states of light exiting the beam-splitter to herald the entanglement of the remote atoms’ spins [@simon:2003], and after a series of microwave transfer pulses, the remote atom entangled state is ${{\left| {01} \right\rangle}} + e^{i\phi_{AB}}{{\left| {10} \right\rangle}}$ with the intermodule phase $\phi_{AB}$ defined in the main text. The phase $\phi_D$ is 0 if coincident photons are detected on PMTs 1 and 2 or 3 and 4 (see Fig. \[fig:network and dream\]a). The phase $\phi_D$ is $\pi$ if coincident photons are detected on PMTs 1 and 3 or 2 and 4.
The remote entanglement rate is limited by the collection and detection efficiency of emitted photons from the atoms. The probability for coincident detection of two emitted photons upon exciting both atoms simultaneously with a resonant laser pulse is $P= p_{Bell}[P_{\pi} P_{S_{1/2}} Q_E T_{fib} T_{opt} \frac{\Omega}{4\pi}]^2 = 9.7 \times 10^{-6}$ where $P_{\pi} = 0.95$ is the probability of exciting the atom with a resonant ${}^{2}S_{1/2} \rightarrow {}^{2}P_{1/2}$ laser pulse, ${}^{2}P_{S_{1/2}} = 0.995$ is the probability to decay from ${}^{2}P_{1/2} \rightarrow {}^{2}S_{1/2}$ (as opposed to the ${}^{2}D_{3/2}$ state), $p_{Bell} = 1/2$ accounts for selecting two of the four possible Bell states of light, $Q_E \approx 0.35$ is the quantum efficiency of the single photon PMT detectors, $T_{fib} \approx 0.14$ is the fiber coupling and transmission probability of a single-mode optical fiber, $T_{opt} = 0.95$ is the photon transmission through optical components, and $\frac{\Omega}{4\pi} = 0.1$ is the fraction of the solid angle each microscope objective subtends. The experimental repetition rate of 470 kHz is limited by the need for Doppler cooling (adding $\sim$500 ns on average to the repetition time), the atomic state lifetime of the ${}^{2}P_{1/2}$ state (necessitating $\sim$1 $\mu$s of optical pumping for state preparation of the pure quantum state ${{\left| {0} \right\rangle}}$), and sound wave propagation time in AOM crystals used in the experiment. These factors result in a measured atom-atom entanglement rate of 4.5 sec$^{-1}$.
The Coulomb entangling gate makes use of Walsh function modulation $W[1]$ to reduce the sensitivity of the gate to detuning and timing errors [@hayes:2012]. We pick a detuning $\delta$ from a transverse mode of motion and set the gate time $t_g = 2 / \delta$ with a $\pi$ phase advance of the sidebands at $t = t_g/2$. We adjust the average Raman laser intensity power to make sideband Rabi frequency $\eta \Omega$ satisfy $\delta = 2^{3/2}\eta \Omega$ to complete the entangling gate ${{\left| {00} \right\rangle}} \rightarrow {{\left| {00} \right\rangle}}-ie^{-i\phi_{A}}{{\left| {11} \right\rangle}}$ in ion trap module A.
Detection error of a single atom in an ion trap module is limited by off-resonant pumping from the F = 1 to the F = 0 manifold of the ${}^{2}S_{1/2}$ ground state through the F = 1 manifold of the ${}^{2}P_{1/2}$ excited state [@olmschenk:2007], and is $\sim1$% in the experiments presented here. Detection error of two qubits in the same module is limited by the use of a single PMT detector where the photon detection histograms of a single qubit in the state ${{\left| {1} \right\rangle}}$ and two qubits in the state ${{\left| {11} \right\rangle}}$ may overlap. This overlap is $\sim8$% in these experiments.
Acknowledgments
===============
We thank Kenneth R. Brown, L.-M. Duan, J. Kim, P. Kwiat, D. N. Matsukevich, P. Maunz, D. L. Moehring, S. Olmschenk, and P. Richerme for helpful discussions. This work was supported by the Intelligence Advanced Research Projects Activity, the Army Research Office MURI Program on Hybrid Quantum Optical Circuits, and the NSF Physics Frontier Center at JQI.
|
---
author:
- 'Alejandra Recio-Blanco'
- 'Emma Fernández-Alvar'
- Patrick de Laverny
- Teresa Antoja
- |
\
Amina Helmi
- Aurélien Crida
bibliography:
- 'biblio.bib'
date: 'Received ...; accepted ...'
title: 'The heavy-elements heritage of the falling sky'
---
[A fundamental element of galaxy formation is the accretion of mass through mergers of satellites or gas. Recent dynamical analysis based on Gaia data have revealed major accretion events in Milky Way’s history. Nevertheless, our understanding of the primordial Galaxy is hindered because the bona fide identification of the most metal-poor and correspondently oldest accreted stars remains challenging.]{} [Galactic Archaeology needs a new accretion diagnostic to understand primordial stellar populations. Contrary to $\alpha$-elements, neutron-capture elements present unexplained large abundance spreads for low metallicity stars, that could result from a mixture of formation sites. ]{} [We have analysed the abundances of yttrium, europium, magnesium and iron in Milky Way satellite galaxies, field halo stars and globular clusters. The chemical information has been complemented with orbital parameters based on Gaia data. In particular, the orbit’s average inclination has been considered. ]{} [The \[Y/Eu\] abundance behaviour with respect to the \[Mg/Fe\] turnovers for satellite galaxies of different masses reveals that higher luminosity systems, for which the \[Mg/Fe\] abundance declines at higher metallicities, present enhanced \[Y/Eu\] abundances, particularly in the \[Fe/H\] regime between -2.25 dex and -1.25 dex. In addition, the analysis has uncovered a chemo-dynamical correlation for both globular clusters and field stars of the Galactic halo, accounting for about half of the \[Y/Eu\] abundance spread. In particular, \[Y/Eu\] under-abundances typical of protracted chemical evolutions, are preferentially observed in polar-like orbits, pointing to a possible anisotropy in the accretion processes. ]{} [Our results strongly suggest that the observed \[Y/Eu\] abundance spread in the Milky Way halo could result from a mixture of systems with different masses. They also highlight that both nature and nurture are relevant to the Milky Way’s formation, since its primordial epochs, opening new pathways for chemical diagnostics of our Galaxy building up.]{}
Introduction
============
The most primitive Galactic stars are essential to understand the Milky Way formation. Nevertheless, the robust identification of accreted objects is particularly challenging for stars with primordial abundances having at most 30 times less metals than the Sun (\[Fe/H\]$\lesssim$-1.5). Kinematical or dynamical indications of accretion are insufficient to reveal ancient mergers [@JeanBaptiste17]. They need to be complemented by chemical diagnostics [@FreemanJoss], as the chemical evolution of a system strongly depends on its mass. Compared to the massive Milky Way, satellite galaxies generally present signs of protracted evolutions, being more metal deficient and showing a variety of chemical patterns that we should retrieve in the accreted populations, now mixed with in situ formed stars. The most commonly used chemical diagnostic of accretion is the $\alpha$-elements (O, Mg, Si, S, Ca, Ti) ratio with respect to iron (\[$\alpha$/Fe\]). Initially enhanced, the \[$\alpha$/Fe\] abundance starts to strongly decline with metallicity after the supernovae Ia explosion rate reaches a maximum [@MatteucciGreggio]. This produces a knee in the \[$\alpha$/Fe\] vs. \[Fe/H\] trend whose location provides constraints on the system total mass: the less massive the system, the more metal-poor is the \[$\alpha$/Fe\] turnover. Unfortunately, this accretion diagnostic is not discriminating enough for stars in the Galactic halo, with metallicities lower than the \[$\alpha$/Fe\] turnover of most satellite galaxies. As a consequence, metal-poor field stars kinematically proposed to be members of ancient accreted satellites, like Gaia-Enceladus/Sausage [@AminaEnceladus; @Sausage], have similar \[$\alpha$/Fe\] abundances as non-members for \[Fe/H\]$\lesssim$-1.5 dex. They only appear as a separate sequence at higher metallicity [@AminaEnceladus], hampering also the detection of low mass mergers. Similarly, the population of clusters in the Galactic halo is mostly homogeneous in their \[$\alpha$/Fe\] abundances [@DualGalaxy18].
Galactic Archaeology thus needs a new accretion diagnostic to understand the primordial stellar populations and, in this work, we have used neutron-capture elements to identify it. Contrary to $\alpha$-elements, neutron-capture elements present unexplained large abundance spreads for low metallicity stars, that could result from a mixture of formation sites. In particular, we have considered the logarithm of the ratio of a star’s yttrium abundance with respect to its europium one, \[Y/Eu\]. Approximately 75% of the solar Yttrium was produced [@Nikos18] by low and intermediate mass asymptotic giant branch (AGB) stars, through slow neutron captures (relatively to the $\beta$-decay rates of unstable nuclei). In addition, first peak s-elements like Y have a larger contribution from low mass stars than second peak elements like Ba. On the other hand, 94% of europium is produced by massive stars through rapid neutron captures [@Bisterzo14]. Proposed Eu production sites are neutron star mergers [@Rosswog99], high energy winds accompanying core collapse supernovae explosions [@Woosley94] or magneto-hydrodynamical explosions of fast rotating stars [@Winteler12]. As a consequence, the \[Y/Eu\] abundance ratio characterizes the relative contribution of low-intermediate mass stars with respect to high mass stars, being therefore a good indicator of the chemical evolution efficiency.
Chemical abundances and orbital parameter estimations
=====================================================
The present study relies on several samples of objects: globular clusters and field stars, both from the Milky Way and its satellites. We have made use of abundances of europium, yttrium and \[Mg/Fe\], collected from different literature works (c.f. Table 1 and further details in the Appendix). Concerning the \[Y/Eu\] uncertainty estimates, we have examined the abundance dispersion of the objects analysed by more than one study, including the dwarf stars database. The mean dispersion in the \[Y/Eu\] ratio is 0.07 dex, indicating a reasonable agreement between different literature sources. To adopt a conservative value, we have multiplied that dispersion by 2, adopting a typical error-bar of 0.15 dex.
The chemical analysis of Milky Way objects has been complemented with orbital parameters based on Gaia data [@GaiaDR2]. For globular clusters, the orbital parameters are taken from Model-2 in [@AminaGaiaDR2]. They have been computed as the average values over 10 Gyr of integration. To this purpose, we used the median values obtained from 1000 orbits for each cluster obtained through Monte Carlo realizations of the initial conditions, considering the observational measurements and their errors. In particular, the orbit’s average inclination has been computed as arccos(Lz/Ltot). In our convention, the orbital inclination is defined from the Galactic plane and comprised between 0$^{\circ}$ and 180$^{\circ}$, with prograde orbits below 90$^{\circ}$. Error bars in the orbital parameters associated to model assumptions, have been estimated by comparing the results obtained with different Galactic potentials [defined as Model-1, -2, -3 in @AminaGaiaDR2]. In particular, the dispersion in the orbital inclination (estimated as the third quantile value of the differences distribution between two models) is 6 degrees. In addition to this main dataset of cluster orbits, we have completed the sample with six additional objects from [@Vasiliev19].
For our field stars samples, we have derived the orbital parameters using the python package galpy [@galpy15]. We assume the MWPotential14 Milky Way mass model included in this package. We derived the action parameters through the action-angle isochrone approximation [@galpy14]. As input parameters we have used the radial velocities gathered in Simbad, the Gaia DR2 proper motions and the distances from [@CorynDist18]. In addition, we have checked the effect of using two different methodologies of the dynamical parameters for clusters and field stars. To this purpose, we have re-computed the clusters orbital inclinations using the field stars methodology calculated the differences with respect to the Model-2 orbital results from Gaia Collaboration et al. 2018. The median absolute deviation of the orbital inclination differences is 2.5 degrees, confirming the consistency of the two approaches.
Finally, we have assessed the impact of the detected Gaia kinematic biases [@ShoenrichBias] in our data. For the field star samples, only 18 targets had a few parameters outside the Schoenrich et al. quality cuts and were excluded from the analysis. Regarding the globular cluster data, the [@AminaGaiaDR2] database is within the quality cuts, and the Vasiliev et al. compilation uses literature distances and line-of-sight velocities not concerned by the Gaia parallax bias.
Chemo-dynamical correlations and abundance spread in the Halo
=============================================================
![Mg abundance with respect to iron as a function of \[Fe/H\] for stars belonging to low mass satellites (Ursa Minor, Draco and Carina; square symbols) and to higher mass satellites (Fornax, Sculptor and Leo I; circles). Points are colour coded by the stars \[Y/Eu\] content. []{data-label=""}](MgFeYEuTrend.pdf){width="9.5cm" height="6cm"}
First of all, we have analysed the \[Y/Eu\] behaviour with respect to the \[Mg/Fe\] turnovers for satellite galaxies of different masses. Figure 1 shows the Mg abundance (an $\alpha$-element) with respect to iron as a function of \[Fe/H\] for stars belonging to low mass satellites (Ursa Minor, Draco and Carina; square symbols) and to higher mass satellites (Fornax, Sculptor and Leo I; circles). The points are colour coded by the stars \[Y/Eu\] content. It can be observed that higher luminosity systems, for which the \[Mg/Fe\] abundance declines at higher metallicities, present enhanced \[Y/Eu\] abundances, particularly in the \[Fe/H\] regime between -2.25 dex and -1.25 dex (see the Appendix for a separate \[Y/Fe\] and \[Eu/Fe\] analysis).
Following the previous result, the observed \[Y/Eu\] abundance spread in our Milky Way could result from a mixture of systems with different masses. If this is the case, the \[Y/Eu\] indicator should be compatible with the commonly used \[Mg/Fe\] accretion diagnostic, also in our Galaxy. This has already been observed in the high metallicity regime [@Fishlock17], but it is difficult to test in the metal-poor one, where the \[$\alpha/$Fe\] spread is very low.
{width="17.5cm" height="9cm"}
![Deviations in \[Y/Eu\] (panel a) and \[Mg/Fe\] (panel b) abundances, with respect to the average, as a function of the orbital inclination. No objects in common to the \[Y/Eu\] and the \[Mg/Fe\] analysis are included. Average values have been defined by a Theil-Sen linear fit for each abundance trend with metallicity. The \[Mg/Fe\] analysis is restricted to -2.0 $\le$ \[M/H\] $\le$-1.2 dex, to reduce the non linear effect of the \[M/Fe\] turnover.[]{data-label=""}](YEuInclinCorrelation.pdf){width="9.5cm" height="9cm"}
Fortunately, since the arrival of precise Gaia astrometric data, dynamical information can be used to break down this degeneracy. Indeed, chemo-dynamical correlations retrieved both in the \[Mg/Fe\] and the \[Y/Eu\] spread could reinforce the \[Y/Eu\] abundance as a good accretion indicator. To test this possibility, our Milky Way objects have been classified into three categories, using the \[Y/Eu\] and the \[Mg/Fe\] criteria independently (upper and lower panels of Figure 2, respectively): first, objects with depleted \[Y/Eu\] values or metal-poor \[Mg/Fe\] turnovers (red targets) compatible with low-mass progenitors; second, objects with intermediate \[Y/Eu\] abundances or intermediate-metallicity \[Mg/Fe\] turnovers (green targets) possibly formed in higher-mass systems; and third, targets with enhanced \[Y/Eu\] values or a metal-rich \[Mg/Fe\] turnover typical of the Milky Way in situ population (blue objects). The \[Mg/Fe\]-selected samples act here as control groups testing the \[Y/Eu\] diagnostic.
Panels [*c*]{} and [*d*]{} show the normalized distribution of orbital inclinations for the three sets of objects, selected either with the \[Y/Eu\] diagnostic or with the \[Mg/Fe\] one, respectively. Although the two chemical diagnostics target different objects (those in common being excluded from panel d histograms) and span different metallicity regimes, the similarities between panels [*c*]{} and [*d*]{} distributions are important. Two-sampled Kolmogorov-Smirnov tests between the nine possible pairs of distributions have been performed to test this similarity. The null hypothesis, assuming that the samples come from a population with the same distribution, is rejected for all the pairs except those having the same colour (targeting therefore the same parent system mass). In particular, depleted \[Y/Eu\] objects tend to present high orbital inclinations, as targets with a metal-poor \[Mg/Fe\] turnover. On the contrary, objects with intermediate \[Y/Eu\] abundances and intermediate metallicity \[Mg/Fe\] turnovers display mainly low inclination retrograde orbits. Finally, targets with high \[Y/Eu\] ratios and metal-rich \[Mg/Fe\] turnovers show primarily low inclination prograde orbits. As expected, adjacent groups in \[Y/Eu\] or \[Mg/Fe\] abundances (red-green and green-blue pairs), partially overlap in their orbital inclination distributions as a result of abundance uncertainties, but also to the fact that no perfectly separated components seem to exist. In particular, in situ formed objects dynamically heated by past mergers [e.g. @Belokurov19; @Paola19] could also blur the orbital inclination distributions.
The above result confirms the coherence of the \[Y/Eu\] diagnostic with the \[Mg/Fe\] one, revealing possible chemo-dynamical correlations with two independent chemical indicators. To quantify those trends, Figure 3 shows the deviations in \[Y/Eu\] and \[Mg/Fe\] abundances with respect to the average, as a function of orbital inclination. Contrary to the analysis of Figure 2, no data subsamples are predefined and the considered metallicity regime spans -2.0 $\le$ \[M/H\] $\le$-1.2 dex in both panels. The two chemical diagnostics show under-abundances around the polar direction (60$^{\circ}$ $\lesssim$ inclination $\lesssim$ 120$^{\circ}$) and over-abundances near the plane (prograde objects with inclination $\lesssim$ 60$^{\circ}$ and retrograde objects with inclination $\gtrsim$120$^{\circ}$). The observed chemo-dynamical correlations, including both globular clusters and field stars, are more pronounced for the \[Y/Eu\] abundances than for the \[Mg/Fe\] ones as expected from their corresponding abundance spreads in this metallicity regime. In particular, the orbital inclination seems to account for about half of the \[Y/Eu\] abundance scatter.
Conclusions
===========
Although Galactic studies need to be constantly validated in the huge parameter space of Milky Way populations, the observed chemo-dynamical correlations open new paths of exploration of our Galaxy formation history. In the light of the previous conclusions, the heavy elements abundance scatter of the primordial Milky Way possibly results from an amalgam of systems with different masses and chemical evolutions.
First, objects in polar-like orbits showing underabundances of \[Y/Eu\] could result from a composite debris from low mass accretions. Interestingly, polar orbits are also found for more recent merger events as the Sagittarius one. This suggests the possible existence of a preferential accretion axis around the polar direction, linking the Milky Way to its satellites and deserving further study. In the metal-poor and intermediate metallicity regime, where the \[Y/Eu\] under-abundances are larger than the \[$\alpha$/Fe\] ones, future large scale heavy-element studies seem crucial to distinguish between low-mass accretions and slow rotating debris from more massive mergers.
Second, satellite merger debris in retrograde orbits was previously suggested by the analysis of several dynamical overdensities [e.g. @AminaEnceladus; @Sausage; @Myeong19], and attributed to high mass progenitors (Gaia Enceladus/Saussage, Sequoia). In our study, the chemical patterns dominating that retrograde regime near the plane are indeed typical of high mass systems, reaching metallicities of -0.5 dex and relatively high \[Y/Eu\] abundances. The interplay of this old retrograde population with the prograde disc and the slow rotating accretion debris is probably an important piece of the Galaxy formation puzzle.
Third, a prograde population, showing \[Y/Eu\] overabundances, seems to be present even in the low metallicity regime. It could be the fossil signature of the primitive collapsed Galaxy, probably occupying prograde orbits near the plane, as the more metal-rich disc. This hypothesis is strengthen by the recent discovery of very metal-poor stars with disc like orbits [@Sestito20] In conclusion, both nature and nurture appear to have played a role to build up the ancient Milky Way, leaving inprints we are starting to decode. Chemical diagnostics, including heavy elements abundances, will certainly be fundamental in the on going Gaia revolution.
This work has made use of data from the European Space Agency (ESA) mission Gaia Data Processing and Analysis Consortium (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium. (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. ARB, PdL and EFA acknowledge financial support from the ANR 14-CE33-014-01. TA has received funding from the European Union’s Horizon 2020 research and innovation programme under Marie Sklodowska-Curie grant agreement number 745617 and also acknowledges funding from the MINECO (Spanish Ministry of Economy) through grants ESP2016-80079-C2-1-R (MINECO/FEDER, UE) and ESP2014-55996-C2-1-R (MINECO/FEDER, UE). AH acknowledges funding from a Vici grant from the Netherlands Organisation for Scientific Research (NWO). We thank E. Vasiliev for providing his orbital parameters for globular clusters. ARB thanks Vanessa Hill, Sebastian Peirani and Oliver Hahn for useful discussions and Chris Wegg for kindly language corrections.
Complementary information on literature abundances
==================================================
The adopted references for the abundances of the different elements and populations analysed in this work are summarized in Table 1.A. The study of globular clusters chemical abundances is currently confined to heterogeneous compilations from different groups. Nevertheless, despite these words of caution, clusters benefit today from several decades of efforts in chemical abundance estimations. The analysed Milky Way field stars abundances come from three different compilations: a photometric selection of metal-poor stars [@Roederer14], a study of heavy-element abundances for high-$\alpha$ and low-$\alpha$ stars at intermediate metallicity [@Fishlock17] and a selection of high transversal velocity stars from the APOGEE survey [@ApogeeDR14]. When considering the field star homogeneous abundances from Roederer et al. 2014, we only take into account stars with abundances estimated from 3 or more lines in order to select a high quality sample. We do not consider stars for which only upper limits were provided. The APOGEE sample is composed of Gaia DR2 stars with parallax $>$ 0.3 mas, G $<$ 15 mag and Vtot$ >$ 180 km/s. Our final sample comprises 972 objects with APOGEE DR14 \[Mg/Fe\] abundances. In addition, the chemical abundances of Milky Way satellites have been analyzed using a compilation with metallicities \[Fe/H\] $<$ -0.5 dex, obtained from the SAGA database [@Saga]. We gather stars with Y, Eu and Mg abundance determinations, excluding those with only upper limits, carbon-enriched stars (defined as \[C/Fe\] $<$ 0.9 dex if \[Fe/H\] $<$ -1.0 dex) and objects reported as binaries.
To better understand the \[Y/Eu\] behaviour, a separate study of \[Eu/Fe\] and \[Y/Fe\] abundance trends with \[Mg/Fe\], for Milky Way satellites of different luminosities can be performed. Figure A.1 shows the Mg abundance with respect to iron as a function of \[Fe/H\] for stars belonging to low mass satellites (Ursa Minor, Draco and Carina; square symbols) and to higher mass satellites (Fornax, Sculptor and Leo I; circles). A colour code on the \[Eu/Fe\] and \[Y/Fe\] abundances is used for panels [$\it a$]{} and [$\it b$]{}, respectively. Stars showing high \[Mg/Fe\] values present lower \[Eu/Fe\] abundances than those of similar metallicity with lower \[Mg/Fe\] values. As a consequence, stars with \[Eu/Fe\] abundances lower than about 0.5 dex display low-\[Mg/Fe\] abundances only for metallicities higher than around -1.75 dex, suggesting a faster chemical evolution of their parent systems. Conversely, at a given metallicity, higher \[Mg/Fe\] stars tend to have slightly higher \[Y/Fe\] values than lower \[Mg/Fe\] stars. This suggests that lower mass systems tend to present higher \[Eu/Fe\] enrichments and lightly lower \[Y/Fe\] abundances than more massive ones, conducting to higher \[Y/Eu\] ratios as shown in Figure 1.
------------------------------------------------------------------- ------------------------------------------------------------------
{width="8cm" height="5.5cm"} {width="8cm" height="5.5cm"}
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**Population & **\[Y/Fe\] & \[Eu/Fe\] references & **\[Mg/Fe\] references\
& &\
& &\
& [@Johnson17] & references in their table 5, [@McWilliam2298], &\
Milky Way clusters & [@Shetrone03], [@Munoz3201], [@Roederer11] , & [@DualGalaxy18]\
& [@Massari6362], [@James6752] &\
& &\
& &\
Milky Way field stars & [@Fishlock17], [@Roederer14] & [@Nissen2010]\
& & [@ApogeeDR14]\
& &\
& &\
Satellites field stars & [@Saga] & [@Saga]\
& &\
******
-------------------------------------------------------------------------------------- -- --
|
---
abstract: 'In this work we present an alternative formulation of the higher eigenvalue problem associated to the infinity Laplacian, which opens the door for numerical approximation of eigenfunctions. A rigorous analysis is performed to show the equivalence of the new formulation to the traditional one. We define consistent numerical schemes for approximating infinity ground states and higher eigenfunctions and perform numerical experiments which also shed light on some open conjectures in the field.'
author:
- 'Farid Bozorgnia[^1]'
- 'Leon Bungert[^2]'
- Daniel Tenbrinck
bibliography:
- 'biblio.bib'
title: |
The Infinity Laplacian eigenvalue problem:\
reformulation and a numerical scheme
---
Infinity Laplacian operator, Infinity ground states, Nonlinear Eigenvalue problems, Monotone schemes.
Introduction {#s:introduction}
============
The infinity Laplacian equation was introduced by G. Aronsson in [@aronsson1965minimization] and has been extensively studied in the following years. It can be categorized as a nonlinear degenerate elliptic partial differential equation (PDE), which has interesting connections to Lipschitz extensions [@crandall2001remark; @crandall2001optimal] and also to probabilistic games [@peres2009tug]. For important contributions to the analysis of the infinity Laplacian we refer to [@aronsson1965minimization; @aronsson2004tour; @crandall2001optimal]. Regarding the uniqueness of Lipschitz extensions and the theory of absolute minimizers we refer the interested reader to the work of Jensen [@jensen1993uniqueness], and further works in [@aronsson2004tour; @crandall2001remark; @juutinen1998minimization]. A numerical approximation of the infinity Laplacian equation is investigated by Oberman in [@oberman2005convergent], where he introduced a convergent finite difference scheme. In the context of finite weighted graphs as discretizations of nonlocal operators the infinity Laplacian operator has been studied and applied for data processing task in [@elmoataz2017nonlocal; @elmoataz2015p].
As we will recap in \[s:background\] below, the infinity Laplacian can be formally understood as limit of a family of $p$-Laplacian operators for $p \rightarrow \infty$. Hence, there is a natural relation between the eigenfunctions of the $p$-Laplacian and the eigenfunctions of the infinity Laplacian. In the last decade nonlinear eigenvalue problems of both the $p$-Laplacian operator as well as the infinity Laplacian operator have gained increasing attention [@gilboa2018nonlinear; @le2006eigenvalue; @lindqvist1990equation]. Horak discussed numerical approximations for the two smallest eigenvalues of the $p$-Laplacian operator for different values of $1<p\leq 10$ in [@horak2011numerical]. For large values of $p$, however, it turns out to be difficult to compute eigenvalues and corresponding eigenfunctions of the $p$-Laplacian due to stiffness of the discretized systems. On the other hand, the eigenvalue problem for the infinity Laplacian operator has been analytically studied for example by Juutinen, Lindqvist, Kawohl, Manfredi, and Saksman in [@juutinen2005higher; @juutinen1999infinity; @juutinen1999eigenvalue; @lindqvist2000superharmonicity; @yu2007some]. A natural approach to numerically approximate eigenfunctions of the infinity Laplacian is to look at eigenfunctions of the $p$-Laplacian for high values of $p$ (cf. [@bozorgnia2016convergence; @horak2011numerical], for instance). To the best of our knowledge a direct numerical approximation of the first and second eigenfunctions of the infinity Laplacian operator has not been investigated so far.
Let $\Omega$ be an open, bounded domain in $\mathbb{R}^d$. In this work we consider the following Dirichlet eigenvalue problem of the infinity Laplacian operator as studied in [@juutinen2005higher]. One looks for a function $u\in W^{1,\infty}_0(\Omega)$ which is a viscosity solution of $$\label{eq:higher_efs}
0\ = \
\begin{cases}
\min(|\nabla u|- \Lambda u, -\Delta_\infty u) & \quad \text{where }u>0, \\
-\Delta_\infty u & \quad \text{where }u=0,\\
\max(- |\nabla u|- \Lambda u, -\Delta_\infty u ) & \quad \text{where }u < 0.
\end{cases}$$ Here, $\Lambda>0$ denotes a corresponding eigenvalue of the infinity Laplacian $\Delta_\infty$. Positive solutions are referred to as ground states and fulfill the simpler equation $$\label{eq:first_ef}
0 \ = \ \min(|\nabla u|-\Lambda u,-\Delta_\infty u).$$ Equation \[eq:higher\_efs\] turns out to be rather challenging from a numerical point-of-view. This is due to the fact that the eigenfunction equation \[eq:higher\_efs\] is formulated by means of a distinction of cases. However, these different cases are based on the unknown sign of the solution itself, and thus one is not able to implement a numerical approximation scheme directly.
The **main contribution** of this work is twofold. First, we give a reformulation of \[eq:higher\_efs\] as *one* equation which avoids the distinction of cases. Second, we define consistent numerical schemes on unstructured grids for approximating solutions of \[eq:higher\_efs\] and \[eq:first\_ef\].
The structure of this paper is as follows: \[s:background\] recalls the mathematical background of the $p$- and infinity Laplacian operators and their respective eigenvalues and eigenfunctions. In \[s:formulation\] we propose the alternative formulation of the infinity Laplacian eigenfunction problem \[eq:higher\_efs\] and rigorously prove equivalence. Based on the reformulation, we define consistent schemes for approximating eigenfunctions in \[s:numerics\]. In \[s:results\] we show numerical results using the proposed approximations.
Mathematical background {#s:background}
=======================
To make this paper more self-contained we begin by recalling the concept of viscosity solutions in \[ss:viscosity\]. This is the suitable solution concept for both the infinity Laplacian equation and the eigenvalue problem \[eq:higher\_efs\]. Furthermore, we recap properties of the infinity Laplacian equation, which is a substantial part of the eigenvalue problem, in \[ss:inf\_L\_eq\]. Finally, we summarize the analytic relationship of the $p$- and infinity Laplacian operators and discuss properties of their respective eigenvalues and eigenfunction in \[ss:p-eigenproblem\_and\_limit\].
Viscosity solutions {#ss:viscosity}
-------------------
We focus on PDEs of the following general form $$\label{eq:elliptic_general}
F(u, \nabla u, D^2u) \ = \ 0$$ for a real-valued function $u \colon \Omega \rightarrow \mathbb{R}$, $F \colon \mathbb{R}\times\mathbb{R}^n\times\mathbb{S}^n \to \mathbb{R}$, and $\mathbb{S}^n$ is the space of real, symmetric $n\times n$-matrices. We further assume that $F$ is *degenerate elliptic*, meaning $$\label{eq:degen_elliptic}
F(u, p, M)\leq F(u, p, N)\quad\text{ if } N\leq M$$ for all $u\in\R$ and $p\in\mathbb{R}^n$. By $N \leq M$ in \[eq:degen\_elliptic\] we denote that the matrix $M - N$ is positive semi-definite. Any equation fulfilling these properties is called *degenerate elliptic*. For a comprehensive overview on the theory of viscosity solutions we refer the interested reader to the seminal paper of Crandall, Ishii, and Lions in [@crandall1992user].
\[visc\_definition\] Any upper (respectively lower) semi-continuous function $u:\Omega\rightarrow\mathbb{R}$ is called a *viscosity subsolution* (respectively *supersolution*) of \[eq:elliptic\_general\] if for all $\phi\in C^2(\Omega)$ and all $x\in\Omega$ such that $u-\phi$ has a local maximum (respectively minimum) at $x$, we have $$F(u, \nabla\phi, D^2 \phi)\leq 0, \quad (\text{respectively } F(u, \nabla\phi, D^2\phi)\geq 0).$$ A continuous function $u \colon \Omega \rightarrow \mathbb{R}$ is said to be a *viscosity solution* if it is both a viscosity sub- and supersolution of $F=0$.
In the following, we consider the Eikonal equation on the interval $\Omega = (-1, 1)$ $$\begin{cases}
|u'(x)| -1 \ = \ 0 \quad & \text{ for } x \in \Omega, \\
u(x) \ = \ 0 \quad & \text{ for } x \in \partial\Omega.
\end{cases}$$ It is clear that there is no classical $C^1(\Omega)$ solution to this problem. However, one can verify that there exists a unique solution in the viscosity sense given by $u(x) \ = \ 1 - |x|$ for $x \in [-1, 1]$. Any $C^1$-function $\phi$ touching $u$ from above in $x=0$ has a slope $|\phi'(0)| \leq 1$ and obviously there is exists no such function touching $u$ from below in $x=0$.
The infinity Laplacian equation {#ss:inf_L_eq}
-------------------------------
One possible definition of the infinity Laplacian operator is $$\label{eq:infLapOp}
\Delta_{\infty} u \ = \ (\nabla u)^T D^2u \nabla u \ = \ \sum_{i,j=1}^{d} \frac{\partial u}{\partial x_i} \frac{\partial u}{\partial x_j}\frac{\partial^2 u}{\partial x_i \partial x_j}.$$ Sometimes the operator in \[eq:infLapOp\] is normalized by $\frac{1}{|\nabla u|^2}$, e.g., cf. [@barron2008infinity]. A function $u$ is said to be infinity harmonic if it solves the homogeneous infinity Laplacian equation in the viscosity sense, i.e., $$\label{eq:infLapEq}
\Delta_{\infty} u \ = \ 0 .$$ The infinity Laplacian equation can be formally understood as the limit of a sequence of $p$-Laplacian equations $\Delta_p u = \div(|\nabla u|^{p-2}\nabla u) = 0$ under certain boundary conditions for $p \rightarrow \infty$.
The infinity Laplacian equation is related to the absolute minimal Lipschitz extension (AMLE) problem [@aronsson1965minimization; @aronsson2004tour; @crandall2001remark; @jensen1993uniqueness]. In this setting one searches for a continuous real-valued function which has the smallest possible Lipschitz constant in every open set whose closure is compactly contained in $\Omega$. This interpretation has some advantages as it directly leads to numerical approximation schemes for solutions of the infinity Laplacian equation \[eq:infLapEq\]. A function $u\in W^{1,\infty}(\Omega)$ is called absolutely minimizing Lipschitz extension of a Lipschitz function $g:\partial\Omega\to\R$ if $u\vert_{\partial\Omega}=g$ and $$\norm{\nabla u}_{L^\infty(\Omega')}\leq\norm{\nabla v}_{L^\infty(\Omega')},$$ for all open sets $\Omega'\subset\Omega$ and all $v$ such that $u-v\in W^{1,\infty}_0(\Omega')$. The relationship between an AMLE and the infinity Laplacian is stated in
If $u\in\mathrm{Lip}(\Omega)$ is an AMLE then $u$ is also solution of the infinity Laplacian equation.
Clearly, one can exploit this relationship to numerically solve the infinity Laplacian equation, i.e., to construct absolutely minimal Lipschitz extensions in a discrete setting [@oberman2005convergent].
Let us finally remark that infinity harmonic functions might not be $C^2$ differentiable in general. A well-known example from [@aronsson1965minimization] is given by $$u(x, y) \ = \ |x|^{\frac{4}{3}} - |y|^{\frac{4}{3}},$$ which is a $C^{1, 1/3}$ infinity harmonic function. To the best of our knowledge it is still an open problem whether all viscosity solutions of the infinity Laplacian equation are in $C^1$. Evans and coauthors proved $C^{1, \alpha}$ regularity of viscosity solutions for the case $d = 2$ in [@evans2008c] and differentiability in general dimension in [@evans2011everywhere].
Eigenfunctions of the p-Laplacian and their limit {#ss:p-eigenproblem_and_limit}
-------------------------------------------------
The eigenvalues and eigenfunctions of the infinity Laplacian can be formally defined as the limit of respective eigenvalues and eigenfunctions of the $p$-Laplacian operator, see [@juutinen1999infinity; @juutinen1999eigenvalue; @lindqvist2000superharmonicity] for details. For this reason we shortly recall the definition of eigenvalues of the $p$-Laplacian in the following, see [@le2006eigenvalue; @lindqvist1990equation] for details.
For $1 \leq p<\infty$ the first (smallest) eigenvalue of the $p$-Laplacian operator has a variational form and is given by the Rayleigh quotient $$\label{eq:first_p_eigenvalue}
\lambda_{1}(p) \ = \ \underset{\varphi \in W^{1,p}_{0} (\Omega)} {\operatorname{inf}} \, \frac{\int_{\Omega}|\nabla \varphi|^{p} dx}{\int_{\Omega}|\varphi|^{p} dx} \ = \ \underset{\varphi \in W^{1,p}_{0} (\Omega)} {\operatorname{inf}} \frac{\| \nabla \varphi\|_{p}^{p}}{\| \varphi\|_{p}^{p}},$$ for which the minimization is performed over all non-zero functions in the Sobolev space $W^{1,p}_{0} (\Omega).$ Any minimizer of \[eq:first\_p\_eigenvalue\] has to satisfy the following Euler-Lagrange equation $$\label{eq:p-Laplac_ground_state}
\left \{
\begin{array}{ll}
-\operatorname{div}(|\nabla u|^{p-2} \nabla u)\ = \ \lambda_1(p) |u|^{p-2} u \ & \quad \text{ in } \Omega,\\
u \ = \ 0 & \quad \text{on } \partial \Omega,\\
\end{array}
\right.$$ which has to be interpreted in the usual weak sense. According to [@juutinen2005higher] the second $p$-eigenvalue $\lambda_{2}(p)$ can be defined as $$\lambda_2(p)=\min{\{ \lambda \in \mathbb{R} \st \lambda \text{ is a $p$-eigenvalue and } \lambda> \lambda_{1} }\}.$$ Analogously to \[eq:first\_p\_eigenvalue\], the first eigenvalue of the infinity Laplacian, denoted by $\Lambda_{1}$, is given by $$\label{eq:var_first_ev}
\Lambda_{1} \ = \ \inf_{\varphi \in W_0^{1,\infty}} \frac{\| \nabla \varphi\|_{\infty}}{\| \varphi\|_{\infty}},$$ where ${\| \varphi \|_{\infty}}= \operatorname{ess}\sup_{x \in \Omega}| \varphi(x)|.$ It is easy to see (e.g., cf. [@juutinen1999eigenvalue]) that the Euclidean distance function $d(x) = \dist(x, \partial \Omega)$ solves the minimization problem \[eq:var\_first\_ev\]. However, solutions to the minimization problem \[eq:var\_first\_ev\] in $W_{0}^{1, \infty}(\Omega)$ are in general not unique (see also \[rem:nonuniqueness\] below). A first eigenfunction of the infinity Laplacian operator can be obtained through the limit of the $p$-Laplacian equation \[eq:p-Laplac\_ground\_state\] for $p\rightarrow \infty$. The limit of these equations as $p \rightarrow \infty$ is found to be $$\min{\{| \nabla u|- \Lambda_{1} u,\, -\Delta_{\infty} u \}} \ = \ 0.$$ All this was proved in [@juutinen1999eigenvalue] and we subsume their results in
\[thm:existence\_ground\_state\] Let $\Omega$ be an open, bounded domain. Then there exists a positive viscosity solution $ u \in W_{0}^{1,\infty}(\Omega)$ of the problem $$\label{eq:first_eigenfct}
\begin{cases}
\min( |\nabla u| -\Lambda_{1} u,\, -\Delta_{\infty} u ) \ = \ 0 & \quad \text{in }\Omega,\\
u \ = \ 0 & \quad \text{on }\partial \Omega,
\end{cases}$$ where $$\Lambda_{1} \ = \ \Lambda_{1}(\Omega) \ = \ \frac{1}{\max_{x \in \Omega}\dist(x, \partial \Omega)}.$$ Moreover, any positive solution $u$ of \[eq:first\_eigenfct\] realizes the minimum in \[eq:var\_first\_ev\]. Such a function $u$ can be constructed as a cluster point for $p \rightarrow \infty $ of a properly normalized sequence of first eigenfunctions of the $p$-Laplacian operator. Furthermore, $$\Lambda_{1} \ = \ \lim_{ p \rightarrow \infty } \lambda_{1}(p)^{\frac{1}{p}},$$ where $\lambda_{1}(p)$ denotes the first eigenvalue of the $p$-Laplacian operator given by \[eq:first\_p\_eigenvalue\].
An important distinction has to be made between those solutions of \[eq:first\_eigenfct\] which are a limit of $p$-Laplacian ground states and those which are not.
\[def:variational\_gs\] A solution of \[eq:first\_eigenfct\] which is a cluster point of normalized solutions of \[eq:p-Laplac\_ground\_state\] is called *variational* ground state. All other solutions are called *non-variational*.
\[rem:nonuniqueness\] In [@hynd2013nonuniqueness] Hynd, Smart, and Yu have shown the non-uniqueness of infinity ground states for a dumbbell domain. However, the ground state which was constructed there is non-variational. Yu in [@yu2007some] proved that on stadium-like domains (as for instance the ball) ground states are unique up to scaling and coincide with the distance function of the domain. Whether uniqueness holds for general convex domains or variational ground states, remains an open problem. However, in \[sss:rectangle\] below we present numerical insights on that topic.
\[thm:existence\_ground\_state\] states that the first eigenvalue can be interpreted geometrically, i.e., $\Lambda_{1}$ is the reciprocal of the radius of the largest ball that fits inside the domain $\Omega$. In general, $\Lambda_1$ cannot be detected in regions where the solution is smooth, i.e., the term $|\nabla u (x_0)| - \Lambda_1 u(x_0)$ in \[eq:first\_eigenfct\] is not active. According to [@yu2007some] if $u \in C^{1}(\Omega)$ in $x_{0} \in \Omega$ then $$\Lambda u(x_{0}) < |\nabla u (x_{0})| \quad \text { and } \quad \Delta_{\infty} u(x_{0}) \ = \ 0.$$ It is known that also the second eigenvalue has a geometric characterization. According to [@juutinen2005higher] we have $$\label{eq:second_eigenvalue}
\Lambda_{2} \ = \ \frac{1}{r_{2}}.$$ where $r_{2}= \sup \{ r > 0 : \text{there are disjoint balls } B_1, B_2 \subset \Omega \text{ with radius } r\}$. Furthermore, one has
Let $\lambda_{2}(p)$ be the second $p$-eigenvalue in $\Omega.$ Then it holds that $$\Lambda_2 \ = \ \underset { p \rightarrow \infty } {\lim} \lambda_2(p)^{\frac{1}{p}}$$ and $\Lambda_{2} \in \mathbb{R}$ is the second eigenvalue of the infinity Laplacian.
According to [@juutinen2005higher] higher eigenfunctions of the infinity Laplacian operator can be obtained as a viscosity solution $u\in W^{1,\infty}_0(\Omega)$ of the equation $F_\Lambda(u,\nabla u,D^2u)=0$, where $F_\Lambda:\R\times\R^n\times\S^{n}\to\R$ is given by $$\label{eq:definition_F}
F_{\Lambda}(u , p, M) \ = \
\begin{cases}
\min(|p|- \Lambda u, -p^T M p ) & \quad \text{ for } \quad u>0 \ , \\
-p^T M p & \quad \text{ for } \quad u=0 \ ,\\
\max(- |p|- \Lambda u, - p^T M p ) & \quad \text{ for } \quad u < 0 \ ,
\end{cases}$$ and $\Lambda$ denotes the corresponding eigenvalue. Since the sign of the solution is unknown a-priori, this is a free boundary problem and hence hard to solve numerically.
The equation of the first eigenfunction in \[eq:first\_eigenfct\] can also be expressed through \[eq:definition\_F\] since the first eigenfunction does not change sign.
Reformulation of the infinity Laplacian eigenvalue problem {#s:formulation}
==========================================================
In the following we present an equivalent formulation of the higher infinity Laplacian eigenvalue problem, which allows us to avoid the distinction of cases in \[eq:definition\_F\]. To this end we introduce the function $H_\Lambda:\R\times\R\times\S^n\to\R$, defined as $$\label{eq:defintion_H}
H_\Lambda(u,p,M)=\min(|p|-\Lambda u, -p^T Mp)+ \max(-|p|-\Lambda u, -p^T M p) + p^T M p,$$ and consider the associated problem of finding a viscosity solution to the equation $H_\Lambda(u,\nabla u,D^2u)=0$. The following is our main theorem and states that the formulations through $F_\Lambda$ and $H_\Lambda$ are equivalent.
\[thm:equivalence\] It holds that $u\in W^{1,\infty}_0(\Omega)$ is a viscosity solution of $F_\Lambda(u,\nabla u,D^2u) = 0$ if and only if it is a viscosity solution of $H_\Lambda(u,\nabla u,D^2u) = 0$, where $F_\Lambda$ and $H_\Lambda$ are given by \[eq:definition\_F\] and \[eq:defintion\_H\], respectively.
Assume $F_{\Lambda}(u,\nabla u,D^2u)=0$. We need to make a case distinction on the sign of the solution $u$.
#### Case 1.1
Let $\varphi$ be a $C^2$ function touching $u$ from above in $x$ such that $u(x)>0$. Then we have $
\min(|\nabla \varphi(x) |- \Lambda \varphi(x), - \Delta_{\infty} \varphi(x) ) \leq 0.
$ If $-\Delta_\infty\varphi(x)\geq 0$ then using $-\Lambda\varphi(x)=-\Lambda u(x)<0$ we infer $$\max(-|\nabla\varphi(x)|-\Lambda\varphi(x),-\Delta_\infty\varphi(x))=-\Delta_\infty\varphi(x)$$ and hence $$H_\Lambda(\varphi(x),\nabla \varphi(x),D^2 \varphi(x))\leq 0 - \Delta_\infty\varphi(x)+\Delta_\infty\varphi(x)=0.$$ If, however, $-\Delta_\infty\varphi(x)<0$ we have to investigate two subcases. Let us first assume that $|\nabla\varphi(x)|-\Lambda\varphi(x)\leq-\Delta_\infty\varphi(x)$. Then we get that $$\max(-|\nabla\varphi(x)|-\Lambda\varphi(x),-\Delta_\infty\varphi(x))=-\Delta_\infty\varphi(x)$$ and hence $$H_\Lambda(\varphi(x),\nabla \varphi(x),D^2 \varphi(x))\leq 0 - \Delta_\infty\varphi(x)+\Delta_\infty\varphi(x)=0.$$ If we assume that $|\nabla\phi(x)|-\Lambda\phi(x)>-\Delta_\infty\phi(x)$ we obtain $$\min(|\nabla\phi(x)|-\Lambda\phi(x),-\Delta_\infty\phi(x))=-\Delta_\infty\phi(x).$$ Furthermore, from $-\Delta_\infty\phi(x)\leq 0$ it follows $$\max(-|\nabla\phi(x)|-\Lambda\phi(x),-\Delta_\infty\phi(x))\leq 0.$$ Combining these two we infer $$H_\Lambda(\varphi(x),\nabla \varphi(x),D^2 \varphi(x))\leq -\Delta_\infty\phi(x) +0 +\Delta_\infty\phi(x)=0.$$ Hence, we have shown that $u$ is a subsolution of $H_\Lambda(u,\nabla u,D^2 u)=0$ and showing that it is a supersolution is straightforward.
#### Case 1.2
For the case $u<0$ the argumentation is analogous to Case 1.1 above.
#### Case 1.3
Let $\varphi$ be a $C^2$ function touching $u$ from above in $x$ such that $u(x)=0$ meaning $\varphi(x) = 0$. Then we have $-\Delta_{\infty} \varphi(x) \leq 0$ which implies $$\begin{split}
H_{\Lambda} (&\varphi(x) ,\nabla \varphi(x) , D^{2} \varphi(x)) \\
= \ &\min(|\nabla \varphi(x) |, -\Delta_{\infty} \varphi(x))+ \max(-|\nabla \varphi(x) |, -\Delta_{\infty} \varphi(x))+ \Delta_{\infty} \varphi(x) \\
= \ &-\Delta_{\infty} \varphi(x) + \max(-|\nabla \varphi(x) |, -\Delta_{\infty} \varphi(x))+ \Delta_{\infty} \varphi(x) \\
= \ & \max(-|\nabla \varphi(x) |, -\Delta_{\infty} \varphi(x)) \leq 0.
\end{split}$$ This means that $u$ is a viscosity subsolution of $H_\Lambda(u,\nabla u,D^2 u)=0$. An analogous argument shows that $u$ is a supersolution.
Now we prove the converse statement and assume that $H_\Lambda(u,\nabla u,D^2 u)=0$ in the viscosity sense. Again, we consider the different possible signs of $u$.
#### Case 2.1
Let $\varphi$ be a $C^2$ function touching $u$ from above in $x$ such that $u(x)>0$. Then it holds that $H_\Lambda(\varphi(x),\nabla\varphi(x),-\Delta_\infty\varphi(x))\leq 0$ and we must show that $$F_\Lambda(\varphi(x),\nabla\varphi(x),D^2\varphi(x))\leq 0.$$ If $-\Delta_{\infty} \varphi(x)\geq -|\nabla \varphi(x)|-\Lambda \varphi(x)$ we conclude $$F_\Lambda(\varphi(x),\nabla\varphi(x),D^2\varphi(x))=H_\Lambda(\varphi(x),\nabla\varphi(x),D^2\varphi(x))\leq 0.$$ On the other hand, if $$-\Delta_{\infty} \varphi(x)\leq -|\nabla \varphi(x)|-\Lambda \varphi(x)=|\nabla \varphi(x)|-\Lambda \varphi(x)\leq 0$$ then obviously $-\Delta_\infty\varphi(x)\leq 0$ and $$\begin{split}
F_\Lambda(\varphi(x),\nabla\varphi(x),D^2\varphi(x))=\min(|\nabla\varphi(x)|-\Lambda\varphi(x),-\Delta_\infty\varphi(x))=-\Delta_\infty\varphi(x)\leq 0.
\end{split}$$ This shows that $u$ is a subsolution of $F_\Lambda(u,\nabla u,D^2u)=0$ and showing that it is a supersolution works analogously.
#### Case 2.2
For the case $u<0$ the argumentation is analogous to Case 2.1 above.
#### Case 2.3
Let $\varphi$ be a $C^2$ function touching $u$ from above in $x$ such that $u(x)=0$ meaning $\varphi(x) = 0$. Then it holds $$\min(|\nabla\varphi(x_0)|,-\Delta_\infty\varphi(x_0))+\max(-|\nabla\varphi(x_0)|,-\Delta_\infty\varphi(x_0))+\Delta_\infty\varphi(x_0)\leq 0.$$ If $-\Delta_\infty\varphi(x_0)\leq 0$ we are done since this implies that $u$ is a viscosity subsolution of $-\Delta_\infty u=0$. If $-\Delta_\infty\varphi(x_0)\geq 0$ we infer that $$\max(-|\nabla\varphi(x_0|,-\Delta_\infty\varphi(x_0))=-\Delta_\infty\varphi(x_0)$$ and hence from above we see that $$\min(|\nabla\varphi(x_0)|,-\Delta_\infty\varphi(x_0))\leq 0.$$ This implies that $|\nabla\varphi(x_0)|=0$ and hence also $-\Delta_\infty\varphi(x_0)=0$. Thus, in both cases $-\Delta_\infty\varphi(x_0)\leq 0$ such that $u$ is a viscosity subsolution. Analogously, one shows that $u$ is a supersolution as well.
For completeness we also prove that the equation $H_\Lambda(u,\nabla u,D^2u)=0$ is degenerate elliptic.
\[prop:deg\_elliptic\_H\] Function $H_\Lambda$ in \[eq:defintion\_H\] is degenerate elliptic as defined in \[eq:degen\_elliptic\].
We have to show that $M \geq N$ for $M,N\in\S^n$ yields $H_\Lambda(u, p, M) \le H_\Lambda(u, p, N)$.
#### [Case 1]{}
First we assume that $
\min(|p|-\Lambda u, -p^{T} M p)=|p|-\Lambda u.
$ Then one has $
\max(-|p|-\Lambda u, -p^{T} M p)=-p^{T} M p,
$ which yields $
H_\Lambda(u, p, M) =|p|-\Lambda u.
$ Now considering the inequality $ -p^{T} M p \le -p^{T} N p $ the following relationships hold $$\min(|p|-\Lambda u, -p^{T}N p)=|p|-\Lambda u, \quad \max(-|p|-\Lambda u, -p^{T} N p)=-p^{T} N p.$$ From here we see that $
H_\Lambda(u, p, M) =|p|-\Lambda u = H_\Lambda(u, p, N).
$
#### [Case 2]{}
Now let us assume $$\min(|p|-\Lambda u, -p^{T} M p)=-p^{T} M p \quad \text{ and } \quad \max(-|p|-\Lambda u, -p^{T} M p)=-|p|-\Lambda u.$$ Then $
H_\Lambda(u, p, M) = -|p|-\Lambda u.
$ For $H_\Lambda(u,p,N)$ we will now consider three different cases.
#### [Case 2.1]{}
If $
\min(|p|-\Lambda u, -p^{T} N p)=|p|-\Lambda u,
$ then $
\max(-|p|-\Lambda u, -p^{T} N p)=-p^{T} N p.
$ Consequently $
H_\Lambda(u, p, N) =|p|-\Lambda u \ge -|p|-\Lambda u = H_\Lambda(u, p, M).
$
#### [Case 2.2.1]{}
If $$\min(|p|-\Lambda u, -p^{T} N p)= -p^{T} N p \quad \text{ and } \quad \max(-|p|-\Lambda u, -p^{T} N p)= -|p|-\Lambda u$$ then $
H_\Lambda(u, p, N) = H_\Lambda(u, p, M).
$
#### [Case 2.2.2]{}
If $$\min(|p|-\Lambda u, -p^{T} N p)= -p^{T} N p \quad \text{and} \quad \max(-|p|-\Lambda u, -p^{T} N p)= - p^{T} N p,$$ then $
H_\Lambda(u, p, M)=-|p|-\Lambda u\le - p^{T} N p = H_\Lambda(u, p, N).
$ A similar case distinction for the complementary cases analogously shows that $H_\Lambda(u,p,M)\le H_\Lambda(u,p,N)$.
Numerical method {#s:numerics}
================
In this section we propose methods to approximate eigenfunctions of the infinity Laplacian. First, we recall the concept of monotone schemes in \[ss:monotone\_schemes\] as these are needed to construct numerical schemes which approximate eigenfunctions of the infinity Laplacian on general unstructured grids. Then, we sketch the approximation of the distance function and the first infinity Laplacian eigenvalue in \[ss:eigenvalues\]. Finally, our main contribution in this section is that we define consistent monotone schemes to approximate ground states and higher eigenfunctions of the infinity Laplacian in \[ss:eigenfunction\_first\] and \[ss:eigenfunction\_second\], respectively.
Monotone schemes {#ss:monotone_schemes}
----------------
In order to numerically compute approximate viscosity solutions to the abstract degenerate elliptic equation \[eq:elliptic\_general\], which in particular allows us to solve the infinity eigenvalue problems, we make use of monotone schemes and follow the description by Oberman in [@oberman2006convergent]. We first define an unstructured grid on the domain $\Omega$ as a graph consisting of a set of vertices $V = \{x_i \in \Omega$, $i = 1,\dots,M\}$ for $M \in \mathbb{N}$. To each point $x_i\in V$ we associat a list of global neighbors indices given by $N_i= \{i_1,\dots,i_{k_i}\}\subset\{1,\dots,M\}$ for some $k_i\in\N$. A grid function $\hat{F} \colon V \rightarrow \mathbb{R}$ is a real-valued function defined on $V$ which is based on values $u_i = u(x_i)$ of a function $u \colon \Omega \rightarrow \mathbb{R}$ and is given by: $$\hat{F}[u](x_i) = \hat{F}_i\left[u_i, \frac{u_i-u_{i_1}}{x_{i}-x_{i_{1}}},\dots,\frac{u_i-u_{i_{k_i}}}{x_{i}-x_{i_{k_i}}}\right] \ , \quad \text{ for } i = 1,\dots,M,$$ where the functions $\hat{F}_i$ on the right are possibly different for every grid point $x_i\in V$. Then, a discrete solution of \[eq:elliptic\_general\] on the unstructured grid introduced above is a grid function $u$ which satisfies $\hat{F}[u](x_i) = 0$ for all $i=1,\dots,M$. Note that the grid function $\hat{F}$ depends on the choice of the neighborhood and, in particular, it comes with intrinsic spatial and directional errors $$\begin{aligned}
\label{eq:errors}
dx_i=\max_{j\in N_i}|v_j|,\quad
d\theta_i=\max_{v\in S^n}\min_{j\in N_i}|v-v_j|,\end{aligned}$$ where $v_{j}=x_i-x_{j}$ for $j\in N_i$ denote the distance to the neighbors and $S^n$ denotes the unit sphere in $\R^n$.
\[def:deg\_elliptic\_scheme\] The grid function $\hat{F}$ is called
- *degenerate elliptic* if for $i=1,\dots,M$ the functions $\hat{F}_i$ are non-decreasing in the variables $2,\dots,k_i$.
- *consistent* with respect to \[eq:elliptic\_general\] in $x_i \in V$ if for every $C^2$ function $\phi$ defined in a neighborhood of $x_i$ we have: $$\hat{F}[\phi](x_i)\to F(\phi(x_i),\nabla\phi(x_i),D^2\phi(x_i))$$ as $dx_i,\,d\theta_i\to 0$. The scheme defined on $\Omega$ is consistent if the above limit holds uniformly for all $x\in\Omega.$
- *stable* if there exists a solution $u$ of $\hat{F}[u]=0$ which is uniformly bounded independently of the grid.
The following theorem gives a convergence criterium for degenerate elliptic schemes. It is a straightforward generalization of the classical Barles-Souganidis theorem to the case where the equation does not admit a comparison principle.
\[thm:barles\_souganidis\] If the grid function is degenerate elliptic, consistent, and stable according to \[def:deg\_elliptic\_scheme\], then (up to a subsequence) its solutions converge locally uniformly to a viscosity solution of \[eq:elliptic\_general\] as $dx_i,\,d\theta_i\to 0$.
Approximation of the distance function and infinity eigenvalues {#ss:eigenvalues}
---------------------------------------------------------------
As we have already seen in \[thm:existence\_ground\_state\] the first eigenvalue $\Lambda_1$ is directly linked to the geometry of the domain, i.e., $$\label{eq:first_ev_repeated}
\Lambda_{1}=\frac{1}{r_{1}}, \quad \text{ with } \, r_{1}=\underset {x \in \Omega} {\text{max}}\text{ dist}(x, \partial \Omega).$$ For simple domains $\Omega \subset \mathbb{R}^2$, such as a circle, square, or triangle, the so-called in-radius $r_1$ and hence also the first eigenvalue $\Lambda_{1}$ can be easily calculated by geometric reasoning. In general, for a more complicated domain $\Omega$ we have to compute the distance function $d(x)=\dist(x,\partial\Omega)$ which is the unique solution of the following Eikonal equation $$\label{Ek}
\left \{
\begin{array}{ll}
|\nabla d |= 1 & \text{in }\ \Omega, \\
d=0 & \text{on } \partial \Omega.\\
\end{array}
\right.$$ From the solution of \[Ek\] we thus obtain the in-radius $r_1$ together with the set of points in $\Omega$ where this maximal distance to the boundary is attained. The solution of the Eikonal equation on a discrete grid can be approximated with different methods, the best-known of which is the fast marching method [@sethian1999fast]. Originally formulated on structured grids, it was generalized to weighted graphs in [@desquesnes2013eikonal]. Alternatively, it was shown in [@zagatti2014maximal], that the solution of \[Ek\] coincides with the solution to the optimization problem $$\label{eq:ground_state}
\max_{\substack{v\in W^{1,\infty}_0(\Omega)\\\norm{\nabla v}_\infty=1}}{\norm{v}_2}$$ and it was characterized as nonlinear eigenfunction of a subdifferential operator in [@bungert2020structural]. There, the same was shown for a graph analogue of \[eq:ground\_state\]. Therefore, one can also use the gradient flow based methods [@bungert2019asymptotic; @bungert2019nonlinear; @feld2019rayleigh] to solve discrete versions of \[eq:ground\_state\] or \[Ek\], respectively.
Hence, by employing any of these methods, one obtains a discrete distance function and an associated first eigenvalue $\Lambda_1$ (cf. \[eq:first\_ev\_repeated\]) subordinate to the discrete grid $V$, defined in \[ss:monotone\_schemes\].
One should remark that the second eigenvalue $\Lambda_2$ cannot be approximated as easily. Remember that it has a geometric characterization as reciprocal of the maximal radius of two equal non-intersecting balls which fit into the domain (cf. \[eq:second\_eigenvalue\]). For many symmetric domains (e.g. circle, square, isosceles right triangle, L-shape, etc.) the solution of this sphere packing problem can be derived using elementary geometric reasoning. However, we could not find a circle / sphere packing algorithm in the literature which works for general domains.
Furthermore, higher infinity-eigenvalues have not yet been characterized. Only in some special cases one knows that they are given by the reciprocal of the maximal radius of $k$ equal non-overlapping spheres which fit into the domain [@juutinen2005higher]. In these cases one can use known solutions of the general sphere packing problem to obtain the eigenvalue.
Approximation of the first eigenfunction {#ss:eigenfunction_first}
----------------------------------------
Let us consider the first eigenfunction problem: $$\label{eq:first_ef_prob}
\begin{cases}
\min ( |\nabla u|- \Lambda_1 u , -\Delta_{\infty} u ) =0 & \text{ in } \Omega,\\
u=0 & \text{ on } \partial \Omega.
\end{cases}$$ We subdivide the set of vertices in the discrete grid into $V=V_\mathrm{inn}\cup\Gamma$ where $V_\mathrm{inn}$ denotes the inner nodes of the grid and $\Gamma$ corresponds to nodes where a value of $u$ is prescribed, for instance at the boundary. We approximate \[eq:first\_ef\_prob\] by the scheme $F[u](x_i)=0$ for all $x_i\in V$, where $$\label{eq:discrete_scheme}
F[u](x_i) \ = \
\begin{cases}
\min ( F_{1}^+[u](x_{i}) , F_{2}[u](x_{i})),\quad&\text{if }x_i\in V_\mathrm{inn},\\
u(x_i)-v_i,\quad&\text{if }x_i\in\Gamma,
\end{cases}$$ and $F_{1}^+$ and $F_{2}$ are degenerate elliptic and consistent grid functions, which implies the same for $F$. Furthermore, the values $v_i\in\R$ corresponds to prescribed values of $u$. For instance, all $v_i$ such that $x_i$ is a boundary node are equal to zero. Since, however, the values of an infinity ground state can also be fixed in the set where the distance function of the domain attains its maximum [@juutinen1999infinity], we reserve the possibility for setting $v_i=r_1$ for all $v_i$ in this set, where $r_1$ denotes the in-radius of the domain. Note that the superscript in $F_1^+$ serves to distinguish this scheme from a similar one for higher eigenfunctions, introduced in the next section.\
\
We first discuss the grid function $F_1^+$ which is the novelty of our approach. Taking \[eq:first\_ef\_prob\] into account we have to approximate the term $|\nabla u|- \Lambda_1 u$. In the following, we fix a vertex $x_i$ and, suppressing the dependency on $i$, denote the distances to its neighbors by $d_j=|x_i-x_j|$ for $j\in N_i$. We define $F_1^+$ as $$\begin{aligned}
\label{eq:def_scheme_F1}
F_1^+[u](x_i)=u_i-u_{i_{\max}}-d_{i_{\max}}\Lambda_1 u_i,\end{aligned}$$ where the index $i_{\max}$ is chosen such that $$\begin{aligned}
\label{eq:idx_max_gradterm}
i_{\max}\in\arg\max_{j\in N_i}\frac{u_i-u_j}{d_j}.\end{aligned}$$ The number $\Lambda_1$ is the reciprocal of the maximal value of the distance function on the grid. Note that \[eq:def\_scheme\_F1\] in fact approximates a multiple of the desired term, which, however, does not change equation \[eq:discrete\_scheme\]. One can see that \[eq:def\_scheme\_F1\] is non-decreasing in the differences $u_i-u_j$. This is the reason why the naïve approximation $|\nabla u(x_i)|\approx |u_i-u_j|/d_j$ is not suitable; it is not monotone.\
\
Next we recap the approximation of the infinity Laplacian due to Oberman in [@oberman2005convergent]. One defines a discrete Lipschitz constant $L(u_i)$ of $u$ in $x_i$ as $$L(u_{i})= \max_{j\in N_i} \frac{|u_i-u_j|}{d_j}.$$ In [@oberman2005convergent Theorem 5] Oberman has proved that the minimizer of this discrete Lipschitz constant with respect to $u_i$ is given by $$\begin{aligned}
u^*_i=\argmin_{u_i} L(u_i)=\frac{d_{s}u_{r}+d_{r}u_{s}}{d_{r}+d_{s}},\end{aligned}$$ where the indices $r,s\in N_i$ are chosen such that $$\begin{aligned}
\label{eq:idx_max_inf_lapl}
(r,s)\in\arg\max_{k,l\in N_i}{\left\{ \frac{|u_{k}-u_{l}|}{d_{k}+d_{l}}\right\}}. \end{aligned}$$ Furthermore, $u^{*}_i$ is non-decreasing as a function of $\{u_{j} \;:\; j\in N_i\}$ and it holds $$-\Delta_{\infty} u(x_{i})=\frac{1}{d_{r}d_{s}} \left(u_i- u^*_i\right) + \mathcal{O}(dx_i+d \theta_i),$$ where $dx_i$ and $d\theta_i$ denote the errors \[eq:errors\]. Hence, we can express the grid function $F_2$ in \[eq:discrete\_scheme\], evaluated in a grid point $x_i$ as $$\begin{aligned}
\label{eq:def_scheme_F2}
F_2[u](x_i)=u_i-u^*_i.\end{aligned}$$ Again, this approximates only a multiple of the infinity Laplacian which is no problem due to the nature of equation \[eq:discrete\_scheme\].
Our main statement of this section is that the grid function $F$ from \[eq:discrete\_scheme\] is degenerate elliptic, consistent, and stable under reasonable conditions on the grid. Together with \[thm:barles\_souganidis\] this implies convergence to viscocity solutions of the infinity Laplacian eigenvalue problem.
\[prop:discrete\_scheme\] Assume that
1. For every $r\in N_i$ there exists $s\in N_i$ such that $x_i-x_r$ and $x_s-x_i$ are parallel vectors,
2. $\max_{j\in N_i}d_j-\min_{j\in N_i} d_j=\mathcal{O}(1)$.
Then the grid function \[eq:discrete\_scheme\] is degenerate elliptic, consistent with respect to \[eq:first\_ef\_prob\], and stable.
The statements for $F_2$ given by \[eq:def\_scheme\_F2\] were proved in [@oberman2005convergent]. In particular the proofs there utilize the grid conditions 1. and 2.\
Monotonicity of $F_1^+$ in the differences is clear from its definition in \[eq:def\_scheme\_F1\]. For consistency one notices that by Taylor expansion one can write a smooth function $u$ as $$u_i-u_j=\nabla u(x_i)\cdot(x_j-x_i)+o(d_i).$$ Hence, maximizing the left hand side corresponds to choosing $j$ such that $x_j-x_i$ is as close to being parallel to $\nabla u(x_i)$ as possible. By Cauchy-Schwarz one obtains that $$\lim_{d_j\to 0}\frac{u_i-u_j}{d_j}\leq |\nabla u(x_i)|,$$ with equality if and only if $x_j-x_i$ is parallel to $\nabla u(x_i)$. Hence, if one chooses $j=i_{\max}$ such that the left hand side is maximized and sends the directional error $d\theta_i$ to zero, we get equality. For stability it suffices to notice that, due to positive homogeneity of \[eq:discrete\_scheme\], any solution can be rescaled to be uniformly bounded.
Note that we do not rigorously prove existence of roots of the grid function \[eq:discrete\_scheme\] since this requires some discrete theory which is beyond the scope of this paper and does not give much insights. Clearly, existence also depends on the set $\Gamma$. Here, we just sketch the proof for the case that $\Gamma$ coincides with the union of boundary nodes of the domain and nodes where the distance function $d$ is maximal. Correspondingly, we set $v_i=0$ or $v_i=r_1$ in , respectively. In this case, for every infinity harmonic function, i.e., $F_2[u_\mathrm{harm}]=0$, it holds $F[u_\mathrm{harm}]\leq 0$. Furthermore, the distance function $d$ meets $F[d]\geq 0$. By continuity of $F$ in every component there has to be $u$ which meets $u_\mathrm{harm}\leq u \leq d$ and satisfies $F[u]=0$.
\[rem:monotonicity\] One might ask whether the grid function $F_1^+$, and hence also the combined function $F$, is monotone in the nodal values $u_i$. In the theory of monotone schemes this would ensure that the scheme $F[u]=0$ possesses a unique solution, which cannot be expected. However, $F_1^+$ can be rewritten as $F_1^+[u](x_i)=(1-d_{i_{\max}}\Lambda_1) u_i-u_{i_{\max}}$ and the coefficient $1-d_{i_{\max}}\Lambda_1$ is non negative due to the definition of $\Lambda_1$. Since the term $u_{i_{\max}}$ does not change for sufficiently small changes in $u_i$, function $F_1^+$ is at least locally monotone. Furthermore, from this representation it can be seen that $1$ is a Lipschitz constant of $F_1^+$.
Approximation of higher eigenfunctions {#ss:eigenfunction_second}
--------------------------------------
Similarly as before, we would like to approximate our reformulation for higher eigenfunctions $$\min(|\nabla u|-\Lambda u, -\Delta_\infty)+ \max(-|\nabla u|-\Lambda u, -\Delta_\infty u) + \Delta_\infty u=0$$ as monotone scheme. Analogously to the previous section we approximate this equation with $F[u]=0$ where $F$ is now given by $$\begin{aligned}
\label{eq:discrete_scheme_2}
F[u](x_i) \ = \
\begin{cases}
\!\begin{aligned}
\min ( F_{1}^+[u](x_{i}) , &F_{2}[u](x_{i}))+\\
&\max ( {F}_{1}^-[u](x_{i}) , F_{2}[u](x_{i}))-F_2[u](x_i),
\end{aligned}
\; &\text{if }x_i\in V_\mathrm{inn},\\
u(x_i)-v_i,\; &\text{if }x_i\in\Gamma,
\end{cases}\end{aligned}$$ and $F_1^+$ and $F_2$ are as in \[eq:def\_scheme\_F1\] and \[eq:def\_scheme\_F2\], respectively. The function $F_1^-$ is given by $$\label{eq:scheme_F1-}
F_1^-[u](x_i)=u_i-u_{i_{\min}}-d_{i_{\min}}\Lambda u_{i}.$$ Here the index $i_{\min}$ is chosen such that $$\begin{aligned}
i_{\min}\in\arg\min_{j\in N_i}\frac{u_i-u_j}{d_j}\end{aligned}$$ and hence \[eq:scheme\_F1-\] approximates a multiple of $-|\nabla u|-\Lambda u$. As for the first eigenfunction, one can easily see the consistency of the grid function. Furthermore, monotonicity in the differences $u_i-u_j$ is proved as in \[prop:deg\_elliptic\_H\], using that the non-monotone term $-F_2[u](x_i)$ in \[eq:discrete\_scheme\_2\] always vanishes and the first two terms are obviously monotone.
Lastly, the set $\Gamma$ together with the values $v_i$ allows us again to fix some known values of the eigenfunction, for instance at the boundary of the domain or at the locations where the eigenfunction attains its global minimum and maximum.
Numerical solution of the schemes
---------------------------------
Now we describe how we solve $F[u]=0$ where $F$ given by \[eq:discrete\_scheme\] or \[eq:discrete\_scheme\_2\]. Due to the non-smoothness of the grid functions $F$, Newton-type methods are not applicable to compute a root of $F$. Also quasi-Newton methods require some degree of (directional) differentiability in the root $x^*$ such that $F[u^*]=0$ in order to converge (cf. e.g. [@martinez1995inexact; @sun1997newton]). Due to the strong non-smoothness of $F$ given by \[eq:discrete\_scheme\] or \[eq:discrete\_scheme\_2\], this is too much of an assumption. Hence, it seems natural to study the simple fixed-point iteration $$\begin{aligned}
\label{eq:fixpoint_it}
u\gets E[u]\end{aligned}$$ where $E[u]=u-\rho F[u]$ is referred to as Euler map. Obviously, roots of $F$ correspond to fixed-points of $E$. The terminology “Euler map” stems from the obvious fact that \[eq:fixpoint\_it\] can be seen as explicit Euler discretization of the ODE $\dot{u}(t)=-F[u(t)]$ with time step size $\rho>0$. Is is well-known (cf. [@oberman2005convergent], for instance) that if $\rho>0$ is smaller than the reciprocal Lipschitz constant of $F$ and $F$ is monotone in the sense that $u\geq v$ implies $F[u]\geq F[v]$ in the partial order in $\R^M$, then the Euler map $E$ is a contraction. Since this would in particular imply a unique fixed point of $E$ and hence a unique root of $F$, we cannot expect this in our case.
However, due to the “local monotonicity” of $F$ (cf. \[rem:monotonicity\]) one can expect that in the proximity of a root the map $F$ is monotone and hence $E$ is a contraction there. In practice, the fixed point iteration \[eq:fixpoint\_it\] converges very reliably on our numerical experiments. For designing a stopping criterion we utilize both the relative changes of the iterates and the accuracy of the root. The detailed algorithm to find a root of $F$, and hence an infinity Laplacian eigenfunction, is given in \[alg:root\].
$u$
Experimental results {#s:results}
====================
In the following, we present numerical results which use the schemes and algorithms from \[s:numerics\]. Many of the experiments deal with open questions and conjectures regarding infinity eigenfunctions and, thereby, we hope to shed some light on the theory.
The computations take place on a regular grid which discretizes the unit square $[-1,1]^2$. In order to compute on more general domains we simply restrict the computations on those grid nodes which belong to the domain of interest (see, for instance, \[sss:domains\] below).
In all experiments apart from the very first one we choose the number of local neighbors $k_i$ of a generic node $x_i\in V_\mathrm{inn}$—which appears in \[eq:idx\_max\_gradterm\] and \[eq:idx\_max\_inf\_lapl\], for instance—as $k_i=120$. In our regular grid this corresponds to a quadratic stencil of $11\times 11$ around the node of interest. If parts of the stencil leave the computational domain—which happens close to the boundary, for instance—we simply reduce the number of neighbors of the corresponding node. We discretize the unit square including its boundary with $97\times 97$ nodes. In most experiments the vertex set $\Gamma$ where we prescribe values of $u$ is chosen to be the boundary of the computational domain, in which case we set $v_i=0$ for all $x_i\in\Gamma$ in the grid functions \[eq:discrete\_scheme\] and \[eq:discrete\_scheme\_2\].
If not stated differently, the inputs in \[alg:root\] were chosen as follows. When computing infinity ground states (cf. \[ss:ground\_state\_numerics\] below) the initial guess $u^0\in\R^M$ is chosen as discrete distance function of the domain[^3]. The constant $\rho>0$—which should be chosen smaller than the reciprocal Lipschitz constant of $F$—is chosen as $\rho = 0.9$. Note that the Lipschitz constants of $F$ given by \[eq:discrete\_scheme\] or \[eq:discrete\_scheme\_2\] equal one. We allow for a maximum of $K=3000$ iterations and choose the tolerance $\mathrm{TOL}=10^{-7}$. Let us remark that in almost all our experiments the algorithm required only a few hundred iterations in order to reach the tolerance. Furthermore, the tolerance should scale with the square of the characteristic grid size which can be seen from \[eq:def\_scheme\_F2\].
Our implementation uses MathWorks MATLAB^^ R2018b and a typical test-case requires a few minutes of computation on a standard laptop computer. [ The code which reproduces all results will be made publicly available upon acceptance of this manuscript.]{}
Infinity ground states {#ss:ground_state_numerics}
----------------------
In this section we perform numerical experiments for infinity ground states, by computing a root of \[eq:discrete\_scheme\]. The eigenvalue occurring in \[eq:def\_scheme\_F1\] is chosen as maximum of the distance function on the grid.
### Influence of the number of local neighborhood size
First, we would like to investigate the influence of the number of local neighbors $k_i$ on the computed ground state. Remember that due to \[prop:discrete\_scheme\] one can expect more accurate results as the number increases. In \[fig:contour\_plots\_neighborhoods\] we show the level lines of the ground state on the unit square $[-1,1]^2$, computed using neighborhoods of size $3\times 3$, $5\times 5$, $7\times 7$, and $11\times 11$. Looking at the level lines, one can observe that the smoothness of the ground state increases as the neighborhood size grows. This can be explained by a more accurate approximation of $\nabla u$ and its norm. Further experiments show the same behavior for the infinity harmonic function on the punctured square $[-1,1]^2\setminus\{0\}$ (see also [@oberman2005convergent] for similar observations). In the following experiments we will use the $11\times 11$ stencil in order to produce accurate results.
[ ]{} [ ]{} [ ]{} [ ]{}
### Infinity ground state and infinity harmonic on the square
In this experiment we investigate the long-standing conjecture that the infinity harmonic function on the punctured square is a ground state (cf. e.g [@juutinen1999infinity]). Note that the analogue of this statement is known to be true on stadium-like domains [@yu2007some] (like for instance the ball) but is false in general [@lindgren2012infty].
In \[fig:comparison\_ef\_harm\] we show the infinity ground state on the square, computed with our method, and the infinity harmonic function on the punctured square. Note that we compute the latter by simply solving the scheme $F_2[u]=0$ together with appropriate boundary conditions, where $F_2$ is given by \[eq:def\_scheme\_F2\]. The two functions differ by an $L^\infty$-error of order $10^{-3}$ which numerically confirms the conjecture that ground state and infinity harmonic coincide on the square.
[ ]{} [ ]{}
### Discrete non-uniqueness on the rectangle {#sss:rectangle}
In this experiment we address the question of uniqueness of infinity ground states, computed with our method. As mentioned in \[rem:nonuniqueness\], it is known that ground states are in general not unique and a non-convex dumbbell domain where uniqueness fails was constructed in [@hynd2013nonuniqueness]. However, for convex domains uniqueness is neither proved nor disproved apart from the case of stadium-like domains [@yu2007some]. In \[fig:nonuniqueness\_rectangle\] we show surface and contour plots of two different ground states on the rectangle $[-1,1]\times[-0.5,0.5]$, computed with our method, and their pointwise difference. Both results fulfill $F[u]=0$ with high accuracy, as enforced through the stopping criterion in \[alg:root\]. In this experiment we choose $\Gamma$ as the boundary of the rectangle together with the point $(0,0)$ and set $u(0,0)=0.5$ there. This is no loss of generality due to the homogeneity of the eigenvalue problem \[eq:discrete\_scheme\]. The first result was computed by initializing \[alg:root\] with the distance function, whereas the second one was initialized with zero. The two results differ significantly: the first one attains its maximum on the so-called high ridge of the rectangle, given by $[-0.5,0.5]\times\{0\}$, and has already been constructed analytically in [@juutinen1999infinity]. It is glued from the ground state of the square an the central part of the distance function of the rectangle. In contrast, the second result attains its maximum only in the point $(0,0)$ and its level lines are not parallel to the long sides of the rectangle as it is the case for the first ground state. Note that the second result in \[fig:nonuniqueness\_rectangle\] stands in contradiction to [@yu2007some Thm. 2.4] which states that every ground state on a convex domain attains its maximum exactly in the set where also the distance function attains its maximum. This suggests that either the Barles-Souganidis \[thm:barles\_souganidis\] is inapplicable in our case, or the results in [@yu2007some] are not entirely correct, or an extremely fine discretization is necessary such that the second result equals the first one. We suspect the latter reason, however, at the time of submission of this manuscript this is still an open question which may spark interesting future discoveries.
[ ]{} [ ]{} [ ]{}\
[ ]{} [ ]{} [ ]{}
### Infinity ground states on different domains {#sss:domains}
With this test-case we demonstrate the aptness of our algorithm to compute infinity ground state also on more complicated and in particular non-convex domains. \[fig:domains\] shows the computed ground states on six different convex (top row) and non-convex (bottom row) domains. The shapes of the ground states are similar to $p$-Laplacian ground states for large values of $p$, see for instance [@horak2011numerical; @bozorgnia2016convergence].
[ ]{} [ ]{} [ ]{}\
[ ]{} [ ]{} [ ]{}
### Regularity of ground states
[r]{}[0.4]{}
[ ]{}
Next we study the regularity of ground states which is still an open problem from the theoretical perspective. Some results on singular sets of ground states were proven in [@yu2007some], among which are the statements that ground states are non-differentiable in the maximal set of the distance function and that singular points of the gradient are not isolated. Furthermore, in two dimensions ground states are $C^1$ away from their maximal set if and only if they are infinity harmonic there. However, general statements on the regularity of ground states outside the maximal set are still pending.
Here we recap the ground state computed for the dumbbell shape (see also bottom center in \[fig:domains\]). \[fig:contour\_dumbbell\] shows the level lines of the ground state and exhibits a non-smoothness along the line segment which connects to the two maxima. Here the level lines show kinks and even touch in the center of the domain. This suggests that ground states are non-differentiable, in general. Furthermore, the second ground state on the rectangle (cf. center in \[fig:nonuniqueness\_rectangle\]) seems to be non-differential on the line segment $\{0\}\times [-0.5,0.5]$ which would imply that non-smoothness is possible even on convex domains.
Higher eigenfunctions {#ss:higher_efs_numerics}
---------------------
There are two main difficulties with the computation of higher eigenfunctions: finding the eigenvalue and a good initialization. While the second eigenvalue has a variational characterization \[eq:second\_eigenvalue\], there is no such interpretation for higher infinity eigenvalues (see [@juutinen2005higher] for an extensive discussion). Another difficulty arises from discretization, since even if an eigenvalue for some domain is explicitly known, it is not necessarily an eigenvalue of the grid function \[eq:discrete\_scheme\_2\]. However, on domains which enjoy some symmetry the discrete eigenvalue can be computed explicitly or using the distance function of the expected nodal domains of the eigenfunction. Here, one should remark that the restriction of an eigenfunction to a nodal domain is a ground state of the domain. This knowledge can also be used by fixing values of the eigenfunction in the set $\Gamma$ (see \[s:numerics\]) where it is expect to attains its maximum or minimum.
Regarding initialization of \[alg:root\] there are several possibilities. If one prescribes values in $\Gamma$ which lie inside the domain, one can simply initialize with zero. Alternatively, if no nodal values are known, one can also initialize with a Laplacian eigenfunction, hoping that infinity eigenfunctions lie sufficiently close. A third possibility is to initialize randomly and slightly modify \[alg:root\], by adding the normalization step $$\begin{aligned}
\label{eq:normalization_step}
u_\mathrm{P} \gets \frac{\max(u,0)}{\norm{\max(u,0)}_\infty},\quad
u_\mathrm{N}\gets \frac{\max(-u,0)}{\norm{\max(-u,0)}_\infty},\quad
u \gets u_\mathrm{P} - u_\mathrm{N}\end{aligned}$$ after the fixed-point iteration. This assures that the maximum of the positive and negative parts of $u$ are equal, which is a necessary condition for second eigenfunctions [@juutinen2005higher]. However, by this modification convergence of \[alg:root\] *does not* imply that one has solved $F[u]=0$. Instead it only implies that $F[u]=cu$ for some $c\in\R$. However, in practice this modification often converges to some $u$ which is very close to an eigenfunction, meaning that $c\approx 0$. If the modulus of $c$ is larger than some tolerance, one can utilize this intermediate solution as initial condition for \[alg:root\], whose convergence implies that $F[u]=0$.
shows second infinity eigenfunctions on three different domains, for which the second eigenvalue can be computed. The algorithm was initialized with zero and the peak values of the eigenfunctions were fixed. Again the results are similar to second eigenfunctions of the $p$-Laplacian for large $p$ (e.g. [@horak2011numerical]).
[ ]{} [ ]{} [ ]{}
In \[fig:higher\_efs\_square\] we show three different eigenfunctions on the square where no peak values were fixed. Here we initialized with the first three eigenfunctions of the standard Laplacian on the square which can be computed with standard linear algebra tools[^4].
[ ]{} [ ]{} [ ]{}
Finally, we also show a result which was computed with the normalizations steps \[eq:normalization\_step\] of the positive and negative parts of the solution. The initialization was chosen as random noise and also this method converges to a second eigenfunction nicely, as visualized in \[fig:triangle\_normalized\], which shows the solution after 0, 300, and 655 iterations of \[alg:root\] with the normalizations \[eq:normalization\_step\].
[ ]{} [ ]{} [ ]{}
Conclusion
==========
In this work we have presented a reformulation of the higher eigenvalue problem for the infinity Laplacian, thereby avoiding the distinction of cases on the sign of the solution. Utilizing this reformulation, we proposed consistent and monotone schemes for approximating ground states and higher eigenfunctions. The schemes are solved by a fixed-point iteration. Our numerical results show the aptness of the schemes to approximate eigenfunctions even on complicated domains. This appears to be the first theoretically profound numerical method in the literature to compute infinity Laplacian eigenfunction.
There are two main open problems related to our work, which will be subject to future research. The first aim is to find an algorithm to compute the second eigenvalue on general domains. While on symmetric domains one can easily compute it using the distance function on the expected nodal domains, for more complicated domains there is currently no reliable approach. We believe that the several variational characterizations of the second eigenvalue in [@juutinen2005higher] can be used to design an algorithm.
Second, the fixed-point iteration, which we currently use to compute roots of the grid functions, can possibly be replaced by a more involved non-smooth Newton-type method. However, currently we are not aware of an alternative method which can handle the strong non-smoothness of the problem.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the European Unions Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 777826 (NoMADS).
[^1]: Department of Mathematics, Instituto Superior Técnico, Lisbon,
[^2]: Department of Mathematics, Friedrich-Alexander University Erlangen-Nürnberg, (, )
[^3]: We used the MATLAB^^ routine `bwdist`, Copyright 1993-2017 The MathWorks, Inc.
[^4]: We used the MATLAB^^ routine `eigs`, Copyright 1984-2018 The MathWorks, Inc.
|
---
abstract: 'It is widely believed that the cool gas clouds traced by Mg [ii]{} absorption, within a velocity offset of 5000 kms$^{-1}$ relative to the background quasar are mostly associated with the quasar itself, whereas the absorbers seen at larger velocity offsets towards us are intervening absorber systems and hence their existence is completely independent of the background quasar. Recent evidence by has seriously questioned this paradigm, by showing that the number density of intervening Mg [ii]{} absorbers towards the 45 blazars in their sample is nearly 2 times the expectation based on the Mg [ii]{} absorption systems seen towards normal QSOs. Given its serious implications, it becomes important to revisit this finding, by enlarging the blazar sample and subjecting it to an independent analysis. Here, we first report the outcome of our re-analysis of the available spectroscopic data for the BBM sample itself. Our analysis of the BBM sample reproduces their claimed factor of 2 excess of $dN/dz$ along blazar sightlines, vis-a-vis normal QSOs. We have also assembled a $\sim$3 times larger sample of blazars, albeit with moderately sensitive optical spectra. Using this sample together with the BBM sample, our analysis shows that the $dN/dz$ of the Mg [ii]{} absorbers statistically matches that known for normal QSO sightlines. Further, the analysis indicates that associated absorbers might be contributing significantly to the estimated $dN/dz$ upto offset speeds $\Delta v \sim 0.2c$ relative to the blazar.'
author:
- |
[S. Mishra$^{1}$[^1], H. Chand$^{1}$, Gopal-Krishna$^{2}$[^2], R. Joshi$^{3,4}$, Y. A. Shchekinov$^{5,6}$, T. A. Fatkhullin$^{7}$]{}\
\
$^{1}$Aryabhatta Research Institute of Observational Sciences (ARIES), Manora Peak, Nainital $-$ 263002, India\
$^{2}$UM-DAE Centre for Excellence in Basic Sciences, University of Mumbai, Mumbai 400098, India\
$^{3}$Inter-University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411007, India\
$^{4}$Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China\
$^{5}$Lebedev Physical Institute, Russian Academy of Sciences, 53 Leninsky Ave., Moscow 119991, Russia\
$^{6}$Raman Research Institute, Sadashiva Nagar, Bangalore 560080, India\
$^{7}$Special Astrophysical Observatory, Russian Academy of Science, Karachai-Cherkessia, 369167, Russia\
bibliography:
- 'references.bib'
date: 'Accepted —. Received —; in original form —'
title: 'On the incidence of Mg [ii]{} absorbers along the blazar sightlines'
---
\[firstpage\]
galaxies: active – galaxies: photometry – galaxies: jet – quasars: general – (galaxies:) BL Lacertae objects: general – (galaxies:) quasars: emission lines
Introduction {#sec:intro_mgiidndz}
============
Analysis of the narrow absorption-line systems (in the spectra of quasars) has emerged as a powerful probe of the physical conditions of the gaseous medium of intervening galaxies, particularly when they lie at extremely large distances and hence too faint for direct imaging/spectroscopy even with the largest telescopes . It is widely held that the cool gas clouds (e.g., Mg [ii]{} absorption systems) with velocities offsets $\beta$c up to $\sim$ 5000 km s$^{-1}$ relative to the background quasar are gravitationally bound to the quasar itself , whereas absorbers showing larger velocity offsets directed towards us are intervening systems probably associated with foreground galaxies and, consequently, their existence should be totally independent of the background quasar. A few recent studies, however, seem to question this canonical view, based on the differing estimates for the incidence rates of intervening systems (like Mg [ii]{} absorption having $\beta c \ge 5000$ km s$^{-1}$) detected towards different types of background sources, such as normal QSOs, gamma-ray bursters (GRBs) and blazars . It has been also claimed by BBM and @2009ApJ...697..345C that associated systems having a significantly relativistic speed relative to the quasar may also be present (BBM), e.g., when the quasar is undergoing powerful jet activity and/or ejecting high speed accretion-disk outflows. A possible way to differentiate between these possibilities would be to check if the incidence rates, $dN/dz$, of intervening absorbers differ depending on whether the background sources are non-blazars, or blazars whose powerful relativistic jets are therefore expected to be pointed close to our direction and, consequently the jet-accelerated potential absorbers would lie along the line-of-sight. Indeed, this expectation is echoed in the unexpected finding of BBM that the $dN/dz$ of Mg [ii]{} absorption systems (for strong absorbers having a rest-frame equivalent width W$_{r} ~ \geq$ 1 Å) towards blazars is $\sim$ 2 times larger (at 3$\sigma$ confidence) than the value established for the sightlines to normal quasars (QSOs). An even greater excess had earlier been reported by Stocke & Rector (1997), albeit using a much smaller sample of blazars. On the other hand, a recent analysis by @Chand2012ApJ...754...38C of the existing high-resolution spectra of a sample of about 115 flat-spectrum radio-loud quasars (FSRQs, of non-blazar type) did not show any excess in the incidence of Mg [ii]{} absorption systems, as compared to QSOs. They reconciled the two seemingly discrepant results by appealing to the jet orientation scenario which lies at the heart of the Unified Scheme for powerful extragalactic radio sources . In their explanation, since the jets in FSRQs are thought to be less closely aligned to the line-of-sight, any gas clouds accelerated outward by the powerful jets are unlikely to appear in the foreground of the quasar’s nucleus and hence escape being detected in absorption against its bright optical emission. Later, @Joshi2013MNRAS.435..346J extended this probe by analyzing a large set of redshift-matched sightlines to 3975 radio core-dominated (CDQs, i.e., FSRQs) and 1583 radio lobe-dominated (LDQs) quasars. While, overall, only a marginal (9% at 1.5$\sigma$ significance) excess of $dN/dz$ was found towards the FSRQ sightlines, as compared to the sightlines to normal QSOs, they showed that the excess becomes quite significant (3.75$\sigma$) when the comparison is restricted to the absorbers having offset speeds i.e., $\beta < 0.1c$ relative to the background quasar. Similarly, @2011ApJ...742...44T have used observations of Fe XXV/XXVI K-shell resonance lines in the X-ray band and found their outflow velocity distribution spans from $\sim$ 10,000 kms$^{-1}$ up to $\sim$ 100,000 kms$^{-1}$ ($\sim$ 0.3c), with a peak and mean value of $\sim$ 42,000 kms$^{-1}$ ($\sim$ 0.14c), for highly ionised gas clouds with column densities of N$_{H} \approx$ 10$^{23}$ cm$^{-2}$ located within the central parsec of the AGN.
In this context, it is important to emphasize that even though BBM’s analysis has employed very high-sensitivity spectral data, their result rests on just 45 blazars, due to which small number statistics might be at work. It is worthwhile recalling that the 4-fold excess of $dN/dz$ along the GRB sightlines, inferred by @Prochter2006ApJ...648L..93P using just 14 GRBs, has subsequently been pronounced as a possible statistical fluke, on the basis of a 3 times larger set of sightlines [@Cucchiara2013ApJ...773...82C]. Given its potentially deep ramifications, it is therefore desirable to revisit the BBM claim of excess $dN/dz$ towards blazars, by enlarging the blazar sample and carrying out an independent analysis. The present study is motivated by this objective.\
The paper is organized as follows, in Section 2 we describe our sample, while in Section 3 our data analysis procedure is outlined. The results are given in Section 4, followed by a brief discussion and conclusions in Section 5.
The Sample {#sample}
==========
Archive Instrument Content Resolution
--------- ------------ --------------------------------- ------------ -- -- --
ESO FORS-1/2 48 blazars found (10 taken[^3]) 900
ESO X-SHOOTER 17 found(8 taken[^4]) 3600
ESO UVES 1 found (taken) 40000
SAO SCORPIO 3 new observations (3 taken) 818
KECK LRIS 2 (both taken) 9800
SDSS BOSS 622 found (196 taken[^5]) 2500
BBM FORS-1 42 (all 42 taken) 900
: The spectral data sourcing for our enlarged sample of blazars.[]{data-label="table:sample"}
The blazar sample employed in our analysis is an amalgamation of 3 sets of blazars extracted from the catalogues published by @Massaro2009yCat..34950691M [hereafter ROMA-BZCAT], , and @Padovani1995MNRAS.277.1477P [hereafter, Padovani-Catalogue]. From the ROMA-BZCAT we selected sources classified as BZB (implying confirmed BL Lac), resulting in a set of 1059 blazars. From the VV catalogue we selected the sources classified either as ‘BL’ (i.e., confirmed BL Lac), or ‘HP’ (a confirmed highly polarized quasar). This resulted in a set of 729 confirmed blazars from this catalogue. Accounting for the 480 sources that are common to these two sets, led to a list of 1308 confirmed blazars. The Padovani-catalogue also classifies BL Lacs objects using homogeneous criteria. It contains a total of 233 blazars, of which 189 were already in the above two sets (among them 169 blazars of the Padovani-catalogue are in BZ-ROMA, while 20 in the VV catalogue). Their exclusion left us with 44 blazars solely contributed by the Padovani-catalogue. Merging these 3 sets resulted in our final ‘parent sample’ of 1352 confirmed blazars.
We then performed an extensive search for optical spectra of our parent sample of blazars, in the archives of the Sloan Digital Sky Survey[^6] (SDSS), the European Southern Observatory[^7] (ESO) and the KECK[^8] Observatory. We applied two main selection filters: (i) median SNR of the entire spectrum should be more than 5, so that false detections of Mg [ii]{} line are minimized, and (ii) the blazar’s redshift should allow at least $10\times(1+z_{em}$)Å wide coverage in the available spectrum, of the region between the Ly$\alpha$ and Mg [ii]{} emission lines; this would ensure that the observed spectrum can be used to search for the Mg [ii]{} doublet due to at least one absorber (given that the two components of the doublet Mg [ii]{} $\lambda$ 2796, 2803, are separated by 8 Å in the rest frame).
In the ESO archive, after excluding the spectra of the 42 BBM blazars, which had been taken using the FOcal Reducer and the low-dispersion Spectrograph (FORS1) at the ESO observatory, we found that for 66 of our blazars (within 1 arcmin search radius) a spectrum with SNR $> 5$ was available either in the reduced form, from the ESO-advanced data product [^9](17 observed using X-shooter spectrograph, 1 observed using the Ultraviolet and the Visual Echelle Spectrograph (UVES) ), or we were at least able to extract the spectra based on their raw images using associated calibration files (48 observed using FORS-1,2). Among these 66 blazars, emission redshift was available for only 31, out of which just 16 were found useful for Mg [ii]{} absorber search after applying our emission redshift constraint mentioned above. For the remaining 35 blazars with unknown emission redshifts we could establish a lower redshift limit for 4 blazars using the redshift of the observed most redshifted Mg [ii]{} absorption doublet. One of these 4 had to be excluded as the Mg [ii]{} derived redshift was not yielding adequate redshift path (i.e., not satisfying the selection criteria-ii). Here, it is also important to clarify that the spectral region containing this most redshifted absorption doublet was excluded for the purpose of computing $dN/dz$, in order to keep the estimate free of bias resulting from exclusion of those blazars with unknown redshift, for which even a lower limit to redshift could not be established (using the absorption doublet).
For another three blazars from our ‘parent-sample’ (see above), viz, J145127$+$635426, J165248$+$363212, J182406$+$565100, we have newly obtained spectra using the SCORPIO spectrograph (using VPHG1200 grism) mounted on the 6-m telescope at the Special Astrophysical Observatory (SAO). Inclusion of another 2 blazars viz, J001937$+$202146, J043337$+$290555, became possible due to the availability of their spectra in the KECK archive, taken with the Low Resolution Imaging Spectrometer (LRIS).
For 622 blazars in our ‘parent sample’ we could find spectra in the SDSS archives (within a tolerance of 2 arcsec) in reduced form, covering a wavelength range 3800-10000Å. Excluding the 69 spectra with SNR $< 5$, left us with good quality spectra for 553 blazars. Among these, emission redshifts were available for 277 blazars, out of which 150 sources were found useful for Mg [ii]{} absorber search, after meeting our aforementioned emission redshift criterion (selection criterion-ii). In addition, from among the 276 blazars with unknown emission redshift, a lower redshift limit could be set for 59 sources, using the detected Mg [ii]{} absorption feature. However, one of these 59 sources (viz, J130008.5$+$175538) had to be excluded as it did not meet our criterion of useful minimum redshift path (i.e., the selection criterion-ii). In addition, after excluding another 12 blazars as they are already included in our above sample from other resources (ESO archive and SAO observations) we are left with 196 blazars solely contributed by the SDSS, which are found satisfactory for the purpose of our Mg [ii]{} absorption-line search.
To recapitulate, we have assembled a sample of 220 blazars (SDSS: 196, ESO: 19, SAO: 3, KECK: 2) as summarized in Table \[table:sample\], to make a search for intervening Mg [ii]{} absorbers. Out of these, only a lower limit to $z_{em}$ is available for 58 blazars (i.e. 54 SDSS, 3 ESO and 1 KECK). We have dicussed three out of them in Appendix \[ap:appendix\_A\] where we have also shown their represtative spectra as well.
Further, as mentioned above, we have also made use of the BBM blazar sample to revisit their conclusion (section 1), by subjecting it to an independent data reduction and analysis procedure, as followed in the present work. The sample employed in their analysis consists of 45 blazars. For 42 of them, we could obtain the raw spectral data from the ESO archive[^10] based on their program ID 080:A-0276, 081:A-0193. The raw data used in the BBM analysis for the remaining 3 (northern sky) blazars were not accessible and hence they could not be included in our analysis.
The $z_{em}$ and SNR distributions for our sample of 262 blazars (including 42 from the BBM sample) are shown in Fig. \[fig:zemi\_snr\_dist\]. However, as described in Section \[subsec:ew\_redshiftpath\], among the 220 non-BBM blazars only 149 blazars were found to contribute to robust redshift path for the case of strong absorbers and only 58 of them have contributed to the robust redshift path for weak absorbers, as well. Thus, for the purpose of the present $dN/dz$ analysis, we are led to a final sample of 191 blazars (149 ours plus 42 BBM blazars) for the strong absorber case. Only 100 out of them also contribute to the $dN/dz$ analysis for weak absorbers. The sample of 191 blazars is listed in Table \[table:full\_sample\].
[@lll lr @]{} Target & $z_{em}$ & Data archive & $\Delta$z$_{strong}^{a}$ & SNR\
\
J001937$+$202146 & 0.858 & KECK & 0.417 & 6.4\
J003514$+$151504 & 0.443 & SDSS & 0.123 & 55.0\
J003808$+$001336 & 0.740 & FORS/ESO & 0.439 & 91.8\
J004054$-$091525 & 5.030 & FORS/ESO & 0.891 & 66.2\
— & — & — & — & —–\
\
\
\
\
\[table:full\_sample\]
{width="80.00000%" height="0.4\textheight"}
{width="80.00000%" height="0.4\textheight"}
Analysis
========
Data Reduction {#subsec:data_reduction}
--------------
Data reduction for the FORS spectra available for our 53 blazars, including the 42 blazars from BBM, was performed using the ESO-FORS pipeline (version 5.1.4), by executing it using the ESOREX [^11] algorithm. The pipeline performs a precise background subtraction on science frames, does master flat-fielding, rejects cosmic-ray impacts by employing an optimal extraction technique and then applies calibrations for wavelength and flux. The Keck/LRIS data were reduced using a publicly available Lris automated reduction Pipeline[^12] (lpipe) written in [idl]{}. The UVES and SDSS spectra (total 197 blazars) were already available in the reduced form. Finally, the spectra taken with the SAO 6-m telescope for 3 of our blazars (viz, J145127$+$635426, J165248$+$363212 and J182406$+$565100) were reduced using the standard IRAF tasks. For post-processing of each one-dimensional spectrum, which involves steps such as air-to-vacuum wavelength conversion, heliocentric correction, combining individual exposures for SNR enhancement and continuum fitting to determine the normalized spectrum, we have followed the procedure described in .
Computation of the EW detection limit and the corresponding useful redshift path {#subsec:ew_redshiftpath}
--------------------------------------------------------------------------------
For identifying absorption features in a given spectrum, proper evaluation of noise plays a crucial role as it defines the detection limit for the features. Therefore, to generate the distribution of rms noise along the spectrum we used the matched-filtering technique employed by @Zhu2013ApJ...770..130Z, which involves the following main steps:\
(i) Subtracting unity from each point/pixel of the normalized spectrum (thus, the mean level of the spectrum becomes zero).\
(ii) Using this residual spectrum, generate its amplitude version, by plotting only the magnitude of the signal at each spectral pixel (i.e., setting the negative sign to positive).\
(iii) This ‘noise amplitude spectrum’ is then subjected to a top-hat smoothing over the ‘effective spectral resolution’, which is taken to be the quadratic sum of the instrumental resolution and the typical absorption line width. For instance, the typical FWHM of the QSOs absorption line convolved with the SDSS instrumental resolution lies in the range $\sim$ 100-400 kms$^{-1}$ ($\sim$ 2-6 pixels). Thus, for the SDSS instrumental resolution of 120 kms$^{-1}$, we have taken an ‘effective spectral resolution’ of about 4 pixels [i.e., $\sim$ 276 kms$^{-1}$, e.g., see @Zhu2013ApJ...770..130Z].\
(iv) This ‘smoothed noise amplitude spectrum’ was then sub-divided into a sequence of 100 Å wide segments. Within each segment all points deviating by more than one $\sigma$ were clipped (including any spectral lines) and substituted with interpolated values. [In this smoothed noise amplitude spectrum the amplitude at a given spectral (i.e., wavelength) pixel ‘$i$’ is the representative noise, $n_i$, for that pixel.]{}\
(v) For each spectral pixel, we then set the 3$\sigma$ detection threshold of a spectral feature. To do this, we model the feature as a Gaussian having a FWHM equal to the afore-mentioned ‘effective spectral resolution’ and then subject it to the same ‘top-hat’ smoothing as mentioned in (iii) above, and finally, we optimize its amplitude to equal 3$n_i$. Equivalent width of the Gaussian satisfying this criterion thus becomes the limiting equivalent width ($W_{i,det}$) of the absorption line that would be accepted as a significant (3$\sigma$) detection at that particular pixel in the spectrum. Only provided the pre-set restframe threshold value ($W_{th}$), which is 0.3 Å for weak and 1.0 Å for strong absorption systems, respectively, exceeds the computed $W_{i,det}$ for that pixel, would that spectral pixel be accepted as contributing to ‘useful’ redshift path, not otherwise. Note that this procedure is very similar to that adopted in @2017arXiv170105624M.
As a result, for our blazar sample, the net useful redshift path at a given redshift $z_{i}$ (so that the absorption line falls in the $i^{th}$ spectral pixel) for detection of the doublet above a designated rest-frame equivalent width threshold, $W_{th}$ would be: $$\begin{aligned}
\begin{aligned}
g\left(W_{th},z_{i} \right) & = \sum_{j=1}^{N_{blazar}}\ H(z_i - z_{j,
min})\times H(z_{j, max} - z_i )\\ & \times H(W_{th} -{W_{j,det}(z_{i})})
\\
\end{aligned}
\label{eq:goz1}\end{aligned}$$
where $H$ is the Heaviside step function, and the summation is taken over all the blazar spectra in our sample, $z_{j,min}$ and $z_{j,max}$ are, respectively, the minimum and maximum expected absorption redshift limits which were used in the search for the doublet for $j^{th}$ quasar (see Section \[subsec:mgii\_iden\_mgiidndz\]). $W_{th}$ is the threshold rest-frame EW of the absorption line which we have set at 1 Å and 0.3 Å, for strong and weak absorption systems, respectively. $W_{j,det}(z_{i})$ is the computed rest-frame EW detection limit at the i$^{th}$ pixel in the spectrum of the $j^{th}$ quasar, as discussed above. In the parent sample of 262 blazar, a non-zero $g\left(W_{th},z_{i} \right)$ was found for only 191 blazars for the strong absorber case, and 100 out of them also contributed a non-zero $g\left(W_{th},z_{i} \right)$ for the case of weak absorption system, as well. Hence only these two subsets have been used in the present $dN/dz$ analysis.
![The distributions of redshift path density for the intervening Mg[ii]{} systems towards blazars, for the strong (W$_r (2796) \geq 1.0$ Å: red dashed curve) and the weak absorption systems ($0.3 \leq $ W$_r (2796) < 1.0$ Å: blue solid curve).[]{data-label="fig:redshift_path_density"}](redshift_path.ps){width="30.00000%" height="0.35\textheight"}
absorption-line identification {#subsec:mgii_iden_mgiidndz}
------------------------------
[@ccccl @]{} & $z_{em}$ & $z_{abs} $ & W$_{r}$ & Associated absorptions\
\
J001937$+$202146 & 0.858 & 0.69616 & 1.0969 &\
J010009$-$333730 & 0.875 & 0.67996 & 0.5110 &\
J014125$-$092843 & 0.730 & 0.50042 & 0.3641 &\
J021748$+$014449 & 1.715 & 1.34428 & 2.0130 & , , ,\
J023838$+$163659 & 0.940 & 0.52379 & 2.3725 & , ,\
— & — & — & — & —\
\
\
\
\[table:mgii\_absorbers\]
Normally, the wavelength coverage may differ from spectrum to spectrum, as it depends on the spectrograph’s specifications and the instrumental settings used for the observations. Additionally, we place two constraints on the redshift path over which the search for the doublet was made. Firstly, the search was restricted to within the range $(1+z_{em}) \times 1216$Å $< \lambda < (1+z_{em}) \times 2803$Å. The lower limit is meant to avoid the Ly$\alpha$ forest, while the upper limit is dictated by the fact that any absorber with $\lambda \ge (1+z_{em}) \times 2803$Å has a strong likelihood of being an associated system falling into the background AGN. We performed the search for the doublet at $z_{em}$ following the steps enumerated in section 3 of @Chand2012ApJ...754...38C. Accordingly, a line profile matching technique was used, such that we first plotted the normalized spectrum of a given blazar and then overplotted the same spectrum by shifting the wavelength axis by a factor of $\lambda2796.3543$/$\lambda2803.5315$ (i.e. 0.997). Then, about 50 Å wide spectral segments were manually examined. The location of perfect overlap between the absorption lines in the shifted and the original (unshifted) spectra were marked as a detected absorption system. As a further corroboration, we looked for the corresponding metal lines (e.g. , , , etc.) in the spectrum. If the redshift/velocity of a absorption doublet was found to be consistent with that of the peak estimated for the metal line(s), this was taken as a further confirmation of the previously identified absorber. Note that the systems having a velocity offset within 5000 kms$^{-1}$ of $z_{em}$ of the blazar were classified as associated systems, following the standard practice. For each detected absorption system, we also performed visually a quality check on the fit to the underlying continuum. If deemed desirable we carried out a local continuum fitting and this improved fit was then used to obtain a better estimate of (). Detailed information on the 43 absorption systems thus identified in the spectra of 34 blazars, out of the total present sample of $191$ blazars, is provided in Table \[table:mgii\_absorbers\].
Computation of $dN/dz$
-----------------------
The incidence rate of Mg [ii]{} absorbers is defined as $\frac{dN}{dz}
= N_{obs}/\Delta z$; where $N_{obs}$ is the total number of the Mg [ii]{} absorbers detected within the entire [*useful*]{} redshift path ($\Delta z$), defined as: $$\Delta z = \int_0^\infty \sum_{j=1}\ g_{j}(W_{min},z_i) dz_{i}
\label{eq:goz}$$ where g$_{j}(W_{min},z_i) = 1$ if W$_{th}$ ( 0.3 Å for weak systems and 1 Å for strong systems) is more than the (3$\sigma$) detection threshold W$_{j,det}(z_j)$ estimated for the $i^{th}$ spectral pixel (see Eq. \[eq:goz1\]), otherwise $g(W_{min},z_i) = 0$. The values of redshift path density for $i^{th}$ redshift pixel, i.e., $g(W_{min},z_i) $, were thus computed using the total 191 blazar sightlines for strong absorption systems and the 100 blazar sightlines for weak absorption systems (Fig. \[fig:redshift\_path\_density\]).
Table \[table:full\_sample\] lists the values of $\Delta z$ calculated for the individual sightline in our entire blazar sample and in its various sub-sets. The errors in the computed $dN/dz$ values were estimated assuming the Poisson small number statistics for $N_{obs} < 50$, within a limit of 1$\sigma$ confidence level of a Gaussian distribution, using the tabulation given by @Gehrels1986ApJ...303..336G.
Results
=======
Incidence of absorbers in the present enlarged sample {#section:enlarge_sample_analysis}
-----------------------------------------------------
[llllclllc]{} Sample type & &\
\
& $N_{obs}$ & $\Delta$z & $\frac{N_{obs}}{\Delta z} \equiv dN/dz$ & $\left(\frac{(N_{obs}/\Delta z)}{(dN/dz)_{qso}} \right)^{\alpha}$ & $N_{obs}$ & $\Delta$z & $\frac{N_{obs}}{\Delta z}$ & $\left(\frac{(N_{obs}/\Delta z)}{(dN/dz)_{qso}} \right)^{\beta}$\
\
Sample I $^{a}$ & 23& 44.21& $ 0.52_{ 0.11}^{ 0.13} $& $ 1.21_{ 0.25}^{ 0.31} $&20& 86.08&$ 0.23_{ 0.05}^{ 0.06} $&$ 1.26_{ 0.28}^{ 0.35} $\
\
Sample II $^{b}$ & 20& 40.06& $ 0.50_{ 0.11}^{ 0.14} $& $ 1.16_{ 0.26}^{ 0.32} $&16& 81.41&$ 0.20_{ 0.05}^{ 0.06} $&$ 1.07_{ 0.26}^{ 0.34} $\
\
Sample III $^{c}$ & 17& 32.69& $ 0.52_{ 0.12}^{ 0.16} $& $ 1.20_{ 0.29}^{ 0.37} $&13& 60.10&$ 0.22_{ 0.06}^{ 0.08} $&$ 1.11_{ 0.30}^{ 0.40} $\
\
Sample IV $^{d}$ & 6& 19.05& $ 0.31_{ 0.12}^{ 0.19} $& $ 0.74_{ 0.29}^{ 0.44} $&8& 58.98&$ 0.14_{ 0.05}^{ 0.07} $&$ 0.80_{ 0.28}^{ 0.39} $\
\
Sample V $^{e}$ & 3& 11.69& $ 0.26_{ 0.14}^{ 0.25} $& $ 0.57_{ 0.31}^{ 0.56} $&5& 37.67&$ 0.13_{ 0.06}^{ 0.09} $&$ 0.72_{ 0.31}^{ 0.49} $\
\
Sample VI $^{f}$ & 4& 15.48& $ 0.26_{ 0.12}^{ 0.20} $& $ 0.59_{ 0.28}^{ 0.47} $&7& 52.21&$ 0.13_{ 0.05}^{ 0.07} $&$ 0.76_{ 0.28}^{ 0.41} $\
\
Sample VII $^{g}$ & 2& 9.29 & $ 0.22_{ 0.14}^{ 0.28} $& $ 0.46_{ 0.29}^{ 0.60} $&5& 32.70&$ 0.15_{ 0.07}^{ 0.10} $&$ 0.76_{ 0.33}^{ 0.51} $\
\
Sample VIII $^{h}$ & 16& 24.58& $ 0.65_{ 0.16}^{ 0.21} $& $ 1.52_{ 0.38}^{ 0.48} $&9& 29.20&$ 0.31_{ 0.10}^{ 0.14} $&$ 1.61_{ 0.53}^{ 0.73} $\
\
Sample IX $^{i}$ & 15& 23.41& $ 0.64_{ 0.16}^{ 0.21} $& $ 1.50_{ 0.38}^{ 0.50} $&8& 27.41&$ 0.29_{ 0.10}^{ 0.14} $&$ 1.52_{ 0.52}^{ 0.75} $\
\
\
\
\
\
\
\
\
\
\
\
\
\
\[Table:all\_dndz\_result\]
[@rccc ccccc @]{} & & $\Delta$z &$\frac{N_{obs}}{\Delta z}$ &$\left(\frac{(N_{obs}/\Delta z)}{(dN/dz)_{qso}} \right)^{\alpha}$\
\
Low SNR & 7& 28.74&$ 0.24_{ 0.09}^{ 0.13} $&$ 1.19_{ 0.44}^{ 0.64} $\
\
High SNR & 13& 57.33&$ 0.23_{ 0.06}^{ 0.08} $&$ 1.29_{ 0.35}^{ 0.47} $\
\
\
\
\[table:low\_high\_snr\_comp\]
As noted in Section \[sec:intro\_mgiidndz\], the BBM result is based on a sample of just 45 blazars and this has motivated us to build an enlarged sample. Our sample of 191 blazars provides nearly a factor of 3 increase in the redshift path (Table \[table:full\_sample\]). Table \[Table:all\_dndz\_result\] summarizes the results for this enlarged sample and its various subsets. Note that, as in BBM, the values of $dN/dz$ for normal QSOs are calculated at the mean value of the redshift path for the corresponding blazar subset (column 1 of Table \[Table:all\_dndz\_result\]). This was done using equation 2 of BBM for the case of weak absorption systems, and equation 6 of BBM, for the case of strong absorption systems. From Table \[Table:all\_dndz\_result\], no significant excess is evident in the $dN/dz$ for the blazar sightlines, vis a vis normal QSOs, both for weak (column 5) and strong (column 9) absorbers. The same is apparent from Fig. \[fig:cummulative\_plot\], which displays the cumulative numbers of absorbers up to different values of absorption redshift. Although, when only the absorption systems having $\Delta v < 5000$ kms$^{-1}$ are deemed as associated systems and therefore excluded, a mild excess may be present for blazar sightlines (Fig. \[fig:cummulative\_plot\] top panel). However, it vanishes for the strong systems if one excludes all absorbers having offset velocities up to $\Delta v < 30000$ kms$^{-1}$ (Fig. \[fig:cummulative\_plot\] middle panel). The mild excess vanishes even for weak systems if the absorbers with offset velocities up to $\Delta v < 60000$ kms$^{-1}$ are excluded (Fig. \[fig:cummulative\_plot\] bottom panel), suggesting the possibility of extension of intrinsic absorbers up to $\Delta v$ = 0.2c for blazars. In any case, our focus here is on the results for strong absorption systems, which are statistically more robust since the majority of our spectra (which have relatively low SNR) have contributed to the useful redshift path only for strong systems and not for weak systems.
In Fig. \[fig:evol\_dndz\_zhu\] we have displayed the redshift dependence of for strong ($W_{\rm r}(\lambda 2796) \ge 1.0$ Å) absorption systems detected in our blazar sample and compared it with that computed for the sightlines towards normal QSOs, using the analytical expression given by @Zhu2013ApJ...770..130Z for strong absorption systems. To quantify the similarity of these two distributions we have applied Kolmogorov-Smirnov (KS) test, which resulted in P$_{null}$=0.997; where P$_{null}$ is the null probability that two distributions are indistinguishable. Similarly a very good statistical agreement is found between the estimates of $dN/dz$ for blazars and QSOs, with a $\chi^{2}$-test giving P$_{null}$ = 0.99, leading us to infer that the distributions of $dN/dz$ for blazars and QSOs are statistically indistinguishable. Recall that it is for the strong absorption systems that BBM had reported a significant excess of $dN/dz$ (compared to QSOs sightlines), based on the high SNR spectroscopic data available for their sample of 45 blazars. To pursue this further, we present in the next section a re-analysis of their data following our data reduction and analysis procedure. Since this has yielded results consistent with the BBM claim, could then the discrepant result we have found here using a much larger sample of blazars (Table \[table:sample\]) have its origin in the substantially lower SNR of the spectra available for most of the present enlarged sample? To check this possibility we divide our blazar sample into (i) a low SNR subset (spectra having SNR between 5 and 15) and (ii) a high SNR subset (SNR $\textgreater$ 15). It is seen from Table \[table:low\_high\_snr\_comp\] that in neither case is a significant excess of $dN/dz$ (vis a vis normal QSOs) detected for the strong absorption systems (the same is found to hold for the weak absorption systems as well ). Thus, the discrepant result found here for the present blazar sample from the BBM sample is unlikely to be on account of the SNR contrast between the spectral data employed in the two studies. An alternative possibility is explored in the next section.
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. \[fig:cummulative\_plot\]
![ Number density evolution of strong (W$_{\rm r}(2796) \geq
1.0$ Å) absorption systems (averaged over redshift bins of 0.5) towards our 191 blazar sightlines (black triangles), and the sightlines towards the QSOs in the SDSS (blue solid line). The absorption systems with $\Delta v < 5000$ kms$^{-1}$ have been excluded. The solid line for the SDSS QSOs has been computed from the analytical expression given by @Zhu2013ApJ...770..130Z for strong absorption systems towards QSOs. The two distributions show an excellent agreement with a P$_{null}$ of $\sim 0.99$ based on the KS and $\chi^{2}$ tests.[]{data-label="fig:evol_dndz_zhu"}](dndz_Strong.ps){width="30.00000%" height="0.35\textheight"}
Re-analysis of the BBM sample {#subsection:reanalysis_bbm}
-----------------------------
[@rccc ccccc @]{} & & & $\Delta$z$_{re-analysis}$ & & $\Delta z_{BBM}$ & $\left(\frac{(dN/dz)_{re-analysis}}{(dN/dz)_{BBM}} \right) $\
\
Strong$^{\gamma}$ & W$_{r} \ge 1.0 ~$Å$ $ & 12& 27.10 & 12 & 28.04 & $1.03\pm0.48$\
\
Strong$^{\beta}$ & W$_{r} \ge 1.0
~$Å$ $ & 8& 22.44 & 10 & 23.44 & $0.83\pm0.46$\
\
Weak$^{\gamma}$ & $~ 0.3 ~$Å$ \le $W$_{r}< 1.0$ & 17& 25.16 & 19 & 25.11 & $0.89\pm0.33$\
\
Weak$^{\beta}$ & $~ 0.3 ~$Å$ \le$ W$_{r}< 1.0$ & 14& 21.01 & 15 & 20.55 & $0.91\pm0.38$\
\
\
\
\
\
\
\[Table:bbm\_excess\_netzor\]
As discussed in Section \[sec:intro\_mgiidndz\], based on a sample of 45 blazars having high-sensitivity (ESO/FORS) spectra BBM found about a factor of 2 excess in the number density of absorbers on the blazar sightlines, as compared to the sightlines to normal QSOs. Since the present analysis of a sample of 191 blazars does not show such a trend, we have carried out a re-analysis of the BBM sample using the same procedure which we have followed here for our sample. As mentioned in Section \[sample\], we have limited the re-analysis to 42 out of the 45 BBM blazars, since we could not access the requisite raw spectral data for the remaining 3 (northern) blazars. For both weak and strong absorption systems, Table \[Table:bbm\_excess\_netzor\] compares our results with the BBM estimates of $N_{obs}$, $\Delta z$ and $dN/dz$. It is clear that the BBM estimates are reasonably well reproduced in our analysis; a few minor discrepancies are noted below.
For strong systems, there is a small difference in the redshift path, our value of 27.1 is slightly lower than the BBM estimate of 28.04. This small difference might be owing to the difference in the methods of determining *‘useful’* redshift path. We also compared the absorption redshifts and equivalent widths of the detected absorption systems and a good match was found, except in two cases: (i) the system at $z_{abs}$ = 0.5592 towards the blazar J0428$-$3756 was classified as ‘weak’ in BBM (= 0.93), but ‘strong’ (= 1.03) in our analysis, and (ii) the $z_{abs}$ = 1.1158 system towards J2031$+$1219 was classified as ‘strong’ (= 1.29 ) in BBM, but ‘weak’ (= 0.94) in our analysis. Coming to the weak systems, we detected a total of 17 absorbers, whereas BBM reported 19 absorbers, with the redshift path being 25.16 in our case, very close to their estimate of 25.11. Two systems, viz. (i) the $z_{abs}$ = 1.1039 system towards J1419$+$0445 (= 0.52) and (ii) the $z_{abs}$ = 0.6236 system towards J1956$-$3225 (with = 0.95), could not be included in our analysis. The former remained undetected and the latter system corresponds to a wavelength of 4539 Å which falls just below the starting wavelength of 4540 Å of the spectrum used in our analysis.
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[llllclllc]{} Sample type$^{\dagger}$ & &\
\
& $N_{obs}$ & $\Delta$z & $\frac{N_{obs}}{\Delta z}$ & $\left(\frac{(N_{obs}/\Delta z)}{(dN/dz)_{qso}} \right)^{\alpha}$ & $N_{obs}$ & $\Delta$z & $\frac{N_{obs}}{\Delta z}$ & $\left(\frac{(N_{obs}/\Delta z)}{(dN/dz)_{qso}} \right)^{\beta}$\
\
Sample I & 6& 18.80&$ 0.32_{ 0.13}^{ 0.19} $&$ 0.76_{ 0.30}^{ 0.45} $& 8& 39.68&$ 0.20_{ 0.07}^{ 0.10} $&$ 1.10_{ 0.38}^{ 0.54} $\
\
Sample II & 5& 16.87&$ 0.30_{ 0.13}^{ 0.20} $&$ 0.70_{ 0.30}^{ 0.47} $& 7& 37.56&$ 0.19_{ 0.07}^{ 0.10} $&$ 1.00_{ 0.37}^{ 0.54} $\
\
Sample III & 5& 15.19&$ 0.33_{ 0.14}^{ 0.22} $&$ 0.77_{ 0.33}^{ 0.52} $& 7& 29.83&$ 0.23_{ 0.09}^{ 0.13} $&$ 1.16_{ 0.43}^{ 0.62} $\
\
Sample IV & 1& 7.38 &$ 0.14_{ 0.11}^{ 0.31} $&$ 0.29_{ 0.24}^{ 0.67} $& 4& 27.56&$ 0.15_{ 0.07}^{ 0.11} $&$ 0.74_{ 0.35}^{ 0.58} $\
\
Sample V & 1& 5.69 &$ 0.18_{ 0.15}^{ 0.40} $&$ 0.36_{ 0.30}^{ 0.84} $& 4& 19.83&$ 0.20_{ 0.10}^{ 0.16} $&$ 0.85_{ 0.41}^{ 0.67} $\
\
Sample VI & 1& 6.37 &$ 0.16_{ 0.13}^{ 0.36} $&$ 0.33_{ 0.27}^{ 0.75} $& 4& 25.12&$ 0.16_{ 0.08}^{ 0.13} $&$ 0.79_{ 0.38}^{ 0.63} $\
\
Sample VII & 1& 5.19 &$ 0.19_{ 0.16}^{ 0.44} $&$ 0.39_{ 0.32}^{ 0.90} $& 4& 18.19&$ 0.22_{ 0.11}^{ 0.17} $&$ 0.91_{ 0.44}^{ 0.72} $\
\
Sample VIII & 4& 10.50&$ 0.38_{ 0.18}^{ 0.30} $&$ 0.92_{ 0.44}^{ 0.73} $& 3& 12.44&$ 0.24_{ 0.13}^{ 0.23} $&$ 1.38_{ 0.75}^{ 1.35} $\
\
Sample IX & 4& 10.00&$ 0.40_{ 0.19}^{ 0.32} $&$ 0.97_{ 0.46}^{ 0.77} $& 3& 11.65&$ 0.26_{ 0.14}^{ 0.25} $&$ 1.48_{ 0.81}^{ 1.44} $\
\
Sample X & 5& 11.42&$ 0.44_{ 0.19}^{ 0.30} $&$ 1.06_{ 0.46}^{ 0.72} $& 4& 12.12&$ 0.33_{ 0.16}^{ 0.26} $&$ 1.89_{ 0.91}^{ 1.50} $\
\
\
\
\
\
\
\[Table:all\_dndz\_result\_60k\]
Discussion and Conclusion {#Section:Discussion_and_conclusions}
=========================
We have presented a new comparison of the incidence rates of absorption systems towards two different classes of AGNs, namely blazars and normal (optically selected) QSOs. A factor of two higher rate towards blazars has earlier been claimed by BBM (Section \[sec:intro\_mgiidndz\]) and similar excess incidence of intervening absorbers has been reported in a few earlier studies of GRBs @Prochter2006ApJ...648L..93P, @Sudilovsky2007ApJ...669..741S, and @Tejos2009ApJ...706.1309T. However, the physical cause of the purported excess relative to normal QSOs still remains to be understood. In fact, BBM have already discounted the possibilities of dust obscuration and gravitational lensing playing a significant role [see also @Cucchiara2013ApJ...773...82C].
On the other hand, no excess in the incidence rate of intervening absorbers towards flat-spectrum radio quasars has been reported in some recent studies based on large samples [@Chand2012ApJ...754...38C; @Joshi2013MNRAS.435..346J]. Therefore, in order to take a fresh look into the BBM finding of excess incidence of absorbers along blazar sightlines, we have assembled a large sample of 191 blazars (including the BBM sample of blazars). An independent sample of sightlines is also intended to provide a check on the possible role of statistical fluctuation arising from small source sample, as indeed turned out to be the case for GRBs [@Cucchiara2013ApJ...773...82C Section \[sec:intro\_mgiidndz\]]
From the results of our analysis of the enlarged blazar sample (Table \[Table:all\_dndz\_result\]), no excess is evident in the $dN/dz$ along the sightlines to blazars, as compared to the sightlines to normal QSOs. Recognizing that the spectral data for our blazar sample have mostly rather modest SNR in comparison to the BBM sample (see Fig. \[fig:zemi\_snr\_dist\]), we have sub-divided the spectra for our blazar sample into two ranges of SNR (SNR between 5 and 15, and SNR $>$ 15). For neither of these SNR ranges did our analysis show a significant difference between the $dN/dz$ estimates towards blazars and normal QSOs (see Table \[Table:all\_dndz\_result\] and Table \[table:low\_high\_snr\_comp\]). Conceivably, the discrepancy between our and BBM estimates of $dN/dz$ may then be rooted in the use of different analysis procedures. However, this too seems unlikely since our independent re-analysis of the BBM blazar sample reproduces the $dN/dz$ excess reported by them (Table \[Table:bbm\_excess\_netzor\]).
To probe this issue further, we compare in Fig. \[fig:beta\_hist\], the $\beta$ distributions of the absorbers for the BBM and our samples of blazars. Here c$\beta$ is the velocity of an absorber measured relative to the background blazar, where: $$\beta \equiv \frac{v}{\rm c} = \frac {(1+z_{\rm em})^2-(1+z_{\rm
abs})^2} {(1+z_{\rm em})^2+(1+z_{\rm abs})^2}
\label{eq:beta}$$ with, z$_{em}$ and z$_{abs}$ are the redshifts of the background AGN and the absorber, respectively. The distributions shown in Fig.\[fig:beta\_hist\] are useful for checking the extent of clustering, if any, of absorbers up to mildly relativistic $\beta$ values, as was noted in some recent studies of other AGN samples (see below). The top two panels in Fig. \[fig:beta\_hist\] show the histograms of $\beta$ values of absorbers for the BBM sample, both for weak (left panel of Fig. \[fig:beta\_hist\]) and strong absorbers (right panel in Fig. \[fig:beta\_hist\]). The lower two panels show the histograms for the absorbers for our blazar sample, after excluding the BBM blazars. For the strong absorbers in the BBM sample, a slight clustering at smaller $\beta$ is hinted, which is consistent with the trend noticed in some earlier studies [@Joshi2013MNRAS.435..346J; @Chand2012ApJ...754...38C also BBM], as well as from Fig. \[fig:cummulative\_plot\] (see above). This might indicate that associated absorbers may still contribute significantly to $dN/dz$ up to offset velocities $\sim 0.2c$, especially for weak systems towards blazars (e.g., see Fig. \[fig:cummulative\_plot\], left panel). Table \[Table:all\_dndz\_result\_60k\] summarizes the $dN/dz$ estimates for the various subsets of our 191 blazar sample, after excluding the systems with $\Delta v< 60000$ kms$^{-1}$ i.e., $\beta < 0.2c$.
There is a hint of discrepancy when the excess for blazar (relative to QSO sightlines) is compared for strong and weak absorption systems, the excess being noticeable for weak absorbers (e.g., see the top and middle panels in Fig. \[fig:cummulative\_plot\] ). Attributing this marginally significant excess to gas clumps accelerated outwards by the powerful blazar jet, (e.g., up to $\Delta v< 60000$ kms$^{-1}$), as also noted in BBM, the hinted excess in the case of weak absorbers could have its origin in a physical cause, or merely an observational bias. For instance, observationally the detection of gas clumps with higher column density (i.e. stronger systems) would be easier as compared to the lower column density clumps (i.e., weak absorbers). On the other hand, occurrence of lower column density clumps is more likely, intuitively. This seems to be the case as dynamical stability of the relativistic jets suggests that external perturbations do not disrupt the jets globally [see, e.g., @2017IAUS..324..141K]. This means, in particular, that most of the clumps (or clouds) impinging on the jet, as it propagates through the mostly diffuse gas, are smaller than the jet radius. Assuming that clumps or clouds in the ambient medium have similar volume densities, those with lower column densities (hence weak systems) are likely to have a less disruption effect on the jets via a slower growth of global instabilities. Hence, lower column density clumps accelerated by the jets should be intuitively more abundant in comparison to higher column density clumps, consistent with the results shown in (Fig. \[fig:cummulative\_plot\], left panel). The reality of the discrepancy, however, remains to be confirmed using larger set of blazar sighlines.
In summary, we conclude that (i) our independent analysis of the spectral data used by BBM for their blazar sample has reproduced the factor two excess claimed by them in $dN/dz$ for absorbers seen towards blazars, vis-a-vis normal QSO; (ii) by using a $\sim 4$ times larger blazar sample (albeit, mostly with a moderate SNR) which includes the BBM sample as well, we have arrived at a statistically more robust and independent estimate of $dN/dz$ of absorbers along blazar sightlines and the present analysis does not show a significant difference from the $dN/dz$ known for the sightlines to normal QSOs; (iii) the agreement improves further when we limit the comparison to offset velocities above 60000 kms$^{-1}$. This would be consistent with the possibility that associated absorbers remain a significant contributor to $dN/dz$ up to $\beta = 0.2c$ measured relative to the background QSO [see @Joshi2013MNRAS.435..346J also BBM].
Finally, in order to firmly settle the issues raised in the present study a significant enlargement of the sample of absorbers towards blazars would be vital. This can be achieved, e.g., by extending the high sensitivity optical spectroscopic coverage to the 71 blazars which had to be excluded from the present analysis because the SNR of their currently available spectra falls below our adopted reasonable threshold (SNR $>$ 5).
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank the anonymous referee for his/her detailed comments to improve our manuscript. G-K thanks the National Academy of Sciences, India for the award of a Platinum Jubilee Senior Scientist fellowship. YS is supported by RFBR (17$-$52$-$45053$\_$IND). We thank the scientific staff and the observing team at SAO for the help in our observations. This research made use of (i) the Keck Observatory Archive (KOA), which is operated by the W. M. Keck Observatory and the NASA Exoplanet Science Institute (NExScI), under contract with the National Aeronautics and Space Administration, using observations made using the LRIS spectrograph at the Keck, Mauna Kea, HI; (ii) ESO Science Archive Facility by using observations made using the UVES, FORS and X-SHOOTER spectrographs at the VLT, Paranal, Chile. Special thanks to John Pritchard from ESO user support for the useful discussions on the FORS pipeline.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
Repersentative spectra of identified absorption systems {#ap:appendix_A}
=======================================================
For 51 out of the 191 blazars that form our sample, emission redshift measurement are not available and therefore we have taken lower limit estimates based on the most redshifted Mg II absorber seen in their spectra. A representative normalized spectrum is shown in Fig. \[fig:asso\_system\_spectra\] for 3 of these 51 blazars, with brief comments as below.
As seen from Fig. \[fig:asso\_system\_spectra\] (top panel) for J001937$+$202146, two absorption systems were identified, (i) at $\lambda = 4743.06$ Å (z$_{\rm abs}=0.69616$, W$_{r}$()=1.0969 Å ) (ii) at $\lambda = 5196.74$Å (z$_{\rm
abs}=0.85840$, W$_{r}$()=2.7312 Å ). For both these systems, corresponding absorptions lines due to ($\lambda$2344,2374,2383,2585,2600) and ($\lambda$2852) are detected.
In J191816$-$411154 (Fig. \[fig:asso\_system\_spectra\] middle panel), two absorption doublets were detected (i) at $\lambda =
6457.92$ Å (z$_{\rm abs}= 1.30941$, W$_{r}$() = 0.9761 Å ) together with the corresponding ($\lambda$2344, 2374, 2383, 2586, 2600) and ($\lambda$2852) absorption lines; and (ii) at $\lambda
= 7243.02$ (z$_{\rm abs}= 1.59017$, W$_{r}$()= 1.5710 Å ) together with the corresponding absorption lines due to ($\lambda$2344, 2374, 2383, 2586, 2600) and ($\lambda$2852), ($\lambda\lambda$1854,1862) and ($\lambda\lambda$1393,1402). The redshift of the more redshifted absorption system was used as the lower limit of the emission redshift for this source.
The blazar J073346$+$411120 (Fig. \[fig:asso\_system\_spectra\] bottom panel), has four absorption systems: (i) at $\lambda =
4555.11$ Å (z$_{\rm abs}= 0.62895$, W$_{r}$() = 0.72 Å ), (ii) at $\lambda =7821.67 $ Å ($z_{\rm abs}= 1.79710$, W$_{r}$()=0.27 Å ), (iii) at $\lambda = 8046.36$ Å ($z_{\rm abs}= 1.87745$, W$_{r}$() = 0.59 Å ) and (iv) at $\lambda =8107.74 $ Å ($z_{\rm abs}= 1.89940$, W$_{r}$() = 0.07 Å ), with weak corresponding ($\lambda$2383) absorption feature detected for all four systems. The absorption redshift of the most redshifted absorption system (i.e., $z_{\rm abs}= 1.89940$) was used as the lower limit of the emission redshift for this source. The systems at $z_{\rm
abs}= 1.87745$ could, however, not be included in the analysis as it falls within 5000 kms$^{-1}$ of the lower limit to the source’s emission redshift, which we have fixed at $z_{\rm abs}= 1.89940$).
{width="100.00000%" height="0.8\textheight"}
\[lastpage\]
[^1]: E-mail: sapna@aries.res.in(SM)
[^2]: Platinum Jubilee Senior Scientist, The National Academy of Sciences, India.
[^3]: Ten blazars do not have useful redshift path (i.e., not satisfying our\
selection criteria-ii), another 28 lack emission redshift ($48-10-28=10$).
[^4]: Two blazars do not have useful redshift path (i.e., not satisfying the\
selection criteria-ii), another 7 lack emission redshift ($17-2-7=8$).
[^5]: Excluded 69 sources with SNR $<$ 5 (i.e. not satisfying the selection\
criteria-i, 207 lack emission redshift, and 138 sources were excluded for\
not meeting the selection criteria-ii), 12 sources were excluded since their\
spectra had been already taken from the other archives listed in this\
table ($622-69-207-138-12=196$).
[^6]: https://dr12.sdss.org/bulkSpectra
[^7]: http://archive.eso.org/eso/eso$\_$archive$\_$main.html
[^8]: https://koa.ipac.caltech.edu/cgi-bin/KOA/nph-KOAlogin
[^9]: http://archive.eso.org/wdb/wdb/adp/phase3\_main/form
[^10]: http://archive.eso.org/eso/eso\_archive\_main.html
[^11]: http://www.eso.org/sci/software/cpl/esorex.html
[^12]: http://www.astro.caltech.edu/$\sim$dperley/programs/lpipe.html
|
---
abstract: 'We demonstrate room temperature chiral strong coupling of valley excitons in a transition metal dichalcogenide monolayer with spin-momentum locked surface plasmons. In this regime, we measure spin-selective excitation of directional flows of polaritons. Operating under strong light-matter coupling, our platform yields robust intervalley contrasts and coherences, enabling us to generate coherent superpositions of chiral polaritons propagating in opposite directions. Our results reveal the rich and easy to implement possibilities offered by our system in the context of chiral optical networks.'
author:
- Thibault Chervy
- Stefano Azzini
- Etienne Lorchat
- Shaojun Wang
- Yuri Gorodetski
- 'James A. Hutchison'
- Stéphane Berciaud
- 'Thomas W. Ebbesen'
- Cyriaque Genet
bibliography:
- 'biblio-new-arxiv-v2.bib'
title: 'Spin-momentum locked polariton transport in the chiral strong coupling regime'
---
Optical spin-orbit (OSO) interaction couples the polarization of a light field with its propagation direction [@bliokh2015spin]. An important body of work has recently described how OSO interactions can be exploited at the level of nano-optical devices, involving dielectric [@bomzon2002space; @Kuipers2014; @Rauschenbeutel2014science; @rafayelyan2016reflective] or plasmonic architectures [@bliokhPRL2008; @gorodetski2013generating; @Capasso2013; @Zayats2013; @spektor2015; @Drezet2016], all able to confine the electromagnetic field below the optical wavelength. Optical spin-momentum locking effects have been used to spatially route the flow of surface plasmons depending on the spin of the polarization of the excitation beam [@Zayats2014] or to spatially route the flow of photoluminescence (PL) depending on the spin of the polarization of the emitter transition [@Rauschenbeutel2014natcomm]. Such directional coupling, also known as chiral coupling, has been demonstrated in both the classical and in the quantum regimes [@Junge2013; @Lodhal2015; @GonzalezPRB2015; @Rauschenbeutel2015; @Young2015; @GonzalezPRA2016; @Coles2016]. Chiral coupling opens new opportunities in the field of light-matter interactions with the design of non-reciprocal devices, ultrafast optical switches, non destructive photon detector, and quantum memories and networks (see [@Zoller2016] and references therein).
In this letter, we propose a new platform consisting of spin-polarized valleys of a transition metal dichalcogenide (TMD) monolayer strongly coupled to a plasmonic OSO mode, at room temperature (RT). In this strong coupling regime, each spin-polarized valley exciton is hybridized with a single plasmon mode of specific momentum. The chiral nature of this interaction generates spin-momentum locked polaritonic states, which we will refer to with the portmanteau [*chiralitons*]{}. A striking feature of our platform is its capacity to induce RT robust valley contrasts, enabling the directional transport of chiralitons over micron scale distances. Interestingly, the strong coupling regime also yields coherent intervalley dynamics whose contribution can still be observed in the steady-state. We hence demonstrate the generation of coherent superpositions (i.e. pairs) of chiralitons flowing in opposite directions. These results, unexpected from the bare TMD monolayer RT properties [@Xu2013; @moody2016exciton; @Li2016], point towards the importance of the strong coupling regime where fast Rabi oscillations compete with TMD valley relaxation dynamics, as recently discussed [@Tartakovskii2016; @Chen2017; @Kleemann2017].
The small Bohr radii and reduced screening of monolayer TMD excitons provide the extremely large oscillator strength required for light matter interaction in the strong coupling regime, as already achieved in Fabry-Pérot cavities [@Menon2015; @Tartakovskii2015; @Imamoglu2016] and more recently in plasmonic resonators [@Agarwal2016; @ShaojunWS2]. In this context, Tungsten Disulfide (WS$_2$) naturally sets itself as a perfect material for RT strong coupling [@ShaojunWS2] due to its sharp and intense $A$-exciton absorption peak, well separated from the higher energy $B$-exciton line (see Fig. \[fig:1\](a)) [@Heinz2014]. Moreover, the inversion symmetry breaking of the crystalline order on a TMD monolayer, combined with time-reversal symmetry, leads to spin-polarized valley transitions at the $K$ and $K'$ points of the associated Brillouin zone, as sketched in Fig. \[fig:1\](b) [@reviewTMDs]. This polarization property makes therefore atomically thin TMD semiconductors very promising candidates with respect to the chiral aspect of the coupling between the excitonic valleys and the plasmonic OSO modes [@LiACS2016; @Gong2017], resulting in the strongly coupled energy diagram shown in Fig. \[fig:1\](c).
![(a) Absorbance spectrum of a WS$_2$ monolayer as obtained from its transmission spectrum. (b) Crystal packing of a tungsten disulfide (WS$_2$) monolayer, and sketch of its electronic band structure around the points $K$ and $K'$ of the Brillouin zone, with the corresponding optical selection rules for band-edge excitons formation under left ($\sigma^+$) and right ($\sigma^-$) circular excitation. (c) Energy level diagram of the $K$ and $K'$ excitons of WS$_2$ strongly coupled to an OSO plasmonic mode at energy $\hbar\omega_{\rm OSO}$ and wavevector $\pm k_{\textrm{\tiny SP}}$. \[fig:1\]](Figure1rev.png){width="1\columnwidth"}
Experimentally, our system, shown in Fig. \[fig:2\](a), consists of a mechanically exfoliated monolayer of WS$_2$ covering a plasmonic OSO hole array, with a $5$ nm thick dielectric spacer (polymethyl methacrylate). The array, imaged in Fig. \[fig:2\](b), is designed on a $(x,y)$ square lattice with a grating period $\Lambda$, and consists of rectangular nano-apertures $(160\times90$ nm$^2)$ rotated stepwise along the $x$-axis by an angle $\phi=\pi/6$. The associated orbital period $6\times\Lambda$ sets a rotation vector $\boldsymbol{\Omega}=(\phi /
\Lambda)\hat{z}$, which combines with the spin $\sigma$ of the incident light into a geometric phase $\Phi_g=- \Omega\sigma x$ [@bliokh2008coriolis]. The gradient of this geometric phase imparts a momentum ${\bf k}_g=-\sigma(\phi / \Lambda)\hat{x}$ added to the matching condition on the array between the plasmonic ${\bf k}_{\rm SP}$ and incidence in-plane ${\bf
k}_{\rm in}$ momenta: ${\bf k}_{\rm
SP}={\bf k}_{\rm in} + (2\pi / \Lambda) (n\hat{x}+m\hat{y})+{\bf
k}_g$. This condition defines different $(n,m)$ orders for the plasmonic dispersions, which are transverse magnetic (TM) and transverse electric (TE) polarized along the $x$ and $y$ axis of the array respectively (see Fig. \[fig:2\](b)). The dispersive properties of such a resonator thus combines two modal responses: plasmon excitations directly determined on the square Bravais lattice of the grating for both $\sigma^+$ and $\sigma^-$ illuminations via $(2\pi / \Lambda) (n\hat{x}+m\hat{y})$, and spin-dependent plasmon OSO modes launched by the additional geometric momentum ${\bf k}_{g}$. It is important to note that the contribution of the geometric phase impacts the TM dispersions only. The period of our structure $\Lambda=480$ nm is optimized to have $n=+1$ and $n=-1$ TM modes resonant with the absorption energy of the $A$-exciton of WS$_2$ at $2.01$ eV for $\sigma^+$ and $\sigma^-$ illuminations respectively. This strict relation between $n=\pm
1$ and $\sigma=\pm 1$ is the OSO mechanism that breaks the left vs. right symmetry of the modal response of the array, which in this sense becomes chiral. Plasmon OSO modes are thus launched in counter-propagating directions along the $x$-axis for opposite spins $\sigma$ of the excitation light. In the case of a bare plasmonic OSO resonator, this is clearly observed in Fig. \[fig:2\] (c). We stress that similar arrangements of anisotropic apertures have previously been demonstrated to allow for spin-dependent surface plasmon launching [@Hasman2011; @SZhang2013; @Capasso2013; @Drezet2016].
![(a) White light (WL) microscope image of the sample and photoluminescence (PL) image of the same area under $2.58$ eV excitation. (b) SEM image of the plasmonic OSO resonator fabricated by sputtering $200$ nm of gold on a glass substrate coated by a $5$ nm thick chromium adhesion layer. The array with $\phi$-rotated rectangular apertures is milled through the metallic layers using a focused ion beam (FIB). (c) Real-space leakage radiation microscope [@Drezet2016] images of the surface plasmons launched by $\sigma^+$ and $\sigma^-$ excitations on a OSO plasmonic resonator similar to the one of panel (b). \[fig:2\]](Figure2rev.png){width="1\columnwidth"}
As explained in the Supporting Information (Sec. A), the low transmission measured through our WS$_2$/plasmonic array sample (Fig. \[fig:2\](a)) enables us to obtain absorption spectra directly from reflectivity spectra. Angle-resolved white light absorption spectra are hence recorded and shown in Fig. \[fig:3\] (a) and (b) for left and right circular polarizations. In each case, two strongly dispersing branches are observed, corresponding to upper and lower chiralitonic states. As detailed in the Supporing Information (Sec. A), a fit of a coupled dissipative oscillator model to the dispersions enables us to extract a branch splitting $2\sqrt{\left(\hbar\Omega_R\right)^2 - \left( \hbar\gamma_{\tiny\textrm{ex}}- \hbar\Gamma_{\tiny\textrm{OSO}}\right)^2}=40$ meV. With measured linewidths of the excitonic mode $\hbar\Gamma_{\tiny\textrm{OSO}}=80$ meV and of the plasmonic mode $\hbar\gamma_{\tiny\textrm{ex}}=26$ meV, this fitting yields a Rabi frequency of $\hbar\Omega_{\rm R}=70$ meV, close to our previous observations on non-OSO plasmonic resonators [@ShaojunWS2]. We emphasize that this value clearly fulfills the strong coupling criterion with a figure-of-merit $\Omega_{\rm R}^2/(\gamma_{\rm exc}^2+\Gamma_{\rm OSO}^2) =0.69$ larger than the $0.5$ threshold that must be reached for strong coupling [@Houdre2005; @TormaBarnes2015]. This demonstrates that our system does operate in the strong coupling regime, despite the relatively low level of visibility of the anti-crossing. This is only due $(i)$ to spatial and spectral disorders which leave, as always for collective systems, an inhomogeneous band of uncoupled states at the excitonic energy, and $(ii)$ to the fact that an uncoupled Bravais plasmonic branch is always superimposed to the plasmonic OSO mode, leading to asymmetric lineshapes clearly seen in Fig. \[fig:3\] (a) and (b). As shown in the Supporting Information (Sec. A), the anti-crossing can actually be fully resolved through a first-derivative analysis of our absorption spectra.
In such strong coupling conditions, the two dispersion diagrams also show a clear mirror symmetry breaking with respect to the normal incident axis ($k_x=0$) for the two opposite optical spins. This clearly demonstrates the capability of our structure to act as a spin-momentum locked polariton launcher. From the extracted linewidth that gives the lifetime of the chiralitonic mode and the curvature of the dispersion relation that provides its group velocity, we can estimate a chiraliton propagation length of the order of $4~\mu$m, in good agreement with the measured PL diffusion length presented in the Supporting Information (Sec. B).
![Angle-resolved absorption spectrum of the sample analyzed in (a) left and (b) right circular polarizations, with the best fit coupled oscillator model drawn. Angle-resolved resonant second harmonic spectrum for (c) right and (d) left circular excitations at $1.01$eV. Corresponding crosscuts are displayed with the angular profile of the SH signal (red curves), of the absorption spectra at $2$ eV (green shades) and of the product (blue shades) between the absorption and the $4^{\rm th}$ power of the excitonic Hopfield coefficient of the chiralitonic state -see details in the Supporting Information (Sec. D) \[fig:3\].](Figure3_New.png){width="1\columnwidth"}
In view of chiral light-chiral matter interactions, we further investigate the interplay between this plasmonic chirality and the valley-contrasting chirality of the WS$_2$ monolayer. A first demonstration of such an interplay is found in the resonant second harmonic (SH) response of the strongly coupled system. Indeed, monolayer TMDs have been shown to give a high valley contrast in the generation of a SH signal resonant with their $A$-excitons [@Seyler2015]. As we show in Supporting Information (Sec. G) such high SH valley contrast are measured on a bare WS$_2$ monolayer. The optical selection rules for SH generation are opposite to those in linear absorption since the process involves two excitation photons, and are more robust since the SH process is instantaneous.
The angle resolved resonant SH signals are shown in Fig. \[fig:3\] (c) and (d) for right and left circularly polarized excitation. The SH signals are angularly exchanged when the spin of the excitation is reversed with a right vs. left contrast (ca. $20\%$) close to the one measured on the reflectivity maps (ca. $15\%$). This unambigously demonstrates the selective coupling of excitons in one valley to surface plasmons propagating in one direction, thus realizing valley-contrasting chiralitonic states with spins locked to their propagation wavevectors: $$\begin{aligned}
&&|P^\pm_{K,\sigma^+,-k_{\textrm{\tiny SP}}}> = |g_K,1_{\sigma^+},-k_{\textrm{\tiny SP}}> \pm |e_K,0_{\sigma^+},-k_{\textrm{\tiny SP}}> \nonumber\\
&&|P^\pm_{K',\sigma^-,+k_{\textrm{\tiny SP}}}> = |g_{K'},1_{\sigma^-},+k_{\textrm{\tiny SP}}> \pm |e_{K'},0_{\sigma^-},+k_{\textrm{\tiny SP}}>, \nonumber \end{aligned}$$ where $e_{i}(g_{i})$ corresponds to the presence (absence) of an exciton in the valley $i=(K,K')$ of WS$_2$, and $1_{j}(0_{j})$ to 1 $(0)$ plasmon in the mode of polarization $j=(\sigma^+,\sigma^-)$ and wavevector $\pm k_{\textrm{\tiny SP}}$.
The detailed features of SH signal (crosscuts in Fig. \[fig:3\] (c) and (c)) reveal within the bandwidth of our pumping laser the contributions of both the uncoupled excitons and the upper chiraliton to the SH enhancement. The key observation, discussed in the Supporting Information (Sec. D), is the angular dependence of the main SH contribution. This contribution, shifted from the anticrossing region, is a feature that gives an additional proof of the strongly coupled nature of our system because it is determined by the excitonic Hopfield coefficient of the spin-locked chiralitonic state. In contrast, the residual SH signal related to the uncoupled (or weakly coupled) excitons simply follows the angular profile of the absorption spectra taken at $2$ eV, thus observed over the anticrossing region. Resonant SH spectroscopy of our system therefore confirms the presence of the chiralitonic states, with the valley contrast of WS$_2$ and the spin-locking property of the OSO plasmonic resonator being imprinted on these new eigenstates of the system.
Revealed by these resonant SH measurements, the spin-locking property of chiralitonic states incurs however different relaxation mechanisms through the dynamical evolution of the chiralitons. In particular, excitonic intervalley scattering can erase valley contrast in WS$_2$ at RT [@Jonker2016] -see below. In our configuration, this would transfer chiraliton population from one valley to the other, generating via optical spin-locking, a reverse flow, racemizing the chiraliton population. This picture however does not account for the possibility of more robust valley contrasts in strong coupling conditions, as recently reported with MoSe$_2$ in Fabry-Pérot cavities [@Tartakovskii2016]. The chiralitonic flow is measured by performing angle resolved polarized PL experiments, averaging the signal over the PL lifetime of ca. $200$ ps (see Supporting Information, Sec. D and E). For these experiments, the laser excitation energy is chosen at $1.96$ eV, slightly below the WS$_2$ band-gap. At this energy, the measured PL results from a phonon-induced up-convertion process that minimizes intervalley scattering events [@Xiaodong2016]. The difference between PL dispersions obtained with left and right circularly polarized excitations is displayed in Fig. \[fig:4\] (a), showing net flows of chiralitons with spin-determined momenta. This is in agreement with the differential white-light reflectivity map $R_{\sigma^-}-R_{\sigma^+}$ of Fig. \[fig:4\] (b). Considering that this map gives the sorting efficiency of our OSO resonator, such correlations in the PL implies that the effect of the initial spin-momentum determination of the chiralitons (see Fig. \[fig:3\] (e) and (f)) is still observed after $200$ ps at RT. After this PL lifetime, a net chiral flow $\mathcal{F}=I_{\sigma^-}-I_{\sigma^+}$ of $\sim 6\%$ is extracted from Fig. \[fig:4\] (a). This is the signature of a chiralitonic valley polarization, in striking contrast with the absence of valley polarization that we report for a bare WS$_2$ monolayer at RT in the Supporting Information, Sec. G. The extracted net flow is however limited by the finite optical contrast $\mathcal{C}$ of our OSO resonator, which we measure at a $15\%$ level from a cross-cut taken on Fig. \[fig:4\] (b) at $1.98$ eV. It is therefore possible to infer that a chiralitonic valley contrast of $\mathcal{F} / \mathcal{C} \simeq 40\%$ can be reached at RT for the strongly coupled WS$_2$ monolayer. As mentioned above, we understand this surprisingly robust contrast by invoking the fact that under strong coupling conditions, valley relaxation is outweighted by the faster Rabi energy exchange between the exciton of each valley and the corresponding plasmonic OSO mode, as described in the Supporting Information (Sec. A). From the polaritonic point of view, the local dephasing and scattering processes at play on bare excitons -that erase valley contrasts on a bare WS2 flake as observed in the Supporting Information (Sec. G)- are reduced by the delocalized nature of the chiralitonic state, a process akin to motional narrowing and recently observed on other polaritonic systems [@Whittaker1996; @Tartakovskii2016; @Chen2017; @Kleemann2017].
![(a) Differential PL dispersion spectrum for left and right circularly polarized excitations. The shaded regions in all panels are removed by the laser line filter, and the cross-cuts are taken at $2$ eV. (b) Differential angle-resolved reflection spectrum for left and right circularly polarized light. (c), (e) Angle-resolved spectrum of the normalized coefficient $S_1^{\textrm{\tiny out}}|_{\textrm{\tiny TM(TE)}}/S_0$ of the PL Stokes vector for a TM(TE) polarized excitation (see text for details). Note that we have put a detection threshold below $100$ photon counts that cuts the signal above $\sim 2.03$ eV in panels (a), (c), (e) and (f). (d) Differential angle-resolved reflection spectrum obtained from analyzed TM and TE measurements. (f) $k_x$-energy dispersion of the degree of chiralitonic inter-valley coherence $m_{11}$ computed from (c) and (e). \[fig:4\]](Figure4.png){width="1\columnwidth"}
As a consequence of this motional narrowing effect, such a strongly coupled system involving atomically thin crystals of TMDs could then provide new ways to incorporate intervalley coherent dynamics [@Xu2013; @Li2016; @Urbaszek2016; @schmidt2016magnetic; @ye2017optical] into the realm of polariton physics. To illustrate this, we now show that two counter-propagating flows of chiralitons can evolve coherently. It is clear from Fig. \[fig:1\] (c) that within such a coherent superposition of counter-propagating chiralitons $$|\Psi> = |P^{\pm}_{K,\sigma^+,-k_{\textrm{\tiny SP}}}> + |P^{\pm}_{K',\sigma^-,+k_{\textrm{\tiny SP}}}>
\label{eq:Psi}$$ flow directions and spin polarizations become non-separable. Intervalley coherence is expected to result in a non-zero degree of linearly polarized PL when excited by the same linear polarization. This can be monitored by measuring the $S_1=I_{\textrm{\tiny TM}}-I_{\textrm{\tiny TE}}$ coefficient of the PL Stokes vector, where $I_{\textrm{\tiny TM}(\textrm{\tiny TE})}$ is the emitted PL intensity analyzed in TM (TE) polarization. This coefficient is displayed in the $k_x$-energy plane in Fig. \[fig:4\](c) for an incident TM polarized excitation at $1.96$ eV. Fig. \[fig:4\](e) displays the same coefficient under TE excitation. A clear polarization anisotropy on the chiraliton emission is observed for both TM and TE excitation polarizations, both featuring the same symmetry along the $k_x=0$ axis as the differential reflectivity dispersion map $R_{\textrm{\tiny TM}}-R_{\textrm{\tiny TE}}$ shown in Fig. \[fig:4\](d). As detailed in the Supporting Information (Sec. F), the degree of chiralitonic intervalley coherence can be directly quantified by the difference $(S_1^{\textrm{\tiny out}}|_{\textrm{\tiny
TM}}-S_1^{\textrm{\tiny out}}|_{\textrm{\tiny TE}})/2$, which measures the PL linear depolarization factor displayed (as $m_{11}$) in Fig. \[fig:4\] (f). By this procedure, we retrieve a chiralitonic intervalley coherence that varies between $5\%$ and $8\%$ depending on $k_x$. Interestingly, these values that we reach at RT have magnitudes comparable to those reported on a bare WS$_2$ monolayer at 10 K [@Xiaodong2014]. This unambiguously shows how such strongly coupled TMD systems can sustain RT coherent dynamics robust enough to be observed despite the long exciton PL lifetimes and plasmonic propagation distances.
In summary, we demonstrate valley contrasting spin-momentum locked chiralitonic states in an atomically thin TMD semiconductor strongly coupled to a plasmonic OSO resonator. Likely, the observation of such contrasts even after 200 ps lifetimes is made possible by the unexpectedly robust RT coherences inherent to the strong coupling regime. Exploiting such robust coherences, we measure chiralitonic flows that can evolve in superpositions over micron scale distances. Our results show that the combination of OSO interactions with TMD valleytronics is an interesting path to follow in order to explore and manipulate RT coherences in chiral quantum architectures [@Coles2016; @low2016polaritons].
We thank David Hagenmüller for fruitful discussions. This work was supported in part by the ANR Equipex “Union” (ANR-10-EQPX-52-01), ANR Grant (H2DH ANR-15-CE24-0016), the Labex NIE projects (ANR-11-LABX-0058-NIE) and USIAS within the Investissement d’Avenir program ANR-10-IDEX-0002-02. Y. G. acknowledges support from the Ministry of Science, Technology and Space, Israel. S. B. is a member of the Institut Universitaire de France (IUF).
[*Author Contributions -*]{} T. C. and S. A. contributed equally to this work.
Supporting Information
======================
A: Linear absorption dispersion analysis
========================================
Angle resolved absorption spectra are given from the measured reflectivity of the WS$_2$ flake on top of the plasmonic grating $R_{\tiny\textrm{sample}}$ with: $$A = 1 - R_{\tiny\textrm{sample}}/R_{\tiny\textrm{substrate}},$$ where $R_{\tiny\textrm{substrate}}$ is the angle resolved reflectivity of the optically thick (200 nm thickness) Au substrate. The $1\%$ max. transmission through the structure can be safely neglected. As we explain in the main text, the resulting dispersion spectra are broadened by the contribution of different plasmonic modes as well as coupled and uncoupled exciton populations.
In order to highlight the polaritonic contribution in the absorption spectra, we calculate the first derivative of the reflectivity dispersions $\textrm{d} \left[
R_{\tiny\textrm{sample}}/R_{\tiny\textrm{substrate}} \right] / \textrm{d}E$. The derivative was approximated by interpolating the reflectivity spectra on an equally spaced wavelength grid of step $\Delta \lambda = 0.55$ nm and using the following finite difference expression valid up to fourth order in the grid step: $$\frac{\textrm{d}R}{\textrm{d}\lambda}(\lambda_0) \simeq \frac{\frac{1}{12}R(\lambda_{-2})
-\frac{2}{3}R(\lambda_{-1}) + \frac{2}{3}R(\lambda_{+1}) -
\frac{1}{12}R(\lambda_{+2})}{\Delta \lambda},$$ where $R(\lambda_{n})$ is the reflectivity evaluated $n$ steps away from $\lambda_{0}$. The resulting first derivative reflectivity spectra were then converted to an energy scale and plotted as dispersion diagrams in Fig. \[fig:S1\] (a) and (b).
In these first derivative reflectivity maps, the zero-crossing points correspond to the peak positions of the modes, and the maxima and minima indicate the inflection points of the reflectivity lineshapes. At the excitonic asymptotes of the dispersion curves, where the polaritonic linewidth is expected to match that of the bare WS$_2$ exciton, we read a linewidth of $26$ meV from the maximum to minimum energy difference of the derivative reflectivity maps. This value is equal to the full width at half maximum (FWHM) $\hbar\gamma_{\rm exc}$ that we measured from the absorption spectrum of a bare WS$_2$ flake on a dielectric substrate.
![First derivative reflectivity maps in (a) left and (b) right circular polarizations, with the best fit coupled oscillator model drawn. \[fig:S1\]](FigureS1.png){width="1\columnwidth"}
On the low energy plasmonic asymptotes, we clearly observe the effect of the Bravais and OSO modes, partially overlapping in an asymmetric broadening of the branches. In this situation, a measure of the mode half-widths can be extracted from the (full) widths of the positive or negative regions of the first differential reflectivity maps. This procedure yields an energy half-width for the plasmonic modes of $\hbar\Gamma_{\rm OSO}/2=40$ meV. This width in energy can be related to an in-plane momentum width of ca. $0.5
~\mu\textrm{m}^{-1}$ via the plasmonic group velocity $v_G
= \partial E/\partial k = 87$ meV$\cdot\mu$m, that we calculate from the branch curvature at $1.85$ eV. This in-plane momentum width results in a plasmonic propagation length of about $4 ~\mu$m. This value is in very good agreement with the measured PL extension above the structure, as discussed in section B below, validating our estimation of the mode linewidth $\hbar\Gamma_{\rm OSO}=80$ meV.
The dispersive modes of the system can be modeled by a dipolar Hamiltonian, where excitons in each valley are selectively coupled to degenerated OSO plasmonic modes of opposite wavevectors $\pm {\bf k}_{\rm SP}$, as depicted in Fig. 2 in the main text: $$\mathcal{H} = \sum_{k_x}\left[\mathcal{H}_{\tiny\textrm{OSO}}(k_x) +
\mathcal{H}_{\tiny\textrm{ex}} +
\mathcal{H}_{\tiny\textrm{int}}(k_x)\right], \label{eq:H}$$ which consists of three different contributions: $$\mathcal{H}_{\tiny\textrm{OSO}}(k_x) =
\hbar\omega_{\tiny\textrm{OSO}}(k_x)\left(a^\dagger_{k_x}a_{k_x} +
a^\dagger_{-k_x}a_{-k_x}\right),$$ $$\mathcal{H}_{\tiny\textrm{ex}}(k_x) =
\hbar\omega_{\tiny\textrm{ex}}\left(b^\dagger_{K-k_x}b_{K-k_x} + b^\dagger_{K'+k_x}b_{K'+k_x}\right),$$ $$\begin{aligned}
\mathcal{H}_{\tiny\textrm{int}}(k_x) &=& \hbar g\left(a^\dagger_{k_x}
+ a_{k_x}\right)\left(b^\dagger_{K'+k_x} + b_{K'+k_x}\right) \nonumber \\
& & + \hbar g\left(a^\dagger_{-k_x}
+ a_{-k_x}\right)\left(b^\dagger_{K-k_x} + b_{K-k_x}\right),\end{aligned}$$ where $a(a^\dagger)$ are the lowering (raising) operators of the OSO plasmonic modes of energy $\hbar\omega_{\tiny\textrm{OSO}}(k_x)$, $b(b^\dagger)$ are the lowering (raising) operators of the exciton fields of energy $\hbar\omega_{\tiny\textrm{ex}}$, and $g = \Omega_R/2$ is the light-matter coupling frequency. In this hamiltonian the chiral light-chiral matter interaction is effectively accounted for by coupling excitons of the valley $K'(K)$ to plasmons propagating with wavevectors $k_x(-k_x)$. Moreover, the dispersion of the exciton energy can be neglected on the scale of the plasmonic wavevector ${\bf k}_{\rm SP}$.
Using the Hopfield procedure [@CiutiPRB2005], we can diagonalize the total Hamiltonian by finding polaritonic normal mode operators $P^\pm_{K(K')}$ associated with each valley, and obeying the following equation of motion at each $k_x$ $$\left[P^\pm_{K(K')},\mathcal{H}\right] = \hbar\omega_\pm
P^\pm_{K(K')}, \label{eq:motion}$$ with $\omega_\pm>0$. In the rotating wave approximation (RWA), justified here by the moderate coupling strength (see below), $P^j_{\lambda}
\simeq \alpha^j_{\lambda}a +\beta^j_{\lambda}b$, $j\in\{+,-\}$ and $\lambda\in\{K,K'\}$. The plasmonic and excitonic Hopfield coefficients $\alpha^j_{\lambda}(k_x)$ and $\beta^j_{\lambda}(k_x)$ are obtained by diagonalizing the following matrix for every $k_x$ $$\left(
\begin{array}{cccc}
\hbar\omega_{\tiny\textrm{OSO}} & i\hbar\Omega_{\rm R} & 0 & 0 \\
-i\hbar\Omega_{\rm R} & \hbar\omega_{\tiny\textrm{ex}} & 0 & 0 \\
0 & 0 & -\hbar\omega_{\tiny\textrm{OSO}} & i\hbar\Omega_{\rm R}\\
0 & 0 & -i\hbar\Omega_{\rm R}& -\hbar\omega_{\tiny\textrm{ex}}
\end{array}
\right). \label{eq:Matrix}$$
The dynamics of the coupled system will be ruled by the competition between the coherent evolution described by the Hamiltonian (\[eq:H\]) and the different dissipative processes contributing to the uncoupled modes linewidths. This can be taken into account by including the measured linewidths as imaginary parts in the excitonic and plasmonic mode energies (Weisskopf-Wigner approach). Under such conditions, we evaluate the eigenvalues $\omega_{\pm}$ of the the matrix (\[eq:Matrix\]). The real parts of $\omega_{\pm}$ are then fitted to the maxima of the angle resolved reflectivity maps presented on Fig. 3 in the main text, or to the zeros of the first derivative reflectivity maps shown here in Fig. \[fig:S1\] (a) and (b). Both procedures give the same best fit that yields the polaritonic branch splitting as [@Houdre2005; @TormaBarnes2015] $$\hbar(\omega_+-\omega_-) = 2\sqrt{\left(\hbar\Omega_R\right)^2 - \left( \hbar\gamma_{\tiny\textrm{ex}}- \hbar\Gamma_{\tiny\textrm{OSO}}\right)^2}
\label{eq:SCsplit}$$ which equals $40$ meV. From the determination (see above) of the FWHM of the excitonic $\hbar\gamma_{\tiny\textrm{ex}}$ and plasmonic $\hbar\Gamma_{\tiny\textrm{OSO}}$ modes, we evaluate a Rabi energy $ \hbar\Omega_R=70$ meV.
These values give a ratio $$\left(\hbar\Omega_R\right)^2 / \left(\left(\hbar\gamma_{\tiny\textrm{ex}}\right)^2 +
\left(\hbar \Gamma_{\tiny\textrm{OSO}}\right)^2\right) = 0.69,$$ above the $0.5$ threshold which is the strong coupling criterion -see [@Houdre2005] for a detailed discussion. This figure-of-merit of $0.69>0.5$ therefore clearly demonstrates that our system is operating in the strong coupling regime.
Interestingly, the intervalley scattering rate $\hbar\gamma_{KK'}$ does not enter in this strong coupling criterion. Indeed, such events corresponding to an inversion of the valley indices $K\leftrightarrow K'$ do not contribute to the measured excitonic linewidth, and are thus not detrimental to the observation of strong coupling. In the $\hbar\Omega_R \ll
\hbar\gamma_{KK'}$ limit, the Hamiltonian (\[eq:H\]) would reduce to the usual RWA Hamiltonian and the valley contrasting chiralitonic behavior would be lost. The results gathered in Fig. 4 in the main text clearly show that this is not the case for our system, allowing us to conclude that the Rabi frequency overcomes such intervalley relaxation rates. Remarkably, strong coupling thus allows us to put an upper bounds to those rates, in close relation with [@Holmes].
B: Chiraliton diffusion length
==============================
The diffusion length of chiralitons can be estimated by measuring the extent of their photoluminescence (PL) under a tightly focused excitation. To measure this extent, we excite a part of a WS$_2$ monolayer located above the plasmonic hole array (Fig. S\[fig:S1\](a) and (b)). This measurement is done on a home-built PL microscope, using a $100\times$ microscope objective of 0.9 numerical aperture and exciting the PL with a HeNe laser at $1.96$ eV, slightly below the exciton band-gap. A diffraction-limited spot of $430$ nm half-width is obtained (Fig. S\[fig:S1\](c)) by bringing the sample in the focal plane of the microscope while imaging the laser beam on a cooled CCD camera. The PL is collected by exciting at $10~\mu$W of optical power, and is filtered from the scattered laser light by a high-energy-pass filter. The resulting PL image is shown in Fig. S\[fig:S1\](d), clearly demonstrating the propagating character of the emitting chiralitons. The logarithmic cross-cuts (red curves in (c) and (d)) reveal a propagation length of several microns.
![(a) White light image of a WS$_2$ flake covering the plasmonic hole array, and (b) its wide field PL. (c) Normalized image of the diffraction-limited laser spot on the structure (indicated by the white dashed rectangle in (c) and (d)). The logarithmic scale cross-cuts (red curves in (c) and (d)) are taken along the vertical axis, through the intensity maximum. (d) Normalized chiraliton PL image. \[fig:S2\]](FigureS2.png){width="0.8\columnwidth"}
We note that the PL of the WS$_2$ monolayer also extends further away from the flake above the OSO array (Fig. S\[fig:S2\](b)). We extract from this result the $1/e$ decay length of the plasmon to be $\sim 3.4 ~\mu$m. This value nicely agrees with that obtained in section A from the linear dispersion analysis.
C: Resonant Second Harmonic generation on a WS$_2$ monolayer
============================================================
As discussed in the main text, TMD monolayers have recently been shown to give a high valley contrast in the generation of a second harmonic (SH) signal resonant with their $A$-excitons. We obain a similar result when measuring a part of the WS$_2$ monolayer sitting above the bare metallic surface, i.e. aside from the plasmonic hole array. In Fig. S\[fig:S3\] we show the SH signal obtained in left and right circular polarization for an incident femto-second pump beam (120 fs pulse duration, 1 kHz repetition rate at 1.01 eV) in (a) left and (b) right circular polarization. This result confirms that the SH signal polarization is a good observable of the valley degree of freedom of the WS$_2$ monolayer, with a contrast reaching ca. $80\%$. In Fig. 3 (c) and (d) in the main text, we show how this valley contrast is imprinted on the chiralitonic states.
![Resonant SH spectrum for left and right circular analysis, for (a) left and (b) right circular excitation at $1.01$eV. \[fig:S3\]](FigureS3.png){width="\columnwidth"}
D: Resonant SH generation in the strong coupling regime
=======================================================
The resonant SH signal writes as [@Heinz1982; @ChervyNano2016]: $$I(2\omega)\propto (\rho_{\omega}I_{\omega})^2\cdot |\chi^{(2)}(2\omega) |^2 \cdot \rho_{2\omega}$$ where $I_{\omega}$ is the pump intensity, $\chi^{(2)}(2\omega)$ the second order susceptibility, $\rho_{\omega}$ the optical mode density of the resonator related to the fraction of the pump intensity that reaches WS$_2$ and $\rho_{2\omega}$ the optical mode density of the resonator that determines the fraction of SH intensity decoupled into the far field. While $\rho_{\omega}$ can safely be assumed to be non-dispersive at $\hbar\omega = 1$ eV, the dispersive nature of the resonator leads to $\rho_{2\omega}$ strongly dependent on the in-plane wave vector $k_x$. The optical mode density being proportional to the absorption, $\rho_{2\omega}(k_x)$ is given by the angular absorption spectrum crosscut at $2\hbar\omega = 2$ eV, displayed in the lower panels of Fig. 3 (e) and (f) in the main text.
Under the same approximations of [@LinPRA1993], the resonant second order susceptibility can be written as $$\chi^{(2)}(2\omega)=\alpha^{(1)}(2\omega)\sum_{n}\frac{K_{eng}}{\omega_{ng}-\omega}$$ where $\sum_n$ sums over virtual electronic transitions, and $K_{eng}=\langle e|{\bf p}|g\rangle\otimes \langle e|{\bf p}|n\rangle\otimes\langle n|{\bf p}|g\rangle$ is a third-rank tensor built on the electronic dipole moments ${\bf p}$ taken between the $e,n,g$ states. The prefactor $\alpha^{(1)}(2\omega)$ is the linear polarizability of the system at frequency $2\omega$, yielding resonantly enhanced SH signal at every allowed $|g\rangle \rightarrow |e\rangle$ electronic transitions of the system.
With two populations of uncoupled and strongly coupled WS$_2$ excitons, the SH signal is therefore expected to be resonantly enhanced when the SH frequency matches the transition frequency of either uncoupled or strongly coupled excitons. When the excited state is an uncoupled exciton associated with a transition energy fixed at frequency $2\hbar\omega=2$ eV for all angles, the tensor $K_{eng}$ is non-dispersive and the SH signal is simply determined and angularly distributed from $\rho_{2\omega}(k_x)$.
When the excited state is a strongly coupled exciton, the resonant second order susceptibility becomes dispersive with $\chi^{(2)}(2\omega,k_x)$. This is due to the fact that the tensor $K_{eng}$ incorporates the excitonic Hopfield coefficient of the polaritonic state involved in the electronic transition $|g\rangle \rightarrow |e\rangle$ when the excited state is a polaritonic state. In our experimental conditions with a pump frequency at $1$ eV, this excited state is the upper polaritonic state with $|e\rangle \equiv |P^+_{K(K'),\sigma^\pm},\mp k_{\rm SP}\rangle$ and therefore $K_{eng}\propto [\beta_{K(K')}^+ (k_x)]^2$. This dispersive excitonic Hopfield coefficient is evaluated by the procedure described in details above, Sec. A. The profile of the SH signal then follows the product between the optical mode density $\rho_{2\omega}(k_x)$ and $|\chi^{(2)}(2\omega,k_x)|^2\propto[\beta_{K(K')}^+ (k_x)]^4$.
These two contributions are perfectly resolved in the SH data displayed in Fig. 3 (e) and (f) in the main text. The angular distribution of the main SH signal clearly departs from $\rho_{2\omega}(k_x)$, revealing the dispersive influence of $\beta_{K(K')}^+ (k_x)$. This is perfectly seen on the crosscuts displayed in the lower panels of Figs. 3 (e) and (f) in the main text. This feature is thus an indisputable proof of the existence of chiralitonic states, i.e. of the strongly coupled nature of our system.
A residual SH signal is also measured which corresponds to the contribution of uncoupled excitons. This residual signal is measured in particular within the anticrossing region, as expected from the angular profile of $\rho_{2\omega}(k_x)$ shown in the lower panels of Figs. 3 (e) and (f) in the main text.
Finally, the angular features of the SH signal exchanged when the spin of the pump laser is flipped from $\sigma^+$ to $\sigma^-$ reveal how valley contrasts have been transferred to the polariton states. These features therefore demonstrate the chiral nature of the strong coupling regime, i.e. the existence of genuine *chiralitons*.
E: PL lifetime measurement on the strongly coupled system
=========================================================
The PL lifetime of the strongly coupled system is measured by time-correlated single photon counting (TCSPC) under pico-second pulsed excitation (instrument response time $120$ ps, 20 MHz repetition rate at 1.94 eV). The arrival time histogram of PL photons, when measuring a part of the WS$_2$ monolayer located above the plasmonic hole array, gives the decay dynamic shown in Fig. S\[fig:S4\](a). On this figure we also display the PL decay of a reference WS$_2$ monolayer exfoliated on a dielectric substrate (polydimethylsiloxane), as well as the instrument response function (IRF) measured by recording the excitation pulse photons scattered by a gold film. Following the procedure detailed in [@Berciaud2015], we define the [*calculated*]{} decay times $\tau_{\textrm{\tiny calc}}$ as the area under the decay curves (corrected for their backgrounds) divided by their peak values. This yields a calculated IRF time constant $\tau_{\textrm{\tiny calc}}^{\textrm{\tiny IRF}} = 157$ ps, and calculated PL decay constants $\tau_{\textrm{\tiny
calc}}^{\textrm{\tiny ref}} = 1.39$ ns and $\tau_{\textrm{\tiny
calc}}^{\textrm{\tiny sample}} = 384$ ps for the reference bare flake and the strongly coupled sample respectively. The [*real*]{} decay time constants $\tau_{\textrm{\tiny
real}}$ corresponding to the calculated ones can then be estimated by convoluting different monoexponential decays with the measured IRF, computing the corresponding $\tau_{\textrm{\tiny
calc}}$ and interpolating this calibration curve (Fig. S\[fig:S3\](b)) for the values of $\tau_{\textrm{\tiny
calc}}^{\textrm{\tiny ref}}$ and $\tau_{\textrm{\tiny
calc}}^{\textrm{\tiny sample}}$. This results in $\tau_{\textrm{\tiny
real}}^{\textrm{\tiny ref}} = 1.06$ ns and $\tau_{\textrm{\tiny
real}}^{\textrm{\tiny sample}} = 192$ ps.
While the long life-time (ns) of the bare WS$_2$ exciton has been attributed to the trapping of the exciton outside the light-cone at RT through phonon scattering [@PollmannNatMat2015; @RobertPRB2016], Fig. S\[fig:S4\] simply shows that the exciton life time is reduced by the presence of the metal or the OSO resonator. Clearly, the competition induced under strong coupling conditions between the intra and inter valley relaxation rates and the Rabi oscillations must act under shorter time scales that are not resolved here.
![(a) TCSPC histogram showing the PL decay dynamic of the strongly coupled WS$_2$ monolayer (red curve), as compared to that of a bare WS$_2$ monolayer on a dielectric substrate (green curve). The IRF of our measurement apparatus is shown in blue. (b) Calibration (blue curve) used to retrieve $\tau_{\textrm{\tiny real}}$ from the measurement of $\tau_{\textrm{\tiny calc}}$, obtained by convoluting an exponential decay of time constant $\tau_{\textrm{\tiny real}}$ with the measured IRF. The calculated IRF time constant $\tau_{\textrm{\tiny calc}}^{\textrm{\tiny IRF}} = 157$ ps is shown by the blue dashed line. $\tau_{\textrm{\tiny calc}}^{\textrm{\tiny ref}}$ and $\tau_{\textrm{\tiny calc}}^{\textrm{\tiny sample}}$ are represented as green and red spots respectively. \[fig:S4\]](FigureS4.png){width="\columnwidth"}
F: Optical setup
================
The optical setup used for PL polarimetry experiments is shown in Fig. S\[fig:S5\]. The WS$_2$ monolayer is excited by a continous-wave HeNe laser at $1.96$ eV (632 nm), slightly below the direct band-gap of the atomic crystal, in order to reduce phonon-induced inter-valley scattering effects at room temperature. The pumping laser beam is filtered by a bandpass filter (BPF) and its polarization state is controlled by a set of polarization optics: a linear polarizer (LP), a half-wave plate (HWP) and a quarter-wave plate (QWP). The beam is focused onto the sample surface at oblique incidence angle by a microscope objective, to a typical spot size of $100~\mu\textrm{m}^2$. This corresponds to a typical flux of $10$ W$\cdot$cm$^{-2}$. In such conditions of irradiation, the PL only comes from the $A-$exciton. The emitted PL signal is collected by a high numerical aperture objective, and its polarization state is analyzed by another set of broadband polarization optics (HWP, QWP, LP). A short-wavelength-pass (SWP) tunable filter is placed on the optical path to stop the laser light scattered. Adjustable slits (AS) placed at the image plane of the tube lens (TL) allow to spatially select the PL signal coming only from a desired area of the sample, whose Fourier-space (or real space) spectral content can be imaged onto the entrance slits of the spectrometer by a Fourier-space lens (FSL), or adding a real-space lens (RSL). The resulting image is recorded by a cooled CCD Si camera.
![Optical setup used for the angle-resolved polarimetric measurements. See the corresponding paragraph for details. \[fig:S5\]](FigureS5.png){width="\columnwidth"}
G: Valley contrast measurements on a bare WS$_2$ monolayer
==========================================================
The valley contrast $\rho^\pm$ of a bare WS$_2$ monolayer exfoliated on a dielectric substrate (polydimethylsiloxane) is computed from the measured room temperature PL spectra obtained for left and right circular excitations, analysed in the circular basis by a combination of a quarter-wave plate and a Wollaston prism: $$\rho^\pm = \frac{I_{\sigma^\pm}(\sigma^+) - I_{\sigma^\pm}(\sigma^-)}{I_{\sigma^\pm}(\sigma^+) + I_{\sigma^\pm}(\sigma^-)},
\label{eq:CPL}$$ where $I_j(l)$ is the measured PL spectrum for a $j=(\sigma^+,\sigma^-)$ polarized excitation and a $l=(\sigma^+,\sigma^-)$ polarized analysis. A typical emission spectrum ($I_{\sigma^-}(\sigma^-)$) is shown in Fig. S\[fig:S6\](a) and the valley contrasts $\rho^\pm$ are displayed in Fig. S\[fig:S6\](b). As discussed in the main text, this emission spectrum consists of a phonon-induced up-converted PL [@Xiaodong2016]. Clearly, there is no difference in the $I_j(l)$ spectra, hence no valley polarization at room temperature on the bare WS$_2$ monolayer. These results are in striking contrast to those reported in the main text for the strongly coupled system, under similar excitation conditions. Note also that the absence of valley contrast on our bare WS$_2$ monolayer differs from the results of [@PoWenChiu2016] reported however on WS$_2$ grown by chemical vapor deposition.
![(a) Emission spectrum $I_{\sigma^-}(\sigma^-)$ obtained by exciting the bare WS$_2$ monolayer at $1.96$ eV with $\sigma^-$ polarized light, and analysing the PL in $\sigma^-$ polarization. The gray area in (a) and (b) corresponds to the spectral region cut by the PL emission filter. (b) The valley contrast $\rho^{+(-)}$ defined in (\[eq:CPL\]) is displayed in blue (green). \[fig:S6\]](FigureS6.png){width="\columnwidth"}
H: Angle-resolved Stokes vector polarimetry
===========================================
The optical setup shown in Fig. S\[fig:S5\] is used to measure the angle-resolve PL spectra for different combinations of excitation and detection polarizations. Such measurements allow us to retrieve the coefficients of the Mueller matrix $\mathcal{M}$ of the system, characterizing how the polarization state of the excitation beam affects the polarization state of the chiralitons PL. As discussed in the main text, the spin-momentum locking mechanism of our chiralitonic system relates such PL polarization states to specific chiraliton dynamics. An incident excitation in a given polarization state is defined by a Stokes vector $\bf{S}^{\textrm{\tiny in}}$, on which the matrix $\mathcal{M}$ acts to yield an output PL Stokes vector $\bf{S}^{\textrm{\tiny out}}$: $$\bf{S}^{\textrm{\tiny out}} = \left(
\begin{array}{c}
I\\
I_{V} - I_{H}\\
I_{45} - I_{-45}\\
I_{\sigma^+} - I_{\sigma^-}
\end{array}
\right)_{\textrm{\tiny out}} = \mathcal{M}\left(
\begin{array}{c}
I_0\\
I_{V} - I_{H}\\
I_{45} - I_{-45}\\
I_{\sigma^+} - I_{\sigma^-}
\end{array}
\right)_{\textrm{\tiny in}}, \label{eq:Mueller}$$ where $I_{(0)}$ is the emitted (incident) intensity, $I_{V} - I_{H}$ is the relative intensity in vertical and horizontal polarizations, $I_{45} - I_{-45}$ is the relative intensity in $+45^o$ and $-45^o$ polarizations and $I_{\sigma^+} - I_{\sigma^-}$ is the relative intensity in $\sigma^+$ and $\sigma^-$ polarizations. We recall that for our specific alignment of the OSO resonator with respect to the slits of the spectrometer, the angle-resolved PL spectra in $V$ and $H$ polarizations correspond to transverse-magnetic (TM) and transverse-electric (TE) dispersions respectively (see Fig. 2 (b) in the main text). Intervalley chiraliton coherences, revealed by a non-zero degree of linear polarization in the PL upon the same linear excitation, are then measured by the $S_1 = I_{V} - I_{H}$ coefficient of the PL output Stokes vector. This coefficient is obtained by analysing the PL in the linear basis, giving an angle-resolved PL intensity $\left(S_0^{\textrm{\tiny out}} +(-) S_1^{\textrm{\tiny out}}\right)/2$, for TM (TE) analysis. In order to obtain the polarization characteristics of the chiralitons, we measure the four possible combinations of excitation and detection polarization in the linear basis: $$\begin{aligned}
I_{\textrm{\tiny TM/TM}} &=& \left(m_{00} + m_{01} + m_{10} + m_{11}\right)/2\\
I_{\textrm{\tiny TM/TE}} &=& \left(m_{00} + m_{01} - m_{10} - m_{11}\right)/2\\
I_{\textrm{\tiny TE/TM}} &=& \left(m_{00} - m_{01} + m_{10} - m_{11}\right)/2\\
I_{\textrm{\tiny TE/TE}} &=& \left(m_{00} - m_{01} - m_{10} + m_{11}\right)/2, \end{aligned}$$ where $I_{p/a}$ is the angle-frequency resolved intensity measured for a pump polarization $p=(\textrm{TE,TM})$ and analysed in $a=(\textrm{TE,TM})$ polarization, and $m_{i,j}$ are the coefficients of the 4x4 matrix $\mathcal{M}$. By solving this linear system of equations, we obtain the first quadrant of the Mueller matrix: $m_{00},
m_{01}, m_{10}$ and $m_{11}$. The $S_1^{\textrm{\tiny out}}|_{\textrm{\tiny TM}}$ coefficient of the output Stokes vector for a TM excitation is then directly given by $m_{10}+m_{11}$ as can be seen from (\[eq:Mueller\]) by setting $I_V=1, I_H=0$ and all the other input Stokes coefficients to zeros. This quantity, normalized to $S_0^{\textrm{\tiny out}}$, is displayed in the $k_x-$energy plane in Fig. 4 (c) in the main text. Similarly, the $S_1^{\textrm{\tiny out}}|_{\textrm{\tiny TE}}$ coefficient is given by $m_{10}-m_{11}$, which is the quantity displayed in Fig. 4 (d) in the main text.
As the dispersion of the OSO resonator is different for TE and TM polarizations, the pixel-to-pixel operations performed to obtain $S_1^{\textrm{\tiny out}}$ do not directly yield the chiraliton inter-valley contrast. In particular, the observation of negative value regions in $S_1^{\textrm{\tiny out}}|_{\textrm{\tiny TM}}$ only reveals that the part of the chiraliton population that lost inter-valley coherence is dominating the total PL in the region of the dispersion where the TE mode dominates over the TM mode (compare Fig. 4(c) and (e)). It [*does not*]{} correspond to genuine anti-correlation of the chiraliton PL polarization with respect to the pump polarization. To correct for such dispersive effects and obtain the degree of chiraliton intervalley coherence, the appropriate quantity is $(S_1^{\textrm{\tiny out}}|_{\textrm{\tiny TM}}-S_1^{\textrm{\tiny out}}|_{\textrm{\tiny TE}})/(2S_0^{\textrm{\tiny out}}) = m_{11}$, resolved in the $k_x-$energy plane in Fig. 4 (f) in the main text. This quantity can also be refered to as a chiraliton linear depolarization factor. For these polarimetry measurements, the base-line noise was determined by measuring the Mueller matrix associated with an empty setup which are expected to be proportional to the identity matrix. With polarizer extinction coefficients smaller than $0.1\%$, white light (small) intensity fluctuations, and positioning errors of the polarization optics, we reach standard deviations from the identity matrix of the order of $0.4\%$. This corresponds to a base-line noise valid for all the polarimetry measurements presented in the main text. The noise level seen in Fig. 4 in the main text is thus mostly due to fluctuations in the WS$_2$ PL intensity.
|
---
abstract: |
The NuTeV collaboration has performed precision measurements of the ratio of neutral current to charged current cross-sections in high rate, high energy neutrino and anti-neutrino beams on a dense, primarily steel, target. The separate neutrino and anti-neutrino beams, high statistics, and improved control of other experimental systematics, allow the determination of electroweak parameters with significantly greater precision than past $\nu
N$ scattering experiments. Our null hypothesis test of the standard model prediction measures $\stwos=0.2277\pm0.0013({\rmt stat})\pm0.0009({\rmt
syst})$, a value which is $3.0\sigma$ above the prediction. We discuss possible explanations for and implications of this discrepancy.
author:
- |
The NuTeV Collaboration , represented by\
Kevin S. McFarland\
[*University of Rochester, Dept. of Physics and Astronomy, Rochester, NY 14627 USA*]{}
title: '**OFF THE MASS SHELL: ELECTROWEAK PHYSICS AT NUTEV**'
---
[[1]{}=0 {height="4.5cm"} ]{}
=14.5pt
Introduction and Motivation
===========================
[ ]{} Neutrino scattering played a key role in establishing the structure of the standard model of electroweak unification, and it continues to be one of the most precise probes of the weak neutral current available experimentally today. With copious data from the production and decay of on-shell $Z$ and $W$ bosons for comparison, contemporary neutrino scattering measurements serve to validate the theory over many orders of magnitude in momentum transfer and provide one of the most precise tests of the weak couplings of neutrinos. In addition, precise measurements of weak interactions far from the boson poles are inherently sensitive to processes beyond our current knowledge, including possible contributions from leptoquark and $Z^\prime$ exchange[@langacker] and new properties of neutrinos themselves[@oscpaper].
The Lagrangian for weak neutral current $\nu$–$q$ scattering can be written as $$\begin{aligned}
{\cal L}&=&-\frac{G_F\rho_0}{\sqrt{2}}(\nubar\gamma^\mu(1-\gamma^5)\nu)
\nonumber\\
&&\times\left( \epsilon^q_L {\qbar}\gamma_\mu(1-\gamma^5){q}+
\epsilon^q_R {\qbar}\gamma_\mu(1+\gamma^5){q}\right) ,
\label{eqn:lagrangian}\end{aligned}$$ where deviations from $\rho_0=1$ describe non-standard sources of SU(2) breaking, and $\epsilon^q_{L,R}$ are the chiral quark couplings [^1] For the weak charged current, $\epsilon^q_L=I_{\rms weak}^{(3)}$ and $\epsilon^q_R=0$, but for the neutral current $\epsilon^q_L$ and $\epsilon^q_R$ each contain an additional term, $-Q\stw$, where $Q$ is the quark’s electric charge in units of $e$.
The ratio of neutral current to charged current cross-sections for either $\nu$ or $\nub$ scattering from isoscalar targets of $u$ and $d$ quarks can be written as[@llewellyn] $$R^{\nu(\nub)} \equiv \frac{\sigma(\nunub N\rightarrow\nunub X)}
{\sigma(\nunub N\rightarrow\ell^{-(+)}X)}
= (g_L^2+r^{(-1)}g_R^2),
\label{eqn:ls}$$ where $$r \equiv \frac{\sigma({\overline \nu}N\rightarrow\ell^+X)}
{\sigma(\nu N\rightarrow\ell^-X)} \sim \frac{1}{2},
\label{eqn:rdef}$$ and $g_{L,R}^2=(\epsilon^u_{L,R})^2+(\epsilon^d_{L,R})^2$. Many corrections to Equation \[eqn:ls\] are required in a real target[@nc-prl], but those most uncertain result from the suppression of the production of charm, which is the CKM-favored final state for charged-current scattering from the strange sea. One way to reduce this source of uncertainty on electroweak parameters is to measure the observable $$\begin{aligned}
R^{-} &\equiv& \frac{\sigma(\nu_{\mu}N\rightarrow\nu_{\mu}X)-
\sigma(\nub_{\mu}N\rightarrow\nub_{\mu}X)}
{\sigma(\nu_{\mu}N\rightarrow\mu^-X)-
\sigma(\nub_{\mu}N\rightarrow\mu^+X)} \nonumber\\
&=& \frac{\Rnu-r\Rnub}{1-r}=(g_L^2-g_R^2),
\label{eqn:rminus}\end{aligned}$$ first suggested by Paschos and Wolfenstein[@Paschos-Wolfenstein] and valid under the assumption of equal momentum carried by the $u$ and $d$ valence quarks in the target. Since $\sigma^{\nu
q}=\sigma^{\nub\, \qbar}$ and $\sigma^{\nub q}=\sigma^{\nu \qbar}$, the effect of scattering from sea quarks, which are symmetric under the exchange $q\leftrightarrow\qbar$, cancels in the difference of neutrino and anti-neutrino cross-sections. Therefore, the suppressed scattering from the strange sea does not cause large uncertainties in $R^-$. $R^-$ is more difficult to measure than $R^\nu$, primarily because the neutral current scatterings of $\nu$ and $\nub$ yield identical observed final states which can only be distinguished through [*a priori*]{} knowledge of the initial state neutrino.
The experimental details and theoretical treatment of cross-sections in the NuTeV electroweak measurement are described in detail elsewhere[@nc-prl]. In brief, we measure the experimental ratio of neutral current to charged current candidates in both a neutrino and anti-neutrino beam. A Monte Carlo simulation is used to express these experimental ratios in terms of fundamental electroweak parameters. This procedure implicitly corrects for details of the neutrino cross-sections and experimental backgrounds. For the measurement of $\stw$, the sensitivity arises in the $\nu$ beam, and the measurement in the $\nubar$ beam is the control sample for systematic uncertainties, as suggested in the Paschos-Wolfenstein $R^-$ of Eqn. \[eqn:rminus\]. For simultaneous fits to two electroweak parameters, e.g., $\stw$ and $\rho$ or left and right handed couplings, this redundant control of systematics cannot be realized.
Result {#sect:results}
======
As a test of the electroweak predictions for neutrino nucleon scattering, NuTeV performs a single-parameter fit to $\stw$ with all other parameters assumed to have their standard values, e.g., standard electroweak radiative corrections with $\rho_0=1$. This fit determines $$\begin{aligned}
\sin^2\theta_W^{({\rms on-shell)}}&=&0.22773\pm0.00135({\rmt stat.})\pm0.00093({\rmt syst.})
\nonumber\\
&-&0.00022\times(\frac{M_{\rms top}^2-(175 \: \mathrm{GeV})^2}{(50 \: \mathrm{GeV})^2})
\nonumber\\
&+&0.00032\times \ln(\frac{M_{\rms Higgs}}{150 \: \mathrm{GeV}}).\end{aligned}$$ The small dependences in $M_{\rms top}$ and $M_{\rms Higgs}$ result from radiative corrections as determined from code supplied by Bardin[@bardin] and from V6.34 of ZFITTER[@zfitter]; however, it should be noted that these effects are small given existing constraints on the top and Higgs masses[@LEPEWWG]. A fit to the precision electroweak data, excluding neutrino measurements, predicts a value of $0.2227\pm0.00037$[@LEPEWWG; @Martin], approximately $3\sigma$ from the NuTeV measurement. Interpretations of the NuTeV data in terms of $M_W$ and $\rho_\nu$ and model-independent neutrino-quark chiral couplings are discussed elsewhere [@nc-prl; @ksm-lathuile].
Interpretation
==============
The NuTeV $\stw$ result is approximately three standard deviations from the prediction of the standard electroweak theory. This by itself is surprising; however, it is not immediately apparent what the cause of this discrepancy might be. We discuss, in turn, the possibility that the NuTeV result is a statistical fluctuation among many precision results, the possibility that unexpected quark flavor asymmetries or nuclear effects influence the result, and finally possibilities for non-standard physics which could be appearing in the anomalous NuTeV value.
Significance in a Global Context
--------------------------------
For fits assuming the validity of the standard model, it is appropriate to consider the [*a priori*]{} null hypothesis test chosen in the proposal of the NuTeV experiment, namely the measurement of $\stwos$. The fit to precision data, including NuTeV, performed by the LEPEWWG has a global the global $\chi^2$ of $28.8$ for $15$ degrees of freedom[@LEPEWWG; @Martin], including significant contributions from NuTeV’s $\stw$ measurement and $A_{FB}^{0,b}$ from LEP I. The probability of the fit $\chi^2$ being above $28.8$ is $1.7\%$. Without NuTeV, this probability of the resulting $\chi^2$ is a plausible $14\%$. This suggests that in the context of all the precision data, as compiled by the LEPEWWG, the NuTeV result is still a statistical anomaly sufficient to spoil, or at least sully, the fit within the standard model.
This large $\chi^2$ is dominated by two moderately discrepant measurements, namely $A_{FB}^{0,b}$ and the NuTeV $\stw$, and if one or both are discarded arbitrarily, then the data is reasonably consistent with the standard model. However, the procedure of merely discarding one or both of these measurements to make the fit “work” is clearly not rigorous. Furthermore, the potential danger of such a procedure has been noted previously in the literature. For example, if $A_{FB}^{0,b}$ were disregarded, then the most favored value of the Higgs mass from the fit would be well below the direct search limits. Constraining the fit to be consistent with mass limits from standard model Higgs boson searches results in still uncomfortably large $\chi^2$[@Chanowitz].
Unexpected QCD Effects
----------------------
As noted above, corrections to Eqns. \[eqn:ls\] and \[eqn:rminus\] are required to extract electroweak parameters from neutrino scattering on the NuTeV target. In particular, these equations assume targets symmetric under the exchange of $u$ and $d$ quarks, and that quark seas consist of quarks and anti-quarks with identical momentum distributions.
The NuTeV analysis corrects for the significant asymmetry of $d$ and $u$ quarks that arises because the NuTeV target, which is primarily composed of iron, has an $\approx 6$% fractional excess of neutrons over protons. However, this correction is exact only with the assumption of isospin symmetry, i.e., $\uubar_p(x)=\ddbar_n(x)$, $\ddbar_p(x)=\uubar_n(x)$. This assumption, if significantly incorrect, could produce a sizable effect in the NuTeV extraction of $\stw$[@Sather; @Thomas; @Cao; @Gambino].
Dropping the assumptions of symmetric heavy quark seas, isospin symmetry and a target symmetric in neutrons and protons, but assuming small deviations in all cases, we calculate the effect of these deviations on $R^-$ is[@nc-asym]: $$\begin{aligned}
\delta R^- & \approx
& + \: \delta N \left( \frac{U_p-D_p}{U_p+D_p}\right) (3\Delta_u^2+\Delta_d^2)
\nonumber\\
&& + \: \frac{(U_p-\Ubar_p-D_n+\Dbar_n)-(D_p-\Dbar_p-U_n+\Ubar_n)}{2(U_p-\Ubar_p+D_p-\Dbar_p)} (3\Delta_u^2+\Delta_d^2) \nonumber\\
&& + \: \frac{S_p-\Sbar_p}{U_p-\Ubar_p+D_p-\Dbar_p} (2\Delta_d^2-3(\Delta_d^2+\Delta_u^2)\epsilon_c),
\label{eqn:deltaR-}\end{aligned}$$ where $\Delta_{u,d}^2 = (\epsilon^{u,d}_L)^2-(\epsilon^{u,d}_R)^2$, $Q_N$ is the total momentum carried by quark type $Q$ in nucleon $N$, and the neutron excess, $\delta N \equiv A-2Z/A$. $\epsilon_c$ denotes the ratio of the scattering cross section from the strange sea including kinematic suppression of heavy charm production to that without kinematic suppression. The first term is the effect of the neutron excess, which is accounted for in the NuTeV analysis; the second is the effect of isospin violation and the third is the effect of an asymmetric strange sea.
NuTeV does not exactly measure $R^-$, in part because it is not possible experimentally to measure neutral current reactions down to zero recoil energy. To parameterize the exact effect of the symmetry violations above, we have numerically evaluated the effects on the NuTeV results of isospin and $s-\sbar$ asymmetries as a function of $x$ [@nc-asym]. This analysis shows that the level of isospin violation required to shift the $\stw$ measured by NuTeV to its standard model expectation would be, e.g., $D_p-U_n\sim0.01$ (about 5% of $D_p+U_n$), and that the level of asymmetry in the strange sea required would be $S-\Sbar\sim +0.007$ (about $30\%$ of $S+\Sbar$).
### Isospin Violations
Several recent classes of non-perturbative models predict isospin violation in the nucleon[@Sather; @Thomas; @Cao]. The earliest estimation in the literature, a bag model calculation[@Sather], predicts large valence asymmetries of opposite sign in $u_p-d_n$ and $d_p-u_n$ at all $x$, which would produce a shift in the NuTeV $\stw$ of $-0.0020$. However, this estimate neglects a number of effects, and a complete bag model calculation by Rodionov [*et al.*]{}[@Thomas] conclude that asymmetries at very high $x$ are larger, but the asymmetries at moderate $x$ are smaller and even of opposite sign at low $x$, thereby reducing the shift in $\stw$ to a negligible $-0.0001$. Finally, the effect is also evaluated in the meson cloud model[@Cao], and there the asymmetries are much smaller at all $x$, resulting in a modest shift in the NuTeV $\stw$ of $+0.0002$.
Models aside, the NuTeV data itself cannot provide a significant independent constraint on this form of isospin violation. However, because PDFs extracted from neutrino data (on heavy targets) are used to separate sea and valence quark distributions which affect observables at hadron colliders[@bodek], global analyses of PDFs including the possibility of isospin violation may be able to constrain this possibility experimentally. At least one author[@Kumano] has begun to consider the experimental isospin constraints in the context of “nuclear PDFs”, and found very small isospin effects, except at very high $x$ and low $Q^2$, a region removed by the visible energy requirement ($E_{\rm calorimeter}>20$ GeV) of the NuTeV analysis.
### Strange Sea Asymmetry
If the strange sea is generated by purely perturbative QCD processes, then neglecting electromagnetic effects, one expects $\sav=\savbar$. However, it has been noted that non-perturbative QCD effects can generate a significant momentum asymmetry between the strange and anti-strange seas[@sNEsbar].
By measuring the processes $\txnunub N\to \mu^+\mu^- X$ the CCFR and NuTeV experiments constrain the difference between the momentum distributions of the strange and anti-strange seas. Within the NuTeV cross-section model, this data implies a [*negative*]{} asymmetry[@nc-asym], $$S-\Sbar = -0.0027 \pm 0.0013,$$ or an asymmetry of $11\pm6$% of $(S+\Sbar)$. Therefore, dropping the assumption of strange-antistrange symmetry results in an [*increase*]{} in the NuTeV value of $\stw$, $$\Delta\stw = +0.0020 \pm 0.0009.$$ The initial NuTeV measurement, which assumes $\sav=\savbar$, becomes $$\stwos=0.2297\pm0.0019.$$ Hence, if we use the experimental measurement of the strange sea asymmetry, the discrepancy with the standard model is increased to $3.7\sigma$ significance.
### Nuclear Effects
Nuclear effects which can be absorbed into process-independent PDFs will not affect the NuTeV result. However, several authors have recently explored the possibility that neutrino neutral and charged current reactions may see different nuclear effects and therefore influence the NuTeV result.
A recent comment in the literature[@Thomas-Miller] has offered a Vector Meson Dominance (VMD) model of low $x$ shadowing in which such an effect might arise. The most precise data that overlaps the low $x$ and $Q^2$ kinematic region of NuTeV comes from NMC[@NMC-arneodo], which observed a logarithmic $Q^2$ dependence of the shadowing effect as predicted by perturbative QCD for $Q^2$ independent shadowing as in Pomeron models. However, models with a mixture of VMD and Pomeron shadowing can be consistent with this high $Q^2$ data [@two-phase; @thomas-melnit-rant].
The NuTeV analysis, which uses $\nu$ and $\nubar$ data at $<Q^2>$ of $25$ and $16$ GeV$^2$, respectively, is far away from the VMD regime, and the effect of this VMD model is significantly smaller than stated in Ref. [@Thomas-Miller]. The most serious flaw in the hypothesis that this accounts for the NuTeV result, however, is that it is not internally consistent with the NuTeV data. Shadowing, a low $x$ phenomenon, largely affects the sea quark distributions which are common between $\nu$ and $\nubar$ cross-sections, and therefore cancel in $R^-$. However, the effects in $\Rnu$ and $\Rnub$ individually are much larger than in $R^-$ and this model [*increases*]{} the prediction for NuTeV’s $\Rnu$ and $\Rnub$ by $0.6\%$ and $1.2\%$, respectively. NuTeV’s $\Rnu$ and $\Rnub$ are both below predictions and the significant discrepancy is in the $\nu$ mode, not the $\nubar$ “control” sample, both in serious contradiction with the prediction of the VMD model.
Another recent paper[@Schmidt] has suggested that there may be little or no EMC effect in the neutrino charged-current but the expected EMC effect suppression at high $x$ in the neutral current. If true, this could have the right behavior and perhaps magnitude to explain the NuTeV data because of the effect at high $x$. Unfortunately, this mechanism would cause large differences between $F_2^\nu$ and $F_2^{\ell}$ on heavy targets at high $x$ which are excluded by the CCFR charged-current cross-section measurements[@unki-pmi].
New Physics
-----------
The primary motivation for embarking on the NuTeV measurement was the possibility of observing hints of new physics in a precise measurement of neutrino-nucleon scattering. NuTeV is well suited as a probe of non-standard physics for two reasons. First, the precision of the measurement is a significant improvement, most noticeably in systematic uncertainties, over previous measurements. Second, NuTeV’s measurement has unique sensitivity to new processes when compared to other precision data. In particular, NuTeV probes weak processes far off-shell, and thus is sensitive to other tree level processes involving exchanges of heavy particles. Also, the initial state particle is a neutrino, and neutrino couplings are the most poorly constrained by the $Z^0$ pole data, since they are primarily accessed via the measurement of the $Z$ invisible width.
In considering models of new physics, a “model-independent” effective coupling measurement [@nc-prl; @ksm-lathuile] is the best guide for evaluating non-standard contributions to the NuTeV measurements. This measurement suggests a large deviation in the left-handed chiral coupling to the target quarks, while the right-handed coupling is as expected. Such a pattern of changes in couplings is consistent with either a hypothesis of loop corrections that affect the weak process itself or another tree level contribution that contributes primarily to the left-handed coupling. Chiral coupling deviations are often parameterized in terms of the mass scale for a unit-coupling “contact interaction” in analogy with the Fermi effective theory of low-energy weak interactions. Assuming a contact interaction described by a Lagrangian of the form $$-{\cal L}=\sum_{H_q\in\{L,R\}}
\frac{\pm 4\pi} {\left( \Lambda^\pm_{LH_q}\right) ^2}\times
\left\{ \overline{l_L}\gamma^\mu l_{L}\overline{q_{H_q}}\gamma_\mu q_{H_q}
+ l_{L}\gamma^\mu \overline{l_L}\overline{q_{H_q}}\gamma_\mu q_{H_q}\right.
\left. + {\rmt C.C.}\right) ,$$ the NuTeV result can be explained by an interaction with mass scale $\Lambda_{LL}^+\approx 4\pm 0.8$ TeV.
### Extra U(1) Interaction
Phenomenologically, an extra $U(1)$ gauge group which gives rise to interactions mediated by a heavy $Z^{\prime}$ boson, $m_{Z^{\prime}}\gg m_Z$, is an attractive model for new physics. In general, the couplings associated with this new interaction are arbitrary, although specific models in which a new $U(1)$ arises may provide predictions or ranges of predictions for these couplings. An example of such a model is an $E(6)$ gauge group, which encompasses the $SU(3)\times SU(2)\times U(1)$ of the standard model and also predicts several additional $U(1)$ subgroups which lead to observable interactions mediated by $Z^{\prime}$ bosons. Before the NuTeV measurement, several authors had suggested in the literature that the other precision electroweak data favored the possibility of a $Z^{\prime}$ boson[@Zprime; @Erler].
We have analyzed the effect of $Z^{\prime}$s in $E(6)$ GUT models[@langacker; @E6-model] on the NuTeV measurement of the chiral couplings. The effect of these bosons when the $Z$ and $Z^{\prime}$ do not mix is primarily on the right-handed coupling. It is possible to reduce the left-handed coupling somewhat by allowing $Z-Z^{\prime}$ mixing; however, this possibility is severely constrained by precision data at the $Z^0$ pole[@Erler].
A $Z^{\prime}$ with coupling magnitudes equal to those of the $Z$ ($Z^{\prime}_{SM}$) but leading to a destructive interference with the $Z$ exchange could explain the NuTeV measurement if the $Z^{\prime}$ mass were in the range $\approx1$–$1.5$ TeV. Current limits from the TeVatron experiments on such $Z^{\prime}_{SM}$ are approximately $0.7$ TeV[@TeV-zprime]. Several authors have also recently discussed other $U(1)$ extensions in the context of the NuTeV result and found significant effects[@Gambino; @Ma-Roy].
### Anomalous Neutrino Neutral Current
![*Measurements of the neutrino current coupling, interpreted as a neutrino neutral current interaction rate ($\propto\rho^{(\nu)}$). The precise measurements, $\Gamma (Z\to\nu\nubar)$ at LEP I and the NuTeV data, interpreted as an overall deviation in the strength of the neutral current coupling to neutrinos, are both below expectation.*[]{data-label="fig:nunc"}](nunc-rate.eps){width="\textwidth"}
There are few precision measurements of neutrino neutral current interactions. Measurements of neutrino-electron scattering from the CHARM II experiment[@Charm2] and the direct measurement of $\Gamma(Z\to\nu\nubar)$ from the observation of $Z\to\nu\nubar\gamma$ at the $Z^0$ pole[@LEPEWWG] provide measurements of a few percent precision. The two most precise measurements come from the inferred $Z$ invisible width[@LEPEWWG] and NuTeV. As is shown in Figure \[fig:nunc\], both of the precise rate measurements are significantly below the expectation. Theoretically, such a deviation is difficult to accomodate. One idea is a mixing of the light neutrinos with another heavy gauge singlet, but this mechanism leads to effects in [*both*]{} $Z\nu\nubar$ and $W\ell\nu$ vertices[@Gambino]. However, Takeuchi and collaborators recently suggested that both of these effects could be accomodated in the precision electroweak data if the Higgs boson were heavy[@Takeuchi].
Summary
=======
The NuTeV experiment has performed a measurement of $\stw$, and finds a deviation of three standard deviations from the null hypothesis which assumes the validity of the standard model of electroweak interactions. Motivated by the significance of this discrepancy, we study both conventional and new physics explanations. Several possibilities exist, although none is theoretically compelling or has sufficient independent supporting evidence to be a clear favorite. Therefore, this result remains a puzzle.
Acknowledgements {#acknowledgements .unnumbered}
================
We gratefully acknowledge support for this work from the U.S.Department of Energy, the National Science Foundation and the Alfred P. Sloan Foundation. The NuTeV experiment benefitted greatly from significant contributions from the Fermilab Particle Physics, Computing, Technical and Beams Divisions. In addition, we thank Stan Brodsky, Jens Erler, Martin Grünewald, Shunzo Kumano, Paul Langackger, Jerry Miller, Michael Peskin, Jon Rosner, Ivan Schmidt and Tony Thomas for useful input and discussions.
[99]{} \#1\#2\#3 [[ Nucl. Phys.]{} [**\#1**]{}, \#2 (\#3). ]{} \#1\#2\#3 [[ Proc. Cam. Phil. Soc.]{} [**\#1**]{}, \#2 (\#3). ]{} \#1\#2\#3 [[ Phys. Lett.]{} [**\#1**]{}, \#2 (\#3). ]{} \#1\#2\#3 [[ Phys. Lett.]{} [**\#1**]{}, \#2 (\#3); ]{} \#1\#2\#3 [[ Phys. Rep.]{} [**\#1**]{}, \#2 (\#3). ]{} \#1\#2\#3 [[ Phys. Rev.]{} [**\#1**]{}, \#2 (\#3). ]{} \#1\#2\#3 [[ Phys. Rev. Lett.]{} [**\#1**]{}, \#2 (\#3). ]{} \#1\#2\#3 [[ Proc. Roy. Soc.]{} [**\#1**]{}, \#2 (\#3). ]{} \#1\#2\#3 [[ Prog. Th. Phys.]{} [**\#1**]{}, \#2 (\#3). ]{} \#1\#2\#3 [[ Rev. Mod. Phys.]{} [**\#1**]{}, \#2 (\#3). ]{} \#1\#2\#3 [[ Rep. Prog. Phys.]{} [**\#1**]{}, \#2 (\#3). ]{} \#1\#2\#3 [[ Zeit. Phys.]{} [**\#1**]{}, \#2 (\#3). ]{} \#1\#2\#3 [[ Eur. Phys. Jour.]{} [**\#1**]{}, \#2 (\#3). ]{} \#1\#2\#3 [[ Nucl. Instr. Meth.]{} [**\#1**]{}, \#2 (\#3). ]{}
P. Langacker [*et al.*]{}, K. S. McFarland, D. Naples [*et al.*]{}, C. H. Llewellyn Smith, G. P. Zeller [*et al.*]{}, . See also G. P. Zeller, Ph.D Thesis, Northwestern University (2002), unpublished. E. A. Paschos and L. Wolfenstein D. Bardin and V. A. Dokuchaeva, JINR-E2-86-260 (1986). D. Bardin [*et al.*]{}, Comp. Phys. Commun. 133 229 (2001). CERN-EP/2001-98, hep-ex/0112021. Updated numbers used in this note are taken from http://lepewwg.web.cern.ch/LEPEWWG/ M. Grünewald, private communication, for the fit of Ref. [@LEPEWWG] without neutrino-nucleon scattering data included. K.S. McFarland [*et al.*]{}, hep-ex/0205080. M. S. Chanowitz, hep-ph/0207123. M. S. Chanowitz, . E. Sather, E. N. Rodionov, A. W. Thomas, and J. T. Londergan, Mod. Phys. Lett. A [**9**]{}, 1799 (1994). F. Cao and A. I. Signal, S. Davidson, S. Forte, P. Gambino, N. Rius, and A. Strumia, hep-ph/0112302. G. P. Zeller [*et al.*]{}, hep-ex/0203004. A. Bodek [*et al.*]{}, S. Kumano, hep-ph/0209200. A.I. Signal and A.W. Thomas, . M. Burkardt and B. J. Warr, . S. Brodsky and B. Ma, . W. Melnitchouk and M. Malheiro, . G. A. Miller and A. W. Thomas, hep-ex/0204007. M. Arneodo [*et al.*]{} \[NMC Collaboration\], J. Kwiecinski and B. Badelek, Phys. Lett. [**B208**]{}, 508 (1988). W. Melnitchouk and A. Thomas, hep-ex/0208016. S. Kovalenko et al., hep-ph/0207158. U. K. Yang [*et al.*]{}, R. Casalbuoni, S. De Curtis, D. Dominici and R. Gatto, . J. L. Rosner, . A. Bodek and U. Baur, . J. Erler and P. Langacker, G. C. Cho, K. Hagiwara and Y. Umeda, . D. Zeppenfeld and K. Cheung, hep-ph/9810277. F. Abe [*et al.*]{} \[CDF Collaboration\], . B. Abbott [*et al.*]{} \[D0 Collaboration\], . E. Ma and D. P. Roy, P. Vilain [*et al*]{}, Phys. Lett. [**B335**]{} (1994) 248. T. Takeuchi, hep-ph/0209109.
[^1]: Note that although we use a process-independent notation here for a tree-level $\rho$, radiative corrections to $\rho$ depend slightly on the particles involved in the weak neutral interaction. In this case, $\rho\equiv \sqrt{\rho^{(\nu)}\rho^{(q)}}$.
|
---
abstract: 'We discuss the process of adiabatic growth of central black holes in the presence of a stationary, pre-existing distribution of collisionless stars. Within the limitations of the assumptions, the resulting models make robust physical predictions for the presence of a central cusp in the stellar and dark matter density, a Keplerian rise in the velocity dispersion, and a significant tangential polarization of the velocity tensor. New generations of numerical models have confirmed and extended previous results, permit the study of axisymmetric and triaxial systems, and promise new insight into the dynamics of the central regions of galaxies. These studies enable detailed comparisons with observations, further our understanding on the fueling processes for AGNs and quiescent black holes, and help elucidate the secular evolution of the inner regions and spheroids of galaxies.'
author:
- |
S. SIGURDSSON\
Department of Astronomy & Astrophysics,\
Pennsylvania State University
---
\[1996/06/01\]
Adiabatic Growth of Massive Black Holes
=======================================
Introduction
------------
Given the premise that the massive central dark objects in normal galaxies in the local Universe are in fact supermassive black holes (Lynden-Bell 1969; Rees 1990; Kormendy & Richstone 1995; Richstone et al. 1998), we can entertain a number of conjectures about the interaction of the central black hole with its environment. Obvious questions to consider include: formation scenarios for the black hole (e.g., Rees 1984; Shapiro in this volume); the demographics of the present population of black holes (Richstone and Ho in this volume); the fueling of active nuclei (Blandford in this volume); the interaction of the active nucleus with its environment (Begelman in this volume); and, the effect of the central object upon the surrounding stellar population and the larger-scale structure of the host galaxy (Burkert, Gebhardt, Haehnelt, and Merritt in this volume). Hence, one can also test whether the inferred effects of the central object are consistent with observations, and whether additional observational constraints can be placed on either the presence or the evolutionary history of the central black hole.
A particular assumption can be made (with the caveat, that, as with all assumptions, it may be false) that a substantial increase in mass of the central black hole takes place after initial formation, and that the mass is in some rigorous sense (to be established) added slowly to the pre-existing seed black hole. This is the assumption of [*adiabatic growth*]{}, which will be reviewed here. It leads to some nontrivial, testable predictions for the effects of black holes on their environments.
A central supermassive black hole dynamically dominates the surrounding stellar population inside some characteristic radius, $r_h =
GM_{\rm BH}/\sigma^2$, where $M_{\rm BH}$ is the mass of the black hole and $\sigma$ is the velocity dispersion of the stars outside the radius of influence. A natural “shortest” timescale for growth of a black hole is the “Salpeter” timescale, $t_S = M_{\rm BH}/\dot M_{\rm Edd} \approx
5\times 10^{7} \ {\rm yr}$, where $\dot M_{\rm Edd}$ is the usual Eddington accretion timescale. The dynamical timescale inside $r_h$ is just $t_{dyn} =
r_h/\sigma$. For the Milky Way, $t_{dyn}(r_h) \sim 10^4 \ {\rm yr}$, for $M_{\rm BH} \sim 2\times 10^6 \,{M_{\odot}\,}$ and $\sigma \approx 66\, \kms$. If the observed correlation between dispersion and black hole mass holds, then $M_{\rm BH} \propto \sigma^4$ (Gebhardt 2000a; Ferrarese & Merritt 2000; Tremaine 2002), and hence $t_{dyn} \propto \sigma $. Hence we conclude that for reasonable black hole masses ($M_{\rm BH} \ltorder 10^{10} \,{M_{\odot}\,}$), the dynamical timescale inside $r_h$ is always much shorter than the Salpeter timescale, and therefore the likely timescale for black hole growth through accretion of baryonic matter is much longer than the dynamical timescale inside the radius at which the black hole dominates the dynamics. We thus conclude that there may be a broad range of situations under which black hole growth is “adiabatic” and the assumptions of these studies hold. The stellar population will generally form a [*density cusp*]{}, $\rho \propto r^{-A}$, inside $r_h$, with the stellar velocity dispersion showing a Keplerian rise $\sigma (r) \propto r^{-1/2}$ inside the cusp (Peebles 1972; Bahcall & Wolf 1976; Young 1980; Quinlan, Hernquist, & Sigurdsson 1995).
### Assumptions
We consider the response of a stellar distribution function () to the slow growth of a massive central black hole. The initial conditions assume there is no central black hole to begin with (or more realistically a seed black hole with a negligible initial mass), and there are usually implicit assumptions that the mass is “magically” added to the black hole — that is to say, the mass of the central object is increased without necessarily withdrawing the mass from some explicit reservoir. The underlying assumption here is that the mass is accreting from some diffuse medium, like cool gas, that is distributed like the stellar population but with a density much lower than that of the stellar mass density, and is replaced by some inflow that is an implicit outer boundary condition (e.g., Young 1980; Quinlan 1995). This is [*not*]{}a necessary assumption; it is just a simplifying assumption. It is trivial to extend it to scenarios where the mass is explicitly withdrawn from some reservoir, with the added complication of having to specify the physical nature of the mass reservoir. In most situations modeled so far, where the mass comes from is not important; the response of the system is robust, independent of the source of the mass. The exception is if all the mass comes from the black hole swallowing the most tightly bound stars only, in which case the conclusions are somewhat different and the process effectively violates our assumptions of adiabatic growth.
An additional implicit assumption is that the black hole is [*central*]{}; that is, it is at the center of mass of the stellar system. In practice, the surrounding stellar system is discrete and the black hole mass is finite, so we expect the black hole to undergo quasi-Brownian motion away from the center (see, e.g., Chatterjee, Hernquist, & Loeb 2001). For masses of astrophysical interest, the displacement is typically much larger than the black hole Schwarzschild radius, $r_S$, but much smaller than $r_h$, and the timescale for wandering is short enough that the outer cusp is not carried with the black hole as it moves; this can lead to modification of the cusp profile at small radii ($r \ltorder 10^{-3} r_h$ for typical $M_{\rm BH}$) as the black hole wandering produces rapid fluctuations in the central potential seen by stars in the inner cusp (see Sigurdsson 2003).
A final fundamental consideration is whether the dynamics of the stellar population are “collisionless” — that is, whether the relaxation timescale for a population of $N$ stars, $t_R \sim N t_{dyn}/8 \ln \Lambda$, is shorter or longer than the evolutionary timescale of the stellar system, usually taken to be the Hubble time, $t_H \sim 10^{10} \ {\rm yr}$ (e.g., Spitzer 1971; Hills 1975). The response of a relaxed stellar system to the presence of a central massive black hole has been extensively considered, primarily in the context of globular clusters, or in the context of initial black hole formation and rapid growth in protogalaxies (Bahcall & Wolf 1976, 1977; Lightman & Shapiro 1977; Cohn & Kulsrud 1978; Shapiro & Marchant 1978; Shapiro 1985; Amaro-Seoane & Spurzem 2001 and Freitag & Benz 2001 and citations therein). For supermassive black holes in normal, evolved, galaxies the relaxation timescales in the inner spheroid, but outside the black hole cusp, are generally longer than the Hubble time; inside the cusp the relaxation time may be constant, increase, or decrease with decreasing radius. For those cases where the relaxation time decreases with decreasing radius, the dynamics of the stellar population surrounding the black hole [*may*]{}undergo a transition to the fully collisional regime in the inner cusp, and the discussion in the papers cited above then becomes appropriate but is beyond the scope of this review. The mean central relaxation time can be approximated as $t_R \sim 2\times 10^9 (\sigma/200 \, {\rm km\, s^{-1}})^3/(\rho /10^6 \,
{M_{\odot}\,}{\rm pc^{-3}})$ (Young 1980). It is not sufficient that $t_R < t_H$ for non-adiabatic growth. For such relaxed cusps the relaxation time at small radii may become shorter, and, if $t_R \ltorder t_S$ at some small radius, which may well occur for a significant fraction of galactic nuclei or proto-nuclei at some point in their evolution, then any central black hole may grow by tidal disruption of stars or by swallowing stars whole, more rapidly then the cusp can dynamically readjust its structure; in such a situation, the growth is definitely non-adiabatic.
The underlying physical assumption of the “adiabatic growth” model is that as the integrals of motion change smoothly in response to the increase in central mass, the action variables for the surrounding stellar population remain invariant (Binney & Tremaine 1987). This is to be contrasted with the opposite extreme assumption of “violent relaxation,” in which the potential is assumed to fluctuate rapidly compared to the dynamical time, and the evolves to some final statistical equilibrium state (Lynden-Bell 1967; Stiavelli 1998). The resulting “final distribution” may then be compared with observations. It should be noted that real galaxies may not have “initial” DFs that are well represented by any of the analytic or numerical distributions assumed in these models, nor is it necessarily the case that significant increase in black hole mass ever takes place under conditions in which the adiabatic approximation holds. In particular, an implicit assumption is that a relaxed stellar population is in place as an initial condition, and that significant increase in black hole mass takes place [*after*]{} (the inner region of) the galaxy is assembled. The adiabatic models are physically distinct from [*ab initio*]{} models, where a including a central black hole, by design, is required to satisfy the Boltzmann equation (e.g., Huntley & Saslaw 1975; Tremaine 1994). The adiabatic models are also distinct from the “orbit assembly” models used to construct kinematic models of observed galaxies (Schwarzschild 1979; Richstone & Tremaine 1984, 1988; Magorrian 1998).
An interesting question is whether any of these models in some sense rigorously represent real stellar systems. Nature need not settle on the analytically or numerically derived solutions of the Boltzmann equation, out of the infinite number that exist. As found by Quinlan (1995), apparently small differences in some phase-space values can lead to large changes in the averaged properties of the evolved system. We may also worry whether the different techniques for constructing stationary solutions of the Boltzmann equation representing collisionless stellar objects surrounding a central black hole are actually equivalent, or whether the different techniques produce wholly distinct families of solutions, as opposed to solutions with an overlap in properties or formally identical for some range of parameters.
Spherical Growth
----------------
The response of a spherical distribution of stars to the adiabatic growth of a central black hole in the collisionless limit was first considered by Peebles (1972) for an isothermal sphere. Young (1980) confirmed the primary result that a density cusp $\rho \propto r^{-3/2}$ would form, with an associated velocity dispersion cusp, $\sigma (r) \propto r^{-1/2}$; he also pointed out that the velocity anisotropy, $\beta(r) = 1 - \langle v_t^2\rangle/\langle 2v_r^2\rangle $, becomes negative (tangentially biased) at small radii, where $v_t$ and $v_r$ is the tangential and radial velocity, respectively. Goodman & Binney (1984) showed that when $\beta (0) = 0$ the distribution is isotropic at the center for an initial isothermal distribution (see also Binney & Petit 1989), and Lee & Goodman (1989) generalised the approximate solution of the problem to axisymmetric rotating distributions. The basic physics of the problem for a spherical system are discussed in Shapiro & Teukolsky (1983 and reference therein), as a simple application of Liouville’s theorem. Their Equation 14.2.9 shows the response of a spherical system, with some initial DF $f(E)$, to a central black hole. The final density $n(R) = 4\pi \int f(E) \sqrt{[2(E - \Phi)]} dE \propto r^{-1/2}\times r^{-1}$ for $f(E) \rightarrow {\rm constant}$, appropriate for the $n=0$ case discussed by Quinlan et al. (1995).
Quinlan (1995) generalised the result to a broad range of initially spherical DFs and found that for different the final cusp slope may be very different, even for near-identical initial spatial density profiles. They also found that, in contrast with the result for initially isothermal distributions, for some initial the polarization of the velocity distribution is generic and always tangentially biased, and that the tangential bias may persist to zero radius. The velocity distribution is in general non-Gaussian, and initially non-Gaussian distributions may evolve to be either closer to or farther from Gaussian in response to the black hole growth (Sigurdsson, Hernquist, & Quinlan 1995). The net results are distinct, but unfortunately not provide a simple or unique prediction for the final spherical distribution of a stellar population responding adiabatically to the growth of a central black hole. The semi-analytic results of Quinlan were confirmed numerically in a companion paper by Sigurdsson (1995), who extended the numerical methodology to a family of non-spherical models.
A major purpose for producing a broad range of adiabatic growth models is for comparison with observations, for example to establish robust estimators for central black hole masses from the observed surface density profiles or spectroscopically determined projected velocity dispersion profiles. In addition to the intrinsic degeneracies between the and the density and dispersion profiles, we are mostly restricted to observing projected quantities, the line integrals of the light density and velocity distribution, which lead to degeneracies in the inversion to the full volume distribution (e.g., Romanowsky & Kochanek 1997 and references therein). We are further restricted to observing the dominant light-emitting population (mainly giant, sub-giant and post-AGB stars), and the mass may be distributed differently, with different stellar populations (or dark matter) having different density profiles. Still, with the use of higher moments of the velocity distribution (van der Marel & Franx 1993; Dehnen & Gerhard 1994; van der Marel 1994a, b; van der Marel 1994) strong constraints can be put on the true stellar ; by making some “natural” assumptions (e.g., the unobserved dark matter distribution is consistent with the light distribution), strong constraints can be put on the total mass of any inferred central dark object.
### Action
In a spherical potential, we consider some initial $f$ specified by the energy $E$ and angular momentum $L$. The quantity $f$ is then also a function of the actions $L$ and $J_r = \oint v_r dr$. As the integrals change under the adiabatic growth of the black hole, the action, by assumption, remains invariant, and $f$ evolves to remain a fixed function of the actions (see Young 1980 and Quinlan 1995 for discussion).
We want to consider some initial with an explicitly specified form and a corresponding density profile (see Binney & Mamon 1982; Binney & Tremaine 1987). Of particular interest is the asymptotic behaviour of the density profile at small radii ($\rho(r) \propto r^{-\gamma}$; as $r\rightarrow 0$) and the corresponding asymptotic behaviour of $f(E)$ in the limit $E\rightarrow \Phi (0)$, which in general is some power law $f(E) \sim [E - \Phi (0)]^{-n}$. (But note that real galaxies need not be nicely monotonic power laws, even asymptotically, but may, for example, have density inversions at small radii (e.g., Peebles 1972; Lauer 2002.) For an isothermal density profile the central density approaches a constant at small radii, as does the at the lowest energies. In general, $0 \leq \gamma \leq 3$, and it is useful to distinguish between models with “analytic” cores (in the nomenclature of Quinlan 1995), which have a density profile $\rho(r) \approx \rho_0 + {1\over 2} \rho'' r^2 + \ldots$, as $r\rightarrow 0$, and non-analytic models, which do not approximate a harmonic potential at the origin. Particular examples of analytic models include a non-singular isothermal sphere, a King model, a Plummer model, or an isochrone. Non-analytic models include (1) a singular isothermal sphere (Cipollina & Bertin 1994); (2) the $\gamma = 2$ Jaffe (1983) or $\gamma = 1$ Hernquist (1990) model; (3) the generalised “gamma” models ($0 \leq \gamma \leq 3$; Dehnen 1993; Tremaine 1994); (4) and further spherical generalisations of these models, such as those of Navarro, Frenk, & White (1997) and Zhao (1997).
It turns out that the final density cusp slope $A$ generally depends both on the initial density slopes $\gamma$ and the asymptotic divergence of the $n$; the relationship among the variables is given analytically by[^1]
$$A = {3\over 2} + n\left ( {{ 2 - \gamma}\over {4 - \gamma}} \right ).$$
As illustrated in Figure 1, which compares the adiabatic growth of a black hole in a $\gamma = 0$ model with an isochrone model (Hénon 1960), analytic and non-analytic models with the same, or very nearly the same, initial density profiles produce qualitatively different final density profiles.
More generally, the response to the adiabatic growth of a black hole produces a cusp with slope as low as $A = 3/2$, as originally found, up to values as steep as $A=3$, although $A=5/2$ is probably the steepest physically sustainable slope before collisional effects in the inner cusp necessarily dominate the dynamics. Table 1.1 lists the values of some initial and final slopes (Quinlan 1995). $C$ is the final cusp slope in the limit of an initially completely tangentially biased .
[Model]{} [$\gamma$]{} [$n$]{} [$A$]{} $C$
---------------- -------------- --------- --------- --------
isochrone $0$ $0$ $3/2$ $9/4$
$\gamma = 0$ $0$ $1$ $2$ $9/4$
$\gamma = 1$ $1$ $5/2$ $7/3$ $7/3$
$\gamma = 3/2$ $3/2$ $9/2$ $12/5$ $12/5$
$\gamma = 2$ $2$ $-$ $5/2$ $5/2$
: Adiabatic density cusps
\[ss-table1\]
Note that the presence of a density cusp by itself is [*not*]{} a robust indicator of a central supermassive black hole; this is clearly so, since, for example, the Jaffe (1983) model or singular isothermal sphere, with no central black holes, have $\gamma = 2$ cusps, steeper than the $A=3/2$ cusps predicted for the response of a non-singular isothermal sphere to a central black hole. On the other hand, the presence of a density cusp, a Keplerian rise in the velocity dispersion, and the kinematic signatures of tangential anisotropy at small radii [*are*]{} robust indicators of a central supermassive black hole.
This does not preclude the possibility that in the absence of a central black hole actual stellar systems tend toward flat, constant density cores, whether through formation or relaxation, and that in practice cusps are in fact signatures of central black holes. We know that for a broad range of formation scenarios, stellar cusps form around central black holes (with cusps as shallow as $A=1/2$ or as steep as $A=5/2$); however, it is possible that in some situations binary black holes completely destroy cusps, leaving density inversions (e.g., Peebles 1972), and we know it is possible for cuspy stellar systems to exist in the absence of central black holes. Assuming that cusps are tracers of black holes, van der Marel (1999) has explored the use of the adiabatic growth models in matching observed density profiles.
### Anisotropy
Quinlan (1995) also experimented with spherical, radially anisotropic distributions (Osipkov 1979; Tonry 1983; Merritt 1985; Dejonghe 1987; Cudderford 1991; Gerhard 1993), but found it impossible to generate physical distributions with significant radial anisotropy persisting to zero radius. The general conclusion is therefore that adiabatic growth induces tangential bias at small radii, and that an initial tangential bias can, but does not necessarily, lead to steeper final cusps, compared to the equivalent isotropic model. The Keplerian velocity cusp is a robust prediction of spherical adiabatic growth models. As noted by Duncan & Wheeler (1980), however, a strong radial velocity anisotropy can mimic a Keplerian rise in velocity in projection, although there are severe concerns about the stability of any such models (Merritt 1987; Palmer & Papaloizou 1988). A robust prediction of a tangential bias induced by any central black hole is therefore potentially important, although the anisotropy is not a directly observable quantity but must be inferred from the projected moments of the velocity distribution.
As shown in Figure 1.1, the final anisotropy, $\beta (r)$, may be either zero at the black hole or remain negative at small radii. The deviation from Gaussianity is conveniently measured by the kurtosis, $\kappa$ (by construction, the skew is zero for these models), or equivalently, the fourth Gauss-Hermite moment, $h_4 \approx (\kappa - 3)/8\sqrt{6}$, for $h_4 \ltorder 0.03$ (van der Marel & Franx 1993; Dehnen & Gerhard 1994; Quinlan 1995).
With $A = 3/2 + p > 3/2$, the relaxation timescale at small radii decreases as $t_R \propto r^p$. The cusps induced in isotropic, analytic models have $p=0$ and constant $t_R$; more generally, $p > 0$ and $t_R$ can be small close to the black hole. Very close to the black hole, relaxation and collision timescales get short for strong cusps, and strong collisional effects may lead to rapid growth of the black hole, with corresponding associated depletion of the stellar population. This is certainly the case for cusps as steep as $A = 3$, and may even be a problem for shallower cusps (Frank & Rees 1975; Quinlan & Shapiro 1990; Quinlan 1995; Sigurdsson & Rees 1997; Freitag & Benz 2001).
### Non-adiabatic growth
Formally, the adiabatic growth model implies an infinitely long timescale for accretion. In practice, of course, any growth in mass occurs on a finite timescale. We can investigate the nature of non-adiabatic growth without losing the predictive power of the adiabatic models.
Sigurdsson (1995; see also Hernquist & Ostriker 1992; Hernquist, Sigurdsson, & Bryan 1995; Sigurdsson 1997a) explored the timescale for adding mass, and concluded that, for timescales $t \gtorder 10\, t_{dyn}(r_h)$, the adiabatic approximation was satisfied for the resolution of the models. The use of $N$-body modeling also showed the final distributions after adiabatic growth was stable; stability is not guaranteed by the adiabatic growth process, nor is there a general analytic criterion for stability of arbitrary .
Adiabatic growth formally also implies reversibility. Sigurdsson & Hernquist (unpublished) experimented with numerical models in which a central black hole grown adiabatically in a spherical stellar distribution was [*removed*]{} adiabatically. The original distribution was in fact recovered to within the resolution of the models.
In general, violent formation can lead to either galaxies with constant-density cores (e.g., Lynden-Bell 1967; van Albada 1982; Norman, May, & van Albada 1985; Burkert, this volume) or singular profiles (e.g., Aarseth 1966; Fillmore & Goldreich 1984; Bertschinger 1985; Navarro et al. 1997).
Stiavelli (1998) and Ullio (2001) explored non-adiabatic growth with a pre-existing black hole and found results that did not deviate strongly from the case of adiabatic growth. More recently, MacMillan & Henriksen (2002) suggested that non-adiabatic accretion of dark matter might account for the $M_{\rm BH}-\sigma$ relation, which is not explained by a simple adiabatic compression of the dark matter halo (Dubinski & Carlberg 1991). Adiabatic compression of the dark matter in the inner regions by the formation of a central black hole is potentially interesting, as it can lead to increased rates of dark matter accretion onto the black hole, and to higher rates of dark matter self-interaction, for models in which such interactions may occur (Gondolo & Silk 1999; Ostriker 2000).
Sigurdsson (1995) also found that steep initial density cusps were vulnerable to violent disruption by the “wandering” of the central black hole. Black hole mergers will also efficiently destroy steep stellar cusps around a black hole (Makino & Ebisuzaki 1996; Quinlan & Hernquist 1997; Faber 1997; Milosavljević & Merritt 2001; Zier & Biermann 2001; Hemsendorf, Sigurdsson, & Spurzem 2002; Ravindranath, Ho, & Filippenko 2002).
Non-spherical Systems
---------------------
The obvious next approximation beyond spherical (isotropic and anisotropic) models is to consider axisymmetric ones. A number of families of two- and three-integral axisymmetric models exist in the literature (e.g., Evans 1993; Hunter & Qian 1993; Kuijken & Dubinski 1994; Qian 1995; Gebhardt 2000b; Lynden-Bell 2002).
Van der Marel (1997a) constructed a detailed model for the central black hole in M32, and van der Marel, Sigurdsson, & Hernquist (1997b) ran a numerical model, using techniques developed for adiabatic growth simulations, to demonstrate its stability. A concern remains that such models may be unstable to $m=1$ modes, which are typically suppressed in numerical simulations (if not, they can arise spontaneously through numerical artifacts, which make it difficult in general to identify physical instabilities). Lee & Goodman (1989) modeled adiabatic growth of black holes in approximate rotating, isothermal axisymmetric models. Rather interestingly, they found that the rotation curve rises more rapidly than the dispersion curve, but not enough to account for the observed high rotation in the inner regions of some systems. A more general exploration of adiabatic growth in non-rotating axisymmetric models was done by Sigurdsson & Hernquist (unpublished, see Fig. 2). They found that axisymmetric models are quite similar to spherical models, particularly in that the tangential anisotropy is induced in the cusp that the density profile becomes rounder at small radii (Fig. 2). Leeuwin & Athanassoula (2000) simulated adiabatic growth in Lynden-Bell (1962) models and obtained results consistent with those of Lee & Goodman (1989), including rounding of the inner density profile and a significant rise in the rotation velocity inside the cusp, consistent with the tangential polarization of the central black hole.
### Triaxial systems
Real galaxies are generally not spherical or axisymmetric, but triaxial (Binney 1976; Franx, Illingworth, & de Zeeuw 1991; Ryden 1992, 1996; Tremblay & Merritt 1995; Bak & Statler 2000). We expect triaxial galaxies to form from general cosmological initial conditions (e.g., Norman 1985; Dubinski & Carlberg 1991) and from galaxy mergers (Barnes 1988, 1992; Hernquist 1992, 1993). Exact analytic models exist for triaxial galaxies with cores (Schwarzschild 1979; de Zeeuw 1985; Statler 1987; van de Ven 2002). Observationally, we also see that the density profiles of the spheroidal component of galaxies generally continue to rise toward the center, with $0.5 \ltorder \gamma \ltorder 2.3$ (e.g., Lauer 1995; Gebhardt 1996; Faber 1997; Ravindranath et al. 2001). There are also observed correlations between the cusp slope $\gamma$ and the global properties of the galaxy, including shape in the form of boxy or disky isophotes (e.g., Faber 1997).
Historically, dynamical arguments suggest that the presence of a strong central cusp ($\gamma > 1$) induces chaos in the orbit families thatb populate the galaxy, driving the system away from strong triaxiality (e.g., Gerhard & Binney 1985; Norman 1985; Merritt & Valluri 1996; Merritt & Quinlan 1998; Merritt 1997, 1999). The argument is that central cusps or central point masses scatter the box orbits that support triaxiality in galaxies, inducing a population of chaotic orbits which drive figure evolution toward axisymmetry (Miralda-Escude & Schwarzschild 1989; Lees & Schwarzschild 1992; Fridman & Merritt 1997; Valluri & Merritt 1998). Hence, central supermassive black holes should preclude the presence of triaxiality at small radii, and might drive global figure evolution of the system (Norman 1985; Merritt & Quinlan 1998). This is potentially very important because triaxial potentials support fueling of the central black hole through material falling into it on box orbits (e.g., Norman & Silk 1983; Valluri & Merritt 1998) or by gas traveling on intersecting orbits that drive dissipation and inflow, thus providing a direct link between the dynamics in the center of the galaxy and its global properties. In the extreme case of disk systems, analogous instabilities exist (e.g., Hasan & Norman 1990; Sellwood & Valluri 1997).
### Adiabatic growth and triaxiality
Holley-Bockelmann (2001; see also Sigurdsson 1997b, 1998) showed that applying numerical adiabatic growth techniques to “squeeze” an initially spherically symmetric cuspy DF could produce a stable, stationary, cuspy triaxial configuration with well-characterised phase-space properties. A key aspect of the models is that they contain a central cusp of near-constant slope and near-constant axis ratios with significant triaxiality at all radii resolved by the models (Holley-Bockelmann et al. 2001, 2002). Galaxies with density cusps support different stellar orbits than, for instance, $\gamma=0$ core models (Gerhard & Binney 1985; Gerhard 1986; Pfenniger & de Zeeuw 1989; Schwarzschild 1993; de Zeeuw 1995; Merritt 1999; Holley-Bockelmann et al. 2001). The set of models thus produced provide a starting point for investigation of the adiabatic growth of central black holes in cuspy, triaxial potentials. A black hole is then grown using the previously developed numerical $N$-body techniques (Sigurdsson 1995; Holley-Bockelmann 2002).
Following Holley-Bockelmann (2001, 2002), consider a black hole grown in a triaxial Hernquist model with initial cusp slope $\gamma = 1$. As the black hole grows, both the cusp slope $\gamma$ and central velocity dispersion $\sigma_p$ increase, as in spherical and axisymmetric models. The cusp settles to an equilibrium value $\gamma \simeq 2.05$, measured at projected ellipsoidal radius $Q = 10^{-1.3}$, with projected central dispersion $\sigma_p \simeq 0.7$, measured at projected ellipsoidal radius $Q = 10^{-2.3}$. These results are characteristic of adiabatic black hole growth in cuspy galaxies and can be compared both to analytic estimates for adiabatic black hole growth in a spherical $\gamma=1.0$ model, which predict $\gamma = 7/3$ and $\sigma_p=0.75$ (Quinlan 1995), and to the results from $N$-body simulations where $\gamma \approx 2.2$ and $\sigma \approx 0.65$ (Sigurdsson 1995). The fact that the measured cusp slope is less than the analytic value is to be expected, since the cusp slope is measured over a finite radial range near the center, and it is not the asymptotic $q=0$ value.
As the black hole grows, the inner regions become rounder (Fig. 3[*b*]{}); the central 10% of the mass, corresponding to an ellipsoidal radius $q$ $=\sqrt{ x^2 + (y/b)^2 + (z/c)^2}< 0.1$, is close to spherical with axis ratios $a:b:c= 1.0:0.95:0.92$. The shape evolution in the outer regions is much less dramatic. Following the growth of the black hole, the model exhibits a marked shape gradient, becoming more strongly triaxial with increasing radius. Despite the nearly axisymmetric shape at the center, the inner region is still triaxial enough to influence the stellar-orbital dynamics (Statler 1987; Hunter & de Zeeuw 1992; Arnold, de Zeeuw, & Hunter 1994).
The final state of this model features several hallmarks of a black hole-embedded triaxial figure. Figure 3 shows the properties of this object as a function of ellipsoidal radius $q$ at $T=40$ (12.8 $t_{\rm dyn}$ at $q=1$), well after the model black hole has stopped growing. Figure 3[*a*]{}shows the $\gamma\approx 2$ density cusp induced by the black hole inside $\log q = -1$. At a larger radii $\log q > -1$, however, this plot demonstrates that the system retains the original Hernquist density profile. Figure 3[*b*]{} shows explicitly the strong shape gradient in the model. Inside $r_h$, both the projected and intrinsic velocity dispersions exhibit a strong central cusp (panels [*c*]{} and [*d*]{}). In the outskirts, where the model maintains its triaxiality, the projected velocity distributions follow $\sigma_x>\sigma_y>\sigma_z$, in accord with a triaxial model where $a>b>c$. However, inside the cusp the projected velocity dispersions are commensurate. Interestingly, the anisotropy parameter, $\beta = 1-\langle v_t^2\rangle / \langle 2 v_r^2\rangle$, becomes negative near the black hole. This is consistent with models of stellar orbits around a black hole that is adiabatically grown, where $\beta=-0.3$ (Goodman & Binney 1983; Quinlan et al. 1995). Exterior to the black hole’s radius of influence, the system is radially anisotropic ($\beta>0$), as expected for a triaxial galaxy.
Poon & Merritt (2001, 2002), using Schwazschild’s orbital-assembly technique, have now also found models with triaxiality at small radii in the presence of a central black hole and a surrounding cusp.
Clearly, it is possible for some significant triaxiality to persist both in the presence of a central cusp, and, more importantly, in the presence of a central black hole.
### Chaos
If a significant fraction of the orbits in triaxial models containing central black holes become chaotic, then by the ergodic theorem the shape of the distribution must evolve toward sphericity (possibly halting when axisymmetry is reached). It is clear that the onset of chaos in the most tightly bound orbit families leads to a rapid change in the inner structure of the model galaxies. In the outer regions, orbits stay regular even after repeated passages near the potential center. It is possible that many of these orbits are actually chaotic orbits that are “sticky” (Siopis & Kandrup 2000), with a very long diffusion timescale. The course grainedness of the numerical model potential seems to argue against this explanation; a course-grained potential effectively creates holes in the Arnold web (Arnold 1964) through which an otherwise confined orbit may escape. Figure 4 illustrates what is probably happening in these models; strong scattering is taking place, inducing some chaos, but triaxiality is sustained by the persistence of resonant boxlet orbits that scatter between each other, rather than into true chaotic orbits.
There are two important issues to be explored here.
- When a spherical (or axisymmetric) model is squeezed adiabatically into a triaxial configuration, one of the implicit assumptions of adiabatic growth is violated. The evolution of the potential has discretely broken a symmetry underlying the second and third integrals of motion, and, incidentally, the reversibility of the process is destroyed. However, the action is still conserved, at least approximately, so the must bifurcate, leaving an excluded region of phase space. In 2-D this region would be forbidden; in 3-D other bifurcating branches can cross-over into the newly created vacant region of phase space, but do not in general fill it. By construction, this technique leaves vacant islands in phase space, and consequently we reach stable and stationary triaxial configurations despite the presence of the black hole. Numerically these are robust solutions, but they are not guaranteed to be robust physically. It is possible that small amounts of relaxation or potential fluctuation could rapidly refill these vacated phase-space regions, leading to boxlet–chaos transitions, breaking the dynamical equilibrium constructed for boxlet–boxlet transitions. Some of this is seen through chaos induced by numerical scattering in the models.
We cannot yet be sure that our triaxial adiabatic solutions are physically robust ones achieved by natural systems, as distinct from mathematical curiosities; nevertheless, they are potentially very interesting solutions for triaxial systems.
- We also do not understand well how a transition to chaos occurs in these systems. The scattering by the central singular potential, in and of itself, need not induce chaos. After all, perturbed orbits in Keplerian potentials are regular. Some insight can be gained by considering a toy 2-D model (to be compared with the dynamics in the principal plane of a triaxial system).
Following Devaney (1982), consider a homogenous, [*anisotropic*]{} potential of degree $k$, $\Phi (r,\psi) \propto r^{-k}$. There are three special values of $k = \{0,1,2\}$, the first two corresponding to the isothermal and Keplerian potential, respectively. We consider the characteristic exponents in the linearised theory for 2-D orbits and explore the effects of varying $k$ and the degree of anisotropy. Solving for the characteristic exponent $\lambda$, we find
$$\lambda (k,r) = {1\over 2} [ ({1\over 2}k - 1)v_r - [ ({1\over 2}k - 1)^2 v_r^2 - 4\Phi''(\psi)]^{1/2} ]$$
where $\Phi''$ is the second angular derivative of the potential, which is by hypothesis self-similar (i.e., the shape of the isopotential contours is independent of radius). Clearly, realistic galaxy models are not homogenous, but at large and small radii they well approximate a homogenous potential, and the dynamics, in particular the orbit divergences, are due to [*local*]{} potential gradients.
Instability formally occurs when $\lambda$ is imaginary, so the critical points occur when $ [({1\over 2}k - 1)^2 v_r^2 - 4\Phi''(\psi)] = 0$. Using Poisson’s equation, $\Phi'' = 4\pi r^2 \rho - 2v_c^2 - k(k-1)\Phi$, where, by hypothesis, $\Phi''(\psi)$ is independent of radius. We therefore conclude, that for Keplerian and isothermal potentials orbits in this model go chaotic at large $r$, not crossing through the center. For $k=2$, appropriate for the outer regions of galaxy models, orbits that are chaotic anywhere are chaotic everywhere.
If the dynamics of the toy 2-D homogenous model are a good indicator, then the onset of chaos in triaxial models occurs not at small radii, but in the transition region between the inner cusp and the region outside $r_h$.
### Uses of triaxial models
Much work remains to be done. The following are some implications worth exploring.
- The gas dynamics on intermediate scales and possibilities for AGN fueling may depend strongly on even mild triaxialities on small scales.
- The dynamical evolution of merging dwarf galaxies and the fate of low-mass black holes merging with massive galaxies may be sensitive to triaxiality persisting as the central black hole becomes more massive. Dynamical friction processes may be expedited through centrophilic orbits in triaxial potentials.
- On small scales, triaxiality inside $r_h$ may promote rapid loss-cone refilling and be critical for sustaining high tidal-disruption rates and the influx of low-mass compact objects coalescing with the central massive black hole (Sigurdsson 2003).
Conclusions
-----------
Adiabatic growth models provide valuable physical insights into the dynamical processes and interactions of central massive black holes with their surroundings. Over the last 30 years, a broad range of physically robust results on the dynamical influence of the black hole on the surrounding stellar population have contributed to our confidence in the reality of supermassive black holes and helped provide strong quantitative constraints on black hole masses. Additional physical insight has been gained in understanding the ever-fascinating subtleties of Newtonian dynamics of many-body systems. New generations of $N$-body models will allow a broader exploration of more realistic, less symmetric systems, more tests of stability and secular evolution, and possibly a deeper understanding of dynamical processes on small and large scales.
[*Acknowledgements:*]{} The author gratefully acknowledges the support of the NSF under grant PHY-0203046, the Center for Gravitational Wave Physics at Penn State, an NSF-supported Physics Frontier Center, and the hospitality of Carnegie Observatories. I would like to thank my collaborators Gerry Quinlan, Lars Hernquist, Roeland van der Marel, Chris Mihos, Colin Norman, and especially Kelly Holley-Bockelmann for her hard work and for letting me use unpublished results.
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|
---
abstract: |
Biclustering involves the simultaneous clustering of objects and their attributes, thus defining local two-way clustering models. Recently, efficient algorithms were conceived to enumerate all biclusters in real-valued datasets. In this case, the solution composes a complete set of maximal and non-redundant biclusters. However, the ability to enumerate biclusters revealed a challenging scenario: in noisy datasets, each true bicluster may become highly fragmented and with a high degree of overlapping. It prevents a direct analysis of the obtained results. Aiming at reverting the fragmentation, we propose here two approaches for properly aggregating the whole set of enumerated biclusters: one based on single linkage and the other directly exploring the rate of overlapping. Both proposals were compared with each other and with the actual state-of-the-art in several experiments, and they not only significantly reduced the number of biclusters but also consistently increased the quality of the solution.
, bicluster enumeration, bicluster aggregation, outlier removal, metrics for biclusters
.
author:
- 'Saullo Oliveira[^1]'
- Rosana Veroneze
- 'Fernando J. Von Zuben'
bibliography:
- 'tese.bib'
title: On bicluster aggregation and its benefits for enumerative solutions
---
Introduction
============
Biclustering techniques aim to simultaneously cluster objects and attributes of a dataset. Each bicluster is represented as a tuple containing a subset of the rows, and a subset of the columns, as long as they exhibit some kind of coherence pattern. There are several kinds of coherence which can be found in a bicluster, and they directly interfere on the mechanism of bicluster identification. As finding all biclusters in a dataset is an NP-hard problem, several heuristics were proposed, such as CC [@Cheng2000] and FLOC [@Yang2003]. Such heuristics may miss important biclusters, and may also return non-maximal biclusters (biclusters that can be further augmented).
In the case of binary datasets, there are a plenty of algorithms for enumerating all maximal biclusters. Some examples are Makino & Uno [@Makino2014], LCM [@Uno2004] and In-Close2 [@Andrews2009]. The enumeration of all maximal biclusters in an integer or real-valued dataset is a much more challenging scenario, but we already have some proposals, such as RIn-Close [@Veroneze2014] and RAP [@Pandey2009].
The drawback of enumerative algorithms, particularly in the context of noisy datasets, is the existence of a large number of biclusters, due to fragmentation of a much smaller number of true biclusters. This is exemplified in one of our experiments, where we take artificial datasets, gradually increment the variance of a Gaussian noise, and get the enumerative result. As shown in Fig. \[fig:qtds\], with enough noise, the enumerative results exhibit an strong increase on the quantity of biclusters. This fragmentation leads to a challenging scenario for the analysis of the results, which can become impractical even in small datasets. In fact, the noise is responsible for fragmenting each true bicluster into many with high overlapping, so that the aggregation of these biclusters is recommended [@Liu04] [@Zhao2005].
We propose a way of aggregating biclusters from a biclustering result that shows a high overlapping among its components, as it is the case when enumerating biclusters in noisy datasets. For this reason, in this paper we will focus on enumerative results, but our proposal can be applied to the result of any algorithm that returns biclusters with high overlapping among them. The formulation is based on the fact that the high overlapping among biclusters may indicate that they are fragments of a true bicluster that should be reconstructed. We propose two different techniques to perform the aggregation, followed by a step that removes elements that should not be part of a bicluster. We performed experiments with three artificial datasets posing different challenges, and two real datasets from distinct backgrounds. We compared our proposals with a bicluster ensemble algorithm, and the merging/deleting steps of MicroCluster [@Zhao2005]. The experimental results show that the aggregation not only severely reduces the quantity of biclusters, but also tends to increase the quality of the solution.
The paper is organized as follows. In Section \[related\], we give the main definitions and discuss the related works in the literature. Section \[proposals\] outlines our proposals. The metrics used to evaluate our proposals will be presented in Section \[metrics\]. In Section \[experiments\], we present the experimental procedure and the obtained results of the experiments. Concluding remarks and future work are outlined in Section \[conclusions\].
Definitions and Related Work {#related}
============================
Consider a dataset $\textbf{A} \in \mathbb{R}^{n \times m}$, with rows $X = \{x_1, x_2, \dots, x_n\}$ and columns $Y = \{y_1, y_2, \dots, y_m\}$. We define a bicluster $B = (B^r, B^c)$, where $B^r \subseteq X$ and $B^c \subseteq Y$, such that the elements in the bicluster show a coherence pattern. A bicluster solution is a set of biclusters represented by $\bar{B} = \{B_i\}_{i=1}^q$, containing $q$ biclusters. A bicluster is maximal if and only if we can not include any other object / attribute without violating the coherence threshold. If a solution contains non-maximal biclusters, the result is redundant because there will be biclusters which are part of larger ones.
Madeira & Oliveira [@Madeira2004] categorized the types of biclusters according to their similarity patterns. They also categorized the biclusters structure in a dataset based on their disposition and level of overlapping. We highlight that biclusters with constant values, constant values on rows, or constant values on columns are special cases of biclusters with coherent values, and we will focus our attention on the latter, due to its generality. For a comprehensive survey of biclustering algorithms, the reader may refer to [@Madeira2004] and [@Tanay2005].
The overlapping between two biclusters $B$ and $C$ is an important concept in this work, and is defined as: $$ov(B, C) = \frac{|B^r \cap C^r \times B^c \cap C^c|}{min(|B^r \times B^c|, |C^r \times C^c|)}.$$
Now we shall proceed to the aggregation proposals in the literature. It is important to highlight that, when aggregating two maximal biclusters, the coherence threshold will be violated. Otherwise, the biclusters would not be maximal.
MicroCluster Aggregation {#cap:ens:aggregation:mc}
------------------------
MicroCluster [@Zhao2005] is an enumerative proposal that has two additional steps after the enumeration. These steps have the task of deleting or merging biclusters which are not covering an area much different from other biclusters. The first is the deleting step. If we find a bicluster such that the ratio of its area that is not covered by any other bicluster, by its total area, is less than a threshold $\eta$, it can be removed. The second step is the merging one. Let us consider two biclusters and generate a third one with the union of rows and columns of the previous two. If the ratio of the area of the third bicluster that is not covered by any of the previous two, by its total area, is less than a threshold $\gamma$, we can aggregate the two biclusters into this third one. In this method of aggregation, non-maximal biclusters will be removed in the deleting step, thus not interfering in the final result. For more details, please refer to Zhao & Zaki [@Zhao2005].
Aggregation Using Triclustering
-------------------------------
Triclustering was proposed by Haczar & Nadif [@Hanczar2012] as a biclustering ensemble algorithm. First, they transform each bicluster into a binary matrix. After that, they propose a triclustering algorithm to find the $k$ most relevant biclusters. As they were able to improve the biological relevance of biclustering for microarray data [@Hanczar2011b], we will use this method as a contender in this paper. One major point in ensemble is that we want to combine the results reinforcing the biclusters that seem to be important for several components, and discarding the ones that may come from noise. Due to the way the triclustering algorithm handles the optimization step, non-maximal biclusters can interfere in the final results.
Bicluster aggregation is slightly different from bicluster ensemble. While on ensemble tasks we discard biclusters that seem unimportant and combine the ones that contribute the most for the solution, in bicluster aggregation we never discard any bicluster. Given this characteristic, the bicluster ensemble solution is expected to show a high *Precision* with an impacted *Recall* (see Section \[metrics\]), as it eliminates biclusters.
Other Aggregation Methods
-------------------------
Gao & Akoglu [@GAO2014] used the principle of Minimum Description Length to propose CoClusLSH, an algorithm that returns a hierarchical set of biclusters. The hierarchical part can be seen as an aggregation step. This step is done based on the LSH technique as a hash function. Candidates hashed to the same bucket are then aggregated until no merging improves the final solution. Their work is focused in finding biclusters in a checkerboard structure, that does not allow overlapping, thus being not suitable for the kind of problem we are dealing with.
Liu *et al*. [@Liu04] proposed OPC-Tree, a deterministic algorithm to mine Order Preserving Clusters (OP-Clusters), a general case of Order Preserving Sub Matrices (OPSM) type of biclusters. They also have an additional step for creating a hierarchical aggregation of the OP-Clusters. The Kendall coefficient is used to determine which clusters should be merged and in which order the objects should participate in the resultant OP-Cluster. The highest the Rank Correlation using the Kendall coefficient, the highest the similarity between two OP-Clusters. The merging is allowed according to a threshold that is reduced in a level-wise way. OPC-Tree considers the order of the rows in the bicluster. In this work, we are dealing with biclusters of coherent values. In this case, a perfect coherent values bicluster keeps the order of its rows and the hierarchical step of OPC-Tree would be able to be used in this case as well. But we are considering noisy datasets, in which this assumption probably will not hold, thus the hierarchical step of OPC-Tree is not suitable for the problem we are dealing with.
New Proposals for Aggregation {#proposals}
=============================
Aggregation with Single Linkage {#aggregation:sl}
-------------------------------
Our first proposal receives as input a biclustering solution $\bar{B}$, from enumeration or from a result presenting high overlapping among its components. With this solution, we transform each bicluster into a binary vector representation as follows: Given the dimensions of the dataset $\textbf{A} \in \mathbb{R}^{n \times m}$, each bicluster will be a binary vector $\textbf{x}$ of length ${n + m}$. For a bicluster $B$ transformed into the binary vector $\textbf{x}$, the first $n$ positions represent the rows of the dataset $\textbf{A}$ and if the bicluster contains the $i$th row, $\textbf{x}_i = 1$, otherwise $\textbf{x}_i = 0$. The last $m$ positions represent the columns of the dataset $\textbf{A}$ and if the bicluster contains the $i$th column, $\textbf{x}_{n+i} = 1$, otherwise $\textbf{x}_{n+i} = 0$. After this transformation, we use the Hamming distance to apply the single linkage clustering on the existing biclusters. Notice that the Hamming distance on this transformation will just count how many rows and columns are different among the two biclusters. In this case, a non-maximal bicluster may be distant from the bicluster that covers its maximal area, thus impacting the quality of the results of this method of aggregation. In this case, it is necessary that this proposal receives a biclustering solution $\bar{B}$ containing only maximal biclusters.
After choosing a cut on the dendrogram, we aggregate all biclusters that belong to a junction using the function *aggreg*, defined as: $$aggreg(B, C) = (B^r \cup C^r, B^c \cup C^c),
\label{eq:aggreg}$$ that is simply the union of rows / columns of the biclusters. It is important to note that the *aggreg* function is associative, since it is based on the union operation. Moreover, we want to highlight that the direct union of rows / columns may include elements that should not be part of a bicluster. In Section \[aggregation:outlier\] we will present a way to remove rows / columns that may be interpreted as outliers.
Aggregation by Overlapping {#aggregation:by_ov}
--------------------------
It seems intuitive to aggregate the biclusters with an overlapping rate above a defined threshold. This proposal is based on the aggregation by pairs: while having two biclusters with an overlapping rate higher than a pre-determined threshold $th$, we remove them from the set of biclusters, and include the result of the function *aggreg*, defined on Eq. \[eq:aggreg\], taking these two biclusters as the arguments.
Let $B, C, D$, and $E$ be biclusters. Note that for $D = aggreg(B,C)$, $ov(D,E) \geq ov(B,E)$ and $ov(D,E) \geq ov(C,E)$. So, for all biclusters $E$ where $ov(B,E) \geq th$ or $ov(C,E) \geq th$, we have $ov(D,E) \geq th$. For this reason, the order of the aggregation does not interfere on the final result. It is also important to note that the new bicluster $D$ can have $ov(D,E) \geq th$, for some bicluster $E$ where $ov(B,E) < th$ and $ov(C,E) < th$. In this aggregation proposal, maximal biclusters will properly merge with non-maximal biclusters.
Outlier Removal {#aggregation:outlier}
---------------
After aggregating the results, we need to process each final bicluster to look for objects and / or attributes that may be interpreted as outliers. In this work, this step will always be executed after the aggregation using any of our two proposals.
Let $B = (B^r, B^c)$ be an aggregated bicluster, with $|B^r| = o, |B^c| = p$. We define a participation matrix $\textbf{P} \in \mathbb{Z}^{o \times p}$, where each element $p_{ij}$ indicates the quantity of biclusters in which this element takes part in $B$. For example, if an element is part of 15 biclusters that compose $B$, then its value on the $\textbf{P}$ matrix will be 15.
So, we will explain the process of outlier removal with the help of Figure \[fig:outlier\]. We have two steps of outlier removal: one for the objects, the other for the attributes. To remove possibly outlier objects, we take the mean and the standard deviation of all columns on the participation matrix **P**. The left side of Figure \[fig:outlier\] illustrates this step. After that, we check the values of each element of the columns. If the value is less than the mean minus one standard deviation, then we check this element as a potential outlier. In Figure \[fig:outlier\], we can see that the entire first row was checked as potential outlier because $1 < 7.75 - 4$. If we mark the entire row as a potential outlier, it is removed from the bicluster. In our example, that is the case.
We execute the same process for the columns, calculating the mean, standard deviation and checking for potential outliers on the rows. We remove the column if it is entirely marked as a potential outlier.
Metrics for Biclustering {#metrics}
========================
In this paper we will use only external metrics, except for the Gene Ontology Enrichment Analysis (GOEA). External metrics compare a given solution with a reference one. For an extensive comparison of external metrics for biclustering solutions, the reader may refer to [@Horta2014].
The Gene Ontology Project [^2] (GO) is an initiative to develop a computational representation of the knowledge of how genes encode biological functions at the molecular, cellular and tissue system levels. The GOEA compares a set of genes with known information. For example, given a set of genes that are up-regulated under certain conditions, an enrichment analysis will find which GO terms are over-represented (or under-represented) using annotations for that gene set[^3]. This method is commonly used to analyze results from biclustering techniques on microarray gene expression datasets.
*Precision*, *Recall* and *F-score* are often used on information retrieval for measuring binary classification [@Salton1971]. If we take pairs of elements, we can extend these metrics to evaluate clustering / biclustering solutions with overlapping. The pairwise definition of *Precision* and *Recall* can be found in [@Menestrina2009]. It is important to highlight that these metrics do not consider the quantity of biclusters. *Pairwise Precision*, or just *Precision* for simplicity, is the fraction of retrieved pairs that are relevant; while *Pairwise Recall*, or just *Recall* for simplicity, is the fraction of relevant pairs that are retrieved. The *F-score* is the harmonic mean of *Precision* and *Recall*.
Clustering Error (*CE*) is an external metric that considers the quantity of biclusters in its evaluation. This metric severely penalizes a solution with more biclusters than the reference, thus not being recommended for evaluating enumerative results. The definition and more details can be found in [@Horta2014].
We propose the difference in coverage, that measures what the reference biclustering solution covers and the found biclustering solution does not cover, and vice versa. Although very similar, when compared with the pairwise definitions of *Precision* and *Recall*, this metric gives a more intuitive idea of how two solutions cover distinct areas of the dataset. It also can be computed much faster. Let $\cup_{\bar{B}} = \bigcup B_i^r \times B_i^c$ be the usual union set of a biclustering solution $\bar{B}$. Let $\bar{B}$ and $\bar{C}$ be the found and the reference biclustering solutions, respectively. Then the difference in coverage is given by: $$dif\_cov(\bar{B}, \bar{C}) = \frac{|\cup_{\bar{B}} - \cup_{\bar{C}}| + |\cup_{\bar{C}} - \cup_{\bar{B}}|}{m \times n}.
\label{eq:diff_cov}$$
We will use this measure to verify how different an aggregated solution is from the enumerative one.
Experiments
===========
In our experiments, we employed three artificial datasets: *art1*, *art2*, and *art3*; and two real datasets: GDS2587 and *FOOD*. We designed the artificial datasets to present different scenarios with increasing difficulty. They have 1000 objects and 15 attributes. Each entry is a random integer, drawn from a discrete uniform distribution on the set {1, 2, ..., 100}. Then we inserted: 5 bicluster arbitrarily positioned and without overlapping on *art1*; 5 bicluster arbitrarily positioned and with a similar degree of overlapping on *art2*; and 15 bicluster arbitrarily positioned and with different degrees of overlapping on *art3*.
For each bicluster, the quantity of objects was randomly drawn from the set $\{50, \dots, 60\}$, and the quantity of attributes was randomly drawn from the set $\{4, 5, 6, 7\}$. To insert a bicluster, we fixed the value of the first attribute and obtained the values of the other attributes by adding a constant value to the first column. This characterizes biclusters of coherent values. This constant value was randomly drawn from the set $\{-10, -9, \dots, -1, 1, \dots, 9, 10\}$.
*GDS2587*[^4] is a microarray gene expression dataset, with 2792 genes and 7 samples, collected from the organism *E. coli*. We removed every gene with missing data in any sample, and the data was normalized by mean centralization, as usual in gene expression data analysis [@Prelic2006]. In this dataset we aim to validate our contribution when devoted to microarray gene expression data analysis, as it is considered a relevant application of biclustering methods.
*FOOD*[^5] is a dataset with 961 objects, which represent different foods, and 7 attributes, which represent nutritional information. As the values of each attribute are in different ranges, we used the same pre-processing as Veroneze *et al*. [@Veroneze2014]. In this dataset our goal is to illustrate the usefulness of bicluster aggregation in a different scenario and to verify if the aggregation leaves uncovered areas that the enumeration has covered at first.
Experiments on Artificial Datasets
----------------------------------
Our goal is to verify the impact of noise in the enumeration of biclusters, and how the aggregation can improve the quality of the final results. To this end, we will add a Gaussian noise with $\mu = 0$ and $\sigma \in \{0, 0.01, \dots, 1\}$, to each dataset, and then run the RIn-Close algorithm. This procedure will be repeated for 30 times and all reported values will be the average of this 30 executions. We will set RIn-Close to mine coherent values biclusters, with at least 50 rows and 4 columns. Also, we will use crescent values for $\epsilon$ due to the importance of the parameter. If $\epsilon$ is too small, we may miss important biclusters expressing more internal variance. If $\epsilon$ is too high, the biclusters may include unexpected objects or attributes.
As we know the biclusters, we will use *Precision*, *Recall* and *F-score* to assess the quality of the results after the enumeration. After that, we will perform the aggregation on the results with the value of $\epsilon$ that led to an initial *Precision* closest to 0.85. This value was chosen because if the *Precision* is too low, it means that the $\epsilon$ value is allowing too many undesired objects or attributes in the enumerated biclusters. In this case, the aggregation may not improve the quality of the final results because their input is not of good quality. If the *Precision* is too high, we will only be able to see improvements in the reduced quantity of biclusters, but the aggregation may increase the *Precision* too.
We will consider the following algorithms as contenders:
- We set $k$ to the true number of biclusters. The authors supplied the code for this algorithm.
- To parameterize this algorithm, we ran a grid search with the values in the set ${0.15, 0.1, 0.05}$, getting 9 results for each run. Also, as the aggregation step of the algorithm is composed of two steps, merging and deleting, we ran each experiment twice: with the merging step first (MD) and with the deleting step first (DM). Unless we want to draw attention to some particular fact, we will report only the best result. The authors supplied the code for this algorithm [^6].
- We cut the dendrogram with the proper quantity of biclusters: for *art1* and *art2*, 5 biclusters; for *art3*, 15 biclusters.
- We tested several values for the rate of overlapping.
After getting the results for all executions of the listed algorithms, we will choose the best result from each one and compare them using the *CE* metric.
Figure \[fig:qtds\] shows the quantity of enumerated biclusters on the artificial datasets, for several values of $\epsilon$.
[ \[fig:qtds:art1\] ]{}
[ \[fig:qtds:art2\] ]{}
[ \[fig:qtds:art3\] ]{}
In all datasets, for every value of $\epsilon$, the behavior is the same: as the noise increases the quantity of enumerated biclusters starts to increase. In Figures \[fig:qtds:art1\] and \[fig:qtds:art2\], we know that the real quantity of biclusters is 5, but when the noise increases, the enumerated quantity reaches approximately 800 biclusters, depending on the value of $\epsilon$. In Figure \[fig:qtds:art3\], we can see that the quantity of biclusters reaches high values too. At some level of noise, the number of biclusters starts to decrease to a point that the algorithm is not able to find any bicluster.
[ \[fig:art1\_precision\] ]{}
[ \[fig:art2\_precision\] ]{}
[ \[fig:art3\_precision\] ]{}
[ \[fig:art1\_recall\] ]{}
[ \[fig:art2\_recall\] ]{}
[ \[fig:art3\_recall\] ]{}
In Figure \[fig:prec\_rec\], we can see the quality of the enumeration without considering the quantity of biclusters.
As we can see in Figure \[fig:art1\_recall\], the noise has almost no interference in the recall for *art1*. It means that this dataset has biclusters very well defined, that even with some noise they are not missed. On the other hand, when the variance of the noise is too low, Figure \[fig:art1\_precision\] shows that the found biclusters contains more elements than expected. It is happening because the parameter $\epsilon$ is high, allowing some elements to be part of the biclusters even without being part of the original solution. As the noise increases, less of these intruder elements are going to satisfy the $\epsilon$ restriction to be thus included in some bicluster. In this dataset, the effect of the noise were not so severe on the quality, given that the recall started to decrease only when the variance of the noise was close to 1.
In dataset *art2* the effect of noise can be better observed. Figure \[fig:art2\_recall\] shows that the noise starts to affect the solutions very early. When $\epsilon = 3$, the recall starts to decrease very soon, with $\sigma \approx 0.5$. However, for more relaxed values of $\epsilon$ we can still see the decrease on the recall. Being the most difficult, dataset *art3* is the most affected by noise. Independently of the value of $\epsilon$, the RIn-Close was not able to find any biclusters after some levels of variance in the noise. For example, when $\epsilon = 2$, after $\sigma \approx 0.4$ the *Precision* gets undefined. This happens because the metric is not defined when the quantity of biclusters is zero. In Figure \[fig:art3\_recall\], we can see that the decline of the recall starts when $\sigma \approx 0.3$ for $\epsilon = 2$.
Now we will discuss the results of the aggregation with the previously listed algorithms. As stated earlier, we will use the results from a value of $\epsilon$ that led to an initial *Precision* close to 0.85. In this case, we have $\epsilon = 6, 4, 3$ for *art1*, *art2* and *art3*, respectively.
[ \[fig:aggreg\_art1:sl\] ]{}
[ \[fig:aggreg\_art1:ov\] ]{}
[ \[fig:aggreg\_art1:ces\] ]{}
Figure \[fig:aggreg\_art1:sl\] shows the quality of the aggregation with single linkage for dataset *art1*. We can see that, with the proper number of biclusters, the aggregation was able to get an almost perfect result. The same thing happened with the aggregation by overlapping, reported in Figure \[fig:aggreg\_art1:ov\]. Figure \[fig:aggreg\_art1:ces\] shows the CE metric for all solutions of aggregation. We can see that our proposals were capable of producing the best performance on this dataset.
[ \[fig:aggreg\_art2:sl\] ]{}
[ \[fig:aggreg\_art2:ov\] ]{}
[ \[fig:aggreg\_art2:ces\] ]{}
Figure \[fig:aggreg\_art2:sl\] shows the quality of the aggregation with single linkage for the dataset *art2*. This time, the solution was close to the maximum achievable performance, but not so close as it was in *art1*. Figure \[fig:aggreg\_art2:ov\] shows the quality of the aggregation by overlapping for the same dataset. The quality of this solution is very similar to the one obtained with single linkage. Figure \[fig:aggreg\_art2:ces\] shows the CE metrics obtained by all the methods of aggregation. Again, our proposals outperformed the other two algorithms.
[ \[fig:aggreg\_art3:sl\] ]{}
[ \[fig:aggreg\_art3:ov\] ]{}
[ \[fig:aggreg\_art3:ces\] ]{}
Figures \[fig:aggreg\_art3:sl\] and \[fig:aggreg\_art3:ov\] show the quality of aggregation with single linkage and by overlapping, respectively. We can see that this dataset is more challenging than the previous ones. However, the aggregation was able to significantly reduce the quantity of biclusters, while keeping a good quality. Figure \[fig:aggreg\_art3:ces\] shows the CE metric for all aggregation methods. Initially MicroCluster had a better performance, but our proposals were more robust to noise, getting a better result when $\sigma \gtrapprox 0.4$.
The aggregation was not only able to reduce the quantity of biclusters of the enumeration, but also improve the quality of the final result. Now we are going to verify the behavior of the aggregation in real datasets.
Experiments on Real Datasets
----------------------------
We will start with the *GDS2587* dataset by running RIn-Close to enumerate its coherent values biclusters. We set $minRow = 50, minCol = 4$. When $\epsilon < 2.8$ no biclusters were found, and when $\epsilon = 3.0$ the quantity of biclusters was already huge. We found 23, 2.825 and 19.649 biclusters when $\epsilon = 2.8, 2.9,$ and $3.0$, respectively.
Proceeding to the aggregation, Figure \[fig:gds:dendrogram\] shows the dendrograms of the aggregation with single linkage. In this case, the cuts are straightforward, having 2, 4, and 5 clusters respectively. The aggregation by overlapping with a rate of $75\%$ reached the same quantity of biclusters. We used these quantities to parameterize the triclustering algorithm. The results of the aggregation with MicroCluster were very similar, and they depended only on the $\gamma$ parameter. We got 7, 8 and 11 biclusters when $\gamma = 0.15, 0.1,$ and $0.05$, respectively. We will now compare the results with the *gene ontology enrichment analysis*. A bicluster is called ’enriched’ when any ontology term gets a p-value less than 0.01.
When $\epsilon = 2.8$, except for triclustering (only the first bicluster was enriched), all the algorithms returned only enriched biclusters. In fact, the four main enriched terms were always the same, sometimes on different orders but with very close p-values.
[|l|l|l|p[7.3cm]{}|]{} GO Term & p-val & counts & definition\
GO:0044464 & 0.00000000 & 39 / 774 & Any constituent part of a cell, the basic structural and functional unit of all organisms...\
GO:0044444 & 0.00000011 & 19 / 608 & Any constituent part of the cytoplasm, all of the contents of a cell excluding the plasma membrane...\
GO:0044424 & 0.00000350 & 19 / 578 & Any constituent part of the living contents of a cell; the matter contained within (but not including) the plasma membrane...\
When $\epsilon = 2.9$, all algorithms returned only enriched biclusters, including triclustering. When $\epsilon = 3$, all algorithms except for triclustering returned only enriched biclusters. Triclustering returned 4 from 5 enriched biclusters.
Table \[tab:go\] shows the main enriched terms of one bicluster from the aggregation by overlapping after outlier removal, when $\epsilon = 2.8$. In this case, the expert should choose which solution fits better the goal of the data analysis.
We will now proceed to the analysis of the *FOOD* dataset. We are going to verify how the aggregation changes the coverage of the dataset when compared to the enumeration. As the aggregation will severely reduce the quantity of final biclusters, it is important to see if it will leave uncovered areas that were previously covered.
We replicated the experiment from Veroneze *et al*. [@Veroneze2014] on this dataset and we will use $\epsilon = 1.25$ as recommended on that work. With $minRow = 48, minCol = 2$ and looking for coherent values biclusters, the quantity of enumerated biclusters for $\epsilon = 1.25$ is 8.676.
Figure \[fig:food:dendrogram\] shows the dendrogram of the aggregation with single linkage. We can see that the cuts between 2 and 7 are acceptable. In fact, cutting in two groups seems the best option, but it may be considered a small quantity of biclusters. As from 4 to 5 the height is more pronounced, for the comparison it seems acceptable to cut the dendrogram on 4 objects. The aggregation by overlapping with a rate of 70% was also able to recover 4 aggregated biclusters.
MicroCluster with the deleting operation first was not able to properly aggregate the biclusters, keeping more than 800 biclusters when $\eta = 0.15$. This behavior is the opposite of what happened with the artificial datasets. There, when the deleting operation came first the results were more effective. Here when the merging operation came first, the aggregation was able to reach 13 to 27 biclusters, depending on the $\gamma$ parameter. As on the artificial datasets the best parameters were $\eta = \gamma = 0.15$, for the comparison we will use this parameterization with the merging operation occurring first, that gives us 13 biclusters. For the triclustering algorithm we set $k = 4$, using insider information from the aggregation by overlapping.
[|l|l|l|l|l|]{} & Single Linkage & MicroCluster & Triclustering & RIn-Close\
By Ov. & 12.50% & 35.50% & 70.31% & 9.1%\
Single Linkage & - & 46.60% & 81.51% & 20.17%\
MicroCluster & - & - & 45.73% & 27.38%\
Triclustering & - & - & - & 61.33%\
Table \[tab:comparison\] shows the comparison of difference in coverage (see Eq. \[eq:diff\_cov\]) between the aggregated solutions with the enumerated solution from RIn-Close. We can see that the triclustering algorithm produces the most distinct solution when compared with the enumerated solution obtained with RIn-Close, exhibiting $\approx 61.33\%$ of difference in coverage. The solutions from the aggregation by overlapping and with single linkage are relatively close to each other, as on the artificial datasets, showing a difference in coverage of $\approx 12.50\%$. At the end, the closest solution to the RIn-Close results was the aggregation by overlapping, with a difference in coverage of $9.1\%$. If we consider that this solution reduced the quantity of biclusters from 8.676 to 4 biclusters, the difference in coverage of only $9.1\%$ seems very promising.
Considering Remarks and Future Work {#conclusions}
===================================
We have compared the performance of our proposals against the most similar proposal in the literature, using artificial and real datasets. The artificial datasets were characterized by a controlled structure of biclusters and were useful to show that the aggregation can severely reduce the quantity of biclusters, while increasing the quality of the final solution. Our proposals outperformed the compared algorithms on the first two artificial datasets, and showed to be more robust to noise on the third artificial dataset.
We also verified if the aggregation could get enriched biclusters in the case of a gene expression dataset. For different values of $\epsilon$ on the RIn-Close algorithm, we could see that the different methods of aggregation reached very similar results. The main challenge of the aggregation with single linkage is to decide where to cut the dendrogram, but as we could see, this task was straightforward on the tested datasets. Except for the triclustering, all aggregations returned only enriched biclusters. And finally, we applied the aggregation methods to the *FOOD* dataset and analyzed how the aggregation changed the coverage area when compared to the enumeration without aggregation. Triclustering led to the most distinct result, and the aggregation by overlapping covered an area very similar to the area covered by the enumeration.
We can conclude that the aggregation is strongly recommended when enumerating all biclusters from a dataset. The aggregation will not only significantly reduce the quantity of biclusters, but will also reduce the fragmentation and increase the quality of the final result. A post-processing step for outlier removal brings additional robustness to the methodology. As a further step of the research, we can adapt our proposals to work on an ensemble configuration. We can also extend this work to deal with time series biclusters, which require contiguous attributes.
The authors would like to thank CAPES and CNPq for the financial support.
[^1]:
[^2]: http://geneontology.org
[^3]: http://geneontology.org/page/about Acessed on 2015, January, 16
[^4]: http://www.ncbi.nlm.nih.gov/sites/GDSbrowser?acc=GDS2587
[^5]: http://www.ntwrks.com/chart1a.htm
[^6]: http://www.cs.rpi.edu/$\sim$zaki/www-new/pmwiki.php/Software/Software
|
---
abstract: 'This paper defines a generalized column subset selection problem which is concerned with the selection of a few columns from a source matrix $A$ that best approximate the span of a target matrix $B$. The paper then proposes a fast greedy algorithm for solving this problem and draws connections to different problems that can be efficiently solved using the proposed algorithm.'
author:
- |
Ahmed K. Farahat, Ali Ghodsi, and Mohamed S. Kamel\
University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada\
`{afarahat,aghodsib,mkamel}@uwaterloo.ca`
bibliography:
- 'References.bib'
title: |
A Fast Greedy Algorithm for\
Generalized Column Subset Selection
---
Generalized Column Subset Selection {#Sec:CSS}
===================================
The Column Subset Selection (CSS) problem can be generally defined as the selection of a few columns from a data matrix that best approximate its span [@Frieze98-Rnd; @Drineas06-Cols; @Boutsidis08-Clust; @Boutsidis09a-CSS; @Boutsidis11a-NearOpt; @Civril12-CSS-Sparse]. We extend this definition to the generalized problem of selecting a few columns from a source matrix to approximate the span of a target matrix. The generalized CSS problem can be formally defined as follows:
[**(Generalized Column Subset Selection)**]{} \[Pr:GenCSSNew\] \[Pr:FS\] Given a source matrix $A\in\mathbb{R}^{m\times n}$, a target matrix $B\in\mathbb{R}^{m\times r}$ and an integer $l$, find a subset of columns ${\ensuremath{\mathcal{L}}}$ from $A$ such that $|{\ensuremath{\mathcal{L}}}| =l$ and $${\ensuremath{\mathcal{L}}}={\arg\min}_{{\ensuremath{\mathcal{S}}}}\:\|B-{P^{\left({\ensuremath{\mathcal{S}}}\right)}}B\|_{F}^{2},$$ where ${\ensuremath{\mathcal{S}}}$ is the set of the indices of the candidate columns from $A$, ${P^{\left({\ensuremath{\mathcal{S}}}\right)}}\in\mathbb{R}^{m\times m}$ is a projection matrix which projects the columns of $B$ onto the span of the set ${\ensuremath{\mathcal{S}}}$ of columns, and ${\ensuremath{\mathcal{L}}}$ is the set of the indices of the selected columns from $A$.
The CSS criterion $\mathbf{F}\left({\ensuremath{\mathcal{S}}}\right)=\|B-{P^{\left({\ensuremath{\mathcal{S}}}\right)}}B\|_{F}^{2}$ represents the sum of squared errors between the target matrix $B$ and its rank-$l$ approximation ${P^{\left({\ensuremath{\mathcal{S}}}\right)}}B$ . In other words, it calculates the Frobenius norm of the residual matrix $F=B-{P^{\left({\ensuremath{\mathcal{S}}}\right)}}B$. Other types of matrix norms can also be used to quantify the reconstruction error [@Boutsidis09a-CSS; @Boutsidis11a-NearOpt]. The present work, however, focuses on developing algorithms that minimize the Frobenius norm of the residual matrix. The projection matrix ${P^{\left({\ensuremath{\mathcal{S}}}\right)}}$ can be calculated as ${P^{\left({\ensuremath{\mathcal{S}}}\right)}}=A_{:{\ensuremath{\mathcal{S}}}} {\left( A_{:{\ensuremath{\mathcal{S}}}}^{T}A_{:{\ensuremath{\mathcal{S}}}} \right)^{-1}} A_{:{\ensuremath{\mathcal{S}}}}^{T} \:,$ where $A_{:{\ensuremath{\mathcal{S}}}}$ is the sub-matrix of $A$ which consists of the columns corresponding to ${\ensuremath{\mathcal{S}}}$. It should be noted that if ${\ensuremath{\mathcal{S}}}$ is known, the term ${\left( A_{:{\ensuremath{\mathcal{S}}}}^{T}A_{:{\ensuremath{\mathcal{S}}}} \right)^{-1}} A_{:{\ensuremath{\mathcal{S}}}}^{T}B$ is the closed-form solution of least-squares problem $T^{*}={\arg\min}_T\left\Vert B-A_{:{\ensuremath{\mathcal{S}}}}T\right\Vert _{F}^{2}$.
A Fast Greedy Algorithm for Generalized CSS
===========================================
Problem \[Pr:GenCSSNew\] is a combinatorial optimization problem whose optimal solution can be obtained in $O{\left(\max{\left(n^lmrl, n^lml^2\right)}\right)}$. In order to approximate this optimal solution, we propose a fast greedy algorithm that selects one column from $A$ at a time. The greedy algorithm is based on a recursive formula for the projection matrix $P^{({\ensuremath{\mathcal{S}}})}$ which can be derived as follows.
\[Lm:Proj\] Given a set of columns ${\ensuremath{\mathcal{S}}}$. For any ${\ensuremath{\mathcal{P}}} \subset {\ensuremath{\mathcal{S}}}$, ${P^{\left({\ensuremath{\mathcal{S}}}\right)}}=P^{\left({\ensuremath{\mathcal{P}}}\right)}+R^{\left({\ensuremath{\mathcal{R}}}\right)}\:, $ where $R^{\left({\ensuremath{\mathcal{R}}}\right)} = E_{:{\ensuremath{\mathcal{R}}}} {\left( E_{:{\ensuremath{\mathcal{R}}}}^{T}E_{:{\ensuremath{\mathcal{R}}}} \right)^{-1}} E_{:{\ensuremath{\mathcal{R}}}}^{T}$ is a projection matrix which projects the columns of $E = A - P^{\left({\ensuremath{\mathcal{P}}}\right)} A$ onto the span of the subset ${\ensuremath{\mathcal{R}}} = {\ensuremath{\mathcal{S}}}\setminus {\ensuremath{\mathcal{P}}}$ of columns.
Define $D=A_{:{\ensuremath{\mathcal{S}}}}^{T}A_{:{\ensuremath{\mathcal{S}}}}$. The projection matrix ${P^{\left({\ensuremath{\mathcal{S}}}\right)}}$ can be written as ${P^{\left({\ensuremath{\mathcal{S}}}\right)}}=A_{:{\ensuremath{\mathcal{S}}}}D^{-1}A_{:{\ensuremath{\mathcal{S}}}}^{T}$. Without loss of generality, the columns and rows of $A_{:{\ensuremath{\mathcal{S}}}}$ and $D$ can be rearranged such that the first sets of rows and columns correspond to ${\ensuremath{\mathcal{P}}}$. Let $S=D_{{\ensuremath{\mathcal{R}}}{\ensuremath{\mathcal{R}}}}-D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{R}}}}^{T}D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{P}}}}^{-1}D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{R}}}}$ be the Schur complement [@lutkepohl1996handbook] of $D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{P}}}}$ in $D$, where $D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{P}}}}=A_{:{\ensuremath{\mathcal{P}}}}^{T}A_{:{\ensuremath{\mathcal{P}}}}$, $D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{R}}}}=A_{:{\ensuremath{\mathcal{P}}}}^{T}A_{:{\ensuremath{\mathcal{R}}}}$ and $D_{{\ensuremath{\mathcal{R}}}{\ensuremath{\mathcal{R}}}}=A_{:{\ensuremath{\mathcal{R}}}}^{T}A_{:{\ensuremath{\mathcal{R}}}}$. Using the block-wise inversion formula [@lutkepohl1996handbook], $D^{-1}$ can be calculated as $$D^{-1}= \left[\begin{array}{cc}
D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{P}}}}^{-1}+D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{P}}}}^{-1}D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{R}}}}S^{-1}D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{R}}}}^{T}D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{P}}}}^{-1} & -D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{P}}}}^{-1}D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{R}}}}S^{-1}\\
-S^{-1}D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{R}}}}^{T}D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{P}}}}^{-1}
& S^{-1}\end{array}\right]$$ Substituting with $A_{:{\ensuremath{\mathcal{S}}}}$ and $D^{-1}$ in ${P^{\left({\ensuremath{\mathcal{S}}}\right)}}=A_{:{\ensuremath{\mathcal{S}}}}D^{-1}A_{:{\ensuremath{\mathcal{S}}}}^{T}$, the projection matrix can be simplified to $$\label{Eq:ProjP2}
\begin{split}
{P^{\left({\ensuremath{\mathcal{S}}}\right)}}=A_{:{\ensuremath{\mathcal{P}}}}D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{P}}}}^{-1}A_{:{\ensuremath{\mathcal{P}}}}^{T}
+\left(A_{:{\ensuremath{\mathcal{R}}}}-A_{:{\ensuremath{\mathcal{P}}}}D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{P}}}}^{-1}D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{R}}}}\right)S^{-1}\left(A_{:{\ensuremath{\mathcal{R}}}}^{T}-D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{R}}}}^{T}D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{P}}}}^{-1}A_{:{\ensuremath{\mathcal{P}}}}^{T}\right) \:.
\end{split}$$ The first term of the right-hand side is the projection matrix $P^{\left({\ensuremath{\mathcal{P}}}\right)}$ which projects vectors onto the span of the subset ${\ensuremath{\mathcal{P}}}$ of columns. The second term can be simplified as follows. Let $E$ be an $m
\times n$ residual matrix which is calculated as: $E=A-P^{\left({\ensuremath{\mathcal{P}}}\right)}A$. The sub-matrix $E_{:{\ensuremath{\mathcal{R}}}}$ can be expressed as $$E_{:{\ensuremath{\mathcal{R}}}}=A_{:{\ensuremath{\mathcal{R}}}}-P^{\left({\ensuremath{\mathcal{P}}}\right)}A_{:{\ensuremath{\mathcal{R}}}} = A_{:{\ensuremath{\mathcal{R}}}}-A_{:{\ensuremath{\mathcal{P}}}}{\left(A_{:{\ensuremath{\mathcal{P}}}}^{T}A_{:{\ensuremath{\mathcal{P}}}}\right)}^{-1}A_{:{\ensuremath{\mathcal{P}}}}^{T}A_{:{\ensuremath{\mathcal{R}}}}=A_{:{\ensuremath{\mathcal{R}}}}-A_{:{\ensuremath{\mathcal{P}}}}D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{P}}}}^{-1}D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{R}}}} \:.$$ Since projection matrices are idempotent, then $P^{\left({\ensuremath{\mathcal{P}}}\right)}P^{\left({\ensuremath{\mathcal{P}}}\right)}=P^{\left({\ensuremath{\mathcal{P}}}\right)}$ and $$E_{:{\ensuremath{\mathcal{R}}}}^{T}E_{:{\ensuremath{\mathcal{R}}}} = {\left(A_{:{\ensuremath{\mathcal{R}}}}-P^{\left({\ensuremath{\mathcal{P}}}\right)}A_{:{\ensuremath{\mathcal{R}}}}\right)}^T {\left(A_{:{\ensuremath{\mathcal{R}}}}-P^{\left({\ensuremath{\mathcal{P}}}\right)}A_{:{\ensuremath{\mathcal{R}}}}\right)}
=A_{:{\ensuremath{\mathcal{R}}}}^TA_{:{\ensuremath{\mathcal{R}}}}-A_{:{\ensuremath{\mathcal{R}}}}^TP^{\left({\ensuremath{\mathcal{P}}}\right)}A_{:{\ensuremath{\mathcal{R}}}}\:.$$ Substituting with $P^{\left({\ensuremath{\mathcal{P}}}\right)}=A_{:{\ensuremath{\mathcal{P}}}}{\left(A_{:{\ensuremath{\mathcal{P}}}}^{T}A_{:{\ensuremath{\mathcal{P}}}}\right)}^{-1}A_{:{\ensuremath{\mathcal{P}}}}^T$ gives $$\begin{split}
E_{:{\ensuremath{\mathcal{R}}}}^{T}E_{:{\ensuremath{\mathcal{R}}}} =A_{:{\ensuremath{\mathcal{R}}}}^TA_{:{\ensuremath{\mathcal{R}}}}-A_{:{\ensuremath{\mathcal{R}}}}^TA_{:{\ensuremath{\mathcal{P}}}}{\left(A_{:{\ensuremath{\mathcal{P}}}}^{T}A_{:{\ensuremath{\mathcal{P}}}}\right)}^{-1}A_{:{\ensuremath{\mathcal{P}}}}^TA_{:{\ensuremath{\mathcal{R}}}}= D_{{\ensuremath{\mathcal{R}}}{\ensuremath{\mathcal{R}}}}-D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{R}}}}^{T}D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{P}}}}^{-1}D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{R}}}} = S \:.
\end{split}$$ Substituting ${\left(A_{:{\ensuremath{\mathcal{P}}}}D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{P}}}}^{-1}A_{:{\ensuremath{\mathcal{P}}}}^{T}\right)}$, ${\left(A_{:{\ensuremath{\mathcal{R}}}}-A_{:{\ensuremath{\mathcal{P}}}}D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{P}}}}^{-1}D_{{\ensuremath{\mathcal{P}}}{\ensuremath{\mathcal{R}}}}\right)}$ and $S$ with $P^{\left({\ensuremath{\mathcal{P}}}\right)}$, $E_{:{\ensuremath{\mathcal{R}}}}$ and $E_{:{\ensuremath{\mathcal{R}}}}^{T}E_{:{\ensuremath{\mathcal{R}}}}$ respectively, Equation (\[Eq:ProjP2\]) can be expressed as $$\begin{split}
{P^{\left({\ensuremath{\mathcal{S}}}\right)}}={P^{\left({\ensuremath{\mathcal{P}}}\right)}} + E_{:{\ensuremath{\mathcal{R}}}}{\left(E_{:{\ensuremath{\mathcal{R}}}}^{T}E_{:{\ensuremath{\mathcal{R}}}}\right)}^{-1}E_{:{\ensuremath{\mathcal{R}}}}^T\:.
\end{split}$$ The second term is the projection matrix $R^{\left({\ensuremath{\mathcal{R}}}\right)}$ which projects vectors onto the span of $E_{:{\ensuremath{\mathcal{R}}}}$. This proves that ${P^{\left({\ensuremath{\mathcal{S}}}\right)}}$ can be written in terms of $P^{\left({\ensuremath{\mathcal{P}}}\right)}$ and $R$ as ${P^{\left({\ensuremath{\mathcal{S}}}\right)}}=P^{\left({\ensuremath{\mathcal{P}}}\right)}+R^{\left({\ensuremath{\mathcal{R}}}\right)}$
------------------------------------------------------------------------
Given the recursive formula for ${P^{\left({\ensuremath{\mathcal{S}}}\right)}}$, the following theorem derives a recursive formula for $\mathbf{F}\left(\mathcal{S}\right)$.
\[Th:RecF\] Given a set of columns ${\ensuremath{\mathcal{S}}}$. For any ${\ensuremath{\mathcal{P}}} \subset {\ensuremath{\mathcal{S}}}$, $ \mathbf{F}\left(\mathcal{S}\right)=\mathbf{F}\left(\mathcal{P}\right)-\left\Vert R^{\left(\mathcal{R}\right)}F\right\Vert _{F}^{2}
\:,$ where $F = B - P^{\left({\ensuremath{\mathcal{P}}}\right)}B$ and $R^{\left({\ensuremath{\mathcal{R}}}\right)}$ is a projection matrix which projects the columns of $F$ onto the span of the subset ${\ensuremath{\mathcal{R}}} = {\ensuremath{\mathcal{S}}}\setminus {\ensuremath{\mathcal{P}}}$ of columns of $E=A - P^{\left({\ensuremath{\mathcal{P}}}\right)}A$
By definition, $\mathbf{F}\left(\mathcal{S}\right)=\left\Vert B-P^{\left(\mathcal{S}\right)}B\right\Vert _{F}^{2}$. Using Lemma \[Lm:Proj\], $P^{\left(\mathcal{S}\right)}B=P^{\left(\mathcal{P}\right)}B+R^{\left(\mathcal{R}\right)}B$. The term $R^{\left(\mathcal{R}\right)}B$ is equal to $R^{\left(\mathcal{R}\right)}F$ as $E_{:{\ensuremath{\mathcal{R}}}}^{T}B = E_{:{\ensuremath{\mathcal{R}}}}^{T}F$. To prove that, multiplying $E_{:{\ensuremath{\mathcal{R}}}}^{T}$ by $F = B -
P^{\left({\ensuremath{\mathcal{P}}}\right)}B$ gives $ E_{:{\ensuremath{\mathcal{R}}}}^{T}F=E_{:{\ensuremath{\mathcal{R}}}}^{T}B-E_{:{\ensuremath{\mathcal{R}}}}^{T}P^{\left({\ensuremath{\mathcal{P}}}\right)}B$. Using $E_{:{\ensuremath{\mathcal{R}}}}=A_{:{\ensuremath{\mathcal{R}}}}-P^{\left({\ensuremath{\mathcal{P}}}\right)}A_{:{\ensuremath{\mathcal{R}}}}$, the expression $E_{:{\ensuremath{\mathcal{R}}}}^{T}P^{\left({\ensuremath{\mathcal{P}}}\right)}$ can be written as $ E_{:{\ensuremath{\mathcal{R}}}}^{T}P^{\left({\ensuremath{\mathcal{P}}}\right)}=A_{:{\ensuremath{\mathcal{R}}}}^{T}P^{\left({\ensuremath{\mathcal{P}}}\right)}-A_{:{\ensuremath{\mathcal{R}}}}^{T}P^{\left({\ensuremath{\mathcal{P}}}\right)}P^{\left({\ensuremath{\mathcal{P}}}\right)}$. This is equal to $0$ as $P^{\left({\ensuremath{\mathcal{P}}}\right)}P^{\left({\ensuremath{\mathcal{P}}}\right)}=P^{\left({\ensuremath{\mathcal{P}}}\right)}$ (an idempotent matrix). Substituting in $\mathbf{F}\left(\mathcal{S}\right)$ and using $F=B-P^{\left(\mathcal{P}\right)}B$ gives $$\mathbf{F}\left(\mathcal{S}\right)=\left\Vert B-P^{\left(\mathcal{P}\right)}B-R^{\left(\mathcal{R}\right)}F\right\Vert _{F}^{2} = \left\Vert F-R^{\left(\mathcal{R}\right)}F\right\Vert _{F}^{2}$$ Using the relation between Frobenius norm and trace, $\mathbf{F}\left(\mathcal{S}\right)$ can be simplified to $$\mathbf{F}\left(\mathcal{S}\right)=\text{tr}\left(\left(F-R^{\left(\mathcal{R}\right)}F\right)^{T}\left(F-R^{\left(\mathcal{R}\right)}F\right)\right)
=\text{tr}\left(F^{T}F-F^{T}R^{\left(\mathcal{R}\right)}F\right)=\left\Vert F\right\Vert _{F}^{2}-\left\Vert R^{\left(\mathcal{R}\right)}F\right\Vert _{F}^{2}$$ Using $\mathbf{F}\left(\mathcal{P}\right)=\left\Vert F\right\Vert _{F}^{2}$ proves the theorem.
------------------------------------------------------------------------
Using the recursive formula for $\mathbf{F}\left(\mathcal{S}\cup\{i\}\right)$ allows the development of a greedy algorithm which at iteration $t$ selects column $p$ such that $$p={\arg\min}_i\:\mathbf{F}\left(\mathcal{S}\cup\{i\}\right)={\arg\max}_i\left\Vert P^{{\left(\left\{ i\right\}\right)}}F\right\Vert _{F}^{2}\:.$$
Let $G=E^TE$ and $H=F^TE$, the objective function $\left\Vert P^{{\left(\left\{ i\right\}\right)}}F\right\Vert _{F}^{2}$ can be simplified to $$\left\Vert E_{:i}\left(E_{:i}^{T}E_{:i}\right)^{-1}E_{:i}^{T}F\right\Vert _{F}^{2}=\text{tr}\left(F^TE_{:i}\left(E_{:i}^{T}E_{:i}\right)^{-1}E_{:i}^{T}F\right)=\frac{\left\Vert F^TE_{:i}\right\Vert ^{2}}{E_{:i}^{T}E_{:i}}=\frac{\left\Vert H_{:i}\right\Vert ^{2}}{G_{ii}}\:.$$ This allows the definition of the following greedy generalized CSS problem.
[**(Greedy Generalized CSS)**]{} \[Pr:GreedyCSS\] At iteration $t$, find column $p$ such that $$p={\arg\max}_i\hspace{1em}\frac{\left\Vert H_{:i}\right\Vert ^{2}}{G_{ii}}$$where $H=F^TE$, $G = E^TE$, $F=B-P^{\left(\mathcal{S}\right)}B$, $E=A-P^{\left(\mathcal{S}\right)}A$ and ${\ensuremath{\mathcal{S}}}$ is the set of columns selected during the first $t-1$ iterations.
For iteration $t$, define ${\ensuremath{\boldsymbol{\delta}}} = G_{:p}$, ${\ensuremath{\boldsymbol{\gamma}}} = H_{:p}$, ${\ensuremath{\boldsymbol{\omega}}} =
G_{:p}/\sqrt{G_{pp}} = {\ensuremath{\boldsymbol{\delta}}}/\sqrt{{\ensuremath{\boldsymbol{\delta}}}_{p}}$ and ${\ensuremath{\boldsymbol{\upsilon}}} =
H_{:p}/\sqrt{G_{pp}} = {\ensuremath{\boldsymbol{\gamma}}}/\sqrt{{\ensuremath{\boldsymbol{\delta}}}_{p}}$ . The vectors ${\ensuremath{\boldsymbol{\delta}}}^{(t)}$ and ${\ensuremath{\boldsymbol{\gamma}}}^{(t)}$ can be calculated in terms of $A$, $B$ and previous ${\ensuremath{\boldsymbol{\omega}}}$’s and ${\ensuremath{\boldsymbol{\upsilon}}}$’s as $$\label{eq:delta_omega}
{\ensuremath{\boldsymbol{\delta}}}^{(t)}=A^{T}A_{:p}-\sum_{r=1}^{t-1}{\ensuremath{\boldsymbol{\omega}}}_{p}^{(r)}{\ensuremath{\boldsymbol{\omega}}}^{(r)}, \:\:\:\:\:\: {\ensuremath{\boldsymbol{\gamma}}}^{(t)}=B^{T}A_{:p}-\sum_{r=1}^{t-1}{\ensuremath{\boldsymbol{\omega}}}_{p}^{(r)}{\ensuremath{\boldsymbol{\upsilon}}}^{(r)}\:.$$ The numerator and denominator of the selection criterion at each iteration can be calculated in an efficient manner without explicitly calculating $H$ or $G$ using the following theorem.
\[Th:Rec\_fg2\] Let ${\ensuremath{\boldsymbol{f}}}_{i}=\left\Vert H_{:i}\right\Vert ^{2}$ and ${\ensuremath{\boldsymbol{g}}}_{i}=G_{ii}$ be the numerator and denominator of the greedy criterion function for column $i$ respectively, ${\ensuremath{\boldsymbol{f}}}=\left[{\ensuremath{\boldsymbol{f}}}_{i}\right]_{i=1..n}$, and ${\ensuremath{\boldsymbol{g}}}=\left[{\ensuremath{\boldsymbol{g}}}_{i}\right]_{i=1..n}$. Then, $$\begin{split}{\ensuremath{\boldsymbol{f}}}^{(t)}&=\Big({\ensuremath{\boldsymbol{f}}}-2\left({\ensuremath{\boldsymbol{\omega}}}\circ\left(A^{T}B{\ensuremath{\boldsymbol{\upsilon}}}-\Sigma_{r=1}^{t-2}\left({\ensuremath{\boldsymbol{\upsilon}}}^{\left(r\right)T}{\ensuremath{\boldsymbol{\upsilon}}}\right){\ensuremath{\boldsymbol{\omega}}}^{^{\left(r\right)}}\right)\right)
+\|{\ensuremath{\boldsymbol{\upsilon}}}\|^{2}\left({\ensuremath{\boldsymbol{\omega}}}\circ{\ensuremath{\boldsymbol{\omega}}}\right)\Big)^{(t-1)},\\
{\ensuremath{\boldsymbol{g}}}^{(t)}
&=\Big({\ensuremath{\boldsymbol{g}}}-\left({\ensuremath{\boldsymbol{\omega}}}\circ{\ensuremath{\boldsymbol{\omega}}}\right)\Big)^{(t-1)}\:,\end{split}$$ where $\circ$ represents the Hadamard product operator.
In the update formulas of Theorem \[Th:Rec\_fg2\], $A^TB$ can be calculated once and then used in different iterations. This makes the computational complexity of these formulas ${O}(nr)$ per iteration. The computational complexity of the algorithm is dominated by that of calculating $A^TA_{:p}$ in (\[eq:delta\_omega\]) which is of ${O}(mn)$ per iteration. The other complex step is that of calculating the initial ${\ensuremath{\boldsymbol{f}}}$, which is ${O}(mnr)$. However, these steps can be implemented in an efficient way if the data matrix is sparse. The total computational complexity of the algorithm is ${O}(\max(mnl, mnr))$, where $l$ is the number of selected columns. Algorithm \[Alg:GenGCSS\] in Appendix A shows the complete greedy algorithm.
Generalized CSS Problems
========================
We describe a variety of problems that can be formulated as a generalized column subset selection (see Table \[tab:GCSS\]). It should be noted that for some of these problems, the use of greedy algorithms has been explored in the literature. However, identifying the connection between these problems and the problem presented in this paper gives more insight about these problems, and allows the efficient greedy algorithm presented in this paper to be explored in other interesting domains.
**Method** **Source**[ ]{} **Target**
--------------------------------------- --------------------- ------------------------------------------------------------------------------------------------------------------------
[Generalized CSS]{} [$A$]{} [$B$]{}
[Column Subset Selection]{} [Data matrix $A$]{} [Data matrix $A$]{}
[Distributed CSS]{} [Data matrix $A$]{} [Random subspace $A\Omega$ ]{}
[SVD-based CSS]{} [Data matrix $A$]{} [SVD-based subspace $U_{k}\Sigma_{k}$]{}
[Sparse Approximation]{} [Atoms $D$]{} [Target vector $\bf{y}$]{}
[Simultaneous Sparse Approximation]{} [Atoms $D$]{} [Target vectors $\left[\bf{y}_{{\left(1\right)}}, \bf{y}_{{\left(2\right)}}, ... \bf{y}_{{\left(r\right)}}\right]$]{}
**Column Subset Selection.** The basic column subset selection [@Frieze98-Rnd; @Drineas06-Cols; @Boutsidis08-Clust; @Boutsidis09a-CSS; @Boutsidis11a-NearOpt] is clearly an instance of the generalized CSS problem. In this instance, the target matrix is the same as the source matrix $B=A$ and the goal is to select a subset of columns from a data matrix that best represent other columns. The greedy algorithm presented in this paper can be directly used for solving the basic CSS problem. A detailed comparison of the greedy CSS algorithm and the state-of-the-art CSS methods can be found at [@Farahat12tt]. In our previous work [@farahat11-icdm; @farahat12], we successfully used the proposed greedy algorithm for unsupervised feature selection which is an instance of the CSS problem. We used the greedy algorithm to solve two instances of the generalized CSS problem: one is based on selecting features that approximate the original matrix $B=A$ and the other is based on selecting features that approximate a random partitioning of the features $B_{:c}=\sum_{j\in{\ensuremath{\mathcal{P}}}_{c}}A_{:j}$. The proposed greedy algorithms achieved superior clustering performance in comparison to state-of-the-art methods for unsupervised feature selection. **Distributed Column Subset Selection.** The generalized CSS problem can be used to define distributed variants of the basic column subset selection problem. In this case, the matrix $B$ is defined to encode a concise representation of the span of the original matrix $A$. This concise representation can be obtained using an efficient method like random projection. In our recent work [@Farahat13css], we defined a distributed CSS based on this idea and used the proposed greedy algorithm to select columns from big data matrices that are massively distributed across different machines.
**SVD-based Column Subset Selection.** [Ç]{}ivril and Magdon-Ismail [@Civril12-CSS-Sparse] proposed a CSS method which first calculates the Singular Value Decomposition (SVD) of the data matrix, and then selects the subset of columns which best approximates the leading singular values of the data matrix. The formulation of this CSS method is an instance of the generalized CSS problem, in which the target matrix is calculated from the leading singular vectors of the data matrix. The greedy algorithm presented in [@Civril12-CSS-Sparse] can be implemented using Algorithm \[Alg:GenGCSS\] by setting $B=U_{k}\Sigma_{k}$ where $U_{k}$ is a matrix whose columns represent the leading left singular vectors of the data matrix, and $\Sigma_{k}$ is a matrix whose diagonal elements represent the corresponding singular values. Our greedy algorithm is however more efficient than the greedy algorithm of [@Civril12-CSS-Sparse]. **Sparse Approximation.** Given a target vector and a set of basis vectors, also called atoms, the goal of sparse approximation is to represent the target vector as a linear combination of a few atoms [@tropp2004greed]. Different instances of this problem have been studied in the literature under different names, such as variable selection for linear regression [@Das2008], sparse coding [@olshausen1997sparse; @lee2007efficient], and dictionary selection [@CevherK11; @DasK11]. If the goal is to minimize the discrepancy between the target vector and its projection onto the subspace of selected atoms, the sparse approximation can be considered an instance of the generalized CSS problem in which the target matrix is a vector and the columns of the source matrix are the atoms. Several greedy algorithms have been proposed for sparse approximation, such as basic matching pursuit [@mallat1993matching], orthogonal matching pursuit [@tropp2007signal], the orthogonal least squares [@chen1989orthogonal]. The greedy algorithm for generalized CSS is equivalent to the orthogonal least squares algorithm (as defined in [@blumensath2007difference]) because at each iteration it selects a new column such that the reconstruction error after adding this column is minimum. Algorithm \[Alg:GenGCSS\] can be used to efficiently implement the orthogonal least squares algorithm by setting $B=\bf{y}$, where $\bf{y}$ is the target vector. However, an additional step will be needed to calculate the weights of the selected atoms as ${\left( A_{:{\ensuremath{\mathcal{S}}}}^{T}A_{:{\ensuremath{\mathcal{S}}}} \right)^{-1}} A_{:{\ensuremath{\mathcal{S}}}}^{T}\bf{y}$.
**Simultaneous Sparse Approximation.** A more general sparse approximation problem is the selection of atoms which represent a group of target vectors. This problem is referred to as simultaneous sparse approximation [@tropp2006algorithms]. Different greedy algorithms have been proposed for simultaneous sparse approximation with different constraints [@tropp2006algorithms; @CevherK11]. If the goal is to select a subset of atoms to represent different target vectors without imposing sparsity constraints on each representation, simultaneous sparse approximation will be an instance of the greedy CSS problem, where the source columns are the atoms and the target columns are the input signals.
Conclusions
===========
We define a generalized variant of the column subset selection problem and present a fast greedy algorithm for solving it. The proposed greedy algorithm can be effectively used to solve a variety of problems that are instances of the generalized column subset selection problem.
Appendix A {#appendix-a .unnumbered}
==========
**Input:** Source matrix $A$, Target matrix $B$, Number of columns $l$\
**Output:** Selected subset of columns ${\ensuremath{\mathcal{S}}}$
Initialize ${\ensuremath{\boldsymbol{f}}}_i^{(0)}=\|B^TA_{:i}\|^{2}$, ${\ensuremath{\boldsymbol{g}}}_i^{(0)}=A_{:i}^TA_{:i}$ for $i=1\:...\:n$ Repeat $t=1\rightarrow l$: $p={\arg\max}_i\ {\ensuremath{\boldsymbol{f}}}_{i}^{(t)}/{\ensuremath{\boldsymbol{g}}}_{i}^{(t)}$, ${\ensuremath{\mathcal{S}}}={\ensuremath{\mathcal{S}}}\cup \{p\}$ ${\ensuremath{\boldsymbol{\delta}}}^{(t)}=A^TA_{:p}-\sum_{r=1}^{t-1}{\ensuremath{\boldsymbol{\omega}}}_{p}^{(r)}{\ensuremath{\boldsymbol{\omega}}}^{(r)}$ ${\ensuremath{\boldsymbol{\gamma}}}^{(t)}=B^TA_{:p}-\sum_{r=1}^{t-1}{\ensuremath{\boldsymbol{\omega}}}_{p}^{(r)}{\ensuremath{\boldsymbol{\upsilon}}}^{(r)}$ ${\ensuremath{\boldsymbol{\omega}}}^{(t)}={\ensuremath{\boldsymbol{\delta}}}^{(t)}/\sqrt{{\ensuremath{\boldsymbol{\delta}}}^{(t)}_{p}}$, ${\ensuremath{\boldsymbol{\upsilon}}}^{(t)}={\ensuremath{\boldsymbol{\gamma}}}^{(t)}/\sqrt{{\ensuremath{\boldsymbol{\delta}}}^{(t)}_{p}}$ Update ${\ensuremath{\boldsymbol{f}}}_i$’s, ${\ensuremath{\boldsymbol{g}}}_i$’s (Theorem \[Th:Rec\_fg2\])
Proof of Theorem \[Th:Rec\_fg2\] {#proof-of-theorem-threc_fg2 .unnumbered}
--------------------------------
Let ${\ensuremath{\mathcal{S}}}$ denote the set of columns selected during the first $t-1$ iterations, $F^{(t-1)}$ denote the residual matrix of $B$ at the start of the $t$-th iteration (i.e., $F^{\left(t-1\right)}=B-P^{\left(\mathcal{S}\right)}B$), and $p$ be the column selected at iteration $t$. From Lemma \[Lm:Proj\], $P^{\left(\mathcal{S}\cup\left\{ p\right\} \right)}=P^{\left(\mathcal{S}\right)}+R^{\left(\left\{ p\right\} \right)}$. Multiplying both sides with $B$ gives $P^{\left(\mathcal{S}\cup\left\{ p\right\} \right)}B=P^{\left(\mathcal{S}\right)}B+R^{\left(\left\{ p\right\} \right)}B$. Subtracting both sides from $B$ and substituting $B-P^{\left(\mathcal{S}\right)}B$, and $B-P^{\left(\mathcal{S}\cup\left\{ p\right\} \right)}B$ with $F^{\left(t-1\right)}$ and $F^{\left(t\right)}$ respectively gives $F^{\left(t\right)}=\left(F-R^{\left(\left\{ p\right\} \right)}B\right)^{\left(t-1\right)}.$
Since $R^{\left(\left\{ p\right\} \right)}B=R^{\left(\left\{ p\right\} \right)}F$ (see the proof of Theorem \[Th:RecF\]), $F^{(t)}$ can be calculated recursively as $$\label{Eq:ERec}
F^{\left(t\right)}=\left(F-R^{\left(\left\{ p\right\} \right)}F\right)^{\left(t-1\right)}.$$ Similarly, $E^{(t)}$ can be expressed as $$E^{\left(t\right)}=\left(E-R^{\left(\left\{ p\right\} \right)}E\right)^{\left(t-1\right)}.$$ Substituting with $F$ and $E$ in $H=F^TE$ gives $$H^{\left(t\right)}=\left(\left(F-R^{\left(\left\{ p\right\} \right)}F\right)^{T}\left(E-R^{\left(\left\{ p\right\} \right)}E\right)\right)^{\left(t-1\right)}=\left(H-F^{T}R^{\left(\left\{ p\right\} \right)}E\right)^{\left(t-1\right)}.$$ Using $R^{\left(\left\{ p\right\} \right)}=E_{:p}\left(E_{:p}^{T}E_{:p}\right)^{-1}E_{:p}^{T}$, and given that $\boldsymbol{\omega}=G_{:p}=E^{T}E_{:p}/\sqrt{E_{:p}^{T}E_{:p}}$ and $\boldsymbol{\upsilon}=H_{:p}=F^{T}E_{:p}/\sqrt{E_{:p}^{T}E_{:p}}$, the matrix $H$ can be calculated recursively as $$H^{\left(t\right)}=\left(H-\boldsymbol{\upsilon}\boldsymbol{\omega}^{T}\right)^{\left(t-1\right)}.$$ Similarly, $G$ can be expressed as $$G^{\left(t\right)}=\left(G-\boldsymbol{\omega}\boldsymbol{\omega}^{T}\right)^{\left(t-1\right)}.$$ Using these recursive formulas, ${\ensuremath{\boldsymbol{f}}}_{i}^{(t)}$ can be calculated as $$\begin{split}
{\ensuremath{\boldsymbol{f}}}_{i}^{\left(t\right)}&=\left(\left\Vert H_{:i} \right\Vert ^{2}\right)^{(t)}=\left(\|H_{:i}-{\ensuremath{\boldsymbol{\omega}}}_i{\ensuremath{\boldsymbol{\upsilon}}}\|^{2}\right)^{\left(t-1\right)}\\
&=\left((H_{:i}-{\ensuremath{\boldsymbol{\omega}}}_i{\ensuremath{\boldsymbol{\upsilon}}})^T(H_{:i}-{\ensuremath{\boldsymbol{\omega}}}_i{\ensuremath{\boldsymbol{\upsilon}}})\right)^{\left(t-1\right)}\\
&=\left(H_{:i}^TH_{:i}-2{\ensuremath{\boldsymbol{\omega}}}_{i}H_{:i}^{T}{\ensuremath{\boldsymbol{\upsilon}}}+{\ensuremath{\boldsymbol{\omega}}}_{i}^{2}\|{\ensuremath{\boldsymbol{\upsilon}}}\|^{2}\right)^{\left(t-1\right)}\\
&=\left({\ensuremath{\boldsymbol{f}}}_{i}-2{\ensuremath{\boldsymbol{\omega}}}_{i}H_{:i}^{T}{\ensuremath{\boldsymbol{\upsilon}}}+{\ensuremath{\boldsymbol{\omega}}}_{i}^{2}\|{\ensuremath{\boldsymbol{\upsilon}}}\|^{2}\right)^{\left(t-1\right)}.
\end{split}$$ Similarly, ${\ensuremath{\boldsymbol{g}}}_{i}^{(t)}$ can be calculated as $$\begin{split}
{\ensuremath{\boldsymbol{g}}}_{i}^{(t)}&=G_{ii}^{(t)}=\left(G_{ii}-{\ensuremath{\boldsymbol{\omega}}}_{i}^{2}\right)^{\left(t-1\right)}=\left({\ensuremath{\boldsymbol{g}}}_{i}-{\ensuremath{\boldsymbol{\omega}}}_{i}^{2}\right)^{\left(t-1\right)}.
\end{split}$$ Let ${\ensuremath{\boldsymbol{f}}}=\left[{\ensuremath{\boldsymbol{f}}}_{i}\right]_{i=1..n}$and ${\ensuremath{\boldsymbol{g}}}=\left[{\ensuremath{\boldsymbol{g}}}_{i}\right]_{i=1..n}$, ${\ensuremath{\boldsymbol{f}}}^{(t)}$ and ${\ensuremath{\boldsymbol{g}}}^{(t)}$ can be expressed as $$\label{eq:f22}
\begin{split}
{\ensuremath{\boldsymbol{f}}}^{(t)}&=\left({\ensuremath{\boldsymbol{f}}}-2\left({\ensuremath{\boldsymbol{\omega}}}\circ
H^T{\ensuremath{\boldsymbol{\upsilon}}}\right)+\|{\ensuremath{\boldsymbol{\upsilon}}}\|^{2}\left({\ensuremath{\boldsymbol{\omega}}}\circ{\ensuremath{\boldsymbol{\omega}}}\right)\right)^{(t-1)},
\\
{\ensuremath{\boldsymbol{g}}}^{(t)}&=\left({\ensuremath{\boldsymbol{g}}}-\left({\ensuremath{\boldsymbol{\omega}}}\circ{\ensuremath{\boldsymbol{\omega}}}\right)\right)^{(t-1)},
\end{split}$$ where $\circ$ represents the Hadamard product operator.
Using the recursive formula of $H$, the term $H^T{\ensuremath{\boldsymbol{\upsilon}}}$ at iteration $(t-1)$ can be expressed as $$\begin{split}
H^T{\ensuremath{\boldsymbol{\upsilon}}}&=\left(A^TB-\Sigma_{r=1}^{t-2}\left({\ensuremath{\boldsymbol{\omega}}}{\ensuremath{\boldsymbol{\upsilon}}}^{T}\right)^{\left(r\right)}\right){\ensuremath{\boldsymbol{\upsilon}}}
=A^{T}B{\ensuremath{\boldsymbol{\upsilon}}}-\Sigma_{r=1}^{t-2}\left({\ensuremath{\boldsymbol{\upsilon}}}^{\left(r\right)T}{\ensuremath{\boldsymbol{\upsilon}}}\right){\ensuremath{\boldsymbol{\omega}}}^{^{\left(r\right)}}
\end{split}$$ Substituting with $H^T{\ensuremath{\boldsymbol{\upsilon}}}$ in (\[eq:f22\]) gives the update formulas for ${\ensuremath{\boldsymbol{f}}}$ and ${\ensuremath{\boldsymbol{g}}}$.
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Kazumi Okuyama
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Department of Physics and Astronomy, University of British Columbia
Vancouver, BC, V6T 1Z1, Canada
kazumi@phas.ubc.ca
We construct a two-dimensional ${\cal N}=(0,4)$ quiver gauge theory on D1-brane probing D5-branes on ALE space, and study its IR behavior. This can be thought of as a gauged linear sigma model for the NS5-branes on ALE space.
The D1-D5-KK system (KK = Kaluza-Klein monopole) is a 1/8 BPS configuration in type IIB string theory. We are interested in the two-dimensional ${\cal N}=(0,4)$ gauge theory on the D1-brane of this system. In the limit where the KK-monopole is replaced by the orbifold ${\Bbb C}^2/{\Bbb Z}_n$, it is straightforward to construct the theory on D1-brane following the standard procedure of Douglas and Moore . Namely, we consider the ${\Bbb Z}_n$ orbifolding of the ${\cal N}=(4,4)$ $U(Q_1)$ gauge theory with one adjoint and $Q_5$ fundamental hypermultiplets coming from the D1-D5 system. The resulting theory is an ${\cal N}=(0,4)$ quiver gauge theory. As compared to the study of gauge theory of the D1-D5 system , the ${\cal N}=(0,4)$ gauge theory of the D1-D5-KK system is less understood. See for some of the works related to this system.
The D1-D5-KK system is related to the various brane configurations by duality. Obviously it is S-dual to the F1-NS5-KK system, and it is also dual to the triple intersection of M5-branes . In general, ${\cal N}=(0,4)$ CFTs appear in many places in string theory; some of these are related to the D1-D5-KK system and some are not. The examples of $(0,4)$ preserving configuration that are not directly related to the D1-D5-KK system include the D1-D5-D9 system described by the $(0,4)$ ADHM sigma model , and the intersecting brane configuration of D1-D5-D5 .
One of the motivation to study the ${\cal N}=(0,4)$ quiver gauge theory is that it can be thought of as a gauged linear sigma model (GLSM) for the NS5-branes on ALE space. In general, GLSM is a quite useful tool to understand the moduli space of CFTs and the relations among them . In , NS5-branes on $\S^1$ and their relation to the T-dual KK-monopoles are studied by using an ${\cal N}=(4,4)$ GLSM. It is very interesting to find GLSMs describing NS5-branes in other backgrounds. Our ${\cal N}=(0,4)$ quiver gauge theory is such an example.
This paper is organized as follows. In section 2, we construct the ${\cal N}=(0,4)$ quiver gauge theory describing the D1-D5 branes on ${\Bbb C}^2/{\Bbb Z}_n$. In section 3, we study the Higgs branch of this theory. In section 4, we compute the 1-loop correction to the Coulomb branch metric. Section 5 is discussions.
In this section, we construct the Lagrangian on the D1-brane probe for the D5-branes on ${\Bbb C}^2/{\Bbb Z}_n$. To find the Lagrangian on the D1-brane, we first summarize the symmetries of the system. Let us consider a configuration of D1-branes and D5-branes extending in the $x^0x^1$ and $x^0x^1\cdots x^5$ directions, respectively. We can perform the orbifolding of the transverse directions ${\Bbb C}^2(=x^6x^7x^8x^9)$ by ${\Bbb Z}_n$ such that the resulting configuration is 1/8 BPS. This can be seen by writing the BPS condition for the supersymmetry generator $\ep_LQ_L+\ep_RQ_R$ where $\ep_L,\ep_R$ are ten-dimensional Majorana-Weyl spinors with the same chirality. From these relations, it follows that the unbroken supersymmetry in the $x^0x^1$ space is chiral $\ep_R=\ep_L=\Ga^0\Ga^1\ep_L$. One can check that it is a two-dimensional ${\cal N}=(0,4)$.
Before orbifolding, the massless open string spectrum on the D1-brane is decomposed into various representations under the symmetry $SO(1,1)_{01}\times SO(4)_{2345}\times SO(4)_{6789}$. Following , we introduce the indices $(A',\til{A}')$ and $(A,Y)$ to represent the fundamental ${\bf 2}$ of various $SU(2)$ groups: The massless modes on the D1-brane coming from the 1-1 string and 1-5 string are summarized as where $A_{\pm\pm}=A_0\pm A_1$. This spectrum represent the ${\cal N}=(4,4)$ $U(Q_1)$ gauge theory with one adjoint hypermultiplet and $Q_5$ fundamental hypermultiplets. $b^{AY}$ is the scalar in the ${\cal N}=(4,4)$ vectormultiplet representing the position of D1-brane in the $(6789)$ directions, while $b^{A'\til{A}'}$ is the scalar in the adjoint hypermultiplet corresponding to the $(2345)$ directions. $SU(2)_{A'}$ is usually denoted $SU(2)_R$ since the scalars in the hypermultiplets $b^{A'\til{A}'}$ and $H^{A'}$ are doublets under this $SU(2)$. The ${\cal N}=(4,4)$ supersymmetry is generated by the supercharges For example, the super transformation of $b^{AY}$ is
Now we consider the orbifolding of ${\Bbb C}^2={\Bbb R}^4_{6789}$ by ${\Bbb Z}_n$. We would like to preserve the right moving $(0,4)$ supersymmetry generated by $Q^{AA'}_+$ and break the left moving one $Q^{A'Y}_-$. Therefore, we embed the orbifold group ${\Bbb Z}_n$ in $SU(2)_Y$: Thus the nontrivial orbifold action is on the $Y$-index where $\om=e^{2\pi i/n}$ and $Y=\pm$, and the rest of the fields in are invariant under the ${\Bbb Z}_n$.
To construct the gauge theory on the D1-brane, we follow the general procedure . We extend the Chan-Paton factors ${\cal H}_{D1}$ and ${\cal H}_{D5}$ associated with the D1-branes and the D5-branes by tensoring ${\Bbb C}^n$. Physically this corresponds to going to the covering space of ${\Bbb C}^2/{\Bbb Z}_n$. Then we extend the orbifold action by acting ${\Bbb Z}_n$ on the Chan-Paton factor ${\Bbb C}^n$ as well. We choose this action on ${\Bbb C}^n$ to be the regular representation of ${\Bbb Z}_n$. We also set ${\cal H}_{D1}={\Bbb C}$ and ${\cal H}_{D5}={\Bbb C}^k$, so that the gauge group becomes product of $U(1)$’s.
The resulting ${\Bbb Z}_n$ invariant spectrum are summarized by a quiver diagram (see Fig. 1). This is a two sets of $\h{A}_{n-1}$ Dynkin diagram connected by links. The inner (outer) quiver corresponds to the D1-brane (D5-brane) . The inner quiver represents the theory on the D1-branes on ${\Bbb C}^2/{\Bbb Z}_n$ constructed in , which is non-chiral and invariant under the ${\cal N}=(4,4)$ supersymmetry. This theory is well-known to lead to the hyperKähler quotient construction of $A_{n-1}$ ALE space. The left-right asymmetry comes from the orbifold action on the 1-5 string. As we can see, $H^{A'}$ and $\chi_+^A$ in are invariant under ${\Bbb Z}_n$ while $\chi_{-}^{Y}$ transforms non-trivially . This difference leads to the peculiar structure of the links connecting the inner and outer quivers as shown in Fig. 1.
To write down the Lagrangian explicitly, it is useful to use the ${\cal N}=(0,2)$ superspace language. We follow the notation in .
Before orbifolding, the theory representing the D1-D5 system is an ${\cal N}=(4,4)$ $U(Q_1)$ gauge theory with one adjoint and $Q_5$ fundamental hypermultiplets. In the language of ${\cal N}=(2,2)$ superfields, the matter content of this theory is the vector multiplet $(\Si,\Phi)$, the adjoint hypermultiplet $(B,\til{B})$ and the fundamental hypermultiplets $(Q,\til{Q})$. All superfields are ${\cal N}=(2,2)$ chiral superfields, except for $\Si$ which is an ${\cal N}=(2,2)$ twisted chiral superfield. These fields correspond to the notation in the previous section as
To proceed, we need to know the orbifold action in the ${\cal N}=(2,2)$ language. Without loss of generality, we can choose the superspace coordinate $\th^-$ to carry the index $Y=+$. Therefore, the ${\Bbb Z}_n$ action in the ${\cal N}=(2,2)$ language is given by This condition can be written down in the ${\cal N}=(0,2)$ language by decomposing the ${\cal N}=(2,2)$ superfields into the ${\cal N}=(0,2)$ superfields: where $\Si$ and $\Phi^i$ are the $(0,2)$ chiral superfields, $\Up$ is the $(0,2)$ gauge superfield, and $\La^i$ is the Fermi superfield.
The resulting theory is an ${\cal N}=(0,4)$ gauge theory having $\prod_{a=1}^nU(1)_a$ as the gauge group and $\prod_{a=1}^nU(k)_a$ as the flavor symmetry. The kinetic term of the orbifolded theory is given by and the ${\cal N}=(0,2)$ superpotential term is given by This $(0,2)$ superpotential is easily obtained by starting from the $(2,2)$ superpotential $W=\til{Q}(\Phi-m)Q+\tr\til{B}[\Phi,B]$, reducing to $(0,2)$ superspace, and then orbifolding by ${\Bbb Z}_n$. In , we introduced the FI-parameters and the theta-angle
Let us explain our notation in and . We use the same letter for the $(0,2)$ chiral superfields as the parent $(2,2)$ chiral superfields, and put the superscript in $\La$ for the corresponding Fermi superfields. $(\Up_a,\La^{\Phi}_a,B_a,\til{B}_a)$ are neutral multiplets living at the $a^{\rm th}$ node in the inner quiver (see Fig. 1). $(\Si_{a+1,a},\La^B_{a+1,a},\La^{\til{B}}_{a+1,a})$ are living on the link connecting the $a^{\rm th}$ node and the $(a+1)^{\rm th}$ node in the inner quiver. They carry charge $(+1,-1)$ under the gauge group $U(1)_{a+1}\times U(1)_a$. $\Phi_{a,a+1}$ is living on the same link, but carries opposite charge $(-1,+1)$. $Q_a^{f_a}$ and $\til{Q}_{f_aa}$ are on the link between the $a^{\rm th}$ node in the inner quiver and the $a^{\rm th}$ node in the outer quiver; they transform as $(+1,{\bf k})$ and $(-1,\b{\bf k})$ under $U(1)_a\times U(k)_a$, respectively. Finally, the Fermi multiplet $\La^{Qf_a}_{a+1}$ is on the link between the $(a+1)^{\rm th}$ inner node and the $a^{\rm th}$ outer node, and $\La^{\til{Q}}_{f_{a+1},a}$ is on the link between the $a^{\rm th}$ inner node and the $(a+1)^{\rm th}$ outer node; they transform as $(+1,{\bf k})$ and $(-1,\b{\bf k})$ under $U(1)_{a+1}\times U(k)_a$ and $U(1)_a\times U(k)_{a+1}$, respectively.
The Fermi superfields are not chiral, but satisfy the constraint of the form $\b{{\cal D}}_+\La^i=\rt{2}E^i$. In our case, the constraints are given by In order for the $(0,2)$ superpotential $-{1\o{\rt{2}}}\int d\th^+\La^iJ_i$ to be chiral, $E^i$ and $J_i$ should be orthogonal For our choice of the superpotential and the constraint , the orthogonality is satisfied by requiring Note that $m^{f_{a+1}}_{f_a}$ and $\til{m}_{f_{a+1}}^{f_a}$ correspond to the complex mass and the twisted mass, respectively, in the language of ${\cal N}=(2,2)$ theory. These parameters naturally live on the links of the outer quiver.
We expect that the theory flows in the IR to the non-linear sigma model with the target space given by the vanishing locus of the bosonic potential. There are three contributions to the potential energy: the $D$-term, the $E$-term, and the $J$-term The $E$-term and the $J$-term can be read off from and . Note that the parameter $s_a$ in enter in the $J$-term potential as Here and hereafter we denote the scalar component of $(0,2)$ chiral superfield by the lower case letter. The $D$-term for the $a^{\rm th}$ $U(1)$ gauge group is given by In the following two sections, we will consider the moduli space of vacua defined by $U=0$.
In this section, we consider the Higgs branch, [*i.e.*]{} the vacuum with non-vanishing $q,\til{q}$. For simplicity, we set $s_a=0$ and also set the mass term to be diagonal This mass term breaks the flavor symmetry $\prod_{a=1}^nU(k)_a$ down to the diagonal $U(k)$. Then the parameter $\mu_a$ in is given by
Instead of setting $D_a=E^i=J_i=0$, we can consider the holomorphic condition $E^i=J_i=0$ and mod out by the complexified gauge group $\prod_{a=1}^n{\Bbb C}^{\times}_a$. Then the equation for vacuum is given by Here we suppressed the flavor index.
When all mass are non-zero, we can solve this equation up to the gauge transformation as Therefore, the conditions for $(q_a,\til{q}_a)$ are reduced to the single equation $\til{q}_1q_1=0$. It is known that the solution space of the equation $\til{q}_1q_1=0$ modded out by ${\Bbb C}^{\times}$ gives the $U(k)$ one-instanton moduli space on ${\Bbb R}^4$ . Therefore, the Higgs branch is given by In the opposite extreme, [*i.e.*]{} when all mass are zero, the vacuum is given by $\si_{a+1,a}=\phi_{a,a+1}=0$ and $\til{q}_aq_a=0$. In this case, $q_a$’s and $\til{q}_a$’s with different $a$ are unrelated. Therefore, the Higgs branch is given by We expect that in general the dimension of the Higgs branch jumps when setting some of the mass $m_a,\til{m}_a$ to zero.
It is natural to expect that the boundary CFT of D1-D5-KK system is given by the Higgs branch CFT of D1-brane theory. It is interesting to observe that the Higgs branch of our $(0,4)$ theory is basically given by the instanton moduli space, which is the same as the Higgs branch of the D1-D5 system . The similarity of the $(0,4)$ moduli space and the $(4,4)$ moduli space is emphasized in . Our result , is consistent with the conjecture that the D1-D5-KK system admits a solvable boundary CFT written as a sigma model on the symmetric product of the manifold corresponding to the (2345) directions.
We should also mention that there is another branch ${\cal M}_B$ corresponding to the expectation value of $(B_a,\til{B}_a)$. By a similar analysis as above, we find This branch represents the center-of-mass motion of D1-brane in the (2345) directions.
In this section, we consider the Coulomb branch of our $(0,4)$ theory and compute the 1-loop correction to the effective metric. Let us first clarify our terminology of Coulomb and Higgs branch. The criterion is the Higgsing of the diagonal $U(1)$ of the gauge group $\prod_{a=1}^nU(1)_a$. In the branch with $q_a,\til{q}_a\not=0$ considered in the previous section, the diagonal $U(1)$ is broken. On the other hand, in the branch with $\si_{a+1,a},\phi_{a,a+1}\not=0$, the diagonal $U(1)$ is unbroken, so we will call this branch Coulomb branch.
Before going to the analysis of $(0,4)$ theory, we first review the $(4,4)$ case.
Let us first consider the Coulomb branch of the system of single D1-brane and $Q_5$ D5-branes. The theory on the D1-brane is an ${\cal N}=(4,4)$ $U(1)$ gauge theory with $Q_5$ hypermultiplets with charge 1. The Coulomb branch is parametrized by the vectormultiplet $(\Si,\Phi)$. The metric on the Coulomb branch is corrected by integrating out the hypermultiplet. The 1-loop corrected metric is given by From the general structure of ${\cal N}=(4,4)$ non-linear sigma model, it can be shown that the metric is 1-loop exact. One can immediately notice that is the metric on the transverse space of $Q_5$ NS5-branes . Near the origin of Coulomb branch, this metric reduces to the familiar throat metric, and the CFT on the Coulomb branch is described by where ${\Bbb R}_\phi$ is the linear dilaton CFT and $SU(2)_{Q_5}$ is the WZW model.
The gauge theory on the D1-brane on ${\Bbb C}^2/{\Bbb Z}_n$ is given by the ${\cal N}=(4,4)$ $\h{A}_{n-1}$ quiver gauge theory . In our notation, this is given by the matter content on the inner quiver in Fig. 1. The fields written in the $(0,2)$ superspace naturally organize themselves into the ${\cal N}=(4,4)$ multiplets. Namely, $(B_a,\til{B}_a,\Up_a,\La^{\Phi}_a)$ is the $(4,4)$ vectormultiplet living on the $a^{\rm th}$ node, and $(\Si_{a+1,a},\Phi_{a,a+1},\La^B_{a+1,a},\La^{\til{B}}_{a+1,a})$ is the $(4,4)$ hypermultiplet living on the link between the $a^{\rm th}$ node and the $(a+1)^{\rm th}$ node. In this case, the diagonal $U(1)$ is decoupled and it never gets Higgsed. Although it is natural to call the branch of non-zero $(\si_{a+1,a},\phi_{a,a+1})$ Coulomb branch in the presence of D5-brane, this branch is usually referred to as the Higgs branch in the case of D1 on ${\Bbb C}^2/{\Bbb Z}_n$, since it is parametrized by the hypermultiplet.
It is known that the metric on the Higgs branch is the ALE metric. This is obtained by the following steps . We first note that the action of the $(4,4)$ quiver theory is reproduced from our $(0,4)$ quiver theory by setting the fields from the 1-5 string to zero. In particular, the equations for the vacuum is the same as those for the Coulomb branch of our $(0,4)$ theory. For the consistency, the FI-parameters have to satisfy the condition Next we introduce the variables $(\vec{r}_a,\varphi_a)$ by The vacuum equations imply that where $\vec{x}_a$ is given by Then we write down the kinetic term of $(\si_{a+1,a},\phi_{a,a+1})$ restricted on the locus where $\vec{\om}_a$ satisfies Finally, by classically integrating out the gauge field $A_{a,\mu}$, we arrive at the effective Lagrangian on the locus where $V,\vec{\om}$ and $\th$ are given by The metric is nothing but the familiar ALE metric. The above procedure is known as the hyperKähler quotient construction of ALE metric.
Now let us go back to the analysis of $(0,4)$ theory. From the discussion in the case 1,2 above, we can expect that the metric on the Coulomb branch of our $(0,4)$ theory reproduces the supergravity solution of D1-D5-KK system, which is given by the usual harmonic function rule Here $ds^2_{KK}$ denotes the metric of KK-monopoles, [*i.e.*]{} Taub-NUT space. The metric seen by the probe D1-brane can be obtained by plugging the solution into the D1-brane Born-Infeld action $S_{D1}=-\int d\tau d\si e^{-\Phi}\rt{-\det g}$, and expanding it around the configuration $X^0=\tau, X^1=\si, X^m=X^m(\tau)$, where $X^m$ is the coordinate on the Taub-NUT space. By looking at the term quadratic in the velocity $\dot{X}^m(\tau)$, we find that the effective metric seen by the D1-brane probe is This is independent of the factor $Z_1$ as expected from the BPS property.
The metric can also be obtained as the string metric on the S-dual F1-NS5-KK system. The S-dual of is given by Therefore, the metric on the transverse space seen by the fundamental string is which agrees with the D1-brane probe metric as expected. In the next subsection, we compute the 1-loop correction to the metric on the Coulomb branch and compare it with the supergravity result .
In this section, we compute the 1-loop correction to the metric on the Coulomb branch. There is no correction from $(B,\til{B})$, since if we turn off the fields coming from the 1-5 string, the theory reduces to the ${\cal N}=(4,4)$ quiver gauge theory and it is known that there is no correction to the Higgs branch metric in this case. Recall that, as we discussed in section 4.2, the role of Coulomb and Higgs branch is exchanged in the D1-brane on ${\Bbb C}^2/{\Bbb Z}_n$ case. Therefore, the 1-loop correction only comes from integrating out the modes of 1-5 string, [*i.e.*]{} the fields $(Q_a^{f_a},\til{Q}_{f_aa},\La^{Qf_a}_{a+1},
\La^{\til{Q}}_{f_{a+1},a})$. In this section, we set $m=\til{m}=s_a=0$ for simplicity. Then the equation for vacuum is given by with $\mu_a=0$.
Let us compute the 1-loop integral in the background of $(\si_{a+1,a},\phi_{a,a+1})$ obeying . The term quadratic in bosonic fields is given by and the fermionic term is given by where $\Psi_a$ and $M_a$ are defined as Here we suppressed the flavor indices for notational simplicity. We can check that the bosons and fermions have the same mass eigenvalues In the last step, we used the vacuum equation with $\mu_a=0$. Clearly, the eigenvalues of $M_aM_a^\dag$ agree with the mass of $q_a,\til{q}_a$.
Now we can compute the 1-loop effective action by integrating out the 1-5 string modes: Here we expanded the result up to two derivatives and the dots denote the higher derivative terms. The 1-loop integral in can be evaluated by using the formula Finally, we arrive at the 1-loop corrected effective Lagrangian The first line in is the tree level term and the second line is the 1-loop correction.
For the general value of the FI parameters $\zeta_a^3$, it is not so easy to rewrite this Lagrangian in terms of the variables $(\vec{r},\th)$ introduced in section 4.2. Here we focus on the orbifold limit corresponding to $\mu_a=\zeta_a^3=0$. Then we can show that This implies that the denominators in the second line in are common for all terms and they can be factored out. Therefore, the 1-loop correction term in becomes proportional to the tree level term. In other words, the metric on the Coulomb branch is given by the ALE metric up to a conformal factor: This is exactly the metric obtained in the supergravity approximation , with the understanding that the KK-monopole metric $ds^2_{KK}$ is replaced by the orbifold limit of the ALE metric. This shows that our $(0,4)$ theory can be thought of as a GLSM for the NS5-branes on ALE space. Near the origin of Coulomb branch, the metric reduces to One can easily show that the last two terms is the metric on the lens space $\S^3/{\Bbb Z}_n$ written as the Hopf fibration of $\S^1$ over $\S^2$. Therefore, the near horizon limit is described by the ${\Bbb Z}_n$ orbifold of throat CFT where $\phi=\log r$ is the linear dilaton direction.
In this paper, we have constructed the ${\cal N}=(0,4)$ quiver gauge theory corresponding to the D1-D5 branes on ${\Bbb C}^2/{\Bbb Z}_n$. The 1-loop correction of the Coulomb branch shows that this theory can be seen as a GLSM of NS5-branes on ALE space. Our result of 1-loop effective action on the Coulomb branch is not proportional to the tree level term for the general FI parameters. This seems to suggest that the naive harmonic function rule breaks down in the string theory. We also expect that the effective metric for the $(0,4)$ theory is not 1-loop exact, although it is constrained by supersymmetry to be the hyperKähler with torsion sigma model. For both Higgs and Coulomb branch, our analysis is limited to the special value of the FI parameters. It would be interesting to study the general parameter case.
[**Acknowledgment:**]{} I would like to thank Yuji Sugawara for discussion in 1998.
|
---
abstract: 'In 2006 A. Gorodetski proved that central fibers of perturbed skew products are continuous with respect to the base point. In the present paper we give an explicit estimate of the exponent mentioned above. Moreover, we extend the Gorodetski theorem from the case when the fiber maps are close to the identity to a much wider class that satisfy the so-called modified dominated splitting condition. In many cases (for example, in the case of skew products over the solenoid or over linear Anosov diffeomorphisms of a torus), the exponent is close to 1. This allows us in a sense to overcome the so called Fubini nightmare. Namely, we prove that the union of central fibers that are strongly atypical from the point of view of the ergodic theory, has Lebesgue measure zero, despite the lack of absolute continuity of the holonomy map for the central foliation. For that we revisit the Hirsch-Pugh-Shub theory, and estimate the contraction constant of the graph transform map.'
author:
- 'Yu. Ilyashenko[^1] [^2] and A. Negut [^3]'
title:
-
- 'properties of perturbed skew products and Fubini regained[^4]'
---
Introduction {#sec:intro}
============
Skew products and continuity {#sub:hofub}
----------------------------
In this paper we study perturbations of over hyperbolic maps in the base. Under the so-called dominated splitting condition, these perturbations have an invariant center lamination (see [@HPS]). It was proven by Gorodetski that this lamination will be continuous, [@G06] (see also [@N] for an earlier particular result). In this paper we estimate the exponent, and prove that in some cases it can be made arbitrarily close to $1$ by making the perturbation small enough.\
Such a center lamination allows us to conjugate any perturbation of a skew product with another skew product, which can be very useful in applications. A priori, the conjugation is only a homeomorphism, and as such it might not behave nicely with the measure. However, the property which we prove gives us some control over this conjugation, and allows to resolve a number of measure-related issues.\
For example, the property allows us to overcome in some sense the Fubini nightmare. In more detail, we prove that perturbations of a skew product over the Smale-Williams solenoid are semiconjugated with the duplication of the circle. We prove that the fibers of this semiconjugacy $q$ (i.e. the manifolds $q^{-1}(y), \ y \in S^1$) are in $y$ with exponent close to $1$. As a corollary, we prove that for a set $A\subset S^1$ of points with “strongly nonergodic orbits” under the duplication of the circle, the inverse image $q^{-1}(A)$ has Lebesgue measure $0$ (Theorem \[thm:fubini\] below). This is proved despite the fact that the foliation by the fibers $q^{-1}(y), y \in S^1$ may not have an absolutely continuous holonomy.\
The entire Section \[sec:intro\] is concerned with statements of results. In Subsection \[sub:perhol\] below, we introduce our Main Theorem \[thm:mt\] for perturbations of skew products over arbitrary hyperbolic maps. In the next Subsection \[sub:st\] we introduce Theorem \[thm:st\], which improves the Main Theorem in some cases (an important example of which being the solenoid). Two applications of Theorem \[thm:st\] are presented in Subsections \[sub:fubini\] and \[sub:kan\].\
The sections after that will be concerned with proofs. In Sections \[sec:grtrans\] and \[sec:hold\] we work with laminations and graph transform operators, culminating with the proof of the Main Theorem \[thm:mt\]. Several of the results we obtain help us in Section \[sec:st\], where we prove Theorem \[thm:st\]. In Section \[sec:fubini\], we *regain the Fubini property of our central leaves*, thus proving Theorem \[thm:fubini\]. Finally, Section \[sec:app\] consists of an appendix where several technical results are proved.
Persistence and property for skew products {#sub:perhol}
------------------------------------------
Throughout this paper a *$C^r-$morphism* will refer to a $C^r$ map with a $C^r$ inverse. We will use this notion both for maps of a manifold (with or without boundary) onto itself, and for maps of a manifold with boundary strictly into itself.\
Given a linear operator $A$ on a normed linear space $V$, when we write $a\leq |A|\leq b$ we mean that $$a|v|\leq |A(v)|\leq b|v|,
\textrm{ }\textrm{ }\textrm{ }\forall v\in V.$$ We will use this convention repeatedly throughout the paper.\
Let $B$ be a compact Riemannian manifold, henceforth called the *base*. Suppose $h:B\rightarrow B$ is a $C^2-$morphism with a hyperbolic invariant subset $\Lambda \subset B$. When $h$ is onto, we can take $B = \Lambda $. When $h$ is into, we can take $\Lambda $ to be the maximal attractor of $h$: $$\Lambda = \bigcap_{n \geq 0}
h^n(B).$$ Being hyperbolic, the map $h$ will have contracting and expanding directions in the vicinity of $\Lambda$. Thus there exist a Riemannian metric $d$ on $B$ and real numbers $0\leq \lambda_-\leq
\lambda<1$ and $0\leq \mu_-\leq \mu<1$, as well as a decomposition of the tangent bundle: $$\label{eqn:splittangent}
TB|_\Lambda=E^s\oplus E^u,$$ such that $$dh:E^s
\rightarrow E^s\textrm{ }\textrm{ }\textrm{ and }\textrm{ }\textrm{
} \lambda_- \leq |dh|\leq \lambda,$$ $$\label{eqn:hypstructure} dh:E^u \rightarrow E^u\textrm{ }\textrm{
}\textrm{ and }\textrm{ }\textrm{ } \mu_-\leq |dh^{-1}|\leq \mu.$$ Note that if $\lambda_-=\mu_-=0$, then we get the standard notion of hyperbolicity.\
We assume that the bundles $E^s$ and $E^u$ are trivialized, i.e. that there exist isomorphisms over $B$: $$\label{eqn:triv} \varphi^s:B \times \rr^{k}
\rightarrow E^s, \textrm{ }\textrm{ }\textrm{ } \varphi^u:B \times
\rr^l \rightarrow E^u$$ for some positive integers $k,l\geq 0$. The above is a technical condition necessary for our proof, but we conjecture that all our results hold without it. Let us note that it holds when $h$ is the Smale-Williams solenoid map or any linear Anosov diffeomorphism of a torus.\
\[def:lmhyp\] An invariant set $\L$ of a map $h$ with the above properties will be called $(\l_-, \l, \mu_-, \mu)$- hyperbolic.
Take another compact manifold $M$, called the *fiber*, and form the Cartesian product $X=B\times M$. A *skew product* over $h$ is defined as any $C^1-$map of the form
$$\label{eqn:skprod} \F:X\rightarrow X,\textrm{ }\textrm{ }\textrm{ }\F(b,m)=(h(b), f_b(m)),$$
where $f_b(m):M \rightarrow M$ is a $C^1$ family of $C^1-$morphisms.
\[def:domsplit\] We say that the skew product satisfies the *modified dominated splitting condition* if
$$\label{eqn:dom} \max\left(\max(\lambda,\mu)+
{\left| \left| \frac {\partial f_b^{\pm 1}}{\partial b}\right |
\right | }_{C^0(X)}, \ {\left| \left| \frac {\partial f_b^{\pm
1}}{\partial m}\right | \right | }_{C^0(X)}\right)=:L <
\min(\lambda^{-1}, \mu^{-1}).$$
Skew products are very useful in constructing dynamical systems with various properties. However, one often wants to study generic phenomena of dynamical systems, and therefore one also has to study small perturbations of skew products.
\[def:rpert\] Given $\rho>0$, a *$\rho-$perturbation* of the skew product is a $C^1-$morphism $\G:X \rightarrow X$ such that $$\label{eqn:perturbation} d(\G^{\pm 1}, \F^{\pm 1})_{C^1(X)}< \rho.$$
Let us make a notational convention. In this paper, we will consider a fixed skew product $\F$ and a neighborhood $\Omega \ni \F$ in the $C^1-$norm. We will often be concerned with small perturbations $\G \in \Omega$ of $\F$, and with various geometric objects related to these perturbations (such as central foliations, Hölder exponents etc). The leaves of central foliations of the perturbed maps $\G$ are graphs of parameter dependent maps $\beta_b$, or in other words, parameter dependent perturbations of the central fibers of $\F$. Whenever we write $||\beta_b|| = O(\rho)$, we mean that there exists a constant $C$ depending only on $\Omega$, such that for any $\rho-$perturbation $\G \in \Omega$ the maps corresponding to all central leaves satisfy the inequality $||\beta_b||\leq C\rho$. Thus the operator that maps $\G$ to $\beta_b$ is Lipshitz at $\F$ with constant $C$ (uniformly in $b$). We will consider other (parameter dependent) operators and functionals defined on $\Omega$; the expression $O(\rho)$ has the same meaning for them.\
Small perturbations of skew products are not necessarily skew products anymore. However, in this paper we will show that they are conjugated to skew products, and moreover the conjugation map satisfies a continuity property.\
We will now state our main result.
\[The Main Theorem\] \[thm:mt\] Consider a skew product $\F $ as in over a $(\l_-, \l, \mu_-, \mu
)-$ hyperbolic map in the base, satisfying the modified dominated splitting condition. Then for small enough $\rho>0$, any $\rho-$ perturbation $\G $ of $\F$ has the following properties:\
a) There exists a $\G-$invariant set $Y\subset X$ and a continuous map $p:Y\rightarrow B$ such that the diagram
$$\label{eqn:semi}
\begin{CD} Y @>{\G}>> Y \\
@V{p}VV @V{p}VV \\
\Lambda @>{h}>> \Lambda
\end{CD}$$
commutes. Moreover, the map $$\label{eqn:homeo1}
H: Y \to \Lambda \times M, \textrm{ }\textrm{ }\textrm{ } H(b,m)=(p(b,m),m)$$ is a homeomorphism.\
b) The fibers $W_b = p^{-1}(b)$ are Lipschitz close to vertical (constant) fibers, and continuous in $b$. This means that $W_b$ is the graph of a Lipschitz map $\widetilde{\beta}_b:M \to B$ such that $$\label{eqn:fibers}
d(\widetilde{\beta}_b,b)_{C^0} \leq O(\rho), \textrm{ }\textrm{ }\textrm{ }\mLip \widetilde{\beta}_b \leq O(\rho)$$
$$\label{eqn:hol}
d(\widetilde{\beta}_b,\widetilde{\beta}_{b'})_{C^0} \le \frac {d(b , b' )^{\a-O(\rho)}}{O(\rho)^\alpha},$$
where $$\label{eqn:defalpha}
\alpha=\min\left(\frac {\ln \lambda}{\ln \lambda_-}, \frac {\ln
\mu}{\ln \mu_-}\right).$$ Moreover, the map $H^{-1}$ is also continuous, with the same $\alpha$.
Let us first make a remark about the exponent $\alpha$. In many cases (e.g. when $h$ is the solenoid map or a linear Anosov diffeomorphism of a torus), it may happen that $\lambda_-=\lambda$ and $\mu_-=\mu$. In that case, in the above theorem we have $\alpha=1$, and thus the exponent can be made arbitrarily close to 1 by making $\rho$ small enough.
If $\Lambda=B$ (which would require $h$ to be surjective) the invariant set $Y$ equals the entire phase space $X$. This may be proven in similar fashion to Proposition \[prop:whole phase space\] below.
Let us explain the usefulness of this Theorem. Quite often, one may use skew products $\F$ to exhibit various dynamical or ergodic phenomena (see [@GI99], [@GI00], [@GIKN], [@IKS08], [@DG]). One would like to prove the same properties for small perturbations $\G$ of $\F$, but $\G$ is a priori not a skew product anymore. However, letting $G=H\circ \G|_Y\circ H^{-1}$, statement a) of the above theorem implies that $G:\Lambda \times M \rightarrow
\Lambda \times M$ is indeed a skew product: $$G(b,m)=(h(b),
g_b(m)).$$ One can then study the dynamical properties of the more mysterious map $\G|_Y$ by studying the dynamical properties of its conjugate skew product $G$.\
The fiber maps $g_b$ of the skew product $G$ are $C^1$-close to those of the skew product $\F|_Y$, in the following sense: $$\label{eqn:close} d(g_b^{\pm 1},f_b^{\pm 1})_{C^1}
\leq O(\rho).$$
But what can be said about the fiber maps $g_b$ for different $b$’s? Since $\F$ is a $C^1-$morphism, the fiber maps $f_b$ depend in a $C^1$ fashion on the point $b\in \Lambda$. Such a result fails for the fiber maps $g_b$, but statement b) of Theorem \[thm:mt\] implies that the fiber maps $g_b$ depend continuously on the point $b\in \Lambda$: $$\label{eqn:holfibermaps}
d(g_b^{\pm 1},g_{b'}^{\pm 1})_{C^0} \leq O(d(b,b')^\a),$$ where $\a$ is given by . A skew product $G$ whose fiber maps satisfy will be called a *skew product*. Thus Theorem \[thm:mt\] can be summarized as follows:
**Let $\G$ be any small perturbation of a $C^2$ skew product $\F$ over a $(\l_-, \l, \mu_-, \mu)-$ hyperbolic map $h$, satisfying the modified dominated splitting condition . Then $\G$ has an invariant set $Y$ such that the restriction of $\G|Y$ is conjugated to a skew product close to $\F|\L \times M$, in the sense of , and .**
The solenoid case {#sub:st}
-----------------
In this section we will present a partial improvement of Theorem \[thm:mt\] that is inspired by the example of the Smale-Williams solenoid. Let us begin by introducing and describing the solenoid map. Fix constants $R\geq 2$ and $\lambda<0.1$, whose particular values will not be important. Let $B$ denote the solid torus $$B = S^1 \times D, \textrm{ where }\ S^1 = \{y \in \rr
/\zz\}, \ D = \{z \in \mathbb C| |z| \le R \}.$$ The *solenoid map* is defined as $$\label{eqn:sol} h:B \rightarrow
B, \textrm{ }\textrm{ }\textrm{ } h(y, z) = (2y, e^{2\pi i y} +
\lambda z).$$ The maximal attractor of the solenoid map: $$\Lambda = \bigcap_{k=0}^\infty h^k(B)$$ is called the *Smale-Williams solenoid*. It is a hyperbolic invariant set with contraction coefficient $\lambda$ and expansion coefficient $\mu^{-1} = 2$ (we take the sup norm in $T_bB$ in the coordinates $y,z$). Moreover, the estimates in hold with $\lambda=\lambda_-$ and $\mu=\mu_-$.\
We can generalize the above to the following setup: let $B=Z\times
F$ be the product of two compact Riemannian manifolds; $F$ and $B$ may be with boundary. We suppose that $h:B \rightarrow B$ is a itself, i.e. there exists a $C^2-$morphism $\zeta:Z\rightarrow Z$ such that the following diagram commutes: $$\label{eqn:CD} \begin{CD} B @>{h}>> B \\ @V{\pi}VV
@V{\pi}VV \\ Z @>{\zeta}>> Z , \end{CD}$$ where $\pi$ is the standard projection. We assume that the map $\zeta$ downstairs is expanding, and that the fibers $\{z\}\times F$ are the stable manifolds of $h$: $$\label{eqn:zetaexp}
\mu_-\leq |d\zeta^{-1}|\leq \mu,$$ $$\lambda_-\leq
|dh|\leq \lambda \textrm{ }\textrm{ }\textrm{ on } \textrm{
}\textrm{ } T(\{z\} \times F) \textrm{ ,}\forall z\in Z.$$ Again, for technical reasons we assume that $E^s=TF$ is trivialized as in . In this setup, Theorem \[thm:mt\] can be partially improved by the following result.
\[thm:st\] Consider a skew product $\F $ as above that also satisfies the modified dominated splitting condition. Then for small enough $\rho>0$, any $\rho-$perturbation $\G $ of $\F$ has the following properties:\
a) There exists a continuous map $q : X \to Z$ such that the diagram $$\label{eqn:semisol}
\begin{CD}
X @>{\G}>> X \\
@V{q}VV @V{q}VV \\
Z @>{\zeta}>> Z
\end{CD}$$ commutes. Moreover, the commutative diagrams and must be compatible, in the sense that $q|_Y=\pi \circ p$.\
b)The fibers $W^s_z=q^{-1}(z)$ are Lipschitz close to the vertical (constant) fibers, and continuous in $z$. This means that $W^s_z$ is the graph of a Lipschitz map $\beta^s_z:F\times M \rightarrow Z$ such that $$\label{eqn:smallcs}
d(\beta^s_z,z)_{C^0}\leq O(\rho), \textrm{ }\textrm{ }\textrm{ }\mLip \beta^s_z \leq O(\rho)$$
$$\label{eqn:holdcs}
d(\beta^s_z ,\beta^s_{z'})_{C^0} \le \frac {d(z,z')^{\a-O(\rho)}}{O(\rho)^\alpha},$$
where $\a=\dsp \frac {\ln \mu}{\ln \mu_-}$.
As was mentioned above, a particularly important case in which the theorem applies is the Smale-Williams solenoid with $Z=S^1$, $F=D$ and $h$ given by .
Fubini revisited {#sub:fubini}
----------------
Let $\Sigma^2_+$ be the set of all sequences $\omega^+ =
\omega_0\omega_1\omega_2 \dots $ of zeroes and ones, infinite to the right with the $(\frac 1 2, \frac 1 2)$ Bernoulli measure. It is known that for almost all such sequences $\omega^+$, any finite word $w$ of any length $n$ is encountered within $\omega^+$ with frequency exactly equal to $2^{-n}$. We are interested in sequences for which this property fails. More precisely, given $\kappa>0$ and $w$ a finite word of length $n$, we say that $\om^+$ is $\kappa,w-$*atypical* if the sequence $$\label{eqn:freq} a_k(\omega^+,w):=\frac {\textrm{number of
occurences of }w\textrm{ among first }k\textrm{ digits of
}\omega^+}{k}$$ has a limit point *outside* $[2^{-n}-\kappa, 2^{-n}+\kappa]$. The sequence $\omega$ defines a point $y=\overline{0,\om^+}\in S^1$ written in base 2. We say that $y$ is $\kappa,w-$*atypical* if $\om^+$ is $\kappa
,w-$atypical. Let $A_{\kappa,w}\subset S^1$ be the set of atypical points. By the ergodic theorem, it has Lebesgue measure 0 in $S^1$.
\[thm:fubini\] Consider a $\F $ over the solenoid map as in Theorem \[thm:st\]. Let $\zeta : Z \to Z$ be the duplication of a circle. For any word $w$ and positive $\ka $ $\rho $ the following holds. Let $A_{\ka ,w}$ be the same as above. Then for any $\rho $ perturbation $\G $ of $\F $ and $q$ as in , we have: $$\label{eqn:semifub2} \emph{mes }q^{-1}(A_{\kappa,w}) = 0.$$ In other words, the union of $\kappa,w-$atypical fibers has Lebesgue measure 0 in $X$.
An analog of this theorem for perturbations of skew products over the Anosov of a two-torus may be proved basing on a recent result of P. Saltykov [@S09].
Attractors with intermingled basins {#sub:kan}
-----------------------------------
The tools developed in this paper allow us to get a new proof of the following phenomenon discovered by I. Kan:
\[thm:kan\] \[[@IKS08], [@KS*]\] The set of maps of an annulus $S^1\times [0,1]$ that have intermingled attracting basins is open in the set of all maps of the annulus into itself that keep the boundary invariant.
*Intermingled attracting basins* means the following thing: the Milnor attractor of the maps mentioned in Theorem \[thm:kan\] consists of the two boundary circles, each one having an attracting basin which is dense and of positive Lebesgue measure.\
In [@IKS08], [@KS*] the theorem above is improved by:\
*The complement to the union of the attracting basins in the perturbed Kan example has Hausdorff dimension smaller than 2.*\
Theorem \[thm:kan\], in a slightly different form, was claimed in [@K94]; as far as we know, the first proof was obtained in [@Bo]. The same tools also allow us to construct diffeomorphisms with intermingled attracting basins [@I08]; the phase space in this case is the product of a solid torus and a circle.
Rate of contraction of the graph transform map {#sec:grtrans}
==============================================
In this section we prove statement $a)$ of Theorem \[thm:mt\], and establish the rate of contraction of the graph transform map, see Lemma \[lem:1\] below. There are two ways to prove statement $a)$. The first one is to establish partial hyperbolicity of the $\F $, and refer to the theory. This theory implies the semiconjugacy statement $a)$, but gives no estimate of the rate of contraction of the graph transform map. The second way is to revisit the graph transform map and to prove simultaneously the fixed point theorem and the rate of contraction estimate for this map. This is done in the present section.
Laminations {#sub:lam}
-----------
Let $B,h,\Lambda$ be as in Subsection \[sub:perhol\]. In the fibers of the bundles $E^s$ and $E^u$ we have the abstract Riemannian metric, while in the fibers of the trivial bundles $B
\times \rr^k$ and $B \times \rr^l$ we have the standard Euclidean metric. The isomorphism $\varphi^s$ of implies that there exist $k$ linearly independent sections of $E^s$. By applying Gram-Schimdt orthonormalization to these sections, it follows that there exist $k$ orthonormal sections of $E^s$. Sending a fixed orthonormal basis of $\rr^k$ to these orthonormal sections will give us a *metric-preserving* isomorphism $B \times \rr^k
\rightarrow E^s$, and it is this isomorphism that we will henceforth denote by $\varphi^s$. The same discussion applies to $\varphi^u$.\
For any $\delta>0$, we define $Q^s(\delta)$ and $Q^u(\delta)$ to be the open balls of radius $\delta$ around the origin of $\rr^k$ and $\rr^l$, respectively. The metric-preserving isomorphisms $\varphi^s$ and $\varphi^u$ induce metric-preserving isomorphisms in each fiber: $$\label{eqn:trivfiber} \varphi^s_b(\delta):Q^s(\delta) \rightarrow
Q^s_b(\delta), \textrm{ }\textrm{ }\textrm{ } \varphi^u_b(\delta):Q^u(\delta) \rightarrow
Q^u_b(\delta),$$ where $Q^s_b(\delta) \subset E^s$ and $Q^u_b(\delta) \subset
E^u$ are the open balls of radius $\delta$ around the origin in the respective fibers.\
The number $\delta$ must be chosen small enough such that for any $b\in B$, the exponential map gives us an open embedding $Q^s_b(\delta) \times Q^u_b(\delta)
\hookrightarrow B$. We write $B_b(\delta)$ for the image of this map. Composing this embedding with the isomorphism $\varphi^s_b(\delta) \times
\varphi^u_b(\delta)$ gives us an open embedding (coordinate chart):
$$\label{eqn:fixiso} \varphi_b(\delta):Q^s(\delta) \times Q^u(\delta) \hookrightarrow B.$$
Let us take $C>\textrm{max}(\lambda_-^{-1},\mu_-^{-1})$, and consider the above constructions for radius $C\delta$. Then we can express the map $h:B\rightarrow B$ locally around $b$ in the domain and around $h(b)$ in the target. Therefore, in coordinates given by the chart , the map $h$ has the form:
$$\label{eqn:hcoord} h_b(\delta) =
(\ph_{h(b)}(C\delta))^{-1} \circ h \circ \ph_b(\delta), \textrm{
}\textrm{ }\textrm{ } {(h^{-1})}_b(\delta) = (\ph_{b}(C\delta))^{-1}
\circ h^{-1} \circ \ph_{h(b)}(\delta).$$
For various values of $\delta$, the maps $h_b(\delta)$ will represent the same germ at $0$, but will have different domains. Similarly, the maps ${(h_b)}^{-1}(\delta)$ and ${(h^{-1})}_b(\delta)$ are representatives of the same germ at $0$, but have different domains.\
Until Section \[sec:hold\], we will work with a single, fixed $\delta$. Therefore, we will often write simply $Q^s,Q^u,Q^s_b,Q^u_b,B_b,\ph^s_b,\ph^u_b,\ph_b,h_b,(h^{-1})_b$ for the notions introduced in the previous paragraphs. By and the fact that are metric-preserving, $dh_b$ has block-diagonal form at $0$: $$dh_b(0) = \textrm{diag} (A^u, A^s),$$ where $\lambda_-\leq |A_s|\leq \lambda$ and $\mu_-\leq
|A_u^{-1}|\leq \mu$. Because the coordinate charts $\ph_b$ are smooth functions, we have the following estimate throughout $Q^s\times Q^u$: $$\label{eqn:esth} {||dh_b -
\textrm{diag} (A^u, A^s)||}_{C^0} \le O(\delta ).$$
Now consider another compact manifold $M$, as in the statement of Theorem \[thm:mt\]. For any domain $A$ and any mapping $\b : A \to B$, we will denote by $\ga (\b )$ the map from $A$ onto the graph:
$$\label{eqn:gamma} \ga(\b):A \rightarrow A \times
B, \textrm{ }\textrm{ }\textrm{ } \ga (\b ): a \mapsto (a, \b (a))
\in A \times B.$$
Statement $a)$ of Theorem \[thm:mt\] provides a correspondence between leaves and base points, so it’s about time we defined these. The leaves of center-stable, center-unstable and center foliations corresponding to $b \in \L$ are represented by Lipschitz maps: $$\label{eqn:leaf} \beta^s_b:Q^s\times M \rightarrow Q^u,\textrm{
}\textrm{ } \textrm{ }\beta^u_b:Q^u\times M \rightarrow Q^s,\textrm{
}\textrm{ } \textrm{ }\beta_b:M \rightarrow Q^u \times Q^s.$$ Then we define the leaves to be simply the graphs of the Lipschitz maps, embedded in $B\times M$ via :
$$\label{eqn:leaf1} W_b^* = \mbox{Im}(\ph_b \times
\textrm{Id}) \circ \ga (\b_b^*)$$
Here and below, $*$ stands for $s, u$ or blank space.\
Intuitively, $W_b^s$ denotes a center-stable leaf, $W_b^u$ denotes a central-unstable leaf, while $W_b$ denotes a central leaf. We will never consider strongly stable or unstable leaves.\
We now define certain functional spaces $\B^*$ of maps $\b^*$. These are, by definition, the spaces of Lipschitz maps that satisfy the condition:
$$\label{eqn:max} \max \left\{ \| \b^* \|_{C^0}, \frac {\mLip \b^*}D \right\}
\le \frac {\delta}2$$
Here, $D$ is a constant that will be picked in the proof of Lemma \[lem:1\]. The norm on the spaces $\B^*$ will always be the $C^0$ norm, and will be denoted by $||\cdot||$.\
Intuitively speaking, a central-stable, central-unstable or central *lamination* is a continuous assignment of leaves as $b$ runs over $\Lambda$. Rigorously speaking, a lamination is a continuous map:
$$\label{eqn:section} S^*:\Lambda \rightarrow \BB^*.$$
The map $S^*$ is completely determined by the continuous collection of maps $\b^*_b=S^*(b)$, as $b$ ranges over $\Lambda$. Equivalently, $S^*$ is completely determined by the leaves $W^*_b$ of these maps.\
The space of continuous sections $S^*$ as above is denoted by $\Ga^*$. The norm in this space is again the $C^0$ norm:
$$\| S^*\| = \max_{b \in \L} \|S^*(b)\| .$$ For any $\delta > 0$ small enough, the metric space $\Ga^*$ with the distance $\rho (S^*_1, S^*_2) = ||S^*_1 - S^*_2||$ is complete. Indeed, if $\b^*_n \to \b^*$ and $\mLip \b^*_n \le D\delta/2 $, then $\mLip \b^* \le D\delta/2$.\
Now consider a map $\G:B\times M \rightarrow B \times M$, like in the setup of Theorem \[thm:mt\]. A central-stable, central-unstable or central lamination is called $\G-$invariant if its leaves $W^*_b$ satisfy: $$\label{eqn:invfol1} \G(W^{s}_{b}) \subset W^s_{h(b)}, \textrm{
}\textrm{ }\textrm{ } W^u_{h(b)} \subset \G(W^u_b)$$ $$\label{eqn:invfol2}\textrm{or }\textrm{ }\textrm{ } W_{h(b)} = \G(W_b).$$ These conditions can all be written in terms of the maps $\beta^*_b$ defining these leaves, and thus in terms of laminations $S^*$ themselves. This will be done in the beginning of Subsection \[sub:graph\].\
Our plan for the proof of Statement $a)$ of Theorem \[thm:mt\] will be the following: we will use the graph transform method described in the following subsection to find $\G-$invariant central-stable and central-unstable laminations. Then the central lamination will be given by $$W_b=W^s_b\cap W^u_b.$$ Property will follow from , so the central lamination will also be invariant under $\G$. Once we have the central lamination, we will define $$Y=\bigsqcup_{b\in \Lambda} W_b.$$ Sending $W_b$ to $b$ defines the desired projection map $p:Y\rightarrow \Lambda$ of . Then the $\G-$invariance of the central lamination is precisely equivalent to the commutativity of diagram . We will follow this plan in the next subsections.
The graph transform map {#sub:graph}
-----------------------
Here we will deal with the $*=s$ case only, since the $*=u$ case is treated similarly. After that, the central case will be treated as described above. We will introduce first a “pointwise” graph transform map:
$$\g_b:\B^s \longrightarrow \B^s$$ that acts on single leaves, and then a “global” graph transform map:
$$\g:\Ga^s \longrightarrow \Ga^s$$ that acts on entire laminations. In both cases, the geometric idea is the same: start with a map $\beta^s:Q^s \times M \longrightarrow Q^u$ as in . Take the corresponding leaf $W_{h(b)}^s \subset B
\times M$, and take its inverse image under $\G$. The claim is that we obtain a different leaf $\overline{W}_{b}^s \subset B \times M$, corresponding to a map $\overline{\beta}^s: Q^s \times M
\longrightarrow Q^u$. Then we define the graph transform map as: $$\g_b(\beta^s)=\overline{\beta}^s.$$ In other words, the graph transform is implicitly defined by the following relation:
$$\label{eqn:implicitdef}
\{\G^{-1}(\ph_{h(b)}(x_s,\beta^s(x_s,m)),m)\}=\{(\ph_b(x_s,\overline{\beta}^s(x_s,m)),m)\}.$$
We will prove in the appendix that the above correctly defines $\overline{\beta}^s$ (in other words, that the Implicit Function Theorem applies). The above definition also works in families. For a lamination $S^s \in \Gamma^s$ with leaves that are graphs of $\beta^s=S^s(h(b))$, define its graph transform as: $$\g(S^s)=(\overline{S}^s),$$ where $\overline{S}^s(b)=\overline{\beta}^s$ is defined by relation .\
Comparing with , we see that a lamination $S^s$ is $\G-$invariant if and only if it is a fixed point of the graph transform map $\g$. Therefore, to show that there exists a unique $\G-$invariant central-stable lamination, we will use the fixed point principle: it is enough to show that $\g$ is well defined and contracting.
\[lem:1\] For $\rho$ small enough and any $\F, \G$ as in Theorem \[thm:mt\], there exists $\delta=O(\rho)$ so that the graph transform $\g$ maps $\Ga^s$ into itself and is contracting with Lipschitz constant $\mu+O(\delta)$. In other words, for any $S^s_0,
S^s_1 \in \Ga^s $ we have: $$\label{eqn:cont1} \|
\g(S_0^s) - \g(S_1^s) \| \le (\mu + O(\delta))\| S_0^s - S_1^s\| .$$ In the pointwise situation, for any $b\in \Lambda$, we claim that $\g_b$ maps $\B^s$ into itself. Furthermore, for any $\b^s_0,\beta^s_1 \in \B^s$, we have: $$\label{eqn:36a}
\| \g_b(\b_0^s) - \g_b(\b^s_1)\| \leq (\mu + O(\delta ))\| {\b^s_0} - {\b^s_1} \|.$$
\[cor:fixed\] For $\delta=O(\rho)$ small enough, the graph transform map $\g$ has a unique fixed point in $\Ga^s$.
The statements about the global graph transform immediately follow from the corresponding statements in the pointwise case. So let us start by proving that $\g_b$ maps $\B^s$ to itself. Take $b\in \La$, $\b^s \in \B^s$ and let $\overline{\beta}^s = \g_b(\b^s)$. We need to prove that: $$\label{eqn:cnorm} \| \overline{\b}^s \| \le \frac {\delta}2,$$ $$\label{eqn:lip} \mLip \overline{\b}^s \le \frac
{D\delta}2.$$ Recall that $\gamma_\b $ is the map of $Q^s \times M$ onto the graph of $\b^s$, see . In the Appendix we prove that for any $\b = \b^s$ that satisfies , there exists a Lipshitz homeomorphism $G_{\bar \b ,
b}: Q^s \times M \to Q^s \times M$, see , such that $$\overline{\b}^s = \pi_u \circ \G_b^{-1} \circ \gamma (\b^s) \circ
G_{\overline\b, b}.$$ Here $\G_b ={(\ph_{h(b)}\times
\textrm{Id})}^{-1} \circ \G \circ (\ph_b \times \textrm{Id})$. Note that $${||\overline{\b}^s||} \le ||\pi_u \circ \G_b^{-1} \circ
\gamma(\b^s)||,$$ because the shift in the argument of the right hand side does not change the $C^0$ norm. Therefore, by , we have: $$||{\overline{\b}}^s|| \le
||\pi_u \circ \F_b^{-1} \circ \gamma(\b^s)|| + O(\rho) = ||\pi_u
\circ (h^{-1})_b \circ \gamma(\b^s)|| + O(\rho).$$ By , we can further estimate the above: $$||{\overline{\b}}^s|| \leq (\mu + O(\delta))||\b^s||+O(\rho).$$ Since $\mu<1$ and $||\b^s||\leq \delta/2$, for appropriately chosen $\rho=O(\delta)$ the above can be made $\leq \delta/2$. This proves . As for , note that $$\label{eqn:estlip} \mLip {\overline{\b}^s} \le
\mLip (\pi_u \circ \G_b^{-1}\circ \gamma(\b^s)) \cdot \mLip
G_{\overline\b, b}.$$ We need to show that the right hand side of the above is $\leq D\delta/2$. It is enough to do this for $\b^s$ and $\overline\beta^s$ of class $C^1$, since these maps are dense in $\B^s$. In this $C^1$ case, we have:
$$d(\pi_u \circ \G_b^{-1} \circ \gamma(\b^s)) = d(\pi_u \circ \G_b^{-1})\circ \gamma(\b^s) \cdot d\gamma(\b^s) \leq$$
$$\leq \left[d(\pi_u \circ \F_b^{-1})\circ \gamma(\b^s) +O(\rho)\right]\cdot d\gamma(\b^s) \leq$$
$$\leq \left[\left( \begin{array}{ccc} \frac {\partial \pi_u \circ
h^{-1}_b}{\partial x_s} & \frac {\pi_u \circ \partial
h^{-1}_b}{\partial x_u} & 0 \\ \end{array} \right) \circ
\gamma(\b^s) +O(\rho) \right] \cdot \left( \begin{array}{cc} 1 & 0
\\ \frac {\partial \beta^s}{\partial x_s} & \frac {\partial
\beta^s}{\partial m} \\ 0 & 1 \\ \end{array} \right) \leq$$
$$\leq \left[\left( \begin{array}{ccc} 0 & \mu & 0 \\ \end{array}
\right)+O(\delta)+O(\rho) \right] \cdot \left( \begin{array}{cc} 1 &
0 \\ \frac {\partial \beta^s}{\partial x_s} & \frac {\partial
\beta^s}{\partial m} \\ 0 & 1 \\ \end{array} \right) \leq$$
$$\label{eqn:ppp} \leq \left( \begin{array}{cc}
\mu\cdot \frac {\partial \beta^s}{\partial x_s} & \mu\cdot \frac
{\partial \beta^s}{\partial m} \\ \end{array} \right)+O(\delta)+
O(\rho)\leq \mu\cdot \mLip \beta^s + O(\delta),$$
$$$$ since $\rho=O(\delta)$. Combining this estimate with Proposition \[prop:composition\] of the Appendix, we see that: $$\mLip {\overline\beta^s} \leq (\mu \cdot \mLip \beta^s +
O(\delta))\cdot (L + O(\delta))\cdot (1+\textrm{Lip }\overline\b^s).$$ Since $\mLip \beta^s \leq D\delta/2$, the above gives us: $$\mLip {\overline\beta^s} \leq \frac {\mu L \cdot D\delta/2+L\cdot
O(\delta)+O(\delta^2)}{1-\mu L \cdot D\delta/2-L\cdot
O(\delta)-O(\delta^2)}.$$ By assumption , we have $\mu L<1$. Therefore, if we pick the constant $D$ large enough (but still requiring that $D\delta<<1$), the right hand side of the above will be $\leq D\delta/2$. This proves .\
Now that we have proved $\g$ and $\g_b$ to be well-defined, let us pass to proving and . As we said before, the second inequality implies the first, so we will only prove the second one. As above, write $\overline\b^s_0=\g_b(\beta^s_0)$ and $\overline\b^s_1=\g_b(\beta^s_1)$. From , we see that:
$$\label{eqn:estimates0}
||\overline\b^s_0-\overline\b^s_1|| \leq T_1+T_2,$$
where:
$$T_1 =||\pi_u \circ \G_{b}^{-1} \circ \ga({\b^s_0})\circ G_{\overline\b_0,b} -\pi_u \circ \G_{b}^{-1}
\circ \ga({\b^s_1})\circ G_{\overline\b_0,b}||,$$
$$T_2=||\pi_u \circ \G_{b}^{-1} \circ \ga({\b^s_1}) \circ G_{\overline\b_0,b} - \pi_u \circ \G_{b}^{-1}
\circ \ga({\b^s_1}) \circ G_{\overline\b_1,b}||.$$ As it will soon be clear, $T_1$ is the dominant term:
$$T_1 \leq \textrm{Lip }(\pi_u \circ \G_{b}^{-1}) \cdot
||\ga({\b^s_0}) -\ga({\b^s_1})||.$$ The second factor in the right hand side is $\leq ||{\b^s_0} -{\b^s_1}||$. As for the first factor, we see that: $$\textrm{Lip }(\pi_u \circ \G_{b}^{-1}) \leq
\textrm{Lip }(\pi_u \circ \F_{b}^{-1}) + O(\rho)=\textrm{Lip }(\pi_u
\circ h_{b}^{-1}) + O(\rho) \leq \mu+O(\rho).$$ Since $\rho=O(\delta)$, we conclude that:
$$\label{eqn:t1}
T_1 \leq (\mu+O(\delta))\cdot ||{\b^s_0} -{\b^s_1}||.$$
As for $T_2$, we see that:
$$T_2\leq \textrm{Lip }(\pi_u \circ \G_{b}^{-1} \circ
\ga({\b^s_1}))\cdot ||G_{\overline\b_0,b} - G_{\overline\b_1,b}||.$$ In , we saw that: $$\mLip (\pi_u \circ
\G_{b}^{-1} \circ \gamma({\b^s_1})) \leq O(\delta).$$ In Proposition \[prop:last\] of the Appendix, we will prove that $$||G_{\overline\b_0,b} - G_{\overline\b_1,b}|| \leq O(1)\cdot
||\overline\beta^s_0-\overline\beta^s_1||.$$ Therefore, we obtain:
$$T_2 \leq O(\delta)\cdot ||\overline\beta^s_0-\overline\beta^s_1||.$$ Together with , this implies: $$||\overline\b^s_0-\overline\b^s_1|| \leq (\mu+O(\delta))\cdot ||{\b^s_0} -{\b^s_1}|| + O(\delta)
\cdot ||\overline\beta^s_0-\overline\beta^s_1|| \Rightarrow$$ $$||\overline\b^s_0-\overline\b^s_1|| \leq (\mu+O(\delta))\cdot ||{\b^s_0} -{\b^s_1}||.$$ This is precisely the desired inequality .
The central lamination {#sub:central}
----------------------
Corollary \[cor:fixed\] tells us that there exists a unique $\G-$invariant central stable lamination $S^s \in \Ga^s$. This can be presented either via the maps $\beta^s_b$, or via the leaves $W^s_b$ (as $b$ ranges over $\Lambda$). Similarly, there exists a unique $\G-$invariant central unstable lamination $S^u\in \Ga^u$. Let us define the central lamination $S$ via its leaves $W_b$, which we define by: $$\label{eqn:defcentral}
W_b=W^s_b\cap W^u_b.$$ This lamination will be $\G-$invariant, in the sense of . Let us describe $W_b$ more explicitly. By definition, $$W^s_b=\textrm{Im
}(\varphi_b \times \textrm{Id})\{(x_s,\beta^s_b(x_s,m),m)|\textrm{
}x_s\in Q^s, m\in M\}$$ $$W^u_b=\textrm{Im }(\varphi_b \times
\textrm{Id})\{(\beta^u_b(x_u,m),x_u,m)|\textrm{ }x_u\in Q^u, m\in
M\},$$ where $\beta_b^s,\beta_b^u$ have Lipschitz norms at most $D\delta/2<<1$. Then, for each $m\in M$, the system of equations $$\label{eqn:system} \left\{
\begin{array}{ll}
x_s=\beta^u_b(x_u,m) \\
x_u=\beta^s_b(x_s,m)
\end{array}
\right.$$ has a unique solution $(x_s,x_u)=:\beta_b(m)\in Q^s\times Q^u$. Indeed, for any fixed $m$ the maps $\b^s_b \circ \b^u_b: Q^u \to Q^u$ and $\b^u_b \circ
\b^s_b: Q^s \to Q^s$ are Lipshitz with constant $\le {(L\delta
)}^2 << 1$. Then each of the two maps is contracting and has a unique fixed point: call these $x_u$ and $x_s$, respectively. Then the pair $(x_s,x_u)$ is the solution of , and the above map $\beta_b$ is well-defined. If we define the map $\widetilde{\beta}_b=\varphi_b(\beta_b):M \rightarrow B$, then its graph is precisely $W_b$: $$W_b=\{(\widetilde{\beta}_b(m),m)|\textrm{ }m\in M\}.$$ Because it is the intersection of an invariant central-stable lamination with an invariant central-unstable lamination, $S=(\beta_b)=(W_b)$ is an invariant central lamination.\
It is not hard to see from that $$\label{eqn:smallb}
||\beta_b||_{C^0} \leq \frac {\delta}2 \textrm{ }\textrm{ }\textrm{ and }\textrm{ }\textrm{ }\frac {\mLip \beta_b}D \leq \delta.$$ Because the chart $\ph_b$ is metric preserving at 0 and smooth in the domain $Q^s\times Q^u$ (which has diameter of order $\delta$), we have: $$\label{eqn:smalltildeb}
d(\widetilde{\beta}_b,b)_{C^0} \leq O(\delta)=O(\rho) \textrm{, }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\mLip
\widetilde{\beta}_b \leq O(\delta)=O(\rho).$$
**of statement a) of Theorem \[thm:mt\]:** Start from the $\G-$invariant central lamination $S$ constructed above, and define $Y=\bigcup_{b\in \Lambda} W_b$. This union is obviously an invariant set of $\G$, and moreover the following proposition implies that it is actually a disjoint union.
\[prop:disjointfibers\] For all $b\neq b' \in
\Lambda$, the corresponding central leaves are disjoint: $$W_b\cap
W_{b'}=\emptyset.$$
Let us assume by contraposition that $W_b\cap
W_{b'} \neq \emptyset$. By the $\G-$ invariance of the lamination, then $$W_{h^k(b)}\cap W_{h^k(b')} \neq \emptyset,$$ for all $k\in
\zz$. Pick a point $(\tilde{b},m)$ in the above non-empty intersection. Then $$\label{eqn:tildeb}
\widetilde{b}=\widetilde{\beta}_{h^k(b)}(m)=\widetilde{\beta}_{h^k(b')}(m)$$ By , the point $\widetilde{\beta}_{h^k(b)}(m)$ is at distance at most $O(\rho)$ from $h^k(b)$. Similarly, $\widetilde{\beta}_{h^k(b')}(m)$ is at distance at most $O(\rho)$ from $h^k(b')$. This implies that $h^k(b)$ and $h^k(b')$ are at most $2\cdot O(\rho)$ apart, for all $k\in \mathbb{Z}$. This is obviously impossible for $\rho$ small enough, because for such $\rho $, the quantity $ O(\rho )$ is smaller than the expansivity constant of $h$.\
Therefore, the map $p:Y \rightarrow \Lambda$ sending $W_b$ to $b$ is well-defined. Moreover, the $\G-$invariance of the lamination $S=(W_b)$ is precisely equivalent to the commutativity of the diagram . The continuity of $p$ follows from the continuity of our laminations, and this also implies that the map $H$ of is continuous.\
Note that the map $H$ is bijective, with inverse given by $H^{-1}(b,m)=(\widetilde{\beta}_b(m),m)$. The map $H^{-1}$ is clearly continuous in $m$, and continuity in $b$ follows from the continuity statement , which will be proved in the next subsection. Therefore $H$ is a homeomorphism, thus concluding the proof of statement $a)$.
Hölder continuity of the central lamination {#sec:hold}
===========================================
This section will be concerned with the proof of statement $b)$ of Theorem \[thm:mt\]. By definition, we have $p^{-1}(b)=W_b=\textrm{Graph}(\widetilde{\beta}_b)$, where $\widetilde{\beta}_b$ satisfies relations . This is precisely the requirement . In this section we will prove the rest of statement $b)$, which refers to continuity.\
First, for any $b\in \Lambda$, we will define its *local central-stable* and *central-unstable manifolds* as $$V^s_b=\{b'\in B|d(h^n(b'),h^n(b))\leq \delta
\textrm{, }\forall n\geq 0\},$$ $$V^u_b=\{b'\in
B|d(h^{-n}(b'),h^{-n}(b))\leq \delta \textrm{, }\forall n\geq 0\}.$$
\[prop:inthyp\] Let $h, \Lambda $ and $d$ be the same as at the beginning of Subsection \[sub:perhol\]. Then the following statements hold for all $b,b'\in \Lambda$:
1. $\emph{if } b'\in V^s_b \emph{ and } d(h^{-1}(b), h^{-1}(b'))\leq \delta \Rightarrow h^{-1}(b')\in V^s_{h^{-1}(b)}$
2. $\emph{if } b'\in V^u_b \emph{ and } d(h(b), h(b'))\leq \delta \Rightarrow h(b')\in V^u_{h(b)}$
3. $\emph{if } b'\in V^s_b \Rightarrow \lambda_- - O(\delta) \leq \dsp \frac {d(h(b),h(b'))}{d(b,b')}\leq \lambda+O(\delta) $
4. $\emph{if } b'\in V^u_b \Rightarrow \mu_- - O(\delta) \leq \dsp \frac {d(h^{-1}(b),h^{-1}(b'))}{d(b,b')} \leq \mu+O(\delta)$
Statements 1 and 2 follow immediately from the definitions of $V^s_b,V^u_b$. We will now prove Statements $3$ and $4$. The map $h$ has invariant stable and unstable laminations; $V^s_b, V^u_b$ are the leaves of these laminations. They are smooth manifolds, and $V^s_b (V^u_b)$ is tangent at $b$ to $E^s (E^u)$. Now Statements $3$ and $4$ follow from and the $C^2$-smoothness of $h$.
We further ask that $h$ has the following *local product structure*: for all $b,b'\in \Lambda$ such that $d(b,b')\leq
\delta$, there exists a unique $b^*\in B$ such that $$\label{eqn:prodstr} V^u_b \cap V^s_{b'}=\{b^*\},$$ and moreover: $$\label{eqn:prod} d(b,b^*)+d(b',b^*)\leq O(d(b,b')).$$ This property is easily seen to hold for linear Anosov diffeomorphisms of the torus, because then $V^s_b$ and $V^u_{b'}$ are just straight lines that meet transversely under a fixed angle independent of $b,b'$. It also holds for the Smale-Williams solenoid, because then $V^s_{b}=\{y(b)\}\times D$ and $V^u_{b'}$ is a curve that intersects $V^s_{b}$ transversely (such that the angle between $V^s_b$ and $V^u_{b'}$ is separated from zero).\
**of statement b) of Theorem \[thm:mt\]:** We have already proved the closeness property in relation above. As for the Hölder property , it is enough to prove it for $b,b'$ which are at most $\delta $ apart. Indeed, for any $\a > 0$ and any $b,b'$ with $d(b,b') > \delta $, we have by default: $$\label{eqn:hhol} d(h(b), h(b')) \le Cd^\a (b,b'),$$ where $C = \frac {\mbox{diam} B}{\delta^\a }$. Therefore, we can restrict attention to $b,b'$ that are such that the unique point $b^*$ of satisfies:
$$\label{eqn:closeness} d(b,b^*) \leq \delta, \textrm{ }d(b',b^*) \leq \delta, \textrm{ }
d(b,b') \leq \delta,$$
For such nearby $b,b'$, we essentially need to estimate the distance between the maps $\widetilde{\beta}_b,\widetilde{\beta}_{b'}:M \rightarrow B$. These maps were defined by the condition that their graphs coincide with $W^s_b \cap W^u_b$ and $W^s_{b'}\cap W^u_{b'}$, respectively.\
However, this is a bit of an issue: *different* leaves $W^s_b$ and $W^s_{b^*}$ (and also their $u$ counterparts) are defined using *different* coordinate charts $\ph_b$ and $\ph_{b^*}$. To resolve this problem in the $s$-case (the $u$-case is treated in the same way), let us write $W^s_{b}$ as the graph of a function $\overline{\b}_b^s:Q^s \times M \rightarrow Q^u$ in the coordinate chart $\ph_{b^*}$: $$\{(\ph_b \times
\textrm{Id}_M)(x_s,\beta^s_b(x_s,m),m)\}
\stackrel{\textrm{def}}{=}W^s_b =\{(\ph_{b^*} \times
\textrm{Id}_M)(x_s,\overline{\beta}^s_b(x_s,m),m)\}$$ The function $\overline{\beta}^s_b$ is defined uniquely and implicitly by the above relation, but we must require the inclusion $\textrm{Im}(\ph_b) \subset \textrm{Im}(\ph_{b^*})$. We certainly cannot ensure this if we define the charts $\ph_b$ and $\ph_{b^*}$ with respect to the same $\delta$ in . But if we define $\ph_{b^*}$ with respect to $3\delta$ instead of $\delta$ (i.e. define the chart on a neighborhood 3 times bigger), then the desired inclusion becomes a consequence of .
With the assumption , we define the distance between the leaves corresponding to $d, b^*$, to be $$d(W^s_b,W^s_{b^*}) := ||\beta^s_{b^*}-\overline{\beta}^s_b||.$$
Implicit in the definition is the fact that the right hand side only makes sense on the domain of $\overline{\beta}^s_b$, which as was said before, is contained in the domain of $\beta^s_{b^*}$. Note that the above definition is *not* symmetric in $b$ and $b^*$.\
Now we must look at what happens with these leaves under the graph transform. Take two leaves $W^s_{h(b)}$ and $W^s_{h(b^*)}$, given in the coordinate chart $\ph_{h(b^*)}$ by maps $\overline{\beta}^s_{h(b)}$ and $\beta^s_{h(b^*)}$, respectively. Then take their images under the graph transform $W^s_{b}$ and $W^s_{b^*}$, given in the coordinate chart $\ph_{b^*}$ by maps $\overline{\beta}^s_{b}$ and $\beta^s_{b^*}$. The inequality of Lemma \[lem:1\] precisely says that: $$\label{eqn:contWs} d(W^s_b, W^s_{b^*}) \leq
(\mu+O(\delta)) \cdot d(W^s_{h(b)}, W^s_{h(b^*)}).$$ Doing the analogous computations for central-unstable foliations, we see that: $$\label{eqn:contWu} d(W^u_{b'}, W^u_{b^*})
\leq (\lambda+O(\delta)) \cdot d(W^u_{h^{-1}(b')},
W^u_{h^{-1}(b^*)}).$$ Now recall that we fixed points $b,b'$ satisfying relation . Let us consider the positive integers: $$k=\left\lfloor \log_{\mu_--O(\rho)} \frac
{d(b,b^*)}{\delta}\right\rfloor, \textrm{ }\textrm{ }\textrm{
}l=\left\lfloor \log_{\lambda_--O(\rho)} \frac
{d(b',b^*)}{\delta}\right\rfloor.$$ Iterate relation $k$ times, and we obtain: $$d(W^s_b, W^s_{b^*})
\leq (\mu+O(\delta))^k \cdot d(W^s_{h^k(b)}, W^s_{h^k(b^*)}).$$ By the definition of $k$ (and property 4 of Proposition \[prop:inthyp\]), $k$ is the biggest positive integer which would ensure that the points $h^k(b)$ and $h^k(b^*)$ remain at most distance $\delta$ apart. Indeed, if they were at a bigger distance apart, the entire discussion above would break down. But since the distance between $h^k(b)$ and $h^k(b^*)$ is at most $\delta$, we infer that the distance between the corresponding leaves is also at most $O(\delta)$. Therefore, the above inequality implies:
$$d(W^s_b, W^s_{b^*}) \leq (\mu+O(\delta))^k \cdot O(\delta) \leq
d(b,b^*)^{\frac {\ln \mu}{\ln \mu_-}-O(\delta)} \cdot O(1).$$ The analogous discussion with $l,\lambda,b',u$ instead of $k,\mu,b,s$ gives us: $$d(W^u_{b'}, W^u_{b^*}) \leq (\lambda+O(\delta))^l \cdot
O (\delta )\leq d(b',b^*)^{\frac {\ln \lambda}{\ln
\lambda-}-O(\delta)} \cdot O(1).$$ Letting $\alpha$ be defined as in , the above relations give us:
$$\label{eqn:aa} d(W^s_b, W^s_{b^*}) \leq
d(b,b^*)^{\alpha-O(\delta)} \cdot O(1),\ \ \textrm{ }d(W^u_{b'},
W^u_{b^*}) \leq d(b',b^*)^{\alpha-O(\delta)} \cdot O(1).$$
Let’s now prove that $$\label{eqn:inc} W^s_{b'}
\subset W^s_{b^*}, \textrm{ }\textrm{ }\textrm{ and analogously
}\textrm{ }\textrm{ } W^u_b \subset W^u_{b^*}.$$ Relation for $b$ replaced with $b'$ becomes: $$d(W^s_{b'}, W^s_{b^*}) \leq (\mu+O(\delta)) \cdot d(W^s_{h(b')},
W^s_{h(b^*)}).$$ However, since $b'\in V^s_{b^*}$, then the map $h$ actually brings the points $b'$ and $b^*$ closer together (by property 3 of Proposition \[prop:inthyp\]). So we can iterate the above inequalities as many times as we want. We see that: $$d(W^s_{b'}, W^s_{b^*}) \leq (\mu+O(\delta))^i \cdot d(W^s_{h^i(b')},
W^s_{h^i(b^*)}),$$ for any $i>0$. As $i \rightarrow \infty$, this implies $d(W^s_{b'}, W^s_{b^*})=0$. This proves in the $s$-case. The proof in the $u$-case is similar.\
We can now turn to the proof of , thus completing the proof of Theorem \[thm:mt\]. Recall that for any $b \in B, \ W_b =
W^s_b \cap W^u_b$. Let us first prove that $$\label{eqn:star} d(W_b, W_{b^*}) \le d(b, b_*)^{\a -O(\delta )}\cdot O(1),
\ d(W_{b'}, W_{b^*}) \le d(b', b_*)^{\a -O(\delta )}\cdot O(1).$$ By we have: $$\label{eqn:center} W_b
= W^s_b \cap W^u_{b}, \ W_{b^*} = W^s_{b^*} \cap W^u_{b^*}.$$ In the chart $\ph_{b^*} \times \textrm{Id}$, the leaves $W^s_{b^*}$, $W^u_{b^*}$, $W^s_b$, $W^u_{b}$, $W^s_{b'}$, $W^u_{b'}$ are given by maps $\beta^s_{b^*}$, $\beta^u_{b^*}$, $\overline{\beta}^s_b$, $\overline{\beta}^u_{b}$, $\overline{\beta}^s_{b'}$, $\overline{\beta}^u_{b'}$. Then gives us: $$||\beta^s_{b^*}-\overline{\beta}^s_b||,
\textrm{ }||\beta^u_{b^*}-\overline{\beta}^u_{b'}|| \leq
d(b,b')^{\alpha-O(\delta)} \cdot O(1),$$ while gives us: $$\beta^s_{b^*} = \overline{\beta}^s_{b'}, \textrm{ }
\beta^u_{b^*}=\overline{\beta}^u_b.$$ Of course, when one reads the above inequalities, one should keep in mind that the maps $\beta^{s,u}_{b^*}$ are defined on a neighborhood 3 times bigger than the maps $\overline{\beta}^{s,u}_{b,b'}$. Actually, the domain of the maps $\beta^{s,u}_{b^*}$ strictly contains the domain of the maps $\overline{\beta}^{s,u}_{b,b'}$. Therefore the above relations should be understood on the smaller domain, on which the maps $\overline{\beta}^{s,u}_{b,b'}$ are actually defined.\
Now, relation is equivalent to $\overline{\b}_b(m) = (x_s, x_u)$ and $\b_{b^*}(m) = (x^*_s, x^*_u)$, where: $$\label{eqn:systems2}\begin{cases} x_s = \overline{\b}^u_{b}(x_u, m) \\ x_u = \overline{\b}^s_b(x_s,
m) \end{cases} \ \begin{cases} x_s^* = \b^u_{b^*}(x^*_u,m) \\
x_u^* = \b^s_{b^*}(x^*_s, m) \end{cases}$$
For fixed $m$, the solutions $(x_s, x_u)$ and $(x^*_s, x^*_u)$ are fixed points of the contracting maps $\overline{\b}^s_b \circ \overline{\b}^u_b \times
\overline{\b}^u_b \circ \overline{\b}^s_b: Q^s \times Q^u \to Q^s \times Q^u$ and $\b^s_{b^*} \circ \b^u_{b^*} \times \b^u_{b^*} \circ \b^s_{b^*}: Q^s
\times Q^u \to Q^s \times Q^u$, respectively. The contraction coefficient is $<<1$, uniformly in $m$ and $b$. Therefore, the systems have a unique solution for each $m$.\
As was shown in and , the maps $\beta^{s,u}_{b^*}$ and $\overline{\beta}^{s,u}_b$ of are continuous in $b$. Therefore, the unique solutions of the systems are also continuous in $b$, and thus so are the maps $\beta_{b^*}$ and $\overline{\beta}_b$. Therefore, we have analogues of :
$$||\overline{\b}_b - \b_{b^*}|| \le d(b,b^*)^{\a-O(\delta)}\cdot O(1),
\textrm{ }\textrm{ }\textrm{ }||\overline{\b}_{b'} - \b_{b^{*}}||
\le d(b',b^{*})^{\a-O(\delta)} \cdot O(1).$$ By the triangle inequality, this implies:
$$||\overline{\b}_b - \overline{\b}_{b'}|| \le (d(b,b^*)^{\a-O(\delta)} + d(b',b^*)^{\a-O(\delta)})\cdot O(1) \le$$ $$\leq 2{(d(b,b^*) + d(b', b^*))}^{\a-O(\delta)}\cdot O(1) \le d(b,b')^{\a-O(\delta)} \cdot O(1),$$ where the last inequality follows from . This proves the desired inequality in the chart $\ph_{b^*} \times
\textrm{Id}$ (that is, for the maps $\overline{\b}_{b}, \overline{\b}_{b'}:M
\longrightarrow Q^s \times Q^u$). On the manifold (that is, for the maps $\widetilde{\b}_{b},\widetilde{\b}_{b'}:M \longrightarrow B$), the analogous relation follows from the fact that the derivative of $\ph_{b^*}$ at $0$ is identity.\
Therefore, relation is proved. Note, that $O(1)$ above is a constant not depending on $b, b'$, but depending on $\delta $ as in . We have put $\rho $ in the denominator of instead of $\delta $, because $\rho = O(\delta )$. Finally, the inverse map $H^{-1}$ of is explicitly given as $$H^{-1}(b,m)=(\widetilde{\beta}_b(m),m).$$ This map is Lipschitz in the variable $m$, and continuous in the variable $b$ by . Therefore $H^{-1}$ is continuous. This concludes the proof of Theorem \[thm:mt\].
continuity of center-stable foliation {#sec:st}
=====================================
In this section we complete the proof of Theorem \[thm:st\]. Recall that in this theorem the map $h$ is a skew product itself, see , whose fibers are globally defined stable manifolds for $h$. In this case, we will see that the central-stable leaves of $\G$ can also be globally defined.\
By analogy with Subsection \[sub:lam\], for $z\in Z$ a *global central-stable leaf* is defined as a Lipschitz function $$\label{eqn:glob} \beta^s_z:F\times M \rightarrow Z,$$ and its graph is defined as $$W^s_z=\gamma(\beta^s_z)=\{(\beta^s_z(f,m),f,m)| (f,m)\in F\times
M\}.$$ We ask that our leaves be Lipschitz close to the constant function $z$, in the sense that: $$\label{eqn:small2}
\max\left\{d(\beta^s_z,z)_{C^0}, \frac {\mLip \beta^s_z}{D}\right\}
\leq \frac {\delta}2.$$ Finally, a *global central-stable lamination* is defined as a continuous assignment $S^s=(\beta^s_z)=(W^s_z)$ of such leaves, as $z$ ranges over $Z$. Such a lamination is called $\G-$invariant if $$\label{eqn:invfolcs} \G(W^s_z) = W^s_{\zeta(z)}, \textrm{ }\forall
z\in Z,$$ where $D$ is so chosen that the estimates in the (sketch of the) proof below work out. All these constructions are analogous to the ones in Subsection \[sub:lam\]. Moreover, the entire machinery of Lemma \[lem:1\] applies to our situation and produces a unique $\G-$invariant lamination $S^s=(\beta^s_z)$ satisfying for $D$ properly chosen. We will henceforth focus solely on this lamination. In particular, since $\zeta$ is expanding we obtain: $$\label{eqn:hold0} d(\beta^s_z
,\beta^s_{z'})_{C^0} \le \frac
{d(z,z')^{\alpha-O(\rho)}}{O(\rho)^{\alpha}}, \textrm{ }\textrm{
}\textrm{ where }\alpha=\frac {\ln \mu}{\ln \mu_-}.$$ This is proven in analogous fashion to statement b) of Theorem \[thm:mt\], which was proved in the previous Section.\
**of Theorem \[thm:st\]:**
\[prop:whole phase space\] The leaves $W^s_z=\emph{Graph}(\beta^s_z)$ are disjoint and they cover the whole of $X$: $$X=\bigsqcup_{z\in Z} W^s_z.$$
The fact that the leaves are disjoint is proven analogously to Proposition \[prop:disjointfibers\]. As for their union being the whole of $X$, this is equivalent to the following claim: for any $z\in Z$ and $y\in F\times M$, there exists $\widetilde{z}\in Z$ such that $\beta^s_{\widetilde{z}}(y)=z$. Let us fix $y$ and $z$, and prove this claim.\
Fix a coordinate neighborhood of radius $2\delta$ of $z$ inside $Z$. Let $D(z,\delta)$, $D(z,2\delta)$, $S(z,\delta)$, $S(z,2\delta)$ be the balls/spheres centered at $z$ of radii $\delta$ and $2\delta $ in $Z$, respectively. The map $f:D(z,\delta) \rightarrow D(z,2\delta)$ given by $f(\widetilde{z}) = \beta_{\widetilde{z}}(y)$ is well-defined, because implies that $d(f(\widetilde{z}),\widetilde{z})\leq \delta/2$. Moreover, implies that the map $f$ is continuous. Therefore, sliding points along a straight line segment gives us an isotopy between the identity map of $D(z,\delta)$ and $f$: $$h_t(\tilde z,y)
= (\tilde z,y) + t((\b_{\tilde z}(y) - \tilde z),0)$$ Because of $d(f(\widetilde{z}),\widetilde{z})\leq \delta/2$, the image of the boundary sphere $h_t(S(z,\delta))$ never touches the center $z$ during this isotopy. Therefore, the index of $z$ with respect to the sphere $h_t(S(z,\delta))$ does not change during the isotopy. Therefore $$z\in \textrm{Im}(f) \Rightarrow \exists
\widetilde{z} \textrm{ such that } \beta^s_{\widetilde{z}}(y)=z.$$
By Proposition \[prop:whole phase space\], the map $q:X\rightarrow Z$ given by sending $W^s_z$ to $z$ is well-defined. Moreover, the $\G-$invariance condition implies that $q$ makes the diagram commute. Part $b)$ of Theorem \[thm:st\] follows immediately from and .\
Finally, let us prove the relation $q|_Y=\pi\circ p$. Take any point $b=(f,z)\in F \times Z$, and recall that we denote $z=\pi(b)$. If we take the map $\beta^s_{\pi(b)}$ defining the global lamination (see ), and restrict it to the $\delta$ neighborhood of $f\in F$, we obtain a map $\overline{\beta}_b^s$ as in . In other words restricting the leaves of the global lamination $W^s_{\pi(b)}$ produces a valid local lamination $\overline{W}^s_b$. Since the global lamination $W^s_{\pi(b)}$ is $\G-$invariant, it is easily seen that the local lamination $\overline{W}^s_b$ will also be $\G-$invariant.\
But local laminations are unique, as proved in Corollary \[cor:fixed\]! Therefore, the local leaves $\overline{W}^s_b$ coincide with the central-stable leaves $W^s_b$ of Section \[sec:grtrans\]. By the very definition of $\overline{W}^s_b$, this implies that $W^s_b \subset W^s_{\pi(b)}$, for all $b$. Since $W_b \subset W^s_b$ by construction, we conclude that:
$$W_b \subset W^s_{\pi(b)}, \textrm{ }\forall \textrm{ }b.$$ Now take any point $x\in Y = \bigsqcup_{b \in \Lambda} W_b$, and assume $x\in W_b$. By the definition of $p$, we have $p(x)=b$. But the above inclusion implies that $x\in W^s_{\pi(b)}$, and then by the definition of $q$, we have $q(x)=\pi(b)$. This precisely amounts to saying that $q|_Y=\pi\circ p$.
Fubini regained {#sec:fubini}
===============
In this section we prove Theorem \[thm:fubini\]. Moreover, at the end of this Section we discuss the “weak ergodic theorem” that appears in [@IKS08].
Measure zero and incomplete Hausdorff dimension
-----------------------------------------------
Let us begin by recalling the concept of Hausdorff dimension, denoted henceforth by $\textrm{dim}_H$.
\[def:haus\] Let $A$ be a subset of a Euclidean space. A *cover* $U$ of $A$ is a finite or countable collection of balls $Q_j$ of radii $r_j$ whose union contains $F$. The $d$-dimensional volume of $U$, denoted by $V_d(U)$, is defined as $$V_d(U) = \sum_j r_j^{d_j}.$$ The of $A$ is defined as the infimum of those $d$ for which there exists a cover of $A$ with arbitrarily small $d-$dimensional volume: $$\emph{dim}_H\textrm{ }A
= \emph{inf} \{ d|\forall \e > 0 \textrm{ }\exists \emph{ a cover
}U\emph{ of }A\emph{ such that } V_d(U) < \e \} .$$
Note that a compact manifold of dimension $d$ also has Hausdorff dimension $d$. The same holds for a set of a positive Lebesgue measure on the Riemannian of dimension $d$. Theorem \[thm:fubini\] immediately follows from the following two propositions:
\[prop:fubini1\] Recall the general setup of Theorem \[thm:st\]. If $A\subset Z$ satisfies $$\emph{dim}_H \textrm{ }A < \frac {\ln \mu}{\ln \mu_-}\cdot \emph{dim } Z,$$ then for $\rho$ small enough, the set $q^{-1}(A)$ has Lebesgue measure 0 in $X$.
Now recall the particular setup of Theorem \[thm:fubini\], which takes place over the solenoid map. Note that in this case we have $Z=S^1$ and $\mu_-=\mu=\frac 12$.
\[prop:fubini2\] For any $\kappa>0$ and finite word $w$, there exists $\e=\e(\kappa,w)$ such that the set $A_{\kappa,w}$ of Subsection \[sub:fubini\] has Hausdorff dimension at most $1-\e$.
Saving Fubini: the proof of Proposition \[prop:fubini1\]
--------------------------------------------------------
We are in the more general setup of Theorem \[thm:st\]. Since $X=Z\times F \times M$, the classical Fubini theorem states that $$\textrm{mes}(q^{-1}(A)\cap Z\times \{x\})=0, \forall x\in F \times M
\Rightarrow \textrm{mes}(q^{-1}(A))=0.$$ So all we need to do is to show that for any fixed $x\in F \times M$, the intersection $q^{-1}(A)\cap Z\times \{x\}$ has measure 0 in $Z$. By the very definition of the map $q$ of , this intersection is nothing but the set $\{\beta^s_z(x)|z\in A\}\subset Z$. Moreover, by statement b) of Theorem \[thm:st\] the map $$\ph:Z\rightarrow Z, \textrm{ }\textrm{ }\textrm{ } \ph(z) =
\beta^s_z(x)$$ is continuous with exponent $\alpha=\frac {\ln
\mu}{\ln \mu_-}-O(\rho)$. All that we need to prove is that the set $\ph(A)$ has measure 0 in $Z$. The following general lemma will do the trick:
\[lem:haus\] Let $Z$ be any Riemannian manifold, and $A \subset Z$ a subset. If $\ph:Z\rightarrow Z$ is a map with exponent $\a$, then $$\emph{dim}_H\textrm{ }\ph (A)
\le \frac {\emph{dim}_H \textrm{ }A} \a$$
The proof of this Lemma can be found in [@Fa]; the proof is straightforward. The above Lemma and the assumptions of Proposition \[prop:fubini1\] imply that for small enough $\rho$, we will have $\textrm{dim}_H\textrm{ }\ph(A)<\textrm{dim }Z$. Therefore, $\ph(A)$ has Lebesgue measure 0 in $Z$, and as we have seen above this implies that $q^{-1}(A)$ has Lebesgue measure 0 in $X$. This concludes the proof of Proposition \[prop:fubini1\].
Large deviations: the proof of Proposition \[prop:fubini2\]
-----------------------------------------------------------
In this section, we must prove that for any $\kappa>0$ and finite word $w$, the set $A_{\kappa,w}\subset S^1$ of Subsection \[sub:fubini\] has Hausdorff dimension at most $1-\e$. Call a finite word of length $N$ a $\kappa,w-$atypical word if the frequency of appearances of $w$ in that word is outside the interval $[2^{-n}-\kappa, 2^{-n}+\kappa]$. Obviously, if a sequence is $\kappa,w-$atypical then infinitely many of its initial parts will be $\kappa,w-$atypical words. Thus for any $N_0$, we have the following inclusion: $$\{\kappa,w-\textrm{atypical sequences}\}
\subset \bigcup^{N\geq N_0}\bigcup^ {\textrm{length }v= N}_{v
\textrm{ is }\kappa,w-\textrm{atypical}} \{\textrm{sequences
starting with }v\}.$$ Looking at the points of $S^1$ that correspond in binary notation to these sequences, we have: $$A_{\kappa,w} \subset \bigcup^{N\geq N_0}\bigcup^{\textrm{length
}v=N}_{v \textrm{ is }\kappa,w-\textrm{atypical}} \{\textrm{ball of
radius }2^{-N} \textrm{ around }\overline{0.v}\}.$$ This produces a covering $U$ of the set $A_{\kappa,w}$, as in Definition \[def:haus\]. Let us compute the $1-\e$ dimensional volume of this covering: $$V_{1-\e}(U)\leq \sum_{N\geq N_0}
2^{-N(1-\e)}\cdot \#\{\kappa,w-\textrm{atypical words of length
}N\}.$$ By Theorem \[thm:largedev\] below, we can estimate the number of $\kappa,w-$atypical words of length $N$, thus obtaining $$V_{1-\e}(U) \leq \sum_{N \geq N_0} 2^{-N(\nu-\e)}=\frac
{2^{-N_0(\nu-\e)}}{1-2^{\e-\nu}}.$$ If we choose $\e<\nu$ and let $N_0\rightarrow \infty$, the above expression can be made arbitrarily small. Therefore, the Hausdorff dimension of the set $A_{\kappa,w}$ is at most $1-\e$. This concludes the proof of Proposition \[prop:fubini2\] and of Theorem \[thm:fubini\], modulo the following estimate:
\[thm:largedev\] There exists $\nu=\nu(\kappa,w)$ such that for any $N$ greater than some $N_0$, the number of $\kappa,w-$ atypical words of length $N$ is at most $2^{N(1-\nu)}$.
Weak ergodic theorem {#subsec:weak}
--------------------
For the sake of completeness, we formulate here the weak ergodic theorem of [@IKS08], whose proof is very closely related to the above material.\
The classical ergodic theorem claims that for a given ergodic map and a continuous function $\ph $, the set of points for which the time average of $\ph $ either does not exist or is not equal to the space average of $\ph $, has measure zero. Here we claim that, for the duplication of a circle, for any fixed continuous function $\ph
\in C(S^1)$ and any $\delta > 0$, the set of points for which the sequence of partial time averages of $\ph $ has a limit point that differs from the space average of $\ph $ by more than $\delta$, has Hausdorff dimension smaller than $1$. We expect that this theorem may be generalized to any ergodic hyperbolic map of a compact Riemannian manifold $M^n$ with a smooth invariant measure.
\[thm:weak\] Let $\zeta$ be the duplication of the circle $S^1 = \rr /\zz$, given by $\zeta(y)=2y$. Let $\ph \in
C(S^1)$ and $\d > 0$ be given. The partial time averages of $\ph$ and its space average are defined as: $$\ph_n(y) = \frac 1n
\sum_{i=0}^{n-1}\ph (\zeta^i(y)), \textrm{ }\textrm{ }\textrm{ } I =
\int_{S^1}\ph.$$ Then the set $$K_{\ph ,\d } = \{ y| \textrm{ the
sequence } \ph_n(y) \textrm{ has a limit point outside } [I - \d , I
+ \d ] \}$$ has Hausdorff dimension smaller than 1.
A similar theorem for Anosov of the two-torus was proved recently by Saltykov [@S09].
Appendix {#sec:app}
========
The graph transform map made explicit
-------------------------------------
Recall that the graph transform map $\g_b:\B^s \rightarrow \B^s$ was defined by $\b^s \rightarrow \overline{\b}^s$, where:
$$\{\G^{-1}(\ph_{h(b)}(x_s,\beta^s(x_s,m)),m)\} \supset
\{(\ph_b(x_s,\overline{\beta}^s(x_s,m)),m)\}.$$ We now want to turn this implicit definition into an explicit formula. Recall our notation $\gamma(\beta)$, under which the above becomes: $$\textrm{Im } \G^{-1}\circ (\ph_{h(b)}\times \textrm{Id}) \circ
\gamma(\beta^s) \supset \textrm{Im } (\ph_{b}\times
\textrm{Id})\circ \gamma(\overline{\beta}^s).$$ If we write $\G_b
={(\ph_{h(b)}(C\delta)\times \textrm{Id})}^{-1} \circ \G \circ
(\ph_b(\delta) \times \textrm{Id})$ as in , then our relation takes the form:
$$\label{eqn:implicit1} \textrm{Im }\G_b^{-1} \circ
\gamma(\beta^s) \supset \textrm{Im } \gamma(\overline{\beta}^s).$$
Write $\pi_u:Q^s \times Q^u \times M \rightarrow Q^u$ and $\pi_{sc}:Q^s\times Q^u\times M \rightarrow Q^s \times M$ for the standard projections, and define:
$$\label{eqn:Gbb} G_{\overline\b ,b} = \pi_{sc} \circ
\G_b \circ \ga (\overline{\b}^s):Q^s \times M \rightarrow Q^s \times
M.$$
Then is equivalent to:
$$\label{eqn:graph} \pi_u \circ \G_b^{-1} \circ
\gamma (\b^s) \circ G_{\overline\b, b} = \overline{\b}^s.$$
\[prop:composition\] The composition is well-defined and $$\emph{Lip }
G_{\overline\beta,b}\leq (L + O(\delta))\cdot (1+\emph{Lip
}\overline\b^s),$$ where $L$ is the constant from Definition \[def:domsplit\]. A similar estimate holds in the central-unstable case.
Define the composition $$F_{0 ,b} = \pi_{sc} \circ
\F_b \circ \ga (0),$$ in analogy with , with $\G$ replaced by $\F$ and $\overline{\beta}^s$ replaced by the zero map $0:Q^s\times M \rightarrow Q^u$. Since $d(\G,\F)_{C^1}\leq \rho$, we see that: $$\label{eqn:qqq} \mLip G_{\beta,b} \leq
(\mLip F_{0,b}+O(\rho))\cdot (1+\textrm{Lip }\overline\b^s)$$ But one can simply unravel the definition of $F_{0,b}$ when $\F$ is a skew product, and obtain $$F_{0,b}(x_s,m)=(\pi_s\circ h_b(x_s,0), f_{(x_s,0)}(m)).$$ From this it is clear that $$\mLip F_{0,b} \leq L + O(\delta).$$ Recalling that we always choose $\delta=O(\rho)$, implies: $$\mLip G_{\overline{\beta},b} \leq (L + O(\delta))\cdot (1+\textrm{Lip
}\overline\b^s).$$
\[prop:last\] For any two central-stable leaves $\overline\b^s_0, \overline\b^s_1 \in \B^s$, we have: $$||G_{\overline\b_0,b} - G_{\overline\b_1,b}|| \leq O(1)\cdot ||\overline\beta^s_0-\overline\beta^s_1||.$$ A similar result holds in the central-unstable case.
We have: $$||G_{\overline\b_0,b} -
G_{\overline\b_1,b}||=||\pi_{sc}\circ \G_{b} \circ
\ga(\overline\b^s_0) - \pi_{sc}\circ \G_{b} \circ
\ga(\overline\b^s_1)||\leq$$ $$\leq \textrm{Lip }(\pi_{sc}\circ
\G_{b})\cdot ||\ga(\overline\b^s_0) - \ga(\overline\b^s_1)|| \leq
O(1)\cdot ||\overline\beta^s_0-\overline\beta^s_1||.$$
Persistence of skew products
----------------------------
The second, independent technical result that we will prove concerns the setup of Theorem \[thm:mt\]: we have a small $\rho-$perturbation $\G$ of the skew product $\F$ from Theorem \[thm:mt\]. This theorem tells us that $\G$ is conjugated to a skew product $G$: $$G(b,m)=(h(b), g_b(m)).$$ In this Subsection, we will prove formulas and . To this end, from the very definition of $G$ we have the following explicit formula for the fiber maps $g_b$: $$\label{eqn:fibermapg} g_b(m)=\pi_m
(\G(\widetilde{\beta}_b(m),m)), \textrm{ }\textrm{ }\textrm{ }
g_b^{-1}(m)=\pi_m(\G^{-1}(\widetilde{\beta}_{h(b)}(m),m))$$ where $\pi_m:X=B\times M \rightarrow M$ is the standard projection. Obviously, we have $$f_b(m)=\pi_m(\F(b,m)), \textrm{ }\textrm{
}\textrm{ } f_b^{-1}(m)=\pi_m(\F^{-1}(h(b),m)).$$ Since $d(\G^{\pm
1}, \F^{\pm 1})_{C^1}<\rho$, it follows from the above formulas that $$d(g_b,f_b)_{C^1} \leq d(\G(\widetilde{\beta}_b(m),m), \F(b,m))_{C^1} \leq$$ $$\leq d(\G(\widetilde{\beta}_b(m),m), \G(b,m))_{C^1}+\rho \leq
||\G||_{C^1}\cdot d(\widetilde{\beta}_b,b)_{C^1}+\rho = O(\rho),$$ and similarly for $d(g_b^{-1}, f_b^{-1})_{C^1}$. This proves . As for the property, we have that $$d(g_b,
g_{b'})_{C^0}\leq ||\G||_{C^1} \cdot d(\widetilde{\beta}_b, \widetilde{\beta}_{b'})_{C^0}
\leq O(d(b,b')^\a),$$ by . The statement concerning $d(g_b^{-1}, g_{b'}^{-1})_{C^0}$ is proved analogously, thus concluding the proof of .
Acknowledgments
---------------
We are grateful to A.Bufetov, A.Klimenko, C.Pugh, M.Shub for fruitful discussions. The second author would like to thank the Max Planck Institut für Mathematik in Bonn for hosting him while a significant portion of this work was written.
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V. Kleptsyn, P. Saltykov, $C^2$-stable example of attractors with intermingled basins for boundary preserving maps of an annulus, in preparation.
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[^1]: The author was supported by part by the grants NSF 0700973, RFBR 07-01-00017-à, RFFI-CNRS 050102801
[^2]: Cornell University, US; Moscow State and Independent Universities, Steklov Math. Institute, Moscow
[^3]: “Simion Stoilow” Institute of Mathematics of the Romanian Academy, Romania; Harvard University, US
[^4]: **AMS Classification**: 37D30, **Keywords**: invariant laminations, partially hyperbolic invariant set, property, Hausdorff dimension
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---
abstract: 'Results of a 4 year X-ray monitoring campaign of the Small Magellanic Cloud using the Rossi X-ray Timing Explorer ([[*RXTE*]{}]{}) are presented. This large dataset makes possible detailed investigation of a significant sample of SMC X-ray binaries. 8 new X-ray pulsars were discovered and a total of 20 different systems were detected. Spectral and timing parameters were obtained for 18. In the case of 10 pulsars, repeated outbursts were observed, allowing determination of candidate orbital periods for these systems. We also discuss the spatial, and pulse-period distributions of the SMC pulsars.'
author:
- 'S. Laycock, R. H. D. Corbet, M. J. Coe, F. E. Marshall, C. Markwardt, J. Lochner'
title: 'Long-Term Behavior of X-ray Pulsars in the Small Magellanic Cloud'
---
The Survey
==========
A program of regular observations of the Small Magellanic Cloud (SMC) using the [*Rossi X-ray Timing Explorer*]{} (RXTE) has been underway since 1997, in conjunction with optical imaging and spectroscopic follow-up observations. New SMC pulsars discovered by [[*RXTE* ]{}]{}have been reported in a number of publications, and this work is intended to present the wealth of long-term data now accumulated for these sources. This paper reports the results of 4 years (November 1997 - February 2002) of weekly monitoring and presents the long-term behavior of most of the known X-ray pulsar systems. During this time, 16 X-ray pulsars were discovered by [[*RXTE*]{}]{}, at the same time many discoveries were made by [[*ASCA* ]{}]{}and [[*BeppoSAX*]{}]{}, bringing the number of known X-ray pulsars in the SMC to 30.
Long term pulsed-flux lightcurves of all pulsars observed during the project are presented. Twenty pulsars were positively detected in the data presented in this paper, and upper limits were placed on 6 other known pulsars that were not detected during this time. In the case of 15 systems, multiple X-ray outbursts were observed, raising the possibility of determining orbital periods. Interpretation of the findings, in terms of the overall SMC population is discussed.
[[*RXTE* ]{}]{}is particularly well suited to monitoring pulsars in the SMC. The Proportional Counter Array (PCA) is able to cover a significant fraction of the SMC at sufficient sensitivity to detect active pulsars with luminosities in the typical range 10$^{36}$-10$^{38}$ . The spectral response is also favorable as the construction of the PCA provides its optimum sensitivity across the peak of the typical pulsar spectrum. The PCA’s full-width zero intensity (FWZI) field of view is 2 and the instrument is non-imaging. However, the PCA can be scanned across a target region and the location of a bright source determined to an accuracy of 1-2$\arcmin$ by fitting the collimator response to the observed count-rate.
The low extinction and unobstructed line of sight to the SMC enable accurate measurements of X-ray luminosity and useful observations of the optical counterparts with 1-2m telescopes [see for example @edge2002]. From the X-ray viewpoint, since the distance to the SMC is significantly greater than its ’depth’ all of the objects in the SMC can be considered to lie at an equal distance for the purposes of luminosity calculations.
Observing Strategy
------------------
The [[*RXTE* ]{}]{}Proportional Counter Array (PCA) [@Jahoda96] was used to make regular observations of the SMC to look for new X-ray pulsars and monitor the outbursts of known systems. The instrument consists of 5 co-aligned Proportional Counter Units (PCU), each has a xenon filled detector volume with 3 layers of anode-grids for photon detection plus a lower xenon veto layer for background rejection. Propane-filled veto detectors are positioned above the PCA to attenuate particle background and non-source direction X-rays. Operated in Good Xenon mode, individual photon arrival times are recovered to 1$\mu$s accuracy, and pulse-heights processed into 256 channels between 2-100 keV. According to the [*ABC of XTE*]{} the top layer of the PCA is the most sensitive, with the majority of source events occurring there, while the background rate is similar in all 3 layers. We verified that the detection significance for pulsars is slightly better using layer 1 only versus layers 1 and 2, when extracting lightcurves in the 2-10 keV band.
The program began after a new X-ray transient was detected during a PCA slew in the vicinity of SMC X-3 [@IAU6777]. A PCA observation was conducted on 1997 November 29 in response to this detection, revealing not SMC X-3 but three previously unknown pulsars with pulse periods of 91.s, 74s and 46.6s [@IAU6803]. Continued observations throughout December and January detected 2 more pulsars with periods of 169s [@IAU6814] and 58.9s [@IAU6818]. At times observations were being made at a rate of 1 per day, giving excellent temporal coverage of the outbursts of these pulsars. Following these observations, a program of regular monitoring was established (see Table \[tab:positions\]), with pointed PCA observations of several ksec duration made on a weekly basis. Pointing positions were carefully chosen to give good coverage of the main body of the SMC, as shown in Figure \[fig:smcpic\]. The “wing” was not regularly observed due to the presence of the luminous supergiant High-Mass X-ray Binary (HMXB) SMC X-1. This source was deliberately excluded from the PCA survey because its persistent emission, very strong pulsations, and timing noise mask the presence of other weaker pulsars when it is in the field of view. Observations were made in 3 phases during which slightly different observing strategies have been employed, as described below.
*Phase 1.* This comprises the series of observations conducted and described by @Lochner99a. The observations were made primarily in two positions designated 1a and 1c. Position 1a was centered on the location of SMC X-3, because the initial source detected was at first thought to be SMC X-3. Position 1b was the position of a single PCA pointing aimed at the location of one of the new pulsars. Position 1c lies close by and was used throughout 1998 to monitor the activity of the 4 newly discovered pulsars mentioned above.
*Phase 2.* During this period we continued to monitor the known pulsars and at the same time search for new sources. Position 1 largely overlaps with the fields of view of the earlier observations. Position 3 was chosen to cover the eastern wing of the SMC which contains SMC X-1. Positions 2 and 4 were chosen to cover the rest of the SMC.
*Phase 3.* After 2 years of monitoring position 1, we broadened the search by shifting south to position 5, also in the main body of the SMC. The supplementary positions 2, 3 and 4 were not observed, as previous observations of these positions caused gaps in coverage of the main position. Position 5 was observed each week for 2.1 years.
Data Reduction Pipeline
-----------------------
Data reduction was performed in two stages, “real-time monitoring” and “survey”: Immediately after each routine PCA observation, quick-look data were searched for new pulsars. Initially a lightcurve was generated at 3-10 keV and 0.1 second time resolution. No filtering criteria or background subtraction were applied at this stage. The power spectrum was inspected visually and with a peak search algorithm. This preliminary data analysis was usually performed within a day or so of the observation taking place, to identify new pulsars and schedule rapid TOO follow-up observations. The long-term survey made use of the production data, and more detailed analysis methods. All data reduction was done with [*Ftools*]{}. Standard filtering criteria were applied to exclude data collected during: periods of high background; times when the source was attenuated by Earth’s atmosphere; during slews on/off source; and times immediately after SAA passage. On 2000 May 20 the upper propane veto-layer of PCU 0 was lost, after that date PCU 0 data were excluded. Filtering by detector/layer was performed with bit-masks generated by [*sefilter*]{}.
Background count-rates were generated using [*pcabackest*]{} and the L7 models for faint sources. Lightcurves and spectra were extracted from the generated background using [*saextrct*]{} with the same filtering criteria and detector/layer combinations as were applied to the science data.
### Lightcurve Extraction
Science lightcurves were extracted using [*seextrct*]{} at 0.01 second time binning for the 3-10 keV Good Xenon data from the top anode layer of each active PCU. This configuration maximizes signal-to-noise for pulsars. Background lightcurves were extracted with 16s binning (the minimum available) and subtracted from the science lightcurve, without modifying the errors on the science count-rates.
The lightcurve was then normalized to , by dividing each flux value by the number of PCUs active at that time. Lightcurve bin-times were corrected for the motion of the satellite and Earth’s orbit using [*faxbary*]{} to give the arrival times at the solar system barycenter.
### Correcting for Collimator Response
For every pulsar with a well known position, the detector sensitivity ($R$) was calculated at each pointing position. The collimator for each PCU are constructed from corrugated sheets with hexagonal cells. The hexagonal cells have small offsets in pointing direction and the angular responses for each PCU are different. For the purposes of examining long term light curves we adopt a simplified model for the collimators of a circular field of view with a simple triangular collimator response of full-width half maximum (FWHM) = 1, full-width zero intensity (FWZI) = 2. This simplified model differs from the true collimator response in being somewhat sharper peaked, lacking extended wings at larger angles (where we do not have significant pulsar detections) and lacking the hexagonal symmetry which is also most pronounced at larger offset angles. The use of this simplified model speeds data analysis and is not a significant factor in, for example, searching for periodic behavior. With our simplified collimator model, for a particular pulsar at known distance from the pointing center, $R$ equals 1.0 minus the net pointing offset in degrees. Values of [*R*]{} for each pulsar in each of the monitoring positions are listed in Table \[tab:responses\]. This table should be consulted when examining the long-term pulsar lightcurves in Section \[sect:lcurves\]. It is apparent that the scatter in flux measurements is correlated with $R$. Blank entries in Table \[tab:responses\] indicate that the pulsar was not in the field of view. In practice, observations at marginal sensitivity were also excluded. A value of $R < 0.1$ was regarded as marginal and a lightcurve not extracted for observations of that particular pulsar/pointing position combination. For spectral analysis in this paper we utilize the full detailed model of collimator response for those sources with accurate positions.
Timing Analysis
---------------
The methods used to measure pulse periods, pulse amplitudes, calculate significance levels and reject false detections are outlined here.
A [Lomb-Scargle periodogram ]{}[@Scargle82] was calculated for each observation, scanning the range 0.02-1000 s at a resolution of $1\times 10^{-5}$ Hz. Also generated by the pipeline were statistical parameters used in assigning significance levels and converting between spectral power and pulsed flux.
The special properties of the [Lomb-Scargle periodogram ]{}were exploited to scan every observation around the period of each known pulsar. We first determined if there was $significant$ power at a given (known pulsar) frequency and then determined its period and amplitude, if however there was non-zero power at a low significance level, an upper limit was placed on the pulse-amplitude of the (presumably inactive) pulsar.
For each known pulsar, (i.e those detected at a high level of confidence in one or more observations, or from the literature) an appropriate period search-range was determined from inspection of occurrences detected at $\geq$ 90% significance in a *blind search*. For “literature” pulsars a $\pm$5% frequency band centered on the known period was used. The [Lomb-Scargle periodogram ]{}of each observation was then scanned in this period range, the maximum power identified and its significance estimated, considering only the independent frequencies in the allowed period range. This is the *prior knowledge* significance.
Given that 30 SMC pulsars are known with periods less than 1000 s, allowing for up to 3 harmonics gives a possible 93 genuine signal frequencies. Once period variability and timing resolution are included, the possibility of contamination requires the adoption of an algorithm to reject false detections of pulsars that lie close to the harmonic frequencies of others.
If the maximum power recorded was not a *peak* inside the selected range, i.e. it lay at edge of the frequency range, then it was assumed to be leakage of power from a nearby and unrelated frequency. Whenever this situation occurred, the detection significance was set to zero since the probability that the power was due to the pulsar being searched for is negligible. For positive detections the pulse period and its uncertainty were also recorded. The uncertainty on the period was calculated using the formulation of [@kovacs1980] which accounts for the strength of the detected signal.
For each pulsar we only considered observations of duration greater than 4 pulsation cycles otherwise there are unacceptably large errors on the pulse period and poor confidence in their identification.
For each positive detection, the measured power $P_{LS}$ was converted into a pulse amplitude using Equation \[eqn:pow2amp\]: where the lightcurve has $N_P$ points and variance $\sigma^2$, the peak-to-valley amplitude (twice the sinusoidal amplitude) $A^{\ast}$ of a signal detected in the [Lomb-Scargle periodogram ]{}is: $$\label{eqn:pow2amp}
A^{\ast} = 4\sqrt {\frac{P_{LS} \sigma ^2}{N_P }}$$
The significance of peaks in the [Lomb-Scargle periodogram ]{}is described by a simple exponential function, related to the number of independent frequencies in the range being searched in the case of periodic signals superimposed on white noise with a Gaussian distribution. The significance calculations in this work follow the prescription of @press93.
The precise nature of the data sampling is relevant in timing analysis. Data gaps are unavoidable and applying barycenter correction to the lightcurve bin times causes a small departure from regular sampling. Random sampling is $recommended$ by @Scargle82 in order to maximize the anti-aliasing properties of the Lomb-Scargle periodogram, but clumpy or gappy data are not handled so efficiently and, in cases involving one or two large gaps in otherwise regular data, aliasing is similar to a Discrete Fourier Transform. We used the fast coding of @press89.
Flux Measurements {#section:flux}
-----------------
Throughout this paper we use units of , these can be approximately compared to the flux in if assumptions are made regarding the X-ray photon spectral index ($\alpha$) and line-of-sight absorption column-density ($N_H$). Taking reasonable values for X-ray pulsars to be $\alpha=1, N_H=1\times10^{22}cm^{-2}$ gives an approximate conversion factor of 1= 1.2$\times10^{-11}$. At the SMC (65 kpc), this corresponds to a luminosity of $L_{X}=5.7\times 10^{36}\ erg\ s^{-1}$. Pulse amplitude is simply related to the pulsed flux (pulsed component summed over a full cycle) by a factor of 2. The total flux can only be determined if the pulse fraction (ratio of pulsed to unpulsed component) is known. We choose to discuss the pulse amplitude as this is the most directly measurable quantity and enables robust measurements to be made even when multiple pulsars are present in a single observation.
Orbital Period Estimation {#sect:sta}
-------------------------
One of our primary goals was to measure orbital periods for the X-ray sources. This was done by analyzing the long-term pulsed-flux lightcurves produced by our pipeline. In prototypical Be-X systems, X-ray outbursts occur in periodic sequences. Many of our lightcurves presented in Section \[sect:lcurves\] show such features, for example XTE J0055-724 (Figure \[fig:59amp\]). In these cases we employed Phase Dispersion Minimization [PDM, @Stellingwerf78], which is the most appropriate technique to search for highly irregular modulations which are also poorly sampled.
Systems such as AX J0049.4-7323 (Figure \[fig:755amp\]) presented two outbursts which, although insufficient to formally claim a “period”, have nonetheless proved to be highly reliable indicators of the orbital period. If two outbursts are seen from the same Be system within a recurrence time of several weeks, we know from the underlying physics that these are highly likely to be separated by one orbital period. Measurement of such “recurrence times” was done by folding the dates of significant detections on a range of trial periods, and selecting the period that produced the minimum scatter in phase. This is similar to taking the mean separation of outbursts, but includes multiple data-points for each outburst, and accounts for the fact that we do not know the date of outburst-maximum very well, due to sparse sampling. The uncertainty is calculated by multiplying the minimum standard deviation in phase by the best period. We term this method “simple timing analysis” (STA). Given the small number of points available (typically 4-20), STA results are a “best estimate" only.
In other cases (e.g. AX J 0051.6-7311, see Figure \[fig:172amp\]) the long-term lightcurve contains many detections of the pulsar on dates that do not form an obvious periodic sequence. The explanations include X-ray emission that is genuinely aperiodic, as well as missed outbursts which could hide an underlying periodic pattern. These cases were analyzed with both PDM and STA and the most likely periods or recurrence times reported, in order to provide a guide for other observers.
Finally a few systems (e.g. SMC X-2) have undergone a single giant outburst during our monitoring project. In these cases no orbital periods could be measured, instead pulse-period variations were used to constrain the contributions from orbital motion (Doppler shifts) and accretion torques.
Results I: Pulsar Monitor Charts {#sect:monitor}
================================
After reduction, the data were analyzed in 2 separate ways to search for variable sources, the first of these is the blind search.
Having generated the [Lomb-Scargle periodogram ]{}for every observation, the resulting database of periodograms was searched for significant peaks. The significance level for each observation was estimated individually. This stage was executed as a “blind search”, the threshold adopted was 90% significance considering all frequencies in the (period) range 0.5 -1000 seconds. We used a simple boxcar peak-search algorithm which identified a peak as any frequency having a greater power than its nearest neighbors, subject to that power being above the 90% significance level. Figure \[fig:peaks12345\] displays the combined results of this analysis in the form of a “pulsar activity monitor”, showing the activity of bright pulsars in all the monitoring positions. The size of the graph markers indicates spectral power on a logarithmic scale, horizontal dashed lines (red) indicate pulse periods of the majority of known pulsars in the SMC, these periods are given on the right of the plot. In addition to detecting pulse periods, this procedure also picks up harmonics and, in principle at least, low frequency quasi-periodic oscillations. In particular it has been observed in the course of this work that at times the $P_{pulse}/2$ harmonic can dominate the power spectrum. Therefore all significant power is included on the plots. The expected $P_{pulse}/2$ harmonics of known pulsars are indicated by dotted (green) lines. As the various pointing positions cover partially overlapping fields of view, different pulsars are visible in subsets of the data. Positions 1 and 5 comprise the bulk of the data.
Figure \[fig:peaks12345\] shows pulsars ranging in period from 0.7s (SMC X-1) to 755s (AX J0049.4-7323). It is immediately apparent that two sources were particularly active throughout the survey. The 172.4s and 323s pulsars appear to have undergone respectively 5 and 7 outbursts over a period of about 700 days. If these outbursts are normal Be/X-ray binary outbursts then the results are suggestive of orbital periods of (very) approximately 700/5=140d and 700/7=100d. The line labeled “2.37 & 2.39” is also of particular interest as the points lying along it actually belong to two different pulsars. The group of points at about MJD 51600 is due to an outburst from SMC X-2, Below these points, 2$^{nd}$, 3$^{rd}$ and 4$^{th}$ harmonics are visible. There then follows a short ($\sim$ 2 week) break in observing coverage after which the pulsations were again detected but visibly weaker and with the relative strength of the harmonics altered. Just after MJD 51900 a third group of points appear on the “2.37 & 2.39” line, these are the first harmonic of XTE J0052-723 which was first discovered in these observations, the fundamental is seen just once on this plot.
No sub-second pulsars were detected, only 3 observations containing significant peaks at P$<$1s were ever seen:
\(1) A single detection of the (P/2) harmonic of SMC X-1; and 2 harmonics of SMC X-2 (P/5 and P/6).
\(2) On MJD 51381, period 0.0329429(5)s, amplitude 0.73 , significance above 99% in one 1050s observation. However two other observations on the same day, of comparable length made $~$1 hr before and $~$1 hr after failed to show any evidence for this period.
\(3) On MJD 51432, period 0.0747731(3)s, amplitude 0.31 , significance 97%. It is expected that a small number of false detections will occur given the large number of observations analyzed and this low significance detection may thus not be real.
Results II: Long Term Pulsar Lightcurves {#sect:lcurves}
========================================
Long term behavior of all known pulsars in the SMC was investigated using the database of [Lomb-Scargle periodogram]{}s generated from the [[*RXTE* ]{}]{}monitoring observations.
In the following section the pulsars are ordered by pulse period. Trends in physical characteristics are likely to follow the pulse period, so it is appropriate to list the systems in a natural sequence that facilitates comparison.
The most important results are shown in a series of 3-panel plots for each source, laid out as follows:
[*Top panel.*]{} Pulsed flux lightcurve, filled symbols indicate positive detections, open symbols are upper limits. Pulse period with uncertainty. Statistical significance of each pulsed flux measurement. Two significance estimators are plotted: Blind search (squares), and prior knowledge of period (circles). The criterion for a positive detection was 99% prior knowledge significance. Blind search significance values are shown in order to convey those detections of outstanding magnitude.
SMC X-2 (2.37s)
---------------
SMC X-2 was discovered by *SAS-3* in 1978 [@Clarke78] and although outbursts were also observed by *HEAO 1* and [[*ROSAT*]{}]{}, pulsations had not been detected until the [[*RXTE* ]{}]{} monitoring data presented here.
A single outburst was detected from SMC X-2, lasting from 2000 January 24 to April 23. During this outburst, the luminosity (2-25 keV) reached a peak of 4.7 $\times 10^{38}{\mbox{${{\rm\thinspace erg}}{{\rm\thinspace s}}^{-1}$}}$, and was detected down to a level of 5.7$\times 10^{37}{\mbox{${{\rm\thinspace erg}}{{\rm\thinspace s}}^{-1}$}}$ before disappearing from view sometime before May 5. The detection of SMC X-2 with the [[*RXTE* ]{}]{}All-Sky Monitor (*ASM*) followed by discovery of 2.37 second pulsations co-incident with the known position of SMC X-2 from *PCA* scans and targeted observations was presented by @Corbet01_x2, along with contemporaneous optical observations confirming the counterpart originally proposed by @Murdin79. An [[*ASCA* ]{}]{}observation on April 24 by @Yokogawa01 confirmed the 2.37 s pulsar to be exactly coincident with the position of SMC X-2 determined by *SAS-3* and [[*ROSAT* ]{}]{}.
From the SMC monitoring data, pulsations with a period of 2.37s were detected with the PCA during 13 observations between MJD 51567 and MJD 51657. Of these, 9 were pointings at the regular monitoring position (position 5, see Table \[tab:positions\]), plus 2 targeted pointings centered on SMC X-2, and 2 sets of scans across the source region. All of these observations are included in the long-term lightcurve (Figure \[fig:smcx2amp\]).
The duration of the outburst was constrained by non-detections before and after the dates given in Table \[tab:x2flux\], on MJD 51560 and 51666-7 and 51670. The upper limits for the end of the outburst are more stringent as these observations were pointed directly at the position of SMC X-2.
Due to changes is the power spectrum of SMC X-2 during the outburst, pulse periods were refined by PDM [@Stellingwerf78]. The refined periods are given in Table \[tab:x2flux\]. The pulse profiles (Figure \[fig:x2pro\]) obtained during the outburst show an interesting trend with luminosity. During the low luminosity observations at the beginning and end of the outburst, the pulse profile was a weakly double peaked shape, with roughly equal peaks separated by half a cycle: typical of that seen in many X-ray pulsars. At high luminosity the shape changed, with peak flux occurring either side of a narrow minimum, the separation between the first and second peak now 0.7 in phase. This change was seen in the power spectra as a reversal in the normal ratio of the fundamental and 1st harmonic: evident from the pulsar monitor (Figure \[fig:peaks12345\]) around MJD 51600. For this reason the pulse amplitudes plotted in Figure \[fig:smcx2amp\] are derived either from the fundamental or harmonic depending on which was the stronger.
The X-ray spectrum of SMC X-2 was extracted for 3 observations, the two targeted pointings [@Corbet01_x2] and observation 5 when the highest flux was observed. Assuming the spectral parameters in Table \[tab:x2spec\] to be reasonably representative of SMC X-2 during the whole outburst ($N_H=2\times10^{22}$ cm$^{-2}$, $\alpha=1$) the fluxes reported in Table \[tab:x2flux\] may be converted to $L_X$ by $1{\hbox{${{\rm\thinspace counts}}{{\rm\thinspace PCU}}^{-1}{{\rm\thinspace s}}^{-1}\,$}}= 1.72\times10^{37}{\mbox{${{\rm\thinspace erg}}{{\rm\thinspace s}}^{-1}$}}$ assuming a 65 kpc distance to the SMC.
### In-outburst Timing Behavior
Period variations evident in Figure \[fig:smcx2amp\] are suggestive of spin-up or orbital modulation, these measurements were investigated and refined by Phase Dispersion Minimization (PDM). The PDM method was used because the shape of the pulse profiles are highly irregular and change between observations. A period range was selected that encompasses all the periods determined from the power spectra, hence a range of 2.371-2.373s at a resolution of 10$^{-6}$ s rather than the fundamental. These periods are given in Table \[tab:x2flux\]. The magnitude of the variations is 1.7$\times10^{-3}$ s, significantly larger than the mean uncertainty (3.9$\times10^{-5}$ s) in the period determination. The actual timing behavior is somewhat surprising as there appear to be two spin-up events on similar timescales, one before the gap in coverage and one after. Observations 4 and 10 seem not to fit this picture, which may be attributable to systematic errors resulting from pulse profile variation.
A problem with the spin-up interpretation is the size of the pulse period changes. Standard accretion torque theory [@GL1979] predicts spin period changes of approximately 2.4 $\times$10$^{-11}$ s s$^{-1}$ L$_{38}^{6/7}\mu_{30}^{2/7}$ which is exceeded by factors of up to $\sim$100 (Table 2). Another contribution to period changes can come from orbital motion. For a circular orbit, and assuming a neutron star mass of 1.4 M$_{\sun}$ and a mass donor mass of 10 M$_{\sun}$ then the orbital velocity semi-amplitude is $\sim$170 (15/P$_{orb}$)$^{1/3}$ sin i km s$^{-1}$. where P$_{orb}$ is in units of days. The observed period changes are of at least of this approximate magnitude although an obvious systematic trend was not detected. Finally, if our period determinations are dependent on pulse profile, although we do not expect this, then an artificial variation of pulse period with luminosity may be produced.
The pulse period discovered for SMC X-2 is among the shortest known for HMXBs especially for those containing a Be star primary. If SMC X-2 follows the loose relation between $P_{pulse}$ and $P_{orbit}$ seen in Be/X-ray binaries [@Corbet86] then its orbital period may be around 15 days. The pulse profiles appear to be correlated with $L_{X}$, similar changes were observed in XTE J0052-723 and also the 51 s XTE pulsar that was initially reported as having a period of 25.5s by @lamb2001. The temporal coverage of the [[*RXTE* ]{}]{}observations of SMC X-2 presented here were sufficient to track the luminosity and pulse period behavior reasonably closely for the duration of an entire outburst. A gap in coverage occurred between MJD 51600-51640 and certain observed properties suggest that 2 outbursts may have occurred. Evidence for this interpretation comes from the flux history and the pulse period variations. If two separate outbursts did in fact occur, then an estimate can be made that the orbital period is less than approximately 70 days, by taking the difference of the dates on which the two periods of spin-up began.
XTE J0052-723 (4.78s)
---------------------
Pulsations with a 4.78 s period were detected on 2000 December 27 during an observation of pointing position 5. An analysis of the X-ray and optical observations was presented by @Laycock02b identifying a possible B0V-B1V counterpart. No additional outbursts were detected, as evidenced by Figure \[fig:4.78amp\]. For most of the outburst the pulse profile was strongly double peaked, causing the $P_{pulse}/2$ harmonic power to be more indicative of the actual pulsed flux, a feature that is evident in the pulsar monitor (Figure \[fig:peaks12345\]). In all cases around the time of the outburst, the amplitude of the $P/2$ harmonic in Figure \[fig:4.78amp\] was plotted if it was greater than the fundamental.
2E 0050.1-7247 (8.88s)
----------------------
The pulsar activity monitor (Figure \[fig:peaks12345\]) shows two strong detections of a $\sim$9 second pulsar which appear to correspond to the 8.88s pulsar 2E 0050.1-7247. The pulsed flux history reveals a number of detections in 3 groups separated by $\approx$ 200 days. Although some of the detections appear to at least roughly coincide with detections of the 16.6s pulsar the uncertainties on our period determinations apparently exclude the possibility that pulsations at 8.88s are harmonics of the 16.6s source. A 2-10 keV pulse profile obtained during the brightest detection of 8.88s pulsations is shown in Figure \[fig:8.88pro\], it is triple peaked.
RX J0052.1-7319 (15s)
---------------------
This pulsar was not conclusively detected in any regular monitoring observation, but pulsations with a 15.7s period were detected in a special deep observation described in Section \[sect:deep\]. The source was very faint and only detected due to the length of the observation, pulse amplitude was 0.12 . No spectrum could be obtained due the many active pulsars in the field of view, including the close period of 16.6s at about 5 times greater amplitude.
XTE 16.6 seconds
----------------
The 16.6 second pulsar was discovered in a deep observation of position 4 (see Section \[sect:deep\]). The pulsar appears in Figure \[fig:16amp\] on 8 occasions, which seem to belong to 6 separate outbursts. Simple timing analysis (Section \[sect:sta\]) was performed on the 99% detections, suggesting a candidate orbital period of 189$\pm18$ days, with $T_0$ = MJD 51393. The folded lightcurve is uninteresting as the fluxes when the pulsar is detected are similar to the upper limits in non-detections. The 2-10 keV pulse profile Figure \[fig:16pro\] is approximately sinusoidal.
An analysis of archival [[*RXTE*]{}]{}, [[*ROSAT* ]{}]{}and [[*ASCA* ]{}]{}data has now appeared in the literature [@lamb2001] and a tentative association made with the [[*ROSAT* ]{}]{}source RX J0051.8-7310 on the basis of a marginal detection of periodicity in data from [[*ROSAT* ]{}]{}and [[*ASCA*]{}]{}. @Yokogawa02 demonstrate this identification is incorrect. From our observations the position of the 16.6 second pulsar is constrained to lie within the overlap of the PCA field of view at positions 4 and 5.
XTE J0111.2-7317 (31s)
----------------------
This source was in the survey field of view on 3 occasions. 31 s pulsations were detected in only the first of these observations on MJD 51220. This detection coincides with the very end of a giant outburst of this system which was simultaneously discovered by [[*RXTE* ]{}]{}and BATSE [@Chakrabarty98a; @WF98]. For the single PCA detection, the pulse period was 30.65$\pm$0.05 s The spectral fit to this observation gave an unabsorbed 2-10 keV luminosity of 4.6$\times$10$^{37}$ , and showed a prominent 6.4 keV iron line, full spectral parameters given in Table \[tab:catalogue\]. The 2-10 keV pulse profile for this observation is shown in Figure \[fig:31pro\], it is highly irregular, featuring three distinct peaks and one deep minimum per cycle.
1WGA J0053.8-7226 (46.6s)
-------------------------
The 46.6 second pulsar was one of three sources discovered in the vicinity of SMC X-3 in 1997 November [@IAU6803] and remained active through December. Its subsequent reappearance on 1998 August 3 was reported by @IAU7007 who suggested an orbital period of 139 days based on these 2 outbursts. After analyzing the long-term monitoring data, pulsations at 46.6 s were positively detected in 54 observations, apparently grouped in 8 separate outbursts. The full dataset presented in Figure \[fig:46.6amp\] was used to determine the orbital period of the system. This was done in two stages because the data quality was not constant due to changes in pointing position. For observations made at positions 1a, 1b, 1c and 1 the pulsar was at or close to the center of the PCA field of view. This subset of the data was analyzed using PDM, only one likely period was discovered at 137$\pm$8 days. In order to include the two later outbursts, the position 5 data were filtered to remove all points at less than 99% significance and those remaining were added to the first dataset and reanalyzed. This procedure was justified because the source was poorly placed in the position 5 field of view and probably only detectable close to the peak of the last 2 outbursts. The addition of these points slightly deepened and narrowed the PDM minimum to give a period of 139$\pm$6 days, the uncertainty is the FWHM of the PDM minimum. Numbering orbital cycles from the first outburst, the observed detections correspond to phase zero of cycles 1, 2, 3, 4, 5, 6, 9 and 12, where the adopted zero-point is MJD 50779. The folded lightcurve is shown in Figure \[fig:46fold\].
Two candidates have been proposed for the optical counterpart [@Buckley01], both lie in the error box determined by [[*ROSAT* ]{}]{}and [[*ASCA* ]{}]{}and both show strong H$\alpha$ emission lines and photometric colors of Be stars. One of the candidates was also reported to exhibit an IR excess and variability.
XTE J0055-724 (59s)
-------------------
Strong pulsations at a period of 59 s were discovered by @IAU6818 during a search for pulsars in the vicinity of SMC X-3. The 59 s pulsar appears to have been emitting approximately regular outbursts throughout the lifetime of [[*RXTE* ]{}]{}. The pulsar was in our field of view for the entire monitoring program although it was only detected in positions 1, 1a, 1b and 1c. XTE J0055-724 was close to the center of the field of view in these observations but only marginally covered by the other positions, as a consequence the upper limits for the pulsed flux are rather large after MJD 51555. Figure \[fig:59amp\] shows just the position 1, 1a, 1b, and 1c results, only this subset of the data was used for determining the orbital period. The 2-10 keV pulse profile (Figure \[fig:59pro\]) is dominated by a single asymmetric peak with some finer structure visible.
Four separate outbursts were observed, each reaching a similar brightness and occurring on similar timescales. Averaging over the latter 3 peak fluxes gives a mean of 2.0 with a spread of 0.1 . The timescale for the flux to go from an approximately quiescent level, up to peak, and back down again was about 40$\pm$5 days. The first outburst had excellent temporal coverage and was possibly brighter than the other 3, although this may just be because more observations were made during the peak of the outburst.
A timing analysis was conducted to identify a possible orbital period in XTE J0055-724 which might be responsible for the regularity of the outbursts. PDM was selected as the most appropriate technique owing to the irregularity of the modulation. The signal-to-noise ratio appears good (the scatter of the open circles in Figure \[fig:59amp\] is small compared to the amplitude of the putative orbital modulation), and 59 second pulsations were detectable down to very low flux levels because the source was in the center of the PCA field of view. The resulting period is 123 $\pm$ 1 days, the folded lightcurve is shown in Figure \[fig:59orbitnobins\] where the zero point is MJD 50841, obtained by placing the peak at $\phi = 0$.
AXJ0049-729 (74s)
-----------------
AXJ0049-729 is one of the 3 pulsars discovered by [[*RXTE* ]{}]{}in November 1997 [@IAU6803]. The low amplitude of the 74 s pulsations made it the weakest source in the discovery observation and slews were unable to constrain its position accurately. An [[*ASCA* ]{}]{}observation on 1997 November 13 detected pulsations at 74.68(2)s [@Yokogawa99] and determined that AX J0049-729 lies in the error circle of the [[*ROSAT* ]{}]{}source RX J0049.1-7250 [@KP98]. According to @Stevens99 there is a single Be star within the 13" error radius of the [[*ROSAT* ]{}]{}position, therefore this star is probably the optical counterpart. Figure \[fig:74amp\] shows three separate outbursts occurred during the [[*RXTE* ]{}]{}monitoring program. Selecting only the 99% significant detections (filled symbols in Figure \[fig:74amp\]) STA indicates a candidate outburst-recurrence (orbital) period of 642$\pm$59 days. The flux measurements were folded at the 642 day period, and appear to be tightly constrained within a range of 0.3 in phase (Figure \[fig:74orbit\]). Spectral parameters were obtained at the peak of the brightest recorded outburst on MJD 52078 and the pulse profile shown in Figure \[fig:74pro\] is also from this observation. The spectrum was well fit by the classic Be/X-ray binary model, parameters are given in Table \[tab:catalogue\], the cutoff energy was 16.2$\pm$1.2 keV and a prominent iron K emission line was present.
XTE J0052-725 (82s)
-------------------
Although pulsations at 82.4 s were regularly detected, the source was always extremely faint and the close similarity of its period to the harmonic of the 169s pulsar XTE J0054-720 made positive detection difficult. A brighter outburst was eventually observed in February 2002 (after the data presented in this paper). During the bright outburst slews were performed in order to localize the pulsar’s position [@IAUC7932]. This position was then used retrospectively to generate the lightcurve shown in Figure \[fig:82amp\].
AX J0051-722 (91s)
------------------
Pulsations with a period of 91.1 s were discovered in November 1997 [@IAU6803] and were detected regularly throughout 1997 to 1999 as shown in Figure \[fig:91amp\] until a change in monitoring position shifted AX J0051-722 out to the edge of the field of view (see Table \[tab:positions\]). @IAU6858 reported that the source re-brightened on 1998 March 25, having faded since its initial discovery, suggesting an orbital period around 110 days. PDM analysis was performed on this subset of the data giving a period of 115 days, there is a secondary minimum at 123 days and this may well be due to some degree of cross-contamination from the nearby 59s pulsar XTE J0055-724 which has this orbital period, although the pulse periods are not harmonics of each other. The folded lightcurve is shown in Figure \[fig:91fold\]. The 2-10 keV pulse profile in Figure \[fig:91pro\] is noisy due to the off-axis angle of the source, and shows a broad peak, roughly twice the duration of the pulse-minimum.
XTE SMC95 (95s) {#sect:95}
---------------
The 95 second pulsar was discovered on 1999 March 11 during the [[*RXTE* ]{}]{}monitoring project. An analysis of the initial discovery observations has been published by @Laycock02a. For reference purposes the source is provisionally designated *SMC95*. According to the additional data presented here (see Figure \[fig:95amp\]), two separate outbursts were observed at $>$99% significance in position 1. The lower panel of Figure \[fig:95amp\] shows that 2 or 3 more groups of observations reached greater than 90% significance and these groupings are spaced at intervals roughly equal to the interval between the two strong outbursts. Since the position of the source is poorly constrained [@Laycock02a], 95 s pulsations were searched for in all observations and at all positions. The data obtained in positions 1a, 1b and 1c had to be excluded due to the very bright 91 second pulsar AX J0051-722. The side-lobes of this pulsar contained so much power at 95 seconds that accurate estimates of *SMC95* were impossible. A candidate orbital period was estimated with STA (Section \[sect:sta\]), giving 280$\pm$8 days, for this period, we estimate the epoch of maximum flux T$_{0}$ = MJD 51248.
AX J0057.4-7325 (101s)
----------------------
This source appears in the long term lightcurve during 6 observations, only one of which attains a high detection significance. This is a result of the fact that AX J0057.4-7325 lies far from the center of the PCA field of view at all of the pointing positions. The strongest detection was during the deep observation described in section \[sect:deep\]. No spectrum was extracted for that observation because several other pulsars were active, and mostly at higher flux levels. No orbital period could be estimated owing to the limited number of observations.
XTE J0054-720 (169s)
--------------------
This source was discovered on 1998 December 17 with [[*RXTE* ]{}]{}[@IAU6814]. Except for its first detected outburst it was a faint source as seen in Figure \[fig:169amp\]. Timing analysis of the lightcurve did not provide conclusive evidence of an orbital period. PDM gave weak minima at 112 and 224 days using the full dataset. Selecting only the 99% significance detections (17 points) and applying the simplified timing analysis produced minimum scatter in the detection dates at 53$\pm$11 and 201$\pm$41 days, the former is seemingly ruled out because it is shorter than the well observed outburst at the beginning of the dataset. This outburst has an apparent duration of about 100 days and is brighter than the other outbursts. and there is a pattern of 3 narrow minima in its significance plot. The large outburst could therefore be anomalous, extending far beyond periastron. The true period could then be close to the lower figure. The result of folding the lightcurve at a period of 224 days is shown in Figure \[fig:169fold\].
AX J0051.6-7311 (172s)
----------------------
Pulsations at 172.4 s were reported by @Torii00a in an [[*ASCA* ]{}]{}observation in April 2000. [[*RXTE* ]{}]{}monitoring has revealed AX J0051.6-7311 to be one of the most frequently active pulsars in the SMC. The source has been identified with the [[*ROSAT* ]{}]{}source RX J0051.9-7311 and a Be star found in the error circle by @Cowley97.
The pulsed flux for AX J0051.6-7311 was generally $<$0.5, placing it at the low luminosity end of the SMC pulsar population. Despite many detections over a long timebase, the orbital period remains elusive. A number of approaches were tried in an effort to determine an approximate value for the recurrence time between outbursts. For this purpose we used only the position 5 data (after MJD 516000), with the pulsar well placed in the PCA field of view.
After removing all points with non-zero detection significance we were left with 52 possible measurements of the pulsed flux. PDM analysis of this lightcurve gave no conclusive period. Under the assumption that detections are more likely to be close to orbital phase zero, and seeing a series of at least 6 equally spaced peaks in Figure \[fig:172amp\], the 99% detections (20 points) were analyzed with STA, giving a period of 64$\pm$16 days. Selecting only the 6 evenly spaced outbursts (MJD 51686 - 52039), the recurrence period is 67$\pm$5 days, T$_0$ = MJD51694. It seems likely that in addition to strings of periodic outbursts this pulsar exhibits some outbursts that are not closely correlated with orbital phase.
RX J0050.8-7316 (323s)
----------------------
323s pulsations were first detected by [[*ASCA* ]{}]{}on 1997 November 13 [@Yokogawa98a] at a position coincident with the [[*ROSAT* ]{}]{}source RX J0050.8-7 316. The optical counterpart is thought to be a Be star identified by @Cowley97. @CoeOrosz have shown that this star is in a binary system with a 1.4 day period. RX J0050.8-7316 is a particularly interesting system as it has recently been proposed as a triple system [@trinary].
Pulsations consistent with a 323 s period were detected frequently with [[*RXTE* ]{}]{}throughout the survey and the source was near the center of the field of view for the majority of the time (MJD 51555 -52333). The long pulse period and low luminosity result in fairly large uncertainties on both period and flux determinations for most observations, however there seem to be no other pulsars with periods that are likely to cause confusion in the 300s - 330s period range. Although AX J0103-722 has a pulse period variously reported as 343s or 348s it was not in the position 5 field of view and therefore cannot be present in the lightcurve (Figure \[fig:323amp\]) after MJD 51555.
The signal to noise level for the long term lightcurve was rather low, especially in the early observations obtained in position 1. During these observations the source was 0.85$\degr$ off-axis and hence our sensitivity was poor. The 99% significance detections obtained during 1998 - 2002 (23 black points in Figure \[fig:323amp\] after MJD 51200) were analyzed using STA, giving a period of 108$\pm$18 days. This procedure did not take direct account of the flux values and is based on the assumption that each detection corresponds to X-ray emission at or close to phase zero (outburst peak). For a faint source this seems a reasonable assumption considering our sensitivity limit is approximately 0.2 for most observations.
Having identified a tentative period, the earlier observations from 1997-1999 were included in the folded lightcurve shown in Figure \[fig:323fold\], the 6 brightest points belong to the earlier position 1 data. Having identified a candidate orbital period we propose the peak of the best observed outburst be used as epoch of phase zero, this is MJD 51651.
@Imanishi99 performed an analysis of archival data from [[*ASCA* ]{}]{}, [[*ROSAT* ]{}]{}and *Einstein* (total 18 observations), finding weak evidence for a periodicity of $~$185 days.
AX J0103-722 (348s)
-------------------
This pulsar appeared in 4 observations in Figure \[fig:348amp\], which could be used to estimate some kind of recurrence timescale. However the source was not near the center of the field of view in any position except position 2 (which was only observed 4 times). The [[*ASCA* ]{}]{}pulsar AX J0103-722 has been identified with a [[*ROSAT* ]{}]{}source lying in a supernova remnant with a Be optical companion. Detections have been found in data from [[*Einstein*]{}]{}, [[*BeppoSAX*]{}]{}, [[*ROSAT*]{}]{}, and [[*Chandra* ]{}]{}all at luminosities of $\sim$10$^{36}$, the infrequent detections by [[*RXTE* ]{}]{}are therefore explainable by the low luminosity of the source. In fact at 10$^{36}$ it would be near the sensitivity limit for most observations.
RX J0101.5-7211 (455s)
----------------------
Pulsations consistent with the 455s pulsar RX J0101.5-7211 were detected at low flux levels. The source was discovered by [[*ROSAT* ]{}]{}[@Haberl00], noted to be highly variable and was identified as a Be/X-ray binary. Although Figure \[fig:455amp\] appears somewhat noisy there is no other known pulsar with a period or harmonics likely to cause confusion. We note the presence of approximately equally spaced points at $>$ 90% significance which are suggestive of an outburst spacing of 200-300 days. Only 2 points reach our nominal detection threshold (99%) and clearly further observations are required.
AX J0049.4-7323 (755s) {#sect:755}
----------------------
This source has the longest pulse period so far seen in the SMC. It was discovered with [[*ASCA* ]{}]{}on 2000 April 11 [@Ueno00b; @Yokogawa00a] and the reported [[*ASCA* ]{}]{}pulse period was 755.5(6)s. @Yokogawa02 described how, in addition to the discovery observation, the source was also detected by [[*ASCA* ]{}]{}on 1997 November 13 and 1999 May 11 but with no detection of pulsations, presumed due to the short duration of these observations. The reported luminosities during the [[*ASCA* ]{}]{}detections of AX J0049.4-7323 all appear to be around 5$\times$10$^{35}$ -well below the [[*RXTE* ]{}]{} sensitivity threshold. According to @Yokogawa02 a revised analysis of the [[*ASCA* ]{}]{}position associates AX J0049.4-7323 with the [[*ROSAT* ]{}]{}source RX J0049.5-7310.
Figure \[fig:755amp\] shows that [[*RXTE* ]{}]{}saw AX J0049.4-7323 in two separate outbursts, centered around MJD 51800 and MJD 52200. The mean pulse period measured by [[*RXTE* ]{}]{}was 751s. Based on the spacing of the outbursts and the scatter of points, a candidate orbital period was estimated, to be 396$\pm$5 days with STA (Section \[sect:sta\]). The zero point to place the observed outburst peak at $\phi$ = 0 is $T_{0}$ = MJD 51800. Obviously a “period” derived from two outbursts would be considered provisional until future observations confirm it. Such confirmation for this outburst interval being the orbital period has indeed now come from optical measurements by @Schmidtke04 who found optical outbursts from this system every $\sim$394 days.
The second outburst is remarkable in its apparent brightness, the pulsed flux was approximately 3.2 on 2001 October 11, making it one of the most luminous pulsars in the SMC. The pulse profile for that observation is shown in Figure \[fig:755pro\]. A spectrum was also extracted for this observation (full parameters in Table \[tab:catalogue\]) implying an unabsorbed $L_{X}^{2-10}$= 1.7$\times$10$^{37}$, a high energy cutoff was required at 12.16$\pm$1 keV and an iron K line was also present.
Archival data from [[*ROSAT* ]{}]{}and [[*Einstein* ]{}]{}were analyzed by @Yokogawa00a, demonstrating that AX J0049.4-7323 has been active at the $\leq$5$\times$10$^{35}$ level for over 20 years. Activity at this level falls below our detection threshold and we are apparently only detecting relatively infrequent large outbursts. It is noted that the pulse profile obtained for the 3-10 keV range (Figure \[fig:755pro\]) bears no resemblance to the [[*ASCA* ]{}]{}pulse profile suggesting that the pulse profile is very luminosity-dependent.
A Very Deep Observation {#sect:deep}
=======================
One very deep observation was made at Position 4 (see Table \[tab:positions\]). Beginning on MJD 51801.2 a total of 106.5 ksec of good time, spanning nearly 3 days was obtained. The standard analysis as described above revealed the presence of 7 pulsars, of which 2 were new discoveries. The dataset was included in the long-term lightcurves presented above. The periodogram is shown in Figure \[fig:longls\] with all of the 99% significant peaks labeled and identified. Because this observation spanned a very long time interval, the periodogram was calculated out to 100 ksec in order to search for hitherto undiscovered long-period pulsars. A number of features are evident at low frequencies and most of these are attributed to systematic effects related to the [[*RXTE* ]{}]{}orbital period and residual diurnal background variations. Pulsars detected are (from left in Figure \[fig:longls\]) RX J0049.7-7323 and several harmonics, RX J0051.9-7311, AX J0057.4-7325, XTE 51s (detected P/2 harmonic), XTE 16.6s, RX J0052.1-7319, 2E 0050.1-7247.
Known SMC Pulsars Not Detected With [[*RXTE* ]{}]{}
===================================================
A table of known SMC X-ray pulsars which were not detected in any of the monitoring observations despite being in the [[*RXTE* ]{}]{}field of view at various times is provided, see Figure \[tab:nondetect\]. In each case a single upper limit on the flux is quoted, based on the observation judged to be the most sensitive in terms of position, duration, and number of detectors functioning.
General Properties of the SMC HMXB Population.
==============================================
Pulse Periods
-------------
We compare the pulse period distributions for X-ray pulsars in the SMC, LMC and the Galaxy. Figure \[fig:dist\_smclmcgal\] shows the 3 populations binned with 2 bins per decade in period. A comparable number of pulsars are known in the SMC (30) and Galaxy (54), and a lesser number in the LMC [@Liu00; @Sakano00; @Marshall00; @Intzand01; @Bamba01].
The pulse periods in Figure \[fig:dist\_smclmcgal\] mainly occupy the 10 - few 100 s range, with the Galactic population skewed toward longer periods. The known range of pulse periods stretches from below 0.1s to over 1000s. We compared the Galactic and SMC populations with the Kolmogorov-Smirnov test. The K-S statistic was 0.3222, giving a probability of 0.028 that the two samples are drawn from the same population. Some caution is required in interpreting this result because the conditions under which the two samples were obtained were far from homogeneous. If one were to try to characterize typical observing conditions for the SMC pulsars the important factors would be low galactic extinction, and constant distance ($\sim$65kpc). For the Galactic pulsars the conditions are varying extinction across as much as several decades in $N_H$ and a wide range of distances which are uncertain by factors of 2 or more.
If systematic differences exist between the pulse period distributions in the two galaxies, physical causes must be disentangled from selection effects. Our results presented here tend to support the link between X-ray luminosity and pulse period. @SWR86 pointed out that the maximum observed luminosity for binary X-ray pulsars is anti-correlated with $P_{pulse}$, although the results presented here suggest that the effect is weak for periods longer than a few seconds. Luminosity selection bias (against fainter long period pulsars) in our survey seems unlikely to seriously affect results. A more serious bias against long period pulsars is observation length: 3 ksec is only long enough to detect pulsars up to periods of about 750 seconds. At very short pulse periods, there is unlikely to be any luminosity bias because these sources are bright (typically several 10$^{37}$). The important factor for short period pulsars is density of observing coverage because these systems rarely go into outburst. It seems unlikely that any other pulsars similar to SMC X-2 and XTE J0052-723 could have been missed because the typical duration of these outbursts is several weeks. With observations on a weekly basis at a sensitivity $\sim$10$^{36}$ any such pulsars would have been detected if they lie within the field of view of the PCA. XTE J0052-723 was easily detected at a collimator response of just 0.2. A look at the Corbet diagram for Galactic and SMC pulsars (Figure \[fig:newcorbet\]) reveals that the cause of the discrepancy in pulse period distributions is in fact the predominance of Be/X-ray binaries in the SMC.
Our luminosity sensitivity of $\sim$10$^{36}$ may be converted into an approximate pulse period detection threshold. [@SWR86] found an inverse correlation between pulse period and maximum X-ray luminosity. From Figure 2 of @SWR86 our luminosity sensitivity implies that we would be able to detect SMC pulsars with periods shorter than $\sim$1000 seconds if SMC pulsars follow the same relationship. We note that @Majid04 do find a comparable relationship between maximum luminosity and pulse period for the SMC and Galactic sources.
For Galactic HMXBs, a significant fraction of long period pulsars are accreting from the winds of supergiant companions. These systems, although variable, are persistent which makes them much easier to detect than transient Be star systems. In spite of this, none of the SMC pulsars under discussion here has the characteristics of a supergiant wind-fed binary. Therefore, one contribution to the difference between the Galactic and SMC pulse period distributions is the apparent lack of supergiant wind-accretion pulsars in the SMC.
Orbital Periods
---------------
Transient Be star X-ray pulsars exhibit two types of outbursts [e.g. @Bildsten97]. Type I outbursts recur, when a system is in an active state, on the orbital period of the system. Activity is confined to limited orbital phases around the time of periastron passage. This activity pattern is the most common type of outburst. In addition, Type II (giant) outbursts may also occur where the X-ray luminosity is much larger and is not modulated on the orbital period. Our observations of SMC pulsars (over a timescale very much longer than the expected orbital periods) shows detections of repeat outbursts in 10 sources. Under the assumption that many of these are the more common Type I outbursts, these should allow the estimation of orbital periods for several systems.
Candidate orbital periods and their uncertainties are listed in Table \[tab:catalogue\]. Under the widely discussed “standard” model of Be/X-ray binaries [e.g. @Negueruela98], it is expected that Type I outbursts will occur periodically as the neutron star’s eccentric orbit intercepts the Be star’s circumstellar disk, while type II outbursts are triggered by large-scale enhancements in the disk. Orbital periods for the SMC pulsars and for all known Galactic HMXB systems are shown in Figure \[fig:newcorbet\], an updated version of the $P_{spin}/P_{orbit}$ diagram [@Corbet86]. Uncertainties in $P_{orbit}$ are indicated, uncertainties in $P_{pulse}$ are smaller than the plot symbols.
The $P_{spin}/P_{orbit}$ diagram demonstrates that X-ray pulsars with massive companions display three different correlations between their pulse and orbital periods. These correlations have been successfully explained by @Corbet86 in terms of the specific mode of mass transfer occurring in the binary. Different regions of the diagram are populated by systems undergoing steady wind-fed accretion, Roche lobe overflow, and transient (often periodic) accretion as is the case for Be systems. A powerful feature of the Corbet diagram is that the optical counterpart to an HMXB can be predicted to a high degree of confidence if only the timing parameters are known. Optical counterparts for around half of the currently known SMC pulsars have been identified and classified. The similarity of the X-ray properties of virtually all SMC pulsars (excepting SMC X-1) suggest the unidentified sources are also Be/X-ray binaries. In summary this evidence comprises: (1) the transient nature of the X-ray emission, (2) the typical luminosities and (3) the spectral parameters. The picture established from X-ray observations is supported by the optical identifications all of which are Be, excluding SMC X-1 which seems to be the lone example of its class. There are 32 HMXBs in the Galaxy for which both orbital *and* pulse periods have been reported, and 2 in the LMC [@Liu00; @Intzand01b; @Delgado01; @Intzand01]. In Figure \[fig:newcorbet\] color coding used to signify the 3 types of high mass accreting systems. Square, triangle, and star symbols indicate Galactic, LMC and SMC sources respectively.
All 8 of the newly determined candidate orbital periods for SMC pulsars lie in the upper half of the Be distribution as defined by the Galactic systems. Despite the pulse period distribution being weighted (although weakly) in the opposite direction. There is no characteristic of our [[*RXTE* ]{}]{}observing strategy that would be expected to cause a bias toward finding pulsars with such long orbital periods, although the weekly sampling makes orbital periods below about 14 days difficult to measure. The apparent lack of short orbital periods for the SMC population is instead due to the small number of pulsars detected with pulse periods below $\sim$10s. For periods approaching 1 second, the minimum accretion-driven luminosity for a 10$^{12}$ G neutron star is close to the Eddington limit. So fast spinning pulsars are expected to spend the majority of the time in the centrifugal inhibition regime (if they are in Be systems). Thus unless a large dense Be star disk is continuously present, regular type I outbursts cannot occur. For such systems the orbital period can only be determined from pulse timing analysis or monitoring of the optical counterpart.
Spectral Parameters
===================
HMXB spectra are in general characterized by a single power law, modified by absorption at low energy and an exponential cut-off at high energy [e.g. @White83]. In systems where a cyclotron scattering resonance feature has been seen, which provides a measurement of the neutron star magnetic field strength, the spectral cutoff energy is found to be correlated with cyclotron energy [e.g. @makishima92; @coburn02]. Measurements of spectral cutoffs thus provide at least estimates of the magnetic field strength. A fluorescence line is also often seen at 6.4 keV corresponding to the iron K line from relatively cool material [@Nagase89].
For each bright pulsar observed during the SMC monitoring, a spectral fit was performed for at least one detection. Spectra were only extracted when no other pulsar was active, to avoid cross-contamination. The resulting spectral parameters given in Table \[tab:catalogue\] show values consistent with those exhibited by Galactic sources [@White83].
Iron lines were detected in most SMC pulsars: typical line widths were around 0.5 keV and the line strengths ($\sigma$) were correlated with overall luminosity. One exception to the above was XTE J0055-724 (period 59 s) which was X-ray bright but showed no trace of an iron K line. Three sources stand out as having peculiar features at the high energy end of their PCA spectra. The 74.7s, 172.4s and 323s pulsars all show very steep cutoffs although their spectral indices below the cut-off are not unusual. In addition these fits are not good representations of the observed spectra above the cutoff and the residuals show the possible presence of either a deep absorption feature around 20 keV or an emission feature at slightly lower energy. The two most notable of these are also the most frequently active sources in the SMC. The 172.4s and 323s pulsars were found to exhibit at least 5 and 7 outburst respectively. The pulsed-lightcurves presented in Section \[sect:lcurves\] show that the situation with these two sources is complex and there may be faint pulsations emitted at scattered times not clearly correlated with orbital phase. This interpretation of the pulsed lightcurves could imply that the pulsars are not being centrifugally inhibited at low accretion rates. Such a situation would be consistent with a lower value for the magnetic field which would be expected to reveal itself as unusually low energy cyclotron lines in the X-ray spectrum.
The Spatial Distribution of HMXB in the SMC
===========================================
HMXB provide tracers of recent star formation activity because they are short lived, make an appearance very soon in the evolution of a star forming region, and are of course highly conspicuous. It has been noted that many X-ray pulsars discovered in the Galaxy are concentrated in the nearby “5 kpc arm”, a region identified by radio, IR, and CO molecular line emissions [@Hayakawa77]. Six pulsars discovered with the *Ginga* satellite are known in this region and all are transient sources having the X-ray characteristics of Be systems. Spiral arms in galaxies are well known to harbor star forming regions and in fact such regions account for the majority of star formation within galaxies. The work of @mgm99 proposes that the ratio of Be to normal B type stars is higher in the SMC than in the Galaxy, although this was based on observations of certain SMC clusters which are probably not representative of the SMC as a whole. A promising line of inquiry would be to carry out a similar survey concentrating on comparing the regions harboring concentrations of HMXB. In order to identify the regions to observe, a large number of HMXB are needed for significant clustering to become apparent. Be stars are themselves relatively conspicuous objects from their photometric colors, and their place as the most common optical counterpart in HMXB undoubtedly has strong implications for the preferred channel for massive binary star evolution.
Recent optical surveys of the SMC [@Zaritsky00; @Cioni00; @Maragoudaki01] show that the young stellar population is concentrated into distinct structures. 30 X-ray pulsars have been identified in the SMC, the majority with positions to the level. These positions are plotted in Figure \[fig:spatial\] and can be compared with the distributions of other SMC constituents. Neutral hydrogen is a natural galactic constituent to compare with the HMXB distribution because large concentrations of hydrogen are a necessary precursor to star formation. Figure \[fig:spatial\] panel 3 shows the HI density distribution as mapped by @S99, superimposed are the positions of all the SMC pulsar with accurately known positions. @Maragoudaki01 produced isodensity contour maps of the SMC showing the spatial distributions of stars of different ages. Perhaps the most interesting panel from the point of view of HMXBs is the distribution of stars whose age is comparable to the evolutionary timescale for HMXB formation. The isochrone map for stars aged 8 - 12.2 My is reproduced here in Figure \[fig:spatial\] panel 2. A catalog of emission-line stars in the SMC has been compiled by @MA93 based on an objective-prism survey. The catalog was searched for all objects with a positively detected H$\alpha$ emission-line, in a 5 degree wide field centered on the SMC, and the results are plotted in Figure \[fig:spatial\] panel 1. The final population distribution plotted here is the X-ray source population as seen by [[*ROSAT* ]{}]{}with the PSPC and HRI. The PSPC catalog is larger due to greater sensitivity however the HRI catalog gives the possibility to restrict the search to point-sources only. This distinction enables at least an approximate restriction to a sample of HMXB. There is some observing bias surrounding the [[*ROSAT* ]{}]{} distribution because the 5$\degr \times$5$\degr$ field was not uniformly observed by either [[*ROSAT* ]{}]{}instrument, it appears that the main body of the SMC was evenly covered while the two apparent clusters around (00$^h$ 40$^m$, -72) and (01$^h$ 20$^m$, -75) are due to two particular pointings. The cluster of sources in the “wing” is real.
Looking at Figure \[fig:spatial\] it is apparent that there is a close correlation between the spatial distributions of the 5 populations. The SMC bar and wing are clearly marked out by the clustering of X-ray sources, emission-line stars, young stars and HI.
Summary
=======
The SMC is an intriguing galaxy because of its surprisingly high abundance of X-ray pulsars. The X-ray binary population of the SMC shows important differences from the LMC and the Galaxy. The SMC appears to have been chemically isolated from the Galaxy and has been sculpted by gravitational and hydrodynamic forces originating in tidal interactions with the LMC and the Galaxy. As such the SMC provides an ideal environment in which to study the effects of dynamic encounters on star formation and the evolution of the stellar population. A number of recent works have revealed patterns in the spatial distributions of different aged stellar populations, and this evidence can be used to constrain various models of stellar evolution and structure formation. The large population of pulsars provides both a probe of the star formation and an ideal sample to investigate the general properties of HMXB.
Catalog of Sources Detected
===========================
Parameters are summarized here for each pulsar detected during the project. For every known pulsar in the SMC, name(s), position and optical counterpart can be found in Table \[tab:pulsars\]. This table should be used to identify the pulsars for which detailed measurements were made during the [[*RXTE* ]{}]{}monitoring project. Parameters determined in this work are summarized in Table \[tab:catalogue\] and concern the subset of 18 pulsars which were observed under particularly good signal to noise conditions. For these pulsars the minimum and maximum pulsed fluxes, spectral parameters, and luminosities are presented.
A complete description of the columns in Table \[tab:catalogue\] is given below:
\(1) *Pulse Period.* All of the pulsars exhibited pulse period variations, this column lists a characteristic value. (2) *Orbital period.* Measured in days by the methods described in the relevant sections of this paper. (3) *$T_0$ Epoch of Phase 0.* Orbital period zero point, where phase zero is taken to correspond with peak X-ray emission. (4) *Minimum pulsed flux.* Units of . Lowest pulsed flux measured for the source at better than 99% significance. Determined from the power spectrum as described in Section \[sect:lcurves\]. (5) *Maximum pulsed flux.* Units of . (6)-(16) *Characteristic Spectral parameters.* Taken from selected observations based on brightness and lack of interfering sources. Fluxes are the absorbed values in 10$^{-11}$. With the exception of (L$_x$) the values have not been corrected for collimator response or distance to the SMC. (9) *Absorption column density* for neutral hydrogen in units of 10$^{22}$cm$^{-2}$ (14) Reduced $\chi^{2}$ for the fit. Fits were performed over the range 3 - 20 keV. (15) *Collimator response* for the pulsar in the observation from which the spectral fit was performed. Numbers in parentheses are the pointing positions. Both parameters refer to Table \[tab:positions\]. (16) *Luminosity* in units of 10$^{37}$ erg s$^{-1}$ The unabsorbed source luminosity assuming a distance of 65kpc after correction for collimator response. The 2-10 keV luminosity is given to facilitate comparison with results from other missions.
{width="15cm"}
![Periodogram for the deep (2 day) observation shows 7 pulsars, two of which were new discoveries. From left: RX J0049.7-7323, RX J0051.9-7311, AX J0057.4-7325, XTE 51sec, XTE16.6sec, RX J0052.1-7319, 2E 0050.1-7247[]{data-label="fig:longls"}](f41.ps){width="10cm"}
![Spatial distributions (clockwise from top left) 1. Emission line stars, 2. Isodensity contours for stars aged 8-12.2 My, 3. Neutral hydrogen map, with positions of X-ray pulsars superimposed, 4. [[*ROSAT* ]{}]{}X-ray Sources, blue +’s= PSPC, red +’s= HRI point sources. Credits: 1. ALADIN/@MA93, 2. adapted from @Maragoudaki01, 3. HI map adapted from @S99, 4. ALADIN/@Haberl00 [@Sasaki00]](f44.ps){width="10cm"}
. \[fig:spatial\]
---------- -------------------- --------------- ------------- --------------
Position $N_{obs}(N_{sec})$ Dates (MJD) $RA(\degr)$ $Dec(\degr)$
1a 9 (13) 50779 - 50802 13.033 -72.429
1b 1 (1) 50825 12.767 -72.229
1c 18 (31) 50833 - 51115 13.728 -72.445
1 41 (54) 51186 - 51555 13.471 -72.445
2 4 (6) 51198 - 51326 16.25 -72.1
3 3 (4) 51220 - 51310 18.75 -73.1
4 6 (4) 51348 - 51417 12.686 -73.268
5 45 (70) 51560 - 52333 12.5 -73.1
---------- -------------------- --------------- ------------- --------------
: Pointing Positions for SMC Monitoring.
\[tab:positions\]
------------ ----------- ------------- ------- -------------
Obs. Date Period Flux Pulsed Flux
(2000) (sec)
1$^{\dag}$ Jan 24 2.3728(1) 0.03 0.16
2 Jan 30 2.37210(5) 10.04 2.91
3 Feb 6 2.3717(1) 11.32 1.34
4 Feb 12 2.37281(1) 16.15 1.44
5 Feb 18 2.371815(5) 26.49 11.97
6 Feb 26 2.371787(5) 25.23 7.99
7 Apr 6 2.372217(5) 8.45 1.96
8a Apr 12-13 2.37157(1) 7.14 1.86
8b$^*$ Apr 12 5.2 -
9 Apr 18 2.3711(1) 7.02 1.62
10$^*$ Apr 22-23 2.371859(5) 3.23 1.19
------------ ----------- ------------- ------- -------------
: [[*RXTE* ]{}]{}PCA Pointed Observations of SMC X-2.\[tab:x2flux\]
---------------------- ------------------------------ ------------------------ ------------------------ -- --
Obs 5 8b 10
$N_{H}\times10^{22}$ 1.6 $\pm$ 0.3 1.6$\pm$0.8 3.3$\pm$0.6
$\alpha$ 1.17$\pm$0.02 0.70$\pm$0.04 1.0$\pm$0.03
$E_{c}$ 12.4$\pm$1
$E_{f}$ 50$\pm$12
$\sigma Fe$ 0.8$\pm$0.2
norm $Fe$ (2.9$\pm$1)$\times 10^{-10}$
$flux_{2-10}$ 1.5$\times 10^{-10}$ 6.93$\times 10^{-11 }$ 5.71$\times 10^{-11 }$
$flux_{10-20}$ 1.6$\times 10^{-10}$
$L_{X}$ 4.4$\times 10^{38}$
$\chi_{\nu}{^2}$ 1.25 1.52 1.12
---------------------- ------------------------------ ------------------------ ------------------------ -- --
: Spectral fits for SMC X-2[]{data-label="tab:x2spec"}
----------------- ------------ --------- ----------------
Pulsar Period (s) Ref.
RX J0059.2-7138 2.8 $<$0.40 [@Hughes1994]
AX J0105-722 3.34 $<$0.16 [@IAU7028]
AX J0049-732 9.13 $<$0.10 [@IAU7040]
RX J0117.6-7330 22 $<$0.29 [@IAU6305]
AX J0058-7203 280 $<$0.17 [@Yokogawa98a]
----------------- ------------ --------- ----------------
: Pulsars not detected with [[*RXTE*]{}]{}. \[tab:nondetect\]
|
---
abstract: 'The kinetics of irreversible adsorption of spherical particles onto a flat surface is theoretically studied. Previous models, in which hydrodynamic interactions were disregarded, predicted a power-law behavior $t^{-2/3}$ for the time dependence of the coverage of the surface near saturation. Experiments, however, are in agreement with a power-law behavior of the form $t^{-1/2}$. We outline that, when hydrodynamic interactions are considered, the assymptotic behavior is found to be compatible with the experimental results in a wide region near saturation.'
author:
- |
P. Wojtaszczyk$^{\dagger}$ and J.B. Avalos$^{*}$\
$^{\dagger}$ Departament de Física Fonamental, Facultat de Física,\
Universitat de Barcelona,\
Avda. Diagonal 647, E-08028 Barcelona (Spain)\
$^{*}$ Departament d’Enginyeria Química, ETSEQ\
Universitat Rovira i Virgili\
Carretera de Salou s/n, E-43006 Tarragona (Spain)
title: '**Influence of Hydrodynamic Interactions on the Kinetics of Colloidal Particle’s adsorption**'
---
2ex
PACS[68.10.Jy,02.50.+s,82.65.-i]{}
[2]{}
The adsorption of colloidal particles or macromolecules such as proteins onto adsorbing surfaces is a very common phenomenon in many fields of Biology, Chemistry and Physics. Deposition of bacteria on teeth or the adsorption of antibodies on living cells are examples of such phenomena which are of great interest in medical sciences. In many cases, the adsorption is irreversible under conditions of practical interest. A model system to study the adsorption process is a suspension of latex spheres put in contact with a suitable adsorbing flat surface. Much work has been done for this system [@lang; @ada1; @ada2; @ada3; @stprl; @sprl; @tar; @woj1; @woj2; @sflu; @sfl2; @senf; @mpar], both on the structural properties of the adsorbed layer and on the kinetics of the process. For irreversible adsorption of sufficiently light spherical particles onto a flat surface, Schaaf [*et al.*]{}[@sprl] predicted a power-law behavior $t^{-2/3}$ for the time-dependence of the coverage of the surface near saturation. This behavior is due to the interplay of the diffusion of particles from the bulk and the [*blocking effect*]{} caused by the saturation of the surface due to the previously adsorbed particles. This result agrees with Brownian dynamics simulation of spheres diffusing from a bulk solution with a constant diffusion coefficient[@sen]. Experiments on the adsorption kinetics of small spherical particles (proteins or small latex particles), however, show a power law behavior of the form $t^{-1/2}$[@kinexp; @lenov] when gravitational effects on the particles can be ignored in front of its pure diffusion. Such a behavior is predicted by kinetic models based on the Random Sequential Adsorption (RSA) filling rules[@stprl]. RSA, however, ignores the physical mechanisms driving the particles to the surface. Thus, it seems that an important physical ingredient has been missed in previous approaches[@sprl]. In this Letter, we will show that hydrodynamic interactions between the particles and the surface substantially modify the predicted asymptotic behavior of the system near saturation. A simple theoretical model taking into account the blocking effect, the diffusion of the particles from the bulk and the hydrodynamic interactions between the particles and the surface, allows us to predict a complex asymptotic behavior for the time-dependence of the surface coverage near saturation. Indeed, near saturation, the model predicts a wide first time-domain where the time-dependence of the coverage is dominated by the hydrodynamic interactions between the free particles and the adsorbing surface. Remarkably, the time-dependence in this region is compatible with the experimental findings of refs. [@kinexp; @lenov]. Furthermore, our model also allows to derive a second time-domain in which the dynamics is dominated by the blocking effect. In this terminal regime, the asymptotic time-dependence of the coverage is in agreement with the preditcions of Schaaf et al.[@sprl]. Nevertheless, we will see that this terminal regime should not be observed for short-range adsorption potentials, whose interaction range is much smaller than the size of the colloidal particles.
Despite their apparent simplicity, the deposition processes are determined by the interplay of various phenomena: the Brownian motion of the free particles, the gravitational force, the dynamic interactions mediated by the solvent (hydrodynamic interactions) and all other kinds of interactions between free particles and the adsorbed ones, as well as between the free particles and the wall. Irreversible adsorption leads to non-equilibrium configurations, thus, it cannot be studied in the framework of equilibrium statistical mecanics. Most of the previous models[@lang; @stprl; @rsa; @bal] have neglected the effect of the solvent. They consider the particles as moving in dry water[@fey] and have focused primarily on the geometric aspects, related to the excluded surface effects. Recently, however, the determinant role played by the hydrodynamic behavior of the solvent was pointed out [@bafa; @igna1]. For instance, the theoretically predicted pair distribution function of the adsorbed layer shows significant deviations from the experimental curves for this function when hydrodynamic interactions are ignored[@woj1; @woj2; @igna1; @igna2; @PhD]. The effect of the hydrodynamic interaction is to increase the frictional force experienced by a particle when it approaches a flat surface. Despite its clear implication in the kinetics of the adsorption process, the effect of the hydrodynamic interaction in the time- dependence of the coverage near saturation has never been analyzed before. In the analysis proposed here we assume that the free particles diffusing from the bulk have to cross an [*entropic*]{} barrier, due to the presence of the previously adsorbed (bound) particles in the layer, before they get trapped by an adsorbing short range potential between the particles and the plane. The density $\rho$ of free particles in the region near the wall is assumed to satisfy a diffusion equation of the form[@nous] $$\frac{\partial}{\partial t} \rho= -\frac{\partial}{\partial \gamma}
J(\gamma,t) =
\frac{\partial}{\partial
\gamma} \frac{D(\gamma,\theta)}{R^2} \left[\frac{\partial}{\partial \gamma}
\rho-\rho \frac{\partial}{\partial \gamma} \ln\Phi(\gamma,\theta) \right]
\label{2}$$ where $\gamma \equiv z/2R$ is the dimensionless coordinate in the $z$-direction orthogonal to the wall, $R$ being the radius of the particles. The origin of coordinates is taken at the center of an adsorbed particle. In this equation, $J(\gamma,t)$ is the flux of free particles in the vicinity of the wall. Thus, it is in the region $0 \leq \gamma \leq 1$ that the effect of the excluded surface due to the presence of adsorbed particles takes place. $\Phi(\gamma,\theta)$ is the available area for a free particle to move in at a height $\gamma$ and at a coverage $\theta$ of the surface[@wid]. $\theta$ is defined as $\theta \equiv
\pi R^2 \rho_s$, where the number of adsorbed particles per unit of area is denoted by $\rho_s$. The diffusion coefficient $D(\gamma,\theta)$ is related to the mobility of the free particles. Far from the wall, the diffusion coefficient of spheres in a dilute solution is given simply by the Stokes-Einstein formula $D = kT/6\pi \eta R$, $\eta$ being the viscosity of the solvent and is constant. Near the wall, however, hydrodynamic interactions modify this behavior. Lubrication theory[@Rus] shows that the mobility in the direction orthogonal to the wall vanishes [*linearly*]{} with the distance between the hydrodynamic surfaces. Thus, the diffusion coefficient behaves as $$D(\gamma,\theta) = (\gamma+\delta) D_0 \label{dif}$$ as $\gamma \rightarrow 0$. $D_0$ is a constant and $\delta \equiv
d/2R$, where $d$ stands for the repulsive range of the adsorption potential (fig. 1). Note that the [*hydrodynamic*]{} wall is then shifted with respect to the [*adsorbing*]{} wall due to the finite range of the adsorption potential considered here. $\delta$ is finite but can be made arbitrarily small in our model, in order to compare with previous analysis[@sprl]. If $\delta \rightarrow 0$ no particles can be adsorbed in a finite time due to the strength of the lubrication forces. The hydrodynamic interaction between the free particles and the adsorbed ones is subdominant for the motion in the direction orthogonal to the wall[@Rus], due to the fact that their surfaces move parallel to each other when $\gamma \rightarrow 0$. As a consequence, the diffusion coefficient is independent of the coverage $\theta$ in this limit.
The diffusion equation (\[2\]) contains all the relevant phenomena driving the kinetics of particle adsorption. In the expression between brackets, the first term stands for pure diffusion of the free particles while the second accounts for the fact that the available area at a given height $\gamma$ and at a given coverage $\theta$ is limited by the presence of the adsorbed particles. Therefore, if the available area is reduced as $\gamma$ decreases, this term acts as an effective entropic potential tending to decrease the flux of free particles. Our model can be applied to a large class of systems provided that they meet the following requirements; (i) the particles are irreversibly adsorbed on the surface and stop moving once trapped; (ii) diffusion dominates over gravitational effects in the dynamics of the free particles.
Notice that, the use of equation (\[2\]) implies that we describe the transport of the free particles across the layer of adsorbed ones without explicit calculation of the structural properties of the layer. Brownian motion permits the particles to explore large regions in space before they get adsorbed. Thus, one expects that the overall adsorption process is not determined by the local inhomogeneities of the layer of adsorbed particles but by its global properties, in the spirit of a mean field approach [@nous].
The saturation coverage $\theta_{\infty}$, or [*jamming limit*]{}, is reached when on the adsorbing surface $(\gamma=0)$, the available surface function becomes equal to zero. For $\theta=\theta_{\infty}$, the entropic barrier becomes infinite, thus the adsorption of new incoming particles is imposible. For spherical particles, the entropic potential near saturation can be written as $$\ln \Phi(\theta,\gamma) \simeq
\ln \left(\theta_{\infty}-\theta (1-\gamma^2) \right)^3 \label{pot}$$ The form $(\theta_{\infty}-\theta)^3$ is the behavior of the available area near saturation for irreversible adsorption of particles, and has been first derived by Pomeau[@Pom]. In addition, we have explicitly indicated the fact that, at a given height, the area excluded by the adsorbed spherical particles is reduced by a factor $(1-\gamma^2)$ for spheres[@nous].
In order to describe the kinetics of the adsorption process, we have to find the incoming flux of free particles arriving at the adsorbing surface $J_s(t) = J(\gamma=0,t)$. Since we are interested only in the kinetics near saturation, where the adsorption process is very slow due to the blocking effect, we can solve eq. (\[2\]) neglecting the explicit time-dependence of $\rho$. We then assume that the variations in the density profile and, thus, in the flux, adiabatically follow the changes in the coverage through $\Phi(\theta,\gamma)$[@nous]. Therefore, we set $\partial\rho/\partial t \simeq 0$ in the left hand side of eq. (\[2\]), implying that $J(\gamma,t)$ is independent of $\gamma$ and equal in fact to $J_s(t)$. We consider here that the density of particles in the bulk $\rho_B$ is the control parameter and thus express $J_s$ in terms of $\rho_B$, with the boundary conditions $$\begin{aligned}
\rho(\gamma = 1) &=& \rho_B, \label{3}\\
\rho(\gamma = 0) &=& 0 \label{4}\end{aligned}$$ The first boundary condition assumes that the density of bulk particles in the vicinity of the adsorbed layer is approximately constant due to the slow adsorption process occuring near saturation. We thus consider a particle’s reservoir located at $\gamma = 1$ with a density $\rho=\rho_B$ constant. The second boundary condition stands for an irreversible adsorption: free particles reaching the wall become irreversibly adsorbed and then the density of free particles is zero at $\gamma = 0$. We can thus obtain the flux of particles reaching the surface in terms of $\rho_B$ by solving the differential equation $$J_s=-\frac{D(\gamma)}{R^2} \left[\frac{\partial}{\partial \gamma}
\rho-\rho \frac{\partial}{\partial \gamma} \ln \Phi \right] \label{5}$$ with boundary conditions specified in eqs. (\[3\]) and (\[4\]). We find the following kinetic equation $$\frac{\partial\rho_s}{\partial t} = -J_s = -\frac{D_0}{R^2} \rho_B I(\theta) \label{jsu2}$$ where $$I(\theta)=\frac {1}{\int_{0}^{1}\frac{D_0}{D(\gamma) \,
\Phi(\theta,\gamma)}d\gamma} \sim \frac
{1}{\int_{0}^{1}\frac{D_0}{D(\gamma)(\theta_{\infty}-\theta
(1-\gamma^2))^3}d\gamma}\;
\label{hjsu}$$ as $\theta \rightarrow \theta_{\infty}$. A closed equation for the time-dependece of the coverage then follows by multiplying both sides by $\pi R^2$, yielding a generalized Langmuir equation[@nous] $$\frac{\partial\theta}{\partial t} = K_a \rho_B I(\theta) \label{lang}$$ where we have defined the kinetic coefficient $K_a = D_0 \pi$. A crucial point in our analysis is that the leading contribution to $I(\theta)$ near saturation depends on the behavior of the integrand for small $\gamma$, which allows us to use the expression of $D(\gamma)$ given in eq. (\[dif\]). Inserting this dependence in the right hand side of eq. (\[hjsu\]) we obtain that the adsorption rate near saturation is proportional to $$I(\theta) \sim \frac
{1}{\int_{0}^{1}\frac{1}{(\gamma+\delta)(\theta_{\infty}-\theta
(1-\gamma^2))^3}d\gamma} \label{eq}$$ The asymptotics of $I(\theta)$ as given by this expression strongly depends on the relative magnitude of $\delta$ and $\Delta \theta \equiv
(\theta_{\infty}-\theta)/\theta_{\infty}$. Clearly, when the coverage approaches saturation, in an initial regime the condition $(\theta_{\infty}-\theta)/\theta_{\infty} \gg \delta$ is satisfied since $\delta$ is a constant that can be taken as arbitrarily small. In this region, the adsorption rate is dominated by the hydrodynamic interactions and takes the assymptotic form $$I(\theta) \sim \frac{2 \Delta \theta^3}{\ln \Delta \theta/\delta^2 - 3/2}.
\label{reg1}$$ The range of validity of this regime is determined by the fact that $I(\theta)$ must be positive since eq. (\[lang\]) describes a relaxation process in which the coverage tends irreversibily to saturation. Effectively, the condition $\ln \Delta
\theta/\delta^2 - 3/2 > 0$ indicates that $\Delta \theta/\delta^2 >
\exp(3/2) \sim 1$. Therefore, the crossover coverage scales as, $\Delta
\theta_c \sim \delta^2$. In this region, the time-dependence of the coverage can be obtained by inserting the asymptotic behavior given in eq. (\[reg1\]) in the right hand side of eq. (\[lang\]). After integration we obtain $$t \sim \frac{\ln \Delta \theta/\delta^2 -1}{4 \Delta \theta^2}. \label{ech}$$ for $\Delta \theta/\delta^2>\exp(3/2)$. Eq. (\[ech\]) gives an implicit relation between the time and $\Delta \theta$. The scaling of the crossover time $t_c$ is obtained by inserting the scaling of the crossover coverage in this expression, giving $t_c \sim 1/\delta^4$. Notice the fact that if $\delta \rightarrow 0$, $t_c \rightarrow \infty$, indicating that this regime must dominate the asymptotic behavior of the coverage near saturation. The numerator on the right hand side of eq. (\[ech\]) is a slowly varying function of $\Delta \theta$. This suggests an iterative procedure to obtain the behavior of $\Delta \theta$ for times $t \ll t_c$. Effectively, one can write $$\Delta \theta \sim \frac{1}{2 t^{1/2}} \sqrt{\ln
\left(\frac{\Delta \theta}{\delta^2} \right)} \sim \frac{1}{2 t^{1/2}}
\sqrt{\ln \left(\frac{1}{2 \delta^2 t} \right)} \label{as}$$ where, in deriving the last expression, a term $\sqrt{\ln \ln (\Delta \theta /
\delta^2)} \sim 1 \ll \ln (1/2\delta^2 t)$ has been neglected. Eq. (\[as\]) predicts a novel behavior for the time-dependence of the coverage near saturation dominated by the hydrodynamic interactions between the free particles and the wall. Such a behavior differs from that found from the RSA model[@Ps] as well as from that predicted by the model incorporating the diffusion of the particles from the bulk[@sprl].
The terminal regime $t \gg t_c$ or, equivalently, $\Delta
\theta/\delta^2 < 1$ can also be obtained from eq. (\[eq\]). The adsorption rate in this case is dominated by the blocking effect and obeys a different asymptotics of the form $$I(\theta) \sim \delta \, (\theta_{\infty}-\theta)^{5/2} \label{reg2}$$ Notice that the right hand side of this equation vanishes as $\delta
\rightarrow 0$. From eq. (\[lang\]) and (\[reg2\]) one arrives at the $\Delta \theta \sim t^{-2/3}$ behavior as found by Schaaf [*et al.*]{}[@sprl]. Therefore, the model proposed here is also able to reproduce Schaaf’s regime[@sprl] when the adsorption kinetics is dominated by the blocking effect. However, due to the fact that the crossover time between the two regimes scales as $\delta^{-4}$ and tends to infinity as $\delta
\rightarrow 0$, it suggests that this regime is never observed for short range adsorption potentials.
In summary, we have explicitly discussed a model where the hydrodynamic interactions are included, in addition to other physical mechanisms like diffusion and blocking effect, which are relevant to describe the adsorbing rate. Hydrodynamic interactions play a crucial role in the adsorption kinetics and cannot be avoided in any experimental work on this process. Previous models ignore the physical mechanisms driving the particles to the surface (see ref.[@stprl] and related). Schaaf [*et al.*]{}[@sprl], by taking into account the diffusion of the particles from the bulk as well as the blocking effect, made a significant step in the description of adsorption kinetics. However, the behavior predicted by Schaaf [*et al.*]{} has never been observed. Experimental results[@kinexp; @lenov] suggest a behavior near saturation compatible with a power law $\Delta \theta \sim
t^{-1/2}$. An important conclusion that can be drawn from the present work is that the regime predicted by Shaaf [*et al.*]{} should not be observed for short range adsorption potential. The most relevant result of this Letter is, however, to have shown that the inclusion of hydrodynamic interactions leads to a behavior near saturation compatible with the experimental findings, in view of the slow behavior of the logarithmic factor in eq. (\[as\]). As it has already been pointed out for structural aspects of colloidal particles’ adsorption onto solid surfaces[@bafa; @igna1], the results of the present work stress the importance of the hydrodynamic interactions also in the kinetics of the process.
P. Wojtaszczyk would like to acknowledge Professor J.M. Rubi, G. Gomila and J.M.G. Vilar for fruitful discussions and suggestions. This work has been supported by the “Ministerio de Educacion y Sciencia” of Spain (contract No. SB95-B41700129).
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W.B. Russel, D.A. Saville and W.R. Schowalter, [*Colloidal Dispersions*]{}, Cambridge University Press (1989).
P. Wojtaszczyk “Proprietes Statistiques d’Ensembles de Particules Adsorbees sur une Surface Solide” Ph.D. Thesis, Universite Louis Pasteur, Strasbourg (1995).
Figure captions {#figure-captions .unnumbered}
===============
Schematic representation of the adsorption process. The adsorbing (free) particles diffuse in the bulk and get finally adsorbed at a distance $d$ (arbitrarily small) from the wall. The origin of the dimensionless coordinate $\gamma$ is taken in the plane defined by the centers of the adsorbed particles.
|
---
abstract: 'In contrast to an infinite family of explicit examples of two-dimensional $p$-harmonic functions obtained by G. Aronsson in the late 80s, there is very little known about the higher-dimensional case. In this paper, we show how to use isoparametric polynomials to produce diverse examples of $p$-harmonic and biharmonic functions. Remarkably, for some distinguished values of $p$ and the ambient dimension $n$ this yields first examples of rational and algebraic $p$-harmonic functions. Moreover, we show that there are no $p$-harmonic polynomials of the isoparametric type. This supports a negative answer to a question proposed in 1980 by J. Lewis.'
address: 'Linköping University, Department of Mathematics, SE-581 83 '
author:
- 'Vladimir G. Tkachev'
title: 'New explicit solutions to the $p$-Laplace equation based on isoparametric foliations'
---
[^1]
Introduction
============
A foliation of a Riemannian submanifold is called isoparametric if its regular leaves have constant mean curvature. The study and classification of isoparametric foliations is an important problem of geometry. Remarkably, isoparametric foliations have been shown to be useful in constructing explicit examples in many problems of analysis, geometry and algebra. We briefly mention their appearance in entire solutions to the minimal surface equation [@BGG], [@Simon89], [@SS], exotic smooth structures [@GeJ2016], Yau conjecture on the first eigenvalue [@TangYan13], nonassociative algebras [@Karcher86], [@Tk10a], [@Tk14], eigenmaps between spheres and Brouwer degrees of gradient maps [@Tang2], [@Tang3], [@GeXie], viscosity solutions to fully nonlinear elliptic PDEs [@NTV], [@NTVbook], Willmore submanifolds [@Xie], integrable structures and mathematical physics [@Savo], [@Ferap].
In this paper, we give yet another application of isoparametric foliations to constructing explicit examples of $p$-harmonic and biharmonic functions. We employ an elementary approach and it is also worth noting that many results of the paper may be easily extended to diverse variational type PDEs. Many of the constructed examples are either rational or algebraic functions of coordinates.
The existence and construction of homogenous (sometimes called quasiradial or separable) $p$-harmonic functions of the form $|x|^\beta\omega(\theta)$, $\theta=x/|x|$ is well-known and exploited extensively, for example, by L. Véron and S. Kichenassamy [@Veron], [@KSVeron]: the orthogonal decomposition of the $p$-Laplace operator yields the characteristic equation for $\beta$ and a homogenous first order differential equation for $\omega(\theta)$ on the unit sphere $S^{n-1}\subset\R{n}$. In the two-dimensional case $n=2$, the equation can be solved explicitly in terms of trigonometric functions [@Krol73], [@Aronsson86], [@Tk06c]. Some explicit examples of $p$-harmonic functions are also available in [@Lind06], [@Lind16a] and for $p=N$ [@BorVer07].
Our approach is somewhat complementary to the aforementioned techniques and relies on the Cartan-Münzner equation: we do not work with the orthogonal decomposition of the $p$-Laplace operator, instead we reduce the original equation to a certain second order quasilinear equation, eq. below. One of the benefits of our approach is that it allows to consider a wider (than homogenous functions) class of solutions. On the other hand, due to specific properties of isoparametric polynomials, our construction works only for specific combinations of the parameter $p$ and the ambient dimension $n$. It would be interesting to clarify whether these particular combinations are indeed distinguished in an appropriate sense, or they are just a consequence of our method.
Even in the homogeneous (quasiradial) case, all explicitly known so far higher-dimensional examples correspond to the lowest values of the isoparametric parameter $m=0$ and $m=1$, see the next section for more details. We extend this on all possible values of the isoparametric parameter $m\in \{1,2,3,4,6\}$.
The paper is organized as follows. In section \[sec54\] we discuss the definition and provide some principal examples of isoparametric polynomials. In section \[sec:plap\] we show that the $p$-Laplace equation for a function $f(s,t)$ depending on the distance function $s=|x|^m$ and an isoparametric polynomial $t=\phi(x)$ becomes a quasilinear second order PDE in $s$ and $t$. While classification of solutions to the resulting equation appears formidably difficult in general, we can report here a complete solution of the problem in some important particular cases, see section \[sec:spec\]. We also classify homogeneous solutions of ; this can be interpreted as a direct generalization of the quasiradial solutions. In section \[sec:pol\], we prove that there are no polynomial solutions of isoparametric type. This supports a negative answer to a question proposed by J. Lewis [@Lewis80] on the existence of $p$-harmonic polynomials. Finally, we outline in section \[sec:some\] some further applications of our method to construct $p$-Laplacian eigenfunctions of the unit sphere and biharmonic functions.
Preliminary facts on isoparametric functions {#sec54}
============================================
In this section we recall some basic concepts and facts on isoparametric functions and isoparametric hypersurfaces, see for example a recent book [@CecilRyan2015]. Let $M$ be a Riemannian manifold, $E\subset M$ be an open subset. A smooth function $u : E \to \R{}$ is called *isoparametric* if there exist smooth functions $f$ and $g$ defined on the range of $f$ such that: $$|Du|^2=f(u), \quad \Delta u=g(u).$$ Recall that a (smooth) hypersurface is called [isoparametric]{} if it is a regular level set of an isoparametric function.
The most interesting case is that of isoparametric hypersurfaces in the real space forms. Then it is well-known that an isoparametric hypersuraface has constant principal curvatures and conversely, any hypersuraface in a space form having constant principal curvatures is a leaf in an isoparametric foliation. Isoparametric hypersurfaces in Euclidean space $M=\R{n}$ were classified by Levi-Civita [@LevCivita] for $n=3$ and Segre [@Segre] for all $n$. At the same time, È. Cartan solved the problem in the case of the hyperbolic space. In both cases the number $m$ of distinct principal curvatures is at most 2, and the hypersurfaces are essentially tubes over a totally geodesic subspace.
The sphere the situation is much more interesting and the problem to classify isoparametric hypersurfaces in the Euclidean spheres $\mathbb{S}^{n-1}\subset \R{n}$ has very deep connections with nonassociative division algebras [@Karcher86] and commutative algebra [@CCC], [@ChiBook]. Cartan itself classified all isoparametric hypersurfaces in the Euclidean spheres with $m\le 3$ distinct principal curvatures. According to a celebrated result of F. Münzner [@Mun1], the number $m$ of distinct principal curvatures of an isoparametric hypersurface in a unit sphere can only be $m=1,2,3,4$ or $6$, and any such a hypersurface is the restriction on the unit sphere of a level set $M=\phi^{-1}(t)\cap \mathbb{S}^{n-1}$ ($t\in [-1,1]$) of a *homogeneous polynomial solution* of $$\begin{aligned}
\label{Muntzer1}
|\nabla \phi(x)|^2&=m^2|x|^{2m-2},\qquad x\in \R{n}
\\ \label{Muntzer2}
\Delta \phi(x)&=\frac{1}{2}(m_2-m_1)\,m^2|x|^{m-2},\qquad x\in \R{n}.
\end{aligned}$$ where $(m_1,m_2)$ are the multiplicities of distinct principal curvatures related to the ambient dimension $n$ and the multiplicity number $m$ by $$\label{obstr}
n=\frac{1}{2}(m_2+m_1)m+2.$$ We emphasize that $m_1$ and $m_2$ are positive integers. As the matter of convention we always assume that $m_1\le m_2$. Notice that $\phi$ is harmonic if and only if $m_1=m_2$.
\[def1\] A homogeneous polynomial $\phi$ is said to be *isoparametric* if it satisfies (\[Muntzer1\])–(\[Muntzer2\]). We write $$\phi\in \mathscr{I}_m(m_1,m_2)$$ if $\phi$ satisfies both (\[Muntzer1\]) and (\[Muntzer2\]), and $\phi\in \mathscr{I}_m,$ if rather $\phi$ satisfies (\[Muntzer1\]).
Notice that a homogeneous polynomial solution to the first Cartan-Münzner equation (\[Muntzer1\]) alone is a composition of some isoparametric form with a Chebyshov polynomial [@Tk14b].
Below we consider some explicit representations of isoparametric polynomials. Let us make some remarks concerning notations. By $\phi_{m,m_1,m_2}$ we denote an isoparametric polynomial of degree $m$ and having the multiplicities $(m_1,m_2)$. Note that this notation may be ambiguous in certain cases when $m=4$ and $m_1\ne m_2$. In the examples below, we give corresponding explicit representations in some specific Euclidean coordinates providing a most optimal form. The reader have to note, however, that an isoparametric polynomial is determined up to an orthogonal transformation of the ambient space.
The case $m=1$ is trivial: any linear function of $x\in \R{n}$ is an isoparametric polynomial. It is also straightforward to verify that for $m=2$ the only possible isoparametric quadratic forms are $$\label{m2}
\phi=x_1^2+\ldots+x_{m_2}^2+x_{m_2+1}^2 -x_{m_2+2}^2-\ldots-x_{m_1+m_2+2}^2, \quad m_1, m_2\in \mathbb{Z}^{+}.$$ If $m_1=m_2$ then the above function is harmonic and the zero level set $\phi(x)=0$ is a Clifford-Simon minimal cone.
By the classical Cartan result [@Cartan38], there are exactly four (up to an isometry) isoparametric polynomials for $m=3$. Furthermore, they are *harmonic* and the multiplicities of the principal curvatures coincides with the dimensions of the classical division algebras: $m_1=m_2\in \{1,2,4,8\}$. More precisely, the corresponding cubic forms in dimensions $n=5,8,14$ and $26$ are given by $$\label{CartanFormula}
\begin{split}
\phi(x)=&x_{n}^3+\frac{3}{2}x_{n}(|z_1|^2+|z_2|^2-2|z_3|^2-2x_{n-1}^2)+ \frac{3\sqrt{3}}{2}x_{n-1}(|z_1|^2\\
&-|z_2|^2)+{3\sqrt{3}}\operatorname{\mathrm{Re}}z_1(z_2z_3)\qquad\qquad d=1,2,4,8,
\end{split}$$ where $z_k=(x_{kd-d+1},\ldots,x_{kd})\in \R{d}\cong\mathbb{A}_d$, $k=1,2,3$, and $\mathbb{A}_d$ denotes the real division algebra of dimension $d$: $\mathbb{A}_1=\mathbb{R}$ (the reals), $\mathbb{A}_2=\mathbb{C}$ (the complexes), $\mathbb{A}_4=\mathbb{H}$ (the quaternions) and $\mathbb{A}_8=\mathbb{O}$ (the octonions). In fact, one can rewrite (\[CartanFormula\]) in a more compact way as follows $$\label{Iso3det}
\phi(x)=-\frac{\sqrt{3}}{2}\operatorname{\mathrm{tr}}T^3,
\quad\quad
T=\left(
\begin{array}{ccc}
\half{1}{\sqrt{3}}x_{n}+x_{n-1} & z_3 & \bar z_2 \\
\bar z_3& \half{1}{\sqrt{3}}x_{n}-x_{n-1} & x_1 \\
z_2 & \bar z_1 & \half{-2}{\sqrt{3}}x_{n} \\
\end{array}
\right),$$ where the trace representation for $d=4,8$ should be understood in the Jordan algebra sense [@BS2], [@Tk14].
When $m=4$, the situation is much more involved. The results of R. Takagi [@Takagi], Ozeki and Takeuchi [@OT1], [@OT2], S. Stolz [@Stoltz], Cecil, Chi and Jensen [@CCC] and Q.-S. Chi [@ChiBook] establish an ultimate classification of isoparametric hypersurfaces with four principal curvatures. In summary, any isoparametric hypersurface with $m=4$ is either from the Ferus-Karcher-Münzner family [@FKM] based on the representations of Clifford algebras (see below), or homogeneous (i.e. the orbits of certain subgroups of the orthogonal group $O(n)$) with $(m_1,m_2)=(2,2)$ or $(4,5)$ in $\R{10}$ and $\R{20}$, respectively. É. Cartan [@Cartan40] was the first to classify all such isoparametric hypersurfaces. He established that $m_1=m_2\in \{1,2\}$. The corresponding explicit representations in $\R{6}$ and $\R{10}$ are given respectively by $$\label{m411}
\phi=|x|^4+|y|^4+8\scal{x}{y}^2-6|x|^2|y|^2,$$ where $x=(x_1,x_2,x_3)$, $y=(x_4,x_5,x_6)$, and $$\label{m422}
\phi=\frac{1}{2}(\operatorname{\mathrm{tr}}X^4-\frac{3}{8}(\operatorname{\mathrm{tr}}X^2)^2),
\quad X=\left(
\begin{array}{rrrrr}
0 & x_1 & x_2 & x_3 & x_4 \\
-x_1 & 0 & x_5 & x_6 & x_7 \\
-x_2 & -x_5 & 0 & x_8 & x_9 \\
-x_3 & -x_6 & -x_8 & 0 & x_{10} \\
-x_4 & -x_7 & -x_9 & -x_{10} & 0 \\
\end{array}
\right),$$ see [@OT1], [@OT2]. The first example is a member of the so-called FKM-isoparamteric (or the Ferus-Karcher-Münzner) polynomials parameterized by symmetric Clifford systems as follows. A finite set $\mathcal{A}=\{A_i\}_{1\le i\le q}$ of real symmetric $2p\times 2p$-matrices $A_i$ is called a [*symmetric Clifford system*]{} if $$\label{Apolar1}
A_iA_j+A_jA_i=2\delta_{ij}I,$$ where $I$ is the unit matrix. The simplest example is for $q=2$, $p=1$: $$A_1=\left(
\begin{array}{cc}
1 & 0 \\
0 & -1\\
\end{array}
\right),
\quad
A_2=\left(
\begin{array}{cc}
0 & 1 \\
1 & 0\\
\end{array}
\right).$$ A symmetric Clifford system of size $q$ in $\R{2p}$ exists if and only if the inequality $$\label{qp}
q\le 1+\rho(p)$$ holds, where the Hurwitz-Radon function $\rho$ is defined by $$\label{foll}
\rho(s)=8a+2^b, \qquad \text{if} \;\,\,s=2^{4a+b}\cdot \mathrm{odd} , \;\; 0\leq b\le 3,$$ see [@Shapiro], [@CecilRyan2015]. Then the quartic form $$\label{FKM}
\phi_{\mathcal{A}}=|x|^4-2\sum_{i=1}^q(x^tA_ix)^2$$ is an isoparametric polynomial in $\R{2p}$, see [@CecilRyan2015] for more details. Then the isoparametric parameters (multiplicities of the principal curvatures) are given by $$\label{m1m2m}
m_1=q-1, \qquad m_2=p-q.$$ Note also a nice relation between FKM-isoparamteric polynomials and cubic minimal cones arising from Clifford algebras, see [@Tk10c]: the zero level set of the cubic form $$\label{FKM4}
f(z)=\sum_{i=1}^q (x^tA_ix)y_i, \qquad z=(x,y)\in \R{2p}\times \R{q},$$ is a minimal cone in $\R{2p+q}$.
Finally, if $m=6$ then by the results of Abresh [@Abresh] we know that $m_1=m_2\in\{1,2\}$, and then Dorfmeister and Neher [@DorfN] and Miyaoka [@Miyaoka13] have proved that there exist only two isoparametric forms of degree $6$ in dimensions $8$ and $14$, respectively and the corresponding isoparametric hypersurfaces are homogeneous (see also [@Siffert3] which simplifies the Dorfmeister and Neher’s result and refines the Miyaoka’s argument). Explicit matrix representations can be found in [@PengHou].
$m$ $m_1$ $n=mm_1+2$
----- ---------------- ------------
$2$ $1,2,3,\ldots$ $2m_1+2$
$3$ $1,2,4,8$ $3m_1+2$
$4$ $1,2$ $4m_1+2$
$6$ $1,2$ $6m_1+2$
: The possible multiplicities $m_1=m_2$ and the ambient dimension $n$[]{data-label="tab1"}
The isoparametric Ansats {#sec:plap}
========================
It is the well-known and frequently exploited fact that radial symmetric solutions encode the most fundamental properties of many variational type PDEs, in particular those invariant under the action of the orthogonal group $O(n)$. It is the also a particular aim of the present paper to extend this observation onto a subclass of solutions based on isoparametric polynomials. As the main model example, we consider the $p$-Laplace operator including the limit case $p=\infty$. The method used below, however, applies also to general $O(n)$-equivariant operators, such as the mean curvature equation or a general Hessian equation.
Given $p\in \R{}$, the operator $$\label{plaplace}
\Delta_p u:=|\nabla u|^2\Delta u+\half{p-2}{2}\nabla u\cdot \nabla |\nabla u|^2=0$$ is called the $p$-Laplacian. Here $u(x)$ is a function defined on a domain $E\subset \R{n}$, $\nabla u$ is its gradient and $\cdot$ denotes the standard inner product in $\R{n}$. A classical $C^2$-solution of is called a $p$-harmonic function. In general, when $p>1$ and $p\ne2$, $u$ should be understood as a weak (in the distributional sense) solution to (\[plaplace\]). Then $u$ is normally in the class $C^{1,\alpha}(E)$ [@Ural68], [@Uhlenbeck], [@Evans82], but need not to be a Hölder continuous or even continuous in a closed domain with nonregular boundary [@KrolMaz72]. On the other hand, if $u(x)$ is a weak solution of (\[plaplace\]) such that $\mathrm{ess} \sup |\nabla u(x)|>0$ holds locally in a domain $E\subset \R{n}$ then $u(x)$ is in fact a real analytic function in $E$ [@Lewis77].
The natural question to ask is then: If there exist (nontrivial) *rational* or *algebraic* examples of $p$-harmonic functions for $p\ne2$ (including the case $p=\infty$)? In $\R{2}$ the answer is ‘yes’ as it follows from the quasiradial examples constructed by G. Aronsson in [@Aronsson86], [@Aronsson89] and its further analysis given by the author in [@Tk06c]. In this section (see also Remark \[rem:rat\]) we present several rational and algebraic examples of $p$-harmonic function.
A very related but much more difficult question is whether there exist nontrivial *homogeneous polynomial* $p$-harmonic functions for $p\ne2$? This problem naturally emerges in connection to the nonvanishing propery of analytic $p$-harmonic functions and was proposed and studied by J. Lewis in [@Lewis80]. In particular, Lewis wa able to establish the negative answer for $n=2$. Very recently some higher-dimensional generalizations for degree $=3,4,5$ were obtained in [@Tk16pLapl], [@Lewis2016]. We refer to section \[sec:pol\] below for some particular results in this direction.
The radial symmetric $p$-harmonic functions are well-known: $$\label{fundam}
E_{p,n}(x)=
\left\{
\begin{array}{cc}
(n-p)|x|^{\frac{p-n}{p-1}} & \text{for $p\ne n$} \\
-\log |x| & \text{for $p= n$}
\end{array}
\right.$$ and play the role of the fundamental solution to (\[plaplace\]). Notice that $E_{p,n}(x)$ is of class $W^{1,p}(\Omega)$ for any domain omitting origin.
In this paper, we study solutions $u(x)$ of having a general form $$\label{uharm}
u(x)=f(|x|^m,\,\phi(x))$$ where $f=f(s,t)$ is a $C^2$-function and $\phi\in \Iso_m(m_1,m_2)$.
It seems natural to relax the Münzner-Cartan equations (\[Muntzer1\])–(\[Muntzer2\]) on $\phi$ and consider instead an *a priori* bigger set of $p$-harmonic functions of the form (\[uharm\]) with $\phi$ satisfying the first Münzner-Cartan equation (\[Muntzer1\]) only. Note, however, that this does not yield some further examples. Indeed, it follows from [@Tk14b] that there exists a Chebyshov polynomial $T_k(t)$ and $\psi\in \Iso_{m'}(m'_1,m'_2)$ such that $\phi(x)=|x|^{m'} T_k\bigl(\psi(x)|x|^{-m'}\bigr)$, which implies that $f(\phi,|x|^m)=F(\psi,|x|^{m'})$ for some suitable $F$ depending on $f$.
\[rem:angle\] Functions given by have a clear geometric interpretation: it is known [@CecilRyan2015 Sec. 3.5] that if $\phi(x)\in \Iso_m$ then there exists a function $\theta(x)$ smooth on $S^{n-1}\setminus M(\phi)$ and $\theta(x):S^{n-1}\to [-1,1]$, such that $$\label{anglem}
\phi(x)=|x|^m\cos m\theta(x)$$ In this setup, $\theta(x)$ is naturally understood as a certain (isoparametric) angle function on the unit sphere. Thus, expresses the fact that $u$ depends only the radius $r$ and the ‘isoparametric angle’ $\theta$. In the trivial case $n=2$, we have the standard polar angle $\theta=\arctan\frac{y}{x}$ which amounts to the quasiradial solutions studied by Aronsson [@Aronsson86].
\[prop1\] Let $\phi\in \Iso_m(m_1,m_2)$ and $$\label{st}
s=|x|^m, \quad t=\phi(x).$$ Then for any $C^2$-function $f=f(s,t)$ $$\label{plap1}
\Delta_p f(|x|^m,\,\phi(x)) =m^{4}s^{1-4/m}(sAC+\half{p-2}{2}sB+\half{(m-2)(p-2)}{2m}Ah),$$ where $$\label{notation}
\begin{split}
h&=f_ss+f_tt, \\
A&= (f_s^2+f_t^2)s+2f_sf_tt,\\
B&= (f_sA_s+f_tA_t)s+(f_sA_t+f_tA_s)t,\\
C&= (f_{ss}+f_{tt})s+2f_{st}t+(\mu+1)f_s+\nu f_t,\\
\mu&=\half1{2}(m_1+m_2),\\
\nu&=\half1{2}(m_2-m_1),\\
\end{split}$$ In particular, $u(x)=f(|x|^m,\,\phi(x))$ is $p$-harmonic if and only if $$\label{plap2}
sAC+\frac{p-2}{2}sB+\frac{(m-2)(p-2)}{2m}Ah=0.$$
Using the Euler homogeneity function theorem and (\[Muntzer1\]) we have $\scal{\nabla s}{\nabla t}=m^2s^{1-2/m}t$ and $|\nabla s|^2=|\nabla t|^2=m^2s^{2-2/m}$. This yields for any two functions $f(t,s)$ and $g(t,s)$ that $$\label{scalgrad}
\scal{\nabla f}{\nabla g}=m^2s^{1-2/m}((f_s^2+f_t^2)s+2f_sf_tt),$$ hence $$\label{Df}
|\nabla f|^2=m^2s^{1-2/m}((f_s^2+f_t^2)s+2f_sf_tt)=m^2s^{1-2/m}A.$$ Applying , one readily obtain in the notation (\[notation\]) that $$\begin{aligned}
\nabla f\cdot \nabla |\nabla f|^2
&=m^4s^{1-4/m}(sB+(1-\frac{2}{m})Ah)\nonumber\\\end{aligned}$$ and similarly that $$\begin{aligned}
\Delta f&
=m^2s^{1-2/m}\left((f_{ss}+f_{tt})s+2f_{st}t+\frac{m+n-2}{m}f_s +\frac{m_2-m_1}{2}f_t\right)\nonumber\\
&=m^2s^{1-2/m}\bigl((f_{ss}+f_{tt})s+2f_{st}t+(\mu+1)f_s +\nu f_t\bigr)\label{laplce}\\
&=m^2s^{1-2/m}C.\nonumber\end{aligned}$$ Combining the obtained relations with (\[plaplace\]) finishes the proof.
A complete analysis of the nonlinear partial differential equation (\[plap2\]) is a rather difficult problem. In this paper, we make the first step and consider the special solutions satisfying the following ansats: $$\label{ansats}
f(s,t)=s^kg(z), \quad z=\frac{t}{s}.$$ In the terminology of [@Aronsson89], [@Porr], describes the so-called *separable* $p$-harmonic functions, i.e. those having the form $$\label{ansats1}
u(x)=|x|^k\cdot g\bigl(\phi(x)|x|^{-m})\equiv |x|^k g(\cos m\theta),$$ where the isoparametric angle is defined as in Remark \[rem:angle\]. By virtue of , $$\label{z1}
|z|\le 1.$$ Then Proposition \[prop1\] readily yields
\[cor:homogen:m1\] The function is $p$-harmonic if and only if $$\label{geq}
\begin{split}
(z^2-1)\biggl(b_1(z^2-1)g'^2g''+b_2g^2g''+ (b_3z+\nu)g'^3+b_4g'^2g\biggr)+(b_5z-\nu)g'g^2 +b_6 g^3=0,
\end{split}$$ where $$\label{alpha}
\begin{split}
&b_1=1-p,\\
&b_2=k^2,\\
&b_3=1-p-\mu,\\
&b_4=k(2kp-3k+\half{n-p}{m}), \\
&b_5=k^2(\mu+1), \\
&b_6=-k^3(kp-k+\half{n-p}{m}).
\end{split}$$
Some integrable cases of {#sec:spec}
=========================
Even the full analysis of the nonlinear equation is out the scope of the paper. We confine ourselves to some particular cases of , more precisely:
- for certain particular values of ‘isoparametric parameters’ $m_1,m_2$ and $m$;
- for some fixed values of $k$;
- some specific ansatz for the function $g$.
Some preliminary remarks are in order. Note that in the simplest case $m=1$ the isoparametric polynomial $\phi(x)$ is essentially one-dimensional and can be written as $\phi(x)=x_1$ in an appropriate orthogonal coordinate system. To be consistent with and –, one obtains that $m_1=m_2=n-2$, hence in the notation of Corollary \[cor:homogen:m1\] this yields $\mu=n-2$ and $\nu=0$. Solutions under this form were first considered and studied by I. Kroll’ in [@Krol73]; some further modifications can be made in the dimension $n=3$, see [@KrolMaz72], [@Veron88 p. 359]. On the other hand, the corresponding differential equations do not yield any explicit solution for $n\ge 3$. We refer to [@Krol73] for more details.
The case of arbitrary $k$ and $g=bz+a$ {#sublinear}
--------------------------------------
This is the first case that yields new nontrivial explicit solutions and works well for all possible isoparametric parameters. More precisely, we consider solutions given by with $g$ satisfying the linear Ansats: $$g(z)=z+a.$$ Note that by the homogeneity, we may without loss generality consider a monic polynomial ($b=1$). Then a simple analysis shows that if $p\ne 2$ then nontrivial solutions emerge only if either $g(z)=z\pm 1$ or $g(z)=z$. In the former case one has $m=2$, $k=1$ and $p=1-m_1$ which is unsatisfactory from the analytic point of view because $p<0$. But the latter case $g(z)=z$ contains some nontrivial solutions with $p>1$ as the proposition below shows.
\[pro:z\] Let $\phi\in \mathscr{I}_m(m_1,m_2)$. Then $u(x)=\phi(x)|x|^{(k-1)m}$ is $p$-harmonic if and only if $$\label{m1m2}
m_1=m_2$$ and $k\in \{-1,0,1\}$. The corresponding solutions are given by $$\begin{array}{lclcl}
\bullet\,\, k=1 &\quad& u=\phi(x) &\quad& p=2;\\\bullet\,\, k=0 &\quad& u=\phi(x)|x|^{-m} &\quad& p=1-m_1\\\bullet\,\, k=-1 &\quad& u=\phi(x)|x|^{-2m} &\quad& p=2+\frac{2m_1m}{m+1}\\
\end{array}$$ When $k=-1$, the distinguished dimensions $n$ and the corresponding rational $p$-harmonic functions are given by
$m$ $2$ $2$ … $2$ … $3$ $3$ $3$ $3$ $4$ $4$ $6$ $6$
----- -------------- -------------- --- ------------------- --- ------------- ----- ------ ------ -------------- -------------- -------------- --------------
$n$ $4$ $6$ … $2k$ … $5$ $8$ $14$ $26$ $6$ $10$ $8$ $14$
$p$ $\frac{10}3$ $\frac{14}3$ … $\frac{2(2k+1)}3$ … $\frac{7}2$ $5$ $8$ $14$ $\frac{18}5$ $\frac{26}5$ $\frac{26}7$ $\frac{38}7$
Setting $g(z)=z$ in yields that $\nu=0$ (implying ) and also that $\beta_3+\beta_4=\beta_5+\beta_6=0$. This implies by virtue of $m\ne0$ that the only solutions are given by $k(1-k^2)=0$ which readily yields the desired conclusion.
\[rem:rat\] From the analytic point of view, the case $k=-1$ is the most interesting because we have $p>2$. In fact, since $$n-p=\frac{m-1}{m+1}m_1m>0$$ the stronger inequality $2<p<n$ holds. The most spectacular observation here is that *$u(x)=\frac{\phi(x)}{|x|^{2m}}$ is a rational $p$-harmonic function.*
We illustrate the case $k=-1$ in Proposition \[pro:z\] with some examples. As we already pointed out, all these examples are rational functions. First consider Clifford cone type isoparametric solutions for $m=2$. Then for an arbitrary $k=1,2,3,\ldots$, the rational function $$u(x)=\frac{x_1^2+\ldots+x_k^2-x_{k+1}^2-\ldots-x_{2k}^2} {(x_1^2+\ldots+x_{2k}^2)^2}$$ is a homogeneous order $-2$ rational solution of the $p$-Laplace in $\R{2k}$ with $p=\frac{4k}{3}+2$. Next, for $m=3$ we have that $$u=\frac{x_{5}^3+\frac{3}{2}x_{5}(x_1^2+x_2^2-2x_3^2-2x_{4}^2)+ \frac{3\sqrt{3}}{2}x_{4}(x_1^2 -x_2^2)+{3\sqrt{3}}x_1x_2x_3}{(x_1^2+\ldots+x_{5}^2)^3}$$ is a homogeneous order $-3$ rational solution of the $p$-Laplace in $\R{5}$ with $p=\frac{7}{2}$. Finally, let us mention an explicit example of homogeneous degree $-4$ quartic in $\R{6}$ with $p=\frac{18}{5}$ corresponding to : $$u=\frac{|x|^4+|y|^4+8\scal{x}{y}^2-6|x|^2|y|^2}{(|x|^2+|y|^2)^4},$$ where $x=(x_1,x_2,x_3)$, $y=(x_4,x_5,x_6)$.
The case $k=0$, $m_2=m_1$ and arbitrary function $g$
----------------------------------------------------
In this case, we have the following complete description of the function $g(z)$.
Let $\phi\in \mathscr{I}_m(m_1,m_1)$. Then $g(\phi(x)/|x|^{m})$ is $p$-harmonic iff $$\label{alpha1}
g'(z)=C(1-z^2)^{-\alpha}, \qquad p=1+\frac{m_1}{2\alpha-1}.
$$
By the assumption, we have $k=0$ in . Then $\beta_2=\beta_4=\beta_5=\beta_6=0$, thus amounts to $$\label{geqk0}
\bigl((1-p)(z^2-1)g''+(1-m_1-p)zg'\bigr)g'^2=0.$$ Eliminating the trivial case $g=\mathrm{const}$, one finds that $
g''/g'=\frac{m_1+p-1}{p-1}\cdot \frac{z}{1-z^2}
$ which implies .
Specializing $\alpha$, one obtains some interesting particular cases. Furthermore, if $\phi\in \mathscr{I}_m(m_1,m_1)$ then one has the following $p$-harmonic functions:
$$\begin{array}{lclcl}
(\mathrm{i})\,\, u=\mathrm{arctanh}\,(\phi(x)|x|^{-m}) &\quad& p=1+m_1 &\quad& n=mm_1+2\\
\\
(\mathrm{ii})\,\, u=\frac{\phi(x)}{\sqrt{|x|^{2m}-\phi(x)^2}} &\quad& p=1+\frac12m_1 && n=mm_1+2\\
\\
(\mathrm{iii})\,\, u=\arcsin \frac{\phi(x)}{|x|^{m}},&\quad& p=\infty && n=mm_1+2,\\
\\
(\mathrm{iv})\,\, u=\ln \frac{|x|}{|y|},\,\, (x,y)\in \R{k}\times \R{k} &\quad& p=k &&n=2k, \,\,k=1,2,3\ldots
\end{array}$$
Specializing $\alpha=1$, $\alpha=\frac32$ and $\alpha=\frac12$ yields (i), (ii) and (iii) respectively. The last case is a corollary of (i) corresponding to the quadratic isoparametric polynomials . In that case, we have $k-1=m_1=m_2$ and using (i), we see that the function $\mathrm{arctanh}\frac{|x|^2-|y|^2}{|x|^2+|y|^2}=\ln \frac{|x|}{|y|}$ is $k$-harmonic in $\R{2k}$.
Two-zone anisotropic isolated singularities
-------------------------------------------
The quasiradial $p$-harmonic functions play a fundamental role in the description of general isolated interior or boundary singularities. In particular, the following famous result [@Serrin65] by Serrin [^2] says that a positive singular solution of the $p$-Laplacian equation should behave as the fundamental solution $E_{p,n}(x)$: If $u$ is a continuous positive solution of $\Delta_p u=0$ in the punctured ball $B\setminus\{0\}$ then either $0$ is a removable singularity (i.e. $u$ can be extended to the whole $B$) or $$u \sim |x|^{\frac{p-n}{p-1}} \quad \text{for }x\rightarrow 0.$$
It is interesting to ask if there is an analogues result for solutions $u$ of the $p$-Laplace equation with anisotropic isolated singularities. In particular, it is natural to consider the following situation: Let $u$ be a solution of the $p$-Laplace equation with isolated singularity at the origin and satisfying the following *two-zone anisotropic property*: for any $\epsilon>0$ small enough, the sets $$U^\pm_\epsilon:=\{x\in \R{n}: \pm u(x)>0 \text{ and }0<|x|<\epsilon\}$$ are connected. What can be said about the asymptotic behavior of $u$? When the origin is a removable singularity? It is also interesting to characterize the geometry of the zero locus $U^0:=\{x:u(x)=0\}$ nearby the origin in this case.
In two dimensions, the situation is well-studied and according to [@Manfredi], any solution of the $p$-Laplace equation in $\R{2}$ with an isolated singularity at the origin behaves as the corresponding quasiradial solution. Also precise asymptotic representation near the singularity has been obtained in [@Manfredi].
In higher dimensions, there exist of plenty of anisotropic singularities, see for example [@Veron], however, little is known so far about analogues of the Serrin result for solutions with anisotropic singularities. Below we suggest some heuristic argument supporting the following conjecture.
**Conjecture.** If $u$ is a solution of the $p$-Laplace equation with a two-zone isolated singularity at the origin such that $$\limsup_{x\to 0}|\phi(x)|\cdot |x|^{\frac{2n-p-2}{p-2}}=0$$ then $0$ is a removable singularity.
Let us explain the appearance of the exponent $\alpha=\frac{2n-p-2}{p-2}$. First note that the constructed in Proposition \[pro:z\] solutions with $k=-1$ satisfy the two-zone property. Indeed, it is well-known fact (see, for example, [@CecilRyan2015 Sec. 3.5]) that given an isoparametric polynomial $u(x)$, its zero locus $$M:=\{x\in\R{n}:|x|=1 \text{ and } u(x)=0\}$$ is a smooth embedded constant mean curvature hypersurface of the unit sphere $S^{n-1}\subset\R{n}$. The complete zero locus $U^0=\{x: u(x)=0\}$ is the cone over $M$. Then it is known that $M$ divides the sphere $S^{n-1}$ into two connected components (respectively, $U_0$ divides $\R{n}$ into two components nearby the origin). In fact, this property readily follows from the eiconal equation . We emphasize that this two-zone property property holds true for all admissible isoparametric parameters $m,m_1,m_2$ and the dimension $n$.
Now, let us consider the corresponding solution $u_\phi(x)=\phi(x)|x|^{-2m}$ with $\phi\in \mathscr{I}_m(m_1,m_1)$. Then $u$ is singular at the origin and satisfies the asymptotic growth condition $$|u_\phi(x)|\cdot |x|^{m}\le C,$$ where the exponent $m$ is sharp. Using the relation $p=2+\frac{2m_1m}{m+1}$ and eliminating $m_1=m_2$ by virtue of we obtain $$m=\frac{2n-p-2}{p-2}.$$ By analogy with the harmonic polynomials in $\R{2}$, it is natural to think of $u_\phi(x)$ as a good candidate for the extremal case for two-zone solutions. Note also that $m$ depend on the dimension $n$ and the parameter $p$ only, but not on a particular choice of $\phi$. This makes plausible to believe that the above Conjecture is true.
Polynomial solutions to {#sec:pol}
========================
The present paper arose in an attempt to construct homogeneous polynomial solutions to the $p$-Laplace equation based on isoparametric polynomials. In 1980, John Lewis asked in [@Lewis80] to characterize all (non-linear) polynomial solutions $u$ to the $p$-Laplace equation for $p>1$ and $p\ne2$. The nontrivial part of the problem is to characterize the possible homogeneous polynomial solutions. If the degree of $u$ is even then one always has radially symmetric polynomial solutions, but in that case always $p<1$. If $p>1$ and $p\ne 2$ then it is known that there are no homogeneous degree $m$ polynomial solutions to the $p$-Laplace equation in the following cases:
- $n=2$ and any $m\ge 2$ [@Lewis80],
- $m=3$ and any $n\ge 2$ [@Tk16pLapl],
- $m=4$ and any $n\ge 2$, $m=5$ and $n=3$ [@Lewis2016].
It is a common believe that the answer to Lewis’ question is negative. Note that, heuristically, if a homogeneous solution would exist it would have some very distinguished (symmetric or extremal) properties. Isoparametric form is a natural candidate. But the result below shows that there are no isoparametric type examples.
Let $p>1$, $p\ne 2$. Then there are no homogenous polynomial solution of satisfying and $\deg u\ge2$.
We argue by contradiction and assume that $u=f(s,t)\not\equiv 0$ is homogenous degree $\deg u=N\ge2$ (in $x$) solution to . Then $m | N$, say $N=mk$ for some integer $k\ge1$ and $f(s,t)$ is itself a homogeneous polynomial in $(s,t)$ of homogeneous degree $k$. Let us denote it by $$f(s,t)=\sum_{j=0}^ka_js^{j}t^{k-j}\equiv s^kg(z),$$ where $g=g(z)$ is a (nontrivial) polynomial in $z=t/s$ of degree $\le k$. Then $g(z)$ satisfies . Factorizing $g(z)=\prod_{i=1}^{r} (z-z_i)^{q_i}$ with $z_i$ pairwise distinct complex numbers, we obtain $$h:=\frac{g'}{g}=\sum_{i=1}^{r} \frac{q_i}{z-z_i},$$ where each $q_i\in \mathbb{Z}^+$ is a positive integer ($=$ the multiplicity of $z_i$). It follows from that $$\label{geq1}
\begin{split}
b_1(z^2-1)^2(h'+h^2)h^2&+b_2(z^2-1)(h^2+1)+(b_3z+\nu)(z^2-1)h^3\\ &+b_4(z^2-1)h^2+(b_5z-\nu)h +b_6 =0,
\end{split}$$
**Claim 1**: The set of zeros $\{z_1,\ldots,z_r\}$ does not contain $\pm1$. Indeed, arguing by contradiction, assume, for example, $z_1=1$. Suppose first that $q_1\ge 2$. Then the principal part of the corresponding Laurent decompositions at $z=1$ are $$\begin{aligned}
(z^2-1)^2(h^2+h')h^2&=\frac{4q_1^3(q_1-1)}{(z-1)^2}+O((z-1)^{-1}),\\
z(z^2-1)h^3&=\frac{2q_1^3}{(z-1)^2}+O((z-1)^{-1}),\\
(z^2-1)(h^2+h')&=\frac{2q_1(q_1-1)}{z-1}+O(1),\\ (z^2-1)h^2&=\frac{2q_1^2}{z-1}+O(1),\\
zh&=\frac{q_1}{z-1}+O(1).\end{aligned}$$ Combining the found relations with and implies that $$q_1^3(4b_1(q_1-1)+2(b_3+\nu))=0$$ hence by the assumption $p>1$ it follows $$q_1=1-\frac{b_3+\nu}{2b_1}=1-\frac{p+m_1-1}{2(p-1)}
<1,$$ a contradiction follows. Thus $q_1=1$ and $h=\frac{1}{z-1}+O(1)$ at $z=1$. Repeating the above argument yields the Laurent expansion $$\begin{aligned}
z(z^2-1)h^3&=\frac{2}{(z-1)^2}+O(\frac{1}{z-1}),
$$ while the other terms in either regular at $z=1$ or have the order $O(\frac{1}{z-1})$. This yields $b_3+\nu=0$, hence $p=1-m_1\le 0$, a contradiction again.
The same argument holds also true for $z=-1$. Thus, $z_i\ne \pm1.$
**Claim 2**: $q_i=1$ for all $i$. Indeed, if some $q_i>1$ then arguing as above we obtain for the Laurent decompositions at $z=z_i$ that $$\begin{aligned}
(z^2-1)^2(h^2+h')h^2&=\frac{q_i^3(q_i-1)(z_i^2-1)^2}{(z-z_i)^4}+O((z-z_i)^{-3}),\end{aligned}$$ while other terms in have singularity of order at most $O((z-z_i)^{-3})$. This yields $b_1=1-p=0$, a contradiction follows.
It follows from Claim 1 and 2 that the polynomial $g(z)$ has only simple roots, all distinct from $\pm1$. Equivalently, $$\label{equiv}
(z_i^2-1)g'(z_i)\ne0, \quad 1\le i\le r.$$ Let us rewrite as a polynomial identity $$\label{PQ}
(z^2-1)g'^2P+gQ=0$$ where $$\begin{aligned}
Q&=b_2(z^2-1)g''g+b_4(z^2-1)g'^2+b_5zg'g +b_6 g^2\\
P&=b_1g''(z^2-1)+(b_3z+\nu)g'.\end{aligned}$$ Setting $z=z_i$ in and taking into account that $g(z_i)=0$ yields by virtue of that $P(z_i)=0$. Thus, the polynomial $P$ vanishes whenever $g$ does, and also $\deg P\le \deg g$. Since the roots of $g$ are simple, we have $P(z)=\lambda g(z)$ for some $\lambda\in \R{}$, i.e. $$\label{Pdiff}
b_1(z^2-1)g''+(b_3z+\nu)g'=\lambda g$$ Substituting this identity into yields $Q=-\lambda (z^2-1)g'^2$, thus after elimination of $(z^2-1)g''$ from the obtained relation by virtue of (\[Pdiff\]) yields $$(b_6b_1+\lambda b_2) g^2+((b_5b_1-b_2 b_3)z-\nu b_2)) gg'+b_1(\lambda+b_4) (z^2-1)g'^2\equiv 0.$$ Setting $z=z_i$ in the latter identity yields by virtue of , $b_1\ne0$ and $g(z_i)=0$ that $\lambda=-b_4$, and therefore $$(b_6b_1-b_4 b_2) g+((b_5b_1-b_2 b_3)z-b_2\nu) g'\equiv 0.$$ Since $$b_5b_1-b_2 b_3=k^2(2-p)\mu\ne0,$$ we obtain $g(z)=C_1(z-a)^\beta$, where $$a=\frac{b_2\nu}{b_5b_1-b_2 b_3}, \quad \beta=\frac{b_4 b_2-b_6b_1}{b_5b_1-b_2 b_3},$$ therefore either $g$ is linear (the case treated in Section \[sublinear\]), or it has a single root of multiplicity $\ge 2$, a contradiction again.
Biharmonic examples {#sec:some}
===================
We finish this paper by a few more curious examples of biharmonic functions based on isoparametric polynomials. Let us consider the isoparametric ansats . Then it follows from that the harmonicity of $f$ is equivalent to the vanishing of the linear operator $C$ in , and biharmonic examples are obtained by the first iteration of . A complete analysis of the obtained equation, though much simpler than in the nonlinear case , is beyond the scope of this article. We confine ourselves to a particular case $$\label{biharm}
u(x):=(|x|^m+\phi(x))^{\alpha}.$$ Interesting that in contrast to the $p$-harmonic case, the obtained below biharmonic examples involve isoparametric polynomials with $m_1\ne m_2$.
\[pro:bihar\] Let $\phi\in \Iso_m(m_1,m_2)$ and $u$ is defined by . Then $\Delta^2 u=0$ but $\Delta u\not\equiv 0$ iff
- $m=2$ and $u$ is the fundamental solution of $\Delta^2$ in $\R{k}$ for some $k\le n$, or
- $m=4$ and $(m_1,m_2)\in \{(1,4), (2,5), (4,7), (6,9)\}$.
In the latter case, the corresponding function $u(x)$ is biharmonic everywhere in $\R{n}$ outside a minimal cone of codimension $m_2+1$.
The first case yields the well-known fundamental solutions, while the four examples obtained in (ii) are new, to the best of our knowledge. Note that for $m_1=1$ the corresponding solution is algebraic, and for $m_2=2,4,6$ yields three *rational* biharmonic functions. It also follows from the KFM table, see [@CecilRyan2015 p. 178], that for each pair $(m_1,m_2)\in \{(1,4), (2,5), (6,9)\}$ there is essentially unique (up to an isometry of the ambient space) isoparametric quartic. In the exceptional case $(m_1,m_2)=(4,7)$ there exist two different isoparametric quartics.
Let $v=|x|^m+\varphi$, such that $u=v^\alpha$. Then using , and we find $$|\nabla v|^2=m^2|x|^6+2m|x|^2\scal{x}{\nabla \phi(x)}+|\nabla \phi|^2=2m^2v|x|^2$$ and $$\Delta v=\bigl(m(n+m-2)+\half{m^2}{2}(m_2-m_1)\bigr)|x|^2=m^2(m_2+1)|x|^2,$$ therefore $$\Delta v^\alpha=\alpha v^{\alpha-1}\Delta v+\alpha(\alpha-1)|\nabla v|^2v^{\alpha-2}=\gamma_\alpha v^{\alpha-1}|x|^2,$$ where $\gamma_\alpha=\alpha m^2(m_2+2\alpha-1)$. By the assumption $\Delta v^\alpha\not\equiv 0$, hence $\gamma_\alpha\ne0$. Then iterating the obtain relation we find $$\frac{1}{\gamma_\alpha}\Delta^2 v^\alpha=\Delta |x|^2v^{\alpha-1}=\biggl((m-2)(n-4+m(2\alpha-1))v +\gamma_{\alpha-1}|x|^4\biggr)v^{\alpha-2}.$$ The latter expression identically vanishes if and only if $$\begin{aligned}
\label{bihh1}
(m-2)(n-4+m(2\alpha-1))&=0 \\
\gamma_{\alpha-1}=\alpha m^2(m_2+2\alpha-3)&=0.\label{bihh2}\end{aligned}$$
First assume that $m=3$ or $6$. Then $m_1=m_2$ (see Section \[sec54\]), therefore yields $m_2=3-2\alpha$, hence $n=mm_2+2=m(3-2\alpha)+2$. On the other hand, gives $n=4+m(1-2\alpha)$. Eliminating $n$ yields $m=1$, a contradiction.
Next consider $m=2$. Then is fulfilled automatically, and gives $m_2=3-2\alpha$. It follows from that $\phi=|\xi|^2-|\eta|^2$, where $x=(\xi,\eta)\in \R{m_2+1}\times \R{m_1+1}$. But in that case, $u=(|x|^2+\phi)^{(3-m_2)/2}=C|\eta|^{3-m_2}$ is a function of $\eta\in \R{m_2+1}$ and it is the fundamental solution of $\Delta^2$ in $\R{m_2+1}$. This yields (i).
Finally, suppose $m=4$. Arguing similarly, we find $m_2=3+m_1$ and $\alpha=-m_1/2$. In particular, $(m_1,m_2)$ is distinct from the two exceptional pairs $(2,2)$, $(4,5)$ and $(7,8)$, thus the corresponding isoparametric quartic is of KFM type, i.e. it is congruent to , see [@ChiBook]. Therefore, combining the obtained relation with yields $p=2q+2$, where the possible values $(p,q)$ are determined from the Hurwitz-Radon obstruction , i.e. $$q\le 1+\rho(2q+2).$$ Since the Hurwitz-Radon function $\rho$ has the logarithmic growth, the latter inequality has only finitely many solutions. A simple analysis shows that the only possible solutions are $q\in\{1,2,3,5,7\}$. By , $m_1=q-1$. If $m_1=0$ then $\alpha=0$, hence $u=const$. If $q\ge 2$ then $m_1\ge1$, thus we arrive at the four possible pairs of isoparametric parameters: $$(m_1,m_2)\in \{(1,4), \,(2,5), \,(4,7), \,(6,9)\},$$ all realizable, see for example Table in section 4.3 in [@FKM] or [@CecilRyan2015 p. 178].
Next, since $\alpha<0$, the function $u(x)$ is well-defined everywhere in $\R{n}\setminus CM^-$, where $CM^-$ is the cone over $$M^-=\{x\in S^{n-1}:u(x)=0\}=\{x\in S^{n-1}:\phi(x)=-1\}.$$ Then the desired claim follows from the fact that the focal varieties $M^\pm:=\{x\in S^{n-1}:\phi(x)=\pm1\}$ are minimal submanifolds of the unit sphere $S^{n-1}$ of codimension $m_1+1$ for $M^+$ and $m_2+1$ for $M^-$, respectively, see [@Nomi].
Then $q=2$, $p=2q+2=6$ and $n=2p=12$. The corresponding Clifford symmetric system in $\R{12}$ is given by $$A_1=\left(
\begin{array}{cc}
1_6 & 0 \\
0 & -1_6\\
\end{array}
\right),
\quad
A_2=\left(
\begin{array}{cc}
0 & 1_6 \\
1_6 & 0\\
\end{array}
\right),$$ where $1_d$ is the $d\times d$-unit matrix. It follows from and that $$u=\frac{1}{2\sqrt{|\xi|^2|\eta|^2-\scal{\xi}{\eta}^2}}, \quad \xi=(x_1,\ldots,x_6),\quad \eta=(x_7,\ldots,x_{12})$$ The obtained biharmonic function is well-defined is degree $-2$ homogeneous and well-defined everywhere in $\R{12}\setminus CM^-$, where the singular set $CM^-$ is the $7$-dimensional minimal cone $$\Gamma=\{(x\cos \theta ,x\sin \theta )\in \R{12}: x\in \R{6} \quad \text{and}\quad \theta\in [0,2\pi]\}.$$
In this case, $(q,p)=(3,8)$ and $n=16$. Then any symmetric Clifford system with $(q,p)=(3,8)$ is geometric equivalent to the triple $$A_1=\left(
\begin{array}{cccc}
1_4 & 0 & 0 &0\\
0 & -1_4 & 0 &0\\
0 & 0 & 1_4 &0\\
0 & 0 & 0 &-1_4
\end{array}
\right),
\quad
A_2=\left(
\begin{array}{cccc}
0 & 1_4 & 0 &0\\
1_4 & 0 & 0 &0\\
0 & 0 & 0 & 1_4\\
0 & 0 & 1_4 & 0
\end{array}
\right),
\quad A_3=\left(
\begin{array}{cccc}
0 & 0 & 0 &1_4\\
0 & 0 & 1_4 &0\\
0 & -1_4 & ´0 &0\\
1_4 & 0 & 0 &0
\end{array}
\right),$$ Then the corresponding biharmonic function is rational: $$u=\frac{1}{(|\xi_1|^2+|\xi_3|^2)(|\xi_2|^2+|\xi_4|^2)- (\scal{\xi_1}{\xi_2}+ \scal{\xi_3}{\xi_4})^2- (\scal{\xi_1}{\xi_4}- \scal{\xi_2}{\xi_3})^2},$$ where $\xi=(x_{i+1},\ldots,x_{i+4})$, $i=1,2,3,4.$ The function $u$ is biharmonic outside the minimal cone $CM^-\subset\R{16}$ of codimension $m_2+1=6$.
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[^1]: Supported by G.S. Magnuson’s Foundation, grant MG 2017-0101
[^2]: This result holds for any quasilinear divergence form equation
|
---
abstract: 'In this paper we express the multiplicities of modular representation theoretic categories of type A in terms of affine p-KL polynomials of Elias and Williamson. The representation theoretic categories we deal with include the category of rational representations of $\operatorname{GL}_n$ and of the quantum group $U_q(\mathfrak{gl}_n)$, representations of (degenerate) cyclotomic Hecke and Schur algebras, and the base field is an algebraically closed field of arbitrary prime characteristics. In order to approach this problem we define Soergel theoretic version of parabolic categories $\mathcal{O}$ in characteristic $p$. We show that these categories have many common features with the classical parabolic categories $\mathcal{O}$, for example, they are highest weight. We produce a homomorphism from a (finite or affine) type A 2-Kac-Moody category to the diagrammatic version of the category of singular Soergel bimodules (again, of finite or affine type A). This leads to a categorical Kac-Moody action on the Soergel theoretic categories $\mathcal{O}$. Then we relate the representation theoretic categories to Soergel theoretic ones by proving a uniqueness result for highest weight categorical actions on Fock spaces.'
author:
- Ben Elias and Ivan Losev
title: Modular representation theory in type A via Soergel bimodules
---
Introduction5.tex
SoergelDiags2.tex
FiniteADiags2.tex
AffineADiags2.tex
Cellular2.tex
SoergelO.tex
FockUnique4.tex
decomp\_pKL.tex
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|
---
abstract: 'KIC8560861 (HD183648) is a marginally eccentric ($e=0.05$) eclipsing binary with an orbital period of $P_\mathrm{orb}=31.973$d, exhibiting mmag amplitude pulsations on time scales of a few days. We present the results of the complex analysis of high and medium-resolution spectroscopic data and [*Kepler*]{} Q0 – Q16 long cadence photometry. The iterative combination of spectral disentangling, atmospheric analysis, radial velocity and eclipse timing variation studies, separation of pulsational features of the light curve, and binary light curve analysis led to the accurate determination of the fundamental stellar parameters. We found that the binary is composed of two main sequence stars with an age of $0.9\pm0.2$Gyr, having masses, radii and temperatures of $M_1=1.93\pm0.12$M$_{\odot}$, $R_1=3.30\pm0.07$R$_{\odot}$, $T_\mathrm{eff1}=7650\pm100$K for the primary, and $M_2=1.06\pm0.08$M$_{\odot}$, $R_2=1.11\pm0.03$R$_{\odot}$, $T_\mathrm{eff2}=6450\pm100$K for the secondary. After subtracting the binary model, we found three independent frequencies, two of which are separated by twice the orbital frequency. We also found an enigmatic half orbital period sinusoidal variation that we attribute to an anomalous ellipsoidal effect. Both of these observations indicate that tidal effects are strongly influencing the luminosity variations of HD183648. The analysis of the eclipse timing variations revealed both a parabolic trend, and apsidal motion with a period of $P_\mathrm{apse}^\mathrm{obs}=10\,400\pm3\,000$y, which is three times faster than what is theoretically expected. These findings might indicate the presence of a distant, unseen companion.'
author:
- |
T. Borkovits$^{1,2,3}$[^1], A. Derekas$^{2,3}$, J. Fuller$^4$, Gy. M. Szabó$^{2,3}$, K. Pavlovski$^5$, B. Csák$^3$, Á. Dózsa$^3$, J. Kovács$^3$, R. Szabó$^{2}$, K. M. Hambleton$^{6,7}$, K. Kinemuchi$^{8}$, V. Kolbas$^5$, D. W. Kurtz$^{6}$, F. Maloney$^{7}$, A. Prša$^{7}$, J. Southworth$^{9}$, J. Sztakovics$^{10}$, I. B. Bíró$^{1}$, I. Jankovics$^3$\
$^{1}$Baja Astronomical Observatory, H-6500 Baja, Szegedi út, Kt. 766, Hungary\
$^{2}$Konkoly Observatory, MTA CSFK, H-1121 Budapest, Konkoly Thege M. út 15-17, Hungary\
$^{3}$ELTE Gothard Astrophysical Observatory, H-9704 Szombathely, Szent Imre herceg út 112, Hungary\
$^{4}$Department of Astronomy, Center for Space Research, Cornell University, Ithaca, NY 14853, USA\
$^{5}$Department of Physics, University of Zagreb, Bijenička cesta 32, 10000 Zagreb, Croatia\
$^{6}$Jeremiah Horrocks Institute, University of Central Lancashire, Preston PR12HE, UK\
$^{7}$Department of Astrophysics and Planetary Science, Villanova University, 800 Lancaster Ave, Villanova PA 19085, USA\
$^{8}$Apache Point Observatory, Sunspot NM 88349, USA\
$^{9}$Astrophysics Group, Keele University, Newcastle-under-Lyme, ST5 5BG, UK\
$^{10}$Astronomical Department of Eötvös University, H-1118 Pázmány Péter stny. 1/A, Budapest, Hungary
date: 'Accepted ??? Received ???; in original form ???'
title: 'HD183648: a ***Kepler*** eclipsing binary with anomalous ellipsoidal variations and a pulsating component'
---
\[firstpage\]
(stars:) binaries: eclipsing – stars: fundamental parameters – stars: oscillations (including pulsations) – stars: individual: HD 183648
Introduction
============
Eclipsing binary stars have long been recognized as key objects for calibrating astronomical observations in terms of fundamental stellar parameters. In fact, binarity has been, until recently, the only way to accurately determine stellar masses. The combination of time-series photometry and spectroscopy of eclipsing binaries enables us to measure the most accurate masses and radii for stars, namely to better than 1percent [e.g. @and91; @cla08; @tor10].
There is a special group of eclipsing binaries that take a very important place in astrophysics: those with pulsating components. Such systems are important laboratories for confronting theories with observations. The mass measured from an eclipsing binary can be compared with those coming from other determinations and models, such as evolutionary or pulsational models [@aer07].
Eclipses can be helpful in mode detection and identification, and pulsations enable us to measure the internal rotational velocity of the pulsating star through the rotational splitting of the non-radial modes [@bap93; @gou00; @gam03; @mik05; @bir11]. In close binary systems it is common that tidal forces induce pulsations [@wil03; @wel11; @tho12; @ful13] and in special cases resonances of the frequency of the pulsation and the orbital period can be detected [@ful13; @ham13 Hambleton 2014, in preparation].
Almost every type of pulsating star has been found as a component in an eclipsing binary system. A good overview of these systems and their distribution is given by @pig06. Since then the number of known systems has grown significantly thanks to large ground base photometric surveys [e.g. @pig07; @mic08] and the unprecedented quality of the photometric light curves delivered by space telescopes CoRoT [@mac13; @dasilvaetal14] and [*Kepler*]{} [@ost10; @der11; @sou11; @tel12; @debosscheretal13; @fra13; @ham13; @bec14; @maceronietal14].
Here we present the analysis of an eclipsing binary with a pulsating component discovered in the [*Kepler*]{} dataset. KIC8560861 (HD183648) is a relatively long-period ($P_\mathrm{orb} = 31.97$d), marginally eccentric ($e = 0.05$) eclipsing binary system which exhibits mmag pulsations with periods on the order of a few days. It has a magnitude of $V = 8.5$, hence it is above the saturation limit of the [*Kepler*]{} space telescope, which was taken into account for the data reduction (see in Sect. \[Sect2\]). It is listed in the catalogues of the first and second releases of the Kepler Eclipsing Binary Catalogue [@prs11; @sla11]. The Kepler Input Catalogue (KIC) lists the following parameters for this target: $r_\mathrm{SDSS}=8.498$, $T_\mathrm{eff} = 7647$K, $\log{g} = 3.532$, $\mathrm{[Fe/H]}=-0.084$.
In the following sections we present the combined analysis of the [*Kepler*]{} photometry and ground based spectroscopic data, which includes (i) analysis of eclipse timing variation (Sect.\[Sect:ETV\]), (ii) determination of atmospheric properties of the primary star from crosscorrelation function spectroscopy (Sect.\[Subsect.:fundparam\]), (iii) a radial velocity study (Sect.\[Subsect:radvelstudy\]), $(iv)$ spectral disentangling and determination of the dynamical masses (Sect.\[sec: spd\]), $(v)$ eclipse light curve analysis (Sect.\[lcanalysis\]), and $(vi)$ determination of the pulsation frequencies (Sect.\[Subsect:freqsearch\]). Finally, in Sect.\[Subsect:tidaloscillation\] we discuss the characteristics and the possible tidal origin of the detected oscillations.
Observations and data reduction {#Sect2}
===============================
[*Kepler*]{} photometry
-----------------------
The photometric analysis is based on photometry from the [*Kepler*]{} space telescope [@bor10; @gil10; @koc10; @jen10a; @jen10b]. HD183648 was observed both in long and short cadence mode between 2009 and 2013. The long cadence (time resolution of 29.4min) dataset covers nearly the whole length of Kepler’s 4-y life-time (Quarters 0 – 16), while it was observed for only 30d in Q3.2 in short cadence (time resolution of 58.9s) mode.
The MAST[^2] database indicates $0.1-0.3$percent contamination, depending on the quarter in question. We have downloaded the target pixel files for all quarters and performed several checks by using PyKE[^3] tools. First, we verified that all signals come from one object; that is, no contamination is seen from a close-by blended object within Kepler’s resolution (4arcsec). This was done by examining the amplitude of the (visible) signals in individual pixels. Any signal coming from a different source would be revealed by a displaced pixel showing a higher amplitude of that signal. We also checked that no signal was lost due to the assigned target aperture mask. Due to the brightness of the star and saturation on the [*Kepler*]{} photometer, the target aperture mask was elongated. Along the elongation axis, we still detect signal from the star, and we assume the flux could be still detected in pixels outside the target aperture. However, we have determined that the contribution of these peripheral pixels outside of the target aperture is negligible. We estimate that the lost flux from the star is less than 0.01percent.
Spectroscopy {#Subsect:Spectroscopy}
------------
We obtained high and medium resolution spectra at five observatories. We took 2 spectra in 2010 at Kitt Peak National Observatory (KPNO), USA, using the Echelle Spectrograph at the Mayall 4-meter telescope with a resolution of $R = 20\,000$ in the spectral range $4700 - 9300$Å. 34 spectra were taken on 11 nights in 2012 with the eShel spectrograph mounted on a 0.5-m Ritchey-Chrétien telescope at the Gothard Astronomical Observatory (GAO), Szombathely, Hungary, in the spectral range $4200 - 8700$Å with a resolution of $R = 11\,000$. The wavelength calibration was done using a ThAr lamp. The same instrument was used at Piszkéstető Observatory (PO), Hungary mounted on the 1-m telescope, where we took 36 spectra on 16 nights in 2012 and 2013. A detailed description of the instrument can be found in @csa14. We obtained 5 spectra at Apache Point Observatory (APO), USA, using the ARCES Echelle spectrograph on the 3.5 m telescope with a resolution of $R = 31\,500$ in the spectral range $3200 - 10\,000$Å. We took 3 spectra at Lick Observatory, USA, using the Hamilton Echelle Spectrograph mounted on the Shane 3-meter Telescope. The resolution was $R = 37\,000$ in the spectral range $4200 - 6850$Å. The journal of observations can be found in Table\[obsjour\].
Observatory wavelength range Res. No. of spectra
------------- ------------------ -------- ----------------
GAO 4200–8700 Å 11 000 34
Piszkéstető 4200–8700 Å 11 000 36
KPNO 4700–9300 Å 20 000 2
APO 3200–10000 Å 31 500 5
LICK 4200–6850 Å 37 000 3
: Journal of observations.[]{data-label="obsjour"}
All spectra were reduced either using IRAF or a dedicated pipeline, then normalised to the continuum level. The radial velocities were determined by cross-correlating the spectra with a well-matched theoretical template spectrum from the extensive spectral library of @mun05. In cases of spectra obtained at Gothard Astronomical Observatory and Piszkéstető Observatory, we co-added those taken in the same night to produce higher signal-to-noise ratios. All radial velocities were corrected to barycentric radial velocities, and are listed in Table\[Tab:radveldata\].
By the use of this conventional cross-correlation technique, HD183648 was found to be a single lined spectroscopic binary (SB1), which was in good agreement with the expectation based on the preliminary light curve fit.
----------------- ------------------- ----------------- --------------
BJD BJD
$-2\,400\,000$ (kms$^{-1}$) $-2\,400\,000$ (kms$^{-1}$)
$56057.48133$ $-23.0(5)$
$55987.67491^*$ $10.5(4)$ $56058.45077$ $-25.8(5)$
$56009.64102$ $11.3(3)$ $56059.49054$ $-30.1(5)$
$56012.60284$ $25.5(5)$ $56514.36738$ $-20.2(3)$
$56015.54585$ $27.6(5)$ $56516.55074$ $-9.7(4)$
$56020.52020$ $5.0(4)$ $56521.53346^*$ $19.0(3)$
$56084.43023$ $6.4(5)$ $56542.35874$ $-30.0(3)$
$56091.37875$ $-32.0(4)$ $56555.37107$ $24.9(3)$
$56104.40849$ $6.5(5)$
$56105.52976$ $15.7(5)$ $55738.95801^*$ $-26.0(2)$
$56106.49568$ $20.4(5)$ $55742.97074^*$ $-32.4(2)$
$56117.45252$ $0.3(5)$
$56126.85215^{*}$ $-29.2(2)$
$55990.65432^*$ $-3.6(4)$ $56204.66367^*$ $29.2(2)$
$55995.67350^*$ $-29.9(3)$ $56224.65775^*$ $-26.2(2)$
$55996.65663$ $-28.2(4)$ $56225.65443^*$ $-22.9(2)$
$55998.65543^*$ $-34.6(3)$ $56228.58654^*$ $-9.9(2)$
$56000.65120$ $-29.0(5)$
$56008.64047$ $9.3(3)$ $56133.00141^*$ $-11.0(2)$
$56048.52003^*$ $25.7(3)$ $56133.99831^*$ $-4.0(2)$
$56053.49436$ $0.9(5)$ $56134.87329^*$ $3.3(2)$
----------------- ------------------- ----------------- --------------
: Radial velocity measurements []{data-label="Tab:radveldata"}
$^*$: measurements used for spectral disentanglement\
---------------- ------- ----------- ---------------- ------- ----------- ---------------- ------- -----------
BJD Cycle std. dev. BJD Cycle std. dev. BJD Cycle std. dev.
$-2\,400\,000$ no. $-2\,400\,000$ no. $-2\,400\,000$ no.
54966.868878 0.0 0.000085 55463.197910 15.5 0.000073 55942.797650 30.5 0.000179
54983.600442 0.5 0.000193 55478.440878 16.0 0.000064 55958.039172 31.0 0.000085
55030.815327 2.0 0.000086 55495.171972 16.5 0.000070 55974.770697 31.5 0.000192
55047.547225 2.5 0.000184 55510.413023 17.0 0.000050 55990.013154 32.0 0.000084
55062.788056 3.0 0.000083 55527.144610 17.5 0.000095 56006.743789 32.5 0.000194
55079.519265 3.5 0.000197 55542.386918 18.0 0.000058 56021.986202 33.0 0.000084
55094.761357 4.0 0.000084 55574.359507 19.0 0.000070 56038.716950 33.5 0.000180
55111.494290 4.5 0.000193 55591.091868 19.5 0.000059 56053.959844 34.0 0.000086
55126.734729 5.0 0.000084 55606.333366 20.0 0.000084 56070.690183 34.5 0.000195
55143.466542 5.5 0.000191 55623.065473 20.5 0.000191 56085.932769 35.0 0.000085
55158.708288 6.0 0.000084 55655.037652 21.5 0.000193 56102.664102 35.5 0.000194
55175.440176 6.5 0.000200 55670.279778 22.0 0.000085 56117.906765 36.0 0.000090
55190.681240 7.0 0.000079 55687.011344 22.5 0.000187 56134.637722 36.5 0.000193
55207.413066 7.5 0.000178 55702.252701 23.0 0.000084 56149.879518 37.0 0.000084
55222.654283 8.0 0.000084 55718.984941 23.5 0.000195 56166.611156 37.5 0.000181
55239.385793 8.5 0.000193 55734.226527 24.0 0.000085 56181.853449 38.0 0.000086
55254.627387 9.0 0.000084 55750.957816 24.5 0.000206 56198.583760 38.5 0.000195
55271.359097 9.5 0.000192 55766.199574 25.0 0.000086 56213.826767 39.0 0.000085
55286.600691 10.0 0.000084 55782.931099 25.5 0.000182 56230.558449 39.5 0.000193
55303.332429 10.5 0.000101 55798.173244 26.0 0.000081 56245.800354 40.0 0.000730
55318.573707 11.0 0.000081 55814.904382 26.5 0.000196 56262.530887 40.5 0.000194
55335.305117 11.5 0.000051 55830.146219 27.0 0.000086 56277.773536 41.0 0.000085
55350.547360 12.0 0.000043 55846.877296 27.5 0.000193 56294.504414 41.5 0.000180
55367.279166 12.5 0.000076 55862.120139 28.0 0.000084 56309.747978 42.0 0.000093
55382.520503 13.0 0.000092 55878.850805 28.5 0.000193 56326.478929 42.5 0.000182
55399.251293 13.5 0.000106 55894.092879 29.0 0.000084 56341.720533 43.0 0.000085
55414.494091 14.0 0.000057 55910.823884 29.5 0.000194 56373.694490 44.0 0.000085
55431.225013 14.5 0.000029 55926.067023 30.0 0.000085 56390.425555 44.5 0.000194
55446.466863 15.0 0.000086
---------------- ------- ----------- ---------------- ------- ----------- ---------------- ------- -----------
Eclipse timing analysis {#Sect:ETV}
=======================
In the case of an eclipsing binary, eclipse timing analysis is the most powerful tool for $(i)$ determining an accurate period, $(ii)$ detecting and identifying any kind of period variation, either physical or apparent, (iii) calculating an accurate value of the $e\cos\omega$ parameter for eccentric systems, and ($iv$) detecting a slow variation in the eclipse times caused by an apsidal advance of the binary’s orbit.
We therefore analysed the eclipse timing variations (ETV) first. The individual times of minima were determined with the following algorithm. First a folded, equally binned and averaged light curve was formed from the whole Q0 – Q16 data set. Then two template minima were calculated with polynomial fits of degree $4-6$ on the primary and secondary eclipses. Finally, these templates were fitted to all individual minima. As an alternative method and check, parabolic and cubic linear least-squares fits and minima determinations to each individual minimum were also applied. These methods are very similar to those used by @rappaportetal13, for example. Furthermore, we estimated the accuracy of minima determinations by calculating the standard deviations for each minimum with bootstrap sampling [see, e.g., @bratetal14].
Our observed times of minima ($O$) were compared with calculated times of minima ($C$) with the following linear ephemeris to give values of $O-C$: $$\mathrm{MIN}_\mathrm{I}=2454966.8687+31.9732\times E,
\label{Eq:linephem}$$ which was determined with the method described above.
The raw $O-C$ diagram revealed a complex nature which was a combination of a cyclic variation with a period of $\sim287$d and a slower, quadratic term (red and blue curves in Fig.\[Fig:ETV\]). The primary and the secondary minima correlated in both features; however, the cyclic variation had a greater amplitude in the secondary curve. In the present situation this variation does not arise from the presence of a third companion (which is the most common interpretation of such $O-C$ curves), nor does it arise from any other real physical or geometric cause. It is purely the result of the pulsational distortion of the light curve. This comes from the fact that the mean pulsational frequency is close to a $165:9$ ratio to the orbital frequency (see Sect.\[lcanalysis\]), and consequently, every ninth primary, and secondary eclipsing minima are distorted in a similar way.
Apparent ETVs forced by light curve variations are also seen in other stars using accurate [*Kepler*]{} data. Recently, @tranetal13 reported quasi-periodic $O-C$ diagrams for almost 400 short period, mostly overcontact binaries, whose phenomena were interpreted as an effect of large, spotted regions on the binary members. A difference, however, is that the spotted stars resulted in anticorrelated primary and secondary ETVs, while in the present case the ETVs are correlated.
![$O-C$ diagram of the Eclipse Timing Variations (ETV) for the primary and secondary minima. (For better visibility the secondary curve is shifted by $0.735$ days, which corresponds to the displacement of secondary minima from phase 0.5.) The parabolic trend seems to be real. The cyclic feature, however, arises from the pulsational distortion of the light curve. The apparent cyclic variation can be eliminated with either a local smoothing of the light curve around each minimum, or the removal of the pulsational variations. (See text for details.)[]{data-label="Fig:ETV"}](MN-14-1385-MJf1.eps){width="84mm"}
This apparent timing variation was eliminated by the removal of the pulsations from the light curve. In Fig.\[Fig:ETV\] we illustrate this in two different ways. First we applied local smoothing around all individual minima. We fitted polynomials of order $4-8$ on short sections of the light curve before the first and after the fourth contacts, and then removed them from the light curves (as was done by @borkovitsetal13). This procedure removed the effect within the errors from the primary $O-C$ curve, but some residuals with moderate amplitude remained in the secondary one. Next, after the light curve analysis, we removed the pulsations from the original light curve in the manner described in Sect.\[lcanalysis\]. Running our code on the latter, prewhitened curve, we obtained both primary and secondary $O-C$ curves without the cyclic variations.
![The sum (red) and the difference (blue) of the primary and secondary $O-C$ curves calculated from data after the pulsations were removed, together with Levenberg-Marquardt fits (black lines). Such a visualization helps to separate the parabolic trend having correlated nature between primary and secondary minima, and apsidal motion which has an approximately anticorrelated effect for primary and secondary minima variations. The parabolic trend in the ‘sum’ curve is evident. The small non-horizontal slope of the ‘difference’ curve is an indication of apsidal motion.[]{data-label="Fig:ETVsumdifffit"}](MN-14-1385-MJf2.eps){width="84mm"}
For the final ETV analysis these latter, pulsation-removed $O-C$ curves were used. These prewhitened curves showed an additional feature. Subtracting the secondary $O-C$ values from the primary ones (which technically was carried out by a cubic spline interpolation of the secondary minima data to the times of the primary minima), we found that the difference curve had a non-zero slope, as can be seen in Fig.\[Fig:ETVsumdifffit\]. This is a likely indication of apsidal motion in the binary.
Therefore, we modelled the ETV in the following mathematical form: $$\begin{aligned}
\Delta&=&T(E)-T(0)-P_\mathrm{s}E \nonumber \\
&=&\sum_{i=0}^2c_iE^i+\frac{P_\mathrm{a}}{2\pi}\left[2\arctan\left(\frac{\mp e\cos\omega}{1+\sqrt{1-e^2}\pm e\sin\omega}\right)\right. \nonumber \\
&&\left.\mp\sqrt{1-e^2}\frac{e\cos\omega}{1\pm e\sin\omega}\right]_0^E,\end{aligned}$$
where $$\omega(E)=\omega(0)+\Delta\omega E.$$
Here, in the first row, $T(E)$ means the observed time of the $E$-th minimum, $T(0)=T_0$ is the same for the reference minimum, $P_\mathrm{s}$ stands for the sidereal (eclipsing) period. Note, the cycle number $E$ takes integer values for primary, and half-integer ones for secondary minima, respectively. In the second and third rows, the $c_0$, $c_1$ coefficients of the quadratic polynomial give corrections in $T_0$, $P_\mathrm{s}$, respectively, while $c_2$ is equal to the half of the constant period variation rate per cycle (i.e. $\Delta P/2$). The last two terms give the apsidal motion contribution. Usually it is given in the form of trigonometric series of $\omega$ [see e.g. @gimenezgarcia83]. The present computational facilities, however, allow us to use its exact, analytic form. In this expression $P_\mathrm{a}\sim P_\mathrm{s}(1+\Delta\omega/2\pi)$ denotes the anomalistic period, $e$ stands for the eccentricity, while $\omega$ refers to the argument fo periastron of the primary’s physical (or spectroscopic) orbit. This latter quantity varies in time. $\omega(0)=\omega_0$ means its value at $T_0$ epoch, and $\Delta\omega$ denotes the apsidal advance rate for one binary revolution. Furthermore, upper signs refer to primary, and lower ones to secondary minima, respectively. Note, we neglect the small effects of the weak inclination dependence on the time of the deepest eclipse in eccentric binaries [see e.g. @gimenezgarcia83], and the intrinsic light-time effect between primary and secondary minima for stars of unequal masses [@fabrycky10][^4].
In order to determine the parameters listed above, the $\Delta$ function was fitted by a Levenberg-Marquardt-based differential correction procedure. For such a short time-scale, however, the $\Delta\omega$ parameter is highly correlated with $e$ and $\omega$ [See @claret98 for details.] Consequently, we decided to fix one of these three parameters, and adjust only the other two (together with the three polynomial coefficients $c_i$-s) in the differential correction process. Therefore, we fixed the eccentricity on its RV analysis obtained value. Then, in order to estimate the uncertainty of the parameters obtained, we repeated the process with slightly modified eccentricities. This refinement allowed us to reduce the uncertainty in the eccentricity an order of magnitude. The results of the complete process are listed in Table\[Tab:ETVresult\]. In Fig.\[Fig:ETVsumdifffit\] we plot our results on the averaged (red) and the difference (blue) $O-C$ curves. The first was calculated by summing the $O-C$ values of primary and secondary minima, while the second by with subtracting them. (The results were divided by two in both cases.) The advantage of such visualization is that it nicely separates quadratic variations and apsidal motion, as the former has correlated nature, while the second one shows primarily anticorrelated behaviour with respect to primary and secondary minima.
------------------------------------ -------------------------
$T_0\mathrm{~(BJD)}$ $2\,454\,966.86896(20)$
$P_\mathrm{s}\mathrm{~(days)}$ $31.973126(18)$
$\Delta P\mathrm{~(days/cycle)}$ $7.2(8)\times10^{-6}$
$e$ $0.0477(1)$
$\omega_0 (\degr)$ $37.260(22)$
$P_\mathrm{apse}\mathrm{~(years)}$ $10\,432(3\,033)$
------------------------------------ -------------------------
: Results of ETV solution (one sigma uncertainties in the last digits are given in parentheses)[]{data-label="Tab:ETVresult"}
Parabolic-shaped ETV curves, corresponding to constant period variations in time (or, more strictly, in cycle number), have been observed in hundreds of eclipsing binaries (see [@sterken05], in general, and [@zhuetal12], for a recent example). However, the most common interpretations, such as mass exchange, mass loss and magnetic interactions, can be neglected in this widely separated and therefore weakly interacting binary. Thus, in our case, the most probable source of the observed small period increase would be a gravitationally bound, distant, third companion. This additional component must be a faint object, as there is no evidence for an additional light source in the spectroscopic or the photometric data (see subsequent sections). There might be, however, weak indirect evidence for the presence of this body in the observed period of $P_\mathrm{apse}^\mathrm{obs}\sim10\,000$y of the apsidal motion. From the orbital and fundamental stellar parameters obtained from our complex analysis we calculated the theoretically expected apsidal motion period (see Sect.\[lcanalysis\]), and found to be $P_\mathrm{apse}^\mathrm{theo}\sim34\,000$y (see Table\[Tab:syntheticfit\]). The insignificant length of 4 years of observations, compared to the ten-thousand-year-long period, could be attributed to be the main cause of the difference. We nevertheless cannot exclude the possibility of perturbations by a third star, which produces in a faster apsidal advance rate. A similar scenario has been detected in several [*Kepler*]{}-discovered hierarchical triple stellar systems (Borkovits et al., 2014, in preparation).
Spectroscopy {#spectroscopy}
============
\[Subsect.:fundparam\]Fundamental parameters
--------------------------------------------
To determine the fundamental parameters, we used the two spectra taken at KPNO in 2011 and co-added to produce higher signal-to-noise ratio spectrum. We chose these two spectra because they cover large wavelength range and they have the highest signal-to-noise ratios among the spectra we had. We used the fitting recipe described in Shporer et al. (2011) based on crosscorrelating model spectra by [@mun05] in the wavelength range of $5000 - 6400$Å. This is a two-step method that first fits $T_\mathrm{eff}$, $\log g$ and $v_\mathrm{rot}\sin i_\mathrm{rot}$ assuming solar metallicity, and then accepting the effective temperature, the metallicity is refitted together with $\log g$ and $v_\mathrm{rot}\sin i_\mathrm{rot}$. This iterative method is stable in the high temperature range ($>7000$K) where $T_\mathrm{eff}$ and $[Fe/H]$ are significantly correlated. We found a preliminary solution of $T_\mathrm{eff}=7400 \pm 150$K, $\log g=3.5 \pm 0.3$, $v_\mathrm{rot}\sin i_\mathrm{rot}=100 \pm 10$kms$^{-1}$ and $[M/H]=-0.5 \pm 0.3$. This preliminary result was re-iterated by combining spectroscopic and photometric data. The most stable parameter is $v_\mathrm{rot}\sin i_\mathrm{rot}$, since its value is practically independent of the other three. This solution is also in good agreement with those in the Kepler Input Catalogue, $T_\mathrm{eff}=7500$ K, $\log g = 3.5$ and $[Fe/H]=-0.08$ [@bro11].
Rapid rotation causes significant gravity darkening. According to the von Zeipel law [@zei24], there is a temperature gradient reaching almost $1000$K on the surface of the primary. The effect of this gradient on the spectrum cannot be handled because we do not know the aspect angle of the spin axis. We think that this temperature gradient is the most important source of systematics in spectral modeling, and therefore the internal errors of fitting algorithms should be considered as indicative values.
Since the geometry is unknown, we fitted a complete set of unique spectra (instead of a weighted average of spectra to describe the temperature gradient), and also introduced stellar evolution tracks into the fitting procedure (Padova evolutionary tracks, [@ber08; @ber09]) to constrain the fit to components with compatible ages. We fitted jointly the stellar spectra, and observed the parameter set in the $T_{\rm eff}$ and $\log g$ isochrone. Moreover, we involved the second component in the fit, since its mass function, relative temperature and relative radius had been constrained from the light curves with acceptable precision at this stage of fitting.
We searched for a solution that satisfied all the following criteria:
- The model is consistent with the measured KPNO spectra, according to a standard [$\chi^2$]{} analysis;
- The model describes a valid position in the $T_\mathrm{eff}$ – $\log g$ evolutionary track;
- The model produces a secondary component which is also consistent with a valid position in the evolutionary track and has a similar age to that of the primary.
This iteration stabilized $T_\mathrm{eff}$ around $7400-7700$K, suggested a $\log g$ between $3.75-4.25$ depending on the age (which is larger than the fit of the spectrum alone), and also confirmed a slightly low metallicity (around $-0.5$). It is worth noting that the criterion of both stars having compatible ages confined the joint parameter set significantly, and resolved much of the known degeneracies of fitting a single spectrum. The new parameter set is consistent with a main sequence $\gamma$Dor star with rapid rotation and a fairly young age (see Table\[fundpar\] for the determined parameters). The determined models within the confidence volume formed the allowed parameter set of the detailed light curve modelling (Sect. \[lcanalysis\]) and describes all spectroscopic and photometric data well. In section 4.2 and 4.3, we will repeat the spectral analysis with disentangling. Although these two methods are based on partly differing input data and different data processing, the resulting stellar models are satisfactory compatible with each other, confirming the validity of the derived stellar parameters.
Parameter Value Error
-------------------- ------- -------
T$_{\rm eff}$ (K) 7500 150
$\log{g}$ (dex) 4.0 0.25
$\rm [M/H]$ (dex) -0.5 0.3
$v \sin{i}$ (km/s) 100 10
Age (Gyr) 0.9 0.2
: \[fundpar\] Fundamental parameters of the main component of HD183648 system adopted from spectrum fitting.
![The position of the two components on the T$_{\rm eff}$ – $\log{g}$ evolution tracks with \[Fe/H\]=-0.5 metallicity. []{data-label="Fig:evoltrack"}](MN-14-1385-MJf3.eps){width="84mm"}
\[Subsect:radvelstudy\]Radial velocity study
--------------------------------------------
It has been usual since the epochal work of @wilson79, that radial velocity (RV) curves and photometric light curves are analysed simultaneously to obtain a combined solution. However, in the present situation we carried out these studies partially independently. The main reasons are as follows: while it is generally said that $e\cos\omega$ is very robustly determined by the light curve, this robustness is chiefly due to the timing, and not from any other parts of the light curve. On the other hand, for small eccentricities, the light curve itself has little dependence on $e\sin\omega$, which is better determined by the radial velocity data. These facts are especially valid for this present low-eccentricity system, where the out-of-eclipse parts of the light curve are strongly modulated by pulsations, which cannot be disentangled satisfactorily from the possibly anomalous ellipsoidal variations (see Sect.\[lcanalysis\]). Therefore, we decided to obtain eccentricity ($e$) and argument of periastron ($\omega$) from the combination of iterative RV and eclipse timing solutions, and then to keep them fixed until the final refinement of the light curve solution. Note that other parameters of the spectroscopic and radial velocity solution (e.g., the spectroscopic mass function, and $v_\mathrm{rot}\sin i_\mathrm{rot}$) were also included in the light curve solution by constraining certain parameters; details of this are given below in Sect.\[lcanalysis\].
The RV analysis was carried out iteratively combined with the ETV analysis. For the first run we used all the available radial velocity points (Table\[Tab:radveldata\]). In this preliminary stage the systemic velocity $V_\gamma$, and the five usual orbital elements were adjusted by a Levenberg-Marquardt algorithm based non-linear least-squares fit, while the orbital period was kept fixed on the period determined in Sect.\[Sect:ETV\]. Then, to check whether the period change that was detected in the ETV analysis manifests itself in the radial velocity curve as a variation in the systemic $V_\gamma$ velocity, an additional parameter, $\dot{V}_\gamma$, was also adjusted in an alternative run.
As a next step, the resulted eccentricity was used to refine the ETV solution, as discussed in Sect.\[Sect:ETV\]. In this way we obtained refined $e$ and $\omega$ parameters that were consistent with the previous RV results, but had substantially smaller formal errors. Finally, we fixed the eccentricity to its ETV-fit value, and reiterated the RV fits. In these runs, although the argument of periastron ($\omega$) kept its large formal error of $>10\degr$, it converged to a value differing only by $\sim1\degr$ from the ETV solution. Our results are plotted in Fig.\[Fig:rvsol\] ($\dot{V_\gamma}\equiv0$ solution), and listed in Table\[tab: RVresult\]. In the last rows we give some additional, derived quantities. As one can see, the apparent period variations ($\Delta P$), which were calculated from $\dot{V}_\gamma$, are slightly higher, yet agree with the result obtained from the ETV analysis.
In Fig.\[Fig:rvsol\] (lower panel) the residual velocity data are also plotted. As one can observe, these values exceed the estimated observational uncertainties for most of the data points. In order to investigate whether these deviations come from stellar pulsation, and/or instrumental effects, we performed a test. We checked whether the residuals of the radial velocity data show correlations with instrumental parameters such as spectral resolution and S/N, or are more likely of non-instrumental origin. The S/N was calculated near the blue wing of the H$\alpha$ line, between 640 and 645 nm, where the spectrum is nearly featureless. We estimated the continuum S/N levels to be between 40–300. The scatter of the radial velocity residuals did not exhibit a correlation neither with S/N nor the resolution of the spectra. The median absolute deviation of the residuals was $700$ms$^{-1}$ for the $\dot{V}_\gamma\neq0$ solution (i.e. observation – model, assuming a long-term component to describe the effect of the assumed third companion), regardless of the instrumental parameters. Moreover, the residuals did not show any periodicity, which could be related to the observed pulsation (see Sect.\[lcanalysis\]). Thus, the origin of this wobbling remains unexplained. Nevertheless, since the full amplitude of the radial velocity curve is over $50$kms$^{-1}$, the velocity wobbling is under 2% and does not influence the dynamical analysis.
![The observed radial velocties and the best fit radial velocity solution for the $\dot{V}_\gamma\equiv0$ case (solid line). The residuals of the fit are shown in the lower panel.[]{data-label="Fig:rvsol"}](MN-14-1385-MJf4.eps){width="84mm"}
Parameters $\dot{V}_\gamma\equiv0$ $\dot{V}_\gamma$ adjusted
------------------------------------- ------------------------- ---------------------------
$T_0$ (BJD)
$P_\mathrm{orb}$ (d)
$(V_\gamma)_0$ (kms$^{-1}$) $-2.6(3)$ $-6.3(17)$
$\dot{V}_\gamma$ (kms$^{-1}$/cycle) $0(-)$ $0.104(46)$
$a_1\sin i$ (R$_{\odot}$) $19.08(24)$ $19.04(24)$
$e$ $0.050(13) $ $0.048(13)$
$\omega$ ($\degr$) $43.9(129)$ $38.4(137)$
$M_0$ ($\degr$) $44.2(129)$ $49.0(136)$
$\tau$ (BJD) $2\,454\,962.9(11)$ $2\,454962.5(12)$
$K_1$ (kms$^{-1}$) $30.25(39)$ $30.18(36)$
$f(m_2)$ (M$_{\odot}$) $0.0911(35)$ $0.0906(35)$
$\Delta P\mathrm{~(d/cycle)}$ $-$ $1.1(5)\times10^{-5}$
: Results of radial velocity solutions, and some derived parameters (probable errors in the last digits). Note, reference epoch ($T_0$) and period $P_\mathrm{orb}$ were kept fixed.[]{data-label="tab: RVresult"}
Detection of the secondary component and dynamic masses {#sec: spd}
-------------------------------------------------------
The spectra of the faint secondary component was not detected in the crosscorrelation function (CCF) (see Sect.\[Subsect.:fundparam\]). Therefore, the direct dynamical determination of the masses of the components has not been possible. Furthermore, the secondary’s spectral properties have also remained unclassified. None of this information is crucial for the complex analysis of the system, as neither the orbital nor the light curve solutions are dependent on the stellar masses. Moreover, the less than 5% contribution of the secondary’s light to the total flux of the system suggests, that the composite spectra, and therefore the CCF solution of the primary is only weakly affected by the contribution of the secondary. However, from astrophysical point of view, stellar masses are the most important parameters. Therefore, in order to obtain dynamical masses and additional information on the secondary we made additional efforts to separate the signal of the secondary from the composite spectra.
The method of spectral disentangling ([spd]{}) enables isolation of the indvidual component spectra simultaneously with the determination of the optimal set of orbital elements [@simonsturm94; @hadrava95]. A time-series of the spectra are needed spread along the orbital cycle. Faint components are detected by [spd]{} in the high-resolution spectra [c.f. @pavlovskietal09; @lehmannetal13; @tkachenkoetal14] but a good phase coverage and a high S/N are needed. This is an important feature of [spd]{} since the disentangled spectra are effectively co-added from the original observed spectra and so have a higher S/N.
Our spectroscopic data sets are of different spectral resolutions and S/N (Sect.\[Subsect:Spectroscopy\]). Several spectra per night were usually obtained at Piszkéstető and Gothard Observatories and we stacked them to enhanced S/N. Still some of these stacked spectra suffered from low S/N and were not used in [spd]{}. We decided to omit these spectra, and after the selection we dealt with 16 spectra suitable for [spd]{} (the observed spectra used in [spd]{} are indicated in Table \[Tab:radveldata\] by asterisk). Fortunately, selected spectra cover a complete orbital cycle and hence fulfill a prerequisite for a stable disentangling. Because of different resolution we re-sampled all spectra to medium resolution of GAO spectra. We assigned the weights according to the S/N, and an initial spectral resolution.
The code [FDBinary]{} [@ilijicetal04] which implements disentangling in the Fourier domain [@hadrava95] was used to perform [spd]{} in spectral regions centred on the [[Mg]{}]{} triplet, $\lambda\lambda 5167-5184$ [Å]{}, covering about 200[Å]{}. In these calculations eccentricity, $e$, and the argument of periastron, $\omega$, were set fixed, as these orbital elements were better constrained from the combined RV+ETV analysis (sects.\[Subsect:radvelstudy\]\[Sect:ETV\]). Then the orbital solution obtained by [spd]{} yielded velocity semi-amplitudes of $K_1 = 34.4\pm1.1$ kms$^{-1}$ and $K_2 = 62.3\pm1.6$ kms$^{-1}$, and thus a mass ratio $q = 0.552\pm0.023$. The quoted errors for the semi-amplitudes derived by [spd]{} were calculated by the ‘jackknife’ method [c.f. @pavlovskisouthworth09]. A comparison with the single-lined RV study reveals that [spd]{} resulted an $\sim13\%$ larger primary semi-amplitude, and consequently, via the spectroscopic mass-function, a higher mass-ratio. The discrepancy might come from two reasons, either (i) the unresolved light contamination of the secondary’s spectra to the primary’s spectral lines in CCF measurements, which acts to reduce the semi-amplitude of the primary RV curve, or (ii) the same effect which causes the radial velocity residual wobbling, discussed in Sect.\[Subsect:radvelstudy\], resulting in slight spectral variations and therefore, slightly biases the disentangling. Note, a thorough analysis of different CCF and disentangling methods were carried out in @southworthclausen07, who also found that [spd]{} gives higher (and more reliable) semi-amplitudes, especially when the spectra were affected by line-blending. A portion of disentangled spectral region is shown in Fig.\[figDisent\]. The spectrum of a mid-F type secondary component is clearly revealed as could be judged by comparison with the syntethic spectrum for its atmospheric parameters and diluted by the factor of 20 to mimic its contribution to the total light of the system. As shown in Fig.\[figDisent\] [spd]{} was performed in the ‘separation’ mode, and with the known light ratio (from the light-curve analysis). These separated spectra are renormalised to the continua of the respective components [@pavlovskihensberge05] for the further detailed spectroscopic analysis. For this latter process we fixed $\log g$s and light factors on the values found from the combined detailed CCF-spectroscopic and light curve analyses (see Table\[Tab:syntheticfit\]), and fitted only temperatures and projected rotational velocities. Then our analysis resulted in $T_\mathrm{eff1}=7510\pm90$K, $\left(v_\mathrm{rot}\sin i_\mathrm{rot}\right)_1=104.2\pm1.5$kms$^{-1}$ and $T_\mathrm{eff2}=6490\pm140$K, $\left(v_\mathrm{rot}\sin i_\mathrm{rot}\right)_2=26.0\pm2.4$kms$^{-1}$ for the primary and the secondary, respectively. Therefore, the effective temperatures were found to be in accordance with the results of the combined analysis within their errors (see Table\[Tab:syntheticfit\]). The main significance of this result lies in the substantially reduced uncertainty of the primary’s projected rotational velocity, and the determination of the same parameter for the secondary component. We note also that the temperature ratio obtained from disentangled spectra of the components was found to be $0.864\pm0.021$ in contrast to the photometrically found value $0.843\pm0.017$, with a difference within the uncertainty limit. This is an additional interdependent verification of the results obtained in different manners.
{width="18cm"}
Light curve analysis {#lcanalysis}
====================
The [*Kepler*]{} light curve reveals at least three different features. The most prominent pattern shows that HD183648 is a relatively long-period ($P_\mathrm{orb}=31.973$d) eclipsing binary on an eccentric orbit. The light curve also shows pulsations with periods near 1.78d. Moreover, the amplitudes of these pulsations shows an obvious beat phenomenon with a period that is equal to half of the orbital period. As a consequence, the maxima and minima of the envelope of the pulsation occur at the same orbital phases during the whole 4-year observational interval. Furthermore, another sinusoidal light variation is also observable with a period equal to half-orbital period, and phased in such a way that the maximum brightnesses occur near orbital phases $0.0$ and $0.5$ (i.e., near the eclipses). Therefore, this enigmatic variation looks like an “inverse” ellipsoidal effect, or resembles a reflection or irradiation effect, although its high amplitude clearly excludes this latter explanation. An additional sinusoidal brightness variation with a period equal to the orbital period is also observable, however, as it will be shown later, this latter feature well can be explained by Doppler boosting. All these light curve features are illustrated in Fig.\[Fig:envelope\].
{width="0.95\linewidth"}
In order to obtain a physically correct binary star model, the different properties of these complex light curve variations were disentangled. As simultaneous eclipsing binary and pulsation modelling methods and programs are not available yet, we followed an iterative procedure, similar to that which was described and applied in the papers of @mac13 and @debosscheretal13. This method is based on the rectification of the light curve with an iterative separation and then, removal of the other light curve variations from the eclipsing binary features, by the use of Fourier space, obtaining an approximately pure eclipsing binary light curve, which can then be fitted by a light curve fitting algorithm. Following the removal of this solution from the original curve, a more accurate pulsation pattern can be obtained. This method can lead to an improved pulsation model that is then removed from the original light curve to obtain a more improved eclipsing binary light curve. In the present situation, however, the presence of the exactly half orbital period extra variation provides a slight complication, as it covers the possibly small “normal” ellipsoidal effect, and modifies eclipse depths and shapes coherently in phase. Note that @sou11 explained the unphysical outputs of their light curve solution for KIC10661783 with such an effect. Fortunately, however, the amplitude of this variation is less than 1mmag, and consequently, it has only minor effect on the eclipses.
For the initial disentangling of the pulsation and eclipse patterns, we calculated the Discrete Fourier Transform amplitude spectrum of the raw data. As one can see in Fig.\[Fig:DFTraw\], a very regular spectrum was obtained which contains harmonics of the orbital frequency almost exclusively. There are two main pulsation peaks separated equally in frequency from $17f_\mathrm{orb}$ and $19f_\mathrm{orb}$.
{width="84mm"}{width="84mm"}
After obtaining these two pulsation frequencies, we carried out a four-frequency linear least-squares fit on the out-of-eclipse parts of the raw, detrended Q0 – Q16 LC light curve. We fit not only the two dominant pulsation frequencies, but also, in accordance with the additional light curve features mentioned above, the frequencies $f_\mathrm{orb}$, $2f_\mathrm{orb}$, too. Then we removed this least-squares solution from the raw curve. It is evident that we possibly removed the ellipsoidal, reflection and Doppler-boosting effects. However, from the preliminary system characteristics and light curve properties, we expected only minor (if any) contributions from ellipsoidal and reflection effects. Regarding Doppler-boosting, the version of the PHOEBE software package [@prsazwitter05] that was used in this preliminary stage does not model it.
The initial values of the fundamental parameters for the primary star were taken from the spectroscopic results, while the orbital elements were taken from the preliminary radial velocity and ETV analyses. The differential correction part of the PHOEBE analysis was applied for three different datasets, namely, for the pulsation-removed (i) Q0 LC data, (ii) Q3.2 SC data and finally, (iii) the binned, averaged Q0 – Q16 LC data.
After the removal of the PHOEBE solution an improved pulsation model was calculated and subtracted in a similar manner. Then, after reaching a quick convergence of this iterative method, we made a final parameter refinement with our own [lightcurvefactory]{} light curve synthesis program [@borkovitsetal13]. As a recent improvement, a linear least-squares based multi-frequency Fourier polynomial fitting subroutine was also built into the code, which made it possible to fit quasi-simultaneously both the eclipsing binary and the pulsation models internally, at every step. Note that such a combination has only practical and time saving advantages, but remains an unphysical solution for the combined investigation of pulsation and binarity effects, and hence suffers from all the disadvantages that were discussed in @wilsonvanhamme10. We used five frequencies for the Fourier fitting procedure, namely, the four higher amplitude pulsation frequencies (see Sect.\[Subsect:freqsearch\]), and $2f_\mathrm{orb}$. As our program also takes Doppler-boosting into account, we removed the $f_\mathrm{orb}$ frequency component from the Fourier fitting. Furthermore, in this refinement process, the rotation synchronization parameter was no longer kept fixed, but was constrained according to the spectroscopically determined values of $v_\mathrm{rot}\sin{i}_\mathrm{rot}$ (assuming that $i_\mathrm{rot}=i_\mathrm{orb}$).
For this next combined refinement, the complete, previously detrended, unaveraged, unbinned Q0–Q16 long cadence light curve was used. This curve contains $64\,528$ data points. In order to reduce computing time, we switched off the computation of the reflection effect (which is by far the most time-consuming part of the calculations), and used four-times coarser stellar grids in the out-of-eclipse phases. A test verified that reflection/irradiation affected the light curve around the secondary minima (where it reaches its maximum), at an insignificant 10ppm level, while the coarser grid was found to have no systematic effect on the goodness of a given parameter set, but resulted in a somewhat higher $\chi^2$ value due to the noisier synthesis curve. Naturally, reaching a convergent solution, the final light curves and residuals were calculated with reflection and finer grids.
Calculating the refined solution in this manner, the residual curve revealed that there were systematic discrepancies in certain Quarters. This manifested itself in non-zero average slopes of some quarterly data. Although it cannot be excluded that these effects have physical origins (e.g., longer time-scale brightening or fading of the system, or a residual effect of the Doppler-boosting caused by the [*Kepler*]{} spacecraft’s motion), from our point of view, they represent additional, systematic noise which should be removed. Therefore, we fitted the residual data with first order polynomials, individually for each quarter, and by the use of them, we detrended again the previously used Q0 – Q16 LC data. Then, we repeated only the last, refining part of our analysis. The residuals of the combined eclipsing and 5-frequency pulsation curve before and after this final detrending are plotted in Fig.\[Fig:resbeforeafter\].
![The residuals of the combined eclipsing + 5-frequency pulsation solution before (red) and after (black) of the final detrending. (See text for details.)[]{data-label="Fig:resbeforeafter"}](MN-14-1385-MJf8.eps){width="84mm"}
[@lll]{}\
$P_\rmn{orb}$ (days) &\
$T_\rmn{MIN I}$ (BJD) &\
$a$ (R$_\odot$) &\
$e$ &\
$\omega$ ($\degr$)&\
$i$ ($\degr$) &\
$\tau$ (BJD) &\
$q$ &\
$P_\rmn{apse}^\rmn{obs}$ (years) &\
\
$P_\rmn{apse}^\rmn{theo}$ (years) &\
$\dot\omega_\mathrm{rel}^\rmn{theo}$ (arcsec/$P_\mathrm{orb}$) &\
$\dot\omega_\mathrm{cl}^\rmn{theo}$ (arcsec/$P_\mathrm{orb}$) &\
\
& Primary & Secondary\
\
$r_\rmn{pole}$ & $0.05240\pm0.00010$ & $0.01810\pm0.00020$\
$r_\rmn{side}$ & $0.05475$ & $0.01813$\
$r_\rmn{point}$ & $0.05476$ & $0.01813$\
$r_\rmn{back}$ & $0.05476$ & $0.01813$\
\
$M$ (M$_\odot$) & $1.93\pm0.12$ & $1.06\pm0.08$\
$R$ (R$_\odot$) & $3.30\pm0.07$ & $1.11\pm0.03$\
$T_\mathrm{eff}$ (K)& $7650\pm100$ & $6450\pm100$\
$L$ (L$_\odot$) & $32.88\pm0.20$ & $1.87\pm0.12$\
$\log g$ (dex) & $3.71\pm0.03$ & $4.38\pm0.04$\
$P_\mathrm{rot}$ (days) & $1.60\pm0.04$ & $2.15\pm0.21$\
Parameter Primary Secondary
---------------------------------------- ---------- -----------
Linear limb darkening (bolometric) $0.6658$ $0.6658$
Logarithmic limb darkening (bol.) $0.2493$ $0.1701$
Linear limb darkening (monochrom.) $0.6121$ $0.6191$
Logarithmic limb darkening (mono.) $0.2350$ $0.1799$
First apsidal motion constant ($k_2$) $0.0020$ $0.0080$
Second apsidal motion constant ($k_3$) $-$ $0.0020$
Bolometric albedo $1.0$ $0.6$
Gravity darkening exponent $1.0$ $0.32$
: Model-dependent (readjusted) and fixed parameters[]{data-label="Tab:syntheticfix"}
{width="168mm"}\
{width="168mm"}
The final results of the combined eclipsing light and radial velocity curve analysis are listed in Tables\[Tab:syntheticfit\] and \[Tab:syntheticfix\], and are shown in Fig.\[Fig:Q0lcsolution\]. Furthermore, Fig.\[Fig:lcsolution\_folded\] gives details on the out-of-eclipse part of the folded and binned solution, which is plotted there both with and without the 5-frequency pulsation. The latter corresponds to the theoretical, pure eclipsing binary light curve solution, i.e. the sum of the “normal” ellipsoidal effect and Doppler boosting (blue curve in the upper panel). The regular half-orbital-period sinusoidal shape of the phased residuals of this theoretical curve (blue curve in the lower panel of Fig.\[Fig:lcsolution\_folded\]) (i.e., the absence of the brightness differences between the two quadratures) demonstrates clearly that the $f_\mathrm{orb}$ component is well described purely with Doppler boosting. This also gives independent evidence for the absence of significant third light in the light curve. If there were significant third light, the additional light contribution would reduce the observable amplitude of Doppler-boosting and, consequently, the theoretical fit would overestimate it. This result suggests that if there is a third companion, it should be probably a low-mass M dwarf star (see Sect.\[Sect:ETV\]).
The oscillatory features of the residual curve will be discussed in the next section. Here we only comment on the small residual discrepancies during the two kinds of minima. What is surprising is not their presence (they occur commonly in the case of very accurate satellite light curves due to the incomplete physics included in the presently available models, see @ham13 for a short discussion), but that their amplitudes do not exceed $300-500$ ppm in relative flux. Taking into account the irregular, and therefore incompletely modelled, ellipsoidal effect (to be discussed in the next Section), we are inclined to take the extraordinay goodness of our fit as a mere coincidence and not the outcome of a serendipitously found accurate physical model.
In Table\[Tab:syntheticfit\] we tabulate some derived quantities, such as the rotational period ($P_\mathrm{rot}$) of the two components, and the theoretical relativistic and classical tidal apsidal motion angular velocities. For this calculation the apsidal motion constants (listed in Table\[Tab:syntheticfix\]) were taken from the tables of @claretgimenez92. Note that for the calculation of the tidally forced apsidal motion we used only the equilibrium tide model [@cowling38; @sterne39], and did not consider the dynamical contribution [see, e.g., @claretwillems02]. A proper calculation of the dynamical tides for the fast rotating primary is beyond the scope of the present paper. It should be stressed, however, that in the case of resonant tidal locking, the contribution of the dynamical tides may exceed the classical ones [@willemsclaret05]. This fact might offer an additional explanation for the discrepancy between the calculated apsidal advance rate and the observed one, which was examined previously in Sect.\[Sect:ETV\]. The role of the fast rotation of the primary in the tidal oscillations will be discussed below in Sect.\[Subsect:tidaloscillation\].
The uncertainties of the parameters were determined with various methods. For the ETV and the radial velocity analysis, the errors given are mostly the formal errors of the differential correction procedures. It is well known, however, that these formal errors underestimate the real uncertainties due to the strongly degenerate nature of the eclipsing binary light curve modelling, with substantial correlations among the parameters, and should not be taken too seriously. Therefore we resorted to the more realistic estimations given by the final refinement to the light curve solution, which was essentially a Monte Carlo simulation. Our experiences are in accordance with those found by @ham13 in a similar situation. Therefore we conclude that, despite the significant correlations, the light curve parameters are relatively well determined for this sort of detached [*Kepler*]{} binary with significantly deep eclipses.
![[*Upper panel:*]{} The folded and binned out-of-eclipse section of the whole Q0 – Q16 detrended LC data (red) together with the combined, simultaneous eclipse and 5-frequency pulsation solution (black) and with the pure eclipsing part of the same solution (blue). [*Bottom panel:*]{} The folded, binned residuals of the solutions above.[]{data-label="Fig:lcsolution_folded"}](MN-14-1385-MJf10.eps){width="84mm"}
Oscillations and tidal effects
==============================
\[Subsect:freqsearch\]Frequency search
--------------------------------------
After subtracting the eclipses, rotation and other binary related variations (see Sect.\[lcanalysis\]), we analysed the remaining nearly continuous data set containing mainly the pulsations. For the period analysis we used [period04]{} [@len05], least-squares fitting of the parameters was also included and the signal-to-noise ratio ($S/N$) of each frequency was calculated following @bre93. The resulting significant peaks are listed in Table\[freqs\], while the Fourier spectrum is shown in Fig.\[fourier\].
We identified the two main pulsation frequencies at $F_{1}={\rm 0.535157 (1)\,d^{-1}}$ and $F_{2}={\rm0.597712(1)\,d^{-1}}$. The most intriguing result is that $F_{2}-F_{1}$ is exactly equal to $2 f_{orb}$ within $0.000003\,d^{-1}$, suggesting tidal origin. A further 6 statistically significant peaks were identified in the data. The $F_8$, $F_3$ and $F_4$ peaks represent the orbital frequency, and its second and third harmonics, respectively. While the less-significant $F_8$ frequency (i.e. the orbital frequency) might purely be the remnants of the light curve solution, the two higher harmonics are thought to be real, and will be discussed below. Furthermore, $F_{5}={\rm 0.766149\,d^{-1}}$ might also be interpreted as an independent oscillation frequency, which is supported by the fact that $F_{7}={\rm 0.230964\,d^{-1}}$ is equal to $F_{5}-F_{1}$ within $0.000028\,d^{-1}$. Finally, $F_{6}={\rm 0.100662\,d^{-1}}$ might be a remnant of the light curve fit or an instrumental effect.
-------- ------------- -------------------------- -------------- ----- -------------------------------------------
Frequency Amplitude phase S/N Orbital
(d$^{-1}$) ($\times 10^{-04}$ flux) (rad) solution
F1$^*$ 0.535157(1) 8.711(11) -0.2200(13) 88 $f_{1}$
F2$^*$ 0.597711(1) 5.880(11) -2.6713(19) 61 $f_{2}$
F3$^*$ 0.062551(1) 4.946(11) -1.3473(22) 40 $2 \cdot f_\mathrm{orb}$ or $f_{2}-f_{1}$
F4 0.093783(1) 0.6805(112) 2.0380(165) 6 $3 \cdot f_\mathrm{orb}$
F5$^*$ 0.766149(1) 0.6267(112) 1.9013(180) 7 $f_{3}$
F6 0.100660(1) 0.6199(112) -2.9061(182) 5
F7$^*$ 0.230942(1) 0.5647(112) -1.2457(199) 5 $f_{3}-f_{1}$
F8 0.031280(1) 0.5630(112) -1.4853(200) 4 $f_\mathrm{orb}$
-------- ------------- -------------------------- -------------- ----- -------------------------------------------
$^*$ denotes the frequencies used for the simultaneous binary light-curve, pulsation curve fitting process (see Sect.\[lcanalysis\]).
{width="168mm"}
Discussion: Oscillations and Tidal Effects {#Subsect:tidaloscillation}
------------------------------------------
The presence of oscillations at integer harmonics of the orbital frequency is not surprising. As described above, these oscillations are produced by a combination of tidal ellipsoidal effects, reflection effects, and Doppler beaming [see @shporeretal11]. In a circular orbit, the reflection and Doppler beaming effects contribute mainly to variation at $f_{\rm orb}$, while the tidal effect contributes mainly to variation at $2 f_{\rm orb}$. For an eccentric orbit, each of these effects contributes variations at every harmonic of the orbital frequency [see @wel11; @burkartetal12]. In HD183648, the low eccentricity implies that these effects only contribute at low harmonics of the orbital frequency. Indeed, we only observe modulation at $f_{\rm orb}$, $2f_{\rm orb}$, and $3f_{\rm orb}$. In what follows, we assume all modulation arises from the primary since its light dominates the luminosity of the system.
The especially odd feature of HD183648 is the phase of the oscillation at $2f_{\rm orb}$. In typical nearly circular eclipsing binaries which show ellipsoidal variations, the eclipses occur near the minima of the ellipsoidal modulations, while the maxima occur one quarter of an orbital period later. The phase of this modulation is intuitive: the maxima occur away from eclipse when the equilibrium tidal distortion[^5] causes the star to present a larger surface area toward the line of sight. However, in HD183648, the oscillation at $2f_{\rm orb}$ shows the opposite phase, with maxima near the eclipses (phases 0 and 0.5).
There are three possible explanations for the strange features of the oscillations at orbital harmonics in HD183648. The first is that non-adiabatic effects near the surface of the star are strongly affecting the temperature perturbation created by the equilibrium tidal distortion of the primary star. Tidal ellipsoidal variations are typically modelled by using Von Zeipel’s theorem to calculate the surface temperature perturbations. In this case, the tidally depressed regions (where the surface gravity is stronger) are hotter, creating a luminosity fluctuation of the same phase as described above (i.e., the luminosity maxima occur away from eclipses). However, as shown in @pfahletal08, non-adiabatic effects can completely alter the temperature perturbations in hot stars with radiative envelopes like the primary in HD183648. For hot stars, the luminosity variation is typically dominated by temperature variations (rather than surface area distortion), and the phase of this variation can be arbitrary. Therefore, non-adiabatic effects may be strongly altering the luminosity variations produced by the equilibrium tidal distortion of HD183648, leading to the strange phase of the oscillation at $2 f_{\rm orb}$.
A second explanation is that dynamical tidal effects are important. In stars with radiative envelopes, dynamical tides are composed of stellar gmodes that are nearly resonant with the tidal forcing frequencies. As with the equilibrium tide, the dynamical tide produces observable oscillations at exact integer harmonics of the orbital frequency [@kumaretal95; @wel11; @fullerlai12; @burkartetal12]. The phase of the luminosity fluctuations produced by dynamical tides can be different from that of the equilibrium tidal distortion [see @olearyburkart14], potentially creating the observed oscillations. However, in most cases the luminosity oscillations produced by the dynamical tide are smaller than that of the equilibrium tide [see @tho12], so a gmode unusually close to resonance may be needed to produce the oscillation at $2 f_{\rm orb}$.
A third possibility is that non-linear interactions are affecting the mode phases and amplitudes. We discuss this in greater detail below.
A full calculation of tidal excitation of non-adiabatic oscillation modes in rotating stars is beyond the scope of this paper. Instead, we simply calculate the expected luminosity fluctuation and phase of the adiabatic equilibrium tidal distortion, using the stellar parameters of Table\[Tab:syntheticfit\]. The tidal distortion causes luminosity fluctuations of form $$\label{eqtide}
\frac{\Delta L}{L} = A_n \cos \bigg(2 \pi n f_{\rm orb} t+ \delta_n \bigg),$$ where the integer $n$ is the orbital harmonic of the oscillation, $A_n$ is its amplitude, and $\delta_N$ is its phase relative to periastron when $t=0$. We calculate the amplitude of the equilibrium tide as described in @burkartetal12, using Von Zeipel’s theorem to calculate the flux perturbation. We also calculate the expected phase of the equilibrium tide luminosity fluctuation, which is[^6] $$\begin{aligned}
\label{phase}
\delta_{n, \rm eq} &= |m| \omega \ \ {\rm for} \ |m|=2 \nonumber \\
&= \pi \ \ {\rm for} \ \ {\rm for} \ m=0 \, ,\end{aligned}$$ where $\omega$ is the argument of periastron listed in Table\[Tab:syntheticfit\]. We plot these results in Fig.\[kic85plot\]. It is evident that although the magnitude of the observed luminosity fluctuations are similar to those expected from an adiabatic equilibrium tide, the phases are completely different. Hopefully, a more in depth investigation of tidally excited oscillations in this system will provide constraints on the tidal dynamics at play.
![\[kic85plot\] Top: Observed luminosity fluctuations and luminosity fluctuations due to the equilibrium tidal distortion (calculated using Von Zeipel’s theorem) as a function of the orbital harmonic $f/f_{\rm orb}$. Bottom: Observed cosine phase of the luminosity fluctuations and the expected phase for the equilibrium tide. Although the observed oscillation amplitude at $2 f_\mathrm{orb}$ is near the expected equilibrium tide amplitude, it is out of phase from the expected equilibrium tide phase.](MN-14-1385-MJf12.eps)
Finally, we emphasize that it is important to consider the rapid rotation frequency of the primary in HD183648 relative to the orbital frequency. If the primary’s spin axis is aligned with the orbit, the stellar rotation period is about 1.6d, implying the rotation frequency is $f_{\rm spin} \simeq 19.98 f_{\rm orb}$. In this scenario, the observed gmodes ($f_1$ and $f_2$) cannot be prograde modes (in the rotating frame of the star) because their minimum frequency in the inertial frame is $|m| f_{\rm spin} > f_1, f_2$. Moreover, the tidally excited oscillations are retrograde oscillations in the rotating frame of the primary (although they are prograde in the inertial frame). Note that in the rotating frame, the tidal forcing frequencies are $f_{\rm tide} = n f_{\rm orb} - |m| f_{\rm spin}$. Since $f_{\rm spin} \gg f_{\rm orb}$ the absolute values of the forcing frequencies are much larger in the rotating frame, allowing for excitation of gmodes with frequencies (in the rotating frame) $f_\alpha \sim |m| f_{\rm spin}$.
### Non-linear Mode Coupling
As described above, the dominant two oscillation frequencies $f_1$ and $f_2$ are separated by exactly $2 f_{\rm orb}$, which is the third largest amplitude oscillation observed. It is well known that combination frequencies of this sort are indicative of non-linear mode coupling [see, e.g., @wugoldreich01]. Indeed, there are now many cases of combination frequencies in close binaries which appear to be caused by non-linear mode coupling with tidally excited modes [see @mukadametal10; @fullerlai12; @burkartetal12; @ham13]. In the case of HD183648, the non-linear coupling causes interactions between two gmodes (corresponding to $f_1$ and $f_2$ in Table\[freqs\]) and a tidally excited oscillation at $2 f_{\rm orb}$. At least one of the gmodes may be self-excited, perhaps because one of the stars is a $\gamma$Dor variable. Indeed, the primary of HD183648 lies near the hot end of the $\gamma$Dor instability strip, while the secondary lies near the cool end. The observed frequencies $f_1$ and $f_2$ are on the low side, but are compatible with $\gamma$Dor pulsations [@balonaetal11]. The primary also lies within the $\delta$-Scuti instability strip, although no p-modes are observed.
The tidally excited oscillation is either composed of the equilibrium tide (which is dominated by the $l=2$, $|m|= 2$ fmodes) or the dynamical tide (which is dominated by an $|m|=2$ gmode). The modes interact via a parametric resonance, redistributing energy amongst the three modes[^7]. This transfer of energy changes the phases of the observed oscillations, and it may be possible that this is affecting the phase of the $2 f_{\rm orb}$ oscillation.
It is also possible that $f_1$ and $f_2$ are not self-excited modes, but instead are non-linearly tidally driven modes. The signature of non-linear tidal excitation is that $f_{\alpha} \pm f_{\beta} = n f_{\rm orb}$ where $n$ is an integer [see @weinbergetal12]. In HD183648, $f_2 - f_1 = 2 f_{\rm orb}$, which is compatible with non-linear excitation of $f_1$ and $f_2$ (although this does not explain the large amplitude of $f_3$). Non-linear tidal driving was observed in a similar system examined by @ham13.
Summary and conclusions
=======================
We have presented a complex photometric and spectroscopic analysis of HD183648, a marginally eccentric ($e = 0.05$), wide ($P_\mathrm{orb} = 31.973$), detached eclipsing binary system with a low amplitude pulsating component. The photometric analysis of the extremely accurate [*Kepler*]{} Q0 – Q16 long cadence photometry incorporated disentangling of the eclipse and pulsation features, an eclipsing light curve solution and extended orbital period study via an ETV analysis. The spectroscopic investigations were based on ground-based high- and medium resolution spectra obtained with various instruments (echelle spectrograph at KPNO, ARCES Echelle spectrograph at APO, Hamilton Echelle Spectrograph at Lick Observatory, and eShel spectrograph of GAO, mounted on two telescopes at Szombathely and Piszkéstető, in Hungary) between 2011 and 2013. The spectroscopic data were mainly used for radial velocity analysis and for determination of stellar atmospheric properties and evolutionary states. Furthermore, the spectral disentangling technique made it also possible to detect the spectral lines of the secondary star despite its small (less than 5%) contribution to the total light of the system. This fact allowed us to calculate dynamical masses and hence, relatively accurate stellar parameters. However, we emphasize that all of the various investigations were carried out in a complex and interdependent manner. Namely, the results and constraints of the radial velocity and ETV analysis were incorporated in the light curve analysis and vice versa, in an iterative manner; similarly, the quantitative spectral analysis was constrained at the same time by the outputs of the light curve analysis. In this way we were able to find a solution consistent with both the observations and the theoretical constraints. We found that the binary is composed of two main sequence stars with an age of $0.9\pm0.2$Gyr, having fundamental parameters of $M_1=1.93 \pm 0.126$M$_{\odot}$, $R_1=3.30 \pm 0.07$R$_{\odot}$ for the primary, and $M_2=1.06\pm 0.08$M$_{\odot}$, $R_2=1.11 \pm 0.03$R$_{\odot}$ for the secondary. Both stars were found to be rapid rotators with $\left(v_\mathrm{rot}\sin i_\mathrm{rot}\right)_1=104$kms$^{-1}$ and $\left(v_\mathrm{rot}\sin i_\mathrm{rot}\right)_2=26$kms$^{-1}$ which in the aligned case correspond to rotation periods $P_\mathrm{rot1}=1.60 \pm 0.04\,{\rm d}\sim19.98f_\mathrm{orb}$ and $P_\mathrm{rot2}=2.15 \pm 0.21\,{\rm d}\sim14.87f_\mathrm{orb}$, respectively.
We have found various types of eclipse timing variations in our analysis. We showed that the short time scale ($\sim\!287$d) periodic variation is a false positive due to an apparent beating between orbital and pulsational frequencies. The parabolic variation, which indicates a period increase with a constant rate, however, is suggested to be a real effect. The most plausible explanation is presence of an additional, distant, third body in the system.
Clear indicators of apsidal motion have been found as well. We found a significant discrepancy between the theoretically computed and observed apsidal advance rates. This can be explained either with the insufficiently short time coverage of the apsidal motion cycle, which has a period of the order of ten thousand years, or perturbations from a tertiary component. Another alternative explanation is precession induced by dynamical tides [@willemsclaret05].
We made efforts to separate the oscillatory features from the binary characteristics, but there are strong connections between binarity and the detected oscillations. First, the difference of the two most dominant oscillation frequencies is equal to the twice of the orbital frequency, which indicates a binary origin. Furthermore, the most enigmatic feature of the out-of-eclipse part of the light curve is a sinusoidal variation with similar frequency and amplitude expected for the ellipsoidal effect, but with a completely opposite phase. Finally, there is a low amplitude oscillation at three times the orbital frequency, in addition to the inverse ellipsoidalvariation at twice the orbital frequency.
These phenomena are likely due to tidal effects. The oscillations at two and three times the orbital frequency are most likely tidally induced oscillations. However, it is unclear whether they are produced by hydrostatic equilibrium tides or by tidally excited g-modes. If they are equilibrium tides, non-adiabatic effects must be strongly altering their observed phase. If they are g-modes, they must be resonantly excited to account for their large amplitudes. Finally, the observed combination frequency $F_2 - F_1 = F_3 = 2 f_{\rm orb}$ indicates that non-linear mode coupling with the tidally excited oscillations is occurring. We are hopeful that a more detailed tidal analysis of HD 183648 may explain these observations and yield constraints on tidal dissipation theories.
Acknowledgements {#acknowledgements .unnumbered}
================
This project has been supported by the Hungarian OTKA Grant K83790, ESA PECS Contract No. 4000110889/14/NL/NDe, the Lendület-2009 Young Researchers Programme of the Hungarian Academy of Sciences and the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 269194 (IRSES/ASK) and no. 312844 (SPACEINN). AD, RSz and GyMSz have been supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. TB, BCs, JK and GyMSz would like to thank City of Szombathely for support under Agreement No. S-11-1027. Based on observations obtained with the Apache Point Observatory 3.5-meter telescope, which is owned and operated by the Astrophysical Research Consortium.
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[^1]: E-mail: borko@electra.bajaobs.hu (TB)
[^2]: http://archive.stsci.edu/kepler/
[^3]: http://keplerscience.arc.nasa.gov/PyKE.shtml
[^4]: Eq.(14) of the cited paper is valid strictly for $i = 90^{\circ}$; otherwise, it should be multiplied by $\sin i$ for the correct value.
[^5]: The equilibrium tide is the hydrostatic tidal bulge raised on the star by the gravitational force of the companion star. The equilibrium tide creates a tidal bulge along the line connecting the center of mass of the two stars. Typically, the tidal bulge is decomposed into spherical harmonics. Here we consider only the dominant components of the equilibrium tide, namely the $l=2$, $|m|=2,0$ components.
[^6]: The phase can change by $\pi$ depending on whether the amplitude $A_n$ is positive or negative, which in turn depends upon, e.g., the sign of the Hansen coefficients used to calculate the tidal potential for eccentric orbits [see @burkartetal12]. We calculate these coefficients, and adjust the phase $\delta_{n, \rm eq}$ such that the amplitude $A_n$ is positive.
[^7]: In the classic parametric resonance discussed by, e.g., @wugoldreich01, a self-excited “parent” mode non-linearly transfers energy to two “daughter” modes. However, it is also possible for two self-excited (or tidally excited) parent modes to transfer energy to a single daughter mode.
|
---
author:
- Antonin Guilloux
bibliography:
- 'biblio.bib'
title: 'Polynomial dynamic and lattice orbits in $S$-arithmetic homogeneous spaces'
---
Introduction
============
Consider an homogeneous space under a locally compact group $G$ and a lattice ${\Gamma}$ in $G$. Then the lattice naturally acts on the homogeneous space. Looking at a dense orbit, one may wonder how to describe its repartition. One then adopt a dynamical point of view and compare the asymptotic distribution of points in the orbits with the natural measure on the space. In the setting of Lie groups and their homogeneous spaces, several results we will present afterwards showed an equidistribution of points in the orbits.
We address here this problem in the setting of $p$-adic and $S$-arithmetic groups.
Historical background
---------------------
Ten years ago, F. Ledrappier [@ledrappier] explained how Ratner’s theory shall be used to understand the asymptotic properties of the action of ${\textrm}{SL}(2,{\mathbb}Z)$ on the euclidean plane ${\mathbb}R^2$. He proved the following:
\[the:led\] Let ${\Gamma}$ be a lattice of ${\textrm}{SL}(2,{\mathbb}R)$ of covolume $c({\Gamma})$, $\|.\|$ the euclidean norm on the algebra of $2\times 2$-matrices $\mathcal M(2,{\mathbb}R)$, and $v\in {\mathbb}R^2$ with non-discrete orbit under ${\Gamma}$.
Then we have the following limit, for all ${\varphi}\in \mathcal C_c({\mathbb}R^2\setminus\{0\})$: $$\frac{1}{T}\sum_{{\gamma}\in {\Gamma}\;, \|{\gamma}\|\leq T} {\varphi}({\gamma}v)\xrightarrow{T\to \infty} \frac{1}{|v|c({\Gamma})}\int_{{\mathbb}R^2\setminus\{0\}} {\varphi}(w) \frac{dw}{|w|} \; .$$
Nogueira [@nogueira] proved also the previous theorem for ${\Gamma}=SL(2,{\mathbb}Z)$ using different techniques.
After that A. Gorodnik develloped the strategy for the space of frames [@gorodnik-frames] and eventually A. Gorodnik and B. Weiss gave an abstract theorem for this problem in Lie groups and then applied it to different situations [@goroweiss].
Recently F. Ledrappier and M. Pollicott [@ledrappier-pollicott], and independently the author in its PhD thesis [@mathese], proved a $p$-adic analog of the first theorem for lattices of ${\textrm}{SL}(2,{\mathbb}Q_p)$ acting on the $p$-adic plane.
In this paper we adapt this strategy to handle the case of homogeneous space under $S$-arithmetic groups. Our work can be viewed as the analog of [@goroweiss] in this setting.
The $S$-arithmetic setting
--------------------------
We will work in the following arithmetic setting: let $K$ be a number field, $\mathcal O$ its integer ring and $\mathcal V$ the set of its places. We fix a finite set $S$ in $\mathcal V$ containing the archimedean ones. For all $\nu \in \mathcal V$, we note $K_\nu$ the completion of $K$ associated to $\nu$ and $K_S$ the module product of all $K_\nu$ for $\nu\in S$. This ring has a set of integer, noted $\mathcal O_S$.
Consider ${{\bf G}}$ a semisimple simply connected $K$-group. We note $G:={\bf G}(K_S)$ its $S$-points, and we fix ${\Gamma}$ an arithmetic lattice - i.e. commensurable to ${{\bf G}}(\mathcal O_S)$. Recall that, according to Margulis superrigidity theorem, as soon has the total rank of $G$ is greater than $2$, any lattice in $G$ is an arithmetical one. Then let $H$ be a subgroup of $G$ which is a product $\displaystyle \prod_{\nu\in S}H_\nu$ of closed subgroups of ${{\bf G}}(K_\nu)$. For example, one can think to the stabilizer of a point for an action of $G$ defined over $K$, i.e. $H=g\bar H g^{-1}$ where $\bar H$ is the $K_S$-points of a $K$-group and $g$ an element in $G$. We will always assume that the subgroup $H$ is unimodular. Some references for these objects are to be found in [@platonov-rapinchuk] and [@margulis].
We are interested in the asymptotic distribution of orbits of ${\Gamma}$ in ${{H \backslash G}}$ so we will always assume this orbit to be dense, or equivalently that $H{\Gamma}$ is dense in $G$. This last asumption is quite different of some recent works in the same area ([@gorodnik-oh], [@venkatesh-ellenberg]...) where $H$ is supposed to have a closed projection in ${{G / {\Gamma}}}$ and the dynamic appears by looking at larger and larger orbits. In particular, there won’t be any adelic arguments in this work.
### Measures and projections
We say that a triple $(G,H,{\Gamma})$ is *under study* if we are in the precedent case, that is if there is a number field $K$, a finite set $S$ of places containing the archimedean ones, and a $K$-group ${{\bf G}}$, $K$-reductive and with simply connected semisimple part, such that :
- $G$ is the $K_S$ points of ${{\bf G}}$,
- ${\Gamma}$ is an arithmetic lattice in $G$,
- $H$ is the product of unimodular $K_\nu$-subgroups of ${{\bf G}}(K_\nu)$ for $\nu \in S$,
- $H{\Gamma}$ is dense in $G$ and $H$ is not compact,
- $H$ is a semidirect product $H^{ss}\rtimes H^u$ of a semisimple part and an unipotent radical.
We now fix some notations for projections and measures : the Haar measure on $G$ is noted $m_G$ ; on $H$, $m_H$ ; and $m$ the probability on ${{G / {\Gamma}}}$ locally proportional to $m_G$. On ${{H \backslash G}}$, as $H$ is unimodular, we have a unique - up to scaling - $G$-invariant measure. We normalize the measure $m_{{H \backslash G}}$ on ${{H \backslash G}}$ such that $m_G$ is locally the product of $m_H$ and $m_{{H \backslash G}}$. The notations for the projections are as shown: $$\begin{matrix}
& G& \\
\tau \swarrow & & \searrow \pi \\
{{H \backslash G}}& & {{G / {\Gamma}}}\\
\end{matrix}$$
### Balls and volume
In order to adopt a dynamical point of view, we need to instillate some evolution in the so far static situation. So we consider families $(G_t)_{t\in R}$ of open and bounded subsets in $G$ (often called balls), and consider the sets ${\Gamma}_t={\Gamma}\cap G_t$. Letting $t$ go to $\infty$, we may now consider the asymptotic distribution of the sets $H\backslash H{\Gamma}_t$ in ${{H \backslash G}}$. Of course we will usually consider family $(G_t)$ that are increasing and exhausting (the union of $G_t$ covers $G$).
We introduce a notation for the intersection of such a family $(G_t)$ and its translates with subsets of $G$:
\[def:skewball\] Fix $(G_t)_{t\in{\mathbb}R}$ a family of open subset $G$, $L$ a subset of $G$ and $g$ an element of $G$. Then for all real $t$, we note $L_t:=L\cap G_t$ the intersection of $G_t$ ad $L$ and $L_t(g)$ the intersection $L\cap G_t g^{-1}$.
As the restriction of the so-called balls of $G$, we call the sets $L_t$ *balls* in $L$, and *skew-balls* the sets $L_t(g)$.
When $L$ is a subgroup, we can compare the growth of volume of its normal subgroup with respect to the sets $(G_t)$. It may happens that a strict subgroup grows as fast as the whole group. Such a subgroup is exhibited in [@goroweiss Section 12.3]. We will call such a subgroup dominant:
Let $L$ be a unimodular subgroup of $G$ and $m_L$ be its Haar measure. Fix $G_t$ a family of open bounded subsets of $G$, increasing and exhausting.
A normal subgroup $L'$ is said to be dominant in $L$ if for some compact $C$ in $L$, the volume of $C.L'_t$ grows as fast as the volume of $L_t$, i.e. $\frac{m_L(C.L'_t)}{m_L(L_t)}$ does not converge to $0$ with $t$.
Eventually we need an explicit way to define balls in ${\Gamma}$. Going back to Ledrappier’s theorem, we see that the balls are constructed considering a norm on the algebra of matrices. Moreover, Gorodnik and Weiss [@goroweiss] defined their balls in the same spirit, first representing the group $G$ and then using a norm on the matrix algebra in which $G$ is embedded. Our strategy is the same, but for technical reasons we assume firstly that the unipotent radical and the semisimple part are somehow orthogonal with respect to the norm and secondly that the norms are “algebraic”.
A *size function* $D$ from $G$ to ${\mathbb}R_+$ is any function constructed in the following way : consider a $K$-representation $\rho$ of ${{\bf G}}$ in a space ${\bf V}$ and for all $\nu\in S$ a norm $|\, .\, |_\nu$ on the space ${\textrm}{End}({\bf V}(K_\nu))$ verifying :
1. for all $h_\nu=(h_\nu^{ss},h_\nu^u)$ in $H$, its norm $|h_\nu|_\nu$ is an increasing function of both $|h^{ss}_\nu|_\nu$ and $|h^u_\nu|_\nu$.
2. If $\nu$ is archimedean, the norm $|.|_\nu$ may be written in a suitable basis as the $L_p$-norm for $p$ in ${\mathbb}N^*\cup \{\infty\}$. If $\nu$ is ultrametric, we assume that it is the max-norm in some basis.
Now define $D$ for all $g=(g_\nu)_{\nu \in S}$ by the formula $D(g)=\max\{|g_\nu|_\nu {\textrm}{ for } \nu \in S\}$.
In this setting given a size function, we have a family of open bounded subsets $G_t := \{g\in G{\textrm{ such that }}F(g)< t\}$ in $G$.
These two assumptions, especially the first one, are annoying. The second one does not seem to be an important one and in numerous applications our work may be applied without it. For the first one, I do not know wether it is necessary or not. The positive point is that for applications we may verify it (see section \[sec:exa\]): e.g. there is no condition when $H$ is either unipotent or semisimple. Morever every example given in the historical section fit into the framework of our article.
Statement of the main result
----------------------------
We prove in this article the following result:
\[the:normball\] Let $(G,H,{\Gamma})$ be a triple under study, $D$ be a size function on $G$ and $(G_t)_{t>0}$ be the associated family of balls. Assume that every dominant subgroup $H'$ verifies $H'{\Gamma}$ is dense in $G$.
Then there is a finite partition $I_1,\ldots, I_l$ of ${\mathbb}R_{>0}$, and, for each $1\leq i\leq l$, a function $\alpha_i \, :\, {{H \backslash G}}\to {\mathbb}R_{>0}$ such that the orbit of the sets ${\Gamma}_t=G_t\cap {\Gamma}$ for $t\in I_i$ becomes distributed in ${{H \backslash G}}$ according to the density $\alpha_i$ with respect to $m_{{H \backslash G}}$. That means, for all $\psi\in \mathcal C_c({{H \backslash G}})$, we have: $$\frac{1}{m_H(H_t)}\sum_{{\gamma}\in{\Gamma}_t}\psi(\tau({\gamma})) \xrightarrow[t\in I_i]{t\to+\infty} \int_{{H \backslash G}}\psi(x)\alpha_i(x)dm_{{H \backslash G}}(x)\;.$$
The partition of the parameter space in a finite number of subspaces is not needed when there is no non-archimedean places as in [@goroweiss] but appears even with very simple examples as soon as ultrametric part is to be taken in consideration. Let us also precise that the densities $\alpha_i$ are explicitely described and effectively computable in examples given afterwards (see theorem \[the:duality\]).
We present here some examples of applications. Of course one may look at numerous situations. I just present here some variations about linear actions of the special linear group on points or subspaces. I believe that these examples show how to apply the previous theorem to specific situations, using algebraic features such as strong approximation in the special linear group. The proofs are postponed to section \[sec:exa\].
### Applications to ${\textrm}{SL}(2)$
Consider the group $G={\textrm}{SL}(2,{\mathbb}R)\times {\textrm}{SL}(2,{\mathbb}Q_p)$ for $p$ a prime number, and fix the lattice ${\Gamma}={\textrm}{SL}(2,{\mathbb}Z[\frac{1}{p}])$. We fix here (for sake of simplicity) the standard euclidean norm $| . |_\infty$ on the matrix algebra $\mathcal M(2,{\mathbb}R)$ and the max-norm $|.|_p$ on $\mathcal M(2,{\mathbb}Q_p)$. For a point $v$ in ${\mathbb}R^2$, we note also $|v|_\infty$ the norm of the matrix whose first column is $v$ and the second one is $0$. We define similarly the norm of a point in ${\mathbb}Q_p^2$. We choose a Haar measure $m=m_\infty\otimes m_p$ on $G$.
First we look at the action on the real plane, proving a result similar to Ledrappier’s theorem but for the action of matrices in ${\Gamma}$ subject to congruence conditions on their coefficients modulo $p$:
\[appl21\] Let $O$ be a bounded open subset of ${\textrm}{SL}(2,{\mathbb}Q_p)$. Note ${\Gamma}^O_T$ the set of elements ${\gamma}\in {\Gamma}$ such that $|{\gamma}|_\infty \leq T$ and ${\gamma}\in O$ as an element of ${\textrm}{SL}(2,{\mathbb}Q_p)$. Let $v$ be a point of the plane ${\mathbb}R^2\setminus \{0\}$ with coordinates independant over ${\mathbb}Q$.
Then we have the following limit, for any function ${\varphi}$ continuous with compact support in ${\mathbb}R^2\setminus \{0\}$: $$\frac{1}{T} \sum_{{\Gamma}_T^O} {\varphi}({\gamma}(v)) \xrightarrow{T\to \infty} \frac{m_p(O)}{m({{G / {\Gamma}}})|v|_\infty}\int_{{\mathbb}R^2} {\varphi}(w)\frac{dw}{|w|_\infty}$$
Another action of ${\Gamma}$ of interest is on the product of real and $p$-adic planes. A precision : on the $p$-adic plane, we normalize the measure such that it gives mass $1$ to ${\mathbb}Z_p^2$. The result is that if your beginning point generates the whole plane among the ${\mathbb}Q$-subspaces, then its orbit is dense and you get a distribution result (the function $E$ appearing is the integer part):
\[appl22\] Let $(v_\infty,v_p)$ be an element of $({\mathbb}R^2\setminus{0})\times({\mathbb}Q_p^2\setminus{0})$. Suppose that any ${\mathbb}Q$-subspace $V$ of ${\mathbb}Q^2$ verifying $v_\infty\in V\otimes_{{\mathbb}Q} {\mathbb}R$ and $v_p \in V\otimes_{{\mathbb}Q} {\mathbb}Q_p$ is ${\mathbb}Q^2$. Denote ${\Gamma}_T$ the set of elements ${\gamma}\in {\Gamma}$ with $|{\gamma}|_\infty\leq T$ and $|{\gamma}|_p\leq T$.
Then, for all function ${\varphi}$ continuous with compact support in $({\mathbb}R^2\setminus{0})\times({\mathbb}Q_p^2\setminus{0})$, we have the following limit: $$\frac{1}{T p^{E(\ln_p(T))}} \sum_{{\Gamma}_T} {\varphi}({\gamma}v_\infty, {\gamma}v_p) \xrightarrow{T\to\infty} \frac{p^2-1}{p^2 m({{G / {\Gamma}}})|v_\infty|_\infty |v_p|_p} \int_{{\mathbb}R^2\times {\mathbb}Q_p^2} {\varphi}(v,w) \frac{dv dw}{|w|_\infty |w|_p}$$
All these results may be extended with the tools presented in the paper for any norm on the matrix algebras and by considering not only a prime number but a finite number of them.
### Applications to ${\textrm}{SL}(n)$
We look here at a generalization in greater dimension. We consider the action of ${\Gamma}={\textrm}{SL}(n,{\mathbb}Z)$ on the $k$-th exterior power $\Lambda^k({\mathbb}R^n)$, or the space of $k$-planes equipped with a volume. Once again we fix the standard euclidean norm $|.|$ on $\mathcal M(n,{\mathbb}R)$, but this time it is necessary to apply our theorem (see section \[sec:exa\]). We consider also the standard euclidean norm $|.|$ on $\Lambda^k({\mathbb}R^n)$. And $m$ is a Haar measure on ${\textrm}{SL}(n,{\mathbb}R)$. We get:
\[appln\] Let $v$ be a non-zero element of $\Lambda^k({\mathbb}R^n)$ such that its corresponding $k$-plane of ${\mathbb}R^n$ contains no rational vector. Denote ${\Gamma}_T$ the set of elements ${\gamma}\in {\Gamma}$ with $|{\gamma}|\leq T$.
Then we have a positive real constant $c$ (independant of ${\Gamma}$ and $v$) such that for all function ${\varphi}$ continuous with compact support on $\Lambda^k({\mathbb}R^n)\setminus \{0\}$: $$\frac{1}{T^{n^2+k^2-nk-n}}\sum_{{\Gamma}_T} {\varphi}({\gamma}v) \xrightarrow{T\to\infty} \frac{c}{m({{G / {\Gamma}}})|v|}\int_{\Lambda^k({\mathbb}R^n)} {\varphi}(v') \frac{dv'}{|v'|}$$
The $S$-arithmetic generalization of the previous result holds of course. I prefer to postpone its statement and its proof to the section \[sec:exa\]. Moreover I do not want to multiply here applications but one may think at examples in special unitary groups or Spin groups instead of the special linear one.
Organization of the paper
-------------------------
The organization of the paper is the following : in the next section we work out the so-called duality phenomenon, reducing the stated theorem to two results : a statement on volume of balls in the group and an analog of a result of Shah about equidistribution of balls of $H$ in ${{G / {\Gamma}}}$. The third section is devoted to the study of volume of balls, using $p$-adic integration. In the fourth section we review some tools we need to prove the analog of Shah theorem : mainly Ratner theorem for unipotent flows in a $p$-adic setting and several results due to G. Tomanov for polynomial dynamics in $S$-arithmetic homogeneous spaces. The fifth section is the devoted to some technical work. We conclude the proof in the sixth section. Eventually we treat the examples in the last section.
Duality
=======
The duality phenomenon, as used by F. Ledrappier [@ledrappier] and A. Gorodnik-B. Weiss [@goroweiss], is a consequence of the following idea : a property of the action of ${\Gamma}$ on ${{H \backslash G}}$ reflects in a property of the action of $H$ on ${{G / {\Gamma}}}$. The simplest example is the density of an orbit : $Hg$ has dense orbit under ${\Gamma}$ in ${{H \backslash G}}$ if and only if $g{\Gamma}$ has dense orbit under $H$ in ${{G / {\Gamma}}}$. This consideration leads to the key point in the proof of Ledrappier : instead of looking at the orbit of the lattice ${\Gamma}$ in the space ${{H \backslash G}}$, we prefer to translate the problem in terms of the action of $H$ in ${{G / {\Gamma}}}$. And then we may use the precise description of unipotent orbits in the space ${{G / {\Gamma}}}$, namely Ratner’s theory (cf section \[sec:ratner\]) to prove some equidistribution results. However, for asymptotic distribution of points, this phenomenon is not granted and requires additional assumptions we will review in this section.
We may remark that if $H$ is symmetric, Y. Benoist and H. Oh used other techniques - i.e. the mixing property - to study asymptotic distribution of orbits [@benoist-oh].
In [@goroweiss Corollary 2.4], Gorodnik and Weiss presented an axiomatic frame for duality. Unfortunately we cannot use directly their statement as we miss some continuity hypothesis on the distance function - once again the ultrametric part has to be handled specifically, even if the final result holds. So we present a slightly adapted version of their result in the theorem \[the:duality\].
In the setting defined in the precedent section, consider an increasing and exhausting family $G_t$ of open bounded subsets in $G$. We need an hypothesis of regularity on this family. We choose to state it using the right action of open subsets of $G$ and asking the sets $G_t$ to be uniformly almost invariant by some open set. As we are interested in the intersections with $H$, the precise (and classical) definition is:
Let $(G_t)_{t\in I}$ be a family of open bounded subsets of $G$. We say that it is *almost (right)-invariant* if for every $\epsilon>0$ one can find on open neighborhood $U_\epsilon$ of $id$ in $G$ such that the two following inequalities hold for every $t \in I$:
- the set $G_t U_\epsilon$ is not too big with respect to $G_t$ inside $H$: $$m_H(H\cap G_t U_\epsilon\setminus G_t) \leq \epsilon m_H(H\cap G_t)\; ,$$
- Not too much points inside $G_t$ are $U_\epsilon$-closed to its complement inside $H$ : $$m_H(H\cap G_t \setminus G_t^cU_\epsilon)\geq (1-\epsilon)m_H(H\cap G_t)\;.$$
One easily checks that the balls $G_t$ defined by a size function on $G$ are almost invariant. Indeed for the archimedean part, any norm on the matrix algebra is continuous. And for the ultrametric part the max-norm is invariant under some open neighborhood of identity.
We also need a result of existence of limits for ratios of volumes of skew-balls in $H$ (Hypothesis D2 in [@goroweiss]). Recall the definition \[def:skewball\] : for $g\in G$ and $t \in I$, $H_t(g)$ is the set $H\cap G_tg^{-1}$.
We say that a family $(G_t)_{t\in I}$ admits *volume ratio limits for $H$* if for all $g$ in $G$ the ratio $\frac{m_H(H_t(g))}{m_H(H_t)}$ admits a limit as $t$ goes to $+\infty$ in $I$.
The corollary 2.4 of [@goroweiss] (and its proof) implies the following one :
\[the:duality\] Let $(G,H,{\Gamma})$ be a triple under study. Let $(G_t)_{t\in I}$ be a family of bounded open subsets of $G$ almost invariant, admitting volume ratio limits for $H$ and such that the volumes of $H_t=H\cap G_t$ go to $+\infty$. Assume moreover that the orbit of $H_t$ in ${{G / {\Gamma}}}$ becomes equidistributed with respect to $m_{{G / {\Gamma}}}$ ; i.e. for all ${\varphi}\in \mathcal C_c({{G / {\Gamma}}})$, we have: $$\frac{1}{m_H(H_t)}\int_{H_t}{\varphi}(\pi(h))dm_H(h)\xrightarrow[t \in I]{t\to +\infty} \int_{{G / {\Gamma}}}{\varphi}dm_{{G / {\Gamma}}}\;.$$
Then the orbit of ${\Gamma}_t=G_t\cap {\Gamma}$ is distributed in ${{H \backslash G}}$ according to a density with respect to $m_{{H \backslash G}}$ ; i.e. for all $\psi\in \mathcal C_c({{H \backslash G}})$, we have: $$\frac{1}{m_H(H_t)}\sum_{{\gamma}\in{\Gamma}_t}\psi(\tau({\gamma})) \xrightarrow[t \in I]{t\to+\infty} \int_{{H \backslash G}}\psi(H.g)\frac{m_H(H_T g)}{m_H(H_T)}dm_{{H \backslash G}}(Hg)\;.$$
Particularly the density of the limit measure is described as limit ratio of volumes of balls. We will see in the next section a proof of existence of these ratios. But we will not in this paper go into precise and general estimates of these volumes. Our theorem still benefits of these estimation when available, e.g. in the applications (see section \[sec:exa\]). Maucourant [@maucourant] get very precise estimations for $H$ real semisimple.
The proof is the same as [@goroweiss Part 3 and 4]: the almost invariance replacing the hypothesis of right continuity of the distance function in their paper.
Now we have to understand the right setting to apply this theorem. There are two difficulties : the existence of volume ratio limits and the equidistribution of $H$-orbits in ${{G / {\Gamma}}}$. The next section address the first problem. We will prove the following theorem:
\[the:volumeratiolimits\] Let $(G,H,{\Gamma})$ be a triple under study, $D$ a size function on $G$. Consider $(G_t)_{t\in {\mathbb}R}$ the family of balls for $F$. Suppose that the volume of $H_t$ goes to $+\infty$.
Then there exists a finite partition of ${\mathbb}R$ in unbounded subsets $I_1,\ldots,\,I_k$ such that for all $1\leq l\leq k$ the family $(G_t)_{t\in I_l}$ admits volume ratio limits for $H$.
We shall exhibit in the following section a very simple example showing that we really need this partition.
The second part of the paper is to prove the equidistribution property under the hypothesis of theorem \[the:normball\]: $H$ is a semidirect product of a semisimple and a unipotent groups and every dominant subgroup has dense orbit in ${{G / {\Gamma}}}$. We will prove in section \[section:equidistribution\] the following theorem:
\[the:Horbit\] Let $(G,H,{\Gamma})$ be a triple under study, $D$ a size function and $H_t$ the induced family of balls in $H$. Assume that every dominant subgroup $H'$ of $H$ has dense orbit in ${{G / {\Gamma}}}$.
Then the orbits of $H_t$ becomes equidistributed in ${{G / {\Gamma}}}$ with respect to $m_{{G / {\Gamma}}}$ ; i.e. for all ${\varphi}\in \mathcal C_c({{G / {\Gamma}}})$, we have: $$\frac{1}{m_H(H_t)}\int_{H_t}{\varphi}(\pi(h))dm_H(h)\xrightarrow{t\to +\infty} \int_{{G / {\Gamma}}}{\varphi}dm_{{G / {\Gamma}}}\;.$$
Theorem \[the:normball\] is then a direct consequence of the three previous results.
Asymptotic developments of volumes
==================================
An example
----------
The following part is a little bit technical and may be misunderstood without any example in mind. Let us show on a very simple example that we have to be careful in describing the asymptotics of volumes of balls.
We will take here $G={\textrm}{SL}(3,{\mathbb}R)\times {\textrm}{SL}(3,{\mathbb}Q_p)$ for some prime $p$ and $H$ the image under the adjoint representation of ${\textrm}{SL}(2)$ of the upper triangular nilpotent subgroup: $$H=\left\{h(t_\infty,t_p)=\left( \begin{pmatrix} 1 & 2t_\infty &t_\infty^2 \\ 0& 1 &t_\infty \\ 0&0&1 \end{pmatrix}, \begin{pmatrix} 1 & 2t_p &t_p^2 \\ 0& 1 &t_p \\ 0&0&1 \end{pmatrix}\right) \; ; \; t_\infty \in {\mathbb}R {\textrm}{ and } t_p \in{\mathbb}Q_p\right\}$$
We choose the max-norm on both $\mathcal M_3({\mathbb}R)$ and $\mathcal M_3({\mathbb}Q_p)$ such that: $$H_{p^n}=\left\{h(s_\infty,s_p) {\textrm}{ for } s_\infty \in {\mathbb}R{\textrm}{ with }|s_\infty^2|\leq p^n{\textrm}{ and }s_p \in {\mathbb}Q_p{\textrm}{ with }|s_p^2|_p\leq p^n\right\}\; .$$ Hence the volume of $H_{p^n}$ is equal to $p^{\frac{n}{2}+E(\frac{n}{2})}$ ($E$ is the *integer part*).
Now let us have a look on a specific skew-ball : $H_{p^n}(Id,\begin{pmatrix} p & 0 & 0\\ 0&1&0\\0&0&p^{-1}\end{pmatrix})$, and we note $g=(Id,\begin{pmatrix} p & 0 & 0\\ 0&1&0\\0&0&p^{-1}\end{pmatrix})$. Then the skew-ball is described by: $$H_{p^n}(g)=\left\{ h(s_\infty,s_p) {\textrm}{ for } |s_\infty|^2\leq p^n {\textrm}{ and }|p^{-1} s_p^2|_p \leq p^n \right\}\; ,$$ hence its volume $m_H(H_{p^n}(g))$ is equal to $p^{\frac{n}{2}+E(\frac{n-1}{2})}$. We see that the ratio $\frac{m_H(H_{p^n}(g))}{m_H(H_{p^n})}$ is equal to $p^{E(\frac{n}{2})-E(\frac{n-1}{2})}$. This sequence does not admit any limit as $n$ goes to $\infty$. But we can split it in two subsequences: $n$ odd or even. And then both subsequences admit a limit (respectively $p$ and $1$).
Keeping this example in mind we will now explain why we are always able to do this: split the space of parameters $t$ in a finite number of subspaces in which the hypothesis of admitting volume ratio limits is fulfilled.
Volume ratio limits
-------------------
We will prove here the theorem \[the:volumeratiolimits\] stated above. We will use the fact that if two functions have an asymptotic development on the same (reasonnable) scale and their ratio is bounded, then this ratio admits a limit.
In order to get this asymptotic behaviour, we use the algebraic hypothesis on the norm. Then, following Benoist-Oh [@benoist-oh Part 16], we get the wanted result as consequence of resolution of singularities in the archimedean case and Denef’s Cell decompostion theorem in the non-archimedean one. These result are the two following propositions:
\[pro:benoist-oh\] Let $H$ be the group of ${\mathbb}R$-points of an algebraic ${\mathbb}R$-group, $\rho\; :\; H \to GL(V)$ a ${\mathbb}R$-representation of $H$, $m_H$ the Haar measure on $H$ and $|\,.\,|$ an algebraic norm on ${\textrm}{End}(V)$.
Then, for all $g\in {\textrm}{GL}(V)$, the volume $m_H(H_t(g))=m_H\{h\in H\, |\rho (h)g|\leq t\}$ has an asymptotic development on the scale $t^a ln(t)^b$ with $a\in {\mathbb}Q^+$ and $b\in {\mathbb}N$.
For the ultrametric part, we do not get exactly an asymptotic development rather a finite number of asymptotic developments. This was already noted in [@benoist-oh] but we need here a slightly more precise result, namely a uniformity on the number of simple functions needed:
\[the:benoist-oh\] Let $k$ be a finite extension of ${\mathbb}Q_p$, $q$ be the norm of an uniformizer, $H$ the group of $k$-points of an algebraic $k$-group, $\rho\; :\; H \to GL(V)$a $k$-representation of $H$, $m_H$ the Haar measure on $H$ and $|\,.\,|$ a ${\textrm}{max}$-norm on ${\textrm}{End}(V)$. Let $S_t(g)$ be the sphere of radius $t$ : $S_t(g):=\{h\in H{\textrm{ such that }}|hg|=t\}$.
Then there exist $N_0$ an integer such that for all $g\in G$ and for each $0\leq j_0\leq N_0$ one of the following holds:
1. $S_{q^j}(g)$ is empty for all $j=j_0 {\textrm}{ mod }N_0$.
2. There exist $d_{j_0}\in {\mathbb}Q_{\geq 0}$, $e_{j_0}$ an integer and $c_{j_0}>0$ such that $m_H(S_{q^j}(g))\sim c_{j_0} q^{d_{j_0} j} j^{e_{j_0}}$ for all $j=j_0 {\textrm}{ mod }N_0$.
I will not go into details as the proof is the same as [@benoist-oh Corollary 16.7]. I will just say that applying a theorem of Denef [@denef Theorem 3.1 and remark below], we get the following:
for any polynomial map $f(x,\lambda)$ from ${\mathbb}Q_p^{m+d}$ to some $GL(V)$, for any semialgebraic measure $\mu$ on a semialgebraic set $S\subset {\mathbb}Q_p^m$, there are some functions $\gamma_i(\lambda,n)$ and $\beta_i(\lambda,n)$ for $1\leq i\leq e$ such that the measure $I(\lambda,n)$ of the set of element $x\in S$ with $|f(x,\lambda)|=q^n$ is of the form : $$I(\lambda,n)=\sum_{i=1}^e \gamma_i(\lambda,n)p^{\beta_i(\lambda,n)}$$
Moreover the functions $\gamma_i$ and $\beta_i$ are simple in the following sense: for any of these functions (hereafter denoted $\alpha$) there exists an integer $N$ such that for all $\lambda$, the map $n\mapsto \alpha(\lambda,n)$ is affine along at most $N$ arithmetic progressions in ${\mathbb}N$ which cover ${\mathbb}N$ up to a finite set.
Now, the above proposition is just this result in the case where $S$ is the image under the representation $\rho$ of $H$, $\mu$ is the Haar measure on $H$ and $f(\lambda,x)=\lambda.x$ for $\lambda\in GL(V)$ and $x\in H$.
We may go on with the proof of theorem \[the:volumeratiolimits\]. Let us write more explicitly the informations we get on the function $m_H(H_t(g))$ from this two results. Fix some $g$ in $G$. Consider the set $S_f$ of finite places in $S$. For each $\nu \in S_f$, we note $q_\nu$ the norm of the uniformiser of $K_\nu$. The previous proposition gives us an integer $N_\nu$ and for all $0\leq j\leq N_\nu-1$ some $d_{\nu,j}\in {\mathbb}Q$, $d_{\nu,j}\in {\mathbb}N$ and $c_{\nu,j}>0$ describing the volume of spheres in the group $H_\nu$. Moreover for the archimedean part, the proposition \[pro:benoist-oh\] gives some triple $d_\infty \in {\mathbb}Q_{>0}$, $e_\infty \in {\mathbb}N$ and $c_\infty >0$ such that the volume of $(H_\infty)_t$ is equivalent to $c_\infty t^{e_\infty} e^{d_\infty t}$. With this data we are able to describe the volume of $H_t$:
\[equivalentvolume\] With the data above, $m_H(H_t(g))$ is equivalent, as $t$ goes to $\infty$, to : $$\begin{aligned}
\label{for:volume}
c_\infty t^{d_\infty}(\ln\, t)^{E_\infty}\prod_{\nu \in S_f} \left(\sum_{j=0}^{E(\ln_{q_\nu}t)} c_{\nu, j[N_\nu]}q_\nu^{d_{\nu, j[N_\nu]} j} j^{e_{\nu, j[N_\nu]}} \right)\; .\end{aligned}$$
Moreover, let $\displaystyle d=d_\infty \times \prod_{\nu\in S_f} {\textrm}{max}_{0\leq j\leq N_\nu} d_{\nu,j}$ and $\displaystyle e=e_\infty \times \prod_{\nu\in S_f} {\textrm}{max}_{0\leq j\leq N_\nu} e_{\nu,j}$. Then $m_H(H_t(g))$ lies between two constants times $t^e e^{dt}$.
By definition of the size function, the ball $H_t(g)$ is the product for all $\nu$ in $S$ of the balls $(H_\nu)_t(g_\nu)$ in the group $H_\nu$. For each of these balls the two previous theorems give us an equivalent for the volume in $H_\nu$ (all functions are positive so there is no trouble summing equivalent). Now the Haar measure on $H$ is the product of the Haar measures on the $H_\nu$’s. And the formula of the previous theorem is just the product of these equivalences.
The second part directly comes from the first one.
The following lemma is the last step:
Under the hypothesis of theorem \[the:volumeratiolimits\] fix an element $g$ in $G$.
Then there exists a constant $c>1$ such that the ratio $\frac{m_H(H_t(g))}{m_H(H_t)}$ lies between $c^{-1}$ and $c$ for all $t$.
The element $g$ acts continuously on the module ${\textrm}{End}({\bf V}(K_S))$ (recall that in order to define balls in $G$ we fixed some representation of ${{\bf G}}$ in a vector space ${\bf V}$). So there are two constants $A$ and $B$ such that we have for all $h$ in $H$ (recall that $D$ denotes the size function) : $$A.D(h)\leq D(hg) \leq B. D(h)$$ That implies that the set $H_t(g)$ contains $H_{At}$ and is contained in $H_{Bt}$.
But the second part of the previous lemma implies that the ratios $\frac{m_H(H_{At})}{m_H(H_t)}$ and $\frac{m_H(H_{Bt})}{m_H(H_t)}$ are bounded. Hence we have proven the lemma.
We now have the tools to proceed with the proof of theorem \[the:volumeratiolimits\]:
Each finite place leads to a finite partition of the space of parameters in the following way: For $\nu \in S_f$ we have $q_\nu$ the norm of the uniformizer and the integer $N_\nu$ given by the theorem \[the:benoist-oh\]. For $0\leq j\leq N_\nu-1$ we call $I_{\nu,j}$ the set of real numbers $t$ such that $E(\ln_{q_\nu}t)$ is equal to $j$ modulo $N_\nu$. The theorem \[the:benoist-oh\] implies that *on the sets* $I_{\nu,j}$ and for all $g\in G$ we have a asymptotic development of the volume of $(H_\nu)_t(g)$ of the form: $m_H((H_\nu)_t(g)) \sim C_{\nu,j}t^{E_{\nu,j}}e^{D_{\nu,j} t}$.
Now consider the finite partition $I_1,\ldots,I_l$ of ${\mathbb}R$ given by the intersection of all these partitions. Then on a set $I_j$ of this partition and for all $g$ in $G$, the volume $m_H(H_t(g))$ is equivalent to some $C_j(g) t^{E_j(g)} e^{D_j(g)t}$. But we know by the previous lemma that the ratio $\frac{m_H(H_t(g))}{m_H(H_t)}$ is bounded.
At this point we are done: since the ratio is bounded, we have $E_j(g)=E_j(Id)$ and $D_j(g)=D_j(Id)$. Hence the ratio admits a limit (depending on the set $I_j$), namely $\frac{C_j(g)}{C_j(Id)}$.
Polynomial dynamic in homogeneous spaces {#sec:ratner}
========================================
We here recall some facts about polynomial dynamic in $S$-arithmetic groups. The result we need can mainly be found in Tomanov [@Tomanov1]. They are also used in [@gorodnik-oh]. The main difference here - which is only a technical one - is that we need to extend all the results to orbit of polynomial in several variables. This does not change deeply the proof of the theorems. The interested reader may refer to the author’s PhD thesis [@mathese] for details.
Measure on ${{G / {\Gamma}}}$ invariant under the action of a unipotent subgroup
--------------------------------------------------------------------------------
### Measure rigidity in an $S$-arithmetic setting {#sssec:ratner}
We need the rigidity theorem for measures invariant under an unipotent group, often called Ratner’s theorem. For $p$-adic groups, it has been proved by Ratner and Margulis-Tomanov. But in an $S$-arithmetic setting a more precise version can be found in [@Tomanov1].
Accordingly to [@Tomanov1], we define the notion of subgroup of class $\mathcal F$ :
Let ${\bf A}$ be a ${\mathbb}Q$-subgroup of ${{\bf G}}$. Then ${\bf A}$ belongs to the class $\mathcal F$ if and only if ${\bf A}(K_S)$ is the Zariski closure of the group generated by the unipotent elements of ${\bf A}(K_S)$.
Recall from [@Tomanov1] that for a class $\mathcal F$-group ${\bf P}$, the subgroup $P\cap {\Gamma}$ is a lattice in $P$. It implies that the projection of $P$ in ${{G / {\Gamma}}}$ is closed.
We can now state the measure rigidity theorem :
\[the:ratner\] Let ${{\bf G}}$ be a ${\mathbb}Q$-group, ${\Gamma}$ an arithmetic subgroup of $G={{{\bf G}}}(K_S)$ and $U$ a subgroup of $G$ generated by its one-parameter unipotent subgroups.
Then for all probability measure $\mu$ on ${{G / {\Gamma}}}$ which is $U$-invariant and $U$-ergodic, there exist a class $\mathcal F$-subgroup ${\bf P}$ of ${{\bf G}}$ and $P'$ a finite index subgroup of $P={\bf P}(K_S)$ such that the probability $\mu$ is the $P'$-invariant probability on a translate of a $P'$-orbit in ${{G / {\Gamma}}}$.
This theorem allows a complete description of $U$-invariant probability measures.
### The non-ergodic case
Let $U$ be a subgroup of $G$ generated by its one-parameter unipotent subgroups and $\mu$ be a $U$-invariant probability measure on ${{G / {\Gamma}}}$.
For each class $\mathcal F$ subgroup of ${{\bf G}}$, the precedent theorem defines a class of $U$-ergodic probability measures. To understand the decomposition of $\mu$ into ergodic components, we have to define some subsets of $G$ :
Let ${\bf P}$ be a class $\mathcal F$ subgroup of ${{\bf G}}$. Then the sets $X(P,U)$ and $S(P,U)$ are defined in the following way : $$\begin{aligned}
X(P,U) &=& \left\{g \in G {\textrm{ such that }}Ug \subset gP\right\} \\
S(P,U) &=& \bigcup_{{\bf P}'\in
\mathcal F \; , \; {\bf P}'\subset {\bf P}} X(P',U)\end{aligned}$$
We remark that $X(P,U)$ is an algebraic subvariety of $G$.
For each class $\mathcal F$ subgroup ${\bf P}$ of ${{\bf G}}$, let $\mu_{\bf P}$ be the restriction of $\mu$ to $\pi(X(P,U)-S(P,U))$. Then each ergodic component of $\mu_{\bf P}$ is of the form given by the precedent theorem for this group ${\bf P}$. Moreover, since the sets $\pi(X(P,U)-S(P,U))$ are disjoint we get the following decomposition of $\mu$ in a denombrable sum : $$\mu = \sum_{{\bf P}\in \mathcal F} \mu_{{\bf P}} \;.$$
This decomposition enlightens the following fact : in order to understand a measure $U$-invariant, we have to understand the behaviour of trajectories near the variety $\pi(X(P,U)-S(P,U))$. The goal of this section is to get a such a result. But first of all, we will define some useful representations of the group $G$.
A suitable representation
-------------------------
We fix here a class $\mathcal F$-subgroup ${\bf P}$. Chevalley’s theorem [@Borel 5.1] grants the existence of a $K$-representation $\rho_P$ of ${{\bf G}}$ such that ${\bf P}$ is the stabilizer of a line ${\bf D}$ in the space ${\bf V}_P$ of the representation.
We fix a point $v_P$ in ${\bf D}(K)$. Moreover we consider $v_P$ as a point of the $K_S$-module $V_P={\bf V}_P(K_S)$. We now get a function $\eta_P$ from $G$ to $ V$ given by the following formula : $$\eta_P(g)=\rho_P(g).v_P \; .$$
The normalizer ${\bf N(P)}$ of ${\bf P}$ fix the line ${\bf D}$ but not the point $v_P$. So we define ${\bf N_1(P)}$ to be the fixator of the point $v_P$.
The following lemma will be useful, as a link between properties of subset in ${{G / {\Gamma}}}$ and in $V_P$ :
- The set $\eta_P({\Gamma})$ is discrete in $V_P$.
- The set $N_1(P){\Gamma}/{\Gamma}$ is closed in ${{G / {\Gamma}}}$.
First the subgroup ${\bf V}_P(\mathcal O_S)$ is discrete in $V_P={\bf V}_P(K_S)$ and $\rho_P$ is a $K$-representation. So the set $\rho_P({\bf G}(\mathcal O_S)).v_P$ is discrete in $V_P$. Moreover ${\Gamma}$ is supposed to be arithmetic, so $\eta_P({\Gamma})$ is contained in a finite number of translates of $\rho_P({\bf G}(\mathcal O_S)).v_P$. Hence it is a discrete set.
Second, let $g_k=n_k {\gamma}_k$ be a sequence of points in $N_1(P){\Gamma}$ and assume that $g_k$ converges to a point $g$. We want to prove that $g{\Gamma}/{\Gamma}$ belongs to $N_1(P){\Gamma}/{\Gamma}$. We rewrite the definition of $g_k$ : ${\gamma}_k^{-1}=g_k^{-1}n_k$. By definition of $N_1(P)$, we then get $\eta_P({\gamma}_k^{-1})=\eta_P(g_k^{-1})$. We just showed that $\eta_P({\Gamma})$ is discrete. So the sequence ${\gamma}_k$ is stationary equal to a ${\gamma}$ for $k$ large enough. Then $g_k {\gamma}^{-1}$ fixes $v_P$ for $k$ large enough. That is $g_k {\gamma}^{-1}$ belongs to $N_1(P)$. So does its limit and we can conclude : $g$ belongs to $N_1(P){\Gamma}$.
We conclude with a last definition involving the group $U$. The set $X(P,U)$ is $N(P)$-invariant hence $N_1(P)$-invariant by right multiplication and it is a Zariski closed set of $G$. So its image by the function $\eta_P$, which is Zariski-open and surjective on $\eta_P(G)$, is Zariski-closed in $\eta_P(G)$. However there is no reason for it to be Zariski-closed as well in $V_P$. So we define $F(P,U)$ as the Zariski-closure of $\eta_P(X(P,U))$ in $V_P$.
To avoid confusion, let us describe the Zariski topology in $K_S$-modules : a polynomial $Q$ of $K_S[X_1,\ldots,X_n]$ is nothing else than a collection of polynomial $Q_\nu$ for all $\nu$ in $S$. A Zariski-closed subset of a $K_S$-module $\displaystyle M=\prod_{\nu \in S} m_\nu$ is then naturally an intersection of products of Zariski-closed subsets of each $M_\nu$
Behavior of polynomial functions
--------------------------------
We now state a theorem allowing to control polynomial dynamics along the sets $\pi(X(P,U) - S(P,U))$. Let us begin by the definition of a polynomial function in the $K_S$-points $G$ of a $K$-group ${{\bf G}}$ with a faithful linear representation $\rho$: a function $f=(f_\nu)_{\nu\in S}$ from $(K_S)^m$ to $G$ is said polynomial of degree $d$ if for all $\nu \in S$, the matrix entries of $\rho\circ f_\nu$ are all polynomial of degree $d$. The set of functions from $K_S^m$ to $G$ polynomial of degree at most $d$ will be noted $\mathcal P_{d,m} (G)$. Moreover we note $\theta=\bigotimes_{\nu in S} \theta_\nu$ the Haar measure on $K_S$ normalized such that the volume of $K_S/\mathcal O_S$ equals $1$ and $\theta_m=\bigotimes^m \theta$ the induced measure on $K_S^m$.
Recall the definition of $\eta$ from $G$ to some $K$-module $V_G$ given by Chevalley’s theorem. Moreover $F(P,U)$ has been defined as the Zariski closure of $\eta(X(P,U))$ inside $V_G$. Hereafter, we call *cube* in $(K_S)^m$ a product of balls $\prod_{i=1}^m\prod_{\nu\in S} B_{i,\nu}$.
\[the:DM\] Let ${\bf G}$ be a $K$-group, ${\Gamma}$ an arithmetic subgroup of $G= {\bf G}(K_S)$, $U$ be a subgroup of $G$ generated by its one-parameter unipotent subgroups and ${\bf P}$ a class $\mathcal F$-subgroup. Let $C$ be a compact subset of $X(P,U) {\Gamma}/ {\Gamma}$, $d$ and $m$ two integers and $\varepsilon >0$.
Then there exists a compact subset $D$ of $F(P,U)$ such that for all relatively compact neighbourhood $W_0$ of $D$ in $V_G$, there exists a neighbourhood $W$ of $C$ in ${{G / {\Gamma}}}$, such that for all $m$, for all cube $B$ in $(K_S)^m$, and all function $f$ in $\mathcal P_{(d,m)} (G)$ we have :
- either we can find ${\gamma}$ in ${\Gamma}$ such that $\eta(f(B){\gamma}) \subset
W_0$
- or $\theta_m(\left\{t \in B {\textrm{ such that }}(f(t){\Gamma}/ {\Gamma}) \in W \right\}) < \varepsilon
\theta_m(B)$.
In [@Tomanov1] the theorem was not stated for functions in $\mathcal P_{d,m}$ but for one parameters unipotent orbits. However there is no conceptual jump in the proof of the above theorem. Moreover the real cases of this theorem (and of all this section) is well known [@shah]. The interested reader may find more technical details in the author’s PhD thesis [@mathese].
Non-divergence of polynomial orbits
-----------------------------------
We need a last result in order to control the divergence of polynomial orbit. The following theorem is a kind of analog of a result of Eskin-Margulis-Shah [@EMS]. However, we won’t need the whole precision of their result, we may just use a slight adaptation of [@kleinbock-tomanov Theorem 8.4 and 9.1] :
\[the:nondiv\] Let ${\bf G}$ be a $K$-group, ${\Gamma}$ an arithmetic subgroup of $G= {\bf G}(K_S)$. Fix $d$ and $m$ two integers.
Then there are a finite number of parabolic subgroups ${\bf P_k}$ of ${\bf G}$ and their associated Chevalley representations $\rho_k$ in a space $V_k$ with a marked point $v_k \in V_k$ in a line stabilized by ${\bf P_k}$ such that :
for all $\varepsilon>0$ there are a compact $D$ in ${{G / {\Gamma}}}$ and compact subsets $D_k$ in each $V_k$ verifying: for all $f$ in $\mathcal P_{(d,m)} (G)$, for all cube $B$ in $(K_S)^m$, one of the following holds :
1. $ \theta_m(\left\{t \in B {\textrm{ such that }}(f(t){\Gamma}/ {\Gamma}) \not\in D \right\}) < \varepsilon \theta_m(B)$.
2. There is an integer $k$ such that there exists $\gamma \in {\Gamma}$ with : $\rho_k(f(B)\gamma).v_k \subset D_k$.
Some tools: Cartan decomposition, decomposition of measures and representations
===============================================================================
Our proof of theorem \[the:Horbit\] requires some technical tools. The first one is more than classical: the Cartan decomposition in the semisimple part, which we recall to settle some notations. The second one is merely a way to note all the measures (and their translates) we will consider in the sequel, together with some basic lemmas. The third and last one is a lemma on representations of $H$. It is an extension of [@shah Part 5] to our setting.
Cartan Decomposition in $H^{ss}$
--------------------------------
The group $H$ is a semidirect product of a semisimple part $H^{ss}$ and a unipotent one $H^u$. For the semisimple part we have a Cartan decomposition: for all $\nu$ in $S$ such that $H_\nu$ is non-compact we choose a maximal $K_\nu$-split torus $A_\nu$ in $H_\nu$. We choose then a system of positive simple restricted roots $\Phi_\nu$ thus defining the associated sub-semigroup $A^+_\nu$ of $A_\nu$. Then there exists maximal compact subgroups $C_\nu$ and finite sets $D_\nu$ in the normalizer of $A_\nu$ such that the following Cartan decomposition holds: $H_\nu$ is the disjoint union of the double class $C_\nu d a C_\nu$ for $a\in A^+_\nu$ and $d\in D_\nu$. For the existence of these objects we refer to [@Tits]. When $H_\nu$ is compact we just choose $C_\nu=H_\nu$, $A_\nu$ and $D_\nu$ are reduced to the identity.
Let $A^+=\prod_{\nu\in S} A^+_\nu$ and similarly $C$ and $D$ are the products of the $C_\nu$’s and $D_\nu$’s. Let $\Phi$ be the union of the $\Phi_\nu$. For $\alpha \in \Phi_\nu \subset \Phi$ and $a=(a_\nu)_{\nu \in S}$ we define $\alpha(a)=\alpha(a_\nu)$.
Consider a sequence $a_n$ of elements of $A^+$.
A sequence $a_n$ of elements of $A^+$ is *simplified* if for all $\alpha$ in $\Phi$ we have the alternative:
- either $\alpha(a_n)$ is bounded,
- or $\alpha(a_n)$ goes to $+\infty$
Associated to such a simplified sequence, we consider the contracted unipotent subgroup of $H^ss$. $$U^+ = \left\{ h\in H^{ss} {\textrm{ such that }}\lim_{n\to +\infty}a_n^{-1} h a_n =e \right\}\, .$$
We did not assume that a simplified sequence $a_n$ is unbounded. So the group $U^+$ associated may be equal to the trivial group.
Decomposition of measures
-------------------------
The idea is simple: given some measure $\mu$ on the ball $(H^{ss})_t$, we want to define a probability measure on the ball $H_t$ which disintegrates (in the product $H=H^{ss}\rtimes H^u$) on $\mu$ and the Haar measure in the fibers. The notations may seem tedious as we must work at each place in parallel. But it will proove useful later.
The assumptions made on the norm ensure the following : for all $h^{ss}$ in $H_\nu^{ss}$, the set of elements in $H_\nu^u$ such that $h^{ss}h^u$ belongs to $(H_\nu)_t$ is a ball of radius some $l_{[\nu,t]}(h^{ss}$) in $H_\nu^u$ and moreover depends continuously on $h^{ss}$ and $t$. So for all $t$, there is a continuous function $l_{[\nu,t]}$ from $H_\nu^{ss}$ to $\mathbb R^+$ such that : $$(H_\nu)_t=\bigcup_{h\in H_\nu^{ss}} \{h\}\times (H_\nu^u)_{l_{[\nu,t]}(h^{ss})}$$ This in turn translates in terms of measures. We note $m^u_\nu(l)$ the restriction of the Haar measure $m_{H_\nu^u}$ to the ball $(H_\nu^u)_l$. And for measure $\mu_\nu$ in $H_\nu^{ss}$, we may define the measure $m_\nu(\mu_\nu,t)$ by the formula, for all ${\varphi}$ continuous with compact support on $H_\nu$ :
$$\int_{H_\nu} {\varphi}dm_\nu(\mu_\nu,t)=\int_{H^{ss}_\nu} \int_{(H_\nu^u)_{l_{[\nu,t]}(o)}} {\varphi}(ob)dm^u_\nu(l_{[\nu,t]}(o))(b)d\mu_\nu(o)$$
For $\mu=\bigotimes \mu_\nu$ a product measure on $H^{ss}$ of finite total mass and $t$ positive, we note $m(\mu,t)$ the product $\bigotimes_{\nu \in S} m_\nu(\mu_\nu,t)$’s. Eventually we note ${\mathbb}P(\mu,t)$ the renormalized probability measure and ${\textrm}{Supp}(\mu,t)$ its support. Remark that, if $\mu$ proportionnal to the Haar measure of some subgroup $S$ in $H^{ss}$, then $m(\mu,t)$ is proportional to the Haar measure in $S\rtimes H^{u}$ restricted to $(S\rtimes H^u)_t$.
Let us immediatly state two lemmas showing that this probability measures behave well with respect to $\mu$ as soon as the support of $\mu$ does not approach the frontier of $H_t$. First look at translations:
\[lem:limtranslate\] Let $\mu_n$ be a sequence of probability measure on $H^{ss}$ and $t_n$ go to $\infty$. Let $h_n$ go to $Id$ in $H^{ss}$. Assume that the support of $\mu_n$ is included in a ball of radius $H^{ss}_{(1-\varepsilon)t_n}$ for some $\varepsilon>0$.
Then the sequence of (signed) measure ${\mathbb}P(((h_n)_*\mu_n),t_n)-{\mathbb}P(\mu_n,t_n)$ converges to $0$.
The assumption on the supports of $\mu_n$ ensures that the supports of $(h_n)_*\mu_n$ are included in $(H^{ss})_{t_n}$ for $n$ big enough. Moreover (by left-uniform continuity of the norms) we have for every sequence $g_n$ in the support of $\mu$ and for all place $\nu$ (here I forget some indices $\nu$ to keep the formula readable): $$\frac{l_{[\nu,t_n]}(h_ng_n)}{l_{[\nu,t_n]}(g_n)}\xrightarrow{n\to \infty} 1\; .$$ As, eventually, the signed measures $(h_n)_*\mu_n-\mu_n$ go to $0$, the lemma is proven by a straightforward calculus.
The second lemma allows to handle also a sequence of measure $\mu_n$:
\[lem:limsupport\] Let $\mu_n$ be a sequence of probability measures on $H^{ss}$ converging to $\mu$ with all these measures supported in a given compact set and absolutely continuous with respect to some $\lambda$. Let $t_n$ be a sequence of real numbers going to $+\infty$ and $h_n$ a sequence of elements in $H^{ss}$.
Then the sequence of (signed) measure ${\mathbb}P(((h_n)_*\mu_n),t_n)-{\mathbb}P((h_n)_*\mu,t_n)$ converge to $0$.
By hypothesis, the signed measure $\mu_n-\mu$ has a density going to zero in $L^1(\lambda)$. But all these densities are supported inside a compact set. Hence $\mu_n-\mu$ has a total variation going to zero : for all $\epsilon$ there is $n$ such that for all function on $H^{ss}$, we get: $$|\int f d\mu_n -\int f d \mu| \leq \epsilon {\textrm}{max}(|f|)$$
This ensures that its translates under $h_n$ go to zero i.e. that ${\mathbb}P((h_n)_*(\mu_n,t_n))-{\mathbb}P(((h_n)_*\mu),t_n)$ go to $0$.
A lemma on linear representation
--------------------------------
The first equidistribution result we will prove is for projections of probability measures of the form ${\mathbb}P((a_n)_* l,t_n)$ where $l$ is a probability measure on $U^+$ absolutely continuous with respect to the Haar measure. But we need a result on the action of the support $S((a_n)_* l,t_n)$ of this measure: it sends every non-invariant point to $\infty$. For technical reasons, we need this property directionnally in $H^u$, i.e. along $1$-parameter subgroups in $H^u$.
The situation of this section is the following one: let $(a_n)$ be a simplified sequence. Let $\Omega$ be an open and relatively compact subset of $U^+$. Let $(t_n)$ be a sequence of real numbers going to $\infty$ such that the sets $a_n\Omega$ are included in balls $H^{ss}_{t_n}$. Let $N^{ss}$ be the smallest normal subgroup of $H$ such that the projection of $a_n$ is bounded in $H/N^{ss}$.
\[lem:representation\] Let $\rho=(\rho_\nu)_{\nu\in S}$ be a $K_S$-representation of $H$ in a finite dimensional $K_S$-module $V=\prod V_\nu$. Let $O$ be a $1$-parameter subgroup of $H^u$, $O_n$ be the set $\{o\in \Omega\times O {\textrm{ such that }}{\textrm}{for some }\omega\in \Omega{\textrm}{, }D(a_n o)\leq t_n\}$. Let $N_O$ be the smallest subgroup of $H$ such that $a_nO_n$ stay in a compact in $H/N_O$.
Let $\Lambda$ be a discret subset of $V$ with no $N_O$-invariant points and $v_n$ a sequence of elements of $\Lambda$.
Then the sequence of sets $\rho (a_n O_n) v_n$ is not contained in any compact subset of $V$.
This whole subsection will be the proof of this lemma.
We split this proof in two cases : whether the sequence $a_n$ is bounded or not.
*Case 1 : $a_n$ is bounded*
We may assume that all $a_n$ equal $1$. Then $U^+$ is trivial, the $O_n$ is the ball $D(o)\leq t_n$ in $O$ and $N_O$ is the group $O$. As $t_n$ go to $\infty$, we may extract a increasing subsequence of balls covering $O$. If $v_n$ is not bounded, as $Id$ belongs to $O_n$, then the lemma is proven. If not, as $\Lambda$ is discrete, we may assume that $v_n$ is constantly equal to some $v$ which is not $N_O$-invariant. Now the exponential function composed with $g\mapsto\rho(g)v$ gives us a polynomial function from the Lie algebra of $O$ to $V$, and $\rho(O)v$ is the image of this polynomial function. That means that this function is constant or unbounded. As it is not constant, it is unbounded, proving the lemma in this case.
*Case 2 : $a_n$ is not bounded*
In this situation, the action of $a_n$ and $U^+$ alone send non-invariant points to $\infty$ (remark that $\Omega$ is included in $O_n$ by definition).
First of all, let $V^{N^{ss}}$ be the $N^{ss}$-invariant sub-module of $V$ and $W$ an $N^{ss}$-invariant complement. Write $v_n=v_n^{N^{ss}}+w_n$. If $w_n$ goes to $0$, by discreteness of $\Lambda$, $v_n^N$ goes to $\infty$. Let $C$ be a compact of $G$ such that $a_n O_n$ is included in $C N_O$. Then, by definition of $U^+$ and semisimplicity of $H^{ss}$, the sets $a_n U^+$ are included in $C N^{ss}$. And for any $\omega\in \Omega$, the sequence $\rho(a_n \omega) v_n=\rho(a_n \omega)(v_n^N)+\rho(a_n\omega)w_n$ belongs to $\rho(C)v_n^N+W$. Hence this sequence goes to $\infty$, proving the lemma in this case.
So we may assume that $w_n$ does not go to zero. Up to a renormalization and an extraction, we assume that $w_n$ converges to some non-zero element $w\in W$. It is enough to prove that the sets $\rho (a_n \Omega) w$ leave every compact of $V$. Making this reduction we loose the discreteness hypothesis on $\Lambda$ but we will not need it anymore.
We now prove the lemma by contradiction: suppose that the above sets stay in some compact. We prove first that $w$ is $N^{ss}$-invariant and then $N$-invariant.
The first step is to show that we may assume that $w$ is $U^+$-invariant: let $V^+$ be the module of $U^+$-invariant points and $V^-$ its $a_n$-invariant complement. Note $p^+$ the projection on $V^+$ in the direction $V^-$. We have the following
Let $\rho=(\rho_\nu)_{\nu\in S}$ be a $K_S$-representation of $H$ in a finite dimensional $K_S$-module $V=\prod V_\nu$. Let $U$ be a non-trivial unipotent subgroup, and $\Omega$ an open subset of $U$.
Then the set $\rho(\Omega) w$ is not contained in any complement of the submodule $V^U$ of $U$-invariant points.
Once again we prove it by contradiction: suppose $\rho(\Omega) w$ generates some submodule $V'$ in direct sum with $V^U$. And let $\omega_1, \ldots , \,\omega_k$ be elements of $\Omega$ such that the $\rho(\omega_i)w$ generate $V'$. Then there is an neighborhood $\Omega'$ of the identity in $U$ such that all the $\Omega' \omega_i$ are included in $\Omega$.
And $V'$ is $\Omega'$ invariant. So it is invariant by the Zariski closure of $\Omega'$ i.e. by $U$. The Lie-Kolchin theorem implies that there is a non-zero $U$-invariant element in the $U$-invariant module $V'$ (to be very precise, you have to apply the Lie-Kolchin theorem at each place, restricting the representation in the obvious way). This is the contradiction: $V'$ cannot be in direct sum with $V^U$.
So, there is some $\omega \in \Omega$ such that $p^+(\rho(\omega)w)$ is not zero. But we know that $\rho(a_n\omega)w$ is bounded. Hence $\rho(a_n) p^+(\rho(\omega)w)$ is bounded. Let us show that it implies that $N^{ss}$ is contained in the kernel of the representation:
Let $v$ be a $U^+$-invariant and non-zero point of $V$ such that $\rho(a_n)v$ is bounded. Then $N^{ss}$ is contained in the kernel of the representation.
We may assume that at each place $\rho_\nu$ is an irreductible representation. First of all, let $W$ be the sub-$K_S$-module of $V$ containing all the vectors $w$ such that $\rho(a_n)w$ is bounded. Consider $P^-$ the opposite parabolic subgroup in $H^{ss}$: $$P^-=\left\{h\in H{\textrm{ such that }}a_n ha_n^{-1}{\textrm}{ remains bounded}\right\}\; .$$ Then it is clear that $\rho(P^-)v$ is included in $W$. By $U^+$ invariance of $v$, we even get that $\rho(P^-U^+)v$ is included in $W$. But $P^-U^+$ is open in $H$; so Zariski-dense. We deduce that $\rho(H)v$ is included in $W$ and by irreducibility that $W=V$.
Let us now prove that all the element of $V$ are $U^+$-invariant. We just have to prove it on eigenvectors for the action of $a_n$ ($V$ is the sum of the eigenspaces for this action). Remind that, as $a_n$ has determinant one and all the vectors have a bounded orbit under the action of $a_n$, all the eigenvalues of this action are of modulus $1$. So let $v'$ be in $V$ with $\rho(a_n)v'=\lambda_n v'$ and $\omega$ be some element of $U^+$. Fix an open neighborhood of the identity $\Omega$ in $U^+$. Then by definition there is some integer $i$ such that $a_i^{-1} \omega a_i$ belongs to $\Omega$. Hence $\rho(\omega)v'$ belongs to $\rho(a_i)(\rho(\Omega)\lambda_i^{-1}v')$. But the latter is included in some compact $B$ independent of $i$ because we have seen that all elements of $V$ have bounded orbit in $V$ and the sets $\rho(\Omega)\lambda_i^{-1}v'$ are contained in some compact. So $\rho(U^+) v'$ is included in $B$. But $U^+$ is an unipotent subgroup hence $\rho(U^+)v'$ is the whole image of a polynomial function. It can be bounded if and only if it is constant. Hence $v'$ is $U^+$-invariant.
We have just proven that every element of $V$ is $U^+$-invariant. Hence the kernel of the representation contains the normal subgroup generated by $U^+$, hence contains $N^{ss}$.
The situation is now simple: we may forget about the semisimple part because it acts trivially. And we just have an element $w$ of $V$ such that $\rho(O_n)w$ is bounded. Now for each $n$, the projection of $O_n$ in $O$ is a ball. If the $O_n$ are bounded, then $N_O=N^{ss}$ is semisimple and we are done. If not we may as in case 1 assume that the projections of $O_n$ on $O$ are increasing balls and $\rho(O_n)w$ may be bounded only if $w$ is $O$-invariant. Here $N_O$ is the subgroup generated by $N^{ss}$ and $O$ and $w$ is $N_O$-invariant.
In both cases we found the contradiction: $w$ is $N_O$-invariant. Hence the lemma \[lem:representation\] is proved.
Equidistribution of dense orbits {#section:equidistribution}
================================
The aim of this section is to prove the theorem \[the:Horbit\]. First recall the theorem:
Let $(G,H,{\Gamma})$ be a triple under study, $D$ a size function and $H_t$ the induced family of balls in $H$. Assume that every dominant subgroup $H'$ of $H$ has dense orbit in ${{G / {\Gamma}}}$.
Then the orbit of $H$ becomes equidistributed in ${{G / {\Gamma}}}$ with respect to $m_{{G / {\Gamma}}}$: $${\textrm}{For all }{\varphi}\in \mathcal C_c({{G / {\Gamma}}}){\textrm}{, }\frac{1}{m_H(H_t)}\int_{H_t}{\varphi}(\pi(h))dm_H(h)\xrightarrow{t\to +\infty} \int_{{G / {\Gamma}}}{\varphi}dm_{{G / {\Gamma}}}\;.$$
We use in this section the rigidity of the dynamic of unipotent flows reviewed in the previous section. The article of Shah [@shah] is the main source of inspiration for this section.
Equidistribution over unipotent subgroups
-----------------------------------------
The first equidistribution result is the following one: if $a_n$ is simplified and $l$ a probability measure on $U^+$, then the projections of ${\mathbb}P((a_n)_*l,t_n)$ in ${{G / {\Gamma}}}$ become equidistributed with respect to the Haar measure $m_{{G / {\Gamma}}}$ if its support ${\textrm}{Supp}((a_n)_*l,t_n)$ does not stay close to a group with closed orbit:
\[pro:unipotent\] Let $(G,H,{\Gamma})$ be a triple under study with a size function $D$. Let $t_n$ be a sequence of positive number going to $+\infty$ and $(a_n)$ be a simplified sequence in $A^+$, $U^+$ the contracted unipotent subgroup of $H^{ss}$ and $l$ a measure on $U^+$ compactly supported and absolutely continuous with respect to the Haar measure. Let $N$ be the smallest normal subgroup of $H$ such that the projections of ${\textrm}{Supp}((a_n)_*l,t_n)$ remain in a compact subset in $H/N$. Assume eventually that $N{\Gamma}$ is dense in $G$.
Then we have the following limit in the space of probability on ${{G / {\Gamma}}}$: $$\lim_{n\to\infty} \pi_*({\mathbb}P((a_n)_*l,t_n)) = m_{{G / {\Gamma}}}\; ,$$ that is, for every function ${\varphi}$ continuous with compact support on ${{G / {\Gamma}}}$, we have: $$\int_{H} {\varphi}(x{\Gamma}/{\Gamma}) d{\mathbb}P((a_n)_*l,t_n)(x) \xrightarrow{n\to\infty}\int_{{G / {\Gamma}}}{\varphi}dm_{{G / {\Gamma}}}\; .$$
The proof of this proposition is the core of the theorem \[the:Horbit\]. We will use here the theory of polynomial orbits and Ratner’s theorem exposed above, together with lemma \[lem:representation\]. The derivation of theorem \[the:Horbit\] from this proposition won’t present any major difficulty.
We first show that any weak limit of the sequence studied in the proposition is a probability invariant by some unipotent subgroup. Then we will use the theory developed and the previous lemma to show that it can only be the Haar probability measure on ${{G / {\Gamma}}}$.
So consider the sequence of probability measures $\pi_*({\mathbb}P((a_n)_*l,t_n))$ and $\mu$ a weak limit. The first step will be to prove that $\mu$ is a probability measure on ${{G / {\Gamma}}}$. The second one will be an invariance of $\mu$ by some unipotent subgroup, thus allowing the use of the tools reviewed. Eventually we will prove that this $\mu$ is the Haar measure on ${{G / {\Gamma}}}$, proving the proposition.
The group $U^+$ is a unipotent subgroup of $G$ of dimension say $m$. Hence the exponential map form its Lie algebra to it is a polynomial map. Up to adding variables, we have a polynomial parametrisation $exp_1$ from $K_S^m$ to $U^+$. The measure $l$ is absolutely continuous with respect to $(exp_+)_*(\theta_m)$.
In the same way, look at the projections of ${\textrm}{Supp}((a_n)_*l,t_n)$ to $H^u$. They are product of balls $(H_\nu^u)_{r_n(\nu)}$ of radius some $r_n(\nu)$. Moreover we have a polynomial parametrisation $exp_2$ from $K_S^r$ to $H^u$ which verifies that $(exp_2)_*(\theta_r)=m_{H^u}$. And the measure $(a_n^{-1})_*[{\mathbb}P((a_n)_*l,t_n)]$ is absolutely continuous with respect to the image under $exp_1\times exp_2$ of the Haar measure $\theta_m\otimes \theta_r$ on $K_S^m\times K_S^r$. We even get a uniformity result on the absolute continuity :
\[lem:abscon\] Let $C$ be a positive real number. There exist a cube $B$ in $K_S^m$, a sequence of subsets $B_n$ in $K_S^r$ and an $\varepsilon>0$ such that for all measurable subset $E$ in ${{G / {\Gamma}}}$ we have :\
$\textrm{If } \frac{1}{\theta_m(B)\theta_r(B_n)}\pi'_*\left((exp_1)_*(\theta_m)\otimes (exp_2)_*(\theta_r)\right)(E)\leq \varepsilon$
then $\pi'_*((a_n^{-1})_*[{\mathbb}P((a_n)_*l,t_n)])(E)\leq \frac{C}{2}$.
We choose $B$ to be a cube in $K_S^m$ such that $\Omega$ is included in $exp_1(B)$ and $B_n$ to be the preimage of $\prod_\nu H^u_{r_n(\nu)}$ under $exp_2$.
We claim that for a set of positive measure of element $\omega$ in $\Omega$, the ball $\{u\in H^u{\textrm{ such that }}D(a_n \omega u)\leq t_n\}$ contains the product of balls of radius $\frac{r_n(\nu)}{2}$. This is a direct consequence of the fact that $\Omega$ is a compact and the hypothesis made on $D$; namely the so-called orthogonality between the semisimple part and the unipotent part.
Hence in the set $exp_1(B)\times exp_2(B_n)$, the set $a_n^{-1}{\textrm}{Supp}((a_n)_*l,t_n)$ is of positive and bounded from $0$ relative measure, i.e for some $C>0$, for all n: $$\frac{\left((exp_1)_*(\theta_m)\otimes (exp_2)_*(\theta_r)\right)[a_n^{-1}{\textrm}{Supp}((a_n)_*l,t_n)]}{\theta_m(B)\theta_r(B_n)}\geq C$$
As $(a_n^{-1})_*[{\mathbb}P((a_n)_*l,t_n)]$ is the restriction of the measure $l\otimes(exp_2)_*(\theta_r)$ to its support ${\textrm}{Supp}((a_n)_*l,t_n)$ renormalized to be a probability measure, and $l$ is absolutely continuous with respect to $(exp_1)_*(\theta_m)$ the conclusion of the lemma follows.
[**Step 1:**]{}
The measure $\mu$ is a probability measure on ${{G / {\Gamma}}}$.
This result is quite classical, at least in the setting of Lie groups. We will of course use the theorem \[the:nondiv\]. Moreover it is enough to prove it for the sequence of measures $(a_n)_*(exp_1)_*(\theta_m)\otimes (exp_2)_*(\theta_r)$ restricted to $B\times B_n$ thanks to the previous lemma.
Consider the functions $\Theta_n(t,s)=a_n exp_1(t)exp_2(s)$. They are polynomials of fixed degree. Fix some $0<\epsilon<1$. We want to find a compact set $D$ in ${{G / {\Gamma}}}$ such that the images of all (but a finite number) the function $\Theta_n$ are included inside this compact except for a set of relative measure at most $\epsilon$.
We claim now that the subset $D$ given to us by theorem \[the:nondiv\] is convenient. The strategy seems clear : apply theorem \[the:nondiv\] and then show that the second part of the alternative is impossible for all but finitely many $n$. But a difficulty appears : the sets $B\times B_n$ on which we look at the functions $\Theta_n$ are not cube. We overcome this difficulty in a somewhat artificial way : we restrict our attention $1$-parameter subgroups $O$ of $H^u$ instead of the whole $H^u$ (exactly those subgroups which appear in lemma \[lem:representation\]). We are then able to recombine these $1$-dimensional estimates to get the wanted result.
So consider $O$ a $1$-parameter subgroup in $H^u$ and $L$ its Lie algebra in the Lie algebra of $H^u$. $L$ is a line, and (up to the choice of a basis vector in $L$) for $n$ big enough, $L_n=L\cup B_n$ is a ball in $K_S$.
Hence we may apply theorem \[the:nondiv\] to the functions $\Theta_n$ restricted to $B\times L_n$ which is a cube. And we know, using lemma \[lem:representation\], that the action of $a_n exp_1(B)\times exp_2(L_n)$ sends the points $v_k$ outside of the compact $D_k$ unless it is invariant by the group $N_O$.
So for $n$ big enough, either all the points $v_k$ appearing in theorem \[the:nondiv\] are invariant under $N^{ss}$ and $O$ or the whole cube $B\times L_n$ but a set of relative measure at most $\epsilon$ is mapped inside $D$. Now the first part of the alternative means that the subgroup $N_O$ is included in the intersection of the parabolic subgroups $P_k$ and as a corollary its orbit in ${{G / {\Gamma}}}$ is closed. So this may happen only along a negligeable set of directions $O$ : as any set $B$ of positive measure of directions $O$ generates $H^u$ itself, the $N_O$’s for $O$ in $B$ generates the subgroup $N$ (smallest normal subgroup such that ${\textrm}{Supp}((a_n)_*l,t_n)$ is bounded in $H/N$). And we assumed the group $N$ has a dense orbit in ${{G / {\Gamma}}}$.
So for $n$ big enough, the total mass of points $(t,s)\in B\times L_n$ such that $\Theta_n(t,s)$ does not belong to $D$ does not exceed $2 \epsilon$ times the mass of $B\times L_n$.
[**Step 2:**]{}
The probability measure $\mu$ is left-invariant by some unipotent subgroup $Z$.
We also handle differently the cases according to the behaviour of $a_n$:
*Case 1 : $a_n$ is bounded*
We may assume that $a_n$ is constantly equal to $Id$ and $U^+$ is restricted to $\{Id\}$. Hence the set $U_n={\textrm}{Supp}(Id,t_n)$ is an increasing sequence of balls of radius $t_n$ in $H^u$ and the probability measure ${\mathbb}P(\omega, t_n)$ is the Haar measure of $H^u$ restricted to $U_n$.
Let $Z$ be the center of the unipotent group $H^u$, $\mathfrak z$ its Lie algebra. Then the polynomial map $P$ given by the composition of the representation choosen to define the norm and the exponential map from the Lie algebra $\mathfrak h^u$ to $H^u$ is proper and verifies for all $z\in \mathfrak z$ and $u\in \mathfrak h^u$: $$P(z+u)=P(z)P(u)$$
Hence, for a fixed $z\in \mathfrak z$, the “norm” $D(\exp(z)\exp(u))$ is equivalent to $D(\exp(u))$, as $P(z+u)=P(u)+O(P'(u))$. This proves that the ratio $\frac{m_{H^u}({\textrm}{exp}(z)U_n\cap U_n)}{m_{H^u}(U_n)}$ tends to $1$, which means that $\mu$ is left-invariant by $Z$.
*Case 2 : $a_n$ is not bounded*
Then, by construction $a_n$ has a contracting action on $U^+$. Moreover $u^+_*\mu$ is the limit of $u^+_*\pi_*({\mathbb}P((a_n)_*l,t_n))$. And the last one may be rewritten $\pi_*\left({\mathbb}P[(a_n)_*(a_n^{-1}u^+a_n)_*l,t_n]\right)$. As $a_n^{-1}u^+a_n$ goes to $Id$, lemma \[lem:limsupport\] implies that $\mu$ is $U^+$ invariant.
Hence we may use all the tools presented: there exists a class $\mathcal F$-subgroup ${\bf P}$ of ${{\bf G}}$ such that $\mu(X(P,Z))$ is positive. We want to show that ${\bf P}={{\bf G}}$. This is the third and final step:
[**Step 3:**]{}
Any class $\mathcal F$-subgroup ${\bf P}$ such that $\mu(X(P,Z))>0$ is the group $G$.
We will naturally use the theorem \[the:DM\]. Fix a compact $C$ of $X(P,V)$ of positive measure.
Using lemma \[lem:abscon\], we get an $\varepsilon$ such that for all measurable subset $E$ in $C$ we have :
$\textrm{If } \frac{1}{\theta_m(B)\theta_r(B_n)}\pi_*\left((exp_1)_*(\theta_m)\otimes (exp_2)_*(\theta_r)\right)(E)\leq \varepsilon$
$\textrm{ then }\pi_*\left((a_n^{-1})_*[{\mathbb}P((a_n)_*l,t_n)]\right)(E)\leq \frac{\mu(C)}{2}\;$
Once again we will apply the theorem \[the:DM\] directionally to the function $\Theta_n(t,s)=a_n exp_1(t)exp_2(s)$, restricted to some $B\times L_n$ ($L_n$ being a ball in the Lie algebra $L$ of a $1$-parameter subgroup $O$ in $H^u$), to the compact $C$ and the $\varepsilon$ just defined. Note $\theta$ a normalized Haar measure on $L$. Then there exists a compact $D$ of $F(P,V)$ such that for all neighborhood $W_0$ of D there exists a neighborhood $W$ of $C$ such that for all $n$ we get the alternative:
- There exists $\gamma_n$ in $\Gamma$ such that $\eta(\Theta_n(B\times L_n) \gamma_n)\subset W_0$
- $\theta_m\otimes \theta(\{t,s\in B\times L_n {\textrm}{ such that } \Theta_n(t,s){\Gamma}/{\Gamma}\in W\})<\varepsilon \theta_m\otimes \theta(B\times L_n)$
Now fix any neighborhood $W_0$ of $D$ and suppose we are in the second case of the previous alternative. Then by construction, we have: $$\frac{1}{\theta_m\otimes \theta(B\times B_n)}\pi_*((exp_1)_*(\theta_m)\otimes (exp_2)_*(\theta))(a_n^{-1})(W)<\varepsilon\; .$$
But we have $\pi_*({\mathbb}P((a_n)_*l,t_n))(W)>\frac{\mu(C)}{2}$ as $W$ contains $C$ and the measures $\pi_*({\mathbb}P((a_n)_*l,t_n))$ converges to $\mu$. By definition of $\varepsilon$, there is a set of $1$-parameter subgroups $O$ of positive measure for which the previous inequality does not hold for all $n$ big enough.
Now we use the lemma \[lem:representation\]. Consider the representation $\rho$ of $G$ in the $K$ module $V$ associated to ${\bf P}$ via Chevalley’s theorem. And restrict it to a representation of $H$ in $V$. Let $\Lambda$ be the discrete set $\eta(h_0\Gamma)$. We want to show that one of this point is invariant under the action of the group $N_O$ (see lemma \[lem:representation\]). But this is a direct application of the lemma \[lem:representation\]: the sets $\rho(a_nB\times L_n)\eta(\gamma_n)=\eta(\Theta_n(B\times B_n))$ are included in $W_0$ hence bounded. The conclusion of lemma \[lem:representation\] being violated, the hypothesis is not fulfilled: one of the points $\gamma_n$ is $N_O$-invariant.
And now we may intervert the quantifiers without loosing everything : there is an integer $n$ such that $\gamma_n$ is invariant under $N_O$ for a set of positive measure of directions $O$. As previously noted, this point is $N$-invariant ($N$ being the smallest normal subgroup of $H$ containing all the sets ${\textrm}{Supp}((a_n)_*l,t_n)$ in a compact neighbourhood).
We have done most of the work. Let us conclude, using notations and results of the section \[sec:ratner\]: $N$ is included in $\gamma_n^{-1} N_1(P) \gamma_n$. So the projection of $N_1(P)$ in ${{G / {\Gamma}}}$ contains a translate of the projection of $N$. But the latter is dense and the first one is closed : $N_1(P)$ projects onto ${{G / {\Gamma}}}$ hence is Zariski-dense in $G$. We conclude that $N_1(P)=G$. That means that ${\bf P}$ is a normal subgroup of ${{\bf G}}$, so is equal to ${{\bf G}}$ by simplicity.
To conclude the proof of the proposition \[pro:unipotent\], note that the rigidity theorem \[the:ratner\] implies that $\mu$ is invariant under some finite index subgroup $P$ of $G$. As $G$ is a simply connected group, $G$ itself is the unique finite index subgroup of $G$. Eventually $\mu$ is $G$-invariant so is the Haar probability measure on ${{G / {\Gamma}}}$.
Equidistribution of spheres
---------------------------
We need a last step before proving theorem \[the:Horbit\] : that is a proposition very similar to proposition \[pro:unipotent\] but more adapted to Cartan decomposition in the group $H^{ss}$. Recall that, at the begining of section \[section:equidistribution\], we defined the Cartan decomposition $H^{ss}=C D A^+ C$. The following proposition holds (compare with [@shah Corollary 1.2]):
\[pro:spheres\] Let $(G,H,{\Gamma})$ be a triple under study. Let $(h_n)$ be a sequence in $H^{ss}$, $t_n$ a sequence of positive number going to $+\infty$ and $\mu$ a probability measure on $C$ absolutely continuous with respect to the Haar probability measure on $C$. We assume that for all $c$ in the support of $\mu$, we have $D(h_n c)\leq (1-\varepsilon)t_n$ for some $\varepsilon>0$. Let $N$ be the smallest normal subgroup of $H$ such that the projection of the support of ${\mathbb}P((a_n)_*\mu, t_n)$ is bounded in $H/N$. Assume that ${\Gamma}N$ is dense in $G$.
Then the projection of probability measures ${\mathbb}P((a_n)_*\mu, t_n)$ in ${{G / {\Gamma}}}$ becomes equidistributed: $$\lim_{n\to\infty} \pi_*({\mathbb}P((a_n)_*\mu, t_n)) = m_{{G / {\Gamma}}}\; ,$$ that is, for every function ${\varphi}$ continuous with compact support on ${{G / {\Gamma}}}$, we have: $$\int_H {\varphi}(h {\Gamma}/{\Gamma}) d{\mathbb}P((a_n)_*\mu, t_n) \xrightarrow{n\to\infty}\int_{{G / {\Gamma}}}{\varphi}dm_{{G / {\Gamma}}}\; .$$
We will prove that any weak limit of this sequence of probability measure is the Haar measure $m_{{G / {\Gamma}}}$.
First of all, we may assume that $h_n$ is an element of $A^+$. Indeed, using Cartan decomposition, we write $h_n=c^1_n d_n a_n c^2_n$, and, up to an extraction, the three sequences $c^1_n$, $c^2_n$ and $d_n$ converge to respectively $c^1$, $c^2$ and $d$. Now, let $\mu'$ be the pushforward of $\mu$ under $c^2$ : $\mu'(c^2A)=\mu(A)$. The lemma \[lem:limsupport\] guarantees that the equidistribution of $\pi_*({\mathbb}P((h_n)_*\mu, t_n))$ is equivalent to the one of $\pi_*({\mathbb}P((a_n)_*\mu', t_n))$. And by construction, $N$ is also the smallest normal subgroup such that the projection of ${\textrm}{Supp}((a_n)_*\mu', t_n)$ is bounded in $H/N$.
Moreover, up to another extraction, we assume that $a_n$ is simplified. Consider now the opposite parabolic subgroup $P^-$ to $U^+$ in $H^{ss}$ and $U^-$ the expanded unipotent subgroup : $$U^- = \left\{ h\in H^{ss} {\textrm{ such that }}\lim_{n\to +\infty}a_n h a_n^{-1} =e \right\}\, .$$ Every neighbourhood of an element $c$ in $C$, contains a neighbourhood which is homeomorphic to a neighbourhood of $Id$ in $P^-\times U^+$ via the application $(p^-,u^+)\mapsto p^-u^+c$. We may split the support of $\mu$ in such sets (up to a negligible set), or in other words, we assume $\mu$ to be supported inside an open set homeomorphic to an open set $\Omega^-\times \Omega^+$ in $P^-\times U^+$. We furthermore assume that both $\Omega^-$ and $\Omega^+$ are product set of the form $\displaystyle \prod_{\nu\in S}\Omega_\nu$. Moreover at the archimedean places, we may “thicken” a little bit $\mu$ to construct a measure absolutely continuous with respect to $m_{H^{ss}}$ : let $\lambda$ be a probability measure on a sufficently small neighbourhood $O$ of $Id$ in $U^-_\infty$ (the archimedean part of $U^-$) absolutely continuous with respect to the Haar measure on $U^-_\infty$. Then $\lambda \otimes \mu$ is absolutely continuous with respect to the Haar measure on $H^{ss}$ (see [@shah Page 15]).
Looking at the action of $a_n$ on $U^-$ and using lemma \[lem:limtranslate\] it is clear that for every function $f$ continuous with compact support in ${{G / {\Gamma}}}$, the integrals of $f$ for the both measures $\pi_*({\mathbb}P((a_n)_*\lambda \otimes\mu,t_n))$ and $\pi_*({\mathbb}P((a_n)_*\mu,t_n))$ are equivalent as $n$ go to $\infty$:\
$\,$\
$
|\int_{{G / {\Gamma}}}f d\pi_*({\mathbb}P((a_n)_*\lambda \otimes\mu,t_n))- \int_{U^+\times H^u} f(x) d\pi_*({\mathbb}P((a_n)_*\mu,t_n))|$\
$
\leq \int_{U^-}|\int_H f(x) d\pi_*\left(({\mathbb}P((a_n o a_n^{-1})a_n)_*\mu,t_n)-({\mathbb}P((a_n o a_n^{-1})a_n)_*\mu,t_n)\right)(x)|d\lambda(o)$\
$$\begin{aligned}
\label{eqn:compare}
\xrightarrow{n\to\infty} 0\end{aligned}$$
The limit is obtained using $a_n o a_n^{-1}\xrightarrow{n\to \infty} Id$, lemma \[lem:limtranslate\] and the dominated convergence theorem.
We work now with $\lambda\otimes \mu$. Remark that, at non-archimedean places, we do not have to modify $\mu$, as maximal compact subgroups are also open.
Now, using [@shah Proposition 6.1], we may decompose this probability measure $\lambda \otimes\mu$ in the product $\Omega^-\times \Omega^+$ : there are a probability measure $\nu^-$ on $\Omega^-$ and for almost all $x$ in $\Omega^-$, a probability measure $\nu^+_x$ on $\Omega^+$ such that :
- $\nu^-$ and all the $\nu^+_\omega$ are absolutely continuous with respect to the Haar measure on $P^-$ and $U^+$ respectively.
- for all ${\varphi}$ continuous with compact support in $H^{ss}$, we have $$\int_{H^{ss}} {\varphi}d(\lambda\otimes \mu)=\int_{\Omega^-}\int_{\Omega^+} {\varphi}(xy) d\nu^+_x(y)d \nu^-(x)\, .$$
Consider now a function $f$ continuous with compact support in ${{G / {\Gamma}}}$. We have: $$\int_{{G / {\Gamma}}}f d\pi_*({\mathbb}P((a_n)_*\lambda \otimes\mu,t_n)) = \int_{\Omega^-}\int_{\Omega^+\times H^u} f(y{\Gamma}/{\Gamma}) d{\mathbb}P((a_nx)_*\nu^+_x, t_n)(y)d\nu^-(x)$$
So the last difficulty that remains is to compare the two probability measures ${\mathbb}P((a_nx)_*\nu^+_x, t_n)$ and ${\mathbb}P((a_n)_*\nu^+_x, t_n)$ : if we prove that they are sufficiently close, then we may use the proposition \[pro:unipotent\] to conclude that the limit is the Haar probability measure $m_{{G / {\Gamma}}}$. But under conjugacy by $a_n$, the elements in $P^-$ remains bounded. So, if we choose the support of $\lambda$ small enough, the lemma \[lem:limtranslate\] ensures that the two measures ${\mathbb}P((a_nx)_*\nu^+_x, t_n)={\mathbb}P((a_n x a_n^{-1})_*(a_n)_*\nu^+_x, t_n)$ and ${\mathbb}P((a_n)_*\nu^+_x, t_n)$ are arbitrarily closed.
Fix $\epsilon>0$ and choose the support $O$ of $\lambda$ such that we have : for all $x\in O$, all $n$ $$\left|\int_{\Omega^+\times H^u} f(y{\Gamma}/{\Gamma}) d{\mathbb}P((a_nx)_*\nu^+_x, t_n)(y)-\int_{\Omega^+\times H^u} f(y{\Gamma}/{\Gamma}) d{\mathbb}P((a_n)_*\nu^+_x, t_n)(y)\right|\leq \epsilon$$
Then, we have : $$\left| \int_{{G / {\Gamma}}}f d\pi_*({\mathbb}P((a_n)_*\lambda \otimes\mu,t_n)) - \int_{\Omega^-}\int_{\Omega^+\times H^u} f(y{\Gamma}/{\Gamma}) d{\mathbb}P((a_n)_*\nu^+_x, t_n)(y)d\nu^-(x)\right|\leq \epsilon$$
Now, the proposition \[pro:unipotent\] states that for all $x$, we have the limit: $$\int_{\Omega^+\times H^u} f(y{\Gamma}/{\Gamma}) d{\mathbb}P((a_n)_*\nu^+_x, t_n)(y)\xrightarrow{n\to\infty}\int_{{G / {\Gamma}}}f dm_{{G / {\Gamma}}}$$ We conclude applying the dominated convergence theorem: $$\left|\int_{{G / {\Gamma}}}f d\pi_*({\mathbb}P((a_n)_*\lambda \otimes\mu,t_n)) - \int_{{{G / {\Gamma}}}}fdm_{{G / {\Gamma}}}\right|\leq \epsilon$$
So the previous inequality together with \[eqn:compare\] leads to (for $n$ big enough): $$|\int_{{G / {\Gamma}}}f d\pi_*({\mathbb}P((a_n)_*\mu,t_n)) - \int_{{{G / {\Gamma}}}}fdm_{{G / {\Gamma}}}|\leq 2\epsilon$$ As this is true for arbitrary $\epsilon$, we have finally obtained the desired result : $$\int_{{G / {\Gamma}}}f d\pi_*({\mathbb}P((a_n)_*\mu,t_n)) \xrightarrow{n\to \infty} \int_{{{G / {\Gamma}}}}fdm_{{G / {\Gamma}}}\, .$$
The proposition is proven
Thanks to this proposition, we are able to define a subset of large relative volume in $H$ such that, basically, as soon as the support of ${\mathbb}P(h_*m_C,t)$ hits this subset, the projection of this measure in ${{G / {\Gamma}}}$ is closed to the Haar probability measure (once again, the statement is a bit more complicated than what I just explained but this will be the exact result needed):
\[coro:spheres\] Let $(G,H,{\Gamma})$ be a triple under study, together with a size function $D$. Assume that every dominant normal subgroup of $H$ has a dense orbit in ${{G / {\Gamma}}}$. Fix $\varepsilon>0$, $f$ a continuous function with compact support in ${{G / {\Gamma}}}$, and $O$ some open subset in $C$.Then there is a finite number of non-dominant normal subgroups $N_1$, $\ldots$, $N_k$ of $H$, a compact subset $B$ in $H$ such that:
For $h$ in $H$, $O' \subset C$ containing $O$ with $\mu$ the probability measure on $O'$ proportional to the Haar measure on $C$ and $t>0$ verifying for all $o$ in $O$, $D(go)\leq \frac{t}{1+\varepsilon}$, we have:
If the support of ${\mathbb}P(h_*\mu,t)$ is not included in any $B N_i$, then $$|\int_{{G / {\Gamma}}}f d \pi_*({\mathbb}P(h_*\mu,t))-\int_{{G / {\Gamma}}}f dm_{{G / {\Gamma}}}|\leq \varepsilon$$
Take the $N_i$’s to be the maximal normal non dominant subgroups. They are in finite number. Suppose they do not verify the corollary. Then we construct a sequence $h_n$, $t_n$, $O_n$ such that the supports of ${\mathbb}P((h_n)_*\mu_n,t_n)$ are not included in any compact neighbourhood of a non-dominant normal subgroup and the difference of integrals is always greater than $\varepsilon$: $$\begin{aligned}
\label{eqn:diff}
|\int_{{G / {\Gamma}}}f d \pi_*({\mathbb}P((h_n)_*\mu_n,t_n))-\int_{{G / {\Gamma}}}f dm_{{G / {\Gamma}}}|> \varepsilon\end{aligned}$$
Up to an extraction, we may assume that $\mu_n$ converges to a measure which is equal the probability measure $\mu_\infty$ on an open $O'_\infty$ containing $O$ and proportional to the Haar measure of $C$.
These supports are yet included in a compact neighbourhood of some normal subgroup $N$ which has to be dominant. By assumption, $N$ has a dense projection in ${{G / {\Gamma}}}$. So we may apply the above proposition to this sequence: the projection $\pi_*({\mathbb}P((h_n)_*\mu_\infty,t_n))$ converges to the Haar probability measure $m_{{G / {\Gamma}}}$. But, using lemma \[lem:limsupport\], letting $n$ go to infinity, the measure $\pi_*({\mathbb}P((h_n)_*\mu_\infty,t_n))$ is arbitrarily closed to $\pi_*({\mathbb}P((h_n)_*\mu_n,t_n))$ This contradicts \[eqn:diff\].
Equidistribution of balls
-------------------------
At last we are able to conclude the proof of equidistribution of balls. Fix a function $f$ continuous with compact support in ${{G / {\Gamma}}}$. Fix $\varepsilon>0$. Let $\eta>0$ be such that $\frac{m_H(H_{(1+\eta) t})}{m_H(H_t)}\leq 1+\varepsilon$ for all $t$.
There is a neighborhood $O$ of $Id$ in $C$ such that for all $h\in H^{ss}$ we have $D(ho)\leq \sqrt{1+\eta}D(h)$. And we may choose $O$ such that $C$ is a disjoint union of translates of $O$ (up to a negligible set): there exist $c_1,\ldots, c_s$ such that $c_iO\cap c_jO$ has measure $0$ and the union $\cup_1^s c_i O$ is of full measure in $C$. Note $\mu_O$ the restriction of the probability Haar measure of $C$ to $O$.
Let $\tilde H_t$ be the union over $c\in CD$, $a\in A^+$, and $1\leq i\leq s$ with $D(cac_i)\leq t$, of the support of $m((cac_i)_*\mu_{O},(1+\eta)t)$. Thanks to the Cartan decomposition, up to a negligible set, $\tilde H_t$ contains $H_t$, is contained in $H_{(1+\eta) t}$ and the restriction of $m_H$ to $\tilde H_t$ may be written: $$(m_H)_{|\tilde H_t} =\sum_1^s \int_{c\in CD ,\; a\in A,\; D(cac_i)\leq t} m((cac_i)_*\mu_O,(1+\eta)t)$$
Let $E_t$ be the union of the supports of measures $m((ca)_*\mu_{ca},(1+\eta)t)$ which are completely included in $\displaystyle B\bigcup_1^k N_i$ (the sets constructed in the above corollary). As none of the $N_i$’s are dominant, for $t$ big enough, the relative mass of $E_t$ in $\tilde H_t$ is less than $\varepsilon$ and the symmetric difference between $H_t$ and $\tilde H_t\setminus E_t$ is almost negligible: $$\frac{m_H(H_t \Delta (\tilde H_t \setminus E_t))}{m_H(H_t)}\leq 2\varepsilon$$
Corollary \[coro:spheres\] implies that for all $a\in A$, $c\in CD$ and $1\leq i\leq s$, if the support ${\textrm}{Supp}(m((cac_i)_*\mu_O, (1+\eta)t))$ is not included in $E_t$, then its projection is pretty well distributed: $$\left|\int_H f\; dm((cac_i)_*\mu_O, (1+\eta)t)- m((cac_i)_*\mu_O, (1+\eta)t)(H)\int_{{G / {\Gamma}}}f \;dm_{{G / {\Gamma}}}\right|\leq \varepsilon$$
Integrating all these approximation over $c$, $a$ and $c_i$ leads to: $$|\frac{1}{m_H(\tilde H_t\setminus E_t)}\int_{\tilde H_t\setminus E_t} fd\pi_*(m_H) -\int_{{G / {\Gamma}}}fdm_{{G / {\Gamma}}}|\leq \varepsilon \;.$$
So going back to the desired integral, we get (for $t$ big enough): $$|\frac{1}{m_H(H_t)}\int_{H_t} fd\pi_*(m_H) -\int_{{G / {\Gamma}}}fdm_{{G / {\Gamma}}}|\leq (1+4\max(|f|))\varepsilon \;.$$ As $\epsilon$ is arbitrarily small, we get the desired result: $$\frac{1}{m_H(H_t)}\int_{H_t} fd\pi_*(m_H) \xrightarrow{t\to \infty}\int_{{G / {\Gamma}}}fdm_{{G / {\Gamma}}}\;.$$
This concludes the proof of theorem \[the:Horbit\].
Applications {#sec:exa}
============
We conclude this text by some explanations on the applications described in the introduction.
In dimension 2
--------------
Recall the framework: we consider the group $G={\textrm}{SL}(2,{\mathbb}R)\times {\textrm}{SL}(2,{\mathbb}Q_p)$ for $p$ a prime number, and the lattice ${\Gamma}={\textrm}{SL}(2,{\mathbb}Z[\frac{1}{p}])$. We fix here (for sake of simplicity) the standard euclidean norm $| . |_\infty$ on the matrix algebra $\mathcal M(2,{\mathbb}R)$ and the max-norm $|.|_p$ on $\mathcal M(2,{\mathbb}Q_p)$. For a point $v$ in ${\mathbb}R^2$, we note also $|v|_\infty$ the norm of the matrix whose first column is $v$ and the second one is $0$. We define similarly the norm of a point in ${\mathbb}Q_p^2$. We choose a Haar measure $m=m_\infty\otimes m_p$ on $G$.
The first result was:
Let $O$ be a bounded open subset of ${\textrm}{SL}(2,{\mathbb}Q_p)$. Note ${\Gamma}^O_T$ the set of elements ${\gamma}\in {\Gamma}$ such that $|{\gamma}|_\infty \leq T$ and ${\gamma}\in O$ as an element of ${\textrm}{SL}(2,{\mathbb}Q_p)$. Let $v$ be a point of the plane ${\mathbb}R^2\setminus \{0\}$ with coordinates independant over ${\mathbb}Q$.
Then we have the following limit, for any function ${\varphi}$ continuous with compact support in ${\mathbb}R^2\setminus \{0\}$: $$\frac{1}{T} \sum_{{\Gamma}_T^O} {\varphi}({\gamma}(v)) \xrightarrow{T\to \infty} \frac{m_p(O)}{m({{G / {\Gamma}}})|v|_\infty}\int_{{\mathbb}R^2} {\varphi}(w)\frac{dw}{|w|_\infty}$$
We work here in the product of ${\mathbb}R^2\setminus\{0\}$ and ${\textrm}{SL}(2,{\mathbb}Q_p)$. We see it as the homogeneous space ${{H \backslash G}}$ with $H={\textrm}{Stab}(v)$ the stabilizator of $v$ for the linear action of ${\textrm}{SL}(2,{\mathbb}R)$ on the plane.
Then it is not difficult to see that the hypothesis on the norm are fulfilled and that $H$ has no dominant subgroup except itself. Moreover the volume of balls are explicitely computed: the ratios of $m_H(H_tg)$ and $m_H(H_t)$ tends to $\frac{1}{|v||w|}$ where $w=g(v)$ [@goroweiss Section 12.4]. Remark that there is no need here to split the parameter space.
It remains to prove that $H.{\textrm}{SL}(2,{\mathbb}Z[\frac{1}{p}])$ is dense in $G$. But it contains $Stab(v).{\textrm}{SL}(2,{\mathbb}Z)$ which is by hypothesis dense in ${\textrm}{SL}(2,{\mathbb}R)$. Now we may use the strong aproximation in ${\textrm}{SL}(2)$ [@platonov-rapinchuk]: the algebraic group ${\textrm}{SL}(2)$ is semisimple simply connected, hence the product ${\textrm}{SL}(2,{\mathbb}R).{\textrm}{SL}(2,{\mathbb}Z[\frac{1}{p}])$ is dense in $G$. This yields the desired property: $H.{\textrm}{SL}(2,{\mathbb}Z[\frac{1}{p}])$ is dense in $G$.
Now, theorem \[the:duality\] implies the stated result.
The second application was the following one. Recall that on the $p$-adic plane, we normalize the measure such that it gives mass $1$ to ${\mathbb}Z_p^2$. The result is that if your beginning point generates the whole plane among the ${\mathbb}Q$-subspaces, then its orbit is dense and you get a distribution result (the function $E$ appearing is the integer part):
Let $(v_\infty,v_p)$ be an element of $({\mathbb}R^2\setminus{0})\times({\mathbb}Q_p^2\setminus{0})$. Suppose that any ${\mathbb}Q$-subspace $V$ of ${\mathbb}Q^2$ verifying $v_\infty\in V\otimes_{{\mathbb}Q} {\mathbb}R$ and $v_p \in V\otimes_{{\mathbb}Q} {\mathbb}Q_p$ is ${\mathbb}Q^2$. Denote ${\Gamma}_T$ the set of elements ${\gamma}\in {\Gamma}$ with $|{\gamma}|_\infty\leq T$ and $|{\gamma}|_p\leq T$.
Then, for all function ${\varphi}$ continuous with compact support in $({\mathbb}R^2\setminus{0})\times({\mathbb}Q_p^2\setminus{0})$, we have the following limit:\
$\frac{1}{T p^{E(\ln_p(T))}} \sum_{{\Gamma}_T} {\varphi}({\gamma}v_\infty, {\gamma}v_p)\xrightarrow{T\to\infty}$
$\frac{p^2-1}{p^2 m({{G / {\Gamma}}})|v_\infty|_\infty |v_p|_p} \int_{{\mathbb}R^2\times {\mathbb}Q_p^2} {\varphi}(v,w) \frac{dv dw}{|w|_\infty |w|_p}$
The proof here is similar to the previous one, the group $H$ being ${\textrm}{Stab}(v_\infty)\times {\textrm}{Stab}(v_p)$. The hypothesis on the norm are fulfilled, as $H$ is unipotent. The volume ratio limits are easy to compute and left to the reader. You just have to be careful with the normalizations of measures, letting appear this constant $\frac{p^2-1}{p^2}$.
So it just remains to prove that $H{\textrm}{SL}(2,{\mathbb}Z[\frac{1}{p}])$ is dense in $G$. The key point is that its closure must be (up to finite index) the ${\mathbb}R\times {\mathbb}Q_p$-points of a ${\mathbb}Q$-subgroup of ${\textrm}{SL}(2)$, by Tomanov theorem : it is a closed subset in ${{G / {\Gamma}}}$ invariant under unipotent subgroups.
Hence, if either $v_\infty$ or $v_p$ has coordinates independant over ${\mathbb}Q$, the argument in previous application show the density. The only remaining case is when both $v_\infty$ and $v_p$ are stabilized by a ${\mathbb}Q$-unipotent group. But the assumption that $v_\infty$ and $v_p$ “generates” ${\mathbb}Q^2$ is then equivalent to the fact that these two stabilizers are different. Now we may conclude, arguing that two different unipotent subgroups of ${\textrm}{SL}(2,{\mathbb}Q)$ generate the whole group. Hence the smallest ${\mathbb}Q$-subgroup of ${\textrm}{SL}(2,{\mathbb}Q)$ such that its real points contains the stabilizer of $v_\infty$ and its $p$-adic the stabilizer of $v_p$ is ${\textrm}{SL}(2)$. And the closure of $H.{\textrm}{SL}(2,{\mathbb}Z[\frac{1}{p}])$ is $G$.
The two previous examples showed how to profit of both the rigidity of orbit closures in an $S$-arithmetic setting and algebraic featurees such as strong approximation in the ambient group $G$. These arguments are also the core of the next case.
In greater dimension
--------------------
Recall that we look at the action of ${\Gamma}={\textrm}{SL}(n,{\mathbb}Z)$ on the $k$-th exterior power $\Lambda^k({\mathbb}R^n)$. And we fix the standard euclidean norm $|.|$ on $\mathcal M(n,{\mathbb}R)$. We consider also the standard euclidean norm on $\Lambda^k({\mathbb}R^n)$ and $m$ is a Haar measure on ${\textrm}{SL}(n,{\mathbb}R)$. We want to prove:
Let $v$ be a non-zero element of $\Lambda^k({\mathbb}R^n)$ such that its corresponding $k$-plane of ${\mathbb}R^n$ contains no rational vector. Denote ${\Gamma}_T$ the set of elements ${\gamma}\in {\Gamma}$ with $|{\gamma}|\leq T$.
Then we have a positive real constant $c$ (independant of ${\Gamma}$ and $v$) such that for all function ${\varphi}$ continuous with compact support on $\Lambda^k({\mathbb}R^n)\setminus \{0\}$: $$\frac{1}{T^{n^2+k^2-nk-n}}\sum_{{\Gamma}_T} {\varphi}({\gamma}v) \xrightarrow{T\to\infty} \frac{c}{m({{G / {\Gamma}}})|v|}\int_{\Lambda^k({\mathbb}R^n)} {\varphi}(v') \frac{dv'}{|v'|}$$
Here we have to be more careful than in previous section. We consider the subgroup $H = {\textrm}{Stab} (v)$. It is a conjugate of the group $H_0$ of the form: $$H_0=\left( \begin{matrix} {\textrm}{SL}(k,{\mathbb}R) & H^u\\ 0 & {\textrm}{SL}(n-k,{\mathbb}R)\end{matrix}\right):=\left( \begin{matrix} H_k & H^u\\ 0 & H_{n-k}\end{matrix}\right)$$
So it is a semidirect product of a semisimple and a unipotent group. Moreover the quotient $H_0\backslash G$ identifies with $\Lambda^k({\mathbb}R^n)\setminus \{0\}$ via the projection associating at an element of ${\textrm}{SL}(n,{\mathbb}R)$ the exterior product of its $k$-first lines.
We have to prove the orthogonality property for the norm on $H=gH_0g^{-1}$ ($g\in {\textrm}{SL}(n,{\mathbb}R)$). The key point is that one may use Iwasawa decomposition to write $g=o a n$ where $o$ belongs to ${\textrm}{SO}(n)$, $a$ is diagonal and $n$ is upper triangular and nilpotent so an element of $H_0$. By bi-invariance of the euclidean norm under ${\textrm}{SO}(n)$, and the fact that $a$ normalizes the semisimple part of $H$ and the unipotent one, we get, for $h=gh_0g^{-1}$ with the obvious notation: $$|h|^2= |a h_0 a^{-1}|^2 =|a h_0^{ss} a^{-1}|^2+|a h_0^{u} a^{-1}|^2=|h^{ss}|^2+|h^u|^2$$.
Now it is clear that $H_0$ has no dominant subgroup except itself, so the same holds for $H$. Let us prove that $H{\Gamma}$ is dense before evaluating the volume ratio limits. The simplest way to see it is to pull back this dynamic on the space of $k$-frames : choose a family of $k$ vectors in ${\mathbb}R^n$ such that their exterior product is $v$. Then the hypothesis on $v$ is that the $k$-plane generated by this family of vector contains no non-zero rational vectors. By a theorem of Dani and Raghavan [@dani-raghavan], it implies that the orbit of this family under ${\Gamma}$ is dense in the space of $k$-frames. This in turn implies by projection that the orbit of $v$ under ${\Gamma}$ is dense in $H_0\backslash G$, i.e. that $H{\Gamma}$ is dense in $G$.
We have compute the volume ratios to get the limiting density. Precisely, let $w=H_0g' =g^{-1}H(g'g^{-1})$ be a non-zero point in $\Lambda^k({\mathbb}R^n)$. Then the limiting density at $w$ given by theorem \[the:duality\] is the ratio: $$\frac{m_H(H_t(g'g^{-1})}{m_H(H_t)}$$
The set $H_t(g'g^{-1})$ is by definition $\{h\in H {\textrm{ such that }}|hgg'|\leq t\}$ ; or the set $\{h_0\in H_0 {\textrm{ such that }}|gh_0g'|\leq t\}$. Hence we have to compute the measure $M_t(g,g')=m_{H_0}(\{h_0\in H_0 {\textrm{ such that }}|gh_0g'|\leq t\})$. The choice of normalization of $m_{H_0}$ has no importance, as we only want to compute ratios. Using the bi-invariance of the norm and the Iwasawa decomposition of $g$ and $g'^{-1}$, we immediatly see that $M_t(g,g)=\frac{1}{|{\textrm}{Vol}(g)| |{\textrm}{Vol}(g')|}M_t(1,1)$, where ${\textrm}{Vol}(g)$ is the determinant of the $k$ first line of $g$. And, by the definition of the exterior product, the absolute value of this determinant is the euclidean norm of their exterior product. So we may rewrite $M_t(g,g')=\frac{1}{|v||w|}M_t(1,1)$. This gives the limiting density.
At this point, we need a last estimation: an equivalent of $M_T(1,1)$ which gives the renormalisation factor $T^{n^2+k^2-nk-n}$. So we want to compute the volume of the set $\{h_0\in H_0 {\textrm{ such that }}|h_0|\leq T\}$ for the standard Haar measure on $H_0$: the product of the standard Haar measure on the three groups ${\textrm}{SL}(k,{\mathbb}R)$, ${\textrm}{SL}(n-k,{\mathbb}R)$ and $H^u$. Using the estimations of Maucourant [@maucourant], we see that the volume of the sphere of radius $T$ in these groups are respectively of order $T^{k^2-k-1}$, $T^{(n-k)^2-(n-k)-1}$ and $T^{k(n-k)-1}$. So the leading term of the volume of the ball of radius $T$ is of order: $$\int_{T_1^2+T_2^2+T_3^2\leq T^2} T_1^{k^2-k-1}T_2^{(n-k)^2-(n-k)-1}T_3^{k(n-k)-1}$$ Hence the leading term is of order: $$T^{k^2-k+(n-k)^2-(n-k)+n(n-k)}=T^{n^2+k^2-nk-n}$$
This concludes the proof of application \[appln\]
I conclude this article with the $S$-arithmetic generalization of the previous result. I leave the proof to the reader. All the arguments are in the three previous proofs except an estimation of the volume of the ball of radius $T$ in ${\textrm}{SL}(k,{\mathbb}Q_p)$ ($p$ being a prime number). Using Cartan decomposition and some basic combinatorics, we get that the leading term of this volume is $(p^{E(\ln_p(T)})^{k^2-k}$. We fix the max norm in the standard basis on $\mathcal M(n,{\mathbb}Q_p)$ and $\Lambda^k({\mathbb}Q_p^n)$. The group ${\Gamma}$ is ${\textrm}{SL}(n,{\mathbb}Z[\frac{1}{p}])$, and we note for an element ${\gamma}\in {\Gamma}$, $|{\gamma}|$ the max of its real euclidean norm and $p$-adic max norm.
Let $v=(v_\infty,v_p)$ be a non-zero element of $\Lambda^k({\mathbb}R^n\times {\mathbb}Q_p^n)$ such that there is no non-zero rational vector belonging to both the real $k$-planes associated to $v_\infty$ and the $p$-adic one associated to $v_p$. Denote ${\Gamma}_T$ the set of elements ${\gamma}\in {\Gamma}$ with $|{\gamma}|\leq T$.
Then we have a positive real constant $c$ (independant of ${\Gamma}$ and $v$) such that for all function ${\varphi}$ continuous with compact support on $\Lambda^k({\mathbb}R^n)\setminus \{0\}$:
$\frac{1}{(Tp^{E(\ln_p(T))})^{n^2+k^2-nk-n}}\sum_{{\Gamma}_T} {\varphi}({\gamma}v)\xrightarrow{T\to\infty}$
$\frac{c}{m({{G / {\Gamma}}})|v_\infty|_\infty}\int_{\Lambda^k({\mathbb}R^n\times {\mathbb}Q_p^n)} {\varphi}(v_\infty',v'_p) \frac{dv_\infty'v_p'}{|v_\infty'|_\infty|v_p|_p}$
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abstract: 'We provide a general and unified combinatorial framework for a number of colored partition identities, which include the five, recently proved analytically by B. Berndt, that correspond to the exceptional modular equations of prime degree due to H. Schröter, R. Russell and S. Ramanujan. Our approach generalizes that of S. Kim, who has given a bijective proof for two of these five identities, namely the ones modulo 7 (also known as the Farkas-Kra identity) and modulo 3. As a consequence of our method, we determine bijective proofs also for the two highly nontrivial identities modulo 5 and 11, thus leaving open combinatorially only the one modulo 23.'
address:
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Department of Mathematics\
MIT\
Cambridge, MA 02139-4307
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Department of Mathematics\
MIT\
Cambridge, MA 02139-4307\
[and]{}
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Department of Mathematical Sciences\
Michigan Tech\
Houghton, MI 49931-1295
author:
- Colin Sandon
- Fabrizio Zanello
title: 'Warnaar’s bijection and colored partition identities, I'
---
Introduction
============
Colored partition identities are a very active research area within the theory of integer partitions. In particular, they provide natural combinatorial interpretations for certain classes of objects coming from other mathematical fields, including equations that involve modular forms or theta functions. The simplest and perhaps best known identity of this family is the so-called “Farkas-Kra identity modulo 7” (see [@FK]), which states that there are as many integer partitions of $2N+1$ into distinct odd parts as there are integer partitions of $2N$ into distinct even parts, provided the multiples of 7 appear in two different copies. A combinatorial proof of this result had been asked for by H.M. Farkas and I. Kra, R. Stanley, B. Berndt and a number of other authors, and was recently given by S. Kim [@Ki].
The Farkas-Kra identity is part of a set of five exceptional colored partition identities, sometimes referred to as “identities of the Schröter, Russell and Ramanujan type”, which correspond to five, conjecturally unique, modular equations of prime degree, discovered independently by H. Schröter [@Sc], R. Russell [@Ru1; @Ru2] and S. Ramanujan [@BR; @Ra]. These modular equations, respectively of degree 3, 5, 7, 11 and 23, as Berndt pointed out in [@Be], appear to be the only ones of such a simple type. See [@Be] for an interesting and detailed discussion of the history of these equations. In fact, in his paper, Berndt determined and proved analytically the five corresponding partition identities. As Berndt remarked, however (see also M.D. Hirschhorn [@Hi]), these five identities remained “manifestly mysterious”, as they still lacked “simple bijective proofs”, which “would be of enormous interest”.
Soon afterwards, S. Kim [@Ki], who employed in a clever fashion a bijection of S.O. Warnaar [@Wa] and generalized one of his results, provided an entirely bijective proof of, among other facts, two of the above identities — the one modulo 7, as we have said, and that modulo 3.
A main goal of this paper is to respond to Berndt’s call for a unified combinatorial framework in which to look at the five identities of the Schröter, Russell and Ramanujan type. In fact, extending Kim’s idea, we prove an equivalence between a very broad family of colored partition identities, which include the above five, and suitable equations in $(\nu_1,\dots,\nu_{t};d_1,\dots,d_{t})$, where $t\geq 1$, the $\nu_i$ are partitions, and the $d_i$ are integers whose sum is odd.
In particular, our approach allows us to prove bijectively two more of the identities of the Schröter, Russell and Ramanujan type, namely those corresponding to the modular equations of degree 5 and 11, whose specific proofs turn out to be highly nontrivial. Unfortunately, we have not been able to show bijectively the last identity, that modulo 23. We state its equivalent equation as Conjecture \[7.2\].
In a sequel to this paper [@CSFZ2], we will prove, again as a consequence of our method, a number of new (and challenging) colored partition identities.
The master bijection
====================
Let us first briefly recall the main definitions from partition theory that we are going to use in this paper. For an introduction, a survey of the main techniques, or a discussion of the philosophy behind this fascinating field, see e.g. [@And; @AE; @Pak], Section I.1 of [@Ma], and Section 1.8 of [@St0].
Given a nonnegative integer $N$, we say that the nonincreasing sequence $\lambda=(\lambda^{(1)},\dots, \lambda^{(s)})$ of nonnegative integers is a *partition* of $N$, and often write $|\lambda|=N$, if $\sum_{i=1}^s\lambda^{(i)}=N$. The $\lambda^{(i)}$ are called the *parts* of $\lambda $, and the number of parts of $\lambda $ is its *length*, denoted by $l(\lambda)$. As usual, we define $p(N)$ to be the number of partitions of $N$ into positive parts; thus $p(a)=0$ for $a<0$, and $p(0)=1$, since we adopt the standard convention that $\emptyset $ is the only partition of $N=0$.
Finally, let $P$ be the set of all partitions into positive parts, $D_0$ the set of partitions into distinct nonnegative parts, and $D=P\cap D_0$ the set of partitions into distinct positive parts. For instance, $\lambda=(6,6,3)\in P$ has length $l(\lambda )=3$, and $\lambda=(7,6,3,0)\in D_0$ has length $l(\lambda )=4$.
We begin with the following crucial bijection due to S.O. Warnaar [@Wa], who generalized an earlier bijection of E.M. Wright [@Wr]. As usual, we set $\binom{d}{2}=d(d-1)/2$, for any $d\in \mathbb Z$.
\[warnaar\] There exists a bijection between the set of triples $(\alpha, \beta, d)$, where $\alpha \in D_0$, $\beta \in D$ and $d=l(\alpha )- l(\beta )$, and the set of pairs $(\nu, d)$, where $\nu \in P$ and $d\in \mathbb Z$, such that $$|\alpha| + |\beta| = |\nu|+\binom{d}{2}.$$
See [@Wa], pages 48–49, for a description of the bijection.
The next theorem is the main general result of this paper. (We present it in a form that suffices for our purposes, even though it could easily be stated in more general terms.) It is an immediate corollary of the following lemma:
\[N\] Fix integers $t\geq 1$, $C_1,\dots,C_t\geq 1$, and $0\leq A_i\leq C_i/2$ for all $i=1,\dots ,t$. Let $S$ be the set containing one copy of all positive integers congruent to $\pm A_i$ modulo $C_i$ for each $i$, and let $D_S(N)$ be the number of partitions of $N$ into distinct elements of $S$, where we require such partitions to have an odd number of parts if no $A_i$ is equal to zero. Finally, set $r=|\{A_i=0\}|-1$, adopting the convention that $|X|=1$ if $X=\emptyset$.
Then, for all $N\geq 1$, $$2^{r}\cdot D_S(N)$$ is the number of solutions $(\nu_1,\dots,\nu_{t};d_1,\dots,d_{t})$ to the equation $$\label{eee}
\sum_{i=1}^{t}C_i|\nu_i|+\sum_{i=1}^{t}C_i{{d_i}\choose{2}} +\sum_{i=1}^{t}A_{i}d_{i}=N,$$ where $\nu_i\in P$ and $d_i\in \mathbb Z$ for all $i$, and $\sum_{i=1}^{t}d_i$ is odd.
This proof will greatly generalize, but proceed for the most part in a similar way to, Kim’s combinatorial proof of the Farkas-Kra identity modulo 7 (cf. [@Ki], second proof of Theorem 2.1). A substantial difference is that we are going to push the bijectivity of this type of argument all the way through, so that Theorem \[main\] below will give us (ii) equivalent to (i), which is going to be the crucial tool in attacking the identities of the Schröter, Russell and Ramanujan type.
Fix $N\geq 1$. We start by assuming that all of the $A_i$ are positive, and consider any partition $\pi$ of $N$ into distinct elements of $S$. We first split $\pi$ into $t$ pairs of partitions $(\lambda_1, \mu_1), \dots, (\lambda_t, \mu_t)$, where, for any $i$, both $\lambda_i$ and $\mu_i$ are in $D$, all parts of $\lambda_i$ come from the copy of the integers of $S$ that are congruent to $ A_i$ (mod $C_i$), and all parts of $\mu_i$ come from the copy of the integers of $S$ that are congruent to $-A_i$ (mod $C_i$).
Let us now construct a new partition $\pi^{\ast}$ from $\pi$, which we split as $(\lambda_1^{\ast}, \mu_1^{\ast}), \dots,(\lambda_t^{\ast}, \mu_t^{\ast})$, where (entrywise) $\lambda_i^{\ast}=(\lambda_i-A_i)/C_i$ and $\mu_i^{\ast}=(\mu_i+A_i)/C_i$, for all $i$. Notice that, clearly, $\lambda_i^{\ast}\in D_0$ and $\mu_i^{\ast}\in D$, for all $i$.
Set $$d_i=l(\lambda_i )-l(\mu_i )=l(\lambda_i^{\ast} )-l(\mu_i^{\ast}).$$
Note that $\sum_{i=1}^{t}d_i \equiv \sum_{i=1}^{t}(l(\lambda_i )+l(\mu_i ))=l(\pi)$ (mod $2$); that is, $\sum_{i=1}^{t}d_i$ is odd if and only if $\pi$ has an odd number of parts.
By Lemma \[warnaar\], the triples $(\lambda_i^{\ast}, \mu_i^{\ast},d_i)$ are in (Warnaar’s) bijection with pairs $(\nu_i, d_i)$, where $\nu_i \in P$ and $|\lambda_i^{\ast}| + |\mu_i^{\ast}| = |\nu_i|+\binom{d_i}{2}$.
Therefore, it is easy to see that $$N=|\pi |=\sum_{i=1}^{t}(|\lambda_i |+|\mu_i |)=\sum_{i=1}^{t}C_i|\lambda_i^{\ast}|+\sum_{i=1}^{t}C_i|\mu_i^{\ast}|+\sum_{i=1}^{t}d_iA_i=\sum_{i=1}^{t}C_i|\nu_i|+\sum_{i=1}^{t}C_i\binom{d_i}{2}+\sum_{i=1}^{t}d_iA_i.$$
Since all previous steps are reversible, this implies that the number of solutions to equation (\[eee\]) is $D_S(N)=2^{0}\cdot D_S(N)$, as desired.
This completes the proof when all of the $A_i$ are positive.
Suppose now that some $A_i=0$. Let us assume, without loss of generality, that $A_1=A_2=\dots =A_{r+1}=0$ for some $r\geq 0$, and that all other $A_j\neq 0$. The proof of this case goes along the same lines, except that now the partitions $\lambda_i^{\ast}$ are in $D$ (not in $D_0$), for all $i\leq r+1$. Therefore, it is easy to see that, for $i\leq r+1$, the same partition $\lambda_i$ corresponds to exactly two solutions to equation (\[eee\]) — one given by operating with Warnaar’s bijection with $\lambda^{\ast}_i$, and the other with $\lambda^{\ast}_i$ to which a 0 is added at the end.
Thus, each of our partitions $\pi$ corresponds bijectively to $2^{r+1}$ solutions to (\[eee\]), when $\sum_{i=1}^{t}d_i$ is arbitrary.
Now, it is immediate to see that, in every solution to (\[eee\]), $d_i$ can be replaced by $1-d_i$, for any $i\leq r+1$. Since the parity of $d_i$ and $1-d_i$ is different, it follows that exactly $2^{r+1}/2=2^r$ of the solutions to (\[eee\]) corresponding to the partition $\pi$ yield an odd value for $\sum_{i=1}^{t}d_i$. This easily concludes the proof of the lemma.
\[main\] Consider the equation $$\label{t}
\sum_{i=1}^{t}C_i|\mu_i|+\sum_{i=1}^{t}C_i{{d_i}\choose{2}} +\sum_{i=1}^{t}A_{i}d_{i}=\sum_{i=1}^{t}C_i|\alpha_i|+\sum_{i=1}^{t}C_i{{e_i}\choose{2}} +\sum_{i=1}^{t}B_{i}e_{i}+m,$$ for given integers $t\geq 1$, $C_1,\dots,C_t\geq 1$, $0\leq A_i\leq C_i/2$ and $0\leq B_i\leq C_i/2$ for all $i$, and $m\geq 0$. Let $S$ be the set containing one copy of all positive integers congruent to $\pm A_i$ modulo $C_i$ for each $i$, and $T$ the set containing one copy of all positive integers congruent to $\pm B_i$ modulo $C_i$ for each $i$. Let $D_S(N)$ (respectively, $D_T(N)$) be the number of partitions of $N$ into distinct elements of $S$ (respectively, $T$), where we require such partitions to have an odd number of parts if no $A_i$ (respectively, no $B_i$) is equal to zero. Finally, set $$p=|\{B_i=0\}|-|\{A_i=0\}|,$$ adopting the convention that $|X|=1$ if $X=\emptyset$.
Then the following are equivalent:
1. For any $N\geq N_0\ge 1$, the number of tuples $(\mu_1,\dots,\mu_{t};d_1,\dots,d_{t})$ such that the left-hand side of (\[t\]) equals $N$, $\mu_i\in P$ and $d_i\in \mathbb Z$ for all $i$, and $\sum_{i=1}^{t}d_i$ is odd, is equal to the number of tuples $(\alpha_1,\dots,\alpha_{t};e_1,\dots,e_{t})$ such that the right-hand side of (\[t\]) equals $N$, $\alpha_i\in P$ and $e_i\in \mathbb Z$ for all $i$, and $\sum_{i=1}^{t}e_i$ is odd;
2. For any $N\geq N_0\ge 1$, $$D_S(N)=2^p\cdot D_T(N-m).$$
Straightforward from Lemma \[N\].
The identities of the Schröter, Russell and Ramanujan type
==========================================================
The object of the rest of this paper is to show bijectively, using Theorem \[main\], four of the five partition identities proved by Berndt in [@Be], which correspond to the five exceptional modular equations of prime degree due to Schröter, Russell and Ramanujan, as we discussed in the introduction. They will be proved in Theorems \[5.222\], \[3.222\], \[4.222\], and \[6.222\]. We have not been able to show the identity modulo 23; we will state an equation equivalent to it via Theorem \[main\] as Conjecture \[7.2\].
We start with the partition identities modulo 7 (i.e., the Farkas-Kra identity) and modulo 3. These are the two of the five for which a bijective proof is already known, thanks to the work of Kim [@Ki] (we will just slightly modify Kim’s bijection here so as to fit our setting).
\[5.2\] Condition (i) of Theorem \[main\] holds for $N_0=1$, $t=4$, $C_1=\dots=C_4=14$, $m=1$, and $$(A_1,A_2,A_3,A_{4})=(1,3,5,7),{\ }{\ }(B_1,B_2,B_3,B_{4})=(0,2,4,6).$$
It is easy to verify that the result follows by associating the tuple $$(\mu_1,\mu_2,\mu_3,\mu_4;d_1=2s+1-k-l+n,d_2=k-n,d_3=l-n,d_4=n),$$ where $n,l,k,$ and $s$ are arbitrary integers, to the tuple $$(\alpha_1=\mu_1,\alpha_2=\mu_2,\alpha_3=\mu_3,\alpha_4=\mu_4;e_1=2n+1-k-l+s,e_2=k-s,e_3=l-s,e_4=-s).$$ (This is exactly Kim’s map of [@Ki], Theorem 1.1, except that here we needed to set $e_4=-s$ in place of $e_4=s$.)
\[fk15\] For any $N\geq 1$, Lemma \[5.2\] puts the tuples $(\mu_1,\dots,\mu_{4};d_1,\dots,d_{4})\in P^4\times\mathbb{Z}
^4$ such that the left-hand side of the following equation equals $N$ and $d_1+d_2+d_3+d_4$ is odd in bijection with the tuples $(\alpha_1,\dots,\alpha_{4};e_1,\dots,e_{4})\in P^4\times\mathbb{Z}^4$ such that the right-hand side equals $N$ and $e_1+e_2+e_3+e_4$ is odd:
$$\label{fk}
14\sum_{i=1}^{4}|\mu_i|+14\sum_{i=1}^{4}{{d_i}\choose{2}} +1d_1+3d_2+5d_3+7d_4=14\sum_{i=1}^{4}|\alpha_i|+14\sum_{i=1}^{4}{{e_i}\choose{2}} +0e_1+2e_2+4e_3+6e_4+1.$$
Let $N=15$. It is easy to check that there are exactly six such tuples. The left-hand side of (\[fk\]) equals 15 for $(\mu_1,\mu_2,\mu_3,\mu_{4};d_1,d_2,d_3,d_{4})$ equal to:
$$((1),\emptyset,\emptyset,\emptyset;1,0,0,0),{\ }{\ }{\ }{\ }{\ }{\ }(\emptyset,(1),\emptyset,\emptyset;1,0,0,0),{\ }{\ }{\ }{\ }{\ }{\ }(\emptyset,\emptyset,(1),\emptyset;1,0,0,0),$$ $$(\emptyset,\emptyset,\emptyset,(1);1,0,0,0),{\ }{\ }{\ }{\ }{\ }{\ }(\emptyset,\emptyset,\emptyset,\emptyset;0,1,1,1),{\ }{\ }{\ }{\ }{\ }{\ }(\emptyset,\emptyset,\emptyset,\emptyset;0,1,1,-1).$$
The bijection given in the proof of Lemma \[5.2\] maps the above solutions, respectively, to the following six tuples $(\alpha_1,\alpha_2,\alpha_3,\alpha_{4};e_1,e_2,e_3,e_{4})$ for which the right-hand side of equation (\[fk\]) equals 15:
$$((1),\emptyset,\emptyset,\emptyset;1,0,0,0),{\ }{\ }{\ }{\ }{\ }{\ }(\emptyset,(1),\emptyset,\emptyset;1,0,0,0),{\ }{\ }{\ }{\ }{\ }{\ }(\emptyset,\emptyset,(1),\emptyset;1,0,0,0),$$ $$(\emptyset,\emptyset,\emptyset,(1);1,0,0,0),{\ }{\ }{\ }{\ }{\ }{\ }(\emptyset,\emptyset,\emptyset,\emptyset;0,1,1,-1),{\ }{\ }{\ }{\ }{\ }{\ }(\emptyset,\emptyset,\emptyset,\emptyset;-1,0,0,0).$$
\[5.222\] Let $S$ be the set containing one copy of the odd positive integers and one more copy of the odd positive multiples of 7, and $T$ the set containing one copy of the even positive integers and one more copy of the even positive multiples of 7. Then, for any $N\geq 1$, $$D_S(N)=D_T(N-1).$$
Straightforward from Theorem \[main\] and Lemma \[5.2\].
Let $N=15$ in Theorem \[5.222\]. By Example \[fk15\] and Lemma \[N\], we have $$D_S(15)=D_T(14)=6.$$
Indeed, it is easy to check that 15 can be partitioned in the following six ways into distinct odd positive integers, where the multiples of 7 appear in two copies, say $7n$ and $\overline{7n}$: $$(15), (11,3,1), (9,5,1), (7,5,3), (\overline{7}, 5,3), (7,\overline{7},1).$$
Similarly, 14 can be partitioned in the following six ways into distinct even positive integers, where the multiples of 7 appear in two copies: $$(14), (\overline{14}), (12,2), (10,4), (8,6), (8,4,2).$$
\[3.2\] Condition (i) of Theorem \[main\] holds for $N_0=1$, $t=4$, $C_1=\dots=C_4=6$, $m=1$, and $$(A_1,A_2,A_3,A_{4})=(1,1,3,3),{\ }{\ } (B_1,B_2,B_3,B_{4})=(0,0,2,2).$$
The exact same bijection as in Lemma \[5.2\] easily gives the result.
\[3.222\] Let $S$ be the set containing 2 copies of the odd positive integers and 2 more copies of the odd positive multiples of 3, and $T$ the set containing 2 copies of the even positive integers and 2 more copies of the even positive multiples of 3. Then, for any $N\geq 1$, $$D_S(N)=2D_T(N-1).$$
Straightforward from Theorem \[main\] and Lemma \[3.2\].
Notice that the two equations we are going to deal with next, which are equivalent, respectively, to the partition identities modulo 5 and 11 of the Schröter, Russell and Ramanujan type, will be:
$$\begin{aligned}
&2|\mu_1|+2|\mu_2|+10|\mu_3|+10|\mu_4|+2{d_1 \choose 2}+2{d_2 \choose 2}+10{d_3 \choose 2}+10{d_4 \choose 2}+d_1+d_2+5d_3+5d_4=\\
&2|\alpha_1|+2|\alpha_2|+10|\alpha_3|+10|\alpha_4|+2{e_1 \choose 2}+2{e_2 \choose 2}+10{e_3 \choose 2}+10{e_4 \choose 2}+0e_1+0e_2+0e_3+0e_4+3\end{aligned}$$ and
$$2|\mu_1|+22|\mu_2|+2{d_1 \choose 2}+22{d_2 \choose 2}+d_1+11d_2=2|\alpha_1|+22|\alpha_2|+2{e_1 \choose 2}+22{e_2 \choose 2}+0e_1+0e_2+3.$$
Therefore, one moment’s thought gives that the type of argument that held for Lemmas \[5.2\] and \[3.2\], where the bijection between the solutions could simply be taken to be the identity on all the partitions $\mu_i$, will not apply here, where $m=3$. For instance, in the first of the two equations, the tuple $$(\mu_1,\dots,\mu_4;d_1,\dots,d_4)=(\emptyset,\emptyset,\emptyset,(1);1,0,0,0),$$ which makes the left-hand side equal 11, must be mapped to a tuple $(\alpha_1,\dots,\alpha_{4};e_1, \dots, e_{4})$ such that the right-hand side equals 11, so we clearly need to have the partition $\alpha_4=\emptyset$. An entirely similar argument holds for the second equation. This is why the two corresponding partition results will be far more difficult to treat bijectively than the previous ones.
The following lemma is a classical application of Euler’s Pentagonal Number Theorem:
\[lemma0\] For any $n>0$, $$\sum_{i\in \mathbb Z} (-1)^ip\left(n-\frac{i(3i-1)}{2}\right)=0.$$
See e.g. [@Pak], formula 5.1.2, or [@St0], equation (1.91).
\[lemma1\] Fix arbitrary $C_1,\dots,C_t,A_1,\dots,A_t,B_1,\dots,B_t$, such that $0\le A_i\le {C_i/2}$ and $0\le B_i\le C_i/2$, for all $i=1,\dots,t$. Let $S_N$ be the set of all tuples of $t$ partitions and $t$ integers $(\mu_1,\dots,\mu_{t};d_1,\dots,d_{t})$ such that $\sum_{i=1}^{t} d_i$ is odd and
$$\sum_{i=1}^tC_i|\mu_i|+\sum_{i=1}^tC_i{{d_i}\choose{2}}+\sum_{i=1}^t A_id_i =N.$$ Similarly, let $T_N$ be the set of all tuples of $t$ partitions and $t$ integers $(\alpha_1,\dots,\alpha_{t};e_1,\dots,e_{t})$ such that $\sum_{i=1}^{t} e_i$ is odd and
$$\sum_{i=1}^{t}C_i|\alpha_i|+\sum_{i=1}^{t}C_i{{e_i}\choose{2}} +\sum_{i=1}^{t} B_ie_i +m=N,$$ where $m$ is an integer chosen so that the smallest value of $N$ for which $T_N\ne \emptyset$ is also the second smallest value of $N$ for which $S_N\ne \emptyset$. Define $k$ to be the smallest value such that $S_k\ne \emptyset$. Further, let $U_N$ be the union of the set of all tuples of $t$ integers $(d_1,\dots,d_{t})$ such that $\sum_{i=1}^{t} d_i$ is odd and
$$\sum_{i=1}^{t}C_i{{d_i}\choose{2}}+\sum_{i=1}^{t} A_id_i =N,$$ with $|S_k|$ copies of the set of all tuples of $t$ integers $(f_1,\dots,f_{t})$ such that $\sum_{i=1}^{t} f_i$ is odd and
$$\sum_{i=1}^{t}C_i \frac{f_i(3f_i-1)}{2}+k=N.$$ Finally, let $V_N$ be the union of the set of all tuples of $t$ integers $(e_1,\dots,e_{t})$ such that $\sum_{i=1}^{t} e_i$ is odd and
$$\sum_{i=1}^{t}C_i{{e_i}\choose{2}}+\sum_{i=1}^{t} B_ie_i+m =N,$$ with $|S_k|$ copies of the set of all tuples of $t$ integers $(f_1,\dots,f_{t})$ such that $\sum_{i=1}^{t} f_i$ is even and
$$\sum_{i=1}^{t} C_i\frac{f_i(3f_i-1)}{2}+k=N.$$ Then $|S_N|=|T_N|$ for all $N>k$ if and only if $|U_N|=|V_N|$ for all $N$.
For every $N$, let $U_N^{\ast}$ be the set of all pairs consisting of a tuple of $t$ partitions $(\mu_1,\dots,\mu_{t})$ and an element of $U_{N-x}$, where $x=\sum_{i=1}^{t} C_i|\mu_i|$. Likewise, let $V_N^{\ast}$ be the set of all pairs consisting of a tuple of $t$ partitions $(\alpha_1,\dots,\alpha_{t})$ and an element of $V_{N-x}$, where $x=\sum_{i=1}^{t} C_i|\alpha_i|$.
Obviously, if $|U_N|=|V_N|$ for all $N$, then $|U_N^{\ast}|=|V_N^{\ast}|$ for all $N$. Conversely, if $|U_N^{\ast}|=|V_N^{\ast}|$ and $|U_x|=|V_x|$ for all $x<N$, then all of the terms of $|U_N^{\ast}|$ and $|V_N^{\ast}|$ in which any of the partitions are nonempty cancel out, leaving $|U_N|=|V_N|$.
Thus, $|U_N|=|V_N|$ for all $N$ if and only if $|U_N^{\ast}|=|V_N^{\ast}|$ for all $N$.
So we need only prove that $|S_N|=|T_N|$ for all $N>k$ if and only if $|U_N^{\ast}|=|V_N^{\ast}|$ for all $N$. By definition, $|S_N|=|T_N|=0$ for all $N<k$, and $|T_k|=0$ as well. Hence, for any $N<k$, $|U_N^{\ast}|=|V_N^{\ast}|=0$, and it is easy to see that $$|U_k^{\ast}|=|S_k|+0 =0+|S_k|\cdot 1=|V_k^{\ast}|.$$
Therefore, it suffices to show that $|U_N^{\ast}|-|S_N|=|V_N^{\ast}|-|T_N|$ for all $N>k$. This is equivalent to the existence of a bijection between the set of all $(\mu_1,\dots,\mu_{t};f_1,\dots,f_{t})$ such that $\sum_{i=1}^{t} f_i$ is odd and $$\sum_{i=1}^{t} C_i|\mu_i|+\sum_{i=1}^{t} C_i f_i(3f_i-1)/2+k=N,$$ and the set of all $(\alpha_1,\dots,\alpha_{t};f_1,\dots,f_{t})$ such that $\sum_{i=1}^{t} f_i$ is even and $$\sum_{i=1}^{t} C_i|\alpha_i|+\sum_{i=1}^{t} C_i f_i(3f_i-1)/2+k=N.$$
We can associate each element of either set with a tuple $(n_1,\dots,n_{t})$, where, for any index $i$, $$n_i=|\mu_i|+f_i(3f_i-1)/2 \phantom{x} \text{ or } \phantom{x} n_i=|\alpha_i|+{f_i(3f_i-1)/2},$$ as appropriate. For any given $(n_1,\dots,n_{t})$, it is easy to see that the difference between the number of elements of the second set associated with $(n_1,\dots,n_{t})$ and the number of elements of the first set associated with it, is $$\prod_{i=1}^{t}\sum_{f_i\in \mathbb Z} (-1)^{f_i}p\left(n_i-\frac{f_i(3f_i-1)}{2}\right).$$
Thus, by Lemma \[lemma0\], unless $(n_1,\dots,n_{t})=(0,\dots,0)$, the last displayed formula is $0$. But we have $ \sum_{i=1}^{t} C_i n_i=N-k$, which implies that $n_i=0$ for all $i$ if and only if $N=k$. This proves the bijection between the two sets for any $N>k$, as desired.
Notice that, given a bijection $f$ between $U_N$ and $V_N$, we can construct a bijection between $S_N$ and $T_N$ as follows. First, create a bijection $f^{\ast}$ between $U_N^{\ast}$ and $V_N^{\ast}$ by having $f^{\ast}$ leave their partition components unchanged and act as $f$ on their integer components. Also, let $g$ be a bijection between $U_N^{\ast}-S_N$ and $V_N^{\ast}-T_N$ (where these set differences are defined in the obvious way). Constructing a bijection between $S_N$ and $T_N$ is now a standard variation of the Garsia-Milne involution principle [@GM] (see e.g. [@St0], formula (2.33)). For instance, construct a graph where every element of $U_N^{\ast}$ or $V_N^{\ast}$ is a vertex, and there is an edge between any pair of elements that are in correspondence through $f^{\ast}$ or $g$. Thus, each element of $S_N$ or $T_N$ has one edge in this graph, and each element of $U_N^{\ast}-S_N$ or $V_N^{\ast}-T_N$ has two, so every element of $S_N$ is the other endpoint of a path ending at an element of $T_N$, and vice-versa. Therefore, the desired bijection between $S_N$ and $T_N$ maps each element of $S_N$ or $T_N$ to the other end of its path.
We are now ready for the bijective proofs of the two partition identities corresponding, respectively, to the modular equations of degree 5 and 11 of the Schröter, Russell and Ramanujan type (see Theorems 4.2 and 6.2 of [@Be]).
\[4.2\] Condition (i) of Theorem \[main\] holds for $N_0=3$, $t=4$, $C_1=C_2=2$, $C_3=C_4=10$, $m=3$, and $$(A_1,A_2,A_3,A_{4})=(1,1,5,5),{\ }{\ } (B_1,B_2,B_3,B_{4})=(0,0,0,0).$$
It is easy to check that, in the notation of Lemma \[lemma1\], we have $k=1$ and $|S_k|=4$. Thus, by Lemma \[lemma1\], proving the lemma is equivalent to showing the existence of a bijection from the union of the set of all quadruples $(d_1,d_2,d_3,d_4)$ with $d_1+d_2+d_3+d_4$ odd and the set containing $4$ copies of each quadruple $(f_1,f_2,f_3,f_4)$ with $f_1+f_2+f_3+f_4$ odd, to the union of the set of all quadruples $(e_1,e_2,e_3,e_4)$ with $e_1+e_2+e_3+e_4$ odd and the set containing $4$ copies of each quadruple $(f_1',f_2',f_3',f_4')$ with $f_1'+f_2'+f_3'+f_4'$ even, such that, for every pair of corresponding elements,
$$2{d_1 \choose 2}+2{d_2 \choose 2}+10{d_3 \choose 2}+10{d_4 \choose 2}+d_1+d_2+5d_3+5d_4$$ $$\text{ or }{\ }{\ } f_1(3f_1-1)+f_2(3f_2-1)+5f_3(3f_3-1)+5f_4(3f_4-1)+1$$ $$=2{e_1 \choose 2}+2{e_2 \choose 2}+10{e_3 \choose 2}+10{e_4 \choose 2}+0e_1+0e_2+0e_3+0e_4+3$$ $$\text{ or }{\ }{\ } f_1'(3f_1'-1)+f_2'(3f_2'-1)+5f_3'(3f_3'-1)+5f_4'(3f_4'-1)+1.$$ Notice that, by replacing $d_i$ with $1/2-e_i$ in the above $d$-formula, for $i=1,\dots,4$, we obtain the $e$-formula. Thus, if we apply the map $d_i=1/2-e_i$, we can view the $d$-tuples and $e$-tuples as all being in the set $D=\{d\in \mathbb{Z}^4\cup(\mathbb{Z}+1/2)^4:d_1+d_2+d_3+d_4\in 2\mathbb{Z}+1\}$. (A tuple $(d_1,d_2,d_3,d_{4})\in D$ will in some sense be considered “of negative type” if the $d_i$ are half-integers, since it will come from the opposite side of the bijection as the tuples in which the $d_i$ are integers.)
Furthermore, if we define a dot product so that $$(d_1,d_2,d_3,d_4)\cdot(d_1',d_2',d_3',d_4')=d_1 d_1'+d_2 d_2'+5d_3 d_3'+5d_4 d_4',$$ then every point in this set corresponds to a quadruple with a value in the previous equation of its “length” squared.
Now, let $$V_{1}=(1,1,1,1), V_{2}=(1,-1,1,-1), V_{3}=(5,5,-1,-1), V_{4}=(5,-5,-1,1).$$
It is easy to check that these vectors are pairwise orthogonal. For each $i=1, \dots, 4$, set $M_i= \|V_i \|^2/12$.
Thus, $M_1=1$, $M_2=1$, $M_3=5$, and $M_4=5$. Also, for arbitrary $d=(d_1,d_2,d_3,d_4)$, we clearly have that $d\cdot V_1$, $d\cdot V_2$, $d\cdot V_3$, and $d\cdot V_4$ are all odd integers, and $d\cdot V_3$ and $d\cdot V_4$ are divisible by $5$. It follows that $d\cdot V_i$ is an odd multiple of $M_i$, for all $d$ and $i$.
Now, for each $d\in D$ and $i$, let $$r_i(d)=d-\frac{d\cdot V_i}{6M_i}V_i.$$
It is easy to check that $\|r_i(d)\|=\|d\|$, for all $d$ and $i$. If $d\cdot V_i\equiv 0\pmod{3M_i}$ then $\frac{d\cdot V_i}{6M_i}$ is a half-integer, so $r_i(d)$ is a vector that corresponds to an $e$-quadruple if $d$ corresponds to a $d$-quadruple, and vice-versa. Hence, we can map every point in $D$ that has a dot product with at least one of the $V_i$ that is divisible by $3M_i$, to a point of the opposite type and the same value in the above $d$-formula, by sending it to $r_i(d)$, where $i$ is the smallest integer such that $d\cdot V_i\equiv 0\pmod{3M_i}$.
Note that $r_i(d)\cdot V_i=-d\cdot V_i$. Also, $r_i(d)$ has the same dot product with $V_j$ as $d$ does, for all $j\ne i$, because of the orthogonality of the vectors. This implies that $r_i(r_i(d))\cdot V_j=d\cdot V_j$ for all $j$, and thus that $r_i(r_i(d))=d$. Therefore, the above map is an involution.
Hence, now we only need to consider the points in $D$ whose dot products with $V_i$ are not divisible by $3M_i$, for any $i$. Let $d\in D$ be any such point. For each $i$, let $x_i$ be the nearest integer to $\frac{d\cdot V_i}{6M_i}$, $y_i=\frac{d\cdot V_i-6M_ix_i}{M_i}$, and $z=d-\sum_{i=1}^{4} \frac{x_i}{2}V_i$.
For any $i$, $d\cdot V_i\equiv \pm M_i\pmod{6M_i}$, so $y_i=\pm 1$. Thus, by the Pythagorean Theorem we obtain: $$\|d\|^2=\sum_{i=1}^{4} \frac{(d\cdot V_i)^2}{\|V_i\|^2}=\sum_{i=1}^{4} \frac{(6M_ix_i+M_iy_i)^2}{12M_i}=1+\sum_{i=1}^{4} M_ix_i(3x_i+y_i).$$
We easily have that $z$ must be either a quadruple of integers or a quadruple of half-integers, and $z\cdot V_i=y_i M_i=\pm M_i$, for each $i$. It is a simple exercise to verify that the only $z$ that fit these criteria are: $(1,0,0,0)$, $(-1,0,0,0)$, $(0,1,0,0)$, and $(0,-1,0,0)$.
Therefore, we can choose a bijection between the $4$ possible values of $z$ and the $4$ copies of each tuple $(f_1,\dots,f_4)$, and then map $d$ to the copy of $(f_1, \dots,f_4)=(-x_1 y_1,\dots,-x_{4} y_{4})$ corresponding to $z$.
It easily follows that $$f_1(3f_1-1)+f_2(3f_2-1)+5f_3(3f_3-1)+5f_4(3f_4-1)+1=\|d\|^2.$$
Also, the $y_i$ are determined by $z$, and for any given choice of $z$, since $x_i=-y_if_i$, the only $d$ that maps to a given tuple $(f_1, \dots,f_4)$ is $z-\sum_{i=1}^{4} \frac{y_i f_i}{2} V_i.$
Furthermore, the entries of such $d$ are all half-integers if $\sum_{i=1}^{4} f_i$ is odd, and integers if it is even. Thus, this map always takes elements of $D$ corresponding to tuples of $d$’s to tuples of $f$’s with an even sum, and elements of $D$ corresponding to tuples of $e$’s to tuples of $f$’s with an odd sum. This completes the proof of the lemma.
\[4.222\] Let $S$ be the set containing 4 copies of the odd positive integers and 4 more copies of the odd positive multiples of 5, and $T$ the set containing 4 copies of the even positive integers and 4 more copies of the even positive multiples of 5. Then, for any $N\geq 3$, $$D_S(N)=8D_T(N-3).$$
Straightforward from Theorem \[main\] and Lemma \[4.2\].
\[6.2\] Condition (i) of Theorem \[main\] holds for $N_0=3$, $t=2$, $C_1=2$, $C_2=22$, $m=3$, and $$(A_1,A_2)=(1,11),{\ }{\ } (B_1,B_2)=(0,0).$$
It is easy to check that, in the notation of Lemma \[lemma1\], we have $k=1$ and $|S_k|=2$. Thus, proving the lemma is equivalent to proving the existence of a bijection from the union of the set of all pairs $(d_1,d_2)$ with $d_1+d_2$ odd and the set containing $2$ copies of each pair $(f_1,f_2)$ with $f_1+f_2$ odd, to the union of the set of all pairs $(e_1,e_2)$ with $e_1+e_2$ odd and the set containing $2$ copies of each pair $(f_1',f_2')$ with $f_1'+f_2'$ even, such that, for every pair of corresponding elements, $$\begin{aligned}
2{d_1 \choose 2}+22{d_2 \choose 2}+d_1+11d_2{\ }{\ }\text{ or }{\ }{\ }f_1(3f_1-1)+11f_2(3f_2-1)+1\\
=2{e_1 \choose 2}+22{e_2 \choose 2}+0e_1+0e_2+3{\ }{\ }\text{ or }{\ }{\ }f_1'(3f_1'-1)+11f_2'(3f_2'-1)+1.\end{aligned}$$
Thus, similarly to what we did in Lemma \[4.2\], we can apply the map $d_1=1/2-e_1$, $d_2=e_2-1/2$, in order to view the $d$-pairs and $e$-pairs as both being in the same set $D=\{d\in \mathbb{Z}^2\cup(\mathbb{Z}+1/2)^2:d_1+d_2\in 2\mathbb{Z}+1\}$. Notice that, for any $(d_1,d_2)\in D$, $d_1+d_2$ is odd.
If $d_1+d_2\equiv 0\pmod{3}$, then the map $$d_1'=d_1-\frac{11(d_1+d_2)}{6}, {\ }{\ }{\ }{\ } d_2'=d_2-\frac{d_1+d_2}{6}$$ yields a pair having the same value as $(d_1,d_2)$, since $(d_1')^2+11(d_2')^2=d_1^2+11d_2^2$. Furthermore, it is easy to see that $(d_1',d_2')$ is an $e$-tuple (i.e., it has half-integer entries) if and only if $(d_1,d_2)$ is a $d$-tuple (i.e., it has integer entries). Thus, this map cancels out all such elements.
If $d_1+d_2\not\equiv 0\pmod{3}$, then let $x$ be the closest integer to $(d_1+d_2)/6$, and let $d_1'=d_1-11x/2$ and $d_2'=d_2-x/2$.
We have $d_1'+d_2'=\pm 1$, so there must exist an integer $y$ such that $d_1'=y/2\pm 1$ and $d_2'=-y/2$. This means that $d_1=y/2+11x/2\pm1$ and $d_2=-y/2+x/2$. It easily follows that in this case $(d_1,d_2)\in D$ has a value of $$d_1^2+11d_2^2=11x(3x\pm1)+y(3y\pm1)+1,$$ and therefore we can map $(d_1,d_2)$ to a copy of $(f_1=\mp y,f_2=\mp x)$.
Finally, it is a standard task to verify that $(d_1,d_2)$ and $(-d_1,-d_2)$ get mapped to the same pair $(f_1,f_2)$, and that, for any $(d_1,d_2)$, $f_1+f_2$ is even if $(d_1,d_2)$ is a $d$-tuple and odd if it is an $e$-tuple. This completes the bijection and the proof of the lemma.
\[6.222\] Let $S$ be the set containing 2 copies of the odd positive integers and 2 more copies of the odd multiples of 11, and $T$ the set containing 2 copies of the even positive integers and 2 more copies of the even multiples of 11. Then, for any $N\geq 3$, $$D_S(N)=2D_T(N-3).$$
Straightforward from Theorem \[main\] and Lemma \[6.2\].
Finally, we state as a conjecture the “missing lemma” of this paper, whose bijective proof eludes us. By Theorem \[main\], such a proof will imply a bijective proof also for the last of the five identities of the Schröter, Russell and Ramanujan type (the one modulo 23, proved analytically in [@Be], Theorem 7.2), and will therefore complete our unified combinatorial approach to the five identities.
\[7.2\] Condition (i) of Theorem \[main\] holds for $N_0=3$, $t=12$, $C_1=\dots=C_{12}=46$, $m=3$, and $$(A_1,\dots,A_{12})=(1,3,5,7,9,11,13,15,17,19,21,23),$$$$(B_1,\dots,B_{12})=(0,2,4,6,8,10,12,14,16,18,20,22).$$
**Corollary to Conjecture \[7.2\].** *Let $S$ be the set containing one copy of the odd positive integers and one more copy of the odd positive multiples of 23, and $T$ the set containing one copy of the even positive integers and one more copy of the even positive multiples of 23. Then, for any $N\geq 3$, $$D_S(N)=D_T(N-3).$$*
Acknowledgements {#acknowledgements .unnumbered}
================
This work, along with the subsequent paper [@CSFZ2], is the result of the first author’s MIT senior thesis, done in Summer and Fall 2011 under the supervision of the second author, and funded by the Institute through two UROP grants. The second author warmly thanks Richard Stanley for his terrific hospitality during the whole year, the MIT Math Department for partial financial support, and Dr. Gockenbach and the Michigan Tech Math Department, from which he was on partial leave, for extra Summer support. The two authors also wish to thank the anonymous referees for comments, and Abhinav Kumar, Joel Lewis, and Richard Stanley for helpful discussions related to the materials of this work. Finally, the second author wants to acknowledge to be quite a distant second: the first author has done the better part of this project.
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abstract: 'A new example of a saturated Kochen-Specker (KS) type configuration of 64 rays in 8-dimensional space (the Hilbert space of a triple of qubits) is constructed. It is proven that this configuration has a tropical dimension 6 and that it contains a critical subconfiguration of 36 rays. A natural multicoloured generalisation of the Kochen-Specker theory is given based on a concept of an entropy of a saturated configuration of rays.'
address: ' Department of Mathematics and Computer Science, University of Antwerp, Middelheim Campus Building G, Middelheimlaan 1, B-2020, Antwerp, Belgium '
author:
- 'Artur E. Ruuge'
title: 'New examples of Kochen-Specker type configurations on three qubits'
---
Introduction
============
The purpose of the present paper is to introduce some new theoretical concepts related to the Bell-Kochen-Specker theory, and to illustrate them on a new example of a *saturated* Kochen-Specker (KS) configuration in eight-dimensional space (the Euclidean space of three qubits). The saturated configurations are of special interest due to their symmetry properties. The present configuration consists of $64 = 2^{6}$ rays and it is the smallest non-trivial *saturated* KS configuration known at this moment in dimension $8$.
The KS configurations are sometimes termed “non-colourable” [@ZimbaPenrose] and the terminology associated to them can slightly vary from author to author. The original construction of S. Kochen and E. P. Specker can be found in [@KochenSpecker] (a configuration of $117$ rays in three dimensions), and a motivation behind their work can be traced back to the results of J. S. Bell [@Bell] (the Bell inequalities in quantum mechanics).
The KS configurations are of interest for the foundations of quantum mechanics [@AbramskyHardy; @BancalGisinPironio; @IshamLinden; @Smith; @Vourdas; @Wilce] since they do not refer directly to the concept of *probability*. An additional reason to study such configurations is due to the links to some issues of quantum gravity [@ButterfieldIsham1; @ButterfieldIsham2; @HamiltonIshamButterfield; @Isham; @IshamButterfield; @IshamLinden]. On the other hand, they are of interest in the quantum information science (QIS). The literature about QIS is quite extensive in our days.
The example of the saturated KS configuration constructed in the present paper (64 rays) is similar to the configurations described, for instance, in [@CabelloEstebaranzGarcia-Alcaine; @Mermin; @Planat] in the sense that the coordinates of the the vectors representing the rays can be chosen to be $0$, $1$, or $-1$. In a recent paper [@WaegellAravind] M. Waegell and P. K. Aravind point out that there exists a rather large class of KS proofs related to the Pauli group on $N$ qubits. The configurations of rays underlying such proofs can be described by vectors with coordinates $0$, $\pm 1$, and $\pm i$, where $i$ is the imaginary unit. I observe that the saturated configuration of 64 rays mentioned admits such a proof. I also compute a *critical* subconfiguration $\mathcal{N} \subset \mathcal{M}$ in it (i.e. the KS configuration $\mathcal{N}$ which cannot be made smaller by deleting the rays) which happens to contain $36$ rays (the same number as in the critical configuration of [@KernaghanPeres], but the configurations are *not* isomorphic).
In the present paper I consider *three* types of KS configurations: saturated, critical, and *tropical*. The first two types (saturated and critical) have already appeared in the literature [@Larsson; @PavicicMerletMcKayMegill; @Ruuge2; @RuugeFVO1; @ZimbaPenrose], and the term “tropical” is new. It is a “critical” configuration, but in a different sense, which is explained in the next section. It is being used as an intermediate step to find a “small” critical subconfiguration of $\mathcal{M}$. There is a sequence of inclusions: $$\mathcal{N} \subset \mathcal{T} \subset \mathcal{M},$$ where $\mathcal{M}$ is a saturated KS configuration (64 rays), $\mathcal{T}$ is tropical (48 rays), and $\mathcal{N}$ is critical (36 rays). The configuration $\mathcal{N}$ is obtained from $\mathcal{T}$ by deleting the rays.
Small critical KS configurations are more easy to test in experiment (they must be less expensive). For instance, there exists an example of a KS configuration in four dimensions consisting of just 18 rays [@CabelloEstebaranzGarcia-Alcaine] which has been tested experimentally [@Experiment1; @Experiment2; @Experiment3; @Experiment4]. The dimension four can be perceived as the dimension of the Euclidean space corresponding to a pair of qubits ($4 = 2 \times 2$). In the present paper the dimension is eight, which is the dimension of a three-qubit system ($8 = 2 \times 2 \times 2$). A triple of qubits is a rather special quantum object on its own. To observe the EPR-type correlations [@EPR] one may work with only two qubits, but to observe the *really* weird features of quantum theory it is better to have at least three qubits. A recent experimental realisation of a triple of qubits on a “single chip” has been reported in [@IBM]. In another work, it has recently been shown [@VertesiBrunner] that a triple of qubits suffices to refute the conjecture of A. Peres about quantum nonlocality and entanglement distillability. One can observe Berry’s phase [@Berry] on a triple of qubits and describe it in terms of quantum groups [@HuWuXue]. There exists a non-trivial link between the $E_8$-root system and the theory of three qubits found in [@Ruuge2; @RuugeFVO1]. It turns out that the rays represented by the roots of an $E_8$-type Lie algebra yield an example of a *saturated* Kochen-Specker type configuration and that this fact can be generalised in several different ways. One can obtain an infinite family of orthoalgebras [@RuugeFVO2] starting from the $E_8$-root system which can be of interest in some approaches to quantum gravity. On the other hand one can scale up and deform this example on multiple qubits [@Ruuge1], as well as one can consider other root systems [@Ruuge2].
The present paper focuses on the case of a three qubit system, although other important KS examples are known in other dimensions [@AravindLee-Elkin; @BadziagBengtssonCabelloPitowsky; @Cabello; @CabelloEstebaranzGarcia-Alcaine; @HarveyChryssanthacopoulos; @KochenSpecker; @Peres; @Planat; @PlanatSaniga; @Ruuge1; @Ruuge2; @WaegellAravind; @ZimbaPenrose]. In the beginning I introduce several new definitions and then I test them on the saturated configuration $\mathcal{M}$ (64 rays) mentioned above. The proof that the configuration is of a Kochen-Specker type can be given analytically (see Appendix A), or checked on a computer. The supplementary files mentioned in the main text can be found on arXiv or on my personal webpage. It is worth to draw attention to the concept of an *entropy* of a configuration of rays: a Kochen-Specker colouring corresonds to a zero entropy $S = 0$. It is possible to consider more general colourings with $S > 0$. This naturally leads to a *multicoloured* generalisation of the Kochen-Specker theory, which is not immediately visible in other approaches (KS proofs in terms of operators). I mention a couple of examples related to multi-colourings, but I leave the rest for another paper.
Terminology and notation
========================
Throughout the paper, $$[n] := \lbrace 0, 1, \dots, n - 1 \rbrace,$$ where $n = 1, 2, \dots$. It is a little more convenient to start counting from $0$ rather than from $1$ having in mind the C programming language.
Let $\mathcal{M}$ be a *finite* collection of rays in a Euclidean space $\mathbb{C}^d$. A Kochen-Specker (KS) colouring on $\mathcal{M}$ is a function $f: \mathcal{M} \to \lbrace 0, 1 \rbrace$, such that 1) for any $x, y \in \mathcal{M}$, if $x$ and $y$ are orthogonal, then at least one of the values $f (x)$ or $f (y)$ is zero; 2) if $x_0, x_1, \dots, x_{d - 1} \in \mathcal{M}$ is a set of mutually orthogonal rays, then $\sum_{i = 0}^{d - 1} f (x_i) = 1$. The collection $\mathcal{M}$ is termed a *KS configuration* of rays iff it does not admit a KS colouring. A KS configuration $\mathcal{M}$ is termed *critical* iff it can not be made smaller by deleting the rays, i.e. iff for any $x \in \mathcal{M}$ the collection $\mathcal{M} \backslash \lbrace x \rbrace$ admits a KS colouring. A collection of rays $\mathcal{M}$ (not necessary a KS configuration) is termed *saturated* iff every collection $x_0, x_1, \dots, x_{k - 1} \in \mathcal{M}$ of $k$ mutually orthogonal rays, where $k = 1, 2, \dots, d - 1$, can be extended by $x_{k}, x_{k + 1}, \dots, x_{d - 1} \in \mathcal{M}$ to a collection of $d$ mutually orthogonal rays ($d$ is the dimension of space).
It is natural to associate to $\mathcal{M}$ an undirected graph as follows: the vertices are the elements of $\mathcal{M}$; a pair of vertices $x, y \in \mathcal{M}$ is connected with an edge iff $x \perp y$. Let us term this graph the *orthogonality graph* of $\mathcal{M}$ and denote it $\Gamma (\mathcal{M})$. A *clique* in $\Gamma (\mathcal{M})$ of size $k$ is a collection of $k$ distinct vertices $x_0, x_1, \dots, x_{k - 1} \in \mathcal{M}$ such that every vertex $x_i$ is connected to every other vertex $x_j$ with an edge, $i \not = j$, $i, j \in [k]$. A *maximal clique* in $\Gamma(M)$ is a clique of the largest possible size $\omega \leqslant d$ ($d$ is the dimension of the Euclidean space). Denote the set of all cliques of size $k$ as $\mathcal{P}_{\perp}^{(k)} (\mathcal{M})$, $k \in \mathbb{Z}_{> 0}$. A *clique covering* of the graph $\Gamma (\mathcal{M})$ is a collection of cliques $U_i \in \mathcal{P}_{\perp}^{(m_i + 1)} (\mathcal{M})$, $m_{i} \in [d]$, $i \in [n]$, such that $\cup_{i \in [n]} U_i = \mathcal{M}$. A *clique partition* of the graph $\Gamma (\mathcal{M})$ is a clique covering $\lbrace U_i \rbrace_{i \in [n]}$ such that $U_i \cap U_j = \emptyset$, $i \not = j$, $i, j \in [n]$. An *anticlique* in $\Gamma (\mathcal{M})$ of size $k$ is a collection of $k$ distinct vertices $x_0, x_1, \dots, x_{k - 1} \in \mathcal{M}$ such that every vertex $x_i$ is *not* connected to any other vertex $x_j$ with an edge, $i \not = j$, $i, j \in [k]$. A maximal anticlique can have a size greater than $d$. Denote the set of all anticliques of size $k$ as $\mathcal{A}_{\perp}^{(k)} (\mathcal{M})$, $k = 1, 2, \dots $.
Configurations of rays $\mathcal{M}$ and $\mathcal{M}'$ are termed *isomorphic* iff their orthogonality graphs $\Gamma (\mathcal{M})$ and $\Gamma (\mathcal{M'})$ are isomorphic.
The concept of an isomorphism of configurations should not be confused with an isomorphism of *proofs* that the configurations do not admit a KS colouring (the *KS proofs*). A KS proof is given by a *hypergraph* with a set of vertices $V \subset \mathcal{M}$, and a set of hyper-edges $U_0, U_1, \dots, U_{n - 1} \in \mathcal{P}_{\perp}^{(d)} (\mathcal{M})$. The property that ensures that $\mathcal{M}$ is a KS configuration is as follows: one cannot construct a function $f: \mathcal{M} \to \lbrace 0, 1 \rbrace$ such that for every $i \in [n]$ there is a unique $x \in U_i$ assigned with $f (x) = 1$. Such hyper-graphs (in case $d = 4$) are studied in the approach of [@McKayMegillPavicic; @MegillKresimirWaegellAravindPavicic; @PavicicMcKayMegillFresl; @PavicicMerletMcKayMegill] using some algorithms from [@McKay]. An *isomorphism of KS proofs* is an isomorphism of these hyper-graphs.
A *signature* of a configuration of rays $\mathcal{M}$ is a pair of functions $(f, g)$, where $f: \mathbb{Z}_{> 0} \to \mathbb{Z}$, $k \mapsto \# \mathcal{P}_{\perp}^{(k)} (\mathcal{M})$, and $g: \mathbb{Z}_{> 0} \to \mathbb{Z}$, $k \mapsto \# \mathcal{A}_{\perp}^{(k)} (\mathcal{M})$.
If the signatures of configurations $\mathcal{M}$ and $\mathcal{M}'$ are different, then the configurations cannot be isomorphic.
Let $\mathcal{M}$ be a set of rays in a $d$-dimensional Euclidean space. Let $\lbrace U_i \rbrace_{i \in [n]}$ be a collection of cliques $U_i \in \mathcal{P}_{\perp}^{(m_i + 1)} (\mathcal{M})$, $m_i \in [d]$, $i \in [n]$. An *anticlique section* of this collection is a function $f: \cup_{i \in [n]} U_i \to \lbrace 0, 1 \rbrace$, such that for every $i \in [n]$ there exists a unique $x \in U_i$ such that $f (x) = 1$.
Notice that if $f$ is an anticlique section of $\lbrace U_i \rbrace_{i \in [n]}$, then $f^{- 1} (\lbrace 1 \rbrace) \in \mathcal{A}_{\perp}^{(k)} (\mathcal{M})$, where $k \leqslant n$. The cardinality $k$ is not necessary equal to $n$ since the cliques $U_i$, $i \in [n]$, can have non-empty intersections. It is also important to point out that even if $\lbrace U_i \rbrace_{i \in [n]}$ admits an anticlique section, the union $\cup_{i \in [n]} U_i$ can still be a KS configuration, since there can be other orthogonalities between the rays which are not described by $U_i$, $i \in [n]$.
Let $\mathcal{M}$ be a KS configuration of rays in a $d$-dimensional Euclidean space, such that every pair of orthogonal rays $x, y \in \mathcal{M}$, $x \perp y$, is contained in a maximal clique $U \in \mathcal{P}_{\perp}^{(d)} (\mathcal{M})$. The smallest integer $n$ such that there exists a collection of maximal cliques $\lbrace U_i \rbrace_{i \in [n]}$, $U_i \in \mathcal{P}_{\perp}^{(d)} (\mathcal{M})$, $i \in [n]$, which does not admit an anticlique section, is termed a *tropical dimension* of $\mathcal{M}$. Notation: $n = \dim (\mathcal{M})$.
Notice that for the collection $\lbrace U_i \rbrace_{i \in [n]}$ mentioned in the definition, where $n = \dim (\mathcal{M})$ is the tropical dimension of $\mathcal{M}$, we can conclude that $\mathcal{T} := \cup_{i \in [n]} U_i$ is a KS configuration of rays. At the same time, the collection $\lbrace U_i \rbrace_{i \in [n]}$ is only a *proper* subset of the hyper-edges of the hyper-graph describing a KS proof for $\mathcal{T}$. The cliques $U_i$, $i \in [n]$, can even be mutually disjoint. In the present paper I compute the tropical dimension of the saturated KS configuration $\mathcal{M}$ of 64 rays in 8-dimensional space mentioned in the introduction (the answer is $\dim (\mathcal{M}) = 6$).
Let $\mathcal{T}$ be a KS configuration of rays in a $d$-dimensional Euclidean space, such that every pair of orthogonal rays $x, y \in \mathcal{T}$, $x \perp y$, is contained in a maximal clique $U \in \mathcal{P}_{\perp}^{(d)} (\mathcal{T})$. The configuration $\mathcal{T}$ is termed *tropical* iff there exists a covering $\lbrace U_i \rbrace_{i \in [n]}$ of $\mathcal{T}$ by $n = \dim (\mathcal{T})$ maximal cliques $U_i \in \mathcal{P}_{\perp}^{(d)} (\mathcal{T})$, $i \in [n]$, which does not admit an anticlique section.
In the present paper I use a tropical configuration $\mathcal{T} \subset \mathcal{M}$ (48 rays) as a starting point to search for a critical subconfiguration of $\mathcal{M}$. The output is a critical configuration $\mathcal{N} \subset \mathcal{T}$ consisting of 36 rays.
To describe $\mathcal{M}$ in more detail, it is of interest to consider a generalisation of the concept of a KS colouring. Let $\mathcal{M}$ be a finite *saturated* collection of rays a Euclidean space of dimension $d$, (not necessary a KS configuration). A function $f: \mathcal{M} \to [0, 1]$ is termed a *probability weight* on $\mathcal{M}$ iff for every $U \in \mathcal{P}_{\perp}^{(d)} (\mathcal{M})$ holds: $$\sum_{x \in U} f (x) = 1.$$ Consider an *entropy*: $$S_{U}^{f} := - \sum_{x \in U} f(x) \log f (x),$$ for every $U \in \mathcal{P}_{\perp}^{(d)} (\mathcal{M})$. If the function $f$ takes only two values $0$ and $1$, then we have a KS colouring of $\mathcal{M}$, and all the entropies $S_{U}^{f} = 0$, $U \in \mathcal{P}_{\perp}^{(d)} (\mathcal{M})$. It is important to notice that this does *not* imply that there is a quantum state with such entropies [@Deutsch; @WehnerWinter]. Associate to every $x \in \mathcal{M}$ an orthogonal projector ${\widehat{\pi}}_{x}$ on $x$ (i.e. a quantum observable). Every $U \in \mathcal{P}_{\perp}^{(d)} (\mathcal{M})$ corresponds to a maximal set of compatible observables $\lbrace {\widehat{\pi}}_{x} \rbrace_{x \in U}$. For every quantum state described by a statistical operator ${\widehat{\rho}}$, we can compute $$S_U [{\widehat{\rho}}] := - \sum_{x \in U} \mathrm{Tr} ({\widehat{\pi}}_{x} {\widehat{\rho}}) \log ( \mathrm{Tr} ({\widehat{\pi}}_{x} {\widehat{\rho}}) ).$$ In general, even if $\mathcal{M}$ admits a KS colouring $f$, one cannot find a quantum state ${\widehat{\rho}}$ such that $S_{U} [{\widehat{\rho}}] = 0$, for all $U \in \mathcal{P}_{\perp}^{(d)} (\mathcal{M})$.
Let us term a probability weight $f: \mathcal{M} \to [0, 1]$ *equientropic* iff there exists a constant $S_{0}^{(f)} \in \mathbb{R}$, such that $S_{U}^{f} = S_{0}^{(f)}$, for every $U \in \mathcal{P}_{\perp}^{(d)} (\mathcal{M})$. Denote the collection of all equientropic probability weights as $\mathcal{E}_{S} (\mathcal{M})$. Notice that $\mathcal{E}_{S} (\mathcal{M})$ is not empty, since it contains at least one function: $f (x) = 1/ d$, for all $x \in \mathcal{M}$. The corresponding entropy $S_{0}^{(f)} = \log (d)$ and this is the maximal possible value of $S_{0}^{(f)}$, $f \in \mathcal{E}_{S} (\mathcal{M})$. Put $$S (\mathcal{M}) := \inf \lbrace S_{0}^{(f)} \,|\, f \in \mathcal{E}_{S} (\mathcal{M}) \rbrace.$$ It is natural to term $S (\mathcal{M})$ an *entropy* of the configuration $\mathcal{M}$. The quantity $D (\mathcal{M}) := \exp (S (\mathcal{M}))$ is a certain “dimension” describing $\mathcal{M}$ which can be termed a *statistical weight* of $\mathcal{M}$.
The link between the concept of an entropy and the concept of a KS colouring provides a non-trivial theoretical insight about the Kochen-Specker topic. Take any $f \in \mathcal{E}_{S} (\mathcal{M})$ and denote the range of its values $w_0 < w_1 < \dots < w_{m - 1}$, $w_i \in [0, 1]$, $i \in [m]$, $m \leqslant d$. For every $U \in \mathcal{P}_{\perp}^{(d)} (\mathcal{M})$, denote $N_{i}^{U}$ the number of times $f (x)$ takes the value $w_i$ as $x$ varies over $U$. Then we have: $$\sum_{i \in [m]} N_{i}^{U} = d, \quad
\sum_{i \in [m]} N_{i}^{U} w_{i} = 1, \quad
\sum_{i \in [m]} N_{i}^{U} w_{i} \log (w_{i}) = S_{0}^{(f)},$$ for every $U \in \mathcal{P}_{\perp}^{(d)} (\mathcal{M})$. If all values $w_i$ are *rational*, then write $w_i = q_i/ \Gamma$, where $q_i \in \mathbb{Z}$, $i \in [m]$, and assume that $\Gamma \in \mathbb{Z}_{> 0}$ is chosen as small as possible. We can perceive $f: \mathcal{M} \to [0, 1] \cap \mathbb{Q}$, $f \in \mathcal{E}_{S} (\mathcal{M})$, as a kind of “multicoloured” generalisation of a KS colouring: $q_i$ can be termed *mixed* colours, and the factors present in their factorisations into primes can be termed *primary* colours. In a particular case where all $q_i$ are *prime*, $i \in [m]$, we obtain just a condition: $N_{i}^{U} = c_i$, where $c_i \in \mathbb{Z}_{> 0}$, $i \in [m]$, for all $U \in \mathcal{P}_{\perp}^{(d)} (\mathcal{M})$. If we are not interested in a particular value of $S_{0}^{(f)}$, then this motivates the following:
Let $\mathcal{M}$ be a collection of rays in a Euclidean space of dimension $d$. Let $d = N_0 + N_1 + \dots + N_{s - 1}$ be a partition of $d$ into a sum of $s \geqslant 2$ positive integers, $N_0 \geqslant N_1 \geqslant \dots \geqslant N_{s - 1}$. A function $h: \mathcal{M} \to \lbrace 0, 1, \dots, s - 1 \rbrace$ is termed a *colouring of $\mathcal{M}$ compatible with this partition* iff for every clique $U \in \mathcal{P}_{\perp}^{(d)} (\mathcal{M})$ and for every $\alpha \in [s]$ holds: $\# \lbrace x \in U \,|\, h (x) = \alpha \rbrace = N_{\alpha}$.
A KS-colouring is just a colouring compatible with the partition $(d - 1, 1) \vdash d$. In the present paper I give a pair of examples of colourings compatible with other partitions, and an example when such a colouring does not exist.
Saturated Kochen-Specker on three qubits
========================================
The dimension of the Hilbert space of a single qubit is 2. The dimension of the Hilbert space $\mathcal{H}$ of a triple of qubits is $d = 8$, $8 = 2 \times 2 \times 2$. In this section we describe a new *saturated* Kochen-Specker configuration $\mathcal{M}$ in $\mathcal{H}$ consisting of $N = 64$ rays: $\mathbb{C} \psi_0, \mathbb{C} \psi_1, \dots, \mathbb{C} \psi_{N - 1}$. Write $\psi_{i}$ as a line of eight numbers (the coordinates of a vector in a selected basis), $$\psi_{i} = (\psi_{i}^{(0)}, \psi_{i}^{(1)}, \dots, \psi_{i}^{(d - 1)}),$$ where $i \in [N]$. The inner product $\langle -, - \rangle$ on $\mathcal{H} = \mathbb{C}^{d}$ is then given by $$\langle \psi_i, \psi_j \rangle =
\sum_{k = 0}^{d - 1} {\overline{\psi}}_{i}^{(k)} \psi_{j}^{(k)},$$ where the bar denotes complex conjugation, $i, j \in [N]$. The collection of vectors $\lbrace \psi_{i} \rbrace_{i \in [64]}$ described below has an additional property: $$\psi_{i}^{(k)} \in \lbrace -1, 0, 1 \rbrace,$$ for $i \in [N]$, and $k \in [d]$, so it makes sense to use a special notation. We write $\bar 1$ instead of $-1$ and skip the spaces and commas between the coordinates of a vector: for example, $\psi = (1 {\bar 1} {\bar 1} 1 0 0 0 0)$ is a vector $\psi = (1, -1, -1, 1, 0, 0, 0, 0)$. Put: $$\begin{array}{lll}
\psi_{0} := (1 {\bar 1} {\bar 1} {\bar 1} {\bar 1} {\bar 1} {\bar 1} 1 ),
&\psi_{1} := (1 {\bar 1} {\bar 1} {\bar 1} {\bar 1} {\bar 1} 1 {\bar 1} ),
&\psi_{2} := (1 {\bar 1} {\bar 1} {\bar 1} 1 1 {\bar 1} 1 ), \\
\psi_{3} := (1 {\bar 1} {\bar 1} {\bar 1} 1 1 1 {\bar 1} ),
&\psi_{4} := (1 {\bar 1} {\bar 1} 1 {\bar 1} {\bar 1} {\bar 1} {\bar 1} ),
&\psi_{5} := (1 {\bar 1} {\bar 1} 1 {\bar 1} {\bar 1} 1 1 ), \\
\psi_{6} := (1 {\bar 1} {\bar 1} 1 1 1 {\bar 1} {\bar 1} ),
&\psi_{7} := (1 {\bar 1} {\bar 1} 1 1 1 1 1 ),
&\psi_{8} := (1 {\bar 1} 1 {\bar 1} {\bar 1} {\bar 1} {\bar 1} {\bar 1} ), \\
\psi_{9} := (1 {\bar 1} 1 {\bar 1} {\bar 1} {\bar 1} 1 1 ),
&\psi_{10} := (1 {\bar 1} 1 {\bar 1} 1 1 {\bar 1} {\bar 1} ),
&\psi_{11} := (1 {\bar 1} 1 {\bar 1} 1 1 1 1 ), \\
\psi_{12} := (1 {\bar 1} 1 1 {\bar 1} {\bar 1} {\bar 1} 1 ),
&\psi_{13} := (1 {\bar 1} 1 1 {\bar 1} {\bar 1} 1 {\bar 1} ),
&\psi_{14} := (1 {\bar 1} 1 1 1 1 {\bar 1} 1 ), \\
\psi_{15} := (1 {\bar 1} 1 1 1 1 1 {\bar 1} ),
&\psi_{16} := (1 1 {\bar 1} {\bar 1} {\bar 1} 1 {\bar 1} 1 ),
&\psi_{17} := (1 1 {\bar 1} {\bar 1} {\bar 1} 1 1 {\bar 1} ), \\
\psi_{18} := (1 1 {\bar 1} {\bar 1} 1 {\bar 1} {\bar 1} 1 ),
&\psi_{19} := (1 1 {\bar 1} {\bar 1} 1 {\bar 1} 1 {\bar 1} ),
&\psi_{20} := (1 1 {\bar 1} 1 {\bar 1} 1 {\bar 1} {\bar 1} ), \\
\psi_{21} := (1 1 {\bar 1} 1 {\bar 1} 1 1 1 ),
&\psi_{22} := (1 1 {\bar 1} 1 1 {\bar 1} {\bar 1} {\bar 1} ),
&\psi_{23} := (1 1 {\bar 1} 1 1 {\bar 1} 1 1 ), \\
\psi_{24} := (1 1 1 {\bar 1} {\bar 1} 1 {\bar 1} {\bar 1} ),
&\psi_{25} := (1 1 1 {\bar 1} {\bar 1} 1 1 1 ),
&\psi_{26} := (1 1 1 {\bar 1} 1 {\bar 1} {\bar 1} {\bar 1} ), \\
\psi_{27} := (1 1 1 {\bar 1} 1 {\bar 1} 1 1 ),
&\psi_{28} := (1 1 1 1 {\bar 1} 1 {\bar 1} 1 ),
&\psi_{29} := (1 1 1 1 {\bar 1} 1 1 {\bar 1} ), \\
\psi_{30} := (1 1 1 1 1 {\bar 1} {\bar 1} 1 ),
&\psi_{31} := (1 1 1 1 1 {\bar 1} 1 {\bar 1} ),
\end{array}
\label{eq:saturated_part1}$$ and $$\begin{array}{lll}
\psi_{32} := (1 {\bar 1} {\bar 1} 1 0 0 0 0 ),
&\psi_{33} := (1 {\bar 1} {\bar 1} {\bar 1} 0 0 0 0 ),
&\psi_{34} := (1 {\bar 1} 0 0 0 0 {\bar 1} 1 ), \\
\psi_{35} := (1 {\bar 1} 0 0 0 0 {\bar 1} {\bar 1} ),
&\psi_{36} := (1 {\bar 1} 0 0 0 0 1 1 ),
&\psi_{37} := (1 {\bar 1} 0 0 0 0 1 {\bar 1} ), \\
\psi_{38} := (1 {\bar 1} 1 1 0 0 0 0 ),
&\psi_{39} := (1 {\bar 1} 1 {\bar 1} 0 0 0 0 ),
&\psi_{40} := (1 1 {\bar 1} 1 0 0 0 0 ), \\
\psi_{41} := (1 1 {\bar 1} {\bar 1} 0 0 0 0 ),
&\psi_{42} := (1 1 0 0 0 0 {\bar 1} 1 ),
&\psi_{43} := (1 1 0 0 0 0 {\bar 1} {\bar 1} ), \\
\psi_{44} := (1 1 0 0 0 0 1 1 ),
&\psi_{45} := (1 1 0 0 0 0 1 {\bar 1} ),
&\psi_{46} := (1 1 1 1 0 0 0 0 ), \\
\psi_{47} := (1 1 1 {\bar 1} 0 0 0 0 ),
&\psi_{48} := (0 0 1 {\bar 1} {\bar 1} 1 0 0 ),
&\psi_{49} := (0 0 1 1 {\bar 1} 1 0 0 ), \\
\psi_{50} := (0 0 0 0 1 {\bar 1} {\bar 1} 1 ),
&\psi_{51} := (0 0 0 0 1 {\bar 1} {\bar 1} {\bar 1} ),
&\psi_{52} := (0 0 0 0 1 {\bar 1} 1 1 ), \\
\psi_{53} := (0 0 0 0 1 {\bar 1} 1 {\bar 1} ),
&\psi_{54} := (0 0 1 1 1 {\bar 1} 0 0 ),
&\psi_{55} := (0 0 1 {\bar 1} 1 {\bar 1} 0 0 ), \\
\psi_{56} := (0 0 1 {\bar 1} {\bar 1} {\bar 1} 0 0 ),
&\psi_{57} := (0 0 1 1 {\bar 1} {\bar 1} 0 0 ),
&\psi_{58} := (0 0 0 0 1 1 {\bar 1} 1 ), \\
\psi_{59} := (0 0 0 0 1 1 {\bar 1} {\bar 1} ),
&\psi_{60} := (0 0 0 0 1 1 1 1 ),
&\psi_{61} := (0 0 0 0 1 1 1 {\bar 1} ), \\
\psi_{62} := (0 0 1 1 1 1 0 0 ),
&\psi_{63} := (0 0 1 {\bar 1} 1 1 0 0 ).
\end{array}
\label{eq:saturated_part2}$$
The collection of rays , , $$\mathcal{M} := \lbrace \mathbb{C} \psi_{0}, \mathbb{C} \psi_1, \dots, \mathbb{C} \psi_{63} \rbrace$$ is a saturated Kochen-Specker configuration in $\mathcal{H} = \mathbb{C}^{8}$.
*Proof.* Denote $N = 64$, $d = 8$. One may use the following strategy to verify that $\mathcal{M}$ is a KS configuration on a computer. First, find all non-empty subsets $I \subseteq [N]$ such that $\mathbb{C} \psi_{i}$ is not orthogonal to $\mathbb{C} \psi_{j}$, for any $i, j \in I$. The function $f_I: \mathcal{M} \to \lbrace 0, 1 \rbrace$, defined as $f_{I} (\mathbb{C} \psi_i) = 1$, if $i \in I$, and $f_{I} (\mathbb{C} \psi_j) = 0$, if $j \in [N] \backslash I$, is a *candidate* for a Kochen-Specker colouring of $\mathcal{M}$. One needs to check, that for every $I$, the collection $\mathcal{M} \backslash \lbrace \mathbb{C} \psi_{i} \rbrace_{i \in I}$ always contains a tuple of $d$ mutually orthogonal rays.
To generate the tuples of mutually orthogonal rays one can proceed as follows. Take $i_0 \in [N]$. Then find $i_1 \in [N]$ such that $i_1 > i_0$ and $\mathbb{C} \psi_{i_1} \perp \mathbb{C} \psi_{i_0}$. After that find $i_2 \in [N]$ such that $i_2 > i_1$ and $\mathbb{C} \psi_{i_2} \perp \mathbb{C} \psi_{i_1}, \mathbb{C} \psi_{i_0}$, and so on. Having a tuple $i_0 < i_1 < \dots < i_{s - 1}$ of mutually orthogonal rays, $1 \leqslant s \leqslant d - 1$, one needs to check that there always exists a $j \in [N] \backslash \lbrace i_0, i_1, \dots, i_{s - 1} \rbrace$ such that $\mathbb{C} \psi_{j} \perp \mathbb{C} \psi_{k}$, for all $k \in \lbrace i_0, i_1, \dots, i_{s - 1} \rbrace$. This establishes the saturation property.
For a realisation of this method in C look at the supplementary file `saturated.c`. The supplementary file `vectors.txt` contains a sequence of $64 \times 8 = 512$ numbers $a_0, a_1, \dots, a_{511}$ separated by a space, such that $\psi_{i}^{(k)} = a_{i*8 + k}$, for $i \in [64]$, $k \in [8]$. An analytical way to prove that the configuration $\mathcal{M}$ is of the Kochen-Specker type is discussed in the Appendix A. The details of how $\mathcal{M}$ had actually been found are given in the Appendix B.
*Remark.* The configuration $\mathcal{M}$ does not admit a colouring compatible with the partition $(7, 1) \vdash 8$, but it admits other types of colourings. Without going into details, one can mention that $\mathcal{M}$ admits colourings compatible with the partitions $(6, 2) \vdash 8$ and $(4, 4) \vdash 8$, but not with the partition $(5, 3) \vdash 8$. An example of a colouring compatible with $(6, 2) \vdash 8$ is an indicator function $h = 1_{\mathcal{A}}$ of $\mathcal{A} = \lbrace \mathbb{C} \psi_i \rbrace_{i \in A}$, where $$A := \lbrace 0, 1, 3, 4, 5, 7, 11, 15, 32, 33, 36, 37, 60, 61, 62, 63 \rbrace,$$ and an example of a colouring compatible with $(4, 4) \vdash 8$ is an indicator function $h = 1_{\mathcal{B}}$ of $\mathcal{B} = \lbrace \mathbb{C} \psi_i \rbrace_{i \in B}$, where $$\begin{gathered}
B := \lbrace 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, \\
32, 33, 34, 35, 36, 37, 38, 39, 56, 57, 58, 59, 60, 61, 62, 63 \rbrace. \end{gathered}$$ For every $U \in \mathcal{P}_{\perp}^{(8)} (\mathcal{M})$ holds: $\# U \cap \mathcal{A} = 2$, and $\# U \cap \mathcal{B} = 4$. $\Diamond$
*Remark.* It is of interest to point out the following property of the configuration $\mathcal{M}$, $\# \mathcal{M} = 64$. If we look at all maximal cliques of the orthogonality graph $\Gamma (\mathcal{M})$ then it turns out that the cardinality of an intersection $\# U \cap U'$, as $U, U' \in \mathcal{P}_{\perp}^{(8)} (\mathcal{M})$, $U \not = U'$, can be $0, 1, \dots, 6$, but is never $7$. At the same time, the configuration $\mathcal{M}$ is saturated, so we conclude: $$\forall W \in \mathcal{P}_{\perp}^{(7)} (\mathcal{M}) \,
\exists ! U \in \mathcal{P}_{\perp}^{(8)} (\mathcal{M}): U \supseteq W.
\label{eq:Steiner}$$ This property is similar to the property required in the definition of a *Steiner system* $S (t, k, n)$, $t = 7$, $k = 8$, $n = 64$, except that in the case of $S (64, 8, 7)$ arbitrary subsets of cardinality 7 are allowed, while in one restricts to subsets of mutually orthogonal elements. $\Diamond$
Critical Kochen-Specker on three qubits
=======================================
In [@RuugeFVO1] one has constructed an example of a *saturated* Kochen-Specker configuration in 8-dimensional space. This configuration contains 120 rays and extends the Kochen-Specker example found by [@KernaghanPeres] (40 rays). It turns out that this 120 rays can be perceived as the rays represented by the 240 elements of the $E_8$-root system (each ray is represented by a pair of roots $v$ and $-v$).
In the previous section we have described a new *saturated* configuration formed by 64 rays in 8 dimensions. It turns out that the number of rays producing the Kochen-Specker property can be reduced at the expense of sacrificing the saturation property. In this section I describe a *critical* subconfiguration $\mathcal{N} \subseteq \mathcal{M}$ consisting of 36 rays. We keep the notation , , for the vectors $\psi_i \in \mathbb{C}^{8}$, $i \in [64]$.
The collection of rays $\mathcal{N} := \lbrace
\mathbb{C} \psi_{I}
\rbrace_{i \in I}$, where $$\begin{gathered}
I := \lbrace
2, 3, 4, 9, 12, 13, 14, 15, 16, 19, 21, 23, 24, 25, 26, 27, 29, 30, \\
32, 33, 34, 37, 39, 40, 41, 43, 46, 48, 51, 52, 55, 58, 59, 60, 61, 62
\rbrace
\label{eq:critical}\end{gathered}$$ is a critical Kochen-Specker configuration in $\mathcal{H} = \mathbb{C}^{8}$, $\# \mathcal{N} = 36$.
*Proof.* The fact that $\mathcal{N}$ does not admit a Kochen-Specker colouring is checked on a personal computer in analogy with the configuration $\mathcal{M}$. One also needs to check that for every $i \in I$, the collection $\mathcal{N} \backslash \lbrace \mathbb{C} \psi_{i} \rbrace$ does admit a Kochen-Specker colouring.
For an implementation in C take the supplementary file `critical.c`. The sequence of numbers $i_0 < i_1 < \dots < i_{35}$ which form the set $I = \lbrace i_{\alpha} \rbrace_{\alpha = 0}^{35}$ is written in a supplementary file `critseq.txt` using space as a separator.
Note that precisely half of the vectors $\psi_{i}$ described by representing the rays in $\mathcal{N}$ have non-zero coordinates $\psi_{i}^{(k)} \in \lbrace \pm 1 \rbrace$, $k \in [8]$, i.e. $i < 32$, while the other half comes from the $32 \leqslant i < 64$ part. To explain how the configuration $\mathcal{N}$ had actually been found we need first to introduce the tropical subconfigurations in $\mathcal{M}$. This is done in the next section and the required details are given in the Appendix B.
Tropical Kochen-Specker on three qubits
=======================================
Let $\mathcal{M}$ be a collection of $n$ rays in a $d$-dimensional Euclidean space $\mathcal{H}$. Let $(C_{\mathcal{M}}, {\widetilde}{C}_{\mathcal{M}})$ denote the *signature* of $\mathcal{M}$. Notice that if $\mathcal{M}$ is a single $d$-tuple of mutually orthogonal rays, then $C_{\mathcal{M}} (k)$, $k \in \mathbb{Z}_{> 0}$, is a binomial coefficient $C_{d}^{k}$. A signature is a convenient concept which allows to formulate a *necessary* condition meant to distinguish a pair of configurations with non-isomorphic orthogonality graphs. If the signatures are different, then the orthogonality graphs cannot be isomorphic. At the same time, intuitively, if the the signatures of two configurations coincide, then there is a “high chance” that their orthogonality graphs are isomorphic, although some additional investigation is necessary to establish this isomorphism.
The signature of the critical Kochen-Specker configuration $\mathcal{N}$ described by is as follows: $$\begin{gathered}
(C_{\mathcal{N}} (1), C_{\mathcal{N}} (2), \dots, C_{\mathcal{N}} (8)) =
(36, 346, 1224, 2063, 1776, 830, 204, 21), \\
({\widetilde}{C}_{\mathcal{N}} (1), {\widetilde}{C}_{\mathcal{N}} (2), {\widetilde}{C}_{\mathcal{N}} (3), {\widetilde}{C}_{\mathcal{N}} (4)) =
(36, 284, 536, 212), \\
\end{gathered}$$ and $C_{\mathcal{N}} (k) = 0$, if $k > 8$, and ${\widetilde}{C}_{\mathcal{N}} (k) = 0$, if $k > 4$.
The signature of the saturated Kochen-Specker configuration $\mathcal{M}$ described by , is as follows: $$\begin{gathered}
(C_{\mathcal{M}} (1),
\dots, C_{\mathcal{M}} (8)) =
(64, 992, 5056, 11504, 13312, 8192, 2560, 320), \\
({\widetilde}{C}_{\mathcal{M}} (1), {\widetilde}{C}_{\mathcal{M}} (2), \dots, {\widetilde}{C}_{\mathcal{M}} (6)) =
(64, 1024, 4864, 8512, 5632, 1536), \\
\end{gathered}$$ and $C_{\mathcal{M}} (k) = 0$, if $k > 8$, and ${\widetilde}{C}_{\mathcal{M}} (k) = 0$, if $k > 6$.
It turns out that one can can compute all tropical subconfigurations in $\mathcal{M}$ and that all of them have the same signatures.
Let $\mathcal{M} = \lbrace \mathbb{C} \psi_{i} \rbrace_{i = 0}^{63}$ be the saturated Kochen-Specker configuration in $\mathcal{H} = \mathbb{C}^{d}$, $d = 8$, described in , . The following holds:
- The tropical dimension $\dim (\mathcal{M}) = 6$.
- The total number of tropical subconfigurations in $\mathcal{M}$ is 32. Their signatures coincide.
*Proof.* The computation of the tropical dimension $\dim_{d} (\mathcal{M})$ can be done on a personal computer. To generate *all* tropical subconfigurations a program written in C needs about a day on a modest machine.
The total number of maximal cliques in the orthogonality graph $\Gamma (\mathcal{M})$ is $n = 320$. Let us denote these cliques $U_0, U_1, \dots, U_{n - 1} \in \mathcal{P}_{\perp}^{(d)} (\mathcal{M})$, $d = 8$. A straightforward computation yields the following fact: any collection of maximal cliques $U_{i_0}, U_{i_1}, \dots, U_{i_{q - 1}} \in \mathcal{P}_{\perp}^{(d)} (\mathcal{M})$, $i_0 < i_1 < \dots < i_{q - 1}$, $i_{\alpha} \in [n]$, $\alpha \in [q]$, containing five elements or less, $q \leqslant 5$, admits an anticlique section. On the other hand (and this is the longest part of the computation), there exist 6-tuples $\lbrace i_0 < i_1 < \dots < i_5 \rbrace$ such that $\lbrace U_{i_{\alpha}} \rbrace_{\alpha \in [6]}$ does not admit an anticlique section. The number of variants of such tuples is $N = 308992$ and we denote the corresponding variants as $\lbrace i_{0}^{(\beta)} < i_{1}^{(\beta)} < \dots < i_{5}^{(\beta)} \rbrace$, $\beta \in [N]$. The corresponding computations can be found in the supplementary file `antisect.c`.
It turns out that for every $\beta \in [N]$ holds: $\# \cup_{\alpha \in [6]} U_{i_{\alpha}^{(\beta)}} = 48$. The dimension of space in our case is $d = 8$, so, since $48 = 6 \times 8$, we immediately conclude that every collection $\lbrace U_{i_{\alpha}^{(\beta)}} \rbrace_{\alpha \in [6]}$ is a collection of *mutually disjoint* maximal cliques, $\beta \in [N]$. Intuitively, one may perceive this observation as an effect of *“repulsion”* of cliques: the corresponding configuration tries to be as large as possible.
The unions $\cup_{\alpha \in [6]} U_{i_{\alpha}^{(\beta)}}$ are Kochen-Specker configurations, since any KS colouring of such a union would induce an anticlique section of $\lbrace U_{i_{\alpha}^{(\beta)}}\rbrace_{\alpha \in [6]}$, $\beta \in [N]$. So our conclusion is as follows: $\dim (\mathcal{M}) = 6$.
If we compute the image of the map $[N] \ni \beta \mapsto \cup_{\alpha \in [6]} U_{i_{\alpha}^{(\beta)}} \in \mathcal{P} (\mathcal{M})$, then it turns out that it contains only 32 different variants. It is straightforward to check that their signatures coincide.
Write the 32 tropical configurations mentioned as $\mathcal{T}_{m} = \lbrace \mathbb{C} \psi_{i} \rbrace_{i \in J_{m}}$, where $J_{m} = \lbrace j_{m}^{(0)} < j_{m}^{(1)} < \dots < j_{m}^{(47)} \rbrace$, for $m \in [32]$. The supplementary file `tropseq.txt` contains a sequence of $32 \times 48 = 1536$ numbers $b_0, b_1, \dots, b_{1535}$, separated by a space, such that $j_{m}^{(s)} = b_{48 m + s}$, $m \in [32]$, $s \in [48]$. The only tropical subconfiguration of $\mathcal{M}$ containing $\mathcal{N}$ is $\mathcal{T}_0$. The corresponding set $J_{0}$ is as follows: $$\begin{gathered}
J_{0} = \lbrace
0, 1, 2, 3, 4, 7, 9, 10,
12, 13, 14, 15, 16, 19, 20, 21, \\
22, 23, 24, 25, 26, 27, 29, 30,
32, 33, 34, 37, 38, 39, 40, 41, \\
43, 44, 46, 47, 48, 50, 51, 52,
53, 55, 57, 58, 59, 60, 61, 62
\rbrace.
\label{eq:tropical}\end{gathered}$$ We have: $\mathcal{N} \subseteq \mathcal{T}_{0} \subseteq \mathcal{M}$ and $\# \mathcal{T}_{0} = 48$. The computations are implemented in the supplementary file `tropical.c`.
The signature of the configuration $\mathcal{T} = \mathcal{T}_{0}$ described by in the proof of the theorem is of the shape: $$\begin{gathered}
(C_{\mathcal{T}} (1), C_{\mathcal{T}} (2), \dots, C_{\mathcal{T}} (8)) =
(48, 600, 2752, 6096, 7008, 4304, 1344, 168), \\
({\widetilde}{C}_{\mathcal{T}} (1), {\widetilde}{C}_{\mathcal{T}} (2), \dots, {\widetilde}{C}_{\mathcal{T}} (5)) =
(48, 528, 1536, 1312, 384), \\
\end{gathered}$$ and $C_{\mathcal{T}} (k) = 0$, if $k > 8$, and ${\widetilde}{C}_{\mathcal{T}} (k) = 0$, if $k > 5$.
Thirty-six rays
===============
There is another known example of a critical Kochen-Specker type configuration due to [@KernaghanPeres]. It so happens that it also contains 36 vectors, just like the configuration $\mathcal{N}$ described by . Are these configurations equivalent or not? In other words, are the corresponding orthogonality graphs isomorphic or not?
The critical configuration discovered in [@KernaghanPeres] can be described as follows. Consider first a configuration $\mathcal{T}' := \lbrace \mathbb{C} \varphi_{i} \rbrace_{i \in [40]}$, represented by 40 vectors of the shape: $$\begin{array}{lll}
\varphi_{0} := (1 0 0 0 0 0 0 0 ),
&\varphi_{1} := (0 1 0 0 0 0 0 0 ),
&\varphi_{2} := (0 0 1 0 0 0 0 0 ), \\
\varphi_{3} := (0 0 0 1 0 0 0 0 ),
&\varphi_{4} := (0 0 0 0 1 0 0 0 ),
&\varphi_{5} := (0 0 0 0 0 1 0 0 ), \\
\varphi_{6} := (0 0 0 0 0 0 1 0 ),
&\varphi_{7} := (0 0 0 0 0 0 0 1 ),
&\varphi_{8} := (1 1 1 1 0 0 0 0 ), \\
\varphi_{9} := (1 1 {\bar 1} {\bar 1} 0 0 0 0 ),
&\varphi_{10} := (1 {\bar 1} 1 {\bar 1} 0 0 0 0 ),
&\varphi_{11} := (1 {\bar 1} {\bar 1} 1 0 0 0 0 ), \\
\varphi_{12} := (0 0 0 0 1 1 1 1 ),
&\varphi_{13} := (0 0 0 0 1 1 {\bar 1} {\bar 1} ),
&\varphi_{14} := (0 0 0 0 1 {\bar 1} 1 {\bar 1} ), \\
\varphi_{15} := (0 0 0 0 1 {\bar 1} {\bar 1} 1 ),
&\varphi_{16} := (1 1 0 0 1 1 0 0 ),
&\varphi_{17} := (1 1 0 0 {\bar 1} {\bar 1} 0 0 ), \\
\varphi_{18} := (1 {\bar 1} 0 0 1 {\bar 1} 0 0 ),
&\varphi_{19} := (1 {\bar 1} 0 0 {\bar 1} 1 0 0 ),
&\varphi_{20} := (0 0 1 1 0 0 1 1 ), \\
\varphi_{21} := (0 0 1 1 0 0 {\bar 1} {\bar 1} ),
&\varphi_{22} := (0 0 1 {\bar 1} 0 0 1 {\bar 1} ),
&\varphi_{23} := (0 0 1 {\bar 1} 0 0 {\bar 1} 1 ), \\
\varphi_{24} := (1 0 1 0 1 0 1 0 ),
&\varphi_{25} := (1 0 1 0 {\bar 1} 0 {\bar 1} 0 ),
&\varphi_{26} := (1 0 {\bar 1} 0 1 0 {\bar 1} 0 ), \\
\varphi_{27} := (1 0 {\bar 1} 0 {\bar 1} 0 1 0 ),
&\varphi_{28} := (0 1 0 1 0 1 0 1 ),
&\varphi_{29} := (0 1 0 1 0 {\bar 1} 0 {\bar 1} ), \\
\varphi_{30} := (0 1 0 {\bar 1} 0 1 0 {\bar 1} ),
&\varphi_{31} := (0 1 0 {\bar 1} 0 {\bar 1} 0 1 ),
&\varphi_{32} := (1 0 0 1 0 1 {\bar 1} 0 ), \\
\varphi_{33} := (1 0 0 {\bar 1} 0 1 1 0 ),
&\varphi_{34} := (1 0 0 1 0 {\bar 1} 1 0 ),
&\varphi_{35} := (1 0 0 {\bar 1} 0 {\bar 1} {\bar 1} 0 ), \\
\varphi_{36} := (0 1 1 0 {\bar 1} 0 0 1 ),
&\varphi_{37} := (0 1 {\bar 1} 0 1 0 0 1 ),
&\varphi_{38} := (0 {\bar 1} 1 0 1 0 0 1 ), \\
\varphi_{39} := (0 {\bar 1} {\bar 1} 0 {\bar 1} 0 0 1 ).
\end{array}$$ After that construct another configuration by excluding four vectors: $$\mathcal{N}' := \lbrace \mathbb{C} \varphi_{i} \rbrace_{i \in [40] \backslash \lbrace 0, 12, 22, 31 \rbrace}.$$
The configuration $\mathcal{N}'$ is a critical Kochen-Specker configuration on three qubits.
*Remark.* There is a typo in the original paper [@KernaghanPeres]: by accident, the authors exclude the vector $\varphi_{27} = (1 0 {\bar 1} 0 {\bar 1} 0 1 0 )$ instead of $\varphi_{31} = (0 1 0 {\bar 1} 0 {\bar 1} 0 1 )$. $\Diamond$.
The configuration $\mathcal{N}'$ is an analogue of our configuration $\mathcal{N}$, and $\mathcal{T}'$ is an analogue of the tropical configuration $\mathcal{T}$. A *saturated* extension $\mathcal{M}'$ of $\mathcal{T}'$ has been constructed in [@RuugeFVO1]. It turns out that $\# \mathcal{M}' = 120$ and that the rays of $\mathcal{M}'$ can be represented by the vectors of the $E_8$ root system (this is an observation related to the question about the symmetry of $\mathcal{N}'$ stated in [@KernaghanPeres]). We have: $$\mathcal{N}' \subset \mathcal{T}' \subset \mathcal{M}'.$$ The configurations $\mathcal{N}$ and $\mathcal{N}'$ have the same cardinalities, $$\# \mathcal{N}' = 36, \quad
\# \mathcal{N} = 36,$$ but their signatures are different: $$\begin{gathered}
(C_{\mathcal{N}'} (1), C_{\mathcal{N}'} (2), \dots, C_{\mathcal{N}'} (8)) =
(36, 374, 1384, 1991, 1120, 416, 96, 11), \\
({\widetilde}{C}_{\mathcal{N}'} (1), {\widetilde}{C}_{\mathcal{N}'} (2), {\widetilde}{C}_{\mathcal{N}'} (3), {\widetilde}{C}_{\mathcal{N}'} (4)) =
(36, 256, 448, 192), \\
\end{gathered}$$ and $C_{\mathcal{N}'} (k) = 0$, if $k > 8$, and ${\widetilde}{C}_{\mathcal{N}'} (k) = 0$, if $k > 4$. Therefore the configuration of rays $\mathcal{N}$ can not be isomorphic to $\mathcal{N}'$.
The signature of the configuration $\mathcal{T}'$ is of the shape: $$\begin{gathered}
(C_{\mathcal{T}'} (1), C_{\mathcal{T}'} (2), \dots, C_{\mathcal{T}'} (8)) =
(40, 460, 1880, 2990, 1880, 780, 200, 25), \\
({\widetilde}{C}_{\mathcal{T}'} (1), {\widetilde}{C}_{\mathcal{T}'} (2), {\widetilde}{C}_{\mathcal{T}'} (3), {\widetilde}{C}_{\mathcal{T}'} (4)) =
(40, 320, 640, 320), \\
\end{gathered}$$ and $C_{\mathcal{T}'} (k) = 0$, if $k > 8$, and ${\widetilde}{C}_{\mathcal{T}'} (k) = 0$, if $k > 4$.
Let us also give (for the reference purposes) the signature of the $E_8$ configuration $\mathcal{M}'$: $$\begin{gathered}
(C_{\mathcal{M}'} (1), C_{\mathcal{M}'} (2), \dots, C_{\mathcal{M}'} (8)) = \\ =
(120, 3780, 37800, 122850, 113400, 56700, 16200, 2025), \end{gathered}$$ $$\begin{gathered}
({\widetilde}{C}_{\mathcal{M}'} (1), {\widetilde}{C}_{\mathcal{M}'} (2), \dots, {\widetilde}{C}_{\mathcal{M}'} (8)) = \\ =
(120, 3360, 31360, 120960, 241920, 241920, 103680, 8640), \end{gathered}$$ and $C_{\mathcal{M}'} (k) = 0$ and ${\widetilde}{C}_{\mathcal{M}'} (k) = 0$, if $k > 8$.
Note that it is of interest to search for multi-qubit generalisations of these configurations [@HarveyChryssanthacopoulos; @PlanatSaniga]. In [@Ruuge1] one can find an infinite family of Kochen-Specker type configurations generalising $\mathcal{T}'$ on any number of qubits $n = 4 m - 1$, $m \geqslant 1$.
Conclusion and discussion
=========================
The present paper describes explicitly a new *saturated* Kochen-Speker (KS) configuration of 64 rays containing a *critical* configuration of 36 rays on three qubits (8-dimensional space). It turns out that this saturated configuration has quite nice properties which can be studied in terms of *tropical* subconfigurations introduced in this paper.
In this section I make some informal remarks about the subject in general. First of all, it is quite natural to expect that the definition of a Kochen-Specker (KS) colouring (the latter is a certain function on a configuration with values 0 and 1) should have a “multicoloured” generalisation. Intuitively, one may think that these colours can be perceived as elements of some finite group or a similar algebraic structure. From the perspective of pure graph theory such a generalisation is certainly possible. The problem is that it is not immediately clear how to construct a generalisation which would still be of interest in quantum mechanics, but not just in pure mathematics. The approach suggested in the present paper is based on the concept of an *entropy* of a *saturated* configuration. The important step is the interpretation of a configuration of rays which admits a Kochen-Specker colouring as a configuration with entropy equal to zero.
At first sight, an extension of a given critical Kochen-Specker configuration to a saturated one is not really necessary since it only complicates things, i.e. adds new measuring devices to an experimental set-up as if there is not enough trouble with decoherence of quantum states. Nonetheless a saturated configuration provides a natural environment where a critical configuration “lives”. A saturated Kochen-Specker configuration is a rather special object with many symmetries and in general one would expect many isomorphic copies of a given critical configuration inside it. There can be different isomorphism classes of critical configurations. Intuitively, a saturated KS configuration, or its orthogonality graph, is a discrete analogue of the space of pure states of a quantum system. One can try to mimic quantum mechanics on such finite undirected graphs.
The objects that exist inside a saturated Kochen-Specker configuration $\mathcal{M}$ (e.g. saturated subconfigurations, critical subconfigurations, etc.) can “move”: if we take an automorphism of the orthogonality graph of $\mathcal{M}$, then a subgraph in it does not need to stay fixed. It can be transferred to an isomorphic copy of itself. A subconfiguration $\mathcal{N}$ in $\mathcal{M}$ has a *signature* (the numbers of cliques and anticliques in the orthogonality graph). It is natural to perceive the number of maximal cliques mentioned in the signature as a “*capacity*” of the configuration $\mathcal{N}$. Intuitively, the higher is the capacity, the higher is the chance that a configuration does not admit a KS colouring. It is tempting to term the number of elements in $\mathcal{N}$ as “*inductivity*” of the configuration. A pair of subconfigurations $\mathcal{N}_1$ and $\mathcal{N}_2$ in $\mathcal{M}$ can “interact”: if the capacity of $\mathcal{N}_1$ is $n_1$, and the capacity of $\mathcal{N}_2$ is $n_2$, then the capacity $n$ of $\mathcal{N}_1 \cup \mathcal{N}_2$ is at least $n_1 + n_2$, but it can be $n > n_1 + n_2$. The maximal value of $n$ corresponds to a “resonance effect”, etc.
The saturated KS configuration of rays $\mathcal{M}$ constructed in the present paper has $64 = 2^{6}$ elements and it corresponds to a system of three qubits. It would be of interest to construct a generalisation of this configuration for any number of qubits (the number $64$ suggests that it might be possible). It would also be of interest to count all kinds of generalised KS colourings of $\mathcal{M}$ corresponding to non-zero entropies. Let’s leave it for another paper.
Appendix A {#appendix-a .unnumbered}
==========
It is of interest to point out a link between the configurations discussed in the present paper and the recent work of M. Waegell and P. K. Aravind [@WaegellAravind], where they make an important observation about the algebraic nature of a certain class of proofs of the Kochen-Specker theorem. Consider the matrices: $$I = \left(
\begin{matrix}
1 &0 \\
0 &1
\end{matrix}
\right), \quad
X = \left(
\begin{matrix}
0 &1 \\
1 &0
\end{matrix}
\right), \quad
Y = \left(
\begin{matrix}
0 &1 \\
-1 &0
\end{matrix}
\right), \quad
Z = \left(
\begin{matrix}
1 &0 \\
0 &-1
\end{matrix}
\right).$$ The matrix $I$ is the $2 \times 2$ identity matrix, and $X$, $Y$, and $Z$, are related to the Pauli matrices as follows: $\sigma_1 = X$, $\sigma_2 = - i Y$, $\sigma_3 = Z$. Write $$X^{(0)} := I, \quad
X^{(1)} := X, \quad
X^{(2)} := Y, \quad
X^{(3)} := Z,$$ and put $$A^{(\alpha)} := X^{(\alpha_0)} \otimes X^{(\alpha_1)} \otimes X^{(\alpha_2)},
\label{eq:tensor}$$ where $\alpha \in [64] = \lbrace 0, 1, \dots, 63 \rbrace$, $\alpha = \alpha_0 + 4 \alpha_1 + 16 \alpha_2$, for $\alpha_0, \alpha_1, \alpha_2 \in [4] = \lbrace 0, 1, 2, 3 \rbrace$. A rather special property of the collection of matrices $\lbrace A^{(\alpha)} \rbrace_{\alpha \in [64]}$ is that there exist quite many variants to choose tuples $\mu_{0} < \mu_{1} < \dots < \mu_{s - 1}$, where $\mu_i \in [64]$, $i \in [s]$, $s = 3, 4, 7$, such that $$[A^{(\mu_i)}, A^{(\mu_j)}] = 0, \quad
\prod_{k \in [s]} A^{(\mu_k)} = \pm 1,
\label{eq:tuples}$$ where $i, j \in [s]$, and $[-, -]$ denotes the commutator, the order of the factors in the product does not matter since the matrices commute.
Suppose we have a hyper-graph with $n$ vertices labelled by $A^{(\nu_i)}$, $\nu_i \in [64]$, $\nu_i \not = \nu_j$, if $i \not = j$, $i, j \in [n]$, and with hyper-edges of cardinalities 3 or 4. Assume that the labels of the vertices in every hyper-edge correspond to the tuples of the shape . A hyper-edge is termed *negative* if the product of the labels of all its vertices is $- 1$, and it is termed *positive* if this product is $+ 1$. In [@WaegellAravind] it is pointed out that every time we have a hyper-graph like this with an *odd* number of negative hyper-edges, then if we have a property that every vertex is contained in an *even* number of hyper-edges, then this yields immediately a proof of the Kochen-Specker theorem. A complete characterization of this class of hyper-graphs is an open mathematical problem. In the paper mentioned the authors provide explicitly a series of examples of the hyper-graphs they have found.
We observe now that the saturated Kochen-Specker configuration $\mathcal{M}$ (64 rays) obtained the present paper admits a proof from the class [@WaegellAravind]. At the same time this *configuration of rays* does not underlie any of the proofs mentioned in [@WaegellAravind]. We also notice that a straightforward search on a computer (i.e. without using any symmetries of the Pauli group) for a saturated Kochen-Specker configuration with $N = 2^6$ rays would require going through $C_{135}^{8} = 2214919483920$ variants (the lower index $135$ in the binomial coefficient $C_{135}^{8}$ is the total number of all tuples of the type of maximal possible length $s = 7$, and the upper index $8$ should be perceived as $N/ d$, where $d = 2^3$ is the dimension of the Euclidian space of three qubits).
It is a common convention to drop the symbol $\otimes$ in the tensor product . If we use the notation $I$, $X$, $Y$, $Z$, then we write, for example, just $\mathit{IXX}$ in place of $X^{(0)} \otimes X^{(1)} \otimes X^{(1)}$, $\mathit{YZX}$ in place of $X^{(2)} \otimes X^{(3)} \otimes X^{(1)}$, etc. Consider a list: $$\begin{matrix}
0: &\mathit{XII} &\mathit{IXI} &\mathit{XXI} &\mathit{IIZ} &\mathit{XIZ} &\mathit{IXZ} &\mathit{XXZ} \\
1: &\mathit{XII} &\mathit{IXX} &\mathit{XXX} &\mathit{IYY} &\mathit{XYY} &\mathit{IZZ} &\mathit{XZZ} \\
2: &\mathit{IXI} &\mathit{ZIX} &\mathit{ZXX} &\mathit{YIY} &\mathit{YXY} &\mathit{XIZ} &\mathit{XXZ} \\
3: &\mathit{XXI} &\mathit{YYX} &\mathit{ZZX} &\mathit{ZYY} &\mathit{YZY} &\mathit{XIZ} &\mathit{IXZ} \\
4: &\mathit{ZXI} &\mathit{YYI} &\mathit{XZI} &\mathit{IIZ} &\mathit{ZXZ} &\mathit{YYZ} &\mathit{XZZ} \\
5: &\mathit{ZXI} &\mathit{ZIX} &\mathit{IXX} &\mathit{XYY} &\mathit{YZY} &\mathit{YYZ} &\mathit{XZZ} \\
6: &\mathit{YYI} &\mathit{XXX} &\mathit{ZZX} &\mathit{YIY} &\mathit{IYY} &\mathit{ZXZ} &\mathit{XZZ} \\
7: &\mathit{XZI} &\mathit{ZXX} &\mathit{YYX} &\mathit{YXY} &\mathit{ZYY} &\mathit{XIZ} &\mathit{IZZ} \\
\end{matrix}$$ Every line of this list contains a set of 7 mutually commuting operators. Each time there is a set of 8 mutually orthogonal one-dimensional joint eigenspaces (rays) associated to it. These sets of rays corresponding to different lines of the list are mutually disjoint, and their union yields a set of 64 rays. This is precisely the *saturated* KS configuration $\mathcal{M}$ introduced in the paper, if we assume that the matrices $A^{(\alpha)}$, $\alpha \in [64]$, act on $x = (x_0, x_1, \dots, x_7)$ as follows: $A^{(\alpha)} .x = y$, $y = (y_0, y_1, \dots, y_7)$: $$y_{i_0 + 2 i_1 + 4 i_2} = \sum_{j_0, j_1, j_2 = 0, 1}
X_{i_0, j_0}^{(\alpha_0)} X_{i_1, j_1}^{(\alpha_1)} X_{i_2, j_2}^{(\alpha_2)}
\, x_{j_0 + 2 j_1 + 4 j_2},$$ for $i_0, i_1, i_2 = 0, 1$, and $\alpha_0 + 4 \alpha_1 + 16 \alpha_2 = \alpha$, where $\alpha_0, \alpha_1, \alpha_2 \in [4]$, and $X_{k, l}^{(m)}$ denotes the element of the matrix $X^{(m)}$, $m \in [4]$, standing in the $(k + 1)$-th row, $k \in [2]$, and the $(l + 1)$-th column, $l \in [2]$.
Let us point out a proof that the configuration of rays $\mathcal{M}$ is of the Kochen-Specker type. The collection of matrices present in the list above contains 26 elements: $$\begin{gathered}
\mathit{XII}, \mathit{IXI}, \mathit{XXI}, \mathit{ZXI}, \mathit{YYI}, \mathit{XZI}, \mathit{ZIX}, \mathit{IXX}, \mathit{XXX}, \\
\mathit{ZXX},
\mathit{YYX},
\mathit{ZZX}, \mathit{YIY}, \mathit{YXY}, \mathit{IYY}, \mathit{XYY}, \mathit{ZYY}, \mathit{YZY}, \\
\mathit{IIZ}, \mathit{XIZ}, \mathit{IXZ}, \mathit{XXZ},
\mathit{ZXZ}, \mathit{YYZ}, \mathit{IZZ}, \mathit{XZZ}.
\label{eq:sat_matrices}\end{gathered}$$ Consider the following subsets of this collection: $$\begin{matrix}
\phantom{*} 0: &\mathit{IXI} &\mathit{ZIX} &\mathit{YXY} &\mathit{XIZ} \\
\phantom{*} 1: &\mathit{IXI} &\mathit{ZXX} &\mathit{YIY} &\mathit{XIZ} \\
\phantom{*} 2: &\mathit{ZXI} &\mathit{YYI} &\mathit{IIZ} &\mathit{XZZ} \\
\phantom{*} 3: &\mathit{ZXI} &\mathit{XZI} &\mathit{IIZ} &\mathit{YYZ} \\
\phantom{*} 4: &\mathit{YYI} &\mathit{XXX} &\mathit{YIY} &\mathit{XZZ} \\
\phantom{*} 5: &\mathit{XZI} &\mathit{ZXX} &\mathit{YXY} &\mathit{IZZ} \\
\phantom{*} 6: &\mathit{ZIX} &\mathit{IXX} &\mathit{YYZ} &\mathit{XZZ} \\
*7: &\mathit{IXX} &\mathit{XXX} &\mathit{IZZ} &\mathit{XZZ} \\
\end{matrix}$$ There are 15 matrices present in the list: $$\begin{gathered}
\mathit{IXI}, \mathit{ZXI}, \mathit{YYI}, \mathit{XZI}, \mathit{ZIX}, \mathit{IXX}, \mathit{XXX}, \\
\mathit{ZXX}, \mathit{YIY}, \mathit{YXY}, \mathit{IIZ}, \mathit{XIZ}, \mathit{YYZ}, \mathit{IZZ}, \mathit{XZZ}. \end{gathered}$$ Construct a hyper-graph on 15 vertices labelled with these matrices. The 4-tuples in the lines of the list above define 8 hyper-edges: the fist seven are *negative* and the hyper-edge corresponding to the last line marked with a star is *positive*. The matrix $\mathit{IZZ}$ occurs 4 times in a hyper-edge, and all other matrices occur twice in a hyper-edge. The underlying configuration of rays is, therefore, of the Kochen-Specker type.
The hyper-graph mentioned contains just one positive hyper-edge and all other hyper-edges are negative. All hyper-edges are of cardinality 4. This is not the only hyper-graph of this shape which corresponds to a proof that $\mathcal{M}$ of the Kochen-Specker type, but, perhaps, the most simple one. The other hyper-graphs with a single positive hyper-edge can be generated on a personal computer as follows. There are 24 negative hyper-edges of size 4, and 54 positive hyper-edges of size 4 which can be constructed from the 26 matrices . $2^{24}$ is not a “huge” number compared to $2^{54}$. Look at all hyper-graphs built only from an *odd* number of negative hyper-edges of size 4. Count for each vertex the number of hyper-edges containing it, and keep only those hyper-graphs which contain exactly 4 vertices corresponding to an odd number of hyper-edges. Then check if this 4-tuple of vertices forms a *positive* hyper-edge of size 4. If this is the case, we obtain a proof that the configuration is of the Kochen-Specker type. Since every hyper-graph contains just one *positive* hyper-edge, it is natural to classify these hyper-graphs by the tuples of numbers $\lbrace n_k \rbrace_{k \in [24]}$, where $n_k$ is the number of vertices contained in exactly $k$ *negative* hyper-edges. The computation yields 33 variants of $\lbrace n_k \rbrace_{k \in [24]}$. In principle, it would be of interest to describe all isomorphism classes of KS proofs for $\mathcal{M}$ corresponding to any number of positive hyper-edges.
Appendix B {#appendix-b .unnumbered}
==========
In the present paper we have encountered three types of Kochen-Specker configurations: saturated (the configuration $\mathcal{M}$, 64 rays), tropical (the configuration $\mathcal{T}$, 48 rays), critical (the configuration $\mathcal{N}$, 36 rays). We have: $\mathcal{N} \subset \mathcal{T} \subset \mathcal{M}$. The search for a configuration $\mathcal{T}$ has been discussed in the main text. Let us say a few words about how to come up with configurations $\mathcal{M}$ and $\mathcal{N}$.
Look at the saturated extension of the example of M. Kernaghan and A. Peres (40 rays) given by the $E_{8}$ Kochen-Specker configuration (120 rays). These rays can be represented by the following vectors: 64 vectors of the shape $$(1, \pm 1, \pm 1, \dots, \pm 1),$$ assuming the number of minus signs is even, and 56 vectors of the shape $$(0, \dots, 0, 1, 0, \dots, 0, \pm 1, 0, \dots, 0),$$ where $1$ stands in the $i$-th position, and $\pm 1$ stands in the $j$-th position, $0 \leqslant i < j \leqslant 7$. We select these vectors in such a way that the first non-zero coordinate is 1. If, for every vector $v$ in the collection, one takes also $- v$, then the whole set is going to be precisely the 240 vectors describing the $E_8$ root system. It is natural to ask the following questions:
- This configuration is a KS configuration of rays in 8-dimensional space (the Euclidean space of three qubits). Is it possible to generalise it for $N > 3$ qubits?
- Does the $E_8$ configuration contain a smaller configuration which is saturated and at the same time Kochen-Specker (i.e. is it simple)?
Let us look at a subconfiguration $\mathcal{A}$ consisting of 64 rays corresponding to the first 64 vectors mentioned above (i.e. no zeros in the lists of coordinates). It turns out that this is a *saturated* subconfiguration, but it is *not* Kochen-Specker. The good thing about it is that $64 = 2^{6}$ (a power of two). This number is intuitively better if we speak about $N$ *qubits*. This subconfiguration is also special in the following sense: one can represent it as a union of two isomorphic saturated configurations with 32 elements each; each of these 32-rays configurations splits into a pair of isomorphic saturated 16-rays configurations; each of these 16-rays configurations can be represented as a union of two disjoint 8-tuples of mutually orthogonal rays (i.e. as $U_0 \cup U_1$, where $U_0, U_1 \in \mathcal{P}_{\perp}^{(8)} (\mathcal{M})$, $U_0 \cap U_1 = \emptyset$).
Consider an orthogonal transformation $T: \mathbb{R}^{8} \to \mathbb{R}^{8}$, $$\begin{gathered}
(1 {\bar 1} {\bar 1} {\bar 1} {\bar 1} {\bar 1} {\bar 1} 1 ) \mapsto (1 {\bar 1} {\bar 1} 1 0 0 0 0 ), \quad
(1 {\bar 1} {\bar 1} {\bar 1} 1 1 1 {\bar 1} ) \mapsto (1 {\bar 1} 1 {\bar 1} 0 0 0 0 ),\\
(1 {\bar 1} 1 1 {\bar 1} {\bar 1} 1 {\bar 1} ) \mapsto (1 1 {\bar 1} {\bar 1} 0 0 0 0 ), \quad
(1 {\bar 1} 1 1 1 1 {\bar 1} 1 ) \mapsto (1 1 1 1 0 0 0 0 ),\\
(1 1 {\bar 1} 1 {\bar 1} 1 {\bar 1} {\bar 1} ) \mapsto (0 0 0 0 1 {\bar 1} {\bar 1} 1 ), \quad
(1 1 {\bar 1} 1 1 {\bar 1} 1 1 ) \mapsto (0 0 0 0 1 {\bar 1} 1 {\bar 1} ),\\
(1 1 1 {\bar 1} {\bar 1} 1 1 1 ) \mapsto (0 0 0 0 1 1 {\bar 1} {\bar 1} ), \quad
(1 1 1 {\bar 1} 1 {\bar 1} {\bar 1} {\bar 1} ) \mapsto (0 0 0 0 1 1 1 1 ).
\end{gathered}$$ It turns out that the configuration $\mathcal{B} := \mathcal{A} \cup T (\mathcal{A})$ is saturated *and* Kochen-Specker (the transformation $T$ is found on a personal computer going though the variants of orthonormal bases with vectors having coordinates $0$, $1$, or $- 1$). The number of elements in $\mathcal{B}$ is $128 = 2^{7}$ (a power of two). It remains to notice that $\mathcal{B}$ splits into two isomorphic copies of a saturated Kochen-Specker configuration of 64 rays. One of these copies is precisely the configuration $\mathcal{M}$ described in the present paper.
The splitting mentioned can be noticed as follows. Call the number of elements in $\mathcal{P}_{\perp}^{(d)} (\mathcal{K})$ the *capacity* of a configuration of rays $\mathcal{K}$, $d$ is the dimension of space (in our case, $d = 8$). Take the configuration $\mathcal{A}$ and look at all pairs $U_0, U_1 \in \mathcal{P}_{\perp}^{(8)} (\mathcal{A})$ such that $U_0 \cap U_1 = \emptyset$. It turns out that $\# \mathcal{P}_{\perp}^{(8)} (U_0 \cup U_1)$ can be 2 or 4. Select the pairs corresponding to the maximal capacity 4. This way we obtain a collection $\lbrace W_{i} \rbrace_{i \in [m]}$ of 16-tuples of rays, $\# W_i = 16$, $i \in [m]$, (a computation yields $m = 420$). Look now at all disjoint pairs $W_{i_0}$, $W_{i_1}$ and try to maximise $\# \mathcal{P}_{\perp}^{(8)} (W_{i_0} \cup W_{i_1})$. It turns out that, for the configuration $\mathcal{A}$, the capacity of $W_{i_0} \cup W_{i_1}$, in case $W_{i_0} \cap W_{i_1} = \emptyset$, can be 8, 16, or 24. Select the combinations corresponding to the maximal capacity 24. The corresponding computation yields a collection of sets $\lbrace Q_{i} \rbrace_{i \in [70]}$, $\# Q_i = 32$, and $\# \mathcal{P}_{\perp}^{(8)} (Q_{i}) = 24$, $i \in [70]$.
We have another isomorphic copy $T (\mathcal{A})$ of the configuration $\mathcal{A}$. Construct in a similar way the 70 subsets $\widetilde{Q}_{i} \subset T (\mathcal{A})$, $i \in [70]$, such that $\# \widetilde{Q}_i = 32$, and $\# \mathcal{P}_{\perp}^{(8)} (\widetilde{Q}_{i}) = 24$, $i \in [70]$. The capacity of the configuration $\mathcal{A}$ is 240, and it can be represented as $\mathcal{A} = Q_{i_0} \cup Q_{i_1}$, where $i_0, i_1 \in [70]$. It turns out that in $\mathcal{A} \cup T (\mathcal{A})$ we can do better: there exist $i, j \in [70]$ such that $\# \mathcal{P}_{\perp}^{(8)} (Q_{i} \cup \widetilde{Q}_{j}) = 320$. The corresponding computation yields 12 variants of such $(i, j)$. One of the configurations $Q_{i} \cup \widetilde{Q}_{j}$ is the configuration $\mathcal{M}$ presented in this paper. As we know, it is saturated *and* Kochen-Specker. In short, $\mathcal{M}$ is constructed following a *“principle of maximal capacity”*.
Let us now look at the critical configuration $\mathcal{N}$. We start with $\mathcal{T}$ (a tropical subconfiguration of $\mathcal{M}$) and obtain $\mathcal{N}$ by a process of deleting rays. A similar problem is considered in a recent paper [@MegillKresimirWaegellAravindPavicic] where the authors try to classify *all* critical subconfigurations of the $H_4$ configuration (this is a KS configuration in four dimensions found in [@AravindLee-Elkin]). It is mentioned that one would need a year on a large computer cluster to compute the instances of them all. The problem with this approach is that it does not seem to use the *symmetry* of the problem. I do not use the corresponding C libraries, and my method exploits actively the concept of a *signature* (see the definition in the main text) of a configuration. Since the aim is just to generate an example of a small critical subconfiguration there is no need for optimisations of the C code: a personal computer does the job in a few minutes.
Given a KS configuration of $n$ rays, look at all $n$ possibilities of deleting a ray. This defines $n$ subconfigurations containing $n - 1$ rays each. Keep only those configurations which are not KS colourable. Denote their number $m$. Compute the signatures of each of the remaining $m$ configurations. The observation is that the number of signatures $k$ is typically “much less” than $m$. Intuitively, a pair of configurations with the same signatures have a high chance to be isomorphic. Choose a random representative of each signature (for example, the first one encountered). This leaves $k$ configurations instead of $m$, and this is much better! After that, repeat the procedure described above for each of the chosen $k$ configurations selecting again the representatives of signatures. Repeat this loop until you are left with only instances of critical subconfigurations.
If we start with the tropical KS configuration $\mathcal{T}$ of 48 rays mentioned, then, for example, in the first iteration: $m = 47$ and $k = 2$. The number $2$ is “much less” than $47$. It so happens, that after 12 iterations, the configuration $\mathcal{N}$ is the only configuration that “survives”. An implementation in C is in the supplementary file `generate.c`. I have also tested the method on the configuration $\mathcal{T}'$ of M. Kernaghan and A. Peres (40 rays in 8 dimensions), and it yields the critical configuration $\mathcal{N}'$ of 36 rays pointed out in their paper. In short, the observation is that the *“method of signatures”* mentioned allows to avoid a rapid growth of the number of variants, i.e. the computer does not “hang up”. If we are a little more general (this is not the problem I consider in the present paper) and say that we can select representatives of the isomorphism classes of orthogonality graphs of configurations instead of representatives of their signatures, then this yields a method to compute all isomorphism classes of critical subconfigurations of $\mathcal{M}$.
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---
abstract: 'The continuum of $^{10}$He nucleus is studied theoretically in a three-body $^{8}$He+$n$+$n$ model basing on the recent information concerning $^9$He spectrum \[Golovkov, *et al.*, Phys. Rev. C **76**, 021605(R) (2007)\]. The $^{10}$He ground state (g.s.) candidate with structure $[p_{1/2}]^2$ for new g.s. energy of $^9$He is predicted to be at about $2.0-2.3$ MeV. The peak in the cross section associated with this state may be shifted to a lower energy (e.g. $\sim 1.2$ MeV) when $^{10}$He is populated in reactions with $^{11}$Li due to peculiar reaction mechanism. Formation of the low-energy ($E< 250$ keV) “alternative” ground state with structure $[s_{1/2}]^2$ is highly probable in $^{10}$He in the case of considerable attraction (e.g. $a<-5$ fm) in the $s$-wave $^9$He channel, which properties are still quite uncertain. This result either questions the existing experimental low-energy spectrum of $^{10}$He or place a limit on the scattering length in $^9$He channel, which contradicts existing data.'
author:
- 'L. V. Grigorenko'
- 'M. V. Zhukov'
date: '. [File: /latex/10he/10he-17.tex ]{}'
title: 'Problems with interpretation of $^{10}$He ground state'
---
Introduction
============
The first, at that moment theoretical, attempt to study $^{10}$He was undertaken in the end of 60-th [@baz69]. In this work a possibility of the nuclear-stable $^{10}$He existence was investigated in the microscopic 10-body hyperspherical harmonic (HH) model. However, until now the $^{10}$He nucleus remains relatively poorly studied system. Since it became clear that $^{10}$He is nuclear unstable [@ste88] and ground state properties of $^9$He were defined [@set87; @boh88], it became possible to predict theoretically the ground state of $^{10}$He as a narrow three-body $^8$He+$n$+$n$ resonance. It was found with $E \sim 0.7-0.9$, $\Gamma \sim 0.1-0.3$ MeV [@kor93], for valence neutrons populating mainly $[p_{1/2}]^2$ configuration (the energy $E$ in the present work is always given relative to the three-body $^8$He+$n$+$n$ threshold). These predictions were soon confirmed experimentally: $E = 1.2(3)$, $\Gamma < 1.2$ MeV [@kor94], $E = 1.07(7)$, $\Gamma = 0.3(2)$ MeV [@ost94; @boh99], and $E = 1.7\pm 0.3 \pm 0.3$ MeV [@kob97].
A new possible theoretical understanding of $^{10}$He was proposed after the existence of a virtual state in $^9$He was demonstrated by Chen *et al*. in Ref. [@che01]. An *upper* limit for scattering length $a<-10$ fm was established in this experimental work. For such an attractive $s$-wave interaction in $^9$He Aoyama predicted in Ref. [@aoy02] the existence of a narrow near-threshold $0^+$ state in $^{10}$He ($E = 0.05$, $\Gamma = 0.21$ MeV) with the $[s_{1/2}]^2$ structure in addition to the $[p_{1/2}]^2$ state (calculated in this work to be at about 1.7 MeV). Concerning evident discrepancy with the experimental data the author of Ref. [@aoy02] suggested that the ground state (g.s.) of $^{10}$He had not been observed so far and the state at $\sim 1.3$ MeV is actually the first excited state. However, no possible explanation was proposed in Ref. [@aoy02] for which reason the $[s_{1/2}]^2$ g.s. was missed in experiments.
In recent experiment by Golovkov *et al*. [@gol07] at Dubna radioactive beam facility ACCULINNA the low-lying spectrum of $^9$He was revised, providing a higher position of the $p_{1/2}$ state than in the previous studies. A broad $p_{1/2}$ state was observed at about 2 MeV instead of the (presumably) $p_{1/2}$-$p_{3/2}$ doublet of narrow states at 1.27 and 2.4 MeV as in Refs. [@set87; @boh88; @boh99]. The experiment [@gol07] also claims a unique spin-parity identification below 5 MeV. The presence of the $s_{1/2}$ contribution is evident in the data [@gol07], but the exact nature of this contribution is still unclear, whether it is a virtual state with considerably large negative scattering length or just a smooth nonresonant background. A relaxed *lower* limit for scattering length $a>-20$ fm was established in this work. These new data should have a strong impact on the calculated properties of $^{10}$He, which inspired us to “revisit” the issue.
We study the question in theoretical models, which are schematic but have a clear relevance to real possible reaction mechanisms of the $^{10}$He continuum population. In a contrast with approach of Ref. [@aoy02], which provided only energies and widths of the states, we are interested in the observable consequences of the $J^{\pi}=0^+$ states with structures $[s_{1/2}]^2$ and $[p_{1/2}]^2$ “coexistence” in the $^{10}$He spectrum. We demonstrate that this problem has a key importance for understanding of observable properties of $^{10}$He. We arrive to a conclusion that the simultaneous consistent understanding of the low-lying spectra of $^9$He and $^{10}$He is still a challenge both from theoretical and experimental sides.
The unit system $\hbar=c=1$ is used in this work.
Theoretical model
=================
To choose the interactions in this work we generally follow the prescription of the three-cluster $^8$He+$n$+$n$ calculations of Ref. [@kor93] with appropriate modifications of potentials. From the set of the core-$n$ potentials tested in [@kor93] we selected one (denoted there as “I2”). Other choices do not change qualitatively the result and quantitatively are quite close. The potential is parameterized by Gaussian formfactor $$V^l_{c,ls}(r) = V^l_{c,ls} \exp[-r^2/r^2_0]$$ with $r_0=3.4$ fm. The depths of the $d$-wave potential $V^2_c = -33$ MeV and the $(ls)$ component in $p$-wave $V^1_{ls} = 10$ MeV, are the same as in original paper. The inverse $(ls)$ forces were used in Ref. [@kor93] in $p$-wave to account for occupied $p_{3/2}$ subshell in the $^8$He core. The interaction in the $s$-wave $^8$He-$n$ channel was pure repulsive in Ref. [@kor93] to account for an occupied $s_{1/2}$ shell in the $^8$He core. Central potential parameters in $s$- and $p$-waves $V^0_c$ and $V^1_c$ are being varied to clarify different aspects of the system dynamics. To manage the occupied $s_{1/2}$ state in $^8$He in this work an additional repulsive core is introduced in the $s$-wave with parameters $r_0(core)=2.35$ fm and $V^0_c(core) = 75$ MeV.
With the above potential the $d$-wave state in $^9$He is found at 4.8 MeV which is consistent with the the experimental data [@set87; @boh99; @gol07] giving values in the range $4.2-4.9$ MeV for the $d_{5/2}$ state. With $V^1_c =
-10$ MeV (the value from the Ref. [@kor93]) the $p_{1/2}$ state is obtained at 0.74 MeV. This value is different from value 1.15 MeV quoted in Ref. [@kor93], where this is the energy at which the phase shift pass $\pi/2$. In this work we have to deal also with broad states, where the phase shift does not reach $\pi/2$. Thus, we define the resonance position for two-body subsystem by “observable value” (peak in the elastic cross section) and define the width as the full width on half maximum (FWHM) for this peak.
The realistic soft-core potential [@gog70] is used in the $n$-$n$ subsystem also following Ref. [@kor93].
![Coordinate sets used in this paper. Panel (b) illustrates a proton removal from $^{11}$Li as a method to populate $^{10}$He.[]{data-label="fig:coor"}](coor){width="47.00000%"}
To study qualitatively a possible influence of the reaction mechanism we follow the approach of paper [@gri03b] to exotic $^5$H system. We introduce a compact source function $\Phi(\rho,\Omega_{\rho})$ in the right hand side of the three-body Schrödinger equation and solve the inhomogeneous system of equations $$\begin{aligned}
\left(\hat{H}-E \right) \Psi_E^{(+)}(\rho,\Omega_{\rho}) =
\Phi(\rho,\Omega_{\rho}) \;,
\label{eq:source} \\
%
\hat{H} = \hat{T} + \hat{V}_{cn}(\mathbf{r}_{cn_1}) +
\hat{V}_{cn}(\mathbf{r}_{cn_2})+ \hat{V}_{nn}(\mathbf{r}_{n_1n_2})\;, \nonumber
\\
%
\rho^2 = \frac{8}{10}\left( r^2_{cn_1}+r^2_{cn_2} \right) +
\frac{1}{10}r^2_{n_1n_2} = \frac{1}{2}X^2+\frac{8}{5}Y^2 \;,
%\end{aligned}$$ for pure outgoing wave boundary conditions, utilizing the hyperspherical harmonic (HH) method. The used coordinates are shown in Fig. \[fig:coor\]. The hyperradial components $\chi^{(+)}_{K\gamma}(\rho)$ of the WF $$\Psi_E^{(+)}(\rho,\Omega_{\rho}) = \rho^{-5/2} \sum ^{K_{\max}}_{K \gamma} \,
\chi^{(+)}_{K \gamma}( \varkappa \rho) \,
\mathcal{J}_{K \gamma}^{JM}(\Omega _{\rho}) \; ,$$ are matched to Riccati-Bessel functions of half-integer index $\mathcal{H}^{(+)}_{K+3/2}$. Functions $\mathcal{H}^{(+)}$ have the asymptotic behavior $\exp[i \varkappa \rho]$, where $\varkappa=\sqrt{2ME}$ ($M$ is a nucleon mass), describing the partial outgoing waves for hyperspherical equations. The value $K_{\max}$ truncates the hyperspherical expansion. The hypermoment $\varkappa$ is expressed via the energies of the subsystems $E_x$, $E_y$ or via Jacobi momenta $k_x$, $k_y$ conjugated to Jacobi coordinates $X$, $Y$: $$\begin{aligned}
%
\mathbf{k}_x & = & \frac {1} {2} \left( \mathbf{k}_{n_1} - \mathbf{k}_{n_2}
\right) \; ,\quad
\mathbf{k}_y = \frac{4}{5} \left( \mathbf{k}_{n_1} + \mathbf{k}_{n_2} \right)
- \frac{1}{5} \mathbf{k}_{c} \;,
\nonumber \\
%
\varkappa^2 & = &2ME = 2M(E_x+E_y) = 2 k_x^2 + \frac{5}{8}\,k_y^2 \;.
\label{eq:momenta}\end{aligned}$$ The Jacobi variables are given in “T” Jacobi system. A more detailed picture of Jacobi coordinates for coordinate and momentum spaces in “T” and “Y” Jacobi systems can be found in Fig.\[fig:corel\].
The set of coupled equations for functions $\chi^{(+)}$ has the form $$\begin{aligned}
%
\left[ \frac{d^{2}}{d\rho^{2}}-\frac{\mathcal{L}(\mathcal{L}+1)}{\rho^{2}}+
2M\left\{ E-V_{K \gamma,K \gamma}(\rho)\right\} \right] \chi^{(+)}_{K \gamma}
(\rho) \qquad \nonumber \\
%
= 2M \sum_{K' \gamma '} V_{K \gamma,K^{ \prime
}\gamma^{\prime}}(\rho)\chi^{(+)}_{K^{\prime}
\gamma^{ \prime }}(\rho) + 2M \,\Phi_{K \gamma}(\rho) \, ,
%
\label{shredl} \\
%
V_{K\gamma,K^{\prime}\gamma^{\prime}}(\rho)=\int \!\! d \Omega_{\rho} \,
\mathcal{J}_{K' \gamma'}^{JM*}(\Omega_{\rho}) \sum_{i<j}
V_{ij}(\mathbf{r}_{ij})\,\mathcal{J}_{K \gamma}^{JM}(\Omega_{\rho})\, ,
\label{hhpot} \\
%
\Phi_{K \gamma}(\rho) = \rho^{5/2} \int d \Omega_{\rho} \,
\mathcal{J}_{K \gamma}^{JM*}(\Omega_{\rho}) \, \Phi(\rho,\Omega_{\rho}) \, ,
\nonumber
%\end{aligned}$$ where ${\cal L}=K+3/2$ and $V_{K\gamma,K^{\prime}\gamma^{\prime}}(\rho)$ are matrix elements of the sum of the pairwise potentials referred to in this work as three-body potentials.
More detailed account of the method can be found e.g. in Ref.[@gri03b]. It is shown there that the method is consistent with “sudden removal” approximation for high energy fragmentation reactions. The development of the technically similar approach in the framework of the DWBA theory, applied to the inelastic processes in the transfer reactions, can be found in Ref. [@asc69].
We used two different sources, consistent with different reaction conditions. One is a “narrow” source with a Gaussian formfactor $$\Phi(\rho,\Omega_{\rho}) = \exp[-\rho^2/\rho_0^2] \;
\sum_{K=0,2 \; S=0} \mathcal{J}_{K \gamma}^{JM}(\Omega _{\rho}) \;,
%
\label{eq:narrow}$$ where $\rho_0=4.1$ fm provides the source rms radius $\langle \rho
\rangle = 5$ fm. This is a typical radius for the “reaction volume” for ordinary nuclei. The source populates only the lowest hyperspherical components of the WF ($K=0,2$). This qualitatively corresponds to the population of the $[s]^2$ and $[p]^2$ shell model configurations in the $^{10}$He nucleus, which are expected to be the most important for the low-energy part of the spectrum. The condition $S=0$ is qualitatively consistent with mechanism of transfer reactions, where the $^{10}$He states are populated by transferring a two-neutron pair (with total spin equal zero) to the $^8$He core. In such reactions the $\Delta S=1$ transfer is strongly suppressed and the $\Delta S=0$ transfer is a very reliable assumption.
The other choice of the source is more reaction specific. When $^{10}$He is produced from $^{11}$Li in a process which can be approximated as a sudden proton removal from $^{9}$Li core, the source term $\Phi(\rho,\Omega_{\rho})$ should contain the Fourier transform of the overlap integral between the $^8$He WF $\Psi_{^{8}\mbox{\scriptsize He}}$, the spin-isospin function of the removed proton $\chi_p$ and the $^{11}$Li wave function over the radius-vector $\mathbf{r}$ between the removed proton and the center-of-mass of $^{10}$He \[see Fig. \[fig:coor\] (b)\]: $$\Phi(\rho,\Omega_{\rho}) =
\int d \mathbf{r} \, e^{i \mathbf{qr}} \langle \Psi_{^{8}\mbox{\scriptsize He}}
\chi_p| \Psi_{^{11}\mbox{\scriptsize Li}} \rangle \, .
%
\label{eq:overlap}$$ In general, this quantity is a complicated function of the recoil momentum vector $\mathbf{q}$, transferred to the $^{10}$He system in the proton removal process. However, if the reaction energy is large and the internal energy of $^{10}$He is small, one can neglect this dependence (see Ref. [@gri03b] for details). It can be shown that in this case partial hyperspherical components of the source function are well approximated by the corresponding components of the $^{11}$Li WF. Thus, this type of calculations is further referred as “$^{11}$Li source”. The $^{11}$Li WF was taken from an analytical parametrization developed in Ref.[@shu06] taking into account broad range of experimental information on this nucleus. The dominant $[s_{1/2}]^2$ and $[p_{1/2}]^2$ configurations are populated by the $^{11}$Li source with almost equal probabilities. The rms radius of such a source function $\langle \rho \rangle = 9.5$ fm is enormous compared to typical nuclear sizes.
In the approach with the source function of Eq. (\[eq:source\]) the cross section for population of the $^{10}$He continuum is proportional to the outgoing flux of the three particles on a hypersphere of some large radius $\rho
= \rho_{\max}$: $$\frac{d \sigma}{dE} \sim \frac{1}{M} \mathop{\rm Im} \!
\int \! d \Omega _{\rho } \left. \Psi_E ^{(+)\dagger}
\rho^{5/2}\frac{d}{d\rho} \, \rho^{5/2} \,
\Psi_E^{(+)}\right| _{\rho =\rho_{\max} } \, .
%
\label{eq:cross}$$ Differentials of this flux on the hypersphere provide angular and energy distributions among the decay products at given decay energy $E$ (see Ref. [@gri03b] for details of correlation calculations).
Calculations
============
Basis size convergence
----------------------
![Convergence of calculations as a function of $K_{\max}$ (the value truncating the hyperspherical basis). Calculations with narrow source. (a) Resonance peak with $[p_{1/2}]^2$ structure. The $^8$He-$n$ potential parameters are: $V^0_c=0$, $V^1_c=-10$ MeV. (b) Resonance peak with $[s_{1/2}]^2$ structure. Parameters are: $V^0_c=-26.93$ MeV (this corresponds to $a=-15$ fm in $^9$He), $V^1_c=-4.5$ MeV.[]{data-label="fig:hh-conv"}](hh-conv-1 "fig:"){width="42.00000%"}\
![Convergence of calculations as a function of $K_{\max}$ (the value truncating the hyperspherical basis). Calculations with narrow source. (a) Resonance peak with $[p_{1/2}]^2$ structure. The $^8$He-$n$ potential parameters are: $V^0_c=0$, $V^1_c=-10$ MeV. (b) Resonance peak with $[s_{1/2}]^2$ structure. Parameters are: $V^0_c=-26.93$ MeV (this corresponds to $a=-15$ fm in $^9$He), $V^1_c=-4.5$ MeV.[]{data-label="fig:hh-conv"}](hh-conv-2 "fig:"){width="42.50000%"}
The HH calculations in our method can be performed with $K_{\max}=24-26$. Such basis sizes could be not sufficient for obtaining a good energy convergence of calculations in some complicated cases. The basis size can be further increased effectively using the adiabatic procedure based on the so called Feshbach reduction (FR) [@gri07]. Feshbach reduction eliminates from the total WF $\Psi=\Psi_{p}+\Psi_{q}$ an arbitrary subspace $q$ using the Green’s function of this subspace: $$H_{p}=T_{p}+V_{p}-V_{pq}G_{q}V_{pq} \;.$$ In a certain adiabatic approximation we can assume that the radial part of kinetic energy is small under the centrifugal barrier in the channels with high centrifugal barriers and can be approximated by a constant. In this approximation the FR procedure is reduced to the construction of effective three-body interactions $V^{\text{eff}}_{K\gamma,K^{\prime}\gamma^{\prime}}$ by matrix operations $$\begin{aligned}
G_{K\gamma,K^{\prime}\gamma^{\prime}}^{-1} & = & (H-E)_{K\gamma,K^{\prime
}\gamma^{\prime}}=V_{K\gamma,K^{\prime}\gamma^{\prime}} \nonumber \\
%
& + & \left[ E_{f}-E+\frac{(K+3/2)(K+5/2)}{2M\rho^{2}}\right]
\delta_{K\gamma,K^{\prime} \gamma^{\prime}}\,, \nonumber \\
%
V^{\text{eff}}_{K\gamma, K^{\prime}\gamma^{\prime}} & = &
V_{K\gamma,K'\gamma'}-\sum V_{K\gamma,\bar{K}\bar{\gamma}}
G_{\bar{K}\bar{\gamma}, \bar{K}^{\prime} \bar{\gamma}^{\prime}}
V_{\bar{K}^{\prime}\bar{\gamma}^{\prime}, K^{\prime
}\gamma^{\prime}}\;.\nonumber
%\end{aligned}$$ Summation over indexes with bar is made for eliminated channels. We take the “Feshbach energy” $E_{f}$ in our calculations as $E_{f}\equiv E$.
![Hyperradius of the classical turning point $\rho_{\min}$ for hyperradial centrifugal barriers in the channels with different $K$ values.[]{data-label="fig:rhomin"}](rhomin_ot_e){width="44.00000%"}
Reliability of the FR procedure can be checked in two ways. We can compare dynamic calculations for some large $K_{\max}$ with the “reduced” calculations $K_{\max} \rightarrow K_{FR}$ (with much smaller dynamic basis size $K_{FR}$) and in principle they should coincide. Calculations show that for $^{10}$He starting from $K_{\max}=26$ we get practically the same result down to $K_{FR}=10$. The other way to make a check is the following. We can also start FR with some quite large fixed $K_{\max}$ (e.g. from $K_{\max}=100$ in this work), make the reduction to different $K_{FR} \leq 26$ and perform dynamic calculations with each of them. Again the results were found to coincide precisely for $K_{FR} \geq 10$. Thus it was found reliable to perform most of the calculations (except those for correlations) with dynamic basis size $K_{FR}
= 12$ varying effective basis size $K_{\max}$ when necessary.
The cross section convergence with the increase of the effective hyperspherical basis size is demonstrated in Fig. \[fig:hh-conv\]. For resonance peak with $[p_{1/2}]^2$ structure the convergence is reliably achieved by $K_{\max}=30$. However, in the case of a state with $[s_{1/2}]^2$ structure more efforts are required to achieve the convergence (very close to the threshold even a minor variation of the energy becomes noticeable). We intentionally demonstrate in Fig. \[fig:hh-conv\] (b) the case, which is numerically more complicated than the others considered in the paper. When the $s$-wave potential in the $^8$He-$n$ subsystem is taken to provide the scattering length $a=-15$ fm, the resonance peak in $^{10}$He is obtained at $E=4$ keV with $\Gamma=0.7$ keV. The basis size $K_{\max}=80$ is required in such a case to obtain the convergence.
Another aspect of the basis size choice is connected with the radial extent of the calculations $\rho_{\max}$. The formulation of the cross section calculations in the form (\[eq:cross\]) implies that the WF residues at $\rho_{\max}$ in the classically allowed region. Taking into account the character of the hyperspherical centrifugal barrier (\[shredl\]) this requires a very large radial extent for large basis sizes. Fig. \[fig:rhomin\] provides the estimates of the minimally required values of $\rho_{\max}$ to satisfy this condition for different $K$ values. So, we used $\rho_{\max} \sim
300-500$ fm for calculations of the $[p_{1/2}]^2$ states and $\rho_{\max} \sim
1000-2000$ fm for the extreme low-energy calculations of the $[s_{1/2}]^2$ states.
![Behavior of the $^{10}$He spectrum with decrease of the $p$-wave potential depth $V^1_c$. The corresponding $p_{1/2}$ state energies $E(p_{1/2})$ in $^9$He relative to the $^8$He-$n$ threshold are shown in the legend. Calculations with narrow source.[]{data-label="fig:p-dec"}](p12dec){width="42.00000%"}
Sensitivity to the $p$-wave in $^9$He
-------------------------------------
The ground state resonance properties were predicted as $E \sim 0.7-0.9$ MeV, $\Gamma \sim 0.1-0.3$ MeV in Ref. [@kor93]. Within the approach used in this work we first of all reproduce the results of previous studies. The calculation with model parameters consistent with these of Ref. [@kor93] is shown in Fig. \[fig:p-dec\], by solid curve. The peak position is somewhat lower than in Ref. [@kor93] ($E=0.6$ MeV, $\Gamma = 0.27$ MeV) which is connected to the larger basis size (see Fig. \[fig:hh-conv\](a)). Note that $K_{\max}=8$ was used in Ref. [@kor93].
{width="34.20000%"} {width="32.80000%"} {width="31.20000%"}\
{width="34.20000%"} {width="32.30000%"} {width="32.30000%"}\
The evolution of the cross section with decrease of the $p$-wave interaction from the value adopted in Ref. [@kor93] ($V^1_c=-10$ MeV, which provided the energy of the $p_{1/2}$ state $E(p_{1/2})=0.74$ MeV) to a value providing the $^9$He g.s. to be at about 2 MeV ($V^1_c=-4.5$ MeV), is shown in Fig. \[fig:p-dec\] for the narrow source function. The new peak position for the $^{10}$He population cross section is at $E=2.3$ MeV. The impact of this change is drastic: the narrow $^{10}$He g.s. peak is practically “dissolved” as the system becomes less bound: e.g.the width of the peak can not be any more well defined as FWHM.
Sensitivity to the reaction mechanism
-------------------------------------
![Partial decomposition of the cross section; dashed, dotted and dash-dotted curves provide contributions of the main WF components. Calculations with $p$-wave resonance in $^9$He at 2 MeV ($V^1_c=-4.5$ MeV). Different panels correspond to: (a) $V^0_c=-25.82$ ($a=-10$ fm), narrow source; (b) $V^0_c=0$ narrow source, and (c) $V^0_c=0$, $^{11}$Li source.[]{data-label="fig:bn-com"}](bncom){width="38.00000%"}
The evolution of cases with different $p$-wave interactions with increase of the $s$-wave interaction is shown in Fig. \[fig:s-inc\] for the narrow and broad source functions, which should simulate different reaction conditions. We first discuss sensitivity of the cross section to the reaction mechanism.
The narrow ground state in $^{10}$He is not significantly sensitive to the reaction mechanism. This can be seen comparing Figs. \[fig:s-inc\] (a) and (d): difference of curves of the same style in the upper and lower panels is quantitative, not qualitative. This is an expected result as the narrow states have a sufficiently large lifetime to “forget” how they were populated and thus loose the sensitivity to the population mechanism.
When the state is above 1 MeV, the width becomes comparable to 1 MeV and the dependence on the source function is considerable \[Figs. \[fig:s-inc\] (b) and (e)\]. In the case of even higher $^{10}$He g.s. the calculations with narrow and broad sources have very little in common \[Figs. \[fig:s-inc\] (c) and (f)\]. According to the recent result [@gol07] the cases (c) and (f) should be regarded as the most realistic. Thus peculiarities of the reaction mechanism could be a problem for interpretation of the $^{10}$He spectra.
Sensitivity to the $s$-wave in $^9$He
-------------------------------------
It can be seen from Fig. \[fig:s-inc\] that for relatively weak $s$-wave attraction the g.s. peak is shifted to lower energies with minimal distortion. However, as the $s$-wave attraction becomes stronger the threshold peculiarity is formed in the spectrum. With the further increase of the $s$-wave interaction this peculiarity is transformed into very sharp low-energy ($E< 300$ keV) peak. The WF at this peak has a practically pure $[s_{1/2}]^2$ structure and we characterize it as a “three-body virtual state”.
Using the term “three-body virtual state” we have two things in mind: this is an $s$-wave state build upon the virtual states in all the subsystems, and this state has distinct properties compared to ordinary resonant three-body states (relevant discussion of “Efimov-like three-body virtual excitations” can be found in Ref. [@dan07]).
The ordinary two-body virtual states are typically characterized in two ways: (i) as a negative energy pole on the second Riemann sheet or (ii) as a threshold peculiarity [^1] preceding the formation of the bound state in the case of absence of the potential barrier. The pole behavior in the three-body systems has been studied in a number of works with the emphasis on the possible similarities with two-body virtual state poles behavior [@glo78; @tan99; @del00; @aoy02; @fre07]. In paper [@fre07] the possibility of such behavior in the three-body $s$-wave system was shown for interactions with certain extreme properties. Observable consequences of such a pole behavior in the three-body systems remain unclear. Our way to think about three-body virtual state is more relevant to the second characteristic of the two-body virtual state, which is connected to its observables.
For relatively strong $s$-wave interaction in the $^8$He-$n$ subsystem (namely such that the scattering length $a<-5$ fm), we unavoidably (means independently on the structure and reaction mechanism details) get a sharp peak in the cross section with energy less than 0.3 MeV and with dominating $[s_{1/2}]^2$ configuration. Stable formation of the low-energy peak at certain strength of attractive $s$-wave interaction in $^9$He is an important dynamical feature of the $^{10}$He system which makes us optimistic about predictive abilities of theoretical models in this situation. The extreme low-energy peaks could hardly be consistent with the experimental data [@kor94], the discussion of the issue is provided below in Section \[sec:exp-consist\].
It can be noticed that in the case of the very narrow three-body virtual state formation, some structure can be seen as a “shoulder” on the right slope of the $[s_{1/2}]^2$ peak. It is possible to understand that this structure corresponds to the state with the $[p_{1/2}]^2$ structure which becomes sufficiently well split from the $[s_{1/2}]^2$ state and even preserves the position typical for $V_c^0=0$ case. The analysis of the partial decomposition of the cross section provided in Fig. \[fig:bn-com\] indicates that this is generally true. However, the $[p_{1/2}]^2$ contribution to WF is considerably broadened and reduced in absolute value compared to the case when there was no $s$-wave attraction. For understanding of Fig. \[fig:bn-com\] it is useful to note that at the “shell model language” the $K=0$ configuration is a pure $[s_{1/2}]^2$, while the $K=2$ components (for $p$-shell nuclei) are mainly decomposed as $$\begin{aligned}
\left| K=2,L=0 \right\rangle & = & \sqrt{1/3} \; [p_{1/2}]^2 + \sqrt{2/3} \;
[p_{3/2}]^2 \;,
\nonumber \\
%
\left| K=2,L=1 \right\rangle & = & \sqrt{2/3} \; [p_{1/2}]^2 - \sqrt{1/3} \;
[p_{3/2}]^2 \;.
\nonumber
%
\label{eq:wf-decomp}\end{aligned}$$ The weight of the $[p_{1/2}]^2$ configuration relative to the total weight of $[p_{1/2}]^2$ and $[p_{3/2}]^2$ configurations varies from 80 to 90 percent in different calculations of the $p$-wave state.
Properties of the three-body virtual state
------------------------------------------
Important feature which differs a three-body virtual state (the one with dominant $[s_{1/2}]^2$ structure) from the ordinary two-body virtual states is evident from the structure of equations (\[shredl\]). This feature has been exploratory discussed in the past (e.g. Ref. [@tan99]), but it seems that in $^{10}$He this kind of physics could become really accessible for observation. The state with $[s_{1/2}]^2$ structure should be characterized by domination of component with lowest possible value of the generalized angular momentum $K=0$. However, the centrifugal barrier $\mathcal{L}(\mathcal{L}+1)/2M\rho^2$ is not zero even in the channel with $K=0$, as it depends on “effective angular momentum” $\mathcal{L}=K+3/2$. This means that the low-energy three-body virtual state may exist in the form of a real resonance peak, not a threshold peculiarity as two-body virtual state. It is also easy to demonstrate that the low-energy behavior of the inelastic cross section for population of the three-body continuum is $$d \sigma / dE \propto E^2 \;,$$ in contrast with the two-body inelastic cross section which has a square root peculiarity in the case of the virtual state $$d \sigma / dE \propto \sqrt{E}\;.$$ Such a behavior should in principle distinctly separate the three-body virtual state peak from zero energy. Such a separation was demonstrated in Ref. [@tan99] for a toy model of the $[s^2]$ state for the “Borromean system” [^2]. Namely, it was shown in the analytical continuation of coupling constant (ACCC) method that the pole trajectories in the case of the $[s^2]$ three-body state are analogous to the trajectories in the system with barriers, while for the two-body virtual states they are qualitatively different.
![Width as a function of resonance energy for $[s_{1/2}]^2$ and $[p_{1/2}]^2$ states. Standard two-body R-matrix estimates are shown for $l=1$ and $l=2$ (channel radius 40 fm) by dashed curves. Possible experimental ranges according to experiments [@kor94; @boh99] are shown by hatched and grey rectangles correspondingly. The result of theoretical prediction [@aoy02] is shown by a small circle. The curve for $[p_{1/2}]^2$ state is calculated with the broad source. Calculations for $[s_{1/2}]^2$ state with broad and narrow sources practically coincide within the shown energy range.[]{data-label="fig:gam-ot-e"}](gam-ot-e){width="48.00000%"}
{width="93.00000%"}
The mentioned features, however, do not mean that a three-body virtual state is an ordinary resonance state; there is an important difference. It is known that the resonance behavior is connected to time delays in the propagation of particles (and corresponding time-dependent theory can be formulated in these terms). Ordinarily, the time delay is connected to the confinement of particles inside the potential barrier and their WF is localized inside the potential well, displaying the “quasibound” nature of such resonances. In the case of virtual state the time delay is not connected with barrier and tight spacial localization of particles close to each other. It is connected with *slow* motion of particles in the volume of sphere with *large* radius (comparable to scattering length). In the three-body case the hyperspherical centrifugal barrier $\mathcal{L}(\mathcal{L}+1)/2M\rho^2$ has an effective collective nature; it is clear that individual nucleons in $[s_{1/2}]^2$ configuration do not “see” any barriers. The time delay is connected therefore to the simultaneous presence of two valence nucleons in the volume around the core, associated with scattering lengths, which means a peripheral nature of such a state.
The peripheral character of the state presumes different character of the dependence of the resonance width on energy than the behavior which could be expected for “typical” barrier penetration. We can take, for example, the calculated properties of the $^{10}$He g.s. in the case $a=-15$ fm ($E=4$ keV, $\Gamma=0.7$ keV, see Fig. \[fig:hh-conv\]) and deduce the “channel radius” $\rho_{\text{ch}}$ using for penetration expression analogous to the single-chanel R-matrix formula. It can be found in Ref. [@gol04a]: $$\Gamma = \frac{1}{M \rho^2_{\text{ch}}} \; \frac{2}{\pi} \;
\frac{1}{J^2_{K+2}(\varkappa \rho_{\text{ch}})+N^2_{K+2}(\varkappa
\rho_{\text{ch}})} \;.
%
\label{gamm-3}$$ Then the value $\rho_{\text{ch}} \approx 40$ fm is obtained. So, the radial range, which can be interpreted as an “internal region” in the case of the virtual three-body state, is huge.
The dependence of the width (defined as FWHM) on resonance energy is shown in Fig. \[fig:gam-ot-e\] for $[s_{1/2}]^2$ and $[p_{1/2}]^2$ resonance peaks. The variation of energy in each case is obtained by respective variation of parameters of the $s$- and $p$-wave interactions. The $[p_{1/2}]^2$ curve is obtained in calculations with a broad source. The curves for $[s_{1/2}]^2$ calculations practically coincide for broad and narrow source calculations; this independence is an expected result for such a narrow structure. It can be seen in Fig. \[fig:gam-ot-e\] that the curve for $[s_{1/2}]^2$ state stays mainly in between standard two-body R-matrix estimates with $l=1$, $l=2$ (channel radius 40 fm) as an “effective angular momentum” for $K=0$ is $\mathcal{L}=3/2$. The obtained dependence is in a good agreement with the $^{10}$He ground state prediction by Aoyama [@aoy02] which gave $E=0.05$ MeV and $\Gamma=0.21$ MeV (small circle in Fig. \[fig:gam-ot-e\]). We think that this agreement is an important fact demonstrating stability of the theoretical results on this issue because very different theoretical models and different $p$-wave interactions were employed in the studies of Ref. [@aoy02].
The point about a peripheral character of the $[s_{1/2}]^2$ state is also confirmed by analysis of the correlation density. The correlation densities $|\Psi^{(+)}|^2$ for the $^{10}$He WFs on the $\{\rho,\theta_{\rho}\}$ plane are shown in Fig. \[fig:wfden\]. The $\theta_{\rho}$ hyperspherical variable describes the distribution between $X$ and $Y$ subsystems. It is a component of the 5-dimensional hyperangle $\Omega_{\rho}=\{\theta_{\rho},\Omega_x, \Omega_y
\}$. For $^{10}$He in the “T” Jacobi system $$X=\sqrt{2} \, \rho \sin \theta_{\rho} \quad ; \quad
Y=\sqrt{5/8} \, \rho \cos \theta_{\rho} \;.$$ Some properties of the three-body virtual state are well illustrated by this plot.
1. The $[s_{1/2}]^2$ and $[p_{1/2}]^2$ configurations demonstrate very different correlations in the internal region and on asymptotic. While in the $[s_{1/2}]^2$ case the distributions are expectedly quite featureless, in the $[p_{1/2}]^2$ case we observe in the internal region the double-humped structures — “dineutron” and “cigar” — in the variable $\theta_{\rho}$, which are connected to so called Pauli focusing \[Fig. \[fig:wfden\] (b), (c); in the case (b) only “dineutron” peak is seen\]. These structures are well known from the studies of the other $p$-shell nuclei [@zhu93]. In the case when there is no attractive $s$-wave interaction in the $^8$He-$n$ channel \[Fig. \[fig:wfden\] (c)\] this double-humped correlation “survives” up to the asymptotic region in somewhat modified form and thus could possibly be observed in experiment (see also the discussion in the Section \[sec:corel\]).
2. Peripheral character of the $[s_{1/2}]^2$ WF. It can be seen from a comparison of Figs. \[fig:wfden\] (b) and (c) that the double-humped structure connected to $[p_{1/2}]^2$ configurations is sharply concentrated in the internal region ($\rho \sim 7$ fm) and rapidly decreases beyond $10-15$ fm. In contrast, the $[s_{1/2}]^2$ WF is peaked at $\rho \sim 15$ fm and extends smoothly to around 50 fm \[Fig. \[fig:wfden\] (a)\]. Distance $\rho \sim 15$ is well beyond the typical nuclear size; for configurations with such typical $\rho$ values the individual valence nucleons have on average 10 fm distance to the core.
3. Radial stabilization of the $[s_{1/2}]^2$ WF (the distances at which the $\sim \exp [i\varkappa \rho]$ behavior is mainly achieved) is taking place at quite large distances \[$\rho \approx 300-400$ fm, see Fig. \[fig:wfden\] (a)\]. This is connected both to very low energy of the peak (4 keV in this particular calculation) and effectively long-range character of interactions in the $s$-wave $^8$He-$n$ channel, responsible for the formation of the three-body virtual state.
4. One can see from Fig. \[fig:s-inc\] (c) that the cross section behavior in the energy region of $[p_{1/2}]^2$ state ($E \approx 2-3$ MeV) is practically not sensitive to the low-energy behavior of the spectrum (presence or absence of the $[s_{1/2}]^2$ state). It is thus possible to think that these configurations are practically independent in this energy region. A comparison of Figs. \[fig:wfden\] (b) and (c) shows that this is not true. Both the internal structure of the WF and correlations for decay products demonstrate strong sensitivity to the presence of $[s_{1/2}]^2$ state, although it is not clearly seen in the total production cross section in this energy range.
{width="96.00000%"}
An important technical insight could be obtained from the behavior of the effective hyperspherical interactions (\[hhpot\]). In Fig. \[fig:poten\] we show the “most attractive” diagonal potential (this appear to to be $K=0$ term in all the cases) and the lowest diagonalized potential. The later can be considered as an effective interaction for some simple adiabatic approximation to the problem, which is qualitatively illustrative, but quantitatively could be not very reliable. The important dynamical aspects which become clear from these plots have already been discussed in our work [@gri03b] on example of broad states of $^5$H system. It can be seen in Fig. \[fig:poten\] that even the “most attractive” diagonal potentials are in reality repulsive and do not even show any sign of “pocket” formation. The structures in the continuum are formed here only by interaction of multiple channels. It is interesting to note that the possibility of this class of states has been discussed many years ago (so called “resonances of the second kind” [@baz76]) but now we seem to face them systematically in the few-body dripline nuclei.
In the broad range of the resonance energies the state is located above the effective barrier top. Formation of the peaks, which could be very narrow (see Figs. \[fig:s-inc\], \[fig:gam-ot-e\]) is presumably connected here not with barrier penetration, but with slow motion above the barrier and reflection from the right slope of the barrier [^3]. Only in the extreme low energy case Fig. \[fig:poten\] (c) the process could be interpreted as penetration through the effective barrier. Typical range of the barrier is consistent in this case with the “channel radius” estimates by Eq. (\[gamm-3\]).
The evolution of the nuclear structure near the threshold is also an interesting question which is briefly discussed below. The short-dash curves ($V^0_c=-25.82$ MeV, $a=-10$ fm) in Fig. \[fig:s-inc\] (a) and (d) show qualitatively different behavior compared to the expected sharpening of the threshold peak. It happens because in this case the bound $0^+$ state of $^{10}$He is formed with binding energy $\approx 60$ keV. Therefore these curves do not represent a valid result in the context of our studies (we should consider only the three-body Hamiltonians which do not lead to the bound $^{10}$He). However, it is interesting to see how the nuclear structure evolves in this case (Table \[tab:struc\]). The virtual three-body state shows strong domination of the $[s_{1/2}]^2$ component in the internal region (the first row in Table \[tab:struc\]). As soon as the state becomes bound, the structure changes drastically with rapid increase of the $[p_{1/2}]^2$ configuration weight (see the second row from Table \[tab:struc\]). If we bind the $^{10}$He even more stronger, so that the binding energy becomes $0.3$ MeV, which corresponds to the binding energy of $^{11}$Li, its structure begins to resemble closely the typical structure of $^{11}$Li with practically equal population of the $[s_{1/2}]^2$ and $[p_{1/2}]^2$ configurations (the third row in Table \[tab:struc\]). So, the bound analogue of the virtual three-body state is expected to be not a state with dominant $[s_{1/2}]^2$ configuration, but a state with strong “competition” between the $s$-wave and $p$-wave configurations. Usually the structure of narrow resonant (or quasibound) states is characterized by a high identity with the structure of the corresponding bound states [^4]. The virtual three-body state demonstrates the behavior, which is qualitatively different in this respect.
Based on the presented results we can probably conclude that in the sense of nuclear dynamics the three-body virtual states are something intermediate between two-body virtual state and an ordinary resonance.
$E$ (MeV) $[s_{1/2}]^2$ $[p_{1/2}]^2$ $[p_{3/2}]^2$ $[d]^2$
----------- --------------- --------------- --------------- ---------
$0.04$ 93.3 2.2 1.8 1.8
$-0.06$ 66.1 23.8 4.7 4.2
$-0.3$ 51.0 35.9 6.1 5.7
: Internal structure of the virtual three-body state and the bound $^{10}$He states close to the $^8$He+$n$+$n$ threshold.
\[tab:struc\]
Consistence with $3 \rightarrow 3 $ scattering calculations
-----------------------------------------------------------
The model cross section calculations for realistic $^9$He energies Fig.\[fig:s-inc\] (c) and (f) show very diverse results in the case of narrow and broad sources. A question can be asked in that case what should be considered as a “real” position of $^{10}$He g.s. and whether it is reasonable at all to speak about such “real” position if diverse experimental responses could be expected. The theoretical approach which is “neutral” with respect to possible reaction mechanism is represented by $3 \rightarrow 3 $ scattering calculations.
Figures \[fig:scat33-1\] and \[fig:scat33-2\] show the results of the $3
\rightarrow 3$ scattering calculations in the cases of a pure $[p_{1/2}]^2$ state and the same in the presence of the low-energy $[s_{1/2}]^2$ state respectively. The details of the approach can be found in Ref. [@shu00] on example of the $^5$H nucleus. Three-body Hamiltonian here is the same as in Figs. \[fig:s-inc\] (c) and (f). Three different values are displayed for these calculations: the diagonal $3 \rightarrow 3$ scattering phase shifts, the first diagonalized phase shift (so called eigenphase), and the diagonal internal normalizations for scattering WFs $$N_{\rho_{\text{int}}}(E) = \frac{1}{\varkappa^5}\sum_{K \gamma}
\int_0^{\rho_{\text{int}}} d \rho \left| \chi_{K \gamma}^{K \gamma} (\varkappa
\rho) \right|^2 \;.
%
\label{int-norm}$$ The size of the “internal region” $\rho_{\text{int}}=6$ fm was taken for this value as in Ref. [@shu00].
![The $3 \rightarrow 3$ scattering calculations, $V_c^0=0$, $V_c^1=-4.5$ MeV. (a) Internal normalizations Eq. (\[int-norm\]) for dominant components of the WF. (b) Diagonalized phase shift (“eigenphase”) is shown by solid curve, while diagonal phase shifts for the lowest hyperspherical components are given by dashed, dotted, and dash-dotted curves.[]{data-label="fig:scat33-1"}](scat33-1){width="42.00000%"}
For $[p_{1/2}]^2$ state the $3 \rightarrow 3$ calculations in Fig. \[fig:scat33-1\] give somewhat different resonant energies for different responses: $E=1.8$ MeV for internal normalization, $E \sim 2$ MeV for the most strongly changing diagonal phase shifts and $E=2.3$ for eigenphase. Such spread is clearly connected to the fact that the phase shifts barely pass 90 degrees. So, we can speak about resonant energy of about $E \sim 2.0-2.3$ MeV, when only scattering is concerned. The agreement of $3 \rightarrow 3$ calculations with narrow source calculations is reasonable. That could be an indication that “ordinary” reactions (simulated in this model) are a preferable tool to access properties of $^{10}$He compared to reactions with exotic nuclei (like $^{11}$Li).
In the case of narrow low-lying $[s_{1/2}]^2$ state (shown in Fig. \[fig:scat33-2\]) the results provided by all $3 \rightarrow 3$ calculations (resonance energy $E=40$ keV) are in excellent agreement with each other and with previous model calculations \[Fig. \[fig:s-inc\] (c) and (f), curves with $a=-10$ fm\]. This state is formed exclusively by $K=0$ WF component. An evidence for the $[p_{1/2}]^2$ state contributions could be found in the phase shift at around 2 MeV, but it is not very expressed. Better evidence is provided by internal normalizations for $K=2$ components of the WF. These show maximum at about 2.3 MeV and provide much broader structures, than in the case of the $[p_{1/2}]^2$ state not affected by $[s_{1/2}]^2$ configuration (Fig. \[fig:scat33-1\]). Again, we can come to a conclusion that the $[p_{1/2}]^2$ state survives in the presence of the $[s_{1/2}]^2$ state in somewhat modified (shifted up and broadened) form, but the population of it is expected to be poor.
![The $3 \rightarrow 3$ scattering calculations, $V_c^0=-25.82$ MeV ($a=-10$ fm), $V_c^1=-4.5$ MeV. See Fig. \[fig:scat33-1\] for details.[]{data-label="fig:scat33-2"}](scat33-2){width="42.50000%"}
Discussion
==========
What is the ground state of $^{10}$He? {#sec:exp-consist}
--------------------------------------
It was proposed in Ref. [@aoy02] that the observed so far state of $^{10}$He is not the ground but the first excited state with $[p_{1/2}]^2$ structure while the ground $[s_{1/2}]^2$ state remains unobserved. We confirm here the finding of Ref. [@aoy02] that for considerable $s$-wave attraction in $^{9}$He subsystem two $0^+$ states with different structures should coexist in the low-energy spectrum of $^{10}$He. However, we also find that population of the $[s_{1/2}]^2$ configuration (in the case of a strong $s$-wave attraction in $^{9}$He and realistic reaction scenario) is always very pronounced compared to the $[p_{1/2}]^2$ configuration. For that reason we can expect that if the $[s_{1/2}]^2$ state *really exists* then the $[p_{1/2}]^2$ component is difficult to observe in experiment (as it is lost on a “nonresonant background” of $[s_{1/2}]^2$ low-energy excitation). It can be found that the energy position of the $[p_{1/2}]^2$ component of $0^+$ state is quite stable when the $s$-wave attraction is increased. However, for extreme cases of the $s$-wave attraction this contribution becomes much broader and in general “lost” on a thick right “tail” of the $[s_{1/2}]^2$ ground state.
Current experimental situation in $^{10}$He is clearly not in favour of the $[s_{1/2}]^2$ state existence. Several theoretical spectra of $^{10}$He are provided in Fig. \[fig:exp-com\] on top of the experimental data Ref. [@kor94]. Theoretical curves are convoluted with energy resolution of the experiment [@kor94] which is parameterized as $\Delta E= 0.7\sqrt{E}$ ($\Delta E$ is FWHM). The calculation with $^9$He subsystem having $p_{1/2}$ g.s. at about 2 MeV reasonably fit the data. The experimental cross section peaked at about 1.2 MeV could be consistent with some range of $p$-wave interactions for $^{11}$Li source \[Fig. \[fig:s-inc\], (e)–(f)\]. This, however, is possible only for quite a weak attractive part of $s$-wave potential: $V^0_c > - 20$ MeV. For such value of parameters the $s$-wave interaction is in general still effectively repulsive (due to a large repulsive core). For that reason, if we completely rely on the data [@kor94] we would impose theoretical limit $a > -5$ fm for $^8$He-$n$ scattering length. The derived theoretical limit is in a strong contradiction with the *upper* limit for scattering length in $^9$He $a<-10$ fm imposed in experiment [@che01]. There is no contradiction between our theoretical limit and data [@gol07] where a *lower* limit $a>-20$ fm for scattering length is given.
The unclear situation with exotic reaction mechanism expected for reactions with $^{11}$Li could have been resolved by a different experimental approach. Such an experiment in principle exists: the ground state of $^{10}$He and two excited states were identified in the complicated $2p$-$2n$ exchange reaction $^{10}$Be($^{14}$C,$^{14}$O)$^{10}$He [@ost94; @boh99]. Unfortunately, the observed peaks rest on a “thick” background and has a low statistical confidence. None of our calculations are consistent with the results of this experiment. Namely, we can not reproduce in any model assumptions the small width of the g.s. obtained in this work ($300 \pm 200$ keV at 1.07 MeV of excitation). For example, in Fig. \[fig:s-inc\] (b) the width of the state found at about 1.1 MeV is $\Gamma \sim 1.1$ MeV (see also Fig. \[fig:gam-ot-e\]). Smaller width of the $^{10}$He g.s. *if it takes place in reality* should mean non single-particle nature of this state \[means not described as $^8$He(g.s.)+$n$+$n$\] and hence a limited applicability of our model.
Prospects of correlation studies {#sec:corel}
--------------------------------
Important structure information about the three-body system could be obtained analysing the correlations among the decay products. The recent examples of such data analysis include successful application to the broad states in the continuum of $^5$H system [@gol05b] and to the two-proton radioactivity decays of $^{19}$Mg [@muk07] and $^{45}$Fe [@mie07]. Such range of application indicates a potential power of the correlation studies.
The partial decompositions of the cross section given in Fig. \[fig:bn-com\] show how the contributions of the $[s_{1/2}]^2$ component (mainly $K=0$) and $[p_{1/2}]^2$ component (mainly $K=2$) change when we add the $s$-wave interaction in $^9$He channel on top of the $p$-wave interaction or switch from the narrow to the broad source. The qualitative differences in these decompositions should be seen as qualitative differences in the correlation patterns.
\
The complete correlation information is provided in Fig. \[fig:corel\] for correlations in the $[s_{1/2}]^2$ state \[$E=4$ keV, panel (a)\] and in the $[p_{1/2}]^2$ state at $E=2.3$ MeV in the calculations with attraction in $s$-wave (b) and no $s$-wave attraction (c). The correlation densities are given in the plane of parameters $\{ \varepsilon, \cos(\theta_k) \}$ both in “T” and “Y” Jacobi coordinate systems. Parameter $\varepsilon=E_x/E$ describes the energy distribution between $X$ and $Y$ Jacobi subsystems ($E_x$ is energy in the $X$ Jacobi subsystem). Parameter $\theta_k$ is angle between Jacobi momenta $k_x$ and $k_y$ in the chosen Jacobi coordinate system: $$\cos(\theta_k) = \frac{(\mathbf{k}_x,\mathbf{k}_y)}{ k_{x}\; k_y}\;.$$ More details can be found in Ref. [@gri03c].
The correlation picture for the virtual three-body state with the $[s_{1/2}]^2$ structure is quite featureless \[Fig. \[fig:corel\] (a)\]. The energy distribution between subsystems is close to the “phase volume” distribution $$\frac{d^2 \sigma}{ dE d E_{x} } \sim E \;\sqrt{E_x(E-E_x)} \;.
\label{eq:ps}$$ There are only minor deviations from flat distribution for $\cos(\theta_k)$ at $\theta_k \sim 0^{\circ}$ and $\theta_k \sim 180^{\circ}$ in “T” Jacobi system (these are configurations when three particles come out in a line).
The predicted correlations for the $[p_{1/2}]^2$ state in “T” Jacobi system \[Fig. \[fig:corel\] (c)\] look very much like those already observed in other $p$-wave systems $^6$Be [@boc89] and $^5$H [@gol05b]. There is a double-hump structure reflecting the $[p_{1/2}]^2$ population. The hump, which corresponds to low-energy motion between neutrons is strongly enhanced due to FSI in the $n$-$n$ cannel.
Fig. \[fig:corel\] (b) provides prediction of correlations which is possibly important for prospective $^{10}$He studies. If the attractive $s$-wave interaction is added it qualitatively changes the picture of correlations at the expected $[p_{1/2}]^2$ state position. Now the energy distribution between subsystems (in the “T” Jacobi system) is close to the phase space distribution Eq. (\[eq:ps\]). Also the angular distribution changes drastically: the correlation density is concentrated in the regions where one of the neutrons is close to the $^8$He core in the momentum space \[$\cos(\theta_k)
\sim \pm 1$ and $E_x/E \sim 5/9$ in the “T” Jacobi system\]. In this case the only expressed feature in the “Y” Jacobi system is the “dineutron” correlation which can be seen as a small peak at $\cos(\theta_k) \sim - 1$ and $E_x/E \sim 1/2$.
The drastic changes between distributions Fig. \[fig:corel\] (b) and (c) mean that in the experimental measurements giving access to such a characteristic there will be no doubts in the structure identification even in the case of a poor population of the low-energy part of the spectrum or technical problems with detection of the low-energy events.
Reliability of the results
--------------------------
It should be mentioned once again that the aspects of the $^{10}$He dynamics, discussed in this work, are only valid if the single-particle $^{8}$He(g.s.)+$n$+$n$ structure of the low-lying $^{10}$He states really takes place. The ground for such an assumption is provided by knowledge of the $^{9}$He spectrum. However, the narrow first resonant states of $^{9}$He, as observed in Refs. [@set87; @boh88] \[$E(p_{1/2})=1.27$, MeV, $\Gamma=0.1$ MeV and $E(p_{3/2})=2.4$ MeV, $\Gamma=0.7$ MeV\], presume that it is not true, because small spectroscopic factors are expected [@bar04]. In the case that the results of Ref. [@gol07] are preferable \[$E(p_{1/2})=2$ MeV, $\Gamma=2$ MeV\], implying that this is a single-particle state, the basis for our model becomes reliable.
Sensitivity of the predictions to the $s$-wave interaction in the $^9$He channel is very high. The experimental results of Refs. [@che01] ($a<-10$ fm) and [@gol07] ($a>-20$ fm) are not contradictory, although not too restrictive. Thus, still no solid experimental ground can be found here. We think that this issue is a key point for understanding of the $^{10}$He structure.
Important conclusion of these studies is that energy spectrum obtained in experiments with $^{11}$Li is strongly affected by the reaction mechanism and we do not reproduce the results of the experiment [@kor94] without taking this effect into account. The question can be raised from theoretical side, how reliable is the statement that for $p_{1/2}$ state in $^9$He at about 2 MeV we can not get a *state* in $^{10}$He at 1.2 MeV straightforwardly. In Table \[tab:paring\] we list paring energy $E_p$ for valence neutrons calculated for $^{10}$He in different theoretical approaches. With $p_{1/2}$ state at 2 MeV the paring energy should be about 2.8 MeV, while in various theoretical approaches it is typically around 1 MeV and never exceeds 2 MeV. It is clear that relatively small paring energy in $^{10}$He is common for different theoretical models and can be considered as a reliable prediction. Also if we have a look on the nearby isotopes, for $p_{3/2}$ subshell nuclei $^{6}$He and $^{8}$He $E_p$ is $2.6$ MeV and $3.04$ MeV respectively. For $^{11}$Li, where $p_{1/2}$ subshell is populated, paring energy is known to be small: $E_p \sim 0.8$ MeV.
Conclusion
==========
In conclusion we would like to emphasize the most important results of our studies:
\(i) Within theoretical model for $p_{1/2}$ state in the $^9$He located at about 2 MeV it is problematic to obtain the $^{10}$He g.s. at about 1.2 MeV straightforwardly. The required for that paring energy is 2.8 MeV, while for $[p_{1/2}]^2$ configuration it is typically obtained $\sim 1-2$ MeV.
\(ii) The attraction in $s$-wave allows to shift state with $[p_{1/2}]^2$ configuration to significantly lower energy. However, for some extreme values of attraction ($a \leq -5$ fm) lead to formation of low-energy $[s_{1/2}]^2$ state which is seen as a sharp peak in the cross section at energies less than 0.3 MeV. The appearance of such a state is in accord with predictions of Ref. [@aoy02].
\(iii) In contrast with approach of Ref. [@aoy02], we study the conditions of “coexistence” of $[s_{1/2}]^2$ and $[p_{1/2}]^2$ states in the $0^+$ continuum for realistic scenarios. It is shown that the state with $[p_{1/2}]^2$ structure is poorly populated (also suffer significant broadening) in the presence of $[s_{1/2}]^2$ ground state and can be easily lost (small on the $s$-wave “background”). For that reason the idea of Ref. [@aoy02] that the ground $[s_{1/2}]^2$ state of the $^{10}$He remains unobserved, while the observed so far state is the first excited one with $[p_{1/2}]^2$ structure, does not get support in our studies.
Concerning comparison with experimental data:
\(i) Observation of quite a broad peak in $^{10}$He at about 1.2 MeV in Ref. [@kor94] could be explained by a specific mechanism of the chosen reaction induced by $^{11}$Li (namely the huge size of neutron halo in $^{11}$Li). This explanation is possible only in the case of absence of virtual state in $^9$He channel. For $^{10}$He ground $[p_{1/2}]^2$ state located at $E
\geq 2$ MeV the mentioned reaction mechanism leads to a strong enhancement of the low-energy transition strength even without any significant attraction in $s$-wave. As a result, the peak in the cross section may be shifted to a lower energy (e.g. $\sim 1.2$ MeV).
\(ii) The provided theoretical model essentially infer the properties of the $^{10}$He system basing on the properties of the $^9$He subsystem. At the moment we can not make the existing data on $^9$He and $^{10}$He consistent within this model. Calculations with large negative scattering length (e.g.$a<-5$ fm; experimental limit [@che01] is $a<-10$ fm) in core-$n$ subsystem necessarily lead to formation of the single narrow peak below 0.3 MeV in the spectrum which should have been seen in the experiment [@kor94].
\(iii) We have to conclude that the existing experimental data do not allow to establish unambiguously the “real” g.s. position for $^{10}$He. Alternative experiments (relative to those utilizing $^{11}$Li beams) are desirable. Further clarification of controversy between the $^9$He and $^{10}$He spectra is indispensable for theoretical understanding of the Helium isobar properties.
Acknowledgements
================
We are grateful to Prof. Yu. Ts. Oganessian for inspiration for this work. We are grateful for careful reading of the manuscript and valuable discussions to Profs. A. A. Korsheninnikov, G. M. Ter-Akopian, and M. S. Golovkov. LVG is supported by the INTAS Grant 05-1000008-8272, Russian RFBR Grants Nos. 05-02-16404 and 05-02-17535, and Russian Ministry of Industry and Science grant NS-1885.2003.2.
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Work [@ste88] [@kor93] [@she96] [@nav98] [@aoy02] [@vol05] This work
----------- ---------- ---------- ---------- ---------- ---------- ---------- ----------------
$-S_{n}$ 1.22 0.74 0.84 2.38 1.27 1.60 2.0
$-S_{2n}$ 1.18 0.6 1.09 2.78 1.68 1.94 $\sim 2.0-2.3$
$E_p$ 1.26 0.88 0.59 1.98 0.86 1.25 $\sim 1.7-2.0$
\[tab:paring\]
[^1]: For example in the inelastic process the virtual state is seen as near threshold peak with non-Lorencian shape.
[^2]: What is called the “Borromean” property of resonances, due to relation to artificially created Borromean states in ACCC method [@tan99], is better characterized as energy condition of a “true three-body decay” or a “democratic decay”. The detailed discussion of three-body decay modes can be found in Ref. [@gri01].
[^3]: The attractive potentials with barriers are not necessarily needed to form resonances in the continuum. They can be formed by pure repulsive potentials with broad “shelves”.
[^4]: Good example is the structure of the bound and quasibound states — isobaric partners. An exception here is the situation of the Thomas-Ehrmann shift, when significant deviations from isobaric symmetry can be observed.
|
---
abstract: |
In this work, we address ergodicity of smooth actions of finitely generated semigroups on an $m$-dimensional closed manifold $M$. We provide sufficient conditions for such an action to be ergodic with respect to the Lebesgue measure. Our results improve the main result in [@DKN], where the ergodicity for one dimensional fiber was proved. We will introduce Markov partition for finitely generated semi-group actions and then we establish ergodicity for a large class of finitely generated semi-groups of $C^{1+\alpha}$-diffeomorphisms that admit a Markov partition.
Moreover, we present some transitivity criteria for semi-group actions and provide a weaker form of dynamical irreducibility that suffices to ergodicity in our setting.
address:
- 'Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran'
- 'Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran'
- 'Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran'
author:
- 'A.Ehsani, F.H.Ghane and M.Zaj'
title: ' on ergodicity of mostly expanding semi-group actions '
---
introduction {#int}
============
One of the main goals in dynamical systems is to describe the typical behavior of orbits and find that how they evolve in time. An interesting approach in this direction is given by ergodic theory, which describes the typical behavior of orbits from a measurable point of view. In this context ergodic measures play a key role in understanding the ergodic behavior of a dynamical system.
In this paper, we discuss ergodic properties of random iterations of a finite family of $C^{1+\alpha}$-diffeomorphisms defined on an $m$-dimensional compact manifold.
Some knowledge of ergodic properties of semi-group or group action of diffeomorphisms is already available, through works of several authors. For instance, see [@S; @DKN] for the case that the ambient space is $S^{1}$ and [@BFS] for general case.\
Motivated by these results, we consider semi-groups or groups of $C^{1+\alpha}$-diffeomorphisms on a closed manifold $M$ exhibiting mostly expanding behavior and establish their ergodicity with respect to Lebesgue measure.
Write $\mathcal{G}^+$ (or $\mathcal{G}$) for the semigroup (or group) generated by a finite family of diffeomorphisms on a compact manifold $M$.
A subset $B\subset M$ is *forward invariant* for $\mathcal{G}^+$ , if $f(B)\subset B$ for all $f\in \mathcal{G}^+$. For the group action case, we can replace it by $f(B)= B$ for all $f\in \mathcal{G}$.
An invariant probability measure $\mu$ for a semigroup (or group) action is *ergodic* if for every measurable forward $\mathcal{G}^+$-invariant (or $\mathcal{G}$-invariant) subset $A\subset M$ either $\mu(A)=0$ or $\mu(M\setminus A)=0$. This definition can be extended to a measure $\mu$, which is *quasi-invariant*, i.e. $f_{*}\mu$ is absolutely continuous with respect to $\mu$ for each element $f$ in the acting semigroup or group. Note that for any $C^1$-map, the Lebesgue measure is quasi-invariant [@DKN]. In this context, the following question is natural [@DKN]:
Under what conditions a smooth action of a semigroup or group on a compact manifold is Lebesgue ergodic?
The main results of [@S] and [@DKN], solve the question in the affirmative for the one dimensional case under some additional assumptions. Indeed, for finitely generated groups of $C^{1+\alpha}$ diffeomorphisms on $S^{1}$, if $\mathcal{G}$ acts minimally and its Lyapunov expansion exponent is positive then the action is ergodic.\
We recall that the action of a semigroup (or group) is *minimal* if every closed invariant subset is either empty or coincides with the whole space, equivalently total orbit of each point is dense in the ambient space.
A point $x\in M$ is said to be non-expandable provided that for every $g\in \mathcal{G}^+$ (or $g \in \mathcal{G}$) one has $m(Dg(x))\leq1$, otherwise is called an expandable point.
In [@DKN], the authors proved that if $\mathcal{G}$ is a finitely generated subgroup of $Diff^{2}(S^{1})$ such that it acts minimally and possesses a finite number of non-expandable points with some aditional assumption, then the action of $\mathcal{G}$ is ergodic with respect to the Lebesgue measure. Several authors also worked on ergodicity of semi-group actions on compact $m$-dimensional manifolds. As shown in [@BFS], ergodicity happens whenever the semi-group action is expanding minimal $C^{1+\alpha}$-conformal.\
The concept of expanding action in [@BFS] has different means: a semi-group action is *expanding* provided that its inverse semigroup behaves locally expandable.
In general, minimality [@F] does not imply ergodicity. In the opposite direction ergodicity does not imply minimality. Indeed, one can easily construct examples of ergodic group actions having a global fixed points.\
Moreover, the main result of this paper illustrates an example of an ergodic semi-group action for which minimality is not a necessary condition.
Here, we improve the results of [@S; @DKN] for the case that the ambient space is $m$-dimensional, $m\geqslant 2$. In our setting, we allow that the non-expandable points exist and they are contained in an open region $V\subset M$.
In fact, we contribute sufficient conditions for ergodicity that applicable to a large class of finitely generated semi-groups of $C^{1+\alpha}$-diffeomorphisms.\
Our hypothesis that formulated in section 2 are conditions of the type that Oliveira and Viana [@OV] introduced for deterministic non-uniformly expanding maps. We extend the approach used by them to semi-groups of $C^{1+\alpha}$-diffeomorphisms on a compact $m$-dimensional manifold. However, this extension involves some difficulties. For instance, Markov partitions are not known to exist for semi-group actions.
Let us note that non-uniformly expanding maps with some random noise was addressed in several works. In particular, the stochastical stability of general classes of non-uniformly expanding maps were established in [@AA], [@AV], [@Y] and [@BY]. Also in [@BFS], the authors introduced a strong form of non-uniform expanding property for random maps. This concept in our setting has slightly different means. Indeed, we will establish the ergodicity of semi-group actions which equip with a weak form of non-uniform expanding property. This property is referred to as *orbital non-uniform expanding property*.
In section 6, some criteria for dynamical irreducibility is obtained. It provides a weak form of irreducibility in comparing with minimality. Also, the concept of weak cycle property will be introduced. This concept is equivalent to density of backward orbits for a subset of points with full Lebesgue measure and suffices to establish ergodicity from local ergodicity. Finally, we construct some examples that fit our assumptions in the last section.
**[Preliminaries and results]{}** {#non}
=================================
Suppose that $X$ is a compact metric space and $\mathcal{F}=\{f_1,\cdots,f_k\}$ is a finite family of homeomorphisms of $X$. Let us consider the *symbol space* $\Sigma_k^+$ which is the set of one sided infinite words over the alphabet $\{1,\ldots, k\}$. For any sequence $\omega=(\omega_0\omega_1\cdots \omega_n\cdots)\in\Sigma_k^+$, we write $f^n_\omega(x)=f_{\omega_{n-1}}\circ f_\omega^{n-1}(x)$; $n\in\mathbb{N}$ and $f_\omega^0=Id$. A fiber orbit corresponding to a one-way infinite word $\omega=(\omega_0\omega_1\cdots)\in\Sigma_k^+$ at a point $x$ is defined by $\mathcal{O}^+(x,\omega)=\{f^n_\omega(x)\}_{n=0}^\infty$.
We say that a finite family $\mathcal{R}=\{R_1,\cdots,R_k\}$ of subsets of $X$ is a topological partition if it satisfies the following:\
(1) each $R_i$ is open in $X$;\
(2) $R_i\cap R_j=\emptyset$, for each $i,j=1,\cdots,k$, with $i\neq j$;\
(3) $X=\overline{R_1}\cup\cdots\cup \overline{R_k}$.
In below, we generalize the concept of Markov partition to finitely semi-group actions.
Take a topological partition $\mathcal{R}=\{R_1,\cdots,R_k\}$ of a compact metric space $X$ with together a finite family $\mathcal{F}=\{f_1,\cdots,f_k\}$ of homeomorphisms of $X$. Let us consider the restriction $f_i\vert_{R_i}$, for each $i=1,\cdots,k$.\
We say that $(\mathcal{R},\mathcal{F})$ satisfies the $(n,\omega)$-fold intersection property for a positive integer $n\geq 3$ and $\omega=( \omega_0\omega_1\omega_2\cdots)\in\Sigma_k^+$ if $R_{\omega_j}\cap f^{-1}_{\omega_j}(R_{\omega_{j+1}})\neq\emptyset$, $0\leq j\leq n-1$, implies that $\cap_{j=0}^{n-1}(f_\omega^j)^{-1}(R_{\omega_{j+1}})\neq\emptyset$.\
A topological partition $\mathcal{R}=\{R_1,\cdots,R_k\}$ is *Markov* for semigroup generated by $\mathcal{F}=\{f_1,\cdots,f_k\}\subset home(X)$, if $(\mathcal{R},\mathcal{F})$ satisfies the $(n,\omega)$-fold intersection property for all $n\geq 3$ and $\omega\in\Sigma_k^+$.
It is not hard to see that $(\mathcal{R},\mathcal{F})$ is Markov provided that it satisfies the following condition: if $f_i(R_i)\cap R_j\neq\emptyset$, then $f_i(R_i)\supset R_j$, for each $i,j=1,\cdots,k$.
**[Statement of the main result]{}** {#non}
------------------------------------
In the rest of the paper we take M to be a compact Riemannian manifold, $m$ denotes the normalized Lebesgue measure and $\{f_1, \ldots, f_p, f_{p+1}, \ldots, f_{p+q} \}$ a finite family of $C^{1+\alpha}$-diffeomorphisms defined on $M$ satisfying the following conditions:
$(A_0)$ There exists a topological partition $\mathcal{R}=\{R_1,\cdots, R_p, R_{p+1}, \ldots, R_{p+q}\}$ such that for each $i=1,\cdots,p+q$, the clouser $\overline{R_i}$ has finite inner diameter, i.e. any two points in $\overline{R_i}$ may be joined by a curve contained in $\overline{R_i}$ whose length is bounded by a constant $L$. Moreover, $m(\partial R_i)=0$, for each $i=1,\cdots,p+q$.
$(A_1)$ If $f_i(R_i)\cap R_j\neq\emptyset$ then $f_i(R_i)\supset R_j$ which implies that $f_i(\overline{R_i})\supset \overline{R_j}$.
$(A_2)$ There exists $\sigma_1, \sigma_2>1$ such that
- $\Vert Df_i(x)^{-1}\Vert^{-1}\geq\sigma_1$ for every $x \in R_i$, $1 \leq i \leq p$,
- $\Vert Df_{p+j}(x)^{-1}\Vert^{-1}\geq\sigma_2^{-1}$ for every $x\in R_{p+j}$, $1 \leq j \leq q$
and $\sigma_2$ is close enough to 1.
$(A_3)$ $|det(Df_{p+j}(x))|> q$ for every $x \in R_{p+j}$, $1 \leq j \leq q$.
In fact, the acting of semi-group $\mathcal{G}^+$ generated by $\{f_1, \ldots, f_{p+q}\}$ is expanding on $R_1 \cup \ldots \cup R_p$ and is never very contacting. According to condition $(A_1)$, the topological partition $\mathcal{R}$ is Markov for $\mathcal{F}$. Also, by $(A_0)$, $m(\partial \mathcal{R})=0$ and therefore, $m(\mathcal{O}_\mathcal{G}^+(\partial \mathcal{R}))=0$, where $\mathcal{O}_\mathcal{G}^+(\partial \mathcal{R})$ is the total orbit of $\partial \mathcal{R}$, i.e. $$\mathcal{O}_\mathcal{G}^+(\partial \mathcal{R}) = \bigcup_{g \in \mathcal{G}^+} g(\partial \mathcal{R}).$$ Moreover, we take $N=M\backslash\mathcal{O}_\mathcal{G}^+(\partial R)$. Then $m(N)=1$.
We say that a semi-group $\mathcal{G}^+$ satisfies the weak cycle property if for each open set $B\subset M$, there exists a sequence $\{h_{i}\}\subset \mathcal{G}^{+}$ such that $M\circeq \bigcup_{i=1}^{\infty}h_{i}(B)$, this means that $m (M\setminus \bigcup_{i=1}^{\infty} h_{i}(B)=0)$.
Now, we state the main result of the paper that illustrates the Lebesgue ergodicity for a large class of finitely generated semigroups of $C^{1+\alpha}$-diffeomorphisms. Here the concept of ergodicity is referred to ergodicity of quasi-invariant measures and our focus is on the Lebesgue measure since it is quasi invariant for $C^1$ maps.
\[main\] Suppose that $\mathcal{G}^+$ is a finitely generated semi-group of $Diff^{1+\alpha}(M)$ for which the following holds:
$(1)$ $\mathcal{G}^+$ satisfies the weak-cycle property;
$(2)$ there exists a finite family $\{f_1, \ldots, f_{p+q}\}\subset \mathcal{G}^+$ satisfies conditions $(A0)$, $(A1)$, $(A2)$ and $(A3)$ above.
Then $\mathcal{G}^+$ is ergodic with respect to the Lebesgue measure.
**[Non-uniform expanding semigroups]{}** {#non}
========================================
Non-uniformly expanding maps introduced in [@A; @Vi]. Then general conclusions for systems exhibiting non-uniform expanding behavior provided in [@ABV]. Let us recall that a local diffeomorphism $f:M \to M$ is *non-uniformly expanding* if there exists $c>0$ such that for Lebesgue almost every point $x\in M$ one has: $$\limsup_{n \to \infty} \frac{1}{n}\sum_{j=0}^{n-1} \log \parallel Df(f^{j}(x))^{-1}\parallel <-c.$$ This approach has been most effective in studying ergodic aspects of systems and extended to the random setting [@V]: the authors considered random perturbations at each iterate a map, that is close to a non-uniformly expanding map, chosen independently according to some probabilistic law $\theta_{\varepsilon}$, where $\varepsilon >0$ is the noise level.
The notion of non-uniformly expanding on random orbits was addressed in [@BFS] with slightly different means.
To be more precise, consider a measure preserving system $(\Omega ,\sigma , \mathbb{P})$, where $\mathbb{P}$ is a Borel measure, $\sigma:\omega \to \omega$ is $\mathbb{P}$-invariant ergodic transformation and $\Omega$ is a compact separable metric space with a random continuous map $F:\Omega \rightarrow C^{r}(M,M)$. We denote $f_{\omega}:= F(\omega)$, $f_{\omega}^n:= f_{\sigma^{n-1}\omega}\circ \dots \circ f_{\omega}$ and $f_{\omega}^{0}:= id$.\
Now, given $x\in M$ and $\omega \in \Omega^{\mathbb{Z}}$, the sequence $(f^{n}_{\omega}(x))_{n\geq 1}$ is called a *random orbit* of $x$.
We say that $F$ is a *non-uniformly expanding on random orbits* if there exists $c>0$ such that for $(\mathbb{P}\times m)$-almost $(\omega,x)\in \Omega \times m$ it holds: $$\limsup_{n \to \infty} \frac{1}{n}\sum_{i=0}^{\infty} \parallel Df_{\sigma^i\omega}(f^{j}_{\omega}(x)^{-1})\parallel<-c.$$ They proved that \[Proposition 4.4, [@BFS]\] there are no non-uniform expanding finitely generated semi-group of diffeomorphisms which acts on random orbits.
In this paper, we deal with random product of finitely many maps and semi-group action generated by these maps and then introduce a weak form of non-uniform expanding property that we addressed here.
Consider a finitely generated semigroup $\mathcal{G}^+$ (or group $\mathcal{G}$) with generators $\{f_0, \ldots, f_{k-1}\}$. We say that $\mathcal{G}^+$ (or $\mathcal{G}$) is orbital non-uniformly expanding if there exists $c>0$ such that for Lebesgue almost every $x\in M$, there is a sequence $\omega \in \Sigma_+^k$ (or $\omega \in \Sigma^k$) satisfying $$\label{1}
\limsup_{n \to \infty} \dfrac{1}{n}\sum_{i=0}^{n-1}\log \parallel Df_{\omega_i}(f_\omega^i(x))^{-1}\Vert \leq -c.$$
In the rest of this section, we show that a large class of finitely generated semi-groups of $C^{1+\alpha}$ diffeomorphisms on a compact manifold $M$ exhibit orbital non-uniformly expanding property.
Let $x\in N= M - \partial \mathcal{R}$ be given. Suppose that $x\in R_{\omega_0}$ and the indices $\omega_0, \omega_1, \cdots, \omega_{j-1} \in\{1, \cdots, p+q\}$ are chosen such that for each $k=0,...,j-2$,
$f_{\omega_k} \circ \cdots \circ f_{\omega_0}(x)\in R_{\omega_{k+1}}.$
Now, we choose an index $\omega_j\in\{1, \cdots, p+q\}$ satisfying
$f_{\omega_{j-1}}\circ \cdots \circ f_{\omega_0}\in R_{\omega_j}.$
In this way, inductively we may associate a sequence $\omega=\omega(x)$, to each $x\in N$, by taking $\omega=(\omega_0,\omega_1,\cdots)$ which is called the *itinerary* of $x$. Let $\pi_n$ denote the projection that maps the sequence $\omega=(\omega_0,\omega_1,\cdots)$ to the finite word $\omega(x, n)=(\omega_0,\omega_1,\cdots, \omega_{n-1})$ which we refer to it as $n$-*itinerary* of $x$.
\[uniform\] Consider semi group $\mathcal{G}^+ \subset Diff^{1+\alpha}(M)$ for which there exist a finite family $\{f_1, \ldots, f_p, f_{p+1}, \ldots, f_{p+q} \}\subset\mathcal{G}^+$ and a topological partition $\mathcal{R}=\{R_1,\cdots, R_p, R_{p+1}, \ldots, R_{p+q}\}$ satisfying conditions $(A0)$, $(A1)$, $(A2)$ and $(A3)$. Then $\mathcal{G}^+$ is orbital non-uniformly expanding.
Consider the subset $\{f_1, \ldots, f_p, \ldots, f_{p+q}\}$ of $\mathcal{G}^+$ and a family $\mathcal{R}$ of open subsets of $M$ satisfying the assumptions. Also, for each $x\in N$, take a sequence $\omega=\omega(x)$ which is the itinerary of $x$. Since $N$ has full Lebesgue measure and condition $(A3)$ ensures that the approach applied by Alves, Bonatti and Viana \[Lemma A1, [@ABV]\] can be extended to our setting, hence the following claim holds.
For Lebesgue almost every point $x\in M$, the $\omega$-orbital branch of $x$ spends a fraction $\epsilon_0>0$ of the time in $R_1\cup\cdots\cup R_p$, for some $\epsilon_0>0$; that is
$\# \{0\leq k< n: f_{\omega}^k(x)\in R_1\cup\cdots\cup R_p\}\geq\epsilon_0n$
for large enough $n$, where $\omega$ is the itinerary of $x$.
Now, we use the claim to show that $\mathcal{G}^+$ is orbital non-uniformly expanding, that is the equality (\[1\]) holds for Lebesgue almost every point $x\in M$. Take $\epsilon_0>0$ as introduced in the claim and $\sigma_2$ close enough to 1 so that $\sigma_1^{- \epsilon_0}\sigma_2 \leq e^{-c}$, for some $c>0$. Let $x$ be any point satisfies the conclusion of the claim and $\omega=\omega(x)$ is the itinerary of $x$. Then $$\begin{aligned}
\prod_{j=0}^{n-1}\Vert Df_{\omega_j}(f_\omega^j(x))^{-1}\Vert\leq\sigma_1^{- \epsilon_0 n} \sigma_2^{(1- \epsilon_0)n}\leq e^{-c n}\end{aligned}$$ for every large enough $n$. This means that $x$ satisfies the conclusion of the proposition.
Hyperbolic times and hyperbolic cyliders {#result}
========================================
Throughout this section, we take $\{f_1, \ldots, f_p, f_{p+1}, \ldots, f_{p+q} \}\subset \mathcal{G}^+$ a finite family of $C^{1+\alpha}$-diffeomorphisms and a topological partition $\mathcal{R}=\{R_1,\cdots, R_p, R_{p+1}, \ldots, R_{p+q}\}$ satisfying the conditions $(A0)$, $(A1)$, $(A2)$, and $(A3)$.
First, we present the concept of *hyperbolic time* of a point $x\in M$ for semigroup action $\mathcal{G}^+$. This concept was introduced in [@ABV] for differentiable deterministic maps.
Given $0<c<1$, we say that $n\in\mathbb{N}$ is a hyperbolic time for $x\in M$ if for every $1\leq k\leq n$, one has $$\label{2}
\prod_{j=n-k}^{n-1}\Vert Df_{\omega_j}(f_\omega^j(x))^{-1}\Vert\leq e^{-c k},$$
where $\omega=\omega(x)=(\omega_0, \omega_1, \ldots, \omega_j, \ldots)$ is the itinerary of $x$.\
Any nonempty set of the form $$C^n=C^n[\omega_0, \ldots, \omega_{n-1}]= \{x \in M : x \in R_{\omega_0}, f_{\omega_0}(x) \in R_{\omega_1}, \ldots, f_\omega^n(x) \in R_{\omega_{n-1}}\}$$ is called a *cylinder* of length $n$. We say that $C^n$ is a *hyperbolic cylinder* provided that $n$ is a hyperbolic time for each point $x \in C^n$ and its itinerary $\omega$.
Let $\mathcal{C}^n$ be the family of all cylinders of length $n$ and $\mathcal{C}^n_h$ the subset of hyperbolic cylinders.
The closure of $\mathcal{C}^{n}[\omega_{0},\dots ,\omega_{n-1}]$ is defined as $$\bar{C}^{n}[\omega_{0},\dots ,\omega_{n-1}]=
\{y\in M : y\in \bar{R}_{\omega_{0}}, f_{\omega
_{0}}(y)\in \bar{R}_{\omega_{1}}, \dots , f_{\omega}^n(y)\in \bar{R}_{\omega_{n-1}} \}.$$ Hence, $f_{\omega_{j-1}}\circ \dots \circ f_{\omega_{0}}(\bar{C}^{n}[\omega_{0},\dots , \omega_{n-1}])=\bar{C}^{n-j}[\omega_{j},\dots , \omega_{n-1}]$, for any $1\leq j<n$.\
So, $f_{\omega_{n-1}}\circ \dots \circ f_{\omega_{0}}(\bar{C}^{n}[\omega_{0},\dots ,\omega_{n-1}])=f_{\omega_{n-1}}(\bar{R}_{\omega_{n-1}})$ and thus its inner diameter is bounded by the constant $$K_{2}=K_{1} max_{1\leq i\leq p+q} \{\parallel Df_{i}(x)\parallel : x\in R_{i}\},$$ where $K_{1}$ is the maximum inner diameter of $\bar{R}_{i}$ over all $i=1,\dots ,p+q$.\
Let $x\in M$ with itinerary $\omega$. We say that the frequency of hyperbolic times for $x$ is greater than $\epsilon_0>0$ if for large $n\in\mathbb{N}$, there are $l\geq\epsilon_0 n$ and integers $1\leq n_1<\cdots<n_l\leq n$ which are hyperbolic times for $x$.
Let us note that if $\omega$ is the itinerary of $x$ then $\sigma^k\omega$ is the itinerary of $f_\omega^k(x)$, where $\sigma$ is the Bernoulli shift transformation.\
If $n$ is a hyperbolic time for $x$ with itinerary $\omega$, then $n-s$ is a hyperbolic time for $f^s_\omega(x)$, for any $1\leq s<n$.\
It is not hard to see that the converse is also true. Indeed, if $k<n$ is a hyperbolic time for $x$ and there exists $1\leq s\leq k$ such that $n-s$ is a hyperbolic time for $f^s_\omega(x)$ then $n$ is a hyperbolic time for $x$.
The next lemma asserts that for points satisfy the orbital non-uniformly expanding property (\[1\]), there are infinitely many hyperbolic times. Moreover, the set of hyperbolic times has positive density at infinity, and its proof is based on a lemma due to Pliss (see e.g. [@M]).
\[frequency\] Suppose that $x\in M$, $\omega$ is its itinerary and $n\geq 1$ is such that $$\label{3}
\frac{1}{n}\sum_{j=0}^{n-1}\log\Vert Df_{\omega_j}(f_{\omega}^j(x))^{-1}\Vert\leq -c<0.$$ Then, there is $\epsilon_0>0$, depending only on $\mathcal{G}^+$ and $c$, and a sequence of hyperbolic times $1\leq n_1<\cdots<n_t\leq n$ for $x$, with $t\geq\epsilon_0 n$; that is the frequency of hyperbolic times for $x$ is larger than $\epsilon_0$.
Similar to Corollary 3.2 of [@ABV]. See also Proposition 4.4 of [@V].
Suppose that $x$ satisfies the conclusion of Proposition \[uniform\], so for large enough $n>0$ $$\begin{aligned}
\sum_{j=0}^{n-1}\log\Vert Df_{\omega_j}(f_\omega^j(x))^{-1}\Vert<-nc,\end{aligned}$$ thus $$\begin{aligned}
\prod_{j=0}^{n-1}\Vert Df_{\omega_j}(f_\omega^j(x))^{-1}\Vert<e^{-nc}.\end{aligned}$$ Then $n$ is a hyperbolic time for $x$.\
Let $\mathcal{H}$ be the set of all points $x\in M$ with itinerary $\omega$ satisfying the orbital non-uniformly expanding property (\[1\]). Clearly $\mathcal{H}$ has full Lebesgue measure.
Here, the ball of radius $r>0$ is meant with respect to the Riemannian distance $dist(x, y)$ on $M$. If $D \subset M$ is any path connected domain, we define the distance $dist_D(x, y)$ between two points $x$ and $y$ in $D$ to be the infimum the lengths of all curves joining $x$ to $y$ inside $D$. In particular, $dist_D(x, y)\geq dist(x, y)$ for every $x$ and $y$ in $D$.
There is $r>0$ such that if $n$ is a hyperbolic time for $x\in\mathcal{H}$ with itinerary $\omega=(\omega_0, \omega_1, \ldots)$, then $$\label{5}
\|Df_{\omega_0}(y)^{-1}\| \leq e^{\frac{c}{2}} \|Df_{\omega_0}(x)^{-1}\|,$$ for any point $y$ in the ball $B(x, r e^{\frac{-nc}{2}})$.
By continuity we can choose $r > 0$ small enough so that the uniform bounds $\sigma_1$ and $\sigma_2$ in conditions $(A1)$ and $(A2)$ hold for the mappings $f_i$ in $r$-neighborhoods of $R_i$, $i=1, \ldots, p+q$. Moreover, we take $r$ small enough so that the inverse of the exponential map $exp_x$ is defined on the $r$-neighborhood of every point $x \in M$ and it is isometry on $B(x, r)$.
Now, by continuity and compactness of $M$ we can take $r > 0$ small enough such that if $n$ is a hyperbolic time for $x$ with itinerary $\omega= (\omega_0, \omega_1, \ldots)$ then $$\|Df_{\omega_0}(y)^{-1}\| \leq e^{\frac{c}{2}} \|Df_{\omega_0}(x)^{-1}\|,$$ whenever $y \in B(x, r e^{\frac{-c}{2}}).$ Since $B(x, r e^{\frac{-nc}{2}}) \subset B(x, r e^{\frac{-c}{2}})$, the statement of the lemma follows by the above inequality.
A *dynamical ball* of center $x$, itinerary $\omega \in \Sigma^k_+$, radius $r$, and length $n \geq 1$ is defined by $$B(\omega, x, n, r)=\{y \in M: dist(f_\omega^j(x), f_\omega^j(y))\leq r, \forall \ 0\leq j \leq n\}.$$
\[preball\] There exists $r>0$ such that if $n$ is a hyperbolic time for $x\in\mathcal{H}$, then there exists a neighborhood $V(x, n)$ of $x$ for which the following hold:\
1) $f_\omega^n$ maps $V(x, n)$ diffeomorphically onto $B(f_\omega^n(x),r)$, where $B(f_\omega^n(x),r)$ is a ball of radius $r$ and with center $f^n_\omega(x)$;\
2) for every $y\in V(x, n)$ and $1\leq j\leq n$, we have
$\Vert D(f^j_{\sigma_{\omega}^{n-j}})^{-1}(z)\| \leq e^ {\frac{-jc}{2}},$
where $z$ belongs to $B(f_\omega^n(x),r)$.
Analogous to the proof of Proposition 4.9 of [@V] and according to Lemma 4.6.
It is not hard to see that the neighborhood $V(x, n)$ is a dynamical ball about $x$.
For every $1\leq j\leq n$ and $x,y$ in the closure of any $C^{n}=C^{n}[\omega_{0},\dots ,\omega_{n-1}]\in \mathcal{C}^{n}_{h}$ we have: $$d_{f^{n-j}_{\omega}(\bar{C}^{n})}(f^{n-j}_{\omega}(x),f^{n-j}_{\omega}(y))\leq e^{\frac{-jc}{2}}d_{f_\omega ^{n}(\bar{C}^{n})}(f^{n}_{\omega}(x),f^{n}_{\omega}(y))\leq K_{2}e^{\frac{-jc}{2}},$$ where $f_\omega ^j= f_{\omega_{j-1}}\circ \ldots \circ f_{\omega_0}$.
Any curve joining $f^{n}_{\omega}(x)$ to $f^{n}_{\omega}(y)$ inside $f^{n}_{\omega}(\bar{C}^{n})$ lifts to a unique curve joining $f^{n-j}_{\omega}(x)$ to $f^{n-j}_{\omega}(y)$ inside $f^{n-j}_{\omega}(\bar{C}^{n})$. Now we conclude the result by the above proposition.
There is $L_1>0$ such that for any $x,y$ in closure of any $C^{n}=C^n[\omega_0, \ldots, \omega_{n-1}] \in \mathcal{C}^{n}_{h}$ $$L^{-1}_{1}\leq \frac{\mid det Df^{n}_{\omega}(x)\mid}{\mid det Df^{n}_{\omega}(y)\mid}\leq L_{1}.$$
Note that by assumption $g_{j}=log \mid det Df_{j{\mid \bar{R}_{j}}}\mid$, for $j=1,\dots ,p+q$, is $\alpha$-Hölder and thus for each $ x,y \in R_{j}$, it holds that $$\mid g_{j}(x)-g_{j}(y)\mid \leq C_0 d(x,y)^{\alpha},$$ for some constants $C_0> 0$ and $\alpha>0$. Now, suppose that $C^{n}= C^n[\omega_0, \ldots, \omega_{n-1}]$ is a hyperbolic cylinder and $x, y \in \overline{C}^n$. This means that $x, y \in R_{\omega_0}, \ f_{\omega_0}(x), f_{\omega_0}(y) \in R_{\omega_1}$ and for each $1 \leq i \leq n$, $f_\omega ^i(x), f_\omega ^i(y) \in R_{\omega_i}$. According to Corollary 4.8, for each $1 \leq j \leq n$ and $x, y \in \overline{C}^n$ it holds that $$d_{f^{n-j}_{\omega}(\bar{C}^{n})}(f^{n-j}_{\omega}(x),f^{n-j}_{\omega}(y))\leq e^{\frac{-jc}{2}}d_{f_\omega^{n}(\bar{C}^{n})}(f^{n}_{\omega}(x),f^{n}_{\omega}(y))\leq K_{2}e^{\frac{-jc}{2}},$$ and therefore, $$log \frac{\mid det Df^{n}_{\omega}(x)\mid}{\mid det Df^{n}_{\omega}(y)\mid}=\sum^{n-1}_{i=0}\mid g_{\omega_{i}}(f^{i}_{\omega}(x))-g_{\omega_{i}}(f^{i}_{\omega}(y))\mid \leq \sum^{n-1}_{i=0}C_0d_{f_\omega^j(\overline{C}^n)}(f^{i}_{\omega}(x),f^{i}_{\omega}(y))^{\alpha}$$ $$\leq C_0 K_2^\alpha \sum^{n-1}_{i=0}e^{\frac{-(n-i)c\alpha}{2}} \leq C_0 K_2^\alpha \sum^{\infty}_{i=0} e^{\frac{-ic \alpha}{2}}.$$ Take $L_{1}=\exp(C_0 K_2^\alpha \sum^{\infty}_{i=0} e^{\frac{-ic \alpha}{2}}).$
There exists a constant $L_2>0$ such that if $C^{n}=C^n[\omega_0, \ldots, \omega_{n-1}]$ is a hyperbolic cylinder and $A_1, A_2$ are two subsets of $\overline{C}^n$ then $$\begin{aligned}
\label{10}
L_2^{-1} \dfrac{m(A_1)}{m(A_2)}\leq\dfrac{m(f_\omega^n(A_1))}{m(f_\omega^n(A_2))}\leq L_2\dfrac{m(A_1)}{m(A_2)}
\end{aligned}$$
We apply the change of variable formula for $f_\omega^n$ and analogous to Corollaries 4.10, 4.11 and 4.12 of [@V], we conclude the result.
**[Proof of Theorem A]{}** {#non}
==========================
Let us take a finite family $\{f_1, \ldots, f_{p+q}\}\subset \mathcal{G}^+$ of $C^{1+\alpha}$ diffeomorphisms on $M$ and a Markov partition $\mathcal{R}=\{R_1, \ldots, R_{p+q}\}$ for which the conditions $(A0)$, $(A1)$, $(A2)$ and $(A3)$ hold. Moreover, suppose that $\mathcal{G}^+$ has weak-cycle property. We will prove that the semi-group action $\mathcal{G}^+$ is ergodic with respect to the Lebesgue measure. We take $\mathcal{A}$ the set of points $x \in M$ that belong to the closure of any hyperbolic cylinder $C^n$ for infinitely many values $n_1(x)< n_2(x) < \ldots< n_k(x) < \ldots$ of $n$. Clearly $\mathcal{H}\subseteq \mathcal{A}$. In particular, each $x \in \mathcal{A}$ has infinitely many hyperbolic times and hence $m(\mathcal{A})=1$.
Now, for each $n \geq 1$, we consider a covering $\mathcal{A}_n$ of $\mathcal{A}$ by pairwise disjoint measurable sets such that $A_n \in \mathcal{A}_n$ satisfying $C^m \subseteq A_n \subseteq \overline{C}^m$ for some $C^m \in \mathcal{C}_h^m$ and $m\geq n$. Indeed, we can take the family $\mathcal{A}_n$ which consists of cylinders $C^m \in \mathcal{C}_h^m$ with $m\geq n$ that are not contained in any $C^k \in \mathcal{C}_h^k$ with $m>k \geq n$.
Suppose that $B \subset M$ is a measurable $\mathcal{G}^+$-forward invariant subset with positive Lebesgue measure. Then there exists an element $R_i$ of the topological partition $\mathcal{R}$ such that $m(R_i \cap B^c)=0$.
Let $B \subset M$ be a measurable $\mathcal{G}^+$-forward invariant subset with positive Lebesgue measure. The next claim is needed.
Given $\delta >0$, there exist $n\geq 1$ and a subset $\{A_{n,i}: i\in I\}$ of $\mathcal{A}_{n}$ such that $$m(B\bigtriangleup \bigcup_{i\in I}A_{n,i})\leq \delta.$$
Indeed, it is enough to apply the technique of \[Lemma 3.11, [@OV]\] to our setting. Take compact subsets $K \subset B$ and $K^{\prime}\subset B^c$ such that $$m(B\bigtriangleup K)\leq\frac{\delta}{3}, \ and \
m(B^c \bigtriangleup K^{\prime})\leq\frac{\delta}{3}.$$ We set $\rho = dist(K, K^{\prime})$. By Corollary 4.8, one has $diam (\mathcal{A}_n)\leq K_2 e^{\frac{-nc}{2}}< \rho$ provided that $n$ is large enough. Since $\mathcal{A}$ is covered by $\mathcal{A}_n$, so there exists a family $\{A_{n,i}: i\in I\}\subset \mathcal{A}_n$ so that $m(K \setminus \bigcup_{i\in I}A_{n,i})\leq \frac{\delta}{3}.$ Since all $A_{i,n}, \ i \in I$, intersect $K$, they are disjoint from $K^{\prime}$. Therefore, $$m(B\bigtriangleup \bigcup_{i\in I}A_{n,i})\leq m(B\setminus K)+m(K \setminus \bigcup_{i\in I}A_{n,i})+
m(B^c \setminus K^{\prime})\leq \delta.$$ Now, according to the above claim and the approach of \[Corollary 3.12, [@OV]\] the following claim holds.
For each $\epsilon >0$, there exist $n\geq 1$ and $A_n \in \mathcal{A}_n$ such that $$\frac{m(B\cap A_n)}{m(A_n)}>1-\epsilon.$$
By Claim 5.3, there exists a sequence $A_{n}$ of measurable sets for which the following holds: $C^{m_{n}}\subset A_n \subset \bar{C}^{m_{n}}$, for some hyperbolic cylinder $C^{m_{n}}$ with $m_{n}\geq n$ and $$\frac{m(A_n \cap B^{c})}{m(A_n)} \to 0, \ n \to \infty.$$ Then, according to the distortion Lemma 4.9, Corollary 4.10 and the assumption that $B$ is forward invariant, one has that $$\label{2}
\frac{m(f^{m_{n}}_{\omega({m_n})}(A_n)\cap B^{c})}{m(f^{m_{n}}_{\omega({m_n})}(A_n))}\to 0, \ when \
n \to \infty,$$ where $\omega({m_n})$ is the itinerary of the cylinder $C^{m_n}= C^{m_n}[\omega_0, \ldots, \omega_{m_n -1}]$ and $f^{m_{n}}_{\omega({m_n})}= f_{\omega_{m_n-1}} \circ \ldots \circ f_{\omega_0}$.\
By Remark 4.6, $$R_{\omega_{m_n -1}}\subset f^{m_{n}}_{\omega(m_n)}(C^{m_{n}}) \subset f^{m_{n}}_{\omega(m_n)}(A_n),$$ for some $\omega_{m_n -1}\in \{1,\dots , p+q\}$. Fix any $j$ such that $\omega_{m_n -1}=j$ for infinitely many values of $n$. We know that $R_{j}$ is an open subset and therefore has positive $m$-measure. Then $(6)$ implies that $m(R_{j}\cap B^{c})=0$.
Now, we will finish the proof of Theorem A. As you have seen in the previous proposition, for forward invariant set $B$ with $m(B) > 0$, there exists a cylinder $R_j$ satisfying $m(R_{j}\cap B^{c})=0$. It is enough to show that $m(B)=1$.
Let us fix the cylinder $R_j$. According to weak-cycle property of $\mathcal{G}^+$, there exists a sequence $\{g_n\}\subset \mathcal{G}^+$ such that $$M \circeq \bigcup_{n\geq1}g_n(R_j).$$ Since $B$ is $\mathcal{G}^+$-forward invariant, $B^c$ is also $\mathcal{G}^+$-forward invariant. Hence, one has that $$g_n(R_j)\setminus B \subset g_n(R_j\setminus B), \ for \ all \ n\geq 1.$$ The Lebesgue measure $m$ is quasi-invariant for $C^1$- diffeomorphisms which imply that $m(g_n(R_j \setminus B))=0$ and therefore, $m(g_n(R_j) \setminus B)=0$. Since $\{g_n(R_j)\}$ is a countable family, $m(B)=1$ which terminates the proof.
**[Transitivity criteria]{}**
=============================
In this section we obtain some results for transitivity of semi-group actions of homeomorphisms defined on a compact manifold $M$. We introduce some kinds of transitive property and finally we provide a strong form of transitivity that suffices to conclude ergodicity in our setting. First, we need to formalize some notions.
Consider a finite family $\{f_1, \ldots, f_k\}$ of homeomorphisms defined on a compact manifold $M$. Let us take $\mathcal{G}^+$ the semi-group generated by these homeomorphisms. For given $x \in M$, the *forward total orbit* of $x$ is defined by $$\mathcal{O}_{\mathcal{G}}^+(x)= \{h(x): h \in \mathcal{G}^+\}.$$ Analogously, the *backward total orbit* of $x$ is defined by $$\mathcal{O}_{\mathcal{G}}^-(x)= \{h^{-1}(x): h \in \mathcal{G}^+\}.$$ We say that $\mathcal{G}^+$ acts *minimally* on $M$ if any point has a dense forward total orbit.\
For any sequence $\omega=(\omega_0, \omega_1, \ldots, \omega_n, \ldots) \in \Sigma_k^+$, we write $$f_\omega^n(x)=f_{\omega_n}\circ f_\omega^{n-1}(x); \ n \in \mathbb{N} \ and \ f_\omega^0=Id.$$ The set $\{f_\omega^n(x): n \geq 0\}$ is called $\omega$-*orbital branch* of $x$. We say that $x$ possesses an *orbital dense orbit* if there exists a sequence $\omega \in \Sigma_k^+$ such that $\omega$-orbital branch of $x$ is dense in $M$.
It is easy to see that if $\mathcal{G}^+$ acts minimally on $M$, then each $x$ has orbital dense orbit.\
In a finitely generated semi-group action, if minimality of action preserves under small perturbation of generators then we say that $\mathcal{G}^+$ is *robustly minimal*. Some examples of robustly minimal semi-group actions are already available, for instance see [@GH] and [@H2]. Let us mention that transitive semi-group actions have a weak form of dynamical irreducible property with respect to minimal semi-groups.
Here, we introduce three kinds of transitive property. The action of $\mathcal{G}^+$ is *weak transitive* if it possesses a point with a dense forward total orbit. We say that $\mathcal{G}^+$ is *transitive* whenever $U$ and $V$ are two open subsets of $M$, there exists $h \in \mathcal{G}^+$ such that $h(U) \cap V \neq \emptyset$. Finally, $\mathcal{G}^+$ is *strong transitive* if it admits a full measure subset $\widetilde{M} \subset M$ such that every point $x \in \widetilde{M}$ has a dense forward orbit.
In bellow, we illustrate the relationship between these different concepts of transitive property. Moreover, to obtain some results about the transitive properties of semi-group actions, we adapt some techniques from deterministic systems [@LP] to our setting.
A semi-group $\mathcal{G}^+$ is *contractive* at point $x \in M$ whenever there exist a sequence $\{ h_n \}\subset \mathcal{G}^+$ and open set $B$ containing $x$ for which $diam(h_n(B)) \to 0$, as $n \to \infty$.
Let $\{f_{1},\dots ,f_{k}\}$ be a finite family of homeomorphisms on a compact manifold $M$ and $\mathcal{G}^+$ be the semi-group generated by them. Suppose that there exist $x\in M$ for which the total backward orbit and total forward orbit are dense in $M$. If the semi-group $\mathcal{G}^+$ is contractive at $x$, then the action of $\mathcal{G}^+$ is transitive, i.e. for each two open subsets $U$ and $V$ of $M$ there exists $h\in <\mathcal{G}>^{+}$ such that $h(U)\cup V\neq\emptyset$.\
In particular, the point $x$ possesses an orbital branch which is dense in $M$.
Suppose that $U$ and $V$ are two open subsets of $M$. Since the total forward orbit of $x$ is dense in $M$, there exist $k_{1},k_{2}\in \mathcal{G}^{+}$ such that $k_{1}(x)\in U$ and $k_{2}(x)\in V$.\
According to the density of $\mathcal{O}^{-}_{\mathcal{G}}(x)$, we can choose two sequences $\{z_{i}\}\subset U$ and $\{h_{i}\}\subset \mathcal{G}^{+}$ that satisfying the following property: $h_{i}(z_{i})=x$ and $z_{i}$ converges to $k_{1}(x)$ when $i$ goes to infinity and therefore, $h_{i}(k_{1}(x))$ is contained in a neighborhood $B$ of $x$, for large enough $i$.\
On the other hand, continuity of $k_{2}$ and this fact that $k_{2}(x)\in V$ imply that $k_{2}(B_{r}(x))\subset V$, for some small $r>0$.\
Now, by contractibility of $\mathcal{G}^{+}$ at $x$, one can find $g\in <\mathcal{G}>^{+}$ such that $g(B)\subset B_{r}(x)$, and therefore $k_{2}(g(h_{i}(k_{1}(x))))\in V.$ Let us take $h:= k_{2}\circ g\circ h_{i}\circ k_{1}$, then $h(U)\cap V\neq\emptyset$.\
Finally, if we apply the above argument for a countable basis of $M$, the second statement follows immediately.
Let us take $\mathcal{G}^+ \subset Diff^{1+\alpha}(M)$ with generators $g_1, \ldots, g_k$ and there exists $\{f_1, \ldots, f_{p+q}\} \subset \mathcal{G}^+$ satisfies the conditions proposed in subsection 2.1. Then Corollary 4.8 ensures that the inverse semi-group $\mathcal{G}^-=<g_1^{-1}, \ldots, g_k^{-1}>$ is contractive at each point $x \in \mathcal{H}$. Hence, Lemma 6.2 implies that if $\mathcal{G}^+$ possesses a point $x \in \mathcal{H}$ with forward and backward dense orbit, then both semi-groups $\mathcal{G}^+$ and $\mathcal{G}^-$ are transitive.
Suppose that $\{f_{1}, \ldots ,f_{k}\}$ is a finite family of homeomorphisms on $M$ and $\mathcal{G}^{+}$ is the semi-group generated by these maps. If $\mathcal{G}^{+}$ has any point with forward orbital dense orbit, then the set of points with total forward dense orbit is a residual subset of $M$.
Suppose that there exists a point $x\in M$ and $\omega \in \Sigma^{+}_{k}$ such that the $\omega$-orbital orbit of $x$, $\mathcal{O}^{+}_{\mathcal{G}}(\omega, x)=\{f^{j}_{\omega}(x): j\in \mathbb{N}\}$, is dense in $M$. Let us take a countable basis $\mathcal{V}= \{ V_j : j \in \mathbb{N}\}$ of $M$ and we set $$\mathcal{A}_n=\{z \in M: \mathcal{O}_{\mathcal{G}}^+(z) \ is \ \frac{1}{n}-dense\}.$$ By density of $\mathcal{O}^{+}_{\mathcal{G}}(\omega, x)$ there exist sequences $\{x_{j_{l}}\}\subset \mathcal{O}^{+}_{\mathcal{G}}(\omega, x)$ and $k_{n,l}\subset \mathbb{N}$, for each $n,l\in \mathbb{N}$, for which the following holds:\
$x_{j_{l}}\in V_{l}$ and the segment orbit $\{x_{j_{l}}=f^{j_{l}}_{\omega}(x), f^{j_{l}+1}_{\omega}(x), \ldots f^{j_{l}+k_{n,l}}_{\omega}(x)\}$ is $\frac{1}{2n}$-dense. By continuity, there exists $\{r_{n,l}>0\}_{n,l\in \mathbb{N}}$ such that $f^{i}_{\sigma^{j_{l}-1}\omega}(B_{r_{n,l}}(x_{j_{l}}))\subset B_{\frac{1}{2n}}(f^{i}_{\sigma^{j_{l}-1}\omega}(x_{j_{l}}))$ for $0\leq i\leq k_{n,l}$. This means that for each $y\in B_{r_{n,l}}(x_{j_{l}})$, the segment orbit $\{y, f_{\sigma^{j_{l}-1}\omega}(y), \ldots , f^{k_{n,l}}_{\sigma^{j_{l}-1}\omega}(y)\}$ is $\frac{1}{n}$-dense.\
Therefore, $\cup_{l \in \mathbb{N}}B_{r_{n,l}}(x_{j_{l}})$ is an open and dense subset. In particular, this proves that $\cap_{n \in \mathbb{N}}\cup_{l \in \mathbb{N}}B_{r_{n,l}}(x_{j_{l}})$ is a residual subset contained in $\cap_{n=1}^{\infty}\mathcal{A}_n$ which consists of the points with dense forward orbit. This terminates the proof of the lemma.
Let $\mathcal{F}$ be a finite family of homeomorphism on $M$. If there exists a point $x\in M$ with forward orbital dense orbit, then $\{x\in M: \overline{\mathcal{O}^{-}_{\mathcal{F}}(x)}=\overline{\mathcal{O}^{+}_{\mathcal{F}}(x)}=M\}$ is a residual subset of $M$.
From the previous lemma, $\{x\in M: \overline{\mathcal{O}^{+}_{\mathcal{F}}(x)}=M\}$ is a residual subset of $M$. Let us take $x\in M$ with forward orbital dense orbit, so there exists $\omega \in \Sigma^{+}_{n}$ for which $\mathcal{O}^{+}_{\mathcal{F}}(x,\omega)$ is dense in $M$, i.e, the set $\{x, f_{\omega_{0}}(x), \dots ,f^{j}_{\omega}(x), \dots\}$ is dense in $M$. Let us take $x_{l}=f^{l}_{\omega}(x)$ and $\mathcal{A}_{n}=\{x\in M: \mathcal{O}^{-}_{\mathcal{F}}(x)\ \ is\ \ \frac{1}{n}-dense\}$. Since $\mathcal{O}^{+}_{\mathcal{F}}(x,\omega)$ is dense, there exists $\{k_{n}\}$ such that $\{x, \ldots ,f^{k_{n}}_{\omega}(x)\}$ is $\frac{1}{2n}$-dense. As $\{x, \ldots , f^{k_{n}}_{\omega}(x)\}\subset \mathcal{O}^{-}(x_{j})$ for all $j\geq k_n$, we get that $x_{j}\in \mathcal{A}_{n}$ for each $j\geq k_{n}$.\
By continuity, one can find $r_{j}>0$ such that $f^{l}_{\omega}(B_{r_{j}}(x))\subset B_{\frac{1}{2n}}(f^{l}_{\omega}(x))$, for all $0\leq l\leq k_{j}$. This proves that $\mathcal{O}^{-}_{\mathcal{F}}(y)$ is $\frac{1}{n}$-dense for all $y\in f^{j}_{\omega}(B_{r_{j}}(x))$ and $j\geq k_{n}$. As $f^{j}_{\omega}(B_{r_{j}}(x))$ is an open neighborhood of $x_{j}$ and $\{x_{j}: j\geq k_{n}\}$ is dense, then $\cup _{j\geq k_{n}}f^{j}_{\omega}(B_{r_{j}}(x))\subset \mathcal{A}_{n}$ is an open and dense subset $M$. Therefore $\cap_{n\geq1}\cup_{j\geq k_{n}}f^{j}_{\omega}(B_{r_{j}}(x))\subset \cap_{n\geq1}\mathcal{A}_{n}$ is a residual subset of $M$. Since $\cap _{n\geq1}\mathcal{A}_{n}$ consists of points with dense backward orbit, then $\{x\in M : \overline{\mathcal{O}^{-}_{\mathcal{F}}(x)}=M\}\cap \{x\in M : \overline{\mathcal{O}^{+}_{\mathcal{F}}(x)}=M\}$ is also residual. This terminates the proof of the lemma.
Let us recall that the a finitely generated semi-group $\mathcal{G}^+$ satisfies the *weak cycle* property if for each open set $B\subset M$, there exists a sequence $\{h_{i}\}\subset \mathcal{G}^{+}$ such that $M\circeq \cup^{\infty}_{i=1}h_{i}(B)$, this means that $m(M\setminus \cup h_{i}(B))=0$.
A finitely generated semi-group $\mathcal{G}^+$ of homeomorphisms on $M$ has weak cycle property if and only if there exists a full measure subset $\tilde{M}\subset M$ such that for each point $x\in \tilde{M}$, the total backward orbit of $x$ is dense in $M$.
First, suppose that $\mathcal{G}^+$ has weak cycle property. Let us take a countable basis $\{V_{j}: j\geq 1\}$ of $M$. Weak cycle property implies that there exists a sequence $\{h_{ij}\}_{i,j\in \mathbb{N}}\subset \mathcal{G}^+$ so that $M\circeq \cup^{\infty}_{i=1}h_{ij}(V_{j})$, for each $j\in \mathbb{N}$.\
Suppose that $F_{j}=M\setminus \cup_{i}h_{ij}(V_{j})$, then $F_{j}$ has zero Lebesgue measure, hence $F=\cup_{j}F_{j}$ has also zero Lebesgue measure. We set $\tilde{M}=M\setminus F$ and we show that each $x\in \tilde{M}$ has a dense backward orbit.\
Indeed, if $x\in \tilde{M}$ then $x$ is not belong to $F$, which implies that $x$ is not belong to $F_{j}$, for each $j$. This means that $x\in h_{ij}(V_{j})$ for each $i$. Therefore $h^{-1}_{ij}(x)\in V_{j}$ and hence the backward orbit of $x$ is dense in $M$.
Conversely, suppose that there exists a full measure subset $\tilde{M}\subset M$ so that each $x\in \tilde{M}$ has dense backward orbit. This means that for each open set $B$ there exists $h_{x}\in \mathcal{G}^{+}$ with $h^{-1}_{x}(x)\in B$. Therefore $x\in h_{x}(B)$. Then, the family $\{h_{x}(B): x\in \tilde{M}\}$ is an open covering of $\tilde{M}$ and hence there exists a countable subcover $\{h_{x_{i}}(B): i\in \mathbb{N}\}$. If we set $h_{i}:= h_{x_{i}}$, then we conclude that $M\circeq \cup_{i}h_{i}(B)$.
**[Examples]{}**
================
In this section, we exhibit some examples of finitely generated semigroups of $Diff^{1+\alpha}(M)$ satisfying our hypothesis in subsection 2.1. To start the construction, we recall some concepts of topology.
The *standard* $m$-*simplex* is the set
$\triangle_m=\{x \in\mathbb{R}^{m+1}:x_i\geq0 \ and \ \sum_{i=0}^{m+1}x_i=1\}.$
A *general* $m$-*simplex* is a subset of $M$ diffeomorphic to the standard $m$-simplex $\triangle_m$ and a *general* $n$-*face* is a subset of $M$ diffeomorphic to the standard $n$-face.
A *triangulation* of a compact $m$- manifold $M$ is a finite collection $\mathcal{T}$ of diffeomorphic images of $\triangle_m$ that cover $M$ and satisfying the following condition: for any general $m$-simplices $S_i,S_j\in \mathcal{T}$, if $S_i\cap S_j\neq\emptyset$, their intersection is a $(m-1)$-face of both $S_i$ and $S_j$.\
The *barycentric subdivision* of an $m$-dimensional simplex $S$ consists of $(m+1)!$ simplices. Each piece, with vertices $v_0,\ldots, v_m$ can be associated with a permutation $p_0,\ldots, p_m$ of the vertices of $S$, in such a way that each vertex $v_i$ is the barycenter of the points $p_0,\ldots, p_i$. Barycentric subdivision is an important tool which is used to obtain finer simplicial complexes. Now, we start by considering any smooth Riemannian manifold $M$ that supports orbital expanding or orbital non-uniformly expanding finitely generated semigroups of $Diff^{1+\alpha}(M)$. We are going to introduce a finite family of $ Diff^{1+\alpha}(M)$ that admits a Markov partition satisfying conditions $(A_0), (A_1), (A_2)$ and $(A_3)$.
It is a well-known fact that $M$ possesses a triangulation $\mathcal{T}=\{S_1, \ldots, S_k\}$ , where $S_i, \ i=1, \ldots, k$, are diffeomorphic images of $\triangle_m$ and cover $M$. Suppose that $g_i, \ i=1, \ldots, k$, are diffemorphisms which map $\triangle_m$ to $S_i$.\
We divide $\triangle_m$ by barycentric subdivision to smaller $m$- dimensional $m$-simplices $\triangle_{m,1}, \ldots, \triangle_{m,l}$. Take affine maps $h_j, j=1, \ldots, l$, which map $\triangle_{m,j}$ to $\triangle_m$.\
Clearly, each $h_j$ is an expanding map. Now, we set $f_{i,j}:= g_i \circ h_j\circ g_i^{-1}$ and extend it to a $C^{1+\alpha}$-diffeomorphism on $M$. Also, take $S_{i.j}:= g_i(\triangle_{m,j})$, then the interiors of $S_{i,j}$, $i=1, \ldots, k$ , $j=1, \ldots, l$ exhibit a Markove partition, that is if we take $R_{i,j}:= int(S_{i,j})$, then $(R_{i,j},f_{i,j})$ satisfying conditions $(A0), \ldots, (A3)$, with $p=kl$ and $q=0$. Clearly, the semigroup generated by $\{f_{i.j}\}$ is an orbital expanding finitely generated $C^{1+\alpha}$ semi-group.
Now, we will give some changes in the subdivision corresponding to one of generalized $m$-simplices that allows us to exhibit a finitely generated semi-group of $C^{1+\alpha}$-diffeomorphisms on any smooth manifold that is orbital non-uniformly expanding.
Again, we start with a triangulation $\mathcal{T}$ on a smooth compact Riemannian manifold $M$ and choose a generalized $m$-simplex $S^*\in \mathcal{T}$. Let $\{\triangle_{m,1}, \ldots, \triangle_{m,l}\}$ be a subdivision of standard $m$-simplex $\triangle_m$ in the following sense: there exists a sub-simplex $\triangle_{m,j}$ which admits the entire of an $(m-1)$-face of $\triangle_m$ as a face but other faces are strictly smaller than the faces of $\triangle_m$. Moreover, other sub-simplicies are smaller than $\triangle_m$ in all direction.
We take $h_*$ an affine transformation which maps $\triangle_{m,j}$ to $\triangle_m$. Also, suppose that $h_i$, $i=1, \ldots, l$, $i\neq j$, are affine transformations which map $\triangle_{m,i}$ to $\triangle_m$.\
Let $g$ be a diffeomorphism which takes $\triangle_m$ diffeomorphically to $S^*$.\
We set $S_j^*:= g(\triangle_{m,j})$, $R_j^*:= int(S_j^*)$ and $f_j^*:= g \circ h^*\circ g^{-1} \mid_{S^*_j}$. Also, we take $S_i^*:= g(\triangle_{m,i})$, $R_i^*:= int(S_i^*)$ and $f_i^*:= g \circ h_i\circ g^{-1} \mid_{S_i^*}$, for $i=1, \ldots, l$, with $i\neq j$.\
Clearly, this subdivision is not barycentric. Other elements of $\mathcal{T}$ admit barycentric subdivisions with the affine transformations which map them to the whole of standard $m$-simplex, analogous to the previous example. It is easy to see that this partition is also Markov and satisfies the conditions $(A0)$, $(A1)$, $(A2)$ and $(A3)$ with $q=1$.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to M. Nassiri, A. Fakhari and M.Saleh for many useful conversations and valuable suggestions.
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M. Viana, *Multidimensional non-hyperbolic attractors*, Publ. Math. IHES 85 (1997), 63–96. L.S. Young, *Stochastic stability of hyperbolic attractors*, Erg. Th. and Dyn. Sys 6 (1986), 311-319.
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abstract: 'New and archival HST images of the globular cluster systems of 17 relatively nearby galaxies and 6 more distant galaxies have been used to establish the characteristics of GC subpopulations over a range of host galaxy morphological type from E5 to Sa. GC color/metallicity, size and luminosity distributions have been examined in the nearby galaxies and color distributions have been determined for the more distant sample. Correlations with parent galaxy properties and trends with galactocentric radius have been explored. Supplemented with Keck spectroscopy, our results are best explained by an [*in situ*]{} formation scenario in which [*both*]{} GC subpopulations formed at early times within the potential well of the protogalaxy, in multiple episodes of star formation. We have also discovered a third population of clusters, fundamentally distinct from the compact red and blue clusters common in early type galaxies.'
author:
- 'Jean P. Brodie'
title: |
Globular Cluster Subpopulations in Early-Type Galaxies:\
Insights into Galaxy Formation
---
epsf \#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} =
\#1 1.25in .125in .25in
Introduction
============
Since their discovery in early-type galaxies nearly a decade ago, globular cluster (GC) subpopulations have been widely recognized as holding important clues about both GC and host galaxy formation. The presence of subpopulations is apparent from the bimodal color distributions in many early-type galaxies, implying multiple mechanisms or epochs of GC formation in a galaxy’s history.
Explanations for bimodality fall into three categories: Mergers, [*in situ*]{}/multi-phase collapse and accretion scenarios. The merger scenario, in which elliptical galaxies form from gaseous mergers of spiral galaxies (Schweizer 1987; Ashman & Zepf 1992), was the only one to actually predict bimodality rather than explaining it after the fact. The GC system of the resulting galaxy consists of a blue (metal poor) population from the progenitor spiral galaxies and a new population of red (metal rich) clusters formed in the merger event itself out of enriched gas. The [*in situ*]{}/multi-phase scenario was put forward by Forbes, Brodie & Grillmair (1997) in light of an increasing body of observational evidence not easily explained under the merger model. In this scenario the blue GCs form first in a clumpy protogalactic medium. Star and cluster formation is halted after most of the blue clusters and a few stars have formed, perhaps because of the ejection of gas from these clumps due to supernova explosions. There follows a dormant period of a few Gyr while the gas expands, cools and falls back into the now more fully formed galaxy potential. At this point star and cluster formation starts again, this time from enriched gas, forming the red GCs and the bulk of the galaxy stars. Harris et al. (1999) came to similar conclusions based on their HST study of GCs and halo stars in NGC 5128. In accretion scenarios (e.g. C[ô]{}t[é]{} et al 1998) the red GCs form [*in situ*]{} in large elliptical galaxies and the blue GCs are accreted along with lower luminosity galaxies or they can be stripped from neighboring galaxies. Some new cluster formation may also be required (Hilker et al. 1999).
Clearly, mergers take place and massive clusters are formed in the process. This is frequently observed (e.g. Schweizer, 1997). Equally clearly, more massive galaxies accrete less massive ones, along with their retinues of more metal poor GCs. These phenomena must influence the final characteristics of a GC system, but to what extent? The questions we need to address are: Is there a dominant mechanism determining the global properties of GC systems in ellipticals? Is that mechanism the same for all galaxies. How do spirals fit into this picture? If GC systems are the result of a mix of processes how does that mix vary with host galaxy properties? A point to remember, though, is that at high enough redshift the distinction between the scenarios is quite blurred.
We have explored the red and blue GC subpopulations of 23 galaxies observed with the WFPC2 camera on HST. Much of the data is from our own HST programs but we have supplemented with data from the HST archive where these are of sufficient depth. The result is a large, [*homogeneous*]{} set of GC system data observed with the same telescope (HST) and instrument (WPFC2) through the same filters (except for two galaxies) and all subjected to the same reduction and analysis techniques. This homogeneity is the key to uncovering subtle correlations which can be masked by systematics when comparing data sets from the literature which have been observed and analyzed in a variety of different ways.
Our approach has been to determine the key properties of the GCs (ages, metallicities, sizes, luminosity function turnover magnitudes), relate them to characteristics of the parent galaxy (luminosity, color, environment) and try to identify trends with galactocentric radius. In addition to the HST data we have used Keck spectroscopy to provide more accurate age and abundance estimates for selected clusters. This information has allowed us to put some useful constraints on the formation history of GC systems and their host galaxies.
Nearby Galaxies
===============
Our nearby sample comprises 17 galaxies out to the distance of the Virgo cluster. It is discussed in detail in Larsen et al. (2001) so I will provide only a summary and a few updates here. In Larsen, Forbes & Brodie (2001) we present additional information on the Sombrero (Sa) galaxy. Included in our nearby sample are 1 Sa, 4 S0, 11 E galaxies and 1 cD galaxy. They cover a range in M$_B$ from $-$18.6 to $-$22.1 mag. Some are found in groups, some in rich clusters and a few are in relatively isolated environments.
Color Magnitude diagrams
------------------------
The examples in Figure 1 illustrate the point that some color-magnitude diagrams are obviously bimodal to the eye, while others are not. Nonetheless, a KMM test (Ashman, Bird & Zepf 1994) indicates, with high probability, that two Gaussians are a better fit to the data than a single one in nearly all cases. Interestingly, the peak $(V-I)_o$ colors are consistent even in galaxies where the CMDs and $(V-I)_o$ histograms show weak/no evidence for bimodality. One galaxy, NGC 4365, appears to have only one (broadened) peak, at $(V-I)_o = 0.98$, typical for the metal poor populations in bimodal systems. This galaxy, a bright elliptical, would be expected to possess a significant metal rich population under the accretion scenario.
The average blue and red peak $(V-I)_o$ colors are 0.95$\pm$0.02 and 1.18$\pm$0.04, corresponding to $[Fe/H]=-1.4$ and $-0.6$ respectively (Kissler-Patig et al. 1998). This is remarkably similar to the metallicities of the Milky Way metal poor (halo) and metal rich (disk/bulge) GCs (Zinn, 1985).
Globular Cluster Luminosity Functions
-------------------------------------
The GC luminosity function (GCLF) is usually assumed to have a Gaussian form (although a students t$_5$ function is actually a better fit) whose peak appears to occur at a constant absolute magnitude, modulo metallicity effects (Ashman, Conti & Zepf 1995). The physical basis for the observed GCLF is unknown as is the degree to which it is universal. Its shape may have been set up at the time of formation or it may be the result of dynamical evolution. For example, dynamical effects could have eroded the faint end of a power-law initial luminosity function, resulting in the GCLF we observe today.
Differences between the turnover magnitudes of the blue and red subpopulations (${\Delta}m_V^{TO}$) would be indicative of differences in their formation and/or subsequent evolution. Assuming the same mass function for the blue and the red GCs, similar old ages and metallicities of \[Fe/H\] $= -$1.6 and $-0$.6, 1996 versions of the Bruzual & Charlot SSP models predict a difference in m$_V^{TO}$ between the blues and the reds of 0.26 mag.
We have examined the GCLF turnover magnitudes for the galaxies in our nearby sample and find $<{\Delta}m_V^{TO}>=$0.47 mag., or 0.30$\pm$0.16 mag. if we exclude systems for which the error on ${\Delta}m_V^{TO}$ is $>$0.25. Note, though that this may introduce a bias because less populous systems belonging to lower-luminosity galaxies tend to be excluded. The fact that our observational result is close to the theoretical expectation suggests that both populations are indeed old [*if*]{} their mass functions are identical. However, if these populations formed at different epochs under different environmental conditions, differences in their mass functions would not be altogether surprising.
The Discovery of a Third Class of Cluster
-----------------------------------------
Observational limitations have meant that little has been known until now about the faint wing of the GCLF, yet this is where the signatures of formation and dynamical evolution should be the strongest. At $m-M\sim$30, the nearby S0 galaxy, NGC 1023 is close enough that the entire faint wing of the GCLF is accessible for high S/N imaging with HST and spectroscopy with 8–10m class telescopes.
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As Figure 2a shows, the faint wing of the GCLF in NGC 1023 deviates significantly from that of the Milky Way. This due to the presence of a third population of clusters in addition to the normal compact blue and red subpopulations commonly found in early type galaxies (Figure 2b). These hitherto undetected faint, red extended objects are aligned with the galaxy isophotes and thus appear to be associated with its disk. These “faint fuzzies” are discussed in detail in Larsen & Brodie (2000). They have no analogs in the Milky Way and it is impossible at present to assess how common they might be. They are obviously difficult to detect, being extended low surface brightness objects fainter than the GCLF turnover and, of the galaxies studied with HST in sufficient depth for accurate size measurements, they would have been detectable in only 4 cases. They do appear to be present in NGC 3384, another lenticular galaxy, but are definitely absent in the populous lenticular NGC 3115.
In speculating about the origin of these objects we note that there is a nearby dwarf companion galaxy containing two brighter blue objects for which we have Keck spectra. These objects are young clusters (ages $\sim$300 Myr) possibly formed as a result of the interaction with NGC 1023. Perhaps the faint fuzzies are older remnants of similar interactions with long-ago digested companion galaxies.
Sizes
-----
Sizes differences between blue and red GC populations had been noticed previously in four galaxies by ourselves (NGC 4472: Puzia et al. 1999; NGC 1023: Larsen & Brodie 2000) and others (NGC 3115: Kundu & Whitmore 1998; M87: Kundu 1999). With our new large sample we could assess how common this phenomenon might be. We found that the blue GCs are always larger than the reds by about 20%. Figure 3 gives examples. Although suitable multiple pointings exist for only a few galaxies, the size difference between the blue and red GCs persists at all radii in these systems. Interestingly, the same size difference is seen in the Milky Way, and at all radii. Note again, the similarities between the early type galaxies and the spirals (Milky Way and Sombrero).
Correlations with Parent Galaxy Properties
------------------------------------------
Brodie & Huchra (1991) showed that GC mean metallicity correlates with parent galaxy luminosity. With the subsequent discovery of multiple GC populations the question naturally arises as to whether the correlation exists for one or both of the subpopulations separately.
In Figure 4 we show that the peak colors of [*both*]{} the red and the blue GCs correlate with parent galaxy M$_B$ at the 2-3 ${\sigma}$ confidence level. A similar relation exists for $(V-I)_{0}$ vs. central velocity dispersion. The slope of the relations is steeper for the red GCs than for the blues. There is a 4 ${\sigma}$ correlation between the peak colors of the red GCs and parent galaxy color[^1] and a 2 ${\sigma}$ correlation for the blues.
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These results are somewhat different from previous findings (Forbes et al. 1997; Burgarella, Kissler-Patig & Buat 2001; Forbes & Forte 2001) in which the red GC colors were found to correlate strongly with parent galaxy properties while the blue ones were thought not to correlate significantly. This may be because of the heterogeneous nature of the data used in these studies which tends to mask more subtle correlations.
The implication of a correlation between the properties of the blue GCs and the parent galaxy is profound. At the time of formation, the blue GCs, or at least a significant fraction of them, must have known about the final galaxy to which they would belong. This may present problems for mergers and/or accretion as dominant mechanisms in the formation history of GC systems.
More Distant Galaxies
=====================
Work is underway on the GCLFs and color distributions of a sample of more distant galaxies. These systems are too far away for accurate size estimates for individual clusters.
Galaxy Location Dist. Mod $(V-I)_{0B}$ $(V-I)_{0R}$
---------- ---------- ----------- -------------- --------------
NGC 3311 Hydra 33.5 0.91 1.09
IC 4051 Coma 35 0.95 1.15
NGC 4881 Coma 35 0.95 -
NGC 5846 Group 32.3 0.96 1.17
NGC 7562 GH166 33 0.97 -
NGC 7619 Peg I 33.5 0.99 1.24
: \[tab:hstpt\] Color Distributions of Galaxies Beyond Virgo.
In Table 1 we summarize our results for the peak colors of the blue and red subpopulations. Notice that these galaxies have peak colors that are typical of those found in the nearby sample. NGC 3311 was previously thought to have an almost exclusively metal-rich GC population (Secker et al. 1995) from ground-based photometry. This would present challenges to both the [*in situ*]{} and merger scenarios. However, Brodie, Larsen & Kissler-Patig (2000) recently showed that this galaxy has an entirely normal color distribution based on our deep HST data. NGC 4881 appears to be the analog of the nearby galaxy, NGC 4365, in having a single (broadened) metal poor peak, as noted by Baum et al. 1995. Too few clusters are assigned to the red peak in NGC 7562 to allow a meaningful estimate of $(V-I)_{0R}$.
Formation Scenarios
===================
As we have seen, any successful formation scenario needs to explain a very wide range of properties of GC color distributions:
$\bullet$ Bimodal with roughly equal $N_{Blue}$ and $N_{Red}$ (NGC 1404, 4649, 4472).
$\bullet$ Bimodal with much reduced $N_{Red}$ (NGC 4406).
$\bullet$ Single (but broadened) metal poor peak (NGC 4365, 4881).
$\bullet$
In addition, a successful model will have to accommodate the increasing evidence that both subpopulations are old. That this is the case is borne out by our studies of the GCLFs of the blue and red subpopulations in that the turnover magnitudes differ only by the amount predicted by the SSP models under the assumption of similar old ages (see Puzia et al. 1999 for a detailed discussion). Evidence for old ages comes most convincingly from spectroscopy but few galaxies GC systems have so far been studied with large enough samples and adequate signal-to-noise to place interesting constraints on age differences. Only NGC 1399 (Kissler-Patig et al. 1998), NGC 4472 (Beasley et al. 2000), M87 (Cohen, Blakeslee & Ryzhov 1998), M81 (Schroder et al. 2001) and M31 (Huchra, Brodie & Kent 1991; Perrett et al. 2001) have good enough spectra for relative age constraints. It is likely to prove extremely diffiult to differentiate ages greater than $\sim$8–10 Gyr, even with superb signal-to-noise data because age contours are so closely spaced (they even cross in some models) in index-index planes generated from SSP models. Similar old ages would be at odds with the standard merger picture but would be compatible with mergers occurring at high redshift where the similarities between the models outweigh their differences.
Particularly interesting constraints to emerge from this work are the correlations between GC colors and host galaxy luminosity (mass) and color for [*both*]{} red and blue GCs. If the correlation for the blue GCs is verified it will be difficult to explain under both the merger and accretion scenarios. Note too that we consistently find roughly equal numbers of red and blue GCs, or a preponderance of blues. This is also a challenge for these scenarios. In particular, in the merger picture, the lower specific frequency of spirals must be increased to the higher (by a factor of $\sim$3 for giant E galaxies) value in ellipticals by the production (in the merger) of numerous red clusters. It is hard to escape the conclusion that the resulting galaxy would have a preponderance of red clusters. The accretion model is challenged only because such a large number of blue clusters must be acquired along with minimal amounts of starlight.
What’s Next?
============
We do not yet understand the origin of the size difference between red and blue GCs, whether it is primordial or the result of dynamical evolution. To produce it by dynamical effects would require constancy in the ratio of the perigalactic distances of the blue and red subpopulations over a wide range of parent galaxy luminosities and environments. Were red GCs formed on predominantly radial orbits and blue GCs formed on predominantly circular orbits? It might be more natural to suppose that red GCs formed from denser protocluster clumps, perhaps under conditions influenced by their higher metallicities.
An answer to the size question will naturally require kinematic information for large samples of GCs. This information will also inevitably assist in differentiating between models for the origins of GC subpopulations but it is not yet clear how to interpret such data. We are accumulating relevant information at an increasing pace from spectroscopic programs on 8–10m class telescopes. What is largely lacking is a theoretical basis for comparison between the scenarios (but see Bridges article in these proceedings for a comparison of current data with merger ideas).
Spectroscopic data also offer the best chance of age discrimination and we can expect additional insights into the timescales for GC formation from studies of individual element abundance ratios, in particular the ratios of $\alpha$ elements to Fe which reflect the relative contributions of SN Types I and II.
Much is still to be learned from photometry, particularly with HST. Outstanding questions relevant to the work described here include: How common are the “faint fuzzies”? Are they always associated with the disks of lenticulars? What else will we find when we study the faint wings of GCLFs? Can we see the signatures of dynamical evolution by observing galaxies with a range of ages?
We are beginning to learn that there are many similarities in the characteristics of the subpopulations in spirals and ellipticals. Since it is not generally thought that spirals themselves form from [*major*]{} mergers of disk systems, this might present another challenge to the standard merger picture. However, more spiral GC systems (especially those of later type galaxies) need to be studied before we will know how widespread these similarities are and to what extent they will influence our ideas on GC and galaxy formation.
Conclusions
===========
Overall then, our results seem to fit better with an [*in situ*]{} scenario in which both GC populations “knew” about the size of the final galaxy to which they would belong. This implies that the initial phase of GC formation in gE galaxies must have taken place after they assembled into individual entities, i.e., both GC populations formed within the potential well of the protogalaxy in multiple episodes of star formation. If the [*in situ*]{} idea is right, accurate age-dating of the red GCs will pinpoint the epoch at which the bulk of the galaxy was formed. Our best estimates suggest this occurred $>$ 8–10 Gyr ago but it is of critical importance to improve this constraint for comparison with models (such as hierarchical clustering) for the formation of structure in the early universe.
I am grateful to my many collaborators, especially S[ø]{}ren Larsen, and to Mike Fall for interesting discussions. This work was supported by National Science Foundation grant number AST9900732.
Ashman, K. M., Bird, C. M., and Zepf, S. E. 1994, , 108, 2348 Ashman, K. M., Conti, A., and Zepf, S. E. 1995, , 110, 1164 Ashman, K. M., and Zepf, S. E. 1992, , 384, 50 Beasley, M.A., Sharples, R.M., Bridges, T.J., et al. 2000, , 318, 1249 Brodie, J.P. & Huchra, J.P. 1991, , 379, 157 Brodie, J.P., Larsen, S.S. & Kissler-Patig, M. 2000, Letters, 543, L19 Burgarella, D., Kissler-Patig, M., Buat, V. 2001, , 121, 2647 Cohen, J., Blakeslee, J., Ryzhov, A., 1998, , 496, 808 C[ô]{}t[é]{}, P., Marzke, R. O., and West, M. J. 1998, , 501, 554 Forbes, D. A., Brodie, J. P., and Grillmair, C. J. 1997, , 113, 1652 Forbes, D. A., and Forte, J. C. 2000, , 322, 257 Harris, G. L. H., Harris, W. E. and Poole, G. B. 1999, , 117, 855 Hilker, M., Infante, L., and Richtler, T. 1999, , 138, 55 Huchra, J.P., Brodie, J.P. & Kent, S. 1991, , 370, 495 Kissler-Patig, M., Brodie, J. P., Schroder, L. et al., 1998, , 115, 105 Kundu, A. 1999, PhD thesis, Univ. of Maryland Kundu, A., and Whitmore, B. C. 1998, , 116, 2841 Larsen, S. S., and Brodie, J. P., 2000, , 120, 2938 Larsen, S. S., Brodie, J.P., Huchra, J.P. et al., 2001 , 121, 2974 Larsen, S. S., Forbes, D. A., & Brodie, J. P., 2001, , submitted Perrett, K., et al. 2001 in preparation Puzia, T. H., Kissler-Patig, M., Brodie, J. P., and Huchra, J. P. 1999, , 118, 2734 Schroder, L., Brodie, J.P., Huchra, J.P., et al. 2001, , in press Secker, J., Geisler, D., McLaughlin, D.E. & Harris, W.E. 1995, , 109, 1033 Schweizer, F. 1987, in Nearly Normal Galaxies, ed. S. M. Faber (New York: Springer), 18 Schweizer, F. 1997, in The Nature of Elliptical Galaxies, ASP Conf. Ser., Vol. 116, eds. M. Arnaboldi, G.S. Da Costa, P. Saha, 447 Tonry, J. L., Dressler, A., Blakeslee, J. P., et al. 2001, , 546, 681 Zinn, R., 1985, , 293, 424
[^1]: The lack of correlation reported previously was due to the use of inaccurate galaxy colors. New galaxy colors from Tonry et al. (2001) are used here and reveal the expected correlation.
|
---
author:
- 'B. Georgeot'
- 'O. Giraud'
date:
-
- 'May 12, 2011'
title: The game of go as a complex network
---
Introduction
============
The study of complex networks has attracted increasing interest in the past decade, fueled in particular by the great recent development of communication and information networks. Tools and models have been created, enabling to describe the growth mechanisms and properties of such systems. In parallel, it has been realized that many important aspects of the physical world or of social interactions can be modelized by such networks. Such tools have been applied to many fields of human activity, such as e.g. languages or friendships [@barabasi].
One of the oldest activities of human beings, board games have been played for millenia. Besides their intrinsic interest, they represent a privileged approach to the working of decision-making in the human brain. Some of them are very difficult to modelize or simulate: only recently were computer programs able to beat world chess champions. The old Asian game of go is even less tractable. The game complexity, that is, the total number of legal positions, is about $10^{171}$, compared to a mere $10^{50}$ for chess [@TroFar07]. It remains an open challenge for computer scientists: while Deep Blue famously beat the world chess champion Kasparov in 1997, no computer program has beaten a very good player even in recent times.
Many studies have been devoted to “computer go”, the simulation of the go game on a computer. They were historically based on deterministic tree search algorithms such as minimax or alpha-beta, which estimate an evaluation function (giving the game-theoretic value of a move) on the tree of allowed moves (see e. g. [@computers; @BouCaz01]). Recently, much progress has been done by introducing Monte-Carlo techniques [@MonteCarloGo; @progressive], which basically estimate the value of a move by playing subsequent moves at random according to the rules of go. Monte-Carlo tree search algorithms are based on a non-uniform probability distribution over legal moves, and explore only the most promising ones. Variations on these techniques allowed computer programs to make significant progress in the last few years, so that professional human players with a large enough handicap could be beaten by a computer [@LeeWang09]. The choice of the evaluation function and the way in which the tree is explored are crucial ingredients for any further progress.
Since go is a popular game with millions of players in the world, many games have been recorded, which enables statistical data to be extracted reliably. A few works have used statistical properties of recorded professional games to optimize performances of Monte-Carlo algorithms. Usually the simplest features of real games are retained, such as local patterns or contiguity to the previous move [@Coulom]; including more real-game features improves noticeably the winning rate of computer programs [@Coulom2].
In this paper, we thus study the game of go from a complex network perspective. We use databases of expert games in order to construct networks from the different sequences of moves, and study the properties of these networks. We based our numerical results on the whole available record, from 1941 onwards, of the most important historical professional Japanese go tournaments: Kisei (143 games), Meijin (259 games), Honinbo (305 games), Judan (158 games) [@database]. To increase statistics and compare with professional tournaments, 4000 amateur games also available from [@database] were used.
Definition of inequivalent moves
================================
The game of go is played by two players (Black and White) on a board (goban) consisting of 19 horizontal and 19 vertical lines. The players alternately place a stone of their own color at an empty intersection on the board. Stones entirely surrounded by the opponent must be removed, and the aim of the game is to delimit large territories. As the game unfolds, local and global properties of stones are involved. A network approach will obviously not be able to capture all features of the game, as the number of possible moves is far too large. Here we follow an approach where only local features are retained. This approach is reminiscent of the one used in the context of language networks [@language].
A move consists in placing a stone at an empty intersection $(h,v)$ with $1\leq h,v\leq 19$. We call ”plaquette” a square of $3\times 3$ intersections, that is, a subset of the board of the form $\{(h+r,v+s),-1\leq r,s\leq 1\}$ (to account for edges and corners of the board one can imagine that there are two additional dummy lines at each side of the board). To define our network we only take into account intersections closest to $(h,v)$. Vertices correspond to the different kinds of plaquettes in which a player can put a stone, irrespective of where it has been played on the board. Since each of the 8 neighboring intersections can be either empty, black or white, there are $\sim 3^8$ different plaquettes. We choose to consider identical plaquettes that transform to each other under any symmetry of the square (rotation or flip). We also identify patterns with color swapped. That is, a move where Black plays in a given plaquette is considered the same as a move where White plays in the same plaquette with colors swapped. An exact computation taking into account borders and symmetries leaves us with 1107 nonequivalent plaquettes with empty centers, which are the vertices of our network. We note that certain computer programs based on knowledge from real professional games also consider similar $3\times 3$ stone patterns [@Coulom; @Coulom2]. Considering larger plaquettes is possible and would convey more relevant information; however, the number of vertices then becomes enormously large ($\approx 3.10^{10}$ for $5 \times 5$ plaquettes).
![(Color online) Normalized integrated frequency distribution of moves $F(n)$ for Honinbo (black), Meijin (red), Judan (green), Kisei (blue) and amateur (violet) tournaments. The normalized number of occurrences of the 500 most frequent moves (among the 1107 moves described in the text) is shown vs the ranks of the moves (rankings slightly depend on the database). Slopes are resp. -1.058, -1.056, -1.065, -1.067, -1.081. Thick dashed line is $y=-x$. Inset: same with moves defined as position of the stone on the board. Log. is decimal. \[zipf\_all\]](fig1.eps){width="7.5cm"}
This definition of inequivalent moves enables us to investigate the first properties of the databases in term of frequencies of moves. Zipf’s law is an empirical characteristics which has been observed in many natural distributions, such as e. g. word frequency in the English language [@zipf], city sizes [@zipf1], income distribution of companies [@zipf3], and chess openings [@chess]. If items are ranked according to their frequency, it predicts a power-law decay of the frequency as a function of the rank. Zipf’s law was observed in the frequency distribution of $5\times 5$ go patterns [@LiuDouLu08]. In Fig. \[zipf\_all\] we display the integrated frequency distribution for our 1107 moves labeled from the most to the least frequent. The integrated distribution of moves is very similar for all databases and clearly follows a Zipf’s law, with an exponent $\approx 1.06$. In contrast, such a law cannot be seen if one simply takes the $361$ possible positions $(h,v)$ as vertices, disregarding local features (inset of Fig. \[zipf\_all\]). Thus Zipf’s law appears when tactical information is taken into account. For all databases the 10 most frequent moves are the same (see Fig. \[frequent\_moves\], upper line), but sometimes in a slightly different order.
The go network
==============
![[(Color online) Integrated frequency distribution of sequences of moves $f(n)$ for (from top to bottom) two to seven successive moves (all databases together), plotted against the ranks of the moves. Moves are the 1107 moves described in the text. Slopes are resp. -1.01, -0.91, -0.86, -0.83, -0.81, -0.77. \[frequencies1\] ]{}](fig2.eps){width="7.2cm"}
![[(Color online) Integrated frequency distribution of sequences of moves $f(n)$ for sequences of two (continuous lines) and three (dashed lines) successive moves for (from bottom to top) case C1 (black, slopes -0.23, -0.4), C2 (red, slopes -0.25, -0.39), same curves as in the main panel (blue), C3 (green, -0.91, -0.70). Inset: distribution of distances between moves $P(d)$; same color code as in Fig. 1. The four professional tournaments are almost undistiguinshable. \[frequencies2\] ]{}](fig3.eps){width="7.2cm"}
The dynamics of the game is built from successive moves. In the game of go, a game often consists in a series of small fights played at different places on the board. A player might put a stone in the vicinity of their opponent’s stones to engage the battle, but the opponent might prefer to first continue a fight occurring somewhere else, in which case two consecutive moves would not be directly related. In order to construct our network, it is thus natural to connect two moves by a directed link only if these moves follow each other in the same region of the board. To be more specific, we connect vertices corresponding to moves $a$ and $b$ played at $(h_a,v_a)$ and $(h_b,v_b)$ respectively if $b$ follows $a$ in a game and $\max\{|h_b-h_a|,|v_b-v_a|\}\leq d$. Each choice of the integer $d$ defines a different network. The choice of $d$ determines the distance beyond which two moves are considered nonrelated. We present in Fig. \[frequencies1\] the frequency distribution for sequences of moves defined in such a way with $d=4$. We observe an algebraic decrease with exponent ranging from $\approx 1$ for short sequences to $\approx 0.8$ for longer sequences. Thus sequences of moves follow Zipf’s law, as was observed for word sequences in languages [@language]. We attribute the decrease of the exponent to the fact that longer sequences reflect more and more individual strategies. As a comparison, Fig. \[frequencies2\] displays the frequency distribution of successive moves for three other definitions of moves and sequences of moves. In cases C1 and C2, moves are defined as positions $(h,v)$ on the board, disregarding local features, and $b$ is considered to follow $a$ if $b$ is played immediately after $a$ (case C1) or if $b$ is the first move played after $a$ and in the vicinity of $a$, with $d=4$ (case C2); in case C3 sequences of vectors between two moves played in the same region (with $d=4$) are considered. These results indicate that move sequences, even long ones, are best hierarchized by our initial definition. In what follows we will thus disregard other choices C1-C3. In the inset of Fig. \[frequencies2\], we also show the distribution $P(d)$ of distances between consecutive moves. Interestingly enough, the amateur database departs significantly from all the professional ones, with a tendency to play more often at shorter distances. This may reflect the fact that professionals are more prone to play several tactical fights in parallel, or play on average shorter local tactical fights.
We now investigate the properties of our networks. We construct a network for each database by playing the games according to the rules of go and adding directed links between the 1107 vertices as indicated above. To each link is assigned a weight given by the number of occurrences in the database.
![(Color online) Normalized integrated distribution of ingoing links $P_{\textrm{in}}$ (lower curves, solid) and outgoing links $P_{\textrm{out}}$ (upper curves, dashed), for networks built with $d=4$. The number of vertices with more than $k$ ingoing (outgoing) links is shown vs the normalized number of ingoing (outgoing) links $k/k_{\textrm{max}}$. Same databases and color code as in Fig. \[zipf\_all\]. Thick solid line is $y=-x$. Inset: $P_{\textrm{in}}$ (solid curves) and $P_{\textrm{out}}$ (dashed curves) for the Honinbo database, $d=2$ (black), 3 (red), 4 (green), 5 (blue) and 6 (violet). \[pinpout\]](fig4.eps){width="7.6cm"}
The distribution of ingoing and outgoing links $P_{\textrm{in}}$ and $P_{\textrm{out}}$ is displayed in Fig. \[pinpout\]. The tails of both distributions are very close to a power-law $1/k^{\gamma}$ with exponent $\gamma=1.0$ for the integrated distribution. The results are stable in the sense that the exponent does not depend on the database considered. The presence of such power laws indicates that the network displays the scale-free property: the distribution of links around a given link frequency is independent of that frequency. Such a property has been seen in many social or biological networks, but is absent in e.g. the famous Erdös-Renyi model of random networks. The symmetry between ingoing and outgoing links is a peculiarity of this network; it is well known for the World Wide Web (WWW) for example that the exponent for $P_{\textrm{out}}$ ($\gamma\approx 1.7$) is much larger than for $P_{\textrm{in}}$ ($\gamma\approx 1.1$) [@donato]. In the case of the WWW, the number of outgoing links is limited by the behavior of each independent webmaster. In our case, the results indicate a symmetry, at least at a statistical level, between moves that often follow others and moves which have many possible following moves. This may correspond to the fact that many short tactical sequences can be played in a different order within several different contexts. In order to analyze the dependence of $P_{\textrm{in}}$ and $P_{\textrm{out}}$ on the choice of $d$, we plot these quantities for a network constructed for various values of $d$ in the inset of Fig. \[pinpout\]. The distribution of ingoing and outgoing links stabilizes at $d=4$. Other databases give similar results (data not shown). We now focus on $d=4$.
Some properties can be extracted from the unweighted adjacency matrix (i. e. without weighing the links according to their frequency). The clustering coefficient (CC) describes the tendency of many real-world networks to form local clusters of highly connected vertices. The CC of a given vertex $i$ is defined as the probability that two neighbors of $i$ be connected to each other, irrespective of the direction of the link. The average CC for our networks is displayed in Fig. \[log\_sorted\_pageranks\] (inset). The CC depends on the number of games $n_g$ included to construct the network, but almost not on the database. For large $n_g$, the CC goes to an asymptotic value which appears to be larger than $0.7$, indicating high clustering, larger than the WWW (where the CC is $\approx 0.11$ [@barabasi]) and comparable to social [@barabasi] or language networks [@language].
Ranking vectors for the go network
==================================
![(Color online) Ranking vectors for matrices $G$ with $\alpha=1$. Same color code as in Fig. \[zipf\_all\], $d=4$. From top to bottom, top bundle: PageRank. Second bundle: CheiRank. Third bundle: Hubs. Fourth (bottom) bundle: Authorities. Straight dashed line is $y=-x$. Inset: Clustering coefficient (CC) as a function of the number of games $n_g$ included to construct the network; blue squares: professional tournaments, all games included; circles: amateur games. \[log\_sorted\_pageranks\]](fig5.eps){width="7.5cm"}
In order to get an insight into how our network captures aspects of the strategy of the game, we now use the weighted adjacency matrix (links are weighted according to their frequency in the database) and apply tools developed to rank vertices in order of importance, used e.g. to determine the order of appearance of answers to queries by search engines. These methods generally build a ranking vector, whose value on each vertex enables to determine its importance. The most famous such vector is the PageRank vector [@brin; @googlebook], which was the basis of the Google search engine. It is built from the Google matrix $G$, defined as $G_{ij}=\alpha S_{ij}+(1-\alpha)\, ^{t}ee/N$, where $e=(1,...,1)$, $N=1107$, $0<\alpha\leq 1$, $S$ is the weighted adjacency matrix with any column of 0 replaced by a column of $1$, and the sum of each column normalized to 1. The PageRank vector is the right eigenvector of $G$ associated with the largest eigenvalue $\lambda=1$, and singles out vertices with many incoming links from important nodes. From its definition, its components are real and nonnegative, and therefore can be used to rank nodes according to the value of these components. Other ranking vectors built from $G$ include the CheiRank vector [@dima] (which is the PageRank of the network with all links inverted), and the Hubs and Authorities of the HITS algorithm [@hits]. They all share the properties of being real nonnegative vectors, and thus can be used to rank the nodes of the network. While PageRanks and Hubs reflect properties of vertices depending on their incoming links, CheiRanks and Authorities are based on outgoing links. In Fig. \[log\_sorted\_pageranks\] we show these ranking vectors for our networks. They follow an algebraic law with slope $\approx -1$ (PageRank and CheiRank) and $\approx -1.5$ (Hubs and Authorities). A similar distribution of the PageRank was observed in e.g. the WWW [@donato; @google], but in contrast with the WWW and other systems there is a marked symmetry between distributions of ranking vectors based on ingoing links and those of vectors based on outgoing links.
![(Color online) K\* vs K where K (resp. K\*) is the rank of a vertex when ordered according to PageRank vector (resp CheiRank) for Honinbo (black squares), Meijin (red circles), Judan (green diamonds), Kisei (blue crosses) and amateur (violet stars) databases. \[ranks\]](fig6.eps){width="7.cm"}
In order to further shed light on this symmetry, we plot in Fig. \[ranks\] the correlation between the PageRank and the CheiRank for the five databases considered. In all these cases, there is a remarkably strong correlation between these rankings based respectively upon ingoing and outgoing links. In the WWW, there is a difference of nature between ingoing and outgoing links: webmasters are free to create as many outgoing links as they wish from their webpage, whereas the ingoing links depend on the cumulative effect of all other webmaster behaviors. In contrast, for the go network, the fact that a link is ingoing or outgoing depends on the chronological order in which the moves are played. The results displayed in Fig. \[ranks\] thus seem to indicate that there is a strong correlation between moves which open many possibilities of new moves and moves that can follow many other moves. However, the symmetry is far from exact, as can be seen in Fig. \[ranks\].
Although there is always some correlation between the different ranking vectors, they usually can be quite different, for example in Wikipedia articles [@dima]. A recent analysis of the world trade network [@trade] showed such a symmetry when all commodities were aggregated, but the symmetry was much less visible when each different commodity was treated separately. It is possible that in our case the symmetry is made more prominent by our definition of moves through $3\times 3$ plaquettes. A more refined approach with larger plaquettes may thus disambiguate the moves and give different results. We nevertheless think that our result indicate a specific feature of the game, such as the existence of many short sequences of tactical moves which can be played at different moments of the game.
Eigenvectors of the Google matrix
=================================
![(Color online) Top left: eigenvalues in the complex plane for matrices $G$, $d=4$, $\alpha=1$; black circles: Honinbo; red crosses: amateur. Bottom: $\lambda_c$ such that from top to bottom $99\%$, $95\%$, $90\%$, $80\%$ of eigenvalues $\lambda$ verify $|\lambda| <\lambda_c$ for amateur games. Top right: $\lambda_c$ for $80\%$ of eigenvalues for our 5 databases, same color code as in Fig. \[zipf\_all\].\[nuagesvp\]](fig7.eps){width="8.cm"}
As can be seen in Fig. \[log\_sorted\_pageranks\], the ranking vectors are distributed according to power laws and thus are mainly localized on few vertices, mainly the most frequent ones according to Zipf’s law (see e. g. Fig. \[frequent\_moves\], upper line). However, these ranking vectors correspond to the eigenvector associated with the largest eigenvalue of different matrices built from the network. We now consider the other eigenvectors of $G$. In particular, the eigenvectors associated with next to leading eigenvalues can describe specific communities inside the network [@google]. The distribution of eigenvalues is also important, as it reflects the structure of the network [@google]. Fig. \[nuagesvp\] shows the complex eigenvalues of the matrix $G$. For the WWW a sizable fraction of eigenvalues are close to zero, while the remaining ones are spread inside the unit circle, with no gap between the first eigenvalue and the bulk [@google]. By contrast, in the case of the go network, there is a huge gap between the first eigenvalue $\lambda=1$ and the next ones. This is reminiscent of what can be seen in some lexical networks [@google]. Such features indicate that the network is well-connected, with few isolated communities, and is consistent with the finding of a high clustering coefficient (see inset of Fig. \[log\_sorted\_pageranks\]). Whereas the WWW contains many communities of webpages which are almost cut off from the rest of the network, this is not the case for the go network, where communities – i.e. sequences of tactical moves preferentially played together – have more connections to the rest of the network, indicating that tactical moves can belong to different strategic groupings. To put these data in perspective, we have constructed a random version of the network, by randomly shuffling the moves inside each game of the databases. This process conserves Zipf’s law and the global characteristics, but eigenvalues of $G$ are all concentrated in the bulk (data not shown), in contrast with the real go network, indicating that communities are destroyed by the randomization process.
![(Color online) Moduli squared of the right eigenvectors associated with the 7 largest eigenvalues $|\lambda_1|=1>|\lambda_2|...>|\lambda_7|$ of $G$ (Honinbo database) for the first 100 moves in decreasing frequency ($|\lambda_1|$ (PageRank): thick black line, $|\lambda_2|$: orange pluses; $|\lambda_3|=|\lambda_4|$: red circles; $|\lambda_5|$: green squares; $|\lambda_6|$: blue triangles; $|\lambda_7|$: violet stars). Inset: Same for amateur database (full black line) and random network (dashed red line, see text). \[eigenv\]](fig8.eps){width="8.cm"}
![Moves corresponding to the 10 largest entries of right eigenvectors of $G$ for eigenvalues $\lambda_1$ (PageRank)(top), $\lambda_3$ (middle) and $\lambda_7$ (bottom), Honinbo database. Black is playing at the cross. Top line coincides with the 10 most frequent moves. \[frequent\_moves\]](fig9.eps){width="7.5cm"}
In Fig. \[nuagesvp\] (bottom), it is shown that the radius of the bulk of eigenvalues changes with the number of games $n_g$ entered in the network. This indicates that as more games are taken into account, rare links appear which break more and more the weakly coupled communities. The next to leading eigenvalues are important, as they indicate the presence of communities of moves which have common features. The distribution of the first 7 eigenvectors (Fig. \[eigenv\]) shows that they are concentrated on particular sets of moves different for each vector. The corresponding moves are displayed in Fig. \[frequent\_moves\] for the Honinbo database. The first eigenvector is mainly localized on the most frequent moves. By contrast, the third one is localized on moves describing captures of the opponent’s stones, and part of them single out the well-known situation of [*ko*]{} (“eternity”), where players repeat captures alternately. The 7th eigenvector singles out moves which appear to protect an isolated stone by connecting it with a chain. These eigenvectors are different for different tournaments and from professional to amateur. Indeed, the inset of Fig. \[frequent\_moves\] shows the distribution of the first seven eigenvectors for amateur database, very different from the one for Honinbo. It also shows for comparison the distribution for the randomized network (see above), which is much less peaked. Systematic studies of these eigenvectors, as well as the frequency of sequences of moves, should enable to group together certain moves, and should help to elaborate efficient go simulators.
Conclusion
==========
In this paper, we have studied the game of go, one of the most ancient and complex board games, from a complex network perspective. We have defined a proper categorization of moves taking into account the local environment, and shown that in this case Zipf’s law emerges from data taken from different tournaments. The network of go moves has some peculiarities, such as a statistical symmetry between ingoing and outgoing links distributions, which reflects itself in a symmetry between rankings based on ingoing on outgoing links, a feature not seen in many other complex directed networks such as the WWW. Differences between professional tournaments and amateur games can be seen. Properties of eigenvalues and eigenvectors of the matrices producing ranking vectors vary between amateur and different professional tournaments. Certain eigenvectors are localized on specific groups of moves which correspond to different strategies. We think that the point of view developed in this paper should allow to better modelize such games and could also help to design simulators which could in the future beat good human players. Our approach could be used for other types of games, and in parallel shed light on the human decision making process.
We thank CalMiP for the use of their supercomputers, and D. Shepelyansky for useful discussions.\
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abstract: 'In this paper, we study positivity phenomena for the $e$-coefficients of Stanley’s chromatic function of a graph. We introduce a new combinatorial object: the [*correct*]{} sequences of unit interval orders, and using these, in certain cases, we succeed to construct combinatorial models of the coefficients appearing in Stanley’s conjecture. Our main result is the proof of positivity of the coefficients $c_{n-k,1^k}$, $c_{n-2,2}$, $c_{n-3,2,1}$ and $c_{2^k,1^{n-2k}}$ of the expansion of the chromatic symmetric function in terms of the basis of the elementary symmetric polynomials for the case of $(3+1)$-free posets.'
author:
- 'Alexander Paunov, András Szenes'
date: February 2016
title: |
A new approach to $e$-positivity for\
Stanley’s chromatic functions
---
Introduction
============
Let $G$ be a finite graph, $V(G)$ - the set of vertices of $G$, $E(G)$ - the set of edges of $G$.
\[coloring\] A [*proper coloring*]{} $c$ of $G$ is a map $$c:V\rightarrow\mathbb{N}$$ such that no two adjacent vertices are colored in the same color.
For each coloring $c$ we define a monomial $$x^c = \prod_{v\in
V}x_{c(v)},$$ where $x_1, x_2, ..., x_n,...$ are commuting variables. We denote by $\Pi(G)$ the set of all proper colorings of $G$, and by $\Lambda$ the ring of symmetric functions in the infinite set of variables $\{x_1, x_2,...\}.$
In [@Stanley95a], Stanley defined the chromatic symmetric function of a graph.
\[chromfunction\] The *chromatic symmetric function $X_G\in\Lambda$ of a graph $G$ is the sum of the monomials $x^c$ over all proper colorings of $G$: $$X_G=\sum\limits_{c\in\Pi(G)}x^c.$$*
\[efunc\] Denote by $e_m$ the $m$-th elementary symmetric function: $$e_m = \sum\limits_{i_1<i_2<...<i_m}x_{i_1}\cdot
x_{i_2}\cdot...\cdot x_{i_m},$$ where $i_1,..,i_k\in \mathbb{N}$. Given a non-increasing sequence of positive integers (we will call these [*partitions*]{}) $$\lambda = (\lambda_1\geq \lambda_2\geq...\geq\lambda_k),\ \lambda_i\in
\mathbb{N},$$ we define the elementary symmetric function $e_{\lambda} = \prod\limits_{i=1}^k e_{\lambda_i}.$ These functions form a basis of $\Lambda.$
For a natural number $k$, we denote by $1^k$ the partition $\lambda$ of length $k$, where $$\lambda_1=\lambda_2=...=\lambda_k=1.$$
\[epos\] A symmetric function $X\in \Lambda$ is *$e$-positive if it has non-negative coefficients in the basis of the elementary symmetric functions.*
\[pfunc\] Denote by $p_m$ the $m$-th power sum symmetric function: $$p_m =
\sum\limits_{i\in\mathbb{N}}x^m_{i}.$$ Given a partition $\lambda = (\lambda_1\geq \lambda_2\geq...\geq\lambda_k)$, we define the power sum symmetric function $p_{\lambda} = \prod\limits_{i=1}^k p_{\lambda_i}.$ These functions also form a basis of $\Lambda.$
\[mfunc\] Given a partition $\lambda = (\lambda_1\geq \lambda_2\geq...\geq\lambda_k)$, we define the monomial symmetric function $$m_\lambda=\sum\limits_{i_1<i_2<...<i_k}\sum\limits_{\lambda'\in S_k(\lambda)}x_{i_1}^{\lambda_{1}'}\cdot
x_{i_2}^{\lambda_{1}'}\cdot...\cdot x_{i_k}^{\lambda_{k}'},$$ where the inner sum is taken over the set of all permutations of the sequence $\lambda$, denoted by $S_k(\lambda)$.
The chromatic symmetric function of $K_n$, the complete graph on $n$ vertices, is $e$-positive: $X_{K_n} = n!\,e_n$.
\[incgraph\] For a poset $P$, the *incomparability graph, $\textnormal{inc}(P)$, is the graph with elements of $P$ as vertices, where two vertices are connected if and only if they are not comparable in $P$.*
\[nplusmfree\] Given a pair of natural numbers $a,b\in\mathbb{N}^2$, we say that a poset $P$ is *(a+b)-free if it does not contain a length-$a$ and a length-$b$ chain, whose elements are incomparable.*
A unit interval order (UIO) is a partially ordered set which is isomorphic to a finite subset of $U\subset{\mathbb{R}}$ with the following poset structure: $$\text{for } u,w\in U:\ u\succ w \text{ iff } u\ge w+1.$$ Thus $u$ and $w$ are incomparable precisely when $|u-w|<1$ and we will use the notation $u\sim w$ in this case.
\[S\_S\] A finite poset $P$ is a UIO if and only if it is $(2+2)$- and $(3+1)$-free.
Stanley [@Stanley95a] initiated the study of incomparability graphs of $(3+1)$-free partially ordered sets. Analyzing the chromatic symmetric functions of these incomparability graphs, Stanley [@Stanley95a] stated the following positivity conjecture.
\[eposconj\] If $P$ is a $(3+1)$-free poset, then $X_{\textnormal{inc}(P)}$ is $e$-positive.
For a graph $G$ let us denote by ${c_\lambda}(G)$ the coefficients of $X_G$ with respect to the $e$-basis. We omit the index $G$ whenever this causes no confusion: $$X_G=\sum\limits_{\lambda}c_{\lambda}e_\lambda.$$
Conjecture \[eposconj\] has been verified with the help of computers for up to 20-element posets [@Guay-Paquet13]. In 2013, Guay-Paquet [@Guay-Paquet13] showed that to prove this conjecture, it would be sufficient to verify it for the case of $(3+1)$- and $(2+2)$-free posets, i.e. for unit interval orders (see Theorem \[S\_S\]). More precisely:
\[G\_P\] Let $P$ be a $(3+1)$-free poset. Then, $X_\mathrm{inc}(P)$ is a convex combination of the chromatic symmetric functions $$\{X_\mathrm{inc}(P')\ |\ P'\ \mathrm{is}\ \mathrm{a}\
(3+1)\mathrm{-}\ \mathrm{and}\ (2+2)\mathrm{-free}\ \mathrm{poset}
\}.$$
The strongest general result in this direction is that of Gasharov [@Gasharov94].
\[sfunc\] For a partition $\lambda = (\lambda_1\geq
\lambda_2\geq...\geq\lambda_k)$, define the Schur functions *$s_{\lambda}=\mathrm{det}(e_{\lambda_i^*+j-i})_{i,j}$, where $\lambda^*$ is the conjugate partition to $\lambda$. The functions $\{ s_{\lambda}\}$ form a basis of $\Lambda$.*
\[spos\] A symmetric polynomial $X$ is *$s$-positive if it has non-negative coefficients in the basis of Schur functions.*
Obviously, a product of $e$-positive functions is $e$-positive. This also holds for $s$-positive functions. Thus, the equality $e_n=s_{1^n}$ implies that $e$-positive functions are $s$-positive, and thus $s$-positivity is weaker than $e$-positivity.
\[sposthm\] If $P$ is a $(3+1)$-free poset, then $X_{\textnormal{inc}(P)}$ is $s$-positive.
Gasharov proved $s$-positivity by constructing so-called $P$-tableau and finding a one-to-one correspondence between these tableau and $s$-coefficients [@Gasharov94]. However, $e$-positivity conjecture \[epos\] is still open. The strongest known result on the $e$-coefficients was obtained by Stanley in [@Stanley95a]. He showed that sums of $e$-coefficients over the partitions of fixed length are non-negative:
For a finite graph $G$ and $j\in\mathbb{N}$, suppose $$X_G=\sum\limits_{\lambda}c_{\lambda}e_\lambda,$$ and let $\text{sink}(G,j)$ be the number of acyclic orientation of $G$ with $j$ sinks. Then $$\text{sink}(G,j)=\sum\limits_{l(\lambda)=j}c_{\lambda}.$$
By taking $j=1$, it follows from the theorem that $c_n$ is non-negative.
Stanley in [@Stanley95a] showed that for $n\in\mathbb{N}$ and the unit interval order $P_n=\{\frac{i}{2}\}_{i=1}^n$, the corresponding $X_{\text{inc}(P_n)}$ is $e$-positive, while $e$-positivity for the UIOs $$P_{n,k}=\bigg\{\frac{i}{k+1}\bigg\}_{i=1}^n$$ with $k>1$ has not yet been proven. It was checked for small $n$ and some $k$ (see [@Stanley95a]).
Next, we introduce [*correct sequences*]{} (abbreviated as [*corrects*]{}), defined below. These play a major role in the article.
Let U be a UIO. We will call a sequence ${\vec w}= (w_1,\dots, w_k)$ of elements of $U$ [ *correct*]{} if
- $w_i\not\succ w_{i+1}$ for $i=1,2,\dots,k-1$
- and for each $j=2,\dots,k$, there exists $i<j$ such that $w_i\not\prec w_j$.
Every sequence of length 1 is correct, and sequence $(w_1,w_2)$ is correct precisely when $w_1\sim w_2$. The second condition (supposing that the first one holds) may be reformulated as follows: for each $j=1,\dots k$, the subset $\{w_1,\dots,w_j\}\subset U$ is connected with respect to the graph structure ${(U,\sim)}$. Using this notation, we prove the following theorems.
\[eposn\] Let $X_{\text{inc}(U)}=\sum\limits_{\lambda}c_\lambda e_\lambda$ be a chromatic symmetric function of the $n$-element unit interval order $U$. Then $c_n$ is equal to the number of corrects of length $n$, in which every element of $U$ is used exactly once.
Let $X_{\text{inc}(P)}=\sum\limits_{\lambda}c_\lambda e_\lambda$ be a chromatic symmetric function of $n$-element $(3+1)$-free poset $P$, then $c_n$ is a nonnegative integer.
Indeed, positivity for the general case follows from Theorem \[G\_P\], which presents the chromatic symmetric function of a $(3+1)$-free poset as a convex combination of the chromatic symmetric functions of unit interval orders.
Stanley [@Stanley95b] and Chow [@Chow95] showed the positivity of $c_n$ for $(3+1)$-free posets using combinatorial techniques, and linked $e$-coefficients with the acyclic orientations of the incomparability graphs. The construction of corrects not only serves this purpose for UIOs (see [@Paunov16b]), but also creates a new approach, which allows us to obtain the following new result:
\[eposn21\] Let $X_{\text{inc}(P)}=\sum\limits_{\lambda}c_\lambda e_\lambda$ be a chromatic symmetric function of the $(3+1)$-free poset $P$, and $k\in\mathbb{N}$. Then $c_{n-k,1^k}$, $c_{n-2,2}$, $c_{n-3,2,1}$ and $c_{2^k,1^{n-2k}}$ are non-negative integers.
The proofs of Theorem \[eposn\] and Theorem \[sposthm\], and positivity of correspondent $G$-power sum symmetric functions and Schur $G$-symmetric functions can be found in [@Paunov16] and [@Paunov16b]. The article is structured as follows: in Section \[Ghom\], we describe the $G$-homomorphism introduced by Stanley in [@Stanley95b], which is essential for our approach. Positivity of $c_{n-k,1^k}$, $c_{n-2,2}$, $c_{n-3,2,1}$ and $c_{2^k,1^{n-2k}}$ (Theorem \[eposn21\]) is proven in Section 3.
**Acknowledgements.** We are grateful to Emanuele Delucchi and Bart Vandereycken for their help and useful discussions.
Stanley’s $G$-homomorphism {#Ghom}
==========================
For a graph $G$, Stanley [@Stanley95b p. 6] defined $G$-analogues of the standard families of symmetric functions. Let $G$ be a finite graph with vertex set $V(G)=\{v_1,...,v_n\}$ and edge set $E(G)$. We will think of the elements of $V(G)$ as commuting variables.
\[eG\] For a positive integer $i$, $1\leq i, \leq n$, we define the *$G$-analogues of the elementary symmetric polynomials, or *the elementary $G$-symmetric polynomials, as follows $$e_i^G =\sum\limits_{\substack{\#S=i\\
S-\mathrm{stable}}}\prod\limits_{v\in S}v,$$ where the sum is taken over all $i$-element subsets $S$ of $V$, in which no two vertices form an edge, i.e. stable subsets. We set $e_0^G=1$, and $e_i^G=0$ for $i<0$.**
Note that these polynomials are not necessarily symmetric.
Let $\Lambda_G\subset\mathbb{R}[v_1,...,v_n]$ be the subring generated by $\{e_i^G\}_{i=1}^{n}$. The map $e_i\mapsto e_i^G$ extends to a ring homomorphism $\phi_G: \Lambda\rightarrow\Lambda_G$, called the [*$G$-homomorphism*]{}. For $f\in \Lambda$, we will use the notation $f^G$ for $\phi_G(f)$.
Given a partition $\lambda = \lambda_1\geq \lambda_2\geq...\geq\lambda_k,\ k\in \mathbb{N},$ we have $$e_{\lambda}^G = \prod\limits_{i=1}^k e_i^G,$$ $$s_{\lambda}^G=\mathrm{det}(e_{\lambda_i^*+j-i}^G).$$
For an integer function $\alpha: V\rightarrow \mathbb{N}$ and $f^G\in\Lambda_G$, let $$v^\alpha = \prod\limits_{v\in V}v^{\alpha(v)},$$ and $[v^\alpha]f^G$ stands for the coefficient of $v^\alpha$ in the polynomial $f^G\in\Lambda_G$.
Let $G^\alpha$ denote the graph, obtained by replacing every vertex $v$ of $G$ by the complete subgraph of size $\alpha(v)$: $K_{\alpha(v)}^v$. Given vertices $u$ and $v$ of $G$, a vertex of $K_{\alpha(v)}^v$ is connected to a vertex of $K_{\alpha(u)}^u$ if and only if $u$ and $v$ form an edge in $G$.
Considering the Cauchy product [@Macdonald79 ch. 4.2], Stanley [@Stanley95b p. 6] found a connection between the $G$-analogues of symmetric functions and $X_G$. Following Stanley [@Stanley95b], we set $$T(x,v) = \sum\limits_\lambda m_\lambda(x)e^G_\lambda(v),$$ where the sum is taken over all partitions. Then
$$\label{gnechrom}
[v^\alpha]T(x,v)\prod\limits_{v\in V}\alpha(v)! =X_{G^\alpha}.$$
Using the Cauchy identity $$\sum\limits_\lambda s_\lambda(x)s_{\lambda^*}(y)=\sum\limits_\lambda m_\lambda(x)e_\lambda(y) = \sum\limits_\lambda e_\lambda(x)m_\lambda(y)$$ and applying the $G$-homomorphism, one obtains: $$\label{GCauchy}
T(x,v) = \sum\limits_\lambda m_\lambda(x)e^G_\lambda(v) = \sum\limits_\lambda s_\lambda(x)s^G_{\lambda^*}(v)=T(v,x) = \sum\limits_\lambda e_\lambda(x)m^G_\lambda(v).$$
An immediate consequence of the formulas and is the following result of Stanley:
\[poscrit\] For every finite graph G
1. $X_{G^\alpha}$ is s-positive for every $\alpha:V(G)\rightarrow\mathbb{N}$ if and only if $s_\lambda^G\in
\mathbb{N}[V(G)]$ for every partition $\lambda$.
2. $X_{G^\alpha}$ is e-positive for every $\alpha:V(G)\rightarrow\mathbb{N}$ if and only if $m_\lambda^G\in \mathbb{N}[V(G)]$ for every partition $\lambda$.
\[c\_m\] If $X_{G^\alpha}=\sum\limits_{\lambda}c^\alpha_{\lambda}e_\lambda,$ then $c_\lambda^\alpha=[v^\alpha]m^G_\lambda.$ Hence, monomial positivity of $m^G_\lambda$ is equivalent to the positivity of $c_\lambda^\alpha$ for every $\alpha$.
The proofs of positivity of $G$-power sum symmetric functions and Schur $G$-symmetric functions for the case of unit interval orders can be found in [@Paunov16].
Proofs of the theorems {#proofs}
======================
It follows from Theorem \[poscrit\] that to prove that the graph $G$ is $e$-positive, it is enough to show the monomial positivity of its monomial $G$-symmetric functions. On the other hand, Guay-Paquet in Theorem \[G\_P\] showed that it is sufficient to check $e$-positivity for unit interval orders, in order to prove it for the general case of $(3+1)$-free posets. Therefore, in the following section \[proofs\] we analyze the functions $m_{\lambda}^G$ for the case $G=\text{inc}(U),$ where $U$ is UIO.
Let us repeat the definition of a central notion for our work, that of correct sequences of elements of a unit interval order.
Let $(U,\prec)$ be a unit interval order, and $G=\text{inc}(U)$. We will call a sequence $\vec{w} = (w_1,\dots, w_k)$ of elements of $U$ [ *correct*]{} if
- $w_i\not\succ w_{i+1}$ for $i=1,2,\dots,k-1$
- and for each $j=2,\dots,k$, there exists $i<j$ such that $w_i\not\prec w_j$.
We denote by $P^U_k$ the set of all correct sequences (abbreviated as [ *corrects*]{}) of length $k$. Since $G$ is uniquely defined by $U$, and we are working only with UIO, here and below we use the $U$-index instead of $G$. The $U$-analogues of symmetric functions will be analyzed.
\[Ppos\] Let $U$ be a unit interval order and $p_k^U$ the Stanley power-sum function of the corresponding incomparability graph. Then, for every natural $k$, we have $$p_k^U=\sum\limits_{\vec{w}\in P^U_k} w_1\cdot...\cdot w_k\ \in N[U],$$ where the sum is taken over all corrects of length $k$.
The proof of this theorem can be found in [@Paunov16].
Below, we prove positivity of $m^U_{{l},1^k}$, $m^U_{{l},2}$, $m^U_{{l},2,1},$ and $m^U_{2^{l},1^k}$. We need the following mild technical generalization of correct sequences: let $\lambda=(\lambda_1\ge\dots\ge \lambda_k)$ be a partition of $|\lambda|=\sum\limits_{i=1}^k\lambda_i$. Then, we will call sequence $(w_1,\dots, w_{|\lambda|})$ [*$\lambda$-correct*]{} if each of the subsequences $(w_1,\dots w_{\lambda_1})$, $(w_{\lambda_1+1},\dots,
w_{\lambda_1+\lambda_2})$,$\dots$ $(w_{|\lambda|-\lambda_k+1},\dots, w_{|\lambda|})$ are correct. Introduce the set $$P_\lambda^U=\{{\vec w}=(w_1,\dots w_{\lambda_1}{|}w_{\lambda_1+1},\dots,
w_{\lambda_1+\lambda_2}{|}\dots{|}w_{|\lambda|-\lambda_k+1},\dots, w_{|\lambda|})\ |\ {\vec w}\text{ is
}\lambda\text{-correct }\}$$ of $\lambda$-correct sequences of length-$|\lambda|$. In particular, $P_{l}^U$ is the set of ${l}$-corrects of $U$. This definition is consistent with Theorem \[Ppos\], and we have: $$p_\lambda^U=\prod_{i=1}^kp_{\lambda_i}^U=\sum_{\vec{w}\in P_\lambda^U}w_1\cdot ...\cdot w_{|\lambda|}.$$ For $\vec{w}=(w_1,\dots w_{{l}})\in P^U_{l}$ and $z\in U$ we write $z\succ\vec{w}$, if $z\succ w_i$ for every $1\leq i\leq{l}.$
\[Thn1\] Let $$M_{{l},1}^U = \{({\vec w}\ |z)\in P_{{l}.1}|\; z\succ{\vec w}\vee z\prec w_{l}\},$$
then $$m^U_{{l},1}=\sum\limits_{({\vec w};z)\in M^U_{{l},1}}w_1\cdot...\cdot w_{l}\cdot z.$$
According to Remark \[c\_m\], this implies $c_{n-1,1}(U)\geq 0$.
Since $P^U_{{l}+1}\subset P^U_{{l}}\times P^U_{1}$, using the following relation $$m^U_{{l},1}=p^U_{l}\cdot p^U_1 - p^U_{{l}+1},$$ we have $$P^U_{{l}}\times P^U_{1}\setminus P^U_{{l}+1}=M_{{l},1}^U,$$ and, as a consequence, $$m^U_{{l},1}=\sum\limits_{({\vec w};z)\in M^U_{{l},1}}w_1\cdot...\cdot w_{l}\cdot z.$$
Next, we introduce the set $$E_k^U=\{\vec{\varepsilon}=(\varepsilon_1,\dots, \varepsilon_{k})|\; \varepsilon_i\prec \varepsilon_{i+1}\text{, for }1\leq i<k \}.$$
\[n1k\] For natural numbers ${l}$ and $k$, let $$M^U_{{l},1^k}=\{(\vec{w}\ |\vec{\varepsilon})\in P^U_{l}\times E_k^U|\; \varepsilon\prec w_{l}\ \vee\ \varepsilon\succ \vec{w}\text{, for every } \varepsilon \in \vec{\varepsilon}\},$$ Then, $$m_{{l},1^k}^U=\sum_{({\vec w};\vec{\varepsilon})\in M^U_{{l},1^k}}w_1\cdot...\cdot w_{{l}}\cdot \varepsilon_1\cdot...\cdot\varepsilon_k.$$
According to Remark \[c\_m\], this implies $c_{n-k,1^k}(U)\geq 0$.
We prove this by induction on $k$. Note that for $k=1$, the definition of $M^U_{{l},1^k}$ coincides with $M^U_{{l},1}$ from Theorem \[Thn1\]. Thus, the case $k=1$ follows from Theorem \[Thn1\].
Assume the statement is true for $k$, and consider the standard equation $$p^U_{{l}}*e^U_{k+1}=m^U_{{l},1^{k+1}}+m^U_{{l}+1,1^{k}}.$$
Below, we construct a pair of inverse maps, $\phi_{{l},1^{k+1}}$ and $\psi_{{l},1^{k+1}}$ from the left part to the right part of the latter equation and vice versa respectively. Every case is followed by a visual illustration.
[**. We define $$\phi_{{l},1^{k+1}}:P^U_{l}\times E_{k+1}^U\to M^U_{{l},1^{k+1}}\sqcup M^U_{{l}+1,1^{k}}$$ as follows:**]{}
Let $(\vec{w}\ |\vec{\varepsilon}\ )\in P^U_{l}\times E_{k+1}^U$
1. If $\varepsilon_i\prec w_{l}\ \vee\ \varepsilon_i\succ \vec{w}$ for $1\leq i\leq k+1$, then $${\color{orange}}\phi_{{l},1^{k+1}}(\vec{w}\ |\vec{\varepsilon}\ )= (\vec{w}\ ;\vec{\varepsilon}\ )\in M^U_{{l},1^{k+1}}.$$
[**. The inverse of map $\psi_{{l},1^{k+1}}$: $$\begin{aligned}
&\psi^1_{{l},1^{k+1}}:M^U_{{l},1^{k+1}}\to P^U_{l}\times E_{k+1}^U;\\ &\psi^2_{{l},1^{k+1}}:M^U_{{l}+1,1^{k}}\to P^U_{l}\times E_{k+1}^U. \end{aligned}$$** ]{} Let $ (\vec{w}\ ;\vec{\varepsilon}\ )\in M^U_{{l},1^{k+1}}$ and $(\vec{w}\ ,z ;\vec{\nu}\ )\in M^U_{{l}+1,1^{k}}$.
1. For $(\vec{w}\ ;\vec{\varepsilon}\ )\in M^U_{{l},1^{k+1}}$, we have: $${\color{red}}\psi^1_{{l},1^{k+1}}(\vec{w}\ ;\vec{\varepsilon}\ )= (\vec{w}\ |\vec{\varepsilon}\ )\in P^U_{l}\times E_{k+1}^U.$$
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{}\]
\(b) [${w}_1$]{}; (d) [$...$]{}; (f) [${w}_{{l}}$]{}; (k) [$\prec$]{}; (l) [$\varepsilon_j$]{}; (zl) [$\prec$]{}; (zl1) [$...$]{}; (zl2) [$\prec$]{}; (zq1) [$\varepsilon_{k+1}$]{};
(b1) [$\prec$]{}; (b2) [$...$]{}; (b3) [$\prec$]{}; (b4) [$\varepsilon_1$]{};
\(m) [$\prec$]{}; (n) [$\prec$]{};
1. If $\exists\ i$, s.t. $\varepsilon_i\nprec w_{l}\ \wedge\ \varepsilon_i\nsucc \vec{w}$, then define $$m=\max(i\ |1\leq i\leq k+1, \varepsilon_i\nprec w_{l}\ \wedge\ \varepsilon_i\nsucc \vec{w}),$$ then we have
$$\begin{aligned}
&{\color{red}}\phi_{{l},1^{k+1}}(\vec{w}\ |\vec{\varepsilon}\ )=\\&{\color{red}}=(\vec{w}\ ,\varepsilon_{m}; \varepsilon_1,...,\varepsilon_{m-1},\varepsilon_{m+1},..,\varepsilon_{k+1})\in M^U_{{l}+1,1^{k}}.\end{aligned}$$
1. For $(\vec{w}\ ,z ;\vec{\nu}\ )\in M^U_{{l}+1,1^{k}}$, we define $$j=\min(i\ |1\leq i\leq k, z\prec \nu_i),$$ then we have
$$\begin{aligned}
&{\color{orange}}\psi^2_{{l},1^{k+1}}(\vec{w}\ ,z ;\vec{\nu}\ )=\\&{\color{orange}}=(\vec{w}\ |\nu_1,...,\nu_{j-1},z,\nu_{j},..,\nu_k)\in P^U_{l}\times E_{k+1}^U.\end{aligned}$$
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{}, sh/.style=[shade,shading=axis,left color=orange!20,right color=blue!20]{}\]
\(b) [${w}_1$]{}; (d) [$...$]{}; (f1) [${w}_{{l}}$]{}; (k1) [$\preceq$]{}; (f) \[sh\] [$\varepsilon_{m}|z$]{}; (k) [$\prec$]{}; (l) [$\varepsilon_{m+1}|\nu_j$]{}; (zl) [$\prec$]{}; (zl1) [$...$]{}; (zl2) [$\prec$]{}; (zq1) [$\varepsilon_{k+1}|\nu_k$]{};
(b1) [$\prec$]{}; (b2) [$...$]{}; (b3) [$\prec$]{}; (b4) [$\varepsilon_1|\nu_1$]{};
\(m) [$\prec$]{}; (n) [$\prec$]{};
This completes the proof.
Given a correct ${\vec w}\in P^U_{{l}}$, let $$\theta= \theta({\vec w})=\max(\{i<{l}|\; w_{i}\sim w_{i+1} \})\in\mathbb{N} \text{ and } J_{{l}-1}=(w_1,...,w_{{l}-1})\in P_{l-1}.$$
\[Thn2\] For natural ${l}\geq 2$, let $$M^U_{{l},2} = \{({\vec w}\ {|}q_0,q_1)\in P^U_{{l}.2}|\; J_{{l}-1}\prec q_0 \text{ and } w_{{l}}\prec q_1 \ \vee\
w_\theta\succ q_0 \text{ and } w_{\theta+1}\succ q_1
\}.$$ Then, $$m_{{l},2}^U=\sum_{({\vec w};q_0,q_1)\in M^U_{{l},2}}w_1\cdot...\cdot w_{{l}}\cdot q_0\cdot q_1.$$
According to Remark \[c\_m\], this implies $c_{n-2,2}(U)\geq 0$.
There is a slightly more elegant version of $M^U_{{l},2},$ which we will use in the future: $$M^U_{{l},2} = \{({\vec w},q_0,q_1)\in P^U_{{l},2}|\; (J_{{l}-1}\prec q_0 \wedge w_{{l}}\prec q_1) \ \vee\
(w_\theta\succ q_0 \wedge w_{\theta+1}\succ q_1)
\}$$
Here is an illustration of an element of $M^U_{{l},2}$:
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{}\]
\(b) [$w_1$]{}; (d) [$...$]{}; (f) [$w_{\theta}$]{}; (ff) [$w_{\theta+1}$]{}; (g) [$\prec$]{}; (h) [$...$]{}; (i) [$\prec$]{}; (ii) [$w_{{l}}$]{};
\(k) [$\prec$]{}; (l) [$q_0$]{}; (kk) [$\prec$]{}; (ll) [$q_1$]{}; (f) – (ff); (l) – (ll);
We can write $$\label{eqpn2}
P^U_{{l},2}\setminus M^U_{{l},2} = \{({\vec w}\ {|}q_0,q_1)\in P^U_{{l}.2}|\; (\vec{w}\prec q_0 \Rightarrow
(w_{{l}}\nprec q_1\wedge J_{{l}-1}\nprec q_1) \ \wedge\
( w_\theta\succ q_0\Rightarrow w_{\theta+1}\not\succ q_1)
\}.$$ The conditions in have the form $A\wedge B$. We begin with a few remarks.
1. Observe that the conditions of $A$ and $B$ are mutually exclusive, so we can consider the two statements independently.
2. Define $\tau=\max\{i\leq{l}|\; q_1\nprec w_{i}\vee w_i\sim w_{i+1}\}$. Note that it could happen that $q_1\prec\theta$, but clearly $\tau\geq\theta$.
3. For $\vec{u}\in P^U_{{l}+2}$, let $U_{{l}-1}=(u_1,...,u_{{l}-1})$ and $\breve{\theta}=\max(\{i<{l}|\; u_{i}\sim u_{i+1} \}).$
To prove the theorem, we consider the following formula $$m^U_{{l},2}=p^U_{l}\cdot p^U_2-p^U_{{l}+2}.$$ and construct two injective maps.
[0.48]{}[0.48]{}
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{},scale=1.5,transform shape\]
\(b) [${\scalebox{0.6}{$w_1\mid u_1$}}$]{}; (d) [$...$]{}; (f) [${\scalebox{0.6}{$q_1\mid u_{{l}}$}}$]{}; (ii) [${\scalebox{0.6}{$w_{{l}}\mid u_{{l}+1}$}}$]{}; (iii) [$\prec$]{}; (j) [${\scalebox{0.6}{$q_0\mid u_{{l}+2}$}}$]{}; (k) [$\prec$]{}; (n) [${\scalebox{0.6}{$\vec{w}\mid U_{{l}-1}$}}$]{}; (f) – (ii); (f) to \[out=-50,in=-130\] (j);
[0.48]{}[0.48]{}
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{},scale=1.5,transform shape\]
\(b) [${\scalebox{0.6}{$w_1\mid u_1$}}$]{}; (d) [$...$]{};
\(dd) [${\scalebox{0.6}{$w_\tau\mid u_{\breve{\theta}-1}$}}$]{}; (k) [$\prec$]{}; (n) [${\scalebox{0.6}{$q_0\mid u_{\breve{\theta}+1}$}}$]{}; (f) [${\scalebox{0.6}{$q_1\mid u_{\breve{\theta}}$}}$]{}; (fff) [${\scalebox{0.6}{$q_0\mid u_{\breve{\theta}+1}$}}$]{}; (g) [$\prec$]{}; (t1) [${\scalebox{0.6}{$w_{\tau+1}\mid u_{\breve{\theta}+2}$}}$]{}; (t2) [$\prec$]{}; (h) [$...$]{};
\(f) – (fff); (dd) – (f);
[0.48]{}[0.48]{}
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{},scale=1.5,transform shape\]
\(b) [${\scalebox{0.6}{$w_1\mid u_1$}}$]{}; (d) [$...$]{};
\(dd) [${\scalebox{0.6}{$w_\tau\mid u_{\breve{\theta}-1}$}}$]{}; (k) [$\nprec$]{}; (n) [${\scalebox{0.6}{$q_1\mid u_{\breve{\theta}+1}$}}$]{}; (dd1) [$\preceq$]{}; (f) [${\scalebox{0.6}{$q_0\mid u_{\breve{\theta}}$}}$]{}; (fff) [${\scalebox{0.6}{$q_1\mid u_{\breve{\theta}+1}$}}$]{}; (g) [$\prec$]{}; (t1) [${\scalebox{0.6}{$w_{\tau+1}\mid u_{\breve{\theta}+2}$}}$]{}; (t2) [$\prec$]{}; (h) [$...$]{};
\(f) – (fff);
To find a combinatorial interpretation of $m^U_{{l},2,1}$, we construct a bijection between the right and left hand sides of the following equality:
$$\label{n21}
p^U_{{l}}*m^U_{2,1}=m^U_{{l}+2,1}+m^U_{{l}+1,2}+m^U_{{l},2,1}.$$
This formula and its proof are similar to the previous one.
\[thmn21\] For natural ${l}$, let $$\begin{aligned}
M^U_{{l},2,1}= &\{ (\vec{w}\ |\vec{q}\ |z)\in P_{{l},2,1}|\; (\vec{w}\ ;\vec{q}\ )\in M^U_{{l},2},\ (\vec{w}\ ;z)\in M^U_{{l},1},\ (\vec{q}\ ;z)\in M^U_{2,1}, \}\\ & \cup \{ (\vec{w}\ |\vec{q}\ |z)\in P_{{l},2,1}|\; s_{l}\succ z\succ\vec{q},\ \exists \gamma \in \mathbb{N}, \text{ such that } \theta < \gamma<{l},\ s_{ \gamma}\sim z, \ s_{ \gamma}\succ q_2 \}\\ & \cup \{ (\vec{w}\ |\vec{q}\ |z)\in P_{{l},2,1}|\; s_{l}\succ z\succ q_2,\ \exists \gamma \in \mathbb{N}, \text{ such that } \theta < \gamma<{l},\ s_{\gamma}\sim z,\ z\sim q_1,\ s_{ \gamma}\succ q_1 \},
\end{aligned}$$ Then, $$m_{{l},2,1}^U=\sum_{({\vec w};q_0,q_1;z)\in M^U_{{l},2,1}}w_1\cdot...\cdot w_{{l}}\cdot q_0\cdot q_1\cdot z.$$
According to Remark \[c\_m\], this implies $c_{n-3,2,1}(U)\geq 0$.
Let us explain the meaning of $M_{{l},2,1}^U$. This theorem states that in addition to combinations of pairwise comparable corrects of lengths ${l}$, 2 and 1 we have two more cases:
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{}\]
(start) [$...$]{}; (a) [$w_{\theta}$]{}; (b) [$w_{\theta+1}$]{}; (c) [$\prec$]{}; (d) [$...$]{}; (e) [$\prec$]{}; (f) [$w_{\gamma}$]{}; (g) [$\prec$]{}; (h) [$...$]{}; (i) [$\prec$]{}; (j) [$w_{{l}}$]{}; (k) [$\prec$]{}; (l) [$z$]{}; (m) [$\prec$]{}; (n) [$q_1$]{}; (o) [$\prec$]{}; (p) [$\prec$]{}; (q) [$q_0$]{}; (b) – (a); (l) – (f); (n) – (q);
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{}\]
(start) [$...$]{}; (a) [$w_{\theta}$]{}; (b) [$w_{\theta+1}$]{}; (c) [$\prec$]{}; (d) [$...$]{}; (e) [$\prec$]{}; (f) [$w_{\gamma}$]{}; (g) [$\prec$]{}; (h) [$...$]{}; (i) [$\prec$]{}; (j) [$w_{{l}}$]{}; (k) [$\prec$]{}; (l) [$q_1$]{};
\(n) [$q_0$]{}; (o) [$\prec$]{}; (p) [$\prec$]{}; (q) [$z$]{}; (b) – (a); (l) – (f); (l) – (n); (n) – (q);
To prove Theorem \[thmn21\] using Formula \[n21\], we construct the maps $\varphi_{{l}|2,1}$ and $\psi_{{l}|2,1}$.
[****]{}. We construct the map from the left hand side to the right hand side $$\phi_{{l}|2,1}: P^U_{{l}}\times M^U_{2,1} \to M_{{l}+2,1}^U\sqcup M_{{l}+1,2}^U\sqcup M^U_{{l},2,1}.$$
Let us take $$(\vec{{w}}\ |q_1,q_2\ ;z)\in P^U_{{l}}\times M^U_{2,1}.$$
Let $$\theta=\max(\{i<{l}|\; {w}_{i}\sim {w}_{i+1} \}).$$
We will use this $\theta$ for ${w}$ on the right hand side as well.
1. If $z\succ\vec{q}$.
1. If $z\succ\vec{{w}}$.
1. If $(\vec{{w}}\ ;\vec{q}\ )\in M^U_{{l},2}$, then $${\color{red}}\phi_{{l}|2,1}(\vec{{w}}\ |\vec{q}\ ;z)=(\vec{{w}}\ ;\vec{q}\ ;z)\in M^U_{{l},2,1}.$$
[**.**]{} We construct $$\psi^1_{{l}|2,1}: M^U_{{l}+2,1} \to P^U_{l}\times M^U_{2,1},$$ $$\psi^2_{{l}|2,1}: M^U_{{l}+1,2} \to P^U_{l}\times M^U_{2,1},$$ $$\psi^3_{{l}|2,1}: M^U_{{l},2,1} \to P^U_{l}\times M^U_{2,1}.$$ We take $$(\vec{{u}}\ ;{\xi}) \in M^U_{{l}+2,1};$$ $$(\vec{{w}},{\xi}\ ;\ {q}_0,{q}_1) \in M^U_{{l}+1,2};$$ $$(\vec{{w}}\ ;{q}_0,{q}_1;z) \in M^U_{{l},2,1}.$$
Let $\breve{\theta}=\max(\{i<{l}+2|\ {u}_i\sim {u}_{i+1}\})$.
1. If $(\vec{{w}}\ ;{q}_0,{q}_1; z) \in M^U_{{l},2,1}$ and $z\succ\vec{{q}}$ and $z\succ\vec{{w}}$, then $${\color{orange}}\psi^3_{{l}|2,1}(\vec{{w}}\ ;{q}_0,{q}_1;z)=(\vec{{w}}\ {|}\ {q}_0,{q}_1\ ; z).$$
In this case we have 2 illustrations:
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{}\]
\(b) [${w}_1$]{}; (d) [$...$]{}; (f) [${w}_{{l}-1}$]{}; (i) [$\preceq$]{}; (j) [${w}_{l}$]{}; (k) [$\prec$]{}; (l) [$q_0$]{}; (m) [$\prec$]{}; (n) [$\prec$]{}; (o) [$\prec$]{}; (p) [$q_1$]{}; (oo) [$\prec$]{}; (pp) [$z$]{}; (nn) [$\prec$]{}; (l) – (p);
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{}\]
\(b) [${w}_1$]{}; (d) [$...$]{}; (f) [${w}_{\theta}$]{}; (ff) [${w}_{\theta+1}$]{}; (g) [$\prec$]{}; (h) [$...$]{}; (i) [$\prec$]{}; (ii) [${w}_{{l}}$]{}; (iii) [$\prec$]{}; (j) [$z$]{}; (m) [$\prec$]{}; (n) [$\prec$]{};
\(k) [$\prec$]{}; (l) [$q_0$]{}; (kk) [$\prec$]{}; (ll) [$q_1$]{}; (f) – (ff); (l) – (ll);
1. If $(\vec{{w}}\ |\vec{q}\ )\notin M^U_{{l},2}$, then using $$\phi_{{l}|2}(\vec{{w}}\ |\vec{q}\ )\in M^U_{{l}+2}$$ we have $${\color{red}}\phi_{{l}|2,1}(\vec{{w}}\ |\vec{q}\ |z)=(\phi_{{l}|2}(\vec{{w}}\ |\vec{q}\ )\ ;z)\in M^U_{{l}+2,1}$$
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1. If $(\vec{{u}}\ ;{\xi}) \in M^U_{{l}+2,1}$ and ${\xi}\succ\vec{{u}}$, then using $$\psi_{{l}|2}(\vec{{u}}\ )\in M^U_{{l},2}$$ we have $${\color{orange}}\psi^1_{{l}|2,1}(\vec{{u}}\ ;z)=(\psi_{{l}|2}(\vec{{u}}\ );\ {\xi}).$$
Here, we provide illustrations for the right hand side, see Theorem \[Thn2\], where $\phi_{{l}|2}$ and $\psi_{{l}|2}$ are defined, for more details.
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{}\]
(start) [${u}_1$]{}; (start1) [$...$]{}; (a) [${u}_{\theta}$]{}; (b) [${u}_{\theta+1}$]{}; (c) [$\prec$]{}; (cc) [${u}_{\theta+2}$]{}; (ccc) [$\prec$]{}; (d) [$...$]{}; (e) [$\prec$]{};
(fff) [${u}_{{l}+1}$]{};
\(i) [$\prec$]{}; (j) [${u}_{{l}+2}$]{}; (k) [$\prec$]{}; (l) [${\xi}$]{}; (m) [$\prec$]{}; (m) [$\prec$]{}; (m) [$\prec$]{}; (b) – (a);
Note that the picture above illustrates most of the cases, except the special one, when ${u}_{{l}+2}~\succ~{u}_{{l}+1}$, ${u}_{{l}+2}\succ J_{{l}-1}$ and ${u}_{{l}+2}\sim {u}_{{l}}$. In this case map $\psi_{{l}|2}$ takes out ${u}_{{l}+2}$ and ${u}_{{l}}$:
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{}\]
\(b) [${u}_1$]{}; (d) [$...$]{}; (f) [${u}_{l}$]{}; (ii) [${u}_{{l}+1}$]{}; (iii) [$\prec$]{}; (j) [${u}_{{l}+2}$]{}; (k) [$\prec$]{}; (l) [${\xi}$]{}; (m) [$\prec$]{}; (m) [$\prec$]{}; (m) [$\prec$]{}; (f) – (ii); (j) to \[out=130,in=50\] (f);
1. If $z\nprec {w}_{l}$ and $z\nsucc\vec{w}$, then denote $$(\hat{w}_\theta,\hat{w}_{\theta+1})=
\begin{cases}
(w_{l},z), & \text{if}\ w_{l}\sim z, \\
({w}_\theta,{w}_{\theta+1}), & z\succ w_{l}.
\end{cases}$$
1. If $\hat{w}_\theta\succ q_0$ and $\hat{w}_{\theta+1}\succ q_1$, then $${\color{red}}\phi_{{l}|2,1}(\vec{{w}}\ |q_0,q_1;z)=(\vec{{w}},z\ ;q_0,q_1)\in M^U_{{l}+1,2}.$$
<!-- -->
1. For $(\vec{{w}},{\xi}\ ;{q}_0,{q}_1) \in M^U_{{l}+1,2}$, such that
(${\xi}\succ w_{l},$ ${w}_{\theta}\succ {q}_0,$ ${w}_{\theta+1}\succ{q}_1,$ and ${\xi}\succ {q}_0$),\
or
(${\xi}\sim w_{l},$ ${w}_{{l}}\succ {q}_0,$ ${\xi}\succ{q}_1,$ and ${\xi}\succ {q}_0$),\
we have: $${\color{orange}}\psi^2_{{l}|2,1}(\vec{{w}},{\xi}\ ;{q}_0,{q}_1)=(\vec{{w}}\ |{q}_0,{q}_1;{\xi}).$$
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{}\]
\(b) [${w}_1$]{}; (d) [$...$]{}; (f) [${w}_{\theta}$]{}; (ff) [${w}_{\theta+1}$]{}; (g) [$\prec$]{}; (h) [$...$]{}; (i) [$\prec$]{}; (ii) [${w}_{{l}}$]{}; (iii) [$\preceq$]{}; (j) [$z\mid{\xi}$]{};
(ar2) [$\prec$]{}; (arr2) [$q_0$]{};
\(k) [$\prec$]{}; (l) [$q_0$]{}; (kk) [$\prec$]{}; (ll) [$q_1$]{}; (f) – (ff); (l) – (ll);
1. If $\hat{w}_{\theta+1}\nsucc q_1$ or $\hat{w}_{\theta}\nsucc q_0$, we take $$\tau =\max(\{i<{l}| {w}_{i}\nsucc q_1 \vee {w}_i\sim {w}_{i+1}\vee {w}_i\sim z\})$$ (note that $\tau\geq\theta$), and insert $q_0$ or $q_1$ after it:
1. If $q_0\prec {w}_\tau$, then
$$\begin{gathered}
{\color{red}}\phi_{{l}|2,1}(\vec{{w}}\ |\vec{q}\ ;z)=\\ {\color{red}}= ({w}_1,...,{w}_\tau,q_1,{w}_{\tau+1},...,{w}_{l},z\ ;q_0)\in M^U_{{l}+2,1} \end{gathered}$$
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1. For $(\vec{{u}}\ ;{\xi})\in M^U_{{l}+2,1},$ such that ${\xi}\prec{u}_{{l}+2}$, define $$\eta=\max(0,\{i\geq\breve{\theta}| {u}_i\sim {\xi}\}).$$
1. If $\eta>0$, then we take out ${u}_{\eta},\ {u}_{{l}+2}$ and ${\xi}$:
1. If $\eta=\breve{\theta}+1$ and ${u}_{\breve{\theta}}\succ {\xi}$, then
$$\begin{gathered}
{\color{orange}}\psi^1_{{l}|2,1}(\vec{{u}}\ ;{\xi}) = ({u}_1,...,{u}_{\eta-1},{u}_{\eta+1},...,{u}_{{l}+1}\ |{\xi},{u}_{\eta};{u}_{{l}+2}).\end{gathered}$$
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{},scale=1.5,transform shape\]
\(b) [${\scalebox{0.6}{${w}_1\mid{u}_1$}}$]{}; (d) [$...$]{}; (e) [${\scalebox{0.6}{${w}_{\tau}\mid{u}_{\breve{\theta}}$}}$]{}; (ee) [$\prec$]{}; (eee) [${\scalebox{0.6}{$q_0\mid{\xi}$}}$]{};
\(f) [${\scalebox{0.6}{$q_1\mid{u}_{\eta}$}}$]{}; (ff) [$\prec$]{}; (fff) [${\scalebox{0.6}{${w}_{\tau+1}|{u}_{\breve{\theta}+2}$}}$]{}; (ffff) [$\prec$]{}; (g) [$\prec$]{}; (h) [$...$]{}; (i) [$\prec$]{}; (ii) [${\scalebox{0.6}{${w}_{l}|{u}_{{l}+1}$}}$]{}; (iii) [$\prec$]{}; (j) [${\scalebox{0.6}{$ z\mid{u}_{{l}+2}$}}$]{}; (k) [$\prec$]{}; (l) [${\scalebox{0.6}{$q_0\mid{\xi}$}}$]{}; (f) – (e); (l) to \[out=190,in=30\] (f);
1. If $q_0\nprec s_\tau$, then
$$\begin{gathered}
{\color{red}}\phi_{{l}|2,1}(\vec{w}\ |\vec{q}\ ; z)=\\{\color{red}}=({w}_1,...,{w}_\tau,q_0,s_{\tau+1},...,{w}_{l},z\ ;q_1)\in M^U_{{l}+2,1}.\end{gathered}$$
1. If ($\eta=\theta$ and ${\xi}\prec {u}_{\theta+1}$) or ($\eta=\theta+1$ and (${u}_{\theta}\sim{u}_{\breve{\theta}+2}$ or ${u}_{\breve{\theta}}\nsucc{\xi}$ ) or $\eta>\breve{\theta}+1$, then $${\color{orange}}\psi^1_{{l}|2,1}(\vec{{u}}\ ;{\xi}) = ({u}_1,...,{u}_{\eta-1},{u}_{\eta+1},...,{u}_{{l}+1}\ |{u}_{\eta}, {\xi}\ ;{u}_{{l}+2}).$$
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{},scale=1.5,transform shape\]
\(b) [${\scalebox{0.6}{${w}_1\mid{u}_1$}}$]{}; (d) [$...$]{}; (e) [${\scalebox{0.6}{${w}_{\tau}\mid{u}_{\eta-1}$}}$]{};
(ee) [$\nprec$]{}; (eee) [${\scalebox{0.6}{$q_1\mid{\xi}$}}$]{}; (ef) [$\preceq$]{}; (f) [${\scalebox{0.6}{$q_0\mid{u}_{\eta}$}}$]{};
(ff) [$\preceq$]{}; (fff) [${\scalebox{0.6}{${w}_{\tau+1}\mid{u}_{\eta+1}$}}$]{}; (ffff) [$\prec$]{}; (g) [$\prec$]{}; (h) [$...$]{}; (i) [$\prec$]{}; (ii) [${\scalebox{0.6}{${w}_{l}|{u}_{{l}+1}$}}$]{}; (iii) [$\prec$]{}; (j) [${\scalebox{0.6}{$z|{u}_{{l}+2}$}}$]{}; (k) [$\prec$]{}; (l) [${\scalebox{0.6}{$q_1|{\xi}$}}$]{};
\(l) to \[out=190,in=30\] (f);
Here we have a special case when $\tau=\theta$. Note that it is possible that $w_\tau\succ q_1$:
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{},scale=1.5,transform shape\]
\(b) [${\scalebox{0.6}{${w}_1\mid{u}_1$}}$]{}; (d) [$...$]{}; (e) [${\scalebox{0.6}{${w}_{\theta}\mid{u}_{\eta-1}$}}$]{}; (f) [${\scalebox{0.6}{$q_0\mid{u}_{\eta}$}}$]{}; (ff) [$\preceq$]{}; (fff) [${\scalebox{0.6}{${w}_{\theta+1}\mid{u}_{\eta+1}$}}$]{}; (ffff) [$\prec$]{}; (g) [$\prec$]{}; (h) [$...$]{}; (i) [$\prec$]{}; (ii) [${\scalebox{0.6}{${w}_{l}|{u}_{{l}+1}$}}$]{}; (iii) [$\prec$]{}; (j) [${\scalebox{0.6}{$z|{u}_{{l}+2}$}}$]{}; (k) [$\prec$]{}; (l) [${\scalebox{0.6}{$q_1|{\xi}$}}$]{}; (f) – (e); (l) to \[out=190,in=30\] (f); (e) to \[out=-50,in=-130\] (fff);
1. If $z\prec {w}_{l}$.
1. If ${w}_{\theta+1}\succ q_1$ and ${w}_\theta\succ q_0$, then $${\color{red}}\phi_{{l}|2,1}(\vec{{w}}\ |\vec{q}\ ;z)=(\vec{{w}}\ ;\vec{q}\ ;z)\in M^U_{{l},2,1}.$$
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1. If $(\vec{{w}}\ ;{q}_0,{q}_1;z) \in M^U_{{l},2,1}$ and $z\succ\vec{{q}},$ and $z\prec{w}_{l},$ and ${w}_{\theta+1}\succ q_1$ and ${w}_\theta\succ q_0$, then
$${\color{orange}}\psi^3_{{l}|2,1}(\vec{{w}}\ ;{q}_0,{q}_1;z)=(\vec{{w}}\ |{q}_0,{q}_1;z).$$
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{}\]
\(b) [${w}_1$]{}; (d) [$...$]{}; (f) [${w}_{\theta}$]{}; (ff) [${w}_{\theta+1}$]{}; (g) [$\prec$]{}; (h) [$...$]{}; (i) [$\prec$]{}; (ii) [${w}_{{l}}$]{}; (iii) [$\prec$]{}; (j) [$z$]{}; (m) [$\prec$]{}; (n) [$\prec$]{};
\(k) [$\prec$]{}; (l) [$q_0$]{}; (kk) [$\prec$]{}; (ll) [$q_1$]{}; (f) – (ff); (l) – (ll);
1. If $w_{\theta+1}\nsucc q_1$ or $w_\theta\nsucc q_0$, then Let $$\gamma=\max(0,\{\theta<i<{l}| w_{i}\sim z\}).$$
1. if $\gamma>\theta$ and $\gamma>\tau$ (i.e. $\exists\ w_\gamma\sim z$), then $${\color{red}}\phi_{{l}|2,1}(\vec{w}\ |\vec{q}\ ;z)=(\vec{w}\ ;\vec{q}\ ;z)\in M^U_{{l},2,1}.$$
This is the first type exceptional element, shown before on the Figure \[Firstn21\] and on the picture below:
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1. If $(\vec{w}\ ;{q}_0,{q}_1;z) \in M^U_{{l},2,1}$ is the first type exceptional element , then\
$${\color{orange}}\psi^3_{{l}|2,1}(\vec{w}\ ;{q}_0,{q}_1;z)=(\vec{w}\ |{q}_0,{q}_1;z).$$
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{}\]
(start) [$...$]{}; (a) [${w}_{\theta}$]{}; (b) [${w}_{\theta+1}$]{}; (c) [$\prec$]{}; (d) [$...$]{}; (e) [$\prec$]{}; (f) [${w}_{\gamma}$]{}; (g) [$\prec$]{}; (h) [$...$]{}; (i) [$\prec$]{}; (j) [${w}_{{l}}$]{}; (k) [$\prec$]{}; (l) [$z$]{}; (m) [$\prec$]{}; (n) [$q_1$]{}; (o) [$\prec$]{}; (p) [$\prec$]{}; (q) [$q_0$]{}; (b) – (a); (l) – (f); (n) – (q);
1. Otherwise (i.e. if $\gamma=0$ or $\gamma<\tau$), we have:\
\
\
\
$${\color{red}}\phi_{{l}|2,1}(\vec{w}\ |\vec{q}\ ;z)=(\phi_{{l}|2}(\vec{w}\ |\vec{q}\ );z).$$
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1. For $(\vec{{u}}\ ;{\xi}) \in M^U_{{l}+2,1}$, such that $${\xi}\prec{u}_{{l}+2} \text{ and } \eta=0,$$ what implies $$({\xi}\succ{u}_{\theta+1} \text{ and } {\xi}\succ{u}_{\theta}),$$ we have $${\color{orange}}\psi^2_{{l}|2,1}(\vec{{u}}\ ;{\xi})=(\psi_{{l}|2}(\vec{{u}}\ );{\xi}).$$
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{},scale=1.5,transform shape\]
\(b) [${\scalebox{0.6}{${w}_1\mid{u}_1$}}$]{}; (d) [$...$]{}; (f)[${\scalebox{0.6}{$q_{01}\mid{u}_{\breve{\theta}}$}}$]{}; (mm1) [$\prec$]{}; (fff) [${\scalebox{0.6}{$q_{01}\mid{u}_{\breve{\theta}+1}$}}$]{}; (mm2) [$\prec$]{}; (g) [$\prec$]{}; (h) [$...$]{}; (i) [$\prec$]{}; (ii) [${\scalebox{0.6}{${w}_{{l}-1}\mid{u}_{{l}+1}$}}$]{}; (iii) [$\prec$]{}; (j) [${\scalebox{0.6}{${w}_{{l}}\mid{u}_{{l}+2}$}}$]{}; (k) [$\prec$]{}; (l) [${\scalebox{0.6}{$z\mid{\xi}$}}$]{}; (f) – (fff);
1. If $q_1\succ z$ and $q_0\sim z$.
1. If $q_0\succ \vec{{w}}$
1. If $z\succ \vec{{w}}$, then
$${\color{red}}\phi_{{l}|2,1}(\vec{{w}}\ |\vec{q}\ ;z)=(\vec{{w}}\ ;\vec{q}\ ;z)\in M^U_{{l},2,1}.$$
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1. If For $(\vec{{w}}\ ;{q}_0,{q}_1;z) \in M^U_{{l},2,1}$, such that $$q_1\succ z\text{ and }q_0\sim z \text{ and } z\succ \vec{{w}},$$ we have $${\color{orange}}\psi^3_{{l}|2,1}(\vec{{w}}\ ;{q}_0,{q}_1;z)=(\vec{{w}}\ |{q}_0,{q}_1;z).$$
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{}\]
\(b) [${w}_1$]{}; (d) [$...$]{}; (f) [${w}_\theta$]{}; (ii) [${w}_{\theta+1}$]{}; (iii) [$\prec$]{}; (i1) [$...$]{}; (i2) [$\prec$]{}; (j) [${w}_{{l}}$]{}; (k) [$\prec$]{}; (l) [$z$]{}; (m) [$\prec$]{}; (m) [$\prec$]{}; (m) [$\prec$]{};
(q11) [$\prec$]{}; (q1) [$q_1$]{}; (q0) [$q_0$]{}; (f) – (ii); (l) – (q0); (q1) – (q0);
1. If $z\nsucc \vec{{w}}$ and $z\nprec {w}_{l}$, then
$${\color{red}}\phi_{{l}|2,1}(\vec{w}\ |\vec{q}\ ;z)=(\vec{w},z\ ;\vec{q}\ )\in M^U_{{l}+1,2}.$$
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1. For $(\vec{{w}},{\xi}\ ; {q}_0,{q}_1) \in M^U_{{l}+1,2}$, such that ${q}_0\succ\vec{{w}},$ ${q}_0\sim{\xi}$ and ${q}_1\succ {\xi}$, we have $${\color{orange}}\psi^2_{{l}|2,1}(\vec{{w}},{\xi}\ ;{q}_0,{q}_1)=(\vec{{w}}\ |{q}_0,{q}_1;{\xi}).$$
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{}\]
\(b) [${w}_1$]{}; (d) [$...$]{}; (f) [${w}_\theta$]{}; (ii) [${w}_{\theta+1}$]{}; (iii) [$\prec$]{}; (i1) [$...$]{}; (i2) [$\prec$]{}; (j) [${w}_{{l}}$]{}; (k) [$\preceq$]{}; (k) [$\prec$]{}; (l) [$z\mid{\xi}$]{};
(q11) [$\prec$]{}; (q1) [$q_1$]{}; (q0) [$q_0$]{}; (m) [$\prec$]{}; (m) [$\prec$]{}; (m) [$\prec$]{}; (f) – (ii); (l) – (q0); (q1) – (q0);
1. If $q_0\nsucc \vec{w}$ and $q_0\nprec {w}_{l}$, then
$${\color{red}}\phi_{{l}|2,1}(\vec{w}\ |\vec{q}\ ;z)=(\vec{w},q_0,q_1\ ;z)\in M^U_{{l}+2,1}.$$
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1. For $(\vec{{u}}\ ;{\xi}) \in M^U_{{l}+2,1}$, such that\
${u}_{{l}+1}\sim{\xi}$, ${u}_{{l}+1}\sim{u}_{{l}+2}$, and ${u}_{{l}+2}\succ{\xi}$, we have $${\color{orange}}\psi^2_{{l}|2,1}(\vec{{u}}\ ;{\xi})=({u}_1,...,{u}_{l}{|}{u}_{{l}+1},{u}_{{l}+2}\;{\xi}).$$
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{}\]
\(b) [${w}_1|{u}_1$]{}; (b1) [$\prec$]{}; (d) [$...$]{}; (i2) [$\prec$]{}; (j) [${w}_{l}|{u}_{l}$]{}; (k) [$\preceq$]{}; (q0) [$q_0|{u}_{{l}+1}$]{}; (q1) [$q_1|{u}_{{l}+2}$]{}; (q11) [$\prec$]{}; (z) [$z\mid{\xi}$]{}; (z) – (q0); (q1) – (q0);
1. If $q_0\prec {w}_{l}$.
1. If $q_1\sim {w}_{l}$, then\
\
\
$${\color{red}}\phi_{{l}|2,1}(\vec{{w}}\ |\vec{q}\ ;z)=(\vec{{w}},q_1;q_0,z)\in M^U_{{l}+1,2}.$$
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1. For $(\vec{{w}},{\xi}\ ;{q}_0,{q}_1) \in M^U_{{l}+1,2}$, such that $${\xi}\sim {w}_{l}\text{ and }{\xi}\sim {q}_0\text{ and }{q}_0\prec {w}_{l}\text{ and }{q}_1\prec {\xi},$$ we have $${\color{orange}}\psi^2_{{l}|2,1}(\vec{{w}},{\xi}\ ;{q}_0,{q}_1)=(\vec{{w}}\ |{q}_0,{\xi}\ ;{q}_1).$$
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{}\]
\(b) [${w}_1$]{}; (d) [$...$]{}; (f) [${w}_{l}$]{}; (j) [$q_1\mid{\xi}$]{}; (k) [$\prec$]{}; (l) [$q_0$]{}; (o) [$\prec$]{}; (p) [$z\mid q_1$]{}; (l) – (p); (f) – (j); (l) – (j);
1. If $q_1\prec {w}_{l}$ and ${w}_{\theta}\succ q_0$ and ${w}_{\theta+1}\succ q_1$, then\
\
\
$${\color{red}}\phi_{{l}|2,1}(\vec{{w}}\ |\vec{q}\ ;z)=(\vec{{w}}\ ;\vec{q}\ ;z)\in M^U_{{l},2,1}.$$
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1. For $(\vec{{w}}\ ;{q}_0,{q}_1;z) \in M^U_{{l},2,1}$, such that\
${w}_\theta\succ{q}_0 \text{ and }{w}_{\theta+1}\succ{q}_1\text{ and }{q}_0\sim z\text{ and }{q}_1\succ z $, we have $${\color{orange}}\psi^3_{{l}|2,1}(\vec{{w}}\ ;{q}_0,{q}_1;z)=(\vec{{w}}\ |{q}_0,{q}_1;z).$$
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{}\]
\(b) [${w}_1$]{}; (d) [$...$]{}; (f) [${w}_{\theta}$]{}; (ff) [${w}_{\theta+1}$]{}; (g) [$\prec$]{}; (h) [$...$]{}; (i) [$\prec$]{}; (ii) [${w}_{{l}}$]{};
\(k) [$\prec$]{}; (l) [$q_0$]{}; (kk) [$\prec$]{}; (ll) [$q_1$]{};
(lla) [$\prec$]{}; (j) [$z$]{};
\(f) – (ff); (l) – (j); (l) – (ll);
1. If $q_1\prec {w}_{l}$ and (${w}_{\theta}\nsucc q_0$ or ${w}_{\theta+1}\nsucc q_1$), let $$\tau=\max(\{i<{l}| {w}_{i}\nsucc q_1\vee {w}_i\sim {w}_{i+1}\}).$$
1. If $q_0\prec {w}_\tau$ ($\Rightarrow\ q_1\sim {w}_\tau$), then $${\color{red}}\phi_{{l}|2,1}(\vec{w}\ |\vec{q}\ ;z)=(\vec{{w}}\ ;\vec{q}\ ;z)\in M^U_{{l},2,1}.$$
This case is isomorphic to the second exceptional element type, shown below:
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1. If $(\vec{{w}}\ ;{q}_0,{q}_1;z) \in M^U_{{l},2,1}$ has the second\
exceptional element type, then\
\
$${\color{orange}}\psi^3_{{l}|2,1}(\vec{{w}}\ ;{q}_0,{q}_1;z)=(\vec{{w}}\ |{q}_0,{q}_1;z).$$
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{}\]
(start) [$...$]{}; (a) [${w}_{\theta}$]{}; (b) [${w}_{\theta+1}$]{}; (c) [$\prec$]{}; (d) [$...$]{}; (e) [$\prec$]{}; (f) [${w}_{\tau}$]{}; (g) [$\prec$]{}; (h) [$...$]{}; (i) [$\prec$]{}; (j) [${w}_{l}$]{}; (k) [$\prec$]{}; (l) [$q_1$]{};
\(n) [$q_0$]{}; (o) [$\prec$]{}; (p) [$\prec$]{}; (q) [$z$]{}; (b) – (a); (l) – (f); (l) – (n); (n) – (q);
1. If $q_0\nprec w_\tau$, then
$$\begin{gathered}
{\color{red}}\phi_{{l}|2,1}(\vec{w}\ |\vec{q}\ ;z)=(\phi_{{l}|2}(\vec{w}\ |q_0,z)\ ;q_1)=\\{\color{red}}=({w}_1,...,w_\tau,q_0,z,w_{\tau+1},...,{w}_{l}\ ;q_1)\in M^U_{{l}+2,1}.\end{gathered}$$
The following 3 pictures illustrate this case:
1. For $(\vec{{u}}\ ;{\xi}) \in M^U_{{l}+2,1}$, such that ${\xi}\prec{u}_{{l}+2}$, $\eta=\theta$ and ${\xi}\succ{u}_{\theta+1}$, we have $${\color{orange}}\psi^1_{{l}|2,1}(\vec{{u}}\ ;{\xi})=({u}_1,...,{u}_{\theta-1},{u}_{\theta+2},...,{u}_{{l}+2}\ |{u}_{\theta},{\xi}\ ;{u}_{\theta+1}).$$ As a reminder, $$\begin{aligned}
& \breve{\theta}=\max(\{i<{l}+2|\ {u}_i\sim {u}_{i+1}\}), \\ & \eta=\max(0,\{i\geq\theta| {u}_i\sim {\xi}\}).\end{aligned}$$
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{},scale=1.5,transform shape\]
\(b) [${\scalebox{0.6}{${w}_1\mid{u}_1$}}$]{}; (d) [$...$]{};
\(dd) [${\scalebox{0.6}{${w}_{\theta}\mid{u}_{\breve{\theta}-1}$}}$]{}; (f) [${\scalebox{0.6}{$q_0\mid{u}_{\breve{\theta}}$}}$]{}; (fff) [${\scalebox{0.6}{$z\mid{u}_{\breve{\theta}+1}$}}$]{}; (ffff) [$\succ$]{}; (g) [$\prec$]{}; (t1) [${\scalebox{0.6}{${w}_{\theta+1}\mid{u}_{\breve{\theta}+2}$}}$]{}; (t2) [$\prec$]{}; (h) [$...$]{}; (i) [$\prec$]{}; (j) [${\scalebox{0.6}{${w}_{l}\mid{u}_{{l}+2}$}}$]{}; (k) [$\prec$]{}; (l) [${\scalebox{0.6}{$q_1\mid{\xi}$}}$]{}; (f) – (fff); (l) – (f); (dd) – (f); (dd) to \[out=-50,in=-130\] (t1);
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{},scale=1.5,transform shape\]
\(b) [${\scalebox{0.6}{${w}_1\mid{u}_1$}}$]{}; (d) [$...$]{};
\(dd) [${\scalebox{0.6}{${w}_{\tau}\mid{u}_{\breve{\theta}-1}$}}$]{}; (f) [${\scalebox{0.6}{$q_0\mid{u}_{\breve{\theta}}$}}$]{}; (fff) [${\scalebox{0.6}{$z\mid{u}_{\breve{\theta}+1}$}}$]{}; (ffff) [$\succ$]{}; (g) [$\prec$]{}; (t1) [${\scalebox{0.6}{${w}_{\tau+1}\mid{u}_{\breve{\theta}+2}$}}$]{}; (tb) [$\prec$]{}; (t2) [$\prec$]{}; (h) [$...$]{}; (i) [$\prec$]{}; (j) [${\scalebox{0.6}{${w}_{l}\mid{u}_{{l}+2}$}}$]{}; (k) [$\prec$]{}; (l) [${\scalebox{0.6}{$q_1\mid{\xi}$}}$]{}; (f) – (fff); (l) – (f); (dd) – (f); (dd) – (l);
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{},scale=1.5,transform shape\]
\(b) [${\scalebox{0.6}{${w}_1\mid{u}_1$}}$]{}; (d) [$...$]{};
\(dd) [${\scalebox{0.6}{${w}_{\tau}\mid{u}_{\breve{\theta}-1}$}}$]{}; (dd1) [$\succ$]{}; (d1) [$\preceq$]{}; (f) [${\scalebox{0.6}{$q_0\mid{u}_{\breve{\theta}}$}}$]{}; (fff) [${\scalebox{0.6}{$z\mid{u}_{\breve{\theta}+1}$}}$]{}; (ffff) [$\succ$]{}; (g) [$\prec$]{}; (t1)[${\scalebox{0.6}{${w}_{\tau+1}\mid{u}_{\breve{\theta}+2}$}}$]{}; (tb) [$\prec$]{}; (t2) [$\prec$]{}; (h) [$...$]{}; (i) [$\prec$]{}; (j) [${\scalebox{0.6}{${w}_{l}\mid{u}_{{l}+2}$}}$]{}; (k) [$\prec$]{}; (l) [${\scalebox{0.6}{$q_1\mid{\xi}$}}$]{}; (f) – (fff); (l) – (f);
1. If $q_1\succ z$ and $q_0\succ z$.
This is the easiest case, since we insert $\vec{q}$ and $z$ independently.
1. If $z\succ\vec{{w}}$, then $${\color{red}}\phi_{{l}|2,1}(\vec{{w}}\ |\vec{q}\ ;z)=(\vec{{w}}\ |\vec{q}\ ;z)\in M^U_{{l},2,1}.$$
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1. For $(\vec{{w}}\ ;{q}_0,{q}_1;z) \in M^U_{{l},2,1}$, such that
$z\succ\vec{{w}}$, ${q}_0\succ z$ and ${q}_1\succ z$,
we have $${\color{orange}}\psi^3_{{l}|2,1}(\vec{{w}}\ ;{q}_0,{q}_1;z)=(\vec{{w}}\ |{q}_0,{q}_1;z).$$
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{}\]
\(b) [${w}_1$]{}; (d) [$...$]{}; (f) [${w}_{{l}}$]{}; (k) [$\prec$]{}; (l) [$z$]{}; (zl) [$\prec$]{}; (zq1) [$q_1$]{}; (zq0) [$q_0$]{};
\(mm) [$\succ$]{}; (m) [$\prec$]{}; (n) [$\prec$]{};
(zq1) – (zq0);
1. If $z\nprec {w}_{l}$ and $z\nsucc\vec{w}$ ($\Rightarrow q_0\nprec {w}_{l}$)
1. If $q_0\succ\vec{w}$, then $${\color{red}}\phi_{{l}|2,1}(\vec{w}\ |\vec{q}\ ;z)=(\vec{w},z\ ;\vec{q}\ )\in M^U_{{l}+1,2}.$$
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1. If $(\vec{{w}},{\xi}\ ;{q}_0,{q}_1) \in M^U_{{l}+1,2}$,\
and ${q}_0\succ\vec{{w}}$, and ${q}_1\succ {\xi}$, then
$${\color{orange}}\psi^2_{{l}|2,1}(\vec{{w}},{\xi}\ ;{q}_0,{q}_1)=(\vec{{w}}\ |{q}_0,{q}_1;{\xi}).$$
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{}\]
\(b) [${w}_1$]{}; (d) [$...$]{}; (f) [${w}_{l}$]{}; (i) [$\preceq$]{}; (j) [$z\mid{\xi}$]{}; (k) [$\prec$]{}; (l) [$q_0$]{}; (mm) [$\succ$]{}; (m) [$\prec$]{}; (n) [$\prec$]{}; (o) [$\prec$]{}; (p) [$q_1$]{}; (l) – (p);
1. If $q_0\nsucc\vec{{w}}$, then\
\
\
\
\
\
$${\color{red}}\phi_{{l}|2,1}(\vec{{w}}\ |\vec{q}\ ;z)=(\vec{{w}}\ ,\vec{q}\ ;z)\in M^U_{{l}+2,1}.$$
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1. For $(\vec{{u}}\ ;{\xi}) \in M^U_{{l}+2,1}$, such that $${\xi}\prec{u}_{{l}+2}\text{ and }{\xi}\prec{u}_{{l}+2},$$ what implies $$({\xi}\prec{u}_{\theta+1} \text{ and } {\xi}\prec{u}_{\theta}),$$ we have $${\color{orange}}\psi^2_{{l}|2,1}(\vec{{u}}\ ;{\xi})=(\psi_{{l}|2}(\vec{{u}}\ );{\xi})=({u}_1,...,{u}_{l}| {u}_{{l}+1},{u}_{{l}+2};{\xi}).$$
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{}\]
\(b) [${w}_1|{u}_1$]{}; (b1) [$\prec$]{}; (d) [$...$]{}; (i2) [$\prec$]{}; (j) [${w}_{l}|{u}_{l}$]{}; (k) [$\preceq$]{}; (q0) [$q_0|{u}_{{l}+1}$]{}; (mm) [$\succ$]{}; (q1) [$q_1|{u}_{{l}+2}$]{}; (q11) [$\prec$]{}; (z) [$z\mid{\xi}$]{}; (q1) – (q0);
1. If $z\prec {w}_{l}$
1. If $(\vec{{w}}\ ;\vec{q}\ )\in M_{{l},2}^U$, then $${\color{red}}\phi_{{l}|2,1}(\vec{{w}}\ |\vec{q}\ ;z)=(\vec{{w}}\ ;\vec{q}\ ;z)\in M^U_{{l},2,1}.$$
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1. If $(\vec{{w}}\ ;{q}_0,{q}_1;z) \in M^U_{{l},2,1}$, and $${q}_0\succ z,\ {q}_1\succ z\text{ and } w_{l}\succ z, \text{ then}$$ $${\color{orange}}\psi^3_{{l}|2,1}(\vec{{w}}\ ;\vec{{q}}\ ;z)=(\vec{{w}}\ |\vec{{q}}\ ;z).$$
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{}\]
\(b) [${w}_1$]{}; (d) [$...$]{}; (f) [${w}_{\theta}$]{}; (ff) [${w}_{\theta+1}$]{}; (g) [$\prec$]{}; (h) [$...$]{}; (i) [$\prec$]{}; (ii) [${w}_{{l}}$]{};
\(k) [$\prec$]{}; (l) [$q_0$]{}; (kk) [$\prec$]{}; (mm) [$\succ$]{}; (ll) [$q_1$]{};
(lla) [$\prec$]{}; (j) [$z$]{};
\(f) – (ff); (l) – (ll);
1. If $(\vec{{w}}\ |\vec{q}\ )\notin M_{{l},2}^U$, then\
\
$${\color{red}}\phi_{{l}|2,1}(\vec{{w}}\ |\vec{q}\ ;z)=( \phi_{{l}|2}(\vec{{w}}\ |\vec{q}\ ) ;z)\in M^U_{{l}+2,1}$$
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1. If $(\vec{{u}}\ |{\xi}) \in M^U_{{l}+2,1}$, ${\xi}\prec{u}_{{l}+2}$ and $\eta=0$, what implies $$({\xi}\prec{u}_{\theta+1}\text{ and }{\xi}\prec{u}_{\theta}),$$ then we have $${\color{orange}}\psi^2_{{l}|2,1}(\vec{{u}}\ ;{\xi})=(\psi_{{l}|2}(\vec{{u}}\ );{\xi}).$$
\[>=latex,every node/.style=[minimum width=3em, node distance=4em]{},scale=1.5,transform shape\]
\(b) [${\scalebox{0.6}{${w}_1\mid{u}_1$}}$]{}; (d) [$...$]{}; (f)[${\scalebox{0.6}{$q_{01}\mid{u}_{\breve{\theta}}$}}$]{}; (mm1) [$\succ$]{}; (fff) [${\scalebox{0.6}{$q_{01}\mid{u}_{\breve{\theta}+1}$}}$]{}; (mm2) [$\succ$]{}; (g) [$\prec$]{}; (h) [$...$]{}; (i) [$\prec$]{}; (ii) [${\scalebox{0.6}{${w}_{{l}-1}\mid{u}_{{l}+1}$}}$]{}; (iii) [$\prec$]{}; (j) [${\scalebox{0.6}{${w}_{{l}}\mid{u}_{{l}+2}$}}$]{}; (k) [$\prec$]{}; (l) [${\scalebox{0.6}{$z\mid{\xi}$}}$]{}; (f) – (fff);
It is easy to see that by construction the left hand side fully describes the set $P^U_{{l}}\times M^U_{2,1}$. Now, we check that the right hand side coincide with $M_{{l}+2,1}^U\sqcup M_{{l}+1,2}^U\sqcup M^U_{{l},2,1}$:
- Let $(\vec{{u}}\ ; {\xi})\in M^U_{{l}+2,1}$. Then the following cases from the right hand side clearly describe $M^U_{{l},2,1}$:
1. ${\xi}\succ\vec{{u}};$
2. ${\xi}\prec{u}_{{l}+2}$ and $\eta>0$;
3. ${\xi}\prec{u}_{{l}+2}$, $\eta=0$ and $({\xi}\succ{u}_{\theta+1} \text{ and } {\xi}\succ{u}_{\theta})$;
4. ${\xi}\prec{u}_{{l}+2}$, $\eta=0$ and $({\xi}\prec{u}_{\theta+1} \text{ and } {\xi}\prec{u}_{\theta})$.
- Let $(\vec{{w}}\ ,{\xi};{q}_0,{q}_1)\in M^U_{{l}+1,2}$. Then the following cases from the right hand side clearly describe $M^U_{{l},2,1}$.:
1. (${\xi}\succ w_{l},$ ${w}_{\theta}\succ {q}_0,$ ${w}_{\theta+1}\succ{q}_1,$ and ${\xi}\succ {q}_0$), or (${\xi}\sim w_{l},$ ${w}_{{l}}\succ {q}_0,$ ${\xi}\succ{q}_1,$ and ${\xi}\succ {q}_0$);
2. $({\xi}\sim w_{l},$ ${w}_{{l}}\succ {q}_0,$ ${\xi}\succ{q}_1,$ and ${\xi}\sim {q}_0);$
3. $q_0\succ\vec{w}$, $q_1\succ{\xi}$ and $q_0\sim{\xi};$
4. $q_0\succ\vec{w}$, $q_1\succ{\xi}$ and $q_0\succ{\xi}.$
- Let $(\vec{{w}}\ ;{q}_0,{q}_1; z)\in M^U_{{l},2,1}$. Then the following cases from the right hand side clearly describe $M^U_{{l},2,1}$.:
1. $z\succ\vec{q}$ and $z\succ{w}$;
2. $z\succ\vec{q}$ and $z\prec{w_{l}}$;
3. $q_0\sim z$, $q_1\succ z$ and $z\succ\vec{w}$;
4. $q_0\sim z$, $q_1\succ z$ and $(w_\theta\succ q_0\text{ and }w_{\theta+1}\succ q_1)$;
5. $q_0\succ z$, $q_1\succ z$ and $z\prec w_{l}$;
6. $q_0\succ z$, $q_1\succ z$ and $z\prec w_{l}$;
7. First exceptional type element;
8. Second exceptional type element.
Since every number from 1 to 19 was used exactly once, this completes the proof.
\[2[l]{}1k\] For natural numbers ${l}$ and $k$, let $$M^U_{2^{l},1^k}=\{(\vec{\xi},\vec{\varepsilon}\ )\in E_{l}^U\times E_{k+{l}}^U|\; \exists\{i_j\}_{j=1}^{l}, \ 0<i_1<i_2<...<i_{l}<k+{l}, \text{ s.t. } \xi_j\sim\varepsilon_{i_j} \text{ for } 1\leq j\leq{l}\},$$ Then $$m_{2^{l},1^k}^U=\sum_{(\vec{\xi},\vec{\varepsilon}\,)\in M^U_{2^{l},1^k}}\xi_1\cdot...\cdot \xi_{{l}}\cdot \varepsilon_1\cdot...\cdot\varepsilon_{{l}+k}.$$
According to Remark \[c\_m\], this implies $c_{2^k,1^{n-2k}}(U)\geq 0$.
The proof is omitted and can be found in [@Paunov16].
[1]{}
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